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HKUST NLP Group 15

# Introduction Black-box optimization problems [@Kimiaei_2022] (or also known as derivative-free optimization problems [@Conn_2009; @Rios_2013]) arise when the gradient computation process is unavailable for some reason, e.g., the objective function $f(x)$ is not smooth [@Polyak_1969; @Dvinskikh_2022; @Huang_2023; @Lobanov_MOTOR] or the process of computing the gradient $\nabla f(x)$ is too "expensive" compared to computing the value of the objective function $f(x)$ (in this case, the functions can be either smooth [@Ajalloeian_2020; @Akhavan_2022] or of higher order of smoothness [@Bach_2016; @Akhavan_2020; @Lobanov_2023]). Moreover, there are often situations in practice [@Bogolubsky_2016] when the oracle returns a noisy value of the objective function (i.e., the value of the function $f(x)$ with some bounded noise $\xi$) at the requested point $x$, where the noise directly affects the "cost" of calling the oracle: the more inaccurate the oracle returns the value of the objective function (i.e., the greater the noise), the cheaper the oracle call. Such an oracle has a common "charactonym" name in the literature, namely a gradient-free oracle or a zero-order oracle [@Rosenbrock_1960]. Since this class of optimization problem has significant interest in settings such as federated learning [@Lobanov_2022; @Patel_2022], distributed learning [@Akhavan_2021; @Yu_2021; @Lobanov_NET], overparameterized models [@Lobanov_overparametrization] (in particular, in application problems such as hyperparameter tuning [@Li_2017; @Hazan_2017], multi-armed bandits [@Flaxman_2004; @Bartlett_2008], and many others [@Nguyen_2023]\...), it is important to know and understand what approaches exist to solve this class of problem. Apparently, the main way to solve the black-box problem is to apply gradient-free algorithms/zero-order methods to this problem. Among such methods there are two classes (or two approaches to the creation of gradient-free algorithms): the first is the class of evolutionary algorithms [@Storn_1997; @Auger_2005; @Hansen_2006], which can often show their efficiency only empirically; the second is the class based on the advantages of first-order algorithms [@Kiefer_1952; @Gasnikov_ICML], whose efficiency is provided in the form of theoretical estimates. Evolutionary algorithms are often used in the class of non-convex multimodal problems, in which the main goal is to find not a local but a global optimum. However, in the class of convex problems, it makes sense to use theoretically based algorithms, which often guarantee faster convergence by taking advantage of first-order algorithms. The basic idea of creating efficient gradient-free algorithms for solving the convex black-box optimization problem is to use instead of the true gradient in first-order optimization algorithms some estimate of the gradient or also known as a gradient approximation [@Gasnikov_2022]. It is this seemingly simple idea that allows gradient-free algorithms to utilize the power of efficient first-order methods to solve the black box problem. However, it is important to correctly choose the algorithm and gradient approximation based on the original problem. Often, accelerated batched methods for solving corresponding optimization problems are chosen as efficient first-order algorithms, but there are also exceptions, e.g., for the class of problems satisfying the Polyak--Lojasiewicz condition, unaccelerated algorithms are already considered efficient (see [@Yue_2022] for more details). Regarding the question of the choice of gradient approximation: in [@Scheinberg_2022] it is shown that central finite difference is a more preferable scheme for constructing a gradient approximation than forward finite difference. Also in [@Lobanov_2023] in the Experiments section, the authors have shown on a model practical experiment the advantage of using randomized approximations, in particular among the randomized approximations, they highlight $l_2$ randomization. However, this approximation is the most preferable for solving a smooth black-box optimization problem, but not for a problem with increased smoothness. In 1990, B. Polyak and A. Tsybakov managed to propose such a gradient approximation, which takes into account the advantage of increased smoothness of the function [@Polyak_1990]. This gradient approximation is called the Kernel approximation. What distinguishes this approximation from $l_2$ randomization is the presence of a kernel by which the information about the increased smoothness of the function is taken into account. And it was this paper [@Polyak_1990] that became the starting point for the study of the solution of the black box problem with the assumption that the function has a high order of smoothness. This paper investigates the improvement of the iteration complexity of gradient-free algorithms for solving a class of convex black-box optimization problems, assuming that the objective function has increased smoothness. To create an optimal zero-order optimization method in terms of iteration complexity, we use the accelerated batched stochastic gradient descent method [@Vaswani_2019] (Nesterov--accelerated) as a basis. Among the gradient approximations that take into account the advantage of increased smoothness, we choose the Kernel approximation because it is the one that requires only two calls to the gradient-free oracle per iteration (which guarantees a better estimate of oracle complexity), unlike the higher-order finite-difference gradient approximation [@Berahas_2022]. However, the Kernel approximation is not a biased gradient estimator, so it is important to understand how noise is accounted for in the first-order algorithm. To this end, we generalize the accelerated first-order algorithm [@Vaswani_2019] to the case with a biased gradient oracle. Thus, to create a gradient-free algorithm, we base on the accelerated first-order method with a biased gradient oracle (see Section [3](#sec:First_order){reference-type="ref" reference="sec:First_order"}), using the Kernel approximation with a one-point zero-order oracle instead of the true gradient. In addition, we explicitly derive the estimate at the maximum noise level at which the desired accuracy can be achieved. Finally, we demonstrate our theoretical results on a model example. ## Our contribution Our contribution to this paper can be summarized as follows: - We generalize the convergence results of the accelerated batched stochastic gradient descent algorithm (Nesterov--accelerated) [@Vaswani_2019] to the case with a biased gradient oracle; - We provide a novel gradient-free optimization algorithm for solving a convex black-box optimization problem under an increased smoothness condition: the zero-order accelerated stochastic gradient descent method (ZO-AccSGD, see Algorithm 1). This algorithm improves existing estimates on iteration complexity by working out the batching technique: $N = \mathcal{O} \left( \varepsilon^{-1/2} \right)$. Moreover, using the well-known analysis of bias and second moment (variance) estimation, we were able to get rid of the smoothness order dependence $\beta$ of the objective function in oracle complexity: $T = \mathcal{O} \left( \frac{d^2}{\varepsilon^{2 + \frac{2}{\beta - 1}}} \right)$; - We provide an analysis that includes an elaboration on the maximum allowable noise level. We show that the noise level at which the desired accuracy is still achieved depends directly on the batch size $B$; - We confirm our theoretical results in Section "Experiments" by considering a model that is often used in machine learning of the NLLLoss function. ## Related works #### Gradient-free oracle. The gradient approximation is typically a finite difference zero-order oracle. Therefore, every work on gradient-free oracle utilizes one or another zero-order oracle concept. For example, the works in [@Ajalloeian_2020] propose an oracle concept that returns the exact value of the objective function at the requested point: $\tilde{f} = f(x)$. This concept is intuitive and widely used, especially in tutorials such as introductions to optimization [@Polyak_1987], etc. The following concept of gradient-free oracle, introduced in [@Dvinskikh_2022; @Lobanov_2022], is oriented towards practical problems and is presented as follows: the oracle returns the value of the objective function at the requested point with some bounded deterministic noise $\tilde{f} = f(x) + \delta (x)$, where $||\delta (x)| \leq \Delta$. This concept applies well to deterministic optimization problems, but we can modernize it for a stochastic optimization problem [@Gasnikov_ICML]: $\tilde{f} = f(x, \xi) + \delta(x)$. This stochastic variant of the concept of a gradient-free oracle with bounded deterministic noise allows us to construct a gradient approximation depending on the availability of feedback. For example, if we can call the gradient-free oracle on one realization of the function twice, then the Kernel approximation with two-point feedback takes the following form: $\gg = \frac{d}{2h} \left( f(x + h r \ee, \xi) + \delta(x + h r \ee) - f(x - h r \ee, \xi) - \delta(x + h r \ee) \right) K(r) \ee$. However, if we only have access to one-point feedback, i.e., we can call the oracle on one realization of the function only once, then the kernel approximation takes the following form: $\gg = \frac{d}{h} \left( f(x + h r \ee, \xi) + \delta(x + h r \ee) \right) K(r) \ee$. In addition, there is another concept of the gradient-free oracle that is quite controversial in gradient approximation, namely the kernel approximation [@Polyak_1990; @Akhavan_2020; @Novitskii_2021; @Lobanov_2023; @Akhavan_2023] is as follows: $\gg = \frac{d}{2h} \left( f(x + h r \ee) + \xi_1 - f(x - h r \ee) - \xi_2 \right) K(r) \ee$. Such a gradient approximation can quite rightly be called a one-point feedback approximation, although at first glance it is not even obvious that a gradient-free oracle that returns the value of the objective function at the requested point with some bounded stochastic noise $\tilde{f} = f(x) + \xi$ is a stochastic gradient-free oracle. However, if we consider the stochastic noise $\xi_1$ and $\xi_2$ as realizations of the function, it will be clear enough that such a gradient approximation can rightfully be called a one-point approximation, since the function is computed though twice in one iteration, but on different realizations. In our work, we also use the concept of a gradient-free oracle with stochastic noise, which generates a gradient approximation with one-point feedback. #### Bounded gradient noises. Currently, there are a series of works [@Woodworth_2021_over; @Rakhlin_2012; @Hazan_2014; @Bertsekas_1996; @Stich_2019; @Lobanov_2023; @Schmidt_2013; @Srebro_2010] that assume different constraints on gradient noise. For example, [@Rakhlin_2012; @Hazan_2014] uses a standard and common in earlier works constraint on gradient noise, namely, they estimate some constant: $\expect{\norms{\nabla f(x,\xi)}^2} \leq \sigma^2$. However, there is some disadvantage of such a constraint, namely the large number constraint, that is, if the norm of the gradient decreases with the number of iterations, the estimate of the second moment will remain as large. To address this problem, some works [@Bertsekas_1996; @Stich_2019; @Lobanov_2023] impose a constraint that is considered more adaptive: $\expect{\norms{\nabla f(x,\xi)}^2} \leq \rho \norms{\nabla f(x)}^2 + \sigma^2$. There are also papers [@Schmidt_2013] that assume the strong growth condition is satisfied: $\expect{\norms{\nabla f(x, \xi)}^2} \leq \rho \norms{\nabla f(x)}^2.$ In the case where the model is overparameterized [@Woodworth_2021_over; @Srebro_2010], it is proposed to estimate the gradient noise as follows: $\expect{\norms{\nabla f(x^*, \xi)}^2} \leq \sigma_*^2$. The essential difference from the previous constraints is that the gradient estimate is evaluated at the solution point and depends directly on the solution of the problem, i.e., if $f^* = \min_x f(x)$ tends to zero, then also $\sigma^2_* \leq L f^*$ decreases. In our work, we use the following constraint on the noise of the biased gradient oracle $\expect{\norms{\gg(x, \xi)}^2} \leq \rho \norms{\nabla f(x)}^2 + \sigma^2$, since this constraint is adaptive and our approach in creating a gradient-free algorithm is based on the approach of the paper [@Vaswani_2019], which also addresses this constraint. The gradient oracle $\gg(x,\xi)$ will be introduced in Subsection [2.2](#subsec:ASS_gradient_oracle){reference-type="ref" reference="subsec:ASS_gradient_oracle"}. #### Iteration complexity. The study of a class of convex black-box optimization problems under the condition of increased smoothness began with a 1990 paper[@Polyak_1990], where a method of gradient estimation through the kernel and lower bound estimates of a gradient-free algorithm was proposed. At present, there already exist "pretty" results in this direction [@Bach_2016; @Akhavan_2020; @Novitskii_2021; @Lobanov_2023; @Akhavan_2023], which can be considered as "state of the art". For example, in [@Akhavan_2020], the authors proposed a Zero-Order Stochastic Projected Gradient algorithm that used a central finite difference kernel approximation and required the following iteration $N$ (as well as oracle $T$) complexity to achieve a given accuracy $\varepsilon$: $T = N = \mathcal{\tilde{O}} \left( \frac{d^{2 + \frac{2}{\beta -1}}}{\varepsilon^{2 + \frac{2}{\beta - 1}}} \right).$ In another paper [@Novitskii_2021], the authors managed to improve this dimensionality estimate by using some "trick" in analyzing the bias of the gradient-free oracle: $T = N = \mathcal{\tilde{O}} \left( \frac{d^{2 + \frac{1}{\beta -1}}}{\varepsilon^{2 + \frac{2}{\beta - 1}}} \right).$ In a recent paper [@Akhavan_2023], the authors have managed to propose an improved analysis for estimating the bias and second moment (variance) of the gradient approximation, getting rid of the smoothness order dependence in the degree of dimensionality (in the strongly convex case). Moreover, it is not difficult to show that with the help of this analysis one can get rid of the dependence in the convex case as well (see [@Novitskii_2021] for the transformation from the strongly convex case to the convex case). However, these works focus on one of the three optimality criteria, namely oracle complexity. In our work, we use an improved analysis from [@Akhavan_2023] for gradient approximation to improve existing estimates of oracle complexity in the convex case, namely getting rid of the dependence of dimensionality on smoothness order, and by using an accelerated version of stochastic gradient descent and working out the batching technique we improve iteration complexity estimation. ## Paper organization This paper has the following structure. Section [2](#sec:Problem_formulation){reference-type="ref" reference="sec:Problem_formulation"} introduces the formulation of the problem considered in this paper, as well as the main idea of its solution. Section [3](#sec:First_order){reference-type="ref" reference="sec:First_order"} presents generalized results for the case of a biased oracle to solve the problem. The main result, which provides a novel gradient-free algorithm, can be found in Section [4](#sec:Main_results){reference-type="ref" reference="sec:Main_results"}. A discussion of the results is given in Section [5](#sec:Discussion){reference-type="ref" reference="sec:Discussion"}. Section [6](#sec:Experiments){reference-type="ref" reference="sec:Experiments"} presents the experiments. While Section [7](#sec:Conclusion){reference-type="ref" reference="sec:Conclusion"} concludes the paper. # Problem formulation {#sec:Problem_formulation} In this section, we introduce the notations, definitions and assumptions used in our analysis to formulate the optimization problem. We also describe the main idea of our approach to solve the black-box optimization problem. #### Notation. We use $\dotprod{x}{y}:= \sum_{i=1}^{d} x_i y_i$ to denote standard inner product of $x,y \in \mathbb{R}^d$, where $x_i$ and $y_i$ are the $i$-th component of $x$ and $y$ respectively. We denote Euclidean norm ($l_2$-norm) in $\mathbb{R}^d$ as $\norms{x} = \| x\|_2 := \sqrt{\dotprod{x}{x}}$. We use the following notation $B_2^d(r):=\left\{ x \in \mathbb{R}^d : \| x \| \leq r \right\}$ to denote Euclidean ball ($l_2$-ball) and $S_2^d(r):=\left\{ x \in \mathbb{R}^d : \| x \| = r \right\}$ to denote Euclidean sphere. Operator $\mathbb{E}[\cdot]$ denotes full mathematical expectation. We consider a standard optimization problem of the following form, which is commonly encountered in the literature, especially at the first acquaintance with optimization methods: $$\label{eq:init_problem} f^* = \min_{x \in Q \subseteq \mathbb{R}^d} f(x),$$ where $f:\mathbb{R}^d \rightarrow \mathbb{R}$ convex function that we want to minimize on the convex set $Q$. This general formulation is a broad class of optimization problems. To narrow down the class of optimization problems, we use a standard formulation of the problem and impose constraints on the function and the gradient oracle in the form of assumptions that will be used in our analysis throughout paper. ## Assumptions on objective function In our analysis presented in Section [3](#sec:First_order){reference-type="ref" reference="sec:First_order"}, we assume that function is $L$-smooth. [\[ass:L_smooth\]]{#ass:L_smooth label="ass:L_smooth"} Function $f$ is $L$-smooth if it holds $$f(y) \leq f(x) + \dotprod{\nabla f(x)}{y - x} + \frac{L}{2} \norms{y - x}^2, \quad \forall x,y \in Q.$$ And already in Section [4](#sec:Main_results){reference-type="ref" reference="sec:Main_results"} our theoretical reasoning assumes that the objective function $f(x)$ is not just smooth, but has a higher order of smoothness. [\[Ass:Higher_order\]]{#Ass:Higher_order label="Ass:Higher_order"} Let $l$ denote maximal integer number strictly less than $\beta$. Let $\mathcal{F}_\beta(L)$ denote the set of all functions $f: \mathbb{R}^d \rightarrow \mathbb{R}$ which are differentiable $l$ times and for all $x,z \in Q$ the Hölder-type condition: $$\left| f(z) - \sum_{0 \leq |n| \leq l} \frac{1}{n!} D^n f(x) (z-x)^n \right| \leq L_\beta \norms{z - x}^\beta,$$ where $L_\beta>0$, the sum is over multi-index $n~=~(n_1, ..., n_d) \in \mathbb{N}^d$, we used the notation $n!~=~n_1! \cdots n_d!$, $|n| = n_1 + \cdots + n_d$, and $\forall v = (v_1, ..., v_d) \in \mathbb{R}^d$ we defined $D^n f(x) v^n = \frac{\partial ^{|n|} f(x)}{\partial^{n_1}x_1 \cdots \partial^{n_d}x_d} v_1^{n_1} \cdots v_d^{n_d}$. The assumptions introduced in this subsection are standard and common in the literature in related works, e.g., see Assumption [\[ass:L_smooth\]](#ass:L_smooth){reference-type="ref" reference="ass:L_smooth"} in [@Nemirovski_2009; @Nesterov_2018], and Assumption [\[Ass:Higher_order\]](#Ass:Higher_order){reference-type="ref" reference="Ass:Higher_order"} in the following works [@Polyak_1990; @Bach_2016; @Akhavan_2023]. Moreover, it is not hard to see the connection between two assumptions, namely, in the case $\beta = 2: L_2 = \frac{L}{2}$. ## Assumptions on gradient oracle {#subsec:ASS_gradient_oracle} Before presenting the assumptions on the gradient oracle, we introduce a formal definition, which is used extensively in the analysis for the convergence results of first-order algorithm in Section [3](#sec:First_order){reference-type="ref" reference="sec:First_order"}. Regarding convergence of zero-order algorithm, already gradient-free oracle will be introduced in Subsection [4.1](#subsec:Gradient_approximation){reference-type="ref" reference="subsec:Gradient_approximation"}. [\[Definition_1\]]{#Definition_1 label="Definition_1"} A map $\mathbf{g}~:~\mathbb{R}^d~\times~\mathcal{D} \rightarrow \mathbb{R}^d$ s.t. $$\label{eq:Definition_1} \mathbf{g}(x,\xi) = \nabla f(x, \xi) + \mathbf{b}(x)$$ for a bias $\mathbf{b}: \mathbb{R}^d \rightarrow \mathbb{R}^d$ and unbiased stochastic gradient $\expect{\nabla f(x, \xi)}=f(x)$. We assume that the bias and gradient noise are bounded. [\[Ass:Bounded_bias\]]{#Ass:Bounded_bias label="Ass:Bounded_bias"} There exists constant $\delta \geq 0$ s.t. $\forall x \in \mathbb{R}^d$ $$\label{eq:Bounded_bias} \norms{\mathbf{b}(x)} = \norms{\expect{\gg(x,\xi)} - \nabla f(x)}\leq \delta.$$ [\[Ass:Gradient_noise\]]{#Ass:Gradient_noise label="Ass:Gradient_noise"} There exists constants $M, \sigma^2 \geq 0$ such that the more general condition of strong growth is satisfied $\forall x \in \mathbb{R}^d$ $$\label{eq:Gradient_noise} \expect{\norms{\gg(x,\xi)}^2} \leq \rho \norms{\nabla f(x)}^2 + \sigma^2.$$ Assumptions [\[Ass:Bounded_bias\]](#Ass:Bounded_bias){reference-type="ref" reference="Ass:Bounded_bias"} and [\[Ass:Gradient_noise\]](#Ass:Gradient_noise){reference-type="ref" reference="Ass:Gradient_noise"} are not uncommon in works studying optimization algorithms with biased oracle (see Definition [\[Definition_1\]](#Definition_1){reference-type="ref" reference="Definition_1"}), such as [@Ajalloeian_2020; @Lobanov_2023; @Lobanov_overparametrization]. ## The main idea of problem solving The problem presented above does not strongly correspond to the black-box optimization problem, which is also stated in the title of the paper. This is done in order to present in Section [3](#sec:First_order){reference-type="ref" reference="sec:First_order"} a first-order algorithm that has access to the noisy value of the gradient (see Definition [\[Definition_1\]](#Definition_1){reference-type="ref" reference="Definition_1"}). However, our approach to solving the optimization problem [\[eq:init_problem\]](#eq:init_problem){reference-type="eqref" reference="eq:init_problem"} when the gradient is still not available to the algorithm (black-box problems) is to create a gradient-free optimization algorithm based on and exploiting the power of the first-order method. Despite the fact that the original problem [\[eq:init_problem\]](#eq:init_problem){reference-type="eqref" reference="eq:init_problem"} is deterministic, we must rely on a first-order optimization algorithm that solves exactly the stochastic optimization problem $f(x) := \expect{f(x,\xi)}$, since "stochasticity" is artificially created in the gradient approximation (see Subsection [4.1](#subsec:Gradient_approximation){reference-type="ref" reference="subsec:Gradient_approximation"}). Moreover, the Kernel approximation is a biased gradient estimator, so it is important to choose an algorithm that accounts for the imprecision in the gradient oracle. Thus, to summarize our approach, in Section [3](#sec:First_order){reference-type="ref" reference="sec:First_order"} we generalize the SOTA results to the case with a biased gradient oracle (see Definition [\[Definition_1\]](#Definition_1){reference-type="ref" reference="Definition_1"}) that satisfies Assumptions [\[Ass:Bounded_bias\]](#Ass:Bounded_bias){reference-type="ref" reference="Ass:Bounded_bias"} and [\[Ass:Gradient_noise\]](#Ass:Gradient_noise){reference-type="ref" reference="Ass:Gradient_noise"}, and use this first-order algorithm to create a gradient-free algorithm (see Section [4](#sec:Main_results){reference-type="ref" reference="sec:Main_results"}) for solving the black-box optimization problem under the condition of increased smoothness of the objective function (see Assumption [\[Ass:Higher_order\]](#Ass:Higher_order){reference-type="ref" reference="Ass:Higher_order"}). # Generalization of convergence results for Accelerated SGD to the biased oracle {#sec:First_order} In this section, we provide the first-order algorithm on which the novel gradient-free method for solving the black-box optimization problem in Section 1 will be based. Since this first-order algorithm must solve a stochastic optimization problem (due to the artificial "stochasticity" in the gradient approximation: $\ee \in S_{2}^d(1)$, which will be introduced later), we reformulate the initial optimization problem as follows [\[eq:init_problem\]](#eq:init_problem){reference-type="eqref" reference="eq:init_problem"}: $$\label{eq:stoc_init_problem} f^* = \min_{x \in Q \subseteq \mathbb{R}^d} \left\{ f(x) := \expect{f(x, \xi)} \right\}.$$ Next, before providing the convergence of the first-order algorithm with the biased gradient oracle, we present the known convergence results of accelerated stochastic gradient descent for solving problem [\[eq:stoc_init_problem\]](#eq:stoc_init_problem){reference-type="eqref" reference="eq:stoc_init_problem"}. ## Background In 2019, the authors of [@Vaswani_2019] provided convergence results of Nesterov-accelerated Stochastic Gradient Descent [@Nesterov_2012] for the problem when the unbiased gradient oracle $\gg(x, \xi) = \nabla f(x, \xi)$ (see Definition [\[Definition_1\]](#Definition_1){reference-type="ref" reference="Definition_1"} with $\delta = 0$ in Assumption [\[Ass:Bounded_bias\]](#Ass:Bounded_bias){reference-type="ref" reference="Ass:Bounded_bias"}) satisfies the strong growth condition (see Assumption [\[Ass:Gradient_noise\]](#Ass:Gradient_noise){reference-type="ref" reference="Ass:Gradient_noise"}). In particular, this algorithm consists of the following update rules: align\* x\_k+1 &= y_k - (y_k, \_k)\ y_k &= \_k z_k + (1-\_k) x_k\ z\_k+1 &= \_k z_k + (1 - \_k) y_k - \_k (y_k, \_k). And has the following convergence result in the case where $\sigma \neq 0$: [\[lem:Vaswani\]]{#lem:Vaswani label="lem:Vaswani"} Let the function $f$ satisfy Assumption [\[ass:L_smooth\]](#ass:L_smooth){reference-type="ref" reference="ass:L_smooth"}, and the unbiased gradient oracle $\gg(x, \xi) = \nabla f(x, \xi)$ satisfies the strong growth condition ( Assumption [\[Ass:Gradient_noise\]](#Ass:Gradient_noise){reference-type="ref" reference="Ass:Gradient_noise"}), then the accelerated Stochastic Gradient Descent by Nesterov with chosen parameters: $$\begin{aligned} \gamma_{k} = \frac{\rho^{-1} + \sqrt{\rho^{-2} + 4\gamma_{k - 1}^{2}}}{2}; \quad a_{k + 1} = \gamma_k \sqrt{\eta \rho}; \quad \alpha_k = \frac{\gamma_{k} \eta}{\gamma_{k} \eta + a_k^2}; \quad \eta = \frac{1}{\rho L} \end{aligned}$$ has the following rate of convergence: $$\boxed{\expect{f(x_{N})} - f^* \lesssim \frac{\rho^2 L R^2}{N^2} + \frac{N \sigma^2}{L \rho^2}.}$$ This result is considered a state of the art for this problem formulation, however, as mentioned earlier, due to the Kernel approximation, which accumulates noise (i.e., has bias), we are not suitable for this algorithm as a basis for creating a gradient-free optimization method. Therefore, in the next subsection, we extend the convergence results of the algorithm (Lemma [\[lem:Vaswani\]](#lem:Vaswani){reference-type="ref" reference="lem:Vaswani"}) to the case where the gradient oracle $\gg(x, \xi)$ (see Definition [\[Definition_1\]](#Definition_1){reference-type="ref" reference="Definition_1"}) can return a noisy gradient value $\nabla f(x, \xi) + \bb(x)$, i.e., Assumption [\[Ass:Bounded_bias\]](#Ass:Bounded_bias){reference-type="ref" reference="Ass:Bounded_bias"} is satisfied. ## Accelerated SGD with biased gradients In this subsection, we present the main result of Section [3](#sec:First_order){reference-type="ref" reference="sec:First_order"}, namely, we provide a first-order algorithm that we will base on in the next section to create a gradient-free algorithm. To achieve one of the main goals of our work, namely to improve and, if necessary, to obtain an optimal estimate of the iteration complexity $N$ of the gradient-free optimization algorithm, we not only generalize the convergence result of Lemma [\[lem:Vaswani\]](#lem:Vaswani){reference-type="ref" reference="lem:Vaswani"} to the case with a biased oracle, but also get rid of the constant $\rho$ from the first term, thus improving the estimate by the number of successive iterations of the accelerated Stochastic Gradient Descent. We improve the results of Lemma [\[lem:Vaswani\]](#lem:Vaswani){reference-type="ref" reference="lem:Vaswani"} in terms of iteration complexity by applying the batching technique. Thus, Accelerated Stochastic Gradient Descent (Nesterov acceleration) with a biased gradient oracle $\gg(x, \xi)$ (see Definition [\[Definition_1\]](#Definition_1){reference-type="ref" reference="Definition_1"}) has the following convergence results presented in Theorem [\[th:First_order\]](#th:First_order){reference-type="ref" reference="th:First_order"}. [\[th:First_order\]]{#th:First_order label="th:First_order"} Let the function $f$ satisfy Assumption [\[ass:L_smooth\]](#ass:L_smooth){reference-type="ref" reference="ass:L_smooth"}, and the gradient oracle $\gg(x, \xi)$ from Definition [\[Definition_1\]](#Definition_1){reference-type="ref" reference="Definition_1"} satisfies Assumptions [\[Ass:Bounded_bias\]](#Ass:Bounded_bias){reference-type="ref" reference="Ass:Bounded_bias"} and [\[Ass:Gradient_noise\]](#Ass:Gradient_noise){reference-type="ref" reference="Ass:Gradient_noise"}, then the accelerated Stochastic Gradient Descent with batching ($B$ is a batch size) by Nesterov with $\rho_B = \max\{1, \frac{\rho}{B}\}$ and chosen parameters: $$\begin{aligned} \gamma_{k} = \frac{\rho_B^{-1} + \sqrt{\rho_B^{-2} + 4\gamma_{k - 1}^{2}}}{2}; \quad a_{k + 1} = \gamma_k \sqrt{\eta \rho_B}; \quad \alpha_k = \frac{\gamma_{k} \eta}{\gamma_{k} \eta + a_k^2}; \quad \eta = \frac{1}{\rho_B L} \end{aligned}$$ has the following rate of convergence: $$\boxed{\expect{f(x_{N})} - f^* \lesssim \frac{\rho_B^2 L R^2}{N^2} + \frac{N \sigma^2}{\rho_B^2 L B } + \delta \tilde{R} + \frac{N}{L} \delta^2.}$$ It is not hard to see that the convergence result of the accelerated batched algorithm presented in Theorem [\[th:First_order\]](#th:First_order){reference-type="ref" reference="th:First_order"} is a generalization to the case when the gradient oracle $\gg(x, \xi)$ (see Definition [\[Definition_1\]](#Definition_1){reference-type="ref" reference="Definition_1"}) returns a noisy gradient value. If we put $\delta = 0$ we get exactly the same convergence as in Lemma [\[lem:Vaswani\]](#lem:Vaswani){reference-type="ref" reference="lem:Vaswani"} up to the constants $\rho$ from the condition of strongly growing (see Assumption [\[Ass:Gradient_noise\]](#Ass:Gradient_noise){reference-type="ref" reference="Ass:Gradient_noise"}) and $B$ the size of the batch. The two terms accounting for noise accumulation are standard for the accelerated algorithm (see, e.g., [@Gorbunov_2019; @Dvinskikh_2021; @Vasin_2023]) and can be found, for example, by using the ($\delta,L$)-oracle technique [@Gasnikov_ICML]. It is through the use of the batched technique in Theorem [\[th:First_order\]](#th:First_order){reference-type="ref" reference="th:First_order"} that we will able to achieve an optimal estimate on the iteration complexity $N = \mathcal{O}\left(\sqrt{\varepsilon^{-1} LR^2}\right)$ that will obtained from the first term, since it dominates the second term for a sufficiently large value of the batch size $B \geq \rho$. This result allows us to use this accelerated batched algorithm to create a gradient-free method for solving the black-box optimization problem under the condition of increased smoothness of the objective function $f$. A detailed proof of Theorem [\[th:First_order\]](#th:First_order){reference-type="ref" reference="th:First_order"} can be found in Appendix [\[app:proof_th1\]](#app:proof_th1){reference-type="ref" reference="app:proof_th1"}. # Main results {#sec:Main_results} In this section, we present the main result of our paper, namely a novel gradient-free method for solving the black-box optimization problem [\[eq:init_problem\]](#eq:init_problem){reference-type="eqref" reference="eq:init_problem"} with the condition that the objective function is not only smooth but also has a higher order of smoothness (i.e., the Assumption [\[Ass:Higher_order\]](#Ass:Higher_order){reference-type="ref" reference="Ass:Higher_order"} is satisfied). Our approach to create a gradient-free algorithm is to choose and use a gradient estimate $\gg(x, \ee)$ (an approximation of the gradient that will account for the increased smoothness of the function) instead of the real gradient oracle $\gg(x, \xi)$ see Definition [\[Definition_1\]](#Definition_1){reference-type="ref" reference="Definition_1"} in the accelerated batched first-order method. ## Gradient approximation {#subsec:Gradient_approximation} To solve a deterministic convex black-box optimization problem [\[eq:init_problem\]](#eq:init_problem){reference-type="eqref" reference="eq:init_problem"}, where the "black box" plays the role of a gradient-free oracle $\tilde{f}$, which is formally defined as follows: we assume that the oracle $\tilde{f}$ can only return the value of the objective function $f(x)$ at the requested point with some stochastic noise $\xi$: $$\label{eq:gradient_free_oracle} \Tilde{f} = f(x) + \xi,$$ where $\xi$ is stochastic, possibly adversarial, noise, $\expect{\xi^2} \leq \Delta^2$. Then we use the so-called "Kernel-based approximation", which was presented in 1990 in the paper [@Polyak_1990] and was recognized years later in a number of papers [@Bach_2016; @Akhavan_2020; @Novitskii_2021; @Lobanov_2023; @Akhavan_2023], as an approximation of the gradient that takes into account the information about the increased smoothness, has the following form: $$\label{eq:gradient_approx} \gg(x,\ee) = d \frac{f(x+ h r \ee) + \xi_1 - f(x - h r \ee) - \xi_2}{2 h} K(r) \ee,$$ where $h>0$ is a smoothing parameter, $\ee \in S_2^d(1)$ is a vector uniformly distributed on the Euclidean unit sphere, $r$ is a vector uniformly distributed on the interval $r \in [0,1]$, $K:~[-1,1]~\rightarrow~\mathbb{R}$ is a kernel function that satisfies $$\begin{gathered} \mathbb{E}[K(u)] = 0, \; \mathbb{E}[u K(u)] = 1, \; \mathbb{E}[u^j K(u)] = 0, \; j=2,...,l, \; \mathbb{E}[|u|^\beta |K(u)|] < \infty. \end{gathered}$$ This conception of noise is often found in the literature [@Akhavan_2020; @Lobanov_2023], where the $\xi_1 \neq \xi_2$ such that $\mathbb{E}[\xi_1^2] \leq \Delta^2$ and $\mathbb{E}[\xi_2^2] \leq \Delta^2$, $\Delta \geq 0$ is level noise, and the random variables $\xi_1$ and $\xi_2$ are independent from $\ee$ and $r$. Also, this concept does not necessarily have to have a zero mean $\xi_1$ and $\xi_2$. It is enough that $\mathbb{E}[\xi_1 \ee] = 0$ and $\mathbb{E}[\xi_2 \ee] = 0$. Moreover, the gradient approximation [\[eq:gradient_approx\]](#eq:gradient_approx){reference-type="eqref" reference="eq:gradient_approx"} may at first glance appear to be an approximation with two-point feedback because of the structure of the central finite difference, but this is not entirely true. Since $\xi_1 \neq \xi_2$ and if we consider $\xi_1$ and $\xi_2$ as concrete realizations of the objective function $f(x)$, it is clear that the function cannot be computed on the same realization twice per iteration. Thus, the approximation with this concept of gradient-free oracle [\[eq:gradient_free_oracle\]](#eq:gradient_free_oracle){reference-type="eqref" reference="eq:gradient_free_oracle"} is an approximation with one-point feedback. ## Zero-order accelerated stochastic gradient descent {#subsec:ZO_AccSGD} Now that we have chosen the gradient approximation and the accelerated batched first-order method, we can present a novel gradient-free algorithm Zero-Order Accelerated Stochastic Gradient Descent (ZO-AccSGD), which is obtained by replacing the real gradient with the gradient approximation [\[eq:gradient_approx\]](#eq:gradient_approx){reference-type="eqref" reference="eq:gradient_approx"}. iteration number $N$, batch size $B$, Kernel $K: [-1, 1] \rightarrow \mathbb{R}$, step size $\eta$, smoothing parameter $h$, $x_0=y_0=z_0~\in~\mathbb{R}^d$, $\alpha_0 = \gamma_0 = 0$.    Sample vectors $\ee_1, \ee_2 ..., \ee_B$ uniformly distributed on the unit sphere $S_2^d(1)$ and [1.]{style="color: white"}    scalars $r_1, r_2, ..., r_B$ uniformly distributed on the interval \[-1, 1\] independently    Define $\gg(x_k,\ee_i) = d \frac{\Tilde{f}(x_k+h r_i \ee_i) - \Tilde{f}(x_k - h r_i \ee_i)}{2 h} K(r_i) \ee_i$ via [\[eq:gradient_free_oracle\]](#eq:gradient_free_oracle){reference-type="eqref" reference="eq:gradient_free_oracle"}    Calculate $\gg_k = \frac{1}{B} \sum_{i=1}^B \gg (x_k,\ee_i)$    $x_{k+1} \gets y_k - \eta \gg_k$    $z_{k+1} \gets z_k - \gamma_k \eta \gg_k$    $y_{k + 1} \gets \alpha_{k + 1} z_{k + 1} + (1-\alpha_{k + 1}) x_{k + 1}$ $x_{N}$ To obtain the convergence rate of the Algorithm [\[algorithm\]](#algorithm){reference-type="ref" reference="algorithm"}, we need to first estimate the bias $\norms{\expect{\gg(x, \ee)} - \nabla f(x)}$ and second moment (variance) $\expect{\norms{\gg(x, \ee)}^2}$ of the gradient approximation $\gg(x, \ee)$ of [\[eq:gradient_approx\]](#eq:gradient_approx){reference-type="eqref" reference="eq:gradient_approx"}. Then, by substituting these estimates into the convergence result of the first-order algorithm we plan to rely on (in our case it is the Biased Accelerated Stochastic Gradient Descent, see Theorem [\[th:First_order\]](#th:First_order){reference-type="ref" reference="th:First_order"}), in particular instead of $\sigma^2$ from the second term we need to substitute the obtained estimate on the second moment (variance) $\expect{\norms{\gg(x, \ee)}^2}$, and instead of $\delta$ of the third and fourth terms we need to substitute the obtained estimate for the bias $\norms{\expect{\gg(x, \ee)} - \nabla f(x)}$, we get the convergence rate of the novel gradient-free algorithm. Then, using the bias and second moment estimates for the gradient approximation that takes into account information about the higher order of smoothness (Kernel approximation [\[eq:gradient_approx\]](#eq:gradient_approx){reference-type="eqref" reference="eq:gradient_approx"}) presented in [@Akhavan_2023] we have the following convergence results for Zero-Order Accelerated Stochastic Gradient Descent (see ZO-AccSGD, Algorithm [\[algorithm\]](#algorithm){reference-type="ref" reference="algorithm"}). [\[th:gradient_free\]]{#th:gradient_free label="th:gradient_free"} Let the function $f$ satisfy Assumption [\[Ass:Higher_order\]](#Ass:Higher_order){reference-type="ref" reference="Ass:Higher_order"} and the gradient approximation $\gg(x, \ee)$ of [\[eq:gradient_approx\]](#eq:gradient_approx){reference-type="eqref" reference="eq:gradient_approx"} satisfies Assumptions [\[Ass:Bounded_bias\]](#Ass:Bounded_bias){reference-type="ref" reference="Ass:Bounded_bias"} and [\[Ass:Gradient_noise\]](#Ass:Gradient_noise){reference-type="ref" reference="Ass:Gradient_noise"}, then Zero-Order Accelerated Stochastic Gradient Descent (see Algorithm [\[algorithm\]](#algorithm){reference-type="ref" reference="algorithm"}) with $\rho_B = \max \{ 1, \frac{4 d \kappa}{B} \}$, and with the chosen algorithm parameters: $$\begin{aligned} \gamma_{k} = \frac{\rho_B^{-1} + \sqrt{\rho_B^{-2} + 4\gamma_{k - 1}^{2}}}{2}; \quad a_{k + 1} = \gamma_k \sqrt{\eta \rho_B}; \quad \alpha_k = \frac{\gamma_{k} \eta}{\gamma_{k} \eta + a_k^2}; \quad \eta = \frac{1}{\rho_B L} \end{aligned}$$ converges to the desired $\varepsilon$ accuracy, $\expect{f(x_N)} - f^* \leq \varepsilon$ - in the case $B \in \left[1, 4d \kappa \right]$, $h \lesssim \varepsilon^{3/4}$ and $\beta \geq \frac{7}{3}$ after\ \ - *in the case* $B > 4d \kappa$ and $h \lesssim \varepsilon^{1/(\beta - 1)}$ after\ From the results of Theorem [\[th:gradient_free\]](#th:gradient_free){reference-type="ref" reference="th:gradient_free"}, it is not difficult to see that at the batch size $B = 1$ the dimensionality factor $d$ comes out in the iteration complexity. The dimensionality can be eliminated by batching, i.e. the larger the batch size $B$, the better the iteration complexity $N$ becomes, in particular, starting from $B = 4 d \kappa$ the iteration complexity completely gets rid of dimensionality and reaches the optimal estimate for the accelerated algorithm. It is worth noting that the maximum noise level when the batch size is $1 \leq B \leq 4d \kappa$, in particular when $\beta \geq \frac{7}{3}$ has, perhaps close to the optimal value for the smooth case (i.e., the case where the Assumption [\[ass:L_smooth\]](#ass:L_smooth){reference-type="ref" reference="ass:L_smooth"} holds): $\Delta \lesssim d^{-1/2} \varepsilon^{3/2}$. However, this estimate is invariant regardless of the order of smoothness. But there is a way to improve the maximum noise level at which the algorithm is still guaranteed to achieve the desired $\varepsilon$ accuracy. This method is called "overbatching". If we take the size of the batches larger than $B > 4 d \kappa$, then the maximum allowable noise level $\Delta$ will improve and, in particular, will depend on the order of smoothness $\beta$. That is, the maximum noise level can be maximized in two ways: taking a larger batch size or using a higher order function. However, improving the maximum noise level entails a deterioration of the oracle complexity, but has no effect on the iteration complexity $N \sim \varepsilon^{-1/2}$. It is not difficult to see that in any case considered, our results outperform all known results (see subsection Related works), in particular, we improve the iterative complexity estimate, as well as get rid of the dimensionality dependence of oracle complexity, and finally we present estimates of the maximum noise level as a function of batch size. For a detailed proof of Theorem [\[th:gradient_free\]](#th:gradient_free){reference-type="ref" reference="th:gradient_free"}, see Appendix [\[App:proof_th2\]](#App:proof_th2){reference-type="ref" reference="App:proof_th2"}. # Discussion and further work {#sec:Discussion} Section [4](#sec:Main_results){reference-type="ref" reference="sec:Main_results"} focuses on solving the convex deterministic black-box optimization problem [\[eq:init_problem\]](#eq:init_problem){reference-type="eqref" reference="eq:init_problem"}, however, when constructing the gradient-free algorithm (see Subsection [4.2](#subsec:ZO_AccSGD){reference-type="ref" reference="subsec:ZO_AccSGD"}) is based on a first-order method that solves the convex stochastic optimization problem [\[eq:stoc_init_problem\]](#eq:stoc_init_problem){reference-type="eqref" reference="eq:stoc_init_problem"} due to the arising of artificial "stochasticity" in the gradient approximation [\[eq:gradient_approx\]](#eq:gradient_approx){reference-type="eqref" reference="eq:gradient_approx"}. It is not difficult to show that the results of Theorem [\[th:gradient_free\]](#th:gradient_free){reference-type="ref" reference="th:gradient_free"} will be robust if the original problem of Section [4](#sec:Main_results){reference-type="ref" reference="sec:Main_results"} is replaced by a stochastic black-box optimization problem, since there will already be two stochasticities in the analysis that can be formally combined into one $(\xi, \ee)$. If we pay attention to the results presented in the works of [@Bach_2016; @Akhavan_2020; @Novitskii_2021] and others, we can see that they "struggle" for oracle complexity $T$. However, in high dimensional problems, it is important to be able to distribute the computational power loads, thereby reducing the time taken to solve a particular problem. Therefore, with the help of a not tricky technique, namely with the help of batching technique and using the accelerated algorithm as a base (in particular, Accelerated Stochastic Gradient Descent with accelerated of Nesterov, see Theorem [\[th:First_order\]](#th:First_order){reference-type="ref" reference="th:First_order"}), we managed to improve the estimate of the number of consecutive iterations $N \sim \varepsilon^{-1/2}$ to achieve the desired accuracy $\varepsilon$ of the solution of the original problem, without worsening the oracle complexity $T$, and moreover improving in terms of dimensionality $d$ for a class of convex optimization problems. It is due to this fact, namely the ability to improve one optimality criterion without compromising the second one, that recently authors of works on gradient-free optimization algorithms have been evaluating the efficiency of their algorithms by three optimality criteria at once [@Gasnikov_2022]: iteration complexity, total number of calls to the gradient-free oracle, and maximum noise level $\Delta$ at which it is still possible to achieve desired accuracy. We see the following directions as the development of this work: obtaining convergence results for a $\mu$-strongly convex black-box optimization problem. For this formulation of the problem there are already some results presented in [@Akhavan_2023], we expect that using similar reasoning, namely generalizing the convergence results of the accelerated algorithm [@Vaswani_2019] for solving a strongly convex stochastic optimization problem to the case with a biased gradient oracle (see Definition [\[Definition_1\]](#Definition_1){reference-type="ref" reference="Definition_1"}) and using the kernel approximation [\[eq:gradient_approx\]](#eq:gradient_approx){reference-type="eqref" reference="eq:gradient_approx"}, we will be able to improve the performance of [@Akhavan_2023] in terms of iteration complexity $N$, achieving the same oracle complexity estimates $T$, and provide an explicit condition on the maximum noise level $\Delta$. Another direction of development of our work is to improve oracle complexity $T$ for convex and strongly convex black-box optimization problem. It is worth noting that in the class of convex functions we managed to improve oracle complexity, but this upper bound does not match the lower bound presented in [@Akhavan_2020; @Novitskii_2021]. Finally, the last direction that looks promising at the moment is the study of the maximum allowable noise level. In this paper, we have provided a maximum noise level at which convergence to the desired accuracy $\varepsilon$ is guaranteed; however, we have not guaranteed the optimality of this estimate since the upper bound on the noise level is not yet known. Furthermore, we expect that this estimator can be improved by using a different concept of a gradient-free oracle, in particular when the oracle can output the objective function value with some bounded adversarial deterministic noise (see [@Dvinskikh_2022] for details). In this case, we can also expect an improvement in oracle complexity, since the gradient approximation with a central finite difference structure will already have access to two-point feedback. # Experiments {#sec:Experiments} In this section, we verify the performance of the proposed gradient-free algorithm in Section [4](#sec:Main_results){reference-type="ref" reference="sec:Main_results"}: Zero-Order Accelerated Stochastic Gradient Descent (see Algorithm 1) on a standard optimization problem that is often encountered in application problems such as binary classification. To demonstrate the effectiveness of our algorithm, we compare it to stochastic gradient descent, and demonstrate the effect of the size of the batches on the convergence rate. Then our optimization problem [\[eq:init_problem\]](#eq:init_problem){reference-type="eqref" reference="eq:init_problem"} takes the following form: $\min_{w \in Q} f(w)$, where $$f(w) := -y \log \left[ \frac{1}{1 + \exp \left( -w^{\text{T}} X \right)}\right] + \left( 1 - y \right) \log \left[ 1 - \frac{1}{1 + \exp \left( -w^{\text{T}} X \right)} \right],$$ with $w \in \mathbb{R}^d$ is vector of weights, $X \in \mathbb{R}^{d \times n}$ is a dataset of $n$ points of dimension $d$, $y \in \mathbb{R}^n$ is vector of labels. For brevity of notation, we assume that the operation of logarithm, exponent, and division is taken coordinate-wise. Then as the Kernel $K(r)$ of gradient approximation [\[eq:gradient_approx\]](#eq:gradient_approx){reference-type="eqref" reference="eq:gradient_approx"} we use the already standard function, namely Legendre polynomials, for which it was proved in the paper [@Bach_2016] that the constants $\kappa$ and $\kappa_\beta$ do not depend on the dimensionality, but only on the smoothness order of $\beta$. We have the following values for different $\beta$: $$\begin{aligned} K(r) &= \frac{15r}{4}(5 - 7r^2) & \text{for } \beta = 3, 4;\\ K(r) &= \frac{195r}{16}(99r^4 - 126r^2 + 35) & \text{for } \beta = 5, 6.\end{aligned}$$ In Figure [\[ris:image1\]](#ris:image1){reference-type="ref" reference="ris:image1"}, a) we compare the performance of the Zero-Order Accelerated Stochastic Gradient Descent proposed in Section [4](#sec:Main_results){reference-type="ref" reference="sec:Main_results"} of this paper, ZO-AccSGD, with the gradient-free Stochastic Gradient Descent method (ZO-SGD). It is not hard to see that Algorithm [\[algorithm\]](#algorithm){reference-type="ref" reference="algorithm"} significantly outperforms its unaccelerated counterpart in terms of the number of iterations, requiring for convergence to accuracy $\varepsilon$, thus confirming the results of Theorem [\[th:gradient_free\]](#th:gradient_free){reference-type="ref" reference="th:gradient_free"}. However, it is not surprising that Algorithm [\[algorithm\]](#algorithm){reference-type="ref" reference="algorithm"} is inferior to the accelerated first-order algorithm. Figure [\[ris:image1\]](#ris:image1){reference-type="ref" reference="ris:image1"},b) shows the effect of the batch size $B$ on the convergence of the gradient-free algorithm ZO-AccSGD. It is not difficult to see that indeed, using a sufficiently large batch size, one can improve the iteration complexity, in particular by getting rid of the constant $\rho$ from the strong growth condition (see Assumption [\[Ass:Gradient_noise\]](#Ass:Gradient_noise){reference-type="ref" reference="Ass:Gradient_noise"}). Following from Theorem [\[th:First_order\]](#th:First_order){reference-type="ref" reference="th:First_order"}, we can see that $\rho$ is proportional to the dimensionality $d$ of the problem, i.e., the iteration complexity will improve until the batch size $B$ reaches the order of the dimensionality. # Conclusion {#sec:Conclusion} In this paper, we proposed a novel gradient-free algorithm (see Algorithm [\[algorithm\]](#algorithm){reference-type="ref" reference="algorithm"}) that improves the iteration $N$, oracle complexities $T$, and explicitly defines the maximum noise level $\Delta$ at which the algorithm can still be guaranteed to converge to the desired $\varepsilon$ accuracy in a class of convex black-box optimization problem where the function is not just smooth but has a higher order of smoothness. Our approach for the gradient-free algorithm was based on the work of [@Vaswani_2019], however, due to the biased gradient approximation (Kernel approximation), we generalized the convergence results of this work to the gradient oracle with bias, and applied a batting technique to improve the first term in the convergence of [@Vaswani_2019] (this result may be of independent interest). In the experiments section, we confirmed our theoretical results obtained in this paper. In addition, we considered possible developments of this paper.

arxiv_math

--- abstract: | Renewed interest in the relationship between artificial and biological neural networks motivates the study of gradient-free methods. Considering the linear regression model with random design, we theoretically analyze in this work the biologically motivated (weight-perturbed) forward gradient scheme that is based on random linear combination of the gradient. If $d$ denotes the number of parameters and $k$ the number of samples, we prove that the mean squared error of this method converges for $k\gtrsim d^2\log(d)$ with rate $d^2\log(d)/k.$ Compared to the dimension dependence $d$ for stochastic gradient descent, an additional factor $d\log(d)$ occurs. author: - Thijs Bos  and Johannes Schmidt-Hieber bibliography: - AlternativeOptimationforRegressionLibrary.bib title: Convergence guarantees for forward gradient descent in the linear regression model --- **Keywords:** Convergence rates, estimation, gradient descent, linear model, zeroth-order methods. **MSC 2020:** Primary: 62L20; secondary: 62J05 # Introduction Looking at the past developments, it is apparent that artificial neural networks (ANNs) became more powerful the more they resembled the brain. It is therefore anticipated that the future of AI is even more biologically inspired. As in the past, the bottlenecks towards more biologically inspired learning are computational barriers. For instance, shallow networks only became computationally feasible after the backpropagation algorithm was proposed. Deep neural networks were proposed for a longer time but deep learning became implementable after the development of large scale GPU computing. Neuromorphic computing aims to imitate the brain on computer chips, but is currently not fully scalable. The mathematics of AI has focused on explaining the state-of-the-art performance of modern machine learning methods and empirically observed phenomena such as the good generalization properties of extreme overparametrization. To shape the future of AI, statistical theory needs more emphasis on anticipating future developments and proposing biologically motivated methods already at a stage before scalable implementations exist. This work aims to analyze a biologically motivated learning rule building on the renewed interest of the differences and similarities between ANNs and biological neural networks (BNNs) [@BackpropAndBrain; @schmidt2023interpreting; @LocalHebbianPlasticity] which are rooted in the foundational literature from the 1980s [@GROSSBERG198723; @ExcitementCrick]. A key difference between ANNs and BNNs is that ANNs are usually trained based on a version of (stochastic) gradient descent, while this seems prohibitive for BNNs. Indeed, to compute the gradient, knowledge of all parameters in the network is required, but biological networks do not posses the capacity to transport this information to each neuron. This suggests that biological networks cannot directly use the gradient to update their parameters [@ExcitementCrick; @BackpropAndBrain; @FundamentalsCompNeuro]. The brain still performs well without gradient descent and can learn tasks with much fewer examples than ANNs. This sparks interest in biologically plausible learning methods that do not require (full) access of the gradient. Such methods are called derivative-free. A simple example of a derivative-free method is to randomly sample in each step a new parameter. If this decreases the loss one keeps the parameter and otherwise discards it. There is a wide variety of derivative-free strategies [@IntroDerivativeFreeConnea; @DerivativeFreeLarsonea; @IntroStochasticSearch]. Among those, so-called zeroth-order methods use evaluations of the loss function to build a noisy estimate of the gradient. This substitute is then used to replace the gradient in the gradient descent routine [@liu2020primer; @ZeroOrderRates]. [@schmidt2023interpreting] establishes a connection between the Hebbian learning underlying the local learning of the brain (see e.g. Chapter 6 of [@FundamentalsCompNeuro]) and a specific zeroth-order method. A statistical analysis of this zeroth-order scheme is provided in the companion article [@SHKoolen2023]. In this article, we study (weight-perturbed) forward gradient descent. This method is motivated by biological neural networks [@baydin2022gradients; @ren2022scaling] and lies between full gradient descent methods and derivative-free methods, as only random linear combination of the gradient are required. The form of the random linear combination is related to zeroth-order methods, see Section [2](#S: Model Description){reference-type="ref" reference="S: Model Description"}. Settings with partial access to the gradient have been studied before. For example, [@GradientFreeMinimization] proposes a learning method based on directional derivatives for convex functions. In this work we specifically derive theoretical guarantees for forward gradient descent in the linear regression model with random design. Theorem [Theorem 1](#T: Expectations){reference-type="ref" reference="T: Expectations"} establishes an expression for the expectation. A bound on the mean squared error is provided in Theorem [Theorem 3](#T:Convergence bound){reference-type="ref" reference="T:Convergence bound"}. The structure of the paper is as follows. In Section [2](#S: Model Description){reference-type="ref" reference="S: Model Description"} we describe the linear regression model and define the update rule. We present the main theorems and some discussion thereof in Section [3](#S: Main Results){reference-type="ref" reference="S: Main Results"}. Proofs can be found in Section [4](#S: Proofs){reference-type="ref" reference="S: Proofs"}. ### Notation {#notation .unnumbered} Vectors are denoted by bold letters and we write $\|\cdot\|_2$ for the Euclidean norm. We denote the largest and smallest eigenvalue of a matrix $A$ by the respective expressions $\lambda_{\max}(A)$ and $\lambda_{\min}(A)$. The spectral norm is $\|A\|_S:=\sqrt{\lambda_{\max}(A^\top A)}.$ The condition number of a positive semi-definite matrix $B$ is $\kappa(B):=\lambda_{\max}(B)/\lambda_{\min}(B).$ For a random variable $U$ we denote the expectation with respect to $U$ by $\mathbb{E}_U.$ The symbol $\mathbb{E}$ stands for an expectation taken with respect to all random variables that are inside that expectation. The (multivariate) normal distribution with mean vector $\mu$ and covariance matrix $\Sigma$ is denoted by $\mathcal{N}(\mu,\Sigma).$ # Weight-perturbed forward gradient descent {#S: Model Description} Suppose we want to learn a parameter vector $\bm{\theta}$ from training data $(\mathbf{X}_1,Y_1),(\mathbf{X}_2,Y_2),\ldots \in \mathbb{R}^{d}\times \mathbb{R}.$ Stochastic gradient descent (SGD) is based on the iterative update rule $$\bm{\theta}_{k+1}=\bm{\theta}_k-\alpha_{k+1}\nabla L(\bm{\theta}_k), \quad k=0,1,\ldots \label{eq.SGD}$$ with $\bm{\theta}_0$ some initial value and $L(\bm{\theta}_k):=L(\bm{\theta}_k,\mathbf{X}_k,Y_k)$ a loss that depends on the data only through the $k$-th sample $(\mathbf{X}_k, Y_k)$. For a standard normal random vector $\bm{\xi}_{k+1}\sim \mathcal{N}(0,\mathbf{I}_d)$ that is independent of all the other randomness, the quantity $(\nabla L(\bm{\theta}_k))^\top \bm{\xi}_{k+1} \bm{\xi}_{k+1}$ is called the (weight-perturbed) forward gradient [@baydin2022gradients; @ren2022scaling]. *(Weight-perturbed) forward gradient descent* is then given by the update rule $$\label{Eq: Update rule} \bm{\theta}_{k+1}=\bm{\theta}_k-\alpha_{k+1}\big(\nabla L(\bm{\theta}_k)\big)^\top \bm{\xi}_{k+1} \bm{\xi}_{k+1}, \quad k=0,1,\ldots$$ Assuming that the exogenous noise has unit variance is sufficient as generalizing to $\bm{\xi}_{k+1}\sim \mathcal{N}(0,\sigma^2 \mathbf{I}_d)$ with variance parameter $\sigma^2$ does not add flexibility to the procedure. Indeed, by rescaling the learning rate $\alpha_{k+1}\to \sigma^{-2}\alpha_{k+1},$ we recover [\[Eq: Update rule\]](#Eq: Update rule){reference-type="eqref" reference="Eq: Update rule"}. Since for a deterministic $d$-dimensional vector $\mathbf{v},$ one has $\mathbb{E}[\mathbf{v}^t\bm{\xi}_{k+1}\bm{\xi}_{k+1}]=\mathbf{v},$ taking the expectation of the weight-perturbed forward gradient descent scheme with respect to the exogenous randomness induced by $\bm{\xi}_1,\bm{\xi}_2,\ldots$ gives $$\begin{aligned} \mathbb{E}_{(\bm{\xi}_i)_{i\geq 1}}[\bm{\theta}_{k+1}]=\mathbb{E}_{(\bm{\xi}_i)_{i\geq 1}}[\bm{\theta}_k]-\alpha_{k+1} \mathbb{E}_{(\bm{\xi}_i)_{i\geq 1}}[\nabla L(\bm{\theta}_k)], \label{eq.fGD_expectation}\end{aligned}$$ resembling the SGD dynamic [\[eq.SGD\]](#eq.SGD){reference-type="eqref" reference="eq.SGD"}. If $\nabla L(\bm{\theta}_k)$ depends on $\bm{\theta}_k$ linearly then also $\mathbb{E}_{(\bm{\xi}_i)_{i\geq 1}}[\nabla L(\bm{\theta}_k)]= \nabla L(\mathbb{E}_{(\bm{\xi}_i)_{i\geq 1}}[\bm{\theta}_k]).$ While in expectation, forward gradient descent is related to SGD, the induced randomness of the $d$-dimensional random vectors $\mathbf{x}_{k+1}$ induces a large amount of noise. To control the high noise level in the dynamic is the main obstacle in the mathematical analysis. One of the implications is that one has to make small steps by choosing a small learning rate to avoid completely erratic behavior. This particularly effects the first phase of the learning. First order multivariate Taylor expansion shows that $L(\bm{\theta}_k+\bm{\xi}_k)-L(\bm{\theta}_k)$ and $(\nabla L(\bm{\theta}_k))^\top \bm{\xi}_{k+1}$ are close. Therefore, forward gradient descent is related to the zeroth-order method $$\bm{\theta}_{k+1}=\bm{\theta}_k-\alpha_{k+1} \big(L(\bm{\theta}_k+\bm{\xi}_k)-L(\bm{\theta}_k)\big)\bm{\xi}_k,$$ [@liu2020primer; @schmidt2023interpreting]. Consequently, forward gradient descent can be viewed as an intermediate step between gradient descent, with full access to the gradient, and zeroth-order methods that are solely based on (randomly) perturbed function evaluations. To complete this section, we briefly compare forward gradient descent with feedback alignment as both methods are motivated by biological learning and are based on additional randomness. As mentioned in the introduction, the brain cannot do gradient descent. Backpropagation is an algorithm to compute the gradient and consists of a forward pass and a backward pass. While biological neural networks can execute a forward pass and evaluate the loss for a training sample, issues arise in the implementation of the backward pass. Inspired by biological learning, feedback alignment proposes to replace the learned weights in the backward pass by random weights chosen at the start of the training procedure [@lillicrap2016random; @BackpropAndBrain]. The so-called direct feedback alignment method goes even further: instead of back-propagating the gradient through all the layers of the network by the chain-rule, layers are updated with the gradient of the output layer multiplied with a fixed random weight matrix [@NIPS2016_d490d7b4; @NEURIPS2020_69d1fc78]. (Direct) feedback alignment causes the forward weights to change in such a way that the true gradient of the network weights and the substitutes used in the update rule become more aligned [@lillicrap2016random; @NIPS2016_d490d7b4; @BackpropAndBrain]. The linear model can be viewed as neural network without hidden layers. The absence of layers means that in the backward step, no weight information is transported between different layers. As a consequence, both feedback alignment and direct feedback alignment collapse in the linear model into standard gradient descent. The conclusion is that feedback alignment and forward gradient descent are not comparable. The argument also shows that to unveil nontrivial statistical properties of feedback alignment, one has to go beyond the linear model. We leave the statistical analysis as an open problem. # Convergence rates in the linear regression model {#S: Main Results} We analyze weight-perturbed forward gradient descent for data generated from the $d$-dimensional linear regression with Gaussian random design. In this framework, we observe i.i.d. pairs $(\mathbf{X}_i,Y_i)\in \mathbb{R}^d\times\mathbb{R},$ $i=1,2,\ldots$ satisfying $$\label{Eq: Regression model} \mathbf{X}_i\sim \mathcal{N}(0,\Sigma), \quad Y_i=\mathbf{X}_i^\top\bm{\theta}_{\star}+\epsilon_i, \quad i=1,2,\ldots$$ with $\bm{\theta}_{\star}$ the unknown $d$-dimensional regression vector, $\Sigma$ an unknown covariance matrix, and independent noise variables $\epsilon_i$ with mean zero and variance one. For the analysis, we consider the squared loss $L(\bm{\theta}_k,\mathbf{X}_k,Y_k)=\tfrac 12 (Y_k- \mathbf{X}_k^\top\bm{\theta}_k)^2.$ The gradient is given by $$\begin{aligned} \nabla L(\bm{\theta}_k) = -\big(Y_k-\mathbf{X}_k^\top \bm{\theta}_k \big)\mathbf{X}_k. \label{eq.grad_loss}\end{aligned}$$ We now analyze the forward gradient estimator assuming that the initial value $\bm{\theta}_0$ can be random or deterministic but should be independent of the data. We employ a similar proving strategy as in the recent analysis of dropout in the linear model in [@2023arXiv230610529C]. In particular, we will derive a recursive formula for $\mathbb{E}\left[(\bm{\theta}_k-\bm{\theta}_{\star})(\bm{\theta}_k-\bm{\theta}_{\star})^\top\right].$ In contrast to this work, we consider a different form of noise and non-constant learning rates. The first result shows that forward gradient descent does gradient descent in expectation. **Theorem 1**. *We have $\mathbb{E}[\bm{\theta}_{k}]-\bm{\theta}_{\star}=\big(\mathbf{I}_d-\alpha_k\Sigma\big)\big(\mathbb{E}[\bm{\theta}_{k-1}]-\bm{\theta}_{\star}\big)$ and thus $$\label{Eq: Expectation single entry squared loss covariance matrix case non recursive} \mathbb{E}[\bm{\theta}_k]=\bm{\theta}_{\star}+\left(\prod_{\ell=1}^k(\mathbf{I}_d-\alpha_{\ell}\Sigma)\right)\big(\mathbb{E}[\bm{\theta}_0]-\bm{\theta}_{\star}\big).$$* The proof does not exploit the Gaussian design and only requires that $\mathbf{X}_{i}$ is centered and has covariance matrix $\Sigma$. The exogenous randomness induced by $\bm{\xi}_1,\bm{\xi}_2,\ldots$ disappears in the expected values but heavily influences the recursive expressions for the squared expectations. **Theorem 2**. *Consider forward gradient descent [\[Eq: Update rule\]](#Eq: Update rule){reference-type="eqref" reference="Eq: Update rule"}. If $A_k:=\mathbb{E}\left[(\bm{\theta}_k-\bm{\theta}_{\star})(\bm{\theta}_k-\bm{\theta}_{\star})^\top\right],$ then $$\begin{aligned} A_k=&(\mathbf{I}_d-\alpha_{k}\Sigma)A_{k-1}(\mathbf{I}_d-\alpha_k\Sigma)\\ &+3\alpha_k^2\Sigma A_{k-1}\Sigma+2\alpha_{k}^2\mathbb{E}\big[(\bm{\theta}_{k-1}-\bm{\theta}_{\star})^\top\Sigma(\bm{\theta}_{k-1}-\bm{\theta}_{\star})\big]\Sigma+2\alpha_{k}^2\Sigma\\ &+2\alpha_k^2\operatorname{tr}\big(\Sigma A_{k-1}\Sigma\big)\mathbf{I}_d+\alpha_{k}^2\mathbb{E}\big[(\bm{\theta}_{k-1}-\bm{\theta}_{\star})^\top\Sigma(\bm{\theta}_{k-1}-\bm{\theta}_{\star})\big]\operatorname{tr}\big(\Sigma\big)\mathbf{I}_d\\ &+\alpha_k^2\operatorname{tr}(\Sigma)\mathbf{I}_d. \end{aligned}$$* Since $A_k$ depends on $\bm{\theta}_k^2$, the fourth moments of the design vectors $\mathbf{X}_i$ and the exogenous random vectors $\bm{\xi}_k$ play a role in this equation. The risk $\mathbb{E}\big[\|\bm{\theta}_k-\bm{\theta}_{\star}\|_2^2\big]$ is the trace of the matrix $A_k$. Setting $$\begin{aligned} \kappa(\Sigma):= \frac{\|\Sigma\|_S}{\lambda_{\min}(\Sigma)}\end{aligned}$$ for the condition number and building on Theorem [Theorem 2](#T:Covariance){reference-type="ref" reference="T:Covariance"}, we can establish the following risk bound for forward gradient descent. **Theorem 3** (Mean squared error). *Consider forward gradient descent [\[Eq: Update rule\]](#Eq: Update rule){reference-type="eqref" reference="Eq: Update rule"} and assume that $\Sigma$ is positive definite. For constant $a>2,$ choosing the learning rate $$\label{Eq: Combined choice of alpha_k} \begin{aligned} \alpha_k=\frac{a\lambda_{\min}(\Sigma)}{k\lambda^2_{\min}(\Sigma)+a\|\Sigma\|_S^2(d+2)^2}, \quad k=1,2,\ldots, \end{aligned}$$ yields $$\begin{aligned} \mathbb{E}\big[\big\|\bm{\theta}_k-\bm{\theta}_{\star}\big\|_2^2\big]&\leq \Bigg(\frac{1+a\kappa^2(\Sigma)(d+2)^2}{k+a\kappa^2(\Sigma)(d+2)^2}\Bigg)^a\mathbb{E}\big[\big\|\bm{\theta}_{0}-\bm{\theta}_{\star}\big\|_2^2\big] +\frac{2ea \kappa(\Sigma) (d+2)^2}{\lambda_{\min}(\Sigma)(k+a\kappa^2(\Sigma)(d+2)^2)}. \end{aligned}$$* Alternatively, the upper bound of Theorem [Theorem 3](#T:Convergence bound){reference-type="ref" reference="T:Convergence bound"} can be written as $$\begin{aligned} \mathbb{E}\big[\|\bm{\theta}_k-\bm{\theta}_{\star}\|_2^2\big]&\leq \Big(1-a^{-1}\lambda_{\min}(\Sigma)(k-1)\alpha_{k}\Big)^a\mathbb{E}\big[\|\bm{\theta}_{0}-\bm{\theta}_{\star}\|_2^2\big]+2e\kappa(\Sigma)(d+2)^2\alpha_{k}. \end{aligned}$$ In the upper bound, the risk $\mathbb{E}\big[\|\bm{\theta}_{0}-\bm{\theta}_{\star}\|_2^2\big]$ of the initial estimate $\bm{\theta}_0$ appears. A realistic scenario is that the entries of $\bm{\theta}_\star$ and $\bm{\theta}_0$ are all of order one. In this case, the inequality $\|\bm{\theta}_{0}-\bm{\theta}_{\star}\|_2^2\leq d\|\bm{\theta}_{0}-\bm{\theta}_{\star}\|_{\infty}^2$ shows that the risk of the initial estimate will scale with the number of parameters $d$. Taking $a=\log(d)$ (for $d\geq 8> e^2$ such that $a>\log(e^2)=2$), Theorem [Theorem 3](#T:Convergence bound){reference-type="ref" reference="T:Convergence bound"} implies that $$\mathbb{E}\big[\|\bm{\theta}_k-\bm{\theta}_{\star}\|_2^2\big]\lesssim d\Bigg(\frac{d^2\log(d)}{k}\Bigg)^{\log(d)}\mathbb{E}\big[\|\bm{\theta}_{0}-\bm{\theta}_{\star}\|_{\infty}^2\big]+\frac{d^2\log(d)}{k}.$$ For $k_\star=e^2d^2\log(d),$ $d^2\log(d)/k_\star=e^{-2}$ and $d(d^2\log(d)/k_\star)^{\log(d)}=1/d$. Since $d>e^2$, this means that $d\big(d^2\log(d)/k_\star\big)^{\log(d)}<d^2\log(d)/k_\star.$ Moreover, $k^{-\log(d)}$ tends faster to zero than $k^{-1}$ as $k\rightarrow\infty$. So, for $k\geq k_\star=e^2d^2\log(d),$ $$d\Bigg(\frac{d^2\log(d)}{k}\Bigg)^{\log(d)}\mathbb{E}\big[\|\bm{\theta}_{0}-\bm{\theta}_{\star}\|_{\infty}^2\big]+\frac{d^2\log(d)}{k}\leq \frac{d^2\log(d)}{k}\Big(1+\mathbb{E}\big[\|\bm{\theta}_{0}-\bm{\theta}_{\star}\|_{\infty}^2\big]\Big).$$ The rate for $k\geq e^2d^2\log(d)$ is thus $d^2\log(d)/k.$ This means that forward gradient descent has dimension dependence $d^2\log(d).$ This is by a factor $d\log(d)$ worse than the minimax rate for the linear regression problem, [@OptimalRatesAggregation; @RandomDesignRidge; @ExactminimaxLinear]. In contrast, methods that have access to the gradient can achieve optimal dimension dependence in the rate, [@AccelerationAveraging; @LSAHowFarConstantGo]. The obtained convergence rate is in line with results for zeroth-order methods, which show that for convex optimization problems these methods have a higher dimension dependence, [@ZeroOrderRates; @liu2020primer; @GradientFreeMinimization]. Assuming that the covariance matrix $\Sigma$ is positive definite is standard for linear regression with random design [@RandomDesignRidge; @ExactminimaxLinear; @GMTheoremRandomDesign]. For $k\gtrsim d^2,$ the decrease of the learning rate $\alpha_k$ is is of the order $1/k$, which is the standard choice [@MR1993642; @AveragedGyorfi; @AdaptiveAlgoStochasticApprox]. A constant learning rate is used for Ruppert-Polyak averaging in [@AccelerationAveraging; @AveragedGyorfi]. For least squares linear regression, it is possible to achieve (near) optimal convergence with a constant (universal) stepsize [@bach2013non]. Conditions under which a constant (universal) stepsize in more general settings than linear least squares works or fails are investigated in [@LSAHowFarConstantGo]. # Proofs {#S: Proofs} *Proof of Theorem [Theorem 1](#T: Expectations){reference-type="ref" reference="T: Expectations"}.* By [\[eq.grad_loss\]](#eq.grad_loss){reference-type="eqref" reference="eq.grad_loss"} and the linear regression model $Y_{k-1}=\mathbf{X}_{k-1}^\top\bm{\theta}_{\star}+\epsilon_{k-1}$, we have $$\begin{aligned} \begin{split} \nabla L(\bm{\theta}_{k-1}) &=-(Y_{k-1}-\mathbf{X}_{k-1}^\top\bm{\theta}_{k-1})\mathbf{X}_{k-1} \\ &=-(\mathbf{X}_{k-1}^\top(\bm{\theta}_\star-\bm{\theta}_{k-1})+\epsilon_{k-1})\mathbf{X}_{k-1} \\ &=- \epsilon_{k-1} \mathbf{X}_{k-1}-2 \mathbf{X}_{k-1} \mathbf{X}_{k-1}^\top(\bm{\theta}_\star-\bm{\theta}_{k-1}). \end{split} \label{eq.grad_Ltheta}\end{aligned}$$ Since $\mathbb{E}[\mathbf{X}_{k-1} \mathbf{X}_{k-1}^\top]=\Sigma,$ $\mathbb{E}[\epsilon_{k-1}]=0,$ and $\mathbf{X}_{k-1},\epsilon_{k-1},\bm{\theta}_{k-1}$ are jointly independent, we obtain $$\begin{aligned} \begin{split} \mathbb{E}\big[\nabla L(\bm{\theta}_{k-1})\, \big| \, \bm{\theta}_{k-1}\big] &= \mathbb{E}\big[ - \epsilon_{k-1} \mathbf{X}_{k-1}- \mathbf{X}_{k-1} \mathbf{X}_{k-1}^\top(\bm{\theta}_\star-\bm{\theta}_{k-1})\, \big| \, \bm{\theta}_{k-1}\big] \\ &= - \Sigma (\bm{\theta}_\star-\bm{\theta}_{k-1}). \end{split} \label{eq.grad_cond}\end{aligned}$$ Combined with [\[eq.fGD_expectation\]](#eq.fGD_expectation){reference-type="eqref" reference="eq.fGD_expectation"}, we find $$\begin{aligned} \mathbb{E}\big[\bm{\theta}_k\big] &=\mathbb{E}\big[\bm{\theta}_{k-1}\big]-\alpha_k \mathbb{E}\big[\nabla L(\bm{\theta}_{k-1})\big] = \mathbb{E}\big[\bm{\theta}_{k-1}\big]+\alpha_k\Sigma \mathbb{E}\big[\bm{\theta}_\star-\bm{\theta}_{k-1}\big].\end{aligned}$$ The true parameter $\bm{\theta}_\star$ is deterministic. Subtracting $\bm{\theta}_\star$ on both sides, yields the claimed identity $\mathbb{E}[\bm{\theta}_{k}]-\bm{\theta}_{\star}=\big(\mathbf{I}_d-\alpha_k\Sigma\big)\big(\mathbb{E}[\bm{\theta}_{k-1}]-\bm{\theta}_{\star}\big).$ ◻ ## Proof of Theorem [Theorem 2](#T:Covariance){reference-type="ref" reference="T:Covariance"} {#proof-of-theorem-tcovariance} **Lemma 4**. *If $\mathbf{Z}\sim \mathcal{N}(0,\Gamma)$ is a $d$-dimensional random vector and $\mathbf{U}$ is a $d$-dimensional random vector that is independent of $\mathbf{Z},$ then $$\mathbb{E}\big[(\mathbf{U}^\top\mathbf{Z})^2\mathbf{Z}\mathbf{Z}^\top\big]=2\Gamma\mathbb{E}\big[\mathbf{U}\mathbf{U}^\top\big]\Gamma+\mathbb{E}\big[\mathbf{U}^\top\Gamma\mathbf{U}\big]\Gamma.$$* *Proof.* Because $\mathbf{U}$ and $\mathbf{Z}$ are independent, the $(i,j)$-th entry of the $d\times d$ matrix $\mathbb{E}\big[(\mathbf{U}^\top\mathbf{Z})^2\mathbf{Z}\mathbf{Z}^\top\big]$ is $$\begin{aligned} \sum_{\ell,m=1}^d\mathbb{E}\big[U_{\ell}U_{m}\big]\mathbb{E}\big[Z_{\ell}Z_mZ_iZ_j\big]. \end{aligned}$$ Since $\mathbf{Z}\sim\mathcal{N}(0,\Gamma),$ $$\begin{aligned} \mathbb{E}\big[Z_{\ell}Z_mZ_iZ_j\big] &=\Gamma_{\ell,m}\Gamma_{i,j}+\Gamma_{\ell,i}\Gamma_{m,j}+\Gamma_{\ell,j}\Gamma_{m,i}, \end{aligned}$$ see for instance the example at the end of Section 2 in [@GaussianMoments]. Thus $$\begin{aligned} &\sum_{\ell,m=1}^d\mathbb{E}\big[U_{\ell}U_{m}\big]\mathbb{E}\big[Z_{\ell}Z_mZ_iZ_j\big]=\sum_{\ell,m=1}^d\mathbb{E}\big[U_{\ell}U_{m}\big]\big(\Gamma_{\ell,m}\Gamma_{i,j}+\Gamma_{\ell,i}\Gamma_{m,j}+\Gamma_{\ell,j}\Gamma_{m,i}\big). \end{aligned}$$ Because of $$\begin{aligned} \sum_{\ell,m=1}^d\mathbb{E}\big[U_{\ell}U_{m}\big]\Gamma_{\ell,m}\Gamma_{i,j} &=\sum_{\ell,m=1}^d\mathbb{E}\big[U_{\ell}\Gamma_{\ell,m}U_{m}\big]\Gamma_{i,j}=\mathbb{E}\left[\mathbf{U}^\top\Gamma\mathbf{U}\Gamma_{i,j}\right],\\ \sum_{\ell,m=1}^d\mathbb{E}\big[U_{\ell}U_{m}\big]\Gamma_{\ell,i}\Gamma_{m,j} &=\sum_{\ell,m=1}^d\mathbb{E}\big[U_{\ell}\Gamma_{\ell,i}U_{m}\Gamma_{m,j}\big]=\mathbb{E}\left[\big(\mathbf{U}^\top\Gamma\big)_i\big(\mathbf{U}^\top\Gamma\big)_j\right],\end{aligned}$$ and $$\sum_{\ell,m=1}^d\mathbb{E}\big[U_{\ell}U_{m}\big]\Gamma_{\ell,j}\Gamma_{m,i}= \sum_{\ell,m=1}^d\mathbb{E}\big[U_{m}\Gamma_{m,i}U_{\ell}\Gamma_{\ell,j}\big]=\mathbb{E}\left[\big(\mathbf{U}^\top\Gamma\big)_i\big(\mathbf{U}^\top\Gamma\big)_j\right],$$ the $(i,j)$-th entry of the matrix $\mathbb{E}\left[(\mathbf{U}^\top\mathbf{Z})^2\mathbf{Z}\mathbf{Z}^\top\right]$ is $$2\mathbb{E}\left[\big(\mathbf{U}^\top\Gamma\big)_i\big(\mathbf{U}^\top\Gamma\big)_j\right]+\mathbb{E}\left[\mathbf{U}^\top\Gamma\mathbf{U}\Gamma_{i,j}\right].$$ For a vector $\mathbf{a}=(a_1,\ldots,a_d)^\top,$ the scalar $a_ia_j$ is the $(i,j)$-th entry of the matrix $\mathbf{a}\mathbf{a}^\top.$ Combined with the previous display, the result follows. ◻ *Proof of Theorem [Theorem 2](#T:Covariance){reference-type="ref" reference="T:Covariance"}.* As Theorem [Theorem 2](#T:Covariance){reference-type="ref" reference="T:Covariance"} only involves one update step, we can simplify the notation by dropping the index $k$ and analyzing $\bm{\theta}''=\bm{\theta}'-\alpha \big(\nabla L(\bm{\theta}')\big)^\top \bm{\xi}\bm{\xi}$ for one data point $(\mathbf{X},Y)$ and independent $\bm{\xi}\sim \mathcal{N}(0,I_d).$ With $A':=\mathbb{E}\big[(\bm{\theta}'-\bm{\theta}_\star)(\bm{\theta}'-\bm{\theta}_\star)^\top\big]$ and $A'':=\mathbb{E}\big[(\bm{\theta}''-\bm{\theta}_\star)(\bm{\theta}''-\bm{\theta}_\star)^\top\big]$, we then have to prove that $$\begin{aligned} A''=&(\mathbf{I}_d-\alpha\Sigma)A'(\mathbf{I}_d-\alpha\Sigma)+3\alpha^2\Sigma A'\Sigma+2\alpha^2\mathbb{E}\big[(\bm{\theta}'-\bm{\theta}_{\star})^\top\Sigma(\bm{\theta}'-\bm{\theta}_{\star})\big]\Sigma+2\alpha^2\Sigma\\ &+2\alpha^2\operatorname{tr}\big(\Sigma A'\Sigma\big)\mathbf{I}_d+\alpha^2\mathbb{E}\big[(\bm{\theta}'-\bm{\theta}_{\star})^\top\Sigma(\bm{\theta}'-\bm{\theta}_{\star})\big]\operatorname{tr}\big(\Sigma\big)\mathbf{I}_d+\alpha^2\operatorname{tr}(\Sigma)\mathbf{I}_d. \end{aligned}$$ Substituting the update rule [\[Eq: Update rule\]](#Eq: Update rule){reference-type="eqref" reference="Eq: Update rule"} in $A_k$ gives by the linearity of the transpose that $$\label{Eq: Equation for A_k} \begin{aligned} A''&=\mathbb{E}\big[(\bm{\theta}''-\bm{\theta}_{\star})(\bm{\theta}''-\bm{\theta}_{\star})^\top\big]\\ &=\mathbb{E}\bigg[\Big(\bm{\theta}'-\alpha\big(\nabla L(\bm{\theta}')\big)^\top \bm{\xi}\bm{\xi}-\bm{\theta}_{\star}\Big)\Big(\bm{\theta}'-\alpha\big(\nabla L(\bm{\theta}')\big)^\top \bm{\xi}\bm{\xi}-\bm{\theta}_{\star}\Big)^\top\bigg]\\ &=A' -\alpha\mathbb{E}\bigg[\Big(\bm{\theta}-\bm{\theta}_{\star}\Big)\Big(\big(\nabla L(\bm{\theta}')\big)^\top \bm{\xi}\bm{\xi}\Big)^\top\bigg]-\alpha\mathbb{E}\bigg[\Big(\big(\nabla L(\bm{\theta}')\big)^\top \bm{\xi}\bm{\xi}\Big)\Big(\bm{\theta}'-\bm{\theta}_{\star}\Big)^\top\bigg]\\ &\quad+\mathbb{E}\bigg[\Big(\alpha\big(\nabla L(\bm{\theta}')\big)^\top \bm{\xi}\bm{\xi}\Big)\Big( \alpha\big(\nabla L(\bm{\theta}')\big)^\top \bm{\xi}\bm{\xi}\Big)^\top\bigg]. \end{aligned}$$ First, consider the terms with the minus sign in the above expression. The random vector $\bm{\xi}$ is independent of all other randomness and hence $\mathbb{E}_{\bm{\xi}}\Big[\big(\nabla L(\bm{\theta}')\big)^\top \bm{\xi}\bm{\xi}\Big]=\nabla L(\bm{\theta}').$ Moreover, together with [\[eq.grad_cond\]](#eq.grad_cond){reference-type="eqref" reference="eq.grad_cond"}, $$\begin{aligned} \mathbb{E}\bigg[\Big(\big(\nabla L(\bm{\theta}')\big)^\top \bm{\xi}\bm{\xi}\Big)\Big(\bm{\theta}'-\bm{\theta}_{\star}\Big)^\top \, \bigg|\, \bm{\theta}'\bigg] = \mathbb{E}\big[\nabla L(\bm{\theta}') \, \big|\, \bm{\theta}'\big] (\bm{\theta}'-\bm{\theta}_{\star})^\top = \Sigma (\bm{\theta}'-\bm{\theta}_\star)(\bm{\theta}'-\bm{\theta}_{\star})^\top.\end{aligned}$$ Taking the transpose and tower rule, we find $$\begin{aligned} \label{Eq: Bound on (I)} \begin{split} &-\alpha\mathbb{E}\bigg[\Big(\bm{\theta}-\bm{\theta}_{\star}\Big)\Big(\big(\nabla L(\bm{\theta}')\big)^\top \bm{\xi}\bm{\xi}\Big)^\top\bigg]-\alpha\mathbb{E}\bigg[\Big(\big(\nabla L(\bm{\theta}')\big)^\top \bm{\xi}\bm{\xi}\Big)\Big(\bm{\theta}'-\bm{\theta}_{\star}\Big)^\top\bigg]\\ &=-\alpha\mathbb{E}\big[(\bm{\theta}'-\bm{\theta}_{\star})(\bm{\theta}'-\bm{\theta}_{\star})^\top\big]\Sigma-\alpha\Sigma\mathbb{E}\big[(\bm{\theta}'-\bm{\theta}_{\star})(\bm{\theta}'-\bm{\theta}_{\star})^\top\big]. \end{split} \end{aligned}$$ In a next step, we derive an expression for $\mathbb{E}\Big[\Big(\alpha\big(\nabla L(\bm{\theta}')\big)^\top \bm{\xi}\bm{\xi}\Big)\Big( \alpha\big(\nabla L(\bm{\theta}')\big)^\top \bm{\xi}\bm{\xi}\Big)^\top\Big]$. Since $\bm{\xi}\sim\mathcal{N}(0,\mathbf{I}_d)$ is independent of $\nabla L(\bm{\theta}')$ we can apply Lemma [Lemma 4](#L: terms with quadratic normal random variables){reference-type="ref" reference="L: terms with quadratic normal random variables"} to derive $$\label{Eq: gradient product term expectation} \begin{aligned} &\mathbb{E}\bigg[\Big(\alpha\big(\nabla L(\bm{\theta}')\big)^\top \bm{\xi}\bm{\xi}\Big)\Big( \alpha\big(\nabla L(\bm{\theta}')\big)^\top \bm{\xi}\bm{\xi}\Big)^\top\bigg]\\ &=\alpha^2\mathbb{E}\bigg[\Big(\big(\nabla L(\bm{\theta}')\big)^\top \bm{\xi}\Big)^2\bm{\xi}\bm{\xi}^\top\bigg]\\ &=2\alpha^2\mathbb{E}\Big[\big(\nabla L(\bm{\theta}')\big)\big(\nabla L_{k-1}(\bm{\theta}')\big)^\top\Big]+\alpha^2\mathbb{E}\Big[\big(\nabla L(\bm{\theta}')\big)^\top\big(\nabla L(\bm{\theta}')\big)\Big]\mathbf{I}_d \\ &= 2\alpha^2\mathbb{E}\Big[\big(\nabla L(\bm{\theta}')\big)\big(\nabla L_{k-1}(\bm{\theta}')\big)^\top\Big]+\alpha^2\operatorname{tr}\bigg(\mathbb{E}\Big[\big(\nabla L(\bm{\theta}')\big)\big(\nabla L_{k-1}(\bm{\theta}')\big)^\top\Big]\bigg)\mathbf{I}_d. \end{aligned}$$ Arguing as for [\[eq.grad_Ltheta\]](#eq.grad_Ltheta){reference-type="eqref" reference="eq.grad_Ltheta"} gives $\nabla L(\bm{\theta}') =- \epsilon \mathbf{X}- \mathbf{X}\mathbf{X}^\top(\bm{\theta}_\star-\bm{\theta}')$ and this yields $$\begin{aligned} \mathbb{E}\Big[\big(\nabla L(\bm{\theta}')\big)\big(\nabla L_{k-1}(\bm{\theta}')\big)^\top\Big]= \mathbb{E}\bigg[\mathbb{E}_{\epsilon}\Big[\big( \epsilon \mathbf{X}+ \mathbf{X}\mathbf{X}^\top(\bm{\theta}_\star-\bm{\theta}')\big)\big( \epsilon \mathbf{X}+ \mathbf{X}\mathbf{X}^\top(\bm{\theta}_\star-\bm{\theta}')\big)^\top\Big]\bigg]. \end{aligned}$$ Because $\epsilon$ has mean zero and variance one and is independent of $(\mathbf{X},\bm{\theta}')$, we conclude that $$\label{Eq: gradient product term post epsilon} \begin{aligned} \mathbb{E}\Big[\big(\nabla L(\bm{\theta}')\big)\big(\nabla L_{k-1}(\bm{\theta}')\big)^\top\Big]&=\mathbb{E}\Big[\big(\mathbf{X}\mathbf{X}^\top(\bm{\theta}_\star-\bm{\theta}')\big)\big(\mathbf{X}\mathbf{X}^\top(\bm{\theta}_\star-\bm{\theta}')\big)^\top+\mathbf{X}\mathbf{X}^\top\Big]\\ &=\mathbb{E}\Big[ \big(\mathbf{X}^\top(\bm{\theta}_\star-\bm{\theta}')\big)^2\mathbf{X}\mathbf{X}\Big]+\Sigma, \end{aligned}$$ where for the last equality we used that $\mathbf{X}^\top(\bm{\theta}_\star-\bm{\theta}')$ is a scalar and that $\mathbf{X}\sim\mathcal{N}(0,\Sigma)$. Since $\mathbf{X}\sim\mathcal{N}(0,\Sigma)$ is independent of $\bm{\theta}'$ we get by Lemma [Lemma 4](#L: terms with quadratic normal random variables){reference-type="ref" reference="L: terms with quadratic normal random variables"} that $$\begin{aligned} &\mathbb{E}\Big[ \big(\mathbf{X}^\top(\bm{\theta}_\star-\bm{\theta}')\big)^2\mathbf{X}\mathbf{X}\Big]=2\Sigma\mathbb{E}\big[(\bm{\theta}'-\bm{\theta}_{\star})(\bm{\theta}'-\bm{\theta}_{\star})^\top\big]\Sigma+\mathbb{E}\big[(\bm{\theta}'-\bm{\theta}_{\star})^\top\Sigma(\bm{\theta}'-\bm{\theta}_{\star})\big]\Sigma. \end{aligned}$$ Substituting this in [\[Eq: gradient product term post epsilon\]](#Eq: gradient product term post epsilon){reference-type="eqref" reference="Eq: gradient product term post epsilon"} and [\[Eq: gradient product term expectation\]](#Eq: gradient product term expectation){reference-type="eqref" reference="Eq: gradient product term expectation"} yields $$\label{Eq: Equation for (II)} \begin{aligned} &\mathbb{E}\bigg[\Big(\alpha\big(\nabla L(\bm{\theta}')\big)^\top \bm{\xi}\bm{\xi}\Big)\Big( \alpha\big(\nabla L(\bm{\theta}')\big)^\top \bm{\xi}\bm{\xi}\Big)^\top\bigg]\\ &=4\alpha^2\Sigma\mathbb{E}\big[(\bm{\theta}'-\bm{\theta}_{\star})(\bm{\theta}'-\bm{\theta}_{\star})^\top\big]\Sigma+2\alpha^2\mathbb{E}\big[(\bm{\theta}'-\bm{\theta}_{\star})^\top\Sigma(\bm{\theta}'-\bm{\theta}_{\star})\big]\Sigma+2\alpha^2\Sigma\\ &+2\alpha^2\operatorname{tr}\Big(\Sigma\mathbb{E}\big[(\bm{\theta}'-\bm{\theta}_{\star})(\bm{\theta}'-\bm{\theta}_{\star})^\top\big]\Sigma\Big)\mathbf{I}_d+\alpha^2\operatorname{tr}\Big(\mathbb{E}\big[(\bm{\theta}'-\bm{\theta}_{\star})^\top\Sigma(\bm{\theta}'-\bm{\theta}_{\star})\big]\Sigma\Big)\mathbf{I}_d\\ &+\alpha^2\operatorname{tr}(\Sigma)\mathbf{I}_d. \end{aligned}$$ Combining [\[Eq: Equation for A_k\]](#Eq: Equation for A_k){reference-type="eqref" reference="Eq: Equation for A_k"} with [\[Eq: Bound on (I)\]](#Eq: Bound on (I)){reference-type="eqref" reference="Eq: Bound on (I)"} and [\[Eq: Equation for (II)\]](#Eq: Equation for (II)){reference-type="eqref" reference="Eq: Equation for (II)"} yields the statement of the theorem. ◻ ## Proof of Theorem [Theorem 3](#T:Convergence bound){reference-type="ref" reference="T:Convergence bound"} {#proof-of-theorem-tconvergence-bound} For two vectors $\mathbf{u},\mathbf{v}$ of the same length, $\operatorname{tr}(\mathbf{u}\mathbf{v}^\top)=\mathbf{u}^\top\mathbf{v}.$ Thus, $\mathbb{E}\big[\|\bm{\theta}_{k}-\bm{\theta}_{\star}\|_2^2\big]=\operatorname{tr}\big(\mathbb{E}\big[(\bm{\theta}_k-\bm{\theta}_{\star})(\bm{\theta}_k-\bm{\theta}_{\star})^\top\big]\big).$ Together with Theorem [Theorem 2](#T:Covariance){reference-type="ref" reference="T:Covariance"}, $\operatorname{tr}(\mathbf{I}_d)=d$ and $\operatorname{tr}(AB)=\operatorname{tr}(BA)$ for square matrices $A$ and $B$ of the same size, this yields $$\label{Eq: Risk as trace of Theorem 3.2} \begin{aligned} \mathbb{E}\big[\|\bm{\theta}_{k}-\bm{\theta}_{\star}\|_2^2\big] &=\operatorname{tr}\Big((\mathbf{I}_d-\alpha_{k}\Sigma)\mathbb{E}\big[(\bm{\theta}_{k-1}-\bm{\theta}_{\star})(\bm{\theta}_{k-1}-\bm{\theta}_{\star})^\top\big](\mathbf{I}_d-\alpha_k\Sigma)\Big)\\ &\quad +3\alpha_k^2\operatorname{tr}\Big(\Sigma \mathbb{E}\big[(\bm{\theta}_{k-1}-\bm{\theta}_{\star})(\bm{\theta}_{k-1}-\bm{\theta}_{\star})^\top\big]\Sigma\Big)\\&\quad +2\alpha_{k}^2\operatorname{tr}\Big(\mathbb{E}\big[(\bm{\theta}_{k-1}-\bm{\theta}_{\star})^\top\Sigma(\bm{\theta}_{k-1}-\bm{\theta}_{\star})\big]\Sigma\Big)+2\alpha_{k}^2\operatorname{tr}\big(\Sigma\big)\\ &\quad +2\alpha_k^2\operatorname{tr}\Big(\Sigma \mathbb{E}\big[(\bm{\theta}_{k-1}-\bm{\theta}_{\star})(\bm{\theta}_{k-1}-\bm{\theta}_{\star})^\top\big]\Sigma\Big)\operatorname{tr}\big(\mathbf{I}_d\big)\\ &\quad +\alpha_{k}^2\mathbb{E}\big[(\bm{\theta}_{k-1}-\bm{\theta}_{\star})^\top\Sigma(\bm{\theta}_{k-1}-\bm{\theta}_{\star})\big]\operatorname{tr}\big(\Sigma\big)\operatorname{tr}\big(\mathbf{I}_d\big)\\ &\quad +\alpha_k^2\operatorname{tr}(\Sigma)\operatorname{tr}\big(\mathbf{I}_d\big)\\ &=\mathbb{E}\big[(\bm{\theta}_{k-1}-\bm{\theta}_{\star})^\top(\mathbf{I}_d-2\alpha_{k}\Sigma)^\top(\bm{\theta}_{k-1}-\bm{\theta}_{\star})\big]\\ &\quad +2(d+2)\alpha_k^2\operatorname{tr}\Big(\Sigma \mathbb{E}\big[(\bm{\theta}_{k-1}-\bm{\theta}_{\star})(\bm{\theta}_{k-1}-\bm{\theta}_{\star})^\top\big]\Sigma\Big)\\&\quad +(d+2)\alpha_k^2\Big(\mathbb{E}\big[(\bm{\theta}_{k-1}-\bm{\theta}_{\star})^\top\Sigma(\bm{\theta}_{k-1}-\bm{\theta}_{\star})\big]\operatorname{tr}\big(\Sigma\big)+\operatorname{tr}\big(\Sigma\big)\Big). \end{aligned}$$ If $\lambda$ is an eigenvalue of $\Sigma$ then $(1-2\alpha_{k}\lambda)$ is an eigenvalue of $\mathbf{I}_d-2\alpha_{k}\Sigma$. By assumption, $0<\alpha_{k}\leq \lambda_{\min}(\Sigma)/\big(2\|\Sigma\|_S^2\big)\leq 1/\big(2\lambda_{\max}(\Sigma)\big)$ and therefore the matrix $\mathbf{I}_d-2\alpha_{k}\Sigma$ is positive semi-definite and $(1-2\alpha_{k}\lambda_{\min}(\Sigma))$ is the largest eigenvalue. For a positive semi-definite matrix $A$ and a vector $\mathbf{v},$ the min-max theorem states that $\mathbf{v}^\top A\mathbf{v}\leq \lambda_{\max}(A)\|\mathbf{v}\|_2^2=\|A\|_S\|\mathbf{v}\|_2^2.$ Using that for a vector $\mathbf{x}$ it holds that $\operatorname{tr}(\mathbf{x}\mathbf{x}^\top)=\mathbf{x}^\top\mathbf{x}$, with $\mathbf{x}=\Sigma(\bm{\theta}_{k-1}-\bm{\theta}_{\star})$ in [\[Eq: Risk as trace of Theorem 3.2\]](#Eq: Risk as trace of Theorem 3.2){reference-type="eqref" reference="Eq: Risk as trace of Theorem 3.2"} and applying $\mathbf{v}^\top A\mathbf{v}\leq\|A\|_S\|\mathbf{v}\|_2^2$ with $\mathbf{v}=\bm{\theta}_{k-1}-\bm{\theta}_{\star}$ and $A\in \{\Sigma,\mathbf{I}_d-2\alpha_k\Sigma,\Sigma^2\}$, yields $$\begin{aligned} \mathbb{E}\big[\|\bm{\theta}_{k}-\bm{\theta}_{\star}\|_2^2\big] &\leq \big(1-2\alpha_{k}\lambda_{\min}(\Sigma)\big) \mathbb{E}\big[\|\bm{\theta}_{k-1}-\bm{\theta}_{\star}\|_2^2\big]\\ &+(d+2)\alpha_k^2\bigg(\operatorname{tr}(\Sigma)\|\Sigma\|_S\mathbb{E}\big[\|\bm{\theta}_{k-1}-\bm{\theta}_{\star}\|_2^2\big]+2\|\Sigma\|_S^2\mathbb{E}\big[\|\bm{\theta}_{k-1}-\bm{\theta}_{\star}\|_2^2\big]+\operatorname{tr}(\Sigma)\bigg). \end{aligned}$$ The spectral norm of a positive semi-definite matrix is equal to the largest eigenvalue and so $\operatorname{tr}(\Sigma)=\sum_{i=1}^d\lambda_i\leq d\lambda_{\max}=d\|\Sigma\|_S.$ Therefore, $$\begin{aligned} &\mathbb{E}\big[\|\bm{\theta}_{k}-\bm{\theta}_{\star}\|_2^2\big]\leq\Big(1-2\alpha_k\lambda_{\min}(\Sigma)+\|\Sigma\|_S^2(d+2)^2\alpha_k^2\Big)\mathbb{E}\big[\|\bm{\theta}_{k-1}-\bm{\theta}_{\star}\|_2^2\big]+\|\Sigma\|_S(d+2)^2\alpha_k^2. \end{aligned}$$ Using that $\alpha_k\leq \lambda_{\min}(\Sigma)/\big(\|\Sigma\|_S^2(d+2)^2\big)$ yields $$\begin{aligned} &\mathbb{E}\big[\|\bm{\theta}_{k}-\bm{\theta}_{\star}\|_2^2\big]\leq \big(1-\alpha_k\lambda_{\min}(\Sigma)\big)\mathbb{E}\big[\|\bm{\theta}_{k-1}-\bm{\theta}_{\star}\|_2^2\big]+\|\Sigma\|_S(d+2)^2\alpha_k^2. \end{aligned}$$ Rewritten in non-recursive, we obtain $$\label{Eq: Nonrecursive Bound on MSE} \begin{aligned} \mathbb{E}\big[\|\bm{\theta}_{k}-\bm{\theta}_{\star}\|_2^2\big]\leq &\mathbb{E}\big[\|\bm{\theta}_{0}-\bm{\theta}_{\star}\|_2^2\big]\prod_{\ell=1}^k \big(1-\alpha_{\ell}\lambda_{\min}(\Sigma)\big)\\ &+\|\Sigma\|_S(d+2)^2\sum_{m=0}^{k-1}\alpha_{k-m}^2\prod_{\ell=k-m+1}^{k}\big(1-\alpha_{\ell}\lambda_{\min}(\Sigma)\big), \end{aligned}$$ where we use the convention that the (empty) product over zero terms is assigned the value $1.$ For ease of notation define $c_d:=a\kappa^2(\Sigma)(d+2)^2,$ with condition number $\kappa(\Sigma)=\|\Sigma\|_S/\lambda_{\min}(\Sigma).$ From the definition of $\alpha_k$, [\[Eq: Combined choice of alpha_k\]](#Eq: Combined choice of alpha_k){reference-type="eqref" reference="Eq: Combined choice of alpha_k"}, it follows that $\alpha_k=\frac{a}{\lambda_{\min}(\Sigma)}\cdot \frac{1}{k+c_d}.$ Using that for all real numbers $x$ it holds that $1+x\leq e^x$, we get that for all integers $k^*< k,$ $$\label{Eq: Bound on the product term alternative} \begin{aligned} &\prod_{\ell=k^*}^{k}\big(1-\alpha_{\ell}\lambda_{\min}(\Sigma)\big)\leq \exp\Bigg(-\lambda_{\min}(\Sigma)\sum_{\ell=k^*}^k\alpha_{\ell}\Bigg)=\exp\Bigg(-a\sum_{\ell=k^*}^k\frac{1}{\ell+c_d}\Bigg). \end{aligned}$$ The function $x\mapsto 1/(x+c)$ is monotone decreasing for $x>0$ and $c\geq 0$ and thus, $$\label{Eq: Lower bound on the sum in the exponential alternative} \begin{aligned} \sum_{\ell=k^*}^k\frac{1}{\ell+c_d} &\geq \sum_{\ell=k^*}^k\int_{\ell}^{\ell+1}\frac{1}{x+c_d}dx \\ &\geq \int_{k^*}^{k+1}\frac{1}{x+c_d}dx\\ &=\log(k+1+c_d)-\log(k^*+c_d)\\ &=\log\Big(\frac{k+1+c_d}{k^*+c_d}\Big). \end{aligned}$$ Using [\[Eq: Bound on the product term alternative\]](#Eq: Bound on the product term alternative){reference-type="eqref" reference="Eq: Bound on the product term alternative"} and [\[Eq: Lower bound on the sum in the exponential alternative\]](#Eq: Lower bound on the sum in the exponential alternative){reference-type="eqref" reference="Eq: Lower bound on the sum in the exponential alternative"} with $k^*=1$ gives $$\label{Eq: Bound on the product for the MSE(btheta0) term Alternative} \begin{aligned} \prod_{\ell=1}^k \big(1-\alpha_{\ell}\lambda_{\min}(\Sigma)\big)\leq \exp\bigg(-a\log\Big(\frac{k+1+c_d}{1+c_d}\Big)\bigg)=\bigg(\frac{1+c_d}{k+1+c_d}\bigg)^a. \end{aligned}$$ Using [\[Eq: Bound on the product term alternative\]](#Eq: Bound on the product term alternative){reference-type="eqref" reference="Eq: Bound on the product term alternative"} and [\[Eq: Lower bound on the sum in the exponential alternative\]](#Eq: Lower bound on the sum in the exponential alternative){reference-type="eqref" reference="Eq: Lower bound on the sum in the exponential alternative"} with $k^*=k-m+1$ gives $$\label{Eq: Bound on the sum product term alternative} \begin{aligned} &\sum_{m=0}^{k-1}\alpha_{k-m}^2\prod_{\ell=k-m+1}^{k}\big(1-\alpha_{\ell}\lambda_{\min}(\Sigma)\big)\\ &\leq \frac{a^2}{\lambda^2_{\min}(\Sigma)}\sum_{m=0}^{k-1}\frac{1}{\big((k-m)+c_d\big)^2}\Bigg(\frac{k-m+1+c_d}{k+1+c_d}\Bigg)^a\\ &=\frac{a^2}{\lambda^2_{\min}(\Sigma)(k+1+c_d)^a}\sum_{m=0}^{k-1}\frac{\big(k-m+1+c_d\big)^a}{\big((k-m)+c_d\big)^2}\\ &=\frac{a^2}{\lambda^2_{\min}(\Sigma)(k+1+c_d)^a}\sum_{m=1}^{k}\frac{\big(m+1+c_d\big)^a}{\big(m+c_d\big)^2}. \end{aligned}$$ Observe that $c_d=a\kappa^2(\Sigma)(d+2)^2\geq a$. This gives us that $c_d+1\leq (1+1/a)c_d$ and thus $m+1+c_d\leq (1+1/a)(m+c_d)$. For all real numbers $x,$ $(1+x)\leq e^x$ and thus $(1+1/a)^a\leq e.$ Therefore, $$\label{Eq: Step in alternative bound extra term} \begin{aligned} &\sum_{m=1}^{k}\frac{\big(m+1+c_d\big)^a}{\big(m+c_d\big)^2}\leq e\sum_{m=1}^{k}\big(m+c_d\big)^{a-2}. \end{aligned}$$ For $p>0,$ the function $x\mapsto (x+c)^p$ is monotone increasing for $x,c>0,$ Hence, $$\begin{aligned} \sum_{\ell=1}^k(\ell+c)^{p} &\leq \sum_{\ell=1}^{k}\int_{\ell}^{\ell+1}(x+c)^{p}dx\\ &=\int_1^{k+1}(x+c)^{p}dx\\ &=\frac{(k+1+c)^{p+1}}{p+1}-\frac{(1+c)^{p+1}}{p+1} \\ &\leq \frac{(k+1+c)^{p+1}}{p+1}. \end{aligned}$$ Since $a>2,$ we can apply this with $p=a-2>0$ to find $$e\sum_{m=1}^{k}\big(m+c_d\big)^{a-2}\leq e\frac{(k+1+c_d)^{a-1}}{a-1}$$ Combining [\[Eq: Nonrecursive Bound on MSE\]](#Eq: Nonrecursive Bound on MSE){reference-type="eqref" reference="Eq: Nonrecursive Bound on MSE"}, [\[Eq: Bound on the product for the MSE(btheta0) term Alternative\]](#Eq: Bound on the product for the MSE(btheta0) term Alternative){reference-type="eqref" reference="Eq: Bound on the product for the MSE(btheta0) term Alternative"}, [\[Eq: Bound on the sum product term alternative\]](#Eq: Bound on the sum product term alternative){reference-type="eqref" reference="Eq: Bound on the sum product term alternative"} and [\[Eq: Step in alternative bound extra term\]](#Eq: Step in alternative bound extra term){reference-type="eqref" reference="Eq: Step in alternative bound extra term"} finally gives $$\begin{aligned} \mathbb{E}[\|\bm{\theta}_k-\bm{\theta}_{\star}\|_2^2]&\leq \Bigg(\frac{1+a\kappa^2(\Sigma)(d+2)^2}{k+1+a\kappa^2(\Sigma)(d+2)^2}\Bigg)^a\mathbb{E}[\|\bm{\theta}_{0}-\bm{\theta}_{\star}\|_2^2]\\ &\quad +\frac{ea^2\kappa(\Sigma)(d+2)^2}{\lambda_{\min}(\Sigma)\big(a-1\big)\big(k+1+a\kappa^2(\Sigma)(d+2)^2\big)}. \end{aligned}$$ Using that $0<a/(a-1)<2$ for $a>2,$ now yields the result. 0◻

arxiv_math

--- abstract: | The $k$-token graph $F_k(G)$ of a graph $G$ is the graph whose vertices are the $k$-subsets of vertices from $G$, two of which being adjacent whenever their symmetric difference is a pair of adjacent vertices in $G$. It is a known result that the algebraic connectivity (or second Laplacian eigenvalue) of $F_k(G)$ equals the algebraic connectivity of $G$. In this paper, we first give results that relate the algebraic connectivities of a token graph and the same graph after removing a vertex. Then, we prove the result on the algebraic connectivity of 2-token graphs for two infinite families: the odd graphs $O_r$ for all $r$, and the multipartite complete graphs $K_{n_1,n_2,\ldots,n_r}$ for all $n_1,n_2,\ldots,n_r$ In the case of cycles, we present a new method that allows us to compute the whole spectrum of $F_2(C_n)$. This method also allows us to obtain closed formulas that give asymptotically exact approximations for most of the eigenvalues of $F_2(\textit{}C_n)$. author: - | M. A. Reyes$^a$, C. Dalfó$^a$, M. A. Fiol$^b$, and A. Messegué$^a$\ \ $^a$Dept. de Matemàtica, Universitat de Lleida, Lleida/Igualada, Catalonia\ `{monicaandrea.reyes,cristina.dalfo,visitant.arnau.messegue}@udl.cat`\ $^{b}$Dept. de Matemàtiques, Universitat Politècnica de Catalunya, Barcelona, Catalonia\ Barcelona Graduate School of Mathematics\ Institut de Matemàtiques de la UPC-BarcelonaTech (IMTech)\ `[email protected]` \ title: "On the spectra of token graphs of cycles and other graphs [^1] " --- Token graph, Laplacian spectrum, Algebraic connectivity, Binomial matrix, Lift graph, Regular partition. 05C15, 05C10, 05C50. # Introduction {#sec:-1https://www.overleaf.com/project/635919a7a2694a23b29b9dab} Let $G=(V,E)$ be a simple graph with vertex set $V=V(G)=\{1,2,\ldots,n\}$ and edge set $E=E(G)$. By convenience, we consider every edge $e=\{u,v\}$ constituted by two opposite arcs $(u,v)$ and $(v,u)$. Let $N(u)$ denote the set of vertices adjacent to $u\in V$, so that the minimum degree of $G$ is $\delta(G)=\min_{u\in V}|N(u)|$. For a given integer $k$ such that $1\leq k \leq n$, the *$k$-token graph* $F_k(G)$ of $G$ is the graph whose vertex set $V (F_k(G))$ consists of the ${n \choose k}$ $k$-subsets of vertices of $G$, and two vertices $A$ and $B$ of $F_k(G)$ are adjacent if and only if their symmetric difference $A \bigtriangleup B$ is a pair $\{a,b\}$ such that $a\in A$, $b\in B$, and $(a,b)\in E(G)$. Then, if $G$ has $n$ vertices and $m$ edges, $F_k(G)$ has ${n \choose k}$ vertices and ${n-2 \choose k-1}m$ edges. (Indeed, for each edge of $G$, there are ${n-2 \choose k-1}$ edges of $F_k(G)$.) We also use the notation $\{a,b\}$, with $a,b\in V$, for a vertex of a 2-token graph. Moreover, we use $ab$ for the same vertex in the figures. The naming 'token graph' comes from an observation in Fabila-Monroy, Flores-Peñaloza, Huemer, Hurtado, Urrutia, and Wood [@ffhhuw12], that vertices of $F_k(G)$ correspond to configurations of $k$ indistinguishable tokens placed at distinct vertices of $G$, where two configurations are adjacent whenever one configuration can be reached from the other by moving one token along an edge from its current position to an unoccupied vertex. The $k$-token graphs are also called symmetric $k$-th power of graphs in Audenaert, Godsil, Royle, and Rudolph [@agrr07], and $k$-tuple vertex graphs in Alavi, Lick, and Liu [@all02]. In Figures [1](#F_2(C_9)){reference-type="ref" reference="F_2(C_9)"}, [2](#F_2(C_{10})){reference-type="ref" reference="F_2(C_{10})"}, and [3](#F_2(C_8)){reference-type="ref" reference="F_2(C_8)"}, we show the 2-token graphs of cycles $C_9$, $C_{10}$, and $C_8$, respectively. Note that if $k=1$, then $F_1(G)\cong G$; and if $G$ is the complete graph $K_n$, then $F_k(K_n)\cong J(n,k)$, where $J(n,k)$ denotes the Johnson graph [@ffhhuw12], which is distance-transitive (and, hence, distance-regular). Moreover, if $G$ is bipartite, so it is $F_k(G)$ for any $k=1,\ldots,|V|-1$. Token graphs have some applications in physics. For instance, a relationship between token graphs and the exchange of Hamiltonian operators in quantum mechanics is given in Audenaert, Godsil, Royle, and Rudolph [@agrr07]. Our interest in the study of token graphs is motivated by some of their applications in mathematics and computer science: Analysis of complex networks, coding theory, combinatorial designs (by means of Johnson graphs), algebraic graph theory, enumerative combinatorics, the study of symmetric functions, etc. Recently, it was conjectured by Dalfó, Duque, Fabila-Monroy, Fiol, Huemer, Trujillo-Negrete, and Zaragoza Martı́nez [@ddffhtz21] that the algebraic connectivity of $F_k(G)$ equals the algebraic connectivity of $G$. After submitting the first version of this paper, the authors learned (from Fabila-Monroy [@f23]) that this conjecture was already known as the *Aldous' spectral gap conjecture*, and it was proved in 2010 by Caputo, Ligget, and Richthammer in [@clr10]. Moreover, Ouyang [@o19] and Lew [@l23] also mentioned that this conjecture was actually solved. Moreover, Cesi [@c16] provided a simpler proof of the so-called 'octopus inequality', which is one of the main ingredients to prove the Aldous' conjecture. These results were obtained in completely different contexts and using distinct techniques. More precisely, they used the theory of continuous Markov chains of random walks and the so-called 'interchange process'. In this paper, we present an algebraic approach to this problem based on voltage graphs, and we give a new method that can be of interest giving an alternative proof. This paper is structured as follows. In Section [2](#resultats-coneguts){reference-type="ref" reference="resultats-coneguts"}, we give the preliminaries and background together with the known results. Moreover, there are the concepts of quotient graph (or digraph) and lift graph (or digraph). In Section [3](#resultats-nous){reference-type="ref" reference="resultats-nous"}, we give results on the algebraic connectivity of a $k$-token graph when we remove one of its vertices. Besides, given that the algebraic connectivities of a $k$-token graph and its original graph are the same, we provide an algebraic proof of this result when $k=2$ for two infinite families of graphs: the odd graphs and the multipartite complete graphs. Section [4](#sec:2-tokens){reference-type="ref" reference="sec:2-tokens"} deals with the $2$-token graph of a cycle $C_n$ and gives an efficient method to compute the whole spectrum of $F_2(C_n)$ for any $n$ by using the theory of lift graphs and a new method called over-lifts. Finally, in the last section, we give some closed formulas that provide asymptotic approximations of the eigenvalues of $F_2(C_n)$. # Preliminaries and background {#resultats-coneguts} ## Some notation and basic facts The notation of this paper is as follows: $\mbox{\boldmath $A$}=\mbox{\boldmath $A$}(G)$ is the adjacency matrix of the graph $G$, $\mbox{\boldmath $L$}=\mbox{\boldmath $L$}(G)$ the Laplacian matrix of the graph $G$, $\mbox{\boldmath $P$}$ a permutation matrix, $\pi=V_1\cup V_2\cup \cdots \cup V_r$ a (regular or not) partition of the vertex set, $G/\pi$ a quotient graph over $\pi$, $\mbox{\boldmath $A$}(G/\pi)$ and $\mbox{\boldmath $L$}(G/\pi)$ the adjacency and Laplacian matrices of $G/\pi$, and $\mbox{\boldmath $S$}$ the characteristic matrix of the partition $\pi$. The transpose of a matrix $\mbox{\boldmath $M$}$ is denoted by $\mbox{\boldmath $M$}^\top$, the identity matrix by $\mbox{\boldmath $I$}$, the all-$1$ vector $(1,\ldots, 1)^{\top}$ by $\mbox{\boldmath $1$}$, the all-$1$ (universal) matrix by $\mbox{\boldmath $J$}$, and the all-$0$ vector and all-$0$ matrix by $\mbox{\boldmath $0$}$ and $\mbox{\boldmath $O$}$, respectively. Let $[n]:=\{1,\ldots,n\}$ and ${[n]\choose k}$ denote the set of $k$-subsets of $[n]$, which is the set of vertices of the $k$-token graph. For our purpose, it is convenient to denote by $W_n$ the set of all column vectors $\mbox{\boldmath $v$}$ with $n$ entries such that $\mbox{\boldmath $v$}^{\top }\mbox{\boldmath $1$} = 0$. Recall that any square matrix $\mbox{\boldmath $M$}$ with all zero row sums has an eigenvalue $0$ with corresponding eigenvector $\mbox{\boldmath $1$}$. Then, given a graph $G=(V,E)$ of order $n$, we say that a vector $\mbox{\boldmath $v$}\in \mathbb{R}^n$ is an *embedding* of $G$ if $\mbox{\boldmath $v$}\in W_n$. Note that if $\mbox{\boldmath $v$}$ is a $\lambda$-eigenvector of $G$, with $\lambda>0$, then it is an embedding of $G$. When $\mbox{\boldmath $M$}=\mbox{\boldmath $L$}(G)$, the Laplacian matrix of a graph $G$, the matrix is positive semidefinite, with eigenvalues $(0=)\lambda_1\le \lambda_2\le \cdots \le \lambda_n$. Its second smallest eigenvalue $\lambda_2$ is known as the *algebraic connectivity* of $G$ (see Fiedler [@fi73]), and we denote it by $\alpha(G)$. The spectral radius $\lambda_{\max}(G)=\lambda_n$ satisfies several lower and upper bounds (see Patra and Sahoo [@ps17] for a survey). For a graph $G$ with Laplacian matrix $\mbox{\boldmath $L$}(G)$ and an embedding $\mbox{\boldmath $v$}$ of $G$, let $$\lambda_G(\mbox{\boldmath $v$}):=\frac{\mbox{\boldmath $v$}^{\top}\mbox{\boldmath $L$}(G)\mbox{\boldmath $v$}}{{\mbox{\boldmath $v$}}^{\top}\mbox{\boldmath $v$}}=\frac{\sum\limits_{(i,j)\in E}[\mbox{\boldmath $v$}(i)-\mbox{\boldmath $v$}(j)]^2}{\sum\limits_{i\in V}\mbox{\boldmath $v$}^2(i)},$$ where $\mbox{\boldmath $v$}(i)$ denotes the entry of $\mbox{\boldmath $v$}$ corresponding to the vertex $i\in V(G)$. The $\lambda_G(\mbox{\boldmath $v$})$ value is known as the *Rayleigh quotient*. If $\mbox{\boldmath $v$}$ is an eigenvector of $G$, then its corresponding eigenvalue is $\lambda(\mbox{\boldmath $v$})$. Moreover, for an embedding $\mbox{\boldmath $v$}$ of $G$, we have $$\label{bound-lambda(v)} \alpha(G)\le \lambda_G(\mbox{\boldmath $v$}),$$ and we have equality when $\mbox{\boldmath $v$}$ is an $\alpha(G)$-eigenvector of $G$. In this paper, we first give some results about the algebraic connectivity of a token graph. Besides, we provide results about the spectrum of a token graph of a cycle graph when we deal with 2 tokens. This study was initiated by Dalfó, Duque, Fabila-Monroy, Fiol, Huemer, Trujillo-Negrete, and Zaragoza Martı́nez in [@ddffhtz21]. One of their results is the following. Given some integers $n$ and $k$ (with $k\in [n]$), we define the $(n;k)$-*binomial matrix* $\mbox{\boldmath $B$}$. This is a ${n \choose k}\times n$ matrix whose rows are the characteristic vectors of the $k$-subsets of $[n]$ in a given order. Thus, if the $i$-th $k$-subset is $A$, then $$(\mbox{\boldmath $B$})_{ij}= \left\lbrace \begin{array}{ll} 1 & \mbox{if } j\in A,\\ 0 & \mbox{otherwise.} \end{array} \right.$$ **Lemma 1** ([@ddffhtz21]). *Let $G$ be a graph on $n$ vertices. For some integers $h,k$ such that $1\le h<k\le \frac{n}{2}$, let $F_h=F_h(G)$ and $F_k=F_k(G)$ be its $h$- and $k$-token graphs with respective Laplacian matrices $\mbox{\boldmath $L$}_h$ and $\mbox{\boldmath $L$}_k$. Then, the following holds:* - *If $\mbox{\boldmath $v$}$ is a $\lambda$-eigenvector of $\mbox{\boldmath $L$}_h$, then $\mbox{\boldmath $B$}\mbox{\boldmath $v$}$ is a $\lambda$-eigenvector of $\mbox{\boldmath $L$}_k$. Thus, the Laplacian spectrum (eigenvalues and their multiplicities) of $\mbox{\boldmath $L$}_h$ is contained in the Laplacian spectrum of $\mbox{\boldmath $L$}_k$.* - *If $\mbox{\boldmath $u$}$ is a $\lambda$-eigenvector of $\mbox{\boldmath $L$}_k$ such that $\mbox{\boldmath $B$}^{\top}\mbox{\boldmath $u$}\neq \mbox{\boldmath $0$}$, then $\mbox{\boldmath $B$}^{\top}\mbox{\boldmath $u$}$ is a $\lambda$-eigenvector of $\mbox{\boldmath $L$}_h$.* From the inclusion property of the successive spectra in $(i)$, we have: $$\label{eq:non-increasing} \alpha(G)\ge \alpha(F_2(G))\ge \alpha(F_3(G))\ge\cdots\ge \alpha(F_{\lfloor n/2\rfloor}(G)).$$ Recall that Caputo, Liggett, and Richthammer [@clr10] proved that all these inequalities actually are equalities. In our context of token graphs, it was shown in [@ddffhtz21] and by Dalfó and Fiol in [@df22] that the conjecture (now, a result) holds for the following infinite families of graphs. **Theorem 2** ([@ddffhtz21; @df22]). *For each of the following classes of graphs, the algebraic connectivity of a token graph $F_k(G)$ equals the algebraic connectivity of $G$.* - *Let $G=K_n$ be the complete graph on $n$ vertices. Then, $\alpha(F_k(G))=\alpha(G)=n$ for every $n$ and $k=1,\ldots,n-1$.* - *Let $G= K_{n_1,n_2}$ be the complete bipartite graph on $n=n_1+n_2$ vertices, with $n_1\le n_2$. Then, $\alpha(F_k(G))=\alpha(G)=n_1$ for every $n_1,n_2$ and $k=1,\ldots,n-1$.* - *Let $T_n$ be a tree on $n$ vertices. Then, $\alpha(F_k(T_n))=\alpha(T_n)$ for every $n$ and $k=1,\ldots,n-1$.* - *Let $G$ be a graph such that $\alpha(F_k(G))=\alpha(G)$. Let $T_G$ be a graph obtained from $G$ by attaching a (possibly empty) $u$-rooted tree $T(u)$ to each vertex $u$ of $G$. Then, $\alpha(F_k(T_G))=\alpha(T_G)$.* All these results were obtained by induction on $n$, and using that for some vertex $i$, $\alpha(G\setminus i)\ge \alpha(G)$. See Table [1](#tab:Data1){reference-type="ref" reference="tab:Data1"}, where we show two particular cases of $(iii)$. Namely, the star graph $S_n$ and the path graph $P_n$, both on $n$ vertices. These two cases, together with the complete bipartite graph $K_{n_1,n_2}$ were proved in [@ddffhtz21 Th. 7.2]. The fact that, for any tree $T_n$, we have $\alpha(T_n\setminus i)\geq \alpha(T_n)$ can be proved by using interlacing (see Bunch, Nielsen, and Sorensen [@bns78]). An alternative proof using Fiedler vectors (eigenvectors with their eigenvalue equal to the algebraic connectivity) was given in Dalfó and Fiol [@df22]. Graph $G$ ** $\alpha(G)$** Vertex $i$ **$\alpha(G\setminus i)$** ----------------------------- ---------------------------- ------------------------------------- ------------------------------ -- $P_n$ $2(1-\cos(\frac{\pi}{n}))$ a leaf $2(1-\cos(\frac{\pi}{n-1}))$ $S_n$ $1$ a leaf $1$ $K_{n_1,n_2}$ ($n_1 < n_2$) $n_1$ $i \in V_2$, $n_2 = \vert V_2\vert$ $n_1$ $T_n$ (see [@df22]) a leaf (see [@df22]) : Some graphs with a vertex $i$ such that $\alpha(G) \leq \alpha (G\setminus i)$. ## Regular partitions and their spectra {#sec:reg-part} Let $G=(V,E)$ be a graph with vertex set $V=V(G)$ and Laplacian matrix $\mbox{\boldmath $L$}$. A partition $\pi=(V_1,\ldots, V_r)$ of $V$ is called *regular* (or *equitable*) whenever, for any $i,j=1,\ldots,r$, the *intersection numbers* $b_{ij}(u)=|N(u)\cap V_j|$, where $u\in V_i$, do not depend on the vertex $u$ but only on the subsets (usually called *classes* or *cells*) $V_i$ and $V_j$. In this case, such numbers are simply written as $b_{ij}$, and the $r\times r$ matrix $\mbox{\boldmath $Q$}_L=\mbox{\boldmath $L$}(G/\pi)$ with entries $$(\mbox{\boldmath $Q$}_L)_{ij}=\left\{ \begin{array}{cl} \!\!\!\!\!\!-b_{ij} & \mbox{if } i\neq j,\\[.2cm] b_{ii}-\displaystyle \sum_{h=1}^r b_{ih} & \mbox{if } i=j , \end{array} \right.$$ is referred to as the *quotient Laplacian matrix* of $G$ with respect to $\pi$. This is also represented by the *quotient (weighted) directed graph* $G/\pi$ (associated with the partition $\pi$), with vertices representing the $r$ cells, and there is an arc with weight $b_{ij}$ from vertex $V_i$ to vertex $V_j$ if and only if $b_{ij}\neq 0$. Of course, if $b_{ii}>0$, for some $i=1,\ldots,r$, the quotient graph (or digraph) $G/\pi$ has loops. Given a partition $\pi$ of $V$ with $r$ cells (or partition sets), let $\mbox{\boldmath $S$}$ be the *characteristic matrix* of $\pi$, that is, the $n\times r$ matrix whose columns are the characteristic vectors of the cells of $\pi$. Then, as in the case of the adjacency matrix (see, for instance, Godsil and Royle [@gr01]), $\pi$ is a regular partition if and only if $\mbox{\boldmath $L$}\mbox{\boldmath $S$}=\mbox{\boldmath $S$}\mbox{\boldmath $Q$}_L$. In the case of the Laplacian matrix, it follows that $$\mbox{\boldmath $Q$}_L=(\mbox{\boldmath $S$}^{\top}\mbox{\boldmath $S$})^{-1}\mbox{\boldmath $S$}^{\top}\mbox{\boldmath $L$}\mbox{\boldmath $S$},$$ and the characteristic polynomial of $\mbox{\boldmath $Q$}_L$ divides the characteristic polynomial of $\mbox{\boldmath $L$}$. Thus, a part of the spectrum of $\mbox{\boldmath $L$}$ can be determined by the spectrum of the (usually much smaller) matrix $\mbox{\boldmath $Q$}_L$. Moreover, if the graph $G$ is bipartite, the maximum eigenvalue of its Laplacian quotient matrix $\mbox{\boldmath $Q$}_L$ equals the spectral radius of $\mbox{\boldmath $L$}$. The reason is that, in bipartite graphs, the Laplacian matrix's characteristic polynomial is equal to the signless Laplacian $\mbox{\boldmath $L$}^+$ (see, for instance, Grone, Merris, and Sunder [@gms90]). Then, the same holds for the quotient Laplacian $\mbox{\boldmath $Q$}_L$ and quotient signless Laplacian $\mbox{\boldmath $Q$}_L^+$ matrices. Thus, since each eigenvector of $\mbox{\boldmath $Q$}_L^+$ gives rise to an eigenvector of $\mbox{\boldmath $L$}^+$, the spectral radius of $\mbox{\boldmath $L$}$ corresponds to the eigenvalue of the Perron vector of $\mbox{\boldmath $Q$}_L^+$ or maximum eigenvalue of $\mbox{\boldmath $Q$}_L$. For more information about quotient (Laplacian) matrices, see Dalfó, Fiol, Pavlíková, and Širán [@df22b]. ## Lift graphs and their spectra {#sec:sp} Let $\cal{G}$ be a group. An (*ordinary*) *voltage assignment* on the (di)graph (that is, graph or digraph) $G=(V,E)$ is a mapping $\beta: E\to \cal{G}$ with the property that $\beta(a^-)=(\beta(a^+))^{-1}$ for every arc $a\in E$. Thus, a voltage assigns an element $g\in \cal{G}$ to each arc of the (di)graph so that a pair of mutually reverse arcs $a^+$ and $a^{-}$, forming an undirected edge, receive mutually inverse elements $g$ and $g^{-1}$. The (di)graph $G$ and the voltage assignment $\beta$ determine a new (di)graph $G^{\beta}$, called the *lift* of $G$, which is defined as follows. The vertex and arc sets of the lift are simply the Cartesian products $V^{\beta}=V\times \cal{G}$ and $E^{\beta}=E\times \cal{G}$, respectively. Moreover, for every arc $a\in E$ from a vertex $u$ to a vertex $v$ for $u,v\in V$ (possibly, $u=v$) in $G$, and for every element $g\in \cal{G}$, there is an arc $(a,g)\in E^{\beta}$ from the vertex $(u,g)\in V^{\beta}$ to the vertex $(v,g\beta(a))\in V^{\beta}$. Let $G=(V,E)$ be a connected graph on $n$ vertices (with loops and multiple edges allowed) and with Laplacian matrix $\mbox{\boldmath $L$}$. Let $\beta$ be a voltage assignment on the arc set $E$ in a group $\cal{G}$ with identity element $e$. Now we show that the spectrum of the Laplacian matrix of the lift $G^{\beta}$ may be computed. To this end, the key idea is to define the so-called *Laplacian base matrix* properly as follows. **Definition 3**. *To the pair $(G,\beta)$, we assign the $n\times n$ *Laplacian base matrix* $\mbox{\boldmath $B$}(\mbox{\boldmath $L$})$ defined by $$\mbox{\boldmath $B$}(\mbox{\boldmath $L$})=-\mbox{\boldmath $B$}(\mbox{\boldmath $A$})+\mbox{\boldmath $B$}(\mbox{\boldmath $D$}),$$ where the matrices $\mbox{\boldmath $B$}(\mbox{\boldmath $A$})$ and $\mbox{\boldmath $B$}(\mbox{\boldmath $D$})$ have entries as follows:* - *$\mbox{\boldmath $B$}(\mbox{\boldmath $A$})_{uv}=\beta(a_1)+\cdots +\beta(a_j)$ if $a_1,\ldots,a_j$ is the set of all the arcs of $G$ from $u$ to $v$, not excluding the case $u=v$, and $\mbox{\boldmath $B$}(\mbox{\boldmath $A$})_{uv}=0$ if $(u,v)\not\in E$.* - *$\mbox{\boldmath $B$}(\mbox{\boldmath $D$})_{uu}=\deg(u)\cdot e$, and $\mbox{\boldmath $B$}(\mbox{\boldmath $D$})_{uv}=0$ if $u\neq v$.* Let $\rho \in \mathop{\mathrm{Irep}}(\cal{G})$ be a unitary irreducible representation of $\cal{G}$ of dimension $d_\rho=\dim(\rho)$. Given a graph $G$ on $n$ vertices, the assignment $\beta$ in $G$, and the Laplacian base matrix $\mbox{\boldmath $B$}=\mbox{\boldmath $B$}(\mbox{\boldmath $L$})$, let $\rho(\mbox{\boldmath $B$})$ be the $d_\rho n\times d_\rho n$ matrix obtained from $\mbox{\boldmath $B$}$ by replacing every nonzero entry $(\mbox{\boldmath $B$})_{u,v} \in \mathbb{C}[\cal{G}]$ as above by the $d_\rho\times d_\rho$ matrix $\rho(\mbox{\boldmath $B$}_{u,v})$. That is, each element $g$ of the group is replaced by $\rho(g)$, and the zero entries of $\mbox{\boldmath $B$}$ are changed to all-zero $d_\rho\times d_\rho$ matrices. We refer to $\rho(\mbox{\boldmath $B$})$ as the *$\rho$-image* of the Laplacian base matrix $\mbox{\boldmath $B$}$. For every $\rho\in \mathop{\mathrm{Irep}}(\cal{G})$, we consider the $\rho$-image $\rho(\mbox{\boldmath $B$})$ of the Laplacian base matrix $\mbox{\boldmath $B$}$, and we let $\mathop{\mathrm{sp}}(\rho(\mbox{\boldmath $B$}))$ denote the spectrum of $\rho(\mbox{\boldmath $B$})$, that is, the multiset of all the $d_\rho n$ eigenvalues of the matrix $\rho(\mbox{\boldmath $B$})$. Finally, the notation $d_\rho\cdot\mathop{\mathrm{sp}}(\rho(\mbox{\boldmath $B$}))$ denotes the multiset obtained by taking each of the $d_\rho n$ entries of the spectrum $\mathop{\mathrm{sp}}(\rho(\mbox{\boldmath $B$}))$ exactly $d_\rho$ times. In particular, if $(d_\rho)=0$, we take $d_\rho\cdot\mathop{\mathrm{sp}}(\rho(B))=\emptyset$. With all these notations, we can state the following result from Dalfó, Fiol, Pavlíková, and Širáň [@df22b], in which, to our knowledge, the theory of lift graphs was applied to the study of the spectrum of token graphs for the first time. This result allows us to compute the spectrum of a (regular) lifted (di)graph from its associated matrix and the irreducible representations of its corresponding group. **Theorem 4** ([@dfs19]). *Let $G=(V,E)$ be a base (di)graph on $n$ vertices, with a voltage assignment $\beta$ in a group $\cal{G}$. For every irreducible representation $\rho\in \mathop{\mathrm{Irep}}(\cal{G})$, let $\rho(\mbox{\boldmath $B$})$ be the complex matrix whose entries are given by $\rho(\mbox{\boldmath $B$}_{u,v})$. Then, $$\mathop{\mathrm{sp}}G^{\beta} = \bigcup_{\rho\in \mathop{\mathrm{Irep}}(\cal G)}d_\rho\cdot\mathop{\mathrm{sp}}(\rho(\mbox{\boldmath $B$})).$$* In Section [4](#sec:2-tokens){reference-type="ref" reference="sec:2-tokens"}, we use this result in the case when the group $\cal{G}$ is cyclic. Then, if $g$ is a generator of $\cal{G}$, with order $n$, the faithful representation $\rho$ such that $\rho(g^r)=\zeta^r$, with $\zeta=e^{i\frac{2\pi}{n}}$, has dimension $1$. Then, we consider the Laplacian base matrix with each entry being a polynomial in $z$ with integer coefficients and represent such a 'polynomial matrix' by $\mbox{\boldmath $B$}(z)$. Thus, Theorem [Theorem 4](#theo-sp){reference-type="ref" reference="theo-sp"} gives $$\mathop{\mathrm{sp}}G^{\beta} = \bigcup_{z\in R(n)}\mathop{\mathrm{sp}}(\mbox{\boldmath $B$}(z)),$$ where $R(n)$ is the set of all $n$-th roots of unity. A simple property of the polynomial matrix is that $\mbox{\boldmath $B$}(1)$ is the quotient matrix of a regular partition of the lift graph $G^{\beta}$. For more information on lift graphs, see Dalfó, Fiol, Miller, Ryan, and Širáň [@dfmrs17]. # The algebraic connectivity of $F_k(G)$ {#resultats-nous} In this section, we give some results on the algebraic connectivity of token graphs. We begin with a known lemma and continue with a new result, from which some of the algebraic connectivity results are derived. Let $G$ be a graph with $k$-token graph $F_k(G)$. For a vertex $a\in V(G)$, let $S_a:=\{A\in V(F_k(G)):a\in A\}$ and $S'_a:=\{B\in V( F_k(G)): a\not\in B\}$. Let $H_a$ and $H'_a$ be the subgraphs of $F_k(G)$ induced by $S_a$ and $S'_a$, respectively. Note that $H_a\cong F_{k-1}(G\setminus \{a\})$ and $H'_a\cong F_k(G\setminus \{a\})$. **Lemma 5** ([@ddffhtz21]). *Given a vertex $a\in G$ and an eigenvector $\mbox{\boldmath $v$}$ of $F_k(G)$ such that $\mbox{\boldmath $B$}^{\top}\mbox{\boldmath $v$}=\mbox{\boldmath $0$}$, let $$\mbox{\boldmath $w$}_a:=\ensuremath{\left.\mbox{\boldmath $v$}\right|_{S_a}} \mbox{ and } \quad \mbox{\boldmath $w$}'_a:=\ensuremath{\left.\mbox{\boldmath $v$}\right|_{S'_a}}.$$ Then, $\mbox{\boldmath $w$}_a$ and $\mbox{\boldmath $w$}'_a$ are embeddings of $H_a$ and $H'_a$, respectively.* **Lemma 6**. *Let $G=(V,E)$ be a graph with token graph $F_k=F_k(G)$ for some integer $k\ge 2$. Let $\xi(G^-)=\min_{i\in V}\alpha(F_{k-1}(G\setminus i))$. If $\alpha(F_k(G))<\alpha(G)$, then the following statements hold.* - *$\alpha(F_k(G))\ge \frac{k}{k-1}\xi(G^-)$,* - *$\alpha(F_k(G))\ge \alpha(F_k(G\setminus i))$ for $i\in V$.* *Proof.* If $\alpha(F_k(G))<\alpha(G)$, we know that the eigenvector $\mbox{\boldmath $v$}$ of $\alpha(F_k(G))$ must satisfy $\mbox{\boldmath $B$}^{\top}\mbox{\boldmath $v$}=\mbox{\boldmath $0$}$. Let $\|\mbox{\boldmath $v$}\|=1$. Given a vertex $i\in V$, let $S_i:=\{A\in V(F_k):i\in A \}$ and $S'_i:=\{B\in V(F_k):i \not\in B \}$. Let $H_i\cong F_{k-1}(G\setminus i)$ and $H'_i\cong F_{k}(G\setminus i)$ be the subgraphs of $F_k(G)$ induced by $S_i$ and $S'_i$, respectively. Let $\mbox{\boldmath $w$}_i=\mbox{\boldmath $v$}|_{S_i}$ and $\mbox{\boldmath $w$}'_i=\mbox{\boldmath $v$}|_{S'_i}$. From $\mbox{\boldmath $B$}^{\top}\mbox{\boldmath $v$}=\mbox{\boldmath $0$}$, from Lemma [Lemma 5](#lem:embedding){reference-type="ref" reference="lem:embedding"}, $\mbox{\boldmath $w$}_i$ and $\mbox{\boldmath $w$}'_i$ are embeddings of $H_i$ and $H'_i$, respectively. Then, by [\[bound-lambda(v)\]](#bound-lambda(v)){reference-type="eqref" reference="bound-lambda(v)"}, we have that their Rayleigh quotients satisfy $$\begin{aligned} \lambda(\mbox{\boldmath $w$}_i)& =\frac{\sum\limits_{(A,B)\in E(H_i)} [\mbox{\boldmath $w$}_i(A)-\mbox{\boldmath $w$}_i(B)]^2}{\sum\limits_{A\in V(H_i)}\mbox{\boldmath $w$}_i(A)^2}\ge \alpha(F_{k-1}(G\setminus i))\ge \alpha(F_{k}(G\setminus i)), \label{eq:Rlambda(w)}\\ \lambda(\mbox{\boldmath $w$}'_i)& =\frac{\sum\limits_{(A,B)\in E(H'_i)} [\mbox{\boldmath $w$}'_i(A)-\mbox{\boldmath $w$}'_i(B)]^2}{\sum\limits_{A\in V(H'_i)}\mbox{\boldmath $w$}'_i(A)^2}\ge \alpha(F_{k}(G\setminus i)), \label{eq:Rlambda(w')}\end{aligned}$$ where, in the last inequality of [\[eq:Rlambda(w)\]](#eq:Rlambda(w)){reference-type="eqref" reference="eq:Rlambda(w)"}, we applied [\[eq:non-increasing\]](#eq:non-increasing){reference-type="eqref" reference="eq:non-increasing"} with $G\setminus i$. In order to prove $(i)$, we use [\[eq:Rlambda(w)\]](#eq:Rlambda(w)){reference-type="eqref" reference="eq:Rlambda(w)"}: $$\begin{aligned} \alpha(F_k) =\lambda(\mbox{\boldmath $v$})&=\sum\limits_{(A,B)\in E(F_k)}[\mbox{\boldmath $v$}(A)-\mbox{\boldmath $v$}(B)]^2 =\frac{1}{k-1}\sum_{i=1}^n \sum\limits_{(A,B)\in E(H_i)}[\mbox{\boldmath $w$}_i(A)-\mbox{\boldmath $w$}_i(B)]^2 \\ & \ge \frac{1}{k-1}\xi(G^-)\sum_{i=1}^n\sum\limits_{A\in V(H_i)}\mbox{\boldmath $v$}(A)^2= \frac{k}{k-1}\xi(G^-),\end{aligned}$$ since, in the double summatory, each edge $(A,B)$ of $F_k$ is considered $k-1$ times, whereas for the last equality, each vertex $A$ of $F_k$ is considered $k$ times. Now, we prove $(ii)$ by using [\[eq:Rlambda(w)\]](#eq:Rlambda(w)){reference-type="eqref" reference="eq:Rlambda(w)"} and [\[eq:Rlambda(w\')\]](#eq:Rlambda(w')){reference-type="eqref" reference="eq:Rlambda(w')"}. Since $V(H_i)\cup V(H'_i)=V(F_k)$, and $\|\mbox{\boldmath $v$}\|=1$, we have: $$\begin{aligned} \alpha(F_k) =\lambda(\mbox{\boldmath $v$})&=\sum\limits_{(A,B)\in E(F_k)}[\mbox{\boldmath $v$}(A)-\mbox{\boldmath $v$}(B)]^2\\ &\ge \sum\limits_{(A,B)\in E(H_i)}[\mbox{\boldmath $w$}_i(A)-\mbox{\boldmath $w$}_i(B)]^2 +\sum\limits_{(A,B)\in E(H'_i)}[\mbox{\boldmath $w$}'_i(A)-\mbox{\boldmath $w$}'_i(B)]^2\\ & \ge \alpha(F_k(G\setminus i))\left[\sum\limits_{A\in V(H_i)}\mbox{\boldmath $w$}_i(A)^2+\sum\limits_{B\in V(H'_i)}\mbox{\boldmath $w$}'_i(B)^2\right]\\ & \ge \alpha(F_k(G\setminus i))\left[\sum\limits_{A\in V(H_i)}\mbox{\boldmath $v$}(A)^2+\sum\limits_{B\in V(H'_i)}\mbox{\boldmath $v$}(B)^2\right]=\alpha(F_k(G\setminus i)). \label{lem:basic}\end{aligned}$$ This completes the proof. ◻ **Theorem 7**. *Let $G$ be a graph with a vertex $i$ such that $\alpha(G)\ge \alpha(G\setminus i)$. Then, $\alpha(F_k(G))\ge \alpha(F_k(G\setminus i)).$* *Proof.* If $\alpha(F_k(G))<\alpha(G)$, Lemma [Lemma 6](#lem:basic){reference-type="ref" reference="lem:basic"}$(ii)$ gives $\alpha(F_k(G))\ge \alpha(F_k(G\setminus i))$. Then, if $\alpha(F_k(G))< \alpha(F_k(G\setminus i))$, we must have, using [\[eq:non-increasing\]](#eq:non-increasing){reference-type="eqref" reference="eq:non-increasing"}, $$\alpha(F_k(G))\ge\alpha(G)\ \Rightarrow\ \alpha(F_k(G))=\alpha(G)\ \Rightarrow\ \alpha(G)<\alpha(F_k(G\setminus i))\le\alpha(G\setminus i).$$ Thus, the result corresponds to the contrapositive statement. ◻ Now we consider the case of 2 tokens and vertex-transitive graphs. In the following result, given a vertex $i$, we use the parameter $\kappa(i)=\frac{\alpha(G\setminus i)}{\alpha(G)}$ introduced by Kirkland in [@k10]. When $G$ is vertex-transitive, we denote $\kappa(G)=\kappa(i)$ for all $i$, so that $\xi(G^-)=\alpha(G\setminus i)=\kappa(G)\alpha(G)$. **Theorem 8**. *Let $G=(V,E)$ be a graph on $n>3$ vertices.* - *If $\displaystyle\min_{i\in V}\kappa(i)\ge \textstyle\frac{1}{2}$, then $\alpha(F_2(G))=\alpha(G)$.* - *If $G$ is vertex-transitive and $\kappa(G)\ge \frac{1}{2}$, then $\alpha(F_2(G))=\alpha(G)$.* *Proof.* Let $\mbox{\boldmath $v$}$ be an eigenvector of $F_2=F_2(G)$ with eigenvalue $\alpha(F_2)$ and norm $\|\mbox{\boldmath $v$}\|=1$. If $\mbox{\boldmath $B$}^{\top}\mbox{\boldmath $v$}\neq \mbox{\boldmath $0$}$, by Lemma [Lemma 1](#coro:LkL1){reference-type="ref" reference="coro:LkL1"}$(ii)$, $\alpha(F_2)=\alpha(G)$, and the results hold. Thus, we can assume that $\mbox{\boldmath $B$}^{\top}\mbox{\boldmath $v$}=\mbox{\boldmath $0$}$ (this could only occur if $F_2\neq G$). To prove $(i)$, we use Lemma [Lemma 6](#lem:basic){reference-type="ref" reference="lem:basic"}$(i)$ with $k=2$ that yields $$\alpha(F_2)\ge 2\cdot \min_{i\in V}\alpha(G\setminus i)=2\alpha(G)\cdot\min_{i\in V}\kappa(i).$$ Thus, if $\min_{i\in V}\kappa(i)\ge \frac{1}{2}$, we have $\alpha(F_2)\ge \alpha(G)$ and the claimed equality is obtained. Now, $(ii)$ is a consequence of $(i)$ since, when $G$ is vertex-transitive, $\alpha(G\setminus i)=\alpha (G\setminus j)$ for every $i,j\in V$ and, then, $\min_{i\in V}\kappa(i)=\kappa(G)$. ◻ In Table [2](#tab:Data2){reference-type="ref" reference="tab:Data2"}, there are some examples of graphs that satisfy Theorem [Theorem 8](#th:G-vt){reference-type="ref" reference="th:G-vt"}$(ii)$. Graph $G$ ** $\alpha(G)$** **$\alpha(G\setminus i)$** --------------------------------- --------------------------------------------------- ----------------------------------------- -- $K_n$ $n$ $n-1$ Petersen $2$ $\approx 1.26$ Heawood $3-\sqrt{2} \approx 1.58$ $1$ Tetrahedron $4$ $3$ Octahedron $4$ $3$ Hexahedron $2$ $2(1-\cos(\frac{2\pi}{5}))\approx1.38$ Dodecahedron $3-\sqrt{5}\approx0.76$ $\approx 0.59$ Icosahedron $5-\sqrt{5}\approx 2.76$ $\frac{422009\pi}{605811} \approx 2.18$ Truncated tetrahedron $1$ $\approx 0.53$ Prism Graph $GP(n,1)$ $(n=3,4)$ $2$ $\approx 1.38$ Prism Graph $GP(n,1)$ $(n>4)$ $2\left(1-\cos\left(\frac{2\pi}{n}\right)\right)$ $-$ Hypercube $Q_n$ 2 $\ge 1$ : Some vertex-transitive graphs, with $\alpha(G) \leq 2\alpha (G\setminus i)$, where $i$ is a vertex. Note that, in most cases of Table [2](#tab:Data2){reference-type="ref" reference="tab:Data2"}, $\alpha(G)\geq2$. In fact, this condition is sufficient because Fiedler [@fi73] proved that, for any graph, $\alpha(G\setminus i)\ge \alpha(G)-1$ or, equivalently, when $G$ is vertex-transitive, $\kappa(G)\ge 1-\frac{1}{\alpha(G)}$. From this, if $\alpha(G)\ge 2$, the condition $\kappa(G)\ge \frac{1}{2}$ holds, as required to have $\alpha(F_2(G))=\alpha(G)$. An alternative, much more involved, proof of this case is given in Dalfó, Fiol, and Messegué [@dfm22]. Besides the hypercube $Q_n$ cited in Table [2](#tab:Data2){reference-type="ref" reference="tab:Data2"}, the following graphs constitute another infinite family of vertex-transitive graphs with algebraic connectivity $2$. The odd graph $O_r$ has vertices labeled with the $(r-1)$-subsets of a $(2r-1)$-set, and two vertices are adjacent if their corresponding subsets have void intersection. Thus, $O_r$ has ${2r-1\choose r-1}$ vertices, and it is regular of degree $r$. For instance, $O_2=K_3$, and $O_3$ is the Petersen graph. **Corollary 9**. *Let $O_r$ be the odd graph of degree $r$. Then, $\alpha(F_2(O_r))=\alpha(O_r)$.* *Proof.* As the second largest (adjacency matrix) eigenvalue of $O_r$ is $r-2$ for $r\ge 3$, its algebraic connectivity is $\alpha(O_r)=r-(r-2)=2$, as claimed. Then, according to the above consequences of Theorem [Theorem 8](#th:G-vt){reference-type="ref" reference="th:G-vt"}, the algebraic connectivities of $F_2(O_r)$ and $O_r$ coincide. ◻ Another infinite family satisfying Theorem [Theorem 8](#th:G-vt){reference-type="ref" reference="th:G-vt"}$(i)$ is the following. **Corollary 10**. *Let $G=K_{n_1,n_2,\ldots,n_r}\neq K_r$ be the multipartite complete graph on $n=\sum_{i=1}^r n_i$ vertices with $n_1\le n_2\le \cdots \le n_r$, and $r\geq3$. Then, $\alpha(F_2(G))=\alpha(G)$.* *Proof.* Notice that $\overline{G}=K_{n_1}\cup \cdots \cup K_{n_r}$. Then, the maximum eigenvalue of $\overline{G}$ coincides with the maximum eigenvalue of $K_{n_r}$, which is $\lambda_{\max}(K_{n_r})=n_r$. Thus, $\alpha(G)=n-\lambda_{\max}(\overline{G})=n-n_r$ (see Fiedler [@fi73]). Thus, by taking any vertex $i\not\in V_r$ (with $|V_r|=n_r$), we have $\xi(G^-)=\alpha(G\setminus i)=n-n_r-1$. Thus, $\min_{i\in V}\kappa(i)=\frac{n-n_r-1}{n-n_r}\ge \frac{1}{2}$, since $r\geq3$ and $n\ge 4$ (recall that $G\neq K_r$). Then, Theorem [Theorem 8](#th:G-vt){reference-type="ref" reference="th:G-vt"}$(i)$ gives the result. ◻ # The spectrum of $F_2(C_n)$ {#sec:2-tokens} In this section, we provide some results about the Laplacian spectrum (and, in particular, the algebraic connectivity) of the $k$-token $F_k(C_n)$ of the cycle $C_n$ on $n$ vertices $0,1,\ldots,n-1$, where $i$ is adjacent to $i+1\ (\textrm{mod}\, n)$. By Lemma [Lemma 1](#coro:LkL1){reference-type="ref" reference="coro:LkL1"}$(i)$, we already know that $F_k(C_n)$ contains all the eigenvalues of $C_n$. Namely, $$\label{tj} \theta_{j} =2\left(1-\cos\left(\frac{j2\pi}{n}\right)\right)=4\sin^2\left(\frac{j\pi}{n}\right),\qquad j=0,1,\ldots, n-1.$$ Next, we begin with a general lower bound for $\alpha(F_k(C_n))$, which is given in terms of the algebraic connectivity of the path $P_{n-1}$ on $n-1$ vertices. **Theorem 11**. *Let $C_n$ be the cycle graph on $n>3$ vertices. Then, $$\alpha(F_k(C_n))\ge \frac{k}{k-1}\alpha(P_{n-1}) =\frac{2k}{k-1}\left(1-\cos\left(\frac{\pi}{n-1}\right)\right) \label{eq:bound-token-cycle}$$ for every $n$ and $k=2,\ldots, \lfloor n/2\rfloor$.* *Proof.* As before, let $\mbox{\boldmath $v$}$ be an eigenvector of $F_k=F_k(C_n)$ with eigenvalue $\alpha(F_k)$ and norm $\|\mbox{\boldmath $v$}\|=1$. If $\mbox{\boldmath $B$}^{\top}\mbox{\boldmath $v$}\neq \mbox{\boldmath $0$}$, by Lemma [Lemma 1](#coro:LkL1){reference-type="ref" reference="coro:LkL1"}$(ii)$, $\alpha(F_k(C_n))=\alpha(C_n)$, and the result holds since $\alpha(C_n)>2\alpha(P_{n-1})$. Thus, since $n\neq 3$, we can assume that $\mbox{\boldmath $B$}^{\top}\mbox{\boldmath $v$}=\mbox{\boldmath $0$}$ and apply again Lemma [Lemma 6](#lem:basic){reference-type="ref" reference="lem:basic"}$(i)$ to get $$\begin{aligned} \alpha(C_n) & \ge \frac{k}{k-1}\xi(C_n^-)=\frac{k}{k-1}\alpha(F_{k-1}(C_n\setminus i))=\frac{k}{k-1}\alpha(F_k(P_{n-1}))\\ &=\frac{k}{k-1}\alpha(P_{n-1}),\end{aligned}$$ where the last equality follows from Theorem [Theorem 2](#theo:alg-connec-antic){reference-type="ref" reference="theo:alg-connec-antic"}$(iii)$. ◻ In particular, for $k=2$, [\[eq:bound-token-cycle\]](#eq:bound-token-cycle){reference-type="eqref" reference="eq:bound-token-cycle"} gives that $\alpha(F_2(C_n))\ge 2\alpha(P_{n-1})$, and equality holds for $n=4$ since $\alpha(F_2(C_4))= 2\alpha(P_{3})=2$. **Proposition 12**. - *If $n=2\nu$ is even, then the 2-token graph $F_2(C_n)$ has the eigenvalues $$\lambda_{r} =8\sin^2 \left(\frac{r\pi}{n-1}\right),\qquad \mbox{$r=0,1,\ldots, \nu-1,$} \label{lr-even}$$ with $\lambda_0=0$, and $\lambda_{\nu-1}=8\sin^2\left(\frac{n-2}{n-1}\frac{\pi}{2}\right)$ being the spectral radius of $F_2(C_n)$.* - *If $n=2\nu+1$ is odd, then the 2-token graph $F_2(C_n)$ has the eigenvalues $$\lambda_{r} =8\cos^2 \left(\frac{r\pi}{n-1}\right),\qquad \mbox{$r=1,2,\ldots, \nu,$} \label{lr-odd}$$ with $\lambda_{\nu}=0$, and $\lambda_{1}=8\cos^2\left(\frac{\pi}{n-1}\right)$ being a lower bound for the spectral radius of $F_2(C_n)$.* *Proof.* Let us see that $F_2=F_2(C_n)$ has a regular 'path-shaped' partition $\pi$ with $r=\lfloor n/2 \rfloor$ classes $V_1,V_2,\ldots,V_{r}$, where $V_i$ consists of the vertices $\{u,v\}$ such that $\mathop{\mathrm{dist}}(u,v)=i$ in $C_n$. - Each vertex $\{u,v\}$ in $V_1$ is adjacent to $2$ vertices $\{u+1,v\}$ and $\{u,v+1\}$ in $V_2$ (all arithmetic is modulo $n$). - Each vertex $\{u,v\}$ in $V_i$, for $i=2,\ldots,r-1$, is adjacent to $2$ vertices $\{u-1,v\}$ and $\{u,v-1\}$ in $V_{i-1}$, and $2$ vertices $\{u+1,v\}$ and $\{u,v+1\}$ in $V_{i+1}$. - Every vertex $\{u,v\}$ in $V_r$ is adjacent to $4$ vertices $\{u\pm 1,v\}$ and $\{u,v\pm 1\}$. If $n$ is even, all these vertices are in $V_{r-1}$. If $n$ is odd, two of them are in $V_{r-1}$ and the other two in $V_r$. For instance, the quotient graphs of the path-shaped regular partitions of $F_2(C_9)$, $F_2(C_{10})$, and $F_2(C_8)$ are shown in Figures [1](#F_2(C_9)){reference-type="ref" reference="F_2(C_9)"}$(c)$, [2](#F_2(C_{10})){reference-type="ref" reference="F_2(C_{10})"}$(c)$, and [3](#F_2(C_8)){reference-type="ref" reference="F_2(C_8)"}$(c)$, respectively. Then, the quotient matrix $\mbox{\boldmath $Q$}_A=\mbox{\boldmath $A$}(F_2/\pi)$ and quotient Laplacian matrix $\mbox{\boldmath $Q$}_L=\mbox{\boldmath $L$}(F_2/\pi)$ are tridiagonal matrices of the form $$\label{Qodd} \mbox{\boldmath $Q$}_A = \left( \begin{array}{ccccc} 0 & 2 & & & \\ 2 & 0 & 2 & & \\ %& -2 & 4 & -2 & \\ & \ddots & \ddots & \ddots & \\ & & 2 & 0 & 2\\ & & & a & b \end{array} \right),\quad \mbox{and}\quad \mbox{\boldmath $Q$}_L = \left( \begin{array}{ccccc} 2 & -2 & & & \\ -2 & 4 & -2 & & \\ %& -2 & 4 & -2 & \\ & \ddots & \ddots & \ddots & \\ & & -2 & 4 & -2\\ & & & -c & c \end{array} \right),$$ where $(a,b,c)=(4,0,4)$ if $n$ even, and $(a,b,c)=(2,2,2)$ if $n$ is odd. Then, [\[lr-even\]](#lr-even){reference-type="eqref" reference="lr-even"} and [\[lr-odd\]](#lr-odd){reference-type="eqref" reference="lr-odd"} correspond to the eigenvalues of the matrix $\mbox{\boldmath $Q$}_L$ for the even and odd cases of $n$, respectively (see Yueh [@y05 Th.4]). Moreover, in the case of even $n$, the maximum eigenvalue in [\[lr-even\]](#lr-even){reference-type="eqref" reference="lr-even"} is obtained when $r=\nu-1$. Then, from the last comments of Subsection [2.2](#sec:reg-part){reference-type="ref" reference="sec:reg-part"}, $\lambda_{\nu-1}$ is the spectral radius of $F_2(C_n)$. ◻ Next, we show that, depending on the parity of $n$, much more can be stated about the $F_k(C_n)$ spectrum. Let us consider first the case when $n$ is odd. {#F_2(C_9) width="14cm"} ## The case of odd $n$ As commented in Section [2](#resultats-coneguts){reference-type="ref" reference="resultats-coneguts"}, the case of odd $n$ was studied in Dalfó, Fiol, Pavlíková, and Širán [@df22b], giving the following result. **Theorem 13** ([@df22b]). *The 2-token graph $F_2(C_n)$ of the cycle with an odd number $n=2\nu+1$ of vertices is the lift $G^{\beta}(P^+_{\nu})$ of the base graph the path $P^+_{\nu}$ with vertex set $\{u_1,u_2,\ldots,u_{\nu}\}$, a loop at $u_{\nu}$, and arcs $a_i = u_iu_{i+1}$ and $a_i^-=u_{i+1}u_i$, for $i = 1,2,\ldots, k-1$. The voltages on the group $\mathbb Z_n$ are as follows: $$\begin{aligned} \beta(u_iu_{i+1})&=-1 \qquad \mathrm{for} \ i=1,\ldots,\nu-1,\\ \beta(u_{i+1}u_i)&=+1 \qquad \mathrm{for} \ i=1,\ldots,\nu-1,\\ \beta(u_{\nu}u_{\nu}) &=\pm \nu.\end{aligned}$$* For example, in the case of $n=9$, the 2-token graph $F_2(C_9)$ and its base graph are shown in Figure [1](#F_2(C_9)){reference-type="ref" reference="F_2(C_9)"}. Then, the whole spectrum of $F_2(C_n)$ can be obtained from its Laplacian base $\nu\times \nu$ matrix $\mbox{\boldmath $B$}(z)$ (see again [@df22b]), with $z=e^{ir\frac{2\pi}{n}}$, or its similar tridiagonal matrix $\mbox{\boldmath $B$}^*(r)$, for $r=0,1,\ldots,n-1$. $$% \begin{align*} % \label{B(z)Alegre} \mbox{\boldmath $B$}(z) = \left( {\small{ \begin{array}{cccccc} 2 & -1-z^{-1} & 0 & 0 & \ldots & 0 \\ -1-z & 4 & -1-z^{-1} & 0 & \ldots & 0 \\ 0 & -1-z & 4 & -1-z^{-1} & \ddots & 0 \\ 0 & 0 & -1-z & \ddots & \ddots & 0 \\ \vdots & \vdots & \ddots &\ddots & 4 & -1-z^{-1}\\ 0 & 0 & \ldots &0 &-1-z & 4-z^\nu-z^{-\nu} \end{array}}} \right)\cong %\end{align*}$$ $$\begin{aligned} \mbox{\boldmath $B$}^*(r)&= \left( \begin{array}{cccccc} 2 & 2\cos(\frac{r\pi}{n}) & 0 & 0& \ldots & 0\\ 2\cos(\frac{r\pi}{n}) & 4 & 2\cos(\frac{r\pi}{n})& 0 & \ldots & 0\\ 0 & 2\cos(\frac{r\pi}{n}) & 4 & 2\cos(\frac{r\pi}{n})& \ddots& 0\\ 0 & 0 & 2\cos(\frac{r\pi}{n}) & \ddots & \ddots & 0\\ \vdots & \vdots & \ddots & \ddots & 4 & 2\cos(\frac{r\pi}{n}) \\[.1cm] 0 & 0 & \ldots & 0 & 2\cos(\frac{r\pi}{n}) & 4 + 2(-1)^{r + 1} \cos(\frac{r\pi}{n}) \end{array} \right). \label{B-star}\end{aligned}$$ The key idea is that, as shown in the following proposition, each eigenvector of $\mbox{\boldmath $B$}(z)$, with eigenvalue $\lambda$, gives rise to an eigenvector of $\mbox{\boldmath $L$}$, the Laplacian matrix of $F_2(C_n)$, with the same eigenvalue $\lambda$. Although this follows from results by Dalfó, Fiol, and Širáň [@dfs19], and Dalfó, Fiol, Pavlíková, and Širán [@df22b], we give here a direct new proof for completeness. To this end, we label the ${n\choose 2}$ vertices of $F_2(C_n)$, with $n=2\nu+1$, with the pairs $(i,j)=(j+h,j)$ where $j=0,1,\ldots,n-1$ and $h=1,\ldots,\nu$, with arithmetic modulo $n$. With this notation, notice that $h=i-j$ is the distance from $i$ to $j$ in the cycle $C_n$. Besides, every vertex $(i,j)$, with $i=j+h$, of $F_2(C_n)$ corresponds to the vertex $(u_{h},j)$, for $j\in \mathbb Z_n$, of the lift $G^{\beta}(P_\nu^{+})$. See Figure [1](#F_2(C_9)){reference-type="ref" reference="F_2(C_9)"}$(a)$-$(b)$ for the case of $F_2(C_9)$ with the simplified notation $ij=(i,j)$. In what follows, and with a slight abuse of notation, we use $\zeta$ with two meanings. First, $\zeta$ refers to a generic $n$-th root of unity (as in the following proposition). Second, $\zeta^r$ refers to the power $r$ of the $n$-th root of unity so that, in this case, $\zeta=e^{i\frac{2\pi}{n}}$ stands for the first $n$-th root of unity different from $1$ (as in Subsection [2.3](#sec:sp){reference-type="ref" reference="sec:sp"}). **Proposition 14**. *Every eigenvalue $\lambda$ of $F_2(C_n)\cong G^{\beta}(P_\nu^{+})$, with $n=2\nu+1$, has an eigenvector $\mbox{\boldmath $y$}\in \mathbb R^{{n\choose 2}}$ with components $$\label{vector-y} y_{(i,j)}=f_{i-j}\zeta^j=f_{h}\zeta^j\qquad j=0,\ldots,n-1,\ h=1,\ldots,\nu,$$ where $\zeta$ is a given $n$-th root of unity, and $\mbox{\boldmath $f$}= (f_1,\ldots,f_{\nu})$ is a $\lambda$-eigenvector of the matrix $\mbox{\boldmath $B$}(\zeta)$.* *Proof.* Let $\mbox{\boldmath $L$}$ be the Laplacian matrix of $F_2(C_n)$. Let $\mbox{\boldmath $x$}$ be an eigenvector of $\mbox{\boldmath $L$}$ with eigenvalue $\lambda$. We show that $\mbox{\boldmath $x$}$ implies the existence of a vector whose entries can be written as claimed. Note first that, since $n$ is odd, all the classes $V_1,\ldots, V_{\nu}$ of the regular partition of $F_2(C_n)$ have the same number $n$ of vertices. Then, from $\mbox{\boldmath $x$}$, we construct the vectors $\mbox{\boldmath $x$}^0(=\mbox{\boldmath $x$}),\mbox{\boldmath $x$}^1,\ldots,\mbox{\boldmath $x$}^{n-1}$ by shifting in the same way the entries of $\mbox{\boldmath $x$}$ corresponding to each class. More precisely, $$\label{vector-xa} x^a_{(i,j)}=x_{(i+a,j+a)}\quad \mbox{ for $a=0,\ldots,n-1$,}$$ (all arithmetic understood modulo $n$). But, in $F_2(C_n)$, the mapping $(i,j)\mapsto (i+a,j+a)$ is an automorphism for every $a=0,\ldots, n-1$. In fact, it is known that $\mathop{\mathrm{Aut}}F_2(C_n)\cong \mathop{\mathrm{Aut}}C_n$, see Ibarra and Rivera [@ir22]. Then, all the vectors $\mbox{\boldmath $x$}^a$ are eigenvectors of $F_2(C_n)$ with eigenvalue $\lambda$. Moreover, from [\[vector-xa\]](#vector-xa){reference-type="eqref" reference="vector-xa"}, there exists a matrix $\mbox{\boldmath $R$}$ of size $n\times n$ such that for every $i,j$ (with $i=j+h\ (\mbox{mod } n)$): $$\label{x-R-x} \left( \begin{array}{c} x^0_{(i+1,j+1)}\\ x^1_{(i+1,j+1)}\\ \vdots \\ x^{n-1}_{(i+1,j+1)} \end{array} \right) = \mbox{\boldmath $R$} \left( \begin{array}{c} x^0_{(i,j)}\\ x^1_{(i,j)}\\ \vdots \\ x^{n-1}_{(i,j)} \end{array} \right),$$ where $$\mbox{\boldmath $R$}=\mathop{\mathrm{circ}}(0,1,0\ldots,0)= \left( \begin{array}{cccccc} 0 & 1 & 0 & \cdots & 0\\ 0 & 0 & 1 & \cdots & 0\\ \vdots & \vdots & \vdots &\ddots & \vdots \\ 0 & 0 & 0 & \cdots & 1\\ 1 & 0 & 0 & \cdots & 0\\ \end{array} \right).$$ Then, with $\mbox{\boldmath $R$}^n=\mbox{\boldmath $I$}$, the $n$ eigenvalues of $\mbox{\boldmath $R$}$ are simple and equal to the $n$-th roots of unity $\zeta^r=e^{i r\frac{2\pi}{n}}$, with $i=\sqrt{-1}$ and $r=0,\ldots,n-1$. Thus, there exists an inversible (and orthogonal as $\mbox{\boldmath $L$}$ is symmetric) matrix $\mbox{\boldmath $Q$}$ such that $\mbox{\boldmath $D$}=\mbox{\boldmath $Q$}^{-1}\mbox{\boldmath $R$}\mbox{\boldmath $Q$}$, where $\mbox{\boldmath $D$}=\mathop{\mathrm{diag}}(1,\zeta,\zeta^2,\ldots,\zeta^{n-1})$. Let $\mbox{\boldmath $X$}$ be the matrix with rows $\mbox{\boldmath $x$}^0,\mbox{\boldmath $x$}^1,\ldots,\mbox{\boldmath $x$}^{n-1}$. From $\mbox{\boldmath $X$}\mbox{\boldmath $L$}=\lambda\mbox{\boldmath $X$}$, we have $\mbox{\boldmath $Q$}^{-1}\mbox{\boldmath $X$}\mbox{\boldmath $L$}=\lambda\mbox{\boldmath $Q$}^{-1}\mbox{\boldmath $X$}$ and, hence, the vectors $\mbox{\boldmath $y$}^0,\mbox{\boldmath $y$}^1,\ldots,\mbox{\boldmath $y$}^{n-1}$ with components $$\label{y-Q-x} \left( \begin{array}{c} y^0_{(i,j)}\\ y^1_{(i,j)}\\ \vdots \\ y^{n-1}_{(i,j)} \end{array} \right) = \mbox{\boldmath $Q$}^{-1} \left( \begin{array}{c} x^0_{(i,j)}\\ x^1_{(i,j)}\\ \vdots \\ x^{n-1}_{(i,j)} \end{array} \right),$$ or rows of the matrix $\mbox{\boldmath $Y$}=\mbox{\boldmath $Q$}^{-1}\mbox{\boldmath $X$}$, are also $\lambda$-eigenvectors of $F_2(G)$, provided that they are different from $\mbox{\boldmath $0$}$. (Notice that the number of such eigenvectors, that is, $\mathop{\mathrm{rank}}\mbox{\boldmath $Y$}$, is at most $m(\lambda)$, the multiplicity of $\lambda$.) Moreover, from [\[y-Q-x\]](#y-Q-x){reference-type="eqref" reference="y-Q-x"} and [\[x-R-x\]](#x-R-x){reference-type="eqref" reference="x-R-x"} with $\mbox{\boldmath $R$}=\mbox{\boldmath $Q$}\mbox{\boldmath $D$}\mbox{\boldmath $Q$}^{-1}$, we get $$\label{y-R-y} \left( \begin{array}{c} y^0_{(i+1,j+1)}\\ y^1_{(i+1,j+1)}\\ \vdots \\ y^{n-1}_{(i+1,j+1)} \end{array} \right) = \mbox{\boldmath $D$} \left( \begin{array}{c} y^0_{(i,j)}\\ y^1_{(i,j)}\\ \vdots \\ y^{n-1}_{(i,j)} \end{array} \right).$$ Now, since $\mbox{\boldmath $x$}\neq \mbox{\boldmath $0$}$, there is at least a non-zero vector $\mbox{\boldmath $y$}^a$ satisfying $y^a_{(i+1,j+1)}=\zeta^a y^a_{(i,j)}$ or, iterating, $y^a_{(i+b,j+b)}=\zeta^{ab} y^a_{(i,j)}$. Therefore, letting $b=-j$, $$y^a_{(i,j)}= y^a_{(i-j,0)}\zeta^{aj}.$$ Then, the result in [\[vector-y\]](#vector-y){reference-type="eqref" reference="vector-y"} follows by taking $\mbox{\boldmath $y$}=\mbox{\boldmath $y$}^a$, $\zeta=\zeta^a$, and $f_h=y^a_{(h,0)}$. To show that $\mbox{\boldmath $f$}(\zeta)$ is an eigenvector of $\mbox{\boldmath $B$}(z)$ with $z=\zeta$, we distinguish three cases: - $h=1$: The vertex $(j+1,j)$, with vector entry $y_{(j+1,j)}=f_1\zeta^j$, is adjacent to both vertices $(j+2,j)$ and $(j+1,j-1)$, with respective vector entries $f_2\zeta^j$ and $f_2\zeta^{j-1}$. Then, the first entry of $\mbox{\boldmath $y$}\mbox{\boldmath $L$}=\lambda\mbox{\boldmath $y$}$ is $2f_1\zeta^j-f_2\zeta^j-f_2\zeta^{j-1}=\lambda f_1\zeta^j$. Hence, $2f_1-f_2-f_2\zeta^{-1}=\lambda f_1$, which corresponds to the first entries of $\mbox{\boldmath $B$}(\zeta)\mbox{\boldmath $f$}=\lambda\mbox{\boldmath $f$}$, as it should. - $2\le h\le \nu-1$: The vertex $(j+h,j)$, with vector entry $y_{(j+h,j)}=f_h\zeta^j$, is adjacent to the four vertices $(j+h+1,j)$, $(j+h,j+1)$, $(j+h-1,j)$, and $(j+h,j-1)$, with respective vector entries $f_{h+1}\zeta^j$, $f_{h-1}\zeta^{j+1}$, $f_{h-1}\zeta^{j}$, and $f_{h+1}\zeta^{j-1}$. Then, the $h$-th entry of $\mbox{\boldmath $y$}\mbox{\boldmath $L$}=\lambda\mbox{\boldmath $y$}$ is $$4f_h\zeta^j-f_{h+1}\zeta^j-f_{h-1}\zeta^{j+1}-f_{h-1}\zeta^{j}-f_{h+1}\zeta^{j-1}=\lambda f_h\zeta^j.$$ Thus, $$4f_h-f_{h+1}-f_{h-1}\zeta-f_{h-1}-f_{h+1}\zeta^{-1}=\lambda f_h,$$ which corresponds to the $h$-th entry of $\mbox{\boldmath $B$}(\zeta)\mbox{\boldmath $f$}=\lambda\mbox{\boldmath $f$}$. - $h=\nu$: The vertex $(j+\nu,j)$, with vector entry $y_{(j+\nu,j)}=f_{\nu}\zeta^j$, is adjacent (according to the used notation) to the four vertices $(j,j+\nu+1)$, $(j+\nu-1,j)$, $(j+\nu,j+1)$, and $(j-1,j+\nu)$, with respective vector entries $f_{-(\nu+1)}\zeta^{j+\nu+1}=f_{\nu}\zeta^{j+\nu+1}$ (the subscripts of $f$ are modulo $n$), $f_{\nu-1}\zeta^{j}$, $f_{\nu-1}\zeta^{j+1}$, and $f_{-(1+\nu)}\zeta^{j+\nu}=f_{\nu}\zeta^{j+\nu}$. Then, the $\nu$-th entry of $\mbox{\boldmath $y$}\mbox{\boldmath $L$}=\lambda\mbox{\boldmath $y$}$ is $$4f_{\nu}\zeta^j-f_{\nu}\zeta^{j+\nu+1}-f_{\nu-1}\zeta^{j}-f_{\nu-1}\zeta^{j+1}-f_{\nu}\zeta^{j+\nu}=\lambda f_{\nu}\zeta^j.$$ Thus, $$4f_{\nu}-f_{\nu}\zeta^{\nu+1}-f_{\nu-1}-f_{\nu-1}\zeta-f_{\nu}\zeta^{\nu}=\lambda f_{\nu},$$ which corresponds to the $\nu$-th entry of $\mbox{\boldmath $B$}(\zeta)\mbox{\boldmath $f$}=\lambda\mbox{\boldmath $f$}$. This completes the proof. ◻ In the case of $F_2(C_9)$, the obtained eigenvalues are shown in Table [3](#taula:C9){reference-type="ref" reference="taula:C9"}. We focus on the matrix $\mbox{\boldmath $B$}^*(r)$ in the following result. **Proposition 15**. *Given $r=0,1,\ldots,2\nu$, let $\lambda_{r,1}\le\lambda_{r,2}\le\cdots \le\lambda_{r,\nu}$ be the eigenvalues of the matrix $\mbox{\boldmath $B$}^*(r)$. Then, the following holds:* - *The eigenvalues of $\mbox{\boldmath $B$}^*(0)$ are $$\lambda_{0,s} =8\cos^2 \left(\frac{s\pi}{2\nu}\right)\quad \mbox{for $s=1,2,\ldots, \nu$}. \label{lj}$$ Then, $\lambda_{0,\nu}=0$. Moreover, the smallest nonzero eigenvalue is obtained when $s=\nu-1$ and satisfies $\lambda_{0,\nu-1}>\alpha(C_{2\nu+1})$.* - *For each $r=1,\ldots,2\nu$, the smallest eigenvalue of $\mbox{\boldmath $B$}^*(r)$ satisfies $$\lambda_{r,1}\ge 4\sin^2\left(\frac{r\pi}{2(2\nu+1)}\right).$$* - *The matrix $\mbox{\boldmath $B$}(\zeta^r)$, with $\zeta=e^{i\frac{2\pi}{2\nu+1}}$, has exactly one eigenvalue of $C_{2\nu+1}$, which is $\lambda_r=2\left(1-\cos\left(\frac{r2\pi}{2\nu+1}\right)\right)$. Besides, the eigenvalues of $\mbox{\boldmath $B$}^*(0)$ have multiplicity one, whereas the eigenvalues of $\mbox{\boldmath $B$}^*(r)$, with $r=1,\ldots, 2\nu$, have multiplicity two.* *Proof.* $(i)$ Notice that, when $r=0$, $\mbox{\boldmath $B$}^*(0)$ is the tridiagonal matrix $\mbox{\boldmath $Q$}_L$ in [\[Qodd\]](#Qodd){reference-type="eqref" reference="Qodd"}, with eigenvalues already given in Proposition [Proposition 12](#propo:path-shaped){reference-type="ref" reference="propo:path-shaped"}. Moreover, the function $$\phi(\nu)=\frac{\alpha(C_{2\nu+1})}{\lambda_{0,\nu-1}}= \frac{\sin^2\left(\frac{\pi}{2\nu+1}\right)}{2\cos^2\left(\frac{(\nu-1)\pi}{2\nu}\right)}$$ satisfies $\phi(\nu)< 1/2$ for $\nu>0$.\ $(ii)$ From the matrix $\mbox{\boldmath $B$}^*(r)$, we have three different Gershgorin circles determining three intervals $I_i(r)$, for $i=1,2,3$, in the real line. Thus, all eigenvalues of $\mbox{\boldmath $B$}^*(r)$ are within $I_1(r)\cup I_2(r)\cup I_3(r)$. The left endpoints of these intervals are $\ell_1(r)=4\sin^2(\frac{r\pi}{2(2\nu+1)})$, $\ell_2(r)=8\sin^2(\frac{r\pi}{4\nu+2})$, and $\ell_3(r)=4$ for $r$ odd and $\ell_3(r)=8\sin^2(\frac{r\pi}{4\nu+2})$ for $r$ even. Then, $$\begin{aligned} \ell_i(0) &=0\quad \mbox{for $i=1,2,3$,}\\ \ell_1(1) &=4\sin^2\left(\frac{\pi}{2(2\nu+1)}\right)=\alpha(P_{2\nu+1}),\\ \ell_1(2) &=4\sin^2\left(\frac{\pi}{2\nu+1}\right)=\alpha(C_{2\nu+1}).\end{aligned}$$ (See Table [4](#taula:limits-C9){reference-type="ref" reference="taula:limits-C9"} for the case $\nu=4$, corresponding to the matrices $\mbox{\boldmath $B$}^*(r)$ of $F_2(C_9)$.) Now, a simple analysis shows that the values of $\ell_i(r)$ are increasing when $r=0,1,\ldots, 2\nu$ when $i=1,2$, and $r=0,2,\ldots, 2\nu$ when $i=3$. Thus, the result follows since $\ell_1(r)<\min\{\ell_2(r),\ell_3(r)\}$ for $r\neq 0$. $(iii)$ We prove that, for every $z=e^{ir\frac{2\pi}{n}}$, for $r=0,\ldots,n-1$, the matrix $\mbox{\boldmath $B$}(z)\cong \mbox{\boldmath $B$}^*(r)$ has exactly one eigenvalue of $C_n$. From Proposition [Proposition 14](#propo:eigenvec-odd){reference-type="ref" reference="propo:eigenvec-odd"}, we know that each of the eigenvectors $\mbox{\boldmath $y$}$ of $F_2(C_{2\nu+1})$ has entries $y_{(i,j)}=f_{i-j}\zeta^j=f_{h}\zeta^j$ for $i=0,\ldots,n-1$, $h=1,\ldots,\nu$, and $\mbox{\boldmath $f$}=(f_1,\ldots,f_{\nu})$ an eigenvector of $\mbox{\boldmath $B$}(\zeta)$. Let $\mbox{\boldmath $B$}$ be the $(n,2)$-binomial matrix. Recall that, from Lemma [Lemma 1](#coro:LkL1){reference-type="ref" reference="coro:LkL1"}$(ii)$, if $\mbox{\boldmath $y$}$ is a $\lambda$-eigenvector of $F_2(G)$ and $\mbox{\boldmath $x$}=\mbox{\boldmath $B$}^{\top}\mbox{\boldmath $y$}\neq \mbox{\boldmath $0$}$, then $\mbox{\boldmath $x$}$ is a $\lambda$-eigenvector of $G$. In our case, notice that, for $j=0,\ldots,n-1$, the $j$-th entry of the vector $\mbox{\boldmath $x$}=\mbox{\boldmath $B$}^{\top}\mbox{\boldmath $y$}$ is obtained by adding all the $n-1$ entries of the vector $\mbox{\boldmath $y$}$ with labels having a common $j$, that is, corresponding to the vertices $$(j+1,j),(j+2,j),\ldots,(j+\nu,j),(j,j-\nu),(j,j-\nu+1),\ldots,(j,j-1).$$ Then, the $j$-th entry of the vector $\mbox{\boldmath $x$}$ turns out to be $$\label{x-from-f} f_1(\zeta^j+\zeta^{j-1})+f_2(\zeta^j+\zeta^{j-2})+\cdots + f_{\nu}(\zeta^j+\zeta^{j-\nu}),\qquad j=0,\ldots,n-1.$$ Let $\mbox{\boldmath $F$}$ be the $\nu\times \nu$ matrix whose columns $\mbox{\boldmath $f$}^1,\ldots,\mbox{\boldmath $f$}^{\nu}$ are the eigenvectors of $\mbox{\boldmath $B$}(\zeta)$, and $\mbox{\boldmath $Z$}$ the $n\times \nu$ matrix with $j$-th row $(\zeta^j+\zeta^{j-1}), (\zeta^j+\zeta^{j-2}),\ldots, (\zeta^j+\zeta^{j-\nu})$. Then, in matrix form, [\[x-from-f\]](#x-from-f){reference-type="eqref" reference="x-from-f"} is $\mbox{\boldmath $X$}=\mbox{\boldmath $Z$}\mbox{\boldmath $F$}$, where $\mbox{\boldmath $X$}$ is the $n\times \nu$ matrix with columns being the putative eigenvectors of $C_n$. But $\mbox{\boldmath $F$}$ has full rank, whereas $\mbox{\boldmath $Z$}$ has rank 1 (every row is a multiple of the first one). Consequently, $\mathop{\mathrm{rank}}\mbox{\boldmath $X$}=1$ and, hence, exactly one $\lambda$-eigenvector of $\mbox{\boldmath $B$}(\zeta)$ gives a $\lambda$-eigenvector of $C_n$. The statement about the multiplicities follows from the fact that if $\zeta\neq 1$, the spectra of $\mbox{\boldmath $B$}(\zeta)$ and $\mbox{\boldmath $B$}(\zeta^{-1})=\mbox{\boldmath $B$}(\overline{\zeta})$ coincide, where $\overline{\zeta}$ is the conjugate of $\zeta$. ◻ $\zeta=e^{i\frac{2\pi}{9}}$, $z=\zeta^r$ $\lambda_{r,1}$ $\lambda_{r,2}$ $\lambda_{r,3}$ $\lambda_{r,4}$ ----------------------------------------------------------------------------------------------------------- ------------------ ----------------- ----------------- ----------------- $\mathop{\mathrm{sp}}(\mbox{\boldmath $B$}(\zeta^0))$ **0** 1.171572876 4 6.828427124 $\mathop{\mathrm{sp}}(\mbox{\boldmath $B$}(\zeta^1))=\mathop{\mathrm{sp}}(\mbox{\boldmath $B$}(\zeta^8))$ **0.4679111136** 2.52079560 5.420264509 7.470414013 $\mathop{\mathrm{sp}}(\mbox{\boldmath $B$}(\zeta^2))=\mathop{\mathrm{sp}}(\mbox{\boldmath $B$}(\zeta^7))$ 0.783324839 **1.65270363** 3.895673125 6.136209510 $\mathop{\mathrm{sp}}(\mbox{\boldmath $B$}(\zeta^3))=\mathop{\mathrm{sp}}(\mbox{\boldmath $B$}(\zeta^6))$ 1.50913638 **3** 4.656620432 5.834243185 $\mathop{\mathrm{sp}}(\mbox{\boldmath $B$}(\zeta^4))=\mathop{\mathrm{sp}}(\mbox{\boldmath $B$}(\zeta^5))$ 1.939683655 3.382489411 **3.87938479** 4.451145779 : All the eigenvalues of the matrices $\mbox{\boldmath $B$}(\zeta^r)$, which yield the eigenvalues of the 2-token graph $F_2(C_9)$. The values in boldface correspond to the eigenvalues of $C_9$. $r=0$ $r=1$ $r=2$ $r=3$ ------------- ------- ------------------ ------------------ ------- $\ell_1(r)$ 0 **0.1206147584** **0.4679111138** 1 $\ell_2(r)$ 0 0.24122951686 0.93582222752 2 $\ell_3(r)$ 0 4 0.93582222752 4 : Left endpoints of the Gershgorin circles of the matrices $\mbox{\boldmath $B$}^*(r)$. The values in boldface correspond to the algebraic connectivities of $P_9$ and $C_9$. Moreover, results from Table [3](#taula:C9){reference-type="ref" reference="taula:C9"} suggest that the minimum and maximum eigenvalues of the matrix $\mbox{\boldmath $B$}^*(1)$ correspond to the algebraic connectivity $\alpha(F_2(C_{2\nu+1}))=\alpha(C_{2\nu+1})$, and spectral (Laplacian) radius $\rho(F_2(C_{2\nu+1}))$, respectively. ## The case of even $n$ and odd $n/2$ In the case of even $n=4r+2$ (so that $n/2$ is odd), the $2$-token $F_2(C_n)$ can also be seen as a lift graph, as shown in the following result. See an example of this kind of token graph in Figure [2](#F_2(C_{10})){reference-type="ref" reference="F_2(C_{10})"}. {#F_2(C_{10}) width="14cm"} Given an integer $r$, let us consider the path graph $G=P_{4r+1}$ with vertices $$u_{-2r},u_{-2r+1},\ldots, u_{-1},u_0,u_1,\ldots,u_{2r-1},u_{2r},$$ (with its corresponding edges) and additional arcs $$\begin{array}{lll} a_i^{+}=u_{-i}u_{i-1}, & a_i^{-}=u_{i-1}u_{-i}, & \mbox{for } i=0,1\ldots,2r,\\ b_i^{+}=u_{i}u_{-i+1}, & b_i^{-}=u_{-i+1}u_{i}, & \mbox{for } i=0,1,\ldots,2r. \end{array}$$ Let $\beta$ be the voltage assignment on $G$ in the cyclic group $\mathbb Z_{2r+1}$ given by $$\label{eq:voltatgesC10} \begin{array}{c} \beta(a_i^+)=\beta(b_i^+)=+r,\\ \beta(a_i^-)=\beta(b_i^-)=-r. \end{array}$$ Figure [2](#F_2(C_{10})){reference-type="ref" reference="F_2(C_{10})"}$(b)$ shows the base graph $G$ for $r=2$. Now, we have the following result. **Lemma 16**. *Given $G=P_{4r+1}$ with the voltage assignment [\[eq:voltatgesC10\]](#eq:voltatgesC10){reference-type="eqref" reference="eq:voltatgesC10"} on $\mathbb Z_{2r+1}$, the 2-token graph of the cycle $C_{n}$ with $n=4r+2$ is the lift graph $G^{\beta}$. That is, $$F_2(C_{4r+2})\cong G^{\beta}.$$* *Proof.* The vertex set $V(G^{\beta})$ of the lift $G^{\beta}$ has elements labeled with the pairs $(u_i,g)$ for $i=-2r,\ldots,2r$ and $g\in \mathbb Z_{2r+1}$. Thus $|V(G^{\beta})|=(4r+1)(2r+1)={4r+2\choose 2}$, which corresponds to the number of vertices of $F_2(C_{4r+2})$. Indeed, such vertices correspond to 2-subsets $\{i,j\}$ of the set $\{1,2,\ldots,4r+2\}$. Thus, we have to show a 1-to-1 mapping between $V(G^{\beta})$ and $V(F_2(C_{4r+2}))$ that must be consequent with the adjacencies of both graphs. Such a mapping is shown in Table [5](#taula:token-lifts){reference-type="ref" reference="taula:token-lifts"}, from where it is easily checked that the adjacency conditions are fulfilled. Let us take an example. The vertex $(u_{-2r+1},1)\equiv \{2,4\}$ (written as 24 in Figure [2](#F_2(C_{10})){reference-type="ref" reference="F_2(C_{10})"}) of $G^{\beta}$ is adjacent to: - The vertices $(u_{-2r},1)$ and $(u_{-2r+2},1)$ of the same 'copy'. - The vertex $(u_{2r-2},r+1)$ by the arc $a_{2r-1}^+$ with voltage $+r$. - The vertex $(u_{2r},r+2)$ by the arc $b^{-}_{2r}$ with voltage $-r$. Then, looking again at Table [5](#taula:token-lifts){reference-type="ref" reference="taula:token-lifts"}, we find the following equivalences: - $(u_{-2r},1)\equiv \{2,3\}$ and $(u_{-2r+2},1)\equiv \{2,5\}$, - $(u_{2r-2},r+1)\equiv \{1,4\}$ and $(u_{2r},r+2)\equiv \{3,4\}$, which correspond to the vertices adjacent to $\{2,4\}$ in $F_2(C_{4r+2}$). $0$ $r+1$ $1$ $r+2$ $\cdots$ $r$ ------------- ----------------- -------------------- -------------------- -------------------- ---------- ----------------- $u_{-2r}$ $\{0,1\}$ $\{2r+2,2r+3\}$ $\mathbf{\{2,3\}}$ $\{2r+4,2r+5\}$ $\cdots$ $\{2r,2r+1\}$ $u_{-2r+1}$ $\{0,2\}$ $\{2r+2,2r+4\}$ $\mathbf{\{2,4\}}$ $\{2r+4,2r+6\}$ $\cdots$ $\{2r,2r+2\}$ $u_{-2r+2}$ $\{0,3\}$ $\{2r+2,2r+5\}$ $\mathbf{\{2,5\}}$ $\{2r+4,2r+7\}$ $\cdots$ $\{2r,2r+3\}$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $u_{-1}$ $\{0,2r\}$ $\{2r+2,0\}$ $\{2,2r+2\}$ $\{2r+4,2\}$ $\cdots$ $\{2r,4r\}$ $u_{0}$ $\{0,2r+1\}$ $\{1,2r+2\}$ $\{2,2r+3\}$ $\{3,2r+4\}$ $\cdots$ $\{2r,4r+1\}$ $u_{1}$ $\{2r+1,4r+1\}$ $\{1,2r+1\}$ $\{2r+3,1\}$ $\{3,2r+3\}$ $\cdots$ $\{4r+1,2r-1\}$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $u_{2r-2}$ $\{2r+1,2r+4\}$ $\mathbf{\{1,4\}}$ $\{2r+3,2r+6\}$ {3,6} $\cdots$ $\{4r+1,2\}$ $u_{2r-1}$ $\{2r+1,2r+3\}$ $\{1,3\}$ $\{2r+3,2r+5\}$ {3,5} $\cdots$ $\{4r+1,1\}$ $u_{2r}$ $\{2r+1,2r+2\}$ $\{1,2\}$ $\{2r+3,2r+4\}$ $\mathbf{\{3,4\}}$ $\cdots$ $\{4r+1,0\}$ : All the vertices of the 2-token graph $F_2(C_n)$ for even $n$ and odd $n/2$. The vertex {2,4} and its adjacent vertices are in boldface, following the example.  ◻ As a consequence, for each $z=\zeta ^{\ell}$, where $\zeta= e^{i\frac{2\pi}{2r+1}}$, with $\ell=0,1,\ldots,2r$, an irreducible representation of the Laplacian base matrix of $G^{\beta}\cong F_2(C_{2r+2})$ is the matrix $\mbox{\boldmath $B$}(z)$ as shown next. Note that matrix $\mbox{\boldmath $B$}(z)$ is tridiagonal with respect to the main and the secondary diagonals. $$\mbox{\boldmath $B$}(z)=%\rho_z(B(L)) \scriptsize \left( \arraycolsep=1pt \begin{array}{ccccccccccccc} 2 & -1 & 0 & \cdots& 0 & 0 & 0 & 0 & 0 & \cdots& 0 & -z^r & 0 \\ -1 & 4 & -1 & \cdots& 0 & 0 & 0 & 0 & 0 & \cdots&-z^{r} & 0 & -z^{-r}\\ 0 &-1 & 4 & \cdots& 0 & 0 & 0 & 0 & 0 &\cdots & 0 &-z^{-r}& 0 \\ \vdots & \vdots& \vdots& \ddots& \vdots& \vdots & \vdots & \vdots & \vdots&\iddots&\vdots & \vdots& \vdots \\ 0 & 0 & 0 & \cdots& 4 & -1 & 0 &-z^{r} & 0 &\cdots & 0 & 0 & 0 \\ 0 & 0 & 0 & \cdots& -1 & 4 &-1-z^{r}& 0 &-z^{-r}& \cdots& 0 & 0 & 0 \\ 0 & 0 & 0 & \cdots& 0 &-1-z^{-r}& 4 &-1-z^{-r}& 0 & \cdots& 0 & 0 & 0\\ 0 & 0 & 0 & \cdots&-z^{-r}& 0 &-1-z^{r}& 4 &-1 & \cdots& 0 & 0 & 0 \\ 0 & 0 & 0 & \cdots& 0 & -z^{r} & 0 & -1 & 4 & \cdots& 0 & 0 & 0 \\ \vdots & \vdots&\vdots &\iddots& \vdots& \vdots & \vdots & \vdots & \vdots& \ddots& \vdots& \vdots& \vdots\\ 0 &-z^{-r}& 0 &\cdots & 0 & 0 & 0 & 0 & 0 & \cdots& 4 & -1 & 0\\ -z^{-r}& 0 &-z^{r} &\cdots & 0 & 0 & 0 & 0 & 0 & \cdots& -1 & 4 & -1 \\ 0 &-z^r & 0 & \cdots& 0 & 0 & 0 & 0 & 0 & \cdots& 0 & -1 & 2 \\ \end{array} \right).$$ For instance, in the case of $F_2(C_{6})$ ($r=1$), we have $$\mbox{\boldmath $B$}(z)=%\rho_z(B(L))= \left( \begin{array}{ccccc} 2 & -1 & 0 & -z & 0 \\ -1 & 4 & -1-z & 0 & -z^{-1} \\ 0 & -1-z^{-1} & 4 & -1-z^{-1}& 0 \\ -z^{-1} & 0 & -1-z & 4 & -1\\ 0 & -z & 0 & -1 & 2 \end{array} \right).$$ In Table [6](#table5){reference-type="ref" reference="table5"}, we show the different eigenvalues of $F_2(C_{6})$, obtained as the eigenvalues of each $\mbox{\boldmath $B$}(z)$ for $z=\zeta^{\ell}$ with $\ell =0,1,2$. $\zeta=e^{i\frac{2\pi}{3}}$, $z=\zeta^r$ $\lambda_{r,1}$ $\lambda_{r,2}$ $\lambda_{r,3}$ $\lambda_{r,4}$ $\lambda_{r,5}$ ------------------------------------------------------- ----------------- ----------------------------------------- --------------------------- ----------------- ----------------------------------------- $\mathop{\mathrm{sp}}(\mbox{\boldmath $B$}(\zeta^0))$ **0** 2 $5-\sqrt{5}\approx 2.764$ **4** $5+\sqrt{5}\approx 7.236$ $\mathop{\mathrm{sp}}(\mbox{\boldmath $B$}(\zeta^1))$ **1** $\frac{1}{2}(7-\sqrt{17})\approx 1.438$ **3** 5 $\frac{1}{2}(7+\sqrt{17})\approx 5.561$ $\mathop{\mathrm{sp}}(\mbox{\boldmath $B$}(\zeta^2))$ **1** $\frac{1}{2}(7-\sqrt{17})\approx 1.438$ **3** 5 $\frac{1}{2}(7+\sqrt{17})\approx 5.561$ : All the eigenvalues of matrices $\mbox{\boldmath $B$}(\zeta^r)$, which yield the eigenvalues the 2-token graph $F_2(C_6)$. The values in boldface correspond to the eigenvalues of $C_6$. ## The case of even $n$ and $n/2$ When we consider the case of cycles $C_n$ with $n$ and $n/2$ even, it is not very useful to represent $F_2(C_n)$ as a lift graph to compute the whole spectrum. The reason is that the base graph has too many vertices with respect to the original graph. Alternatively, besides the spectrum of $C_n$, we can easily find another part of the spectrum by means of regular partitions. In fact, we can use the regular path-partition and U-partition with the same structure as in the previous subsection. As an example, the 2-token graph of $C_8$ is shown in Figure [3](#F_2(C_8)){reference-type="ref" reference="F_2(C_8)"}$(a)$ and $(b)$, together with its regular partitions $(c)$ (the path $P_{n/2}$) and $(d)$ (the U-shaped graph). Compare such partitions with those in Figure [2](#F_2(C_{10})){reference-type="ref" reference="F_2(C_{10})"}$(b)$ and $(c)$. Then, the quotient $7\times 7$ Laplacian matrix of the regular U-shaped partition $\tau$ of $F_2=F_2(C_8)$ and its spectrum are $$\begin{aligned} \mbox{\boldmath $Q$}(F_2/\tau) &= \left( \begin{array}{ccccccc} 2 & -1 & 0 & 0 & 0 & -1& 0\\ -1 & 4 & -1 & 0 & -1 & 0& -1\\ 0 & -1 & 4 & -2 & 0 & -1 & 0\\ 0 & 0 & -2 & 4 & -2 & 0 & 0\\ 0 & -1 & 0 & -2 & 4 & -1& 0\\ -1 & 0 & -1 & 0 & -1 & 4& -1\\ 0 & -1 & 0 & 0 & 0 & -1 & 2 \end{array} \right),\\ \mathop{\mathrm{sp}}\mbox{\boldmath $Q$}(F_2/\tau) &=\{0,1.5060,2,4^2,4.8900,7.6038\} \ \subset \mathop{\mathrm{sp}}F_2(C_{8}).\end{aligned}$$ With respect to the path-shaped partition $\pi$, Proposition [Proposition 12](#propo:path-shaped){reference-type="ref" reference="propo:path-shaped"} gives that the quotient Laplacian matrix and its spectrum are $$\begin{aligned} \mbox{\boldmath $Q$}(F_2/\pi) &= \left( \begin{array}{cccc} 2 & -2 & 0 & 0 \\ -2 & 4 & -2 & 0 \\ 0 & -2 & 4 & -2\\ 0 & 0 & -4 & 4\\ \end{array} \right),\\ \mathop{\mathrm{sp}}\mbox{\boldmath $Q$}(F_2/\pi) &=\{0,1.5060,4.8900,7.6038\} \ \subset \mathop{\mathrm{sp}}\mbox{\boldmath $Q$}(F_2/\tau) \ \subset \mathop{\mathrm{sp}}F_2(C_8).\end{aligned}$$ Notice that the inclusion $\mathop{\mathrm{sp}}\mbox{\boldmath $Q$}(F_2/\pi) \subset \mathop{\mathrm{sp}}\mbox{\boldmath $Q$}(F_2/\tau)$ is due to the fact that $\tau$ can be seen as a regular partition of the quotient graph $F_2/\pi$. Moreover, according to the same proposition, the largest eigenvalue $7.6038$ of $\mbox{\boldmath $Q$}(F_2/\pi)$ (or $\mbox{\boldmath $Q$}(F_2/\tau)$) is the spectral radius $\rho(F_2(C_8))$. Compare these results with the whole spectrum of $F_2(C_8)$, which is $$\begin{aligned} \mathop{\mathrm{sp}}F_2(C_8) =\{&0,0.5857^2,0.9486^2,1.5060,1.7117^2,2^3,3.1259^2,3.4142^2, \nonumber\\ &4^3,4.5173^2,4,8740^2,4.8900,6.2882^2,6.5340^2,7.6038\}. \label{spF2(C8)}\end{aligned}$$ ## The new method of over-lifts In this subsection, we use a new method called *over-lifts*, which allows us to unify the cases of cycles with even $n$ and compute the whole spectrum of $F_2(C_n)$. (For instance, as a result of such a method, all the eigenvalues in [\[spF2(C8)\]](#spF2(C8)){reference-type="eqref" reference="spF2(C8)"} are shown in Table [7](#taula:C8){reference-type="ref" reference="taula:C8"}.) This is accomplished by means of a new polynomial matrix $\mbox{\boldmath $B$}(z)$ that does **not** correspond to the base graph of a lift. By its characteristics, we say that $\mbox{\boldmath $B$}(z)$ is associated with an over-lift. The basic difference is that such a matrix has dimension $\nu\times\nu$ (recall that $n=2\nu$), and there are $n$ possible values for $z$ ($n$-th roots of unity). Thus, the total number of eigenvalues obtained is $\nu n$. However, $F_2(C_n)$ has ${n\choose 2}=\nu(n-1)$ vertices, which is the number of eigenvalues of $\mbox{\boldmath $L$}$. We will see that, in fact, the $\nu$ 'extra' eigenvalues provided by $\mbox{\boldmath $B$}(z)$ are all equal to $4$. {#F_2(C_8) width="14cm"} **Theorem 17**. *Let $\mbox{\boldmath $L$}$ be the Laplacian matrix of $F_2(C_n)$, with $n=2\nu$. Let $\Lambda$ be the multiset with elements $4,4,\stackrel{(\nu)}{\ldots},4$. Then, the spectrum of $\mbox{\boldmath $L$}$ can be obtained from the spectrum of the $\nu\times\nu$ matrix $\mbox{\boldmath $B$}(z)$ or, equivalently, from the spectrum of its similar matrix $\mbox{\boldmath $B$}^*(r)$: $$% \begin{align*} % \label{B(z)Alegre} \mbox{\boldmath $B$}(z) = \left( \small{ \begin{array}{cccccc} 2 & -1-z^{-1} & 0 & 0 & \ldots & 0 \\ -1-z & 4 & -1-z^{-1} & 0 & \ldots & 0 \\ 0 & -1-z & 4 & -1-z^{-1} & \ddots & 0 \\ 0 & 0 & -1-z & \ddots & \ddots & 0 \\ \vdots & \vdots & \ddots &\ddots & 4 & -1-z^{-1}\\ 0 & 0 & \ldots &0 &-1-z-z^{\nu}-z^{\nu+1} & 4 \end{array} } \right) %\end{align*}$$ for $z=\zeta^r=e^{ir\frac{2\pi}{n}}$, $r=0,1,\ldots,n-1$, and* * $$\begin{aligned} \mbox{\boldmath $B$}^*(r)&= \left( \begin{array}{cccccc} 2 & 2\cos(\frac{r\pi}{n}) & 0 & 0& \ldots & 0\\ 2\cos(\frac{r\pi}{n}) & 4 & 2\cos(\frac{r\pi}{n})& 0 & \ldots & 0\\ 0 & 2\cos(\frac{r\pi}{n}) & 4 & 2\cos(\frac{r\pi}{n})& \ddots& 0\\ 0 & 0 & 2\cos(\frac{r\pi}{n}) & \ddots & \ddots & 0\\ \vdots & \vdots & \ddots & \ddots & 4 & 2\cos(\frac{r\pi}{n}) \\[.1cm] 0 & 0 & \ldots & 0 & 2\cos(\frac{r\pi}{n})+2\cos(\frac{r(n-1)\pi}{n}) & 4 \end{array} \right) \label{B-star-even}\end{aligned}$$ for $r=0,1,\ldots,n-1$. Formally, $$\label{spB(z)-spL} \bigcup_{z\in R(n)}\mathop{\mathrm{sp}}\mbox{\boldmath $B$}(z)=\bigcup_{r=0}^{n-1}\mathop{\mathrm{sp}}\mbox{\boldmath $B$}^*(r)= \Lambda \cup\mathop{\mathrm{sp}}\mbox{\boldmath $L$},$$ where $R(n)$ denotes the set of the $n$-th roots of unity, * *Proof.* For convenience, we will use indistinctly $\mbox{\boldmath $B$}(z)$ or $\mbox{\boldmath $B$}^*(r)$ because of $z=\zeta^r$. First, let us prove that, as in the case of odd $n$, most of the eigenvectors $\mbox{\boldmath $f$}=(f_1,f_2,\ldots,f_{\nu})$ of $\mbox{\boldmath $B$}(z)$, with eigenvalue $\lambda$, give rise to an eigenvector of $\mbox{\boldmath $L$}$, with the same eigenvalue $\lambda$. Indeed, with the same notation for the vertices $(i,j)=(i+h,j)$ of $F_2(C_n)$ as in Proposition [Proposition 14](#propo:eigenvec-odd){reference-type="ref" reference="propo:eigenvec-odd"}, let us consider the vector $\mbox{\boldmath $x$}\in \mathbb R^{{n\choose 2}}$ with components $$x_{(i,j)}=f_{h}\zeta^j,\qquad i=0,\ldots,n-1, \ h=1,\ldots,\nu,$$ where $\zeta$ is a given $n$-th root of unity. Now, to show that, under some conditions, $\mbox{\boldmath $f$}$ is an eigenvector of $\mbox{\boldmath $B$}(z)$ with $z=\zeta$, we distinguish four cases: $(i)$ $h=1$; $(ii)$ $2\le h\le \nu-2$; $(iii)$ $h=\nu-1$; and $(iv)$ $h=\nu$. Since the cases $(i)$ and $(ii)$ are proved as in Proposition [Proposition 14](#propo:eigenvec-odd){reference-type="ref" reference="propo:eigenvec-odd"}, we only consider $(iii)$ and $(iv)$. Since case $(iii)$ is the most involved, we begin with $(iv)$. - $h=\nu$: The vertex $(j+\nu,j)=(j,j+\nu)$, with vector entry $x_{(j+\nu,j)}=f_{\nu}\zeta^j$, is adjacent to the four vertices $(j+\nu-1,j)$, $(j+\nu,j+1)$, $(j-1,j+\nu)$, and $(j, j+\nu+1)$. (Notice that the two entries $i,j$ of each vertex have been chosen in such a way that $h=i-j\ (\!\!\!\mod n)$ is not greater than $\nu$, as required.) These vertices have respective vector entries $f_{\nu-1}\zeta^j$, $f_{\nu-1}\zeta^{j+1}$, $f_{\nu-1}\zeta^{j+\nu}$, and $f_{\nu-1}\zeta^{j+\nu+1}$. Then, the $\nu$-th entry of $\mbox{\boldmath $L$}\mbox{\boldmath $x$}=\lambda\mbox{\boldmath $x$}$ is $$4f_{\nu}\zeta^j-f_{\nu-1}\zeta^j-f_{\nu-1}\zeta^{j+1}-f_{\nu-1}\zeta^{j+\nu}-f_{\nu-1}\zeta^{j+\nu-1}=\lambda f_{\nu}\zeta^j.$$ Thus, $$4f_{\nu}-f_{\nu-1}(1+\zeta+\zeta^{\nu}+\zeta^{\nu+1})=\lambda f_{\nu},$$ which corresponds to the $\nu$-th entry of $\mbox{\boldmath $B$}(\zeta)\mbox{\boldmath $f$}=\lambda\mbox{\boldmath $f$}$. - $h=\nu-1$: The vertex $(j+\nu-1,j)$, with vector entry $x_{(j+\nu-1,j)}=f_{\nu-1}\zeta^j$, is adjacent to the four vertices $(j+\nu-2,j)$, $(j+\nu,j)=(j,j+\nu)$, $(j+\nu-1,j-1)=(j-1,j+\nu-1)$, and $(j+\nu-1,j+1)$. To be consistent with the notation, these vertices must have respective vector entries $$\label{eq:critic} f_{\nu-2}\zeta^j,\quad f_{\nu}\zeta^{j}=f_{\nu}\zeta^{j+\nu},\quad f_{\nu}\zeta^{j-1}=f_{\nu}\zeta^{j-1+\nu},\quad \mbox{and}\quad f_{\nu-2}\zeta^{j+1}.$$ Then, if the above equalities hold, the $(\nu-1)$-th entry of $\mbox{\boldmath $L$}\mbox{\boldmath $x$}=\lambda\mbox{\boldmath $x$}$ is $$4f_{\nu-1}\zeta^j-f_{\nu-2}\zeta^j-f_{\nu}\zeta^{j}-f_{\nu}\zeta^{j-1}-f_{\nu-2}\zeta^{j+1}=\lambda f_{\nu-1}\zeta^j.$$ Thus, $$4f_{\nu-1}-f_{\nu-2}(1+\zeta)-f_{\nu}(1+\zeta^{-1})=\lambda f_{\nu-1},$$ which corresponds to the $(\nu-1)$-th entry of $\mbox{\boldmath $B$}(\zeta)\mbox{\boldmath $f$}=\lambda\mbox{\boldmath $f$}$. Now, the second and third equalities in [\[eq:critic\]](#eq:critic){reference-type="eqref" reference="eq:critic"} hold if, either $$f_{\nu}=0 \quad \mbox{ or } \quad \zeta^{\nu}=1. \label{eq:f=0oznu=1}$$ Let us show that one or the other condition happens in the two following subcases: ```{=html} <!-- --> ``` - $r$ even: When $\zeta$ is an even power of $e^{i\frac{2\pi}{n}}$, we have that $\zeta^{\nu}=1$, and [\[eq:f=0oznu=1\]](#eq:f=0oznu=1){reference-type="eqref" reference="eq:f=0oznu=1"} holds. Thus, each eigenvector of $\mbox{\boldmath $B$}^*(r)$ gives rise to an eigenvector of $\mbox{\boldmath $L$}$. In particular, when $r=0$, the matrix $\mbox{\boldmath $B$}^*(0)$ is similar to $\mbox{\boldmath $B$}(1)$, which equals the quotient matrix $\mbox{\boldmath $Q$}_L$ in [\[Qodd\]](#Qodd){reference-type="eqref" reference="Qodd"} of the path-shaped regular partition of $F_2(C_n)$. This is the special case when we know, in advance, that all eigenvalues of $\mbox{\boldmath $B$}^*(0)$ are eigenvalues of $\mbox{\boldmath $L$}$. - $r$ odd: In this case $\cos(\frac{r\pi}{n})+\cos\left(\frac{r(n-1)\pi}{n}\right)=0$. Thus, the last row of $\mbox{\boldmath $B$}^*(r)$ is $(0,\ldots,0,4)$ and, hence, the matrix has one eigenvalue $\lambda=4$, with corresponding eigenvector $\mbox{\boldmath $f$}$ having $f_{\nu}\neq 0$. Moreover, since $\zeta$ is an odd power of $e^{i\frac{2\pi}{n}}$, we get $\zeta^{\nu}\neq 1$. Consequently, $\mbox{\boldmath $f$}$ does **not** yield an eigenvector of $\mbox{\boldmath $L$}$ since [\[eq:f=0oznu=1\]](#eq:f=0oznu=1){reference-type="eqref" reference="eq:f=0oznu=1"} does not hold. Apart from this eigenvalue $4$, the other eigenvalues of $\mbox{\boldmath $B$}^*(r)$ are those of the principal submatrix $\mbox{\boldmath $B$}^-$ of the first $\nu-1$ rows and columns. Thus, the corresponding eigenvectors of $\mbox{\boldmath $B$}^*(r)$ have the last component $f_{\nu}=0$ to satisfy [\[eq:f=0oznu=1\]](#eq:f=0oznu=1){reference-type="eqref" reference="eq:f=0oznu=1"} and, then they yield an eigenvector of $\mbox{\boldmath $L$}$. From the above discussion, we see that the eigenvalues not present in $\mbox{\boldmath $L$}$ are all equal to $4$ and belong to the spectrum of $\mbox{\boldmath $B$}^*(r)$ when $r$ is odd. More precisely, for every odd $r(\neq n/2)$, there is an eigenvalue $4$ not in the spectrum of $\mbox{\boldmath $L$}$, obtaining $n/2$ of such eigenvalues. Besides, when $r=\nu=n/2$, all the off-diagonal entries of $\mbox{\boldmath $B$}^*(r)$ are zero, and $\mbox{\boldmath $B$}^*(r)=\mathop{\mathrm{diag}}(2,4,\ldots,4)$. If $\nu$ is even, this provides one eigenvalue $2$ and $n/2-1$ eigenvalues $4$ in $\mbox{\boldmath $L$}$. Otherwise, if $\nu$ is odd, the condition in [\[eq:critic\]](#eq:critic){reference-type="eqref" reference="eq:critic"} is not fulfilled for one $4$-eigenvector of $\mbox{\boldmath $B$}^*(r)$ and, hence, we only obtain one eigenvalue $2$ and $n/2-2$ eigenvalues $4$. We obtain, in this way, a total of $n\nu=2\nu^2$ eigenvalues (including repetitions), which is the number $2\nu(\nu-1)$ of eigenvalues of the matrix $\mbox{\boldmath $L$}$ plus $\nu=|\Lambda|$ eigenvalues equal to $4$. Now, to complete the proof, we need to show that the eigenvalues of $\mbox{\boldmath $B$}(z)$ for $z\in R(n)$ (not belonging to $\Lambda$) constitute the spectrum of $\mbox{\boldmath $L$}$. With this aim, we first use that, because of the properties of the polynomial matrix (see Dalfó, Fiol, Miller, Ryan, and Širáň [@dfmrs17]), if $(\mbox{\boldmath $B$}(z)^{\ell})_{ii}=\alpha_{i0}^{(\ell)}+\alpha_{i1}^{(\ell)}z+\alpha_{i2}^{(\ell)}z^2+\cdots$ (for $\ell\geq0$), then $$\mathop{\mathrm{tr}}(\mbox{\boldmath $L$}^{\ell})+\nu 4^{\ell}=\sum_{\lambda\in\mathop{\mathrm{sp}}\mbox{\boldmath $L$}\cup\Lambda}\lambda^{\ell} =n\sum_{i=1}^{\nu}\alpha_{i0}^{(\ell)},$$ where we have taken into account that the matrices $\mbox{\boldmath $B$}(z)$, for $z\in R(n)$, have $\nu$ eigenvalues $4$ not in $\mbox{\boldmath $L$}$. Since $\sum_{z\in R(n)}z^{\ell}=0$ for every $z\neq 1$ and $\ell\neq 0$, we have that $$\alpha_{i0}^{(\ell)}=\frac{1}{n}\sum_{z\in R(n)}(\mbox{\boldmath $B$}(z)^{\ell})_{ii}.$$ Hence, $$\begin{aligned} \sum_{\lambda\in\mathop{\mathrm{sp}}\mbox{\boldmath $L$}}\lambda^{\ell} +\nu 4^{\ell} &=\sum_{i=1}^{\nu}\sum_{z\in R(n)}(\mbox{\boldmath $B$}(z)^{\ell})_{ii}= \sum_{z\in R(n)}\mathop{\mathrm{tr}}(\mbox{\boldmath $B$}(z)^{\ell})\\ &=\sum_{z\in R(n)} \sum_{\mu}\in\mathop{\mathrm{sp}}\mbox{\boldmath $B$}(z)\mu^\ell. \end{aligned}$$ Since this holds for every $\ell\ge 0$, both multisets of eigenvalues in [\[spB(z)-spL\]](#spB(z)-spL){reference-type="eqref" reference="spB(z)-spL"} must coincide (see Gould [@go99]). This completes the proof. ◻ By way of example, when $\mbox{\boldmath $L$}$ and $\mbox{\boldmath $B$}(z)$ are the matrices associated to $F_2(C_8)$, the equality $\mathop{\mathrm{tr}}(\mbox{\boldmath $L$}^{\ell})+\nu 4^{\ell}=\sum_{z\in R(n)}\mathop{\mathrm{tr}}(\mbox{\boldmath $B$}(z)^{\ell})$ for $\ell=0,\ldots,8$, give the values 32, 112, 512, 2656, 14976, 9792, 564032, 3670464, so that the corresponding traces $\mathop{\mathrm{tr}}(\mbox{\boldmath $L$}^{\ell})=\displaystyle\sum_{\lambda\in\mathop{\mathrm{sp}}\mbox{\boldmath $L$}} \lambda^{\ell}$ are 28, 96, 448, 2400, 13952, 85696, 547648, 3604928, as can be checked by using the values in [\[spF2(C8)\]](#spF2(C8)){reference-type="eqref" reference="spF2(C8)"} or Table [7](#taula:C8){reference-type="ref" reference="taula:C8"}. $\zeta=e^{i\frac{2\pi}{8}}$, $z=\zeta^r$ $\lambda_{r,1}$ $\lambda_{r,2}$ $\lambda_{r,3}$ $\lambda_{r,4}$ ----------------------------------------------------------------------------------------------------------- ------------------ ----------------- ----------------- ----------------- $\mathop{\mathrm{sp}}(\mbox{\boldmath $B$}(\zeta^0))$ **0** 1.506040792 4.890083735 7.603875471 $\mathop{\mathrm{sp}}(\mbox{\boldmath $B$}(\zeta^1))=\mathop{\mathrm{sp}}(\mbox{\boldmath $B$}(\zeta^7))$ **0.5857864376** 3.12596795 4.0 6.288245611 $\mathop{\mathrm{sp}}(\mbox{\boldmath $B$}(\zeta^2))=\mathop{\mathrm{sp}}(\mbox{\boldmath $B$}(\zeta^6))$ 0.9486257582 **2.0** 4.517304045 6.534070196 $\mathop{\mathrm{sp}}(\mbox{\boldmath $B$}(\zeta^3))=\mathop{\mathrm{sp}}(\mbox{\boldmath $B$}(\zeta^5))$ 1.711754388 **3.414213562** 4.0 4.87403204 $\mathop{\mathrm{sp}}(\mbox{\boldmath $B$}(\zeta^4))$ 2.0 **4.0** 4.0 4.0 : All the eigenvalues of the matrices $\mbox{\boldmath $B$}(\zeta^r)$, which yield the eigenvalues of the 2-token graph $F_2(C_8)$. The values in boldface correspond to the eigenvalues of $C_8$. # Asymptotic results In this last section, we derive closed formulas that give asymptotic approximations of the eigenvalues of $F_2(C_n)$ when $n$ is large. **Theorem 18**. - *For $n$ odd, $n=2\nu+1$, and fixed odd $r<n$, the eigenvalues of $F_2(C_n)$, in the matrix $\mbox{\boldmath $B$}^*(r)$ of [\[B-star\]](#B-star){reference-type="eqref" reference="B-star"}, are asymptotically equal to $$\label{asymp:odd-odd} \lambda_k=4+4\cos\left(\frac{r\pi}{n}\right)\cos\left(\frac{2k-1}{n-1}\pi\right),\quad k=1,2,\ldots,\nu.$$* - *For $n$ odd, $n=2\nu+1$, and fixed even $r<n$, the eigenvalues of the matrix $\mbox{\boldmath $B$}^*(r)$ in [\[B-star\]](#B-star){reference-type="eqref" reference="B-star"} are asymptotically equal to $$\label{asymp:odd-even} \lambda_k=4+4\cos\left(\frac{r\pi}{n}\right)\cos\left(\frac{k-1}{n-1}2\pi\right)\quad k=1,2,\ldots,\nu.$$* - *For $n$ even, $n=2\nu$, and fixed odd $r<n$ or $r=\nu$ even, the eigenvalues of $F_2(C_n)$, in the matrix $\mbox{\boldmath $B$}^*(r)$ of [\[B-star-even\]](#B-star-even){reference-type="eqref" reference="B-star-even"}, are asymptotically equal to $$\label{asymp:even} \lambda_k=4+4\cos\left(\frac{r\pi}{n}\right)\cos\left(\frac{2k-1}{n-1}\pi\right),\quad k=1,2,\ldots,\nu-1.$$* *Proof.* We begin with the case $(ii)$. For odd $r$ or $r=\nu$ even, the last row of $\mbox{\boldmath $B$}^*(r)$ is $(0,\ldots,0,4)$. Hence, apart from the eigenvalue 4, the other eigenvalues of $\mbox{\boldmath $B$}^*(r)$ are those of the principal submatrix $\mbox{\boldmath $B$}^-$ of the first $\nu-1$ rows and columns. Moreover, since the function $\cos(\frac{r\pi}{x})$ is continuous for $x>1$ and tend to $1$ when $x\rightarrow \infty$, we can work with the approximation $$\begin{aligned} \mbox{\boldmath $B$}^-&\approx \mbox{\boldmath $C$}_e= \left( \begin{array}{cccccc} 4-2\cos(\frac{r\pi}{n}) & 2\cos(\frac{r\pi}{n})& 0 & 0 & \ldots & 0\\ 2\cos(\frac{r\pi}{n}) & 4 & 2\cos(\frac{r\pi}{n})& 0 & \ldots & 0\\ 0 & 2\cos(\frac{r\pi}{n}) & 4 & 2\cos(\frac{r\pi}{n}) & \ldots & 0\\ \vdots & \vdots & \ddots & \ddots & \ddots & \vdots\\ 0 & 0 & \ldots & 2\cos(\frac{r\pi}{n}) & 4 & 2\cos(\frac{r\pi}{n})\\ 0 & 0 & \ldots & 0 & 2\cos(\frac{r\pi}{n}) & 4 \end{array} \right). \label{B-star-even-matrix}\end{aligned}$$ Then, from the results by Yueh, see [@y05 Th.2], the eigenvalues of $\mbox{\boldmath $C$}_e$ are those in [\[asymp:even\]](#asymp:even){reference-type="eqref" reference="asymp:even"}.\ For the cases $(i.1)$ and $(i.2)$ of odd $n=2\nu+1$, we proceed similarly. Now, the approximation $\mbox{\boldmath $B$}'$ of the matrix $\mbox{\boldmath $B$}^*(r)$ in [\[B-star\]](#B-star){reference-type="eqref" reference="B-star"} is $$\begin{aligned} \mbox{\boldmath $B$}'&\approx \mbox{\boldmath $C$}_o= \left( \begin{array}{cccccc} 4-2\cos(\frac{r\pi}{n}) & 2\cos(\frac{r\pi}{n})& 0 & 0 & \ldots & 0\\ 2\cos(\frac{r\pi}{n}) & 4 & 2\cos(\frac{r\pi}{n})& 0 & \ldots & 0\\ 0 & 2\cos(\frac{r\pi}{n}) & 4 & 2\cos(\frac{r\pi}{n}) & \ldots & 0\\ \vdots & \vdots & \ddots & \ddots & \ddots & \vdots\\ 0 & 0 & \ldots & 2\cos(\frac{r\pi}{n}) & 4 & 2\cos(\frac{r\pi}{n})\\ 0 & 0 & \ldots & 0 & 2\cos(\frac{r\pi}{n}) & 4\pm 2\cos(\frac{r\pi}{n}) \end{array} \right), \label{B-star-odd-matrix}\end{aligned}$$ where in $(\nu,\nu)$-entry, we must take the plus sign when $r$ is odd and the minus when $r$ is even. Then, by using the results of Yueh, [@y05 Th.3] and [@y05 Th.5], respectively, the eigenvalues of $\mbox{\boldmath $C$}_o$ are the claimed ones in [\[asymp:odd-odd\]](#asymp:odd-odd){reference-type="eqref" reference="asymp:odd-odd"} and [\[asymp:odd-even\]](#asymp:odd-even){reference-type="eqref" reference="asymp:odd-even"}. ◻ By using Gershgorin circles as in Proposition [Proposition 15](#propo:B*(r)){reference-type="ref" reference="propo:B*(r)"}, we can prove that the minimum eigenvalue of $F_2(C_n)$ coincides with the minimum eigenvalue of the matrices $\mbox{\boldmath $B$}^*(r)$ in [\[B-star\]](#B-star){reference-type="eqref" reference="B-star"} and [\[B-star-even\]](#B-star-even){reference-type="eqref" reference="B-star-even"} when $r=1$. Moreover, the minimum eigenvalue in [\[asymp:odd-odd\]](#asymp:odd-odd){reference-type="eqref" reference="asymp:odd-odd"} and [\[asymp:even\]](#asymp:even){reference-type="eqref" reference="asymp:even"} is obtained when $k=\nu$ and $k=\nu-1$, respectively, giving $$\label{aprox-alpha-odd} \alpha(F_2(C_n))\approx\lambda_{\nu}=4+4\cos\left(\frac{\pi}{n}\right)\cos\left(\frac{n-2}{n-1}\pi\right),$$ and $$\label{aprox-alpha-even} \alpha(F_2(C_n))\approx \lambda_{\nu-1}=4+4\cos\left(\frac{\pi}{n}\right)\cos\left(\frac{n-3}{n-1}\pi\right).$$ Notice that, as $n$ increases, the expressions in [\[aprox-alpha-odd\]](#aprox-alpha-odd){reference-type="eqref" reference="aprox-alpha-odd"} and [\[aprox-alpha-even\]](#aprox-alpha-even){reference-type="eqref" reference="aprox-alpha-even"} tend to $2-2\cos(\frac{2\pi}{n})$, which is the exact value of $\alpha(C_n)$, as expected. 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Appl.* **11** (1990) 218--238. S. Ibarra and L. M. Rivera, The automorphism groups of some token graphs, `arXiv:1907.06008v3[math.CO]` S. Kirkland, Algebraic connectivity for vertex-deleted subgraphs, and a notion of vertex centrality, *Discrete Math.* **310** (2010) 911--921. A. Lew, Garland's method for token graphs, `https://arxiv.org/abs/2305` `.02406v1`, 2023. Y. Ouyang, Computing spectral bounds of the Heisenberg ferromagnet from geometric considerations, *J. Math. Physics* **60** (2019) 071901. K. L. Patra and B. K. Sahoo, Bounds for the Laplacian spectral radius of graphs, *Electron. J. Graph Theory and Appl.* **5** (2017), no. 2, 276--303. W.-C. Yueh, Eigenvalues of several tridiagonal matrices, *Appl. Math. E-Notes* **5** (2005) 66--74. [^1]: This research has been supported by AGAUR from the Catalan Government under project 2021SGR00434 and MICINN from the Spanish Government under project PID2020-115442RB-I00. The research of M. A. Fiol was also supported by a grant from the Universitat Politècnica de Catalunya with references AGRUPS-2022 and AGRUPS-2023.

arxiv_math

--- abstract: | In this paper, we present a novel approach for computing the large genus asymptotics of intersection numbers. Our strategy is based on a resurgent analysis of the $n$-point functions of such intersection numbers, which are computed via determinantal formulae, and relies on the presence of a quantum curve. With this approach, we are able to extend the recent results of Aggarwal for Witten--Kontsevich intersection numbers with the computation of all subleading corrections, proving a conjecture of Guo--Yang, and to obtain new results on $r$-spin and Theta-class intersection numbers. address: - Université Paris-Saclay, CNRS, CEA, Institut de Physique Théorique, Gif-sur-Yvette, France and CRM, Centre de Recherches Mathématiques de Montréal, Université de Montréal, QC, Canada - Sorbonne Université, CNRS, Institut de Mathématiques de Jussieu--Paris Rive Gauche, Paris, France - Université Paris-Saclay, CNRS, CEA, Institut de Physique Théorique, Gif-sur-Yvette, France - Université Paris-Saclay, CNRS, CEA, Institut de Physique Théorique, Gif-sur-Yvette, France - Università degli Studi di Trieste, Sezione di Matematica, Trieste, Italy and Université de Genève, Section de Mathématiques, Genève, Switzerland author: - B. Eynard - E. Garcia-Failde - A. Giacchetto - P. Gregori - D. Lewański bibliography: - BibliographyLargeGenus.bib title: Resurgent large genus asymptotics of intersection numbers --- -1.2 # Introduction ## Motivations and methods {#motivations-and-methods .unnumbered} The aim of this paper is to study the large genus asymptotics of certain intersection numbers on the moduli space $\overline{{\mathcal{M}}}_{g,n}$ of genus $g$ stable complex curves with $n$ marked points. The main class of intersection numbers under consideration is that of $\psi$-class intersection numbers [@Wit91; @Kon92]: $$\label{eqn:WK:int:numbrs} \braket{\tau_{d_1} \cdots \tau_{d_n}} = \int_{\overline{{\mathcal{M}}}_{g,n}} \prod_{i=1}^n \psi_i^{d_i} \,.$$ Here $|d| = d_1 + \cdots + d_n = 3g - 3 + n$ uniquely determines the genus, and $\psi_i = c_1(\mathcal{L}_i)$ is the first Chern class of the line bundle $\mathcal{L}_i$ whose fibre over $[C,p_1\dots,p_n] \in \overline{{\mathcal{M}}}_{g,n}$ is the cotangent line $T_{p_i}^{\ast} C$. Such intersection numbers can be thought of as the intersection of $\psi$-classes with the fundamental class of the moduli space of curves. Another possibility, recently considered by Norbury [@Nor23], is to intersect $\psi$-classes with a different cohomology class called the $\Theta$-class: $$\label{eqn:Theta:int:numbrs} \braket{\tau_{d_1} \cdots \tau_{d_n}}^{\Theta} = \int_{\overline{{\mathcal{M}}}_{g,n}} \Theta_{g,n} \prod_{i=1}^n \psi_i^{d_i} \,,$$ where $|d| = g - 1$ due to $\Theta_{g,n}$ being of pure complex degree $2g-2+n$. Yet another collection of intersection numbers is that of $r$-spin intersection numbers [@Wit93]: for every $r \ge 2$, $$\label{eqn:rspin:int:numbrs} \braket{\tau_{d_1,a_1} \cdots \tau_{d_n,a_n}}^{r\textup{-spin}} = \int_{\overline{{\mathcal{M}}}_{g,n}} W^r_{g,n}(a_1,\dots,a_n) \prod_{i=1}^n \psi_i^{d_i} \,,$$ where $W^r_{g,n}(a_1,\dots,a_n)$ is called the Witten $r$-spin class. Here $a_i \in \set{1,\dots,r-1}$, and $r|d| + |a| = (r+1)(2g-2+n)$. For $r = 2$, they coincide with $\psi$-class intersection numbers. In the last few decades, there has been a growing realisation that many fundamental invariants in physics and geometry can be expressed through intersection numbers. The first connection was established by Witten [@Wit91], who conjectured that the partition functions of topological $2d$ quantum gravity coincides with the generating series of $\psi$-class intersection numbers. Similarly, $r$-spin intersection numbers emerge in the context of topological $2d$ quantum gravity coupled to a certain gauge theory [@Wit92]. More recently, $\psi$-class intersection numbers have regained attention within the physics community due to new discoveries linking them to Jackiw--Teitelboim gravity [@SSS]. In similar fashion, super Jackiw--Teitelboim gravity has been linked to $\Theta$-class intersection numbers [@SW20]. Geometrically, $\psi$-class intersection numbers represent the easiest possible instance of Gromov--Witten invariants, namely for the target space being a point. They appear in the works of Kontsevich [@Kon92] and Mirzakhani [@Mir07b] on the symplectic volumes of the moduli space of metric ribbon graphs and hyperbolic Riemann surfaces, as well as in the the asymptotic counting of geodesic curves in flat and hyperbolic random geometry and in the theory of Masur--Veech volumes [@Mir08b; @DGZZ21; @And+23]. All such intersection numbers also appear in the context of combinatorics of maps and random matrix theory [@Eyn16; @BCEG], as well as in the theory of integrable systems [@Kon92; @FSZ10; @CGG]. Given their ubiquitous presence in mathematics and physics, a natural question arises: [How does one compute such intersection numbers?]{.sans-serif} The connection with integrable systems has in many instances been the key for the evaluation of these invariants. Both $\psi$- and $\Theta$-class intersection numbers are specific solutions of the Korteweg--De Vries (KdV) integrable hierarchy [@Kon92; @CGG], while $r$-spin intersection numbers constitute a solution of the $r$-KdV hierarchy [@FSZ10]. Concretely, their generating series obey an infinite set of partial differential equations in infinitely many variables that recursively determine the intersection numbers uniquely after fixing initial conditions. An equivalent way of specifying these solutions of integrable hierarchies is by means of Virasoro constraints [@DVV91; @GN92; @AvM92]: the generating series of intersection numbers is annihilated by an infinite set of linear differential operators, forming a representation of the so-called Virasoro (or more generally $W$-) algebra. Alternative recursive methods include the topological recursion [@EO07], the cut-and-join equation [@Ale11], a recursion by Liu--Xu [@LX14], as well as some exact formulae for the generating series such as the determinantal formula [@BE; @BDY16], integral representations by Okounkov [@Oko02], Fock space inspired generating series by Buryak [@Bur17], and the symmetric functions approach of Eynard--Mitsios [@EM]. While the methods mentioned above theoretically allow for an explicit evaluation of intersection numbers, it is widely acknowledged that all these quantities are typically intricate. With a few exceptional cases, no simple closed-form expression is expected to exist. Even if on the one hand it might be hopeless to control these invariants exactly, on the other it might be possible to gain an understanding of their behaviour as the genus increases. Thus, another natural question arises: [Is it possible to access intersection numbers asymptotically in the large genus limit?]{.sans-serif} For $\psi$-class intersection numbers, it was predicted by Delecroix--Goujard--Zograf--Zorich [@DGZZ21] that in the large genus limit the intersection numbers simplify considerably: $$\braket{\tau_{d_1} \cdots \tau_{d_n}} \sim \frac{(6g-5+2n)!!}{24^g \, g! \, \prod_{i=1}^{n} (2d_i + 1)!!}$$ uniformly in $d_1,\dots,d_n$. Special cases of the asymptotic formula were proved in [@LX14] and in [@DGZZ20] when the genus is mostly concentrated in $d_1$ and in $d_1 + d_2$ respectively. The above asymptotic formula has been recently proved by Aggarwal [@Agg21], whose strategy employs a careful combinatorial and probabilistic analysis of the associated Virasoro constraints (more precisely, Aggarwal proved the asymptotic formula under the more stringent regime $n = \mathrm{o}(g^{1/2})$). An alternative proof has been later proposed by Guo--Yang [@GY22], whose approach is based on a combinatorial analysis of the determinantal formula. The large genus asymptotics of $r$-spin intersection numbers was considered by Dubrovin--Yang--Zagier in [@DYZ] in the specific case of $n = 1$. The study of intersection numbers in the large genus limit holds significance for several reasons. Firstly, the intricate nature of intersection numbers simplifies enormously in the large genus limit, leading to closed-form asymptotic evaluations. Secondly, many interesting quantities associated to several geometric models appear to be exclusively attainable in the asymptotic regime. For instance, the length of the shortest geodesic on a fixed hyperbolic surface is notoriously hard to compute, but its average value in the large genus limit can be explicitly computed as $\approx 1.61498\ldots$ [@MP19]. This computation crucially relies on the large genus behaviour of Weil--Petersson volumes predicted by Mirzakhani--Zograf [@MZ15]. Thirdly, in the large genus limit many universality phenomena can be unveiled. For instance, all intersection numbers studied so far manifest a $(2g)!$ factorial growth, as well as a (conjectural) polynomiality structure [@GY22]. Another example of universality regards the length spectra of random hyperbolic and combinatorial geodesics, both of which converge to a Poisson distribution [@MP19; @JL23]. The large genus regime is also interesting from the physics point-of-view, as it is intimately linked to non-perturbative effects in quantum theory with new phenomena that are not captured by Feynman diagram expansions appear (see for instance [@CESV16; @GM; @GKKM; @GS; @EGGLS; @IM]). The aforementioned methods used to derive the large genus asymptotics of $\psi$-class intersection numbers heavily depend on ad hoc estimates for various quantities involved in the specific problem. Therefore, it would be desirable to have a new, more general strategy for approaching large genus asymptotics. In this paper, we aim to address this issue by introducing a novel method that utilises two main techniques: 1. The *Borel transform method*, that is the computation of large-order asymptotics through the study of the singularity structure in the Borel plane. 2. The existence of *determinant formulae* (a.k.a. the matrix resolvent method) for computing the $n$-point functions of the corresponding enumerative problem, with building blocks being the solutions of a quantum curve (a.k.a. topological ODE). The Borel transform method essentially combines classical results from Borel (concerning resummation of divergent series) and from Darboux (concerning asymptotic analysis), and can be naturally placed within Écalle's theory of resurgence [@Eca81; @MS16]. On the other hand, determinantal formulae have been established in connection with topological recursion [@BE; @BBE15] and integrable systems [@BDY16]. In the present paper, we employ the proposed strategy to address the three enumerative problems mentioned earlier: $\psi$-, $\Theta$-, and $r$-spin intersection numbers. Many more enumerative problems are solved by determinantal formulae. These include higher Weil--Petersson volumes [@BDY16], GUE correlators [@DY17], Gromov--Witten theory of $\mathbb{P}^1$ [@DYZ20], Fan--Jarvis--Ruan--Witten theory [@BDY21], and more. Given this broad applicability, we anticipate that our approach will pave the way for further investigations into large genus phenomena in the future. The strategy outlined here offers multiple advantages. Firstly, it applies to several enumerative problems with minimal modifications, such as the study of the structure of the Borel singularities of the building blocks appearing in the determinantal formula. A second, and perhaps more noteworthy, advantage is that it enables the computation of subleading and exponentially subleading corrections through an implementable algorithm, revealing the polynomiality phenomena conjectured in [@GY22]. Thirdly, it sheds light on the universal features of the large genus regime, as well as the model-dependent ones, as exemplified in [\[thm:intro:psi:Theta,thm:intro:rspin\]](#thm:intro:psi:Theta,thm:intro:rspin){reference-type="ref" reference="thm:intro:psi:Theta,thm:intro:rspin"}. ## Results {#results .unnumbered} As explained above, the geometry of the quantum curve underlying the different enumerative problems under consideration propagates to their large genus asymptotics. Consider, for example, the case of $\psi$- and $\Theta$-class intersection numbers. The underlying quantum curves are the Airy and the Bessel ODE respectively: $$\biggl( \biggl(\hslash\frac{d}{dx}\biggl)^2 - \; x \biggr) \psi_{\textup{Airy}}(x;\hslash) = 0 \,, \qquad\qquad \biggl( \biggl(x \, \hslash\frac{d}{dx}\biggr)^2 - \; x \biggr) \psi_{\textup{Bessel}}(x;\hslash) = 0 \,.$$ For the corresponding enumerative problems, the large genus formula is given as follows (see [\[thm:large:g:WK,thm:large:g:Nor\]](#thm:large:g:WK,thm:large:g:Nor){reference-type="ref" reference="thm:large:g:WK,thm:large:g:Nor"} for more precise statements). In the following, we denote $(x)^{\underline{m}} = x (x-1)\cdots (x-m+1)$ as the falling factorial. **Theorem 1**. *For any given $n \ge 1$ and $K \ge 0$, uniformly in $d_1,\dots,d_n$ as $g \to \infty$: $$\begin{aligned} % psi \label{eqn:intro:psi:large:g} \begin{split} \braket{\tau_{d_1} \cdots \tau_{d_n}} = \mathsf{S}_{\mathsf{A}} \, \frac{2^n}{4\pi} \, \frac{\Gamma(2g-2+n)}{\mathsf{A}^{2g-2+n} \, \prod_{i=1}^n (2d_i + 1)!!} \bigg( 1 & + \frac{\mathsf{A}}{2g-3+n} \, \alpha_{1} + \cdots \\ & + \frac{\mathsf{A}^K}{(2g-3+n)^{\underline{K}}} \, \alpha_{K} + \mathrm{O}\Bigl( \frac{1}{g^{K+1}} \Bigr) \bigg) \,, \end{split} \\ % Theta \label{eqn:intro:Theta:large:g} \begin{split} \braket{\tau_{d_1} \cdots \tau_{d_n}}^{\Theta} = \mathsf{S}_{\mathsf{B}} \, \frac{2^n}{4\pi} \, \frac{\Gamma(2g-2+n)}{\mathsf{B}^{2g-2+n} \, \prod_{i=1}^n (2d_i + 1)!!} \bigg( 1 & + \frac{\mathsf{B}}{2g-3+n} \, \beta_{1} + \cdots \\ & + \frac{\mathsf{B}^K}{(2g-3+n)^{\underline{K}}} \, \beta_{K} + \mathrm{O}\Bigl( \frac{1}{g^{K+1}} \Bigr) \bigg) \,. \end{split} \end{aligned}$$ Moreover, the constants $\mathsf{A}$, $\mathsf{B}$, $\mathsf{S}_{\mathsf{A}}$, $S_\mathsf{B}$ and the sequences $(\alpha_k)_{k \ge 0}$ and $(\beta_k)_{k \ge 0}$ have the following values and geometric interpretations:* - *The constants $\mathsf{A} = \frac{2}{3}$ and $\mathsf{B} = 2$ determine the exponential growth of the Airy and Bessel functions respectively: $$\psi_{\textup{Airy}}(x;\hslash) \sim \frac{1}{\sqrt{2} \, x^{1/4}} e^{\pm \frac{\mathsf{A}}{\hslash} x^{3/2}} \,, \qquad\quad \psi_{\textup{Bessel}}(x;\hslash) \sim \frac{1}{\sqrt{2} \, x^{1/4}} e^{\pm \frac{\mathsf{B}}{\hslash} x^{1/2}} \,, \qquad\quad \text{as } x \to \infty \,.$$* - *The constants $\mathsf{S}_{\mathsf{A}} = 1$ and $\mathsf{S}_{\mathsf{B}} = 2$ are the Stokes constants appearing in the Stokes phenomenon underlying the Airy and Bessel ODEs.* - *Each term $\alpha_k$ and $\beta_k$ appearing in the asymptotic expansions is a polynomial function of $n$ and the multiplicities $p_m = \#\set{ d_i = m }$: $$\alpha_k = \alpha_k\bigl( n, p_0, \ldots, p_{\floor{\frac{3}{2}k} - 1} \bigr) \,, \qquad\qquad \beta_k = \beta_k\bigl( n, p_0, \ldots, p_{\floor{\frac{1}{2}k} - 1} \bigr) \,,$$ The values $3/2$ and $1/2$ are the exponents of $x$ in the exponential growth of the Airy and Bessel functions respectively. Moreover, we provide an algorithm to effectively compute $\alpha_k$ and $\beta_k$ from the coefficients of the asymptotic expansion of the Airy and Bessel functions respectively (see [2](#table:intro){reference-type="ref" reference="table:intro"} for the first few values).* $k$ $\alpha_{k}$ ----- ------------------------------------------------------------------- $0$ $1$ $1$ $- \tfrac{17 - 15n + 3n^2}{12} - \tfrac{(3 - n)(n - p_0)}{2} - \tfrac{(n - p_0)^{\underline{2}}}{4}$ $2$ $\tfrac{1225 - 1632 n + 741 n^2 - 138 n^3 + 9 n^4}{288} + \tfrac{(105 - 98 n + 30 n^2 - 3 n^3)(n - p_0)}{24} + \tfrac{3(10 - 7 n + n^2)(n - p_0 - p_1)}{8} + \tfrac{(59 - 51 n + 9 n^2)(n - p_0)^{\underline{2}}}{48} + \tfrac{5(n - p_0 - p_1 - p_2)}{8} + \tfrac{3(4 - n)(n - p_0 - 1)(n - p_0 - p_1)}{4} + \tfrac{(7 - 3 n)(n - p_0)^{\underline{3}}}{24} + \tfrac{3(n - p_0 - 1)^{\underline{2}}(n - p_0 - p_1)}{48} + \tfrac{3(n - p_0)^{\underline{4}}}{96}$ : The subleading corrections $\alpha_k$ and $\beta_k$ for the first few values of $k$. $k$ $\beta_k$ ----- -------------------------------------------------- $0$ $1$ $1$ $- \frac{1}{4}$ $2$ $\tfrac{9 - 4 n}{8} + \tfrac{n - p_0}{8}$ $3$ $- \tfrac{57 - 44 n + 8 n^2}{128} - \tfrac{(13 - 4 n)(n - p_0)}{32} - \tfrac{(n - p_0)^{\underline{2}}}{16}$ : The subleading corrections $\alpha_k$ and $\beta_k$ for the first few values of $k$. It can be easily shown that formula for $K = 0$ reduces to Aggarwal's result. The new insight that we provide is the geometric interpretation of the constants $\mathsf{S}_{\mathsf{A}} = 1$ and $\mathsf{A} = 2/3$. The subleading correction and their polynomiality behaviour is a novel result, and provides a proof of a conjecture by Guo--Yang [@GY22 conjecture 1]. The case of $\Theta$-class intersection numbers is completely new. As noted previously, our proof strategy highlights the universality of the large genus asymptotics, as well as the the model-dependent ingredients. Moreover, it provides a concrete algorithm to compute the subleading corrections $\alpha_k$ and $\beta_k$ appearing in the asymptotic expansions. We also provide the large genus asymptotics for $r$-spin intersection numbers, which were previously unknown for $n>1$. In this case, the underlying quantum curve is the $r$-Airy ODE: $$\biggl( \biggl(\hslash\frac{d}{dx}\biggl)^r - x \biggr) \psi_{r\textup{-Airy}}(x;\hslash) = 0 \,.$$ One notable distinction compared to the previous case is the increased order of the differential equation, which is now $r$. This feature introduces a new element in the asymptotic analysis, specifically the appearance of exponentially subleading terms. Nonetheless, these terms can be effectively addressed using the Borel transform method. See [Theorem 36](#thm:large:g:rspin){reference-type="ref" reference="thm:large:g:rspin"} for a more precise statement. In the following, we denote by $$\label{eqn:r:fact} m!_{(r)} = \begin{cases} m (m - r)!_{(r)} & \text{if } m > r \,, \\ m & \text{if } 0 < m \le r \,, \\ \end{cases}$$ the $r$-factorial, which generalises the double factorial for $r = 2$. **Theorem 2**. *For any given $n \ge 1$, $K \ge 0$, and $a_1,\dots,a_n$, uniformly in $d_1,\dots,d_n$ as $g \to \infty$: $$\label{eqn:intro:rspin:large:g} \begin{split} \braket{\tau_{d_1,a_1} \cdots \tau_{d_n,a_n}}^{r\textup{-spin}} = \, & \frac{2^n}{2\pi} \frac{\Gamma(2g-2+n)}{r^{g-1-|d|} \, \prod_{i=1}^n (r d_i + a_i)!_{(r)}} \\ \times \Bigg[ & \frac{\mathsf{S}_{r,1}}{|\mathsf{A}_{r,1}|^{2g-2+n}} \biggl( \gamma^{(r,1)}_0 + \cdots + \frac{|\mathsf{A}_{r,1}|^K}{(2g-3+n)^{\underline{K}}} \, \gamma^{(r,1)}_K + \mathrm{O}\Bigl( \frac{1}{g^{K+1}} \Bigr) \biggr) \\[1.2ex] + \, & \cdots \\ + \, & \frac{\mathsf{S}_{r,\floor{\frac{r-1}{2}}}}{|\mathsf{A}_{r,\floor{\frac{r-1}{2}}}|^{2g-2+n}} \biggl( \gamma^{(r,\floor{\frac{r-1}{2}})}_0 + \cdots + \frac{|\mathsf{A}_{r,\floor{\frac{r-1}{2}}}|^K}{(2g-3+n)^{\underline{K}}} \, \gamma^{(r,\floor{\frac{r-1}{2}})}_K + \mathrm{O}\Bigl( \frac{1}{g^{K+1}} \Bigr) \biggr) \\ + \, & \frac{\delta_{r}^{\textup{even}}}{2} \frac{\mathsf{S}_{r,\frac{r}{2}}}{|\mathsf{A}_{r,\frac{r}{2}}|^{2g-2+n}} \biggl( \gamma^{(r,\frac{r}{2})}_0 + \cdots + \frac{|\mathsf{A}_{r,\frac{r}{2}}|^K}{(2g-3+n)^{\underline{K}}} \, \gamma^{(r,\frac{r}{2})}_K + \mathrm{O}\Bigl( \frac{1}{g^{K+1}} \Bigr) \biggr) \Bigg] \,. \end{split}$$ Here $\delta_{r}^{\textup{even}}$ gives one if $r$ is even and zero otherwise. Moreover, the constants $\mathsf{A}_{r,\alpha}$, $\mathsf{S}_{r,\alpha}$, and the sequences $(\gamma^{(r,\alpha)}_k)_{k \ge 0}$ have the following values and geometric interpretations:* - *The constants $\mathsf{A}_{r,\alpha} = \frac{r}{r+1} (1 - e^{2\pi\mathrm{i}\frac{\alpha}{r}})$ are determined by the exponential growth of the $r$-Airy functions: $$\psi_{r\textup{-Airy}}(x;\hslash) \sim \frac{1}{\sqrt{r} \, x^{\frac{r-1}{2r}}} e^{\frac{1}{\hslash} \frac{r}{r+1} e^{2\pi\mathrm{i}\frac{\alpha}{r}} x^{\frac{r+1}{r}}} \,, \qquad\quad \alpha = 1,\dots,r \,, \quad \text{ as } x \to \infty \,.$$ Furthermore, the various exponentially asymptotic behaviours are arranged in ascending order of distance from the origin, coupled together via the symmetry relation $|\mathsf{A}_{r,\alpha}| = |\mathsf{A}_{r,r-\alpha}|$. The behaviour depends on the parity of $r$, as pictured in [\[fig:actions:rspin\]](#fig:actions:rspin){reference-type="ref" reference="fig:actions:rspin"}.* - *The constants $\mathsf{S}_{r,\alpha} = 1$ are the Stokes constants appearing in the Stokes phenomenon underlying the $r$-Airy ODE.* - *Each term $\gamma^{(r,\alpha)}_k$ is a function of $n$ and the multiplicities $p_m = \#\set{ r d_i + a_i = m }$. Moreover, we provide an algorithm to effectively compute $\gamma_k$ from the coefficients of the asymptotic expansion of the $r$-Airy function. The leading coefficient is explicitly given by $$\gamma^{(r,\alpha)}_0 = (-1)^{(\alpha-1)(|d|+n)} \frac{\prod_{i=1}^n \sin\left(\frac{\alpha a_i}{r}\pi \right)}{\sin(\frac{\alpha}{r}\pi)} \,.$$ We refer to [Theorem 36](#thm:large:g:rspin){reference-type="ref" reference="thm:large:g:rspin"} for a more precise statement about its polynomial dependence on the multiplicities.* ## Acknowledgements {#acknowledgements .unnumbered} The authors would like to thank Raphaël Belliard, Philip Boalch, Veronica Fantini, Kohei Iwaki, Maxim Kontsevich, and David Sauzin for useful discussions. We are also grateful to IMJ-PRG, IPhT, IHES, SISSA, and the University of Trieste for their kind hospitality. B. E., A. G., and P. G. are supported by the ERC-SyG project "Recursive and Exact New Quantum Theory" (ReNewQuantum) which received funding from the European Research Council under the European Union's Horizon 2020 research and innovation programme under grant agreement No 810573. E. G.-F. was funded by the public grant "Jacques Hadamard" as part of the "Investissement d'Avenir" project, reference ANR-11-LABX-0056-LMH, LabEx LMH, and supported by the ERC-StG project "New Interactions of Combinatorics Through Topological Expansions" (CombiTop) which received funding from the European Union's Horizon 2020 research and innovation programme under the grant agreement No 716083. D. L. is funded by the University of Trieste, by the Swiss National Foundation Ambizione project "Resurgent Topological Recursion, Enumerative Geometry and Integrable Hierarchies" (EnTIRe) under the grant agreement No PZ00P2-202123, by the INdAM group GNSAGA and by the Trieste node of the INFN project MMNLP. # The Borel transform method In this section we present the *Borel transform method*, namely a technique which allows to obtain the large-order asymptotics of the coefficients of a formal power series in terms of the singularity structure of its Borel transform. In order to do so, we first need to introduce several basic concepts from the theory of resurgence. Those are mostly taken from [@MS16; @Mar], to which we refer the interested reader for more details (see also [@ABS19] for a physicist-oriented comprehensive review of resurgence). ## The Borel transform and resurgent functions Define the *Borel transform* as the following linear invertible operator acting on formal power series[^1] $$\mathfrak{B} \colon \mathbb{C}\llbracket \hslash \rrbracket \longrightarrow \mathbb{C}\llbracket s \rrbracket \,, \qquad \hslash^m \longmapsto \frac{s^m}{m!}\,.$$ Throughout the text we denote (when possible) quantities in the $\hslash$-plane with a tilde and the corresponding quantities in the $s$-plane (also called Borel plane) with a hat. For instance, writing the starting series as $$\widetilde{\varphi}(\hslash) = \sum_{m \ge 0} \varphi_m \hslash^m \,,$$ its Borel transform is given by $$\widehat{\varphi}(s) = \sum_{m \ge 0} \varphi_m \frac{s^m}{m!} \,.$$ Due to the division by the factorial, $\widehat{\varphi}$ is "more likely" to be convergent. This observation leads to the definition of Gevrey-1 series. **Definition 1**. A formal power series $\widetilde{\varphi}(\hslash) = \sum_{m \ge 0} \varphi_m \hslash^m$ is *Gevrey-1* if there exists a constant $a > 0$ such that the large $m$ behaviour of the coefficients $\varphi_m$ is given by $\varphi_m = \mathrm{O}(m! \, a^{-m})$. The above definition is designed in such a way that the Borel transform maps Gevrey-1 series to holomorphic functions at the origin (and vice versa, every holomorphic function at the origin is the image of a Gevrey-1 series). It is then natural to study the analytic continuation of the Borel transform of Gevrey-1 series. The obstruction to analytic continuation is dictated by singularities in the Borel plane. As we are going to see shortly, singularities in the Borel plane play a crucial role in the theory. Among other things, they control the large order asymptotics of the coefficients of the original series. In order to simplify our discussion, we restrict ourselves to a specific class of functions: that of simple resurgent functions. **Definition 2**. A *resurgent function* is a function $\widehat{\varphi}$ which is holomorphic at the origin and satisfies the following property: along every ray issuing from the origin, there is a finite set of singular points such that $\widehat{\varphi}$ can be analytically continued along any path that follows the ray, while circumventing (from above or from below) the singular points. We call a resurgent function *simple* if its singularities are either simple poles or logarithmic branch cuts. A Gevrey-1 series $\widetilde{\varphi}$ is called a simple resurgent series if its Borel transform $\widehat{\varphi}$ is a simple resurgent function. For simplicity, we restrict ourselves to simple resurgent functions with logarithmic singularities only. Including simple poles in the discussion does not pose any particular difficulties, as shown in [@MS16]. For a fixed simple resurgent series $\widetilde{\varphi}$ whose Borel transform has a logarithmic singularity at $A \in \mathbb{C}^{\times}$, we can write locally at $s = A$ $$\widehat{\varphi}(s) = - \frac{S_A}{2\pi} \widehat{\varphi}_A(s - A) \, \log(s - A) + \text{holomorphic at } A \,,$$ where $\widehat{\varphi}_A(s)$, called the *minor* at $A$, is a holomorphic function at $s = 0$ (so that $\widehat{\varphi}_A(s - A)$ is holomorphic at $s = A$). Notice that we might want a specific choice of normalisation for $\widehat{\varphi}_A$, which motivates the introduction of an additional constant $S_A \in \mathbb{C}$ called the *Stokes constant*. In what follows, we adopt the physics jargon and refer to the singularities $A$ as *instanton actions*. Since $\widehat{\varphi}_A$ is holomorphic at the origin, we can consider its Taylor expansion: $$\widehat{\varphi}_A(s) = \sum_{m \ge 0} \varphi_{A,m} \frac{s^m}{m!} \,.$$ We consider $\widehat{\varphi}_A$ as the Borel transform of a formal power series $$\widetilde{\varphi}_A(\hslash) = \sum_{m \ge 0} \varphi_{A,m} \hslash^m \,.$$ The key point so far is that, given a simple resurgent series $\widetilde{\varphi}$, we can associate to it the data of its instanton actions, Stokes constants, and minors: $$\widetilde{\varphi} \longrightarrow \bigl\{ \, (S_A, \widetilde{\varphi}_A) \, \bigr\}_{A \in \mathcal{A}} \,.$$ Here $\mathcal{A}$ denotes the set of singularities of the Borel transform of $\widetilde{\varphi}$. We call this set of data the *Borel plane singularity structure* of $\widetilde{\varphi}$. ## Exponential integrals {#subsec:exp:int} A common source of simple resurgent functions, both in mathematics and in physics, is that of exponential integrals. We refer the reader to [@ABS19 section 3] for an in-depth review of the main results and the available literature on the topic, and to [@KS22] for a short treatment. The result presented here essentially rely on results from [@DH02]. Let $X = \mathbb{C}\setminus \set{ p_1,\dots,p_M }$ be the punctured complex plane, $V \colon X \to \mathbb{C}$ an algebraic function (the potential) and $\mu$ an algebraic $1$-form on $X$. We are interested in computing integrals of the form $$\varphi(\hslash; \gamma) = \int_{\gamma} e^{-\frac{1}{\hslash} V(t)} \mu(t)$$ for $\gamma$ a cycle in $X$. We assume that $V$ is Morse, i.e. has only finitely many critical points $(t_{\alpha})_{\alpha \in I}$ with $V''(t_{\alpha}) \ne 0$, and we assume that the critical values $V_{\alpha} = V(t_{\alpha})$ are pairwise distinct. The examples we have in mind are the $r$-Airy integrals, for an integer $r \ge 2$, for which we have $$X = \mathbb{C}\,, \qquad\quad V(t) = \frac{t^{r+1}}{r+1} - t \,, \qquad\quad \mu(t) = dt \,.$$ We are going to associate to each critical point a formal power series as follows. Fix a critical point $t_{\alpha}$. From the general theory of Morse functions, one can show that for all values of $\theta = \arg(\hslash)$ such that $\theta \ne \arg(V_{\beta} - V_{\alpha})$ there is a well-defined integral cycle $\gamma_{\alpha,\theta} \subset X$, called Lefschetz thimble or steepest descent path, passing through $t_{\alpha}$ along which the integral $\varphi(\hslash;\gamma_{\alpha,\theta})$ converges. We define the normalised integrals $$\varphi^{(\alpha)}(\hslash) \coloneqq \frac{e^{\frac{V_{\alpha}}{\hslash}}}{\sqrt{2\pi \hslash}} \int_{\gamma_{\alpha,\theta}} e^{-\frac{1}{\hslash} V(t)} \mu(t) \,.$$ The function $\varphi^{(\alpha)}$ is holomorphic in $\hslash$ in each sector defined by the Stokes rays, that is the ray defined by $\arg(\hslash) = \arg(V_{\beta} - V_{\alpha})$. Again from the general theory of Morse functions and Laplace method, the above integral has a well-defined divergent asymptotic expansion series: $$\varphi^{(\alpha)}(\hslash) \sim \sum_{m \ge 0} \varphi_{m}^{(\alpha)} \, \hslash^{m} \eqqcolon \widetilde{\varphi}^{(\alpha)}(\hslash) \,,$$ where the coefficients $\varphi_{m}^{(\alpha)}$ are independent of $\hslash$. The first coefficient is given by $\varphi_0^{(\alpha)} = V''(t_{\alpha})^{-1/2}$. Notice that the asymptotics is the same in every sector. As usual, let $\widehat{\varphi}^{(\alpha)}$ denote the Borel transform of $\widetilde{\varphi}^{(\alpha)}$. The analytic properties of $\widehat{\varphi}^{(\alpha)}$ are summarised as follows. **Theorem 3** (Singularity structure for exponential integrals). *The asymptotic expansion series $\widetilde{\varphi}^{(\alpha)}$ are simple resurgent. More precisely, the Borel transform $\widehat{\varphi}^{(\alpha)}$ has finitely many singularities at $A_{\alpha,\beta} = V_{\beta} - V_{\alpha}$ of logarithmic type, with behaviour given by $$\label{eqn:sing:exp:int} \widehat{\varphi}^{(\alpha)}(s) = \frac{\mathsf{S}_{\alpha,\beta}}{2\pi\mathrm{i}} \, \widehat{\varphi}^{(\beta)}(s - A_{\alpha,\beta}) \, \log(s - A_{\alpha,\beta}) + \textup{holomorphic at } A_{\alpha,\beta} \,.$$ Here $\mathsf{S}_{\alpha,\beta}$ are the integers defined by the intersection of two Lefschetz thimbles: $\mathsf{S}_{\alpha,\beta} = \gamma_{\alpha,\theta_+} \cdot \gamma_{\beta,\theta_- + \pi}$ for $\theta = \arg(A_{\alpha,\beta})$.* In other words, exponential integrals naturally generate simple resurgent series $\widetilde{\varphi}^{(\alpha)}$ labelled by critical points of the potential. Moreover, the Borel plane singularity structure of $\widetilde{\varphi}^{(\alpha)}$ is well understood in terms of: - *instanton actions* -- difference of critical values for the location of singularities: $A_{\alpha,\beta} = V_{\beta} - V_{\alpha}$, - *Stokes constants* -- intersection numbers of Lefschetz thimbles: $\mathsf{S}_{\alpha,\beta}$, - *minors* -- the asymptotic expansion $\widetilde{\varphi}^{(\beta)}$ for the series attached to the singularity at $A_{\alpha,\beta}$. As an example, let us apply the above result to the aforementioned generalised Airy integrals. The critical points are given by $V'(t) = t^r - 1 = 0$, that is $t_{\alpha} = \zeta^{\alpha}$ for $\zeta = e^{\frac{2\pi\mathrm{i}}{r}}$ and $\alpha = 1,\dots, r$. The critical values are $V_{\alpha} = - \frac{r}{r+1} \zeta^{\alpha}$. According to the above theorem, the Borel transforms $\widehat{\varphi}^{(\alpha)}$ have logarithmic singularities at $A_{\alpha,\beta} = \frac{r}{r+1} (\zeta^{\alpha} - \zeta^{\beta})$ and Stokes constants $$\label{eqn:Stokes:rAiry} \mathsf{S}_{\alpha,\beta} = \begin{cases} +1 & \text{if } \alpha > \beta \,,\\ -1 & \text{if } \alpha < \beta \,. \end{cases}$$ See [@BEH03 section 3.2] for a detailed derivation of the intersection numbers for generalised Airy integrals. It should also be noted that the problem of the Stokes phenomenon for the higher Airy functions was addressed in the literature also beyond the exponential integral context, starting directly from the associated ODE. See for instance [@Tur50; @Ohy95; @Suz01]. **Remark 4** (Normalisation conventions). In the theory of exponential integrals, it is customary to normalise [\[eqn:sing:exp:int\]](#eqn:sing:exp:int){reference-type="ref" reference="eqn:sing:exp:int"} with $\frac{\mathsf{S}_{\alpha,\beta}}{2\pi\mathrm{i}}$, so that the numerator coincide with the (integral) intersection number of two Lefschetz thimbles. As we are going to see in the next section, in the theory of large-order asymptotics an alternative normalisation is more convenient. Specifically, we adopt the normalisation $- \frac{S_{\alpha,\beta}}{2\pi}$. Throughout the rest of the paper, we employ the latter normalisation while utilising [\[eqn:sing:exp:int\]](#eqn:sing:exp:int){reference-type="ref" reference="eqn:sing:exp:int"} to calculate the Stokes constants of interest. ## Large-order asymptotics from the singularity structure We are now ready to establish the main theorem of this section, which gives the large-order asymptotics of the coefficients of a simple resurgent function $\widetilde{\varphi}$ in terms of the singularity structure of its Borel transform. The following result justifies the term "resurgence" too. Indeed, we have seen that the singularities of the Borel transform lead to new power series. These new series resurge in the original series through the large-order asymptotics of its coefficients. This is essentially an old theorem of Darboux, which relates the large-order asymptotics of the coefficients of an analytic function at the origin to the behaviour near the closest singularity. It is a well-known fact in the physics community, and perhaps not as widely known in the mathematics community. For the reader's convenience, we provide the proof here, which essentially relies on the application of Cauchy's theorem in the Borel plane. **Theorem 5** (Borel transform method). *Assume that $\widetilde{\varphi}$ is a simple resurgent series starting at order $\hslash^{\beta_0}$, with $\beta_0 \in \mathbb{N}$ called the critical exponent: $$\widetilde{\varphi}(\hslash) = \sum_{m \ge 0} \varphi_{m} \hslash^{m + \beta_0} \,.$$ Assume that its Borel transform $\widehat{\varphi}$ satisfies the following assumptions.* - *Polynomial growth at infinity. There exists $\nu \in \mathbb{R}$ such that $\widehat{\varphi}(s) = \mathrm{O}(s^{\nu})$ as $s \to \infty$.* - *Singularity structure. The function $\widehat{\varphi}$ has finitely many logarithmic singularities $A_1, \dots, A_n \in \mathbb{C}^{\times}$ around which it behaves as $$\label{eqn:behaviour:sing} \widehat{\varphi}(s) = - \frac{S_i}{2\pi} \, \widehat{\varphi}^{(i)}(s - A_i) \, \log(s - A_i) + \textup{holomorphic at } A_i\,,$$ for some constants $S_i$ and functions $\widehat{\varphi}^{(i)}$, the minors, which are holomorphic at the origin.* - *Behaviour of the minors. The functions $\widehat{\varphi}^{(i)}$ have a polynomial growth at infinity and have a finite number of singularities avoiding the ray starting at the origin and passing through $A_i$. Moreover, their Taylor expansion at the origin have critical exponent $\beta_i \in \mathbb{N}$, i.e. $\widehat{\varphi}^{(i)}(s) = \sum_{k = 0}^{\infty} \varphi^{(i)}_{k} \frac{s^{k + \beta_i}}{(k + \beta_i)!}$.* *Then the large $m$ asymptotics of $\varphi_m$ is given by $$\label{eqn:large:order} \varphi_m = \sum_{i = 1}^n \frac{S_i}{2\pi} \frac{\Gamma(m + \beta_0 - \beta_i)}{A_i^{m + \beta_0 - \beta_i}} \left( \sum_{k = 0}^{K} \frac{A_i^{k}}{(m + \beta_0 - \beta_i - 1)^{\underline{k}}} \, \varphi^{(i)}_{k} + \mathrm{O}\biggl( \frac{1}{m^{K+1}} \biggr) \right) \,.$$ Here $(x)^{\underline{k}} \coloneqq x (x-1) \cdots (x-m+1)$ denotes the falling factorial.* Before proceeding with a proof of the above result, it is instructive to write down explicitly the first terms of [\[eqn:large:order\]](#eqn:large:order){reference-type="ref" reference="eqn:large:order"}. Without loss of generality, suppose that the singularities are ordered as $|A_1| \le \cdots \le |A_n|$ from the closest to farthest distance from the origin. Then $$\begin{split} \varphi_m & = \phantom{+} \frac{S_1}{2\pi} \frac{\Gamma(m + \beta_0 - \beta_1)}{A_1^{m + \beta_0 - \beta_1}} \left( \varphi^{(1)}_{0} + \frac{A_1}{(m + \beta_0 - \beta_1 - 1)} \, \varphi^{(1)}_{1} + \frac{(A_1)^2}{(m + \beta_0 - \beta_1 - 1)^{\underline{2}}} \, \varphi^{(1)}_{2} + \cdots \right) \\ & \phantom{=} + \cdots \\ & \phantom{=} + \frac{S_n}{2\pi} \frac{\Gamma(m + \beta_0 - \beta_n)}{A_n^{m + \beta_0 - \beta_n}} \left( \varphi^{(n)}_{0} + \frac{A_n}{(m + \beta_0 - \beta_n - 1)} \, \varphi^{(n)}_{1} + \frac{(A_n)^2}{(m + \beta_0 - \beta_n - 1)^{\underline{2}}} \, \varphi^{(n)}_{2} + \cdots \right)\,. \end{split}$$ We can see that the asymptotics presents an overall factorial growth, $\Gamma(m + \beta_0 - \beta_i) = \mathrm{O}(m!)$, and it is "organised" into different exponential growths corresponding to the rows governed by $1/A_i^{m + \beta_0 - \beta_1} = \mathrm{O}(A_i^{-m})$ with overall constant given by $S_i$. Moreover, the $i$-th row contains an asymptotic series in $1/m$ which scales as $A_i$ and whose coefficients are the Taylor coefficients of the corresponding minors. *Proof of [Theorem 5](#thm:large:order){reference-type="ref" reference="thm:large:order"}.* Since $\widehat{\varphi}$ is holomorphic at the origin, we can extract its expansion coefficients through Cauchy's formula: $$\frac{\varphi_{m}}{(m + \beta_0)!} = \frac{1}{2\pi\mathrm{i}} \oint_{\mathcal{C}_0} \frac{\widehat{\varphi}(s)}{s^{m + \beta_0 + 1}} \, ds \,.$$ Here $\mathcal{C}_0$ is a small contour around the origin oriented counter-clockwise. We can deform the contour, avoiding the logarithmic cuts starting at each singularity $A_i$ with partial Hankel contours $\mathcal{H}^{(R)}_i$ starting at $A_i$ in the direction $\theta_i \coloneqq \arg(A_i)$, connected by arcs whose union is denoted by $\gamma^{(R)}$ (see [\[fig:deformation:contour\]](#fig:deformation:contour){reference-type="ref" reference="fig:deformation:contour"}). Here $R$ is the radius of the arcs, and is assumed to be greater than $\max_{i=1,\dots,n}{|A_i|}$. Taking the limit $R \to +\infty$ gives zero contribution from the contours $\gamma^{(R)}$. Indeed, assuming $R$ large enough so that $|\widehat{\varphi}(s)| < C s^{\nu}$, we find $$\left| \frac{1}{2\pi \mathrm{i}} \int_{\gamma_R} \frac{\widehat{\varphi}(s)}{s^{m + \beta_0 + 1}} \, ds \right| \le \frac{C}{R^{m + \beta_0 - \nu}}$$ which tends to zero for $R \to +\infty$ for $m \gg 0$. Thus we only have to consider the Hankel contours $\mathcal{H}_i$ obtained in the limit $R \to +\infty$. A simple manipulation of the integral shows that $$\begin{gathered} \frac{\varphi_{m}}{(m + \beta_0)!} = \frac{1}{2\pi\mathrm{i}} \sum_{i=1}^n \int_{\mathcal{H}_i} \frac{\widehat{\varphi}(s)}{s^{m + \beta_0 + 1}} \, ds = -\frac{1}{2\pi\mathrm{i}} \sum_{i=1}^n \frac{S_i}{2\pi} \int_{A_i}^{e^{\mathrm{i}\theta_i }\infty} \mathop{\mathrm{Disc}}\left[ \widehat{\varphi}^{(i)}(s - A_i) \, \log(s - A_i) \right] \frac{ds}{s^{m + \beta_0 + 1}} \\ = \sum_{i=1}^n \frac{S_i}{2\pi} \int_{A_i}^{e^{\mathrm{i}\theta_i }\infty} \frac{\widehat{\varphi}^{(i)}(s - A_i)}{s^{m + \beta_0 + 1}} \, ds = \sum_{i=1}^n \frac{S_i}{2\pi} e^{-\mathrm{i}\theta_i (m+\beta_0)} \int_{0}^{+\infty} \frac{\widehat{\varphi}^{(i)}(e^{\mathrm{i}\theta_i} s)}{(s + |A_i|)^{m + \beta_0 + 1}} \, ds \,. \end{gathered}$$ In the second equality, we used the behaviour of $\widehat{\varphi}$ at $A_i$ given in [\[eqn:behaviour:sing\]](#eqn:behaviour:sing){reference-type="ref" reference="eqn:behaviour:sing"}. In the third equality, we used the fact that the discontinuity is completely governed by the logarithm, and gives a contribution of $-2\pi\mathrm{i}$. The last equality is simply a change of variable in the integral. The goal now is to approximate $\widehat{\varphi}^{(i)}$ with the first $K$ terms of its Taylor expansion, and estimate the error using Watson's lemma. Define the tail of $\widehat{\varphi}^{(i)}$ by removing the first $K$ terms of its Taylor expansion around the origin: $\widehat{\varphi}^{(i)}(s; K) \coloneqq \widehat{\varphi}^{(i)}(s) - \sum_{k=0}^{K} \varphi^{(i)}_{k} \frac{s^{k + \beta_i}}{(k + \beta_i)!}$. Then $$\begin{gathered} \label{eqn:to:estimate} \frac{\varphi_{m}}{(m + \beta_0)!} - \sum_{i=1}^n \frac{S_i}{2\pi} e^{-\mathrm{i}\theta_i (m + \beta_0)} \sum_{k=0}^K \frac{\varphi^{(i)}_k e^{\mathrm{i}\theta_i (k + \beta_i)}}{(k + \beta_i)!} \int_{0}^{+\infty} \frac{s^{k + \beta_i}}{(s + |A_i|)^{m + \beta_0 + 1}} \, ds \\ = \sum_{i=1}^n \frac{S_i}{2\pi} e^{-\mathrm{i}\theta_i (m + \beta_0)} \int_{0}^{+\infty} \frac{\widehat{\varphi}^{(i)}(e^{\mathrm{i}\theta_i} s;K)}{(s + |A_i|)^{m + \beta_0 + 1}} \, ds \,. \end{gathered}$$ The left-hand side of the equation above can be explicitly computed using the Euler-type integral $\frac{1}{\Gamma(\mu + 1)} \int_{0}^{+\infty} \frac{s^{\mu}}{(s + A)^{\nu + 1}} ds = A^{\mu-\nu} \, \frac{ \Gamma(\nu - \mu) }{ \Gamma(\nu + 1) }$ valid for $A > 0$ and $\nu > \mu > 0$. We thus find $$\frac{\varphi_{m}}{(m + \beta_0)!} - \frac{1}{(m + \beta_0)!} \sum_{i=1}^n \frac{S_i}{2\pi} \sum_{k=0}^K \varphi^{(i)}_k \frac{\Gamma(m + \beta_0 - \beta_i - k)}{A_i^{m + \beta_0 - \beta_i - k}} \,.$$ We are now left with an estimate of the right-hand side of [\[eqn:to:estimate\]](#eqn:to:estimate){reference-type="eqref" reference="eqn:to:estimate"}. In the $i$-th term of the sum, let us set $s = |A_i|(e^t - 1)$. We obtain $$\sum_{i=1}^n \frac{S_i}{2\pi} e^{-\mathrm{i}\theta_i (m+\beta_0)} \int_{0}^{+\infty} \frac{\widehat{\varphi}^{(i)}(e^{\mathrm{i}\theta_i} s;K)}{(s + |A_i|)^{m + \beta_0 + 1}} \, ds = \sum_{i=1}^n \frac{S_i}{2\pi} \frac{1}{A_i^{m + \beta_0}} \int_{0}^{+\infty} e^{-t(m + \beta_0)} \widehat{\varphi}^{(i)}( A_i (e^t - 1) ;K) \, dt \,.$$ We can now apply Watson's [Lemma 6](#Watson:lemma){reference-type="ref" reference="Watson:lemma"} to estimate the integral: since $\widehat{\varphi}^{(i)}( A_i(e^t - 1) ;K) = \mathrm{O}(t^{K+\beta_i+1})$ as $t \to 0^+$ (recall that $\widehat{\varphi}^{(i)}(s;K)$ is $\widehat{\varphi}^{(i)}(s)$ pruned of its first $K$ expansion coefficients at the origin) and $\widehat{\varphi}^{(i)}( A_i(e^t - 1) ;K)$ behaves at most exponentially at infinity, we find $$\int_{0}^{+\infty} e^{-t(m + \beta_0)} \widehat{\varphi}^{(i)}( A_i(e^t - 1) ;K) \, dt = \mathrm{O}\biggl( \frac{1}{(m + \beta_0)^{K+\beta_i+2}} \biggr) = \mathrm{O}\biggl( \frac{1}{m^{K+\beta_i+2}} \biggr) \,.$$ After multiplication by $(m+\beta_0)!$ and some algebraic manipulation, we find the thesis. ◻ The estimate for the error term in the asymptotics was a consequence of Watson's lemma for Laplace-type integrals, which we report here for the reader's convenience. See [@Olv97] for more details. **Lemma 6** (Watson). *Let $f \colon \mathbb{R}_{+} \to \mathbb{C}$ be a continuous function satisfying the following assumptions.* - *Polynomial growth at the origin. There exists an integer $K$ such that $f(t) = \mathrm{O}( t^{d} )$ as $t \to 0^+$.* - *Exponential growth at infinity. There exists $\nu \in \mathbb{R}$ such that $f(t) = \mathrm{O}( e^{t\nu} )$ as $t \to + \infty$.* *Then, as $x \to +\infty$: $$\int_{0}^{+\infty} e^{-tx} \, f(t) \, dt = \mathrm{O}\biggl( \frac{1}{x^{d+1}} \biggr) \,.$$* **Remark 7** (Gevrey-2 series). In the next sections, we are going to apply the Borel transform method to formal power series with vanishing even or odd coefficients. In other words, up to relabelling of the coefficients, we have $$\widetilde{\varphi}(\hslash) = \sum_{g \ge 0} \varphi_{g} \hslash^{2g + \beta_0} \,.$$ Formal power series with $\varphi_{g} = \mathrm{O}((2g)! a^{-g})$ are also known as *Gevrey-2* series. In this case, the large $g$ asymptotics, that is [\[eqn:large:order\]](#eqn:large:order){reference-type="ref" reference="eqn:large:order"}, reads $$\varphi_{g} = \sum_{i = 1}^n \frac{S_i}{2\pi} \frac{\Gamma(2g + \beta_0 - \beta_i)}{A_i^{2g + \beta_0 - \beta_i}} \left( \sum_{k = 0}^{K} \frac{A_i^{k}}{(2g + \beta_0 - \beta_i - 1)^{\underline{k}}} \, \varphi^{(i)}_{k} + \mathrm{O}\biggl( \frac{1}{g^{K+1}} \biggr) \right) \,.$$ We also remark that the above theorem can be generalised to the case of $\widehat{\varphi}$ admitting simple poles. Moreover, one can allow for the critical exponents $\beta_0$ and $\beta_i$ to be any real numbers by suitably adapting the definition of the Borel transform. ## Algebraic properties of resurgent series {#subsec:Borel:properties} So far we have established that the large-order asymptotics of the coefficients of a simple resurgent series is entirely determined by its Borel singularity structure. One advantage of working within the class of simple resurgent functions is that they exhibit favourable properties under addition and multiplication. As will become evident in the subsequent sections, this aspect is of particular significance to us, since our goal is to deduce the singularity structure of specific power series. These series are derived as sums of products of simple resurgent "building blocks", whose singularity structure is completely under control. For all these reasons, let us discuss some algebraic properties of simple resurgent series. The Borel transform is by definition a linear operation. On the other hand, the Borel transform does not respect the Cauchy product of formal power series, but it rather sends it to the convolution product. **Definition 8**. Given $\widehat{\phi}, \widehat{\psi} \in \mathbb{C}\llbracket s \rrbracket$, their *convolution product* is $$\widehat{\phi} \ast \widehat{\psi} \coloneqq \mathfrak{B} \bigl[ \widetilde{\phi} \cdot \widetilde{\psi} \bigr] \,,$$ with $\widehat{\phi} = \mathfrak{B}[ \widetilde{\phi} ]$ and $\widehat{\psi} = \mathfrak{B}[ \widetilde{\psi} ]$. It is easy to check that, if both $\widehat{\phi}$ and $\widehat{\psi}$ converge in a disc of radius $R$ around the origin, then for all $s$ in such a disc we find $(\widehat{\phi} \ast \widehat{\psi})(s) = \frac{d}{ds} \int_{0}^{s} \widehat{\phi}(\sigma) \widehat{\psi}(s-\sigma) \, d\sigma$. In other words, $\widehat{\phi} \ast \widehat{\psi}$ is (the derivative of) the usual convolution product of functions, hence the name. The next two theorems precisely determine how the analytic properties of simple resurgent functions transform under convolution. **Lemma 9** (Analyticity of the convolution product [@MS16 lemma 5.54]). *Let $\Omega$ be a star-shaped open subset of $\mathbb{C}$ (that is, for every $s\in\Omega$, the line segment $[0,s]$ is contained in $\Omega$), and $\widehat{\phi},\widehat{\psi}$ be holomorphic in $\Omega$. Then their convolution product is also holomorphic in $\Omega$.* **Theorem 10** (Product of simple resurgent series [@MS16 theorem 6.83]). *Let $\mathcal{A}$ and $\mathcal{B}$ be non-empty closed discrete subsets of $\mathbb{C}^{\times}$, and denote $\mathcal{C} \coloneqq \mathcal{A} \cup \mathcal{B} \cup \left( \mathcal{A} + \mathcal{B} \right)$, with $\mathcal{A} + \mathcal{B} \coloneqq \set{ A + B | A \in \mathcal{A}, \, B \in \mathcal{B} }$. Let $\widetilde{\phi}$ and $\widetilde{\psi}$ be simple resurgent series with Borel plane singularities in $\mathcal{A}$ and $\mathcal{B}$ respectively, such that $$\begin{aligned} \widehat{\phi}(s) & = - \frac{S_{A}}{2\pi} \widehat{\phi}_{A}(s - A) \, \log(s - A) + \textup{holomorphic at } A && \qquad \forall A \in \mathcal{A} \,, \\ \widehat{\psi}(s) & = - \frac{S_{B}}{2\pi} \widehat{\psi}_{B}(s - B) \, \log(s - B) + \textup{holomorphic at } B &&\qquad \forall B \in \mathcal{B} \,. \end{aligned}$$ If $\mathcal{C}$ is closed and discrete, then $\widetilde{\phi} \cdot \widetilde{\psi}$ is also a simple resurgent series with Borel plane singularities in $\mathcal{C}$ and $$\begin{aligned} \bigl( \widehat{\phi} \ast \widehat{\psi} \bigr)(s) & = - \frac{S_{A}}{2\pi} \bigl( \widehat{\phi}_{A} \ast \widehat{\psi} \bigr)(s - A) \, \log(s - A) + \textup{holomorphic at } A && \qquad \forall A \in \mathcal{A} \,, \\ \bigl( \widehat{\phi} \ast \widehat{\psi} \bigr)(s) & = - \frac{S_{B}}{2\pi} \bigl( \widehat{\phi} \ast \widehat{\psi}_{B} \bigr)(s - B) \, \log(s - B) + \textup{holomorphic at } B && \qquad \forall B \in \mathcal{B} \,. \end{aligned}$$* **Remark 11** (Singularities on the principal sheet). A priori, [Theorem 10](#theorem:prod){reference-type="ref" reference="theorem:prod"} implies that the set of Borel plane singularities of $\widetilde{\phi} \cdot \widetilde{\psi}$ not only contain elements of $\mathcal{A}$ and $\mathcal{B}$, but also of $\mathcal{A} + \mathcal{B}$. Nonetheless, [Lemma 9](#lemma:conv){reference-type="ref" reference="lemma:conv"} ensures that, if all elements of $\mathcal{A} \cup \mathcal{B}$ lay on distinct rays issuing from the origin, then all singularities from $\mathcal{A} + \mathcal{B}$ will not be on the principal sheet of the Borel plane. Thus, according to [Theorem 5](#thm:large:order){reference-type="ref" reference="thm:large:order"}, they do not contribute to the large-order asymptotic behaviour of the coefficients of $\widetilde{\phi} \cdot \widetilde{\psi}$. # Large genus asymptotics of $\psi$-class intersection numbers {#sec:WK} The goal of this section is to prove the large genus asymptotics of the $\psi$-class intersection numbers, including all subleading corrections. To this end, we apply the Borel transform method to the $n$-point function, which in turn is built out of formal solutions of the Airy differential equations through the determinantal formulae. Thus, all terms appearing in Aggarwal's asymptotic formula can be explained in terms of data appearing in such formal solutions of the Airy ODE. We start by recalling some facts about the Airy functions and their Borel transform, as well as the determinantal formulae for the $\psi$-class intersection numbers. We then proceed with the resurgent analysis of $n$-point functions, i.e. their singularity structure on the Borel plane. ## Airy functions Consider the ($\hslash$-dependent) *Airy ODE*: $$\biggl( \biggl( \hslash\frac{d}{dx} \biggr)^2 - x \biggr) \psi(x;\hslash) = 0 \,.$$ A general solution is given by the Airy integral: $\int_{\gamma} e^{-\frac{1}{\hslash}V(t,x)} dt$, with $V(t,x) \coloneqq \frac{t^3}{3} - xt$ and $\gamma$ a properly chosen integration contour. In this section, we are interested in a basis of formal (or WKB) solutions, i.e. the asymptotic expansion of the (properly normalised) Airy integrals with $\gamma$ being a Lefschetz thimble. In the context of Lax operator formalism these are called the *wave functions* or Baker--Akhiezer functions. Following the prescription outlined in [2.2](#subsec:exp:int){reference-type="ref" reference="subsec:exp:int"}, such formal solutions and their derivatives are given by $$\begin{split} \psi_{\pm}(x;\hslash) &= \frac{e^{\mp \frac{V(x)}{\hslash}}}{\sqrt{2}} x^{-1/4} \sum_{k = 0}^{\infty} \frac{1}{864^k}\frac{(6k)!}{(2k)!(3k)!} \left(\mp \frac{\hslash}{V(x)} \right)^{k} \,, \\ \psi_{\pm}'(x;\hslash) &= \pm \frac{e^{\mp \frac{V(x)}{\hslash}}}{\sqrt{2}} x^{1/4} \sum_{k = 0}^{\infty} \frac{1}{864^k}\frac{(6k)!}{(2k)!(3k)!} \frac{1+6k}{1-6k} \left(\mp \frac{\hslash}{V(x)} \right)^{k} \,, \\ \end{split}$$ where $\pm V(x) \coloneqq \mp \frac{2}{3} x^{3/2}$ are the critical values of the potential at the critical point $t = \pm x^{1/2}$. The reader can recognise $\psi_{-}$ and $\psi_{+}$ as the asymptotic expansions in the appropriate regions of the Airy and Bairy functions respectively. The normalisation is fixed so that the Wronskian is worth $\psi_- \psi_+' - \psi_-' \psi_+ = 1$. Consider the Borel transform of the Airy functions. To start with, let us separate the exponential part by setting $\psi_{\pm}(x;\hslash) \eqqcolon \exp(\mp \frac{1}{\hslash} V(x)) \, \widetilde{\psi}(x;\hslash)$, so that $\widetilde{\psi}_{\pm}$ is a formal power series in $\hslash$. Similarly for the derivatives. Then the Borel transforms of $\widetilde{\psi}_{\pm}$ and $\widetilde{\psi}_{\pm}'$ are expressed in terms of Gauss hypergeometric functions as: $$\widehat{\psi}_{\pm}(x;s) = \frac{x^{-1/4}}{\sqrt{2}} \, \vphantom{F}_{2}F_{1} \left( \tfrac{5}{6},\tfrac{1}{6} ; 1 ; \pm \frac{s}{A(x)} \right) \,, \qquad \widehat{\psi}_{\pm}'(x;s) = \pm \frac{x^{1/4}}{\sqrt{2}} \, \vphantom{F}_{2}F_{1} \left( \tfrac{7}{6},-\tfrac{1}{6} ; 1 ; \pm \frac{s}{A(x)} \right) \,,$$ where $\pm A(x) \coloneqq \pm \frac{4}{3} x^{3/2}$ are the instanton actions. From the above explicit expression, we deduce that the Borel transforms $\widehat{\psi}_{\pm}$ and $\widehat{\psi}_{\pm}'$ converge in a disc of radius $|A(x)|$ centred at the origin, and they can be extended analytically to functions with logarithmic singularities at $s = \pm A(x)$. From a direct analysis of the integral representation of the Gauss hypergeometric functions, one can prove the following behaviour at the singularities: $$\label{eqn:Borel:Airy} \begin{aligned} \widehat{\psi}_{\pm}(x;s) & = - \frac{S}{2\pi} \, \widehat{\psi}_{\mp}\bigl( x;s \mp A(x) \bigr) \log\bigl( s \mp A(x) \bigr) + \text{holomorphic at} \pm A(x)\,, \\ \widehat{\psi}_{\pm}'(x;s) & = - \frac{S}{2\pi} \, \widehat{\psi}_{\mp}'\bigl( x;s \mp A(x) \bigr) \log\bigl( s \mp A(x) \bigr) + \text{holomorphic at} \pm A(x) \,, \end{aligned}$$ where $S = 1$ is the Stokes constant. In other words, $\widetilde{\psi}_{\pm}$ and $\widetilde{\psi}_{\pm}'$ are simple resurgent functions with finitely many logarithmic singularities, with Stokes constants and minors that are fully under control. Notice that the above behaviour also follows from [Theorem 3](#thm:sing:exp:int){reference-type="ref" reference="thm:sing:exp:int"} and the (properly normalised) integral representation of $\psi_{\pm}$ and $\psi_{\pm}'$, for which the explicit expressions of the Borel transforms is not needed. In the remaining part of this section, we often keep the generic notation $S$ and $A(x)$ to emphasise how their actual values are not crucial in the following analysis. This fact will allow us to compute the large genus asymptotics of $\Theta$-class intersection numbers in [4](#sec:Norbury){reference-type="ref" reference="sec:Norbury"} by simply changing the values of $S$ and $A(x)$ to the appropriate ones. ## Airy correlators The Airy ODE can be re-written as a first order $2 \times 2$ system in terms of the companion matrix $\mathcal{D}$: $$\hslash\frac{d}{dx} \Psi(x;\hslash) = \mathcal{D}(x) \Psi(x;\hslash) \,, \qquad \mathcal{D}(x) \coloneqq \begin{pmatrix} 0 & 1 \\ x & 0 \end{pmatrix} \,, \qquad \Psi(x;\hslash) = \begin{pmatrix} \psi_-(x;\hslash) & \psi_+(x;\hslash) \\ \psi_-'(x;\hslash) & \psi_+'(x;\hslash) \end{pmatrix} \,.$$ We refer to $\Psi$ as the *wave matrix*. Notice that with the chosen normalisation of the Wronskian, we have $\det{\Psi} = 1$. We are now interested in computing the correlators associated to the above differential system. To this end, define the matrix $$\label{eqn:M:matrix} M(x;\hslash) \coloneqq \Psi(x;\hslash) E \Psi^{-1}(x;\hslash) \,, \qquad\qquad E \coloneqq \frac{1}{2} \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} .$$ Notice that $E$ is a generator of the Cartan subalgebra of $\mathfrak{sl}_2(\mathbb{C})$. More explicitly $M$ is given by $$M = \begin{pmatrix} \frac{1}{2}(\psi_+' \psi_- + \psi_+ \psi_-') & \psi_+ \psi_- \\ \psi_+' \psi_-' & - \frac{1}{2}(\psi_+' \psi_- + \psi_+ \psi_-') \end{pmatrix} = \begin{pmatrix} \frac{1}{2}(\widetilde{\psi}_+' \widetilde{\psi}_- + \widetilde{\psi}_+ \widetilde{\psi}_-') & \widetilde{\psi}_+ \widetilde{\psi}_- \\ \widetilde{\psi}_+' \widetilde{\psi}_-' & - \frac{1}{2}(\widetilde{\psi}_+' \widetilde{\psi}_- + \widetilde{\psi}_+ \widetilde{\psi}_-') \end{pmatrix} .$$ In the second equation, we used the simple but crucial fact that the exponential factors $e^{\pm \frac{1}{\hslash}V(x)}$ cancel out in quadratic expressions involving functions labelled by '$+$' and '$-$'. In other words, we can substitute the $\psi$'s with their respective $\widetilde{\psi}$'s. Thus, $M$ is a matrix-valued formal power series in $\hslash$. We remark that, from a more geometric point-of-view, $\Psi$ can be identified with a formal flat section for the $\mathfrak{sl}_2(\mathbb{C})$-connection $\nabla = \hslash d - \mathcal{D} dx$ on the trivial bundle over the projective line. Besides, $M$ can be seen as a flat section of the adjoint bundle, i.e. it satisfies the differential system $\hslash dM - [\mathcal{D},M] dx = 0$. We are now ready to define the correlators associated to the Airy differential system. To the best of our knowledge, correlators in the context of ODEs appeared for the first time in [@BE], inspired by the analogy with random matrix models (see for instance [@Dys70; @Met04]). Later, they were re-discovered in the context of tau-functions of the KdV hierarchy in [@BDY16]. **Definition 12** (Airy correlators). Define the $n$-point *Airy correlators* $W_n$ as[^2] $$\label{eqn:n:pnt:Airy} \begin{aligned} & W_1(x_1;\hslash) \coloneqq - \frac{1}{\hslash} \mathop{\mathrm{Tr}}{\bigl( \mathcal{D}(x_1) M(x_1;\hslash) \bigr)} \,, \\ & W_n(x_1,\dots,x_n;\hslash) \coloneqq (-1)^{n-1} \sum_{\sigma \in S_n^{\textup{cyc}}} \frac{\mathop{\mathrm{Tr}}{ \bigl( \prod_{i=1}^n M(x_{\sigma^i(1)};\hslash) } \bigr) }{\prod_{i=1}^n ( x_{i} - x_{\sigma(i)} )} \qquad\quad \text{for }n \ge 2 \,. \end{aligned}$$ We refer to the above formula as the *determinantal formula*. Here $S_n^{\textup{cyc}}$ denotes the set of cyclic permutations, that is the subset of $S_n$ consisting of all permutations with a single cycle of length $n$. It can be shown that $W_n$ is regular along $x_i = x_j$ (see for instance [@BBE15]). Moreover, as $M$ is a formal power series in $\hslash$, so is $W_{n}$. The next theorem makes the $\hslash$-dependence more precise, and most importantly it establishes an enumerative-geometric interpretation of the expansion coefficients: they correspond to $\psi$-class intersection numbers. For this reason, the $\hslash$-expansion of $W_{n}$ is also called a *genus expansion*. A proof of the theorem can be deduced from the original work of Kontsevich [@Kon92], together with the fact that correlators of matrix models are computed by determinantal formulae (fact already hinted in [@EO07 section 10.5]). The theorem was later re-discovered in [@BDY16] in the context of solutions of the KdV hierarchy. **Theorem 13** (Genus expansion of the Airy correlators). *The $n$-point Airy correlators $W_{n}$ admit the following $\hslash$-expansion: $$\label{eqn:genus:expns:Airy} W_{n}(\bm{x};\hslash) = \sum_{g = 0}^{\infty} \hslash^{2g-2+n} \, W_{g,n}(\bm{x}) \,.$$ Moreover, $W_{g,n}$ stores $\psi$-class intersection numbers: if $2g-2+n>0$, $$W_{g,n}(\bm{x}) = (-2)^{-(2g-2+n)} \sum_{\substack{ d_{1},\dots,d_n \ge 0 \\ |d| = 3g-3+n }} \braket{ \tau_{d_1} \cdots \tau_{d_n} } \prod_{i=1}^n \frac{(2d_i+1)!!}{2 \, x_i^{d_i+3/2}} \,.$$* **Example 14** ($1$-point correlators). A direct computation shows that $$\begin{split} W_1(x;\hslash) &= \frac{x \, \widetilde{\psi}_- \widetilde{\psi}_+ - \widetilde{\psi}_-' \widetilde{\psi}_+'}{\hslash} = \frac{1}{\hslash} x^{1/2} - \hslash\tfrac{1}{32} x^{-5/2} - \hslash^3 \tfrac{105}{2048} x^{-11/2} - \hslash^5 \tfrac{25025}{65536} x^{-17/2} + \mathrm{O}(\hslash^7) \,. \end{split}$$ In particular, the above expansion agrees with the intersection numbers: $$\begin{aligned} {4} W_{0,1}(x) &= x^{1/2} \,, \qquad & W_{1,1}(x) &= - \tfrac{1}{32} x^{-5/2} \,, \qquad & W_{2,1}(x) &= - \tfrac{105}{2048} x^{-11/2} \,, \qquad & W_{3,1}(x) &= - \tfrac{25025}{65536} x^{-17/2} \,, \\ % & & \braket{\tau_1} &= \tfrac{1}{24} \,, & \braket{\tau_4} &= \tfrac{1}{1152} \,, & \braket{\tau_7} &= \tfrac{1}{82944} \,. \end{aligned}$$ More generally, an explicit computation of the expansion coefficients of $W_1$ from those of the Airy and Bairy functions shows the well-known closed formula $\braket{\tau_{3g-2}} = \frac{1}{24^g g!}$ for $1$-point intersection numbers. **Remark 15** (Connection with topological recursion and the spectral curve). The reader familiar with topological recursion can recognise $W_{g,n}$ as essentially the topological recursion correlators computed from the Airy spectral curve $(\mathbb{P}^1, x(z) = z^2, y(z) = z, \omega_{0,2}(z_1,z_2) = \frac{dz_1 dz_2}{(z_1 - z_2)^2})$. More precisely: $$\omega_{g,n}(\bm{z}) = W_{g,n}(\bm{x}) \, dx_1 \cdots dx_n \big|_{x_i = z_i^2} \,.$$ This can be explained by the fact that both correlators defined through determinantal formulae and topological recursion satisfy the same loop equations [@BE; @BEO15]. We also emphasise that the wave functions, and consequently the correlators, are defined up to a choice of square root of $x$. In other words, they are well-defined on the spectral curve $\set{x = y^2}$. Throughout the rest of the section, we will work with a fixed choice of square root of $x$, often denoted as $\sqrt{x} = z$. The next result gives an alternative expression for the $n$-point Airy correlators in terms of the Airy kernel, which plays an important role in the Tracy--Widom law and in extreme values statistics [@TW94a]. **Lemma 16** (Determinantal formula in kernel form). *For $n \ge 2$, the $n$-point Airy correlators are given by the two equivalent expressions $$W_n(\bm{x};\hslash) = (-1)^{n-1} \sum_{\sigma \in S_n^{\textup{cyc}}} \prod_{i = 1}^n K_{+,-}(x_i,x_{\sigma(i)};\hslash) = (-1)^{n-1} \sum_{\sigma \in S_n^{\textup{cyc}}} \prod_{i = 1}^n K_{-,+}(x_i,x_{\sigma(i)};\hslash)\,,$$ where $K_{\pm,\mp}(x,y;\hslash)$, known as the Airy kernels, are defined as $$K_{\pm,\mp}(x,y;\hslash) \coloneqq \frac{\widetilde{\psi}_{\pm}'(x;\hslash) \widetilde{\psi}_{\mp}(y;\hslash) - \widetilde{\psi}_{\pm}(x;\hslash) \widetilde{\psi}_{\mp}'(y;\hslash)}{x-y} \,. \\ $$ Moreover, the kernels are related by the parity relation $K_{-,+}(x,y;\hslash) = - K_{+,-}(x,y;-\hslash)$.* *Proof.* The two expression for $W_n$ are equivalent, due to the cyclicity of the sum and the symmetry property $K_{+,-}(x,y;\hslash) = K_{-,+}(y,x;\hslash)$. Thus, we only need to prove one equality. Let us start from the definition of the $n$-point correlators, [\[eqn:n:pnt:Airy\]](#eqn:n:pnt:Airy){reference-type="ref" reference="eqn:n:pnt:Airy"}. Since the formula is invariant under shifts of $M$ by matrices proportional to the identity [@EM appendix A], we can substitute $M$ by $$M - \tfrac{1}{2} \, \mathrm{Id} = \begin{pmatrix} \widetilde{\psi}_+ \widetilde{\psi}_-' & - \widetilde{\psi}_+ \widetilde{\psi}_- \\ \widetilde{\psi}_+' \widetilde{\psi}_-' & - \widetilde{\psi}_+' \widetilde{\psi}_- \end{pmatrix} = \begin{pmatrix} \widetilde{\psi}_+ \\ \widetilde{\psi}_+' \end{pmatrix} \otimes \begin{pmatrix} \widetilde{\psi}_-' \\ - \widetilde{\psi}_- \end{pmatrix} \,.$$ Here $u \otimes v$ denotes the outer product of two vectors. Inserting the above expression in the determinantal formula and using the identity $\mathop{\mathrm{Tr}}( \prod_{k=1}^{n} u_{k} \otimes v_k ) = \prod_{k=1}^{n} \braket{v_k,u_{k+1}}$ (with the identification $u_{n+1} = u_1$) yields the thesis. To conclude, the relations $\psi_{-}(x;\hslash) = \psi_{+}(x;-\hslash)$ and $\psi_{-}'(x;\hslash) = - \psi_{+}'(x;-\hslash)$ imply the parity property. ◻ ## Singularity structure and large genus asymptotics The goal of this section is to understand the large genus behaviour of the coefficients in the $\hslash$-expansion of the correlators via the Borel transform method. Since the singularity structure of the wave functions is well understood (see [\[eqn:Borel:Airy\]](#eqn:Borel:Airy){reference-type="ref" reference="eqn:Borel:Airy"}), one can deduce a similar statement for the correlator $W_n$ using the algebraic properties of the Borel transform, as explained in [2.4](#subsec:Borel:properties){reference-type="ref" reference="subsec:Borel:properties"}. In the following analysis we assume $n \ge 2$. The case $n = 1$ can be considered separately and leads to the same result. Moreover, we assume $(x_1,\dots,x_n)$ to be in a generic position, so that the instanton actions lay on distinct rays issuing from the origin, therefore realising the conditions of [Remark 11](#rem:princ:sheet){reference-type="ref" reference="rem:princ:sheet"}. **Proposition 17** (Singularity structure of the Airy correlators). *The Borel transform $\widehat{W}_{n}(\bm{x};s)$ of the $n$-point Airy correlator is simple resurgent. More precisely, on the principal sheet $\widehat{W}_{n}$ has $2n$ logarithmic singularities located at $s = \pm A(x_i)$ and such that $$\widehat{W}_{n}(\bm{x};s) = - \frac{S}{2\pi} \, \widehat{W}^{(\pm,i)}_{n}\bigl( \bm{x}; s \mp A(x_i) \bigr) \, \log\bigl( s \mp A(x_i) \bigr) + \textup{holomorphic at } \pm A(x_i) \,,$$ where:* - *$A(x) = \frac{4}{3} x^{3/2}$ and $S = 1$ are the instanton action and the Stokes constant associated to the Airy functions,* - *the minor $\widehat{W}^{(\pm,i)}_{n}$ is the Borel transform of the formal power series $$\label{eqn:1inst:corr} W^{(\pm,i)}_{n}( \bm{x}; \hslash) \coloneqq (-1)^{n-1} \sum_{\sigma \in S_n^{\textup{cyc}}} K_{\mp,\mp}(x_i,x_{\sigma(i)};\hslash) \prod_{j \neq i} K_{\pm,\mp}(x_j,x_{\sigma(j)};\hslash) \,,$$ where the kernels $K_{\pm,\pm}$ are given by $$K_{\pm,\pm}(x,y;\hslash) \coloneqq \frac{\widetilde{\psi}_{\pm}'(x;\hslash) \widetilde{\psi}_{\pm}(y;\hslash) - \widetilde{\psi}_{\pm}(x;\hslash) \widetilde{\psi}_{\pm}'(y;\hslash)}{x-y} \,.$$* *Moreover, the kernels are related by the parity relation $K_{-,-}(x,y;\hslash) = - K_{+,+}(x,y;-\hslash)$. Hence, the analogous relation holds for the correlators: $W^{(-,i)}_{n}( \bm{x}; \hslash) = (-1)^n \, W^{(+,i)}_{n}( \bm{x}; -\hslash)$.* *Proof.* The proof simply follows from the behaviour of the Borel transform of products, [Lemma 9](#lemma:conv){reference-type="ref" reference="lemma:conv"} and [Remark 11](#rem:princ:sheet){reference-type="ref" reference="rem:princ:sheet"}, together with the definition of the $n$-point function through the determinantal formula. The parity relations follow again from those satisfied by the Airy functions and their derivatives: $\psi_{-}(x;\hslash) = \psi_{+}(x;-\hslash)$ and $\psi_{-}'(x;\hslash) = - \psi_{+}'(x;-\hslash)$. ◻ **Remark 18** (Minors in matrix form and $\mathbb{Z}/2\mathbb{Z}$-symmetry). In matrix formulation, the minors are expressed as $$W^{(\pm,i)}_{n}( \bm{x}; \hslash) = (-1)^{n-1} \sum_{\sigma \in S_n^{\textup{cyc}}} \frac{\mathop{\mathrm{Tr}}{ \bigl( M_{\mp}(x_i;\hslash) \prod_{j=1}^{n-1} M(x_{\sigma^j(i)};\hslash) } \bigr) }{\prod_{j=1}^n ( x_{j} - x_{\sigma(j)} )} \,,$$ where $M$ and $M_{\pm}$ are given by $$M = \begin{pmatrix} \frac{1}{2}(\widetilde{\psi}_+' \widetilde{\psi}_- + \widetilde{\psi}_+ \widetilde{\psi}_-') & \widetilde{\psi}_+ \widetilde{\psi}_- \\ \widetilde{\psi}_+' \widetilde{\psi}_-' & - \frac{1}{2}(\widetilde{\psi}_+' \widetilde{\psi}_- + \widetilde{\psi}_+ \widetilde{\psi}_-') \end{pmatrix} , \qquad M_{\pm} = \begin{pmatrix} \widetilde{\psi}'_{\pm}\widetilde{\psi}_{\pm} & - ( \widetilde{\psi}_{\pm} )^2 \\ ( \widetilde{\psi}'_{\pm} )^2 & - \widetilde{\psi}'_{\pm}\widetilde{\psi}_{\pm} \end{pmatrix} .$$ From these expression, it is easy to show that $x_j^{1/2} W^{(\pm,i)}_{n}$ is an even function of $x_j^{1/2}$ for all $j \ne i$. This follows from the fact that $x^{1/4} \psi_{\pm}$ is mapped to $x^{1/4} \psi_{\mp}$ under $x^{1/2} \mapsto -x^{1/2}$, and similarly for the derivatives. Thus $x^{1/2} M$ is even in $x^{1/2}$. With a similar argument, $x_i^{1/2} W^{(\pm,i)}_{n}$ is mapped to $x_i^{1/2} W^{(\mp,i)}_{n}$ under $x_i^{1/2} \mapsto -x_i^{1/2}$. Since $W^{(\pm,i)}_{n}$ are related by the parity relation, we simply consider $W_n^{(+,i)}$ in what follows. To simplify the notation, we drop the '$+$' symbol from the superscript. We can now apply the Borel transform method to deduce the large genus asymptotics of the expansion coefficients of $W_n$ from the knowledge of its Borel singularity structure. **Proposition 19** (Large genus asymptotics of the Airy correlators). *For every $K$, the large genus asymptotics of the expansion coefficients of the $n$-point Airy correlators is given by $$\label{eqn:large:g:corr:WK} W_{g,n}(\bm{x}) = \frac{S}{\pi} \sum_{i=1}^n \frac{\Gamma(2g-2+n)}{A(x_i)^{2g-2+n}} \left( \sum_{k = 0}^K \frac{A(x_i)^{k}}{(2g-3+n)^{\underline{k}}} W^{(i)}_{k,n}(\bm{x}) + \mathrm{O}\biggl( \frac{1}{g^{K+1}} \biggr) \right) \,,$$ where:* - *$A(x) = \frac{4}{3} x^{3/2}$ and $S = 1$ are the instanton action and the Stokes constant associated to the Airy functions,* - *$W^{(i)}_{k,n}$ is the coefficient of $\hslash^k$ in the expansion of $W_n^{(i)}$.* *Moreover, the leading term is explicitly given by $$\label{eqn:large:g:corr:leading:WK} W_{g,n}(\bm{x}) = \frac{(-1)^n}{4\pi} \, \frac{\Gamma(2g-2+n)}{\bigl( \frac{4}{3} \bigr)^{2g-2+n}} \Biggl( \sum_{\substack{ d_1,\dots,d_n \ge 0 \\ |d| = 3g-3+n}} \prod_{i=1}^n \frac{1}{x_i^{d_i+3/2}} + \mathrm{O}\bigl( g^{-1} \bigr) \Biggr) \,.$$* *Proof.* As a consequence of the Borel transform method [Theorem 5](#thm:large:order){reference-type="ref" reference="thm:large:order"} (see also [Remark 7](#rem:Gevrey:2){reference-type="ref" reference="rem:Gevrey:2"}) and the proposition above, we find $$W_{g,n}(\bm{x}) = \frac{S}{2\pi} \sum_{i=1}^n \frac{\Gamma(2g-2+n)}{A(x_i)^{2g-2+n}} \left( \sum_{k = 0}^K \frac{A(x_i)^k}{(2g-3+n)^{\underline{k}}} \left( W^{(+,i)}_{k,n}(\bm{x}) + (-1)^k W^{(-,i)}_{k,n}(\bm{x}) \right) + \mathrm{O}\biggl( \frac{1}{g^{K+1}} \biggr) \right) .$$ The parity relation implies $W^{(-,i)}_{k,n}(\bm{x}) = (-1)^{n+k} \, W^{(+,i)}_{k,n}(\bm{x})$, hence the first part of the proposition. As for the leading term, it is determined by the coefficient of $\hslash^0$ in the kernels. From the definition of the kernels and the asymptotic expansion of the Airy functions, we see that the first term in the $\hslash$-expansion is given by $$K_{\pm,-}(x,y;\hslash) = \pm \frac{1}{2 (x y)^{1/4}} \frac{1}{x^{1/2} \mp y^{1/2}} + \mathrm{O}(\hslash) \,.$$ Inserting the above expressions in the formula for the $W_{n}^{(i)}$, we find $$\begin{split} W_{0,n}^{(i)}(\bm{x}) &= \frac{(-1)^n}{2^n \, (x_1 \cdots x_n)^{1/2}} \sum_{\sigma \in S_n^{\textup{cyc}}} \frac{1}{x_i^{1/2} + x_{\sigma(i)}^{1/2}} \prod_{j \ne i} \frac{1}{x_j^{1/2} - x_{\sigma(j)}^{1/2}} = - \frac{1}{4} \frac{1}{(x_1 \cdots x_n)^{1/2}} \frac{x_i^{n/2 - 1}}{ \prod_{i \ne j} (x_i - x_j) } \,. \end{split}$$ In the last equation, we applied [Lemma 38](#tech:lemma){reference-type="ref" reference="tech:lemma"}. As a consequence, recalling that $S = 1$ and $A(x_i) = \frac{4}{3} x^{3/2}$, we find $$W_{g,n}(\bm{x}) = -\frac{1}{4\pi} \, \frac{\Gamma(2g-2+n)}{\bigl( \frac{4}{3} \bigr)^{2g-2+n}} \Biggl( \frac{1}{(x_1 \cdots x_n)^{1/2}} \sum_{i=1}^n \frac{x_i^{-(3g-3+n)-1}}{ \prod_{i \ne j} (x_i - x_j) } + \mathrm{O}\bigl( g^{-1} \bigr) \Biggr) \,.$$ Using the relation $\sum_{i=1}^n \frac{x_{i}^{-D-1}}{\prod_{i \ne j} (x_i - x_j)} = \frac{(-1)^{n-1}}{x_1 \cdots x_n} h_D(\bm{x}^{-1})$, with $h_D$ being the complete homogeneous symmetric polynomial of degree $D$, we find the thesis. This relation is a special case of [Lemma 39](#lemma:poly){reference-type="ref" reference="lemma:poly"}, but it can be easily proved by taking the generating series on both sides. ◻ As a consequence, we obtain the leading large genus asymptotics of $\psi$-class intersection numbers. **Corollary 20** (Leading large genus asymptotics for $\psi$-class intersection numbers). *For any given $n \ge 1$, uniformly in $d_1,\dots,d_n$ as $g \to \infty$: $$\braket{\tau_{d_1} \cdots \tau_{d_n}} \prod_{i=1}^n (2d_i + 1)!! \\ = \frac{2^n}{4\pi} \frac{\Gamma(2g-2+n)}{( \frac{2}{3} )^{2g-2+n}} \Bigl( 1 + \mathrm{O}\bigl( g^{-1} \bigr) \Bigr) \,.$$* The most challenging aspect of proving the corollary lies in the uniformity statement. We postpone the proof to the more general [Theorem 24](#thm:large:g:WK){reference-type="ref" reference="thm:large:g:WK"}, which accounts for all subleading corrections. In fact, the next subsection is devoted to the study of the subleading terms $W_{k,n}^{(i)}$ for arbitrary $k$. ### Subleading contributions From [\[eqn:large:g:corr:leading:WK\]](#eqn:large:g:corr:leading:WK){reference-type="ref" reference="eqn:large:g:corr:leading:WK"}, it is clear that both $W_{g,n}$ and its leading large genus behaviour are polynomials in $x_1^{-1},\dots,x_n^{-1}$ of degree $3g-3+n$ (up to a prefactor of $(x_1 \cdots x_n)^{-3/2}$). It is natural to ask whether the same property holds for the subleading corrections. More precisely, from the analysis of the kernel we see that the leading term $W_{0,n}^{(i)}$ has, a priori, simple poles along $\sqrt{x_i} = - \sqrt{x_j}$ for all $j \ne i$ and along $\sqrt{x_k} = \sqrt{x_l}$ for all $k \ne l$. However, the only poles appearing are those at $\sqrt{x_i} = \pm \sqrt{x_j}$. Furthermore, after summing over all contributions, the poles remarkably disappear, yielding a polynomial of the expected form. The next lemmas show that a similar structure holds for all subleading terms. **Lemma 21** (Pole structure of the minors). *The minor $W_n^{(i)}$ is regular along $\sqrt{x_k} = \sqrt{x_l}$ for all $k \ne l$ with $k,l \ne i$. Moreover, it has simple poles at $\sqrt{x_i} = \pm \sqrt{x_j}$ for all $j \neq i$, with residue $$\mathop{\mathrm{Res}}_{\sqrt{x_j} \to \pm \sqrt{x_i}} W_n^{(i)}(x_1,\dots,x_n;\hslash) = - \frac{1}{2 \sqrt{x_i}} \, W_{n-1}^{(i)}(x_1,\dots,\widehat{\,x_j\,},\dots,x_{n};\hslash) \, .$$ On the right-hand side, the superscript $(i)$ means that the "special variable" is $x_i$.* *Proof.* Throughout the proof, we denote $z_m = \sqrt{x_m}$. Consider the poles at $z_k = z_l$ for all $k \ne l$ with $k,l \ne i$. The only terms in the definition of $W_n^{(i)}$ contributing to the residue are those permutations $\sigma \in S_n^{\textup{cyc}}$ such that $\sigma(k) = l$ or $\sigma(l) = k$. If both $k$ and $l$ are different from $i$, we find $$\begin{gathered} \mathop{\mathrm{Res}}_{z_k \to z_l} W_n^{(i)}(\bm{x};\hslash) = (-1)^{n-1} \mathop{\mathrm{Res}}_{z_k \to z_l} \Biggl( \sum_{\substack{\sigma \in S_n^{\textup{cyc}} \\ \sigma(k) = l}} K_{-,-}(x_i,x_{\sigma(i)};\hslash) K_{+,-}(x_k,x_l;\hslash) \prod_{j \ne i,k} K_{+,-}(x_j,x_{\sigma(j)};\hslash) \\ + \sum_{\substack{\sigma \in S_n^{\textup{cyc}} \\ \sigma(l) = k}} K_{-,-}(x_i,x_{\sigma(i)};\hslash) K_{+,-}(x_l,x_k;\hslash) \prod_{j \ne i,l} K_{+,-}(x_j,x_{\sigma(j)};\hslash) \Biggr) . \end{gathered}$$ The pole at $z_k \to z_l$ is given by $K_{+,-}(x_k,x_l;\hslash)$ in the first sum and by $K_{+,-}(x_l,x_k;\hslash)$ in the second one. Moreover, the residues are $$\mathop{\mathrm{Res}}_{z_k \to z_l} K_{+,-}(x_k,x_l;\hslash) = \frac{1}{2z_l} \,, \qquad\qquad \mathop{\mathrm{Res}}_{z_k \to z_l} K_{+,-}(x_l,x_k;\hslash) = - \frac{1}{2z_l} \,,$$ since they coincide (up to a sign) with the Wronskian of the differential system. After taking the residue, the two terms cancel out. This proves the first part of the statement. For the second part, assume without loss of generality that $j = n$ (and $i < n$). Consider the pole at $z_n = z_i$. With the same computation as before, we find $$\begin{gathered} \mathop{\mathrm{Res}}_{z_n \to z_i} W_n^{(i)}(\bm{x};\hslash) = (-1)^{n-1} \mathop{\mathrm{Res}}_{z_n \to z_i} \Biggl( \sum_{\substack{\sigma \in S_n^{\textup{cyc}} \\ \sigma(n) = i}} K_{-,-}(x_i,x_{\sigma(i)};\hslash) K_{+,-}(x_n,x_i;\hslash) \prod_{j \ne i,n} K_{+,-}(x_j,x_{\sigma(j)};\hslash) \\ + \sum_{\substack{\sigma \in S_n^{\textup{cyc}} \\ \sigma(i) = n}} K_{-,-}(x_i,x_n;\hslash) \prod_{j \ne i} K_{+,-}(x_j,x_{\sigma(j)};\hslash) \Biggr) . \end{gathered}$$ Now the residues from the two sums give different contributions: $$\mathop{\mathrm{Res}}_{z_n \to z_i} K_{+,-}(x_n,x_i;\hslash) = \frac{1}{2z_i} \,, \qquad\qquad \mathop{\mathrm{Res}}_{z_n \to z_i} K_{-,-}(x_i,x_n;\hslash) = 0 \,.$$ The first residue is again the Wronskian, while the second one is simply a cancellation of the numerator of $K_{-,-}$ in the limit. Thus, the second sum gives no contribution, while the first sum can be re-written as a sum over $S_{n-1}^{\textup{cyc}}$ recovering $- W_{n-1}^{(i)}$. Finally, consider the pole at $z_n = - z_i$. With the same computation as before, but with the residue contributions $$\mathop{\mathrm{Res}}_{z_n \to -z_i} K_{+,-}(x_n,x_i;\hslash) = 0 \,, \qquad\qquad \mathop{\mathrm{Res}}_{z_n \to -z_i} K_{-,-}(x_i,x_n;\hslash) = \frac{\mathrm{i}}{2z_i} \,.$$ and the property $K_{+,-}(x_n,x_{\sigma(n)};\hslash) \big|_{z_n \mapsto -z_i} = - \mathrm{i}\, K_{-,-}(x_i,x_{\sigma(n)};\hslash)$, we conclude the proof. ◻ We now turn our attention to the $k$-th subleading term in the asymptotic expansion of the $n$-point Airy correlators. For simplicity, denote $$U_{k,g,n}(\bm{x}) \coloneqq (-1)^n \, 2^{k+2} \, (x_1 \cdots x_n)^{3/2} \sum_{i=1}^n x_i^{-\frac{3}{2}(2g-2+n-k)} \, W_{k,n}^{(i)}(\bm{x}) \,.$$ With this notation, [\[eqn:large:g:corr:WK\]](#eqn:large:g:corr:WK){reference-type="eqref" reference="eqn:large:g:corr:WK"} reads $$\label{eqn:large:g:corr:Y} W_{g,n}(\bm{x}) = \frac{(-1)^n}{4\pi} \frac{\Gamma(2g-2+n)}{(\frac{4}{3})^{2g-2+n}} \left( \sum_{k = 0}^K \frac{(\frac{2}{3})^{k}}{(2g-3+n)^{\underline{k}}} \frac{U_{k,g,n}(\bm{x})}{(x_1 \cdots x_n)^{3/2}} + \mathrm{O}\biggl( \frac{1}{g^{K+1}} \biggr) \right) \,.$$ is equivalent to $U_{0,g,n}(\bm{x}) = h_{3g-3+n}(\bm{x}^{-1})$, i.e. the leading term is the complete homogeneous symmetric polynomial of degree $3g-3+n$ in the variables $x_i^{-1}$. The next lemma extends the result to subleading corrections, showing that the genus dependence is completely captured by a complete homogeneous polynomial of degree $3g-3+n$ minus a small defect which depends on $k$. We refer to [5.5](#app:symmetric:fncts){reference-type="ref" reference="app:symmetric:fncts"} and [@Mac98] for more details and notations on symmetric functions. **Lemma 22** (Polynomial properties of subleading corrections). *For every $k \ge 0$, $n \ge 1$ and $g$ large enough, $U_{k,g,n}$ is a homogeneous symmetric polynomial in $x_i^{-1}$ of degree $3g-3+n$ of the form $$\label{eqn:subleading:poly} U_{k,g,n}(\bm{x}) = \sum_{d=0}^{\min\{ \floor{\frac{3k+n-1}{2}},\, 3k-1 \}} P_{k,n}^{(d)}(\bm{x}^{-1/2}) \, h_{3g-3+n-d}(\bm{x}^{-1}) \,,$$ where $P_{k,n}^{(d)}$ is a homogeneous symmetric polynomial of degree $2d$, both even and of degree $\le 3k$ in each individual variable. The sequence of polynomials $(P_{k,n})_{n \ge 1}$, with[^3] $$P_{k,n} \coloneqq \sum_{d=0}^{\min\{ \floor{\frac{3k+n-1}{2}},\, 3k-1 \}} P_{k,n}^{(d)} \,,$$ satisfies the following two specialisation properties: $$\label{eqn:spec} P_{k,n+1}(u_1,\dots, u_n, u_{n+1}) \big|_{u_{n+1} = \pm 1} = P_{k,n}(u_1,\dots,u_n) \,.$$ In turn, the degree condition and the specialisation properties uniquely determine $P_{k,n}$ for all $n$ by an explicit algorithm from the first terms of the sequence, namely $P_{k,1},\dots, P_{k,3k-1}$. Moreover, the coefficient of $m_{2\nu}$ in the expansion of $P_{k,n}$ in the monomial basis is a polynomial of $n$ of degree $\le 3k-1 - |\nu|$.* $k$ $P_{k,n}$ $\alpha_k(n,\bm{p})$ ----- ------------------------------------------------------------------------- ------------------------------------------------------------------- $0$ $1$ $1$ $1$ $- \tfrac{17 - 15n + 3n^2}{12} m_{\varnothing} $- \tfrac{17 - 15n + 3n^2}{12} - \tfrac{3 - n}{2} \, m_{(1)} - \tfrac{(3 - n)(n - p_0)}{2} - \tfrac{1}{2} \, m_{(1^2)}$ - \tfrac{(n - p_0)^{\underline{2}}}{4}$ $2$ $\tfrac{1225 - 1632 n + 741 n^2 - 138 n^3 + 9 n^4}{288} m_{\varnothing} $\tfrac{1225 - 1632 n + 741 n^2 - 138 n^3 + 9 n^4}{288} + + \tfrac{105 - 98 n + 30 n^2 - 3 n^3}{24} m_{(1)} \tfrac{(105 - 98 n + 30 n^2 - 3 n^3)(n - p_0)}{24} + + \tfrac{3(10 - 7 n + n^2)}{8} m_{(2)} \tfrac{3(10 - 7 n + n^2)(n - p_0 - p_1)}{8} + + \tfrac{59 - 51 n + 9 n^2}{24} m_{(1^2)} \tfrac{(59 - 51 n + 9 n^2)(n - p_0)^{\underline{2}}}{48} + + \tfrac{5}{8} m_{(3)} \tfrac{5(n - p_0 - p_1 - p_2)}{8} + + \tfrac{3(4 - n)}{4} m_{(2,1)} \tfrac{3(4 - n)(n - p_0 - 1)(n - p_0 - p_1)}{4} + + \tfrac{7 - 3 n}{4} m_{(1^3)} \tfrac{(7 - 3 n)(n - p_0)^{\underline{3}}}{24} + + \tfrac{3}{4} m_{(2,1^2)} \tfrac{3(n - p_0 - 1)^{\underline{2}}(n - p_0 - p_1)}{48} + + \tfrac{3}{4} m_{(1^4)}$ \tfrac{3(n - p_0)^{\underline{4}}}{96}$ : The polynomials $P_{k,n}(u_1,\dots,u_n)$ and the coefficients $\alpha_k$ for $k = 0,1,2$. Here $m_{\lambda}$ denotes the monomial symmetric polynomial in the variables $u_1^2,\dots,u_n^2$. As consequence of the above lemma, we get a polynomiality statement for the coefficients of $U_{k,g,n}$ in terms of $n$ and the multiplicities of the exponents. **Corollary 23** (Polynomial properties of subleading corrections: the coefficients). *For every $k \ge 0$, $n \ge 1$, $g$ large enough, and $d_1, \dots, d_n \ge 0$ satisfying $d_1 + \dots + d_n = 3g-3+n$, the coefficients of $U_{k,g,n}$ are of the form $$\label{eqn:subleading:coeffs} \bigl[ x_1^{-d_1} \cdots x_n^{-d_n} \bigr] \, U_{k,g,n}(\bm{x}) = \alpha_k\bigl( n, p_0, \dots, p_{\floor{\frac{3k}{2}}-1} \bigr) \,,$$ where $\alpha_k$ is a polynomial in $n$ and $p_m \coloneqq \#\set{ d_i = m}$ for $m = 1, \dots, \floor{\frac{3k}{2}}-1$ that can be effectively computed. Moreover, the degree is $\le 3k-1$, under the assignment $\deg{n} = 1$ and $\deg{p_m} = m+1$.* The first few polynomials $P_{k,n}$ and coefficients $\alpha_k$ are given in [3](#table:Pkn:ck){reference-type="ref" reference="table:Pkn:ck"}. The intuitive explanation of the above statement can be given as follows. shows that the $k$-th subleading correction in the large genus asymptotics of the correlators is proportional to $$\sum_{i=1}^n x_i^{-\frac{3}{2}(2g-2+n-k)} \, W_{k,n}^{(i)} \,.$$ In the large genus limit, most of the degree is then coming from the factors $x_i^{-\frac{3}{2}(2g-2+n)}$, which in turn give rise to complete homogeneous polynomials of large degree times some polynomials of small degree. Thus, the extraction of coefficients depends only on the number of variables with a small degree, as all monomials appear in the complete homogeneous polynomial with coefficient $1$. We also emphasise that [Lemma 22](#lemma:poly:subleading){reference-type="ref" reference="lemma:poly:subleading"} gives a concrete algorithm to compute the subleading corrections. Indeed the proof shows that, for fixed $k$, the computation of $P_{k,n}$ for $n \le 3k-1$ gives a closed, explicit expression for $P_{k,n}$ for arbitrary $n$. In turn, this gives a closed, explicit expression for the coefficients $\alpha_k$, see [\[fig:algorithm\]](#fig:algorithm){reference-type="ref" reference="fig:algorithm"}. *Proof of [Lemma 22](#lemma:poly:subleading){reference-type="ref" reference="lemma:poly:subleading"}.* We first claim that the functions $W_{k,n}^{(i)} = [\hslash^k]W_{n}^{(i)}$ are of the form $$W_{k,n}^{(i)}(\bm{x}) = - \frac{2^{-k-2}}{(x_1 \cdots x_n)^{1/2}} \frac{x_i^{n/2 - 1 - 3k/2}}{\prod_{j \ne i} (x_i - x_j)} \, P_{k,n-1} \biggl( \Bigl( \frac{x_i}{x_1} \Bigr)^{1/2}, \dots, \widehat{ \Bigl( \frac{x_i}{x_i} \Bigr)^{1/2} }, \dots, \Bigl( \frac{x_i}{x_n} \Bigr)^{1/2} \biggr) \,,$$ where $P_{k,n-1}$ is a symmetric polynomial in $n-1$ variables of total degree $\le \min\set{ 3k + n - 2, 2(3k - 1) }$, both even and of degree $\le 3k$ in each individual variable, and satisfying the two specialisation properties $P_{k,n+1}( \,\cdot\,, \pm 1) = P_{k,n}(\,\cdot\,)$. Indeed, from the homogeneity property and the parity relation satisfied by the Airy functions, i.e. $\psi_{\pm}(x;\hslash) = x^{-1/4} \, \psi_{\pm}(1;\frac{\hslash}{2A(x)})$, $\psi_{+}(x;\hslash) = \psi_{-}(x;-\hslash)$, and similarly for the derivatives, we deduce that $$K_{\pm,-}(x,y;\hslash) = \pm \frac{1}{(x y)^{1/4}} \frac{1}{x^{1/2} \mp y^{1/2}} \sum_{m \ge 0} A_m\bigl( \mp x^{-1/2},y^{-1/2} \bigr) \hslash^m \,,$$ where $A_m$ is a homogeneous symmetric polynomial of degree $3m$. From [Lemma 21](#lemma:poles:minors){reference-type="ref" reference="lemma:poles:minors"} and a degree consideration, we conclude that $$(x_1 \cdots x_n)^{1/2} \, x_i^{3k/2 - n/2 + 1} \Biggl( \prod_{j \ne i} (x_i - x_j) \Biggr) W_{k,n}^{(i)}(\bm{x})$$ is a symmetric polynomial in $t_j = \sqrt{x_i}/\sqrt{x_j}$ for $j \neq i$ of degree $\le 3k + n - 2$. In the $t_j$ variables the above polynomial is given by $$(-1)^n \prod_{j \neq i} (1 - t_j^2) \sum_{m \neq i} \sum_{\substack{k_1 + \dots + k_{n} = k \\ \sigma \in S_n^{\textup{cyc}}, \, \sigma(m) = i}} \frac{A_{k_i}(1, t_{\sigma(i)})}{1 + t_{\sigma(i)}} \frac{A_{k_m}(-t_{m},1)}{t_{m} - 1} \prod_{\substack{j \ne i, \, m}} \frac{A_{k_j}(-t_j, t_{\sigma(j)})}{t_j - t_{\sigma(j)}} \,.$$ The polynomiality and the specialisation properties, which are not obvious from the above expression, are guaranteed by [Lemma 21](#lemma:poles:minors){reference-type="ref" reference="lemma:poles:minors"}. It is clear that the degree in each individual variable is $\le 3k$. Moreover, from [Remark 18](#rem:minors:Z2symmetry){reference-type="ref" reference="rem:minors:Z2symmetry"} we deduce that the above expression is even in $t_j$. Thus, after normalising by $-2^{-k-2}$, we have that $$W_{k,n}^{(i)}(\bm{x}) = - \frac{2^{-k-2}}{(x_1 \cdots x_n)^{1/2}} \frac{x_i^{n/2 - 1 - 3k/2}}{\prod_{j \ne i} (x_i - x_j)} \, P_{k,n-1} \biggl( \Bigl( \frac{x_i}{x_1} \Bigr)^{1/2}, \dots, \widehat{ \Bigl( \frac{x_i}{x_i} \Bigr)^{1/2} }, \dots, \Bigl( \frac{x_i}{x_n} \Bigr)^{1/2} \biggr) \,,$$ where $P_{k,n-1}$ is a symmetric polynomial of total degree $\le 3k + n - 2$, both even and of degree $\le 3k$ in each individual variable. Our next goal is to prove that there exists a value $N_k$ such that $$\deg(P_{k,n+1}) = \deg(P_{k,n}) \qquad\quad \text{ for all } n \geq N_k \,.$$ To this end, consider for $P_{k,n}(u_1, \dots, u_n)$ the two changes of variables (which naturally follows from the specialisation properties) $$\hat{u}_j \coloneqq u_j - 1 \,, \quad \hat{e}_s \coloneqq e_s(\hat{u}_1, \dots, \hat{u}_n) \,, \qquad\qquad \check{u}_j \coloneqq u_j + 1 \,, \quad \check{e}_s \coloneqq e_s(\check{u}_1, \dots, \check{u}_n) \,.$$ We also refer to $\hat{P}_{k,n}$ (resp. $\check{P}_{k,n}$) as to $P_{k,n}$ expanded in the basis of elementary symmetric polynomials $\hat{e}_s$ (resp. $\check{e}_s$). From the specialisation properties and the fact that setting any of the $\hat{u}_j$ (resp. $\check{u}_j$) to zero is the same as setting $\hat{e}_{n+1}$ (resp. $\check{e}_{n+1}$) to zero, we obtain the two decompositions $$\label{eqn:decomp} \hat{P}_{k,n+1} = \hat{P}_{k,n} + \hat{e}_{n+1}\hat{Q}_{k,n} \,, \qquad \qquad \check{P}_{k,n+1} = \check{P}_{k,n} + \check{e}_{n+1}\check{Q}_{k,n} \,,$$ for some $\hat{Q}_{k,n}$ and $\check{Q}_{k,n}$, polynomials in $\hat{e}_s$ and $\check{e}_s$ respectively. With the natural degree assignment $\deg{\hat{e}_s} = \deg{\check{e}_s} = s$, we find $\deg(\hat{Q}_{k,n}) = \deg(\check{Q}_{k,n}) \leq 3k - 1$. Define the linear unitriangular ring isomorphism $\mathsf{T}_n$ performing the change of basis from $\hat{e}$ to $\check{e}$: $$\begin{aligned} \mathsf{T}_n \colon \mathbb{Q}[\hat{e}] \longrightarrow \mathbb{Q}[\check{e}] \,, \qquad \hat{e}_s \longmapsto \sum_{m=0}^s \binom{n-s+m}{m} (-2)^m \check{e}_{s-m} \,. \end{aligned}$$ The factor $-2$ arises from the difference $\hat{u}_j - \check{u}_j$. Applying $\mathsf{T}_{n+1}$ to the first decomposition of [\[eqn:decomp\]](#eqn:decomp){reference-type="ref" reference="eqn:decomp"} we obtain $$\check{P}_{k,n+1} - \mathsf{T}_{n+1}(\hat{P}_{k,n}) = \mathsf{T}_{n+1}(\hat{e}_{n+1} \hat{Q}_{k,n})\,.$$ Here we used the fact that $\mathsf{T}_{n+1}(\hat{P}_{k,n+1}) = \check{P}_{k,n+1}$. However, $\mathsf{T}_{n+1}(\hat{P}_{k,n})$ does not necessarily coincide with $\check{P}_{k,n}$, as the change of basis depends on the number of variables. Adding and subtracting $\check{P}_{k,n}$ to the above equation and applying the second decomposition of [\[eqn:decomp\]](#eqn:decomp){reference-type="ref" reference="eqn:decomp"} yields $$\check{P}_{k,n} - \mathsf{T}_{n+1}(\hat{P}_{k,n}) - \mathsf{T}_{n+1}(\hat{e}_{n+1}) \mathsf{T}_{n+1}(\hat{Q}_{k,n}) = \check{e}_{n+1}\check{Q}_{k,n} \,.$$ Notice that, as the right-hand side is divisible by $\check{e}_{n+1}$, so does the left-hand side. The only factors that (might) contain $\check{e}_{n+1}$ are $\mathsf{T}_{n+1}(\hat{e}_{n+1})$ and $\mathsf{T}_{n+1}(\hat{Q}_{k,n})$. However, for $n \geq 3k - 1$ there are no such terms in $\mathsf{T}_{n+1}(\hat{Q}_{k,n})$ for degree reasons. Therefore, after removing the leading term from $\mathsf{T}_{n+1}(\hat{e}_{n+1})$, we find $$\check{P}_{k,n} - \mathsf{T}_{n+1}(\hat{P}_{k,n}) = \left(\mathsf{T}_{n+1}(\hat{e}_{n+1}) - \check{e}_{n+1}\right) \mathsf{T}_{n+1}(\hat{Q}_{k,n}) \,.$$ By the unitriagularity of the operator $\mathsf{T}_n$, the degree of $\check{P}_{k,n} - \mathsf{T}_{n+1}(\hat{P}_{k,n})$ is lower by one compared to the degree of $\check{P}_{k,n}$, as the leading terms cancel each other out. Explicitly, we obtain that $\deg(\check{P}_{k,n} - \mathsf{T}_{n+1}(\hat{P}_{k,n})) \leq 3k + n - 2,$ so that $\deg(\hat{Q}_{k,n}) \leq 3k - 2$, which in turns forces $$\deg(P_{k,n+1}) = \deg(P_{k,n}) \qquad \text{for } n \geq 3k-1 \,.$$ In other words, we proved that for $n \ge 3k-1$ the degree of $P_{k,n}$ is independent on $n$, which in turn implies that $\deg{( P_{k,n} )} \le \min\{ 3k + n - 1, 2(3k - 1) \}$. Being even in all variables, we find that $\deg{( P_{k,n} )} \le 2 \min\{ \floor{\frac{3k + n - 1}{2}}, 3k - 1 \}$[^4]. From the above proof it follows that, for any fixed $k$, the first few values of $n$ determine the whole sequence $(P_{k,n})_{n \ge 1}$. Moreover, the coefficients of $P_{k,n}$ in the monomial basis are polynomials of $n$. Indeed, as the degree of $P_{k,n}$ is constant in $n$ for $n$ large enough, we deduce from [\[eqn:decomp\]](#eqn:decomp){reference-type="ref" reference="eqn:decomp"} that eventually $\hat{P}_{k,n+1} = \hat{P}_{k,n}$. The change of basis from $\hat{e}_s = e_s(u_1-1,\dots,u_n-1)$ to $e_s = e_s(u_1,\dots,u_n)$, which is analogous to the transformation $\mathsf{T}_n$, depends polynomially in $n$. Furthermore, the change of basis from elementary to complete homogeneous polynomials is independent of $n$ (it involves Kostka numbers, see [@Mac98]). Hence, the coefficients of $P_{k,n}$ in the monomial basis are polynomials of $n$. As for the number of polynomials $P_{k,n}$ that determine the whole sequence, one can show that $P_{k,1}, \dots, P_{k,3k - 1}$ are sufficient by degree considerations and the parity property. Moreover, as polynomials of $n$, it can be shown that the degree of $m_{2\nu}$ is bounded by $3k-1-|\nu|$ by the explicit change of basis from $\hat{e}$ to $e$. We now proceed with the proof of [\[eqn:subleading:poly\]](#eqn:subleading:poly){reference-type="ref" reference="eqn:subleading:poly"} for $U_{k,g,n}$. From the definition of $U_{k,g,n}$ and the definition of $P_{k,n-1}$, we find $$U_{k,g,n}(\bm{x}) = (-1)^{n-1} \, x_1 \cdots x_n \sum_{i=1}^n \frac{ x_i^{-(3g-3+n)-1} }{ \prod_{j \ne i} (x_i - x_j) } \, P_{k,n-1} \biggl( \Bigl( \frac{x_i}{x_1} \Bigr)^{1/2}, \dots, \widehat{ \Bigl( \frac{x_i}{x_i} \Bigr)^{1/2} }, \dots, \Bigl( \frac{x_i}{x_n} \Bigr)^{1/2} \biggr) \,.$$ We claim that, for $g$ large enough, $$\begin{gathered} \sum_{i=1}^n \frac{ x_i^{-(3g-3+n)-1} }{ \prod_{j \ne i} (x_i - x_j) } \, P_{k,n-1} \biggl( \Bigl( \frac{x_i}{x_1} \Bigr)^{1/2}, \dots, \widehat{ \Bigl( \frac{x_i}{x_i} \Bigr)^{1/2} }, \dots, \Bigl( \frac{x_i}{x_n} \Bigr)^{1/2} \biggr) \\ = \frac{(-1)^{n-1}}{x_1 \cdots x_n} \sum_{d=0}^{\min\{ \floor{\frac{3k+n-1}{2}},\, 3k-1 \}} P_{k,n}^{(d)} ( \bm{x}^{-1/2} ) \, h_{3g-3+n-d} ( \bm{x}^{-1} )\,, \end{gathered}$$ where $P_{k,n}^{(d)}$ is the homogeneous component of degree $2d$ in $P_{k,n}$ (notice the shift in $n$). This would complete the proof of the lemma. In order to prove the claimed formula, consider the function $$f(q) \coloneqq \frac{P_{k,n}(\frac{q}{\sqrt{x_1}}, \dots, \frac{q}{\sqrt{x_n}})}{\prod_{i=1}^n (q^2 - x_i)} \,.$$ We omit its dependence on $x_1^{1/2},\dots,x_n^{1/2}$ for simplicity. As a function of $q$, it has simple poles at $q = \pm \sqrt{x_i}$ with residues $$\mathop{\mathrm{Res}}_{q = \pm\sqrt{x_i}} f(q) = \frac{P_{k,n}\bigl( \pm \frac{\sqrt{x_i}}{\sqrt{x_1}}, \dots, \pm \frac{\sqrt{x_i}}{\sqrt{x_i}}, \dots, \pm \frac{\sqrt{x_i}}{\sqrt{x_n}} \bigr)}{\pm 2\sqrt{x_i} \prod_{j \ne i} (x_i - x_j)} = \pm \frac{P_{k,n-1}\bigl( \frac{\sqrt{x_i}}{\sqrt{x_1}}, \dots, \widehat{ \frac{\sqrt{x_i}}{\sqrt{x_i}} }, \dots, \frac{\sqrt{x_i}}{\sqrt{x_n}} \bigr)}{2\sqrt{x_i} \prod_{j \ne i} (x_i - x_j)} \,.$$ Here we used the fact that $P_{k,n}$ is even and it satisfies $P_{k,n}( \,\cdot\,, \pm 1) = P_{k,n-1}(\,\cdot\,)$. As a consequence, we obtain $$\begin{split} \bigl[ q^{6g-6+2n} \bigr] f(q) & = \bigl[ q^{6g-6+2n} \bigr] \sum_{i=1}^n \left( \frac{P_{k,n-1}\bigl( \frac{\sqrt{x_i}}{\sqrt{x_1}}, \dots, \widehat{ \frac{\sqrt{x_i}}{\sqrt{x_i}} }, \dots, \frac{\sqrt{x_i}}{\sqrt{x_n}} \bigr)}{(q - \sqrt{x_i}) \, 2\sqrt{x_i} \prod_{j \ne i} (x_i - x_j)} - \frac{P_{k,n-1}\bigl( \frac{\sqrt{x_i}}{\sqrt{x_1}}, \dots, \widehat{ \frac{\sqrt{x_i}}{\sqrt{x_i}} }, \dots, \frac{\sqrt{x_i}}{\sqrt{x_n}} \bigr)}{(q + \sqrt{x_i}) \, 2\sqrt{x_i} \prod_{j \ne i} (x_i - x_j)} \right) \\ & = \bigl[ q^{6g-6+2n} \bigr] \sum_{i=1}^n \frac{1}{q^2 - x_i} \frac{P_{k,n-1}\bigl( \frac{\sqrt{x_i}}{\sqrt{x_1}}, \dots, \widehat{ \frac{\sqrt{x_i}}{\sqrt{x_i}} }, \dots, \frac{\sqrt{x_i}}{\sqrt{x_n}} \bigr)}{\prod_{j \ne i} (x_i - x_j)} \\ & = - \sum_{i=1}^n \frac{x_i^{-(3g-3+n)-1}}{\prod_{j \ne i} (x_i - x_j)} \, P_{k,n-1} \biggl( \Bigl( \frac{x_i}{x_1} \Bigr)^{1/2}, \dots, \widehat{ \Bigl( \frac{x_i}{x_i} \Bigr)^{1/2} }, \dots, \Bigl( \frac{x_i}{x_n} \Bigr)^{1/2} \biggr) \, . \end{split}$$ On the other hand, extracting the coefficient of $q^{6g-6+2n}$ from the definition of $f$ gives $$\bigl[ q^{6g-6+2n} \bigr] f(q) = \frac{(-1)^{n}}{x_1 \cdots x_n} \sum_{d=0}^{\min\{ \floor{\frac{3k+n-1}{2}},\, 3k-1 \}} P_{k,n}^{(d)} ( \bm{x}^{-1/2} ) \, h_{3g-3+n-d} ( \bm{x}^{-1} ) \,.$$ This completes the proof. ◻ *Proof of [Corollary 23](#cor:poly:subleading){reference-type="ref" reference="cor:poly:subleading"}.* From [Lemma 22](#lemma:poly:subleading){reference-type="ref" reference="lemma:poly:subleading"}, we see that $P_{k,n}$ in the monomial basis reads $$P_{k,n}(u_1,\dots,u_n) = \sum_{\substack{ |\nu| \le \min\{ \floor{\frac{3k+n-1}{2}},\, 3k-1 \} \\ \nu_1 \le \floor{\frac{3k}{2}},\, \ell(\nu) \le n }} C_{k,n,\nu} m_{\mu}(\bm{u}^2) \,.$$ The condition $|\nu| \le \min\{ \floor{\frac{3k+n-1}{2}},\, 3k-1 \}$ follows from the bound on the total degree, while $\nu_1 \le \floor{\frac{3k}{2}}$ follows from the bound on the degrees in each individual variable. Notice that we also employed the parity condition, together with the property $m_{2\nu}(\bm{u}) = m_{\nu}(\bm{u}^2)$, so that only monomial symmetric polynomials in the squared variables appear. Moreover, we know that $C_{k,n,\nu}$ is an (explicit) polynomial of $n$ of degree $\le 3k-1-|\nu|$. Inserting the above expansion in the formula for $U_{k,g,n}$, we obtain: $$U_{k,g,n}(\bm{x}) = \sum_{\substack{ |\nu| \le \min\{ \floor{\frac{3k+n-1}{2}},\, 3k-1 \} \\ \nu_1 \le \floor{\frac{3k}{2}} ,\, \ell(\nu) \le n }} C_{k,n,\nu} \, m_{\nu}(\bm{x}^{-1}) \, h_{3g-3+n-|\nu|}(\bm{x}^{-1}) \,.$$ Since we are interested in extracting coefficients, it is natural to decompose $U_{k,g,n}$ in the basis of monomial symmetric polynomials. For products of the form $m_{\nu} h_{D - |\nu|}$, this is discussed in [5.5](#app:symmetric:fncts){reference-type="ref" reference="app:symmetric:fncts"}: $$m_{\nu} \, h_{D - |\nu|} = \sum_{|\mu| = D} M_{n,\mu,\nu} \, m_{\mu} \,,$$ where $M_{n,\mu,\nu}$ are (explicit) polynomials of $n$ and $p_m = \#\set{ \mu_i = m}$ for $m = 0,\dots, \nu_1 - 1$. For a fixed tuple $d_1, \dots, d_n \ge 0$ satisfying $|d| = 3g-3+n$, consider the associated partition $\mu$ (since $U_{k,g,n}$ is symmetric, the order does not matter). Then $$\alpha_k\bigl( n, p_0, \dots, p_{\floor{\frac{3k}{2}}-1} \bigr) = \bigl[ x_1^{-d_1} \cdots x_n^{-d_n} \bigr] \, U_{k,g,n}(\bm{x}) = \sum_{\substack{ |\nu| \le \min\{ \floor{\frac{3k+n-1}{2}},\, 3k-1 \} \\ \nu_1 \le \floor{\frac{3k}{2}} ,\, \ell(\nu) \le n} } C_{k,n,\nu} \, M_{n,\mu,\nu}$$ is a polynomial of $n$ and $p_m = \#\set{ d_i = m}$ for $m = 1, \dots, \floor{\frac{3k}{2}}-1$. This concludes the proof. ◻ The three main ingredients that entered in the proof of the above results are: 1. the Wronskian equation: $\psi'_+ \psi_- - \psi_+ \psi_-' = 1$; 2. the homogeneity property and the parity relation satisfied by the formal Airy functions: $\psi_{\pm}(x;\hslash) = x^{-1/4} \, \psi_{\pm}(1;\frac{\hslash}{2A(x)})$, $\psi_{+}(x;\hslash) = \psi_{-}(x;-\hslash)$ and similarly for the derivatives; 3. the exponent $3/2$ in the instanton action $A(x) = \frac{4}{3} x^{3/2}$, giving rise to the value $\floor{3k/2}$. This observation will allow us to obtain a similar result for $\Theta$-class and $r$-spin intersection numbers for which, mutatis mutandis, the same properties hold. We can now extract coefficients from both sides of [\[eqn:large:g:corr:Y\]](#eqn:large:g:corr:Y){reference-type="ref" reference="eqn:large:g:corr:Y"} to get the large genus asymptotics of Witten--Kontsevich intersection numbers. The leading term, already computed in [Corollary 20](#cor:leading:large:g:WK){reference-type="ref" reference="cor:leading:large:g:WK"}, recovers Aggarwal's result. The polynomiality structure of the subleading corrections proved in the previous proposition confirms a conjecture of Guo--Yang [@GY22 conjecture 1]. **Theorem 24** (Large genus asymptotics for $\psi$-class intersection numbers). *For any given $n \ge 1$ and $K \ge 0$, uniformly in $d_1,\dots,d_n$ as $g \to \infty$: $$\begin{gathered} \braket{\tau_{d_1} \cdots \tau_{d_n}} \prod_{i=1}^n (2d_i + 1)!! \\ = \frac{2^n}{4\pi} \frac{\Gamma(2g-2+n)}{( \frac{2}{3} )^{2g-2+n}} \left( \sum_{k = 0}^K \frac{( \frac{2}{3} )^{k}}{(2g-3+n)^{\underline{k}}} \, \alpha_k\bigl( n, p_0, \dots, p_{\floor{\frac{3k}{2}}-1} \bigr) + \mathrm{O}\biggl( \frac{1}{g^{K+1}} \biggr) \right) \,. \end{gathered}$$ Each term $\alpha_k$ in the asymptotic expansion is a polynomial in $n$ and $p_m = \#\set{ d_i = m}$ for $m = 1, \dots, \floor{\frac{3k}{2}}-1$ of degree $\le 3k-1$, with the assignment $\deg{n} = 1$ and $\deg{p_m} = m+1$. Moreover, the term $\alpha_k$ can be effectively computed from [\[eqn:subleading:coeffs\]](#eqn:subleading:coeffs){reference-type="ref" reference="eqn:subleading:coeffs"}.* *Proof.* Our starting point is the application of the Borel transform method, [Theorem 5](#thm:large:order){reference-type="ref" reference="thm:large:order"}, to the $n$-point function. From the proof of [Theorem 5](#thm:large:order){reference-type="ref" reference="thm:large:order"}, after the application of Cauchy's theorem and a deformation of the contour around the origin into several Hankel contours along the logarithmic branch-cuts, we find $$\begin{gathered} \label{eqn:asympt:WK:start} \frac{W_{g,n}(\bm{x})}{\Gamma(2g-2+n)} - \sum_{i=1}^n \frac{S}{\pi} \frac{1}{A(x_i)^{2g-2+n}} \sum_{k=0}^K \frac{A(x_i)^{2g-2+n-k}}{(2g-3+n)^{\underline{k}}} \, W_{k,n}^{(i)}(\bm{x}) \\ = (2g - 2 + n) \sum_{i=1}^n \frac{S}{\pi} \, \frac{1}{A(x_i)^{2g-2+n}} \int_{0}^{+\infty} e^{-t(2g-2+n)} \, \widehat{W}_{n}^{(i)} \bigl(\bm{x};A(x_i)(e^t-1);K \bigr) \, dt \,. \end{gathered}$$ Here we recall that $S=1$ and $A(x) = \frac{4}{3} x^{3/2}$ are the Stokes constant and the instanton action of the Airy functions, and $\widehat{W}_{n}^{(i)}(\bm{x};s;K)$ denotes the correlator $\widehat{W}_{n}^{(i)}(\bm{x};s)$ minus its Taylor expansion at $s = 0$ up to order $K$. We also notice that the application of the above theorem requires all singularities to be distinct. Since the singularities are located at $\pm A(x_i)$, we require $x_i \neq \pm x_j$ for all $i \ne j$. From [\[eqn:asympt:WK:start\]](#eqn:asympt:WK:start){reference-type="ref" reference="eqn:asympt:WK:start"}, it is clear that the dominant contribution to the asymptotics comes from the closest singularity to the origin (see [5.6.4](#app:sings){reference-type="ref" reference="app:sings"} for a visualisation of the phenomenon). Without loss of generality, we can suppose that $R_1 < \cdots < R_n$, so that $i = 1$ is the dominant contribution. On the other hand, after multiplication by $(x_1 \cdots x_n)^{3/2}$, the left-hand side is a homogeneous symmetric polynomial in $\bm{x}^{-1}$ of degree $3g-3+n$ (thanks to [\[thm:genus:expns:Airy,lemma:poly:subleading\]](#thm:genus:expns:Airy,lemma:poly:subleading){reference-type="ref" reference="thm:genus:expns:Airy,lemma:poly:subleading"}). Our goal is to extract the coefficient of a fixed monomial $x_1^{-d_1} \cdots x_{n}^{-d_n}$ and show that the resulting value has the appropriate asymptotics as $g \to \infty$, uniform in $d_1,\dots,d_n$. As the dominant contribution comes from $i = 1$, it is natural to normalise the left-hand side of [\[eqn:asympt:WK:start\]](#eqn:asympt:WK:start){reference-type="ref" reference="eqn:asympt:WK:start"} as: $$\begin{gathered} \Biggl( \prod_{i=1}^{n} 2x_i^{3/2} \Biggr) (-2A(x_1))^{2g-2+n} \, x_1^{n/2} \Bigg[ \frac{W_{g,n}(\bm{x})}{\Gamma(2g-2+n)} \\ - \sum_{i=1}^n \frac{S}{\pi} \frac{1}{A(x_i)^{2g-2+n}} \sum_{k=0}^K \frac{A(x_i)^{2g-2+n-k}}{(2g-3+n)^{\underline{k}}} \, W_{k,n}^{(i)}(\bm{x}) \Bigg] . \end{gathered}$$ Indeed, a simple degree counting shows that the left-hand side is a homogeneous polynomial of degree $3g-3+n$ in $x_1,x_2^{-1},\dots,x_n^{-1}$. Extracting the coefficient of $x_1^{d_1} x_2^{-d_2} \cdots x_n^{-d_n}$ yields $$\mathcal{C}_{g,d,K} \coloneqq \frac{( \frac{2}{3} )^{2g-2+n}}{\Gamma(2g-2+n)} \braket{\tau_{d_1} \cdots \tau_{d_n}} \prod_{i=1}^n (2d_i + 1)!! - \frac{2^n}{4\pi} \sum_{k = 0}^K \frac{( \frac{2}{3} )^{k}}{(2g-3+n)^{\underline{k}}} \, \alpha_k\bigl( n, p_0, \dots, p_{\floor{\frac{3k}{2}}-1} \bigr) \,.$$ The thesis is equivalent to showing that $\mathcal{C}_{g,d,K} = \mathrm{O}( g^{-K-1} )$, uniformly in $d$. In other words, there must exist $g_0 \ge 0$ and $C > 0$, depending on $n$ and $K$ but not on $d$, such that for $g \ge g_0$ we have $|\mathcal{C}_{g,d,K}| \le C \, g^{-K-1}$. In order to prove such a claim, let us consider the appropriately normalised right-hand of [\[eqn:asympt:WK:start\]](#eqn:asympt:WK:start){reference-type="ref" reference="eqn:asympt:WK:start"}. The coefficient extraction can be achieved by applying the operator $$\mathcal{O}_{d} \colon Z \longmapsto \left( \prod_{i=1}^n \oint_{|x_i| = R_i} \frac{dx_i}{2\pi\mathrm{i}} \, x_i^{\widetilde{d}_i-1} \right) Z(\bm{x}) \,.$$ Here $Z$ is any continuous function on $\set{ \bm{x} \in \mathbb{C}^n | |x_i| = R_i }$, $\widetilde{d}_1 = -d_1$ and $\widetilde{d}_j = d_j$ for all $j = 2,\dots,n$. Thus, looking at the right-hand side of [\[eqn:asympt:WK:start\]](#eqn:asympt:WK:start){reference-type="ref" reference="eqn:asympt:WK:start"}, we find $$\mathcal{C}_{g,d,K} = (2g-2+n) \, \mathcal{O}_{d} \biggl[ \sum_{i=1}^n \left( \frac{x_1}{x_i} \right)^{3g-3+n} \int_{0}^{+\infty} e^{-t(2g-2+n)} \, \widehat{Z}_{n}^{(i)}(\bm{x};t;K) \, dt \biggr] \,,$$ where we have set $$\widehat{Z}_{n}^{(i)}(\bm{x};t;K) \coloneqq (-1)^n \frac{S}{\pi} \, \Biggl( \prod_{i=1}^{n} 2x_i^{3/2} \Biggr) \, \widehat{W}_{n}^{(i)} \bigl(\bm{x};A(x_i)(e^t-1);K \bigr) \,.$$ Notice that summing over $n$ we find a polynomial in $x_1,x_2^{-1},\dots,x_n^{-1}$, though the $j$-th summand has simple poles at $x_i = x_j$. However, since we are integrating along circles of different radii $R_1 < \cdots < R_n$, there is no issue with the application of $\mathcal{O}_{d}$. Furthermore, it should be noted that $\widehat{Z}_{n}^{(i)}$ satisfies the following properties as functions of $t$ for $|x_i| = R_i$ fixed: - Polynomial growth at the origin. $\widehat{Z}_{n}^{(i)}(\bm{x};t;K) = \mathrm{O}(t^{K+1})$ as $t \to 0^+$, since the left hand-side is obtained from the truncation of the Taylor series of $\widehat{W}_{n}^{(i)}$ at the origin. - Exponential growth at infinity. $\widehat{Z}_{n}^{(i)}(\bm{x};t;K) = \mathrm{O}(e^{t\nu})$ as $t \to +\infty$ for some exponent $\nu \in \mathbb{R}_+$, since $\widehat{W}_{n}^{(i)}(\bm{x};A(x_i)s)$ is a sum of convolutions of wave functions with polynomial growth at $s = \infty$. The latter property can be easily checked from the integral representation of the Airy functions. From the homogeneity properties of the Airy function, it can be shown that both estimates are uniform in $R_i$, as long as all $R_i$ are fixed in a compact set, say $R_i \in [1,2]$. In order to prove the claim, let us analyse separately the term corresponding to $i = 1$ (the leading term), and the one corresponding to $i > 1$ (the exponentially subleading terms). Leading term. Consider $$\mathcal{C}_{g,d,K}^{\textup{lead}} \coloneqq (2g-2+n) \, \mathcal{O}_{d} \biggl[ \int_{0}^{+\infty} e^{-t(2g-2+n)} \, \widehat{Z}_{n}^{(1)}(\bm{x};t;K) \, dt \biggr] \,.$$ As the integrand is analytic in the integration domain, we can exchange the integral in $t$ with the one in $\bm{x}$ to get $$| \mathcal{C}_{g,d,K}^{\textup{lead}} | \le (2g-2+n) \int_{0}^{+\infty} e^{-t(2g-2+n)} \left| \mathcal{O}_{d} \Bigl[ \widehat{Z}_{n}^{(1)}(\bm{x};t;K) \Bigr] \right| dt \,.$$ Notice that the integrand $| \mathcal{O}_{d} [ \widehat{Z}_{n}^{(1)}(\bm{x};t;K) ] |$ can be bounded by $$\left| \mathcal{O}_{d} \Bigl[ \widehat{Z}_{n}^{(1)}(\bm{x};t;K) \Bigr] \right| \le \frac{R_{2}^{d_2} \cdots R_{n}^{d_n}}{R_1^{d_1}} \, f_{n,K}(t) \,,$$ where $f_{n,K}$ satisfies the hypothesis of Watson's [Lemma 6](#Watson:lemma){reference-type="ref" reference="Watson:lemma"}: it behaves like $\mathrm{O}(t^{K+1})$ as $t \to 0^+$ and it has exponential growth as $t \to + \infty$. Both estimates depend on $n$ and $K$, but not on $R_i$ and $d_i$. Applying Watson's [Lemma 6](#Watson:lemma){reference-type="ref" reference="Watson:lemma"}, we find $$\mathcal{C}_{g,d,K}^{\textup{lead}} = (2g - 2 + n) \frac{R_{2}^{d_2} \cdots R_{n}^{d_n}}{R_1^{d_1}} \, \mathrm{O}\biggl( \frac{1}{(2g - 2 + n)^{K+2}} \biggr) = \frac{R_{2}^{d_2} \cdots R_{n}^{d_n}}{R_1^{d_1}} \, \mathrm{O}\biggl( \frac{1}{g^{K+1}} \biggr) \,.$$ Again, $\mathrm{O}( g^{-K-1} )$ depends on $n$ and $K$ but not on $d$. Exponentially subleading terms. With the same argument as before, we deduce that $$\begin{split} \mathcal{C}_{g,d,K}^{\textup{sub}} & \coloneqq (2g-2+n) \, \mathcal{O}_{d} \Biggl[ \sum_{i=2}^n \left( \frac{x_1}{x_i} \right)^{3g-3+n} \int_{0}^{+\infty} e^{-t(2g-2+n)} \, \widehat{Z}_{n}^{(i)}(\bm{x};t;K) \, dt \Biggr] \\ & = \frac{R_{2}^{d_2} \cdots R_{n}^{d_n}}{R_1^{d_1}} \, \sum_{i=2}^n \left( \frac{R_1}{R_i} \right)^{3g-3+n} \mathrm{O}\biggl( \frac{1}{g^{K+1}} \biggr) \,. \end{split}$$ Notice that $(R_1/R_i)^{3g-3+n}$ are exponentially suppressed terms. Conclusion. To conclude, let us choose the radii $R_i = 1 + \frac{i-1}{n-1} \frac{1}{g}$. In the limit $g \to + \infty$ all radii coincide. It is then natural to expect that all terms would give a contribution behaving as $\mathrm{O}(g^{-K-1})$. Indeed, since $|d| = 3g-3+n$, we find $$\begin{split} \mathcal{C}_{g,d,K}^{\textup{lead}} & = \left(1 + \frac{1}{g} \right)^{3g-3+n} \mathrm{O}\biggl( \frac{1}{g^{K+1}} \biggr) = \mathrm{O}\biggl( \frac{1}{g^{K+1}} \biggr) \,, \\ \mathcal{C}_{g,d,K}^{\textup{sub}} & = \left(1 + \frac{n-1}{g} \right)^{3g-3+n} \, (n-1) \left( \frac{(n-1)g}{1 + (n-1)g} \right)^{3g-3+n} \mathrm{O}\biggl( \frac{1}{g^{K+1}} \biggr) = \mathrm{O}\biggl( \frac{1}{g^{K+1}} \biggr) \,. \end{split}$$ Here we used the fact that $(1 + \frac{1}{g} )^{3g-3+n} = (1 + \frac{n-1}{g} )^{3g-3+n} = (n-1) ( \frac{(n-1)g}{1 + (n-1)g} )^{3g-3+n} = \mathrm{O}(1)$. All estimates depend on $n$ and $K$, but not on $d$. All together, we find the thesis: $\mathcal{C}_{g,d,K} = \mathrm{O}(g^{-K-1})$ uniformly in $d$. ◻ ## Trans-series structure {#subsec:transseries} For the reader familiar with resurgence and trans-series analysis, it can be shown that the correlators have a full trans-series expansion that can be constructed as follows. Consider the matrix $$\mathsf{M}(x;\hslash) = \Psi(x;\hslash) \,\mathsf{E}\, \Psi^{-1}(x;\hslash) \,, \qquad\qquad \mathsf{E} = \begin{pmatrix} \frac{\sigma}{2} & -\sigma_- \\ \sigma_+ & -\frac{\sigma}{2} \end{pmatrix} ,$$ which depends on trans-series parameters $\sigma, \sigma_+, \sigma_-$ (but we omit its dependence for the sake of notation simplicity). Notice that $\mathsf{E}$ is a generic element of $\mathfrak{sl}_2(\mathbb{C})$, rather than a Cartan element as in [\[eqn:M:matrix\]](#eqn:M:matrix){reference-type="ref" reference="eqn:M:matrix"}. We decompose $\mathsf{M}$ as $$\mathsf{M} = \sigma \, M + \sigma_{+} \, e^{\frac{A(x)}{\hslash}} M_+ + \sigma_{-} \, e^{-\frac{A(x)}{\hslash}} M_- \,,$$ where $M$ and $M_{\pm}$ are the formal power series in $\hslash$ previously encountered. We can then define the $n$-point correlators with the exact same determinantal formula as in [\[eqn:n:pnt:Airy\]](#eqn:n:pnt:Airy){reference-type="ref" reference="eqn:n:pnt:Airy"}: $$\mathsf{W}_1(x_1;\hslash) \coloneqq - \frac{1}{\hslash} \mathop{\mathrm{Tr}}{\bigl( \mathcal{D}(x_1) \mathsf{M}(x_1;\hslash) \bigr)} \,, \qquad \mathsf{W}_n(\bm{x};\hslash) \coloneqq (-1)^{n-1} \sum_{\sigma \in S_n^{\textup{cyc}}} \frac{\mathop{\mathrm{Tr}}{ \bigl( \prod_{i=1}^n \mathsf{M}(x_{\sigma^i(1)};\hslash) } \bigr) }{\prod_{i=1}^n ( x_{i} - x_{\sigma(i)} )} \,.$$ In the above definition we have chosen different constants $\sigma_{i}$, $\sigma_{i,+}$, $\sigma_{i,-}$ for different $i=1,\dots,n$. Thus, we have the trans-series expansion $$\label{eqn:transseries} \mathsf{W}_n(\bm{x};\hslash) = \sum_{I_{+} \sqcup I_{-} \subseteq [n]} \prod_{i \not \in I_{+} \sqcup I_{-} } \sigma_{i} \prod_{i \in I_{+} } \sigma_{i,+} \prod_{i \in I_{-} } \sigma_{i,-} \, e^{\frac{1}{\hslash} \bigl( \sum_{i \in I_{+}} A(x_i) - \sum_{i \in I_{-}} A(x_i) \bigr)} \, \mathsf{W}^{(I_{+},I_{-})}_n(\bm{x};\hslash) \,.$$ Notice that, by definition, the perturbative sector $\mathsf{W}^{(\varnothing,\varnothing)}_{n}$ coincides with the $W_n$ defined in [\[eqn:n:pnt:Airy\]](#eqn:n:pnt:Airy){reference-type="ref" reference="eqn:n:pnt:Airy"} and stores $\psi$-class intersection numbers. Moreover, the $(1,0)$- and $(0,1)$-instanton sectors coincide with the correlators defined in [\[eqn:1inst:corr\]](#eqn:1inst:corr){reference-type="ref" reference="eqn:1inst:corr"}: $\mathsf{W}^{(\{i\},\varnothing)}_n = W^{(+,i)}_n$ and $\mathsf{W}^{(\varnothing,\{i\})}_n = W^{(-,i)}_n$. In particular, they govern the large genus asymptotics of the intersection numbers, but they do not have (to the best of our knowledge) an enumerative-geometric interpretation. It would be interesting to explore whether such a geometric interpretation actually exists for all multi-instanton sectors. Finally, we remark that the trans-series expansion is in accordance with the fact that $\mathsf{W}_n$ satisfies a linear ODE of order $3^n$ (see [@EMO] for a proof). For instance, the $1$-point correlator satisfies $$\mathsf{W}^{'''}_1 - 4x \, \mathsf{W}'_1 + 2 \hslash\, \mathsf{W}_1 = 0 \,,$$ where $f'$ stands for $\hslash\frac{d}{dx}f$. In particular, a perturbative ansatz of the form $W_1 = \sum_{g \ge 0} w_g \frac{\hslash^{2g-1}}{x^{3g-1/2}}$ would give rise to a recursive relation for the coefficients $w_g$ that can be explicitly solved, and is equivalent to $\braket{\tau_{3g-2}}_{g} = \frac{1}{24^g g!}$ after setting $w_0 = 1$. # Large genus asymptotics of $\Theta$-class intersection numbers {#sec:Norbury} The goal of this section is to prove the large genus asymptotics of $\Theta$-class intersection numbers. To the best of our knowledge, their asymptotics is not known in the literature. The strategy is, mutatis mutandis, the same as the one employed in the previous section: we apply the Borel transform method to the $n$-point function, which in turn is built out of formal solutions of the Bessel differential equations through the determinantal formulae. We are going to skip most of the details, and highlight only the substantial changes compared to the previous section. Before proceeding with the details of the determinantal formula, let us briefly recall the definition of these intersection numbers. The *$\Theta$-class*, introduced by Norbury in [@Nor23], is a collection of cohomology classes (more precisely, a cohomological field theory) of pure degree: $$\Theta_{g,n} \in H^{2(2g-2+n)}(\overline{{\mathcal{M}}}_{g,n},\mathbb{Q}) \,.$$ Their construction goes through the moduli space of spin curves: let $\overline{{\mathcal{M}}}_{g,n}^{\textup{spin}}$ be the moduli space parametrising roots (also known as spin structures) of the form $$L^{\otimes 2} \cong \omega_C \left( \sum_{i=1}^n p_i \right) \,,$$ where $[C,p_1,\dots,p_n] \in \overline{{\mathcal{M}}}_{g,n}$ and $\omega_C$ denotes the canonical line bundle on $C$. There is a forgetful map $\pi \colon \overline{{\mathcal{M}}}_{g,n}^{\textup{spin}} \to \overline{{\mathcal{M}}}_{g,n}$ that forgets the spin structure. Consider the vector bundle $\mathcal{E}_{g,n} \to \overline{{\mathcal{M}}}_{v,n}^{\textup{spin}}$ whose fibre over $[C,p_1,\dots,p_n,L]$ is $H^1(C,L^{\ast})^{\ast}$. The $\Theta$-class is then defined as the push-forward of the (normalised) top Chern class of $\mathcal{E}_{g,n}$: $$\Theta_{g,n} \coloneqq 2^{g-1+n} \, \pi_{\ast} c_{\textup{top}}\left( \mathcal{E}_{g,n} \right) .$$ Define the $\Theta$-class intersection numbers as $$\braket{\tau_{d_1} \cdots \tau_{d_n}}^{\Theta} \coloneqq \int_{\overline{{\mathcal{M}}}_{g,n}} \Theta_{g,n} \prod_{i=1}^n \psi_i^{d_i} \,.$$ They are non-zero only for $|d| = g-1$. In [@Nor23], Norbury conjectures that the partition function associated the $\Theta$-class coincides with Brézin--Gross--Witten solution of the KdV hierarchy, discovered in the '80s in the context of unitary matrix models [@BG80; @GW80]. Since then, a proof of Norbury's conjecture has been proposed in [@CGG] through Givental formalism and its connection with topological recursion. In this section we are interested in yet another method for computing $\Theta$-class intersection numbers, namely the determinantal formula with building blocks being the asymptotic solutions of the ($\hslash$-dependent) *Bessel ODE*: $$\left( \hslash^2 \frac{d}{dx} x \frac{d}{dx} - 1 \right) \psi(x;\hslash) = 0 \,.$$ The general solution is given by the Bessel integral: $\int_{\gamma} e^{-\frac{1}{\hslash}V(t,x)} \frac{dt}{t}$ with $V(t,x) \coloneqq -\frac{1}{t} - xt$ and $\gamma$ a properly chosen integration contour. The asymptotic solutions constructed through Lefschetz thimbles are given by $$\psi_{\pm}(x;\hslash) \coloneqq \frac{e^{\mp \frac{V(x)}{\hslash}}}{\sqrt{2}} x^{-1/4} \sum_{k = 0}^{\infty} \frac{(\tfrac{1}{2})^{\overline{k}} (\tfrac{1}{2})^{\overline{k}}}{2^k k!} \left(\mp \frac{\hslash}{V(x)} \right)^{k} \,,$$ where $\pm V(x) \coloneqq \mp 2 x^{1/2}$ are the critical values of the potential. Here $(x)^{\overline{k}} \coloneqq x (x+1) \cdots (x+k-1)$ denotes the rising factorial. Notice that $\psi_{\pm}$ correspond to the (properly renormalised) asymptotic expansions of the modified Bessel functions $\mathrm{K}_0$ and $\mathrm{I}_0$. The Bessel ODE can be re-written as a $2 \times 2$ system: $$\hslash\frac{d}{dx} \Psi_0(x;\hslash) = \mathcal{D}_0(x;\hslash) \Psi_0(x;\hslash) \,, \qquad \mathcal{D}_0(x;\hslash) = \begin{pmatrix} 0 & 1 \\ \tfrac{1}{x} & -\tfrac{\hslash}{x} \end{pmatrix} \,.$$ However, we notice that $\det(\Psi_0(x;\hslash)) = x^{-1}$ is not constant. We can gauge-transform the above system so that the determinant of the wave matrix is normalised to $1$. Setting $\Psi(x;\hslash) \coloneqq \left(\begin{smallmatrix} 1 & 0 \\ 0 & x \end{smallmatrix}\right) \Psi_0(x;\hslash)$, the system becomes $$\hslash\frac{d}{dx} \Psi(x;\hslash) = \mathcal{D}(x) \Psi(x;\hslash) \,, \qquad\qquad \mathcal{D}(x) = \begin{pmatrix} 0 & \tfrac{1}{x} \\ 1 & 0 \end{pmatrix} \,.$$ Again, we denote the rows of $\Psi$ as $\psi_{\pm}$ and $\psi_{\pm}'$. Because of the gauge transformation, we have $\psi_{\pm}' = x \hslash\frac{d}{dx}\psi_{\pm}$. After separating the exponential part, denoting the resulting formal power series with a tilde, and taking the Borel transforms, we find $$\widehat{\psi}_{\pm}(x;s) = \frac{x^{-1/4}}{\sqrt{2}} \, \vphantom{F}_{2}F_{1} \left( \tfrac{1}{2},\tfrac{1}{2} ; 1 ; \pm \frac{s}{A(x)} \right) \,, \qquad \widehat{\psi}_{\pm}'(x;s) = \pm \frac{x^{1/4}}{\sqrt{2}} \, \vphantom{F}_{2}F_{1} \left( \tfrac{3}{2},-\tfrac{1}{2} ; 1 ; \pm \frac{s}{A(x)} \right) \,,$$ where $\pm A(x) \coloneqq \pm 4 x^{1/2}$ are the instanton actions for the Bessel ODE. In particular, they converge in a disc of positive radius centred at the origin, and can be extended analytically to functions with logarithmic singularities at $s = \pm A(x)$. One can check the following behaviour at the singularities: $$\begin{aligned} \widehat{\psi}_{\pm}(x;s) & = - \frac{S}{2\pi} \, \widehat{\psi}_{\mp}\bigl( x;s \mp A(x) \bigr) \log\bigl( s \mp A(x) \bigr) + \text{holomorphic at } \pm A(x) \,, \\ \widehat{\psi}_{\pm}'(x;s) & = - \frac{S}{2\pi} \, \widehat{\psi}_{\mp}'\bigl( x;s \mp A(x) \bigr) \log\bigl( s \mp A(x) \bigr) + \text{holomorphic at } \pm A(x) \,, \end{aligned}$$ where the Stokes constant is $S = 2$. Again, $\widehat{\psi}_{\pm}$ and $\widehat{\psi}_{\pm}'$ are simple resurgent functions whose singularity structure is fully under control. Consider now the matrix $M \coloneqq \frac{1}{2} \, \Psi \left(\begin{smallmatrix} 1 & 0 \\ 0 & -1 \end{smallmatrix} \right) \Psi^{-1}$, and define the $n$-point *Bessel correlators* via determinantal formulae: $$\label{eqn:n:pnt:Bessel} W_1(x_1;\hslash) \coloneqq - \frac{1}{\hslash} \mathop{\mathrm{Tr}}{\bigl( \mathcal{D}(x_1) M(x_1;\hslash) \bigr)} \,, \qquad W_n(\bm{x};\hslash) \coloneqq (-1)^{n-1} \sum_{\sigma \in S_n^{\textup{cyc}}} \frac{\mathop{\mathrm{Tr}}{ \bigl( \prod_{i=1}^n M(x_{\sigma^i(1)};\hslash) } \bigr) }{\prod_{i=1}^n ( x_{i} - x_{\sigma(i)} )} \,.$$ Again, one finds a genus expansion of the correlators with the expansion coefficients storing interesting enumerative invariants, namely $\Theta$-class intersection numbers. The determinantal formula for such intersection numbers was deduced in [@DYZ21] assuming the validity of Norbury's conjecture [@Nor23], later proved in [@CGG]. **Theorem 25** (Genus expansion of the Bessel correlators [@DYZ21]). *The $n$-point Bessel correlators $W_{n}$ admit the following $\hslash$-expansion: $$\label{eqn:genus:expns:Bessel} W_{n}(\bm{x};\hslash) = \sum_{g = 0}^{\infty} \hslash^{2g-2+n} \, W_{g,n}(\bm{x}) \,.$$ Moreover, $W_{g,n}$ stores $\Theta$-class intersection numbers: if $2g-2+n>0$, $$W_{g,n}(\bm{x}) = (-2)^{-(2g-2+n)} \sum_{\substack{ d_{1},\dots,d_n \ge 0 \\ d_{1}+\cdots+d_n = g-1 }} \braket{ \tau_{d_1} \cdots \tau_{d_n} }^{\Theta} \prod_{i=1}^n \frac{(2d_i+1)!!}{2 \, x_i^{d_i+3/2}} \,.$$* **Remark 26** (Connection with topological recursion). For the reader familiar with topological recursion, $W_{g,n}$ coincides with topological recursion differentials computed from the Bessel spectral curve $( \mathbb{P}^1, \, x(z) = z^2, \, y(z) = z^{-1}, \, B(z_1,z_2) = \frac{dz_1 dz_2}{(z_1 - z_2)^2} )$ -- see [@DN19]: $$\omega_{g,n}(\bm{z}) = W_{g,n}(\bm{x}) \, dx_1 \cdots dx_n \big|_{x_i = z_i^2} \,.$$ Using the exact same strategy as before, we find the large genus behaviour of $\Theta$-class intersection numbers. In a nutshell: - Write the correlators in terms of Bessel kernels (see also [@TW94b]). - Study the Borel plane singularity structure of the Bessel correlators. One finds that, on the principal sheet, the Borel transform of $W_n$ has $2n$ logarithmic singularities located at $s = \pm A(x_i)$ and such that $$\widehat{W}_{n}(\bm{x};s) = - \frac{S}{2\pi} \, \widehat{W}^{(\pm,i)}_{n}\bigl( \bm{x}; s \mp A(x_i) \bigr) \, \log\bigl( s \mp A(x_i) \bigr) +\text{holomorphic at } \pm A(x_i) \,.$$ Here $A(x) = 4 x^{1/2}$ and $S = 2$ are the instanton action and the Stokes constant associated to the Bessel functions respectively, and $\widehat{W}^{(\pm,i)}_{n}$ are defined through determinantal formulae as in [\[eqn:1inst:corr\]](#eqn:1inst:corr){reference-type="ref" reference="eqn:1inst:corr"}. - Deduce the large genus asymptotics of the correlators through the Borel transform method, and get the large genus asymptotics of the intersection numbers by extracting coefficients. The final result is the following asymptotic formula. We omit the proof, which is completely analogous to that of [Theorem 24](#thm:large:g:WK){reference-type="ref" reference="thm:large:g:WK"}. **Theorem 27** (Large genus asymptotics for $\Theta$-class intersection numbers). *For any given $n \ge 1$ and $K \ge 0$, uniformly in $d_1,\dots,d_n$ as $g \to \infty$: $$\begin{gathered} \braket{\tau_{d_1} \cdots \tau_{d_n}}^{\Theta} \prod_{i=1}^n (2d_i + 1)!! \\ = \frac{2^n}{2\pi} \frac{\Gamma(2g-2+n)}{2^{2g-2+n}} \left( \sum_{k = 0}^K \frac{2^{k}}{(2g-3+n)^{\underline{k}}} \, \beta_k \bigl( n, p_0, \dots, p_{\floor{\frac{k}{2}}-1} \bigr) + \mathrm{O}\biggl( \frac{1}{g^{K+1}} \biggr) \right) \,. \end{gathered}$$ Each term $\beta_k$ in the asymptotic expansion is a polynomial in $n$ and $p_m = \#\set{ d_i = m}$ for $m = 1, \dots, \floor{\frac{k}{2}}-1$ of degree $\le k-1$, with the assignment $\deg{n} = 1$ and $\deg{p_m} = m+1$, that can be effectively computed.* The polynomials $Q_{k,n}$ (defined in complete analogy to the Witten--Kontsevich case) and the coefficients $\beta_k$ for the first few values of $k$ are given in [4](#table:Pkn:ck:Theta){reference-type="ref" reference="table:Pkn:ck:Theta"}. Notice that the complexity of $\beta_k$ is much lower than the corresponding one for the Witten--Kontsevich case. This is due to the fact that $\Theta$-class intersection numbers are computationally less involved that $\psi$-class intersection numbers. Indeed, the $\Theta$-class fills $2g-2+n$ cohomological degrees, which in turn are spared to $\psi$-classes. $k$ $Q_{k,n}$ $\beta_k(n,\bm{p})$ ----- ------------------------------------------------------------- -------------------------------------------------- $0$ $1$ $1$ $1$ $- \frac{1}{4}$ $- \frac{1}{4}$ $2$ $\tfrac{9 - 4 n}{8} m_{\varnothing} + \tfrac{1}{8} m_{(1)}$ $\tfrac{9 - 4 n}{8} + \tfrac{n - p_0}{8}$ $3$ $- \tfrac{57 - 44 n + 8 n^2}{128} m_{\varnothing} $- \tfrac{57 - 44 n + 8 n^2}{128} - \tfrac{13 - 4 n}{32} m_{(1)} - \tfrac{(13 - 4 n)(n - p_0)}{32} - \tfrac{1}{8} m_{(1^2)}$ - \tfrac{(n - p_0)^{\underline{2}}}{16}$ : The polynomials $Q_{k,n}(u_1,\dots,u_n)$ and the coefficients $\beta_k$ for $k = 0,1,2,3$. Here $m_{\lambda}$ denotes the monomial symmetric polynomial in the variables $u_1^2,\dots,u_n^2$. # Large genus asymptotics of $r$-spin intersection numbers {#sec:rspin} The goal of this section is to prove the large genus asymptotics of $r$-spin intersection numbers, introduced for the first time by Witten [@Wit92; @Wit93] in the context of topological gravity coupled to a Wess--Zumino--Witten theory. Compared to the previous sections, this case contains novel effects due to the underlying differential system being of order $r \times r$ (rather than $2 \times 2$). As a consequence, the number of singularities in the Borel plane for the wave function is $r-1$, which translates into $2(r-1)n$ singularities for the $n$-point function. Thus, the large genus asymptotics of $r$-spin intersection numbers manifests the novel feature of exponentially suppressed contributions coming from the singularities that are further away from the origin. In the literature, an asymptotic analysis of $r$-spin intersection numbers was recently considered only for $n = 1$ by Dubrovin--Yang--Zagier in [@DYZ theorem 5 (vii)]. We emphasise that their asymptotic formula only considers the exponentially dominant contribution and is valid at first order in $g$. In this section, we generalise their result to arbitrary $n$, and taking into account all subleading and exponentially subleading contributions. Once again, the strategy consists in analysing the determinantal formula through the Borel transform method. Before proceeding with the large genus asymptotics, let us briefly recall the definition of these intersection numbers. Fix $r \ge 2$. The *Witten $r$-spin class* is a collection of pure-dimensional cohomology classes (to be precise, a cohomological field theory) depending on some parameters $a = (a_1,\dots,a_n) \in \set{1,\dots,r-1}^n$ called *primary fields*[^5]: $$W_{g,n}^{r}(a) \in H^{2 D_{g,n}^{r}(a)}(\overline{{\mathcal{M}}}_{g,n},\mathbb{Q}) \,.$$ Here $D_{g,n}^{r}(a) = \frac{(r-2)(g-1)-n+|a|}{r}$, and the class is set to be zero if $(r-2)(g-1)-n+|a| \not\equiv 0 \pmod{r}$. In genus $0$, the construction was first carried out by Witten using $r$-spin structures, and we briefly recall it here. Let $\overline{{\mathcal{M}}}_{0,n}^{r}(a)$ be the moduli space parametrising $r$-th roots of the form $$L^{\otimes r} \cong \omega_{C} \biggl( - \sum_{i=1}^n a_i p_i \biggr) ,$$ where $[C,p_1,\dots,p_n] \in \overline{{\mathcal{M}}}_{0,n}$ and $\omega_C$ denotes the canonical line bundle on $C$. There is a forgetful map $\pi \colon \overline{{\mathcal{M}}}_{0,n}^{r}(a) \to \overline{{\mathcal{M}}}_{0,n}$ that forgets the $r$-th root. Consider the vector bundle $\mathcal{V}_{0,n}^{r}(a) \to \overline{{\mathcal{M}}}_{0,n}^{r}(a)$ whose fibre over $[C,p_1,\dots,p_n,L]$ is $H^1(C,L)^{\ast}$. The Witten $r$-spin class[^6] is defined as the push-forward of the (normalised) top Chern class of $\mathcal{V}_{0,n}^{r}(a)$: $$W_{0,n}^{r}(a) \coloneqq r \cdot \pi_{\ast} c_{\textup{top}}\left( \mathcal{V}_{0,n}^{r}(a) \right) .$$ The definition of the Witten class in higher genera is much more complicated, since the fibres $H^1(C,L)^{\ast}$ no longer glue together to form a vector bundle. The construction was first carried out by Polishchuk--Vaintrob [@PV00], and was later simplified by Chiodo [@Chi06]. We refer the reader to [@PPZ15] for more details on the Witten classes, and its connection to (the conjecturally full set of) relations on the tautological ring of the moduli space of curves. The $r$-spin intersection numbers are defined as $$\braket{\tau_{d_1,a_1} \cdots \tau_{d_n,a_n}}^{r\textup{-spin}} \coloneqq \int_{\overline{{\mathcal{M}}}_{g,n}} W_{g,n}^{r}(a_1,\dots,a_n) \prod_{i=1}^n \psi_i^{d_i} \,.$$ Notice that the numbers $\braket{\tau_{d_1,a_1} \cdots \tau_{d_n,a_n}}^{r\textup{-spin}}$ are set to be zero unless $D_{g,n}^{r}(a) + |d| = 3g-3+n$. Similar to the cases of $\psi$- and $\Theta$-class intersection numbers, $r$-spin intersection numbers are also connected to integrable hierarchies. Specifically, the associated partition function is the unique tau function of the $r$-KdV hierarchy satisfying a particular initial condition, called the string equation, as demonstrated by Faber--Shadrin--Zvonkine in [@FSZ10]. Another approach to computing $r$-spin intersection numbers is through the determinantal formula [@BDY18; @BDY21], which relies on the asymptotic solution of the $r$-Airy ODE as its fundamental component. ## Higher Airy functions Consider the ($\hslash$-dependent) *$r$-Airy ODE*: $$\left( \left( \hslash\frac{d}{dx} \right)^r - x \right) \psi(x;\hslash) = 0 \,.$$ The general solution is given by the $r$-Airy integral: $\int_{\gamma} e^{-\frac{1}{\hslash}V(t,x)} dt$ with $V(t,x) \coloneqq \frac{t^{r+1}}{r+1} - xt$ and $\gamma$ a properly chosen integration contour. Again, we are interested in asymptotic solutions constructed through Lefschetz thimbles: $\psi_{\alpha}(x;\hslash)$ for $\alpha = 1, \dots, r$. Their expression, together with that of their derivatives, is given by $$\psi_{\alpha}^{(m)}(x;\hslash) \coloneqq (-1)^{\frac{r-\alpha+2}{2}} \zeta^{\alpha(m + \frac{1}{2})} \frac{e^{-\frac{V_{\alpha}(x)}{\hslash}}}{\sqrt{r}} x^{-\frac{r-1-2m}{2r}} \sum_{k = 0}^{\infty} a_k^{(m)} \left( - \frac{\hslash}{V_{\alpha}(x)} \right)^{k} \,,$$ where $V_{\alpha}(x) \coloneqq - \zeta^{\alpha} \frac{r}{r+1} x^{(r+1)/r}$ and $\zeta \coloneqq e^{\frac{2\pi\mathrm{i}}{r}}$. The coefficients $a_{k}^{(m)}$ are independent of $\alpha$ and are computed recursively from the differential equation (see for instance [@CCGG]) as $$\begin{cases} a_{k}^{(m)} = a_{k}^{(m-1)} - \left( k - \frac{1}{2} - \frac{m}{r+1} \right) a_{k-1}^{(m-1)} & \text{if $m = 1,\dots, r$} \\ a_{k}^{(0)} = a_{k}^{(r)} \end{cases}$$ with $a_{0}^{(m)} = 1$. With the above normalisation, we can see that the Wronskian is constantly $1$. Indeed, the ODE implies that the Wronskian is constant, and we can compute its value at $x \to \infty$ as $$\det\left( (-1)^{\frac{r-\alpha+2}{2}} \zeta^{\alpha(m + \frac{1}{2})} \frac{ e^{-\frac{V_{\alpha}(x)}{\hslash}} }{ \sqrt{r} } x^{-\frac{r-1-2m}{2r}} \right)_{\substack{ m = 0,\dots,r-1 \\ \alpha = 1,\dots,r}} = \frac{(-1)^{\frac{(r-1)(r-2)}{4}}}{r^{r/2}} \det\bigl( \zeta^{\alpha m} \bigr)_{\substack{ m = 0,\dots,r-1 \\ \alpha = 1,\dots,r}} \,.$$ The Vandermonde determinant on roots of unity was computed by Schur [@Sch21] and it reads $\det( \zeta^{\alpha m} ) = r^{r/2} (-1)^{\frac{(r-1)(3r+2)}{4}}$, so that the Wronskian simplifies to $1$. Following the prescription of [2.2](#subsec:exp:int){reference-type="ref" reference="subsec:exp:int"} we find that, after separating the exponential part as $\psi_{\alpha}^{(m)} \eqqcolon e^{-V_{\alpha}/\hslash} \, \widetilde{\psi}_{\alpha}^{(m)}$, the Borel transform of $\widetilde{\psi}_{\alpha}^{(m)}$ is simple resurgent with logarithmic singularities at $s = A_{\alpha,\beta}(x)$ for $\beta \neq \alpha$, where $$A_{\alpha,\beta}(x) \coloneqq V_{\beta}(x) - V_{\alpha}(x) = \frac{r}{r+1} \bigl( \zeta^{\alpha} - \zeta^{\beta} \bigr) x^{(r+1)/r} \,.$$ The behaviour at the singularities is given by $$\begin{aligned} \widehat{\psi}_{\alpha}(x;s) & = - \frac{S_{\alpha,\beta}}{2\pi} \, \widehat{\psi}_{\beta}\bigl( x;s - A_{\alpha,\beta}(x) \bigr) \log\bigl( s - A_{\alpha,\beta}(x) \bigr) + \text{holomorphic at } A_{\alpha,\beta}(x) \,, \end{aligned}$$ where the Stokes constants $S_{\alpha,\beta}$ are given in terms of constants $\mathsf{S}_{\alpha,\beta}$ computed as Lefschetz thimble intersection numbers (see [\[eqn:Stokes:rAiry\]](#eqn:Stokes:rAiry){reference-type="ref" reference="eqn:Stokes:rAiry"}) and are worth $$S_{\alpha,\beta} = (-1)^{\frac{\beta-\alpha+1}{2}} \, \mathsf{S}_{\alpha,\beta} = \begin{cases} +(-1)^{\frac{\beta-\alpha+1}{2}} & \text{if } \alpha > \beta \,,\\ -(-1)^{\frac{\beta-\alpha+1}{2}} & \text{if } \alpha < \beta \,. \end{cases}$$ The above ODE can be re-written as a first order $r \times r$ system: $$\hslash\frac{d}{dx} \Psi(x;\hslash) = \mathcal{D}(x) \Psi(x;\hslash) \,, \qquad\qquad \mathcal{D}(x;\hslash) = \begin{pmatrix} 0 & 1 & & \\ & \ddots & \ddots & \\ & & \ddots & 1 \\ x & & & 0 \end{pmatrix} \,.$$ A simple computation shows that the matrix $\Phi \coloneqq \Psi^{-t}$, called the dual wave matrix, satisfies the differential system $-\hslash\frac{d}{dx}\Phi = \mathcal{D}^{t} \Phi$. In other words, $\Phi$ is a (permutation of the) companion matrix associated to the higher Airy ODE, with $\hslash\mapsto - \hslash$. In order to fix the normalisation, it is sufficient to compute $\Phi$ at $x \to \infty$. We find that $$\Phi(x;\hslash) = \left( (-1)^{r-\alpha} \phi_{\alpha}^{(r-m-1)} \right)_{\substack{ m = 0,\dots,r-1 \\ \alpha = 1,\dots,r}} , \qquad \phi_{\alpha}^{(k)}(x;\hslash) \coloneqq \psi_{\alpha}^{(k)}(x;-\hslash) \,.$$ Since the $\phi$'s are obtained from the $\psi$'s simply by substituting $\hslash$ with $-\hslash$, they enjoy similar properties. In particular we can deduce that, after separating the exponential part as $\phi_{\alpha}^{(m)} \eqqcolon e^{+V_{\alpha}/\hslash} \, \widetilde{\phi}_{\alpha}^{(m)}$, the Borel transform of $\widetilde{\psi}_{\alpha}^{(m)}$ is simple resurgent with logarithmic singularities at $s = - A_{\alpha,\beta}$ for $\beta \neq \alpha$. Its behaviour is given by $$\begin{aligned} \widehat{\phi}_{\alpha}(x;s) & = - \frac{S_{\alpha,\beta}}{2\pi} \, \widehat{\phi}_{\beta}\bigl( x;s + A_{\alpha,\beta}(x) \bigr) \log\bigl( s + A_{\alpha,\beta}(x) \bigr) + \text{holomorphic at } -A_{\alpha,\beta}(x) \,, \end{aligned}$$ where the Stokes constants are the same as before. ## Higher Airy correlators The Lie algebra $\mathfrak{sl}_r(\mathbb{C})$ admits the root space decomposition into traceless diagonal matrices, upper-diagonal matrices, and lower-diagonal matrices: $\mathfrak{sl}_r(\mathbb{C}) = \mathfrak{h} \oplus \mathfrak{n}_+ \oplus \mathfrak{n}_-$. A basis of $\mathfrak{h}$ is given in terms of elementary matrices $e_{i,j} = (\delta_{i,k} \delta_{j,l})$ by $$E_a \coloneqq e_{a,a} - e_{a+1,a+1} \in \mathfrak{h} \,, \qquad\qquad a = 1,\dots,r-1 \,.$$ Define the matrix $$M(x;\hslash) \coloneqq \Psi(x;\hslash) E \Psi^{-1}(x;\hslash) \,, \qquad\qquad E \coloneqq \frac{1}{r} \sum_{a=1}^{r-1} a E_a \,,$$ which is an $\mathfrak{h}$-valued formal power series in $\hslash$. Indeed, as in the Airy case, $M$ contains only quadratic expressions in the functions $\psi$'s and $\phi$'s with opposite exponential parts. The result is then a formal power series in $\hslash$. As before, one can consider the $n$-point *$r$-Airy correlators* defined through determinantal formulae: $$\label{eqn:n:pnt:rspin} \begin{aligned} & W_1(x_1;\hslash) \coloneqq - \frac{1}{\hslash} \mathop{\mathrm{Tr}}{\bigl( \mathcal{D}(x_1) M(x_1;\hslash) \bigr)} \,, \\ & W_n(x_1,\dots,x_n;\hslash) \coloneqq (-1)^{n-1} \sum_{\sigma \in S_n^{\textup{cyc}}} \frac{\mathop{\mathrm{Tr}}{ \bigl( \prod_{i=1}^n M(x_{\sigma^i(1)};\hslash) } \bigr) }{\prod_{i=1}^n ( x_{i} - x_{\sigma(i)} )} \qquad\quad \text{for }n \ge 2 \,. \end{aligned}$$ Again, the genus expansion of the correlators stores interesting enumerative invariants, the $r$-spin intersection numbers. The determinantal formula for such intersection numbers was proved by Bertola--Dubrovin--Yang [@BDY18; @BDY21] as a consequence of the Faber--Shadrin--Zvonkine result [@FSZ10]. Adjusted to our chosen normalisation, and with the insertion of $\hslash$ by homogeneity, the result of Bertola--Dubrovin--Yang reads as follows. **Theorem 28** (Genus expansion of the higher Airy correlators [@BDY18; @BDY21]). *The higher Airy correlators $W_{n}$ admit the following $\hslash$-expansion: $$\label{eqn:genus:expns:rAiry} W_{n}(\bm{x};\hslash) = \sum_{g = 0}^{\infty} \hslash^{2g-2+n} \, W_{g,n}(\bm{x}) \,.$$ Moreover, $W_{g,n}$ stores $r$-spin intersection numbers: if $2g-2+n>0$, $$W_{g,n}(\bm{x}) = \!\!\! \sum_{\substack{ d_{1},\dots,d_n \ge 0 \\ a_1,\dots,a_n \in \{1,\dots,r-1\} \\ r|d|+|a| = (r+1)(2g-2+n) }} \!\!\! (-r)^{g-1-|d|} \braket{ \tau_{d_1,a_1} \cdots \tau_{d_n,a_n} }^{r\textup{-spin}} \prod_{i=1}^n \frac{(rd_i+a_i)!_{(r)}}{r \, x_i^{d_i+\frac{a_i}{r}+1}} \,.$$ Here $m!_{(r)}$ denotes the $r$-factorial, see [\[eqn:r:fact\]](#eqn:r:fact){reference-type="ref" reference="eqn:r:fact"}. The degree condition is equivalent to $D_{g,n}^r(a) + |d| = 3g-3+n$, where $D_{g,n}^r(a)$ is the complex cohomological degree of the Witten class.* **Remark 29** (Connection with topological recursion). For the reader familiar with topological recursion, $W_{g,n}$ coincides with the topological recursion differentials computed from the $r$-Airy spectral curve $( \mathbb{P}^1, \, x(z) = z^r, \, y(z) = z, \, B(z_1,z_2) = \frac{dz_1 dz_2}{(z_1 - z_2)^2} )$ -- see [@BE17]: $$\omega_{g,n}(\bm{z}) = W_{g,n}(\bm{x}) \, dx_1 \cdots dx_n \big|_{x_i = z_i^r} \,.$$ Throughout this section, we will work with a fixed choice of $r$-th root of $x$. The $n$-point correlators can be expressed through determinantal formulae in the kernel form. We omit the proof, which is completely analogous to [Lemma 16](#lemma:kernel){reference-type="ref" reference="lemma:kernel"}. **Lemma 30**. *For $n \ge 2$, the $n$-point correlators are given by the two equivalent expressions $$W_n(\bm{x};\hslash) = (-1)^{n-1} \sum_{\sigma \in S_n^{\textup{cyc}}} \prod_{i=1}^n K_{+,-}(x_i,x_{\sigma(i)};\hslash) = (-1)^{n-1} \sum_{\sigma \in S_n^{\textup{cyc}}} \prod_{i=1}^n K_{-,+}(x_i,x_{\sigma(i)};\hslash) \,,$$ where the kernels $K_{\pm}$ are defined by $$\begin{aligned} K_{+,-}(x,y;\hslash) & \coloneqq \frac{ \sum_{m=0}^{r-1} \widetilde{\psi}_{r}^{(m)}(x;\hslash) \, \widetilde{\phi}_{r}^{(r-1-m)}(y;\hslash) }{x-y} \,, \\ K_{-,+}(x,y;\hslash) & \coloneqq - \frac{ \sum_{m=0}^{r-1} \widetilde{\phi}_{r}^{(m)}(x;\hslash) \, \widetilde{\psi}_{r}^{(r-1-m)}(y;\hslash) }{x-y} \,. \end{aligned}$$ Moreover, the kernels are related by the parity relation $K_{-,+}(x,y;\hslash) = - K_{+,-}(x,y;-\hslash)$.* ## Singularity structure and large genus asymptotics From the usual properties of the Borel transform, we immediately find the singularity structure of $W_n$ in the Borel plane from the knowledge of the singularity structure of the functions $\widehat{\psi}$ and $\widehat{\phi}$. Again, we assume $(x_1,\dots,x_n)$ in a generic position. **Proposition 31** (Singularity structure of the higher Airy correlators). *The Borel transform $\widehat{W}_n(\bm{x};s)$ of the $n$-point $r$-Airy correlator is simple resurgent. More precisely, on the principal sheet, $\widehat{W}_n$ has $2(r-1)n$ logarithmic singularities located at $s = \pm A_{r,\alpha}(x_i)$ (with $\alpha = 1,\dots,r-1$ and $i = 1,\dots,n$) and such that $$\widehat{W}_{n}(\bm{x};s) = - \frac{S_{r,\alpha}}{2\pi} \, \widehat{W}_{n}^{(\pm\alpha,i)}\bigl( \bm{x}; s \mp A_{r,\alpha}(x_i) \bigr) \log\bigl( s \mp A_{r,\alpha}(x_i) \bigr) + \text{holomorphic at } \pm A_{r,\alpha}(x_i) \,,$$ where:* - *$A_{r,\alpha}(x) = \frac{r}{r+1} (1 - \zeta^{\alpha}) x^{(r+1)/r}$ and $S_{r,\alpha} = (-1)^{\frac{\alpha-r+1}{2}}$ are the instanton actions and the Stokes constants associated to the $r$-Airy functions;* - *the minor $\widehat{W}^{(\pm \alpha,i)}_{n}$ is the Borel transform of the formal power series $$\label{eqn:1inst:corr:rspin} W_{n}^{(\pm \alpha,i)}( \bm{x}; \hslash) \coloneqq (-1)^{n-1} \sum_{\sigma \in S_n^{\textup{cyc}}} K_{\pm\alpha,\mp}(x_i,x_{\sigma(i)};\hslash) \prod_{j \ne i} K_{\pm,\mp}(x_j,x_{\sigma(j)};\hslash) \,,$$ where the kernels $K_{\pm \alpha,\mp}$ are given by $$\begin{aligned} K_{+\alpha,-}(x,y;\hslash) & \coloneqq \frac{ \sum_{m=0}^{r-1} \widetilde{\psi}_{\alpha}^{(m)}(x;\hslash) \, \widetilde{\phi}_{r}^{(r-1-m)}(y;\hslash) }{x-y} \,, \\ K_{-\alpha,+}(x,y;\hslash) & \coloneqq - \frac{ \sum_{m=0}^{r-1} \widetilde{\phi}_{\alpha}^{(m)}(x;\hslash) \, \widetilde{\psi}_{r}^{(r-1-m)}(y;\hslash) }{x-y} \,. \end{aligned}$$* *Moreover, the kernels are related by the parity relation $K_{-\alpha,-}(x,y;\hslash) = - K_{+\alpha,+}(x,y;-\hslash)$. Hence, the analogous relation holds for the correlators: $W^{(-\alpha,i)}_{n}( \bm{x}; \hslash) = (-1)^n \, W^{(+\alpha,i)}_{n}( \bm{x}; -\hslash)$.* Again, since $W^{(\pm\alpha, i)}_{n}$ are related by the parity relation, we can simply use $W_n^{(+\alpha,i)}$ and drop the '$+$' symbol from the superscript. From the analysis of the singularity structure of $W_n$ in the Borel plane, we find the large genus asymptotics of its coefficients through the Borel transform method. **Proposition 32** (Large genus asymptotics of the $r$-Airy correlators). *The large genus asymptotics of the expansion coefficients of the $n$-point $r$-Airy correlators is given by $$\label{eqn:large:g:corr:rspin} W_{g,n}(\bm{x}) = \sum_{\alpha=1}^{r-1} \frac{S_{r,\alpha}}{\pi} \sum_{i=1}^n \frac{\Gamma(2g-2+n)}{A_{r,\alpha}(x_i)^{2g-2+n}} \left( \sum_{k = 0}^K \frac{A_{r,\alpha}(x_i)^{k}}{(2g-3+n)^{\underline{k}}} W^{(\alpha,i)}_{k,n}(\bm{x}) +\mathrm{O}\biggl( \frac{1}{g^{K+1}} \biggr) \right) \,.$$ Moreover, the leading term is explicitly given by $$\begin{gathered} \label{eqn:large:g:corr:leading:rspin} W_{g,n}(\bm{x}) = \frac{(-1)^{g-1+n}}{\pi} \frac{ \Gamma(2g-2+n) }{ \left( \frac{r}{r+1} \right)^{2g-2+n} } \frac{2^n}{r^n x_1 \cdots x_n} \\ \times \Bigg[ \sum_{\alpha = 1}^{\floor{\frac{r-1}{2}}} \frac{(-1)^{\alpha n}}{\left( 2\sin(\frac{\alpha}{r} \pi) \right)^{2g-1+n}} \Bigl( h_{(r+1)(2g-2+n)}^{(r,\alpha)}(\bm{x}^{-1/r}) + \mathrm{O}\bigl( g^{-1} \bigr) \Bigr) \\ + \frac{\delta_{r}^{\textup{even}}}{2} \, \frac{(-1)^{\frac{r}{2}n}}{2^{2g-1+n}} \Bigl( h_{(r+1)(2g-2+n)}^{(r,\frac{r}{2})}(\bm{x}^{-1/r}) + \mathrm{O}\bigl( g^{-1} \bigr) \Bigr) \Bigg] \,. \end{gathered}$$ Here $\delta_{r}^{\textup{even}}$ gives one if $r$ is even and zero otherwise, and $h_{D}^{(r,\alpha)}(\bm{u})$ denotes the polynomial $$\label{eqn:h:rspin} h_{D}^{(r,\alpha)}(\bm{u}) \coloneqq \sum_{ \substack{k_1,\dots,k_n \ge 0 \\ |k| = D} } \prod_{i=1}^n \sin\Bigl( \frac{\alpha k_i}{r}\pi \Bigr) u_i^{k_i} = \sum_{ \substack{d_1,\dots,d_n \ge 0 \\ a_1,\dots,a_n \in \{1,\dots,r-1\} \\ r|d|+|a| = D} } (-1)^{\alpha|d|} \prod_{i=1}^n \sin\left( \frac{\alpha a_i}{r}\pi \right) u_i^{r d_i+a_i} \,.$$* *Proof.* follows from [Theorem 5](#thm:large:order){reference-type="ref" reference="thm:large:order"} and the parity relation. For the leading term, notice that $\widetilde{\psi}_r^{(m)}(x;\hslash) = \frac{1}{\sqrt{r}} x^{-\frac{r-1-2m}{2r}} + \mathrm{O}(\hslash)$ and $\widetilde{\phi}_r^{(r-1-m)}(y;\hslash) = \frac{1}{\sqrt{r}} y^{\frac{r-1-2m}{2r}} + \mathrm{O}(\hslash)$. Thus, $$K_{+,-}(x,y;\hslash) = \frac{1}{r} \frac{1}{x-y} \left( \frac{x}{y} \right)^{\frac{1-r}{2m}} \sum_{m=0}^{r-1} \left( \frac{x}{y} \right)^{\frac{m}{r}} + \mathrm{O}(\hslash) = \frac{1}{r} \frac{(xy)^{\frac{1-r}{2r}}}{x^{1/r} - y^{1/r}} + \mathrm{O}(\hslash) \,.$$ A similar computation shows that $K_{+\alpha,-}(x,y;\hslash) = - \frac{(-1)^{\frac{r-\alpha}{2}}}{r} \frac{(xy)^{\frac{1-r}{2r}}}{\zeta^{\alpha/2} x^{1/r} - \zeta^{-\alpha/2} y^{1/r}} + \mathrm{O}(\hslash)$. Inserting the above expressions in the definition of $W^{(\alpha,i)}_{n}$, we find $$\begin{split} W^{(\alpha,i)}_{0,n}(\bm{x}) &= (-1)^{n-1} \frac{- (-1)^{\frac{r-\alpha}{2}}}{r^n (x_1 \cdots x_n)^{\frac{r-1}{r}}} \sum_{\sigma \in S_n^{\textup{cyc}}} \frac{1}{\zeta^{\alpha/2} x_i^{1/r} - \zeta^{-\alpha/2} x_{\sigma(i)}^{1/r}} \prod_{j \ne i} \frac{1}{ x_j^{1/r} - x_{\sigma(j)}^{1/r}} \\ &= (-1)^{n-1} \frac{(-1)^{\frac{r-\alpha}{2}}}{r^n (x_1 \cdots x_n)^{\frac{r-1}{r}}} \frac{ \zeta^{\frac{\alpha}{2}} (1-\zeta^{\alpha})^{n-2} \, x_i^{\frac{n-2}{r}} }{ \prod_{j \ne i} (x_i^{1/r} - x_j^{1/r})(\zeta^{\alpha} x_i^{1/r} - x_j^{1/r}) } \,. \end{split}$$ In the second line, we applied [Lemma 38](#tech:lemma){reference-type="ref" reference="tech:lemma"} to simplify the sum over permutations. Recalling the values of $S_{r,\alpha} = (-1)^{\frac{\alpha-r+1}{2}}$ and $A_{r,\alpha}(x) = \frac{r}{r+1} (1 - \zeta^{\alpha}) x^{(r+1)/r}$, we find $$\begin{aligned} W_{g,n}(\bm{x}) & = \frac{(-1)^{n}}{\pi\mathrm{i}\, r^n (x_1 \cdots x_n)^{\frac{r-1}{r}}} \frac{\Gamma(2g-2+n)}{\left( \frac{r}{r+1} \right)^{2g-2+n}} \sum_{\substack{\alpha = 1,\dots,r-1 \\ i = 1,\dots,n}} \left( \frac{ \zeta^{\frac{\alpha}{2}} (1-\zeta^{\alpha})^{-2g} \, x_i^{-\frac{(r+1)(2g-2+n)+n-2}{r}} }{ \prod_{j \ne i} (x_i^{1/r} - x_j^{1/r})(\zeta^{\alpha} x_i^{1/r} - x_j^{1/r}) } + \mathrm{O}\bigl( g^{-1} \bigr) \right) \\ & = \frac{(-1)^n}{\pi \mathrm{i}\, r^n (x_1 \cdots x_n)^{\frac{r-1}{r}}} \frac{\Gamma(2g-2+n)}{\left( \frac{r}{r+1} \right)^{2g-2+n}} \sum_{i=1}^n \Bigg( \sum_{\alpha = 1}^{\floor{\frac{r-1}{2}}} \Bigg( \frac{ \zeta^{\frac{\alpha}{2}} (1-\zeta^{\alpha})^{-2g} }{ \prod_{j \ne i} (x_i^{1/r} - x_j^{1/r})(\zeta^{\alpha} x_i^{1/r} - x_j^{1/r}) } \\ &\qquad\qquad\qquad\qquad - \frac{ \zeta^{-\frac{\alpha}{2}} (1-\zeta^{-\alpha})^{-2g} }{ \prod_{j \ne i} (x_i^{1/r} - x_j^{1/r})(\zeta^{-\alpha} x_i^{1/r} - x_j^{1/r}) } \Bigg) x_i^{-\frac{(r+1)(2g-2+n)+n-2}{r}} \\ &\qquad\qquad\qquad\qquad + \delta_{r}^{\textup{even}} \frac{ \mathrm{i}\, 2^{-2g} }{ \prod_{j \ne i} (x_i^{1/r} - x_j^{1/r})(- x_i^{1/r} - x_j^{1/r}) } x_i^{-\frac{(r+1)(2g-2+n)+n-2}{r}} + \mathrm{O}\bigl( g^{-1} \bigr) \Bigg) \,. \end{aligned}$$ In the second line, we have re-arranged the sum as $$\sum_{\alpha=1}^{r-1} w_{\alpha} = \sum_{\alpha=1}^{\floor{\frac{r-1}{2}}} ( w_{\alpha} + w_{r-\alpha}) + \delta_{r}^{\textup{even}} \, w_{\frac{r}{2}} \,.$$ The reason behind it is that the singularities corresponding to $(1-\zeta^{\alpha})$ and $(1-\zeta^{r-\alpha})$ are complex conjugate, hence at the same distance from the origin (cf. [\[fig:actions:rspin\]](#fig:actions:rspin){reference-type="ref" reference="fig:actions:rspin"}). It is natural to combine them, since they contribute to the same leading asymptotic. We can now perform the sum over $i$ employing [Lemma 39](#lemma:poly){reference-type="ref" reference="lemma:poly"} to get the thesis. ◻ As a consequence, we obtain the following asymptotic formula for $r$-spin intersection numbers, expressing all $\mathrm{O}(1)$ terms for each exponential contribution. **Corollary 33** (Leading large genus asymptotics for $r$-spin intersection numbers). *For any given $n \ge 1$ and $a_1,\dots,a_n \in \set{1,\dots,r-1}$, uniformly in $d_1,\dots,d_n$ as $g \to \infty$: $$\begin{gathered} \braket{\tau_{d_1,a_1} \cdots \tau_{d_n,a_n}}^{r\textup{-spin}} \prod_{i=1}^n (rd_i + a_i)!_{(r)} \\ = \frac{2^n}{2\pi} \, \frac{\Gamma(2g-2+n)}{r^{g-1-|d|}} \Bigg[ \sum_{\alpha = 1}^{\floor{\frac{r-1}{2}}} \frac{ (-1)^{(\alpha-1)(|d|+n)} }{ \left( \frac{2r}{r+1} \sin(\frac{\alpha}{r}\pi) \right)^{2g-2+n} } \Biggl( \frac{\prod_{i=1}^n \sin(\frac{\alpha a_i}{r}\pi)}{\sin(\frac{\alpha}{r}\pi)} + \mathrm{O}\bigl( g^{-1} \bigr) \Biggr) \\ + \frac{\delta_{r}^{\textup{even}}}{2} \, \frac{ (-1)^{(\frac{r}{2} - 1)(|d|+n)} }{ \left( \frac{2r}{r+1} \right)^{2g-2+n} } \Biggl( \prod_{i=1}^n \sin\left( \tfrac{a_i}{2}\pi \right) + \mathrm{O}\bigl( g^{-1} \bigr) \Biggr) \Bigg] \,. \end{gathered}$$* **Remark 34**. At a practical level, only the term corresponding to $\alpha = 1$ contributes in the asymptotic formula (for $r > 2$), since all other values of $\alpha$ give exponentially suppressed terms. In this case, the formula simplifies to $$\begin{gathered} \braket{\tau_{d_1,a_1} \cdots \tau_{d_n,a_n}}^{r\textup{-spin}} \prod_{i=1}^n (rd_i + a_i)!_{(r)} \\ = \frac{2^n}{2\pi \sin(\frac{\pi}{r})} \, \frac{ \Gamma(2g-2+n) }{ r^{g-1-|d|} \, \left( \frac{2r}{r+1} \sin(\frac{\pi}{r}) \right)^{2g-2+n} } \Biggl( \prod_{i=1}^n \sin\left( \tfrac{a_i}{r}\pi \right) + \mathrm{O}\bigl( g^{-1} \bigr) \Biggr) \,. \end{gathered}$$ Specialised to $n = 1$, the above asymptotic formula retrieves the one proved by Dubrovin--Yang--Zagier in [@DYZ theorem 5 (vii)]. We also emphasise that, despite being exponentially suppressed, the terms corresponding to $\alpha > 1$ can be detected numerically as explained in [5.6.3](#app:exp){reference-type="ref" reference="app:exp"}. ### Subleading contributions Similarly to the $r = 2$ case, the minors exhibit a distinct pole structure and polynomiality structure. We provide a brief outline of the proof, as it resembles the $r = 2$ case with a few minor variations that we outline below. **Proposition 35**. *For any $k \ge 0$, $n \ge 2$, and $\alpha = 1,\dots, r-1$: $$\begin{gathered} \label{eqn:minors:rspin} W_{k,n}^{(\alpha,i)}(\bm{x}) = \frac{(-1)^{n-1+\frac{r-\alpha}{2}}}{r^n (x_1 \cdots x_n)^{\frac{r-1}{r}}} \frac{ \zeta^{\alpha/2} (1-\zeta^{\alpha})^{n-2} \, x_i^{\frac{n-2-(r+1)k}{r}} }{ \prod_{j \ne i} (x_i^{1/r} - x_j^{1/r})(\zeta^{\alpha}x_i^{1/r} - x_j^{1/r}) } \\ \times R_{k,n-1}^{(\alpha)}\left( \Big( \frac{x_i}{x_1} \Big)^{1/r} , \dots, \widehat{ \Big( \frac{x_i}{x_i} \Big)^{1/r} } , \dots, \Big( \frac{x_i}{x_n} \Big)^{1/r} \right), \end{gathered}$$ where $R_{k,n-1}^{(\alpha)}$ is a symmetric polynomial of degree $\le \min\set{ (r+1)k + n - 2, 2((r+1)k - 1)}$ and degree $\le (r+1)k$ in each individual variable satisfying the $\mathbb{Z}/r\mathbb{Z}$-symmetry $$(-\zeta^{\alpha})^k \cdot R_{k,n}^{(\alpha)}(\zeta^{-\alpha}u_1, \dots, \zeta^{-\alpha}u_n) = R_{k,n}^{(r-\alpha)}(u_1, \dots, u_n) \,.$$ The sequence of polynomials $(R_{k,n}^{(\alpha)})_{n \ge 1}$ satisfies the following specialisation properties: $$\label{eqn:spec:rspin} \begin{aligned} & R_{k,n+1}^{(\alpha)}(u_1,\dots, u_n, u_{n+1}) \big|_{u_{n+1} = 1} = R_{k,n}^{(\alpha)}(u_1,\dots,u_n) \,, \\ & R_{k,n+1}^{(\alpha)}(u_1,\dots, u_n, u_{n+1}) \big|_{u_{n+1} = \zeta^{-\alpha}} = R_{k,n}^{(\alpha)}(u_1,\dots,u_n) \,. \end{aligned}$$ In turn, the degree condition and the specialisation properties uniquely determine $R_{k,n}^{(\alpha)}$ for all $n$ by an explicit algorithm from the first terms of the sequence[^7]. Moreover, the coefficient of $R_{k,n}^{(\alpha)}$ in the monomial basis is a polynomial of $n$.* *Sketch of the proof.* The Wronskian condition appearing in the residue computation of the kernels (cf. [Lemma 21](#lemma:poles:minors){reference-type="ref" reference="lemma:poles:minors"}) is no longer the key fact. This is because, for $\Psi \in \mathrm{SL}(2,\mathbb{C})$, matrix elements of $\Psi^{-1}$ are linear in the matrix elements of $\Psi$. However, this linearity does not hold for elements of $\mathrm{SL}(r,\mathbb{C})$ with $r > 2$. Nonetheless, we can employ the relation $\Phi^t \Psi = \mathrm{Id}$, which indeed yields quadratic relations that generalise the Wronskian condition. This, together with homogeneity arguments, yields [\[eqn:minors:rspin\]](#eqn:minors:rspin){reference-type="ref" reference="eqn:minors:rspin"} and the specialisation properties. From the specialisation properties, one can prove that the degree of $R_{k,n}^{(\alpha)}$ is eventually independent of $n$, and more precisely $$\deg{( R_{k,n}^{(\alpha)} )} \le \min\Set{ (r+1)k + n - 1, 2(r+1)k - 2 } \,.$$ Thus, from the degree bound and the specialisation properties, we get an algorithm for computing $(R_{k,n}^{(\alpha)})_{n \ge 1}$ from the first few values of $n$, together with the polynomiality statement for the expansion coefficients in the monomial basis. As for the $\mathbb{Z}/r\mathbb{Z}$-symmetry, which substitutes evenness for the corresponding polynomials in the $r=2$ case, it follows from analogous relations between the wave functions. ◻ As a consequence of the above result, one can prove that the subleading corrections to $W_{g,n}$ exhibit the following structure. Firstly, let us group together subleading corrections in accordance with the absolute value of the instanton actions. In other words, let us group together terms corresponding to $\alpha$ and $r-\alpha$: $$\begin{gathered} W_{g,n}(\bm{x}) = \frac{(-1)^n}{\pi} \Gamma(2g-2+n) \Bigg[ \sum_{\alpha=1}^{\floor{\frac{r-1}{2}}} \Bigg( \sum_{k=0}^K \frac{1}{(2g-3+n)^{\underline{k}}} \frac{V_{k,g,n}^{(\alpha)}(\bm{x})}{x_1 \cdots x_n} + \mathrm{O}\biggl( \frac{1}{|A_{r,\alpha}(x_i)|^{2g} g^{K+1}} \biggr) \Bigg) \\ + \frac{\delta_{r}^{\textup{even}}}{2} \Bigg( \sum_{k=0}^K \frac{1}{(2g-3+n)^{\underline{k}}} \frac{V_{k,g,n}^{(\frac{r}{2})}(\bm{x})}{x_1 \cdots x_n} + \mathrm{O}\biggl( \frac{1}{|A_{r,\frac{r}{2}}(x_i)|^{2g} g^{K+1}} \biggr) \Bigg) \Bigg],\end{gathered}$$ where we have set $$V_{k,g,n}^{(\alpha)}(\bm{x}) \coloneqq (-1)^n \, x_1 \cdots x_n \sum_{i=1}^n \left( S_{r,\alpha} \frac{W_{k,n}^{(\alpha,i)}(\bm{x})}{A_{r,\alpha}(x_i)^{2g-2+n-k}} + S_{r,r-\alpha} \frac{W_{k,n}^{(r-\alpha,i)}(\bm{x})}{A_{r,r-\alpha}(x_i)^{2g-2+n-k}} \right) \,.$$ With the residue strategy of [\[lemma:poly:subleading,lemma:poly\]](#lemma:poly:subleading,lemma:poly){reference-type="ref" reference="lemma:poly:subleading,lemma:poly"}, but taking $f_{\alpha}(q) = \frac{1}{\pi\mathrm{i}} \frac{\zeta^{\alpha/2} (1-\zeta^{\alpha})^{k+1-2g}}{\prod_{i=1}^n (q-z_i)(\zeta^{\alpha}q-z_i)} R_{k,n}^{(\alpha)}(q \bm{z}^{-1})$, we find $$\begin{gathered} V_{k,g,n}^{(\alpha)}(\bm{x}) = (-1)^{g-1+\alpha n} \frac{2^n}{2 \sin(\frac{\alpha}{r}\pi)} \frac{1}{r^n \left( \frac{2r}{r+1} \sin(\frac{\alpha}{r}\pi) \right)^{2g-2+n-k}} \\ \sum_{d= 0}^{\min{\set{ (r+1)k + n - 1, 2(r+1)k - 2 }} } \bar{R}_{k,n}^{(\alpha,d)}(\bm{x}^{-1/r}) \, h^{(r,\alpha)}_{(r+1)(2g-2+n)-d}(\bm{x}^{-1/r})\,,\end{gathered}$$ where $h^{(r,\alpha)}_{D}$ is the polynomial defined in [\[eqn:h:rspin\]](#eqn:h:rspin){reference-type="ref" reference="eqn:h:rspin"}, $\bar{R}_{k,n}^{(\alpha,d)}$ is the homogeneous component of $R_{k,n}^{(\alpha)}$ of degree $d$, normalised according to the following formula: $$R_{k,n}^{(\alpha)}(q\bm{u}) = (-1)^{\frac{k}{2}} \sum_{d \ge 0} q^d \, \zeta^{\frac{\alpha (d-k)}{2}} \bar{R}_{k,n}^{(\alpha,d)}(\bm{u}) \,.$$ Except for having to group together instanton actions according to their distance from the origin, we highlight two more differences between the $r = 2$ case and the general $r$ case. Firstly, the $\mathbb{Z}/r\mathbb{Z}$-symmetry, which substitutes the evenness property, is a much weaker constraint. Thus, some simplifications no longer occur. Secondly, the complete homogeneous polynomial is essentially substituted by the polynomials $h^{(r,\alpha)}_D$. While in the complete homogeneous polynomial $h_D$ all monomials of degree $D$ appear with coefficient $1$, in $h^{(r,\alpha)}_D$ all monomials appear with coefficient $\sin(\frac{\alpha k}{r}\pi)$, where $k$ is the exponent of the corresponding variable. The latter depends on the (parity of the) quotient and on the reminder of the Euclidean division of $k$ by $r$: writing $k = r[k] + \braket{k}$ for $[k] \ge 0$ and $\braket{k} \in \set{ 0,\dots,r-1 }$, $$\sin\left( \frac{\alpha k}{r}\pi \right) = (-1)^{\alpha [k]} \sin\left( \frac{\alpha \! \braket{k}}{r}\pi \right) \,.$$ This last difference complicates the extraction of coefficients from products of the form $m_{\nu} h^{(r,\alpha)}_{D}$. Nonetheless, the strategy employed in [5.5](#app:symmetric:fncts){reference-type="ref" reference="app:symmetric:fncts"} can still be adapted to obtain the following formula. For all partitions $\mu = (\mu_1,\dots,\mu_n)$ and $\nu = (\nu_1,\dots,\nu_n)$ (zero parts are allowed): $$\bigl[ u_1^{\mu_1} \cdots u_n^{\mu_n} \bigr] m_{\nu} h^{(r,\alpha)}_{|\mu| - |\nu|} = (-1)^{\alpha ([\mu] - [\nu])} \sum_{ A \in \{-(r-1),\dots,r-1\}^n } M_{n,\mu,\nu}^{(r,\alpha)}(A) \, \prod_{i=1}^n \sin\left( \frac{\alpha A_i}{r}\pi \right) \,.$$ Here $[\mu] = \sum_{i=1}^n [\mu_i]$ is the sum of the quotients of the Euclidean division of the parts $\mu_i$ by $r$, the quantity $M_{n,\mu,\nu}^{(r,\alpha)}(A)$ is the explicit polynomial in the multiplicities $p_{m}(\mu)$ given by $$\label{eqn:M:mu:nu:rspin} M_{n,\mu,\nu}^{(r,\alpha)} \coloneqq \frac{1}{z_{\nu}} \prod_{k=0}^{\nu_1} \prod_{s = -(r-1)}^{r-1} \left( n - \sum_{i=0}^{k-1} p_i(\mu) - \sum_{\substack{ i=k, \dots, \mu_1 \\ i \not\equiv s-k \pmod{r}}} p_{i}(\mu) - \sum_{j = k+1}^{\nu_1} p_{j,s-k-j}(\nu,A) \right)^{\underline{p_{k,s}(\nu,A)}} \,,$$ and $p_{k,s}(\nu,A) \coloneqq \#\set{ i | \nu_i = k \,\wedge\, A_i = s }$ is the joint multiplicity of $(k,s)$ in $(\nu,A)$. As a consequence, we obtain the following asymptotic formula for the $r$-spin intersection numbers. We omit the proof, which is parallel to that of [Theorem 24](#thm:large:g:WK){reference-type="ref" reference="thm:large:g:WK"}. **Theorem 36** (Large genus asymptotics for $r$-spin intersection numbers). *For any given $n \ge 1$ and $a_1,\dots,a_n \in \set{1,\dots,r-1}$, uniformly in $d_1,\dots,d_n$ as $g \to \infty$: $$\begin{gathered} \braket{\tau_{d_1,a_1} \cdots \tau_{d_n,a_n}}^{r\textup{-spin}} \prod_{i=1}^n (rd_i + a_i)!_{(r)} \\ = \frac{2^n}{2\pi} \, \frac{\Gamma(2g-2+n)}{r^{g-1-|d|}} \Bigg[ \sum_{\alpha = 1}^{\floor{\frac{r-1}{2}}} \frac{ 1 }{ \left( \frac{2r}{r+1} \sin(\frac{\alpha}{r}\pi) \right)^{2g-2+n} } \Biggl( \sum_{k=0}^K \frac{ ( \frac{2r}{r+1} \sin(\frac{\alpha}{r}\pi) )^{k} }{(2g-3+n)^{\underline{k}}} \, \gamma^{(r,\alpha)}_{k} + \mathrm{O}\biggl( \frac{1}{g^{K+1}} \biggr) \Biggr) \\ + \frac{\delta_{r}^{\textup{even}}}{2} \, \frac{ 1 }{ \left( \frac{2r}{r+1} \right)^{2g-2+n} } \Biggl( \sum_{k=0}^K \frac{ ( \frac{2r}{r+1} )^{k} }{(2g-3+n)^{\underline{k}}} \, \gamma^{(r,\frac{r}{2})}_{k} + \mathrm{O}\biggl( \frac{1}{g^{K+1}} \biggr) \Biggr) \Bigg] \,. \end{gathered}$$ Each $\gamma^{(r,\alpha)}_{k}$ is a function of $n$ and the multiplicities $p_{m} = \#\set{i | rd_i + a_i = m}$. Moreover, it depends polynomially in the multiplicities, up to an overall factor of $(-1)^{\alpha |d|}$.* **Remark 37**. Contrary to the case $r = 2$, it is not clear whether the subleading corrections $\gamma^{(r,\alpha)}_{k}$ have a polynomiality behaviour in $n$. Moreover, from [\[eqn:M:mu:nu:rspin\]](#eqn:M:mu:nu:rspin){reference-type="ref" reference="eqn:M:mu:nu:rspin"}, there seems to be no bound on the number of multiplicities appearing in $\gamma^{(r,\alpha)}_{k}$ (unless some vanishing condition would hold). Nonetheless, our approach provides an algorithm to compute the subleading corrections for any value of $k$. # Appendices {#appendices .unnumbered} ## Some technical lemmas **Lemma 38**. *Let $n$ be a positive integer and let $z_1, \dots, z_n$ be formal variables. Let $\tau$ be any function. Then: $$\sum_{\sigma \in S_n^{\textup{cyc}}} \frac{1}{z_{\sigma(1)} - \tau(z_{1})} \prod_{i=2}^n \frac{1}{z_{i} - z_{\sigma(i)}} = \frac{(z_1 - \tau(z_1))^{n-2} }{\prod_{i=2}^{n}(z_1 - z_i)(\tau(z_1) - z_i)} \,.$$* *Proof.* We prove this by induction on $n$. The base case is easily checked. Call the left-hand side of the statement $P_n(\bm{z})$. The statement is equivalent to proving that $$P_{n+1}(\bm{z}) = P_n(\bm{z}) \frac{z_1 - \tau(z_1)}{(z_1 - z_{n+1})(\tau(z_1) - z_{n+1})}.$$ Observe that $P_{n+1}(\bm{z})$ can also be written as $$\begin{gathered} \sum_{\sigma \in S_n^{\textup{cyc}}} \Biggl( \frac{1}{(z_{n+1} - \tau(z_1)) (z_{n+1} - z_{\sigma(1)})} \prod_{i=2}^{n} \frac{1}{z_{i} - z_{\sigma(i)}} \\ + \sum_{j=2}^n \frac{1}{(z_{\sigma(1)} - \tau(z_1)) (z_j - z_{n+1}) (z_{n+1} - z_{\sigma(j)})} \prod_{\substack{i=2,\dots,n \\ i \ne j}} \frac{1}{z_i - z_{\sigma(i)}} \Biggr), \end{gathered}$$ since every cyclic permutation of $n+1$ elements can be obtained from a cyclic permutation of $n$ elements by inserting the $(n+1)$-st element in all possible positions. Factoring out $\frac{1}{z_{\sigma(1)} - \tau(z_1)} \prod_{i=2}^{n} \frac{1}{z_{i} - z_{\sigma(i)}}$, we find $$\begin{gathered} \sum_{\sigma \in S_n^{\textup{cyc}}} \frac{1}{z_{\sigma(1)} - \tau(z_1)} \prod_{i=2}^{n} \frac{1}{z_{i} - z_{\sigma(i)}} \left( \frac{z_{\sigma(1)} - \tau(z_1)}{(z_{n+1} - z_{\sigma(1)}) (z_{n+1} - \tau(z_1))} + \sum_{j=2}^{n} \frac{z_j - z_{\sigma(j)}}{(z_{n+1} - z_{\sigma(j)}) (z_j - z_{n+1})} \right) \\ = \sum_{\sigma \in S_n^{\textup{cyc}}} \frac{1}{z_{\sigma(1)} - \tau(z_1)} \prod_{i=2}^{n} \frac{1}{z_{i} - z_{\sigma(i)}} \left( \frac{1}{z_{n+1} - z_{\sigma(1)}} - \frac{1}{z_{n+1} - \tau(z_1)} + \sum_{j=2}^{n} \left( \frac{1}{z_{n+1} - z_{\sigma(j)}} - \frac{1}{z_{n+1} - z_j} \right) \right). \end{gathered}$$ Finally, notice that $$\frac{1}{z_{n+1} - z_{\sigma(1)}} - \frac{1}{z_{n+1} - \tau(z_1)} + \sum_{j=2}^{n} \left( \frac{1}{z_{n+1} - z_{\sigma(j)}} - \frac{1}{z_{n+1} - z_j} \right) = \frac{z_1 - \tau(z_1)}{(z_1 - z_{n+1})(\tau(z_1) - z_{n+1})}$$ does not depend on $\sigma$ and can therefore be brought outside the sum over permutations, providing the wanted prefactor. This concludes the proof of the lemma. ◻ **Lemma 39**. *Let $n$ be a positive integer and let $z_1, \dots, z_n$ be formal variables. Set $\zeta = e^{\frac{2\pi\mathrm{i}}{r}}$ for $r \ge 2$. Then for every $\alpha = 1,\dots, r-1$: $$\begin{gathered} \frac{1}{\pi\mathrm{i}} \sum_{i=1}^n \left( \frac{\zeta^{\alpha/2} (1-\zeta^{\alpha})^{-2g} }{\prod_{j \ne i} (z_i - z_j)(\zeta^{\alpha} z_i - z_j) } - \frac{\zeta^{-\alpha/2} (1-\zeta^{-\alpha})^{-2g} }{\prod_{j \ne i} (z_i - z_j)(\zeta^{-\alpha} z_i - z_j) } \right) z_i^{-(r+1)(2g-2+n)+n-2} \\ = (-1)^{g-1+\alpha n} \frac{2^n}{2\pi \sin(\frac{\alpha}{r} \pi)} \frac{1}{\left( 2 \sin(\frac{\alpha}{r} \pi) \right)^{2g-2+n}} \sum_{ \substack{k_1,\dots,k_n \ge 0 \\ |k| = (r+1)(2g-2+n)}} \prod_{i=1}^n \frac{\sin(\frac{\alpha k_i}{r}\pi)}{z_i^{k_i+1}} \,. \end{gathered}$$* *Proof.* Consider the function $f_{\alpha}(q) \coloneqq \frac{1}{\pi\mathrm{i}} \frac{\zeta^{\alpha/2} (1-\zeta^{\alpha})^{1-2g}}{\prod_{i=1}^n (q - z_i)(\zeta^{\alpha} q - z_i)}$. We omit its dependence on $z_1, \dots, z_n$ for simplicity. As a function of $q$, it has simple poles at $q = z_i$ and $q = \zeta^{-\alpha} z_i$ for $i = 1, \dots, n$ and no other poles. Moreover, its residues are computed as $$\begin{aligned} \mathop{\mathrm{Res}}_{q = z_i} f_{\alpha}(q) & = - \frac{1}{\pi\mathrm{i}} \frac{\zeta^{\alpha/2} (1-\zeta^{\alpha})^{1-2g}}{(1-\zeta^{\alpha}) \prod_{j \ne i}(z_i - z_j)(\zeta^{\alpha} z_i - z_j)} z_i^{-1} \,, \\ \mathop{\mathrm{Res}}_{q = \zeta^{-\alpha} z_i} f_{\alpha}(q) & = - \frac{1}{\pi\mathrm{i}} \frac{\zeta^{-\alpha/2} (1-\zeta^{\alpha})^{1-2g}}{(1-\zeta^{-\alpha}) \prod_{j \ne i}(z_i - z_j)(\zeta^{-\alpha} z_i - z_j)} z_i^{-1} \,. \end{aligned}$$ Thus we find $$\begin{split} f_{\alpha}(q) & = \frac{1}{\pi\mathrm{i}} \sum_{i=1}^n \left( \frac{1}{z_i-q} \frac{\zeta^{\alpha/2} (1-\zeta^{\alpha})^{1-2g}}{(1-\zeta^{\alpha}) \prod_{j \ne i}(z_i - z_j)(\zeta^{\alpha} z_i - z_j) } \right. \\ & \qquad \qquad \qquad \left. + \frac{1}{\zeta^{-\alpha}z_i-q} \frac{\zeta^{-\alpha/2} (1-\zeta^{\alpha})^{1-2g}}{(1-\zeta^{-\alpha}) \prod_{j \ne i}(z_i - z_j)(\zeta^{-\alpha} z_i - z_j) } \right) z_i^{-1} \,. \end{split}$$ Let $D \coloneqq (r+1)(2g-2+n)$. Expanding around $q = 0$ and taking the coefficient of $q^{D-n}$, we obtain $$\begin{split} \bigl[ q^{D-n} \bigr] f_{\alpha}(q) = % \frac{1}{\pi\iu} % \sum_{i=1}^n \left( % \frac{\zeta^{\alpha/2} (1-\zeta^{\alpha})^{1-2g}}{(1-\zeta^{\alpha}) \prod_{j \ne i}(z_i - z_j)(\zeta^{\alpha} z_i - z_j) } % \right. \\ % & \qquad \qquad \qquad \left. % + % \frac{\zeta^{-\alpha/2} (1-\zeta^{\alpha})^{1-2g} \zeta^{\alpha(D-n+1)}}{(1-\zeta^{-\alpha}) \prod_{j \ne i}(z_i - z_j)(\zeta^{-\alpha} z_i - z_j) } % \right) z_i^{-D+n-2} \\ % = \frac{1}{\pi\mathrm{i}} \sum_{i=1}^n \left( \frac{\zeta^{\alpha/2} (1-\zeta^{\alpha})^{-2g}}{\prod_{j \ne i}(z_i - z_j)(\zeta^{\alpha} z_i - z_j) } - \frac{\zeta^{-\alpha/2} (1-\zeta^{-\alpha})^{-2g}}{\prod_{j \ne i}(z_i - z_j)(\zeta^{-\alpha} z_i - z_j) } \right) z_i^{-D+n-2} \, . \end{split}$$ On the other hand, we can compute the expansion of $f_{\alpha}$ around $q = 0$ starting from its definition. Start by considering the expansion $$\prod_{i=1}^n \frac{1}{(q - z_i)(\zeta^{\alpha} q - z_i)} = q^{-n} z_1^{-1} \cdots z_n^{-1} \sum_{k_1,\dots,k_n \ge 0} \prod_{i=1}^n \frac{1-\zeta^{\alpha k_i}}{1-\zeta^{\alpha}} \left( \frac{q}{z_i} \right)^{k_i} \, .$$ Taking the coefficient of $q^{D-n}$, we obtain $$\begin{gathered} \bigl[ q^{D-n} \bigr] f_{\alpha}(q) = \frac{1}{\pi\mathrm{i}} \zeta^{\alpha/2} (1-\zeta^{\alpha})^{1-2g} \sum_{ \substack{k_1,\dots,k_n \ge 0 \\ |k| = D}} \prod_{i=1}^n \frac{1-\zeta^{\alpha k_i}}{1-\zeta^{\alpha}} \, z_i^{-k_i-1} \\ = - \frac{1}{\pi\mathrm{i}} \frac{1}{(\zeta^{\alpha/2} - \zeta^{-\alpha/2})^{2g-1+n}} \sum_{ \substack{k_1,\dots,k_n \ge 0 \\ |k| = D}} \zeta^{\alpha\frac{|k|- (2g-2+n)}{2}} \prod_{i=1}^n (\zeta^{\alpha k_i/2} - \zeta^{-\alpha k_i/2}) \, z_i^{-k_i-1} \,. \end{gathered}$$ Notice that $\zeta^{\beta/2} - \zeta^{-\beta/2} = 2 \mathrm{i}\sin(\frac{\beta}{r}\pi)$. Moreover, the condition $|k| = D =(r+1)(2g-2+n)$ implies that $\zeta^{\alpha \frac{|k|- (2g-2+n)}{2}} = (-1)^{\alpha n}$. Simplifying the remaining powers of $\mathrm{i}$ yields the thesis. ◻ ## Symmetric functions and polynomiality properties {#app:symmetric:fncts} The goal of this appendix is to prove some polynomiality results through the theory of symmetric functions [@Mac98]. Throughout this section, we consider partitions $\lambda = (\lambda_1, \dots, \lambda_n)$ consisting of $n$ ordered parts $\lambda_1 \ge \cdots \ge \lambda_n \ge 0$ (notice that we allow empty parts). Define the weight and the length of a partition $\lambda$ as $$|\lambda| \coloneqq \sum_{i=1}^n \lambda_i \,, \qquad\qquad \ell(\lambda) \coloneqq \max\Set{ i | \lambda_i > 0 } \,.$$ Define the multiplicities and the automorphism factor as $$p_k(\lambda) \coloneqq \#\set{ \lambda_i = k } \,, \qquad\qquad z_{\lambda} \coloneqq \prod_{k = 0}^{\lambda_1} p_k(\lambda)! \,.$$ In this paper, we consider three different bases of symmetric polynomials in the variables $\bm{u} = (u_1, \dots, u_n)$. For any integer $k \ge 0$, the elementary and complete homogeneous symmetric polynomials are defined as $$\label{eqn:elementary:complete} e_k(\bm{u}) \coloneqq \sum_{1 \le i_1 < \cdots < i_k \le n} u_{i_1} \cdots u_{i_k} \,, \qquad\qquad h_k(\bm{u}) \coloneqq \sum_{1 \le i_1 \le \cdots \le i_k \le n} u_{i_1} \cdots u_{i_k} \,,$$ respectively. For any partition $\lambda$ as above, set $e_{\lambda} \coloneqq \prod_{i=1}^n e_{\lambda_i}$ and $h_{\lambda} \coloneqq \prod_{i=1}^n h_{\lambda_i}$. We also define the monomial symmetric polynomial as $$\label{eqn:monomial} m_{\lambda}(\bm{u}) \coloneqq \frac{1}{z_{\lambda}} \sum_{\sigma \in S_n} \prod_{i=1}^n u_i^{\lambda_{\sigma(i)}} \,.$$ Let $\mu = (\mu_1,\dots,\mu_n)$ and $\nu = (\nu_1,\dots,\nu_n)$ be partitions with $|\mu| \ge |\nu|$. The goal of this appendix is to study the coefficients of a generic monomial in the product of a monomial and a complete homogeneous polynomial: $$M_{n,\mu,\nu} \coloneqq \bigl[ u_1^{\mu_1} \cdots u_n^{\mu_n} \bigr] m_{\nu} \, h_{|\mu| - |\nu|} \,.$$ **Proposition 40**. *$M_{n,\mu,\nu}$ is a polynomial in $n$ and the multiplicities $p_0(\mu), \dots, p_{\nu_1 - 1}(\mu)$, explicitly given by $$\label{eqn:M:n:mu:nu} M_{n,\mu,\nu} = \frac{1}{z_{\nu}} \prod_{k=0}^{\nu_1} \Biggl( n - \sum_{i=0}^{k-1} p_i(\mu) - \sum_{j = k+1}^{\nu_1} p_j(\nu) \Biggr)^{\!\! \underline{p_k(\nu)}} \,.$$ Here $x^{\underline{m}} = x (x-1) \cdots (x-m+1)$ denotes falling factorial. Moreover, the degree is equal to $|\nu|$ if we set $\deg{n} = 1$ and $\deg{p_k(\mu)} = k + 1$.* *Proof.* Notice that extracting the coefficient of $u_1^{\mu_1} \cdots u_n^{\mu_n}$ means selecting a submonomial $u_1^{\nu_1} \cdots u_n^{\nu_n}$ from $m_{\nu}$ and complete it to $u_1^{\mu_1} \cdots u_n^{\mu_n}$ by extracting the missing powers from $h_{|\mu| - |\nu|}$. From the definition of complete homogeneous and monomial symmetric polynomials, [\[eqn:elementary:complete,eqn:monomial\]](#eqn:elementary:complete,eqn:monomial){reference-type="ref" reference="eqn:elementary:complete,eqn:monomial"} respectively, we find $$M_{n,\mu,\nu} = \frac{1}{z_{\nu}} \sum_{\sigma \in S_n} \prod_{i=1}^n \Theta(\mu_{i} - \nu_{\sigma(i)}) \,,$$ where $\Theta$ is the Heaviside function. We can re-interpret combinatorially the formula above as follows: $M_{n,\mu,\nu}$ is the number of rearrangements $(\nu_{\sigma(1)},\dots,\nu_{\sigma(n)})$ of $\nu$, for $\sigma \in S_n$, such that $\mu_i \geq \nu_{\sigma(i)}$ for all $i = 1,\dots,n$ (up to the automorphism factor $z_{\nu}$). Intuitively speaking, we are counting in how many ways the Young tableau $\mu$ can contain some rearrangement of the Young tableau $\nu$ (where both tableaux have $n$ rows with zero parts allowed). For instance, when $\mu = (4,3,2,1)$ and $\nu = (3,2,1^2)$ there are $8$ such rearrangements, up to the symmetry factor $z_{\nu} = 2!$ that permutes the last two rows of $\nu$. In other words: $M_{n,\mu,\nu} = \frac{8}{2!} = 4$. The $8$ rearrangements are shown below. is obtained by counting the number of embeddings of the parts of $\nu$ into $\mu$, starting from the parts with the highest value of $\nu$ and proceeding in decreasing order. This concludes the proof. ◻ **Remark 41**. The above counting can be performed in a dual way, namely by counting the number of coverings of $\nu$ by parts of $\mu$, starting by the smallest parts of $\mu$ and proceeding in increasing order. From this point-of-view, we obtain $$M_{n,\mu,\nu} = \frac{1}{z_{\nu}} \prod_{k=0}^{\nu_1} \Biggl( \sum_{i = 0}^{k} p_i(\nu) - \sum_{j=0}^{k-1} p_j(\mu) \Biggr)^{\!\! \underline{\tilde{p}_k(\mu)}} %\in \Q\bigl[ p_0(\nu), \dots, p_{\nu_1}(\nu) \bigr] ,$$ where and $\tilde{p}_k(\mu) = p_k(\mu)$ for all $k$, with the only exception of $k = \nu_1$ for which one sets $\tilde{p}_{\nu_1}(\mu) = \sum_{m \geq \nu_1} {p}_m(\mu)$. **Example 42**. From [\[eqn:M:n:mu:nu\]](#eqn:M:n:mu:nu){reference-type="ref" reference="eqn:M:n:mu:nu"}, one can show that $$M_{n,\mu,(1^m,0^{n-m})} = \frac{(n - p_0(\mu))^{\underline{m}}}{m!} \,, \quad M_{n,\mu,(2,1^m,0^{n-m-1})} = \frac{(n - p_0(\mu) - 1)^{\underline{m}} (n - p_0(\mu) - p_1(\mu))}{m!} \,.$$ ## Visualising the large genus asymptotics {#app:visual} In the main body of the paper, we have highlighted several features of the large genus asymptotics of intersection numbers. On the one hand, thanks to determinantal formulae, we were able to provide an algorithm for the computation of subleading corrections to the leading asymptotics of $\psi$-class intersection numbers as functions of the multiplicities $p_i$. On the other hand, we have shown how, in the presence of multiple Borel plane singularities, the resurgent large-order asymptotics organise themselves according to the distance from the origin of the various singularities. This results in the leading asymptotics being determined by the singularity that is the closest to the origin, while the remaining singularities yield exponentially subleading contributions. In the case of the $W_{g,n}(\bm{x})$ correlators, because of the competition between the Borel plane singularities $\pm A(x_i)$, the values of the $x_i$ determine which singularity yields the leading asymptotics. This kind of effect manifests itself also in the case of the $r$-spin intersection numbers with $r \geq 3$, where we observe exponentially suppressed corrections to the leading asymptotics associated to Borel singularities which are at higher distances from the origin than the leading ones. Most of the above-mentioned effects are subleading in nature, and as such hidden by the leading asymptotics. However, as it will be made clear below, one can always construct sequences out of intersection numbers which converge to various quantities highlighting subleading corrections to the leading asymptotics. Plotting such sequences provides instructive visualisations of various effects. ### Subleading corrections to the asymptotics of $\psi$-class intersection numbers The large genus asymptotics for $\psi$-class intersection numbers is captured by the following sequence $G_{d,K}$, whose behaviour is dictated by [\[eqn:intro:psi:large:g\]](#eqn:intro:psi:large:g){reference-type="ref" reference="eqn:intro:psi:large:g"}: $$G_{d,K} \coloneqq \mathsf{S}_{\mathsf{A}}^{-1} \, \frac{4\pi}{2^n} \, \frac{\mathsf{A}^{2g-2+n} \, \prod_{i=1}^n (2d_i + 1)!!}{\Gamma(2g-2+n)} \, \braket{\tau_{d_1} \cdots \tau_{d_n}} - \sum_{k=1}^{K}\frac{\mathsf{A}^k}{(2g-3+n)^{\underline{k}}} \, \alpha_{k} = 1 + \mathrm{O}\Bigl( \frac{1}{g^{K+1}} \Bigr) \,.$$ Notice that the sum starts from $k = 1$, i.e. it leaves out the leading term. By construction, the higher $K$ is (that is, the more subleading terms encoded by the coefficients $\alpha_k$ we include in the sequence), the faster the sequence converges to $1$. This is shown for $n = 1$ in [\[fig:subleading\]](#fig:subleading){reference-type="ref" reference="fig:subleading"}, which illustrates how the asymptotics of intersection numbers are substantially improved by including more and more subleading corrections. ### Subleading corrections: dependence on the multiplicities {#app:subleading:mult} The dependence of the coefficients $\alpha_k$ on the multiplicities $p_m = \#\set{ d_i = m }$ can also be illustrated quite effectively. This is achieved by constructing the following sequences $H_{d,K}$, whose large genus asymptotics is again given by [\[eqn:intro:psi:large:g\]](#eqn:intro:psi:large:g){reference-type="ref" reference="eqn:intro:psi:large:g"}: $$H_{d,K} \coloneqq \frac{(2g-3+n)^{\underline{k}}}{\mathsf{A}^K} \, G_{d,K-1} = \alpha_K \bigl( n, p_0, \ldots, p_{\floor{\frac{3}{2}K} - 1} \bigr) + \mathrm{O}\bigl( g^{-1} \bigr) \,.$$ The dependence on the multiplicities $p_m$ means that, according to the family of intersection numbers which are used to construct $H_{d,K}$, the latter converges to different values. This indeed can be observed in [\[fig:multip:dep\]](#fig:multip:dep){reference-type="ref" reference="fig:multip:dep"}. We consider $2$-point intersection numbers and construct the sequences $H_{d,K}$ for four distinct families of intersection numbers, corresponding to $d_1 = 0,1,2,3$, and for three different values of $K$. In [\[fig:multip:dep:K0\]](#fig:multip:dep:K0){reference-type="ref" reference="fig:multip:dep:K0"} ($K=0$) we see that they all converge to the same value, namely $\alpha_0 = 1$, which indeed does not depend on the the multiplicities. In [\[fig:multip:dep:K1\]](#fig:multip:dep:K1){reference-type="ref" reference="fig:multip:dep:K1"} ($K=1$), instead, we see that the sequence constructed from intersection numbers with $d_1 = 0$ converges to a different value compared to the other three sequences, due to the dependence on $p_0$ of $\alpha_1$. Finally, in [\[fig:multip:dep:K2\]](#fig:multip:dep:K2){reference-type="ref" reference="fig:multip:dep:K2"} ($K=2$) we see all four sequences converge to distinct values, as expected from the dependence on $p_0$, $p_1$ and $p_2$ of $\alpha_2$. ### Exponentially subleading corrections {#app:exp} The Borel plane singularity structure of $r$-spin intersection numbers with $r \geq 3$ leads to large genus asymptotics which sharply differ from those of $\psi$-class and $\Theta$-class intersection numbers. First of all, the leading contribution to the asymptotics is given by pairs of complex conjugate instanton actions, which leads to the oscillating asymptotics given in [\[eqn:intro:rspin:large:g\]](#eqn:intro:rspin:large:g){reference-type="eqref" reference="eqn:intro:rspin:large:g"}. Moreover, for $r \geq 4$ the asymptotics are characterised by the presence of exponentially subleading corrections, due to the remaining Borel plane singularities. The leading oscillating behaviour is particularly easy to visualise. One simply has to introduce the sequences $I^{r\textup{-spin}}_{d,a}$, whose asymptotics is given by [\[eqn:intro:rspin:large:g\]](#eqn:intro:rspin:large:g){reference-type="ref" reference="eqn:intro:rspin:large:g"}: $$\begin{split} I^{r\textup{-spin}}_{d,a} & \coloneqq \mathsf{S}_{r,1}^{-1} \, \frac{2\pi}{2^n} \, \frac{(-r)^{g-1-|d|} |\mathsf{A}_{r,1}|^{2g-2+n} \prod_{i=1}^n (r d_i + a_i)!_{(r)}}{\Gamma(2g-2+n)} \, \braket{\tau_{d_1,a_1} \cdots \tau_{d_n,a_n}}^{r\textup{-spin}} \\ & = (-1)^{|d|-g+1} \, \gamma^{(r,1)}_0 + \mathrm{O}\bigl(g^{-1}\bigr) \,. \end{split}$$ This is shown in [\[fig:spin4leading\]](#fig:spin4leading){reference-type="ref" reference="fig:spin4leading"} for $r=4$, $n=1$. The contributions to the asymptotics due to the remaining Borel plane singularities are exponentially suppressed, and therefore a bit trickier to visualise. Nevertheless, this can be achieved by constructing the following sequences: $$J^{r\textup{-spin}}_{d,a,K} \coloneqq \frac{|\mathsf{A}_{r,2}|^{2g-2+n}}{|\mathsf{A}_{r,1}|^{2g-2+n}} \left( I^{r\textup{-spin}}_{d,a} - \sum_{k=0}^{K} \frac{|\mathsf{A}_{r,1}|^k}{(2g-3+n)^{\underline{k}}} \,(-1)^{|d|-g+1}\, \gamma^{(r,1)}_{k} \right) \,.$$ For $K \gg 0$, we are removing a large number of subleading corrections from the first line of [\[eqn:intro:rspin:large:g\]](#eqn:intro:rspin:large:g){reference-type="ref" reference="eqn:intro:rspin:large:g"}, allowing for the second line to show up. This is illustrated , again for $r=4$, $n=1$, in [\[fig:spin4subleading\]](#fig:spin4subleading){reference-type="ref" reference="fig:spin4subleading"}. It is important to note that, for $r=4$, there is only one exponentially subleading correction to the asymptotics, corresponding to the last line of [\[eqn:intro:rspin:large:g\]](#eqn:intro:rspin:large:g){reference-type="ref" reference="eqn:intro:rspin:large:g"}. Such a term, which is specific to the even $r$ case, does not come from a pair of complex-conjugate instanton actions, but rather from a single one. As such, it does not lead to oscillating asymptotics, much like in the $r=2$ case. ### Competition between singularities in the Borel plane {#app:sings} The competition between singularities in the Borel plane is particularly well illustrated by the large genus asymptotics of the $n$-point correlators $W_{g,n}(\bm{x})$. In order to show this effect, let us consider the case of Airy correlators, whose large genus asymptotics were given in [\[eqn:large:g:corr:WK\]](#eqn:large:g:corr:WK){reference-type="ref" reference="eqn:large:g:corr:WK"}. One immediately realises that the large genus asymptotics are dominated by the instanton action $A(x_i)$ with the smallest absolute value. Therefore, as the values of the $x_i$ change, so does the dominance in the large genus asymptotics. This can be seen quite distinctly for two-point correlators by introducing the sequence $L_g$, whose asymptotic is given by [\[eqn:large:g:corr:WK\]](#eqn:large:g:corr:WK){reference-type="ref" reference="eqn:large:g:corr:WK"}: $$L_g(x_1,x_2) \coloneqq 2g\sqrt{\frac{W_{g,2}(x_1,x_2)}{W_{g+1,2}(x_1,x_2)}} = \min\set{ \left|A(x_1)\right|, \, \left|A(x_2)\right| } + \mathrm{O}\bigl( g^{-1} \bigr) \,,$$ with $A(x) = \frac{4}{3}x^{3/2}$. Hence, if for example $x_1$ and $x_2$ are allowed to vary on the positive real axis, one expects $L_g(x_1,x_2)$ to converge to $A(x_1)$ as long as $x_1 \leq x_2$, and to $A(x_2)$ as long as $x_2 \leq x_1$. This is indeed what happens, as it can be seen in [\[fig:W2\]](#fig:W2){reference-type="ref" reference="fig:W2"}. [^1]: Note that for our convenience we choose a convention for the Borel transform that is slightly different from the usual one used in the resurgence literature, which acts as $\hslash^{m+1} \mapsto \frac{s^m}{m!}$. This leads to no substantial differences with respect to the literature we cite in this section. [^2]: In the literature, a different convention is used for $n = 2$. Since the the difference is of order $\hslash^0$ and we are interested in the $\hslash$-asymptotic expansion, we can ignore the correction term. Another difference in the literature is the convention for the matrix $E$, which is sometimes shifted by a matrix proportional to the identity $\lambda \mathrm{Id}$. Thus, $M$ is also shifted by the same matrix $\lambda \mathrm{Id}$. One can prove that such a shift leaves $W_n$ invariant, see [@EM appendix A]. [^3]: *Experimentally, we can see $P_{k,n}$ has degree $2 \min\{ \floor{\frac{3k+n-1}{2}}, 2k \}$, so that the sum in [\[eqn:subleading:poly\]](#eqn:subleading:poly){reference-type="ref" reference="eqn:subleading:poly"} terminates earlier than expected. This would slightly improve the algorithm for computing the subleading corrections, since the improved bound would lower the number of quantities to compute. Moreover, we notice that the expansion of $P_{k,n}$ in the basis of elementary symmetric does not contain all possible partitions of $d$. This underlying structure of intersection numbers was noticed in [@EL; @EM] as well.* [^4]: We also remark that there is also a shortcut proof that applies for the Airy and Bessel cases, though it does not extend to the $r$-spin setting. Consider the homogeneous component of maximal degree in $P_{k,n}(u_1,\dots,u_n)$, denoted by $P_{k,n}^{\textup{max}}(u_1,\dots,u_n)$. Suppose that $u_n$ divides $P_{k,n}^{\textup{max}}$. By symmetry and parity, $u_1^2 \cdots u_n^2$ must divide $P_{k,n}^{\textup{max}}$. Hence, $2n \le \deg{( P_{k,n}^{\textup{max}} )} = \deg{( P_{k,n} )} \le 3k+n-1$. This gives a contradiction as soon as $n \ge 3k$. Thus, $u_n$ cannot divide $P_{k,n}^{\textup{max}}$ for $n \ge 3k$. In this case, we obtain that $\deg{( P_{k,n}^{\textup{max}} )} = \deg{( P_{k,n}^{\textup{max}}\big|_{u_n = 1} )} = \deg{( P_{k,n-1}^{\textup{max}} )}$ from the recursive relation $P_{k,n}( \,\cdot\,, 1) = P_{k,n-1}(\,\cdot\,)$, from which we conclude that $\deg(P_{k,n+1}) = \deg(P_{k,n}) \text{ for any } n \ge 3k-1 \,.$ [^5]: In the literature, a different convention is often found, with all primary fields shifted by a unit: $a_i \mapsto a_i -1$. [^6]: The reader might notice the similarity to the definition of the $\Theta$-class when $r$ is set to $2$. The main difference is in the definition of the fibres of the vector bundle: $H^1(C,L)^{\ast}$ for Witten, $H^1(C,L^{\ast})^{\ast}$ for Norbury. This similarity has been extensively studied in [@CGG], where "higher spin" $\Theta$-classes have been defined. [^7]: *Experimentally, we can see that the degree of $R_{k,n}^{(\alpha)}$ is actually $\le \min\set{ (r+1)k + n - 1, (r+2)k}$. As in the Witten--Kontsevich case, this lower bound would improve the efficiency of the algorithm.*

arxiv_math

--- bibliography: - bib.bib date: - today - title: Introduction --- **New methods for quasi-interpolation approximations: resolution of odd-degree singularities** *Martin Buhmann, Justus-Liebig University, Mathematics, 35392 Giessen, and\ Janin Jäger, Katholische Universität Eichstätt-Ingolstadt, 85049 Ingolstadt, Germany,\ Joaquín Jódar, Department of Mathematics, University of Jaén, and\ Miguel L. Rodríguez, Department of Mathematics, University of Granada, Spain.* *Abstract:* In this paper, we study functional approximations where we choose the so-called radial basis function method and more specifically, quasi-interpolation. From the various available approaches to the latter, we form new quasi-Lagrange functions when the orders of the singularities of the radial function's Fourier transforms at zero do not match the parity of the dimension of the space, and therefore new expansions and coefficients are needed to overcome this problem. We develop explicit constructions of infinite Fourier expansions that provide these coefficients and make an extensive comparison of the approximation qualities and - with a particular focus - polynomial precision and uniform approximation order of the various formulae. One of the interesting observations concerns the link between algebraic conditions of expansion coefficients and analytic properties of localness and convergence. # Introduction {#rem} In this paper, we study functional approximations in one or more variables to approximands that are at a minimum continuous. A variety of approximation methods are available by splines, multivariable polynomials, trigonometric polynomials etc., but here we choose the so-called radial basis function method. Among the approximants that can be formed from the linear spaces spanned by shifts $\varphi(\|\cdot-x_j\|)$, where the norm is usually Euclidean and $\varphi:{\mathbb{R}}_+\to{\mathbb{R}}$ is the so-called radial basis function, are mostly interpolants (see [@rbf]) and quasi-interpolants (see [@qi]). Those in turn can be formed for finitely many scattered data or gridded data, often infinitely many. We will study quasi-interpolants in this article because they are relatively simple to form and have excellent convergence properties. Those originate from the spaces spanned by shifts $\psi(\cdot-x_j)$ containing polynomials of some low total degree *and* the $\psi$ themselves coming from the span of shifts $\varphi(\|\cdot-x_j\|)$ decaying quickly. Also, collocation if often explicitly not desired. Those two properties together avail us to establish convergence theorems if the approximands $f$ are smooth enough, the basis ingredients coming from local Taylor expansions of course. All this rests on the "quasi-Lagrange functions" satisfying the famous Strang and Fix conditions, which we cite in an appropriate form for the convenience of the reader at the end of the introduction. A basic tool is that of Fourier transforms, in one or more dimensions, of a function $f$, $$\hat{f}(\xi)=\int_{\mathbb{R}^n} f(x) e^{-i \xi \cdot x} dx,\quad \xi\in\mathbb{R}^n.$$ Since the $\psi$ are to be linear combinations of the $\varphi(\|\cdot-x_j\|)$ satisfying these conditions depends on orders of singularities of the generalised Fourier transforms of the radial basis functions $\varphi$. In order to resolve those singularities at the origin, the coefficients forming the quasi-Lagrange functions from the $\varphi$s are designed such that there is a high-order contact between the functions at zero. This works very well if the parities of the singularities and the orders of the zero of the trigonometric polynomials in Fourier space that are formed from the aforementioned coefficients match and are even, but if the orders of the singularities are odd, extra work has to be done. Roughly speaking, it is no longer possible to have trigonometric *polynomials* coming from the said coefficients of $\psi$ as a linear combination of shifts $\varphi(\|\cdot-x_j\|)$, but they end up in infinite series in order to have the required high order contact absorbing the singularities of different parities. We aim to undertake an analysis of this phenomenon, our prime examples being multiquadrics (which ought to be used in odd dimensions so that the order of the singularity at the origin is even) and thin-plate splines (which ought to be used in even dimensions so that the order of the singularity at the origin is even). And we intend to employ these radial functions just in the spaces of "wrong" dimensions, i.e., even and odd respectively and develop and compare different methods to resolve the mentioned problems (cardinal functions, infinite expansions for quasi-Lagrange functions and others). The specific examples we study are the thin-plate spline in dimension one and then compare to the generalised multiquadric in one dimension, which is in the 'right' dimension. For the latter we give an explicit expression of the quasi-Lagrange function. We put particular emphasis not only on comparisons but on the order of decay (localness) of the $\psi$s and of course on the polynomial precision of the generated vector spaces. We shall need the following famous result of Strang and Fix several times. (See, among many other possible references, [@qi Theorem 2.2]). **Theorem 1**. *\[Strang and Fix conditions\]* *Let $\psi:\mathbb{R}^n \rightarrow \mathbb{R}$ be a continuous function such that* 1. *there exists a positive $\ell$ such that for some nonnegative integer $m$, when $\|x\|\rightarrow \infty$, $|\psi(x)|={O}(\Vert x\Vert^{-n-m-\ell})$, which immediately implies $m$-fold differentiability of the Fourier transform,* 2. *$D^{\alpha} \hat{\psi}(0)=0$, $\forall \alpha \in \mathbb{Z}_+^n$, $1\leq \vert \alpha\vert\leq m$, and $\hat{\psi}(0)=1$,* 3. *$D^{\alpha}\hat{\psi}(2\pi j)=0,\ \forall j \in \mathbb{Z}^n\setminus\{0\}$ and $\forall \alpha \in \mathbb{Z}_+^n$ with $\vert\alpha\vert \leq m$.* *Then the quasi-interpolant $$Q_hf(x)=\sum_{j\in\mathbb{Z}^n}f(hj)\psi(x/h-j),\qquad x\in{\mathbb{R}}^n,$$ is well-defined and exact on $\mathbb{P}_m$. The approximation error can be estimated by $$\Vert Q_hf-f\Vert_{\infty}=\begin{cases}{O}(h^{m+\ell}), & \text{when } 0<\ell<1,\\ {O}(h^{m+1}\log (1/h)), & \text{when } \ell=1,\\ {O}(h^{m+1}), & \text{when } \ell>1,\\ \end{cases}$$ for $h\rightarrow 0$ and a bounded function $f\in C^{m+1}({\mathbb{R}}^n)$ with bounded derivatives.* Here and below $\mathbb{P}_m$ denotes the space of polynomials of degree at most $m$. # The thin-plate spline in one-dimension using infinitely many coefficients {#sect2} Along this section and as outlined in the introduction, we consider the radial basis function: $\varphi(r)=r^2\log (r)$. Therefore the relevant generalised Fourier transform is (see [@jon Chapter 4.6]): $$\hat{\varphi}(r)=-2\pi\left(\left(1+\frac12-\gamma\right)\delta^{''}(r)- r^{-3}\right),$$ where, as usual, $\delta$ denotes the Delta-distribution and $\gamma$ is Euler's constant ($\gamma \equiv 0.577216$). At this step, we define $\mu$ as the real number (if it exists) satisfying $\hat{\varphi}(r) \doteq r^{-\mu}+O(r^{-\mu+1})$, $r\rightarrow 0^+.$ Here, the $\doteq$ means equality up to a nonzero constant multiple. We have $\mu=3$ in this particular case. The quasi-Lagrange functions will take the form $$\psi(x)=\sum_{j\in \mathbb{Z}} \lambda_j \varphi(|x-j|),\qquad x\in\mathbb{R}.$$ We study the resulting schemes for different choices of the $\lambda$ coefficients in this "wrong" (odd) dimension, where we cannot achieve the Strang-Fix conditions using only finitely many coefficients because the trigonometric expansions with odd-order zeros at the origin (resolving odd order singularities) will always be infinite unlike the even power singularities (we can use powers and tensor-products of $1-\cos\xi$). ## Cardinal interpolation, infinite expansions from shifts of the radial function An Ansatz that always works in forming of Lagrange functions (no longer quasi-Lagrange functions) from equally spaced shifts $$\psi(x)=\chi(x)=\sum_{j\in \mathbb{Z}} \lambda_j \varphi(|x-j|),\qquad x\in\mathbb{R},$$ which satisfy $\psi(k)=\chi(k)=\delta_{0k}$ for all integers $k$, where we use the standard notation $\chi$ for the cardinal functions. The coefficients $\lambda_j$ come from the Fourier expansion within the Wiener algebra of the reciprocal of the so-called symbol, $\sigma(\vartheta)=\sum_{\ell \in {\mathbb{Z}}} \hat{\varphi}(\vartheta+2\pi \ell)$, so that $$\lambda_j= \displaystyle \frac1{2\pi}\int_{\mathbb{T}}\dfrac{e^{i \vartheta j}}{\sigma(\vartheta)} {\rm d}\vartheta,\qquad j\in{\mathbb{Z}},$$ and $$\psi(x)= \displaystyle \sum_{j\in \mathbb{Z}} \lambda_j \varphi(|x-j|),\qquad x\in{\mathbb{R}}.$$ The localness or asymptotic decay of $\psi(x)$ is identified as $|\psi(x)|=O((1+|x|)^{-4}))$ for all (in particular large in modulus) $x$. For a suitable result see [@rbf Theorem 4.3], and here $\mu=3$. We use $\int_{\mathbb{T}}$ for $\int_{-\pi}^\pi$ throughout. As we stated in the introduction, polynomial precision $$\sum_{k\in\mathbb{Z}} p(k)\psi(\cdot-k)\equiv p$$ for some (usually, low-degree) polynomials $p$ is crucial. In this case we get the polynomial precision/reproduction: $\mathbb{P}_2$-reproduction because $\mu=3$. (See [@rbf Theorem 4.4]). A by now standard theorem then delivers a uniform approximation error for suitably smooth approximands $f\in C^{4}(\mathbb{R})$ with bounded derivatives $$\Bigl\| f- \sum_{k\in\mathbb{Z}} f(kh)\psi(h^{-1}\cdot-k))\Bigr\|_\infty=O(h^3 |\log h|)$$ for $h\to 0$. (See [@rbf Theorem 4.6] or the convergence results in [@BDai].) We have to apply the remark in the proof, and not Theorem 4.6 itself, because $\mu$ is an integer or we can apply Theorem [Theorem 1](#SF){reference-type="ref" reference="SF"} with $m=2, \ell=1, n=1$. While cardinal interpolation always works (note the absence of demands for certain parities of dimensions and orders of singularities) because no polynomials of trigonometric type but infinite expansions are used, we now turn to "genuine" quasi-interpolations where no cardinal conditions are demanded. ## Quasi-interpolation without cardinality conditions {#subsect22} We now wish to go away from the well-known cardinal function approach and use straight quasi-interpolation instead. That brings us to the problem, when the parity of the radial function's generalised Fourier transform's singularity at zero is odd, we can no longer form a trigonometric *polynomial* $q$, say, that matches the degree of the said singularity. This comes from the fact that only even powers of trigonometric expansions have finitely many coefficients when written as Fourier series expansions. So we need to use expansions of periodic series with infinitely many coefficients $\lambda_j$ coming from the Fourier coefficients of $(2-2\cos x)^{3/2}$ or for instance $|\sin x|^3$. These are of course by no means unique, but mere examples. Therefore we end up in expressions $$\psi(x)= \displaystyle \sum_{j\in \mathbb{Z}} \lambda_j \varphi(|x-j|),\qquad x\in{\mathbb{R}}.$$ We like to apply the famous Strang and Fix conditions to check the polynomial precision's degree of quasi-interpolation with thin-plate splines in one dimension $$\sum_{k\in\mathbb{Z}} p(k)\psi(\cdot-k)\equiv p.$$ They depend on the properties of the (classical) Fourier transforms of $\psi$. By straightforward computations (and dividing by $2\pi$ in order to normalise) it can be shown that $$\hat{\psi} (0)=1, \qquad \dfrac{d\, \hat{\psi}}{d \xi}(0)=0, \qquad \dfrac{d^{k}\hat{\psi}}{d \xi^k}(2\pi j)=0, \: \forall j \in \mathbb{Z}\setminus \{0\}, \: k=0, 1, \quad \dfrac{d^2\, \hat{\psi}(\xi)}{d \xi^2}(0)\neq 0.$$ As the Strang-Fix condition for first degree derivative is satisfied but the second derivative is not satisfied at $\xi=0$, we have $\mathbb{P}_1-$reproduction at a maximum (but not any higher). We arrive at **Theorem 2**. *Let $\varphi$ be the thin-plate spline radial basis function. With $n=1$ and the Fourier coefficients of $(2-2\cos x)^{3/2}$ being the $\lambda_j$, the quasi-interpolation $$\sum_{k\in\mathbb{Z}} p(k)\psi(\cdot-k)\equiv p$$ is exact for linear polynomials $p$.* The next question is the decay of the quasi-Lagrange functions $\psi$: We expect $|\psi(x)|=O((1+|x|)^{-4})$ (see Theorem [Theorem 11](#th1){reference-type="ref" reference="th1"} for $\underline n=3$). This is one order better than the routinely required third order decay which would suffice for absolute convergence of the quasi-interpolant when at most linearly growing approximands (in particular linear polynomials) are inserted. This leads us to the question of approximation error; a routine result gives us from the identified polynomial precision and the order of decay of the quasi-Lagrange functions $O(h^2 )$ for $h\to 0$. (See Theorem [Theorem 1](#SF){reference-type="ref" reference="SF"} with $m=1, \ell=2, n=1$.) Notice the absence of the logarithmic term due to the one order faster decay of the quasi-Lagrange function. ## An intermediate formulation of $\psi$ and its Fourier transform {#subsect23} Another scheme is the separation of the Fourier transform's singularity into two factors: one that resolves the odd singularities degree separately and leaving an even negative power, and then using the classical approach for the high even order contact at the origin. So we begin in trying to improve the polynomial reproduction using the above scheme and the situation at zero. We set therefore $$\hat{\psi}(\xi)=P(\xi) |\sin(\xi)| \hat{\varphi}(|\xi|),\qquad\xi\in{\mathbb{R}},$$ being $P(\xi)= \displaystyle \sum_{k=-N}^N \mu_k e^{ik\xi}$ a suitable trigonometric polynomial. In summary, we set this new scheme in this way: We have to fix the coefficients of the quasi-Lagrange functions. This always begins with setting the coefficients $\lambda_k$ to be the (infinitely many) Fourier coefficients of the expansion of $$P(\xi)|\sin (\xi)|,\qquad -\pi\leq \xi\leq\pi,$$ -- this is by no means unique, we could use for instance $$P(\xi)(1-\cos (\xi))^{1/2},\qquad -\pi\leq \xi\leq\pi.$$ Many other choices of roots of trigonometric functions are possible. Depending on those coefficients, especially which trigonometric expansions they form and which orders their zeros have, we will arrive at a reproduction of polynomials $\mathbb{P}_2-$precision (see below). In order to compute the terms and derivatives that will serve to verify up to which order the Strang-Fix conditions hold at zero, we will use $$\hat{\psi}(\xi)=2\pi\dfrac{P(\xi) |\sin(\xi)|}{|\xi|^3}\equiv 2\pi\dfrac{P(\xi) \sin(\xi)}{\xi^3}$$ as $\xi$ goes to 0 because the singular term in $\hat{\varphi}(r)$ is a constant multiple of $r^{-3}$ at the origin. In order to satisfy the conditions at the zero for $\mathbb{P}_2$-reproduction we demand according to the Strang and Fix approach $$\label{strang} \hat{\psi}(\xi)=1+ O(\xi^3).$$ Therefore, close to the origin, we can expand $$2\pi\dfrac{P(\xi) \sin(\xi)}{\xi^3}=2\pi \displaystyle \sum_{k=-N}^N \mu_k e^{ik\xi}\left (\xi - \dfrac{\xi^3}{3!}+ \dfrac{\xi^5}{5!}- \dfrac{\xi^7}{7!} \pm \cdots \right)\times \xi^{-3}$$ which is $$2\pi\displaystyle \sum_{k=-N}^N \sum_{j=0}^{\infty} \mu_k \dfrac{(ik\xi)^j}{j!}\left (\xi - \dfrac{\xi^3}{3!}+ \dfrac{\xi^5}{5!}- \dfrac{\xi^7}{7!} \pm \cdots \right)\times \xi^{-3}.$$ Now, by imposing the condition ([\[strang\]](#strang){reference-type="ref" reference="strang"}) we have $$1+ O(\xi^3)= 2\pi\displaystyle \sum_{k=-N}^N \sum_{j=0}^{\infty} \mu_k \dfrac{(ik\xi)^j}{j!}\left (\xi^{-2} - \dfrac{1}{3!}+ \dfrac{\xi^2}{5!}- \dfrac{\xi^4}{7!} \pm \cdots \right).$$ The above equation will give conditions on the $\mu_k$s (they will not be unique, of course). Specifically, for $N=2$ we obtain $$\displaystyle \sum_{k=-2}^2 \mu_k=0,\: \displaystyle \sum_{k=-2}^2 k \mu_k=0, \: - \pi\displaystyle \sum_{k=-2}^2 \left(k^2+ \dfrac13\right) \mu_k = 1, \:$$ and further $$- \dfrac16\displaystyle \sum_{k=-2}^2 \left(k+ k^3\right) \mu_k = 0,\: \dfrac{1}{12}\displaystyle \sum_{k=-2}^2 \left(\dfrac{1}{10}+ k^2+ \dfrac{k^4}{2}\right) \mu_k = 0.$$ This can be formulated equivalently as $$\label{mu} \displaystyle \sum_{k=-2}^2 \mu_k= \displaystyle \sum_{k=-2}^2 k \mu_k=0, \quad \displaystyle \sum_{k=-2}^2 k^2 \mu_k = -\dfrac{1}{\pi}, \quad \displaystyle \sum_{k=-2}^2 k^3 \mu_k = 0, \quad \displaystyle \sum_{k=-2}^2 k^4 \mu_k = \dfrac{2}{\pi}.$$ We will then choose $$\mu_{-2}= \dfrac{1}{8\pi}, \: \mu_{-1} = -\dfrac{1}{\pi}, \: \mu_0 = \dfrac{7}{4\pi}, \: \mu_1 = -\dfrac{1}{\pi}, \: \mu_2= \dfrac{1}{8\pi}$$ or $$P(\xi)= \dfrac{1}{8\pi} e^{-2 i \xi} -\dfrac{1}{\pi} e^{-i \xi}+ \dfrac{7}{4\pi} -\dfrac{1}{\pi} e^{i \xi}+\dfrac{1}{8\pi} e^{2 i \xi},$$ i.e. $$P(\xi)= \dfrac{7}{4\pi}-\dfrac{2}{\pi} \cos(\xi) + \dfrac{1}{4\pi}\cos(2\xi).$$ We can easily verify that the upper bound on the right-hand side of ([\[strang\]](#strang){reference-type="ref" reference="strang"}) holds for $\hat{\psi}(\xi)=P(\xi)|\sin (\xi)|\hat{\varphi}(|\xi|)$. Now, let us study the behaviour of the derivatives $\hat{\psi}^{(\ell)} (2\pi j)$, $j\in\mathbb{Z}\setminus \{0\}$. The purpose of this is of course checking the Strang and Fix conditions: 1. For $\ell=0$ we have $$\hat{\psi}(2\pi j)=0,\quad j\in\mathbb{Z}\setminus \{0\},$$ because of the $|\sin (\xi)|$-term and the continuity of $\hat{\varphi}(\xi)$ away from the origin. 2. For $\ell=1$ $$\label{s1} \dfrac{d\hat{\psi}}{d \xi}(\xi)=P'(\xi)|\sin (\xi)|\hat{\varphi}(\xi) +P(\xi) \dfrac{\sin(\xi) \cos(\xi)}{|\sin (\xi)|}\hat{\varphi}(\xi)+ P(\xi)|\sin (\xi)|\hat{\varphi}'(\xi)$$ which vanishes for all $2\pi j$, $j \in \mathbb{Z}\setminus \{0\}$: the first and the third term clearly vanish; the second term of the right-hand part of ([\[s1\]](#s1){reference-type="ref" reference="s1"}) vanishes due to $\displaystyle \sum_{k=-2}^2 \mu_k=0$. 3. And for $\ell=2$ the derivatives of the first and third terms of ([\[s1\]](#s1){reference-type="ref" reference="s1"}) vanish at $2\pi j$ with $j \in \mathbb{Z}\setminus \{0\}$ (we now have to use that $\displaystyle \sum_{k=-2}^2k \mu_k=0$). In fact, problems could come from the derivative of the second term of ([\[s1\]](#s1){reference-type="ref" reference="s1"}) which we will therefore have to compute explicitly. It is $$\begin{split} &P'(\xi) \dfrac{\sin(\xi) \cos(\xi)}{|\sin (\xi)|}\hat{\varphi}(\xi) +P(\xi) \dfrac{\cos^2(\xi)}{|\sin (\xi)|}\hat{\varphi}(\xi) -P(\xi) |\sin (\xi)|\hat{\varphi}(\xi)\\ & +P(\xi) \dfrac{\sin(\xi) \cos(\xi)}{|\sin (\xi)|}\hat{\varphi}'(\xi) -P(\xi) \dfrac{\cos^2(\xi)}{|\sin (\xi)|}\hat{\varphi}(\xi)\\ &=P'(\xi) \dfrac{\sin(\xi) \cos(\xi)}{|\sin (\xi)|}\hat{\varphi}(\xi) -P(\xi) |\sin (\xi)|\hat{\varphi}(\xi) +P(\xi) \dfrac{\sin(\xi) \cos(\xi)}{|\sin (\xi)|}\hat{\varphi}'(\xi). \end{split}$$ In summary, the second order derivatives of $\hat{\psi}$ vanish at the points $2\pi j$, $j\in\mathbb{Z}\setminus\{0\}$ and we have all the requirements for getting $\mathbb{P}_2-$reproduction. We sum our findings up in the following result. **Theorem 3**. *Let $\varphi$ be the thin-plate spline radial basis function. With $n=1$ and the Fourier coefficients of $P(\cdot)\times|\sin(\cdot)|$, $P$ as above, being the quasi-Lagrange function's coefficients $\lambda_j$, the quasi-interpolation $$\sum_{k\in\mathbb{Z}} p(k)\psi(\cdot-k)\equiv p$$ is exact for quadratic polynomials $p$.* **Remark 4**. *At the points $x=2\pi j, j \in \mathbb{Z}\setminus \{0\}$ the third derivative of $\hat{\psi}$ has a jump discontinuity, so we will not be able to satisfy high enough degree Strang-Fix conditions that we could successfully impose in order to arrive at $\mathbb{P}_3-$reproduction.* We note that a convenient way to show that the second order Strang-and-Fix conditions, and only those degrees, are satisfied at the $\xi=2\pi j, j \in \mathbb{Z}\setminus \{0\}$ (while they would hold at zero up to order three, although that does not help) is to notice the following: if we define $g(\xi)=\frac12\sin|\xi|+\frac12\sin|\xi-\pi|$, then $$|\sin \xi| = \sum_{k=-\infty}^\infty g(\xi-k\pi),\qquad \xi\in\mathbb{R}.$$ Therefore, using $P$ as above and expanding about the origin we get near zero $$\begin{aligned} P(\xi)|\sin \xi|\hat\varphi(|\xi|)&= 2 \pi P(\xi)|\sin \xi| \times |\xi|^{-3} =(\sin \xi) 2 \pi \xi^{-3} \Bigl( \frac{\xi^2}{2 \pi } +\frac{\xi^4}{12 \pi } -\frac{7\xi^6}{360 \pi } +\cdots\Bigr) \\& = \Bigl( \xi- \frac{\xi^3}{6} + \cdots \Bigr) \times \Bigl( \frac{1}{\xi} +\frac{\xi}{6}-\frac{7\xi^3}{180} +\cdots\Bigr)= \end{aligned}$$ which is $$1-\frac{7}{120}\xi^4 + O(\xi^6)$$ near zero (so third order SF-conditions possible) but since $$P(\xi)|\sin \xi|\hat\varphi(|\xi|)=|\xi|^{-3} \left( |\xi-2\pi j|^{3} + O( |\xi-2\pi j|^{7})\right)$$ near $\xi=2\pi j, j \in \mathbb{Z}\setminus \{0\}$, only second order Strang-and-Fix conditions are satisfied. The mentioned third degree derivative discontinuity can be seen as well. In fact, using the above form of $|\sin|$ we can compute its generalised Fourier transform which could then be used to compute some $\psi$ explicitly. The details of the computation are given in Appendix A.1 Lemma [Lemma 14](#sin){reference-type="ref" reference="sin"}, where we show that $${\cal F}^{-1}|\sin|(x)=\frac1{\pi}\times\frac{1+\exp(ix\pi)}{1-x^2}\times{\cal D}_2(x).$$ Here ${\cal D}_2$ is the Dirac comb $${\cal D}_2=\sum_{k=-\infty}^\infty \delta(\cdot-2k).$$ We always have to study the decay of $\psi$, because the resolution of the singularities at the origin will not deliver the desired polynomial precision of $$\sum_{k\in\mathbb{Z}} p(k)\psi(\cdot-k)\equiv p$$ unless the series above converge absolutely. For this we will need at least an asymptotic decay of $O((1+|x|)^{-3-\varepsilon})$ for the quasi-Lagrange function in order to get the summability of the series for the aforementioned quadratic polynomial reproduction. The asymptotic decay is fairly easily established (see, for instance, [@rbf] or [@qi]) by exploiting the differentiability properties of the quasi-Lagrange function's Fourier transform. At even order multiples of $\pi$ this $\hat\psi$ is infinitely smooth, but at odd multiples of $\pi$, we observe the following behaviours. We get about $\xi=(2j+1)\pi$ (but not about $\xi=2j\pi$) $$\lim_{\xi \to -\pi^-} \hat{\psi}'(\xi)=-\dfrac{8}{\pi^3}, \quad \lim_{\xi \to -\pi^+} \hat{\psi}'(\xi)=\dfrac{8}{\pi^3}.$$ As $\hat{\psi}(\xi) \notin C^1(\mathbb{R})$, then the maximum decay we can obtain for $\psi(x)$ will be $O((1+|x|)^{-2})$ according to our previous remark. Finally we note that the approximation order (with the established decay and [@qi Theorem 2.2] with $m=0$) will be $O(h\, |\log h|)$. **Remark 5**. *So far, we have only considered pointwise function evaluation as a means to put the approximand's information into the quasi-interpolant. It is entirely possible to formulate the latter with different linear functionals applied to the approximation: $$Q f(x)= \displaystyle \sum_{j\in \mathbb{Z}} \lambda_j(f) \psi(x-j)$$ where $\lambda_j(f)$ are suitable functionals. For example, they could be related to the polynomial $P(x)$, i.e. $$\lambda_j(f)=\sum_{k=-M}^{M} \mu_k f(j-k) \quad {\rm or} \quad \displaystyle \lambda_j(f)=\sum_{k=0}^{M}\mu_j f^{(k)}(j)$$ being $\mu_k$ the coefficients of $P(x)$ in order to get polynomial reproduction; we may also take local integrals.* *That Ansatz would amount to a preconditioning of the approximand before it is fed into the quasi-interpolation procedure.* *As soon as the decay of $\psi$ in absolute terms is not sufficient for the absolute convergence (summability) of the series in $$\sum_{k\in\mathbb{Z}} p(k)\psi(\cdot-k)\equiv p,$$ the functionals must help in order to improve the decay of the function $\psi$ because we need the summability of the series so that the polynomial precision is formed in a well-defined way. However, we do not provide any further detailed analysis of this aspect in this work (but see [@qi]).* # Finitely many coefficients for the quasi-Lagrange functions {#sect3} ## The classical scheme with the "natural degree" radial basis function in $\mathbb{R}$ {#2s3} In odd dimension the "natural degree" (i.e., even order singularity at the origin) radial basis function using multiquadrics and their ilk will be $\varphi(r)=(r^2+ c^2)^{3/2}$ instead $\varphi(r)=r^2\log (r)$. In this case we have the following scheme. The radial basis function is in the first place $\varphi(r)=(r^2+ c^2)^{3/2}$ which we shall call the generalised multiquadric function. Its Fourier transform has the property that near zero we have that $\hat{\varphi}(r)\doteq r^{-4}$, as required (even order), and therefore in our notation above $\mu=4$. The quasi-Lagrange function's now available *finite number* of coefficients $\lambda_j$ are the Fourier coefficients of $(1-\cos (\xi))^{2}$. The latter is a trigonometric polynomial with an even order zero at the origin. Thus, the $\lambda_j$ will have finite support with respect to their indices. One way of choosing the Fourier transform is, as a consequence, $\hat{\psi}(\xi)=(1-\cos (\xi))^{2} \hat{\varphi}(|\xi|)$. It is interesting to verify the polynomial precision results for the cases in the classical way of analysis. Taking into account that the (distributional) Fourier transform, $\hat{\Phi}(\xi)$, of the generalised multiquadric $\Phi(x)=(c^2+\|x\|^2)^{\beta}$, $x\in\mathbb{R}^n$, $c>0$, $\beta\in\mathbb{R}\setminus\mathbb{N}_0$ is $$\hat{\Phi}(\xi)=(2\pi)^{n/2}\frac{2^{1+\beta}}{\Gamma(-\beta)} \left( \frac{c}{\|\xi\|} \right)^{\beta+\frac{n}{2}}\times K_{\beta+\frac{n}{2}} (c\|\xi\|),\quad \xi \neq 0,$$ according to [@jon] we have, in our case ($n=1, \beta=3/2$), $$\hat{\varphi}(\xi)=(2\pi)^{1/2} \frac{2^{5/2}}{\Gamma (-3/2)} \frac{c^2}{\xi^{2}} K_2(c|\xi|).$$ Furthermore, from [@olv] we read $$\begin{aligned} \label{Olver} \frac{c^s}{\|\xi\|^{s}} K_s(c\|\xi\|) =& 2^{s-1}\frac{1}{\|\xi\|^{2s}} \sum_{k=0}^{s-1} \dfrac{ (s-k-1)!}{k!(-4)^k}(c\|\xi\|)^{2k}+{}\\ &{} -\left(\frac{-c^2}{2}\right)^s \log \|\xi\| \sum_{k=0}^{\infty} \frac{(c\|\xi\|)^{2k}}{4^kk!\Gamma(s+ k+1)} +{} \\&{}- \left(\frac{-c^2}{2}\right)^s\log c \sum\limits_{k=0}^{\infty} \frac{(c\|\xi\|)^{2k}}{4^kk!\Gamma(s+ k+1)} +{}\\&+{} \left(\frac{-c^2}{2}\right)^s \sum_{k=0}^{\infty} \left(\frac{\log 2}{4^kk!\Gamma(s+ k+1)}+\frac12\dfrac{\Psi(k+1)+\Psi(s+k+1)}{4^k(s+k)!k!}\right)\times{}\\ &{}\times(c\|\xi\|)^{2k},\end{aligned}$$ where $\displaystyle\Psi(z)=\frac{\Gamma '(z)}{\Gamma (z)}$ is the Digamma function, and $\Gamma$ is the Gamma function. We summarise it in the following result. **Theorem 6**. *The quasi-Lagrange function satisfies the bound $|\psi(x)|=O((1+|x|)^{-5})$.* This localness, i.e., decay of $\psi$, is identified by differentiating its Fourier transform and because $\hat{\psi}(\xi)$ can be expanded, about $\xi=0$, as $$\tilde{a}+a\xi^2+b\xi^4\log|\xi|+\cdots, \quad \tilde{a},a,b\in\mathbb{R},$$ where the dots mean higher order terms (higher powers of $\xi$ and/or logarithms) in the expansion. Indeed, this comes from the previous expansion of $\frac{c^2}{\|\xi\|^{2}} K_2(c\|\xi\|)$ and the expansion of $(1-\cos (\xi))^{2}$ given by $\frac{\xi^4}{4}$, around $\xi=0$. Therefore, by applying the (inverse) Fourier transform we would obtain the above-mentioned decay (see [@jon], for instance). In order to prove that the polynomial reproduction is $\mathbb{P}_1$, we make some explicit computations. First we derive the suitable normalisation constant, by noting that $$\begin{aligned} \hat{\varphi}(|\xi|)&=(2\pi)^{1/2} \frac{2^{5/2}}{\Gamma (-3/2)} \frac{c^2}{\xi^{2}} K_2(c|\xi|)\\ &=\dfrac{12}{\xi^4}-\dfrac{3c^2}{\xi^2}+6\left(\dfrac{1}{16}c^4\left (\dfrac{3}{2}-2\gamma\right)+\dfrac{1}{8}c^4\log(2)-\dfrac{1}{8}c^4\log(c)-\dfrac{1}{8}c^4\log(|\xi|)\right)+\cdots.\end{aligned}$$ Since we multiply by $(1-\cos(\xi))^2=\frac{\xi^4}{4}{\rm plus\ H.O.T.}$, $\xi\rightarrow 0$, we would have $\hat{\psi}(0)=\frac{12}{4}=3$. Therefore, in this case, we should divide by 3. After dividing by 3, it can be shown that $$\hat{\psi} (0)=1, \qquad \dfrac{d\, \hat{\psi}}{d \xi}(0)=0, \qquad \dfrac{d^{k}\hat{\psi}}{d \xi^k}(2\pi j)=0, \: \forall j \in \mathbb{Z}\setminus \{0\}, \: k=0, 1.$$ As $\dfrac{d^2\, \hat{\psi}(\xi)}{d \xi^2}(0)\neq 0$ the Strang-Fix condition for the second derivative is not satisfied at $\xi=0$. Therefore we will have $\mathbb{P}_1$ reproduction at most. **Theorem 7**. *Let $\varphi$ be the generalised multiquadric radial basis function. With $n=1$ and the Fourier coefficients of $\dfrac{1}{3}(2-2\cos x)^{2}$ being the $\lambda_j$, the quasi-interpolation $$\sum_{k\in\mathbb{Z}} p(k)\psi(\cdot-k)\equiv p$$ is exact for linear polynomials $p$.* The approximation error will have by Theorem [Theorem 1](#SF){reference-type="ref" reference="SF"} with $m=1, \ell=3, n=1$ the asymptotic order $O(h^2)$. **Remark 8**. *As $$(1-\cos\xi)^2=\frac{1}{4}e^{-i2\xi}-e^{-i\xi}+\frac{3}{2}-e^{i\xi}+\frac{1}{4} e^{i2\xi},\qquad -\pi\leq \xi\leq\pi,$$ the kernel $\psi$ is given by the finite sum $$\psi(x)=\frac{1}{4}\varphi(|x+2|)-\varphi(|x+1|)+\frac{3}{2}\varphi(|x|)-\varphi(|x-1|)+\frac{1}{4} \varphi(|x-2|),\;x\in{\mathbb{R}}.$$ We can also use Mathematica to check that this finite sum satisfies $$|\psi (x)|=O((1+|x|)^{-5}),\quad |x|\to \infty.$$* ## Improving the reproduction Starting with the above explicit construction, we can improve the above scheme to $\mathbb{P}_3$-reproduction changing $(1-\cos (\xi))^2$ by a suitable trigonometric polynomial $P(\xi)$. We consider - as the radial basis function we take the generalised multiquadric function defined by $\varphi(r)=(r^2+ c^2)^{3/2}$, - as the finite sum of exponentials in order to resolve the generalised multiquadrics' singularity $P(\xi)=\displaystyle\sum_{k=-N}^{N} \mu_k e^{ik\xi}$, - and therefore, finally, as Fourier transform $\hat{\psi}(\xi)=P(\xi) \hat{\varphi}(|\xi|)$. We adjust the coefficients $\mu_k$ of $P(\xi)$ in such a way that the expansion of $\hat{\psi}(\xi)$ around $\xi=0$ is $1+O\left(\xi^4\log(|\xi|\right)$. To do that, we impose that, in that expansion, the coefficient of $\xi^0$ is 1 and the coefficients of $\xi^{-4},\xi^{-3},\xi^{-2},\xi^{-1},\xi,\xi^2,\xi^3$ vanish (even the vanishing of the coefficient of $\xi^4$ may be added, but not the one of $\xi^4 \log (|\xi|)$, which is not compatible with the previous conditions due to the log-term). The system to be solved collecting all of these requirements is $$\begin{split} \sum_{k=-N}^N \mu_k=\sum_{k=-N}^N k\mu_k=\sum_{k=-N}^N k^2\mu_k= \sum_{k=-N}^N k^3\mu_k=0, \; \sum_{k=-N}^N k^4\mu_k=2, \quad \sum_{k=-N}^N k^5\mu_k=0, \; \\ \sum_{k=-N}^N k^6\mu_k=-15c^2, \; \sum_{k=-N}^N k^7\mu_k=0, \; \sum_{k=-N}^N k^8\mu_k=\frac{105}{2}c^4(1+4\gamma-4\log 2+4\log c). \end{split}$$ Its solution for $N=4$ is: $$\begin{split} \mu_{-4}=\mu_4=&\frac{1}{11520}(28+60c^2+15c^4+60c^4\gamma-60c^4\log 2+60c^4\log c), \\ \mu_{-3}=\mu_3=&\frac{1}{480}(-16-30c^2-5c^4-20c^4\gamma+20c^4\log 2 -20c^4\log c),\\ \mu_{-2}=\mu_2=&\frac{1}{2880} (676+780c^2+105c^4+420c^4\gamma-420c^4\log 2+420c^4\log c), \\ \mu_{-1}=\mu_1=&\frac{1}{1440}(-976-870c^2-105c^4-420c^4\gamma+420c^4\log 2-420c^4\log c),\\ \mu_0=&\frac{1}{384} (364+300c^2+35c^4+140c^4\gamma-140c^4\log 2+140 c^4 \log c). \end{split}$$ By inserting these coefficients into $P(\xi)$ we obtain that the expansion of $\hat{\psi}(\xi)$ around $\xi=0$ is $$1-\frac{c^4}{16}\xi^4\log |\xi|+\cdots.$$ Therefore, by applying the (inverse) Fourier transform (see, e.g., [@jon]), we obtain a *decay of order -5 for $\psi$: $$|\psi (x)|=O((1+|x|)^{-5}),\quad |x|\to \infty.$$* On the other hand, for this choice of $P(\xi)$, the (distributional) Fourier transform of $\psi$, $$\hat{\psi}(\xi)=(2\pi)^{1/2} \frac{2^{5/2}}{\Gamma (-3/2)} \frac{c^2}{\xi^{2}} K_2(c|\xi|) P(\xi),\qquad \xi\in{\mathbb{R}},$$ satisfies the conditions $$\begin{split} \hat{\psi}(0)=1,\quad \frac{d^k \hat{\psi}}{d\xi^k}(2\pi j)=0, \, \forall j\in\mathbb{Z},\, k=1,2,3. \end{split}$$ Therefore, according to the Strang-Fix conditions, $\mathbb{P}_3$-reproduction is reached by the quasi-interpolant. As $\frac{d^4 \hat{\psi}}{d\xi^4}(0)=\infty$ it is not possible $\mathbb{P}_4$-reproduction in this case. **Theorem 9**. *Let $\varphi$ be the generalised multiquadric function. With $n=1$ and the explicit $\lambda_j$ as above for general parameters $c$, the quasi-interpolation $$\sum_{k\in\mathbb{Z}} p(k)\psi(\cdot-k)\equiv p$$ is exact for cubic polynomials $p$.* Putting all together, the *approximation order of the quasi-interpolant is $h^4 |\log h|$* (Theorem [Theorem 1](#SF){reference-type="ref" reference="SF"} with $m=3, \ell=1, n=1$). **Remark 10**. *The kernel $\psi$ is given by the finite sum $$\psi(x)=\sum_{k=-4}^4 \mu_k \varphi (|x-k|),\qquad x\in{\mathbb{R}},$$ where $\mu_k$, $k=-4,\ldots,4$ are the ones as above. Using Mathematica we are also able to check that this finite sum satisfies $$|\psi (x)|=O((1+|x|)^{-5}),\quad |x|\to \infty.$$* ## Scheme with the cubic B-spline The basic ideas of forming quasi-interpolants, quasi-Lagrange functions and providing polynomial precision come of course from B-spline quasi-interpolation. This one has the same behaviour than the function of the previous section at $r=0,$ i.e., we set - as a radial basis function $\varphi(r)=r^3$ and therefore we have the Fourier transform $\hat{\varphi}(r)$ to be a constant multiple of $r^{-4}$, - the quasi-Lagrange functions $\psi$ are of course the compactly supported normalised B-splines that have well-known analytic Fourier transforms $\hat{N_k}(\xi) = \left( \hat{N_0}(\xi) \right)^{k+1}$ being $\hat{N_0}(\xi)=\dfrac{1-e^{-i \xi}}{i \xi}$, - and our well-known coefficients $\lambda_j$ are the (finitely supported) Fourier coefficients of $(1-\cos (\xi))^{2}$. - the decay of $\psi(x)$ is clear, we notice that it is faster than $O((1+ |x|)^{-k})$ for any $k\in \mathbb{N}$. Studying this setup using Theorem [Theorem 1](#SF){reference-type="ref" reference="SF"} with $\ell > 1, n=1$ gives - as far as polynomial precision is concerned, $\mathbb{P}_1$-reproduction. The same case as in the first part of Subsection [3.1](#2s3){reference-type="ref" reference="2s3"}. By the way, we could improve the polynomial exactness to $\mathbb{P}_3$ by adding a suitable polynomial $P(x)$. - Finally, the approximation error is $O(h^{m+1})$, the exponent being $m=1$ or $m=3$ depending on polynomial reproduction order in our particular cases. # A further construction in the Fourier domain {#sec4} Looking closely again at the Strang-Fix conditions we notice that the second and third set of conditions can be easily satisfied starting the construction in the Fourier domain. When we start in the Fourier domain the more important part is to ensure that the first condition is satisfied. In order to deduce the decay conditions of the function from the Fourier transform we establish the following result. **Theorem 11**. *Let $f$ be a symmetric real valued function satisfying* 1. *$f\in C^{\underline n-1}(\mathbb{R})$ with all derivatives absolutely integrable and* 2. *$f^{(\underline n)}$ is a locally absolutely continuous function and it has bounded variation on $\mathbb{R},$* *then $|\hat{f}(\omega)|= o(|\omega|^{-\underline n-1})$, $\omega \to \pm \infty$.* *Proof.* It follows immediately from the Riemann-Lebesgue Lemma and by integration by parts that $|\hat f(\omega)|=o(\omega^{-\underline n})$, $\omega\to\pm\infty$. Also, $f^{(\underline n)}$ has a well-defined Fourier transform with bounded $L^1$-norm (see the first display in [@lif Theorem 1]) and it even has an integrable derivative, so we have once more by the Riemann-Lebesgue Lemma that even $|\hat f(\omega)|=o(|\omega^{-\underline n-1}|)$. Notice for the argument of partial differentiation where we integrate the exponential and differentiate the other factor in the integrand, integrability is sufficient. This is what we wanted to show. ◻ ## A first example By Theorem [Theorem 11](#th1){reference-type="ref" reference="th1"}, it will be useful to have a function $\psi$ with Fourier transform of type $\hat{\psi}(\xi)=\dfrac{Q(\xi)}{|\xi|^3},$ with $Q(\xi)$ such that $\hat{\psi}(\xi)$ satisfies the following conditions: 1. It belongs to $C^2(\mathbb{R})$ and $\hat{\psi}'''(\xi)$ is a piecewise continuous function, and it has bounded variation in $\mathbb{R}$. Moreover $\hat{\psi}(\xi)$ and its derivatives up to the third order have to be integrable. 2. The Strang-Fix conditions must be satisfied in order to get $\mathbb{P}_2-$reproduction which requires that $Q(\xi)$ must have a zero of order 3 at the origin. We think that the point is that we must avoid jumps in the Fourier transform. More in detail, if $\hat{f}^{(\underline n)}(\omega)$ has a jump then it will have a behaviour like a multiple and shift of the Heaviside function. So, we will obtain for $f(x)$ an asymptotic decay of $O(|x|^{-\underline{n}-1})$ at maximum (although we also need some other conditions as absolute integrability and bounded variation of the function and its derivatives). The reason is that the derivative picks up a Dirac delta, which has a Fourier transform of constant modulus. With the above conditions, we have found the example $$\hat{\psi}(\xi)= e^{-a \xi^4+ \frac{\xi^2}{2}}\left |\dfrac{\sin \xi}{\xi}\right|^3, \quad a>0.$$ The function in the Fourier domain is plotted in Figure 1. This function satisfies all our requirements. Moreover, as the third derivative of $\hat{\psi}(\xi)$ has a jump at $\xi=\pi$, visualised in Figure 2, we would obtain an asymptotic decay of $(1+|y|)^{-4}$. In fact, computations with Mathematica indicate that this is so. We have the following properties: 1. Decay of $\psi(x)$ turns out to be $\psi(x)=o((1+|x|)^{-4})$: to find that, we apply Theorem [Theorem 11](#th1){reference-type="ref" reference="th1"} with $\underline n=3$. 2. Polynomial precision: $\mathbb{P}_2-$reproduction. 3. Approximation error: $O(h^3 |\log h|)$, (use Theorem [Theorem 1](#SF){reference-type="ref" reference="SF"} with $m=2, \ell=1, n=1$). [\[invGau\]]{#invGau label="invGau"} {#invGau width="40%"} {#invGau width="40%"} [\[Jump\]]{#Jump label="Jump"} {#Jump width="30%"} {#Jump width="30%"} {#Jump width="30%"} ## Generalisation of the construction We now give a general description of a method to derive similar Quasi-Lagrange functions with higher order polynomial reproduction. We look for a $\psi$ with Fourier transform of type $$\label{Construct} \hat{\psi}(\xi)=\dfrac{Q(\xi)\vert \sin (\xi)\vert ^{m+1}}{|\xi|^{m+1}},$$ with $Q(\xi)=e^{p(\xi)}$, for a polynomial $p$. Further $\hat{\psi}(\xi)$ should satisfy the following conditions: 1. It belongs to $C^m(\mathbb{R})$. Moreover, $\hat{\psi}^{m+1}(\xi)$ is a piecewise continuous function and has bounded variation on $\mathbb{R}$. Furthermore, $\hat{\psi}(\xi)$ and its derivatives up to order $m$ have to be integrable. 2. The Strang-Fix conditions cited in Theorem [Theorem 1](#SF){reference-type="ref" reference="SF"} must be satisfied in order to get $\mathbb{P}_m-$reproduction. The properties can be translated into conditions on $Q$, from [\[Construct\]](#Construct){reference-type="eqref" reference="Construct"}. Since for $m$ even $$g(\xi)=\vert \sin (\xi)\vert^{m+1}=\sin(\xi)^{m+1} \operatorname{sign}(\sin(\xi))$$ is in $C^{\infty}({\mathbb{R}}\setminus \pi {\mathbb{Z}})$, it is further in $C^{m}({\mathbb{R}})$ since $g^{(j)}(k\pi)=0$ for all $k\in {\mathbb{Z}}$ and $j\leq m$. Combining this with the higher order product rules and continuity of $\vert \xi \vert^{-m-1}$ outside zero, $\hat{\psi}(\xi)$ satisfies the third condition of Theorem [Theorem 1](#SF){reference-type="ref" reference="SF"} if $Q(\xi)$ is $C^ {\infty}({\mathbb{R}})$. Part 2 of the second condition is satisfied if $Q(0)=1$, which is equivalent to assuming $p(0)=0$. For the first part of the second condition of Theorem [Theorem 1](#SF){reference-type="ref" reference="SF"}, we need to compute the derivative of $\hat{\psi}(\xi)$ near zero we find that for $\vert \xi \vert< \pi$: $$\begin{split} \hat{\psi}'(\xi)=&Q'(\xi) \dfrac{\vert \sin(\xi)\vert^{m+1}}{|\xi|^{m+1}} +Q(\xi) \dfrac{(m+1)\vert \sin(\xi)\vert^{m} \cos(\xi) \operatorname{sign}(\sin(\xi))}{|\xi|^{m+1}}\\ & +Q(\xi) \dfrac{\vert \sin(\xi)\vert^{m+1} (-m-1) \operatorname{sign}(\xi)}{|\xi|^{m+2}}\\ =&Q(\xi) \dfrac{p'(\xi) \sin(\xi)^{m+1}\xi + (m+1)\sin(\xi)^{m} \cos(\xi) \xi +\sin(\xi)^{m+1} (-m-1)}{\xi^{m+2}}, \end{split}$$ where we used that for $\vert \xi \vert < \pi$, $\operatorname{sign}(\sin \xi)=\operatorname{sign}(\xi)$. The condition $\hat{\psi}'(0)=0$ is therefore satisfied if $$p'(\xi) \sin(\xi)^{m+1}\xi + (m+1)\sin(\xi)^{m} \left( \cos(\xi) \xi -\sin(\xi) \right)= {O}(\xi^{m+3}),$$ taking into account that $\lim \frac{\sin(x)}{x}=1$ for $x\rightarrow 0$. It remains to show that $$p'(\xi) \sin(\xi)\xi + (m+1) \left( \cos(\xi) \xi - \sin(\xi)\right)= {O}(\xi^{3}).$$ Using the Taylor series of the trigonometric functions we can show $$p'(\xi) \left( \xi^2-\frac{\xi^4}{6}\right) + (m+1) \left( -\frac{1}{3}\xi^3+\frac{1}{30}\xi^5 \right) + H.O.T. ={O}(\xi^{3}),$$ which is true if $$p'(\xi) = {O}(\xi).$$ This condition can be satisfied for $\ell=1$ if $p(\xi)=\sum_{k=2}^{j} a_k \xi^k$ for any choice of coefficients $a_k$ with $a_j<0$. For higher order derivatives one needs to ensure that the Taylor expansion near zero of $$\label{conditionQ} Q(\xi)\sin(\xi)^{m+1}=\xi^{m+1}+O(\xi^{2m+2}).$$ **Lemma 12**. *Let $\hat{\psi}$ be of the form $\eqref{Construct}$ and $Q(\xi)=e^{p(\xi)}$ satisfies [\[conditionQ\]](#conditionQ){reference-type="eqref" reference="conditionQ"} for an even $m$. Then the quasi-interpolation $$\sum_{k\in\mathbb{Z}} p(k)\psi(\cdot-k)\equiv p$$ is exact for polynomials $p$ of degree $m$.* **Remark 13**. 1. *For the case of $m=2$ this gives $$\begin{aligned} Q(\xi)\sin(\xi)^{m+1}&=e^{p(\xi)}\left(\frac{\xi}{1}-\frac{\xi^3}{6}+...\right)^3\\ &=\left( 1+ p'(0)\xi+\left(p''(0)+p'(0)^2\right)\frac{\xi^2}{2}+\cdots \right)\left(\frac{\xi}{1}-\frac{\xi^3}{6}+\cdots\right)^3\\ &=\left( 1+ p''(0)\frac{\xi^2}{2}\right)\left(\xi^3-\frac{\xi^5}{2}+\cdots \right)\\\end{aligned}$$ which is equal to $\xi^{3}+O(\xi^{6})$ if we choose $p(\xi)=-\xi^4+\frac{\xi^2}{2}$.* 2. *In order to compute the function $\psi$ it is helpful to note that the inverse Fourier transform of $e^{-x^{4}}$ has been computed in [@Boyd2014] and a series representation of the Fourier transform of $e^{-\Vert x\Vert^{\beta}}$ is given in the Appendix A.2 to also compute higher order polynomial reproduction properties.* # Summary of the results In Section [2](#sect2){reference-type="ref" reference="sect2"} we compared two options of constructing quasi-interpolants using thin-plate spline and an infinite number of linear combinations to cardinal interpolation. The results are summarised in Table [1](#TPS){reference-type="ref" reference="TPS"}. In Section [3](#sect3){reference-type="ref" reference="sect3"} we studied multiquadric and polyharmonic-spline based quasi-interpolants. The result for the use of these basis functions with and without forming adequate finite linear combinations are displayed in Table [2](#MQ){reference-type="ref" reference="MQ"}. Two alternative constructions can be found in Section [4](#sec4){reference-type="ref" reference="sec4"} - where a new radial basis function characterised in the Fourier domain which gives good approximation properties is introduced - (see Table [3](#New){reference-type="ref" reference="New"}). For the general construction described in this section the approximation order can even be increased to order $h^{m+1} |\log h|$. Method $\lambda$s Reproduction Decay Approximation order -------------------------------------------------------------------------- ------------ ---------------- ------------------- --------------------- Cardinal Infinite $\mathbb{P}_2$ $O((1+|x|)^{-4})$ $O(h^3 |\log h|)$ Subsection [2.2](#subsect22){reference-type="ref" reference="subsect22"} Infinite $\mathbb{P}_1$ $O((1+|x|)^{-4})$ $O(h^2)$ Subsection [2.3](#subsect23){reference-type="ref" reference="subsect23"} Infinite $\mathbb{P}_2$ $O((1+|x|)^{-2})$ $O(h |\log h|)$ : $\varphi(r)= r^2\log r$ RBF $\varphi(r)$ $\lambda$'s Reproduction Decay Approximation order -------------------- --------------- ---------------- ---------------------------------------------- --------------------- $(r^2+ c^2)^{3/2}$ Finite $\mathbb{P}_1$ $O((1+|x|)^{-5})$ $O(h^2)$ $(r^2+ c^2)^{3/2}$ Finite$^\ast$ $\mathbb{P}_3$ $O((1+|x|)^{-5})$ $O(h^4 |\log h|)$ Cubic B-spline Finite $\mathbb{P}_1$ $O((1+|x|)^{-k}), \, \forall k\in\mathbb{N}$ $O(h^2)$ Cubic B-spline Finite$^\ast$ $\mathbb{P}_3$ $O((1+|x|)^{-k}), \, \forall k\in\mathbb{N}$ $O(h^4)$ : The asterisk means that we included a suitable polynomial. Function $\lambda$'s Reproduction Decay Approximation order ------------------- ------------- ---------------- ------------------- --------------------- $\hat{\psi}(\xi)$ Infinite $\mathbb{P}_2$ $O((1+|x|)^{-4})$ $O(h^3 |\log h|)$ : The function $\hat{\psi}(\xi)= e^{-a x^4+ \frac{x^2}{2}}\left |\dfrac{\sin \xi}{\xi}\right|^3, \quad a>0$. # Appendix {#appendix .unnumbered} ## A.1 Computation of the inverse Fourier transform of $\vert \sin(\xi)\vert$ {#a.1-computation-of-the-inverse-fourier-transform-of-vert-sinxivert .unnumbered} **Lemma 14**. *The inverse Fourier transform of $|sin(x)|$ is $${\cal F}^{-1}|\sin|(x)=\frac1{\pi}\times\frac{1+\exp(ix\pi)}{1-x^2}\times{\cal D}_2(x).$$ Here ${\cal D}_2$ is the Dirac comb $${\cal D}_2=\sum_{k=-\infty}^\infty \delta(\cdot-2k).$$* *Proof.* In Section [2](#sect2){reference-type="ref" reference="sect2"} we used that if $g(\xi)=\frac12\sin|\xi|+\frac12\sin|\xi-\pi|$, then $$|\sin \xi| = \sum_{k=-\infty}^\infty g(\xi-k\pi),\qquad \xi\in\mathbb{R}.$$ In order to give the inverse Fourier transform of $|\sin \xi|$ we note first that the generalised inverse Fourier transform ${\cal F}^{-1}$ of $\sin|\xi|$ is $$\begin{aligned} {\cal F}^{-1}\sin|\cdot|(x) &=\frac{1}{2\pi}\int_{-\infty}^{\infty} \sin|\xi| \exp(i\xi x) \,d\xi =\frac{1}{\pi}\int_{0}^{\infty}\sin \xi \cos(\xi)\,d\xi \\ %&= \frac{1}{2\pi}\lim_{\epsilon\to0_+}\int_{-\infty}^{\infty}\exp(-\epsilon \xi)\sin |\xi| \exp(i\xi x)\,d\xi \\ &= \frac{1}{\pi}\lim_{\epsilon\to0_+}\int_0^\infty\exp(-\epsilon \xi)\sin \xi\cos \xi x\,d\xi \\ &= \frac{1}{\pi}\sqrt{\frac{\pi}{2}}\lim_{\epsilon\to0_+}\sqrt{\frac{2}{\pi}}\int_0^\infty\exp(-\epsilon \xi)\sin \xi\cos \xi x\,d\xi \\ &= \frac{1}{\pi}\sqrt{\frac{\pi}{2}}\lim_{\epsilon\to0_+}{\cal F}_{cos}\left(\exp(-\epsilon \xi)\sin \xi\right) (x)\\ &=\frac{1}{\sqrt{2\pi}}\lim_{\epsilon\to0_+}\frac{1}{\sqrt{2\pi}} \left( \frac{1+x}{\epsilon^2+(1+x)^{2}} +\frac{1-x}{\epsilon^2+(1-x)^2}\right)\\ &=\frac{1}{2\pi} \frac{2}{1-x^2}=\frac{1}{\pi}\frac1{1-x^2}. \end{aligned}$$ where we used the cosine transform, ${\cal F}_{cos}$, given in [@GR], ($17.34.22^{7}$). This gives $${\cal F}^{-1}\sin|\cdot|(x)=\frac1\pi\frac1{1-x^2}.$$ Therefore, the generalised inverse Fourier transform of $g$ is $${\cal F}^{-1}g(x)=\frac1{2\pi}\biggl(\frac1{1-x^2}+\frac{\exp(ix\pi)}{1-x^2}\biggr).$$ This gives according to [@SW] $${\cal F}^{-1}|\sin|(x)=\frac1{\pi}\times\frac{1+\exp(ix\pi)}{1-x^2}\times{\cal D}_2(x).$$ First, we have $$|\sin (x)|=\sum_{k=-\infty}^\infty g(x-k\pi)=\left(\sum_{k=-\infty}^{\infty} \delta(\cdot- k\pi)\ast g\right)(x)$$ Now, from $\mathcal{F}^{-1}f(x) = \frac{1}{(2\pi)^n} (\mathcal{F}f)(-x)$ and $\mathcal{F}^{-1}(f*g)(x)= (2\pi)^n \, \mathcal{F}^{-1}f(x) \cdot \mathcal{F}^{-1}g(x)$ it follows: $$\begin{aligned} {\cal F}^{-1} |\sin (\cdot)|(x)=&2\pi{\cal F}^{-1} \left(\sum_{k=-\infty}^{\infty} \delta(\cdot- k\pi)\right)(x) \times {\cal F}^{-1} g(x) \\=& 2\pi \biggl(\frac{1}{\pi} \left(\sum_{k=-\infty}^{\infty} \delta(\cdot- 2k)\right)(x) \biggr) \times \biggl( \frac1{2\pi}\biggl(\frac1{1-x^2}+\frac{\exp(ix\pi)}{1-x^2}\biggr) \biggr) \\=& {\cal D}_2(x) \times \frac1{\pi}\biggl(\frac{1+\exp(ix\pi) }{1-x^2}\biggr). \end{aligned}$$ ◻ ## A.2 Inverse Fourier transform of $\exp(-\Vert x\Vert^{\beta})$ {#a.2-inverse-fourier-transform-of-exp-vert-xvertbeta .unnumbered} We now want to investigate the class of inverse $n$-dimensional Fourier transforms of the functions $$Q(\xi)=e^{-\Vert \xi\Vert ^{\beta}},$$ which are integrable for $\beta>0$. The presented results are based on the results given in the thesis of one of the authors [@Jaeger2018 Chapter 3.3]. We start by gathering informations about the special choices of $\beta$ which have already been considered. - $\beta=1$: In this case the function is $$Q( \xi )=e^{-\Vert \xi \Vert},$$ which is the Poisson kernel. Its Fourier transform is $$\mathcal{F}^{-1}{Q}(\xi)=\frac{1}{2\pi} \Gamma\left(\frac{n}{2}+\frac{1}{2}\right)\frac{1}{(1+\Vert \xi \Vert^2)^{\frac{n}{2}+\frac{1}{2}}},$$ which is a special case of the generalised inverse multiquadric, $\varphi(r)=(1+r^2)^{\alpha/2}$, with $\alpha=-n-1$, - $\beta=2$: The function is the Gaussian basis function $Q(\xi)=e^{-\Vert \xi \Vert ^2}$, which has the inverse Fourier transform $\mathcal{F}^{-1}{Q}(\xi)=(1/4\pi)^{n/2}e^{-\Vert \xi \Vert^2/4}$ which is also a Gaussian basis function, - $\beta=2N$: The function is ${Q}(\xi)=e^{-\Vert \xi \Vert^{2N}}$; its Fourier transform was considered, for the case $n=1$, in [@Boyd2014]. The Fourier transforms of $Q(\xi)=e^{-A\vert \xi \vert ^{2N}}$ have therein been approximated without giving a representation different from the obvious integral description. For the special case $\beta=4$ the resulting radial basis function is called the inverse quartic Gaussian ($\beta=4$). A series representation has been computed using Matlab by Boyd in [@Boyd2013] and takes the form $$\begin{gathered} \label{eq:Boyding} \mathcal{F}^{-1}{Q}(\xi)=\frac{1}{2^{3/2}}\sum_{k=0}^{\infty} \frac{\Gamma(1/2)}{\Gamma(1/2+N)\Gamma(3/4+k)}\frac{\left(\frac{\vert \xi\vert}{4}\right)^{4k}}{k!}\\ -\frac{1}{8\pi}\Gamma(3/4)\vert \xi\vert^2 \sum_{k=0}^{\infty} \frac{\Gamma(5/4)\Gamma(3/2)}{\Gamma(3/2+k)\Gamma(5/4+k)}\frac{\left(\frac{\vert \xi\vert}{4}\right)^{4k}}{k!}.\end{gathered}$$ We now give a representation of the inverse Fourier transform of $Q(\xi)=e^{-\Vert \xi\Vert^{\beta}}$. We focus on the case $\beta > 1$ using the series representation of the Bessel function. However, to be able to compute the Fourier transform we need to prove this additional lemma first. **Lemma 15**. *The series $$\sum_{k\geq 0}(-1)^ka^{2k}\frac{\Gamma\left( \frac{n+2k}{\beta}\right)}{\Gamma(k+1)\Gamma(k+ \frac{n}{2})}, \quad a\in \mathbb{R},$$ is absolutely convergent for every $\beta >1.$* *Proof.* We are going to prove that by applying the root test to the series the resulting limit is 0. First, we have $$\label{lim} 0 \leq \lim_{k\to \infty} \left | (-1)^ka^{2k}\frac{\Gamma\left( \frac{n+2k}{\beta}\right)}{\Gamma(k+1)\Gamma(k+ \frac{n}{2})}\right |^{\frac1k} =\lim_{k\to \infty} \left (a^{2k}\frac{\Gamma\left( \frac{n+2k}{\beta}\right)}{\Gamma(k+1)\Gamma(k+ \frac{n}{2})}\right )^{\frac1k}.$$ Applying formula 8.327, $1^{\ast}$ of [@GR] we have that for large $k$ $$\begin{aligned} \left( \frac{ a^{2k}\Gamma\left( \frac{n+2k}{\beta}\right)}{\Gamma(k+1)\Gamma(k+ \frac{n}{2} )}\right)^{\frac1k} &\leq \left( \frac{a^{2k}\sqrt{2\pi} \sqrt{\frac{n+2k}{\beta}-1} \left(\frac{n+2k}{\beta}-1\right)^{\frac{n+2k}{\beta}-1} e^k e^{k+ \frac{n}{2} -1}c}{ \sqrt{2\pi} \sqrt{k} k^{k} \sqrt{2\pi} \sqrt{k+ \frac{n}{2}-1} \left(k+ \frac{n}{2}-1\right)^{k+ \frac{n}{2} -1} e^{\frac{n+2k}{\beta}-1}}\right)^{\frac1k} \\ &= \left (\frac{c e^{n(\frac12- \frac{1}{\beta})} \left(\frac{n+2k}{\beta}-1\right)^{\frac{n}{\beta}-\frac12} }{ \sqrt{k} \sqrt{2\pi} \left(k+ \frac{n}{2}-1\right)^{ \frac{n}{2}-\frac12}}\times \dfrac{ a^{2k}e^{2k(1- \frac{1}{\beta})} \left(\frac{n+2k}{\beta}-1\right)^{\frac{2k}{\beta}} }{k^k (k+ \frac{n}{2}-1)^{k}}\right)^{\frac1k} \\ &=\left (\frac{c e^{n(\frac12- \frac{1}{\beta})} \left(\frac{n+2k}{\beta}-1\right)^{\frac{n}{\beta}-\frac12} }{ \sqrt{k} \sqrt{2\pi} \left(k+ \frac{n}{2}-1\right)^{ \frac{n}{2}-\frac12}}\right)^{\frac1k} \times\left ( \dfrac{ a^{2}e^{2(1- \frac{1}{\beta})} \left(\frac{n+2k}{\beta}-1\right)^{\frac{2}{\beta}} }{k (k+ \frac{n}{2}-1)}\right) \end{aligned}$$ where $c$ is a constant that comes from the remainder of the series in the numerator. The last expression, for large $k$ involves a (first) fraction that tends to one and a second fraction that is of order $O(k^{\frac{2}{\beta}-2}).$ As $\beta>1,$ the absolute convergence of the series for any $a\in \mathbb{R}$ follows. ◻ **Lemma 16**. *The inverse Fourier transform of $Q(x )=e^{-\Vert x\Vert^{\beta}}$, $x\in \ensuremath{\mathbb{R}}^n$, $\beta > 1$ is $$\mathcal{F}^{-1}Q(\xi)=\frac{2^{1-n}}{\pi^{n/2}} \frac{1}{\beta}\sum_{k=0}^{\infty}\frac{(-1)^k\left(\frac{\Vert \xi \Vert}{2}\right)^{2k}}{k!\Gamma(k+\frac{n}{2})}\Gamma\left(\frac{n+2k}{\beta}\right).$$* *Proof.* We use the formula for Fourier transforms of radial functions to compute the inverse Fourier transform; this is applicable because $Q \in L^1(\ensuremath{\mathbb{R}}^n)$, for all $\beta>1$, and $n\in \ensuremath{\mathbb{N}}$. We then use the series representation of the Bessel function ([@abr] (9.1.10)) $$\begin{aligned} \mathcal{F}^{-1}Q(\xi)&=\frac{1}{\sqrt{2\pi}^{n}}\Vert \xi \Vert^{-(\frac{n-2}{2})}\int_0^{\infty}e^{-t^{\beta}}t^{n/2} J_{\frac{n-2}{2}}(\Vert \xi \Vert t)\,dt\\ &=\frac{1}{\sqrt{2\pi}^{n}}\Vert \xi \Vert^{-(\frac{n-2}{2})}\int_0^{\infty}e^{-t^{\beta}}t^{n/2}\sum_{k=0}^{\infty} \frac{(-1)^k(\Vert \xi \Vert t/2)^{2k+\frac{n}{2}-1}}{k!\Gamma(k+\frac{n}{2})}\,dt\\ &=\frac{1}{\sqrt{2\pi}^{n}}2^{-\frac{n}{2}+1}\int_0^{\infty}e^{-t^{\beta}}t^{n-1}\sum_{k=0}^{\infty} \frac{(-1)^k\left(\frac{\Vert \xi \Vert t}{2}\right)^{2k}}{k!\Gamma(k+\frac{n}{2})}\,dt\\ &=\frac{1}{\sqrt{2\pi}^{n}}2^{-\frac{n}{2}+1}\underset{u\rightarrow \infty}{\lim}\int_0^{u}e^{-t^{\beta}}t^{n-1}\underset{n\rightarrow \infty}{\lim}\sum_{k=0}^{n} \frac{(-1)^k\left(\frac{\Vert \xi \Vert t}{2}\right)^{2k}}{k!\Gamma(k+\frac{n}{2})}\,dt.\\\end{aligned}$$ The sum inside the integrand can be bounded as follows: $$\left \vert \sum_{k=0}^{n} \frac{\left(-1\right)^k\left(\frac{\Vert \xi \Vert t}{2}\right)^{2k}}{k!\Gamma(k+\frac{n}{2})}\right \vert \leq\sum_{k=0}^{\infty}\left\vert \frac{\left(\frac{\Vert \xi \Vert t}{2}\right)^{2k}}{k!\Gamma(k+\frac{n}{2})}\right \vert \leq e^{\frac14 (\Vert \xi \Vert t)^{2}}+c,\quad c>0,$$ with $\frac{1}{\Gamma(k+n/2)}<1$, for $(k+n/2)>2$, which gives an integrable majorant on $[0,u].$ Thereby we get $$\begin{aligned} \mathcal{F}^{-1}Q(\xi)&=\frac{1}{\sqrt{2\pi}^{n}}2^{-\frac{n}{2}+1}\underset{u\rightarrow \infty}{\lim}\sum_{k=0}^{\infty}\frac{\left(-1\right)^k\left(\frac{ \Vert \xi \Vert}{2}\right)^{2k}}{k!\Gamma\left(k+\frac{n}{2}\right)}\int_0^{u}e^{-t^{\beta}}t^{n-1+2k}\,dt\\ &=\frac{1}{\sqrt{2\pi}^{n}}2^{-\frac{n}{2}+1}\underset{u\rightarrow \infty}{\lim}\sum_{k=0}^{\infty}\frac{\left(-1\right)^k\left(\frac{ \Vert \xi \Vert}{2}\right)^{2k}}{k!\Gamma(k+\frac{n}{2})}\int_0^{u^{\beta}}e^{-z}z^{\frac{n+2k}{\beta}-1}\frac{1}{\beta}\, dz\\ &=\frac{1}{\sqrt{2\pi}^{n}}2^{-\frac{n}{2}+1}\frac{1}{\beta}\underset{u \rightarrow \infty}{\lim}\sum_{k=0}^{\infty}\frac{\left(-1\right)^k\left(\frac{ \Vert \xi \Vert}{2}\right)^{2k}}{k!\Gamma\left(k+\frac{n}{2}\right)}\gamma\left(\frac{n+2k}{\beta},u^{\beta}\right),\end{aligned}$$ where we have used the expression 8.350.1 of [@GR] in the last equality. Here $\gamma(\cdot,\cdot)$ is the incomplete $\Gamma$-function. We know that $\gamma\left(\frac{n+2k}{\beta},u^{\beta}\right)\leq \Gamma\left(\frac{n+2k}{\beta}\right)$ for all $\beta>1$ and applying Lemma [Lemma 15](#leInvGaus2){reference-type="ref" reference="leInvGaus2"} we get a convergent majorant. So, we have $$\begin{aligned} \mathcal{F}^{-1}Q(\xi)&=\frac{2^{1-n}}{\pi^{n/2}} \frac{1}{\beta}\sum_{k=0}^{\infty}\frac{(-1)^k\left(\frac{ \Vert \xi \Vert}{2}\right)^{2k}}{k!\Gamma\left(k+\frac{n}{2}\right)}\Gamma\left(\frac{n+2k}{\beta}\right).\end{aligned}$$ ◻ The last series is absolutely convergent for $\beta>1$ and can be further simplified for many values of $\beta$ by applying the doubling or tripling formulas for the Gamma function. For the application in Section [4](#sec4){reference-type="ref" reference="sec4"} we are specifically interested in the case $n=1$ and $Q(x)=\exp(-|x|^{\beta})$. In this case, and if $\beta=2N, N\in \mathbb{N}$ our formula simplifies to: $$\mathcal{F}^{-1}Q(\xi)=\frac{1}{2N\sqrt{\pi}}\sum_{k=0}^{\infty}\frac{(-1)^k\left(\frac{| \xi |}{2}\right)^{2k}}{k!\Gamma(k+\frac{1}{2})}\Gamma\left(\frac{1+2k}{2N}\right).$$

arxiv_math

--- abstract: | A deformed Donaldson--Thomas (dDT) connection is a Hermitian connection of a Hermitian line bundle over a $G_2$-manifold $X$ satisfying a certain nonlinear PDE. This is considered to be the mirror of a (co)associative cycle in the context of mirror symmetry. It can also be considered as an analogue of a $G_2$-instanton. In this paper, we see that some important observations that appear in other geometric problems are also found in the dDT case as follows. \(1\) A dDT connection exists if a 7-manifold has full holonomy $G_2$ and the $G_2$-structure is "sufficiently large\". (2) The dDT equation is described as the zero of a certain multi-moment map. (3) The gradient flow equation of a Chern-Simons type functional of Karigiannis and Leung, whose critical points are dDT connections, agrees with the ${\rm Spin}(7)$ version of the dDT equation on a cylinder with respect to a certain metric on a certain space. This can be considered as an analogue of the observation in instanton Floer homology for 3-manifolds. address: - Yanqi Lake Beijing Institute of Mathematical Sciences and Applications, No. 544, Hefangkou Village, Huaibei Town, Huairou District, Beijing, 101408, China - Department of Mathematics, Osaka Metropolitan University, 3-3-138, Sugimoto, Sumiyoshi-ku, Osaka, 558-8585, Japan author: - Kotaro Kawai bibliography: - references.bib title: Some observations on deformed Donaldson--Thomas connections --- [^1] # Introduction Let $X^7$ be a $7$-manifold with a $\rm G_2$-structure $\varphi\in\Omega^3(X)$. For the definition of $G_2$-structures, see for example [@kawai2021mirror Section 2.2]. We use the same sign convention there. Denote by $g$, $\mathrm{vol}$ and $\ast$ the induced Riemannian metric, volume form and Hodge star operator, respectively. Let $(L,h)\to X$ be a smooth complex Hermitian line bundle over $X$. We denote by $\mathcal{A}_0$ the affine space of Hermitian connections on $(L,h)$. Given $\nabla\in\mathcal{A}_0$, we regard its curvature $F_{\nabla}$ as a $\sqrt{-1}\mathbb{R}$-valued closed $2$-form on $X$. **Definition 1**. A Hermitian connection $\nabla\in\mathcal{A}_0$ satisfying $$\label{eq: dDT} \frac{1}{6}F_{\nabla}^3 + F_{\nabla}\wedge\ast\varphi = 0$$ is called a **deformed Donaldson-Thomas (dDT) connection**. DDT connections appeared in the context of mirror symmetry. They were introduced in [@lee2009geometric] as "mirrors\" of calibrated (associative) submanifolds. Historically, deformed Hermitian Yang--Mills (dHYM) connections were introduced first in [@lyz2000FM] as "mirrors\" of special Lagrangian submanifolds. There is also a similar notion of dDT connections for a manifold with a ${\rm Spin}(7)$-structure ([@lee2009geometric; @kawai2021FM]). As the names indicate, dDT connections can also be considered as analogues of Donaldson--Thomas connections ($G_2$-instantons). Thus it is natural to expect that dDT connections would have similar properties to associative submanifolds and $G_2$-instantons. We show that it is indeed the case in [@kawai2020deformation; @kawai2021mirror]. For example, the moduli space of dDT connections is $b^1$-dimensional and canonically orientable if we perturb the $G_2$-structure. Any dDT connection on a compact $G_2$-manifold is a global minimizer of the "mirror volume\" and its value is topological by the "mirror" of associator equality. We could also prove similar statements in the ${\rm Spin}(7)$ case in [@kawai2021deformationSpin(7); @kawai2021mirror]. Moreover, dDT connections are given by critical points of the Chern-Simons type functional in [@karigiannis2009hodge Theorem 5.13]. The variational characterization is known only for the $G_2$ case, and no such characterization is known for the ${\rm Spin}(7)$ case.\ This paper is organized as follows. In Section [2](#sec:lrl){reference-type="ref" reference="sec:lrl"}, we study the existence of a dDT connection. Known examples of dDT connections are either trivial or constructed in [@lotay2020examples; @fowdar2022examples], and are very few in number. So it would be important to consider the existence problem. We first see that the formal "large radius limit\" of the defining equation of dDT connections is that of $G_2$-instantons. Thus it is natural to expect that dDT connections for a "sufficiently large\" $G_2$-structure will behave like $G_2$-instantons. Moreover, it is known that any complex Hermitian line bundle admits a $G_2$-instanton on a compact holonomy $G_2$-manifold. Then we show the following from these facts. **Theorem 2** (Theorem [Theorem 6](#thm:exist dDT){reference-type="ref" reference="thm:exist dDT"}). *Suppose that $(X, \varphi)$ is a compact holonomy $G_2$-manifold. Let $(L,h) \to X$ be a smooth complex Hermitian line bundle over $X$. If the $G_2$-structure is "sufficiently large\", there exists a dDT connection.* In Section [3](#sec:mmm){reference-type="ref" reference="sec:mmm"}, we formulate the dDT equation in terms of a multi-moment map. The multi-moment map is a generalization of the moment map introduced in [@Madsen2012multi; @Madsen2013closed]. The dHYM equation is described as the zero of a certain moment map on a infinite dimensional symplectic manifold ([@collins2021moment Section 2], [@collins2021survey Section 2.1]). Analogously, we show that the dDT equation is described as the zero of a certain multi-moment map. **Theorem 3** (Theorem [Theorem 11](#thm:mmm){reference-type="ref" reference="thm:mmm"}). *The dDT equation is described as the zero of a certain multi-moment map.* In Section [4](#sec:grad){reference-type="ref" reference="sec:grad"}, we study the gradient flow of the Karigiannis-Leung functional introduced in [@karigiannis2009hodge] whose critical points are dDT connections. It is known that the gradient flow equation of the Chern-Simons functional on an oriented 3-manifold $X^3$ agrees with the ASD equation on $\mathbb{R}\times X^3$. This is an important observation in instanton Floer homology for 3-manifolds. We show that there is an analogous relation between dDT equations for $G_2$- and ${\rm Spin}(7)$-manifolds using the Karigiannis-Leung functional. This will establish a new link between 3, 4-manifold theory and $G_2$-, ${\rm Spin}(7)$-geometry, and we might define analogues of instanton Floer homology using dDT connections. **Theorem 4** (Theorem [Theorem 14](#thm:gradient flow){reference-type="ref" reference="thm:gradient flow"}). *The gradient flow equation of a Chern-Simons type functional of Karigiannis and Leung, whose critical points are dDT connections, agrees with the ${\rm Spin}(7)$ version of the dDT equation on a cylinder with respect to a certain metric on a certain space.* # Large radius limit {#sec:lrl} In this section, we show the existence of a dDT connection if a 7-manifold has full holonomy $G_2$ and the $G_2$-structure is "sufficiently large\". Suppose that $(X, \varphi)$ is a compact holonomy $G_2$-manifold. Let $(L,h) \to X$ be a smooth complex Hermitian line bundle over $X$. Set $$\begin{aligned} \mathcal{A}_{0}=\{\, \mbox{Hermitian connections of }(L,h) \,\} = \nabla_0 + {\sqrt{-1}}\Omega^1 \cdot \mathrm{id}_L, \end{aligned}$$ where $\nabla_0 \in \mathcal{A}_{0}$ is any fixed connection and $\Omega^1$ is the space of 1-forms on $X$. Denote by $\mathcal{G}_U$ the group of unitary gauge transformations of $(L,h)$, which acts on $\mathcal{A}_{0}$. Explicitly, $$\mathcal{G}_U= \{\, f \cdot \mathrm{id}_L \mid f \in \Omega^0_{\mathbb{C}}, \ |f|=1 \,\} \cong C^\infty(X, S^1),$$ where $\Omega^0_{\mathbb{C}}$ is the space of $\mathbb{C}$-valued smooth functions, and the action $\mathcal{G}_U \times \mathcal{A}_0 \rightarrow \mathcal{A}_0$ is defined by $(\lambda, \nabla) \mapsto \lambda^* \nabla:= \lambda^{-1} \circ \nabla \circ \lambda$. When $\lambda=f \cdot \mathrm{id}_L$ for $f \in C^\infty(X, S^1)$, we have $$\begin{aligned} \label{eq:Gu action} \lambda^* \nabla = \lambda^{-1} \circ \nabla \circ \lambda = \nabla + f^{-1}df \cdot \mathrm{id}_L. \end{aligned}$$ Thus the $\mathcal{G}_U$-orbit through ${\nabla}\in \mathcal{A}_0$ is given by ${\nabla}+ \mathcal{K}_U \cdot \mathrm{id}_L$, where $$\label{eq:Gu orbit G2} \mathcal{K}_U:= \mathopen{}\mathclose\bgroup\left\{ f^{-1} d f \in {\sqrt{-1}}\Omega^1 \; \middle| \; f \in \Omega^{0}_{\mathbb{C}}, \ |f|=1 \aftergroup\egroup\right\}.$$ Note that the curvature 2-form $F_\nabla$ is invariant under the action of $\mathcal{G}_U$.\ Consider the family of $G_2$-structures $$\{ \varphi_r:=r^3 \varphi \}_{r>0},$$ all of which induce holonomy $G_2$ metrics. The defining equation of dDT connections with respect to $\varphi_r$ is given by $$0=\mathcal{F}_r({\nabla}) :=\frac{1}{6} F_{\nabla}^3 + r^4 F_{\nabla}\wedge * \varphi.$$ Thus, formally taking the \"large radius limit\", which means the leading behaviour of $\mathcal{F}_r({\nabla})$ as $r \to \infty$, we obtain $$F_{\nabla}\wedge * \varphi =0.$$ This is exactly the defining equation of $G_2$-instantons. Thus it is natural to expect that dDT connections for a sufficiently large $G_2$-structure will behave like $G_2$-instantons. The following would be well-known for $G_2$-instantons on a smooth complex Hermitian line bundle, but we give the proof for completeness. **Lemma 5**. *On a compact holonomy $G_2$-manifold $(X^7,\varphi)$, there is a unique $G_2$-instanton on a smooth complex Hermitian line bundle $L\to X$ up to the action of $\mathcal{G}_U$.* *Proof.* For any ${\nabla}\in \mathcal{A}_0$, we have $d F_{\nabla}=0$. So it defines a cohomology class $[F_{\nabla}] \in {\sqrt{-1}}H^2(X, \mathbb{R})$, which is known to be equal to $-2 \pi {\sqrt{-1}}c_1(L)$. Then there exists a 1-form $\alpha \in {\sqrt{-1}}\Omega^1$ such that $F_{\nabla}+ d \alpha$ is harmonic by Hodge theory. Denote by $\Omega^k_\ell \subset \Omega^k$ the subspace of the space of $k$-forms corresponding to the $\ell$-dimensional irreducible representation of $G_2$. For more details, see for example [@kawai2021mirror Section 2.2]. Denote by $\mathcal{H}^k$ the space of harmonic $k$-forms on $X$ and set $\mathcal{H}^k_\ell= \mathcal{H}^k \cap \Omega^k_\ell$. Then by [@joyce2000compact Theorem 10.2.4], we have $\mathcal{H}^2_7 \cong \mathcal{H}^1_7 = \mathcal{H}^1 = \{ 0 \}$. Thus we have $$F_{{\nabla}+ \alpha \cdot \mathrm{id}_L} = F_{\nabla}+ d \alpha \in {\sqrt{-1}}\mathcal{H}^2 = {\sqrt{-1}}\mathcal{H}^2_7 \oplus \mathcal{H}^2_{14} = {\sqrt{-1}}\mathcal{H}^2_{14},$$ which implies that $F_{{\nabla}+ \alpha \cdot \mathrm{id}_L} \wedge * \varphi =0$. If ${\nabla}'={\nabla}+ (\alpha + \alpha') \cdot \mathrm{id}_L$ for $\alpha' \in {\sqrt{-1}}\Omega^1$ is also a $G_2$-instanton, we have $0=F_{{\nabla}'} \wedge * \varphi = d \alpha' \wedge * \varphi$, which is equivalent to $$\begin{aligned} \label{eq:ex G2inst} -d \alpha' = * (d \alpha' \wedge \varphi) = * d (\alpha' \wedge \varphi). \end{aligned}$$ Since $d \Omega^1 \cap d^* \Omega^3 = \{ 0 \}$, we have $d \alpha'=0$. Since $H^1(X,\mathbb{R})= \{ 0 \}$ by [@joyce2000compact Theorem 10.2.4] again and ${\sqrt{-1}}\mathbb{R}$-valued exact 1-forms are contained in $\mathcal{K}_U$, the $G_2$-instanton is unique up to the action of $\mathcal{G}_U$. ◻ Using this, we can show the following. **Theorem 6**. *Suppose that $(X, \varphi)$ is a compact holonomy $G_2$-manifold. Let $(L,h) \to X$ be a smooth complex Hermitian line bundle over $X$. Then for sufficiently large $r>0$, there exists a dDT connection with respect to $\varphi_r$.* *Proof.* Define a map $\mathcal{F}:[0,1] \times \mathcal{A}_0 \to {\sqrt{-1}}d \Omega^5$ by $$\mathcal{F}(s,{\nabla}) = \frac{s^4}{6} F_{\nabla}^3+F_{\nabla}\wedge * \varphi.$$ Then $\mathcal{F}(0, \cdot)^{-1} (0)/\mathcal{G}_U$, which is a point by Lemma [Lemma 5](#lem:ex G2inst){reference-type="ref" reference="lem:ex G2inst"}, is the moduli space of $G_2$-instantons with respect to $\varphi$ and $\mathcal{F}(s, \cdot)^{-1} (0)/\mathcal{G}_U$ for $s \neq 0$ is the moduli space of dDT connections with respect to $\varphi_{1/s}$. We want to apply the implicit function theorem to show the statement. Fix a $G_2$-instanton ${\nabla}_0 \in \mathcal{F}(0, \cdot)^{-1} (0)$, whose existence is guaranteed by Lemma [Lemma 5](#lem:ex G2inst){reference-type="ref" reference="lem:ex G2inst"}. Denote by the linearization $(d \mathcal{F})_{(0,{\nabla}_0)}: \mathbb{R}\oplus {\sqrt{-1}}\Omega^1 \to {\sqrt{-1}}d \Omega^5$ of $\mathcal{F}$ at $(0,{\nabla}_0)$. Then we have $$(d \mathcal{F})_{(0,{\nabla}_0)} (0, {\sqrt{-1}}b) = {\sqrt{-1}}db \wedge * \varphi.$$ **Lemma 7**. *We have $$\ker (d \mathcal{F})_{(0,{\nabla}_0)} = \mathbb{R}\oplus {\sqrt{-1}}d \Omega^0, \qquad \mathrm{Im}{(d \mathcal{F})_{(0,{\nabla}_0)}} = {\sqrt{-1}}d \Omega^5.$$* *Proof.* The first equation is proved as in [[]{.upright}](#{eq:ex G2inst}). For the second equation, the Hodge decomposition implies that $d^* \Omega^2=d^* d \Omega^1$. For any $b \in \Omega^1$, we have $$d^* d b = d^* (d b + * (\varphi \wedge d b)) \in d^* \Omega^2_7,$$ where we use the fact that $\varphi$ is closed. This implies that $d^* \Omega^2=d^* \Omega^2_7$. Then $$d \Omega^5=*d^* \Omega^2=*d^* \Omega^2_7=d \Omega^5_7.$$ Since $\Omega^5_7$ is spanned by $b \wedge * \varphi$ for $b \in \Omega^1$, the proof is completed. ◻ By the Hodge decomposition and $H^1(X,\mathbb{R})= \{ 0 \}$, we have $\Omega^1=d \Omega^0 \oplus d^* \Omega^2$. By this and Lemma [Lemma 7](#lem:ex ker im){reference-type="ref" reference="lem:ex ker im"}, we see that $(d \mathcal{F})_{(0,{\nabla}_0)}|_{{\sqrt{-1}}d^* \Omega^2}: {\sqrt{-1}}d^* \Omega^2 \to {\sqrt{-1}}d \Omega^5$ is an isomorphism. Hence, we can apply the implicit function theorem (after the Banach completion) and we see that $\mathcal{F}(s, \cdot)^{-1} (0) \neq \emptyset$ for sufficiently small $s$. Finally, we explain how to recover the regularity of elements in $\mathcal{F}(s, \cdot)^{-1} (0)$ after the Banach completion. Since the curvature is invariant under the addition of closed 1-forms, there exists $a_s \in \Omega^1$ such that $$\begin{aligned} \mathcal{F}(s, {\nabla}_0 + {\sqrt{-1}}a_s \cdot \mathrm{id}_L)=0, \qquad d^* a_s=0 \tag{$*_s$}\end{aligned}$$ for sufficiently small $s$. In particular, $(*_0)$ is given by $d a_0 \wedge * \varphi=d^* a_0=0$, which is an overdetermined elliptic equation. To be overdetermined elliptic is an open condition, so we see that $(*_s)$ is also overdetermined elliptic for sufficiently small $s$. Hence we can find a smooth element in $\mathcal{F}(s, \cdot)^{-1} (0)$ around $(0,{\nabla}_0)$ and the proof is completed. ◻ # The multi-moment map {#sec:mmm} It is known that there is a moment map picture in the dHYM case. In particular, the dHYM equation is described as the zero of a certain moment map on a infinite dimensional symplectic manifold. See for example [@collins2021moment Section 2] or the survey article [@collins2021survey Section 2.1]. Analogously, we show that the dDT equation is described as the zero of a certain multi-moment map. First, recall the definition of the multi-moment map in [@Madsen2012multi; @Madsen2013closed]. **Definition 8**. Let $X$ be a smooth manifold and $c \in \Omega^3$ be a closed 3-form on $X$. Suppose that a Lie group $G$ acts on $X$ preserving $c$. Denote by ${\mathfrak g}$ the Lie algebra of $G$ and set $$\mathcal{P}_{\mathfrak g}=\ker (L: \Lambda^2 {\mathfrak g}\to {\mathfrak g}) \subset \Lambda^2 {\mathfrak g},$$ where $L$ is the linear map induced by the Lie bracket. (Note that $\mathcal{P}_{\mathfrak g}= \Lambda^2 {\mathfrak g}$ if $G$ is abelian.) Denote by $u^*$ the vector field on $X$ generated by $u \in {\mathfrak g}$. For a two vector $p = \sum_j u_j \wedge v_j \in \Lambda^2 {\mathfrak g}$, set $$p^* = \sum_j u_j^* \wedge v_j^*, \qquad i(p^*) c = \sum_j c(u_j^*, v_j^*, \cdot).$$ Denote by $\langle \cdot, \cdot \rangle: \Lambda^2 {\mathfrak g}^* \times \Lambda^2 {\mathfrak g}\to \mathbb{R}$ the canonical pairing. Then a map $\nu:M \to \mathcal{P}_{\mathfrak g}^*$ is called a **multi-moment map** if it is $G$-equivariant and satisfies $$d \langle \nu, p \rangle = i(p^*) c$$ for any $p \in \mathcal{P}_{\mathfrak g}$. Let $X$ be a compact 7-manifold with a coclosed $G_2$-structure $\varphi$ ($d * \varphi=0$) and $(L,h) \to X$ be a smooth complex Hermitian line bundle over $X$. Let $\mathcal{A}_0$ be the space of Hermitian connections of $(L,h)$. Define a map $\mathcal{F}_{G_2}: \mathcal{A}_0 \to {\sqrt{-1}}\Omega^6$ by $$\mathcal{F}_{G_2} ({\nabla})= \frac{1}{6} F_{\nabla}^3 + F_{\nabla}\wedge * \varphi.$$ Then the space of dDT connections is given by $\mathcal{F}_{G_2}^{-1}(0)$. Denote by $\mathcal{G}_U$ the group of unitary gauge transformations of $(L,h)$ acting $\mathcal{A}_0$ canonically as in [[]{.upright}](#{eq:Gu action}). Since $\mathcal{G}_U = C^\infty(X,S^1)$, the Lie algebra ${\mathfrak g}_U$ of $\mathcal{G}_U$ is identified with the space ${\sqrt{-1}}\Omega^0$ of ${\sqrt{-1}}\mathbb{R}$-valued functions on $X$. Note that $\mathcal{P}_{\mathfrak g}=\Lambda^2 {\mathfrak g}$ since $\mathcal{G}_U$ is abelian. Define a 3-form $\Theta \in \Omega^3(\mathcal{A}_0)$ on $\mathcal{A}_0$ by $$\Theta_{\nabla}(\alpha_1,\alpha_2,\alpha_3) = {\sqrt{-1}}\int_X \alpha_1 \wedge \alpha_2 \wedge \alpha_3 \wedge \mathopen{}\mathclose\bgroup\left( \frac{1}{2} F_{\nabla}^2 + * \varphi \aftergroup\egroup\right),$$ where ${\nabla}\in \mathcal{A}_0$ and $\alpha_1, \alpha_2, \alpha_3 \in {\sqrt{-1}}\Omega^1 = T_{\nabla}\mathcal{A}_0$. We first show the following as required in Definition [Definition 8](#def:mmm){reference-type="ref" reference="def:mmm"}. **Lemma 9**. *The 3-form $\Theta$ is $\mathcal{G}_U$-invariant and closed.* *Proof.* Take any ${\nabla}\in \mathcal{A}_0$, $\lambda = f \cdot \mathrm{id}_L \in \mathcal{G}_U$, where $f \in C^\infty (X, S^1)$, and $\alpha_1, \alpha_2, \alpha_3, \alpha_4 \in {\sqrt{-1}}\Omega^1 \cong T_{\nabla}\mathcal{A}_0$. Identify $\alpha_j$ with a vector field on $\mathcal{A}_0$ by $$(\alpha_j)_{\widetilde {\nabla}} = \mathopen{}\mathclose\bgroup\left. \frac{d}{dt} \mathopen{}\mathclose\bgroup\left( \widetilde {\nabla}+ t \alpha_j \cdot \mathrm{id}_L \aftergroup\egroup\right) \aftergroup\egroup\right|_{t=0}$$ for $\widetilde {\nabla}\in \mathcal{A}_0$. We first show the $\mathcal{G}_U$-invariance of $\Theta$. That is, $$\begin{aligned} \label{eq:Gu inv Theta} \Theta_{\lambda^* {\nabla}} \mathopen{}\mathclose\bgroup\left(\lambda_* (\alpha_1), \lambda_* (\alpha_2), \lambda_* (\alpha_3) \aftergroup\egroup\right) = \Theta_{{\nabla}} \mathopen{}\mathclose\bgroup\left( \alpha_1, \alpha_2, \alpha_3 \aftergroup\egroup\right). \end{aligned}$$ By [[]{.upright}](#{eq:Gu action}), we compute $$\begin{aligned} \lambda_* (\alpha_j)_{\nabla} = \lambda_* \mathopen{}\mathclose\bgroup\left. \frac{d}{dt} \mathopen{}\mathclose\bgroup\left({\nabla}+ t \alpha_j \cdot \mathrm{id}_L \aftergroup\egroup\right) \aftergroup\egroup\right|_{t=0} = \mathopen{}\mathclose\bgroup\left. \frac{d}{dt} \mathopen{}\mathclose\bgroup\left({\nabla}+ (t \alpha_j+ f^{-1} df) \cdot \mathrm{id}_L \aftergroup\egroup\right) \aftergroup\egroup\right|_{t=0} = (\alpha_j)_{\lambda^* {\nabla}}. \end{aligned}$$ Since $F_{\lambda^* {\nabla}}=F_{\nabla}$, we obtain [[]{.upright}](#{eq:Gu inv Theta}). Next, we show the closedness of $\Theta$. Note that $[\alpha_i, \alpha_j]=0$. Then it follows that $$\begin{aligned} d \Theta (\alpha_1, \alpha_2, \alpha_3, \alpha_4) =& \alpha_1 \mathopen{}\mathclose\bgroup\left(\Theta (\alpha_2, \alpha_3, \alpha_4)\aftergroup\egroup\right) - \alpha_2 \mathopen{}\mathclose\bgroup\left(\Theta (\alpha_1, \alpha_3, \alpha_4)\aftergroup\egroup\right) \\ &+ \alpha_3 \mathopen{}\mathclose\bgroup\left(\Theta (\alpha_1, \alpha_2, \alpha_4)\aftergroup\egroup\right) - \alpha_4 \mathopen{}\mathclose\bgroup\left(\Theta (\alpha_1, \alpha_2, \alpha_3)\aftergroup\egroup\right). \end{aligned}$$ Since $$\begin{aligned} \alpha_i \mathopen{}\mathclose\bgroup\left(\Theta (\alpha_j, \alpha_k, \alpha_\ell)\aftergroup\egroup\right)_{\nabla} =& {\sqrt{-1}} \mathopen{}\mathclose\bgroup\left. \frac{d}{dt} \int_X \alpha_j \wedge \alpha_k \wedge \alpha_\ell \wedge \mathopen{}\mathclose\bgroup\left( \frac{1}{2} F_{{\nabla}+t \alpha_i \cdot \mathrm{id}_L}^2 + * \varphi \aftergroup\egroup\right) \aftergroup\egroup\right|_{t=0} \\ =& {\sqrt{-1}}\int_X \alpha_j \wedge \alpha_k \wedge \alpha_\ell \wedge d \alpha_i \wedge F_{\nabla}, \end{aligned}$$ we have $$(d \Theta)_{\nabla}(\alpha_1, \alpha_2, \alpha_3, \alpha_4) = {\sqrt{-1}}\int_X d( \alpha_1 \wedge \alpha_2 \wedge \alpha_3 \wedge \alpha_4 \wedge F_{\nabla}) =0,$$ which implies that $d \Theta =0$. ◻ We also need the following lemma. **Lemma 10**. *We have $$\Omega^1 = \mathopen{}\mathclose\bgroup\left\{ \sum_{j=1}^N f^j_1 d f^j_2 \; \middle| \; N \in \mathbb{N}, f^j_1, f^j_2 \in \Omega^0 \aftergroup\egroup\right\}.$$* *Proof.* Take any 1-form $\alpha \in \Omega^1$. We first show that for any $x\in X$, there exists an open neighborhood $U_x$ of $x$ and smooth functions $\{ \widetilde f_{x, j}^{1}, \widetilde f_{x, j}^{2} \}_{j=1}^{7}$ on $X$ such that $$\begin{aligned} \label{eq:alpha local} \alpha |_{U_x}= \sum ^{7}_{j=1} \widetilde f_{x,j}^{1} \ d \widetilde f_{x,j}^{2}|_{U_x}. \end{aligned}$$ Indeed, take any local coordinates $(V, (x^1, \cdots, x^7))$ of $x$ and set $$\alpha|_{V}=\sum^{7}_{j=1}\alpha _{j}dx^{j}.$$ We can take a cutoff function $h$ such that $h$ has compact support in $V$ and $h=1$ on an open neighborhood $U_x$ of $x$. Then setting $\widetilde f_{x,j}^1= h \alpha_{j}$ and $\widetilde f_{x,j}^2= h x_j$, which are smooth functions on $X$, we obtain [[]{.upright}](#{eq:alpha local}). Since $\{ U_x \}_{x \in X}$ is an open cover of $X$ and $X$ is compact, there exists $x_1, \cdots, x_N \in X$ such that $\{ U_{x_p} \}_{p=1}^N$ covers $X$. Denote by $\{ h_p \}_{p=1}^N$ the partition of unity subordinate to $\{ U_{x_p} \}_{p=1}^N$. Set $$f_{p,j}^{1}= h_{p} \widetilde {f}_{x_p, j}^1 \qquad f_{p,j}^{2}= \widetilde f_{x_p, j}^{2}.$$ Then we have $\alpha =\sum ^{N}_{p=1}\sum ^{7}_{j=1}f_{p,j}^{1}df_{p,j}^{2}$. Indeed, take any $x \in X$. We may assume that $x \in U_{x_1} \cap \cdots \cap U_{x_k}$ and $x \not\in U_p$ for $p=k+1, \cdots, N$. Then $\sum ^{7}_{j=1} \mathopen{}\mathclose\bgroup\left(\widetilde f_{x_p,j}^{1} \ d \widetilde f_{x_p, j}^{2} \aftergroup\egroup\right)_x = \alpha_x$ for $p=1, \cdots, k$ by [[]{.upright}](#{eq:alpha local}) and $h_p (x)=0$ for $p=k+1, \cdots, N$. Hence $$\sum ^{N}_{p=1}\sum ^{7}_{j=1} \mathopen{}\mathclose\bgroup\left(f_{p,j}^{1}df_{p,j}^{2} \aftergroup\egroup\right)_x = \sum ^{k}_{p=1} h_p (x) \sum ^{7}_{j=1} \mathopen{}\mathclose\bgroup\left( \widetilde f_{x_p,j}^{1} \ d \widetilde f_{x_p, j}^{2} \aftergroup\egroup\right)_x = \sum ^{k}_{p=1} h_p (x) \alpha_x = \sum ^{N}_{p=1} h_p (x) \alpha_x = \alpha_x.$$ ◻ Denote by $Z^6$ the space of closed 6-forms on $X$. Define a map $\iota_{Z^6}: Z^6 \to \Lambda^2 {\mathfrak g}_U^*$ by $$\iota_{Z^6}(\xi) (f_1,f_2) = \int_X \xi \wedge \frac{1}{2} (f_1 d f_2 -f_2 d f_1) = \int_X \xi \wedge f_1 d f_2$$ for $\xi \in Z^6$ and $f_1,f_2 \in {\sqrt{-1}}\Omega^0 = {\mathfrak g}_U$. **Theorem 11**. *Define a $\mathcal{G}_U$-invariant map $\nu: \mathcal{A}_0 \to \Lambda^2 {\mathfrak g}_U^*$ by $$\nu ({\nabla})= \iota_{Z^6} ({\sqrt{-1}}\mathcal{F}_{G_2}({\nabla})).$$ Then we have $d \langle \nu, p \rangle = i(p^*) \Theta$ for any $p \in \Lambda^2 {\mathfrak g}_U$.* Since we assume that $d * \varphi =0$, we see that ${\sqrt{-1}}\mathcal{F}_{G_2}({\nabla}) \in Z^6$ for any ${\nabla}\in \mathcal{A}_0$. By Lemma [Lemma 10](#lem:omega1 fn){reference-type="ref" reference="lem:omega1 fn"}, $\iota_{Z^6}$ is injective. Hence we have $\nu^{-1} (0)= \mathcal{F}_{G_2}^{-1} (0)$. In this sense, we can regard the dDT equation as the zero of a multi-moment map. *Proof.* First note that the vector field $f^*$ generated by $f \in {\sqrt{-1}}\Omega^0 = {\mathfrak g}_U$ is given by $$f^*_{\nabla} = \mathopen{}\mathclose\bgroup\left. \frac{d}{dt} (e^{t f})^* {\nabla}\aftergroup\egroup\right|_{t=0} = \mathopen{}\mathclose\bgroup\left. \frac{d}{dt} \mathopen{}\mathclose\bgroup\left( {\nabla}+ e^{-t f} d e^{t f} \cdot \mathrm{id}_L \aftergroup\egroup\right) \aftergroup\egroup\right|_{t=0} = df$$ at ${\nabla}\in \mathcal{A}_0$. Hence for any $f_1, f_2 \in {\sqrt{-1}}\Omega^0 = {\mathfrak g}_U$ and $\alpha \in {\sqrt{-1}}\Omega^1 = T_{\nabla}\mathcal{A}_0$, we have $$\begin{aligned} &\Theta_{\nabla}((f_1^*)_{\nabla}, (f_2^*)_{\nabla}, \alpha) \\ =& {\sqrt{-1}}\int_X d f_1 \wedge d f_2 \wedge \alpha \wedge \mathopen{}\mathclose\bgroup\left( \frac{1}{2} F_{\nabla}^2 + * \varphi \aftergroup\egroup\right)\\ =& {\sqrt{-1}}\int_X f_1 d f_2 \wedge d\alpha \wedge \mathopen{}\mathclose\bgroup\left( \frac{1}{2} F_{\nabla}^2 + * \varphi \aftergroup\egroup\right) = {\sqrt{-1}}\int_X f_1 d f_2 \wedge (d \mathcal{F}_{G_2})_{\nabla}(\alpha), \end{aligned}$$ where $(d \mathcal{F}_{G_2})_{\nabla}: {\sqrt{-1}}\Omega^1 \to {\sqrt{-1}}\Omega^6$ is the linearization of $\mathcal{F}_{G_2}$ at ${\nabla}\in \mathcal{A}_0$. Hence we obtain $$\begin{aligned} \Theta_{\nabla}((f_1^*)_{\nabla}, (f_2^*)_{\nabla}, \alpha) =& \iota_{Z^6} ({\sqrt{-1}}(d \mathcal{F}_{G_2})_{\nabla}(\alpha)) (f_1, f_2)\\ =& \mathopen{}\mathclose\bgroup\left. \frac{d}{d t} \iota_{Z^6} ({\sqrt{-1}}\mathcal{F}_{G_2} ({\nabla}+t \alpha \cdot \mathrm{id}_L)) \aftergroup\egroup\right|_{t=0} (f_1, f_2) = (d \langle \nu, f_1 \wedge f_2 \rangle)_{\nabla}(\alpha). \end{aligned}$$ ◻ In the dHYM case, the "$\mathcal{J}$ functional\" defined in [@collins2021moment Remark 2.15] or [@collins2021survey Lemma 2.6 (ii)] is convex along geodesics and the critical points are solutions of the dHYM equation. Hence it plays an important role in the existence problem. In the dDT case, there is a functional whose critical points are dDT connections. See Section [4.2](#sec:KL func){reference-type="ref" reference="sec:KL func"}. However, no metric has yet been found that makes the functional convex along geodesics. Since no such results have been found for associative submanifolds, it might be difficult to relate the functional to the existence problem. However, as we see in the next section, we have an observation as in the case of instanton Floer homology for 3-manifolds by using the functional in Section [4.2](#sec:KL func){reference-type="ref" reference="sec:KL func"}. We might develop the theory like instanton Floer homology using dDT connections. # Gradient flow of the Karigiannis-Leung functional {#sec:grad} It is known that the gradient flow equation of the Chern-Simons functional on an oriented 3-manifold $X^3$ agrees with the ASD equation on $\mathbb{R}\times X^3$. See for example [@Donaldson2002 Section 2.5.3]. This is an important observation in instanton Floer homology for 3-manifolds. We show that there is an analogous relation between dDT equations for $G_2$- and ${\rm Spin}(7)$-manifolds. Let $X^7$ be a 7-manifold with a $G_2$-structure $\varphi$ and $(L,h) \to X^7$ be a smooth complex Hermitian line bundle over $X^7$. Let $\{ {\nabla}_t \}_{t \in \mathbb{R}}$ be a family of Hermitian connections of $(L,h) \to X^7$. We identify this with a connection $\widetilde {\nabla}$ of $\pi^*L \to \mathbb{R}\times X^7$, where $\pi: \mathbb{R}\times X^7 \to X^7$ is the projection. If we set $${\nabla}_t={\nabla}_0+ {\sqrt{-1}}a_t \cdot \mathrm{id}_L,$$ where $a_t \in \Omega^1(X^7)$, we have $\widetilde {\nabla}= \pi^* {\nabla}_0 + {\sqrt{-1}}\pi^* a_t \cdot \mathrm{id}_{\pi^* L}$ and the curvature $F_{\widetilde {\nabla}}$ of $\widetilde {\nabla}$ is given by $$F_{\widetilde {\nabla}} = \sqrt{-1} dt \wedge \frac{\partial \pi^* a_t}{\partial t} + \pi^* F_{{\nabla}_t}.$$ ## The ${\rm Spin}(7)$-dDT condition on $\mathbb{R}\times X^7$ {#sec:Spin7 cyl} The product $\mathbb{R}\times X^7$ admits a canonical ${\rm Spin}(7)$-structure. We write down the condition that $F_{\widetilde {\nabla}}$ is a ${\rm Spin}(7)$-dDT connection, a dDT connection for a manifold with a ${\rm Spin}(7)$-structure. For simplicity, set $$\dot a_t:=\frac{\partial \pi^* a_t}{\partial t}, \qquad E_t:= - \sqrt{-1} \pi^* F_{{\nabla}_t}.$$ **Lemma 12**. *The connection $\widetilde {\nabla}$ is a ${\rm Spin}(7)$-dDT connection if and only if $$\begin{aligned} -* \varphi \wedge E_t + \frac{1}{6} E_t^3 - \mathopen{}\mathclose\bgroup\left( 1-\frac{1}{2} *(\varphi \wedge E_t^2) \aftergroup\egroup\right) * \dot a_t + * (\dot a_t \wedge E_t \wedge \varphi) \wedge * E_t &=0 \label{eq:Spin7cyl 1}\\ \frac{1}{2} \varphi \wedge * E_t^2 - \dot a_t \wedge E_t \wedge \varphi &=0. \label{eq:Spin7cyl 2}\end{aligned}$$* *Proof.* Denote by $*_8$ and $*=*_7$ the Hodge star operators on $\mathbb{R}\times X^7$ and $X^7$, respectively. Then, $\widetilde {\nabla}$ is a ${\rm Spin}(7)$-dDT connection (in the sense of [@kawai2021FM Definition 1.3]) if and only if $$\begin{aligned} \label{eq:Spin7cyl 3} \mathopen{}\mathclose\bgroup\left\langle F_{\widetilde {\nabla}} + \frac{1}{6} *_8 F_{\widetilde {\nabla}}^3, \ dt \wedge b + i(b^\sharp) \varphi \aftergroup\egroup\right\rangle=0,\qquad \mathopen{}\mathclose\bgroup\left\langle F_{\widetilde {\nabla}}^2, \ dt \wedge i(b^\sharp) * \varphi - b \wedge \varphi \aftergroup\egroup\right\rangle=0\end{aligned}$$ for any $b \in \Omega^1(X^7)$ by [@kawai2021FM Lemma 3.4]. Since $$\frac{1}{6} *_8 F_{\widetilde {\nabla}}^3 = - \frac{\sqrt{-1}}{6} *_8 (3 dt \wedge \dot a_t \wedge E_t^2 +E_t^3) = \sqrt{-1} \mathopen{}\mathclose\bgroup\left( -\frac{1}{2} * \mathopen{}\mathclose\bgroup\left( \dot a_t \wedge E_t^2 \aftergroup\egroup\right) - \frac{1}{6} dt \wedge * E_t^3 \aftergroup\egroup\right),$$ [[]{.upright}](#{eq:Spin7cyl 3}) is equivalent to $$\begin{aligned} \mathopen{}\mathclose\bgroup\left\langle \dot a_t - \frac{1}{6} * E_t^3, b \aftergroup\egroup\right\rangle + \mathopen{}\mathclose\bgroup\left\langle E_t - \frac{1}{2} * \mathopen{}\mathclose\bgroup\left( \dot a_t \wedge E_t^2 \aftergroup\egroup\right), i(b^\sharp) \varphi \aftergroup\egroup\right\rangle =0, \label{eq:Spin7cyl 1 0}\\ \mathopen{}\mathclose\bgroup\left\langle 2 \dot a_t \wedge E_t, i(b^\sharp) * \varphi \aftergroup\egroup\right\rangle - \mathopen{}\mathclose\bgroup\left\langle E_t^2, b \wedge \varphi \aftergroup\egroup\right\rangle =0. \label{eq:Spin7cyl 2 0}\end{aligned}$$ We compute $$\mathopen{}\mathclose\bgroup\left\langle E_t, i(b^\sharp) \varphi \aftergroup\egroup\right\rangle = * (E_t \wedge * (i(b^\sharp) \varphi)) = * (E_t \wedge b \wedge * \varphi) = \langle * \varphi \wedge E_t, * b \rangle$$ and $$\begin{aligned} \mathopen{}\mathclose\bgroup\left\langle - \frac{1}{2} * \mathopen{}\mathclose\bgroup\left( \dot a_t \wedge E_t^2 \aftergroup\egroup\right), i(b^\sharp) \varphi \aftergroup\egroup\right\rangle =& - \frac{1}{2} * \mathopen{}\mathclose\bgroup\left( \dot a_t \wedge E_t^2 \wedge i(b^\sharp) \varphi \aftergroup\egroup\right) \\ =& - \frac{1}{2} * \mathopen{}\mathclose\bgroup\left( i(b^\sharp) (\dot a_t \wedge E_t^2) \wedge \varphi \aftergroup\egroup\right) \\ =& - \frac{1}{2} * \mathopen{}\mathclose\bgroup\left( E_t^2 \wedge \varphi \aftergroup\egroup\right) \cdot \langle \dot a_t, b \rangle + * \mathopen{}\mathclose\bgroup\left( \dot a_t \wedge E_t \wedge (i(b^\sharp) E_t) \wedge \varphi \aftergroup\egroup\right). \end{aligned}$$ Since $i(b^\sharp) E_t = -* (b \wedge * E_t)$, we have $$\begin{aligned} * \mathopen{}\mathclose\bgroup\left( \dot a_t \wedge E_t \wedge (i(b^\sharp) E_t) \wedge \varphi \aftergroup\egroup\right) = \langle \dot a_t \wedge E_t \wedge \varphi, b \wedge * E_t \rangle = - \langle * (\dot a_t \wedge E_t \wedge \varphi) \wedge * E_t, * b \rangle. \end{aligned}$$ Then, we see that [[]{.upright}](#{eq:Spin7cyl 1 0}) is equivalent to [[]{.upright}](#{eq:Spin7cyl 1}). Similarly, since $$\begin{aligned} \mathopen{}\mathclose\bgroup\left\langle 2 \dot a_t \wedge E_t, i(b^\sharp) * \varphi \aftergroup\egroup\right\rangle &=- 2 * (\dot a_t \wedge E_t \wedge b \wedge \varphi) = 2 \langle \dot a_t \wedge E_t \wedge \varphi, * b \rangle, \\ - \mathopen{}\mathclose\bgroup\left\langle E_t^2, b \wedge \varphi \aftergroup\egroup\right\rangle &=- *(b \wedge \varphi \wedge * E_t^2) =- \langle \varphi \wedge * E_t^2, * b \rangle, \end{aligned}$$ we see that [[]{.upright}](#{eq:Spin7cyl 2 0}) is equivalent to [[]{.upright}](#{eq:Spin7cyl 2}). ◻ Hence, eliminating $* (\dot a_t \wedge E_t \wedge \varphi)$ from [[]{.upright}](#{eq:Spin7cyl 1}) by [[]{.upright}](#{eq:Spin7cyl 2}), we obtain $$\begin{aligned} \label{eq:Spin7cyl 4} -* \varphi \wedge E_t + \frac{1}{6} E_t^3 + \frac{1}{2} *(\varphi \wedge * E_t^2) \wedge * E_t = \mathopen{}\mathclose\bgroup\left( 1-\frac{1}{2} *(\varphi \wedge E_t^2) \aftergroup\egroup\right) * \dot a_t. \end{aligned}$$ **Remark 13**. If $1-*(\varphi \wedge E_t^2)/2 \neq 0$, [[]{.upright}](#{eq:Spin7cyl 1}) and [[]{.upright}](#{eq:Spin7cyl 2}) are equivalent to [[]{.upright}](#{eq:Spin7cyl 4}) by Proposition [Proposition 17](#prop:equiv Spin7 ptwise){reference-type="ref" reference="prop:equiv Spin7 ptwise"}. ## The Karigiannis-Leung functional {#sec:KL func} Karigiannis and Leung [@karigiannis2009hodge] introduced the functional whose critical points are dDT connections. We first review it. Let $X^7$ be a compact 7-manifold with a coclosed $G_2$-structure $\varphi$ ($d * \varphi =0$) and let $(L, h) \to X^7$ be a smooth complex Hermitian line bundle. Denote by $\mathcal{A}_0$ the space of Hermitian connections of $(L,h)$. Define a 1-form $\Theta$ on $\mathcal{A}_0$ by $$\Theta_{\nabla}({\sqrt{-1}}b) = \int_X {\sqrt{-1}}b \wedge \mathopen{}\mathclose\bgroup\left( \frac{1}{6} F_\nabla^3 + F_\nabla \wedge * \varphi \aftergroup\egroup\right)$$ for ${\nabla}\in \mathcal{A}_0$ and ${\sqrt{-1}}b \in {\sqrt{-1}}\Omega^1 = T_{\nabla}\mathcal{A}_0$. Then we see that $\Theta_{\nabla}=0$ if and only if ${\nabla}$ is a dDT connection. We can show that $\Theta$ is closed as in the proof of Lemma [Lemma 9](#lem:3-form closed){reference-type="ref" reference="lem:3-form closed"}. Since $\mathcal{A}_0$ is contractible, there exists $\mathcal{F}: \mathcal{A}_0 \to \mathbb{R}$ such that $d \mathcal{F}= \Theta$. Hence we see that dDT connections are critical points of $\mathcal{F}$.\ Now, we study the relation between ${\rm Spin}(7)$-dDT connections on $\mathbb{R} \times X^7$ and the Karigiannis-Leung functional $\mathcal{F}$. Set $$\mathcal{A}_{ac}:=\mathopen{}\mathclose\bgroup\left\{ {\nabla}\in \mathcal{A}_0 \; \middle| \; 1+\frac{1}{2} *(\varphi \wedge F_{\nabla}^2) > 0 \aftergroup\egroup\right\}.$$ This type of the subset is also considered in the dHYM case. For example, see the survey article [@collins2021survey Definition 2.1]. By the mirror of the associator equality in [@kawai2021mirror Theorem 5.1], it will be natural to call a Hermitian connection ${\nabla}$ satisfying $1 + *(\varphi \wedge F_{\nabla}^2)/2 > 0$ **almost calibrated** as in the dHYM case. Define a metric $\mathcal{G}$ on $\mathcal{A}_{ac}$ by $$\mathcal{G}_{\nabla}(\sqrt{-1} a, \sqrt{-1} b) = \int_X \langle a, b \rangle_{\nabla} \mathopen{}\mathclose\bgroup\left( 1+\frac{1}{2} *(\varphi \wedge F_{\nabla}^2) \aftergroup\egroup\right) \mathrm{vol}$$ where ${\nabla}\in \mathcal{A}_{ac}, \sqrt{-1} a, \sqrt{-1} b \in {\sqrt{-1}}\Omega^1 = T_{\nabla}\mathcal{A}_{ac}$, $\mathrm{vol}$ is the induced volume form from $\varphi$, and $\langle \cdot, \cdot \rangle_{\nabla}$ is the induced metric on the space of differential forms from $(\mathrm{id}_{TX} + (- {\sqrt{-1}}F_{\nabla})^\sharp)^* \varphi$. Here, $(- {\sqrt{-1}}F_{\nabla})^\sharp$ is an endomorphism of $TX$ defined by $g((- {\sqrt{-1}}F_{\nabla})^\sharp (u),v) = - {\sqrt{-1}}F_{\nabla}(u,v)$ for $u,v \in TX$, where $g$ is the induced metric (on $TX$) from $\varphi$. Note that $(- {\sqrt{-1}}F_{\nabla})^\sharp$ is skew-symmetric with respect to $g$. Explicitly, if we denote by $g_{\nabla}$ the induced metric (on $TX$) from $(\mathrm{id}_{TX} + (- {\sqrt{-1}}F_{\nabla})^\sharp)^* \varphi$, we have $g_{\nabla}=(\mathrm{id}_{TX} + (- {\sqrt{-1}}F_{\nabla})^\sharp)^* g$ and $\langle \cdot, \cdot \rangle_{\nabla}$ is the induced metric from $g_{\nabla}$. The following is the main theorem of this paper. **Theorem 14**. *The gradient flow equation of $\mathcal{F}$ with respect to $\mathcal{G}$ on $\mathcal{A}_{ac}$ agrees with the ${\rm Spin}(7)$-dDT equation on $\mathbb{R} \times X^7$.* *Proof.* We first deduce the gradient flow equation and compare it with the computation in Section [4.1](#sec:Spin7 cyl){reference-type="ref" reference="sec:Spin7 cyl"}. Take any ${\nabla}\in \mathcal{A}_{ac}$ and $b \in \Omega^1$. Set $$E_{\nabla}=-{\sqrt{-1}}F_{\nabla}\in \Omega^2.$$ Denote by $\langle \cdot, \cdot \rangle$ the induced metric on the space of differential forms from $\varphi$. Then we compute $$\begin{aligned} \label{eq:gradient flow 1} (d \mathcal{F})_{\nabla}({\sqrt{-1}}b) = \int_X {\sqrt{-1}}b \wedge \mathopen{}\mathclose\bgroup\left( \frac{1}{6} F_\nabla^3 + F_\nabla \wedge * \varphi \aftergroup\egroup\right) = \int_X \mathopen{}\mathclose\bgroup\left\langle b, *\mathopen{}\mathclose\bgroup\left( \frac{1}{6} E_\nabla^3 - E_\nabla \wedge * \varphi \aftergroup\egroup\right) \aftergroup\egroup\right\rangle \mathrm{vol}.\end{aligned}$$ By Proposition [Proposition 15](#prop:pb xi){reference-type="ref" reference="prop:pb xi"}, we have $$*\mathopen{}\mathclose\bgroup\left( \frac{1}{6} E_\nabla^3 - E_\nabla \wedge * \varphi \aftergroup\egroup\right) = \mathopen{}\mathclose\bgroup\left(\mathopen{}\mathclose\bgroup\left(\mathrm{id}_{TX} - (E_{\nabla}^\sharp)^2 \aftergroup\egroup\right)^{-1} \aftergroup\egroup\right)^* \eta_{\nabla},$$ where $$\begin{aligned} \eta_{\nabla} = * \mathopen{}\mathclose\bgroup\left( -* \varphi \wedge E_{\nabla}+ \frac{1}{6} E_{\nabla}^3 + \frac{1}{2} *(\varphi \wedge * E_{\nabla}^2) \wedge * E_{\nabla}\aftergroup\egroup\right) \in \Omega^1. \end{aligned}$$ Since $$\mathrm{id}_{TX} - (E_{\nabla}^\sharp)^2 = (\mathrm{id}_{TX} - E_{\nabla}^\sharp) (\mathrm{id}_{TX} + E_{\nabla}^\sharp) = {}^t (\mathrm{id}_{TX} + E_{\nabla}^\sharp) (\mathrm{id}_{TX} + E_{\nabla}^\sharp),$$ where ${}^t (\mathrm{id}_{TX} + E_{\nabla}^\sharp)$ is the transpose of $\mathrm{id}_{TX} + E_{\nabla}^\sharp$ with respect to $g$, we have $$\label{eq:gradient flow 2} \begin{split} \mathopen{}\mathclose\bgroup\left\langle b, *\mathopen{}\mathclose\bgroup\left( \frac{1}{6} E_\nabla^3 - E_\nabla \wedge * \varphi \aftergroup\egroup\right) \aftergroup\egroup\right\rangle =& \mathopen{}\mathclose\bgroup\left\langle b, \mathopen{}\mathclose\bgroup\left(\mathopen{}\mathclose\bgroup\left(\mathrm{id}_{TX} - (E_{\nabla}^\sharp)^2 \aftergroup\egroup\right)^{-1} \aftergroup\egroup\right)^* \eta_{\nabla}\aftergroup\egroup\right\rangle \\ =& \mathopen{}\mathclose\bgroup\left\langle \mathopen{}\mathclose\bgroup\left( \mathopen{}\mathclose\bgroup\left(\mathrm{id}_{TX} + E_{\nabla}^\sharp \aftergroup\egroup\right)^{-1} \aftergroup\egroup\right)^* b, \mathopen{}\mathclose\bgroup\left( \mathopen{}\mathclose\bgroup\left(\mathrm{id}_{TX} + E_{\nabla}^\sharp \aftergroup\egroup\right)^{-1} \aftergroup\egroup\right)^* \eta_{\nabla}\aftergroup\egroup\right\rangle = \mathopen{}\mathclose\bgroup\left\langle b, \eta_{\nabla}\aftergroup\egroup\right\rangle_{\nabla}. \end{split}$$ Then by [[]{.upright}](#{eq:gradient flow 1}) and [[]{.upright}](#{eq:gradient flow 2}), the gradient vector field of $\mathcal{F}$ with respect to $\mathcal{G}$ is given by $$\mathcal{A}_{ac} \ni {\nabla}\mapsto \frac{{\sqrt{-1}}\eta_{\nabla}} {1-\frac{1}{2} *(\varphi \wedge E_{\nabla}^2)} \in {\sqrt{-1}}\Omega^1.$$ Thus a family $\{ {\nabla}_t \}_{t \in \mathbb{R}} \subset \mathcal{A}_{ac}$ satisfies the gradient flow of $\mathcal{F}$ with respect to $\mathcal{G}$ if and only if $$\begin{aligned} \label{eq:gradient flow 3} \dot a_t = \frac{\eta_{{\nabla}_t}} {1-\frac{1}{2} *(\varphi \wedge E_{{\nabla}_t}^2)} = \frac{* \mathopen{}\mathclose\bgroup\left( -* \varphi \wedge E_t + \frac{1}{6} E_t^3 + \frac{1}{2} *(\varphi \wedge * E_t^2) \wedge * E_t \aftergroup\egroup\right)}{1-\frac{1}{2} *(\varphi \wedge E_t^2)}, \end{aligned}$$ where ${\nabla}_t={\nabla}_0+{\sqrt{-1}}a_t \cdot \mathrm{id}_L$, $a_t \in \Omega^1$, $\dot a_t = \partial a_t/ \partial t$ and $E_t=E_{{\nabla}_t}=-{\sqrt{-1}}F_{{\nabla}_t}$. Then we see that [[]{.upright}](#{eq:gradient flow 3}) is equivalent to [[]{.upright}](#{eq:Spin7cyl 4}). By Remark [Remark 13](#rem:equiv Spin7){reference-type="ref" reference="rem:equiv Spin7"}, this is equivalent to the ${\rm Spin}(7)$-dDT equation on $\mathbb{R} \times X^7$. ◻ By Theorem [Theorem 14](#thm:gradient flow){reference-type="ref" reference="thm:gradient flow"}, we will have to consider the deformation theory of the ${\rm Spin}(7)$-dDT connections on $\mathbb{R} \times X^7$ next for the analogue of instanton Floer homology for 3-manifolds. Deformations of ${\rm Spin}(7)$-dDT connections on a compact manifold with a ${\rm Spin}(7)$-structure are studied in [@kawai2021deformationSpin(7) Theorem 1.2], but there are some technical assumptions. We will have to deal with more technical issues, including these, to develop the deformation theory on a cylinder. # Algebraic Computations In this appendix, we give some algebraic computations needed in the proof of Theorem [Theorem 14](#thm:gradient flow){reference-type="ref" reference="thm:gradient flow"}. Set $V =\mathbb{R}^7$ and let $g$ be the standard inner product on $V$. Denote by $*$ the standard Hodge star operator on $V$. For a 2-form $F \in \Lambda^2 V^*$, define $F^\sharp \in {\rm End} (V)$ by $$g(F^\sharp (u), v) = F(u, v)$$ for $u,v \in V$. Then, $F^\sharp$ is skew-symmetric, and hence, $\mathop\mathrm{det}\nolimits(I + F^\sharp) >0$, where $I$ is the identity matrix. We also have $$\mathop\mathrm{det}\nolimits(I - (F^\sharp)^2) = \mathop\mathrm{det}\nolimits(I+F^\sharp) \mathop\mathrm{det}\nolimits(I-F^\sharp) = \mathop\mathrm{det}\nolimits(I+F^\sharp) \mathop\mathrm{det}\nolimits(I+{}^t F^\sharp) = (\mathop\mathrm{det}\nolimits(I+F^\sharp))^2 >0,$$ where we ${}^t F^\sharp$ is the transpose of $F^\sharp$ with respect to $g$. Define a 3-form $\varphi \in \Lambda^3 V^*$ by $$\varphi = e^{123} + e^{145} + e^{167} + e^{246} - e^{257} - e^{347} - e^{356},$$ where $\{ e_i \}_{i=1}^7$ is a standard oriented basis of $V$ with the dual basis $\{ e^i \}_{i=1}^7$ of $V^\ast$ and $e^{i_1 \dots i_k}$ is short for $e^{i_1} \wedge \cdots \wedge e^{i_k}$. The stabilizer of $\varphi$ is known to be the exceptional $14$-dimensional simple Lie group $G_2$. The elements of $G_2$ preserve the standard inner product $g$ and volume form $\mathrm{vol}$. The group $G_2$ acts canonically on $\Lambda^k V^*$, and $\Lambda^2 V^*$ is decomposed as $\Lambda^2 V^* = \Lambda^2_7 V^* \oplus \Lambda^2_{14} V^*$, where $\Lambda^2_\ell V^*$ is a $\ell$-dimensional irreducible subrepresentation of $G_2$ in $\Lambda^2 V^*$. For more details, see for example [@kawai2021mirror Section 2.2]. Set $$F=F_7 + F_{14} =i(u)\varphi + F_{14} \in \Lambda^2_7 V^* \oplus \Lambda^2_{14} V^*$$ for $u \in V$. **Proposition 15**. *For a 2-form $F \in \Lambda^2 V^*$, set $\xi= -* \varphi \wedge F + F^3/6 \in \Lambda^6 V^*$. Then we have $$\begin{aligned} (I - (F^\sharp)^2)^* * \xi = * \mathopen{}\mathclose\bgroup\left( \xi + \frac{1}{2} *(\varphi \wedge * F^2) \wedge * F \aftergroup\egroup\right). \end{aligned}$$* *Proof.* Since $(I - (F^\sharp)^2)^* * \xi = * \xi - ((F^\sharp)^2)^* * \xi$, we only have to compute $((F^\sharp)^2)^* * \xi$. Set $$F_{i j} =F(e_i, e_j).$$ We have $F^\sharp=\sum_{i,j}F_{i j} e^i \otimes e_j$, which implies that $(F^\sharp)^2 = \sum_{i, j, k} F_{i j} F_{j k} e^i \otimes e_k$. Then we compute $$\begin{aligned} ((F^\sharp)^2)^* * \xi = \sum_{i, j, k} F_{i j} F_{j k} * \xi (e_k) \cdot e^i = -\sum_j \langle i(e_j)F, * \xi \rangle \cdot i(e_j) F. \end{aligned}$$ Since $$\begin{aligned} \langle i(e_j)F, * \xi \rangle =& * \mathopen{}\mathclose\bgroup\left( * \xi \wedge * (i(e_j)F) \aftergroup\egroup\right) = -* \mathopen{}\mathclose\bgroup\left( * \xi \wedge e^j \wedge * F \aftergroup\egroup\right) = \langle e^j, *(* \xi \wedge * F) \rangle, \\ i(e_j) F =& -* (e^j \wedge * F), \end{aligned}$$ we have $$\begin{aligned} \label{eq:pb xi 1} ((F^\sharp)^2)^* * \xi = * \mathopen{}\mathclose\bgroup\left( * \mathopen{}\mathclose\bgroup\left( * \xi \wedge * F \aftergroup\egroup\right) \wedge *F \aftergroup\egroup\right) = * \mathopen{}\mathclose\bgroup\left( * \mathopen{}\mathclose\bgroup\left( * \mathopen{}\mathclose\bgroup\left( -*\varphi \wedge F + \frac{F^3}{6} \aftergroup\egroup\right) \wedge * F \aftergroup\egroup\right) \wedge *F \aftergroup\egroup\right). \end{aligned}$$ **Lemma 16**. *We have $$\begin{aligned} (*F^3) \wedge *F &=0, \\ *(\varphi \wedge * F^2) &= -6 i(u) F. \end{aligned}$$* *Proof.* We can prove the first equation as in [@kawai2020deformation Lemma C.2]. For any $v \in V$, set $$v^\flat = g(v, \cdot) \in V^*.$$ We compute $$\begin{aligned} v^\flat \wedge (*F^3) \wedge *F = (*F^3) \wedge * (i(v) F) =F^3 \wedge i(v) F = i(v)(F^4/4)=0, \end{aligned}$$ which implies the first equation. Similarly, for any $v \in V$, we have $$v^\flat \wedge \varphi \wedge * F^2 =* (v^\flat \wedge \varphi) \wedge F^2 =-i(v) * \varphi \wedge F^2 =* \varphi \wedge i(v) F^2 = 2 i(v) F \wedge F \wedge * \varphi.$$ Since $F \wedge * \varphi=i(u) \varphi \wedge * \varphi = 3 * u^\flat$ by for example [@kawai2020deformation Lemma B.1], we obtain $$v^\flat \wedge \varphi \wedge * F^2 = 6 \langle u^\flat, i(v) F \rangle \mathrm{vol} = 6 \langle v^\flat \wedge u^\flat, F \rangle \mathrm{vol} = -6 \langle v^\flat, i(u) F \rangle \mathrm{vol},$$ which implies the second equation. ◻ Then by [[]{.upright}](#{eq:pb xi 1}), Lemma [Lemma 16](#lem:pb xi 1){reference-type="ref" reference="lem:pb xi 1"} and the equation $F \wedge * \varphi=i(u) \varphi \wedge * \varphi = 3 * u^\flat$, we obtain $$\begin{aligned} ((F^\sharp)^2)^* * \xi =& * \mathopen{}\mathclose\bgroup\left( * \mathopen{}\mathclose\bgroup\left( * \mathopen{}\mathclose\bgroup\left( -*\varphi \wedge F \aftergroup\egroup\right) \wedge * F \aftergroup\egroup\right) \wedge *F \aftergroup\egroup\right)\\ =& * \mathopen{}\mathclose\bgroup\left( * \mathopen{}\mathclose\bgroup\left( -3 u^\flat \wedge * F \aftergroup\egroup\right) \wedge *F \aftergroup\egroup\right)\\ =& 3 * \mathopen{}\mathclose\bgroup\left( (i(u) F) \wedge * F \aftergroup\egroup\right) = - \frac{1}{2} * \mathopen{}\mathclose\bgroup\left( *(\varphi \wedge * F^2) \wedge * F \aftergroup\egroup\right) \end{aligned}$$ and the proof is completed. ◻ **Proposition 17**. *For a 1-form $a \in V^*$ and a 2-form $F \in \Lambda^2 V^*$ such that $1-*(\varphi \wedge F^2)/2 \neq 0$, $$\begin{aligned} -* \varphi \wedge F + \frac{1}{6} F^3 - \mathopen{}\mathclose\bgroup\left( 1-\frac{1}{2} *(\varphi \wedge F^2) \aftergroup\egroup\right) * a + * (a \wedge F \wedge \varphi) \wedge * F &=0, \label{eq:appSpin7cyl 1}\\ \frac{1}{2} \varphi \wedge * F^2 - a \wedge F \wedge \varphi &=0 \label{eq:appSpin7cyl 2}\end{aligned}$$ if and only if $$\begin{aligned} \label{eq:appSpin7cyl 4} -* \varphi \wedge F + \frac{1}{6} F^3 + \frac{1}{2} *(\varphi \wedge * F^2) \wedge * F = \mathopen{}\mathclose\bgroup\left( 1-\frac{1}{2} *(\varphi \wedge F^2) \aftergroup\egroup\right) * a. \end{aligned}$$* *Proof.* Eliminating $a \wedge F \wedge \varphi$ from [[]{.upright}](#{eq:appSpin7cyl 1}) by [[]{.upright}](#{eq:appSpin7cyl 2}), we obtain [[]{.upright}](#{eq:appSpin7cyl 4}). Conversely, [[]{.upright}](#{eq:appSpin7cyl 4}) implies [[]{.upright}](#{eq:appSpin7cyl 2}) by the following Lemma [Lemma 18](#lem:lem gfl){reference-type="ref" reference="lem:lem gfl"}. By [[]{.upright}](#{eq:appSpin7cyl 4}), the left hand side of [[]{.upright}](#{eq:appSpin7cyl 1}) is computed as $$\begin{aligned} - \frac{1}{2} *(\varphi \wedge * F^2) \wedge * F + * (a \wedge F \wedge \varphi) \wedge * F = * \mathopen{}\mathclose\bgroup\left(- \frac{1}{2} \varphi \wedge * F^2 + a \wedge F \wedge \varphi \aftergroup\egroup\right) \wedge * F, \end{aligned}$$ which vanishes by [[]{.upright}](#{eq:appSpin7cyl 2}). ◻ **Lemma 18**. *For any 2-form $F \in \Lambda^2 V^*$, we have $$* \mathopen{}\mathclose\bgroup\left( -* \varphi \wedge F + \frac{1}{6} F^3 + \frac{1}{2} *(\varphi \wedge * F^2) \wedge * F \aftergroup\egroup\right) \wedge F \wedge \varphi = \frac{1}{2} \mathopen{}\mathclose\bgroup\left(1-\frac{1}{2} *(\varphi \wedge F^2) \aftergroup\egroup\right) \varphi \wedge * F^2.$$* *Proof.* Fix any $v \in V$ and set $$J_1= v^\flat \wedge * \mathopen{}\mathclose\bgroup\left( -* \varphi \wedge F + \frac{1}{6} F^3 \aftergroup\egroup\right) \wedge F \wedge \varphi, \qquad J_2= v^\flat \wedge * \mathopen{}\mathclose\bgroup\left( \frac{1}{2} *(\varphi \wedge * F^2) \wedge * F \aftergroup\egroup\right) \wedge F \wedge \varphi.$$ We compute $J_1$ and $J_2$. We have $$\begin{aligned} J_1=& *\mathopen{}\mathclose\bgroup\left( i(v) \mathopen{}\mathclose\bgroup\left( * \varphi \wedge F - \frac{1}{6} F^3 \aftergroup\egroup\right) \aftergroup\egroup\right) \wedge *(2F_7 -F_{14})\\ =& i(v) \mathopen{}\mathclose\bgroup\left( * \varphi \wedge F - \frac{1}{6} F^3 \aftergroup\egroup\right) \wedge (2F_7 -F_{14}) = \mathopen{}\mathclose\bgroup\left( -3 * (v^\flat \wedge u^\flat) - \frac{1}{2} i(v)F \wedge F^2 \aftergroup\egroup\right) \wedge (2F_7 -F_{14}), \end{aligned}$$ where we use $* \varphi \wedge F=3*u^\flat$. We also have $$-3 * (v^\flat \wedge u^\flat) \wedge (2F_7 -F_{14}) = -3 \langle v^\flat \wedge u^\flat, 2F_7 -F_{14} \rangle \mathrm{vol} = -3 \langle v^\flat, i(u)F \rangle \mathrm{vol}$$ as $i(u) F_7= i(u) i(u) \varphi=0$, and $$\begin{aligned} &\mathopen{}\mathclose\bgroup\left(- \frac{1}{2} i(v)F \wedge F^2 \aftergroup\egroup\right) \wedge (2F_7 -F_{14}) \\ =& - \frac{1}{2} i(v)F \wedge (F_7^2+2F_7 \wedge F_{14} +F_{14}^2) \wedge (2F_7 -F_{14})\\ =& - \frac{1}{2} \mathopen{}\mathclose\bgroup\left(i(v)F_7 + i(v)F_{14} \aftergroup\egroup\right) \wedge (2 F_7^3+3 F_7^2 \wedge F_{14} - F_{14}^3)\\ =& - \frac{1}{2} \mathopen{}\mathclose\bgroup\left\{ i(v)F_7 \wedge (3 F_7^2 \wedge F_{14} - F_{14}^3) + i(v)F_{14} \wedge (2 F_7^3+3 F_7^2 \wedge F_{14}) \aftergroup\egroup\right\}, \end{aligned}$$ where we use $i(v)F_7 \wedge F_7^3=i(v)(F_7^4/4)=0$ and $i(v)F_{14} \wedge F_{14}^3=i(v)(F_{14}^4/4)=0$. By [@kawai2020deformation (B.7)], we have $$\begin{aligned} \label{eq:equiv Spin7 ptwise 1} F_7^3=6|u|^2 * u^\flat. \end{aligned}$$ Then $$\begin{aligned} 3 i(v)F_7 \wedge F_7^2 \wedge F_{14} =& i(v) F_7^3 \wedge F_{14} =-6|u|^2 *(v^\flat \wedge u^\flat) \wedge F_{14} =6|u|^2 \langle v^\flat, i(u) F \rangle \mathrm{vol}, \\ 2 i(v)F_{14} \wedge F_7^3 =& 12|u|^2 i(v)F_{14} \wedge * u^\flat = 12|u|^2 \langle F_{14}, v^\flat \wedge u^\flat \rangle \mathrm{vol} = - 12|u|^2 \langle v^\flat, i(u) F \rangle \mathrm{vol}. \end{aligned}$$ Hence we obtain $$\begin{aligned} \label{eq:equiv Spin7 ptwise 2} J_1= (-3+3|u|^2) \langle v^\flat, i(u) F \rangle \mathrm{vol} + \frac{1}{2} \mathopen{}\mathclose\bgroup\left( i(v)F_7 \wedge F_{14}^3 -3 i(v)F_{14} \wedge F_7^2 \wedge F_{14} \aftergroup\egroup\right). \\\end{aligned}$$ Next, we compute $J_2$. By Lemma [Lemma 16](#lem:pb xi 1){reference-type="ref" reference="lem:pb xi 1"}, we have $$\begin{aligned} J_2 =& v^\flat \wedge * \mathopen{}\mathclose\bgroup\left(-3 i(u)F \wedge * F \aftergroup\egroup\right) \wedge *(2F_7-F_{14}) \\ =& 3 * \mathopen{}\mathclose\bgroup\left( i(u)F \wedge * F \aftergroup\egroup\right) \wedge *\mathopen{}\mathclose\bgroup\left(i(v) (-2F_7+F_{14}) \aftergroup\egroup\right) = 3 i(u)F \wedge *F \wedge i(v) (-2F_7+F_{14}). \end{aligned}$$ Since $$\begin{aligned} i(u)F \wedge *F =& i(u)F_{14} \wedge \mathopen{}\mathclose\bgroup\left( \frac{1}{2} F_7 \wedge \varphi - F_{14} \wedge \varphi \aftergroup\egroup\right)\\ =& \frac{1}{2} \mathopen{}\mathclose\bgroup\left( i(u) (F_{14} \wedge F_7 \wedge \varphi) -F_{14} \wedge F_7 \wedge i(u) \varphi \aftergroup\egroup\right) - \frac{1}{2} i(u) F_{14}^2 \wedge \varphi \\ =& - \frac{1}{2} F_7^2 \wedge F_{14} - \frac{1}{2} \mathopen{}\mathclose\bgroup\left( i(u) (F_{14}^2 \wedge \varphi)-F_{14}^2 \wedge i(u) \varphi \aftergroup\egroup\right)\\ =& \frac{1}{2} \mathopen{}\mathclose\bgroup\left( |F_{14}|^2 * u^\flat - F_7^2 \wedge F_{14} + F_7 \wedge F_{14}^2 \aftergroup\egroup\right), \end{aligned}$$ we have $$\begin{aligned} J_2= \frac{3}{2} \mathopen{}\mathclose\bgroup\left(- F_7^2 \wedge F_{14} + F_7 \wedge F_{14}^2 \aftergroup\egroup\right) \wedge i(v) (-2F_7+F_{14}) + \frac{3}{2} |F_{14}|^2 * u^\flat \wedge i(v) (-2F_7+F_{14}). \end{aligned}$$ We compute $$\begin{aligned} &\mathopen{}\mathclose\bgroup\left(- F_7^2 \wedge F_{14} + F_7 \wedge F_{14}^2 \aftergroup\egroup\right) \wedge i(v) (-2F_7+F_{14}) \\ =& 2 i(v) F_7 \wedge F_7^2 \wedge F_{14} - 2 i(v) F_7 \wedge F_7 \wedge F_{14}^2 - i(v) F_{14} \wedge F_7^2 \wedge F_{14} + i(v) F_{14} \wedge F_7 \wedge F_{14}^2. \end{aligned}$$ By [[]{.upright}](#{eq:equiv Spin7 ptwise 1}), it follows that $$2 i(v) F_7 \wedge F_7^2 \wedge F_{14} = \frac{2}{3} i(v) F_7^3 \wedge F_{14} = 4|u|^2 \langle v^\flat, i(u)F \rangle \mathrm{vol}.$$ Since $- 2 i(v) F_7 \wedge F_7 \wedge F_{14}^2 = -i(v) F_7^2 \wedge F_{14}^2 = F_7^2 \wedge i(v) F_{14}^2 = 2 i(v) F_{14} \wedge F_7^2 \wedge F_{14}$, we have $$- 2 i(v) F_7 \wedge F_7 \wedge F_{14}^2 - i(v) F_{14} \wedge F_7^2 \wedge F_{14} = i(v) F_{14} \wedge F_7^2 \wedge F_{14}.$$ We also have $$i(v) F_{14} \wedge F_7 \wedge F_{14}^2 = \frac{1}{3} i(v) F_{14}^3 \wedge F_7 = - \frac{1}{3} F_{14}^3 \wedge i(v) F_7$$ and $$\frac{3}{2} |F_{14}|^2 * u^\flat \wedge i(v) (-2F_7+F_{14}) = \frac{3}{2} |F_{14}|^2 \langle -2F_7+F_{14}, v^\flat \wedge u^\flat \rangle \mathrm{vol} = -\frac{3}{2} |F_{14}|^2 \langle v^\flat, i(u)F \rangle \mathrm{vol}.$$ Hence we obtain $$\begin{aligned} \label{eq:equiv Spin7 ptwise 3} J_2= \mathopen{}\mathclose\bgroup\left(6|u|^2-\frac{3}{2} |F_{14}|^2 \aftergroup\egroup\right) \langle v^\flat, i(u)F \rangle \mathrm{vol} +\frac{3}{2} i(v) F_{14} \wedge F_7^2 \wedge F_{14} -\frac{1}{2} i(v) F_{7} \wedge F_{14}^3. \\\end{aligned}$$ Then by [[]{.upright}](#{eq:equiv Spin7 ptwise 2}) and [[]{.upright}](#{eq:equiv Spin7 ptwise 3}), we obtain $$\begin{aligned} J_1+J_2 = 3 \mathopen{}\mathclose\bgroup\left(-1+3|u|^2-\frac{1}{2} |F_{14}|^2 \aftergroup\egroup\right) \langle v^\flat, i(u)F \rangle \mathrm{vol} = 3 \mathopen{}\mathclose\bgroup\left(-1+\frac{1}{2} *(\varphi \wedge F^2) \aftergroup\egroup\right) \langle v^\flat, i(u)F \rangle \mathrm{vol}, \end{aligned}$$ where we use $*(\varphi \wedge F^2) =* \mathopen{}\mathclose\bgroup\left(F \wedge *(2F_7-F_{14}) \aftergroup\egroup\right) = 2|F_7|^2 - |F_{14}|^2 = 6|u|^2-|F_{14}|^2$ by [@kawai2020deformation Lemma B.1]. Then it follows that $$* \mathopen{}\mathclose\bgroup\left( -* \varphi \wedge F + \frac{1}{6} F^3 + \frac{1}{2} *(\varphi \wedge * F^2) \wedge * F \aftergroup\egroup\right) \wedge F \wedge \varphi = 3 \mathopen{}\mathclose\bgroup\left(-1+\frac{1}{2} *(\varphi \wedge F^2) \aftergroup\egroup\right) *(i(u)F).$$ Since $\varphi \wedge * F^2 = -6 * (i(u)F)$ by Lemma [Lemma 16](#lem:pb xi 1){reference-type="ref" reference="lem:pb xi 1"}, the proof is completed. ◻ [^1]: This work was supported by JSPS KAKENHI Grant Number JP21K03231 and MEXT Promotion of Distinctive Joint Research Center Program JPMXP0723833165.

arxiv_math

--- abstract: | We show that the toric locus of a reaction network is a smoothly embedded submanifold of the Euclidean space. More precisely, we prove that the toric locus of a reaction network is the image of an embedding and it is diffeomorphic to the product space between the affine invariant polyhedron of the network and its set of complex-balanced flux vectors. Moreover, we prove that within each affine invariant polyhedron, the complex-balanced equilibrium depends smoothly on the parameters (i.e., reaction rate constants). We also show that the complex-balanced equilibrium depends smoothly on the initial conditions. author: - "Gheorghe Craciun[^1], Jiaxin Jin[^2], Miruna-Ştefana Sorea[^3]" bibliography: - biblioEmbedding.bib title: The toric locus of a reaction network is a smooth manifold --- # Introduction Reaction networks and interaction networks are very important in many different settings, such as chemistry, biochemistry, cell biology, population dynamics, the study of species dynamics in an ecosystem, or the study of the spread of infectious diseases. The most common type of mathematical models for analyzing the dynamics of concentrations or populations in these systems are *mass-action systems*, i.e., are based on the *principle of mass-action kinetics*, which simply says that the rate of each reaction or interaction is proportional to the concentrations of its interacting species [@feinberg; @HJ; @cy]. Mass-action systems are finite-dimensional dynamical systems with polynomial right-hand-sides. They can be rigorously defined by using one of the several (equivalent) mathematical representations of a reaction network; here we use the mathematical definition of reaction networks that is based on *Euclidean embedded graphs*, also called *E-graphs*, see Section [2](#sec:preliminaryNotions){reference-type="ref" reference="sec:preliminaryNotions"}. For other uses of Euclidean embedded graphs see also [@MR3920470]. It may seem at first that the polynomial dynamical systems that result from mass-action kinetics are somehow special; it actually turns out that, except for the fact that they are restricted to the positive orthant[^4], they can give rise to absolutely *all* the dynamics that fully general polynomial dynamical systems can exhibit on a compact set [@Brunner_Craciun_2018; @cy]. In particular, mass-action systems can give rise to multiple equilibria, oscillations, and chaotic dynamics. For example, recall that the second part of Hilbert's 16th problem involves limit cycles of polynomial dynamical systems in the plane. It is not hard to see that if this Hilbert problem could be solved for mass-action systems, then it would also immediately be solved in full generality, i.e., for all polynomial systems. Here we focus on the most studied class of mass-action systems, called *complex-balanced systems*. This class of dynamical systems has its origins in some important connections between reaction networks and thermodynamics; indeed, complex-balanced systems can be regarded as generalizations of finite-dimensional versions of the Boltzmann equation [@cy]. In particular, the *complex balance condition* is a natural generalization of Boltzmann's *detailed balance condition*, which is a consequence of his principle of microscopic reversibility [@cy]. Complex-balanced systems are known to be exceptionally stable: they have a unique positive equilibrium within each linear invariant subspace, and this equilibrium (which is called a *complex-balanced equilibrium*) is locally stable; furthermore, complex-balanced systems *cannot* give rise to oscillations or chaotic dynamics [@feinberg; @HJ; @cy]. Moreover, it has actually been conjectured that any complex-balanced equilibrium is a *globally attracting point* within its linear invariant subspace, and this conjecture (called the *global attractor conjecture*) has already been proved in many cases [@Anderson; @cdss2009; @Craciun_Nazarov_Pantea_2013; @Gopalkrishnan_Miller_Shiu]. The complex balance condition also has very strong consequences for the dynamics of associated *stochastic* mass-action systems. In particular, it implies that such systems (if considered for the parameter values that satisfy complex balance) have a stationary distribution, which takes on an explicit form, as a product of Poisson distributions [@Anderson_Craciun_Kurtz_2010]. Moreover, the same complex balance condition has very strong consequences for large classes of *reaction-diffusion systems*; for example, for systems without boundary equilibria, it can be shown that such (nonlinear) reaction-diffusion systems admit classical solutions, and these solutions converge exponentially fast to the complex-balanced equilibrium [@Desvillettes_Fellner_Tang_2017; @Craciun_Jin_Pantea_Tudorascu_2021] ## Main results We consider a class of nonlinear dynamical systems, introduced by Horn and Jackson (see [@HJ]) and called *complex-balanced dynamical systems* or *toric dynamical systems* (see [@cdss2009]). They are dynamical systems associated to reaction networks, under the assumption of mass-action kinetics, such that they can give rise to *complex-balanced equilibria* (see Definition [Definition 10](#def:CB){reference-type="ref" reference="def:CB"}). For a detailed introduction to these topics we refer the reader to [@feinberg; @cy]. Given a reaction network, its *toric locus* (see Definition [Definition 11](#def:VkG){reference-type="ref" reference="def:VkG"}) is the set of positive parameters giving rise to toric dynamical systems associated to this network. Recently, researchers have shown increasing interest in the study of the toric locus (see for instance [@haque2022disguised], [@bcs]). It was shown in [@Connected] that the toric locus is connected. Moreover, a construction in [@Connected] exhibits a homeomorphism from a product space $\mathcal{P}$ to the toric locus; here $\mathcal{P}$ denotes the product space between the affine invariant polyhedron of the reaction network and its set of complex-balanced flux vectors. Using this homeomorphism, we construct a smooth embedding and prove that the toric locus is a smoothly embedded submanifold in the ambient space. In other words, we show that the toric locus is diffeomorphic to the product space $\mathcal{P}$ mentioned above, see Theorem [Theorem 1](#thm:submanifold){reference-type="ref" reference="thm:submanifold"} and Corollary [Corollary 23](#cor:diffeomorphicToTheProduct){reference-type="ref" reference="cor:diffeomorphicToTheProduct"}. Moreover, we prove that the complex-balanced equilibria depend smoothly on the parameters belonging to the toric locus (Theorem [Theorem 2](#thm:smoothDependanceEquil){reference-type="ref" reference="thm:smoothDependanceEquil"}), respectively on the initial condition (Theorem [Theorem 3](#thm:smoothDependanceinitial){reference-type="ref" reference="thm:smoothDependanceinitial"}). We now state the main results obtained in this paper. Our first theorem shows that given a Euclidean embedded graph, the associated toric locus is a smooth manifold. **Theorem 1**. *Let $G=(V, E)$ be a weakly reversible Euclidean embedded graph. Then the toric locus $\mathcal{V}(G)$ is a smoothly embedded submanifold of $\mathbb{R}_{>0}^{\vert E \vert}$.* The second and third theorems show that complex-balanced equilibria of the mass-action system vary smoothly with the parameters in the toric locus and with the initial conditions, respectively. **Theorem 2**. *Let $G=(V, E)$ be a weakly reversible Euclidean embedded graph, let $\mathcal{V}(G)$ be its toric locus, and $\mathcal{S}$ be its stoichiometric subspace. Consider also some positive state ${\boldsymbol{x}_0} \in \mathbb{R}_{>0}^n$, and for any $\boldsymbol{k}\in \mathcal{V}(G)$ denote by $\boldsymbol{x}^*(\boldsymbol{k})$ the complex-balanced equilibrium of the mass-action system $(G, \boldsymbol{k})$ within the affine invariant subspace $({\boldsymbol{x}_0} + \mathcal{S})\cap \mathbb{R}_{>0}^n$. Then $\boldsymbol{x}^*(\boldsymbol{k})$ depends smoothly on $\boldsymbol{k}$.* **Theorem 3**. *Let $G=(V, E)$ be a weakly reversible Euclidean embedded graph and fix some $\boldsymbol{k}\in \mathcal{V}(G)$, where $\mathcal{V}(G)$ is the toric locus of $G$. For any $\boldsymbol{x}_0 \in \mathbb{R}^n_{>0}$ denote by $\boldsymbol{x}^*(\boldsymbol{x}_0)$ the complex-balanced equilibrium of the mass-action system $(G, \boldsymbol{k})$ within the affine invariant subspace $({\boldsymbol{x}_0} + \mathcal{S})\cap \mathbb{R}_{>0}^n$. Then $\boldsymbol{x}^*(\boldsymbol{x}_0)$ depends smoothly on $\boldsymbol{x}_0$.* ## Structure of the paper In Section [2](#sec:preliminaryNotions){reference-type="ref" reference="sec:preliminaryNotions"}, we present terminology and notations specific to dynamical systems arising from reaction networks, with an emphasis on mass-action complex-balanced dynamical systems. The latter are also referred to as toric systems. In Section [3](#sec:smooth_embedding){reference-type="ref" reference="sec:smooth_embedding"}, we prove our first main result which says that the toric locus is a smoothly embedded submanifold of Euclidean space. Based on this result, in Section [4](#sec:smooth_dependent_toric_locus){reference-type="ref" reference="sec:smooth_dependent_toric_locus"} we show that the complex-balanced equilibrium depends smoothly on the parameter values in the toric locus. Finally, in Section [5](#sec:smooth_dependent_initial){reference-type="ref" reference="sec:smooth_dependent_initial"}, we prove Theorem [Theorem 3](#thm:smoothDependanceinitial){reference-type="ref" reference="thm:smoothDependanceinitial"} showing that the complex-balanced equilibrium depends smoothly on the initial conditions. # Preliminaries {#sec:preliminaryNotions} In this preliminary section, we present classical and relevant terminology and notations from the field of reaction network theory. We mostly follow the presentation from [@Connected] (see also [@cy]). More precisely, we recall notions and results about deterministic dynamical systems generated by reaction networks, under the assumption of mass-action kinetics, with a focus on complex-balanced systems. Let us first begin with some useful notations. **Notation 4**. *(a) Let $\mathbb{R}_{\geq 0}^n$, respectively $\mathbb{R}_{>0}^n$ denote the sets of vectors with non-negative, respectively positive real components. Similarly, $\mathbb{Z}_{\geq 0}^n$ denotes the set of vectors with non-negative integer components. The cardinality of a set $S$ is denoted by $\vert S \vert$. Let $M \sqcup N$ denote the disjoint union of the sets $M$ and $N$.* *(b) Consider two vectors $\boldsymbol{x}= (x_1, \ldots, x_n)^{\intercal} \in \mathbb{R}_{>0}^n$ and $\boldsymbol{y}= (y_1, \ldots, y_n)^{\intercal} \in \mathbb{R}_{\ge 0}^n$. Throughout this paper we will use the following multivariate notations:* *$$\notag \begin{split} \boldsymbol{x}\circ \boldsymbol{y}& := (x_1 y_1, \ldots, x_n y_n)^{\intercal}, \\ {\boldsymbol{x}}^{\boldsymbol{y}} & := x_1^{y_{1}} x_2^{y_{2}} \ldots x_n^{y_{n}}, \\ \log (\boldsymbol{x}) & := (\log(x_1), \ldots, \log(x_n))^{\intercal}, \\ \exp (\boldsymbol{x}) & := (\exp(x_1), \ldots, \exp(x_n))^{\intercal}. \end{split}$$* ## Mass-action dynamical systems and reaction networks Classically, in the definition of a reaction network, there are some important sets: the set of species, the set of complexes, and the set of reactions (see, for instance, [@feinberg Definition 3.1.1.]). In this paper, we define a reaction network to be a directed graph embedded in the Euclidean space, as it was first presented in [@MR3920470]. Due to the combinatorial and geometric aspects of this way of defining a reaction network, this equivalent definition has appeared frequently in the recent research literature (see for example [@cy] and references therein), since it is very advantageous for algebraic computations. **Definition 5**. 1. Let $n$ denote the **number of species** involved in the reaction network. The **species** are denoted by $X_1,\ldots, X_n$. 2. The **concentration** of the species $X_i$, for $i=1,\ldots, n$ is denoted by $x_i=x_i(t)$. Note that $x_i$ is a function of time $t$. At any fixed time $t \geq 0$, we obtain a vector $\boldsymbol{x}(t) = (x_1(t), \ldots, x_n(t))^\intercal \in \mathbb{R}^n$, that we call a **state** of the system at time $t$. 3. A **complex** is a formal linear combination of species $\{ X_i\}^n_{i=1}$, having non-negative real coefficients. A directed edge between two distinct complexes is called a **reaction**. **Definition 6** ([@MR3920470]). Consider a finite directed graph $G=(V, E)$. We say that $G$ is a **reaction network** (or a **Euclidean embedded graph**) if it has a finite set of **vertices** $V\subset \mathbb{R}^n$ and a finite set of **edges** $E\subseteq V\times V$. Throughout this paper, we suppose that $G=(V, E)$ does not have isolated vertices, nor self-loops. 1. Let $m$ denote the **number of vertices**, and let $V = \{ \boldsymbol{y}_1, \ldots, \boldsymbol{y}_m\}$ be the set of vertices. There is a one-to-one correspondence between the vertices and the complexes of the reaction network, namely each ${\boldsymbol{y}_i}\in V$ corresponds to a unique complex: the entries of the vertex ${\boldsymbol{y}_i}\in V$ are exactly the corresponding coefficients of the associated formal linear combination. 2. We represent a reaction in the network by a directed edge in the graph, connecting two vertices ${\boldsymbol{y}_i}\in V$ to ${\boldsymbol{y}_j}\in V$. We denote it by ${\boldsymbol{y}_i}\rightarrow {\boldsymbol{y}_j}\in E$. The difference vector ${\boldsymbol{y}_j}- {\boldsymbol{y}_i}\in\mathbb{R}^n$ is called a **reaction vector**. Here ${\boldsymbol{y}_i}$, respectively ${\boldsymbol{y}_j}$ denote the **source vertex**, respectively the **target vertex**. **Definition 7**. Consider a Euclidean embedded graph, $G=(V, E)$. 1. The connected components (or **linkage classes**) of $G=(V, E)$ partition the set of vertices $V$. Denote by $V = V_1 \sqcup V_2 \cdots \sqcup V_\ell$, where each $V_i$ represents a connected component of $G$ and the **number of connected components** is $\ell$. 2. If every edge of a connected component is part of an oriented cycle, then we say that the component is **strongly connected**. 3. If every connected component of a graph $G=(V, E)$ is strongly connected, we say that $G$ is **weakly reversible**. If we work in the context of mass-action kinetics (see [@feinberg page 28]), we obtain the ODE system of the form ([\[eq:massAction\]](#eq:massAction){reference-type="ref" reference="eq:massAction"}) (see Definition [Definition 8](#def:massAction){reference-type="ref" reference="def:massAction"} below). This is due to the fact that, under this assumption, the reaction rates are proportional to the product of the concentrations of the reactant species. **Definition 8**. Let $G=(V, E)$ be a Euclidean embedded graph. Under mass-action kinetics, we decorate each edge ${\boldsymbol{y}_i}\rightarrow {\boldsymbol{y}_j}$ with a positive **reaction rate constant**, denoted by $k_{ij}$ or $k_{{\boldsymbol{y}_i}\rightarrow {\boldsymbol{y}_j}}$. Then the **vector of reaction rate constants** is ${\boldsymbol{k}}:=(k_{ij})\in\mathbb{R}_{>0}^{E}$ and the **mass-action system associated to** $(G,{\boldsymbol{k}})$ on $\mathbb{R}^n_{>0}$ is the following: $$\label{eq:massAction} \frac{\mathrm{d}\boldsymbol{x}}{\mathrm{d} t}= \sum_{{\boldsymbol{y}_i}\rightarrow {\boldsymbol{y}_j}\in E}k_{{\boldsymbol{y}_i}\rightarrow {\boldsymbol{y}_j}} \boldsymbol{x}^{\boldsymbol{y}_i}({\boldsymbol{y}_j}-{\boldsymbol{y}_i}).$$ **Definition 9**. Consider a Euclidean embedded graph $G = (V, E)$. Then the **stoichiometric subspace** of $G$ is the following: $$\mathcal{S} = \text{span}\{ {\boldsymbol{y}_j}- {\boldsymbol{y}_i}: {\boldsymbol{y}_i}\rightarrow {\boldsymbol{y}_j}\in E \}.$$ It is known that any solution to [\[eq:massAction\]](#eq:massAction){reference-type="eqref" reference="eq:massAction"} with initial condition ${\boldsymbol{x}_0}\in \mathbb{R}_{>0}^n$ and $V \subset \mathbb{Z}_{\geq 0}^n$, is confined to $({\boldsymbol{x}_0} + \mathcal{S})\cap \mathbb{R}_{>0}^n$, see [@cy]. Hence, we say that $({\boldsymbol{x}_0} + \mathcal{S})\cap \mathbb{R}_{>0}^n$ is the **affine invariant polyhedron** of ${\boldsymbol{x}_0}$. In the sequel, we use the shorter notation: $\mathcal{S}_{{\boldsymbol{x}_0}} := ({\boldsymbol{x}_0} + \mathcal{S})\cap \mathbb{R}_{>0}^n$. ## The toric locus of a reaction network In what follows we recall the definition of *complex-balanced dynamical systems* and of the associated *toric locus*, which are key concepts in this paper. **Definition 10**. Consider the mass-action system $(G, {\boldsymbol{k}})$ given in ([\[eq:massAction\]](#eq:massAction){reference-type="ref" reference="eq:massAction"}). A state ${\boldsymbol{x}^*} \in \mathbb{R}_{>0}^n$ satisfying $$\label{eq:ss} \frac{\mathrm{d}\boldsymbol{x}}{\mathrm{d} t}= \sum_{{\boldsymbol{y}_i}\rightarrow {\boldsymbol{y}_j}\in E}k_{{\boldsymbol{y}_i}\rightarrow {\boldsymbol{y}_j}} ({\boldsymbol{x}^*})^{\boldsymbol{y}_i}({\boldsymbol{y}_j}-{\boldsymbol{y}_i}) = \mathbf{0}$$ is called a **positive steady state**. Moreover, if at each vertex $\boldsymbol{y}_0 \in V$ we have $$\label{eq:cB} \sum_{\boldsymbol{y}_0 \to {\boldsymbol{y}^\prime}\in E} k_{\boldsymbol{y}_0 \to {\boldsymbol{y}^\prime}} ({\boldsymbol{x}^*})^{\boldsymbol{y}_0} = \sum_{\boldsymbol{y}\to \boldsymbol{y}_0 \in E}k_{\boldsymbol{y}\to \boldsymbol{y}_0} ({\boldsymbol{x}^*})^{\boldsymbol{y}},$$ then the positive steady state ${\boldsymbol{x}^*} \in \mathbb{R}_{>0}^n$ is called a **complex-balanced steady state**. Given a mass-action system $(G, \boldsymbol{k})$, we say that it is a **complex-balanced system** (also called **toric dynamical system** [@cdss2009]), if every steady state of the system is a complex-balanced steady state. Definition [Definition 11](#def:VkG){reference-type="ref" reference="def:VkG"} below was introduced in [@bcs Definition 2.2] (see also [@Connected Definition 2.14]) and it was due to the results in [@cdss2009]. **Definition 11**. Consider a Euclidean embedded graph $G=(V, E)$. The set of parameters ${\boldsymbol{k}}\in\mathbb{R}_{>0}^{|E|}$, for which the mass-action system $(G, \boldsymbol{k})$ is toric or complex-balanced is called the **toric locus** of the mass-action system given by the Euclidean embedded graph $G$. We denote the toric locus by $\mathcal{V}(G) \subseteq \mathbb{R}_{>0}^{|E|}$. Besides the nice algebraic and combinatorial structure of toric dynamical systems, it is also well-known that they have strong stability properties that are desirable in many applications, as the following classical theorem has shown. **Theorem 12** ([@HJ]). *Let $(G, \boldsymbol{k})$ be a complex-balanced dynamical system, and let $\mathcal{S}$ be its associated stoichiometric subspace. Denote by $\boldsymbol{x}^* \in \mathbb{R}^n_{>0}$ a steady state of $(G, \boldsymbol{k})$. Then:* 1. *All the positive steady states are complex-balanced and there is exactly one steady state within each invariant polyhedron.* 2. *For any complex-balanced steady state $\boldsymbol{x}$ we have: $\ln \boldsymbol{x}\in \ln \boldsymbol{x}^* + \mathcal{S}^\perp$.* 3. *Every complex-balanced steady state is locally asymptotically stable within its invariant polyhedron.* In the sequel we only work in the context of $\mathcal{V}(G)$ being non-empty. Thus we assume that $G=(V, E)$ is a weakly reversible graph throughout the paper. # The toric locus is a smoothly embedded submanifold {#sec:smooth_embedding} This section aims to prove Theorem [Theorem 1](#thm:submanifold){reference-type="ref" reference="thm:submanifold"}. For this purpose, we define the *set of complex-balanced flux vectors* (see Definition [Definition 13](#def:fluxVectors){reference-type="ref" reference="def:fluxVectors"} and [@Connected Section 4.1]) and two maps in Theorem [\[thm:mapTildePhi\]](#thm:mapTildePhi){reference-type="ref" reference="thm:mapTildePhi"} and Definition [Definition 17](#def:mapTildeTildePhi){reference-type="ref" reference="def:mapTildeTildePhi"} below. ## The set of complex-balanced flux vectors $\mathcal{B}(G)$ {#subsec:CBFluxVectors} **Definition 13**. Let $G=(V, E)$ be a Euclidean embedded graph. 1. A vector $\boldsymbol\beta= (\beta_{{\boldsymbol{y}_i}\to {\boldsymbol{y}_j}})_{{\boldsymbol{y}_i}\to {\boldsymbol{y}_j}\in E}$ **satisfies the complex-balanced condition** on $G$, if at each vertex $\boldsymbol{y}_0 \in V$ we have $$\label{def: CBF} \sum_{\boldsymbol{y}\to \boldsymbol{y}_0 \in E} \beta_{\boldsymbol{y}\to \boldsymbol{y}_0} = \sum_{\boldsymbol{y}_0 \to {\boldsymbol{y}^\prime}\in E} \beta_{\boldsymbol{y}_0 \to {\boldsymbol{y}^\prime}}.$$ We denote the set of all vectors satisfying condition [\[def: CBF\]](#def: CBF){reference-type="eqref" reference="def: CBF"} by $$\notag % \label{def: tBG} \tilde{\mathcal{B}} (G) := \{\boldsymbol\beta\in \mathbb{R}^{E} \mid \boldsymbol\beta\text{ satisfies the complex-balanced condition on $G$}\}.$$ 2. Further, a vector $\boldsymbol\beta= (\beta_{{\boldsymbol{y}_i}\to {\boldsymbol{y}_j}})_{{\boldsymbol{y}_i}\to {\boldsymbol{y}_j}\in E}$ is called a **complex-balanced flux vector**, if $\boldsymbol\beta\in \mathbb{R}^{|E|}_{>0}$ and it satisfies the complex-balanced condition on $G$. We denote the set of all complex-balanced flux vectors by $$\notag % \label{def: BG} \mathcal{B}(G) := \tilde{\mathcal{B}} (G) \cap \mathbb{R}^{|E|}_{>0}.$$ **Lemma 14**. *([@Connected Lemma 4.4]) [\[lem:weak_reversible_BG\]]{#lem:weak_reversible_BG label="lem:weak_reversible_BG"} Let $G=(V, E)$ be a Euclidean embedded graph. Then $\mathcal{B}(G) \neq \emptyset$ if and only if $G=(V, E)$ is weakly reversible.* Suppose $G = (V, E)$ is a weakly reversible Euclidean embedded graph. Lemma [\[lem:weak_reversible_BG\]](#lem:weak_reversible_BG){reference-type="ref" reference="lem:weak_reversible_BG"} shows that both sets $\mathcal{B}(G)$ and $\tilde{\mathcal{B}} (G)$ are non-empty. Since the set $\tilde{\mathcal{B}} (G)$ is formed by finitely many linear restrictions given in [\[def: CBF\]](#def: CBF){reference-type="eqref" reference="def: CBF"}, the set $\tilde{\mathcal{B}} (G)$ is a linear subspace of $\mathbb{R}^{|E|}$. Note that the set $\mathcal{B}(G)$ is obtained by intersecting $\tilde{\mathcal{B}} (G)$ with the positive orthant $\mathbb{R}_{>0}^{E}$, therefore $\mathcal{B}(G)$ forms a pointed cone in $\mathbb{R}_{>0}^{|E|}$ (see [@Connected Lemma 4.5]). **Lemma 15**. *Let $G=(V, E)$ be a weakly reversible Euclidean embedded graph and let $\boldsymbol\beta= (\beta_{\boldsymbol{y}\rightarrow \boldsymbol{y}'})_{\boldsymbol{y}\rightarrow \boldsymbol{y}' \in E}$ be a (non-zero) complex-balanced flux vector in $\mathcal{B}(G)$. Suppose for every vertex $\boldsymbol{y}\in V$, there exists a corresponding weight $c(\boldsymbol{y}) \in \mathbb{R}$, such that $$\label{tilde_beta} \widetilde{\beta}_{\boldsymbol{y}\rightarrow \boldsymbol{y}'} = c(\boldsymbol{y}) \beta_{\boldsymbol{y}\rightarrow \boldsymbol{y}'}.$$ If $\widetilde{\boldsymbol\beta} = (\widetilde{\beta}_{\boldsymbol{y}\rightarrow \boldsymbol{y}'})_{\boldsymbol{y}\rightarrow \boldsymbol{y}' \in E} \in \tilde{\mathcal{B}} (G)$, then the weights $c ( \cdot )$ are constant within each connected component of $G$, i.e., for any two vertices $\boldsymbol{y}, \boldsymbol{y}'$ in the same connected component of $G$, $c(\boldsymbol{y}) = c (\boldsymbol{y}')$.* *Proof.* First, we consider the case when $G=(V, E)$ has a single connected component. In this case, it is equivalent to showing that $c(\boldsymbol{y})$ is a constant function with respect to any vertex $\boldsymbol{y}\in V$. Let us argue by contradiction. Suppose that there exist two distinct vertices $\boldsymbol{y}, \boldsymbol{y}' \in V$, such that $c(\boldsymbol{y})\neq c(\boldsymbol{y}')$. Assume that $M = \max\limits_{\boldsymbol{y}\in V} c(\boldsymbol{y})$, we define the following set as $$V_M = \{ \boldsymbol{y}\in V \mid c (\boldsymbol{y}) = M \}.$$ We note that $\emptyset \neq V_M \subsetneq V$. Since $G=(V, E)$ is weakly reversible, there are some reactions between vertices in $V_M$ and vertices in $V \setminus V_M$. Thus we can find two vertices $\boldsymbol{y}_0 \in V_M$ and $\hat{\boldsymbol{y}} \in V \setminus V_M$, such that $$\label{vertex y0} M = c(\boldsymbol{y}_0) > c(\hat{\boldsymbol{y}}) \ \text{ and } \ \hat{\boldsymbol{y}} \rightarrow \boldsymbol{y}_0 \in E.$$ Since $\widetilde{\boldsymbol\beta} \in \widetilde{\mathcal{B}}(G)$, for each vertex $\boldsymbol{y}\in V$ we have $$\label{eq:tilde_beta_1} \sum_{\boldsymbol{y}\rightarrow \boldsymbol{y}' \in E} \widetilde{\beta}_{\boldsymbol{y}\rightarrow \boldsymbol{y}'} = \sum_{\boldsymbol{y}''\rightarrow \boldsymbol{y}\in E} \widetilde{\beta}_{\boldsymbol{y}'' \rightarrow \boldsymbol{y}}.$$ From $\boldsymbol\beta\in \mathcal{B}(G)$ and [\[tilde_beta\]](#tilde_beta){reference-type="eqref" reference="tilde_beta"}, we deduce $$\label{eq:tilde_beta_2} \sum_{\boldsymbol{y}_0\rightarrow \boldsymbol{y}' \in E}\widetilde{\beta}_{\boldsymbol{y}_0\rightarrow \boldsymbol{y}'} % = \sum_{\by_0\rightarrow \by' \in E} c(\by_0) \beta_{\by_0\rightarrow \by'} = c(\boldsymbol{y}_0) \sum_{\boldsymbol{y}_0\rightarrow \boldsymbol{y}' \in E} \beta_{\boldsymbol{y}_0\rightarrow \boldsymbol{y}'} = c(\boldsymbol{y}_0) \sum_{\boldsymbol{y}'' \rightarrow \boldsymbol{y}_0 \in E} \beta_{\boldsymbol{y}'' \rightarrow \boldsymbol{y}_0}.$$ Using [\[vertex y0\]](#vertex y0){reference-type="eqref" reference="vertex y0"} and [\[eq:tilde_beta_2\]](#eq:tilde_beta_2){reference-type="eqref" reference="eq:tilde_beta_2"}, we obtain $$\notag \sum_{\boldsymbol{y}_0\rightarrow \boldsymbol{y}' \in E}\widetilde{\beta}_{\boldsymbol{y}_0\rightarrow \boldsymbol{y}'} = \sum_{\boldsymbol{y}'' \rightarrow \boldsymbol{y}_0 \in E} c(\boldsymbol{y}_0) \beta_{\boldsymbol{y}'' \rightarrow \boldsymbol{y}_0} > \sum_{\boldsymbol{y}'' \rightarrow \boldsymbol{y}_0 \in E} c(\boldsymbol{y}'') \beta_{\mathbf{y''}\rightarrow \boldsymbol{y}_0} = \sum_{\boldsymbol{y}'' \rightarrow \boldsymbol{y}_0 \in E}\widetilde{\beta}_{\boldsymbol{y}'' \rightarrow \boldsymbol{y}_0},$$ which contradicts with [\[eq:tilde_beta_1\]](#eq:tilde_beta_1){reference-type="eqref" reference="eq:tilde_beta_1"}. This concludes the proof in the case of a single connected component. Next, we consider the case when $G=(V, E)$ has $\ell > 1$ connected components $\{ V_i \}^{\ell}_{i=1}$. We claim that $c(\boldsymbol{y})$ is a constant function with respect to vertices in the same connected component. Let us argue by contradiction. Suppose that there exists a connected component $V_1$ with two distinct vertices $\boldsymbol{y}, \boldsymbol{y}' \in V_1$, such that $c(\boldsymbol{y}) \neq c(\boldsymbol{y}')$. Follow the steps in the single connected component case and assume $M = \max\limits_{\boldsymbol{y}\in V_1} c(\boldsymbol{y})$. Then we can find two vertices $\boldsymbol{y}_0, \hat{\boldsymbol{y}} \in V_1$, such that $$\label{vertex y0_L1} M = c(\boldsymbol{y}_0) > c(\hat{\boldsymbol{y}}) \ \text{ and } \ \hat{\boldsymbol{y}} \rightarrow \boldsymbol{y}_0 \in E.$$ Since $\widetilde{\boldsymbol\beta} \in \widetilde{\mathcal{B}}(G)$, by Definition [Definition 13](#def:fluxVectors){reference-type="ref" reference="def:fluxVectors"}, for each vertex $\boldsymbol{y}\in V$ we have: $$\label{tilde B G L1} \sum_{\boldsymbol{y}\rightarrow \boldsymbol{y}' \in E} \widetilde{\beta}_{\boldsymbol{y}\rightarrow \boldsymbol{y}'} = \sum_{\boldsymbol{y}'' \rightarrow \boldsymbol{y}\in E} \widetilde{\beta}_{\boldsymbol{y}'' \rightarrow \boldsymbol{y}}.$$ Using $\boldsymbol\beta\in \mathcal{B}(G)$, [\[tilde_beta\]](#tilde_beta){reference-type="eqref" reference="tilde_beta"} and [\[vertex y0_L1\]](#vertex y0_L1){reference-type="eqref" reference="vertex y0_L1"}, we have $$\notag \sum_{\boldsymbol{y}_0\rightarrow \boldsymbol{y}' \in E}\widetilde{\beta}_{\boldsymbol{y}_0\rightarrow \boldsymbol{y}'} = c(\boldsymbol{y}_0) \sum_{\boldsymbol{y}_0\rightarrow \boldsymbol{y}' \in E} \beta_{\boldsymbol{y}_0\rightarrow \boldsymbol{y}'} = c(\boldsymbol{y}_0) \sum_{\boldsymbol{y}\rightarrow \boldsymbol{y}_0 \in E} \beta_{\boldsymbol{y}\rightarrow \boldsymbol{y}_0}.$$ Note that $\boldsymbol{y}\rightarrow \boldsymbol{y}_0 \in E$ if and only if $\boldsymbol{y}\in V_1$. Thus we derive $$\notag \sum_{\boldsymbol{y}_0\rightarrow \boldsymbol{y}' \in E}\widetilde{\beta}_{\boldsymbol{y}_0\rightarrow \boldsymbol{y}'} > \sum_{\boldsymbol{y}\rightarrow \boldsymbol{y}_0 \in E} c(\boldsymbol{y}) \beta_{\boldsymbol{y}\rightarrow \boldsymbol{y}_0 \in E} =\sum_{\boldsymbol{y}\rightarrow \boldsymbol{y}_0} \widetilde{\beta}_{\boldsymbol{y}\rightarrow \boldsymbol{y}_0},$$ which contradicts [\[tilde B G L1\]](#tilde B G L1){reference-type="eqref" reference="tilde B G L1"}. Therefore, we conclude this lemma. ◻ ## A smooth map to the toric locus We start by defining a map $\varphi$ from the product space $\mathcal{S}_{\boldsymbol{x}_0} \times \mathcal{B}(G)$ to the toric locus $\mathcal{V}(G)$, which was first introduced in [@Connected Definition 4.9]. **Theorem 16**. *([@Connected Theorem 4.8]) [\[thm:mapTildePhi\]]{#thm:mapTildePhi label="thm:mapTildePhi"} Let $G = (V, E)$ be a weakly reversible Euclidean embedded graph. Then for any $\boldsymbol{x}_0\in\mathbb{R}^n_{>0}$, there exists a topological embedding: $$\varphi: \mathcal{S}_{\boldsymbol{x}_0} \times \mathcal{B}(G) \rightarrow \mathbb{R}^{|E|}$$ such that for $(\boldsymbol{x}, \boldsymbol\beta) \in \mathcal{S}_{\boldsymbol{x}_0} \times \mathcal{B}(G)$, $$\label{eq:mapPhi} \varphi(\boldsymbol{x}, \boldsymbol\beta):=\bigg (\frac{\beta_{\boldsymbol{y}\rightarrow \boldsymbol{y}'}}{\boldsymbol{x}^{\boldsymbol{y}}}\bigg )_{\boldsymbol{y}\rightarrow \boldsymbol{y}' \in E}.$$ Moreover, $\varphi$ yields a homeomorphism between the product space $\mathcal{S}_{\boldsymbol{x}_0} \times \mathcal{B}(G)$ and the toric locus $\mathcal{V}(G)$.* Note that in [@Connected Definition 4.9] the target of the map $\varphi$ was $\mathcal{V}(G)$, whereas in Theorem [\[thm:mapTildePhi\]](#thm:mapTildePhi){reference-type="ref" reference="thm:mapTildePhi"} above we use $\mathbb{R}^{|E|}$. Recall that our goal is to prove Theorem [Theorem 1](#thm:submanifold){reference-type="ref" reference="thm:submanifold"}. To this end, we will show in Proposition [Proposition 21](#lemma:smoothImmersion){reference-type="ref" reference="lemma:smoothImmersion"} that the map $\varphi$ is a smooth immersion. Then by Theorem [\[thm:mapTildePhi\]](#thm:mapTildePhi){reference-type="ref" reference="thm:mapTildePhi"} we will conclude that $\varphi$ is a smooth embedding, in Proposition [Proposition 22](#prop:PhiIsEmbedding){reference-type="ref" reference="prop:PhiIsEmbedding"}. Next, let us introduce the map $\hat{\varphi}$, which is an extension of the map $\varphi$ from Theorem [\[thm:mapTildePhi\]](#thm:mapTildePhi){reference-type="ref" reference="thm:mapTildePhi"} and it will be used in the proof of Proposition [Proposition 21](#lemma:smoothImmersion){reference-type="ref" reference="lemma:smoothImmersion"}. **Definition 17**. Let $G = (V, E)$ be a weakly reversible Euclidean embedded graph. Define the following map: $$\hat{\varphi} : \mathbb{R}^n_{>0} \times \mathbb{R}^{|E|} \rightarrow \mathbb{R}^{|E|}$$ such that for $(\boldsymbol{x}, \boldsymbol\beta)\in \mathbb{R}^n_{>0} \times \mathbb{R}^{|E|}$, $$\label{hat phi} \hat{\varphi} (\boldsymbol{x}, \boldsymbol\beta) := \bigg (\frac{\beta_{\boldsymbol{y}\rightarrow \boldsymbol{y}'}}{\boldsymbol{x}^{\boldsymbol{y}}}\bigg )_{\boldsymbol{y}\rightarrow \boldsymbol{y}' \in E}.$$ *Remark 18*. Note that $\mathcal{S}_{\boldsymbol{x}_0} \times \mathcal{B}(G) \subset \mathbb{R}^n_{>0} \times \mathbb{R}^{|E|}$, thus we have $\text{Dom} (\varphi) \subset \text{Dom} (\hat{\varphi})$. It is also clear that $\hat{\varphi} \big|_{\mathcal{S}_{\boldsymbol{x}_0} \times \mathcal{B}(G)} = \varphi$, therefore $\hat{\varphi}$ is an extension of $\varphi$. *Example 19*. The toric locus of the reversible Euclidean embedded graph (reaction network) depicted in Figure [\[fig:4thchamber\]](#fig:4thchamber){reference-type="ref" reference="fig:4thchamber"} is the non-negative real part of the *Segre variety*, given by the equation $k_{34}k_{21}=k_{43}k_{12}$, $k_{ij}\in\mathbb{R}$, $k_{ij}>0$; see [@shiu10] and [@bcs Section 5]. The Segre variety is the classical well-known embedding of the product of the projective line with itself in $\mathbb{P}^3$. It is a *smooth* quadric (a determinantal surface); see [@MR1288523 page 478]. Note that Segre embeddings appear frequently in Applied Algebraic Geometry, for example in Algebraic Statistics (see the independence model for two binary random variables in [@MR4423369 Example 8.27]) and in Quantum Mechanics and Quantum Information Theory (see [@gharahi2020fine]). ![The toric locus of the mass-action system given by the Euclidean embedded graph $G$ from Figure [\[fig:4thchamber\]](#fig:4thchamber){reference-type="ref" reference="fig:4thchamber"}. See Example [Example 19](#ex:Segre){reference-type="ref" reference="ex:Segre"}.](Segre_08.png "fig:"){#fig:segreAffine height="6cm"} ![The toric locus of the mass-action system given by the Euclidean embedded graph $G$ from Figure [\[fig:4thchamber\]](#fig:4thchamber){reference-type="ref" reference="fig:4thchamber"}. See Example [Example 19](#ex:Segre){reference-type="ref" reference="ex:Segre"}.](Segre_09.png "fig:"){#fig:segreAffine height="6cm"} ## Proof of Theorem [Theorem 1](#thm:submanifold){reference-type="ref" reference="thm:submanifold"} {#proof-of-theorem-thmsubmanifold} We first recall some definitions in differential topology, which appear in the main theorems. **Definition 20**. Given two smooth manifolds $A$ and $B$, let $f: A \to B$ be a smooth map. Then $f$ is an **immersion** if its derivative is everywhere injective. See [@leeSmooth page 77]. A **topological embedding** is a homeomorphism onto its image. See [@leeTopological page 54]. A **smooth embedding** is an immersion which is also a topological embedding. Further, a **(smoothly) embedded submanifold** is the image of a smooth embedding. See [@leeSmooth page 85]. For a textbook in differential topology we refer the reader to [@leeSmooth] (see also [@MR2680546]). The following proposition is the most important step towards proving Theorem [Theorem 1](#thm:submanifold){reference-type="ref" reference="thm:submanifold"}. **Proposition 21**. *The map $\varphi$ from Theorem [\[thm:mapTildePhi\]](#thm:mapTildePhi){reference-type="ref" reference="thm:mapTildePhi"} is a smooth immersion.* *Proof.* For simplicity, throughout the proof we use the following notation: $$M := \text{Dom} (\varphi) = \mathcal{S}_{\boldsymbol{x}_0}\times \mathcal{B}(G)$$ Recall that $\mathcal{S}_{\boldsymbol{x}_0} = ({\boldsymbol{x}_0} + \mathcal{S})\cap \mathbb{R}_{>0}^n$ and $\mathcal{B}(G) = \tilde{\mathcal{B}} (G) \cap \mathbb{R}^{|E|}_{>0}$. Since $\mathcal{S}$ and $\widetilde{\mathcal{B}}(G)$ are linear subspaces of $\mathbb{R}^n$ and $\mathbb{R}^{|E|}$ respectively, we derive that both $\mathcal{S}_{\boldsymbol{x}_0}$ and $\mathcal{B}(G)$ are smooth manifolds, and thus $M$ is a smooth manifold. Note that for any $\boldsymbol{x}\in \mathcal{S}_{\boldsymbol{x}_0} \subset \mathbb{R}_{>0}^n$ and $\boldsymbol{y}\in \mathbb{R}_{\geq 0}^n$, we obtain $\boldsymbol{x}^{\boldsymbol{y}} > 0$. Hence $\varphi$ is a smooth map on $M$. By Definition [Definition 20](#def:embedding){reference-type="ref" reference="def:embedding"}, it suffices to show that the derivative of $\varphi$ is injective. Since both $\mathcal{S}$ and $\widetilde{\mathcal{B}}(G)$ are linear subspaces, the tangent space of $M$ at the point $(\boldsymbol{x}, \boldsymbol\beta) \in M$ is $$\label{tangent space} \mathrm{T}_{(\boldsymbol{x}, \boldsymbol\beta)} M = \mathcal{S}\times \widetilde{\mathcal{B}}(G).$$ Note that both $\mathcal{S}$ and $\widetilde{\mathcal{B}}(G)$ are independent with respect to $\boldsymbol{x}$ and $\boldsymbol\beta$, hence we let $\mathcal{T} := \mathrm{T}_{(\boldsymbol{x}, \boldsymbol\beta)} M$ denote the tangent space of $M$ at any point. Then we consider the differential of $\varphi$ at the point $(\boldsymbol{x}, \boldsymbol\beta) \in M$ as follows: $$\mathrm{d} \varphi_{(\boldsymbol{x}, \boldsymbol\beta)}: \mathcal{T} \rightarrow \mathbb{R}^{|E|}.$$ We claim $\mathrm{d} \varphi$ at any point in $M$ has only trivial kernel in $\mathcal{T}$, that is, for any $(\boldsymbol{x}, \boldsymbol\beta) \in M$ $$\label{eq:ker} \mathrm{ker}(\mathrm{d} \varphi_{(\boldsymbol{x}, \boldsymbol\beta)}) \cap \mathcal{T} = \{ \mathbf{0} \}.$$ Assuming the claim, we deduce that the kernel of $\mathrm{d} \varphi$ at any point in $M$ only contains the zero vector, thus the derivative of $\varphi$ is injective. Since $\varphi$ is also smooth, by Definition [Definition 20](#def:embedding){reference-type="ref" reference="def:embedding"} we conclude that $\varphi$ is a smooth immersion. Thus it remains to prove the claim ([\[eq:ker\]](#eq:ker){reference-type="ref" reference="eq:ker"}). To simplify our computations, we consider $\hat{\varphi}$ in Definition [Definition 17](#def:mapTildeTildePhi){reference-type="ref" reference="def:mapTildeTildePhi"} and work with the differential of $\hat{\varphi}$, such that for any $(\boldsymbol{x}, \boldsymbol\beta) \in M$ $$\mathrm{d} \hat{\varphi}_{(\boldsymbol{x}, \boldsymbol\beta)} : \mathbb{R}^n \times \mathbb{R}^{|E|}\rightarrow \mathbb{R}^{|E|}.$$ From Remark [Remark 18](#rmk:extension_map){reference-type="ref" reference="rmk:extension_map"}, $\hat{\varphi}$ is an extension of $\varphi$ with $\hat{\varphi} \big|_{M} = \varphi$, which implies $\mathrm{d} \hat{\varphi}\big|_{\mathcal{T}} = \mathrm{d} \varphi$ at any point in $M$. Thus we only need to prove that for any $(\boldsymbol{x}, \boldsymbol\beta) \in M$ and any $\boldsymbol{v}\in \mathcal{T} \ \backslash \ \{\mathbf{0} \}$, $$\notag \mathrm{d} \hat{\varphi}_{(\boldsymbol{x}, \boldsymbol\beta)} (\boldsymbol{v}) \neq \mathbf{0}.$$ By contradiction, suppose there exists some $\boldsymbol{v}\in \mathcal{T} \ \backslash \ \{\mathbf{0} \}$ such that $\mathrm{d} \hat{\varphi} (\boldsymbol{v}) = \mathbf{0}$. Here we write $\boldsymbol{v}$ as $$\label{eq:v=tvtb} \boldsymbol{v}= (\tilde{\boldsymbol{v}}, \widetilde{\boldsymbol\beta}) \in \mathcal{T} =\mathcal{S} \times \widetilde{\mathcal{B}}(G),$$ and we denote $\tilde{\boldsymbol{v}} = (v_1, \ldots, v_n) \in \mathcal{S}$ and $\widetilde{\boldsymbol\beta} = (\widetilde{\beta}_1, \ldots, \widetilde{\beta}_{\vert E\vert}) \in \widetilde{\mathcal{B}}(G)$. Let us focus on $\mathrm{d} \hat{\varphi}_{(\boldsymbol{x}, \boldsymbol\beta)}$, that is the Jacobian matrix of $\hat{\varphi}$ at $(\boldsymbol{x}, \boldsymbol\beta)$. By Definition [Definition 17](#def:mapTildeTildePhi){reference-type="ref" reference="def:mapTildeTildePhi"}, the Jacobian matrix is given by $$\boldsymbol{J}_{\hat{\varphi}} (\boldsymbol{x}, \boldsymbol\beta) = \left[ \frac{\partial \hat{\varphi}}{\partial \boldsymbol{x}}, \frac{\partial \hat{\varphi}}{\partial \boldsymbol\beta} \right].$$ Consider a fixed reaction $\boldsymbol{y}\rightarrow \boldsymbol{y}' \in E$ and let us compute the corresponding row vector $$\label{eq:yy'row} \left( \frac{\partial \hat{\varphi}_{\boldsymbol{y}\rightarrow \boldsymbol{y}'}}{\partial \boldsymbol{x}}, \ \frac{\partial \hat{\varphi}_{\boldsymbol{y}\rightarrow \boldsymbol{y}'}}{\partial \boldsymbol\beta} \right)$$ in the Jacobian matrix $\boldsymbol{J}_{\hat{\varphi}} (\boldsymbol{x}, \boldsymbol\beta)$. For the first $n$ entries in the row given by [\[eq:yy\'row\]](#eq:yy'row){reference-type="eqref" reference="eq:yy'row"}, which correspond to the derivatives of $\hat{\varphi}_{\boldsymbol{y}\longrightarrow \boldsymbol{y}'}$ with respect to $\boldsymbol{x}$, we obtain $$\label{diff first n} \frac{\partial \hat{\varphi}_{\boldsymbol{y}\longrightarrow \boldsymbol{y}'}}{\partial x_j} = \frac{\beta_{\boldsymbol{y}\rightarrow \boldsymbol{y}'}}{\boldsymbol{x}^{\boldsymbol{y}}}\frac{-y_j}{x_j} \ \text{ for } j = 1, \ldots, n,$$ where $\boldsymbol{x}= (x_1, \ldots, x_n) \in \mathbb{R}^n_{>0}$ and $\boldsymbol{y}= (y_1, \ldots, y_n) \in \mathbb{R}^n_{\geq 0}$. Furthermore, the last $\vert E \vert$ entries of the row in [\[eq:yy\'row\]](#eq:yy'row){reference-type="eqref" reference="eq:yy'row"} are obtained by taking the derivatives of $\hat{\varphi}_{\boldsymbol{y}\longrightarrow \boldsymbol{y}'}$ with respect to $\boldsymbol\beta$. Thus we compute for $k = 1, \ldots, \vert E \vert$, $$\label{diff last E} \frac{\partial \hat{\varphi}_{\boldsymbol{y}\rightarrow \boldsymbol{y}'}}{\partial \beta_k} = \begin{cases} \frac{1}{{\boldsymbol{x}}^{\boldsymbol{y}}}, & \text{ if } {k = {\boldsymbol{y}} \rightarrow {\boldsymbol{y}}'}, \\ 0, & \text{ otherwise.} \end{cases}$$ Using the assumption $\mathrm{d} \hat{\varphi} (\boldsymbol{v})= \mathbf{0}$, [\[eq:v=tvtb\]](#eq:v=tvtb){reference-type="eqref" reference="eq:v=tvtb"}, [\[diff first n\]](#diff first n){reference-type="eqref" reference="diff first n"} and [\[diff last E\]](#diff last E){reference-type="eqref" reference="diff last E"}, we get that $$\label{dv = 0} \begin{split} \frac{\partial \hat{\varphi}_{\boldsymbol{y}\rightarrow \boldsymbol{y}'}} {\partial (\boldsymbol{x}, \bm\beta)} \cdot \boldsymbol{v} & = \sum^{n}_{j=1} \frac{\partial \hat{\varphi}_{\boldsymbol{y}\rightarrow \boldsymbol{y}'}}{\partial x_j} v_j + \sum^{\vert E \vert}_{k=1} \frac{\partial \hat{\varphi}_{\boldsymbol{y}\rightarrow \boldsymbol{y}'}}{\partial \beta_{k}} v_{n+k} \\& = \frac{\beta_{\boldsymbol{y}\rightarrow \boldsymbol{y}'}}{\boldsymbol{x}^{\boldsymbol{y}}}\frac{-y_1}{x_1} v_1 + \ldots + \frac{\beta_{\boldsymbol{y}\rightarrow \boldsymbol{y}'}}{\boldsymbol{x}^{\boldsymbol{y}}}\frac{-y_n}{x_n} v_n + \frac{1}{\boldsymbol{x}^{\boldsymbol{y}}} \widetilde{\beta}_{\boldsymbol{y}\rightarrow \boldsymbol{y}'} = 0. \end{split}$$ Multiplying ${\boldsymbol{x}}^{\boldsymbol{y}} > 0$ on both sides of [\[dv = 0\]](#dv = 0){reference-type="eqref" reference="dv = 0"}, we obtain for any $\boldsymbol{y}\rightarrow \boldsymbol{y}'\in E$ $$\label{eq:betaBeta} \beta_{\boldsymbol{y}\rightarrow \boldsymbol{y}'} \big ( \sum_{i=1}^n \frac{y_i}{x_i} v_i \big ) = \widetilde{\beta}_{\boldsymbol{y}\rightarrow \boldsymbol{y}'}.$$ Here we introduce the weight functions $c (\boldsymbol{y})$ as follows: $$\label{def: c} c(\boldsymbol{y}) := \sum_{i=1}^n \frac{y_i}{x_i} v_i \ \text{ for any } \boldsymbol{y}\in V.$$ Applying [\[eq:betaBeta\]](#eq:betaBeta){reference-type="eqref" reference="eq:betaBeta"} on all reactions in $G = (V, E)$, we have $$\label{eq:betaBeta 1} c(\boldsymbol{y}) \beta_{\boldsymbol{y}\rightarrow \boldsymbol{y}'} = \widetilde{\beta}_{\boldsymbol{y}\rightarrow \boldsymbol{y}'} \ \text{ for any } \boldsymbol{y}\rightarrow \boldsymbol{y}'\in E.$$ By [\[eq:v=tvtb\]](#eq:v=tvtb){reference-type="eqref" reference="eq:v=tvtb"}, we have $\widetilde{\boldsymbol\beta} = (\widetilde{\beta}_1, \ldots, \widetilde{\beta}_{\vert E\vert}) \in \widetilde{\mathcal{B}}(G)$. This shows that, under the weight function $c(\boldsymbol{y})$, the vector $\boldsymbol\beta\in \mathcal{B}(G)$ is mapped to $\tilde{\mathcal{B}} (G)$. Suppose $G = (V, E)$ has $\ell \geq 1$ connected components $\{ V_1, \cdots, V_{\ell} \}$. By Lemma [Lemma 15](#lem:constant_weight_flux){reference-type="ref" reference="lem:constant_weight_flux"}, we obtain that for any $1 \leq i \leq \ell$, the weight function $c(\cdot)$ is constant within each connected component $V_i \subset V$. Hence there is a set of real numbers $\{ k_i \}^{l}_{i=1} \in \mathbb{R}$, such that $$\label{c constant Li} c(\boldsymbol{y}) = k_i \ \text{ for any } \boldsymbol{y}\in V_i \subset V.$$ From [\[def: c\]](#def: c){reference-type="eqref" reference="def: c"} and [\[c constant Li\]](#c constant Li){reference-type="eqref" reference="c constant Li"}, for any $\boldsymbol{y}= (y_1, \cdots, y_n) \in V_i$ we get $$\label{c constant Li_2} \sum_{j=1}^n \frac{y_j}{x_j} v_j = k_i.$$ Then we pick one vertex in each connected component, denoted by $\boldsymbol{y}^i = (y^i_{1}, \cdots, y^i_{n}) \in V_i$ with $1 \leq i \leq \ell$. From [\[c constant Li_2\]](#c constant Li_2){reference-type="eqref" reference="c constant Li_2"}, for any $1 \leq i \leq \ell$ and $\boldsymbol{y}\in V_i$, $$\label{delta c = 0 Li} c(\boldsymbol{y}) - c(\boldsymbol{y}^i) = \sum_{j=1}^n \frac{y_j}{x_j} v_j - \sum_{j=1}^n \frac{y^i_{j}}{x_j} v_j = \sum_{j=1}^n \frac{(y_j - y^i_{j})}{x_j} v_j = 0.$$ We also note that $\mathcal{S} = \text{span}\{ \boldsymbol{y}- \boldsymbol{y}^i : \boldsymbol{y}\in V_i \ \text{with} \ 1 \leq i \leq \ell \}$. By [\[eq:v=tvtb\]](#eq:v=tvtb){reference-type="eqref" reference="eq:v=tvtb"}, we have $\tilde{\boldsymbol{v}} = (v_1, \ldots,v_n) \in \mathcal{S}$ and thus $\tilde{\boldsymbol{v}}$ can be written as $$\label{tilde v Li} \tilde{\boldsymbol{v}} = \sum\limits^{\ell}_{i=1} \sum\limits_{\boldsymbol{y}\in V_i} w_{\boldsymbol{y}} (\boldsymbol{y}- \boldsymbol{y}^i) \ \text{ with } \ w_{\boldsymbol{y}} \in \mathbb{R}.$$ For every vertex $\boldsymbol{y}\in V$, we multiply $c(\boldsymbol{y}) - c(\boldsymbol{y}^i)$ in [\[delta c = 0 Li\]](#delta c = 0 Li){reference-type="eqref" reference="delta c = 0 Li"} by $w_{\boldsymbol{y}}$, and then do the summation of all such multiplications. Hence we derive that $$\label{weight d c = 0 Li 0} 0 = \sum\limits^{\ell}_{i=1} \sum\limits_{\boldsymbol{y}\in V_i} w_{\boldsymbol{y}} \big( c(\boldsymbol{y}) - c(\boldsymbol{y}^i) \big) = \sum\limits^{\ell}_{i=1} \sum\limits_{\boldsymbol{y}\in V_i} w_{\boldsymbol{y}} \bigg( \sum_{j=1}^n \frac{(y_j - y^i_{j})}{x_j} v_j \bigg).$$ Inputting [\[tilde v Li\]](#tilde v Li){reference-type="eqref" reference="tilde v Li"} into [\[weight d c = 0 Li 0\]](#weight d c = 0 Li 0){reference-type="eqref" reference="weight d c = 0 Li 0"}, we have $$\label{weight d c = 0 Li} \begin{split} 0 & = \sum\limits^{\ell}_{i=1} \sum\limits_{\boldsymbol{y}\in V_i} w_{\boldsymbol{y}} \bigg( \sum_{j=1}^n \frac{(y_j - y^i_{j})}{x_j} \Big( \sum\limits^{\ell}_{i=1} \sum\limits_{\boldsymbol{y}\in V_i} w_{\boldsymbol{y}} (y_{j} - y^i_{j}) \Big) \bigg) \\& = \sum\limits^{\ell}_{i=1} \sum\limits_{\boldsymbol{y}\in V_i} \bigg( \sum_{j=1}^n \frac{w_{\boldsymbol{y}} (y_j - y^i_{j})}{x_j} \Big( \sum\limits^{\ell}_{i=1} \sum\limits_{\boldsymbol{y}\in V_i} w_{\boldsymbol{y}} (y_{j} - y^i_{j}) \Big) \bigg) \\& = \sum_{j=1}^n \Bigg( \sum\limits^{\ell}_{i=1} \sum\limits_{\boldsymbol{y}\in V_i} \frac{w_{\boldsymbol{y}} (y_{j} - y^i_{j})}{x_j} \Big( \sum\limits^{\ell}_{i=1} \sum\limits_{\boldsymbol{y}\in V_i} w_{\boldsymbol{y}} (y_{j} - y^i_{j}) \Big) \Bigg) \\& = \sum_{j=1}^n \frac{\Big( \sum\limits^{\ell}_{i=1} \sum\limits_{\boldsymbol{y}\in V_i} w_{\boldsymbol{y}} (y_{j} - y^i_{j}) \Big)^2 }{x_j}. \end{split}$$ Since $\boldsymbol{x}= (x_{1}, \cdots, x_{n}) \in \mathbb{R}_{>0}^n$, [\[weight d c = 0 Li\]](#weight d c = 0 Li){reference-type="eqref" reference="weight d c = 0 Li"} implies that $$v_j = \sum\limits^{\ell}_{i=1} \sum\limits_{\boldsymbol{y}\in V_i} w_{\boldsymbol{y}} (y_{j} - y^i_{j}) = 0 \ \text{ for } \ j = 1, \ldots, n.$$ This shows $\tilde{\boldsymbol{v}} = (v_1, \ldots,v_n) = \mathbf{0}$. Then we apply $\tilde{\boldsymbol{v}} = \mathbf{0}$ on [\[eq:betaBeta 1\]](#eq:betaBeta 1){reference-type="eqref" reference="eq:betaBeta 1"}, and obtain $$\widetilde{\beta}_{\boldsymbol{y}\rightarrow \boldsymbol{y}' } = c(\boldsymbol{y}) \beta_{\boldsymbol{y}\rightarrow \boldsymbol{y}'} = \big( \sum_{i=1}^n \frac{y_i}{x_i} v_i \big) \beta_{\boldsymbol{y}\rightarrow \boldsymbol{y}'} = 0 \ \text{ for any } \ \boldsymbol{y}\rightarrow \boldsymbol{y}' \in E,$$ which indicates $\widetilde{\boldsymbol\beta} = (\widetilde{\beta}_1, \ldots, \widetilde{\beta}_{\vert E\vert}) = \mathbf{0}$. However, this contradicts with $\boldsymbol{v}= (\tilde{\boldsymbol{v}}, \widetilde{\boldsymbol\beta}) \neq \mathbf{0}$ and we prove the claim. ◻ **Proposition 22**. *The map $\varphi: \mathcal{S}_{\boldsymbol{x}_0} \times \mathcal{B}(G) \rightarrow \mathbb{R}^{|E|}$ from Theorem [\[thm:mapTildePhi\]](#thm:mapTildePhi){reference-type="ref" reference="thm:mapTildePhi"} is a smooth embedding.* *Proof.* By Definition [Definition 20](#def:embedding){reference-type="ref" reference="def:embedding"}, we need to prove that $\varphi$ is a smooth immersion and a topological embedding. Theorem [\[thm:mapTildePhi\]](#thm:mapTildePhi){reference-type="ref" reference="thm:mapTildePhi"} shows the map $\varphi$ is a topological embedding, as well as a homeomorphism onto its image, $\mathcal{V}(G)$. Finally, by Proposition [Proposition 21](#lemma:smoothImmersion){reference-type="ref" reference="lemma:smoothImmersion"}, we get that $\varphi$ is a smooth immersion. ◻ Now we are ready to prove the most important result of our paper: *Proof of Theorem [Theorem 1](#thm:submanifold){reference-type="ref" reference="thm:submanifold"}.* The proof follows directly from Definition [Definition 20](#def:embedding){reference-type="ref" reference="def:embedding"} and Proposition [Proposition 22](#prop:PhiIsEmbedding){reference-type="ref" reference="prop:PhiIsEmbedding"}. ◻ **Corollary 23**. *The toric locus is diffeomorphic to the product space between the affine invariant polyhedron and the set of complex-balanced flux vectors.* *Proof.* This follows directly from Theorem [Theorem 1](#thm:submanifold){reference-type="ref" reference="thm:submanifold"} and [@leeSmooth Proposition 5.2] (Images of Embeddings as Submanifolds). ◻ # Complex-balanced equilibria depend smoothly on the parameters {#sec:smooth_dependent_toric_locus} In this section, we prove Theorem [Theorem 2](#thm:smoothDependanceEquil){reference-type="ref" reference="thm:smoothDependanceEquil"}, i.e., we show that the complex-balanced equilibria vary smoothly with the parameters (also called reaction rate constants) $\boldsymbol{k}$ in the toric locus. Let us introduce the map $\hat{Q}$ in Definition [Definition 24](#def:mapStep1){reference-type="ref" reference="def:mapStep1"}. We will show that $\hat{Q}$ is a smooth map. **Definition 24**. Let $G=(V, E)$ be a weakly reversible Euclidean embedded graph and let $\mathcal{S}$ be its associated stoichiometric subspace. Define the following map: $$\hat{Q}: \mathcal{V}(G) \rightarrow \mathbb{R}^n,$$ such that $$\label{def:hat_Q} \hat{Q}(\boldsymbol{k}) := \boldsymbol{X}^*,$$ where $\boldsymbol{X}^* \in \mathcal{S}$ and $\exp (\boldsymbol{X}^*)$ is a complex-balanced equilibrium. We first show that the map $\hat{Q}$ in Definition [Definition 24](#def:mapStep1){reference-type="ref" reference="def:mapStep1"} is well-defined and smooth. To this end, we introduce the Kirchoff matrix and Proposition [Proposition 25](#prop:iff){reference-type="ref" reference="prop:iff"} below to explain the connection between complex-balanced equilibria and the reaction rates. Consider a mass-action system $(G, \boldsymbol{k})$. Given a vertex $\boldsymbol{y}_i \in V$, suppose $\boldsymbol{y}_i$ belongs to a connected component $V_1$ with $|V_1| = m_1$. We construct the $m_1 \times m_1$ **Kirchoff matrix** $A_{\boldsymbol{k}}$ as follows: $$\label{def: Ak} [A_{\boldsymbol{k}}]_{ji} := \begin{cases} k_{{\boldsymbol{y}_i}\rightarrow {\boldsymbol{y}_j}}, & \ \text{if } \ \boldsymbol{y}_i, \boldsymbol{y}_j \in V_1 \ \text{and } {\boldsymbol{y}_i}\rightarrow {\boldsymbol{y}_j}\in E \\[5pt] -\sum\limits_{{\boldsymbol{y}_i}\rightarrow {\boldsymbol{y}_j}\in E} k_{{\boldsymbol{y}_i}\rightarrow {\boldsymbol{y}_j}}, & \ \text{if } \ i =j, \\[5pt] 0, & \ \text{otherwise}. \end{cases}$$ Denote by $K_i$ the minor of the entry in the $i$-th row and the $i$-th column of $A_{{\boldsymbol{k}}}$. The following proposition (see [@cdss2009 Section 2] and [@Connected Proposition 3.9]) gives a characterization of the complex-balanced equilibria. **Proposition 25** ([@cdss2009 Section 2],[@Connected Proposition 3.9]). *Let $(G,{\boldsymbol{k}})$ be a weakly reversible mass-action system, with $\ell$ connected components. For any two vertices ${\boldsymbol{y}_i}$ and ${\boldsymbol{y}_j}$ in $G$, consider the equation: $$\label{eq:binom} K_i \boldsymbol{x}^{{\boldsymbol{y}_j}}-K_j \boldsymbol{x}^{{\boldsymbol{y}_i}}=0.$$ Then the following are equivalent:* *(i) Equations ([\[eq:binom\]](#eq:binom){reference-type="ref" reference="eq:binom"}) are satisfied for every pair of vertices belonging to the same connected component of $G$.* *(ii) $\boldsymbol{x}$ is a complex-balanced equilibrium for the reaction rate vector ${\boldsymbol{k}}$.* The next Lemma [Lemma 26](#Q hat smooth){reference-type="ref" reference="Q hat smooth"} shows the smoothness of the map $\hat{Q}$ defined in [\[def:hat_Q\]](#def:hat_Q){reference-type="eqref" reference="def:hat_Q"}. A similar conclusion was obtained in [@Connected Corollary 3.13]. **Lemma 26** ([@Connected]). *The map $\hat{Q}$ from Definition [Definition 24](#def:mapStep1){reference-type="ref" reference="def:mapStep1"} is well-defined and smooth.* *Proof.* Note that, for the completeness of this paper, we explain here the proof of Lemma [Lemma 26](#Q hat smooth){reference-type="ref" reference="Q hat smooth"}, which was given in the proof of [@Connected Theorem 3.5]. Suppose $\boldsymbol{x}^* \in \mathbb{R}^n_{>0}$ is a complex-balanced steady state of $(G, \boldsymbol{k})$. Then by Theorem [Theorem 12](#thm:HJ){reference-type="ref" reference="thm:HJ"} (b), every complex-balanced steady state $\boldsymbol{x}\in \mathbb{R}^n_{>0}$ of the complex-balanced system $(G, \boldsymbol{k})$ satisfies the following $$\ln (\boldsymbol{x}) \in \ln \boldsymbol{x}^* + \mathcal{S}^\perp.$$ Hence, in particular, for any $\boldsymbol{k}\in \mathcal{V}(G)$, there exists a unique complex-balanced steady state $\boldsymbol{x}$, such that $$\ln (\boldsymbol{x}) \in \mathcal{S}.$$ Therefore, the map $\hat{Q}$ is well-defined. Next, we show the smoothness of the map $\hat{Q}$. From Definition [\[def: Ak\]](#def: Ak){reference-type="ref" reference="def: Ak"}, it is clear that the vector ${\boldsymbol{K}} = (K_i) \in \mathbb{R}_{>0}^m$ depends smoothly on the reaction rate vector ${\boldsymbol{k}} = (k_{ij}) \in \mathbb{R}_{>0}^{E}$. Hence, it suffices to show that $\hat{Q}(\boldsymbol{k}) = \boldsymbol{X}^*$ depends smoothly on ${\boldsymbol{K}}$. By Proposition [Proposition 25](#prop:iff){reference-type="ref" reference="prop:iff"}, a state $\boldsymbol{x}$ is a complex-balanced equilibrium if and only if for any two vertices ${\boldsymbol{y}_i}, {\boldsymbol{y}_j}$ in the same connected component of $G$ we have $$\label{eq:binom2} K_i \boldsymbol{x}^{{\boldsymbol{y}_j}} = K_j \boldsymbol{x}^{{\boldsymbol{y}_i}}.$$ Taking the log of both sides in Equation [\[eq:binom2\]](#eq:binom2){reference-type="eqref" reference="eq:binom2"}, we derive $$\label{eq:logMultivar1} \ln (K_i) + {\boldsymbol{y}_j}^\intercal \cdot \ln (\boldsymbol{x}) =\ln (K_j) + {\boldsymbol{y}_i}^\intercal \cdot \ln (\boldsymbol{x}).$$ By setting $\boldsymbol{X}= \ln (\boldsymbol{x})$, we can rewrite ([\[eq:logMultivar1\]](#eq:logMultivar1){reference-type="ref" reference="eq:logMultivar1"}) as $$\label{eq:logMultivar2} \ln (K_i / K_j) = ({\boldsymbol{y}_i}^\intercal - {\boldsymbol{y}_j}^\intercal) \cdot \boldsymbol{X},$$ where ${\boldsymbol{y}_i}$ and ${\boldsymbol{y}_j}$ are two vertices belonging to the same connected component of $G$. The proof consists of two steps, as it was shown in the proof of [@Connected Theorem 3.5] (see also [@Connected Corollary 3.13]). First, we prove the smoothness in the case where the graph $G$ has only one connected component. Second, we generalize the result for any number of connected components. #### Step 1. Suppose that the graph $G$ has a single connected component. In this case, the vertices $\{\boldsymbol{y}_1, \ldots, {\boldsymbol{y}_m}\}$ belong to the same connected component of $G$. Note that Equations ([\[eq:logMultivar2\]](#eq:logMultivar2){reference-type="ref" reference="eq:logMultivar2"}) are equivalent to the following system of linear equations in the unknowns $\boldsymbol{X}$: $$\label{system 1} \begin{bmatrix} \ln ( K_1 / K_2) \\ \ln ( K_2 / K_3) \\ \vdots \\ \ln ( K_{m-1} / K_m) \end{bmatrix} = \underbrace{ \begin{bmatrix} \boldsymbol{y}_1^\intercal - \boldsymbol{y}_2^\intercal \\ \boldsymbol{y}_2^\intercal - \boldsymbol{y}_3^\intercal \\ \vdots \\ \boldsymbol{y}_{m-1}^\intercal - \boldsymbol{y}_{m}^\intercal \end{bmatrix} }_{\Delta \boldsymbol{y}} \begin{bmatrix} X_1\\ X_2\\ \vdots \\ X_n \end{bmatrix}.$$ The graph $G$ is strongly connected, hence its stoichiometric subspace (recall Definition [Definition 9](#def:stoichiom){reference-type="ref" reference="def:stoichiom"}) is given by $$\notag \mathcal{S} = \text{span}\{ \boldsymbol{y}_1^\intercal -\boldsymbol{y}_2^\intercal, \boldsymbol{y}_2^\intercal -\boldsymbol{y}_3^\intercal, \ldots, \boldsymbol{y}_{m-1}^\intercal -\boldsymbol{y}_{m}^\intercal \}.$$ If we denote the dimension of $\mathcal{S}$ by $s$, then we obtain $s \leq \min \{ m-1, n\}$. In addition, the matrix $\Delta \boldsymbol{y}$ has exactly $s$ linearly independent rows. We may assume without loss of generality that the first $s$ rows of $\Delta \boldsymbol{y}$ are linearly independent. Hence, we have: $$\label{S span} \mathcal{S} = \text{span}\{ \boldsymbol{y}_1^\intercal -\boldsymbol{y}_2^\intercal, \boldsymbol{y}_2^\intercal -\boldsymbol{y}_3^\intercal, \ldots, \boldsymbol{y}_{s}^\intercal -\boldsymbol{y}_{s+1}^\intercal \}.$$ Let us use the following notations: $$\notag \Delta_s \boldsymbol{y} := \begin{bmatrix} \boldsymbol{y}_1^\intercal - \boldsymbol{y}_2^\intercal \\ \boldsymbol{y}_2^\intercal - \boldsymbol{y}_3^\intercal \\ \vdots \\ \boldsymbol{y}_s^\intercal - \boldsymbol{y}_{s+1}^\intercal \end{bmatrix}, \ \text{and } \Delta_s {\boldsymbol{K}}:=\begin{bmatrix} K_1 / K_2 \\ K_2 / K_3 \\ \vdots \\ K_{s} / K_{s+1} \end{bmatrix}.$$ Consider now the system of equations ([\[system 2\]](#system 2){reference-type="ref" reference="system 2"}): $$\label{system 2} \ln (\Delta_s {\boldsymbol{K}})=(\Delta_s \boldsymbol{y}) \boldsymbol{X}.$$ By Theorem [Theorem 12](#thm:HJ){reference-type="ref" reference="thm:HJ"}, for any $\boldsymbol{k}\in \mathcal{V}(G)$, there exists a complex-balanced steady state $\boldsymbol{x}^* \in \mathbb{R}^n_{>0}$ of the complex-balanced system $(G, \boldsymbol{k})$. In addition, we can write the solutions to [\[system 1\]](#system 1){reference-type="eqref" reference="system 1"} as follows: $\boldsymbol{X}= \ln \boldsymbol{x}^* + \mathcal{S}^\perp$ and we obtain that the dimension of the set of solutions to ([\[system 1\]](#system 1){reference-type="ref" reference="system 1"}) is $n-s$. Moreover, one can check that the solutions of the system ([\[system 1\]](#system 1){reference-type="ref" reference="system 1"}) are also solutions of the system ([\[system 2\]](#system 2){reference-type="ref" reference="system 2"}) and the set of solutions to ([\[system 2\]](#system 2){reference-type="ref" reference="system 2"}) is also of dimension $n-s$. In conclusion, when solving for $\boldsymbol{X}$, the systems ([\[system 1\]](#system 1){reference-type="ref" reference="system 1"}) and ([\[system 2\]](#system 2){reference-type="ref" reference="system 2"}) are equivalent. In the sequel, let us construct $\boldsymbol{X}^*$, the special solution to the system [\[system 2\]](#system 2){reference-type="eqref" reference="system 2"}, where $\boldsymbol{X}^* \in \mathcal{S}$. Remember that the dimension of the stoichiometric subspace $s \leq \min \{ m-1, n\}$. We consider the following two cases: **Case 1:** $s = n$. In this case, we have that the stoichiometric subspace $\mathcal{S} = \mathbb{R}^n$ and the matrix $\Delta_s \boldsymbol{y}$ is invertible. We compute a solution of ([\[system 2\]](#system 2){reference-type="ref" reference="system 2"}) as follows: $$\label{X* 1} \boldsymbol{X}^* = (\Delta_s \boldsymbol{y})^{-1} \ln (\Delta_s {\boldsymbol{K}}).$$ Then $\boldsymbol{X}^* \in \mathcal{S} = \mathbb{R}^n$, and $\exp (\boldsymbol{X}^*)$ is a complex-balanced equilibrium. **Case 2:** $s < n$. Remember that we denote the orthogonal complement of $\mathcal{S}$ by $\mathcal{S}^{\perp}$. Then we have: $$\label{S perp} 0 < \dim (\mathcal{S}^{\perp}) = n - \dim (\mathcal{S}) = n - s,$$ since the stoichiometric subspace $\mathcal{S} \subset \mathbb{R}^n$. Let us consider $B = \{ \boldsymbol{v}_1, \boldsymbol{v}_2, \ldots, \boldsymbol{v}_{n-s} \} \subset \mathbb{R}^n$, a basis of $\mathcal{S}^{\perp}$. Next, let us construct a new matrix and vector as follows: $$\label{tilde matirx} \tilde{\Delta} \boldsymbol{y} := \begin{bmatrix} \Delta_s \boldsymbol{y} \\ \boldsymbol{v}_1^\intercal \\ \vdots \\ \boldsymbol{v}_{n-s}^\intercal \end{bmatrix}, \ \text{and } \tilde{\Delta} {\boldsymbol{K}} := \begin{bmatrix} \Delta_s {\boldsymbol{K}} \\ 0 \\ \vdots \\ 0 \end{bmatrix}.$$ We now focus on system ([\[system 3\]](#system 3){reference-type="ref" reference="system 3"}): $$\label{system 3} \ln (\tilde{\Delta} {\boldsymbol{K}}) = (\tilde{\Delta} \boldsymbol{y}) \boldsymbol{X}.$$ From [\[S span\]](#S span){reference-type="eqref" reference="S span"} and the fact that $\{ \boldsymbol{v}_1, \ldots, \boldsymbol{v}_{n-s} \}$ forms a basis of $\mathcal{S}^{\perp}$, we obtain that $\tilde{\Delta} \boldsymbol{y} \in \mathbb{R}_{n \times n}$ is an invertible matrix. Therefore, we get a solution of system ([\[system 3\]](#system 3){reference-type="ref" reference="system 3"}) as $$\label{X* 2} \boldsymbol{X}^* = (\tilde{\Delta} \boldsymbol{y})^{-1} \ln (\tilde{\Delta} {\boldsymbol{K}}).$$ We have that $\boldsymbol{X}^*$ solves ([\[system 2\]](#system 2){reference-type="ref" reference="system 2"}) and for $i = 1, \cdots, n-s$, $$\notag \boldsymbol{v}_i^\intercal \cdot \boldsymbol{X}^* = 0.$$ Therefore $\boldsymbol{X}^* \in \mathcal{S}$. Moreover, by construction, $\exp (\boldsymbol{X}^*)$ is a complex-balanced equilibrium. To sum up, in both cases we computed the vector $\boldsymbol{X}^* \in \mathcal{S}$, such that $\exp (\boldsymbol{X}^*)$ is a complex-balanced equilibrium of the system $(G, \boldsymbol{k})$. Since the map $\hat{Q}$ is well-defined, we have that $\hat{Q}(\boldsymbol{k}) = \boldsymbol{X}^*$. In addition, since $(\Delta_s \boldsymbol{y})^{-1}$ and $(\tilde{\Delta} \boldsymbol{y})^{-1}$ are fixed real matrices, we conclude that $\boldsymbol{X}^*$ depends smoothly on the vector ${\boldsymbol{K}}$. #### Step 2. Let us suppose that the graph $G$ has multiple connected components, denoted by $V_1, \ldots, V_\ell$ with $\ell > 1$. By relabeling the vertices according to the connected components of $G$ we have: $$V_p = \{ \boldsymbol{y}_{m_{p-1}+1}, \ldots, \boldsymbol{y}_{m_p}\}, \text{ for } 1 \leq p \leq \ell.$$ The system [\[eq:logMultivar2\]](#eq:logMultivar2){reference-type="eqref" reference="eq:logMultivar2"} is equivalent to the following system of equations in the unknown $\boldsymbol{X}$: $$\label{system 1 l>1} \underbrace{ \begin{bmatrix} \ln ( K_1 / K_2) \\ \vdots \\ \ln ( K_{m_1 - 1} / K_{m_1}) \\ \hdashline[1.5pt/1.5pt] \ln ( K_{m_1 + 1} / K_{m_1 + 2}) \\ \vdots \\ \ln ( K_{m_2 - 1} / K_{m_2}) \\ \hdashline[1.5pt/1.5pt] \vdots \\ \ln ( K_{m_{\ell}-1} / K_{m_{\ell}}) \end{bmatrix} }_{\ln (\Delta {\boldsymbol{K}})} = \underbrace{ \begin{bmatrix} \boldsymbol{y}_1^\intercal - \boldsymbol{y}_2^\intercal \\ \vdots \\ \boldsymbol{y}_{m_1 - 1}^\intercal - \boldsymbol{y}_{m_1}^\intercal \\ \hdashline[1.5pt/1.5pt] \boldsymbol{y}_{m_1 + 1}^\intercal - \boldsymbol{y}_{m_1 + 2}^\intercal \\ \vdots \\ \boldsymbol{y}_{m_2 - 1}^\intercal - \boldsymbol{y}_{m_2}^\intercal \\ \hdashline[1.5pt/1.5pt] \vdots \\ \boldsymbol{y}_{m_{\ell}-1}^\intercal - \boldsymbol{y}_{m_{\ell}}^\intercal \end{bmatrix} }_{\Delta \boldsymbol{y}} % \begin{bmatrix} X_1 \\ X_2 \\ \vdots \\ X_n \end{bmatrix}.$$ By Definition [Definition 9](#def:stoichiom){reference-type="ref" reference="def:stoichiom"}, the stoichiometric subspace of $G$ is $$\notag \mathcal{S} = \text{span}\{ \boldsymbol{y}_1^\intercal -\boldsymbol{y}_2^\intercal, \ldots, \boldsymbol{y}_{m_1 - 1}^\intercal - \boldsymbol{y}_{m_1}^\intercal, \boldsymbol{y}_{m_1 + 1}^\intercal - \boldsymbol{y}_{m_1 + 2}^\intercal, \ldots, \boldsymbol{y}_{m_{\ell}-1}^\intercal -\boldsymbol{y}_{m_{\ell}}^\intercal \}.$$ As before, letting $s$ denote the dimension of $\mathcal{S}$, we have $s \leq \min \{ m-\ell, n\}$ and the matrix $\Delta \boldsymbol{y}$ has exactly $s$ linearly independent rows. Then we choose $s$ linearly independent rows in $\Delta \boldsymbol{y}$, and these rows belonging to $\Delta \boldsymbol{y}$ provide a matrix $\Delta_s \boldsymbol{y}$ which is full row rank. The vector $\ln (\Delta_s {\boldsymbol{K}})$ is given by the corresponding rows in $\ln (\Delta {\boldsymbol{K}})$. The system ([\[system 1 l\>1\]](#system 1 l>1){reference-type="ref" reference="system 1 l>1"}) is equivalent to the following one in the unknowns $\boldsymbol{X}$: $$\label{system 2 l>1} \ln (\Delta_s {\boldsymbol{K}})=(\Delta_s \boldsymbol{y}) \boldsymbol{X}.$$ Subsequently, let us construct $\boldsymbol{X}^* \in \mathcal{S}$, the special solution where $\exp (\boldsymbol{X}^*)$ is a complex-balanced equilibrium. As we did in the first step we consider two cases, in function of $s$. First, if $s = n$, then the matrix $\Delta_s \boldsymbol{y}$ is invertible. Hence, we compute a solution of system ([\[system 2 l\>1\]](#system 2 l>1){reference-type="ref" reference="system 2 l>1"}) as follows: $$\notag \boldsymbol{X}^* = (\Delta_s \boldsymbol{y})^{-1} \ln (\Delta_s {\boldsymbol{K}}) \in \mathcal{S} = \mathbb{R}^n,$$ and $\exp (\boldsymbol{X}^*)$ is a complex-balanced equilibrium. Second, if $s < n$, let $B = \{ \boldsymbol{v}_1, \ldots, \boldsymbol{v}_{n-s} \}$ be a basis of $\mathcal{S}^{\perp}$. Analogously to Equations [\[tilde matirx\]](#tilde matirx){reference-type="eqref" reference="tilde matirx"}-[\[X\* 2\]](#X* 2){reference-type="eqref" reference="X* 2"}, we start by adding $\boldsymbol{v}_1^\intercal, \ldots, \boldsymbol{v}_{n-s}^\intercal$ on the bottom of the matrix $\Delta_s \boldsymbol{y}$, and add $n-s$ zeros to the vector $\ln (\Delta_s {\boldsymbol{K}})$. Next, we get the desired solution $\boldsymbol{X}^* \in \mathcal{S}$ of [\[system 2 l\>1\]](#system 2 l>1){reference-type="eqref" reference="system 2 l>1"}, and $\exp (\boldsymbol{X}^*)$ is a complex-balanced equilibrium. As in the case of Step 1 (the single connected component case), we conclude that $\boldsymbol{X}^* = \hat{Q}(\boldsymbol{k})$ depends smoothly on the vector ${\boldsymbol{K}}$. ◻ # Complex-balanced equilibria depend smoothly on the initial conditions {#sec:smooth_dependent_initial} Here our goal is to prove Theorem [Theorem 3](#thm:smoothDependanceinitial){reference-type="ref" reference="thm:smoothDependanceinitial"}, i.e., to show that the complex-balanced equilibria vary smoothly with the initial condition vector $\boldsymbol{x}_0$. This section is divided into the two logical steps of the proof. First, we introduce the map $\phi$ in Definition [Definition 27](#def:mapStep2){reference-type="ref" reference="def:mapStep2"} and we show that $\phi$ is a smooth map in Lemma [Lemma 28](#phi smmoth){reference-type="ref" reference="phi smmoth"}. Second, we give another map $\Phi$ in Definition [Definition 29](#def:mapStep3){reference-type="ref" reference="def:mapStep3"}. Lemma [Lemma 30](#Phi smmoth){reference-type="ref" reference="Phi smmoth"} is dedicated to proving that $\Phi$ is also smooth. **Definition 27**. Let $G=(V, E)$ be a weakly reversible Euclidean embedded graph and let $\mathcal{S}$ be its associated stoichiometric subspace. Given a fixed state $\boldsymbol{x}_0 \in \mathbb{R}_{>0}^n$, define the following map: $$\phi: \mathbb{R}^n \rightarrow \boldsymbol{x}_0 + \mathcal{S},$$ such that $$\phi (\boldsymbol{X}^*) := \boldsymbol{x}^*,$$ where $\boldsymbol{x}^* = \exp (\boldsymbol{X}^* + \mathcal{S}^\perp) \cap ({\boldsymbol{x}_0} + \mathcal{S})$. **Lemma 28**. *The map $\phi$ from Definition [Definition 27](#def:mapStep2){reference-type="ref" reference="def:mapStep2"} is well-defined and smooth.* *Proof.* Using the well-known Birch Theorem (see, for instance [@cdss2009 Proposition 10], [@pachterStumrfels Theorem 1.10]), for any $\boldsymbol{X}^* \in \mathbb{R}^n$ and $\boldsymbol{x}_0 \in \mathbb{R}_{>0}^n$, then $(\boldsymbol{x}_0 + \mathcal{S})$ and $\exp (\boldsymbol{X}^* + \mathcal{S}^{\perp})$ have a unique intersection point. Therefore, the map $\phi$ is well-defined. Next, we show the smoothness of the map $\phi$. Let $s$ be the dimension of $\mathcal{S}$, we have $$\dim (\mathcal{S}) = s \leq n, \ \text{ and } \ \dim (\mathcal{S}^{\perp}) = n - \dim (\mathcal{S}) = n - s.$$ Then we consider $s$ in two cases: $s = n$ and $s < n$. **Case 1:** $s = n$. This shows $\mathcal{S} = \mathbb{R}^n$ and $\mathcal{S}^{\perp} = \emptyset$. Thus we derive $\boldsymbol{x}_0 + \mathcal{S} = \mathbb{R}^n$ and $$\phi (\boldsymbol{X}^*) = \exp (\boldsymbol{X}^*+\mathcal{S}^\perp ) \cap (\boldsymbol{x}_0+\mathcal{S}) = \exp(\boldsymbol{X}^*).$$ It is clear that the map $\phi$ is smooth in this case. **Case 2:** $s < n$. Then $\mathcal{S}^{\perp} \neq \emptyset$ and $\dim (\mathcal{S}^{\perp}) = n - s > 0$. Then we pick a basis of $\mathcal{S}^{\perp}$, denoted by $B = \{ \boldsymbol{v}_1, \boldsymbol{v}_2, \ldots, \boldsymbol{v}_{n-s} \} \subset \mathbb{R}^n$. Given $\boldsymbol{x}_0 \in \mathbb{R}_{>0}^n$, the Birch Theorem shows that for any $\boldsymbol{X}^* \in \mathbb{R}^n$, there exists a unique real vector $\boldsymbol{w}= ( w_1, \ldots, w_{n-s} )$, such that $$\label{eq:w} \phi (\boldsymbol{X}^*) = \exp (\boldsymbol{X}^* + \sum^{n-s}_{i=1} w_i \boldsymbol{v}_i ) \in \boldsymbol{x}_0 + \mathcal{S}.$$ Thus each $w_i$ can be considered as a real function of $\boldsymbol{X}^*$, i.e., $w_i = w_i (\boldsymbol{X}^*) \in \mathbb{R}$ for $1 \leq i \leq n-s$. Since $\{ \boldsymbol{v}_1, \boldsymbol{v}_2, \ldots, \boldsymbol{v}_{n-s} \}$ forms a basis of $\mathcal{S}^{\perp}$, [\[eq:w\]](#eq:w){reference-type="eqref" reference="eq:w"} is equivalent to the following: for any $\boldsymbol{X}^* \in \mathbb{R}^n$, the real vector function $( w_1 (\boldsymbol{X}^*), \ldots, w_{n-s} (\boldsymbol{X}^*))$ satisfies $$\label{eq:w 2} \exp (\boldsymbol{X}^* + \sum^{n-s}_{i=1} w_i (\boldsymbol{X}^*) \boldsymbol{v}_i ) \cdot \boldsymbol{v}_i = \boldsymbol{x}_0 \cdot \boldsymbol{v}_i, \ \text{for} \ 1 \leq i \leq n-s.$$ Now we construct $n-s$ functions $\{ f_1, \ldots, f_{n-s} \}$ as follows: given $\boldsymbol{x}_0 \in \mathbb{R}_{>0}^n$ and suppose $\{ \boldsymbol{v}_1, \boldsymbol{v}_2, \ldots, \boldsymbol{v}_{n-s} \} \subset \mathbb{R}^n$ is a basis of $\mathcal{S}^{\perp}$, then for $1 \leq i \leq n-s$ $$f_i: \mathbb{R}^n \times \mathbb{R}^{n-s} \rightarrow \mathbb{R},$$ such that for $(\boldsymbol{X}, \boldsymbol{y}) \in \mathbb{R}^n \times \mathbb{R}^{n-s}$, $$f_i (\boldsymbol{X}, \boldsymbol{y}) := \exp (\boldsymbol{X}+ \sum^{n-s}_{i=1} y_i \boldsymbol{v}_i ) \cdot \boldsymbol{v}_i - \boldsymbol{x}_0 \cdot \boldsymbol{v}_i.$$ It is clear that the solution to [\[eq:w 2\]](#eq:w 2){reference-type="eqref" reference="eq:w 2"} is equivalent to say that for any $\boldsymbol{X}^* \in \mathbb{R}^n$, $$f_1 = \cdots = f_{n-s} = 0, \ \text{at} \ (\boldsymbol{X}, \boldsymbol{y}) = (\boldsymbol{X}^*, w_1 (\boldsymbol{X}^*), \cdots, w_{n-s} (\boldsymbol{X}^*) ).$$ Here we claim that for any $\boldsymbol{X}^* \in \mathbb{R}^n$, we have $$\label{det non-zero} \det \Big( \frac{\partial f_i}{\partial y_j} \Big) \neq 0, \ \text{at} \ (\boldsymbol{X}, \boldsymbol{y}) = (\boldsymbol{X}^*, w_1 (\boldsymbol{X}^*), \cdots, w_{n-s} (\boldsymbol{X}^*) ).$$ By direct computation, we get for $1 \leq i, j \leq n-s$, $$\notag \boldsymbol{J}_{\mathbf{f}} := \Big( \frac{\partial f_i}{\partial y_j} \Big)_{i, j} = \exp (\boldsymbol{X}+ \sum^{n-s}_{i=1} y_i \boldsymbol{v}_i ) \cdot ( \boldsymbol{v}_i \boldsymbol{v}_j ),$$ where $\boldsymbol{v}_i \boldsymbol{v}_j := (v_{i, 1} v_{j, 1}, v_{i, 2} v_{j, 2}, \cdots, v_{i, n} v_{j, n})$. We can rewrite the matrix $\boldsymbol{J}_{\mathbf{f}}$ as $$\label{Jf} \boldsymbol{J}_{\mathbf{f}} = \begin{bmatrix} \boldsymbol{v}_1 \\ \vdots \\ \boldsymbol{v}_{n-s} \end{bmatrix} \begin{bmatrix} \tilde{x}_{1} & & \\ & \ddots & \\ & & \tilde{x}_{n} \end{bmatrix} \begin{bmatrix} \boldsymbol{v}^{\intercal}_1, \cdots, \boldsymbol{v}^{\intercal}_{n-s} \\ \end{bmatrix},$$ where $(\tilde{x}_{1}, \cdots, \tilde{x}_{n}) := \exp (\boldsymbol{X}+ \sum\limits^{n-s}_{i=1} y_i \boldsymbol{v}_i )$. Supposing $\det (\boldsymbol{J}_{\mathbf{f}}) = 0$ at a point $(\hat{\boldsymbol{X}}, \hat{\boldsymbol{y}})$, then there exists a non-zero column vector $\boldsymbol{\lambda}\in \mathbb{R}^{n-s}$, such that $$\notag \mathbf{J}_{\mathbf{f}} (\hat{\boldsymbol{X}}, \hat{\boldsymbol{y}}) \cdot \boldsymbol{\lambda}= \mathbf{0}.$$ This implies that $$\label{lambda J} \boldsymbol{\lambda}^{\intercal} \cdot \mathbf{J}_{\mathbf{f}} (\hat{\boldsymbol{X}}, \hat{\boldsymbol{y}}) \cdot \boldsymbol{\lambda}= 0.$$ From [\[Jf\]](#Jf){reference-type="eqref" reference="Jf"}, [\[lambda J\]](#lambda J){reference-type="eqref" reference="lambda J"} and $\exp (\boldsymbol{X}+ \sum\limits^{n-s}_{i=1} y_i \boldsymbol{v}_i ) \in \mathbb{R}^n_{>0}$, we obtain $$\notag \begin{bmatrix} \boldsymbol{v}^{\intercal}_1, \cdots, \boldsymbol{v}^{\intercal}_{n-s} \\ \end{bmatrix} \cdot \boldsymbol{\lambda}= \mathbf{0}.$$ Since $\{ \boldsymbol{v}_1, \cdots, \boldsymbol{v}_{n-s} \}$ forms a basis for $\mathcal{S}^{\perp}$ and they are linearly independent, we get $\boldsymbol{\lambda}= \mathbf{0}$ and this contradicts the assumption. Thus, we proved the claim. Note that $\{ f_1, \ldots, f_{n-s} \}$ are all smooth functions. Using [\[det non-zero\]](#det non-zero){reference-type="eqref" reference="det non-zero"} and the Implicit Function Theorem, we obtain that there exists a unique smooth function $$g: \mathbb{R}^n \rightarrow \mathbb{R}^{n-s}$$ such that for any $\boldsymbol{X}\in \mathbb{R}^n$, $$f_i (\boldsymbol{X}, g(\boldsymbol{X})) = 0, \ \text{for} \ i = 1, \cdots, n-s.$$ Using the Birch Theorem, we deduce that given a state $\boldsymbol{x}_0 \in \mathbb{R}^n_{>0}$, for any $\boldsymbol{X}^* \in \mathbb{R}^n$ $$g(\boldsymbol{X}^*) = (w_1 (\boldsymbol{X}^*), \ldots, w_{n-s} (\boldsymbol{X}^*)).$$ This shows that $w_1 (\boldsymbol{X}^*), \ldots, w_{n-s} (\boldsymbol{X}^*)$ are smooth functions with respect to $\boldsymbol{X}^*$. Therefore, we conclude that the map $\phi (\boldsymbol{X}^*) = \exp (\boldsymbol{X}^* + \sum^{n-s}_{i=1} w_i \boldsymbol{v}_i )$ is smooth. ◻ Now we are ready to prove Theorem [Theorem 2](#thm:smoothDependanceEquil){reference-type="ref" reference="thm:smoothDependanceEquil"}. *Proof of Theorem [Theorem 2](#thm:smoothDependanceEquil){reference-type="ref" reference="thm:smoothDependanceEquil"}.* Given an initial state ${\boldsymbol{x}_0} \in \mathbb{R}_{>0}^n$, we define the following map, introduced in [@Connected Definition 3.1]: $$\label{equ:q} Q_{{\boldsymbol{x}_0}}: \mathcal{V}(G) \rightarrow ({\boldsymbol{x}_0}+\mathcal{S}) \cap \mathbb{R}_{>0}^n,$$ such that for any $\boldsymbol{k}\in \mathcal{V}(G)$, $Q_{{\boldsymbol{x}_0}}({\boldsymbol{k}})$ is the unique complex-balanced equilibrium in the invariant polyhedron $\mathcal{S}_{\boldsymbol{x}_0}$, under the mass-action system $(G, \boldsymbol{k})$. The map $Q_{{\boldsymbol{x}_0}}$ is well-defined for any state ${\boldsymbol{x}_0} \in \mathbb{R}_{>0}^n$ and $\boldsymbol{k}\in \mathcal{V}(G)$, since Theorem [Theorem 12](#thm:HJ){reference-type="ref" reference="thm:HJ"} shows every complex-balanced system admits a unique equilibrium within each invariant polyhedron. Using Lemma [Lemma 26](#Q hat smooth){reference-type="ref" reference="Q hat smooth"}, we obtain that for any $\boldsymbol{k}\in \mathcal{V}(G)$, we have $$\hat{Q} (\boldsymbol{k}) = \boldsymbol{X}^* \in \mathcal{S},$$ and $\exp (\boldsymbol{X}^*)$ is a complex-balanced equilibrium of the mass-action system $(G, \boldsymbol{k})$. On the other hand, from Lemma [Lemma 28](#phi smmoth){reference-type="ref" reference="phi smmoth"}, for any $\boldsymbol{X}^* \in \mathbb{R}^n$ we have $$\notag \phi (\boldsymbol{X}^*) = \exp (\boldsymbol{X}^* + \mathcal{S}^\perp) \cap ({\boldsymbol{x}_0} + \mathcal{S}).$$ From Theorem [Theorem 12](#thm:HJ){reference-type="ref" reference="thm:HJ"}, every complex-balanced equilibrium can be written as $$\exp (\boldsymbol{X}^* + \mathcal{S}^\perp ).$$ Furthermore, $\exp(\boldsymbol{X}^* + \mathcal{S}^\perp )\cap (\boldsymbol{x}_0+\mathcal{S})$ is the unique positive complex-balanced equilibrium in the invariant polyhedron $\mathcal{S}_{\boldsymbol{x}_0}$. Thus, given $\boldsymbol{x}_0 \in \mathbb{R}_{>0}^{n}$ and for any $\boldsymbol{k}\in \mathcal{V}(G)$, we deduce $$Q_{\boldsymbol{x}_0} (\boldsymbol{k}) = \phi \circ \hat{Q} (\boldsymbol{k}).$$ By Lemma [Lemma 26](#Q hat smooth){reference-type="ref" reference="Q hat smooth"} and Lemma [Lemma 28](#phi smmoth){reference-type="ref" reference="phi smmoth"}, we get that both $\hat{Q}$ and $\phi$ are smooth functions. Therefore, we conclude the smoothness of the map $Q_{x_0}$ and prove the theorem. ◻ To prove Theorem [Theorem 3](#thm:smoothDependanceinitial){reference-type="ref" reference="thm:smoothDependanceinitial"}, we introduce another map in Definition [Definition 29](#def:mapStep3){reference-type="ref" reference="def:mapStep3"} and show it is smooth. **Definition 29**. Let $(G, \boldsymbol{k})$ be a weakly reversible mass-action system with $\boldsymbol{k}\in \mathcal{V}(G)$, and let $\mathcal{S}$ be its associated stoichiometric subspace. Given fixed $\boldsymbol{X}^* \in \mathbb{R}^n$, define the following map: $$\Phi: \mathbb{R}_{>0}^n \rightarrow \mathbb{R}_{>0}^n,$$ such that $$\Phi (\boldsymbol{x}_0) := \boldsymbol{x}^*,$$ where $\boldsymbol{x}^* = \exp (\boldsymbol{X}^* + \mathcal{S}^\perp) \cap ({\boldsymbol{x}_0} + \mathcal{S})$. **Lemma 30**. *The map $\Phi$ from Definition [Definition 29](#def:mapStep3){reference-type="ref" reference="def:mapStep3"} is well-defined and smooth.* *Proof.* Similar to the proof of Lemma [Lemma 28](#phi smmoth){reference-type="ref" reference="phi smmoth"}, the Birch Theorem shows that the map $\Phi$ is well-defined. To show the smoothness of the map $\Phi$, we let $s$ be the dimension of $\mathcal{S}$, and we consider $s$ in two cases: $s = n$ and $s < n$. **Case 1:** $s = n$. This shows $\mathcal{S} = \mathbb{R}^n$ and $\mathcal{S}^{\perp} = \emptyset$. Thus we derive $$\phi (\boldsymbol{X}^*) = \exp (\boldsymbol{X}^*+\mathcal{S}^\perp ) \cap (\boldsymbol{x}_0+\mathcal{S}) = \exp(\boldsymbol{X}^*).$$ Since $\boldsymbol{X}^*$ is a fixed vector, the map $\Phi$ is a constant function and it is smooth in this case. **Case 2:** $s < n$. In this case, $\mathcal{S}^{\perp} \neq \emptyset$ and $\dim (\mathcal{S}^{\perp}) = n - s > 0$. Then we pick a basis of $\mathcal{S}^{\perp}$, denoted by $B = \{ \boldsymbol{v}_1, \boldsymbol{v}_2, \ldots, \boldsymbol{v}_{n-s} \} \subset \mathbb{R}^n$. Given $\boldsymbol{X}^* \in \mathbb{R}^n$, the Birch Theorem shows that for any $\boldsymbol{x}_0 \in \mathbb{R}^n_{>0}$, there exists a unique real vector $\boldsymbol{z}= ( z_1, \ldots, z_{n-s} )$, such that $$\label{eq:z} \Phi (\boldsymbol{x}_0) = \exp (\boldsymbol{X}^* + \sum^{n-s}_{i=1} z_i \boldsymbol{v}_i ) \in \boldsymbol{x}_0 + \mathcal{S}.$$ Thus each $z_i$ is a real function of $\boldsymbol{x}_0$, i.e., $z_i = z_i (\boldsymbol{x}_0) \in \mathbb{R}$ for $1 \leq i \leq n-s$. Since $\{ \boldsymbol{v}_1, \boldsymbol{v}_2, \ldots, \boldsymbol{v}_{n-s} \}$ forms a basis of $\mathcal{S}^{\perp}$, [\[eq:z\]](#eq:z){reference-type="eqref" reference="eq:z"} is equivalent to the following: for any $\boldsymbol{x}_0 \in \mathbb{R}^n_{>0}$, the real vector function $( z_1 (\boldsymbol{x}_0), \ldots, z_{n-s} (\boldsymbol{x}_0))$ satisfies $$\label{eq:z 2} \exp (\boldsymbol{X}^* + \sum^{n-s}_{i=1} z_i (\boldsymbol{x}_0) \boldsymbol{v}_i ) \cdot \boldsymbol{v}_i = \boldsymbol{x}_0 \cdot \boldsymbol{v}_i, \ \text{for} \ 1 \leq i \leq n-s.$$ Now we construct $n-s$ functions $\{ f_1, \ldots, f_{n-s} \}$ as follows: given $\boldsymbol{X}^* \in \mathbb{R}^n$ and suppose $\{ \boldsymbol{v}_1, \boldsymbol{v}_2, \ldots, \boldsymbol{v}_{n-s} \} \subset \mathbb{R}^n$ is a basis of $\mathcal{S}^{\perp}$, then for $1 \leq i \leq n-s$ $$f_i: \mathbb{R}^n_{>0} \times \mathbb{R}^{n-s} \rightarrow \mathbb{R},$$ such that for $(\boldsymbol{x}, \boldsymbol{y}) \in \mathbb{R}^n \times \mathbb{R}^{n-s}$, $$f_i (\boldsymbol{x}, \boldsymbol{y}) := \exp (\boldsymbol{X}^* + \sum^{n-s}_{i=1} y_i \boldsymbol{v}_i ) \cdot \boldsymbol{v}_i - \boldsymbol{x}\cdot \boldsymbol{v}_i.$$ It is clear that the solution to [\[eq:z 2\]](#eq:z 2){reference-type="eqref" reference="eq:z 2"} is equivalent to say that for any $\boldsymbol{x}_0 \in \mathbb{R}^n_{>0}$, $$f_1 = \cdots = f_{n-s} = 0, \ \text{at} \ (\boldsymbol{x}, \boldsymbol{y}) = (\boldsymbol{x}_0, z_1 (\boldsymbol{x}_0), \cdots, z_{n-s} (\boldsymbol{x}_0) ).$$ By the proof of Lemma [Lemma 28](#phi smmoth){reference-type="ref" reference="phi smmoth"}, we obtain that for any $\boldsymbol{x}_0 \in \mathbb{R}^n_{>0}$, $$\label{det non-zero_b} \det \Big( \frac{\partial f_i}{\partial y_j} \Big) \neq 0, \ \text{at} \ (\boldsymbol{x}, \boldsymbol{y}) = (\boldsymbol{x}_0, z_1 (\boldsymbol{x}_0), \cdots, z_{n-s} (\boldsymbol{x}_0) ).$$ Since $\{ f_1, \ldots, f_{n-s} \}$ are all smooth functions, we apply [\[det non-zero_b\]](#det non-zero_b){reference-type="eqref" reference="det non-zero_b"} on the Implicit Function Theorem. Then we derive that there exists a unique smooth function $$g: \mathbb{R}^n \rightarrow \mathbb{R}^{n-s}$$ such that for any $\boldsymbol{x}_0 \in \mathbb{R}^n_{>0}$, $$f_i (\boldsymbol{X}, g(\boldsymbol{X})) = 0, \ \text{for} \ i = 1, \cdots, n-s.$$ The Birch Theorem implies that for any $\boldsymbol{x}_0 \in \mathbb{R}^n_{>0}$, $$g(\boldsymbol{x}_0) = (z_1 (\boldsymbol{x}_0), \ldots, z_{n-s} (\boldsymbol{x}_0)).$$ This shows that $z_1 (\boldsymbol{x}_0), \ldots, z_{n-s} (\boldsymbol{x}_0)$ are smooth functions of $\boldsymbol{x}_0$. Therefore, we conclude that given $\boldsymbol{X}^* \in \mathbb{R}^n$, the map $\Phi (\boldsymbol{x}_0) = \exp (\boldsymbol{X}^* + \sum^{n-s}_{i=1} z_i (\boldsymbol{x}_0) \boldsymbol{v}_i )$ is smooth. ◻ Finally, we are ready to prove Theorem [Theorem 3](#thm:smoothDependanceinitial){reference-type="ref" reference="thm:smoothDependanceinitial"}. *Proof of Theorem [Theorem 3](#thm:smoothDependanceinitial){reference-type="ref" reference="thm:smoothDependanceinitial"}.* Given the vector of reaction rate constants $\boldsymbol{k}\in \mathcal{V}(G)$, we define the following map: $$\label{equ:p} P: \mathbb{R}_{>0}^n \rightarrow \mathbb{R}_{>0}^n,$$ such that for any $\boldsymbol{x}_0 \in \mathbb{R}^n_{>0}$, we define $P (\boldsymbol{x}_0)$ to be the unique complex-balanced equilibrium in the invariant polyhedron $\mathcal{S}_{\boldsymbol{x}_0}$, under the mass-action system $(G, \boldsymbol{k})$. By Theorem [Theorem 12](#thm:HJ){reference-type="ref" reference="thm:HJ"}, the map $P$ is well-defined and $P (\boldsymbol{x}_0) = \exp(\boldsymbol{X}^* + \mathcal{S}^\perp )\cap (\boldsymbol{x}_0+\mathcal{S})$. By Lemma [Lemma 26](#Q hat smooth){reference-type="ref" reference="Q hat smooth"}, we obtain that for any $\boldsymbol{k}\in \mathcal{V}(G)$, $\hat{Q} (\boldsymbol{k}) = \boldsymbol{X}^* \in \mathcal{S}$ and $\exp (\boldsymbol{X}^*)$ is a complex-balanced equilibrium of the mass-action system $(G, \boldsymbol{k})$. Then Lemma [Lemma 30](#Phi smmoth){reference-type="ref" reference="Phi smmoth"} shows that, given $\boldsymbol{X}^* \in \mathbb{R}^n$ and for any $\boldsymbol{x}_0 \in \mathbb{R}^n_{>0}$, we have $$\notag \Phi (\boldsymbol{x}_0) = \exp (\boldsymbol{X}^* + \mathcal{S}^\perp) \cap ({\boldsymbol{x}_0} + \mathcal{S}).$$ Therefore, given $\boldsymbol{k}\in \mathcal{V}(G)$ and for any $\boldsymbol{x}_0 \in \mathbb{R}_{>0}^{n}$, we have $$P (\boldsymbol{x}_0) = \Phi \circ \hat{Q} (\boldsymbol{x}_0).$$ By Lemma [Lemma 26](#Q hat smooth){reference-type="ref" reference="Q hat smooth"} and Lemma [Lemma 30](#Phi smmoth){reference-type="ref" reference="Phi smmoth"}, both $\hat{Q}$ and $\Phi$ are smooth functions. Hence we conclude the smoothness of the map $P$ and prove the theorem. ◻ # Ackowledgements {#ackowledgements .unnumbered} G. Craciun and M.-Ş. Sorea are thankful for the support of the Nonlinear Algebra research group of Bernd Sturmfels at the Max Planck Institute for Mathematics in the Sciences in Leipzig, Germany. M.-Ş. Sorea is grateful to Antonio Lerario for the very supportive working conditions during her postdoc at SISSA, in Trieste, Italy. G. Craciun's work was supported in part by the National Science Foundation grant DMS-2051568. **Authors:** Gheorghe Craciun\ University of Wisconsin-Madison, USA\ `[email protected]` Jiaxin Jin\ The Ohio State University, USA\ `[email protected]` Miruna-Ştefana Sorea\ SISSA (Scuola Internazionale Superiore di Studi Avanzati), Trieste, Italy and Lucian Blaga University, Sibiu, Romania\ `[email protected]` [^1]: University of Wisconsin-Madison, USA [^2]: Ohio State University, USA [^3]: SISSA (Scuola Internazionale Superiore di Studi Avanzati), Trieste, Italy and RCMA Lucian Blaga University, Sibiu, Romania [^4]: because the variables of interest are concentrations or populations, so they cannot take on negative values

arxiv_math

--- abstract: | We prove that for any compact toric symplectic manifold, if a Hamiltonian diffeomorphism admits more fixed points, counted homologically, than the total Betti number, then it has infinitely many simple periodic points. This provides a vast generalization of Franks' famous two or infinity dichotomy for periodic orbits of area-preserving diffeomorphisms on the two-sphere, and establishes a conjecture attributed to Hofer--Zehnder in the case of toric manifolds. The key novelty is the application of gauged linear sigma model and its bulk deformations to the study of Hamiltonian dynamics of symplectic quotients. address: - Columbia University, 2990 Broadway, New York, NY 10027, USA - Department of Mathematics, Rutgers University, Hill Center--Busch Campus, 110 Frelinghuysen Road, Piscataway, NJ 08854-8019, USA author: - Shaoyun Bai - Guangbo Xu bibliography: - reference.bib title: Hofer--Zehnder conjecture for toric manifolds --- # Introduction ## Result The main theorem of this paper is concerned with the existence of infinitely many periodic points of Hamiltonian diffeomorphisms of symplectic manifolds. **Theorem 1**. *Let $(X^{2n}, \omega)$ be a compact toric symplectic manifold. Suppose $\phi: X \to X$ is a Hamiltonian diffeomorphism whose set of fixed points ${\rm Fix}(\phi)$ is finite. For any $x \in {\rm Fix}(\phi)$, denote by ${\it HF}^{\rm loc}(\phi, x; {\mathbb Q})$ the local Floer homology group of the fixed point $x$ over the rationals. If the inequality $$\label{eqn:exceeds} N(\phi, {\mathbb Q}) := \sum_{x \in {\rm Fix}(\phi)} \dim_{{\mathbb Q}} {\it HF}^{\rm loc}(\phi, x; {\mathbb Q}) > \sum_{i=0}^{2n} \dim_{\mathbb Q} H_i(X;{\mathbb Q})$$ holds, then $\phi$ has infinitely many simple periodic points.* Here a point $x \in X$ is called a simple periodic point (of period $k$) if $\phi^k(x) = x$ for some positive integer $k$ and $\phi^l(x) \neq x$ for all $l < k$. The number $N(\phi, {\mathbb Q})$ can be viewed as a quantity which measures the number of fixed points of a generic small Hamiltonian perturbation of $\phi$. In particular, the following statement holds because $\dim_{{\mathbb Q}} {\it HF}^{\rm loc}(\phi, x; {\mathbb Q}) = 1$ if $x$ is nondegenerate. **Corollary 1**. *If all the fixed points of $\phi$ are nondegenerate, i.e., for any $x \in {\rm Fix}(\phi)$, we have $\det(D\phi_x - id) \neq 0$, and the inequality $$\# {\rm Fix}(\phi) > \sum_{i=0}^{2n} \dim_{\mathbb Q} H_i(X;{\mathbb Q})$$ holds, then $\phi$ has infinitely many simple periodic points.* Theorem [Theorem 1](#thm:main){reference-type="ref" reference="thm:main"} is a high-dimensional generalization of a famous result due to Franks [@Franks_1; @Franks_2]: any area-preserving homeomorphism of $S^2$ has either two or infinitely many simple periodic points. Just as in Franks' theorem, the assumption [\[eqn:exceeds\]](#eqn:exceeds){reference-type="eqref" reference="eqn:exceeds"} is necessary. Indeed, for a toric manifold $(X, \omega)$ with a Hamiltonian $T^n$-action, a generic element of the torus $T^n$, which can be regarded as a higher-dimensional analogue of an irrational rotation on $S^2$, has exactly $\sum_{i=0}^{2n} \dim_{\mathbb Q} H_i(X;{\mathbb Q})$ many simple periodic points given by the fixed points of the $T^n$-action. In fact, using the Anosov--Katok method, the paper [@leroux_seyfaddini] constructed a Hamiltonian diffeomorphism on $X$ with exactly $1 + \sum_{i=0}^{2n} \dim_{\mathbb Q} H_i(X;{\mathbb Q})$ ergodic measures, which come from the measure induced by the volume form $\omega^n$ and the Dirac measures supported at the toric fixed points. Theorem [Theorem 1](#thm:main){reference-type="ref" reference="thm:main"} broadens the scope of the recent investigation of generalizations of Franks' dichotomy to higher dimensional symplectic manifolds initiated by Shelukhin [@Shelukhin_2022], who proved a similar result on the existence of infinitely many periodic orbits of Hamiltonian diffeomorphisms defined over monotone symplectic manifolds with semisimple quantum homology. Our main theorem resolves a visionary conjecture set forth by Hofer and Zehnder [@hofer-zehnder-book Page 263], which asserts the existence of infinitely many simple periodic points of a Hamiltonian diffeomorphism $\phi$ of a compact symplectic manifold if $\# {\rm Fix}(\phi)$ exceeds the lower bound provided by the Arnold conjecture, in the case of toric manifolds. ## Context from Hamiltonian dynamics and symplectic geometry As mentioned in the statement of the Hofer--Zehnder conjecture, one rigidity aspect of Hamiltonian diffeomorphisms is governed by the *Arnold conjecture* [@Arnold_conjecture] (see various proofs in [@Conley_Zehnder_1983] [@Floer_CMP] [@Ono_1995] [@Hofer_Salamon] [@Fukaya_Ono] [@Liu_Tian_Floer] [@Abouzaid_Blumberg] [@Bai_Xu_Arnold]), which implies that the inequality $N(\phi, {\mathbb Q}) \geq \sum_{i=0}^{2n} \dim_{\mathbb Q} H_i(X;{\mathbb Q})$ is always true. Notice that one does not detect simple periodic points of higher periods by applying the Arnold conjecture to $\phi^k$, as fixed points of $\phi$ are automatically fixed points of $\phi^k$. On the other hand, Hamiltonian diffeomorphisms "tend" to have infinitely many simple periodic points. The *Conley conjecture* asserts the infinitude of the number of simple periodic points for all Hamiltonian diffeomorphisms on a certain class of symplectic manifolds (originally only conjectured for torus [@Conley_conjecture]). This conjecture has been proved for wider and wider ranges of manifolds, see [@Salamon_Zehnder][@Franks_Handel_2003][@Hingston][@Ginzburg_2010][@Hein_2012_Conley][@Ginzburg_Gurel_2009; @Ginzburg_Gurel_2012; @Ginzburg_Gurel_2019] and the survey article [@GG19]. However, the Conley conjecture does not hold for all symplectic manifolds, as easily seen from the case of irrational rotations on $S^2$ (or general toric manifolds). While Conley conjecture is an unconditional statement about Hamiltonian diffeomorphisms on certain symplectic manifolds, the Hofer--Zehnder conjecture observes a simple condition on the Hamiltonian diffeomorphism responsible for the infinitude of periodic points: the number of fixed points (counted homologically) is strictly greater than the "Arnold lower bound\". A broader interpretation of this condition is the existence of "unnecessary" fixed points, such as non-contractible ones. Distinguished from the intensive research activities surrounding the Conley conjecture, the Hofer--Zehnder conjecture has only been understood in limited cases. Except for the cases covered in [@Shelukhin_2022] (including projective spaces, Grassmannians, and their monotone products), the Hofer--Zehnder conjecture is known for weighted projective spaces [@Allais], some cases related to non-contractible orbits ([@Gurel_2013][@Ginzburg_Gurel_2016][@Orita_2017; @Orita_2019][@Sugimoto]) or hyperbolic fixed points ([@Ginzburg_Gurel_2014]). Our Theorem [Theorem 1](#thm:main){reference-type="ref" reference="thm:main"} covers a large family of new instances of the original homological version of the Hofer--Zehnder conjecture. Very interestingly, our proof actually indicates a surprising connection between mirror symmetry and Hamiltonian dynamics, see the discussions in Section [1.3](#subsec:intro-2){reference-type="ref" reference="subsec:intro-2"}. The holomorphic curve method, most notably the package of Floer homology, has been a dominant tool in the study of Hamiltonian dynamics. The developments of Conley conjecture and Hofer--Zehnder conjecture have shown that holomorphic curves have more subtle influence on Hamiltonian dynamics than being merely a tool. For example, Conley conjecture is true for Calabi--Yau or negatively monotone manifolds. These manifolds do not have "very many" holomorphic curves. Recent progress [@CGG20][@shelukhin21][@CGG] further reveals a close connection between the failure of Conley conjecture and the effect of holomorphic curves (which is also related to the *Chance--McDuff conjecture*). The semisimplicity condition in the proof of the Hofer--Zehnder conjecture [@Shelukhin_2022] can also be viewed as a characterization of the abundance of holomorphic curves. Note that toric manifolds underlies rational algebraic varieties, so they provide examples demonstrating this phenomenon. It will be very interesting to have a more precise and systematic formulation of such a mechanism for general symplectic manifolds. ## Key ingredient: GLSM {#subsec:intro-2} While inspired by the method of Shelukhin [@Shelukhin_2022], our resolution of the Hofer--Zehnder conjecture for toric symplectic manifolds is largely based on introducing a new player in Hamiltonian dynamics: *gauged linear sigma model (GLSM)*. Gauged linear sigma model was originally introduced by Witten [@Witten_LGCY] in the physics context in much greater generality. In the current situation, the basic usage of the GLSM is to replace holomorphic curves in a toric manifold $X$ by certain gauge-theoretic objects called *vortices*. This is possible because $X$ is the symplectic quotient (or GIT quotient) of a vector space $V\cong {\mathbb C}^N$ by a torus $K \cong (S^1)^{N-n}$ with a moment map $\mu$. In this situation, a vortex over any Riemann surface $\Sigma$ consists of a principal $K$-bundle $P \to \Sigma$, a connection $A \in {\mathcal A}(P)$, and a section $u$ of the associated vector bundle $P(V):= (P\times V)/ K$, solving the *vortex equation* $$\begin{aligned} &\ \overline\partial_A u = 0,\ &\ * F_A + \mu(u) = 0.\end{aligned}$$ Mathematically, the general symplectic vortex equation was firstly introduced by Cieliebak--Gaio--Salamon [@Cieliebak_Gaio_Salamon_2000] and Mundet [@Mundet_thesis; @Mundet_2003], with many related technical works by Cieliebak--Gaio--Mundet--Salamon [@Cieliebak_Gaio_Mundet_Salamon_2002], Ott [@Ott_compactness], Mundet--Tian [@Mundet_Tian_2009], the second author [@Guangbo_compactness], Zilter [@Ziltener_Decay; @Ziltener_thesis; @Ziltener_book], Venugopalan [@Venugopalan_quasi], etc. In particular, one can use Hamiltonian perturbed vortex equation over surfaces with cylindrical ends to develop the *vortex Floer theory* (see Frauenfelder [@Frauenfelder_thesis; @Frauenfelder_2004] and the second author [@Xu_VHF]). Many aspects of ordinary Hamiltonian Floer theory has a counterpart in vortex Hamiltonian Floer theory, including continuation maps and energy filtration. Accordingly, recent advances on quantitative Floer theory, especially the theory of persistence modules [@Polterovich_Shelukhin_2016; @Usher_Zhang_2016], can be adapted to the vortex context. For readers who are not familiar with this variant of Floer theory, just keep in mind that the chain complex underlying the vortex Hamiltonian Floer homology is still freely generated by $1$-periodic orbits of the given Hamiltonian diffeomorphism, and the differentials are defined by counting solutions to Hamiltonian-perturbed vortex equations instead of Floer equations, provided that all the $1$-periodic orbits are nondegenerate. For the general isolated degenerate case, the theory of *local Floer homology* carries over to the vortex context without much difficulty, therefore all of our results hold in such a generality. One remarkable feature of the vortex Floer theory is that we can define Floer theories over *integers* in our setting. Indeed, as the target space $V$ is a symplectic vector space, the Uhlenbeck--Gromov--Floer compactification of moduli spaces of solutions to vortex equations do not acquire adding configurations with sphere bubbles, much as in the case of symplectically aspherical manifolds. Except for simplifying the technical arguments for achieving transversality, the ability of reducing to characteristic $p$ allows us to extend the scope of applicability of *symplectic Smith theory* [@Seidel_pants; @Shelukhin-Zhao] beyond the exact or semi-positive setting. With the above explanation, our main results concerning the structural aspects of (filtered) vortex Hamiltonian Floer theory can be summarized as follows. Given a commutative ring $R$, let $\Lambda = \Lambda_R$ be the upward Novikov ring $$\Lambda_R = \Big\{ \sum_{i=1}^\infty a_i T^{g_i}\ |\ g_i \in {\mathbb R},\ a_i \in R,\ \lim_{i \to \infty} g_i = +\infty \Big\}.$$ Denote by ${\bf z}_1, \dots, {\bf z}_N$ the $K$-equivariant degree $2$ cohomology classes dual to the coordinate hyperplanes $V_1, \cdots, V_N$ in $V$. **Theorem 2**. *There exists a bulk-deformation of the form $$\label{eqn:bulk} {\mathfrak b} = \sum_{j=1}^N \log c_j \cdot {\bf z}_j\ {\rm where}\ c_j \in {\mathbb Z}[{\bf i}]$$ satisfying the following properties.* 1. *The ${\mathfrak b}$-deformed vortex quantum homology algebra ${\it VHF}_\bullet^{\mathfrak b}(V; \Lambda_{\overline{\mathbb Q}})$ is semisimple over $\Lambda_{\overline{\mathbb Q}}$, with the number of idempotent summands, all of which are 1-dimensional, equal to $\sum_{i=0}^{2n} \dim_{\mathbb Q} H_i(X;{\mathbb Q})$.* 2. *The operator ${\mathbb E}_{\mathfrak b}: {\it VHF}_\bullet^{\mathfrak b} ( V; \Lambda_{\overline{\mathbb Q}}) \to {\it VHF}_\bullet^{\mathfrak b} ( V; \Lambda_{\overline{\mathbb Q}})$ given by the quantum multiplication with the equivariant first Chern class has distinct nonzero eigenvalues.* *Remark 2*. The reader may wonder about the legitimacy of taking the logarithm of elements of ${\mathbb Z}[{\bf i}]$. In reality, we will take the exponential of the intersection number between the bulk ${\mathfrak b}$ and Riemann surfaces to deform Floer-theoretic operations in the spirit of the divisor axiom in Gromov--Witten theory, and we take the above formal expression for the sake of conciseness. We explain the central ingredient in the proof of Theorem [Theorem 2](#thm:main-2){reference-type="ref" reference="thm:main-2"}: the GLSM version of the *closed string mirror symmetry*. To find such a bulk and calculate the quantum homology ring, we develop *closed-open field theory*, in particular, the *closed-open map* $${\rm CO}^{\mathfrak b}: {\it VHF}_\bullet^{\mathfrak b}(V; \Lambda_{\overline{\mathbb Q}}) \to {\it HH}^\bullet( {\mathcal F}_{\mathfrak b}^K (V), {\mathcal F}_{\mathfrak b}^K(V))$$ where ${\mathcal F}_{\mathfrak b}^K(V)$ is an equivariant version of the Fukaya category: its objects are roughly Lagrangians in the toric manifold $X$ and its structural maps are defined via equivariant counts of holomorphic disks upstairs in $V$. Hence we reduce the calculation of the quantum homology algebra to the determination of Hochschild cohomology by showing that the closed-open map is a unital ring isomorphism. Moreover, the structural coefficients of the $A_{\infty}$ operations are governed by the mirror superpotential, which is a Laurent polynomial $W: ({\mathbb C}^*)^N \to {\mathbb C}$. The Hochschild cohomology can be accordingly computed as the Jacobian ring of $W$, and the bulk-deformations can be understood as taking certain unfoldings of $W$ by adding elements from the Jacobi ring. It is well-known that a generic unfolding of $W$ only has nondegenerate critical points, and such unfolding can be realized by adjusting the bulk-deformation, implying that the quantum homology is generically semisimple. As for the statement on the first Chern class, it is the application of the folklore principle, usually attributed to Auroux--Kontsevich--Seidel (see [@Auroux_2007 Section 6] and [@Sheridan_2016 Lemma 2.7]), that generalized eigenvalues of quantum multiplication with the first Chern class have a one-to-one correspondence with critical values of the mirror superpotential . Such a closed-string mirror symmetry statement has been established using ordinary pseudo-holomorphic curves and Floer theory. For general toric manifolds, the symplectic version of the mirror superpotential is defined by counting *stable* pseudoholomorphic disks, usually having infinitely many terms (see [@Cho_Oh; @FOOO_toric_1; @FOOO_toric_2; @FOOO_mirror]). To Morsify such a superpotential, one usually needs very general bulk deformations. This in turn demonstrates another advantage of the GLSM: the mirror superpotential takes a rather simple form. As shown by Woodward [@Woodward_toric], the mirror superpotential in GLSM agrees with the mirror superpotential given by Givental [@Givental_potential] and Hori--Vafa [@Hori_Vafa]. One can Morsify this superpotential (called the Givental--Hori--Vafa potential) by only using "small" bulk deformation, i.e., divisor classes. ## Proof of the main theorem Once the crucial Theorem [Theorem 2](#thm:main-2){reference-type="ref" reference="thm:main-2"} is established, the rest of the proof can be streamlined in the same way as [@Shelukhin_2022]. Many key arguments are in fact algebraic while other geometric arguments need nontrivial, but straightforward extensions to the vortex setting. The first step is to take mod $p$ reductions of the vortex quantum homology algebra. Notice that as the coefficients of ${\mathfrak b}$ are integral, the mod $p$ deformed counts also define a vortex Hamiltonian Floer homology ${\it VHF}_\bullet^{\mathfrak b}(V; \Lambda_{\overline{\mathbb F}_p})$, where $\overline{\mathbb F}_p$ is the algebraic closure of ${\mathbb F}_p \cong {\mathbb Z}/ p{\mathbb Z}$. By a purely algebraic argument (see Theorem [Theorem 15](#thm_semisimple_finite_characteristic){reference-type="ref" reference="thm_semisimple_finite_characteristic"}), one obtains the following corollary of Theorem [Theorem 2](#thm:main-2){reference-type="ref" reference="thm:main-2"}. **Corollary 3**. *There exists $p_0>0$ such that for all primes $p \geq p_0$, the vortex quantum homology algebra ${\it VHF}_\bullet^{\mathfrak b}(V; \Lambda_{\overline{\mathbb F}_p})$ is semisimple with the number of idempotent summands, all of which are 1-dimensional, equal to $\sum_{i=1}^{2n} {\rm dim}_{\mathbb Q} H_i(X; {\mathbb Q})$.* We also need two quantitative results about the filtered theories in finite characteristics. Recall that to each Floer--Novikov complex such as the bulk-deformed vortex Floer complex, one can associate a **barcode**. In our case one needs the general formulation by Usher--Zhang [@Usher_Zhang_2016]. We consider two associated numerical invariants: the **boundary depth**, which is the length of the longest finite bar, and the **total bar length**, which is the sum of lengths of all finite bars. **Theorem 3**. *Let ${\mathfrak b}$ be a bulk-deformation satisfying Theorem [Theorem 2](#thm:main-2){reference-type="ref" reference="thm:main-2"} and $p_0$ be the one from Corollary [Corollary 3](#cor13){reference-type="ref" reference="cor13"}. Then there exists $C>0$ satisfying the following condition. Let ${\it VCF}_\bullet^{\mathfrak b}(H; \Lambda_{\overline{\mathbb F}_p})$ be the ${\mathfrak b}$-deformed filtered vortex Floer chain complex associated to a nondegenerate Hamiltonian $H$ on $X$[^1] and let $\beta_{(p)}^{\mathfrak b}(H)$ be its boundary depth, then for all $p \geq p_0$, $$\label{eqn:uniform-bound} \beta_{(p)}^{\mathfrak b}(H) \leq C.$$* The barcodes (with bottleneck distance) have a Lipschitz dependence on Hamiltonian diffeomorphisms (with the Hofer metric). Hence the above uniform bound on boundary depth extends to all Hamiltonians on the toric manifold $X$. On the other hand, the total bar length, denoted by $\tau^{\mathfrak b}_{(p)}(-)$ in characteristic $p$, can be extended to Hamiltonian diffeomorphisms with isolated fixed points. In particular, the barcode of a possibly degenerate Hamiltonian diffeomorphism $\phi$ with isolated fixed points is still finite, and the number of bar ends agrees with the homological counts of fixed points $N(\phi; \overline{\mathbb F}_p)$ (see Theorem [Theorem 75](#thm_degenerate_barcode){reference-type="ref" reference="thm_degenerate_barcode"}). The last key input is about the growth of total bar length of prime iterations of Hamiltonian diffeomorphisms. Suppose $\phi: X \to X$ is a Hamiltonian diffeomorphism such that all prime iterations of $\phi$ have isolated fixed points. If $\phi$ is the time-1 map of a Hamiltonian $H: S^1 \times X \to {\mathbb R}$, then the $p$-fold iteration $\phi^p$ is the time-1 map of $H^{(p)} := H_{pt}$. **Theorem 4**. *For any bulk ${\mathfrak b}$ for the form [\[eqn:bulk\]](#eqn:bulk){reference-type="eqref" reference="eqn:bulk"} and any odd prime $p$, we have the inequality $$\label{eqn:linear-grow} \tau_{(p)}^{\mathfrak b} (H^{(p)}) \geq p \cdot \tau_{(p)}^{\mathfrak b} (H).$$* *Remark 4*. The above equality should also hold for $p=2$, but we do not give a full treatment for this case because this would introduce extra notations in the discussion of equivariant Floer theory. In fact, for the proof, we do not need the $p=2$ version of [\[eqn:linear-grow\]](#eqn:linear-grow){reference-type="eqref" reference="eqn:linear-grow"} anyway. With the above technical ingredients, establishing the Hofer--Zehnder conjecture for toric manifolds is a matter of elementary arguments. *Proof of Theorem [Theorem 1](#thm:main){reference-type="ref" reference="thm:main"}.* Let $\phi: X \to X$ be a Hamiltonian diffeomorphism satisfying [\[eqn:exceeds\]](#eqn:exceeds){reference-type="eqref" reference="eqn:exceeds"}. By Proposition [Proposition 68](#prop:iso-local){reference-type="ref" reference="prop:iso-local"}, one can replace the ordinary local Floer homology by the (bulk-deformed) local vortex Floer homology. One also knows from Proposition [Proposition 53](#computation){reference-type="ref" reference="computation"} and Theorem [\[thm_vhf_bulk\]](#thm_vhf_bulk){reference-type="ref" reference="thm_vhf_bulk"} that the total rank of rational homology of $X$ agrees with the rank of the bulk-deformed vortex Floer homology of $V$. Hence [\[eqn:exceeds\]](#eqn:exceeds){reference-type="eqref" reference="eqn:exceeds"} can be rewritten as $$N(\phi, {\mathbb Q}) = \sum_{x \in {\rm Fix} (\phi)} {\rm dim}_{\overline{\mathbb Q}} {\it VHF}^{\rm loc}(\phi, x; {\mathbb Q}) > {\rm dim}_{\Lambda_{\mathbb Q}} {\it VHF}_\bullet^{\mathfrak b}(V; \Lambda_{\mathbb Q}).$$ Because the (local) vortex Floer homology are defined over the integers (see Section [5](#sec:local-Floer){reference-type="ref" reference="sec:local-Floer"}), by the universal coefficient theorem, for $p$ sufficiently large, $$\sum_{x \in {\rm Fix}(\phi)} \dim_{\overline{\mathbb F}_p} {\it VHF}^{\rm loc}(\phi, x; \overline{\mathbb F}_p) > {\rm dim}_{\Lambda_{\overline{\mathbb F}_p}} {\it VHF}_\bullet^{\mathfrak b}(V; \Lambda_{\overline{\mathbb F}_p}).$$ Suppose on the contrary that $\phi$ only has only finitely many periodic points. Then for any sufficiently large prime $p$, for all $k \geq 1$, ${\rm Fix}(\phi^{p^k}) = {\rm Fix}(\phi)$. Then by Theorem [Theorem 67](#thm:property-vortex-local){reference-type="ref" reference="thm:property-vortex-local"}, one has $$\sum_{x \in {\rm Fix}(\phi^{p^k})} {\rm dim}_{\overline{\mathbb F}_p} {\it VHF}^{\rm loc}(\phi^{p^k}, x; \overline{\mathbb F}_p) = \sum_{x\in {\rm Fix}(\phi)} {\rm dim}_{\overline{\mathbb F}_p} {\it VHF}^{\rm loc}(\phi, x; \overline{\mathbb F}_p) > {\rm dim}_{\Lambda_{\overline{\mathbb F}_p}} {\it VHF}_\bullet^{\mathfrak b}(V; \Lambda_{\overline{\mathbb F}_p}).$$ Consider the barcode of $\phi^{p^k}$ coming from the bulk-deformed vortex Floer theory (over the Novikov field $\Lambda_{\overline{\mathbb F}_p}$). The above implies that the number of finite bars is positive and independent of the iteration $p^k$. The uniform bound on the boundary depth (length of the longest finite bar) given by Theorem [Theorem 3](#thmc){reference-type="ref" reference="thmc"} implies that the total bar length $\tau^{\mathfrak b}_{(p)}(\phi^{p^k})$ is uniformly bounded. On the other hand, by Theorem [Theorem 4](#thm_total_length_growth){reference-type="ref" reference="thm_total_length_growth"}, for any $k \geq 1$, the total bar length grows as $$\tau^{\mathfrak b}_{(p)} (\phi^{p^k}) \geq p^k \cdot \tau^{\mathfrak b}_{(p)} (\phi) \geq C p^k > 0.$$ This is a contradiction. Hence $\phi$ must have infinitely many periodic points. ◻ Because the above argument works for any $p \geq p_0$, we know that the number of periodic points of $\phi$ grows like $\frac{k}{\log(k)}$ as $k \to \infty$, as a result of the prime number theorem. *Remark 5*. Arguments of the above form first appeared in [@Shelukhin_2022 Section 8], which we reproduce in our context for completeness. As noted above, Shelukhin's result on the Hofer--Zehnder conjecture relies on the assumptions that the ambient symplectic manifold is monotone and that the quantum homology is semisimple, which respectively account for the inequalities [\[eqn:linear-grow\]](#eqn:linear-grow){reference-type="eqref" reference="eqn:linear-grow"} (the monotonicity condition allows one to define Floer theory integrally using classical methods) and [\[eqn:uniform-bound\]](#eqn:uniform-bound){reference-type="eqref" reference="eqn:uniform-bound"} (which will be discussed in more detail in the body part of this paper). For general toric symplectic manifolds, traditional Hamiltonian Floer homology is only defined over the rationals, which sets difficulties for establishing symplectic Smith-type inequalities. Moreover, the quantum homology of toric symplectic manifolds fails to be semisimple in general, which is already the case even for Fano/monotone toric manifolds [@Ostrover-Tyomkin]. ## Outlook and speculations We are very surprised to find out that classical considerations from mirror symmetry can be quite useful for investigations in Hamiltonian dynamics. We expect such a connection could open up new avenues for future research. As mentioned above, GLSM can more generally be used to study symplectic topology and Hamiltonian dynamics of other symplectic/GIT quotients or complete intersections in them, the latter of which requires studing the gauged Witten equation (see [@Tian_Xu; @Tian_Xu_geometric]) with Hamiltonian perturbations. It is conceivable that one could resolve the Hofer--Zehnder conjecture for a broader class of symplectic quotients, provided that certain form of closed string mirror symmetry can be established. On a different note, except for deploying tools like GLSM, there are some recent advances [@Bai_Xu_2022; @Bai_Xu_Arnold] on defining Hamiltonian Floer theory over integers for general symplectic manifolds. The methods from *loc. cit.* are general enough for us to expect that a version of symplectic Smith-type inequality should hold using such a theory. Deriving dynamical applications using such a toolkit, including proving the Hofer--Zehnder conjecture in more general settings, is another topic for future research. ## Outline of the paper The following provides an outline of this paper. - Basic notions related to toric manifolds are recalled in Section [2](#sec:prelim){reference-type="ref" reference="sec:prelim"}, which also includes an introduction to the symplectic vortex equations arising from GLSM. - In Section [3](#sec:alg-prelim){reference-type="ref" reference="sec:alg-prelim"}, various algebraic preliminaries relevant for our purpose, including semisimple algebras over Novikov rings defined over fields with possibly positive characteristics, abstract setups for filtered Floer theories, persistence modules, and $A_\infty$ algebras and their Hochschild cohomology, are recalled systematically. - A (filtered) Hamiltonian Floer theory package in the vortex setting is recorded in Section [4](#sec:ham-package){reference-type="ref" reference="sec:ham-package"}. Most notably, we introduce bulk deformations in vortex Hamiltonian Floer theory which allow us to incorporate ideas from generic semisimplicity of quantum homology to derive applications in quantitative symplectic topology. - In Section [5](#sec:local-Floer){reference-type="ref" reference="sec:local-Floer"}, we introduce local Floer theory in the vortex setting in order to establish Theorem [Theorem 1](#thm:main){reference-type="ref" reference="thm:main"} for Hamiltonian diffeomorphisms with isolated but degenerate fixed points. - The main purpose of Section [6](#sec:beta){reference-type="ref" reference="sec:beta"} is to prove Theorem [Theorem 80](#thm_boundary_depth){reference-type="ref" reference="thm_boundary_depth"} = Theorem [Theorem 3](#thmc){reference-type="ref" reference="thmc"}, which ensures a uniform upper bound on the boundary depth of the bulk-deformed vortex Hamiltonian Floer persistence module of any Hamiltonian diffeomorphism provided that the bulk-deformed vortex quantum homology is semisimple. - In Section [7](#sec:equiv){reference-type="ref" reference="sec:equiv"}, we develop ${\mathbb Z}/p$-equivariant vortex Hamiltonian Floer theory by adapting the work [@Seidel_pants; @Shelukhin-Zhao] in the GLSM setting. Theorem [Theorem 4](#thm_total_length_growth){reference-type="ref" reference="thm_total_length_growth"} = Theorem [Theorem 89](#thm:smith){reference-type="ref" reference="thm:smith"} is proven as a consequence by appealing to the work of Shelukhin [@Shelukhin_2022]. - We turn our attention to Lagrangian Floer theory in Section [8](#section8){reference-type="ref" reference="section8"}. The key result is to demonstrate the existence of a "convenient\" bulk deformation (cf. Definition [Definition 114](#defn:convenient){reference-type="ref" reference="defn:convenient"}) whose associated Fukaya category (in the GLSM setting) takes a very simple form, such that its Hochschild cohomology is a semisimple algebra. - Lastly, in Section [9](#section9){reference-type="ref" reference="section9"}, Theorem [Theorem 122](#thm_CO){reference-type="ref" reference="thm_CO"} = Theorem [Theorem 2](#thm:main-2){reference-type="ref" reference="thm:main-2"} is proven by showing that the closed-open string map is a unital ring isomorphism. ## Acknowledgements {#acknowledgements .unnumbered} : We thank Marcelo Atallah, Hiroshi Iritani, Han Lou, Egor Shelukhin, Nick Sheridan, Michael Usher, and Chris Woodward for useful discussions and email correspondences. The first-named author is grateful to the Simons Center for Geometry and Physics for its warm hospitality during Spring 2023. # Geometric Preliminaries {#sec:prelim} We recall basic notions about toric symplectic manifolds and symplectic vortex equations. ## Toric manifolds as symplectic quotients We recall the notion of symplectic reduction/quotients. Let $K$ be a compact Lie group with Lie algebra ${\mathfrak k}$. Let $(V, \omega_V)$ be a symplectic manifold. A smooth $K$-action on $V$ is called a **Hamiltonian action** if there exists a **moment map** $$\mu: V \to {\mathfrak k}^*$$ satisfying 1. $\mu$ is $K$-equivariant (with respect to the co-adjoint action on ${\mathfrak k}^*$). 2. For each $\xi\in {\mathfrak k}$, let the infinitesimal action of $\xi$ be ${\mathcal X}_\xi$. Then $$\omega_V( {\mathcal X}_\xi, \cdot) = d \langle \mu, \xi \rangle.$$ It follows that the level set $\mu^{-1}(0)$ is $K$-invariant. Define the **symplectic reduction** of $V$ (with respect to the $K$-action and the moment map) to be $$X:= \mu^{-1}(0)/K.$$ We always assume that $0$ is a regular value of $\mu$ and the $K$-action on $\mu^{-1}(0)$ is free. This assumption implies that $X$ is a smooth manifold. In this case, $X$ carries a canonically induced symplectic form $\omega_X$. When $V$ has a $K$-invariant integrable almost complex structure $J_V$, the $K$-action can be extended to its complexification $K^{\mathbb C}$ as a holomorphic action. When this is the case, (under certain extra conditions), the Kempf--Ness theorem says that the symplectic reduction can be identified with the geometric invariant theory (GIT) quotient. ### Compact symplectic toric manifolds {#subsec:fan} Symplectic toric manifolds can be realized as symplectic quotients of a vector space. We provide a minimal description of symplectic toric manifolds necessary for this paper. An $2n$-dimensional compact symplectic toric manifold $X$ is described by a convex polytope $P \subset {\mathbb R}^n$ satisfying the following conditions. 1. For each face $\partial_j P$ of $P$, there are ${\bf v}_j \in {\mathbb Z}^n$ and $\lambda_j\in {\mathbb R}$ such that ${\bf v}_j$ is an inward normal vector and the face is defined by $$\partial_j P = \{ {\bf u}\in {\mathbb R}^n\ |\ \langle {\bf v}_j, {\bf u}\rangle = \lambda_j \}.$$ 2. For each vertex of $P$, the normal vectors ${\bf v}_{j_1}, \ldots, {\bf v}_{j_n}$ of all adjacent faces form a ${\mathbb Z}$-basis of ${\mathbb Z}^n$. In this paper, denote by $N$ the number of faces of $P$. We can realize $X$ as the symplectic quotient of ${\mathbb C}^N$ (with the standard symplectic form) by the $N-n$ dimensional torus $K = T^{N-n} = (S^1)^{N-n}$. The collection of vectors ${\bf v}_1, \ldots, {\bf v}_N$ defines a linear map $$\tilde \pi_K: {\mathbb R}^N \to {\mathbb R}^n$$ which sends ${\mathbb Z}^N$ onto ${\mathbb Z}^n$. Hence it induces a surjective group homomorphism $$\pi_K: T^N \to T^n.$$ Let $K$ be the kernel of $\pi_P$. Hence $K$ acts on ${\mathbb C}^N$ as a subgroup of $T^N$. Notice that for the standard $\widehat K = T^N$-action, the moment map can be written as $$\widehat\mu(x_1, \ldots, x_N) = \left( \pi |x_1|^2 - \lambda_1, \ldots, \pi |x_N|^2 - \lambda_N \right) \in {\mathbb R}^N \cong \widehat{\mathfrak k}^*.$$ Then the moment map of the $K$-action is simply the composition $$\mu = (d\pi_K)^* \circ \widehat\mu: V \to {\mathfrak k}^*.$$ On the other hand, there is a residual torus action on $X$ by the quotient $T^n$ which is the torus action usually appear in the discussion of toric manifolds. The associated moment map is denoted by $$\pi_X: X \to {\mathbb R}^n$$ whose range is actually the moment polytope $P$. Notice that if one translates the moment polytope $P$ in ${\mathbb R}^N$ by a vector in ${\mathbb R}^n$, then it does not change the moment map $\mu$ and hence the symplectic form on $X$. ## Symplectic vortex equation The symplectic vortex equation was originally introduced by Cieliebak--Gaio--Salamon [@Cieliebak_Gaio_Salamon_2000] and Mundet [@Mundet_thesis]. It is a generalization of the pseudoholomorphic curve equation to the equivariant setting. Here we briefly recall its setup and some basic analytical result. ### Gauged maps and vortex equation Let $V$ be the complex vector space acted on by the complex torus $G = K^{\mathbb C}$ with moment map $\mu$ under the $K$-action and symplectic quotient the toric manifold $X$. Let $\Sigma$ be a Riemann surface. A **gauged map** from $\Sigma$ to $V$ is a triple ${\mathfrak u} = (P, A, u)$ where $P\to \Sigma$ is a principal $K$-bundle, $A\in {\mathcal A}(P)$ is a connection on $P$, and $u$ is a section of the associated vector bundle $P(V):= P\times_K V$. The group of gauge transformations ${\mathcal G}(P)$, which, in the abelian case, is the group of smooth maps $$g: \Sigma \to K,$$ which acts on gauged maps by $$g^* {\mathfrak u} = g^* (P, A, u) = (P, g^* A, g^* u) = (P, A + g^{-1} dg, g^{-1} u).$$ We need three quantities to define the vortex equation. First the covariant derivative of $u$ is a section $$d_A u \in \Omega^1(P, u^* TV)$$ which descends to an element in $\Omega^1(\Sigma, u^* TV / K)$. There are also the curvature and the moment potential $$\begin{aligned} &\ F_A \in \Omega^2(\Sigma, {\rm ad} P),\ &\ \mu(u) \in \Omega^0(\Sigma, {\rm ad}P^* ).\end{aligned}$$ By choosing an invariant inner product on the Lie algebra ${\mathfrak k}$ one can identify ${\rm ad}P\cong {\rm ad} P^*$; by choosing a volume form $\nu_\Sigma$ one can identify $\Omega^2 \cong \Omega^0$. The gauged map ${\mathfrak u}$ is called a **vortex** if $$\begin{aligned} \label{vortex_equation} &\ \overline\partial_A u = 0,\ &\ *F_A + \mu(u) = 0.\end{aligned}$$ Here $\overline\partial_A u$ is the $(0,1)$-part of the covariant derivative $d_A u$. Both equations are invariant under gauge transformations. The energy of a vortex is defined to be $$E({\mathfrak u}) = \frac{1}{2} \int_\Sigma \left( \| d_A u \|^2 + \| F_A \|^2 + \|\mu(u)\|^2 \right) \nu_\Sigma.$$ Analogous to pseudoholomorphic curves, vortices satisfy an energy identity. Suppose $\Sigma$ is closed. Then each gauged map ${\mathfrak u}$ represents an equivariant homology class $[{\mathfrak u}] \in H_2^K(V; {\mathbb Z})$ defined as follows. The section $u: \Sigma \to P(V)$ can be identified with a $K$-equivariant map $\tilde u: P \to V$. Let $EK \to BK$ be the universal $K$-bundle. The classifying map of $P \to \Sigma$ is a map $\iota: \Sigma \to BK$ which is covered by a bundle map $\tilde \iota: P \to EK$. Then the equivariant map $(\tilde \iota, \tilde u): P \to EK \times V$ descends to a continuous map from $\Sigma$ to $(EK \times V)/K$, which represents a class $[{\mathfrak u}] \in H_2^K(V; {\mathbb Z})$. In the toric case, this class is just the degree of the principal bundle $P\to \Sigma$. Then for any gauged map ${\mathfrak u} = (P, A, u)$, one has $$E({\mathfrak u}) = \langle \omega^K, [{\mathfrak u}] \rangle + \| \overline\partial_A u \|_{L^2(\Sigma)}^2 + \| * F_A + \mu(u) \|_{L^2(\Sigma)}^2.$$ Here $\omega^K\in H_K^2(V; {\mathbb R})$ is the equivariant class represented by the equivariant 2-form $\omega - \mu$ (see[@Cieliebak_Gaio_Salamon_2000 Proposition 3.1] and [@Mundet_2003 Lemma 14]). *Remark 6*. An important feature of the symplectic vortex equation in the toric setting is that no bubbling happens as the space $V$ is symplectically aspherical. In general, energy concentration could cause bubbling of holomorphic spheres as shown in [@Mundet_thesis; @Mundet_2003; @Cieliebak_Gaio_Mundet_Salamon_2002; @Ott_compactness]. One can introduce Hamiltonian perturbations. Given a 1-form $${\mathcal H} \in \Omega^1(\Sigma, C^\infty(V)^K)$$ with coefficients in the space of $K$-invariant smooth functions on $V$, we can define a family of Hamiltonian vector fields $$X_{\mathcal H} \in \Gamma( \Sigma \times V, \pi_{\Sigma}^* T^* \Sigma \otimes TV)$$ which is $K$-invariant, where $\pi_{\Sigma}: \Sigma \times V \to \Sigma$ is the projection to the first factor. Hence for any principal $K$-bundle $\pi_P: P \to \Sigma$, the vector field $X_{\mathcal H}$ induces a section on the total space of the vector bundle $\pi_{P(V)}: P(V)\to \Sigma$ $$X_{\mathcal H} \in \Gamma( P(V), \pi_{P(V)}^* T^* \Sigma \otimes P(TV)),$$ where $P(TV) := P \times_K u^* TV$. The perturbed symplectic vortex equation is $$\begin{aligned} &\ \overline\partial_{A, {\mathcal H}} u = 0,\ &\ * F_A + \mu(u) = 0.\end{aligned}$$ where $$\overline\partial_{A, {\mathcal H}} u = (d_A u)^{0,1} + (X_{\mathcal H}(u))^{0,1}.$$ For our applications, ${\mathcal H}$ is obtained by extending the pullback of Hamiltonian connections in $\Omega^1(\Sigma, C^{\infty}(X)) = \Omega^1(\Sigma, C^{\infty}(\mu^{-1}(0) / K))$. ### Compactness Although in aspherical targets vortices cannot bubble off holomorphic spheres, in general holomorphic curves can bubble off. It is the case when one considers Lagrangian boundary conditions. Let $L \subset V$ be a $K$-invariant Lagrangian submanifold. One can impose the Lagrangian boundary condition for gauged maps ${\mathfrak u} = (P, A, u)$ from $\Sigma$ to $V$ with $u|_{\partial \Sigma} \subset P(L)$. Given a sequence of solutions ${\mathfrak u}_i$ to the vortex equation on $\Sigma$ subject to the Lagrangian boundary condition, if $u_i$ has uniformly bounded image and $E({\mathfrak u}_i)$ is uniformly bounded, the energy density could blow up near a boundary point. The boundedness of the images of $u_i$ implies that the curvatures $F_{A_i}$ do not blowup. Moreover, if one scales by the rate of energy concentration, the sequence of connections $A_i$ converge subsequentially (up to gauge transformation) to a flat connection. All Hamiltonian perturbations and variations of almost complex structures will also be scaled off. Hence a subsequece can bubble off a (stable) holomorphic disk in $V$ with boundary in $L$ with respect to a fixed almost complex structure. See details in [@Wang_Xu]. # Algebraic preliminaries {#sec:alg-prelim} ## Novikov rings We set up the notations for our coefficient rings. In this paper, $R$ always denotes a commutative ring with a unit, hence comes with a canonical ring map $${\mathbb Z} \to R.$$ Let $\Lambda = \Lambda_R$ be the (upward) **Novikov ring** $$\Lambda_R = \Big\{ \sum_{i=1}^\infty a_i T^{g_i}\ |\ g_i \in {\mathbb R},\ a_i \in R,\ \lim_{i \to \infty} g_i = +\infty \Big\}$$ The **valuation** on ${\mathfrak v}: \Lambda \to {\mathbb R} \cup \{+\infty\}$ is defined by $${\mathfrak v}\left( \sum_{i=1}^\infty a_i T^{g_i} \right) = \inf \big\{ g_i\ |\ a_i \neq 0 \big\}\ \ \ {\rm and}\ \ \ {\mathfrak v}(0) = +\infty.$$ We will also need the following version $$\Lambda_{0, R} = \Big\{ \sum_{i=1}^\infty a_i T^{g_i}\ |\ g_i \in {\mathbb R}_{\geq 0},\ a_i \in R,\ \lim_{i \to \infty} g_i = +\infty \Big\},$$ which also comes with a valuation by restricting the above valuation. When $R$ is a field, $\Lambda_R$ is also a field, and it is the field of fraction of $\Lambda_{0, R}$. In many cases we can restrict to a Novikov ring of series $\sum a_i T^{g_i}$ where $g_i$ are restricted to a finitely generated additive group $\Gamma \subsetneq {\mathbb R}$. In this paper $\Gamma$ is fixed and actually determined by the GIT presentation of a toric manifold. Indeed, the discrete monoid $\Gamma$ associated with the toric manifold $X(\Sigma)$ is defined to be the image of effective $1$-cycles in ${\mathbb R}$ defined from pairing with the cohomology class represented by the symplectic form, see Section [2.1.1](#subsec:fan){reference-type="ref" reference="subsec:fan"}. Denote $$\Lambda_R^\Gamma:= \Big\{ \sum_{i=1}^\infty a_i T^{g_i}\in \Lambda_R \ |\ g_i \in \Gamma \Big\}$$ and $$\Lambda_{0, R}^\Gamma:= \Lambda_{0, R} \cap \Lambda_R^\Gamma.$$ However $\Lambda_R^\Gamma$ does not enjoy certain algebraic properties of $\Lambda_R$. For example, when $R = {\mathbb K}$ is an algebraically closed field, $\Lambda_{\mathbb K}$ is algebraically closed but $\Lambda_{\mathbb K}^\Gamma$ is not. ### Modules and algebras over Novikov rings **Definition 7**. A **non-Archimedean normed free module** over $\Lambda_R$ is a pair $(C, \ell)$ where $C$ is a free $\Lambda_R$-module endowed with a function $\ell: C \to {\mathbb R} \cup \{-\infty\}$ satisfying 1. **(Nondegeneracy)** $\ell(x) = -\infty$ if and only if $x = 0$. 2. **(Homogeneity)** For all $\lambda \in \Lambda_R$ and $x \in C$, $\ell(\lambda x) = \ell(x) - {\mathfrak v}(\lambda)$. 3. **(Subadditivity)** For $x, y \in C$, $\ell(x +y ) \leq \max \{ \ell(x), \ell(y)\}$; if $\ell(x) \neq \ell(y)$, then $\ell(x + y) = \max \{ \ell(x), \ell(y)\}$.[^2] Now suppose ${\mathbb K}$ is an $R$-module which is also a field. Then one can extend the function $\ell$ to $C \otimes_{\Lambda_R} \Lambda_{\mathbb K}$ via **(Homogeneity)**. The then obtained pair is a non-Archimedean normed vector space in the sense of [@Usher_Zhang_2016 Definition 2.2] (except that the coefficient field was $\Lambda_{\mathbb K}^\Gamma$ rather than $\Lambda_{\mathbb K}$). We also need to consider multiplicative structures compatible with the non-Archimedean norm. **Definition 8**. A **non-Archimedean normed algebra** over $\Lambda_R$ is a non-Archimedean normed free module $(C, \ell)$ together with a $\Lambda_R$-algebra structure satisfying the - **(Triangle inequality)** For all $x, y \in C$, $$\ell( xy) \leq \ell(x) + \ell(y).$$ ### Specific coefficients and mod $p$ reductions In this paper we need to use certain non-traditional coefficient ring or fields. Here we briefly summarize them and set up the notations. First, let $\overline{\mathbb Q}$ be the algebraic closure of ${\mathbb Q}$, which is viewed as a subfield of ${\mathbb C}$. Inside $\overline{\mathbb Q}$ there is the subring of algebraic integers $\overline{\mathbb Z}$, which is the set of algebraic numbers which are solutions to monic polynomials with integer coefficients. Further, in characteristic $p$ (where $p$ is an odd prime), let ${\mathbb F}_p\cong {\mathbb Z}/ p {\mathbb Z}$ be the smallest field with characteristic $p$. Let $\overline{\mathbb F}_p$ be the algebraic closure of ${\mathbb F}_p$, which is only well-defined up to isomorphism of field extensions. Notice that the notion of non-Archimedean normed algebras can be transferred between different coefficient rings via tensor products. A crucial feature of the geometric construction of this paper is that, as long as one has a counting theory over ${\mathbb Z}$ (or $\overline{\mathbb Z}$), then it automatically induces a theory over any ring $R$ (or $\overline{\mathbb Z}$-algebra). In particular, one needs to perform the "mod $p$ reduction" which is roughly associated to the ring map ${\mathbb Z} \to {\mathbb F}_p$. In our situation, one needs the corresponding extension to the algebraic closure of ${\mathbb F}_p$. **Lemma 9**. *For each prime $p$, there exists a unital ring map $$\overline\pi_p: \overline{\mathbb Z} \to \overline{\mathbb F}_p.$$* *Proof.* (Following the mathoverflow post [@Fp_closure]) $p\in {\mathbb Z} \subset \overline{\mathbb Z}$ is not an invertible element. Hence there exists a maximal ideal ${\mathfrak m} \subset \overline{\mathbb Z}$ containing $p$. Consider the quotient field $\overline{\mathbb Z}/ {\mathfrak m}$, which is an extension of ${\mathbb F}_p$. We prove that it is an algebraic closure of ${\mathbb F}_p$. First, each element $x\in \overline{\mathbb Z} / {\mathfrak m}$ has a lift $\tilde x\in \overline{\mathbb Z}$ which is the solution to an algebraic equation with integer coefficients. Hence $x$ is an algebraic element over ${\mathbb F}_p$. Therefore $\overline{\mathbb Z}/{\mathfrak m}$ is algebraic over ${\mathbb F}_p$. Second, for any monic polynomial $f$ with ${\mathbb F}_p$-coefficients, one can find a monic integral lift $\tilde f$ and all roots of $\tilde f$ are in $\overline{\mathbb Z}$. Hence $f$ has roots in the field $\overline{\mathbb Z}/ {\mathfrak m}$. Therefore $\overline{\mathbb Z}/ {\mathfrak m}$ is algebraically closed. Hence $\overline{\mathbb Z}/ {\mathfrak m}$ is an algebraic closure of ${\mathbb F}_p$. Therefore, $\overline{\mathbb Z}/ {\mathfrak m} \cong \overline{\mathbb F}_p$ as fields. Precomposing with $\overline{\mathbb Z} \to \overline{\mathbb Z}/ {\mathfrak m}$ one obtains the desired ring map. ◻ *Remark 10*. The map $\overline\pi_p$ is not unique. But we fix one for each prime $p$. The "mod $p$ reduction" also makes sense for any rational number as long as $p$ is greater than the denominator. We need the following lemma to extend this simple fact to the algebraic closure. **Lemma 11**. *For each $y \in \overline{\mathbb Q}$, there exists $m \in {\mathbb Z}\setminus \{ 0\}$ such that $my \in \overline{\mathbb Z}$.* *Proof.* If $y\in \overline{\mathbb Q}$ is an algebraic number, then it is a root of a polynomial with integer coefficients $P = a_k u^k + \cdots + a_1 u + a_0$ with $a_k \neq 0$. Then $$0 = a_k^{k-1} P(y) = (a_k y)^k + a_{k-1} (a_k y )^{k-1} + a_{k-2} a_k (a_k y)^{k-2} + \cdots + a_1 a_k^{k-2} (a_k u) + a_0 a_k^{k-1}.$$ So $a_k y$ is a root of a monic polynomial with integer coefficients, hence is in $\overline{\mathbb Z}$. ◻ ## Semisimple algebras over Novikov fields In this paper we use a more restrictive notion of semisimplicity of algebras over Novikov fields. **Definition 12**. Let ${\mathbb F}$ be a field. A unital ${\mathbb F}$-algebra $(A, \ast)$ is called **semisimple** if it splits as a direct sum of rings $$A = F_1 \oplus \cdots \oplus F_k$$ where $F_i \cong {\mathbb F}$ as a ring. Each summand $F_i$ is called an **idempotent summand** of $A$ and the splitting is called the **idempotent splitting**.[^3] *Remark 13*. In many papers such as [@Entov_Polterovich_1; @Shelukhin_2022; @FOOO_toric_1], the meaning of semisimplicity is more general: for example, each summand $F_i$ is allowed to be a finite extension of the field ${\mathbb F}$. The number of idempotent summands also depends on the choice of the field. In our situation, one can achieve the above stronger semisimplicity of a version of quantum cohomology algebra by turning on bulk deformations and taking a sufficiently large field. Suppose $A$ is semisimple. Then for each idempotent summand $F_i$, there is a unique generator $e_i \in F_i$ such that $e_i \ast e_i = e_i$. We call $e_i$ the **idempotent generator**. Then $(e_1, \ldots, e_k)$ is a basis of $A$. Given any element $\alpha = \lambda_1 e_1 + \cdots + \lambda_k e_k$, one can see that the linear map $$\alpha \ast: A \to A$$ has eigenspace decomposition $F_1 \oplus \cdots \oplus F_k$ with eigenvalues $\lambda_1, \ldots, \lambda_k$. The following statement shows that the converse also holds under additional assumptions. **Lemma 14**. *Let $A$ be a $k$-dimensional commutative unital ${\mathbb F}$-algebra and $\alpha \in A$. Suppose $\alpha\ast:A \to A$ has $k$ distinct *nonzero* eigenvalues $\lambda_1, \ldots, \lambda_k$. Then $A$ is semisimple.* *Proof.* Let $(\varepsilon_1, \ldots, \varepsilon_k)$ be an eigen-basis of $\alpha \ast$. Write $$\alpha = \sum_{i=1}^k \mu_i \varepsilon_i.$$ Then we see $$\alpha \ast ( \varepsilon_i \ast \varepsilon_j ) = \lambda_i \varepsilon_i \ast \varepsilon_j = \lambda_j \varepsilon_i \ast \varepsilon_j.$$ As $\lambda_i$ are all distinct, one has $\varepsilon_i \ast \varepsilon_j = 0$ whenever $i \neq j$. Then one obtains $$\alpha \ast \varepsilon_i = \mu_i \varepsilon_i \ast\varepsilon_i = \lambda_i \varepsilon_i.$$ As $\lambda_i \neq 0$, one can see $\mu_i \neq 0$. Define $e_i = \lambda_i^{-1} \mu_i \varepsilon_i$. Then $$e_i \ast e_i = (\lambda_i^{-1} \mu_i)^2 \varepsilon_i \ast \varepsilon_i = \lambda_i^{-1} \mu_i \varepsilon_i = e_i.$$ Hence $A$ is semisimple. ◻ ### Semi-simplicity and different characteristics Here we prove a useful algebraic fact which allows us to derive semi-simplicity in finite characteristics from semi-simplicity in characteristic zero. We set up the problem as follows. Let $(A, \ell)$ be a non-Archimedean normed (free) algebra over the Novikov ring $\Lambda_{\overline{\mathbb Z}}$. Denote $$A_{(0)}:= A \otimes_{\Lambda_{\overline{\mathbb Z}}} \Lambda_{\overline{\mathbb Q}}$$ and for each prime $p$ $$A_{(p)}:= A \otimes_{\Lambda_{\overline{\mathbb Z}}} \Lambda_{\overline{\mathbb F}_p}.$$ Denote the induced valuations by $$\begin{aligned} &\ \ell_0: A_{(0)} \to {\mathbb R} \cup \{-\infty\},\ &\ \ell_p: A_{(p)} \to {\mathbb R} \cup \{ -\infty \}.\end{aligned}$$ Moreover, let ${\mathcal U} \in A$ be a distinguished nonzero element (which will be the first Chern class in quantum homology in our later discussions), and let ${\mathcal U}_{(0)} \in A_{(0)}$, ${\mathcal U}_{(p)} \in A_{(p)}$ be the corresponding induced element. They induce linear operators $$E_{(m)}: A_{(m)} \to A_{(m)},\ x \mapsto {\mathcal U}_{(m)} \ast x,\ m = 0, p.$$ **Theorem 15**. *Suppose $A_{(0)}$ is semisimple over $\Lambda_{\overline{\mathbb Q}}$ and all eigenvalues of $E_{(0)}$ are nonzero and distinct. Then there exist $p_0>0$ and $C>0$ such that for all prime $p \geq p_0$, the following conditions hold.* 1. *$A_{(p)}$ is semisimple over $\Lambda_{\overline{\mathbb F}_p}$.* 2. *If $e_{1, p}, \ldots, e_{m, p}$ are idempotent generators of $A_{(p)}$, then $$\ell(e_{l, p}) \leq C.$$* *Proof of Theorem [Theorem 15](#thm_semisimple_finite_characteristic){reference-type="ref" reference="thm_semisimple_finite_characteristic"} (1).* Consider the operator $E: A \to A$ and its characteristic polynomial $f_E$. Notice that $\Lambda_{\overline{\mathbb Q}}$ is the field of fraction of $\Lambda_{\overline{\mathbb Z}}$. Hence $f_E$ has $m$ distinct roots in $\Lambda_{\overline{\mathbb Q}}$ and so the discriminant of $f_E$, denoted by $D(f_E)\in \Lambda_{\overline{\mathbb Z}}$, is nonzero. Hence for sufficiently large prime $p$, the discriminant of $f_{E_{(p)}}$, which is the mod $p$ reduction of $D(f_E)$, is nonzero. It follows that $E_{(p)}$ also has $m$ distinct eigenvalues. Moreover, as all eigenvalues of $E_{(0)}$ are nonzero, $f_E(0) \neq 0$. Hence $f_{E_{(p)}}(0) \neq 0$ when $p$ is sufficiently large. Hence $E_{(p)}$ is invertible and has no zero eigenvalue. By Lemma [Lemma 14](#semisimple_criterion){reference-type="ref" reference="semisimple_criterion"}, $A_{(p)}$ is semisimple for sufficiently large $p$. ◻ ### Proof of Theorem [Theorem 15](#thm_semisimple_finite_characteristic){reference-type="ref" reference="thm_semisimple_finite_characteristic"} (2) {#proof-of-theorem-thm_semisimple_finite_characteristic-2} To prove the quantitative statement of Theorem [Theorem 15](#thm_semisimple_finite_characteristic){reference-type="ref" reference="thm_semisimple_finite_characteristic"}, we introduce the notion of truncation. First, given an element $$\lambda = \sum_{i =1}^\infty a_i T^{g_i} \in \Lambda_{\overline{\mathbb Q}},$$ and $Z \in {\mathbb R}$, define the $Z$-truncation of $\lambda$ to be the element $$\lambda^Z:= \sum_{g_i \leq Z} a_i T^{g_i},$$ which has only finitely many terms. Then it follows easily $$\label{truncation_1} {\mathfrak v} ( \lambda - \lambda^Z) \geq Z.$$ For an element in a module over $\Lambda_{\overline{\mathbb Q}}$, its truncations are not canonically defined. We fix, throughout the proof, a basis ${\mathfrak x}_1, \ldots, {\mathfrak x}_m$, of the $\Lambda_{\overline{\mathbb Z}}$-module $A$. Without loss of generality, we can choose the basis such that $$\ell( {\mathfrak x}_1) = \cdots = \ell({\mathfrak x}_m) = 0.$$ By abuse of notations, denote the induced basis of $A_{(0)}$ and $A_{(p)}$ still by ${\mathfrak x}_1, \ldots, {\mathfrak x}_m$. Then for each $\alpha \in A_{(0)}$, we can write $$\alpha = \sum_{j=1}^m \alpha_j {\mathfrak x}_j$$ where $\alpha_j \in \Lambda_{\overline{\mathbb Q}}$. Then define the $Z$-truncation $$\alpha^Z = \sum_{j=1}^B \alpha_j^Z {\mathfrak x}_j.$$ Then by [\[truncation_1\]](#truncation_1){reference-type="eqref" reference="truncation_1"} we have the estimate $$\begin{gathered} \label{truncation_2} \ell_0 ( \alpha - \alpha^Z) = \ell_0 \left( \sum_{l=1}^m (\alpha_l - \alpha_l^Z) {\mathfrak x}_l \right) \leq \max_{1 \leq l \leq m} \ell_0 \left( (\alpha_l - \alpha_l^Z) {\mathfrak x}_l \right) \\ = \max_{1\leq l \leq m} \left( \ell_0 ({\mathfrak x}_l) - {\mathfrak v}(\alpha_l - \alpha_l^Z) \right) \leq \max_{1 \leq j \leq m} \ell ({\mathfrak x}_j) - Z = - Z.\end{gathered}$$ **Running convention.** Within this proof, $Z$ is a large real number which can be fixed from the beginning. The lower bound of $p_0$ which is valid for the statement of Theorem [Theorem 15](#thm_semisimple_finite_characteristic){reference-type="ref" reference="thm_semisimple_finite_characteristic"} depends on the choice of $Z$. The letter $C>0$ denotes a real number which is independent of $Z$ and $p\geq p_0$ but whose value is allowed to change from line to line. **Lemma 16**. *Suppose $e_{1,(0)}, \dots, e_{m, (0)}$ constitute the idempotent generators of $A_{(0)}$, with valuations $\lambda_{1, (0)}, \dots, \lambda_{m, (0)}$. Then for $Z$ sufficiently large, $e_{1,(0)}^Z, \dots, e_{m, (0)}^Z$ form a basis of $A_{(0)}$, and $\lambda_{1, (0)}^Z, \dots, \lambda_{m, (0)}^Z$ are all nonzero and distinct.* *Proof.* With respect to the basis $({\mathfrak x}_{1, (0)}, \ldots, {\mathfrak x}_{m, (0)})$ of $A_{(0)}$, we identify $e_{l, (0)}$ with its coordinate vector in $(\Lambda_{\overline{\mathbb Q}})^m$. Then the $m \times m$ matrix with columns $e_{l, (0)}$ is invertible with a nonzero determinant. Then when $Z$ is sufficiently large, the corresponding determinant with $e_{l, (0)}$ replaced by $e_{l, (0)}^Z$ is also nonzero. On the other hand, as all $\lambda_{l,(0)}$ are nonzero, $\lambda_{l, (0)}^Z\neq 0$ when $Z$ is large. ◻ We would like to construct, for large primes $p$, eigenvectors and eigenvalues of ${\mathbb E}_{(p)}$ over the field $\Lambda_{\overline{\mathbb F}_p}$. The basic idea is to take some truncation $e_{l, (0)}^Z$ of the idempotent generators and their mod $p$ reductions as an appropriate eigenbasis and then to apply certain corrections. By Lemma [Lemma 11](#lemma46){reference-type="ref" reference="lemma46"}, for each $Z\in {\mathbb R}$, there exists $m^Z \in {\mathbb Z}$ such that $$\begin{aligned} &\ m^Z \lambda_{l, (0)}^Z \in \Lambda_{\overline{\mathbb Z}},\ & m^Z e_{l, (0)}^Z \in A_{\overline{\mathbb Z}}.\end{aligned}$$ This allows us to define the "mod $p$ reduction" of $\lambda_{l, (0)}^Z$ and $e_{l, (0)}^Z$ as follows. Fixing $m^Z$, by choosing a sufficiently large $p$ so that it cannot divide $m^Z$, the quantity $m^Z$ has a nonzero reduction $[m^Z]_p \in {\mathbb F}_p$. Moreover, $m^Z \lambda_{l, (0)}^Z$ has a mod $p$ reduction $[m^Z \lambda_{l, (0)}^Z]_p \in \Lambda_{\overline{\mathbb F}_p}$ and $m^Z e_{l, (0)}^Z$ has a mod $p$ reduction $[m^Z e_{l, (0)}^Z]_p \in A_{(p)}$ (defined via the integral basis ${\mathfrak x}_1, \ldots, {\mathfrak x}_m$). Then define $$\begin{aligned} &\ \lambda_{l, (p)}^Z:= [m^Z]_p^{-1} [m^Z \lambda_{l, (0)}^Z]_p,\ &\ e_{l, (p)}^Z:= [m^Z]_p^{-1} [m^Z e_{l, (0)}^Z]_p.\end{aligned}$$ **Lemma 17**. *There exists $C>0$ such that for any sufficiently large $Z$, upon choosing $m^Z$ as above, there exists $p^Z > 0$ such that whenever $p \geq p^Z$, $e_{l, (p)}^Z$ is a basis of $A_{(p)}$ and all $\lambda_{l, (p)}^Z$ are nonzero and distinct. Moreover, for some constant $C>0$ one has $$\begin{aligned} &\ \ell_p ( e_{l, (p)}^Z) \geq - C,\ &\ {\mathfrak v}(\lambda_{l, (p)}^Z) \leq C.\end{aligned}$$ Moreover, for all $k \neq l$ $${\mathfrak v}( \lambda_{l, (p)}^Z - \lambda_{k, (p)}^Z) \leq C.$$* *Proof.* Straightforward. ◻ **Proposition 18**. *There exists $C>0$ such that given any sufficiently large $Z$, for all sufficiently large prime $p$, there exist eigenvectors $\varepsilon_{l, (p)}$ of $E_{(p)}$ with corresponding distinct eigenvalues $\lambda_{l, (p)} \in \Lambda_{\overline{\mathbb F}_p}$ such that $$\ell_p ( e_{l, (p)}^Z - \varepsilon_{l, (p)} ) \leq -Z + C$$ and $${\mathfrak v} ( \lambda_{l,(p)}^Z - \lambda_{l, (p)} ) \geq Z - C.$$* *Proof.* In $A_{(0)}$, one has $$(m^Z)^{-1} E_{(0)} ( m^Z e_{l, (0)}) = ((m^Z)^{-1} \lambda_{l, (0)})(m^Z e_{l, (0)}).$$ Using [\[truncation_2\]](#truncation_2){reference-type="eqref" reference="truncation_2"}, it follows that $$\label{approx} \begin{split} &\ \ell_0 \Big( (m^Z)^{-1} E_{(0)} (m^Z e_{l, (0)}^Z) - ((m^Z)^{-1} \lambda_{l, (0)}^Z) (m^Z e_{l, (0)}^Z) \Big)\\ = &\ \ell_0 \Big( E_{(0)} (e_{l, (0)}^Z) - \lambda_{l, (0)}^Z e_{l, (0)}^Z \Big)\\ = &\ \ell_0 \Big( E_{(0)} \big(e_{l, (0)}^Z - e_{l, (0)} \big) + ( \lambda_{l, (0)} - \lambda_{l, (0)}^Z) e_{l, (0)} + \lambda_{l, (0)}^Z (e_{l, (0)} - e_{l, (0)}^Z) \Big) \\ \leq &\ \max \Big\{ \ell_0 ( c_{1, (0)} \ast (e_{l, (0)}^Z - e_{l, (0)})),\ \ell_0 ( (\lambda_{l, (0)} - \lambda_{l, (0)}^Z) e_{l, (0)} ),\ \ell_0 (\lambda_{l, (0)}^Z (e_{l, (0)} - e_{l, (0)}^Z) ) \Big\}\\ \leq &\ \max \Big\{ \ell_0 (c_{1, (0)}) + C - Z,\ \ell_0(e_{l, (0)}) - Z,\ - {\mathfrak v}(\lambda_{l, (0)}) + C - Z \Big\}\\ \leq &\ C- Z. \end{split}$$ Now take a sufficiently large $p$. Notice that the left hand side of [\[approx\]](#approx){reference-type="eqref" reference="approx"} is the valuation of an integral element, hence descends to $\overline{\mathbb F}_p$. Then [\[approx\]](#approx){reference-type="eqref" reference="approx"} implies $$\label{approximate_eigenvalue} \ell_p \left( E_{(p)} (e_{l, (p)}^Z) - \lambda_{l, (p)}^Z e_{l, (p)}^Z \right) \leq C-Z.$$ We would like to correct $e_{l, (p)}^Z$ by adding higher order terms. With respect to the basis $(e_{1, (p)}^Z, \ldots, e_{m, (p)}^Z)$, let the matrix of $E_{(p)}$ be $T_{(p)}$. For a matrix $S = (S_{ij})$ with entries in $\Lambda_{\mathbb K}$, we write $${\mathfrak v}(S) = \min_{i, j} {\mathfrak v}(S_{ij}) \in {\mathbb R}.$$ **Lemma 19**. *One has $${\mathfrak v} \left( T_{(p)} - \left[ \begin{array}{cccc} \lambda_{1, (p)}^Z & 0 & \cdots & 0 \\ 0 & \lambda_{2, (p)}^Z & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & \lambda_{m, (p)}^Z \end{array} \right] \right) \geq Z - C.$$* *Proof.* We can write $$E_{(p)}(e_{l, (p)}^Z) - \lambda_{l, (p)}^Z e_{l, (p)}^Z = \sum_{l=1}^m a_l e_{l, (p)}^Z.$$ By [\[approximate_eigenvalue\]](#approximate_eigenvalue){reference-type="eqref" reference="approximate_eigenvalue"} and the subadditivity property of the function $\ell$ (see Definition [Definition 7](#defn_nonarchimedean_module){reference-type="ref" reference="defn_nonarchimedean_module"}), one has ${\mathfrak v}(a_l) \geq Z- C$ for some appropriate $C$. ◻ For each $l$, consider the following equation for $$\begin{aligned} &\ x = \sum_{k \neq l} x_k e_{k, (p)}^Z,\ &\ \delta \in \Lambda_{\overline{\mathbb F}_p}\end{aligned}$$ which is $$E_{(p)} \big( e_{l, (p)}^Z + x \big) = \big( \lambda_{l,(p)}^Z + \delta \big) \big( e_{l, (p)}^Z + x \big)$$ which is equivalent to $$E_{(p)} \left( \sum_{k \neq l} x_k e_{k, (p)}^Z \right) - \sum_{k\neq l} \lambda_{l, (p)}^Z x_k e_{k, (p)}^Z - \delta e_{l, (p)}^Z = \delta \sum_{k \neq l} x_k e_{k, (p)}^Z + \rho_l.$$ Here $\rho_l$ is the error term with ${\mathfrak v}(\rho_l) \leq -Z + C$ by [\[approximate_eigenvalue\]](#approximate_eigenvalue){reference-type="eqref" reference="approximate_eigenvalue"}. To simplify notations, assume $l = 1$. Then using the basis $e_{1, (p)}^Z, \ldots, e_{m, (p)}^Z$, this equation is equivalent to the linear system $$\left( \left[ \begin{array}{cccc} 1 & 0 & \cdots & 0 \\ & & T_{(p)}' & \end{array}\right] - \left[ \begin{array}{cccc} 0 & 0 & \cdots & 0 \\ 0 & \lambda_{1, (p)}^Z & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & \lambda_{1, (p)}^Z \end{array} \right] \right) \left[ \begin{array}{c} \delta \\ x_2 \\ \vdots \\ x_m \end{array} \right] = Q( \delta, x_2, \ldots, x_m) + \rho_1.$$ Here the left hand side is linear and $Q$ is quadratic. Let the matrix on the left hand side be $F_1$. Lemma [Lemma 19](#lemma_matrix_valuation){reference-type="ref" reference="lemma_matrix_valuation"} implies that $F_1$ is invertible with $${\mathfrak v}(F_1) \leq C$$ where $C$ is independent of $Z$ and $p$. Then one can use an iteration argument to solve the equation term by term. The correction term has valuation at least $Z - C$ for some constant $C$. So the theorem follows. ◻ We continue the proof of (2) of Theorem [Theorem 15](#thm_semisimple_finite_characteristic){reference-type="ref" reference="thm_semisimple_finite_characteristic"}. By the proof of Lemma [Lemma 14](#semisimple_criterion){reference-type="ref" reference="semisimple_criterion"}, each idempotent generator of $A_{(p)}$ is a multiple of $\varepsilon_{l, (p)}$. Indeed, if $$\varepsilon_{l, (p)} \ast \varepsilon_{l, (p)} = \mu_l \varepsilon_{l, (p)}$$ then the corresponding idempotent generator is $$e_{l, (p)} = \mu_l^{-1} \varepsilon_{l, (p)}.$$ So we need to estimate the valuation of $\mu_l$. In characteristic zero one has $$e_{l, (0)} \ast e_{l, (0)} = e_{l, (0)}.$$ Taking truncation at $Z$ one has $$\ell_0 \big( e_{l, (0)}^Z \ast e_{l, (0)}^Z - e_{l, (0)}^Z \big) \leq C-Z.$$ Taking mod $p$ reduction, one obtains $$\ell_p \big( e_{l, (p)}^Z \ast e_{l, (p)}^Z - e_{l, (p)}^Z \big) \leq C-Z.$$ Then $$\begin{gathered} \ell_p \big( \mu_l \varepsilon_{l, (p)} - e_{l, (p)}^Z \big) = \ell_p \big( \varepsilon_{l, (p)} \ast \varepsilon_{l, (p)} - e_{l, (p)}^Z \ast e_{l, (p)}^Z + e_{l, (p)}^Z \ast e_{l, (p)}^Z - e_{l, (p)}^Z \big) \\ \leq \max \Big\{ \ell_p ( \varepsilon_{l, (p)} - e_{l, (p)}^Z ) + \ell_p (\varepsilon_{l, (p)} + e_{l, (p)}^Z), C-Z \Big\} \leq C-Z.\end{gathered}$$ As we have $\ell_p( e_{l, (p)}^Z ) \geq -C$, it follows that $\ell_p( \mu_l \varepsilon_{l, (p)}) = \ell_p (e_{l, (p)}^Z) = \ell_p( \varepsilon_{l, (p)})$. Hence ${\mathfrak v}(\mu_l) = 0$ and hence $$\ell_p( e_{l, (p)}) = \ell_p ( \mu_l^{-1} \varepsilon_{l, (p)}) = \ell_p (\varepsilon_{l, (p)}) + {\mathfrak v}(\mu_l) = \ell_p (e_{l,(p)}^Z) = \ell_0 (e_{l, (0)})$$ which is independent of $p$. This finishes the proof of Theorem [Theorem 15](#thm_semisimple_finite_characteristic){reference-type="ref" reference="thm_semisimple_finite_characteristic"}. ## Floer--Novikov complexes Let $\Gamma \subsetneq {\mathbb R}$ be a proper additive subgroup. **Definition 20** (Floer--Novikov complex). (cf. [@Usher_2008 Definition 1.1])[\[defn_fn\]]{#defn_fn label="defn_fn"} A **${\mathbb Z}_2$-graded Floer--Novikov package** over a commutative unital ring $R$ consists of data $${\mathfrak c} = \Big( P, {\mathcal A}, gr, n_R \Big)$$ where 1. $P$ is a $\Gamma$-torsor with $\underline{\smash{P}}:= P/ \Gamma$ finite. 2. ${\mathcal A}: P \to {\mathbb R}$ is the "action functional" and $gr: P \to {\mathbb Z}_2$ is the "grading." 3. For $p \in P$ and $g \in \Gamma$, one has $$\begin{aligned} &\ {\mathcal A}(g p) = {\mathcal A}(p) - g,\ &\ gr(g p) = gr (p) \end{aligned}$$ 4. $n_R: P \times P \to R$ is a function such that - $n_R (p, q) \neq 0 \Longrightarrow gr(p) = gr(q) + 1,\ {\mathcal A}(p) > {\mathcal A}(q)$; - for all $p \in P$ and $C\in {\mathbb R}$, the set $$\{ q \in P\ |\ n_R (p, q) \neq 0,\ {\mathcal A}(q) \geq C \}$$ is finite; - for any $g \in \Gamma$, we have $n_R (gp, gq) = n_R (p,q)$; - the $\Lambda_R$-linear map $\partial$ defined in [\[eqn:boundary\]](#eqn:boundary){reference-type="eqref" reference="eqn:boundary"} satisfies $\partial^2 = 0$. Given a Floer--Novikov package one can construct the associated Floer chain complex. First, define $$CF_\bullet ({\mathfrak c}) = \Big\{ \sum_{p \in P} a_p p\ |\ a_p \in R,\ \forall C\in {\mathbb R}, \# \{ p \in P \ |\ a_p \neq 0,\ {\mathcal A}(p)\geq C \} < \infty \Big\}$$ which is ${\mathbb Z}_2$-graded. The $\Lambda_R^\Gamma$-module structure is induced from the $\Gamma$-action on $P$. Define the differential $$\partial: CF_\bullet({\mathfrak c}) \to CF_{\bullet -1} ({\mathfrak c})$$ by $$\label{eqn:boundary} \partial \left( \sum_{p \in P} a_p p \right) = \sum_{q\in P} \left( \sum_{ p \in P} a_p n_R (p, q)\right) q.$$ We also define the function $$\label{valuation_function} \ell: CF_\bullet({\mathfrak c}) \to {\mathbb R}\cup \{-\infty\},\ \ell\left( \sum_{p \in P} a_p p \right) = \sup \big\{ {\mathcal A}(p)\ |\ a_p \neq 0 \Big\}.$$ Given a Floer--Novikov package ${\mathfrak c}$ over $R$, if $\iota: R \to \widetilde R$ is a ring map, then one can extend ${\mathfrak c}$ to a Floer--Novikov package ${\mathfrak c}\otimes_R \widetilde R$ by simply defining $n_{\widetilde R}:= \iota \circ n: P \times P \to \widetilde R$ **Proposition 21**. *If $R = {\mathbb K}$ is a field, the triple $(CF_\bullet({\mathfrak c}), \partial, \ell)$ is a Floer-type complex over $\Lambda_{\mathbb K}^\Gamma$ in the sense of [@Usher_Zhang_2016 Definition 4.1].* *Proof.* It follows directly from the definitions of Floer-type complexes. The proof serves rather as a brief clarification about this concept. First, for each $k \in {\mathbb Z}_2$, the pair $(CF_k({\mathfrak c}), \ell|_{CF_k({\mathfrak c})})$ is a non-Archimedean normed vector space over $\Lambda_{\mathbb K}^\Gamma$ (see [@Usher_Zhang_2016 Definition 2.2]. In addition it is an orthogonalizable $\Lambda_{\mathbb K}^\Gamma$-space (see [@Usher_Zhang_2016 Definition 2.7]. The last requirement for being a Floer-type complex is the inequality $$\ell(\partial(x)) \leq \ell(x)\ \forall x \in CF_\bullet({\mathfrak c}),$$ which is a consequence of the property of the function $n_{\mathbb K}$ in the data ${\mathfrak c}$. ◻ ### Spectral invariants Following Usher [@Usher_2008], one can also define spectral invariants in an abstract way. First, define the "energy filtration" on the complex $CF_\bullet({\mathfrak c})$: for each $\tau \in {\mathbb R}$, define $$CF_\bullet^{\leq \tau}({\mathfrak c}):= \left\{ \sum_{p \in P} a_p p \in CF_\bullet ({\mathfrak c})\ |\ a_p \neq 0 \Longrightarrow {\mathcal A}(p) \leq \tau \right\}.$$ Then since the differential decreases the action, it is a subcomplex with homology $$HF_\bullet^{\leq \tau}({\mathfrak c})$$ and natural maps when $\tau \leq \kappa$ $$\label{eqn:iota-map} \iota^{\tau, \kappa}: HF_\bullet^{\leq \tau}({\mathfrak c}) \to HF_\bullet^{\leq \kappa}({\mathfrak c}).$$ For $\alpha \in HF_\bullet ({\mathfrak c})$, define $$\rho (\alpha):= \inf \Big\{ \tau \in {\mathbb R}\ |\ \alpha \in {\rm Im} \left( \iota^\tau: HF_\bullet^{\leq \tau}({\mathfrak c}) \to HF_\bullet ({\mathfrak c}) \right) \Big\}\in {\mathbb R} \cup \{-\infty\}$$ **Theorem 22**. *[@Usher_2008 Theorem 1.3, 1.4][\[Usher_thm\]]{#Usher_thm label="Usher_thm"} Given a Floer--Novikov package ${\mathfrak c}$ (over a Noetherian ring $R$) and $\alpha \in HF({\mathfrak c}) \setminus \{0\}$, $\rho (\alpha) > -\infty$ and $\alpha \in {\rm Im} (\iota_{\rho(\alpha)})$.* ### Boundary depth **Definition 23**. [@Usher_boundary_depth][\[defn_boundary_depth_1\]]{#defn_boundary_depth_1 label="defn_boundary_depth_1"} Let ${\mathfrak c}$ be a Floer--Novikov package and let $CF^{\leq \lambda}({\mathfrak c})$ be the associated filtered Floer--Novikov complex over $\Lambda_{\mathbb K}^\Gamma$. Then the **boundary depth** of the filtered complex is the infimum of $\beta > 0$ such that for all $\lambda \in {\mathbb R}$ $$CF^{\leq \lambda} ({\mathfrak c}) \cap {\rm Im} \partial \subset \partial( CF^{\leq \lambda + \beta} ({\mathfrak c}) ).$$ **Theorem 24**. *[@Usher_2008 Theorem 1.3] Given a Floer--Novikov package ${\mathfrak c}$, the boundary depth of the associated Floer--Novikov complex is finite.* ### Quasiequivalence distance We rephrase the notion of quasiequivalences between Floer--Novikov complexes, which was originally introduced in [@Usher_Zhang_2016] for the more general situation of Floer-type complexes. **Definition 25**. (cf. [@Usher_Zhang_2016 Definition 1.3]) [\[defn_quasiequivalence\]]{#defn_quasiequivalence label="defn_quasiequivalence"} Let $(CF_\bullet({\mathfrak c}_i), \partial_i)$, $i = 1, 2$, be two Floer--Novikov complexes associated to Floer--Novikov data ${\mathfrak c}_i$ over a field ${\mathbb K}$. Let $\ell_i$ be the valuation function on the two complexes defined by [\[valuation_function\]](#valuation_function){reference-type="eqref" reference="valuation_function"}. Let $\delta \geq 0$. A **$\delta$-quasiequivalence** between $CF_\bullet({\mathfrak c}_1)$ and $CF_\bullet({\mathfrak c}_2)$ is a quadruple $(\Phi, \Psi, K_C, K_D)$ where 1. $\Phi: CF_\bullet({\mathfrak c}_1) \to CF_\bullet({\mathfrak c}_2)$ and $\Psi: CF_\bullet({\mathfrak c}_2) \to CF_\bullet({\mathfrak c}_1)$ are chain maps with $$\begin{aligned} &\ \ell_2(\Phi(x_1)) \leq \ell_1(x_1) +\delta,\ &\ \ell_1(\Psi(x_2)) \leq \ell_2(x_2) + \delta \end{aligned}$$ for all $x_1 \in CF_\bullet({\mathfrak c}_1)$ and $x_2 \in CF_\bullet({\mathfrak c}_2)$. 2. $K_i: CF_\bullet({\mathfrak c}_i) \to CF_{\bullet+1}({\mathfrak c}_i)$, $i = 1, 2$, obey the homotopy equations $$\begin{aligned} &\ \Psi \circ \Phi - {\rm Id}_{CF_\bullet({\mathfrak c}_1)} = \partial_1 K_1 + K_1 \partial_1,\ &\ \Phi \circ \Psi - {\rm Id}_{CF_\bullet({\mathfrak c}_2)} = \partial_2 K_2 + K_2 \partial_2 \end{aligned}$$ and for all $x_i \in CF_\bullet({\mathfrak c}_i)$, $i = 1, 2$, one has $$\ell_i(K_i(x_i)) \leq \ell_i(x_i) + 2\delta.$$ The **quasiequivalence distance** between $CF_\bullet({\mathfrak c}_1)$ and $CF_\bullet({\mathfrak c}_2)$, denoted by $d_Q(CF_\bullet({\mathfrak c}_1), CF_\bullet({\mathfrak c}_2))$, is the infimum of $\delta$ such that there exists a $\delta$-quasiequivalence between them. ## Persistence modules and stability of boundary depth {#subsection_persistence} **Definition 26**. Let ${\mathbb K}$ be a field. 1. A **persistence module** ${\bm V}$ is a family of ${\mathbb K}$-vector spaces $${\bm V} = (V^s)_{s \in {\mathbb R}}$$ together with linear maps (called the **structural maps** of ${\bm V}$) $$\iota^{s, t}:= \iota^{s, t}_{\bm V}: V^s \to V^t\ \forall s \leq t$$ such that $\iota^{s, s} = {\rm Id}_{V^s}$ for all $s$ and $\iota^{t, r} \circ \iota^{s, t} = \iota^{s, r}$ for all $s \leq t \leq r$. 2. Let ${\bm V}$ be a persistence module and $\delta\in {\mathbb R}$. The $\delta$-shift of ${\bm V}$ is the persistence module ${\bm V}[\delta]$ with $V[\delta]^s = V^{s + \delta}$ and $\iota[\delta]^{s, t} = \iota^{s + \delta, t + \delta}$. 3. Let ${\bm V}$ and ${\bm W}$ be two persistence modules. A morphism from ${\bm V}$ to ${\bm W}$ is a collection of linear maps ${\bm f} = (f^s: V^s \to W^s)_{s \in{\mathbb R}}$ such that for all $s \leq t$ the following diagram commutes. $$\xymatrix{ V^s \ar[r]^{f^s} \ar[d]_{\iota^{s, t}_{\bm V}} & W^s \ar[d]^{\iota^{s, t}_{\bm W}} \\ V^t \ar[r]_{f^t} & W^t }$$ 4. The direct sum of persistence modules is defined in a natural way. *Remark 27*. The notion of persistence modules first appeared in topological data analysis and then was adopted to symplectic topology (see [@Polterovich_Shelukhin_2016; @Usher_Zhang_2016]). In many papers, notably in the symplectically aspherical or monotone setting such as [@Polterovich_Shelukhin_2016; @Shelukhin_2022],the Floer persistence modules are usually finite-dimensional and hence generate *barcodes* as in the original situation of topological data analysis. However, in more general situations such as [@Usher_Zhang_2016] where the Floer persistence modules are infinite-dimensional, the notion of barcodes becomes more complicated. See Subsection [3.5](#subsection_barcodes){reference-type="ref" reference="subsection_barcodes"} for more details. ### Interleaving distance **Definition 28**. Let $\delta \geq 0$. Two persistence modules ${\bm V}$, ${\bm W}$ are called **$\delta$-interleaved** if there are ${\mathbb K}$-linear maps $$\begin{aligned} &\ f^s: V^s \to W^{s+\delta},\ &\ g^s: W^s \to V^{s+\delta}\end{aligned}$$ for all $s \in {\mathbb R}$ such that for all $s \leq t$ the following diagram commutes. $$\xymatrix{ V^{s-\delta} \ar[r]_{f^{s-\delta}} \ar[d] \ar@/^2.0pc/[rr]^{\iota^{s-\delta, s+\delta}_{\bm V}} & W^s \ar[r]_{g^s} \ar[d] & V^{s+\delta} \ar[r]_{f^{s + \delta}} \ar[d] & W^{s + 2\delta} \ar[d]\\ V^{t-\delta} \ar[r]_{f^{t-\delta}} & W^t \ar[r]_{g^{t}} \ar@/_2.0pc/[rr]_{\iota^{t, t+2\delta}_{\bm W}} & V^{t+\delta} \ar[r]_{f^{t+\delta}} & W^{ t+ 2\delta} }$$ Here all vertical arrows are the structural maps in the persistence modules. Define the **interleaving distance** between ${\bm V}$ and ${\bm W}$ to be the infimum of all $\delta\geq 0$ such that ${\bm V}$ and ${\bm W}$ are $\delta$-interleaved; if such $\delta$ does not exist, define the interleaving distance to be $+\infty$. Here ends this definition. ### Boundary depth of persistence modules and stability We observe that one can generalize the notion of boundary depth to persistence modules. **Definition 29**. Let ${\bm V}$ be a persistence module over ${\mathbb K}$. The **boundary depth** of ${\bm V}$, denoted by $\beta({\bm V})$, is the infimum of $\beta>0$ such that for all $s \in {\mathbb R}$, $x \in V^s$, if $\iota^{s, t}(x) = 0$ for some $t > s$, then $\iota^{s, s+\beta}(x) = 0$. As we allow persistent modules to be infinite-dimensional, we reprove the stability result of boundary depth. **Proposition 30**. *Suppose ${\bm V}$, ${\bm W}$ are $\delta$-interleaved persistence modules. Suppose ${\bm V}$ has finite boundary depth. Then ${\bm W}$ has finite boundary depth and $$\beta({\bm W}) \leq \beta({\bm V}) + 2\delta.$$* *Proof.* Suppose on the contrary that $\beta({\bm W}) \geq \beta({\bm V}) + 2\delta + 2\epsilon$ for some $\epsilon>0$. Then there exist $s \in {\mathbb R}$ and $x \in W_s$ such that $\iota^{s, s + \beta({\bm V}) + 2\delta + \epsilon}(x) \neq 0$. Then by the definition of $\delta$-interleaving, one has $y: = f^{s, s+ \delta}(x) \neq 0$ and $$\iota^{s+\delta, s+\delta + \beta({\bm V}) + \epsilon}(y) \neq 0$$ but $y$ cannot survive eventually. This contradicts the definition of $\beta({\bm V})$. ◻ ### Persistence modules associated to filtered Floer--Novikov complexes {#subsubsec:floer-persistent} Fix a field ${\mathbb K}$. Let ${\mathfrak c}$ be a Floer--Novikov package (see Definition [\[defn_fn\]](#defn_fn){reference-type="ref" reference="defn_fn"}) and $CF_\bullet({\mathfrak c})$ be the associated filtered Floer--Novikov complex. Then the collection of homology groups $$V^s({\mathfrak c}):= HF_\bullet^{\leq s}({\mathfrak c}; \Lambda_{\mathbb K}^\Gamma)$$ together with the natural maps $\iota^{s, t}$ (cf. Equation [\[eqn:iota-map\]](#eqn:iota-map){reference-type="eqref" reference="eqn:iota-map"}) is a persistence module over ${\mathbb K}$, denoted by ${\bm V}({\mathfrak c})$. It is easy to derive from definitions the following stability results of persistence modules coming from Floer--Novikov complexes. **Proposition 31**. *Let $CF_\bullet({\mathfrak c}_i)$, $i = 1, 2$ be two Floer--Novikov complexes over a field ${\mathbb K}$ and ${\bm V}({\mathfrak c}_i)$ be the associated persistence module. Then the interleaving distance between ${\bm V}({\mathfrak c}_1)$ and ${\bm V}({\mathfrak c}_2)$ is no greater than the quasiequivalence distance between $CF_\bullet({\mathfrak c}_1)$ and $CF_\bullet({\mathfrak c}_2)$.* Moreover, the two notions of boundary depths (Definition [\[defn_boundary_depth_1\]](#defn_boundary_depth_1){reference-type="ref" reference="defn_boundary_depth_1"} and Definition [Definition 29](#defn_boundary_depth_2){reference-type="ref" reference="defn_boundary_depth_2"}) agree. **Proposition 32**. *Let ${\mathfrak c}$ be a Floer--Novikov package over ${\mathbb K}$. Then the boundary depth of the filtered Floer--Novikov complex $CF_\bullet({\mathfrak c})$ and the boundary depth of the persistence module ${\bm V}({\mathfrak c})$ coincide.* *Proof.* Let $\beta_1$ be the boundary depth of $CF_\bullet({\mathfrak c})$ and $\beta_2$ be the boundary depth of ${\bm V}({\mathfrak c})$. Suppose $[x]\in HF^{\leq s}_\bullet({\mathfrak c})$ which does not survive eventually. Let $x\in CF^{\leq s}_\bullet({\mathfrak c})$ be a representative. Then $x$ is exact. Then by Definition [\[defn_boundary_depth_1\]](#defn_boundary_depth_1){reference-type="ref" reference="defn_boundary_depth_1"}, for all $\epsilon>0$, one has $$x \in \partial ( CF^{\leq s+\beta_1 + \epsilon}_\bullet({\mathfrak c})).$$ Hence $\iota^{s, s+\beta_1 + \epsilon}([x]) = 0$. As $\epsilon$ is arbitrary, this implies that $\beta_2 \leq \beta_1$. On the other hand, for all $s \in {\mathbb R}$ and all exact $x \in CF_\bullet^{\leq s} ({\mathfrak c})$, the class $[x]\in HF_\bullet^{\leq s}({\mathfrak c})$ does not survive eventually. Then by Definition [Definition 29](#defn_boundary_depth_2){reference-type="ref" reference="defn_boundary_depth_2"}, for any $\epsilon>0$, one has $\iota^{s, s+\beta_2 + \epsilon}([x]) = 0$. This implies that $$x \in \partial ( CF^{\leq s + \beta_2 + \epsilon}_\bullet({\mathfrak c})).$$ It follows that $\beta_1 \leq \beta_2$. Hence $\beta_1 = \beta_2$. ◻ ## Barcodes and reduced barcodes {#subsection_barcodes} In the symplectically aspherical or monotone case, the notion of barcodes is the same as the one used in topological data analysis. In more general situations, Usher--Zhang [@Usher_Zhang_2016] gave a modification for any Floer-type complexes (in particular Floer--Novikov complexes) over any Novikov field $\Lambda_{\mathbb K}^\Gamma$. **Definition 33**. (cf. [@Usher_Zhang_2016 Definition 8.13, 8.14]) Fix a finitely generated subgroup $\Gamma \subsetneq {\mathbb R}$. 1. A **barcode** is a finite multiset $\tilde {\mathcal B}$ of elements of $({\mathbb R}/\Gamma) \times (0, +\infty]$. A member of $\tilde {\mathcal B}$, which is usually called a **bar**, is denoted by $([a], L)$ where $[a]\in {\mathbb R}/\Gamma$ and $L \in (0, +\infty]$. 2. A **reduced barcode** is a finite multiset ${\mathcal B}$ of elements of $(0, +\infty]$. Although ${\mathcal B}$ is not a set in general and a member $L \in {\mathcal B}$ may appear multiple times, we still use the same notations as if $L$ is an element of a set ${\mathcal B}$, such as $L \in {\mathcal B}$, without confusion. Let ${\mathcal B}_{\rm finite}\subset {\mathcal B}$ denote the submultiset of finite bars, i.e., those with $L < +\infty$. Let ${\mathcal B}_{\rm finite} \subset {\mathcal B}$ be the submultiset of finite bars. Notice that a barcode $\tilde {\mathcal B}$ induces a reduced barcode ${\mathcal B}$ by forgetting the first coordinates. 3. The **total bar length** of a reduced barcode ${\mathcal B}$ is $$\tau ({\mathcal B} ):= \sum_{L_i \in {\mathcal B}_{\rm finite}} L_i.$$ 4. The **reduced bottleneck distance** between two reduced barcodes ${\mathcal B}$ and ${\mathcal B}'$, denoted by $d_B({\mathcal B}, {\mathcal B}')$, is the infimum of $\delta>0$ such that, after removing certain submultisets ${\mathcal B}_{\rm short}\subset {\mathcal B}$ and ${\mathcal B}_{\rm short}' \subset {\mathcal B}'$ whose members all have length at most $2\delta$, there is a bijection between ${\mathcal B}\setminus {\mathcal B}_{\rm short}$ and ${\mathcal B}' \setminus {\mathcal B}_{\rm short}'$ such that the differences of the corresponding bar lengths are all bounded by $\delta$. The bottleneck distance is symmetric and satisfies the triangle inequality. It is not a metric in the usual sense as it may take infinite value. Indeed, $d_B({\mathcal B}, {\mathcal B}')< \infty$ if and only if ${\mathcal B}$ and ${\mathcal B}'$ has the same number of infinite bars. **Proposition 34**. *(cf. [@Roux_Seyfaddini_Viterbo_2021 Proposition 20]) For any $k \geq 0$, the completion of the set of reduced barcodes having $k$ infinite bars is the set of possibly infinite reduced barcodes (with $k$ infinite bars) such that for all $\epsilon>0$, the number of finite bars with length greater than $\epsilon$ is finite.* ### Barcodes associated to Floer--Novikov complexes Usher--Zhang [@Usher_Zhang_2016] defined for each ${\mathbb Z}_2$-graded Floer-type complexes over $\Lambda_{\mathbb K}^\Gamma$ and each $k \in {\mathbb Z}_2$ the associated barcodes (which allows bars of length zero). As Floer--Novikov complexes are all Floer-type complexes, one has an associated barcode (and hence a reduced barcode). Let the reduced barcode associated to a Floer--Novikov complex $CF_\bullet({\mathfrak c})$ be ${\mathcal B}({\mathfrak c})$. As the differential strictly decreases the action, there are no bars of length zero (which was allowed in the abstract setting of [@Usher_Zhang_2016]). We do not recall the detail of the definition here. **Proposition 35**. *Let ${\bm V}({\mathfrak c})$ be the persistence module induced from a filtered Floer--Novikov complex $CF_\bullet({\mathfrak c})$ over $\Lambda_{\mathbb K}^\Gamma$. Then the boundary depth of ${\bm V}({\mathfrak c})$ (see Definition [\[defn_boundary_depth_1\]](#defn_boundary_depth_1){reference-type="ref" reference="defn_boundary_depth_1"} and Definition [Definition 29](#defn_boundary_depth_2){reference-type="ref" reference="defn_boundary_depth_2"}) coincides with the length of the longest finite bar in ${\mathcal B}({\mathfrak c})$. In particular, the boundary depth is zero if and only if ${\mathcal B}({\mathfrak c})$ has no finite bar.* *Proof.* It follows from the definitions of boundary depth and barcodes (via singular value decompositions, see [@Usher_Zhang_2016]). The details are left to the reader. ◻ ### Stability of barcodes **Theorem 36**. *[@Usher_Zhang_2016 Theorem 8.17][\[thm:bottle-neck-Hofer\]]{#thm:bottle-neck-Hofer label="thm:bottle-neck-Hofer"} Let $(CF_\bullet({\mathfrak c}_1), \partial_1, \ell_1)$ and $(CF_\bullet({\mathfrak c}_2), \partial_2, \ell_2)$ be two Floer--Novikov complexes associated to Floer--Novikov data ${\mathfrak c}_1$, ${\mathfrak c}_2$ over a field ${\mathbb K}$. Suppose the quasiequivalence distance between $CF_\bullet({\mathfrak c}_1)$ and $CF_\bullet({\mathfrak c}_2)$ is finite. Then $$d_B ({\mathcal B}({\mathfrak c}_1), {\mathcal B} ({\mathfrak c}_2)) \leq 2 d_Q ( CF_\bullet({\mathfrak c}_1), CF_\bullet({\mathfrak c}_2)).$$* ## $A_\infty$ algebras and Hochschild cohomology Let ${\mathbb K}$ be a field of characteristic zero. We recall the notion of ${\mathbb Z}_2$-graded $A_\infty$ algebras over the Novikov field $\Lambda_{\mathbb K}$. **Definition 37** (Curved $A_\infty$ algebra). 1. A ${\mathbb Z}_2$-graded **curved $A_\infty$ algebra** over $\Lambda_{\mathbb K}$ consists of a ${\mathbb Z}_2$-graded $\Lambda_{\mathbb K}$-vector space ${\mathcal A}$ (the degree of a homogeneous element $a$ is denoted by $|a|$) and for all positive integers $k \geq 0$ **higher composition maps** $$m_k: {\mathcal A}^{\otimes k} \to {\mathcal A}\ ({\rm where}\ m_0: \Lambda_{\mathbb K} \to A)$$ (which are $\Lambda_{\mathbb K}$-linear and have degree $k\ {\rm mod}\ 2$). The higher composition maps need to satisfy the following $A_\infty$ composition law: for all $k \geq 1$ and $a_k, \ldots, a_1 \in {\mathcal A}$,[^4] $$\sum_{i=0}^k \sum_{j=0}^{k-i} (-1)^{\maltese_1^j} m_{k-i+1} \left( a_k, \ldots, a_{i+j+1}, m_i ( a_{i+j}, \ldots, a_{j+1}), a_j, \ldots, a_1 \right) = 0$$ where the symbol $\maltese_a^b$ for all $a < b$ is defined as $$\label{maltese} \maltese_a^b = \sum_{a \leq i \leq b} \| a_i \| \quad {\rm where}\ \| a_i \| = |a_i| + 1.$$ 2. The **curvature** of a curved $A_\infty$ algebra is the element $$m_0(1) \in {\mathcal A}.$$ If $m_0 = 0$, then we say that the $A_\infty$ algebra is **flat**. 3. Given a (curved or flat) $A_\infty$ algebra ${\mathcal A}$, a **cohomological unit** is an even element $e\in {\mathcal A}$ such that $m_1(e) = 0$ and that for all homogeneous $x\in {\mathcal A}$ $$(-1)^{|x|} m_2(e, x) = m_2(x, e) = x.$$ $e$ is called a **strict unit** if in addition $$m_k(\ldots, e, \ldots) = 0\ \forall k \geq 3.$$ In these two cases we call $({\mathcal A}, e)$ a cohomologically unital (resp. strictly unital) $A_\infty$ algebra. 4. When ${\mathcal A}$ is flat, $A_\infty$ composition law implies that $m_1\circ m_1 = 0$. The **cohomology algebra** of $A$, denoted by $H^\bullet({\mathcal A})$, is the ${\mathbb Z}_2$-graded associative $\Lambda_{\mathbb K}$ algebra whose underlying space is $H^\bullet({\mathcal A}) = {\rm ker} m_1/ {\rm Im} m_1$ and whose multiplication is induced from $m_2$. Because of bubbling of holomorphic disks, $A_\infty$ algebras associated to a Lagrangian brane is generally curved. There is a way to turn certain curved $A_\infty$ algebras to flat ones. **Definition 38**. Let $({\mathcal A}, e)$ be a strictly unital $A_\infty$ algebra. A **weakly bounding cochain** of $({\mathcal A}, e)$ is an odd element $b \in {\mathcal A}^{\rm odd}$ such that $$m(b):= \sum_{k \geq 0} m_k( b, \ldots, b) = W(b) e\ {\rm where}\ W(b) \in \Lambda_{\mathbb K}.$$ Suppose $b$ is a weakly bounding cochain of $({\mathcal A}, e)$. Then define ${\mathcal A}^\flat$ (which depends on the weakly bounding cochain $b$) to be the flat $A_\infty$ algebra whose underlying space is the same as ${\mathcal A}$ and whose composition maps $m_k^\flat$ is defined by $$m_k^\flat( x_k, \ldots, x_1) := \sum_{l_0, \ldots, l_k \geq 0} m_{k + l_0 + \cdots + l_k} (\underbrace{b, \ldots, b}_{l_k}, x_k, \cdots, x_1, \underbrace{b,\ldots, b}_{l_0} ).$$ **Lemma 39**. *${\mathcal A}^\flat$ is a flat $A_\infty$ algebra. 0◻* ### Hochschild cohomology for associative algebras Let $A$ be a ${\mathbb Z}_2$-graded associative algebra over ${\mathbb K}$. Hochschild cohomology ${\it HH}^\bullet(A, M)$ can be defined for all ${\mathbb Z}_2$-graded bimodules $M$ of $A$. Here we only consider the case when $M = A$. The Hochschild cochain complex (with coefficients in $A$ itself) is defined by $${\it CC}^{\bullet, n}(A):= {\it CC}^{\bullet, n}(A, A):= {\rm Hom}_{\Lambda_{\mathbb K}}^\bullet (A^{\otimes n}, A[n]).$$ Here the bullet is the ${\mathbb Z}_2$-grading on linear maps and $A[n]$ is the ${\mathbb Z}_2$-graded vector space $A$ with the ${\mathbb Z}_2$-grading shifted by $n$ (modulo $2$). Denote the ${\mathbb Z}_2$-degree of a homogeneous element $\phi \in {\it CC}^{\bullet, \bullet}(A)$ by $|\phi| \in {\mathbb Z}_2$ and the *reduced* degree by $$\| \phi \|:= |\phi| + 1 \in {\mathbb Z}_2.$$ A Hochschild cochain is represented by a sequence $\tau = (\tau_n)_{n \geq 0}$ of such multi-linear maps. The differential $\delta_{{\it CC}}$, which raises the length grading $n$ by $1$, is defined by $$\begin{gathered} \label{dcc} (\delta_{{\it CC}}(\tau))(x_{n+1}, \ldots, x_1) = x_{n+1} \tau_n (x_n, \ldots, x_1) + (-1)^{\| \tau \| \| x_1 \|} \tau_n(x_{n+1}, \ldots, x_2)x_1\\ - \sum_{0 \leq i < n} (-1)^{\| \tau \| + \maltese_1^i } \tau_n(x_{n+1}, \ldots, x_{i+2} x_{i+1}, x_i, \ldots, x_1).\end{gathered}$$ The cohomology defined by $\delta_{{\it CC}}$ is called the **Hochschild cohomology** of $A$ (with coefficients in $A$). As the simplest example, via a straightforward calculation one obtains (for $A = {\mathbb K}$ trivially graded) $${\it HH}^{\bullet, n} ({\mathbb K}) = \left\{ \begin{array}{cc} {\mathbb K},\ &\ n = 0 \text{ and } n \text{ even},\\ 0,\ &\ \text{ otherwise}, \end{array}\right.$$ where the superscript $n$ comes from the length filtration of Hochschild cochains. *Remark 40*. The formula [\[dcc\]](#dcc){reference-type="eqref" reference="dcc"} differs from the usual version of Hochschild differential, see for example [@Loday_cyclic (1.5.1.1)]. Indeed, suppose $A$ is ungraded, i.e. all elements are even. Then the ${\mathbb Z}_2$-grading of a length $n$ cochain is $n$ mod $2$. In this case [\[dcc\]](#dcc){reference-type="eqref" reference="dcc"} reduces to $$\begin{gathered} (\delta_{{\it CC}}(\tau))(x_{n+1}, \ldots, x_1) = x_{n+1} \tau_n(x_n, \ldots, x_1) + (-1)^{n+1} \tau_n(x_{n+1}, \ldots, x_2) x_1 \\ + (-1)^{n+i} \tau_n(x_{n+1}, \ldots, x_{i+2} x_{i+1}, x_i, \ldots, x_1).\end{gathered}$$ If we replace $A$ by the opposite algebra $A^{\rm op}$ (i.e. the same set with multiplication reversed), and identify a length $n$ Hochschild cochain $\tau$ on $A$ with $\tau^{\rm op}$ on $A^{\rm op}$ defined by $\tau^{\rm op}(x_1, \ldots, x_n) =\tau(x_n, \ldots, x_1)$. Then the above formula differ from the standard Hochschild differential on $\tau^{\rm op}$ up to a sign $(-1)^{n+1}$. ### Hochschild cohomology for $A_\infty$ algebras Now let ${\mathcal A}^\flat$ be a flat ${\mathcal A}_\infty$ algebra. Define the length $n$-part of Hochschild cochain complex of $A^\flat$ to be $${\it CC}^{\bullet, n} ({\mathcal A}^\flat) = {\it CC}^{\bullet, n} ({\mathcal A}^\flat, {\mathcal A}^\flat) = {\rm Hom}_{\Lambda_{\mathbb K}}^\bullet ( ({\mathcal A}^\flat)^{\otimes n}, {\mathcal A}^\flat[n]).$$ Here $\bullet$ denotes the ${\mathbb Z}_2$-grading and ${\mathcal A}^\flat[n]$ denote the super vector space ${\mathcal A}^\flat$ with grading shifted by $n$ (mod $2$). On the Hochschild cochain complex there is the **Gerstenhaber product** (which is graded with respect to the reduced grading $\| \cdot \|$) defined by $$(\phi \circ \psi)(x_s, \ldots, x_1) = \sum_{i+j+k = s} (-1)^{\| \psi \| \cdot \maltese_1^i} \phi(x_s, \ldots, \psi(x_{i+j}, \cdots, ), x_i, \ldots, x_1)$$ as well as the **Gerstenhaber superbracket** $$:= \phi \circ \psi - (-1)^{\| \phi \| \cdot \| \psi \| } \psi \circ \phi.$$ Then the $A_\infty$-structure on ${\mathcal A}^\flat$ is equivalent to an even Hochschild cochain $m^\flat$ with $m^\flat_0 = 0$ with the $A_\infty$ relation being equivalent to $$= 2 m^\flat \circ m^\flat = 0.$$ We define the Hochschild differential $\delta_{\it CC}$ on $${\it CC}^\bullet({\mathcal A}^\flat) = \prod_{n \geq 0} {\it CC}^{\bullet, n} ({\mathcal A}^\flat)$$ by the formula $$\delta_{\it CC}(\phi):= [ m^\flat, \phi].$$ Notice that if $m^\flat_k \neq 0$ only when $k = 2$, ${\mathcal A}^\flat$ is a ${\mathbb Z}_2$-graded associative algebra with the Hochschild differential reduces to the differential [\[dcc\]](#dcc){reference-type="eqref" reference="dcc"}. The Hochschild cohomology of ${\mathcal A}^\flat$ is defined by $${\it HH}^\bullet({\mathcal A}^\flat ):= {\rm ker} \delta_{\it CC}/ {\rm im} \delta_{\it CC}.$$ On ${\it CC}^\bullet({\mathcal A}^\flat )$ there is also an $A_\infty$ structure whose composition maps start with $\delta_{\it CC}$. We only need the 2-fold composition map, i.e., the Yoneda product. **Definition 41**. The **Yoneda product** on ${\it CC}^\bullet({\mathcal A}^\flat)$, denoted by $\star$, is defined by $$\begin{gathered} \label{Yoneda} (\phi \star \psi) (a_k, \ldots, a_1) \\ = \sum (-1)^\clubsuit m_k^\flat \left( a_k, \ldots, \phi_r (a_{i+r}, \ldots, a_{i+1}), \cdots, \psi_s (a_{j+s}, \ldots, a_{j+1}), \ldots, a_1 \right)\end{gathered}$$ where the sum is taken over all $i, j, r, l$ such that each summand makes sense. The sign is defined by (see [\[maltese\]](#maltese){reference-type="eqref" reference="maltese"} for the definition of $\maltese$) $$\label{clubsuit} \clubsuit:= \|\phi\| \cdot \left( \maltese_1^i + |\psi| \right) + \| \psi\| \cdot \maltese_1^j.$$ As there are many inconsistent conventions in literature (see for example [@Ganatra_thesis; @Sheridan_ncHodge; @Mescher_ainfinity]), we verify that the Yoneda product indeed reduces a product on the cohomology. As recalled from above, the Yoneda product can be extended to define an $A_{\infty}$ structure on ${\it CC}^\bullet({\mathcal A}^\flat)$, so the induced product on the Hochschild cohomology ${\it HH}^\bullet({\mathcal A}^\flat )$ is associative. **Proposition 42**. *The map $\star: {\it CC}^\bullet({\mathcal A}^\flat) \otimes {\it CC}^\bullet({\mathcal A}^\flat) \to {\it CC}^\bullet({\mathcal A}^\flat)$ is a cochain map of even degree.* *Proof.* The fact that $\star$ has even degree follows directly from the definition of the ${\mathbb Z}_2$-grading on the Hochschild cochain complex and the fact that $m^\flat$ is even. We verify that $\star$ is a chain map. To save notations, we assume that both $\phi$ and $\psi$ are odd; the general situation can be verified similarly. Then in this case, we need to prove that $$\label{chain_verification} \begin{split} &\ \big( m^\flat \circ ( \phi \star \psi) \big) (a_k, \ldots, a_1) + \big( (\phi \star \psi) \circ m^\flat\big) (a_k, \ldots, a_1) \\ =&\ \big( (m^\flat \circ \phi - \phi \circ m^\flat) \ast \psi\big) (a_k, \ldots, a_1) \\ &\ + (-1)^{|\phi|} \big( \phi \star ( m^\flat \circ \psi - \psi \circ m^\flat) \big) (a_k, \ldots, a_1). \end{split}$$ First we compute the left hand side, in which the involved sign $\clubsuit$ (see [\[clubsuit\]](#clubsuit){reference-type="eqref" reference="clubsuit"}) in $\phi \star \psi$ always vanishes and in which $\| \phi \star \psi \| = 1$. Then $$\begin{split} &\ \big( m^\flat \circ (\phi \star \psi)\big)( a_k, \ldots, a_1) \\ = &\ \sum (-1)^{\maltese_1^i} m^\flat \big( a_k, \ldots, (\phi \star \psi) (a_{i+r}, \cdots, a_{i+1}), a_i, \ldots, a_1 \big)\\ = &\ \sum (-1)^{\maltese_1^i} m^\flat\big( a_k, \ldots, m^\flat ( \cdots, \phi ( \cdots), \cdots, \psi (\cdots), a_i, \ldots, a_1 \big). \end{split}$$ Notice that this is a sum of 4-fold compositions using two $m^\flat$ with $\phi$ and $\psi$ such that $\phi$ and $\psi$ are contained in the interior $m^\flat$, which can be abbreviated as $$m^\flat( -, m^\flat( -, \phi(-), -, \psi(-), -), - ).$$ We will make similar abbreviations in the following computations; moreover, we always assume that after the interior $m^\flat$ the inputs start with $a_i$. Then similar to above (remember $\clubsuit = 0$), one has $$\begin{split} &\ \big( (\phi \star \psi) \circ m^\flat \big) (a_k, \ldots, a_1) \\ = &\ \sum (-1)^{\maltese_1^i} ( \phi \star \psi) (-, m^\flat(-), -)\\ = &\ \sum (-1)^{\maltese_1^i} \left( m^\flat \big(-, \phi(-), -, \psi(-), -, m^\flat(-), -\big) \right. \\ &\ \left. + m^\flat \big( -, \phi(-), -, m^\flat(-), -, \psi(-), - \big) + m^\flat\big( -, m^\flat(-),-, \phi(-), -, \psi(-), -\big) \right)\\ &\ + \sum(-1)^{\maltese_1^i} \left( m^\flat \big( -, \phi(-, m^\flat(-), -), -, \psi(-), - \big) + m^\flat\big( -, \phi(-), -, \psi(-, m^\flat(-), -), -\big) \right). \end{split}$$ On the right hand side, the first part is a sum of 4-fold compositions such that neither $\phi$ nor $\psi$ are contained in the interior $m^\flat$ and the second part is a sum of 4-fold compositions such that either $\phi$ or $\psi$ contain the interior $m^\flat$. Now we can observe that the chain map property should be a consequence of the $A_\infty$ relation $m^\flat \circ m^\flat = 0$. Notice that to match the signs of the $A_\infty$ relation, we see $$\| \phi(a_{i+r}, \cdots, a_{i+1}) \| = |\phi| + |a_{i+r}| + \cdots + |a_{i+1}| + r + 1 = \| \phi\| + \maltese_{i+1}^{i+r} = \maltese_{i+1}^{i+r}$$ as $\phi$ is assumed to be odd. Now compute $(m^\flat \circ \phi) \star \psi$. Notice that in this computation, because $m^\flat \circ \phi$ is even and $\psi$ is odd, the sign $\clubsuit$ (see [\[clubsuit\]](#clubsuit){reference-type="eqref" reference="clubsuit"}) is $$\clubsuit = \| m^\flat \circ \phi \| \cdot \big( \maltese_1^i + | \psi| \big) + \| \psi \| \cdot \maltese_1^j = 1 + \maltese_1^i.$$ Moreover, as $\phi$ is odd, the signs appearing in $m^\flat \circ \phi$ (in fact, all signs in all Gernstenhaber products until the end of the proof) vanish. Then $$\begin{split} &\ \big( (m^\flat \circ \phi) \star \psi \big)(a_k, \ldots, a_1)\\ = &\ \sum(-1)^{\clubsuit} m^\flat \big( -, (m^\flat \circ \phi)(-), -, \psi(-), - \big)\\ = &\ \sum (-1)^{1 + \maltese_1^i} m^\flat \big( -, m^\flat(-, \phi(-), -), -, \psi(-), -\big). \end{split}$$ This is the sum of 4-fold compositions with the interior $m^\flat$ only contains $\phi$. Similarly, for computing $\phi \star (m^\flat \circ \psi)$, one has $$\clubsuit = \| \phi \| \cdot \big( \clubsuit_1^i + | m^\flat \circ \psi |\big) + \| m^\flat \circ \psi\| \cdot \maltese_1^j = \maltese_1^j.$$ As our running convention is that after the second $m^\flat$ the inputs start with $a_i$, this sign is rewritten as $\maltese_1^i$ below. Then $$\begin{split} &\ (-1)^{|\phi|} \big( \phi \star (m^\flat \circ \psi) \big) (a_k, \ldots, a_1) \\ =&\ \sum (-1)^{1 + \clubsuit} m^\flat \big( -, \phi(-), -, (m^\flat \circ \psi) (-), a_i, \cdots, a_1 \big)\\ = &\ \sum(-1)^{1 + \maltese_1^i} m^\flat \big( -, \phi(-), m^\flat(-, \psi(-), -), - \big). \end{split}$$ Now compute $- (\phi \circ m^\flat) \star \psi$. $$\begin{split} &\ - \big( ( \phi \circ m^\flat) \star \psi \big) (a_k, \ldots, a_1 ) \\ = &\ \sum (-1)^{\maltese_1^j} m^\flat \big( -, (\phi \circ m^\flat) (-), a_j, \cdots, \psi(-), -\big)\\ = &\ \sum (-1)^{\maltese_1^j + \maltese_{j+1}^i} m^\flat \big( -, \phi( -, m^\flat(-), a_i, \cdots, a_{j+1}), a_j, \cdots, \psi(-), - \big)\\ = &\ \sum (-1)^{\maltese_1^i} m^\flat( \big(-,\phi(-, m^\flat(-), -), -, \psi(-), - \big). \end{split}$$ Lastly we compute $-(-1)^{|\phi|} \phi \star (\psi \circ m^\flat) = \phi \star (\psi \circ m^\flat)$, which is $$\begin{split} &\ \big(\phi \star (\psi \circ m^\flat)\big) (a_k, \ldots, a_1) \\ = &\ \sum (-1)^{\maltese_1^j} m^\flat \big(-, \phi(-), -, (\psi \circ m^\flat) (-), a_j, \ldots, a_1 \big)\\ = &\ \sum (-1)^{\maltese_1^j + \maltese_{j+1}^i} m^\flat \big(-, \phi(-), -, \psi(-, m^\flat(-), a_i, \ldots, a_{j+1}), a_j, \ldots, a_1 \big)\\ = &\ \sum (-1)^{\maltese_1^i} m^\flat \big( -, \phi(-), -, \psi(-, m^\flat(-), -), - \big). \end{split}$$ Gathering all computations together, we see that [\[chain_verification\]](#chain_verification){reference-type="eqref" reference="chain_verification"} follows from the $A_\infty$ relation $m^\flat \circ m^\flat = 0$. ◻ Therefore the Yoneda product descends to Hochschild cohomology. We still call the induced one by Yoneda product and denote it by the same symbol $\ast$. The Yoneda product has a chain-level unit in the strictly unital case. **Proposition 43**. *Suppose ${\mathcal A}^\flat$ has a strict unit $e$. Then the Hochschild cochain ${\bm 1}_{{\mathcal A}^\flat}$ defined by $${\bm 1}_{{\mathcal A}^\flat}(x_k, \ldots, x_1) = \left\{ \begin{array}{cc} 0,\ &\ k \geq 1,\\ e,\ &\ k = 0. \end{array}\right.$$ is a unit with respect to the Yoneda product.* *Proof.* By the definition of strict unit and Yoneda product, for any Hochschild cochain $\phi$, one has $$\begin{split} &\ ({\bm 1}_{{\mathcal A}^\flat} \star \phi)(a_k, \ldots, a_1) \\ = &\ \sum (-1)^\clubsuit m^\flat \big(a_k, \cdots, e, a_i, \cdots, \phi (a_{j+l}, \cdots, a_{j+1}), \cdots \big)\\ = &\ (-1)^{ \maltese_1^k + |\phi|} m_2^\flat\big(e, \phi (a_k, \ldots, a_1) \big)\\ = &\ (-1)^{|\phi(a_k, \ldots, a_1)|} m_2^\flat \big(e, \phi (a_k, \ldots, a_1) \big)\\ = &\ \phi (a_k, \ldots, a_1). \end{split}$$ Similarly $$\begin{split} &\ (\phi \star {\bm 1}_{{\mathcal A}^\flat} )(a_k, \ldots, a_1) \\ = &\ \sum (-1)^\clubsuit m^\flat \big(a_k, \cdots, \phi (a_{j+l}, \cdots, a_{j+1}),\cdots, e, a_i, \cdots \big)\\ = &\ (-1)^\clubsuit m_2^\flat\big(\phi (a_k, \ldots, a_1), e \big)\\ = &\ m_2^\flat \big(\phi (a_k, \ldots, a_1),e \big)\\ = &\ \phi (a_k, \ldots, a_1).\qedhere \end{split}$$ ◻ Finally, we remark that the Yoneda product on ${\it HH}^\bullet({\mathcal A}^\flat )$ is graded commutative. It is compatible with the Gerstenhaber bracket, which makes ${\it HH}^\bullet({\mathcal A}^\flat )$ into a Gerstenhaber algebra. ### Clifford algebras The Lagrangian Floer cohomology ring of a torus is often isomorphic to a Clifford algebra. Hence the Hochschild cohomology of Clifford algebras are one of the most important cases related to symplectic geometry and mirror symmetry. Recall that given a finite-dimensional ${\mathbb K}$-vector space $W$ equipped with a quadratic form $q$, the Clifford algebra $Cl(W, q)$ is the tensor algebra of $W$ modulo the relation $$w \otimes w' + w' \otimes w + 2 q(w, w') {\rm Id} = 0.$$ We only care about the case when $q$ is nondegenerate and the case when ${\mathbb K}$ is algebraically closed. In this case, all nondegenerate quadratic forms are equivalent to the standard one. When $W$ has dimension $n$, we abbreviate $Cl(W, q)$ by $Cl_n$. **Proposition 44**. *For all $n \geq 0$, the Hochschild cohomology of $Cl_n$ is $${\it HH}^k( Cl_n, Cl_n) = \left\{ \begin{array}{cc} {\mathbb K},\ &\ k = 0,\\ 0,\ &\ k \geq 1. \end{array} \right.$$ In particular, ${\it HH}^0(Cl_n, Cl_n)$ is generated by the identity.* *Proof.* The calculation was provided by Sheridan [@Sheridan_2016] and we recall it here. First, Hochschild cohomology is Morita invariant (see [@Loday_cyclic 1.5.6]). Second, there are only two Morita equivalence classes among Clifford algebras, the even ones and the odd ones (Bott periodicity). Hence we only need to calculate for $n = 0$ and $n=1$. When $n= 0$, $Cl_0 \cong {\mathbb K}$, giving ${\it HH}^\bullet({\mathbb K}, {\mathbb K}) = {\mathbb K}$. When $n = 1$ [^5], the calculation can be deduced from the more general case of J. Smith [@Smith_2019 Section 5] using reduced Hochschild cohomology. ◻ When the Floer cohomology algebra of a Lagrangian brane is isomorphic to a Clifford algebra, the argument via *formality* shows that the Hochschild cohomology of the corresponding $A_\infty$ algebra is the same as the Hochschild cohomology of the cohomology algebra. Recall that an $A_\infty$ algebra is called *formal* if it is $A_\infty$ quasi-isomorphic to its cohomology algebra. An associated algebra $A$ is called *intrinsically formal* if any ${\mathbb Z}_2$-graded $A_\infty$ algebra whose cohomology algebra is isomorphic to $A$ is formal. It was shown in [@Sheridan_2016 Corollary 6.4] that $Cl_n$ is intrinsically formal. Due to the Morita invariance of Hochschild cohomology, the following statement is immediate. **Corollary 45**. *If ${\mathcal A}^\flat$ is a flat $A_\infty$ algebra over ${\mathbb K}$ whose cohomology algebra is isomorphic to $Cl_n$, then $${\it HH}^\bullet ({\mathcal A}^\flat) = {\mathbb K}.$$0◻* Notice that if in addition ${\mathcal A}^\flat$ is strictly unital, $1_{{\mathcal A}^\flat} \neq 0$ and it generates the Hochschild cohomology. # Vortex Hamiltonian Floer theory {#sec:ham-package} We review the construction of vortex Hamiltonian Floer theory developed by the second author [@Xu_VHF] following the proposal of Cieliebak--Gaio--Salamon [@Cieliebak_Gaio_Salamon_2000]. ## Floer chain complexes ### Equivariant action functional Our convention for Hamiltonian vector field is fixed as follows. Let $(M, \omega)$ be a symplectic manifold and $H: M \to {\mathbb R}$ be a smooth function. The associated Hamiltonian vector field $X_H$ is specified by $$dH = \omega( X_H, \cdot).$$ We would like to consider the Hamiltonian dynamics upstairs in the gauged linear sigma model. Let $X$ be the toric manifold we are considering. Let $H: S^1 \times X \to {\mathbb R}$ be a smooth Hamiltonian function. Let ${\rm Per}(H)$ be the set of $1$-periodic orbits [^6] of $H$ whose elements are maps $x: S^1 \to X$. The Hamiltonian $H$ lifts to a $K$-invariant function on $S^1\times \mu^{-1}(0)$. Choose an arbitrary $K$-invariant extension $\widehat H: S^1 \times V \to {\mathbb R}$ whose support is compact and disjoint from the unstable locus $V^{\rm us}$ under the $K^{\mathbb C}$-action. Consider the set of **equivariant loops** $$L^K (V):= \Big\{ {\mathfrak x} = (\widehat x, \zeta): S^1 \to V \times {\mathfrak k}. \Big\}$$ Here function $\zeta: S^1 \to {\mathfrak k}$ can be viewed as a gauge field on $S^1$. Notice that as $V$ is contractible, the loop $x$ is contractible and there is only one homotopy class of cappings. The loop group $L K$ acts on the set of capped equivariant orbits by $$g\cdot {\mathfrak x} = ( g\cdot \widehat x, g\cdot \xi)\ {\rm where}\ (g \cdot \widehat x)(t) = g(t) \widehat x (t),\ (g \cdot \xi)(t) = \zeta (t) - \frac{d}{dt} \log g(t).$$ Define the action functional $$\widehat{\mathcal A}_H: L^K(V) \to {\mathbb R},\ {\mathfrak x}\mapsto - \int_{\mathbb D} u^* \omega_V + \int_{S^1} \left( \langle \mu( \widehat x (t)), \zeta (t) \rangle - \widehat H_t( \widehat x(t)) \right) dt$$ where $u: {\mathbb D}^2 \to V$ is any capping. Critical points are solutions $$\begin{aligned} &\ \mu(\widehat x (t)) \equiv 0,\ &\ \widehat x'(t) = X_{\widehat H_t}(\widehat x(t)) - {\mathcal X}_{\zeta (t)}(\widehat x(t)).\end{aligned}$$ Here ${\mathcal X}_{\zeta (t)}$ is the Hamiltonian vector field of the function $\langle \mu, \zeta (t) \rangle$. The action functional satisfies the following transformational law with respect to the loop group action. Indeed, for $g \in LK$ and ${\mathfrak x} \in L^K(V)$, one has $$\label{deck} \widehat{\mathcal A}_H ( g {\mathfrak x}) = - \omega^K (g) + {\mathcal A}_H ({\mathfrak x}).$$ Denote $${\mathfrak L}^K(V):= L^K(V)/ {\rm ker} \omega^K.$$ Its elements are denoted by $[{\mathfrak x}]$. Then $\Gamma \cong LK/ {\rm ker} \omega^K$ acts on ${\mathfrak L}^K(V)$. We denote the action by $g \cdot [{\mathfrak x}]$. Then $\widehat{\mathcal A}_H$ induces a functional on ${\mathfrak L}^K(V)$, denoted by $${\mathcal A}_H: {\mathfrak L}^K(V) \to {\mathbb R}.$$ Each critical point of ${\mathcal A}_H$ is called an **equivariant 1-periodic Hamiltonian orbit**. There is a correspondence between ordinary Hamiltonian orbits downstairs and equivariant Hamiltonian orbits upstairs. More precisely, let $\widetilde {\rm Per}(H)$ be the covering of ${\rm Per}(H)$ consisting of equivalence classes $[u, x]$ of capped 1-periodic orbits of $H$: the equivalence relation $(u, x) \sim (u', x')$ is defined by the equality of action values: $$(u, x) \sim (u', x') \Longleftrightarrow x = x' \in {\rm Per}(H)\ {\rm and}\ \int_{{\mathbb D}^2} u^* \omega_X = \int_{{\mathbb D}^2} (u')^* \omega_X.$$ Then there is a map $$\label{downtoup} \iota: \widetilde{\rm Per}(H) \to {\rm crit} {\mathcal A}_H \subset {\mathfrak L}^K(V).$$ Indeed, suppose $x: S^1 \to X$ is a contractible 1-periodic orbit of the Hamiltonian flow of $H$ and $u: {\mathbb D}^2 \to X$ is a capping of $x$. View $\mu^{-1}(0) \to X$ as a principal $K$-bundle $P$. The Euclidean metric on $V$ induces a connection on $P$ whose horizontal distribution is the orthogonal complement of tangent planes of $K$-orbits; equivalently, this gives a connection 1-form $\theta \in \Omega^1(\mu^{-1}(0)) \otimes {\mathfrak k}$. The pullback $u^* P \to {\mathbb D}^2$ is trivial and different trivializations differ by a smooth map $g: {\mathbb D}^2 \to K$. Any trivialization of this pullback bundle induces a connection matrix $u^* \theta$ whose boundary restriction is $\zeta (t) dt$. A trivialization also induces a map $\widehat u: {\mathbb D}^2 \to \mu^{-1}(0)$ lifting $u$. Let the boundary restriction of $\widehat u$ be $\widehat x$. Then ${\mathfrak x} = (\widehat x, \zeta)$ is an equivariant 1-periodic orbit, well-defined up to $L_0 K$-actions. Furthermore, if $u'$ is a different capping with the same resp. different action value, then the correspondence we just described gives the same resp. different element in ${\mathfrak L}^K(V)$. **Lemma 46**. *In the toric case the map [\[downtoup\]](#downtoup){reference-type="eqref" reference="downtoup"} is bijective.* *Proof.* Given any equivariant Hamiltonian orbit ${\mathfrak x}$ upstairs, the map $\widehat x: S^1 \to \mu^{-1}(0)$ projects down to a 1-periodic orbit $x: S^1 \to X$. As $X$ is simply connected, $x$ is contractible. Choose a capping $u: {\mathbb D}^2 \to X$ and let ${\mathfrak x}' = (\widehat x', \zeta')$ be equivariant Hamiltonian orbit lifting $[u, x]$. As the $K$-action on $\mu^{-1}(0)$ is free, there is a gauge transformation on the circle making $\widehat x' = \widehat x$. The condition $$\widehat x'(t) = X_{H_t}(\widehat x (t)) - {\mathcal X}_{\zeta(t)} (\widehat x(t))$$ implies that $\zeta = \zeta'$. ◻ **Definition 47**. The **Conley--Zehnder index** of an equivariant 1-periodic orbit ${\mathfrak x} \in {\rm crit} {\mathcal A}_H$ is the usual Conley--Zehnder index of the capped 1-periodic orbit $\iota^{-1}({\mathfrak x}) \in \widetilde{\rm Per}(H)$, denoted by ${\rm CZ}({\mathfrak x}) \in {\mathbb Z}$. ### Floer trajectories Similar to the standard Hamiltonian Floer theory, one considers the equation for the gradient flow of the equivariant action functional. Choose a 1-periodic $K$-invariant $\omega_V$-compatible almost complex structure $\widehat J_t$ on $V$. Formally the negative gradient flow equation of $\widehat{\mathcal A}_H$ is the following equation for pairs $(u, \eta): {\mathbb R}\times S^1 \to V \times {\mathfrak k}$ $$\begin{aligned} &\ \partial_s u + \widehat J_t \left( \partial_t u + {\mathcal X}_\eta (u) - X_{\widehat H_t}(u) \right) = 0,\ &\ \partial_s \eta + \mu(u) = 0.\end{aligned}$$ This is in fact the symplectic vortex equation on the cylinder ${\mathbb R}\times S^1$ for the trivial $K$-bundle and the standard cylindrical volume form, written in temporal gauge $A = d + \eta dt$. In general, for $A = d + \xi ds + \eta dt$, the vortex equation [\[vortex_equation\]](#vortex_equation){reference-type="eqref" reference="vortex_equation"} reads $$\begin{aligned} \label{vortex_equation_2} &\ \partial_s u + {\mathcal X}_{\xi}(u) + \widehat J_t \left( \partial_t u + {\mathcal X}_{\eta} (u) - X_{\widehat H_t}(u) \right) = 0,\ &\ \partial_s \eta - \partial_t \xi + \mu(u) = 0.\end{aligned}$$ It was shown in [@Xu_VHF] that any finite energy solution converges up to gauge transformation to critical points of ${\mathcal A}_H$. **Theorem 48**. *[@Xu_VHF Theorem 3.1, Corollary 4.3]* 1. *Given a bounded solution ${\mathfrak u} = (u, \xi, \eta)$ (i.e. finite energy solution with $u({\mathbb R}\times S^1)$ bounded) to [\[vortex_equation_2\]](#vortex_equation_2){reference-type="eqref" reference="vortex_equation_2"}, there exist a gauge equivalent solution, still denoted by $(u, \xi, \eta)$, as well as equivariant 1-periodic orbits ${\mathfrak x}_\pm = (\widehat x_\pm, \zeta_\pm)$ such that uniformly for $t \in S^1$ $$\label{asymptotic} \lim_{s \to \pm \infty} (u(s, \cdot), \xi(s, \cdot), \eta(s, \cdot)) = (\widehat x_\pm, 0, \zeta_\pm).$$* 2. *If ${\mathfrak x}_\pm'$ are another pair of equivariant 1-periodic orbits satisfying [\[asymptotic\]](#asymptotic){reference-type="eqref" reference="asymptotic"} with ${\mathfrak u}$ replaced by any gauge equivalent solution, then there exists $g_\pm \in LK$ with $g_- g_+^{-1} \in L_0 K$ such that ${\mathfrak x}_\pm' = g_\pm {\mathfrak x}_\pm$.* 3. *If $H$ is a nondegenerate Hamiltonian downstairs, then one can make the convergence [\[asymptotic\]](#asymptotic){reference-type="eqref" reference="asymptotic"} exponentially fast by choosing suitable gauge equivalent solutions. More precisely, there exist $C>0$ and $\delta>0$ such that $$d_V(u(s, t), \widehat x_\pm(t)) + | \xi(s, t)| + |\eta(s, t) - \zeta(t)| \leq C e^{-\delta|s|}.$$ Here $d_V$ is the Euclidean distance on $V$. Similar exponential decay estimates hold for covariant derivatives of arbitrary higher orders.[^7]* Therefore, one can use a pair of elements ${\mathfrak x}_\pm\in {\rm crit}{\mathcal A}_H \subset {\mathfrak L}^K(V)$ to label solutions. Let $${\mathcal M}({\mathfrak x}_-, {\mathfrak x}_+)$$ be the set of gauge equivalence classes of bounded solutions ${\mathfrak u}$ to [\[vortex_equation_2\]](#vortex_equation_2){reference-type="eqref" reference="vortex_equation_2"} modulo the ${\mathbb R}$-translation. One has the energy identity ([@Xu_VHF Proposition 3.8]) $$E({\mathfrak u}) = {\mathcal A}_H ({\mathfrak x}_-) - {\mathcal A}_H ({\mathfrak x}_+).$$ *Remark 49*. To achieve transversality, one has to avoid certain "bad" $K$-equivariant lifts of a given Hamiltonian $H$ downstairs and choose almost complex structures appropriately. In [@Xu_VHF Section 6] the second author used the notion of *admissible* almost complex structures and *admissible* $K$-invariant lifts of a Hamiltonian downstairs. We briefly recall the precise meanings of them adapted to the toric case. First, in the stable locus $V^{\rm st}$ there is the projection $\pi: V^{\rm st} \to X$ which is invariant under the complex torus $G$. Hence there is a splitting $$TV|_{V^{\rm st}} \cong \pi^* TX \oplus ({\mathfrak k}\otimes {\mathbb C}).$$ Throughout this paper we fix a $K$-invariant (small) open neighborhood $U$ of $\mu^{-1}(0)$ and consider only $K$-invariant, $\omega_V$-compatible almost complex structures $\widehat J$ on $V$ which agrees with $J_V$ outside $U$ (this is necessary to guarantee the $C^0$-compactness in [@Xu_VHF]). Moreover, given a *nondegenerate* Hamiltonian $H: S^1 \times X \to {\mathbb R}$, an $S^1$-family of almost complex structures $\widehat J_t$ is said to be *admissible* with respect to the $H$ downstairs if for any loop $\widehat x: S^1 \to \mu^{-1}(0)$ that projects to a 1-periodic orbit downstairs, one imposes some conditions on the $1$-jet of $\widehat J_t$ along $\widehat x$ (see [@Xu_VHF Definition 6.2]. Then the notion of admissibility of $K$-invariant lifts of $H$ was defined (see [@Xu_VHF Definition 6.5]), which is a condition on the infinitesimal behavior of the lifts $\widehat H_t$ along 1-periodic orbits given in terms of the Hessian of the equivariant action functional. **Theorem 50**. *Given a nondegenerate Hamiltonian $H_t$ downstairs, for a generic admissible pair $(\widehat H_t, \widehat J_t)$, the following is true.* 1. *Each moduli space ${\mathcal M}({\mathfrak x}, {\mathfrak y})$ is regular and has dimension ${\rm CZ}({\mathfrak x}) - {\rm CZ}({\mathfrak y}) - 1$.* 2. *Moduli spaces with bounded energy are compact up to breaking as in the usual setting for the Uhlenbeck--Gromov--Floer compactification.* 3. *If ${\rm CZ}({\mathfrak x}) - {\rm CZ}({\mathfrak y}) = 1$, then the moduli space consists of finitely many points.* 4. *When ${\rm CZ}({\mathfrak x}) - {\rm CZ}({\mathfrak y}) = 2$, the compactified moduli space is a compact 1-dimensional manifold with boundary.* We briefly explain the reason for transversality. Indeed, as the total energy is finite and the volume on the cylinder is infinite, near infinity any solution is contained in the neighborhood $U$ of $\mu^{-1}(0)$ fixed in Remark [Remark 49](#rem_admissible){reference-type="ref" reference="rem_admissible"}. Therefore, there is a nonempty open subset of the cylinder whose image is contained in the free locus of the $K$-action. Then using an equivariant version of the argument of Floer--Hofer--Salamon [@Floer_Hofer_Salamon] one can achieve transversality by perturbing $\widehat J_t$ in a neighborhood of $\mu^{-1}(0)$. It is a standard procedure to construct a coherent system of orientations on the moduli spaces (see [@Floer_Hofer_Orientation]). Then for $R = {\mathbb Z}$, there is a well-defined count $$n({\mathfrak x}, {\mathfrak y}) \in {\mathbb Z}$$ which is the signed count of the number of Floer trajectories in $0$-dimensional components of ${\mathcal M}({\mathfrak x}, {\mathfrak y})$. When $R$ is any commutative ring with a unit, $n({\mathfrak x}, {\mathfrak y})$ induces an element $$n_R({\mathfrak x}, {\mathfrak y}) \in R$$ by the induced count via the map ${\mathbb Z} \to R$. ### Floer homology We first define the Floer chain group for a smaller Novikov ring. Recall that one has the finitely generated abelian group $$\Gamma:= LK/ {\rm ker} \omega^K$$ which naturally embeds into ${\mathbb R}$. For any commutative ring $R$, introduce $$\Lambda_R^\Gamma:= \Big\{ \sum_{i=1}^\infty a_i T^{g_i}\in \Lambda_R\ |\ g_i \in \Gamma \Big\}.$$ We define the Floer chain group ${\it VCF}_\bullet(\widehat H)$ to be the "downward" completion: $${\it VCF}_\bullet(\widehat H; \Lambda_R^\Gamma ) = \Big\{ \sum_{i=1}^\infty b_i {\mathfrak x}_i\ |\ b_i \in R,\ {\mathfrak x}_i \in {\rm crit} {\mathcal A}_H,\ \lim_{i \to \infty} {\mathcal A}_H ({\mathfrak x}_i) = -\infty \Big\}.$$ It is graded by the Conley--Zehnder index (modulo 2). The $\Lambda_R^\Gamma$-module structure is defined by $$\Big( \sum_{i=1}^\infty a_i T^{g_i} \Big) \Big( \sum_{j=1}^\infty b_j {\mathfrak x}_j \Big) = \sum_{i, j = 1}^\infty a_i b_j (g_i \cdot {\mathfrak x}_j).$$ By [\[deck\]](#deck){reference-type="eqref" reference="deck"}, the right hand side is in ${\it VCF}_\bullet(\widehat H; \Lambda_R^\Gamma)$ and this is a well-defined action. Define $${\it VCF}_\bullet(\widehat H; \Lambda_R):= {\it VCF}_\bullet( \widehat H; \Lambda_R^\Gamma) \otimes_{\Lambda_R^\Gamma} \Lambda_R.$$ The Floer differential $\partial_{\widehat J}: {\it VCF}_\bullet (\widehat H; \Lambda_R^\Gamma) \to {\it VCF}_{\bullet -1}(\widehat H; \Lambda_R^\Gamma)$ is defined by the counts $n_R ({\mathfrak x}, {\mathfrak y})$. More precisely, on generators, $$\partial_{\widehat J} {\mathfrak x} = \sum_{{\mathfrak y}} n_R ({\mathfrak x}, {\mathfrak y}) {\mathfrak y}.$$ One has $\partial_{\widehat J}^2 = 0$, resulting in the **vortex Floer homology** $${\it VHF}_\bullet( \widehat H, \widehat J; \Lambda_R^\Gamma).$$ Notice that the differential $\partial_{\widehat J}$ *decreases* the action. ### Adiabatic limit The adiabatic limit argument allows us to relate the gauged linear sigma model with holomorphic curves in the symplectic quotient. While we do not need a complete analysis of such a correspondence, we do need to consider the family of vortex equations related to the adiabatic limit argument. Indeed, if on the infinite cylinder we choose, instead of the standard area form $ds dt$, a rescaled one $\lambda^2 ds dt$, then the corresponding vortex Floer equation reads $$\begin{aligned} \label{vortex_lambda} &\ \partial_s u + {\mathcal X}_\xi (u) + \widehat J_t \left( \partial_t u + {\mathcal X}_\eta(u) - X_{\widehat H_t}(u) \right) = 0, &\ \partial_s \eta - \partial_t \xi + \lambda^2 \mu(u) = 0.\end{aligned}$$ One can define a vortex Floer chain complex for the triple $(\lambda, \widehat H, \widehat J)$ in completely the same way as the $\lambda = 1$ case, once transversality holds, which can be achieved via perturbation. We denote the vortex Floer chain complex by ${\it VCF}_\bullet^\lambda( \widehat H, \widehat J; \Lambda_R^\Gamma)$. The corresponding homology is denoted by $${\it VHF}^\lambda_\bullet(\widehat H, \widehat J; \Lambda_R^\Gamma).$$ There are a few subtleties. First, given a nondegenerate Hamiltonian $H$ downstairs on $X$, the notion of admissible almost complex structures ([@Xu_VHF Definition 6.2]) is independent of $\lambda$; the notion of admissible lifts, however, depends on $\lambda$. **Definition 51**. A triple $(\lambda, \widehat H, \widehat J)$ is called a **regular triple** if 1. The descent Hamiltonian $H$ on $X$ is nondegenerate. 2. $(\widehat H, \widehat J)$ is admissible with respect to $H$. 3. Moduli spaces of gauge equivalence classes of finite energy solutions to [\[vortex_lambda\]](#vortex_lambda){reference-type="eqref" reference="vortex_lambda"} are all regular. ### Continuation map Given two regular triples $(\lambda_\pm, \widehat H_\pm, \widehat J_\pm)$, one compares the two associated vortex Floer complexes via continuation maps. By an **interpolation** between these two triples, we mean a triple $(\lambda_s, \widehat H_s, \widehat J_s)$ where $\lambda_s \in {\mathbb R}_+$ is a smooth function in $s\in {\mathbb R}$ which agrees with $\lambda_\pm$ near $\pm \infty$, $\widehat H_s$ is a smooth family of $K$-invariant compactly supported functions parametrized by $(s, t) \in {\mathbb R} \times S^1$ which agrees with $\widehat H_\pm$ near $\pm \infty$, and $\widehat J_s$ is a smooth family of $K$-invariant $\omega_V$-compatible almost complex structures parametrized by $(s, t)\in {\mathbb R}\times S^1$ which agrees with $\widehat J_\pm$ near $\pm \infty$. Choosing a generic interpolation $(\lambda_s, \widehat H_s, \widehat J_s)$, by considering moduli spaces of gauge equivalence classes of solutions to the equation $$\begin{aligned} \label{vortex_continuation} &\ \partial_s u + {\mathcal X}_\xi + \widehat J_{s, t} \left( \partial_t u + {\mathcal X}_\eta - X_{\widehat H_{s, t}}(u) \right) = 0,\ &\ \partial_s \eta - \partial_t \xi + \lambda_s^2 \mu(u) = 0,\end{aligned}$$ one can define a continuation map $$\mathfrak{cont}: {\it VCF}_\bullet^{\lambda_-} (\widehat H_-, \widehat J_-; \Lambda_R^\Gamma) \to {\it VCF}_\bullet^{\lambda_+} ( \widehat H_+, \widehat J_+; \Lambda_R^\Gamma)$$ completely analogous to the case of classical Hamiltonian Floer theory. The map $\mathfrak{cont}$ is a chain homotopy equivalence, inducing an isomorphism on Floer homology $${\it VHF}_\bullet^{\lambda_-} ( \widehat H_-, \widehat J_-; \Lambda_R^\Gamma) \cong {\it VHF}_\bullet^{\lambda_+} ( \widehat H_+, \widehat J_+; \Lambda_R^\Gamma).$$ Completely analogous to the classical situation, these isomorphisms are natural, hence the resulting homology groups define a common object called the **vortex Hamiltonian Floer homology** of $V$, denoted by $${\it VHF}_\bullet(V; \Lambda_R^\Gamma).$$ Define $${\it VHF}_\bullet(V; \Lambda_R):= {\it VHF}_\bullet( V; \Lambda_R^\Gamma) \otimes_{\Lambda_R^\Gamma} \Lambda_R.$$ In order to consider effects on the filtered theories, one needs to estimate the energy of solutions contributing to the continuation maps. **Proposition 52**. *Given any solution ${\mathfrak u} = (u, \xi, \eta)$ to [\[vortex_continuation\]](#vortex_continuation){reference-type="eqref" reference="vortex_continuation"} which converges to ${\mathfrak x}_\pm\in {\rm crit} {\mathcal A}_{H_\pm}$ at $\pm \infty$, one has $$\label{continuation_energy_identity} \int_{{\mathbb R}\times S^1} \Big( |\partial_s u + {\mathcal X}_\xi(u)|^2 + \lambda_s^2 |\mu(u)|^2 \Big) ds dt = {\mathcal A}_{H_-}({\mathfrak x}_-) - {\mathcal A}_{H_+}({\mathfrak x}_+) - \int_{{\mathbb R}\times S^1} \frac{\partial \widehat H_{s, t}}{\partial s}(u) ds dt.$$ In particular, if $\widehat H_{s, t} = (1-\chi(s)) \widehat H_- + \chi(s) \widehat H_+$ for some non-decreasing function $\chi: {\mathbb R} \to [0, 1]$, then one has $$\label{continuation_energy_inequality} {\mathcal A}_{H_+}({\mathfrak x}_+) \leq {\mathcal A}_{H_-}({\mathfrak x}_-) + \int_0^1 \max_V \left( \widehat H_- - \widehat H_+ \right) dt.$$* *Proof.* When $\lambda_s$ is a constant, [\[continuation_energy_identity\]](#continuation_energy_identity){reference-type="eqref" reference="continuation_energy_identity"} is [@Xu_VHF Proposition 7.5]. The general case is the same as the area form on the domain does not affect the topological nature of the energy. As the left-hand-side of [\[continuation_energy_identity\]](#continuation_energy_identity){reference-type="eqref" reference="continuation_energy_identity"} is nonnegative, [\[continuation_energy_inequality\]](#continuation_energy_inequality){reference-type="eqref" reference="continuation_energy_inequality"} follows. ◻ ### Computation of ${\it VHF}$ It is expected that the vortex Floer homology is isomorphic to the Hamiltonian Floer homology of the symplectic quotient, and hence its singular homology (in appropriate coefficients). However, such a calculation relies involved technical constructions. The Piunikhin--Salamon--Schwarz (PSS) approach forces one to deal with multiple covers of equivariant Floer cylinders with $H \equiv 0$ which may have negative equivariant Chern number. The adiabatic limit approach (similar to [@Gaio_Salamon_2005]) requires the study of affine vortices for a general toric manifold. In particular, for general symplectic quotients both approaches require the use of the virtual technique. However, in the toric case, even without having the PSS map, it is rather easy to compute the rank of ${\it VHF}(V)$ as one can find a perfect Morse function. **Proposition 53**. *For any commutative ring $R$, as $\Lambda_R^\Gamma$-modules, ${\it VHF}_\bullet (V; \Lambda_R^\Gamma)$ is isomorphic to $H_\bullet (X; \Lambda_R^\Gamma)$ (with the reduced ${\mathbb Z}_2$-grading) up to a degree shifting.* *Proof.* Recall that the $2n$-dimensional toric manifold $X$ carries a Hamiltonian $T^n$-action. For a generic circle $S^1 \subset T^n$, the induced moment map $f: X \to {\mathbb R}$ is a perfect Morse function whose critical points are the toric fixed points. In particular, the Morse indices are all even. Then for $\epsilon$ small, $\epsilon f$ is a nondegenerate time-independent Hamiltonian. After a small perturbation and $K$-invariant lift to $V$, the corresponding vortex Floer chain complex has no two generators with adjacent degrees. Hence the ${\it VHF}_\bullet (V; \Lambda_R^\Gamma)$ has the same rank as $H_\bullet(X; \Lambda_R^\Gamma)$. Lastly, the usual normalization of the Conley--Zehnder index is taken in such a way that if $x$ is a critical point of $\epsilon f$ viewed as a 1-periodic orbit with a constant capping, then $${\rm CZ}(x) = n - {\rm index}_f(x)$$ where $2n = {\rm dim} X$ and ${\rm index}_f(x)$ is the Morse index of $x$ (see [@McDuff_Salamon_2004 (12.1.7)]). ◻ ## Small bulk deformations Here we define a family of deformations of the vortex Floer homology parametrized by "small" bulk deformations. Recall that the toric manifold $X$ has $N$ toric divisors $D_j$ corresponding to the $N$ faces of the moment polytope. These divisors are GIT quotients of the coordinate hyperplanes $$V_j= \{ (x_1, \ldots, x_N) \in V\ |\ x_j = 0 \}.$$ Introduce a small bulk deformation of the form $$\label{small_bulk_deformation_general} {\mathfrak b} = \sum_{j=1}^N c_j V_j\ {\rm where}\ c_j \in \Lambda_{0, R}.$$ The ${\mathfrak b}$-deformed vortex Floer complex is the complex generated by equivariant 1-periodic orbits upstairs whose differential counts gauge equivalence classes of solutions to the vortex equation in a different way: for each rigid (modulo gauge transformation) solution ${\mathfrak u} = (u, \xi, \eta)$, we weight the count by the factor $$\exp \left( \sum_{j=1}^N c_j (u \cap V_j) \right) \in \Lambda_R$$ where $u \cap V_j$ is the intersection number between the cylinder $u$ and the divisor $V_j$. Formally, this count coincides with the count of solutions on the cylinder with markings mapped to $V_j$. *Remark 54*. The use of bulk deformations in Lagrangian Floer theory was invented by Fukaya--Oh--Ohta--Ono [@FOOO_toric_2; @FOOO_mirror] which resembles the notion of *big quantum cohomology* in Gromov--Witten theory. Bulk deformations are adapted to Hamiltonian Floer theory in [@Usher_2011] and [@FOOO_spectral]. In gauged linear sigma model it was discussed in [@Woodward_toric]. The term "small" used here comes from the terminology in Gromov--Witten theory where small means deforming Gromov--Witten invariants by divisor classes and "big" means deforming by classes with arbitrary degrees. ### Bulk-avoiding Hamiltonians One can only have a well-defined topological intersection number between Floer cylinders and the divisors if periodic orbits do not intersect these toric divisors. We introduce the following type of Hamiltonians on the toric manifold. **Definition 55** (Bulk-avoiding Hamiltonians). 1. A Hamiltonian $H$ on the toric manifold $X$ is called **bulk-avoiding** if all 1-periodic orbits of $nH$ for all $n\geq 1$ do not intersect the divisor $D_1 \cup \cdots \cup D_N$. 2. Denote by $${\mathcal H}_K^{**}(V) \subset {\mathcal H}_K^*(V)$$ the space of admissible $K$-invariant Hamiltonians on $V$ whose reductions are bulk-avoiding. 3. A bulk-avoiding admissible pair is an admissible pair $(\widehat H, \widehat J)$ such that $\widehat H$ descends to a bulk-avoiding Hamiltonian downstairs. It is easy to see that a $C^2$-small perturbation of any Hamiltonian is bulk-avoiding. Now we can define the topological intersection numbers. Let ${\mathfrak u} = (u, \xi, \eta)$ be a solution to [\[vortex_equation_2\]](#vortex_equation_2){reference-type="eqref" reference="vortex_equation_2"} which converges to equivariant 1-periodic orbits ${\mathfrak x}$ resp. ${\mathfrak y}$ at $-\infty$ resp. $+\infty$. Then a generic compactly supported perturbation $\tilde u$ intersects transversely with $V_j$. Define $$\cap V_j = \tilde u \cap V_j \in {\mathbb Z}$$ which counts transverse intersection points with signs. Notice that this number is well-defined: first, if $\tilde u'$ is another perturbation, then $\tilde u \cap V_j = \tilde u' \cap V_j$; second, if ${\mathfrak u}' = (u', \xi', \eta')$ is gauge equivalent to ${\mathfrak u}$ via a gauge transformation $g$, then $\tilde u':= g \tilde u$ is a perturbation of $u'$. As $V_j$ is $K$-invariant, $\tilde u'$ still intersect transversely with $V_j$ and the intersection number is the same. ### Bulk-deformed vortex Floer complex For our application, we only consider small bulk deformations of the form $${\mathfrak b} = \sum_{j=1}^N \log c_j\ V_j\ {\rm where}\ c_j \in {\mathbb Z}[{\bf i}] = {\mathbb Z}\oplus {\bf i}{\mathbb Z}.$$ Here ${\bf i} = \sqrt{-1}$ and one can regard ${\mathbb Z}[{\bf i}]\subset {\mathbb C}$. The weighted counts eventually only depend on $c_j$ so we allow $c_j$ to be zero and the ambiguity of taking logarithm does not affect further discussions. Consider the vortex Floer chain complex $${\it VCF}_\bullet( \widehat H, \widehat J; \Lambda_{{\mathbb Z}[{\bf i}]}).$$ Due to the special behavior of the bulk ${\mathfrak b}$, the weighted counts of cylinders are still integral. Define the bulk-deformed vortex differential $$\partial^{\mathfrak b}: {\it VCF}_\bullet (\widehat H, \widehat J; \Lambda_{{\mathbb Z}[{\bf i}]}) \to {\it VCF}_\bullet ( \widehat H, \widehat J; \Lambda_{{\mathbb Z}[{\bf i}]})$$ by $$\label{bulk_differential} \partial^{\mathfrak b}({\mathfrak x}) = \sum_{{\mathfrak y}\atop {\rm CZ}({\mathfrak x})- {\rm CZ}({\mathfrak y}) = 1} \left( \sum_{[{\mathfrak u}]\in {\mathcal M}^{\rm cyl}({\mathfrak x}, {\mathfrak y})} \epsilon([{\mathfrak u}]) \exp \left( \sum_{j=1}^N \log c_j\ [{\mathfrak u}] \cap V_j \right) \right) {\mathfrak y}.$$ Here $\epsilon([{\mathfrak u}]) \in \{\pm 1\}$ is the sign of the rigid solution $[{\mathfrak u}]$. In particular, when ${\mathfrak b} = 0$, the above coincides with the original differential map $\partial$. **Lemma 56**. *$\partial_{\mathfrak b}$ is a legitimate linear map and $(\partial^{\mathfrak b})^2 = 0$.* *Proof.* First, as $c_j \in {\mathbb Z}[{\bf i}]$, the weights $$\exp \left( \sum_{j=1}^N \log c_j \ [{\mathfrak u}] \cap V_j \right) = \prod_{j=1}^N c_j^{[{\mathfrak u}] \cap V_j}\in {\mathbb Z}[{\bf i}].$$ Hence the coefficients on the right hand side of [\[bulk_differential\]](#bulk_differential){reference-type="eqref" reference="bulk_differential"} are still in ${\mathbb Z}[{\bf i}]$. Second, by Gromov compactness, the sum [\[bulk_differential\]](#bulk_differential){reference-type="eqref" reference="bulk_differential"} is still in the module ${\it VCF}_\bullet ( \widehat H, \widehat J; \Lambda_{{\mathbb Z}[{\bf i}]})$. Hence $\partial^{\mathfrak b}$ is a well-defined linear map. To prove that its square is zero, consider for each ${\mathfrak x}$ and ${\mathfrak z}$ with Conley--Zehnder indices differing by $2$ and consider the $1$-dimensional components of the moduli space $\overline{\mathcal M}{}^{\rm cyl}({\mathfrak x}, {\mathfrak z})$. It can be further decomposed into connected components. Within each connected components, the topological intersection number for each cylinder with each $V_j$ is a constant. Moreover, for the concatenation of two cylinders $[{\mathfrak u}_1]$ and $[{\mathfrak u}_2]$ which is in the boundary of such a component, this intersection number with $V_j$ is equal to the sum $[{\mathfrak u}_1]\cap V_j + [{\mathfrak u}_2] \cap V_j$. It follows that $(\partial^{\mathfrak b})^2 = 0$. ◻ Hence for each regular admissible bulk-avoiding pair $(\widehat H, \widehat J)$, one can define the ${\mathfrak b}$-deformed vortex Floer homology by $${\it VHF}_\bullet^{\mathfrak b}(\widehat H, \widehat J; \Lambda_{{\mathbb Z}[{\bf i}]}):= {\rm ker} \partial^{\mathfrak b}/ {\rm im} \partial^{\mathfrak b}.$$ Below we summarize its properties. **Theorem 57** (Properties of bulk-deformed vortex Floer complex). *[\[thm_vhf_bulk\]]{#thm_vhf_bulk label="thm_vhf_bulk"}* 1. *For each regular bulk-avoiding admissible pair $(\widehat H, \widehat J)$, the complex ${\it VCF}_\bullet (\widehat H, \widehat J; \Lambda_{{\mathbb Z}[{\bf i}]})$ with differential $\partial^{\mathfrak b}$ is a ${\mathbb Z}_2$-graded filtered Floer--Novikov complex (see Definition [\[defn_fn\]](#defn_fn){reference-type="eqref" reference="defn_fn"}).* 2. *For each two regular admissible bulk-avoiding pairs $(\widehat H_1, \widehat J_1)$ and $(\widehat H_2, \widehat J_2)$, there is a continuation map $$\mathfrak{cont}: {\it VCF}_\bullet^{\mathfrak b} (\widehat H_1, \widehat J_1; \Lambda_{{\mathbb Z}[{\bf i}]} ) \to {\it VCF}_\bullet^{\mathfrak b} (\widehat H_2, \widehat J_2; \Lambda_{{\mathbb Z}[{\bf i}]} )$$ which is canonical up to chain homotopy. Hence there is a ${\mathbb Z}_2$-graded $\Lambda_{{\mathbb Z}[{\bf i}]}$-module ${\it VHF}_\bullet^{\mathfrak b} (V; \Lambda_{{\mathbb Z}[{\bf i}]})$, called the ${\mathfrak b}$-deformed vortex Floer homology, with canonical isomorphisms $${\it VHF}_\bullet^{\mathfrak b} (\widehat H, \widehat J; \Lambda_{{\mathbb Z}[{\bf i}]}) \cong {\it VHF}_\bullet^{\mathfrak b} (V; \Lambda_{{\mathbb Z}[{\bf i}]})$$ for all regular admissible bulk-avoiding pairs $(\widehat H, \widehat J)$.* 3. *There is a linear isomorphism $${\it VHF}_\bullet^{\mathfrak b} (V; \Lambda_{{\mathbb Z}[{\bf i}]}) \cong H_\bullet(X; \Lambda_{{\mathbb Z}[{\bf i}]}).$$* ### Poincaré duality In Morse--Floer theory one can define the Poincaré duality on the chain-level by "reversing" the Morse function or the symplectic action functional. We recall this construction in the setting of vortex Floer theory. If $\widehat H: S^1\times V \to {\mathbb R}$ is a $K$-invariant Hamiltonian, define $\widehat H^{\rm op}: S^1 \times V \to {\mathbb R}$ by $$\widehat H^{\rm op}(t, v) = - \widehat H(-t, v).$$ Then similar to the case of the ordinary Floer homology (see [@McDuff_Salamon_2004 Section 12.3]), there is a one-to-one correspondence between ${\rm crit} {\mathcal A}_H$ and ${\rm crit} {\mathcal A}_{H^{\rm op}}$. More precisely, if ${\mathfrak x} = (\widehat x, \eta) \in L^K(V)$ is an equivariant 1-periodic orbit, then $${\mathfrak x}^{\rm op}:= (\widehat x^{\rm op}, \eta^{\rm op})\ {\rm where}\ \widehat x^{\rm op}(t) = \widehat x(-t),\ \eta^{\rm op}(t) = - \eta(-t)$$ solves $$\frac{d}{dt} \widehat x^{\rm op} (t) + {\mathcal X}_{\eta^{\rm op}(t)}(\widehat x^{\rm op}(t)) - X_{\widehat H^{\rm op}}(\widehat x^{\rm op}(t)) = 0$$ and hence is an equivariant 1-periodic orbits for $H^{\rm op}$. The map ${\mathfrak x} \mapsto {\mathfrak x}^{\rm op}$ induces a one-to-one correspondence $${\rm crit} {\mathcal A}_H \cong {\rm crit} {\mathcal A}_{H^{\rm op}}$$ with critical values and Conley--Zehnder indices reversed. Similarly, if $\widehat J_t$ is an $S^1$-family of $K$-invariant almost complex structures on $V$, then define $$(\widehat J^{\rm op})_t = \widehat J_{-t}.$$ One can verify easily that if $(\widehat H, \widehat J)$ is admissible, so is $(\widehat H^{\rm op}, \widehat J^{\rm op})$. Now we define a Poincaré pairing on the vortex Floer homology. Let $(\widehat H_1, \widehat J_1)$ and $(\widehat H_2, \widehat J_2)$ be two regular bulk-avoiding admissible pairs on $V$. Consider the genus zero curve with two incoming cylindrical ends, denoted by $\Sigma_{\supset}$. Choose an area form with cylindrical ends on $\Sigma_{\supset}$. Define a $K$-invariant Hamiltonian perturbation $\widehat H_{\supset}$ on $\Sigma_{\supset}$ which is equal to $\widehat H_1 dt$ on the first cylindrical end and which is equal to $\widehat H_2^{\rm op} dt$ on the second cylindrical end. Choose a domain-dependent $K$-invariant almost complex structure $\widehat J_{\supset}$ which agrees on $\widehat J_1$ on the first cylindrical end and which is equal to $\widehat J_2^{\rm op}$ on the second cylindrical end. Consider the $\widehat H_{\supset}$-perturbed symplectic vortex equation on $\Sigma_{\supset}$ with respect to the family of almost complex structures $\widehat J_{\supset}$. Finite energy solutions converge to critical points of ${\mathcal A}_{H_1}$ resp. ${\mathcal A}_{H_2^{\rm op}}$ at the two cylindrical ends. Then given ${\mathfrak x} \in {\rm crit}{\mathcal A}_{H_1}$ and ${\mathfrak y}^{\rm op} \in {\rm crit} {\mathcal A}_{H_2^{\rm op}} \cong {\rm crit} {\mathcal A}_{H_2}$, one can obtain a well-defined count $${\mathfrak n}_{\supset}^{\mathfrak b} ({\mathfrak x}, {\mathfrak y}) \in {\mathbb Z}$$ by looking at rigid solutions. Define a bilinear pairing $$\langle \cdot, \cdot \rangle^{\mathfrak b}: {\it VCF}_\bullet^{\mathfrak b} ( \widehat H_1, \widehat J_1; \Lambda_R^\Gamma) \otimes {\it VCF}_\bullet^{\mathfrak b} ( \widehat H_2^{\rm op}, \widehat J_2^{\rm op}; \Lambda_R^\Gamma) \to R$$ by $$\langle \sum_{i=1}^\infty a_i {\mathfrak x}_i, \sum_{j=1}^\infty b_j {\mathfrak y}_j^{\rm op} \rangle^{\mathfrak b}:= \sum_{i, j} a_i b_j {\mathfrak n}_{\supset}({\mathfrak x}_i, {\mathfrak y}_j^{\rm op}).$$ An argument via energy inequality shows that the above form is finite and well-defined; by considering 1-dimensional moduli spaces one can show that the above pairing descends to homology $$\langle \cdot, \cdot \rangle^{\mathfrak b}: {\it VHF}_\bullet^{\mathfrak b} ( \widehat H_1, \widehat J_1; \Lambda_R^\Gamma) \otimes {\it VHF}_\bullet^{\mathfrak b} ( \widehat H_2, \widehat J_2; \Lambda_R^\Gamma) \to R.$$ One can also show that the pairing is compatible with respect to the continuation map. Hence it induces a pairing $$\langle \cdot, \cdot \rangle^{\mathfrak b}: {\it VHF}_\bullet^{\mathfrak b} (V; \Lambda_R^\Gamma) \otimes {\it VHF}_\bullet^{\mathfrak b} (V; \Lambda_R^\Gamma) \to R.$$ Now we specialize to the case when $\widehat H_1 = \widehat H_2 = \widehat H$ and $\widehat J_1 = \widehat J_2 = \widehat J$. In this case the pairing takes a simple form on the chain level. Indeed, if we choose $\widehat H_\supset$ and $\widehat J_\supset$ to be the trivial ones, then the countings $n_\supset^{\mathfrak b} ({\mathfrak x}, {\mathfrak y}^{\rm op})$ is 1 if ${\mathfrak x} = {\mathfrak y}$ and zero otherwise. Then if $$\begin{aligned} &\ \alpha = \sum a_i {\mathfrak x}_i \in {\it VCF}_\bullet^{\mathfrak b} (\widehat H, \widehat J; \Lambda_R^\Gamma),\ &\ \beta = \sum b_j {\mathfrak x}_j^{\rm op} \in {\it VCF}_\bullet^{\mathfrak b} (\widehat H^{\rm op}, \widehat J^{\rm op}; \Lambda_R^\Gamma)\end{aligned}$$ one has $$\langle \alpha, \beta \rangle^{\mathfrak b} = \sum_i a_i b_i \in R.$$ This sum is finite as ${\mathcal A}_H({\mathfrak x}_i) \to -\infty$ and ${\mathcal A}_{H^{\rm op}}({\mathfrak x}_j^{\rm op}) = - {\mathcal A}_H({\mathfrak x}_j) \to -\infty$. ### Pair-of-pants products A TQFT type construction allows us to define a multiplicative structure on the vortex Floer homology. In particular, using any volume form on the pair-of-pants with cylindrical ends, one can define the pair-of-pants product $$\ast_{\mathfrak b}: {\it VHF}_\bullet^{\mathfrak b} (V; \Lambda_R^\Gamma ) \otimes {\it VHF}_\bullet^{\mathfrak b} ( V; \Lambda_R^\Gamma ) \to {\it VHF}_\bullet^{\mathfrak b} (V; \Lambda_R^\Gamma )[n]$$ which is associative. Here $2n = {\rm dim} X$. The details were given in [@Wu_Xu]. There is also an identity element in the vortex Floer homology. Fix a regular bulk-avoiding admissible pair $(\widehat H, \widehat J)$. Consider a once-punctured sphere $\Sigma_{\rm cigar}$ which is biholomorphic to the complex plane. View the puncture as an output. Equip $\Sigma_{\rm cigar}$ with a cylindrical volume form $\nu_{\rm cigar}$ so that one has has the isometric identification $${\mathbb C} \setminus B_1 \cong [0, +\infty) \times S^1.$$ Turn on the Hamiltonian perturbation on this cylindrical end, meaning that one has a Hamiltonian perturbation $${\mathcal H} \in \Omega^1(\Sigma_{\rm cigar}, C^\infty_c (V)^K)\ {\rm s.t.}\ {\mathcal H}|_{[S, +\infty) \times S^1} = H_t dt\ {\rm for}\ S \gg 0.$$ Choose a domain-dependent $K$-invariant $\omega_V$-compatible almost complex structure ${\mathcal J}$ parametrized by $z \in \Sigma_{\rm cigar}$ such that over the cylindrical end it agrees with $J_t$. Consider the Hamiltonian perturbed symplectic vortex equation $$\begin{aligned} &\ \overline\partial_{A, {\mathcal H} } u = 0,\ &\ F_A + \mu(u) \nu^{\rm cigar} = 0.\end{aligned}$$ Each finite energy solution ${\mathfrak u} = (A, u)$ converges to an equivariant 1-periodic orbit and hence represents an element ${\mathfrak x} \in {\rm crit} {\mathcal A}_H$. Hence for each ${\mathfrak x}$ there is a moduli space $${\mathcal M}^{\rm cigar}({\mathfrak x}).$$ Elements in this moduli space have a uniform energy bound by $- {\mathcal A}_H ({\mathfrak x}) + C$ where $C$ depends on the perturbation data on the cigar which is uniformly bounded. The virtual dimension is $n - {\rm CZ}({\mathfrak x})$. Counting elements (with signs) of index zero moduli spaces ${\mathcal M}^{\rm cigar}({\mathfrak x})$ defines an element $${\bm 1}_{{\mathfrak b}, \widehat H}^{{\rm GLSM}} = \sum_{\mathfrak x} {\mathfrak n}_{\rm cigar}^{\mathfrak b} ({\mathfrak x}) {\mathfrak x} \in {\it VCF}_n^{\mathfrak b} (\widehat H, \widehat J; \Lambda_R^\Gamma ).$$ Standard TQFT argument shows that ${\bm 1}_{{\mathfrak b}, \widehat H}^{{\rm GLSM}}$ is closed, induces a well-defined element in ${\it VHF}_\bullet^{\mathfrak b} (V; \Lambda_R^\Gamma)$, and is the multiplicative identity of ${\it VHF}_\bullet^{\mathfrak b} (V; \Lambda_R^\Gamma)$. Denote this element by $${\bm 1}_{\mathfrak b}^{{\rm GLSM}} \in {\it VHF}_n^{\mathfrak b} (V; \Lambda_R^\Gamma).$$ **Lemma 58**. *The element ${\bm 1}_{\mathfrak b}^{\rm GLSM}$ is nonzero.* *Proof.* In the undeformed case this was proved using the closed-open map in [@Wu_Xu] and the fact that some Lagrangian Floer theory is nontrivial. Here as we know that the algebra ${\it VHF}_\bullet^{\mathfrak b} (V; \Lambda_R^\Gamma)$ is nonzero (see Lemma [Proposition 53](#computation){reference-type="ref" reference="computation"}) for any ring $R$, one must have ${\bm 1}_{\mathfrak b}^{\rm GLSM}\neq 0$. ◻ **Lemma 59**. *One has $$\langle \alpha, \beta \rangle^{\mathfrak b} \neq 0 \Longrightarrow \langle \alpha \ast_{\mathfrak b} \beta, {\bm 1}_{\mathfrak b}^{\rm GLSM}\rangle^{\mathfrak b} \neq 0.$$* *Proof.* This theorem follows from the standard TQFT and cobordism argument. See Figure [1](#figure1){reference-type="ref" reference="figure1"}. The details are left to the reader. {#figure1} ◻ Before we end this part, we state a major step towards our proof of the Hofer--Zehnder conjecture. **Theorem 5**. *There exists a bulk-deformation ${\mathfrak b}$ of the form $${\mathfrak b} = \sum_{j=1}^N \log c_j V_j$$ with $c_j \in {\mathbb Z}[{\bf i}]$ such that the algebra ${\it VHF}_\bullet^{\mathfrak b}(V; \Lambda_{\overline{\mathbb Q}})$ is semisimple in the sense of Definition [Definition 12](#defn_semisimple){reference-type="ref" reference="defn_semisimple"}.* The proof occupies Section [8](#section8){reference-type="ref" reference="section8"} and Section [9](#section9){reference-type="ref" reference="section9"}, using the closed-open string map in the vortex setting. ## Bulk-deformed spectral invariants, persistence modules, and barcodes We fit the bulk-deformed vortex Floer theory into the abstract packages developed by Usher etc. Let ${\mathfrak b}$ be a bulk-deformation of the form [\[small_bulk_deformation_general\]](#small_bulk_deformation_general){reference-type="eqref" reference="small_bulk_deformation_general"}. **Proposition 60**. *Given a regular bulk-avoiding pair $(\widehat H, \widehat J)$, the quadruple $${\mathfrak c}^{\mathfrak b}(\widehat H, \widehat J):= ( P_H, {\mathcal A}_H, {\rm CZ}_{(2)}, n^{\mathfrak b})$$ is a ${\mathbb Z}_2$-graded Floer--Novikov package over $R$ (see Definition [\[defn_fn\]](#defn_fn){reference-type="ref" reference="defn_fn"}).* *Proof.* Straightforward. ◻ Next we consider the quantitative dependence of the vortex Floer chain complex on the Hamiltonian. We restrict to the case where $R = {\mathbb K}$ is a field. The vortex Floer chain complex ${\it VCF}_\bullet^{\mathfrak b}(\widehat H, \widehat J; \Lambda_{\mathbb K}^\Gamma)$ is the associated Floer--Novikov complex. **Proposition 61**. *Given two regular bulk-avoiding pairs $(\widehat H_1, \widehat J_1)$ and $(\widehat H_2, \widehat J_2)$, the quasi-equivalence distance (see Definition [\[defn_quasiequivalence\]](#defn_quasiequivalence){reference-type="ref" reference="defn_quasiequivalence"}) between ${\it VCF}_\bullet^{\mathfrak b}(\widehat H_1, \widehat J_1; \Lambda_{\mathbb K}^\Gamma)$ and ${\it VCF}_\bullet^{\mathfrak b}( \widehat H_2, \widehat J_2; \Lambda_{\mathbb K}^\Gamma)$ is no greater than the Hofer distance between the induced Hamiltonians $H_1, H_2$ downstairs, i.e. $$d_Q \Big( {\it VCF}_\bullet^{\mathfrak b}(\widehat H_1, \widehat J_1; \Lambda_{\mathbb K}^\Gamma), {\it VCF}_\bullet^{\mathfrak b}( \widehat H_2, \widehat J_2; \Lambda_{\mathbb K}^\Gamma) \Big) \leq \max \Big\{ \int_0^1 \max_X (H_2 - H_1) dt,\ \int_0^1 \max_X (H_1 - H_2) dt \Big\}.$$* *Proof.* This follows from the quantitative analysis of the continuation map. As the bulk ${\mathfrak b}$ and the coefficient field are fixed, we drop it from notations. To show that the complex only depends on the induced Hamiltonian downstairs (measured by quasiequivalence distance), we need to introduce the parameter $\lambda$ (see [\[vortex_lambda\]](#vortex_lambda){reference-type="eqref" reference="vortex_lambda"}). For each regular bulk-avoiding triple $(\lambda, \widehat H, \widehat J)$, there is a Floer--Novikov package ${\mathfrak c}^\lambda( \widehat H, \widehat J)$ defined from (${\mathfrak b}$-deformed) counts of solutions to [\[vortex_lambda\]](#vortex_lambda){reference-type="eqref" reference="vortex_lambda"}. Denote the associated Floer--Novikov complex by ${\it VCF}_\bullet^\lambda(\widehat H, \widehat J; \Lambda_{\mathbb K}^\Gamma)$ with valuation denoted by $\ell^\lambda$. **Lemma 62**. *The quasi-equivalence distance between ${\it VCF}_\bullet^{\lambda_1}(\widehat H_1, \widehat J_1)$ and ${\it VCF}_\bullet^{\lambda_2}( \widehat H_2, \widehat J_2)$ is bounded by $$\widehat d_{\rm Hofer}( \widehat H_1, \widehat H_2):= \max \left\{ \int_0^1 \max_V ( \widehat H_2 - \widehat H_1) dt,\ \int_0^1 \max_V (\widehat H_1 - \widehat H_2) \right\}.$$* *Proof.* Indeed, this follows from the energy calculation for the continuation maps (see Proposition [Proposition 52](#prop_continuation_energy){reference-type="ref" reference="prop_continuation_energy"}). One can construct chain homotopy equivalences $\Phi$, $\Psi$ between these two complexes and maps $K_1$, $K_2$ as in the diagram $$\xymatrix{ {\it VCF}_\bullet^{\lambda_1} (\widehat H_1, \widehat J_1) \ar@(ul,ur)^{K_1} \ar@/^2.0pc/[r]^{\Phi} & {\it VCF}_\bullet^{\lambda_2} ( \widehat H_2, \widehat J_2) \ar@/^2.0pc/[l]_{\Psi} \ar@(ul,ur)^{K_2}}.$$ The first item of Definition [\[defn_quasiequivalence\]](#defn_quasiequivalence){reference-type="ref" reference="defn_quasiequivalence"} follows directly from [\[continuation_energy_inequality\]](#continuation_energy_inequality){reference-type="eqref" reference="continuation_energy_inequality"}. Using the same method, the second item of Definition [\[defn_quasiequivalence\]](#defn_quasiequivalence){reference-type="ref" reference="defn_quasiequivalence"} can be verified for the maps $K_1$, $K_2$. ◻ We fix the two regular bulk-avoiding pairs $(\widehat H_\pm, \widehat J_\pm)$. For each $\epsilon>0$, one can find a $K$-invariant cut-off function $\rho_\epsilon: V \to [0, 1]$ supported near $\mu^{-1}(0)$ such that if we define $\widehat H_\pm^\epsilon:= \rho_\epsilon \widehat H_\pm$, then $$\widehat d_{\rm Hofer}( \widehat H_-^\epsilon, \widehat H_+^\epsilon) \leq d_{\rm Hofer}(H_-, H_+) + \epsilon.$$ Hence in view of Lemma [Lemma 62](#quasi-equivalence_weak){reference-type="ref" reference="quasi-equivalence_weak"} above, we only need to prove the following. **Lemma 63**. *Suppose $(\widehat H_\pm, \widehat J_\pm)$ are two regular bulk-avoiding pairs such that $\widehat H_+$ and $\widehat H_-$ descend to the same Hamiltonian $H$ downstairs. Then $$d_Q ( {\it VCF}_\bullet( \widehat H_-, \widehat J_-), {\it VCF}_\bullet( \widehat H_+, \widehat J_+)) = 0.$$* *Proof.* We prove that the quasi-equivalence distance is less than $\epsilon$ for all $\epsilon>0$. Notice that the potential failure of this assertion comes from the difference between $\widehat H_-$ and $\widehat H_+$ which is *a priori* large outside $\mu^{-1}(0)$. We use the adiabatic limit argument to push solutions contributing to the continuation maps near the level set $\mu^{-1}(0)$. Choose a sequence $\lambda_i \to \infty$. For each $\lambda_i$, one can choose a $\lambda_i$-admissible lift $\widehat H_\pm^{\lambda_i}$ of $H$. As the admissible condition is only about the infinitesimal behaviors of the lifts $\widehat H_\pm^{\lambda_i}$ near lifts of 1-periodic orbits of $H$, we may require that $$\| \widehat H_\pm^{\lambda_i} - \widehat H_\pm \|_{C^0} \leq \epsilon.$$ Hence by Lemma [Lemma 62](#quasi-equivalence_weak){reference-type="ref" reference="quasi-equivalence_weak"}, one only needs to consider the quasi-equivalence $$d_Q \left( {\it VCF}_\bullet^{\lambda_i} ( \widehat H_-^{\lambda_i}, \widehat J_-^{\lambda_i}), {\it VCF}_\bullet^{\lambda_i}( \widehat H_+^{\lambda_i}, \widehat J_+^{\lambda_i}) \right).$$ We claim that the above sequence (in $i$) converges to zero. We set up the moduli spaces for the continuation maps. Choose a cut-off function $\chi: {\mathbb R} \to [0, 1]$ which is non-decreasing, equals zero on $(-\infty, 0]$, and equals $1$ on $[1, +\infty]$. Consider the equation with $$\widehat H_{s, t}^{\lambda_i} = ( 1- \chi(s)) \widehat H_-^{\lambda_i} + \chi(s) \widehat H_+^{\lambda_i}.$$ We claim that, for all $\epsilon>0$, there exists $i_\epsilon>0$ such that when $i \geq i_\epsilon$, for all finite energy solutions to [\[vortex_continuation\]](#vortex_continuation){reference-type="eqref" reference="vortex_continuation"}, if the limit at $\pm \infty$ is ${\mathfrak x}_\pm$, then one has $${\mathcal A}_H({\mathfrak x}_{+, i}) - {\mathcal A}_H({\mathfrak x}_{-, i}) \leq \epsilon.$$ This would establish item (1) of Definition [\[defn_quasiequivalence\]](#defn_quasiequivalence){reference-type="ref" reference="defn_quasiequivalence"}. Suppose on the contrary that this is not true. Then there exist $\delta>0$, a subsequence (still indexed by $i$), a sequence of solutions ${\mathfrak u}_i = (u_i, \xi_i, \eta_i)$ to the equation connecting ${\mathfrak x}_{-, i}$ and ${\mathfrak x}_{+, i}$ such that $${\mathcal A}_H({\mathfrak x}_{+, i}) - {\mathcal A}_H({\mathfrak x}_{-, i}) \geq \delta > 0.$$ By the energy identity [\[continuation_energy_identity\]](#continuation_energy_identity){reference-type="eqref" reference="continuation_energy_identity"}, one has a uniform bound which is independent of $\lambda_i$: $$E_{\lambda_i} ({\mathfrak u}_i) = {\mathcal A}_H({\mathfrak x}_{-, i}) - {\mathcal A}_H({\mathfrak x}_{+, i}) - \int_{[0, 1]\times S^1} \partial_s \widehat H_{s, t}^{\lambda_i} (u) ds dt \leq C.$$ Now one can apply the adiabatic limit argument. Notice that although we cannot guarantee the convergence of $\widehat H_{s, t}^{\lambda_i}$, but we may require that $\widehat J_{s, t}^{\lambda_i}$ converges in sufficiently high order to a fixed almost complex structure $\widehat J$ outside a compact subset of $V$. In the $\lambda_i \to \infty$ limit, *a priori* there are three types of bubbles (see [@Gaio_Salamon_2005 Section 11]): holomorphic spheres in $V$, holomorphic spheres in $X$, and *affine vortices*, which are solutions to the vortex equation over ${\mathbb C}$ (without Hamiltonian term). The three kind of bubbles can be classified by the rate of energy concentration compared to the rate of the divergence $\lambda_i \to \infty$. As there is a lower bound on the energy of these bubbles, the uniform bound on energy implies that, after passing to a subsequence (still indexed by $i$), except near a finite subset $Z \subset [0, 1]\times S^1=:Q$ at which bubbling could occur, the energy density $$|\partial_s u_i + {\mathcal X}_{\xi_i}(u_i)|^2 + \lambda_i^2 |\mu(u_i)|^2$$ stays bounded. In particular, the map $u_i|_Q$ stays arbitrarily close to $\mu^{-1}(0)$ except near $Z$ as $i \to \infty$. More precisely, for any $r>0$, there exists $i_r>0$ such that for all $i \geq i_r$, $$\sup_{z\in [0, 1]\times S^1 \setminus B_r(Z)} |\mu(u_i(z))| \leq r.$$ Then one has $$\begin{gathered} {\mathcal A}_H({\mathfrak x}_{+, i}) -{\mathcal A}_H({\mathfrak x}_{-, i}) \leq \int_{Q} |\partial_s \widehat H_{s, t}^{\lambda_i} (u_i)| ds dt\\ \leq \int_{Q \setminus B_r(Z)} |\partial_s \widehat H_{s, t}^{\lambda_i}(u_i)| ds dt + \int_{B_r(Z)} |\partial_s \widehat H_{s, t}^{\lambda_i}(u_i)| ds dt.\end{gathered}$$ As $\widehat H_-^{\lambda_i} = \widehat H_+^{\lambda_i}$ on $\mu^{-1}(0)$, the first item is bounded by $Cr$; the second term is bounded by $C{\rm Area}(B_r(Z))$ which can be arbitrarily small. This contradicts the assumption that ${\mathcal A}_H({\mathfrak x}_{+, i}) - {\mathcal A}_H({\mathfrak x}_{-, i}) \geq \delta > 0$. Therefore, we established item (1) of Definition [\[defn_quasiequivalence\]](#defn_quasiequivalence){reference-type="ref" reference="defn_quasiequivalence"}. The case of item (2) is similar and hence omitted. ◻ Now the proof of Proposition [Proposition 61](#prop_quasiequivalence_distance){reference-type="ref" reference="prop_quasiequivalence_distance"} is complete. ◻ ### Spectral invariants Spectral numbers of Hamiltonian diffeomorphisms were introduced by Oh [@Oh_spectral_1], Schwarz [@Schwarz_spectral] and enhanced by Entov--Polterovich [@Entov_Polterovich_1; @Entov_Polterovich_2; @Entov_Polterovich_3]. In [@Wu_Xu] Wu and the second author constructed the analogue in the vortex Floer theory. By Theorem [\[Usher_thm\]](#Usher_thm){reference-type="ref" reference="Usher_thm"} and Proposition [Proposition 60](#prop_obvious){reference-type="ref" reference="prop_obvious"}, one can define the spectral numbers $$\rho^{\mathfrak b} (\alpha; \widehat H, \widehat J):= \rho_{{\mathfrak c}^{\mathfrak b}(\widehat H, \widehat J)}(\alpha)\in {\mathbb R} \cup \{-\infty\},\ \forall \alpha \in {\it VHF}_\bullet^{\mathfrak b}(V; \Lambda_{{\mathbb Z}[{\bf i}]}^\Gamma).$$ One can establish the following properties of these spectral numbers, which were proved in [@Wu_Xu] in the undeformed (${\mathfrak b} = 0$) case. **Theorem 64**. *(cf. [@Wu_Xu Proposition 3.6]) The spectral numbers $\rho^{\mathfrak b} (\alpha; \widehat H, \widehat J)$ have the following properties.* 1. ***(Independence of lifting and almost complex structure)** The number $\rho^{\mathfrak b} (\alpha; \widehat H, \widehat J)$ only depends on the induced Hamiltonian $H$ downstairs. Denote this number by $$c^{\mathfrak b} (\alpha, H) \in {\mathbb R}.$$* 2. ***(Homogeneity)** Given $\alpha \in {\it VHF}(V; \Lambda_{{\mathbb Z}[{\bf i}]}^\Gamma)$ and $\lambda \in \Lambda_{{\mathbb Z}[{\bf i}]}^\Gamma$, for any $H$, one has $$c^{\mathfrak b} (\lambda \alpha, H) = c^{\mathfrak b} (\alpha, H) - {\mathfrak v}(\lambda).$$ One uses this formula to extend the spectral numbers to classes in $${\it VHF}_\bullet^{\mathfrak b} (V; \Lambda_{{\mathbb Z}[{\bf i}]} ) = {\it VHF}_\bullet^{\mathfrak b} ( V; \Lambda_{{\mathbb Z}[{\bf i}]}^\Gamma) \otimes_{\Lambda_{{\mathbb Z}[{\bf i}]}^\Gamma} \Lambda_{{\mathbb Z}[{\bf i}]}.$$* 3. ***(Lipschitz continuity)** Given any two nondegenerate Hamiltonians $H_1, H_2$ downstairs, one has $$\int_{S^1} \min_{X} ( H_1 - H_2) dt \leq c^{\mathfrak b} (\alpha, H_1) - c^{\mathfrak b} (\alpha, H_2) \leq \int_{S^1} \max_X (H_1 - H_2) dt.$$ This implies that $c^{\mathfrak b} (\alpha, H)$ is defined for all Hamiltonians.* 4. ***(Invariance)** $c^{\mathfrak b} (\alpha, H)$ only depends on the homotopy class of the Hamiltonian path $\tilde \phi_H$ on $X$. Let ${\rm Ham}(X)$ be the group of Hamiltonian diffeomorphisms on $X$ and let ${\rm H}{\widetilde{\rm am}}(X) \to {\rm Ham}(X)$ be the covering of homotopy classes of Hamiltonian isotopies on $X$. Then we can define $$c^{\mathfrak b} (\alpha, \tilde \phi ) \in {\mathbb R} \cup \{-\infty\}\ \forall \alpha \in {\it VHF}_\bullet^{\mathfrak b}(V; \Lambda_R),\ \tilde\phi \in {\rm H}{\widetilde{\rm am}}(X).$$* 5. ***(Triangle inequality)** For any $\alpha_1, \alpha_2 \in {\it VHF}(V; \Lambda_R)$ and $\tilde \phi_1, \tilde \phi_2\in {\rm H}{\widetilde{\rm am}}(X)$ one has $$c^{\mathfrak b} ( \alpha_1 \ast \alpha_2, \tilde \phi_1 \tilde \phi_2) \leq c^{\mathfrak b} ( \alpha_1, \tilde \phi_1) + c^{\mathfrak b} ( \alpha_2, \tilde \phi_2).$$* **Definition 65**. The **valuation** of a class $\alpha \in {\it VHF}_\bullet^{\mathfrak b} (V; \Lambda_R)$ is defined to be $${\mathcal A}^{\mathfrak b} (\alpha):= c^{\mathfrak b} (\alpha, \tilde {\rm Id})\in {\mathbb R} \cup \{+\infty\}.$$ ### Poincaré duality One useful property of the spectral numbers is related to the Poincaré duality map. **Proposition 66**. *Let ${\mathbb K}$ be a field.* 1. *For any $\alpha \in {\it VHF}_\bullet^{\mathfrak b}(V; \Lambda_{\mathbb K})$ and $\tilde \phi \in {\rm H}{\widetilde{\rm am}}(X)$, there holds $$c^{\mathfrak b} (\alpha, \tilde \phi) = - \inf \Big\{ c^{\mathfrak b} ( \beta, \tilde \phi^{-1} )\ |\ \langle \alpha, \beta \rangle^{\mathfrak b} \neq 0 \Big\}.$$* 2. *If $\langle \alpha, \beta \rangle^{\mathfrak b} \neq 0$, then $${\mathcal A}^{\mathfrak b} (\alpha) + {\mathcal A}^{\mathfrak b} (\beta)\geq 0.$$* *Proof.* Notice that one only needs to prove this proposition for coefficient field being $\Lambda_{\mathbb K}^\Gamma$. In the case of ordinary Hamiltonian Floer theory, the proof of (1) uses the PSS map and the correspondence between the pairing $\langle \cdot, \cdot \rangle$ and the intersection pairing on the singular homology of the manifold (see [@Entov_Polterovich_1][@Ostrover_2006][@FOOO_spectral]). It was pointed in [@Usher_2010] that (1) holds for abstract filtered Floer--Novikov complexes. As the complex ${\it VHF}_\bullet^{\mathfrak b} ( \widehat H, \widehat J; \Lambda_{\mathbb K}^\Gamma)$ is an abstract filtered Floer--Novikov complex over $\Lambda_{\mathbb K}^\Gamma$ (see Proposition [Proposition 60](#prop_obvious){reference-type="ref" reference="prop_obvious"}), (1) follows. For (2), take $\tilde \phi = {\rm Id}$. Then $${\mathcal A}^{\mathfrak b}(\alpha) = c^{\mathfrak b} (\alpha, {\rm Id}) = - \inf \Big\{ {\mathcal A}^{\mathfrak b}(\beta) \ |\ \langle \alpha, \beta \rangle^{\mathfrak b} \neq 0 \Big\}.$$ Hence if $\langle \alpha, \beta \rangle^{\mathfrak b} \neq 0$, ${\mathcal A}^{\mathfrak b}(\beta) \geq - {\mathcal A}^{\mathfrak b}(\alpha)$. ◻ ### Persistence modules and barcodes Recall that (see Subsection [3.4](#subsection_persistence){reference-type="ref" reference="subsection_persistence"}) to any filtered Floer--Novikov complex $CF_\bullet({\mathfrak c})$ over the Novikov field $\Lambda_{\mathbb K}^\Gamma$ one can associate a persistence module ${\bm V}({\mathfrak c})$. In particular, for each regular bulk-avoiding admissible pair $(\widehat H, \widehat J)$, the bulk-deformed vortex Floer complex ${\it VCF}_\bullet^{\mathfrak b}(\widehat H, \widehat J; \Lambda_{\mathbb K}^\Gamma)$ gives a persistence module, denoted by $${\bm V}^{\mathfrak b} (\widehat H, \widehat J; \Lambda_{\mathbb K}^\Gamma).$$ We omit the dependence on the bulk deformation ${\mathfrak b}$ most of the time. One can check easily that we can extend the coefficient field to the universal Novikov field $\Lambda_{\mathbb K}$, obtaining a persistence module ${\bm V} (\widehat H, \widehat J; \Lambda_{\mathbb K})$ with $$V^s(\widehat H, \widehat J; \Lambda_{\mathbb K}):= HF_\bullet^{\leq s} (\widehat H, \widehat J; \Lambda_{0, {\mathbb K}}^\Gamma) \otimes_{\Lambda_{0, {\mathbb K}}^\Gamma} \Lambda_{0, {\mathbb K}}.$$ When the ground field ${\mathbb K}$ is clear from the context, we often abbreviate this persistence module by ${\bm V}(\widehat H, \widehat J)$. One can prove, using the continuation map, that up to isomorphism, this persistence module is independent of the choice of the almost complex structure $\widehat J$. Hence denote the persistence module by ${\bm V}(\widehat H)$. One can also use the same idea of proving Proposition [Proposition 61](#prop_quasiequivalence_distance){reference-type="ref" reference="prop_quasiequivalence_distance"} that, for different lifts $\widehat H_1, \widehat H_2$ of the same Hamiltonian $H$ downstairs, the interleaving distance between ${\bm V}(\widehat H_1)$ and ${\bm V}(\widehat H_2)$ is zero. By identifying persistence modules with zero interleaving distance, the persistence module only depends on the Hamiltonian path $\tilde \phi \in {\rm H}{\widetilde{\rm am}}(X)$ generated by $H$. Hence we loosely denote the object by ${\bm V}(\tilde\phi)$. Recall also that to any Floer--Novikov complex one can associate a barcode (and hence a reduced barcode). The reduced barcode corresponding to a regular bulk-avoiding admissible pair $(\widehat H, \widehat J)$ is denoted by ${\mathcal B}(\widehat H, \widehat J)$. One can prove that (similar to the case of ordinary Floer barcodes, see [@Usher_2013 Proposition 5.3]) the reduced barcode only depends on the time-1 map $\phi = \phi_H$ on the toric manifold $X$. Hence we also denote it by ${\mathcal B}(\phi)$. # Local Floer theory {#sec:local-Floer} To extend the Hofer--Zehnder conjecture to degenerate Hamiltonian diffeomorphisms, one needs to have a good notion of counts of fixed points. Following [@Shelukhin_2022], we will use the rank of a local version of the vortex Floer homology (with bulk deformation), which is ultimately isomorphic to the local Floer homology in the classical sense, to define a homological count of fixed points. This section can be skipped at first reading, especially if the reader is mainly interested in the nondegenerate case. The following statements will be proved in this section. **Theorem 67**. *Let ${\mathbb K}$ be a field. Let $\phi: X \to X$ be a Hamiltonian diffeomorphism and $p \in X$ be an isolated fixed point. Then there is a ${\mathbb Z}_2$-graded ${\mathbb K}$-vector space $${\it VHF}^{\rm loc}(\phi, p; {\mathbb K})$$ satisfying the following properties.* 1. *If $p$ is a nondegenerate fixed point, then ${\it VHF}^{\rm loc}(\phi, p; {\mathbb K})$ has rank $1$ graded by the Conley--Zehnder index of $p$ (modulo $2$).* 2. *If $\phi^s$, $s \in [0, 1]$ is a smooth family of Hamiltonian diffeomorphisms such that $p$ is a uniformly isolated fixed point, i.e., there exists an open neighborhood of $p$ in which $\phi_s$ has $p$ as the only fixed point for all $s$. Then ${\it VHF}^{\rm loc}(\phi^0, p; {\mathbb K}) \cong {\it VHF}^{\rm loc} ( \phi^1, p; {\mathbb K})$.* 3. *If $\phi'$ is a generic $C^2$ small perturbation of $\phi$ supported near $p$, then the number of fixed points of $\phi'$ near $p$ is at least ${\rm rank} {\it VHF}^{\rm loc}(\phi, p; {\mathbb K})$.* 4. *If $\phi^k$ is an admissible iteration of $\phi$ at $p$, meaning that $\lambda^k \neq 1$ for all eigenvalues $\lambda \neq 1$ of $D\phi_x$, which implies that $p$ is also an isolated fixed point of $\phi^k$, then $${\rm rank} {\it VHF}^{\rm loc}(\phi, p; {\mathbb K}) = {\rm rank} {\it VHF}^{\rm loc} (\phi^k, p; {\mathbb K})$$* The homology group ${\it VHF}^{\rm loc}(\phi, p; {\mathbb K})$ is constructed via generating Hamiltonians of $\phi$. For a Hamiltonian $H_t: S^1 \times X \to {\mathbb R}$ with $x: S^1 \to X$ being an isolated $1$-periodic orbit, we will define the local Floer homology group $${\it VHF}^{\rm loc}(H, x; {\mathbb K}).$$ It turns out that the vortex version of local Floer homology is in fact isomorphic to the classical one. **Proposition 68**. *There is an isomorphism $${\it VHF}^{\rm loc}(H, x; {\mathbb K}) \cong HF^{\rm loc}(H, x; {\mathbb K}).$$* As we will use bulk-deformed vortex Floer theory, the right notion of local Floer homology may *a priori* depend on the bulk deformation, when the fixed point is contained in the bulk divisor. However, we will prove (see Proposition [Proposition 73](#prop:indep-bulk){reference-type="ref" reference="prop:indep-bulk"}) that the bulk-deformed local vortex Floer homology is (non-canonically) isomorphic to the undeformed one. Hence the homological count we defined here does not see the effect of bulk deformation *a posteri*. Now we use the rank of local Floer homology to define the so-called homological count of the number of fixed points. **Definition 69**. Let ${\mathbb K}$ be a field. Given a Hamiltonian diffeomorphism $\phi$ on $X$ with only isolated fixed points, the **homological count** (over coefficient field ${\mathbb K}$) of the number of fixed points of $\phi$ is $$\label{homological_count} N(\phi, {\mathbb K}):= \sum_{p \in {\rm Fix} \phi} {\rm dim}_{\mathbb K} {\it VHF}^{\rm loc}(\phi, p; {\mathbb K}) = \sum_{p \in {\rm Fix} \phi} {\rm dim}_{\mathbb K} {\it HF}^{\rm loc}(\phi, p; {\mathbb K}).$$ By definition, when $\phi$ is nondegenerate, we have $$N(\phi, {\mathbb K}) = \# {\rm Fix} (\phi).$$ On the other hand, if $\phi$ be a Hamiltonian diffeomorphism on $X$ with only isolated fixed points, then for $\phi^k$ being an admissible iteration at all the fixed points of $\phi$ with all fixed points isolated, we know that $$N(\phi, {\mathbb K}) \leq N( \phi^k, {\mathbb K}).$$ ## Local Morse and Floer homology ### Local Morse homology We follow the treatment of local Morse and Floer homology by Ginzburg [@Ginzburg_2010]. First, let $M$ be a smooth manifold and $f: M \to {\mathbb R}$ be a smooth function. Suppose $x$ is an isolated (but not necessarily nondegenerate) critical point. Then for any coefficient ring $R$, there is an invariant, the *local Morse homology* $$HM^{\rm loc}(f, x)$$ defined by taking the homology of a Morse-type complex over $R$ of a small generic perturbation of $f$ in a sufficiently small neighborhood of $x$, which only takes into account critical points and gradient trajectories (for a generic Riemannian metric) contained in that neighborhood. We recall the details of the construction as this is the prototype of the local version of the (vortex) Floer homology. First, choose a small neighborhood $U$ of $p$ which contains no other critical points of $f$. Fix a reference Riemannian metric $g_U$ on $U$ to measure relevant norms. Let $U'$ be a smaller neighborhood of $p$ whose closure is contained in $U$. Let $f_1$ be an $\epsilon$-small perturbation supported in $U'$, i.e. $$\begin{aligned} &\ {\rm supp}(f_1 - f) \subset U',\ &\ \| f_1 - f \|_{C^2} < \epsilon.\end{aligned}$$ A generic such perturbation $f_1$ is Morse inside $U$ and ${\rm crit}(f_1|_U)$ is contained in $U'$. Then consider the Morse complex of $f_1: U \to {\mathbb R}$, which is freely generated by critical points of $f_1 |_U$ over $R$, graded by the Morse index. To define the differential, consider an arbitrary Riemannian metric $g_1$ on $U$. Consider, for each two critical points $p_1, q_1$ of $f_1$ the moduli space of negative gradient trajectories of $f_1$ (with respect to $g_1$) that connect $p_1$ and $q_1$. Then by a compactness argument, for $\epsilon$ sufficiently small, all trajectories connecting critical points of $f_1|_U$ must stay in $U'$. Then by a small perturbation of the Riemannian metric $g_1$ to achieve transversality, one can count rigid negative gradient trajectories over ${\mathbb Z}_2$; choosing an orientation of $U$ and orientations on unstable manifolds of critical points of $f_1|_U$ one can define integral counts. Hence one obtains a chain complex whose homology is defined to be the local Morse homology $HM^{\rm loc}(f, p)$. Using continuation map one can prove that the local Morse homology is an invariant, which only depends on the infinitesimal behavior of $f$ at $p$. Indeed, fix $U$, $U'$, $g_U$ as above. Let $f_2$ be another $\epsilon$-small perturbation supported in $U'$. Let $g_2$ be another Riemannian metric on $U$ for which the local Morse complex of $(f_2, g_2)$ is defined. Choose a homotopy $(f_{\chi(s)}, g_{\chi(s)})$ between $(f_1, g_1)$ and $(f_2, g_2)$ using a fixed cut-off function $\chi: (-\infty, +\infty) \to [1, 2]$. By possibly shrinking the value of $\epsilon$, one can show that in this case all solutions to the continuation equation $$\dot x(s) + \nabla^{g_{\chi(s)}} f_{\chi(s)} (x(s)) = 0$$ are contained in $U'$. By slightly perturbing the data $(f_{\chi(s)}, g_{\chi(s)})$ one can achieve transversality for the moduli spaces of this equation and hence define a continuation map. The same type of argument can be applied to show that the continuation map is uniquely determined up to chain homotopy. As a result, in the same vein of classical arguments in Floer theory, one can prove that the local Morse homology $HM^{\rm loc}(f, x)$ is independent of the small perturbation of the function and the Riemannian metric. One can also see that the homology is neither dependent on the neighborhoods $U$, $U'$ nor the reference metric $g_U$. Hence $HM^{\rm loc}(f, x)$ is an invariant of the germ of $f$ at $x$. ### Local Floer homology One can similarly define local homology groups in the Floer setting which admits extensions to the vortex setting. Note that the explicit construction depends on the 1-periodic family of Hamiltonians but we will prove that it only depends on the time-1 map. Let $(M, \omega)$ be a symplectically aspherical manifold and $H$ be a 1-periodic family of Hamiltonians on $M$ with time-1 map $\varphi_H: M \to M$. Choose a reference Riemannian metric $g_M$ which induces a distance function $d_M$ on $M$. **Lemma 70**. *If $q \in {\rm Fix} (\varphi_H)$ is an isolated fixed point corresponding to a 1-periodic orbit $x: S^1 \to M$, then there exists $r>0$ such that, for all smooth loop $y: S^1 \to M$ with $$\label{epsilon_close} \sup_{t \in S^1} d_M( y (t), x(t)) < r,$$ if $y$ is a 1-periodic orbit of $H$, then $y \equiv x$.* *Proof.* This follows from the definition of isolated fixed point. ◻ We say that a loop $y: S^1 \to M$ is $r$-close to $x$ if $y(t)$ satisfies [\[epsilon_close\]](#epsilon_close){reference-type="eqref" reference="epsilon_close"}. We may choose $r$ smaller than the injectivity radius of $g_M$. Now we review the definition of the local Floer homology. Fix $r$ as in Lemma [Lemma 70](#lemma_epsilon_close){reference-type="ref" reference="lemma_epsilon_close"}. Let $R$ be a coefficient ring such as ${\mathbb Z}_2$ or ${\mathbb Z}$. The local Floer homology $HF^{\rm loc} (H, x)$ is a ${\mathbb Z}_2$-graded $R$-module. To define it, consider a perturbation $H_{1,t}: S^1 \times M \to {\mathbb R}$ satisfying 1. For all $t \in S^1$, ${\rm supp} (H_{1, t} - H_t) \subset B_{r/2} (x(t))$. 2. $\| H_1 - H \|_{C^2(S^1 \times M)} \leq \delta$. We call such perturbations $\delta$-small perturbations. **Lemma 71**. *For any $\rho>0$, there exists $\delta>0$ such that for all $\delta$-small perturbation $H_1$, if $y$ is a 1-periodic orbit of $H_1$ which is $r$-close to $x$, then $y$ is $\rho$-close to $x$.* *Proof.* Suppose this is not true, then there exist $\rho>0$, a sequence $\delta_i \to 0$ and a sequence of $\delta_i$-small perturbations $H_i$, and a sequence of 1-periodic orbits $y_i$ of $H_i$ with $d_M(y_i(t_i), x(t_i)) \geq \rho$ for some $t_i \in S^1$. Then $H_i \to H$ in $C^2$. By choosing a subsequence, one may assume that $y_i$ converges to a 1-periodic orbit $y_\infty$ of $H$. As $y_i(0)$ is $r$-close to $x(0)$, one can see that $y_\infty(0)$ is $r$-close to $x(0)$, hence is a fixed point of $\varphi$ which is $r$-close to $x(0)$. As $x(0)$ is an isolated fixed point, $y_\infty(0) = x(0)$ and hence $y_\infty(t) \equiv x(t)$. This contradicts the assumption. ◻ The following construction is analogous to the Morse case. For each loop $\overline x: S^1 \to M$ which is $\frac{r}{2}$-close to $x$, one can define the action functional $${\mathcal A}_{H_1}^{\rm loc}(\overline{x}) = - \int_{[0,1]\times S^1} u^* \omega + \int_{S^1} H_1(\overline{x}(t)) dt.$$ Here $u: [0, 1]\times S^1 \to M$ is a "small" cobordism connecting $\overline{x}$ and $x$ whose homotopy class is canonical as $\overline{x}$ is sufficiently close to $x$. Then for each pair of critical points $x_1, y_1$ of ${\mathcal A}_{H_1}^{\rm loc}$, the difference $${\mathcal A}_{H_1}^{\rm loc}(x_1) - {\mathcal A}_{H_1}^{\rm loc}(y_1)$$ is sufficiently small; this is needed to run the compactness argument. Consider the Floer complex generated over $R$ by critical points of ${\mathcal A}_{H_1}^{\rm loc}$, graded by the Conley--Zehnder index modulo $2$. Notice that for any two generators $x_1, y_1$, there is a canonical homotopy class of (short) cylinders connecting them. To define the differential, choose a 1-periodic family of $\omega$-compatible almost complex structures $J_1$ and consider Floer trajectories (using $J_1$) connecting generators in the canonical homotopy class. Then the energy identity for Floer trajectories guarantees that when $\rho$ and $\delta$ are sufficiently small, the total energy of Floer trajectories can be arbitrarily small. More precisely, we consider Floer trajectories $u: {\mathbb R}\times S^1 \to M$ for $(H_1, J_1)$ satisfying $$\sup_{s \in {\mathbb R}} \sup_{t \in S^1} d_M( u(s, t), x(t)) < r.$$ The smallness of total energy guarantee that the above supremum can be arbitrarily small. One can hence guarantee compactness (up to breaking) of such Floer trajectories which are sufficiently close to $x(t)$ and define a chain complex. Coherent orientations can also be chosen if one would like to define the complex over ${\mathbb Z}$. One can prove using the continuation map that the local Floer homology is independent of the pair $(H_1, J_1)$. We omit the details. We denote the local Floer homology defined in this way as $HF^{\rm loc}(H, x;R)$. Moreover, we can prove that the local Floer homology only depends on the fixed point $q$ and the time-1 map $\phi \in {\rm Ham}(X)$. Hence we denote the local Floer homology by $$HF^{\rm loc} (\phi, q; R).$$ Among various properties of the local Floer homology we only recall the following one. **Proposition 72**. *[@Ginzburg_Gurel_2010 Theorem 1.1] Let ${\mathbb K}$ be a field. If $q$ is an isolated fixed point of $\phi$, and $\phi^k$ is an iteration admissible at $p$, then $${\rm rank}_{\mathbb K} {\it HF}^{\rm loc} (\phi^k, q; {\mathbb K}) = {\rm rank}_{\mathbb K} {\it HF}^{\rm loc} (\phi, q; {\mathbb K}).$$* ## Local vortex Floer homology with bulk We adapt the definition of local Floer homology in the vortex setting, possibly with bulk deformations, and establish analogues of the statements in Proposition [Proposition 72](#prop:local-Floer){reference-type="ref" reference="prop:local-Floer"} as listed in Theorem [Theorem 67](#thm:property-vortex-local){reference-type="ref" reference="thm:property-vortex-local"}. Let ${\mathfrak b}$ be a small bulk deformation. Let $\phi: X \to X$ be a Hamiltonian diffeomorphism and $q \in X$ be an isolated fixed point. We would like to define a local invariant $${\it VHF}^{\mathfrak b}_{\rm loc}(\phi, q; {\mathbb K}).$$ Indeed, let $H$ be a 1-periodic family of Hamiltonian on $X$ generating the Hamiltonian isotopy $\phi_t$ with $\phi_1 = \phi$. Let $x(t) = \phi_t(p)$ be the corresponding 1-periodic orbit of $H$. Notice that even if $x(t)$ is nondegenerate, it may intersect the bulk divisor $D\subset X$. Choose a small perturbation $H_1$ of $H$ supported near $x(t)$ such that all nearby 1-periodic orbits are nondegenerate and are disjoint from the bulk divisor $D$. Let $\widehat H$ be a $K$-invariant lift of $H$ and $\widehat H_1$ be a $K$-invariant admissible lift of $H_1$. Then the 1-periodic orbit $x(t)$ lifts to a gauge equivalence class of equivariant 1-periodic orbits. Let ${\mathfrak x}(t) = (x(t), \eta(t))$ be a representative. There are also gauge equivalence classes of equivariant 1-periodic orbits of $H_1$ which are near ${\mathfrak x}$. Indeed, fixing the $L_0 K$-orbit of ${\mathfrak x}(t)$, there are well-defined $L_0 K$-orbits of equivariant 1-periodic orbits which are nearby. Then for each pair of nearby equivariant 1-periodic orbits ${\mathfrak x}_1$, ${\mathfrak y}_1$ of $\widehat H_1$, there is a canonical homotopy class of (small) cylinders connecting ${\mathfrak x}_1$ and ${\mathfrak y}_1$. Consider the moduli space of solutions to the vortex equation over the cylinder connecting ${\mathfrak x}_1$ and ${\mathfrak y}_1$. The energy of these solutions is $${\mathcal A}_{H_1}({\mathfrak x}_1) - {\mathcal A}_{H_2}({\mathfrak y}_1)$$ which are arbitrarily small. Then similar to the case of ordinary local Floer homology, these moduli spaces can be used to define a chain complex over any coefficient filed ${\mathbb K}$. As the orbits are disjoint from $D$, one can also use topological intersection numbers with the bulk divisor and associated weighted counts to define the bulk-deformed version. Denote the resulting homology by $${\it VHF}^{\mathfrak b}_{\rm loc}(H, x; {\mathbb K}).$$ The continuation argument shows that the homology is independent of the data $(\widehat H_1, \widehat J_1)$. On the other hand, *a priori* the homology depends on the bulk ${\mathfrak b}$. When ${\mathfrak b} = 0$, denote this homology by ${\it VHF}_{\rm loc}(H, x; {\mathbb K})$. **Proposition 73**. *One has $$\label{local_independent} {\it VHF}^{\mathfrak b}_{\rm loc}( H, x; {\mathbb K}) \cong {\it VHF}^{\rm loc}( H, x; {\mathbb K}).$$* *Proof.* First, suppose $x$ does not intersect the bulk divisor $D \subset X$. Then for a small perturbation of $H$, all cylinders contributing to the definition of the local Floer homology have zero topological intersection number with the divisor upstairs. Hence [\[local_independent\]](#local_independent){reference-type="eqref" reference="local_independent"} holds in this case. Now suppose $x$ intersects the bulk divisor $D$. One can find a loop of Hamiltonians $\psi(t)$ supported near $x(t)$ such that $y(t):= \psi(t) (x(t))$ is disjoint from $D$. Moreover, define $$y(t) = (\psi(t) \phi(t) \psi(0)^{-1})(\psi(0) (q)) = (\psi(t) \phi(t) \psi(0)^{-1})(y(0)),$$ then $y(t)$ is a 1-periodic orbit of the Hamiltonian isotopy $\psi(t) \phi(t) \psi(0)^{-1}$. Let the generating Hamiltonian function of this new family be $G$, which can be made sufficiently close to $H$. Then $y(t)$ is also an isolated 1-periodic orbit of $G$. Then a generic perturbation of $G$ also serves as a perturbation of $H$. Hence $${\it VHF}^{\mathfrak b}_{\rm loc}(H, x; {\mathbb K}) \cong {\it VHF}^{\mathfrak b}_{\rm loc}(G, y; {\mathbb K}).$$ However, as $y$ is disjoint from $D$, the right hand side is isomorphic to ${\it VHF}^{\rm loc}(G, y; {\mathbb K})$, which is also isomorphic to ${\it VHF}^{\rm loc}(H, x; {\mathbb K})$. ◻ Now we prove that the local vortex Floer homology is isomorphic to the local Floer homology inside the symplectic quotient. *Proof of Proposition [Proposition 68](#prop:iso-local){reference-type="ref" reference="prop:iso-local"}.* It follows from the adiabatic limit argument in the same spirit as in [@Dostoglou_Salamon; @Gaio_Salamon_2005] and [@Lagrange_multiplier]. Let $H_1$ be a nondegenerate Hamiltonian on $X$ which is arbitrarily close to $H$. Let $\widehat H_1$ be an admissible lift and $\widehat J_1$ is a generic time-dependent almost complex structure. Consider the local vortex Floer homology defined by critical points of ${\mathcal A}_{H_1}$ which are close to the fixed point $x \in {\rm Fix}(\varphi_H)$ whose differential counts rigid solutions to the equation [\[vortex_lambda\]](#vortex_lambda){reference-type="eqref" reference="vortex_lambda"} (with $(\widehat H, \widehat J)$ replaced by $(\widehat H_1, \widehat J_1)$). Using continuation maps we can show that the resulting homology is independent of $\lambda$. Moreover, the energy of relevant solutions can be arbitrarily small. Then consider the $\lambda \to \infty$ limit. For any sequence $\lambda_i \to \infty$ and any sequence of solutions to [\[vortex_lambda\]](#vortex_lambda){reference-type="eqref" reference="vortex_lambda"} for $\lambda = \lambda_i$ which contributes to the local vortex Floer differential, there is an upper bound of the energy of these solutions. Then by the adiabatic limit compactness theorem (see [@Gaio_Salamon_2005][@Wang_Xu] in similar settings) a subsequence converges to a possibly broken ordinary Floer trajectory inside $X$ modulo bubbling. As there is a lower bound for the energy of bubbles, we can choose the perturbation $H_1$ sufficiently close to $H$ so that bubbles can be ruled out. Moreover, we may assume that the pair $(H_1, J_1)$ on the symplectic quotient $X$ induced from the pair $(\widehat H_1, \widehat J_1)$ makes the local Floer complex well-defined (i.e. moduli spaces are transverse). Then if we are considering the zero-dimensional moduli spaces, then the possible limits must be unbroken trajectories in $X$. Now we claim that for $\lambda$ sufficiently large, there is an orientation-preserving bijection between index zero solutions to [\[vortex_lambda\]](#vortex_lambda){reference-type="eqref" reference="vortex_lambda"} (modulo gauge transformation) and index zero solutions to the ordinary Floer equation in $X$. Indeed, using the same kind of estimates as in [@Dostoglou_Salamon][@Gaio_Salamon_2005] (and the much simpler case in [@Lagrange_multiplier]) one can construct a gluing map from the limiting moduli space to the vortex moduli space with sufficiently large parameter $\lambda$. The compactness result explained above shows that the gluing map is surjective, while via the implicit function theorem one can show that the gluing map is injective. The fact that the gluing map preserves orientation follows from the explicit comparison of the linearized Fredholm operators (they differ by, roughly speaking, an invertible operator). ◻ In view of Proposition [Proposition 68](#prop:iso-local){reference-type="ref" reference="prop:iso-local"} and the properties of local Floer homology as proved in, e.g., [@Ginzburg_2010], the assertions in Theorem [Theorem 67](#thm:property-vortex-local){reference-type="ref" reference="thm:property-vortex-local"} are straightforward. The following is also immediate. **Corollary 74**. *The local vortex Floer homology has the following properties.* 1. *(Up to isomorphism) ${\it VHF}^{\rm loc}(H, x; {\mathbb K})$ only depends on the fixed point $q$ and the time-1 map $\phi\in {\rm Ham}(X)$. Hence we denote the (bulk-deformed) local vortex Floer homology by $${\it VHF}_{\rm loc}^{\mathfrak b}(\phi, q; {\mathbb K}).$$* 2. *If $\phi^k$ is an admissible iteration of $\phi$ at $q$, then $${\it VHF}_{\rm loc}^{\mathfrak b}( \phi, q; {\mathbb K}) \cong {\it VHF}_{\rm loc}^{\mathfrak b}(\phi^k, q; {\mathbb K}).$$* ## Barcodes of degenerate Hamiltonians Recall that one can associate to each nondegenerate Hamiltonian on a closed symplectic manifold a (finite) barcode. As this association is Lipschitz continuous with respect to the bottleneck distance for barcodes and Hofer metric for Hamiltonians, we hope one can define barcodes for all Hamiltonians using this Lipschitz continuity. However, the bottleneck distance is not complete. Therefore, *a priori*, the barcode for a general Hamiltonian only exists in the completion. **Theorem 75**. *Let ${\mathbb K}$ be a field. Let $\phi \in {\rm Ham}(X)$ be a Hamiltonian diffeomorphism with isolated fixed points. Let ${\mathcal B} (\phi)$ be the (*a priori* infinite) reduced barcode of $\phi$ (in coefficient field $\Lambda_{\mathbb K}^\Gamma$). Then ${\mathcal B} (\phi)$ has finitely many bars whose number of end points is equal to $N(\phi)$.* **Corollary 76**. *The total bar length is defined for all $\phi\in {\rm Ham}(X)$ with isolated fixed points.* Now we prove Theorem [Theorem 75](#thm_degenerate_barcode){reference-type="ref" reference="thm_degenerate_barcode"}. Suppose $\phi \in {\rm Ham}(X)$ has only isolated fixed points. Let $H$ be a Hamiltonian whose time one map is $\phi$. Let $\widehat H$ be any $K$-invariant lift of $H$ and let $\widehat J$ be a $K$-invariant $\omega_V$-compatible almost complex structure. Notice that in general $(\widehat H, \widehat J)$ is not an admissible pair so does not have a vortex Floer complex. However, one can still consider the vortex equation with the data $(\widehat H, \widehat J)$. **Lemma 77**. *There exists $\delta>0$ which only depends on $(\widehat H, \widehat J)$ satisfying the following condition. Let $x(t) \neq y(t)$ be two different 1-periodic orbits of $H$ downstairs. Let ${\mathfrak u}$ be a possibly broken solution to [\[vortex_equation_2\]](#vortex_equation_2){reference-type="eqref" reference="vortex_equation_2"} which connects $x(t)$ and $y(t)$ (without conditions on capping). Then the energy of ${\mathfrak u}$ is at least $\delta$.* *Proof.* For admissible $(\widehat H, \widehat J)$ this statement is proved as [@Xu_VHF Proposition 5.5] using a compactness argument. Notice that to run the compactness argument and to have the notion of converging to a 1-periodic orbit, one does not really need to require that the Hamiltonian is nondegenerate or the pair $(\widehat H, \widehat J)$ is admissible. ◻ **Corollary 78**. *The lengths of all bars in ${\mathcal B}(\phi)$ are no less than $\delta$.* *Proof.* Suppose on the contrary ${\mathcal B}(\phi)$ has a finite bar whose length is positive and smaller than $\delta$. Let $(\widehat H_k, \widehat J_k)$ be a sequence of regular bulk-avoiding pairs such that $(\widehat H_k, \widehat J_k)$ converges to $(\widehat H, \widehat J)$. Consider the reduced barcode associated to $\phi_{H_k}$. By the continuous dependence of barcodes on the Hamiltonian, for $k$ sufficiently large, there exists a finite bar in ${\mathcal B}(\phi_{H_k})$ whose length is between $\frac{\delta}{2}$ and $\delta - \epsilon$ for some small $\epsilon$. By the definition of barcodes by Usher--Zhang, there exists a rigid solution ${\mathfrak u}_k$ to [\[vortex_equation_2\]](#vortex_equation_2){reference-type="eqref" reference="vortex_equation_2"} with data $(\widehat H_k, \widehat J_k)$ whose energy is between $\frac{\delta}{2}$ and $\delta-\epsilon$. Via the compactness argument, there is a subsequence, still indexed by $k$, such that ${\mathfrak u}_k$ converges to a possibly broken trajectory with data $(\widehat H, \widehat J)$ whose total energy is between $\frac{\delta}{2}$ and $\delta -\epsilon$. This contradicts Lemma [Lemma 77](#no_short_trajectory){reference-type="ref" reference="no_short_trajectory"}. ◻ *Proof of Theorem [Theorem 75](#thm_degenerate_barcode){reference-type="ref" reference="thm_degenerate_barcode"}.* Choose a sequence of regular bulk-avoiding admissible pair $(\widehat H_k, \widehat J_k)$ converging to $(\widehat H, \widehat J)$. Consider the complex ${\it VCF}_\bullet^{\mathfrak b}(\widehat H_k, \widehat J_k; \Lambda_{\mathbb K}^\Gamma)$. One can write $$\partial = \partial_{\rm short} + \partial_{\rm long}$$ where $\partial_{\rm short}$ counts rigid trajectories whose energy is smaller than $\delta$ and $\partial_{\rm long}$ counts rigid trajectories whose energy is bigger than $\delta$. Then $\partial_{\rm short}^2 = 0$ and its homology coincides with the direct sum of all local vortex Floer homology of $\phi$. Moreover, one can decompose the reduced barcode of $\phi_{H_k}$ as $${\mathcal B}(\phi_{H_k}) = {\mathcal B}_{\rm short}( \phi_{H_k}) \sqcup {\mathcal B}_{\rm long}( \phi_{H_k}) \sqcup {\mathcal B}_\infty(\phi_{H_k})$$ where the first component consists of finite bars of lengths at most $\delta$ and the second component consists of other finite bars. As $\partial_{\rm short}^2 = 0$, one can also define a barcode ${\mathcal B}_{\rm local}(\phi_{H_k})$ by modifying the definition of Usher--Zhang, whose finite part coincides with ${\mathcal B}_{\rm short}(\phi_{H_k})$. Then by the definition, $$\begin{split} N(\phi_{H_k}) = &\ \# {\rm End}({\mathcal B}_{\rm short}(\phi_{H_k})) + \sum_{x \in {\rm Fix}(\phi)} {\rm dim} {\it VHF}^{\rm loc}(\phi, x)\\ = &\ \# {\rm End}({\mathcal B}_{\rm short}(\phi_{H_k})) + \# {\rm End}({\mathcal B}_{\rm long}(\phi_{H_k})) + {\rm dim} {\it VHF}_\bullet(V). \end{split}$$ As in the limit, all short bars disappear and long bars survive with respect to the bottleneck distance, the theorem follows. ◻ # Boundary depth {#sec:beta} In this section we prove Theorem [Theorem 3](#thmc){reference-type="ref" reference="thmc"}, namely, under the semisimple condition, the boundary depth of the vortex Floer complex of any Hamiltonian diffeomorphism is uniformly bounded from above. ## Vortex Floer persistence modules Recall that from Section [3.4.3](#subsubsec:floer-persistent){reference-type="ref" reference="subsubsec:floer-persistent"} we know that any Floer--Novikov complex over a Novikov field $\Lambda_{\mathbb K}^\Gamma$ induces a persistence module over ${\mathbb K}$. Given a regular bulk-avoiding admissible pair $(\widehat H, \widehat J)$ and a bulk deformation $${\mathfrak b} = \sum_{j=1}^N \log c_j V_j\ {\rm where}\ c_j \in {\mathbb Z}[{\bf i}],$$ the persistence module induced from the complex ${\it VCF}_\bullet^{\mathfrak b}(\widehat H, \widehat J; \Lambda_{\overline{\mathbb F}_p}^\Gamma)$ is denoted by $${\bm V}_{(p)} (\widehat H, \widehat J).$$ Recall that each filtered Floer--Novikov complex has a finite boundary depth which coincides with the boundary depth of the associated persistence module. We denote the boundary depth of ${\bm V}_{(p)}(\widehat H, \widehat J)$ by $$\beta_{(p)}( \widehat H, \widehat J)\in [0, +\infty).$$ It is also equal to the length of the longest finite bar in the associated barcode (cf. Proposition [Proposition 35](#prop:beta-equal){reference-type="ref" reference="prop:beta-equal"}). **Proposition 79**. *Given any two regular bulk-avoiding admissible pairs $(\widehat H_1, \widehat J_1)$ and $(\widehat H_2, \widehat J_2)$, for any prime $p$, one has $$\label{boundary_depth_continuity} | \beta_{(p)}( \widehat H_1, \widehat J_1) - \beta_{(p)} (\widehat H_2, \widehat J_2)| \leq 2 d_{\rm Hofer} (H_1, H_2).$$ In particular, the boundary depth only depends on the descent Hamiltonian downstairs.* *Proof.* This is a consequence of the stability of the persistence module and the boundary depth. Indeed, Proposition [Proposition 61](#prop_quasiequivalence_distance){reference-type="ref" reference="prop_quasiequivalence_distance"} implies that the quasi-equivalence distance between ${\it VCF}_\bullet^{\mathfrak b}(\widehat H_1, \widehat J_1; \Lambda_{\overline{\mathbb F}_p})$ and ${\it VCF}_\bullet^{\mathfrak b}(\widehat H_2, \widehat J_2; \Lambda_{\overline{\mathbb F}_p})$ is at most equal to the Hofer distance $d_{\rm Hofer}(H_1, H_2)$. Using Theorem [\[thm:bottle-neck-Hofer\]](#thm:bottle-neck-Hofer){reference-type="ref" reference="thm:bottle-neck-Hofer"}, it implies that the interleaving distance between the two associated persistence modules is no greater than the same bound. By Proposition [Proposition 30](#prop_boundary_depth_stability){reference-type="ref" reference="prop_boundary_depth_stability"}, one can conclude [\[boundary_depth_continuity\]](#boundary_depth_continuity){reference-type="eqref" reference="boundary_depth_continuity"}. ◻ Using typical arguments, one can also show that the boundary depth only depends on the induced (nondegenerate) Hamiltonian isotopy $\tilde \phi_H$ on the toric manifold $X$. Then Proposition [Proposition 30](#prop_boundary_depth_stability){reference-type="ref" reference="prop_boundary_depth_stability"} implies that $\beta_{(p)}$ descends to a Hofer continuous function $$\beta_{(p)}: {\rm H}{\widetilde{\rm am}}(X) \to [0, +\infty).$$ Below is the main theorem of this section. **Theorem 80**. *Suppose there exist $p_0>0$ and $C_0>0$ such that for all prime $p \geq p_0$, the algebra ${\it VHF}_\bullet^{\mathfrak b}(V; \Lambda_{\overline{\mathbb F}_p})$ is a semisimple $\Lambda_{\overline{\mathbb F}_p}$-algebra with idempotent generators $e_{l, (p)}, \ldots, e_{m, (p)}$ satisfying $$\ell_p( e_{l, (p)}) \leq C_0.$$ Then there exists $C>0$ such that for all prime $p \geq p_0$ and all ${\tilde\phi} \in {\rm H}{\widetilde{\rm am}}(X)$, one has $$\beta_{(p)} (\tilde \phi) \leq C.$$* ## Action by quantum multiplication Recall how we define pair-of-pants product on the vortex Floer homology (see [@Wu_Xu]). On the pair-of-pants $\Sigma^{\rm pop}$, equip the two inputs bulk-avoiding admissible pairs $(\widehat H_1, \widehat J_1)$ and $(\widehat H_2, \widehat J_2)$ and equip the output another bulk-avoiding admissible pair $(\widehat H_3, \widehat J_3)$. Extend these data to a domain-dependent Hamiltonian perturbation and a domain-dependent almost complex structure on $\Sigma^{\rm pop}$. By counting solutions to the Hamiltonian perturbed vortex equation on $\Sigma^{\rm pop}$ (with appropriate weights coming from the bulk deformation ${\mathfrak b}$), one can define a chain map $${\it VCF}_\bullet^{\mathfrak b}(\widehat H_1, \widehat J_1; \Lambda_{\mathbb K}) \otimes {\it VCF}_\bullet^{\mathfrak b} (\widehat H_2, \widehat J_2; \Lambda_{\mathbb K}) \to {\it VCF}_\bullet^{\mathfrak b}(\widehat H_3, \widehat J_3; \Lambda_{\mathbb K}).$$ We fix a class $\alpha \in {\it VHF}_\bullet^{\mathfrak b}( V; \Lambda_{\mathbb K})$. For each $\delta>0$, let $\widehat H_\delta$ be a bulk-avoiding admissible Hamiltonian on $V$ with $\| \widehat H_\delta \|_{C^2} \leq \delta$. We temporarily omit the dependence on the almost complex structure and the coefficient field from the notations. For notational simplicity, we also omit the bulk ${\mathfrak b}$ in the formulas at the moment. Consider the chain-level map $${\it VCF}_\bullet (\widehat H_\delta) \otimes {\it VCF}_\bullet (\widehat H) \to {\it VCF}_\bullet (\widehat H).$$ By using the energy inequality, one can show that there exists a constant $C > 0$, such that for all $s, \tau \in {\mathbb R}$, the above multiplication induces a bilinear map $$\label{chain_multiplication} {\it VHF}_\bullet^{\leq \tau} (\widehat H_\delta) \otimes {\it VHF}_\bullet^{\leq s}(\widehat H) \to {\it VHF}_\bullet^{\leq s + \tau + C \delta} (\widehat H).$$ Denote $${\mathcal A}^{\mathfrak b} (\alpha): = c^{\mathfrak b} (\alpha, 0) = \lim_{\delta \to 0} c^{\mathfrak b} (\alpha, H_\delta).$$ Then one has the linear map for all $\epsilon>0$, one can choose $\delta$ sufficiently small so that by setting $\tau = {\mathcal A}^{\mathfrak b}(\alpha) + \delta$ and inserting a representative of $\alpha$ in ${\it VHF}_\bullet^{\leq \tau}(\widehat H_\delta)$ in [\[chain_multiplication\]](#chain_multiplication){reference-type="eqref" reference="chain_multiplication"}, one obtains a well-defined map $$m_\epsilon (\alpha): {\it VHF}^{\leq s}_\bullet ( \widehat H) \to {\it VHF}_\bullet^{\leq s + {\mathcal A}^{\mathfrak b} (\alpha) + \epsilon} ( \widehat H).$$ Using the standard argument one can show that this map only depends on the class $\alpha$. By applying any positive shift, the above operation defines a family of operations which are recorded in the following statement. **Proposition 81**. *For all $\epsilon>0$, the maps $m_{\epsilon}(\alpha)$ define a morphism of persistence modules $$m_\epsilon (\alpha): {\bm V}({\tilde\phi}) \to {\bm V}({\tilde \phi})[{\mathcal A}(\alpha) + \epsilon]\ \forall \epsilon>0$$ satisfying for all $\epsilon < \epsilon'$, one has $$m_{\epsilon'}(\alpha) = {\rm shift}_{\epsilon'-\epsilon} \circ m_{\epsilon}(\alpha).$$* **Definition 82**. Given $\alpha \in {\it VHF}_\bullet^{\mathfrak b}(V; \Lambda_{\mathbb K}) \setminus \{0\}$ and $\epsilon>0$, the persistence module ${\bm W}_\alpha(\tilde \phi)_{\epsilon}$ is defined by $$W_\alpha(\tilde \phi)_{\epsilon}^s = {\rm Im} \left( m_{\epsilon}(\alpha): {\it VHF}_\bullet^{\leq s - {\mathcal A}^{\mathfrak b} (\alpha)}(\tilde H) \to {\it VHF}_\bullet^{\leq s +\epsilon} (\tilde H) \right) \subset V(\tilde \phi)^{s+ \epsilon}.$$ *Remark 83*. Our notion of persistence modules (Definition [Definition 26](#defn_persistence){reference-type="ref" reference="defn_persistence"}) is very different from the traditionally used ones (see for example [@Polterovich_Shelukhin_Stojisavljevic] where similar operators were firstly defined for Floer persistence modules in the monotone case); notably we allow each piece $V^s$ of a persistence module ${\bm V}$ to be infinite-dimensional. Hence it is not straightforward, though not necessarily difficult, to prove that when $\epsilon \to 0$, the above persistence modules "converges," giving a limiting object similar to the one used in [@Shelukhin_2022]. However, we could also carry the $\epsilon$ everywhere as we are doing here. ## Proof of Theorem [Theorem 80](#thm_boundary_depth){reference-type="ref" reference="thm_boundary_depth"} {#proof-of-theorem-thm_boundary_depth} We prove Theorem [Theorem 80](#thm_boundary_depth){reference-type="ref" reference="thm_boundary_depth"} following the strategy of [@Shelukhin_2022]. This theorem is the consequence of the following three lemmas (Lemma [Lemma 84](#lemma_interleaving_bound){reference-type="ref" reference="lemma_interleaving_bound"}, Lemma [Lemma 85](#lemma_split_boundary_depth){reference-type="ref" reference="lemma_split_boundary_depth"}, and Lemma [Lemma 86](#spectral_norm_estimate){reference-type="ref" reference="spectral_norm_estimate"}). We first introduce and simplify the notations. As we work with an individual prime, we drop the dependence on the prime $p$ in most notations. Let $e_1, \ldots, e_m$ be the idempotent generators of ${\it VHF}_\bullet^{\mathfrak b}(V; \Lambda_{\overline{\mathbb F}_p})$. For each nondegenerate $\tilde \phi \in {\rm H}{\widetilde{\rm am}}(X)$, consider the direct sum persistence module $${\bm W} (\tilde \phi)_{\epsilon} = \bigoplus_{l=1}^m {\bm W}_{e_l} (\tilde \phi)_{\epsilon}.$$ **Lemma 84**. *The interleaving distance between ${\bm V} (\tilde \phi)$ and ${\bm W} (\tilde \phi)_{\epsilon}$ is at most $C_0 + \epsilon$.* For all ${\tilde\phi}\in {\rm H}{\widetilde{\rm am}}(X)$, define $$\gamma ({\tilde\phi}):= \max_{1\leq l \leq m} \gamma_{e_l} ({\tilde\phi}) := \max_{1 \leq l \leq m} \left( c^{\mathfrak b} ( e_l, {\tilde\phi}) + c^{\mathfrak b} ( e_l, {\tilde \phi}^{-1}) \right).$$ Temporarily let ${\rm pr}: {\rm H}{\widetilde{\rm am}}(X) \to {\rm Ham}(X)$ be the canonical projection. Define for $\phi \in {\rm Ham}(X)$ $$\gamma (\phi):= \inf_{{\rm pr}({\tilde \phi}) = \phi} \gamma ({\tilde\phi}).$$ The following is an analogue of [@Shelukhin_2022 Proposition 12]. **Lemma 85**. *The boundary depth of the persistence module ${\bm W}_{e_l} ({\tilde\phi})_{\epsilon}$ is finite. Moreover, given nondegenerate ${\tilde \phi}, {\tilde \psi} \in {\rm H}{\widetilde{\rm am}}(X)$, for each $l = 1, \ldots, m$, one has $$\label{boundary_depth_variation} \left| \beta({\bm W}_{e_l} ({\tilde\phi})_\epsilon ) - \beta({\bm W}_{e_l} ({\tilde\psi})_\epsilon ) \right| \leq \gamma_{e_l} ( \tilde\phi \tilde\psi^{-1}).$$* The following is analogous to [@Usher_2011 Proposition 5.4] and [@Shelukhin_2022 Proposition 13]. **Lemma 86**. *For all ${\tilde \phi}\in {\rm H}{\widetilde{\rm am}}(X)$, one has $$c^{\mathfrak b} (e_l, {\tilde\phi}) + c^{\mathfrak b} (e_l, {\tilde\phi}^{-1}) \leq 4C_0.$$* *Proof of Theorem [Theorem 80](#thm_boundary_depth){reference-type="ref" reference="thm_boundary_depth"}.* As the boundary depth depends continuously on the Hamiltonian isotopy $\tilde \phi$, one only needs to prove the theorem for nondegenerate ones. First, by Lemma [Lemma 84](#lemma_interleaving_bound){reference-type="ref" reference="lemma_interleaving_bound"}, the interleaving distance between ${\bm V} (\tilde \phi)$ and ${\bm W} (\tilde \phi)_{\epsilon}$ is bounded by $C_0$. Hence by Proposition [Proposition 30](#prop_boundary_depth_stability){reference-type="ref" reference="prop_boundary_depth_stability"}, it suffices to bound the boundary depth of ${\bm W}(\tilde\phi)_{\epsilon}$. As ${\bm W} (\tilde \phi)_{\epsilon}$ is the direct sum of ${\bm W}_{e_l}(\tilde \phi)_{\epsilon}$, it suffices to bound the boundary depth of ${\bm W}_{e_l}(\tilde \phi)_{\epsilon}$ for all idempotent generators $e_l$. Then applying Lemma [Lemma 85](#lemma_split_boundary_depth){reference-type="ref" reference="lemma_split_boundary_depth"}, one obtains $$\beta({\bm W}_{e_l}(\tilde \phi)_{\epsilon} ) \leq \gamma_{e_l} (\tilde\phi \tilde\psi^{-1}) + \beta( {\bm W}_{e_l}(\tilde \psi)_{\epsilon}) \leq 4 C_0 + \beta( {\bm W}_{e_l}(\tilde \psi)_{\epsilon})$$ where $\tilde\psi \in {\rm H}{\widetilde{\rm am}}(X)$ is an arbitrary fixed nondegenerate Hamiltonian isotopy on $X$. Then the right hand side is finite and independent of $\tilde\phi$. ◻ ## Proofs of the technical lemmas In this subsection we drop all dependence on the bulk deformation from notations. *Proof of Lemma [Lemma 84](#lemma_interleaving_bound){reference-type="ref" reference="lemma_interleaving_bound"}.* We construct maps between persistence modules ${\bm f}_{\epsilon}: {\bm V}(\tilde \phi) \to {\bm W}(\tilde \phi)_{\epsilon}[C_0]$ and ${\bm g}_{\epsilon}: {\bm W}(\tilde \phi)_{\epsilon} \to {\bm V}(\tilde \phi)[C_0]$ as follows. For $s \in {\mathbb R}$, define $$f_{\epsilon}^s: V(\tilde\phi)^s \to \bigoplus_{l=1}^m W_{e_l}(\tilde\phi)^{s+C_0}_{\epsilon}$$ to be the composition of $$\begin{aligned} &\ V(\tilde\phi)^s \to \bigoplus_{l=1}^m W_{e_l}(\tilde \phi)_{\epsilon}^{s + {\mathcal A}(e_l)},\ &\ \alpha \mapsto (e_1 \ast \alpha, \ldots, e_m \ast \alpha)\end{aligned}$$ and the natural map $$\bigoplus_{l=1}^m W_{e_l}(\tilde\phi)_{\epsilon}^{s + {\mathcal A} (e_l)} \to \bigoplus_{l=1}^m W_{e_l}(\tilde\phi)_{\epsilon}^{s + C_0}.$$ Define $$\begin{aligned} &\ g_{+\epsilon}^s: \bigoplus_{l=1}^m W_{e_l}(\tilde\phi)_{\epsilon}^s \to V(\tilde \phi)^{s + C_0},\ &\ (\alpha_1, \ldots, \alpha_m) \mapsto \iota^{s + \epsilon, s+C_0} (\alpha_1 + \cdots + \alpha_m).\end{aligned}$$ It is straightforward to check, using the fact that $e_1 + \cdots + e_m = {\bm 1}^{{\rm GLSM}}_{\mathfrak b}$ and that $e_l$ are idempotent generators, that ${\bm f}_{\epsilon}$, ${\bm g}_{\epsilon}$ provide $C_0$-interleaving between ${\bm V}(\tilde \phi)$ and ${\bm W}(\tilde\phi)_{\epsilon}$. ◻ *Proof of Lemma [Lemma 85](#lemma_split_boundary_depth){reference-type="ref" reference="lemma_split_boundary_depth"}.* The detailed proof would be almost identical to the part of the proof of [@Shelukhin_2022 Proposition 12] corresponding to this lemma. Hence we only briefly sketch it. First we show the finiteness of the boundary depth. The boundary depth of ${\bm V}(\tilde\phi)$ is finite because it coincides of the boundary depth of the associated Floer--Novikov complex (see Proposition [Proposition 32](#prop_boundary_depth_equivalence){reference-type="ref" reference="prop_boundary_depth_equivalence"}). Hence by Lemma [Lemma 84](#lemma_interleaving_bound){reference-type="ref" reference="lemma_interleaving_bound"} and Proposition [Proposition 30](#prop_boundary_depth_stability){reference-type="ref" reference="prop_boundary_depth_stability"}, ${\bm W}(\tilde\phi)_{\epsilon}$ has finite boundary depth. Therefore each summand ${\bm W}_{e_l}(\tilde\phi)_{\epsilon}$ has finite boundary depth. Now we prove the inequality [\[boundary_depth_variation\]](#boundary_depth_variation){reference-type="eqref" reference="boundary_depth_variation"}. Let $F, G$ be Hamiltonians downstairs with time-1 maps $\tilde \phi$ and $\tilde \psi$ respectively. Choose bulk-avoiding admissible lifts $\widehat F$, $\widehat G$ upstairs and let $(\widehat F, \widehat J_F)$, $(\widehat G, \widehat J_G)$ be regular pairs. Let $\ell_F$ resp. $\ell_G$ be the non-Archimedean valuation on the complex ${\it VCF}_\bullet ( \widehat F, \widehat J_F)$ resp. ${\it VCF}_\bullet ( \widehat G, \widehat J_G)$. Let $\Delta_{\widehat F, \widehat G} = G \# \overline{F}$ be the difference Hamiltonian upstairs with descent difference Hamiltonian $\Delta_{F, G}$ downstairs. Let $\widehat J_{F, G}$ be an admissible almost complex structure so that the pair $(\Delta_{\widehat F, \widehat G}, \widehat J_{F, G})$ is regular. One can obtain a pair $(\Delta_{\widehat G, \widehat F}, \widehat J_{G, F})$ with the roles of $\widehat F$ and $\widehat G$ reversed. Now fix $\epsilon>0$. Choose a cycle $c_{\widehat F, \widehat G, \epsilon}\in {\it VCF}_\bullet (\Delta_{\widehat F, \widehat G}, \widehat J_{F, G})$ representing $e_l$ such that $$\ell ( c_{\widehat F, \widehat G, \epsilon}) \leq c (e_l, \Delta_{\widehat F, \widehat G}) + \epsilon.$$ We also choose a cycle $c_{\widehat G, \widehat F, \epsilon}\in {\it VCF}_\bullet ( \Delta_{\widehat G, \widehat F}, \widehat J_{G, F})$ representing $e_l$ with $$\ell( c_{\widehat G, \widehat F, \epsilon}) \leq c ( e_l, \Delta_{\widehat G, \widehat F}) + \epsilon.$$ Now after choosing perturbation data on the pair-of-pants, one can define a chain map $$C_{\widehat F, \widehat G, \epsilon}: {\it VCF}_\bullet ( \widehat F, \widehat J_F) \to {\it VCF}_\bullet ( \widehat G, \widehat J_G),\ x \mapsto c_{\widehat F, \widehat G, \epsilon} \ast x.$$ satisfying $$\ell_G ( C_{\widehat F, \widehat G, \epsilon}(x)) \leq c (e_l, \Delta_{\widehat F, \widehat G}) + \ell_F(x) + 2 \epsilon.$$ Similarly, by using the cycle $c_{\widehat G, \widehat F, \epsilon}$ one can also define a chain map $$C_{\widehat G, \widehat F, \epsilon}: {\it VCF}_\bullet ( \widehat G, \widehat J_G) \to {\it VCF}_\bullet (\widehat F, \widehat J_F)$$ satisfying $$\ell_F( C_{\widehat G, \widehat F, \epsilon}(y)) \leq c (e_l, \Delta_{G, F}) + \ell_G(y) + 2\epsilon.$$ The lemma will follow from the following claim. **Claim.** $C_{\widehat F, \widehat G, \epsilon}$ and $C_{\widehat G, \widehat F, \epsilon}$ induce a $\frac{1}{2} \gamma_{e_l}(\tilde \phi \tilde \psi^{-1}) + 4\epsilon$-interleaving between ${\bm W}_{e_l}(\widehat F)_{\epsilon}$ and ${\bm W}_{e_l}(\widehat G)_{\epsilon}$. The detailed proof would also be almost identical to that of [@Shelukhin_2022] except for notations. We omit the details. ◻ *Remark 87*. As one can infer from the above proof, the inequality "$\beta \leq \gamma$\" is a consequence of studying filtered continuations maps in terms of taking the pair-of-pants product with the filtered continuation elements, which in particular does not depend on the semi-simplicity assumption. *Proof of Lemma [Lemma 86](#spectral_norm_estimate){reference-type="ref" reference="spectral_norm_estimate"}.* Using Proposition [Proposition 66](#prop413){reference-type="ref" reference="prop413"}, Lemma [Lemma 59](#lem:pairing-product){reference-type="ref" reference="lem:pairing-product"} and the triangle inequality for spectral invariants, one has $$\begin{split} - c (e_l, {\tilde\phi}^{-1}) = &\ \inf \Big\{ c (\alpha, {\tilde \phi})\ |\ \langle e_l, \alpha \rangle \neq 0 \Big\}\\ \geq &\ - {\mathcal A} (e_l) + \inf \Big\{ c ( e_l \ast \alpha, {\tilde \phi})\ |\ \langle e_l \ast \alpha, {\bm 1}^{\rm GLSM}\rangle \neq 0 \Big\}\\ \geq &\ - {\mathcal A} (e_l) + \inf \Big\{ c (e_l, {\tilde\phi}) - {\mathcal A} ( (e_l \ast \alpha)^{-1}) \ |\ \langle e_l \ast \alpha, {\bm 1}^{\rm GLSM}\rangle \neq 0 \Big\}\\ \geq &\ - {\mathcal A} (e_l) + c(e_l, {\tilde \phi}) + \inf \Big\{ - {\mathcal A} ( e_l \ast \alpha) - {\mathcal A} ( (e_l \ast \alpha)^{-1}) \ |\ e_l \ast \alpha \neq 0 \Big\} \\ &\ + \inf \Big\{ {\mathcal A} ( e_l \ast \alpha) \ |\ \langle e_l \ast \alpha, {\bm 1}^{\rm GLSM}\rangle \neq 0 \Big\} \end{split}$$ Here the quantum product and the Poincaré pairing are both the bulk-deformed versions. Notice that as $e_l$ is an idempotent generator, $e_l \ast \alpha = \lambda(\alpha) e_l$ and $(e_l \ast \alpha)^{-1} = \lambda(\alpha)^{-1} e_l$. Hence $${\mathcal A} ( e_l \ast \alpha) + {\mathcal A} ( (e_l \ast \alpha)^{-1}) = 2 {\mathcal A} (e_l) - {\mathfrak v}(\lambda(\alpha)) - {\mathfrak v}(\lambda(\alpha)^{-1}) = 2 {\mathcal A} (e_l)$$ which is uniformly bounded. Moreover, by Proposition [Proposition 66](#prop413){reference-type="ref" reference="prop413"} $$\inf \Big\{ {\mathcal A} ( e_l \ast \alpha) \ |\ \langle e_l \ast \alpha, {\bm 1}^{\rm GLSM}\rangle \neq 0 \Big\} \geq - {\mathcal A} ( {\bm 1}^{\rm GLSM}).$$ Therefore $$c (e_l, {\tilde\phi}) + c (e_l, {\tilde\phi}^{-1}) \leq 3 {\mathcal A} (e_l ) + {\mathcal A} ( {\bm 1}^{\rm GLSM}).$$ Lemma [Lemma 86](#spectral_norm_estimate){reference-type="ref" reference="spectral_norm_estimate"} follows by using the assumption ${\mathcal A} (e_l) \leq C_0$ and noticing $${\mathcal A}({\bm 1}^{\rm GLSM}) = {\mathcal A} ( e_1 + \cdots + e_m) \leq \max_{1 \leq l \leq m } {\mathcal A} (e_l) \leq C_0.\qedhere$$ ◻ *Remark 88*. The above argument crucially relies on the semi-simplicity assumption, which allows us to take advantage of the feature that any nonzero element in a field summand of the quantum homology is invertible. Note that such a phenomenon is ultimately due to the abundance of rational curves in toric manifolds. # ${\mathbb Z}/p$-equivariant vortex Floer theory {#sec:equiv} Following [@Seidel_pants; @Shelukhin-Zhao], we develop ${\mathbb Z}/p$-equivariant Hamiltonian Floer theory in the vortex setting. Using equivariant pair of pants operations, we show that the following analogue of [@Shelukhin_2022 Theorem D] about the total bar length holds in our setting. **Theorem 89**. *Let $\phi$ be a Hamiltonian diffeomorphism on the toric symplectic manifold $(X, \omega)$ with lift $\tilde\phi \in {\rm H}{\widetilde{\rm am}}(X)$. Then for any odd prime $p$, if ${\rm Fix}(\phi)$ and ${\rm Fix}(\phi^p)$ are finite, then $$\tau_{(p)}^{\mathfrak b}( \tilde\phi^p) \geq p \cdot \tau_{(p)}^{\mathfrak b}( \tilde\phi)$$* Here we work over $\Lambda_{\overline{\mathbb F}_p}$, which is omitted from the notations above. Given the arguments from [@Shelukhin_2022 Section 6], the only missing ingredient for establishing Theorem [Theorem 89](#thm:smith){reference-type="ref" reference="thm:smith"} is the package of ${\mathbb Z}/p$ Borel equivariant vortex Floer theory with bulk deformation. As demonstrated in other parts of the paper, one salient feature of vortex Floer theory is the absence of sphere bubbles due to the contractibility of symplectic vector space, which allows us to achieve transversality in many settings by only perturbing the almost complex structure. Specializing to the Borel equivariant theory, except for the necessity to deal with the symplectic vortex equations and the appearance of Novikov coefficients, our theory is quite similar to the exact setting as from the original reference [@Seidel_pants; @Shelukhin-Zhao], at least for bulk-avoiding Hamiltonians, which suffice for our purpose via a limiting argument. Therefore, unless there is anything special in our situation, we will be brief and refer the reader to the original references for full proofs. In this section the bulk deformation ${\mathfrak b}$ is fixed. All curve counts are weighted by the bulk term. We often drop it in order to shorten the notations. ## The Borel construction We take the following model of $E {\mathbb Z}/p$: the ambient space is $$S^{\infty} := \{ (z_0, z_1, \dots) \ | \ z_k \in {\mathbb C} \text{ for } k \in {\mathbb Z}_{\geq 0}, \sum |z_k|^2 = 1, \text{ only finitely many } z_k \text{'s are nonzero}\},$$ and the group ${\mathbb Z} / p$ freely acts on $S^{\infty}$ by multiplying each coordinate by $p$-th roots of unity. The quotient space of $S^\infty$ under this ${\mathbb Z}/p$-action is a model for the classifying space $B {\mathbb Z}/p$. The group cohomology of ${\mathbb Z}/p$ over ${\mathbb F}_p$ is recovered as the (graded-commutative) cohomology ring $${\it H}^*(B{\mathbb Z}/p; {\mathbb F}_p) = {\mathbb F}_p \llbracket u\rrbracket \langle \theta\rangle, \mathrm{deg}(u)=2 \text{ and } \mathrm{deg}(\theta)=1.$$ For $\epsilon > 0$ sufficiently small, $E {\mathbb Z}/p$ admits a ${\mathbb Z}/p$-invariant Morse function $$\tilde{F}(z) = \sum k|z_k|^2 + \epsilon \sum \mathrm{re}(z_k^p)$$ obtained by perturbing the standard Morse--Bott function $\sum k|z_k|^2$ on $S^{\infty}$ along the critical submanifolds. The function $\tilde{F}(z)$ has the following properties: 1. defining the map $$\label{borel_translation} \begin{aligned} \tilde{\tau}: S^{\infty} &\to S^{\infty} \\ (z_0, z_1, \dots) &\mapsto (0, z_0, z_1, \dots), \end{aligned}$$ then we have $\tilde{F} \circ \tilde{\tau} = \tilde{F} + 1$; 2. for any $l \in {\mathbb Z}_{\geq 0}$, the critical points of $\tilde{F}$ obtained from perturbing the critical submanifold $\{ |z_l| = 1 \}$ of $\sum k|z_k|^2$ can be indexed by $$Z_{2l}^{0}, \dots, Z_{2l}^{p-1}, \text{ and }, Z_{2l+1}^{0}, \dots, Z_{2l+1}^{p-1},$$ where each $Z_{2l}^{i}$ has Morse index $2l$ and each $Z_{2l+1}^{i}$ has Morse index $2l+1$; 3. the sets $\{Z_{2l}^{0}, \dots, Z_{2l}^{p-1}\}$ and $\{Z_{2l+1}^{0}, \dots, Z_{2l+1}^{p-1} \}$ respectively form an ${\mathbb Z}/p$-orbit of the ${\mathbb Z}/p$-action on $S^\infty$; 4. there exists a ${\mathbb Z}/p$-equivariant Riemannian metric $\tilde{g}$ on $S^\infty$ such that $(\tilde{f}, \tilde{g})$ is Morse--Smale, and the differential on the corresponding Morse cochain complex is $$\begin{aligned} Z_{2l}^{m} &\mapsto Z_{2l+1}^{m} - Z_{2l+1}^{m+1}, \\ Z_{2l+1}^{m} &\mapsto Z_{2l+2}^0 + \cdots + Z_{2l+2}^{p-1}, \end{aligned}$$ where the index $m \in {\mathbb Z}/p$ is read cyclically. ## The Tate construction Next, we review the Tate construction for cyclic groups of prime order. Let $R$ be a unital commutative ring which is an ${\mathbb F}_p$-algebra (later $R$ will become $\overline{\mathbb F}_p$). Suppose $(\hat{C}_\bullet, d_{\hat{C}})$ is a ${\mathbb Z}_2$-graded chain complex defined over the Novikov ring $\Lambda_{0, R}$. Note that $\Lambda_{0, R}$ is a module over $\Lambda_{0, {\mathbb F}_p}$. Introduce the graded field $${\mathcal K} = {\mathbb F}_p [u^{-1}, u\rrbracket, \ \mathrm{deg}(u)=2.$$ Then the ${\mathbb Z} / p$-equivariant Tate complex $$C_{\mathrm{Tate}}({\mathbb Z} / p, \hat{C}_\bullet^{\otimes p})$$ is a module over $\Lambda_{0, \mathcal K}\langle \theta \rangle$ where $\mathrm{deg}(\theta)=1, \theta^2=0$, explicitly given by $$\hat{C}_\bullet^{\otimes p} \otimes_{\Lambda_{0, {\mathbb F}_p}} \Lambda_{0, \mathcal K} \langle \theta \rangle.$$ The differential $d_{\mathrm{Tate}}$ is $\Lambda_{0,R} \otimes_{\Lambda_{0, {\mathbb F}_p}} \Lambda_{0, \mathcal K}$-linear, such that for $x_0 \otimes \cdots \otimes x_{p-1} \in \hat{C}_\bullet^{\otimes p}$, we have $$\begin{aligned} d_{\mathrm{Tate}}(x_0 \otimes \cdots \otimes x_{p-1}) &= d_{\hat{C}}^{\otimes p}(x_0 \otimes \cdots \otimes x_{p-1}) + \theta (id - \zeta) (x_0 \otimes \cdots \otimes x_{p-1}), \\ d_{\mathrm{Tate}}(\theta (x_0 \otimes \cdots \otimes x_{p-1})) &= - \theta d_{\hat{C}}^{\otimes p}(x_0 \otimes \cdots \otimes x_{p-1}) + u (id + \zeta + \cdots + \zeta^{p-1})(x_0 \otimes \cdots \otimes x_{p-1}), \end{aligned}$$ in which the $\zeta$ is the automorphism on $\hat{C}_\bullet^{\otimes p}$ defined by $$x_0 \otimes \cdots \otimes x_{p-1} \mapsto (-1)^{|x_{p-1}|(|x_0| + \cdots + |x_{p-2}|)} x_{p-1} \otimes x_0 \otimes \cdots \otimes x_{p-2}.$$ In other words, the Tate complex $(C_{\mathrm{Tate}}({\mathbb Z} / p, \hat{C}_\bullet^{\otimes p}), d_{\mathrm{Tate}})$ is obtained from the ${\mathbb Z}/p$ group cohomology of the chain complex $(\hat{C}_\bullet^{\otimes p}, d_{\hat{C}}^{\otimes p})$ by inverting the equivariant parameter $u$. Denote the homology of the Tate complex by $$H_{\rm Tate}({\mathbb Z}/p, \hat C_\bullet^{\otimes p}).$$ The following algebraic statement will be used in establishing the localization result proved later. **Lemma 90**. *[@Shelukhin_2022 Lemma 21] Denote the homology of $(\hat{C}_\bullet, d_{\hat{C}})$ by $\hat{H}_\bullet$. The $p$-th power map $$\label{eqn:p-power} \begin{aligned} \hat{C}_\bullet &\to \hat{C}_\bullet^{\otimes p} \\ x &\mapsto x \otimes \cdots \otimes x \end{aligned}$$ induces an isomorphism of $\Lambda_{0,R} \otimes_{\Lambda_{0, {\mathbb F}_p}} \Lambda_{0,\mathcal K}$-modules $$r_p^* (\hat{H}_\bullet \otimes_{\Lambda_{0,{\mathbb F}_p}} \Lambda_{0,\mathcal K}) \to H_{\mathrm{Tate}}({\mathbb Z} / p, \hat{C}_\bullet^{\otimes p})$$ where $r_p$ is the operator on $\Lambda_{0,R} \otimes_{\Lambda_{0, {\mathbb F}_p}} \Lambda_{0, \mathcal K}$ defined by mapping the Novikov variable $T$ to $T^{1/p}$.* This is referred to as the quasi-Frobenius isomorphism in [@Shelukhin_2022 Section 7]. ## ${\mathbb Z}/p$-equivariant vortex Floer theory {#mathbb-zp-equivariant-vortex-floer-theory} Given a 1-periodic Hamiltonian $H_t$ on $X$, its $p$-th iteration is the family $H^{(p)}_t:= H_{pt}$ If $\phi: X \to X$ is the time-1 map of $H$, then the time-1 map of $H^{(p)}$ is the iteration $\phi^p$. Following [@Seidel_pants; @Shelukhin-Zhao], we define the ${\mathbb Z}/p$-equivariant vortex Hamiltonian Floer homology for $H^{(p)}$ by using the family Floer homology coming from the Borel construction. For all the Floer-theoretic constructions involving moduli spaces, we always assume that the Hamiltonians involved in the discussion are nondegenerate. Recall that the toric divisors of $X$ are given by $D_1, \cdots, D_N$, which are obtained as the symplectic quotient of the coordinate hyperplanes $V_1, \cdots, V_N$ in the symplectic vector space $V$. As in the definition of bulk-deformed Floer homology, we assume that the Hamiltonian $H$ is bulk-avoiding; in particular, for any odd prime $p$, $1$-periodic orbits of $H$ and $H^{(p)}$ are disjoint from $V_1\cup \cdots \cup V_N$. We also assume that both $H$ and $H^{(p)}$ are nondegenerate. Let $\widehat H$ be an admissible lift of $H$ and $\widehat H^{(p)}$ an admissible lift of $H^{(p)}$ (see Remark [Remark 49](#rem_admissible){reference-type="ref" reference="rem_admissible"}). Let $\widehat J^{(p)} = \{ \widehat J_t^{(p)}\}_{t\in S^1}$ be a $1$-periodic family of compatible almost complex structures on $V$ such that the pair $(\widehat H^{(p)}, \widehat J^{(p)})$ is admissible and the Floer chain complex ${\it VCF}_\bullet^{\mathfrak b}( \widehat H^{(p)}, \widehat J^{(p)}; \Lambda_{0, R})$ has a well-defined bulk-deformed differential $\partial^{(p)}_{\mathfrak b}$, where ${\mathfrak b} = \sum_{i=1}^N \log c_i\ V_i$ is a chosen bulk in which $c_i \in {\mathbb Z}[{\bf i}]$. Note that we work over $\Lambda_{0,R}$ instead of $\Lambda_R$, which does not introduce any further difficulty due to the fact that $\partial^{(p)}_{\mathfrak b}$ preserves the energy filtration on ${\it VCF}_\bullet^{\mathfrak b} ( \widehat H^{(p)}, \widehat J^{(p)}; \Lambda_{0, R})$. To define equivariant differentials, we include more parameters from the Borel construction. We choose an $S^{\infty} = E {\mathbb Z}/p$ family of time-dependent compatible *admissible* almost complex structures $$\widehat J_\infty^{(p)} = \{ \widehat J^{(p)}_{t,z} \}_{t \in S^1, z \in S^{\infty}}$$ satisfying the following requirements: 1. Near each critical point $Z^0_{i}, i \in {\mathbb Z}_{\geq 0}$ of the Morse function $\tilde{F}(z)$ on $S^{\infty}$, we have $\widehat J^{(p)}_{t,z} = \widehat J^{(p)}_{t}$; 2. Regard ${\mathbb Z}/p \subset S^1$. For any $m \in {\mathbb Z}/p$ and $z \in S^{\infty}$, there holds the equivariance relation $$\widehat J^{(p)}_{t - m,z} = \widehat J^{(p)}_{t,m \cdot z};$$ 3. $\widehat J^{(p)}_{t,z}$ is invariant under the translation [\[borel_translation\]](#borel_translation){reference-type="eqref" reference="borel_translation"}. Namely $$\widehat J^{(p)}_{t,\tilde{\tau}(z)} = \widehat J^{(p)}_{t,z}.$$ After making such a choice, we can write down the following version of parametrized vortex Floer equation. Let ${\mathfrak x}_\pm = (x_\pm, \eta_\pm) \in {\rm crit}{\mathcal A}_{H^{(p)}}$ be a pair of equivariant $1$-periodic orbits of $H^{(p)}$ (which do not depend on the lift $\widehat H^{(p)}$). Given $i \in {\mathbb Z}_{\geq 0}, m \in {\mathbb Z}/p$ and $\alpha \in \{0,1\}$, the moduli space $${\mathcal M}^{i.m}_{\alpha}({\mathfrak x}_-, {\mathfrak x}_+)$$ consists of gauge equivalence classes of pairs of smooth maps (the gauge transformations act on the $(u, \phi, \psi)$-component) $$(u, \phi, \psi): {\mathbb R}_s\times S^1_t \to V \times {\mathfrak k} \times {\mathfrak k}, \quad \quad \quad w: {\mathbb R}_s \to S^{\infty}$$ which satisfy the equations and asymptotic conditions $$\label{eqn:equiv-Floer} \left\{ \begin{array}{llll} \displaystyle \partial_s u + {\mathcal X}_{\phi}(u) + \widehat J_{w(s), t}^{(p)} (\partial_t u + {\mathcal X}_\psi (u) - X_{\widehat H_t^{(p)}}(u)) = 0,\ &\ \partial_s \psi - \partial_t \phi + \mu(u) = 0, \\ \partial_s w(s) - \nabla \tilde{F}(w) = 0, \\ \displaystyle \lim_{s \to -\infty}(u(s,\cdot), \phi(s,\cdot), \psi(s,\cdot), w(s)) = (x_-, 0, \eta_-, Z_{\alpha}^0), \\ \displaystyle \lim_{s \to + \infty}(u(s,\cdot), \phi(s,\cdot), \psi(s,\cdot), w(s)) = (x_+, 0, \eta_+, Z_i^m), \end{array} \right.$$ modulo the ${\mathbb R}$-translation action given by $$(u(s,\cdot), \phi(s,\cdot), \psi(s,\cdot), w(s)) \mapsto (u(s+r,\cdot), \phi(s+r,\cdot), \psi(s+r,\cdot), w(s+r)), \quad \quad \quad r \in {\mathbb R}.$$ Because of the absence of sphere bubbles, as the capped orbits impose an upper bound on energy, the moduli space ${\mathcal M}^{i.m}_{\alpha}({\mathfrak x}_-, {\mathfrak x}_+)$ admits a Uhlenbeck--Gromov--Floer compactification $\overline{{\mathcal M}}{}^{i.m}_{\alpha}({\mathfrak x}_-, {\mathfrak x}_+)$ by adding equivalence classes of solutions the above coupled equations defined over broken configurations. On the other hand, for a generic choice of $\{ J^{(p)}_{t,z} \}_{t \in S^1, z \in S^{\infty}}$, the moduli space ${\mathcal M}^{i.m}_{\alpha}({\mathfrak x}_-, {\mathfrak x}_+)$ is transversely cut out, and the dimension of the moduli space satisfies $$\dim {\mathcal M}^{i.m}_{\alpha}({\mathfrak x}_-, {\mathfrak x}_+) = {\rm CZ}({\mathfrak x}_-) - {\rm CZ}({\mathfrak x}_+) + i -\alpha - 1.$$ For a more detailed discussion of these facts, the reader may consult [@Seidel_pants Section 4], [@Shelukhin-Zhao Section 6], whose arguments apply to our case after using the setup from [@Xu_VHF Section 6]. After achieving transversality, for each triple $i \in {\mathbb Z}_{\geq 0}, m \in {\mathbb Z}/p$ and $\alpha \in \{0,1\}$, we can define a $\Lambda_{0,R}$-linear map $\partial_{\alpha,{\mathfrak b}}^{i,m}$ on ${\it VCF}_\bullet (\widehat H^{(p)}, \widehat J^{(p)}; \Lambda_{0, R})$ of the form $$\partial_{\alpha,{\mathfrak b}}^{i,m}({\mathfrak x}) = \sum_{{\mathfrak y}\atop {\rm CZ}({\mathfrak x}) - {\rm CZ}({\mathfrak y}) + i -\alpha = 1} \left( \sum_{[({\mathfrak u},w)]\in {{\mathcal M}^{i.m}_{\alpha}({\mathfrak x}, {\mathfrak y})}} \epsilon([({\mathfrak u},w)]) \exp \left( \sum_{i=1}^N \log c_i\ [{\mathfrak u}] \cap V_i \right) \right) {\mathfrak y},$$ where $\epsilon([({\mathfrak u},w)]) \in \{\pm 1\}$ is the sign of the rigid solution $[{\mathfrak u}]$, which is well-defined due to the existence of coherent orientations, and $[{\mathfrak u}] \cap V_i$ is defined as before, coming from the topological intersection number. We further introduce the notation $$\partial_{\alpha,{\mathfrak b}}^{i} = \partial_{\alpha,{\mathfrak b}}^{i,0} + \cdots + \partial_{\alpha,{\mathfrak b}}^{i,p-1}.$$ **Definition 91**. The ${\mathbb Z}/p$-equivariant ${\mathfrak b}$-deformed vortex Floer chain complex $${\it VCF}_\bullet^{{\mathbb Z}/p}( \widehat H^{(p)}, \widehat J^{(p)}_\infty; \Lambda_{0, R})$$ is the ${\mathbb Z}_2$-graded $\Lambda_{0, R}$-module given by $${\it VCF}_\bullet ( \widehat H^{(p)}, \widehat J^{(p)}; \Lambda_{0, R})\llbracket u \rrbracket \langle \theta \rangle, \mathrm{deg}(u)=2, \mathrm{deg}(\theta)=1$$ with $\Lambda_{0, R}\llbracket u \rrbracket$-linear differential $$\begin{aligned} \partial^{(p)}_{eq, {\mathfrak b}}({\mathfrak x} \otimes 1) &= \sum_{i \geq 0} \partial_{0,{\mathfrak b}}^{2i}({\mathfrak x}) \otimes u^i + \sum_{i \geq 0} \partial_{0,{\mathfrak b}}^{2i+1}({\mathfrak x}) \otimes u^i \theta, \\ \partial^{(p)}_{eq, {\mathfrak b}}({\mathfrak x} \otimes \theta) &= \sum_{i \geq 0} \partial_{1,{\mathfrak b}}^{2i+1}({\mathfrak x}) \otimes u^i \theta + \sum_{i \geq 1} \partial_{1,{\mathfrak b}}^{2i}({\mathfrak x}) \otimes u^i. \end{aligned}$$ The statement that $(\partial^{(p)}_{eq, {\mathfrak b}})^2 = 0$ follows from the signed count of boundaries of the compactified $1$-dimensional moduli spaces $\overline{{\mathcal M}}{}^{i.m}_{\alpha}({\mathfrak x}_-, {\mathfrak x}_+)$. The differential is well-defined over $\Lambda_R^0$ because we only perturb the almost complex structure to achieve transversality. By continuation map considerations, the resulting homology group $${\it VHF}_\bullet^{{\mathbb Z}/p}( \widehat H^{(p)}, \widehat J_\infty^{(p)}; \Lambda_{0, R})$$ is independent of the choice of $\widehat J_\infty^{(p)}$, and it is a module over $\Lambda_{0, R} \llbracket u \rrbracket \langle \theta \rangle$. By inverting $u$, we can define $${\it VCF}_{\mathrm{Tate}}( \widehat H^{(p)}, \widehat J_\infty^{(p)}; \Lambda_{0, R}) = {\it VCF}^{{\mathbb Z}/p} ( \widehat H^{(p)}, \widehat J^{(p)}_\infty; \Lambda_{0, R})[u^{-1}, u \rrbracket \langle \theta \rangle$$ for which the differential is the $\Lambda_{0,R}[u^{-1}, u \rrbracket$-linear extension of $\partial^{(p)}_{eq, {\mathfrak b}}$. The homology group is written as $${\it VHF}_{\mathrm{Tate}}( \widehat H^{(p)}; \Lambda_{0, R}),$$ which is a module over $\Lambda^{0, R} \otimes_{\Lambda_{0, {\mathbb F}_p}} \Lambda^{0, {\mathcal K}}\langle \theta \rangle$. Here is some explanation of the definition of the equivariant differential. By definition, the leading order term $\partial_{0, {\mathfrak b}}^0$ agrees with the differential $\partial^{(p)}_{\mathfrak b}$ on the complex ${\it VCF}_\bullet (\widehat H^{(p)}, \widehat J^{(p)}; \Lambda_{0, R})$, so does $\partial^1_{1,{\mathfrak b}}$. The space of equivariant loops $L^{K}(V)$ admits an $S^1$-action by shifting the domain parameter, and the natural inclusion ${\mathbb Z}/ p \subset S^1$ defines a ${\mathbb Z}/ p$-action on $L^{K}(V)$ such that the action functional ${\mathcal A}_{H^{(p)}}$ is invariant under such an action. More concretely, the reparametrization $${\mathfrak x}(t) = (x(t), \eta(t)) \mapsto (x(t+\frac1p), \eta(t+\frac1p))$$ generates a ${\mathbb Z}/p$-action on the Floer homology $$R_{1/p}: {\it VHF}_\bullet^{\mathfrak b} (\widehat H^{(p)}; \Lambda_{0, R}) \to {\it VHF}_\bullet^{\mathfrak b} ( \widehat H^{(p)}; \Lambda_{0, R})$$ which is realized by the composition $${\it VCF}_\bullet ( \widehat H^{(p)}, \widehat J^{(p)}; \Lambda_{0, R}) \xrightarrow[\text{pullback}]{\sim} {\it VCF}_\bullet ( \widehat H^{(p)}, \widehat J^{(p)}_{\cdot - 1/p}; \Lambda_{0, R}) \xrightarrow{\text{continuation}} {\it VCF}_\bullet ( \widehat H^{(p)}, \widehat J^{(p)}; \Lambda_{0, R})$$ after passing to homology. Here $\widehat J_{\cdot - \frac{1}{p}}^{(p)}$ is the $S^1$-family of almost complex structures whose value at moment $t$ is $\widehat J_{t- \frac{1}{p}}^{(p)}$. The action $R_{1/p}$ generates a ${\mathbb Z}/p$-action on homology; we denote $$R_{m/p}:= (R_{1/p})^m: {\it VHF}_\bullet^{\mathfrak b}( \widehat H^{(p)}; \Lambda_{0, R}) \to {\it VHF}_\bullet^{\mathfrak b}( \widehat H^{(p)}; \Lambda_{0, R}).$$ Then the map $\partial^1_{0,{\mathfrak b}}$ descends to $$id - R_{1/p}: {\it VHF}_\bullet^{\mathfrak b} ( \widehat H^{(p)}; \Lambda_{0, R}) \to {\it VHF}_\bullet^{\mathfrak b}( \widehat H^{(p)}; \Lambda_{0, R})$$ on homology, while the map $\partial^2_{1,{\mathfrak b}}$ descends to $id + R_{1/p} + \cdots + R_{(p-1)/p}$. The higher order terms encodes the chain homotopies realizing relations of the form $(R_{1/p})^p = id$ on homology, and higher homotopy relations. Finally, we observe that the degree filtration on the chain complex ${\it VCF}_\bullet^{{\mathbb Z}/p}(\widehat H^{(p)}, \widehat J_\infty^{(p)}; \Lambda_R^0)$ induced from variables $u$ and $\theta$ is preserved by the equivariant differential $\partial^{(p)}_{eq, {\mathfrak b}}$, and such a filtration is complete and exhaustive. Therefore, we have a spectral sequence converging to ${\it VHF}^{{\mathbb Z}/p}( \widehat H^{(p)}; \Lambda_{0, R})$, whose first page can be identified with ${\it VHF}_\bullet^{\mathfrak b} ( \widehat H^{(p)}; \Lambda_{0, R} )\llbracket u \rrbracket \langle \theta \rangle$. The same holds for the Tate version, which inverts the variable $u$. ## Equivariant $p$-legged pants operations In this subsection, we define equivariant "$p$-legged\" pants operations on vortex Hamiltonian Floer theory, which generalizes the constructions from [@Seidel_pants; @Shelukhin-Zhao] to our situation. We use the homological convention, so the roles of the positive and negative cylindrical ends are the opposite of those from *loc. cit.*. We will continue the setup from the previous subsection, and keep using the notations $H$, $\widehat H$, $H^{(p)}$, $\widehat H^{(p)}$, and $\widehat J_{t, z}^{(p)}$. Furthermore, we choose a $1$-parameter family of compatible almost complex structures $\widehat J$ on $V$ such that $(\widehat H, \widehat J)$ is regular and the Floer chain complex ${\it VCF}_\bullet^{\mathfrak b}(\widehat H, \widehat J; \Lambda_{0, R})$ is well-defined. The equivariant pants operation is defined over a particularly designed domain. Let $\pi: S_{\mathcal P} \to {\mathbb R} \times S^1$ be the $p$-fold branched cover with unique branch point $(0,0) \in {\mathbb R} \times S^1$ whose ramification point has maximal ramification order. Then $S_{\mathcal P}$ has $p+1$ punctures, regarded as $p$ negative ends and one positive ends. Suppose $S_{\mathcal P}$ are equipped with cylindrical ends $$\epsilon_i^{-}: (-\infty, -1] \times S^1 \to S_{\mathcal P}, \quad \epsilon_i^+: [1,\infty) \times S^1_p \to S_{\mathcal P}, \quad i \in {\mathbb Z} / p,$$ subject to the conditions $$\begin{aligned} \pi(\epsilon_i^{-}(s,t)) &= (s,t), \quad m \cdot (\epsilon_{i}^{-}(s,t)) = \epsilon_{i+m}^{-}(s,t) \\ \pi(\epsilon_i^{+}(s,t)) &= (s,t), \quad m \cdot (\epsilon_i^{+}(s,t)) = \epsilon_{i+m}^{+}(s,t) = \epsilon_i^{+}(s,t+m), \quad \text{ for } m \in {\mathbb Z}/p, \end{aligned}$$ where $S^1_p := {\mathbb R} / p {\mathbb Z}$ is the $p$-fold cover of $S^1 = {\mathbb R} / {\mathbb Z}$. Note that all $\epsilon_i^+$ are obtained from shifting from each other by certain $m \in {\mathbb Z}/p$. The domain-dependent almost complex structure needs to have particular symmetry. We consider almost complex structures $\widehat J^+_\infty$ on $V$ parametrized by $z \in S^\infty$, $t \in S^1$, and $s \geq -1$, such that: 1. for $s \geq 2$ and $z \in S^{\infty}$, we have $\widehat J^+_{s,t,z} = \widehat J^{(p)}_{t,z}$; 2. for any $m \in {\mathbb Z}/p$ and $z \in S^{\infty}$, there holds the equivariance relation $$\widehat J^+_{s,t - \frac{m}{p},z} = \widehat J^+_{s,t,m \cdot z};$$ 3. $\widehat J^+_{s,t,z}$ is invariant under the translation: $$\widehat J^+_{s,t,\tilde{\tau}(z)} = \widehat J^+_{s,t,z}.$$ Given such a choice, we further look at almost complex structures $\widehat J^{-,i}_\infty$ parametrized by with $s \leq 1$, $t\in S^1$, $z \in S^\infty$, and indexed by $i \in {\mathbb Z}/p$ (the label of negative ends) satisfying: 1. for $s \leq -2$ and any $w \in S^{\infty}$, we have $\widehat J^{-,i}_{s,t,z} = \widehat J_t$ for any $i \in {\mathbb Z}/p$; 2. for any $i \in {\mathbb Z}/p$ and $w \in S^{\infty}$, we have the equality $\widehat J^{-,i}_{s,t,z} = \widehat J^{+}_{s,t,z}$ hold for $-1 \leq s \leq 1$; 3. for any $m,i \in {\mathbb Z}/p$ and $z \in S^{\infty}$, there holds the equivariance relation $$\widehat J^{-,i}_{s,t - \frac{m}{p},z} = \widehat J^{-,i+m}_{s,t,z};$$ 4. $\widehat J^{-,i}_{s,t,z}$ is invariant under the translation: $$\widehat J^{-,i}_{s,t,\tilde{\tau}(z)} = \widehat J^{-,i}_{s,t,z}.$$ If $w: {\mathbb R} \to S^{\infty}$ is a parametrized negative gradient flow line of $\tilde{F}$, the above data specify a family of almost complex structures $\{ \widehat J^{\mathcal P}_{v,w} \}_{v \in S_{\mathcal P}}$ given by: 1. $\widehat J^{\mathcal P}_{v,w} = \pi^* \widehat J^{-,i}_{s,t,w(s)} = \pi^*J^{+}_{s,t,w(s)}$ for $v \in \pi^{-1}([-1,1] \times S^1)$ and $\pi(v) = (s,t)$; 2. over the negative ends, $\widehat J^{\mathcal P}_{v,w} = \pi^* \widehat J^{-,i}_{s,t,w(s)}$ if $v = \epsilon_i^-(s,t)$ for all $i = 0,1, \dots, p-1$; 3. over the positive end, $\widehat J^{\mathcal P}_{v,w} = \widehat J^+_{s,t, m \cdot w(s)}$ for all $m \in {\mathbb Z}/p$ and $z = \epsilon_m^+(s,t)$. We need to further introduce a Hamiltonian perturbation term $$\widehat{\mathcal H}^{\mathcal P} \in \Omega^1(S_{\mathcal P}, C^{\infty}(V)^K)$$ satisfying the following conditions. 1. For any $i \in {\mathbb Z}/p$, we have $\widehat{\mathcal H}^{\mathcal P} (\epsilon_i^-(s,t)) = \widehat H_t \otimes dt$; 2. On the positive end, for any $i \in {\mathbb Z}/p$, there holds $\widehat{\mathcal H}^{\mathcal P} (\epsilon_i^+(s,t)) = \widehat H_{t+i}^{(p)} \otimes dt$; 3. The ${\mathbb Z}/p$-equivariance condition $\widehat{\mathcal H}^{\mathcal P}(m \cdot v) = \widehat{\mathcal H}^{\mathcal P}(v)$ holds; 4. Let ${\mathcal H}^{\mathcal P}\in \Omega^1( S_{\mathcal P}, C^\infty(X))$ be the induced Hamiltonian perturbation term on $X$. Then the curvature of the Hamiltonian connection ${\mathcal H}^{\mathcal P}$ on $S_{\mathcal P}$ is $0$. Consider moduli spaces of perturbed vortex equation over the surface $S_{\mathcal P}$. Let $P \to S_{\mathcal P}$ be the trivial $K$-bundle. Given ${\mathfrak x}_+ = (x_+, \eta_+) \in {\rm crit}{\mathcal A}_{H^{(p)}}$ and ${\mathfrak x}_0 = (x_0, \eta_0), \dots, {\mathfrak x}_{p-1} = (x_{p-1}, \eta_{p-1}) \in {\rm crit}{\mathcal A}_H$, for any $i \in {\mathbb Z}_{\geq 0}, m \in {\mathbb Z}/p$ and $\alpha \in \{0,1\}$ we can introduce the moduli space $${\mathcal M}^{i.m}_{{\mathcal P}, \alpha}({\mathfrak x}_0, \dots, {\mathfrak x}_{p-1}; {\mathfrak x}_+)$$ which parametrizes gauge equivalence classes of pairs $$(u, A) \in C^{\infty}(S_{\mathcal P}, V) \times {\mathcal A}(P), \quad \quad \quad w: {\mathbb R}_s \to S^{\infty}$$ which satisfy the equations and asymptotic conditions $$\left\{ \begin{array}{llll} \displaystyle \overline{\partial}_{A, \widehat{\mathcal H}^{\mathcal P}, \widehat J^{\mathcal P}_{v,w}} u= 0, \quad \quad \quad *F_A + \mu(u) = 0, \\ w' (s) - \nabla \tilde{F}(w) = 0, \\ \displaystyle \lim_{s \to -\infty}(u(\epsilon_j^-(s,\cdot)), A(\epsilon_j^-(s,\cdot)), w(\epsilon_j^-(s,\cdot))) = (x_j, 0, \eta_j, Z_{\alpha}^0), \quad \forall j \in {\mathbb Z}/p, \\ \displaystyle \lim_{s \to \infty}(u(\epsilon_0^+(s,\cdot)), A(\epsilon_0^+(s,\cdot)), w(\epsilon_0^+(s,\cdot))) = (x_+, 0, \eta_+, Z_i^m).\end{array} \right.$$ As expected, the moduli space ${\mathcal M}^{i.m}_{{\mathcal P}, \alpha}({\mathfrak x}_0, \dots, {\mathfrak x}_{p-1}; {\mathfrak x}_+)$ admits an Uhlenbeck--Gromov--Floer compactification $\overline{{\mathcal M}}{}^{i.m}_{{\mathcal P}, \alpha}({\mathfrak x}_0, \dots, {\mathfrak x}_{p-1}; {\mathfrak x}_+)$, whose detailed description can be found in [@Seidel_pants Section (4c)]. For a generic choice of almost complex structures and Hamiltonian connections, the moduli space ${\mathcal M}^{i.m}_{{\mathcal P}, \alpha}({\mathfrak x}_0, \dots, {\mathfrak x}_{p-1}; {\mathfrak x}_+)$ is cut out transversely, whose dimension is given by $${\rm CZ}({\mathfrak x}_0) + \cdots + {\rm CZ}({\mathfrak x}_{p-1}) - {\rm CZ}({\mathfrak x}_+) + i -\alpha.$$ We define the pants operations using the above moduli spaces. For each $i \in {\mathbb Z}_{\geq 0}$, $m \in {\mathbb Z}/p$, and $\alpha \in \{0,1\}$, define $$\begin{aligned} {\mathcal P}^{i,m}_{\alpha, {\mathfrak b}}: {\it VCF}_\bullet ( \widehat H, \widehat J; \Lambda_{0, R})^{\otimes p} &\to {\it VCF}_\bullet ( \widehat H^{(p)}, \widehat J^{(p)}; \Lambda_{0, R}) \\ {\mathfrak x}_0 \otimes \cdots \otimes {\mathfrak x}_{p-1} \mapsto \sum_{{\mathfrak x}\atop {\rm CZ}({\mathfrak x}_0) + \cdots + {\rm CZ}({\mathfrak x}_{p-1}) - {\rm CZ}({\mathfrak x}) + i -\alpha = 0} &\left( \sum_{[({\mathfrak u},w)]\in {{\mathcal M}^{i.m}_{\alpha}({\mathfrak x}_0, \dots, {\mathfrak x}_{p-1}; {\mathfrak x})}} \epsilon([({\mathfrak u},w)]) \exp \left( \sum_{i=1}^N \log c_i\ [{\mathfrak u}] \cap V_i \right) \right) {\mathfrak x}, \end{aligned}$$ where a discussion on the sign $\epsilon([({\mathfrak u},w)])$ can be found in [@Shelukhin-Zhao Appendix A]. **Definition 92**. Let ${\mathcal P}^i_{\alpha, {\mathfrak b}} = {\mathcal P}^{i,0}_{\alpha, {\mathfrak b}} + \cdots + {\mathcal P}^{i,p-1}_{\alpha, {\mathfrak b}}$. The ${\mathbb Z}/p$-equivariant product is defined to be $$\begin{aligned} {\mathcal P}: {\it VCF}_\bullet (\widehat H,\widehat J; \Lambda_{0, R})^{\otimes p}\llbracket u \rrbracket \langle \theta \rangle &\to {\it VCF}^{{\mathbb Z}/p}( \widehat H^{(p)}, \widehat J^{(p)}_\infty; \Lambda_{0, R} ) \\ {\mathcal P}(- \otimes 1) &= \sum_{i \geq 0} {\mathcal P}^{2i}_{0,{\mathfrak b}} \otimes u^i + \sum_{i \geq 0} {\mathcal P}^{2i+1}_{0, {\mathfrak b}} \otimes u^i \theta, \\ {\mathcal P}(- \otimes \theta) &= \sum_{i \geq 1} {\mathcal P}^{2i}_{1, {\mathfrak b}} \otimes u^i + \sum_{i \geq 0} {\mathcal P}^{2i+1}_{1,{\mathfrak b}} \otimes u^i \theta. \end{aligned}$$ We can apply the Tate construction to the complex ${\it VCF}_\bullet( \widehat H, \widehat J; \Lambda_{0, R})$, obtaining the Tate complex $$(C_{\mathrm{Tate}}({\mathbb Z} / p, {\it VCF}_\bullet ( \widehat H, \widehat J; \Lambda_{0, R})^{\otimes p}), \partial_{\mathrm{Tate}}).$$ By inverting $u$, the ${\mathbb Z}/p$-equivariant product ${\mathcal P}$ induces a map $$C_{\mathrm{Tate}}({\mathbb Z} / p, {\it VCF}_\bullet ( \widehat H, \widehat J; \Lambda_{0, R})^{\otimes p}) \to {\it VCF}_{\mathrm{Tate}}( \widehat H^{(p)}, \widehat J^{(p)}_\infty; \Lambda_{0, R}),$$ which is also denoted by ${\mathcal P}$. By [@Shelukhin-Zhao Section 8.3], ${\mathcal P}$ defines a chain map on the Tate chain complexes. In fact, the chain map property holds without inverting $u$, but we will not need such a statement. We also need to define a ${\mathbb Z}/p$-equivariant coproduct operation $${\mathcal C}: {\it VCF}^{{\mathbb Z}/p}( \widehat H^{(p)}, \widehat J^{(p)}_\infty; \Lambda_{0, R}) \to {\it VCF}_\bullet (\widehat H, \widehat J; \Lambda_{0, R})^{\otimes p}\llbracket u \rrbracket \langle \theta \rangle$$ by counting solutions to Hamiltonian-perturbed vortex equations defined over the Riemann surface $S_{\mathcal C}$ which is a $p$-fold branched cover of ${\mathbb R} \times S^1$ with unique branch point $(0,0) \in {\mathbb R} \times S^1$ whose ramification point has maximal ramification order, coupled with negative gradient trajectory equations of $\tilde{F}: S^{\infty} \to {\mathbb R}$, but this time $S_{\mathcal C}$ has a single negative cylindrical end and $p$ positive cylindrical ends. By inverting the $u$-variable, ${\mathcal C}$ induces a map $${\it VCF}_{\mathrm{Tate}} (\widehat H^{(p)}, \widehat J^{(p)}_\infty; \Lambda_{0, R}) \to C_{\mathrm{Tate}}({\mathbb Z} / p, {\it VCF}_\bullet (\widehat H, \widehat J; \Lambda_{0, R})^{\otimes p}).$$ ## Local theories and equivariant localization In this subsection, we sketch the ingredients necessary for the proof of the following statement: **Theorem 93**. *The equivariant ${\mathbb Z}/p$-equivariant product on homology $$\label{eqn:pants-iso} {\mathcal P}: {\it H}_{\rm{Tate}}({\mathbb Z}/p, {\it VCF}_\bullet^{\mathfrak b}(\widehat H, \widehat J; \Lambda_{0, R})^{\otimes p}) \to {\it VHF}_{\mathrm{Tate}} ( \widehat H^{(p)}, \widehat J_\infty^{(p)}; \Lambda_{0, R})$$ is an isomorphism.* First, one needs an equivariant version of the local vortex Floer theory as discussed in Section [5](#sec:local-Floer){reference-type="ref" reference="sec:local-Floer"}. Namely, if $x$ is a $1$-periodic orbit of the Hamiltonian $H$, and if we denote by $x^{(p)}$ the $p$-th iteration of $x$, which is necessarily a $1$-periodic orbit of $H^{(p)}$, then there is a well-defined (bulk-deformed) ${\mathbb Z}/p$-equivariant local Floer homology group $${\it VHF}^{{\mathbb Z}/p}_{\rm loc}(H^{(p)}, x^{(p)}; \Lambda_{0, R}),$$ which is defined by looking at contributions to $\partial^{(p)}_{eq, {\mathfrak b}}$ from solutions to [\[eqn:equiv-Floer\]](#eqn:equiv-Floer){reference-type="eqref" reference="eqn:equiv-Floer"} which are contained in a $C^2$-small neighborhood of the equivariant lift of $x^{(p)}$. By inverting $u$, the Tate version is denoted by ${\it VHF}_{\rm Tate}^{\rm loc}( H^{(p)}, x^{(p)}; \Lambda_{0, R})$. One can similarly define the local version of the ${\mathbb Z}/p$-equivariant product and coproduct operation localized near $x$ $$\begin{aligned} {\mathcal P}_{x}^{\rm loc}: H_{\rm{Tate}}({\mathbb Z}/p, {\it VCF}^{\rm loc}_\bullet ( H, x; \Lambda_{0,R})^{\otimes p}) \to {\it VHF}_{\rm{Tate}}^{\rm loc}( H^{(p)}, x^{(p)}; \Lambda_{0,R}), \\ {\mathcal C}_{x}^{\rm loc}: {\it VHF}_{\rm{Tate}}^{\rm loc}( H^{(p)}, x^{(p)}; \Lambda_R^0) \to H_{\rm{Tate}}({\mathbb Z}/p, {\it VCF}^{\rm loc}(H,x; \Lambda_{0, R} )^{\otimes p}). \end{aligned}$$ Note that just as in the non-equivariant setting, equivariant local Floer theories can be defined for isolated but not necessarily nondegenerate iterations. Second, the main result of [@Shelukhin-Zhao Section 10] shows that if $x$ and $x^{(p)}$ are nondegenerate, the composition satisfies $${\mathcal C}_{x}^{\rm loc} \circ {\mathcal P}_{x}^{\rm loc} = (-1)^n u^{n(p-1)} \cdot id,$$ which is an isomorphism as $u$ is invertible in the ground ring of the Tate version, thus ${\mathcal P}_{x}^{\rm loc}$ is an isomorphism by rank considerations. The proof in *loc. cit.* goes through an auxiliary operation ${\mathcal Z}_{x}^{\rm loc}$ satisfying ${\mathcal C}_{x}^{\rm loc} \circ {\mathcal P}_{x}^{\rm loc} = {\mathcal Z}_{x}^{\rm loc}$, which can be defined in our setting following [@Shelukhin-Zhao Definition 10.1]. On the other hand, the calculation ${\mathcal Z}_{x}^{\rm loc} = (-1)^n u^{n(p-1)}$ is based on reducing to the case of local Morse theory by a deformation argument, which is also legitimate in the vortex setting. Then by virtue of the proof of Proposition [Proposition 68](#prop:iso-local){reference-type="ref" reference="prop:iso-local"}, when the Hamiltonian $H$ is a $C^2$-small Morse function, we can match the upstairs and downstairs moduli spaces, so that the calculation also works in our setting. Note that by Proposition [Proposition 73](#prop:indep-bulk){reference-type="ref" reference="prop:indep-bulk"}, the presence of bulk ${\mathfrak b}$ does not affect the argument. Finally, we can write $${\mathcal P} = \sum_{x} {\mathcal P}_{x}^{\rm loc} + O(T^{\delta}), \quad \quad \quad \delta > 0,$$ in which $x$ ranges over all $1$-periodic orbits of $H$ and $O(T^{\delta})$ denotes an operation with positive valuation. Because the local operations ${\mathcal P}_{x}^{\rm loc}$ are isomorphisms, and the contributions of the simple (i.e., non-iterated) $1$-periodic orbits of $H^{(p)}$ to the Tate construction are trivial, we see that ${\mathcal P}$ is an isomorphism over $\Lambda_{0, R} [u^{-1}, u \rrbracket = \Lambda_{0, R} \otimes_{\Lambda_{0, {\mathbb F}_p}} \Lambda_{0, \mathcal K}$. This finishes the sketch of the proof of the equivariant localization isomorphism. ## Growth of total bar length After demonstrating the existence of the equivariant Hamiltonian Floer package in the vortex setting, we are in the right position to prove the inequality of total bar length. *Proof of Theorem [Theorem 89](#thm:smith){reference-type="ref" reference="thm:smith"}.* With equivariant Hamiltonian Floer theory and local Floer homology in our hands, the arguments from [@Shelukhin_2022 Section 7] can easily be adapted to the current situation without much modification. Consequently, we will only provide a sketch of the proof, and refer the reader to *loc. cit.* for complete arguments. Firstly, we recall the following alternative characterization of the total bar length. Given a field ${\mathbb K}$, if we define the vortex Hamiltonian Floer homology over the Novikov ring $$\Lambda_{0, \mathbb K} := \Big\{ \sum_{i=1}^\infty a_i T^{g_i}\ |\ g_i \in {\mathbb R}_{\geq 0},\ a_i \in {\mathbb K},\ \lim_{i \to \infty} g_i = +\infty \Big\}$$ instead of its field of fractions $\Lambda_{\mathbb K}$, then finite bars are reflected as nontrivial *torsion components*, which is the language used in [@FOOO-torsion]. If we denote the direct sum of the torsion components of the Floer homology ${\it VHF}_\bullet^{\mathfrak b}(\tilde\phi; \Lambda_{0, \mathbb K})$ by $$\label{eqn:torsion} \Lambda_{0, \mathbb K} / T^{g_1} \Lambda_{0,\mathbb K} \oplus \cdots \oplus \Lambda_{0, \mathbb K} / T^{g_s} \Lambda_{0, \mathbb K}, \quad \quad \quad \text{ with }\ g_1 \geq \cdots \geq g_s \geq 0,$$ then we can write the total bar length of $\tilde\phi$ over $\Lambda_{\mathbb K}$ as $$\tau_{(p)}^{\mathfrak b}( \tilde\phi, \Lambda_{\mathbb K}) = g_1 + \cdots + g_s,$$ and the boundary depth is given by $g_1$, c.f. [@Shelukhin_2022 Section 4.4.4]. Note that these torsion exponents correspond to *verbose bar-length spectrum* in the sense of [@Usher_Zhang_2016], which means that $g_i$ can be $0$, due to that fact that the Floer differential in our discussion may not strictly decrease the energy. The claim is more easily to be proved when $\tilde\phi^p$ is nondegenerate and bulk-avoiding. We choose a generating Hamiltonian $H$ for the Hamiltonian isotopy $\tilde\phi$. The comparison between $\tau_{(p)}^{\mathfrak b}( \tilde\phi^p)$ and $\tau_{(p)}^{\mathfrak b}( \tilde\phi)$ is established in the following three steps. 1. Using the quasi-Frobenius isomorphism from Lemma [Lemma 90](#lemma:quasi-frob){reference-type="ref" reference="lemma:quasi-frob"}, one can show that the total bar length of $$(C_{\mathrm{Tate}}({\mathbb Z} / p, {\it VCF}_\bullet (\widehat H, \widehat J; \Lambda_{0, R})^{\otimes p}), \partial_{\mathrm{Tate}}),$$ i.e., the sum of torsion exponents of the homology group ${\it H}_{\rm{Tate}}({\mathbb Z}/p, {\it VCF}_\bullet (\widehat H, \widehat J; \Lambda_{0, R})^{\otimes p})$, is equal to $p$ times the quantity $\tau_{(p)}^{\mathfrak b}( \tilde\phi)$. 2. By appealing to the isomorphism in Equation [\[eqn:pants-iso\]](#eqn:pants-iso){reference-type="ref" reference="eqn:pants-iso"} and an application of the homological perturbation lemma, it is shown in [@Shelukhin_2022 Section 7.3.1] the total boundary depth, i.e., the sum of torsion exponents of ${\it H}_{\rm{Tate}}({\mathbb Z}/p, {\it VCF}_\bullet (\widehat H, \widehat J; \Lambda_{0, R})^{\otimes p})$ agrees with that of ${\it VHF}_{\mathrm{Tate}}( \widehat H^{(p)}; \Lambda_{0, R})$. 3. Using [@Shelukhin_2022 Proposition 17, Lemma 18], one can prove that the total boundary depth of ${\it VHF}_{\mathrm{Tate}}( \widehat H^{(p)}; \Lambda_{0, R})$ is bounded from above by $\tau_{(p)}^{\mathfrak b}( \tilde\phi^p)$, which is a reminiscent of the Borel spectral sequence in the context of filtered Floer theory. Finally, to establish Theorem [Theorem 89](#thm:smith){reference-type="ref" reference="thm:smith"} for $\tilde\phi$ and $\tilde\phi^p$ which are not necessarily bulk-avoiding and may admit isolated degenerate fixed points, an approximation argument and multiple applications of the homological perturbation lemma as in [@Shelukhin_2022 Section 7.4] suffice. ◻ *Remark 94*. To prove Theorem [Theorem 89](#thm:smith){reference-type="ref" reference="thm:smith"} for degenerate Hamiltonian diffeomorphisms with isolated fixed points assuming the corresponding result for nondegenerate ones, one can alternatively use the following more elementary argument. Suppose $H$ is not necessarily bulk-avoiding and may have isolated but degenerate fixed points and periodic points of period $p$. Let $H_i$ be a sequence of nondegenerate and bulk-avoiding Hamiltonians on $X$ which converges to $H$ under $C^2$-norm. We can choose the perturbations $H_i$ to be supported in an arbitrarily small neighborhood of the one period orbits of $H$ and $H^{(p)}$, over which the perturbation is modeled on $\epsilon_i f$ where $f$ is a Morse function. Then the above implies that $$\tau_{(p)}^{\mathfrak b}(H_i^{(p)}) \geq p \tau_{(p)}^{\mathfrak b}( H_i).$$ Notice that the reduced barcode of $H_i$ resp. $H_i^{(p)}$ is a Cauchy sequence with respect to the bottleneck distance with a uniformly bounded number of bars. Our choice of perturbation also guarantees a uniform upper bound for the short bars. Moreover, we know that the barcode of the limit $H$ resp. $H^{(p)}$ is finite. Hence the total bar length of $H_i$ resp. $H_i^{(p)}$ converges to that of $H$ resp. $H^{(p)}$, which implies the desired result. # Open string theory I. Quasimap Floer theory {#section8} In this section we recall the construction of quasimap Lagrangian Floer theory developed by Woodward [@Woodward_toric]. The basic idea agrees with the philosophy of gauged linear sigma model [@Witten_LGCY]: one replaces the count of holomorphic curves in the toric manifold $X$ by an equivariant count of holomorphic curves upstairs. There are two significant consequences: first, one can achieve transversality of moduli spaces at a very low cost; second, the counts of curves are all integer-valued. We use the Morse--Bott model for Lagrangians Floer theory to construct open-string theories and closed-open maps. We extend the use of domain-dependent perturbations for bulk-deformed vortex Floer cohomology to the open-string situation. We first need to fix certain notions and notations to describe the combinatorial data of various moduli spaces. ## Trees and treed disks {#subsec:trees} We first set up the convention for the notion of trees used in this paper. *Convention 95* (Convention for trees). A tree, usually denoted by $\Gamma$, consists of a nonempty set of vertices $V_\Gamma$ and a nonempty set of edges $E_\Gamma$. The set of vertices is decomposed into the set of finite vertices and the set of vertices at infinity, and the decomposition is denote by $$V_\Gamma = V_\Gamma^{\rm finite} \sqcup V_\Gamma^\infty.$$ We always assume 1. $V_\Gamma^\infty$ contains a distinguished vertex $v_{\rm root}$ called the *root*. 2. The valence (degree) of any $v\in V_\Gamma^\infty$ is either one or two. The set $V_\Gamma$ is partially ordered in the following way: we denote by $v_\alpha \succ v_\beta$ if $v_\alpha$ and $v_\beta$ are adjacent and $v_\beta$ is closer to the root. In this way vertices at infinities are either *incoming* (called *inputs*) or *outgoing* (called *outputs*); in particular the output $v_{\rm root}$ is outgoing. Edges are decomposed into four groups: the set of finite edges $E_\Gamma^{\rm finite}$ consisting of edges connecting two finite vertices, the set of incoming semi-infinite edges $E_\Gamma^{\rm in}$ consisting of edges connecting $v_\alpha \in V_\Gamma^\infty$ with $v_\beta \in V_\Gamma^{\rm finite}$ with $v_\alpha \succ v_\beta$, the set of outgoing semi-infinite edges $E_\Gamma^{\rm out}$ consisting edges connecting $v_\alpha \in V_\Gamma^{\rm finite}$ and $v_\beta \in V_\Gamma^\infty$ with $v_\alpha \succ v_\beta$, and the set of infinite edges $E_\Gamma^\infty$ connecting two vertices at infinity. We also call incoming resp. outgoing semi-infinite edges inputs resp. outputs. A tree $\Gamma$ is called *unbroken* if all vertices $v \in V_\Gamma^\infty$ has valence $1$. A vertex $v \in V_\Gamma^\infty$ of valence $2$ is called a *breaking* of the tree $\Gamma$. Breakings separate $\Gamma$ into *unbroken components*. A *ribbon tree* is a tree $\Gamma$ together with an isotopy class of embeddings $\Gamma \hookrightarrow {\mathbb R}^2$. Equivalently, it means for each vertex $v \in V_\Gamma$ the adjacent edges are cyclically ordered. As $\Gamma$ is rooted, it follows that all incoming edges are strictly ordered. A ribbon tree is *stable* if the valence of each finite vertex is at least three. ### Metric ribbon trees A *metric* on a ribbon tree $\Gamma$ is a function $${\bm l}: E_\Gamma^{\rm finite} \to [0, +\infty).$$ The underlying decomposition $$E_{\Gamma}^{{\rm finite}}= E_{\Gamma}^{{\rm finite}, 0} \sqcup E_{\Gamma}^{{\rm finite}, +} = {\bm l}^{-1}(\{0\})\sqcup {\bm l}^{-1}((0, +\infty))$$ is called a *metric type*, denoted by $[{\bm l}]$. We often call the pair $(\Gamma, [{\bm l}])$ a **domain type**. A **metric ribbon tree** of type $(\Gamma, [{\bm l}])$ is a pair $(\Gamma, {\bm l})$ such that ${\bm l}$ has the metric type $[{\bm l}]$. As in [@Woodward_toric Section 3.3], one needs to work with unstable trees. We hence replace the usual stability condition by another *minimality* condition. We say that a metric ribbon trees $(\Gamma, {\bm l})$ (resp. domain type $(\Gamma, [{\bm l}])$) is **minimal** if it has no finite edges of length zero or infinite edges. Hence for each domain type $\Gamma$, there is a canonical minimal one $\Gamma^{\rm min}$ obtained from $\Gamma$ by shrinking edges violating the minimality condition. We define perturbations over the universal trees. Consider a minimal domain type $\Gamma = (\Gamma, [{\bm l}])$ (which is not necessarily stable). Then there is a moduli space of metric trees of type $\Gamma$, denoted by $\mathcal{MT}_\Gamma$, which is homeomorphic to $(0, +\infty)^{\# E_{\underline{\smash{\Gamma}}}^{{\rm finite}, +}}$, whose elements parametrize the lengths of finite edges with positive lengths. There is also a *universal tree* $$\mathcal{UT}_\Gamma \to \mathcal{MT}_\Gamma$$ whose fiber over a point $p \in \mathcal{MT}_\Gamma$ is homeomorphic to a metric tree representing $p$ (the infinities of semi-infinite or infinite edges are regarded as points in the metric tree). The above moduli spaces have natural compactifications. In fact, we can define a partial order among all minimal domain types. We say that a minimal domain type $\Gamma$ **degenerates** to another minimal domain type $\Pi$, denoted by $\Pi \preceq \Gamma$, if $\Pi$ is obtained from $\Gamma$ by composing the following types of operations 1. Shrinking the length of a finite edge in $\Gamma$ to zero and collapse this edge. 2. Breaking a finite edge of positive length to a pair of semi-infinite edges joined at a new vertex at infinity. Notice that if $\Pi\preceq \Gamma$, then there is a canonical surjective map $\rho: V_\Gamma^{\rm finite} \to V_\Pi^{\rm finite}$. Then $\mathcal{MT}_\Gamma$ has the natural compactification $$\overline{\mathcal{MT}}_\Gamma:= \bigsqcup_{\Pi \preceq\Gamma} \mathcal{MT}_\Pi.$$ The universal tree is also extended to the compactification, which is denoted by $$\overline{\mathcal{UT}}_\Gamma \to \overline{\mathcal{MT}}_\Gamma.$$ There is a special closed subset $\overline{\mathcal{UT}}{}_\Gamma^{\rm node} \subset \overline{\mathcal{UT}}_\Gamma$ corresponding to infinities or vertices. Notice that the complement of $\overline{\mathcal{UT}}{}_\Gamma^{\rm node}$ inside the interior $\mathcal{UT}_\Gamma$ is a smooth manifold. ### Treed disks **Definition 96**. Given a domain type $\Gamma = (\Gamma, [{\bm l}])$. A *treed disk* of type $\Gamma$, denoted by $C = S \cup T$, is the configuration given by the union of disk components $S_\alpha \cong {\mathbb D}$ for all vertices $v_\alpha\in V_{\Gamma}^{\rm finite}$, a metric ${\bm l}$ on $\Gamma$ of type $[{\bm l}]$ and an interval $I_e$ of length ${\bm l}(e)$ for each finite edge $e \in E_\Gamma^{\rm finite}$. The notion of isomorphisms between treed disks is standard and omitted. ## Quasimap Floer theory for Lagrangians We recall the quasimap Floer theory developed by Woodward [@Woodward_toric]. Let ${\bf u} \in {\rm Int} P \subset {\mathbb R}^n$ be an interior point of the moment polytope $P$ of the toric manifold $X$. Recall that the number of faces $N$ of $P$ coincides with the dimension of $V$. Let $L = L({\bf u}) \subset X$ be the torus fiber over ${\bf u}$. Let $\widehat L = \widehat L ({\bf u}) \subset \mu^{-1}(0) \subset V$ be the lift of $L({\bf u})$, which is a $K$-invariant Lagrangian torus in $V$. Explicitly, we have $$\widehat L = \prod_{i=1}^N \Big\{ z_i \in {\mathbb C}\ |\ |z_i|^2 = \tau_i \Big\}$$ where $\tau_i$ are determined by ${\bf u}$ and the constant term in the moment map $\mu$. A **holomorphic quasidisk** is an ordinary holomorphic map $u: ({\mathbb D}, \partial {\mathbb D}) \to (V, \widehat L)$ (with respect to the standard complex structure $\widehat J_V$). Two holomorphic quasidisks $u$ and $u'$ are $K$-equivalent if there exists $g \in K$ such that $g u = u'$. Each $K$-equivalence class of holomorphic quasidisks represents a disk class $$\beta \in H_2( V, \widehat L)/K \cong H_2(V, \widehat L).$$ Each such class has a well-defined energy $$\omega(\beta) = \omega_V (\beta) \in {\mathbb R}$$ and a well-defined Maslov index $$i(\beta) \in 2{\mathbb Z}.$$ Given $k$ and $\beta \in H_2(V, \widehat L)$, let ${\mathcal M}_{k+1}^{\rm disk} (\beta)$ be the moduli space of $K$-equivalence classes of holomorphic quasidisks of class $\beta$ with $k+1$ boundary marked points, and let $\overline{\mathcal M}{}_{k+1}^{\rm disk}(\beta)$ be its compactification. Notice that as $V$ is aspherical, configurations in $\overline{\mathcal M}{}_{k+1}^{\rm disk} (\beta)$ have only disks bubbles but not sphere bubbles. The evaluation of a $K$-equivalence class of quasidisks at the last boundary marked point is well-defined as a point in the quotient Lagrangian $L \subset X$. Hence there is a continuous map $${\rm ev}: \overline{\mathcal M}{}_{k+1}^{\rm disk}(\beta) \to L.$$ **Theorem 97** (Blaschke product). *Let $u: {\mathbb D}^2 \to V$ be a holomorphic quasidisk. Then there exist $\theta_1, \ldots, \theta_N \in [0, 2\pi)$ and $(a_{i, k})_{k = 1, \ldots, d_i} \subset {\mathbb D}^2 \subset {\mathbb C}$ for $i = 1, \ldots, N$ such that $$\label{Blaschke_formula} u(z) = \left( \sqrt{\tau_1} e^{i \theta_1} \prod_{k=1}^{d_1} \frac{ z- a_{1, k}}{ 1 - \overline{a_{1,k}} z}, \ldots, \sqrt{\tau_N} e^{i \theta_N} \prod_{k=1}^{d_N} \frac{ z- a_{N, k}}{ 1 - \overline{a_{N,k}} z} \right).$$ Moreover, the Maslov index of $u$ is $2 (d_1 + \cdots + d_N)$.* In particular, there are $N$ "basic" Maslov two disk classes $\beta_1, \ldots, \beta_N \in H_2(V, L)$ where each $\beta_i$ is represented by a quasidisk given as above with $d_j = \delta_{ij}$. These Maslov two classes form a basis of $H_2(V, \widehat L)$. **Theorem 98**. *The moduli space ${\mathcal M}_{k+1}^{\rm disk}(\beta)$ is regular of dimension $n + 2 i (\beta) + k-2$ and the evaluation map $${\rm ev}: {\mathcal M}_{k+1}^{\rm disk}(\beta) \to L$$ is a smooth submersion.* *Proof.* See [@Cho_Oh Section 6]. ◻ A consequence is that each stratum of the compactification $\overline{\mathcal M}{}_{k+1}^{\rm disk}(\beta)$ is regular. To be more precise, let $\Gamma$ denote a ribbon tree representing the combinatorial type of a nodal disk (with $k$ inputs and $1$ output) with each vertex labelled by a disk class whose sum is equal to $\beta$. Then there is a stratum ${\mathcal M}_\Gamma^{\rm disk}\subset \overline{\mathcal M}{}_{k+1}^{\rm disk}(\beta)$. **Corollary 99**. *Each stratum ${\mathcal M}{}_\Gamma^{\rm disk} \subset \overline{\mathcal M}_{k+1}(\beta)$ is regular and the evaluation map ${\rm ev}: {\mathcal M}_\Gamma^{\rm disk} \to L$ is a submersion.* *Proof.* See [@Woodward_toric Corollary 6.2]. ◻ ### Treed holomorphic quasimaps The idea of treed holomorphic disks goes back to Cornea--Lalonde [@Cornea_Lalonde_2005; @Cornea_Lalonde]. We recall the adaptation by Woodward [@Woodward_toric] in order to define the quasimap $A_\infty$ algebras. Throughout our discussion, we fix a smooth perfect Morse function $f_L: L \to {\mathbb R}$ defined over the Lagrangian torus $L \subset X$, which has exactly $2^n$ critical points. Given a treed disk $C = S \cup T$ of type $\Gamma$, suppose we have a domain-dependent perturbation $f$ of the Morse function $f_L: L \to {\mathbb R}$ parametrized by points $t$ on the tree part $T$, a **treed holomorphic quasimap** on $C$ is a collection of objects $$\Big( (u_v)_{v \in V_\Gamma^{\rm finite}}, (x_e)_{e \in E_{\Gamma}} \Big)$$ where for each finite vertex $v \in V_\Gamma^{\rm finite}$, we assign a smooth map $u_v: S_v \to V$ satisfying $$\overline\partial u_v = 0,\ u_v(\partial S_v) \subset \widehat L,$$ $x_e: I_e \to L$ is a smooth map satisfying $$x_e'(t) + \nabla f(x_e(t)) = 0;$$ moreover, the matching condition requires 1) for each node joining a boundary point $z$ of some surface component $S_v$ and a finite end of an edge $e$, the value of $x_e(z)$ lies in the same $K$-orbit as the value of $u_v(z)$; 2) for each infinite vertex $v\in V_\Gamma^\infty$ joining two (semi-)infinite edges $e_1$ and $e_2$, the limits of $x_{e_1}$ and $x_{e_2}$ at the corresponding infinities agree. Here to ensure the convergence of the maps $x_e$, we require that the perturbation $f$ is supported away from the infinities. Two treed holomorphic quasimaps are regarded as **equivalent** if after identifying domains, the maps on corresponding surfaces parts are $K$-equivalent (recall $K$ is the gauge group). To define the $A_\infty$ structure (or other structures) one would like to regularize the moduli spaces of equivalence classes of treed holomorphic quasimaps and their boundaries. One first needs to use coherent systems of perturbations to describe such moduli spaces. ### Perturbations for the $A_\infty$ algebra To achieve transversality relevant for defining the $A_\infty$ algebra, we only need to perturb the Morse function on edges. Hence for a given *minimal* metric type $\Gamma$, a domain-dependent perturbation can be viewed as a map $$P_\Gamma: \overline{\mathcal{UT}}_\Gamma \to C^\infty(L).$$ We require any such perturbation to vanish near infinities, i.e., vanish near the closed subset $$\overline{\mathcal{UT}}{}_\Gamma^\infty \subset \overline{\mathcal{UT}}_\Gamma$$ corresponding to positions of vertices at infinity. Notice that if $\Gamma$ is not necessarily stable, a perturbation $P_{\Gamma^{\rm min}}$ for the minimal form is enough to determine the treed holomorphic map on any treed disks $C$ of type $\Gamma$. Indeed, on any infinite edges of $C$ (if any) the negative gradient flow equation is taken for the unperturbed Morse function $f_L$. In order to establish the $A_\infty$ relation, we also need to require that, if $\Gamma$ degenerates to $\Pi$, then the restriction of $P_\Gamma$ to the stratum $\overline{\mathcal{UT}}_\Pi\subset \overline{\mathcal{UT}}_\Gamma$ must agree with the perturbation $P_\Pi$ which have been chosen for the minimal domain type $\Pi$. Hence we need to construct a *coherent* system of perturbations indexed for all minimal domain types $\Gamma$. To use the Sard--Smale theorem to prove that generic perturbations are regular, we also need to specify the neighborhood of $\overline{\mathcal{UT}}{}_\Gamma^\infty$ where we require the perturbation to vanish; such choices of neighborhoods need also be coherent. Another complexity in this procedure is that we need to work with unstable domains (as in [@Woodward_toric], see also [@Abouzaid_plumbing]), unlike the cases of [@Charest_Woodward_2015][@Woodward_Xu][@Venugopalan_Woodward_Xu] where domains are always stable. Here we give a different way of writing Woodward's perturbation scheme for unstable trees (see Section [@Woodward_toric Section 3]). Given a minimal domain type $\Gamma$, an **indexing function** is a map $\vec{n}: V_\Gamma^{\rm finite} \to {\mathbb Z}_{\geq 0}$, whose values are denoted by $n_v$, satisfying that $n_v \geq 1$ when $v$ is an unstable vertex. One should regard the values of $\vec{n}$ as one half of the Maslov indices of disk components. We consider perturbations which depend also on such indexing functions. **Definition 100**. A **coherent family of domain-dependent perturbations** is a collection of continuous maps $$P_{\Gamma, \vec{n}}^{\rm qd}: \overline{\mathcal{UT}}_\Gamma \to C^\infty(L)$$ indexed by all minimal domain types $\Gamma$ and all indexing functions $\vec{n}: V_\Gamma^{\rm finite} \to {\mathbb Z}_{\geq 0}$ satisfying the following conditions. 1. For $\Gamma$ the tree with a single vertex, no input, and one output, the Morse function on the outgoing edge is the unperturbed function $f_L$. 2. When $\Gamma$ degenerates to $\Pi$, there is a canonical surjective map $\rho: V_\Pi^{\rm finite} \to V_\Gamma^{\rm finite}$. Hence any indexing function $\vec{n}_\Pi: V_\Pi \to {\mathbb Z}_{\geq 0}$ induces a partition $\vec{n}_\Gamma: V_\Gamma \to {\mathbb Z}_{\geq 0}$. We require that $$P_{\Gamma, \vec{n}_\Gamma}^{\rm qd}|_{\overline{\mathcal U}_\Pi} = P_{\Pi, \vec{n}_\Pi}^{\rm qd}.$$ 3. When $\Gamma$ is broken with unbroken components $\Gamma_1,\ldots, \Gamma_s$, the partition $\vec{n}$ on $\Gamma$ is defined by assembling the partitions $\vec{n}_1, \ldots, \vec{n}_s$ on $\Gamma_1, \ldots, \Gamma_s$. Then $P_{\Gamma, \vec{n}}^{\rm qd}$ should be naturally induced from $P_{\Gamma_i, \vec{n}_i}$. ### Compactification and transversality Let $\Gamma$ be a possibly unstable, non-minimal domain type. A **map type** over $\Gamma$, denoted by ${\bm \Gamma}$, assigns to each finite vertex $v_\alpha \in V_\Gamma^{\rm finite}$ a disk class $\beta_v$ (with nonnegative Maslov index) and to each vertex at infinity $v_\beta \in V_\Gamma^\infty$ a critical point $x_\beta \in {\rm crit} f_L$. A map type ${\bm \Gamma}$ induces an indexing function $\vec{n}$ on the minimal form $\Gamma^{\rm min}$ by setting $n_v$ to be half of the Maslov index of $\beta_v$ and adding together if several vertices are connected by finite edges of length zero. Then use the perturbation $P_{\Gamma^{\rm min}, \vec{n}}^{\rm qd}$ to define a moduli space ${\mathcal M}_{\bm \Gamma}$ of treed holomorphic disks. The topology of ${\mathcal M}_{\bm\Gamma}$ is defined in the usual way. Given a perturbation, the moduli space ${\mathcal M}_{\bm \Gamma}$ is the zero locus a Fredholm section on certain Banach manifold. We say that the moduli space ${\mathcal M}_{\bm \Gamma}$ is regular if the Fredholm section is transverse (it is independent of the corresponding Sobolev completions of the space of smooth maps). We say that a coherent system of perturbations is **regular** if all moduli spaces ${\mathcal M}_{\bm \Gamma}$ are regular. Now we consider possible degenerations of treed holomorphic disks. In general, a sequence of treed holomorphic disks of a fixed map type ${\bm \Gamma}$ can converge to a limit by breaking an edge, shrinking an edge to zero, or bubbling off holomorphic disks. Notice that because $V$ is a vector space and we do not have interior markings, there cannot be any sphere bubbles in the limit. The notion of convergence is standard and its definition is omitted here. As the perturbation system is coherent, any limiting object (of a possibly different map type ${\bm \Pi}$) is also a treed holomorphic disk defined using a corresponding perturbation $P_{\Pi^{\rm min}, \vec{n}}^{\rm qd}$, hence an element in ${\mathcal M}_{\bm \Pi}$. We denote $$\overline{\mathcal M}_{\bm \Gamma}:= \bigsqcup_{{\bm \Pi} \preceq {\bm \Gamma}} {\mathcal M}_{\bm \Pi}$$ where by abuse of notation, $\preceq$ is the natural partial order among map types induced from the notion of convergence. **Proposition 101**. *There exists a coherent system of perturbation data such that every moduli space ${\mathcal M}_{\bm \Gamma}$ is regular.* *Proof.* The proof is an inductive construction with respect to the partial order $\Pi \preceq \Gamma$ among minimal domain types and the indexing function $\vec{n}$. First one can easily check, by the Blaschke formula Theorem [Theorem 97](#thm_Blaschke){reference-type="ref" reference="thm_Blaschke"} that the specification of item (1) in Definition [Definition 100](#defn_coherent_perturbation){reference-type="ref" reference="defn_coherent_perturbation"} can make the relevant configurations transverse. Then once regular perturbations on all boundary strata of $\mathcal{UT}_\Gamma$ have been fixed, one can use the Sard--Smale theorem to find regular extensions to the interior. See details in [@Woodward_toric Corollary 6.2]. ◻ Now we consider the compactification of moduli spaces. A map type ${\bm \Gamma}$ is called **essential** if it is unbroken and has no boundary edges of length zero. Given a collection ${\bm x} = (x_1, \ldots, x_k; x_\infty)$ of critical points of the Morse function $f_L$, for $i = 0, 1$, let $${\mathcal M}^{qd}(x_1, \ldots, x_k; x_\infty)_i := \bigcup_{{\bm \Gamma}} {\mathcal M}_{\bm \Gamma}$$ where the union is taken over all essential map types of index $i$ whose vertices at infinities are labelled by ${\bm x}$. **Lemma 102**. *If $i = 0$, the moduli space ${\mathcal M}^{qd}(x_1, \ldots, x_k; x_\infty)_0$ is discrete and has finitely many points below any given energy bound. If $i = 1$, the compactified moduli space $\overline{\mathcal M}{}^{qd}(x_1, \ldots, x_k; x_\infty)_1$ is a 1-dimensional (topological) manifold with boundary, which is compact below any given energy bound.* *Proof.* For the zero-dimensional moduli space, the claimed finiteness follows from the compactness argument and the transversality. For the one-dimensional moduli space, the fact that it is a 1-dimensional manifold with boundary follows from the transversality, compactness, as well as the standard gluing construction. ◻ Moreover, the moduli spaces are all oriented. The orientation depends on choices of orientations of unstable manifolds of critical points of $f_L$ and the orientations of moduli spaces of quasidisks; the latter depends on the orientation of the Lagrangian torus and the spin structure, which we fix from the beginning. Notice that these choices can be made independent of the position ${\bf u}\in {\rm Int} P$ in the interior of the moment polytope. ### Quasimap Fukaya $A_\infty$ algebra We would like to define a (family of) cohomologically unital $A_\infty$ algebra(s) over $\Lambda_{\overline{\mathbb Q}}$ from the moment Lagrangian tori. Given a Lagrangian torus $L = L({\bf u}) \subset X$, a **local system** on $L$ is a homomorphism $${\bf y}: H_1(L; {\mathbb Z}) \to \exp(\Lambda_{0, \overline{\mathbb Q}}).$$ Introduce the notation ${\bm L}= (L, {\bf y})$. We denote the corresponding bulk-deformed $A_\infty$ algebra of ${\bm L}$ by ${\mathcal F}_{\mathfrak b}({\bm L})$, which is defined as follows. First, the underlying ${\mathbb Z}_2$-graded vector space is $${\it QCF}_{\mathfrak b}^\bullet ( {\bm L}; \Lambda_{\overline{\mathbb Q}}):= {\rm Span}_{\Lambda_{\overline{\mathbb Q}}} {\rm crit} f_L \cong (\Lambda_{\overline{\mathbb Q}})^{2^n}$$ where the degree of a critical point $x \in {\rm crit} f_L$ is $|x| = n - {\rm index} (x)\ {\rm mod}\ 2$. Given critical points $x_1, \ldots, x_k$, define $$\label{composition_defn} m_k (x_k, \ldots, x_1) = \sum_{x_\infty} (-1)^\heartsuit \left( \sum_{ [u]\in {\mathcal M}^{qd}(x_1, \ldots, x_k; x_\infty)_0} {\mathfrak b}([u]) T^{E([u])} {\bf y}^{\partial[u]} \epsilon([u]) \right) x_\infty.$$ We explain the terms below. 1. The sign $\heartsuit$ is defined as $$\label{heartsuit} \heartsuit:= \sum_{i=1}^k i |x_i|\in {\mathbb Z}_2.$$ 2. For each disk $u$ with boundary on $\widehat L$, as $\widehat L$ does not intersect the bulk, there is a well-defined topological intersection number $${\mathfrak b}([u]):= \prod_{j=1}^N c_j^{u \cap V_j}$$ which only depends on $K$-equivalence class $[u]$. Notice that if $c_j \in {\mathbb Z}[{\bf i}]$, so is ${\mathfrak b}(u)$. 3. $E([u])\in {\mathbb R}$ is the energy of $[u]$. 4. ${\bf y}^{\partial [u]} \in \exp (\Lambda_{0, \overline{\mathbb Q}})$ is the value of the local system ${\bf y}$ on the loop $\partial [u]\subset L$. 5. $\epsilon([u]) \in \{\pm 1\}$ is determined by the orientation of the zero-dimensional moduli space. Similar to previous cases involving bulk deformations, the expression [\[composition_defn\]](#composition_defn){reference-type="eqref" reference="composition_defn"} is a legitimate element of ${\it QCF}_{\mathfrak b}^\bullet ({\bm L}; \Lambda_{\overline{\mathbb Q}})$. Extending linearly, one obtains a linear map $$m_k: {\it QCF}_{\mathfrak b}^\bullet ({\bm L}; \Lambda_{\overline{\mathbb Q}} )^{\otimes k} \to {\it QCF}_{\mathfrak b}^\bullet ({\bm L}; \Lambda_{\overline{\mathbb Q}}).$$ Notice that when $k = 0$, this is a linear map $$m_0: \Lambda_{\overline{\mathbb Q}} \to {\it QCF}_{\mathfrak b}^\bullet ({\bm L}; \Lambda_{\overline{\mathbb Q}}).$$ **Theorem 103** ([@Woodward_toric]). *The collection of linear maps $m_0, m_1, \ldots$ defines a curved $A_\infty$ algebra structure on ${\it QCF}_{\mathfrak b}^\bullet ({\bm L}; \Lambda_{\mathbb Q})$, denoted by ${\mathcal F}_{\mathfrak b}({\bm L})$. Moreover, if $x_{\max}$ is the unique maximal point of $f_L$, then ${\bf e} = x_{\rm min}$ is a cohomological unit of ${\mathcal F}_{\mathfrak b}({\bm L})$, namely $m_1 ({\bf e}) = 0$ and $$(-1)^{|x|} m_2( {\bf e}, x) = m_2( x, {\bf e}) = x,\ \forall x\in {\it QCF}_{\mathfrak b}^\bullet({\bm L}; \Lambda_{\overline{\mathbb Q}}).$$* *Proof.* See [@Woodward_toric Theorem 3.6] for the case without bulk deformation. One can verify that the case with bulk deformation can be proved in the same way. ◻ *Remark 104*. The $A_\infty$ algebra can be defined over ${\mathbb Z}$ as long as the bulk deformation has integer coefficients, though we do not need such a fact in our discussion. ### Potential function and nontrivial Floer cohomology Although the quasimap Fukaya algebra is only cohomologically unital, one can still define the potential function. **Proposition 105**. *For the quasimap $A_\infty$ algebra ${\it QCF}({\bm L}; \Lambda_{\overline{\mathbb Q}})$, $m_0(1)$ is a multiple of ${\bf e}$.* *Proof.* See [@Woodward_toric Proposition 3.7] for the case with ${\mathfrak b}= 0$. When we use a nontrivial (small) bulk deformation, as we only change the weights in counting but do not modify the perturbation method, the same proof goes through. ◻ **Definition 106**. Define $W_{\mathfrak b}(u): H_1(L( {\bf u} ); \exp (\Lambda_{0, \overline{\mathbb Q}}) ) \to \Lambda$ by $$m_0(1) = W_{\mathfrak b}(u)({\bf y}) {\bf e}$$ and call it the **potential function** of the brane ${\bm L}= (L({\bf u}), y)$. By abuse of terminology, we also call $W_{\mathfrak b}$ the bulk-deformed potential function of the Lagrangian $L({\bf u})$ or the toric manifold. Let $({\mathbb C}^*)^n \cong X^* \subset X$ be the complement of toric divisors. Choose a trivialization $$\tau_X: {\rm Int}P \times T^n \to X^*$$ which is unique up to isotopy, which induces a well-defined trivialization $$\bigsqcup_{u \in {\rm Int} P} H_1(L({\bf u}); \exp (\Lambda_{0, \overline{\mathbb Q}})) = {\rm Int} P \times (\exp (\Lambda_{0, \overline{\mathbb Q}}))^n.$$ The bulk-deformed **quasimap disk potential** of the toric manifold $X$ is defined by $$\begin{split} W_{\mathfrak b}: {\rm Int} P \times (\exp \Lambda_0)^n & \to \Lambda\\ ({\bf u}, {\bf y}) & \mapsto W_{\mathfrak b}({\bf u})({\bf y}). \end{split}$$ Now we can define the quasimap Floer cohomology. By the $A_\infty$ relation, for any $x \in {\it QCF}({\bm L}; \Lambda_{\overline{\mathbb Q}})$, $$m_1(m_1(x)) + (-1)^{\|x\|} m_2( m_0(1), x) + m_2(x,m_0(1)) = 0.$$ By Theorem [Theorem 103](#cohomological_unit){reference-type="ref" reference="cohomological_unit"}, the last two terms cancel. Hence $m_1^2 = 0$. Hence one can define the ${\mathfrak b}$-deformed **quasimap Floer cohomology** of the brane ${\bm L}$ to be $${\it QHF}_{\mathfrak b}^\bullet ({\bm L}; \Lambda_{\overline{\mathbb Q}}):= {\rm ker} m_1/ {\rm im} m_1.$$ Following [@Cho] [@Cho_Oh] [@Woodward_toric], to find nontrivial Floer cohomology, one needs to establish a version of the divisor equation. Recall that $L \cong (S^1)^n$ with $H_1 (L; {\mathbb Z}) \cong {\mathbb Z}^n$. The perfect Morse function $f_L$ has exactly $n$ critical points of Morse index $1$, whose homology classes are identified with the $n$ standard generators of $H_1(L; {\mathbb Z})$. If $x_1, \ldots, x_n$ are these generators, then any local system ${\bf y}$ is determined by the values $$y_1 = {\bf y}(x_1), \ldots, y_n = {\bf y}(x_n).$$ **Theorem 107**. *If $x$ is a generator of $H_1(L; {\mathbb Z})$, then $$m_1(x) = \partial_x W_{\mathfrak b}({\bf u})(y_1, \ldots, y_n)$$* *Proof.* In the absence of bulk deformation, this is established in [@Woodward_toric Section 3.6], which also carries over in our case. ◻ Lagrangian branes with nontrivial Floer cohomology can be identified with critical points of the potential function. **Theorem 108**. *(cf. [@Woodward_toric Theorem 6.6]) If ${\bf y} = (y_1, \ldots, y_n)$ is a critical point of $W_{\mathfrak b}({\bf u})$, then the Floer cohomology of ${\bm L}({\bf u}) = (L({\bf u}),{\bf y})$ is isomorphic to $H^\bullet( L({\bf u}); \Lambda_{\overline{\mathbb Q}})$.* *Proof.* The case with ${\mathfrak b} = 0$ is given by [@Woodward_toric Theorem 6.6]. When we have a nonzero small bulk deformation, it is still a consequence of the divisor equation (Theorem [Theorem 107](#thm_divisor_eqn){reference-type="ref" reference="thm_divisor_eqn"}). ◻ ## Critical points of the Givental--Hori--Vafa potential In this subsection we study various properties of the deformed Givental--Hori--Vafa potential which arises from disk counting in gauged linear sigma model. We first recall the expression of the Givental--Hori--Vafa potential in terms of the data of the moment polytope and explain its relation with the quasimap disk potential. Let $\Delta\subset {\mathbb R}^n$ be the moment polytope of $X$, described by $$\Delta = \Big\{ u \in {\mathbb R}^n\ |\ l_j(u) = \langle u, v_j \rangle - \lambda_j \geq 0,\ j = 1, \ldots, N \Big\}.$$ Here $v_j = (v_{j, 1}, \ldots, v_{j, n}) \in {\mathbb Z}^n$, $j = 1, \ldots, N$ are the inward normal vectors of each codimension 1 face of $\Delta$ coming from the toric data and $\lambda_j \in {\mathbb R}$. The **Givental--Hori--Vafa potential** of $X$ (or rather its moment polytope) is the element $$W_0 = \sum_{j=1}^N T^{-\lambda_j} y^{v_j}:= \sum_{j=1}^N T^{-\lambda_j} y_1^{v_{j, 1}}\cdots y_n^{v_{j, n}}\in \Lambda[y_1, \ldots, y_n, y_1^{-1}, \ldots, y_n^{-1}].$$ More generally, given any small bulk deformation ${\mathfrak b} = \sum_{j=1}^N \log c_j V_j$, the deformed Givental--Hori--Vafa potential is defined to be $$W_{\mathfrak b} = \sum_{j=1}^N c_j T^{-\lambda_j} y^{v_j}.$$ Without loss of generality, we assume that the origin $0 \in {\mathbb R}^n$ is contained in the interior of $\Delta$. Hence all $\lambda_j$ are positive. **Definition 109**. A point ${\bm \eta} = (\eta_1, \ldots, \eta_n) \in (\Lambda \setminus \{0\})^n$ is called a **critical point** of $W_{\mathfrak b}$ if $$\left( y_1 \frac{\partial W_{\mathfrak b}}{\partial y_1} \right) (\eta_1, \ldots, \eta_n) = \cdots = \left( y_n \frac{\partial W_{\mathfrak b}}{\partial y_n} \right) (\eta_1, \ldots, \eta_n) = 0.$$ A critical point ${\bm \eta}$ is called **nondegenerate** if $$\det \left( \eta_i \eta_j \frac{\partial^2 W_{\mathfrak b}}{\partial y_i \partial y_j}({\bm \eta}) \right) \neq 0.$$ $W_{\mathfrak b}$ is called a **Morse function** if all the critical points are nondegenerate. Observe that the Givental--Hori--Vafa potential is very similar to the quasidisk potential; the latter has a dependence on $u\in {\rm Int}\Delta$. Indeed, the disk potential of the Lagrangian $L(u)$ with a local system ${\bm y}\in (\exp(\Lambda_0))^n$ is $$W_{\mathfrak b}(T^{u_1} y_1, \ldots, T^{u_n} y_n).$$ This is proved by [@Woodward_toric Corollary 6.4] in the absence of bulk deformations, and the bulk-deformed version follows from the same argument by the Blaschke formula. Hence a critical point of $W_{\mathfrak b}$ corresponds to a Floer nontrivial Lagrangian if the valuation of the coordinates of the critical point is in the interior of the moment polytope. On the other hand, in view of mirror symmetry, the Jacobian ring of the Givental--Hori--Vafa potential, or formally the ring of functions on the critical locus, is closely to related to the quantum cohomology under mirror symmetry. However, their ranks agree only in the Fano case. In general, certain critical points fall outside the moment polytope and do not correspond to cohomology classes of the toric manifold. *Example 110*. Consider the $n$-th Hirzebruch surface $F_n$ ($n \geq 1$) whose moment polytope is $$\Delta = \left\{ u = (u_1, u_2) \in {\mathbb R}^2\ \left| \ \begin{array}{c} l_1(u) = u_1 \geq 0,\\ l_2(u) = u_2 \geq 0,\\ l_3(u) = 1-\alpha - u_2 \geq 0,\\ l_4 (u) = n -u_1 - n u_2 \geq 0. \end{array}\right. \right\}$$ Here $\alpha\in (0, 1)$ is a parameter. The (undeformed) Givental--Hori--Vafa potential is $$W_0 (y_1, y_2) = y_1 + y_2 + T^{1-\alpha} y_2^{-1} + T^n y_1^{-1} y_2^{-n}.$$ The equations for critical points are $$\begin{aligned} &\ y_1 = T^n y_1^{-1} y_2^{-n},\ &\ y_2 = T^{1-\alpha} y_2^{-1} + n T^n y_1^{-1} y_2^{-n}.\end{aligned}$$ Assume $n$ is even to simplify notations. Solving $y_1$ one obtains $$y_1 = \pm T^{\frac{n}{2}} y_2^{-\frac{n}{2}}$$ and hence $$\label{Hirzebruch_surface_critical_point} y_2 = T^{1-\alpha} y_2^{-1} \pm n T^{\frac{n}{2}} y_2^{-\frac{n}{2}} \Longrightarrow y_2^{\frac{n}{2}-1}( y_2^2 - T^{1-\alpha} ) = \pm T^{\frac{n}{2}}.$$ Each of the two equations has $\frac{n}{2}+1$ roots, providing $n+2$ critical points, much larger than the rank of homology (which is $4$). Notice that there are two solutions to [\[Hirzebruch_surface_critical_point\]](#Hirzebruch_surface_critical_point){reference-type="eqref" reference="Hirzebruch_surface_critical_point"} of the form $$y_2 = \pm T^{\frac{1-\alpha}{2}} + {\rm higher\ order\ terms}.$$ They give 4 critical points whose "tropical" positions are inside the moment polytope $\Delta$. There are also $n-2$ roots of [\[Hirzebruch_surface_critical_point\]](#Hirzebruch_surface_critical_point){reference-type="eqref" reference="Hirzebruch_surface_critical_point"} whose valuations are $$\frac{\frac{n}{2}-(1-\alpha)}{ \frac{n}{2}-1}> 1-\alpha.$$ They correspond to critical points which are outside the moment polytope. This ends the example. **Definition 111**. We say that a critical point ${\bf \eta} = (\eta_1, \ldots, \eta_n)$ of $W_{\mathfrak b}$ is **inside the moment polytope** $\Delta$ if $$\vec{\mathfrak v}_T({\bm \eta}) = ({\mathfrak v}_T(\eta_1), \ldots, {\mathfrak v}_T(\eta_n)) \in {\rm Int} \Delta \subset {\mathbb R}^n.$$ Denote by $${\rm Crit}_X W_{\mathfrak b} \subset {\rm Crit} W_{\mathfrak b}$$ the set of critical points of $W_{\mathfrak b}$ that are inside the moment polytope of $X$. **Proposition 112**. *Let ${\mathfrak b}$ be an arbitrary small bulk deformation. When $W_{\mathfrak b}$ is a Morse function, one has $$\# {\rm Crit}_X W_{\mathfrak b} = {\rm dim} H_\bullet(X).$$* *Proof.* We use a result of Fukaya *et. al.* [@FOOO_mirror Theorem 2.8.1 (2)]. First, Fukaya *et. al.* defined their bulk-deformed Lagrangian Floer disk potential $\mathfrak{PO}_{\mathfrak b}$ by counting (stable) holomorphic disks inside the toric manifold (using $T^n$-equivariant Kuranishi structures). For our bulk-deformed Givental--Hori--Vafa potential function $W_{\mathfrak b}$, their theorem shows that there exists a bulk deformation ${\mathfrak b}'$ and a "change of coordinate" $y\mapsto y'$ such that $$W_{\mathfrak b}(y') = \mathfrak{PO}_{{\mathfrak b}'}(y).$$ Notice that the change of coordinate does not change the Morse property and the tropical positions of the critical points. Hence one has $$\# {\rm Crit}_X (W_{\mathfrak b}) = \# {\rm Crit}_X (\mathfrak{PO}_{{\mathfrak b}'}).$$ On the other hand, by [@FOOO_mirror Theorem 1.1.3], this number of critical points coincides with the rank of homology. ◻ Lastly we prove the following fact. **Theorem 113**. *There exists a small bulk deformation ${\mathfrak b} = \sum_{j=1}^N \log c_j V_j$ with $c_j \in {\mathbb Z}[{\bf i}]$ such that $W_{\mathfrak b}$ is a Morse function and all critical values are distinct.* *Proof.* We first show that the statement is true for generic ${\mathfrak b}$ with complex coefficients. First, we relate the Givental--Hori--Vafa potential to a complex Laurent polynomial by evaluation at $T = t$ for some complex number $t$. To consider convergence issue, introduce $$\Lambda^{\rm conv}_{0, \overline{\mathbb Q}}:= \left\{ \sum_{i=1}^\infty a_i T^{\lambda_i} \in \Lambda_{0, \overline{\mathbb Q}} \ |\ \sum_{i=1}^\infty |a_i| |t|^{\lambda_i}\ {\rm converges\ for\ } |t|\leq \epsilon\ {\rm for\ some\ }\epsilon>0 \right\}.$$ Let $\Lambda^{\rm conv}_{\overline{\mathbb Q}}$ be its field of fractions. By [@FOOO_toric_1 Proposition 8.5], $\Lambda_{\overline{\mathbb Q}}^{\rm conv}$ is algebraically closed. On the other hand, critical points of $W_{\mathfrak b}$ are solutions to algebraic equations with coefficients in $\Lambda_{\overline{\mathbb Q}}^{\rm conv}$, as the convergence holds due to the fact that $W_{\mathfrak b}$ has only finitely many terms. Hence critical points are in $(\Lambda_{\overline{\mathbb Q}}^{\rm conv})^n$. On the other hand, if we regard $T$ as a complex number, then by Kouchnirenko's theorem [@Koushnirenko_1975], there is a proper analytic subset $S \subset {\mathbb C}^n$ (which in particular has positive codimension) such that when $$c(t) = (c_1 t^{-\lambda_1}, \ldots, c_N t^{-\lambda_N}) \notin S$$ the function $W_{\mathfrak b}^t:=\sum_{j=1}^N c_j t^{-\lambda_j} y^{v_j}$ has finitely many critical points and the number of them is bounded by $n!$ times the volume of the Newton polytope of this Laurent polynomial (which only depends on the moment polytope). As proved by Iritani [@Iritani_2009 Proposition 3.10], we can also guarantee that all critical points are nondegenerate. Now take a generic point $(c_1, \ldots, c_N)$ [^8] so that $c(1) \notin S$. We claim that such a point satisfies our requirement. Indeed, the map $$c: {\mathbb C} \setminus (-\infty, 0] \to {\mathbb C}^n$$ is an analytic map. Hence the complement of $c^{-1}(S)$ contains points arbitrarily close to $0$. We first show that the number of critical points of $W_{\mathfrak b}$ is no greater than Kouchnirenko's bound, temporarily denoted by $N_\Delta$. Indeed, if there are $N_\Delta +1$ critical points, then as the coordinates of them are in $\Lambda_{\overline{\mathbb Q}}^{\rm conv}$, we can evaluate them at $T = t$ with $|t|$ sufficiently small and $c(t)\notin S$, obtaining more critical points of $W_{\mathfrak b}^t$ than possible. Similarly, as we can evaluate critical points at $|t|$ small, all critical points have to be nondegenerate. Lastly, we prove that for generic ${\mathfrak b}$ all critical values of $W_{\mathfrak b}$ are distinct. First notice that the complex monomials $W_1, \ldots, W_N$ separate points, i.e., given $y', y'' \in ({\mathbb C}^*)^n$, $y' \neq y''$, for some $W_j$, $W_j(y') \neq W_j(y'')$. This is because a subset of $n$ monomials among $W_1, \ldots, W_N$ are coordinates on the torus of $y_1, \ldots, y_n$. Now consider the universal critical locus $$\widetilde{\rm Crit} W:= \big\{ (c_1, \ldots, c_N, y_1, \ldots, y_n)\ |\ dW_{\mathfrak b}(y_1, \ldots, y_n) = 0 \big\}.$$ Over the nondegenerate locus it is a smooth $N$-dimensional complex manifold and $c_1, \ldots, c_N$ are local parameters. Given a nondegenerate $c_1, \ldots, c_N$, let $y^{(1)}, y^{(2)}$ be two different critical points. Suppose $W_j(y^{(1)}) \neq W_j(y^{(2)})$. Then deforming $c$ along $(c_1, \ldots, c_j + s, \ldots, c_N)$ and let the two critical points deform as $y^{(1)}(s)$, $y^{(2)}(s)$. Then $$\frac{d}{ds} \left( W_s(y^{(1)}(s)) - W_s(y^{(2)}(s)) \right) = W_j(y^{(1)}) - W_j(y^{(2)}) \neq 0.$$ This means that the locus of $c$ where two critical values coincide is cut out transversely. Now we have shown that for generic complex ${\mathfrak b}$, $W_{\mathfrak b}$ satisfies the requirement. As the set of such complex ${\mathfrak b}$ is open and dense, one can actually find ${\mathfrak b}$ such that $c_j \in {\mathbb Q}[\sqrt{-1}]$. Then by rescaling one can find the desired bulk deformation. ◻ **Definition 114**. A bulk-deformation ${\mathfrak b} = \sum_{j=1}^N \log c_j V_j$ with $c_j \in {\mathbb Z}[{\bf i}]$ is called **convenient** if $W_{\mathfrak b}$ is a Morse function and all critical values are distinct. ## Homotopy units {#subsection_homotopy_unit} The $A_\infty$ algebra constructed using our perturbation scheme only has cohomological units. In order to establish strict unitality one needs the system of perturbations to satisfy an additional property with respect to the operation of forgetting any boundary inputs and stabilize. This is difficult to achieve (in contrast to the case of [@FOOO_toric_1]). Here we use a typical method of constructing a homotopy unit which appeared in [@FOOO_Book][@Ganatra_thesis][@Sheridan_2016][@Charest_Woodward_2015][@Woodward_Xu][@Venugopalan_Woodward_Xu] etc. **Definition 115**. [@Sheridan_2016 Section 4.3] Let $(A, {\bf e})$ be a cohomological unital $A_\infty$ algebra over $\Lambda_{\mathbb K}$. A **homotopy unit structure** on $(A, {\bf e})$ is an $A_\infty$ structure on the $\Lambda_{\mathbb K}$-module $$A^+ = A \oplus \Lambda_{\mathbb K} {\bf f}[1] \oplus \Lambda_{\mathbb K} {\bf e}^+$$ such that the $A_\infty$ composition maps on $A^+$ restrict to the $A_\infty$ composition maps on $A$, $m_1({\bf f}) = {\bf e}^+ - {\bf e}$, and such that ${\bf e}^+$ is a strict unit, i.e. $$\begin{aligned} &\ (-1)^{|a|} m_2({\bf e}^+, a) = m_2(a, {\bf e}^+) = a,\ &\ m_k(\cdots, {\bf e}^+, \cdots) = 0\ \forall k \neq 2.\end{aligned}$$ To construct a homotopy unit, one needs to include a collection of extra moduli spaces. Consider **weighted ribbon trees** $\Gamma$ whose vertices at infinity $v\in V_\Gamma^\infty$ are either **unweighted** or **weighted**. We require that when $v$ is an output or a breaking, it must be unweighted. Each weighted boundary input carries an additional parameter $\rho\in [0, 1]$. Therefore a moduli space of weighted metric ribbon trees has additional parameters from weighted inputs. We require that the perturbations $P_{\Gamma, \vec{n}}^{\rm disk}$ on any (minimal) tree $\Gamma$ also depend on these parameters. Moreover, we require that 1. When all inputs are unweighted, the perturbation on this tree coincides with the perturbation we have chosen to define the cohomologically unital Fukaya algebra ${\mathcal F}_{\mathfrak b} ({\bm L})$. 2. For each weighted input, when the parameter $\rho = 0$, the perturbation on this tree agrees with the perturbation for the tree $\Gamma'$ obtained by changing this weighted input to an unweighted input. 3. For each weighted input $v\in V_\Gamma^\infty$, when the parameter $\rho = 1$, the perturbation $P_{\Gamma, \vec{n}}^{\rm disk}$ on this tree agrees with the perturbation obtained by pulling back a perturbation $P_{\Gamma', \vec{n}'}^{\rm disk}$ via the forgetful map. Here $\Gamma'$ is defined as follows. Suppose $v$ is attached to a finite vertex $v'$. If $n_{v'} > 0$ or after forgetting $v$, $v'$ is still stable, then $\Gamma'$ is just obtained by $\Gamma$ by removing $v$; if $n_{v'} = 0$ and $v'$ becomes unstable after removing $v$, then $\Gamma'$ is obtained from $\Gamma$ by removing $v$ and contracting $v'$ to the next adjacent finite vertex. See Figure [2](#figure_forget_weighted){reference-type="ref" reference="figure_forget_weighted"} for illustration of this operation. {#figure_forget_weighted} Now we need to define the additional composition maps $m_k^+$ on $A^+$ when involves the new generators $f$ and $e^+$, and prove the $A_\infty$ relation for this enlarged set of compositions. We first define $$m_k^+(\cdots, {\bf e}^+, \cdots)$$ according to the requirement of strict unit. Then we need to define $m_k^+$ for variables being either the original generators of $A$ or the element ${\bf f}$. To define this, we require that the incoming edges corresponding to weighted inputs converge to the unique maximal point of the Morse function $f_L: L \to {\mathbb R}$, and count $0$-dimensional moduli spaces. A consequence of the fact that all quasidisks have positive Maslov index is that $$m_k^+({\bf f}, \cdots, {\bf f}) = 0\ \forall k \geq 2.$$ We need to verify the $A_\infty$ relation for all $m_k^+$. Recall that the $A_\infty$ relation reads $$\sum_{j=0}^k \sum_{i=0}^{k-j} (-1)^{\maltese_1^i} m_{k-j+1}^+ (x_k, \cdots, m_j^+ (x_{i+j+1}, \ldots, x_{i+1}), x_i, \ldots, x_1) = 0.$$ We only needs to verify for the case when all variables are generators of $A^+$. When all of them are old generators of $A$, this is the same as the original $A_\infty$ relation for $m_k$; when some variable is ${\bf e}^+$, this can be verified from the requirement that ${\bf e}^+$ satisfies the equations for a strict unit. Now assume that all variables are either old generators or ${\bf f}$. Consider $1$-dimensional moduli spaces with this fixed sequence of inputs and consider its boundary strata. In addition to the strata corresponding to boundary edge breakings, additional boundary strata corresponding to parameters $\rho$ on weighted inputs turn to $0$ or $1$. These strata correspond to the terms $m_k^+( \cdots, m_1^+({\bf f}), \cdots)$ in the $A_\infty$ relation. Hence the $A_\infty$ relation for $m_k^+$ is verified. We summarize the above discussion as follows. **Proposition 116**. *There exists a homotopy unit structure on the cohomologically unit $A_\infty$ algebra ${\mathcal F}_{\mathfrak b} ({\bm L})$. Denote the corresponding strictly unital $A_\infty$ algebra by ${\mathcal F}_{\mathfrak b}^+({\bm L})$. Moreover, if we denote the element whose coboundary relates ${\bf e}$ and ${\bf e}^+$ by ${\bf f}_{{\bm L}}$, then one has $$m_k^+\Big( \underbrace{{\bf f}_{{\bm L}}, \ldots, {\bf f}_{{\bm L}}}_{k} \Big) = 0,\ \forall k \geq 2.$$* ### Canonical weakly bounding cochain Recall that a weakly bounding cochain is an odd element $b \in {\mathcal F}_{\mathfrak b}^+ ({\bm L})$ solving the weak Maurer--Cartan equation $$\sum_{k\geq 0} m_k^+(b, \cdots, b) \in \Lambda {\bf e}^+.$$ In general, worrying about convergence, we require that $b$ has a positive Novikov valuation. In our case, we only use a special weakly bounding cochain. **Definition 117**. The **canonical** weakly bounding cochain of the strictly unital $A_\infty$ algebra ${\mathcal F}_{\mathfrak b}^+({\bm L})$ is $$b_{{\bm L}} = W_{\mathfrak b} {\bf f}_{{\bm L}}.$$ We check that, by the fact that $m_k^+( {\bf f}_{{\bm L}}, \cdots, {\bf f}_{{\bm L}} ) = 0$ for $k \geq 2$ and $m_1^+ ( {\bf f}_{{\bm L}}) = {\bf e}_{{\bm L}}^+ - {\bf e}_{{\bm L}}$, one has $$\sum_{k \geq 0} m_k^+(b_{{\bm L}}, \cdots, b_{{\bm L}}) = m_0^+(1) + m_1^+( W_{\mathfrak b} {\bf f}_{{\bm L}}) = W_{\mathfrak b} {\bf e}_{{\bm L}} + W_{\mathfrak b} ({\bf e}_{{\bm L}}^+ - {\bf e}_{{\bm L}}) = W_{\mathfrak b} {\bf e}_{{\bm L}}^+.$$ Hence indeed $b_{{\bm L}}$ is a weakly bounding cochain. Now we can define the flat $A_\infty$ algebra ${\mathcal F}_{\mathfrak b}^\flat({\bm L})$ with compositions being (for $k \geq 1$) $$m_k^\flat (x_k, \ldots, x_1) = \sum_{l_0, \ldots, l_k \geq 0} m_{k+l_0 + \cdots + l_k}^+ \Big( \underbrace{ b_{{\bm L}}, \ldots, b_{{\bm L}}}_{l_k}, x_k, \cdots, x_1, \underbrace{ b_{{\bm L}}, \ldots, b_{{\bm L}}}_{l_0} \Big).$$ In particular, $m_1^\flat \circ m_1^\flat = 0$ and the cohomology of ${\mathcal F}_{\mathfrak b}^\flat({\bm L})$ agrees with the quasimap Floer cohomology ${\it QHF}_{\mathfrak b}^\bullet({\bm L}; \Lambda_{\overline{\mathbb Q}})$. ### Multiplicative structure We need to identify the multiplicative structures on the quasimap Floer cohomology. The second composition $m_2^\flat$ on ${\mathcal F}_{\mathfrak b}^\flat ({\bm L})$ induces a multiplication on ${\it QHF}_{\mathfrak b}^\bullet( {\bm L}; \Lambda_{\overline{\mathbb Q}})$. **Proposition 118**. *When ${\bf y}$ is a critical point of $W_{\mathfrak b}({\bf u})$ and the Hessian of $W_{\mathfrak b}({\bf u})$ is nondegenerate at ${\bf y}$, i.e. $$\det \left( \frac{\partial^2 W_{\mathfrak b}({\bf u})}{\partial x_i \partial x_j} ({\bf y}) \right) \neq 0,$$ the quasimap Floer cohomology algebra ${\it QHF}_{\mathfrak b}^\bullet({\bm L}; \Lambda_{\overline{\mathbb Q}})$ is isomorphic to a Clifford algebra over $\Lambda_{\overline{\mathbb Q}}$ associated to a nondegenerate quadratic form on an $n$-dimensional space.* Note that the above nondegeneracy condition coincides with the one from Definition [Definition 109](#defn:log-derivative){reference-type="ref" reference="defn:log-derivative"} because we are considering Laurent polynomials. The computation of the ring structure is carried out in a similar situation in [@Venugopalan_Woodward_Xu]. Here we only sketch it. The key of the computation is to establish another divisor equation $$\label{divisor_equation_2} m_2^\flat (x_i,x_j) + m_2^\flat (x_j, x_i) = \frac{\partial^2 W_{\mathfrak b}}{\partial x_i \partial x_j} {\bf e}.$$ on cohomology. When the corresponding critical point of $W_{\mathfrak b}$ is nondegenerate, it follows that the Floer cohomology is isomorphic to a Clifford algebra induced from the Hessian of the critical point. *Remark 119*. We explain why the divisor equation [\[divisor_equation_2\]](#divisor_equation_2){reference-type="eqref" reference="divisor_equation_2"} fails on the chain level if one uses the naive way of perturbation. Consider $X = {\mathbb P}^1$. Fix a torus action. The (undeformed) potential function is $$W = T^u y + T^{1-u} \frac{1}{y}.$$ The two terms come from the contribution of two disks, one through the north pole and the other through the south pole. If the divisor equation [\[divisor_equation_2\]](#divisor_equation_2){reference-type="eqref" reference="divisor_equation_2"} holds, then there should be two configurations with two inputs labelled by the index $1$ critical point, however, once the perturbation is chosen, one can only see one configurations exist in the moduli space. This is because the perturbation is not symmetric with respect to flipping the two incoming semi-infinite edges. *Proof of Proposition [Proposition 118](#prop_ring_structure){reference-type="ref" reference="prop_ring_structure"}.* Once the divisor equation [\[divisor_equation_2\]](#divisor_equation_2){reference-type="eqref" reference="divisor_equation_2"} is established, the calculation of the ring structure follows immediately. Hence we only explain how to achieve the divisor equation following the same idea as [@Venugopalan_Woodward_Xu]. Notice that the $A_\infty$ structure is independent of the perturbation up to homotopy equivalence. Hence the ring structure on the Floer cohomology is independent of the perturbation. Now we broaden the class of perturbations by considering multi-valued ones in order to achieve some symmetry, and use such perturbations to establish Equation [\[divisor_equation_2\]](#divisor_equation_2){reference-type="eqref" reference="divisor_equation_2"} on the chain level. A multi-valued perturbation is just a (finite) multi-set of perturbations on each tree. We consider a coherent family of multi-valued perturbations which still satisfy Definition [Definition 100](#defn_coherent_perturbation){reference-type="ref" reference="defn_coherent_perturbation"}. We say that a multi-valued perturbation is symmetric, if, when restricted to the tree $\Gamma_0$ with two inputs, one output, and one finite vertex, the perturbation $P_{\Gamma_0, \vec{n}}$ (where $\vec{n}$ on the only finite vertex is $1$, corresponding to Maslov index two disks) is invariant under the ${\mathbb Z}_2$-action on the universal tree $\overline{\mathcal{UT}}_{\Gamma_0}$ induced by switching the two incoming semi-infinite edges. One can follow the same inductive argument to construct a symmetric coherent system of multi-valued perturbations and achieve transversality. Now when defining the counts, we need to count for each member of the multi-valued perturbation and then take an average. This still defines an $A_\infty$ algebra and it is homotopy equivalent to any one defined using single-valued perturbations, provided that we work over the rationals. Moreover, for any two critical points $x_i, x_j$ of Morse index $n-1$, the divisor equation [\[divisor_equation_2\]](#divisor_equation_2){reference-type="eqref" reference="divisor_equation_2"} holds. For details, see [@Venugopalan_Woodward_Xu Lemma 5.12]. ◻ ### Hochschild cohomology Now consider the Hochschild cohomology of the $A_\infty$ algebra ${\mathcal F}_{\mathfrak b}^\flat ({\bm L})$. **Proposition 120**. *When ${\bm L}$ corresponds to a nondegenerate critical point of $W_{\mathfrak b}$, one has $${\it HH}^\bullet( {\mathcal F}_{\mathfrak b}^\flat({\bm L})) \cong \Lambda_{\overline{\mathbb Q}}$$ where the Hochschild cohomology is generated by the identity ${\bm 1}_{{\mathcal F}^\flat_{\mathfrak b}({\bm L})}$.* *Proof.* We know that the cohomology of ${\mathcal F}_{\mathfrak b}^\flat({\bm L})$ is isomorphic to a Clifford algebra over $\Lambda_{\overline{\mathbb Q}}$. This proposition follows from Proposition [Proposition 44](#HH_computation_1){reference-type="ref" reference="HH_computation_1"}. ◻ *Remark 121*. When the bulk-deformation ${\mathfrak b}$ is convenient, we can formally define the quasimap Fukaya category as the disjoint union of the $A_\infty$ algebras ${\mathcal F}_{\mathfrak b}^\flat({\bm L})$ for ${\bm L}$ corresponding to all critical points of $W_{\mathfrak b}$ inside the moment polytope. However, what we need is only the direct sum of these Hochschild cohomology. # Open string theory II. Closed-open maps {#section9} In this section, we prove Theorem [Theorem 2](#thm:main-2){reference-type="ref" reference="thm:main-2"}. It is the consequence of the following theorem. **Theorem 122**. *Let ${\mathfrak b}$ be a convenient bulk deformation ${\mathfrak b}$.* 1. *There is an isomorphism of $\Lambda_{\overline{\mathbb Q}}$-algebras $${\rm CO}_{\mathfrak b}: {\it VHF}_\bullet^{\mathfrak b} (V; \Lambda_{\overline{\mathbb Q}}) \to \bigoplus_{{\bm L}\in {\rm Crit}_X W_{\mathfrak b}} {\it HH}^\bullet ({\mathcal F}_{\mathfrak b}^\flat ({\bm L})) \cong (\Lambda_{\overline{\mathbb Q}})^{{\rm Crit}_X W_{\mathfrak b}}.$$* 2. *The operator on ${\it VHF}_\bullet^{\mathfrak b}( V; \Lambda_{{\mathbb Z}[{\bf i}]} )$ defined by the pair-of-pants product with the (bulk-deformed) first Chern class (see Definition [Definition 135](#defn_first_Chern_class){reference-type="ref" reference="defn_first_Chern_class"}) has distinct eigenvalues in $\Lambda_{\overline{\mathbb Q}}$.* *Remark 123*. A closed-open map on the level of Floer cohomology, also in the setting of vortex Floer theory, was constructed in [@Wu_Xu]. The method of using quilted objects to prove the multiplicative property was learned from Woodward, see [@Venugopalan_Woodward_Xu]. ## Moduli spaces for the closed-open map ### Based trees and closed-open domain types Recall our conventions about trees and ribbon trees given in the last section. To model curves with spherical components or Floer cylinders, we consider a broader class of trees called *based trees*. A **based tree** is a pair $(\Gamma, \underline{\smash{\Gamma}})$ where $\underline{\smash{\Gamma}}$ is a subtree with a ribbon structure containing the root $v_{\rm root}$ and adjacent semi-infinite edge. In a based tree, vertices in $V_{\underline{\smash{\Gamma}}}$ are called a *boundary vertex*, and other vertices are called *interior vertices*. Similarly, an edge is either an interior edge or a boundary edge. A **metric based tree** is a based tree $\Gamma$ together with a metric on its base $\underline{\smash{\Gamma}}$. Now specify domains responsible for the definition of the closed-open map on the chain level. Consider based trees with a distinguished interior vertex at infinity $v_{\rm Ham}^\infty \in V_\Gamma^\infty\setminus V_{\underline{\smash{\Gamma}}}$. For each such tree $\Gamma$, let $v_{\rm Ham}\in V_{\underline{\smash{\Gamma}}}^{\rm finite}$ be the distinguished vertex in the base $\underline{\smash{\Gamma}}$ which is closest to $v_{\rm Ham}^\infty$. We also assume that such trees always have exactly one boundary output $v_{\rm out}$. We call such a tree $\Gamma$ (with a metric type on the base $\underline{\smash{\Gamma}}$) a **closed-open domain type**. We say $\Gamma$ is minimal if its base is minimal (see Subsection [8.1](#subsec:trees){reference-type="ref" reference="subsec:trees"}), i.e., the base has no finite edges of length zero or infinite edges. For a minimal $\Gamma$, the base $\underline{\smash{\Gamma}}$ has a moduli space $\mathcal{MT}_{\underline{\smash{\Gamma}}}^{\rm CO}$ and universal tree $\mathcal{UT}_{\underline{\smash{\Gamma}}}^{\rm CO}$. One also has a compactification (see Section [8.1](#subsec:trees){reference-type="ref" reference="subsec:trees"}) denoted by $\overline{\mathcal{MT}}{}_{\underline{\smash{\Gamma}}}^{\rm CO}$. Given a closed-open domain $\Gamma$, a closed-open domain of type $\Gamma$ is a treed disk $C = S \cup T$ which has a distinguished "parametrized component" $C_{\rm Ham}$ corresponding to the vertex $v_{\rm Ham}$ which has a nonempty boundary. See Figure [3](#figure:CO_domain){reference-type="ref" reference="figure:CO_domain"} for an illustration of a closed-open domain. {#figure:CO_domain} We define a type of "mixed equation" on closed-open domains. Fix an admissible bulk-avoiding pair $(\widehat H, \widehat J)$ for which the bulk-deformed vortex Floer chain complex ${\it VCF}_\bullet^{\mathfrak b} (\widehat H, \widehat J; \Lambda_{\overline{\mathbb Q}})$ is defined. Let $C= S \cup T$ be a closed-open domain with distinguished component $C_{\rm Ham}$. Because there is at least one boundary output, $C_{\rm Ham}$ together with the interior puncture and boundary nodes is stable. Hence can always identify $C_{\rm Ham}\cong {\mathbb D} \setminus \{0\} \cong (-\infty, 0]\times S^1$ and equip it with the cylindrical metric. Using a cut-off function supported in $(-\infty, -1]$, one can homotope the pair $(\widehat H, \widehat J)$ with the pair $(0, J_V)$ where $J_V$ is the standard complex structure on the vector space $V \cong {\mathbb C}^N$, giving rise to a domain-dependent pair $(\widehat H_z, \widehat J_z)$ for $z \in C_{\rm Ham}$. Given the above data, we consider tuples $$\Big( (u_v)_{v\in V_\Gamma}, (x_e)_{e\in E_\Gamma} \Big)$$ where 1. For each vertex $v$ belong to the path connecting $v_{\rm Ham}^\infty$ and $v_{\rm Ham}$ (not included), $u_v = [u_v, \xi_v, \eta_v]$ is a gauge equivalence class of solutions to the vortex equation $$\begin{aligned} &\ \partial_s u_v + {\mathcal X}_{\xi_v} + \widehat J_t( \partial_t u_v + {\mathcal X}_{\eta_v} - X_{\widehat H_t}(u_v)) = 0,\ &\ \partial_s \eta_v - \partial_t \xi_v + \mu(u_v) = 0. \end{aligned}$$ 2. For $v = v_{\rm Ham}$, $u_v = [u_v, \xi_v, \eta_v]$ is a gauge equivalence class of solutions to $$\begin{aligned} \label{CO_equation} &\ \partial_s u_v + {\mathcal X}_{\xi_v} + \widehat J_z ( \partial_t u_v + {\mathcal X}_{\eta_v} - X_{\widehat H_z}(u_v)) = 0,\ &\ \partial_s \eta_v - \partial_t \xi_v + \mu(u_v) = 0. \end{aligned}$$ Moreover, $u_v$ satisfies the Lagrangian boundary condition $$\label{CO_boundary_condition} u_v (\partial C_{\rm Ham}) \subset \widehat L.$$ 3. For all other $v$, $u_v$ is a $K$-orbit of quasidisk with boundary in $\widehat L$. 4. For each edge $e \in E_\Gamma$, $x_e$ is a (perturbed) negative gradient line/ray/segment of the Morse function $f_L: L \to {\mathbb R}$. 5. These objects must have finite energy and must satisfy the obvious matching condition at interior and boundary nodes. The finite energy condition forces the component $u_v$ whose domain $C_v$ has the distinguished input $v_{\rm Ham}^\infty$ to converge to an equivariant 1-periodic orbit of $\widehat H$. Given a closed-open domain type $\Gamma$, a **closed-open map type** over $\Gamma$, denoted by ${\bm \Gamma}$, consists of topological types of objects for each component. A closed-open map type is called **essential** if there is no interior node and all finite boundary edges have positive length and there is no breaking. ### Transversality Given a closed-open domain type $\Gamma$, a domain-dependent perturbation consists of a domain-dependent smooth function $f_\Gamma$ depending on positions on the universal tree $\overline{\mathcal{UT}}_\Gamma$ and a domain-dependent almost complex structure $\widehat J^{\rm CO}$ depending only on positions on the component $C_{\rm Ham}\cong (-\infty, 0]\times S^1$. In other words, we keep using the standard complex structure over disk components without interior marked point. As before, the perturbation function $f_\Gamma$ also depends on a function $\vec{n}: V_{\underline{\smash{\Gamma}}}^{\rm finite} \setminus \{ v_{\rm Ham}\} \to {\mathbb Z}_{\geq 0}$. To achieve transversality, one can first fix $\widehat J^{\rm CO}$ which is equal to the given $\widehat J_t$ near $-\infty$. Next we need to extend the perturbation we have chosen to define the (bulk-deformed) quasimap $A_\infty$ algebra of $L$. Notice that for any closed-open domain type $\Gamma$, the base $\underline{\smash{\Gamma}}$ has a distinguished finite vertex $v_{{\rm Ham}}$. The tree $\Gamma$ degenerates to another tree $\Pi$ which has an unbroken component $\Pi'$ that does not contain the distinguished vertex. For such unbroken components $\Pi'$, the domain-dependent perturbation has been chosen as before to define the $A_\infty$ structure. Hence we look for a system of domain-dependent perturbations $$P_{\Gamma, \vec{n}}^{\rm CO}: \overline{\mathcal{UT}}_{\underline{\smash{\Gamma}}} \to C^\infty(L)$$ which respect similar conditions as Definition [Definition 100](#defn_coherent_perturbation){reference-type="ref" reference="defn_coherent_perturbation"}. We omit the complete definition here. Moreover, we require that, once $\Gamma$ has an unbroken component $\Gamma'$ which does not contain $v_{\rm Ham}$, the perturbation on this component agrees with the existing one chosen before. Now we consider relevant moduli spaces. Given a closed-open map type ${\bm \Gamma}$. Let $\vec{n}: V_\Gamma^{\rm finite}\setminus \{v_{\rm Ham}\} \to {\mathbb Z}_{\geq 0}$ be the function whose value on $v$ is half of the Maslov index of the disk class $\beta_v$ contained in the data ${\bm\Gamma}$. The moduli space ${\mathcal M}_{\bm \Gamma}^{\rm CO}$ is the space of solutions to the mixed equation described above for the complex structure $\widehat J_z$ in [\[CO_equation\]](#CO_equation){reference-type="eqref" reference="CO_equation"}, and the negative gradient flow equation with the Morse function $f_L$ perturbed by $P_{\Gamma, \vec{n}}^{\rm CO}$. Then as before, one can find a coherent family of perturbations making all such moduli spaces regular. We omit the details. Furthermore, one can incorporate the perturbations used for defining the homotopy units. For this we allow that the inputs of an closed-open domain type to be weighted or unweighted and require similar properties of perturbations on domains with weighted inputs as in Subsection [8.4](#subsection_homotopy_unit){reference-type="ref" reference="subsection_homotopy_unit"} (the almost complex structure $\widehat J^{\rm CO}$ is independent of the weighting parameters $\rho$). ## The closed-open map Having regularized all relevant moduli spaces, we define the relevant counts for the closed-open maps. A closed-open map type ${\bm \Gamma}$ is called **essential** if it is stable, has no breakings and no sphere bubbles, no boundary edges of length zero. Given a $k+1$-tuple of generators ${\bf x} = (x_1, \ldots, x_k; x_\infty)$[^9], an equivariant 1-periodic orbit ${\mathfrak x}$ of the bulk-avoiding Hamiltonian $\widehat H$ and a disk class $\beta$, denote by $${\mathcal M}_\beta^{\rm CO}({\mathfrak x}, {\bf x})_i,\ i = 0, 1$$ the union of moduli spaces ${\mathcal M}_{\bm \Gamma}^{\rm CO}$ of essential closed-open map types ${\bm \Gamma}$ whose boundary inputs/output are labelled by ${\bf x}$, whose (only) interior input $v_{\rm Ham}^\infty$ is labeled by ${\mathfrak x}$, and whose total disk class is $\beta$, and whose virtual dimension is $i$. Given $E \geq 0$, let $${\mathcal M}_\beta^{\rm CO}({\mathfrak x}, {\bf x})_i^{\leq E}\subset {\mathcal M}_\beta^{\rm CO}({\mathfrak x}, {\bf x})_i$$ be the subset of configurations whose (analytic) energy is at most $E$. It is standard to prove the following theorem. **Theorem 124**. 1. *${\mathcal M}{}_\beta^{\rm CO}({\mathfrak x}, {\bf x})_i$ is an oriented topological manifold of dimension $i$.* 2. *For all $E\geq 0$, ${\mathcal M}{}_\beta^{\rm CO}({\mathfrak x}, {\bf x})_0^{\leq E}$ is a finite set.* 3. *For all $E \geq 0$, ${\mathcal M}{}_\beta^{\rm CO}({\mathfrak x}, {\bf x})_1^{\leq E}$ is compact up to at most one 1) interior breaking, 2) boundary breaking, 3) bubbling of holomorphic disks, or 4) the length of a finite boundary edge shrinks to zero.* 4. *By the standard gluing construction and identifying fake boundary strata, one can compactify the $1$-dimensional moduli space to $\overline{\mathcal M}{}_\beta^{\rm CO}({\mathfrak x}, {\bf x})_1$ which is an oriented topological 1-manifold with boundary whose cut-off at any energy level $E$ is compact.* Now given a local system ${\bf y}$, denote the brane with this local system by ${\bm L}= (L, {\bf y})$. We define a count $$n_{{\bm L}, {\mathfrak b}}^{\rm CO} (\beta, {\mathfrak x}, {\bf x}) = \sum_{[{\mathfrak u}] \in {\mathcal M}_\beta^{\rm CO}({\mathfrak x}, {\bf x})_0} \exp \left( \sum_{j=1}^N \log c_j\ [{\mathfrak u}] \cap V_j \right) T^{E(\beta)} {\bf y}^{\partial \beta} \epsilon([{\mathfrak u}]) \in \Lambda_{\overline{\mathbb Q}}$$ where ${\mathfrak b} = \sum_{j=1}^N \log c_j V_j$. By Gromov compactness one has the following result. **Lemma 125**. *$n_{{\bm L}, {\mathfrak b}}^{\rm CO}(\beta, {\mathfrak x}, {\bf x})$ converges in $\Lambda_{\overline{\mathbb Q}}$.* Then define a sequence of linear map $$\widetilde {\rm CO}{}_{{\bm L}, {\mathfrak b}}^n: {\it VCF}_\bullet^{\mathfrak b}(V; \Lambda_{\overline{\mathbb Q}} ) \to {\rm Hom}_{\Lambda_{\overline{\mathbb Q}}} \left( {\mathcal F}_{\mathfrak b}^+({\bm L})^{\otimes n}, {\mathcal F}_{\mathfrak b}^+({\bm L}) \right), n = 0, 1, \ldots$$ by $$\widetilde {\rm CO}{}_{{\bm L},{\mathfrak b}}^n ({\mathfrak x}) (x_n, \ldots, x_1) = \sum_{x_\infty} n_{{\bm L},{\mathfrak b}}^{\rm CO} ({\mathfrak x}, {\bf x}) x_\infty$$ and linear extension. We use the canonical weakly bounding cochain $b_{\bm L}$ to turn it into a chain map. Define $${\rm CO}_{{\bm L}, {\mathfrak b}}^n: {\it VCF}_\bullet^{\mathfrak b}(V; \Lambda_{\overline{\mathbb Q}} ) \to {\rm Hom}_{\Lambda_{\overline{\mathbb Q}}} \left( {\mathcal F}_{\mathfrak b}^+({\bm L})^{\otimes n}, {\mathcal F}_{\mathfrak b}^+({\bm L}) \right), n = 0, 1, \ldots$$ by $${\rm CO}_{{\bm L},{\mathfrak b}}^n ({\mathfrak x}) (x_n, \ldots, x_1) = \sum_{l_n, \ldots, l_0} \widetilde{\rm CO}{}_{{\bm L}, {\mathfrak b}}^{n+l_0+ \cdots + l_n} \left( \underbrace{b_{\bm L}, \ldots, b_{\bm L}}_{l_n}, x_n, \cdots, x_1, \underbrace{b_{\bm L}, \ldots, b_{\bm L}}_{l_0} \right).$$ The whole sequence $\{{\rm CO}_{{\bm L}, {\mathfrak b}}^n\}_{n= 0, \ldots}$ is then a linear map $${\rm CO}_{L, {\mathfrak b}}: {\it VCF}_\bullet^{\mathfrak b}(V; \Lambda_{\overline{\mathbb Q}}) \to CC^\bullet ( {\mathcal F}_{\mathfrak b}^+({\bm L})).$$ **Proposition 126**. *${\rm CO}_{{\bm L}, {\mathfrak b}}$ is a chain map.* *Proof.* We analyze the boundary of 1-dimensional moduli spaces ${\mathcal M}{}_\beta^{\rm CO}({\mathfrak x}, x)_1$. Given any map type ${\bm \Gamma}$ contributing to this moduli space, the true boundaries of ${\mathcal M}_{\bm \Gamma}^{\rm CO}$ consists of configurations where either there is exactly one interior breaking (at an equivariant 1-periodic orbit) or exactly one boundary breaking (see Figure [4](#Figure_CO_chain){reference-type="ref" reference="Figure_CO_chain"}). {#Figure_CO_chain} The configurations with interior breakings contribute to the composition ${\rm CO}_{{\bm L}, {\mathfrak b}} \circ \delta_{{\it VCF}}$ (the upper left in Figure [4](#Figure_CO_chain){reference-type="ref" reference="Figure_CO_chain"}). On the other hand, there are three types of configurations with boundary breakings, described as follows. 1. The first (corresponding to the upper right in Figure [4](#Figure_CO_chain){reference-type="ref" reference="Figure_CO_chain"}) is where the breaking separates off a treed disk with no interior puncture or boundary insertions except for an arbitrary number of the weakly bounding cochain $b$. As we have $$\sum_{k \geq 0} m_k^+(b, \ldots, b) = W_{\mathfrak b} {\bf e}^+.$$ Such configuration contributes by a multiple of the counting of a closed-open moduli space with a boundary insertion ${\bf e}^+$, which vanishes by the forgetful property of the perturbation. 2. The second (corresponding to the lower left in Figure [4](#Figure_CO_chain){reference-type="ref" reference="Figure_CO_chain"}) is where the interior puncture and the output are separated by the breaking. This kind of broken configuration contributes to the Gersternhaber product $m^\flat \circ {\rm CO}_{{\bm L}, {\mathfrak b}}(-)$ (up to a sign). 3. The third (corresponding to the lower right in Figure [4](#Figure_CO_chain){reference-type="ref" reference="Figure_CO_chain"}) is where the interior puncture and the output are not separated by the breaking. This kind of broken configuration contributes to the Gernstenhaber product ${\rm CO}_{{\bm L}, {\mathfrak b}}(-) \circ m^\flat$ (up to a sign). Therefore, up to sign verifications which we skip here, ${\rm CO}_{{\bm L}, {\mathfrak b}}$ is a chain map. ◻ Standard TQFT type argument shows that up to chain homotopy the closed-open map is well-defined, i.e., independent of the pair $(\widehat H, \widehat J)$ defining the vortex Floer chain complex and independent of the choice of all relevant perturbations. There is another map on the cohomology level which we also need. Namely, if we do not use any boundary inputs, by counting treed vortices over closed-open domains one can obtain a linear map $$\label{CO2} {\rm CO}_{{\bm L}, {\mathfrak b}}^0: {\it VHF}_\bullet^{\mathfrak b} (V; \Lambda_{\overline{\mathbb Q}}) \to {\it QHF}^\bullet_{\mathfrak b} ({\bm L}; \Lambda_{\overline{\mathbb Q}}).$$ It was firstly defined in [@Wu_Xu] in a slightly different way. Here we can easily generalize to the bulk-deformed case. Moreover, this map sends the identity ${\bm 1}_{\mathfrak b}^{\rm GLSM}$ to the identity in the Lagrangian Floer cohomology. Summing over all Floer-nontrivial Lagrangian branes, we define the **closed-open map** $${\rm CO}_{\mathfrak b}:= \bigoplus_{{\bm L}\in {\rm Crit}_X W_{\mathfrak b}} {\rm CO}_{{\bm L}, {\mathfrak b}} : {\it VHF}_\bullet^{\mathfrak b}(V;\Lambda_{\overline{\mathbb Q}}) \to \bigoplus_{{\bm L}\in {\rm Crit}_X W_{\mathfrak b}} {\it HH}^\bullet( {\mathcal F}_{\mathfrak b}^+ ({\bm L})).$$ ## The closed-open map is multiplicative Now we establish the following important property of the closed-open map. **Theorem 127**. *The map ${\rm CO}_{{\bm L}, {\mathfrak b}}$ is multiplicative and maps the unit to the unit.* We use an analogue of "quilted" moduli spaces to prove the multiplicativity, in the same way as in [@Venugopalan_Woodward_Xu Section 3.6]. **Definition 128** (Balanced marked disks and balanced treed disks). 1. A stable marked disk $S \cong {\mathbb D}$ with two interior markings $z', z''\in {\rm Int} S$ and $k+1$ boundary markings $\underline{\smash{z}} = (z_0, \ldots, z_k)$ is called **balanced** if $z', z'', z_0$ lies on a circle in ${\mathbb D}$ tangent to $\partial {\mathbb D}$ at $z_0$. 2. A treed disk with two interior leaves $z', z''$, $k$ boundary inputs and one boundary output is called **balanced** if the following conditions are satisfied. 1. $z', z''$ are contained in the same spherical component. 2. If $z', z''$ are contained in the same disk component $S_v$. Let $z_0'\in \partial S_v$ be the boundary node connecting $S_v$ to the output. Then $(S_v, z', z'', z_0')$ is a balanced marked disk. 3. If $z', z''$ are contained in two different disk components, $S_{v'}$ and $S_{v''}$ respectively. Let $e_1, \ldots, e_l$ be the unique path connecting $v'$ and $v''$ in the tree, then $$\sum_{i=1}^l \pm {\bm l} (e_i) = 0$$ where the sign is positive resp. negative if the edge $e_i$ is oriented toward resp. against the output. We call the unique path $e_1, \ldots, e_l$ the **bridge**. Consider any stable domain type $\Gamma$ with two interior inputs, $k$ boundary inputs and one boundary output. Consider the moduli space ${\mathcal M}_\Gamma^{\rm balanced}$ of balanced treed disks of type $\Gamma$. The list of codimension one boundary strata is different from the unbalanced case, as the balanced condition cuts down the dimension by 1. See Figure [5](#figure:CO_balanced){reference-type="ref" reference="figure:CO_balanced"}. ![The moduli space of balanced treed disks with two interior inputs and one boundary inputs. This moduli space is parametrized by one variable $\rho \in [-1, 1]$.](CO_balanced.pdf){#figure:CO_balanced} Notice that a real boundary ${\mathcal M}_\Pi^{\rm balanced} \subset \partial {\mathcal M}_\Gamma^{\rm balanced}$ could be the product of several other moduli spaces whose types may have either one interior input or zero interior inputs. We have chosen surface metrics with cylindrical ends for stable closed-open domains (with one interior inputs); hence we can extend the choices to a family of surface metrics with cylindrical ends for the moduli space of stable closed-open domains with two interior inputs. We omit the details. Now we can consider the following mixed equation for domains with two interior cylindrical ends. Choose two bulk-avoiding admissible pairs $(\widehat H_t', \widehat J_t')$ and $(\widehat H_t'', \widehat J_t'')$. Turn on the Hamiltonian perturbation on cylindrical ends. Consider the mixed equation similar to that for the closed-open map. We can extend the existing perturbation to this new type of moduli spaces to achieve transversality. *Proof of Theorem [Theorem 127](#thm_CO_multiplicative){reference-type="ref" reference="thm_CO_multiplicative"}.* Choose two Floer cycles ${\mathfrak x}_1$ and ${\mathfrak x}_2$. We only need to show that $$\label{CO_multiplicative_chain} {\rm CO}_{\mathfrak b}( {\mathfrak x}_1 \ast_{\mathfrak b} {\mathfrak x}_2) - {\rm CO}_{\mathfrak b}({\mathfrak x}_1) \star {\rm CO}_{\mathfrak b}({\mathfrak x}_2) \in {\rm Im} \delta_{{\it CC}}.$$ As one can choose perfect Morse functions on toric manifolds, we can assume that ${\mathfrak x}_1$ and ${\mathfrak x}_2$ are two single equivariant 1-periodic orbits. Consider 1-dimensional moduli spaces of treed disks with two cylindrical ends labelled by ${\mathfrak x}_1$ and ${\mathfrak x}_2$ and arbitrary boundary output $x_\infty$ and inputs $$\underbrace{b_{\bm L}, \ldots, b_{\bm L}}_{j_k}, x_k, \cdots, x_1, \underbrace{b_{\bm L}, \ldots, b_{\bm L}}_{j_0}.$$ We call $x_k, \ldots, x_1$ **regular inputs**. Consider the true boundaries of such moduli spaces. *a priori* There are five types of them, listed as below. We count their contributions (weighted by the bulk deformation), whose sum should be zero. 1. Breaking of Floer cylinders at one interior input. As ${\mathfrak x}_1$ and ${\mathfrak x}_2$ are cycles, the contribution of this type of boundary points is zero. 2. Two cylindrical ends merge together to form a pair of pants. The contribution of this type of boundary is $${\rm CO}_{\mathfrak b}( {\mathfrak x}_1 \ast_{\mathfrak b} {\mathfrak x}_2).$$ 3. One boundary edge not belonging to the bridge breaks and the piece broken off is not a disk without regular input. The contribution of this type of boundary is a Hochschild coboundary. 4. One boundary edge not belonging to the bridge breaks and the piece broken off is a disk without regular input. The broken off piece sums to a multiple of the strict unit $e_{\bm L}^+$. By the property of the perturbation data, the contribution of this type of boundary is zero. 5. A pair of boundary edges belonging to the bridge break. The contribution of this type of boundary is the Yoneda product $${\rm CO}_{\mathfrak b}({\mathfrak x}_1) \star {\rm CO}_{\mathfrak b}({\mathfrak x}_2).$$ Therefore, one obtains [\[CO_multiplicative_chain\]](#CO_multiplicative_chain){reference-type="eqref" reference="CO_multiplicative_chain"}. Now we prove the unitality. By the choice of the small bulk deformation, the Hochschild cohomology of the quasimap Fukaya category is semisimple and splits as the direct sum of 1-dimensional pieces. Moreover, each piece is the Hochschild cohomology of the $A_\infty$ algebra ${\mathcal F}_{\mathfrak b}^+({\bm L})$, which is linearly spanned by the identity. Hence we only need to prove that the linear map [\[CO2\]](#CO2){reference-type="eqref" reference="CO2"} sends the identity ${\bm 1}_{\mathfrak b}^{\rm GLSM}\in {\it VCF}_\bullet^{\mathfrak b}(V; \Lambda_{\overline{\mathbb Q}} )$ to the identity element of ${\it QHF}_{\mathfrak b}({\bm L})$. This verification can be found in [@Wu_Xu Theorem 6.11] (this verification does not need to consider the homotopy unit and weakly bounding cochains). ◻ ## The Kodaira--Spencer map To prove the first item of Theorem [Theorem 122](#thm_CO){reference-type="ref" reference="thm_CO"}, it remains to show that the closed-open map is a linear isomorphism. Proposition [Proposition 112](#prop_same_rank){reference-type="ref" reference="prop_same_rank"} shows that the domain and the codomain of ${\rm CO}_{\mathfrak b}$ have the same rank $${\rm dim}_{\Lambda_{\overline{\mathbb Q}}} {\it VHF}_\bullet^{\mathfrak b}(V; \Lambda_{\overline{\mathbb Q}}) = {\rm dim} H^\bullet(X) = \# {\rm Crit}_X W_{\mathfrak b}.$$ Hence we only need to show that ${\rm CO}_{\mathfrak b}$ is either injective or surjective. Following [@FOOO_mirror], we define another closed-open type map which we call the *Kodaira--Spencer map* at ${\mathfrak b}$, denoted by $$\mathfrak{ks}_{\mathfrak b}: \Lambda_{\overline{\mathbb Q}}[{\bf z}_1, \ldots, {\bf z}_N] \to (\Lambda_{\overline{\mathbb Q}})^{{\rm Crit}_X W_{\mathfrak b}}.$$ It is formally the derivative of the bulk-deformed potential function taken at the bulk ${\mathfrak b}$ evaluated at critical points of the Morse function. We only need to use the standard complex structure to define this map. ### Moduli spaces of quasidisks with tangency conditions We go toward the definition of the Kodaira--Spencer map. Fix a Lagrangian $L = L({\bf u})$ for a moment. Let $I = (\alpha_1, \ldots, \alpha_N)$ be a multiindex of nonnegative integers, which defines a monomial $${\bf z}^I = {\bf z}_1^{\alpha_1} \cdots {\bf z}_N^{\alpha_N}.$$ Consider a holomorphic disk $u: ({\mathbb D}, \partial {\mathbb D}) \to (V, \widehat L)$, which can be classified by Theorem [Theorem 97](#thm_Blaschke){reference-type="ref" reference="thm_Blaschke"}. We write $u = (u_1, \ldots, u_N)$ in coordinates. We say that $u$ satisfies the $I$-tangency condition at $z\in {\rm Int}{\mathbb D}$ if $u_i$ vanishes to the order of $\alpha_i$ at $z$, for all $i = 1, \ldots, N$. In particular, when $\alpha_i = 0$, there is no restriction to $u_i$. Given a multiindex $I$ and a disk class $\beta$, denote the moduli space of quasidisks with boundary in $\widehat L$ (with one output) satisfying the $I$-tangency condition at the origin by $${\mathcal M}_{I,1}^{qd}(\beta).$$ Its virtual dimension is $${\rm dim}^{\rm vir} {\mathcal M}_{I,1}^{qd}(\beta) = n + m(\beta) - 2|I| -2.$$ *Remark 129*. We can put the above moduli space into an infinitely dimensional Banach space where we can specify the tangency conditions for arbitrary maps with sufficiently high regularity. For example, using the setup of Cieliebak--Mohnke [@Cieliebak_Mohnke Section 6]. Hence we can examine whether the moduli space of quasididks subject to tangency conditions is regular or not. **Proposition 130**. *Suppose $\beta = \sum_{j=1}^N d_j \beta_j$ with $d_j \in {\mathbb Z}$. Then ${\rm dim} {\mathcal M}_{I, 1}^{qd}(\beta) \neq \emptyset$ only if $d_j \geq \alpha_j$ for all $j$. Moreover, the moduli space ${\mathcal M}_{I, 1}^{qd}(\beta)$ is smooth and the evaluation map at the boundary marking is a submersion.* *Proof.* By Theorem [Theorem 97](#thm_Blaschke){reference-type="ref" reference="thm_Blaschke"}, the $j$-th coordinate of the map $u$ of the form [\[Blaschke_formula\]](#Blaschke_formula){reference-type="eqref" reference="Blaschke_formula"} needs to vanish at least to the order $\alpha_j$ at the origin. Hence $d_j \geq \alpha_j$. To prove the regularity of the moduli space ${\mathcal M}_{I, 1}^{qd}(\beta)$, one only needs to prove the regularity of the corresponding moduli space of holomorphic disks in $V$ with boundary in $\widehat L$ (before quotienting the $K$-action) as the $K$-action is free. Since the complex structure on $V\cong {\mathbb C}^N$ is the standard one, and the tangency condition is imposed on each coordinate independently, one only needs to prove the Fredholm regularity for the $N= 1$ case. In this case, we consider holomorphic disks in ${\mathbb C}$ with boundary contained in the unit circle, which also vanish to a given order $k$ at the origin. Choose $p>2$ and $m$ sufficiently large, so that one has the Sobolev embedding $W^{m, p} \hookrightarrow C^k$ in dimension two. Now fix the disk class $\beta$. Consider the Banach space $W(\beta)$ of maps from $({\mathbb D}, \partial {\mathbb D})$ to $({\mathbb C}, S^1)$ of regularity $W^{m+1, p}$. Let $W_0(\beta) \subset W(\beta)$ be the subspace of maps which vanish at $0$ to the order $k+1$. Let $E(\beta) \to W(\beta)$ be the Banach space bundle, whose fiber over $u$ is the space of $(0, 1)$-forms of regularity $W^{m, p}$, and let $E_0(\beta)\subset E(\beta)$ be the subbundle of those forms which vanish at $0$ to the order $k$. Suppose $u_0: {\mathbb D} \to {\mathbb C}$ is a holomorphic disk in $W_0(\beta)$. Then there is a commutative diagram (see [@Cieliebak_Mohnke Section 6]) $$\xymatrix{ T_{u_0} W(\beta) \ar[r]^{F} & E(\beta)|_{u_0} \\ T_{u_0} W_0(\beta) \ar[r]_{F_0} \ar[u] & E_0(\beta)|_{u_0} \ar[u] }$$ where $F$ resp. $F_0$ is the standard Cauchy--Riemann operator, restricted to corresponding Banach spaces. One needs to prove that $F_0$ is surjective. Notice that by Cho--Oh's theorem [@Cho_Oh Theorem 6.1], $F$ is surjective. Hence for each $\eta_0 \in E_0(\beta)|_{u_0}$, there exists $\xi \in T_{u_0}W(\beta)$ such that $F(\xi) = \eta_0$. One only needs to modify $\xi$ to some $\xi_0 \in T_{u_0} W_0(\beta)$ with $F(\xi) = F(\xi_0)$. Indeed, as $u_0$ vanishes up to order $k+1$ at the origin, the disk class $\beta$, which is only the degree of the map $u_0$, is at least $k+1$. Then by the Blaschke formula [\[Blaschke_formula\]](#Blaschke_formula){reference-type="eqref" reference="Blaschke_formula"}, one can easily deform $u_0$ by $k+1$-jet data. Such deformations are in the kernel of $F$. Hence we can obtain the desired $\xi_0$. This proves the Fredholm regularity of the moduli spaces. The fact that the evaluation map at the output is a submersion onto $L$ follows easily from the Blaschke formula. ◻ ### The derivative of the potential Now we can define the Kodaira--Spencer map. For each critical point ${\bm L}\in {\rm Crit}_X W_{\mathfrak b}$ (lying inside the moment polytope), we will define a linear map $$\widetilde{\mathfrak{ks}}_{{\bm L}, {\mathfrak b}}: \Lambda_{\overline{\mathbb Q}}[{\bf z}_1, \ldots, {\bf z}_N] \to {\it QCF}_\bullet^+ ({\bm L}; \Lambda_{\overline{\mathbb Q}})$$ using the counts of certain zero-dimensional moduli spaces. It will turn out that the value of this map is always a multiple of the unique maximum ${\bf e}_{\bm L}= x_{\rm max}\in {\rm Crit} f_L$, hence descends to a map $$\widetilde{\mathfrak{ks}}_{{\bm L}, {\mathfrak b}}: \Lambda_{\overline{\mathbb Q}}[{\bf z}_1, \ldots, {\bf z}_N] \to {\it QHF}_\bullet^{\mathfrak b}({\bm L}; \Lambda_{\overline{\mathbb Q}}).$$ We define the coefficients to be ${\mathfrak{ks}}_{{\bm L}, {\mathfrak b}}$, i.e., $$\widetilde{\mathfrak{ks}}_{{\bm L}, {\mathfrak b}}({\bf z}^I) = \mathfrak{ks}_{{\bm L}, {\mathfrak b}}( {\bf z}^I) [{\bf e}_{\bm L}].$$ We first fix a multiindex $I = (\alpha_1, \ldots,\alpha_N)$. Denote $$\beta^I = \alpha_1 \beta_1 + \cdots + \alpha_N \beta_N \in H_2(V, \widehat L).$$ For each disk class $\beta \in H_2(V, \widehat L)$ and each critical point $x \in {\rm Crit} f_{L({\bf u})}$ of the Morse function $f_{L({\bf u})}: L({\bf u}) \to {\mathbb R}$, consider the moduli space $${\mathcal M}_{I, 1}^{qd}(\beta; x)$$ where we require that the output converges to the critical point $x$. Proposition [Proposition 130](#prop_tangency_regularity){reference-type="ref" reference="prop_tangency_regularity"} implies that this moduli space is regular. Moreover, $${\mathcal M}_{I, 1}^{qd}(\beta; x) \neq \emptyset\ {\rm and}\ {\rm dim}{\mathcal M}_{I, 1}^{qd}(\beta; x) = 0 \Longrightarrow \beta = \beta^I\ {\rm and}\ x = x_{\rm max}.$$ Moreover, in this case, the moduli space has exactly one point because of the Blaschke formula. We count the unique element weighted by the bulk deformation and the local system. *Remark 131*. *A priori* we should consider treed holomorphic disks with one boundary output and one interior marking with certain tangency condition. It is similar to the case of proving that $m_0$ is a multiple of ${\bm e}_{\bm L}$ that one can prove for zero-dimensional moduli spaces, only those treed disks with one disk component contribute. The count of the above moduli spaces (with a single point) defines the Kodaira--Spencer map. More explicitly, define $$\widetilde{\mathfrak{ks}}_{{\bm L}, {\mathfrak b}}: \Lambda_{\overline{\mathbb Q}}[{\bf z}_1, \ldots, {\bf z}_N] \to {\it QHF}_{\mathfrak b}^\bullet( {\bm L}; \Lambda_{\overline{\mathbb Q}})$$ by $$\widetilde{\mathfrak{ks}}_{{\bm L}, {\mathfrak b}}({\bf z}^I)= {\mathfrak b}^I T^{E(\beta^I)} {\bf y}^{\partial \beta^I} [{\bf e}_{\bm L}] = \mathfrak{ks}_{{\bm L}, {\mathfrak b}}({\bf z}^I) [{\bf e}_{\bm L}].$$ Here for ${\mathfrak b} = \sum_{j=1}^N \log c_j V_j$, the notation ${\mathfrak b}^I$ denotes the quantity $$c_1^{\alpha_1} \cdots c_N^{\alpha_N},$$ which is the exponential of the intersection number between the above unique quasidisk in ${\mathcal M}_{I, 1}^{qd}(\beta^I; x_{\rm max})$ and the bulk ${\mathfrak b}$. The Kodaira--Spencer map takes a very simple form. Recall that we have written $$W_{\mathfrak b} = W_{{\mathfrak b}, 1} + \cdots + W_{{\mathfrak b}, N} = c_1 W_1 + \cdots + c_N W_N.$$ **Proposition 132**. *For each multiindex $I$, one has $$\label{ks_formula} \mathfrak{ks}_{\mathfrak b}( {\bf z}^I ) = W_{\mathfrak b}^I:= W_{{\mathfrak b}, 1}^{\alpha_1} \cdots W_{{\mathfrak b}, N}^{\alpha_N}.$$* *Proof.* The calculation is carried out in a straightforward way. The area of a disk in class $\beta^I$ is $$E(\beta^I) = \alpha_1 l_1({\bf u}) + \cdots + \alpha_N l_N({\bf u}).$$ The contribution of the local system is $${\bf y}^{\partial \beta^I} = \prod_{j=1}^N (y_1^{v_{j, 1}} \cdots y_n^{v_{j, n}})^{\alpha_j}.$$ Hence the formula [\[ks_formula\]](#ks_formula){reference-type="eqref" reference="ks_formula"} follows. ◻ Define the Kodaira--Spencer map by $$\mathfrak{ks}_{\mathfrak b}:= \bigoplus_{{\bm L}\in {\rm Crit}_X W_{\mathfrak b}} \mathfrak{ks}_{{\bm L}, {\mathfrak b}}.$$ **Theorem 133**. *The Kodaira--Spencer map $\mathfrak{ks}_{\mathfrak b}$ is surjective.* *Proof.* By [@FOOO_toric_2 Lemma 3.12], the monomials $W_1, \ldots, W_N$ generate (over the ring $\Lambda_{0, \overline{\mathbb Q}}$) the ring $$\Lambda_0^P {\langle\hspace{-0.2pt}\langle}y_1^\pm, \ldots, y_n^\pm {\rangle\hspace{-0.2pt}\rangle}$$ which is a ring of formal Laurent series satisfying a particular valuation condition determined by the moment polytope $P$. Let ${\bm \eta}_1, \ldots, {\bm \eta}_s$ be the critical points of $W_{\mathfrak b}$ inside the moment polytope. Using the notion of convergent Novikov field $\Lambda_{\overline{\mathbb Q}}^{\rm conv}$, we see that for $T = t$ being a sufficiently small nonzero complex number, ${\bm \eta}_1^t, \ldots, {\bm \eta}_s^t$ are distinct points in $({\mathbb C}^*)^n$. Then there exist $s$ complex Laurent polynomials $$F_1, \ldots, F_s \in {\mathbb C}[y_1, \ldots, y_n, y_1^{-1}, \ldots, y_n^{-1}]$$ such that the matrix $\left[ F_a({\bm \eta}_b^t) \right]_{1\leq a, b \leq s}$ is invertible. Regard $F_1, \ldots, F_s$ as Laurent polynomials with Novikov coefficients, we see the determinant of the matrix $$\det \left[ F_a({\bm \eta}_b) \right]_{1 \leq a, b \leq s} \neq 0 \in \Lambda_{\overline{\mathbb Q}}.$$ The above is still true if we replace $F_a$ by $T^A F_a$ for any $A\in {\mathbb R}$. On the other hand, for $A$ sufficiently large, $T^A F_a \in \Lambda_0^P {\langle\hspace{-0.2pt}\langle}y_1^\pm, \ldots, y_n^\pm {\rangle\hspace{-0.2pt}\rangle}$. This implies that, the restriction of $\mathfrak{ks}_{\mathfrak b}$ to the finite-dimensional subspace spanned by $T^A F_a$ is subjective due to the generation property of the monomials $W_1, \dots, W_N$. Hence $\mathfrak{ks}_{\mathfrak b}$ is also surjective. ◻ ## A quantum Kirwan map The set of small bulk-deformations is contained in the larger set of equivariant cohomology upstairs. Classically, there is the Kirwan map $$\kappa^{\rm classical}: H_K^\bullet(V) \to H^\bullet(X).$$ In principle, by incorporating vortices one can define a quantization of the Kirwan map. This has been pursued by Ziltener [@Ziltener_book] in the symplectic setting and worked out by Woodward [@Woodward_15] in the algebraic setting. Here we define a variant of the quantum Kirwan map, denoted by $$\label{qkirwan} \kappa_{\mathfrak b}: \Lambda_{{\mathbb Z}[{\bf i}]}[{\bf z}_1, \ldots, {\bf z}_N] \to {\it VHF}_\bullet^{\mathfrak b} (V; \Lambda_{{\mathbb Z}[{\bf i}]})$$ such that the image of the unit $1$ is the identity ${\bm 1}_{\mathfrak b}^{\rm GLSM}$. We define the above map by imposing tangency conditions at the origin of the cigar. Fix a regular bulk-avoiding admissible pair $(\widehat H_\infty, \widehat J_\infty)$ which defines a bulk-deformed vortex Floer complex ${\it VHF}_\bullet^{\mathfrak b}( \widehat H, \widehat J; \Lambda_{{\mathbb Z}[{\bf i}]})$. Consider a domain-dependent almost complex structure $\widehat J$ (resp. Hamiltonian perturbation $\widehat H$) parametrized by points on the cigar $\Sigma^{\rm cigar} \cong {\mathbb C}$ which is equal to the standard almost complex structure $\widehat J_V$ (resp. vanishes) in a specified neighborhood of $0\in \Sigma^{\rm cigar}$ and which agrees with $\widehat J_\infty$ (resp. $\widehat H_\infty$) near infinity. Consider the vortex equation with the data $(\widehat H, \widehat J)$ on the cigar. Any finite energy solution should converge to a critical point of ${\mathcal A}_{H_\infty}$. Moreover, as the almost complex structure is standard near $0$, one can impose the tangency condition corresponding to $I$ at the origin. Such a tangency condition is gauge invariant. Then for each critical point ${\mathfrak x} \in {\rm Crit} {\mathcal A}_{H_\infty}$, there is a moduli space $${\mathcal M}{}_I^{\rm cigar}({\mathfrak x}) \subset {\mathcal M}{}^{\rm cigar}({\mathfrak x}).$$ By using domain-dependent perturbations, one can achieve transversality for such a moduli space. Then one has $${\rm dim} {\mathcal M}_I^{\rm cigar}({\mathfrak x}) = {\rm dim} {\mathcal M}{}^{\rm cigar}({\mathfrak x}) - 2 |I|.$$ On the other hand, as the Hamiltonian is bulk-avoiding, each solution has well-defined topological intersection numbers with $V_j$. Then define $$\kappa_{\mathfrak b}({\bf z}^I) = \sum_{{\mathfrak x}\atop {\rm dim}{\mathcal M}_I^{\rm cigar}({\mathfrak x}) = 0 } \left( \sum_{[{\mathfrak u}] \in {\mathcal M}_I^{\rm cigar} ({\mathfrak x})} \left( \prod_{j=1}^N c_j^{[{\mathfrak u}] \cap V_i} \right) \epsilon ([{\mathfrak u}]) \right) {\mathfrak x}.$$ **Theorem 134** (Properties of the bulk-deformed quantum Kirwan map). 1. *The element $\kappa_{\mathfrak b}({\bf z}^I)$ is a legitimate element of ${\it VCF}_\bullet^{\mathfrak b} (\widehat H_\infty, \widehat J_\infty; \Lambda_{{\mathbb Z}[{\bf i}]})$ and is $\partial^{\mathfrak b}$-closed. Moreover, its homology class is independent of the choice of perturbation and its corresponding element in ${\it VHF}_\bullet^{\mathfrak b}(V; \Lambda_{{\mathbb Z}[{\bf i}]})$ is well-defined.* 2. *$\kappa_{\mathfrak b}(1) = {\bm 1}_{\mathfrak b}^{\rm GLSM}$.* *Proof.* The first conclusion follows from the standard argument and the second one follows from the definition of ${\bf 1}_{\mathfrak b}^{\rm GLSM}$. ◻ We define another element in the vortex Floer homology which can be viewed as the first Chern class in the bulk-deformed Hamiltonian Floer homology, or the image of thee first Chern class under the bulk-deformed PSS map. Recall that the first Chern class of a toric manifold is naturally represented by the union of toric divisors. Upstairs, they are the union of all coordinate hyperplanes. **Definition 135**. The **${\mathfrak b}$-deformed first Chern class** is the element $$\kappa_{\mathfrak b}({\bf z}_1 + \cdots + {\bf z}_N) \in {\it VHF}_\bullet^{\mathfrak b}(V; \Lambda_{{\mathbb Z}[{\bf i}]})$$ Denote the operator on ${\it VHF}_\bullet^{\mathfrak b}(V; \Lambda_{{\mathbb Z}[{\bf i}]})$ defined by the pair-of-pants product with the ${\mathfrak b}$-deformed first Chern class by $${\mathbb E}_{\mathfrak b}: {\it VHF}_\bullet^{\mathfrak b}(V; \Lambda_{{\mathbb Z}[{\bf i}]}) \to {\it VHF}_\bullet^{\mathfrak b}(V; \Lambda_{{\mathbb Z}[{\bf i}]}).$$ ## The commutative diagram We prove the following proposition. **Proposition 136**. *When the bulk deformation ${\mathfrak b}$ is convenient, the following diagram commutes. $$\label{commutative_diagram} \vcenter{\xymatrix{ \Lambda_{\overline{\mathbb Q}} [{\bf z}_1, \ldots, {\bf z}_N] \ar[rr]^-{\mathfrak{ks}_{\mathfrak b}} \ar[d]_{\kappa_{\mathfrak b}} & & (\Lambda_{\overline{\mathbb Q}})^{{\rm Crit}_X W_{\mathfrak b}} \ar[d] \\ {\it VHF}_{\mathfrak b} (V; \Lambda_{\overline{\mathbb Q}}) \ar[rr]_{{\rm CO}_{\mathfrak b}} & & \displaystyle \bigoplus_{{\bm L}\in {\rm Crit}_X W_{\mathfrak b}} HH^\bullet( {\mathcal F}_{\mathfrak b}^\flat( {\bm L})) }}$$ Here the right vertical arrow is the natural identification induced by the individual isomorphisms ${\it HH}^\bullet( {\mathcal F}_{\mathfrak b}^\flat( {\bm L})) \cong \Lambda_{\overline{\mathbb Q}}$.* *Proof.* We turn on Hamiltonian perturbations on disks to construct a homotopy between the Kodaira--Spencer map and the closed-open map composed with the quantum Kirwan map. Fix a critical point of $W_{\mathfrak b}$ lying in the interior of the moment polytope with the corresponding Lagrangian brane ${\bm L}= (L({\bf u}), {\bf y})$. We claim that the following diagram commutes. $$\label{commutative_diagram_2} \vcenter{ \xymatrix{ \Lambda_{\overline{\mathbb Q}} [{\bf z}_1, \ldots, {\bf z}_N] \ar[rrr]^-{\widetilde{\mathfrak{ks}}_{{\bm L}, {\mathfrak b}}} \ar[d]_{\kappa_{\mathfrak b}} & & & {\it QHF}_{\mathfrak b}^\bullet( {\bm L}; \Lambda_{\overline{\mathbb Q}}) \ar[d] \\ {\it VHF}_{\mathfrak b} (V; \Lambda_{\overline{\mathbb Q}}) \ar[rrr]_{{\rm CO}_{{\bm L}, {\mathfrak b}}^0} & & & {\it QHF}_{\mathfrak b}^\bullet( {\bm L}; \Lambda_{\overline{\mathbb Q}}) } }$$ Once this is established, it follow that the image ${\rm CO}_{{\bm L}, {\mathfrak b}}^0 \circ \kappa_{\mathfrak b}$ is contained in the line spanned by the identity element of ${\it QHF}_{\mathfrak b}^\bullet({\bm L};\Lambda_{\overline{\mathbb Q}})$. Hence on the chain level, one has $${\rm CO}_{{\bm L}, {\mathfrak b}}^0 ( \kappa_{\mathfrak b}({\bf z}^I)) - \mathfrak{ks}_{{\bm L}, {\mathfrak b}}({\bf z}^I) {\bf e}_{\bm L}^+ \in {\rm Im} (m_1^\flat).$$ As the Hochschild cohomology of ${\bm L}$ is spanned by the identity element, it follows that the diagram [\[commutative_diagram\]](#commutative_diagram){reference-type="eqref" reference="commutative_diagram"} also commutes. Now we prove that [\[commutative_diagram_2\]](#commutative_diagram_2){reference-type="eqref" reference="commutative_diagram_2"} commutes. Consider closed-open domains with one interior marking. Define a 1-parameter family of equations parametrized by $\nu \in [0, 1]$ such that when $\nu = 0$, the equation is the quasidisk equation with tangency condition at the marking. When $\nu$ is positive, we stretch a neighborhood of the interior marking and turn on a Hamiltonian perturbation by a bulk-avoiding pair $(\widehat H, \widehat J)$. We always require the tangency condition at the interior marking. As for boundary insertions, we only allow the boundary inputs to be labelled by the canonical weakly bounding cochain $b_{\bm L}$, while the boundary output can be labelled by any critical point of $f_L$. We can consider such moduli spaces with the tangency condition corresponding to multiindex $I$, total disk class $\beta$, and the output labelled by $x \in {\rm Crit} f_L$. One can use similar arguments as before to regularize relevant moduli spaces using perturbations which naturally extend existing perturbations defining the $A_\infty$ structure, the closed-open map, and the quantum Kirwan map. Then by counting elements in zero-dimensional moduli spaces, one can define a linear map $$R_{\bm L}: \Lambda_{\overline{\mathbb Q}}[{\bf z}_1, \ldots, {\bf z}_N] \to {\it QCF}^\bullet( {\bm L}; \Lambda_{\overline{\mathbb Q}}) \subset {\it QCF}^\bullet( {\bm L}; \Lambda_{\overline{\mathbb Q}})^+.$$ Now we consider boundaries of 1-dimensional moduli spaces. There are the following types of boundary strata. 1. The boundary at $\nu = 0$. This side of the boundary consists of points in zero-dimensional moduli spaces used to define the Kodaira--Spencer map. The contribution of these boundary points is equal to $\widetilde{\mathfrak{ks}}_{{\bm L}, {\mathfrak b}}$. 2. The boundary at $\nu = 1$. This side of the boundary consists of configurations having exactly one interior breaking at certain equivariant 1-periodic orbit of the Hamiltonian $\widehat H$. As the perturbation extends the perturbations chosen for the closed-open map and the quantum Kirwan map, the contribution of these boundary points is equal to $${\rm CO}_{{\bm L}, {\mathfrak b}}^0 \circ \kappa_{\mathfrak b}.$$ 3. Boundary points at $\nu \in (0, 1)$. These configurations have exactly one boundary breakings. There are two possibilities. First, the interior puncture and the boundary output are in the same unbroken component. In this case, the other unbroken component is a treed quasidisk with only boundary insertions being the canonical weakly bounding cochain $b_{\bm L}$. As the perturbation satisfies the forgetful property when one input is unweighted (the strict unit ${\bf e}^+$), the contribution of this kind of boundary points is zero. Second, the interior puncture and the boundary output are in two different unbroken component. The contribution of such configurations is $$m_1^\flat ( R_{\bm L}( {\bf z}^I))$$ which is exact. Therefore, it follows that on the chain level, one has $$\widetilde{\mathfrak{ks}}_{{\bm L}, {\mathfrak b}} ({\bf z}^I) - {\rm CO}_{{\bm L}, {\mathfrak b}}^0( \kappa_{\mathfrak b}({\bf z}^I)) \in {\rm Im} (m_1^\flat).$$ Hence on the cohomology level the diagram [\[commutative_diagram_2\]](#commutative_diagram_2){reference-type="eqref" reference="commutative_diagram_2"} commutes. ◻ Because the Kodaira--Spencer map is surjective, this finishes the proof of item (1) of Theorem [Theorem 122](#thm_CO){reference-type="ref" reference="thm_CO"}. ## Quantum multiplication by the first Chern class Now we prove item (2) of Theorem [Theorem 122](#thm_CO){reference-type="ref" reference="thm_CO"}. We prove the following theorem. **Theorem 137**. *When ${\mathfrak b}$ is a convenient small bulk deformation, the operator ${\mathbb E}_{\mathfrak b}$ on ${\it VHF}_\bullet^{\mathfrak b} (V; \Lambda_{\overline{\mathbb Q}})$ has an eigenspace decomposition $${\it VHF}_\bullet^{\mathfrak b} (V; \Lambda_{\overline{\mathbb Q}}) = \bigoplus_{{\bm L}\in {\rm Crit}_X (W_{\mathfrak b})} {\it VHF}_\bullet^{\mathfrak b} (V; \Lambda_{\overline{\mathbb Q}})_{W_{\mathfrak b}({\bm L})}.$$* *Proof.* By item (1) of Theorem [Theorem 122](#thm_CO){reference-type="ref" reference="thm_CO"}, ${\it VHF}_\bullet^{\mathfrak b} (V; \Lambda_{\overline{\mathbb Q}})$ is semisimple. Hence ${\mathbb E}_{\mathfrak b}$ is diagonalizable, and let the eigenvalues be $\lambda_1, \ldots, \lambda_m$. Now take an eigenvalue $\lambda = \lambda_i$ and a critical point ${\bm L}= (L({\bf u}), {\bf y})\in {\rm Crit}_X W_{\mathfrak b}$. We consider the restriction of ${\rm CO}_{{\bm L}, {\mathfrak b}}^0: {\it VHF}_\bullet^{\mathfrak b}(V;\Lambda_{\overline{\mathbb Q}}) \to {\it QHF}_{\mathfrak b}^\bullet({\bm L}; \Lambda_{\overline{\mathbb Q}})$ to the $\lambda$-eigenspace. We prove that this map is nonzero only when $\lambda$ coincides with the critical value. Consider closed-open domains with two interior markings, one boundary output, and arbitrarily many boundary inputs (to be labelled by the canonical weakly bounding cochain of ${\bm L}$). We distinguish the two interior markings. The first one is $v_{{\rm Ham}}$, which will be labelled by an equivariant 1-periodic orbit. The second one is denoted by $v_{\rm Chern}$, which will be labelled by components of the equivariant toric divisor. Given any such closed-open domain $C = S \cup T$ where $S$ is the surface part and $T$ is the tree part, the marking corresponding to $v_{\rm Ham}$ becomes a puncture while the marking corresponding to $v_{\rm Chern}$ is denoted by $z_{\rm Chern} \in {\rm Int} S$. We would like to include one more constraints on the position of $v_{\rm Chern}$. In the same way as defining the closed-open map, there is a distinguished component $C_{\rm Ham}$ of such domains $C = S \cup T$. Because the domain $C$ has a distinguished output, we can identify $C_{\rm Ham}$ with ${\mathbb D}\setminus \{0\}$ canonically such that the boundary node on $C_{\rm Ham}$ leading towards the output is the point $1 \in S^1 \cong \partial {\mathbb D}$. Define the **offset angle** of $z_{\rm Chern}$ as follows. 1. If $z_{\rm Chern}$ is in a cylindrical component, it does not have an offset angle. 2. If $z_{\rm Chern}$ is on $C_{\rm Ham}\cong {\mathbb D} \setminus \{0\}$, then the offset angle is the angular coordinate of $z_{\rm Chern}$. 3. If $z_{\rm Chern}$ is not on $C_{\rm Ham}$ or any cylindrical component, then there is a unique boundary node on $C_{\rm Ham}$ connecting $C_{\rm Ham}$ to $z_{\rm Chern}$. The offset angle is the angular coordinate of this boundary node. We fix $\theta \in S^1 \setminus \{1\}$ and only consider closed-open domains described as above such that the offset angle of $z_{\rm Chern}$ is equal to $\theta$ or does not have an offset angle. Consider the same equation defining the closed-open maps on such domains with possibly different perturbation data, where on the cylindrical end one has the Hamiltonian perturbation by a regular bulk-avoiding pair $(\widehat H, \widehat J)$, and along the boundary one imposes the Lagrangian boundary condition from ${\bm L}$. We analyze the true boundaries of 1-dimensional such moduli spaces. We assume that the cylindrical end is labelled by a cycle $a$ in ${\it VCF}_\bullet^{\mathfrak b}(V; \Lambda_{\overline{\mathbb Q}})$. The true boundary components corresponding to configurations which have exactly one breaking, either an interior one or a boundary one. See Figure [6](#Figure_CO_spectrality){reference-type="ref" reference="Figure_CO_spectrality"}. {#Figure_CO_spectrality} 1. The breaking is interior and the special marking $z_{\rm Chern}$ is not on a cylindrical component. The sum of this kind of contributions is zero as the interior input is a cycle. Note that as we are counting treed holomorphic disks, the line segment connecting the component on which $z_{\rm Chern}$ lies and $C_{\rm Ham}$ is not meant to be a breaking. 2. The breaking is boundary at the offset angle $1 \in S^1$ (which is different from $\theta$) hence separates $C_{\rm Ham}$ and the output. The sum of this kind of configuration is a coboundary in ${\it QCF}_{\mathfrak b}({\bm L})$, which is zero in cohomology. 3. The breaking is boundary at an offset angle different from $1 \in S^1$ and $\theta$ hence does not separate $C_{\rm Ham}$ and the output. The disk bubble contributes to a multiple of the strict unit ${\bf e}_{\bm L}^+$. Hence by the forgetful property of the perturbation data, the contribution of such configurations is zero. 4. The breaking is boundary at the specified offset angle $\theta$ which separates the special marking $z_{\rm Chern}$ and the component $C_{\rm Ham}$. The disk bubble always has Maslov index 2, hence the interior constraint imposed at $z_{\rm Chern}$ gives a factor $1$. Hence the disk bubble contributes to $W_{\mathfrak b}({\bm L}) e_{\bm L}^+$. However, as the offset angle is fixed, there are such rigid configurations, and the counting is equal to $$W_{\mathfrak b}({\bm L}) \cdot {\rm CO}_{{\bm L}, {\mathfrak b}}^0 (a).$$ 5. The breaking is interior and the special marking $z_{\rm Chern}$ is on the cylindrical component that breaks off. This kind of configuration contributes to $${\rm CO}_{{\bm L}, {\mathfrak b}}^0 ({\mathbb E}_{\mathfrak b}(a)) = \lambda\cdot {\rm CO}_{{\bm L}, {\mathfrak b}}^0 (a),$$ due to the appearance of the pair-of-pants product in the upper component. The analysis above shows that in cohomology, one has $$\lambda\cdot {\rm CO}_{{\bm L}, {\mathfrak b}}^0 (a) = W_{\mathfrak b}({\bm L}) \cdot {\rm CO}_{{\bm L}, {\mathfrak b}}^0 (a).$$ Hence if $\lambda \neq W_{\mathfrak b}({\bm L})$, the map ${\rm CO}_{{\bm L}, {\mathfrak b}}^0$ vanishes on this eigenspace. On the other hand, the linear map $$\bigoplus_{{\bm L}\in {\rm Crit}_X W_{\mathfrak b}} {\rm CO}_{{\bm L},{\mathfrak b}}^0: {\it VHF}_\bullet^{\mathfrak b}(V; \Lambda_{\overline{\mathbb Q}}) \to \bigoplus_{{\bm L}\in {\rm Crit}_X W_{\mathfrak b}} {\it QHF}_{\mathfrak b}^\bullet ({\bm L}; \Lambda_{\overline{\mathbb Q}})$$ is injective, because when we take the component generated by the identity elements of ${\it QHF}_{\mathfrak b}^\bullet( {\bm L}; \Lambda_{\overline{\mathbb Q}})$, it descends to the isomorphism ${\rm CO}_{\mathfrak b}$ onto the direct sum of the Hochschild cohomology. Therefore, one has $$\label{eqn_spectrality} {\rm Spec} ({\mathbb E}_{\mathfrak b}) \subset W_{\mathfrak b} ({\rm Crit}_X W_{\mathfrak b}).$$ On the other hand, for each critical point ${\bm L}\in {\rm Crit}_X (W_{\mathfrak b})$, the closed-open map ${\rm CO}_{{\bm L}, {\mathfrak b}}^0$ is unital hence nonzero. This implies that $W_{{\bm L}, {\mathfrak b}} \in \Lambda$ is also an eigenvalue of ${\mathbb E}_{\mathfrak b}$. Hence [\[eqn_spectrality\]](#eqn_spectrality){reference-type="eqref" reference="eqn_spectrality"} is an identity. As when ${\mathfrak b}$ is convenient, all critical values are distinct, it follows that all eigenspaces of ${\mathbb E}_{\mathfrak b}$ are 1-dimensional. ◻ [^1]: *In fact the chain complex depends on an "admissible lift" upstairs as well as an almost complex structure. But the boundary depth only depends on the Hamiltonian downstairs.* [^2]: The second part is a consequence of the first, see [@Usher_Zhang_2016 Proposition 2.3]. [^3]: It is easy to see that if $A$ is semisimple, then the idempotent splitting is unique up to permuting idempotent summands. [^4]: There are two different conventions: the variables are either ordered as $a_1, \ldots, a_k$ or ordered as $a_k, \ldots, a_1$. [^5]: The original calculation of [@Sheridan_2016] for the $n=1$ case has a mistake resulting in a false result. We thank Nick Sheridan for clarifying it. [^6]: They must be contractible as $X$ is simply connected. [^7]: *Exponential decay type estimates for vortices can also be found in [@Ziltener_Decay][@Venugopalan_quasi][@Chen_Wang_Wang].* [^8]: Within this proof, being generic means being in the complement of a proper complex analytic subset. [^9]: Notice that among $x_1, \ldots, x_k$ some of them could be the weighted element ${\bf f}$.

arxiv_math

--- abstract: | We consider a model that approximates vortex rings in the axisymmetric 3D Euler equation by the movement of almost rigid bodies described by Newtonian mechanics. We assume that the bodies have a circular cross-section and that the fluid is irrotational and interacts with the bodies through the pressure exerted at the boundary. We show that this kind of system can be described through an ODE in the positions of the bodies and that in the limit, where the bodies shrink to massless filaments, the system converges to an ODE system similar to the point vortex system. In particular, we can show that in a suitable set-up, the bodies perform a leapfrogging motion. author: - David Meyer bibliography: - rings.bib title: A model for the approximation of vortex rings by almost rigid bodies --- # Introduction and main results Consider the three-dimensional Euler equation $$\begin{aligned} &\partial_t u+(u\cdot\nabla)u+\nabla p\,=\,0\\ &\operatorname{div}u\,=\,0.\end{aligned}$$ We are interested in axisymmetric solutions with no swirl, that is $u$ of the form $$\begin{aligned} u\,=\,(u_r(r,z,t),u_z(r,z,t),0)\label{no swirl}\end{aligned}$$ in standard axisymmetric coordinates $(r,z,\theta)$, defined through $x=(r\cos\theta,r\sin\theta,z)$ and $r>0$, $\theta\in [0,2\pi)$. The vorticity then takes the form $$\begin{aligned} \omega\,=\,\mathop{\mathrm{curl}}u\,=\,(\partial_zu_r-\partial_ru_z)e_\theta\end{aligned}$$ (note that the sign is different compared to the two-dimensional vorticity). A vortex ring is an axisymmetric solution where the vorticity is concentrated on some torus. These occur in real life for instance as smoke or bubble rings. In axisymmetric coordinates, the equation for the vorticity reads as $$\begin{aligned} &\partial_t\left(\frac{\omega}{r}\right)+u\cdot\nabla\left(\frac{\omega}{r}\right)\,=\,0\\ &\operatorname{div}(ru)\,=\,0\\ &\mathop{\mathrm{curl}}u\,=\,\omega.\end{aligned}$$ The velocity can be recovered from the vorticity through the Biot-Savart law: $$\begin{aligned} u(x)\,=\,\int \frac{y-x}{4\pi|y-x|^3}\times\omega(y)\mathrm{d}y.\end{aligned}$$ Asymptotically, for rings of inner radius $\epsilon$, with center around $(R,Z)$ and total vorticity $\Gamma$, one has in axisymmetric coordinates $$\begin{aligned} u(x)\,\approx\, -\frac{\Gamma\left(x-\binom{R}{Z}\right)^\perp}{2\pi\left|x-\binom{R}{Z}\right|^2}+\frac{\Gamma\left|\log \left|x-\binom{R}{Z}\right|\right|}{4\pi R}e_z+O(1)\label{asymptotic velocity}\end{aligned}$$ (here "$\vphantom{a}^\perp$" is defined in axisymmetric coordinates as the linear map with $e_r^\perp=e_z$ and $e_z^\perp=-e_r$). As the relative vorticity $\frac{\omega}{r}$ is transported, one would expect that asymptotically, only the second term changes the position of a ring, while the main interaction of two different rings is given through the first term. Hence in the asymptotic limit, one would expect that multiple rings of inner radius $\epsilon$ with centers at $q_i=\binom{R_i}{Z_i}$ and strengths $\Gamma_i$ are well approximated through the system $$\begin{aligned} \label{point vortex system} \binom{R_i}{Z_i}'\,=\,\sum_{j\neq i}\frac{\Gamma_j(q_j-q_i)^\perp}{2\pi|q_i-q_j|^2}+\frac{\Gamma_i|\log\epsilon|}{4\pi R_i}e_Z.\end{aligned}$$ Note that in this system, the second term goes to $\infty$, while the first one is bounded, unless the pairwise distances go to $0$. The rigorous justification of this system in such a regime where the interaction between different rings and the self-interaction of each ring are of the same order is currently an open problem. We will derive this system in two regimes in a simpler model. Roughly speaking, our model consists of replacing the vortex rings with "almost rigid" toroidal bodies with a circular crosssection, immersed in an axisymmetric fluid with no swirl and no vorticity, which is described by the Euler equations. However, we do allow for non-zero circulations around the bodies. These circulations will take the role of the vorticity in the limit. This model is simpler than the full Euler equations, as the crosssections are fixed and the possible instability of the rings is not an issue. The system does however still feature the critical interaction between the rings. To get a nontrivial motion of the bodies, one needs to allow them to change their shape because otherwise, they can only move in one direction, which does not allow for any nontrivial dynamics. For this, we allow the bodies to change the ratio between their outer and inner radius. We assume that the bodies interact with the fluid only through the pressure exerted at the boundary. We will describe this model in more detail in Sections [1.1](#StateSpace){reference-type="ref" reference="StateSpace"} and [1.2](#Main results){reference-type="ref" reference="Main results"}.\ In Section [2](#Section2){reference-type="ref" reference="Section2"} we will then be able to describe the fluid velocity through potentials and streamfunctions, which are completely determined by the positions and velocities of the bodies. As a result, we can reduce the system to a second order ODE, with coefficients determined by the potentials and streamfunctions, for the positions of the bodies. This lets us show that our system is well-posed. The proof of the convergence of the system for shrinking bodies then requires analyzing the limit of this system. For this, we study the asymptotics of the potentials and the streamfunctions in Sections [3](#Section3){reference-type="ref" reference="Section3"} and [4](#Section4){reference-type="ref" reference="Section4"}. We will be able to show that these converge to the corresponding two-dimensional functions in the zero radius limit and that the interaction of the different parts of the stream functions produces the same terms as the Biot-Savart law in the limit.\ In Section [5](#Section5){reference-type="ref" reference="Section5"}, we will study the convergence in the zero radius limit of the ODE, which is quite intricate, as the equation degenerates due to the vanishing mass. In order to still get estimates, we use a modulated version of the kinetic energy of the system, whose evolution only depends on the degenerating terms, which allows one to obtain uniform estimates on the velocity and to pass to the limit.\ In a two-dimensional setting, the similar convergence of fluid-body systems with shrinking bodies to point vortex systems in a bounded domain has been studied in [@GlassMunnierSueur1] and [@GlassSueur3], while further work on fluid-body systems has been done e.g. in [@GlassMunnierSueur2], [@HouotMunnier4], [@Munnier5] and [@glass2015uniqueness], see also the references therein. The simpler problem of a stationary shrinking obstacle has been studied e.g. in [@iftimie2003two; @he2019small; @iftimie2009incompressible]. Somewhat similar problems for filaments immersed in 3D Stokes flow have been considered e.g. in [@mori2020theoretical; @MR4373666]. The stability of vortex rings and their approximation by the System [\[point vortex system\]](#point vortex system){reference-type="eqref" reference="point vortex system"} has e.g. been studied in [@ButtaMarchioro; @butta2022vanishing; @benedetto2000motion]. Special solutions which behave like the System [\[point vortex system\]](#point vortex system){reference-type="eqref" reference="point vortex system"} have been constructed in [@Davila2022leapfrogging] and in [@JerrardSmets] for the similar Gross-Pitiaevskii equation. Non-axisymmetric vortex filaments have been considered e.g. in [@jerrard2017vortex]. ## The model {#StateSpace} We use axisymmetric coordinates $(r,z)$ and denote the right halfplane by $\mathbb{H}$. Fix some $k\in \mathbb{N}_{>0}$ and numbers $v_1,\dots, v_k>0$, which we interpret as the volumes of the bodies and which should remain constant along the evolution. We set $$\begin{aligned} \rho_i\,:=\,\sqrt{\frac{v_i}{\pi R_i}}\qquad B_i(R_i,Z_i)\,:=\,B_{\rho_i}((R_i,Z_i))\subset\mathbb{H} \text{ for $i\,=\,1,\dots,k$.}\label{def rho}\end{aligned}$$ With respect to the measure $r\mathrm{d}r\mathrm{d}z$, which corresponds to the three-dimensional volume, the $B_i$ then have the fixed volume $v_i=\pi\rho_i^2R_i$. See also the figure below. Formally, we describe the configuration of the bodies through the manifold $M\subset (\mathbb{R}^2)^k$ of all $(R_1,Z_1)\dots (R_k,Z_k)$ such that the bodies $B_i$ all have positive distance from each other and from $\partial\mathbb{H}$. For $q\in M$, we shall write $q_i$ for the $i$-th component and $q_{R_i}$ and $q_{Z_i}$ for the two components of $q_i$. We shall also write $B_i(q)$ for clarity instead of $B_i$ sometimes. Let $n$ denote the outer normal of $\bigcup_i\partial B_i$.  In this setup, there is a natural correspondence between tangent vectors and normal velocities on the $\partial B_i$, as every $C^1$-curve in $M$ corresponds to a continuous movement of each $B_i$. For a tangent vector $t^*$, let $t_{R_1}^*,t_{Z_1}^*,t_{R_2}^*,\dots$ denote its components. We say that a tangent vector is associated to $B_i$ if only its $R_i$- and $Z_i$-component are non-zero and write $T_{q_i}M$ for the subspace of those tangent vectors. Then for a $C^1$-curve $q$, the normal velocity is given by $$\begin{aligned} u(\dot{q})\,:=\,\dot{q}_{Z_i}n\cdot e_Z+\dot{q}_{R_i}n\cdot e_R-\frac{\dot{q}_{R_i}\rho_i}{2R_i}\text{ on $\partial B_i$.}\label{normal velo}\end{aligned}$$ Here $e_R$ and $e_Z$ are the unit vectors and the last summand is due to the fact that if the outer radius $q_{R_i}$ changes, then the inner radius must also change because the volume is fixed. Let $\mathcal{F}:=\mathbb{H}\backslash \bigcup_i B_i$ denote the (time-dependent) domain of the fluid. Let $L_R^2$ and $H_R^1$ denote the $L^2$ resp. $H^1$ spaces with respect to the measure $r\mathrm{d}r\mathrm{d}z$. We assume that: **Condition 1**. *In $\mathcal{F}(t)$ the fluid fulfills the axisymmetric Euler equations with zero vorticity and no swirl for $t\in [0,\infty)$, i.e. $$\begin{aligned} \partial_t u+(u\cdot\nabla)u+\nabla p\,=\,0\label{Euler1}\\ \operatorname{div}(ru)\,=\,0\label{Euler2}\\ \mathop{\mathrm{curl}}(u)\,=\,0\label{Euler3}\\ u_r\,=\,0 \text{ \normalfont{on} $\partial\mathbb{H}$.}\label{Euler4}\end{aligned}$$* *here the $\operatorname{div}$ and $\mathop{\mathrm{curl}}$ are taken with respect to the variables $(r,z)$ and $u$ is $\mathbb{R}^2$-valued.* These equations are equivalent to the usual Euler equation for $u$ of the form [\[no swirl\]](#no swirl){reference-type="eqref" reference="no swirl"} with no vorticity after going back to three-dimensional coordinates. We define $$\begin{aligned} \gamma_i\,:=\,\int_{\partial B_i}u\cdot\tau\,\mathrm{d}x,\end{aligned}$$ where $\tau=n^\perp$. This is a conserved quantity by Kelvin's circulation law. For technical reasons, we will assume: **Condition 2**. *None of the $\gamma_i$'s is $0$.* **Condition 3**. *[\[strongsol\]]{#strongsol label="strongsol"}We assume that this solution is strong in the sense that $u,\nabla u,\partial_t u\in L_R^2\cap C^1(\mathcal{F})$ and that $q$ is $C^2$ in time. In particular this should hold for the initial data.[\[strongsol\]]{#strongsol label="strongsol"}* We assume that the normal velocity of the boundary of each $B_i$ matches the velocity of $u$ in the corresponding direction: **Condition 4**. *For all $i$ we have $$\begin{aligned} u\cdot n\,=\,u(\dot{q}) \text{ on $\partial B_i$}\label{Euler5},\end{aligned}$$* *where we use the identification between tangent vectors and normal velocities mentioned above.* For simplicity we will assume that all bodies and the fluid have the constant density $1$, though all the arguments still work for different densities. **Condition 5**. *We assume that in the $z$-direction, the momentum is (formally) preserved, which yields the condition $$\begin{aligned} v_i\ddot{q}_{{Z}_i}\,=\,-\int_{\partial B_i}rpn\cdot e_Z\,\mathrm{d}x\label{eq z}\end{aligned}$$* *for all $i$.* It remains to derive a condition for the interaction of the fluid and the solids in the $r$-component. As the solids can change their shape, we make the Ansatz of prescribing an interior velocity field and assuming that the kinetic energy of each $B_i$ only changes through the force exerted by the pressure at the boundary. We associate to each normal velocity $t_i^*$ associated to $B_i$ such an interior velocity field $u_{i,int}(t_i^*)\in H^1(B_i)$ with $$\begin{aligned} &\operatorname{div}ru_{i,int}\,=\,0 \text{ in $B_i$}\label{Int1}\\ &\mathop{\mathrm{curl}}u_{i,int}\,=\,0 \text{ in $B_i$}\label{Int2}\\ &u_{i,int}\cdot n\,=\,u(t_i^*)\text{ on $\partial B_i$.}\label{Int3}\end{aligned}$$ Existence and uniqueness can be obtained by standard elliptic theory [@evans2022partial]\[Chapter 6\], as $u_{i,int}(t_i^*)$ can be written as $\nabla \phi$ with $\operatorname{div}(r\nabla \phi)=0$ by the assumption of curl-freeness. Note that this is linear in $t_i^*$, and that $$\begin{aligned} \label{Ztrivial}u_{i,int}(t_{Z_i}^*)=t_{Z_i}^*e_Z.\end{aligned}$$ We can use this to associate a quadratic form on $T_{q_i}M$ by $$\begin{aligned} (t_i^*)^TE_{q_i}t_i^+\,:=\,\int_{B_i}r\langle u_{i,int}(t_i^*),u_{i,int}(t_i^+) \rangle\,\mathrm{d}x, \label{def E}\end{aligned}$$ for tangent vectors $t_i^*,t_i^+$, associated to $B_i$. Clearly, this is symmetric and positive definite. Because of the explicit form [\[Ztrivial\]](#Ztrivial){reference-type="eqref" reference="Ztrivial"}, we know that $$\begin{aligned} (e_Z)_i^TE_{q_i}(e_Z)_i\,=\,v_i,\end{aligned}$$ where $(e_Z)_i\in (\mathbb{R}^2)^k$ is the vector with $e_Z$ in the $i$-th component. We set $$\begin{aligned} (e_R)_i^TE_{q_i}(e_R)_i\,=:\, f_i(q_{R_i})\label{def f},\end{aligned}$$ here the function $f_i$ depends on $v_i$ and $(e_R)_i$ is the vector with $e_R$ in the $i$-th component. In summary, the quadratic form can be written in components as $$\begin{aligned} (t_i^*)^T\begin{pmatrix} f_i(q_{R_i}) & 0\\ 0 & v_i\end{pmatrix}(t_i^+),\label{def f 2}\end{aligned}$$ as one can see from the explicit formula [\[Ztrivial\]](#Ztrivial){reference-type="eqref" reference="Ztrivial"} and the fact that $u_{i,int}(t_{R_i}^*)$ must be asymmetric in $z$ in the second component and hence $t_{R_i}^*$ and $t_{Z_i}^*$ are orthogonal to each other with respect to $E_{q_i}$. We define the kinetic energy of $B_i$ as $$\begin{aligned} \label{intEnergy} \mathcal{E}_{B_i}\,:=\,\frac{1}{2}\dot{q}_i^TE_{q_i}\dot{q}_i.\end{aligned}$$ We assume that the kinetic energy of each $B_i$ only changes through the force exerted at the boundary, that is $$\begin{aligned} \mathcal{E}_{B_i}'\,=\,-\int_{\partial B_i} rpu(\dot{q}_i)\cdot n\,\mathrm{d}x.\label{energyBalance}\end{aligned}$$ After using the Decomposition [\[def f 2\]](#def f 2){reference-type="eqref" reference="def f 2"} and subtracting the Condition [\[eq z\]](#eq z){reference-type="eqref" reference="eq z"}, one obtains that $$\begin{aligned} f_i(q_{R_i})\ddot{q}_{R_i}\dot{q}_{R_i}+\frac{1}{2}\partial_{q_{R_i}}f_i(q_{R_i})(\dot{q}_{R_i})^3\,=\,-\int_{\partial B_i}r p\dot{q}_{R_i}\left(n\cdot e_R -\frac{\rho_i}{2R_i}\right)\label{eq r}\,\mathrm{d}x.\end{aligned}$$ We make the extra assumption that one can divide out $\dot{q}_{R_i}$, which is equivalent to saying that whenever $\dot{q}_{R_i}=0$ and the force at the boundary is nonzero, then $\ddot{q}_{R_i}$ is not zero, which rules out bodies with fixed $R_i$-coordinate. This gives the final equation: **Condition 6**. *For all $i$ we have $$\begin{aligned} \label{eq R} f_i(q_{R_i})\ddot{q}_{R_i}+\frac{1}{2}\partial_{q_{R_i}}f_i(q_{R_i})(\dot{q}_{R_i})^2\,=\,-\int_{\partial B_i}r p\left(n\cdot e_R -\frac{\rho_i}{2R_i}\right)\,\mathrm{d}x.\end{aligned}$$* We can also write [\[eq z\]](#eq z){reference-type="eqref" reference="eq z"} and [\[eq R\]](#eq R){reference-type="eqref" reference="eq R"} as a single equation $$\begin{aligned} \label{2ndDeri} (t_i^*)^TE_{q_i}\ddot{q}_i+\frac{1}{2}t_i^*(\partial_{\dot{q}_i} E_{q_i}\cdot \dot{q}_i)\dot{q}_i\,=\,-\int_{\partial B_i} rpu(t_i^*)\,\mathrm{d}x,\end{aligned}$$ where $t_i^*$ is an arbitrary tangent vector associated to $B_i$. ## Main results {#Main results} Our first main result is wellposedness: **Theorem 7** (Informal). *For every initial datum $q(0),$ $\dot{q}(0)$ the system detailed in the previous section has a unique solution up to some time $T>0$. If $T<\infty$, then $q$ blows up at $T$ in the sense that some of the bodies either collide with each other, the boundary or escape to $\infty$.* *Furthermore, the system preserves energy.* The more precise statements can be found in Corollary [\[Cor:Exist\]](#Cor:Exist){reference-type="ref" reference="Cor:Exist"}, Lemma [\[Lem:Energy\]](#Lem:Energy){reference-type="ref" reference="Lem:Energy"} and Thm. [\[Thm:exist\]](#Thm:exist){reference-type="ref" reference="Thm:exist"}. For the zero-radius limit, we shall first introduce some notation. We will use a rescaling parameter $\epsilon$ and denote the manifold of configurations associated to the bodies with the "volumes" $v_1\epsilon^2,\dots, v_k\epsilon^2$ by $\tilde{M}_\epsilon$ (recall that the "volumes" were defined in [\[def rho\]](#def rho){reference-type="eqref" reference="def rho"}). We still denote the inner radii with $\rho_1,\dots\rho_k$. We write $\tilde{\rho}_i$ for the unrescaled radii $\frac{\rho_i}{\epsilon}$. We fix some $\binom{R_0}{Z_0}$ with $R_0>0$. There are two different regimes that one can consider: ### The first regime The first one is also considered in [@JerrardSmets] and [@Davila2022leapfrogging]. We set all $\gamma_i$'s to be equal to one and set the centers to be $$\begin{aligned} \label{def regime1} q_i\,=\,\left(R_0+\frac{\tilde{q}_{R_i}}{\sqrt{|\log\epsilon|}},Z_0+\frac{\tilde{q}_{Z_i}}{\sqrt{|\log\epsilon|}}\right),\end{aligned}$$ where the rescaled positions $\tilde{q}(0):=(\tilde{q}_{R_1}(0),\tilde{q}_{Z_1}(0),\dots)$, should be independent of $\epsilon$. We will further rescale time by a factor $|\log\epsilon|$ and work with the rescaled positions $\tilde{q}_i:=(\tilde{q}_{R_i},\tilde{q}_{Z_i})$. Formally, the vorticity hidden in the rotation is $-1$ for each body (where the minus comes from opposite sign of the axisymmetric vorticity). Hence, in this regime, the main part of the self-induced velocity (in rescaled time and space) of a vortex ring, that is $\frac{-1}{4\pi R_0}|\log\epsilon|^\frac{1}{2}e_Z$, is the same for all rings, hence we may neglect this part. The next order part is of the form $\frac{\tilde{q}_{R_i}}{4\pi R_0^2}e_Z$ in rescaled time and space by Taylor's theorem. The velocity induced by the $i$-th ring on the $j$-th ring is of the form $\frac{1}{2\pi}\frac{(q_j-q_i)^\perp}{|q_i-q_j|^2}$. We hence expect that in the limit $\epsilon\rightarrow 0$, the velocities $\tilde{q}_i$'s should solve the system $$\begin{aligned} \label{PointSystem1} \tilde{q}_i'\,=\,\frac{1}{2\pi}\sum_{j\neq i}\frac{(q_i-q_j)^\perp}{|q_i-q_j|^2}+\frac{\tilde{q}_{R_i}}{4\pi R_0^2}e_Z\end{aligned}$$ up to the subtracted term $-\frac{1}{4\pi R_0}|\log\epsilon|^\frac{1}{2}e_Z$. **Theorem 8**. *[\[Limit1\]]{#Limit1 label="Limit1"} Assume that the solution of [\[PointSystem1\]](#PointSystem1){reference-type="eqref" reference="PointSystem1"} exist until time $T$ (in the sense that no components of the solution go to $\infty$ and the distance between the different components stays positively bounded from below). Assume that the shifted initial velocity $\tilde{q}_i'+\frac{|\log\epsilon|^\frac{1}{2}}{4\pi R_0^2}e_Z$ is bounded uniformly in $\epsilon$.* *Then $\tilde{q}+t\frac{1}{4\pi R_0}|\log\epsilon|^\frac{1}{2}e_Z$ converges to the solution of [\[PointSystem1\]](#PointSystem1){reference-type="eqref" reference="PointSystem1"} weakly^\*^ in $W_{loc}^{1,\infty}([0,T))$ in rescaled time.* This ODE system has been studied e.g. in [@JerrardSmets], where the existence of periodic solutions for two rings has been shown. Such periodic solutions correspond to a "leapfrogging" motion of the rings, which was predicted already by Helmholtz in his famous work [@helmholtz1858integrale; @helmholtz1867lxiii]. ### The second regime We can also consider the regime where the self-induced motion and the motion induced by other rings is of the same order. For this we set $$\begin{aligned} \label{def regime2} q_i\,=\,\left(R_0+\frac{\tilde{q}_{R_i}}{|\log\epsilon|},Z_0+\frac{\tilde{q}_{Z_i}}{|\log\epsilon|}\right).\end{aligned}$$ Here the rescaled initial position $\tilde{q}(0)$ should again be independent of $\epsilon$. We will further rescale time by a factor $|\log\epsilon|^2$ and will again work with the rescaled positions $\tilde{q}$. In this regime, all expected velocities are of order $1$ in rescaled time and space. We hence expect that in the limit $\epsilon\rightarrow 0$ and in the rescaled time, the $\tilde{q}$ should solve the system $$\begin{aligned} \label{PointSystem2} \tilde{q}_i'\,=\,\frac{1}{2\pi}\sum_{j\neq i}\gamma_j\frac{(\tilde{q}_i-\tilde{q}_j)^\perp}{|\tilde{q}_i-\tilde{q}_j|}-\frac{\gamma_i}{4\pi R_0}e_Z\end{aligned}$$ This system was studied in [@MarchioroNegrini2] and one can show that if all $\gamma_i$'s have the same sign, then solutions cannot blow up in finite time [@MarchioroNegrini2]\[Thm. 2.1\]. **Theorem 9**. *[\[Limit2\]]{#Limit2 label="Limit2"} Assume that the solution of [\[PointSystem2\]](#PointSystem2){reference-type="eqref" reference="PointSystem2"} exist until some time $T$ in the same sense as in the previous theorem. Further assume that the initial velocities $\tilde{q}_i'(0)$ (in the rescaled time) are bounded uniformly in $\epsilon$.* *Then $\tilde{q}$ converges to the solution of [\[PointSystem2\]](#PointSystem2){reference-type="eqref" reference="PointSystem2"} weakly^\*^ in $W_{loc}^{1,\infty}([0,T))$ in rescaled time.* **General notation**: We will use the notation $\tilde{q}$ for the rescaled positions in both regimes, as most estimates work completely similarly for both regimes. We also denote the manifold of $\tilde{q}$'s for which the corresponding $q$ is in $\tilde{M}_\epsilon$ by $M_\epsilon$. We write $A\lesssim B$ if there is a constant $C>0$ such that $A\leq CB$, where the constant $C$ is allowed to depend on the number $k$ and on $q$ resp. $\tilde{q}$ and on which of the regimes we are in, but not on any other quantities. Similarly, we write $\ell$ for irrelevant (finite) exponents, which are allowed to depend on which regime we are in, but not on any other quantities and are allowed to change their value from line to line. If $\Omega\subset \mathbb{H}$ we write $L_0^2(\Omega)$ for the space of all functions $f\in L^2(\Omega)$ such that for every connected component $\Omega_i$ of $\Omega$ we have $\int_{\Omega_i}f\,\mathrm{d}x=0$. If $S\subset \mathbb{H}$, we write $S^{\mathbb{R}^3}$ for its figure of revolution in $\mathbb{R}^3$. # Wellposedness of the system {#Section2} We will follow the approach in [@GlassMunnierSueur2] to show that our system can be reduced to an ODE in $q$ only. The main additional difficulty is that we are in an unbounded domain and hence need decay estimates to justify partial integrations. ## Representation of $u$ {#repU} In this subsection, we show that $u$ can be recovered from a potential and a streamfunction. **Definition 1**. *For $t^*\in T_{q_i}M$, let $\phi_{i,t^*}=\phi_{i,t^*}(t)$ be defined as the unique solution of the Neumann problem$$\begin{aligned} &\operatorname{div}(r\nabla\phi_{i,t^*})\,=\,0 \text{ \normalfont{in} $\mathcal{F}(t)$}\\ &\partial_n\phi_{i,t^*}\,=\,u(t^*) \text{ \normalfont{on} $\partial B_i$}\\ &\partial_n\phi_{i,t^*}\,=\,0 \text{ \normalfont{on} $\partial B_j$ \normalfont{for} $j\neq i$ \normalfont{and on} $\partial\mathbb{H}$}\\ &\phi_{i,t^*}\in \dot{H}_R^1.\\ &\phi_{i,t^*}\rightarrow 0\text{ at $\infty$}\end{aligned}$$* We first need to check that this is well-defined. **Lemma 10**. *[\[ExistencePhi\]]{#ExistencePhi label="ExistencePhi"} Let $b\in L^2(\bigcup_i \partial B_i)$ be such that $\int_{\bigcup_i\partial B_i}rb\,\mathrm{d}x=0$, then the system $$\begin{aligned} &\operatorname{div}(r\nabla\phi)\,=\,0 \text{ \normalfont{in} $\mathcal{F}(t)$}\\ &\partial_n\phi_{}\,=\,b \text{ \normalfont{on} $\bigcup_i \partial B_i$}\\ &\partial_n\phi_{}\,=\,0 \text{ \normalfont{on} $\partial\mathbb{H}$}\\ &\phi_{}\in \dot{H}_R^1.\\ &\phi_{}\rightarrow 0\text{ at $\infty$}\end{aligned}$$* *has a unique solution. Furthermore $$\begin{aligned} \label{decayphi} |\nabla^m\phi(x)|\,\lesssim\,\frac{\left\lVert b \right\rVert_{H^{m}}}{1+|x|^{2+m}} \text{ \normalfont{for all} $m\in\mathbb{N}_{\geq 0}$}\end{aligned}$$* *where the implicit constant is bounded locally uniformly in $q$.* This implies the welldefinedness of $\phi_{i,t^*}$ and that the Estimate [\[decayphi\]](#decayphi){reference-type="eqref" reference="decayphi"} holds locally uniformly in $q$ for $\phi_{i,t^*}$, as only can directly check that $\int_{\bigcup_i\partial B_i} ru(t^*)\,\mathrm{d}x=0$. *Proof.* We go back to three-dimensional coordinates and set $\phi^{\mathbb{R}^3}(r,z,0)=\phi(r,z)$, where $\phi^{\mathbb{R}^3}$ is axisymmetric. Then $\phi$ solves the system above iff $\phi^{\mathbb{R}^3}$ solves the corresponding Neumann problem for $\Delta$ in $\mathbb{R}^3\backslash \bigcup_j B_j^{\mathbb{R}^3}$. By standard techniques (see e.g. [@Amrouche]), we obtain a unique solution $\phi^{\mathbb{R}^3}\in \dot{H}^{1}(\mathbb{R}^3\backslash \bigcup B_j^{\mathbb{R}^3})$. We furthermore obtain from this that $\partial_n\phi=0$ on $\partial\mathbb{H}$.\ The decay rate then directly follows from the Lemma below. ◻ **Lemma 11**. *[\[decay1\]]{#decay1 label="decay1"}* *Let $\zeta\in \dot{H}^1(\mathcal{F}^{\mathbb{R}^3})$ be axisymmetric and such that $\Delta\zeta=0$ in $\mathcal{F}^{\mathbb{R}^3}$ and $\zeta\rightarrow 0$ at $\infty$. Then it holds that $$\begin{aligned} \left|\nabla^m\zeta(x)\right|\,\lesssim\,\frac{\left\lVert \partial_n\zeta \right\rVert_{H^{m}(\partial\mathcal{F}^{\mathbb{R}^3})}}{1+|x|^{1+|m|}}\quad \forall m\in\mathbb{N}_{\geq 0}.\end{aligned}$$* *The implicit constant is bounded locally uniformly in $q$.* *Let $\zeta$ be as in a). If additionally $$\begin{aligned} \int_{\bigcup_j \partial B_j^{\mathbb{R}^3}}\partial_n\zeta\,\mathrm{d}x\,=\,0,\end{aligned}$$ then $$\begin{aligned} \left|\nabla^m\zeta(x)\right|\,\lesssim\,\frac{\left\lVert \partial_n\zeta \right\rVert_{H^{m}(\partial\mathcal{F}^{\mathbb{R}^3})}}{1+|x|^{2+|m|}}\quad \forall m\in\mathbb{N}_{\geq 0},\end{aligned}$$* *where the implicit constant is controlled as in a).* *Proof.* If we extend $\zeta$ to $\mathbb{R}^3$ by solving the Dirichlet problem for $\zeta|_{\partial B_j^{\mathbb{R}^3}}$ on each $B_j^{\mathbb{R}^3}$, then the (distributional) Laplacian of this extension is of the form $[\partial_n\zeta]\mathcal{H}^2\mathbin{\vrule height 1.6ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.3ex}{ \bigcup \partial B_j^{\mathbb{R}^3}}$ where $[\cdot]$ denotes the jump across the boundary, as a direct calculation shows. By elliptic regularity (cf. [@grisvard2011elliptic]\[Chapter 2\]), this is a finite measure. We claim that we can recover this extension by convoluting the distributional Laplacian with the Newtonian potential, indeed for any $f\in C_c^\infty(\mathbb{R}^3)$, we have $$\begin{aligned} \int_{\mathbb{R}^3}\int_{\bigcup_i\partial B_i^{\mathbb{R}^3}}\frac{-[\partial_n\zeta]}{4\pi|x-y|}\Delta f(x)\,\text{d}\mathcal{H}^2(y)\,\mathrm{d}x\,=\,\int_{\bigcup_i\partial B_i^{\mathbb{R}^3}}[\partial_n\zeta]f(y)\,\text{d}\mathcal{H}^2(y)\,=\,\int_{\mathbb{R}^3}\zeta \Delta f\,\mathrm{d}x,\end{aligned}$$ where in the second step, we used that the Newtonian potential is the inverse Laplacian. Hence the difference $[\partial_n\zeta]\mathcal{H}^2\mathbin{\vrule height 1.6ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.3ex}(\bigcup_i\partial B_i^{\mathbb{R}^3})*\frac{-1}{4\pi|x|}-\zeta$ is a harmonic tempered distribution, i.e. a polynomial. Now $$\begin{aligned} \mathcal{H}^2\mathbin{\vrule height 1.6ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.3ex}(\bigcup_i\partial B_i^{\mathbb{R}^3})*\frac{-1}{4\pi|x|}-\zeta\,\rightarrow\, 0,\end{aligned}$$ by the assumption on $\zeta$ and because $[\partial_n\zeta]\mathcal{H}^2\mathbin{\vrule height 1.6ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.3ex}(\bigcup_i\partial B_i^{\mathbb{R}^3})$ is a finite measure, so this difference is zero, which shows the claim. We hence obtain that $|\nabla^m\zeta(x)|\lesssim\frac{\left\lVert \partial_n\zeta \right\rVert_{L^2(\partial\mathcal{F}^{\mathbb{R}^3})}}{1+|x|^{1+m}}$ for all $m\geq0$ and for $\mathop{\mathrm{dist}}(x,\bigcup B_i^{\mathbb{R}^3})\geq 1$. For $x$ close to the $B_j$'s the estimate follows from elliptic regularity theory. This shows the estimate in part a), part b) works exactly the same way, except that the integral of $[\partial_n\zeta]$ vanishes by the assumption and partial integration, which gives one order more of decay.\ To see that these bounds are locally uniform in $q$, we need uniform estimates on $\left\lVert [\partial_n\zeta] \right\rVert_{L^1(\bigcup_i\partial B_i)}$, note that for this we only need up-to-the-boundary estimates for a neighborhood of each $B_i$, locally uniform in $q$ and a locally uniform $L^2$ estimate. By going back to axisymmetric coordinates, one obtains the former, as the geometries of these neighborhoods (in axisymmetric coordinates) only change by rescaling with a bounded factor. On the other hand, one can obtain from the energy equality (which is justified by the decay estimates we have already proven) in axisymmetric coordinates that $$\begin{aligned} \label{energy eq} \int_{\mathcal{F}} r|\nabla\zeta|^2\,\mathrm{d}x\,=\,-\int_{\bigcup_i \partial B_i}r\zeta\partial_n \zeta\,\mathrm{d}x\gtrsim \left\lVert \zeta \right\rVert_{H^1(\bigcup_i (B_i+B_1(0)\backslash B_i))}^2.\end{aligned}$$ Here we used the (three-dimensional) Sobolev inequality to control the $L^2$-norm on the right-hand side. The constant of the Sobolev inequality is locally uniform in $q$, as one can e.g. see by using a diffeomorphism between the different instances of $\mathcal{F}$. Now we can use the trace inequality from $H^1(B_i+B_1(0)\backslash B_i)$ to $L^2(\partial B_i)$ in [\[energy eq\]](#energy eq){reference-type="eqref" reference="energy eq"}, whose constant is also locally uniform in $q$, as the geometry only changes through rescaling by a bounded factor. This implies the desired locally uniform estimate $$\begin{aligned} \left\lVert \partial_n\zeta \right\rVert_{L^2(\bigcup_i\partial B_i)}\,\gtrsim \,\left\lVert \zeta \right\rVert_{L^2(\bigcup_i (B_i+B_1(0)\backslash B_i))}.\end{aligned}$$ ◻ The argument for the existence of the stream function is more complicated, as there is no easy algebraic relation between the three-dimensional stream function and the axisymmetric stream function. **Definition 2**. *[\[def psi\]]{#def psi label="def psi"} Let $\psi_{i}=\psi_i(t)$ be the solution of the elliptic equation $$\begin{aligned} &\operatorname{div}\left(\frac{1}{r}\nabla\psi_{i}\right)\,=\,0 \text{ in $\mathcal{F}(t)$}\\ &\psi_{i}|_{\partial B_j} \text{ is constant } \forall j\\ &\int_{\partial B_j}\frac{1}{r}\partial_{n}\psi_{i}\,\mathrm{d}x\,=\,\delta_{ij}\label{Flow}\\ &\psi_i|_{\{r=0\}}\,=\,0\\ &\lim_{r\rightarrow 0}\frac{1}{r}\partial_z\psi_i\,=\,0\end{aligned}$$* *We refer to the constant boundary values on $\partial B_j$ as $C_{ij}$.* **Lemma 12**. *[\[ExistencePsi\]]{#ExistencePsi label="ExistencePsi"} Such a $\psi_i$ always exists and is unique under the constraint $\frac{1}{\sqrt{r}}\nabla\psi_i\in L^2$. Furthermore $\frac{1}{r}\nabla\psi_i$ is continuous at $r=0$ and we have the estimate $$\begin{aligned} \left|\nabla^m\left(\frac{1}{r}\nabla\psi_i(x)\right)\right|\,\lesssim\,&\frac{1}{1+|x|^{2+m}}\quad \forall m\in\mathbb{N}_{\geq 0}.\end{aligned}$$ The implicit constant in the estimate is locally uniformly bounded in $q$.* *Proof.* We first show that an auxiliary function $u_{2,i}$ can be constructed by going back to three-dimensional coordinates and later show that one can recover $\psi_i$ from it.\ Step 1. We define $u_{2,i}$ on $\mathcal{F}^{\mathbb{R}^3}$ as the axisymmetric vector field with no azimuthal component which solves (in three-dimensional variables) the system $$\begin{aligned} \label{DivCurl} &\operatorname{div}\, u_{2,i}\,=\,0\\ & \mathop{\mathrm{curl}}\, u_{2,i}\,=\,0\\ & u_{2,i}\cdot n\,=\,0\text{ on $\partial B_j^{\mathbb{R}^3}$ for all $j$}\\ & u_{2,i}\in L^2(\mathcal{F}^{\mathbb{R}^3}).\end{aligned}$$ We make the Ansatz $\mathop{\mathrm{curl}}\Psi_i=u_{2,i}$ for a purely azimuthal field $\Psi_i=\tilde{\Psi}_ie_\theta$, where $e_\theta$ is the unit vector in the $\theta$-direction. This gives the equations $$\begin{aligned} (u_{2,i})_r\,=\,-\partial_z\tilde{\Psi}_i,\quad (u_{2,i})_z\,=\,\partial_r\tilde{\Psi}_i+\frac{1}{r}\partial_r \tilde{\Psi}_i,\end{aligned}$$ using that the last term can be rewritten as $\frac{1}{r}\partial_r(r\Psi_i)$, this turns the System [\[DivCurl\]](#DivCurl){reference-type="eqref" reference="DivCurl"} into the system $$\begin{aligned} &\Delta \Psi_i\,=\,0\text{ in $\mathcal{F}^{\mathbb{R}^3}$}\label{BigPhi1}\\ &r\tilde{\Psi}_i|_{\partial B_j^{\mathbb{R}^3}}\text{ is constant for all $j$.}\label{BigPhi2}\end{aligned}$$ For each fixed set of constant boundary values $(\tilde{C}_{ij})_{j=1,\dots,k}$ for $r\tilde{\Psi}_i$, this system has a unique solution $\Psi_i(\tilde{C}_{ij})\in H^1$ by standard techniques. Also, because $\Psi_i$ is purely azimuthal, it must vanish at $r=0$ and hence it holds that $$\begin{aligned} \label{u at 0} (u_{2,i})_r\,=\,0\end{aligned}$$ at $r=0$.\ Step 2. To show uniqueness of $u_{2,i}$ for given $(\tilde{C}_{ij})$, we note that we can recover a $\Psi_i$ fulling the system [\[BigPhi1\]](#BigPhi1){reference-type="eqref" reference="BigPhi1"},[\[BigPhi2\]](#BigPhi2){reference-type="eqref" reference="BigPhi2"} from $u_{2,i}$. Indeed we may extend $u_{2,i}$ to the full space by zero as $\overline{u}_{2,i}$, which preserves the divergence-freeness, as the distributional divergence on the boundary equals $[\overline{u}_{2,i}\cdot n]=0$. It is well known that on the full space, every divergence-free field can be written as a curl, indeed we have $-\mathop{\mathrm{curl}}\Delta^{-1}\mathop{\mathrm{curl}}g=g$ for every divergence-free $g$, as a straightforward calculation shows. Since we also have $\mathop{\mathrm{curl}}\overline{u}_{2,i}\in H^{-1}$ and the fundamental solution of the Laplacian maps $H^{-1}$ to $H^{1}$, we see that the field obtained this way lies in $H^1$. Hence two different solutions $u_{2,i}$ for the same boundary values would give rise to two different $\Psi_i$'s in $H^1$ with the same boundary values, which is impossible.\ Step 3. Now we can view $u_{2,i}$ as a function in $(r,z)$ again, it fulfills $\operatorname{div}(ru_{2,i})=0$ (with the two-dimensional divergence). Then we can find a $\psi_i(\tilde{C}_{ij})$ such that $$\begin{aligned} u_{2,i}\,=\,\frac{1}{r}\nabla^\perp\psi_i,\end{aligned}$$ because $ru_{2,i}^\perp$ is curl-free. This can be done with the usual path integral construction, it is easy to check that the condition $u_{2,i}\cdot n=0$ ensures that even paths which are not homotopy equivalent yield the same values. One can then check by direct calculation that $\operatorname{div}(\frac{1}{r}\nabla\psi_i(\tilde{C}_{ij}))=0$ holds and by the boundary condition for $u_{2,i}$, we see that $\psi_i(\tilde{C}_{ij})$ must be constant on each $\partial B_j$. Furthermore, we have that $u_{2,i}$ is continuous at $r=0$ by elliptic regularity and hence $\frac{1}{r}\nabla\psi_i(\tilde{C}_{ij})$ is continuous at $r=0$ and $\frac{1}{r}\partial_z\psi_i=0$ at $r=0$ by [\[u at 0\]](#u at 0){reference-type="eqref" reference="u at 0"}. This $\psi_i(\tilde{C}_{ij})$ is unique up to an additive constant under the condition $\frac{1}{\sqrt{r}}\nabla\psi_i(\tilde{C}_{ij})\in L^2$ (here one gets an additional factor $r$ from the coordinate change), as one can recover the unique $u_{2,i}$ from it.\ Next, we argue that we uniquely can pick the boundary values $(\tilde{C}_{ij})$ such that the Condition [\[Flow\]](#Flow){reference-type="eqref" reference="Flow"} holds. It suffices to show that the linear map that sends the boundary values $(\tilde{C}_{ij})$ to the integrals $\int_{\partial B_j}\frac{1}{r}\partial_{n}\psi_i(\tilde{C}_{ij})\,\mathrm{d}x$ is invertible for each fixed $i$. First note that the $\tilde{C}_{ij}$'s are also the boundary values of $\psi_i$ (up to an additive constant) because we have that $ru_{2,i}=\nabla^\perp r\tilde{\Psi}_i$ (in $(r,z)$-coordinates) and hence by applying the fundamental theorem of calculus, we see that $(r\tilde{\Psi})(x)-(r\tilde{\Psi})(y)=\psi_i(x)-\psi_i(y)$ (in axisymmetric coordinates). Assume there is a nonzero vector of $\tilde{C}_{ij}$'s such that all integrals vanish. Without loss of generality, we may assume that $\tilde{C}_{i1}$ is the biggest one of the $\tilde{C}_{ij}$'s. Then the normal derivative of $\psi_i$ on $\partial B_1$ must be nonpositive by the maximum principle and hence must be zero everywhere on $\partial B_1$. Since the tangential derivative also vanishes, the constant extension of $\psi_i(\tilde{C}_{ij})$ to $B_1$ still fulfills $\operatorname{div}(\frac{1}{r}(\psi_i(\tilde{C}_{ij}))=0$. But this extension is then a locally, but not globally constant solution of an elliptic equation, which is a contradiction.\ We have $\partial_z\psi_i=0\cdot u_{2,i}=0$ at $r=0$ because $u_{2,i}$ is continuous by elliptic regularity. We may choose the additive factor that we have leftover such that $\psi_i=0$ at $r=0$ holds. The uniqueness of $\psi_i$ follows from the uniqueness of the $\psi_i(\tilde{C}_{ij})$.\ Step 4. By elliptic regularity, it is easy to see that the $C_{ij}$ are locally uniformly bounded in $q$, and hence $\left\lVert \partial_n\Psi_i(C_{ij}) \right\rVert_{H^m}$ is locally uniformly bounded in $q$ by elliptic regularity theory. By Lemma [\[decay1\]](#decay1){reference-type="ref" reference="decay1"} a), we see that $$\begin{aligned} |\nabla^m\Psi_i(x)|\,\lesssim\, \frac{1}{1+|x|^{1+m}},\end{aligned}$$ for all $m\in \mathbb{N}_{\geq 0}$ which implies the decay estimate. ◻ It is known that the system $$\begin{aligned} &\operatorname{div}\left(\frac{1}{r}\nabla f\right)\,=\,g\text{ in $\mathbb{H}$}\\ &f\,=\,\partial_nf\,=\,0\text{ on $\partial\mathbb{H}$}\end{aligned}$$ has a fundamental solution $K$ such that $$\begin{aligned} \label{definition K} f(y)\,=\,\int_{\mathbb{H}}g(x)K(x,y)\,\mathrm{d}x\end{aligned}$$ is the unique solution under suitable decay assumptions on $f$ for e.g. $g\in C_c^\infty(\mathbb{H})$, see e.g. [@GallaySverak]\[Section 2\]. **Lemma 13**. *[\[rep psi\]]{#rep psi label="rep psi"}* - *Let $\operatorname{div}(\frac{1}{r}\nabla\zeta)=0$ in $\mathcal{F}$ with $\frac{1}{\sqrt{r}}\nabla\zeta\in L^2(\mathcal{F})$, assume that $\frac{1}{r}\nabla\zeta$ is continuous for $r\rightarrow 0$ and that $\zeta|_{r=0}=0$. Furthermore, assume that $\zeta|_{\partial\mathcal{F}}$ is sufficiently smooth, then there is a constant $C$, depending on $\zeta$ such that $$\begin{aligned} |\zeta(x)|\,\leq\, \frac{C}{1+|x|}.\end{aligned}$$ In particular this holds for $\psi_i$.* - *$\psi_i$ can be represented as $$\begin{aligned} \psi_i(y)\,=\,\int_{\bigcup_i\partial B_i}\frac{1}{r}K(x,y)\partial_n\psi_i\,\mathrm{d}x.\end{aligned}$$* *Proof.* a) and b) We claim that if we extend $\zeta$ to $\mathbb{H}$ by solving the Dirichlet problem for $\operatorname{div}(\frac{1}{r}\nabla\cdot)$ with boundary values $\zeta$ in each $B_i$, then it holds that $$\begin{aligned} \zeta\,=\,\int_{\bigcup_i\partial B_i}\frac{1}{r}K(x,y)[\partial_n\zeta]\,\mathrm{d}x.\end{aligned}$$ This directly shows b) because for $\psi_i$ the extension to each $B_i$ is constant. It also shows a) by the fact that $[\partial_n\zeta]\mathcal{H}^1\mathbin{\vrule height 1.6ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.3ex}\partial B_i$ is a finite measure by elliptic regularity and the fact that the fundamental solution $K(x,y)$ decays like $\frac{1}{1+|y|}$ at $\infty$ locally uniformly in $x$ (see [@GallaySverak]\[Lemma 2.1 ii)\]). The same argument as in the proof of Lemma [\[decay1\]](#decay1){reference-type="ref" reference="decay1"} shows that $$\begin{aligned} g(y)\,:=\,\int_{\partial\mathcal{F}}\frac{1}{r}K(x,y)[\partial_n\zeta]\,\mathrm{d}x\end{aligned}$$ fulfills $\operatorname{div}(\frac{1}{r}\nabla (g-\zeta))=0$. Furthermore by the aforementioned decay estimates for $K$ we see that $|g(x)|\lesssim \frac{1}{|1+|x|}$, that $g=0$ at $r=0$ and that $\frac{1}{r}\nabla g$ is continuous at $0$. By elliptic regularity, it holds that $g-\zeta$ is smooth in the interior of $\mathbb{H}$. Now let $h(r,z,\theta)=(\frac{1}{r}\nabla(\zeta-g)(r,z))^\perp$ (in axisymmetric coordinates). This function is divergence-free (with respect to three-dimensional variables) as a direct calculation shows for $r>0$, at $r=0$ it is also divergence-free by continuity. Hence there is an $H$ with $\mathop{\mathrm{curl}}H=h$, where the gradient is taken with respect to three-dimensional variables. Then it holds that $\Delta H=0$ and $H$ is a tempered distribution and hence is a polynomial. Hence we know that $\zeta-g$ is a polynomial as well, however, we have that $\zeta-g=0$ at $r=0$, hence it is also a polynomial in $r$ only. However, $g\rightarrow 0$ for $r\rightarrow \infty$ and hence if $\zeta-g$ does not vanish, then $\zeta$ would have to grow at least linearly in $r$. However for all $a$ and large enough $R$ we would then have $$\begin{aligned} 1\,\lesssim\,\frac{1}{R}\left|\zeta(R,a)-\zeta\left(\frac{R}{2},a\right)\right|\,\lesssim\, \frac{1}{R}\int_\frac{R}{2}^R|\partial_r\zeta(s,a)|\mathrm{d}s\,\lesssim\, \left(\int_\frac{R}{2}^R\frac{1}{s}|\partial_r\zeta(s,a)|^2\mathrm{d}s\right)^\frac{1}{2}.\end{aligned}$$ By taking a square root and using Fubini, we obtain that $\frac{1}{\sqrt{r}}\nabla\zeta\notin L^2$, which is a contradiction. ◻ **Lemma 14**. *[\[rot pre\]]{#rot pre label="rot pre"} The Euler equation [\[Euler1\]](#Euler1){reference-type="eqref" reference="Euler1"} holds if [\[Euler2\]](#Euler2){reference-type="eqref" reference="Euler2"} and [\[Euler3\]](#Euler3){reference-type="eqref" reference="Euler3"} hold and the circulations $\gamma_i=\int_{\partial B_i}\tau\cdot u\,\mathrm{d}x$ are conserved in time.* In the two-dimensional setting this statement is well-known, see e.g. [@filho2007vortex]. *Proof.* Indeed the vorticity equation always holds and therefore we have $\mathop{\mathrm{curl}}(\partial_t u+(u\cdot \nabla)u)=0,$ however not every curl-free field has to be a gradient in $\mathcal{F}$, as the domain has holes. Using the usual path integral construction, it is easy to see that it is a gradient if $$\begin{aligned} \int_\Gamma (\partial_tu+(u\cdot \nabla)u)\cdot \tau_\Gamma\,\mathrm{d}x\,=\,0\end{aligned}$$ along every closed, non-selfintersecting path $\Gamma\subset \mathcal{F}$ with normalized tangent $\tau_\Gamma$, which has winding number $1$ with respect to exactly one $B_i$ and $0$ with respect to all others. If $\Phi_t$ is the flow induced by $u$, then by direct calculation one sees that $$\begin{aligned} \partial_s\int_{\Phi_s(\Gamma)}u\cdot\tau_{\Phi_s(\Gamma)}\,\mathrm{d}x\big|_{s=0}\,=\,\int_\Gamma (\partial_tu+(u\cdot \nabla)u))\cdot \tau_\Gamma\,\mathrm{d}x.\end{aligned}$$ On the other hand we see that for such $\Gamma$ it holds $$\begin{aligned} \int_\Gamma u\cdot\tau_\Gamma\,\mathrm{d}x\,=\,\int_{\partial B_i} u\cdot\tau\,\mathrm{d}x,\end{aligned}$$ which is constant by assumption, and hence we see that there is a $p$ such that $\partial_t u+(u\cdot\nabla)u=-\nabla p$. ◻ **Proposition 15**. *The function $$\begin{aligned} \label{u1u2} u(t)\,=\,\sum_{i=1}^k\left(\nabla\phi_{i,\dot{q}_i}+\gamma_i\frac{1}{r}\nabla^\perp\psi_i\right)\,=:\,u_1+u_2\end{aligned}$$* *is a solution to the axisymmetric Euler equations [\[Euler1\]](#Euler1){reference-type="eqref" reference="Euler1"}-[\[Euler5\]](#Euler5){reference-type="eqref" reference="Euler5"}.* *Proof.* A direct calculation reveals that $u$ is $\mathop{\mathrm{curl}}$-free and fulfills $\operatorname{div}(ru)=0$ and hence $u$ fulfills [\[Euler2\]](#Euler2){reference-type="eqref" reference="Euler2"} and [\[Euler3\]](#Euler3){reference-type="eqref" reference="Euler3"} in $\mathcal{F}$. We further observe that $$\begin{aligned} &n\cdot\sum_{i=1}^k\nabla\phi_{i,\dot{q}_i}\,=\,u(\dot{q}) \text{ on $\bigcup_j\partial B_j$}\\ &n\cdot\frac{1}{r}\nabla^\perp\psi_i\,=\,0 \text{ on $\bigcup_j\partial B_j$}\\ &\int_{\partial B_j}\nabla\phi_{i,\dot{q}_i}\cdot\tau\,\mathrm{d}x\,=\,\int_{\partial B_j}\partial_\tau\phi_{i,\dot{q}_i}\,\mathrm{d}x\,=\,0\\ &\int_{\partial B_j}\frac{1}{r}\nabla^\perp\psi_{i}\cdot\tau\,\mathrm{d}x\,=\,\int_{\partial B_j}\frac{1}{r}\partial_n\psi_{i}\,\mathrm{d}x\,=\,\delta_{ij},\\ &\frac{1}{r}\partial_z\psi_i\,=\,\partial_r\phi_{i,\dot{q}_i}\,=\,0\text{ on $\partial\mathbb{H}$}\end{aligned}$$ where $\tau=n^\perp$. Hence this $u$ has the prescribed circulations and boundary velocities, which shows the statement by Lemma [\[rot pre\]](#rot pre){reference-type="ref" reference="rot pre"}. ◻ We shall refer to both the $\phi_{i,t^*}$'s and their sum as potentials of $u$. We shall refer to both the $\psi_i$'s and their weighted sum as streamfunctions of $u$. **Remark 16**. *[\[uunique\]]{#uunique label="uunique"} This $u$ is uniquely determined (in $L_R^2$) by $q,\,\dot{q}$ and the $\gamma_i$'s. Indeed if there would be two such $u$'s, then their difference would give rise to a nonzero streamfunction with zero circulation (by the same argument as in the proof of Lemma [\[ExistencePsi\]](#ExistencePsi){reference-type="ref" reference="ExistencePsi"}, Step 3), which is impossible by the uniqueness of the $\psi_i$'s.* ### Representation of $\partial_t u$ We will need to show that the potential and stream function are differentiable in $q$ to be able to represent $\partial_tu$. The differentiability of solutions to elliptic equations with respect to changes of the underlying domain is a classical topic and we refer the reader to [@sokolowski1992introduction] for further reading. **Lemma 17**. *[\[SmoothnessInq\]]{#SmoothnessInq label="SmoothnessInq"}* - *The function $\phi_{i,\bullet}$ is smooth as a map from the tangent bundle $TM$ to $H_R^1$ (here differentiability can be understood in both the $L_{loc}^2$-sense and the pointwise sense).* - *The derivatives $\partial_q \phi_{i,t^*},\partial_q^2\phi_{i,t^*}$ lie in $H_R^1\cap C^\infty(\mathcal{F})$, furthermore their $H_{R}^1$-norm is bounded locally uniformly in $q$ and $t^*$.* - *$\operatorname{div}(r\nabla\partial_q \phi_{i,t^*})=0$.* - *$$\begin{aligned} |\nabla^m\partial_q\phi_{i,t^*}(x)|\,\lesssim\,\frac{1}{1+|x|^{2+m}}\quad \forall m\in\mathbb{N}_{\geq 0}.\end{aligned}$$ here the implicit constant is bounded locally uniformly in $q$ and $t^*$.* *Proof.* We can identify the tangent space at every point with $\mathbb{R}^2$ and $\phi_{i,\bullet}$ is linear in $t^*$, hence it suffices to show smoothness in $q$ for a fixed $t^*$.\ Step 1. We first want to apply the implicit function theorem to obtain that a derivative of $\phi_{i,t^*}$ with respect to $q$ exists. Fix some $q^0$. We use the three-dimensional $\phi^{\mathbb{R}^3}=\phi_{i,t^*}(r,z)$ (in axisymmetrical coordinates) again. We set $$\begin{aligned} V\,:=\, (\dot{H}^1\cap L^6)(\mathbb{R}^3\backslash \bigcup_j B_j^{\mathbb{R}^3}(q^0)),\end{aligned}$$ and equip this with the standard inner product of $\dot{H}^1$. By the Sobolev embedding this is a Hilbert space. In order to fit different configurations of the bodies into one space, we introduce $C^\infty$ diffeomorphisms $\Xi:\mathbb{R}^3\backslash\bigcup_j B_j(q^0)\rightarrow\mathbb{R}^3\backslash \bigcup_j B_j(q^1)$ which map each $\partial B_j(q^0)$ to $\partial B_j(q^1)$. We can assume that the family $\Xi$ is smooth in the parameter $q^1$ since the $B_i$'s are. We may also assume that $\Xi$ is the identity outside a large ball depending on $q$, but bounded locally uniformly in $q$. Then $\phi^{\mathbb{R}^3}$ is harmonic on $\mathbb{R}^3\backslash\bigcup_j B_j(q^1)$ with Neumann boundary values $u(t^*,q^1)$ iff the function $\hat{\phi}:=\phi\circ \Xi$ fulfills $$\begin{aligned} \label{PullbackEq} \int_{\mathcal{F}(q^0)^{\mathbb{R}^3}}\langle \nabla\hat{\phi}(\text{D}\Xi)^{-1},\nabla\eta(\text{D}\Xi)^{-1} \rangle|\det \text{D}\Xi|\,\mathrm{d}x\,=\,-\int_{\bigcup \partial B_j(q^0)^{\mathbb{R}^3}}\frac{\rho_j(q^1)}{\rho_j(q^0)}u(t^*,q^1)\eta\,\mathrm{d}x,\end{aligned}$$ for all $\eta\in V$, where we have written the inner radius $\rho_j$ as a function of $q$. We may interpret the difference of the left- and right-hand side as a map $\mathcal{G}:M\times V\rightarrow V^*$. Since $\Xi$ is smooth in $q$ and compactly supported, we obtain that this map is Fréchet-smooth. Furthermore we have that $$\begin{aligned} D_V\mathcal{G}(q^0,\hat{\phi})\cdot\delta\phi\,=\,\int_{\mathcal{F}(q^0)^{\mathbb{R}^3}} \langle \nabla\delta\phi,\nabla\cdot \rangle\,\mathrm{d}x.\end{aligned}$$ This is an isomorphism by the Riesz representation theorem. Hence we see that $\hat{\phi}$ is Fréchet-smooth by the implicit function theorem. This implies that a function $\partial_q\phi^{\mathbb{R}^3}$ exists in $V$ by the smoothness of $\Xi$ in $q$. Similarly, higher derivatives must exist. This shows a).\ Step 2. Clearly, $\partial_q \phi^{\mathbb{R}^3}$ must be harmonic and hence smooth away from the boundary. To see smoothness up to the boundary we differentiate [\[PullbackEq\]](#PullbackEq){reference-type="eqref" reference="PullbackEq"} with respect to $q$ at $q^0$ and obtain that $$\begin{aligned} \label{EquationDeri} &\int_{\mathcal{F}(q^0)^{\mathbb{R}^3}}\langle \nabla\partial_q\hat{\phi}(\text{D}\Xi)^{-1},\nabla\eta(\text{D}\Xi)^{-1} \rangle|\det D\Xi|\,\mathrm{d}x+\notag\\ &\int_{\mathcal{F}(q^0)^{\mathbb{R}^3}}\langle \nabla\hat{\phi}\partial_q((\text{D}\Xi)^{-1}),\nabla\eta(\text{D}\Xi)^{-1} \rangle|\det \text{D}\Xi|\,\mathrm{d}x+\\ &\int_{\mathcal{F}(q^0)^{\mathbb{R}^3}}\langle \nabla\hat{\phi}(\text{D}\Xi)^{-1},\partial_q\left(\nabla\eta(\text{D}\Xi)^{-1}|\det \text{D}\Xi|\right) \rangle\,\mathrm{d}x\,=\,-\int_{\bigcup \partial B_j(q^0) }\langle \partial_q\left(\frac{\rho_j(q)}{\rho_j(q^0)}u(t^*,q^0)\right),\eta \rangle\,\mathrm{d}x\notag,\end{aligned}$$ for all $\eta\in V$, the differentiation of this equation is justified by the differentiability of $\hat{\phi}$ and by $\Xi$ being compactly supported and smooth in $q$. This is an elliptic equation for $\partial_q\hat{\phi}$ with Neumann boundary conditions and a smooth and compactly supported term given by the second and third summand on the left hand side. Hence $\partial_q\hat{\phi}$ is smooth up the boundary. The same argument can be used to show regularity of higher derivatives in $q$. Again this also shows that $\partial_q\phi^{\mathbb{R}^3}$ (and hence also $\phi_{i,t^*}$) is smooth up to the boundary and the same is true for higher derivatives in $q$. By the pointwise smoothness that follows from this, it is obvious that c) holds.\ Step 3. To obtain the decay estimate for the derivative, we note that it is enough to show these estimates for $\partial_q\phi^{\mathbb{R}^3}$. Clearly, it holds that $\Delta\partial_q\phi^{\mathbb{R}^3}=0$.\ We again employ Lemma [\[decay1\]](#decay1){reference-type="ref" reference="decay1"}, by e.g. going back to $\hat{\phi}$ and using Equation [\[EquationDeri\]](#EquationDeri){reference-type="eqref" reference="EquationDeri"}, it is easy to see that the boundary values are locally uniformly controlled by $q$ and $t^*$. To see that the integral over the Neumann boundary values of $\partial_q\phi^{\mathbb{R}^3}$ is $0$, we introduce some compact $B_j'$ with smooth boundary, in which $B_j^{\mathbb{R}^3}$ is compactly contained and which intersects no other $B_{j'}^{\mathbb{R}^3}$. Then we rewrite them as $$\begin{aligned} \int_{\partial B_j^{\mathbb{R}^3}}\partial_n\partial_q\phi^{\mathbb{R}^3}\,\mathrm{d}x\,=\,-\int_{\partial B_j'}\partial_n\partial_q\phi^{\mathbb{R}^3}\,\mathrm{d}x\,=\,-\partial_q\int_{\partial B_j'}\partial_n\phi^{\mathbb{R}^3}\,\mathrm{d}x\,=\,0.\end{aligned}$$ Here pulling out the derivative is justified by the regularity of $\partial_q\phi^{\mathbb{R}^3}$. In particular, the decay estimate also implies that the derivative is in $H^1$. ◻ **Lemma 18**. *The functions $f_i$ are smooth in $R_i$, in particular, $E_{q_i}$ is smooth with respect to $q$.* *Proof.* This can be shown as in the previous Lemma by using a similar smooth family of diffeomorphisms. ◻ **Lemma 19**. *[\[PsiDeri\]]{#PsiDeri label="PsiDeri"}* - *The derivative of $\psi_i$ with respect to $q$ exists and is smooth up to the boundary (here the derivative can e.g. be taken as a classical pointwise derivative or in the $L_{loc}^2$ sense).* - *We have that $\frac{1}{r}\nabla\partial_q\psi_i,\,\frac{1}{r}\nabla\partial_q^2\psi_i\in L_R^2\cap C^\infty$. Furthermore the $L^2$-norm of these derivatives is bounded locally uniformly in $q$.* - *It holds that $$\begin{aligned} \left|\partial_q\psi_i(x)\right|\,\lesssim\,\frac{1}{1+|x|}\end{aligned}$$* *and $$\begin{aligned} \left|\nabla^m\partial_q\frac{1}{r}\nabla^\perp\psi_i(x)\right|\,\lesssim\,\frac{1}{1+|x|^{2+m}}\quad \forall m\in\mathbb{N}_{\geq0}.\end{aligned}$$ In the second estimate the implicit constant is locally uniformly bounded in $q$.* - *The $C_{ij}$'s are differentiable with respect to $q$.* *Proof.* a) and d) The argument uses a similar technique as the existence proof for Lemma [\[ExistencePsi\]](#ExistencePsi){reference-type="ref" reference="ExistencePsi"}. First, we again consider the three-dimensional $\Psi_i$'s as in said proof, for fixed boundary values $(\tilde{C}_{ij})$, they have arbitrarily many derivatives in $q$, which are smooth up to the boundary by the same argument as in the previous proof and the derivatives are in $\dot{H}^1\cap L^6$. This also shows that for fixed $(\tilde{C}_{ij})$ there is a (smooth) derivative of $u_{2,i}$ and $\psi_{i}(\tilde{C}_{ij})$. It remains to argue that the $C_{ij}$'s are differentiable. To see this note that the linear map from the $(\tilde{C}_{ij})$ to the integrals $\int_{\partial B_l}\frac{1}{r}\partial_n\psi_i(\tilde{C}_{ij})\,\mathrm{d}x$, which was used to show existence of the $C_{ij}$'s, is differentiable in $q$ as well. Indeed we may again introduce some compact $B_l'$, which compactly contains $B_l$ and intersects no other $B_{j}$. Then we have $$\begin{aligned} \int_{\partial B_l'}\frac{1}{r}\partial_n\partial_q\psi_i(\tilde{C}_{ij})\,\mathrm{d}x\,=\,\partial_q\int_{\partial B_l'}\frac{1}{r}\partial_n\psi_i(\tilde{C}_{ij})\,\mathrm{d}x\,=\,\partial_q \int_{\partial B_l}\frac{1}{r}\partial_n\psi_i(\tilde{C}_{ij})\,\mathrm{d}x,\end{aligned}$$ which shows differentiability.\ As the $\dot{H}^1$-norm of $\Psi_i$ corresponds to the $L^2$-norm of $\frac{1}{r}\nabla^\perp\psi_i(\tilde{C}_{ij})$, we see the boundedness statement b). The decay of $\partial_q\psi_i$ again follows from the fact that the derivative fulfills $\operatorname{div}(\frac{1}{r}\nabla\partial_q\psi_i)=0$ and is smooth up to the boundary by using Lemma [\[rep psi\]](#rep psi){reference-type="ref" reference="rep psi"}. The decay of $\partial_q \frac{1}{r}\nabla^\perp\psi_i$ follows from the fact that the derivative of the three-dimensional stream function is harmonic and smooth up to the boundary as in the previous proof by Lemma [\[decay1\]](#decay1){reference-type="ref" reference="decay1"}, and can be controlled locally uniformly in $q$. ◻ **Remark 20**. *Note that if $q$ is $C^2$ in time and $u$ is a solution of the Euler equations [\[Euler1\]](#Euler1){reference-type="eqref" reference="Euler1"}-[\[Euler5\]](#Euler5){reference-type="eqref" reference="Euler5"}, then we must have $$\begin{aligned} \label{eqDu} \partial_t u\,=\,\sum_{i=1}^k\partial_q \phi_{i,\dot{q}}\cdot\dot{q}+\phi_{i,\ddot{q}_i}+\gamma_i\frac{1}{r}\nabla^\perp\partial_q\psi_i\cdot \dot{q}.\end{aligned}$$* *Indeed this follows from the fact that $u$ is uniquely determined through $q,\dot{q}$ and the $\gamma_i$'s (Remark [\[uunique\]](#uunique){reference-type="ref" reference="uunique"}). In particular, $u$ has the regularity required in Condition [\[strongsol\]](#strongsol){reference-type="ref" reference="strongsol"}.* **Lemma 21**. *[\[Decayp\]]{#Decayp label="Decayp"} Assume that $u$ solves the Euler equations [\[Euler1\]](#Euler1){reference-type="eqref" reference="Euler1"}-[\[Euler5\]](#Euler5){reference-type="eqref" reference="Euler5"} with pressure $p$. Then we have that $$\begin{aligned} |\nabla p(x)|\,\lesssim\,\frac{1}{1+|x|^2},\end{aligned}$$ and there is a constant $C$ which may be choosen as $0$ such that $$\begin{aligned} |p(x)-C|\,\lesssim\, \frac{1}{1+|x|}.\end{aligned}$$ The implicit constants in these estimates are bounded locally uniformly in $q,\dot{q},\ddot{q}$.* *Proof.* By the construction of $u$ in [\[u1u2\]](#u1u2){reference-type="eqref" reference="u1u2"} and the decay estimates in the Lemmata [\[ExistencePhi\]](#ExistencePhi){reference-type="ref" reference="ExistencePhi"} and [\[ExistencePsi\]](#ExistencePsi){reference-type="ref" reference="ExistencePsi"} we have that $$\begin{aligned} |(u\cdot\nabla)u(x)|\,\lesssim\, \frac{1}{1+|x|^5}.\end{aligned}$$ By Equation [\[eqDu\]](#eqDu){reference-type="eqref" reference="eqDu"} and the Lemmata [\[ExistencePhi\]](#ExistencePhi){reference-type="ref" reference="ExistencePhi"},[\[SmoothnessInq\]](#SmoothnessInq){reference-type="ref" reference="SmoothnessInq"} and [\[PsiDeri\]](#PsiDeri){reference-type="ref" reference="PsiDeri"}, we see that $$\begin{aligned} |\partial_tu(x)|\,\lesssim\,\frac{1}{1+|x|^2}.\end{aligned}$$ Hence $|\nabla p(x)|\lesssim \frac{1}{1+|x|^2}$ and the estimate is locally uniform in $q,\dot{q},\ddot{q}$, because the estimates for $u$ and its derivatives are. Since $\int_{\partial B_R(0)\cap \mathbb{H}}|\nabla p|\,\mathrm{d}x\rightarrow 0$ for $R\rightarrow\infty$ we have $$\begin{aligned} \max_{x\in \partial B_R(0)} p(x)-\min_{x\in \partial B_R(0)} p(x)\,\rightarrow\, 0.\end{aligned}$$ Furthermore $\int_1^\infty |\nabla p(x,a)|\,\mathrm{d}x<\infty$ for all $a$ for which no $B_i$ intersects this line, hence we obtain that $p$ converges to some finite value at infinity, which then gives the decay statement for $p$ by the fundamental theorem of calculus. ◻ ## Derivation of an ODE for the system We reduce the motion of the bodies $B_i$ to an ODE whose coefficients depend on the $\psi$'s and $\phi$'s. This will also yield existence and uniqueness of solutions to the system. In two-dimensional bounded domains, a very similar calculation can be found e.g. in [@GlassMunnierSueur2]. We first introduce some additional terminology. We set $\phi(t^*)=\sum \phi_{i,t_i^*}$ if $t^*=t_1^*+\dots+ t_k^*$. Furthermore we set $\psi=\sum_i\gamma_i\psi_i$. **Definition 3**. *[\[main def\]]{#main def label="main def"} Let $t^*=t_1^*+\dots+ t_k^*$; $s^*=s_1^*+\dots+ s_k^*$ and $w^*=w_1^*+\dots+w_k^*$ for $t_i^*,s_i^*$ and $w_i^*$ associated to $B_i$. We define $$\begin{aligned} &G_i(q,\gamma)\cdot t_i^*\,=\,\int_{\partial B_i} \frac{1}{2r}\left((\partial_n\psi)^2\partial_n\phi_{i,t_i^*}\right)\,\mathrm{d}x\\ &(\mathcal{M}_{ij}(q)t_i^*)\cdot s_j^*\,=\,\int_{\mathcal{F}}r\nabla\phi_{i,t_i^*}\nabla \phi_{j,s_j^*}\,\mathrm{d}x\\ &\langle \Gamma(q),t^*,s^*\rangle\cdot w^*\,=\,\frac{1}{2}\sum_{ij}\Bigl( \left(\left(\partial_q\mathcal{M}_{ij}\cdot s^*\right)t^*\right)\cdot w^*+\left(\left(\partial_q\mathcal{M}_{ij}\cdot t^*\right)s^*\right)\cdot w^*\nonumber\\ &-\left(\left(\partial_q\mathcal{M}_{ij}\cdot w^*\right)s^*\right)\cdot t^*\Bigr)\\ &(A(q,\gamma)t^*)\cdot s^*\,=\,\sum_i\int_{\partial B_i}\Bigl(-\partial_\tau\phi(s^*)\partial_n\phi(t^*)+\partial_\tau\phi(t^*)\partial_n\phi(s^*)\Bigr)\partial_n\psi\,\mathrm{d}x,\end{aligned}$$* *where in the definition of $\Gamma$, the inner dot product refers to the derivative in that direction.* *Furthermore, $\mathcal{M}$ shall be the matrix made up of the blocks $\mathcal{M}_{ij}$ and $E$ shall be the diagonal matrix made up of the blocks $E_{q_i}$. Let $G\in (\mathbb{R}^2)^k\simeq \mathbb{R}^{2k}$ be the vector with the entries $G_1,\dots G_k$.* **Theorem 22**. *The system detailed in Section [1.1](#StateSpace){reference-type="ref" reference="StateSpace"} is equivalent to the system of ODEs given by $$\begin{aligned} \label{MainODE} &E(q)\ddot{q}+\frac{1}{2}\dot{q}(\partial_{q}E(q)\cdot\dot{q})+\mathcal{M}(q)\ddot{q}+\langle\Gamma(q),\dot{q},\dot{q}\rangle \\ &\,=\,G(q,\gamma)+(A(q,\gamma)\dot{q})\notag.\end{aligned}$$* **Remark 23**. *The equation can be interpreted as the geodesic equation for the metric given by $\mathcal{M}+E$, with extra terms due to the calculation on the right hand side. The matrix $\mathcal{M}$ describes the "added inertia", which encodes the fact that to accelerate one of the bodies, one also has to accelerate the surrounding fluid.* *Proof.* We argued in Remark [\[uunique\]](#uunique){reference-type="ref" reference="uunique"} that $u$ is uniquely determined by $q,\dot{q}$, hence it suffices to show that the family of Equations in [\[2ndDeri\]](#2ndDeri){reference-type="eqref" reference="2ndDeri"} is equivalent to this system. Let $t_i^*$ be an arbitary tangent vector associated to $B_i$. We set $u_i^*=\nabla(\phi_{i,t_i^*})$. Then by Equation [\[2ndDeri\]](#2ndDeri){reference-type="ref" reference="2ndDeri"} it holds that $$\begin{aligned} (t_i^*)^TE_{q_i}\ddot{q}_i+\frac{1}{2}\dot{q}_i^T(\partial_{q_i}E_{q_i}\cdot\dot{q}_i)t_i^*\,=\,-\int_{\partial B_i} rpu_i^*\cdot n\,\mathrm{d}x.\end{aligned}$$ By the equation for $p$, partial integration and the identity $u\nabla u=\frac{1}{2}\nabla|u|^2+u\cdot \mathop{\mathrm{curl}}u$ it follows that $$\begin{aligned} &(t_i^*)^TE_{q_i}\ddot{q}_i+\frac{1}{2}\dot{q}_i^T(\partial_{q_i}E_{q_i}\cdot\dot{q}_i)t_i^*\,=\,\int_{\mathcal{F}}r\nabla p\cdot u_i^*\,\mathrm{d}x\\ &\,=\,-\int _{\mathcal{F}}r\left(\partial_t(u_1+u_2)+\frac{1}{2}\nabla|u_1+u_2|^2\right)\cdot u_i^*\,\mathrm{d}x.\notag\end{aligned}$$ It follows from the decay estimates in the Lemmata [\[ExistencePhi\]](#ExistencePhi){reference-type="ref" reference="ExistencePhi"}, [\[ExistencePsi\]](#ExistencePsi){reference-type="ref" reference="ExistencePsi"} and [\[Decayp\]](#Decayp){reference-type="ref" reference="Decayp"} that there are no boundary terms from $\infty$ in this partial integration. We now split this into the different contributions and use the proposition below to obtain the equation in the theorem, tested against $t_i^*$. Since $t_i^*$ was arbitrary this implies the statement. ◻ **Proposition 24**. *[\[Coefficients\]]{#Coefficients label="Coefficients"}* - *We have $$\begin{aligned} -\frac{1}{2}\int_\mathcal{F} r\nabla|u_2|^2\cdot u_i^*\,\mathrm{d}x\,=\,G_i(q,\gamma)\cdot t_i^*.\end{aligned}$$* - *It holds that $$\begin{aligned} -\int_\mathcal{F} r\left(\partial_tu_2+\nabla(u_1\cdot u_2)\right)\cdot u_i^*\,\mathrm{d}x\,=\,(A(q,\gamma)\dot{q})\cdot t_i^*.\end{aligned}$$* - *We have that $$\begin{aligned} \int_\mathcal{F} r\left(\partial_t u_1+\frac{1}{2}\nabla|u_1|^2\right)\cdot u_i^*\,\mathrm{d}x\,=\,\ddot{q}^T\mathcal{M}(q)t_i^*+\langle \Gamma(q),\dot{q},\dot{q} \rangle\cdot t_i^*.\end{aligned}$$* *Proof.* a) Using that both $u_2$ and $u_i^*$ decay like $\frac{1}{|x|^2}$ by Lemma [\[ExistencePhi\]](#ExistencePhi){reference-type="ref" reference="ExistencePhi"} and Lemma [\[ExistencePsi\]](#ExistencePsi){reference-type="ref" reference="ExistencePsi"} we may partially integrate the left-hand side to obtain equality with $$\begin{aligned} \int_{\partial\mathcal{F}}\frac{1}{2}r|u_2|^2\partial_n \phi_{i,t^*}\,\mathrm{d}x.\end{aligned}$$ To see that this equals the definition of $G$ we note that $\partial_n \phi_{i,t_i^*}$ vanishes on every boundary except $\partial B_i$ and that $|u_2|=\frac{1}{r}|\nabla^\perp\sum_j \gamma_j\psi_j|=\frac{1}{r}|\partial_n \sum_j \gamma_j\psi_j|$ since the tangential derivative of the $\psi$'s vanishes.\ b) We have that $$\begin{aligned} -\int_\mathcal{F} r\nabla(u_1\cdot u_2)u_i^*\,\mathrm{d}x\,=\,\int_{\partial\mathcal{F}}r(u_1\cdot u_2)(u_i^*\cdot n)\,\mathrm{d}x,\end{aligned}$$ (this partial integration is justified by the decay estimates from the Lemmata [\[ExistencePhi\]](#ExistencePhi){reference-type="ref" reference="ExistencePhi"} and [\[ExistencePsi\]](#ExistencePsi){reference-type="ref" reference="ExistencePsi"}) which by the construction of $u$ in [\[u1u2\]](#u1u2){reference-type="eqref" reference="u1u2"} equals $$\begin{aligned} \sum_{l}\int_{\partial B_i}r\left(\frac{1}{r}\partial_n\sum_j\gamma_j\psi_j\right)(\partial_\tau \phi_{l,\dot{q}_l})(\partial_n \phi_{i,t_i^*})\,\mathrm{d}x,\end{aligned}$$ because $u_2$ has no normal component on the boundary.\ It holds that $\partial_tu_2=\frac{1}{r}\nabla^\perp\partial_t\psi$. We have that $\operatorname{div}(\frac{1}{r}\nabla\partial_t\psi)=0$ and $\partial_t\psi$ has the boundary values $$\begin{aligned} \partial_t\psi\,=\,\sum_j\gamma_j\partial_{q} C_{jl}\cdot\dot{q}-(u_1\cdot n)\partial_n\psi\quad\text{on $\partial B_l$}\end{aligned}$$ as one can see by differentiating the identity $C_{jl}(q)=\psi_j(x_q)(q)$ where $x_q$ is some fixed point on $\partial B_l$ whose derivative in $t$ equals $u_1\cdot n$. Then a partial integration, which can again be justified by the decay estimates in the Lemmata [\[ExistencePhi\]](#ExistencePhi){reference-type="ref" reference="ExistencePhi"} and [\[PsiDeri\]](#PsiDeri){reference-type="ref" reference="PsiDeri"}, reveals that $$\begin{aligned} -\int_{\mathcal{F}}ru_i^*\cdot\partial_tu_2\,\mathrm{d}x\,=\,\sum_l\sum_j\int_{\partial B_l} \partial_\tau\phi_{i,t_i^*}\left(\gamma_j\partial_{q} C_{jl}\cdot\dot{q}-(u_1\cdot n)\partial_n\psi\right)\,\mathrm{d}x\end{aligned}$$ The first summand vanishes because $\partial_q C_{jl}$ is a constant on each $\partial B_l$ and this proves b).\ c) We introduce an energy functional for the potential part of the fluid velocity:$$\begin{aligned} \mathcal{E}_{u_1}\,:=\,\frac{1}{2}\int_{\mathcal{F}} r|u_1|^2\,\mathrm{d}x.\end{aligned}$$ Following the approach in [@Munnier5], we will show that $$\begin{aligned} \label{Claim1} (\partial_t\partial_{\dot{q}}-\partial_{q})\mathcal{E}_{u_1}\cdot t_i^*\,=\,\int_{\mathcal{F}}u_i^*\cdot(\partial_tu_1+\frac{1}{2}\nabla|u_1|^2)\,\mathrm{d}x.\end{aligned}$$ To prove this claim we shall use the following Lemma which can be proven exactly as in [@Munnier5]\[Lemma 5.1\]: **Lemma 25**. *[\[deriForm\]]{#deriForm label="deriForm"} For $\eta\in \dot{H}_R^1$ let $$\begin{aligned} &\Lambda(\eta)\,=\,\int_{\mathcal{F}}r\langle \nabla\phi(\dot{q}),\nabla \eta \rangle \,\mathrm{d}x\end{aligned}$$* *Then it holds that $$\begin{aligned} (\partial_t\partial_{\dot{q}}-\partial_q)(\Lambda)\,=\,0.\end{aligned}$$* We now note that $$\begin{aligned} \mathcal{E}_{u_1}\,=\,\frac{1}{2}\Lambda(\phi(\dot{q}))\end{aligned}$$ and that $$\begin{aligned} \partial_{\dot{q}}\mathcal{E}_{u_1}\cdot t_i^*\,=\,\frac{1}{2}\bigl((\partial_{\dot{q}}\Lambda)(\phi(\dot{q}))\cdot t_i^*+\Lambda(\phi(t_i^*))\bigr).\end{aligned}$$ Because $\phi(t_i^*)$ also equals $\partial_{\dot{q}}\phi(\dot{q})\cdot t_i^*$, we see that $$\begin{aligned} \label{eq 5} \partial_{\dot{q}}\mathcal{E}_{u_1}\cdot t_i^*\,=\,(\partial_{\dot{q}}\Lambda)(\phi(\dot{q}))\cdot t_i^*.\end{aligned}$$ Hence we obtain that $$\begin{aligned} &(\partial_t\partial_{\dot{q}}-\partial_{q})\mathcal{E}_{u_1}\cdot t_i^*\,=\,\\ &\left(\partial_t\partial_{\dot{q}}\cdot t_i^*-\partial_q\cdot t_i^*\right)\Lambda(\phi) +(\partial_{\dot{q}}\Lambda)(\partial_t\phi(\dot{q}))\cdot t_i^* +\frac{1}{2}(\partial_q\Lambda)(\phi(\dot{q}))\cdot t_i^* -\frac{1}{2}\Lambda(\phi^\dagger),\end{aligned}$$ where $\phi^\dagger=\partial_q\phi(\dot{q})\cdot t_i^*$ and we made use of [\[eq 5\]](#eq 5){reference-type="eqref" reference="eq 5"}. By Lemma [\[deriForm\]](#deriForm){reference-type="ref" reference="deriForm"}, the first term is $0$. By definition, the second term equals $$\begin{aligned} \partial_{\dot{q}}\Lambda(\partial_t\phi(\dot{q}))\cdot t_i^*\,=\,\int_{\mathcal{F}}r\partial_tu_1\nabla \phi(t_i^*)\,\mathrm{d}x.\end{aligned}$$ By Reynolds transport theorem (whose usage is justified by Lemma [\[SmoothnessInq\]](#SmoothnessInq){reference-type="ref" reference="SmoothnessInq"} b)) we have that twice the third term equals $$\begin{aligned} (\partial_q\Lambda\cdot t_i^*)(\phi)\,=\,-\int_{\partial B_i}r|u_1|^2u_i^*\cdot n\,\mathrm{d}x+\Lambda(\phi^\dagger).\end{aligned}$$ This yields the claim [\[Claim1\]](#Claim1){reference-type="eqref" reference="Claim1"} after another partial integration.\ Now we use that $\mathcal{E}_{u_1}=\frac{1}{2}\mathcal{M}(q)\dot{q}\cdot\dot{q}$, which follows directly from the definition of $\mathcal{M}$. Then we may compute the Euler-Lagrange equation of this as $$\begin{aligned} (\partial_t\partial_{\dot{q}}-\partial_{q})\mathcal{E}_{u_1}\cdot t_i^*\,=\,\mathcal{M}(q)\ddot{q}\cdot t_i^*+((\partial_q\mathcal{M}(q)\cdot \dot{q})\dot{q})\cdot t_i^*-\frac{1}{2}((\partial_q\mathcal{M}(q)\cdot t_i^*)\dot{q})\cdot\dot{q}.\end{aligned}$$ The last two summands equal the Christoffel symbol $\Gamma$ as one can directly see by writing them out in components. ◻ ### Uniqueness and Existence In this subsection, we show that the system is actually well-posed and that energy conservation will imply that solutions exist for all times if $q$ does not blow up. **Lemma 26**. *The coefficients $\mathcal{M},G,A, \Gamma$ are all continuously differentiable in $q$.* *Proof.* One can use the definition of all these terms and the Lemmata [\[SmoothnessInq\]](#SmoothnessInq){reference-type="ref" reference="SmoothnessInq"} and [\[PsiDeri\]](#PsiDeri){reference-type="ref" reference="PsiDeri"} to obtain that they are smooth in $q$. We leave the details to the reader. ◻ **Corollary 27**. *[\[Cor:Exist\]]{#Cor:Exist label="Cor:Exist"} For every initial datum $q,\dot{q}$, there is some $T>0$ such that the System [\[MainODE\]](#MainODE){reference-type="eqref" reference="MainODE"} and hence also the system introduced in Section [1.1](#StateSpace){reference-type="ref" reference="StateSpace"} has a unique solution up to time $T$, which is $C^2$ in $q$.* *Proof.* By the lemma above and Picard-Lindelöf, we have local existence and uniqueness if the matrix $\mathcal{M}+E$ is invertible, which follows from the fact that both $\mathcal{M}$ and $E$ are positive definite by definition. ◻ The total energy of the system is conserved: **Lemma 28**. *[\[Lem:Energy\]]{#Lem:Energy label="Lem:Energy"} The kinetic energy $$\begin{aligned} \int_{\mathcal{F}}\frac{1}{2}r|u|^2\,\mathrm{d}x+\sum_i\mathcal{E}_{B_i}\end{aligned}$$ ($\mathcal{E}$ was defined in [\[intEnergy\]](#intEnergy){reference-type="eqref" reference="intEnergy"}) is constant in time.* *Proof.* By Reynolds we have that $$\begin{aligned} &\frac{\text{d}}{\text{d}t}\int_{\mathcal{F}}\frac{1}{2}r|u|^2\,\mathrm{d}x\,=\,\int_{\mathcal{F}}ru\cdot\partial_t u\,\mathrm{d}x-\int_{\partial\mathcal{F}}\frac{1}{2}r|u|^2u\cdot n\,\mathrm{d}x.\end{aligned}$$ Here differentiating under the integral sign is justified by the $L_R^2$-differentiability from the Lemmata [\[SmoothnessInq\]](#SmoothnessInq){reference-type="ref" reference="SmoothnessInq"} and [\[PsiDeri\]](#PsiDeri){reference-type="ref" reference="PsiDeri"}. The first integral also equals $$\begin{aligned} \int_{\mathcal{F}}ru\cdot\partial_t u\,\mathrm{d}x\,=\,\int_{\mathcal{F}}ru\cdot(-u\cdot\nabla u-\nabla p)\,\mathrm{d}x\,=\,\int_{\mathcal{F}}-\frac{1}{2}\operatorname{div}(ru|u|^2)-\operatorname{div}(rpu)\,\mathrm{d}x.\end{aligned}$$ Applying Gauss to this and adding the second integral from the first equation we obtain the statement, as we have by Equation [\[energyBalance\]](#energyBalance){reference-type="eqref" reference="energyBalance"} $$\begin{aligned} \int_{\partial B_i}rpu(\dot{q}_i)\,\mathrm{d}x\,=\,-\frac{\text{d}}{\text{d}t}\mathcal{E}_{B_i}.\end{aligned}$$ ◻ **Lemma 29**. *[\[EnergySplitting\]]{#EnergySplitting label="EnergySplitting"} The kinetic energy of the fluid decomposes into the energies $\int_{\mathcal{F}} \frac{1}{2}r(|u_1|^2+|u_2|^2)\,\mathrm{d}x$.* *Proof.* We have that $$\begin{aligned} \int_{\mathcal{F}}ru_1\cdot u_2\,\mathrm{d}x\,=\,\sum_i\gamma_i\int_{\mathcal{F}}r\nabla\phi\frac{1}{r}\nabla^\perp\psi_i\,\mathrm{d}x\,=\,\sum_{i,j}\gamma_i\int_{\partial B_j}\partial_\tau\phi C_{ij}\,\mathrm{d}x\,=\,0\end{aligned}$$ where we abbreviated the potential of $u_1$ with $\phi$. ◻ **Theorem 30**. *[\[Thm:exist\]]{#Thm:exist label="Thm:exist"} Solutions of the system exist until $q$ leaves any compact set, i.e. until either some of the bodies collide with each other or the boundary or escape to infinity.* *Proof.* By the energy conservation, we see that $\int r|u|^2\,\mathrm{d}x$ is bounded uniformly in time and hence by Lemma [\[EnergySplitting\]](#EnergySplitting){reference-type="ref" reference="EnergySplitting"} we also have that $\int r|u_1|^2\,\mathrm{d}x$ is bounded uniformly in time. Now note that there is no $q$ such that for some $t^*\neq 0$ it holds that $\nabla\phi(t^*)=0$. Hence by the continuity of the coefficients we have on compact sets $$\begin{aligned} \int r|u_1|^2\,\mathrm{d}x\,\gtrsim\, |\dot{q}|^2.\end{aligned}$$ This implies that the only way the solution can blow up is if $q$ leaves any compact set. ◻ # Convergence of the potential part of the velocity {#Section3} In this section, we will consider the limit of the potential velocity and of the interior field in order to compute the limit of the coefficients of the equation. We will show that all relevant main quantities converge to the corresponding two-dimensional quantities for a single body, which can be explicitly written down, and that the error is an order $\epsilon|\log\epsilon|^\ell$ smaller. Furthermore, we will show that quantities that only exist for multiple bodies are even smaller. We will also show that derivatives with respect to $q$ are an order $\epsilon|\log\epsilon|^\ell$ smaller as well. In Subsections [3.3](#subsec3.3){reference-type="ref" reference="subsec3.3"} and [4.2](#subsec4.2){reference-type="ref" reference="subsec4.2"} we will see that $\mathcal{M}$ and $A$ converge to the corresponding two-dimensional quantities for a single body and that $\Gamma$ and $\partial_q A$ are neglectable.\ We omit the indices of $B,C,q, R, Z, u_{int},$ etc. when dealing with only a single body. We identify the tangent space of $\mathcal{M}$ with $(\mathbb{R}^2)^k$ via the map $t^*\rightarrow (t_{R_1}^*,t_{Z_1}^*,\dots)$. ## The interior field For the kinetic energy of each body, we only need to consider a single body as the definition of $E_{q_i}$ (see [\[def E\]](#def E){reference-type="eqref" reference="def E"}) only depends on $B_i$. Therefore we drop the indices in this subsection. We write $f_\epsilon$ for the function $f_i$, defined with the rescaling parameter $\epsilon$. **Lemma 31**. *[\[InteriorField\]]{#InteriorField label="InteriorField"} Consider the energy function $f_\epsilon$ defined in [\[def f\]](#def f){reference-type="eqref" reference="def f"}:* - *We have$$\begin{aligned} \left|f_\epsilon(R)-\pi R\tilde{\rho}^2\epsilon^2\right|\,\lesssim\, \epsilon^3,\end{aligned}$$* *where the implicit constant depends locally uniformly on $R$.* - *$f_\epsilon(R)$ is lipschitz in $R$ with constant $\lesssim \epsilon^3$, locally uniformly in $R$.* In particular this implies that we have $|E_{q_i}|\approx \epsilon^2$ and $|\nabla_q E_{q_i}|\lesssim \epsilon^3$. *Proof.* a) We compare the potential of $u_{int}$ with the one of the constant speed movement. Set $\phi_1(x)=r$, which solves the Neumann problem $\Delta \phi_1=0$, $\partial_n\phi_1=e_R\cdot n$. Similarly, $u_{int}(e_R)$ can by definition (see [\[Int1\]](#Int1){reference-type="eqref" reference="Int1"}-[\[Int3\]](#Int3){reference-type="eqref" reference="Int3"}) be written as $\nabla \phi_2$, where $\operatorname{div}(r\nabla\phi_2)=0$ and $\partial_n\phi_2=u(e_R)$. Testing these equations with $\phi_1-\phi_2$ we obtain that $$\begin{aligned} &\int_{B} \langle \nabla \phi_1,\nabla(\phi_1-\phi_2) \rangle\,\mathrm{d}x\,=\,\int_{\partial B} e_R\cdot n(\phi_1-\phi_2)\,\mathrm{d}x\\ &\int_{B}r\langle \nabla \phi_2,\nabla(\phi_1-\phi_2) \rangle\,\mathrm{d}x\,=\,\int_{\partial B}r\left(e_R\cdot n-\frac{\rho}{2R}\right)(\phi_1-\phi_2)\,\mathrm{d}x.\end{aligned}$$ We multiply the first equation with $R$ and subtract the second from it, this yields that $$\begin{aligned} &\int_{B} r\langle \nabla(\phi_1-\phi_2),\nabla(\phi_1-\phi_2) \rangle+(R-r)\langle \nabla \phi_1,\nabla(\phi_1-\phi_2) \rangle\,\mathrm{d}x\,=\,\\ &\int_{\partial B} \left(Re_R\cdot n-r\left(e_R\cdot n-\frac{\rho}{2R}\right)\right)(\phi_1-\phi_2)\,\mathrm{d}x.\notag\end{aligned}$$ Note that we may add a constant to $\phi_1-\phi_2$ in the last integral because the other factor is mean-free. Applying the Cauchy-Schwarz inequality we obtain that $$\begin{aligned} \label{L2bound1} &\int_{B}r\left|\nabla(\phi_1-\phi_2)\right|^2\,\mathrm{d}x\leq \rho\left\lVert \nabla\phi_1 \right\rVert_{L^2(B)}\left\lVert \nabla(\phi_1-\phi_2) \right\rVert_{L^2(B)}\\ &+\left\lVert Re_R\cdot n-r\left(e_R\cdot n-\frac{\rho}{2R}\right) \right\rVert_{L^2(\partial B)}\left\lVert \phi_1-\phi_2 \right\rVert_{L^2(\partial B)/constants}.\notag\end{aligned}$$ The last factor can be estimated by $c_{trace}\left\lVert \nabla(\phi_1-\phi_2) \right\rVert_{L^2(B)}$, where $c_{trace}$ is the operator norm of the trace from $\dot{H}^1$ to $L^2(\partial B)/constants$. By scaling one can see that this constant is $\lesssim \epsilon^\frac12$.\ This gives us an upper bound on the right-hand side of [\[L2bound1\]](#L2bound1){reference-type="eqref" reference="L2bound1"} of $$\begin{aligned} \label{L2bound2} \rho\left\lVert \nabla(\phi_1-\phi_2) \right\rVert_{L^2}\left\lVert \nabla\phi_1 \right\rVert_{L^2}+c_{trace}\left\lVert \nabla(\phi_1-\phi_2) \right\rVert_{L^2(B)}\left\lVert \rho+\frac{\rho}{2R} \right\rVert_{L^2(\partial B)}\,\lesssim\, \epsilon^2\left\lVert \nabla(\phi_1-\phi_2) \right\rVert_{L^2}.\end{aligned}$$ Together with the observation that $$\begin{aligned} \int_{B}r|\nabla(\phi_1-\phi_2)|^2\,\mathrm{d}x\,\lesssim\, \left\lVert \nabla(\phi_1-\phi_2) \right\rVert_{L^2}^2\end{aligned}$$ we obtain from [\[L2bound1\]](#L2bound1){reference-type="eqref" reference="L2bound1"} and [\[L2bound2\]](#L2bound2){reference-type="eqref" reference="L2bound2"} that $$\begin{aligned} \left\lVert \nabla(\phi_1-\phi_2) \right\rVert_{L^2}\,\lesssim\, \epsilon^2.\end{aligned}$$ Now by definition $$\begin{aligned} |f_\epsilon-\pi R\tilde{\rho}^2\epsilon^2|\,=\,\left|\int_B r\left(|\nabla\phi_2|^2-|\nabla\phi_1|^2\right)\,\mathrm{d}x\right|\,\lesssim\, \left\lVert \nabla(\phi_1-\phi_2) \right\rVert_{L^2}\left(\left\lVert \nabla\phi_1 \right\rVert_{L^2}+\left\lVert \nabla\phi_2 \right\rVert_{L^2}\right)\,\lesssim\, \epsilon^3.\end{aligned}$$ b\) We first estimate the derivative of the potential of $u_{int}$ with respect to $R$ and then use this to estimate the Lipschitz constant. We fix some $q^0=(Z^0,R^0)$ with inner radius $\rho^0$ and use the family of maps $$\begin{aligned} \Xi_q(x)\,:=\,\frac{\rho^0}{\rho}\left(x-\binom{R}{Z}\right)+\binom{R^0}{Z^0},\end{aligned}$$ which map $B(q)$ to $B(q^0)$. Let $\phi^{q}$ be defined by $\nabla \phi^q=u_{int}^{q}(e_R)$, where the $q$ in the superscript denotes the $q$-dependence and we use the identification between the tangent space and $\mathbb{R}^2$ mentioned above. Let $$\begin{aligned} \hat{\phi}\,:=\,\frac{\rho^0}{\rho}\phi^{q}\circ \Xi_q^{-1}.\end{aligned}$$ Then a direct calculation shows that this fulfills the system $$\begin{aligned} &\operatorname{div}\left(\frac{R^0}{R}\left(R+\frac{\rho}{\rho^0}\left(r-R^0\right)\right)\nabla\hat{\phi}\right)\,=\,0 \text{ in $B(q^0)$}\\ &\partial_n \hat{\phi}\,=\,e_R\cdot n-\frac{\rho}{2R} \text{ on $\partial B(q^0)$}\label{Compeq2}.\end{aligned}$$ Using e.g. the implicit function as in the proof of Lemma [\[SmoothnessInq\]](#SmoothnessInq){reference-type="ref" reference="SmoothnessInq"}, one can easily see that one can differentiate the solution of this equation in $R$ (w.r.t. the $H^1$-norm) and that the derivative fulfills the system $$\begin{aligned} &\operatorname{div}\left(\frac{R^0}{R}\left(R+\frac{\rho}{\rho^0}\left(r-R^0\right)\right)\nabla\partial_{R_1}\hat{\phi}\right)+\operatorname{div}\left(\partial_{R_1}\left(\frac{R^0}{R}\left(R+\frac{\rho}{\rho^0}\left(r-R^0\right)\right)\right)\nabla\hat{\phi}\right)\,=\,0\text{ in $B(q^0)$}\label{Compeq1}\\ &\partial_n \partial_{R_1}\hat{\phi}\,=\,\partial_{R_1}\left(e_R\cdot n-\frac{\rho}{2R}\right)\text{ on $\partial B(q^0)$}\label{Compeq2},\end{aligned}$$ here we write $\partial_{R_1}\hat{\phi}$ for the derivative with respect to the parameter $R=R_1$ in order to prevent confusion with the spatial derivative in the $R$-direction. Now one can easily check that $$\begin{aligned} &\left|\partial_{R_1}\left(\frac{R^0}{R}\left(R+\frac{\rho}{\rho^0}\left(r-R^0\right)\right)\right)\right|\,\lesssim\, \epsilon\label{Deriest1}\\ &\left\lVert \nabla\hat{\phi} \right\rVert_{L^2}\,\lesssim\, \epsilon\label{Deriest2}\\ &\left|\partial_{R_1}\left(e_R\cdot n-\frac{\rho}{2R}\right)\right|\,\lesssim\, \epsilon.\label{Deriest3}\end{aligned}$$ We can now test the equations [\[Compeq1\]](#Compeq1){reference-type="eqref" reference="Compeq1"},[\[Compeq2\]](#Compeq2){reference-type="eqref" reference="Compeq2"} with $\partial_{R_1}\hat{\phi}$ and obtain after using the Cauchy-Schwarz inequality similarly as in part a) and the bounds [\[Deriest1\]](#Deriest1){reference-type="eqref" reference="Deriest1"}-[\[Deriest3\]](#Deriest3){reference-type="eqref" reference="Deriest3"} that $$\begin{aligned} \int_{B(q^0)}\frac{R^0}{R}\left(R+\frac{\rho}{\rho^0}(r-R^0)\right)|\nabla\partial_{R_1}\hat{\phi}|^2\,\mathrm{d}x\,\lesssim \,\epsilon^2\left\lVert \nabla \hat{\phi} \right\rVert_{L^2}^2+\epsilon^{\frac{3}{2}}\left\lVert \partial_{R_1}\hat{\phi} \right\rVert_{L^2(\partial B(q^0))/constants}\label{Compeq3},\end{aligned}$$ where we again used the mean-freeness of the boundary values to take the $L^2$-norm modulo constants. Clearly, the prefactor in the integral on the left-hand side is $\simeq 1$. Again the operator norm of the trace operator from $\dot{H}^1$ to $L^2(\partial B(q^0))/constants$ is $\simeq \epsilon^{\frac{1}{2}}$ by scaling, hence we obtain from [\[Compeq3\]](#Compeq3){reference-type="eqref" reference="Compeq3"} that $$\begin{aligned} \left\lVert \nabla \partial_{R_1}\hat{\phi} \right\rVert_{L^2}\,\lesssim\, \epsilon^2\end{aligned}$$ and this bound is locally uniform in $R$.\ By definition it holds that $$\begin{aligned} f_\epsilon(R)\,=\,\int_{B(q)} r|\nabla \phi^q|^2\,\mathrm{d}x\,=\,\int_{B(q^0)}\left(\frac{\rho}{\rho^0}\right)^2\left(R+\frac{\rho}{\rho_0}(r-R^0)\right) |\nabla\hat{\phi}|^2\,\mathrm{d}x.\end{aligned}$$ The prefactor in the second integral is differentiable in $R$ with a derivative $\lesssim \epsilon$. Now we can differentiate the right-hand side under the integral by the $H^1$-differentiability of $\hat{\phi}$ and obtain from the product rule that $$\begin{aligned} |\partial_Rf_\epsilon(R)|\,\lesssim\, \epsilon^2\left\lVert \nabla\hat{\phi} \right\rVert_{L^2}+\epsilon\left\lVert \nabla\hat{\phi} \right\rVert_{L^2}^2\,\lesssim\, \epsilon^3.\end{aligned}$$ This is locally uniform in $R$ as all the used estimates are. ◻ ## The potential part of the velocity We show that the boundary values of the potential converge to the boundary values of the corresponding "two-dimensional" potential. ### The case of a single body **Definition 4**. *[\[def flat func\]]{#def flat func label="def flat func"} Let $t^*,q\in \mathbb{R}^2$. Let $\rho>0$. Let $n$ denote the outer normal vector of $\partial B_\rho(q)$. We define a "two-dimensional" potential $$\begin{aligned} \widecheck{\phi}_{t^*}\,=\,\widecheck{\phi}_{t^*}(q,\rho)\,:=\,-\rho^2\frac{t^*\cdot e_1(x-q)+t^*\cdot e_2(y-q)}{(x-q)^2+(y-q)^2}.\end{aligned}$$* *One can check that this is the solution of $$\begin{aligned} &\Delta \widecheck{\phi}_{t^*}\,=\,0\text{ in $\mathbb{R}^2\backslash B_\rho(q)$}\label{Pot1}\\ &\partial_n\widecheck{\phi}_{t^*}\,=\,t^*\cdot n \text{ on $\partial B_\rho(q)$}\label{Pot2}\\ &\widecheck{\phi}_{t^*}\rightarrow 0 \text{ at $\infty$}\label{Pot3}\end{aligned}$$* *(uniqueness of this can be found e.g. in [@Amrouche]\[Thm. 3.1\]).* In order to estimate the potentials for $\epsilon\rightarrow 0$, we first use only a single body and again drop the indices. Fix some $q$ and $t^*$, where we again make use of the identification of the tangent space with $\mathbb{R}^2$ as in the previous subsection. Furthermore, fix some $\tilde{\rho}>0$ such that $\epsilon\tilde{\rho}=\rho$. We will prove a more general statement for arbitrary normal velocities, which will be useful later to estimate derivatives with respect to $q$. It will simplify the argument to rescale everything by a factor of $\epsilon$. Therefore we let $\grave{B}:=B_{\tilde{\rho}}(\frac{1}{\epsilon}q)$ and first prove our estimates around this rescaled body. **Proposition 32**. *[\[PotentialGeneral\]]{#PotentialGeneral label="PotentialGeneral"} Let $b_1,b_2$ be smooth functions on $\partial\grave{B}$. Further, assume that $$\begin{aligned} \int_{\partial\grave{B}}rb_1\,\mathrm{d}x\,=\,\int_{\partial\grave{B}}b_2\,\mathrm{d}x\,=\,0.\end{aligned}$$* *Let $\widecheck{\phi}\in \dot{H}^1$ and $\grave{\phi}\in \dot{H}_R^1$ be the solutions of $$\begin{aligned} &\Delta\widecheck{\phi}\,=\,0\text{ in $\mathbb{R}^2\backslash \grave{B}$}\label{def chek 1}\\ &\partial_n\widecheck{\phi}\,=\,b_2 \text{ on $\partial\grave{B}$}\label{def chek 2}\\ &\widecheck{\phi}\,\rightarrow\, 0 \text{ at $\infty$}\end{aligned}$$* *and $$\begin{aligned} &\operatorname{div}(r\nabla\grave{\phi})\,=\,0\text{ in $\mathbb{H}\backslash \grave{B}$}\\ &\partial_n\grave{\phi}\,=\,b_1 \text{ on $\partial\grave{B}$}\\ &\partial_n\grave{\phi}\,=\,0 \text{ on $\partial\mathbb{H}$}\\ &\grave{\phi}\,\rightarrow\, 0 \text{ at $\infty$}.\end{aligned}$$* *Then for all $m\in \mathbb{N}_{>0}$ it holds that $$\begin{aligned} &\left\lVert \nabla^m(\grave{\phi}-\widecheck{\phi}) \right\rVert_{L^2(\partial\grave{B})}\\ &\lesssim_m\, \sqrt{|\log\epsilon|}\left(\epsilon\left(\left\lVert b_2 \right\rVert_{H^{m-1}(\partial\grave{B})}+\left\lVert b_1 \right\rVert_{H^{m-1}(\partial\grave{B})}\right)+\left\lVert b_1-b_2 \right\rVert_{H^{m-1}(\partial\grave{B})}\right). \end{aligned}$$ The implicit constant in these estimates is bounded locally uniform in $q$.* **Remark 33**. - *Existence and uniqueness of $\grave{\phi}$ follows by Lemma [\[ExistencePhi\]](#ExistencePhi){reference-type="ref" reference="ExistencePhi"} and existence and uniqueness of $\widecheck{\phi}$ are shown in [@Amrouche]\[Thm. 3.1\].* - *The author strongly believes that the factor $\sqrt{|\log\epsilon|}$ is an artifact of the proof and can be removed by estimating $\left\lVert \Delta(\widecheck{\phi}-\grave{\phi}) \right\rVert_{L^2}$ instead of $\left\lVert \sqrt{r}\nabla(\widecheck{\phi}-\grave{\phi}) \right\rVert_{L^2}$ in the proof, which would require more effort.* **Corollary 34**. *[\[Potential1\]]{#Potential1 label="Potential1"} For all $m\in \mathbb{N}_{>0}$ it holds that $$\begin{aligned} \left\lVert \nabla^m(\phi_{t^*}-\widecheck{\phi}_{t^*}) \right\rVert_{L^2(\partial B)}\,\lesssim_m\, \epsilon^{\frac{5}{2}-m}\sqrt{|\log\epsilon|}|t^*|.\end{aligned}$$ These implicit constant is bounded locally uniformly in $q$.* *Proof of the Corollary.* Observe that if we set $b_1=\epsilon u(t^*)(\epsilon\cdot)$ and $b_2=\epsilon t^*\cdot n$, then it holds that $$\begin{aligned} \phi_{t^*}(\epsilon\cdot)\,=\,\grave{\phi}\text{ and } \widecheck{\phi}_{t^*}(\epsilon\cdot)\,=\,\widecheck{\phi},\end{aligned}$$ because these fulfill the same elliptic equation. One easily sees that $$\begin{aligned} \left\lVert b_1 \right\rVert_{H^m(\partial B)}\,\lesssim \, \epsilon|t^*|,\quad \left\lVert b_2 \right\rVert_{H^m(\partial B)}\,\lesssim \, \epsilon|t^*|,\quad \left\lVert b_1-b_2 \right\rVert_{H^m(\partial B)}\,\lesssim \, \epsilon^2|t^*|.\end{aligned}$$ The statement then follows from applying the proposition and rescaling. ◻ Our strategy to prove the proposition is to again apply $L^2$ estimates as in the previous section, as the coefficients are similar close to $\grave{B}$, together with decay estimates for the far away behavior. **Lemma 35**. *[\[basic est\]]{#basic est label="basic est"} Let $b_2$ and $\widecheck{\phi}$ be as in the Proposition, then for all $m\in \mathbb{N}_{\geq 0}$ it holds that* - *$$\begin{aligned} \left\lVert \nabla^m\widecheck{\phi} \right\rVert_{L^2(\partial\grave{B})}\,\lesssim_m\,\left\lVert b_2 \right\rVert_{{H}^{\max(0,m-1)}}.\end{aligned}$$* - *$$\begin{aligned} |\nabla^m\widecheck{\phi}|(x)\,\lesssim_m\, \frac{\left\lVert b_2 \right\rVert_{L^2}}{\mathop{\mathrm{dist}}(x,\grave{B})^{1+m}}.\end{aligned}$$* - *$$\begin{aligned} \left\lVert \widecheck{\phi} \right\rVert_{\dot{H}^m(\grave{B}+B_1(0)\backslash \grave{B})}\,\lesssim_m \, \left\lVert b_2 \right\rVert_{H^{\max(0,m-1)}}.\label{L2 est flat}\end{aligned}$$* *Here all the implicit constants are bounded locally uniformly in $q$.* *Proof.* All three statements are well-known, we sketch the proof here for the convenience of the reader. One can first repeat the argument in the proof of Lemma [\[decay1\]](#decay1){reference-type="ref" reference="decay1"} to show that $\phi$ and $\nabla\phi$ must decay like $|x|^{-1}$ resp. $|x|^{-2}$. Then by testing the PDE [\[def chek 1\]](#def chek 1){reference-type="eqref" reference="def chek 1"},[\[def chek 2\]](#def chek 2){reference-type="eqref" reference="def chek 2"} with $\widecheck{\phi}$ and partially integrating, we see that $$\begin{aligned} \int_{\partial\grave{B}}b_2\widecheck{\phi}\,\mathrm{d}x\,=\,-\left\lVert \widecheck{\phi} \right\rVert_{\dot{H}^1(\mathbb{R}^2\backslash \grave{B})}^2,\end{aligned}$$ where the partial integration is justified by the decay of $\phi$. Now $b_2$ is mean-free, so by using the trace in $\grave{B}+B_1(0)\backslash \grave{B}$, we see that $$\begin{aligned} \left\lVert \widecheck{\phi} \right\rVert_{\dot{H}^1(\grave{B}+B_1(0)\backslash \grave{B})}\,\lesssim\, \left\lVert b_2 \right\rVert_{L^2(\partial\grave{B})}.\end{aligned}$$ By using elliptic regularity estimates in $\grave{B}+B_1(0)\backslash \grave{B}$, we see that for $m>0$ we have $$\begin{aligned} \left\lVert \nabla^m\widecheck{\phi} \right\rVert_{L^2(\partial\grave{B})}\,\lesssim\, \left\lVert b_2 \right\rVert_{H^{m-1}(\partial\grave{B})}.\end{aligned}$$ This lets us control $\left\lVert \nabla \widecheck{\phi} \right\rVert_{L^2(\partial\grave{B})}$ and we can repeat the argument in the proof of Lemma [\[decay1\]](#decay1){reference-type="ref" reference="decay1"} to show b), where we get one order of decay less from using the two-dimensional Newtonian potential. c\) follows for $m>0$ similarly by using elliptic regularity. The estimates for $m=0$ in a) and c) follows by using the estimate b) for $\mathop{\mathrm{dist}}(x,\partial\grave{B})\geq 1$ and combining the estimate on the gradient in c) with e.g. the Poincare inequality. ◻ **Remark 36**. *In particular, by rescaling, we see that we have $$\begin{aligned} \left\lVert \partial_\tau\widecheck{\phi}_{t^*} \right\rVert_{L^2(\partial B)}\simeq |t^*|\epsilon^{\frac{1}{2}}\end{aligned}$$* *and the same estimate holds for $\phi_{i,t^*}$ by the Proposition.* **Lemma 37**. *[\[Lem:Decay1\]]{#Lem:Decay1 label="Lem:Decay1"} We have the following estimates:* - *$$\begin{aligned} \left\lVert \nabla^m\grave{\phi} \right\rVert_{L^2(\partial\grave{B})}\,\lesssim_m\, \left\lVert b_1 \right\rVert_{H^{\max(0,m-1)}}\end{aligned}$$ for all $m\geq 0$.* - *It holds that $$\begin{aligned} |\nabla^m\grave{\phi}|(x)\,\lesssim_m\, \min\left(\frac{\left\lVert b_1 \right\rVert_{H^{m}}}{1+\mathop{\mathrm{dist}}(x,\grave{B})^{1+m}},\,\frac{\left\lVert b_1 \right\rVert_{H^{m}}}{\epsilon(1+\mathop{\mathrm{dist}}(x,\grave{B})^{2+m})}\right),\label{ decay est b}\end{aligned}$$ for all $m\in \mathbb{N}_{\geq 0}$.* - *For $\mathop{\mathrm{dist}}(x,\grave{B})\geq 1$ and $m\in\mathbb{N}_{\geq 0}$ it holds that $$\begin{aligned} |\nabla^m\grave{\phi}|(x)\,\lesssim_m\, \min\left(\frac{\left\lVert b_1 \right\rVert_{L^2}}{\mathop{\mathrm{dist}}(x,\grave{B})^{1+m}},\,\frac{\left\lVert b_1 \right\rVert_{L^2}}{\epsilon\mathop{\mathrm{dist}}(x,\grave{B})^{2+m}}\right).\end{aligned}$$* *The implicit constant in these estimates is bounded locally uniformly in $R$.* *Proof.* a) follows from the same argument as the previous Lemma, where we again use the decay estimate from b) or c) for the case $m=0$. b\) and c) Our strategy is to quantify the argument of Lemma [\[decay1\]](#decay1){reference-type="ref" reference="decay1"}. If we extend $\grave{\phi}$ to $\grave{B}$ by solving the Dirichlet problem with boundary data $\grave{\phi}$, then by a) and elliptic regularity $$\begin{aligned} \left\lVert [\partial_n\grave{\phi}] \right\rVert_{\mathcal{M}(\partial\grave{B})}\,\lesssim\, \left\lVert b_1 \right\rVert_{L^2},\label{est jump 1}\end{aligned}$$ where $[\cdot]$ denotes the jump across the boundary. We can then proceed as in the proof of Lemma [\[decay1\]](#decay1){reference-type="ref" reference="decay1"} and set $\phi^{\mathbb{R}^3}(R,\theta,Z)=\grave{\phi}(R,Z)$, where $(R,\theta,Z)$ are axisymmetric coordinates in $\mathbb{R}^3$, then as argued there it holds that $$\begin{aligned} \phi^{\mathbb{R}^3}\,=\,\frac{-1}{4\pi|\cdot|}*[\partial_n\grave{\phi}]\mathcal{H}^2\mathbin{\vrule height 1.6ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.3ex}\partial\grave{B}^{\mathbb{R}^3}.\label{repSol}\end{aligned}$$ This gives the second estimate in c) for $m=0$, since $[\partial_n \grave{\phi}]$ is mean-free, where the factor $\frac{1}{\epsilon}$ comes from the fact that $\partial\grave{B}$ is of size $\approx\frac{1}{\epsilon}$. For the first estimate in c) for $m=0$ we may restrict ourselves to the case $1\leq \mathop{\mathrm{dist}}(x,\grave{B})\leq\frac{1}{\epsilon}$, as otherwise it follows from the second one. We use three-dimensional axisymmetric coordinates $(R,\theta,Z)$ again and fix some $x$. Then we split $\grave{B}^{\mathbb{R}^3}$ into parts $S,T_{-n},\dots T_n$ where we take $T_{-n}\dots T_n$ as the sets $$\begin{aligned} T_i\,:=\,\grave{B}^{\mathbb{R}^3}\cap\left\{(R',\theta',Z')|\theta'-\theta_x\in \left[\frac{\pi}{6n}\left(i-\frac{1}{2}\right),\frac{\pi}{6n}\left(i+\frac{1}{2}\right)\right)\right\},\end{aligned}$$ where $\theta_x$ denotes the azimuthal angle of $x$ and the difference is taken modulo $2\pi$. If we set $n=\lfloor\frac{1}{\epsilon}\rfloor$, then each such piece has diameter $\lesssim 1$. We set $S:=\grave{B}^{\mathbb{R}^3}\backslash\bigcup_i T_i$. By the estimate [\[est jump 1\]](#est jump 1){reference-type="eqref" reference="est jump 1"}, the mean-freeness and the fact that the distance between $x$ and $S$ is $\gtrsim \frac{1}{\epsilon}+\mathop{\mathrm{dist}}(x,\grave{B})$ we have $$\begin{aligned} \left|\left(\frac{1}{4\pi|\cdot|}*[\partial_n\grave{\phi}]\mathcal{H}^2\mathbin{\vrule height 1.6ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.3ex}\partial S\right)(x)\right|\,\lesssim\ \frac{\left\lVert b_1 \right\rVert_{L^2}}{\epsilon\left(\frac{1}{\epsilon^2}+\mathop{\mathrm{dist}}(x,\grave{B})^2\right)}.\end{aligned}$$ For every $i$ we have $$\begin{aligned} \left|\left(\frac{1}{4\pi|\cdot|}*[\partial_n\grave{\phi}]\mathcal{H}^2\mathbin{\vrule height 1.6ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.3ex}\partial T_i\right)(x)\right|\,\lesssim\,\frac{\left\lVert b_1 \right\rVert_{L^2}}{1+|i|^2+\mathop{\mathrm{dist}}(x,\hat{B})^2},\end{aligned}$$ where we exploited the facts that due to the rotational symmetry, the integral of the boundary values over each $T_i$ is $0$ and that $\mathop{\mathrm{dist}}(x,T_i)\gtrsim |i|+\mathop{\mathrm{dist}}(x,\grave{B})$. Summing up and using [\[repSol\]](#repSol){reference-type="eqref" reference="repSol"} gives $$\begin{aligned} |\grave{\phi}(x)|\,\lesssim\, \frac{\left\lVert b_1 \right\rVert_{L^2}}{\epsilon\left(\frac{1}{\epsilon^2}+\mathop{\mathrm{dist}}(x,\grave{B})^2\right)}+\sum_{|i|\leq\frac{1}{\epsilon}}\frac{\left\lVert b_1 \right\rVert_{L^2}}{1+|i|^2+\mathop{\mathrm{dist}}(x,\grave{B})^2}\,\lesssim\, \frac{\left\lVert b_1 \right\rVert_{L^2}}{\mathop{\mathrm{dist}}(x,\grave{B})},\end{aligned}$$ where we estimated the sum with the integral and used the assumption $\mathop{\mathrm{dist}}(x,\grave{B})\leq \frac{1}{\epsilon}$. This shows c) if $m=0$. For $m>0$ the estimates in c) follow from making the same argument with the derivatives of the fundamental solution. We set $$\begin{aligned} \label{DefinitionD} D\,:=\,(\grave{B}+B_1(0))\backslash \grave{B}.\end{aligned}$$ In $D$ the estimates in b) for $m>0$ follow from using elliptic regularity and a) to obtain that $$\begin{aligned} \left\lVert \nabla^m\grave{\phi} \right\rVert_{L^\infty(D)}\,\lesssim\, \left\lVert \nabla^m\grave{\phi} \right\rVert_{H^\frac{4}{3}(D)}\,\lesssim\, \left\lVert b_1 \right\rVert_{H^{m}}.\end{aligned}$$ The case $\mathop{\mathrm{dist}}(x,\grave{B})<1$ and $m=0$ follows from this by using the estimate for $\mathop{\mathrm{dist}}(x,\grave{B})=1$ and the fundamental theorem of calculus. ◻ *Proof of Proposition [\[PotentialGeneral\]](#PotentialGeneral){reference-type="ref" reference="PotentialGeneral"}.* By subtracting both PDEs and rearranging we obtain that $$\begin{aligned} \label{Elliptic1} &\int_{\mathbb{H}\backslash\grave{B}} r\langle \nabla\grave{\phi}-\nabla\widecheck{\phi},\nabla \xi \rangle+\left(r-\frac{R}{\epsilon}\right)\langle \nabla\widecheck{\phi}, \nabla\xi \rangle\,\mathrm{d}x\,=\,-\int_{\partial\grave{B}} \xi\left(rb_1-\frac{R}{\epsilon}b_2\right)\,\mathrm{d}x,\end{aligned}$$ for $\xi\in H^1$ compactly supported in $\mathbb{H}\backslash\grave{B}$.\ In order to be able to use both $\widecheck{\phi}$ and $\grave{\phi}$ as test functions for each others equation even though one is defined on the half-space and one on the full space, we introduce smooth cutoff functions $\eta_l$, supported in $(\mathbb{H}+\frac{R}{2\epsilon}e_R)\cap B_{l+1}(0)$, which equal $1$ in $(\mathbb{H}+(\frac{R}{2\epsilon}+1)e_R)\cap B_l(0)$ and whose derivatives have absolute value $\leq 2$ everywhere. By testing with $\eta_l(\grave{\phi}-\widecheck{\phi})$ we obtain that for large enough $l$ it holds that $$\begin{aligned} &\int_{\mathbb{H}\backslash\grave{B}} \eta_l r|\nabla\grave{\phi}-\nabla\widecheck{\phi}|^2+\eta_l \left(\frac{R}{\epsilon}-r\right)\nabla\widecheck{\phi}\cdot \nabla\left(\grave{\phi}-\widecheck{\phi}\right)+\nabla\eta_l\cdot\left(r\nabla\grave{\phi}-\frac{R}{\epsilon}\nabla\widecheck{\phi}\right)\left(\grave{\phi}-\widecheck{\phi}\right) \,\mathrm{d}x\\ &\,=\,-\int_{\partial\grave{B}} \left(\grave{\phi}-\widecheck{\phi}\right)\left(rb_1-\frac{R}{\epsilon}b_2\right)\,\mathrm{d}x. \end{aligned}$$ By rearranging and using the Cauchy-Schwarz inequality, we obtain the inequality $$\begin{aligned} &\left\lVert \sqrt{r\eta_l}\nabla\left(\grave{\phi}-\widecheck{\phi}\right) \right\rVert_{L^2}^2\,\leq\, \left\lVert \sqrt{\eta_l}\frac{|r-\frac{R}{\epsilon}|}{\sqrt{r}}\nabla\widecheck{\phi} \right\rVert_{L^2}\left\lVert \sqrt{r\eta_l}\nabla\left(\grave{\phi}-\widecheck{\phi}\right) \right\rVert_{L^2}+\\ &2\int_{\left([\frac{R}{2\epsilon},\frac{R}{2\epsilon}+1]\times \mathbb{R}\right)\cup B_{l+1}\left(0\right)\backslash B_l\left(0\right)}\left(\left|\grave{\phi}\right|+\left|\widecheck{\phi}\right|\right)\left(\left|r\nabla\grave{\phi}\right|+\left|\frac{R}{\epsilon}\nabla\widecheck{\phi}\right|\right)\,\mathrm{d}x\,+\\ &2R^{-\frac{1}{2}}\epsilon^\frac{1}{2}c_{trace}\left\lVert \sqrt{r\eta_l}\nabla\left(\grave{\phi}-\widecheck{\phi}\right) \right\rVert_{L^2}\left\lVert rb_1-\frac{R}{\epsilon}b_2 \right\rVert_{L^2\left(\partial\grave{B}\right)}\\ &=:\, (I+III)\cdot \left\lVert \sqrt{r\eta_l}\nabla\left(\grave{\phi}-\widecheck{\phi}\right) \right\rVert_{L^2}+II \end{aligned}$$ where $c_{trace}$ denotes the operator norm of the trace from $\dot{H}^{1}(\grave{B}+B_1(0)\backslash \grave{B})$ to $L^2(\partial\grave{B})/constants$, which is $\lesssim 1$ and we have estimated $r^{-\frac{1}{2}}$ with $2R^{-\frac{1}{2}}\epsilon^{\frac{1}{2}}$ in the last summand. Here $I,II$ and $III$ stand for the factors in the first, second, and third lines of the right-hand side.\ Using Lemma [\[basic est\]](#basic est){reference-type="ref" reference="basic est"}, we estimate the first term as $$\begin{aligned} \label{term I} &I\,\lesssim\, \left(\int_{\mathbb{H}\backslash\grave{B}}\eta_l\frac{1}{1+\mathop{\mathrm{dist}}(x,\grave{B})^4}\frac{(r-\frac{R}{\epsilon})^2}{r}\,\mathrm{d}x\right)^\frac{1}{2}\left\lVert b_2 \right\rVert_{L^2}\end{aligned}$$ We split into the regions $r\in [\frac{R}{2\epsilon},\frac{R}{\epsilon}-1]\cup [\frac{R}{\epsilon}+1,\frac{3R}{2\epsilon}],r\in [ \frac{R}{\epsilon}-1,\frac{R}{\epsilon}+1]$ and $r\geq \frac{3R}{2\epsilon}$, for other $r$ the integrand is $0$. This gives that [\[term I\]](#term I){reference-type="eqref" reference="term I"} is $$\begin{aligned} &\lesssim\,\left(\int_{[1,\frac{R}{2\epsilon}]\times\mathbb{R}}\frac{1}{|x|^4}\frac{|x_1|^2}{\frac{R}{\epsilon}}\,\mathrm{d}x+\int_{1}^\infty\frac{1}{|x|^4}{\frac{\epsilon}{R}}\,\mathrm{d}x+\int_{[\frac{R}{\epsilon},\infty)\times \mathbb{R}}\frac{1}{|x|^4}x_1\,\mathrm{d}x\right)^\frac{1}{2}\left\lVert b_2 \right\rVert_{L^2}\\ &\lesssim\, \epsilon^\frac{1}{2}\sqrt{|\log \epsilon|}\left\lVert b_2 \right\rVert_{L^2}.\end{aligned}$$ We use Lemma [\[Lem:Decay1\]](#Lem:Decay1){reference-type="ref" reference="Lem:Decay1"} and Lemma [\[basic est\]](#basic est){reference-type="ref" reference="basic est"} to obtain that for $l>>\frac{1}{\epsilon^2}$, we have $$\begin{aligned} &II\,\lesssim\, \frac{1}{\epsilon}\int_{\left([\frac{R}{2\epsilon},\frac{R}{2\epsilon}+1]\times \mathbb{R}\right)\cup B_{l+1}(0)\backslash B_l(0)}\frac{1}{\mathop{\mathrm{dist}}(x,\grave{B})}\frac{1}{\mathop{\mathrm{dist}}(x,\grave{B})^2}\,\mathrm{d}x\left\lVert b_1 \right\rVert_{L^2}\left\lVert b_2 \right\rVert_{L^2}\\ &\lesssim\, \frac{1}{\epsilon}\left(\int_{B_{l+1}(0)\backslash B_l(0)}\frac{1}{|x|^3}\,\mathrm{d}x+\int_\mathbb{R}\frac{1}{\left(|x|+\frac{1}{\epsilon}\right)^3}\,\mathrm{d}x\right)\left\lVert b_1 \right\rVert_{L^2}\left\lVert b_2 \right\rVert_{L^2}\\ &\lesssim\, \left(\frac{1}{l\epsilon}+\epsilon\right)\left\lVert b_1 \right\rVert_{L^2}\left\lVert b_2 \right\rVert_{L^2}\,\lesssim\,\epsilon\left\lVert b_1 \right\rVert_{L^2}\left\lVert b_2 \right\rVert_{L^2}.\end{aligned}$$ The third term can be estimated as $$\begin{aligned} III\,\lesssim\, \epsilon^\frac{1}{2}\left\lVert rb_1-\frac{R}{\epsilon}b_2 \right\rVert_{L^2(\partial\grave{B})}\,\lesssim\, \epsilon^\frac12\left(\left\lVert b_1 \right\rVert_{L^2(\partial\grave{B})}+\frac{1}{\epsilon}\left\lVert b_1-b_2 \right\rVert_{L^2(\partial\grave{B})}\right).\end{aligned}$$ Hence we obtain that $$\begin{aligned} \left\lVert \sqrt{r\eta_l}\nabla(\grave{\phi}-\widecheck{\phi}) \right\rVert_{L^2}^2\,\lesssim\,& \epsilon^\frac{1}{2}\sqrt{|\log\epsilon|}\left\lVert \sqrt{r\eta_l}\nabla(\grave{\phi}-\widecheck{\phi}) \right\rVert_{L^2}\Bigl(\left\lVert b_2 \right\rVert_{L^2(\partial\grave{B})}+\left\lVert b_1 \right\rVert_{L^2(\partial\grave{B})}+\\ &\frac{1}{\epsilon}\left\lVert b_1-b_2 \right\rVert_{L^2(\partial\grave{B})}\Bigr) +\epsilon\left\lVert b_1 \right\rVert_{L^2(\partial\grave{B})}\left\lVert b_2 \right\rVert_{L^2(\partial\grave{B})}. \end{aligned}$$ This implies that $$\begin{aligned} \left\lVert \sqrt{r\eta_l}\nabla(\grave{\phi}-\widecheck{\phi}) \right\rVert_{L^2}\,\lesssim\, &\epsilon^\frac{1}{2}\sqrt{|\log\epsilon|}\left(\left\lVert b_2 \right\rVert_{L^2(\partial\grave{B})}+\left\lVert b_1 \right\rVert_{L^2(\partial\grave{B})}+\frac{1}{\epsilon}\left\lVert b_1-b_2 \right\rVert_{L^2(\partial\grave{B})}\right).\label{L2 est} \end{aligned}$$ Now we can apply elliptic regularity estimates around $\partial\grave{B}$ to the elliptic equation [\[Elliptic1\]](#Elliptic1){reference-type="eqref" reference="Elliptic1"} to obtain that for $m>0$ it holds that $$\begin{aligned} &\left\lVert \nabla^m(\grave{\phi}-\widecheck{\phi}) \right\rVert_{L^2(\partial\grave{B})}\,\lesssim\, \left\lVert \frac{(r-\frac{R}{\epsilon})}{\frac{R}{\epsilon}}\nabla\widecheck{\phi} \right\rVert_{H^{m}(D)}+\epsilon\left\lVert rb_1-\frac{R}{\epsilon}b_2 \right\rVert_{H^{m-1}(\partial\grave{B})}+\left\lVert \nabla(\grave{\phi}-\widecheck{\phi}) \right\rVert_{L^2(D)}\\ &\lesssim\, \epsilon\sqrt{|\log\epsilon|}\left(\left\lVert b_2 \right\rVert_{H^{m-1}}+\left\lVert b_1 \right\rVert_{H^{m-1}}+\frac{1}{\epsilon}\left\lVert b_1-b_2 \right\rVert_{H^{m-1}}\right),\end{aligned}$$ where the neighborhood $D$ was defined in [\[DefinitionD\]](#DefinitionD){reference-type="eqref" reference="DefinitionD"}, for the first term we used that $\left\lVert \widecheck{\phi} \right\rVert_{H^m(D)}\lesssim \left\lVert b_2 \right\rVert_{H^{m-1}}$ , which follows from [\[L2 est flat\]](#L2 est flat){reference-type="eqref" reference="L2 est flat"} and in the last step we used the Estimate [\[L2 est\]](#L2 est){reference-type="eqref" reference="L2 est"} and the fact that $r\approx \epsilon^{-1}$. ◻ ### Multiple bodies We remind the reader of the convention of writing $\ell$ for irrelevant exponents. We will work in the rescaled setting and keep $\tilde{q}$ (defined in [\[def regime1\]](#def regime1){reference-type="eqref" reference="def regime1"} and [\[def regime2\]](#def regime2){reference-type="eqref" reference="def regime2"}) fixed independently of $\epsilon$. We set $\grave{B}_i=\frac{1}{\epsilon}B_i$. We will use the method of reflections to construct the potentials for multiple bodies from the potential of a single body. Fix some sufficiently smooth normal velocity $b_i$ on $\partial\grave{B}_i$ with $\int_{\partial\grave{B}_i} rb_i\,\mathrm{d}x=0$ and let $\grave{\phi}_i^1\in H_R^1(\mathbb{H}\backslash \grave{B}_i)$ solve $$\begin{aligned} &\operatorname{div}(r\nabla\grave{\phi}_i^1)\,=\,0\text { in $\mathbb{H}\backslash \grave{B}_i$}\\ &\partial_n \grave{\phi}_i^1\,=\,b_i\text{ on $\partial\grave{B}_i$}\\ &\partial_n\grave{\phi}_i^1\,=\,0 \text{ on $\partial\mathbb{H}$}\end{aligned}$$ and let $\grave{\phi}_i\in H_R^1(\mathbb{H}\backslash\bigcup_j\grave{B}_j)$ solve $$\begin{aligned} &\operatorname{div}(r\nabla\grave{\phi}_i)\,=\,0\text { in $\mathbb{H}\backslash \bigcup_j \grave{B}_j$}\\ &\partial_n \grave{\phi}_i\,=\,b_i\text{ on $\partial\grave{B}_i$}\\ &\partial_n \grave{\phi}_i\,=\,0 \text{ on $\partial\grave{B}_j$ for $j\neq i$}\\ &\partial_n \grave{\phi}_i\,=\,0 \text{ on $\partial\mathbb{H}$}.\end{aligned}$$ We then add corrector functions $\grave{\phi}_{1}^2,\grave{\phi}_{2}^2\dots$ to $\grave{\phi}_i^1$, which for $j\neq i$ fulfill the equations $$\begin{aligned} &\operatorname{div}(r\nabla\grave{\phi}_{j}^2)\,=\,0 \text{ in $\mathbb{H}\backslash \grave{B}_j$}\\ &\partial_n \grave{\phi}_{j}^2\,=\,-\partial_n \grave{\phi}_i^1 \text{ on $\partial\grave{B}_j$}\\ &\partial_n\grave{\phi}_j^2\,=\,0 \text{ on $\partial\mathbb{H}$}\\ & \grave{\phi}_j^2\in H_R^1.\end{aligned}$$ Existence and uniqueness of these follows from Lemma [\[ExistencePhi\]](#ExistencePhi){reference-type="ref" reference="ExistencePhi"}. We set $\grave{\phi}_i^2=0$. These correction terms then change the normal trace at all other $\partial B_l$. For this new error we can again construct corrector functions $\grave{\phi}_{1}^3,\grave{\phi}_{2}^3\dots$ with normal boundary values $-\sum_{l\neq j} \partial_n \grave{\phi}_l^2$ and so on. If the sums of the errors and the corrector functions converge, the limit will be $\grave{\phi}_i$, since it is unique. **Proposition 38**. *[\[MultiplePotentials\]]{#MultiplePotentials label="MultiplePotentials"} This iteration scheme converges for small enough $\epsilon$ to the solution $\grave{\phi}_i$ in both the regimes [\[def regime1\]](#def regime1){reference-type="eqref" reference="def regime1"} and [\[def regime2\]](#def regime2){reference-type="eqref" reference="def regime2"}, the convergence is in $H_R^1(\mathbb{R}^3)$. Furthermore, for all $m\in \mathbb{N}_{>0}$ we have the following estimates, if $\epsilon$ is small enough:* - *For $i\neq j$ it holds that $$\begin{aligned} \left\lVert \nabla^m\grave{\phi}_i \right\rVert_{L^2(\partial\grave{B}_j)}\,\lesssim_m\, \epsilon^{2}|\log\epsilon|^{\ell(1+m)}\left\lVert b_i \right\rVert_{L^2}\end{aligned}$$* - *For all $i$ it holds that $$\begin{aligned} \left\lVert \nabla^m(\grave{\phi}_i^1-\grave{\phi}_i) \right\rVert_{L^2(\partial\grave{B}_i)}\,\lesssim_m\,\epsilon^{4}|\log\epsilon|^{\ell(1+m)} \left\lVert b_i \right\rVert_{L^2}.\end{aligned}$$* *The implicit constant in these estimates and the requirements on $\epsilon$ are locally uniform in $\tilde{q}$ in both regimes [\[def regime1\]](#def regime1){reference-type="eqref" reference="def regime1"} and [\[def regime2\]](#def regime2){reference-type="eqref" reference="def regime2"}. In particular the estimates for the single body case in Proposition [\[PotentialGeneral\]](#PotentialGeneral){reference-type="ref" reference="PotentialGeneral"} and Lemma [\[Lem:Decay1\]](#Lem:Decay1){reference-type="ref" reference="Lem:Decay1"} a) are still true.* *Proof.* The rescaled bodies have pairwise distances $\gtrsim\frac{1}{\epsilon|\log\epsilon|}$ resp. $\gtrsim\frac{1}{\epsilon\sqrt{|\log\epsilon|}}$, hence by Lemma [\[Lem:Decay1\]](#Lem:Decay1){reference-type="ref" reference="Lem:Decay1"} c), for $i\neq j$ we have $$\begin{aligned} \left\lVert \nabla^m\grave{\phi}_i^1 \right\rVert_{L^{\infty}(\partial\grave{B}_j)}\,\lesssim_m\, \epsilon^{1+m}|\log\epsilon|^{\ell(1+m)} \left\lVert b_i \right\rVert_{L^2},\end{aligned}$$ for every $m\geq 0$. This lets us estimate the decay of the correctors $\grave{\phi}_j^2$ by the same Lemma, which can again be used to estimate the second-order correctors and so on. Iteratively, we obtain from Lemma [\[Lem:Decay1\]](#Lem:Decay1){reference-type="ref" reference="Lem:Decay1"} that $$\begin{aligned} \label{Normal corrector} \left\lVert \nabla^m\grave{\phi}_j^l \right\rVert_{L^2(\partial\grave{B}_j)}\,\lesssim_m\, C^l\epsilon^{2l-2}|\log\epsilon|^{\ell (l+m+1)} \left\lVert b_i \right\rVert_{L^2}\end{aligned}$$ here $C$ is a numerical factor, depending on $k$, but not on $l$ or $m$, coming from the implicit constant in Lemma [\[Lem:Decay1\]](#Lem:Decay1){reference-type="ref" reference="Lem:Decay1"} and the fact that we have to sum over $k$ correctors in each step. By integrating over the decay estimate in Lemma [\[Lem:Decay1\]](#Lem:Decay1){reference-type="ref" reference="Lem:Decay1"} we obtain that $$\begin{aligned} \label{L2 corrector} \left\lVert \grave{\phi}_j^l \right\rVert_{H_R^1(\mathbb{H}\backslash\bigcup_j \grave{B}_j)}\,\lesssim\, \frac{|\log\epsilon|}{\epsilon}\left\lVert \partial_n\grave{\phi}_j^l \right\rVert_{H^1(\partial\grave{B}_j)}.\end{aligned}$$ Therefore the scheme converges in $H_R^1$ if $\epsilon$ is small enough. By Lemma [\[Lem:Decay1\]](#Lem:Decay1){reference-type="ref" reference="Lem:Decay1"}, we obtain from [\[Normal corrector\]](#Normal corrector){reference-type="eqref" reference="Normal corrector"} that$$\begin{aligned} |\nabla^m \grave{\phi}_j^l|\, \lesssim_m\, C^l\epsilon^{2l+m-1}|\log\epsilon|^{\ell (m+l+1)}\left\lVert b_i \right\rVert_{L^2}\text{ on $\grave{B}_n$ with $n\neq j$}\end{aligned}$$ and for $l\geq2$ we have $$\begin{aligned} \left\lVert \nabla^m \grave{\phi}_j^l \right\rVert_{L^2(\partial\grave{B}_j)}\,\lesssim_m\, \left\lVert \partial_n \grave{\phi}_j^l \right\rVert_{H^{m-1}(\partial\grave{B}_j)}\,\lesssim_m\, C^l\epsilon^{2l-2}|\log\epsilon|^{\ell (m+l+1)}\left\lVert b_i \right\rVert_{L^2},\end{aligned}$$ where we made use of Lemma [\[Lem:Decay1\]](#Lem:Decay1){reference-type="ref" reference="Lem:Decay1"} a). Hence by summing up, we see that for $j\neq i$ and small enough $\epsilon$ we have $$\begin{aligned} &\left\lVert \nabla^m\grave{\phi}_i \right\rVert_{L^2(\partial\grave{B}_j)}\,\lesssim_m\, \epsilon^{2}|\log\epsilon|^{\ell(1+m)}\left\lVert b_i \right\rVert_{L^2} \\ &\left\lVert \nabla^m(\grave{\phi}_i-\phi_i^1) \right\rVert_{L^2(\partial\grave{B}_i)}\,\lesssim_m\, \epsilon^{4}|\log\epsilon|^{\ell(1+m)}\left\lVert b_i \right\rVert_{L^2}. \end{aligned}$$ After rescaling back to the original balls, we obtain the statement. ◻ **Corollary 39**. *[\[MultPotSpec\]]{#MultPotSpec label="MultPotSpec"} Fix some $t^*\in T_{q_i}M$ and let $\phi_{i,t^*}^1$ be the potential for $t^*$ if there is only a single body, let $\phi_{i,t^*}$ be the potential for $k$ bodies, in one of the two regimes [\[def regime1\]](#def regime1){reference-type="eqref" reference="def regime1"} or [\[def regime2\]](#def regime2){reference-type="eqref" reference="def regime2"}. Then we have the following bounds for all $m\in \mathbb{N}_{>0}$, with implicit constant bounded locally uniformly in $\tilde{q}$ in both regimes:* - *For $i\neq j$ it holds that $$\begin{aligned} \left\lVert \nabla^m\phi_{i,t^*} \right\rVert_{L^2(\partial B_j)}\,\lesssim\, \epsilon^{\frac{7}{2}-m}|\log\epsilon|^{\ell(1+m)}|t^*|.\end{aligned}$$* - *For all $i$ it holds that $$\begin{aligned} \left\lVert \nabla^m(\phi_{i,t^*}^1-\phi_{i,t^*}) \right\rVert_{L^2(\partial B_{i})}\,\lesssim\, \epsilon^{\frac{11}{2}-m}|\log\epsilon|^{\ell(1+m)}|t^*|.\end{aligned}$$* *All these estimates hold locally uniformly in $\tilde{q}$. In particular, this implies that the estimates from the single body case in Corollary [\[MultPotSpec\]](#MultPotSpec){reference-type="ref" reference="MultPotSpec"} still hold for multiple bodies.* *Proof.* This follows directly by rescaling the potentials as in the proof of Corollary [\[Potential1\]](#Potential1){reference-type="ref" reference="Potential1"} and using Proposition [\[MultiplePotentials\]](#MultiplePotentials){reference-type="ref" reference="MultiplePotentials"}. ◻ ### The derivative with respect to $q$ {#sec q deri} In the following, we want to obtain bounds on the derivative of the potential with respect to $q$. As we have already shown that these are smooth in $q$ in Lemma [\[SmoothnessInq\]](#SmoothnessInq){reference-type="ref" reference="SmoothnessInq"}, it is enough to estimate partial derivatives with respect to the $R_i$'s and $Z_i$'s. In order to compare boundary data at different instances of one $B_j$, we fix some $q$ and use the affine maps $$\begin{aligned} \label{def c} c_0(x,s)\,=\,\frac{\rho_j(R_j+s)}{\rho_j(R_j)}(x-q_j)+q_j+se_R\end{aligned}$$ and $$\begin{aligned} \label{def d} d_0(x,s)\,=\,x+se_Z\end{aligned}$$ where $\rho_j(\cdot)$ denotes $\rho_j$ as a function $R_j$. As we do not want to move the other bodies, we define $c$ and $d$ as smooth maps which equal $c_0$ and $d_0$ in a neighborhood of $B_j$ and which equal the identity in a neighborhood of the other bodies. For technical reasons, will assume that the neighborhood in which $c$ and $d$ equal the terms in [\[def c\]](#def c){reference-type="eqref" reference="def c"} resp. [\[def d\]](#def d){reference-type="eqref" reference="def d"} has a size $>>|\log\epsilon|^{-2}$, which is not restrictive. We also use the convention of writing $\phi_{i,t^*,R_j}$ resp. $\phi_{i,t^*,R_j+s}$ for $\phi_{i,t^*}$, defined for the position $(R_j,Z_j)$ resp. $(R_j+s,Z_j)$ and also use the same notation with $Z_j$ for the $Z_j$-derivative. **Proposition 40**. *[\[prop deri\]]{#prop deri label="prop deri"} Fix some $t^*\in T_{q_i}M$, then for all $j,l$ we have $$\begin{aligned} \left\lVert \partial_s \left(\left(\partial_{\tau}\phi_{i,t^*,R_j+s}\right)\circ c(\cdot,s)\right) \right\rVert_{L^2(\partial B_l(q))}\,\lesssim\, \epsilon^{\frac{3}{2}}|\log\epsilon|^\ell\end{aligned}$$ and $$\begin{aligned} \left\lVert \partial_s \left(\left(\partial_{\tau}\phi_{i,t^*,Z_j+s}\right)\circ d(\cdot,s)\right) \right\rVert_{L^2(\partial B_l(q))}\,\lesssim\, \epsilon^{\frac{3}{2}}|\log\epsilon|^\ell.\end{aligned}$$* *Here $\tau=n^\perp$ refers to the tangent both on $B_l(q)$ and on the translated body. The implicit constant is bounded locally uniformly in $\tilde{q}$ in both the regimes [\[def regime1\]](#def regime1){reference-type="eqref" reference="def regime1"} and [\[def regime2\]](#def regime2){reference-type="eqref" reference="def regime2"} in a neighborhood of $s=0$.* *Proof.* We assume $|t^*|=1$ and omit the indices $i$ and $t^*$. We first show the statement for the derivative in the $R$-direction and explain in the end why the $Z$-derivative works with the same argument. Without loss of generality, we may also assume that $Z_j=0$, as the system is invariant under translation. We rescale by a factor $\frac{1}{\epsilon}$ again and use the same notation as in the previous proofs. Clearly, we have $\partial_{R_j}\grave{\phi}=\partial_{R_j}\phi(\epsilon\cdot)$, in particular, the derivative exists and it holds that $$\begin{aligned} &\operatorname{div}(r\nabla\partial_{R_j}\grave{\phi})\,=\,0.\label{eq deri}\\ &\partial_n\partial_{R_j}\grave{\phi}\,=\,0 \text{ on $\partial\grave{B}_l$ for $l\neq j$ and on $\partial\mathbb{H}$}.\label{boundary deri 1}\end{aligned}$$ We first want to compute the remaining normal derivative. For this we introduce $$\begin{aligned} \grave{c}(x,s)\,=\,\frac{1}{\epsilon}c(\epsilon x,s).\end{aligned}$$ For $x\in \partial\grave{B}_j$ we have $$\begin{aligned}\label{s deri 1} &\partial_s\left(\partial_n\grave{\phi}_{R_j+s}(\grave{c}(x,s))\right)\big|_{s=0}\,=\,\epsilon\delta_{ij}\partial_s\left(t^*\cdot n+\frac{\epsilon}{2(R_j+s)}\sqrt{\frac{\tilde{\rho}_j^2R_j}{R_j+s}}t^*\cdot e_R\right)\bigg|_{s=0}\,=\,\\ &-\epsilon^2\delta_{ij}\frac{3\tilde{\rho}_j}{4R_j^2}t^*\cdot e_R. \end{aligned}$$ Here the expressions for the normal boundary values follow from Definition [\[normal velo\]](#normal velo){reference-type="eqref" reference="normal velo"} and Assumption [\[def rho\]](#def rho){reference-type="eqref" reference="def rho"} after rearranging. This is $\lesssim \epsilon^2$ in any $H^m$-norm. On the other hand, by the product rule, on $\partial\grave{B}_j$ we also have $$\begin{aligned}\label{s deri 2} &\partial_s\left(\partial_n\grave{\phi}_{R_j+s}(\grave{c}(x,s))\right)\big|_{s=0}\\ &\,=\,\partial_s n(\grave{c}(x,s))\big|_{s=0}\cdot\nabla\grave{\phi}_{R_j}(x)+n\cdot\partial_{s}\nabla\grave{\phi}_{R_j+s}(x)\big|_{s=0}+n\cdot \nabla^2\grave{\phi}_{R_j}(x)\partial_s\grave{c}(x,s)\big|_{s=0}. \end{aligned}$$ Here $n(\grave{c}(x,s))$ stands for the normal at $\grave{c}(x,s)$ of the rescaled body centered at $\frac{1}{\epsilon}(R_j+s)$, which is in fact constant, and hence the first summand drops out. We have by definition $$\begin{aligned} \partial_s\grave{c}(x,s)\big|_{s=0}\,=\,\frac{1}{\epsilon}e_R-\frac{1}{2R_j}(x-q_j).\end{aligned}$$ Combining [\[s deri 1\]](#s deri 1){reference-type="eqref" reference="s deri 1"} and [\[s deri 2\]](#s deri 2){reference-type="eqref" reference="s deri 2"}, we see that on $\partial\grave{B}_j$ we have $$\begin{aligned} \partial_n\partial_{s}\grave{\phi}_{R_j+s}\big|_{s=0}\,=\,- \frac{1}{\epsilon}\partial_n\partial_r\grave{\phi}_{R_j}+\frac{\tilde{\rho}_j}{2R_j}\partial_n^2\grave{\phi}_{R_j}-\epsilon^2\delta_{ij}\frac{3\tilde{\rho}_j}{4R_j^2}t^*\cdot e_R.\label{boundary deri 2}\end{aligned}$$ We now estimate the derivatives of the boundary values on $\partial\grave{B}_j$ and on $\partial\grave{B}_l$ for $l\neq j$ differently.\ 1. Case: $l\neq j$. Here $\grave{c}$ is the identity on $\partial\grave{B}_l$ and hence we only need to estimate $\nabla\partial_s\grave{\phi}_{R_j+s}$. By rescaling we know from Corollary [\[MultPotSpec\]](#MultPotSpec){reference-type="ref" reference="MultPotSpec"} and Lemma [\[Lem:Decay1\]](#Lem:Decay1){reference-type="ref" reference="Lem:Decay1"} a) that $$\begin{aligned} \left\lVert \nabla^m\grave{\phi} \right\rVert_{L^2(\partial\grave{B}_j)}\,\lesssim_m\, \epsilon.\end{aligned}$$ By [\[boundary deri 2\]](#boundary deri 2){reference-type="eqref" reference="boundary deri 2"} it follows that $$\begin{aligned} \left\lVert \partial_n\partial_{s}\grave{\phi}_{R_j+s}\big|_{s=0} \right\rVert_{L^2(\partial\grave{B}_j)}\,\lesssim\, 1.\label{bd boundary deri}\end{aligned}$$ Hence we conclude from [\[eq deri\]](#eq deri){reference-type="eqref" reference="eq deri"},[\[boundary deri 1\]](#boundary deri 1){reference-type="eqref" reference="boundary deri 1"},[\[bd boundary deri\]](#bd boundary deri){reference-type="eqref" reference="bd boundary deri"} and by Proposition [\[MultiplePotentials\]](#MultiplePotentials){reference-type="ref" reference="MultiplePotentials"} that for $m\in \mathbb{N}_{>0}$ it holds$$\begin{aligned} \left\lVert \nabla^m\partial_s\grave{\phi}_{R_j+s}\big|_{s=0} \right\rVert_{L^2(\partial\grave{B}_l)}\,\lesssim_m\, \epsilon^{2}|\log\epsilon|^{\ell(m+1)}.\end{aligned}$$ By rescaling back to the original $B$'s, one obtains the statement.\ 2. Case $l=j$.\ We build an auxiliary function that equals the desired derivative. Consider $\partial_r\grave{\phi}$. It fulfills the PDE $$\begin{aligned} \operatorname{div}(r\nabla\partial_r\grave{\phi})\,=\,\operatorname{div}\left(-\nabla\grave{\phi}+\frac{r\epsilon}{R_j}\nabla\grave{\phi}\right) \label{r eq}\end{aligned}$$ and has the Neumann boundary values $\partial_n\partial_r\grave{\phi}$ on $\partial\grave{B}_j$. Consider the function $x\cdot\nabla\grave{\phi}$, a direct calculation shows that $$\begin{aligned} &\operatorname{div}(r\nabla(x\cdot\nabla\grave{\phi}))\,=\,0\label{n eq}\\ &\partial_n(x\cdot\nabla\grave{\phi})\,=\,\partial_n\grave{\phi}+\frac{R_j}{\epsilon}\partial_n\partial_r\grave{\phi}+\tilde{\rho}_j\partial_n^2\grave{\phi} \text{ on $\partial\grave{B}_j$}.\end{aligned}$$ Here we used the assumption that $Z_j=0$. Now consider the function $$\begin{aligned} \label{def tilde} \tilde{\phi}\,:=\,\partial_{s}\grave{\phi}_{R_j+s}\big|_{s=0}-\frac{1}{2R_j}x\cdot\nabla\grave{\phi}+\frac{1}{2R_j}\grave{\phi}+\frac{3}{2\epsilon}\partial_r\grave{\phi}.\end{aligned}$$ and set $$\begin{aligned} \hat{\phi}_s(x)\,=\,\sqrt{\frac{R_j+s}{R_j}}\grave{\phi}_{R_j+s}(\grave{c}(s,x)).\end{aligned}$$ Then in a neighborhood of $\grave{B}_j$ it holds that $\partial_s\hat{\phi}_s|_{s=0}=\tilde{\phi}$ and hence $$\begin{aligned} \partial_s\left((\nabla\grave{\phi}_{R_j+s})(\grave{c}(x,s)\right)\big|_{s=0}\,=\,\partial_s\nabla\hat{\phi}_s(x)\big|_{s=0} \,=\,\nabla\tilde{\phi}(x).\label{rewritting phi}\end{aligned}$$ Hence to prove the proposition it suffices to estimate $\nabla\tilde{\phi}$.\ Let us further reduce this to the case in which $k=1$. First consider the case where $i\neq j=l$. Then by [\[boundary deri 2\]](#boundary deri 2){reference-type="eqref" reference="boundary deri 2"} and by rescaling the estimates on $\phi$ in Corollary [\[MultPotSpec\]](#MultPotSpec){reference-type="ref" reference="MultPotSpec"} we have $$\begin{aligned} \left\lVert \partial_n\partial_s\grave{\phi}_{R_j+s}\big|_{s=0} \right\rVert_{H^m(\partial\grave{B}_j)}\,\lesssim\, \frac{1}{\epsilon}\left\lVert \nabla^2\grave{\phi} \right\rVert_{H^{m}(\partial\grave{B}_j)}\,\lesssim_m\,\epsilon^{2}|\log\epsilon|^\ell. \end{aligned}$$ Hence we conclude from Proposition [\[MultiplePotentials\]](#MultiplePotentials){reference-type="ref" reference="MultiplePotentials"} and [\[eq deri\]](#eq deri){reference-type="eqref" reference="eq deri"} and [\[boundary deri 1\]](#boundary deri 1){reference-type="eqref" reference="boundary deri 1"} that $$\begin{aligned} \left\lVert \partial_s\grave{\phi}_{R_j+s}\big|_{s=0} \right\rVert_{H^m(\partial\grave{B}_j)}\,\lesssim_m\, \epsilon^{2}|\log\epsilon|^\ell.\end{aligned}$$ Hence by the definition of $\tilde{\phi}$ (see [\[def tilde\]](#def tilde){reference-type="eqref" reference="def tilde"}) and Corollary [\[MultPotSpec\]](#MultPotSpec){reference-type="ref" reference="MultPotSpec"} we have $$\begin{aligned} \left\lVert \nabla\tilde{\phi} \right\rVert_{H^m(\partial\grave{B}_j)}\,\lesssim_m\, \left\lVert \partial_s\grave{\phi}_{R_j+s}\big|_{s=0} \right\rVert_{H^m(\partial\grave{B}_j)}+\frac{1}{\epsilon}\left\lVert \nabla^2\grave{\phi} \right\rVert_{H^m(\partial\grave{B}_j)}+\left\lVert \nabla\grave{\phi} \right\rVert_{H^m(\partial\grave{B}_j)}\,\lesssim_m\,\epsilon^{2}|\log\epsilon|^\ell.\end{aligned}$$ After rescaling back to the original $B$'s, this shows the statement in the case $i\neq j=l$.\ Next, consider the case $i=j=l$ and $k>1$, and let $\grave{\phi}^1$ be the rescaled potential for a single body and let $\tilde{\phi}^1$ be the version of $\tilde{\phi}$ from the single body case. As the formulas [\[boundary deri 1\]](#boundary deri 1){reference-type="eqref" reference="boundary deri 1"} and [\[boundary deri 2\]](#boundary deri 2){reference-type="eqref" reference="boundary deri 2"} hold for both $\grave{\phi}$ and $\grave{\phi}^1$, we note that we have $$\begin{aligned} \left\lVert \partial_n\partial_{s}(\grave{\phi}_{R_j+s}-\grave{\phi}_{R_j+s}^1)\big|_{s=0} \right\rVert_{H^m(\partial\grave{B}_j)}\,\lesssim\, \frac{1}{\epsilon}\left\lVert \nabla^2(\grave{\phi}-\grave{\phi}^1) \right\rVert_{H^m(\partial\grave{B}_j)}\,\lesssim_m \, \epsilon^{4}|\log\epsilon|^\ell,\end{aligned}$$ here we used Proposition [\[MultiplePotentials\]](#MultiplePotentials){reference-type="ref" reference="MultiplePotentials"} b). Hence we conclude that $$\begin{aligned} \left\lVert \nabla^m\partial_{s}(\grave{\phi}_{R_j+s}-\grave{\phi}_{R_j+s}^1)\big|_{s=0} \right\rVert_{L^2(\partial\grave{B}_j)}\,\lesssim_m\,\epsilon^{4}|\log\epsilon|^\ell.\end{aligned}$$ By rescaling Corollary [\[MultPotSpec\]](#MultPotSpec){reference-type="ref" reference="MultPotSpec"} b) and the definition of $\tilde{\phi}$ we conclude that $$\begin{aligned} &\left\lVert \nabla(\tilde{\phi}-\tilde{\phi}^1) \right\rVert_{H^m(\partial\grave{B}_j)}\\ &\lesssim_m\, \left\lVert \nabla^m\partial_{s}(\grave{\phi}_{R_j+s}-\grave{\phi}_{R_j+s}^1)\big|_{s=0} \right\rVert_{L^2(\partial\grave{B}_j)}+ \frac{1}{\epsilon}\left\lVert \nabla^2(\grave{\phi}-\grave{\phi}^1) \right\rVert_{H^m(\partial\grave{B}_j)}+\left\lVert \nabla(\grave{\phi}-\grave{\phi}^1) \right\rVert_{H^m(\partial\grave{B}_j)}\\ &\lesssim_m\, \epsilon^{4}|\log\epsilon|^\ell.\end{aligned}$$ This shows the desired estimate once we have proven the case $k=1$ after rescaling back to the original bodies.\ It remains to deal with the case $k=1$. We drop the indices on the $\grave{B}$'s. By [\[boundary deri 2\]](#boundary deri 2){reference-type="eqref" reference="boundary deri 2"},[\[r eq\]](#r eq){reference-type="eqref" reference="r eq"}-[\[def tilde\]](#def tilde){reference-type="eqref" reference="def tilde"} we have $$\begin{aligned} &\operatorname{div}(r\tilde{\phi})\,=\,-\frac{3}{2}\operatorname{div}\left(\left(\frac{1}{\epsilon}-\frac{r}{R_j}\right)\nabla\grave{\phi}\right)\text{ in a neighborhood of $\grave{B}$}\label{tilde eq 1}\\ &\partial_n\tilde{\phi}\,=\,-\epsilon^2\frac{3\tilde{\rho}_j}{4R_j^2}t^*\cdot e_R\label{tilde eq 2}.\end{aligned}$$ Set $\tilde{B}:=B_{\frac{1}{\epsilon|\log\epsilon|^2}}(0)+\grave{B}\backslash\grave{B}$. By assumption the cutoff in the definition of $\grave{c}$ happens outside of $\tilde{B}$ and hence [\[tilde eq 1\]](#tilde eq 1){reference-type="eqref" reference="tilde eq 1"} holds in all of $\tilde{B}$. The crucial observation is that $\partial_{R_j}\grave{\phi}$ will decay an order faster than expected. **Lemma 41**. *[\[Decay deri\]]{#Decay deri label="Decay deri"} For $x$ with $\mathop{\mathrm{dist}}(x,\grave{B})\geq 1$ it holds that $$\begin{aligned} |\partial_s\phi_{R_j+s}(x)|\,\lesssim\, \frac{\epsilon+\left\lVert \nabla\tilde{\phi} \right\rVert_{L^2(\tilde{B})}}{\epsilon\mathop{\mathrm{dist}}(x,\grave{B})^2}+\frac{1}{\epsilon\mathop{\mathrm{dist}}(x,\grave{B})^3}.\end{aligned}$$* *and $$\begin{aligned} |\nabla\partial_s\tilde{\phi}_{R_j+s}(x)|\,\lesssim\, \frac{\epsilon+\left\lVert \nabla\tilde{\phi} \right\rVert_{L^2(\tilde{B})}}{\epsilon\mathop{\mathrm{dist}}(x,\grave{B})^3}+\frac{1}{\epsilon\mathop{\mathrm{dist}}(x,\grave{B})^4}.\end{aligned}$$ The implicit constant is bounded locally uniformly in $\tilde{q}$.* *Proof.* We only show the first estimate, the second works completely similarly, using the derivative of the fundamental solution. The proof builds on the idea of the proof of Lemma [\[decay1\]](#decay1){reference-type="ref" reference="decay1"}. Recall that in the proof there, we extended the solution to the full space $\mathbb{R}^3$ and used the fundamental solution. We first need to show an estimate on the Neumann boundary values of the solution to the interior problem. As in the proof of Lemma [\[decay1\]](#decay1){reference-type="ref" reference="decay1"}, we extend $\grave{\phi}$ to $\mathbb{H}$ by solving the Dirichlet problem for $\operatorname{div}(r\nabla\cdot)$ in $\grave{B}$. We also set $$\begin{aligned} \hat{\phi}_s(x)\,:=\,\sqrt{\frac{R_j+s}{R_j}}\grave{\phi}_{R_j+s}(\grave{c}(x,s))\end{aligned}$$ inside of $\grave{B}$. We claim that $$\begin{aligned} \left\lVert \partial_s((\partial_{n_{int}}\grave{\phi}_{R_j+s})\circ c) \right\rVert_{L^2(\partial\grave{B})}\,\lesssim\, \left\lVert \tilde{\phi} \right\rVert_{L^2(\tilde{B})}+\epsilon^2,\label{inner bd}\end{aligned}$$ here $\partial_{n_{int}}$ denotes the normal derivative from the inside.\ By applying elliptic regularity estimates to the equations [\[tilde eq 1\]](#tilde eq 1){reference-type="eqref" reference="tilde eq 1"}-[\[tilde eq 2\]](#tilde eq 2){reference-type="eqref" reference="tilde eq 2"} and using the identity $\partial_s\hat{\phi}_s=\tilde{\phi}$, we see that $$\begin{aligned} &\left\lVert \nabla\partial_s\hat{\phi}_s\big|_{s=0} \right\rVert_{L^2(\partial\grave{B})}\,=\,\left\lVert \nabla\tilde{\phi} \right\rVert_{L^2(\partial\grave{B})}\\ &\lesssim\, \left\lVert \nabla\tilde{\phi} \right\rVert_{L^2(\tilde{B})}+\left\lVert \partial_n\tilde{\phi} \right\rVert_{L^2(\partial\grave{B})}+\epsilon\left\lVert \left(\frac{1}{\epsilon}-\frac{r}{R_j}\right)\nabla\grave{\phi} \right\rVert_{H^1(\grave{B}+B_1(0)\backslash\grave{B})}\,\lesssim\,\epsilon^2+\left\lVert \nabla\tilde{\phi} \right\rVert_{L^2(\tilde{B})}.\label{deri dir bd} \end{aligned}$$ In $\grave{B}$ we have $$\begin{aligned} \operatorname{div}\left(\frac{\grave{c}(x,s)_R}{R_j+s}\nabla\hat{\phi}_s\right)\,=\,0,\end{aligned}$$ as one can see from a direct computation, where $\grave{c}(x,s)_R$ denotes the $R$-component of $\grave{c}$. Using for instance the implicit function theorem as in the proof of Lemma [\[SmoothnessInq\]](#SmoothnessInq){reference-type="ref" reference="SmoothnessInq"}, one can justify that this equation can be differentiated in $s$. Hence we obtain that $\partial_s\hat{\phi}_s(x)$ fulfills the equation $$\begin{aligned} &\operatorname{div}\left(r\nabla\partial_s\hat{\phi}_s\right)\,=\,-\operatorname{div}\left(\partial_s\frac{\grave{c}(x,s)_R}{R_j+s}\nabla\grave{\phi}\right).\end{aligned}$$ Hence by elliptic regularity for this equation and [\[deri dir bd\]](#deri dir bd){reference-type="eqref" reference="deri dir bd"} we obtain that $$\begin{aligned} &\left\lVert \partial_{n_{int}}\partial_s\hat{\phi}_s \right\rVert_{L^2(\partial\grave{B})}\,\lesssim\, \left\lVert \nabla\partial_s\hat{\phi}_s \right\rVert_{L^2(\partial\grave{B})}+\epsilon\left\lVert \partial_s\frac{\grave{c}(x,s)_R}{R_j+s}\nabla\grave{\phi} \right\rVert_{H^1(\grave{B})}\\ &\lesssim\, \epsilon^2+\left\lVert \nabla\tilde{\phi} \right\rVert_{L^2(\tilde{B})}+\epsilon\left\lVert \partial_n\grave{\phi} \right\rVert_{H^1(\partial\grave{B})}\,\lesssim\,\epsilon^2+\left\lVert \nabla\tilde{\phi} \right\rVert_{L^2(\tilde{B})}. \end{aligned}$$ The Claim [\[inner bd\]](#inner bd){reference-type="eqref" reference="inner bd"} now follows from the fact that $\nabla\hat{\phi}_s=\left(\nabla\grave{\phi}_{R_j+s}\right)\circ\grave{c}(\cdot,s)$. Now fix some $y\in \tilde{B}$. Then as argued in the proof of Lemma [\[decay1\]](#decay1){reference-type="ref" reference="decay1"} we have $$\begin{aligned} \label{deri conv} \partial_{s}\grave{\phi}_{R_j+s}(y)\,=\,\partial_s\left([\partial_n\grave{\phi}_{R_j+s}]\mathcal{H}^2\mathbin{\vrule height 1.6ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.3ex}\partial\grave{B}(R_j+s)^{\mathbb{R}^3}*\frac{-1}{4\pi|\cdot|}\right)(y),\end{aligned}$$ where the convolution is taken with respect to three-dimensional coordinates and $\grave{B}(R_j+s)^{\mathbb{R}^3}$ is the axisymmetric torus corresponding to $\grave{B}$ defined with respect to $R_j+s$ and the factor $r$ in the jump disappears due to the coordinate change. We shall also view the curves $\grave{c}(x,s)$ as curves in $\mathbb{R}^3$, by setting (in axisymmetric coordinates) $$\begin{aligned} \grave{c}(x,s)\,=\,(\grave{c}(x,s)_R,x_\theta,\grave{c}(x,s)_Z),\end{aligned}$$ where $x_\theta$ is the azimuthal angle of $x$. Using these curves, the convolution in [\[deri conv\]](#deri conv){reference-type="eqref" reference="deri conv"} can be written as $$\begin{aligned} -\partial_s\int_{\partial\grave{B}^{\mathbb{R}^3}}[\partial_n\grave{\phi}_{R_j+s}](\grave{c}(x,s))\frac{\grave{c}(x,s)_R}{r}\sqrt{\frac{R_j}{R_j+s}}\frac{1}{4\pi|y-\grave{c}(x,s)|}\,\mathrm{d}x,\end{aligned}$$ here the factor in the middle is the determinant due to the coordinate change. Now we can differentiate under the integral, as all derivatives are smooth by Lemma [\[SmoothnessInq\]](#SmoothnessInq){reference-type="ref" reference="SmoothnessInq"} and use the product rule. We estimate the derivative of the first three factors and of the last factor separately. First, we deal with $$\begin{aligned} -\int_{\partial\grave{B}^{\mathbb{R}^3}}\partial_s\left([\partial_n\grave{\phi}_{R_j+s}](\grave{c}(x,s))\frac{\grave{c}(x,s)_R}{r}\sqrt{\frac{R_j}{R_j+s}}\right)\bigg|_{s=0}\frac{1}{4\pi|x-y|}\,\mathrm{d}x.\end{aligned}$$ Observe that $$\begin{aligned} &\int_{\partial\grave{B}^{\mathbb{R}^3}}\partial_s\left([\partial_n\grave{\phi}_{R_j+s}](\grave{c}(x,s))\frac{\grave{c}(x,s)_R}{r}\sqrt{\frac{R_j}{R_j+s}}\right)\bigg|_{s=0}\,\mathrm{d}x\\ &=\, \partial_s \int_{\partial\grave{B}(R_j+s)^{\mathbb{R}^3}}[\partial_n\grave{\phi}_{R_j+s}]\,\mathrm{d}x\,=\,0. \end{aligned}$$ Furthermore, we have by [\[inner bd\]](#inner bd){reference-type="eqref" reference="inner bd"} and [\[s deri 1\]](#s deri 1){reference-type="eqref" reference="s deri 1"} $$\begin{aligned} \left\lVert \partial_s\left([\partial_n\grave{\phi}_{R_j+s}](\grave{c}(x,s))\right)\big|_{s=0} \right\rVert_{L^2(\partial\grave{B})}\,\lesssim\, \epsilon^2+\left\lVert \nabla\tilde{\phi} \right\rVert_{L^2(\tilde{B})}\end{aligned}$$ and $$\begin{aligned} \left|\partial_s\frac{\grave{c}(x,s)_R}{r}\sqrt{\frac{R_j}{R_j+s}}\right|\,\lesssim\, 1.\end{aligned}$$ Since the boundary values of $\grave{\phi}$ are $\lesssim \epsilon$ in any $H^m$-norm we have $$\begin{aligned} \left\lVert [\partial_n\grave{\phi}] \right\rVert_{L^2(\partial\grave{B})}\,\lesssim\, \left\lVert \partial_n\grave{\phi} \right\rVert_{L^2(\partial\grave{B})}+\left\lVert \grave{\phi} \right\rVert_{H^1(\partial\grave{B})}\,\lesssim\, \epsilon.\label{est jump}\end{aligned}$$ Hence we have $$\begin{aligned} \left\lVert \partial_s\left([\partial_n\grave{\phi}_{R_j+s}](\grave{c}(x,s))\frac{\grave{c}(x,s)_R}{r}\sqrt{\frac{R_j}{R_j+s}}\right)\bigg|_{s=0} \right\rVert_{L^2(\partial\grave{B})}\,\lesssim\, \epsilon+\left\lVert \nabla\tilde{\phi} \right\rVert_{L^2(\partial\grave{B})}.\end{aligned}$$ Now we combine this with the mean-freeness to estimate $$\begin{aligned} \left|\int_{\partial\grave{B}^{\mathbb{R}^3}}\partial_s\left([\partial_n\grave{\phi}_{R_j+s}](\grave{c}(x,s))\frac{\grave{c}(x,s)_R}{r}\sqrt{\frac{R_j}{R_j+s}}\right)\bigg|_{s=0}\frac{1}{4\pi|x-y|}\,\mathrm{d}x\right|\,\lesssim\, \frac{\epsilon+\left\lVert \nabla\tilde{\phi} \right\rVert_{L^2(\partial\grave{B})}}{\epsilon\mathop{\mathrm{dist}}(y,\grave{B})^2}.\end{aligned}$$ It remains to estimate the second summand, which is $$\begin{aligned} \int_{\partial\grave{B}^{\mathbb{R}^3}}[\partial_n\grave{\phi}]\nabla_x\frac{1}{4\pi|y-x|}\cdot \partial_s\grave{c}(x,0)\,\mathrm{d}x.\end{aligned}$$ Note that $[\partial_n\grave{\phi}]$ is mean-free and one easily sees that $|\partial_s\grave{c}(x,0)|\lesssim \frac{1}{\epsilon}$ and $|\nabla\partial_s\grave{c}(x,0)|\lesssim 1$. Hence by exploiting the mean-freeness of $[\partial_n\grave{\phi}]$ and using [\[est jump\]](#est jump){reference-type="eqref" reference="est jump"}, we see that $$\begin{aligned} &\left|\int_{\partial\grave{B}^{\mathbb{R}^3}}[\partial_n\grave{\phi}]\nabla_x\frac{1}{4\pi|y-x|}\cdot \partial_s\grave{c}(x,0)\,\mathrm{d}x\right|\,\lesssim\, \frac{\left\lVert \partial_s\grave{c}(0) \right\rVert_{\sup}}{\mathop{\mathrm{dist}}(y,\grave{B})^3}+\frac{\left\lVert \nabla_x\partial_s\grave{c}(0) \right\rVert_{\sup}}{\mathop{\mathrm{dist}}(y,\grave{B})^2}\\ &\lesssim\, \frac{1}{\epsilon\mathop{\mathrm{dist}}(y,\grave{B})^3}+\frac{1}{\mathop{\mathrm{dist}}(y,\grave{B})^2}.\end{aligned}$$ The statement follows because $\mathop{\mathrm{dist}}(y,\grave{B})<<\epsilon^{-1}$. ◻ We continue with the proof of Proposition [\[prop deri\]](#prop deri){reference-type="ref" reference="prop deri"}. By testing the elliptic equation [\[tilde eq 1\]](#tilde eq 1){reference-type="eqref" reference="tilde eq 1"},[\[tilde eq 2\]](#tilde eq 2){reference-type="eqref" reference="tilde eq 2"} with $\tilde{\phi}$ in $\tilde{B}$ and using partial integration and the Cauchy-Schwarz inequality, we obtain that $$\begin{aligned} \label{est tilde} &\left\lVert \sqrt{r}\nabla\tilde{\phi} \right\rVert_{L^2(\tilde{B})}^2\,\leq\, \frac{3}{2}\left\lVert \frac{\left(\frac{1}{\epsilon}-\frac{r}{R_j}\right)}{\sqrt{r}}\nabla\grave{\phi} \right\rVert_{L^2(\tilde{B})}\left\lVert \sqrt{r}\nabla\tilde{\phi} \right\rVert_{L^2(\tilde{B})}\\ &-\int_{\partial\grave{B}}\left(\frac{3}{2}\left(\frac{1}{\epsilon}-\frac{r}{R_j}\right)\partial_n\grave{\phi}+r\partial_n\tilde{\phi}\right)\tilde{\phi}\,\mathrm{d}x+\int_{\partial\tilde{B}\backslash\partial\grave{B}}\left(\frac{3}{2}\left(\frac{1}{\epsilon}-\frac{r}{R_j}\right)\partial_n\grave{\phi}+r\partial_n\tilde{\phi}\right)\tilde{\phi}\,\mathrm{d}x\notag\\ &:=\,I+II+III,\end{aligned}$$ where the normal $n$ on $\partial\tilde{B}\backslash\partial\grave{B}$ is taken as the outer normal and $I$, $II$ and $III$ are defined in the obvious way. Using the decay estimate from Lemma [\[Lem:Decay1\]](#Lem:Decay1){reference-type="ref" reference="Lem:Decay1"}, we can estimate $$\begin{aligned} &I\,\lesssim\, \epsilon\left(\int_{B_\frac{1}{\epsilon|\log\epsilon|^2}\backslash B_1(0)}\frac{|x_1|^2}{(|x_1|+\frac{1}{\epsilon})|x|^4}\,\mathrm{d}x\right)^\frac{1}{2}\left\lVert \sqrt{r}\nabla\tilde{\phi} \right\rVert_{L^2(\tilde{B})}\\ &\lesssim\, \epsilon\left(\int_{1}^{\frac{1}{\epsilon|\log\epsilon|^2}}\frac{1}{x(x+\frac{1}{\epsilon})}\,\mathrm{d}x\right)^\frac{1}{2}\left\lVert \sqrt{r}\nabla\tilde{\phi} \right\rVert_{L^2(\tilde{B})}\,\lesssim\, \epsilon^\frac{3}{2}|\log\epsilon|^\ell\left\lVert \sqrt{r}\nabla\tilde{\phi} \right\rVert_{L^2(\tilde{B})}.\end{aligned}$$ To estimate the second term, we note that by partial integration we have $$\begin{aligned} &\int_{\partial\grave{B}}\left(\frac{3}{2}\left(\frac{1}{\epsilon}-\frac{r}{R_j}\right)\partial_n\grave{\phi}+r\partial_n\tilde{\phi}\right)\,\mathrm{d}x\\ &\,=\,\lim_{R\rightarrow \infty}\int_{\partial\left(\mathbb{H}\cap B_R(0)\right)}\left(\frac{3}{2}\left(\frac{1}{\epsilon}-\frac{r}{R_j}\right)\partial_n\grave{\phi}+r\partial_n\tilde{\phi}\right)\,\mathrm{d}x\,=\,0. \end{aligned}$$ Here the last equality follows from the fact that the integrand is $0$ on $\partial\mathbb{H}$ and by using the decay estimates from Lemma [\[Lem:Decay1\]](#Lem:Decay1){reference-type="ref" reference="Lem:Decay1"} and Lemma [\[Decay deri\]](#Decay deri){reference-type="ref" reference="Decay deri"} together with the definition of $\tilde{\phi}$. We can then use this mean-freeness and the explicit form of the normal derivatives to estimate $$\begin{aligned} II\,\lesssim\, \left\lVert \frac{3}{2}\left(\frac{1}{\epsilon}-\frac{r}{R_j}\right)\partial_n\grave{\phi}+r\partial_n\tilde{\phi} \right\rVert_{L^2(\partial\grave{B})}\left\lVert \nabla\tilde{\phi} \right\rVert_{L^2(\tilde{B})}\,\lesssim\, \epsilon^{\frac{3}{2}}\left\lVert \sqrt{r}\nabla\tilde{\phi} \right\rVert_{L^2(\tilde{B})}.\end{aligned}$$ To estimate the third term, we use Lemma [\[Lem:Decay1\]](#Lem:Decay1){reference-type="ref" reference="Lem:Decay1"} and Lemma [\[Decay deri\]](#Decay deri){reference-type="ref" reference="Decay deri"} and obtain that $$\begin{aligned} &III\,\lesssim\,\frac{1}{\epsilon|\log\epsilon|^2}\sup_{x\in \partial\tilde{B}\backslash\partial\grave{B}}|\tilde{\phi}(x)|\left(\frac{1}{\epsilon}|\nabla\tilde{\phi}(x)|+\frac{1}{\epsilon}|\nabla\grave{\phi}(x)|\right)\\ &\lesssim\,\frac{1}{\epsilon} \left(\epsilon^2+\epsilon\left\lVert \nabla\tilde{\phi} \right\rVert_{L^2(\tilde{B})}\right)\left(\epsilon^2+\epsilon\left\lVert \nabla\tilde{\phi} \right\rVert_{L^2(\tilde{B})}\right)|\log\epsilon|^\ell\\ &\lesssim\, \epsilon^3|\log\epsilon|^\ell+\epsilon|\log\epsilon|^\ell\left\lVert \nabla\tilde{\phi} \right\rVert_{L^2(\tilde{B})}.\end{aligned}$$ Putting these back into the estimate [\[est tilde\]](#est tilde){reference-type="eqref" reference="est tilde"}, we obtain $$\begin{aligned} \left\lVert \sqrt{r}\nabla\tilde{\phi} \right\rVert_{L^2(\tilde{B})}\,\lesssim\, \epsilon^{\frac{3}{2}}|\log\epsilon|^\ell.\end{aligned}$$ Now we can apply elliptic regularity estimates to [\[tilde eq 1\]](#tilde eq 1){reference-type="eqref" reference="tilde eq 1"},[\[tilde eq 2\]](#tilde eq 2){reference-type="eqref" reference="tilde eq 2"} and obtain together with the previous estimates on $\grave{\phi}$ that $$\begin{aligned} \left\lVert \nabla\tilde{\phi} \right\rVert_{H^m(\partial\grave{B})}\,\lesssim_m\, \epsilon^2|\log\epsilon|^\ell.\end{aligned}$$ This finishes the proof of the estimate for the $R_j$-derivative. The $Z$-derivative proceeds in the same way, except that one uses $$\begin{aligned} \tilde{\phi}\,=\,\partial_s\grave{\phi}_{Z_j+s}+\frac{1}{\epsilon}\partial_Z\grave{\phi}; \quad \hat{\phi}_s\,=\,\grave{\phi}_{Z_j+s}(\grave{d}(s,x)).\end{aligned}$$ ◻ ## The Christoffel Symbol and the added Inertia {#subsec3.3} **Lemma 42**. *[\[conv Inertia\]]{#conv Inertia label="conv Inertia"} If we identify the tangent space $T_qM$ with $\mathbb{R}^{2k}$, then it holds that $$\begin{aligned} \left|\mathcal{M}-\pi\begin{pmatrix}R_1\epsilon^2\tilde{\rho}_1^2 &0 &\dots\\0& R_1\epsilon^2\tilde{\rho}_1^2& \dots\\0& 0 & R_2\epsilon^2\tilde{\rho}_2 &0\\ \dots\end{pmatrix}\right|\,\lesssim\, \epsilon^3|\log\epsilon|^\ell,\end{aligned}$$* *where $\mathcal{M}$ was defined in [\[main def\]](#main def){reference-type="ref" reference="main def"}.* *Here the implicit constant is bounded locally uniformly in $\tilde{q}$.* *Proof.* By partial integration we have for $t^*\in T_{q_i}M$ and $s^*\in T_{q_j}M$ that $$\begin{aligned} (\mathcal{M}t^*)\cdot s^*\,=\,-\int_{\partial B_j}r \partial_n\phi_{i,t^*}\phi_{j,s^*}\,\mathrm{d}x.\end{aligned}$$ If $i\neq j$ this is $\lesssim \epsilon^3|\log\epsilon|^\ell|s^*||t^*|$ by Corollary [\[MultPotSpec\]](#MultPotSpec){reference-type="ref" reference="MultPotSpec"} a). If $i=j$, then it holds that $$\begin{aligned} &\left|\int_{\partial B_i} r\partial_n\phi_{i,t^*}\phi_{j,s^*}\,\mathrm{d}x-R_i\int_{\partial B_i}t^*\cdot n\widecheck{\phi}_{s^*}\,\mathrm{d}x\right|\\ &\lesssim\, \left\lVert ru(t^*) \right\rVert_{L^2}\left\lVert \phi_{i,s^*}-\widecheck{\phi}_{s^*} \right\rVert_{L^2(\partial B_i)/constants}+\left\lVert ru(t^*)-R_it^*\cdot n \right\rVert_{L^2}\left\lVert \widecheck{\phi}_{s^*} \right\rVert_{L^2(\partial B_i)}, \end{aligned}$$ where we used the fact that $ru(t^*)$ is mean-free and $\widecheck{\phi}$ was defined in [\[def flat func\]](#def flat func){reference-type="ref" reference="def flat func"}. Now by the Corollaries [\[Potential1\]](#Potential1){reference-type="ref" reference="Potential1"} and [\[MultPotSpec\]](#MultPotSpec){reference-type="ref" reference="MultPotSpec"} and the definition of $u(t^*)$ this is $\lesssim \epsilon^3|\log\epsilon|^\ell|s^*||t^*|$. Now observe that if $t^*=e_1$ and $s^*=e_2$ then $t^*\cdot n$ is orthogonal to $\widecheck{\phi}_{s^*}$ as the former is symmetric with respect to the $e_2$-direction while the latter is antisymmetric in that direction. If $t^*=s^*$ then it follows from the explicit form of $\widecheck{\phi}_{s^*}$ that $$\begin{aligned} \int_{\partial B_i}t^*\cdot n\widecheck{\phi}_{s^*}\,\mathrm{d}x\,=\,-\pi|t^*|^2\rho_i^2.\end{aligned}$$ ◻ **Lemma 43**. *[\[conv Gamma\]]{#conv Gamma label="conv Gamma"} It holds that $$\begin{aligned} |\Gamma|\,\lesssim\,\epsilon^3|\log\epsilon|^\ell,\end{aligned}$$* *where the Christoffel symbol $\Gamma$ was defined in [\[main def\]](#main def){reference-type="ref" reference="main def"} and the implicit constant is bounded locally uniformly in $\tilde{q}$.* *Proof.* By the definition of $\Gamma$, it suffices to estimate the derivative of $\mathcal{M}$. If we are differentiating $\mathcal{M}$ with respect to $q_l$ for $l\neq i$, then it holds that $$\begin{aligned} \left|-\partial_{q_l}\int_{\partial B_i}r\partial_n\phi_{i,t^*}\phi_{j,t^*}\,\mathrm{d}x\right|\,\lesssim\, \epsilon\left\lVert ru(t^*) \right\rVert_{L^2}\left\lVert \partial_{q_l}\partial_\tau\phi_{j,s^*} \right\rVert_{L^2}\,\lesssim\,\epsilon^3|\log\epsilon|^\ell|t^*||s^*|,\end{aligned}$$ where we used that $u(t^*)$ does not depend on $q_l$, and used the mean-freeness of $ru(t^*)$ to estimate $\partial_{q_l}\phi_{j,s^*}$ with its derivative, and in the last step we used Proposition [\[prop deri\]](#prop deri){reference-type="ref" reference="prop deri"}. If $i=l$ we can switch the roles of $i$ and $j$ unless $i=j=l$. For simplicity we only consider the derivative with respect to $R_i$ in this case, the other derivative is easier. We again use the diffeomorphism $c$, defined at the beginning of Subsection [3.2.3](#sec q deri){reference-type="ref" reference="sec q deri"}. Setting $\mathcal{M}_{R_i+s}$ for $\mathcal{M}$ defined with respect to $R_i+s$ we have $$\begin{aligned} \label{inertia fixed} (\mathcal{M}_{R_i+s}t^*)\cdot s^*\,=\,- \int_{\partial B_i}c(x,s)_R\frac{\rho_i(R_i+s)}{\rho_i(R_i)}u(t^*)\circ c\phi_{i,s^*,R_i+s}\circ c\,\mathrm{d}x,\end{aligned}$$ where $\rho_i(R_i)$ refers to $\rho_i$ as a function of $R_i$ and $c(x,s)_R$ is the $R$-component of $c$. Using the fundamental theorem of calculus and Proposition [\[prop deri\]](#prop deri){reference-type="ref" reference="prop deri"}, we see that $$\begin{aligned} \left\lVert \partial_s \left(\frac{\rho_i(R_i)}{\rho_i(R_i+s)}\phi_{i,s^*,R_i+s}\circ c\right) \right\rVert_{L^2/constants}\,\lesssim\, \epsilon^\frac{5}{2}|\log\epsilon|^\ell|s^*|.\label{M deri 2}\end{aligned}$$ Furthermore, we have $$\begin{aligned} \left\lVert \partial_su(t^*)\circ c \right\rVert_{L^2}\,\lesssim\, \epsilon^\frac{3}{2},\label{M deri 1}\end{aligned}$$ as one sees by rescaling [\[s deri 1\]](#s deri 1){reference-type="eqref" reference="s deri 1"}. It is easy to see that $$\begin{aligned} \left|\partial_s\left( c(x,s)_R\frac{\rho_i(R_i+s)^2}{\rho_i(R_i)^2}\right)\right|\,\lesssim\, \epsilon,\label{M deri 3}\end{aligned}$$ uniformly in $x$. We can now use this to estimate the derivative of [\[inertia fixed\]](#inertia fixed){reference-type="eqref" reference="inertia fixed"} by the product rule: $$\begin{aligned} &\left|\partial_s \int_{\partial B_i}c(x,s)_R\frac{\rho_i(R_i+s)}{\rho_i(R_i)}u(t^*)\circ c\phi_{i,s^*,R_i+s}\circ c\,\mathrm{d}x\bigg|_{s=0}\right|\\ &\leq\, \left\lVert ru(t^*) \right\rVert_{L^2}\left\lVert \partial_s \left(\frac{\rho_i(R_i)}{\rho_i(R_i+s)}\phi_{i,s^*,R_i+s}\circ c\right)\bigg|_{s=0} \right\rVert_{L^2/constants}\\ &\:\:\:+\left\lVert \partial_s \left(c(x,s)_R\frac{\rho_i(R_i+s)^2}{\rho_i(R_i)^2}\right) \right\rVert_{L^\infty}\left\lVert u(t^*) \right\rVert_{L^2}\left\lVert \phi_{i,s^*} \right\rVert_{L^2(\partial B_i)}\\ &\:\:\:+\left\lVert r\phi_{i,s^*} \right\rVert_{L^2(\partial B_i)}\left\lVert \partial_s(u(t^*)\circ c)\big|_{s=0} \right\rVert_{L^2}\,\lesssim\, \epsilon^3|\log\epsilon|^\ell. \end{aligned}$$ Here we used [\[M deri 1\]](#M deri 1){reference-type="eqref" reference="M deri 1"},[\[M deri 2\]](#M deri 2){reference-type="eqref" reference="M deri 2"},[\[M deri 3\]](#M deri 3){reference-type="eqref" reference="M deri 3"} and used Lemma [\[Lem:Decay1\]](#Lem:Decay1){reference-type="ref" reference="Lem:Decay1"} a) (after rescaling) to estimate $\left\lVert \phi_{i,s^*} \right\rVert_{L^2(\partial B_i)}\leq \epsilon^{\frac{3}{2}}$. ◻ # The stream function {#Section4} In the section, we want to compute the asymptotic of $G$ and $A$ and their derivative with respect to $q$, which requires us to compute the asymptotic of the streamfunction. The streamfunction will, up to lower order terms, resemble the asymptotic of the Biot-Savart law [\[asymptotic velocity\]](#asymptotic velocity){reference-type="eqref" reference="asymptotic velocity"}. Plugged in the definition of $G$, the leading order term of the stream function gives $0$, so for a direct computation, one would need a higher order expansion of the stream function. As we need to compute a derivative of the streamfunction anyway we take the alternative approach of expressing $G$ as the derivative of the energy of the streamfunction, which gives the asymptotic of $G$ just from the highest order term of the stream function at the expense of requiring an estimate on the second derivative, which is not much more complicated than just computing the first derivative. Unlike for the potential function, the interaction between the different bodies will matter and we will obtain an interaction term in $G$, which can be computed from the highest order parts alone. The computation of $A$ on the other hand will be more straightforward. Another difficulty is that the limiting object $\ln|x|$ does not lie in $H^1$, so we can not expect $L^2$-based estimates to work for the streamfunction. As we are only interested in the boundary data anyway, we will directly characterize it in terms of an integral equation on the boundary.\ In this section, we will make massive use of the fundamental solution $K$ of the operator $\operatorname{div}(\frac{1}{r}\nabla\cdot)$, as introduced in [\[definition K\]](#definition K){reference-type="eqref" reference="definition K"}. By an abuse of notation, we will also denote the linear operator $$\begin{aligned} f\rightarrow \int_{\bigcup_i\partial B_i}K(x,y)f(x)\,\mathrm{d}x\end{aligned}$$ by $K$. Similarly, we will write $$\begin{aligned} \overline{K}_R(x,y)=\frac{R}{2\pi}(\log(|x-y|)-\log(8)+2-\log(R))\end{aligned}$$ and also write $\overline{K}_R$ for the associated integral operator. Recall that in Lemma [\[rep psi\]](#rep psi){reference-type="ref" reference="rep psi"}, we showed that: The function $\frac{1}{r}\partial_n \psi_i$ is a solution of the system $$\begin{aligned} &\int_{\bigcup_j\partial B_j}K(x,y)\mu(x)\,\mathrm{d}x\text{ is constant on each $B_j$}\\ &\int_{\partial B_j}\mu(x)\,\mathrm{d}x\,=\,\delta_{ij}.\end{aligned}$$ Our goal is to show that for a single body $\frac{1}{r}\partial_n\psi$ converges to a constant by showing that the kernel $K$ converges to $\overline{K}_R$ (for which the solution of the analogous system is constant). For multiple bodies we will show that the "cross-terms" in $K$ are an order lower and that the corresponding lower order terms in $\frac{1}{r}\partial_n\psi_i$ are essentially given through the derivatives of $K$ itself. Recall that the kernel $K$ can be written as $$\begin{aligned} \label{exp K} K(x,y)\,=\,\frac{-1}{2\pi}\sqrt{x_Ry_R}F\left(\frac{|x-y|^2}{x_Ry_R}\right),\end{aligned}$$ where $x_R$ and $y_R$ stand for the $R$-component and $$\begin{aligned} F(s)\,=\,\int_0^\pi \frac{\cos(t)}{\sqrt{2(1-\cos(t))+s}}\mathrm{d}t\end{aligned}$$ (see [@GallaySverak]\[Section 2\]). This integral cannot be elementarily evaluated, however it has a series expansion at $0$, which we will make use of: **Lemma 44**. *[\[Series F\]]{#Series F label="Series F"} For small enough $s>0$ there is an expansion $$\begin{aligned} F(s)=-\frac{1}{2}\log(s)+\log(8)-2+\sum_{j\geq 1} a_js^j+b_js^j\log(s).\label{exp F}\end{aligned}$$* *This series has a positive radius of convergence, in particular, we also have the corresponding asymptotic for the derivatives of $F$.* *Proof.* The statement is known, see e.g. [@Sverak]\[Footnote 101\] and the computation of the explicit terms can be found there. We provide the proof of the convergence of the expansion here as we were not able to find it in the literature. By elementary manipulations, one sees that $$\begin{aligned} F(s)\,=\,(1+\frac{s}{2})\int_0^\frac{\pi}{2}\frac{1}{\sqrt{1+\frac{s}{4}-\sin^2(t)}}\mathrm{d}t-2\int_0^\frac{\pi}{2}\sqrt{1+\frac{s}{4}-\sin^2(t)}\mathrm{d}t.\end{aligned}$$ These integrals can be rewritten by using complete elliptic integrals of the first and second kind [@byrd2013handbook]. These are defined as $$\begin{aligned} &K_{elliptic}(m)=\int_0^\frac{\pi}{2}(1-m^2\sin^2(t))^{-\frac{1}{2}}\mathrm{d}t,\\ &E_{elliptic}(m)=\int_0^\frac{\pi}{2}(1-m^2\sin^2(t))^{\frac{1}{2}}\mathrm{d}t.\end{aligned}$$ With this it holds that $$\begin{aligned} F(s)\,=\,(1+\frac{s}{2})\frac{1}{\sqrt{1+\frac{s}{4}}}K_{elliptic}(\sqrt{1-\frac{s}{4+s}})-2\sqrt{1+\frac{s}{4}}E_{elliptic}(\sqrt{1-\frac{s}{4+s}}).\end{aligned}$$ It suffices to show that the functions $K_{elliptic}(\sqrt{1-t^2})$ and $E_{elliptic}(\sqrt{1-t^2})$ have an expansion of the type $$\begin{aligned} \sum_{j\geq 0}c_jt^{2j}+d_jt^{2j}\log (t)\label{log analytic}\end{aligned}$$ for small enough $t$, because close to $s=0$ the functions $\frac{1}{\sqrt{1+\frac{s}{4}}}$; $\sqrt{1+\frac{s}{4}}$ and $\sqrt{\frac{s}{4+s}}$ are analytic and one easily sees that the class of functions with an expansion of the type [\[log analytic\]](#log analytic){reference-type="eqref" reference="log analytic"} with a positive radius of convergence are stable under composition and multiplication with analytic functions. It is known [@schwarz1893formeln]\[Page 53\] that there is an expansion $$\begin{aligned} K_{elliptic}(\sqrt{1-t^2})\,=\,\log\left(\frac{4}{t}\right)-2\left(\sum_{j\geq 1}\frac{1}{2j(2j-1)}\sum_{l=j}^\infty \frac{(2l)!}{2^{2l}(l!)^2}t^{2l}\right)\label{expansion K}\end{aligned}$$ By the facts that $\sum_j\frac{1}{2j(2j-1)}<\infty$ and $\frac{(2l)!}{2^{2l}(l!)^2}<1$, we see that this converges for $|t|<1$. By using the definition and elementary calculations, one can see that $$\begin{aligned} E_{elliptic}(m)\,=\,m(1-m^2)K_{elliptic}'(m)+mK_{elliptic}(m).\end{aligned}$$ This implies $$\begin{aligned} E_{elliptic}(\sqrt{1-t^2})\,=\,t^2\sqrt{1-t^2}(K_{elliptic}(\sqrt{1-t^2}))'\frac{\sqrt{1-t^2}}{t}+\sqrt{1-t^2}K_{elliptic}(\sqrt{1-t^2})\label{identity E}\end{aligned}$$ One then obtains the desired expansion by combining [\[expansion K\]](#expansion K){reference-type="eqref" reference="expansion K"}, [\[identity E\]](#identity E){reference-type="eqref" reference="identity E"} and using a binomial series for the prefactor $\sqrt{1-t^2}$. ◻ We set $$\begin{aligned} &h(s)\,:=\,F(s)+\frac{1}{2}\log(s)-\log 8+2\\ &g(x,y)\,:=\,- h\left(\frac{|x-y|^2}{x_Ry_R}\right)\end{aligned}$$ for the remainder. For all $n$ and small enough $|x-y|$ it holds that $$\begin{aligned} \label{est g} |\nabla^ng(x,y)|\,\lesssim_n\, |x-y|^{2-n}|\log|x-y||,\end{aligned}$$ locally uniformly in $x_R$ and $y_R$ by Lemma [\[Series F\]](#Series F){reference-type="ref" reference="Series F"} above. ### The case of a single body In this subsection we drop the index $i$. **Lemma 45**. *[\[K bd\]]{#K bd label="K bd"}* - *The linear map $K$ is invertible from $L_0^2(\partial B)$ to $\dot{H}^1(\partial B)$ with operator norm $\lesssim 1$ for small enough $\epsilon$.* - *We have $$\begin{aligned} \left\lVert K \right\rVert_{L^2(\partial B)\rightarrow L^2(\partial B)}\,\lesssim\, \epsilon|\log\epsilon|.\end{aligned}$$* - *We have that $$\begin{aligned} \left\lVert K-\overline{K}_R \right\rVert_{L^2(\partial B)\rightarrow H^1(\partial B)}\,\lesssim\, \epsilon|\log\epsilon|.\end{aligned}$$ and $$\begin{aligned} \left\lVert K-\overline{K}_R \right\rVert_{L^2(\partial B)\rightarrow L^2(\partial B)}\,\lesssim\, \epsilon^2|\log\epsilon|.\end{aligned}$$* *All these estimates are locally uniform in $q$.* *Proof.* a) and b) Observe that for a) it is enough to show c) and to show that $\overline{K}_R$ is invertible with operator norm $\lesssim 1$, as one sees e.g. by using a geometric series. Similarly, for b) it is enough to show that $$\begin{aligned} \left\lVert \overline{K}_R \right\rVert_{L^2(\partial B)\rightarrow L^2(\partial B)}\,\lesssim\, \epsilon|\log\epsilon|.\end{aligned}$$ Let $\theta\in\mathbb{T}:= [0,1)$ parametrize $\partial B$, then we claim that the kernel $\overline{K}_R$ acts as $$\begin{aligned}\label{action K flat} e^{2\pi\theta in}&\,\rightarrow\, -\frac{R}{2|n|}\epsilon\tilde{\rho} e^{2\pi\theta in} \text{ for $n\neq 0$}\\ 1&\,\rightarrow\, -R\epsilon\tilde{\rho}(-\log(\epsilon\tilde{\rho})+\log(8)-2+\log(R)), \end{aligned}$$ which clearly has the desired operator norm and is invertible with norm $\lesssim 1$ from $L^2(\partial B)$ to $H^1(\partial B)$, as one loses the factor $\epsilon$ again due to the derivative. To show the Claim [\[action K flat\]](#action K flat){reference-type="eqref" reference="action K flat"}, we first observe that the constants $\frac{1}{2\pi}(\log(8)-2+\log(R))$ act as multiplication with $\epsilon\tilde{\rho}(\log(8)-2+\log(R))$ on constants and maps other frequencies to $0$. The Claim [\[action K flat\]](#action K flat){reference-type="eqref" reference="action K flat"} then follows from the Lemma below. **Lemma 46**. *[\[action log\]]{#action log label="action log"} The kernel $\log|x-y|$ acts as the Fourier multiplier $$\begin{aligned} &1\rightarrow 2\pi\epsilon\tilde{\rho}\log(\epsilon\tilde{\rho})\\ &e^{2\pi in\theta}\rightarrow -\frac{\epsilon\tilde{\rho}\pi}{|n|}e^{2\pi in\theta},\end{aligned}$$* *here $\theta\in \mathbb{T}=[0,1)$ is a constant speed parametrization of the boundary $\partial B$.* *Proof.* Note that when parametrizing the boundary with $\theta\in\mathbb{T}$, the action of the kernel corresponds to convolution with $$\begin{aligned} 2\pi\epsilon\tilde{\rho}\left(\log(\epsilon\tilde{\rho})+\log\left|1-e^{2\pi i\theta}\right|\right).\end{aligned}$$ We have that $$\begin{aligned} \log|1-e^{2\pi i\theta}|\,=\,\operatorname{Re}\log1-e^{2\pi i\theta}\end{aligned}$$ and this can be approximated in $L^2$ by $\operatorname{Re}\log(1-(1-\delta)e^{2\pi i\theta})$ for $\delta\searrow 0$ by e.g. dominated convergence. Now it holds that $$\begin{aligned} \operatorname{Re}\log(1-(1-\delta)e^{2\pi i\theta})\,=\,-\operatorname{Re}\sum_{j=1}^\infty\frac{(1-\delta)^j}{j}e^{2\pi ij\theta}\,=\,-\sum_{j=1}^\infty\frac{(1-\delta)^j}{j}\cos\left(2\pi j\theta\right),\end{aligned}$$ where we used the Taylor series of the logarithm around $1$. By Plancherel, we can take the limit $\delta\searrow 0$ in this Fourier series and obtain the statement by the wellknown formula $\widehat{f_1*f_2}=\widehat{f_2}\widehat{f}_2$. ◻ Proof of part c) of Lemma [\[K bd\]](#K bd){reference-type="ref" reference="K bd"}: It suffices to show that the kernels $K-\overline{K}_R$ and $\partial_y(K-\overline{K}_R)$ are bounded on $L^2(\partial B)$. We can write $$\begin{aligned}\label{Exp diff} &2\pi(K-\overline{K}_R)(x,y)\,=\,-(\sqrt{x_Ry_R}-R)(-\log(|x-y|)+\log(8)-2+\log(R)) \\ &-\frac{1}{2}\sqrt{x_Ry_R}\log\left(\frac{x_Ry_R}{R^2}\right)+\sqrt{x_Ry_R}g(x,y), \end{aligned}$$ where we used the Expansion [\[exp F\]](#exp F){reference-type="eqref" reference="exp F"} and the definition of $\overline{K}_R$. It is easy to see that $$\begin{aligned} \left|\sqrt{x_Ry_R}-R\right|\,\lesssim\, \epsilon\quad\text{and}\quad \left|\log\frac{x_Ry_R}{R^2}\right|\,\lesssim\, \epsilon,\end{aligned}$$ hence one obtains $L^2$-boundedness from Schur's Lemma (cf. [@Grafakos]\[Apprendix A.1\]). Taking a $y$-derivative in [\[Exp diff\]](#Exp diff){reference-type="eqref" reference="Exp diff"}, we get $$\begin{aligned} &2\pi\partial_y (K-\overline{K}_R)(x,y)\,=\,-\partial_y (\sqrt{x_Ry_R}-R) \left(-\log(|x-y|)+\log(8)-2+\log(R)\right)\\ &+(\sqrt{x_Ry_R}-R)\partial_y\log(|x-y|)-\frac{1}{2}\partial_y\left(\sqrt{x_Ry_R}\log \frac{x_Ry_R}{R^2}\right)+O(|x-y||\log|x-y|). \end{aligned}$$ Clearly, the $O$-term is bounded from $L^2$ to $L^2$ and of the desired order. It is easy to check that $$\begin{aligned} \left|\partial_y (\sqrt{x_Ry_R}-R)\right|\,\lesssim\, 1\quad\text{and}\quad\left|\partial_y\sqrt{x_Ry_R}\log \frac{x_Ry_R}{R^2}\right|\,\lesssim\, 1.\end{aligned}$$ This shows boundedness of all terms by Schur's Lemma except for $$\begin{aligned} (\sqrt{x_Ry_R}-R)\partial_y\log|x-y|\end{aligned}$$ by direct estimates. For this we use the Lemma below to conclude. ◻ **Lemma 47**. *[\[perturbed log\]]{#perturbed log label="perturbed log"} Let $j\in C^1(\partial B\times \partial B)$, then for all $f\in L^2(\partial B)$ it holds that $$\begin{aligned} \left\lVert \int_{\partial B} j(x,y)f(x)\partial_y\log|x-y|\,\mathrm{d}x \right\rVert_{L_y^2(\partial B)}\,\lesssim\, \left(\left\lVert j \right\rVert_{\sup}+\epsilon\left\lVert j \right\rVert_{C^1}\right)\left\lVert f \right\rVert_{L^2}.\end{aligned}$$* *This estimate holds locally uniformly in $q$.* *Proof.* We can write $j(x,y)=p_1(x)+p_2(x,y)|x-y|$ with $\left\lVert p_1 \right\rVert_{\sup}\lesssim \left\lVert j \right\rVert_{\sup}$ and $\left\lVert p_2 \right\rVert_{\sup}\lesssim \left\lVert j \right\rVert_{C^1}$. Then the kernel $p_1(x)\partial_y\log|x-y|$ has operator norm $\lesssim \left\lVert p_1 \right\rVert_{\sup}$ by applying the Lemma [\[action log\]](#action log){reference-type="ref" reference="action log"} to the function $p_1f$. The other part has operator norm $\lesssim \epsilon\left\lVert p_2 \right\rVert_{\sup}$, as one can easily check that $$\begin{aligned} |(x-y)\partial_y\log|x-y||\,\lesssim\, 1.\end{aligned}$$ ◻ **Proposition 48**. *[\[est single body\]]{#est single body label="est single body"} It holds that $$\begin{aligned} \left\lVert \frac{1}{r}\partial_n\psi-\frac{1}{2\pi \epsilon\tilde{\rho}} \right\rVert_{L^2(\partial B)}\,\lesssim\, \epsilon^\frac{1}{2}|\log\epsilon|,\end{aligned}$$ where the implicit constant is bounded locally uniformly in $q$.* *Proof.* We have that $$\begin{aligned} K\frac{1}{r}\partial_n\psi\quad\text{and}\quad\overline K_R\frac{1}{2\pi\epsilon\tilde{\rho}}\end{aligned}$$ are constant on $\partial B$ and $\frac{1}{r}\partial_n\psi-\frac{1}{2\pi \epsilon\tilde{\rho}}$ is mean-free by the definition of $\psi$, hence we may subtract these two identities from each other and obtain that $$\begin{aligned} \left\lVert \frac{1}{r}\partial_n\psi-\frac{1}{2\pi \epsilon\tilde{\rho}} \right\rVert_{L_0^2(\partial B)}\,\lesssim\, \left\lVert (K-\overline{K}_R)\frac{1}{2\pi \epsilon\tilde{\rho}} \right\rVert_{\dot{H}^1(\partial B)}\,\lesssim\, \epsilon^\frac{1}{2}|\log\epsilon|,\end{aligned}$$ here we made use of Lemma [\[K bd\]](#K bd){reference-type="ref" reference="K bd"} a) in the first estimate and of c) in the second. ◻ ### Multiple bodies Next we consider multiple bodies again. Recall that we defined $$\begin{aligned} L_0^2(\bigcup_i \partial B_i)\,:=\,\left\{f\in L^2(\bigcup_i\partial B_i)\,|\,\int_{\partial B_i} f\,\mathrm{d}x\,=\,0\:\:\:\forall i\right\}. \end{aligned}$$ We denote the space $H^1(\bigcup_i \partial B_i)$ modulo *locally* constant functions with $\dot{H}^1(\bigcup_i\partial B_i)$ with the norm $\left\lVert \partial_\tau\cdot \right\rVert_{L^2(\bigcup_i\partial B_i)}$ where $\tau=n^\perp$. We set $$\begin{aligned} \tilde{K}(x,y)\,=\,K(x,y)I_{\{\exists i \text{ with } x,y\in \partial B_i\}}.\end{aligned}$$ and also denote the associated linear operator with $\tilde{K}$. **Lemma 49**. *[\[comp tilde K\]]{#comp tilde K label="comp tilde K"} We have $$\begin{aligned} \left\lVert K-\tilde{K} \right\rVert_{L^2(\partial B_i)\rightarrow \dot{H}^1(\partial B_j)}\,\lesssim\, \epsilon|q_i-q_j|^{-1}\end{aligned}$$* *locally uniformly in $\tilde{q}$ (in both regimes [\[def regime1\]](#def regime1){reference-type="eqref" reference="def regime1"} and [\[def regime2\]](#def regime2){reference-type="eqref" reference="def regime2"}).* *Proof.* The statement is nontrivial only for $i\neq j$. By the Expansions [\[exp K\]](#exp K){reference-type="eqref" reference="exp K"} and [\[exp F\]](#exp F){reference-type="eqref" reference="exp F"}, we have $$\begin{aligned} |\partial_yK(x,y)|\,=\,\left|\partial_y\sqrt{x_Ry_R}F\left(\frac{|x-y|^2}{x_Ry_R}\right)+\sqrt{x_Ry_R}F'\left(\frac{|x-y|^2}{x_Ry_R}\right)\partial_y\frac{|x-y|^2}{x_Ry_R}\right|\,\lesssim\, \frac{1}{|x-y|}.\end{aligned}$$ The statement immediately follows since the bodies have pairwise distance $\approx |q_i-q_j|$. ◻ **Corollary 50**. *[\[K inv mult\]]{#K inv mult label="K inv mult"} The operator $K$ is invertible from $L_0^2(\bigcup_i \partial B_i)$ to $\dot{H}^1(\bigcup_i \partial B_i)$ for small enough $\epsilon$ with operator norm $\lesssim 1$, where the implicit constant and the smallness requirement for $\epsilon$ are locally uniform in $\tilde{q}$. Furthermore, for $i\neq j$, it holds that $$\begin{aligned} \left\lVert K^{-1} \right\rVert_{\dot{H}^1(\partial B_j)\rightarrow L_0^2(\partial B_i)}\,\lesssim\, \epsilon|q_i-q_j|^{-1}.\label{double bd}\end{aligned}$$* *Proof.* By Lemma [\[K bd\]](#K bd){reference-type="ref" reference="K bd"} a), invertibility holds for the operator $\tilde{K}$. By using e.g. a geometric series, this implies invertibility and by Lemma [\[comp tilde K\]](#comp tilde K){reference-type="ref" reference="comp tilde K"}, we have that $$\begin{aligned} \left\lVert K^{-1}-\tilde{K}^{-1} \right\rVert_{\dot{H}^1(\bigcup_i\partial B_i)\rightarrow L_0^2(\bigcup_i\partial B_i)}\,\lesssim\, \epsilon|q_i-q_j|^{-1}.\end{aligned}$$ This shows the statement, since $\left\lVert \tilde{K}^{-1} \right\rVert_{\dot{H}^1(\partial B_j)\rightarrow L_0^2(\partial B_i)}=0$ by definition. ◻ Let $\psi_i^1$ denote $\psi_i$ in case $B_i$ is the only body present. **Proposition 51**. *[\[comp mult\]]{#comp mult label="comp mult"} For all $i$ we have $$\begin{aligned} \left\lVert \frac{1}{r}\partial_n(\psi_i^1-\psi_i) \right\rVert_{L^2(\partial B_i)}\,\lesssim\, \epsilon^\frac{3}{2}|\log\epsilon|^\ell,\end{aligned}$$* *where the implicit constant is bounded locally uniformly in $\tilde{q}$. In particular the estimate from the single body case in Proposition [\[est single body\]](#est single body){reference-type="ref" reference="est single body"} still holds.* *Proof.* We have that $$\begin{aligned} K\frac{1}{r}\partial_n\psi_i\,=\,const \text{ on all $\partial B_j$}\end{aligned}$$ and $$\begin{aligned} &\tilde{K}\left(\frac{1}{r}\partial_n\psi_i^1\big|_{\partial B_i}\right)\,=\,const \text{ on all $\partial B_j$}.\end{aligned}$$ By subtracting the two equations, we see that $$\begin{aligned} K\left(\frac{1}{r}\left(\partial_n\psi_i-\partial_n\psi_i^1\big|_{\partial B_i}\right)\right)+\left(K-\tilde{K}\right)\left(\frac{1}{r}\partial_n\psi_i^1\big|_{\partial B_i}\right)\,=\,const.\end{aligned}$$ Now we can use Corollary [\[K inv mult\]](#K inv mult){reference-type="ref" reference="K inv mult"} and that $\frac{1}{r}\left(\partial_n\psi-\partial_n\psi_i^1\big|_{\partial B_i}\right)$ is mean-free and [\[double bd\]](#double bd){reference-type="eqref" reference="double bd"} and that $$\begin{aligned} (K-\tilde{K}) \left(\frac{1}{r}\partial_n\psi_i^1\big|_{\partial B_i}\right)\,=\,0\end{aligned}$$ on $\partial B_i$ by definition to obtain that $$\begin{aligned} \left\lVert \frac{1}{r}\left(\partial_n\psi_i-\partial_n\psi_i^1\big|_{\partial B_i}\right) \right\rVert_{L_0^2(\partial B_i)}\,\lesssim\,\epsilon|\log\epsilon|^\ell\left\lVert (K-\tilde{K}) \left(\frac{1}{r}\partial_n\psi_i^1\big|_{\partial B_i}\right) \right\rVert_{\dot{H}^1(\bigcup_l\partial B_l)}\,\lesssim\, \epsilon^\frac{3}{2}|\log\epsilon|^\ell\end{aligned}$$ ◻ **Proposition 52**. *[\[1st order exp\]]{#1st order exp label="1st order exp"} For $i\neq j$ we have that $$\begin{aligned} \left\lVert \frac{1}{r}\partial_n\psi_i-\frac{2}{r}n\cdot\nabla_yK(q_i,q_j) \right\rVert_{L^2(\partial B_j)}\,\lesssim\,\epsilon^\frac{3}{2}|\log\epsilon|^\ell,\end{aligned}$$* *where $''\nabla_y''$ refers to the gradient in the second variable and the implicit constant is bounded locally uniformly in $\tilde{q}$.* *In particular, it follows from the Asymptotics [\[exp K\]](#exp K){reference-type="eqref" reference="exp K"} and [\[exp F\]](#exp F){reference-type="eqref" reference="exp F"} that $\left\lVert \partial_n\psi_i \right\rVert_{L^2(\partial B_j)}\,\lesssim\, \epsilon^\frac{1}{2}|q_i-q_j|^{-1}$ for $i\neq j$, locally uniformly in $\tilde{q}$.* *Proof.* Let $\psi_i^1$ be the potential in case there is only a single body $B_i$. Let $\mu\in L_0^2(\partial B_j)$ be such that $$\begin{aligned} K\mu+K\left(\frac{1}{r}\partial_n\psi_i^1|_{\partial B_i}\right)\,=\,const \text{ on $\partial B_j$}.\end{aligned}$$ This is well-defined by the Lemmata [\[K bd\]](#K bd){reference-type="ref" reference="K bd"} and [\[comp tilde K\]](#comp tilde K){reference-type="ref" reference="comp tilde K"} and it holds that $\left\lVert \mu \right\rVert_{L^2(\partial B_j)}\lesssim \epsilon^\frac{1}{2}|q_i-q_j|^{-1}$. Then for $m\neq j$ by Lemma [\[comp tilde K\]](#comp tilde K){reference-type="ref" reference="comp tilde K"}, it holds that $$\begin{aligned} \left\lVert K\mu \right\rVert_{\dot{H}^1(\partial B_m)}\,\lesssim\, \epsilon^\frac{3}{2}|\log\epsilon|^\ell\end{aligned}$$ and hence $$\begin{aligned} \left\lVert K\mu+K\left(\frac{1}{r}\partial_n\psi_i^1|_{\partial B_i}\right) \right\rVert_{\dot{H}^1(\partial B_m)}\,\lesssim\, \epsilon^\frac{1}{2}|q_i-q_j|^{-1}.\end{aligned}$$ Hence by [\[double bd\]](#double bd){reference-type="eqref" reference="double bd"} we conclude that $$\begin{aligned} \label{comp mu} \left\lVert \mu-\frac{1}{r}\partial_n\psi_i \right\rVert_{L_0^2(\partial B_j)}\,\lesssim\, \epsilon^\frac{3}{2}|\log\epsilon|^\ell,\end{aligned}$$ and hence it suffices to compute $\mu$. We first estimate $\nabla\psi_i^1$. We know from the definition of $\psi_i^1$ and the maximum principle that $\frac{1}{r}\partial_n\psi_i^1\geq 0$ on $\partial B_i$. Hence we can use the mean value theorem to estimate $$\begin{aligned} \label{est psi 1} \left\lVert \nabla K\left(\frac{1}{r}\partial_n\psi_i^1|_{\partial B_i}\right)-\nabla_y K(q_i,\cdot) \right\rVert_{C^0(\partial B_j)}\,\lesssim\,\epsilon\sup_{z\in B_i,y\in \partial B_j}\nabla_{z,y}^2 K(z,y)\,\lesssim\, \epsilon|\log\epsilon|^\ell\end{aligned}$$ where we used the Asymptotics [\[exp K\]](#exp K){reference-type="eqref" reference="exp K"} and [\[exp F\]](#exp F){reference-type="eqref" reference="exp F"}. Similarly we have $$\begin{aligned} \left\lVert \nabla K(q_i,q_j)-\nabla_yK\left(q_i,\cdot\right) \right\rVert_{C^0(\partial B_j)}\,\lesssim\, \epsilon\sup_{y\in B_j}|\nabla_{y}^2K(q_i,y)|\,\lesssim\,\epsilon|\log\epsilon|^\ell.\end{aligned}$$ Hence we have that $$\begin{aligned} \label{inv psi 1} \left\lVert \psi_i^1-x\cdot\nabla_y K(q_i,q_j) \right\rVert_{\dot{H}^1(\partial B_j)}\,\lesssim\, \epsilon^{\frac{3}{2}}|\log\epsilon|^\ell.\end{aligned}$$ We have $$\begin{aligned} \label{inv x} \overline{K}_{R_j}^{(-1)}x\,=\,-\frac{2}{R_j}n+const\end{aligned}$$ by [\[action K flat\]](#action K flat){reference-type="eqref" reference="action K flat"} (where $n$ denotes the normal as usual). By Lemma [\[K bd\]](#K bd){reference-type="ref" reference="K bd"} and the definition of $\mu$ we have $$\begin{aligned} \label{mu est} \left\lVert \mu+\overline{K}_{R_j}^{(-1)}\left(\psi_i^1\right) \right\rVert_{L_0^2(\partial B_j)}\,=\,\left\lVert \left(\overline{K}_{R_j}^{(-1)}-K^{-1}\right)\left(\psi_i^1\right) \right\rVert\,\lesssim\,\epsilon^\frac{3}{2}|\log\epsilon|^\ell.\end{aligned}$$ Together [\[inv psi 1\]](#inv psi 1){reference-type="eqref" reference="inv psi 1"},[\[inv x\]](#inv x){reference-type="eqref" reference="inv x"} and [\[mu est\]](#mu est){reference-type="eqref" reference="mu est"} imply that $$\begin{aligned} \left\lVert \mu-\frac{2}{R_j}n\cdot\nabla_y K(q_i,q_j) \right\rVert_{L_0^2(\partial B_j)}\,\lesssim\, \epsilon^\frac{3}{2}|\log\epsilon|^\ell.\end{aligned}$$ Finally we can replace the $R_j$ in the denominator by $r$ as $\left\lVert n\cdot \nabla_y K(q_i,q_j) \right\rVert_{L^2(\partial B_j)}\lesssim \epsilon^\frac{1}{2}|q_i-q_j|^{-1}$ by [\[exp K\]](#exp K){reference-type="eqref" reference="exp K"} and [\[exp F\]](#exp F){reference-type="eqref" reference="exp F"}. ◻ ### The derivative with respect to $q$ {#the-derivative-with-respect-to-q} To compute derivatives with respect to $q$, we only need to consider partial derivatives with respect to a single $R_i$ or $Z_i$, as everything is smooth by Lemma [\[PsiDeri\]](#PsiDeri){reference-type="ref" reference="PsiDeri"}. In the following we write $\psi_{i,R_j+s}$ instead of $\psi_i$ to emphasize with respect to which $R_j$ the function $\psi_i$ is defined and analogously write $\psi_{i,Z_j+s}$. For mixed derivatives we write $\psi_{i,R_j+s_1,R_m+s_2}$ if we want to indicate multiple positions. We set $$\begin{aligned} \delta_{j}^s(x)\,:=\,\begin{cases} \frac{\rho_j(R_j+s)}{\rho_j(R_j)}& \text{if } x\in \partial B_j\\ 1 & \text{else,} \end{cases}\end{aligned}$$ where we again write $\rho_j(\cdot)$ for $\rho_j$ as a function of $R_j$. **Proposition 53**. *[\[deri mu\]]{#deri mu label="deri mu"} For all $i,j,l$ it holds that $$\begin{aligned} \left\lVert \partial_s\left(\delta_j^s\left(\frac{1}{r}\partial_n\psi_{i,R_j+s}\right)\circ c\right) \right\rVert_{L^2(\partial B_l)}\,\lesssim\, \epsilon^\frac{1}{2}\left(\min_{a\neq b}|q_a-q_b|\right)^{-2},\end{aligned}$$* *where the diffeomorphism $c$ was introduced in the beginning of Subsection [3.2.3](#sec q deri){reference-type="ref" reference="sec q deri"} and corresponds to the change of $R_j$.* *Similarly it holds that $$\begin{aligned} \left\lVert \partial_s\left(\left(\frac{1}{r}\partial_n\psi_{i,Z_j+s}\right)\circ d\right) \right\rVert_{L^2(\partial B_l)}\,\lesssim\, \epsilon^\frac{1}{2}\left(\min_{a\neq b}|q_a-q_b|\right)^{-2},\end{aligned}$$* *where the diffeomorphism $d$ was introduced in the beginning of Subsection [3.2.3](#sec q deri){reference-type="ref" reference="sec q deri"} and corresponds to the change of $Z_j$.* *The implicit constant in both estimates is locally uniform in $\tilde{q}$.* **Proposition 54**. *[\[2nd deri mu\]]{#2nd deri mu label="2nd deri mu"}* - *For all $i,j,l$ we have $$\begin{aligned} \left\lVert \partial_s^2\left(\delta_j^s\left(\frac{1}{r}\partial_n\psi_{i,R_j+s}\right)\circ c\right) \right\rVert_{L^2(\partial B_l)}\,\lesssim\, \epsilon^\frac{1}{2}|\log\epsilon|^\ell\end{aligned}$$* *and the same holds for the second derivative with respect to $Z_j$ and the mixed second derivative. The implicit constant is bounded locally uniformly in $\tilde{q}$.* - *For all $i,j,l,m$ with $j\neq m$ we have $$\begin{aligned} \left\lVert \partial_{s_1}\partial_{s_2}\left(\delta_j^{s_1}\delta_m^{s_2}\left(\frac{1}{r}\partial_n\psi_{i,R_j+s_1,R_m+s_2}\right)\circ c_{jm}(s_1,s_2)\right) \right\rVert_{L^2(\partial B_l)}\,\lesssim\, \epsilon^\frac{1}{2}|\log\epsilon|^\ell\end{aligned}$$* *where $c_{jm}$ is the composition of the map $c$ defined for $j$ and $m$ with arguments $s_1$ and $s_2$ respectively. The same estimate holds for the derivatives with respect to $Z$'s or mixed derivatives. The implicit constant is bounded locally uniformly in $\tilde{q}$.* We shall only prove the statements for the $R_j$-derivative and focus on the first derivative and occasionally comment on the slight changes needed for the $Z_j$-derivative, which is generally easier. The second derivatives can be handled with the same technique, but the involved calculations become a lot more tedious, so we omit them most of the calculations for them. We set $$\begin{aligned} K_s^{R_j}(x,y)\,=\,\left(1+I_{\partial B_j}(x)\left(\frac{R_j}{R_j+s}-1\right)\right)K(c(x,s),c(y,s)),\end{aligned}$$ and similarly $$\begin{aligned} K_s^{Z_j}\,:=\,K(d(x,s),d(y,s)).\end{aligned}$$ We also write $K_s^{R_j}$ for the associated linear map. Note that for $f$ supported on $\partial B_j$ it holds that $$\begin{aligned} K_s^{R_j}f(y)\,=\,\frac{R_j\rho_j}{(R_j+s)\rho_j(R_j+s)}K(f\circ c^{-1})(c(y,s)),\end{aligned}$$ where we again write $\rho_j(\cdot)$ to denote $\rho_j$ as a function of $R_j$ and where the additional prefactor comes from the change of the inner radius. For $f$ supported on any other $\partial B_i$ it holds $$\begin{aligned} K_s^{R_j}f(y)\,=\,K(f)(c(y,s)).\end{aligned}$$ Similar for mixed second derivatives with respect to different indices, one would use the kernel $$\begin{aligned} &K_{s_1,s_2}^{R_j,R_m}(x,y)\,=\,\left(1+I_{\partial B_j}(x)\left(\frac{R_j}{R_j+s}-1\right)+I_{\partial B_m}(x)\left(\frac{R_m}{R_m+s}-1\right)\right)\\ &\times K(c_{jm}(x,s_1,s_2),c_{jm}(y,s_1,s_2)).\nonumber\end{aligned}$$ **Lemma 55**. *[\[deri K\]]{#deri K label="deri K"} The linear operator $K_s^{R_j}$ is Fréchet differentiable in $s$ as a map from $L^2(\partial B_j)$ to $\dot{H}^1(\partial B_j)$ for all $i$ and we have $$\begin{aligned} \left\lVert \partial_sK_s^{R_j} \right\rVert_{L^2(\partial B_j)\rightarrow \dot{H}^1(\partial B_j)},\left\lVert \partial_s^2K_s^{R_j} \right\rVert_{L^2(\partial B_j)\rightarrow \dot{H}^1(\partial B_j)}\,\lesssim\, \epsilon|\log\epsilon|,\end{aligned}$$* *furthermore, the Fréchet derivative is given by integration against the pointwise derivative in $s$.* Note that the corresponding derivatives of $K_s^{Z_j}$ are trivially $0$ by the explicit form of $K$ in [\[exp K\]](#exp K){reference-type="eqref" reference="exp K"}. The statement for mixed second derivatives also trivially reduces to the derivative with respect to a single index, as only the change of $R_j$ matters. *Proof.* It is easy to see that the kernel is pointwise smooth in $s$ for $x\neq y$ by using the Expansions [\[exp K\]](#exp K){reference-type="eqref" reference="exp K"} and [\[exp F\]](#exp F){reference-type="eqref" reference="exp F"} and the differentiability of $c$. We shall estimate the operator norm of the first pointwise derivative. A similar, but tedious calculation, which we omit here can be made to show that the second and third pointwise derivatives are bounded, which by the mean value theorem justifies that the first two pointwise derivatives agree with the Fréchet derivatives.\ Let us estimate the first derivative w.r.t. $s$ of the different parts of the kernel: $$\begin{aligned} &\partial_s \log |c(x,s)-c(y,s)|\,=\,\partial_s\log\left(\frac{\rho_j(R_j+s)}{\rho_j}\right)\,=\,-\frac{1}{2(R_j+s)}\label{log c}\\ &\left|\partial_s\frac{R_j}{R_j+s}\sqrt{c(x,s)_Rc(y,s)_R}\right|\,\lesssim\, \epsilon\label{deri prefactor 0}\\ &\left\lVert \partial_s\frac{R_j}{R_j+s}\sqrt{c(x,s)_Rc(y,s)_R} \right\rVert_{C_{x,y}^1(\partial B_j\times \partial B_j)}\,\lesssim\, 1\label{deri prefactor}\\ &\partial_s\log(R_j+s)\,=\,\frac{1}{R_j+s}\label{deri log R 0}\\ &\left|\partial_s\log \frac{(R_j+s)^2}{c(x,s)_Rc(y,s)_R}\right|\,\lesssim\, \epsilon\label{deri log R 1}\\ &\left\lVert \partial_s\log \frac{(R_j+s)^2}{c(x,s)_Rc(y,s)_R} \right\rVert_{C_{x,y}^1(\partial B_j\times \partial B_j)}\,\lesssim\,1\label{deri log R}\end{aligned}$$ Furthermore we have $$\begin{aligned} \left|\partial_s\frac{(c(x,s)-c(y,s))^2}{c(x,s)_Rc(y,s)_R}\big|_{s=0}\right| \,=\,\left|\partial_s|x-y|^2\frac{\rho_j(R_j+s)^2}{\rho_j^2c(x,s)_Rc(y,s)_R}\big|_{s=0}\right|\,\lesssim\, |x-y|^2\label{s deri bd}\end{aligned}$$ and similarly it holds $$\begin{aligned} &\left|\partial_y\partial_s\frac{(c(x,s)-c(y,s))^2}{c(x,s)_Rc(y,s)_R}\big|_{s=0}\right|\,\lesssim\, |x-y|.\label{s deri bd 2}\end{aligned}$$ Now we may use the Expansions [\[exp K\]](#exp K){reference-type="eqref" reference="exp K"} and [\[exp F\]](#exp F){reference-type="eqref" reference="exp F"} to write $$\begin{aligned} &2\pi\partial_y\partial_sK_s^{R_j}(x,y)\big|_{s=0}\,=\,\notag\\ & \partial_s\partial_y\left(\frac{R_j}{R_j+s}\sqrt{c(x,s)_Rc(y,s)_R}\right)\left(\log(|x-y|)-\log(8)+2-\frac{1}{2}\log(x_Ry_R)+g(x,y)\right)\notag\\ &+\partial_s\left(\frac{R_j}{R_j+s}\sqrt{c(x,s)_Rc(y,s)_R}\right)\partial_y\left(\log(|x-y|)-\frac{1}{2}\log\left(\frac{x_Ry_R}{R_j^2}\right)+g(x,y)\right)\notag\\ &+\partial_y\left(\sqrt{x_Ry_R}\right)\partial_s\left(\log(|c(x,s)-c(y,s)|)-\frac{1}{2}\log\left(\frac{c(x,s)_Rc(y,s)_R}{(R_j+s)^2}\right)+\log( R_j+s)\right)\notag\\ &+\sqrt{x_Ry_R}\partial_y\partial_s\left(\log(|c(x,s)-c(y,s)|)-\frac{1}{2}\log\frac{c(x,s)_Rc(y,s)_R}{(R_j+s)^2}\right)\notag\\ &-\partial_y\left(\sqrt{x_Ry_R}\right)\partial_sh\left(\frac{(c(x,s)-c(y,s))^2}{c(x,s)_Rc(y,s)_R}\right)\notag\\ &-\sqrt{x_Ry_R}\partial_s\partial_yh\left(\frac{(c(x,s)-c(y,s))^2}{c(x,s)_Rc(y,s)_R}\right).\notag\\ &=\,I+II+III+IV+V+VI.\end{aligned}$$ Here $I-VI$ stand for the obvious terms and we dropped some constants whose derivative vanishes. The boundedness of the terms $I$ and $II$ follows from the estimates [\[deri prefactor 0\]](#deri prefactor 0){reference-type="eqref" reference="deri prefactor 0"} and [\[deri prefactor\]](#deri prefactor){reference-type="eqref" reference="deri prefactor"} above and Schur's Lemma and further from using Lemma [\[perturbed log\]](#perturbed log){reference-type="ref" reference="perturbed log"} for the derivative of the logarithm. The boundedness of $III$ follows directly from [\[log c\]](#log c){reference-type="eqref" reference="log c"} and [\[deri log R 1\]](#deri log R 1){reference-type="eqref" reference="deri log R 1"} by using Schur's Lemma. The boundedness of $IV$ follows from [\[log c\]](#log c){reference-type="eqref" reference="log c"} and [\[deri log R\]](#deri log R){reference-type="eqref" reference="deri log R"}. The boundedness of $V$ and $VI$ follows from [\[s deri bd\]](#s deri bd){reference-type="eqref" reference="s deri bd"} resp. [\[s deri bd 2\]](#s deri bd 2){reference-type="eqref" reference="s deri bd 2"} and the estimate [\[est g\]](#est g){reference-type="eqref" reference="est g"} on $g$.\  ◻ **Lemma 56**. *[\[deri K 2\]]{#deri K 2 label="deri K 2"} The kernel $K_s^{R_j}$ is Fréchet-differentiable in $s$ as a map from $L^2(\partial B_i)$ to $\dot{H}^1(\partial B_l)$ for $i\neq l$ and, locally uniformly in $\tilde{q}$, we have that $$\begin{aligned} \left\lVert \partial_s K_s^{R_j} \right\rVert_{L_0^2(\partial B_i)\rightarrow \dot{H}^1(\partial B_l)}\,\lesssim\,\epsilon|q_i-q_l|^{-2}\quad\text{and}\quad\left\lVert \partial_s^2 K_s^{R_j} \right\rVert_{L_0^2(\partial B_i)\rightarrow \dot{H}^1(\partial B_l)}\,\lesssim\, \epsilon|q_i-q_l|^{-3}.\end{aligned}$$* *Furthermore, the Fréchet derivative is given by integration against the pointwise derivative in $s$. The same estimates also hold for the Kernel $K_s^{Z_j}$ and the second derivatives of $K_{s_1,s_2}^{R_j,R_m}$.* *Proof.* We only consider the case $l=j$ and the first derivative, the other cases and the second derivative are very similar. In this case, the kernel is smooth in $(x,y,s)$ and hence the Fréchet- and pointwise derivative agree by e.g. the mean value theorem. By [\[exp K\]](#exp K){reference-type="eqref" reference="exp K"} and [\[exp F\]](#exp F){reference-type="eqref" reference="exp F"}, we have that $$\begin{aligned} &-2\pi\partial_s\partial_yK_s^{R_j}(x,y)\,=\,\partial_s\partial_y\sqrt{x_Rc(y,s)_R}F\left(\frac{|x-y|^2}{x_Ry_R}\right)+\partial_s\sqrt{x_Rc(y,s)_R}F'\left(\frac{|x-y|^2}{x_Ry_R}\right)\partial_y\frac{|x-y|^2}{x_Ry_R}\notag\\ &+\partial_y\sqrt{x_Ry_R}F'\left(\frac{|x-c(y,s)|^2}{x_Rc(y,s)_R}\right)\partial_s\frac{|x-c(y,s)|^2}{x_Rc(y,s)_R}+\\ &\sqrt{x_Ry_R}\biggl(F'\left(\frac{|x-c(y,s)|^2}{x_Rc(y,s)_R}\right)\partial_y\partial_s\frac{|x-c(y,s)|^2}{x_Rc(y,s)_R}\notag\\ &+F''\left(\frac{|x-c(y,s)|^2}{x_Rc(y,s)_R}\right)\partial_s\frac{|x-c(y,s)|^2}{x_Rc(y,s)_R}\partial_y\frac{|x-y|^2}{x_Ry_R}\biggr).\notag\end{aligned}$$ It is easy to see that all relevant derivatives of the prefactor are $\lesssim 1$ and that $|\partial_s c|\lesssim 1$. Hence the absolute value of this is $$\begin{aligned} \lesssim\, \left|F\left(\frac{|x-y|^2}{x_Ry_R}\right)+F'\left(\frac{|x-y|^2}{x_Ry_R}\right)(1+|x-y|)+F''\left(\frac{|x-y|^2}{x_Ry_R}\right)|x-y|^2\right|.\end{aligned}$$ From the asymptotic of $F$ (see [\[exp F\]](#exp F){reference-type="eqref" reference="exp F"}), we see that for $|x-y|$ small this is $$\begin{aligned} \lesssim\, \frac{1}{|x-y|^2}.\end{aligned}$$ Hence we see that $$\begin{aligned} \left\lVert \partial_sK_s^{R_j} \right\rVert_{L^2(\partial B_i)\rightarrow \dot{H}^1(\partial B_j)}\,\lesssim\,\epsilon|q_i-q_l|^{-2}.\end{aligned}$$ ◻ *Proof of Propsition [\[deri mu\]](#deri mu){reference-type="ref" reference="deri mu"}.* We know from Lemma [\[PsiDeri\]](#PsiDeri){reference-type="ref" reference="PsiDeri"} that $\frac{1}{r}\psi_{j,R_j+s}$ is differentiable, hence we may differentiate the equation $$\begin{aligned} K_s^{R_j}\delta_j^s\frac{1}{r}\partial_n\psi_{i,R_j+s}\circ c\,=\,const \text{ on each $\partial B_m$}\end{aligned}$$ with respect to $s$, as $K_s^{R_j}$ is differentiable by Lemma [\[deri K\]](#deri K){reference-type="ref" reference="deri K"}. This yields that $$\begin{aligned} \partial_sK_s^{R_j}\left(\frac{1}{r}\partial_n\psi_{i}\right)+K\left(\partial_s\left(\delta_j^s\left(\frac{1}{r}\partial_n\psi_{i,R_j+s}\right)\circ c\right)\right)\,=\,const \text{ on each $\partial B_m$}.\end{aligned}$$ Note that $\partial_s\delta_j^s\frac{1}{r}\partial_n\psi_{i,R_j+s}\circ c$ is mean-free on each $\partial B_m$, because the integral of $\delta_j^s\frac{1}{r}\partial_n\psi_{i,R_j+s}\circ c$ over $\partial B_m$ is either $1$ or $0$ for all $s$ by definition. Furthermore by Lemma [\[deri K\]](#deri K){reference-type="ref" reference="deri K"} and Lemma [\[deri K 2\]](#deri K 2){reference-type="ref" reference="deri K 2"}, we have $$\begin{aligned} \left\lVert \partial_sK_s^{R_j}\left(\frac{1}{r}\partial_n\psi_{i}\right) \right\rVert_{\dot{H}^1(\bigcup_m \partial B_m)}\,\lesssim\,\epsilon^\frac{1}{2}\left(|\log\epsilon|+\min_{a\neq b}|q_a-q_b|^{-2}\right).\end{aligned}$$ We can absorb the logarithm into the second summand by definition of the regimes. By Corollary [\[K inv mult\]](#K inv mult){reference-type="ref" reference="K inv mult"}, we conclude. ◻ *Proof of Proposition [\[2nd deri mu\]](#2nd deri mu){reference-type="ref" reference="2nd deri mu"}.* We only consider the second derivative with respect to $R_j$, all others work the same because one has the same or better estimates. We have $$\begin{aligned} \partial_s^2K_s^{R_j}\left(\frac{1}{r}\partial_n\psi_i\right)+2\partial_sK_s^{R_j}\left(\partial_s\delta_j^s\left(\frac{1}{r}\partial_n\psi_{i,R_j+s}\right)\circ c\right)+K\left(\partial_s^2\delta_j^s\left(\frac{1}{r}\partial_n\psi_{i,R_j+s}\right)\circ c\right)\,=\,const \end{aligned}$$ on all $\partial B_m$. Note that by the Lemmata [\[deri K\]](#deri K){reference-type="ref" reference="deri K"}, [\[deri K 2\]](#deri K 2){reference-type="ref" reference="deri K 2"} and Proposition [\[deri mu\]](#deri mu){reference-type="ref" reference="deri mu"} it holds that $$\begin{aligned} \left\lVert \partial_s^2K_s^{R_j}\frac{1}{r}\partial_n\psi_i \right\rVert_{\dot{H}^1(\bigcup_m\partial B_m)}+\left\lVert \partial_sK_s^{R_j}\left(\partial_s\delta_j^s\left(\frac{1}{r}\partial_n\psi_{i,R_j+s}\right)\circ c\right) \right\rVert_{\dot{H}^1(\bigcup_m\partial B_m)}\,\lesssim \,\epsilon^\frac{1}{2}|\log\epsilon|^\ell.\end{aligned}$$ Hence from Corollary [\[K inv mult\]](#K inv mult){reference-type="ref" reference="K inv mult"} we conclude the statement. ◻ ### Further estimates on the derivative of $K$ To compute the force $G$, we will also need estimates in the $L^2\rightarrow L^2$-topology. **Lemma 57**. *[\[deri K L2\]]{#deri K L2 label="deri K L2"} For all $j$ it holds that $$\begin{aligned} \left\lVert \partial_s K_s^{R_j} \right\rVert_{L^2(\partial B_j)\rightarrow L^2(\partial B_j)}\,\lesssim\,\epsilon\end{aligned}$$ and $$\begin{aligned} \left\lVert \partial_s^2 K_s^{R_j} \right\rVert_{L^2(\partial B_j)\rightarrow L^2(\partial B_j)}\,\lesssim\, \epsilon,\end{aligned}$$* *locally uniformly in $\tilde{q}$ and furthermore, the pointwise and Fréchet derivatives agree.* Here we only need the estimate in the $R_j$-direction, because the derivative in the $Z_j$-direction is $0$. Also note that changes of the other $R_m$'s trivially give a derivative of zero here. *Proof.* We only show this for the first derivative with respect to $R_j$, the second derivative is very similar. We may expand the kernel as $$\begin{aligned} &2\pi\left(\partial_sK_s^{R_j}(x,y)\big|_{s=0}\right)\\ &\,=\,\partial_s\left(\frac{R_j}{R_j+s}\sqrt{c(x,s)_Rc(y,s)_R}\right)\left(\log(|x-y|)-\log(8)+2-\frac{1}{2}\log\left(x_Ry_R\right)+g(x,y)\right)\notag\\ &+\sqrt{x_Ry_R}\partial_s\left(\log\left(|c(x,s)-c(y,s)|\right)-\frac{1}{2}\log(c(x,s)_Rc(y,s)_R)-h\left(\frac{|c(x,s)-c(y,s)|^2}{c(x,s)_Rc(y,s)_R}\right)\right).\notag\end{aligned}$$ It is easy to see that $$\begin{aligned} \left|\partial_s\left(\frac{R_j}{R_j+s}\sqrt{c(x,s)_Rc(y,s)_R}\right)\right|\,\lesssim\, \epsilon\end{aligned}$$ and by reusing [\[log c\]](#log c){reference-type="eqref" reference="log c"} and [\[deri log R 1\]](#deri log R 1){reference-type="eqref" reference="deri log R 1"} and [\[deri log R 0\]](#deri log R 0){reference-type="eqref" reference="deri log R 0"} one sees that $$\begin{aligned} \label{bd sqrt} \left|\partial_s\left(\log(|c(x,s)-c(y,s)|)-\frac{1}{2}\log(c(x,s)_Rc(y,s)_R)-h\left(\frac{|c(x,s)-c(y,s)|^2}{c(x,s)_Rc(y,s)_R}\right)\right)\right|\,\lesssim \,1.\end{aligned}$$ The boundedness follows by Schur's Lemma. The Fréchet differentiability follows from the boundedness of the second derivative and the mean value theorem. ◻ ## The force $G$ We compute the asymptotic of $G$, defined in [\[main def\]](#main def){reference-type="ref" reference="main def"}. The crucial Lemma for the "self-interaction" terms is the following: **Lemma 58**. *[\[Lem:EnergyDerivative\]]{#Lem:EnergyDerivative label="Lem:EnergyDerivative"} We have $$\begin{aligned} \int_{\mathcal{F}} \frac{1}{2}r\langle \frac{1}{r}\nabla^\perp\psi_i,\frac{1}{r}\nabla^\perp\psi_j \rangle\,\mathrm{d}x\,=\,-\frac{1}{2}C_{ij}\end{aligned}$$ and for any $t_l^*$ associated to $B_l$ it holds that $$\begin{aligned} \partial_{q}\frac{1}{2}C_{ij}\cdot t_l^*\,=\,\frac{1}{2}\int_{\partial B_l}\frac{1}{r}\partial_n\phi_{l,t_l^*}\langle \nabla^\perp\psi_i,\nabla^\perp\psi_j \rangle\,\mathrm{d}x,\end{aligned}$$ where the $C$'s were defined in [\[def psi\]](#def psi){reference-type="ref" reference="def psi"}.* *Proof.* We have that $$\begin{aligned} \frac{1}{2}\int_\mathcal{F} r\frac{1}{r^2}\langle \nabla^\perp\psi_i,\nabla^\perp\psi_j \rangle\,\mathrm{d}x\,=\,-\sum_l\frac{1}{2}\int_{\partial B_l}\frac{1}{r}C_{il}\partial_n\psi_j\,\mathrm{d}x\,=\,-\frac{1}{2}C_{ij}.\end{aligned}$$ Here the partial integration is justified by the Lemmata [\[ExistencePsi\]](#ExistencePsi){reference-type="ref" reference="ExistencePsi"} and [\[rep psi\]](#rep psi){reference-type="ref" reference="rep psi"}. Note that it holds $$\begin{aligned} \label{eq 6} \partial_q\psi_j\cdot t_l^*\,=\,\partial_q C_{jm}\cdot t_l^*-\partial_n\psi_j u(t_l^*) \end{aligned}$$ on $\partial B_m$ as one can see by differentiating $C_{jm}=\psi_j(x_l)$ by $q$ for some point $x_l$ on $\partial B_m$ moving with normal velocity $u(t_l^*)$. Now by Reynolds (which can be used by the integrability statement in Lemma [\[PsiDeri\]](#PsiDeri){reference-type="ref" reference="PsiDeri"}) the derivative of this with respect to $q$ in direction $t_l^*\in T_{q_l}M$ equals $$\begin{aligned} &\frac{1}{2}\partial_q C_{ij}\cdot t_l^*\,=\,-\int_\mathcal{F} \frac{1}{r}\langle \nabla^\perp\psi_i,\nabla^\perp\partial_q\psi_j\cdot t_l^* \rangle\,\mathrm{d}x+\frac{1}{2}\int_{\partial B_l}\frac{1}{r}\langle \nabla^\perp\psi_i,\nabla^\perp\psi_j \rangle u(t_l^*)\,\mathrm{d}x\\ &\,=\,\sum_{m}\int_{\partial B_m}\frac{1}{r}\partial_q\psi_j\cdot t_l^*\partial_n\psi_i\,\mathrm{d}x+\frac{1}{2}\int_{\partial B_l}\frac{1}{r}\langle \nabla^\perp\psi_i,\nabla^\perp\psi_j \rangle u(t_l^*)\,\mathrm{d}x\\ &\,=\,\sum_m\int_{\partial B_m}\frac{1}{r}(\partial_q C_{jm}\cdot t_l^*-\partial_n\psi_j u(t_l^*))\partial_n\psi_i\,\mathrm{d}x+\frac{1}{2}\int_{\partial B_l}\frac{1}{r}\langle \nabla^\perp\psi_i,\nabla^\perp\psi_j \rangle u(t_l^*)\,\mathrm{d}x\\ &\,=\,\partial_qC_{ji}\cdot t_l^*-\int_{\partial B_l}\frac{1}{r}\langle \nabla^\perp\psi_i,\nabla^\perp\psi_j \rangle u(t_l^*)\,\mathrm{d}x+\frac{1}{2}\int_{\partial B_l}\frac{1}{r}\langle \nabla^\perp\psi_i,\nabla^\perp\psi_j \rangle u(t_l^*)\,\mathrm{d}x,\end{aligned}$$ where we have made use of equation [\[eq 6\]](#eq 6){reference-type="eqref" reference="eq 6"} in the third line and of the facts that $\partial_q C_{jm}$ is a constant function and that the matrix $C$ is symmetric by the first statement. ◻ **Proposition 59**. *[\[diagonal A\]]{#diagonal A label="diagonal A"} For every tangent vector $t^*$, we have that $$\begin{aligned} \left|G(q,e_i)\cdot t^*-\frac{1}{4\pi}\log(\epsilon\tilde{\rho})(t_i^*)\cdot e_R\right|\,\lesssim\, 1\end{aligned}$$* *and $$\begin{aligned} \left|\partial_qG(q,e_i)\right|\,\lesssim\, 1.\end{aligned}$$* *These estimate are locally uniform in $\tilde{q}$.* *Proof.* By Lemma [\[Lem:EnergyDerivative\]](#Lem:EnergyDerivative){reference-type="ref" reference="Lem:EnergyDerivative"} and the definition of $G(q,e_i)$ in [\[main def\]](#main def){reference-type="ref" reference="main def"}, it equals $\frac{1}{2}$ times the derivative of the energy $$\begin{aligned} \int_{(\bigcup_l \partial B_l)^2}K(x,y)\frac{1}{r^2}\partial_n\psi_i(x)\partial_n\psi_i(y)\,\mathrm{d}x\mathrm{d}y\end{aligned}$$ with respect to $q$. We first consider the partial derivative in the direction $R_i$. Note that $K\frac{1}{r}\partial_n\psi_i$ is constant and that $\frac{1}{r}\partial_n\psi_i$ is mean-free on all boundaries except $\partial B_i$, hence the integral over all boundaries except $(\partial B_i)^2$ is zero. Using the diffeomorphism $c$, we can rewrite the energy with respect to $R_i+s$ as $$\begin{aligned} \int_{(\partial B_i)^2}\frac{R_i+s}{R_i}K_s^{R_i}(\delta_i^s)^2\left(\frac{1}{r}\partial_n\psi_{i,R_i+s}\right)\circ c\left(\frac{1}{r}\partial_n\psi_{i,R_i+s}\right)\circ c\,\mathrm{d}x\mathrm{d}y,\end{aligned}$$ here the factor $\delta_i^s$ is the determinant due to the change of coordinates. We can differentiate this under the integral as everything is smooth by Lemma [\[PsiDeri\]](#PsiDeri){reference-type="ref" reference="PsiDeri"}. We first show that the parts where a derivative falls on $\frac{1}{r}\partial_n\psi_i$ are small. Indeed by Lemma [\[K bd\]](#K bd){reference-type="ref" reference="K bd"} and Proposition [\[deri mu\]](#deri mu){reference-type="ref" reference="deri mu"} we have $$\begin{aligned} \left\lVert K\partial_s\left(\delta_i^s\left(\frac{1}{r}\partial_n\psi_{i,R_i+s}\right)\circ c\right) \right\rVert_{L^2(\partial B_i)}\,\lesssim\, \epsilon|\log\epsilon|^\ell \left\lVert \partial_s\left(\delta_i^s\left(\frac{1}{r}\partial_n\psi_{i,R_i+s}\right)\circ c\right) \right\rVert_{L^2(\partial B_i)}\,\lesssim\, \epsilon^{\frac{3}{2}}|\log\epsilon|^\ell.\end{aligned}$$ Hence $$\begin{aligned} &\left|\int_{(\partial B_i)^2}K(x,y)\partial_s\left(\delta_i^s\left(\frac{1}{r}\partial_n\psi_{i,R_i+s}\right)\circ c\right)\frac{1}{r}\partial_n\psi_i(y)\,\mathrm{d}x\mathrm{d}y\right|\,\lesssim\, \epsilon^{\frac{3}{2}}|\log\epsilon|^\ell\left\lVert \frac{1}{r}\partial_n\psi_i \right\rVert_{L^2(\partial B_i)}\\ &\lesssim\, \epsilon|\log\epsilon|^\ell.\notag\end{aligned}$$ The same argument can also be made for all terms involving one or two derivatives of $\frac{1}{r}\partial_n\psi_i$ or $K_s^{R_j}$ by the estimates in the Lemmata [\[K bd\]](#K bd){reference-type="ref" reference="K bd"}, [\[deri K L2\]](#deri K L2){reference-type="ref" reference="deri K L2"} and Propositions [\[deri mu\]](#deri mu){reference-type="ref" reference="deri mu"} and [\[2nd deri mu\]](#2nd deri mu){reference-type="ref" reference="2nd deri mu"} and also for derivatives with respect to other $R_j$'s or $Z_j$'s. As the second derivative of $\frac{R_i+s}{R_i}$ vanishes, this shows the estimate for the derivative, in the direction $R_j$. Hence we are left with the main contribution where the derivative falls on $\frac{R_i+s}{R_i}$, which is $$\begin{aligned} \int_{(\partial B_i)^2}\frac{1}{R_i}K(x,y)\frac{1}{r^2}\partial_n\psi_i(x)\partial_n\psi_i(y)\,\mathrm{d}x\mathrm{d}y.\end{aligned}$$ It can be rewritten as $$\begin{aligned} &\strokedint_{(\partial B_i)}\frac{1}{R_i}\overline{K}_{R_i}(x,y)\,\mathrm{d}x\mathrm{d}y+\\ &O\left(\epsilon^{-1}\left\lVert K-\overline{K}_{R_i} \right\rVert_{L^2\rightarrow L^2}+\left\lVert K \right\rVert_{L^2\rightarrow L^2}\left\lVert \frac{1}{r}\partial_n\psi_i-\frac{1}{2\pi\epsilon\tilde{\rho_i}} \right\rVert_{L^2}\left\lVert \frac{1}{r}\partial_n\psi_i \right\rVert_{L^2}\right)\notag.\end{aligned}$$ The $O$-term is $\lesssim \epsilon|\log\epsilon|^\ell$ by Lemma [\[K bd\]](#K bd){reference-type="ref" reference="K bd"} and Proposition [\[comp mult\]](#comp mult){reference-type="ref" reference="comp mult"}. We computed in the Claim [\[action K flat\]](#action K flat){reference-type="eqref" reference="action K flat"} that the main integral equals $$\begin{aligned} \frac{1}{2\pi}\left(\log\left(\epsilon\tilde{\rho}_i\right)-\log(8)-2-\log R_i\right).\end{aligned}$$ ◻ Next, we consider the "cross-terms" in $G$, given by the interaction between $\psi_i$ and $\psi_j$. **Proposition 60**. *[\[crossterms G\]]{#crossterms G label="crossterms G"} Let $t^*=(t_1^*,\dots t_k^*)$ be a tangent vector, identified with a vector in $\mathbb{R}^{2k}$ as usual, then for $i\neq j$ we have that $$\begin{aligned} \left|\int_{\bigcup_m\partial B_m}\frac{1}{r}u(t^*)\partial_n\psi_i\partial_n\psi_j\,\mathrm{d}x-t_j^*\cdot\nabla_yK(q_i,q_j)-t_i^*\cdot\nabla_yK(q_j,q_i)\right|\,\lesssim\, \epsilon|\log\epsilon|^\ell,\end{aligned}$$* *where the normal velocity $u(t^*)$ was defined in [\[normal velo\]](#normal velo){reference-type="eqref" reference="normal velo"}. Furthermore it holds that $$\begin{aligned} \left|\partial_q G(q,\gamma)\right|\,\lesssim\, \max_{i,j}|q_i-q_j|^{-2}|\gamma|^2.\end{aligned}$$* *Both of these estimates are locally uniform in $\tilde{q}$.* *Proof.* We first prove the first statement for the contribution of $\partial B_j$, which also covers the contribution of $\partial B_i$ by symmetry. We have$$\begin{aligned} \label{1st step} &\int_{\partial B_j}\frac{1}{r}\left(t_j^*\cdot n-\frac{\rho_j}{2R_j}t_j^*\cdot e_R\right)\partial_n\psi_j\partial_n\psi_i\,\mathrm{d}x\notag\\ &\,=\,\int_{\partial B_j} \frac{1}{r}t_j^*\cdot n\partial_n \psi_i\partial_n\psi_j\,\mathrm{d}x+O\left(\epsilon\left\lVert \partial_n\psi_i \right\rVert_{L^2(\partial B_j)}\left\lVert \partial_n\psi_j \right\rVert_{L^2(\partial B_j)}\right).\end{aligned}$$ By Propositions [\[comp mult\]](#comp mult){reference-type="ref" reference="comp mult"} and [\[1st order exp\]](#1st order exp){reference-type="ref" reference="1st order exp"} the error term here is $\lesssim \epsilon|\log\epsilon|^\ell$. We can now use Proposition [\[1st order exp\]](#1st order exp){reference-type="ref" reference="1st order exp"} and [\[1st step\]](#1st step){reference-type="eqref" reference="1st step"} to obtain that $$\begin{aligned} \int_{\partial B_j}\frac{1}{r}u(t^*)\partial_n\psi_i\partial_n\psi_j\,\mathrm{d}x\,=\,2\int_{\partial B_j}(t^*\cdot n)n\cdot\nabla_yK(q_i,q_j)\frac{1}{r}\partial_n\psi_j\,\mathrm{d}x+O\left(\epsilon|\log\epsilon|^\ell\right).\end{aligned}$$ We further have $$\begin{aligned} &2\int_{\partial B_j}(t^*\cdot n)n\cdot\nabla_yK(q_i,q_j)\frac{1}{r}\partial_n\psi_j\,\mathrm{d}x\,=\,2\strokedint_{\partial B_j} (t^*\cdot n)n\cdot\nabla_yK(q_i,q_j)\,\mathrm{d}x\\ &+O\left(\epsilon^\frac{1}{2}|\nabla_yK(q_i,q_j)|\left\lVert \frac{1}{r}\partial_n\psi_j-\frac{1}{2\pi\epsilon\tilde{\rho}_j} \right\rVert_{L^2(\partial B_j)}\right).\notag\end{aligned}$$ By Proposition [\[comp mult\]](#comp mult){reference-type="ref" reference="comp mult"}, the error term is $\lesssim \epsilon|\log\epsilon|^\ell$. The main integral equals $t^*\cdot\nabla_yK(q_i,q_j)$.\ For $m$ with $m\neq i,j$, one can directly see by Proposition [\[1st order exp\]](#1st order exp){reference-type="ref" reference="1st order exp"} that the integral is $\lesssim \epsilon|\log\epsilon|^\ell$. It remains to estimate the derivative. Note that $G$ is a quadratic form in $\gamma$ and that we have already shown the statement for $\gamma=e_l$ in Proposition [\[diagonal A\]](#diagonal A){reference-type="ref" reference="diagonal A"}, and that the "off-diagonal" coefficients in $G$ are exactly the integrals we estimated in the first step, so we need to estimate their derivatives. For notational simplicity we only consider the derivative of the integral on $\partial B_j$ with respect to $R_l$, as the derivative with respect to $Z_l$ enjoys the same estimates, this is not restrictive. We begin with the derivative with respect to $R_l$ for $l\neq j$. By Propositions [\[deri mu\]](#deri mu){reference-type="ref" reference="deri mu"} it holds that $$\begin{aligned} \left\lVert \partial_s\frac{1}{r}\partial_n\psi_{i,R_l+s} \right\rVert_{L^2(\partial B_j)}\,\lesssim\, \epsilon^\frac{1}{2}|q_i-q_j|^{-2}\quad\text{and}\quad \left\lVert \partial_s\frac{1}{r}\partial_n\psi_{j,R_i+s} \right\rVert_{L^2(\partial B_j)}\,\lesssim\, \epsilon^\frac{1}{2}|\log\epsilon|^\ell,\end{aligned}$$ which by the Cauchy-Schwarz inequality and Propositions [\[comp mult\]](#comp mult){reference-type="ref" reference="comp mult"} and [\[1st order exp\]](#1st order exp){reference-type="ref" reference="1st order exp"} implies the statement.\ Finally, consider the derivative with respect to $R_j$. We can rewrite the integral as $$\begin{aligned} \int_{\partial B_j(R_j)}\left(t^*\cdot n-\frac{\rho_j(R_j+s)}{2(R_j+s)}t^*\cdot e_R\right)\left(\delta_j^s\frac{1}{r}\partial_n\psi_{j,R_j+s}\right)\circ c\left(\partial_n\psi_{i,R_j+s}\right)\circ c\,\mathrm{d}x.\end{aligned}$$ Here the factor $\delta_j^s$ is the Jacobian due to the coordinate change. Using Proposition [\[deri mu\]](#deri mu){reference-type="ref" reference="deri mu"}, we see that $$\begin{aligned} \left\lVert \partial_s\left(\delta_j^s\left(\frac{1}{r}\partial_n\psi_{j,R_j+s}\right)\circ c\right) \right\rVert_{L^2(\partial B_j)}\,\lesssim\, \epsilon^{\frac{1}{2}}|\log\epsilon|^\ell.\end{aligned}$$ Furthermore by Proposition [\[deri mu\]](#deri mu){reference-type="ref" reference="deri mu"} we have $$\begin{aligned} \left\lVert \partial_s\left(\left(\frac{1}{r}\partial_n\psi_{i,R_j+s}\right)\circ c\right) \right\rVert_{L^2(\partial B_j)}\,\lesssim\, \epsilon^\frac{1}{2}|q_i-q_j|^{-2}\end{aligned}$$ Hence by the Cauchy-Schwarz inequality we conclude. ◻ ## The mixed term $A$ {#subsec4.2} We estimate the force $A$ (defined in [\[main def\]](#main def){reference-type="ref" reference="main def"}), which contains both the stream function and the potentials. **Proposition 61**. *[\[conv A\]]{#conv A label="conv A"} For all $s^*,t^*\in T_qM$ we have that $$\begin{aligned} (A(q,\gamma)t^*)\cdot s^*\rightarrow (t^*)^T\begin{pmatrix} 0 & R_1\gamma_1 &0 &\dots\\ -R_1\gamma_1 & 0 & \dots\\ \dots\\ & & \dots & 0 & R_k\gamma_k\\ & &\dots & -R_k\gamma_k & 0\end{pmatrix}s^*\end{aligned}$$* *with a rate of $O(\epsilon|\log\epsilon|^\ell)$ locally uniformly in $\tilde{q}$.* *Furthermore, it holds that $$\begin{aligned} \left|\partial_q A\right|\,\lesssim\,\epsilon|\log\epsilon|^\ell,\end{aligned}$$* *locally uniformly in $\tilde{q}$.* In particular, $A$ is invertible for small enough $\epsilon$ with an inverse of order $\lesssim 1$ by the assumption that all $\gamma_i$'s are $\neq 0$. *Proof.* Recall that $A$ was defined as $$\begin{aligned} (A(q,\gamma)t^*)\cdot s^*\,=\,\sum_l\int_{\partial B_l}\Bigl(-\partial_\tau\phi(s^*)\partial_n\phi(t^*)+\partial_\tau\phi(t^*)\partial_n\phi(s^*)\Bigr)\partial_n\sum_j\gamma_j\psi_j\,\mathrm{d}x,\end{aligned}$$ where $\phi(s^*)$ and $\phi(t^*)$ are the summed up potentials. Without loss of generality, we may assume $|t^*|=|s^*|=1$, $t^*\in T_{q_i}M$, $s^*\in T_{q_j}M$, and that only $\gamma_l$ is nonzero. We first show the convergence for $i\neq j$. By definition we have $$\begin{aligned} \left\lVert \partial_n\phi_{i,t^*} \right\rVert_{L^\infty(\partial B_i)}\,\lesssim\, 1\quad\text{and}\quad \left\lVert \partial_n\phi_{j,s^*} \right\rVert_{L^\infty(\partial B_j)}\,\lesssim\, 1\end{aligned}$$ and on all other boundaries the normal derivatives vanish. By Corollary [\[MultPotSpec\]](#MultPotSpec){reference-type="ref" reference="MultPotSpec"}, we have $$\begin{aligned} \left\lVert \partial_\tau\phi_{i,t^*} \right\rVert_{L^2(\partial B_j)}\,\lesssim\, \epsilon^\frac{5}{2}|\log\epsilon|^\ell\end{aligned}$$ and vice versa. Furthermore we have $$\begin{aligned} \left\lVert \partial_n\psi_l \right\rVert_{L^2(\partial B_i)},\left\lVert \partial_n\psi_l \right\rVert_{L^2(\partial B_j)}\,\lesssim\, \epsilon^{-\frac{1}{2}}\end{aligned}$$ for all $l$ by the Propositions [\[comp mult\]](#comp mult){reference-type="ref" reference="comp mult"} and [\[1st order exp\]](#1st order exp){reference-type="ref" reference="1st order exp"}. By the Cauchy-Schwarz inequality we conclude convergence to $0$. Similarly, we can directly estimate the derivative. By the Propositions [\[prop deri\]](#prop deri){reference-type="ref" reference="prop deri"} and [\[deri mu\]](#deri mu){reference-type="ref" reference="deri mu"}, we know that all derivatives of the boundary values enjoy estimates which are at worst an order $|\log\epsilon|^\ell$ worse, hence these derivatives are small by the Cauchy-Schwarz inequality and the product rule. Next, consider the case $i=j\neq l$. Here we again have $$\begin{aligned} \left\lVert \partial_\tau\phi_{i,t^*} \right\rVert_{L^2(\partial B_i)}\,\lesssim\, \epsilon^\frac{1}{2}\end{aligned}$$ by Corollary [\[MultPotSpec\]](#MultPotSpec){reference-type="ref" reference="MultPotSpec"} and the same holds for the potential with respect to $s^*$. On the other hand we also have $$\begin{aligned} \left\lVert \partial_n\psi_l \right\rVert_{L^2(\partial B_i)}\,\lesssim\,\epsilon^\frac{1}{2}|\log\epsilon|^\ell\end{aligned}$$ by Proposition [\[1st order exp\]](#1st order exp){reference-type="ref" reference="1st order exp"}. By the Cauchy-Schwarz inequality, this implies that $$\begin{aligned} \left|\int_{\partial B_i}\partial_n\psi_l\left(-\partial_\tau\phi_{i,s^*}\partial_n\phi_{i,t^*}+\partial_\tau\phi_{i,t^*}\partial_n\phi_{i,s^*}\right)\,\mathrm{d}x\right|\,\lesssim\,\epsilon|\log\epsilon|^\ell.\end{aligned}$$ The smallness of the derivative of this term again follows from the fact that all derivatives have estimates which are at worst an order $|\log\epsilon|^\ell$ worse by the Propositions [\[prop deri\]](#prop deri){reference-type="ref" reference="prop deri"} and [\[deri mu\]](#deri mu){reference-type="ref" reference="deri mu"}.\ It remains to consider the case $i=j=l$. In this case we have the same estimates as above for the tangential derivatives and furthermore by Corollary [\[MultPotSpec\]](#MultPotSpec){reference-type="ref" reference="MultPotSpec"} we have $$\begin{aligned} \left\lVert \partial_\tau(\phi_{i,t^*}-\widecheck{\phi}_{t^*}) \right\rVert_{L^2(\partial B_i)}\,\lesssim\, \epsilon^\frac{3}{2}|\log\epsilon|^\ell\quad\text{and}\quad \left\lVert \partial_n(\phi_{i,t^*}-\widecheck{\phi}_{t^*}) \right\rVert_{L^2(\partial B_i)}\,\lesssim\, \epsilon^\frac{3}{2}\end{aligned}$$ where the "two-dimensional" potential $\widecheck{\phi}$ was defined in [\[def flat func\]](#def flat func){reference-type="ref" reference="def flat func"} and the same holds for the potentials with respect to $s^*$. Also by Proposition [\[comp mult\]](#comp mult){reference-type="ref" reference="comp mult"}, we have $$\begin{aligned} \left\lVert \frac{1}{r}\partial_n\psi_i-\frac{1}{2\pi\tilde{\rho}_i\epsilon} \right\rVert_{L^2(\partial B_i)}\,\lesssim\,\epsilon^\frac{1}{2}|\log\epsilon|^\ell.\end{aligned}$$ Finally, we clearly have $\left\lVert r-R_i \right\rVert_{L^2(\partial B_i)}\lesssim \epsilon^\frac{3}{2}$. Hence by the Cauchy-Schwarz inequality we see that $$\begin{aligned} &\int_{\partial B_i}\partial_n\psi_i\left(-\partial_\tau\phi_{i,s^*}\partial_n\phi_{i,t^*}+\partial_\tau\phi_{i,t^*}\partial_n\phi_{i,s^*}\right)\,\mathrm{d}x\notag\\ &\,=\,R_i\strokedint_{\partial B_i}\left(-\partial_\tau\widecheck{\phi}_{s^*}\partial_n\widecheck{\phi}_{t^*}+\partial_\tau\widecheck{\phi}_{t^*}\partial_n\widecheck{\phi}_{s^*}\right)\,\mathrm{d}x+O\left(\epsilon|\log\epsilon|^\ell\right).\end{aligned}$$ By the antisymmetry of these integrals with respect to $t^*$ and $s^*$ it suffices to consider the case $t^*=e_1$ and $s^*=e_2$. In this case, we can use the explicit form of $\widecheck{\phi}_{t^*}$ in [\[def flat func\]](#def flat func){reference-type="ref" reference="def flat func"} to see that $$\begin{aligned} \strokedint_{\partial B_i}\left(-\partial_\tau\widecheck{\phi}_{s^*}\partial_n\widecheck{\phi}_{t^*}+\partial_\tau\widecheck{\phi}_{t^*}\partial_n\widecheck{\phi}_{s^*}\right)\,\mathrm{d}x\,=\,\strokedint \tau\cdot e_2n\cdot e_1-\tau\cdot e_1n\cdot e_2\,\mathrm{d}x\,=\,1.\end{aligned}$$ The smallness of the derivative follows again from the fact that all the derivatives of the boundary values have estimates which are an order $\epsilon|\log\epsilon|^\ell$ better by Propositions [\[prop deri\]](#prop deri){reference-type="ref" reference="prop deri"} and [\[deri mu\]](#deri mu){reference-type="ref" reference="deri mu"}. Finally, all these estimates are locally uniform in $\tilde{q}$ because all the used estimates for the boundary values are. ◻ **Definition 5**. *We let $J_\gamma^1$ and $J_\gamma^2$ be the velocities in [\[PointSystem2\]](#PointSystem2){reference-type="eqref" reference="PointSystem2"} and [\[PointSystem1\]](#PointSystem1){reference-type="eqref" reference="PointSystem1"}, i.e. $$\begin{aligned} &(J_\gamma^1(\tilde{q}))_i\,=\,\frac{1}{2\pi}\sum_{j\neq i}\gamma_j\frac{(\tilde{q}_i-\tilde{q}_j)^\perp}{|\tilde{q}_i-\tilde{q}_j|^2}-\frac{\gamma_i}{4\pi R_0}e_Z\\ &(J_\gamma^2(\tilde{q}))_i\,=\,\frac{1}{2\pi}\sum_{j\neq i}\gamma_j\frac{(\tilde{q}_i-\tilde{q}_j)^\perp}{|\tilde{q}_i-\tilde{q}_j|^2}+\frac{\tilde{q}_{R_i}\gamma_i}{4\pi R_0^2}e_Z\end{aligned}$$* **Corollary 62**. *[\[conv J1\]]{#conv J1 label="conv J1"} In the Regime [\[def regime2\]](#def regime2){reference-type="eqref" reference="def regime2"} (=distances $\approx|\log\epsilon|$), we have that $$\begin{aligned} \frac{A^{-1}G}{|\log\epsilon|}\rightarrow -J_\gamma^1(\tilde{q})\end{aligned}$$* *locally uniformly in $\tilde{q}$.* *Proof.* By the Propositions [\[diagonal A\]](#diagonal A){reference-type="ref" reference="diagonal A"} and [\[conv A\]](#conv A){reference-type="ref" reference="conv A"} have that $$\begin{aligned} \frac{{A(q,\gamma)}^{-1}G(q,e_i)}{|\log\epsilon|}\rightarrow \frac{1}{4\pi R_0\gamma_i}e_{Z,i},\end{aligned}$$ where $e_{Z,i}\in (\mathbb{R}^2)^k\simeq \mathbb{R}^{2k}$ denotes the vector which has an $e_Z$ in the $i$-th component and no other entries. As $G$ is a quadratic form in $\gamma$ by definition, it remains to show the statement for the "off-diagonal" terms in $G$. By the Propositions [\[crossterms G\]](#crossterms G){reference-type="ref" reference="crossterms G"} and [\[conv A\]](#conv A){reference-type="ref" reference="conv A"} we have that $$\begin{aligned} &\lim_{\epsilon\rightarrow 0}\frac{A(q,\gamma)^{-1}\left(G(q,e_i+e_j)-G(q,e_i)-G(q,e_j)\right)}{|\log\epsilon|}\\ &=\,- \lim_{\epsilon\rightarrow 0}\frac{1}{R_0|\log\epsilon|} \left(\frac{P_j}{\gamma_j}\nabla_y^\perp K(q_i,q_j)+\frac{P_i}{\gamma_i}\nabla_y^\perp K(q_j,q_i)\right)\\ &=\, \frac{1}{2\pi}\left(\frac{P_i(\tilde{q}_j-\tilde{q}_i)^\perp}{\gamma_i|\tilde{q}_i-\tilde{q}_j|^2}+\frac{P_j(\tilde{q}_i-\tilde{q}_j)^\perp}{\gamma_j|\tilde{q}_i-\tilde{q}_j|^2}\right).\end{aligned}$$ Here $P_l:\mathbb{R}^2\rightarrow (\mathbb{R}^2)^k$ is the map to the $l$-th coordinate and in the last step we used the Asymptotics [\[exp K\]](#exp K){reference-type="eqref" reference="exp K"} and [\[exp F\]](#exp F){reference-type="eqref" reference="exp F"}. ◻ **Corollary 63**. *[\[conv J2\]]{#conv J2 label="conv J2"} In the Regime [\[def regime1\]](#def regime1){reference-type="eqref" reference="def regime1"} (=distances $\approx|\log\epsilon|^\frac{1}{2}$), we have that $$\begin{aligned} \frac{A^{-1}G}{|\log\epsilon|^\frac{1}{2}}-\frac{|\log\epsilon|^\frac{1}{2}}{4\pi R_0}v_Z\rightarrow -J_\gamma^2(\tilde{q})\end{aligned}$$* *locally uniformly in $\tilde{q}$, where $v_Z\in (\mathbb{R}^2)^k$ is the vector $(e_Z,e_Z,\dots)$.* *Proof.* The calculation of the "off-diagonal" terms is the same as in the previous proof. For the diagonal terms, we have by the Propositions [\[diagonal A\]](#diagonal A){reference-type="ref" reference="diagonal A"} and [\[conv A\]](#conv A){reference-type="ref" reference="conv A"} that $$\begin{aligned} \frac{A(q,\gamma)^{-1}G(q,e_i)}{|\log\epsilon|^\frac{1}{2}}-\frac{|\log\epsilon|^\frac{1}{2}}{4\gamma_i\pi R_i}\rightarrow 0.\end{aligned}$$ The statement then follows from the definition of $\tilde{q}$ and the fact that e.g. by the mean value theorem we have $$\begin{aligned} \frac{|\log\epsilon|^\frac{1}{2}}{4\pi R_i}-\frac{|\log\epsilon|^\frac{1}{2}}{4\pi R_0}+\frac{(R_i-R_0)|\log\epsilon|^\frac{1}{2}}{4\pi R_0^2}\rightarrow 0.\end{aligned}$$ ◻ # Passage to the limit {#Section5} In this section, we write $\tilde{q}_\epsilon$ instead of $\tilde{q}$ to emphasize the $\epsilon$-dependence. *Proof of Thm. [\[Limit1\]](#Limit1){reference-type="ref" reference="Limit1"}.* We write the system [\[MainODE\]](#MainODE){reference-type="eqref" reference="MainODE"} in the rescaled time $s=t|\log\epsilon|^2$ and the rescaled position $\tilde{q}_\epsilon$, defined as in [\[def regime2\]](#def regime2){reference-type="eqref" reference="def regime2"}, it then reads as $$\begin{aligned} \label{rescaled ODE} &|\log\epsilon|^3\left(E(q)\tilde{q}_{\epsilon}''+\frac{1}{2}\tilde{q}_{\epsilon}'(\nabla_{\tilde{q}_\epsilon}E(q)\cdot\tilde{q}_{\epsilon}')+\mathcal{M}(q)\tilde{q}_\epsilon''+|\log\epsilon|^{-1}\langle\Gamma(q),\tilde{q}_{\epsilon}',\tilde{q}_{\epsilon}'\rangle\right)\\ &\,=\,G(q,\gamma)+|\log\epsilon|(A(q,\gamma)\tilde{q}'),\notag\end{aligned}$$ where the time derivatives are denoted with a $'$ and all derivatives taken with respect to the rescaled time and space. We first show that the velocity in rescaled time and space is bounded, until either we approach the boundary or some component of $\tilde{q}$ goes to infinity. We take $C$ as some large compact subset of $M_\epsilon$ containing $\tilde{q}_\epsilon(0)$. If $\tilde{q}_\epsilon\in C$, then this implies that each $q_i$ lies in a compact subset of $\mathbb{H}$. It follows from the definition of $M_\epsilon$ that it we view the $M_\epsilon$'s as subsets of $\mathbb{R}^{2k}$ that for small enough $\epsilon$ such a set $C$ is also a subset of $M_{\epsilon'}$ for $\epsilon'<\epsilon$. Hence $C$ can be chosen as the same set for all small enough $\epsilon$. Recall further that the matrix $A$ is invertible by Proposition [\[conv A\]](#conv A){reference-type="ref" reference="conv A"} and that its inverse is $\lesssim 1$ as long as $\tilde{q}_\epsilon\in C$. We may hence rewrite the equation as $$\begin{aligned} &|\log\epsilon|^3\bigg(\left(E(q)+\mathcal{M}(q)\right)\left(\frac{\text{d}}{\text{d}s}\left(\tilde{q}_\epsilon'+\frac{A^{-1}G}{|\log\epsilon|}\right)\right)+\frac{1}{2}\left(\nabla_{\tilde{q}_\epsilon}E\cdot\tilde{q}_\epsilon'\right)\left(\tilde{q}_\epsilon'+\frac{A^{-1}G}{|\log\epsilon|}\right)\\ &\quad\quad+\frac{1}{2}\left(\nabla_{\tilde{q}_\epsilon}\mathcal{M}\cdot\tilde{q}_\epsilon'\right)\left(\tilde{q}_\epsilon'+\frac{A^{-1}G}{|\log\epsilon|}\right)\bigg)\\ &-|\log\epsilon|^3\bigg(\left(E\left(q\right)+\mathcal{M}\left(q\right)\right)\frac{\text{d}}{\text{d}s}\frac{A^{-1}G}{|\log\epsilon|}+\frac{1}{2}\left(\nabla_{\tilde{q}_\epsilon}E\cdot\tilde{q}_\epsilon'\right)\frac{A^{-1}G}{|\log\epsilon|}+|\log\epsilon|^{-1}\left\langle \Gamma\left(q\right),\tilde{q}_\epsilon',\frac{A^{-1}G}{|\log\epsilon|}\right\rangle\\ &\quad\quad-\left\langle |\log\epsilon|^{-1}\Gamma\left(q\right)-\frac{1}{2}\nabla_{\tilde{q}_\epsilon}\mathcal{M}\left(q\right),\tilde{q}_\epsilon',\tilde{q}_\epsilon'+\frac{A^{-1}G}{|\log\epsilon|}\right\rangle\bigg)\\ &=\,\frac{1}{|\log\epsilon|}A\left(\tilde{q}_\epsilon'+\frac{A^{-1}G}{|\log\epsilon|}\right) \end{aligned}$$ where we used the notation $\langle N,a,b\rangle=Na\cdot b$ in the penultimate line. By testing against $\tilde{q}_\epsilon'+\frac{A^{-1}G}{|\log\epsilon|}$ and dividing out the $|\log\epsilon|^3$ we obtain that from the antisymmetry of $A$ that $$\begin{aligned} \label{mod energy} &\frac{\text{d}}{\text{d}s}\frac{1}{2}\left(\left(\tilde{q}_\epsilon'+\frac{A^{-1}G}{|\log\epsilon|}\right)^T\left(E(q)+\mathcal{M}(q)\right)\left(\tilde{q}_\epsilon'+\frac{A^{-1}G}{|\log\epsilon|}\right)\right)\notag\\ &=\,\left(\tilde{q}_\epsilon'+\frac{A^{-1}G}{|\log\epsilon|}\right)^T\left(E\left(q\right)+\mathcal{M}\left(q\right)\right)\frac{\text{d}}{\text{d}s}\frac{A^{-1}G}{|\log\epsilon|}\notag\\ &+\frac{1}{2}\left(\tilde{q}_\epsilon'+\frac{A^{-1}G}{|\log\epsilon|}\right)^T\left(\nabla_{\tilde{q}_\epsilon}E\cdot\tilde{q}_\epsilon'\right)\frac{A^{-1}G}{|\log\epsilon|}\\ &+|\log\epsilon|^{-1}\left\langle \Gamma\left(q\right),\tilde{q}_\epsilon',\frac{A^{-1}G}{|\log\epsilon|}\right\rangle\left(\tilde{q}_\epsilon'+\frac{A^{-1}G}{|\log\epsilon|}\right)\notag\\ &-\left\langle|\log\epsilon|^{-1} \Gamma\left(q\right)-\frac{1}{2}\nabla_{\tilde{q}_\epsilon}\mathcal{M}\left(q\right),\tilde{q}_\epsilon',\tilde{q}_\epsilon'+\frac{A^{-1}G}{|\log\epsilon|}\right\rangle\left(\tilde{q}_\epsilon'+\frac{A^{-1}G}{|\log\epsilon|}\right)\notag\\ &=:\, I+II+III+IV\notag\end{aligned}$$ where $I-IV$ stand for the terms in each line. Our goal is to show that each of these terms is $\lesssim \epsilon^2(1+|\tilde{q}_\epsilon'|^2+\epsilon|\log\epsilon|^\ell|\tilde{q}_\epsilon'|^3)$ as long as $\tilde{q}_\epsilon\in C$. For $\tilde{q}_\epsilon\in C$ we have the following estimates: By the Lemmata [\[InteriorField\]](#InteriorField){reference-type="ref" reference="InteriorField"} and [\[conv Inertia\]](#conv Inertia){reference-type="ref" reference="conv Inertia"}, we have $$\begin{aligned} \label{bd EM} \epsilon^2\,\lesssim\, E+\mathcal{M}\,\lesssim\, \epsilon^2\end{aligned}$$ (in the sense that the smallest and highest eigenvalues have these bounds). Furthermore by Proposition [\[conv A\]](#conv A){reference-type="ref" reference="conv A"} we have $$\begin{aligned} \label{bd A} |A|,|\nabla_{\tilde{q}}A|, |A^{-1}|,|\nabla_{\tilde{q}}A^{-1}|\,\lesssim\, 1.\end{aligned}$$ By the Propositions [\[diagonal A\]](#diagonal A){reference-type="ref" reference="diagonal A"} and [\[crossterms G\]](#crossterms G){reference-type="ref" reference="crossterms G"}, we have $$\begin{aligned} \label{bd G} |G|, |\nabla_{\tilde{q}} G|\,\lesssim\, |\log\epsilon|,\end{aligned}$$ finally by the Propositions [\[InteriorField\]](#InteriorField){reference-type="ref" reference="InteriorField"} b) and [\[conv Gamma\]](#conv Gamma){reference-type="ref" reference="conv Gamma"}, we have $$\begin{aligned} \label{bd deri M} |\nabla_{\tilde{q}_\epsilon}E|,|\Gamma|\,\lesssim\, \epsilon^3|\log\epsilon|^\ell.\end{aligned}$$ Hence we conclude for $\tilde{q}_\epsilon\in C$ that $$\begin{aligned} &|I|\,\lesssim\,\epsilon^2(1+|\tilde{q}_\epsilon'|^2)\\ &|II|,|III|\,\lesssim\,\epsilon^3|\log\epsilon|^\ell(1+|\tilde{q}_\epsilon'|^3).\end{aligned}$$ We have that $IV=0$. Indeed if we set $p:=\tilde{q}_\epsilon'+\frac{A^{-1}G}{|\log\epsilon|}$ then, by the definition of $\Gamma$ ([\[main def\]](#main def){reference-type="ref" reference="main def"}), it holds $$\begin{aligned} &\langle\Gamma(q),\dot{q},p\rangle p\,=\,\sum_{i,j,k}\frac{1}{2}(\partial_j\mathcal{M}(q)_{ik}+\partial_i\mathcal{M}(q)_{jk}-\partial_k\mathcal{M}(q)_{ij})\dot{q}_ip_jp_k\\ &=\,\sum_{i,j,k}\frac{1}{2}\partial_i\mathcal{M}(q)_{jk}\dot{q}_ip_jp_k\,=\,\frac{1}{2|\log\epsilon|}(\nabla_{\tilde{q}_\epsilon}\mathcal{M}\cdot\dot{q})p\cdot p.\end{aligned}$$ Hence, plugging these estimates into [\[mod energy\]](#mod energy){reference-type="eqref" reference="mod energy"}, we obtain by [\[bd EM\]](#bd EM){reference-type="eqref" reference="bd EM"} that as long as $\tilde{q}_\epsilon\in C$, we have $$\begin{aligned} \label{1st Gronwall} &\frac{\text{d}}{\text{d}s}\left(\left(\tilde{q}_\epsilon'+\frac{A^{-1}G}{|\log\epsilon|}\right)^T\left(E(q)+\mathcal{M}(q)\right)\left(\tilde{q}_\epsilon'+\frac{A^{-1}G}{|\log\epsilon|}\right)\right)\,\lesssim\\ &\epsilon^2+\left(\left(\tilde{q}_\epsilon'+\frac{A^{-1}G}{|\log\epsilon|}\right)^T\left(E(q)+\mathcal{M}(q)\right)\left(\tilde{q}_\epsilon'+\frac{A^{-1}G}{|\log\epsilon|}\right)\right)\notag\\ &+|\log\epsilon|^\ell\left(\left(\tilde{q}_\epsilon'+\frac{A^{-1}G}{|\log\epsilon|}\right)^T\left(E(q)+\mathcal{M}(q)\right)\left(\tilde{q}_\epsilon'+\frac{A^{-1}G}{|\log\epsilon|}\right)\right)^\frac{3}{2}\notag.\end{aligned}$$ As long as we have $$\begin{aligned} \left(\left(\tilde{q}_\epsilon'+\frac{A^{-1}G}{|\log\epsilon|}\right)^T\left(E(q)+\mathcal{M}(q)\right)\left(\tilde{q}_\epsilon'+\frac{A^{-1}G}{|\log\epsilon|}\right)\right)\leq \epsilon^\frac{1}{2}\end{aligned}$$ the last term in [\[1st Gronwall\]](#1st Gronwall){reference-type="eqref" reference="1st Gronwall"} can be absorbed in the first two and by Gronwall we obtain that $$\begin{aligned} &\left(\left(\tilde{q}_\epsilon'+\frac{A^{-1}G}{|\log\epsilon|}\right)^T\left(E(q)+\mathcal{M}(q)\right)\left(\tilde{q}_\epsilon'+\frac{A^{-1}G}{|\log\epsilon|}\right)\right)(s)\\ &\lesssim\, e^s\left(\epsilon^2+\left(\tilde{q}_\epsilon'+\frac{A^{-1}G}{|\log\epsilon|}\right)^T(E(q)+\mathcal{M}(q))\left(\tilde{q}_\epsilon'+\frac{A^{-1}G}{|\log\epsilon|}\right)(0)\right).\end{aligned}$$ By [\[bd EM\]](#bd EM){reference-type="eqref" reference="bd EM"} and the assumption about the initial velocities, this implies that $$\begin{aligned} \left|\tilde{q}_\epsilon'+\frac{A^{-1}G}{|\log\epsilon|}\right|\,\lesssim\, e^s\end{aligned}$$ and hence $$\begin{aligned} |\tilde{q}_\epsilon'|\,\lesssim\, e^s\end{aligned}$$ until either $\tilde{q}_\epsilon$ leaves the set $C$ or up to a time of order $|\log\epsilon|$. By [\[bd A\]](#bd A){reference-type="eqref" reference="bd A"}, [\[bd G\]](#bd G){reference-type="eqref" reference="bd G"} and [\[rescaled ODE\]](#rescaled ODE){reference-type="eqref" reference="rescaled ODE"}, this implies that $$\begin{aligned} |\log\epsilon|^3\left(E(q)\tilde{q}_{\epsilon}''+\frac{1}{2}\tilde{q}_{\epsilon}'(\nabla_{\tilde{q}_\epsilon}E(q)\cdot\tilde{q}_{\epsilon}')+\mathcal{M}(q)\tilde{q}_{\epsilon}''+|\log\epsilon|^{-1}\langle\Gamma(q),\tilde{q}_{\epsilon}',\tilde{q}_{\epsilon}'\rangle\right)\rightarrow 0 \end{aligned}$$ in $W^{-1,\infty}$ up to a time of order $|\log\epsilon|$ or until $\tilde{q}_\epsilon$ leaves $C$. Hence we obtain that $$\begin{aligned} A\tilde{q}_\epsilon'+\frac{G}{|\log\epsilon|}\overset{\ast}{\rightharpoonup} 0 \text{ in $L^\infty$}.\end{aligned}$$ Because $A^{-1}$ and $\frac{G}{|\log\epsilon|}$ converge strongly, we see by Corollary [\[conv J1\]](#conv J1){reference-type="ref" reference="conv J1"} that $$\begin{aligned} \tilde{q}_\epsilon'- J_\gamma^1(\tilde{q}_\epsilon)\overset{\ast}{\rightharpoonup}0 \text{ in $L^\infty$}\end{aligned}$$ until $\tilde{q}_\epsilon$ leaves $C$, which takes at least $\gtrsim 1$ time, as $\tilde{q}_\epsilon'$ is bounded. Hence we have that $$\begin{aligned} \tilde{q}_\epsilon(v)-\tilde{q}_\epsilon(0)-\int_0^v J_\gamma^1(\tilde{q}_\epsilon(s))\mathrm{d}s\rightarrow 0\end{aligned}$$ for all times $v\lesssim 1$ and $\tilde{q}_\epsilon$ converges locally uniformly to some $\tilde{q}_0$ by compactness, as long as long as $\tilde{q}_\epsilon$ lies in $C$. Therefore we see that $\tilde{q}_0$ must be a solution of $\tilde{q}_0'=J_\gamma^1(\tilde{q}_0)$ because $J_\gamma^1$ is locally uniformly continuous.\ Finally, we may remove the condition that $\tilde{q}_\epsilon$ lies in a compact set $C$, by taking $C$ so large that the solution of $q'=J_\gamma^1(q)$ lies in the interior of $C$ until some time $T$, which is possible for small enough $\epsilon$ whenever $q$ does not blow until time $T$. Then we have uniform convergence of $\tilde{q}_\epsilon$ as long as it lies in $C$. As the limit lies in the interior of $C$, the solution $\tilde{q}_\epsilon$ also lies in $C$ for small enough $\epsilon$ up to time $T$. Hence we have convergence, as long as the limiting solution does not blow up. ◻ *Proof of Thm. [\[Limit2\]](#Limit2){reference-type="ref" reference="Limit2"}.* The proof is quite similar to the previous one. In the rescaled time $s=|\log\epsilon|t$, and the rescaled spatial variable $\tilde{q}_\epsilon$, defined as in [\[def regime1\]](#def regime1){reference-type="eqref" reference="def regime1"} the System [\[MainODE\]](#MainODE){reference-type="eqref" reference="MainODE"} reads as $$\begin{aligned} \label{rescaled ODE 2} &|\log\epsilon|^\frac{3}{2}\left(E(q)\tilde{q}_{\epsilon}''+\frac{1}{2}\tilde{q}_{\epsilon}'(\nabla_{\tilde{q}_\epsilon}E(q)\cdot\tilde{q}_{\epsilon}')+\mathcal{M}(q)\tilde{q}_\epsilon''+|\log\epsilon|^{-\frac{1}{2}}\langle\Gamma(q),\tilde{q}_{\epsilon}',\tilde{q}_{\epsilon}'\rangle\right)\\ &\,=\,G(q,\gamma)+|\log\epsilon|^\frac{1}{2}(A(q,\gamma)\tilde{q}_{\epsilon}').\notag\end{aligned}$$ Similarly as in the previous proof we can rewrite the equation, tested against $\tilde{q}_\epsilon'+\frac{A^{-1}G}{|\log\epsilon|^\frac{1}{2}}$ as $$\begin{aligned} &\frac{\text{d}}{\text{d}s}\frac{1}{2}\left(\left(\tilde{q}_\epsilon'+\frac{A^{-1}G}{|\log\epsilon|^\frac{1}{2}}\right)^T\left(E(q)+\mathcal{M}(q)\right)\left(\tilde{q}_\epsilon'+\frac{A^{-1}G}{|\log\epsilon|^\frac{1}{2}}\right)\right)\notag\\ &=\,\left(\tilde{q}_\epsilon'+\frac{A^{-1}G}{|\log\epsilon|^\frac{1}{2}}\right)^T\left(E\left(q\right)+\mathcal{M}\left(q\right)\right)\frac{\text{d}}{\text{d}s}\left(\frac{A^{-1}G}{|\log\epsilon|^\frac{1}{2}}\right)\notag\\ &+\frac{1}{2}\left(\tilde{q}_\epsilon'+\frac{A^{-1}G}{|\log\epsilon|^\frac{1}{2}}\right)^T\left(\nabla_{\tilde{q}_\epsilon}E\cdot\tilde{q}_\epsilon'\right)\left(\frac{A^{-1}G}{|\log\epsilon|^\frac{1}{2}}\right)\\ &+|\log\epsilon|^{-\frac{1}{2}}\left\langle \Gamma\left(q\right),\tilde{q}_\epsilon',\frac{A^{-1}G}{|\log\epsilon|^\frac{1}{2}}\right\rangle\left(\tilde{q}_\epsilon'+\frac{A^{-1}G}{|\log\epsilon|^\frac{1}{2}}\right)\notag\\ &-\left\langle |\log\epsilon|^{-\frac{1}{2}}\Gamma\left(q\right)-\frac{1}{2}\nabla_{\tilde{q}_\epsilon}\mathcal{M}\left(q\right),\tilde{q}_\epsilon',\tilde{q}_\epsilon'+\frac{A^{-1}G}{|\log\epsilon|^\frac{1}{2}}\right\rangle\left(\tilde{q}_\epsilon'+\frac{A^{-1}G}{|\log\epsilon|^\frac{1}{2}}\right)\notag\\ &=:\, I+II+III+IV\notag.\end{aligned}$$ Let $v_Z$ denote the vector $(e_Z,e_Z,\dots)\in \mathbb{R}^{2k}$. We would like to estimate the shifted velocity $\tilde{q}_\epsilon'+\frac{|\log\epsilon|^\frac{1}{2}}{4\pi R_0}v_Z$. We again use a compact set $C\subset M_\epsilon$ containing $\tilde{q}_\epsilon(0)$. Then we let $\tilde{C}:=C+v_Z\mathbb{R}$. On this set we still have uniform estimates because the system is invariant in the $v_Z$ direction. If $\tilde{q}_\epsilon\in \tilde{C}$, then we clearly we still have the estimates [\[bd EM\]](#bd EM){reference-type="eqref" reference="bd EM"} and [\[bd deri M\]](#bd deri M){reference-type="eqref" reference="bd deri M"}. Furthermore by the Propositions [\[diagonal A\]](#diagonal A){reference-type="ref" reference="diagonal A"} and [\[crossterms G\]](#crossterms G){reference-type="ref" reference="crossterms G"} we then also have $$\begin{aligned} |G|\,\lesssim\, |\log\epsilon|\end{aligned}$$ and $$\begin{aligned} |\nabla_{\tilde{q}_\epsilon}G|\,\lesssim\, |\log\epsilon|^\frac{1}{2}.\end{aligned}$$ Furthermore by Proposition [\[conv A\]](#conv A){reference-type="ref" reference="conv A"} we have $$\begin{aligned} |A|\,\lesssim\, 1,\quad |\nabla_{\tilde{q}_\epsilon}A|\,\lesssim\, |\log\epsilon|^{-\frac{1}{2}}\end{aligned}$$ and from Corollary [\[conv J2\]](#conv J2){reference-type="ref" reference="conv J2"} one sees that $$\begin{aligned} \left|\frac{A^{-1}G}{|\log\epsilon|^\frac{1}{2}}-\frac{|\log\epsilon|^\frac{1}{2}}{4\pi R_0}v_Z\right|\,\lesssim\, 1.\end{aligned}$$ Hence, we directly see that $$\begin{aligned} |II|,|III|\,\lesssim\, \epsilon^3|\log\epsilon|^\ell\left(1+\left|\tilde{q}_\epsilon'+\frac{|\log\epsilon|^\frac{1}{2}}{4\pi R_0}v_Z\right|^3\right).\end{aligned}$$ The term $IV$ again drops out by the same calculation as in the previous proof. Note that we have $$\begin{aligned} \frac{\text{d}}{\text{d}s}\left(\frac{A^{-1}G}{|\log\epsilon|^\frac{1}{2}}\right)\,=\,\frac{1}{|\log\epsilon|^\frac{1}{2}}\left(\left(\nabla_{\tilde{q}_\epsilon}A^{-1}\right)G+A^{-1}\nabla_{\tilde{q}_\epsilon}G\right)\left(\tilde{q}_\epsilon'+\frac{|\log\epsilon|^\frac{1}{2}}{4\pi R_0}v_Z\right)\end{aligned}$$ because the derivative of $A^{-1}G$ in the $v_Z$ direction is $0$, as the system is invariant in that direction. Hence we see that $$\begin{aligned} &|I|\,\lesssim\,\frac{\epsilon^2}{|\log\epsilon|^\frac{1}{2}}\left(\left|\tilde{q}_\epsilon'+\frac{|\log\epsilon|^\frac{1}{2}}{4\pi R_0}v_Z\right|+\left|\frac{A^{-1}G}{|\log\epsilon|^\frac{1}{2}}-\frac{|\log\epsilon|^\frac{1}{2}}{4\pi R_0}v_Z\right|\right)\\ &\quad\times\left(|A^{-1}|^2|\nabla_{\tilde{q}_\epsilon}A||G|+|A^{-1}||\nabla_{\tilde{q}_\epsilon}G|\right)\left|\tilde{q}_\epsilon'+\frac{|\log\epsilon|^\frac{1}{2}}{4\pi R_0}v_Z\right|\\ &\lesssim\,\epsilon^2\left(1+\left|\tilde{q}_\epsilon'+\frac{|\log\epsilon|^\frac{1}{2}}{4\pi R_0}v_Z\right|^2\right).\end{aligned}$$ From the assumption that $$\begin{aligned} \left|\tilde{q}_\epsilon'(0)+\frac{|\log\epsilon|^\frac{1}{2}}{4\pi R_0}v_Z\right|\,\lesssim\, 1\end{aligned}$$ we see by the same Gronwall argument as in the previous proof that $$\begin{aligned} \left|\tilde{q}_\epsilon'(s)+\frac{|\log\epsilon|^\frac{1}{2}}{4\pi R_0}v_Z\right|\,\lesssim\, e^s\end{aligned}$$ until $\tilde{q}_\epsilon$ leaves $\tilde{C}$ or until a time of order $|\log\epsilon|$. From this, we conclude that $\tilde{q}_\epsilon''$ is bounded in $W^{-1,\infty}$ and by [\[bd EM\]](#bd EM){reference-type="eqref" reference="bd EM"} and [\[bd deri M\]](#bd deri M){reference-type="eqref" reference="bd deri M"} we see that $$\begin{aligned} |\log\epsilon|^\frac{3}{2}\left(E(q)\tilde{q}_{\epsilon}''+\frac{1}{2}\tilde{q}_{\epsilon}'(\nabla_{\tilde{q}_\epsilon}E(q)\cdot\tilde{q}_{\epsilon}')+\mathcal{M}(q)\tilde{q}_\epsilon''+|\log\epsilon|^{-\frac{1}{2}}\langle\Gamma(q),\tilde{q}_{\epsilon}',\tilde{q}_{\epsilon}'\rangle\right)\rightarrow 0\end{aligned}$$ in $W^{-1,\infty}$ until $\tilde{q}_\epsilon$ leaves $\tilde{C}$. Hence we see again that $$\begin{aligned} \tilde{q}_\epsilon'+\frac{|\log\epsilon|^\frac{1}{2}}{4\pi R_0}v_Z+\left( \frac{A^{-1}G}{|\log\epsilon|^\frac{1}{2}}-\frac{|\log\epsilon|^\frac{1}{2}}{4\pi R_0}v_Z\right)\,\overset{\ast}{\rightharpoonup}\, 0 \text{ in $L^\infty$}.\end{aligned}$$ This implies the statement by the same argument as in the previous proof and Corollary [\[conv J2\]](#conv J2){reference-type="ref" reference="conv J2"}. ◻ **Acknowledgment:** This work is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy EXC 2044 --390685587, Mathematics Münster: Dynamics--Geometry--Structure. The author would like to thank Christian Seis for introducing him to the problem and both Christian Seis and Franck Sueur for some useful discussions and advice.

arxiv_math

--- abstract: | We extend previous results due to Ding and Zhuang in order to prove that a phase transition occurs for the long range Ising model in lower dimensions. By making use of a recent argument due to Affonso, Bissacot and Maia from 2022 which establishes that a phase transition occurs for the long range, random-field Ising model, from a suggestion of the authors we demonstrate that a phase transition also occurs for the long range Ising model, from a set of appropriately defined contours for the long range system, and a Peierls' argument. [^1] author: - P. Rigas title: Phase transition of the long range Ising model in lower dimensions, for $d < \alpha \leq d + 1$, with a Peierls' argument --- # Introduction ## Overview The random-field Ising model, RFIM, is a model of interest in statistical mechanics, not only for connections with the celebrated Ising model, through the phenomena of ferromagnetism \[8\], but also for connections with the random-field, long-range Ising model which was shown to exhibit a phase transition \[1\], correlation length lower bounds with the greedy lattice animal \[3\], a confirmation of the same scaling holding for the correlation length of the random-field Potts model \[8\], long range order \[4\], Monte Carlo studies \[9\], community structure \[11\], supersymmetry \[13\], and the computation of ground states \[14\]. To extend previous methods for proving that a phase transition occurs in the random-field, long-range Ising model besides only one region of $\alpha$ parameters dependent on the dimension $d$ of the lattice, we implement the argument for analyzing contours, provided in \[1\], for the contours provided in \[2\]. In comparison to arguments for proving that the phase transition occurs in \[1\], in which a variant of the classical Peierls' argument is implemented by reversing the direction of the spins contained within contours $\gamma$, the contours described in \[2\] can be of use for proving that the phase transition for the random-field, long-range Ising model occurs for another range of $\alpha$ parameters, in which $d < \alpha \leq d+1$. Beginning in the next section, after having defined the model, as well as connections that it shares with the random-field, and long-range Ising model, we introduce contour systems for the long range, random-field, and long range Ising models, from which we conclude with a Peierls' argument for proving that a phase transition occurs. ## Long range, random-field Ising model objects To introduce the probability measure for the long range, random-field Ising model, first consider, for a finite volume $\Lambda \subsetneq \textbf{Z}^d$, with $\big| \Lambda \big| < + \infty$, $$\begin{aligned} \mathcal{H}^{\mathrm{LR}, \eta}_{\Lambda} \big( \sigma \big) = - \underset{x,y \in \Lambda}{\sum} J_{x,y} \sigma_x \sigma_y - \underset{y \in \Lambda^c}{\underset{x \in \Lambda}{\sum}} J_{x,y} \sigma_x \eta_y \text{ } \text{ , } \end{aligned}$$ corresponding to the Hamiltonian for the long-range Ising model, in which the spins in the first, and second summation, have coupling constants $\big\{ J_{xy} \big\}_{x,y \in \textbf{Z}^d}$, spins $\sigma_x$ and $\sigma_y$ in $\Lambda$, spin $\eta_y$ in $\Lambda^c$ for the boundary conditions, each of which is drawn from the spin-sample space $\Omega \equiv \big\{ -1 , 1 \big\}^{\textbf{Z}^d}$, with coupling constants, $$\begin{aligned} J_{xy} \equiv J \big| x - y \big|^{-\alpha} \text{ } \text{ , } x \neq y \text{ } \text{ , } \end{aligned}$$ for some strictly positive $J$, $\alpha > d$, and $J_{xy} = 0$ otherwise. The couplings for the Hamiltonian, in both the long-range, and random-field case introduced below, are also intended to satisfy, $$\begin{aligned} \underset{|x|>1}{\underset{x \in \textbf{Z}^d}{\sum}} \big| x_i \big| J_{0,x} < J_{0,e_i} \text{ } \text{ , } \end{aligned}$$ in which the couplings are translation invariant, for every $1 \leq i \leq d$. In the presence of disorder through an external field, specifically through the iid family of Gaussian variables $\big\{ h_x \big\}_{x \in \textbf{Z}^d}$, the long-range, random-field Ising model Hamiltonian takes the form, $$\begin{aligned} \mathcal{H}^{\mathrm{LR-RF}, \eta}_{\Lambda} \equiv \mathcal{H}^{\mathrm{LR-RF}, \eta}_{\Lambda} \big( \sigma , h \big) = \mathcal{H}^{\mathrm{LR}, \eta}_{\Lambda} \big( \sigma) - \underset{x \in \Lambda}{\sum} \epsilon h_x \sigma_x \text{ } \text{ , } \end{aligned}$$ which is also taken under boundary conditions $\eta$, for some strictly positive $\epsilon$. In the summation over $x \in \Lambda$ above besides the long-range Hamiltonian terms, the external field takes the form, $$h_x \equiv \text{ } \left\{\!\begin{array}{ll@{}>{{}}l} h^{*} & \text{, if } x = 0 \text{ } \text{ , } \\ h^{*} \big| x \big|^{-\delta} & \textit{, if } x \neq 0 \text{ , } \\ \end{array}\right.$$ for $\delta, h^{*}>0$. The corresponding Gibbs measure, $$\begin{aligned} \textbf{P}_{\Lambda,\beta} \big( \sigma , h \big) \equiv \textbf{P}^{\mathrm{LR-RF},\eta}_{\Lambda,\beta } \big( \sigma , h \big) \equiv \frac{\mathrm{exp} \big( \beta \mathcal{H}^{\mathrm{LR-RF}, \eta}_{\Lambda}\big) }{Z^{\mathrm{LR-RF},\eta}_{\Lambda ,\beta} \big( h \big) } \text{ } \text{ , } \end{aligned}$$ at inverse temperature $\beta >0$, has the partition function as the normalizing constant so that $\textbf{P} \big( \cdot \big)$ is a probability measure, with, $$\begin{aligned} Z^{\mathrm{LR-RF},\eta}_{\Lambda ,\beta } \big( h \big) \equiv \underset{x \in \Omega^{\eta}_{\Lambda} }{\sum}\mathrm{exp} \big( \beta \mathcal{H}^{\mathrm{LR-RF}, \eta}_{\Lambda}\big) \text{ } \text{ , } \end{aligned}$$ over the sample space $\Omega^{\eta}_{\Lambda}$ of spins with boundary condition $\eta$ over $\Lambda$. Similarly, for the long range Ising model, $$\begin{aligned} \textbf{P}_{\Lambda,\beta} \big( \sigma , h \big) \equiv \textbf{P}^{\eta}_{\Lambda} \big( \sigma , h \big) \equiv \textbf{P}^{\mathrm{LR},\eta}_{\Lambda,\beta } \big( \sigma , h \big) \equiv \frac{\mathrm{exp} \big( \beta \mathcal{H}^{\mathrm{LR}, \eta}_{\Lambda}\big) }{Z^{\mathrm{LR},\eta}_{\Lambda ,\beta} \big( h \big) } \equiv \frac{\mathrm{exp} \big( \beta \mathcal{H}_{\Lambda}\big) }{Z^{\eta}_{\Lambda ,\beta} \big( h \big) } \text{ } \text{ , } \end{aligned}$$ with, $$\begin{aligned} Z^{\mathrm{LR},\eta}_{\Lambda ,\beta } \big( h \big) \equiv Z^{\eta}_{\Lambda ,\beta } \big( h \big) \equiv \underset{x \in \Omega^{\eta}_{\Lambda} }{\sum}\mathrm{exp} \big( \beta \mathcal{H}^{\mathrm{LR}, \eta}_{\Lambda}\big) \text{ } \text{ . } \end{aligned}$$ Equipped with $\textbf{P} \big( \cdot \big)$, the joint probability measure for the pair $\big( \sigma , h \big)$ is, $$\begin{aligned} \textbf{Q}^{\mathrm{LR-RF},{\eta}}_{\Lambda , \beta} \big( \sigma \in A , h \in B \big) \equiv \underset{B}{\int} \textbf{P}^{\mathrm{LR-RF},\eta}_{\Lambda_,\beta } \big( A \big) \text{ } \mathrm{d} \textbf{P}^{\mathrm{LR-RF}}_{\Lambda , \beta} \big( h \big) \text{ } \text{ , } \\ \textbf{Q}^{\mathrm{LR},{\eta}}_{\Lambda , \beta} \big( \sigma \in A , h \in B \big) \equiv \underset{B}{\int} \textbf{P}^{\mathrm{LR},\eta}_{\Lambda_,\beta } \big( A \big) \text{ } \mathrm{d} \textbf{P}^{\mathrm{LR},\eta}_{\Lambda , \beta} \big( h \big) \equiv \underset{B}{\int} \textbf{P}^{\eta}_{\Lambda_,\beta } \big( A \big) \text{ } \mathrm{d} \textbf{P}^{\eta}_{\Lambda , \beta} \big( h \big) \text{ } \text{ , } \end{aligned}$$ under boundary conditions $\eta$, for measurable $A \subsetneq \Omega$ and $B \subsetneq \textbf{R}^{\Lambda}$ borelian, with density, $$\begin{aligned} \mathcal{D}^{\mathrm{LR-RF},\eta}_{\Lambda , \beta} \big( \sigma , h \big) = \underset{u \in \Lambda}{\prod} \frac{1}{\sqrt{2 \pi}} \text{ } \mathrm{exp} \big( - \frac{h^2_u }{2} \big) \text{ } \textbf{P}^{\mathrm{LR-RF}, \eta}_{\Lambda_,\beta } \big( \sigma , h \big) \text{ } \text{ , } \\ \mathcal{D}^{\mathrm{LR},\pm}_{\Lambda , \beta} \big( \sigma , \eta \big) \equiv \mathcal{D}^{\pm}_{\Lambda , \beta} \big( \sigma , \eta \big) = \underset{u \in \Lambda}{\prod} \frac{1}{\sqrt{2 \pi}} \text{ } \mathrm{exp} \big( - \frac{\eta^2_u }{2} \big) \text{ } \textbf{P}^{\eta}_{\Lambda_,\beta } \big( \sigma , \eta \big) \text{ } \text{ , } \end{aligned}$$ under $+$ boundary conditions. As a sequence of finite volumes $\Lambda_n$, with $\Lambda_n \subsetneq \Lambda$ and $\big| \Lambda_n \big| < + \infty$, tends to $\textbf{Z}^d$ via a weak limit, $$\begin{aligned} \textbf{P}^{\mathrm{LR-RF},\eta}_{\beta } \big[ \omega \big] = \underset{n \longrightarrow + \infty }{\mathrm{lim}} \textbf{P}^{\mathrm{LR-RF},\eta}_{\Lambda_n,\beta } \big[ \omega \big] \text{ } \text{ , } \\ \textbf{P}^{\mathrm{LR},\eta}_{\beta } \big[ \omega \big] = \underset{n \longrightarrow + \infty }{\mathrm{lim}} \textbf{P}^{\mathrm{LR},\eta}_{\Lambda_n,\beta } \big[ \omega \big] \text{ } \text{ , } \end{aligned}$$ for a random-field, long-range Ising configuration $\omega \in \Omega^{\eta}_{\Lambda}$. From seminal work in \[3\], the authors of \[1\] extend work for proving that the phase transition for the random-field Ising model occurs, introduced in \[3\], surrounding a *Peierls type argument* for demonstrating that the random-field, long-range Ising model for $\alpha > d+1$, for dimensions $d \geq 3$, undergoes a phase transition, by making use of contours of the form, $$\begin{aligned} \Gamma_0 \big( n \big) \equiv \big\{ \text{paths } \gamma \in \Gamma : 0 \in I \big( \gamma \big) , \big| \gamma \big| = n \big\} \text{ } \text{ , } \end{aligned}$$ which denotes each possible contour $\gamma$, of length $n$, in which the interior of the contour contains the origin $0$, and is of length $n$, which are the maximal connected components of the union of faces $C_x \cap C_y$, for which $\sigma_x \neq \sigma_y$ from the set of all possible contours $\Gamma$. Within each $\gamma$, the *Perierls type argument* entails reversing the direction of the spins contained within the contour, ie flipping the spins to $-\sigma_i$ and otherwise setting all of the spins outside of the contour as $\sigma_i$, in which, for $\big( \tau_A \big( \sigma \big) \big)_i : \textbf{R}^{\textbf{Z}^d} \longrightarrow \textbf{R}^{\textbf{Z}^d}$, $$\big( \tau_A \big( \sigma \big) \big)_i \equiv \text{ } \left\{\!\begin{array}{ll@{}>{{}}l} - \sigma_i & \text{, if } i \in A \text{ } \text{ , } \\ \sigma_i & \text{ otherwise } \text{ . } \\ \end{array}\right.$$ With $\Gamma$, $\Gamma_0 \big( n \big)$ and $\big( \tau_A \big( \sigma \big) \big)_i \text{ }$, a portion of previous results for demonstrating that the phase transition occurs for the random-field, long-range Ising model are captured with the following **Proposition**. **Proposition** *1* (*the impact of reversing spins inside contours for the long range, random field Ising model Hamiltonian under plus boundary conditions*, \[1\], **Proposition** *2.1*). For $\alpha > d+1$, there exists a constant $c>0$ such that, for the random-field, long-range Ising model at inverse temperature $\beta > 0$, $$\begin{aligned} \mathcal{H}^{\mathrm{LR-RF}, +}_{\Lambda} \big( \tau_{\gamma} \big( \sigma \big) \big) - \mathcal{H}^{\mathrm{LR-RF}, +}_{\Lambda} \big( \sigma \big) \leq - J c \big| \gamma \big| \text{ } \text{ . } \end{aligned}$$ The **Proposition** above demonstrates the impact of reversing the spins contained within $\gamma$, under $\tau_{\gamma} \big( \sigma \big)$, with the spins $\sigma$ before $\tau_{\gamma} \big( \cdot \big)$ is applied. Along similar lines, from the density introduced previously under $+$ boundary conditions with $\mathcal{D}^{\mathrm{LR-RF}+}_{\Lambda , \beta} \big( \cdot , \cdot \big) \equiv \mathcal{D}^{+}_{\Lambda , \beta} \big( \cdot , \cdot \big)$, the equality, $$\begin{aligned} \frac{\mathcal{D}^{+}_{\Lambda , \beta} \big( \sigma , h \big) Z^{+ }_{\Lambda ,\beta } \big( h \big) }{\mathcal{D}^{+}_{\Lambda , \beta} \big( \tau_{\gamma} \big( \sigma \big) , \tau_{\gamma} \big( h \big) \big) Z^{+}_{\Lambda ,\beta } \big( \tau \big( h \big) \big) } = \mathrm{exp} \big[ \beta \mathcal{H}^{\mathrm{LR-RF}, +}_{\Lambda} \big( \tau_{\gamma} \big( \sigma \big) \big) - \beta \mathcal{H}^{\mathrm{LR-RF}, +}_{\Lambda} \big( \sigma \big) \big] \text{ } \text{ } \text{ , } \end{aligned}$$ between the ratio of the product of the density $\mathcal{D}^{+}_{\Lambda , \beta} \big( \sigma , h \big)$, and $Z^{+ }_{\Lambda ,\beta } \big( h \big)$, with the product of $\mathcal{D}^{+}_{\Lambda , \beta} \big( \tau_{\gamma} \big( \sigma \big) , \tau_{\gamma} \big( h \big) \big)$, and $Z^{\eta}_{\Lambda ,\beta } \big( \tau \big( h \big) \big)$, is equivalent to the exponential of the difference between the long-range, random-field Ising model under $\tau_{\gamma} \big( \sigma \big)$ and $\sigma$, respectively. Similarly, under the probability measure and distributions functions for the long range Ising model, instead for $\mathcal{D}^{\mathrm{LR}+}_{\Lambda , \beta} \big( \cdot , \cdot \big) \equiv \mathcal{D}^{+}_{\Lambda , \beta} \big( \cdot , \cdot \big)$, $$\begin{aligned} \frac{\mathcal{D}^{+}_{\Lambda , \beta} \big( \sigma , \eta \big) Z^{+ }_{\Lambda ,\beta } \big( \eta \big) }{\mathcal{D}^{+}_{\Lambda , \beta} \big( \tau_{\gamma} \big( \sigma \big) , \tau_{\gamma} \big( \eta \big) \big) Z^{+}_{\Lambda ,\beta } \big( \tau \big( \eta \big) \big) } = \mathrm{exp} \big[ \beta \mathcal{H}^{\mathrm{LR}, +}_{\Lambda} \big( \tau_{\gamma} \big( \sigma \big) \big) - \beta \mathcal{H}^{\mathrm{LR}, +}_{\Lambda} \big( \sigma \big) \big] \text{ } \text{ } \text{ . } \end{aligned}$$ Under a random, external field introduced with iid, Gaussian $\big\{ h_x \big\}$, it is possible for the partition function $Z^{+}_{\Lambda , \beta} \big( \tau \big( h \big) \big)$ to exceed $Z^{+}_{\Lambda , \beta} \big( h \big)$. If this were the case, the parameter, $$\begin{aligned} \Delta_A \big( h \big) \equiv - \frac{1}{\beta} \mathrm{log} \big[ \frac{Z^{+}_{\Lambda, \beta} \big( h \big) }{Z^{+}_{\Lambda, \beta} \big( \tau_A \big( h \big) \big) } \big] \text{ } \text{ , } \end{aligned}$$ captures the probability of such an event occurring, in which there exists a path, sampled from $\Gamma$, for which, $$\begin{aligned} \underset{\gamma \in \Gamma_0 }{\mathrm{sup}} \frac{\big| \Delta_{I ( \gamma )} \big( h \big) \big|}{c_1 \big| \gamma \big| } < \frac{1}{4} \text{ } \text{ , } \end{aligned}$$ which we denote with the 'bad' event, $\mathcal{B}$. Hence the complementary event for a bad event is given by, $$\begin{aligned} \mathcal{B}^c \equiv \big\{ \underset{\gamma \in \Gamma_0 }{\mathrm{sup}} \frac{\big| \Delta_{I ( \gamma )} \big( h \big) \big|}{c_1 \big| \gamma \big| } > \frac{1}{4} \big\} \text{ } \text{ . }\end{aligned}$$ From the supremum introduced above, of a term inversely proportional to the length, and directly proportional to the interior of each such $\gamma$, several bounds leading up to the *Peierls' argument* incorporate $\tau_A \big( \sigma \big)$, one of which is first introduced below. From the probability measures $\textbf{P}^{\mathrm{LR-RF}}_{\Lambda} \big( \cdot \big)$, and $\textbf{P}^{\mathrm{LR}}_{\Lambda} \big( \cdot \big)$, denote $\textbf{P}^{\mathrm{RF}}_{\Lambda} \big( \cdot \big)$ as the probability measure for the random field Ising model. **Lemma** *1* (*constant times an exponential upper bound for the random field Ising model* \[1\], **Lemma** *3.4*). For $A , A^{\prime} \subsetneq \textbf{Z}^d$, with $A \cap A^{\prime} \neq \emptyset$ and $\big| A \big| , \big| A^{\prime} \big| < + \infty$, $$\begin{aligned} \textbf{P}^{\mathrm{RF},+}_{\Lambda} \big[ \big| \Delta_A \big( h \big) \big| \geq \lambda \big| h_{A^c} \big] \leq 2 \text{ } \mathrm{exp} \big[ - \frac{\lambda^2}{8 e^2 \big| A \big|} \big] \text{ } \text{ , } \end{aligned}$$ and also that, $$\begin{aligned} \textbf{P}^{\mathrm{RF},+}_{\Lambda} \big[ \big| \Delta_A \big( h \big) - \Delta_{A^{\prime}} \big( h \big) \big| > \lambda \big| h_{(A \cup A^{\prime})^c} \big] \leq 2 \text{ } \mathrm{exp} \big[ - \frac{\lambda^2}{8 e^2 \big| A \Delta A^{\prime} \big|} \big] \text{ } \text{ , } \end{aligned}$$ for the symmetric difference between the sets $A$ and $A^{\prime}$, $A \Delta A^{\prime}$. Besides the result above, we must also make use of a coarse-graining procedure. For the procedure, as described in \[1\] and \[2\], introduce a coarse grained renormalization of $\textbf{Z}^d$, $$\begin{aligned} C_m \big( x \big) \equiv \bigg[ \overset{d}{\underset{i=1}{\prod}} \big[ 2^m x_i - 2^{m-1} , 2^m x_i + 2^{m-1} \big] \bigg] \cap \textbf{Z}^d \text{ } \text{ , } \end{aligned}$$ corresponding to the cube over the hypercube, with center at $2^m x$, with side length $2^m-1$, an *m-cube*, which is a restatement of the coarse-graining approach of \[5\]. From the object above, we make use of the convention that $C_0 \big( 0 \big)$ denotes the point about $0$. Additionally, denote, $$\begin{aligned} \mathcal{P}_i \big( A \cap \mathcal{R} \big) \equiv \big\{ x \in \mathcal{R}_i : l^i_x \cap A \neq \emptyset \big\} \text{ } \text{ , } \end{aligned}$$ which also satisfies, $$\begin{aligned} \mathcal{P}_i \big( A \cap \mathcal{R} \big) \supsetneq \underset{1 \leq i \leq d}{\bigcup} \big( \mathcal{P}^{\mathrm{G}}_i \big( A \cap \mathcal{R} \big) \cup \mathcal{P}^{\mathrm{B}}_i \big( A \cap \mathcal{R} \big) \big) \text{ } \text{ , } \end{aligned}$$ for a rectangle $\mathcal{R} \equiv \overset{n}{\underset{i=1}{\prod}} \big[ 1 , r_i \big]$, with $\mathcal{R}_i \cap \big[ 1 , r_i \big] \neq \emptyset$ for every $i$, which is given by, $$\begin{aligned} \mathcal{R} \supsetneq \mathcal{R}_i \equiv \big\{ x \in \mathcal{R} : x_i = 1 \big\} \text{ } \text{ ,} \end{aligned}$$ and $l^i_x \equiv \big\{ x + k e_i : 1 \leq k \leq r_i \big\}$, satisfying $\mathcal{R} \cap \mathcal{R}_i \neq \emptyset$ for each $i$, as the set of points for which $l^i_x \cap A \neq \emptyset$. From this, denote the *good* set of points in the plane, $$\begin{aligned} \mathcal{P}^{\mathrm{G}}_i \big( A \cap \mathcal{R} \big) \equiv \big\{ \forall \text{ rectangles } \mathcal{R}_i \text{ , } \exists \text{ } \text{countably many } x \in \mathcal{P}_i \big( A \cap \mathcal{R} \big) : l^i_x \cap \big( A \backslash \mathcal{R} \big) \neq \emptyset \big\} \text{ } \text{ , } \end{aligned}$$ and, similarly, denote the set of bad points, $$\begin{aligned} \mathcal{P}^{\mathrm{B}}_i \big( A \cap \mathcal{R} \big) \equiv \big( \mathcal{P}^{\mathrm{G}} \big( A \cap \mathcal{R} \big) \big)^c \text{ } \text{ , } \end{aligned}$$ for which $l^i_x \cap \big( A \backslash \mathcal{R} \big) \equiv \emptyset$. In comparison to the contours discussed in \[2\] which are used to implement a *Peirels' argument*, related to the projections $\mathcal{P}_i$, that, $$\begin{aligned} \big| \mathcal{P}^{\mathrm{G}}_i \big( A \cap \mathcal{R} \big) \big| \leq \big| \partial_{\mathrm{ex}} A \cap \mathcal{R} \big| \text{ } \text{ , } \end{aligned}$$ in which the set of *good* points has cardinality less than, or equal to, the cardinality of $\partial_{\mathrm{ex}} A \cap \mathcal{R}$, where, $$\begin{aligned} \partial_{\mathrm{ex}} A \equiv \big\{ \forall v \in A^c \cup \partial A , \exists \text{ } v^{\prime} \in \partial A : v \cap v^{\prime} \neq \emptyset \big\} \text{ } \text{ , } \end{aligned}$$ and, $$\begin{aligned} \big| \mathcal{P}^{\mathrm{B}}_i \big( A \cap \mathcal{R} \big) \big| \leq C \big| \mathcal{R}_d \big| \text{ } \text{ , } \end{aligned}$$ in which the set of *bad* points has cardinality less than, or equal to, the cardinality of a rectangular subset of the hypercube, $\mathcal{R}_d$, for a real parameter $C \equiv \frac{\lambda}{r_j}$, while finally, that, $$\begin{aligned} \overset{d}{ \underset{i=1}{\sum}} \big| \mathcal{P}_i \big( A \cap \mathcal{R} \big) \big| \leq c \big| \partial_{\mathrm{ex}} A \cap \mathcal{R} \big| \text{ } \text{ , } \end{aligned}$$ where the exterior boundary of a path is given by, $$\begin{aligned} \partial_{\mathrm{ex}} \big( \Lambda \big) \equiv \big\{ \forall x \in \Lambda^c \text{ } , \text{ } \exists y \in \Lambda : \big| x - y \big| = 1 \big\} \text{ } \text{ . } \end{aligned}$$ Similarly, the interior boundary of a path is given by, $$\begin{aligned} \partial_{\mathrm{int}} \big( \Lambda \big) \equiv \big\{ \forall x \in \Lambda \text{ } , \text{ } \exists y \in \Lambda^c : \big| x - y \big| = 1 \big\} \text{ } \text{ . } \end{aligned}$$ Above, the summation of the cardinality of the set of *all* points in the projection $\mathcal{P}_i$ is less than, or equal to, $\partial_{\mathrm{ex}} A \cap \mathcal{R}$, for every $1 \leq i \leq d$, and some $c>0$. Following a description of the paper organization in the next section, we distinguish between the types of contours discussed in \[1\], and in \[2\]. ## Paper organization With the definition of the long range, random-field, and long range, random-field Ising models, in the next section we differentiate between contours discussed in \[1\] and \[2\], from which the existence of a phase transition can be provided for the long range, random-field Ising model for $d < \alpha \leq d+1$. In order to adapt the argument provided in \[1\] with the contours described in \[2\], we implement several steps of the argument for the long range contour system surrounding the coarse graining procedure. To exhibit that a phase transition occurs for lower dimensions in the long range Ising model, we prove the following result. **Theorem PT** (*the long range Ising model undergoes a phase transition in lower dimensions*). Over a finite volume $\Lambda$, for $d \geq 3$, there exists a critical parameter $\beta_c$, with $\beta_c \equiv \beta_c \big( \alpha , d \big)$, and another parameter $\epsilon$, with $\epsilon \equiv \epsilon \big( \alpha , d \big)$, so that for parameters $\beta \geq \beta_c$ and $\epsilon \leq \epsilon_c$, $$\begin{aligned} \textbf{P}^{\mathrm{LR},+}_{\Lambda,\beta, \epsilon} \neq \textbf{P}^{\mathrm{LR},-}_{\Lambda,\beta, \epsilon} \text{ } \text{ , } \end{aligned}$$ $\textbf{P}$-almost surely, in which the long range measures under $+$ and $-$ boundary conditions are not equal. # Contours in the long range, random-field Ising model for the Peirels' argument We introduce long range contours below. ## Contours for the long range Ising model To introduce another family of contours for the *Peierls' argument*, consider the following. **Definition** *1* (*new contours for the Peierls' argument*, \[2\]). For the long range Ising model, real $M,a,r >0$, and a configuration $\sigma \in \Omega^{\mathrm{LR}}$, the sample space of all long range Ising model configurations, from the boundary $\partial \sigma$, the set of all $\big( M , a , r \big)$-partitions, $\Gamma \big( \sigma \big) \equiv \big\{ \bar{\gamma} : \bar{\gamma} \subset \partial \sigma \big\} \neq \emptyset$, satisfies: - [Property 1]{.ul} (*partition equality*): Given $\Gamma \big( \sigma \big)$, there exists countably many $\bar{\gamma}$ which partition each $\partial \sigma$, in which $\underset{\bar{\gamma} \in \Gamma ( \sigma )}{\cup} \bar{\gamma} \equiv \partial \sigma$, such that for another path $\bar{\gamma}^{\prime}$, with $\bar{\gamma} \cap \bar{\gamma}^{\prime} \neq \emptyset$, $\bar{\gamma}^{\prime}$ is contained in the connected component of $( \bar{\gamma} )^c$. - [Property 2]{.ul} (*decomposing each* $\bar{\gamma}$). For all $\bar{\gamma} \in \Gamma \big( \sigma \big)$, $\exists$ $1 \leq n \leq 2^r - 1$ such that: - [Property 2A]{.ul}: $\bar{\gamma}$ can be expressed with the union $\bar{\gamma} \equiv \underset{1 \leq k \leq n}{\bigcup} \bar{\gamma}_k$, for $\bar{\gamma}_k$ such that $\bar{\gamma}_k \cap \bar{\gamma} \neq \emptyset$ for every $k$. - [Property 2B]{.ul}: For $\bar{\gamma}, \bar{\gamma}^{\prime} \in \Gamma \big( \sigma \big)$ such that $\bar{\gamma} \cap \bar{\gamma}^{\prime} \neq \emptyset$, there exists two strictly positive $n \neq n^{\prime}$, for which, $$\begin{aligned} \mathrm{d} \big( \bar{\gamma} , \bar{\gamma}^{\prime} \big) > M \text{ } \mathrm{min} \big\{ \underset{1 \leq k \leq n}{\mathrm{max}} \mathrm{diam} \big( \bar{\gamma}_k \big) , \underset{1 \leq j \leq n^{\prime}}{\mathrm{max}}\mathrm{diam} \big( \bar{\gamma}^{\prime}_j \big) \big\}^a \text{ } \text{ , } \end{aligned}$$ with respect to the metric $\mathrm{d} \big( \cdot , \cdot \big)$ between paths belonging to $\Gamma \big( \sigma \big)$, where, $$\begin{aligned} \mathrm{d} \big( \gamma_1 , \gamma_2 \big) \equiv \big\{ \forall n \in \textbf{Z}_{\geq 0} \text{ } \text{ , } \text{ } \exists \text{ } \gamma_1 , \gamma_2 \in \Gamma : \big\| \gamma_1 - \gamma_2 \big\|_1 = n \big\} \text{ } \text{ . } \end{aligned}$$ With **Definition** *1*, we also denote the set of all *connected components* of some $\sigma$ in finite volume, below. **Definition** *2* (*connected components in a finite volume*). For any $m_1 \neq m_2 >0$, and two vertices $x \neq x^{\prime}$, there exists two *m-cubes*, $C_{m_1} \big( x \big)$ and $C_{m_2} \big( x^{\prime} \big)$, such that the edge set, $$\begin{aligned} V_n \equiv v \big( G_n \big( \Lambda \big) \big) \equiv \big\{ v \in C_m \big( x \big) : v \cap V \big( \Lambda \big) \neq \emptyset \big\} \text{ } \text{ , } \end{aligned}$$ is comprised of the minimum number of cubes for which the union of *m-cubes* covers the set of *connected components*, while the *edge set*, $$\begin{aligned} E_n \equiv e \big( G_n \big( \Lambda \big) \big) \equiv \big\{ e \in E \big( \Lambda \big) : \big| e \cap E \big( \Lambda \big) \cap C_m \big( x \big) \big| \leq M d^a 2^{a n} \big\} \text{ } \text{ , } \end{aligned}$$ is comprised of the number of edges that have nonempty intersection with $E \big( \Lambda \big)$ and $C_m \big( x \big)$, for $G_n \big( \Lambda \big) \equiv \big( V_G , E_G \big)$. Denote the set of *connected components*, $\mathscr{G}_n \big( \Lambda\big)$, associated with some configuration, and contained with some *m-cube*, as, $$\begin{aligned} \gamma_{G} \big( \Lambda , C_m \big( x \big) \big) \equiv \gamma_G \equiv \underset{G_i \cap \Lambda \cap C_m ( x) \neq \emptyset}{\underset{G_i \subsetneq G}{\bigcup}} \gamma_{G_i} \equiv \underset{\forall C_m ( x ) v \in V_G : C_m ( x ) \cap v \neq \emptyset }{\bigcup} \big( \Lambda \cap C_m \big( x \big) \big) \text{ } \text{ , } \end{aligned}$$ corresponding to the *connected components* with nonempty intersection with an *m-cube*. With the set of *connected components* from **Definition** *2*, denote a set of partitions, $\big\{ \mathscr{P}_i \big\}_{i \in \mathcal{I}}$ for some countable index set $\mathcal{I}$, such that $\mathscr{P}_i \cap G_n \big( \Lambda \big) \neq \emptyset$ for every $i$, as the set of finite subvolumes of $\Lambda$ for which, $$\mathscr{P}_i \equiv \text{ } \left\{\!\begin{array}{ll@{}>{{}}l} \big\{ \forall G \in \mathscr{G}_n \big( \Lambda \big) , \exists \sigma_i , r > 0 : \mathscr{G}_n \big( \sigma_i \big) \cap \Lambda \neq \emptyset , \big| v \big( G \big) \big| \leq 2^r - 1 \big\} & \text{, if } i > 0 \text{ } \text{ , } \\ \big\{ \forall G \in \mathscr{G}_n \big( \Lambda \big) , \exists \sigma_i , r > 0 : \mathscr{G}_n \big( \sigma_i \big) \cap \Lambda \neq \emptyset , 1 \leq \big| v \big( G \big) \big| \leq 2^r - 1 \big\} & \text{, if } i \equiv 0 \text{ } \text{ . } \\ \end{array}\right.$$ $\mathscr{P}_i$ is otherwise assumed to be equal to $\emptyset$ if $\partial \sigma_i = \emptyset$. From **Proposition** *3.5* in \[2\], the collection $\big\{ \mathscr{P}_i \big\}$ satisfies [Property 1]{.ul}, and [Property 2]{.ul}. Finally, below, introduce the *inner boundary* and the set of edges that are exactly incident with the boundary configuration. **Definition** *3* (*inner and incident boundaries of edges to the boundary configuration*). Denote the *inner boundary of edges* to $\partial \sigma_i$ with, $$\begin{aligned} \partial_{\mathrm{in}} \big( \Lambda , \partial \sigma_i \big) \equiv \partial_{\mathrm{in}} \Lambda \equiv \big\{ \forall \sigma_i , \exists m > 0 : \big( \mathscr{G}_n \big( \Lambda \big) \cap C_m \big( x \big) \big) \cap \partial \sigma_i \equiv \emptyset \big\} \text{ } \text{ , } \end{aligned}$$ and the *incident boundary of edges to* $\partial \sigma_i$ with, $$\begin{aligned} \mathcal{B} \big( \partial \sigma_i \big) \equiv \big\{ \forall \sigma_i , \exists m > 0 : \big| \mathscr{G}_n \big( \Lambda \big) \cap C_m \big( x \big) \big| \equiv \big| \partial \big( \mathscr{G}_n \big( \Lambda \big) \big) \big| \big\} \text{ } \text{ , } \end{aligned}$$ under the assumption that $\partial_{\mathrm{in}} \Lambda, \mathcal{B} \big( \partial \sigma_i \big) \neq \emptyset$. From quantities from **Definiton** *3*, the isoperimetric inequality states, $$\begin{aligned} \big| \Lambda \big|^{1 - \frac{1}{d}} \leq \big| \partial_{\mathrm{in}} \Lambda \big| \text{ } \text{ , } \end{aligned}$$ for the dimension $d$. ## Long range, versus long range, random-field Ising model contours From contours for the long range Ising model of the previous section, the procedure for reversing the orientation of spins differs. First, fix the *m-cube* of side length $m$ about the point $0$, $$\begin{aligned} C_0 \big( m \big) \equiv \big\{ \mathrm{sp} \big( \gamma \big) \subsetneq \textbf{Z}^d , \big| \mathrm{sp} \big( \gamma \big) \big| < + \infty : \gamma \in \mathcal{E}^{-}_{\Lambda} , 0 \in V \big( \gamma \big) , \big| \gamma \big| = m \big\} \text{ } \text{ . } \end{aligned}$$ As opposed to $\big( \tau_A \big( \sigma \big) \big)_i$ for countours in the long range, random-field Ising model, the flipping procedure is, for the set $\Gamma$ at each $x$, given by the map $\big( \tau_{\Gamma} \big( \sigma \big) \big)_x: \Omega \big( \Gamma \big) \longrightarrow \Omega^{-}_{\Lambda}$, where the target space of the mapping is, $$\begin{aligned} \Omega^{-}_{\Lambda} = \big\{ \text{collection of all paths contained in } \Lambda \text{ with -} 1 \text{ labels} \big\} \equiv \big\{ \gamma \in \Lambda : \gamma \cap \Lambda \neq \emptyset \text{ } , \text{ } \mathrm{lab} \big( \gamma \big) \equiv - 1 \big\} \text{ } \text{ , } \end{aligned}$$ as, $$\big( \tau^{\mathrm{LR}}_{\Gamma} \big( \sigma \big) \big)_x \equiv \big(\tau^{\mathrm{LR}} \big( \sigma \big)\big)_x \equiv \big( \tau_{\Gamma} \big( \sigma \big) \big)_x \equiv \text{ } \left\{\!\begin{array}{ll@{}>{{}}l} \sigma_x & \text{, if } x \in I_{-} \big( \Gamma \big) \cup V \big( \Gamma \big)^c \text{ } \text{ , } \\ - \sigma_x & \text{, if } x \in I_{+} \big( \Gamma \big) \text{ } \text{ , } \\ - 1 & \text{ , if } x \in \mathrm{sp} \big(\Gamma \big) \text{ } \text{ , } \\ \end{array}\right.$$ which can be expressed with the following over all $n$ components of $\gamma$, with, $$\begin{aligned} \big( \tau_{\Gamma} \big( \sigma \big) \big)_x = \big( \tau_{ \{ \gamma_1 , \cdots , \gamma_n \} } \big( \sigma \big) \big)_x \text{ } \text{ . } \end{aligned}$$ Also, given the support, collection of edges with $-$ labels, the set of all labels, vertices of $G$, and interior of each $\gamma$, each of which are respectively given by, $$\begin{aligned} \big| \gamma \big| \equiv \mathrm{sp} \big( \gamma \big) \equiv \big\{ \text{support of paths } \gamma \big\} \text{ } \text{ , } \\ \mathcal{E}^{-}_{\Lambda} \equiv \big\{ \forall \Gamma \equiv \big\{ \gamma_1 , \cdots , \gamma_n \big\} \text{ } , \text{ } \exists V \big( \Gamma \big) \subset \Lambda : \textit{compatible} \text{ } \Gamma , \textit{external} \text{ } \gamma_i , \mathrm{lab} \big( \gamma_i \big) = - 1 \big\} \text{ } \text{ , } \\ \mathrm{lab}_{\bar{\gamma}} \equiv \big\{ \text{labels of paths } \gamma \big\} \equiv \underset{n\geq 0}{\underset{\text{paths } \gamma}{\bigcup} } \big\{ \forall i > 0 , \bar{\gamma} \equiv \big( \bar{\gamma}^0 , \cdots , \bar{\gamma}^n \big) \in \Gamma , \exists 1 < i < n : \bar{\gamma}^i \longrightarrow \big\{ - 1 , + 1 \big\} \big\} \text{ } \text{ , } \\ V \big( G \big) \supsetneq V \big( \Gamma \big) \equiv \big\{ v \in v \big( G \big) : v \cap G \cap \Lambda \neq \emptyset \big\} \text{ } \text{ , } \\ I_{\pm} \big( \gamma \big) \equiv \underset{k \geq 1, 1 \leq k \leq n}{\bigcup} I_{\pm} \big( \gamma_k \big) \equiv \underset{\mathrm{lab}_{\bar{\gamma}} ( I ) = \pm 1 }{\underset{k\geq 1}{\bigcup}} I \big( \mathrm{sp} \big( \gamma \big) \big)^{k} \text{ } \text{ , } \end{aligned}$$ in addition to the two quantities, $$\begin{aligned} V \big( \gamma \big) \equiv \mathrm{sp} \big( \gamma \big) \cup I \big( \gamma \big) \equiv \mathrm{sp} \big( \gamma \big) \cup \underbrace{ \big( {I_{+} \big( \gamma \big) \cup I_{-} \big( \gamma \big)} \big) }_{I ( \gamma ) \equiv I_{+} ( \gamma ) \cup I_{-} ( \gamma )} \text{ } \text{ , } \end{aligned}$$ where in the definition of $\mathcal{E}^{-}_{\Lambda}$, paths are considered *compatible* from the set of all paths $\Gamma$ if there exists a configuration from the long range sample space, $\sigma$, whose contours coincide with those of $\Gamma$. Similarly, for paths with $+1$ labels, introduce the collection of *compatible* paths over $\Lambda$, $$\begin{aligned} \mathcal{E}^{+}_{\Lambda} \equiv \big\{ \forall \Gamma \equiv \big\{ \gamma_1 , \cdots , \gamma_n \big\} \text{ } , \text{ } \exists V \big( \Gamma \big) \subset \Lambda : \textit{compatible} \text{ } \Gamma , \textit{external} \text{ } \gamma_i , \mathrm{lab} \big( \gamma_i \big) = + 1 \big\} \text{ } \text{ , } \text{ } \text{ . } \end{aligned}$$ From the quantities introduced above that are assocated with the flipping procedure $\big( \tau_{\Gamma} \big( \sigma \big) \big)_x$, it is also important to state the difference in $\mathcal{H}^{\mathrm{LR-RF}, +}_{\Lambda} \big( \tau_{\gamma} \big( \sigma \big) \big) - \mathcal{H}^{\mathrm{LR-RF}, +}_{\Lambda} \big( \sigma \big)$ between $\tau_{\gamma} \big( \sigma \big)$ and $\sigma$. For the long range Ising model with the contour system defined in *2.1*, the long range Hamiltonian instead satisfies, (**Proposition** *4.5*, \[2\]), $$\begin{aligned} \mathcal{H}^{\mathrm{LR},-}_{\Lambda} \big( \tau \big( \sigma \big) \big) - \mathcal{H}^{\mathrm{LR},-}_{\Lambda} \big( \sigma \big) \leq - c_1 \big| \gamma \big| - c_2 F_{I_{+}( \gamma) } - c_3 F_{\mathrm{sp}(\gamma)} \text{ } \text{ , } \end{aligned}$$ for a long range configuration $\sigma$, strictly positive $c_1, c_2, c_3$, and for the functions, $$\begin{aligned} F_{I_{\pm} ( \gamma) } \equiv \underset{y \in (I_{\pm}( \gamma))^c}{\underset{x \in I_{\pm} ( \gamma)}{\sum}} J_{x,y} \text{ } \text{ , } \\ F_{\mathrm{sp}(\gamma)} \equiv \underset{y \in (\mathrm{sp}(\gamma))^c}{\underset{x \in \mathrm{sp}(\gamma)}{\sum}} J_{x,y} \text{ } \text{ . } \end{aligned}$$ Long range contours differ from long range, random-field contours to a similar condition as raised in the isoperimetric inequality, in which, (**Lemma** *4.3*, \[2\]), $$\begin{aligned} \mathrm{diam} \big( \Lambda \big) \geq k_d \big| \Lambda \big|^{\frac{1}{d}} \text{ } \text{ , } \end{aligned}$$ in which the diameter of each such path is bound below by some strictly positive prefactor times the cardinality of the finite volume, $\Lambda$, in addition to the fact that the paths for the long range Ising model, in comparison to those from the long range, random-field Ising model, do not satisfy, $$\begin{aligned} \mathscr{C}_l \big( \gamma \big) \equiv \underset{l \in \textbf{N}}{\bigcup} \big\{ C_l : \big| C_l \cap I \big( \gamma \big) \big| \geq \frac{1}{2} \big| C_l \big| \big\} \text{ } \text{ , } \end{aligned}$$ introduced as the $C_l$ admissibility condition \[1\], which has boundary, $$\begin{aligned} \partial \mathscr{C}_l \big( \gamma \big) \equiv \big\{ \big( C_l , C^{\prime}_l \big) : C^{\prime}_l \not\in \mathscr{C}_l \big( \gamma \big) , \big| C^{\prime}_l \cap C_l \big| = 1 \big\} \text{ } \text{ . } \end{aligned}$$ # Phase transition for the long-range Ising model The argument for proving that a phase transition occurs for the long range, random field Ising model can be applied to demonstrate that a phase transition occurs for the long range Ising model, beginning with the following. ## Beginning the argument First, we must determine the upper bound for the behavior of the long range Ising model Hamiltonian under the flipping procedure given in the previous section with $\big( \tau^{\mathrm{LR}} \big( \sigma \big) \big)_x$. For a new range of parameters $\alpha$ satisfying $d < \alpha \leq d+1$, instead of upper bounding the difference $\mathcal{H}^{\mathrm{LR}, }_{\Lambda} \big( \big( \tau_{\Gamma} \big( \sigma \big) \big)_x \big) - \mathcal{H}^{\mathrm{LR}, -}_{\Lambda} \big( \sigma \big)$, under $-$ boundary conditions in the $\alpha > d+1$ regime, we upper bound the difference $\mathcal{H}^{\mathrm{LR}, +}_{\Lambda} \big( \big( \tau_{\Gamma} \big( \sigma \big) \big)_x \big) - \mathcal{H}^{\mathrm{LR}, +}_{\Lambda} \big( \sigma \big)$, under $+$ boundary conditions in the $d < \alpha \leq d+1$ regime. **Proposition** *1* (*upper bound of the flipping procedure of the long range Ising model Hamiltonian with* + *boundary conditions*). For a long range Ising configuration $\sigma \sim \textbf{P}_{\Lambda,\beta} \big( \cdot , \cdot \big)$, with energy $\mathcal{H}^{\mathrm{LR}, \eta}_{\Lambda} \big( \sigma)$, the difference of the Hamiltonian under $\big( \tau^{\mathrm{LR}} \big( \sigma \big) \big)_x$ with the Hamiltonian under $\sigma$ satisfies, $$\begin{aligned} \mathcal{H}^{\mathrm{LR}, +}_{\Lambda} \big( \big( \tau_{\Gamma} \big( \sigma \big) \big)_x \big) - \mathcal{H}^{\mathrm{LR}, +}_{\Lambda} \big( \sigma \big) \leq - c^{\prime}_1 \big| \gamma \big| - c^{\prime}_2 F_{I_{-}( \gamma) } - c^{\prime}_3 F_{\mathrm{sp}(\gamma)} \text{ } \text{ , } \end{aligned}$$ for strictly positive $c^{\prime}_1, c^{\prime}_2, c^{\prime}_3$. *Proof sketch of Proposition 1*. The argument strongly resembles the strategy used in **Proposition** *4.5*, \[2\], in which the authors express each term in the Hamiltonian of the configuration $\sigma$ acted on by the flipping procedure $\big(\tau_{\Gamma} \big( \sigma \big) \big)_x$ for long range contours. Write out the first long range Hamiltonian on the LHS, denoting $\tau_{\Gamma} \big( \sigma_x \big) \equiv \big( \tau_{\Gamma} \big( \sigma \big)_x \big)$, $\tau_{\Gamma} \big( \sigma_y \big) \equiv \big( \tau_{\Gamma} \big( \sigma \big)_y \big)$, $\gamma^1 \equiv \gamma \equiv \{ \gamma^1 , \cdots , \gamma^n \}$, and $\Gamma \big( \sigma \big) \equiv \Gamma$, in which contributions from the Hamiltonian arise from the nonempty regions $I_{-} \big( \gamma \big) \cup V \big( \Gamma \big)^c$, $I_{+} \big( \gamma \big)$, and $\mathrm{sp} \big( \Gamma \big)$, as, $$\begin{aligned} \mathcal{H}^{\mathrm{LR}, +}_{\Lambda} \big( \big( \tau_{\Gamma} \big( \sigma \big) \big)_x \big) = - \underset{\Gamma \equiv \cup_i \{ \gamma^i_1 , \cdots , \gamma^i_n \} }{\underset{x,y \in I_{-} ( \gamma ) \cup V ( \Gamma )^c }{\sum}} J_{x,y} \big[ \tau_{\Gamma} \big( \sigma_x \big) \tau_{\Gamma} \big( \sigma_y \big) \big] - \underset{\Gamma \equiv \cup_i \{ \gamma^i_1 , \cdots , \gamma^i_n \} }{\underset{x,y \in ( I_{-} ( \gamma ) \cup V ( \Gamma )^c )^c }{\sum}} J_{x,y} \big[ \tau_{\Gamma} \big( \sigma_x \big) \tau_{\Gamma} \big( \sigma_y \big) \big] - \cdots \end{aligned}$$ $$\begin{aligned} \underset{\Gamma \equiv \cup_i \{ \gamma^i_1 , \cdots , \gamma^i_n \}}{\underset{ x \in I_{-} ( \gamma ) \cup V ( \Gamma )^c }{\underset{ y \in ( I_{-} ( \gamma ) \cup V ( \Gamma )^c )^c }{\sum}}} J_{x,y} \big[ \tau_{\Gamma} \big( \sigma_x \big) \tau_{\Gamma} \big( \sigma_y \big) \big] - \underset{\Gamma \equiv \cup_i \{ \gamma^i_1 , \cdots , \gamma^i_n \} }{\underset{x,y \in I_{+} ( \gamma )}{\sum}} J_{x,y} \big[ \tau_{\Gamma} \big( \sigma_x \big) \tau_{\Gamma} \big( \sigma_y \big) \big] - \cdots \\ \underset{\Gamma \equiv \cup_i \{ \gamma^i_1 , \cdots , \gamma^i_n \} }{\underset{x , y \in ( I_{+} ( \gamma ) )^c }{\sum}} J_{x,y} \big[ \tau_{\Gamma} \big( \sigma_x \big) \tau_{\Gamma} \big( \sigma_y \big) \big] - \underset{\Gamma \equiv \cup_i \{ \gamma^i_1 , \cdots , \gamma^i_n \} }{\underset{ y \in (I_{+} (\gamma ))^c }{\underset{x \in I_{+} ( \gamma ) }{\sum}}} J_{x,y} \big[ \tau_{\Gamma} \big( \sigma_x \big) \tau_{\Gamma} \big( \sigma_y \big) \big] - \cdots \\ \underset{\Gamma \equiv \cup_i \{ \gamma^i_1 , \cdots , \gamma^i_n \} }{\underset{x,y \in \mathrm{sp} ( \gamma ) }{\sum}} J_{x,y} \big[ \tau_{\Gamma} \big( \sigma_x \big) \tau_{\Gamma} \big( \sigma_y \big) \big] - \underset{\gamma \equiv \{ \gamma_1 , \cdots , \gamma_n\}}{\underset{y \in \mathrm{sp} ( \gamma )^c}{\underset{x \in \mathrm{sp} ( \gamma )}{\sum}} } J_{x,y} \big[ \tau_{\Gamma} \big( \sigma_x \big) \tau_{\Gamma} \big( \sigma_y \big) \big] - \cdots \\ \underset{\gamma \equiv \{ \gamma_1 , \cdots , \gamma_n\}}{\underset{\Gamma \equiv \cup_i \{ \gamma^i_1 , \cdots , \gamma^i_n \}}{\underset{y \in I_{-} ( \gamma ) \cup V ( \Gamma )^c}{\underset{x \in \mathrm{sp} ( \gamma )}{\sum} }} }J_{x,y} \big[ \tau_{\Gamma} \big( \sigma_x \big) \tau_{\Gamma} \big( \sigma_y \big) \big] \text{ } \text{ . } \end{aligned}$$ From the summation above, before evaluating each instance of $\tau_{\Gamma} \big( \sigma_x \big)$ and $\tau_{\Gamma} \big( \sigma_y \big)$, observe, $$\begin{aligned} \underset{\Gamma \equiv \cup_i \{ \gamma^i_1 , \cdots , \gamma^i_n \}}{\underset{ x \in I_{-} ( \gamma ) \cup V ( \Gamma )^c }{\underset{ y \in ( I_{-} ( \gamma ) \cup V ( \Gamma )^c )^c }{\sum}}} J_{x,y} \big[ \tau_{\Gamma} \big( \sigma_x \big) \tau_{\Gamma} \big( \sigma_y \big) \big] = \underset{\Gamma \equiv \cup_i \{ \gamma^i_1 , \cdots , \gamma^i_n \}}{\underset{ x \in I_{-} ( \gamma ) \cup V ( \Gamma )^c }{\underset{ y \in I_{+} ( \gamma ) \cup \mathrm{sp} ( \gamma ) }{\sum}}} J_{x,y} \big[ \tau_{\Gamma} \big( \sigma_x \big) \tau_{\Gamma} \big( \sigma_y \big) \big] \text{ } \text{ , } \end{aligned}$$ corresponding to the summation over $y \in I_{+} \big( \gamma \big) \cup \mathrm{sp} \big( \gamma)$ and $x \in I_{-} \big( \gamma \big) \cup V \big( \Gamma \big)^c$, $$\begin{aligned} \underset{\gamma \equiv \{ \gamma_1 , \cdots , \gamma_n \} }{\underset{y \in (I_{+ } ( \gamma ) )^c}{\underset{x \in I_{+} ( \gamma )}{\sum}}} J_{xy} \big[ \tau_{\Gamma} \big( \sigma_x \big) \tau_{\Gamma} \big( \sigma_y \big) \big] = \underset{\gamma \equiv \{ \gamma_1 , \cdots , \gamma_n\}}{\underset{\Gamma \equiv \cup_i \{ \gamma^i_1 , \cdots , \gamma^i_n \} }{\underset{y \in I_{-} ( \gamma ) \cup V ( \Gamma ) ^c}{\underset{x \in I_{+} ( \gamma )}{\sum}}} } \big[ \tau_{\Gamma} \big( \sigma_x \big) \tau_{\Gamma} \big( \sigma_y \big) \big] \text{ } \text{ , } \end{aligned}$$ corresponding to the summation over $x \in I_{+} \big( \gamma \big)$ and $y \in I_{-} \big( \gamma \big) \cup V \big( \Gamma \big)^c$, $$\begin{aligned} \underset{\gamma \equiv \{ \gamma_1 , \cdots , \gamma_n\} }{\underset{y \in \mathrm{sp} ( \gamma )}{\underset{x \in \mathrm{sp} ( \gamma )}{\sum}} } J_{x,y} \big[ \tau_{\Gamma} \big( \sigma_x \big) \tau_{\Gamma} \big( \sigma_y \big) ] = \underset{\gamma \equiv \{ \gamma_1 , \cdots , \gamma_n\} }{\underset{y \in I_{+} ( \gamma ) }{\underset{x \in \mathrm{sp} ( \gamma )}{\sum}} } J_{x,y} \big[ \tau_{\Gamma} \big( \sigma_x \big) \tau_{\Gamma} \big( \sigma_y \big) ] + \underset{\gamma \equiv \{ \gamma_1 , \cdots, \gamma_n \}}{\underset{\Gamma \equiv \cup_i \{ \gamma_1 , \cdots , \gamma_n\} }{\underset{y \in I_{-} ( \gamma ) \cup V ( \Gamma )^c }{\underset{x \in \mathrm{sp} ( \gamma )}{\sum}} }} J_{x,y} \big[ \tau_{\Gamma} \big( \sigma_x \big) \tau_{\Gamma} \big( \sigma_y \big) ] + \cdots \\ \underset{\gamma \equiv \{ \gamma_1 , \cdots , \gamma_n\} }{\underset{y \in \textbf{Z}^d}{\underset{x \in \mathrm{sp} ( \gamma )} {\sum}}} J_{x,y} \big[ \tau_{\Gamma} \big( \sigma_x \big) \tau_{\Gamma} \big( \sigma_y \big) \big] \text{ } \text{ , } \end{aligned}$$ corresponding to the summation over $x \in \mathrm{sp} \big( \gamma \big)$, $y \in I_{+} \big( \gamma \big)$, $y \in I_{-} \big( \gamma \big)$, and $y \in \textbf{Z}^d$. From each of the three terms in the summation above, $$\underset{\gamma \equiv \{ \gamma_1 , \cdots , \gamma_n\} }{\underset{y \in I_{+} ( \gamma ) }{\underset{x \in \mathrm{sp} ( \gamma )}{\sum}} } J_{x,y} \big[ \tau_{\Gamma} \big( \sigma_x \big) \tau_{\Gamma} \big( \sigma_y \big) ] \equiv \text{ } \left\{\!\begin{array}{ll@{}>{{}}l} \underset{\gamma \equiv \{ \gamma_1 , \cdots , \gamma_n \}}{\underset{y \in I_{+} ( \gamma ) }{\underset{x \in \mathrm{sp} ( \gamma )}{\sum}} } J_{x,y} & \text{ if } \tau_{\Gamma} \big( \sigma_y \big) = 1 \text{ , } \\ 0 & \text{otherwise , } \\ \end{array}\right.$$ corresponding to the first term, $$\underset{\Gamma \equiv \cup_i \{ \gamma^i_1 , \cdots , \gamma^i_n \}}{ \underset{\gamma \equiv \{ \gamma_1 , \cdots , \gamma_n\} }{\underset{y \in I_{-} ( \gamma ) \cup V ( \Gamma )^c }{\underset{x \in \mathrm{sp} ( \gamma )}{\sum}} }} J_{x,y} \big[ \tau_{\Gamma} \big( \sigma_x \big) \tau_{\Gamma} \big( \sigma_y \big) ] \equiv \text{ } \left\{\!\begin{array}{ll@{}>{{}}l} \underset{\Gamma \equiv \cup_i \{ \gamma^i_1 , \cdots , \gamma^i_n \}}{\underset{\gamma_1 \equiv \{ \gamma_1 , \cdots , \gamma_n\}}{\underset{ y \in I_{-} ( \gamma ) \cup V ( \Gamma )^c }{ \underset{x \in \mathrm{sp} ( \gamma )}{\sum}} }} J_{x,y} & \text{ if } \tau_{\Gamma} \big( \sigma_y \big) = - 1 \text{ , } \\ 0 & \text{ otherwise,} \end{array}\right.$$ corresponding to the second term, and, $$\underset{\gamma \equiv \{ \gamma_1 , \cdots , \gamma_n\} }{\underset{y \in \textbf{Z}^d }{\underset{x \in \mathrm{sp} ( \gamma )}{\sum}} } J_{x,y} \big[ \tau_{\Gamma} \big( \sigma_x \big) \tau_{\Gamma} \big( \sigma_y \big) ] \equiv \text{ } \left\{\!\begin{array}{ll@{}>{{}}l} \underset{\gamma \equiv \{ \gamma_1 , \cdots , \gamma_i \}}{{\underset{y \in \textbf{Z}^d }{\underset{x \in \mathrm{sp} ( \gamma )}{\sum}} }} J_{x,y} & \text{ if } \tau_{\Gamma} \big( \sigma_x \big) \neq \tau_{\Gamma} \big( \sigma_y \big) \text{ , } \\ 0 & \text{otherwise , } \\ \end{array}\right.$$ corresponding to the third term. For the remaining terms rather than those considered above for $x \in \mathrm{sp} \big( \Gamma \big)$ and $y \in \mathrm{sp} \big( \Gamma \big)$, $$\underset{\Gamma \equiv \{ \gamma_1 , \cdots , \gamma_n\} }{\underset{y \in \mathrm{sp} ( \gamma ) }{\underset{x \in \mathrm{sp} ( \Gamma )}{\sum}} } J_{x,y} \big[ \tau_{\Gamma} \big( \sigma_x \big) \tau_{\Gamma} \big( \sigma_y \big) ] \leq \text{ } \left\{\!\begin{array}{ll@{}>{{}}l} \underset{\gamma \equiv \{ \gamma_1 , \cdots , \gamma_i \}}{{\underset{y \in \mathrm{sp} ( \gamma ) }{\underset{x \in \mathrm{sp} ( \gamma )}{\sum}} } } J_{x,y} & \text{ if } \tau_{\Gamma} \big( \sigma_x \big) \neq \tau_{\Gamma} \big( \sigma_y \big) \text{ , } \\ 0 & \text{otherwise . } \\ \end{array}\right.$$ On the other hand, for the Hamiltonian of the unflipped configuration $\sigma$ that is not acted on by the mapping $\big( \tau_{\Gamma} \big( \sigma \big) \big)_x$, $$\begin{aligned} \mathcal{H}^{\mathrm{LR}, +}_{\Lambda} \big( \sigma \big) = - \underset{x,y \in \Lambda}{\sum} J_{x,y} \sigma_x \sigma_y - \underset{y \in \Lambda^c}{\underset{x \in \Lambda}{\sum}} J_{x,y} \sigma_x \eta_y \text{ } \text{ , } \end{aligned}$$ from the difference, $$\begin{aligned} \mathcal{H}^{\mathrm{LR}, +}_{\Lambda} \big( \tau_{\Gamma} \big( \sigma_x \big) \big) - \mathcal{H}^{\mathrm{LR}, +}_{\Lambda} \big( \sigma \big) = \underset{x,y\in \Lambda}{\sum} J_{xy} \big( \tau_{\Gamma} \big( \sigma_x \big) \tau_{\Gamma} \big( \sigma_y \big) - \sigma_x \sigma_y \big) - \underset{y \in \Lambda^c}{\underset{x \in \Lambda}{\sum} } J_{xy} \big( \tau_{\Gamma} \big( \sigma_x \big) \eta_y - \sigma_x \eta_y \big) \text{ } \text{ , } \end{aligned}$$ with $\mathcal{H}^{\mathrm{LR}, +}_{\Lambda} \big( \tau_{\Gamma} \big( \sigma \big) \big)$ can be upper bounded with a summation over couplings, $$\begin{aligned} \underset{y \in \mathcal{A}^{\prime}}{\underset{x \in \mathrm{sp} ( \gamma )}{\sum}} J_{x,y} + \underset{y \in \mathcal{B}^{\prime}}{\underset{x \in I_{-} ( \gamma )}{\sum}} J_{x,y} + \underset{y \in \mathcal{C}^{\prime}}{ \underset{x \in V ( \Gamma_1 ) }{\sum} } J_{x,y} \text{ } \text{ , } \end{aligned}$$ which itself can be further upper bounded, as desired, by implementing the remaining argument, from **Proposition** *4.5* of \[2\], where $\mathcal{A}^{\prime} \equiv B \big( \gamma \big)$, $\mathcal{B}^{\prime} \equiv V \big( Y_4 \big)$, $\mathcal{C}^{\prime} \equiv B \big( \gamma \big) \backslash V \big( \Gamma_2 \big)$, $\Gamma_1 \subsetneq \Gamma$, $\Gamma_2 \equiv \Gamma \backslash \Gamma_1$, and $Y_4 \equiv \Gamma_2 \backslash \big\{ \gamma^{\prime} \in \Gamma_2 : \underset{1 \leq k \leq n}{\mathrm{sup} }\big( \mathrm{diam} \big( \gamma_k \big) \big) \leq \underset{1 \leq j \leq n }{\mathrm{sup}} \big( \mathrm{diam} \big( \gamma^{\prime}_j \big) \big) \big\}$, in which the desired constants for the prefactor of $F_{I_{-}(\gamma)}$ are obtained from the observation that, $$\begin{aligned} \underset{y \in V ( \Gamma_{\mathrm{ext}} ( \sigma , I_{-} ( \gamma ) \backslash \{ \gamma \} )}{\underset{x \in I_{-}(\gamma)}{\sum}} J_{x,y} \text{ } + \text{ } \underset{y \in V ( \Gamma_{\mathrm{int}} ( \sigma , I_{-} ( \gamma ) )}{\underset{x \in I_{-}(\gamma)}{\sum}} J_{x,y} \leq F_{I_{-}(\gamma)} \underset{> c^{\prime}_2}{\underbrace{\bigg( \frac{2}{M^{(\alpha - d ) \wedge 1}} + \frac{1}{M} \bigg) \kappa }} \text{ } \text{ , } \end{aligned}$$ for realizations of exterior and interior paths, respectively given by $\Gamma_{\mathrm{ext}}$ and $\Gamma_{\mathrm{int}}$, and suitable $M,\kappa > 0$ from **Corollary** *2.12* of \[1\], and, $$\begin{aligned} \underset{Y \in V ( \Gamma ( \sigma ) \backslash \{ \gamma \}}{ \underset{x \in \mathrm{sp} ( \gamma)}{\sum} } J_{x,y} \leq \text{ } \underset{> c^{\prime}_3}{\underbrace{ 2 \kappa }} \text{ } F_{\mathrm{sp}(\gamma)} \text{ } \text{ , } \end{aligned}$$ from **Proposition** *2.13* of \[1\], while for the remaining term, the desired upper bound takes the form, $$\begin{aligned} c^{\prime}_1 \propto \frac{J c_{\alpha} }{\big( 2 d +1 \big) 2^{\alpha}} \text{ } \text{ , } \end{aligned}$$ for suitable $c_{\alpha}>0$. Hence an upper bound for the three summations above takes the form given in the proposition statement. ## Implementing the Ding and Zhuang approach from the upper bound in the previous section, and the coarse graining procedure Equipped with the upper bound of the previous section, we proceed to implement the Ding and Zhuang approach for the long range Ising model, for $d < \alpha \leq d+1$ \[4\], by making use of concentration results for Gaussian random variables \[7\]. With the results from this approach, we can upper bound the probability of bad events occurring for the long range Ising model, in the same way that bad events are upper bounded for the long range, random-field Ising model. In order to show that the probability of such bad events occurring is exponentially unlikely, we implement a three-pronged approach, consisting of steps in a Majorizing measure theorem, Dudley's entropy bound, and upper bounding the probability, with an exponential. **Theorem** (*it is exponentially unlikely for the complement of bad events to occur*, \[6\]). There exists a strictly positive constants, $C_1 \equiv C_1 \big( \alpha , d \big)$ and $\epsilon$ sufficiently large, for which, $$\begin{aligned} \textbf{P}_{\Lambda} \big[ \mathcal{B}^c \big] \leq \mathrm{exp} \big( - C_1 \epsilon^{-2} \big) \text{ } \text{ . } \end{aligned}$$ *Proof of Theorem*. Refer to **Proposition** *3.7* of \[1\]. To demonstrate that a result similar to the **Theorem** above holds, introduce similar quantities to those for the long range, random-field Ising model, namely, $$\begin{aligned} \Delta^{\mathrm{LR}}_A \big( h \big) \equiv - \frac{1}{\beta} \mathrm{log} \big[ \frac{Z^{+}_{\Lambda, \beta} \big( \eta \big) }{Z^{+}_{\Lambda, \beta} \big( \tau^{\mathrm{LR}}_A \big( \eta \big) \big) } \big] \text{ } \text{ , } \end{aligned}$$ for $\big( \tau^{\mathrm{LR}}_A \big( \eta \big) \big)_{\partial A} \equiv \tau^{\mathrm{LR}}_A \big( \eta \big)$, corresponding to the log-transform of the ratio of the partition functions from the long range flipping procedure applied to the boundary field $\eta$, $$\begin{aligned} \mathcal{B}^{\mathrm{LR}} \equiv \big\{ \underset{\gamma \in \Gamma_0 }{\mathrm{sup}} \frac{\big| \Delta_{I_{-} ( \gamma )} \big( \eta \big) \big|}{c^{\prime}_1 \big| \gamma \big| } < 1 \big\} \text{ } \text{ , } \end{aligned}$$ corresponding to the supremum of paths for which the ratio above is $< 1$, and, $$\begin{aligned} \big( \mathcal{B}^{\mathrm{LR}} \big)^c \equiv \big\{ \underset{\gamma \in \Gamma_0 }{\mathrm{sup}} \frac{\big| \Delta_{I_{-} ( \gamma )} \big( \eta \big) \big|}{c^{\prime}_1 \big| \gamma \big| } > 1 \big\} \text{ } \text{ , } \end{aligned}$$ corresponding to the complement of bad events. With these quantities, to demonstrate that a result similar to the **Theorem** above holds, we make use of an entropy bound and Dudley's argument \[5\]. For these components of the argument, define, $$\begin{aligned} \gamma_{\theta} \big( T , d \big) \equiv \underset{( A_n )_{n \geq 0}}{\mathrm{inf}} \text{ } \underset{t \in T}{\mathrm{sup}} \text{ } \sum_{n \geq 0} 2^{\frac{n}{\theta}} \text{ } \mathrm{diam} \big( A_n \big( t \big) \big) \text{ } \text{ , } \end{aligned}$$ corresponding to the infimum-supremum of the summation over diameters of $A_n \big( t \big)$ for $n \geq 0$, where $A_n \big( t\big)$ denotes a partition of time, $T$, satisfying the properties: - [Property 1]{.ul}: The cardinality of the first partition is $\big| A_0 \big| \equiv 1$, - [Property 2]{.ul}: The upper bound for the cardinality of the n th partition is $\big| A_n \big| \leq 2^{2^n}$, - [Property 3]{.ul}: The sequence of partitions $\big( A_n \big( t \big) \big)_{n \geq 0}$ is increasing, in which $A_{n+1} \big( t \big) \subsetneq A_n \big( t \big)$ for all $n$. We will restrict our attention of the quantity above, $\gamma_{\theta} \big( T , d \big)$, for $\theta \equiv 2$. In addition to these components, we implement, in order, a series of results consisting of the Majorizing measure theorem \[12\] (restated as **Theorem** *3.9* in \[1\]), Dudley's entropy bound \[5\] (restated as **Proposition** *3.10* in \[1\]), as well as an upper bound for the probability of the process $X_t$ obtaining a supremum exceeds a factor dependent upon $\gamma_2 \big( T , d \big)$, and on $\mathrm{diam} \big( T \big)$ \[12\] (restated as **Theorem** *3.11* in \[1\]). Before implementing these three steps, we argue that a version of $\textbf{Lemma}$ *1* holds for the long range Ising model, from arguments originally implemented in the case of the long range, random field Ising model. **Lemma** *2* (*an adaptation of Lemma 1 from the Ding-Zhuang approach for the long range Ising model*, \[4\]). For $A , A^{\prime} \subsetneq \textbf{Z}^d$, with $A \cap A^{\prime} \neq \emptyset$ and $\big| A \big| , \big| A^{\prime} \big| < + \infty$, $$\begin{aligned} \textbf{P}^{\mathrm{LR},+}_{\Lambda} \big[ \big| \Delta^{\mathrm{LR}}_A \big( h \big) \big| \geq \lambda \big| h_{A^c} \big] \leq 2 \text{ } \mathrm{exp} \big[ - \frac{\lambda^2}{8 e^2 \big| A \big|} \big] \text{ } \text{ , } \end{aligned}$$ and also that, $$\begin{aligned} \textbf{P}^{\mathrm{LR},+}_{\Lambda} \big[ \big| \Delta^{\mathrm{LR}}_A \big( h \big) - \Delta^{\mathrm{LR}}_{A^{\prime}} \big( h \big) \big| > \lambda \big| h_{(A \cup A^{\prime})^c} \big] \leq 2 \text{ } \mathrm{exp} \big[ - \frac{\lambda^2}{8 e^2 \big| A \Delta A^{\prime} \big|} \big] \text{ } \text{ , } \end{aligned}$$ for the symmetric difference between the sets $A$ and $A^{\prime}$, $A \Delta A^{\prime}$. *Proof of Lemma 2*. The argument directly mirrors that of **Lemma** *4.1* in \[4\]. Initially, the primary difference arises from the fact that the $\Delta$ parameter for the long range Ising model, implying, $$\begin{aligned} \big| \frac{\partial}{\partial h_{i,v}} \Delta^{\mathrm{LR}}_A \big( h \big) \big| = \big| - \frac{\sum_{\sigma} \epsilon \sigma_v \mathrm{exp} \big( - \beta \mathcal{H}^{\mathrm{LR}} \big( \sigma \big) \big) }{Z^{+} \big( h \big) } - \frac{\sum_{\sigma} \epsilon \sigma_v \mathrm{exp} \big( -\beta \mathcal{H}^{\mathrm{LR}} \big( \sigma \big) \big) }{Z^{+} \big( h^A \big) } \big| \equiv \big| \epsilon \textbf{E}^{\mathrm{LR},+}_{\Lambda_N , \epsilon h} \big[ \sigma_v \big] - \epsilon \textbf{E}^{\mathrm{LR},+}_{\Lambda_N , \epsilon h^A } \big[ \sigma_v \big] \big| \\ \equiv \big| \epsilon \big| \big| \textbf{E}^{\mathrm{LR},+}_{\Lambda_N , \epsilon h} \big[ \sigma_v \big] + \textbf{E}^{\mathrm{LR},+}_{\Lambda_N , \epsilon h^A } \big[ \sigma_v \big] \big| \\ \leq 2 \epsilon \text{ } \text{ , } \end{aligned}$$ from which the Gaussian concentration inequality, from \[7\], implies the desired result for strictly positive $\epsilon$. The second inequality above can be provided with similar arguments. Besides the result above, in order to implement the steps of the Majorizing measure theorem, Dudley's entropy bound, and an upper bound for the probability of the supremum of the process $X_t$, we provide a statement of each item used in the argument, below. **Theorem** *MMT* (*Majorizing measure theorem*). For a metric space $\big( T , d \big)$, and $\big( X_t \big)_{t \in T}$ with $\textbf{E} \big( X_t \big) = 0$ for every $t$, there exists some universal, strictly positive, constant $L$ for which, $$\begin{aligned} L^{-1} \gamma_2 \big( T , d \big) \leq \textbf{E} \big[ \mathrm{sup}_{t \in T} X_t \big] \leq L \gamma_2 \big( T , d \big) \text{ } \text{ . } \end{aligned}$$ **Proposition** *DEB* (*Dudley's entropy bound*). For a family of random variables $\big( X_t \big)_{t \in T}$ satisfying, $$\begin{aligned} \textbf{P}^{\mathrm{LR},+} \big[\text{ } \big| X_t - X_s \big| \geq \lambda \text{ } \big] \leq 2 \text{ } \mathrm{exp} \bigg( - \big( \frac{\lambda}{\sqrt{2}} \big)^2 \big( d \big( s , t \big) \big)^{-2} \bigg) \text{ } \text{ , } \end{aligned}$$ there exists a universal, strictly positive, constant $L$ for which, $$\begin{aligned} \textbf{E} \big[ \mathrm{sup}_{t \in T} X_t \big] \leq L \int_0^{+\infty} \sqrt{\mathrm{log} \big[ N \big( T , d , \epsilon \big) \big] } \text{ } \mathrm{d} \epsilon \text{ } \text{ . } \end{aligned}$$ **Theorem** *S* (*upper bounding the probability of obtaining a supremum of the process* $X_t$). For the metric space $\big( T , d \big)$, and collection $\big( X_t \big)_{t \in T}$, there exists a universal, strictly positive, constant $L$ for which, $$\begin{aligned} \textbf{P} \bigg[ \mathrm{sup}_{t \in T} X_t > L \big( \gamma_2 \big( T , d \big) + u \text{ } \mathrm{diam} \big( T \big) \big) \bigg] \leq \mathrm{exp} \big( - u ^2 \big) \text{ } \text{ , } \end{aligned}$$ for any $u > 0$. The three items above will be used to establish that the following conjecture, stated in \[1\], holds, which we state as another result following the next one below. Below, we state the conjecture, and use it to prove the $\textbf{Theorem}$ for establishing that the complement of bad events occur with exponentially small probability. **Conjecture** (*upper bounding the probability of the complement of a bad event occurring with an exponential*, \[1\]). For the set of contours $\Gamma_0$ containing the origin, for any $\alpha > d$, and $d \geq 3$, there exists a constant $C_2 \equiv C_2 \big( \alpha , d \big)$ for which, $$\begin{aligned} \textbf{P} \big[ \text{ } \underset{\gamma \in \Gamma_0}{\mathrm{sup}} \frac{\big| \Delta_{I_{-} ( \gamma )} \big( \eta \big) \big|}{ \big| \gamma \big| } > 1 \text{ } \big] \leq \mathrm{exp} \big( - C^{\prime}_2 \epsilon^{-2} \big) \text{ } \text{ . } \end{aligned}$$ To prove the item above, we must introduce new counting arguments for the long range contour system. To this end, we must adapt two components of the argument for proving that a phase transition occurs in the long range, random-field Ising model from \[1\]. Recall, from the end of *2*, that the first component that the authors employ for demonstrating that the phase transition occurs is upper bounding the cardinality of, $$\begin{aligned} \mathscr{C}_l \big( \gamma \big) \equiv \underset{l \in \textbf{N}}{\bigcup} \big\{ C_l : \big| C_l \cap I \big( \gamma \big) \big| \geq \frac{1}{2} \big| C_l \big| \big\} \text{ } \text{ , } \end{aligned}$$ which represents the set of *admissible* cubes. Besides upper bounding the number of possible cubes satisfying the admissibility criteria above, the authors also upper bound the total number of paths, containing the origin and of length $n$, which is given by, $$\begin{aligned} \big| B_l \big( \Gamma_0 \big( n \big) \big) \big| \equiv \# \big\{ \forall C_l \text{ } ,\text{ } \exists \gamma \in \Gamma_0 \big( n \big) : C_l \cap B_l \neq \emptyset \text{ } , \text{ } C_l \cap \gamma \neq \emptyset \big\} \text{ } \text{ , } \end{aligned}$$ corresponding to the number of boxes covering the set of all paths containing the origin, $0$, and with length $n$. For contours that are not connected, such as those arising in long range contours, an alternative counting argument presented in \[1\] allows for a phase transition to be shown to occur in the long range Ising model in lower dimensions. For contours in the long range, random-field system, it was shown that an exponential upper bound on the possible number of paths can be obtained by analyzing, $$\begin{aligned} \bar{\mathscr{C}_l \big( \gamma \big) } \equiv \big\{ \forall C_l \in \partial \mathscr{C}_l \big( \gamma \big) \text{ } \exists C^{\prime}_l : C_l \sim C^{\prime}_l \big\} \text{ } \text{ . } \end{aligned}$$ Below, we describe a variant of the argument provided by the authors of ${\color{blue}[1]}$, from **Proposition** *3.5*, **Proposition** *3.18*, **Lemma** *3.14* and **Lemma** *3.17*, which we incorporate into the Dudley's entropy bound. **Lemma** *3* (*admissibility conditions on the number of l-cubes*, **Lemma** *3.14*, \[1\]). Fix some $A \subsetneq \textbf{Z}^d$ and $l \geq 0$. The set of admissibility criteria on the number of *l-cubes*, is comprised of the two conditions, $$\begin{aligned} \frac{1}{2} \big| C_l \big| \leq \big| C_l \cap A \big| \text{ } \text{ , } \\ \big| C^{\prime}_l \cap A \big| < \frac{1}{2} \big| C^{\prime}_l \big| \text{ } \text{ , } \end{aligned}$$ for the two faces $C_l$ and $C^{\prime}_l$ which overlap on exactly one face, the following lower bound holds, $$\begin{aligned} 2^{l(d-1)} \leq b \big| \partial_{\mathrm{ex}} A \cap U \big| \text{ } \text{ , } \end{aligned}$$ for some strictly positive $b \equiv b \big( d \big) \geq 1$. In comparison to the $l$ admissiblity condition presented above from \[1\], a similar notion of admissiblity, $rl$ admissibility, can be used for counting the possible number of contours in the long range Ising model. For completeness, we also provide this alternate notion of admissibility below. **Lemma** *4* (*admissibility conditions on the number of rl-cubes*, **Lemma** *3.17*, \[1\]). Fix some $A \subsetneq \textbf{Z}^d$, and $l \geq 0$. For the set $U \equiv C_{rl} \cup C_{r^{\prime}l}$, with $C_{rl}$ and $C_{r^{\prime}l}$ being two rl-cubes sharing exactly one face. The set of admissibility criteria is the number of $\textit{rl-cubes}$, is comprised of the two conditions, $$\begin{aligned} \frac{1}{2} \big| C_{rl} \big| \leq \big| C_{rl} \cap A \big| \text{ } \text{ , } \\ \big| C^{\prime}_{rl} \cap A \big| < \frac{1}{2} \big| C^{\prime}_{rl} \big| \text{ } \text{ , } \end{aligned}$$ for the two faces $C_{rl}$ and $C^{\prime}_{rl}$ which overlap on exactly one face, the following lower bound holds, $$\begin{aligned} 2^{rl(d-1)} \leq b^{\prime} \big| \partial_{\mathrm{ex}} A \cap U \big| \text{ } \text{ , } \end{aligned}$$ for some strictly positive $b^{\prime} \equiv b^{\prime} \big( d \big) \geq 1$. **Proposition** *1* (*Proposition 3.5 from* \[1\]). For functions the $B_0 , \cdots , B_k$, any one of which is given by, $$\begin{aligned} B_i \big( A , \textbf{Z}^d \big) \equiv B_i \equiv \big\{ \forall A \subsetneq \textbf{Z}^d \text{ } , \text{ } \exists \text{ } B_{\mathscr{C}_m} \equiv {\cup}_{C \in \mathscr{C}_m} C \text{ } : \text{ } A \cap C \neq \emptyset \big\} \text{ } \text{ , } \end{aligned}$$ for each $1 \leq i \leq k$, there exists real constants, $b_1$ and $b_2$, with $b_1 \equiv b_1 \big( d , r \big)$, and $b_2 \equiv b_2 \big( d , r \big)$ so that, $$\begin{aligned} \big| \partial \mathscr{C}_l \big( \gamma \big) \big| \leq b_1 \frac{\big| \partial_{\mathrm{ex}} I \big( \gamma \big) \big|}{2^{l(d-1)}} \leq b_1 \frac{\big| \gamma \big|}{2^{l(d-1)}} \text{ } \text{ , } \end{aligned}$$ and so that, $$\begin{aligned} \big| B_l \big( \gamma \big) \Delta B_{l+1} \big( \gamma \big) \big| \leq b_2 2^l \big| \gamma \big| \text{ } \text{ . } \end{aligned}$$ The same notions of admissibility $\textit{rl-cubes}$ can be extended to obtain an identical set of inequalities (see **Proposition** *3.18* of \[1\]). Besides the propositions above, we introduce another Proposition below for adapting Proposition *3.18* from \[1\]. This is juxtaposed with the Entropy bound which is used to count the number of possible countours for the long rang contour system. **Proposition** *2* (*Propoisitioon 3.18*, \[1\]). There exists a constant $b_4 \equiv b_4 \big( d \big)$ so that, for any natural $n$, $$\begin{aligned} \big| B_l \big( \Gamma_0 \big( n \big) \big) \big| \leq \mathrm{exp} \big( b_4 \frac{ln}{2^{l(d-1)}} \big) \text{ } \text{ , } \end{aligned}$$ in which the number of coarse-grained contours contained within $B_l \big( \Gamma_0 \big( n \big) \big)$ is bounded above by an exponential. For countours in the long range system, in comparison to upper bounding $B_l \big( \Gamma_0 \big( n \big) \big)$, a more complicated exponential bound, of the form stated below, also directly applies for lower dimensions of the long range Ising model. For the exponential upper bound, in comparison to the notation for $B_l \big( \Gamma_0 \big( n \big) \big)$, the upper bound is for $\big| B_l \big( \mathcal{C}_0 \big( n , j \big) \big) \big|$, the number of boxes covering the set of paths, $$\begin{aligned} \mathcal{C}_0 \big( n , j \big) \equiv \big\{ \gamma \in \mathcal{E}^{+}_{\Lambda} : 0 \in V \big( \gamma \big) , \big| \gamma \big| = n \big\} \text{ } \text{ . }\end{aligned}$$ **Proposition** *3* (*Proposition 3.31*, \[1\]). Fix $n,j,l \geq 0$. From the set $\mathcal{C}_0 \big( n , j\big)$ defined above, there exists a constant $c_4 \equiv c_4 \big( \alpha , d \big)$ for which, $$\begin{aligned} \big| B_l \big( \mathcal{C}_0 \big( n , j \big) \big| \leq \mathrm{exp} \bigg( c_4 l^k \bigg[ \frac{n}{2^{rl ( d - 1 - \frac{2\mathrm{log}_2(a) }{r-d-1-\mathrm{log}_2(a)})}} + \frac{n}{2^{2^{rl}}} + 1 \bigg] \bigg) \text{ } \text{ , } \end{aligned}$$ for a suitable, strictly positive constant $a$. Equipped with the counting argument for contours of the long range system, we implement the steps of the argument relying on Dudley's entropy bound, from the admissibility conditions on $\textit{rl-cubes}$. *Proof of Theorem and Conjecture, using Theorem S*. Applied to $\Delta_{I_{-} ( \gamma )} \big( \eta \big)$, rearranging terms after applying **Proposition** *DEB* implies, for $N \equiv \mathcal{C}_0 \big( n , j \big)$, $$\begin{aligned} \textbf{E} \big[ \text{ } \underset{_{\gamma \in \Gamma_0 ( n ) } }{\mathrm{sup}}\Delta_{I_{-} ( \gamma )} \big( \eta \big) \text{ } \big] \leq L \int_0^{+\infty} \sqrt{\mathrm{log} \big[ N \big( \mathcal{C}_0 \big( n , j \big) , d_2 , \epsilon \big) \big] } \text{ } \mathrm{d} \epsilon \leq \mathcal{C} \text{ } \overset{+\infty}{\underset{l=1}{\sum} } \big( 2^{\frac{rl}{2}} - 2^{\frac{rl-1}{2}} \big) \sqrt{\mathrm{log} \big[ N \big( \mathcal{C}_0 \big( n , j \big) , d_2 , l^{\prime} \big) \big] } \text{ } \text{ , }\end{aligned}$$ for strictly positive $\mathcal{C}$ satisfying, $$\begin{aligned} \mathcal{C} = 2 \epsilon b_3 \sqrt{n} \text{ } \text{ , }\end{aligned}$$ and, for $l^{\prime} \equiv \epsilon b_3 \sqrt{2^l n}$, given in **Corollary** *3.16* of \[1\]. From the upper bound above, we proceed to upper bound, $$\begin{aligned} \sqrt{\mathrm{log} \big[ N \big( \mathcal{C}_0 \big( n , j \big) , d_2 , l^{\prime} \big) \big] } \text{ } \text{ , } \end{aligned}$$ in which, from the counting argument for countours of the long range system that are not connected, $$\begin{aligned} \sqrt{\mathrm{log} \big[ \big| B_l \big( \mathcal{C}_0 \big( n , j \big) \big) \big| \big] } \equiv \sqrt{\mathrm{log} \bigg[ \mathrm{exp} \bigg( c_4 l^k \bigg[ \frac{n}{2^{rl ( d - 1 - \frac{2\mathrm{log}_2(a) }{r-d-1-\mathrm{log}_2(a)})}} + \frac{n}{2^{2^{rl}}} + 1 \bigg] \bigg) \bigg] } \text{ } \text{ . } \end{aligned}$$ The fact that the exponential and natural logarithm are inverse functions implies that the final expression above is equal to, $$\begin{aligned} \sqrt{ c_4 l^k \bigg[ \frac{n}{2^{rl ( d - 1 - \frac{2\mathrm{log}_2(a) }{r-d-1-\mathrm{log}_2(a)})}} + \frac{n}{2^{2^{rl}}} + 1 \bigg] } \text{ } \text{ , } \end{aligned}$$ hence implying, $$\begin{aligned} \mathcal{C} \text{ } \overset{+\infty}{\underset{l=1}{\sum} } \big( 2^{\frac{rl}{2}} - 2^{\frac{rl-1}{2}} \big) \sqrt{\mathrm{log} \big[ \mathcal{C}_0 \big( n , j \big) , d_2 , l^{\prime} \big) \big] } \leq \mathcal{C} \overset{+\infty}{\underset{l=1}{\sum} } \big( 2^{\frac{rl}{2}} - 2^{\frac{rl-1}{2}} \big) \sqrt{\mathrm{log} \big[ \big| B_l \big( \mathcal{C}_0 \big( n , j \big) \big) \big| \big] } \text{ } \text{ , } \end{aligned}$$ which, in light of the previous expression obtained for $\sqrt{\mathrm{log} \big[ \big| B_l \big( \mathcal{C}_0 \big( n , j \big) \big) \big| \big] }$, can be further upper bounded with, $$\begin{aligned} \overset{+\infty}{\underset{l=1}{\sum} } \big( 2^{\frac{rl}{2}} - 2^{\frac{rl-1}{2}} \big) \sqrt{ c_4 l^k \bigg[ \frac{n}{2^{rl ( d - 1 - \frac{2\mathrm{log}_2(a) }{r-d-1-\mathrm{log}_2(a)})}} + \frac{n}{2^{2^{rl}}} + 1 \bigg] } \text{ } \text{ . } \end{aligned}$$ To remove the factors $2^{\frac{rl}{2}} - 2^{\frac{rl-1}{2}}$ for $1 \leq l \leq + \infty$ in each term of the summation in the upper bound above, observe, $$\begin{aligned} \overset{+\infty}{\underset{l=1}{\sum} } \big( 2^{\frac{rl}{2}} - 2^{\frac{l-1}{2}} \big) \equiv \big( \sqrt{2} - \frac{1}{\sqrt{2}} \big) + \big( 2 - \sqrt{2} \big) + \cdots \equiv 1 - \frac{\sqrt{2}}{2} < 1 \text{ } \text{ . } \end{aligned}$$ This implies, $$\begin{aligned} \overset{+\infty}{\underset{l=1}{\sum} } \big( 2^{\frac{rl}{2}} - 2^{\frac{rl-1}{2}} \big) \sqrt{ c_4 l^k \bigg[ \frac{n}{2^{rl ( d - 1 - \frac{2\mathrm{log}_2(a) }{r-d-1-\mathrm{log}_2(a)})}} + \frac{n}{2^{2^{rl}}} + 1 \bigg] } \leq \overset{+\infty}{\underset{l=1}{\sum} } \sqrt{ c_4 l^k \bigg[ \frac{n}{2^{rl ( d - 1 - \frac{2\mathrm{log}_2(a) }{r-d-1-\mathrm{log}_2(a)})}} + \frac{n}{2^{2^{rl}}} + 1 \bigg] } \text{ } \text{ . } \end{aligned}$$ Furthermore, from the upper bound above, $$\begin{aligned} \overset{+\infty}{\underset{l=1}{\sum} } \sqrt{ c_4 l^k \bigg[ \frac{n}{2^{rl ( d - 1 - \frac{2\mathrm{log}_2(a) }{r-d-1-\mathrm{log}_2(a)})}} + \frac{n}{2^{2^{rl}}} + 1 \bigg] } \leq \sqrt{c_4} \bigg[ \overset{+\infty}{\underset{l=1}{\sum} } \sqrt{l^k \bigg[ \frac{n}{2^{rl ( d - 1 - \frac{2\mathrm{log}_2(a) }{r-d-1-\mathrm{log}_2(a)})}} + \frac{n}{2^{2^{rl}}} \bigg] } + \overset{+\infty}{\underset{l=1}{\sum} } \sqrt{l^k} \bigg] \text{ } \text{ . } \end{aligned}$$ From these rearrangements, one has, $$\begin{aligned} \textbf{E} \big[ \text{ } \underset{\gamma \in \Gamma_0 ( n )}{\mathrm{sup}} \Delta_{I_{-} ( \gamma)} \big( h \big) \text{ } \big] \leq \textbf{E} \bigg[ \sqrt{c_4} \bigg[ \overset{+\infty}{\underset{l=1}{\sum} } \sqrt{l^k \bigg[ \frac{n}{2^{rl ( d - 1 - \frac{2\mathrm{log}_2(a) }{r-d-1-\mathrm{log}_2(a)})}} + \frac{n}{2^{2^{rl}}} \bigg] } + \overset{+\infty}{\underset{l=1}{\sum} } \sqrt{l^k} \bigg] \bigg] \leq b_5 \big( b_4 \big) \epsilon n \text{ } \text{ . } \end{aligned}$$ Before finishing the argument, first observe, $$\begin{aligned} \textbf{P} \big[ \underset{\gamma \in \Gamma_0}{\mathrm{sup}} \frac{\big| \Delta_{I_{-} ( \gamma )} \big( \eta \big) \big|}{ c^{\prime}_1 \big| \gamma \big| } > 1 \big] \approx \textbf{P} \big[ \underset{\gamma \in \Gamma_0}{\mathrm{sup}} \frac{\big| \Delta_{I_{-} ( \gamma )} \big( \eta \big) \big|}{ \big| \gamma \big| } > 1 \big] \text{ } \text{ , } \end{aligned}$$ from which, $$\begin{aligned} \textbf{P} \big[ \text{ } \underset{\gamma \in \Gamma_0 ( n )}{\mathrm{sup}} \frac{ \Delta_{I_{-} ( \gamma)} \big( \eta \big)}{\big| \gamma \big| } \geq \frac{c^{\prime}_2}{2} \text{ } \big] \equiv \textbf{P} \big[ \text{ } \underset{\gamma \in \Gamma_0 ( n )}{\mathrm{sup}} \Delta_{I_{-} ( \gamma)} \big( \eta \big) \geq \frac{c^{\prime}_2}{2} n \text{ } \big] \leq \textbf{P} \bigg[ \underset{\gamma \in \Gamma_0 ( n )}{\mathrm{sup}} \Delta_{I_{-} ( \gamma)} \big( \eta \big) \geq L \big( b_5 \big( b_4 \big) \epsilon n + \big) \bigg] \text{ } \text{ , } \end{aligned}$$ for a suitable, strictly positive, $b_5$, dependent upon $b_4$, which we achieve by applying the result, $$\begin{aligned} \textbf{P} \bigg[ \mathrm{sup}_{t \in T} X_t > L \big( \gamma_2 \big( T , d \big) + u \text{ } \mathrm{diam} \big( T \big) \big) \bigg] \leq \mathrm{exp} \big( - u ^2 \big) \text{ } \text{ , } \end{aligned}$$ implying the desired upper bound, upon substituting an upper bound for $\gamma_2 \big( T , d \big)$, and also for $\mathrm{diam} \big( T \big)$, $$\begin{aligned} \textbf{P} \bigg[ \underset{\gamma \in \Gamma_0 ( n )}{\mathrm{sup}} \Delta_{I_{-} ( \gamma)} \big( \eta \big) \geq L \big( b_5 \big( b_4 \big) \epsilon n - \frac{\sqrt{\mathscr{C}_2}}{\epsilon} \big) \bigg] \text{ } \text{ , } \end{aligned}$$ where, $$\begin{aligned} \mathrm{diam} \big( T \big) \equiv \mathrm{diam} \big( \mathcal{C}_0 \big( n , j \big) \big) \equiv \underset{\gamma_1 , \gamma_2 \in \mathcal{C}_0 ( n , j )}{\mathrm{sup}} d \big( \gamma_1 , \gamma_2 \big) \equiv \underset{\gamma_1 , \gamma_2 \in \mathcal{C}_0 ( n , j )}{\mathrm{sup}} \big\{ M > 0 : d \big( \gamma_1 , \gamma_2 \big) \equiv M \big\} \text{ } \text{ , } \end{aligned}$$ where, $$\begin{aligned} \underset{\gamma_1 , \gamma_2 \in \mathcal{C}_0 ( n , j )}{\mathrm{sup}} \big\{ M > 0 : d \big( \gamma_1 , \gamma_2 \big) \equiv M \big\} \propto C \big( n , j , \epsilon , M \big) \big| \big| \gamma_1 - \gamma_2 \big|\big|_1 \big| I \big( \gamma_1 \big) \cap I \big( \gamma_2 \big) \big| \text{ } \text{ . }\end{aligned}$$ Therefore, $$\begin{aligned} \textbf{P} \bigg[ \text{ } \underset{\gamma \in \Gamma_0 ( n )}{\mathrm{sup}} \Delta_{I_{-} ( \gamma)} \big( \eta \big) \geq L \bigg( b_5 \big( b_4 \big) \epsilon n - \frac{\sqrt{\mathscr{C}_2} C^{\prime} C }{\epsilon} \big| \big| \gamma_1 - \gamma_2 \big|\big|_1 \big| I \big( \gamma_1 \big) \cap I \big( \gamma_2 \big) \big| \bigg) \bigg] \leq \mathrm{exp} \big( - \mathscr{C}_2 \epsilon^{-2} \big) \text{ } \text{ , } \end{aligned}$$ from which we conclude the argument, for suitable $\mathscr{C}_2 \equiv \mathscr{C}_2 \big( \alpha , d \big)$, and some $C \equiv C \big( n , j , \epsilon , M \big)$, $C>0$. We conclude with the arguments in the next section with the Peierls' argument. ## Concluding with the classical Peierls' argument In the final section, we state the inequality for executing the Peierls' argument. **Theorem** (*Peierls' argument for the long range contour system, a conjecture raised in* \[1\]). For $d \geq 3$ and $d < \alpha \leq d+1$, there exists a suitable constant $C \equiv C \big( \alpha , d \big)$, such that, $$\begin{aligned} \textbf{P}^{\mathrm{LR},+}_{\Lambda} \big[ \sigma_0 \equiv - 1 \big] \leq \mathrm{exp} \big( - C^{\prime} \beta \big) + \mathrm{exp} \big( - C^{\prime} \epsilon^{-2} \big) \text{ } \text{ , } \end{aligned}$$ for the event, $$\begin{aligned} \big\{ \sigma_0 \equiv - 1 \big\} \text{ } \text{ , } \end{aligned}$$ for all $\beta >0$, $e \leq C^{\prime}$ and $N \geq 1$, has $\textbf{P}$-probability less than, or equal to, $$\begin{aligned} 1 - \mathrm{exp} \big( - C^{\prime} \beta \big) + \mathrm{exp} \big( - C^{\prime} \epsilon^{-2} \big) \text{ } \text{ . } \end{aligned}$$ Hence, for $\beta > \beta_c$, the long range Ising model undergoes a phase transition, in which, $$\begin{aligned} \textbf{P}^{\mathrm{LR}, +}_{\Lambda , \beta, \epsilon} \neq \textbf{P}^{\mathrm{LR},-}_{\Lambda, \beta , \epsilon} \text{ } \text{ , } \end{aligned}$$ with $\textbf{P}$-probability $1$, as stated in $\textbf{Theorem PT}$. *Proof of Theorem and Theorem PT*. Under the long rang Ising model probability measure $\textbf{P}^{\mathrm{LR},+}_{\Lambda} \big( \cdot \big) \equiv \textbf{P}^{+}_{\Lambda} \big( \cdot \big)$, to demonstrate that the desired inequality holds, along the lines of the argument for **Theorem** *4.1* in \[1\], write, from the joint probability measure, $$\begin{aligned} \textbf{Q}^{\mathrm{LR},+}_{\Lambda , \beta} \big( \sigma \in A , h \in B \big) \equiv \textbf{Q}^{+}_{\Lambda , \beta} \big( \sigma \in A , h \in B \big) \equiv \underset{B}{\int} \textbf{P}^{\mathrm{LR},+}_{\Lambda_,\beta } \big( A \big) \text{ } \mathrm{d} \textbf{P}^{\mathrm{LR},+}_{\Lambda , \beta} \big( h \big) \equiv \underset{B}{\int} \textbf{P}^{+}_{\Lambda_,\beta } \big( A \big) \text{ } \mathrm{d} \textbf{P}^{+}_{\Lambda , \beta} \big( h \big) \text{ } \text{ , } \end{aligned}$$ under $+$ boundary conditions, from which the joint probability of $\big\{ \sigma_0 \equiv -1 \big\}$, $$\begin{aligned} \textbf{Q}^{+}_{\Lambda , \beta} \big( \sigma_0 \equiv -1 \big) = \textbf{Q}^{+}_{\Lambda , \beta} \big( \big\{ \sigma_0 \equiv -1 \big\} \cap \mathcal{B} \big) + \textbf{Q}^{+}_{\Lambda , \beta} \big( \big\{ \sigma_0 \equiv -1 \big\} \cap \mathcal{B}^c \big) \leq \textbf{Q}^{+}_{\Lambda , \beta} \big( \big\{ \sigma_0 \equiv -1 \big\} \cap \mathcal{B} \big) + \mathrm{exp} \big( - C^{\prime}_1 \epsilon^{-2} \big) \text{ } \text{ , } \end{aligned}$$ where in the last inequality, we upper bound one of the joint probability terms under $+$ boundary conditions from the fact that, $$\begin{aligned} \textbf{Q}^{+}_{\Lambda , \beta} \big( \big\{ \sigma_0 \equiv -1 \big\} \cap \mathcal{B}^c \big) \leq \textbf{Q}^{+}_{\Lambda , \beta} \big( \mathcal{B}^c \big) \leq \mathrm{exp} \big( - C^{\prime}_1 \epsilon^{-2} \big) \text{ } \text{ . } \end{aligned}$$ Next, write, $$\begin{aligned} \textbf{Q}^{+}_{\Lambda , \beta} \big( \sigma_0 \equiv -1 \big) \leq \underset{\gamma \in \mathcal{C}_0}{\sum} \textbf{Q}^{+}_{\Lambda , \beta} \big( \Omega \big( \gamma \big) \big) \text{ } \text{ , } \end{aligned}$$ corresponding to the summation over all contours $\gamma$ with $0 \in V \big( \gamma \big)$, for the collection of spins satisfying, $$\begin{aligned} \Omega \big( \gamma \big) \equiv \big\{ \sigma \in \Omega : \gamma \subset \Gamma \big( \sigma \big) \big\} \text{ } \text{ . } \end{aligned}$$ From the computations thus far with the joint measure $\textbf{Q}^{+}_{\Lambda , \beta} \big( \cdot \big)$, we proceed to write a decomposition for, $$\begin{aligned} \textbf{Q}^{+}_{\Lambda , \beta} \big( \big\{ \sigma_0 \equiv -1 \big\} \cap \mathcal{B} \big) \text{ } \text{ , } \end{aligned}$$ with the integral over all possible bad events, which admits the upper bound, for, $$\begin{aligned} \int_{\mathcal{B}} \underset{\sigma: \sigma_0 \equiv -1}{\sum} \mathcal{D}^{\mathrm{LR}, +}_{\Lambda , \beta} \big( \sigma , \eta \big) \mathrm{d} \eta \equiv \int_{\mathcal{B}} \underset{\sigma: \sigma_0 \equiv -1}{\sum} \mathcal{D}^{+}_{\Lambda , \beta} \big( \sigma , \eta \big) \mathrm{d} \eta \end{aligned}$$ with, denoting $\tau^{\mathrm{LR}}_{I_{-} ( \gamma )} \big( \eta \big) \equiv \tau^{\mathrm{LR}} \big( \eta \big)$, $$\begin{aligned} \underset{\mathcal{C}_0}{\sum} \int_{\mathcal{B}} \underset{ \gamma \in \sigma \in \Omega ( \gamma )}{\sum} \mathcal{D}^{\mathrm{LR},+}_{\Lambda , \beta} \big( \sigma , \eta \big) \mathrm{d} \eta \equiv \underset{\mathcal{C}_0}{\sum} \int_{\mathcal{B}} \underset{ \gamma \in \sigma \in \Omega ( \gamma )}{\sum} \mathcal{D}^{+}_{\Lambda , \beta} \big( \sigma , \eta \big) \mathrm{d} \eta \leq \underset{\gamma \in \mathcal{C}_0}{\sum} \frac{2^{| \gamma |} \int_{\mathcal{B}} \underset{ \gamma \in \sigma \in \Omega ( \gamma )}{\sum} \mathcal{D}^{+}_{\Lambda , \beta} \big( \sigma , \eta \big) \mathrm{d} \eta }{\int_{\mathcal{B}} \underset{ \gamma \in \sigma \in \Omega ( \gamma )}{\sum} \mathcal{D}^{+}_{\Lambda , \beta} \big( \tau^{\mathrm{LR}} \big( \sigma \big) , \tau^{\mathrm{LR}} \big( \eta \big) \big) \mathrm{d} \eta } \\ \leq \underset{\gamma \in \mathcal{C}_0}{\sum} 2^{| \gamma |} {\underset{\eta \in \mathcal{B} , \sigma \in \Omega ( \gamma ) }{\mathrm{sup}}} \frac{\mathcal{D}^{+}_{\Lambda , \beta} \big( \sigma , \eta \big)}{\mathcal{D}^{+}_{\Lambda , \beta} \big( \tau^{\mathrm{LR}} \big( \sigma \big) , \tau^{\mathrm{LR}} \big( \eta \big) \big)} \text{ } \text{ . } \end{aligned}$$ In the rearrangements above, the $2^{| \gamma |}$ arises from the fact that, $$\begin{aligned} \int_{\mathcal{B}} \underset{\omega \in \Omega ( \gamma )}{\sum} \mathcal{D}^{+}_{\Lambda , \beta} \big( \tau^{\mathrm{LR}} \big( \sigma \big) , \tau^{\mathrm{LR}} \big( \eta \big) \big) \mathrm{d} \eta \leq 2^{| \gamma |} \text{ } \text{ . } \end{aligned}$$ Next, recall the identity, $$\begin{aligned} \frac{\mathcal{D}^{+}_{\Lambda , \beta} \big( \sigma , \eta \big) Z^{+ }_{\Lambda ,\beta } \big( \eta \big) }{\mathcal{D}^{+}_{\Lambda , \beta} \big( \tau_{\gamma} \big( \sigma \big) , \tau_{\gamma} \big( \eta \big) \big) Z^{+}_{\Lambda ,\beta } \big( \tau \big( \eta \big) \big) } = \mathrm{exp} \big[ \beta \mathcal{H}^{\mathrm{LR}, +}_{\Lambda} \big( \tau_{\gamma} \big( \sigma \big) \big) - \beta \mathcal{H}^{\mathrm{LR}, +}_{\Lambda} \big( \sigma \big) \big] \text{ } \text{ } \text{ , } \end{aligned}$$ and the definition of bad events $\mathcal{B}$, we proceed in the computations by upper bounding the following supremum, $$\begin{aligned} \underset{\sigma \in \Omega ( \gamma )}{ \underset{\eta \in \mathcal{B}}{ \mathrm{sup}} } \frac{\mathcal{D}^{+}_{\Lambda , \beta} \big( \sigma , \eta \big)}{\mathcal{D}^{+}_{\Lambda , \beta} \big( \tau^{\mathrm{LR}} \big( \sigma \big) , \tau^{\mathrm{LR}} \big( \eta \big) \big)} \leq \mathrm{exp} \big( - \beta c^{\prime}_2 \big| \gamma \big| \big) \text{ } \underset{\sigma \in \Omega ( \gamma )}{ \underset{\eta \in \mathcal{B}}{ \mathrm{sup}} } \frac{Z^{+}_{\Lambda , \beta , \eta} \big( \tau \big( \eta \big)}{Z^{+}_{\Lambda , \beta , \eta} \big( \eta \big) } \equiv \underset{\sigma \in \Omega ( \gamma )}{ \underset{\eta \in \mathcal{B}}{ \mathrm{sup}} } \big[ \mathrm{exp} \big( - \beta c^{\prime}_2 \big| \gamma \big| \big) \text{ } \mathrm{exp} \big( \beta \Delta_{\gamma} \big( h \big) \big) \big] \\ \overset{\Delta_{\gamma} ( h ) \leq \frac{1}{2} c^{\prime}_2 | \gamma | , \forall h \in \mathcal{B} }{\leq} \mathrm{exp} \big( - \frac{\beta}{2} c^{\prime}_2 \big| \gamma \big| \big) \text{ } \text{ . } \end{aligned}$$ From the upper bound above, previous computations imply the following upper bound, $$\begin{aligned} \textbf{Q}^{+}_{\Lambda , \beta} \big( \sigma_0 \equiv -1 \big) \leq \underset{0 \in V ( \gamma )}{\underset{\gamma \in \mathcal{C}_0}{\sum}} 2^{| \gamma |} \mathrm{exp} \big( - \frac{\beta}{2} c^{\prime}_2 \big| \gamma \big| \big) + \mathrm{exp} \big( - c_0 \epsilon^{-2} \big) \equiv \underset{0 \in V ( \gamma )}{\underset{\gamma \in \mathcal{C}_0}{\sum}} \mathrm{exp} \big( - \frac{\beta}{2} c^{\prime}_2 \big| \gamma \big| + \mathrm{log} 2 \big| \gamma \big| \big) + \mathrm{exp} \big( - c_0 \epsilon^{-2} \big) \\ \leq \underset{n \geq 1}{\underset{0 \in V ( \gamma )}{\underset{\gamma \in \mathcal{E}^{+}_{\Lambda}, | \gamma | \equiv n }{\sum}}} \mathrm{exp} \big( - \frac{\beta}{2} c^{\prime}_2 n + \big( \mathrm{log} 2 \big) n \big) + \mathrm{exp} \big( - c_0 \epsilon^{-2} \big) \\ \leq \underset{n \geq 1}{\sum} \big| \mathcal{C}_0 \big( n \big) \big| \mathrm{exp} \big( - \frac{\beta}{2} c^{\prime}_2 n + \big( \mathrm{log} 2 \big) n \big) + \mathrm{exp} \big( - c_0 \epsilon^{-2} \big) \end{aligned}$$ from which the final upper bound, $$\begin{aligned} \underset{n \geq 1}{\sum} \text{ } \mathrm{exp} \bigg( \big( C_1 - \frac{\beta}{2} c^{\prime}_2 + \mathrm{log} 2 \big) n \bigg) + \mathrm{exp} \big( - c_0 \epsilon^{-2} \big) \text{ } \text{ , } \end{aligned}$$ holds, from the existence of a constant for which, $$\begin{aligned} C_1 \geq \frac{1}{n} \mathrm{log} \bigg[ \bigg| \underset{n \geq 1}{\sum} \big| \mathcal{C}_0 \big( n \big) \big| \bigg| \bigg] \text{ } \text{ . } \end{aligned}$$ Proceeding, for $\beta$ sufficiently large, $$\begin{aligned} \mathrm{exp} \big( - \frac{\beta}{2} c^{\prime}_2 \big) \leq \mathrm{exp} \big( - 2 \beta C \big) \text{ } \text{ , } \end{aligned}$$ the ultimate term in the upper bound implies the following upper bound, $$\begin{aligned} \textbf{Q}^{+}_{\Lambda , \beta} \big( \sigma_0 \equiv -1 \big) \leq \mathrm{exp} \big( - 2 \beta C \big) + \mathrm{exp} \big( - c_0 \epsilon^{-2} \big) \text{ } \text{ , } \end{aligned}$$ for a constant satisfying, $$\begin{aligned} C \leq \frac{c^{\prime}_2}{4} \text{ } \text{ . } \end{aligned}$$ Altogether, we conclude the argument with the $\textbf{P}$-probability statement, in which, $$\begin{aligned} \textbf{P} \bigg[ \mathcal{D}^{+}_{\Lambda , \beta} \big( \sigma_0 \equiv -1 \big) \geq \mathrm{exp} \big( - C \beta \big) + \mathrm{exp} \big( - C \epsilon^{-2} \big) \bigg] \overset{(\mathrm{Markov})}{\leq} \frac{\textbf{Q}^{+}_{\Lambda , \beta} \big( \sigma_0 \equiv -1 \big)}{\mathrm{exp} \big( - C \beta \big) + \mathrm{exp} \big( - C \epsilon^{-2} \big) } \\ \leq \frac{\mathrm{exp} \big( - 2 \beta C \big) + \mathrm{exp} \big( - 2 C \epsilon^{-2} \big) }{\mathrm{exp} \big( - C \beta \big) + \mathrm{exp} \big( - C \epsilon^{-2} \big) } \\ \leq \bigg( \mathrm{exp} \big( - C \beta \big) + \mathrm{exp} \big( - C \epsilon^{-2} \big) \bigg)^{-1} \text{ } \text{ . } \end{aligned}$$ Hence the desired phase transition holds with $\textbf{P}$-probability $1$, from which we conclude the argument. # References Affonso, L., Bissacot, R., Maia, J.V. Phase Transition for Long-Range Random Field Ising Model in Higher Dimensions. *arXiv: 2307.14150 v2* (2023). Affonso, L., Bissacot, R., Endo, E.O., Handa, S. Long-Range Ising Models: Contours, Phase Transitions and Decaying Fields. *arXiv: 2105.06103 v4* (2022). Ding, J., Wirth, W. Correlation length of the two-dimensional random field Ising model via greedy lattice animal. arXiv: 2011.08768 v3 (2022). Ding, J., Zhuang, Z. Long range order for the random field Ising and Potts models. *arXiv: 2210.0453 v1* (2021). Dudley, R.M. The sizes of compact subsets of Hilbert space and continuity of Gaussian processes. *Journal of Functional Analysis*, **1**:290 (1967). Fisher, D.S., Fröhlich, J., and Spencer,T. The Ising model in a random magnetic field. *Journal of Statistical Physics*, **34**(5-6): 863--870 (1984). Ledoux, M., Talagrand, M. Probability in Banach Spaces: Isoperimetry and Processes. *A Series of Modern Surveys in Mathematics Series*. Springer (1991). Peierls, R. On Ising's model of ferromagnetism. In Mathematical Proceedings of the Cambridge Philosophical Society **32**, pages 477--481, 1936. Reed, P. The Potts model in a random field: a Monte Carlo study. Journal of Physics c: Solid State Physics, 18, 20 (1985). Rigas, P. Correlation length lower bound for the random-field Potts model with the greedy lattice animal. *arXiv: 2211.06795, v2* (2022). Son, S-W., Jeong, H. Noh, J.D. Random field Ising model and community structure in complex networks. *arXiv: 0502672 v1* (2005). Talagrand, M. Upper and lower bounds for stochastic processes, **60**. Springer (2014). Tissier, M., Tarjus, G. Supersymmetry and its spontaneous breaking in the random field Ising model. *arXiv: 1103.4812 v2* (2011). Wu, Y., Machta, J. Ground states and thermal states of the random field Ising model. *arXiv: 0501619 v2* (2005). [^1]: ***Keywords***: Phase transition, long range Ising model, random field Ising model, long range, random field Ising model, Peierls' argument, contour system

arxiv_math

--- abstract: | A multiparameter quantum affine space of rank $n$ is the $\mathbb F$-algebra generated by indeterminates $X_1, \cdots, X_n$ satisfying $X_iX_j = q_{ij} X_jX_i \ (1 \le i < j \le n)$ where $q_{ij}$ are nonzero scalars in $\mathbb F^\ast$. The corresponding quantum torus is generated by the $X_i$ and together with their inverses subject to the same relations. So far the automorphisms of a quantum affine space have been considered mainly in the uniparameter case, that is, $q_{ij} = q$. We remove this restriction here. Necessary and sufficient conditions are obtained for the quantum affine space to be rigid, that is, the only automorphisms are the trivial ones arising from the action of the torus $(\mathbb F^\ast)^n$. These conditions are based on the multiparameters $q_{ij}$ and also on the subgroup of $\mathbb F^\ast$ generated by these multiparameters. We employ the results in J. Alev and M. Chamarie, Derivations et automorphismes de quelques algebras quantiques, Communications in Algebra, 1992 (20), 1787-1802, and point out a small error in a main theorem in this paper which however remains valid with a small modification. We also note that a quantum affine space whose corresponding quantum torus has dimension one necessarily has a trivial automorphism group. This is a consequence of a result of J. M. Osborne, D. S. Passman, Derivations of Skew Polynomial Rings, J. Algebra, 1995, 176, 417--448. We expand the known list of examples of quantum tori that have dimension one and are thus hereditary noetherian domains. **Keywords.** quantum torus, quantum affine space, automorphism, dimension, hereditary ring\ **2010 Math. Subj. Class.**: 16S38; 16S35; 16S36; 16W20 address: - | Ashish Gupta, Department of Mathematics\ Ramakrishna Mission Vivekananda Educational and Research Institute (Belur Campus)\ Howrah, WB 711202\ India - | Sugata Mandal, Department of Mathematics\ Ramakrishna Mission Vivekananda Educational and Research Institute (Belur Campus)\ Howrah, WB 711202\ India author: - Ashish Gupta and Sugata Mandal title: "Triviality of the automorphism group of the multiparameter quantum affine $n$-space" --- # Introduction {#intro} Quantum affine spaces and their localizations (known as quantum tori) are known to play a key role in the theory of quantum groups [@ART1997; @BrGo] and also in non-commutative geometry [@Man]. The quantum tori arise also in Lie theory as coordinate structures of extended affine Lie algebras [@NeebKH2008] and in the representation theory of nilpotent groups [@Br2000]. The automorphisms of these algebras have been considered in [@AC1992], [@OP1995], [@ART1997] and [@NeebKH2008]. The substantial paper [@MY12] considers the automorphisms of certain completions of a quantum torus algebra in connection with the automorphisms of quantum enveloping algebras. In this last paper it is noted that the automorphisms of a quantum affine space of certain types extend to bifinte unipotent automorphisms of the completion. The automorphisms of quantum division rings are studied in [@VA2000]. Let us briefly recall the definitions. Let $\mathbb{F}$ be a field and $\mathfrak q = (q_{ij})$ be a multiplicatively anti-symmetric $n \times n$-matrix with entries in $\mathbb F^\ast$. This means that $q_{ii} = 1$ and $q_{ji} = q_{ij}^{-1}.$ A (rank-$n$) quantum affine space $\mathcal O_{\mathfrak q} = \mathcal O_{\mathfrak q}(\mathbb F^n)$ over the field $\mathbb F$ can is defined (as a quotient of the free algebra $\mathbb F\{ X_1, X_2, \cdots, X_n \}$) as follows $$\begin{aligned} \label{quantum_space_def} \mathcal O_{\mathfrak q} &:= \mathbb F\{ X_1, X_2, \cdots, X_n \}/\langle X_i X_j - q_{ij}X_j X_i \mid 1 \le i < j \le n \rangle, \qquad q_{ij} \in \mathbb F^\ast.\end{aligned}$$ This ring can also be presented as an iterated skew polynomial ring. Localizing a quantum affine space $\mathcal O_{\mathfrak q}$ at the multiplicative subset generated the variables $X_1, \cdots, X_n$ yields the Laurent version of this algebra which is known as the rank-$n$ quantum torus. We denote this quantum torus as $\widehat {\mathcal O}_{\mathfrak q}$. A (rank-$n$) quantum torus can thus be defined as an algebra generated by the variables $X_1, \cdots, X_n$ together with their inverses subject to relations $$X_i X_j = q_{ij}X_j X_i.$$ **Definition 1**. For a quantum affine space $\mathcal O_{\mathfrak q}$ and the quantum torus $\widehat{\mathcal O}_{\mathfrak q}$ the $\lambda$-group (denoted $\Lambda$) is the subgroup of $\mathbb F^\ast$ generated by the multiparameters $q_{ij}$. For an abelian group $A$ by the *rank* of $A$ we mean its torsion-free rank, that is, $$\mathop{\mathrm{\mathrm{rk}}}(A): = \dim_{\mathbb Q}(A \otimes_{\mathbb Z} \mathbb Q).$$ It was shown in [@MP] that the Krull and the global dimension for the quantum tori coincide and so we will write $\dim(\widehat {\mathcal O}_{\mathfrak q})$ to denote either of these. In the study of the quantum torus this dimension plays an important role [@MP; @Br2000]. Let $\mathop{\mathrm{\mathrm{Aut}}}_{\mathbb F}(\mathcal O_{\mathfrak q})$ denote the $\mathbb F$-automorphism group of the $\mathbb F$-algebra $\mathcal O_{\mathfrak q}$. We show the following. **Theorem 1 1**. *Suppose that $\mathop{\mathrm{\mathrm{char}}}(\mathbb F) = 0$ and $n \ge 3$. Let $\mathfrak q = (q_{ij})$ be a multiplicatively antisymmetric matrix such that for $i<j$ atmost one entry $q_{ij}$ equals to $1$. Then* - *if no $q_{ij}$ equals to $1$ for all $i < j$, then the automorphism group $$\mathop{\mathrm{\mathrm{Aut}}}_{\mathbb F}(\mathcal O_{\mathfrak q}) = {(\mathbb F^\ast)}^n$$ if and only if the following conditions hold* - *$E= 0$[\[E=01\]]{#E=01 label="E=01"} and* - *there does not exist any non-identity permutation matrix $m_{\sigma}$ satisfying $$\label{commute permutation1} \mathfrak{q}\mathfrak{m_{\sigma}}=\mathfrak{m_{\sigma}}\mathfrak{q}.$$* - *if exactly one entry, say $q_{i'j'}$ with $i'<j'$ is $1$, then the automorphism group $$\mathop{\mathrm{\mathrm{Aut}}}_{\mathbb F}(\mathcal O_{\mathfrak q}) = {(\mathbb F^\ast)}^n$$ if and only if the following conditions hold* - *[\[2nd E=01\]]{#2nd E=01 label="2nd E=01"} $E= 0$ and* - *there does not exist any permutation $\pi_{1}$ on the set $\{i',j'\}$ and non-identity permutation $\pi_{2}$ on the set $\mathcal{J}= \{1,2,\cdots,n\} \setminus \{i',j'\}$ satisfying $$\label{2nd commute permutation1} q_{i'r}=q_{\pi_{1}(i')\pi_{2}(r)}, \quad q_{j'r}=q_{\pi_{1}(j')\pi_{2}(r)}, \qquad \forall r \in \mathcal{J}$$ such that $\mathfrak{q'}\mathfrak{m}_{\pi_{2}}=\mathfrak{m}_{\pi_{2}}\mathfrak{q'}$ where $\mathfrak q'$ is a multiplicatively anti-symmetric matrix obtained from $\mathfrak{q}$ by removing the $i',j'$ rows and columns simultaneously.* **Theorem 2 1**. *A quantum affine space $\mathcal O_{\mathfrak q} = \mathcal O_{\mathfrak q}(\mathbb F^n) \ (n \ge 3)$ whose $\lambda$-group is torsion-free and has rank no smaller than $c: = \binom{n - 1}{2} + 1$ satisfies $$\mathop{\mathrm{\mathrm{Aut}}}_{\mathbb F}(\mathcal O_{\mathfrak q})= (\mathbb{F^{\ast}})^n.$$ Moreover, $c$ is minimal with respect to this property and for each $n \ge 3$ and each $r < \binom{n - 1}{2} + 1$ there exists an algebra $\mathcal O_{\mathfrak q}$ with $\lambda$-group equal to $\mathbb Z^r$ but whose automorphism group embeds $(\mathbb F^\ast)^n \rtimes \mathbb F^+$.* This paper is organized as follows: Section 2 treats the facts concerning the quantum torus that we will be needing. Sections 3 and 4 are devoted to the proofs of Theorems 1 and Theorem 2 respectively. Our final section addresses the question of obtaining quantum tori of rank $n$ with dimension one as the automorphism group will be always trivial in this case. # The Quantum Torus {#aut-qtor} ## Twisted Group Algebra Structure Let $\Gamma : = \mathbb Z^n$. A twisted group algebra $\mathbb F \ast \Gamma$ is an $\mathbb F$-algebra which has a copy $\bar \Gamma : = \{\bar \gamma \mid \gamma \in \Gamma \}$ for an $\mathbb F$-basis satisfying $$\bar \gamma \bar {\gamma'} = f(\gamma, \gamma')\overline{\gamma \gamma'}, \qquad \gamma, \gamma' \in \Gamma.$$ for a suitable $2$-cocycle $f: \Gamma \times \Gamma \rightarrow \mathbb F^\ast$. A rank-$n$ quantum torus $\widehat {\mathcal O}_{\mathfrak q}$ has such a structure. Indeed if we define a map $\Gamma \rightarrow \widehat {\mathcal O}_{\mathfrak q}$ via $$(\gamma_1, \cdots, \gamma_n) = :\gamma \mapsto \mathbf X^{\gamma} := X_1^{\gamma_1}\cdots X_n^{\gamma_n}$$ then it can be checked that the above conditions are satisfied. Clearly each element $\alpha$ of a twisted group algebra $\mathbb F \ast \Gamma$ may be expressed as a finite sum $\alpha = \sum_{\gamma \in \Gamma} a_\gamma \bar \gamma\ (a_\gamma \in \mathbb F)$. The subset of $\gamma \in \Gamma$ such that $a_\gamma \ne 0$ is called the support of $\alpha$ and denoted as $\mathop{\mathrm{\mathrm{Supp}}}(\alpha)$. For a subgroup $B \le \Gamma$ the subset $$\{ \alpha \in \mathbb F \ast \Gamma \mid \mathop{\mathrm{\mathrm{Supp}}}(\alpha) \subseteq B \}$$ is a twisted group algebra of $B$ over $\mathbb F$ and is denoted as $\mathbb F \ast B$. [\[sec2.2\]]{#sec2.2 label="sec2.2"} ## The commutator map $\lambda$ In a quantum torus the monomials $\mathbf X^{\gamma} := X_1^{\gamma_1}\cdots X_n^{\gamma_n}$ are units and the group-theoretic commutator $[\mathbf X^{\gamma}, \mathbf X^ {\gamma'}]$ defined as $$[\mathbf X^{\gamma}, \mathbf X^ {\gamma'}] := \mathbf X^{\gamma} \mathbf X^{\gamma'}{(\mathbf X^{\gamma})}^{-1}{(\mathbf X^{\gamma'})}^{-1}$$ yields an alternating bi-character ((e.g., [@OP1995 Section 1])) $$\label{lmbdadefn} \lambda: \Gamma \times \Gamma \rightarrow \mathbb F^\ast, \qquad \lambda(\gamma, \gamma') = [\mathbf X^{\gamma},\mathbf X^{\gamma'}], \qquad \mathbf \gamma, \gamma' \in \Gamma.$$ In particular, $$\lambda(\mathbf e_i, \mathbf e_j) = [X_i, X_j] = q_{ij}, \qquad \forall 1 \le i,j \le n,$$ where $\mathbf e_1, \cdots \mathbf e_n$ are the standard basis vectors of the $\mathbb Z$-module $\Gamma$. ## Dimension {#dim-of-qtorus} As noted above a quantum torus $\widehat{O}_{\mathfrak q}$ is a twisted group algebra $\mathbb F \ast \Gamma$. It was shown in [@Br2000 Theorem A] that the dimension of a quantum torus equals the supremum of the ranks of subgroups $B \le \Gamma$ such that the subalgebra $\mathbb F \ast B$ is commutative (Note that $\mathbb F \ast C$ is commutative for any cyclic subgroup $C \le \Gamma$). It follows that $\dim(\widehat{O}_{\mathfrak q})$ equals the cardinality of a maximal system of independent commuting monomials in $\widehat{O}_{\mathfrak q}$. ## The automorphism group of a quantum torus It is known (e.g., [@MP]) that the units of a quantum torus algebra are trivial, that is, are of the form $a \mathbf X^{\gamma}$, where $a \in \mathbb F^\ast$. By $\mathop{\mathrm{\mathrm{Aut}}}_{\mathbb F}(\widehat{\mathcal O}_{\mathfrak q})$ we denote the group of all $\mathbb F$-automorphisms of the quantum torus $\widehat {\mathcal O}_{\mathfrak q}$. It is easily seen (e.g., [@OP1995]) that the action of the group $\mathop{\mathrm{\mathrm{Aut}}}_{\mathbb F}(\widehat{\mathcal O}_{\mathfrak q})$ on the quantum torus $\widehat{\mathcal O}_{\mathfrak q}$ induces an action of this same group on the group $\mathscr U$ of trivial units fixing $\mathbb F^\ast$ element-wise. There is thus an action of this same group $\mathop{\mathrm{\mathrm{Aut}}}_{\mathbb F}(\widehat{\mathcal O}_{\mathfrak q})$ on the quotient $\mathscr U/\mathbb F^\ast \cong \Gamma$ yielding a homomorphism $$\label{actn-Gma} \mathop{\mathrm{\mathrm{Aut}}}_{\mathbb F}(\widehat{\mathcal O}_{\mathfrak q}) \longrightarrow \mathop{\mathrm{\mathrm{Aut}}}\Gamma = \mathop{\mathrm{GL}}(n, \mathbb Z)$$ whose kernel is the group of all *scalar automorphisms* defined by $\psi(\mathbf X^{\gamma}) = \phi(\gamma)(\mathbf X^{\gamma})$ for $\phi \in \mathop{\mathrm{Hom}}(\Gamma, \mathbb F^\ast)$ [@OP1995]. Clearly this kernel can be identified with the algebraic torus $(\mathbb F^\ast)^n$. By [@OP1995 Lemma 3.3(iii)] the image (in $\mathop{\mathrm{GL}}(n, \mathbb Z)$) of the map in [\[actn-Gma\]](#actn-Gma){reference-type="eqref" reference="actn-Gma"} is the subgroup of all $\sigma \in \mathop{\mathrm{GL}}(n, \mathbb Z)$ such that $$\label{form_prsvng} \lambda (\sigma \gamma, \sigma \gamma') = \lambda(\gamma, \gamma' ) \qquad\ \ \ \forall \gamma, \gamma' \in \Gamma.$$ This subgroup is denoted $\mathop{\mathrm{\mathrm{Aut}}}(\mathbb Z^n, \lambda)$ is known as the *nonscalar automorphism group*. By the foregoing discussion we obtain the following exact sequence noted in [@NeebKH2008]: $$\label{Neeb-Key-Result1} 1 \rightarrow (\mathbb F^\ast)^n \rightarrow \mathop{\mathrm{\mathrm{Aut}}}_{\mathbb F}(\widehat{\mathcal O}_{\mathfrak q}) \rightarrow \mathop{\mathrm{\mathrm{Aut}}}(\mathbb Z^n, \lambda) \rightarrow 1.$$ # Proof of Theorem 1 **Definition 2** (Section 1.4 of [@AC1992]). An automorphism $\sigma$ of $\mathcal O_{\mathfrak q}$ is called linear if it has the form $$\label{lin_aut_def} \sigma(X_i)= \sum_{j = 1}^n \alpha_{ij}X_j \qquad \forall i \in \{1, \cdots, n\}, \qquad (\alpha_{ij}) \in \mathop{\mathrm{GL}}(n, \mathbb F).$$ For a matrix $(\alpha_{ij}) \in \mathop{\mathrm{GL}}(n, \mathbb F)$ to define an automorphism as in [\[lin_aut_def\]](#lin_aut_def){reference-type="eqref" reference="lin_aut_def"} the following necessary and sufficient conditions must hold ([@AC1992]): $$\label{AC-cond-lin-aut} \alpha_{ik} \alpha_{jl}(1 - q_{ij} q_{lk}) = \alpha_{il} \alpha_{jk}(q_{ij} - q_{lk}) \qquad\ \ \forall i < j , \ \ \forall k \le l.$$ The last equation may be re-written as $$\label{AC-cond-lin-aut-re} \alpha_{ik} \alpha_{jl}(q_{kl} - q_{ij}) = \alpha_{il} \alpha_{jk}(q_{kl}q_{ij} - 1) \ \qquad\forall i < j , \ \ \forall k \le l.$$ Setting $k = l$ in the last equation we obtain $$\label{setkeql} \alpha_{ik} \alpha_{jk} (1 - q_{ij}) = \alpha_{ik} \alpha_{jk}(q_{ij} - 1) \qquad \ \forall i < j, \ \ \forall k \in \{1, \cdots, n\} .$$ **Observation 1**. Clearly, the last equation means that if $\mathop{\mathrm{\mathrm{char}}}(\mathbb F) \ne 2$ and none of the multiparameters $q_{ij}\ (i < j)$ equals to unity then at least one of the coefficients $\alpha_{ik}$ and $\alpha_{jk}$ vanishes. It is immediate that in this case the nonsingular matrix $(\alpha_{ij})$ has exactly one nonzero entry in each row and each column. The next proposition is an easy consequence of the preceding observation. **Proposition 1**. *Suppose that $\mathop{\mathrm{\mathrm{char}}}(\mathbb F) \ne 2$ and the entries of $\mathfrak q$ satisfy $$\label{sim-cond1} q_{ij} \ne 1,\qquad \qquad \forall 1 \le i < j \le n.$$ Then each linear automorphism $\sigma$ of $\mathcal O_{\mathfrak q}$ has the form $X_i \rightarrow k_{i} X_{\sigma(i)}$ for a suitable permutation $\sigma$ of the variables $X_{i}$ where $k_{i} \in \mathbb F^\ast$. Thus, $$\label{lin_aut} \mathop{\mathrm{\mathrm{Aut}}}_{\mathrm L}(\mathcal O_{\mathfrak q}) \cong ({\mathbb F^\ast})^n \rtimes \mathcal{P}$$ for a subgroup $\mathcal P$ of $S_n$.* For a permutation $\sigma$ we denote by $\mathfrak{m_{\sigma}}$ the corresponding permutation matrix. **Proposition 2**. **([@OP1995 Remark 3.2])*[\[rem\]]{#rem label="rem"} Suppose that $n \ge 3$. Let $\mathfrak q = (q_{ij})$ be a multiplicatively antisymmetric matrix. For a permutation $\sigma$ the corresponding change of variables $x_{i} \rightarrow x_{\sigma(i)}$ is an automorphism of $\mathcal O_{\mathfrak q}$ if and only if $$\label{automorphism condition} q_{ij} = q_{\sigma(i) \sigma(j)}, \qquad \qquad \forall 1 \le i < j \le n.$$ Equivalently, $$\mathfrak{q}\mathfrak{m_{\sigma}}=\mathfrak{m_{\sigma}}\mathfrak{q}.$$* **Remark 1**. *In the situation of Proposition [Proposition 1](#simple-cse-q(ij)-not-1){reference-type="ref" reference="simple-cse-q(ij)-not-1"} if $q_{ij} \neq -1$ and $\sigma$ is a non-identity permutation in $\mathcal{P}$ and $\sigma$ is decomposed into cycles then any cycle in this decomposition of $\sigma$ must have odd length. Indeed if $(i_1 i_2)$ is a 2-cycle in the decomposition of $\sigma$ then in view of [\[automorphism condition\]](#automorphism condition){reference-type="ref" reference="automorphism condition"} we have $({q_{i_1 i_2}})^{2} = 1$. Again if $(i_1i_2i_3 \cdots i_m)$ is an $m$-cycle ($m \geq 4)$ where $m$ is even then by [\[automorphism condition\]](#automorphism condition){reference-type="eqref" reference="automorphism condition"} we have $$(q_{i_{m/2},i_{m}})^{-1} = q_{i_{1},i_{m/2+1}}=q_{i_{2},i_{m/2 +2}} = \cdots = q_{i_{m/2-1},i_{m/2+m/2-1}} =q_{i_{m/2},i_{m}}$$ whence $(q_{i_{m/2},i_{m}})^2=1$.* Let $\mathop{\mathrm{Der}}(\mathcal O_{\mathfrak q})$ denote the module of derivations of $\mathcal O_{\mathfrak q}$. We recall the submodule $E$ of $\mathop{\mathrm{Der}}(\mathcal O_{\mathfrak q})$ in [@AC1992]. Suppose $$\Lambda_{i} = \{ \nu \in {\mathbb{N}}^{n} : \nu_{i} = 0~~\text{and}~~\prod_{k} q_{kj}^{{\nu}_{k}} = q_{ij}~~ \text{for all}~~ j ~~ \text{such that}~~ j\neq i \}.$$ For all $\nu \in \Lambda_{i}$ there exists a derivation $D_{i\nu}$ of $\mathcal O_{\mathfrak q}$ defined by $$D_{i \nu}(x_{j})= \delta_{ij} {\bf{x}}^{\nu}$$ for all $j$. Note that $D_{i \nu}^{2} = 0$. We then define $$E = \oplus_{i}(\oplus_{\nu \in \Lambda_{i}} \mathcal O_{\mathfrak q} D_{i\nu}).$$ **Lemma 3**. *Suppose that $\mathop{\mathrm{\mathrm{char}}}(\mathbb F) \ne 2$ and $n \ge 3$. Let $\mathfrak q = (q_{ij})$ be a multiplicatively antisymmetric matrix. Then no two rows of $\mathfrak{q}$ are equal if $E = 0$.* *Proof.* Suppose that $i$-th and the $j$-th rows of $\mathfrak{q}$ be equal, that is, $q_{ir} = q_{jr}$ for all $r\in \{1,2,\cdots,n\}$. Let $k \neq i$ and $\nu:=(0,\cdots,1_j,\cdots,0)$. We claim that $\nu \in \Lambda_{i}$. Indeed from the definition of $\Lambda_{i}$ we must have $$\prod_{p} q_{ip}^{{\nu}_{p}} = q_{ik}.$$ But this is true as $q_{ik} = q_{jk}$ for all $k \neq i$. Thus $E \neq 0$ - a contradiction. ◻ **Theorem 4**. **([@AC1992 Proposition 1.4.3])*[\[rect\]]{#rect label="rect"} Suppose that $\mathop{\mathrm{\mathrm{char}}}(\mathbb F) = 0$ and $n \ge 3$. Let $\mathfrak q = (q_{ij})$ be a multiplicatively antisymmetric matrix such that for all $i$ there exist $j$ such that $q_{ij} \neq 1$. If $E = 0$ then $$\mathop{\mathrm{\mathrm{Aut}}}_{\mathbb F}(\mathcal O_{\mathfrak q})=\mathop{\mathrm{\mathrm{Aut}}}_L(\mathcal O_{\mathfrak q})$$* The above theorem is a rectification of the proposition of [@AC1992 Section 1.4.3]. There is a small error in the proof of this proposition. They used the fact that $E = 0$ implies for all $i$ there exist $j$ such that $q_{ij} \neq 1$. This is not true in general: for example, consider $n = 3$ and $q_{12}=q_{13}= 1$, $q_{23} = q$ where $q$ is not a root of unity. Then clearly $E = 0$ but there is an automorphism $\phi_{b}$ for each $b \in {\mathbb F}^{\times}$ defined by $$\phi_b(X_{i})= \begin{cases} X_i &\text{if } i \neq 1\\ X_i + b, & \text{if } i=1\end{cases}$$ Clearly $\phi_{b}$ is not a linear automorphism. **Theorem 1 2**. *Suppose that $\mathop{\mathrm{\mathrm{char}}}(\mathbb F) = 0$ and $n \ge 3$. Let $\mathfrak q = (q_{ij})$ be a multiplicatively antisymmetric matrix such that for $i<j$ atmost one entry $q_{ij}$ equals to $1$. Then* - *if no $q_{ij}$ equals to $1$ for all $i < j$, then the automorphism group $$\mathop{\mathrm{\mathrm{Aut}}}_{\mathbb F}(\mathcal O_{\mathfrak q}) = {(\mathbb F^\ast)}^n$$ if and only if the following conditions hold* - *$E= 0$[\[E=0\]]{#E=0 label="E=0"} and* - *there does not exist any non-identity permutation matrix $m_{\sigma}$ satisfying $$\label{commute permutation} \mathfrak{q}\mathfrak{m_{\sigma}}=\mathfrak{m_{\sigma}}\mathfrak{q}.$$* - *if exactly one entry, say $q_{i'j'}$ with $i'<j'$ is $1$, then the automorphism group $$\mathop{\mathrm{\mathrm{Aut}}}_{\mathbb F}(\mathcal O_{\mathfrak q}) = {(\mathbb F^\ast)}^n$$ if and only if the following conditions hold* - *[\[2nd E=0\]]{#2nd E=0 label="2nd E=0"} $E= 0$ and* - *there does not exist any permutation $\pi_{1}$ on the set $\{i',j'\}$ and non-identity permutation $\pi_{2}$ on the set $\mathcal{J}= \{1,2,\cdots,n\} \setminus \{i',j'\}$ satisfying $$\label{2nd commute permutation} q_{i'r}=q_{\pi_{1}(i')\pi_{2}(r)}, \quad q_{j'r}=q_{\pi_{1}(j')\pi_{2}(r)}, \qquad \forall r \in \mathcal{J}$$ such that $\mathfrak{q'}\mathfrak{m}_{\pi_{2}}=\mathfrak{m}_{\pi_{2}}\mathfrak{q'}$ where $\mathfrak q'$ is a multiplicatively anti-symmetric matrix obtained from $\mathfrak{q}$ by removing the $i',j'$ rows and columns simultaneously.* *Proof.* $(1)$ We first show the necessity. By definiition if $E \neq 0$ then there exists an $i$ such that $\Lambda_{i}\neq 0$. Let $\nu=(\nu_{1},\cdots,\nu_{n})\in \Lambda_{i}$ such that $\nu \neq 0$. It is easy to check that the automorphism $\exp(D_{i\nu})$ is not a member of $\mathbb({F^{*}})^n$ where $D_{i\nu}$ is the locally nilpotent derivation defined by $D_{i\nu}(X_{j})=\delta_{ij}X^{\nu}$. Suppose there exist a non-identity permutation $\sigma$ satisfying [\[commute permutation\]](#commute permutation){reference-type="eqref" reference="commute permutation"} then by Proposition [\[rem\]](#rem){reference-type="ref" reference="rem"} the map $X_{i} \rightarrow X_{\sigma(i)}$ is a nontoric automorphism. Conversely, suppose the conditions in (1) hold. Now $$\mathop{\mathrm{\mathrm{Aut}}}_{\mathbb F}(\mathcal O_{\mathfrak q})=\mathop{\mathrm{\mathrm{Aut}}}_L(\mathcal O_{\mathfrak q})$$ by the Theorem [\[rect\]](#rect){reference-type="ref" reference="rect"}. Also from Observation [Observation 1](#avd-1){reference-type="ref" reference="avd-1"} any linear automorphism must be of the form $X_{i} \rightarrow k_i X_{\sigma(i)}$ for some permutation $\sigma$ where $k_i \in F^{\times}$. By Proposition [\[rem\]](#rem){reference-type="ref" reference="rem"} $\sigma$ must be the identity permutation. $(2)$ The proof of the necessity of the condition $E= 0$ is same as in part $(1)$. Again $(b)$ is necessary as otherwise the map $\phi$ defined by $$\phi(X_{i})= \begin{cases} X_{\pi_{1}(i)} &\text{if } i \in \{i',j'\}\\ X_{\pi_{2}(i)}, & \text{if } i\in\mathcal{J}\end{cases}$$ is a nontoric automorphism of $\mathcal O_{\mathfrak q}$ by proposition [\[rem\]](#rem){reference-type="ref" reference="rem"}. Conversely, the conditions supposed that 2(a) and 2(b) hold.\ As seen above in this case, we have $$\mathop{\mathrm{\mathrm{Aut}}}_{\mathbb F}(\mathcal O_{\mathfrak q})=\mathop{\mathrm{\mathrm{Aut}}}_L(\mathcal O_{\mathfrak q}).$$\ Suppose that $$A = (\alpha_{ij}) \in \mathop{\mathrm{GL}}(n, \mathbb F)$$ induces a linear automorphism $\alpha$ of the given quantum affine space. Using [\[setkeql\]](#setkeql){reference-type="eqref" reference="setkeql"} it is easily seen that in any column of $A$ at most two entries can be nonzero and in the case there are two nonzero entries, these must be in the $i'$-th and $j'$-th rows. We claim that there can be at most two columns in $A$ that have two nonzero entries. Indeed suppose that $2 + s$ columns have two nonzero entries necessarily in the $i'$-th and $j'$-th rows. As just noted the remaining $n - 2 - s$ columns each has exactly one non-zero entry. Clearly, if $s > 0$ these $n - 2 - s$ nonzero entries in $A$ cannot fulfill the requirement of a non-zero entry in each of the $n - 2$ rows other than the $i'$-th and $j'$-th rows. This shows that $s = 0$. Next we note that at least one of the entries $\alpha_{i'i'}$ and $\alpha_{j'i'}$ is non-zero. To see this we pick $m < p$ in the range $1, \cdots,n$. By [\[AC-cond-lin-aut-re\]](#AC-cond-lin-aut-re){reference-type="eqref" reference="AC-cond-lin-aut-re"} we have noting that $q_{i'j'} = 1$ $$\begin{aligned} \label{apln_fundl_rln1} \alpha_{i'm}\alpha_{j'p}(q_{mp} - 1) &= \alpha_{i'p}\alpha_{j'm}(q_{mp} - 1). %\alpha_{i'p}\alpha_{j'm}(q_{pm} - 1) &= \alpha_{j'p}\alpha_{i'm}(q_{pm} -1) \ \ \ \ m > p.\end{aligned}$$ If $(m,p) \ne (i',j')$ then by the hypothesis on the part $(2)$ $q_{mp} \ne 1$ and [\[apln_fundl_rln1\]](#apln_fundl_rln1){reference-type="eqref" reference="apln_fundl_rln1"} means that the minor of $A$ corresponding to the $2 \times 2$ submatrix formed by the $i'$-th and $j'$-th rows and the $m$-th and $p$-th columns is equal to zero. If $\alpha_{i'i'} = \alpha_{j'i'} = 0$ then the minor corresponding to the $2 \times 2$ submatrix $K$ defined by the $i'$-th and $j'$-th rows and the $i'$-th and $j'$-th columns is also equal to zero. This would mean that a row of the exterior square $\wedge^2 A$ of $A$ is the zero row contradicting the assumption $A$ is non-singular. By the same token at least one of the entries $\alpha_{i'j'}$ and $\alpha_{j'j'}$ in column $j'$ is nonzero. Moreover, the non-zero entries in the two columns, namely, $i'$ and $j'$ cannot be in only one of the rows $i'$ or $j'$ as in this case the determinant of $K$ will be zero. We now claim that if a column of $A$ has two non-zero entries then it must be the $i'$-th or the $j'$-th column. Indeed let $h$ be a column of $A$ having two non-zero entries. As noted above these non-zero entries of $h$ must be in rows $i'$ and $j'$. Thus the three columns $i'$, $j'$ and $h$ have non-zero entries only in rows $i'$ and $j'$. Consequently there can be at most $n -3$ nonzero entries in columns other than $i',j'$ and $h$ that are contained in the $n - 2$ rows other than $i'$ and $j'$. But this means there is a row with no non-zero entry contradicting the assumption that $A$ is non-singular. It follows that $\alpha(X_{i'})$ and $\alpha(X_{j'})$ both lie in the $k$-subspace spanned by $X_{i'}$ and $X_{j'}$ while $\alpha(X_r)=k_{r}X_{\pi_{2}(r)}$ for all $r \in \mathcal{J}$ where $k_{r}\in \mathbb{F}^{\times}$ and $\pi_{2}$ is a permutation on the set $\mathcal{J}$. We claim that either $\alpha(X_{i'}) \in \mathbb F^{\times} X_{i'}$ or $\alpha(X_{i'}) \in \mathbb F^{\times} X_{j'}$. Indeed let $$\alpha(X_{i'})=aX_{i'}+ bX_{j'}$$ where $a,b \in \mathbb{F}^{\times}$. Applying $\alpha$ to the relations $$X_{i'}X_{r}= q_{i'r} X_{r}X_{i'}, \qquad \forall r \in \mathcal{J}$$ we obtain $$(aX_{i'}+ bX_{j'})X_{\pi_{2}(r)}=q_{i'r}X_{\pi_{2}(r)}(aX_{i'}+ bX_{j'})$$ Comparing both sides we have $$q_{i'\pi_{2}(r)}= q_{i'r}= q_{j'\pi_{2}(r)}, \qquad \forall r \in \mathcal{J}.$$ Moreover the hypothesis $q_{i'j'}=1$ means that $1=q_{i'i'}=q_{j'i'}$ and $1= q_{i'j'}=q_{j'j'}$. It follows that the $i'$-th and $j'$-th row of the matrix $\mathfrak{q}$ coincide, which is contrary the Lemma [Lemma 3](#equal rows){reference-type="ref" reference="equal rows"}. Similarly $\alpha(X_{j'})\in \mathbb F^{\times} X_{i'}$ or $\alpha(X_{j'})\in \mathbb F^{\times} X_{j'}$. It follows that $\alpha(X_{i'})\in \mathbb F^{\times} X_{\pi_1 (i')}$ and $\alpha(X_{j'})\in \mathbb F^{\times} X_{\pi_1 (j')}$ for some permutation $\pi_{1}$ on the set $\{1,2\}$. Now we claim that $\pi_{2}$ is identity. If not from the relations $X_{r}X_{s} = q_{rs} X_{s}X_{r}$ where $r,s \in \mathcal{J}$ we have $\mathfrak{m}_{\pi_{2}}\mathfrak{q'} = \mathfrak{q'} \mathfrak{m_{\pi_{2}}}$.\ Applying $\alpha$ to the relations $$X_{i'}X_{r}= q_{i'r} X_{r}X_{i'},\qquad \forall r \in \mathcal{J}$$ and $$X_{j'}X_{r}= q_{j'r} X_{r}X_{j'}, \qquad \forall r \in \mathcal{J}$$ we have $q_{i'r}=q_{\pi_{1}(i')\pi_{2}(r)}$, $q_{j'r}=q_{\pi_{1}(j')\pi_{2}(r)}$ respectively $\forall r \in \mathcal{J}$ . But this contradicts the theorem hypothesis. Thus $\pi_{2}$ is an identity permutation.\ \ If $\alpha(X_{i'})\in \mathbb F^{\times} X_{j'}$ and $\alpha(X_{j'})\in \mathbb F^{\times} X_{i'}$ then applying $\alpha$ to the relation $$X_{i'}X_{r}= q_{i'r} X_{r}X_{i'}, \qquad \forall r \in \mathcal{J}$$ we have $q_{i'r}=q_{j'r}$ $\forall r \in \mathcal{J}$ which contradicting the hypothesis $E=0$ in view of Lemma [Lemma 3](#equal rows){reference-type="ref" reference="equal rows"}. ◻ # Proof of Theorem 2 The following fact shown in [@OP1995] reduces the question of automorphisms to the case where the group $\Lambda$ (Definition [Definition 1](#lam-grp){reference-type="ref" reference="lam-grp"}) is torsion-free. **Lemma 5** ([@OP1995]). *Let $p$ denote the size of the torsion subgroup of $\Lambda$. The subalgebra $\widehat{\mathcal O'}$ of $\widehat{\mathcal O}_{\mathfrak q}$ generated by the powers $X_i^{\pm p}$ of the indeterminates $X_i$ is a characteristic sub-algebra of the same rank. Moreover $\widehat {\mathcal O}_{\mathfrak q}$ is free left $\widehat{\mathcal O'}$-module of finite rank and the corresponding $\lambda$-group $\Lambda'$ associated with $\widehat{\mathcal O'}$ is torsion free.* We may thus assume that $\Lambda \cong \mathbb Z^l$ for some natural number $l$. Fixing a $\mathbb Z$-basis $p_1, \cdots, p_l$ in $\Lambda$ we have $$\label{e_i-forms} \lambda(\gamma, \gamma') = p_1^{e_1(\gamma,\gamma')}p_2^{e_2(\gamma, \gamma')}\cdots p_l^{e_l(\gamma, \gamma')}, \qquad \gamma, \gamma' \in \Gamma.$$ **Notation 1**. In view of $\eqref{e_i-forms}$ let $$\label{rel-matrx} \lambda(\mathbf e_i, \mathbf e_j) = p_1^{ m_{(ij), 1}} \cdots p_l^{m_{(ij), l}} , \qquad 1 \le i < j \le n,$$ where $\mathbf e_i, \mathbf e_j$ are standard basis vectors of the free $\mathbb Z$-module $\Gamma$. On the $\binom{n}{2}$ pairs $(ij),\ (i < j)$ we assume the lexicographic order. Let $\mathsf M \in \mathrm{Mat}_{\binom{n}{2} \times l}(\mathbb Z)$ be the matrix whose $((ij), s)$ entry is the exponent $m_{(ij), s}$ of $p_s$ in [\[rel-matrx\]](#rel-matrx){reference-type="eqref" reference="rel-matrx"} ($s = 1, \cdots, l$). We recall that for a given matrix $A \in \mathop{\mathrm{GL}}(n, \mathbb Z)$ the exterior square $\wedge^2 A$ of $A$ is the $\binom{n}{2} \times \binom{n}{2}$-matrix whose rows and columns are indexed by the pairs $(ij)\ (1 \le i < j \le n)$ ordered lexicographically and whose $((ij), (kl))$ entry is the $2 \times 2$-minor corresponding to rows $i,j$ and columns $k,l$. With $\mathsf M$ as defined above we have the following. **Proposition A 1**. *Set $N = \binom{n}{2}$ and let $\mathsf M$ be as defined in Notation [Notation 1](#not1){reference-type="ref" reference="not1"} above. Then $$\mathop{\mathrm{\mathrm{Aut}}}(\mathbb Z^n, \lambda) = \bigl ( \mathop{\mathrm{Stab}}_{\mathop{\mathrm{GL}}(n, \mathbb Z)}(\mathsf M) \bigr )^t$$ where $t$ denotes transposition and $\mathop{\mathrm{Stab}}_{\mathop{\mathrm{GL}}(n, \mathbb Z)}(\mathsf M)$ the stabilizer of $\mathsf M$ in $\mathop{\mathrm{GL}}(n, \mathbb Z)$ with respect to the bivector representation $$\displaystyle\@ifnextchar^\@extp{\mathop{\bigwedge\nolimits^{\!\,}}}^2 : \mathop{\mathrm{GL}}(n, \mathbb Z) \rightarrow \mathop{\mathrm{GL}}(N, \mathbb Z), \qquad \ A \rightarrow \wedge^2 A$$ of $\mathop{\mathrm{GL}}(n, \mathbb Z)$, that is, $$\mathop{\mathrm{Stab}}_{\mathop{\mathrm{GL}}(n, \mathbb Z)}(\mathsf M) = \{ A \in \mathop{\mathrm{GL}}(n,Z) \mid (\wedge^2 A)\mathsf M = \mathsf M \}.$$* *Proof.* Writing the group $\Lambda \le \mathbb F^{\ast}$ additively, in view of Notation [Notation 1](#not1){reference-type="ref" reference="not1"} we have $$\label{expn_in_gens} \lambda(\mathbf e_i, \mathbf e_j) = \sum_{s = 1}^l {m_{(ij),s}} p_s, \qquad\ \ \ \forall 1 \le i < j \le n,$$ where $m_{ (ij),s} \in \mathbb Z$. Now let $$A = (a_{ij}) \in \mathop{\mathrm{GL}}(n, \mathbb Z)$$ be such that $A^t \in \mathop{\mathrm{\mathrm{Aut}}}(\mathbb Z^n, \lambda)$. Setting $$\mathbf e_j' = A^t \mathbf e_j= \sum_{t = 1}^{n} a_{jt}\mathbf e_t$$ we note that since $\lambda$ is an alternating function therefore $\lambda(\mathbf e_i', \mathbf e_j')$ may be expressed as follows: $$\label{transf_commtr} \lambda(\mathbf e_i', \mathbf e_j') = \sum_{(uv)} a_{(ij), (uv)}\lambda(\mathbf e_u, \mathbf e_v),$$ where the coefficients appearing in the RHS of the above expression constitute row (ij) of the matrix $\wedge^2 A$. Since $A^t$ is $\lambda$-preserving, by [\[form_prsvng\]](#form_prsvng){reference-type="eqref" reference="form_prsvng"} we have $$\lambda(\mathbf e_i', \mathbf e_j') = \lambda(\mathbf e_i, \mathbf e_j) \qquad\ \ \forall 1 \le i < j \le n.$$ Expanding and comparing the coefficients of $p_s \ (s = 1, \cdots, l)$ in both sides of the last equation we get $$\label{coeff_compr} \sum_{uv} a_{(ij), (uv)} m_{(uv),s} = m_{(ij),s} \qquad\forall s = 1, \cdots, l.$$ It follows that $$\wedge^2(A)\mathsf M = \mathsf M.$$ Clearly the above reasoning is reversible. This establishes the assertion of the theorem. ◻ **Notation 2**. As before, let $\{\mathbf e_1, \cdots, \mathbf e_n\}$ denote the standard basis in $\Gamma = \mathbb Z^n$. Then $\{\mathbf e_i \wedge \mathbf e_j \mid 1 \le i < j \le n \}$ is a basis in $\wedge^2 \Gamma$. As usual, for a permutation $\pi \in S_n$ let $P \in \mathop{\mathrm{GL}}(n, \mathbb Z)$ denote the corresponding permutation matrix. Clearly $S_n$ acts on the $\mathbb Z$-module $\wedge^2 \Gamma$ via $$\label{actn-Sn} \pi(\mathbf e_i \wedge \mathbf e_j) = \wedge^2 P (\mathbf e_i \wedge \mathbf e_j) = \mathbf e_{\pi(i)} \wedge \mathbf e_{\pi(j)}, \qquad \forall \pi \in S_n.$$ By restriction we get an action of $S_n$ on the subset $$\bar B = \{ \epsilon \mathbf e_i \wedge \mathbf e_j \mid i < j \ \mathrm{and} \ \epsilon \in \{-1, 1\} \}.$$ Restricting the above action of $S_n$ to $C_\pi := \langle \pi \rangle$ we consider the $C_\pi$-orbits. **Definition 3**. We define an *orbit-sum* $\mathscr O_{ij} \ (1 \le i \ne j \le n)$ as $$\mathscr O_{ij} = \wedge^2 P (\mathbf e_i \wedge \mathbf e_j) + (\wedge^2 P)^2 (\mathbf e_i \wedge \mathbf e_j) + \cdots + (\wedge^{2} P)^{m}(\mathbf e_i \wedge \mathbf e_j),$$ where $m$ stands for the order of $\wedge^2 P$. **Lemma 6**. *Let $\pi \in S_n \ (n \ge 3)$ be a non-identity permutation and let $P$ denote the corresponding permutation matrix. Let $\mathrm{Fix}(\wedge^2 P)$ stand for the sub-module of $\wedge^2 \Gamma = \mathbb Z^N \ (N = \binom{n}{2})$ left fixed by the $\mathbb Z$-linear map $\wedge^2 P \in \mathop{\mathrm{GL}}(N,\mathbb Z)$. Then $$\mathop{\mathrm{\mathrm{rk}}}(\mathrm{Fix}(\wedge^2 P)) < \binom{n - 1}{2} + 1.$$* *Proof.* We begin by observing that the submodule $\mathrm{Fix}(\wedge^2 P)$ is generated by the orbit-sums $\mathscr O_{ij}$ of Definition [Definition 3](#orb-sum){reference-type="ref" reference="orb-sum"} where we note that either $\mathscr O_{ij} = - \mathscr O_{ji}$ or else $\mathscr O_{ij} = \mathscr O_{ji} = 0$. In other words if $w \in \mathrm{Fix}(\wedge^2 P)$ then $$w = \sum_{ij} \gamma_{ij} \mathscr O_{ij}, \qquad \gamma_{ij} \in \mathbb Z.$$ Denoting by $\mathcal N_\pi$ the number of $C_{\pi}$-orbits it follows from the above that $\mathrm{Fix}(\wedge^2 P)$ is generated by at most $\floor{\frac{N_\pi}{2}}$ orbit sums $\mathscr O_{ij}$ and thus $$\label{rnk-bnd} \mathop{\mathrm{\mathrm{rk}}}(\mathrm{Fix}(\wedge^2 P)) \le \frac{N_\pi}{2}.$$ Evidently for any non-identity permutation $\pi \in S_n$ the number of fixed points $\mathrm{Fix}(\pi)$ is bounded above by $n - 2$ and this bound in attained only by a transposition $(ij)$. Using this fact and the Burnside formula an upper bound for $\mathcal N_\pi$ was obtained in [@Sil] as follows $$\begin{aligned} \mathcal N_\pi &= \frac{1}{\envert{C_\pi}}\Biggl ( n(n - 1) + \sum_{\phi \in C_\pi , \phi \ne 1}2\binom{\mathrm{Fix}(\pi)}{2} \Biggr ) \le \frac{ n(n - 1)}{\envert{C_\pi}} + \frac{(\envert{C_\pi} - 1)}{\envert{C_\pi}}2\binom{n - 2}{2} \\ &= (n - 2)(n -3) + \frac{4n -6}{\envert{C_\pi}} \le (n - 2)(n -3) + \frac{4n -6}{2} = n^2 - 3n + 3. \end{aligned}$$ Noting [\[rnk-bnd\]](#rnk-bnd){reference-type="eqref" reference="rnk-bnd"} We thus obtain $$\mathop{\mathrm{\mathrm{rk}}}(\mathrm{Fix}(\wedge^2 P) \le \frac{N_\pi}{2} = \frac{n^2 - 3n + 3}{2} < \frac{n^2 - 3n + 4}{2} = \binom{n - 1}{2} + 1.$$ ◻ **Lemma 7**. *For a quantum torus $\widehat{\mathcal O}_{\mathfrak q}$ suppose that the $\lambda$-group $\Lambda$ is torsion-free with rank equal to $l$. Let $\mathsf M$ be the matrix defined in Notation [Notation 1](#not1){reference-type="ref" reference="not1"} (with respect to some choice of a basis in the $\lambda$-group). The $l$ columns of $\mathsf M$ generate a free submodule of $\wedge^2 \Gamma = \mathbb Z^{\binom{n}{2}}$ of rank $l$.* *Proof.* We recall that for a matrix over a commutative ring $R$ the *row rank* is defined as the maximum number of linearly independent rows of $R$ and *column rank* has a parallel definition. If $R$ is a domain then these two ranks coincide (e.g., [@AW Chapter 4, Corollary 2.29]). The choice of a basis $\{p_1, \cdots, p_l \}$ in $\Lambda$ induces an isomorphism $\Lambda \cong \mathbb Z^l$. Since $$\{ \lambda(\mathbf e_i, \mathbf e_j) \mid 1 \le i < j \le n \}$$ is a generating set for the $\mathbb Z$-module $\Lambda$ therefore the rows of $\mathsf M$ generate $\Gamma_l : = \mathbb Z^l$. Let $s \le l$ be the row-rank of $\mathsf M$ and let $U$ be the free submodule spanned by some $s$ linearly independent rows. We note that $\Gamma_l/U$ is a torsion $\mathbb Z$-module and hence finite implying that $$l = \mathop{\mathrm{\mathrm{rk}}}(\Gamma_l) = \mathop{\mathrm{\mathrm{rk}}}(U) = s.$$ The assertion in the lemma is now immediate. ◻ **Lemma 8**. *A quantum torus $\widehat{\mathcal O}_{\mathfrak q}$ whose $\lambda$-group is torsion-free and has rank at least $\binom{n -1}{2} + 1$ has center equal to $\mathbb F$.* *Proof.* To this end we recall (e.g., [@MP]) that a quantum torus $\widehat{\mathcal O}_{\mathfrak q}$ has the structure of a twisted group algebra $\mathbb F \ast \Gamma$ for $\Gamma = \mathbb Z^n$. By [@OP1995 Lemma 1.1(i)] the center of such an algebra is itself a twisted group algebra $\mathbb F \ast Z$ for a subgroup $Z \le \Gamma$. We claim that $\Gamma/ Z$ is torsion-free. Indeed let $\gamma \in \Gamma$ be such that $k \gamma \in Z$ for some $k \in \mathbb N$. In view of [\[lmbdadefn\]](#lmbdadefn){reference-type="eqref" reference="lmbdadefn"} $$\lambda(k\gamma, \gamma') = \lambda(\gamma, \gamma')^k = 1 \qquad \forall \gamma' \in \Gamma.$$ But as the group $\Lambda$ is torsion free (by hypothesis) therefore $\lambda(\gamma, \gamma') = 1$. Thus $\gamma \in Z$ and it follows that $\Gamma / Z$ is torsion-free. Let $\rho := \mathop{\mathrm{\mathrm{rk}}}(\Gamma/Z)$. As the map $\lambda$ of [\[lmbdadefn\]](#lmbdadefn){reference-type="eqref" reference="lmbdadefn"} is constant on the cosets of $Z$ in $\Gamma$ it induces an alternating bicharacter $\bar \lambda$ on $\Gamma/Z$ such that $$\bar{\lambda}(\gamma + Z, \gamma' + Z) = \lambda(\gamma, \gamma'), \qquad \forall \gamma, \gamma' \in \Gamma.$$ As the group $\Lambda$ is generated by the $$q_{ij} = \lambda(\mathbf e_i, \mathbf e_j) = \bar {\lambda}(\mathbf e_i + Z, \mathbf e_j + Z ),$$ letting $u_1 + Z, \cdots u_\rho + Z$ be a basis of $\Gamma/Z$ we must have $$\Lambda \le \langle \bar{\lambda}(u_r + Z, u_s + Z) \mid 1 \le r < s \le \rho \rangle = : \Lambda_1.$$ Clearly the group $\Lambda_1$ has rank at most $\binom {\rho}{2} = \binom{n - \mathop{\mathrm{\mathrm{rk}}}(Z)}{2}$. If $Z$ is nontrivial we thus obtain $\mathop{\mathrm{\mathrm{rk}}}(\Lambda) \le \binom{n - 1}{2}$. But $\mathop{\mathrm{\mathrm{rk}}}(\Lambda) = \binom{n - 1}{2} + 1$ by the hypothesis. It follows that $\mathop{\mathrm{\mathrm{rk}}}(Z) = 0$ and consequently the center of $\widehat{\mathcal O}_{\mathfrak q}$ must be $\mathbb F$. ◻ **Proposition 9**. *Let $\mathfrak q$ be a multiplicatively antisymmetric matrix such that $q_{kj} = 1 \ (j = 1, \cdots, n)$. Then $\mathop{\mathrm{\mathrm{Aut}}}_{\mathbb F}(\mathcal O_{\mathfrak q})=$ contains the group $(\mathbb F^\ast)^n \mathbb F^+ \rtimes \mathbb F^+$.* *Proof.* Let $b\in \mathbb F$. We define a map $\phi_{b} : \mathcal O_{\mathfrak q} \rightarrow \mathcal O_{\mathfrak q}$ via $$\phi_b(X_{i})= \begin{cases} X_i &\text{if } i \neq k\\ X_i + b, & \text{if } i=k\end{cases}$$ It is easily checked that $\phi_{b} \in \mathop{\mathrm{\mathrm{Aut}}}_{\mathbb F}(\mathcal O_{\mathfrak q})$ and that $b \mapsto \phi_b$ is an embedding of $\mathbb F^+$ into $\mathop{\mathrm{\mathrm{Aut}}}_{\mathbb F}(\mathcal O_{\mathfrak q})$. If $\tau \in \mathop{\mathrm{\mathrm{Aut}}}_{\mathbb F}(\mathcal O_{\mathfrak q})$ defined by $\tau(X_i) = t_iX_i$ is a scalar automorphism in $(\mathbb F^\ast)^n$ then it easy to check that $$\tau^{-1}\phi_{b}\tau=\phi_{t_k b_1}.$$ In other words the image of $\mathbb F^+$ is normalized by the subgroup $(\mathbb F^\ast)^n$ of scalar automorphisms. The assertion of the proposition now follows. ◻ **Theorem 2 2**. *A quantum affine space $\mathcal O_{\mathfrak q} = \mathcal O_{\mathfrak q}(\mathbb F^n) \ (n \ge 3)$ whose $\lambda$-group is torsion-free and has rank no smaller than $\binom{n - 1}{2} + 1$ satisfies $$\mathop{\mathrm{\mathrm{Aut}}}_{\mathbb F}(\mathcal O_{\mathfrak q})= (\mathbb F^\ast)^n.$$ Moreover, for each $n \ge 3$ and each $r < \binom{n - 1}{2} + 1$ there exists an algebra $\mathcal O_{\mathfrak q}$ with $\lambda$-group equal to $\mathbb Z^r$ but whose automorphism group embeds $(\mathbb F^\ast)^n \rtimes \mathbb F^+$.* *Proof.* By Lemma [Lemma 8](#simpl-crit){reference-type="ref" reference="simpl-crit"} the corresponding quantum torus $\widehat {\mathcal O}_{\mathfrak q}$ has center equal to $\mathbb F$. Hence by [@OP1995 Propositon 1.5] each $\mathbb F$-automorphism $\sigma$ of $\mathcal O_{\mathfrak q}$ lifts to an $\mathbb F$-automorphism $\hat{\sigma}$ of $\widehat {\mathcal O}_{\mathfrak q}$. Consequently there is permutation $\pi \in S_n$ such that $$\hat{\sigma}(X_i) = \sigma (X_i) = a_iX_{\pi(i)}, \quad i \in \{ 1,\cdots, n\}, a_i \in \mathbb F^\ast.$$ Clearly, the image of $\hat \sigma$ in $\mathop{\mathrm{\mathrm{Aut}}}(\mathbb Z^n, \lambda)$ under the map in [\[actn-Gma\]](#actn-Gma){reference-type="eqref" reference="actn-Gma"} is the permutation matrix $P$ corresponding to a permutation $\pi \in S_n$. Since $P^t = P^{-1}$ hence $P^t \in \mathop{\mathrm{\mathrm{Aut}}}(\mathbb Z^n, \lambda)$ as well. Proposition A now tells us that $P \in \mathop{\mathrm{Stab}}_{\mathop{\mathrm{GL}}(n, \mathbb Z)}(\mathsf M)$ where $\mathsf M$ is a relations matrix for $\widehat {\mathcal O}_{\mathfrak q}$ as defined in Notation [Notation 1](#not1){reference-type="ref" reference="not1"}. In other words $(\wedge^2 P) \mathsf M = \mathsf M$. Thus $\wedge^2 P$ fixes each column of $\mathsf M$ and since $\wedge^2 P$ is $\mathbb Z$-linear it fixes the submodule $W$ of $\wedge^2 \Gamma$ spanned by the columns of $\mathsf M$. By Lemma [Lemma 7](#sbmd-fxd-by-M){reference-type="ref" reference="sbmd-fxd-by-M"} $$\mathop{\mathrm{\mathrm{rk}}}(W) = \binom{n - 1}{2} + 1.$$ If $\pi$ is a non-identity permutation this contradicts Lemma [Lemma 6](#fxd-sbmod-per){reference-type="ref" reference="fxd-sbmod-per"}. The first part of the theorem now follows. For the second part let $r \in \{ 1, \cdots , \binom{n - 1}{2} \}$. Clearly in this case we can find a (multiplicatively antisymmetric) matrix $\mathfrak q$ such that $q_{1j} = 1 \ (j = 2, \cdots, n)$ and the subgroup $\langle q_{ij} \mid i \ge 2, i < j \le n \rangle$ is torsion free with rank $r$. Noting Proposition [Proposition 9](#non-trivial-autog){reference-type="ref" reference="non-trivial-autog"} we are done. ◻ # $\mathop{\mathrm{\mathrm{Aut}}}_{\mathbb F}(\mathcal O_{\mathfrak q})$ when $\mathop{\mathrm{dim}}(\widehat{\mathcal O}_{\mathfrak q})=1$ **Theorem 10**. *A quantum affine space $\mathcal O_{\mathfrak q}$ such that the corresponding quantum torus $\widehat{\mathcal O}_{\mathfrak q}$ has dimension one has a trivial automorphism group, that is, $\mathop{\mathrm{\mathrm{Aut}}}_{\mathbb F}(\mathcal O_{\mathfrak q})= (\mathbb F^\ast)^n$.* *Proof.* Viewing the corresponding quantum torus $\widehat {\mathcal O}_{\mathfrak q}$ as a twisted group algebra $\mathbb F \ast \Gamma$ we recall ([@OP1995 Lemma 1.1(i)]) that the center of this algebra has the form $\mathbb F \ast Z$ for a subgroup $Z \le \Gamma$. We recall (Section [2.3](#dim-of-qtorus){reference-type="ref" reference="dim-of-qtorus"}) that the dimension of the algebra $\widehat {\mathcal O}_{\mathfrak q}$ equals the cardinality of a maximal independent system of commuting monomials. Clearly by the hypothesis such a subgroup $B$ must have rank one. This easily implies that $Z$ is the trivial subgroup and thus $F \ast \Gamma$ has center equal to $\mathbb F$. By the proposition of [@MP Section 1.3] such an algebra is simple. In this case by [@OP1995 Proposition 3.2] each $\mathbb F$-automorphism of $\mathcal O_{\mathfrak q}$ is of the form $X_i \mapsto a_i X_{\sigma(i)}$ where $a_i \in \mathbb F^{\ast}$ and $\sigma$ is a permutation of the subscripts $\{1, \cdots, n\}$. Furthermore, the permutation $\sigma$ occurs if and only if $$\label{admperm} q_{ij} = q_{\sigma(i) \sigma(j)}, \qquad \forall \le 1 \le i < j\le n.$$ We consider the decomposition of $\sigma$ into disjoint cycles. If there is a transposition $(ij)$ in this decomposition then by [\[admperm\]](#admperm){reference-type="eqref" reference="admperm"} $$q_{ij} = q_{ji} = q_{ij}^{-1}$$ and so $q^{2}_{ij}=1$. It follows that $$[X_{i}^{2},X_{j}]=[X_{i},X_{j}]^{2}=q_{ij}^{2}=1.$$ But this means that $\dim(\widehat {\mathcal O}_{\mathfrak q}) \ge 2$ (para 1) and we thus get a contradiction to the hypothesis. Now let $(i_1i_2i_3 \cdots i_r)$ be an $r$-cycle ($r \ge 3$) in the decomposition of $\sigma$. In view of [\[admperm\]](#admperm){reference-type="eqref" reference="admperm"} we have $q_{i_{r-1} i_{r}} = q_{i_{r} i_{1}}$ whence $$1 = q_{i_{r-1} i_{r}}q_{i_{1} i_{r}} = [X_{i_{r-1}} X_{i_{1}}, X_{i_r}].$$ But again (in view of para 1) this is a contradiction to the assumption on the dimension of the corresponding quantum torus $\widehat {\mathcal O}_{\mathfrak q}$. ◻ For the quantum tori the case of dimension one means that no two independent monomials commute and so this may seem to be a rather restrictive. We may think that such a case can only occur if the $\lambda$-group has a relatively large rank, possibly even the maximal possible rank $\frac{1}{2}n(n-1)$. However an example was given in [@MP Section 3.11] of a quantum tori of rank $4$ that has dimension one and whose $\lambda$-group has rank equal to $5$. Using the results of [@GQ06] we will now show that there exist rank $n$ quantum tori having dimension one and whose $\lambda$-group has rank as low as $n$. **Example 1**. *Let $\mathbf q$ be the matrix* *$$\mathbf q = \begin{pmatrix} 1 & {\zeta}^{-1} & {\mu}^{-1} & {\eta}^{-1} \\ {\zeta} & 1 & {\eta}^{-1} & {\mu} \\ {\mu} & {\eta} & 1 & {\zeta}^{-1} \\ {\eta} & {\mu}^{-1} & {\zeta} & 1 \\ \end{pmatrix},$$ where $\zeta, \mu, \nu \in \mathbb K^\ast$ are assumed to be multiplicatively independent. Then the $\lambda$-group of the quantum torus ${\widehat{\mathcal{O}}}_{\mathfrak q}$ has rank $3$ but $\dim \widehat {\mathcal O}_{\mathfrak q} = 1$.* *Proof.* We consider the three alternating forms $\alpha_{i} \ (i = 1,\cdots,3)$ defined on the $\mathbb Z$-module $\mathbb Z^4$ given by $$\begin{aligned} \alpha_{1}(x,y) &=x_{2}y_{1} - x_{1}y_{2} + x_{4}y_{3} - x_{3}y_{4} \\ \alpha_{2}(x,y) &=x_{3}y_{1} - x_{1}y_{3} + x_{2}y_{4} - x_{4}y_{2} \\ \alpha_{3}(x,y) &=x_{4}y_{1} - x_{1}y_{4} + x_{3}y_{2} - x_{2}y_{3} \end{aligned}$$ We claim that these three forms have no common isotropic $\mathbb{Z}$-submodule of $\mathbb Z^4$ of rank greater than one by which we mean a $\mathbb Z$-submodule $B$ with $\mathop{\mathrm{\mathrm{rk}}}(B) \ge 2$ such that the restriction of $\alpha_i$ to $B$ is trivial for all $i = 1\cdots 3$. Indeed we may regard $\alpha_i$ as a form on the $\mathbb Q$-space $V: = \mathbb Q^4 = \mathbb Z^4 \otimes_{\mathbb Z} \mathbb Q$. If $B$ is an isotropic submodule as in the preceding paragraph then clearly $B \otimes_{\mathbb Z} \mathbb Q$ is $\mathbb Q$-subspace of $\mathbb Q^4$ with dimension at least two that is a common isotropic subspace for $\alpha_i \ (i = 1\cdots 3)$. Let $\mathbb H$ denote the quaternion algebra over $\mathbb Q$. We know that $\mathbb H$ is a division algebra with the usual $\mathbb Q$-basis $\{1, i, j , k \}$. It can be easily checked that the matrix images of $i$, $j$ and $k$ in the regular representation $\mathbb H \rightarrow \mathop{\mathrm{End}}_{\mathbb Q} \mathbb H$ of $\mathbb H$ are the gram matrices $M_i$ of the forms $\alpha_1, \alpha_2$ and $\alpha_3$ respectively. Since $\mathbb H$ is a division algebra the nonzero matrices in the $\mathbb Q$-subspace of $\mathop{\mathrm{End}}_{\mathbb Q} \mathbb H$ spanned by the $M_i\ (i = 1 \cdots 3)$ are non-singular. Then by [@GQ06 Corollary 4] the alternating forms $\alpha_i \ (i = 1, \cdots, 3)$ on the $\mathbb{Q}$-space ${\mathbb{Q}}^{4}$ have no common isotropic subspace of dimension greater than one. Thus there cannot exist a common isotropic submodule $B$ of rank greater than one for the given alternating forms $\alpha_i \ (i = 1, \cdots, 3)$. Let $\{\mathbf e_1, \cdots, \mathbf e_4 \}$ be the standard basis elements of $\mathbb Z^4$ and set $$q_{ij}= \zeta^{\alpha_{1}(\mathbf e_{i}, \mathbf e_{j})}\mu^{\alpha_{2}(\mathbf e_{i}, \mathbf e_{j})}\nu^{\alpha_{3}(\mathbf e_{i}, \mathbf e_{j})}, \qquad \forall 1\leq i < j \leq 4.$$ Then $\mathbf q = (q_{ij})$ and the commutator map $\lambda$ (Section[\[sec2.2\]](#sec2.2){reference-type="ref" reference="sec2.2"}) of the quantum torus $\widehat {\mathcal O}_{\mathfrak q}$ has the form $$\lambda(\gamma, \gamma') = \zeta^{\alpha_{1}(\gamma,\gamma')}\mu^{\alpha_{2}(\gamma,\gamma')}\nu^{\alpha_{3}(\gamma,\gamma')}, \qquad \forall 1\leq i < j \leq 4.$$ If there exist two independent commuting monomials $X^{\gamma},X^{\gamma '} \in \widehat {\mathcal O}_{\mathfrak q}$ then clearly $\alpha_{i}(\gamma,\gamma')= 0$ for all $i$, which is a contradiction as we have seen that there does not exist common isotropic submodule $B \le \mathbb Z^4$ with $\mathop{\mathrm{\mathrm{rk}}}(B) \ge 2$ for the three forms $\alpha_i\ (i = 1, \cdots, 3)$. The theorem now follows in view of Section [2.3](#dim-of-qtorus){reference-type="ref" reference="dim-of-qtorus"}. ◻ **Remark 2**. *Similarly using the (non-associative) octonion algebra over $\mathbb Q$ we may define a quantum torus $\widehat{\mathcal{O}_{\mathfrak{q}}}$ of rank $8$ and dimension $1$ whose $\lambda$-group has rank $7$.* Finally we comment on the general case of quantum tori of rank $n$. It is shown in [@BGH; @GQ06] that it is always possible to find $n$ alternating forms on the space $\mathbb Q^n$ for which there is no common isotropic subspace of dimension greater than one. We may follow an approach similar to that of the preceding proposition to come up with examples of dimension one quantum tori of rank $n$ whose $\lambda$-group has rank equal to $n$. This is much smaller than the maximal possible rank $\frac{1}{2}n(n - 1)$. # Acknowledgements {#acknowledgements .unnumbered} The first author thanks the National Board of Higher Mathematics (NBHM) for financial support (DAE/MATH/2015/059). The second author gratefully acknowledges support from an NBHM research award. 99 L. Silberman Action of symmetric group on the second exterior power. (MathOverflow,2020) Developers, T. SageMath, the Sage Mathematics Software System (Version v9.2). (https://www.sagemath.org,2020) Greenhill, C. An algorithm for recognising the exterior square of a matrix. *Linear And Multilinear Algebra Volume*. **46** pp. 213-244 (1999) E. Kirkman, C. & Small, L. A q-analog for the virasoro algebra. *Communications In Algebra*. **22** pp. 3755-3774 (1994) Alev, J. & Chamarie, M. Derivations et automorphismes de quelques algebras quantiques. *Communications In Algebra*. **20** pp. 1787-1802 (1992) Gupta, A. 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arxiv_math

--- abstract: | We investigate $k$-superirreducible polynomials, by which we mean irreducible polynomials that remain irreducible under any polynomial substitution of positive degree at most $k$. Let $\mathbb F$ be a finite field of characteristic $p$. We show that no $2$-superirreducible polynomials exist in $\mathbb F[t]$ when $p=2$ and that no such polynomials of odd degree exist when $p$ is odd. We address the remaining case in which $p$ is odd and the polynomials have even degree by giving an explicit formula for the number of monic 2-superirreducible polynomials having even degree $d$. This formula is analogous to that given by Gauss for the number of monic irreducible polynomials of given degree over a finite field. We discuss the associated asymptotic behaviour when either the degree of the polynomial or the size of the finite field tends to infinity. address: - School of Mathematics, University of Bristol, Fry Building, Woodland Road, Bristol BS8 1UG, UK and the Heilbronn Institute for Mathematical Research, Bristol, UK - Wolverine Pathways, University of Michigan, 610 E. University Ave, Suite 2339, Ann Arbor, MI 48109, USA - Department of Mathematics and Statistics, Fylde College, Lancaster University, Lancaster LA1 4YF, UK - Department of Mathematics, Louisiana State University, 378 Lockett Hall, Baton Rouge, LA 70803, USA - Department of Mathematics, Purdue University, 150 N. University Street, West Lafayette, IN 47907-2067, USA author: - J. W. Bober - L. Du - D. Fretwell - G. S. Kopp - T. D. Wooley bibliography: - references.bib title: On $2$-superirreducible polynomials over finite fields --- [^1] # Introduction Superirreducible polynomials are polynomials that resist factorization under polynomial substitutions. Let $R$ be a commutative domain with unity having field of fractions $F$, and consider a polynomial $f\in R[t]$. For each natural number $k$, we say that $f$ is *weakly $k$-superirreducible* over $R$ if $f(g(t))$ is irreducible over $F[t]$ for all polynomials $g\in R[t]$ having degree $k$. If the polynomial $f$ is weakly $k$-superirreducible over $R$ for $1\leq k\leq K$, then we say that $f$ is *$K$-superirreducible*. In this hierarchy, a $1$-superirreducible polynomial is simply an irreducible polynomial. It transpires that a polynomial may be weakly $k$-superirreducible, and yet not weakly $(k-1)$-superirreducible (and consequently, not $k$-superirreducible). For example, one may check that $x^6+x^5+x^3+x^2+1$ is weakly $3$-superirreducible over $\mathbb{F}_2$, yet not weakly $2$-superirreducible. When $d\geq 2$ and $f\in R[t]$ has degree $d$, a consideration of the polynomial $f(t+f(t))$ reveals that $f$ cannot be $k$-superirreducible whenever $k\geq d$. The situation when $2\leq k<d$ is more subtle, however, and our focus in this paper lies on the simplest situation here in which $k=2$ and $R$ is a finite field. Let $\mathbb F_q$ denote the finite field of $q$ elements, and when $1\leq k<d$, denote by $s_k(q,d)$ the number of monic weakly $k$-superirreducible polynomials lying in $\mathbb F_q[t]$ having degree $d$. In , we provide an explicit formula for $s_2(q,d)$ analogous to the formula given by Gauss for the number of monic irreducible polynomials of given degree over $\mathbb F_q$. A consequence of this formula delivers the asymptotic formula recorded in our first theorem. **Theorem 1**. *One has $s_2(q,d)=0$ when either $q$ is a power of $2$ or $d$ is odd. Furthermore, one has $s_2(q,d)=0$ also when $q>(d-1)^2$. Meanwhile, when $q$ is odd and $d\rightarrow \infty$ through the even integers, one has the asymptotic formula $$\label{1.1} s_2(q,d)=\frac{q^d}{d2^q}+O\Bigl( \frac{1}{d}q^{d/2}\Bigr).$$* Superirreducibility has in fact been studied in the past, although not by name. Strengthening the above pedestrian observation concerning $f(t+f(t))$, it follows from work of Schinzel [@schinzel Lemma 10] that a polynomial of degree $d\geq 3$ lying in $\mathbb Q[t]$ cannot be $(d-1)$-superirreducible. More recently, Bober et al. [@BFMW] have considered superirreducibility as a potential limitation to the understanding of smooth integral values of polynomials. These authors show, inter alia, that $2$-superirreducible polynomials exist in $\mathbb Q[t]$ having degree $6$ (see [@BFMW §6]). Moreover, in work contemporaneous with that reported on herein, Du [@Du2023 Theorem 1.3] (see also [@thesis]) has exhibited $2$-superirreducible polynomials in $\mathbb Q[t]$ of degree $4$, such as the simple example $t^4+2$. With a potential local-global principle in mind, it might be expected that insights into the superirreducibility of polynomials over $\mathbb{Z}$ and over $\mathbb{Q}$ might be gained by examining corresponding superirreducibility properties over the $p$-adic integers $\mathbb{Z}_p$ and $p$-adic numbers $\mathbb{Q}_p$. Such considerations lead in turn to an investigation of the superirreducibility of polynomials over finite fields. We finish our paper by disappointing the reader in §4 with the news that if $k\ge 2$ and $p$ is any prime number, then $k$-superirreducible polynomials exist over neither $\mathbb{Z}_p$ nor $\mathbb{Q}_p$. # Basic lemmas In this section we prove the basic lemmas that provide the infrastructure for our subsequent discussions concerning superirreducibility. Recall the definition of $k$-superirreducibility provided in our opening paragraph. We begin by expanding on the observation that there are no weakly $k$-superirreducible polynomials of degree $k$ or larger. **Lemma 2**. *Let $R$ be a commutative domain with unity, and let $f\in R[t]$ be a polynomial of degree $d\ge 2$. Then $f(t)$ is not weakly $k$-superirreducible for any $k\geq d$.* *Proof.* For each non-negative integer $r$, consider the degree $d+r$ substitution $g(t)=t+t^rf(t)$. We have $$f(g(t))=f(t+t^rf(t))\equiv f(t)\equiv 0\,\,(\text{mod}\,\,f(t)).$$ Thus, we see that $f(g(t))$ is divisible by $f(t)$, and it is hence reducible. It follows that $f$ is not weakly $k$-superirreducible for $k\ge d$. ◻ The next lemma is a mild generalization of [@BFMW Proposition 3.1] to arbitrary fields. The latter proposition is restricted to the rational field $\mathbb Q$, and we would be remiss were we not to record that Schinzel [@Sch2000 Theorem 22] attributes this conclusion to Capelli. **Lemma 3**. *Let $K$ be a field. Suppose that $f(x)\in K[x]$ is a monic irreducible polynomial, let $\alpha$ be a root of $f$ lying in a splitting field extension for $f$ over $K$, and put $L=K(\alpha)$. Then, for any non-constant polynomial $g(t)\in K[t]$, the polynomial $f(g(t))$ is reducible in $K[t]$ if and only if $g(t)-\alpha$ is reducible in $L[t]$.* *Proof.* We consider the $K$-algebra $A = K[x,t]/(f(x),g(t)-x)$ from two perspectives. First, on noting that $f(x)$ is irreducible over $K[x]$, we find that $K[x]/(f(x))\cong K[\alpha]=K(\alpha)=L$. Thus, on the one hand, $$A\cong\frac{K[x,t]/(f(x))}{(g(t)-x)}\cong L[t]/(g(t)-\alpha).$$ Here, of course, we view $(g(t)-x)$ as being an ideal in $K[x,t]/(f(x))$. On the other hand, similarly, $$A\cong\frac{K[x,t]/(g(t)-x)}{(f(x))}\cong K[t]/(f(g(t))).$$ Thus, we obtain a $K$-algebra isomorphism $$\label{eq:Kalgisom} K[t]/(f(g(t)))\cong L[t]/(g(t)-\alpha).$$ Hence $K[t]/(f(g(t)))$ is a field if and only if $L[t]/(g(t)-\alpha)$ is a field, and thus $f(g(t))$ is irreducible in $K[t]$ if and only if $g(t)-\alpha$ is irreducible in $L[t]$. The desired conclusion follows. ◻ We take the opportunity to record a further consequence of the relation [\[eq:Kalgisom\]](#eq:Kalgisom){reference-type="eqref" reference="eq:Kalgisom"}, since it may be of use in future investigations concerning superirreducibility. **Lemma 4**. *Let $K$ be a field. Suppose that $f(x)\in K[x]$ is a monic irreducible polynomial, and let $g(t)\in K[t]$ be any non-constant polynomial. Then, for any polynomial divisor $h(t)$ of $f(g(t))$, we have $\deg(f)|\deg(h)$.* *Proof.* The relation [\[eq:Kalgisom\]](#eq:Kalgisom){reference-type="eqref" reference="eq:Kalgisom"} shows that $K[t]/(f(g(t))$ has the structure of an $L$-algebra. Any ring quotient of an $L$-algebra is still an $L$-algebra. Thus, we see that $K[t]/(h(t))$ is an $L$-algebra, and in particular a vector space over $L$. Consequently, one has $$\deg(h)=\dim_K K[t]/(h(t))=[L:K]\left(\dim_L K[t]/(h(t))\right) =\deg(f)\left( \dim_L K[t]/(h(t)) \right),$$ and thus $\deg(f)|\deg(h)$. ◻ We also provide a trivial lemma explaining the relationship between our definitions of superirreducibility and weak superirreducibility for different values of $k$. **Lemma 5**. *Let $R$ be a commutative domain with unity, and let $f(x) \in R[x]$ and $k \in \mathbb{N}$. The polynomial $f(x)$ is $k$-superirreducible if and only if it is weakly $\ell$-superirreducible for all natural numbers $\ell \leq k$. The polynomial $f(x)$ is weakly $k$-superirredcubible if and only if it is weakly $\ell$-superirreducible for all natural numbers $\ell$ dividing $k$.* *Proof.* All of the implications follow formally from the definitions except for the statement that, if $f(x)$ is weakly $k$-superirreducible and $\ell|k$, then $f(x)$ is weakly $\ell$-superirreducible. To prove this, write $k=\ell m$ and consider a polynomial $g(t)$ of degree $\ell$. The substitution $f(g(t^m))$ is thus irreducible, and hence so is $f(g(t))$. ◻ It follows that "$2$-superirreducible" and "weakly $2$-superirreducible" are synonyms. # Counting $2$-superirreducible polynomials over finite fields Recall that when $1\le k<d$, we write $s_k(q,d)$ for the number of monic weakly $k$-superirreducible polynomials lying in $\mathbb{F}_q[t]$ having degree $d$. In particular, $s_2(q,d)$ is the number of monic $2$-superirreducible polynomials in $\mathbb{F}_q[t]$ having degree $d$, because "$2$-superirreducible" and "weakly $2$-superirreducible" are equivalent conditions by . Our goal in this section is to establish formulae for $s_2(q,d)$ that deliver the conclusions recorded in . ## Elementary cases We begin by confirming that when $q$ is a power of $2$, and also when $d$ is odd, one has $s_2(q,d)=0$. In fact, rather more is true, as we now demonstrate. **Proposition 6**. *Let $p$ be a prime. Then for all natural numbers $\ell$ and $d$, one has $s_p(p^\ell,d)=0$.* *Proof.* Consider a polynomial $f\in \mathbb{F}_{p^\ell}[t]$ having degree $d$. Write $f(x)=\sum_{j=0}^d a_jx^j$, and note that $a_j=a_j^{p^\ell}$ for each index $j$. Thus, we have $$f(t^p)=\sum_{j=0}^d a_j^{p^\ell}t^{pj}= \Biggl(\sum_{j=0}^d a_j^{p^{\ell-1}} t^j\Biggr)^p,$$ and it follows that $f(x)$ is not weakly $p$-superirreducible. Consequently, one has $s_p(p^\ell,d)=0$. ◻ The special case $p=2$ of shows that $s_2(q,d)=0$ when $q$ is a power of $2$. Next, we turn to polynomials of odd degree over $\mathbb{F}_q$. **Proposition 7**. *When $d$ is an odd natural number, one has $s_2(q,d)=0$.* *Proof.* In view of the case $p=2$ of , there is no loss of generality in assuming that $q$ is odd. Let $f(x)\in \mathbb{F}_q[x]$ be a monic irreducible polynomial of degree $d$. The polynomial $f$ has a root $\alpha$ lying in $\mathbb{F}_{q^d}$, and $\mathbb{F}_{q^d}=\mathbb{F}_q(\alpha)$. By virtue of , if we are able to find a quadratic polynomial $g(t)\in \mathbb{F}_q[t]$ having the property that $g(t)-\alpha$ has a root in $\mathbb{F}_{q^d}$, then we may infer that $f(g(t))$ is reducible. This will confirm that $f(x)$ is not $2$-superirreducible, delivering the desired conclusion. We may divide into two cases: - Suppose first that $\alpha=\beta^2$ for some $\beta\in\mathbb{F}_{q^d}$. Then we put $g(t)=t^2$, and observe that the polynomial $g(t)-\alpha$ has the root $\beta\in \mathbb{F}_{q^d}$. - In the remaining cases, we may suppose that $\alpha$ is not the square of any element of $\mathbb{F}_{q^d}$. Since $q\ne 2$, there exists an element $b\in \mathbb{F}_q$ which is not the square of any element of $\mathbb{F}_q$. On recalling our assumption that $d$ is odd, we find that $b$ is not the square of any element in $\mathbb{F}_{q^d}$. Thus, we may infer that $b^{-1}\alpha=\beta^2$ for some $\beta\in \mathbb{F}_{q^d}$. We now put $g(t)=bt^2$ and observe that the polynomial $g(t)-\alpha$ has the root $\beta\in \mathbb{F}_{q^d}$. In either case, our previous discussion shows that $f(x)$ is not $2$-superirreducible, and this implies the desired conclusion. ◻ The conclusion of combines with that of to confirm the first assertion of . These cases of help to explain the example noted in the introduction demonstrating that weak $(k-1)$-superirreducibility is not necessarily inherited from the corresponding property of weak $k$-superirreducibility. Expanding a little on that example, we observe that by making use of commonly available computer algebra packages, one finds the following examples of polynomials weakly $3$-superirreducible over $\mathbb{F}_2[x]$ yet not $2$-superirreducible over $\mathbb{F}_2[x]$: $$\begin{aligned} x^6+x^5+x^3+x^2\;&+1,\\ x^8+x^6+x^5+x^3\;&+1,\\ x^{10}+x^9+x^7+x^2\;&+1,\\ x^{10}+x^9+x^8+x^4+x^3+x^2\;&+1,\\ x^{10}+x^9+x^7+x^6+x^5+x^4+x^3+x^2\;&+1.\end{aligned}$$ In each of these examples of a polynomial $f\in \mathbb{F}_2[x]$, the failure of $2$-superirreducibility follows from . Meanwhile, a direct computation confirms that the polynomial $f(g(t))$ is irreducible over $\mathbb{F}_2[t]$ for each of the $8$ possible monic cubic polynomials $g(t)$ lying in $\mathbb{F}_2[t]$. No analogous odd degree examples are available, of course, by virtue of , though examples of larger even degrees are not too difficult to identify. ## Heuristics {#ssec:heuristic} We next address the problem of determining a formula for the number $s_k(q,d)$ of monic weakly $k$-superirreducible polynomials of degree $d$ over $\mathbb{F}_q$. The simplest situation here with $k=1$ is completely resolved by celebrated work of Gauss, since $1$-superirreducibility is equivalent to irreducibility. Thus, as is well-known, it follows from Gauss [@gauss page 602] that $$s_1(q,d)=\frac{1}{d}\sum_{e|d}\mu \biggl(\frac{d}{e}\biggr) q^{e},$$ whence, as $d\rightarrow \infty$, one has the asymptotic formula $$s_1(q,d)=\frac{q^d}{d}+O\biggl( \frac{1}{d}q^{d/2}\biggr) .$$ The corresponding situation with $k\geq 2$ is more subtle. We now motivate our proof of an asymptotic formula for $s_2(q,d)$ with a heuristic argument that addresses the cases remaining to be considered, namely those where $d$ is even and $q$ is odd. The heuristic argument is based on the following lemma, which will also be used in the proof. **Lemma 8**. *Let $q$ be an odd prime power, and let $f(x)\in \mathbb{F}_q[x]$ be a monic irreducible polynomial of even degree $d$. Let $\alpha \in \mathbb{F}_{q^d}$ be a root of $f(x)$. The polynomial $f(x)$ is $2$-superirreducible if and only if $\alpha+c$ is not a square in $\mathbb{F}_{q^d}$ for all $c\in \mathbb{F}_q$.* *Proof.* As a consequence of , the polynomial $f(x)$ is $2$-superirreducible in $\mathbb{F}_q[x]$ if and only if $g(t)-\alpha$ is irreducible in $\mathbb{F}_{q^d}[t]$ for all quadratic polynomials $g\in \mathbb{F}_q[t]$. Since this condition is invariant under all additive shifts mapping $t$ to $t+v$, for $v\in \mathbb{F}_q$, it suffices to consider only the quadratic polynomials of the shape $g(t)=at^2-b$, with $a,b\in \mathbb{F}_q$. Moreover, the assumption that $d$ is even ensures that $a$ is a square in $\mathbb{F}_{q^d}$, and hence we may restrict our attention further to polynomials of the shape $g(t)=t^2-c$ with $c\in \mathbb{F}_q$. So $f(x)$ is $2$-superirreducible if and only if the equation $t^2-c=\alpha$ has no solution in $\mathbb{F}_{q^d}$ for any $c\in\mathbb{F}_q$. ◻ For heuristic purposes, we now model the behaviour of these elements $\alpha+c$ as if they are randomly distributed throughout $\mathbb{F}_{q^d}$. Since roughly half the elements of $\mathbb{F}_{q^d}$ are squares, one should expect that the condition that $\alpha+c$ is not a square is satisfied for a fixed choice of $c$ with probability close to $\frac{1}{2}$. Treating the conditions for varying $c\in \mathbb{F}_q$ as independent events, we therefore expect that $f(x)$ is $2$-superirreducible with probability close to $1/2^q$. Multiplying this probability by the number of choices for monic irreducible polynomials $f(x)$ of degree $d$, our heuristic predicts that when $d$ is even and $q$ is odd, one should have $$s_2(q,d)\approx \frac{q^d}{d 2^q}.$$ We shall see in the next subsection that this heuristic accurately predicts the asymptotic behaviour of $s_2(q,d)$ as $d\to \infty$ through even integers $d$. ## The large $d$ limit {#ssec:larged} The asymptotic formula predicted by the heuristic described in the previous subsection will follow in the large $d$ limit from Weil's resolution of the Riemann hypothesis for curves over finite fields. We make use, specifically, of the Weil bound for certain higher autocorrelations of the quadratic character generalizing Jacobi sums. Our goal in this subsection is the proof of the estimate for $s_2(q,d)$ supplied by the following theorem, an immediate consequence of which is the asymptotic formula [\[1.1\]](#1.1){reference-type="eqref" reference="1.1"} supplied by Theorem [Theorem 1](#theorem1.1){reference-type="ref" reference="theorem1.1"}. **Theorem 9**. *When $q$ is odd and $d$ is even, one has $$\Bigl| s_2(q,d)-\frac{q^d}{d 2^q}\Bigr|<\frac{q}{2d}q^{d/2}.$$* The proof of this estimate is based on a more rigorous version of the heuristic argument given in , and it employs character sums that we now define. **Definition 10**. Let $q$ be an odd prime power, and write $\chi_q$ for the nontrivial quadratic character $\chi_q : \mathbb{F}_q^\times \to \{1,-1\}$, extended to $\mathbb{F}_q$ by setting $\chi_q(0)=0$. We define the *order $n$ autocorrelation of $\chi_q$* with offsets $u_1,\ldots,u_n \in \mathbb{F}_q$ to be the sum $$a_q(u_1,\ldots,u_n)=\sum_{\beta \in \mathbb{F}_q} \chi_q(\beta+u_1)\cdots\chi_q(\beta+u_n).$$ Noting that this definition is independent of the ordering of the arguments, when $U=\{u_1,\ldots ,u_n\}$ is a subset of $\mathbb{F}_q$, we adopt the convention of writing $a_q(U)$ for $a_q(u_1,\ldots ,u_n)$. Note that $a_q(U)\in \mathbb{Z}$ for all subsets $U$ of $\mathbb{F}_q$. When $\left| U \right|=1$ it is apparent that $a_q(U)=0$. Meanwhile, in circumstances where $\left| U \right|=2$, so that $U=\{u_1,u_2\}$ for some elements $u_1,u_2\in \mathbb{F}_q$ with $u_1\neq u_2$, the autocorrelation $a_q(U)=a_q(u_1,u_2)$ is a quadratic Jacobi sum. Thus, in this situation, we have $a_q(u_1,u_2)=\pm 1$ (see [@irelandrosen Chapter 8]). The higher-order correlations become more complicated, but we will see that they can easily be bounded. First, we relate the autocorrelations of $\chi_q$ to the number $s_2(q,d)$ of monic $2$-superirreducible polynomials of degree $d$ in $\mathbb{F}_q[x]$. **Proposition 11**. *Let $q$ be an odd prime power and $d$ be even. Then $$s_2(q,d)=\frac{1}{d 2^q}\sum_{\substack{e|d \\ {\text{$d/e$ odd}}}} \mu\Bigl( \frac{d}{e}\Bigr)\biggl(q^{e} +\sum_{\emptyset \neq U\subseteq \mathbb{F}_q} (-1)^{\left| U \right|}a_{q^{e}}(U)\biggr).$$* *Proof.* Consider a monic irreducible polynomial $f(x)\in \mathbb{F}_q[x]$ of degree $d$, and let $\alpha$ be a root of $f(x)$ in $\mathbb{F}_{q^d}$. It follows from that $f(x)$ is $2$-superirreducible if and only if $\alpha+c$ is not a square in $\mathbb{F}_{q^d}$ for each $c\in\mathbb{F}_{q}$. Since the latter condition is equivalent to the requirement that $\chi_{q^d}(\alpha + c)=-1$ for all $c\in \mathbb{F}_q$, we see that $$\prod_{c\in \mathbb{F}_q}\frac{1}{2}\left(1-\chi_{q^d}(\alpha +c)\right)=\begin{cases}1,& \text{if $f$ is $2$-superirreducible},\\ 0,&\text{otherwise}. \end{cases}$$ This relation provides an algebraic formulation of the indicator function for $2$-superirreducibility. Instead of summing this quantity over monic irreducible polynomials, we can instead sum over elements $\alpha\in\mathbb{F}_{q^d}$ not lying in any proper subfield, dividing by $d$ to account for overcounting. Thus, we find that $$s_2(q,d)=\frac{1}{d}\sum_{\substack{\alpha \in \mathbb{F}_{q^d}\\ \text{$\alpha \notin\mathbb{F}_{q^{e}}$ (\text{$e<d$ and $e|d$})}}} \prod_{c\in \mathbb{F}_q}\frac{1}{2} \left(1-\chi_{q^d}(\alpha + c)\right).$$ The condition on $\alpha$ in the first summation of this relation may be encoded using the Möbius function. Thus, we obtain $$s_2(q,d)=\frac{1}{d 2^q}\sum_{e|d}\mu\Bigl( \frac{d}{e}\Bigr) \sum_{\alpha \in \mathbb{F}_{q^{e}}} \prod_{c\in \mathbb{F}_q}\left( 1-\chi_{q^d}(\alpha + c)\right).$$ When $d/e$ is even, the quadratic character $\chi_{q^d}$ on $\mathbb{F}_{q^d}$ restricts to the trivial character on $\mathbb{F}_{q^{e}}$, and when $d/e$ is odd it instead restricts to $\chi_{q^{e}}$. We therefore deduce that $$s_2(q,d)=\frac{1}{d2^q}\sum_{\substack{e|d\\ \text{$d/e$ odd}}} \mu\Bigl( \frac{d}{e}\Bigr) \sum_{\alpha \in \mathbb{F}_{q^{e}}}\prod_{c\in \mathbb{F}_q}\left(1-\chi_{q^{e}}(\alpha + c)\right) ,$$ and the desired formula for $s_2(q,d)$ follows on observing that $$\sum_{\alpha \in \mathbb{F}_{q^{e}}}\prod_{c\in \mathbb{F}_q}\left(1-\chi_{q^{e}}(\alpha + c)\right) =q^{e} + \sum_{\emptyset \neq U \subseteq \mathbb{F}_q} (-1)^{\left| U \right|} a_{q^{e}}(U).$$ ◻ We next establish a bound on the autocorrelations $a_{q^e}(U)$. **Lemma 12**. *Let $q$ be an odd prime power. Suppose that $U$ is a non-empty subset of $\mathbb{F}_q$ with $\left| U \right|=n$. Then for each positive integer $e$, one has $\left| a_{q^e}(U) \right|\leq (n-1)q^{e/2}$.* *Proof.* Observe that $$a_{q^e}(U)=\sum_{\beta\in\mathbb{F}_{q^e}}\chi_{q^e}(h(\beta)),$$ where $h(t)=(t+u_1)\cdots (t+u_n)$ is a polynomial in $\mathbb{F}_q[t]$ having roots $-u_1,\ldots ,-u_n$. Since $u_1,\ldots ,u_n$ are distinct and $\chi_{q^e}$ is a multiplicative character of order $2$, it follows from a version of Weil's bound established by Schmidt that $\left| a_{q^e}(U) \right|\leq (n-1)q^{e/2}$ (see, for example, Theorem 2C' on page 43 of [@Sch1976 Chapter 2]). ◻ Now we complete the proof of . In this proof, we expend a little extra effort to achieve a more attractive conclusion. *Proof of .* We begin by observing that, in view of , one has $$\begin{aligned} \biggl| \sum_{\emptyset \neq U \subseteq \mathbb{F}_q} (-1)^{\left| U \right|} a_{q^{e}}(U)\biggr| &\leq \sum_{n=1}^q \binom{q}{n}(n-1)q^{e/2}\notag \\ &=q^{e/2}\biggl( q\sum_{n=2}^q \binom{q-1}{n-1}-\sum_{n=2}^q \binom{q}{n}\biggr) \notag \\ &=q^{e/2}\bigl( q(2^{q-1}-1)-(2^q-q-1)\bigr).\label{tw1}\end{aligned}$$ We note next that since $d$ is assumed to be even, then whenever $e$ is a divisor of $d$ with $d/e$ odd, then $e$ is even. Moreover, if it is the case that $e<d$, then $e\le d/3$. The first constraint on $e$ here conveys us from [\[tw1\]](#tw1){reference-type="eqref" reference="tw1"} to the upper bound $$\begin{aligned} \sum_{\substack{e|d\\ \text{$d/e$ odd}}}\left| \sum_{\emptyset \neq U \subseteq \mathbb{F}_q} (-1)^{\left| U \right|} a_{q^{e}}(U) \right| &\leq \bigl( 2^{q-1}(q-2)+1\bigr) \sum_{m=0}^{d/2}q^m \\ &<\frac{q}{q-1}\bigl( 2^{q-1}(q-2)+1\bigr) q^{d/2}.\end{aligned}$$ Meanwhile, making use also of the second constraint on $e$, we obtain the bound $$\sum_{\substack{e|d\\ \text{$e<d$ and $d/e$ odd}}}q^{e}\le \sum_{0\le m\le d/3}q^m< \frac{q}{q-1}q^{d/2}.$$ By applying these bounds in combination with , we deduce that $$\left| s_2(q,d)-\frac{q^d}{d 2^q} \right|<\frac{1}{d 2^q} \bigl( (q-1)2^{q-1}-2^{q-1}+2\bigr) \frac{q}{q-1}q^{d/2}\le \frac{q}{2d}q^{d/2}.$$ This completes the proof of . ◻ ## Vanishing in the large $q$ limit We turn our attention next to the behaviour of $s_2(q,d)$ when $d$ is fixed and $q$ is large. It transpires that $s_2(q,d)=0$ for large enough prime powers $q$. This conclusion follows from once we confirm that for every primitive element $\alpha \in \mathbb{F}_{q^d}$, there exists an element $c\in \mathbb{F}_q$ for which $\chi_{q^d}(\alpha+c)=1$. **Lemma 13**. *Suppose that $q$ is an odd prime power and $\alpha \in \mathbb{F}_{q^d}$ is a primitive element. Then, whenever $q>(d-1)^2$, one has $$\biggl| \sum_{c\in\mathbb{F}_q}\chi_{q^d}(\alpha+c)\biggr| <q.$$* *Proof.* Consider the $d$-dimensional commutative $\mathbb{F}_q$-algebra $\mathbb{F}_{q^d}=\mathbb{F}_q[\alpha]$. Put $\beta=-\alpha$, and observe that the character $\chi_{q^d}$ is not trivial on $\mathbb{F}_q[\beta]=\mathbb{F}_{q^d}$. Then it follows from Wan [@wan Corollary 2.2] that $$\biggl| \sum_{c\in \mathbb{F}_q}\chi_{q^d}(c-\beta)\biggr| \leq (d-1)q^{1/2}.$$ Provided that $q>(d-1)^2$, one has $(d-1)q^{1/2}<q$, and thus the desired conclusion follows. ◻ We are now equipped to establish the final conclusion of . **Theorem 14**. *Let $d$ be an even integer, and suppose that $q$ is an odd prime power with $q>(d-1)^2$. Then $s_2(q,d)=0$.* *Proof.* Suppose that $f(x)\in \mathbb{F}_q[x]$ is a $2$-superirreducible polynomial of degree $d$ over $\mathbb{F}_q$, and consider a root $\alpha \in \mathbb{F}_{q^d}$ of $f$. By , we must have $\chi_{q^d}(\alpha +c)=-1$ for every $c\in\mathbb{F}_q$, and hence $$\sum_{c\in \mathbb{F}_q}\chi_{q^d}(\alpha +c)=-q.$$ This contradicts the estimate supplied by , since we have assumed that $q>(d-1)^2$. Consequently, there can be no $2$-superirreducible polynomials of degree $d$ over $\mathbb{F}_q$. ◻ # Relationship to rational and $p$-adic superirreducibility Fix a rational prime number $p$. Then, any monic polynomial $f\in \mathbb{Z}[x]$ that is irreducible modulo $p$ is also irreducible over $\mathbb{Q}[x]$. One might guess that this familiar property extends from irreducibility to superirreducibility. Thus, if the monic polynomial $f(x)$ reduces to a weakly $k$-superirreducible polynomial modulo $p$, one might expect that $f(x)$ is itself weakly $k$-superirreducible over $\mathbb{Z}$, and perhaps also over $\mathbb{Q}$. We find that such an expectation is in fact excessively optimistic. Indeed, there are $2$-superirreducible polynomials over $\mathbb{F}_3$ with integral lifts that are not $2$-superirreducible over $\mathbb{Z}$. **Example 15**. Consider the polynomial $f(x) \in \mathbb{Z}[x]$ given by $$f(x)=x^4 -12x^3 +2x^2 -39x +71.$$ Then, we have $f(x)\equiv x^4 -x^2 -1\,\,\left(\operatorname{mod}\, 3\right)$, and it is verified by an exhaustive check that $x^4 -x^2 -1$ is $2$-superirreducible in $\mathbb{F}_3[x]$. However, one has $$f(3t^2 +t) =(t^4+3t^3+2t^2-1)(81t^4-135t^3-27t^2+39t-71),$$ so that $f(x)$ is not $2$-superirreducible over $\mathbb{Z}$. Despite examples like the one above, one may still hope that the assumption of additional congruential properties involving higher powers of $p$ might suffice to exclude such problematic examples, thereby providing a means to lift superirreducible polynomials over $\mathbb{Z}_p$ to superirreducible polynomials over $\mathbb{Z}$. The following proposition reveals a major obstruction to any such lifting process, since it shows that for each natural number $k\ge 2$, there are no $p$-adic weakly $k$-superirreducible polynomials. **Proposition 16**. *Let $p$ be a prime number. When $k\ge 2$, there are no weakly $k$-superirreducible polynomials over $\mathbb{Z}_p$ or over $\mathbb{Q}_p$.* *Proof.* Suppose, if possible, that $f\in \mathbb{Q}_p[x]$ is a weakly $k$-superirreducible polynomial. There is no loss of generality in assuming that $f$ is an irreducible polynomial lying in $\mathbb{Z}_p[x]$. Let $\alpha$ be a root of $f$ lying in a splitting field extension for $f$ over $\mathbb{Q}_p$, and let $e=1+|v_p(\alpha)|$, where $v_p(\alpha)$ is defined in such a manner that $|\alpha|_p=p^{-v_p(\alpha)}$. Let $h\in \mathbb{Z}_p[t]$ be any polynomial of degree $k$, put $g(t)=p^e h(t)+t$, and consider the equation $g(\beta)=\alpha$. Since $|g(\alpha)-\alpha|_p<1$ and $|g'(\alpha)|_p=|1+p^e h'(\alpha)|_p=1$, an application of Hensel's lemma demonstrates that the equation $g(\beta)=\alpha$ has a solution $\beta\in \mathbb{Q}_p(\alpha)$. Thus, the equation $\alpha =p^eh(\beta)+\beta$ has a solution $\beta\in \mathbb{Q}_p(\alpha)$, and by appealing to , we conclude that the polynomial $f(p^e h(t)+t)$ is reducible over $\mathbb{Q}_p[t]$. Since $p^eh(t)+t\in \mathbb{Z}_p[t]$, we see that $f$ is neither weakly $k$-superirreducible over $\mathbb{Z}_p$ nor over $\mathbb{Q}_p$, and we arrive at a contradiction. The desired conclusion follows. ◻ The discussion of this section appears to show, therefore, that superirreducibility over $\mathbb{F}_p$, and indeed superirreducibility over $\mathbb{Z}_p$ and $\mathbb{Q}_p$, is not closely connected to corresponding superirreducibility over $\mathbb{Z}$ and $\mathbb{Q}$. 99 J. W. Bober, D. Fretwell, G. Martin and T. D. Wooley, *Smooth values of polynomials*, J. Austral. Math. Soc. **108** (2020), no. 2, 245--261. L. Du, *Superirreducibility of polynomials, binomial coefficient asymptotics and stories from my classroom*, Ph.D. Thesis, University of Michigan, 2020. L. Du, *$2$-superirreducibility of univariate polynomials over $\mathbb Q$ and $\mathbb Z$*, submitted. C. F. Gauss, *Untersuchungen über höhere Arithmetik*, Chelsea, New York, 1965. K. Ireland and M. Rosen, *A classical introduction to modern number theory*, Springer, New York, 1982. A. Schinzel, *On two theorems of Gelfond and some of their applications*, Acta Arith. **13** (1967), no. 2, 177--236. A. Schinzel, *Polynomials with special regard to reducibility*, Encyclopedia of Mathematics and its Applications, vol. 77, Cambridge University Press, Cambridge, 2000. W. M. Schmidt, *Equations over finite fields. An elementary approach*, Lecture Notes in Math., vol. 536, Springer, New York, 1976. D. Wan, *Generators and irreducible polynomials over finite fields*, Math. Comp. **66** (1997), no. 219, 1195--1212. [^1]: The fourth author is supported by NSF grant DMS-2302514. The fifth author is supported by NSF grants DMS-1854398 and DMS-2001549. The first, third, and fourth authors are also supported by the Heilbronn Institute for Mathematical Research

arxiv_math

--- abstract: | Let $R$ be a unital $*$-ring. For any $a,w,b\in R$, we apply the defined $w$-core inverse to define a new class of partial orders in $R$, called the $w$-core partial order. Suppose $a,b\in R$ are $w$-core invertible. We say that $a$ is below $b$ under the $w$-core partial order, denoted by $a\overset{\tiny{\textcircled{\#}}}\leq_w b$, if $a_w^{\tiny{\textcircled{\#}}} a=a_w^{\tiny{\textcircled{\#}}} b$ and $awa_w^{\tiny{\textcircled{\#}}} =bwa_w^{\tiny{\textcircled{\#}}}$, where $a_w^{\tiny{\textcircled{\#}}}$ denotes the $w$-core inverse of $a$. Characterizations of the $w$-core partial order are given. Also, the relationships with several types of partial orders are considered. In particular, we show that the core partial order coincides with the $a$-core partial order, and the star partial order coincides with the $a^*$-core partial order. address: School of Mathematics, Hefei University of Technology, Hefei 230009, China. author: - Huihui Zhu - Liyun Wu title: "**A new class of partial orders**" --- the $w$-core inverse ,the inverse along an element ,the sharp partial order ,the star partial order ,the core partial order ,rings with involution 06A06 ,15A09 ,16W10 # Introduction There are many types of partial orders based on generalized inverses in mathematical literature, such as the minus partial order [@Hartwig1980], the plus partial order [@Hartwig1980], the sharp partial order [@Mitra1987], the star partial order [@Drazin1978], the diamond partial order [@Lebtahi2014], the core partial order [@Baksalary2010] and so on. These partial orders are investigated in different settings such as complex matrices and rings. In this paper, we aim to introduce the $w$-core partial order based on our defined $w$-core inverses [@Zhu2020] and to give its several characterizations and properties. The paper is organized as follows. In Section 2, we define a class of partial order and establish its characterizations. In Section 3, the relationships between the $w$-core partial order and other partial orders are considered. It is proved that the $w$-core partial order is between the core partial order and the diamond partial order. We show in Theorem [Theorem 23](#three class partial orders){reference-type="ref" reference="three class partial orders"} that the star partial order and the core partial order are both instances of the $w$-core partial order. More precisely, the core partial order coincides with the $a$-core partial order, and the star partial order coincides with the $a^*$-core partial order. Then, the equivalence between $a\overset{\tiny{\textcircled{\#}}}\leq_w b$ and $b-a\overset{\tiny{\textcircled{\#}}}\leq_w b$ is derived, under certain conditions. In the end, the reverse order law for the $w$-core inverse is given. Let us now recall several notions of generalized inverses in rings. Let $R$ be an associative ring with the identity 1. An element $a\in R$ is called (von Neumann) regular if there is some $x\in R$ such that $a=axa$. Such an $x$ is called an inner inverse of $a$, and is denoted by $a^{-}$. If in addition, $xax=x$, then $a$ is called reflexive or $\{1,2\}$-invertible. Such an $x$ is called a reflexive inverse of $a$, and is denoted by $a^+$. Further, an element $a\in R$ is called group invertible if there exists a reflexive inverse $a^+$ of $a$ which commutes with $a$. Such an element $a^+$ is called a group inverse of $a$. It is unique if it exists, and is denoted by $a^\#$. We denote by $R^\#$ the set of all group invertible elements in $R$. Given any $a,d\in R$, $a$ is invertible along $d$ [@Mary2011] if there exists some $b\in R$ such that $bad=d=dab$ and $b\in dR \cap Rd$. Such an element $b$ is called the inverse of $a$ along $d$. It is unique if it exists, and is denoted by $a^{\parallel d}$. As usual, by the symbol $R^{\parallel d}$ we denote the set of all invertible elements along $d$. It is known from [@Mary2013 Theorem 2.2] that $a\in R^{\parallel d}$ if and only if $d\in dadR \cap Rdad$. Mary in [@Mary2011 Theorem 11] proved that $a\in R^\#$ if and only if $a\in R^{\parallel a}$ if and only if $1\in R^{\parallel a}$. Moreover, $a^\#=a^{\parallel a}$ and $1^{\parallel a}=aa^\#$. More results on the inverse along an element can be referred to [@Mary2013; @Zhu2016; @Zhu20171]. Throughout this paper, we assume that $R$ is a unital $*$-ring, that is a ring with unity 1 and an involution $*$ satisfying $(x^*)^*=x$, $(xy)^*=y^*x^*$ and $(x+y)^*=x^*+y^*$ for all $x,y\in R$. An element $a\in R$ (with involution) is Moore-Penrose invertible [@Penrose1955] if there is some $x\in R$ such that \(i\) $axa=a$, (ii) $xax=x$, (iii) $(ax)^*=ax$, (iv) $(xa)^*=xa$. Such an $x$ is called a Moore-Penrose inverse of $a$. It is unique if it exists, and is denoted by $a^\dag$. We denote by $R^\dag$ the set of all Moore-Penrose invertible elements in $R$. It was proved in [@Mary2011; @Zhu20171] that $a\in R^\dag$ if and only if $a^{\parallel a^*}$ exists if and only if $(a^*)^{\parallel a}$ exists. In this case, $a^\dag=a^{\parallel a^*}=((a^*)^{\parallel a})^*$. If $a$ and $x$ satisfy the equations (i) $axa=a$ and (iii) $(ax)^*=ax$, then $x$ is called a $\{1,3\}$-inverse of $a$, and is denoted by $a^{(1,3)}$. If $a$ and $x$ satisfy the equations (i) $axa=a$ and (iv) $(xa)^*=xa$, then $x$ is called a $\{1,4\}$-inverse of $a$, and is denoted by $a^{(1,4)}$. We denote by $R^{(1,3)}$ and $R^{(1,4)}$ the sets of all $\{1,3\}$-invertible and $\{1,4\}$-invertible elements in $R$, respectively. It is well known that $a\in R^\dag$ if and only if $a\in R^{(1,3)} \cap R^{(1,4)}$ if and only if $a\in aa^*R \cap Ra^*a$ if and only if $a\in aa^*aR$ if and only if $a\in Raa^*a$. In this case, $a^\dag=a^{(1,4)}aa^{(1,3)}$. The core inverse of complex matrices was firstly introduced by Baksalary and Trenkler [@Baksalary2010]. In 2014, Rakić et al. [@Rakic2014] extended the core inverse of a complex matrix to an element in a unital $*$-ring. They showed that the core inverse of $a\in R$ is the solution of the following five equations (1) $axa=a$, (2) $xax=x$, (3) $ax^2=x$, (4) $xa^2=a$, (5) $(ax)^*=ax$. The core inverse of $a\in R$ is unique if it exists, and is denoted by $a^{\tiny{\textcircled{\#}}}$. By $R^{\tiny{\textcircled{\#}}}$ we denote the set of all core invertible elements in $R$. In [@Xu2017 Theorem 2.6], Xu et al. showed that $a\in R^{\tiny{\textcircled{\#}}}$ if and only if $a\in R^\# \cap R^{(1,3)}$. In this case, the expression of the core inverse can be given as $a^{\tiny{\textcircled{\#}}}=a^\#aa^{(1,3)}$. Recently, the present authors [@Zhu2020] defined the $w$-core inverse in a ring $R$. Given any $a,w\in R$, we say that $a$ is $w$-core invertible if there exists some $x\in R$ such that $awx^2=x$, $xawa=a$ and $(awx)^*=awx$. Such an $x$ is called a $w$-core inverse of $a$. It is unique if it exists, and is denoted by $a_w^{\tiny{\textcircled{\#}}}$. Also, the $w$-core inverse $x$ of $a$ satisfies $awxa=a$ and $xawx=x$ (see [@Zhu2020 Lemma 2.2]). By $R_w^{\tiny{\textcircled{\#}}}$ we denote the set of all $w$-core invertible elements in $R$. It was proved in [@Zhu2020 Theorem 2.11] that $a \in R_w^{\tiny{\textcircled{\#}}}$ if and only if $w\in R^{\parallel a}$ and $a\in R^{(1,3)}$. Moreover, $a_w^{\tiny{\textcircled{\#}}}=w^{\parallel a}a^{(1,3)}$. Several well known partial orders on a ring $R$ are given below. \(1\) The minus partial order: $a\overset{-}\leq b$ if and only if there exists an inner inverse $a^-\in R$ of $a$ such that $a^-a =a^-b$ and $aa^- = ba^-$. \(2\) The plus partial order: $a\overset{+}\leq b$ if and only if there exists a reflexive inverse $a^+\in R$ of $a$ such that $a^+a =a^+b$ and $aa^+ = ba^+$. \(3\) The sharp partial order: $a\overset{\#}\leq b$ if and only if there exists the group inverse $a^\#\in R$ of $a$ such that $a^\#a=a^\#b$ and $aa^\#=ba^\#$. \(4\) The star partial order: $a\overset{*}\leq b$ if and only if $a^*a=a^*b$ and $aa^*=ba^*$. In particular, if $a\in R^\dag$, then $a\overset{*}\leq b$ if and only if $a^\dag a=a^\dag b$ and $aa^\dag=ba^\dag$. \(5\) The diamond partial order: $a\overset{\diamond}\leq b$ if and only if $aa^*a=ab^*a$, $aR\subseteq bR$ and $Ra\subseteq Rb$. \(6\) The core partial order: $a\overset{\tiny{\textcircled{\tiny\#}}}\leq b$ if and only if there exists the core inverse $a^{\tiny{\textcircled{\#}}}\in R$ of $a$ such that $a^{\tiny{\textcircled{\#}}}a=a^{\tiny{\textcircled{\#}}}b$ and $aa^{\tiny{\textcircled{\#}}}=ba^{\tiny{\textcircled{\#}}}$. # The $w$-core partial order In this section, we aim to define a class of partial orders and to give its characterizations in a ring $R$. **Definition 1**. *Let $a,b,w\in R$ with $a,b\in R_w^{\tiny{\textcircled{\#}}}$. We say that $a$ is below $b$ under the $w$-core relation and write $a\overset{\tiny{\textcircled{\#}}}\leq_w b$ if $a_w^{\tiny{\textcircled{\#}}} a=a_w^{\tiny{\textcircled{\#}}} b$ and $awa_w^{\tiny{\textcircled{\#}}} =bwa_w^{\tiny{\textcircled{\#}}}$.* We next show that the relation $a\overset{\tiny{\textcircled{\#}}}\leq_w b$ is a partial order. First, an auxiliary lemma is given below. **Lemma 2**. *Let $a,b,w\in R$ with $a,b\in R_w^{\tiny{\textcircled{\#}}}$. If $a_w^{\tiny{\textcircled{\#}}} a=a_w^{\tiny{\textcircled{\#}}} b$ and $awa_w^{\tiny{\textcircled{\#}}} =bwa_w^{\tiny{\textcircled{\#}}}$, then we have* **(i)* $a_w^{\tiny{\textcircled{\#}}}a=b_w^{\tiny{\textcircled{\#}}}a$.* **(ii)* $awa_w^{\tiny{\textcircled{\#}}}=awb_w^{\tiny{\textcircled{\#}}}$.* **(iii)* $awb_w^{\tiny{\textcircled{\#}}}a=a$.* **(iv)* $b_w^{\tiny{\textcircled{\#}}}a_w^{\tiny{\textcircled{\#}}}=(a_w^{\tiny{\textcircled{\#}}})^2$.* *Proof.* (i) As $awa_w^{\tiny{\textcircled{\#}}}a=a$ and $aw(a_w^{\tiny{\textcircled{\#}}})^2=a_w^{\tiny{\textcircled{\#}}}$, then we have $$\begin{aligned} b_w^{\tiny{\textcircled{\#}}}a&=&b_w^{\tiny{\textcircled{\#}}}awa_w^{\tiny{\textcircled{\#}}}a =b_w^{\tiny{\textcircled{\#}}}bwa_w^{\tiny{\textcircled{\#}}}a =b_w^{\tiny{\textcircled{\#}}}bwaw(a_w^{\tiny{\textcircled{\#}}})^2a\\ &=&b_w^{\tiny{\textcircled{\#}}}bwbw(a_w^{\tiny{\textcircled{\#}}})^2a =bw(a_w^{\tiny{\textcircled{\#}}})^2a=aw(a_w^{\tiny{\textcircled{\#}}})^2a\\ &=&a_w^{\tiny{\textcircled{\#}}}a.\end{aligned}$$ \(ii\) We have $awa_w^{\tiny{\textcircled{\#}}}=awb_w^{\tiny{\textcircled{\#}}}$. Indeed, $$\begin{aligned} awb_w^{\tiny{\textcircled{\#}}}&=&(awa_w^{\tiny{\textcircled{\#}}}a)wb_w^{\tiny{\textcircled{\#}}} =aw(a_w^{\tiny{\textcircled{\#}}}b)wb_w^{\tiny{\textcircled{\#}}} =(awa_w^{\tiny{\textcircled{\#}}})^*(bwb_w^{\tiny{\textcircled{\#}}})^*\\ &=&(bwb_w^{\tiny{\textcircled{\#}}}awa_w^{\tiny{\textcircled{\#}}})^* =(bwb_w^{\tiny{\textcircled{\#}}}bwa_w^{\tiny{\textcircled{\#}}})^*=(bwa_w^{\tiny{\textcircled{\#}}})^* =(awa_w^{\tiny{\textcircled{\#}}})^*\\ &=&awa_w^{\tiny{\textcircled{\#}}}.\end{aligned}$$ \(iii\) By (ii), $awb_w^{\tiny{\textcircled{\#}}}a=awa_w^{\tiny{\textcircled{\#}}}a=a$. \(iv\) Note that $a_w^{\tiny{\textcircled{\#}}}a=b_w^{\tiny{\textcircled{\#}}}a$ in (i). Then $b_w^{\tiny{\textcircled{\#}}}a_w^{\tiny{\textcircled{\#}}} =b_w^{\tiny{\textcircled{\#}}}aw(a_w^{\tiny{\textcircled{\#}}})^2 =a_w^{\tiny{\textcircled{\#}}}aw(a_w^{\tiny{\textcircled{\#}}})^2 =(a_w^{\tiny{\textcircled{\#}}})^2$. $\Box$ ◻ **Theorem 3**. *The relation $a\overset{\tiny{\textcircled{\#}}}\leq_w b$ of Definition $\ref{w core relation}$ is a partial order on $R$.* *Proof.* To prove that $a\overset{\tiny{\textcircled{\#}}}\leq_w b$ is a partial order. It suffices to show (1) reflexivity, i.e., $a\overset{\tiny{\textcircled{\#}}}\leq_w a$, (2) antisymmetry, i.e., $a\overset{\tiny{\textcircled{\#}}}\leq_w b$ and $b\overset{\tiny{\textcircled{\#}}}\leq_w a$ imply $a=b$, (3) transitivity, i.e., $a\overset{\tiny{\textcircled{\#}}}\leq_w b$ and $b\overset{\tiny{\textcircled{\#}}}\leq_w c$ give $a\overset{\tiny{\textcircled{\#}}}\leq_w c$. \(1\) The reflexivity is clear. \(2\) Suppose $a\overset{\tiny{\textcircled{\#}}}\leq_w b$, i.e., $a_w^{\tiny{\textcircled{\#}}} a=a_w^{\tiny{\textcircled{\#}}} b$ and $awa_w^{\tiny{\textcircled{\#}}} =bwa_w^{\tiny{\textcircled{\#}}}$. Then $a=awa_w^{\tiny{\textcircled{\#}}}a=bwa_w^{\tiny{\textcircled{\#}}}a=bwb_w^{\tiny{\textcircled{\#}}}a$ by Lemma [Lemma 2](#w-core par Lemma){reference-type="ref" reference="w-core par Lemma"}(i). Suppose in addition that $b\overset{\tiny{\textcircled{\#}}}\leq_w a$, i.e., $b_w^{\tiny{\textcircled{\#}}} b=b_w^{\tiny{\textcircled{\#}}} a$ and $bwb_w^{\tiny{\textcircled{\#}}} =awb_w^{\tiny{\textcircled{\#}}}$. Then $b=bwb_w^{\tiny{\textcircled{\#}}}b=bwb_w^{\tiny{\textcircled{\#}}}a$, which together with $a=bwb_w^{\tiny{\textcircled{\#}}}a$ give $a=b$. \(3\) Assume $a\overset{\tiny{\textcircled{\#}}}\leq_w b$ and $b \leq_w^{\tiny{\textcircled{\#}}} c$, i.e., $a_w^{\tiny{\textcircled{\#}}} a=a_w^{\tiny{\textcircled{\#}}} b$, $awa_w^{\tiny{\textcircled{\#}}} =bwa_w^{\tiny{\textcircled{\#}}}$, $b_w^{\tiny{\textcircled{\#}}} b=b_w^{\tiny{\textcircled{\#}}} c$ and $bwb_w^{\tiny{\textcircled{\#}}} =cwb_w^{\tiny{\textcircled{\#}}}$. Then, by the equality $awa_w^{\tiny{\textcircled{\#}}}=awb_w^{\tiny{\textcircled{\#}}}$ of Lemma [Lemma 2](#w-core par Lemma){reference-type="ref" reference="w-core par Lemma"}(ii), we get $a_w^{\tiny{\textcircled{\#}}}a=a_w^{\tiny{\textcircled{\#}}}b=a_w^{\tiny{\textcircled{\#}}}bwb_w^{\tiny{\textcircled{\#}}}b =a_w^{\tiny{\textcircled{\#}}}bwb_w^{\tiny{\textcircled{\#}}}c=a_w^{\tiny{\textcircled{\#}}}awb_w^{\tiny{\textcircled{\#}}}c =a_w^{\tiny{\textcircled{\#}}}awa_w^{\tiny{\textcircled{\#}}}c=a_w^{\tiny{\textcircled{\#}}}c$. Similarly, we have $$\begin{aligned} awa_w^{\tiny{\textcircled{\#}}}&=&bwa_w^{\tiny{\textcircled{\#}}}=bwb_w^{\tiny{\textcircled{\#}}}bwa_w^{\tiny{\textcircled{\#}}} =cwb_w^{\tiny{\textcircled{\#}}}bwa_w^{\tiny{\textcircled{\#}}}=cwb_w^{\tiny{\textcircled{\#}}}awa_w^{\tiny{\textcircled{\#}}} =cwa_w^{\tiny{\textcircled{\#}}}awa_w^{\tiny{\textcircled{\#}}}\\ &=&cwa_w^{\tiny{\textcircled{\#}}}.\end{aligned}$$ The proof is completed. $\Box$ ◻ From now on, the partial order $\overset{\tiny{\textcircled{\#}}}\leq_w$ is called the $w$-core partial order. The $w$-core partial order can be seen as an extension of the core partial order [@Baksalary2010]. However, the $w$-core partial order may not imply the core partial order in general. See Example [Example 4](#ex1){reference-type="ref" reference="ex1"} below. Specially, by fixing the element $w\in R$, we in Theorem [Theorem 23](#three class partial orders){reference-type="ref" reference="three class partial orders"} below show that the $w$-core partial order $\overset{\tiny{\textcircled{\#}}}\leq_w$ coincides with the classical star partial order $\overset{*}\leq$ and the core partial order $\overset{\tiny{\textcircled{\#}}}\leq$, respectively. **Example 4**. *Let $R=M_2(\mathbb{C})$ be the ring of all $2 \times 2$ complex matrices and let the involution $*$ be the conjugate transpose. Take, for example, $a=\begin{bmatrix} 1 & 1 \\ 0 & 0 \\ \end{bmatrix}$, $b=\begin{bmatrix} 1 & 1 \\ 2 & -2 \\ \end{bmatrix}$, $w=\begin{bmatrix} 1 & 0 \\ 1 & 0 \\ \end{bmatrix} \in R$, then $a_w^{\tiny{\textcircled{\#}}}= \begin{bmatrix} \frac{1}{2} & 0 \\ 0 & 0 \\ \end{bmatrix}$ and $a^{\tiny{\textcircled{\#}}}= \begin{bmatrix} 1 & 0 \\ 0 & 0 \\ \end{bmatrix}$. We have $a_w^{\tiny{\textcircled{\#}}}a=a_w^{\tiny{\textcircled{\#}}}b= \begin{bmatrix} \frac{1}{2} & \frac{1}{2} \\ 0 & 0 \\ \end{bmatrix}$ and $awa_w^{\tiny{\textcircled{\#}}}=bwa_w^{\tiny{\textcircled{\#}}}= \begin{bmatrix} 1 & 0 \\ 0 & 0 \\ \end{bmatrix}$. Hence, $a\overset{\tiny{\textcircled{\#}}}\leq_w b$. However, $aa^{\tiny{\textcircled{\#}}}= \begin{bmatrix} 1 & 0 \\ 0 & 0 \\ \end{bmatrix} \neq \begin{bmatrix} 1 & 0 \\ 2 & 0 \\ \end{bmatrix}=ba^{\tiny{\textcircled{\#}}}$.* We next give the characterization of the $w$-core partial order $a\overset{\tiny{\textcircled{\#}}}\leq_w b$ in $R$. **Proposition 5**. *Let $a,b,w\in R$ with $a,b\in R_w^{\tiny{\textcircled{\#}}}$. Then the following conditions are equivalent:* **(i)* $a\overset{\tiny{\textcircled{\#}}}\leq_w b$.* **(ii)* $a_w^{\tiny{\textcircled{\#}}}b=b_w^{\tiny{\textcircled{\#}}}a$, $bwa_w^{\tiny{\textcircled{\#}}}=awb_w^{\tiny{\textcircled{\#}}}$ and $awb_w^{\tiny{\textcircled{\#}}}a=a$.* *Proof.* (i) $\Rightarrow$ (ii) It follows from Lemma [Lemma 2](#w-core par Lemma){reference-type="ref" reference="w-core par Lemma"}. \(ii\) $\Rightarrow$ (i) We have $a_w^{\tiny{\textcircled{\#}}}a=a_w^{\tiny{\textcircled{\#}}}awb_w^{\tiny{\textcircled{\#}}}a =a_w^{\tiny{\textcircled{\#}}}awa_w^{\tiny{\textcircled{\#}}}b=a_w^{\tiny{\textcircled{\#}}}b$, and $awa_w^{\tiny{\textcircled{\#}}}=(awb_w^{\tiny{\textcircled{\#}}}a)wa_w^{\tiny{\textcircled{\#}}} =(bwa_w^{\tiny{\textcircled{\#}}})awa_w^{\tiny{\textcircled{\#}}} =bw(a_w^{\tiny{\textcircled{\#}}}awa_w^{\tiny{\textcircled{\#}}})=bwa_w^{\tiny{\textcircled{\#}}}$, as required. $\Box$ ◻ **Proposition 6**. *Let $a,b,w\in R$ with $a,b\in R_w^{\tiny{\textcircled{\#}}}$. If $a\overset{\tiny{\textcircled{\#}}}\leq_w b$, then* **(i)* $a_w^{\tiny{\textcircled{\#}}}bwb_w^{\tiny{\textcircled{\#}}}=b_w^{\tiny{\textcircled{\#}}}bwa_w^{\tiny{\textcircled{\#}}} =a_w^{\tiny{\textcircled{\#}}}bwa_w^{\tiny{\textcircled{\#}}}=a_w^{\tiny{\textcircled{\#}}}$.* **(ii)* $a_w^{\tiny{\textcircled{\#}}}awb_w^{\tiny{\textcircled{\#}}}=b_w^{\tiny{\textcircled{\#}}}awa_w^{\tiny{\textcircled{\#}}}= b_w^{\tiny{\textcircled{\#}}}awb_w^{\tiny{\textcircled{\#}}}=a_w^{\tiny{\textcircled{\#}}}$.* *Proof.* (i) Given $a\overset{\tiny{\textcircled{\#}}}\leq_w b$, then $a_w^{\tiny{\textcircled{\#}}}b=a_w^{\tiny{\textcircled{\#}}} a=b_w^{\tiny{\textcircled{\#}}} a$ and $bwa_w^{\tiny{\textcircled{\#}}}=awa_w^{\tiny{\textcircled{\#}}}=awb_w^{\tiny{\textcircled{\#}}}$ by Lemma [Lemma 2](#w-core par Lemma){reference-type="ref" reference="w-core par Lemma"}. So, $a_w^{\tiny{\textcircled{\#}}}bwa_w^{\tiny{\textcircled{\#}}}=a_w^{\tiny{\textcircled{\#}}}awb_w^{\tiny{\textcircled{\#}}} =a_w^{\tiny{\textcircled{\#}}}awa_w^{\tiny{\textcircled{\#}}}=a_w^{\tiny{\textcircled{\#}}}=b_w^{\tiny{\textcircled{\#}}}awa_w^{\tiny{\textcircled{\#}}} =b_w^{\tiny{\textcircled{\#}}}bwa_w^{\tiny{\textcircled{\#}}}=b_w^{\tiny{\textcircled{\#}}}awb_w^{\tiny{\textcircled{\#}}}= a_w^{\tiny{\textcircled{\#}}}bwb_w^{\tiny{\textcircled{\#}}}$. \(ii\) By (i) and Lemma [Lemma 2](#w-core par Lemma){reference-type="ref" reference="w-core par Lemma"}. $\Box$ ◻ An element $e\in R$ is idempotent if $e=e^2$. If in addition, $e=e^*$, then $e$ is called a projection. It follows from [@Zhu2020] that if $a\in R_w^{\tiny{\textcircled{\#}}}$ then $a_w^{\tiny{\textcircled{\#}}}=w^{\parallel a}a^{(1,3)}$. So, $a_w^{\tiny{\textcircled{\#}}}aw=w^{\parallel a}a^{(1,3)}aw=w^{\parallel a}w$, $wa_w^{\tiny{\textcircled{\#}}}a=ww^{\parallel a}a^{(1,3)}a=ww^{\parallel a}$ and $awa_w^{\tiny{\textcircled{\#}}}=aww^{\parallel a}a^{(1,3)}=aa^{(1,3)}$. Clearly, $a_w^{\tiny{\textcircled{\#}}}aw$ and $wa_w^{\tiny{\textcircled{\#}}}a$ are both idempotents, and $awa_w^{\tiny{\textcircled{\#}}}$ is a projection. In [@Marovt2016 Lemma 2.1], Marovt derived several characterizations for the idempotent $aa^\#$ in a ring. Inspired by Marovt's result, we establish several characterizations for the projection $awa_w^{\tiny{\textcircled{\#}}}$, the idempotents $a_w^{\tiny{\textcircled{\#}}}aw$ and $wa_w^{\tiny{\textcircled{\#}}}a$, respectively. Given any $a\in R$, the symbol $a^{0}=\{x\in R:ax=0\}$ denotes all right annihilators of $a$. Dually, $^{0}a=\{x\in R:xa=0\}$ denotes all left annihilators of $a$. It should be noted that (see, e.g., [@Rakic2014 Lemmas 2.5 and 2.6]) $aR=bR$ implies ${^0}a={^0}b$, and $Ra=Rb$ implies $a^0=b^0$ for any $a,b\in R$. **Lemma 7**. *Let $a,w\in R$ with $a\in R_w^{\tiny{\textcircled{\#}}}$. Then the following conditions are equivalent:* **(i)* $p=awa_w^{\tiny{\textcircled{\#}}}$.* **(ii)* $aR=pR$ for some projection $p\in R$.* **(iii)* ${^0}a={^0}p$ for some projection $p\in R$.* **(iv)* $a=pa$, ${^0}a\subseteq{^0}p$ for some projection $p\in R$.* *Proof.* (i) $\Rightarrow$ (ii) Given $p=awa_w^{\tiny{\textcircled{\#}}}$, then $pR=awa_w^{\tiny{\textcircled{\#}}}R\subseteq aR$. Also, $aR=(awa_w^{\tiny{\textcircled{\#}}}a)R=(pa)R\subseteq pR$. \(ii\) $\Rightarrow$ (iii) is a tautology. \(iii\) $\Rightarrow$ (iv) As $(1-p)p=0$ and ${^0}p={^0}a$, then $(1-p)a=0$ and hence $a=pa$. \(iv\) $\Rightarrow$ (i) Note that $(1-awa_w^{\tiny{\textcircled{\#}}})a=0$. Then we get $(1-awa_w^{\tiny{\textcircled{\#}}})p=0$ and $p=awa_w^{\tiny{\textcircled{\#}}}p=(awa_w^{\tiny{\textcircled{\#}}}p)^*=pawa_w^{\tiny{\textcircled{\#}}}=awa_w^{\tiny{\textcircled{\#}}}$. $\Box$ ◻ **Theorem 8**. *Let $a,b,w\in R$ with $a\in R_w^{\tiny{\textcircled{\#}}}$. Then the following conditions are equivalent:* **(i)* $a_w^{\tiny{\textcircled{\#}}}a=a_w^{\tiny{\textcircled{\#}}}b$.* **(ii)* $w^{\parallel a}=w^{\parallel a}a^{(1,3)}b$.* **(iii)* $a^*a=a^*b$.* **(iv)* $a=awa_w^{\tiny{\textcircled{\#}}}b$.* **(v)* There exists a projection $p\in R$ such that $aR=pR$ and $pa=pb$.* **(vi)* There exists a projection $p\in R$ such that ${^0}p={^0}a$ and $pa=pb$.* *Proof.* (i) $\Rightarrow$ (ii) Note that $a_w^{\tiny{\textcircled{\#}}}=w^{\parallel a}a^{(1,3)}$ and $w^{\parallel a}\in Ra$. Then, there exists some $x\in R$ such that $w^{\parallel a}=xa=xaa^{(1,3)}a=w^{\parallel a}a^{(1,3)}a=a_w^{\tiny{\textcircled{\#}}}a=a_w^{\tiny{\textcircled{\#}}}b=w^{\parallel a}a^{(1,3)}b$. \(ii\) $\Rightarrow$ (iii) Given $w^{\parallel a}=w^{\parallel a}a^{(1,3)}b$, then $a^*a=a^*(aww^{\parallel a})=a^*aw(w^{\parallel a}a^{(1,3)}b)=a^*(aww^{\parallel a})a^{(1,3)}b=a^*aa^{(1,3)}b=(aa^{(1,3)}a)^*b=a^*b$. \(iii\) $\Rightarrow$ (iv) Since $a=(awa_w^{\tiny{\textcircled{\#}}})^*a=(wa_w^{\tiny{\textcircled{\#}}})^*a^*a$, we have $a=(wa_w^{\tiny{\textcircled{\#}}})^*a^*b=(awa_w^{\tiny{\textcircled{\#}}})^*b=awa_w^{\tiny{\textcircled{\#}}}b$. \(iv\) $\Rightarrow$ (v) Write $p=awa_w^{\tiny{\textcircled{\#}}}$, then $p^2=p=p^*$ and $pR=awa_w^{\tiny{\textcircled{\#}}}R\subseteq aR=awa_w^{\tiny{\textcircled{\#}}}aR\subseteq pR$. Also, $pa=a=awa_w^{\tiny{\textcircled{\#}}}b=pb$. \(v\) $\Rightarrow$ (vi) is a tautology. \(vi\) $\Rightarrow$ (i) It follows from Lemma [Lemma 7](#projection){reference-type="ref" reference="projection"} that $p=awa_w^{\tiny{\textcircled{\#}}}$. Then $a=pa=pb=awa_w^{\tiny{\textcircled{\#}}}b$ and consequently $a_w^{\tiny{\textcircled{\#}}}a=a_w^{\tiny{\textcircled{\#}}}awa_w^{\tiny{\textcircled{\#}}}b=a_w^{\tiny{\textcircled{\#}}}b$. $\Box$ ◻ Given any $a,w\in R$ and $a\in R_w^{\tiny{\textcircled{\#}}}$, we next present several characterizations for $a_w^{\tiny{\textcircled{\#}}}aw$ and $wa_w^{\tiny{\textcircled{\#}}}a$ in a ring, respectively. **Lemma 9**. *Let $a,w,e\in R$ with $a\in R_w^{\tiny{\textcircled{\#}}}$. Then the following conditions are equivalent:* **(i)* $e=a_w^{\tiny{\textcircled{\#}}}aw$.* **(ii)* $Re=Raw$ and $ea=a$.* **(iii)* $e^0=(aw)^0$ and $ea=a$.* **(iv)* $(aw)^0\subseteq e^0$ and $ea=a$.* *Proof.* (i) $\Rightarrow$ (ii) Since $e=a_w^{\tiny{\textcircled{\#}}}aw$, we have $ea=a_w^{\tiny{\textcircled{\#}}}awa=a$ and $Re\subseteq Raw=Rawa_w^{\tiny{\textcircled{\#}}}aw=Rawe \subseteq Re$. \(ii\) $\Rightarrow$ (iii) and (iii) $\Rightarrow$ (iv) are clear. \(iv\) $\Rightarrow$ (i) Note that $ea=a$ concludes $ea_w^{\tiny{\textcircled{\#}}}=a_w^{\tiny{\textcircled{\#}}}$ by $aw(a_w^{\tiny{\textcircled{\#}}})^2=a_w^{\tiny{\textcircled{\#}}}$. Also, from $aw(1-a_w^{\tiny{\textcircled{\#}}}aw)=0$ and $(aw)^0\subseteq e^0$, it follows that $e=ea_w^{\tiny{\textcircled{\#}}}aw=a_w^{\tiny{\textcircled{\#}}}aw$. $\Box$ ◻ A similar characterization for the idempotent $wa_w^{\tiny{\textcircled{\#}}}a$ can also be obtained, whose proof is omitted. **Lemma 10**. *Let $a,w,f\in R$ with $a\in R_w^{\tiny{\textcircled{\#}}}$. Then the following conditions are equivalent:* **(i)* $f=wa_w^{\tiny{\textcircled{\#}}}a$.* **(ii)* $Ra=Rf$ and $fwa=wa$.* **(iii)* $a^0=f^0$ and $fwa=wa$.* **(iv)* $a^0\subseteq f^0$ and $fwa=wa$.* The characterization for $awa_w^{\tiny{\textcircled{\#}}}=bwa_w^{\tiny{\textcircled{\#}}}$ can also be derived similarly. First, a preliminary lemma is given. **Lemma 11**. *[@Mary2013 Theorem 2.1] Let $a,w\in R$. Then the following conditions are equivalent:* **(i)* $w\in R^{\parallel a}$.* **(ii)* $a\in awR$ and $aw\in R^\#$.* **(iii)* $a\in Rwa$ and $wa\in R^\#$.* *In this case, $w^{\parallel a}=a(wa)^\#=(aw)^\# a$.* **Theorem 12**. *Let $a,b,w\in R$ with $a\in R_w^{\tiny{\textcircled{\#}}}$. Then the following conditions are equivalent:* **(i)* $awa_w^{\tiny{\textcircled{\#}}}=bwa_w^{\tiny{\textcircled{\#}}}$.* **(ii)* $a=bww^{\parallel a}$.* **(iii)* $awa=bwa$.* **(iv)* $a(wa)^\#=b(wa)^\#$.* **(v)* $a=bwa_w^{\tiny{\textcircled{\#}}}a$.* **(vi)* There exists some $e\in R$ such that $Re=Raw$, $ea=a$ and $awe=bwe$.* **(vii)* There exists some $e\in R$ such that $e^0=(aw)^0$, $ea=a$ and $awe=bwe$.* **(viii)* There exists some $e\in R$ such that $(aw)^0 \subseteq e^0$, $ea=a$ and $awe=bwe$.* **(ix)* There exists some $f\in R$ such that $Ra=Rf$, $af=bf$ and $fwa=wa$.* **(x)* There exists some $f\in R$ such that $a^0=f^0$, $af=bf$ and $fwa=wa$.* **(xi)* There exists some $f\in R$ such that $a^0\subseteq f^0$, $af=bf$ and $fwa=wa$.* *Proof.* (i) $\Rightarrow$ (ii) Given $awa_w^{\tiny{\textcircled{\#}}}=bwa_w^{\tiny{\textcircled{\#}}}$, then $a=awa_w^{\tiny{\textcircled{\#}}}a=bwa_w^{\tiny{\textcircled{\#}}}a=bww^{\parallel a}a^{(1,3)}a=bww^{\parallel a}$. \(ii\) $\Rightarrow$ (iii) As $a=bww^{\parallel a}$, then $awa=(bww^{\parallel a})wa=bw(w^{\parallel a}wa)=bwa$. \(iii\) $\Rightarrow$ (iv) Note that $a\in R_w^{\tiny{\textcircled{\#}}}$ implies $wa\in R^\#$ by Lemma [Lemma 11](#group result){reference-type="ref" reference="group result"} (i) $\Rightarrow$ (iii). Post-multiplying $awa=bwa$ by $((wa)^\#)^2$ gives $a(wa)^\#=b(wa)^\#$. \(iv\) $\Rightarrow$ (v) As $a\in R_w^{\tiny{\textcircled{\#}}}$, then $w^{\parallel a}=a(wa)^\#$ in terms of Lemma [Lemma 11](#group result){reference-type="ref" reference="group result"}. Note that $w^{\parallel a}\in aR \cap Ra$. Then $a_w^{\tiny{\textcircled{\#}}}a=w^{\parallel a}a^{(1,3)}a=w^{\parallel a}$. Thus, $a=w^{\parallel a}wa=a(wa)^\#wa=b(wa)^\#wa=bwa(wa)^\#=bww^{\parallel a}=bwa_w^{\tiny{\textcircled{\#}}}a$. \(v\) $\Rightarrow$ (vi) Write $e=a_w^{\tiny{\textcircled{\#}}}aw$, then $Re\subseteq Raw=R(awa_w^{\tiny{\textcircled{\#}}}a)w=Raw(a_w^{\tiny{\textcircled{\#}}}aw)=Rawe\subseteq Re$ and $ea=a_w^{\tiny{\textcircled{\#}}}awa=a$. Also, $awe=awa_w^{\tiny{\textcircled{\#}}}aw=aw=bwa_w^{\tiny{\textcircled{\#}}}aw=bwe$. \(vi\) $\Rightarrow$ (vii) and (vii) $\Rightarrow$ (viii) are obvious. \(viii\) $\Rightarrow$ (i) It follow from Lemma [Lemma 9](#idempotent){reference-type="ref" reference="idempotent"} that $e=a_w^{\tiny{\textcircled{\#}}}aw$ and $awa_w^{\tiny{\textcircled{\#}}}=awa_w^{\tiny{\textcircled{\#}}}awa_w^{\tiny{\textcircled{\#}}} =awea_w^{\tiny{\textcircled{\#}}}=bwea_w^{\tiny{\textcircled{\#}}}=bwa_w^{\tiny{\textcircled{\#}}}$. \(i\) $\Rightarrow$ (ix) Write $f=wa_w^{\tiny{\textcircled{\#}}}a$, then $Rf\subseteq Ra=Raf\subseteq Rf$, $af=awa_w^{\tiny{\textcircled{\#}}}a=bwa_w^{\tiny{\textcircled{\#}}}a=bf$ and $fwa=(wa_w^{\tiny{\textcircled{\#}}}a)wa=w(a_w^{\tiny{\textcircled{\#}}}awa)=wa$. \(ix\) $\Rightarrow$ (x) and (x) $\Rightarrow$ (xi) are clear. \(xi\) $\Rightarrow$ (i) Given $a^0\subseteq f^0$ and $fwa=wa$, then, by Lemma [Lemma 10](#idempotent0){reference-type="ref" reference="idempotent0"}, $f=wa_w^{\tiny{\textcircled{\#}}}a$ and $fwa_w^{\tiny{\textcircled{\#}}}=fw(aw(a_w^{\tiny{\textcircled{\#}}})^2)=(fwa)w(a_w^{\tiny{\textcircled{\#}}})^2=(wa)w(a_w^{\tiny{\textcircled{\#}}})^2 =w(aw(a_w^{\tiny{\textcircled{\#}}})^2) =wa_w^{\tiny{\textcircled{\#}}}$. Thus, $awa_w^{\tiny{\textcircled{\#}}}=a(fwa_w^{\tiny{\textcircled{\#}}})=(af)wa_w^{\tiny{\textcircled{\#}}}=(bf)wa_w^{\tiny{\textcircled{\#}}}=b(fwa_w^{\tiny{\textcircled{\#}}})=bwa_w^{\tiny{\textcircled{\#}}}$. $\Box$ ◻ Combining with Theorems [Theorem 8](#projection1){reference-type="ref" reference="projection1"} and [Theorem 12](#idempotent1){reference-type="ref" reference="idempotent1"}, we get several characterizations for the $w$-core partial order $a\overset{\tiny{\textcircled{\#}}}\leq_w b$. **Theorem 13**. *Let $a,b,w\in R$ with $a\in R_w^{\tiny{\textcircled{\#}}}$. Then the following conditions are equivalent:* **(i)* $a\overset{\tiny{\textcircled{\#}}}\leq_w b$.* **(ii)* $w^{\parallel a}=w^{\parallel a}a^{(1,3)}b$ and $a=bww^{\parallel a}$.* **(iii)* $a^*a=a^*b$ and $bwa=awa$.* **(iv)* $a^*a=a^*b$ and $b(wa)^\#=a(wa)^\#$.* **(v)* $a=awa_w^{\tiny{\textcircled{\#}}}b=bwa_w^{\tiny{\textcircled{\#}}}a$.* **(vi)* There exist a projection $p\in R$ and an element $e\in R$ such that $aR=pR$, $pa=pb$, $Re=Raw$, $ea=a$ and $awe=bwe$.* **(vii)* There exist a projection $p\in R$ and an element $e\in R$ such that ${^0}p={^0}a$, $pa=pb$, $e^0=(aw)^0$, $ea=a$ and $awe=bwe$.* **(viii)* There exist a projection $p\in R$ and an element $f\in R$ such that $aR=pR$, $pa=pb$, $Ra=Rf$, $af=bf$ and $fwa=wa$.* **(ix)* There exist a projection $p\in R$ and an element $f\in R$ such that ${^0}p={^0}a$, $pa=pb$, $a^0=f^0$, $af=bf$ and $fwa=wa$.* **(x)* There exist a projection $p\in R$ and an element $f\in R$ such that ${^0}p={^0}a$, $pa=pb$, $a^0\subseteq f^0$, $af=bf$ and $fwa=wa$.* **(xi)* There exist a projection $p\in R$ and an element $e\in R$ such that $pb=a=ea$ and $bwe=aw$.* **(xii)* There exist a projection $p\in R$ and an element $e\in R$ such that $pb=a=ea$ and $bwe=awe$.* *Proof.* (i)-(x) are equivalent by Theorems [Theorem 8](#projection1){reference-type="ref" reference="projection1"} and [Theorem 12](#idempotent1){reference-type="ref" reference="idempotent1"}. It next suffices to prove (i) $\Leftrightarrow$ (xi) $\Leftrightarrow$ (xii). \(i\) $\Rightarrow$ (xi) Set $p=awa_w^{\tiny{\textcircled{\#}}}$ and $e=a_w^{\tiny{\textcircled{\#}}} aw$, then $p=p^2=p^*$. It is obvious that $ea=a_w^{\tiny{\textcircled{\#}}} awa=a=awa_w^{\tiny{\textcircled{\#}}}a=awa_w^{\tiny{\textcircled{\#}}}b=pb$. Also, $bwe=bwa_w^{\tiny{\textcircled{\#}}} aw=awa_w^{\tiny{\textcircled{\#}}} aw=aw$. \(xi\) $\Rightarrow$ (xii) is obvious. \(xii\) $\Rightarrow$ (i) It follows from $a=pb$ that $pa=ppb=pb=a$ and hence $a^*b=(pa)^*b=a^*pb=a^*a$. Also, $bwa=bw(ea)=(bwe)a=(awe)a=aw(ea)=awa$. So, $a\overset{\tiny{\textcircled{\#}}}\leq_w b$ by (iii) $\Rightarrow$ (i). $\Box$ ◻ Note the fact that $1^{\parallel a}=aa^\#$ provided that $a\in R^\#$. Set $w=1$ in Theorem [Theorem 13](#char w core){reference-type="ref" reference="char w core"}, then the condition (ii) can be reduced to $aa^\#=aa^\#a^{(1,3)}b$ and $a=ba^\#a$, which are equivalent to $a=aa^{(1,3)}b=ba^\#a$. We hence get several characterizations for the core partial order, some of which were given in [@Rakic2015 Theorems 2.4 and 2.6]. **Corollary 14**. *Let $a,b\in R$ with $a\in R^{\tiny{\textcircled{\#}}}$. Then the following conditions are equivalent:* **(i)* $a\overset{\tiny{\textcircled{\#}}}\leq b$.* **(ii)* $a=aa^{(1,3)}b=ba^\#a$.* **(iii)* $a^*a=a^*b$ and $ba=a^2$.* **(iv)* $a^*a=a^*b$ and $ba^\#=aa^\#$.* **(v)* $a=aa^{\tiny{\textcircled{\#}}}b=ba^{\tiny{\textcircled{\#}}}a$.* **(vi)* There exist a projection $p\in R$ and an element $e\in R$ such that $aR=pR$, $pa=pb$, $Re=Ra$, $ea=a$ and $ae=be$.* **(vii)* There exist a projection $p\in R$ and an element $e\in R$ such that ${^0}p={^0}a$, $pa=pb$, $e^0=(a)^0$, $ea=a$ and $ae=be$.* **(viii)* There exist a projection $p\in R$ and an element $f\in R$ such that $aR=pR$, $pa=pb$, $Ra=Rf$, $af=bf$ and $fa=a$.* **(ix)* There exist a projection $p\in R$ and an element $f\in R$ such that ${^0}p={^0}a$, $pa=pb$, $a^0=f^0$, $af=bf$ and $fa=a$.* **(x)* There exist a projection $p\in R$ and an element $f\in R$ such that ${^0}p={^0}a$, $pa=pb$, $a^0\subseteq f^0$, $af=bf$ and $fa=a$.* **(xi)* There exist a projection $p\in R$ and an element $e\in R$ such that $pb=a=ea$ and $be=a$.* **(xii)* There exist a projection $p\in R$ and an element $e\in R$ such that $pb=a=ea$ and $be=ae$.* # Connections with other partial orders For any $a,b\in R$, recall that the left star partial order $a ~*\leq b$ is defined as $a^*a=a^*b$ and $aR \subseteq bR$. The right sharp partial order $a\leq_\#b$ is defined by $aa^\#=ba^\#$ and $Ra\subseteq Rb$ for $a\in R^\#$. It is known from [@Baksalary2010] that $A\overset{\tiny{\textcircled{\#}}}\leq B$ if and only if $A~*\leq B$ and $A\leq_\#B$, where $A,B$ are $n \times n$ complex matrices of index 1. The characterization for the core partial order indeed holds in a $*$-ring, namely, for any $a,b\in R$ and $a\in R^{\tiny{\textcircled{\#}}}$, then $a\overset{\tiny{\textcircled{\#}}}\leq b$ if and only if $a~*\leq b$ and $a\leq_\#b$. As stated in Section 1, $a \in R_w^{\tiny{\textcircled{\#}}}$ if and only if $w\in R^{\parallel a}$ and $a\in R^{(1,3)}$. In terms of Lemma [Lemma 11](#group result){reference-type="ref" reference="group result"}, one knows that $a\in R_w^{\tiny{\textcircled{\#}}}$ implies $wa\in R^\#$. It is natural to ask whether the $w$-core partial order also has a similar characterization, i.e., if $a\in R_w^{\tiny{\textcircled{\#}}}$, whether $a\overset{\tiny{\textcircled{\#}}}\leq_w b$ is equivalent to $a~*\leq b$ and $wa\leq_\#wb$. The following result gives the implication $a\overset{\tiny{\textcircled{\#}}}\leq_w b \Rightarrow a~*\leq b$ and $wa\leq_\#wb$. For the converse part, there is, of course, a counterexample (see Example [Example 16](#ex2){reference-type="ref" reference="ex2"}) to illustrate that it is not true in general. **Proposition 15**. *For any $a,b,w\in R$ and $a\in R_w^{\tiny{\textcircled{\#}}}$, if $a\overset{\tiny{\textcircled{\#}}}\leq_w b$, then $a~*\leq b$ and $wa\leq_\#wb$.* *Proof.* Given $a\overset{\tiny{\textcircled{\#}}}\leq_w b$, i.e., $a_w^{\tiny{\textcircled{\#}}} a=a_w^{\tiny{\textcircled{\#}}} b$ and $awa_w^{\tiny{\textcircled{\#}}} =bwa_w^{\tiny{\textcircled{\#}}}$, then $a=awa_w^{\tiny{\textcircled{\#}}}a=bwa_w^{\tiny{\textcircled{\#}}}a$ and hence $aR\subseteq bR$. It follows from Theorem [Theorem 13](#char w core){reference-type="ref" reference="char w core"} (i) $\Rightarrow$ (iii) that $a^*a=a^*b$. So, $a~*\leq b$. From $a_w^{\tiny{\textcircled{\#}}} a=a_w^{\tiny{\textcircled{\#}}} b$, we have $R(wa)=R(wawa_w^{\tiny{\textcircled{\#}}}a)=R(wawa_w^{\tiny{\textcircled{\#}}}b)=R(wawa_w^{\tiny{\textcircled{\#}}}(b_w^{\tiny{\textcircled{\#}}}bwb)) \subseteq R(wb)$. Note also that $awa_w^{\tiny{\textcircled{\#}}} =bwa_w^{\tiny{\textcircled{\#}}}$. Then, by Theorem [Theorem 13](#char w core){reference-type="ref" reference="char w core"}, $a(wa)^\#=b(wa)^\#$ and consequently $wa(wa)^\#=wb(wa)^\#$. So, $wa\leq_\#wb$. $\Box$ ◻ **Example 16**. *Let $R$ and the involution be the same as that of Example [Example 4](#ex1){reference-type="ref" reference="ex1"}. Set $a=\begin{bmatrix} 1 & 1 \\ 0 & 0 \\ \end{bmatrix}$, $w=\begin{bmatrix} 1 & 0 \\ 0 & 0 \\ \end{bmatrix}$, $b=\begin{bmatrix} 1 & 1 \\ 2 & 0 \\ \end{bmatrix} \in R$, then $a_w^{\tiny{\textcircled{\#}}}= \begin{bmatrix} 1 & 0 \\ 0 & 0 \\ \end{bmatrix}$. By a direct check, $a^*a=\begin{bmatrix} 1 & 0 \\ 1 & 0 \\ \end{bmatrix}\begin{bmatrix} 1 & 1 \\ 0 & 0 \\ \end{bmatrix}=\begin{bmatrix} 1 & 1 \\ 1 & 1 \\ \end{bmatrix}=\begin{bmatrix} 1 & 0 \\ 1 & 0 \\ \end{bmatrix}\begin{bmatrix} 1 & 1 \\ 2 & 0 \\ \end{bmatrix}=a^*b$ and $aR\subseteq bR$. So, $a~*\leq b$. Note that $wa=wb=\begin{bmatrix} 1 & 1 \\ 0 & 0 \\ \end{bmatrix}$ are idempotent. Then $Rwa\subseteq Rwb$, $(wa)^\#=(wb)^\#=\begin{bmatrix} 1 & 1 \\ 0 & 0 \\ \end{bmatrix}$ and $wa(wa)^\#=\begin{bmatrix} 1 & 1 \\ 0 & 0 \\ \end{bmatrix}=wb(wa)^\#$. So, $wa\leq_\#wb$. However, $awa_w^{\tiny{\textcircled{\#}}}=\begin{bmatrix} 1 & 0 \\ 0 & 0 \\ \end{bmatrix} \neq \begin{bmatrix} 1 & 0 \\ 2 & 0 \\ \end{bmatrix}=bwa_w^{\tiny{\textcircled{\#}}}$. So, $a\overset{\tiny{\textcircled{\#}}}\leq_w b$ does not hold.* It is of interest to consider, under what conditions, $a~*\leq b$ and $wa\leq_\#wb$ can imply $a\overset{\tiny{\textcircled{\#}}}\leq_w b$, provided that $a\in R_w^{\tiny{\textcircled{\#}}}$. The result below shows that the implication is true under the hypothesis $w\in U(R)$, where $U(R)$ denotes the group of all units in $R$. **Theorem 17**. *For any $a,b,w\in R$ and $a\in R_w^{\tiny{\textcircled{\#}}}$, if $w\in U(R)$, then $a\overset{\tiny{\textcircled{\#}}}\leq_w b$ if and only if $a~*\leq b$ and $wa\leq_\#wb$.* *Proof.* We only need to prove the "if" part. Note that $a~*\leq b$ implies $a^*a=a^*b$ and hence $a_w^{\tiny{\textcircled{\#}}}a=a_w^{\tiny{\textcircled{\#}}}b$ in terms of Theorem [Theorem 8](#projection1){reference-type="ref" reference="projection1"}. To show $a\overset{\tiny{\textcircled{\#}}}\leq_w b$, it suffices to prove $awa=bwa$ by Theorem [Theorem 13](#char w core){reference-type="ref" reference="char w core"}. Since $wa(wa)^\#=wb(wa)^\#$, we have $wa=wb(wa)^\#wa=wbwa(wa)^\#=wbww^{\parallel a}$ and hence $a=w^{-1}wa=w^{-1}wbww^{\parallel a}=bww^{\parallel a}$. Post-multiplying $a=bww^{\parallel a}$ by $wa$ implies $awa=bw(w^{\parallel a}wa)=bwa$, as required. $\Box$ ◻ From the proof of Theorem [Theorem 17](#w-core left star right sharp){reference-type="ref" reference="w-core left star right sharp"}, one knows that if $w\in R$ is left invertible, then we also have the equivalence that $a\overset{\tiny{\textcircled{\#}}}\leq_w b$ if and only if $a~*\leq b$ and $wa\leq_\#wb$. With the next theorem we will present more relationship between the $w$-core partial order and the left star partial order in a $*$-ring $R$. **Theorem 18**. *Let $a,b,w\in R$ with $a,b\in R_w^{\tiny{\textcircled{\#}}}$. Then the following conditions are equivalent:* **(i)* $a\overset{\tiny{\textcircled{\#}}}\leq_w b$.* **(ii)* $a~*\leq b$ and $a=bwa_w ^{\tiny{\textcircled{\#}}}b$.* **(iii)* $a~*\leq b$ and $a_w ^{\tiny{\textcircled{\#}}}=b_w ^{\tiny{\textcircled{\#}}}awa_w ^{\tiny{\textcircled{\#}}}$.* **(iv)* $a~*\leq b$ and $a_w ^{\tiny{\textcircled{\#}}}=b_w ^{\tiny{\textcircled{\#}}}awb_w ^{\tiny{\textcircled{\#}}}$.* *Proof.* (i) $\Rightarrow$ (ii) follows from Theorem [Theorem 13](#char w core){reference-type="ref" reference="char w core"} and Proposition [Proposition 15](#w-core imply){reference-type="ref" reference="w-core imply"}. (i) $\Rightarrow$ (iii) and (i) $\Rightarrow$ (iv) by Propositions [Proposition 6](#new add pro 2.6){reference-type="ref" reference="new add pro 2.6"} and [Proposition 15](#w-core imply){reference-type="ref" reference="w-core imply"}. \(ii\) $\Rightarrow$ (i) Given $a~*\leq b$, then $a^*a=a^*b$ and hence $a_w ^{\tiny{\textcircled{\#}}}a=a_w ^{\tiny{\textcircled{\#}}}b$ by Theorem [Theorem 8](#projection1){reference-type="ref" reference="projection1"}. Also, one has $awa_w ^{\tiny{\textcircled{\#}}}=(bwa_w ^{\tiny{\textcircled{\#}}}b)wa_w ^{\tiny{\textcircled{\#}}}=bw(a_w ^{\tiny{\textcircled{\#}}}a)wa_w ^{\tiny{\textcircled{\#}}}=bw(a_w ^{\tiny{\textcircled{\#}}}awa_w ^{\tiny{\textcircled{\#}}})=bwa_w ^{\tiny{\textcircled{\#}}}$. So, $a\overset{\tiny{\textcircled{\#}}}\leq_w b$. \(iii\) $\Rightarrow$ (i) As $a~*\leq b$, i.e., $a^*a=a^*b$ and $aR\subseteq bR$, then there exists some $x\in R$ such that $a=bx=bwb_w ^{\tiny{\textcircled{\#}}}bx=bwb_w ^{\tiny{\textcircled{\#}}}a$. Hence, $awa_w ^{\tiny{\textcircled{\#}}}=(bwb_w ^{\tiny{\textcircled{\#}}}a)wa_w ^{\tiny{\textcircled{\#}}}=bw(b_w ^{\tiny{\textcircled{\#}}}awa_w ^{\tiny{\textcircled{\#}}})=bwa_w ^{\tiny{\textcircled{\#}}}$. Since $a^*a=a^*b$, $a_w ^{\tiny{\textcircled{\#}}}a=a_w ^{\tiny{\textcircled{\#}}}b$ by Theorem [Theorem 8](#projection1){reference-type="ref" reference="projection1"}. \(iv\) $\Rightarrow$ (i) By (ii) $\Rightarrow$ (i), $a_w ^{\tiny{\textcircled{\#}}}a=a_w ^{\tiny{\textcircled{\#}}}b$. Next, it is only need to show $awa_w ^{\tiny{\textcircled{\#}}}=bwa_w ^{\tiny{\textcircled{\#}}}$. Note that $aR \subseteq bR$ implies $a=bwb_w ^{\tiny{\textcircled{\#}}}a$. So, $bwa_w ^{\tiny{\textcircled{\#}}}=bw(b_w ^{\tiny{\textcircled{\#}}}awb_w ^{\tiny{\textcircled{\#}}})=(bwb_w ^{\tiny{\textcircled{\#}}}a)wb_w ^{\tiny{\textcircled{\#}}}=awb_w ^{\tiny{\textcircled{\#}}}$. Hence, we have $$\begin{aligned} awa_w ^{\tiny{\textcircled{\#}}}&=&aw(b_w ^{\tiny{\textcircled{\#}}}awb_w ^{\tiny{\textcircled{\#}}})=awb_w ^{\tiny{\textcircled{\#}}}aw(b_w ^{\tiny{\textcircled{\#}}}bwb_w ^{\tiny{\textcircled{\#}}})\\ &=&aw(b_w ^{\tiny{\textcircled{\#}}}awb_w ^{\tiny{\textcircled{\#}}})bwb_w ^{\tiny{\textcircled{\#}}}=awa_w ^{\tiny{\textcircled{\#}}}bwb_w ^{\tiny{\textcircled{\#}}}\\ &=&aw(a_w ^{\tiny{\textcircled{\#}}}a)wb_w ^{\tiny{\textcircled{\#}}}=awb_w ^{\tiny{\textcircled{\#}}}\\ &=&bwa_w ^{\tiny{\textcircled{\#}}}.\end{aligned}$$ The proof is completed. $\Box$ ◻ Similarly, the connection between the $w$-core partial order and the right sharp partial order can be given. The proof is left to the reader. **Theorem 19**. *Let $a,b,w\in R$ with $a,b\in R_w^{\tiny{\textcircled{\#}}}$. If $w\in U(R)$, then the following conditions are equivalent:* **(i)* $a\overset{\tiny{\textcircled{\#}}}\leq_w b$.* **(ii)* $wa\leq_\# wb$ and $a=bwa_w ^{\tiny{\textcircled{\#}}}b$.* **(iii)* $wa\leq_\# wb$ and $a_w ^{\tiny{\textcircled{\#}}}=a_w ^{\tiny{\textcircled{\#}}}awb_w ^{\tiny{\textcircled{\#}}}$.* As is noted in Example [Example 4](#ex1){reference-type="ref" reference="ex1"}, the $w$-core partial order may not imply the core partial order. We next consider under what conditions the $w$-core partial order gives a core partial order. Herein, a lemma is presented, which was indeed given in [@Zhu2020 Theorem 2.25]. For completeness, we give its proof. **Lemma 20**. *Let $a,w\in R$. Then $a\in R_w^{\tiny\textcircled{\tiny{\#}}}$ if and only if $aR= awR$ and $aw\in R^{\tiny\textcircled{\tiny{\#}}}$. In this case, $a_w^{\tiny\textcircled{\tiny{\#}}}=(aw)^{\tiny\textcircled{\tiny{\#}}}$.* *Proof.* Suppose $a\in R_w^{\tiny\textcircled{\tiny{\#}}}$. Then $w\in R^{\parallel a}$ and hence $a\in awaR\subseteq awR$. Also, there exists some $x\in R$ such that $xawa=a$, $awx^2=x$ and $(awx)^*=awx$, which guarantee $xawaw=aw$, $awx^2=x$ and $(awx)^*=awx$. So $aw\in R^{\tiny\textcircled{\tiny{\#}}}$. Conversely, as $aw\in R^{\tiny\textcircled{\tiny{\#}}}$, then there exists some $y\in R$ such that $awyaw=aw$, $yawy=y$, $y(aw)^2=aw$, $awy^2=y$ and $awy=(awy)^*$. Since $aR=awR$, $a=awt$ for some $t\in R$ and hence $a=awt=(awyaw)t=awya=(awy)^*a=(wy)^*a^*a\in Ra^*a$, i.e., $a\in R^{(1,3)}$. Note the fact that $(aw)^{\tiny\textcircled{\tiny{\#}}}=(aw)^\#aw(aw)^{(1,3)}=w^{\parallel a}w(aw)^{(1,3)}$. To show that $w^{\parallel a}w(aw)^{(1,3)}$ is the $w$-core inverse of $a$, it suffices to prove that $z=w(aw)^{(1,3)}$ is a $\{1,3\}$-inverse of $a$. Since $a\in awR$, it follows that $a=awt$ for some $t\in R$ and $a=aw(aw)^{(1,3)}awt=aw(aw)^{(1,3)}a=aza$. Also, $az=aw(aw)^{(1,3)}=(az)^*$, as required. $\Box$ ◻ **Theorem 21**. *For any $a,b,w\in R$ and $a\in R_w^{\tiny{\textcircled{\#}}}$, if $w\in U(R)$, then the following conditions are equivalent:* **(i)* $a\overset{\tiny{\textcircled{\#}}}\leq_w b$.* **(ii)* $aw\overset{\tiny{\textcircled{\#}}}\leq bw$.* *Proof.* (i) $\Rightarrow$ (ii) Suppose $a\overset{\tiny{\textcircled{\#}}}\leq_w b$, i.e., $a_w^{\tiny{\textcircled{\#}}} a=a_w^{\tiny{\textcircled{\#}}} b$ and $awa_w^{\tiny{\textcircled{\#}}} =bwa_w^{\tiny{\textcircled{\#}}}$. As $a\in R_w^{\tiny{\textcircled{\#}}}$, then $aw\in R^{\tiny{\textcircled{\#}}}$ and $a_w^{\tiny\textcircled{\tiny{\#}}}=(aw)^{\tiny\textcircled{\tiny{\#}}}$ by Lemma [Lemma 20](#Zhulemma){reference-type="ref" reference="Zhulemma"}. So, $aw(aw)^{\tiny{\textcircled{\#}}} =bw(aw)^{\tiny{\textcircled{\#}}}$, $(aw)^{\tiny{\textcircled{\#}}} a=(aw)^{\tiny{\textcircled{\#}}} b$ and hence $(aw)^{\tiny{\textcircled{\#}}} aw=(aw)^{\tiny{\textcircled{\#}}} bw$. So, $aw\overset{\tiny{\textcircled{\#}}}\leq bw$. \(ii\) $\Rightarrow$ (i) As $aw\overset{\tiny{\textcircled{\#}}}\leq bw$, then $(aw)^{\tiny{\textcircled{\#}}} aw=(aw)^{\tiny{\textcircled{\#}}} bw$. Pre-multiplying $(aw)^{\tiny{\textcircled{\#}}} aw=(aw)^{\tiny{\textcircled{\#}}} bw$ by $aw$ gives $aw=aw(aw)^{\tiny{\textcircled{\#}}} bw$. Post-multiplying $aw=aw(aw)^{\tiny{\textcircled{\#}}} bw$ by $w^{\parallel a}$ implies $a=aww^{\parallel a}=aw(aw)^{\tiny{\textcircled{\#}}} bww^{\parallel a}\in awR$. So, $a_w^{\tiny\textcircled{\tiny{\#}}}=(aw)^{\tiny\textcircled{\tiny{\#}}}$ by Lemma [Lemma 20](#Zhulemma){reference-type="ref" reference="Zhulemma"}. Thus, $aw(aw)^{\tiny\textcircled{\tiny{\#}}}=bw(aw)^{\tiny\textcircled{\tiny{\#}}}$ guarantees that $awa_w^{\tiny\textcircled{\tiny{\#}}}=bwa_w^{\tiny\textcircled{\tiny{\#}}}$, and $(aw)^{\tiny{\textcircled{\#}}} aw=(aw)^{\tiny{\textcircled{\#}}} bw$ gives $a_w^{\tiny{\textcircled{\#}}} aw=a_w^{\tiny{\textcircled{\#}}} bw$. Since $w\in U(R)$, $a_w^{\tiny{\textcircled{\#}}} a=a_w^{\tiny{\textcircled{\#}}} b$, as required. $\Box$ ◻ Combining with Theorems [Theorem 17](#w-core left star right sharp){reference-type="ref" reference="w-core left star right sharp"} and [Theorem 21](#w-core and core){reference-type="ref" reference="w-core and core"}, we have the following result. **Theorem 22**. *For any $a,b,w\in R$ and $a\in R_w^{\tiny{\textcircled{\#}}}$, if $w\in U(R)$, then the following conditions are equivalent:* **(i)* $a\overset{\tiny{\textcircled{\#}}}\leq_w b$.* **(ii)* $aw\overset{\tiny{\textcircled{\#}}}\leq bw$.* **(iii)* $a~*\leq b$ and $wa\leq_\#wb$.* We next show that star partial order $\overset{*}\leq$ and the core partial order $\overset{\tiny{\textcircled{\#}}}\leq$ are instances of the $w$-core partial order $\overset{\tiny{\textcircled{\#}}}\leq_w$. **Theorem 23**. *Let $a,b\in R$. Then we have* **(i)* If $a\in R^{\tiny{\textcircled{\#}}}$, then $a\overset{\tiny{\textcircled{\#}}}\leq b$ if and only if $a\overset{\tiny{\textcircled{\#}}}\leq_a b$ if and only if $a\overset{\tiny{\textcircled{\#}}}\leq_1 b$.* **(ii)* If $a\in R^\dag$, then $a\overset{*}\leq b$ if and only if $a\overset{\tiny{\textcircled{\#}}}\leq_{a^*} b$.* *Proof.* (i) It is known that $a\in R_a^{\tiny{\textcircled{\#}}}$ if and only if $a\in R^{\tiny{\textcircled{\#}}}$ if and only if $a\in R_1^{\tiny{\textcircled{\#}}}$. We first show that $a\overset{\tiny{\textcircled{\#}}}\leq b$ if and only if $a\overset{\tiny{\textcircled{\#}}}\leq_a b$. Suppose $a\overset{\tiny{\textcircled{\#}}}\leq b$, i.e., $a^{\tiny{\textcircled{\#}}}a=a^{\tiny{\textcircled{\#}}}b$ and $aa^{\tiny{\textcircled{\#}}}=ba^{\tiny{\textcircled{\#}}}$. Then $a^\#a=a^\#aa^{(1,3)}b$ and $aa^{(1,3)}=ba^\#aa^{(1,3)}$, and consequently $a_a^{\tiny{\textcircled{\#}}}a=a^\#a^{(1,3)}a=(a^\#)^2aa^{(1,3)}a=a^\#a^\#a=a^\#a^\#aa^{(1,3)}b=a^\#a^{(1,3)}b=a_a^{\tiny{\textcircled{\#}}}b$. Similarly, $a^2a_a^{\tiny{\textcircled{\#}}}=a^2a^\#a^{(1,3)}=aa^{(1,3)}=ba^\#aa^{(1,3)}=baa^\#a^{(1,3)}=baa_a^{\tiny{\textcircled{\#}}}$. Conversely, if $a\overset{\tiny{\textcircled{\#}}}\leq_a b$, then $a_a^{\tiny{\textcircled{\#}}}a=a_a^{\tiny{\textcircled{\#}}}b$ and $a^2a_a^{\tiny{\textcircled{\#}}}=baa_a^{\tiny{\textcircled{\#}}}$, that is, $a^\#a^{(1,3)}a=a^\#a^{(1,3)}b$ and $a^2a^\#a^{(1,3)}=baa^\#a^{(1,3)}$. So, $a^{\tiny{\textcircled{\#}}}a=a^\#aa^{(1,3)}a=aa^\#a^{(1,3)}a=aa^\#a^{(1,3)}b=a^{\tiny{\textcircled{\#}}}b$ and $aa^{\tiny{\textcircled{\#}}}=aa^{(1,3)}=a^2a^\#a^{(1,3)}=baa^\#a^{(1,3)}=ba^{\tiny{\textcircled{\#}}}$. So, $a\overset{\tiny{\textcircled{\#}}}\leq b$. Note that $a^{\tiny{\textcircled{\#}}}=a_1^{\tiny{\textcircled{\#}}}$. Then $a\overset{\tiny{\textcircled{\#}}}\leq b$ if and only if $a\overset{\tiny{\textcircled{\#}}}\leq_1 b$. So, the result follows. \(ii\) One knows that $a\in R^\dag$ if and only if $(a^*)^{\parallel a}$ exists if and only if $a\in R_{a^*}^{\tiny{\textcircled{\#}}}$. Moreover, $a_{a^*}^{\tiny{\textcircled{\#}}}=(a^*)^{\parallel a}a^{(1,3)}=(a^\dag)^*a^{(1,3)}=(a^\dag)^*a^\dag$. It is also known that, for any $a\in R^\dag$, $a\overset{*}\leq b$ if and only if $a^\dag a=a^\dag b$ and $aa^\dag=ba^\dag$. We next only show that $a\overset{\tiny{\textcircled{\#}}}\leq_{a^*} b$ if and only if $a^\dag a=a^\dag b$ and $aa^\dag=ba^\dag$. Suppose $a^\dag a=a^\dag b$ and $aa^\dag=ba^\dag$. We hence have $a_{a^*}^{\tiny{\textcircled{\#}}}a=(a^\dag)^*a^\dag a=(a^\dag)^*a^\dag b=a_{a^*}^{\tiny{\textcircled{\#}}}b$, and $aa^*a_{a^*}^{\tiny{\textcircled{\#}}}=aa^*(a^\dag)^*a^\dag=a(a^\dag a)^*a^\dag=aa^\dag=ba^\dag=ba^*(a^\dag)^*a^\dag=ba^*a_{a^*}^{\tiny{\textcircled{\#}}}$. For the converse part, given $a_{a^*}^{\tiny{\textcircled{\#}}}a=a_{a^*}^{\tiny{\textcircled{\#}}}b$ and $aa^*a_{a^*}^{\tiny{\textcircled{\#}}}=ba^*a_{a^*}^{\tiny{\textcircled{\#}}}$, i.e., $(a^\dag)^*a^\dag a=(a^\dag)^*a^\dag b$ and $aa^*(a^\dag)^*a^\dag=ba^*(a^\dag)^*a^\dag$, then $a^\dag a=a^\dag aa^\dag a=(a^\dag a)^*a^\dag a=a^*(a^\dag)^*a^\dag a=a^*(a^\dag)^*a^\dag b=(a^\dag a)^*a^\dag b=a^\dag b$. Similarly, $aa^\dag=a(a^\dag a)^*a^\dag=aa^*(a^\dag)^*a^\dag=ba^*(a^\dag)^*a^\dag=b(a^\dag a)^*a^\dag=ba^\dag$, as required. $\Box$ ◻ An element $a\in R$ is called EP if $a\in R^\#\cap R^\dag$ and $a^\#=a^\dag$. A well known characterization for EP elements is that $a$ is EP if and only if $a\in R^\dag$ and $aa^\dag=a^\dag a$. According to Theorem [Theorem 23](#three class partial orders){reference-type="ref" reference="three class partial orders"}, we have the following result, of which (iii) $\Leftrightarrow$ (iv) $\Leftrightarrow$ (v) were essentially given in [@Baksalary2010]. **Theorem 24**. *Let $a,b\in R$. If $a$ is EP, then the following conditions are equivalent:* **(i)* $a\overset{\tiny{\textcircled{\#}}}\leq_a b$.* **(ii)* $a\overset{\tiny{\textcircled{\#}}}\leq_{a^*} b$.* **(iii)* $a\overset{\tiny{\textcircled{\#}}}\leq b$.* **(iv)* $a\overset{*}\leq b$.* **(v)* $a\overset{\#}\leq b$.* Given any $a,b\in R$ with $a\in R_w^{\tiny{\textcircled{\#}}}$, the $w$-core partial order gives the diamond partial order, i.e., $a\overset{\tiny{\textcircled{\#}}}\leq_w b \Rightarrow a\overset{\diamond}\leq b$. Indeed, given $a\overset{\tiny{\textcircled{\#}}}\leq_w b$, by Theorem [Theorem 13](#char w core){reference-type="ref" reference="char w core"}, we have $a^*a=a^*b=b^*a$ and hence $aa^*a=ab^*a$. Moreover, $a=bwa_w^{\tiny\textcircled{\tiny{\#}}}a=awa_w^{\tiny\textcircled{\tiny{\#}}}b$ imply $aR \subseteq bR$ and $Ra\subseteq Rb$, respectively. For any $a,b,w\in R$ and $a\in R^{\tiny{\textcircled{\#}}} \cap R_w^{\tiny{\textcircled{\#}}}$, we claim that the $w$-core partial order is between the core partial order and the diamond partial order, that is, $a\overset{\tiny{\textcircled{\#}}}\leq b \Rightarrow a\overset{\tiny{\textcircled{\#}}}\leq_w b \Rightarrow a\overset{\diamond}\leq b$. However, the converse implications may not be true in general, i.e., $a\overset{\diamond}\leq b \nRightarrow a\overset{\tiny{\textcircled{\#}}}\leq_w b \nRightarrow a\overset{\tiny{\textcircled{\#}}}\leq b$ (see Examples [Example 4](#ex1){reference-type="ref" reference="ex1"} and [Example 16](#ex2){reference-type="ref" reference="ex2"}). The equivalences between $a\overset{x}\leq b$ and $b-a\overset{x}\leq b$ were considered by several scholars [@Ferreyra2020; @Marovt2016; @Mitra1987], where $\overset{x}\leq$ denotes the minus partial order $\overset{-}\leq$, the star partial order $\overset{*}\leq$ or the sharp partial order $\overset{\#}\leq$. The characterization fails to hold for the core partial order $\overset{\tiny{\textcircled{\#}}}\leq$ (see, e.g., [@Baksalary2010 p. 695]). Recently, Ferreyra and Malik [@Ferreyra2020 Theorem 4.2] derived the equivalence between $A\overset{\tiny{\textcircled{\#}}}\leq B$ and $B-A\overset{\tiny{\textcircled{\#}}}\leq B$ in the ring of all $n\times n$ complex matrices, under certain conditions. More precisely, if $A$, $B$ and $B-A$ are group invertible complex matrices of $n$ by $n$ size, then $A\overset{\tiny{\textcircled{\#}}}\leq B$ and $AB=BA$ if and only if $B-A\overset{\tiny{\textcircled{\#}}}\leq B$ and $AB=BA$ if and only if $A~*\leq B$ and $A\overset{\#}\leq B$. We next give a similar characterization for the $w$-core inverse by a pure algebraic method in a $*$-ring. **Theorem 25**. *For any $a,b,w\in R$ with $awb=bwa$, let $a,b\in R_w^{\tiny{\textcircled{\#}}}$ such that $a-b\in R_w^{\tiny{\textcircled{\#}}}$. Then the following conditions are equivalent:* **(i)* $a\overset{\tiny{\textcircled{\#}}}\leq_w b$.* **(ii)* $b-a\overset{\tiny{\textcircled{\#}}}\leq_w b$.* **(iii)* $a~*\leq b$ and $wa\overset{\#}\leq wb$.* *Proof.* (i) $\Rightarrow$ (ii) Suppose $a\overset{\tiny{\textcircled{\#}}}\leq_w b$. Then, by Theorem [Theorem 13](#char w core){reference-type="ref" reference="char w core"}, $a^*a=a^*b$ and $bwa=awa$. To show $b-a\overset{\tiny{\textcircled{\#}}}\leq_w b$, it suffices to prove $(b-a)^*(b-a)=(b-a)^*b$ and $(b-a)w(b-a)=bw(b-a)$. As $(b-a)^*a=0$, then $(b-a)^*(b-a)=(b-a)^*b-(b-a)^*a=(b-a)^*b$. Similarly, we get $(b-a)w(b-a)=bw(b-a)-aw(b-a)=bw(b-a)$. \(ii\) $\Rightarrow$ (iii) Given $b-a\overset{\tiny{\textcircled{\#}}}\leq_w b$, then $(b-a)^*(b-a)=(b-a)^*b$, which implies $(b-a)^*a=0$ and so $a^*a=a^*b$. Also, $(b-a)w(b-a)=bw(b-a)$ gives $awb=awa$ and $bwa=awa$. Post-multiplying $bwa=awa$ by $w(a_w^{\tiny{\textcircled{\#}}})^2$ gives $bwa_w^{\tiny{\textcircled{\#}}}=awa_w^{\tiny{\textcircled{\#}}}$. Hence, $a=awa_w^{\tiny{\textcircled{\#}}}a=bwa_w^{\tiny{\textcircled{\#}}}a\in bR$ and $a~*\leq b$. Again, it follows from $awa=awb=bwa$ that $wawa=wawb=wbwa$. We hence have $(wa)^\#wa=(wa)^\#wb$ and $wa(wa)^\#=wb(wa)^\#$. So, $wa\overset{\#}\leq wb$. \(iii\) $\Rightarrow$ (i) Note that $a~*\leq b$ implies $a^*a=a^*b$. To prove $a\overset{\tiny{\textcircled{\#}}}\leq_w b$, we only need to verify that $bwa=awa$ by Theorem [Theorem 13](#char w core){reference-type="ref" reference="char w core"}. Since $wa\overset{\#}\leq wb$, $(wa)^\#wa=(wa)^\#wb$ and $(wa)^2=wawb$. Pre-multiplying $(wa)^2=wawb$ by $w^{\parallel a}$ implies $awa=(w^{\parallel a}wa)wa=(w^{\parallel a}wa) wb=awb$, which together with $awb=bwa$ to guarantee $bwa=awa$, as required. $\Box$ ◻ **Remark 26**. *It should be noted that, in Theorem [Theorem 25](#difference w-core){reference-type="ref" reference="difference w-core"} above, the condition (iii) cannot imply the condition (i) without the assumption $awb=bwa$ in general. Indeed, Example [Example 16](#ex2){reference-type="ref" reference="ex2"} can illustrate this fact. However, for the case of $w=1$, the implication (iii) $\Rightarrow$ (i) is clear by the fact that $a\overset{\#}\leq b$ gives $a^2=ab=ba$. * As is given in Theorem [Theorem 23](#three class partial orders){reference-type="ref" reference="three class partial orders"} above, for any $a\in R$, $a\in R_a^{\tiny{\textcircled{\#}}}$ if and only if $a\in R^{\tiny{\textcircled{\#}}}$ if and only if $a\in R_1^{\tiny{\textcircled{\#}}}$. In terms of Theorem [Theorem 23](#three class partial orders){reference-type="ref" reference="three class partial orders"} and Remark [Remark 26](#rmk2){reference-type="ref" reference="rmk2"}, we get the characterization for the core partial order $a\overset{\tiny{\textcircled{\#}}}\leq b$. **Corollary 27**. *For any $a,b\in R$, let $a,b\in R^{\tiny{\textcircled{\#}}}$ such that $a-b\in R^{\tiny{\textcircled{\#}}}$. Then the following conditions are equivalent:* **(i)* $a\overset{\tiny{\textcircled{\#}}}\leq b$ and $ab=ba$.* **(ii)* $b-a\overset{\tiny{\textcircled{\#}}}\leq b$ and $ab=ba$.* **(iii)* $a~*\leq b$ and $a\overset{\#}\leq b$.* Set $w=a$ in Theorem [Theorem 25](#difference w-core){reference-type="ref" reference="difference w-core"}, another characterization for the core partial order $a\overset{\tiny{\textcircled{\#}}}\leq b$ can be obtained as follows. **Corollary 28**. *For any $a,b\in R$ with $a^2b=ba^2$, let $a,b\in R^{\tiny{\textcircled{\#}}}$ such that $a-b\in R^{\tiny{\textcircled{\#}}}$. Then the following conditions are equivalent:* **(i)* $a\overset{\tiny{\textcircled{\#}}}\leq b$.* **(ii)* $b-a\overset{\tiny{\textcircled{\#}}}\leq b$.* **(iii)* $a~*\leq b$ and $a^2\overset{\#}\leq ab$.* For any $a,b\in U(R)$, it is well known that $(ab)^{-1}=b^{-1}a^{-1}$, the formula above is well known as the reverse order law. Reverse order laws for the group inverse, the Moore-Penrose inverse and the core inverse do not hold in general. For the case of the the reverse order for the core inverse, a counterexample was constructed in [@Cohen2012] to show that $(ab)^{\tiny{\textcircled{\#}}}= b^{\tiny{\textcircled{\#}}}a^{\tiny{\textcircled{\#}}}$ does not hold. Later, Malik et al. [@Malik2014] showed the reverse order law for the core inverse of $AB$, under the core partial order $A\overset{\tiny{\textcircled{\#}}}\leq B$, where $A$ and $B$ are two $n \times n$ complex matrices. A natural question is that whether the $w$-core inverse shares the reverse order law property under the $w$-core partial order, i.e., whether $(ab)_w^{\tiny{\textcircled{\#}}}= b_w^{\tiny{\textcircled{\#}}}a_w^{\tiny{\textcircled{\#}}}$ under the $w$-core partial order $a\overset{\tiny{\textcircled{\#}}}\leq_w b$. Example [Example 29](#counterexample reverse){reference-type="ref" reference="counterexample reverse"} below illustrates that the hypothesis is not accurate in general. **Example 29**. *Let $R$ and the involution be the same as that of Example [Example 4](#ex1){reference-type="ref" reference="ex1"}. Take $a=b=\begin{bmatrix} 1 & 1 \\ 0 & 0 \\ \end{bmatrix}$, $w= \begin{bmatrix} 1 & 0 \\ 1& 0 \\ \end{bmatrix}\in R$, then $(ab)_w^{\tiny{\textcircled{\#}}}=a_w^{\tiny{\textcircled{\#}}}=b_w^{\tiny{\textcircled{\#}}}= \begin{bmatrix} \frac{1}{2} & 0 \\ 0& 0 \\ \end{bmatrix}$ by Example [Example 4](#ex1){reference-type="ref" reference="ex1"}. By a direct check, $a_w^{\tiny{\textcircled{\#}}}a=a_w^{\tiny{\textcircled{\#}}}b= \begin{bmatrix} \frac{1}{2} & \frac{1}{2} \\ 0& 0 \\ \end{bmatrix}$ and $awa_w^{\tiny{\textcircled{\#}}}=bwa_w^{\tiny{\textcircled{\#}}}=\begin{bmatrix} 1 & 0 \\ 0& 0 \\ \end{bmatrix}$, i.e., $a\overset{\tiny{\textcircled{\#}}}\leq_w b$. However, $\begin{bmatrix} \frac{1}{2} & 0 \\ 0& 0 \\ \end{bmatrix}=(ab)_w^{\tiny{\textcircled{\#}}} \neq b_w^{\tiny{\textcircled{\#}}} a_w^{\tiny{\textcircled{\#}}}= \begin{bmatrix} \frac{1}{2} & 0 \\ 0& 0 \\ \end{bmatrix}\begin{bmatrix} \frac{1}{2} & 0 \\ 0& 0 \\ \end{bmatrix}= \begin{bmatrix} \frac{1}{4} & 0 \\ 0& 0 \\ \end{bmatrix}$.* It is natural to ask what types of the product has the reverse order law property, under the assumption $a\overset{\tiny{\textcircled{\#}}}\leq_w b$. The following result illustrates that the $w$-core inverse of $awb$ has the reverse order law property. **Theorem 30**. *Let $a,b,w\in R$ with $a,b\in R_w^{\tiny{\textcircled{\#}}}$. If $a\overset{\tiny{\textcircled{\#}}}\leq_w b$, then $awb\in R_w^{\tiny{\textcircled{\#}}}$ and $(awb)_w^{\tiny{\textcircled{\#}}}= b_w^{\tiny{\textcircled{\#}}}a_w^{\tiny{\textcircled{\#}}}$.* *Proof.* It follows from Lemma [Lemma 2](#w-core par Lemma){reference-type="ref" reference="w-core par Lemma"} that $b_w^{\tiny{\textcircled{\#}}}a_w^{\tiny{\textcircled{\#}}}=(a_w^{\tiny{\textcircled{\#}}})^2$. We next show that $x=(a_w^{\tiny{\textcircled{\#}}})^2$ is the $w$-core inverse of $awb$ by the following three steps. \(1\) Note that $bwa=awa$ and $a_w^{\tiny{\textcircled{\#}}}awa=a$. Then $x(awb)w(awb)=(a_w^{\tiny{\textcircled{\#}}})^2aw(bwa)wb=(a_w^{\tiny{\textcircled{\#}}})^2aw(awa)wb =a_w^{\tiny{\textcircled{\#}}}(a_w^{\tiny{\textcircled{\#}}}awa)wawb=(a_w^{\tiny{\textcircled{\#}}}awa)wb=awb$. \(2\) $(awb)wx=awbw(a_w^{\tiny{\textcircled{\#}}})^2=awaw(a_w^{\tiny{\textcircled{\#}}})^2=awa_w^{\tiny{\textcircled{\#}}}=((awb)wx)^*$. \(3\) $(awb)wx^2=(awbwx)x=awa_w^{\tiny{\textcircled{\#}}}(a_w^{\tiny{\textcircled{\#}}})^2=aw(a_w^{\tiny{\textcircled{\#}}})^2 a_w^{\tiny{\textcircled{\#}}}=x$. $\Box$ ◻ **ACKNOWLEDGMENTS** The authors are highly grateful to the referees for their valuable comments and suggestions which greatly improved this paper. This research is supported by the National Natural Science Foundation of China (No. 11801124) and China Postdoctoral Science Foundation (No. 2020M671068). **References** 00 O.M. Baksalary, G. Trenkler, Core inverse of matrices, Linear Multilinear Algebra 58 (2010) 681-697. N. Cohen, E.A. Herman, S. 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Patrício, Generalized inverses modulo $\mathcal{H}$ in semigroups and rings, Linear Multilinear Algebra 61 (2013) 886-891. S.K. Mitra, On group inverses and the sharp order, Linear Algebra Appl. 92 (1987) 17-37. R. Penrose, A generalized inverse for matrices, Proc. Camb. Phil. Soc. 51 (1955) 406-413. D.S. Rakić, N.C. Dinčić, D.S. Djordjević, Group, Moore-Penrose, core and dual core inverse in rings with involution, Linear Algebra Appl. 463 (2014) 115-133. D.S. Rakić, D.S. Djordjević, Star, sharp, core and dual core partial orders in rings with involution, Appl. Math. Comput. 259 (2015) 800-818. S.Z. Xu, J.L. Chen, X.X. Zhang, New characterizations for core inverses in rings with involution, Front. Math. China 12 (2017) 231-246. H.H. Zhu, J.L. Chen, P. Patrício, Further results on the inverse along an element in semigroups and rings, Linear Multilinear Algebra 64 (2016) 393-403. H.H. Zhu, J.L. Chen, P. Patrício, Reverse order law for the inverse along an element, Linear Multilinear Algebra 65 (2017) 166-177. H.H. Zhu, L.Y. Wu, J.L. Chen, $W$-core inverses in rings with involution, (2021), submitted for publication.

arxiv_math

--- abstract: | We study the space $H(\mathcal{SO})$ of all homomorphisms of the vector lattice of all slowly oscillating functions on the half-line $\mathbb{H}=[0,\infty)$. In contrast to the case of homomorphisms of uniformly continuous functions, it is shown that a homomorphism in $H(\mathcal{SO})$ which maps the unit to zero must be zero-homomorphism. Consequently, we show that the space $H(\mathcal{SO})$ without zero-homomorphism is homeomorphic to $\mathbb{H}\times (0, \infty)$. By describing a neighborhood base of zero-homomorphism, we show that $H(\mathcal{SO})$ is homeomorphic to the space $\mathbb{H}\times (0, \infty)$ with one point added. address: Faculty of Fundamental Science, National Institute of Technology (KOSEN), Niihama College, Niihama, 792-8580, Japan author: - Yutaka Iwamoto title: Homomorphisms of the lattice of slowly oscillating functions on the half-line --- # Introduction The aim of this note is to describe the real-valued homomorphisms of the lattice of all slowly oscillating functions on the half-line $\mathbb{H}=[0, \infty)$. Slowly oscillating functions are used to define Higson compactifications [@Keesling] and are functions that appears frequently in coarse geometry. By analyzing slowly oscillating functions on $\mathbb{H}$, it follows that its Higson corona $\nu\mathbb{H}$ is a non-metrizable indecomposable continuum. Although this fact is topologically interesting in its own right, in the context of geometric group theory, it is applied to characterize the number of ends of finitely generated groups by whether the components of its Higson corona are decomposable or not [@Iwa2]. Let $\mathcal{U}$ be the vector lattice of all uniformly continuous functions on $\mathbb{H}$ and $\mathcal{U}^{\ast}$ the sublattice of bounded functions. In [@CS], Félix Cabello Sánchez analyzed the space $H(\mathcal{U})$ of all homomorphisms of $\mathcal{U}$ and gave a fine description of it as follows: $H(\mathcal{U})$ is homeomorphic to a quotient space[^1] obtained from $[1,2]\times \beta \mathbb{N}\times (0,\infty)$ with one point added, where $\beta \mathbb{N}$ denotes the Stone-Čech-compactification of natural numbers. Also, by considering $H(\mathcal{U}^{\ast})$, he gave a description of Samuel-Smirnov compactification of $\mathbb{H}$ (cf. [@FJCS], [@Woods]). Inspired by his work, we study the space $H(\mathcal{SO})$ of all homomorphisms of the vector lattice of slowly oscillating functions on $\mathbb{H}$. In contrast to the case of homomorphisms of uniformly continuous functions, it is shown that a homomorphism in $H(\mathcal{SO})$ which maps the unit to zero must be zero-homomorphism (Proposition [Proposition 11](#3-9){reference-type="ref" reference="3-9"}). Consequently, we show that the space $H(\mathcal{SO})$ without zero-homomorphism is homeomorphic to $\mathbb{H}\times (0, \infty)$. By describing a neighborhood base of zero-homomorphism, we show that $H(\mathcal{SO})$ is homeomorphic to the space $\mathbb{H}\times (0, \infty)$ with one point added (Theorem [Theorem 12](#3-10){reference-type="ref" reference="3-10"}). # Preliminaries Throughout this note, $\mathbb{H}$ denotes the half-line $[0, \infty)$ with the metric given by the absolute value $|x-y|,\ x,y\in \mathbb{H}$, and $\mathbb{N}$ denotes the space of natural numbers with the subspace metric. Also, $X=(X, d)$ is assumed to be a metric space. Let $\mathcal{L}\subset C(X)$ be a unital vector lattice, that is, $\mathcal{L}$ contains the unit $\mathbf{1}: X\to \mathbb{R}$. The sublattice of all bounded functions of $\mathcal{L}$ is denoted by $\mathcal{L}^{\ast}$. A function $\phi :\mathcal{L}\to \mathbb{R}$ is called a *homomorphism* if it is a linear map preserving joins and meets, that is, $\phi$ satisfies 1. $\phi(f\vee g)=\phi(f)\vee \phi(g)$,  $\phi(f\wedge g)=\phi(f)\wedge \phi(g)$, and 2. $\phi (\lambda \cdot f+ \mu \cdot g)= \lambda\cdot\phi (f) +\mu \cdot\phi(g)$ for all $f,g\in \mathcal{L},~\lambda, \mu \in \mathbb{R}$. Note that (i) and (ii) implies 1. $\phi(|f|)=|\phi (f)|$ for all $f\in \mathcal{L}$. Indeed, the formulation $$|f|=f\vee \mathbf{0}-f\wedge \mathbf{0}$$ implies that $$\begin{aligned} \phi(|f|) &=\phi(f)\vee \phi(\mathbf{0})-\phi(f)\wedge\phi(\mathbf{0})\\ &=\phi(f)\vee \mathbf{0}-\phi(f)\wedge \mathbf{0}\\ &=|\phi(f)|.\end{aligned}$$ Recall that join and meet induce a partial order $\leq$ on $H(\mathcal{L})$, that is, $$f\leq g \Longleftrightarrow f=f\wedge g$$ or equivalently, $$f\leq g \Longleftrightarrow g=f\vee g.$$ Then (i) implies that 1. $\phi(f)\leq \phi(g)$ whenever $f\leq g$. Besides, (iii) implies that a homomorphism $\phi$ is *positive*, that is, 1. $\phi(f)\geq 0$ whenever $f\in \mathcal{L}$ satisfies $f\geq 0$. The set of all homomorphisms $\phi :\mathcal{L}\to \mathbb{R}$ is denoted by $H(\mathcal{L})$. Note that $H(\mathcal{L})$ is a subset of $\mathbb{R}^{\mathcal{L}}$. We consider the topology on $H(\mathcal{L})$ inherited from $\mathbb{R}^{\mathcal{L}}$. Hence, a basic neighborhood of $\phi \in H(\mathcal{L})$ is given by $$V(\phi; f_1, \dots, f_n; \varepsilon) =\{\varphi\in H(\mathcal{L}) : |\varphi(f_i)-\phi(f_i)|<\varepsilon,~\forall i=1,\dots, n \},$$ where $\varepsilon >0$ and $f_i \in \mathcal{L}$, $i=1,\dots, n$. Put $$K(\mathcal{L}) =\{\phi \in H(\mathcal{L}): \phi (\mathbf{1})=1\}.$$ Then it is easy to see that $K(\mathcal{L})\subset H(\mathcal{L})$, and $H(\mathcal{L})$ and $K (\mathcal{L})$ are closed subspaces of $\mathbb{R}^{\mathcal{L}}$. In particular, $H(\mathcal{L}^{\ast})$ and $K (\mathcal{L}^{\ast})$ are compact spaces. Indeed, they are closed subspaces of the Cartesian product $$\prod_{f\in \mathcal{L}^{\ast}}\bigl[\inf f, \sup f\bigr].$$ For each $x\in X$, let $\delta_x : \mathcal{L}\to \mathbb{R}$ be the evaluation homomorphism defined by $\delta_x (f)=f(x)$ for every $f\in \mathcal{L}$. We note that $\delta_x (\mathbf{1})=1$ for every $x\in X$, i.e., $\delta_{x}\in K(\mathcal{L})$. Then define $$\delta : X\to K(\mathcal{L})$$ by $\delta(x)=\delta_x$ for each $x\in X$. When we treat $\mathcal{L}^{\ast}$, consider the map $$e_{\mathcal{L}^{\ast}}: X\to \prod_{f\in \mathcal{L}^{\ast}}\bigl[\inf f, \sup f\bigr],$$ defined by $e_{\mathcal{L}^{\ast}}(x)=(f(x))_{f\in \mathcal{L}^{\ast}}$ for every $x\in X$. One should note that two maps $\delta : X\to K(\mathcal{L}^{\ast})\subset \mathbb{R}^{\mathcal{L}}$ and $e_{\mathcal{L}^{\ast}}: X\to \prod_{f\in \mathcal{L}^{\ast}}[\inf f, \sup f]\subset \mathbb{R}^{\mathcal{L}}$ are essentially the same correspondence. A unital vector lattice $\mathcal{L}\subset C(X)$ is said to *separate points and closed sets* in $X$ provided that, for each close set $F\subset X$ and each point $p\in X\setminus F$, there exists $f\in \mathcal{L}$ such that $f(p)\not\in \mbox{cl}_X F$. The following is a fundamental fact concerning $K(\mathcal{L})$ (see [@Garrido-Jaramillo pp. 129--130], [@PW 1.7 (j)]). **Proposition 1**. *If $\mathcal{L}$ separates points and closed sets in $X$, then $\delta : X\to K(\mathcal{L})$ is a dense topological embedding.* Though $K(\mathcal{L})$ is not compact in general, it can be considered as a realcompactification of $X$ by Proposition [Proposition 1](#2-1){reference-type="ref" reference="2-1"}. See [@Garrido-Jaramillo] for more information about realcompactifications. Let $\mathcal{U}(X)$ denote the lattice of all real-valued uniformly continuous functions on $X$. We write $\mathcal{U}$ (resp. $\mathcal{U}^{\ast}$) instead of $\mathcal{U}(\mathbb{H})$ (resp. $\mathcal{U}(\mathbb{H})^{\ast}$) for notational simplicity. The family $\mathcal{U}^{\ast}(X)$ has a ring structure with respect to $\mathbb{R}$, but $\mathcal{U}(X)$ does not. Therefore, when considering unbounded vector lattices, we need to consider lattice homomorphisms instead of ring homomorphisms. Let $\alpha X$ and $\gamma X$ be compactifications of $X$. We say $\alpha X \succeq \gamma X$ provided that there is a continuous map $f:\alpha X \to \gamma X$ such that $f|_{X}=\mbox{id}_X$. If $\alpha X \preceq \gamma X$ and $\alpha X \succeq \gamma X$ then we say that $\alpha X$ and $\gamma X$ are *equivalent compactifications* of $X$. Of course, two equivalent compactifications of $X$ are homeomorphic. It is easy to check that $\mathcal{U}^{\ast}(X)$ contains all constant maps, separates points from closed sets, and is a closed subring of $C^{\ast} (X)$ with respect to the sup-metric, i.e., $\mathcal{U}^{\ast}(X)$ is a *complete ring on functions* (see [@Eng 3.12.22(e)]). Hence, $\mathcal{U}^{\ast}(X)$ uniquely determines a compactification $u X$ of $X$ (see [@Eng 3.12.22 (e)], [@PW 4.5]), which is called the *Samuel-Smirnov compactification* of $X$ (see [@CS], [@Woods]). We note that $uX$ is equivalent to $K(\mathcal{U}^{\ast}(X))= \mbox{cl}_{\mathbb{R}^{\mathcal{U}^{\ast}}}\delta (X)$ because of the equivalence of two maps $\delta :X\to K(\mathcal{U}^{\ast})$ and $e_{\mathcal{U}^{\ast}}: X\to \prod_{f\in \mathcal{L}^{\ast}}\bigl[\inf f, \sup f\bigr]$. Let $(X, d_X)$ be a metric space and let $B_{d_X}(x,r)$ be the closed ball of radius $r$ centered at $x\in X$. A metric $d_X$ on $X$ is called *proper* if $B_{d_X}(x, r)$ is compact for every $x\in X$ and $r>0$. Let $(X,d_X )$ and $(Y,d_Y)$ be proper metric spaces. A map $f: X\to Y$ is said to be *slowly oscillating* provided that, given $R>0$ and $\varepsilon>0$, there exists a compact subset $K\subset X$ such that $$\operatorname{diam}_{d_Y} f(B_{d_X}(x,R))<\varepsilon$$ for every $x\in X\setminus K$, where $\operatorname{diam}_{d} A=\sup \{ d(x,y): x,y\in A\}$. Let $\mathcal{SO}(X)$ denote the lattice of all real-valued slowly oscillating continuous functions on a proper metric space $X$. The sublattice of all bounded functions of $\mathcal{SO}(X)$ is denoted by $\mathcal{SO}(X)^{\ast}$. When $X=\mathbb{H}$ we just write $\mathcal{SO}$ (resp. $\mathcal{SO}^{\ast}$) instead of $\mathcal{SO}(\mathbb{H})$ (resp. $\mathcal{SO}(\mathbb{H})^{\ast}$) for notational simplicity. It is easy to check that $\mathcal{SO}^{\ast}(X)$ is a closed subring of $C^{\ast}(X)$ with respect to the sup-metric, namely, a complete ring on functions. Hence, $\mathcal{SO}^{\ast}(X)$ uniquely determines a compactification $hX$ of $X$, which is called the *Higson compactification* of $X$. The remainder $\nu X=hX \setminus X$ is called the *Higson corona* of $X$ (cf. [@Roe], [@Keesling]). We note that $\nu X$ is compact and that $hX$ and $K(\mathcal{SO}^{\ast}(X))$ are equivalent compactifications of $X$. **Proposition 2**. *If $(X, d)$ is a proper metric space, then $\mathcal{SO}(X)\subset \mathcal{U}(X)$.* *Proof.* Let $f\in \mathcal{SO}(X)$. Given $\varepsilon>0$, there exists a compact subset $K\subset X$ such that $\operatorname{diam}f(B(x, 1))<\varepsilon$ whenever $x\in X\setminus K$. Put $K'=\mbox{cl}\, B(K,1)$. Since $X$ is a proper metric space, $K'$ is compact. Consider a family $$\mathscr{U}=\{ f^{-1}(B(f(x), \varepsilon/2)): x\in K'\}.$$ Since $K'\subset \bigcup \mathscr{U}$, we can take a Lebesgue number $\delta_0 >0$ of $\mathscr{U}$, that is, every $\delta_0$-neighborhood of $x\in K'$ is contained in some element of $\mathscr{U}$. Let $\delta=\min\{\delta_0, 1\}$. Then $d(x,y)<\delta$ implies that $x,y\in K'$ or $x,y\in X\setminus K$. Hence, $d(f(x), f(y))<\varepsilon$ whenever $d(x,y)<\delta$. ◻ # Homomorphisms of the lattice of slowly oscillating functions on the half-line Let $\tau : \mathbb{H}\to \mathbb{R}$ be the map defined by $$\tau (x)=x+1$$ for every $x\in \mathbb{H}$. One should note that $\tau^{\alpha}\in \mathcal{SO}$ for every $0<\alpha<1$. For each $f\in \mathcal{SO}$, we consider the map $f_{\ast}: H(\mathcal{SO})\to \mathbb{R}$ defined by $$f_{\ast}(\phi)=\phi (f)$$ for every $\phi\in H(\mathcal{SO})$. Recall that a basic neighborhood of $\phi \in H(\mathcal{SO})$ is of the form $$V(\phi; f_1, \dots, f_n; \varepsilon) =\{\varphi\in H(\mathcal{SO}) : |\varphi(f_i)-\phi(f_i)|<\varepsilon,~\forall i=1,\dots, n \},$$ where $\varepsilon >0$ and $f_i \in \mathcal{SO}$, $i=1,\dots, n$. Now it is easy to see that $f_{\ast}$ is continuous. **Proposition 3**. *$K(\mathcal{SO})=\delta(\mathbb{H})$.* *Proof.* It is obvious that $\delta(\mathbb{H})\subset K(\mathcal{SO})$. We shall show that $K(\mathcal{SO})\subset \delta(\mathbb{H})$. Let $\phi\in K(\mathcal{SO})$. Note that $\delta(\mathbb{H})$ is dense in $K(\mathcal{SO})$ by Proposition [Proposition 1](#2-1){reference-type="ref" reference="2-1"}. Thus we can take a net $(x_{\alpha})_{\alpha}$ in $\mathbb{H}$ such that $(\delta_{x_{\alpha}})_{\alpha}$ converges to $\phi$. For each $f\in \mathcal{SO}$, the net $(f_{\ast}(\delta_{x_{\alpha}}))_{\alpha}=(\delta_{x_{\alpha}}(f))_{\alpha}=(f(x_{\alpha}))_{\alpha}$ converges to $f_{\ast}(\phi)=\phi(f)$ because $f_{\ast}$ is continuous, that is, $$\phi (f)=\lim_{\alpha} f(x_{\alpha}).$$ Taking $f=\sqrt{\tau}\in \mathcal{SO}$, we have $$\phi(\sqrt{\tau})= \lim_{\alpha} \sqrt{x_{\alpha}+1}.$$ Put $x_{\phi}=(\phi (\sqrt{\tau}))^2 -1$. Then we have $x_{\phi}=\lim_{\alpha}x_{\alpha}$. Hence, we conclude that $\phi =\delta_{x_{\phi}}\in \delta(\mathbb{H})$, i.e., $K(\mathcal{SO})\subset \delta(\mathbb{H})$. ◻ **Corollary 4**. *For each $\phi\in H(\mathcal{SO})$, $\phi(\mathbf{1} )>0$ if and only if there exist $x_{\phi}\in \mathbb{H}$ and $c>0$ such that $\phi =c\cdot\delta_{x_{\phi}}$. In particular, if $\phi(\mathbf{1} )>0$ then the point $x_{\phi}\in\mathbb{H}$ is uniquely determined.* *Proof.* If $\phi (\mathbf{1})>0$ then $\phi(\mathbf{1})^{-1}\cdot\phi \in K(\mathcal{SO})$. By Proposition [Proposition 3](#3-1){reference-type="ref" reference="3-1"}, there exists $x_{\phi}\in \mathbb{H}$ such that $\phi(\mathbf{1})^{-1}\cdot\phi =\delta_{x_{\phi}}$, i.e., $\phi =\phi(\mathbf{1})\cdot\delta_{x_{\phi}}$. The reverse implication is trivial. Suppose that $\phi(\mathbf{1})>0$ and $\phi=\phi(\mathbf{1})\cdot\delta_{s}=\phi(\mathbf{1})\cdot\delta_{t}$ for some $s, t\in \mathbb{H}$ then the equation $\phi(\tau)=\phi(\mathbf{1})\cdot(s+1)=\phi(\mathbf{1})\cdot(t+1)$ implies that $s=t$. ◻ The following two lemmas are modifications of those stated in [@CS 2.2]. **Lemma 5**. *Let $f \in \mathcal{SO}$ be a map such that $f\geq \mathbf{1}$. If there exits $\phi\in H(\mathcal{SO})$ such that $\phi(f)=1$ and $\phi(\mathbf{1})=0$, then $\phi$ is contained in the closure of $\{ f(n)^{-1} \cdot \delta_n : n \in \mathbb{N}\}$ in $H(\mathcal{SO})$.* *Proof.* Suppose that there exists $\phi\in H(\mathcal{SO})$ such that $\phi(f)=1$ and $\phi(\mathbf{1})=0$ but which is not contained in the closure of $\{ f(n)^{-1}\cdot \delta_n : n \in \mathbb{N}\}$ in $H(\mathcal{SO})$. Then there exist $\varepsilon >0$ and $g_1,\dots, g_k \in\mathcal{SO}$ such that $f(n)^{-1} \cdot \delta_n \not\in V(\phi ; g_1, \dots, g_k; \varepsilon)$ for every $n\in \mathbb{N}$. So, for each $n\in \mathbb{N}$, there exists $i\in \{1,\dots, k\}$ such that $$\left| \phi (g_i )-f(n)^{-1} \cdot g_i(n)\right|\geq \varepsilon.$$ Hence, we have $$\bigvee_{i=1}^{k} \left| \phi (g_i)\cdot f(n) -g_i (n) \right| \geq \varepsilon \cdot f(n)$$ for every $n\in \mathbb{N}$. Let $c_i =\phi(g_i)$ for each $i=1,\dots, k$. Put $$h=0\wedge \left(\bigvee_{i=1}^{k} \left|c_i \cdot f -g_i \right|-\varepsilon \cdot f \right).$$ Then $h\in \mathcal{SO}\subset \mathcal{U}$ and $h(n)=0$ for every $n\in \mathbb{N}$. It follows from uniformity that $h$ is a bounded function. So, there exists $c>0$ such that $|h|\leq c\cdot \mathbf{1}$. Thus, we have $|\phi(h)|=\phi(|h|)\leq c\cdot \phi (\mathbf{1})=0$, i.e., $\phi(h)=0$. We note that $$\bigvee_{i=1}^{k} \left| c_i \cdot f -g_i \right| \geq h+\varepsilon \cdot f$$ and $$\phi\left(\bigvee_{i=1}^{k} \left|c_i \cdot f -g_i \right|\right) =\bigvee_{i=1}^{k} \left|c_i \cdot \phi(f) -\phi(g_i) \right| =0.$$ However, we have $\phi(h+\varepsilon \cdot f )=\phi(h)+\varepsilon \cdot \phi(f)=\varepsilon>0$, a contradiction. ◻ Let $\mathscr{F}$ be an ultrafilter on $\mathbb{N}_{0}=\mathbb{N}\cup\{0 \}$. Then we define the operation $\lim_{\mathscr{F}(n)}$ by $$\lim_{\mathscr{F}(n)} f(n)=\bigcap_{F\in \mathscr{F}} \mbox{cl}\left\{ f(n): n\in F\right\}$$ for every $f\in C(\mathbb{H})$ (cf. [@CS]). If the set $\displaystyle\lim_{\mathscr{F}(n)} f(n)$ consists of a single point then we will consider it to be the value of $\mathbb{R}$. Recall that every element of the Stone-Čech compactification $\beta\mathbb{N}_{0}$ of $\mathbb{N}_{0}$ can be considered as the space of all ultrafilters on $\mathbb{N}_{0}$. **Lemma 6**. *Let $f \in \mathcal{SO}$ be a map such that $f\geq \mathbf{1}$. Suppose that there exists a homomorphism $\phi\in H(\mathcal{SO})$ such that $\phi( f)=1$ and $\phi(\mathbf{1})=0$. Then there exists a free ultrafilter $\mathscr{F}$ such that the functions $\phi_{\mathscr{F}}^{f} : \mathcal{SO}\to \mathbb{R}$ defined by $$\phi_{\mathscr{F}}^{f}(g) =\lim_{\mathscr{F}(n)} \frac{g(n)}{ f(n)} , ~~~(g\in \mathcal{SO})$$ is a well-defined homomorphism with $\phi_{\mathscr{F}}^{f} =\phi$.* *Proof.* Suppose that there exists a homomorphism $\phi\in H(\mathcal{SO})$ such that $\phi( f)=1$ and $\phi(\mathbf{1})=0$ for some $f\in \mathcal{SO}$ with $f\geq \mathbf{1}$. For each neighborhood $V$ of $\phi$ in $H(\mathcal{SO})$, let $N_{V}=\left\{ n\in \mathbb{N}: f(n)^{-1} \cdot \delta_n \in V \right\}$. Then $N_{V}\neq \emptyset$ by Lemma [Lemma 5](#3-3){reference-type="ref" reference="3-3"}. Put $$\mathscr{G}=\{N_{V} : V~\mbox{is a neiborhood of}~\phi\}.$$ Then $\mathscr{G}$ becomes a filter on $\mathbb{N}$. Let $\mathscr{F}$ be an ultrafilter on $\mathbb{N}$ refining $\mathscr{G}$. We note that $\mathscr{F}$ must be a free ultrafilter since $\phi(\mathbf{1})=0$. Indeed, if $\mathscr{G}$ is a fixed ultrafilter, say $\lim \mathscr{G}=n_0$, then $\phi = f(n_0)^{-1}\delta_{n_0}$. Hence, we have $\phi(\mathbf{1})=f(n_0)^{-1}\neq 0$, a contradiction. Given $\varepsilon>0$ and $g\in \mathcal{SO}$, we consider a neighborhood $V_{\varepsilon}=V(\phi;g;\varepsilon)$ of $\phi$ in $H(\mathcal{SO})$, that is, $$V_{\varepsilon} =\{\varphi\in H(\mathcal{SO}) : |\varphi(g)-\phi(g)|<\varepsilon \}.$$ Since $\left\{ n\in \mathbb{N}: f(n)^{-1} \cdot \delta_n \in V_{\varepsilon} \right\} \in \mathscr{G}\subset \mathscr{F}$, we have $$\phi_{\mathscr{F}}^{f}(g) =\bigcap_{F\in \mathscr{F}} \mbox{cl}\left\{ \frac{g(n)}{ f(n)} : n\in F\right\} \subset \bigcap_{G\in \mathscr{G}} \mbox{cl}\left\{ \frac{g(n)}{ f(n)} : n\in G\right\}\subset B(\phi(g), \varepsilon).$$ Then $\phi_{\mathscr{F}}^{f}(g)$ is not empty by the compactness of $B(\phi(g), \varepsilon)$ and it is uniquely determined because $\mathscr{F}$ is an ultrafilter. Since $\varepsilon$ is arbitrary, it follows that $\phi_{\mathscr{F}}^{f}(g)=\phi(g)$, that is, $\phi_{\mathscr{F}}^{f}$ is a well-defined homomorphism with $\phi_{\mathscr{F}}^{f} =\phi$. ◻ **Lemma 7**. *Let $\phi \in H(\mathcal{SO})$. If there are two maps $f, g \in \mathcal{SO}$ such that $\mathbf{1}\leq f\leq g$ and $\displaystyle\lim_{n\to \infty}f(n)^{-1} \cdot g(n)=\infty$, then the condition $\phi(\mathbf{1})=0$ implies that $\phi (f)=0$.* *Proof.* Let $f, g \in \mathcal{SO}$ be such that $\mathbf{1}\leq f\leq g$ and $\lim_{n\to \infty}f(n)^{-1} \cdot g(n)=\infty$. Let $\phi\in H(\mathcal{SO})$ be such that $\phi (\mathbf{1})=0$. Suppose that $\phi(f)\neq 0$. Replacing $\phi$ by $\phi(f)^{-1} \cdot \phi$, we may assume that $\phi (f)=1$. Then, by Lemma [Lemma 6](#3-4){reference-type="ref" reference="3-4"}, there exists a free ultrafilter $\mathscr{F}$ such that $\phi=\phi_{\mathscr{F}}^{f}$. However, since $\mathscr{F}$ is a free ultrafilter, we have $$\phi (g)=\phi_{\mathscr{F}}^{f}(h) =\lim_{\mathscr{F}(n)} \frac{g(n)}{ f(n)} =\infty ,$$ a contradiction. ◻ **Definition 8**. A sequence $\mathfrak{a}=(a_n)\subset \mathbb{N}$ is called a *strictly increasing sequence* provided that $a_n <a_{n+1}$ for every $n\in \mathbb{N}$. Note that if $\mathfrak{a}=(a_n)$ is a strictly increasing sequence then $\displaystyle\lim_{n\to \infty}a_n =\infty$ since $\mathfrak{a}\subset \mathbb{N}$. Let $\mathfrak{a}$ be a strictly increasing sequence. Let $\eta_{\mathfrak{a}}^{0}=\tau: \mathbb{H}\to \mathbb{R}$. Suppose that $\eta_{\mathfrak{a}}^{n-1}$ has been defined for $n\geq 1$. Then we define $\eta_{\mathfrak{a}}^n : \mathbb{H}\to \mathbb{R}$ by $$\eta_{\mathfrak{a}}^{n}(x)=\left\{ \begin{array}{ll} \eta_{\mathfrak{a}}^{n-1}(x),& 0\leq x< a_{n} ,\\[1mm] \eta_{\mathfrak{a}}^{n-1}(a_{n})+\frac{1}{n}\left(x-a_{n}\right), & a_{n}\leq x, \end{array} \right.$$ for every $x\in \mathbb{H}$ (see Figure [\[fig1\]](#fig1){reference-type="ref" reference="fig1"}). Note that $\eta_{\mathfrak{a}}^{n-1}\geq \eta_{\mathfrak{a}}^{n} \geq 1$ for every $n\in \mathbb{N}$. We define $\eta_{\mathfrak{a}}:\mathbb{H}\to \mathbb{R}$ by $$\eta_{\mathfrak{a}} (x)= \lim_{n\to \infty} \eta_{\mathfrak{a}}^{n} (x)$$ for every $x\in \mathbb{H}$. We note that if $x\leq a_n$ then $$\eta_{\mathfrak{a}}(x)=\eta_{\mathfrak{a}}^{n}(x)=\eta_{\mathfrak{a}}^{n-1}(x).$$ It is easy to see that $\eta_{\mathfrak{a}} :\mathbb{H}\to \mathbb{R}$ is a well-defined slowly oscillating continuous function with $\eta_{\mathfrak{a}}\geq 1$. We call $\eta_{\mathfrak{a}}$ the *slowly oscillating function with respect to $\mathfrak{a}$*. **Proposition 9**. *For each $f\in \mathcal{SO}$, there exists a strictly increasing sequence $\mathfrak{a}\subset\mathbb{N}$ and $L>0$ such that $|f|\leq L\cdot \eta_{\mathfrak{a}}$.* *Proof.* Since $f\in \mathcal{SO}$, we can take a strictly increasing sequence $\mathfrak{a}=(a_n)\subset\mathbb{N}$ such that 1. $\operatorname{diam}f(B(x, 1))<(n+1)^{-4}$ for every $x\geq a_n$. Let $L=1+\sup\{ |f(x)|: x\leq a_1\}$. Then we have $$|f(x)|+1 \leq L\leq L\cdot \tau(x) =L\cdot \eta_{\mathfrak{a}}^{0}(x)$$ for every $x\leq a_{1}$. Suppose that we have shown that 1. $|f(x)|+n^{-2}\leq L\cdot \eta_{\mathfrak{a}}^{n-1}(x)$ for every $x\leq a_{n}$. If $x\leq a_{n}$ then $(2)_{n+1}$ follows from $(2)_n$ since $|f(x)|+(n+1)^{-2}\leq|f(x)|+n^{-2}$ and $\eta_{\mathfrak{a}}^{n}(x) =\eta_{\mathfrak{a}}^{n-1}(x)$. Now suppose that $a_{n}\leq x\leq a_{n+1}$. Then we have $$\begin{aligned} |f(x)|+\frac{1}{(n+1)^2} &\leq |f(a_{n})|+\frac{x-a_n}{(n+1)^4} +\frac{1}{(n+1)^4}+\frac{1}{(n+1)^2}\qquad (\mbox{by }(1))\\ &< |f(a_{n})|+\frac{x-a_n}{n+1} +\frac{1}{n^2}\\ &\leq L\cdot \eta_{\mathfrak{a}}^{n}(a_{n})+\frac{x-a_n}{n+1} \qquad (\mbox{by }(2)_n )\\ &\leq L\cdot \eta_{\mathfrak{a}}^{n-1}(a_{n})+\frac{x-a_n}{n} \qquad (\because~\eta_{\mathfrak{a}}^{n}(a_{n})=\eta_{\mathfrak{a}}^{n-1}(a_{n}))\\ &\leq L\cdot \left( \eta_{\mathfrak{a}}^{n-1}(a_{n})+\frac{x-a_n}{n} \right) \qquad (\because~ L\geq 1)\\ &=L\cdot \eta_{\mathfrak{a}}^{n}(x). \end{aligned}$$ Thus $(2)_{n+1}$ holds. Consequently, we have $|f|\leq L\cdot \eta_{\mathfrak{a}}$ since $\displaystyle\lim_{n\to \infty}\eta_{\mathfrak{a}}^{n} =\eta_{\mathfrak{a}}$ and $\displaystyle\lim_{n\to \infty} a_n =\infty$. ◻ **Proposition 10**. *For each strictly increasing sequence $\mathfrak{a}\subset \mathbb{N}$, there exists a strictly increasing sequence $\mathfrak{b}\subset \mathbb{N}$ such that $\eta_{\mathfrak{a}}\leq \eta_{\mathfrak{b}}$ and $\displaystyle\lim_{n\to \infty}\eta_{\mathfrak{a}}(n)^{-1} \cdot \eta_{\mathfrak{b}}(n)=\infty$.* *Proof.* Let $\mathfrak{a}=(a_n)\subset \mathbb{N}$ be a strictly increasing sequence. Let $\mathfrak{b}=(b_n)$ be a strictly increasing sequence such that 1. $b_0 =a_1$ and 2. $b_n \geq n^2 \cdot \eta_{\mathfrak{a}}(b_{n-1})+b_{n-1} +a_{(n+1)^3}$ for each $n\in \mathbb{N}$. We shall show that $\eta_{\mathfrak{b}}(x)\geq n\cdot \eta_{\mathfrak{a}}(x)$ for every $x\in [b_{n}, b_{n+1}]$. Since $b_i > a_i$ for $i=1,2$, we have $\eta_{\mathfrak{b}}(x)\geq 1\cdot\eta_{\mathfrak{a}}(x)$ for every $x\in [b_1, b_2]$. Suppose that we have shown that 1. $\eta_{\mathfrak{b}}(x)\geq (n-1)\cdot\eta_{\mathfrak{a}}(x)$ whenever $x\in [b_{n-1}, b_{n}]$ for $n\geq 2$. Let $x\in [b_n, b_{n+1}]$. We write $x=b_{n-1}+t$, $t> 0$ for some technical reason. Then we have $$\frac{t}{n^2}>\eta_{\mathfrak{a}}(b_{n-1}) \tag{4}$$ Indeed, since $x=b_{n-1}+t \geq b_n$, we have $$\begin{aligned} t&\geq b_n -b_{n-1}\\ &> n^2 \cdot \eta_{\mathfrak{a}}(b_{n-1})+a_{(n+1)^3} \qquad (\mbox{by }(2))\\ &>n^2 \cdot \eta_{\mathfrak{a}}(b_{n-1}).\end{aligned}$$ Then we have $$\begin{aligned} \eta_{\mathfrak{b}}(x) &=\eta^{n}_{\mathfrak{b}}(x) =\eta_{\mathfrak{b}}^{n-1}(b_n) +\frac{1}{n}(x-b_n)\\ &=\eta_{\mathfrak{b}}^{n-2}(b_{n-1})+\frac{1}{n-1}(b_{n}-b_{n-1})+\frac{1}{n}(x-b_n)\\ &\geq \eta_{\mathfrak{b}}^{n-2}(b_{n-1})+\frac{1}{n}(b_{n}-b_{n-1}+x-b_n)\\ &= \eta_{\mathfrak{b}}^{n-2}(b_{n-1})+\frac{1}{n}(x-b_{n-1})\\ &=\eta^{n-2}_{\mathfrak{b}}(b_{n-1})+\frac{t}{n}\\ &=\eta_{\mathfrak{b}}(b_{n-1})+\frac{t}{n}\\ &\geq (n-1)\cdot \eta_{\mathfrak{a}}(b_{n-1})+\frac{t}{n}.\tag{5} \end{aligned}$$ The last inequality follows from $(3)$. Since $b_{n-1} > a_{n^3}$, there exists $k\geq n^3$ such that $a_{k} \leq b_{n-1}<a_{k+1}$. Then we have $$\begin{aligned} \eta_{\mathfrak{a}}(x) &\leq \eta^{k}_{\mathfrak{a}}(x) =\eta^{k-1}_{\mathfrak{a}}(a_k) +\frac{1}{k}(x-a_{k})\\ &=\eta^{k-1}_{\mathfrak{a}}(a_k) + \frac{1}{k}(b_{n-1}-a_{k})+\frac{1}{k}(x-b_{n-1})\\ &=\eta^{k}_{\mathfrak{a}}(b_{n-1})+\frac{1}{k}(x-b_{n-1})\\ &=\eta^{k}_{\mathfrak{a}}(b_{n-1})+\frac{t}{k}\\ &\leq \eta_{\mathfrak{a}}(b_{n-1})+\frac{t}{n^3}.\tag{6} \end{aligned}$$ Hence, we have $$\begin{aligned} \eta_{\mathfrak{b}}(x)-n\cdot \eta_{\mathfrak{a}}(x) &\geq (n-1)\cdot \eta_{\mathfrak{a}}(b_{n-1})+\frac{t}{n} -n\cdot \eta_{\mathfrak{a}}(x) \qquad (\mbox{by }(5))\\ &\geq (n-1)\cdot \eta_{\mathfrak{a}}(b_{n-1})+\frac{t}{n} -n\cdot \left(\eta_{\mathfrak{a}}(b_{n-1})+\frac{t}{n^3}\right) \qquad (\mbox{by }(6))\\ &=(n-1)\cdot\frac{t}{n^2}-\eta_{\mathfrak{a}}(b_{n-1})\\ &> (n-1)\cdot \eta_{\mathfrak{a}}(b_{n-1})-\eta_{\mathfrak{a}}(b_{n-1}) \qquad (\mbox{by }(4))\\ &=(n-2)\cdot \eta_{\mathfrak{a}}(b_{n-1})\\ &\geq 0.\end{aligned}$$ Thus we conclude that $$\lim_{n\to\infty}\frac{\eta_{\mathfrak{b}}(n)}{\eta_{\mathfrak{a}}(n)} \geq \lim_{n\to\infty}n =\infty.$$ ◻ **Proposition 11**. *If $\phi\in H(\mathcal{SO})$ satisfies $\phi(\mathbf{1})=0$ then $\phi =\mathbf{0}$.* *Proof.* Let $f\in \mathcal{SO}$. Suppose that $\phi(\mathbf{1})=0$. By Proposition [Proposition 9](#3-7){reference-type="ref" reference="3-7"}, there exists a strictly increasing sequence $\mathfrak{a}\in \mathbb{N}$ and $L>0$ such that $|f|\leq L\cdot \eta_{\mathfrak{a}}$. Recall that $\eta_{\mathfrak{a}}\geq \mathbf{1}$. By Proposition [Proposition 10](#3-8){reference-type="ref" reference="3-8"}, there exists a strictly increasing sequence $\mathfrak{b}\in \mathbb{N}$ so that $\mathbf{1} \leq \eta_{\mathfrak{a}}\leq \eta_{\mathfrak{b}}$ and $$\lim_{n\to \infty}\frac{\eta_{\mathfrak{b}}(n)}{\eta_{\mathfrak{a}}(n)}=\infty.$$ Thus, the condition $\phi(\mathbf{1})=0$ implies that $\phi (\eta_{\mathfrak{a}})=0$ by Lemma [Lemma 7](#3-5){reference-type="ref" reference="3-5"}. Hence, we have $$|\phi(f)|=\phi (|f|)\leq L\cdot \phi(\eta_{\mathfrak{a}})=0,$$ i.e., $\phi(f)=0$. Since $f$ can be taken arbitrary, $\phi$ must be zero-homomorphism. ◻ By Lemma [Proposition 11](#3-9){reference-type="ref" reference="3-9"}, it follows that the structure of $H(\mathcal{SO})$ is very simple in contrast to the case of uniformly continuous functions [@CS] (see also [@FJCS]). **Theorem 12**. *The space $H(\mathcal{SO})$ of all homomorphisms of the vector lattice of all slowly oscillating functions on the half-line $\mathbb{H}$ is homeomorphic to the space $\left(\mathbb{H}\times(0,\infty)\right)\cup \{\mathbf{0}\}$ where a neighborhood base of the point $\mathbf{0}$ consists of sets of the form: $$\{(x,y)\in \mathbb{H}\times(0,\infty) : y\leq \varepsilon \cdot\eta_{\mathfrak{a}}(x)^{-1}\}\cup \{\mathbf{0}\}$$ for some $\varepsilon >0$ and the slowly oscillating function $\eta_{\mathfrak{a}}$ with respect to some strictly increasing sequence $\mathfrak{a}$.* *Proof.* By Proposition [Proposition 11](#3-9){reference-type="ref" reference="3-9"}, every non-zero homomorphism $\phi\in H(\mathcal{SO})$ satisfies $\phi(\mathbf{1})>0$ since $\phi$ is positive. Hence, every $\phi\in H(\mathcal{SO})\setminus \{\mathbf{0}\}$ is uniquely expressed as $\phi =\phi(\mathbf{1})\cdot \delta_{x_{\phi}}$ for some $x_{\phi}\in \mathbb{H}$ by Corollary [Corollary 4](#3-2){reference-type="ref" reference="3-2"}. Therefore, the function $\Phi: H(\mathcal{SO})\setminus \{\mathbf{0}\} \to \mathbb{H}\times (0, \infty)$ defined by $$\Phi (\phi)=(x_{\phi}, \phi(\mathbf{1}))$$ is a well-defined bijection. The function $\Phi$ is continuous. To see this, fix $\phi \in H(\mathcal{SO})$ and let $\varepsilon>0$. We consider following two neighborhoods of $\phi$: $$\begin{aligned} V(\phi; \mathbf{1}; \varepsilon_1) &=\{\varphi \in H(\mathcal{SO}) : |\varphi (\mathbf{1})-\phi(\mathbf{1})|<\varepsilon_1\},\\ V(\phi; \tau ; \varepsilon_2) &=\{\varphi \in H(\mathcal{SO}) : |\varphi (\tau)-\phi(\tau)|<\varepsilon_2\}, \end{aligned}$$ where 1. $\varepsilon_1 <\min\left\{ \varepsilon/2, \frac{\phi(\mathbf{1})\varepsilon}{2(x_{\phi}+1)}\right\}$ and 2. $\varepsilon_2 <\frac{\phi(\mathbf{1})\varepsilon}{2}-\varepsilon_1 (x_{\phi}+1)$. Then for each $\varphi \in U=V(\phi; \mathbf{1}; \varepsilon_1)\cap V(\phi; \tau ; \varepsilon_2)$, we have $$\begin{aligned} \phi(\mathbf{1})|x_{\phi} -x_{\varphi}| &=|\phi (\mathbf{1}) (x_{\phi}+1)-\phi(\mathbf{1})(x_{\varphi}+1)|\\ &\leq |\phi (\mathbf{1}) (x_{\phi}+1)-\varphi(\mathbf{1})(x_{\phi}+1)|\\ & \qquad + |\varphi (\mathbf{1}) (x_{\phi}+1)-\phi(\mathbf{1})(x_{\varphi}+1)|\\ &=|\phi (\mathbf{1}) - \varphi (\mathbf{1})|(x_{\varphi}+1) +|\varphi(\tau) -\phi(\tau)|\\ &<\varepsilon_1 (x_{\varphi}+1) + \varepsilon_2.\\ \therefore\ |x_{\phi} -x_{\varphi}| &\leq \frac{\varepsilon_1 (x_{\phi}+1)+\varepsilon_2}{\phi(\mathbf{1})} <\varepsilon/2. \end{aligned}$$ Thus we have $$\begin{aligned} d(\Phi(\phi), \Phi(\varphi)) &\leq d((x_{\phi}, \phi(\mathbf{1})), (x_{\phi}, \varphi(\mathbf{1}))) +d((x_{\phi}, \varphi(\mathbf{1})), (x_{\varphi}, \varphi(\mathbf{1})))\\ &=|\varphi (\mathbf{1})-\phi(\mathbf{1})|+|x_{\varphi} -x_{\phi}|\\ &<\varepsilon/2 + \varepsilon/2 =\varepsilon.\end{aligned}$$ Next we shall show that $\Phi^{-1}: \mathbb{H}\times (0, \infty)\ni (x,s)\mapsto s\cdot \delta_x \in H(\mathcal{SO})\setminus \{\mathbf{0}\}$ is continuous. Given $(x,s)\in \mathbb{H}\times (0,\infty)$ and $\varepsilon >0$, let $f\in \mathcal{SO}$ and consider a basic neighborhood $$V(s\cdot\delta_x ; f; \varepsilon) =\{\varphi \in H(\mathcal{SO}) : |\varphi(f) -s\cdot f(x)|<\varepsilon\}$$ of $\Phi^{-1}(x,s)=s\cdot\delta_x$. We take $\lambda_1 >0,$ $\lambda_2 >0$ and $\lambda_0 >0$ so that 1. $\lambda_1 \cdot |f(x)|<\varepsilon /2$, 2. $(s +\lambda_1)\cdot \lambda_2 <\varepsilon /2$, 3. $\lambda_0 <\min \{\lambda_1, \lambda_2\}$ and 4. $|f(x)-f(y)|<\lambda_2$ whenever $|x-y|<\lambda_0$. Suppose that $(y,t)\in \mathbb{H}\times (0,\infty)$ satisfies $d((x,s), (y,t))<\lambda_0$. Then $|x-y|<\lambda_0$ and $|s-t|<\lambda_0 \leq \lambda_1$, in particular, $t\leq s+\lambda_1$. Hence, we have $$\begin{aligned} |\Phi^{-1}(x,s)(f)-\Phi^{-1}(y,t)(f)| &=|s\cdot f(x) -t\cdot f(y)|\\ &\leq |s\cdot f(x) -t\cdot f(x)|+|t\cdot f(x) -t\cdot f(y)|\\ &=|s-t|\cdot |f(x)|+t\cdot |f(x)-f(y)|\\ &\leq \lambda_1 \cdot |f(x)| + (s +\lambda_1)\cdot \lambda_2\\ &\leq \varepsilon /2 + \varepsilon /2 =\varepsilon.\end{aligned}$$ Therefore, $\Phi^{-1}(y,t) \in V(s\cdot\delta_x ; f; \varepsilon)$. Consequently, $\Phi$ is a homeomorphism. Finally, we shall consider neighborhoods of $\mathbf{0}\in H(\mathcal{SO})$. We can take a subbase of neighborhoods of $\mathbf{0}$ in $H(\mathcal{SO})$ as a family which consists of sets of the form: $$\begin{aligned} V(\mathbf{0}; f; \varepsilon) &=\{ \varphi\in H(\mathcal{SO}): |\varphi(f)-\mathbf{0}(f)|<\varepsilon \}\\ &=\{ \varphi\in H(\mathcal{SO}): |\varphi(\mathbf{1})\cdot\delta_{x_{\varphi}}(f)|<\varepsilon \}\\ &=\{ \varphi\in H(\mathcal{SO}): |\varphi(\mathbf{1})\cdot f(x_{\varphi})|<\varepsilon \}\end{aligned}$$ for some $f\in \mathcal{SO}$ and $\varepsilon >0$. Let $f\in \mathcal{SO}$ and $\varepsilon >0$. By Propodition [Proposition 9](#3-7){reference-type="ref" reference="3-7"}, there exists $L>0$ and a strictly increasing sequence $\mathfrak{a}$ such that $|f|\leq L\cdot \eta_{\mathfrak{a}}$. So, if $\varphi\in V(\mathbf{0}; L\cdot\eta_{\mathfrak{a}}; \varepsilon)$ then $|\varphi(\mathbf{1})\cdot f(x_{\varphi})|<|\varphi(\mathbf{1})\cdot L\cdot\eta_{\mathfrak{a}}(x_{\varphi})| <\varepsilon$, that is, $$V(\mathbf{0}; L\cdot\eta_{\mathfrak{a}}; \varepsilon) \subset V(\mathbf{0}; f; \varepsilon).$$ Since $V(\mathbf{0}; L\cdot\eta_{\mathfrak{a}}; \varepsilon) =V(\mathbf{0}; \eta_{\mathfrak{a}}; \varepsilon\cdot L^{-1})$, we can take a subbase of neighborhoods of $\mathbf{0}$ in $H(\mathcal{SO})$ as a family which consists of sets of the form: $$V(\mathbf{0}; \eta_{\mathfrak{a}}; \varepsilon) =\{ \varphi\in H(\mathcal{SO}): \varphi(\mathbf{1}) \cdot\eta_{\mathfrak{a}}(x_{\varphi}) <\varepsilon\}$$ for some $\varepsilon >0$ and an increasing sequence $\mathfrak{a}\subset \mathbb{N}$. Note that for any two increasing sequences $\mathfrak{a}=(a_n)$ and $\mathfrak{b}=(b_n)$, if we take an increasing sequence $\mathfrak{c}=(c_n)$ such that $c_n =\max \{a_n, b_n\}$ then we have $\eta_{\mathfrak{c}}\geq \eta_{\mathfrak{a}}\vee \eta_{\mathfrak{b}}$, i.e., $$V(\mathbf{0}; \eta_{\mathfrak{c}}; \varepsilon) \subset V(\mathbf{0}; \eta_{\mathfrak{a}}, \eta_{\mathfrak{b}}; \varepsilon) .$$ Thus, we can take a base of neighborhoods of $\mathbf{0}$ in $H(\mathcal{SO})$ as a family which consists of sets of the form $V(\mathbf{0}; \eta_{\mathfrak{a}}; \varepsilon)$ for some $\varepsilon >0$ and an increasing sequence $\mathfrak{a}\subset \mathbb{N}$. Consequently, we can take a base of neighborhoof of $\mathbf{0}$ in $\mathbb{H}\times (0,\infty)\cup \{\mathbf{0}\}$ as a family which consists of sets of the form $$\begin{aligned} \Phi (V(\mathbf{0}; \eta_{\mathfrak{a}}; \varepsilon))\cup \{\mathbf{0}\} &=\{(x_{\varphi}, \varphi(\mathbf{1}) )\in \mathbb{H}\times(0,\infty) : \varphi(\mathbf{1}) \cdot\eta_{\mathfrak{a}} (x_{\varphi})<\varepsilon \} \cup \{\mathbf{0}\}\\ &=\{(x,y)\in \mathbb{H}\times(0,\infty) : y\leq \varepsilon \cdot\eta_{\mathfrak{a}}(x)^{-1}\}\cup \{\mathbf{0}\}\end{aligned}$$ for some $\varepsilon >0$ and an increasing sequence $\mathfrak{a}\subset \mathbb{N}$. ◻ 99 Félix Cabello Sánchez, *Fine structure of the homomorphisms of the lattice of uniformly continuous functions on the line*, Positivity 24 (2020), no. 2, 415--426. Félix Cabello Sánchez, Javier Cabello Sánchez, *Quiz your maths: Do the uniformly continuous functions on the line form a ring?*, Proceedings of the American Mathematical Society 147, Issue 10 (2019), 4301-4313. R. Engelking, *General topology*, Second edition. Sigma Series in Pure Mathematics, 6. Heldermann Verlag, Berlin, 1989. M. I. Garrido, J. A. Jaramillo, *Homomorphisms on function lattices*, Monatsh. Math. 141 (2004), no. 2, 127--146. Y. Iwamoto, *Indecomposable continua as Higson coronae*, Topology Appl. 283 (2020), 107334, 16 pp. J. Keesling, *The one-dimensional Čech cohomology of the Higson compactification and its corona*, Topology Proc. 19 (1994), 129--148. Jack R. Porter, R. Grant Woods, *Extensions and absolutes of Hausdorff spaces*, Springer-Verlag, New York, 1988. J. Roe, *Lectures on coarse geometry*, University Lecture Series, 31. American Mathematical Society, Providence, RI, 2003. R. Grant Woods, *The minimum uniform compactification of a metric spaces*, Fund. Math. 147(1995), no. 1, 39--59. [^1]: Detailed equivalence relations in the quotient space are not described here because they require preparation that is not needed in this note. See [@CS] for details.

arxiv_math

--- abstract: | We introduce a new notion of embeddability between Banach spaces. By studying the classical Mazur map, we show that it is strictly weaker than the notion of coarse embeddability. We use the techniques from metric cotype introduced by M. Mendel and A. Naor to prove results about cotype preservation and complete our study of embeddability between $\ell_p$ spaces. We confront our notion with nonlinear invariants introduced by N. Kalton, which are defined in terms of concentration properties for Lipschitz maps defined on countably branching Hamming or interlaced graphs. Finally, we address the problem of the embeddability into $\ell_\infty$. address: - IMPA, Estrada Dona Castorina 110, 22460-320, Rio de Janeiro, Brazil. - Laboratoire de Mathématiques de Besançon, Université de Franche-Comté, CNRS UMR-6623, 16 route de Gray, 25030 Besançon Cédex, Besançon, France author: - Bruno M. Braga - Gilles Lancien title: On the expansiveness of coarse maps between Banach spaces and geometry preservation --- [^1] # Introduction This article deals with a new notion of nonlinear embeddability between Banach spaces and how their geometries are preserved under this new notion. More precisely, the notion considered herein will be large scale in nature and even weaker than the usual coarse embeddability. Before presenting it, we start by recalling the basics of coarse geometry. Given metric spaces $(X,d)$ and $(Y,\partial)$, and a map $f\colon X\to Y$, one defines a modulus $$\omega_f(t)=\sup\{\partial(f(x),f(z))\mid d(x,z)\leq t\},\ \text{ for }\ t\geq 0,$$ and call $f$ *coarse* if $\omega_f(t)<\infty$ for all $t\geq 0$. In words, $f$ is coarse if it sends bounded sets to bounded sets in a uniform manner. Coarse maps are the usual morphisms considered in the study of the large scale geometry of metric spaces and, in particular, of Banach spaces. In order to deal with embeddings, one defines a modulus $$\rho_f(t)=\inf\{\partial (f(x),f(z))\mid d(x,z)\geq t\},\ \text{ for } \ t\geq 0,$$ and call $f$ *expanding* if $\lim_{t\to \infty}\rho_f(t)=\infty$. In words, $f$ is expanding if it sends elements far apart to elements likewise uniformly. The map $f$ is then called a *coarse embedding* if it is both coarse and expanding. Despite its seemingly weak definition, coarse embeddability is known to capture the geometry of Banach spaces in several remarkable ways; to cite a few, we mention the cotype preservation under coarse embeddability into Banach spaces with nontrivial type proved in the seminal paper of M. Mendel and A. Naor ([@MendelNaor2008 Theorem 1.11]) and the preservation of asymptotic-$c_0$-ness$+$reflexivity proved by the second named author together with F. Baudier, P. Motakis, and Th. Schlumprecht ([@BaudierLancienSchlumprecht2018 Theorem A]). Functional analysts working in the nonlinear geometry of Banach spaces are interested in knowing the minimal requirements needed for maps between Banach spaces to still generate an interesting notion of embeddability; here the word "interesting" should be broadly interpreted as "it is strictly weaker than a previously studied notion of embeddability but still strong enough to impose geometric restrictions". For instance, C. Rosendal has started in [@Rosendal2017Sigma] with the program of weakening the notion of expansiveness of a coarse map $f$ by properties such as $f$ being *uncollapsed* in the sense that there are $\Delta,\delta>0$ such that $$\|x-z\|\geq \Delta\ \text{ implies }\ \|f(x)-f(z)\|>\delta,$$ or $f$ being *solvent*, meaning that there is an increasing sequence $(R_n)_n$ in $\mathbb{N}$ such that $$\|x-z\|\in [R_n,R_n+n]\ \text{ implies }\ \|f(x)-f(z)\|>n.$$ Even maps $f$ satisfying only that $$\|x-z\|=\Delta\ \text{ implies }\ \|f(x)-f(z)\|>\delta$$ have already been studied; those are called *almost uncollapsed* (see [@Braga2017JFA]). Inspired by a recent work by the two authors (see [@BragaLancien2023Equiv]), this paper initiates a yet new approach of weakening the expansiveness condition. Indeed, all the weakenings mentioned above are *not* localized: the positions of $x$ and $z$ in $X$ do not matter, but only the distance $\|x-z\|$. However, in [@BragaLancien2023Equiv], the authors started the study of an equivalence between metric spaces called *asymptotic coarse equivalence* and this takes into account the asymptotic behavior of elements $x$ in $X$ as they approach infinity. In particular, those maps are not necessarily expanding anymore, but only satisfy expansiveness as $x,z\to \infty$. This motivates the main definition of these notes: **Definition 1**. Let $X$ and $Y$ be Banach spaces and $\alpha\in [0,1]$. A map $f\colon X\to Y$ is called *expanding at rate $\alpha$* if for all $L>0$ there is a map $\rho\colon [0,\infty)\to [0,\infty)$ with $\lim_{t\to \infty}\rho(t)=\infty$ such that $$\|x-z\|\geq L\max\{\|x\|^\alpha,\|z\|^\alpha\}+L\ \text{ implies }\ \|f(x)-f(z)\|\geq \rho(\|x-z\|).$$ In case $\rho$ can always be chosen to be of the form $\frac{t}{C}-C$ for some $C>0$, we say that $f$ is *linearly expanding at rate $\alpha$*. A few comments are in place here. Firstly, notice that a coarse map $X\to Y$ is expanding if and only if it is expanding at rate $0$. Also, we restrict ourselves to $\alpha\leq 1$ since the condition of $\|x-z\|$ being at least of the order of $\max\{\|x\|^\alpha,\|z\|^\alpha\}$ will not happen (up to a bounded subset) if $\alpha>1$. We say that a coarse map $f$ has *nontrivial coarse expansion* if it is expanding at rate $\alpha$ for some $\alpha\in [0,1]$. Finally, we recall that if a coarse map $f$ between Banach spaces satisfies $\rho_f(t)\geq \frac{t}{C}-C$ for some $C>0$ and all $t\geq 0$, then $f$ is called a *coarse Lispchitz embedding*; a stronger notion than coarse embedding. Hence, the notion of a coarse map $f$ being linearly expanding at some rate should be seen as a weakening of $f$ being a coarse Lipschitz embedding. We now describe the main findings of this paper. ## Mazur maps and metric cotype We first show that the existence of coarse maps which are expanding at rate $\alpha$, for $\alpha\in (0,1]$, is strictly weaker than coarse embeddability. In fact, as it turned out, there are several well-studied maps which witness that: the Mazur maps. Recall, given $p,q\in [1,\infty)$, the *Mazur map* $M_{p,q}\colon \ell_p\to \ell_q$ is the homogeneous extension of the canonical map which adjusts elements in the unit sphere of $\ell_p$ so that they fall into the unit sphere of $\ell_q$; for brevity, we postpone to Section [2](#SectionMazur){reference-type="ref" reference="SectionMazur"} its formal definition. While $\ell_p$ coarsely embeds into $\ell_q$ if and only if either $p\in [1,2]$ or $p\leq q$ ([@MendelNaor2008 Corollary 7.3]), we show that $M_{p,q}$ is coarse and has nontrivial expansion as long as $p>q$. Precisely: **Theorem 2**. *Let $1\le q<p$ and $\alpha\in ( \frac{p-q}{p},1]$. Then $M_{p,q}\colon \ell_p\to \ell_q$ is a coarse map, which is expanding at rate $\alpha$. Moreover, $M_{p,q}$ is linearly expanding at rate $1$.* If $p>2$ and $p>q$, we know from the aforementioned result of M. Mendel and A. Naor that $\ell_p$ does not coarsely embeds into $\ell_q$; in our terminology just introduced, this means that there is no coarse map $\ell_p\to \ell_q$ which is expanding at rate $0$. By Theorem [Theorem 2](#ThmMazurMapACE){reference-type="ref" reference="ThmMazurMapACE"}, we are then left to understand what happens for $\alpha$'s in the interval $(0,\frac{p-q}{p}]$. We show, using techniques from metric cotype of [@MendelNaor2008], that there is no such map for all $\alpha$'s in $(0,\frac{p-q}{p}]$, when $2\le q< p<\infty$, and that there is no such map for all $\alpha$'s in $(0,\frac{p-2}{p}]$, when $1\le q<2<p$ (see Corollary [Corollary 16](#CorEmblpIntolq){reference-type="ref" reference="CorEmblpIntolq"}). We refer to Section [3](#SectionCotype){reference-type="ref" reference="SectionCotype"} for all relevant definitions. Let us just say for this introduction that, for a Banach space $X$, we denote $$q_X=\inf\{q\in [2,\infty]\mid X \text{ has cotype q}\}.$$ We also prove the following general result about cotype preservation, which generalizes [@MendelNaor2008 Theorem 1.11]: **Theorem 3**. *Let $X$ and $Y$ be Banach spaces and suppose $Y$ has nontrivial type. Let $\alpha\in [0,1]$ and suppose there is a coarse map $X\to Y$ which is expanding at rate $\alpha$. Then $q_X\leq \frac{q_Y}{1-\alpha}$.* We stress here that our results do more than generalizing [@MendelNaor2008 Theorem 1.11]. Indeed, Theorem [Theorem 3](#ThmCotypeACE){reference-type="ref" reference="ThmCotypeACE"} together with Theorem [Theorem 2](#ThmMazurMapACE){reference-type="ref" reference="ThmMazurMapACE"} give us optimal results on cotype preservation (see Corollary [Corollary 16](#CorEmblpIntolq){reference-type="ref" reference="CorEmblpIntolq"}). ## Embeddings of metric graphs into Banach spaces After our study of cotype preservation and the embeddability of the $\ell_p$'s, we turn our attention to the embeddability of certain metric graphs into Banach spaces. Recall, given $k\in\mathbb{N}$, we let $[\mathbb{N}]^k$ denote the set of all subsets of $\mathbb{N}$ with $k$ elements and, given $\bar n\in[\mathbb{N}]^k$, we write $\bar n=(n_1,\ldots, n_k)$ where $n_1<\ldots< n_k$. As initiated by N. Kalton (see [@Kalton2007; @Kalton2013AsymptoticStructure]), the study of the embeddability of the sequence $([\mathbb{N}]^k)_k$ (endowed with appropriate metrics) is extremely useful when looking for coarse (or coarse Lipschitz) invariants of Banach spaces. For instance, given $k\in\mathbb{N}$, let $d_{\mathbb H}=d_{\mathbb H,k}$ denote the *Hamming metric* on $[\mathbb{N}]^k$, i.e., $$d_{\mathbb H}(\bar n, \bar m)=|\{i\in \{1,\ldots, k\}\mid n_i\neq m_i\}|, \ \text{ for }\ \bar n,\bar m\in [\mathbb{N}]^k.$$ The following important concentration property was introduced in [@KaltonRandrianarivony2008] and later formalized by A. Fovelle in [@Fovelle] in the format presented below: **Definition 4**. Let $p\in (1,\infty]$. A Banach space $X$ is said to have *Hamming full concentration property $p$*, abbreviated *HFC*$_p$, if there is $C\geq 1$ such that for all $k\in\mathbb{N}$ and all $1$-Lipschitz maps $\phi\colon ([\mathbb{N}]^k,d_{\mathbb H})\to X$, there is an infinite $\mathbb M\subseteq \mathbb{N}$ such that $$\mathrm{diam}(\phi([\mathbb M]^k))\leq Ck^{1/p}$$ (here we use the convention $1/\infty=0$ if $p=\infty$). As shown in [@KaltonRandrianarivony2008 Theorem 4.2], reflexive spaces with a $p$-asymptotically uniformly smooth renorming have the HFC$_p$; see Example [Example 22](#example HFC){reference-type="ref" reference="example HFC"} for definitions. Moreover, as shown in [@BaudierLancienMotakisSchlumprecht2018 Theorems A and B], having HFC$_\infty$ is equivalent to $X$ being asymptotic-$c_0$ and reflexive. We postpone to Example [Example 23](#ExaAsympc_0){reference-type="ref" reference="ExaAsympc_0"} the formal definition of asymptotic-$c_0$-ness, for now, we simply say that $X$ has such property if copies of the finite dimensional subspaces of $c_0$ can be found in the finite codimensional subspaces of $X$ in a uniform manner. It was known that for $p\in (1,\infty)$, HFC$_p$ is stable under coarse-Lipschitz embeddings and that HFC$_\infty$ is even stable under coarse embeddings. We show that our weaker notions of embeddability are already enough for the HFC$_p$ properties to be preserved in the following sense: **Theorem 5**. *Let $X$ and $Y$ be Banach spaces and suppose $Y$ has HFC$_p$ for some $p\in (1,\infty]$.* 1. *[\[ThmPreservationHFCpByLinearExpansionItem1\]]{#ThmPreservationHFCpByLinearExpansionItem1 label="ThmPreservationHFCpByLinearExpansionItem1"} Suppose $p\in (1,\infty)$. If there is a coarse map $f\colon X\to Y$ which is linearly expanding at rate $1$, then $X$ must have HFC$_p$.* 2. *[\[ThmPreservationHFCpByLinearExpansionItem2\]]{#ThmPreservationHFCpByLinearExpansionItem2 label="ThmPreservationHFCpByLinearExpansionItem2"} Suppose $p=\infty$. If there is a coarse map $f\colon X\to Y$ which is expanding at rate $1$, then $X$ must have HFC$_\infty$.* In particular, applying Theorem [Theorem 5](#ThmPreservationHFCpByLinearExpansion){reference-type="ref" reference="ThmPreservationHFCpByLinearExpansion"}[\[ThmPreservationHFCpByLinearExpansionItem2\]](#ThmPreservationHFCpByLinearExpansionItem2){reference-type="eqref" reference="ThmPreservationHFCpByLinearExpansionItem2"} together with the characterization of HFC$_\infty$ mentioned above ([@BaudierLancienMotakisSchlumprecht2018 Theorems B]), yields immediately the following corollary. **Corollary 6**. *If a Banach space $X$ can be mapped by a coarse map which is also expanding at rate $1$ into a reflexive Banach space which is asymptotic-$c_0$, then $X$ must be also reflexive and asymptotic-$c_0$.* We also study interlacing pairs in Hamming graphs in Subsection [4.2](#SubsectionInterlacingHamming){reference-type="ref" reference="SubsectionInterlacingHamming"} and use this to obtain results about the embeddability of the James spaces $J_p$ (Example [Example 27](#ExampleJamesSpace){reference-type="ref" reference="ExampleJamesSpace"}); see Theorem [Theorem 28](#ThmInterlacingHamming){reference-type="ref" reference="ThmInterlacingHamming"} and Corollary [Corollary 31](#CorInterlacingHammingJames){reference-type="ref" reference="CorInterlacingHammingJames"} for details. Another important metric we can endow each $[\mathbb{N}]^k$ with is the *interlaced metric*: we set distinct elements $\bar n,\bar m\in [\mathbb{N}]^k$ to be adjacent if either $$n_1\leq m_2\leq \ldots\leq n_k\leq m_k\ \text{ or }\ m_1\leq n_1\leq \ldots\leq m_k\leq n_k$$ and then we let $d_{\mathbb I}=d_{\mathbb I,k}$ be the shortest path metric on $[\mathbb{N}]^k$ given by this graph structure. The study of those metric spaces was fundamental for N. Kalton to rule out the coarse embeddability of $c_0$ into reflexive spaces and gave rise to the so-called *property $\mathcal Q$'s*: **Definition 7**. Let $p\in (1,\infty]$. A Banach space $X$ is said to have *property $\mathcal Q_p$*, if there is $C\geq 1$ such that for all $k\in\mathbb{N}$ and all $1$-Lipschitz maps $\phi\colon ([\mathbb{N}]^k,d_{\mathbb I})\to X$, there is an infinite $\mathbb M\subseteq \mathbb{N}$ such that $$\mathrm{diam}(\phi([\mathbb M]^k)\leq Ck^{1/p}$$ (here we use the convention $1/\infty=0$ if $p=\infty$). If $p=\infty$, we simply say $X$ has *property $\mathcal Q$*. We prove that our weakenings of coarse embeddability are also strong enough to ensure the preservation of property $\mathcal Q_p$. Precisely, we prove the following: **Theorem 8**. *Let $X$ and $Y$ be Banach spaces and suppose $Y$ has property $\mathcal Q_p$ for some $p\in (1,\infty]$.* 1. *[\[ThmPreservationPropertyQpByLinearExpansionItem1\]]{#ThmPreservationPropertyQpByLinearExpansionItem1 label="ThmPreservationPropertyQpByLinearExpansionItem1"} Suppose $p\in (1,\infty)$. If there is a coarse map $f\colon X\to Y$ which is linearly expanding at rate $1$, then $X$ must have property $\mathcal Q_p$.* 2. *[\[ThmPreservationPropertyQpByLinearExpansionItem2\]]{#ThmPreservationPropertyQpByLinearExpansionItem2 label="ThmPreservationPropertyQpByLinearExpansionItem2"} Suppose $p=\infty$. If there is a coarse map $f\colon X\to Y$ which is expanding at rate $1$, then $X$ must have property $\mathcal Q_\infty$, i.e., property $\mathcal Q$.* In Section [5](#SectionlInfty){reference-type="ref" reference="SectionlInfty"}, we characterize Lipschitz embeddability into $\ell_\infty$ in terms of our new notion of embeddability; this extends a result of N. Kalton (see [@Kalton2011FundMath Theorem 5.3]). Precisely, we prove the following. **Theorem 9**. *The following are equivalent for a Banach space $X$:* 1. *[\[ThmlInftyItem1\]]{#ThmlInftyItem1 label="ThmlInftyItem1"} $X$ Lipschitzly embeds into $\ell_\infty$.* 2. *[\[ThmlInftyItem2\]]{#ThmlInftyItem2 label="ThmlInftyItem2"} There is a coarse map $f\colon X\to \ell_\infty$ which is linearly expanding at rate $\alpha$, for some $\alpha\in (0,1)$.* # Revisiting the Mazur map {#SectionMazur} The purpose of this section is to prove some estimates for the classic Mazur map (see Lemma [Lemma 11](#LemmaMazupMapEstimate){reference-type="ref" reference="LemmaMazupMapEstimate"}) and to deduce the proof of Theorem [Theorem 2](#ThmMazurMapACE){reference-type="ref" reference="ThmMazurMapACE"}. Recall, the Mazur map between the unit spheres of two Lebesgue sequence spaces, say $\ell_p$ and $\ell_q$, is the canonical map which adjusts $p$-summable sequences so that they are $q$-summable. Precisely, given a Banach space $X$, $B_X$ denotes its closed unit ball and $\partial B_X$ its unit sphere. Then, given $p,q\in [1,\infty)$, the *Mazur map* is the map $M_{p,q}\colon \partial B_{\ell_p}\to\partial B_{\ell_q}$ given by $$M_{p,q}((x_n)_n)=(\mathrm{sign}(x_n)|x_n|^{p/q})_n$$ for all $(x_n)_n\in \partial B_{\ell_p}$. It is evident that $M_{p,q}$ is a bijection with inverse $M_{q,p}$. Moreover, it is well-known that this map is a uniform equivalence. Precisely, the following estimates hold (see [@BenyaminiLindenstraussBook Theorem 9.1] for a proof and [@MazurStudia1929] for its first appearance). **Theorem 10** (Mazur Map). *Let $p,q\in [1,\infty)$ with $q<p$ and let $M_{p,q}\colon \ell_p\to \ell_q$ denote the Mazur map. There is a constant $C=C(p,q)>0$ such that $$C\|x-y\|^{p/q}\leq \|M_{p,q}(x)-M_{p,q}(y)\|\leq \frac{p}{q}\|x-y\|$$ for all $x,y\in \partial B_{\ell_p}$.[\[ThmMazurMap\]]{#ThmMazurMap label="ThmMazurMap"}* As it is usually done, we extend the map $M_{p,q}$ to the whole $\ell_p$ by homogeneity; by a abuse of notation, we still denote this extension by $M_{p,q}$. Precisely, for each $x\in \ell_p$, we let $$M_{p,q}(x)=\left\{\begin{array}{ll} \|x\|M_{p,q}\Big(\frac{ x}{\| x\|}\Big),& \text{ if }\ x\neq 0,\\ 0,& \text{ if }\ x=0. \end{array} \right.$$ We emphasize some important properties of the Mazur map below: - $M_{p,q}^{-1}=M_{q,p}$ for all $p,q\in [1,\infty)$, - $\|M_{p,q}(x)\|=\|x\|$ for all $x\in \ell_p$, and - $M_{p,q}\Big(\frac{x}{\|x\|}\Big)=\frac{M_{p,q}(x)}{\|M_{p,q}(x)\|}$ for all $x\in \ell_p\setminus \{0\}$. The following is the main technical result of this section. **Lemma 11**. *Let $p,q\in [1,\infty)$ with $q<p$.* 1. *[\[Item1PropMazupMapEstimate\]]{#Item1PropMazupMapEstimate label="Item1PropMazupMapEstimate"} For all $x,y\in \ell_p$, we have $$\|M_{p,q}(x)-M_{p,q}(y)\|\leq \Big(\frac{2p}{q}+1\Big)\|x-y\|.$$* 2. *[\[Item3PropMazupMapEstimate\]]{#Item3PropMazupMapEstimate label="Item3PropMazupMapEstimate"} For all $\varepsilon>0$ and all $\alpha\in (0,1]$, there is $L=L(p,q,\varepsilon,\alpha)>0$ so that, for all $x,y\in \ell_p$ with $\|x-y\|\geq \varepsilon\max\{\|x\|^\alpha,\|y\|^\alpha\}$, we have $$\frac{1}{L}\|x-y\|^{\frac{p}{q}-\frac{1}{\alpha}\big(\frac{p}{q}-1\big)}\leq \|M_{p,q}(x)-M_{p,q}(y)\|.$$* Before proving Lemma [Lemma 11](#LemmaMazupMapEstimate){reference-type="ref" reference="LemmaMazupMapEstimate"}, we isolate some simple estimates for future use (cf. [@Kalton2013Examples Lemma 3.1]). **Lemma 12**. *Let $X$ be a Banach space and $x,y\in X$ with $\|x\|\geq \|y\|>0$. The following holds* 1. *$\big\|\frac{x}{\|x\|}-\frac{y}{\|y\|}\big\|\leq 2\frac{\|x-y\|}{\|x\|}$ and* 2. *$\|x-y\|\leq \|x\|-\|y\|+\|y\| \big\|\frac{x}{\|x\|}-\frac{y}{\|y\|}\big\|$.0◻* *Proof of Lemma [Lemma 11](#LemmaMazupMapEstimate){reference-type="ref" reference="LemmaMazupMapEstimate"}.* [\[Item1PropMazupMapEstimate\]](#Item1PropMazupMapEstimate){reference-type="eqref" reference="Item1PropMazupMapEstimate"} Fix $x,y\in \ell_p$. Without loss of generality, suppose $\|x\|\geq \|y\|>0$. Then Theorem [\[ThmMazurMap\]](#ThmMazurMap){reference-type="ref" reference="ThmMazurMap"} and Lemma [Lemma 12](#LemmaEstimates){reference-type="ref" reference="LemmaEstimates"} imply $$\begin{aligned} \|M_{p,q}(x)-M_{p,q}(y)\|= & \Big\| \|x\|M_{p,q}\Big(\frac{ x}{\| x\|}\Big)- \|y\|M_{p,q}\Big(\frac{ y}{\| y\|}\Big)\Big\| \\ \leq &\|x\|\Big\| M_{p,q}\Big(\frac{ x}{\| x\|}\Big)- M_{p,q}\Big(\frac{ y}{\| y\|}\Big)\Big\|\\ + & \Big(\|x\|-\|y\|\Big)\Big\| M_{p,q}\Big(\frac{ y}{\| y\|}\Big)\Big\|\\ \leq & \Big(\frac{2p}{q}+1\Big)\|x-y\|.\end{aligned}$$ [\[Item3PropMazupMapEstimate\]](#Item3PropMazupMapEstimate){reference-type="eqref" reference="Item3PropMazupMapEstimate"} Fix $\varepsilon>0$, $\alpha\in (0,1]$, and $x,y\in \ell_p$ with $\|x-y\|\geq \varepsilon\max\{\|x\|^\alpha,\|y\|^\alpha\}$. It is easily checked that we may assume that $\|x\|\geq \|y\|>0$. Suppose first that $\|x\|-\|y\|>\frac{\|x-y\|}{2}$, then $$\begin{aligned} \|M_{p,q}(x)-M_{p,q}(y)\|&\geq \|M_{p,q}(x)\|-\|M_{p,q}(y)\|\\ &= \|x\|-\|y\|\\ &\geq \frac{\|x-y\|}{2}\end{aligned}$$ (here we use that $\|M_{p,q}(z)\|=\|z\|$ for all $z\in \ell_p$). Suppose now that $0\le \|x\|-\|y\|\le \frac{\|x-y\|}{2}$. Then, by Lemma [Lemma 12](#LemmaEstimates){reference-type="ref" reference="LemmaEstimates"}, we have $$\label{Eq.Lemma.Est1} \|x-y\|\leq 2\|y\|\Big\|\frac{x}{\|x\|}-\frac{y}{\|y\|}\Big\|\leq 2\|x\|\Big\|\frac{x}{\|x\|}-\frac{y}{\|y\|}\Big\|.$$ Using Theorem [\[ThmMazurMap\]](#ThmMazurMap){reference-type="ref" reference="ThmMazurMap"} and the fact that $\|M_{p,q}(z)\|=\|z\|$ for all $z\in \ell_p$ again, we have $$\begin{aligned} \|x\|\Big\|\frac{x}{\|x\|}-\frac{y}{\|y\|}\Big\|&\leq \frac{\|x\|}{C^{q/p}}\Big\|M_{p,q}\Big(\frac{x}{\|x\|}\Big)-M_{p,q}\Big(\frac{y}{\|y\|}\Big)\Big\|^{q/p}\\ &= \frac{\|x\|}{C^{q/p}}\Big\|\frac{M_{p,q}(x)}{\|M_{p,q}(x)\|}-\frac{M_{p,q}(y)}{\|M_{p,q}(y)\|}\Big\|^{q/p} \notag\\ &\leq \Big(\frac{2}{C}\Big)^{q/p}\|x\|^{1-q/p}\|M_{p,q}(x)-M_{p,q}(y)\|^{q/p}. \notag \end{aligned}$$ As $q<p$ and $\|x-y\|\geq\varepsilon\max\{\|x\|^\alpha,\|y\|^\alpha\}=\varepsilon\|x\|^\alpha$, [\[Eq.Lemma.Est1\]](#Eq.Lemma.Est1){reference-type="eqref" reference="Eq.Lemma.Est1"} and the inequality above imply that $$\begin{aligned} \label{Eq.1} \|x-y\|\leq 2\Big(\frac{2}{C}\Big)^{q/p}\varepsilon^{\frac{q/p-1}{\alpha}}\|x-y\|^{\frac{1-q/p}{\alpha}}\|M_{p,q}(x)-M_{p,q}(y)\|^{q/p}.\notag \end{aligned}$$ Simplifying the above, we conclude that $$\|x-y\|^{\frac{p}{q}-\frac{1}{\alpha}\big(\frac{p}{q}-1\big)}\leq \frac{2^{1+p/q}}{C} \varepsilon^{\frac{1-p/q}{\alpha}}\|M_{p,q}(x)-M_{p,q}(y)\|.$$ The result then follows taking $L$ to be the maximum of $2$ and $C^{-1}2^{1+p/q} \varepsilon^{(1-p/q)/\alpha}$. ◻ *Proof of Theorem [Theorem 2](#ThmMazurMapACE){reference-type="ref" reference="ThmMazurMapACE"}.* The first claim is immediate from Lemma [Lemma 11](#LemmaMazupMapEstimate){reference-type="ref" reference="LemmaMazupMapEstimate"} since $\alpha$ being larger than $\frac{p-q}{p}$ implies that $\frac{p}{q}-\frac{1}{\alpha}\big(\frac{p}{q}-1\big)$ is positive. The second claim follows immediately from Lemma [Lemma 11](#LemmaMazupMapEstimate){reference-type="ref" reference="LemmaMazupMapEstimate"} also since $\frac{p}{q}-\frac{1}{\alpha}\big(\frac{p}{q}-1\big)=1$ if $\alpha=1$. ◻ As a consequence of Theorem [Theorem 2](#ThmMazurMapACE){reference-type="ref" reference="ThmMazurMapACE"}, we obtain that the existence of a coarse map between Banach spaces which is expanding at rate $\alpha$, for $\alpha\in (0,1]$, is strictly weaker than coarse embeddability. Indeed, it contrasts with the well-known fact that $\ell_p$ coarsely embeds into $\ell_q$ if and only if either $p\leq q$ or $p,q\in [1,2]$ (see [@MendelNaor2008 Corollary 7.3]). # Expansion and cotype preservation {#SectionCotype} In Section [2](#SectionMazur){reference-type="ref" reference="SectionMazur"}, we showed that, for $p,q\in [1,\infty)$ with $q<p$, there are coarse maps with nontrivial expanding properties from $\ell_p$ to $\ell_q$; which contrasts with the known results about the coarse embeddability of $\ell_p$ into $\ell_q$. Precisely, Theorem [Theorem 2](#ThmMazurMapACE){reference-type="ref" reference="ThmMazurMapACE"} shows there are coarse maps $\ell_p \to \ell_q$ which are expanding at rate $\alpha$ as long as $\alpha> \frac{p-q}{p}$. Since we know that $\ell_p$ coarsely embeds into $\ell_q$ if and only if either $p\leq q$ or $p,q\in [1,2]$ (see [@MendelNaor2008 Corollary 7.3]), this result is not true for $\alpha=0$. The main goal of the current section is to understand what happens for $\alpha$'s in the interval $(0,\frac{p-q}{p}]$. Using techniques developed by M. Mendel and A. Naor in their seminal paper about metric cotype ([@MendelNaor2008]), we show that the results of Section [2](#SectionMazur){reference-type="ref" reference="SectionMazur"} do not hold for $2\le q<p<\infty$ and $\alpha\le \frac{p-q}{p}$, nor for $1\le q<2<p$ and $\alpha \le \frac{p-2}{p}$. We start this section recalling the necessary background on metric cotype. Given $m\in\mathbb{N}$, $\mathbb Z_m$ denotes the set of integers modulo $m$. Given $n,m\in\mathbb{N}$, $\mu=\mu_{m,n}$ denotes the normalized counting measure on $\mathbb Z^n_m$ and $\sigma=\sigma_n$ denotes the normalized counting measure on $\{-1,0,1\}^n$. For each $j\in \{1,\ldots, n\}$, $e_j$ denotes the vector in $\mathbb Z^n_m$ whose $j$-th coordinate is $1$ and all others are $0$. **Definition 13** (Metric cotype). Let $(X,d)$ be a metric space and $q,\Gamma>0$. We say that $X$ has *metric cotype $q$ with constant $\Gamma$* if for all $n\in\mathbb{N}$ there is an even $m\in\mathbb{N}$ such that for all $f\colon \mathbb Z^n_m\to X$ we have $$\begin{aligned} \label{Eq.MetricCotype} \sum_{j=1}^n\int_{\mathbb Z^n_m}d\Big(f\big(x &+\frac{m}{2}e_j\big),f(x)\Big)^qd\mu(x)\\ &\leq \Gamma^qm^q \int_{\{-1,0,1\}^n}\int_{\mathbb Z^n_m}d\big(f(x+\varepsilon),f(x)\big)^qd\mu(x)d\sigma(\varepsilon).\notag\end{aligned}$$ Given $n\in\mathbb{N}$ and $\Gamma>0$, we let $m_q(X, n, \Gamma)$ be the smallest even integer $m$ such that [\[Eq.MetricCotype\]](#Eq.MetricCotype){reference-type="eqref" reference="Eq.MetricCotype"} holds for all $f \colon \mathbb Z^n_m\to X$. If no such $m$ exists, we set $m_q(X, n,\Gamma) = \infty$. The next lemma is the main technical result of this section and is a refinement of [@MendelNaor2008 Lemma 7.1]. **Lemma 14**. *Let $(X,d)$ be a metric space, $n\in\mathbb{N}$, $q,s,\Gamma>0$, and $r\in (0,\infty]$. Let $\alpha\in [0,\frac{rs}{rs+1}]$, $\rho\colon [0,\infty)\to [0,\infty)$, and maps $f_n\colon \ell_r^n(\mathbb C)\to X$, for $n\in \mathbb{N}$, be such that $$\|x-z\|\geq \max\{\|x\|^\alpha,\|z\|^\alpha\}+1\ \text{ implies }\ \|f_n(x)-f_n(z)\|\geq \rho(\|x-z\|),$$ for all $n\in \mathbb{N}$ and $x,z \in \ell_r^n(\mathbb C)$. Then, we have that for all $n\in \mathbb{N}$: $$n^{1/q}\rho(2n^s)\leq \Gamma\cdot m_q(X,n,\Gamma)\cdot \omega_{f_n}\Big(\frac{2\pi n^{s+1/r}}{m_q(X,n,\Gamma)}\Big)$$ (if $r=\infty$, we use the conventions $1/\infty=0$ and $\infty/\infty=1$).* *Proof.* To simplify notation, let $m=m_q(X,n,\Gamma)$. By a slight abuse of notation, we let $e_1,\ldots, e_n$ denote the standard basis of both $\mathbb Z^n_m$ and $\ell_r^n(\mathbb C)$. Define a map $h\colon \mathbb Z_m^n\to \ell_r^n(\mathbb C)$ by letting $$h(x)=n^s\cdot\sum_{j=1}^n e^{\frac{2\pi i x_j}{m}}e_j$$ for all $x=(x_j)_{j=1}^n\in \mathbb Z_m^n$. Note first that, since $t\mapsto e^{it}$ is $1$-Lipschitz on $\mathbb{R}$, $$\|h(x+\varepsilon)-h(x)\|_r \le n^s\Big\|\Big(\frac{2\pi |\varepsilon_j|}{m}\Big)_{j=1}^n\Big\|_r \le \frac{2\pi n^{s+1/r}}{m},$$ for all $\varepsilon=(\varepsilon_j)_{j=1}^n\in \{-1,0,1\}^n$ and all $x=(x_j)_{j=1}^n\in Z_m^n$. Let $g_n=f_n\circ h$, then $$\|g_n(x+\varepsilon)-g_n(x)\|\leq \omega_{f_n}\Big(\frac{2\pi n^{s+1/r}}{m}\Big)$$ for all $\varepsilon=(\varepsilon_j)_{j=1}^n\in \{-1,0,1\}^n$ and all $x=(x_j)_{j=1}^n\in Z_m^n$. Therefore, integrating with respect to $x$ and $\varepsilon$, we have $$\label{Eq.cotype1} \int_{\{-1,0,1\}^n}\int_{\mathbb Z_m^n}\|g_n(x+\varepsilon)-g_n(x)\|^qd\mu(x)d\sigma(\varepsilon)\leq \omega_{f_n}\Big(\frac{2\pi n^{s+1/r}}{m}\Big)^q.$$ We now use the hypothesis on $f_n$. As $\alpha\le \frac{rs}{rs+1}$, we have that $$2n^s=2(n^{s+1/r})^{\frac{rs}{rs+1}}\geq (n^{s+1/r})^\alpha+1,$$ for all $n\in \mathbb{N}$. Hence, $$\begin{aligned} \Big\|h\Big( x+\frac{m}{2}e_j\Big)-h(x)\Big\|_r=2n^{s} \ge (n^{s+1/r})^\alpha+1, \end{aligned}$$ for all $x\in \mathbb Z_m^n$ and all $j\in \{1,\ldots, n\}$. Therefore, as $\|h(y)\|_r=n^{s+1/r}$ for all $y\in \mathbb Z_m^n$, it follows that $$\Big\|g_n\Big(x+\frac{m}{2}e_j\Big)-g_n(x)\Big\|\geq \rho(2n^{s})$$ for all $x\in \mathbb Z_m^n$ and all $j\in \{1,\ldots, n\}$. Hence, $$\label{Eq.cotype2}\sum_{j=1}^n\int_{\mathbb Z_m^n}\Big\|g_n\Big(x+\frac{m}{2}e_j\Big)-g_n(x)\Big\|^qd\mu(x)\geq n\rho(2n^{s})^q.$$ By the definition of $m=m_q(M,n,\Gamma)$, [\[Eq.cotype1\]](#Eq.cotype1){reference-type="eqref" reference="Eq.cotype1"} and [\[Eq.cotype2\]](#Eq.cotype2){reference-type="eqref" reference="Eq.cotype2"} show that $$n\rho(2n^{s})^q\leq \Gamma^qm^q \omega_{f_n}\Big(\frac{2\pi n^{s+1/r}}{m}\Big)^q.$$ Taking the $q$-th root at both sides above finishes the proof. ◻ We now turn to our result about preservation of cotype by coarse maps satisfying some weak expanding conditions. For completeness, we quickly recall the notions of type and cotype. Let $X$ be a Banach space and $p\in (1,2]$. We say that $X$ has *type $p$* if there is $C>0$ such that $$\frac{1}{2^n} \sum_{(\varepsilon_i)_{i=1}^n\in \{-1,1\}^n}\Big\|\sum_{i=1}^n\varepsilon_ix_i\Big\|^p\leq C\sum_{i=1}^n\|x_i\|^p$$ for all $n\in\mathbb{N}$ and all $x_1,\ldots, x_n\in X$. If $X$ does not have type $p$ for any $p\in (1,2]$, $X$ is said to have *trivial type*. If $q\in [2,\infty)$, we say that $X$ has *cotype $q$* if there is $C>0$ such that $$\frac{1}{2^n}\sum_{(\varepsilon_i)_{i=1}^n\in \{-1,1\}^n}\Big\|\sum_{i=1}^n\varepsilon_ix_i\Big\|^q\geq C\sum_{i=1}^n\|x_i\|^q$$ for all $n\in\mathbb{N}$ and all $x_1,\ldots, x_n\in X$. We let $$q_X=\inf\{q\in [2,\infty]\mid X \text{ has cotype q}\},$$ where $q_X$ is taken to be infinity if $X$ has no cotype in $[2,\infty)$. Let $p\in [1,\infty]$ and $C\ge 1$. We say that a Banach space $X$ *contains the $\ell_p^n$'s $C$-uniformly* if for all $n\in \mathbb{N}$, $\ell_p^n$ linearly embeds into $X$ with distortion at most $C$ and that $X$ *uniformly contains the $\ell_p^n$'s* if it contains the $\ell_p^n$'s $C$-uniformly, for some $C\ge 1$. It is known that if $X$ uniformly contains the $\ell_p^n$'s, then it contains the $\ell_p^n$'s $C$-uniformly for all $C>1$. We shall also use the following fundamental result of B. Maurey and G. Pisier [@MaureyPisier1976Studia]: a Banach space $X$ uniformly contains the $\ell_{q_X}^n$'s. We refer the reader to [@MaureyHandbook Theorem 6] for a precise statement and a self contained proof of this result, as well as for a complete survey on these notions. We are now ready to prove the following statement, which is slightly more precise than Theorem [Theorem 3](#ThmCotypeACE){reference-type="ref" reference="ThmCotypeACE"}. **Theorem 15**. *Let $2\le q<p<\infty$. Suppose that $X$ is a Banach space uniformly containing the $\ell_p^n$'s and that $Y$ has cotype $q$ and nontrivial type and suppose also that $0\le \alpha\le\frac{p-q}{p}$. Then there is no coarse map from $X$ to $Y$ that is expanding at rate $\alpha$.* *Proof.* Fix a coarse map $f\colon X\to Y$ which is expanding at rate $\alpha$. In particular, there exists $\rho\colon [0,\infty)\to [0,\infty)$ increasing with $\lim_{t\to \infty}\rho(t)=\infty$ for which $$\label{f-embedding} \|x-z\|\geq \frac12\max\{\|x\|^\alpha,\|z\|^\alpha\}+1\ \text{ implies }\ \|f(x)-f(z)\|\geq \rho(\|x-z\|).$$ Suppose towards a contradiction that $0<\alpha\le\frac{p-q}{p}$, that is, $p\ge \frac{q}{1-\alpha}$. Treating $\ell_{p}^n(\mathbb C)$ as a real Banach space, we have, by assumption, that for each $n\in\mathbb{N}$, there exists an isomorphic embedding $g_n\colon \ell_p^n(\mathbb C)\to X$ with distortion at most $2$. Without loss of generality, assume $$\label{g_n-embedding} \|x\|\leq \|g_n(x)\|\leq 2\|x\|$$ for all $n\in\mathbb{N}$ and all $x\in \ell_p^n(\mathbb C)$. For each $n\in\mathbb{N}$, let $h_n= f\circ g_n$. It easily follows from [\[f-embedding\]](#f-embedding){reference-type="eqref" reference="f-embedding"} and [\[g_n-embedding\]](#g_n-embedding){reference-type="eqref" reference="g_n-embedding"} that for $x,z \in \ell_p^n(\mathbb C)$, $$\|x-z\|\geq \max\{\|x\|^\alpha,\|z\|^\alpha\}+1\ \text{ implies }\ \|h_n(x)-h_n(z)\|\geq \rho(\|x-z\|).$$ Note also that $$\label{EqThmCotypeACE} \omega_{h_n}(t)\leq \omega_f(2t)\ \text{ for all } \ n\in\mathbb{N}\ \text{ and all }\ t\geq 0.$$ It follows from [@MendelNaor2008 Theorem 4.1] that there exists $\Gamma>0$ such that $m_q(Y,n,\Gamma)=O(n^{1/q})$, i.e., there exists $A>0$ such that $m_q(Y,n,\Gamma)\leq An^{1/q}$ for all $n \in \mathbb{N}$ (here is there the hypothesis of $Y$ having nontrivial type in used). On the other hand, by [@MendelNaor2008 Lemma 2.3], we have that $m_q(Y,n,\Gamma)\geq \frac{n^{1/q}}{\Gamma}$. We now apply Lemma [Lemma 14](#LemmaCotypeInequality){reference-type="ref" reference="LemmaCotypeInequality"} with $q=q$, $r=p$ and $s=\frac1q-\frac1p$. Then $0\le \alpha\le \frac{p-q}{p}=\frac{rs}{rs+1}$. It then follows from Lemma [Lemma 14](#LemmaCotypeInequality){reference-type="ref" reference="LemmaCotypeInequality"} and [\[EqThmCotypeACE\]](#EqThmCotypeACE){reference-type="eqref" reference="EqThmCotypeACE"} that $$\rho\Big(2n^{\frac{1}{q}-\frac{1}{p}}\Big) \leq \Gamma A \omega_{h_n}\Big(\frac{2\pi n^{1/q}}{m_q(Y,n,\Gamma)}\Big) \leq\Gamma A \omega_f(4\pi\Gamma)$$ for all $n\in \mathbb{N}$. As $\frac1q-\frac{1}{p}>0$ and $\lim_{t\to \infty}\rho(t)=\infty$, we obtain the expected contradiction. ◻ **Corollary 16**. *Let $1\le q<p<\infty$ and $\alpha\in [0,1]$.* 1. *[\[CorEmblpIntolqItem1\]]{#CorEmblpIntolqItem1 label="CorEmblpIntolqItem1"} If $q\geq 2$, then there is a coarse map $f\colon \ell_p\to \ell_q$ which is also expanding at rate $\alpha$ if and only if $\alpha>\frac{p-q}{p}$.* 2. *[\[CorEmblpIntolqItem2\]]{#CorEmblpIntolqItem2 label="CorEmblpIntolqItem2"} If $q\leq 2<p$, then there is a coarse map $f\colon \ell_p\to \ell_q$ which is also expanding at rate $\alpha$ if and only if $\alpha>\frac{p-2}{p}$.* *Proof.* [\[CorEmblpIntolqItem1\]](#CorEmblpIntolqItem1){reference-type="eqref" reference="CorEmblpIntolqItem1"} Assume first that $\alpha \in (\frac{p-q}{p},1]$. The existence of a coarse map $f\colon \ell_p\to \ell_q$ which is expanding at rate $\alpha$ is ensured by Theorem [Theorem 2](#ThmMazurMapACE){reference-type="ref" reference="ThmMazurMapACE"}. Assume now that $\alpha \in [0,\frac{p-q}{p}]$. Since $\ell_p$ obviously contains the $\ell_p^n$'s uniformly and $\ell_q$ has cotype $q$, Theorem [Theorem 15](#ThmCotype){reference-type="ref" reference="ThmCotype"} implies that there is no coarse map from $\ell_p$ to $\ell_q$ that is expanding at rate $\alpha$. [\[CorEmblpIntolqItem2\]](#CorEmblpIntolqItem2){reference-type="eqref" reference="CorEmblpIntolqItem2"} Since $q\leq 2$, $\ell_q$ has cotype 2. Hence, proceeding as in the previous item, we have that there is no coarse map $f\colon \ell_p\to \ell_q$ which is also expanding at rate $\alpha$ for $\alpha\in [0,\frac{p-2}{p}]$. If $\alpha\geq \frac{p-q}{p}$, the Mazur map $M_{p,q}\colon \ell_p\to \ell_q$ is also expanding at rate $\alpha$ (Theorem [Theorem 2](#ThmMazurMapACE){reference-type="ref" reference="ThmMazurMapACE"}). Finally, if $q\in (\frac{p-2}{p},\frac{p-q}{p}]$, the result follows since the Mazur map $M_{p,2}\colon \ell_p\to \ell_2$ is expanding at rate $\alpha$. Therefore, if $f\colon \ell_2\to \ell_q$ is a coarse embedding (see [@Nowak2006FundMath Theorem 5]), then $f\circ M _{p,2}$ has the required property. ◻ **Corollary 17**. *Let $1\le q<2<p<\infty$, and let $\alpha\in [0,\frac{p-2}{p}]$. Then there is no coarse map $f\colon \ell_p\to \ell_q$ which is also expanding at rate $\alpha$.* *Proof.* The proof is identical to the proof of the second implication in the previous corollary, except that we are now using that $\ell_q$ has cotype $2$. ◻ *Proof of Theorem [Theorem 3](#ThmCotypeACE){reference-type="ref" reference="ThmCotypeACE"}.* If $\alpha=0$, this is [@MendelNaor2008 Theorem 1.11] and if $\alpha=1$, this is obvious. Assume now that $\alpha \in (0,1)$. We may also assume that $q_Y<\infty$. As we have recalled, by [@MaureyPisier1976Studia] (alternatively, see [@MaureyHandbook Theorem 6]), $X$ uniformly contains the $\ell_{q_X}^n$'s. On the other hand, for any $q>q_Y$, $Y$ has cotype $q$. It then follows from Theorem [Theorem 15](#ThmCotype){reference-type="ref" reference="ThmCotype"} that $\alpha >\frac{q_X-q}{q_X}$. Taking the infimum over all $q>q_Y$ yields the conclusion. ◻ **Problem 18**. The following question remains open. Assume that $1\le q<2<p<\infty$ and $\alpha \in (\frac{p-2}{p},\frac{p-q}{p}]$. Does there exists a coarse map from $\ell_p$ to $\ell_q$ that is expanding at rate $\alpha$? All other situations are settled, either by [@MendelNaor2008 Corollary 7.3], or in the present paper. # Embedding of certain graphs and expansion In this section we will show how to use some classical countably branching metric graphs to rule out the existence of coarse maps which are linearly expanding (or just expanding) at rate $1$ or or less between Banach spaces. Our statements will actually be about the stability of some concentration properties for Lipschitz maps on certain graphs under these generalizations of coarse Lipschitz (or just coarse) embeddings. We will deal with two different distances that are defined on the following sets. **Definition 19**. Let $k\in\mathbb{N}$. We denote the set of all subsets of $\mathbb{N}$ with $k$ elements by $[\mathbb{N}]^k$ and, given $\bar n\in[\mathbb{N}]^k$, we write $\bar n=(n_1,\ldots, n_k)$ where $n_1<\ldots< n_k$. ## Hamming graphs We start this section by recalling the definition of the Hamming metric and of the associated concentration property HFC$_p$. **Definition 20**. Let $k\in\mathbb{N}$. The *Hamming metric* $d_{\mathbb H}=d_{\mathbb H,k}$ on $[\mathbb{N}]^k$ is defined as follows: given $\bar n=(n_1,\ldots, n_k),\bar m=(m_1,\ldots, m_k)\in [\mathbb{N}]^k$, we let $$d_{\mathbb H}(\bar n, \bar m)=|\{i\in \{1,\ldots, k\}\mid n_i\neq m_i\}|.$$ Note that $([\mathbb{N}]^k,d_{\mathbb H})$ is a metric graph in the sense that if two elements of $[\mathbb{N}]^k$ are declared to be adjacent if and only if they are at distance $1$, then the distance between arbitrary elements of $[\mathbb{N}]^k$ is the length of the shortest path joining them. In particular, if $X$ is a Banach space and $\phi:([\mathbb{N}]^k,d_{\mathbb H}) \to X$ is a Lipschitz map, then the Lipschiz constant of $\phi$ is $\mathrm{Lip}(\phi)=\omega_\phi(1)$. We now recall the following definition, due to A. Fovelle (see [@Fovelle Subsection 2.5]). **Definition 21** (Definition [Definition 4](#DefHCF){reference-type="ref" reference="DefHCF"}). Let $p\in (1,\infty]$. We say that a Banach space $X$ has the *Hamming full concentration property for $p$* (abbreviated as *HFC$_p$*) if there exists a constant $C\ge 1$ such that for any $k\in \mathbb{N}$ and any $1$-Lipschitz map $\phi\colon ([\mathbb{N}]^k,d_{\mathbb H}) \to X$, there exists an infinite subset $\mathbb M$ of $\mathbb{N}$ such that $$\mathrm{diam}(\phi([\mathbb{N}]^k))\le Ck^{1/p}.$$ In the above situation, we say that $X$ has the *HFC$_p$ with constant $C$*. **Example 22**. N. Kalton and L. Randrianarivony proved in [@KaltonRandrianarivony2008 Theorem 4.2] that, given $p\in (1,\infty)$, any reflexive Banach space which is also $p$-asymptotically uniformly smooth (abbreviated as $p$-AUS) must have HFC$_p$. Although this will not play an important role in these notes, we recall the definition of $p$-AUSness here for the readers convenience: the *modulus of asymptotic uniform smoothness* of a Banach space $X$ is given by $$\bar\rho_X(t)=\sup_{x\in \partial B_X}\inf_{Y\in \mathrm{cof}(X)}\sup_{y\in \partial B_Y}\|x+ty\|-1,$$ where $\mathrm{cof}(X)$ denotes the set of all closed finite codimensional subspaces of $X$. Then, for $p\in (1,\infty)$, $X$ is called *$p$-AUS* if there is $C\geq 1$ such that $\bar \rho_X(t)\leq Ct^p$ for all $t\geq 0$. Clearly, for $p\in (1,\infty)$, $\ell_p$ is $p$-AUS, hence, the cited result above implies that $\ell_p$ has HFC$_p$. But, if $q<p$, $\ell_q$ fails HFC$_p$ as it is witnessed by the maps $$\bar n=(n_1,\ldots,n_k)\in [\mathbb{N}]^k\mapsto \sum_{i=1}^ke_i\in \ell_q, \ k\in\mathbb{N},$$ where $(e_i)_{i=1}^\infty$ is the canonical unit basis of $\ell_q$. **Example 23**. A Banach space $X$ is said to be *asymptotic-$c_0$* if the following happens: $$\begin{gathered} \exists C\geq 1,\ \forall n\in\mathbb{N},\ \exists X_1\in \mathrm{cof}(X), \forall x_1\in B_{X_1},\ \ldots,\ \exists X_n\in \mathrm{cof}(X), \forall x_n\in B_{X_n}\\ \text{ such that } \ \Big\|\sum_{i-1}^na_ix_i\Big\|\leq C \max_{1\leq i\leq n}|a_i|\ \text{ for all } \ (a_i)_{i=1}^n\in \mathbb{R}^\mathbb{N}. \end{gathered}$$ The second named author together with F. Baudier, P. Motakis, and Th. Schlumprecht proved that the property of a Banach space having HFC$_\infty$ is equivalent to it being reflexive and asymptotic-$c_0$ (see [@BaudierLancienMotakisSchlumprecht2018 Theorem B]). *Proof of Theorem [Theorem 5](#ThmPreservationHFCpByLinearExpansion){reference-type="ref" reference="ThmPreservationHFCpByLinearExpansion"}[\[ThmPreservationHFCpByLinearExpansionItem1\]](#ThmPreservationHFCpByLinearExpansionItem1){reference-type="eqref" reference="ThmPreservationHFCpByLinearExpansionItem1"}.* Let $f\colon X\to Y$ be a coarse map which is linearly expanding at rate $1$. We start by fixing some constants: As $f$ is coarse, $\omega_f(1)<\infty$ and it follows that there is $K>1$ such that $$\omega_f(t)\le Kt+K, \text{ for all } t\ge 0$$ (this easy consequence of the triangle inequality and the metric convexity of Banach spaces can be found, for instance, in [@KaltonSurvey Lemma 1.4]). As $f$ is linearly expanding at rate $1$, there is $L>1$ such that $$\label{Eq1ThmPreservationHFCpByLinearExpansion}\|x-z\|\geq \frac{1}{8}\max\{\|x\|,\|z\|\}+1\ \text{ implies }\ \|f(x)-f(z)\|\geq \frac{1}{L}\|x-z\|-L.$$ Let $C_Y\ge 1$ be a constant such that $Y$ has HFC$_p$ with constant $C_Y$ and fix $C>6KLC_Y+6L^2$. Assume towards a contradiction that $X$ does not have HFC$_p$. Then there exist $k\in \mathbb{N}$ and a $1$-Lipschitz map $\phi\colon ([\mathbb{N}]^k,d_{\mathbb H})\to X$ such that $$\mathrm{diam}(\phi([\mathbb M]^k))\ge Ck^{1/p}$$ for all infinite $\mathbb M\subseteq \mathbb{N}$. Let $$\label{eqlambda} \lambda=\inf\{k^{-1/p}\mathrm{diam}(\phi([\mathbb M]^k))\mid \mathbb M\subseteq\mathbb{N}\text{ and }|\mathbb M|=\infty\};$$ so, $\lambda\ge C$ (in particular, $\lambda>12$). Pick now an infinite $\mathbb M\subseteq \mathbb{N}$ such that $$\lambda \le \mathrm{diam}(\phi([\mathbb M]^k))k^{-1/p} \le 2\lambda.$$ In particular, there exists $x\in X$ such that $\phi([\mathbb M]^k)$ is included in the closed ball of radius $2\lambda k^{1/p}$ centered at $x$. By replacing $\phi$ by $\phi-x$, we may assume that $\phi([\mathbb M]^k)$ is included in $2\lambda k^{1/p} B_X$. By the definition of $\lambda$, $\mathrm{diam}(\phi([\mathbb D]^k))\ge \lambda k^{1/p}$ for all infinite $\mathbb D\subseteq \mathbb M$. So, for any such $\mathbb D$, we can find $\bar{n},\bar{m}\in [\mathbb D]^k$ with $\bar n< \bar m$ such that $\|\phi(\bar{n})-\phi(\bar{m})\|\ge \frac{\lambda}{3}k^{1/p}$. Indeed, for all $\bar{n},\bar{m}\in [\mathbb D]^k$, we can find $\bar{p}\in [\mathbb D]^k$ such that $\bar{n}<\bar{p}$ and $\bar{m}<\bar{p}$. Therefore, if $\|\phi(\bar{n})-\phi(\bar{m})\|$ were smaller than $\frac{\lambda}{3}k^{1/p}$ for all $\bar n<\bar m$ in $[\mathbb D]^k$, the triangle inequality would imply that $\mathrm{diam}(\phi([\mathbb D]^k))< \lambda k^{1/p}$; contradiction. Moreover, identifying a pair $(\bar n,\bar m)$, where $\bar n,\bar m\in [\mathbb D]^k$ and $\bar{n}<\bar{m}$, with an element of $[\mathbb M]^{2k}$ and applying Ramsey's theorem, we may furthermore assume, by passing to an infinite subset of $\mathbb M$ if necessary, that $$\label{EqHFCp} \|\phi(\bar{n})-\phi(\bar{m})\|\ge \frac{\lambda}{3}k^{1/p}\text{ for all }\bar n,\bar m\in [\mathbb M]^k\text{ with }\bar{n}<\bar{m}.$$ As $\phi([\mathbb M]^k)$ is included in $2\lambda k^{1/p} B_X$, [\[EqHFCp\]](#EqHFCp){reference-type="eqref" reference="EqHFCp"} and the fact that $\lambda>12$ imply that $$\begin{aligned} \frac{1}{8}\max\Big\{\|\phi(\bar{n})\|,{\|\phi(\bar{m})\|}\Big\}+1 \le \frac{\lambda k^{1/p}}{4}+1 \le \frac{\lambda k^{1/p}}{3} \le \|\phi(\bar{n})-\phi(\bar{m})\| \end{aligned}$$ for all $\bar n,\bar m\in [\mathbb M]^k$ with $\bar{n}<\bar{m}$. Therefore, by [\[Eq1ThmPreservationHFCpByLinearExpansion\]](#Eq1ThmPreservationHFCpByLinearExpansion){reference-type="eqref" reference="Eq1ThmPreservationHFCpByLinearExpansion"}, we must have $$\label{Eq.HFCp2} \|(f\circ\phi)(\bar{n})-(f\circ\phi)(\bar{m})\|\ge \frac{\lambda k^{1/p}}{3L}-L \ge \frac{C k^{1/p}}{3L}-L$$ for all $\bar n,\bar m\in [\mathbb M]^k$ with $\bar{n}<\bar{m}$. On the other hand, as $\phi$ is 1-Lipschitz and $[\mathbb M]^k$ is a metric graph, our choice of $K$ implies that $\omega_{f\circ \phi}(1)\le 2K$ and therefore that $f\circ \phi$ is $2K$-Lipschitz. As $Y$ has HFC$_p$ with constant $C_Y$, an homogeneity argument implies the existence of an infinite subset $\mathbb M'$ of $\mathbb M$ satisfying $$\label{Eq.HFCp3} \mathrm{diam}(f(\phi([\mathbb M']^k)))\le 2KC_Y k^{1/p}.$$ Therefore, [\[Eq.HFCp2\]](#Eq.HFCp2){reference-type="eqref" reference="Eq.HFCp2"} and [\[Eq.HFCp3\]](#Eq.HFCp3){reference-type="eqref" reference="Eq.HFCp3"} imply that $$\label{Eq.HFCp4}\frac{C}{3L}\leq 2KC_Y+L.$$ This contradicts our choice of $C$. ◻ Combining this result and Example [Example 22](#example HFC){reference-type="ref" reference="example HFC"}, we obtain immediately the following. **Corollary 24**. *Let $p,q \in [1,\infty)$ and assume that $p<q$. Then there is no coarse map from $\ell_p$ to $\ell_q$ that is linearly expanding at rate $1$.* **Remark 25**. It is important to recall here that if $p>q$, then the Mazur map $M_{p,q}:\ell_p \to \ell_q$ is coarse and linearly expanding at rate $1$ (Theorem [Theorem 2](#ThmMazurMapACE){reference-type="ref" reference="ThmMazurMapACE"}). In case $p=\infty$, a stronger version of Theorem [Theorem 5](#ThmPreservationHFCpByLinearExpansion){reference-type="ref" reference="ThmPreservationHFCpByLinearExpansion"} holds. Precisely, HFC$_\infty$ is preserved by coarse maps which are expanding at rate $1$; no need for linear expansion here. Since the argument is completely analogous, we only indicate the mild differences in the proof below. *Proof of Theorem [Theorem 5](#ThmPreservationHFCpByLinearExpansion){reference-type="ref" reference="ThmPreservationHFCpByLinearExpansion"}[\[ThmPreservationHFCpByLinearExpansionItem2\]](#ThmPreservationHFCpByLinearExpansionItem2){reference-type="eqref" reference="ThmPreservationHFCpByLinearExpansionItem2"}.* The proof follows almost verbatim the one of Theorem [Theorem 5](#ThmPreservationHFCpByLinearExpansion){reference-type="ref" reference="ThmPreservationHFCpByLinearExpansion"}[\[ThmPreservationHFCpByLinearExpansionItem1\]](#ThmPreservationHFCpByLinearExpansionItem1){reference-type="eqref" reference="ThmPreservationHFCpByLinearExpansionItem1"}. Precisely, let $f\colon X\to Y$ be a coarse map which is expanding at rate $1$ and $K$ and $C_Y$ be chosen as in the proof of Theorem [Theorem 5](#ThmPreservationHFCpByLinearExpansion){reference-type="ref" reference="ThmPreservationHFCpByLinearExpansion"}[\[ThmPreservationHFCpByLinearExpansionItem1\]](#ThmPreservationHFCpByLinearExpansionItem1){reference-type="eqref" reference="ThmPreservationHFCpByLinearExpansionItem1"}. The expansion property of $f$ gives us $\rho\colon [0,\infty)\to [0,\infty)$ with $\lim_{t\to \infty}\rho(t)=\infty$ and such that $$\|x-z\|\geq \frac{1}{8}\max\{\|x\|,\|z\|\}+1 \ \text{ implies }\ \|f(x)-f(z)\|\geq \rho(\|x-z\|).$$ Since $\lim_{t\to \infty}\rho(t)=\infty$, we can choose $C>0$ such that $\rho(C /3)>2KC_Y$ and, assuming that $X$ does not have HFC$_\infty$, we obtain $k\in\mathbb{N}$ and a $1$-Lipschitz map $\phi\colon ([\mathbb{N}]^k,d_{\mathbb H})\to X$ such that $$\mathrm{diam}(\phi([\mathbb M]^k))\geq C$$ for all infinite $\mathbb M\subseteq \mathbb{N}$. Defining $\lambda$ as in [\[eqlambda\]](#eqlambda){reference-type="eqref" reference="eqlambda"} (here $1/\infty=0$) and letting $\mathbb M\subseteq \mathbb{N}$ be as in the proof of Theorem [Theorem 5](#ThmPreservationHFCpByLinearExpansion){reference-type="ref" reference="ThmPreservationHFCpByLinearExpansion"}, the proof then proceeds verbatim until [\[Eq.HFCp2\]](#Eq.HFCp2){reference-type="eqref" reference="Eq.HFCp2"} which in this case is replaced by $$\label{InftyEq.HFCp2} \|(f\circ\phi)(\bar{n})-(f\circ\phi)(\bar{m})\|\ge \rho\Big(\frac{\lambda}{3}\Big) > 2KC_Y$$ for all $\bar n,\bar m\in [\mathbb M]^k$ with $\bar{n}<\bar{m}$. Then, [\[Eq.HFCp3\]](#Eq.HFCp3){reference-type="eqref" reference="Eq.HFCp3"} becomes $$\mathrm{diam}(f(\phi([\mathbb M']^k)))\le 2KC_Y.$$ and both those inequalities put together give us a contradiction. ◻ *Proof of Corollary [Corollary 6](#CorollaryHFCInfty){reference-type="ref" reference="CorollaryHFCInfty"}.* By [@BaudierLancienMotakisSchlumprecht2018 Theorem B], we know that having HFC$_\infty$ is equivalent to being reflexive and asymptotic-$c_0$. The result is then an immediate consequence of Theorem [Theorem 5](#ThmPreservationHFCpByLinearExpansion){reference-type="ref" reference="ThmPreservationHFCpByLinearExpansion"}[\[ThmPreservationHFCpByLinearExpansionItem2\]](#ThmPreservationHFCpByLinearExpansionItem2){reference-type="eqref" reference="ThmPreservationHFCpByLinearExpansionItem2"}. ◻ ## Interlacing pairs in Hamming graphs {#SubsectionInterlacingHamming} Property HFC$_p$, for $p\in (1,\infty]$, implies reflexivity (this follows from [@BaudierKaltonLancien2010Studia Theorem 4.1]). Therefore this property is not relevant to study embeddings between non reflexive Banach spaces. With the goal of addressing this problem, the following weakening of the HFC$_p$ was introduced in [@LancienRaja2017Houston] and formalized in [@Fovelle Subsection 2.5]. **Definition 26**. Let $\mathbb M$ be an infinite subset of $\mathbb{N}$ and let $k\in \mathbb{N}$. The set of *strictly interlacing* pairs in $[\mathbb M]^k$ is given by $$I_k(\mathbb M)=\{(\bar n,\bar m)\in [\mathbb M]^k\times [\mathbb M]^k\mid \ n_1< m_1< n_2< m_2< \cdots < n_k < m_k\}.$$ Given a Banach space $X$ and $p\in (1,\infty]$, we say that $X$ has the *Hamming interlaced concentration property for $p$*, abbreviated as *HIC*$_p$, if there exists a constant $C\ge 1$ such that for any $k\in \mathbb{N}$ and any $1$-Lipschitz map $\phi\colon ([\mathbb{N}]^k,d_{\mathbb H}) \to X$, there exists an infinite subset $\mathbb M$ of $\mathbb{N}$ such that $$\|f(\bar n)-f(\bar m)\|\le Ck^{1/p},\ \text{for all}\ (\bar n,\bar m) \in I_k(\mathbb M).$$ **Example 27**. Recall, given $p\in (1,\infty)$, that the James sequence space $J_p$ is defined by $$\begin{aligned} J_p=\Big\{(x(n))_n\in \mathbb{R}^\mathbb{N}\mid & \lim_nx(n)=0\ \text{ and }\\ &\|x\|_{J_p}=\sup_{p_1<\ldots<p_n}\Big(\sum_{i=2}^n|x(p_{i})-x(p_{i-1})|^p\Big)^{1/p}<\infty\Big\} .\end{aligned}$$ So, the classic James sequence space $J$ is simply $J_2$ and, just as $J$, each $J_p$ has codimension $1$ in its bidual; which makes them quasi-reflexive Banach spaces. M. Raja and the second named author proved in [@LancienRaja2017Houston Theorem 2.2] that, for $p\in (1,\infty)$, James space $J_p$ has HIC$_p$ and that, for $q<p$, $J_q$ fails HIC$_p$. We can prove the following variant of Theorem [Theorem 5](#ThmPreservationHFCpByLinearExpansion){reference-type="ref" reference="ThmPreservationHFCpByLinearExpansion"}. Notice that the assumption on the rate of expansion has been weakened. **Theorem 28**. *Let $X$ and $Y$ be Banach spaces and suppose $Y$ has HIC$_p$ for some $p\in (1,\infty)$. If there is a coarse map $f\colon X\to Y$ which is linearly expanding at rate $\frac1p$, then $X$ must have HIC$_p$.* *Proof.* The proof is similar to the one of Theorem [Theorem 5](#ThmPreservationHFCpByLinearExpansion){reference-type="ref" reference="ThmPreservationHFCpByLinearExpansion"}, but we need to detail some of the minor modifications. So, let $f\colon X\to Y$ be as in the statement and $C_Y\ge 1$ be such that $Y$ has HIC$_p$ with constant $C_Y$. There exists $K\ge 1$ so that $\omega_f(t)\le Kt+K$ and $L\ge 1$ such that $$\label{Eq2ThmPreservationHFCpByLinearExpansion}\|x-z\|\geq \max\{\|x\|^{1/p},\|z\|^{1/p}\}+1\ \text{ implies }\ \|f(x)-f(z)\|\geq \frac{1}{L}\|x-z\|-L.$$ Assume that $X$ fails HIC$_p$ and fix $C>3$ (to be precised later). Then there exist $k\in \mathbb{N}$ and a $1$-Lipschitz map $\phi\colon ([\mathbb{N}]^k,d_{\mathbb H})\to X$ such that for any infinite $\mathbb M\subseteq \mathbb{N}$ there exists $(\bar n,\bar m) \in I_k(\mathbb M)$ so that $\|\phi(\bar n)-\phi(\bar m)\|> Ck^{1/p}$. Since $\Phi$ is 1-Lipschitz and $\mathrm{diam}([\mathbb{N}]^k)=k$, we may assume that $\phi([\mathbb{N}]^k)\subseteq kB_X$. Let $$\label{eqlambdabis} \lambda=\inf\Big\{ \sup_{(\bar n,\bar m)\in I_k(\mathbb M)}\|\phi(\bar n)-\phi(\bar m)\|k^{-1/p}\mid\ \mathbb M\subseteq\mathbb{N}\text{ and }|\mathbb M|=\infty\Big\}.$$ So, $\lambda\ge C$. Pick now an infinite $\mathbb M\subseteq \mathbb{N}$ such that $$\lambda \le \sup_{(\bar n,\bar m)\in I_k(\mathbb M)}\|\phi(\bar n)-\phi(\bar m)\|k^{-1/p} < 2\lambda.$$ By the definition of $\lambda$, for all infinite $\mathbb D\subseteq \mathbb M$, there exists $(\bar n,\bar m)\in I_k(\mathbb D)$ such that $\|\phi(\bar n)-\phi(\bar m)\|> \frac23 \lambda k^{1/p}$. Identifying $I_k(\mathbb D)$ with $[\mathbb D]^{2k}$ and using Ramsey's theorem, we can therefore assume that $\|\phi(\bar n)-\phi(\bar m)\|> \frac23 \lambda k^{1/p}$ for all $(\bar n,\bar m) \in I_k(\mathbb M)$. It follows that for all $(\bar n,\bar m) \in I_k(\mathbb M)$: $$\begin{aligned} \|\phi(\bar n)-\phi(\bar m)\|\ge \frac{2C}{3} k^{1/p} \ge \frac{C}{3} k^{1/p}+1 \ge \max\{\|\phi(\bar n)\|^{1/p},\|\phi(\bar m)\|^{1/p}\}+1.\end{aligned}$$ It then follows from ([\[Eq2ThmPreservationHFCpByLinearExpansion\]](#Eq2ThmPreservationHFCpByLinearExpansion){reference-type="ref" reference="Eq2ThmPreservationHFCpByLinearExpansion"}) that $$\|(f \circ \phi)(\bar n)-(f \circ \phi)(\bar m)\|\ge \frac{2C}{3L}k^{1/p}-L,$$ for all $(\bar n,\bar m) \in I_k(\mathbb M)$. On the other hand $f \circ \phi$ is $2K$-Lipschitz and $Y$ has property HIC$_p$ with constant $C_Y$, so there exists an infinite subset $\mathbb M'$ of $\mathbb M$ so that $\|(f \circ \phi)(\bar n)-(f \circ \phi)(\bar m)\|\le 2KC_Yk^{1/p}$, for all $(\bar n,\bar m) \in I_k(\mathbb M')$. This yields a contradiction for an initial large enough choice of $C$ (depending on $K,L,C_Y$). ◻ **Remark 29**. In the case $p=\infty$, the analogous statement is that HIC$_\infty$ is preserved by coarse maps expanding at rate $0$, in other words by coarse embeddings. This was already noticed in [@Fovelle]. **Problem 30**. Let $p\in (1,\infty]$. We do not know if the property HIC$_p$ is preserved by coarse maps that are linearly expanding at rate 1. As an immediate application of Theorem [Theorem 28](#ThmInterlacingHamming){reference-type="ref" reference="ThmInterlacingHamming"} and Example [Example 27](#ExampleJamesSpace){reference-type="ref" reference="ExampleJamesSpace"}, we get the following. **Corollary 31**. *Let $p,q \in [1,\infty)$ and assume that $p<q$. Then there is no coarse map from $J_p$ to $J_q$ that is linearly expanding at rate $1/q$. 0◻* ## Interlacing graphs We now deal with the interlacing metric of $[\mathbb{N}]^k$, which was introduced by N. Kalton in [@Kalton2007] to rule out the coarse embeddability of $c_0$ into reflexive spaces. We start by recalling the definition of the interlaced metric and of property $\mathcal Q_p$. **Definition 32**. Let $k\in\mathbb{N}$. The *interlacing metric* $d_{\mathbb I}=d_{\mathbb I,k}$ on $[\mathbb{N}]^k$ is defined as follows: we endow $[\mathbb{N}]^k$ with a graph structure by letting distinct $\bar n=(n_1,\ldots, n_k),\bar m=(m_1,\ldots, m_k)\in [\mathbb{N}]^k$ be adjacent if either $$n_1\le m_1\le \ldots \le n_k\le m_k\ \text{ or }\ m_1\le n_1\le \ldots\le m_k\le n_k.$$ The metric $d_{\mathbb I}$ is then the shortest path metric on $[\mathbb{N}]^k$ with respect to this graph structure. **Definition 33** (Definition [Definition 7](#DefiPropQ){reference-type="ref" reference="DefiPropQ"}). Let $p\in (1,\infty]$. We say that a Banach space $X$ has property $\mathcal Q_p$ if there exists a constant $C\ge 1$ such that for any $k\in \mathbb{N}$ and any $1$-Lipschitz map $\phi\colon ([\mathbb{N}]^k,d_{\mathbb I}) \to X$, there exists an infinite subset $\mathbb M$ of $\mathbb{N}$ such that $$\mathrm{diam}(\phi([\mathbb{N}]^k))\le Ck^{1/p}.$$ (here if $p=\infty$, we use the convention $1/\infty=0)$. If $p=\infty$, we simply say, following [@Kalton2007], that $X$ has *property $\mathcal Q$*. **Example 34**. N. Kalton proved in [@Kalton2007 Corollary 4.3] that reflexive Banach spaces have property $\mathcal Q$. In [@BragaLancienPetitjeanProchazka2023JTopAna], it is shown that, for $p\in (1,\infty)$, a dual of a $p$-AUS Banach space has $\mathcal Q_{p'}$, where $p'$ is the conjugate of $p$ and that the dual of an asymptotic-$c_0$ space has $\mathcal Q$ ([@BragaLancienPetitjeanProchazka2023JTopAna Theorem 4.1]). We now extend the result insuring the stability of property $\mathcal Q_p$ under coarse Lipschitz embeddings. *Proof of Theorem [Theorem 8](#ThmPreservationPropertyQpByLinearExpansion){reference-type="ref" reference="ThmPreservationPropertyQpByLinearExpansion"}[\[ThmPreservationPropertyQpByLinearExpansionItem1\]](#ThmPreservationPropertyQpByLinearExpansionItem1){reference-type="eqref" reference="ThmPreservationPropertyQpByLinearExpansionItem1"}.* Again the proof follows the lines of the argument for Theorem [Theorem 5](#ThmPreservationHFCpByLinearExpansion){reference-type="ref" reference="ThmPreservationHFCpByLinearExpansion"}[\[ThmPreservationHFCpByLinearExpansionItem1\]](#ThmPreservationHFCpByLinearExpansionItem1){reference-type="eqref" reference="ThmPreservationHFCpByLinearExpansionItem1"}. let $f\colon X\to Y$ be as in the statement and $C_Y$ be such that $Y$ has $\mathcal Q_p$ with constant $C_Y$. Then the constants $K,L,C$ are defined as in the proof of Theorem [Theorem 5](#ThmPreservationHFCpByLinearExpansion){reference-type="ref" reference="ThmPreservationHFCpByLinearExpansion"}[\[ThmPreservationHFCpByLinearExpansionItem1\]](#ThmPreservationHFCpByLinearExpansionItem1){reference-type="eqref" reference="ThmPreservationHFCpByLinearExpansionItem1"} and assume that $X$ fails $\mathcal Q_p$. Then there exist $k\in \mathbb{N}$ and a $1$-Lipschitz map $\phi\colon ([\mathbb{N}]^k,d_{\mathbb I})\to X$ such that for any infinite $\mathbb M\subseteq \mathbb{N}$, $$\mathrm{diam}(\phi([\mathbb M]^k))\geq C$$ Let $$\lambda=\inf\{\mathrm{diam}(\phi([\mathbb M]^k))k^{-1/p}\mid \mathbb M\subseteq\mathbb{N}\text{ and }|\mathbb M|=\infty\};$$ and we pick an infinite $\mathbb M\subseteq \mathbb{N}$ such that $$\lambda \le \mathrm{diam}(\phi([\mathbb M]^k))k^{-1/p} < 2\lambda.$$ So assume, as we may, that $\phi([\mathbb M]^k) \subseteq 2\lambda k^{1/p}B_X$. By the definition of $\lambda$ and arguing as in the proof of Theorem [Theorem 5](#ThmPreservationHFCpByLinearExpansion){reference-type="ref" reference="ThmPreservationHFCpByLinearExpansion"}[\[ThmPreservationHFCpByLinearExpansionItem1\]](#ThmPreservationHFCpByLinearExpansionItem1){reference-type="eqref" reference="ThmPreservationHFCpByLinearExpansionItem1"}, we may assume that for all $\bar n < \bar m \in [\mathbb M]^k$, $\|\phi(\bar{n})-\phi(\bar{m})\|\ge \frac{\lambda}{3}k^{1/p}$. As $\phi([\mathbb M]^k)$ is included in $2\lambda k^{1/p}\cdot B_X$, we deduce similarly that $$\|(f\circ\phi)(\bar{n})-(f\circ\phi)(\bar{m})\|\ge \frac{\lambda k^{1/p}}{3L}-L \ge \frac{C k^{1/p}}{3L}-L$$ for all $\bar n,\bar m\in [\mathbb M]^k$ with $\bar{n}<\bar{m}$. The contradiction then follows exactly as in the proof of Theorem [Theorem 5](#ThmPreservationHFCpByLinearExpansion){reference-type="ref" reference="ThmPreservationHFCpByLinearExpansion"}. ◻ *Proof of Theorem [Theorem 8](#ThmPreservationPropertyQpByLinearExpansion){reference-type="ref" reference="ThmPreservationPropertyQpByLinearExpansion"}[\[ThmPreservationPropertyQpByLinearExpansionItem2\]](#ThmPreservationPropertyQpByLinearExpansionItem2){reference-type="eqref" reference="ThmPreservationPropertyQpByLinearExpansionItem2"}.* We just have to adapt similarly the proof of Theorem [Theorem 5](#ThmPreservationHFCpByLinearExpansion){reference-type="ref" reference="ThmPreservationHFCpByLinearExpansion"}[\[ThmPreservationHFCpByLinearExpansionItem2\]](#ThmPreservationHFCpByLinearExpansionItem2){reference-type="eqref" reference="ThmPreservationHFCpByLinearExpansionItem2"}. We leave the details to the reader. ◻ Asymptotic uniform convexity is often used, together with the approximate midpoint principle to find obstructions to the coarse Lipschitz embeddability. This is how it is shown that $\ell_p$ does not coarse Lipschitz embed into $\ell_q$ for $p>q$ (see [@JohnsonLindenstraussSchechtman1996GAFA]). We cannot find an approximate midpoint principle for coarse maps that are linearly expanding at rate $1$ and, as we already emphasized, there is a good reason for that: it follows from Theorem [Theorem 2](#ThmMazurMapACE){reference-type="ref" reference="ThmMazurMapACE"} that $M_{p,q}:\ell_p \to \ell_q$ is a coarse map from that is linearly expanding at rate $1$ when $p>q$. However, the situation is different for the family of James spaces $(J_p)_p$ and the use of property $Q$ and its variants can serve as a substitute to obtain some preservation of asymptotic uniform convexity. **Corollary 35**. *Let $p,q \in (1,\infty)$ and assume that $p>q$. Then there is no coarse map from $J_p$ to $J_q$ that is linearly expanding at rate $1$.* *Proof.* It is proved by the authors, C. Petitjean and A. Procházka in [@BragaLancienPetitjeanProchazka2023JTopAna Corollaries 3.4 and 3.8] that, for $p\in (1,\infty)$, $J_p$ has property $\mathcal Q_{p'}$, where $p'$ is the conjugate exponent of $p$, but fails $\mathcal Q_{r}$ for all $r>p$. Then the conclusion follows from Theorem [Theorem 8](#ThmPreservationPropertyQpByLinearExpansion){reference-type="ref" reference="ThmPreservationPropertyQpByLinearExpansion"}[\[ThmPreservationPropertyQpByLinearExpansionItem2\]](#ThmPreservationPropertyQpByLinearExpansionItem2){reference-type="eqref" reference="ThmPreservationPropertyQpByLinearExpansionItem2"}. ◻ # Embeddings into $\ell_\infty$ {#SectionlInfty} In this last section, we prove that Lipschitz embeddability into $\ell_\infty$ is equivalent to the existence of a coarse map which is linearly expanding at some rate strictly smaller than $1$. We start with an intermediate result. **Proposition 36**. *Let $\alpha\in (0,1)$. Let $X$ be a Banach space and suppose there is a Lipschitz map $f\colon X\to \ell_\infty$ which is also linearly expanding at rate $\alpha$. Then $X$ Lipschitzly embeds into $\ell_\infty$.* *Proof.* Fix $L>0$ such that, for all $x,z \in X$, $$\|x-z\|\geq L(\max\{\|x\| ,\|z\|\})^\alpha +L\ \text{ implies }\ \|f(x)-f(z)\|\geq \frac{1}{L}\|x-z\|-L.$$ Denote $\mathbb{Q}_+$ the set of positive rational numbers and define a map $F\colon X\to \ell_\infty(\mathbb{Q}_+,\ell_\infty)$ by letting $$F(x)=(q^{-1}f(qx))_{q\in \mathbb{Q}_{+}},\ \ x\in X.$$ Since, $$\|F(x)-F(z)\|=\sup_{q\in \mathbb{Q}_{+}}q^{-1}\|f(qx)-f(qz)\|\leq \mathrm{Lip}(f)\|x-z\|,$$ for all $x,z \in X$, we have that $\mathrm{Lip}(F)\leq \mathrm{Lip}(f)$. Fix now $x,z\in X$ with $x\neq z$. As $\alpha\in (0,1)$, there is $t>0$ large enough so that $$\Big\|\frac{tx}{\|x-z\|}-\frac{tz}{\|x-z\|}\Big\|=t\geq L \Big( \max\Big\{\frac{\|tx\|}{\|x-z\|},\frac{\|tz\|}{\|x-z\|}\Big\}\Big)^\alpha+L.$$ Taking an even larger $t$ if necessary, we can also assume that $L<\frac{t}{2L}$. We may also assume that $q=\frac{t}{\|x-z\|} \in \mathbb{Q}_+$. We obtain that $$\begin{aligned} \|F(x)-F(z)\|&\geq \frac{\|x-z\|}{t}\Big\|f\Big(\frac{tx}{\|x-z\|}\Big)-f\Big(\frac{tz}{\|x-z\|}\Big)\Big\|\\ &\geq \frac{\|x-z\|}{t}\Big(\frac{t}{L}-L\Big)\geq \frac{1}{2L}\|x-z\| \end{aligned}$$ As $x$ and $z$ were arbitrary, this shows that $F$ is a Lipschitz embedding from $X$ into $\ell_\infty(\mathbb{Q}_+,\ell_\infty)$, which is clearly isometric to $\ell_\infty$. ◻ *Proof of Theorem [Theorem 9](#ThmlInfty){reference-type="ref" reference="ThmlInfty"}.* The implication [\[ThmlInftyItem1\]](#ThmlInftyItem1){reference-type="eqref" reference="ThmlInftyItem1"}$\Rightarrow$[\[ThmlInftyItem2\]](#ThmlInftyItem2){reference-type="eqref" reference="ThmlInftyItem2"} is immediate. Suppose [\[ThmlInftyItem2\]](#ThmlInftyItem2){reference-type="eqref" reference="ThmlInftyItem2"} holds and let $f\colon X\to \ell_\infty$ be such a map. Let $N\subseteq X$ be a net of $X$, i.e., for some $\delta,\varepsilon>0$ the set $N$ is $\delta$-separated and $\varepsilon$-dense in $X$. Since $f$ is coarse and $X$ is metrically convex, as we have already seen (cf [@KaltonSurvey Lemma 1.4]), $f$ is coarse-Lipschitz, in fact, we have that $$\|f(x)-f(z)\|\leq \omega_f(1)\|x-z\|+\omega_f(1)$$ for all $x,z\in X$. Therefore, as $N$ is $\delta$-separated, $f\restriction N$, the restriction of $f$ to $N$, is Lipschitz with $\mathrm{Lip}(f\restriction N)\leq \omega_f(1)(1+1/\delta)=C$. Since any Lipschitz map into $\ell_\infty$ can be extended to larger subsets without increasing its Lipschitz constant (see [@KaltonSurvey Section 3.3]), there is a Lipschitz map $h\colon X\to \ell_\infty$ such that $h\restriction N=f\restriction N$ and $\mathrm{Lip}(h)=C$. As $N$ is $\varepsilon$-dense in $X$ and $f$ and $h$ coincide on $N$, we easily deduce that $\|f-h\|\le C\varepsilon+ \omega_f(\varepsilon)$ on $X$. It then follows that $h$ is also linearly expanding at rate $\alpha$. Then, by Proposition [Proposition 36](#PropositionlInfty){reference-type="ref" reference="PropositionlInfty"}, there exists a Lipschitz embedding from $X$ into $\ell_\infty$ and [\[ThmlInftyItem1\]](#ThmlInftyItem1){reference-type="eqref" reference="ThmlInftyItem1"} follows. ◻ **Problem 37**. We do not know whether we can take $\alpha=1$ in the statement of Theorem [Theorem 9](#ThmlInfty){reference-type="ref" reference="ThmlInfty"}. **Problem 38**. It is proved in [@Kalton2011FundMath Theorem 5.3] that the Lipschitz embeddability of a Banach space $X$ into $\ell_\infty$ is in fact equivalent to its coarse embeddability. We do not know whether it is also equivalent to the existence of coarse map from $X$ to $\ell_\infty$ that is expanding at some nontrivial rate $\alpha \in (0,1]$; notice that Theorem [Theorem 9](#ThmlInfty){reference-type="ref" reference="ThmlInfty"} assumes *linear* expansion at some rate $\alpha\in (0,1)$. BLMS21 F. Baudier, N. Kalton, and G. Lancien. A new metric invariant for Banach spaces. , 199(1):73--94, 2010. Y. Benyamini and J. Lindenstrauss. , volume 48 of *American Mathematical Society Colloquium Publications*. American Mathematical Society, Providence, RI, 2000. B. M. Braga and G. Lancien. . , page arXiv:2302.12016, February 2023. F. Baudier, G. Lancien, P. Motakis, and Th. Schlumprecht. A new coarsely rigid class of Banach spaces. , 20(5):1729--1747, 2021. B. M. Braga, G. Lancien, C. Petitjean, and A. Procházka. On Kalton's interlaced graphs and nonlinear embeddings into dual Banach spaces. , 15(2):467--494, 2023. F. Baudier, G. Lancien, and Th. Schlumprecht. The coarse geometry of Tsirelson's space and applications. , 31(3):699--717, 2018. B. M. Braga. Coarse and uniform embeddings. , 272(5):1852--1875, 2017. A. Fovelle. Hamming graphs and concentration properties in non-quasi-reflexive banach spaces. , 48(3):539--579, 2022. W. Johnson, J. Lindenstrauss, and G. Schechtman. Banach spaces determined by their uniform structures. , 6(3):430--470, 1996. N. Kalton. Coarse and uniform embeddings into reflexive spaces. , 58(3):393--414, 2007. N. Kalton. The nonlinear geometry of Banach spaces. , 21(1):7--60, 2008. N. Kalton. Lipschitz and uniform embeddings into $\ell_\infty$. , 212(1):53--69, 2011. N. Kalton. Examples of uniformly homeomorphic banach spaces. , 104:151--182, 2013. N. Kalton. Uniform homeomorphisms of Banach spaces and asymptotic structure. , 365(2):1051--1079, 2013. N. Kalton and L. Randrianarivony. The coarse Lipschitz geometry of $l_p\oplus l_q$. , 341(1):223--237, 2008. G. Lancien and M. Raja. Asymptotic and coarse Lipshitz structures of quasi-reflexive Banach spaces. , 44(3):927--940, 2018. B. Maurey. Type, cotype and $K$-convexity. In *Handbook of the geometry of Banach spaces, Vol. 2*, pages 1299--1332. North-Holland, Amsterdam, 2003. S. Mazur. Une remarque sur l'homéomorphie des champs fonctionnels. , 1(1):83--85, 1929. M. Mendel and A. Naor. Metric cotype. , 168(1):247--298, 2008. B. Maurey and G. Pisier. Séries de variables aléatoires vectorielles indépendantes et propriétés géométriques des espaces de Banach. , 58(1):45--90, 1976. P. Nowak. On coarse embeddability into $l_p$-spaces and a conjecture of Dranishnikov. , 189(2):111--116, 2006. C. Rosendal. Equivariant geometry of Banach spaces and topological groups. , 5:Paper No. e22, 62, 2017. [^1]: B. M. Braga was partially supported by FAPERJ (Proc. E-26/200.167/2023) and by CNPq (Proc. 303571/2022-5).

arxiv_math

--- abstract: | We prove that for any possibly-punctured surface with non-empty boundary $\mathbf{\Sigma}=(\Sigma, \mathbb{M}, \mathbb{P})$, and any tagged triangulation $T$ of $\mathbf{\Sigma}$ in the sense of Fomin--Shapiro--Thurston, the coefficient-free bangle functions of Musiker--Schiffler--Williams coincide with the coefficient-free generic Caldero--Chapoton functions arising from the Jacobian algebra of the quiver with potential $(Q(T), W(T))$ associated to $T$ by Cerulli Irelli and the second author. When the set of boundary marked points $\mathbb{M}$ has at least two elements, Schröer and the first two authors have shown, relying heavily on results of Mills, Muller and Qin, that the generic coefficient-free Caldero-Chapoton functions form a basis of the coefficient-free (upper) cluster algebra $\mathcal{A}(\mathbf{\Sigma})=\mathcal{U}(\mathbf{\Sigma})$. So, the set of bangle functions proposed by Musiker--Schiffler--Williams over ten years ago is indeed a basis. address: - Christof Geiss Instituto de Matemáticas, UNAM, Mexico - Daniel Labardini-Fragoso Instituto de Matemáticas, UNAM, Mexico - Jon Wilson Jeremiah Horrocks Institute, University of Central Lancashire, UK author: - Christof Geiss - Daniel Labardini-Fragoso - Jon Wilson bibliography: - Biblio_bases.bib date: October 5, 2023 title: Bangle functions are the generic basis for cluster algebras from punctured surfaces with boundary --- # Introduction In just over 20 years, the cluster algebras of Sergey Fomin and Andrei Zelevinsky [@fomin2002cluster; @fomin2003cluster; @fomin2007cluster] have pervaded Mathematics and enjoyed significant impact even in Theoretical Physics. The theory has been accompanied since the beginning by the search for natural bases. Naturalness has come to mean that whatever set is proposed as a basis, it should contain the set of all cluster monomials, shown in [@cerulli2013linear] to be indeed linearly independent in the skew-symmetric case. Several authors have proposed various sets as bases for cluster algebras, with different levels of generality [@Berenstein2014triangular; @dupont2013atomic; @felikson2017bases; @geiss2012generic; @gross2018canonical; @lee2016theExistence; @musiker2013bases; @qin2017triangular]. Among these proposals, three have attracted particular attention: **Generic Caldero-Chapoton functions.** Together with Leclerc and Schröer, the first author explored in [@geiss2012generic] the algebraic geometry of (affine) varieties of representations of Jacobi-finite non-degenerate quivers with potential $(Q,S)$ which appear in algebraic Lie theory. They isolated a class of irreducible components of these representation spaces with good geometric-homological properties and called them *strongly reduced* irreducible components. Later on, the adjective used was changed to *generically $\tau$-reduced*, or simply, *$\tau$-reduced*, cf. [@geiss2022schemes]. Roughly and informally speaking, a component $Z\subseteq \operatorname{rep}(\mathcal{P}(Q,S),\mathbf{d})$ is $\tau$-reduced if the codimension in $Z$ of a highest-dimensional $\operatorname{GL}_{\mathbf{d}}(\mathbb{C})$-orbit $\mathcal{O}\subseteq Z$ is equal to the minimum value taken on $Z$ by Derksen-Weyman-Zelevinsky's $E$-invariant. It turns out that each $\tau$-reduced irreducible component $Z$ has an open dense subset where the *Caldero-Chapoton function* $M\mapsto CC_{\mathcal{P}(Q,S)}(M)$ takes a constant value, which is then denoted $CC_{\mathcal{P}(Q,S)}(Z)$. Resorting to decorated representations and spaces of decorated representations in order to be able to hit initial cluster variables through the Caldero-Chapoton function, the set of *generic Caldero-Chapoton functions* is $$\mathcal{B}_{\mathcal{P}(Q,S)}:=\{CC_{\mathcal{P}(Q,S)}(Z)\ | \ \text{$Z$ is a $\tau$-reduced irreducible component}\}.$$ If this set is a basis of the (coefficient-free) upper cluster algebra $\mathcal{U}(Q)$, then it is called the *generic basis* of such algebra. One of the main results of [@geiss2012generic] is, that in the setting of unipotent cells for symmetric Kac-Moody groups, the generic Caldero--Chapoton functions can be identified with the dual of Lusztig's semicanonical basis [@lusztig2000semicanonical], and that this basis contains all cluster monomials. It is in fact quite easy to see with the help of [@derksen2010quivers] that for each Jacobi-finite non-degenerate quiver with potential $(Q,S)$ the set $\mathcal{B}_{\mathcal{P}(Q,S)}$ contains all cluster monomials of the cluster algebra, and that all elements of $\mathcal{B}_{\mathcal{P}(Q,S)}$ belong to the upper cluster algebra corresponding to $Q$. This led the authors of [@geiss2012generic] to conjecture that the set $\mathcal{B}_{\mathcal{P}(Q,S)}$ might be a basis of the upper cluster algebra in more situations beyond the algebraic Lie theoretic context. The set $\mathcal{B}_{\mathcal{P}(Q,S)}$ does not change under mutations as a subset of $\mathcal{U}(Q)$, as shown by Plamondon when $(Q,S)$ is Jacobi-finite non-degenerate [@plamondon2013generic], and by Schröer and the first two authors when $(Q,S)$ is arbitrary non-degenerate [@geiss2020generic]. Furthermore, Schröer and the first two authors showed, relying heavily on results of Mills [@mills2017maximal], Muller [@muller2013locally; @muller2014A=U] and Qin [@qin2019bases], that when $(Q,S)=(Q(T),S(T))$ for some triangulation $T$ of a surface $\mathbf{\Sigma}=(\Sigma,\mathbb{M},\mathbb{P})$ with at least two marked points on the boundary, $\mathcal{B}_{\mathcal{P}(Q,S)}$ is indeed a basis for the (upper) cluster algebra, where $S(T)$ is the potential defined by Cerulli Irelli and the second author in [@cerulli2012quivers]. **Bangle and bracelet functions.** A class of cluster algebras that, for many different reasons, has turned out to be very prominent, is formed by those that arise from surfaces with marked points thanks to works of Fock--Goncharov [@fock2007dual], Fomin--Shapiro--Thurston [@fomin2008cluster], Fomin--Thurston [@fomin2018cluster], Gekhtman--Shapiro--Vainshtein [@gekhtman2005cluster] and Penner [@penner2012decorated]. A *surface with marked points* is a triple $\mathbf{\Sigma}=(\Sigma,\mathbb{M},\mathbb{P})$ consisting of a compact, connected, oriented two-dimensional real differentiable manifold $\Sigma$ with (possibly empty) boundary $\partial\Sigma$, a finite set $\mathbb{M}\subseteq\partial\Sigma$ containing at least one point from each boundary component of $\Sigma$, and a finite set $\mathbb{P}\subseteq\Sigma\setminus\partial\Sigma$. It is required that $\mathbb{M}\cup\mathbb{P}\neq\varnothing$. The elements of $\mathbb{M}\cup\mathbb{P}$ are called *marked points* and the elements of $\mathbb{P}$ are called *punctures*. If $\mathbb{P}=\varnothing$, it is said that $\mathbf{\Sigma}$ is *unpunctured*, whereas if $\mathbb{P}\neq\varnothing$, one says that $\mathbf{\Sigma}$ is punctured. To associate a (coefficient-free) cluster algebra to $\mathbf{\Sigma}$, one considers (tagged) triangulations of $\mathbf{\Sigma}$; these are (tagged) triangulations of $\Sigma$ that have $\mathbb{M}\cup\mathbb{P}$ as prescribed set of vertices. Each such triangulation $T$ has an associated skew-symmetric matrix $B_{\operatorname{FST}}(T)$, and every time two tagged triangulations are related by the flip of a tagged arc, their associated matrices are related by the corresponding matrix mutation. This already enables us to take any tagged triangulation $T_0$, and any $T_0$-tuple of algebraically independent rational functions $\mathbf{x}=(x_\gamma)_{\gamma\in T_0}$, and produce a cluster algebra $\mathcal{A}(B_{\operatorname{FST}}(T_0),\mathbf{x})$ that, in a well-defined sense, is independent of the triangulation $T_0$ chosen. A fundamental theorem of Fomin--Shapiro--Thurston [@fomin2008cluster Theorems 5.6 and 7.11] states that providing $\mathbf{\Sigma}$ is not a closed surface with at most two punctures, there is a bijection $\gamma\mapsto x_\gamma$ between the set of tagged arcs on $\mathbf{\Sigma}$ and the set of cluster variables of $\mathcal{A}(B_{\operatorname{FST}}(T_0),\mathbf{x})$, such that the induced assignment $T\mapsto(B_{\operatorname{FST}}(T),(x_\gamma))_{\gamma_\in T}$ is a bijection between the set of triangulations of $\mathbf{\Sigma}$ and the set of (unlabelled) seeds of $\mathcal{A}(B_{\operatorname{FST}}(T_0),\mathbf{x})$, such that whenever two tagged triangulations are related by the flip of a tagged arc, the seeds corresponding to them are related by the mutation of seeds corresponding to the arc being flipped. Given a surface $\mathbf{\Sigma}=(\Sigma,\mathbb{M},\mathbb{P})$ and a tagged triangulation $T$ of $\mathbf{\Sigma}$, Musiker--Schiffler--Williams associated to each tagged arc (resp. simple closed curve) $\alpha$ on $\mathbf{\Sigma}$, a bipartite graph $G(T,\alpha)$, which they called *snake graph* (resp. *band graph*) [@musiker2011positivity; @musiker2013bases], with edges and *tiles* naturally labelled by the arcs in $T$ (and the boundary segments of $\mathbf{\Sigma}$). Using these labels, they assigned a *weight* monomial $w(P):=x(P)y(P)$ to each perfect (resp. *good*) matching $P$ of $G(T,\alpha)$, thus producing a generating function $\sum_{P}w(P)$ for the perfect matchings (resp. *good* matchings) of $G(T,\alpha)$. Dividing this generating function by the monomial $\operatorname{cross}(T,\alpha)$ that records the number of crossings of $\alpha$ with each arc in $T$, they define the *bangle function* $$\label{eq:general-MSW-expression-intro} x_\alpha:=\frac{\sum_{P}w(P)}{\operatorname{cross}(T,\alpha)}.$$ Furthermore, they define the bangle function associated to a lamination $L$ of $\mathbf{\Sigma}$ to be $x_L:=\prod_{\alpha\in L}x_\alpha^{m_{\alpha}},$ where $m_\alpha$ is the multiplicity of the single laminate $\alpha$ as a member of $L$. The set of bangle functions is $$\mathcal{B}^\circ(\mathbf{\Sigma}):=\{x_L\ | \ \text{$L$ is a lamination of $\mathbf{\Sigma}$}\}.$$ The set of *bracelet functions* $\mathcal{B}(\mathbf{\Sigma})$ is defined similarly, except that if a simple closed curve $\gamma$ appears with multiplicity $m_\gamma$ in a lamination $L$, then in order to form the Laurent polynomial $x_L$ one does not take $x_\gamma^{m_\gamma}$ as factor corresponding to $\gamma$, but rather the expression [\[eq:general-MSW-expression-intro\]](#eq:general-MSW-expression-intro){reference-type="eqref" reference="eq:general-MSW-expression-intro"} corresponding to the *bracelet* $\alpha$ obtained by concatenating $\gamma$ with itself $m_\gamma$ times as a closed curve. Musiker--Schiffler--Williams show in [@musiker2013bases] that if the underlying surface $(\Sigma,\mathbb{M},\mathbb{P})$ does not have punctures whatsoever and has at least two marked points (i.e., $\mathbb{P}=\varnothing$ and $|\mathbb{M}|\geq 2$), then for the coefficient-free cluster algebra associated to $(\Sigma,\mathbb{M},\mathbb{P})$, and for any cluster algebra associated to $(\Sigma,\mathbb{M},\mathbb{P})$ under full-rank extended exchange matrices $\left[\begin{array}{c}B_{\operatorname{FST}}(T)\\ B'\end{array}\right]$, both the set of bangle functions and the set of bracelet functions form bases, each containing the cluster monomials by [@musiker2011positivity]. They conjecture in [@musiker2013bases] that $\mathcal{B}^{\circ}$ and $\mathcal{B}$ form bases also in the punctured case. **Theta functions.** To each skew-symmetrizable matrix $B$ satisfying certain *injectivity assumption* related to the full-rankness of exchange matrices, Gross--Hacking--Keel--Kontsevich associate in the highly influencial [@gross2018canonical] a *consistent scattering diagram* $\mathfrak{D}$, and through the consideration of certain algebraic-combinatorial objects called *broken lines*, define a set of *theta functions* and prove that it is a basis for the cluster algebra(s) associated to $B$, with the extra property that the Laurent expansion of any cluster monomial with respect to any chosen initial seeds has only non-negative coefficients, thus providing a full proof of Fomin--Zelevinsky's *positivity conjecture* from [@fomin2007cluster]. Recently, Mandel--Qin [@mandel2023bracelets] have shown that when $B=B(T)$ for some triangulation of a surface $\mathbf{\Sigma}=(\Sigma,\mathbb{M},\mathbb{P})$, Gross--Hacking--Keel--Kontsevich's *theta basis* is equal to the set of bracelet functions $\mathcal{B}(\mathbf{\Sigma})$, thus proving, through a combination with [@gross2018canonical], that the set of bracelet functions is indeed a basis. Our main result in the present paper, Theorem [Theorem 52](#thm:bangle-equals-generic-basis){reference-type="ref" reference="thm:bangle-equals-generic-basis"}, states that for surfaces with non-empty boundary, the set of generic coefficient-free Caldero-Chapoton functions $\mathcal{B}_{\mathcal{P}(Q(T),S(T))}$ coincides with the set of bangle functions, thus proving, through a combination with [@geiss2020generic], that for surfaces with at least two marked points on the boundary, the set of bangle functions is indeed a basis of the coefficient-free (upper) cluster algebra $\mathcal{A}(Q(T))=\mathcal{U}(Q(T))$. Our proof relies heavily on [@geiss2023laminations]. The paper is organized as follows. In Section [2](#sec:preliminaries){reference-type="ref" reference="sec:preliminaries"} we provide the basics about mutations and cluster algebras from surfaces. In Section [3](#sec:laminations-as-components){reference-type="ref" reference="sec:laminations-as-components"} we summarize the main constructions and results from [@geiss2023laminations] upon which we will rely in Section [6](#sec:bangle-functions-are-the-generic-basis){reference-type="ref" reference="sec:bangle-functions-are-the-generic-basis"}, the most relevant one being [@geiss2023laminations Proposition 2.8] (stated as Proposition [\[prp:JaMut\]](#prp:JaMut){reference-type="ref" reference="prp:JaMut"} below), which gives a precise recursive formula for the change that the generic projective $g$-vector of any $\tau$-reduced component undergoes under mutation of $\tau$-reduced components. In Section [4](#base case section){reference-type="ref" reference="base case section"} we show that for simple closed curve $\alpha$, there exists a tagged $T_\alpha$ for which one can explicitly compute a (decorated) representation $\mathcal{M}(T_\alpha,\alpha)$ such that - the projective $g$-vector $\mathbf{g}_{A(T_\alpha)}(\mathcal{M}(T_\alpha,\alpha))$ is equal to the vector of dual shear coordinates $\operatorname{Sh}_{T_\alpha}(\alpha)$ and to the snake $g$-vector $\mathbf{g}_{G(T_\alpha,\alpha)}$ coming from the snake or band graph $G(T_\alpha,\alpha)$; - the representation-theoretic (dual) $F$-polynomial $F_{M(T_\alpha,\alpha)}$ is equal to the snake $F$-polynomial $F_{G(T_\alpha,\alpha)}$ coming from the snake or band graph $G(T_\alpha,\alpha)$; - the orbit closure of $\mathcal{M}(T_\alpha,\alpha)$ is a $\tau$-reduced component $Z_{T_\alpha,\alpha}$ and any decorated representation mutation-equivalent to $\mathcal{M}(T_\alpha,\alpha)$ is generic, for the values of the projective $g$-vector, the representation-theoretic (dual) $F$-polynomial and Derksen-Weyman-Zelevinsky's $E$-invariant, in the corresponding $\tau$-reduced component. In Section [5](#CKL){reference-type="ref" reference="CKL"} we show that every time a flip of tagged triangulations is applied, the changes undergone by the snake $g$-vectors and snake $F$-polynomials of the band graphs constructed from $\alpha$, obey the same recursive formulas that govern the changes undergone by the projective $g$-vectors and representation-theoretic (dual) $F$-polynomials. In Section [6](#sec:bangle-functions-are-the-generic-basis){reference-type="ref" reference="sec:bangle-functions-are-the-generic-basis"} we combine the results from Sections [4](#base case section){reference-type="ref" reference="base case section"} and [5](#CKL){reference-type="ref" reference="CKL"} to deduce our main result. In Section [7](#sec:remarks-and-problems){reference-type="ref" reference="sec:remarks-and-problems"} we make some remarks about the extent to which the representations that yield the bangle functions have been really explicitly computed, and state a couple of open problems. # Preliminaries {#sec:preliminaries} ## Cluster algebras This section provides a brief review of (skew-symmetric) cluster algebras of geometric type. Let $n \leq m$ be positive integers. Furthermore, let $\mathcal{F}$ be the field of rational functions in $m$ independent variables. Fix a collection $X_1,\ldots,X_n, x_{n+1},\ldots, x_m$ of algebraically independent variables in $\mathcal{F}$. We define the *coefficient ring* to be $\mathbb{ZP}:=\mathbb{Z}[x_{n+1}\ldots x_m]$. **Definition 1**. A *(labelled) seed* consists of a pair, $(\mathbf{x},\mathbf{y}, B)$, where - $\mathbf{x} = (x_1,\ldots x_n)$ is a collection of variables in $\mathcal{F}$ which are algebraically independent over $\mathbb{ZP}$, - $\mathbf{y} = (y_1,\ldots y_n)$ where $y_k = \displaystyle \prod_{j=n+1}^{m} x_j^{b_{jk}}$ for some $b_{jk} \in \mathbb{Z}$, - $B = (b_{jk})_{j,k \in \{1,\ldots, n\}}$ is an $n \times n$ skew-symmetric integer matrix. The variables in any seed are called *cluster variables*. The variables $x_{n+1},\ldots,x_m$ are called *frozen variables*. We refer to $\mathbf{y}$ as the *choice of coefficients*. **Definition 2**. Let $(\mathbf{x},\mathbf{y},B)$ be a seed and let $k \in \{1,\ldots,n\}$. We define a new seed $\mu_{k}(\mathbf{x},\mathbf{y},B) := (\mathbf{x}',\mathbf{y}',B')$, called the *mutation* of $(\mathbf{x},\mathbf{y},B)$ at $k$ where: - $\mathbf{x}' = (x'_1,\ldots x'_n)$ is defined by $$x'_i = \frac{\displaystyle \prod_{b_{ik} >0} x_k^{b_{ik}} + \prod_{b_{ik} <0} x_k^{-b_{ik}}}{x_i}$$ and setting $x_j' = x_j$ when $j \neq k$; - $\mathbf{y}'$ and $B' = (b'_{ij})$ are defined by the following rule: $$b'_{ij} = \begin{cases} -b_{ij},& \text{if $i=k$ or $j=k$,}\\ b_{ij} + \max(0,-b_{ik})b_{kj} + b_{ik}\max(0,b_{kj}),& \text{otherwise.}\\ \end{cases}$$ A *quiver* is a finite directed (multi) graph $Q= (Q_0,Q_1)$ where $Q_0$ is the set of vertices and $Q_1$ is the set of directed edges. It will often be convenient to encode (extended) skew-symmetric matrices as quivers. We describe this simple relationship in the definition below, which follows the conventions set out in the work of Derksen, Weyman and Zelevinsky [@derksen2010quivers]. **Definition 3**. Given a skew-symmetric $n\times n$ matrix $B$ we define a quiver $Q(B)= (Q_0, Q_1)$ by setting $Q_0 =\{1,\ldots, n\}$ and demanding that for any $i,j \in Q_0$ there are $[b_{ij}]_{+}$ arrows from $j$ to $i$ in $Q_1$. We generalise the definition to extended $m\times n$ skew-symmetric matrices $\tilde{B}$ in the obvious way. **Definition 4**. Fix an $(\mathbf{x},\mathbf{y}, B)$. If we label the initial cluster variables of $\mathbf{x}$ from $1,\ldots, n$ then we may consider the labelled n-regular tree $\mathbb{T}_n$. Each vertex in $\mathbb{T}_n$ has $n$ incident edges labelled $1, \ldots, n$. Vertices of $\mathbb{T}_n$ represent seeds and the edges correspond to mutation. In particular, the label of the edge indicates which direction the seed is being mutated in. Let $\mathcal{X}$ be the set of all cluster variables appearing in the seeds of $\mathbb{T}_n$. The *cluster algebra* of the seed $(\mathbf{x},\mathbf{y}, B)$ is defined as $\mathcal{A}(\mathbf{x},\mathbf{y}, B ) := \mathbb{ZP}[\mathcal{X}]$. We say $\mathcal{A}(\mathbf{x},\mathbf{y}, B)$ is the *cluster algebra with principal coefficients* if $m = 2n$ and $\mathbf{y} = (y_1, \ldots, y_n)$ satisfies $y_k = x_{n+k}$ for any $k \in \{1,\ldots, n\}$. ## Cluster algebras from surfaces {#Cluster algebras from surfaces} In this subsection we recall the work of Fomin, Shapiro and Thurston [@fomin2008cluster], which establishes a cluster structure for triangulated orientable surfaces. Let $\Sigma$ be a compact (connected) Riemann surface. Fix a finite set $\mathbb{M}\subset\partial\Sigma$ of marked points on the boundary, such that $\mathbb{M}$ contains at least one element from each connected component of $\partial\Sigma$. Moreover we fix a finte set $\mathbb{P}\subset\Sigma\setminus\partial\Sigma$ of *punctures* in the interior of $\Sigma$. We refer to the triple $\mathbf{\Sigma}=(\Sigma, \mathbb{M}, \mathbb{P})$ as a *bordered surface (with marked points)*. For technical reasons we exclude the cases where $\mathbf{\Sigma}$ is an unpunctured or once-punctured monogon; a digon; a triangle; or a once, twice or thrice punctured sphere. **Definition 5**. An *arc* of $\mathbf{\Sigma}$ is a simple curve in $\Sigma$ connecting two marked points of $\mathbb{M}\cup\mathbb{P}$, which is not homotopic to a boundary segment or a marked point. We consider arcs up to homotopy (relative to its endpoints) and up to inversion of the orientation. **Definition 6**. An *arc* $\gamma$ is an arc whose endpoints have been 'tagged' in one of two ways; *plain* or *notched*. Moreover, this tagging must satisfy the following conditions: if the endpoints of $\gamma$ share a common marked point, they must receive the same tagging; and an endpoint of $\gamma$ lying on the boundary $\partial S$ must always receive a plain tagging. In this paper we shall always consider tagged arcs up to the equivalence induced from the equivalence relation for plain arcs. **Definition 7**. Let $\alpha$ and $\beta$ be two tagged arcs of $\mathbf{\Sigma}$. We say $\alpha$ and $\beta$ are *compatible* if and only if the following conditions are satisfied: - There exist homotopic representatives of $\alpha$ and $\beta$ that don't intersect in the interior of $\Sigma$. - Suppose the untagged versions of $\alpha$ and $\beta$ do not coincide. If $\alpha$ and $\beta$ share an endpoint $p$ then the ends of $\alpha$ and $\beta$ at $p$ must be tagged in the same way. - Suppose the untagged versions of $\alpha$ and $\beta$ do coincide. Then precisely one end of $\alpha$ must be tagged in the same way as the corresponding end of $\beta$. A *tagged triangulation* of $\mathbf{\Sigma}$ is a maximal collection of pairwise compatible tagged arcs of $\mathbf{\Sigma}$. Moreover, this collection is forbidden to contain any tagged arc that enclose a once-punctured monogon. An *ideal triangulation* of $\mathbf{\Sigma}$ is a maximal collection of pairwise compatible plain arcs. Note that ideal triangulations decompose $\mathbf{\Sigma}$ into triangles, however, the sides of these triangles may not be distinct; two sides of the same triangle may be glued together, resulting in a *self-folded triangle*. **Remark 8**. To each tagged triangulation $T$ we may uniquely assign an ideal triangulation $T^{\circ}$ as follows: - If $p$ is a puncture with more than one incident notch, then replace all these notches with plain taggings. - If $p$ is a puncture with precisely one incident notch, and this notch belongs to $\beta \in T$, then replace $\beta$ with the unique arc $\gamma$ of $\mathbf{\Sigma}$ which encloses $\beta$ and $p$ in a monogon. Conversely, to each ideal triangulation $T$ we may uniquely assign a tagged triangulation $\iota(T)$ by reversing the second procedure described above. **Definition 9**. Let $T$ be a tagged triangulation, and consider its associated ideal triangulation $T^{\circ}$. We may label the arcs of $T^{\circ}$ from $1,\ldots, n$ (note this also induces a natural labelling of the arcs in $T$). We define a function, $\pi_{T} : \{1,\ldots,n\} \rightarrow \{1,\ldots,n\}$, on this labelling as follows: $$\pi_{T}(i) = \begin{cases} j & \text{if $ i $ is the folded side of a self-folded triangle in $ T^{\circ}$,}\\ &\text{and $j$ is the remaining side;}\\ i & \text{otherwise.}\\ \end{cases}$$ For each non-self-folded triangle $\Delta$ in $T^{\circ}$, as an intermediary step, define the matrix $B_T^{\Delta} = (b^{\Delta}_{jk})$ by setting $$b^{\Delta}_{jk} = \begin{cases} 1 & \text{if $\Delta$ has sides $ \pi_{T}(j)$ and $ \pi_{T}(k)$, and $ \pi_{T}(k) $ follows $ \pi_{T}(j) $}\\ & \text{in the clockwise sense;}\\ -1 & \text{if $ \Delta$ has sides $ \pi_{T}(j)$ and $\pi_{T}(k) $, and $ \pi_{T}(k) $ precedes $\pi_{T}(j)$}\\ & \text{in the clockwise sense;}\\ 0 & \text{otherwise.}\\ \end{cases}$$ The *adjacency matrix* $B_T = (b_{ij})$ of $T$ is then defined to be the following summation, taken over all non-self-folded triangles $\Delta$ in $T^{\circ}$: $$B_{\operatorname{FST}}(T):= \sum\limits_{\Delta} B_T^{\Delta}$$ **Definition 10**. Let $T$ be a tagged triangulation of a bordered surface $\mathbf{\Sigma}$. Consider the initial seed $(\mathbf{x},\mathbf{y},B_T)$, where: $\mathbf{x}$ contains a cluster variable for each arc in $T$; $B_T$ is the matrix defined in Definition [Definition 9](#adjacency matrix){reference-type="ref" reference="adjacency matrix"}; and $\mathbf{y}$ is any choice of coefficients. We call $\mathcal{A}(\mathbf{x}, \mathbf{y}, B_T)$ a *surface cluster algebra*. **Proposition 11** (Theorem 7.9, [@fomin2008cluster]). *Let $T$ be a tagged triangulation of a bordered surface $\mathbf{\Sigma}$. Then for any $\gamma \in T$ there exists a unique tagged arc $\gamma'$ on $\mathbf{\Sigma}$ such that $f_{\gamma}(T) := (T\setminus \{\gamma\}) \cup \{\gamma'\}$ is a tagged triangulation. We call $f_{\gamma}(T)$ the *flip* of $T$ with respect to $\gamma$.* **Theorem 12** (Theorem 6.1, [@fomin2008cluster]). *Let $\mathbf{\Sigma}$ be a bordered surface. If $\mathbf{\Sigma}$ is not a once punctured closed surface, then in the cluster algebra $\mathcal{A}(\mathbf{x}, \mathbf{y}, B_T)$, the following correspondence holds: $$\begin{aligned} &\hspace{23mm} \mathbf{\mathcal{A}(\mathbf{x}, \mathbf{y}, \text{$B_T$})} & & &\mathbf{\Sigma}\hspace{32mm}& \\ &\hspace{18mm} \emph{Cluster variables} &\longleftrightarrow& &\emph{Tagged arcs} \hspace{23mm} & \\ &\hspace{25mm}\emph{Clusters} &\longleftrightarrow& &\emph{Tagged triangulations} \hspace{14mm}& \\ &\hspace{24mm} \emph{Mutation} &\longleftrightarrow& &\emph{Flips of tagged arcs} \hspace{16mm}& \\\end{aligned}$$* *When $\mathbf{\Sigma}$ is a once-punctured closed surface then cluster variables are in bijection with all plain arcs or all notched arcs depending on whether $T$ consists solely of plain arcs or notched arcs, respectively.* # Laminations as $\tau$-reduced components {#sec:laminations-as-components} ## Tame partial KRS-monoids The simple-minded notion of *partial Krull-Remak-Schmidt monoid* was introduced in [@geiss2023laminations] as a convenient abstract framework to state the naturalness of the bijection between laminations and $\tau$-reduced components proved therein. **Definition 13**. [@geiss2023laminations Definition 2.1] A *partial monoid* is a triple $(X, e, \oplus)$ consisting of a set $X$, a symmetric function $e: X\times X\rightarrow\mathbb{N}$ and a partially-defined sum $\oplus: \{(x,y)\in X\times X\mid e(x,y)=0\}\rightarrow X$ such that: - if $e(x,y)=0$ we have $x\oplus y =y\oplus x$; - there exists a unique element $0\in X$ with $e(0,x)=0$ and $0\oplus x=x$ for all $x\in X$; - if $e(y,z)=0$ we have $e(x, y\oplus z)=e(x,y)+e(x,z)$ for all $x\in Z$; - $(x\oplus y)\oplus z= z\oplus (y\oplus z)$ whenever one side of the equation is defined. A *morphism* of partial monoids from $X=(X, e,\oplus)$ to $X'=(X',e',\oplus')$ is a function $f: X\rightarrow Y$, such that $e'(f(x), f(y))=e(x,y)$ for all $x,y\in X$ and $f(x\oplus y)= f(x)\oplus' f(y)$ whenever $e(x,y)=0$. **Remark 14**. (1) Suppose $(x_1\oplus x_2)\oplus x_3$ is defined. By $(d)$, we have $e(x_i, x_j)=0$ for $1\leq i<j\leq 3$, hence $x_1\oplus (x_2\oplus x_3)$ is defined. \(2\) Suppose that we have $x_1, x_2, \ldots, x_n\in X$ with $e(x_i, x_j)=0$ for all $i<j$, then $x_1\oplus x_2\oplus\cdots\oplus x_n\in X$ is well-defined, and for each permutation $\sigma\in\mathfrak{S}_n$ we have $$x_1\oplus x_2\oplus\cdots\oplus x_n= x_{\sigma(1)}\oplus x_{\sigma(2)}\oplus\cdots\oplus x_{\sigma(n)}.$$ **Definition 15**. [@geiss2023laminations Definition 2.3] [\[def:KRS2\]]{#def:KRS2 label="def:KRS2"} Suppose $X=(X, e, \oplus)$ is a partial monoid. - The elements of the set $$X_{\mathrm{ind}}:=\{x\in X\setminus\{0\} \mid x= y\oplus z \text{ implies } y=0 \text{ or } z=0\}$$ are called *indecomposable elements* of $X$. - We say that $X$ is a *partial KRS-monoid* if every $x\in X$ is equal to a finite direct sum of indecomposable elements, and whenever $$x_1\oplus x_2\oplus\cdots\oplus x_m= y_1\oplus y_2\oplus\cdots\oplus y_n$$ with $x_1, \ldots, x_m, y_1, \ldots, y_m\in X_{\mathrm{ind}}$, necessarily $m=n$ and, moreover, there exists a permutation $\sigma\in\mathfrak{S}_n$ such that $y_i=x_{\sigma(i)}$ for all $i=1,2,\ldots, n$. - We say that $X$ is *tame* if $e(x,x)=0$ for all $x\in X$. - A *framing* for $X$ is a map $\mathbf{g}: X\rightarrow\mathbb{Z}^n$ (for some non-negative integer $n$) such that $\mathbf{g}(x\oplus y)=\mathbf{g}(x)+\mathbf{g}(y)$ for all $x, y\in C$ with $e(x,y)=0$. A framing is *faithful* if it is an injective function. - A framed partial monoid $X=(X, e, \oplus,\mathbf{g})$ is *free of rank* $n$ if the framing $\mathbf{g}: X\rightarrow\mathbb{Z}^n$ is bijective. **Example 16**. [@geiss2023laminations Example 2.4][\[expl:KRS1\]]{#expl:KRS1 label="expl:KRS1"} Suppose $C$ is a set equipped and $e: C\times C\rightarrow\mathbb{N}$ is a function such that $e(c,c)=0$ for all $c\in C$. Then $$\operatorname{KRS}(C,e):=\{f: C\rightarrow\mathbb{N}\mid c_1, c_2\in\operatorname{supp}(f)\Rightarrow e(c_1, c_2)=0 \text{ and } \lvert \operatorname{supp}(f)\rvert<\infty \}$$ is a tame partial KRS-monoid with $$e(f,g):= \sum_{c,d\in C} f(c)\cdot g(d)\cdot e(c,d), \qquad \text{and} \qquad (f\oplus g)(c):= f(c)+g(c).$$ Notice that $$\operatorname{KRS}(C,e)_{\mathrm{ind}}=\{\delta_c:C\rightarrow\mathbb{N}\mid c\in C\},\quad\text{where}\quad \delta_c(d):=\delta_{c,d} \quad\text{(Kronecker's \emph{delta})}.$$ With this notation we have $f=\oplus_{c\in C} \delta_c^{\oplus f(c)}$ for all $f\in\operatorname{KRS}(C,e)$. **Remark 17**. For a tame partial KRS-monoid $X=(X, e, \oplus)$ we have $X\cong\operatorname{KRS}(X_{\mathrm{ind}}, e')$, where $e'$ is the restriction of $e$ to $X_{\mathrm{ind}}\times X_{\mathrm{ind}}$. In this situation, any map $\mathbf{g}: C\rightarrow\mathbb{Z}^n$ can be extended to a framing $\mathbf{g}:\operatorname{KRS}(C, e)\rightarrow\mathbb{Z}^n$ by simply setting $\mathbf{g}(f):=\sum_{c\in C} f(c)\mathbf{g}(c)$. ## KRS-monoids of $\tau$-reduced components {#ssec:FD-KRS} Let $A$ be a basic finite dimensional algebra over an algebraically closed field $\Bbbk$. We may identify the Grothendieck group $K_0(A)$ of $A$ with $\mathbb{Z}^n$ for some non-negative integer $n$. The set $\operatorname{DecIrr}^\tau(A)$ of decorated, generically $\tau$-reduced, irreducible components of the representation varieties of $A$ becomes a partial KRS-monoid under the function $$e_A: \operatorname{DecIrr}^\tau(A)\times \operatorname{DecIrr}^\tau(A)\rightarrow\mathbb{N}$$ that to each pair $(X,Y)\in \operatorname{DecIrr}^\tau(A)\times \operatorname{DecIrr}^\tau(A)$ associates the generic value $e_A(X,Y)$ that on the Zariski product $X\times Y$ takes the symmetrized $E$-invariant $$\label{eq:def-of-sym-proj-E-inv} E_A((M,\mathbf{v}),(N,\mathbf{w})):=\dim\operatorname{Hom}_A(M,\tau N)+\dim\operatorname{Hom}_A(N,\tau M)+ \underline{\dim}(M)\cdot\mathbf{w}+ \mathbf{v}\cdot\underline{\dim}(N);$$ and the partially-defined sum $$\oplus: \{(X,Y)\in \operatorname{DecIrr}^\tau(A)\times \operatorname{DecIrr}^\tau(A)\mid e(x,y)=0\}\rightarrow\operatorname{DecIrr}^\tau(A)$$ given by the direct sum of $e_A$-orthogonal irreducible components, $X\oplus Y:=\overline{X\oplus Y}$. The function $\mathbf{g}_A:\operatorname{DecIrr}^\tau(A)\rightarrow\mathbb{Z}^n$ that to each $X\in \operatorname{DecIrr}^\tau(A)$ associates the generic value taken on $X$ by the $g$-vector $$\mathbf{g}_A(M,\mathbf{v}):= (\dim\operatorname{Hom}(S_i,\tau M) -\dim(M, S_i)+ v_i)_{i\in Q_0}$$ is a framing for the partial KRS-monoid $(\operatorname{DecIrr}^\tau(A),e_A,\oplus)$. More precisely: **Theorem 18**. *Let $A$ a finite-dimensional basic algebra over an algebraically closed field. Let $n$ be the rank of the Grothendieck group of $A$.* - *$\operatorname{DecIrr}^\tau(A)=(\operatorname{DecIrr}^\tau(A), e_A, \oplus, \mathbf{g}_A)$ is a framed, free KRS-monoid of rank $n$. The subset $\operatorname{DecIrr}^\tau_{\mathrm{ind}}(A)$ of components, which contain a dense set of indecomposable representations is precisely the set of indecomposable elements in the sense of Definition [\[def:KRS2\]](#def:KRS2){reference-type="ref" reference="def:KRS2"}.* - *With the framing from the generic $g$-vector, $\operatorname{DecIrr}^\tau(A)$ is a free partial KRS-monoid of rank $n$.* - *If $A$ is of tame representation type, then $\operatorname{DecIrr}^\tau(A)$ is tame in the sense of Definition [\[def:KRS2\]](#def:KRS2){reference-type="ref" reference="def:KRS2"}. Consequently, there is an isomorphism of partial KRS-monoids $$\operatorname{DecIrr}^\tau(A)\cong \operatorname{KRS}(\operatorname{DecIrr}^\tau_{\mathrm{ind}}(A), e_A).$$ Moreover in this case each $Z\in\operatorname{DecIrr}^\tau_{\mathrm{ind}}(A)$ contains either a dense orbit, or a one-parameter family of bricks.* **Remark 19**. 1. In the form it is stated, Theorem [Theorem 18](#thm:tau-red-comps-form-a-KRS-monoid){reference-type="ref" reference="thm:tau-red-comps-form-a-KRS-monoid"} appeared in [@geiss2023laminations]. Its assertions are restatements of previous results by various authors: Part (a) is a well-known combination of [@crawley2002irreducible Theorem 1.2] and [@cerulli2015caldero-chapoton Theorems 1.3 and 1.5]. Part (b) is a theorem of Plamondon [@plamondon2013generic]. Part (c) is [@geiss2023semicontinuous Corollary 1.7] and [@geiss2022schemes Theorem 3.2]. 2. In the context of cluster algebras it is often convenient to work also with the set $\operatorname{DecIrr}^{\tau^-}(A)$ of decorated, generically $\tau^-$-reduced, irreducible components of the representation varieties of $A$, which becomes a framed partial KRS-monoid under the function defined by the generic values of the symmetrized injective $E$-invariant $$\begin{aligned} E_A^{\operatorname{inj}}((M,\mathbf{v}), (N,\mathbf{w}))&:=\dim\operatorname{Hom}_A(\tau^{-1}M, N) + \dim\operatorname{Hom}_A(\tau^{-1}N,M)+ \\ &\quad +\mathbf{v}\cdot\underline{\dim}(N)+ \underline{\dim}(M)\cdot\mathbf{w},\end{aligned}$$ with framing defined by the generic values of the injective $g$-vector $$\mathbf{g}_A^{\operatorname{inj}}(M,\mathbf{v}):= (\dim\operatorname{Hom}_A(\tau^{-1}M, S_i)-\dim\operatorname{Hom}_A(S_i,M) + v_i)_{i\in Q_0}$$ Duality over the ground field $\Bbbk$ induces a natural isomorphism of framed partial KRS-monoids $$(\operatorname{DecIrr}^{\tau^-}(A), e_{A}^{\operatorname{inj}}, \oplus, \mathbf{g}_{A}^{\operatorname{inj}})\cong (\operatorname{DecIrr}^\tau(A^{\operatorname{op}}), e_{A^{\operatorname{op}}}, \oplus, \mathbf{g}_{A^{\operatorname{op}}}).$$ Suppose now that the finite-dimensional $\Bbbk$-algebra $A$ is the Jacobian algebra of a non-degenerate quiver with potential $(Q,W)$. Let $B=(b_{ij})_{i,j}\in\mathbb{Z}^{Q_0\times Q_0}$ be the matrix with entries $$b_{ij}= \lvert \{\alpha\in Q_1\mid \alpha:i\rightarrow j \ \text{in} \ Q\}\rvert -\lvert \{ \beta\in Q_0\mid \beta:j\rightarrow i \ \text{in} \ Q\}\rvert.$$ Thus, $B$ is the transpose of the skew-symmetric matrix $B_{\operatorname{DWZ}}(Q)$ associated by Derksen--Weyman--Zelevinsky to $Q$ in [@derksen2010quivers Equation (1.4)]. Because of this, later on we will have to consider *dual Caldero-Chapoton functions* as in [@geiss2022schemes Sec. 11.3]. Following [@derksen2008quivers Section 10], for each representation $M$ of $A=\mathcal{P}_{\Bbbk}(Q, W)$ and each vertex $k\in Q_0$ we consider the triangle of linear maps $$\xymatrix{ & M(k) \ar[rd]^{M(\beta_k)}\\ M_{\operatorname{in}}(k)\ar[ru]^{M(\alpha_k)}&&\ar[ll]^{M(\gamma_k)} M_{\operatorname{out}}(k) }.$$ In view of [@derksen2010quivers Proposition 10.4 and Remark 10.8], we have $$\begin{aligned} \label{eq:gvec} \begin{split} \mathbf{g}_A(M,\mathbf{0}) &= (\dim\operatorname{coker}(M(\gamma_k))-\dim M(k))_{k\in Q_0}\\ = \mathbf{g}_{A^{\operatorname{op}}}^{\operatorname{inj}}(DM,\mathbf{0}) &=(\dim\operatorname{ker}(DM(\gamma_k))-\dim DM(k))_{k\in Q_0}=:\mathbf{g}_{A^{\operatorname{op}}}^{\mathrm{DWZ}}(DM,\mathbf{0}). \end{split}\end{aligned}$$ Here, $DM$ denotes the $\Bbbk$-linear dual of $M$, which we interpret as a representation of $A^{\operatorname{op}}=\mathcal{P}_{\Bbbk}(Q^{\operatorname{op}}, W^{\operatorname{op}})$, and $\mathbf{g}_{A^{\operatorname{op}}}^{\mathrm{DWZ}}(DM)$ denotes the $g$-vector of a QP-representation in the sense of [@derksen2010quivers (1.13)]. This can be easily extended to decorated QP-representations. Now, write $A':=\mathcal{P}_{\Bbbk}(\mu_k(Q,W))$, where $\mu_k(Q,W)$ is the QP-mutation defined in [@derksen2008quivers], and consider the piecewise linear transformation of integer vectors $$\label{eq:gproj-transf} \gamma_k^B: \mathbb{Z}^{Q_0}\rightarrow\mathbb{Z}^{Q_0}\quad\text{with}\quad \gamma_k^B(\mathbf{g})_i = \begin{cases} -g_i &\text{if } i=k,\\ g_i + \text{sgn}(g_k)[b_{ik}\cdot g_k]_+ &\text{else}. \end{cases}$$ **Proposition 20**. *[@geiss2023laminations Proposition 2.8] [\[prp:JaMut\]]{#prp:JaMut label="prp:JaMut"} For each $Z\in\operatorname{DecIrr}^\tau(A)$ there exists a dense open subset $U_Z\subset Z$, a unique irreducible component $\tilde{\mu}_k(Z)\in\operatorname{DecIrr}^\tau(A')$ and a regular map $\nu_Z: U_Z\rightarrow\tilde{\mu}_k(Z)$ with the following properties:* - *For each $X\in U_Z$ we have $\nu_Z(X)\cong\mu_k(X)$, where $\mu_k$ denotes the mutation of the decorated QP-representation $X$ in direction $k$, as defined in [@derksen2008quivers].* - *The morphism of affine varieties $G_{\mathbf{d}(\tilde{\mu}_k(Z)}\times U_Z \rightarrow\tilde{\mu}_k(Z), (g, X)\mapsto g.\nu_Z(X)$ is dominant.* - *For each $Z\in\operatorname{DecIrr}^\tau(A)$ we have $$\mathbf{g}_{A'}(\tilde{\mu}_k(Z))=\gamma_k^B(\mathbf{g}_A(Z)).$$* - *$\tilde{\mu}_k(\tilde{\mu_k}(Z))=Z$ for all $Z\in\operatorname{DecIrr}^\tau(A)$.* - *The map $$\tilde{\mu}_k:\operatorname{DecIrr}^\tau(A)\rightarrow\operatorname{DecIrr}^\tau(A')$$ is an isomorphism of partial KRS-monoids. In particular, $Z\in\operatorname{DecIrr}^\tau(A)$ is indecomposable if and only if $\tilde{\mu}_k(Z)\in\operatorname{DecIrr}^\tau(A')$ is indecomposable.* **Remark 21**. As above, let $B_{\operatorname{DWZ}}(Q)$ be the skew-symmetric matrix associated by Derksen--Weyman--Zelevinsky to $Q$ in [@derksen2010quivers Equation (1.4)]. Thus, $B_{\operatorname{DWZ}}(Q)$ has entries $\beta_{ij}:=b_{ji}$ for all $i,j\in Q_0$. Then the transformation rule for $g$-vectors in Part (c) of Proposition [\[prp:JaMut\]](#prp:JaMut){reference-type="ref" reference="prp:JaMut"} can be rewritten as follows. Let $\mathbf{g}_A(Z)=(g_i)_{i\in Q_0}$ and $\mathbf{g}_A'(\tilde{\mu}_k(Z))=(g'_i)_{i\in Q_0}$. Then $$g'_i = \begin{cases} -g_i &\text{if } i=k,\\ g_i + \text{sgn}(g_k)[g_k\cdot \beta_{ki}]_+ &\text{else}, \end{cases}$$ i.e., we have the equality $$\left[\begin{array}{c}\mu_k(B_{\operatorname{DWZ}}(Q))\\ \mathbf{g}_A'(\tilde{\mu}_k(Z))\end{array}\right] =\mu_k\left( \left[\begin{array}{c}B_{\operatorname{DWZ}}(Q)\\ \mathbf{g}_A(Z)\end{array}\right]\right)$$ of $(n+1)\times n$ matrices, where $\mu_k$ is Fomin-Zelevinsky's matrix mutation. Thus, Proposition [\[prp:JaMut\]](#prp:JaMut){reference-type="ref" reference="prp:JaMut"} is a generalization of Nakanishi-Zelevinsky's result [@nakanishi2012on Equation (4.1)]. ## KRS-monoids of laminations Let $\mathbf{\Sigma}=(\Sigma,\mathbb{M},\mathbb{P})$ be a possibly-punctured surface with marked points. In [@geiss2023laminations Section 4], we considered the set $\mathcal{LC}^*(\mathbf{\Sigma})$ of homotopy classes of marked curves and loops on $\mathbf{\Sigma}$ that have no kinks. Inspired by [@qiu2017cluster], we introduce a symmetric marked intersection number $$\operatorname{Int}^*_\mathbf{\Sigma}: \mathcal{LC}^*(\mathbf{\Sigma})\times\mathcal{LC}^*(\mathbf{\Sigma})\rightarrow\mathbb{Z}_{\geq 0}.$$ Given a marked curve (or loop) $(\gamma, c)\in\mathcal{LC}^*(\mathbf{\Sigma})$, let $(\gamma,c)^{-1}$ denote the inversely oriented curve $\gamma$, with accordingly swapped decoration $c$. The involution $(\gamma, c)\mapsto (\gamma, c)^{-1}$ induces an equivalence relation $\simeq$ on $\mathcal{LC}^*(\mathbf{\Sigma})$. For us, as in [@qiu2017cluster], a *simple marked curve* is a marked curve $(\gamma, c)\in\mathcal{LC}^*(\mathbf{\Sigma})$ which has self-intersection number $0$. We let $\mathcal{LC}^*_{\tau}(\mathbf{\Sigma})$ denote the set of all simple marked curves, and define the tame partial KRS-monoid of laminations $$\operatorname{Lam}(\mathbf{\Sigma}):= \operatorname{KRS}(\mathcal{LC}^*_{\tau}(\mathbf{\Sigma})/_{\simeq}, \operatorname{Int}^*_\mathbf{\Sigma}).$$ See Example [\[expl:KRS1\]](#expl:KRS1){reference-type="ref" reference="expl:KRS1"} and [@geiss2023laminations Section 4.3] for more details. The set $\mathcal{LC}^*_{\tau}(\mathbf{\Sigma})/_{\simeq}$ can be identified with the set of laminations $\mathcal{C}^\circ(\mathbf{\Sigma})$ considered in [@musiker2013bases] (two marked curves having intersection number equal to $0$ corresponds to them being *$\mathcal{C}^\circ$-compatible* in Musiker-Schiffler-Williams' nomenclature). This is also compatible with the treatment of laminations in [@fomin2018cluster]. The tame partial KRS-monoid $\operatorname{Lam}(\mathbf{\Sigma})$ can be equipped with a framing coming from (dual) shear coordinates. We shall describe this framing only for the indecomposable elements of $\operatorname{Lam}(\mathbf{\Sigma})$, which we call *laminates*. For the reader's convenience, we fully recall from [@geiss2023laminations] the definition of the vector of dual shear coordinates of a laminate with respect to an arbitrary tagged triangulation. Our treatment essentially follows [@fomin2018cluster], but with slight changes in conventions --hence the adjective *dual* in the term *dual shear coordinates*. Let $T$ be a tagged triangulation of $(\Sigma,\mathbb{M},\mathbb{P})$, and let $\lambda$ be either a tagged arc or a simple closed curve on $(\Sigma,\mathbb{M},\mathbb{P})$. We define the vector $\operatorname{Sh}_T(\lambda)=(\operatorname{Sh}_T(\lambda)_{\gamma})_{\gamma\in T}$ of dual shear coordinates of $\lambda$ with respect to $T$ in steps as follows. **Case 1**. Assume that $T$ has non-negative signature $\delta_T:\mathbb{P}\rightarrow\{1,0,-1\}$. Then, at any given puncture $p$, either all tagged arcs in $T$ incident to $p$ are tagged plain, or exactly two tagged arcs in $T$ are incident to $p$, their underlying ordinary arcs being isotopic to each other, their tags at $p$ differing from one another, and the tags at their other endpoint being plain. Following [@fomin2008cluster], represent $T$ through an ideal triangulation $T^\circ$ defined as follows. For each $p\in\mathbb{P}$ such that $\delta_T(p)=0$, let $i_p,j_p,$ be the two tagged arcs in $T$ that are incident to $p$, with $i_p$ tagged *plain* at $p$, and $j_p$ tagged *notched* at $p$. Then $T^\circ:=\{\gamma^\circ\ | \ \gamma\in T\}$ is obtained from $T$ by setting $\gamma^\circ:=\gamma$ for every $\gamma\in T$ both of whose ends are tagged *plain*, and by replacing each $j_p$ with a loop $j_p^\circ$ closely surrounding $i_p^\circ$ for $i$ and $j$ as above. Thus, the corresponding ordinary arcs $i_p^\circ,j_p^\circ\in T^\circ$ form a self folded triangle that has $i_p^\circ$ as the folded side, and $j_p^\circ$ as the loop closely surrounding $i_p^\circ$. See [@fomin2008cluster Sections 9.1 and 9.2]. If $\lambda$ is a simple closed curve, set $L:=\lambda$. Otherwise, let $L$ be the curve obtained from $\lambda$ by modifying its two ending segments according to the following rules: - any endpoint incident to a marked point in the boundary is slightly slided along the boundary segment lying immediately to its right as in Figure [1](#Fig_rhohalf){reference-type="ref" reference="Fig_rhohalf"} (here, we stand upon the surface using its orientation, and look from the marked point towards the interior of surface, note that we use the orientation of $\Sigma$ to determine what is right and what is left); {#Fig_rhohalf} - any ending segment of $\lambda$ tagged *plain* at a puncture $q$ is replaced with a non-compact curve segment spiralling towards $q$ in the clockwise sense; - any ending segment of $\lambda$ tagged *plain* at a puncture $q$ is replaced with a non-compact curve segment spiralling towards $q$ in the counter-clockwise sense. Take an arc $\gamma\in T$. In order to define the shear coordinate $\operatorname{Sh}_T(\lambda)_{\gamma}$ we need to consider two subcases. **Subcase 1**. Suppose that the ordinary arc $\gamma^{\circ}\in T^\circ$ is not the folded side of a self-folded triangle of $T^\circ$. Then $\gamma^\circ$ is contained in exactly two ideal triangles of $T^\circ$, and the union $\overline{\lozenge}_{\gamma^\circ}$ of these two triangles is either a quadrilateral (if $\gamma^\circ$ does not enclose a self-folded triangle) or a digon (if $\gamma^\circ$ encloses a self-folded triangle). In any of these two situations, the complement in $\overline{\lozenge}_{\gamma^\circ}$ of the union of the arcs belonging to $T^\circ\setminus\{\gamma^\circ\}$ can be thought to be an open quadrilateral $\lozenge_{\gamma^\circ}$ in which $\gamma^\circ$ sits as a diagonal. The shear coordinate $\operatorname{Sh}_T(\lambda)_{\gamma}$ is defined to be the number of segments of $\lozenge_{\gamma^\circ}\cap L$ that form the shape of a letter $Z$ when crossing $\gamma^\circ$ minus the number of segments of $\lozenge_{\gamma^\circ}\cap L$ that form the shape of a letter $S$ when crossing $\gamma^\circ$. **Subcase 2**. Suppose that the ordinary arc $\gamma^{\circ}\in T^\circ$ is the folded side of a self-folded triangle of $T^\circ$. Then there is a puncture $p\in\mathbb{P}$ such that $\gamma=i_p$ and $\gamma^\circ=i_p^\circ$. Define $$\operatorname{Sh}_T(\lambda)_{i_p}:=\operatorname{Sh}_T(\lambda')_{j_p}.$$ where $\lambda'$ is obtained from $\lambda$ by switching the tags of $\lambda$ at the puncture $p$. **Case 2**. Assume now that $T$ is an arbitrary tagged triangulation of $(\Sigma,\mathbb{M},\mathbb{P})$. The set of punctures at which $T$ has negative signature is the inverse image $\delta_T^{-1}(-1)$. Set $T'$ to be the tagged triangulation obtained from $T$ by changing from notched to plain all the tags incident to punctures in $\delta_T^{-1}(-1)$. Thus, $T'$ is a tagged triangulation of signature zero, so dual shear coordinates with respect to $T'$ have already been defined. Set $$\operatorname{Sh}_T(\lambda):=\operatorname{Sh}_{T'}(\lambda'),$$ where $\lambda'$ is obtained from $\lambda$ by switching all the tags of $\lambda$ at the punctures belonging to the set $\delta_T^{-1}(-1)$. **Example 22**. In Figure [2](#Fig_dualShearCoords_vs_FSTShearCoords){reference-type="ref" reference="Fig_dualShearCoords_vs_FSTShearCoords"}, {#Fig_dualShearCoords_vs_FSTShearCoords} the reader can glimpse the relation between the dual shear coordinates we have defined above and the shear coordinates used by Fomin--Thurston, namely, $$\label{eq:relation-shear-coord-vs-dual-shear-coord} \operatorname{Sh}_T(\lambda)=\operatorname{Sh}_{\overline{T}}^{\operatorname{FST}}(\overline{\lambda}),$$ where $\operatorname{Sh}_T(\lambda)$ is the vector of dual shear coordinates we have defined above, $\overline{T}$ and $\overline{\lambda}$ are the images of $T$ and $\lambda$ in the surface obtained as the *mirror image* of $(\Sigma,\mathbb{M},\mathbb{P})$, and $\operatorname{Sh}_{\overline{T}}^{\operatorname{FST}}(\overline{\lambda})$ is the vector of shear coordinates of Fomin--Thurston, [@fomin2018cluster Definition 13.1]. Denote by $B_{\operatorname{FST}}(T)$ the skew-symmetric matrix associated to $T$ by Fomin--Shapiro--Thurston [@fomin2008cluster Definitions 4.1 and 9.6], and denote by $B_{\operatorname{DWZ}}(Q)$ the skew-symmetric matrix associated by Derksen--Weyman--Zelevinsky to an arbitrary 2-acyclic quiver $Q$ [@derksen2010quivers Equation (1.4)] and [@derksen2008quivers Section 2 and Equation (7.1)]. Since we have chosen to define $Q(T)$ by drawing the arrows in the clockwise sense within each triangle of $T$ (i.e., following the convention of [@labardini2010quivers]), we have $$B_{\operatorname{DWZ}}(Q(T))=-B_{\operatorname{FST}}(T).$$ Furthermore, for dual shear coordinates, [@fomin2018cluster Theorem 13.5] can be restated as follows. **Theorem 23**. *Let $T$ and $T'$ be tagged triangulations of $(\Sigma,\mathbb{M},\mathbb{P})$ related by the flip of a tagged arc $k\in T$, and let $\lambda$ be either a tagged arc or a simple closed curve on $(\Sigma,\mathbb{M},\mathbb{P})$. Then $$\left[\begin{array}{c}-B_{\operatorname{FST}}(T')\\ \operatorname{Sh}_{T'}(\lambda)\end{array}\right] =\mu_k\left( \left[\begin{array}{c}-B_{\operatorname{FST}}(T)\\ \operatorname{Sh}_T(\lambda)\end{array}\right]\right).$$* Denote the entries of $B_{\operatorname{FST}}(T)$ as $b_{ij}$, and the entries of $-B_{\operatorname{FST}}(T)$ as $\beta_{ij}$. Since $B_{\operatorname{FST}}(T)$ and $-B_{\operatorname{FST}}(T)$ are transpose to each other, Theorem [Theorem 23](#thm:behavior-of-dual-shear-coords-under-matrix-mutation){reference-type="ref" reference="thm:behavior-of-dual-shear-coords-under-matrix-mutation"} states in particular that $$\begin{aligned} \label{eq:unraveled-recursion-satisfied-by-dual-shear-coords} \operatorname{Sh}_{T'}(\lambda)_{i} &= \begin{cases} -\operatorname{Sh}_T(\lambda)_{i} & \text{if $i=k$};\\ \operatorname{Sh}_T(\lambda)_{i}+\text{sgn}(\operatorname{Sh}_T(\lambda)_k)[\operatorname{Sh}_T(\lambda)_k\beta_{ki}]_+ & \text{if $i\neq k$}; \end{cases}\\ \nonumber &= \begin{cases} -\operatorname{Sh}_T(\lambda)_{i} & \text{if $i=k$};\\ \operatorname{Sh}_T(\lambda)_{i}+\text{sgn}(\operatorname{Sh}_T(\lambda)_k)[b_{i k}\operatorname{Sh}_T(\lambda)_k ]_+ & \text{if $i\neq k$}; \end{cases}\\ \nonumber &= \gamma_k^{B_{\operatorname{FST}}(T)}(\operatorname{Sh}_T(\lambda))_i.\end{aligned}$$ ## The isomorphism of framed KRS-monoids $\operatorname{Lam}(\mathbf{\Sigma})\cong\operatorname{DecIrr}^\tau(A)$ Generalizing [@geiss2022schemes Theorem 10.13 and Proposition 10.14] from the unpunctured to the punctured case, the main result of [@geiss2023laminations] states that: **Theorem 24**. *[@geiss2023laminations Theorem 1.1.][\[thm:Lams-vs-taured-comps-iso\]]{#thm:Lams-vs-taured-comps-iso label="thm:Lams-vs-taured-comps-iso"} Let $\mathbf{\Sigma}=(\Sigma,\mathbb{M},\mathbb{P})$ be a surface with non-empty boundary. For each tagged triangulation $T$ of $\mathbf{\Sigma}$ there is a unique isomorphism of partial KRS-monoids $$\pi_T: (\operatorname{Lam}(\mathbf{\Sigma}),\operatorname{Int}^*,+) \rightarrow (\operatorname{DecIrr}^\tau(A(T)), e_{A(T)},\oplus),$$ such that the diagram of functions and sets $$\xymatrix{ \operatorname{Lam}(\mathbf{\Sigma}) \ar[rr]^{\pi_T} \ar[dr]_{\operatorname{Sh}_T} & & \operatorname{DecIrr}^\tau(A(T)) \ar[dl]^{\mathbf{g}_{A(T)}} \\ & \mathbb{Z}^{T} & }$$ where $A(T)$ is the Jacobian algebra of the quiver with potential associated to $T$ in [@cerulli2012quivers Definition 4.1], [@labardini2016quivers Definitions 3.1 and 3.2].* Very roughly speaking, the proof strategy followed in [@geiss2023laminations] consists of two steps: 1. the statement of the theorem is shown to hold tagged triangulations of signature zero, exploiting very heavily the fact that for those triangulations the Jacobian algebra is skewed-gentle; 2. given a tagged triangulation $T$ for which the theorem holds, and another tagged triangulation $T'$ obtained by flipping an arc $k\in T$, Proposition [\[prp:JaMut\]](#prp:JaMut){reference-type="ref" reference="prp:JaMut"} and Theorem [Theorem 23](#thm:behavior-of-dual-shear-coords-under-matrix-mutation){reference-type="ref" reference="thm:behavior-of-dual-shear-coords-under-matrix-mutation"} are applied to produce a commutative diagram $$\xymatrix{ & & & & \operatorname{DecIrr}^\tau(A(T)) \ar[dl]^{\mathbf{g}_{A(T)}} \ar[dddd]^{\cong}_{\widetilde{\mu}_k}\\ & & & \mathbb{Z}^{T} \ar[dd]|-{\qquad\gamma_k^{B_{\operatorname{FST}}(Q(T))}} & \\ \operatorname{Lam}(\mathbf{\Sigma}) \ar@/^2.5pc/[uurrrr]|-{\pi_T}^{\cong\quad} \ar@/_2.5pc/@{.>}[ddrrrr]|-{\quad \pi_{T'}:=\widetilde{\mu}_k\circ \pi_T} \ar[urrr]_{\operatorname{Sh}_T} \ar[drrr]^{\operatorname{Sh}_{T'}} & & & & \\ & & & \mathbb{Z}^{T} &\\ & & & & \operatorname{DecIrr}^\tau(A(T')) \ar[ul]_{\mathbf{g}_{A(T')}} }$$ and this allows to deduce that the theorem holds then for $T'$ as well. Since any two tagged triangulations of a surface with non-empty boundary are related by a finite sequence of flips, and since any such surface certainly admits at least one triangulation of signature zero, these two steps yield a proof of Theorem [\[thm:Lams-vs-taured-comps-iso\]](#thm:Lams-vs-taured-comps-iso){reference-type="ref" reference="thm:Lams-vs-taured-comps-iso"}. # Triangulations adapted to closed curves {#base case section} Let $\alpha$ be an *essential loop* on $\mathbf{\Sigma}=(\Sigma,\mathbb{M},\mathbb{P})$, i.e., a non-contractible simple closed curve that is furthermore not contractible to a puncture, and let $T=\{\tau_1,\ldots,\tau_n\}$ be a tagged triangulation of $\mathbf{\Sigma}$. The *band graph* associated to $\alpha$ with respect to $T$, cf. [@musiker2013bases Definition 3.4 and §8.3], will be denoted $G(T,\alpha)$. The polynomial $F_\zeta^T$ defined in [@musiker2013bases Definition 3.14] (see also [@musiker2013bases §8.3]), with $\zeta:=\alpha$, will be called *snake $F$-polynomial* and denoted $F_{G(T,\alpha)}$. Similarly, the integer vector from [@musiker2013bases Definition 6.1] will be called the *snake $g$-vector* and denoted $\mathbf{g}_{G(T,\alpha)}$. Notice that $$F_{G(T,\alpha)} = \displaystyle \sum_{P \in \mathcal{P}(G(T,\alpha))} y(P)$$ where $\mathcal{P}(G(T,\alpha))$ is the collection of all good matchings of the band graph $G(T,\alpha)$. Furthermore, by [@musiker2013bases Proposition 6.2], we have $$\mathbf{g}_{G(T,\alpha)}=\deg\left(\frac{x(P_-)}{\operatorname{cross}(T,\alpha)}\right),$$ where $P_-$ is the *minimal matching* of the band graph $G(T,\alpha)$, cf. [@musiker2013bases Definition 3.7], and $\deg:\mathbb{Z}[x_1^{\pm1},\ldots,x_n^{\pm1},y_1,\ldots,y_n]\rightarrow \mathbb{Z}^n$ is the grading defined by $$\deg(x_i)=\mathbf{e}_i \qquad \text{and} \qquad \deg(y_j):=-\mathbf{b}_{\operatorname{FST}}(T)_j$$ as in [@fomin2007cluster Proposition 6.1], $\mathbf{b}_{\operatorname{FST}}(T)_j$ being the $j^{\operatorname{th}}$ column of $B_{\operatorname{FST}}(T)$, cf. the paragraph preceding Theorem [Theorem 23](#thm:behavior-of-dual-shear-coords-under-matrix-mutation){reference-type="ref" reference="thm:behavior-of-dual-shear-coords-under-matrix-mutation"} above. **Proposition 25**. *For each non-contractible simple closed curve $\alpha$ on $\mathbf{\Sigma}$ there exists a triangulation $T_{\alpha}$ without self-folded triangles such that:* - *the Jacobian algebra of the restriction $(Q(T_\alpha)|_I,S(T_\alpha)|_I)$ is gentle, where $I$ is the set of arcs crossed by $\alpha$;* - *the band module $M(T_\alpha,\alpha,\lambda,1)$ over $\Lambda_I:=\mathcal{P}(Q(T_\alpha)|_I,S(T_\alpha)|_I)$ of quasi-length $1$ associated to the parameter $\lambda\in\Bbbk\setminus\{0\}$ is a module over the Jacobian algebra $A(T):=\mathcal{P}(Q(T),S(T))$ as well and satisfies: $$\begin{aligned} F_{M(T_\alpha,\alpha,\lambda,1)} &= F_{G(T,\alpha)} & E_{A(T_\alpha)}(M(T_\alpha,\alpha,\lambda,1))&=1 \\ \mathbf{g}_{A(T_\alpha)}(M(T_\alpha,\alpha,\lambda,1)) &= \operatorname{Sh}_{T_\alpha}(\alpha) = \mathbf{g}_{T_\alpha}(G(T_\alpha,\alpha)).\end{aligned}$$* *Proof.* Cutting $\mathbf{\Sigma}$ along $\alpha$ we are left with at most two marked surfaces $(S_1,M_1)$ and $(S_2,M_2)$ -- note that if $\alpha$ is non-separating then one $S_i$ will be empty. To construct $T_{\alpha}$ we split the task into two cases. **Case 3**. If $M_i \neq \emptyset$ for any $i \in \{1,2\}$ then there exist compatible arcs $\gamma_1$ and $\gamma_2$ in $\mathbf{\Sigma}$ which bound $\alpha$ in a cylinder $C_{1,1}$. Define $T_{\alpha}$ to be any triangulation containing both $\gamma_1$ and $\gamma_2$, with the extra property of not having self-folded triangles. See Figure [3](#case1triangulation){reference-type="ref" reference="case1triangulation"}. {#case1triangulation width="13cm"} **Case 4**. Otherwise, $\alpha$ is a separating curve where $S_i \neq \emptyset$ and $M_i = \emptyset$ for some $i \in \{1,2 \}$. In this case there exists an arc $\gamma \in \mathbf{\Sigma}$ bounding $\alpha$ in a bordered surface $\mathbf{\Sigma}_{g,1}$ of genus $g$ with one boundary component and one marked point. Define $T_{\alpha}$ to be any triangulation containing $\gamma$, with the extra property of not having self-folded triangles. See Figures [4](#case2triangulation){reference-type="ref" reference="case2triangulation"} and [6](#bandgraphform2){reference-type="ref" reference="bandgraphform2"}. {#case2triangulation width="13cm"} In both cases, it is clear that $\Lambda_I:=\mathcal{P}(Q(T_\alpha)|_I,S(T_\alpha)|_I)$ is a gentle algebra. That $M(T_\alpha,\alpha,\lambda,1)$ is a module also over $A(T)$ follows immediately from [@derksen2008quivers Proposition 8.9] or by direct inspection. It is easy to check that $A(\overline{T_\alpha})=A(T_\alpha)^{\operatorname{op}}$ and $M(\overline{T_\alpha},\overline{\alpha},\lambda,1)\cong D(M(T_\alpha,\alpha,\lambda,1))$ as representations of $A(\overline{T_\alpha})$. Therefore, by [\[eq:gvec\]](#eq:gvec){reference-type="eqref" reference="eq:gvec"}, [@labardini2010quivers proof of Theorem 10.0.5] and [\[eq:relation-shear-coord-vs-dual-shear-coord\]](#eq:relation-shear-coord-vs-dual-shear-coord){reference-type="eqref" reference="eq:relation-shear-coord-vs-dual-shear-coord"}, we have $$\mathbf{g}_{A(T_\alpha)}(M(T_\alpha,\alpha,\lambda,1)) = \mathbf{g}_{A(\overline{T_\alpha})}^{{\operatorname{inj}}}(M(\overline{T_\alpha},\overline{\alpha},\lambda,1))= \operatorname{Sh}_{\overline{T_\alpha}}^{\operatorname{FST}}(\overline{\alpha}) = \operatorname{Sh}_{T_\alpha}(\alpha).$$ The equality $\operatorname{Sh}_{T_\alpha}(\alpha)=\mathbf{g}_{G(T_\alpha,\alpha)}$ follows from [@geiss2022schemes Proposition 10.14]. **Lemma 26**. *The band graph $G(T_\alpha,\alpha)$ is a 'zig-zag' band graph whose corresponding band is given in Figure [5](#bandgraph){reference-type="ref" reference="bandgraph"}.* {#bandgraph width="13cm"} *Proof.* When $T_{\alpha}$ is defined via case $1$, then a direct computation obtains the $n=2$ band graph in Figure [5](#bandgraph){reference-type="ref" reference="bandgraph"}. So, suppose $T_{\alpha}$ is defined via case $2$. Then there exists a unique triangle $\Delta$ of $T_{\alpha}$, lying inside $\mathbf{\Sigma}_{g,1}$, such that $\gamma$ is an edge of $\Delta$. Let us denote the remaining two edges of $\Delta$ by $\gamma_1$ and $\gamma_2$. For some $\beta$ and $\delta$ in $T$ we see that $\alpha$ has the following sequences of 'zig-zag' intersections: $\gamma_1, \gamma_2, \beta$ and $\delta, \gamma_1, \gamma_2$. All other sequences of intersections (of $\alpha$ with $T$) of length $3$ are 'fan' intersections. This completes the proof. ◻ {reference-type="ref" reference="bandgraphform"}.](Fig_bandgraphform2.pdf){#bandgraphform2 width="8cm"} **Lemma 27**. *For $\lambda\in\mathbb{C}^*$, the endomorphism ring $\operatorname{End}_{A(T_\alpha)}(M(T_\alpha,\alpha,\lambda,1))$ of the regular quasi-simple band module $M(T_\alpha, \alpha,\lambda,1)$ is isomorphic to $\mathbb{C}$.* *Proof.* The band graph of $\alpha$ with respect to $T_\alpha$ is of 'zig-zag' type by construction. As shown in Lemma [Lemma 26](#bandgraphform){reference-type="ref" reference="bandgraphform"}, each 'zig-zag' band graph gives rise to a band $b$ of the form shown in [\[band\]](#band){reference-type="eqref" reference="band"}. Moreover, the \"support\" $\Lambda_I$ of $b$ is a special biserial (in fact hereditary) algebra. Thus we have trivially $\operatorname{End}_{A(T_\alpha)}(M(T_\alpha,\alpha,\lambda,1))= \operatorname{End}_{\Lambda_I}(M(T_\alpha,\alpha,\lambda,1))$ We shall follow the setup and terminology used by Butler and Ringel in [@butler1987auslander] for representations of band modules. Let $M= M(b,\lambda, 1)$ be a regular simple band module corresponding to the following band $b$: $$\label{band} \begin{tikzpicture} \matrix(m)[matrix of math nodes, row sep=3em, column sep=2em, text height=1.5ex, text depth=0.25ex]{t_{i_n} &t_{i_{n-1}} & t_{i_{n-2}} & \ldots &t_{i_3} & t_{i_2} & t_{i_1} \\}; \path[->](m-1-2) edge node [above]{$\alpha_{n-1}$} (m-1-1); \path[->](m-1-3) edge node [above]{$\alpha_{n-2}$} (m-1-2); \path[->](m-1-6) edge node [above]{$\alpha_{2}$} (m-1-5); \path[->](m-1-7) edge node [above]{$\alpha_{1}$} (m-1-6); \path[->](m-1-7) edge [bend right=-20] node [below]{$\alpha_{n}$} (m-1-1); \end{tikzpicture}$$ Here, we may think of vertices $t_{i_j}\in Q_0(T_\alpha)$ as the arcs of the triangulation $T_\alpha$ which intersect consecutively with the loop $\alpha$. Note that possibly $t_{i_j}=t_{i_k}$ for certain $i\neq k$. However, $t_{i_1}\neq t_{i_n}$ since the arrow $\alpha_n$ is not a loop. Specifically, $M$ is defined by the following collection of vector spaces and maps: Now, we can use Krause's formalism from [@krause1991maps]. With the notation from *loc. cit.* p. 191 we denote by $\mathcal{A}(b,b)$ the set of isomorphism classes of admissable (*sic*) triples for the pair of bands $(b,b)$. Since the band $b$ has a unique source, which is mapped to $t_{i_1}$ and a unique sink, which is mapped to $t_{i_n}\neq t_{i_1}$ we have $\mathcal{A}(b,b)=\{[(b, \operatorname{id}_b,\operatorname{id}_b)]\}$. Thus, $\operatorname{End}_{\Lambda_I}(M)\cong\mathbb{C}$ follows directly from the Theorem on [@krause1991maps p. 191]. ◻ By Lemma [Lemma 27](#lem:TrivEnd){reference-type="ref" reference="lem:TrivEnd"} we have $\operatorname{End}_{A(T_\alpha)}(M(T_\alpha,\alpha,\lambda,1))=\mathbb{C}$. On the other hand, one can directly check that $\underline{\dim}(M(T_\alpha,\alpha,\lambda,1))\cdot\mathbf{g}_{A(T_\alpha)}(M(T_\alpha,\alpha,\lambda,1))=0$. Hence $$\begin{aligned} E_{A(T_\alpha)}(M(T_\alpha,\alpha,\lambda,1)) &= \dim(\operatorname{End}_{A(T_\alpha)}(M(T_\alpha,\alpha,\lambda,1)))+\\ &\quad +\underline{\dim}(M(T_\alpha,\alpha,\lambda,1))\cdot\mathbf{g}_{A(T_\alpha)}(M(T_\alpha,\alpha,\lambda,1))=1.\end{aligned}$$ The equality $F_{M(T_\alpha,\alpha,\lambda,1)} = F_{G(T,\alpha)}$ follows from [@geiss2022schemes Lemma 11.4 and Remark 11.7] and [@haupt2012euler Theorem 1.2]. This finishes the proof of Proposition [Proposition 25](#prop:existence-ad-hoc-triangulation-for-closed-curve){reference-type="ref" reference="prop:existence-ad-hoc-triangulation-for-closed-curve"}. ◻ **Remark 28**. In the proof of Proposition [Proposition 25](#prop:existence-ad-hoc-triangulation-for-closed-curve){reference-type="ref" reference="prop:existence-ad-hoc-triangulation-for-closed-curve"} we have partially used the explicit definition of the potential $S(T)$: we know that the restriction of $S(\tau)$ to the set $I$ is a gentle algebra. **Corollary 29**. *In the situation of Proposition [Proposition 25](#prop:existence-ad-hoc-triangulation-for-closed-curve){reference-type="ref" reference="prop:existence-ad-hoc-triangulation-for-closed-curve"},* - *the Zariski closure $$Z_{T_\alpha,\alpha}:=\overline{\bigcup_{\lambda\in\mathbb{C}\setminus\{0\}}\operatorname{GL}_{\mathbf{d}}(\mathbb{C})\cdot M(T_\alpha,\alpha,\lambda,1)}$$ is a generically $\tau$-reduced indecomposable irreducible component of the representation space $\operatorname{rep}(A(T_\alpha),\mathbf{d})$, where $\mathbf{d}:=\underline{\dim}(M(T_\alpha,\alpha,\lambda,1))$;* - *the generic projective $g$-vector $\mathbf{g}_{A(T_\alpha)}(Z_{T_\alpha,\alpha})$ is equal to the snake $g$-vector $\mathbf{g}_{G(T_\alpha,\alpha)}$;* - *the generic value $CC_{A(T_\alpha)}(Z_{T_\alpha,\alpha})$ is equal to Musiker-Schiffler-Williams' expansion $\operatorname{MSW}(G(T_\alpha,\alpha))$.* *Proof.* By [@geiss2016the Theorem 7.1] (see also [@labardini2016on Theorem 3.5]), the algebra $A(T_\alpha)$ is tame. Furthermore, for $\lambda_1\neq\lambda_2$, the band modules $M(T_\alpha,\alpha,\lambda_1,1)$ and $M(T_\alpha,\alpha,\lambda_2,1)$ are non-isomorphic, and by Proposition [Proposition 25](#prop:existence-ad-hoc-triangulation-for-closed-curve){reference-type="ref" reference="prop:existence-ad-hoc-triangulation-for-closed-curve"}, for every point $M$ in the dense open subset $\bigcup_{\lambda\in\mathbb{C}\setminus\{0\}}\operatorname{GL}_{\mathbf{d}}(\mathbb{C})\cdot M(T_\alpha,\alpha,\lambda,1)$ we have $E_{A(T_\alpha)}(M)=1$. Hence $Z_{T_\alpha,\alpha}$ is a generically $\tau$-reduced indecomposable irreducible component by [@geiss2022schemes Lemma 3.1 and Theorem 3.2] (see also [@carroll2015on Section 2.2]). By Proposition [Proposition 25](#prop:existence-ad-hoc-triangulation-for-closed-curve){reference-type="ref" reference="prop:existence-ad-hoc-triangulation-for-closed-curve"}, for every point $M\in \bigcup_{\lambda\in\mathbb{C}\setminus\{0\}}\operatorname{GL}_{\mathbf{d}}(\mathbb{C})\cdot M(T_\alpha,\alpha,\lambda,1)$ we have $$F_{M} = F_{G(T_\alpha,\alpha)} \qquad \text{and} \qquad \mathbf{g}_{A(T_\alpha)}(M) = \mathbf{g}_{G(T_\alpha,\alpha)}.$$ This implies the second and third assertions. ◻ # The Combinatorial Key Lemma {#CKL} ## Derksen--Weyman--Zelevinsky's representation-theoretic Key Lemma The following result is the Key Lemma of Derksen, Weyman and Zelevinsky. **Theorem 30** (Lemma 5.2, [@derksen2010quivers]). *Let $\mathcal{M}$ be a finite-dimensional decorated representation of a non-degenerate quiver with potential $(Q,S)$, and define $\overline{\mathcal{M}}$ to be the mutation of $\mathcal{M}$ in direction $k \in \{1,\ldots, n\}$. Then* *$$\label{Fmutation} (y_k +1)^{h_k}F_{\mathcal{M}}(y_1,\ldots, y_n) = (y_k' +1)^{h_k'}F_{\overline{\mathcal{M}}}(y_1',\ldots, y_n')$$* *where* - *$(\mathbf{y}',B') \in \mathbb{Q}(y_1,\ldots, y_n)$ is obtained from $(\mathbf{y},B_{\operatorname{DWZ}}(Q))$ by $Y$-seed mutation at $k$,* - *$h_k$ and $h_k'$ are the $k^{th}$ components of the $\mathbf{h}$-vectors $\mathbf{h}_{\mathcal{M}}$ and $\mathbf{h}_{\overline{\mathcal{M}}}$, respectively, defined by [@derksen2010quivers Equations (1.8) and (3.2)].* *Moreover, the $\mathbf{g}$-vector $\mathbf{g}_{\mathcal{P}(Q,S)}^{\operatorname{DWZ}}(\mathcal{M}) = (g_1,\ldots, g_n)$ defined by [@derksen2010quivers Equation (1.13)] satisfies $$\label{kth gvector} g_k = h_k - h_k'$$ and is related to the $\mathbf{g}$-vector $\mathbf{g}_{\mathcal{P}(\mu_k(Q,S))}^{\operatorname{DWZ}}(\overline{\mathcal{M}}) = (g_1',\ldots, g_n')$ via $$\label{gvector mutation} g_j' := \left\{ \begin{array}{ll} -g_k, &\text{if $j =k$}; \\ \\ g_j + [b_{jk}]_{+}g_k - b_{jk}h_k, &\text{if $j \neq k$}.\\ \end{array} \right.$$* **Remark 31**. By [@derksen2010quivers Proposition 10.4 and Remark 10.8], $\mathbf{g}_{\mathcal{M}}^{\operatorname{DWZ}}=\mathbf{g}_{\mathcal{M}}^{\operatorname{inj}}$ (see also [\[eq:gvec\]](#eq:gvec){reference-type="eqref" reference="eq:gvec"}). ## Statement of the Combinatorial Key Lemma **Theorem 32** (Combinatorial Key Lemma). *Let $T=\{\tau_1,\ldots,\tau_n\}$ be a tagged triangulation of $\mathbf{\Sigma}$, $\tau_k$ a tagged arc belonging to $T$, and $\alpha$ a simple closed curve. We set $T'$ to be the tagged triangulation obtained from $T$ by flipping $\tau_k$. The following equality holds:* *$$\label{combinatorial Fmutation} (y_k +1)^{h_k}F_{G(T,\alpha)}(y_1,\ldots, y_n) = (y_k' +1)^{h_k'}F_{G(T',\alpha)}(y_1',\ldots, y_n')$$* *where* - *$(\mathbf{y}',B') \in \mathbb{Q}(y_1,\ldots, y_n)$ is obtained from $(\mathbf{y},B_{\operatorname{FST}}(T))$ by mutation at $k$,* - *the *snake $\mathbf{h}$-vector* $\mathbf{h}_{G(T,\alpha)}$ is defined by setting $$u^{h_i} := {F_{G(T,\alpha)}}_{{\vert}_{\operatorname{Trop}(u)}}(u^{[-b_{i1}]_{+}}, \ldots, u^{-1}, \ldots, u^{[-b_{in}]_{+}}),$$ where $u^{-1}$ is in the $i^{th}$ component. The snake $\mathbf{h}$-vector $\mathbf{h}_{G(T',\alpha)}$ is defined analogously with respect to $T'$ and $\mu_{k}(B_{\operatorname{FST}}(T))=B_{\operatorname{FST}}(T')$.* *Moreover, the snake $\mathbf{g}$-vector $\mathbf{g}_{G(T,\alpha)} = (g_1,\ldots, g_n)$ satisfies $$\label{comb kth gvector} g_k = h_k - h_k'$$ and $\mathbf{g}_{G(T,\alpha)} = (g_1,\ldots, g_n)$ is related to $\mathbf{g}_{G(T',\alpha)} = (g_1',\ldots, g_n')$ by the following rule: $$\label{comb gvector mutation} g_j' := \left\{ \begin{array}{ll} -g_k, &\text{if $j =k$}; \\ \\ g_j + [b_{jk}]_{+}g_k - b_{jk}h_k, &\text{if $j \neq k$}.\\ \end{array} \right.$$* **Remark 33**. Since the band graph $G(T,\alpha)$ always admits a minimal matching, the snake $F$-polynomial $F_{G(T,\alpha)}$ has constant term $1$, hence every entry of the snake $h$-vector $\mathbf{h}_{G(T,\alpha)}$ is non-positive. **Example 34**. As shown in Figure [7](#exmp: cylinder){reference-type="ref" reference="exmp: cylinder"}, let $T$ and $T'$ be triangulations of the cylinder $C_{1,1}$, where $T'$ is obtained from $T$ by the flip of $\tau_1$. Furthermore, let $\alpha$ be the unique simple closed curve of $C_{1,1}$. Directly from the definition, let us compute the snake $F$-polynomials $F_{G(T,\alpha)}$ and $F_{G(T',\alpha)}$ of $\alpha$. Namely, considering the good matchings associated to the band graphs $G(T,\alpha)$ and $G(T',\alpha)$ shown in Figure [7](#exmp: cylinder){reference-type="ref" reference="exmp: cylinder"} we get: $$F_{G(T,\alpha)}(y_1,y_2) = 1 + y_2 + y_1y_2 \hspace{10mm} \text{and} \hspace{10mm} F_{G(T',\alpha)}(y_1,y_2) = 1 + y_1 + y_1y_2.$$ Now we wish to calculate the components $h_1$ and $h_1'$ corresponding to $\tau_1$ in the respective $\mathbf{h}$-vectors $\mathbf{h}_{\alpha, T}$ and $\mathbf{h}_{\alpha, T'}$. Since $b_{12} = -2$ and $b_{12}' = 2$ we get: $${F_{G(T,\alpha)}}_{\vert_{Trop(u)}}(u^{-1},u^2) = 1\oplus u^2\oplus u = 1$$ and $${F_{G(T',\alpha)}}_{\vert_{\operatorname{Trop}(u)}}(u^{-1},1) = 1\oplus u^{-1} \oplus u^{-1} = u^{-1}.$$ Consequently, we have $h_1 = 0$ and $h_1' = -1$. As such, [\[combinatorial Fmutation\]](#combinatorial Fmutation){reference-type="eqref" reference="combinatorial Fmutation"} now reduces to: $$F_{G(T,\alpha)}(y_1,y_2) = (y_1'+1)^{-1}F_{G(T',\alpha)}(y_1',y_2'),$$ which follows directly from the definition of $Y$-seed mutation. Indeed, $y_1' := y_1^{-1}$ and $y_2' := y_2(1+y_1)^2$. {#exmp: cylinder width="14cm"} The rest of Section is devoted to proving Theorem [Theorem 32](#thm: comb key lemma){reference-type="ref" reference="thm: comb key lemma"}. ## Local segments and their local flips {#subsec:local-segments} Note that, in general, the band graphs $G(T,\alpha)$ and $G(T',\alpha)$ will have very different shapes, consequently, their collections of good matchings seem to differ considerably. This is due to the fact that $\alpha$ may intersect the flip region multiple times, and in many different combinatorial ways. Nevertheless, we shall show that the relationships between the snake $\mathbf{g}$-vectors, snake $\mathbf{h}$-vectors and snake $F$-polynomials of $G(T,\alpha)$ and $G(T',\alpha)$ are governed by 'local' considerations. Let us (cyclically) label the intersections points between $\alpha$ and $T^{\circ}$ by $p_1, p_2, \ldots, p_d$. By convention, indices of these intersection points will always be taken modulo $d$. **Definition 35**. Consider a subcurve $\alpha_{ij}$ of $\alpha$ with intersection points $p_i, \ldots, p_j$. We call $\alpha_{ij}$ a **local segment** of $\alpha$ **with respect to $\gamma \in T$** if: - $\alpha_{ij}$ intersects $\gamma$ or $\gamma'$ - neither $p_{i-1}$ or $p_{j+1}$ lie on $\gamma$ or $\gamma'$ - $\alpha_{ij}$ is minimal with the above two properties (with respect to subcurve inclusion). We list all possible local segments of $\alpha$, with respect to a tagged arc $\gamma \in T$, in Figure [8](#flips){reference-type="ref" reference="flips"}. {#flips width="12cm"} {width="12cm"} {width="12cm"} {width="12cm"} {width="12cm"} {width="12cm"} {width="12cm"} {width="12cm"} **Remark 36**. Note that in all of the configurations listed in Figure [8](#flips){reference-type="ref" reference="flips"}, we have $$a_1,a_2,b_1,b_2,c_1,c_2 \notin \{\gamma, \gamma'\}.$$ **Remark 37**. We have not included the case that $\alpha$ is a closed curve enclosed by $T$ in a cylinder $C_{1,1}$, but this is an exceptional case and can easily be checked separately. We now list the snake graphs corresponding to the flips illustrated in Figure [8](#flips){reference-type="ref" reference="flips"}. {reference-type="ref" reference="flips"}.](Fig_SG1.pdf){#SG width="17cm"} {width="17cm"} {width="17cm"} {width="17cm"} {width="17cm"} {width="17cm"} {width="17cm"} {width="17cm"} As mentioned above, the strategy in proving Theorem [Theorem 32](#thm: comb key lemma){reference-type="ref" reference="thm: comb key lemma"} is to reduce the problem to checking that the theorem holds on the flips listed in Figure [8](#flips){reference-type="ref" reference="flips"}. To do this, we shall need a better understanding of the associated $\mathbf{h}$-vectors. ## Local $\mathbf{h}$-vectors **Definition 38**. Let $\alpha_{ij}$ be a subcurve of $\alpha$ present in Figure [8](#flips){reference-type="ref" reference="flips"}. One can define a snake graph corresponding to the configuration, and thus obtain a *local* snake $F$-polynomial $F_{G(T,\alpha_{ij})}$. We call the $\mathbf{h}$-vector $\mathbf{h}_{G(T,\alpha_{ij})}$ defined through $F_{G(T,\alpha_{ij})}$ the *local snake $\mathbf{h}$-vector* of $\alpha$ with respect to $\alpha_{ij}$ and $T$. The idea is to show that local snake $\mathbf{h}$-vectors naturally determine the whole snake $\mathbf{h}$-vector. **Proposition 39**. *Let $\mathcal{L}$ be the set of all local segments of $\alpha$ with respect to $\gamma \in T$. Then for each $k \in \{1,\ldots, n\}$ we have the following equality: $$h_{\alpha; k} = \displaystyle \sum_{\alpha_{ij} \in \mathcal{L}} h_{\alpha_{ij}; k}.$$* To prove Proposition [Proposition 39](#local hsum){reference-type="ref" reference="local hsum"}, we first show: **Lemma 40**. *Keeping the terminology used in Proposition [Proposition 39](#local hsum){reference-type="ref" reference="local hsum"}, the following inequality holds.* *$$\displaystyle \sum_{\alpha_{ij} \in \mathcal{L}} h_{\alpha_{ij}; k} \leq h_{\alpha; k}.$$* *Proof.* Let $\gamma$ be any arc in $T$. Note that if $d$ is a diagonal of $G(T,\alpha)$ such that $b_{d\gamma} \neq 0$ then that diagonal appears as the diagonal of some tile in $G(T,\alpha_{ij})$. The lemma then follows by Remarks [Remark 33](#rem:snake-h-vector-is-non-positive){reference-type="ref" reference="rem:snake-h-vector-is-non-positive"} and [Remark 36](#disjointness){reference-type="ref" reference="disjointness"}. ◻ By Lemma [Lemma 40](#leq){reference-type="ref" reference="leq"}, to prove Proposition [Proposition 39](#local hsum){reference-type="ref" reference="local hsum"} we need to show that there exists a good matching $P$ of $G(T,\alpha)$ such that the restriction of $P$ to any $G(T,\alpha_{ij})$ of $G(T,\alpha)$ achieves the corresponding local $h_k$. Namely, we shall introduce the idea of *gluing* together perfect matchings. With respect to this notion of gluing, the basic idea is show that for any $G(T,\alpha_{ij})$ and $G(T,\alpha_{lm})$, there exists perfect matchings $P_{ij}$ and $P_{lm}$, respectively, which can be glued together and which achieve their corresponding local $h_k$'s. ## Extendability of perfect matchings **Definition 41**. Let $\mathcal{H} = [H_1,\ldots, H_s]$ and $\mathcal{K} = [K_1,\ldots, K_t]$ be snake graphs contained in a band graph $\mathcal{G}$, such that $H_i \neq K_j$ for all $i,j$. Let $P_{\mathcal{H}}$ and $P_{\mathcal{K}}$ be perfect matchings of $\mathcal{H}$ and $\mathcal{K}$, respectively. We say that $P_{\mathcal{H}}$ is *extendable* to $P_{\mathcal{K}}$ in $\mathcal{G}$ if there exists a perfect matching $P$ of the union $[H_1, H_2, \ldots, K_{t-1}, K_t]$ such that the orientation induced by $P$ on the diagonals of $\mathcal{H}$ and $\mathcal{K}$ agrees with the orientation induced by $P_{\mathcal{H}}$ and $P_{\mathcal{K}}$, respectively. {reference-type="ref" reference="extend"}, the tiles of $\mathcal{H}$ and $\mathcal{K}$ are shaded green and blue, respectively.](Fig_extendability.pdf){#glueability width="13.5cm"} **Definition 42**. Let $P$ be a perfect (resp. good) matching of a snake (resp. band) graph $\mathcal{G}$, and let $T$ be a tile of $\mathcal{G}$. We say that the diagonal of $T$ is *positive* with respect to $P$ if one of the following holds: - $T$ is an odd tile and the diagonal is oriented downwards; - $T$ is an even tile and the diagonal is oriented upwards. Otherwise we say that the diagonal of $T$ is *negative*. **Lemma 43**. *If $P_{\mathcal{H}}$ and $P_{\mathcal{K}}$ induce negative diagonals on the tiles $H_s$ and $K_1$, respectively, then $P_{\mathcal{H}}$ is extendable to $P_{\mathcal{K}}$.* *Proof.* This follows from the existence of a minimal matching on $[H_s, \ldots, K_1]$. ◻ *Proof of Proposition [Proposition 39](#local hsum){reference-type="ref" reference="local hsum"}.* Lemma [Lemma 43](#negative diagonals){reference-type="ref" reference="negative diagonals"} enables us to prove Proposition [Proposition 39](#local hsum){reference-type="ref" reference="local hsum"} with the following strategy. We shall show that for each $G(T,\alpha_i)$ appearing in Figure [9](#SG){reference-type="ref" reference="SG"} there exists a perfect matching $P_i$ such that: - $P_i$ achieves the corresponding local $h_k$; - $P_i$ induces a negative diagonal on the first and last tile of $G(T,\alpha_i)$. [**Case 1a)**]{.ul}: Here $b_{\gamma a} = -1$ and $b_{\gamma b} =1$. If the diagonal $\gamma$ is positive then so is the diagonal $a$. Therefore $h_k = 0$. An alternative way to achieve $h_k = 0$ is taking $P_i$ to be the minimal matching. [**Case 1b)**]{.ul}: There is no diagonal labeled by $\gamma'$ so $h_k \geq 0$. Taking $P_i$ to be the minimal matching achieves $h_k = 0$. [**Case 2a)**]{.ul}: Here $b_{\gamma a} = b_{\gamma c} = -1$. If the diagonal $\gamma$ is positive then so are the diagonals $a$ and $c$, hence $h_k = 0$. So take $P_i$ to be the minimal matching. [**Case 2b)**]{.ul}: Here $b_{\gamma' a} = b_{\gamma' c} = 1$. Hence $h_k = -1$ is achieved by taking $P_i$ such that the diagonal $\gamma'$ is positive, and all diagonals are negative. [**Case 3a)**]{.ul}: Here $b_{\gamma a} = -2$ and $b_{\gamma b} = 1$. If the diagonal $\gamma$ is positive then so is $a$, hence $h_k \geq 0$. Taking $P_i$ to be the minimal matching achieves $h_k = 0$. [**Case 3b)**]{.ul}: Here $b_{\gamma' a} = 2$ and $b_{\gamma' b} = -1$. If the diagonal $\gamma'$ is positive then so is the subsequent diagonal $b$, hence $h_k \geq 0$. Taking $P_i$ to be the minimal matching achieves $h_k = 0$. [**Case 4a)**]{.ul}: Here $b_{\gamma a} = -2$ and $b_{\gamma b} = b_{\gamma c} = 1$. If any diagonal $\gamma$ is positive then so is the adjacent diagonal $a$, hence $h_k \geq 0$. Taking $P_i$ to be the minimal matching achieves $h_k = 0$. [**Case 4b)**]{.ul}: Here $b_{\gamma' a} = 2$ and $b_{\gamma' b} = b_{\gamma' c} = -1$. Taking $P_i$ such that diagonals $b$ and $c$ are negative, and all other diagonals are positive shows $h_k \leq 1-k$. Moreover, since there are only $k-1$ diagonals labelled by $\gamma'$ then $h_k = 1-k$. [**Case 5a)**]{.ul}: There is no diagonal labelled $\gamma$ so $h_k = 0$, and we can take $P_i$ to be the minimal matching. [**Case 5b)**]{.ul}: Here $b_{\gamma' b} = -1$ and $b_{\gamma' a} = 1$. If $\gamma'$ is positive then so is $b$, hence $h_k = 0$, and we can take $P_i$ to be the minimal matching. [**Case 6a)**]{.ul}: Here $b_{\gamma a} = -1$. If $\gamma$ is positive then so is the subsequent diagonal $a$, hence $h_k = 0$, and we can take $P_i$ to be the minimal matching. [**Case 6b)**]{.ul}: If both diagonals $\gamma$ are positive, then so is the diagonal $c$. Thus, recalling the definition of the height function, we have that $h_k \geq -1$. Taking $P_i$ such that both diagonals $a$ are negative, and all other diagonals are positive achieves $h_k = -1$. [**Case 7a)**]{.ul}: There is no diagonal labelled $\gamma$ so $h_k = 0$, and we can take $P_i$ to be the minimal matching. [**Case 7b)**]{.ul}: Here $b_{\gamma' b} = -1$ and $b_{\gamma' a} = 1$. If the diagonal $\gamma'$ is positive then so is the diagonal $b$, hence $h_k = 0$ and we can take $P_i$ to be the minimal matching. [**Case 8a)**]{.ul}: Here $b_{\gamma a} = -1$. If the diagonal $\gamma$ is positive then so is the subsequent diagonal $a$, hence $h_k = 0$ and we can take $P_i$ to be the minimal matching. [**Case 8b)**]{.ul}: Here $b_{\gamma' a},b_{\gamma' c} \leq 0$. Moreover, there is precisely one tile with diagonal $\gamma'$, so $h_k = -1$. Taking $P_i$ such that the diagonal $\gamma'$ and the subsequent diagonal $c$ are positive, and all other diagonals negative achieves $h_k = -1$. This completes the proof of Proposition [Proposition 39](#local hsum){reference-type="ref" reference="local hsum"}. ◻ ## Proof of equation [\[combinatorial Fmutation\]](#combinatorial Fmutation){reference-type="eqref" reference="combinatorial Fmutation"} in the Combinatorial Key Lemma **Theorem 44**. *Equation [\[combinatorial Fmutation\]](#combinatorial Fmutation){reference-type="eqref" reference="combinatorial Fmutation"} in Theorem [Theorem 32](#thm: comb key lemma){reference-type="ref" reference="thm: comb key lemma"} holds true.* *Proof.* Without loss of generality, suppose that the signature of $T$ is non-negative, and identify $T$ with the ideal triangulation $T^\circ$ representing it. Consider all the local segments of $\alpha$ with respect to the flipping arc $\gamma:=\tau_k$, cf. Subsection [5.3](#subsec:local-segments){reference-type="ref" reference="subsec:local-segments"}. Some of the arcs lying on the region surrounding these local segments, cf. Figure [8](#flips){reference-type="ref" reference="flips"}, may be folded sides of self-folded triangles, even the dotted ones. Suppose first that $n>0$ such arcs are folded sides. We let $d_1, \ldots, d_m$ denote these folded sides, allowing repetition. For convenience, we also order $d_1, \ldots, d_m$ in the order that $\alpha$ intersects them (up to cyclic permutation). For each $j=1,\ldots,m$, we let $d_j^\circ\in T^\circ$ denote the loop that together with $d_j$ forms the corresponding self-folded triangle of $T^\circ$. The sum $$\displaystyle \sum_{P \in \mathcal{P}(G(T,\alpha))} \overline{y}(P)$$ decomposes as the sum of $2^{m}$ collections of perfect matchings -- each of these sums corresponding to a choice of orientation on each of the diagonals $d_1, \ldots, d_m$. Some of these sums may be empty; by convention we define an empty sum to be $0$. Specifically, if we let $(d_{i_1}, \ldots, d_{i_k})$ denote the collection of good matchings which induce positive orientation on $d_{i_1}, \ldots, d_{i_k}$ and negative orientation on the remaining diagonals, then we have the equality $$\displaystyle \sum_{P \in \mathcal{P}(G(T,\alpha))} \overline{y}(P) = \displaystyle \sum_{(d_{i_1}, \ldots, d_{i_k})} \Big( \sum_{P \in (d_{i_1}, \ldots, d_{i_k})} \overline{y}(P) \Big)$$ Moreover for any $(d_{i_1}, \ldots, d_{i_k})$ we have $$\label{product T} \displaystyle \sum_{P \in (d_{i_1}, \ldots, d_{i_k})} \overline{y}(P) = \Big(\prod_{i=1}^{m} \Big(\sum_{P \in (d_{i_1}, \ldots, d_{i_k})_{i,i+1}} y(P)\Big)\Big) \frac{y_{d_{i_1}} \ldots y_{d_{i_k}}}{y_{d_{i_1}^{\circ}} \ldots y_{d_{i_k}^{\circ}}},$$ where $(d_{i_1}, \ldots, d_{i_k})_{i,i+1}$ denotes the collection of perfect matchings, between the tiles $d_i$ and $d_{i+1}$, which induce the orientations on $d_i$ and $d_{i+1}$ dictated by $(d_{i_1}, \ldots, d_{i_k})$. By convention we define $(d_{i_1}, \ldots, d_{i_k})_{m,m+1} := (d_{i_1}, \ldots, d_{i_k})_{m,1}$. Analogously, if we look at $T'$ we get $$\label{product T'} \displaystyle \sum_{P \in (d_{i_1}, \ldots, d_{i_k})'} \overline{y}'(P) = \Big(\prod_{i=1}^{m} \Big(\sum_{P \in (d_{i_1}, \ldots, d_{i_k})'_{i,i+1}} y'(P)\Big)\Big) \frac{y_{d_{i_1}}' \ldots y_{d_{i_k}}'}{y_{d_{i_1}^{\circ}}' \ldots y_{d_{i_k}^{\circ}}'}$$ where $(d_{i_1}, \ldots, d_{i_k})'$ is defined in the same way as above, but now with respect to $\mathcal{P}(G(T',\alpha))$, rather than $\mathcal{P}(G(T,\alpha))$. By Proposition [Proposition 39](#local hsum){reference-type="ref" reference="local hsum"} and the explicit computations carried out in Figure [11](#fig: SGmatching){reference-type="ref" reference="fig: SGmatching"}, the proof of the identity $$(y_k+1)^{h_k}\cdot\displaystyle \sum_{P \in (d_{i_1}, \ldots, d_{i_k})} \overline{y}(P) = (y_k'+1)^{h_k'}\cdot\displaystyle \sum_{P \in (d_{i_1}, \ldots, d_{i_k})'} \overline{y}'(P)$$ is reduced to showing that $\frac{y'_{d_{i_1}} \ldots y'_{d_{i_k}}}{y'_{d_{i_1}^{\circ}} \ldots y'_{d_{i_k}^{\circ}}} = \frac{y_{d_{i_1}} \ldots y_{d_{i_k}}}{y_{d_{i_1}^{\circ}} \ldots y_{d_{i_k}^{\circ}}}$. This follows from the equality $b_{\gamma \hspace{0.2mm} d_i} = b_{\gamma \hspace{0.2mm} d_i^{\circ}}$. !["Local" verification of equation [\[combinatorial Fmutation\]](#combinatorial Fmutation){reference-type="eqref" reference="combinatorial Fmutation"} in the proof of Theorem [Theorem 44](#full comb key lemma){reference-type="ref" reference="full comb key lemma"}. ](Fig_SG1matchings.pdf){#fig: SGmatching width="15cm"} {width="15cm"} {width="15cm"} {width="15cm"} {width="15cm"} {width="15cm"} {width="15cm"} {width="15cm"} When none of the boundary edges of the flip regions encountered corresponds to self-folded triangles, the proof of equation [\[combinatorial Fmutation\]](#combinatorial Fmutation){reference-type="eqref" reference="combinatorial Fmutation"}, is easier, because, in that case, the snake subgraphs of the band graphs $G(T,\alpha)$ that arise from the local segment do not interfere with each other at all. The details for that case are left to the reader. ◻ ## Local $\mathbf{g}$-vectors In this section we show that there are six fundamental configurations that dictate how snake $\textbf{g}$-vectors change under mutation -- these are listed below in Figure [12](#local gvector flips){reference-type="ref" reference="local gvector flips"}. {#local gvector flips width="11cm"} {width="11cm"} {width="11cm"} {width="11cm"} {width="11cm"} {width="11cm"} **Lemma 45**. *Let $\alpha$, $T$ and $T'$ be as in the statment of Theorem [Theorem 32](#thm: comb key lemma){reference-type="ref" reference="thm: comb key lemma"}, and let $\mathcal{L}_k$ denote the set of all local curves of $\alpha$ with respect to the flipping/mutating index $k$. Then the following equality holds: $$\label{gvector sum} g_{G(T,\alpha)} - \displaystyle \sum_{\beta \in \mathcal{L}_k} g_{G(T,\beta)} = g_{G(T',\alpha)} - \displaystyle \sum_{\beta \in \mathcal{L}_k} g_{G(T',\beta)}$$ Moreover, the $k^{th}$ component of $g_{G(T,\alpha)} - \displaystyle \sum_{\beta \in \mathcal{L}_k} g_{G(T,\beta)}$ is $0$.* *Proof.* By direct inspection of Figure [12](#local gvector flips){reference-type="ref" reference="local gvector flips"} we see that the graphs $$G(T,\alpha) \setminus \big(\displaystyle \coprod_{\beta \in \mathcal{L}_k} G(T,\beta)\big) \qquad \text{and} \qquad G(T,\alpha) \setminus \big(\displaystyle \coprod_{\beta \in \mathcal{L}_k} G(T,\beta)\big)$$ are isomorphic. The validity of Equation ([\[gvector sum\]](#gvector sum){reference-type="ref" reference="gvector sum"}) immediately follows. Furthermore, let $\gamma \in T$ be the arc corresponding to $k$. Note that $\gamma$ is an edge label of $G(T,\alpha)$ *if and only if* this edge belongs to $G(T,\beta)$ for some $\beta \in \mathcal{L}_k$. Consequently, the $k^{th}$ component of $g_{G(T,\alpha)} - \displaystyle \sum_{\beta \in \mathcal{L}_k} g_{G(T,\beta)}$ is $0$. ◻ **Proposition 46**. *Let $\alpha$, $T$ and $T'$ be as in the statement of Theorem [Theorem 32](#thm: comb key lemma){reference-type="ref" reference="thm: comb key lemma"}. Then: $$\label{comb gvector mutation eq 2} \left[\begin{array}{c}-B_{\operatorname{FST}}(T')\\ \mathbf{g}_{G(T',\alpha)}\end{array}\right] =\mu_k\left( \left[\begin{array}{c}-B_{\operatorname{FST}}(T)\\ \mathbf{g}_{G(T,\alpha)}\end{array}\right]\right).$$* *Proof.* By Lemma [Lemma 45](#gvector difference){reference-type="ref" reference="gvector difference"} it suffices to show equation ([\[comb gvector mutation eq 2\]](#comb gvector mutation eq 2){reference-type="ref" reference="comb gvector mutation eq 2"}) holds for the six configurations listed in Figure [12](#local gvector flips){reference-type="ref" reference="local gvector flips"}. The result then follows by direct inspection. Note that a little extra care is needed when treating the scenario that boundary edges of the flip region are labelled by some $\ell_p$. To this end, recall that $x_{\ell_p} = x_{\beta}x_{\beta^{(p)}}$ for some $\beta \in T$, and let us denote the arc corresponding to $k$ by $\gamma$. The validity of equation ([\[comb gvector mutation eq 2\]](#comb gvector mutation eq 2){reference-type="ref" reference="comb gvector mutation eq 2"}) then follows from the equality $b_{\beta \gamma} = b_{\beta^{(p)} \gamma}$. ◻ **Theorem 47**. *Suppose $\mathbf{\Sigma}=(\Sigma,\mathbb{M},\mathbb{P})$ is a surface with non-empty boundary. Let $\alpha$ be a simple closed curve on $\mathbf{\Sigma}$, and $T$ a tagged triangulation of $\mathbf{\Sigma}$. Then $$\mathbf{g}_{G(T,\alpha)} = \operatorname{Sh}_T(\alpha).$$* *Proof.* By Proposition [Proposition 25](#prop:existence-ad-hoc-triangulation-for-closed-curve){reference-type="ref" reference="prop:existence-ad-hoc-triangulation-for-closed-curve"}, there exists a tagged triangulation $T_\alpha$ such that $\operatorname{Sh}_{T_\alpha}(\alpha) = \mathbf{g}_{T_\alpha}(G(T_\alpha,\alpha))$. Since $T$ can be obtained from $T_\alpha$ by a finite sequence of flips, the theorem follows from Proposition [Proposition 46](#comb gvector mutation 2){reference-type="ref" reference="comb gvector mutation 2"} and Theorem [Theorem 23](#thm:behavior-of-dual-shear-coords-under-matrix-mutation){reference-type="ref" reference="thm:behavior-of-dual-shear-coords-under-matrix-mutation"}. ◻ **Remark 48**. Theorem [Theorem 47](#thm:shear-coords-equal-snake-g-vector-for-closed-curves){reference-type="ref" reference="thm:shear-coords-equal-snake-g-vector-for-closed-curves"} should be compared to [@musiker2013bases Corollary 6.15-(2)], though the reader is advised to be wary of signs and conventions. Our techniques provide a new, independent proof of Theorem [Theorem 47](#thm:shear-coords-equal-snake-g-vector-for-closed-curves){reference-type="ref" reference="thm:shear-coords-equal-snake-g-vector-for-closed-curves"}. ## Proof of equations ([\[comb kth gvector\]](#comb kth gvector){reference-type="ref" reference="comb kth gvector"}) and ([\[comb gvector mutation\]](#comb gvector mutation){reference-type="ref" reference="comb gvector mutation"}) in the Combinatorial Key Lemma **Lemma 49**. *For each $k \in \{1,\ldots, n\}$ the snake $g$- and $h$-vectors $\mathbf{g}_{G(T,\alpha)} = (g_{\alpha; 1},\ldots, g_{\alpha; n})$ and $\mathbf{h}_{G(T,\alpha)} = (h_{\alpha; 1},\ldots, h_{\alpha; n})$ of a simple closed curve $\alpha$ satisfy:* *$$h_{\alpha; k} = \min(0, g_{\alpha; k})$$* *Proof.* From Proposition [Proposition 39](#local hsum){reference-type="ref" reference="local hsum"} we have: $$h_{\alpha; k} = \displaystyle \sum_{\alpha_{ij} \in \mathcal{L}} h_{\alpha_{ij}; k}.$$ Directly from the definition of shear coordinates, for any (tagged) triangulation $T$ we have: $$\operatorname{Sh}_T(\alpha) = \displaystyle \sum_{\alpha_{ij} \in \mathcal{L}} \operatorname{Sh}_T(\alpha_{ij}).$$ Moreover, for any $k \in \{1,\ldots, n\}$ the following two statements hold: - $g_{\alpha; k} = \operatorname{Sh}_T(\alpha; k)$, - $\operatorname{Sh}_T(\alpha_{ij}; k) \geq 0$ for all $\alpha_{ij} \in \mathcal{L}$, or $\operatorname{Sh}_T(\alpha_{ij}; k) \leq 0$ for all $\alpha_{ij} \in \mathcal{L}$. Therefore, to prove the lemma it suffices to show $$h_{\alpha_{ij}; k} = \min\big(0, \operatorname{Sh}_T(\alpha_{ij}; k)\big)$$ for every $\beta \in \mathcal{L}$. This follows by direct inspection, and from the explicit computations used in the proof of Proposition [Proposition 39](#local hsum){reference-type="ref" reference="local hsum"}. ◻ *Proof of ([\[comb kth gvector\]](#comb kth gvector){reference-type="ref" reference="comb kth gvector"}) and ([\[comb gvector mutation\]](#comb gvector mutation){reference-type="ref" reference="comb gvector mutation"}) from Theorem [Theorem 32](#thm: comb key lemma){reference-type="ref" reference="thm: comb key lemma"}..* The validity of these two equations follow directly from Proposition [Proposition 46](#comb gvector mutation 2){reference-type="ref" reference="comb gvector mutation 2"} and Lemma [Lemma 49](#hk min){reference-type="ref" reference="hk min"}. ◻ # Bangle functions are the generic basis {#sec:bangle-functions-are-the-generic-basis} ## The bangle function of a tagged arc belongs to the generic basis **Theorem 50**. *Let $\mathbf{\Sigma}=(\Sigma,\mathbb{M},\mathbb{P})$ be a surface with non-empty boundary, and let $\alpha$ be a tagged arc on $\mathbf{\Sigma}$. For every tagged triangulation $T$ of $\mathbf{\Sigma}$, the bangle function $\operatorname{MSW}(G(T,\alpha))$ is equal to the generic value taken by the coefficient-free Caldero-Chapoton function $CC_{A(T)}$ on the irreducible component $\pi_T(\alpha)\in \operatorname{DecIrr}^\tau(A(T))$.* *Proof.* Since the boundary of $\Sigma$ is not empty, by [@fomin2008cluster Proposition 7.10] there exists a sequence of flips that transforms $T$ in a tagged triangulation $T_\alpha=f_{k_m}\cdots f_{k_1}(T)$ containing $\alpha$. The negative simple representation $\mathcal{S}_\alpha^-(A(T_\alpha))$ (see [@derksen2008quivers the paragraph preceding Proposition 10.15] or [@derksen2010quivers Equation (1.15)]) is the unique point in an irreducible component $Z_{T_\alpha,\alpha}$ for the Jacobian algebra $A(T_\alpha)$. This component $Z_{T_\alpha,\alpha}$ is obviously $\tau$-rigid, i.e., it is a $\tau$-reduced component with $E$-invariant zero. By Proposition [\[prp:JaMut\]](#prp:JaMut){reference-type="ref" reference="prp:JaMut"} and the invariance of the $E$-invariant under mutations [@derksen2010quivers Theorem 7.1], $Z_{T,\alpha}:=\widetilde{\mu}_{k_m}\cdots\widetilde{\mu}_{k_1}(Z_{T_\alpha,\alpha})$ is a $\tau$-reduced component with $E$-invariant zero for $A(T)$, and the generic value $CC_{A(T)}(Z_{T,\alpha})$ is given by $CC_{A(T)}(\mathcal{M}(T,\alpha))$, where $\mathcal{M}(T,\alpha):=\mu_{k_m}\cdots\mu_{k_1}(\mathcal{S}_\alpha^-(A(T_\alpha)))$. Now, by [@fomin2007cluster Corollary 6.3] and [@derksen2010quivers Equation (2.14) and Theorem 5.1], $CC_{A(T)}(\mathcal{M}(T,\alpha))$ is the cluster variable that corresponds to $\alpha$ according to [@fomin2008cluster Theorem 7.11]. On the other hand, by [@musiker2011positivity Theorems 4.9, 4.16 and 4.20], $CC_{A(T)}(\mathcal{M}(T,\alpha))=\operatorname{MSW}(G(T,\alpha))$. ◻ ## The bangle function of a closed curve belongs to the generic basis **Theorem 51**. *Let $\mathbf{\Sigma}=(\Sigma,\mathbb{M},\mathbb{P})$ be a surface with non-empty boundary, and let $\alpha$ be a simple closed curve on $\mathbf{\Sigma}$. For every tagged triangulation $T$ of $\mathbf{\Sigma}$, the bangle function $\operatorname{MSW}(G(T,\alpha))$ is equal to the generic value taken by the coefficient-free Caldero-Chapoton function $CC_{A(T)}$ on the irreducible component $\pi_T(\alpha)\in \operatorname{DecIrr}^\tau(A(T))$.* *Proof.* Let $T$ be an arbitray tagged triangulation of $\mathbf{\Sigma}$, and let $T_\alpha$, and $M(T_\alpha,\alpha,\lambda,1)$ be as in Proposition [Proposition 25](#prop:existence-ad-hoc-triangulation-for-closed-curve){reference-type="ref" reference="prop:existence-ad-hoc-triangulation-for-closed-curve"}. By Corollary [Corollary 29](#coro:generic-values-on-adhoc-irred-comp-coincide-with-MSW){reference-type="ref" reference="coro:generic-values-on-adhoc-irred-comp-coincide-with-MSW"}, the set $$Z_{T_\alpha,\alpha}:=\overline{\bigcup_{\lambda\in\mathbb{C}\setminus\{0\}}\operatorname{GL}_{\mathbf{d}}(\mathbb{C})\cdot M(T_\alpha,\alpha,\lambda,1)}$$ is a generically $\tau$-reduced indecomposable irreducible component of the representation space $\operatorname{rep}(A(T_\alpha),\mathbf{d})$, where $\mathbf{d}:=\underline{\dim}(M(T_\alpha,\alpha,\lambda,1))$, with $$\label{eq:g-vector-matches-and-CC=MSW-for-closed-curve-and-ad-hoc-triangulation} \mathbf{g}_{A(T_\alpha)}(Z_{T_\alpha,\alpha})=\mathbf{g}_{G(T_\alpha,\alpha)} \qquad \text{and} \qquad CC_{A(T_\alpha)}(Z_{T_\alpha,\alpha})=\operatorname{MSW}(G(T_\alpha,\alpha)).$$ Thus, $Z_{T_\alpha,\alpha}=\pi_{T_\alpha}(\alpha)$. Moreover, any point in $\bigcup_{\lambda\in\mathbb{C}\setminus\{0\}}\operatorname{GL}_{\mathbf{d}}(\mathbb{C})\cdot M(T_\alpha,\alpha,\lambda,1)$ achieves the values [\[eq:g-vector-matches-and-CC=MSW-for-closed-curve-and-ad-hoc-triangulation\]](#eq:g-vector-matches-and-CC=MSW-for-closed-curve-and-ad-hoc-triangulation){reference-type="eqref" reference="eq:g-vector-matches-and-CC=MSW-for-closed-curve-and-ad-hoc-triangulation"}. Since the boundary of $\Sigma$ is non-empty, there exists a finite sequence $(T_0,T_1,\ldots,T_m)$ of tagged triangulations, with $T_0=T_\alpha$ and $T_m=T$, such that for each $i=1,\ldots,n$, $T_i$ is obtained from $T_{i-1}$ by flipping an arc $k_i\in T_{i-1}$. By [@labardini2016quivers Theorem 8.1] and [@derksen2008quivers Proposition 3.7], the Jacobian algebra $A(T):=\mathcal{P}(Q(T),S(T))$ is isomorphic to the Jacobian algebra of $\mu_{k_m}\cdots\mu_{k_1}(Q(T_\alpha),S(T_\alpha))$. By Proposition [\[prp:JaMut\]](#prp:JaMut){reference-type="ref" reference="prp:JaMut"}, $$Z_{T,\alpha}:=\widetilde{\mu}_{k_m}\cdots\widetilde{\mu}_{k_1}(Z_{T_\alpha,\alpha})$$ is a $\tau$-reduced indecomposable irreducible component of the representation varieties of $\mathcal{P}(\mu_{k_m}\cdots\mu_{k_1}(Q(T_\alpha),S(T_\alpha)))\cong A(T)$. By Theorem [\[thm:Lams-vs-taured-comps-iso\]](#thm:Lams-vs-taured-comps-iso){reference-type="ref" reference="thm:Lams-vs-taured-comps-iso"}, we have $Z_{T,\alpha}=\pi_{T}(\alpha)$. Since $CC_{A(T_\alpha)}(M(T_\alpha,\alpha,\lambda,1))$ is the generic value $CC_{A(T_\alpha)}(Z_{T_\alpha,\alpha})$, Proposition [\[prp:JaMut\]](#prp:JaMut){reference-type="ref" reference="prp:JaMut"} implies that $CC_{A(T)}(M(T,\alpha,\lambda,1))$ is the generic value $CC(Z_{T,\alpha})$, where $$M(T,\alpha,\lambda,1):=\mu_{k_m}\cdots\mu_{k_1}(M(T_\alpha,\alpha,\lambda,1)).$$ By Proposition [Proposition 25](#prop:existence-ad-hoc-triangulation-for-closed-curve){reference-type="ref" reference="prop:existence-ad-hoc-triangulation-for-closed-curve"} and Corollary [Corollary 29](#coro:generic-values-on-adhoc-irred-comp-coincide-with-MSW){reference-type="ref" reference="coro:generic-values-on-adhoc-irred-comp-coincide-with-MSW"} we have $$\mathbf{g}_{A(T_\alpha)}(M(T_\alpha,\alpha,\lambda,1))=\mathbf{g}_{A(T_\alpha)}(Z_{T_\alpha,\alpha})=\operatorname{Sh}_{T_\alpha}(\alpha)=\mathbf{g}_{G(T_\alpha,\alpha)},$$ so Proposition [\[prp:JaMut\]](#prp:JaMut){reference-type="ref" reference="prp:JaMut"} and Theorems [\[thm:Lams-vs-taured-comps-iso\]](#thm:Lams-vs-taured-comps-iso){reference-type="ref" reference="thm:Lams-vs-taured-comps-iso"} and [Theorem 47](#thm:shear-coords-equal-snake-g-vector-for-closed-curves){reference-type="ref" reference="thm:shear-coords-equal-snake-g-vector-for-closed-curves"} imply that $$\mathbf{g}_{A(T)}(M(T,\alpha,\lambda,1))=\mathbf{g}_{A(T)}(Z_{T,\alpha})=\operatorname{Sh}_{T}(\alpha)=\mathbf{g}_{G(T,\alpha)}.$$ At this point, we have shown that the identity $\mathbf{g}_{A(T')}(Z_{T',\alpha})=\mathbf{g}_{G(T,\alpha)}$ holds for every tagged triangulation, which allows us to see that Theorem [Theorem 32](#thm: comb key lemma){reference-type="ref" reference="thm: comb key lemma"} and [@derksen2010quivers Theorem 5.1 and Lemma 5.2] and the particular equality $F_{M(T_\alpha,\alpha,\lambda,1)}=F_{G(T_\alpha,\alpha)}$ that was established in Proposition [Proposition 25](#prop:existence-ad-hoc-triangulation-for-closed-curve){reference-type="ref" reference="prop:existence-ad-hoc-triangulation-for-closed-curve"}, imply that $F_{M(T,\alpha,\lambda,1)}=F_{G(T,\alpha)}$. We deduce that $CC_{A(T)}(Z_{T,\alpha})=CC_{A(T)}(M(T,\alpha,\lambda,1))=\operatorname{MSW}(G(T,\alpha))$ as elements of the Laurent polynomial ring $\mathbb{Z}[x_j^{\pm 1}\ | \ j\in T]$. ◻ ## Main result **Theorem 52**. *Let $\mathbf{\Sigma}=(\Sigma,\mathbb{M},\mathbb{P})$ be a surface with non-empty boundary such that $|\mathbb{M}|\geq 2$. For each tagged triangulation $T$ of $\mathbf{\Sigma}$, the set $\mathcal{B}^\circ(\mathbf{\Sigma})$ of coefficient-free bangle functions is equal to the generic basis $\mathcal{B}_{A(T)}$ of the coefficient-free (upper) cluster algebra $\mathcal{A}(\mathbf{\Sigma})=\mathcal{U}(\mathbf{\Sigma})$.* *Proof.* By Theorem [\[thm:Lams-vs-taured-comps-iso\]](#thm:Lams-vs-taured-comps-iso){reference-type="ref" reference="thm:Lams-vs-taured-comps-iso"}, we know that there is a bijection between the set of single laminates on $\mathbf{\Sigma}$ and the set of indecomposable $\tau$-reduced components of the representation spaces of the Jacobian algebra $A(T)$. Furthermore, we have proved in Theorems [Theorem 50](#thm:bangle-of-tagged-arc-belongs-to-generic-basis){reference-type="ref" reference="thm:bangle-of-tagged-arc-belongs-to-generic-basis"} and [Theorem 51](#thm:bangle-of-simple-closed-curve-belongs-to-generic-basis){reference-type="ref" reference="thm:bangle-of-simple-closed-curve-belongs-to-generic-basis"} that for the indecomposable $\tau$-reduced component $Z_{T,\alpha}$ associated to each single laminate $\alpha$, the coefficient-free generic Caldero-Chapoton function $CC_{A(T)}(Z_{T,\alpha})$ is equal to $\operatorname{MSW}(G(T,\alpha))$. The theorem follows by combining these observations with [@geiss2023semicontinuous Corollary 1.7] (see also [@plamondon2023tame Theorem 3.8]), [@geiss2016the Theorem 7.1] (see also [@labardini2016on Theorem 3.5]), [@cerulli2015caldero-chapoton Theorem 5.11 and Proposition 6.7], and [@musiker2013bases Definition 3.19]. ◻ # A couple of remarks and open problems {#sec:remarks-and-problems} During the proofs of Theorems [Theorem 50](#thm:bangle-of-tagged-arc-belongs-to-generic-basis){reference-type="ref" reference="thm:bangle-of-tagged-arc-belongs-to-generic-basis"} and [Theorem 51](#thm:bangle-of-simple-closed-curve-belongs-to-generic-basis){reference-type="ref" reference="thm:bangle-of-simple-closed-curve-belongs-to-generic-basis"}, we have shown that, given $\alpha$, one can find a (tagged) triangulation $T_\alpha$ for which one can further find a very concrete (decorated) representation $\mathcal{M}(T_\alpha,\alpha)$ of the Jacobian algebra $A(T_\alpha)=\mathcal{P}(Q(T_\alpha),S(T_\alpha))$ such that Indeed, when $\alpha$ was a tagged arc, we picked $T_\alpha$ to be any tagged triangulation containing it, and $\mathcal{M}(T_\alpha,\alpha)$ to be the negative simple representation of $(Q(T_\alpha),S(T_\alpha))$ corresponding to $\alpha$; when $\alpha$ was a simple closed curve, we picked $T_\alpha$ to be a triangulation such that the Jacobian algebra $\Lambda_I$ of the restriction of $(Q(T_\alpha),S(T_\alpha))$ to the set $I$ of arcs crossed by $\alpha$ is gentle (see Proposition [Proposition 25](#prop:existence-ad-hoc-triangulation-for-closed-curve){reference-type="ref" reference="prop:existence-ad-hoc-triangulation-for-closed-curve"}), and $\mathcal{M}(T_\alpha,\alpha)$ to be the positive representation of $(Q(T_\alpha),S(T_\alpha))$ given by any quasi-simple band module $M(T_\alpha,\alpha,\lambda,1)$ arising from interpreting $\alpha$ as a band on $\Lambda_I$. Now, although $\mathcal{M}(T_\alpha,\alpha)$ is a very explicit representation, we have not provided an explicit computation of $\mathcal{M}(T,\alpha)$ in general. Of course, when $\mathbb{P}=\varnothing$, the Jacobian algebra $A(T)$ is gentle, and $\mathcal{M}(T,\alpha)$ can be computed explicitly. When $\mathbb{P}\neq\varnothing$ but the signature of $T$ is zero, the Jacobian algebra $A(T)$ is skewed-gentle, and $\mathcal{M}(T,\alpha)$ can be computed explicitly as well, cf. [@crawley1989functorial; @geiss2023onhomomorphisms; @geiss2023laminations]. But when $\mathbb{P}\neq\varnothing$ and $T$ is arbitrary, the decorated representation $\mathcal{M}(T,\alpha)$ still remains to be explicitly computed in general. It should be noticed that the naive candidate for $\mathcal{M}(T,\alpha)$, namely, the obvious string or band representation of the quiver $Q(T)$ induced by $\alpha$, typically fails to be annihilated by the cyclic derivatives of the potential $S(T)$, see e.g. [@labardini2010quivers Example 6.2.7]. Explicit computation of $\mathcal{M}(T,\alpha)$ has been carried out by the second author in [@labardini2010quivers] in the following situations: - when $T$ is a tagged triangulation of positive signature and $\alpha$ is a tagged arc with at most one notch; - when $T$ is a tagged triangulation of positive signature and $\alpha$ is a simple closed curve (this is only implicit in [@labardini2010quivers], but one can check that the results proved therein apply in this situation). **Problem 53**. Compute the decorated representation $\mathcal{M}(T,\alpha)$ of $(Q(T),S(T))$ in general. On an arguably more important matter, one of the reasons why in this paper we have not considered surfaces with empty boundary whatsoever, is that the proof given in [@geiss2020generic] of the linear independence of the set of generic Caldero-Chapoton functions can definitely not be applied for such surfaces. **Problem 54**. For a tagged triangulation $T$ of a punctured surface with empty boundary $(\Sigma,\mathbb{P})$, is the set of generic Caldero-Chapoton functions over $\mathcal{P}(Q(T),S(T))$ linearly independent? Is it a basis for the Caldero-Chapoton algebra of $\mathcal{P}(Q(T),S(T))$, or better, for the (upper, coefficient-free) cluster algebra of $\Sigma$? What is its relation to Musiker--Schiffler--Williams' bangle functions? Here, $S(T)$ is the potential defined in [@labardini2016quivers]. ## Acknowledgments {#acknowledgments .unnumbered} We are grateful to Jan Schröer for many illuminating discussions. The first author acknowledges partial support from PAPIIT grant IN116723 (2023-2025). The work of this paper originated during the third author's back-to-back visits at IMUNAM courtesy of the first author's CONACyT-239255 grant and a DGAPA postdoctoral fellowship. He is grateful for the stimulating environment IMUNAM fostered, and for the additional generous support received from the second author's grants: CONACyT-238754 and a Cátedra Marcos Moshinsky.

arxiv_math

--- abstract: | The increase of renewables in the grid and the volatility of the load create uncertainties in the day-ahead prices of electricity markets. Adaptive robust optimization (ARO) and stochastic optimization have been used to make commitment and dispatch decisions that adapt to the load and capacity uncertainty. These approaches have been successfully applied in practice but current pricing approaches used by US Independent System Operators (marginal pricing) and proposed in the literature (convex hull pricing) have two major disadvantages: a) they are deterministic in nature, that is they do not adapt to the load and capacity uncertainty, and b) require uplift payments to the generators that are typically determined by ad hoc procedures and create inefficiencies that motivate self-scheduling. In this work, we extend pay-as-bid and uniform pricing mechanisms to propose the first adaptive pricing method in electricity markets that adapts to the load and capacity uncertainty, eliminates post-market uplifts and deters self-scheduling, addressing both disadvantages. author: - "Dimitris Bertsimas,  Angelos Georgios Koulouras,  [^1]" bibliography: - ref.bib title: Adaptive Pricing in Unit Commitment Under Load and Capacity Uncertainty --- Shell *et al.*: A Sample Article Using IEEEtran.cls for IEEE Journals Pricing, Robust Optimization, Adaptive Optimization, Unit Commitment, Energy Markets, Uncertainty # Introduction The future of electricity markets is expected to prominently feature renewable energy sources, which bring volatility and unpredictability to the markets [@pinson2023may; @morales2013integrating]. With the increase of wind and solar power comes an increase in the price volatility, among other things, which creates issues for both the market operators and the market participants. Specifically, [@oren2004pay] shows that as the penetration of wind power increases, the market-clearing price becomes more volatile and uncertain, making it difficult for wind power producers to forecast and bid their power accurately. In addition, changes are also expected on the consumer side. For example, the introduction of electric vehicles could change demand patterns as well as available storage options [@hannan2022vehicle; @tan2016integration]. Also, consumers could simultaneously play the role of producers thereby becoming [@khorasany2020new; @morstyn2018using; @zhang2021strategic]. So far, most energy markets do not directly address these challenges as they have been set up using deterministic approaches [@silva2022short]. However, there has been recent work in robust and stochastic optimization with promising results in addressing the unit commitment (UC) problem under uncertainty [@bertsimas2012adaptive; @lorca2016multistage; @xiong2016distributionally; @zheng2014stochastic; @jiang2011robust]. Adaptive robust optimization (ARO) minimizes the cost under the worst-case scenario in an uncertainty set, while stochastic optimization accounts for the probability distribution of the uncertain parameters by minimizing the expected cost. In general, ARO has been used to protect against different types of uncertainty, from contingencies and demand planning to wind energy output and has lower average dispatch and total costs, indicating better economic efficiency and significantly reduces the volatility of the total costs [@bertsimas2012adaptive]. In [@bertsimas2012adaptive], the authors use ARO to find solutions in the UC problem that are robust to uncertain nodal injections. This work is extended in [@lorca2016multistage], which deals with multistage UC and dynamic uncertainty sets related to the capacity of renewable energy sources. The authors of [@xiong2016distributionally] also address wind uncertainty. They propose a distributionally robust approach, where they define a family of wind power distributions and minimize the expected cost under the worst-case distribution. Our approach is based on ARO but there has also been promising work on stochastic optimization. See [@zheng2014stochastic; @reddy2017review] for an introduction to these methods. While the previous approaches have been successful in dealing with the volatility in energy systems and markets, the pricing methods available are still mostly deterministic [@dutta2017literature; @mazzi2017price]. One of the more popular schemes is uniform marginal-cost pricing or , which may result in losses for some generators [@o2005efficient; @o2016dual]. Therefore, side-payments or uplifts are provided to these generators to make them whole, which may modify their incentives. Alternative mechanisms have been proposed, including removing the negative uplifts, such that no generator incurs a loss, and raising the commodity price above marginal cost to reduce uplifts [@bjorndal2008equilibrium; @galiana2003reconciling]. One principled version of that is the pricing scheme, which minimizes the uplifts by langragifying the energy balance constraint and maximizing over its dual price [@hua2016convex; @andrianesis2021computation]. Possibly only [@ye2016uncertainty] and [@fang2019introducing] discuss pricing in robust UC. However, they do not use ARO and [@fang2019introducing] does not offer extended theoretical results. For a detailed review of the pricing methods, see [@liberopoulos2016critical]. Current pricing approaches used by Independent System Operators (ISOs) (marginal pricing) and proposed in the literature (convex hull pricing) have two major disadvantages: a) they are deterministic in nature, that is they do not adapt to the load and capacity uncertainty, and b) require uplift payments to the generators that are typically determined by ad hoc procedures and create inefficiencies that motivate self-scheduling. In contrast, we extend pay-as-bid and uniform pricing mechanisms to propose the first adaptive pricing method in electricity markets that adapts to the load and capacity uncertainty, eliminates post-market uplifts and deters self-scheduling, addressing both disadvantages. ## Contributions In this work, we propose the first pricing method for ARO in energy markets with load and capacity uncertainty by offering contracts contingent to the uncertainty in the data of the day-ahead problem. We summarize the main contributions: - We introduce adaptive pay-as-bid and marginal pricing contracts for UC problems featuring load and capacity uncertainty. We specify fully the day-ahead payments and provide an upper-bound on the next-day or intra-day payments based on the day-ahead commitments. The payments are functions of the load and capacity uncertainty. - We show that the adaptive day-ahead pay-as-bid scheme is equivalent to the adaptive uniform pricing scheme. Also, if the worst-case uncertainty is realized in the next day, the forecasted intra-day payments are optimal and the generators are indifferent between the market schedule and their optimal schedule, eliminating self-scheduling. - We display the adaptive pricing in detail on the adaptive robust version of the Scarf example, see [@o2005efficient], and on realistic UC problems featuring ramp constraints. We compare it to deterministic marginal and convex hull pricing and find that adaptive pricing eliminates the corrections that are necessary in deterministic day-ahead problems. The paper is organized as follows: in Section [2](#sec:sec_method){reference-type="ref" reference="sec:sec_method"}, we introduce the ARO formulations for UC. In Section [3](#sec:adamarkets){reference-type="ref" reference="sec:adamarkets"}, we present the adaptive pricing and its theoretical properties on UC with load and capacity uncertainty. In Section [4](#sec:sec_load_example){reference-type="ref" reference="sec:sec_load_example"} and in Section [5](#sec:sec_load_cap_example){reference-type="ref" reference="sec:sec_load_cap_example"} we provide adaptive pricing on examples with load and capacity uncertainty. In Section [6](#sec:sec_multi){reference-type="ref" reference="sec:sec_multi"}, we demonstrate our method on a realistic example with ramp constraints and compare it to convex hull pricing. Finally, Section [7](#sec:ada_conclusion){reference-type="ref" reference="sec:ada_conclusion"} summarizes our conclusions. The notation that we use is as follows: we use bold faced characters such as $\boldsymbol{p}$ to represent vectors and capital letters such as $\boldsymbol{V}$ to represent matrices. Also, $\boldsymbol{p}^{T}$ denotes the transpose of the column vector $\boldsymbol{p}$ and $\boldsymbol{e}_{i}$ is the $i$th unit vector. We define $[I] = \{1, \dots, I\}$. The ${\cal L}_{1}$ norm of a vector refers to the norm $\| \boldsymbol{x} \|_{1} = \sum_{i=1}^{I} |x_i|$, the ${\cal L}_{2}$ norm refers to $\| \boldsymbol{x} \|_{2} = \sqrt{\sum_{i=1}^{I} x_{i}^{2}}$ and ${\cal L}_{\infty}$ refers to $\| \boldsymbol{x} \|_{\infty} = \max_{i \in [I]} |x_i|$. # Adaptive Robust UC {#sec:sec_method} In this section, we introduce the ARO formulation for an energy market robust to load and capacity uncertainty. The formulation is based on the popular Scarf example, see [@o2005efficient], but considers uncertainty in the load and capacity parameters. Reserves, transmission and ramp constraints can be added using linear constraints with small changes. Consider the following example which tries to minimize the total cost of meeting a fixed level of demand. $${\everymath{\displaystyle} \scalebox{0.99}{ $ \begin{array}{rlr} \min_{x, p} & \sum_{i=1}^{I} F_{i} x_{i} + C_{i} p_{i} \\ \text{s.t.} & \sum_{i=1}^{I} p_{i} = \sum_{j=1}^{J} \bar{q}_{j}, \\ & p_{i} \leq \bar{p}^{\max}_{i} x_{i}, \quad \forall i \in [I], \\ & p_{i} \geq 0, \quad \forall i \in [I], \\ & x_{i} \in \{0, 1\}, \quad \forall i \in [I]. \\ \end{array} $ } }$$ We have $I$ generators, each with a turn-on cost of $F_{i}$ and a unit production cost of $C_{i}$. The expected maximum production levels of generator $i$ are $\bar{p}_{i}^{\max}$. We also have $J$ demand nodes, with expected demand or load $\bar{q}_{j}$ at each node $j$. The binary variable $x_{i}=1$, if generator $i$ is turned on, otherwise $x_{i}=0$. Note, the variable $p_{i}$ represents the dispatch of generator $i$. We can expand on this model by considering uncertainty in the load $\boldsymbol{q}$ and in the capacity $\boldsymbol{p}^{\max}$. In ARO, we describe the uncertainty in the parameters with an uncertainty set that contains all scenarios against which we want to be robust. Essentially, the constraints in our problem should be satisfied for all possible realizations of $\boldsymbol{q}$ and $\boldsymbol{p}^{\max}$ in their uncertainty sets [@ben2009robust; @bertsimas2011theory; @bertsimas2022robust]. We consider uncertainty sets where the norm of the residuals from the expected load $\boldsymbol{d} = \boldsymbol{q} - \bar{\boldsymbol{q}}$ is at most $\Gamma_{q}$ and where the norm of residuals $\boldsymbol{r} = \boldsymbol{p}^{\max}-\bar{\boldsymbol{p}}^{\max}$ from the expected capacity is at most $\Delta_{p}$. $${\everymath{\displaystyle} \mathcal{D} = \{\|\boldsymbol{d}\|_{\ell} \leq \Gamma_{q}\}, \; \; \; \mathcal{U} = \{\|\boldsymbol{r}\|_{\ell} \leq \Delta_p\}. }$$ with $\ell \in \{1, 2, \infty \}$ corresponding to the budget, ellipsoidal and box uncertainty sets. The values of $\Gamma_{q}$ and $\Delta_{p}$ control the conservativeness of our formulation [@guan2013uncertainty]. In ARO, the second-stage decisions $\boldsymbol{p}$ are functions of both uncertain parameters $\boldsymbol{d}, \boldsymbol{r}$ and the problem that minimizes the worst-case commitment and dispatch cost is $${\everymath{\displaystyle} \scalebox{0.99}{ $ \begin{array}{rlr} \min_{\boldsymbol{x}, \boldsymbol{p}} \max_{\boldsymbol{d} \in \mathcal{D}, \boldsymbol{r} \in \mathcal{U}} & \sum_{i=1}^{I} F_{i} x_{i} + C_{i} p_{i}(\boldsymbol{d}, \boldsymbol{r}) \\ \text{s.t.} & \sum_{i=1}^{I} p_{i}(\boldsymbol{d}, \boldsymbol{r}) \geq \sum_{j=1}^{J} (d_{j} + \bar{q}_{j}), \quad \forall\boldsymbol{d} \in \mathcal{D},\\ & p_{i}(\boldsymbol{d}, \boldsymbol{r}) \leq (r_{i} + p^{\max}_{i}) x_{i},~ \forall i \in [I], ~\boldsymbol{r} \in \mathcal{U}, \\ & p_{i}(\boldsymbol{d}, \boldsymbol{r}) \geq 0, \quad \forall i \in [I], \\ & x_{i} \in \{0, 1\}, \quad \forall i \in [I]. \\ \end{array} $ } }$$ The optimization variable $p_{i} = p_{i}(\boldsymbol{d}, \boldsymbol{r})$, called a decision rule, is in fact a vector function [@bertsimas2022robust]. In this paper, to make the problem tractable, we restrict $p_{i}(\boldsymbol{d}, \boldsymbol{r})$ to linear functions or linear decision rules (LDR). Such a decision rule may not be optimal, because of the restriction to a certain class, but LDR have shown very good performance in practice and are optimal in many settings [@dehghan2017adaptive; @bertsimas2022robust; @jabr2017linear]. Alternatively, we can use decomposition schemes to learn the decision rule implicitly [@bertsimas2012adaptive; @zeng2013solving]. Using LDR, the second-stage decisions $\boldsymbol{p}$ are linear functions of the uncertain parameters $\boldsymbol{d}, \boldsymbol{r}$. So, we have an $I$-dimensional vector $\boldsymbol{u}$, an $I \times J$ matrix $\boldsymbol{V}$, an $I \times I$ matrix $\boldsymbol{Z}$ and $\boldsymbol{p}(\boldsymbol{d}, \boldsymbol{r}) = \boldsymbol{u} + \boldsymbol{V} \boldsymbol{d} + \boldsymbol{Z} \boldsymbol{r}$ or, for each $i$, $p_{i}(\boldsymbol{d}, \boldsymbol{r}) = u_{i} + \sum_{j=1}^{J} V_{ij} {d}_{j} + \sum_{k=1}^{I} Z_{ik} r_{k}$. The dispatch $\boldsymbol{p}$ contains a non-adaptive part $\boldsymbol{u}$ and an adaptive part that depends on $\boldsymbol{d}$ and $\boldsymbol{r}$. We will use these terms for the rest of the paper. If we set $\boldsymbol{V}=\boldsymbol{0}$ and $\boldsymbol{Z}=\boldsymbol{0}$, the problem becomes an RO problem. Using LDR, the previous formulation is equivalent to $$\label{centralized1_capacity} {\everymath{\displaystyle} \scalebox{0.85}{ $ \begin{array}{rlr} \min_{\boldsymbol{x}, \boldsymbol{u}, \boldsymbol{V}, \boldsymbol{Z}} \max_{\boldsymbol{d} \in \mathcal{D}, \boldsymbol{r} \in \mathcal{U}} & \sum_{i=1}^{I} F_{i} x_{i} + C_{i} (u_{i} + \sum_{j=1}^{J} V_{ij} {d}_{j} + \sum_{k=1}^{I} Z_{ik} r_{k}) \\ \text{s.t.} & \sum_{i=1}^{I} (u_{i} + \sum_{j=1}^{J} V_{ij} {d}_{j} + \sum_{k=1}^{I} Z_{ik} r_{k}) \geq \sum_{j=1}^{J} (d_{j} + \bar{q}_{j}), \\ & \sum_{i=1}^{I} u_{i} = \sum_{j=1}^{J} \bar{q}_{j}, \\ & u_{i} + \sum_{j=1}^{J} V_{ij} {d}_{j} + \sum_{k=1}^{I} Z_{ik} r_{k} \leq (r_{i} + \bar{p}^{\max}_{i}) x_{i}, \quad \forall i, \\ & u_{i} + \sum_{j=1}^{J} V_{ij} {d}_{j} + \sum_{k=1}^{I} Z_{ik} r_{k} \geq 0, \quad \forall i, \\ & x_{i} \in \{0, 1\}, \quad \forall i, \\ \end{array} $ } }$$ where constraints involving $\boldsymbol{d}, \boldsymbol{r}$ hold for all $\boldsymbol{d} \in \mathcal{D}$ and $\boldsymbol{r} \in \mathcal{U}$. Let the objective function of Problem [\[centralized1_capacity\]](#centralized1_capacity){reference-type="eqref" reference="centralized1_capacity"} be $\xi^*$. The solution $\boldsymbol{x}^{*}$ and $\boldsymbol{p}^{*}(\boldsymbol{d}, \boldsymbol{r}) = \boldsymbol{u}^{*} + \boldsymbol{V}^{*} \boldsymbol{d} + \boldsymbol{Z}^{*} \boldsymbol{r}$ satisfies the constraints for any realization of $\boldsymbol{d} \in \mathcal{D}$ and of $\boldsymbol{r} \in \mathcal{U}$. For example, the second constraint ensures that the total production will be more than the total demand for all scenarios in the uncertainty sets. In addition, to make pricing more intuitive, we have included the third constraint, which ensures that the total non-adaptive dispatch $\sum_{i=1}^{I} u_{i}$ is equal to the expected load. The previous formulation is equivalent to: $${\everymath{\displaystyle} \scalebox{0.85}{ $ \begin{array}{rlr} \min_{\boldsymbol{x}, \boldsymbol{u}, \boldsymbol{V}, \boldsymbol{Z}, \eta} & \sum_{i=1}^{I} F_{i} x_{i} + \eta \\ \text{s.t.} & \sum_{i=1}^{I} C_{i}u_{i} + \max_{\boldsymbol{d} \in \mathcal{D}} \sum_{i=1}^{I} C_{i} \sum_{j=1}^{J} V_{ij} d_{j} + \max_{\boldsymbol{r} \in \mathcal{U}} \sum_{i=1}^{I} C_{i} \sum_{k=1}^{I} Z_{ik} r_{k} \leq \eta, \\ & \max_{\boldsymbol{d} \in \mathcal{D}} \sum_{j=1}^{J} (1 - \sum_{i=1}^{I} V_{ij}){d}_{j} + \max_{\boldsymbol{r} \in \mathcal{U}} \sum_{i=1}^{I} \sum_{k=1}^{I} - Z_{ik} r_{k} \leq 0, \\ & \sum_{i=1}^{I} u_{i} = \sum_{j=1}^{J} \bar{q_{j}}, \\ & u_{i} + \max_{\boldsymbol{d} \in \mathcal{D}} \sum_{j=1}^{J} V_{ij} {d}_{j} + \max_{\boldsymbol{r} \in \mathcal{U}} \{ - x_{i} r_{i} + \sum_{k=1}^{I} Z_{ik} r_{k} \} \leq \bar{p}^{\max}_{i} x_{i}, \forall i, \\ & - u_{i} + \max_{\boldsymbol{d} \in \mathcal{D}} \sum_{j=1}^{J} -V_{ij} d + \max_{\boldsymbol{r} \in \mathcal{U}} \sum_{k=1}^{I} - Z_{ik} r_{k} \leq 0, \quad \forall i, \\ & x_{i} \in \{0, 1\}, \quad \forall i, \\ \end{array} $ } }$$ or, by using the robust counterpart with $\mathcal{D} = \{\|\boldsymbol{d}\|_{\ell} \leq \Gamma_{q}\}$ and $\mathcal{U} = \{\|\boldsymbol{r}\|_{\ell} \leq \Delta_{p}\}$, where $\ell^{*}$ is the dual norm of $\ell$, and $\boldsymbol{V}_{i}$ is the $i$-th row of $\boldsymbol{V}$ and $\boldsymbol{Z}_{i}$ is the $i$-th row of $\boldsymbol{Z}$, $${\everymath{\displaystyle} \scalebox{0.85}{ $ \begin{array}{rlr} \min_{\boldsymbol{x}, \boldsymbol{u}, \boldsymbol{V}, \boldsymbol{Z}, \eta} & \sum_{i=1}^{I} F_{i} x_{i} + \eta \\ \text{s.t.} & \sum_{i=1}^{I} C_{i}u_{i} + \Gamma_{q} \|\sum_{i=1}^{I} C_{i} \boldsymbol{V}_{i}\|_{\ell^{*}} + \Delta_{p} \|\sum_{i=1}^{I} C_{i} \boldsymbol{Z}_{i}\|_{\ell^{*}} \leq \eta, \\ & \Gamma_{q} \|\boldsymbol{1} - \sum_{i=1}^{I} \boldsymbol{V}_{i} \|_{\ell^{*}} + \Delta_{p} \| - \sum_{i=1}^{I} \boldsymbol{Z}_{i} \|_{\ell^{*}} \leq 0, \\ & \sum_{i=1}^{I} u_{i} = \sum_{j=1}^{J} \bar{q_{j}}, \\ & u_{i} + \Gamma_{q} \| \boldsymbol{V}_{i} \|_{\ell^{*}} + \Delta_{p} \| x_{i} \boldsymbol{e}_{i} - \boldsymbol{Z}_{i} \|_{\ell^{*}} \leq p^{\max}_{i} x_{i}, \quad \forall i, \\ & - u_{i} + \Gamma_{q} \| \boldsymbol{V}_{i} \|_{\ell^{*}} + \Delta_{p} \|\boldsymbol{Z}_{i} \|_{\ell^{*}} \leq 0, \forall i, \\ & x_{i} \in \{0, 1\}, \quad \forall i. \\ \end{array} $ } }$$ We are going to work with the following version of the previous formulation, because we want to use the dual problem as well. Following the example of [@o2005efficient], we also set the binary variables to their optimal values $\boldsymbol{x}^{*}$. $$\label{centralized2_capacity} {\everymath{\displaystyle} \scalebox{0.85}{ $ \begin{array}{rlrr} \min_{\boldsymbol{x}, \boldsymbol{u}, \boldsymbol{V}, \boldsymbol{Z}, \eta} & \sum_{i=1}^{I} F_{i} x_{i} + \eta \\ \text{s.t.} & \sum_{i=1}^{I} C_{i}u_{i} + \Gamma_{q} \| \boldsymbol{\omega} \|_{\ell^{*}} + \Delta_{p} \| \bar{\boldsymbol{\omega}} \|_{\ell^{*}} \leq \eta, & \nu \\ & \boldsymbol{\omega} = \sum_{i=1}^{I} C_{i} \boldsymbol{V}_{i}, \; \; \bar{\boldsymbol{\omega}} = \sum_{i=1}^{I} C_{i} \boldsymbol{Z}_{i}, & \boldsymbol{\alpha}, \bar{\boldsymbol{\alpha}} \\ & \Gamma_{q} \|\boldsymbol{\tau}\|_{\ell^{*}} + \Delta_{p} \|\bar{\boldsymbol{\tau}}\|_{\ell^{*}} \leq 0, & \lambda \\ & \boldsymbol{\tau} = \boldsymbol{1} - \sum_{i=1}^{I} \boldsymbol{V}_{i}, \; \; \bar{\boldsymbol{\tau}} = - \sum_{i=1}^{I} \boldsymbol{Z}_{i}, & \boldsymbol{\theta}, \bar{\boldsymbol{\theta}}\\ & \sum_{i=1}^{I} u_{i} = \sum_{j=1}^{J} \bar{q_{j}}, & \mu \\ & u_{i} + \Gamma_{q} \| \boldsymbol{\psi}_{i} \|_{\ell^{*}} + \Delta_{p} \| \bar{\boldsymbol{\psi}}_{i} \|_{\ell^{*}} \leq p^{\max}_{i} x_{i}, \quad \forall i, & \sigma_{i} \\ & \boldsymbol{\psi}_{i} = \boldsymbol{V}_{i}, \; \; \bar{\boldsymbol{\psi}}_{i} = x_{i} \boldsymbol{e}_{i} - \boldsymbol{Z}_{i}, & \boldsymbol{\beta}_{i}, \bar{\boldsymbol{\beta}}_{i} \\ & - u_{i} + \Gamma_{q} \| \boldsymbol{\phi}_{i} \|_{\ell^{*}} + \Delta_{p} \| \bar{\boldsymbol{\phi}}_{i} \|_{\ell^{*}} \leq 0, \quad \forall i, & \zeta_{i} \\ & \boldsymbol{\phi}_{i} = \boldsymbol{V}_{i}, \; \; \bar{\boldsymbol{\phi}}_{i} = \boldsymbol{Z}_{i}, & \boldsymbol{\gamma}_{i}, \bar{\boldsymbol{\gamma}}_{i} \\ & x_{i} = x_{i}^{*}, \quad \forall i, & \rho_{i} \\ & x_{i} \geq 0, \quad \forall i, \\ \end{array} $ } }$$ where the right column contains the dual variables for the corresponding constraints. The Lagrangian of the previous problem is $${\everymath{\displaystyle} \scalebox{0.90}{ $ \begin{array}{llr} & \mathcal{L} = \mu \sum_{j=1}^{J} \bar{q_{j}} + \sum_{j=1}^{J} \theta_{j} + \sum_{i=1}^{I} x_{i}^{*} \rho_{i} \\ & + (1-\nu) \eta \\ & + \sum_{i=1}^{I} (F_{i} - \sigma_{i} p^{\max}_{i} - \rho_{i} + \bar{\beta}_{ii}) x_{i} \\ & + \sum_{i=1}^{I} (\nu C_{i} - \mu + \sigma_{i} - \zeta_{i}) u_{i} \\ & + \sum_{i=1}^{I}(C_{i} \boldsymbol{\alpha} - \boldsymbol{\theta} + \boldsymbol{\beta}_{i} + \boldsymbol{\gamma}_{i})^{T} \boldsymbol{V}_{i} \\ & + \sum_{i=1}^{I}(C_{i} \bar{\boldsymbol{\alpha}} - \bar{\boldsymbol{\theta}} - \bar{\boldsymbol{\beta}}_{i} + \bar{\boldsymbol{\gamma}}_{i})^{T} \boldsymbol{Z}_{i} \\ & - (\boldsymbol{\alpha}^{T} \boldsymbol{\omega} - \nu \Gamma_{q} \| \boldsymbol{\omega} \|_{\ell^{*}}) - (\bar{\boldsymbol{\alpha}}^{T} \bar{\boldsymbol{\omega}} - \nu \Delta_{p} \| \bar{\boldsymbol{\omega}} \|_{\ell^{*}}) \\ & - (\boldsymbol{\theta}^{T} \boldsymbol{\tau} - \lambda \Gamma_{q} \| \boldsymbol{\tau} \|_{\ell^{*}}) - (\bar{\boldsymbol{\theta}}^{T} \bar{\boldsymbol{\tau}} - \lambda \Delta_{p} \| \bar{\boldsymbol{\tau}} \|_{\ell^{*}})\\ & \sum_{i=1}^{I} - (\boldsymbol{\beta}_{i}^{T} \boldsymbol{\psi}_{i} - \sigma_{i} \Gamma_{q} \| \boldsymbol{\psi}_{i} \|_{\ell^{*}}) + \sum_{i=1}^{I} - (\bar{\boldsymbol{\beta}}_{i}^{T} \bar{\boldsymbol{\psi}}_{i} - \sigma_{i} \Delta_{p} \| \bar{\boldsymbol{\psi}}_{i} \|_{\ell^{*}}) \\ & \sum_{i=1}^{I} - (\boldsymbol{\gamma}_{i}^{T} \boldsymbol{\phi}_{i} - \zeta_{i} \Gamma_{q} \| \boldsymbol{\phi}_{i} \|_{\ell^{*}}) + \sum_{i=1}^{I} - (\bar{\boldsymbol{\gamma}}_{i}^{T} \bar{\boldsymbol{\phi}_{i}} - \zeta_{i} \Delta_{p} \| \bar{\boldsymbol{\phi}}_{i} \|_{\ell^{*}}). \\ \end{array} $ } }$$ By taking the minimum of the Lagrangian $\mathcal{L}$ over the primal variables and then maximizing over the dual variables, we obtain the dual problem [@bertsekas1997nonlinear]. Note that $\min_{\boldsymbol{\omega}} -(\boldsymbol{\alpha}^{T} \boldsymbol{\omega} - \nu \Gamma_{q} \| \boldsymbol{\omega} \|_{\ell^{*}}) = 0$ if $\| \boldsymbol{\alpha} \|_{\ell} \leq \Gamma_{q} \nu$, otherwise it is $-\infty$. The same holds for the other similar terms in the Lagrangian. So, at optimality, for all $i$, $$\label{dual_norm_alpha} {\everymath{\displaystyle} \scalebox{0.85}{ $ (\boldsymbol{\alpha}^{*})^{T} \boldsymbol{\omega}^{*} = \nu^{*} \Gamma_{q} \| \boldsymbol{\omega}^{*} \|_{\ell^{*}}, \; \; (\bar{\boldsymbol{\alpha}}^{*})^{T} \bar{\boldsymbol{\omega}}^{*} = \nu^{*} \Delta_{p} \| \bar{\boldsymbol{\omega}}^{*} \|_{\ell^{*}}, $ } }$$ $$\label{dual_norm_theta} {\everymath{\displaystyle} \scalebox{0.90}{ $ (\boldsymbol{\theta}^{*})^{T} \boldsymbol{\tau}^{*} = \lambda^{*} \Gamma_{q} \| \boldsymbol{\tau}^{*} \|_{\ell^{*}}, \; \; (\bar{\boldsymbol{\theta}}^{*})^{T} \bar{\boldsymbol{\tau}}^{*} = \lambda^{*} \Delta_{p} \| \bar{\boldsymbol{\tau}}^{*} \|_{\ell^{*}}, $ } }$$ $$\label{dual_norm_beta} {\everymath{\displaystyle} \scalebox{0.85}{ $ (\boldsymbol{\beta}_{i}^{*})^{T} \boldsymbol{\psi}_{i}^{*} = \sigma_{i}^{*} \Gamma_{q} \| \boldsymbol{\psi}_{i}^{*} \|_{\ell^{*}}, \; \; (\bar{\boldsymbol{\beta}}_{i}^{*})^{T} \bar{\boldsymbol{\psi}}_{i}^{*} = \sigma_{i}^{*} \Delta_{p} \| \bar{\boldsymbol{\psi}}_{i}^{*} \|_{\ell^{*}}, $ } }$$ $$\label{dual_norm_gamma} {\everymath{\displaystyle} \scalebox{0.85}{ $ (\boldsymbol{\gamma}_{i}^{*})^{T} \boldsymbol{\phi}_{i}^{*} = \zeta_{i}^{*} \Gamma_{q} \| \boldsymbol{\phi}_{i}^{*} \|_{\ell^{*}}, \; \; (\bar{\boldsymbol{\gamma}}_{i}^{*})^{T} \bar{\boldsymbol{\phi}}_{i}^{*} = \zeta_{i}^{*} \Delta_{p} \| \bar{\boldsymbol{\phi}}_{i}^{*} \|_{\ell^{*}}. $ } }$$ The corresponding dual problem is $$\label{centralizeddual_capacity} {\everymath{\displaystyle} \scalebox{0.95}{ $ \begin{array}{rlr} \max & \mu \sum_{j=1}^{J} \bar{q_{j}} + \sum_{j=1}^{J} \theta_{j} + \sum_{i=1}^{I} x_{i}^{*} \rho_{i}, \\ \text{s.t.} & F_{i} \geq \rho_{i} + \sigma_{i} p^{\max}_{i} - \bar{\beta}_{ii}, \quad \forall i, & x_{i} \\ & \nu = 1, & \eta \\ & \nu C_{i} = \mu - \sigma_{i} + \zeta_{i}, \quad \forall i, & u_{i} \\ & C_{i} \alpha_{j} = \theta_{j} - \beta_{ij} - \gamma_{ij}, \quad \forall i,j, & V_{ij} \\ & C_{i} \bar{\alpha}_{k} = \bar{\theta}_{k} + \bar{\beta}_{ik} - \bar{\gamma}_{ik}, \quad \forall i,k, & Z_{ik} \\ & \| \boldsymbol{\alpha} \|_{\ell} \leq \Gamma_{q} \nu, \quad \| \bar{\boldsymbol{\alpha}} \|_{\ell} \leq \Delta_{p} \nu \\ & \| \boldsymbol{\theta}\|_{\ell} \leq \Gamma_{q} \lambda, \quad \| \bar{\boldsymbol{\theta}} \|_{\ell} \leq \Delta_{p} \lambda \\ & \| \boldsymbol{\beta}_{i} \|_{\ell} \leq \Gamma_{q} \sigma_{i}, \quad \| \bar{\boldsymbol{\beta}}_{i} \|_{\ell} \leq \Delta_{p} \sigma_{i}, \quad \forall i, \\ & \| \boldsymbol{\gamma}_{i} \|_{\ell} \leq \Gamma_{q} \zeta_{i}, \quad \| \bar{\boldsymbol{\gamma}}_{i} \|_{\ell} \leq \Delta_{p} \zeta_{i} \quad \forall i, \\ & \lambda, \sigma_{i}, \zeta_{i} \geq 0, \quad \forall i. \end{array} $ } }$$ We use $(\cdot)^{*}$ to denote the optimal solutions to Problem [\[centralized2_capacity\]](#centralized2_capacity){reference-type="eqref" reference="centralized2_capacity"} and Problem [\[centralizeddual_capacity\]](#centralizeddual_capacity){reference-type="eqref" reference="centralizeddual_capacity"}. We will use this notation for the rest of this work. # Adaptive Pricing {#sec:adamarkets} In this section, we introduce adaptive pricing and provide its theoretical properties. We suggest that the day-ahead payments include only the commitment and non-adaptive dispatch costs. So, the day-ahead payments reflect the cost of meeting the expected load while planning for the worst-case scenario. The adaptive part of the dispatch serves as an upper bound on the intra-day payments, which take place the following day based on the economic dispatch problems. This upper bound is also equal to the optimal intra-day payments, when the worst-case uncertainty is realized. **Uncertainty Set** $\mathcal{D}$ $\max_{d \in \mathcal{D}} \sum_{j=1}^{J} V_{ij}^{*} d_{j}$ **Pay-as-bid** **Uniform** --------------------- ---------------------------------------------------- ------------------------------------------------------------ ------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------- Budget $\{\|\boldsymbol{d}\|_{1} \leq \Gamma_{q}\}$ $\Gamma_{q} \| V_{i}^{*}\|_{\infty}$ $F_{i} x_{i}^{*} + C_{i} u_{i}^{*}$ $\mu^{*} u_{i}^{*} + \rho_{i}^{*} x_{i}^{*} + \sigma_{i}^{*} \Gamma_{q} \| V_{i}^{*}\|_{\infty} + \zeta_{i}^{*} \Gamma_{q} \| V_{i}^{*}\|_{\infty}$ Ellipsoidal $\{\|\boldsymbol{d}\|_{2} \leq \Gamma_{q}\}$ $\Gamma_{q} \| V_{i}^{*}\|_{2}$ $F_{i} x_{i}^{*} + C_{i} u_{i}^{*}$ $\mu^{*} u_{i}^{*} + \rho_{i}^{*} x_{i}^{*} + \sigma_{i}^{*} \Gamma_{q} \| V_{i}^{*} \|_{2} + \zeta_{i}^{*} \Gamma_{q} \| V_{i}^{*} \|_{2}$ Box $\{ \|\boldsymbol{d}\|_{\infty} \leq \Gamma_{q}\}$ $\Gamma_{q} \| V_{i}^{*}\|_{1}$ $F_{i} x_{i}^{*} + C_{i} u_{i}^{*}$ $\mu^{*} u_{i}^{*} + \rho_{i}^{*} x_{i}^{*} + \sigma_{i}^{*} \Gamma_{q} \| V_{i}^{*} \|_{1} + \zeta_{i}^{*} \Gamma_{q} \| V_{i}^{*} \|_{1}$ [\[tab:tab_unc\]]{#tab:tab_unc label="tab:tab_unc"} #### Pay-as-bid pricing The day-ahead payments to generator $i$ are based on their bids $F_{i}, C_{i}$ and the non-adaptive part of the dispatch. Specifically, each generator is paid $${\everymath{\displaystyle} \scalebox{0.9}{ $ F_{i} x_{i}^{*} + C_{i} u_{i}^{*}. $ } }$$ #### Marginal pricing The price for the non-adaptive dispatch is $\mu$ and each generator is paid some uplift in a discriminatory way. So, each generator $i$ is paid $${\everymath{\displaystyle} \scalebox{0.9}{ $ \begin{array}{cc} \vspace{5pt} & \mu^{*} u_{i}^{*} + (\rho_{i}^{*} - \bar{\beta}_{ii}^{*}) x_{i}^{*} \\ \vspace{5pt} & + \sigma_{i}^{*} \; (\Gamma_{q} \| \boldsymbol{V}_{i}^{*}\|_{\ell^{*}} + \Delta_p \|x_{i}^{*} \boldsymbol{e}_{i} - \boldsymbol{Z}_{i}^{*}\|_{\ell^{*}}) \\ & + \zeta_{i}^{*} \; (\Gamma_{q} \| \boldsymbol{V}_{i}^{*} \|_{\ell^{*}} + \Delta_p \| \boldsymbol{Z}_{i}^{*} \|_{\ell^{*}}). \end{array} $ } }$$ In contrast to deterministic pricing, we also price the uncertainty by introducing payments based on sizes $\Gamma_{q}$ and $\Delta_{p}$ of the uncertainty sets. Note that if we set $\Delta_{p} = 0$ in the uncertainty set $\mathcal{U}$, we do not consider the capacity uncertainty. Table [\[tab:tab_unc\]](#tab:tab_unc){reference-type="ref" reference="tab:tab_unc"} summarizes the payments when there is only load uncertainty or $\Delta_{p}=0$. Similarly, if we set $\Gamma_{q}=0$, we do not consider uncertainty in the load. ## Pay-As-Bid and Uniform Pricing Equivalence One important theoretical property of our approach is that the day-ahead pay-as-bid and marginal pricing payments are the same. This result is presented more formally in the following theorem. **Theorem 1**. *The pay-as-bid payment $F_{i} x_{i}^{*} + C_{i} u_{i}^{*}$ and the uniform price payment to each generator $i$ $${\everymath{\displaystyle} \scalebox{0.9}{ $ \begin{array}{cc} \vspace{5pt} & \mu^{*} u_{i}^{*} + (\rho_{i}^{*} - \bar{\beta}_{ii}^{*}) x_{i}^{*} \\ \vspace{5pt} & + \sigma_{i}^{*} \; (\Gamma_{q} \| \boldsymbol{V}_{i}^{*}\|_{\ell^{*}} + \Delta_p \|x_{i}^{*} \boldsymbol{e}_{i} - \boldsymbol{Z}_{i}^{*}\|_{\ell^{*}}) \\ & + \zeta_{i}^{*} \; (\Gamma_{q} \| \boldsymbol{V}_{i}^{*} \|_{\ell^{*}} + \Delta_p \| \boldsymbol{Z}_{i}^{*} \|_{\ell^{*}}) \end{array} $ } }$$ are equal.* *Proof.* Using complementary slackness between Problems [\[centralized2_capacity\]](#centralized2_capacity){reference-type="eqref" reference="centralized2_capacity"} and [\[centralizeddual_capacity\]](#centralizeddual_capacity){reference-type="eqref" reference="centralizeddual_capacity"}, for all $i \in [I]$, $$\scalebox{0.9}{ $ F_{i} x_{i}^{*} = \rho_{i}^{*} x_{i}^{*} + \sigma_{i}^{*} p^{\max}_{i} x_{i}^{*} - \bar{\beta}_{ii}^{*} x_{i}^{*}, $ }$$ $$\scalebox{0.9}{ $ C_{i} u_{i}^{*} = \mu^{*} u_{i}^{*} - \sigma_{i}^{*} u_{i}^{*} + \zeta_{i}^{*} u_{i}^{*}, $ }$$ $$\scalebox{0.9}{ $ \zeta_{i}^{*} u_{i}^{*} - \zeta_{i}^{*} \Gamma_{q} \| \boldsymbol{\phi}_{i}^{*} \|_{\ell^{*}} - \zeta_{i}^{*} \Delta_{p} \| \bar{\boldsymbol{\phi}}_{i}^{*} \|_{\ell^{*}} = 0, $ }$$ $$\scalebox{0.9}{ $ \sigma_{i}^{*} p^{\max}_{i} x_{i}^{*} - \sigma_{i}^{*} u_{i}^{*} - \sigma_{i}^{*} \Gamma_{q} \| \boldsymbol{\psi}_{i}^{*} \|_{\ell^{*}} - \sigma_{i}^{*} \Delta_{p} \| \bar{\boldsymbol{\psi}}_{i}^{*} \|_{\ell^{*}} = 0. $ }$$ So, for all $i \in [I]$, $${\everymath{\displaystyle} \scalebox{0.9}{ $ \begin{array}{l} \vspace{10pt} F_{i}x_{i}^{*} + C_{i} u_{i}^{*} \\ \vspace{10pt} = \rho_{i}^{*} x_{i}^{*} + \sigma_{i}^{*} p^{\max}_{i} x_{i}^{*} - \bar{\beta}_{ii}^{*} x_{i}^{*} + \mu^{*} u_{i}^{*} - \sigma_{i}^{*} u_{i}^{*} + \zeta_{i}^{*} u_{i}^{*} \\ \vspace{10pt} = (\rho_{i}^{*} - \bar{\beta}_{ii}^{*}) x_{i}^{*} + \mu u_{i}^{*} + \sigma_{i}^{*} (p^{\max}_{i}x_{i}^{*} - u_{i}^{*}) + \zeta_{i}^{*} u_{i}^{*} \\ \vspace{10pt} = (\rho_{i}^{*} - \bar{\beta}_{ii}^{*}) x_{i}^{*} + \mu u_{i}^{*} \\ \vspace{10pt} + \sigma_{i}^{*} \Gamma_{q} \| \boldsymbol{\psi}_{i}^{*} \|_{\ell^{*}} + \sigma_{i}^{*} \Delta_{p} \| \bar{\boldsymbol{\psi}}_{i}^{*} \|_{\ell^{*}} + \zeta_{i}^{*} \Gamma_{q} \| \boldsymbol{\phi}_{i}^{*} \|_{\ell^{*}} + \zeta_{i}^{*} \Delta_{p} \| \bar{\boldsymbol{\phi}}_{i}^{*} \|_{\ell^{*}} \\ \vspace{10pt} = \mu^{*} u_{i}^{*} + (\rho_{i}^{*} - \bar{\beta}_{ii}^{*}) x_{i}^{*} \\ + \sigma_{i}^{*} \Gamma_{q} \| \boldsymbol{V}_{i}^{*}\|_{\ell^{*}} + \sigma_{i}^{*} \Delta_{p} \| x_{i}^{*} \boldsymbol{e}_{i} - \boldsymbol{Z}_{i}^{*}\|_{\ell^{*}} + \zeta_{i}^{*} \Gamma_{q} \| \boldsymbol{V}_{i}^{*}\|_{\ell^{*}} + \zeta_{i}^{*} \Delta_{p} \| \boldsymbol{Z}_{i}^{*}\|_{\ell^{*}}. \end{array} $ } }$$ The last equality follows from the constraints $\boldsymbol{\psi}_{i} = \boldsymbol{V}_{i}$, $\boldsymbol{\phi}_{i} = \boldsymbol{V}_{i}$, $\bar{\boldsymbol{\psi}}_{i} = x_{i}\boldsymbol{e}_{i} - \boldsymbol{Z}_{i}$ and $\bar{\boldsymbol{\phi}}_{i} = \boldsymbol{Z}_{i}$ of Problem [\[centralized2_capacity\]](#centralized2_capacity){reference-type="eqref" reference="centralized2_capacity"}. ◻ ## Total Worst-Case Cost Equivalence In the previous section, we considered only the day-ahead payments. In this section, we show that when the worst-case uncertainty is realized, the pay-as-bid and marginal pricing payments are the same. Let $\boldsymbol{d}^{*}= \arg \max_{\boldsymbol{d} \in \mathcal{D}} \sum_{i=1}^{I} C_{i} \sum_{j=1}^{J} V_{ij}^* d_{j}$ and $\boldsymbol{r}^{*}= \arg \max_{\boldsymbol{r} \in \mathcal{U}} \sum_{i=1}^{I} C_{i} \sum_{k=1}^{I} Z_{ik}^{*} r_{k}$ be the worst case realization of the residual load $\boldsymbol{d}$ and the residual capacity $\boldsymbol{r}$ respectively for the objective function of Problem [\[centralized1_capacity\]](#centralized1_capacity){reference-type="eqref" reference="centralized1_capacity"}. **Theorem 2**. *If the worst-case uncertainty $\boldsymbol{d}^{*}, \boldsymbol{r}^{*}$ is realized, then the total pay-as-bid payment to generator $i$ $$f_i:=F_{i} x_{i}^{*} + C_{i} u_{i}^{*} + C_{i} \sum_{j=1}^{J} V_{ij}^{*} d_{j}^{*} + C_{i} \sum_{k=1}^{I} Z_{ik}^{*} r_{k}^{*},$$ and the uniform price payment to generator $i$ $$g_i:= \rho_{i}^{*} x_{i}^{*} + \mu^{*} u_{i}^{*} + \sum_{j=1}^{J} \theta_{j}^{*} V_{ij}^{*} + \sum_{k=1}^{I} \bar{\theta}_{k}^{*} Z_{ik}^{*}.$$ are equal and satisfy $\sum_{i=1}^{I} f_i=\sum_{i=1}^{I} g_i = \xi^*$, the optimal solution value of Problem [\[centralized1_capacity\]](#centralized1_capacity){reference-type="eqref" reference="centralized1_capacity"}.* *Proof.* By complementary slackness between Problems [\[centralized2_capacity\]](#centralized2_capacity){reference-type="eqref" reference="centralized2_capacity"} and [\[centralizeddual_capacity\]](#centralizeddual_capacity){reference-type="eqref" reference="centralizeddual_capacity"}, for all $i \in [I], j \in [J]$, we have $${\everymath{\displaystyle} \scalebox{0.9}{ $ \begin{array}{l} \vspace{10pt} C_{i} \alpha_{j}^{*} V_{ij}^{*} = (\theta_{j}^{*} - \beta_{ij}^{*} - \gamma_{ij}^{*}) V_{ij}^{*}, \\ \vspace{10pt} C_{i} \bar{\alpha}_{k}^{*} Z_{ik}^{*} = (\theta_{k}^{*} + \beta_{ik}^{*} - \gamma_{ik}^{*}) Z_{ik}^{*}, \\ \vspace{10pt} C_{i} \sum_{j=1}^{J} \alpha_{j}^{*} V_{ij}^{*} = \sum_{j=1}^{J} (\theta_{j}^{*} - \beta_{ij}^{*} - \gamma_{ij}^{*}) V_{ij}^{*}, \\ C_{i} \sum_{k=1}^{I} \bar{\alpha}_{k}^{*} Z_{ik}^{*} = \sum_{k=1}^{I} (\theta_{k}^{*} + \beta_{ik}^{*} - \gamma_{ik}^{*}) Z_{ik}^{*}. \end{array} $ } }$$ Also, using equations [\[dual_norm_beta\]](#dual_norm_beta){reference-type="eqref" reference="dual_norm_beta"} and [\[dual_norm_gamma\]](#dual_norm_gamma){reference-type="eqref" reference="dual_norm_gamma"} and the constraints $\boldsymbol{\psi}_{i} = \boldsymbol{V}_{i}$, $\boldsymbol{\phi}_{i} = \boldsymbol{V}_{i}$, $\bar{\boldsymbol{\psi}}_{i} = x_{i}\boldsymbol{e}_{i} - \boldsymbol{Z}_{i}$ and $\bar{\boldsymbol{\phi}}_{i} = \boldsymbol{Z}_{i}$ of Problem [\[centralized2_capacity\]](#centralized2_capacity){reference-type="eqref" reference="centralized2_capacity"}, for all $i$, $${\everymath{\displaystyle} \scalebox{0.9}{ $ \begin{array}{l} \vspace{10pt} (\boldsymbol{\beta}_{i}^{*})^{T} \boldsymbol{V}_{i}^{*} = (\boldsymbol{\beta}_{i}^{*})^{T} \boldsymbol{\psi}_{i}^{*} = \sigma_{i}^{*} \Gamma_{q} \| \boldsymbol{\psi}_{i}^{*} \|_{\ell^{*}}, \\ \vspace{10pt} (\boldsymbol{\gamma}_{i}^{*})^{T} \boldsymbol{V}_{i} = (\boldsymbol{\gamma}_{i}^{*})^{T} \boldsymbol{\phi}_{i}^{*} = \zeta_{i}^{*} \Gamma_{q} \| \boldsymbol{\phi}_{i}^{*} \|_{\ell^{*}}, \\ \vspace{10pt} (\bar{\boldsymbol{\beta}}_{i}^{*})^{T} (x_{i}^{*} \boldsymbol{e}_{i} - \boldsymbol{Z}_{i}^{*}) = (\bar{\boldsymbol{\beta}}_{i}^{*})^{T} \bar{\boldsymbol{\psi}}_{i}^{*} = \sigma_{i}^{*} \Delta_{p} \| \bar{\boldsymbol{\psi}}_{i}^{*} \|_{\ell^{*}}, \\ (\bar{\boldsymbol{\gamma}}_{i}^{*})^{T} \boldsymbol{Z}_{i} = (\bar{\boldsymbol{\gamma}}_{i}^{*})^{T} \bar{\boldsymbol{\phi}}_{i}^{*} = \zeta_{i}^{*} \Delta_{p} \| \bar{\boldsymbol{\phi}}_{i}^{*} \|_{\ell^{*}}. \end{array} $ } }$$ So, for all $i \in [I]$, using the same complementary slackness conditions as Theorem 1, $${\everymath{\displaystyle} \scalebox{0.9}{ $ \begin{array}{l} F_{i} x_{i}^{*} + C_{i} u_{i}^{*} + C_{i} \sum_{j=1}^{J} \alpha_{j}^{*} V_{ij}^{*} + C_{i} \sum_{k=1}^{I} \bar{\alpha}_{k}^{*} Z_{ik}^{*} \\ = \rho_{i}x_{i}^{*} + \sigma_{i}^{*} p^{\max}_{i} x_{i}^{*} - \bar{\beta}_{ii}^{*} x_{i}^{*} + \mu^{*} u_{i}^{*} - \sigma_{i}^{*} u_{i}^{*} + \zeta_{i}^{*} u_{i}^{*} \\ + \sum_{j=1}^{J} (\theta_{j}^{*} - \beta_{ij}^{*} - \gamma_{ij}^{*}) V_{ij}^{*} \\ + \sum_{k=1}^{I} (\bar{\theta}_{k}^{*} + \bar{\beta}_{ik}^{*} - \bar{\gamma}_{ik}^{*}) Z_{ik}^{*} \\ = \rho_{i}^{*} x_{i}^{*} + \mu^{*} u_{i}^{*} + \sum_{j=1}^{J} \theta_{j}^{*} V_{ij}^{*} \\ + (\zeta_{i}^{*} u_{i}^{*} - \zeta_{i}^{*} \Gamma_{q} \| \boldsymbol{\phi}_{i}^{*} \|_{\ell^{*}} - \zeta_{i}^{*} \Delta_{p} \| \bar{\boldsymbol{\phi}}_{i}^{*} \|_{\ell^{*}}) \\ + (\sigma_{i}^{*} p^{\max}_{i} x_{i}^{*} - \sigma_{i}^{*} u_{i}^{*} - \sigma_{i}^{*} \Gamma_{q} \| \boldsymbol{\psi}_{i}^{*} \|_{\ell^{*}} - \sigma_{i}^{*} \Delta_{p} \| \bar{\boldsymbol{\psi}}_{i}^{*} \|_{\ell^{*}}) \\ = \rho_{i}^{*} x_{i}^{*} + \mu^{*} u_{i}^{*} + \sum_{j=1}^{J} \theta_{j}^{*} V_{ij}^{*} + \sum_{k=1}^{I} \bar{\theta}_{k}^{*} Z_{ik}^{*} = g_{i}. \end{array} $ } }$$ Next, we show that $${\everymath{\displaystyle} \scalebox{0.9}{ $ \begin{array}{l} F_{i} x_{i}^{*} + C_{i} u_{i}^{*} + C_{i} \sum_{j=1}^{J} \alpha_{j}^{*} V_{ij}^{*} + C_{i} \sum_{k=1}^{I} \bar{\alpha}_{k}^{*} Z_{ik}^{*} \\ = F_{i} x_{i}^{*} + C_{i} u_{i}^{*} + C_{i} \sum_{j=1}^{J} V_{ij}^{*} d_{j}^{*} + C_{i} \sum_{k=1}^{I} Z_{ik}^{*} r_{k}^{*} = f_{i}. \end{array} $ } }$$ Using equation [\[dual_norm_alpha\]](#dual_norm_alpha){reference-type="eqref" reference="dual_norm_alpha"} and the constraints $\boldsymbol{\omega} = \sum_{i=1}^{I} C_{i} \boldsymbol{V}_{i}$ and $\bar{\boldsymbol{\omega}} = \sum_{i=1}^{I} C_{i} \boldsymbol{Z}_{i}$ of Problem [\[centralized2_capacity\]](#centralized2_capacity){reference-type="eqref" reference="centralized2_capacity"} and $\nu=1$ of Problem [\[centralizeddual_capacity\]](#centralizeddual_capacity){reference-type="eqref" reference="centralizeddual_capacity"}, $${\everymath{\displaystyle} \scalebox{0.9}{ $ \begin{array}{l} \sum_{i=1}^{I} C_{i} \sum_{j=1}^{J} \alpha_{j}^{*} V_{ij}^{*} = (\boldsymbol{\alpha}^{*})^{T} \boldsymbol{\omega}^{*} \\ = \nu^{*} \Gamma_{q} \| \boldsymbol{\omega}^{*} \|_{\ell^{*}} = \Gamma_{q} \|\boldsymbol{\omega}^{*} \|_{\ell^{*}} \\ = \Gamma_{q} \|\sum_{i=1}^{I} C_{i} \boldsymbol{V}_{i}^{*} \|_{\ell^{*}} = \max_{\boldsymbol{d} \in \mathcal{D}} \sum_{i=1}^{I} C_{i} \sum_{j=1}^{J} V_{ij}^{*} d_{j} \\ = \sum_{i=1}^{I} C_{i} \sum_{j=1}^{J} V_{ij}^{*} d_{j}^{*}, \end{array} $ } } % \end{equation}$$ and $${\everymath{\displaystyle} \scalebox{0.9}{ $ \begin{array}{l} \sum_{i=1}^{I} C_{i} \sum_{k=1}^{I} \bar{\alpha_{k}}^{*} Z_{ik}^{*} = (\bar{\boldsymbol{\alpha}}^{*})^{T} \bar{\boldsymbol{\omega}}^{*} \\ = \nu^{*} \Delta_{p} \| \bar{\boldsymbol{\omega}}^{*} \|_{\ell^{*}} = \Delta_{p} \|\bar{\boldsymbol{\omega}}^{*} \|_{\ell^{*}} \\ = \Delta_{p} \|\sum_{i=1}^{I} C_{i} \boldsymbol{Z}_{i}^{*} \|_{\ell^{*}} = \max_{\boldsymbol{r} \in \mathcal{U}} \sum_{i=1}^{I} C_{i} \sum_{k=1}^{I} Z_{ik}^{*} r_{k} \\ = \sum_{i=1}^{I} C_{i} \sum_{k=1}^{I} Z_{ik}^{*} r_{k}^{*}. \end{array} $ } }$$ Then, $\boldsymbol{\alpha}$ is the worst-case load uncertainty, because $\boldsymbol{\alpha}^{*} = \arg \max_{\boldsymbol{d} \in \mathcal{D}} \sum_{i=1}^{I} C_{i} \sum_{j=1}^{J} V_{ij}^* d_{j}$ and $\boldsymbol{\alpha}^{*}$ is in the uncertainty set $\mathcal{D}$ or $\| \boldsymbol{\alpha}^{*} \|_{\ell} \leq \Gamma_{q}$ by the constraints of Problem [\[centralizeddual_capacity\]](#centralizeddual_capacity){reference-type="eqref" reference="centralizeddual_capacity"}. Similarly, $\bar{\boldsymbol{\alpha}}$ is the worst-case capacity uncertainty, because $\bar{\boldsymbol{\alpha}}^{*} = \arg \max_{\boldsymbol{r} \in \mathcal{U}} \sum_{i=1}^{I} C_{i} \sum_{k=1}^{I} Z_{ik}^{*} r_{k}$. Note, $\boldsymbol{d}^{*}$ and $\boldsymbol{r}^{*}$ may not be unique. So, $f_{i} = g_{i}$ for all $i$ and $\sum_{i=1}^{I} f_i=\sum_{i=1}^{I} g_i = \xi^*$. ◻ ## Absence of Self-Scheduling In this section, we show that when the worst-case uncertainty is realized, the generators are indifferent between the market schedule and their optimal dispatch, so they do not have an incentive to self-schedule. The ISO solves the Problem [\[centralized2_capacity\]](#centralized2_capacity){reference-type="eqref" reference="centralized2_capacity"} and sets prices $\mu$ for the non-adaptive dispatch, $\boldsymbol{\theta}$ and $\bar{\boldsymbol{\theta}}$ for the adaptive dispatch, and $\boldsymbol{\rho}$ for the commitment. Also, each generator $i$ has commitment costs $F_{i}$ and dispatch costs $C_{i}$. The cost of the adaptive dispatch is a linear function of $\boldsymbol{d}^{*}$ and $\boldsymbol{r}^{*}$, as defined earlier and specified by the ISO. So, generator $i$ has revenue $\mu u_{i} + \sum_{j=1}^{J} \theta_{j} V_{ij} + \sum_{k=1}^{I} \bar{\theta}_{k} Z_{ik} + \rho_{i} x_{i}$, while it has a cost $F_{i} x_{i} + C_{i} u_{i} + C_{i} \sum_{j=1}^{J} V_{ij} d_{j}^{*} + C_{i} \sum_{k=1}^{I} Z_{ik} r_{k}^{*}$. Each generator $i$ decides if they will self-schedule by maximizing the difference between the revenue and the costs. They solve the following problem which maximizes their individual profit: $$\label{decentralized_capacity} {\everymath{\displaystyle} \scalebox{0.95}{ $ \begin{array}{rl} \max_{\boldsymbol{x}, \boldsymbol{u}, \boldsymbol{V}, \boldsymbol{Z}} & \mu u_{i} + \sum_{j=1}^{J} \theta_{j} V_{ij} + \sum_{k=1}^{I} \bar{\theta}_{k} Z_{ik} + \rho_{i} x_{i} \\ & - (F_{i} x_{i} + C_{i} u_{i} + C_{i} \sum_{j=1}^{J} V_{ij} d_{j}^{*} + C_{i} \sum_{k=1}^{I} Z_{ik} r_{k}^{*}) \\ \text{s.t.} & u_{i} + \Gamma_{q} \|\boldsymbol{V}_{i} \|_{\ell^{*}} + \Delta_{p} \| x_{i} \boldsymbol{e}_{i} - \boldsymbol{Z}_{i} \|_{\ell^{*}} \leq p^{\max}_{i} x_{i}, \\ & u_{i} - \Gamma_{q} \|\boldsymbol{V}_{i} \|_{\ell^{*}} - \Delta_{p} \|\boldsymbol{Z}_{i} \|_{\ell^{*}} \geq 0, \\ & x_{i} \in \{0, 1\}, \\ \end{array} $ } }$$ The constraints are the robust counterparts of the capacity and non-negativity constraints for each $i$, which means that the dispatch of the decentralized problem will be non-negative and less than the maximum capacity of generator $i$ for all $\boldsymbol{d} \in \mathcal{D}$ and $\boldsymbol{r} \in \mathcal{U}$. Let $h_{i}(x_{i}, u_{i}, \boldsymbol{V}_{i}, \boldsymbol{Z}_{i})$ be the objective of the decentralized problem [\[decentralized_capacity\]](#decentralized_capacity){reference-type="eqref" reference="decentralized_capacity"} for commitment $x_{i}$ and dispatch $u_{i}, \boldsymbol{V}_{i}, \boldsymbol{Z}_{i}$. Also, let $\boldsymbol{x}^{*}, \boldsymbol{u}^{*}, \boldsymbol{V}^{*}, \boldsymbol{Z}^{*}$ be an optimal solution to the centralized problem [\[centralized2_capacity\]](#centralized2_capacity){reference-type="eqref" reference="centralized2_capacity"} and $x_{i}^{**}, u_{i}^{**}, \boldsymbol{V}_{i}^{**}, \boldsymbol{Z}_{i}^{**}$ be a solution to the decentralized problem [\[decentralized_capacity\]](#decentralized_capacity){reference-type="eqref" reference="decentralized_capacity"} for generator $i$. Generator $i$ has no incentive to self-schedule only if $${\everymath{\displaystyle} h_{i}(x_{i}^{**}, u_{i}^{**}, \boldsymbol{V}_{i}^{**}, \boldsymbol{Z}_{i}^{**}) \leq h_{i}(x_{i}^{*}, u_{i}^{*}, \boldsymbol{V}_{i}^{*}, \boldsymbol{Z}_{i}^{*}). }$$ **Theorem 3**. *Suppose generator $i$ solves the decentralized problem [\[decentralized_capacity\]](#decentralized_capacity){reference-type="eqref" reference="decentralized_capacity"} using the prices of the centralized problem [\[centralized2_capacity\]](#centralized2_capacity){reference-type="eqref" reference="centralized2_capacity"}. Then, they cannot obtain a schedule giving greater profit than the centralized market schedule or $${\everymath{\displaystyle} h_{i}(x_{i}^{**}, u_{i}^{**}, \boldsymbol{V}_{i}^{**}, \boldsymbol{Z}_{i}^{**}) \leq h_{i}(x_{i}^{*}, u_{i}^{*}, \boldsymbol{V}_{i}^{*}, \boldsymbol{Z}_{i}^{*}). }$$* *Proof.* Let $\mu^{*}, \boldsymbol{\theta}^{*}, \boldsymbol{\rho}^{*}$ be the prices determined by the dual variables of the Problem [\[centralizeddual_capacity\]](#centralizeddual_capacity){reference-type="eqref" reference="centralizeddual_capacity"}. Using the results of Theorem [Theorem 2](#thm:thm2_capacity){reference-type="ref" reference="thm:thm2_capacity"}, $h_{i}(x_{i}^{*}, u_{i}^{*}, \boldsymbol{V}_{i}^{*}) = 0$, because $f_{i} = g_{i}$. Consider $h_{i}(x_{i}^{**}, u_{i}^{**}, \boldsymbol{V}_{i}^{**})$, $${\everymath{\displaystyle} \scalebox{0.9}{ $ \begin{array}{llr} & h_{i}(x_{i}^{**}, u_{i}^{**}, \boldsymbol{V}_{i}^{**}) = \mu^{*} u_{i}^{**} + \sum_{j=1}^{J} \theta_{j}^{*} V_{ij}^{**} + \sum_{k=1}^{I} \bar{\theta}_{k}^{*} Z_{ik}^{**} + \rho_{i}^{*} x_{i}^{**} \\ & - (F_{i} x_{i}^{**} + C_{i} u_{i}^{**} + C_{i} \sum_{j=1}^{J} V_{ij}^{**} d_{j}^{*} + C_{i} \sum_{k=1}^{I} Z_{ik}^{**} r_{k}^{*}) \\ & \leq (\mu^{*} - C_{i}) u_{i}^{**} + (\rho_{i}^{*} - F_{i}) x_{i}^{**} \\ \vspace{5pt} & + \sum_{j=1}^{J} (\theta_{j}^{*} - C_{i} \alpha_{j}^{*}) V_{ij}^{**} - \sum_{j=1}^{J} (\bar{\theta}_{k}^{*} - C_{i} \bar{\alpha}_{k}^{*}) Z_{ik}^{**} \\ \vspace{5pt} & + \sigma_{i}^{*} (p^{\max}_{i} x_{i}^{**} - u_{i}^{**} - \Gamma_{q} \|V_{i}^{**} \|_{\ell^{*}} - \Delta_{p} \| x_{i}^{**} \boldsymbol{e}_{i} - \boldsymbol{Z}_{i}^{**} \|_{\ell^{*}}) \\ \vspace{5pt} & + \zeta_{i}^{*} (u_{i}^{**} - \Gamma_{q} \|V_{i}^{**} \|_{\ell^{*}} - \Delta_{p} \|\boldsymbol{Z}_{i} \|_{\ell^{*}}) \end{array} $ } }$$ $${\everymath{\displaystyle} \scalebox{0.9}{ $ \begin{array}{llr} & \leq (\mu^{*} - \sigma_{i}^{*} + \zeta_{i}^{*} - C_{i}) u_{i}^{**} + (\rho_{i}^{*} + \sigma_{i}^{*} p^{\max}_{i} - \beta_{ii}^{*} - F_{i}) x_{i}^{**} \\ & + \sum_{j=1}^{J} (\theta_{j}^{*} - \beta_{ij}^{*} - \gamma_{ij}^{*} - C_{i} \alpha_{j}^{*}) V_{ij}^{**} \\ & + \sum_{k=1}^{I} (\theta_{k}^{*} + \beta_{ik}^{*} - \gamma_{ik}^{*} - C_{i} \alpha_{k}^{*}) Z_{ik}^{**} \leq 0. \\ \end{array} $ } }$$ The first inequality is valid because $p^{\max}_{i} x_{i}^{**} - u_{i}^{**} - \Gamma_{q} \|V_{i}^{**} \|_{\ell^{*}} - \Delta_{p} \| x_{i}^{**} \boldsymbol{e}_{i} - \boldsymbol{Z}_{i}^{**} \|_{\ell^{*}} \geq 0$, $u_{i}^{**} - \Gamma_{q} \|V_{i}^{**} \|_{\ell^{*}} - \Delta_{p} \|\boldsymbol{Z}_{i}^{**} \|_{\ell^{*}} \geq 0$ by the constraints of Problem [\[decentralized_capacity\]](#decentralized_capacity){reference-type="eqref" reference="decentralized_capacity"} and $\sigma_{i}^{*} \geq 0$ and $\zeta_{i}^{*} \geq 0$ by Problem [\[centralizeddual_capacity\]](#centralizeddual_capacity){reference-type="eqref" reference="centralizeddual_capacity"}. The third inequality is valid because $(\mu^{*} - \sigma_{i}^{*} + \zeta_{i}^{*} - C_{i}) = 0$, $(\theta_{j}^{*} - \beta_{ij}^{*} - \gamma_{ij}^{*} - C_{i} \alpha_{j}^{*})=0$, $(\theta_{k}^{*} + \beta_{ik}^{*} - \gamma_{ik}^{*} - C_{i} \alpha_{k}^{*}) = 0$ and $(\rho_{i}^{*} + \sigma_{i}^{*} p^{\max}_{i} - \beta_{ii}^{*} - F_{i}) \leq 0$ by the constraints of Problem [\[centralizeddual_capacity\]](#centralizeddual_capacity){reference-type="eqref" reference="centralizeddual_capacity"}. Also, the second inequality is valid, because $$\scalebox{0.85}{ $ (\boldsymbol{\beta}_{i}^{*})^{T} \boldsymbol{V}_{i}^{**} \leq \| \boldsymbol{\beta}_{i}^{*} \|_{\ell} \; \; \| \boldsymbol{V}_{i}^{**} \|_{\ell^{*}} \leq \sigma_{i}^{*} \Gamma_{q} \| \boldsymbol{V}_{i}^{**} \|_{\ell^{*}}, $ }$$ $$\scalebox{0.85}{ $ (\boldsymbol{\gamma}_{i}^{*})^{T} \boldsymbol{V}_{i}^{**} \leq \| \boldsymbol{\gamma}_{i}^{*} \|_{\ell} \; \; \|\boldsymbol{V}_{i}^{**} \|_{\ell^{*}} \leq \zeta_{i}^{*} \Gamma_{q} \|\boldsymbol{V}_{i}^{**} \|_{\ell^{*}}, $ }$$ $$\scalebox{0.85}{ $ (\bar{\boldsymbol{\beta}}_{i}^{*})^{T} (x_{i}^{**} \boldsymbol{e}_{i} - \boldsymbol{Z}_{i}^{**}) \leq \| \bar{\boldsymbol{\beta}}_{i}^{*} \|_{\ell} \; \; \| x_{i}^{**} \boldsymbol{e}_{i} - \boldsymbol{Z}_{i}^{**} \|_{\ell^{*}} \leq \sigma_{i}^{*} \Delta_{p} \|x_{i}^{**} \boldsymbol{e}_{i} - \boldsymbol{Z}_{i}^{**} \|_{\ell^{*}}, $ }$$ $$\scalebox{0.85}{ $ (\bar{\boldsymbol{\gamma}}_{i}^{*})^{T} \boldsymbol{Z}_{i}^{**} \leq \| \bar{\boldsymbol{\gamma}}_{i}^{*} \|_{\ell} \; \; \|\boldsymbol{Z}_{i}^{**} \|_{\ell^{*}} \leq \zeta_{i}^{*} \Delta_{p} \|\boldsymbol{Z}_{i}^{**} \|_{\ell^{*}}, $ }$$ by the dual norm properties and the constraints of Problem [\[centralizeddual_capacity\]](#centralizeddual_capacity){reference-type="eqref" reference="centralizeddual_capacity"}. Specifically, if $\boldsymbol{x}, \boldsymbol{z}$ are two $n$-dimensional vectors, then $\| \boldsymbol{z}\|_{\ell^{*}} = \{ \max_{\boldsymbol{x}} \boldsymbol{z}^{T} \boldsymbol{x}: \| \boldsymbol{x}\|_{\ell} \leq 1 \}$ and $\boldsymbol{z}^{T} \boldsymbol{x} \leq \| \boldsymbol{x}\|_{\ell} \| \boldsymbol{z}\|_{\ell^{*}}$ for all $\boldsymbol{x}, \boldsymbol{z}$ [@bertsekas1997nonlinear]. So, $h_{i}(x_{i}^{**}, u_{i}^{**}, \boldsymbol{V}_{i}^{**}, \boldsymbol{Z}_{i}^{**}) \leq h_{i}(x_{i}^{*}, u_{i}^{*}, \boldsymbol{V}_{i}^{*}, \boldsymbol{Z}_{i}^{*})$, i.e., no generator $i$ can obtain a solution giving a greater profit than the centralized solution. ◻ Also, $\boldsymbol{x}^{*}, \boldsymbol{u}^{*}, \boldsymbol{V}^{*}, \boldsymbol{Z}^{*}$ are market-clearing or $\sum_{i=1}^{I} (u_{i} + \sum_{j=1}^{J} V_{ij} {d}_{j} + \sum_{k=1}^{I} Z_{ik} r_{k}) \geq \sum_{j=1}^{J} (d_{j} + \bar{q}_{j})$ for all $\boldsymbol{d} \in \mathcal{D}$ and $\boldsymbol{r} \in \mathcal{U}$, so the generators do not have incentive to self-schedule. # Example with load uncertainty {#sec:sec_load_example} In this section, we demonstrate adaptive pricing in detail on the Scarf example with load uncertainty. The day-ahead payments and prices in the ARO problem are larger than those in the deterministic problem, because the commitments and dispatch are more conservative. However, the ARO approach reduces the need for ad hoc corrections, so the total cost may be significantly lower once the uncertainty is realized. The Scarf example, after fixing the binary variables to their optimal values, is based on the following formulation $${\everymath{\displaystyle} \scalebox{0.9}{ $ \begin{array}{rlr} \min_{\boldsymbol{x}, \boldsymbol{p}} & \sum_{i=1}^{I} F_{i} x_{i} + C_{i} p_{i} \\ \text{s.t.} & \sum_{i=1}^{I} p_{i} = \sum_{j=1}^{J} \bar{q}_{j}, \\ & p_{i} \leq p^{\max}_{i} x_{i}, \quad \forall i \in [I], \\ & x_{i} = x_{i}^{*}, \quad \forall i \in [I], \\ & x_{i}, p_{i} \geq 0, \quad \forall i \in [I]. \\ % & p_{i} \geq 0, \quad \forall i \in [I]. \end{array} $ } }$$ There are only two types of generators, whose costs and capacities are asymmetrical. Their characteristics are summarized in Table [1](#tab:tab1){reference-type="ref" reference="tab:tab1"}. **Type** $F_{i}$ $C_{i}$ $p_{i}^{\max}$ ---------- --------- --------- ---------------- 1 53 3 16 2 30 2 7 : The two types of generators in the Scarf example and their commitment costs, dispatch costs and maximum capacity. [\[tab:tab1\]]{#tab:tab1 label="tab:tab1"} Suppose we have two generators of Type 1 and 6 generators of type 2. We also have five consumers with expected load $[8, 8, 3, 5, 16]$, so the total expected load is $\sum_{j=1}^{J} \bar{q}_{j} = 40$. The results for the deterministic problem are summarized in Table [2](#tab:tab2){reference-type="ref" reference="tab:tab2"}. - Objective: \$ 260, - Dual price of the load: 2 \$ / unit. **Type** 1 1 2 2 2 2 2 2 ---------------- --- --- --- --- --- --- --- --- **Commitment** 0 0 1 1 1 1 1 1 **Dispatch** 0 0 5 7 7 7 7 7 : Commitment and dispatch in the deterministic Scarf example. We choose to turn on all generators of type 2, because they meet the expected demand and have the least cost. [\[tab:tab2\]]{#tab:tab2 label="tab:tab2"} Table [3](#tab:tab3){reference-type="ref" reference="tab:tab3"} summarizes the pay-as-bid payments, which are $F_{i} x_{i}^{*} + C_{i} p_{i}^{*}$ for each $i$. **Type** 1 1 2 2 2 2 2 2 ------------------- --- --- ---- ---- ---- ---- ---- ---- $F_{i} x_{i}^{*}$ 0 0 30 30 30 30 30 30 $C_{i} p_{i}^{*}$ 0 0 10 14 14 14 14 14 **Payments** 0 0 40 44 44 44 44 44 : Pay-as-bid payments in the Scarf example. The bids consist of commitment and dispatch costs. [\[tab:tab3\]]{#tab:tab3 label="tab:tab3"} The uniform price payments are $2 p_{i}^{*} + \rho_{i}^{*} x_{i}^{*}$ for each $i$, where $\rho_{i}^{*}$ is the dual variable of the $x_{i} = x_{i}^{*}$ constraint. They are are summarized in Table [4](#tab:tab4){reference-type="ref" reference="tab:tab4"}. In general, $F_{i} x_{i}^{*}$ is not the same as $\rho_{i}^{*} x_{i}^{*}$. **Type** 1 1 2 2 2 2 2 2 --------------- --- --- ---- ---- ---- ---- ---- ---- $2 p_{i}^{*}$ 0 0 10 14 14 14 14 14 Uplifts 0 0 30 30 30 30 30 30 **Payments** 0 0 40 44 44 44 44 44 : Uniform price payments with uplifts in the Scarf example. [\[tab:tab4\]]{#tab:tab4 label="tab:tab4"} In the adaptive problem we also need to select the budget of uncertainty. Suppose we use a budget uncertainty set with $\Gamma_{q} = 20$ , which means that we are protected from an increase of the expected demand from 40 to 60 or a decrease from 40 to 20. In this case, $\Delta_{p}=0$. The results for the ARO problem are summarized in Table [5](#tab:tab5){reference-type="ref" reference="tab:tab5"}. - Objective: \$ 378, - Dual price of the load: 3 \$ / unit. **Type** 1 1 2 2 2 2 2 2 ---------------- --- ------ --- --- --- ----- ----- ----- **Commitment** 1 1 1 0 0 1 1 1 $u_{i}^{*}$ 8 14.5 7 0 0 3.5 3.5 3.5 : Commitment and non-adaptive dispatch in the Scarf example. Generator 3, which is of Type 2, will have only deterministic dispatch, which does not depend on the uncertain load. [\[tab:tab5\]]{#tab:tab5 label="tab:tab5"} The LDR for each generator $i$ is $p_{i}(\boldsymbol{d}) = u_{i} + \sum_{j=1}^{J} V_{ij} d_{j}$. The matrix $\boldsymbol{V}^{*}$ is $$\scalebox{0.8}{ $ \begin{bmatrix} 0.4 & 0.4 & 0.4 & 0.4 & 0.4 \\ 0.075 & 0.075 & 0.075 & 0.075 & 0.075 \\ 0.0 & 0.0 & 0.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 0.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 0.0 & 0.0 & 0.0 \\ 0.175 & 0.175 & 0.175 & 0.175 & 0.175 \\ 0.175 & 0.175 & 0.175 & 0.175 & 0.175 \\ 0.175 & 0.175 & 0.175 & 0.175 & 0.175 \\ \end{bmatrix}. $ }$$ For each $i$, $V_{ij}^{*}$ is the same for all consumers $j$, because the load $\sum_{j=1}^{J} \bar{q}_{j}$ bundles consumers together. This can change if we add different coefficients for each $j$ in the uncertainty set. Table [6](#tab:tab6){reference-type="ref" reference="tab:tab6"} summarizes the day-ahead payments in an adaptive pay-as-bid scheme, where each generator $i$ is paid $F_{i} x_{i}^{*} + C_{i} u_{i}^{*}$ and the total payments are \$ 328.5. **Type** 1 1 2 2 2 2 2 2 ------------------- ---- ------ ---- --- --- ---- ---- ---- $F_{i} x_{i}^{*}$ 53 53 30 0 0 30 30 30 $C_{i}u_{i}^{*}$ 24 43.5 14 0 0 7 7 7 **Payments** 77 96.5 44 0 0 37 37 37 : Pay-as-bid payments in the adaptive Scarf example. The bids consist of commitment and non-adaptive dispatch costs. [\[tab:tab6\]]{#tab:tab6 label="tab:tab6"} Table [7](#tab:tab7){reference-type="ref" reference="tab:tab7"} summarizes the day-ahead payments in the adaptive marginal pricing scheme. They payments are the same as the pay-as-bid scheme. Note that the uplifts all less than the deterministic case for each generator of Type 2. **Type** 1 1 2 2 2 2 2 2 --------------------- ---- ------ ---- --- --- ------ ------ ------ $\mu^{*} u_{i}^{*}$ 24 43.5 21 0 0 10.5 10.5 10.5 Uplifts 53 53 23 0 0 26.5 26.5 26.5 **Payments** 77 96.5 44 0 0 37 37 37 : Marginal price payments in the adaptive Scarf example. [\[tab:tab7\]]{#tab:tab7 label="tab:tab7"} Both the deterministic problem and the commitments and non-adaptive dispatch in the ARO problem try to meet a load of 40. However, the deterministic cost is \$ 280, while the non-adaptive ARO cost is \$ 328.5. This is because the ARO commitment is also feasible for an increase in the load up to 20, which is the worst-case scenario in our uncertainty set. The deterministic problem cannot meet such an increase, because Type 2 generators are used at capacity, except the first one, which can provide only two more units of power. To meet the demand of $40+20$, we would need to turn on more generators the following day, which would be very costly and would require large payments. ## Intra-day dispatch and payments In this section, we consider the intra-day economic dispatch problem and the pricing implications of uncertain scenarios. Suppose that the following day the uncertainty $\boldsymbol{d} \geq 0$ is realized, so the realized load is $\boldsymbol{q} = \bar{\boldsymbol{q}} + \boldsymbol{d}$. We have made some commitments in the day-ahead market, so we can solve the following linear optimization problem (LP) to find the optimal dispatch, based on the new data. $${\everymath{\displaystyle} \scalebox{0.9}{ $ \begin{array}{rlr} \min_{\boldsymbol{p}} & \sum_{i=1}^{I} C_{i} p_{i} \\ \text{s.t.} & \sum_{i=1}^{I} p_{i} = \sum_{j=1}^{J} d_{j}, \\ & p_{i} \leq p^{\max}_{i} x_{i}^{*} - u_{i}^{*}, \quad \forall i \in [I], \\ & p_{i} \geq 0, \quad \forall i \in [I], \\ \end{array} $ } }$$ where $x_{i}^{*}$ and $u_{i}^{*}$ are the solution for generator $i$ based on the ARO problem. So, we want to meet only the additional realized load $\boldsymbol{d}$, while we have already committed to produce $u_{i}^{*}$ to meet the expected load. The intra-day electricity price is 2 for loads smaller than 10 and 3 for loads larger than 10, which is equal to the ARO price. Using ARO, we can provide the intra-day electricity price for all realizations of the uncertain parameters. As a result, the pricing is more transparent and predictable and the market participants can plan their bids. {#fig:fig1 width="40%"} In Figure [1](#fig:fig1){reference-type="ref" reference="fig:fig1"}, we plot the cost of the LP and compare it to the adaptive part of the ARO cost $\sum_{i=1}^{I} C_{i} \sum_{j=1}^{J} V_{ij}^{*} d_{j}$. We gradually increase the total realized load $\sum_{j=1}^{J} d_{j}$ from zero to $\Gamma_{q}=20$, which is the worst-case scenario. The adaptive dispatch is an upper bound to the optimized intra-day dispatch. However, we have optimized for the worst-case uncertainty, so the adaptive problem and the LP will have the same cost, if the worst-case load is realized. # Example with load and capacity uncertainty {#sec:sec_load_cap_example} We use the example of Section [4](#sec:sec_load_example){reference-type="ref" reference="sec:sec_load_example"} with uncertain load and consider the uncertainty in the capacity of each generator. Again, we set $\Gamma_{q} = 20$, which means that we are protected from a load increase from 40 to 60. Also, we choose $\Delta_p = 0.5$, which means that we are protected from a decrease of the expected capacity from $\bar{p}_{i}^{\max}$ to $\bar{p}_{i}^{\max}-0.5$ for each generator $i$. The results for the ARO problem are summarized in Table [8](#tab:tab8){reference-type="ref" reference="tab:tab8"}. - Objective: \$ 402.25, - Dual price of the load: 3 \$ / unit. **Type** 1 1 2 2 2 2 2 2 ----------------- ----- ----- ----- ----- ----- ------ ------ --- **Commitments** 1 1 1 1 1 1 1 0 $u_{i}^{*}$ 8.0 8.0 3.5 3.5 3.5 6.75 6.75 0 : Commitment and non-adaptive dispatch in the adaptive Scarf example with load and capacity uncertainty. [\[tab:tab8\]]{#tab:tab8 label="tab:tab8"} The LDR is $p_{i}(\boldsymbol{d}, \boldsymbol{r}) = u_{i} + \sum_{j=1}^{J} V_{ij} d_{j} + \sum_{k=1}^{I} Z_{ik} r_{k}$. The matrix $\boldsymbol{V}^{*}$ is $$\scalebox{0.8}{ $ \begin{bmatrix} 0.25625 & 0.25625 & 0.25625 & 0.25625 & 0.25625 \\ 0.25625 & 0.25625 & 0.25625 & 0.25625 & 0.25625 \\ 0.1625 & 0.1625 & 0.1625 & 0.1625 & 0.1625 \\ 0.1625 & 0.1625 & 0.1625 & 0.1625 & 0.1625 \\ 0.0 & 0.0 & 0.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 0.0 & 0.0 & 0.0 \\ 0.1625 & 0.1625 & 0.1625 & 0.1625 & 0.1625 \\ 0.0 & 0.0 & 0.0 & 0.0 & 0.0 \end{bmatrix}. $ }$$ The matrix $\boldsymbol{Z}^{*}$ is $$\scalebox{0.8}{ $ \begin{bmatrix} 5.75 & 4.75 & 5.75 & -5.75 & -5.75 & -5.75 & 5.75 & -5.75 \\ -5.75 & -4.75 & -5.75 & 5.75 & 5.75 & 5.75 & -5.75 & 5.75 \\ 0.0 & -0.5 & 0.5 & -0.5 & 0.0 & 0.5 & 0.0 & 0.5 \\ 0.5 & 0.0 & 0.0 & 0.5 & 0.5 & 0.0 & 0.5 & -0.5 \\ -0.5 & -0.5 & -0.5 & 0.0 & 0.5 & -0.5 & -0.5 & 0.5 \\ -0.5 & 0.5 & 0.5 & -0.5 & -0.5 & 0.5 & -0.5 & -0.5 \\ 0.5 & 0.5 & -0.5 & 0.5 & -0.5 & -0.5 & 0.5 & 0.0 \\ 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 \end{bmatrix}. $ }$$ Note that the sum of each column of $\boldsymbol{V}^{*}$ is one and of $\boldsymbol{Z}^{*}$ is zero, because of the robust constraint $\Gamma_{q} \|\boldsymbol{1} - \sum_{i=1}^{I} \boldsymbol{V}_{i} \|_{\ell^{*}} + \Delta_{p} \| - \sum_{i=1}^{I} \boldsymbol{Z}_{i} \|_{\ell^{*}} \leq 0$. The norms are non-negative, so $\|\boldsymbol{1} - \sum_{i=1}^{I} \boldsymbol{V}_{i} \|_{\ell^{*}} = 0$ and $\| \sum_{i=1}^{I} \boldsymbol{Z}_{i} \|_{\ell^{*}} = 0$. Table [9](#tab:tab9){reference-type="ref" reference="tab:tab9"} summarizes the day-ahead payments in a pay-as-bid scheme, where the total payments are \$ 352. **Type** 1 1 2 2 2 2 2 2 ------------------- ---- ---- ---- ---- ------ ------ ------ --- $F_{i} x_{i}^{*}$ 53 53 30 30 30 30 30 0 $C_{i} u_{i}^{*}$ 24 24 7 7 13.5 13.5 7 0 **Payments** 77 77 37 37 43.5 43.5 37.0 0 : Pay-as-bid payments in the adaptive Scarf example with load and capacity uncertainty. [\[tab:tab9\]]{#tab:tab9 label="tab:tab9"} Table [10](#tab:tab10){reference-type="ref" reference="tab:tab10"} summarizes the day-ahead payments in the adaptive marginal pricing scheme. The payments are the same as the pay-as-bid scheme. The uplifts for Type 2 generators are still smaller than the deterministic uplifts, while they are close to the adaptive uplifts with load uncertainty. **Type** 1 1 2 2 2 2 2 2 --------------------- ---- ---- ------ ------ ------- ------- ------ --- $\mu^{*} u_{i}^{*}$ 24 24 10.5 10.5 20.25 20.25 10.5 0 Uplifts 53 53 26.5 26.5 23.25 23.25 26.5 0 **Payments** 77 77 37 37 43.5 43.5 37.0 0 : Marginal price payments in the adaptive Scarf example with load and capacity uncertainty. [\[tab:tab10\]]{#tab:tab10 label="tab:tab10"} In this case, we also protect against drops in the available capacity, so we are more conservative. As a result, the prices and payments increase from \$ 328.5 to \$ 352. We turn on all generators, as the commitments in the deterministic problem and in the ARO problem with load capacity are not robust to a decrease of 0.5 in the capacity of the generators. # Multiperiod Pricing {#sec:sec_multi} In this section, we present our method on a realistic multiperiod formulation with ramp constraints and compare it to deterministic marginal and convex hull pricing. Again, the ARO commitments and dispatch are more conservative, so there is an increase in the payments and the prices compared to the deterministic case. However, there are no uplifts related to optimality gaps that are present in convex hull pricing. Consider the following example by Chen et al. [@chen2020unified], which includes two generators and a 3-hour horizon with expected load $\bar{q}_{t}$ 95, 100, and 130 MW. G1 has a maximum of 100 MW, and energy offer $\$10$ /MWh. G2 has a 20 MW minimum, 35 MW maximum, energy offer $\$50$ /MWh, start-up cost $\$1000$, no-load cost $\$30$, ramp rate 5 MW/hour, start-up rate 22.5 MW/hour, shut-down rate 35 MW/hour, minimum up/down times of one hour, and is initially offline. The UC formulation is provided below. $${\everymath{\displaystyle} \scalebox{0.95}{ $ \begin{array}{rlr} \min_{\boldsymbol{x}, \boldsymbol{p}} & \sum_{t=1}^{3} 10 p_{1, t}+30 x_{2, t}^{ON} +50 p_{2, t}+1000 x_{2, t}^{RU} \\ \text{s.t.}& p_{1, t}+p_{2, t}= \bar{q}_{t}, & 1 \leq t , \\ & 0 \leq p_{1, t} \leq 100, & 1 \leq t , \\ & 20 x_{2, t}^{ON} \leq p_{2, t} 5 x_{2, t}^{ON}, & 1 \leq t , \\ & p_{2, t}-p_{2, t-1} \leq 5 x_{2, t-1}^{ON} +22.5 x_{2, t}^{RU}, & 1 \leq t , \\ & p_{2, t-1}-p_{2, t} \leq 5 x_{2, t}^{ON} +35 x_{2, t}^{RD}, & 2 \leq t , \\ & x_{2, t}^{ON} - x_{2, t-1}^{ON} = x_{2, t}^{RU} - x_{2, t}^{RD}, & 1 \leq t , \\ & x_{2, t}^{RU} \leq x_{2, t}^{ON}, \quad x_{2, t}^{RU} \leq 1-x_{2, t-1}^{ON}, & 1 \leq t , \end{array} $ } }$$ with $\boldsymbol{p} \geq 0$ and $x_{2, t}^{ON}, x_{2, t}^{RU}, x_{2, t}^{RD} \in \{0, 1\} \; \forall t$ representing the status, start-up and shut-down variables respectively. The results for the deterministic problem are summarized in Table [11](#tab:tab11){reference-type="ref" reference="tab:tab11"}. We turn on G2 at $t=1$ and both generators are on for all time periods. The objective is \$ 7340 and the electricity price is $\boldsymbol{\mu} = [10, 10, 90]$ \$ / MW. **Generator** 1 2 --------------- ----------------- ---------------- -- -- -- -- -- -- **Dispatch** \[75, 75, 100\] \[20, 25, 30\] : Dispatch solution for each generator at each time period in the deterministic multiperiod problem. [\[tab:tab11\]]{#tab:tab11 label="tab:tab11"} The convex hull payments are summarized in Table [12](#tab:tab12){reference-type="ref" reference="tab:tab12"}. The convex hull prices are $\boldsymbol{\mu}^{CH} = [10, 10, 276]$ \$ / MW, so G1 is paid $\$$ 2500, while G2 is paid $\$$ 4445 and an uplift of $\$$ 365 [@andrianesis2021computation]. **Generator** 1 2 -------------------------------- --------------------- -------------------- -- -- -- -- -- -- $\mu^{CH}_{t} \; p_{i, t}^{*}$ \[750, 750, 27600\] \[200, 250, 8250\] Uplifts -26600 -4255 **Payments** 2500 4445 : Convex hull payments and uplifts for each generator at each time period in the multiperiod example. [\[tab:tab12\]]{#tab:tab12 label="tab:tab12"} The objective of the convex hull method is $\$$ 365 less than the optimal objective, because of the duality gap. So, there is an additional uplift of $\$$ 365 that is paid to G2. The deterministic pay-as-bid payments are summarized in Table [13](#tab:tab13){reference-type="ref" reference="tab:tab13"}. They are are the same as the convex hull payments but do not feature a duality gap. In addition, these payments are equivalent to a marginal pricing scheme by [@o2005efficient]. **Generator** 1 2 ----------------- -------------------- ---------------------- -- -- -- -- -- -- Commitment Cost \[0, 0, 0\] \[1030, 30, 30\] Dispatch Cost \[750, 750, 1000\] \[1000, 1250, 1500\] Total Cost \[750, 750, 1000\] \[2030, 1280, 1530\] **Payments** 2500 4840 : Pay-as-bid payments for each generator at each time period in the multiperiod example. [\[tab:tab13\]]{#tab:tab13 label="tab:tab13"} In the ARO problem we use budget uncertainty sets with $\Gamma_{q} = [10, 10, 2]$ and $\Delta_{p} = [0, 7.5, 0.5]$ at each time period. The objective increases to \$ 7860 and the electricity price is $\boldsymbol{\mu}^{*} = [10, 10, 130]$ \$ / MW. Again, we turn on G2 at $t=1$ and both generators are on for all time periods. **Generator** 1 2 --------------- ---------------------- ---------------------- -- -- -- -- -- -- **Dispatch** \[72.5, 72.5, 97.5\] \[22.5, 27.5, 32.5\] : Non-adaptive dispatch solution for each generator at each time period in the ARO multiperiod problem. [\[tab:tab14\]]{#tab:tab14 label="tab:tab14"} The LDR is $p_{i, t}(\boldsymbol{d}, \boldsymbol{r}) = u_{it} + \sum_{j=1}^{J} V_{ijt} d_{jt} + \sum_{k=1}^{I} Z_{ikt} r_{kt}$. The non-adaptive dispatch results are summarized in Table [14](#tab:tab14){reference-type="ref" reference="tab:tab14"}. The matrix $\boldsymbol{V}_{t}^{*}$ at each time period is $$\scalebox{0.8}{ $ \begin{array}{cc} \vspace{8pt} \begin{bmatrix} 1.0 & 1.0 & 1.0 & 1.0 & 1.0 \\ 0.0 & 0.0 & 0.0 & 0.0 & 0.0 \end{bmatrix} & t = 1, 2, 3 \end{array} $ }$$ with the rows corresponding to the G1 and G2 and the columns corresponding to three consumers that share the load. Also, $\boldsymbol{Z}^{*}$ contains only zeros. The day-ahead payments to the generators are \$ 7640. They pay-as-bid payments are summarized in Table [15](#tab:tab15){reference-type="ref" reference="tab:tab15"} and each generator is paid $F_{i}^{ON} x_{i, t}^{ON} + F_{i}^{RU} x_{i, t}^{RU} + F_{i}^{RD} x_{i, t}^{RD} + C_{i, t} u_{i, t}^{*}$, where the coefficients correspond to the commitment and dispatch costs. **Generator** 1 2 ----------------- ------------------- ---------------------- Commitment Cost \[0, 0, 0\] \[1030, 30, 30\] Dispatch Cost \[725, 725, 975\] \[1125, 1375, 1625\] Total Cost \[725, 725, 975\] \[2155, 1405, 1655\] **Payments** 2425 5215 : Pay-as-bid payments for each generator at each time period in the ARO multiperiod example. [\[tab:tab15\]]{#tab:tab15 label="tab:tab15"} The adaptive marginal price payments are summarized in Table [16](#tab:tab16){reference-type="ref" reference="tab:tab16"}. **Generator** 1 2 ---------------------------- --------------------- -------------------- $\mu_{t}^{*} u_{i, t}^{*}$ \[725, 725, 12675\] \[225, 275, 4225\] Uplifts -11700 490 **Payments** 2425 5215 : Marginal pricing payments for each generator at each time period in the ARO multiperiod example. [\[tab:tab16\]]{#tab:tab16 label="tab:tab16"} The non-adaptive dispatch in the ARO problem and the dispatch in the deterministic problem try to meet the same level of demand, namely \[95, 100, 130\]. However, the cost is higher in the ARO problem, because we protect against uncertainty in the load and in the capacity. For example, the commitments and dispatch are still feasible for a drop of \[0, 7.5, 0.5\] in the capacity of each generator and an increase of \[10, 10, 2\] in the load. So, the payments to the generators and the price $\boldsymbol{\mu}^{*}$ of electricity are higher compared to the deterministic case. In addition, the adaptive payment mechanisms we use do not feature a duality gap. # Conclusion {#sec:ada_conclusion} In this work, we introduce the first pay-as-bid and marginal pricing methods for energy markets with non-convexities under uncertainty. We consider ARO formulations that protect against uncertainty in the load and capacity parameters and we provide the corresponding adaptive pricing schemes. We apply our method to realistic examples with increasing degrees of complexity and show, both theoretically and empirically, that it eliminates uplifts and corrections that are necessary in deterministic approaches. [^1]: D. Bertsimas and A. G. Koulouras are with the Operations Research Center at the Massachusetts Institute of Technology: dbertsim\@mit.edu, angkoul\@mit.edu. Research partially supported by the Advanced Research Projects Agency with award number DE-AR0001282.

arxiv_math

--- address: - Collège Calvin, Geneva, Switzerland 1211 - Department of Mathematical and Physical Sciences, La Trobe University, Melbourne, Australia 3086 author: - Christian Aebi and Grant Cairns title: Equable Triangles on the Eisenstein Lattice --- # Introduction This paper concerns triangles whose vertices lie on the *Eisenstein lattice*, which is the lattice in the complex plane generated by the elements $1$ and $\omega=-\frac12+i\frac{\sqrt3}2$. We investigate triangles that are *equable*, that is, they have equal perimeter and area. Our study is inspired by the classification of equable Heron triangles, which have integer sides and area and are known to be realisable on the integer lattice [@Yiu]. There are precisely five triangles of the preceding type, up to Euclidean motions; see [@Fo] and [@Br]. By comparison, on the Eisenstein lattice, we find the following result. **Theorem 1**. *There are only two equable triangles having vertices on the Eisenstein lattice, up to Euclidean motions. They are realized by the following vertices.* 1. *$A=8+4\omega,\ B= 4+8\omega,\ C=0$,* 2. *$A=6+3\omega,\ B= 8+16\omega,\ C=0$.* Figure [\[F\]](#F){reference-type="ref" reference="F"} shows the two triangles, with the first (equilateral) triangle translated by $5$ to the right. # Proof of the Theorem Consider an equable triangle $T$ with vertices $A=a_1+a_2\omega, B=b_1+b_2\omega$ and $C=0$, where $a_1,a_2,b_1,b_2\in\mathbb N\cup\{0\}$. Let $a,b,c$ denote the lengths of the sides $AC,BC,AB$, respectively. Notice that the squares of the side lengths are integers; for example, $a^2=a_1^2-a_1a_2+a_2^2$. Moreover, the signed area of $T$ is $\frac{\sqrt3}4(a_1b_2-a_2b_1)$. **Lemma 1**. *The side lengths $a,b,c$ are each of the form $\sqrt3n$, for some $n\in \mathbb N$.* *Proof.* From above, as $T$ is equable, $a+b+c=\frac{\sqrt3}4(a_1b_2-a_2b_1)$, so $$\sqrt{3a^2}+\sqrt{3b^2}+\sqrt{3c^2}=\sqrt3(a+b+c)=\frac{3}4(a_1b_2-a_2b_1),$$ which is rational. But it is well known that if $\sum_{i=1}^n\sqrt{m_i}$ is rational for integers $m_1,\dots,m_n$, then $\sqrt{m_i}$ is rational for each $i$; see for example [@art] or [@Yuan]. So $\sqrt{3a^2},\sqrt{3b^2},\sqrt{3c^2}$ are each rational. Hence, as $3a^2,3b^2,3c^2$ are integers, it follows that $\sqrt{3a^2},\sqrt{3b^2},\sqrt{3c^2}$ are also integers. So the side lengths $a,b,c$ are each of the form $\frac{n}{\sqrt3}$, for some $n\in \mathbb N$. Thus, since the squares of the side lengths are integers, the required result follows. ◻ We follow the reasoning used in the proof of the equable Heron triangle theorem given in the Appendix in [@AC1]. For equable triangles with sides $a,b,c$, Heron's formula gives $$\label{E} (a+b+c)(-a+b+c)(a-b+c)(a+b-c)=16(a+b+c)^2.$$ Let $u=\frac{-a+b+c}{\sqrt3}, v=\frac{a-b+c}{\sqrt3},w=\frac{a+b-c}{\sqrt3}$, so that $u,v,w\in \mathbb N$ by the above lemma, and $a=\frac{\sqrt3(v+w)}{2},b=\frac{\sqrt3(u+w)}{2},c=\frac{\sqrt3(u+v)}{2}$. Then Equation [\[E\]](#E){reference-type="eqref" reference="E"} gives $$\label{E:1} 3uvw=16(u+v+w),$$ and we may assume without loss of generality that $u\le v\le w$. Note that by construction, $u,v,w$ have the same parity, and from [\[E:1\]](#E:1){reference-type="eqref" reference="E:1"}, $u,v,w$ are necessarily even. Let $u=2x,v=2y,w=2z$, so $a=\sqrt3(y+z),b=\sqrt3(x+z),c=\sqrt3(x+y)$. Then $3xyz=4(x+y+z)$. Thus $$y\le z=\frac{4(x+y)}{3xy-4},$$ so $3xy^2-8y-4x\le 0$. Hence $$x\le y\le \frac{4+\sqrt{16+12x^2}}{3x}$$ so $3x^2\le 4+\sqrt{16+12x^2}$. Hence $(3x^2-4)^2\le (16+12x^2)$. Thus $9x^4-36x^2\le 0$, which gives $x\le 2$. Then $$y\le \frac{4+\sqrt{16+12x^2}}{3x} \le \frac{4+\sqrt{16+12}}{3},$$ since the function $\frac{4+\sqrt{16+4x^2}}{3x}$ is decreasing for positive $x$. So, as $y$ is an integer, $y\le 3$. Then, considering the values $x\le 2$, $y\le 3$ and $z=\frac{4(x+y)}{3xy-4}$, we find there are only two solutions, which have the following integer values for $x,y,z$: $$1,2,6\quad\text{and}\quad 2,2,2,$$ which imply finally the values for $a,b,c$: $$8\sqrt3,7\sqrt3,3\sqrt3\quad\text{and}\quad 4\sqrt3,4\sqrt3,4\sqrt3.$$ Paul Yiu, Heronian triangles are lattice triangles, *Amer. Math. Monthly* **108** (2001), no. 3, 261--263. Arthur H. Foss, Integer-sided triangles, *Math. Teacher* **73** (1980), no. 5, 390--392. Christopher J. Bradley, *Challenges in geometry*, Oxford University Press, Oxford, 2005. Victor Wang, *The Art of problem Solving*, <https://artofproblemsolving.com/community/c1461h1035155>. Qiaochu Yuan, *Annoying Precision*, <https://qchu.wordpress.com/2009/07/02/square-roots-have-no-unexpected-linear-relationships/>. Christian Aebi and Grant Cairns, *Lattice equable quadrilaterals I: parallelograms*, L'Enseign. Math. **67** (2021), no. 3/4, 369--401.

arxiv_math

--- abstract: | -- -- -- -- author: - "**A.V. Rukavishnikov**" title: "**On the area of optimal parameters choice for the numerical method of non-stationary hydrodynamics problem with feature**" --- Computing Center of the Far Eastern Branch of the Russian Academy of Sciences, 680000, Khabarovsk, Russia, e-mail: `[email protected]` #### Introduction. The purpose of this study is to find the area of optimal parameters choice for the numerical method of non-stationary Navier-Stokes equations with feature. The search for a solution to a nonlinear problem is reduced to a sequence of approximate linear problems using Runge-Kutta methods of 1st and 2nd order. The peculiarity of the study lies in the fact that the domain is a non-convex polygon with a incoming angle at the boundary. Using classical approximate methods, the error arising in the vicinity of the feature point propagates into the inner part of the domain, where the solution has sufficient smoothness. In this case, the convergence rate of the approximate solution to the exact one is significantly less than for convex domains. The proposed numerical method overcomes these difficulties. It is based on two ideas, namely, the introduction into the variational formulation of problems a weight function to some order and special basis functions. In [@Ruk2018] the concept of an $R_{\nu}$-generalized solution in weighted sets for the Stokes problem is defined. The main feature of the variational formulation of this problem, in contrast to the classical formulation [@Brezzi], is that it is asymmetrical. In [@Math2] the existence and uniqueness of an $R_{\nu}$-generalized solution for the Stokes problem is established. In the course of numerical experiments, an area of optimal parameters of the method was determined. The order of convergence of the approximate solution to the exact one of the nonlinear problem is the same for angles more than $\pi$ and significantly greater than using classical approaches. Part of the optimal set of free parameters of the proposed method doesn't depend on the value of the incoming angle. The optimal convergence rate is achieved without using a mesh refinement in the vicinity of the feature point. #### 1. The problem statement. Consider the flow of a viscous incompressible fluid in a 2-dimensional non-convex polygonal domain $\Omega$ with an incoming angle $\omega$ on its boundary $\partial \Omega$ with a vertex at the origin ${\cal O}=(0,0)$. Let ${\bf x}=(x_1, x_2)$ be an element in $R^2,$ $t$ be an element in time and $Q=\Omega\times (0, T)$. Given fields ${\bf u}_0={\bf u}_0({\bf x})$ in $\Omega, {\bf f}={\bf f}({\bf x},t)=\{ f_i({\bf x},t)\}_{i=1}^2$ in $Q$ and ${\bf g}={\bf g}({\bf x},t)=\{g_i({\bf x},t)\}_{i=1}^2$ in $Q$ such that $\int\limits_{\Omega}{\mbox{div}\,{\bf g} d{\bf x}}=0$ at each time $t\in(0,T),$ it is required to find the fields ${\bf u}={\bf u}({\bf x},t)=\{u_i({\bf x},t)\}_{i=1}^2$ and $P=P ({\bf x},t)$ such that the following identities hold: $$\begin{gathered} \frac{\partial {\bf u}}{\partial t}-\triangle {\bf u} + \mbox{ curl } {\bf u}\times {\bf u}+ \mbox{ grad } P={\bf f}, \qquad\, \,\mbox{ div } {\bf w} ={ 0}\, \qquad\mbox{ in } \,\,\qquad Q, \label{eq:1}\\ \qquad\qquad\,\,\,\,\, {\bf u}({\bf x},0)={\bf u}_0 \qquad \mbox{ in } \qquad\Omega,\qquad\qquad {\bf u}={\bf g} \qquad \mbox{ on } \qquad \partial \Omega\times (0,T). \label{eq:2}\end{gathered}$$ As a time discretization of problem (1)-(2), we use the Runge-Kutta schemes of the 1st and 2nd orders. To do this, we first introduce the notation ${\bf v}^n={\bf v}^n({\bf x})$ to approximate the function ${\bf v}({\bf x}, n\triangle t), n=0,1,2,...,N$ and ${\bf v}^{n+\gamma}$ to approximate the function ${\bf v}({\bf x}, (n+\gamma)\triangle t), \gamma\in (0,1), n=0,1,2,...,N-1$. Parameter $\triangle t$ is such that $T=N\cdot \triangle t.$ Moreover, let ${\bf v}^{-1}:={\bf v}^0$ and $\bar{\bf v}^{n+1}:=0.5({\bf v}^ {n+1}+{\bf v}^n)$ and ${\bf U}^n$ a suitable approximation to **u** at time $n \triangle t.$ **1st order scheme**. Given ${\bf u}^n, {\bf U}^n:=\frac{3}{2} {\bf u}^n -\frac{1}{2} {\bf u}^ {n-1}, P^n, \bar{\bf f}^{n+1}$ and ${\bf g}^{n+1}$: find ${\bf u}^{n+1}$ and $P^{n+1}$ as a solution to the system of equations: $$\begin{gathered} (\triangle t)^{-1}{\bf u}^{n+1}-\triangle \bar{\bf u}^{n+1} + \mbox{ curl } {\bf U}^n \times \bar{\bf u}^{n+1}+ \mbox{ grad } \bar P^{n+1}=\bar{\bf f}^{n+1}+(\triangle t) ^{-1}{\bf u}^{n}\qquad \mbox{ in } \,\,\qquad \Omega, \label{eq:3}\\ \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \qquad\, \,\mbox{ div } {\bf u}^{n+1} ={ 0} \,\qquad \mbox{ in } \,\,\qquad \Omega, \label{eq:4}\\ \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad{\bf u}^{n+1}={\bf g}^{n+1 } \qquad \mbox{ on } \qquad \partial \Omega. \label{eq:5}\end{gathered}$$ **2nd order scheme**. This scheme consists of two steps. *Step 1*. Given ${\bf u}^n, {\bf U}^n:=\frac{3}{2} {\bf u}^n -\frac{1}{2} {\bf u}^ {n-1}, {\bf f}^{n+\gamma}$ and ${\bf g}^{n+\gamma}$: find ${\bf u}^{n+\gamma}$ and $P^{n+\gamma} (\gamma\in (0,1))$ as a solution to the system of equations: $$\begin{gathered} (\gamma \triangle t)^{-1}{\bf u}^{n+\gamma}-\triangle {\bf u}^{n+\gamma} + \mbox{ curl } {\bf U}^n\times {\bf u}^{n+\gamma}+ \mbox{ grad } P^{n+\gamma}={\bf f}^{n+\gamma}+(\gamma\triangle t)^{-1}{\bf u}^{n}\qquad \mbox{ in } \,\,\qquad \Omega, \label{eq:6}\\ \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\, \,\mbox{ div } {\bf u}^{n+\gamma} ={ 0}\,\qquad \mbox{ in } \,\,\qquad \Omega, \label{eq:7}\\ \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad{\bf u}^{n+\gamma}={\bf g}^{n+\gamma} \qquad \mbox{ on } \qquad \partial \Omega. \label{eq:8}\end{gathered}$$ *Step 2*. Given ${\bf u}^n, {\bf u}^{n+\gamma}, P^{n+\gamma}, {\bf U}^n:=\frac{3}{2} {\bf u}^n -\frac{1}{2} {\bf u}^{n-1}, {\bf f} ^{n+1}, {\bf f}^{n+\gamma}$ and ${\bf g}^{n+1}$: find ${\bf u}^{n+1}$ and $P^{n+1}$ as a solution to the system of equations: $$\begin{gathered} (\triangle t)^{-1}{\bf u}^{n+1}+\gamma (-\triangle {\bf u}^{n+1} + \mbox{ curl } {\bf U}^n\times {\bf u}^{n+1}+ \mbox{ grad } P^{n+1})=\\\nonumber =(\triangle t)^{-1}{\bf u}^{n}+\gamma {\bf f}^{n+1}+(1-\gamma){\bf f}^{n+\gamma} -(1-\gamma) (-\triangle {\bf u}^{n+\gamma} + \mbox{ curl } {\bf U}^n\times {\bf u}^{n+\gamma}+ \mbox{ grad } P^{n+\gamma}) \quad \mbox{ in } \quad \Omega, \label{eq:9}\\ \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \qquad\, \,\mbox{ div } {\bf u}^{n+1} ={ 0}\,\qquad \mbox{ in } \,\,\qquad \Omega, \label{eq:10}\\ \qquad\qquad\qquad\quad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad{\bf u}^{n+1}={\bf g}^{n+1} \qquad \mbox{ on } \qquad \partial \Omega. \label{eq:11}\end{gathered}$$ At each step of both schemes, it is necessary to be able to solve the following problem: find the fields ${\bf v}=(v_1,v_2)$ and $q$ such that $$\begin{gathered} \theta {\bf v}-\triangle {\bf v} + W \times {\bf v}+ \mbox{ grad } q={\bf F} \qquad\mbox{ in } \,\,\qquad \Omega, \label{eq:12}\\ \qquad\qquad\qquad\qquad\qquad\, \,\mbox{ div } {\bf v} ={ 0}\, \qquad\mbox{ in } \,\,\qquad \Omega, \label{eq:13}\\ \qquad\qquad\qquad\qquad\qquad\qquad {\bf v}={\bf G} \qquad \mbox{ on } \qquad \partial \Omega, \label{eq:14}\end{gathered}$$ where ${\bf F}$ and $W$ are given functions on $\Omega$ and ${\bf G}$ is given on $\partial \Omega.$ Let us define an $R_{\nu}$-generalized solution of problem (12)-(14) in the domain $\Omega,$ having an incoming angle on the $\partial \Omega$ with a vertex at the origin. To do this, we define the necessary weight sets. First, we introduce the concept of a weight function $\rho({\bf x}): \rho({\bf x})=\{\sqrt{x_1^2+x^2_2}, \mbox{ if } {\bf x}\in \Omega_{\delta} \mbox{ and } \delta, \mbox{ if } {\bf x}\in \bar{\Omega}\diagdown \Omega_{\delta}\},$ where $\Omega_{\delta}=\{{\bf x}\in \bar{\Omega}: \sqrt{x_1^2+x^2_2}\leq \delta\}$ and $\delta\ll 1.$ Denote by $L_{2,\alpha}(\Omega,\delta)$ the set of functions $s({\bf x})$ satisfying the conditions:\ 1) $\|\rho^{\alpha}s\|_{L_2(\Omega\diagdown \Omega_{\delta})}\geq C_1>0;$\ 2) $|s({\bf x})|\leq C_2 \delta^{\alpha-\varepsilon} \rho^{\varepsilon-\alpha}({\bf x}),{\bf x}\in \Omega_{\delta},$\ where $C_2$ is a positive constant independents $s({\bf x})$, $\varepsilon$ is a small positive parameter independent of $\delta, \alpha, s({\bf x}),$ with bounded norm $\|s\|_{L_{2,\alpha}(\Omega)}:=\|\rho^{\alpha} s\|_{L_2(\Omega)}$ of the space $L_{2,\alpha}(\Omega).$ $L^0_{2,\alpha}(\Omega,\delta)$ subset of $L_{2,\alpha}(\Omega)$ such that $s\in L^0_{2,\alpha}(\Omega,\delta),$ if $s\in L_{2,\alpha}(\Omega)$ and $\|\rho^{ \alpha}s\|_{L_1(\Omega)}=0.$ Denote by $W^1_{2,\alpha}(\Omega,\delta)$ the set of functions $s({\bf x})$ satisfying the conditions:\ 1) $\|\rho^{\alpha}s\|_{L_2(\Omega\diagdown \Omega_{\delta})}\geq C_1>0;$\ 2) $|s({\bf x})|\leq C_2 \delta^{\alpha-\varepsilon} \rho^{\varepsilon-\alpha}({\bf x}),{\bf x}\in \Omega_{\delta},$\ 3) $|D^1 s({\bf x})|\leq C_2 \delta^{\alpha-\varepsilon} \rho^{\varepsilon-\alpha-1}({\bf x}), { \bf x}\in \Omega_{\delta},$\ where $C_2$ is a positive constant independent $s({\bf x})$, $\varepsilon$ is a small positive parameter, independent of $\delta, \alpha, s({\bf x}),$ with bounded norm $\|s\|_{W^1_{2,\alpha}(\Omega)}:=\sqrt{\sum\limits_{|k|\leq 1}{\|\rho^{\alpha} | D^k s|\|^2_{L_2(\Omega)}}}$ of the space $W^1_{2,\alpha}(\Omega).$ $\hat W^1_{2,\alpha}(\Omega,\delta)$ subset $W^1_{2,\alpha}(\Omega)$ such that $s\in \hat W^1_{2,\alpha}(\Omega,\delta),$ if $s\in W^1_{2,\alpha}(\Omega, \delta)$ and $s=0$ on $\partial \Omega.$ Let $L_{\infty,\alpha}(\Omega,C_3)$ be the set of functions $s({\bf x})$ with a norm $\|s\|_{L_{\infty,\alpha}(\Omega,C_3)}= \mbox{vrai} \max\limits_{{\bf x}\in \Omega} {|\rho^{\alpha}({\bf x})s({\bf x})|}\leq C_3,$ where $C_3>0$ independent of $s({\bf x})$. Let us define an $R_{\nu}$-generalized solution of problem (12)-(14). **Definition 1**. *Pair $({\bf v}_{\nu}, q_{\nu})\in {\bf W}^{1}_{2,\nu}(\Omega, \delta)\times L^0_{2,\nu}(\Omega,\delta)$ is called $R_{\nu}$-generalized solution of problem (12)-(14) if ${\bf w}_{\nu}$ satisfies (14) on $\partial \Omega$ and identities* $$\begin{gathered} a({\bf v}_{\nu},{\bf z})+b({\bf z}, q_{\nu})=l({\bf z}), \label{eq:15}\\ c({\bf v}_{\nu},s)=0\qquad\qquad\qquad \label{eq:16}\end{gathered}$$ *hold, for all $({\bf z}, s)\in \hat{\bf W}^{1}_{2,\nu}(\Omega, \delta)\times L^0_{2,\nu}(\Omega,\delta)$. Here $({\bf F},W,{\bf G})\in {\bf L}_{2,\alpha}(\Omega,\delta)\times L_{\infty,\beta}(\Omega,C_3)\times {\bf W}^{1/2}_{2,\alpha}(\partial \Omega, \delta), \nu\geq\alpha\geq 0, \beta\leq 2.$* We have $a({\bf w},{\bf z})=\int\limits_{\Omega}{\Bigl[\theta {\bf w}\cdot (\rho^{2 \nu} {\bf z})+\nabla{\bf w}:\nabla(\rho^{2 \nu} {\bf z})+(W\times {\bf w})\cdot (\rho^{2 \nu} {\bf z})\Bigr] d{\bf x}},$\ $b({\bf z}, p)=-\int\limits_{\Omega}{p \mbox { div } (\rho^{2 \nu} {\bf z})}d{\bf x},\,\,\, c({\bf w},s)=-\int\limits_{\Omega}{(\rho^{2 \nu} s) \mbox{ div } {\bf w}} d{\bf x},\,\,\, l({\bf z})=\int\limits_{\Omega}{{\bf F} \cdot (\rho^{2\nu} {\bf z})} d{\bf x}.$ **Remark 1**. *$b(\cdot,\cdot)\neq c(\cdot,\cdot)$. Consequently, the variational problem (15), (16) is not symmetric, in contrast to the standard setting (see [@Brezzi]).* **Remark 2**. *If $W\in L_{\infty,\beta}(\Omega, C_3), \beta\leq 2, {\bf g=0}$ on $\partial \Omega,$ then there exists a unique $R_{\nu}$-generalized solution $({\bf w}_{\nu},q_{\nu})$ problem (12)-(14) in the asymmetric formulation (15), (16) (see Theorem 5 in [@RukJCAM2023]).* **Remark 3**. *Scheme 2 (6)-(11) can be applied when $\gamma$ equal to $1-\frac{\sqrt{2}}{2}$ (strongly L-stable method [@Alex]). Scheme 1 can be applied due to the validity of Theorem 6 [@RukJCAM2023].* #### 2. Creation of an approximate approach. We will build a quasi-uniform fragmentation $I^h$ of the domain $\bar{\Omega}$ into triangles $K_i$. Their sides of order $h$ which are the essential elements. We split each of them into 3 using the center of mass, which are finite elements $L_{i_j}$ that make up the fragmentation $J^h$. We define the main finite element spaces. 1\. *For the velocity components*. We will use Lagrangian elements of the 2nd order with nodes at the vertices and midpoints of the sides of $L_{i_j}$. The linear span of basic functions $\chi_k({\bf x})$ will be denote by $W_h$. 2\. *For the pressure*. As approximation nodes, we use the vertices of finite elements, and the vertices of neighboring finite elements are different nodes. On each $L_{i_j}$ we define 1st order basis functions $\theta_l({\bf x})$ whose support is only one finite element. The linear span of such basis functions forms a space $S_h$, consisting of functions discontinuous in the domain under consideration. The pair ${\bf W}_h \times S_h ({\bf W}_h=W_h \times W_h)$ is a Scott-Vogelius 2nd order one [@Scott]. Let's multiply the basis functions of the spaces $W_h$ and $S_h$ by the weight function $\rho({\bf x})$ in some powers ($-\nu^{\ast}$) and (-$\mu^{\ast}$). The values of the powers will be determined later. We define new basis functions $\phi_k({\bf x})=\chi_k({\bf x}) \cdot \rho^{-\nu^{\ast}}({\bf x}), \psi_l({\bf x})=\theta_l({\bf x}) \cdot \rho^{-\mu^{\ast}}({\bf x})$. Their linear spans form finite-dimensional spaces $V_h$ and $Q_h$ respectively. $\hat V_h=\{u_h\in V_h: u_h(M_i)=0 \mbox{ where } M_i \mbox{ are nodes on } \partial \Omega\}$. Having found the solution $\hat v_i$ and $\hat q_j$ at the nodes $M_i$ and $N_j$ for the velocity components and pressure of system of equations (presented below) it is necessary to restore the true values at the nodes $M_i$ and $N_j$ using the formulas $v_i=\hat v_i \cdot\rho^{-\nu^{\ast}}(M_i)$ and $q_j=\hat q_j \cdot\rho^{-\mu^{\ast}}(N_j).$ We have ${\bf V}_h=V_h\times V_h\subset {\bf W}^1_{2,\nu}(\Omega, \delta), \hat{\bf V}_h=\hat V_h\times \hat V_h\subset \hat{\bf W}^{1}_{2,\nu}(\Omega, \delta), Q_h\subset L^0_{2,\nu}(\Omega, \delta).$ We are all set to determine an approximate $R_{\nu}$-generalized solution of the problem (12)-(14). **Definition 2**. *We will say that a pair of functions $({\bf v}^h_{\nu}, \, q^h_{\nu})$ from the spaces ${\bf V}_{h} \times Q_{h},$ satisfying condition (14) at the nodes on $\partial \Omega$, is an approximate $R_{\nu}$-generalized solution of the problem (12)-(14) if for all pairs of functions $({\bf z}^h, s^h)$ from the spaces $\hat{\bf V}_{h} \times Q_{h}$ the following relations $$\begin{gathered} a({\bf v}^h_{\nu}, {\bf z}^h)+b({\bf z}^h, q^h_{\nu})=l({\bf z}^h), \label{eq:17}\\ c({\bf v}^h_{\nu}, s^h)=0\qquad\quad\,\,\,\qquad\quad \label{eq:18}\end{gathered}$$ hold*. **Remark 4**. *How to solve the system (17), (18) by the iterative method see [@Ruk_Vich; @Tech].* **Remark 5**. *If $\nu = \nu^{\ast} = \mu^{\ast} = 0$ then we have an approximate generalized solution $({\bf v}^h, \, q^h)$ of the problem (12)-(14).* ![Optimal parameters of the weighted FEM for $(\nu, \nu^{\ast}), \delta\in[0.025, 0.035], \omega=\frac{9 \pi}{8}$.](1.125.eps) #### 3. The results of numerical experiments. Let's carry out a number of numerical experiments to find an approximate solution to problem (1)-(2) as a sequence of solving problem (12)-(14) of both formulation schemes (17)-(18). Consider domains $\Omega_k, k=1,2,3$ with incoming angle $\omega_k,$ where $$\Omega_0=\{(x_1, x_2): -1<x_1<1, 0<x_2<1\},$$ $$\bar{\Omega}_1=\bar{\Omega}_0\cup \{(x_1, x_2): -1 \leq x_1\leq 0, -1\leq x_2\leq 0\},$$ $$\bar{\Omega}_2=\bar{\Omega}_0\cup \{(x_1, x_2): -1 \leq x_1\leq 0, x_1\leq x_2\leq 0\},$$ $$\bar{\Omega}_3=\bar{\Omega}_0\cup \{(x_1, x_2): -1 \leq x_1\leq 0, \frac{1}{2} x_1\leq x_2\leq 0\}.$$ In such cases $\omega_k=(1+2^{-k}) \pi, k=1,2,3.$ Denote by ${\bf u}^{h_i}_{\nu}$ and ${\bf u}^{h_i}$ an approximate $R_{\nu}$-generalized and generalized solutions (velocity field) of the solution of the problem (1)-(2) at each moment of time. In the second case $(\nu=\nu^{\ast}=\mu^{\ast}=0, \delta=1).$ In the first case $(\nu,\nu^{\ast},\mu^{\ast}, \delta)$ is the set of free parameters of the weighted finite element method. The exact solution of the problem (1)-(2) of the velocity and pressure fields depend on the value of the incoming angle $\omega_k$ in each moment of time do not belong to the spaces ${\bf W}^2_2(\Omega_k)$ and ${\bf W}^1_2(\Omega_k)$, respectively, and have the form in polar coordinates $(r,\theta):$\ $u_1(r,\theta,t)=e^t\Bigl(r^{\lambda_k}\chi_1(\theta)+\psi_1(r,\theta) \Bigr),$ $u_2(r,\theta,t)=e^t\Bigl(r^{\lambda_k}\chi_2(\theta)+\psi_2(r,\theta) \Bigr),$ $P(r,\theta,t)=e^t r^{\lambda_k-1} \gamma(\theta),$\ where $\psi_i(r, \theta)$ is the regular part of $u_i(r,\theta,t),$ i.e. a function belonging to the space ${\bf W}^2_2(\Omega_k),$ and $r^{\lambda_k}\chi_i(\theta)$ and $r^{\lambda_k-1} \gamma(\theta)$ are the singular solution parts of the velocity and pressure fields. The exponent $\lambda_k$ is such that it coincides with the smallest real positive solution of the equation $\sin(\lambda \omega_k)+\lambda \sin \omega_k=0$. Thus $(\lambda_1,\lambda_2,\lambda_3)$ take the following approximate values $(0.5445, 0.6736, 0.8008).$ We have $$\chi_1(\theta)=\cos(\theta)\, \Xi'_k(\theta)+(1+\lambda_k)\, \sin(\theta)\, \Xi_k(\theta),$$ $$\chi_2(\theta)=(\lambda_k-1)\, \cos(\theta)\, \Xi_k(\theta)+ \sin(\theta)\, \Xi'_k(\theta),$$ $$\gamma(\theta)=(\lambda_k-1)^{-1}\, \Bigl( \Xi'''_k(\theta)+ (1+\lambda_k)^2\, \Xi'_k(\theta)\Bigr).$$ Here $$\Xi_k(\theta)=\Bigl[(1+\lambda_k)^{-1} \sin((1+\lambda_k)\theta)-(1-\lambda_k)^{-1} \sin((1-\lambda_k)\theta) \Bigr] \cos(\lambda_k \omega_k)+\cos((1-\lambda_k)\theta)-\cos((1+\lambda_k)\theta),$$ $\Xi'_k(\theta)$ and $\Xi'''_k(\theta)$ are its the first and third derivatives with respect to $\theta$, respectively. ![Optimal parameters of the weighted FEM for $(\nu, \nu^{\ast}), \delta\in[0.025, 0.035], \omega=\frac{5 \pi}{4}$.](1.25.eps) In test cases, consider different steps $h_j=2^{1-j}\cdot h, h=0.025, j=1,2,3.$ Time step $\triangle t=0.01, T=0.5.$ Shown earlier as for stationary [@Vich_Mex22] and non-stationary problems [@RukJCAM2023] problems, it is possible to determine sets of optimal parameters for which the order of convergence of the approximate solution to exact solution is equal to ${\cal O}(h),$ independent of the incoming angle $\omega_k$ in the norm ${\bf W}^1_{2,\nu}(\Omega_k).$ To determine the range of choice of optimal approach parameters, we fix the range $\delta \sim h_1: \delta\in[0.025, 0.035].$ Let $\nu^{\ast}=\mu^{\ast}$ and will take non-negative values. Moreover, each $\omega_k$ will have its own range of value change. The parameter $\nu$ is positive and not exceed the value 2. For the first scheme, consider the case $$\psi_1(r,\theta)=\psi_2(r,\theta)=0$$ (the solution contains only singular components). For the second scheme $$\psi_1(r,\theta)=\sin (r \cos(\theta))\cdot \cos(r \cos(\theta)), \psi_2(r,\theta)=- \cos (r \cos (\theta))\cdot \sin (r \sin (\theta)).$$ We will assume that the set $(\nu^{\ast}, \nu)$ falls into the area of choice of optimal parameters of the numerical method for solving problem (1)-(2), if it differs by no more than 5 percent from the optimal value in terms of convergence in each moment of time for all $h_j, j=1,2,3.$ The Figures 1-3 show the area of choice of optimal parameters in the corresponding ranges for the first scheme for all values of the angle $\omega_k, k=1,2,3.$ For the second scheme, the results similar in structure. ![Optimal parameters of the weighted FEM for $(\nu, \nu^{\ast}), \delta\in[0.025, 0.035], \omega=\frac{3 \pi}{2}$.](1.5.eps) #### Conclusions. It is also necessary to investigate other values of the incoming angle $\omega$ and determine the intersection areas of choice the optimal parameters of the method for each $\omega$. In order to establish the area of optimal parameters of the method for which the required order of convergence is guaranteed regardless of the incoming angle value. #### Acknowledgments. The reported study was supported by Russian Science Foundation, project No. 21-11-00039, https://rscf.ru/en/project/21-11-00039/. The results were obtained using the equipment of SRC "Far Eastern Computing Resource" IACP FEB RAS (https://cc.dvo.ru). 99 **Rukavishnikov V.A., Rukavishnikov A.V.** Weighted finite element method for the Stokes problem with corner singularity. J. Comput. Appl. Math. 2018; 341:144--156. DOI: 10.1016/j.cam.2018.04.014. **Brezzi F., Fortin M.** Mixed and hybrid finite element methods. New York: Springer-Verlag; 1991: 350. **Rukavishnikov V.A., Rukavishnikov A.V.** On the existence and uniqueness of an $R_{\nu}$-generalized solution to the Stokes problem with corner singularity. Mathematics. 2022; 10(10): 1752. DOI: 10.3390/math10101752. **Alexander R.** Diagonally implicit Runge-Kutta methods for stiff odes. SIAM J. Numer. Anal., 1977; 14: 1006--1021. **Rukavishnikov V.A., Rukavishnikov A.V.** Theoretical analysis and construction of numerical method for solving the Navier-Stokes equations in rotation form with corner singularity. J. Comput. Appl. Math., 2023; 429: 115218. **Scott L.R. and Vogelius M.** Norm estimates for a maximal right inverse of the divergence operator in spaces of piecewise polynomials. Mathematical Modeling and Numerical Analysis, 1985; 19: 111--143. **Rukavishnikov A.V.** On the optimal set of parameters for an approximate method for solving stationary nonlinear Navier-Stokes equations with singularity. Computational Technologies, 2022; 27(6): 70--87. **Rukavishnikov A.V.** A numerical approach for solving one nonlinear problem of hydrodynamics in a non-convex polygonal domain. Computational Continuum Mechanics, 2022; 15(1): 19--30.

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--- abstract: | In this article, we employ the construction of the time-marching Discontinuous Petrov-Galerkin (DPG) scheme we developed for linear problems to derive high-order multistage DPG methods for non-linear systems of ordinary differential equations. The methodology extends to abstract evolution equations in Banach spaces, including a class of nonlinear partial differential equations. We present three nested multistage methods: the hybrid Euler method and the two- and three-stage DPG methods. We employ a linearization of the problem as in exponential Rosenbrock methods, so we need to compute exponential actions of the Jacobian that change from time steps. The key point of our construction is that one of the stages can be post-processed from another without an extra exponential step. Therefore, the class of methods we introduce is computationally cheaper than the classical exponential Rosenbrock methods. We provide a full convergence proof to show that the methods are second, third, and fourth-order accurate, respectively. We test the convergence in time of our methods on a 2D + time semi-linear partial differential equation after a semidiscretization in space. address: - Basque Center for Applied Mathematics (BCAM), Bilbao, Spain - | Oden Institute for Computational Engineering and Sciences,\ The University of Texas at Austin, Austin, USA author: - Judit Muñoz-Matute - Leszek Demkowicz bibliography: - references.bib title: Multistage DPG time-marching scheme for nonlinear problems --- multistage DPG method ,hybrid Euler method ,ultraweak formulation ,optimal test functions ,exponential Rosenbrock integrators ,nonlinear problems # Introduction {#Sec:Intro} Exponential time integrators [@hochbruck2010exponential] are a class of time-marching schemes for solving stiff systems of non-linear or semi-linear Ordinary Differential Equations (ODEs), usually arising from the spatial discretization of Partial Differential Equations (PDEs). These methods are of particular interest as they allow the use of large time steps because of their capacity to handle the stiffness of the problem. There exist different types of exponential integrators, including exponential Runge-Kutta methods [@hochbruck2005explicit; @hochbruck2005exponential; @luan2014explicit], exponential Rosenbrock methods [@hochbruck2009exponential; @caliari2009implementation; @luan2014exponential], exponential multistep methods [@hochbruck2011exponential; @calvo2006class], or Lawson methods [@ostermann2020lawson; @krogstad2005generalized], among others. All these integrators require to compute the action of exponential-related functions (called $\varphi$-functions) over vectors. Although the first articles about exponential integrators can be traced back to the 70's [@friedli2006verallgemeinerte], they have attracted more attention in the last decade due to the recent developments in numerical linear algebra on efficient algorithms to compute these actions [@berland2007expint; @higham2020catalogue; @al2011computing; @higham2005scaling; @niesen2012algorithm], making these types of methods significantly more competitive. On the other hand, the Discontinuous Petrov-Galerkin (DPG) method with optimal test functions [@demkowicz2014overview; @demkowicz2010class; @demkowicz2011class] is a well-established method within the class of stabilized Finite Element Methods [@zienkiewicz2005finite] to approximate the solution of challenging PDEs. This approach has presented an impressive performance in approximating challenging problems in engineering like advection-dominated diffusion [@demkowicz2013robust], Stokes flows [@roberts2014dpg], linear elasticity [@bramwell2012locking], or time-dependent problems [@demkowicz2017spacetime], among many others. The main idea behind the DPG method is to construct optimal test functions such that the discrete stability of the method is guaranteed. In our previous work [@munoz2021adpg; @munoz2021equivalence; @munoz2022error; @munoz2022combining], we applied the DPG theory to stiff systems of linear ODEs arising from the semidiscretization of linear transient PDEs, and we derived a DPG-based time-marching scheme. We proved that if we employ an ultraweak variational formulation in the time variable, the optimal test functions that ensure the stability of the scheme are exponential-related functions that can be expressed in terms of the $\varphi$-functions. In our construction, we have two types of variables in time: traces (point values) and fields (piecewise polynomials). We proved that the DPG time-marching scheme is equivalent to classical exponential integrators for the trace variables, and additionally, it delivers the $L^2$-projection of the exact solution in the interior of the time intervals. In summary, the DPG time-marching scheme can be viewed as the hybridization of classical exponential integrators for linear problems. In this article, we employ the structure of the DPG time-marching scheme to construct a new class of higher-order methods for non-linear problems. We derive three methods: the Hybrid Euler method and the two- and three-stage DPG. These methods are of orders two, three, and four, respectively. For that, we first consider a linearization of the problem at the initial time-step $t_n$ of each time interval as in Rosebrock-type methods [@hochbruck2009exponential]. We realize that if we approximate the nonlinear reminder with the known value of the solution at $t_n$, we are in the linear setting. Therefore, we can apply the lowest order (piecewise constants) DPG-time marching scheme, and we prove that it is the hybridization of the classical exponential Euler method. Here, we compute the interior variable with an exponential step, and then we post-process the trace variable without computing an extra action of any $\varphi$-function. The hybrid exponential Euler method is second-order and has the same cost as the classical exponential Euler method. Then, we construct methods up to order four employing a hybrid Euler step for defining the internal stages. We define the update for the two-stage DPG method as in classical exponential Runge-Kutta methods, and we derive the so-called stiff order conditions for the coefficient functions to obtain a third-order method. However, for the three-stage DPG method, it is not enough to derive the stiff order conditions in the classical way to obtain a fourth-order method. In addition, we need to add a term to the update that depends upon the Jacobian to obtain the desired order. The key point of the construction of the fourth-order method is that one of the stages is computed as a postprocessing of the other one without an extra exponential action, being therefore cheaper than the classical exponential Rosenbrock methods. Moreover, the third- and fourth order methods are nested so they can be employed to adapt the time-step size. The article is organized as follows: Section [2](#Sec:Model){reference-type="ref" reference="Sec:Model"} describes the model problem, the linearization we consider, and the assumptions of the solution and the non-linear term. Section [3](#Sec:Linear){reference-type="ref" reference="Sec:Linear"} summarizes the idea of applying the DPG method as a time-integrator for linear problems. Section [4](#Sec:Methods){reference-type="ref" reference="Sec:Methods"} introduces the construction of the hybrid Euler method and the two and three-stage DPG methods. In Section [5](#Sec:Stiff){reference-type="ref" reference="Sec:Stiff"}, we derive the Stiff order conditions by analyzing the local truncation error, and in Section [6](#Sec:Convergence){reference-type="ref" reference="Sec:Convergence"}, we provide the final convergence proofs. In Section [7](#Sec:Results){reference-type="ref" reference="Sec:Results"}, we test the convergence of the methods on a benchmark 2D+time nonlinear problem. Section [8](#Sec:Conclusions){reference-type="ref" reference="Sec:Conclusions"} discusses the conclusions and future work. Finally, in [9](#App:Summarize){reference-type="ref" reference="App:Summarize"} we provide a summary of the notation we employed throughout the article. # Model problem, assumptions, and linearization {#Sec:Model} Let $I=[0,T]\subset\mathbb{R}$, we consider the following first-order nonlinear autonomous system of Ordinary Differential Equations (ODEs) $$\label{ODE1} \displaystyle{ \left\{ \begin{split} u'(t)&=F(u(t)),\\ u(0)&=u_{0}.\\ \end{split} \right.}$$ We consider the framework of strongly continuous semigroups [@pazy2012semigroups] for the convergence analysis. Therefore, let $X$ be a Banach space with norm $\Vert\cdot\Vert$, we consider the following usual assumptions [@hochbruck2009exponential] on the solution $u:[0,T)\longrightarrow X$ and the nonlinear term in ([\[ODE1\]](#ODE1){reference-type="ref" reference="ODE1"}). **Assumption 1**. *We assume that $u\in C([0,T);X)\cap C^1((0,T);X)$ and that $F:X\longrightarrow X$ is sufficiently many times Fréchet differentiable in a strip along the exact solution with all derivatives uniformly bounded. Therefore, the Jacobian operator $J(u)=\frac{\partial F}{\partial u}(u)$ satisfies the Lipschitz condition $$\label{Ass1for} \Vert J(u)-J(v)\Vert_{_{\mathcal{L}(X)}}\leq C\Vert u-v\Vert,$$ in a neighborhood of the exact solution.* **Assumption 2**. *The linear operator $J$ is the generator of a strongly continuous semigroup on $X$, i.e., there exist constants $C$ and $\omega$ such that $$\label{Ass2for} \lVert e^{tJ}\rVert_{_{\mathcal{L}(X)}}\leq Ce^{\omega t},\;\;\;t\geq0,$$ holds uniformly in a neighborhood of the exact solution of ([\[ODE1\]](#ODE1){reference-type="ref" reference="ODE1"}).* For the construction of the methods, we first consider a uniform partition of the time interval $$\label{TimePartition} 0=t_{0}<t_{1}<\ldots<t_{N-1}<t_{N}=T,$$ we define $I_{n}=(t_{n},t_{n+1})$ and $h=t_{n+1}-t_{n},\;\forall n=0,\ldots,N-1$. Let $u_n$ be the numerical approximation to the solution of ([\[ODE1\]](#ODE1){reference-type="ref" reference="ODE1"}) at $t_n$, i.e. $u_n\approx u(t_n)$, we rewrite ([\[ODE1\]](#ODE1){reference-type="ref" reference="ODE1"}) as a semilinear system as follows $$\label{linODE1} \displaystyle{ \left\{ \begin{split} u'(t)&=J_nu(t)+g_n(u(t)),\\ u(0)&=u_{0},\\ \end{split} \right.}$$ where $$\label{JacRem} J_n=\frac{\partial F}{\partial u}(u_n),\;\;\;g_n(u)=F(u)-J_nu,$$ are the Jacobian of $F$ evaluated at $u_n$ and the nonlinear remainder, respectively. **Remark 1**. *If $F$ is already semilinear, i.e., $F(u)=Au+f(u)$, with $A$ being a linear operator, we have $J_n=A+\frac{\partial f}{\partial u}(u_n)$ and $g_n(u)=f(u)-\frac{\partial f}{\partial u}(u_n)u$. Note that the framework presented here also covers the case where $A$ is a differential operator, and $f$ is an algebraic function of $u$. Therefore, the methods developed in this article can be applied to a wide range of Partial Differential Equations (PDEs). In the numerical results in Section [7](#Sec:Results){reference-type="ref" reference="Sec:Results"}, we consider this case being $A$ a discretization of the Laplacian operator.* **Remark 2**. *If the system is non-autonomous, i.e., $u'(t)=F(t,u(t))$, we can easily reduce it to be autonomous with the following change of variables $$U=\begin{bmatrix}t\\u\end{bmatrix},\;\;\mathcal{F}(U)=\begin{bmatrix}1\\F(t,u)\end{bmatrix}.$$ The linearized system then becomes $U'=\mathcal{J}_nU+\mathcal{G}_n(U)$, where the Jacobian is defined as $$\mathcal{J}_n=\begin{bmatrix}0&0\\\frac{\partial F}{\partial t}(t_n,u_n)&\frac{\partial F}{\partial u}(t_n,u_n)\end{bmatrix},$$ and $\mathcal{G}_n(U)=\mathcal{F}(U)-\mathcal{J}_nU$ is the nonlinear remainder.* **Remark 3**. *Note that $$\label{vanishgn} \frac{\partial g_n}{\partial u}(u_n)=\frac{\partial F}{\partial u}(u_n)-J_n=0.$$ As we will see in Section [5](#Sec:Stiff){reference-type="ref" reference="Sec:Stiff"}, this property implies that all the approximation methods derived for ([\[linODE1\]](#linODE1){reference-type="ref" reference="linODE1"}) are at least second order by construction.* The integral representation of the solution of ([\[ODE1\]](#ODE1){reference-type="ref" reference="ODE1"}) at $t_{n+1}$, also known as the *variation-of-constants* formula, reads $$\label{VOC} u_{n+1}=e^{hJ_n}u_n+h\int_0^1e^{(1-\theta)hJ_n}g_n(u(t_n+h\theta))d\theta,$$ and different approximations of the nonlinear term in ([\[VOC\]](#VOC){reference-type="ref" reference="VOC"}) lead to different exponential time integration methods. All these methods are expressed in terms of the so-called *$\varphi$-functions* defined as $$\label{PhiFunctions} \displaystyle{\left\{ \begin{split} \varphi_{0}(z)&=e^{z},\\ \varphi_{p}(z)&=\int_{0}^{1}e^{(1-\theta)z}\frac{\theta^{p-1}}{(p-1)!}d\theta,\;\forall p\geq1, \end{split} \right.}$$ which satisfy the following recurrence relation $$\label{Recurrence} \varphi_{p+1}(z)=\frac{1}{z}\left(\varphi_{p}(z)-\frac{1}{p!}\right).$$ # Overview of the DPG time-marching scheme for linear problems {#Sec:Linear} In our previous works on the DPG method for linear transient problems [@munoz2021adpg; @munoz2021equivalence; @munoz2022error; @munoz2022combining], we proved that from an ultraweak variational formulation in time, we obtain a decoupled time-marching scheme where: (a) we obtain the variation-of-constants formula for the solution at each time instant $t_n$ (traces), and (b) we additionally compute the solution at each time interval $I_n$ (fields). We will employ this hybrid approximation in time to construct the internal stages of a Runge-Kutta-like method in the next section. To be self-contained, we present here a brief overview of the DPG-based time-marching scheme with the basic concepts we will employ in this article. ## DPG time-marching scheme for linear systems We consider the following linear system of ODEs $$\label{ODE2} \displaystyle{ \left\{ \begin{split} u'(t)+Au(t)&=f(t),\\ u(0)&=u_{0}.\\ \end{split} \right.}$$ In [@munoz2021equivalence], we considered an ultraweak variational formulation for problem ([\[ODE2\]](#ODE2){reference-type="ref" reference="ODE2"}). For that, we multiply the equation by suitable test functions, integrate by parts, and introduce additional variables for the traces in time (hybridization). We also proved that the optimal test functions to consider are the ones satisfying the adjoint equation (exponential-related functions). Finally, for each time interval $I_n$, we obtain an approximation $\{\tilde{u}_n(t),u_{n+1}\}$ like in Figure [\[SolutionPG\]](#SolutionPG){reference-type="ref" reference="SolutionPG"}, where $\tilde{u}_n(t)=\displaystyle{\sum_{j=0}^p\tilde{u}^j_n\left(\frac{t-t_n}{h}\right)^j}$ is a polynomial of order $p$ and $u_{n+1}$ is the point-value (trace) approximation at $t_{n+1}$. Finally, we obtained the following decoupled time-marching scheme to compute $u_{n+1}$ and the coefficients of $\tilde{u}_n(t)$ $$\label{DPGpSys} \displaystyle{\left\{ \begin{split} \sum_{j=0}^{p}\frac{h}{r+j+1}\tilde{u}_n^j&=\tilde{v}_{n}^r(A,t_{n}) u_{n}+\int_{I_n}\tilde{v}^r_{n}(A,t)f(t)dt,\;\forall r=0,\ldots,p,\\ u_{n+1}&=v_n(A,t_{n})u_{n}+\int_{I_{n}}v_n(A,t)f(t)dt,\\ \end{split} \right.}$$ where $\{v_n,\tilde{v}_n^r,\;r=0,\ldots,p\}$ are the optimal test function. Here, functions $\tilde{v}_{n}^r(z,t)$ and $v_n(z,t)=e^{z(t-t_{n+1})}$, satisfy the following recurrence formula $$\label{OptTest} \begin{split} \tilde{v}_{n}^r(z,t)&=\frac{1}{z}\left(\left(\frac{t-t_{n}}{h}\right)^{r}+\frac{r}{h}\tilde{v}^{r-1}_{n}(z,t)-v_n(z,t)\right),\;\forall r=0,\ldots,p. \end{split}$$ In ([\[DPGpSys\]](#DPGpSys){reference-type="ref" reference="DPGpSys"}), we obtain the variation-of-constants formula in the second equation and a scheme that delivers the $L^2-$projection of the analytical solution in the element interior in time. ## Post-processing of the trace variables Note that from ([\[OptTest\]](#OptTest){reference-type="ref" reference="OptTest"}) with $r=0$, we have $$v_n(z,t)=1-z\tilde{v}_{n}^0(z,t),$$ and we can express system ([\[DPGpSys\]](#DPGpSys){reference-type="ref" reference="DPGpSys"}) as $$\label{DPGpSysPost} \displaystyle{\left\{ \begin{split} \sum_{j=0}^{p}\frac{h}{r+j+1}\tilde{u}_n^j&=\tilde{v}_{n}^r(A,t_{n})u_{n}+\int_{I_n}\tilde{v}^r_{n}(A,t)f(t)dt,\;\forall r=0,\ldots,p,\\ u_{n+1}&=u_{n}-\sum_{j=0}^p\frac{h}{j+1}A\tilde{u}_n^j+\int_{I_{n}}f(t)dt.\\ \end{split} \right.}$$ Therefore, we can first compute the field variables and then we can post-process the trace variable without an extra action of any exponential-reated function [^1]. The resulting scheme ([\[DPGpSysPost\]](#DPGpSysPost){reference-type="ref" reference="DPGpSysPost"}) delivers an approximation of the solution that is equivalent to classical exponential integrators in the trace variables plus an approximation in the interiors. ## Relation between the optimal test functions and the $\varphi$-functions In order to implement scheme ([\[DPGpSysPost\]](#DPGpSysPost){reference-type="ref" reference="DPGpSysPost"}), we showed in [@munoz2021equivalence] the relation between the optimal test functions and the $\varphi-$functions. Given the optimal test functions over the master element (0,1), $\{v,\tilde{v}^r,\;r=0,\ldots,p\}$, we have $$\label{ToMaster} \begin{split} v_n(z,t_{n}+\theta h)&=v(zh,\theta),\\ \tilde{v}^{r}_{n}(z,t_{n}+\theta h)&=h\tilde{v}^r(zh,\theta),\;\forall r=0,\ldots,p, \end{split}$$ and also $$\label{RelationOptPhi} \begin{split} \tilde{v}^{r}(z,0)&=\sum_{j=0}^{r}\frac{r!}{j!}(-1)^{r-j}\varphi_{r-j+1}(-z),\\ \int_{0}^{1}\tilde{v}^{r}(z,t)\frac{t^{q}}{q!}dt&=\sum_{j=0}^{r}\frac{r!}{j!}(-1)^{r-j}\varphi_{r-j+q+2}(-z). \end{split}$$ We can now easily integrate ([\[DPGpSysPost\]](#DPGpSysPost){reference-type="ref" reference="DPGpSysPost"}) over the master element and then employ ([\[ToMaster\]](#ToMaster){reference-type="ref" reference="ToMaster"}) and ([\[RelationOptPhi\]](#RelationOptPhi){reference-type="ref" reference="RelationOptPhi"}) to express the DPG time-marching scheme in terms of $\varphi$-functions. In general, if we select $s$ integration points to approximate the source term in ([\[DPGpSysPost\]](#DPGpSysPost){reference-type="ref" reference="DPGpSysPost"}), we set $p=s-1$ to obtain optimal convergence rates. # Multistage DPG time-marching scheme for nonlinear problems {#Sec:Methods} In this section, we introduce several multistage Runge-Kutta-like methods (up to order $4$) for approximating the semilinear problem ([\[linODE1\]](#linODE1){reference-type="ref" reference="linODE1"}) employing the lowest-order ($p=0$) DPG construction for the internal stages. ## Hybrid exponential Euler method The simplest choice is to approximate the nonlinear remainder in ([\[linODE1\]](#linODE1){reference-type="ref" reference="linODE1"}) and ([\[VOC\]](#VOC){reference-type="ref" reference="VOC"}) by its value at $u_n$, which is known from the previous time step, i.e., $$g_n(u(t))\approx g_n(u_n),\;\forall t\in I_n,$$ and we are within the framework of linear problems introduced in Section [3](#Sec:Linear){reference-type="ref" reference="Sec:Linear"}. Therefore, we can employ the lowest order DPG method ($p=0$) and we obtain from ([\[DPGpSysPost\]](#DPGpSysPost){reference-type="ref" reference="DPGpSysPost"}) and properties ([\[Recurrence\]](#Recurrence){reference-type="ref" reference="Recurrence"}) and ([\[RelationOptPhi\]](#RelationOptPhi){reference-type="ref" reference="RelationOptPhi"}) the following scheme $$\label{ExpEuler} \displaystyle{\left\{ \begin{split} \tilde{u}^0_{n}&=u_{n}+h\varphi_2(hJ_n)(g_n(u_{n})+J_nu_{n}),\\ u_{n+1}&=u_{n}+hJ_n\tilde{u}^0_n+hg_n(u_{n}). \end{split} \right.}$$ Method ([\[ExpEuler\]](#ExpEuler){reference-type="ref" reference="ExpEuler"}) delivers the solution for the trace variable $u_{n+1}$ that coincides with the classical exponential Euler method and, additionally, a constant approximation of the solution $\tilde{u}_n^0$ in the interior of the time interval (see Figure [\[SolExpEuler\]](#SolExpEuler){reference-type="ref" reference="SolExpEuler"}). Here, we only perform a single action of a $\varphi-$function at each time interval. Therefore, with this DPG method, we compute a hybrid approximation in the whole time interval with the same computational cost as for the classical exponential Euler method. We will show in the next section that method ([\[ExpEuler\]](#ExpEuler){reference-type="ref" reference="ExpEuler"}) is second order accurate due to property ([\[vanishgn\]](#vanishgn){reference-type="ref" reference="vanishgn"}). In the following subsections, we employ approximation ([\[ExpEuler\]](#ExpEuler){reference-type="ref" reference="ExpEuler"}) for the internal stages to construct higher-order multistage methods. ## Two-stage DPG method To construct a two-stage method, we employ the constant approximation we introduced in the first equation of ([\[ExpEuler\]](#ExpEuler){reference-type="ref" reference="ExpEuler"}) (denoted here by $u_{n_2}$). Then, we approximate the solution at the next time step employing a linear combination of the nonlinear remainder evaluated at both $u_n$ and $u_{n_2}$ as follows $$\label{generalDPG2} \displaystyle{\left\{ \begin{split} u_{n+1}&=e^{hJ_n}u_n+hb_1(hJ_n)g_n(u_{n})+hb_2(hJ_n)g_n(u_{n_2}),\\ u_{n_2}&=u_n+h\varphi_2(hJ_n)(g_n(u_{n})+J_nu_n).\\ \end{split} \right.}$$ In the next section, we will derive the so-called stiff order conditions for coefficient functions $b_1$ and $b_2$ to obtain a third-order method. Similar to the classical exponential Runge-Kutta methods, the coefficients $b_1$ and $b_2$ will be expressed in terms of the $\varphi-$functions. ## Three-stage DPG method For the three-stage method, we employ the whole hybrid approximation ([\[ExpEuler\]](#ExpEuler){reference-type="ref" reference="ExpEuler"}) as internal stages (denoted by $u_{n_{2}}$ and $u_{n_{3}}$). In this case, however, it is not possible to build a fourth-order method employing solely a linear combination of evaluations of $g_n$. We need to introduce a "correction factor" denoted by $C_{n_2}$ below that will be given in terms of the Fréchet derivative of $g_n$ in the next section $$\label{generalDPG3} \displaystyle{\left\{ \begin{split} u_{n+1}&=e^{hJ_n}u_n+hb_1(hJ_n)g_n(u_{n})+hb_2(hJ_n)(g_n(u_{n_2})+C_{n_2})+hb_3(hJ_n)g_n(u_{n_3}),\\ u_{n_2}&=u_n+h\varphi_2(hJ_n)(g_n(u_{n})+J_nu_n),\\ u_{n_3}&=u_n+hJ_nu_{n_2}+hg_n(u_n).\\ \end{split} \right.}$$ # Stiff order conditions and local truncation error {#Sec:Stiff} In order to derive the stiff order conditions for the two and three-stage DPG methods, we employ a Taylor expansion in the variations-of-constant formula ([\[VOC\]](#VOC){reference-type="ref" reference="VOC"}) and compare it against the Taylor expansions for expressions ([\[generalDPG2\]](#generalDPG2){reference-type="ref" reference="generalDPG2"}) and ([\[generalDPG3\]](#generalDPG3){reference-type="ref" reference="generalDPG3"}). ## Taylor expansion of the analytical solution Let $\hat{u}_n=u(t_n)$ be the exact solution of $u$ at $t_n$, the linearization of ([\[ODE1\]](#ODE1){reference-type="ref" reference="ODE1"}) at $\hat{u}_n$ now reads $$u'(t)=\hat{J}_nu(t)+\hat{g}_n(u(t))$$ where $$\hat{J}_n=\frac{\partial F}{\partial u}(\hat{u}_n),\;\;\hat{g}_n(u)=F(u)-\hat{J}_nu.$$ The Taylor expansion of $\hat{g}_n$ at $\hat{u}_n$ is $$\label{Taylorf} \hat{g}_n(u(t_n+h\theta))=\hat{g}_n(\hat{u}_n)+\frac{1}{2!}\hat{g}''_n(\hat{u}_n)(w_n,w_n)+\frac{1}{3!}\hat{g}'''_n(\hat{u}_n)(w_n,w_n,w_n)+\mathcal{R},$$ where $w_n:=u(t_n+h\theta)-\hat{u}_n$ and we used from ([\[vanishgn\]](#vanishgn){reference-type="ref" reference="vanishgn"}) that $\hat{g}'_n(\hat{u}_n)=0$. Equivalently, the Taylor expansion of $u$ at $t_n$ is $$\label{Tayloru} u(t_n+h\theta)=\hat{u}_n+\hat{u}'_nh\theta+\frac{1}{2!}\hat{u}''_n(h\theta)^2+\frac{1}{3!}\hat{u}'''_n(h\theta)^3+\mathcal{R}.$$ Substituting ([\[Tayloru\]](#Tayloru){reference-type="ref" reference="Tayloru"}) into ([\[Taylorf\]](#Taylorf){reference-type="ref" reference="Taylorf"}), we obtain $$\label{Taylorfu} \begin{split} \hat{g}_n(u(t_n+h\theta))&=\hat{g}_n(\hat{u}_n)+\frac{(h\theta)^2}{2!}\hat{g}''_n(\hat{u}_n)(\hat{u}'_n,\hat{u}'_n)+\frac{(h\theta)^3}{2!}\hat{g}''_n(\hat{u}_n)(\hat{u}'_n,\hat{u}''_n)\\ &+\frac{(h\theta)^3}{3!}\hat{g}'''_n(\hat{u}_n)(\hat{u}'_n,\hat{u}'_n,\hat{u}'_n)+\mathcal{R}, \end{split}$$ then substituting ([\[Taylorfu\]](#Taylorfu){reference-type="ref" reference="Taylorfu"}) into ([\[VOC\]](#VOC){reference-type="ref" reference="VOC"}) $$\label{Taylorfu2} \begin{split} \hat{u}_{n+1}=e^{h\hat{J}_n}\hat{u}_n&+h\int_0^1e^{(1-\theta)h\hat{J}_n}\hat{g}_n(\hat{u}_n)d\theta+h^3\int_0^1e^{(1-\theta)h\hat{J}_n}g_n''(\hat{u}_n)(\hat{u}'_n,\hat{u}'_n)\frac{\theta^2}{2!} d\theta\\ &+h^4\int_0^1e^{(1-\theta)h\hat{J}_n}\left(3\hat{g}_n''(\hat{u}_n)(\hat{u}'_n,\hat{u}''_n)+\hat{g}'''_n(\hat{u}_n)(\hat{u}'_n,\hat{u}'_n,\hat{u}'_n)\right)\frac{\theta^3}{3!} d\theta+\mathcal{R}.\\ \end{split}$$ From the definition of the $\varphi$-functions ([\[PhiFunctions\]](#PhiFunctions){reference-type="ref" reference="PhiFunctions"}), we have $$\nonumber \begin{split} \hat{u}_{n+1}=e^{h\hat{J}_n}\hat{u}_n&+h\varphi_1(h\hat{J}_n)\hat{g}_n(\hat{u}_n)+h^3\varphi_3(h\hat{J}_n)\hat{g}''_n(\hat{u}_n)(\hat{u}'_n,\hat{u}'_n)\\ &+h^4\varphi_4(h\hat{J}_n)\left(3\hat{g}_n''(\hat{u}_n)(\hat{u}'_n,\hat{u}''_n)+\hat{g}'''_n(\hat{u}_n)(\hat{u}'_n,\hat{u}'_n,\hat{u}'_n)\right)+\mathcal{O}(h^5), \end{split}$$ and equivalently employing recurrence formula ([\[Recurrence\]](#Recurrence){reference-type="ref" reference="Recurrence"}) $$\label{TaylorExact} \begin{split} \hat{u}_{n+1}=\hat{u}_n&+h\varphi_1(h\hat{J}_n)(\hat{g}_n(\hat{u}_n)+\hat{J}_n\hat{u}_n)+h^3\varphi_3(h\hat{J}_n)\hat{g}''_n(\hat{u}_n)(\hat{u}'_n,\hat{u}'_n)\\ &+h^4\varphi_4(h\hat{J}_n)M(\hat{u}_n)+\mathcal{O}(h^5), \end{split}$$ where we denote by $$\label{Mun} M(\hat{u}_n)=3\hat{g}_n''(\hat{u}_n)(\hat{u}'_n,\hat{u}''_n)+\hat{g}'''_n(\hat{u}_n)(\hat{u}'_n,\hat{u}'_n,\hat{u}'_n).$$ ## Stiff order conditions Here, we give the order conditions of methods ([\[generalDPG2\]](#generalDPG2){reference-type="ref" reference="generalDPG2"}) and ([\[generalDPG3\]](#generalDPG3){reference-type="ref" reference="generalDPG3"}) to obtain a local truncation error of orders three and four, respectively. We define the local truncation error as $$\hat{e}_{n+1}=\bar{u}_{n+1}-\hat{u}_{n+1},$$ where $\hat{u}_{n+1}=u(t_{n+1})$ as in the previous section and $\bar{u}_{n+1}$ is the numerical solution obtained in ([\[ExpEuler\]](#ExpEuler){reference-type="ref" reference="ExpEuler"})-([\[generalDPG3\]](#generalDPG3){reference-type="ref" reference="generalDPG3"}) considering $\hat{u}_{n}$ as an initial condition. Similarly, we denote by $\bar{u}_{n_2}$ and $\bar{u}_{n_3}$ the internal stages computed in ([\[generalDPG2\]](#generalDPG2){reference-type="ref" reference="generalDPG2"}) and ([\[generalDPG3\]](#generalDPG3){reference-type="ref" reference="generalDPG3"}) with $\hat{u}_{n}$ on the right-hand side. **Propositon 1**. *The hybrid Euler method defined in ([\[ExpEuler\]](#ExpEuler){reference-type="ref" reference="ExpEuler"}) is second order, i.e., $$\hat{e}_{n+1}=\mathcal{O}(h^3).$$* **Proof.* From ([\[ExpEuler\]](#ExpEuler){reference-type="ref" reference="ExpEuler"}) we have that $$\bar{u}_{n+1}=\hat{u}_{n}+h^2\hat{J}_n\varphi_2(h\hat{J}_n)(\hat{g}_n(\hat{u}_{n})+\hat{J}_n\hat{u}_{n})+h(\hat{g}_n(\hat{u}_{n})+\hat{J}_n\hat{u}_{n}),$$ and from the recurrence formula ([\[Recurrence\]](#Recurrence){reference-type="ref" reference="Recurrence"}) $$\label{ExpEuler2} \bar{u}_{n+1}=\hat{u}_{n}+h\varphi_1(h\hat{J_n})(\hat{g}_n(\hat{u}_n)+\hat{J}_n\hat{u}_n),$$ which is the classical Exponential Euler method. Finally, subtracting ([\[ExpEuler2\]](#ExpEuler2){reference-type="ref" reference="ExpEuler2"}) from ([\[TaylorExact\]](#TaylorExact){reference-type="ref" reference="TaylorExact"}), we obtain the stated result. ◻* **Theorem 1**. *The two-stage DPG method defined in ([\[generalDPG2\]](#generalDPG2){reference-type="ref" reference="generalDPG2"}) with the coefficient functions satisfying $$\label{OrdCondDPG2} \displaystyle{\left\{ \begin{split} b_1(hJ_n)+b_2(hJ_n)&=\varphi_1(hJ_n),\\ \frac{1}{8}b_2(hJ_n)&=\varphi_3(hJ_n).\\ \end{split} \right.}$$ is third order, i.e., $$\hat{e}_{n+1}=\mathcal{O}(h^4).$$* **Proof.* First, we need to verify that method ([\[generalDPG2\]](#generalDPG2){reference-type="ref" reference="generalDPG2"}) is stable for equilibrium points, i.e., it needs to reproduce constant solutions [^2]. If we suppose that $u\equiv \hat{u}_n$ then $\hat{g}_n(\hat{u}_n)=-\hat{J}_n\hat{u}_n$. We easily see that the second equation in ([\[generalDPG2\]](#generalDPG2){reference-type="ref" reference="generalDPG2"}) reproduces constant solutions by construction as in this case $\bar{u}_{n_{2}}=\hat{u}_n$. We need to now enforce $\bar{u}_{n+1}=\hat{u}_n$, so from the first equation in ([\[generalDPG2\]](#generalDPG2){reference-type="ref" reference="generalDPG2"}), we obtain $$\nonumber \begin{split} \hat{u}_n&=e^{h\hat{J}_n}\hat{u}_n-h\hat{J}_n\hat{u}_n(b_1(h\hat{J}_n)+b_2(h\hat{J}_n))\Longrightarrow(h\hat{J}_n)^{-1}(e^{h\hat{J}_n}-I)=b_1(h\hat{J}_n)+b_2(h\hat{J}_n)\\ &\Longrightarrow b_1(h\hat{J}_n)+b_2(h\hat{J}_n)=\varphi_1(h\hat{J}_n), \end{split}$$ which is the first condition in ([\[OrdCondDPG2\]](#OrdCondDPG2){reference-type="ref" reference="OrdCondDPG2"}). We now rewrite method ([\[generalDPG2\]](#generalDPG2){reference-type="ref" reference="generalDPG2"}) employing ([\[Recurrence\]](#Recurrence){reference-type="ref" reference="Recurrence"}) as $$\label{RewDPG2} \displaystyle{\left\{ \begin{split} \bar{u}_{n+1}&=\hat{u}_n+h\varphi_1(h\hat{J}_n)(\hat{g}_n(\hat{u}_{n})+\hat{J}_n\hat{u}_n)+hb_2(h\hat{J}_n)(\hat{g}_n(\bar{u}_{n_2})-\hat{g}_n(\hat{u}_n)),\\ \bar{u}_{n_2}&=\hat{u}_n+h\varphi_2(h\hat{J}_n)(\hat{g}_n(\hat{u}_{n})+\hat{J}_n\hat{u}_n).\\ \end{split} \right.}$$ Next, we write the Taylor expansion of the numerical method to compare it with the analytical one ([\[TaylorExact\]](#TaylorExact){reference-type="ref" reference="TaylorExact"}). Employing the Taylor expansion of $\hat{g}_n$ at $\hat{u}_n$, we have that $$\hat{g}_n(\bar{u}_{n_2})-\hat{g}_n(\hat{u}_n)=\frac{1}{2!}\hat{g}''_n(\hat{u}_n)(\bar{u}_{n_2}-\hat{u}_n,\bar{u}_{n_2}-\hat{u}_n)+\mathcal{R},$$ therefore, in the first equation of ([\[RewDPG2\]](#RewDPG2){reference-type="ref" reference="RewDPG2"}) we obtain $$\label{TaylorNumDPG2} \begin{split} \bar{u}_{n+1}&=\hat{u}_n+h\varphi_1(hJ_n)(\hat{g}_n(\hat{u}_{n})+\hat{J}_n\hat{u}_n)+hb_2(h\hat{J}_n)\frac{1}{2!}\hat{g}''_n(\hat{u}_n)(\bar{u}_{n_2}-\hat{u}_n,\bar{u}_{n_2}-\hat{u}_n)+\mathcal{R}.\\ \end{split}$$ We need to express $\bar{u}_{n_2}-\hat{u}_n$ in terms of $\hat{u}'_n=\hat{g}_n(\hat{u}_{n})+\hat{J}_n\hat{u}_n$. For that, we employ that $$\label{ChainRule} \hat{J}_nu^{(k)}=u^{(k+1)}(t)-\frac{d^k}{dt^k}\hat{g}_n(u(t)),\;\forall k=0,1\ldots$$ therefore, from the chain rule we have $\hat{J}_n\hat{u}'_n=\hat{u}''_n-\hat{g}'_n(\hat{u}_n)\hat{u}'_n=\hat{u}''_n$. From the second equation in ([\[RewDPG2\]](#RewDPG2){reference-type="ref" reference="RewDPG2"}) and ([\[Recurrence\]](#Recurrence){reference-type="ref" reference="Recurrence"}), we have that $$\label{Un2DPG2} \begin{split} \bar{u}_{n_2}-\hat{u}_n&=h\varphi_2(h\hat{J}_n)\hat{u}'_n=h\left(\frac{1}{2!}+h\hat{J}_n\varphi_3(h\hat{J}_n)\right)\hat{u}'_n=\frac{h}{2!}\hat{u}'_n+h^2\varphi_3(h\hat{J}_n)\hat{u}''_n\\ &=\frac{h}{2!}\hat{u}'_n+h^2(\frac{1}{3!}+h\hat{J}_n\varphi_4(h\hat{J}_n))\hat{u}''_n=\frac{h}{2!}u'_n+\frac{h^2}{3!}u''_n+\mathcal{O}(h^3).\\ \end{split}$$ where we have employed that matrices $\hat{J}_n$ and $\varphi_i(h\hat{J}_n)$ commute for all $i$. Finally, substituting ([\[Un2DPG2\]](#Un2DPG2){reference-type="ref" reference="Un2DPG2"}) into ([\[RewDPG2\]](#RewDPG2){reference-type="ref" reference="RewDPG2"}) $$\bar{u}_{n+1}=\hat{u}_n+h\varphi_1(hJ_n)(\hat{g}_n(\hat{u}_{n})+\hat{J}_n\hat{u}_n)+h^3b_2(h\hat{J}_n)\frac{1}{2!2!2!}\hat{g}''_n(\hat{u}_n)(u'_n,u'_n)+\mathcal{O}(h^4),$$ and subtracting from ([\[TaylorExact\]](#TaylorExact){reference-type="ref" reference="TaylorExact"}), we obtain that the local truncation error is of order $h^4$ provided that $b_2(h\hat{J_n})=8\varphi_3(h\hat{J}_n)$. ◻* **Theorem 2**. *The three-stage DPG method defined in ([\[generalDPG3\]](#generalDPG3){reference-type="ref" reference="generalDPG3"}) with the coefficient functions satisfying $$\label{OrdCondDPG3} \displaystyle{\left\{ \begin{split} &b_1(hJ_n)+b_2(hJ_n)+b_3(hJ_n)=\varphi_1(hJ_n),\\ &\frac{1}{4}b_2(hJ_n)+b_3(hJ_n)=2\varphi_3(hJ_n),\\ &\frac{1}{8}b_2(hJ_n)+b_3(hJ_n)=6\varphi_4(hJ_n).\\ \end{split} \right.}$$ and the correction factor $C_{n_2}:=-\frac{1}{4}g'_n(u_{n_2})(u_{n_3}-2u_{n_2}+u_n)$, is fourth order, i.e., $$\hat{e}_{n+1}=\mathcal{O}(h^5).$$* **Proof.* If we suppose that the solution is constant, i.e., $u\equiv \hat{u}_n$ and $\hat{g}_n(\hat{u}_n)=-\hat{J}_n\hat{u}_n$, then it is easy to verify that $\bar{u}_{n_3}=\bar{u}_{n_2}=\hat{u}_n$ in ([\[generalDPG3\]](#generalDPG3){reference-type="ref" reference="generalDPG3"}) by construction. Moreover, in this particular case, $\bar{C}_{n_2}=-\frac{1}{4}\hat{g}'_n(\bar{u}_{n_2})(\bar{u}_{n_3}-2\bar{u}_{n_2}+\hat{u}_n)=0$ and, in order to reproduce constant solutions in ([\[generalDPG3\]](#generalDPG3){reference-type="ref" reference="generalDPG3"}), we need to enforce that $b_1(h\hat{J}_n)+b_2(h\hat{J}_n)+b_3(h\hat{J}_n)=\varphi_1(h\hat{J}_n)$, which is the first condition in ([\[OrdCondDPG3\]](#OrdCondDPG3){reference-type="ref" reference="OrdCondDPG3"}). We now rewrite method ([\[generalDPG2\]](#generalDPG2){reference-type="ref" reference="generalDPG2"}) employing ([\[Recurrence\]](#Recurrence){reference-type="ref" reference="Recurrence"}) as $$\label{RewDPG3} \displaystyle{\left\{ \begin{split} \bar{u}_{n+1}&=\hat{u}_n+h\varphi_1(h\hat{J}_n)(\hat{g}_n(\hat{u}_{n})+\hat{J}_n\hat{u}_n)+hb_2(h\hat{J}_n)(\hat{g}_n(\bar{u}_{n_2})-\hat{g}_n(\hat{u}_n)+\bar{C}_{n_2})\\ &+hb_3(h\hat{J}_n)(\hat{g}_n(\bar{u}_{n_3})-\hat{g}_n(\hat{u}_n)),\\ \bar{u}_{n_2}&=\hat{u}_n+h\varphi_2(h\hat{J}_n)(\hat{g}_n(\hat{u}_{n})+\hat{J}_n\hat{u}_n),\\ \bar{u}_{n_3}&=\hat{u}_n+h\hat{J}_n\bar{u}_{n_2}+h\hat{g}_n(\hat{u}_n).\\ \end{split} \right.}$$ We again employ the Taylor expansion of $\hat{g}_n$ and $\hat{u}_n$ as follows $$\hat{g}_n(\bar{u}_{n_i})-\hat{g}_n(\hat{u}_n)=\frac{1}{2!}\hat{g}''_n(\hat{u}_n)(\bar{u}_{n_i}-\hat{u}_n,\bar{u}_{n_i}-\hat{u}_n)+\frac{1}{3!}\hat{g}'''_n(\hat{u}_n)(\bar{u}_{n_i}-\hat{u}_n,\bar{u}_{n_i}-\hat{u}_n,\bar{u}_{n_i}-\hat{u}_n)+\mathcal{R},\;\ i=2,3,$$ so the first equation in ([\[RewDPG3\]](#RewDPG3){reference-type="ref" reference="RewDPG3"}) now reads $$\label{TaylorNumDPG3} \begin{split} \bar{u}_{n+1}&=\hat{u}_n+h\varphi_1(h\hat{J}_n)(\hat{g}_n(\hat{u}_{n})+\hat{J}_n\hat{u}_n)\\ &+hb_2(h\hat{J}_n)\left(\frac{1}{2!}\hat{g}''_n(\hat{u}_n)(\bar{u}_{n_2}-\hat{u}_n,\bar{u}_{n_2}-\hat{u}_n)+\frac{1}{3!}\hat{g}'''_n(\hat{u}_n)(\bar{u}_{n_2}-\hat{u}_n,\bar{u}_{n_2}-\hat{u}_n,\bar{u}_{n_2}-\hat{u}_n)+\bar{C}_{n_2}\right)\\ &+hb_3(h\hat{J}_n)\left(\frac{1}{2!}\hat{g}''_n(\hat{u}_n)(\bar{u}_{n_3}-\hat{u}_n,\bar{u}_{n_3}-\hat{u}_n)+\frac{1}{3!}\hat{g}'''_n(\hat{u}_n)(\bar{u}_{n_3}-\hat{u}_n,\bar{u}_{n_3}-\hat{u}_n,\bar{u}_{n_3}-\hat{u}_n)\right)+\mathcal{R}. \end{split}$$* *We employ the following identities derived from ([\[ChainRule\]](#ChainRule){reference-type="ref" reference="ChainRule"}) by applying the chain rule $$\nonumber \displaystyle{\left\{ \begin{split} \hat{J}_n\hat{u}'_n&=\hat{u}''_n-\hat{g}'_n(\hat{u}_n)\hat{u}'_n=\hat{u}''_n,\\ \hat{J}_n\hat{u}''_n&=\hat{u}'''_n-\hat{g}''_n(\hat{u}_n)(\hat{u}'_n,\hat{u}'_n)-\hat{g}'_n(\hat{u}_n)\hat{u}''_n=\hat{u}'''_n-\hat{g}''_n(\hat{u}_n)(\hat{u}'_n,\hat{u}'_n),\\ \end{split} \right.}$$ and also that $\hat{u}'_n=\hat{g}_n(\hat{u}_{n})+\hat{J}_n\hat{u}_n$. From the second equation in ([\[RewDPG3\]](#RewDPG3){reference-type="ref" reference="RewDPG3"}) and ([\[Recurrence\]](#Recurrence){reference-type="ref" reference="Recurrence"}), we obtain $$\label{Un2DPG3} \begin{split} \bar{u}_{n_2}-\hat{u}_n&=h\varphi_2(hJ_n)u'_n=h\left(\frac{1}{2!}+h\hat{J}_n\varphi_3(h\hat{J}_n)\right)\hat{u}'_n=\frac{h}{2!}\hat{u}'_n+h^2\varphi_3(h\hat{J}_n)\hat{u}''_n\\ &=\frac{h}{2!}\hat{u}'_n+h^2\left(\frac{1}{3!}+h\hat{J}_n\varphi_4(h\hat{J}_n)\right)\hat{u}''_n\\ &=\frac{h}{2!}\hat{u}'_n+\frac{h^2}{3!}\hat{u}''_n+h^3\varphi_4(h\hat{J}_n)\left(\hat{u}'''_n-\hat{g}''_n(\hat{u}_n)(\hat{u}'_n,\hat{u}'_n)\right),\\ \end{split}$$ and from the third equation of ([\[RewDPG3\]](#RewDPG3){reference-type="ref" reference="RewDPG3"}) $$\label{Un3DPG3} \begin{split} \bar{u}_{n_3}-\hat{u}_n&=h\hat{J}_n\bar{u}_{n_2}+h\hat{g}_n(\hat{u}_n)=h\hat{J}_n\left(\hat{u}_n+h\varphi_2(h\hat{J}_n)\hat{u}'_n\right)+h\hat{g}_n(\hat{u}_n)\\ &=h\hat{u}'_n+h^2\hat{J}_n\varphi_2(h\hat{J}_n)\hat{u}'_n=h\hat{u}'_n+h^2\left(\frac{1}{2!}+h\hat{J}_n\varphi_3(h\hat{J}_n)\right)\hat{u}''_n\\ &=h\hat{u}'_n+\frac{h^2}{2!}\hat{u}''_n+h^3\varphi_3(h\hat{J}_n)(\hat{u}'''_n-\hat{g}''_n(\hat{u}_n)(\hat{u}'_n,\hat{u}'_n)).\\ \end{split}$$ Finally, substituting ([\[Un2DPG3\]](#Un2DPG3){reference-type="ref" reference="Un2DPG3"}) and ([\[Un3DPG3\]](#Un3DPG3){reference-type="ref" reference="Un3DPG3"}) in ([\[TaylorNumDPG3\]](#TaylorNumDPG3){reference-type="ref" reference="TaylorNumDPG3"}), we obtain $$\label{TaylorNumDPG3New} \begin{split} \bar{u}_{n+1}&=\hat{u}_n+h\varphi_1(h\hat{J}_n)(\hat{g}_n(\hat{u}_{n})+\hat{J}_n\hat{u}_n)\\ &+hb_2(h\hat{J}_n)\left(\frac{h^2}{2!2!2!}\hat{g}''_n(\hat{u}_n)(\hat{u}'_n,\hat{u}'_n)+\frac{h^3}{2!3!}\hat{g}''_n(\hat{u}_n)(\hat{u}''_n,\hat{u}'_n)+\frac{h^3}{3!2!2!2!}\hat{g}'''_n(\hat{u}_n)(\hat{u}'_n,\hat{u}'_n,\hat{u}'_n)+\bar{C}_{n_2}\right)\\ &+hb_3(h\hat{J}_n)\left(\frac{h^2}{2!}\hat{g}''_n(\hat{u}_n)(\hat{u}'_n,\hat{u}'_n)+\frac{h^3}{2!}\hat{g}''_n(\hat{u}_n)(\hat{u}''_n,\hat{u}'_n)+\frac{h^3}{3!}\hat{g}'''_n(\hat{u}_n)(\hat{u}'_n,\hat{u}'_n,\hat{u}'_n)\right)+\mathcal{O}(h^5).\\ \end{split}$$* *We employ $\bar{C}_{n_2}$ to correct the coefficients in ([\[TaylorNumDPG3New\]](#TaylorNumDPG3New){reference-type="ref" reference="TaylorNumDPG3New"}) and factor out the term $M(\hat{u}_n)$ defined in ([\[Mun\]](#Mun){reference-type="ref" reference="Mun"}). First, note that from ([\[Un2DPG3\]](#Un2DPG3){reference-type="ref" reference="Un2DPG3"}) and ([\[Un3DPG3\]](#Un3DPG3){reference-type="ref" reference="Un3DPG3"}), we have that $$\label{OrderVarespilon} \bar{u}_{n_3}-2\bar{u}_{n_2}+\hat{u}_n=\frac{h^2}{3!}\hat{u}''_{n}+\mathcal{O}(h^3),\;\;\;\bar{u}_{n_2}-\hat{u}_n=\frac{h}{2!}\hat{u}'_n+\frac{h^2}{3!}\hat{u}''_n+\mathcal{O}(h^3),$$ and employing Taylor expansions $$\begin{split} \bar{C}_{n_2}&=-\frac{1}{4}\hat{g}'_n(\bar{u}_{n_2})(\bar{u}_{n_3}-2\bar{u}_{n_2}+\hat{u}_n)=-\frac{1}{4}\frac{h^2}{3!}\hat{g}'_n(\bar{u}_{n_2})\hat{u}''_n+\mathcal{O}(h^3)\\ &=-\frac{1}{4}\frac{h^2}{3!}\hat{g}'_n(\hat{u}_n)\hat{u}''_n-\frac{1}{4}\frac{h^2}{3!}\hat{g}''(\hat{u}_n)(\bar{u}_{n_2}-\hat{u}_n,\hat{u}''_n)-\frac{1}{4}\frac{h^2}{3!}\hat{g}'''(\hat{u}_n)(\bar{u}_{n_2}-\hat{u}_n,\bar{u}_{n_2}-\hat{u}_n,\hat{u}''_n)+\mathcal{O}(h^3)\\ &=-\frac{1}{4}\frac{h^3}{2!3!}g''(\hat{u}_n)(\hat{u}'_n,\hat{u}''_n)+\mathcal{O}(h^4). \end{split}$$ Therefore, we can express ([\[TaylorNumDPG3New\]](#TaylorNumDPG3New){reference-type="ref" reference="TaylorNumDPG3New"}) in terms of $M(\hat{u}_n)$ and subtracting it from ([\[TaylorExact\]](#TaylorExact){reference-type="ref" reference="TaylorExact"}), we obtain that the local truncation error is of order $h^5$ provided that the coefficients $b_2(h\hat{J}_n)$ and $b_3(h\hat{J}_n)$ satisfy ([\[OrdCondDPG3\]](#OrdCondDPG3){reference-type="ref" reference="OrdCondDPG3"}). ◻* ## Final methods. In Table [1](#FinalCoeff){reference-type="ref" reference="FinalCoeff"}, we summarize the coefficients for the updates in methods ([\[generalDPG2\]](#generalDPG2){reference-type="ref" reference="generalDPG2"}) and ([\[generalDPG3\]](#generalDPG3){reference-type="ref" reference="generalDPG3"}). width=1 method $b_1(hJ_n)$ $b_2(hJ_n)$ $b_3(hJ_n)$ ----------------- ------------------------------------------------------- --------------------------------------- -------------------------------------- two-stage DPG $\varphi_1(hJ_n)-8\varphi_3(hJ_n)$ $8\varphi_3(hJ_n)$ $-$ three-stage DPG $\varphi_1(hJ_n)-14\varphi_3(hJ_n)+36\varphi_4(hJ_n)$ $16\varphi_3(hJ_n)-48\varphi_4(hJ_n)$ $12\varphi_4(hJ_n)-2\varphi_3(hJ_n)$ : Coefficients of the two-stage ([\[generalDPG2\]](#generalDPG2){reference-type="ref" reference="generalDPG2"}) and three-stage ([\[generalDPG3\]](#generalDPG3){reference-type="ref" reference="generalDPG3"}) DPG methods. Note that the second and third-order methods ([\[ExpEuler\]](#ExpEuler){reference-type="ref" reference="ExpEuler"}) and ([\[generalDPG2\]](#generalDPG2){reference-type="ref" reference="generalDPG2"}) have the same cost as classical exponential Rosenbrock methods of one and two stages, respectively. However, the fourth-order method ([\[generalDPG3\]](#generalDPG3){reference-type="ref" reference="generalDPG3"}) requires one less exponential step than the classical three-stage methods, being therefore cheaper. Finally, the extension to variable time-step size of the methods presented here is straightforward; we have considered a uniform partition for simplicity. Moreover, methods ([\[generalDPG2\]](#generalDPG2){reference-type="ref" reference="generalDPG2"}) and ([\[generalDPG3\]](#generalDPG3){reference-type="ref" reference="generalDPG3"}) are nested because the internal stage that requires an exponential step is the same in both methods so they can be employed to adapt the time step size. # Convergence {#Sec:Convergence} In this section, we prove the convergence of the methods described in Section [4](#Sec:Methods){reference-type="ref" reference="Sec:Methods"}. We focus on methods ([\[generalDPG2\]](#generalDPG2){reference-type="ref" reference="generalDPG2"}) and ([\[generalDPG3\]](#generalDPG3){reference-type="ref" reference="generalDPG3"}) as method ([\[ExpEuler\]](#ExpEuler){reference-type="ref" reference="ExpEuler"}) is equivalent to the standard exponential Euler method, and its convergence proof is stated in [@hochbruck2009exponential; @luan2014exponential]. We will follow the strategy presented in [@luan2014exponential], and for that, we define the following errors, $$\begin{split} e_{n+1}=u_{n+1}-\hat{u}_{n+1},&\;\;\bar{e}_{n+1}=u_{n+1}-\bar{u}_{n+1},\\ \bar{E}_{n_2}=u_{n_2}-\bar{u}_{n_2},&\;\;\bar{E}_{n_3}=u_{n_3}-\bar{u}_{n_3}. \end{split}$$ We need to bound the total error $e_{n+1}$. Note that $e_{n+1}=\bar{e}_{n+1}+\hat{e}_{n+1}$, where $\hat{e}_{n+1}$ is the local error truncation error we estimated in Section [5](#Sec:Stiff){reference-type="ref" reference="Sec:Stiff"} (See [9](#App:Summarize){reference-type="ref" reference="App:Summarize"} for a summary of notation.) ## Preliminary error bounds and stability assumptions In order to prove the final convergence, we need the preliminary error bounds that are presented in the following lemma. **Lemma 1**. *Under Asummptions [Assumption 1](#Ass1){reference-type="ref" reference="Ass1"} and [Assumption 2](#Ass2){reference-type="ref" reference="Ass2"}, there exists a constant $C\geq0$ such that the following estimates hold* *$$\label{pre1} \left\lVert g'_n(\hat{u}_n)\right\rVert_{_{\mathcal{L}(X)}}=\Vert J(\hat{u}_n)-J_n\Vert_{_{\mathcal{L}(X)}} \leq C\Vert e_n\Vert ,$$ $$\label{pre2} \Vert g_n(u_n)-g_n(\hat{u}_n)\Vert \leq C\Vert e_n\Vert ^2,$$ $$\label{pre3} \Vert \varphi_j(hJ_n)-\varphi_j(h\hat{J}_n)\Vert_{_{\mathcal{L}(X)}} \leq Ch\Vert e_n\Vert ,$$ $$\label{pre4} \Vert b_j(hJ_n)-b_j(h\hat{J}_n)\Vert_{_{\mathcal{L}(X)}} \leq Ch\Vert e_n\Vert ,$$* *for all $i$ and $j$.* **Proof.* The first estimate ([\[pre1\]](#pre1){reference-type="ref" reference="pre1"}) is a direct consequence of the linearization ([\[JacRem\]](#JacRem){reference-type="ref" reference="JacRem"}) and the Lipschitz condition given in Assumption [Assumption 1](#Ass1){reference-type="ref" reference="Ass1"}. Estimate ([\[pre2\]](#pre2){reference-type="ref" reference="pre2"}) is obtained from ([\[pre1\]](#pre1){reference-type="ref" reference="pre1"}) the following Taylor expansion $$g_n(u_n)-g_n(\hat{u}_n)=g'_n(\hat{u}_n)e_n+\mathcal{O}(\Vert e_n\Vert ^2).$$ Finally, ([\[pre3\]](#pre3){reference-type="ref" reference="pre3"}) and ([\[pre4\]](#pre4){reference-type="ref" reference="pre4"}) are obtained employing the definition of the $\varphi-$functions, the fact that the coefficients $b_j(hJ_n)$ are linear combinations of $\varphi-$functions, and inequalities ([\[Ass1for\]](#Ass1for){reference-type="ref" reference="Ass1for"}) and ([\[Ass2for\]](#Ass2for){reference-type="ref" reference="Ass2for"}) (see Lemmas 4.2 and 4.3 in [@luan2014exponential]). ◻* Finally, as $J_n$ varies from time steps, we will also employ the following stability assumption from [@luan2014exponential]. **Assumption 3**. *We assume that the following stability condition holds $$\left\lVert\prod_{k=0}^{n-\nu} e^{hJ_{n-k}} \right\rVert_{\mathcal{L}(X)}\leq C_S,\;\;\; 0\leq t_\nu \leq t_n\leq T,$$ where $C_S$ is a constant that is uniform in $\nu$ and $n$.* **Remark 4**. *Both in the construction of the methods and the convergence proofs we present in this section, we consider a uniform time step for simplicity. However, the extension to a variable time-step size is straightforward under the usual mild restrictions on the step size sequence (see Section 4 in [@hochbruck2009exponential] for details).* ## Convergence of the two-stage DPG method Subtracting ([\[RewDPG2\]](#RewDPG2){reference-type="ref" reference="RewDPG2"}) from ([\[generalDPG2\]](#generalDPG2){reference-type="ref" reference="generalDPG2"}) and employing the identity $$F(u)=g_n(u)+J_n(u)=\hat{g}_n(u)+\hat{J}_n(u),$$ we have that $$\label{errorDPG2} \begin{split} \bar{e}_{n+1}&=e^{hJ_n}e_n+h\varphi_1(hJ_n)(g_n(u_{n})-g_n(\hat{u}_n))+h(\varphi_1(hJ_n)-\varphi_1(h\hat{J}_n))F(\hat{u}_n)\\ &+hb_2(hJ_n)(D_{n_2}-\hat{D}_{n_2})+h(b_2(hJ_n)-b_2(h\hat{J}_n))\hat{D}_{n_2}, \end{split}$$ where we denote by $$D_{n_2}:=g_n(u_{n_2})-g_n(u_n),\;\;\hat{D}_{n_2}:=\hat{g}_n(\bar{u}_{n_2})-\hat{g}_n(\hat{u}_n).$$ Therefore, form ([\[errorDPG2\]](#errorDPG2){reference-type="ref" reference="errorDPG2"}) we obtain $$\label{Recursion} e_{n+1}=e^{hJ_n}e_n+hq_n+hQ_{n}+\hat{e}_{n+1},$$ where $$q_n=\varphi_1(hJ_n)(g_n(u_{n})-g_n(\hat{u}_n))+(\varphi_1(hJ_n)-\varphi_1(h\hat{J}_n))F(\hat{u}_n),$$ $$Q_{n}=b_2(hJ_n)(D_{n_2}-\hat{D}_{n_2})+(b_2(hJ_n)-b_2(h\hat{J}_n))\hat{D}_{n_2}.$$ **Propositon 2**. *Under Assumptions ([Assumption 1](#Ass1){reference-type="ref" reference="Ass1"}) and ([Assumption 2](#Ass2){reference-type="ref" reference="Ass2"}) we have that* *$$\label{Err1DPG2} \Vert q_n\Vert \leq Ch\Vert e_n\Vert +C\Vert e_n\Vert ^2,$$ $$\label{Err2DPG2} \Vert Q_{n}\Vert \leq Ch\Vert e_n\Vert +C\Vert e_n\Vert ^2+C(h+\Vert e_n\Vert +\Vert \bar{E}_{n_2}\Vert )\Vert \bar{E}_{n_2}\Vert ,$$ $$\label{Err3DPG2} \Vert \bar{E}_{n_2}\Vert \leq C\Vert e_n\Vert +Ch||e_n||^2.$$* **Proof.* Estimate ([\[Err1DPG2\]](#Err1DPG2){reference-type="ref" reference="Err1DPG2"}) is a direct consequence of ([\[pre2\]](#pre2){reference-type="ref" reference="pre2"}) and ([\[pre3\]](#pre3){reference-type="ref" reference="pre3"}). To obtain ([\[Err1DPG2\]](#Err1DPG2){reference-type="ref" reference="Err1DPG2"}) we need to bound $D_{n_2}-\hat{D}_{n_2}$. For that, we rewrite $$D_{n_2}-\hat{D}_{n_2}=(g_n(u_{n_2})-g_n(\bar{u}_{n_2}))+(g_n(\bar{u}_{n_2})-\hat{g}_n(\bar{u}_{n_2}))+(\hat{g}_n(\hat{u}_{n})-g_n(\hat{u}_{n}))+(g_n(\hat{u}_{n})-g_n(u_{n})),$$ and we bound each term separately. We employ the Taylor expansion $$g_n(u_{n_2})-g_n(\bar{u}_{n_2})=g'_n(\bar{u}_{n_2})\bar{E}_{n_2}+\mathcal{O}(\Vert \bar{E}_{n_2}\Vert ^2)=(J(\bar{u}_{n_2})-J_n)\bar{E}_{n_2}+\mathcal{O}(\Vert \bar{E}_{n_2}\Vert ^2),$$ and also the identity $$g_n(u)-\hat{g}_n(u)=F(u)-J_nu-F(u)+\hat{J}_nu=(\hat{J}_n-J_n)u,$$ so we have $$g_n(\bar{u}_{n_2})-\hat{g}_n(\bar{u}_{n_2})=(\hat{J}_n-J_n)\bar{u}_{n_2},\;\;\hat{g}_n(\hat{u}_{n})-g_n(\hat{u}_{n})=(J_n-\hat{J}_n)\hat{u}_{n},$$ and therefore, $$\nonumber D_{n_2}-\hat{D}_{n_2}=(J(\bar{u}_{n_2})-J_n)\bar{E}_{n_2}+(\hat{J}_n-J_n)\bar{u}_{n_2}+(J_n-\hat{J}_n)\hat{u}_{n}+g_n(\hat{u}_n)-g_n(u_n)+\mathcal{O}(\Vert \bar{E}_{n_2}\Vert ^2).$$ From the Lipschitz condition ([\[Ass1for\]](#Ass1for){reference-type="ref" reference="Ass1for"}) and ([\[pre3\]](#pre3){reference-type="ref" reference="pre3"}) we get $$\Vert D_{n_2}-\hat{D}_{n_2}\Vert\leq C\Vert\bar{u}_{n_2}-u_n\Vert\Vert\bar{E}_{n_2}\Vert+C\Vert e_n\Vert\Vert\bar{u}_{n_2}-\hat{u}_n\Vert+C\Vert e_n\Vert^2+C\Vert\bar{E}_{n_2}\Vert^2,$$ from ([\[Un2DPG2\]](#Un2DPG2){reference-type="ref" reference="Un2DPG2"}) we know that $\bar{u}_{n_2}-\hat{u}_n=\mathcal{O}(h)$, therefore $$\Vert \bar{u}_{n_2}-\hat{u}_n+\hat{u}_n-u_n\Vert\leq Ch+C\Vert e_n\Vert,$$ and employing ([\[pre4\]](#pre4){reference-type="ref" reference="pre4"}), we obtain estimate ([\[Err2DPG2\]](#Err2DPG2){reference-type="ref" reference="Err2DPG2"}). For the last estimate, we have that $$\bar{E}_{n_2}=e_n+h\varphi_2(hJ_n)(F(u_n)-F(\hat{u}_n))+h(\varphi_2(hJ_n)-\varphi_2(h\hat{J}_n))F(\hat{u}_n).$$ Employing the identity $$F(u_n)-F(\hat{u}_n)=J_ne_n+g_n(u_n)-g_n(\hat{u}_n),$$ and recurrence formula ([\[Recurrence\]](#Recurrence){reference-type="ref" reference="Recurrence"}), we obtain $$\bar{E}_{n_2}=\varphi_1(hJ_n)e_n+h\varphi_2(hJ_n)(g_n(u_n)-g_n(\hat{u}_n))+h(\varphi_2(hJ_n)-\varphi_2(h\hat{J}_n))F(\hat{u}_n).$$ Finally, from Lemma [Lemma 1](#LemmaPre){reference-type="ref" reference="LemmaPre"} we obtain ([\[Err3DPG2\]](#Err3DPG2){reference-type="ref" reference="Err3DPG2"}). ◻* We provide the final convergence result for the two-stage DPG method in the following theorem. **Theorem 3**. *Under Assumptions [Assumption 1](#Ass1){reference-type="ref" reference="Ass1"}, [Assumption 2](#Ass2){reference-type="ref" reference="Ass2"}, and [Assumption 3](#Ass3){reference-type="ref" reference="Ass3"}, we conclude that the two-stage DPG method ([\[generalDPG2\]](#generalDPG2){reference-type="ref" reference="generalDPG2"}) with coefficients satisfying ([\[OrdCondDPG2\]](#OrdCondDPG2){reference-type="ref" reference="OrdCondDPG2"}), is third-order, i.e., the numerical solution satisfies the error bound $$\label{FinalErrorDPG2} ||e_n||\leq Ch^{4}$$ uniformly in $t_n$ and with constant $C$ independent of the time-step size.* **Proof.* Solving recursion ([\[Recursion\]](#Recursion){reference-type="ref" reference="Recursion"}) with $e_0=0$, we obtain $$e_n=\sum_{j=0}^{n-1}h\prod_{k=1}^{n-j-1}e^{hJ_{n-k}}(q_j+Q_j+h^{-1}\hat{e}_{j+1}).$$ Employing Proposition ([Propositon 2](#ErrsDPG2){reference-type="ref" reference="ErrsDPG2"}) and Theorem ([Theorem 1](#LocalTrunErrDPG2){reference-type="ref" reference="LocalTrunErrDPG2"}) we have the bound $$||q_j||+||Q_j||+h^{-1}||\hat{e}_{j+1}||\leq C(h||e_j||+||e_j||^2+h^{3}),$$ and employing Assumption [Assumption 3](#Ass3){reference-type="ref" reference="Ass3"} we have $$||e_n||\leq C\sum_{j=0}^{n-1}h(h||e_j||+||e_j||^2+h^{3}).$$ Finally, the application of a discrete Gronwall lemma (see [@hochbruck2010exponential], Lemma 2.15) yields to the final error bound ([\[FinalErrorDPG2\]](#FinalErrorDPG2){reference-type="ref" reference="FinalErrorDPG2"}). ◻* ## Convergence of the three-stage DPG method Subtracting ([\[RewDPG3\]](#RewDPG3){reference-type="ref" reference="RewDPG3"}) from ([\[generalDPG3\]](#generalDPG3){reference-type="ref" reference="generalDPG3"}), we have that $$\label{errorDPG3} \begin{split} \bar{e}_{n+1}&=e^{hJ_n}e_n+h\varphi_1(hJ_n)(g_n(u_{n})-g_n(\hat{u}_n))+h(\varphi_1(hJ_n)-\varphi_1(h\hat{J}_n))F(\hat{u}_n)\\ &+hb_2(hJ_n)(D_{n_2}+C_{n_2}-\hat{D}_{n_2}-\bar{C}_{n_2})+h(b_2(hJ_n)-b_2(h\hat{J}_n))(\hat{D}_{n_2}+\bar{C}_{n_2}),\\ &+hb_3(hJ_n)(D_{n_3}-\hat{D}_{n_3})+h(b_3(hJ_n)-b_3(h\hat{J}_n))\hat{D}_{n_3}, \end{split}$$ where $$D_{n_i}=g_n(u_{n_i})-g_n(u_n),\;\;\hat{D}_{n_i}=\hat{g}_n(\bar{u}_{n_i})-\hat{g}_n(\hat{u}_n),\;\;i=2,3.$$ Therefore, form ([\[errorDPG2\]](#errorDPG2){reference-type="ref" reference="errorDPG2"}) we have that $$e_{n+1}=e^{hJ_n}e_n+hq_n+hQ_{n}+\hat{e}_{n+1},$$ where $$q_n=\varphi_1(hJ_n)(g_n(u_{n})-g_n(\hat{u}_n))+(\varphi_1(hJ_n)-\varphi_1(h\hat{J}_n))F(\hat{u}_n),$$ $$\nonumber \begin{split} Q_{n}&=b_2(hJ_n)(D_{n_2}+C_{n_2}-\hat{D}_{n_2}-\bar{C}_{n_2})+(b_2(hJ_n)-b_2(h\hat{J}_n))(\hat{D}_{n_2}+\bar{C}_{n_2})\\ &+b_3(hJ_n)(D_{n_3}-\hat{D}_{n_3})+h(b_3(hJ_n)-b_3(h\hat{J}_n))\hat{D}_{n_3}. \end{split}$$ **Propositon 3**. *Under Assumptions ([Assumption 1](#Ass1){reference-type="ref" reference="Ass1"}) and ([Assumption 2](#Ass2){reference-type="ref" reference="Ass2"}) we have that* *$$\label{Err1DPG3} ||q_n||\leq Ch||e_n||+C||e_n||^2,$$ $$\label{Err2DPG3} ||Q_{n}||\leq Ch||e_n||+C||e_n||^2+C\sum_{j=2}^3(h+||e_n||+||\bar{E}_{n_j}||)||\bar{E}_{n_j}||,$$ $$\label{Err3DPG3} \Vert \bar{E}_{n_j}\Vert \leq C\Vert e_n\Vert +Ch||e_n||^2,\;\;j=2,3.$$* **Proof.* Estimate ([\[Err1DPG3\]](#Err1DPG3){reference-type="ref" reference="Err1DPG3"}) is the same as ([\[Err1DPG2\]](#Err1DPG2){reference-type="ref" reference="Err1DPG2"}). To obtain estimate ([\[Err2DPG3\]](#Err2DPG3){reference-type="ref" reference="Err2DPG3"}), we proceed as in ([\[Err2DPG2\]](#Err2DPG2){reference-type="ref" reference="Err2DPG2"}) to bound $D_{n_i}-\hat{D}_{n_i},\;i=2,3$, and additionally, we have to bound $C_{n_2}-\bar{C}_{n_2}$. For that, we rewrite $$C_{n_2}-\bar{C}_{n_2}=\frac{1}{4}g'_n(u_{n_2})(\varepsilon_n-\bar{\varepsilon}_n)+\frac{1}{4}(\hat{g}'_n(\bar{u}_{n_2})-g'_n(u_{n_2}))\bar{\varepsilon}_n,$$ where we have denoted $$\varepsilon_n:=u_{n_3}-2u_{n_2}+u_n,\;\;\bar{\varepsilon}_n:=\bar{u}_{n_3}-2\bar{u}_{n_2}+\bar{u}_n.$$ We employ that $\varepsilon_n-\bar{\varepsilon}_n=E_{n_3}-2E_{n_2}+e_n$ and from ([\[OrderVarespilon\]](#OrderVarespilon){reference-type="ref" reference="OrderVarespilon"}), we know that $\bar{\varepsilon}_n=\mathcal{O}(h^2)$. Now, from the following estimates $$||g'_n(u_{n_2})||_{_{\mathcal{L}(X)}}=||J(u_{n_2})-J_n||_{_{\mathcal{L}(X)}}\leq C||u_{n_2}-u_n||\leq Ch,$$ $$||\hat{g}'_n(\bar{u}_{n_2})-g'_n(u_{n_2})||_{_{\mathcal{L}(X)}}=||J(u_{n_2})-J_n-J(\bar{u}_{n_2})+\hat{J}_n||_{_{\mathcal{L}(X)}}\leq C||E_{n_2}||+C||e_n||,$$ we obtain that $$||C_{n_2}-\bar{C}_{n_2}||\leq Ch||e_n||+Ch^2||e_n||+Ch||E_{n_2}||+Ch^2||E_{n_2}||+Ch||E_{n_3}||.$$ This last estimate together with the bounds for $D_{n_i}-\hat{D}_{n_i}$, we obtain ([\[Err2DPG3\]](#Err2DPG3){reference-type="ref" reference="Err2DPG3"}). Finally, estimate ([\[Err3DPG3\]](#Err3DPG3){reference-type="ref" reference="Err3DPG3"}) for $j=2$ is the same as ([\[Err3DPG2\]](#Err3DPG2){reference-type="ref" reference="Err3DPG2"}). For $j=3$, we have $$\bar{E}_{n_3}=e^{hJ_n}e_n+h\varphi_1(hJ_n)(g_n(u_n)-g_n(\hat{u}_n))+h(\varphi_1(hJ_n)-\varphi_1(h\hat{J}_n))F(\hat{u}_n).$$ and from Lemma [Lemma 1](#LemmaPre){reference-type="ref" reference="LemmaPre"} we obtain ([\[Err3DPG3\]](#Err3DPG3){reference-type="ref" reference="Err3DPG3"}). ◻* **Theorem 4**. *Under Assumptions [Assumption 1](#Ass1){reference-type="ref" reference="Ass1"}, [Assumption 2](#Ass2){reference-type="ref" reference="Ass2"}, and [Assumption 3](#Ass3){reference-type="ref" reference="Ass3"}, we conclude that the three-stage DPG method ([\[generalDPG3\]](#generalDPG3){reference-type="ref" reference="generalDPG3"}) with coefficients satisfying ([\[OrdCondDPG3\]](#OrdCondDPG3){reference-type="ref" reference="OrdCondDPG3"}) is fourth-order, i.e., the numerical solution satisfies the error bound $$\label{FinalErrorDPG2} ||e_n||\leq Ch^{5},$$ uniformly in $t_n$ and with constant $C$ independent of the time-step size.* **Proof.* Employing the preliminary error bounds from Proposition [Propositon 3](#ErrsDPG3){reference-type="ref" reference="ErrsDPG3"}, the final convergence proof is analoge to the one presented in Theorem [Theorem 3](#TheoDPG2){reference-type="ref" reference="TheoDPG2"}. ◻* # Numerical experiment: Hochbruck-Ostermann equation {#Sec:Results} We consider the Hochbruch-Ostermann equation [@hochbruck2005explicit] in $\Omega=(0,1)^2$ $$\label{HochOster} u_t-\Delta u=\frac{1}{1+u^2}+f(x,y,t),$$ with homogeneous Dirichlet boundary conditions and final time $T=1$. We select the linear source $f$ and the initial condition $u_0$ in such a way that the exact solution is $u(x,y,t)=x(1-x)y(1-y)e^t$. We discretize ([\[HochOster\]](#HochOster){reference-type="ref" reference="HochOster"}) in space by standard second-order central finite differences with a fixed grid of $2^6$ points in each dimension. Figure [\[ConvergenceHochOster\]](#ConvergenceHochOster){reference-type="ref" reference="ConvergenceHochOster"} shows the convergence of the error in time by solving the resulting system of ODEs by the Hybrid exponential Euler method ([\[ExpEuler\]](#ExpEuler){reference-type="ref" reference="ExpEuler"}), the two-stage DPG method ([\[generalDPG2\]](#generalDPG2){reference-type="ref" reference="generalDPG2"}), and the three-stage DPG method ([\[generalDPG3\]](#generalDPG3){reference-type="ref" reference="generalDPG3"}). Note that this problem is semilinear and non-autonomous, so we implemented the DPG methods taking into account Remarks [Remark 1](#rmk:1){reference-type="ref" reference="rmk:1"} and [Remark 2](#rmk:2){reference-type="ref" reference="rmk:2"}. We measure the infinity norm at the final time and observe that the convergence rates are the ones predicted in Section [6](#Sec:Convergence){reference-type="ref" reference="Sec:Convergence"}. # Conclusions and future work {#Sec:Conclusions} In this article, we derive three multistage methods up to order four employing the DPG time-marching scheme we developed in [@munoz2021equivalence; @munoz2021adpg]. We consider general nonlinear systems of ODEs and perform a linearization of the problem at each time step as in exponential Rosebrock methods. We first introduce the hybridization of the classical exponential Euler method, consisting of point values at each time step and piecewise constants inside the time interval. We observe that whereas we need an action of a $\varphi-$function to compute the interior, we only need a matrix-vector product to compute the trace variable. We then employ this hybrid Euler method for the internal stages to construct the third and fourth-order methods and derive the stiff-order conditions to determine the values of the coefficient functions. We observe that for the fourth-order method, we need to additionally introduce a correction term that depends upon the Jacobian. Finally, we provide a convergence proof under classical stability conditions on the exponential of the Jacobian matrix that changes at each time step. For the convergence proof, we ensure that the correction factor is bounded by the error at the internal stages. We also test the convergence numerically on the so-called 2D+time Hochbruck-Ostermann equation. In the literature, the highest-order exponential Rosenbrock method is of order five [@luan2014exponential]. We are interested in seeing if we can derive multistage methods employing the DPG construction for the internal stages to go beyond this threshold. Therefore, we will study in future work the use of higher-order polynomials, for example, piecewise linears or quadratics as we introduced in [@munoz2021equivalence], to derive multistage DPG methods of order higher than four. # Summary of notation {#App:Summarize} The following tables summarize the meaning of each symbol and errors employed throughout the convergence proofs in Sections [5](#Sec:Stiff){reference-type="ref" reference="Sec:Stiff"} and [6](#Sec:Convergence){reference-type="ref" reference="Sec:Convergence"}. $u_n$ numerical approximation of the solution at $t_n$ ---------------------------------- ------------------------------------------------------------------------------------------------ -- -- $u_{n_2}$, $u_{n_3}$ internal stages computed with $u_n$ $u_{n+1}$ update of the numerical scheme computed with $u_n$, $u_{n_2}$ and $u_{n_3}$ $J_n$ Jacobian evaluated in $u_n$ $g_n$ nonlinear reminder after linearization at $u_n$ $C_{n_2}$ correction factor computed with $g'_n$, $u_n$, $u_{n_2}$ and $u_{n_3}$ $\hat{u}_n$, $\hat{u}_{n+1}$ analytical solution at $t_n$ and $t_{n+1}$ $\bar{u}_{n_2}$, $\bar{u}_{n_3}$ internal stages computed with $\hat{u}_n$ $\bar{u}_{n+1}$ update of the numerical scheme computed with $\hat{u}_n$, $\bar{u}_{n_2}$ and $\bar{u}_{n_3}$ $\hat{J}_n$ Jacobian evaluated in $\hat{u}_n$ $\hat{g}_n$ nonlinear reminder after linearization at $\hat{u}_n$ $\bar{C}_{n_2}$ correction factor computed with $\hat{g}'_n$, $\hat{u}_n$, $\bar{u}_{n_2}$ and $\bar{u}_{n_3}$ : Summary of symbols. $e_n=u_n-\hat{u}_n$ ------------------------------------------------------------------ -- -- -- $\bar{e}_{n+1}=u_{n+1}-\bar{u}_{n+1}$ $\hat{e}_{n+1}=\bar{u}_{n+1}-\hat{u}_{n+1}$ $e_{n+1}=\bar{e}_{n+1}+\hat{e}_{n+1}=u_{n+1}-\hat{u}_{n+1}$ $E_{n_2}=u_{n_2}-\bar{u}_{n_2}$, $E_{n_3}=u_{n_3}-\bar{u}_{n_3}$ : Summary of errors. # Acknowledgements {#acknowledgements .unnumbered} Judit Muñoz-Matute has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie individual fellowship grant agreement No. 101017984 (GEODPG). Leszek Demkowicz was partially supported with NSF grant No. 1819101. # References {#references .unnumbered} [^1]: For example, for $p=0$ and approximating the source with a constant value $f(t)\approx f(t_n)$, we compute the field variable with one exponential step: $\tilde{u}_n^0=\varphi_1(-hA)u_n+h\varphi_2(-hA)f(t_n)$. Then, we post-process for the trace variable with the equation $u_{n+1}=u_n-hA\tilde{u}_n^0+hf(t_n)$, without an extra action of a $\varphi-$function. [^2]: *From ([\[TaylorExact\]](#TaylorExact){reference-type="ref" reference="TaylorExact"}), it is easy to see that the numerical scheme should reproduce constant solutions to be at least first-order accurate. In classical exponential Runge-Kutta methods, enforcing the scheme to reproduce constants leads to order conditions for the coefficient functions in the internal stages. In the DPG methods we present here, this is satisfied by construction. We include this step here as a sanity check.*

arxiv_math

--- abstract: | A homologically trivial part of any Turaev -- Viro invariant odd order $r$ is a Turaev -- Viro type invariant order $\frac{r + 1}{2}$. In this paper we find an explicit formulas for this Turaev -- Viro type invariant, corresponding to the invariant order $r = 7$. Our formulas express $6j$-symbols and colour weights in terms of $\gamma$, where $\gamma$ is a root of the polynomial $\mathcal{T}(x) = x^3 - 2x^2 - x + 1$. address: Chelyabinsk State University, Chelyabinsk, Russia; N.N. Krasovsky Institute of Mathematics and Meckhanics, Ekaterinburg, Russia author: - Korablev Ph. G. title: Homologically trivial part of the Turaev -- Viro invariant order 7 --- # Introduction Turaev -- Viro invariants for 3-manifolds were proposed by V. Turaev and O. Viro in 1992 in the paper [@TuraevViro]. They constructed an infinite family of invariants, each of which is determined by the choice of its order $r\geqslant 2$. Let $TV_r$ denote the Turaev -- Viro invariant of order $r$. For each $r\geqslant 2$ define the set $C_r = \{0, 1, \ldots, r - 2\}$. It is convenient to think of this set $C_r$ as the set of colours used to colour 3-manifold spine cells when computing the values of the invariant $TV_r$. For the purposes of this article, the topological meaning of the elements of this set is not important. One way to define the invariant $TV_r$ is to fix the complex values $w_i$ for all $i\in C_r$, called colour weights, and so-called $6j$-symbols, denoted by $\left|\begin{array}{lll} i & j & k \\ l & m & n \end{array}\right|$, for all $i, j, k, l, m, n\in C_r$. The values of all $6j$-symbols and the colour weights that define the $TV_r$ invariant are not arbitrary. They are described by formulas which are given, for example, in the books [@Matveev section 8.1.4] and [@Turaev section XII.8.5], article [@TuraevViro section 7] and below in the second paragraph. Quantum integers and quantum factorials play a central role in these formulas. Let $q$ be a root of unity of degree $2r$ such that $q^2$ is a primitive root of unity of degree $r$. Let us denote $$[n]_r = \dfrac{q^n - q^{-n}}{q - q^{-1}}.$$ The value $[n]_r$ is called a quantum integer. This value is always real, but it depends on the choice of the root $q$. Obviously $[1]_r = 1$ and $[r]_r = 0$. The quantum factorial is the value $$[n]_r! = [n]_r\cdot [n - 1]_r\cdot \ldots\cdot [1]_r.$$ There are invariants of 3-manifolds given by other values of $6j$-symbols and colour weights. According to the terminology proposed in the book [@Matveev remark 8.1.20], such invariants are called Turaev -- Viro type invariants. To construct a Turaev -- Viro type invariant of order $r$, it suffices to choose values of the colour weights $w_i$ and $6j$-symbols such that they satisfy the equations $$\left|\begin{array}{lll} i & j & k \\ l & m & n \end{array}\right|\cdot \left|\begin{array}{lll} i & j & k \\ l' & m' & n' \end{array}\right| = \sum\limits_{z\in C_r}w_z\cdot \left|\begin{array}{lll} i & m & n \\ z & n' & m' \end{array}\right|\cdot \left|\begin{array}{lll} j & l & n \\ z & n' & m' \end{array}\right|\cdot \left|\begin{array}{lll} k & l & m \\ z & m' & l' \end{array}\right|$$ for all $i, j, k, l, m, n, l', m', n' \in C_r$. An example of a non-trivial Turaev -- Viro type invariant of order 3 is the $\varepsilon$-invariant (see [@TInvariant]). It is given by the following set of colour weights and $6j$-symbols (see [@Matveev section 8.1.2]): $w_0 = 1$, $w_1 = \varepsilon$ and --------------------------------------------------- ---------------------------------------------------------- --------------------------------------------------- $\left|\begin{array}{lll} $\left|\begin{array}{lll} $\left|\begin{array}{lll} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 0 1 & 1 & 1 0 & 1 & 1 \end{array}\right| = 1$, \end{array}\right| = \dfrac{1}{\sqrt{\varepsilon}}$, \end{array}\right| = \dfrac{1}{\varepsilon}$, $\left|\begin{array}{lll} $\left|\begin{array}{lll} 0 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 1 & 1 & 1 \end{array}\right| = \dfrac{1}{\varepsilon}$, \end{array}\right| = -\dfrac{1}{\varepsilon^2}$, --------------------------------------------------- ---------------------------------------------------------- --------------------------------------------------- where $\varepsilon$ is a root of the equation $x^2 - x - 1 = 0$. All other $6j$-symbols are zero. It is well known that each Turaev -- Viro invariant $TV_r$ of order $r$ is the sum of three invariants $TV_{r, 0}$, $TV_{r, 1}$ and $TV_{r, 2}$ (see [@ThreeParts]). The invariant $TV_{r, 0}$ is called the homologically trivial part of the Turaev -- Viro invariant. For odd $r$ it is a Turaev -- Viro type invariant of order $\frac{r + 1}{2}$. Let's denote it $TH_{\frac{r + 1}{2}}$. Let us denote the weights of the colours defining the invariant $TH_{\frac{r + 1}{2}}$ by $w_i'$, $i\in C_{\frac{r + 1}{2}} = \{0, 1, \ldots, \frac{r - 3}{2}\}$, and $6j$-symbols by $\left|\begin{array}{lll} i & j & k \\ l & m & n \end{array}\right|'$, $i, j, k, l, m, n\in C_{\frac{r + 1}{2}}$. There is a simple way to find the weights of the colours and $6j$-symbols that define the $TH_{\frac{r + 1}{2}}$ invariant, knowing the weights of the colours and $6j$-symbols that define the $TV_r$ invariant: $w_i' = w_{2i}$ for all $i\in C_{\frac{r + 1}{2}}$, $\left|\begin{array}{lll} i & j & k \\ l & m & n \end{array}\right|' = \left|\begin{array}{lll} 2i & 2j & 2k \\ 2l & 2m & 2n \end{array}\right|$ for all $i, j, k, l, m, n\in C_{\frac{r + 1}{2}}$. It can be shown that the $\varepsilon$-invariant coincides with the $TH_{3}$ invariant, i.e. it is the homologically trivial part of the Turaev -- Viro invariant of order 5 (see [@Matveev theorem 8.1.26] and [@TInvariant]). The purpose of this article is to find formulas that define the $TH_{4}$ invariant, i.e. the homologically trivial part of the $TV_7$ invariant, similar to formulas that define the $\varepsilon$-invariant. The main result of the article is formulated in the theorem [Theorem 5](#Theorem:GammaInvariant){reference-type="ref" reference="Theorem:GammaInvariant"}. In this theorem the values of the colour weights and $6j$-symbols that define the $TH_{4}$ invariant are expressed by the root $\gamma$ of the equation $x^3 - 2x^2 - x + 1 = 0$. This description is completely similar to the description of the $\varepsilon$-invariant by the root $\varepsilon$ of the equation $x^2 - x - 1 = 0$. Therefore, the $TH_{4}$ invariant is called the $\gamma$-invariant. The structure of the article is as follows. In the second section we write explicit formulas for colour weights and $6j$-symbols, which define the invariant $TV_{7, 0}$. These formulas are written in terms of quantum integers $[n]_7$. The third section is devoted to the simplest properties of the polynomial $\mathcal{T}(x) = x^3 - 2x^2 - x + 1$, which plays a central role in the construction of the $\gamma$-invariant. In the fourth section, the main theorem [Theorem 5](#Theorem:GammaInvariant){reference-type="ref" reference="Theorem:GammaInvariant"} is proved. In the fifth section we compute polynomials corresponding to the homologically trivial parts of the Turaev -- Viro invariants of odd orders $5\leqslant r\leqslant 21$. # Turaev -- Viro invariant order 7 As before, denote $C_r = \{0, \ldots, r - 2\}$. The books [@Matveev section 8.1.4], [@Turaev section XII.8.5] and the article [@TuraevViro section 7] contain explicit formulas for calculating the values of the weights $w_i$ and $6j$-symbols $\left|\begin{array}{lll} i & j & k \\ l & m & n \end{array}\right|$, $i, j, k, l, m, n \in C_r$, which define the Turaev-Viro invariant of order $r\geqslant 2$. Let's present these formulas. $w_i = (-1)^i [i + 1]_r$ for all $i\in C_r$. Let us say that the triplet $(x, y, z)\in C_r^3$ is admissible if and only if 1. $x + y\geqslant z$, $y + z\geqslant x$ and $z + x\geqslant y$; 2. $x + y + z$ is even; 3. $x + y + z \leqslant 2\cdot r - 4$. The value of the $6j$-symbol $\left|\begin{array}{lll} i & j & k \\ l & m & n \end{array}\right|$ is 0 if at least one of the triples $(i, j, k)$, $(k , l, m)$, $(m, n, i)$, $(j, l, n)$ is not admissible. The values of the remaining $6j$-symbols are calculated as follows $$\left|\begin{array}{lll} i & j & k \\ l & m & n \end{array}\right| = \sum\limits_{z = \alpha}^{\beta}\frac{(-1)^z\cdot [z + 1]_r!\cdot A(i, j, k, l, m, n)}{B\left(z, \left(\begin{array}{lll} i & j & k \\ l & m & n \end{array}\right)\right)\cdot C\left(z, \left(\begin{array}{lll} i & j & k \\ l & m & n \end{array}\right)\right)},$$ where $A(i, j, k, l, m, n) = \emph{\textbf{i}}^{i+j+k+l+m+n}\cdot \Delta(i, j, k)\cdot \Delta(i, m, n)\cdot \Delta(j, l, n)\cdot \Delta(k, l, m),$ $\emph{\textbf{i}}\in\mathbb{C}$ --- imaginary unit, $$B\left(z, \left(\begin{array}{lll} i & j & k \\ l & m & n \end{array}\right)\right) = [z - \widehat{i} - \widehat{j} - \widehat{k}]_r!\cdot [z - \widehat{i} - \widehat{m} - \widehat{n}]_r! \cdot [z - \widehat{j} - \widehat{l} - \widehat{n}]_r!\cdot [z - \widehat{k} - \widehat{l} - \widehat{m}]_r!,$$ $$C\left(z, \left(\begin{array}{lll} i & j & k \\ l & m & n \end{array}\right)\right) = [\widehat{i} + \widehat{j} + \widehat{l} + \widehat{m} - z]_r!\cdot [\widehat{i} + \widehat{k} + \widehat{l} + \widehat{n} - z]_r!\cdot [\widehat{j} + \widehat{k} + \widehat{m} + \widehat{n}] - z]_r!,$$ $$\Delta(i, j, k) = \sqrt{\frac{[\widehat{i} + \widehat{j} - \widehat{k}]_r!\cdot [\widehat{j} + \widehat{k} - \widehat{i}]_r!\cdot [\widehat{k} + \widehat{j} - \widehat{i}]_r!}{[\widehat{i} + \widehat{j} + \widehat{k} + 1]_r!}},$$ $\widehat{x} = \frac{x}{2}$ for all $x\in C_r$, $\alpha = \max(\widehat{i} + \widehat{j} + \widehat{k}, \widehat{i} + \widehat{m} + \widehat{n}, \widehat{j} + \widehat{l} + \widehat{n}, \widehat{k} + \widehat{l} + \widehat{m})$, $\beta = \min(\widehat{i} + \widehat{j} + \widehat{l} + \widehat{m}, \widehat{i} + \widehat{k} + \widehat{l} + \widehat{n}, \widehat{j} + \widehat{k} + \widehat{m} + \widehat{n})$. Since in this article we are mainly talking about the Turaev -- Viro invariant of order 7, we will write $[n]$ instead of $[n]_7$ for simplicity. **Theorem 1**. *The homologically trivial part $TV_{7, 0}$ of the Turaev -- Viro invariant of order 7 is defined by the following values of the weights $w_i$, $i\in\{0, 2, 4\}$ and $6j$-symbols $\left|\begin{array}{lll} i & j & k \\ l & m & n \end{array}\right|$, $i, j, k, l, m, n\in\{0, 2, 4\}$: $w_0 = 1$, $w_2 = [3]$, $w_4 = [5]$ and* ------------------------------------------------------------------------------------------ --------------------------------------------------------------------------------------- *$\left|\begin{array}{lll} *$\left|\begin{array}{lll} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 2 & 2 & 2 \end{array}\right| = 1$* \end{array}\right| = -\dfrac{1}{\sqrt{[3]}}$* *$\left|\begin{array}{lll} *$\left|\begin{array}{lll} 0 & 0 & 0 \\ 0 & 2 & 2 \\ 4 & 4 & 4 0 & 2 & 2 \end{array}\right| = \dfrac{1}{\sqrt{[5]}}$* \end{array}\right| = \dfrac{1}{[3]}$* *$\left|\begin{array}{lll} *$\left|\begin{array}{lll} 0 & 2 & 2 \\ 0 & 2 & 2 \\ 2 & 2 & 2 2 & 4 & 4 \end{array}\right| = \dfrac{1}{[3]}$* \end{array}\right| = -\dfrac{1}{\sqrt{[3]\cdot [5]}}$* *$\left|\begin{array}{lll} *$\left|\begin{array}{lll} 0 & 2 & 2 \\ 0 & 2 & 2 \\ 4 & 2 & 2 4 & 4 & 4 \end{array}\right| = \dfrac{1}{[3]}$* \end{array}\right| = -\dfrac{1}{\sqrt{[3]\cdot [5]}}$* *$\left|\begin{array}{lll} *$\left|\begin{array}{lll} 0 & 4 & 4 \\ 0 & 4 & 4 \\ 0 & 4 & 4 2 & 4 & 4 \end{array}\right| = \dfrac{1}{[5]}$* \end{array}\right| = \dfrac{1}{[5]}$* *$\left|\begin{array}{lll} *$\left|\begin{array}{lll} 2 & 2 & 2 \\ 2 & 2 & 2 \\ 2 & 2 & 2 2 & 2 & 4 \end{array}\right| = \dfrac{[5] - 1}{[2]\cdot [3]\cdot [4]}$* \end{array}\right| = -\dfrac{[2]}{[3]\cdot [4]}$* *$\left|\begin{array}{lll} *$\left|\begin{array}{lll} 2 & 2 & 2 \\ 2 & 2 & 2 \\ 2 & 4 & 4 4 & 4 & 4 \end{array}\right| = -\dfrac{1}{[4]}\cdot\sqrt{\dfrac{[2]\cdot [6]}{[3]\cdot [5]}}$* \end{array}\right| = \dfrac{[3]}{[4]\cdot\sqrt{[2]\cdot [3]\cdot [5]\cdot [6]}}$* *$\left|\begin{array}{lll} *$\left|\begin{array}{lll} 2 & 2 & 4 \\ 2 & 2 & 4 \\ 2 & 2 & 4 2 & 4 & 4 \end{array}\right| = \dfrac{[2]}{[3]\cdot [4]\cdot [5]}$* \end{array}\right| = \dfrac{[2]}{[4]\cdot [5]}$* *$\left|\begin{array}{lll} 2 & 4 & 4 \\ 2 & 4 & 4 \end{array}\right| = -\dfrac{1}{[4]\cdot [5]\cdot [6]}$* ------------------------------------------------------------------------------------------ --------------------------------------------------------------------------------------- *All other $6j$-symbols and weights $w_i$, $i\in\{1, 3, 5\}$ are zero.* *Proof.* The statement of the theorem is obtained by a careful calculation using the formulas given above. Since we are only interested in the homologically trivial part of the invariant, all weights $w_i$ for odd $i\in C_r$ are zero. The values of all $6j$-symbols $\left|\begin{array}{lll} i & j & k \\ l & m & n \end{array}\right|$ are zero if at least one of the numbers $i, j, k, l, m, n\in C_7$ is odd. ◻ # Polynomial $\mathcal{T}(x) = x^3 - 2x^2 - x + 1$ A simple analysis shows that the polynomial $\mathcal{T}(x)$ has three real roots: one is greater than 2, let's denote it $\gamma_1$, the second belongs to the interval $(0, 1)$, let's denote it $\gamma_2$, and the third root is negative, let's denote it $\gamma_3$ (see figure [\[Figure:TRoots\]](#Figure:TRoots){reference-type="ref" reference="Figure:TRoots"}). **Theorem 2**. *Let $\gamma$ be a root of the polynomial $\mathcal{T}(x)$. Then $1 - \frac{1}{\gamma}$ is also a root of the polynomial $\mathcal{T}(x)$.* *Proof.* First, note that $\gamma\neq 0$. Furthermore, since $$\gamma^3 - 2\gamma^2 - \gamma + 1 = 0,$$ dividing both sides of this equality by $\gamma^3$, we get $$1 - \frac{2}{\gamma} - \frac{1}{\gamma^2} + \frac{1}{\gamma^3} = 0.$$ Calculate $\mathcal{T}(1 - \frac{1}{\gamma})$. $$\begin{gathered} \mathcal{T}\left(1 - \frac{1}{\gamma}\right) = \left(1 - \frac{1}{\gamma}\right)^3 - 2\left(1 - \frac{1}{\gamma}\right)^2 - \left(1 - \frac{1}{\gamma}\right) + 1 = \\ = -1 + \frac{2}{\gamma} + \frac{1}{\gamma^2} - \frac{1}{\gamma^3} = -\left(1 - \frac{2}{\gamma} - \frac{1}{\gamma^2} + \frac{1}{\gamma^3}\right) = 0. \end{gathered}$$ ◻ Consider the function $\tau(x)\colon\mathbb{R}\to\mathbb{R}$, which acts as follows: $$\tau(x) = 1 - \frac{1}{x}.$$ It's easy to see that $\gamma_2 = \tau(\gamma_1)$, $\gamma_3 = \tau(\gamma_2)$ and $\gamma_1 = \tau(\gamma_3)$. The last equality follows from the fact that the function $\tau$ has order 3, i.e. $\tau\circ\tau\circ\tau = id$. # Turaev -- Viro type invariant $TH_4$ **Theorem 3**. *$\mathcal{T}([3]) = 0$.* *Proof.* It follows from the definition that $[3] = q^2 + 1 + q^{-2}$, where $q\neq \pm 1$ is a root of unity of degree 14. Let's calculate $\mathcal{T}(q^2 + 1 + q^{-2})$ explicitly: $$\begin{gathered} \mathcal{T}(q^2 + 1 + q^{-2}) = (q^2 + 1 + q^{-2})^3 - 2(q^2 + 1 + q^{-2})^2 - (q^2 + 1 + q^{-2}) + 1 = \\ = q^6 + q^4 + q^2 + 1 + q^{-2} + q^{- 4} + q^{-6}. \end{gathered}$$ Note that if $\omega = q^2$, then $\omega$ is a primitive root of unity of degree 7. Then $$\mathcal{T}([3]) = \omega^6 + \omega^5 + \omega^4 + \omega^3 + \omega^2 + \omega + 1 = 0.$$ ◻ When calculating the value $[3]$, we can take $q$ to be any root of unity of degree 14 (except $\pm 1$). Figure [\[Figure:Roots\]](#Figure:Roots){reference-type="ref" reference="Figure:Roots"} shows different roots of the polynomial $\mathcal{T}(x)$ corresponding to different choices of $q$. As before, $\gamma_1$ is the largest root of the polynomial, $\gamma_2$ is the middle, and $\gamma_3$ is the smallest. **Theorem 4**. *The following identities hold:* 1. *$[5]\cdot [3] = [3] + [5]$;* 2. *$[2]\cdot [4] = [3]\cdot [5]$;* 3. *$[2]\cdot [3] = [4]\cdot [5]$;* 4. *$[2]\cdot [6] = [5]$;* 5. *$[4]\cdot [6] = [3]$;* 6. *$[3]^2 = [4]^2$.* *Proof.* Each relation is proved by substituting $[2] = q + q^{-1}$, $[3] = q^2 + 1 + q^{-2}$, $[4] = q^3 + q + q^{-1} + q^{-3}$, $[5] = q^4 + q^2 + 1 + q^{-2} + q^{-4}$, $[6] = q^5 + q^3 + q + q^{-1} + q^{-3} + q^{-5}$ into the required equations, opening brackets, and bringing similar. The only additional relation used is that if $\omega = q^2$, then $$1 + \omega + \omega^2 + \omega^3 + \omega^4 + \omega^5 + \omega^6 = 0.$$ ◻ From the first statement of the theorem [Theorem 4](#Theorem:Identities){reference-type="ref" reference="Theorem:Identities"} it follows that if $[3] = \gamma$ is an arbitrary root of the polynomial $\mathcal{T}(x)$, then $$[5] = \dfrac{\gamma}{\gamma - 1}.$$ The homologically trivial part of any Turaev -- Viro invariant of order $r$ for odd $r$ is a Turaev -- Viro type invariant of order $\frac{r + 1}{2}$. To distinguish $6j$-symbols of this Turaev -- Viro type invariant from the original invariant of order $r$, we denote them by $\left|\begin{array}{lll} i & j & k \\ l & m & n \end{array}\right|'$ and weights $w_i'$. **Theorem 5**. *Let $\gamma$ be an arbitrary root of the polynomial $\mathcal{T}(x)$. Then the invariant $TH_{4}$ of the Turaev -- Viro type of order 4, which coincides with the homologically trivial part $TV_{7, 0}$ of the Turaev -- Viro invariant of order 7, is given by the following values of weights $w_i'\in\{0, 1, 2\}$ and $6j$-symbols $\left|\begin{array}{lll} i & j & k \\ l & m & n \end{array}\right|'$, $i, j, k, l, m, n\in\{0, 1, 2\}$: $w_0' = 1$, $w_1' = \gamma$, $w_2' = \frac{\gamma}{\gamma - 1}$ and* ---------------------------------------------------------------------- -------------------------------------------------------------------------- *$\left|\begin{array}{lll} *$\left|\begin{array}{lll} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 1 & 1 & 1 \end{array}\right|' = 1$* \end{array}\right|' = -\dfrac{1}{\sqrt{\gamma}}$* *$\left|\begin{array}{lll} *$\left|\begin{array}{lll} 0 & 0 & 0 \\ 0 & 1 & 1 \\ 2 & 2 & 2 0 & 1 & 1 \end{array}\right|' = \dfrac{\sqrt{\gamma - 1}}{\sqrt{\gamma}}$* \end{array}\right|' = \dfrac{1}{\gamma}$* *$\left|\begin{array}{lll} *$\left|\begin{array}{lll} 0 & 1 & 1 \\ 0 & 1 & 1 \\ 1 & 1 & 1 1 & 2 & 2 \end{array}\right|' = \dfrac{1}{\gamma}$* \end{array}\right|' = -\dfrac{\sqrt{\gamma - 1}}{\gamma}$* *$\left|\begin{array}{lll} *$\left|\begin{array}{lll} 0 & 1 & 1 \\ 0 & 1 & 1 \\ 2 & 1 & 1 2 & 2 & 2 \end{array}\right|' = \dfrac{1}{\gamma}$* \end{array}\right|' = -\dfrac{\sqrt{\gamma - 1}}{\gamma}$* *$\left|\begin{array}{lll} *$\left|\begin{array}{lll} 0 & 2 & 2 \\ 0 & 2 & 2 \\ 0 & 2 & 2 1 & 2 & 2 \end{array}\right|' = \dfrac{\gamma - 1}{\gamma}$* \end{array}\right|' = \dfrac{\gamma - 1}{\gamma}$* *$\left|\begin{array}{lll} *$\left|\begin{array}{lll} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 1 & 1 & 2 \end{array}\right|' = \dfrac{1}{\gamma^3}$* \end{array}\right|' = -\dfrac{1}{\gamma\cdot (\gamma - 1)}$* *$\left|\begin{array}{lll} *$\left|\begin{array}{lll} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 2 & 2 2 & 2 & 2 \end{array}\right|' = -\dfrac{1}{\gamma\cdot\sqrt{\gamma}}$* \end{array}\right|' = \dfrac{\gamma - 1}{\gamma\cdot\sqrt{\gamma}}$* *$\left|\begin{array}{lll} *$\left|\begin{array}{lll} 1 & 1 & 2 \\ 1 & 1 & 2 \\ 1 & 1 & 2 1 & 2 & 2 \end{array}\right|' = \dfrac{1}{\gamma^2}$* \end{array}\right|' = \dfrac{1}{\gamma}$* *$\left|\begin{array}{lll} 1 & 2 & 2 \\ 1 & 2 & 2 \end{array}\right|' = -\dfrac{\gamma - 1}{\gamma^2}$* ---------------------------------------------------------------------- -------------------------------------------------------------------------- *All other $6j$-symbols are zero.* *Proof.* Let the value of the root $q$ be chosen so that $[3] = \gamma$. Let us calculate the values of the required colour weights and $6j$-symbols, using the theorems [Theorem 1](#Theorem:Symbols){reference-type="ref" reference="Theorem:Symbols"} and [Theorem 4](#Theorem:Identities){reference-type="ref" reference="Theorem:Identities"}, and the fact that $[3] = \gamma$ and $[5] = \dfrac{\gamma}{\gamma - 1}$. First calculate the colour weights $w_i'$, $i\in\{0, 1, 2\}$. $w_0' = w_0 = 1$, $w_1' = w_2 = [3] = \gamma$ and $w_2' = w_4 = [5] = \dfrac{\gamma}{\gamma - 1}$. Next values of $6j$-symbols. $\left|\begin{array}{lll} 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right|' = \left|\begin{array}{lll} 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right| = 1$. $\left|\begin{array}{lll} 0 & 0 & 0 \\ 1 & 1 & 1 \end{array}\right|' = \left|\begin{array}{lll} 0 & 0 & 0 \\ 2 & 2 & 2 \end{array}\right| = -\dfrac{1}{\sqrt{[3]}} = -\dfrac{1}{\sqrt{\gamma}}$. $\left|\begin{array}{lll} 0 & 0 & 0 \\ 2 & 2 & 2 \end{array}\right|' = \left|\begin{array}{lll} 0 & 0 & 0 \\ 4 & 4 & 4 \end{array}\right| = \dfrac{1}{\sqrt{[5]}} = \dfrac{\sqrt{\gamma - 1}}{\sqrt{\gamma}}$. $\left|\begin{array}{lll} 0 & 1 & 1 \\ 0 & 1 & 1 \end{array}\right|' = \left|\begin{array}{lll} 0 & 2 & 2 \\ 0 & 2 & 2 \end{array}\right| = \dfrac{1}{[3]} = \dfrac{1}{\sqrt{\gamma}}$. $\left|\begin{array}{lll} 0 & 1 & 1 \\ 1 & 1 & 1 \end{array}\right|' = \left|\begin{array}{lll} 0 & 2 & 2 \\ 2 & 2 & 2 \end{array}\right| = \dfrac{1}{[3]} = \dfrac{1}{\sqrt{\gamma}}$. $\left|\begin{array}{lll} 0 & 1 & 1 \\ 1 & 2 & 2 \end{array}\right|' = \left|\begin{array}{lll} 0 & 2 & 2 \\ 2 & 4 & 4 \end{array}\right| = -\dfrac{1}{\sqrt{[3]\cdot [5]}} = -\dfrac{\sqrt{\gamma - 1}}{\gamma}$. $\left|\begin{array}{lll} 0 & 1 & 1 \\ 2 & 1 & 1 \end{array}\right|' = \left|\begin{array}{lll} 0 & 2 & 2 \\ 4 & 2 & 2 \end{array}\right| = \dfrac{1}{[3]} = \dfrac{1}{\sqrt{\gamma}}$. $\left|\begin{array}{lll} 0 & 1 & 1 \\ 2 & 2 & 2 \end{array}\right|' = \left|\begin{array}{lll} 0 & 2 & 2 \\ 4 & 4 & 4 \end{array}\right| = -\dfrac{1}{\sqrt{[3]\cdot [5]}} = -\dfrac{\sqrt{\gamma - 1}}{\gamma}$ . $\left|\begin{array}{lll} 0 & 2 & 2 \\ 0 & 2 & 2 \end{array}\right|' = \left|\begin{array}{lll} 0 & 4 & 4 \\ 0 & 4 & 4 \end{array}\right| = \dfrac{1}{[5]} = \dfrac{\gamma - 1}{\gamma}$. $\left|\begin{array}{lll} 0 & 2 & 2 \\ 1 & 2 & 2 \end{array}\right|' = \left|\begin{array}{lll} 0 & 4 & 4 \\ 2 & 4 & 4 \end{array}\right| = \dfrac{1}{[5]} = \dfrac{\gamma - 1}{\gamma}$. $\left|\begin{array}{lll} 1 & 1 & 1 \\ 1 & 1 & 1 \end{array}\right|' = \left|\begin{array}{lll} 2 & 2 & 2 \\ 2 & 2 & 2 \end{array}\right| = \dfrac{[5] - 1}{[2]\cdot [3]\cdot [4]}$. Use the second statement of the theorem [Theorem 4](#Theorem:Identities){reference-type="ref" reference="Theorem:Identities"} and replace in the expression $[2]\cdot [4]$ with $[3]\cdot [5]$. As a result we get the value $\dfrac{[5] - 1}{[3]^2\cdot [5]} = \dfrac{1}{\gamma^3}$. $\left|\begin{array}{lll} 1 & 1 & 1 \\ 1 & 1 & 2 \end{array}\right|' = \left|\begin{array}{lll} 2 & 2 & 2 \\ 2 & 2 & 4 \end{array}\right| = -\dfrac{[2]}{[3]\cdot [4]}$. Use the third statement of the theorem [Theorem 4](#Theorem:Identities){reference-type="ref" reference="Theorem:Identities"} and replace $\dfrac{[2]}{[4]}$ with $\dfrac{[5]}{[3]}$. We get the value $-\dfrac{[5]}{[3]^2} = -\dfrac{1}{\gamma\cdot (\gamma - 1)}$. $\left|\begin{array}{lll} 1 & 1 & 1 \\ 1 & 2 & 2 \end{array}\right|' = \left|\begin{array}{lll} 2 & 2 & 2 \\ 2 & 4 & 4 \end{array}\right| = -\dfrac{1}{ [4]}\cdot\sqrt{\dfrac{[2]\cdot [6]}{[3]\cdot [5]}}$. Use the fourth relation from the theorem [Theorem 4](#Theorem:Identities){reference-type="ref" reference="Theorem:Identities"} and replace $[2]\cdot [6]$ with $[5]$. We get the expression $-\dfrac{1}{[4]\cdot\sqrt{[3]}}$. From the sixth relation of the theorem [Theorem 4](#Theorem:Identities){reference-type="ref" reference="Theorem:Identities"} it follows that $[4] = \pm\gamma$, but for different choices of $q$ the value of $[4]$ can coincide with both $\gamma$ and $-\gamma$. As noted in [@Matveev remarks 8.1.17 and 8.1.18], the correctness and the value of the invariant does not depend on the choice of sign when extracting the square root. Therefore, for definiteness, we can choose $[4] = \gamma$, and then the value of the $6j$-symbol is equal to $-\dfrac{1}{\gamma\cdot\sqrt{\gamma}}$. $\left|\begin{array}{lll} 1 & 1 & 1 \\ 2 & 2 & 2 \end{array}\right|' = \left|\begin{array}{lll} 2 & 2 & 2 \\ 4 & 4 & 4 \end{array}\right| = \dfrac{[3]}{[4]\cdot\sqrt{[2]\cdot [3]\cdot [5]\cdot [6]}}$. Use the fourth relation from the theorem [Theorem 4](#Theorem:Identities){reference-type="ref" reference="Theorem:Identities"} and replace $[2]\cdot [6]$ with $[5]$ and, similar to the previous case, choose for definiteness $[4] = \gamma$. We get the value $\dfrac{\gamma - 1}{\gamma\cdot\sqrt{\gamma}}$. $\left|\begin{array}{lll} 1 & 1 & 2 \\ 1 & 1 & 2 \end{array}\right|' = \left|\begin{array}{lll} 2 & 2 & 4 \\ 2 & 2 & 4 \end{array}\right| = \dfrac{[2]}{[3]\cdot [4]\cdot [5]}$. Use the third statement of the theorem [Theorem 4](#Theorem:Identities){reference-type="ref" reference="Theorem:Identities"} and replace $\dfrac{[2]}{[4]}$ with $\dfrac{[5]}{[3]}$. We get the value $\dfrac{1}{[3]^2} = \dfrac{1}{\gamma^2}$. $\left|\begin{array}{lll} 1 & 1 & 2 \\ 1 & 2 & 2 \end{array}\right|' = \left|\begin{array}{lll} 2 & 2 & 4 \\ 2 & 4 & 4 \end{array}\right| = \dfrac{[2]}{[4]\cdot [5]}$. Use the third statement of the theorem [Theorem 4](#Theorem:Identities){reference-type="ref" reference="Theorem:Identities"} and replace $\dfrac{[2]}{[4]}$ with $\dfrac{[5]}{[3]}$. We get the value $\dfrac{1}{[3]} = \dfrac{1}{\gamma}$. $\left|\begin{array}{lll} 1 & 2 & 2 \\ 1 & 2 & 2 \end{array}\right|' = \left|\begin{array}{lll} 2 & 4 & 4 \\ 2 & 4 & 4 \end{array}\right| = -\dfrac{1}{[4]\cdot [5]\cdot [6]}$. Use the fifth statement of the theorem [Theorem 4](#Theorem:Identities){reference-type="ref" reference="Theorem:Identities"} and replace $[4]\cdot [6]$ with $[3]$. We get the value $-\dfrac{1}{[3]\cdot [5]} = -\dfrac{\gamma - 1}{\gamma}$. ◻ # Polynomials for homologically trivial parts of Turaev -- Viro invariants As already noted, the homologically trivial part of the Turaev -- Viro invariant of odd order $r$ is a Turaev -- Viro type invariant of order $\frac{r + 1}{2}$. For $r = 5$ the corresponding invariant of order 3 is the $\varepsilon$-invariant, and for $r = 7$ it is the $\gamma$-invariant described in the previous paragraph. Both invariants are expressed in terms of the roots of two special polynomials. For the $\varepsilon$-invariant this polynomial is equal to $x^2 - x - 1$ and for the $\gamma$-invariant it is equal to $x^3 - 2x^2 - x + 1$. It is natural to assume that any Turaev -- Viro type invariant $TH_{\frac{r + 1}{2}}$ can be expressed in terms of the roots of a suitable polynomial. This is consistent with the fact that the values of the Turaev -- Viro invariants are algebraic integers ([@AlgebraicNumbers]). Looking at the formulas defining the Turaev -- Viro invariants of order $r$, it is difficult to guess which polynomials they should be. The procedure for choosing them is based on the observation proved in Theorem [Theorem 3](#Theorem:Three){reference-type="ref" reference="Theorem:Three"}. To do this, we need to compute the values of $[3]_r$ for all possible $q \neq\pm 1$, which are the roots of unity of degree $2r$. Then we need to construct a polynomial whose roots are all the different computed values. The table [1](#Table:Polynomials){reference-type="ref" reference="Table:Polynomials"} lists the first few polynomials constructed using this rule. The first column of the table contains the order of the Turaev -- Viro type invariant, which coincides with the homologically trivial part of the Turaev -- Viro invariant $TV_{r, 0}$ of odd order $5\leqslant r\leqslant 21$. The second column gives the corresponding polynomial. Order Polynomial ------- --------------------------------------------------------------------------- $3$ $x^2 - x - 1$ $4$ $x^3 - 2x^2 - x + 1$ $5$ $x^4 - 3x^3 + 3 x$ $6$ $x^5 - 4x^4 + 2x^3 + 5x^2 - 2x - 1$ $7$ $x^6 - 5x^5 + 5x^4 + 6x^3 - 7x^2 - 2x + 1$ $8$ $x^7 - 6x^6 + 9x^5 + 5x^4 - 15x^3 + 5 x$ $9$ $x^8 - 7x^7 + 14x^6 + x^5 - 25x^4 + 9x^3 + 12x^2 - 3x - 1$ $10$ $x^9 - 8x^8 + 20x^7 - 7x^6 - 35 x^5 + 29 x^4 + 18 x^3 - 15 x^2 - 3x + 1$ $11$ $x^{10} - 9x^9 + 27x^8 - 20 x^7 - 42 x^6 + 63 x^5 + 14 x^4 - 42 x^3 + 7x$ : [\[Table:Polynomials\]]{#Table:Polynomials label="Table:Polynomials"}Polynomials for invariants $TH_{r}$. We do not formulate or prove any explicit statements about the polynomials in the table [1](#Table:Polynomials){reference-type="ref" reference="Table:Polynomials"}. The procedure described above should be regarded as an empirical rule. It can be used to construct required polynomials. 1 V.G. Turaev, O.Ya. Viro, *State sum invariants of 3-manifolds and quantum 6j-symbols*, Topology, **31**:4 (1992), 865--902. S. Matveev, *Algorithmic Topology and Classification of 3-Manifolds*, Springer Science & Business Media, 2007. V.G. Turaev, *Quantum invariants of knots and 3-manifolds*, de Gruyter Studies in Mathematics, vol. 18, Walter de Gruyter, Berlin, 1994. S.V. Matveev, M.V. Sokolov, *On a simple invariant of Turaev-Viro type*, Journal of Mathematical Sciences, **94** (1999), 1226---1229. M.V. Sokolov, *The Turaev-Viro invariant for 3-manifolds is a sum of three invariants*, Canad. Math. Bull., **39**:4 (1996), 468--475. G. Masbaum, J.D. Roberts, *A simple proof of integrality of quantum invariants at prime roots of unity*, Mathematical Proceedings of the Cambridge Philosophical Society, **121**:3 (1997), 443--454.

arxiv_math

--- abstract: | Let $\pi'$ be a fixed unitary cuspidal representation of $\mathrm{GL}(n)/\mathbb{Q}.$ - We establish a subconvex bound in the $t$-aspect $$L(1/2+it,\pi\times\pi')\ll_{\pi,\pi',\varepsilon}(1+|t|)^{\frac{n(n+1)}{4}-\frac{1}{4\cdot (4n^2+2n-1)}+\varepsilon},$$ for any unitary **pure isobaric** automorphic representation $\pi$ of $\mathrm{GL}(n+1)/\mathbb{Q}.$ Moreover, the bound improves in the standard $L$-function case $$\begin{aligned} L(1/2+it, \pi')\ll_{\pi',\varepsilon}(1+|t|)^{\frac{n}{4}-\frac{1}{4(n+1)(4n-1)}+\varepsilon}.\end{aligned}$$ - We prove an explicit lower bound $$\begin{aligned} \sum_{\pi\in\mathcal{A}_0}\textbf{1}_{L(1/2,\pi\times\pi')\neq 0}\gg_{\varepsilon}|\mathcal{A}_0|^{\frac{1}{n(n+1)(4n^2+2n-1)}-\varepsilon},\end{aligned}$$ for a suitable finite family $\mathcal{A}_0$ of unitary cuspidal representations of $\mathrm{GL}(n+1)/\mathbb{Q}.$ - More generally, we address the spectral side subconvexity in the case of uniform parameter growth, and a quantitative form of simultaneous nonvanishing of central $L$-values for $\mathrm{GL}(n+1)\times\mathrm{GL}(n)$ (over $\mathbb{Q}$) in both level and eigenvalue aspects. Among other ingredients, our proofs employ a new relative trace formula in conjunction with P. Nelson's construction of archimedean test functions in [@Nel21] and volume estimates in [@Nel20]. address: Fine Hall, 304 Washington Rd, Princeton, NJ 08544, USA author: - Liyang Yang bibliography: - SC.bib title: Relative Trace Formula, Subconvexity and Quantitative Nonvanishing of Rankin-Selberg $L$-functions for $\mathrm{GL}(n+1)\times\mathrm{GL}(n)$ --- # Introduction Let $\pi'$ be a *fixed* cuspidal representation of $\mathrm{GL}(n)/\mathbb{Q}.$ Let $\pi=\pi_1\boxplus\cdots\boxplus\pi_m$ be a unitary pure isobaric automorphic representation of $\mathrm{GL}(n+1)/\mathbb{Q}$ (cf. [@Lan79]), where each $\pi_j$ is a unitary cuspidal representation of $\mathrm{GL}(n_j)/\mathbb{Q},$ with $n_1+\cdots+n_m=n+1.$ In particular, if $m=1,$ then $\pi$ is cuspidal; if $m=n+1,$ then $n_1=\cdots=n_m=1$ and each $\pi_j$ is a Hecke character, in which case $\pi$ is a minimal Eisenstein series. We have a decomposition of the complete $L$-function $$\label{1.1} \Lambda(s,\pi\times\pi')=\Lambda(s,\pi_1\times\pi')\cdots \Lambda(s,\pi_m\times\pi'),$$ where $\Lambda(s,\pi_j\times\pi'),$ $1\leq j\leq m,$ is the Rankin-Selberg $L$-function introduced by Jacquet--Piatetski-Shapiro--Shalika [@JPSS83]. In the case that $m=n+1$ and $\pi_1=\cdots=\pi_m=\textbf{1},$ the trivial character, the formula [\[1.1\]](#1.1){reference-type="eqref" reference="1.1"} becomes $\Lambda(s,\pi\times\pi')=\Lambda(s,\pi')^{n+1},$ where $\Lambda(s,\pi')$ is Godemen--Jacquet's standard $L$-function associated with $\pi'$ (cf. [@GJ72]). Conjectures of Langlands [@Lan70] predict that "all $L$-functions" arise as shifted products of such standard L-functions. We shall use $L(s,\pi\times\pi')$ to denote the finite part of $\Lambda(s,\pi\times\pi'),$ i.e., excluding archimedean factors from the definition. So $L(s,\pi\times\pi')$ is defined by an Euler product over rational primes in $\mathop{\mathrm{Re}}(s)>1$ and its meromorphic continuation elsewhere. ## Subconvexity of $L(s,\pi\times\pi')$ in the $t$-aspect {#subc} Bounding $L$-functions on the critical line $\mathop{\mathrm{Re}}(s)=1/2,$ which is known as the *subconvexity problem* (**ScP**), is a far-reaching problem in number theory. See [@Fri95], [@IS00], [@Mic07] and [@Mun18a] for an overview of the history on this problem. In this paper we consider bounds of Rankin-Selberg $L$-functions of the form $$\label{1.4} L(1/2+it,\pi\times\pi')\ll_{\pi,\pi',\varepsilon} (1+|t|)^{\frac{(1-\delta)n(n+1)}{4}+\varepsilon},$$ where $\delta\in [0,1],$ and the implied constant depends on $\varepsilon,$ $\pi$ and $\pi'.$ It follows from the functional equation and the convex bound of Phragmen--Lindelöf that $\delta=0$ is valid in [\[1.4\]](#1.4){reference-type="eqref" reference="1.4"}, while the generalized Lindelöf hypothesis asserts that $\delta=1$ should be admissible. The *$t$-aspect* **ScP** for $\mathrm{GL}(n+1)\times\mathrm{GL}(n)$ (over $\mathbb{Q}$) is to show that [\[1.4\]](#1.4){reference-type="eqref" reference="1.4"} holds for some $\delta>0,$ which relies at most on $n.$ The *$t$-aspect* **ScP** has been known for $n=1$ and $n=2$, see for instance [@Goo81], [@Goo82], [@Li09], [@MV10], [@Mun18b], [@BB20], and [@LS21]. Recently Nelson [@Nel21] resolved the general case of standard $L$-functions for $\mathrm{GL}(n)/\mathbb{Q}.$ Note that under Langlands' conjectures, [\[1.4\]](#1.4){reference-type="eqref" reference="1.4"} reduces to Nelson's theorem, as $L(s,\pi\times\pi')$ factors into a product of *stand $L$-functions*. Unconditionally, Nelson's result verifies [\[1.4\]](#1.4){reference-type="eqref" reference="1.4"} (with $\delta\asymp n^{-5}$) in the case that $\pi=\textbf{1}\boxplus\cdots\boxplus\textbf{1}.$ The first achievement of this paper confirms [\[1.4\]](#1.4){reference-type="eqref" reference="1.4"} (with $\delta\asymp n^{-4}$) for general pure isobaric automorphic representation $\pi.$ **Theorem 1**. *Let $n\geq 2.$ Let $\pi$ be a unitary **pure isobaric** automorphic representation of $\mathrm{GL}(n+1)/\mathbb{Q}.$ Let $\pi'$ be a unitary cuspidal representation of $\mathrm{GL}(n)/\mathbb{Q}.$ Then $$\label{1} L(1/2+it,\pi\times\pi')\ll_{\pi,\pi',\varepsilon}(1+|t|)^{\frac{n(n+1)}{4}-\frac{1}{4\cdot (4n^2+2n-1)}+\varepsilon},$$ where the implied constant depends on $\varepsilon,$ and the conductors of $\pi, \pi'.$ Moreover, if $\pi$ is tempered, $$\label{1.4*} L(1/2+it,\pi\times\pi')\ll_{\pi,\pi',\varepsilon}(1+|t|)^{\frac{n(n+1)}{4}-\frac{1}{4\cdot(4n-1)}+\varepsilon}.$$* Taking $\pi=\textbf{1}\boxplus\cdots\boxplus\textbf{1}$ in [\[1.4\*\]](#1.4*){reference-type="eqref" reference="1.4*"} in Theorem [Theorem 1](#A){reference-type="ref" reference="A"} we then derive the following. **Corollary 1** (Standard $L$-functions). *Let $n\geq 2.$ Let $\delta=\frac{1}{n(n+1)(4n-1)}.$ Let $\pi'$ be a unitary cuspidal representation of $\mathrm{GL}(n)/\mathbb{Q}.$ Then $$\label{1.} L(1/2+it, \pi')\ll_{\pi',\varepsilon}(1+|t|)^{\frac{(1-\delta)n}{4}+\varepsilon},$$ where the implied constant depends on $\varepsilon$ and the conductor of $\pi'.$* *Remark 2*. The subconvex bound [\[1.\]](#1.){reference-type="eqref" reference="1."} improves Theorem 1.1 in [@Nel21] from $\delta=\frac{1}{n^2(3n^3-2n^2-1)}\asymp n^{-5}$ to $\delta=\frac{1}{n(n+1)(4n-1)}\asymp n^{-3}.$ **Corollary 3** (The case of $\mathrm{GL}(n)\times\mathrm{GL}(m)$). *Let $n\geq 2.$ Let $\pi'$ be a unitary cuspidal representation of $\mathrm{GL}(n)/\mathbb{Q}.$ Let $\sigma$ be a unitary cuspidal representation of $\mathrm{GL}(m)/\mathbb{Q}$ with $m\mid (n+1).$ Then $$\begin{aligned} L(1/2+it, \sigma\times \pi')\ll_{\sigma,\pi',\varepsilon}(1+|t|)^{\frac{(1-\delta)nm}{4}+\varepsilon} \end{aligned}$$ for $\delta=\frac{1}{n(n+1)(4n^2+2n-1)},$ where the implied constant depends on $\pi',$ $\sigma$ and $\varepsilon.$* ## Uniform Parameter Growth Case {#subc.} Let $\boldsymbol{\lambda}_{\pi_{\infty}}=\{\lambda_{\pi_{\infty},1},\cdots,\lambda_{\pi_{\infty},n+1}\}$ be the Langlands class of $\pi_{\infty},$ namely, the archimedean $L$-factor of $\pi$ may be written $$\Lambda_{\infty}(s,\pi)=\prod_{1\leq j\leq n+1}\Gamma_{\mathbb{R}}(s+\lambda_{\pi_{\infty,j}})$$ for parameters $\lambda_{\pi_{\infty},j}\in\mathbb{C},$ $1\leq j\leq n+1.$ Let $T\geq 1$ and $C_{\infty}>c_{\infty}>0.$ We say that $\pi$ (or its archimedean component $\pi_{\infty}$) has *uniform parameter growth of size $(T;c_{\infty},C_{\infty})$* if $$\label{1.6} c_{\infty}T\leq |\lambda_{\pi_{\infty},j}|\leq C_{\infty}T,\ \ 1\leq j\leq n+1.$$ Then the bound [\[1\]](#1){reference-type="eqref" reference="1"} in Theorem [Theorem 1](#A){reference-type="ref" reference="A"} is a special case of the following. **Theorem 2**. *Let $n\geq 2.$ Let $\pi=\pi_1\boxplus\cdots\boxplus\pi_m$ be a unitary pure isobaric automorphic representation of $\mathrm{GL}(n+1)/\mathbb{Q}.$ Suppose that $\pi$ has arithmetic conductor $M$ and has uniform parameter growth of size $(T;c_{\infty},C_{\infty}).$ Let $\pi'$ be a unitary cuspidal representation of $\mathrm{GL}(n)/\mathbb{Q}.$ Let $$\begin{aligned} \mathbf{L}:= \begin{cases} M^{\frac{n}{2\cdot (4n^2+2n-1)}}T^{\frac{1}{4\cdot (4n^2+2n-1)}},\ \ & \text{if $M\leq T^{\frac{n+1}{2n^2-1}}$,}\\ M^{\frac{1}{4n^2+4n+2}}T^{\frac{1}{4\cdot (2n^2+2n+1)}},\ \ & \text{if $M>T^{\frac{n+1}{2n^2-1}}.$} \end{cases}\end{aligned}$$ Then we have the hybrid bound $$\label{bur} L(1/2,\pi\times\pi')\ll T^{\frac{n(n+1)}{4}+\varepsilon}M^{\frac{n}{2}+\varepsilon}\mathbf{L}^{-1}\prod_{j=1}^m\sqrt{L(1,\pi_j,\operatorname{Ad})},$$ where the implied constant depends on $\varepsilon,$ parameters $c_{\infty}$ and $C_{\infty}$ defined in [3.1.4](#sec3.14){reference-type="ref" reference="sec3.14"}, and the conductor of $\pi'.$ In particular, $$\label{b.} L(1/2,\pi\times\pi')\ll T^{\frac{n(n+1)}{4}-\frac{1}{4\cdot (4n^2+2n-1)}+\varepsilon},$$ where the implied constant depends on $\varepsilon,$ parameters $c_{\infty}$ and $C_{\infty}$, and the conductors of $\pi_{\operatorname{fin}}$ and $\pi'.$* *Remark 4*. Nelson [@Nel20] proves a subconvex bound for $L(1/2,\sigma_E\times\sigma_E'),$ where $\sigma$ (resp. $\sigma'$) is a tempered cuspidal representation of a unitary group $\mathrm{U}(n+1)$ (resp. $\mathrm{U}(n)$) with uniform parameter growth for $\sigma\boxtimes\sigma'$. Here $\sigma_E$ (resp. $\sigma_E'$) is a quadratic base change of $\sigma$ (resp. $\sigma'$), and $\mathrm{U}(n)$ is anisotropic. The estimate [\[b.\]](#b.){reference-type="eqref" reference="b."} can be regarded as an analogue of Nelson's theorem in the general linear group case (with $\sigma'$ fixed, but without the cuspidality of $\sigma$). *Remark 5*. For $n=1$, the hybrid bound [\[bur\]](#bur){reference-type="eqref" reference="bur"} can be further refined to the Burgess bound for twisted $L$-functions, without assuming the Ramanujan-Selberg conjecture (cf. [@Yan23b], extending [@BH08] and [@BH14a]). ## Quantitative Nonvanishing of Rankin-Selberg $L$-functions {#nonv} Nonvanishing of Rankin-Selberg $L$-functions for $\mathrm{GL}(n+1)\times\mathrm{GL}(n)$ plays an important role in various aspects of number theory, e.g., Landau-Siegel zeros (cf. [@IS00a]), Langlands functorial lifts (cf. [@GJR04]), the Gan-Gross-Prasad conjecture (cf. [@Zha14b], [@Zha14a]), Whittaker periods (cf. [@GH16]), and the Bloch-Kato conjecture (cf. [@LTXZZ22]). Various nonvanishing results have been achieved in lower ranks ($n\leq 2$), see for instance [@Duk95], [@IS00a], [@MRY22], [@ST22] for a far from exhaustive list. Moreover, the existence of simultaneous nonvanishing of central $L$-values has been proved in a few of other cases in higher ranks (e.g., cf. [@Li09], [@Tsu21], [@JN21], [@Yan22]). Nevertheless, finding a quantitative form of these results in higher ranks has remained open so far. Our second main achievement of this paper establishes for the first time a *quantitative simultaneous* nonvanishing result on central $L$-values for $\mathrm{GL}(n+1)\times\mathrm{GL}(n),$ $n\geq 2.$ **Theorem 3**. *Let $\pi_j'=\otimes_{p\leq \infty}\pi_{j,p}'$ be a unitary cuspidal representation of $\mathrm{GL}_n(\mathbb{A})$ with arithmetic conductor $M_j',$ $j=1,2.$ Suppose that $M_1'M_2'>1,$ $(M_1',M_2')=1,$ $\pi_{1,\infty}'\simeq \pi_{2,\infty}'$ and $\pi_{1,p_*}'\simeq \pi_{2,p_*}'$ at some prime $p_*\nmid M_1'M_2'.$ Let $T\geq 1$ and $M\geq 1$ with $(M,M_1'M_2'p_*)=1.$ Fix an unitary irreducible admissible representation $\pi_{\infty}$ of $\mathrm{PGL}_{n+1}(\mathbb{R}),$ which has uniform parameter growth of size $(T;c_{\infty},C_{\infty})$ (cf. [\[1.6\]](#1.6){reference-type="eqref" reference="1.6"}). Fix a supercuspidal representation $\pi_{p_*}$ of $\mathrm{GL}_{n+1}(\mathbb{Q}_{p_*}).$ Let $\mathcal{A}_0(T,M;\pi_{\infty},\pi_{p_*},\pi_1',\pi_2')$ be the set of cuspidal representations $\sigma=\otimes_{p\leq \infty}\sigma_p$ of $G(\mathbb{A})$ such that* - *$\sigma_{\infty}\simeq\pi_{\infty},$ $\sigma_{p_*}\simeq \pi_{p_*},$* - *$\sigma_p$ is right $K(MM_1')I(M_2')$-invariant, where $K(MM_1')$ is the Hecke congruence of level $MM_1',$ and $I(M_2')$ is the Iwahori subgroup of level $M_2'.$* *See Definiiton [Definition 66](#defn11.1){reference-type="ref" reference="defn11.1"} in [11.1](#sec12.1){reference-type="ref" reference="sec12.1"} for the precise description. Then for all $\varepsilon>0,$ $$\begin{aligned} \sum_{\substack{\pi\in \mathcal{A}_0(T,M;\pi_{\infty},\pi_{p_*},\pi_1',\pi_2')\\ L(1/2,\pi\times\pi_1')L(1/2,\pi\times\pi_2')\neq 0}}1\gg \begin{cases} M^{\frac{n}{(4n^2+2n-1)}}T^{\frac{1}{2\cdot (4n^2+2n-1)}}, & \text{if $M\leq T^{\frac{n+1}{2n^2-1}}$}\\ M^{\frac{1}{2n^2+2n+1}}T^{\frac{1}{2\cdot (2n^2+2n+1)}}, & \text{if $M>T^{\frac{n+1}{2n^2-1}}$} \end{cases}, \end{aligned}$$ where the implied constant depends on $\varepsilon,$ parameters $c_{\infty}$ and $C_{\infty}$, $\pi_{p_*},$ $\pi_1',$ and $\pi_2'.$ In particular, as $T+M\to \infty,$ we have $$\begin{aligned} \sum_{\substack{\pi\in \mathcal{A}_0(T,M;\pi_{\infty},\pi_{p_*},\pi_1',\pi_2')\\ L(1/2,\pi\times\pi_1')L(1/2,\pi\times\pi_2')\neq 0}}1\gg |\mathcal{A}_0(T,M;\pi_{\infty},\pi_{p_*},\pi_1',\pi_2')|^{\frac{1}{n(n+1)(4n^2+2n-1)}-\varepsilon},\end{aligned}$$ where the implied constant relies on $\varepsilon,$ $c_{\infty},$ $C_{\infty},$ $\pi_{p_*},$ $\pi_1',$ and $\pi_2'.$* *Remark 6*. When $n=1,$ the set $\mathcal{A}_0(T,M;\pi_{\infty},\pi_{p_*},\pi_1',\pi_2')$ can be roughly interpreted as the collection of Maass cusp forms on $\mathrm{GL}(2)$ with spectral parameter $\asymp T+O(1)$ and level $\asymp M.$ **Corollary 7**. *Let notation be as before. Let $\pi'$ be a unitary cuspidal representation of $\mathrm{GL}(n)/\mathbb{Q}$ with arithmetic conductor $M'\geq 1.$ Let $M\geq M'$ and $(M,6M')=1.$ Fix an unitary irreducible admissible representation $\pi_{\infty}$ of $\mathrm{PGL}(n+1,\mathbb{R}),$ which has uniform parameter growth of size $(T;c_{\infty},C_{\infty})$ (cf. [\[1.6\]](#1.6){reference-type="eqref" reference="1.6"}). Let $\mathcal{A}_0(T,M;\pi_{\infty},\pi')$ be the set of cuspidal representations $\sigma$ of $\mathrm{GL}(n+1)/\mathbb{Q}$ whose archimedean component $\sigma_{\infty}\simeq\pi_{\infty}$ and whose arithmetic conductor divides $2^{n}\cdot 3\cdot MM'.$ Then $$\begin{aligned} \sum_{\substack{\pi\in \mathcal{A}_0(T,M;\pi_{\infty},\pi')\\ L(1/2,\pi\times\pi')\neq 0}}1\gg \begin{cases} M^{\frac{n}{(4n^2+2n-1)}}T^{\frac{1}{2\cdot (4n^2+2n-1)}}, & \text{if $M\leq T^{\frac{n+1}{2n^2-1}}$}\\ M^{\frac{1}{2n^2+2n+1}}T^{\frac{1}{2\cdot (2n^2+2n+1)}}, & \text{if $M>T^{\frac{n+1}{2n^2-1}}$} \end{cases}, \end{aligned}$$ where the implied constant relies on $\varepsilon,$ $c_{\infty},$ $C_{\infty},$ and $\pi'.$ In particular, $$\begin{aligned} \sum_{\substack{\pi\in \mathcal{A}_0(T,M;\pi_{\infty},\pi')\\ L(1/2,\pi\times\pi')\neq 0}}1\gg |\mathcal{A}_0(T,M;\pi_{\infty},\pi')|^{\frac{1}{n(n+1)(4n^2+2n-1)}-\varepsilon},\ \ T+M\to\infty,\end{aligned}$$ where the implied constant relies on $\varepsilon,$ $c_{\infty},$ $C_{\infty},$ and $\pi'.$* *Remark 8*. Note that, in Theorem [Theorem 3](#thmC){reference-type="ref" reference="thmC"} and Corollary [Corollary 7](#1.10){reference-type="ref" reference="1.10"}, for $\varepsilon>0,$ $$T^{\frac{n(n+1)}{2}-\varepsilon}M^n\ll |\mathcal{A}_0(T,M;\pi_{\infty},\pi_{p_*},\pi_1',\pi_2')|\asymp |\mathcal{A}_0(T,M;\pi_{\infty},\pi')|\ll T^{\frac{n(n+1)}{2}+\varepsilon}M^n,$$ where the implied constants depend on $\varepsilon,$ $c_{\infty},$ $C_{\infty},$ $\pi_{p_*},$ $\pi',$ $\pi_1'$ and $\pi_2'.$ ## Discussion of the Proofs The strategy employed to prove the main results in this paper relies heavily on a key structural component: - the relative trace formula on $\mathrm{GL}(n+1)$ developed in [@Yan22] (see [2](#sec2){reference-type="ref" reference="sec2"} below), which allows us to handle the continuous spectrum and central $L$-values, and several crucial technical ingredients: - Nelson's construction of test functions at the archimedean place (cf. [@Nel21], Part 2 and Part 3), and his volume bounds (cf. [@Nel20], -16), - the arithmetic amplification in the spirit of Duke-Friedlander-Iwaniec (cf. [@DFI02] and references). The proofs of Theorems [Theorem 1](#A){reference-type="ref" reference="A"} and [Theorem 2](#B){reference-type="ref" reference="B"} build upon the approaches presented in [@Yan23b] and [@Yan23c], with a focus on the generalization to the case of $n>1$. These earlier works specifically addressed the special case of $n=1$. The proof of Theorem [Theorem 3](#thmC){reference-type="ref" reference="thmC"} follows a similar line of reasoning as presented in [@MRY22], where a simple relative trace formula on $\mathrm{U}(3)$ was established to derive quantitative nonvanishing results for central $L$-values associated with $\mathrm{U}(3)\times\mathrm{U}(2)$. To illustrate the basic idea of subconvexity (e.g., Theorem [Theorem 2](#B){reference-type="ref" reference="B"}), we briefly recall the (formal) pre-trace formula $$\label{39} J_{\mathop{\mathrm{Spec}}}(f,\varphi')=\int_{[\mathrm{GL}(n)]}\int_{[\mathrm{GL}(n)]}\operatorname{K}(x,y)\varphi'(x)\overline{\varphi'(y)}dxdy=J_{\operatorname{Geo}}(f,\varphi'),$$ where $\operatorname{K}(x,y)$ is the kernel function associated with a function $f\in C_c^{\infty}(\mathrm{GL}_{n+1}(\mathbb{A})),$ $[\mathrm{GL}(n)]:= \mathrm{GL}(n,\mathbb{Q})\backslash \mathrm{GL}(n,\mathbb{A}),$ and $\varphi'$ is an automorphic form on $\mathrm{GL}(n)/\mathbb{Q}$. Here the $J_{\mathop{\mathrm{Spec}}}(f,\varphi')$ (resp. $J_{\operatorname{Geo}}(f,\varphi')$) comes from the spectral (resp. geometric) expansion of $\operatorname{K}(\cdot,\cdot).$ Then $$\label{1.8.} J_{\mathop{\mathrm{Spec}}}(f,\varphi')=J_0(f,\varphi')+J_{\operatorname{ER}}(f,\varphi'),$$ where $J_0(f,\varphi')$ (resp. $J_{\operatorname{ER}}(f,\varphi')$) is the contribution from the cuspidal (resp. non-cuspidal) spectrum. The motivation of considering [\[39\]](#39){reference-type="eqref" reference="39"} is that, by Rankin-Selberg convolution, $$\label{1.9.} J_0(f,\varphi')=\sum_{\sigma}|L(1/2,\sigma\times\sigma')|^2\cdot \mathcal{Z}(f;\sigma,\varphi'),$$ where $\sigma$ ranges over cuspidal representations of $\mathrm{GL}(n+1)/\mathbb{Q},$ and $\mathcal{Z}(f;\sigma,\varphi')$ is certain weight. Here $\sigma'$ is the representation generalized by $\varphi'.$ For a fixed $\sigma=\pi$ which has uniform parameter growth of size $(T;c_{\infty},C_{\infty}),$ it is then expected to find $f$ and $\varphi'$ such that the following holds: 1. $J_{\operatorname{ER}}(f,\varphi')\geq 0,$ and $\mathcal{Z}(f;\sigma,\varphi')\geq 0$ for all cuspidal $\sigma;$ 2. $\mathcal{Z}(f;\pi,\varphi')\gg T^{-\alpha};$ 3. $J_{\operatorname{Geo}}(f,\varphi')\ll T^{\beta},$ where $\alpha$ and $\beta$ are constants. As a consequence, $$\label{1.10.} |L(1/2,\pi\times\sigma')|^2\leq \frac{J_0(f,\varphi')}{\mathcal{Z}(f;\pi,\varphi')}\ll T^{\alpha}J_{\mathop{\mathrm{Spec}}}(f,\varphi')=T^{\alpha}J_{\operatorname{Geo}}(f,\varphi')\ll T^{\alpha+\beta}.$$ If $\alpha+\beta=\frac{n(n+1)}{2}+\varepsilon,$ one obtains the convex bound for $L(1/2,\pi\times\sigma')$ (cf. [\[1.4\]](#1.4){reference-type="eqref" reference="1.4"} with $\delta=0$); moreover, with a further utilization of the amplification, one may obtain a subconvex bound (i.e., [\[1.4\]](#1.4){reference-type="eqref" reference="1.4"} with some $\delta>0$). However, the formula [\[39\]](#39){reference-type="eqref" reference="39"} diverges for *all* automorphic forms $\varphi',$ which presents significant difficulties. Note that if one replaces the pair $(\mathrm{GL}(n+1),\mathrm{GL}(n))$ by $(\mathrm{U}(n+1),\mathrm{U}(n))$ with the unitary $\mathrm{U}(n)$ being anisotropic, then [\[39\]](#39){reference-type="eqref" reference="39"} does converge and the corresponding subconvexity problem has been worked out along the above tacitics by Nelson [@Nel20]. To overcome the divergence issue in the $\mathrm{GL}$-case, Nelson ([@Nel21], ) uses certain *wave packet* $\varphi'$ rather than genuine automorphic forms. In this case, the main term of $J_{\operatorname{Geo}}(f,\varphi')$ is similar to the unitary case, and after elaborate estimates of the error terms, Nelson [@Nel21] proves the subconvexity in the $t$-aspect for stand $L$-functions for $\mathrm{GL}(n+1)/\mathbb{Q}.$ As distinct from Nelson's strategy, we shall use the relative trace formula (**RTF**) developed in [@Yan22], which is a fundamentally different methodology to handle the divergence problem in [\[39\]](#39){reference-type="eqref" reference="39"}. The formula (cf. Theorem [Theorem 4](#C){reference-type="ref" reference="C"} in [2](#sec2){reference-type="ref" reference="sec2"}) is of the form $$\label{1.8} J_{\mathop{\mathrm{Spec}}}^{\operatorname{Reg},\heartsuit}(f,\textbf{0})=J_{\operatorname{Geo}}^{\operatorname{Reg},\heartsuit}(f,\textbf{0}).$$ There are multiple advantages of using the relative trace formula [\[1.8\]](#1.8){reference-type="eqref" reference="1.8"}: ### In the Spectral Side: {#in-the-spectral-side .unnumbered} making use of Rankin-Selbeg convolution, we may write $$\label{1.11} J_{\mathop{\mathrm{Spec}}}^{\operatorname{Reg},\heartsuit}(f,\mathbf{0})=\int_{\widehat{G(\mathbb{A})}_{\operatorname{gen}}}|L(1/2,\sigma\times\pi')|^2\cdot \mathcal{Z}(f;\sigma,\phi') d\mu_{\sigma},$$ where $\pi'$ is a cuspidal representation of $\mathrm{GL}(n)/\mathbb{Q}$ and $\phi$ is a cusp form in $\pi'$. - On the RHS of [\[1.11\]](#1.11){reference-type="eqref" reference="1.11"} the weighted central Rankin-Selberg $L$-values are over the *full spectrum*, through which we can study the subconvexity problem for $L(1/2,\pi\times\pi'),$ where $\pi$ is *pure isobaric* and has uniform parameter growth of size $(T;c_{\infty},C_{\infty})$. Nevertheless, the noncuspidal part $J_{\operatorname{ER}}(f,\varphi')$ in the pre-trace formula [\[1.8.\]](#1.8.){reference-type="eqref" reference="1.8."} does not have a simple expansion (as [\[1.9.\]](#1.9.){reference-type="eqref" reference="1.9."}, parallel to [\[1.11\]](#1.11){reference-type="eqref" reference="1.11"}) into central $L$-values -- there are many other incidental complicated terms (e.g., the contribution from the non-generic parts of Eisenstein series, multiple residues from the meromorphic continuation, etc). In [@Nel21] (resp. typical applications of the pre-trace formula in higher ranks) the integral $J_{\operatorname{ER}}(f,\varphi')$ (resp. contributions from the non-cuspidal spectrum) is dropped by positivity (cf. [\[1.10.\]](#1.10.){reference-type="eqref" reference="1.10."}). - By the expression [\[1.11\]](#1.11){reference-type="eqref" reference="1.11"}, a lower bound of $J_{\mathop{\mathrm{Spec}}}^{\operatorname{Reg},\heartsuit}(f,\mathbf{0})$ should yield nonvanishing of $L(1/2,\sigma\times\pi'),$ as $\sigma$ ranges over a family of cuspidal representations of $\mathrm{GL}(n+1)/\mathbb{Q},$ which makes Theorem [Theorem 3](#thmC){reference-type="ref" reference="thmC"} possible. Note that if $\varphi'$ is a wave packet (cf. [@Nel21]), the spectral side $J_0(f,\varphi')$ is an average of *integrals of $L$-functions along vertical lines* (rather than the central $L$-values) that are close to $\mathop{\mathrm{Re}}(s)=1/2,$ which is not appliable to nonvanishing problems. ### In the Geometric Side: {#in-the-geometric-side .unnumbered} the bulk of this paper is to handle $J_{\operatorname{Geo}}^{\operatorname{Reg},\heartsuit}(f,\mathbf{0})$ under various $f.$ For instance, we have (cf. [10.2](#sec11.2){reference-type="ref" reference="sec11.2"}), for $f$ being the test function defined in [3.6](#testfunction){reference-type="ref" reference="testfunction"}, $$\label{1.13} \frac{J_{\operatorname{Geo}}^{\operatorname{Reg},\heartsuit}(f,\mathbf{0})}{\langle\phi',\phi'\rangle }\ll T^{\frac{n}{2}+\varepsilon}M^{n+\varepsilon} \mathcal{N}_f^{-1+\varepsilon}+(TM)^{\varepsilon}T^{\frac{n-1}{2}}(M^{n-1}\mathcal{N}_f^{2n}+\mathcal{N}_f^{4n-2}),$$ where $M$ is the level structure equipped with $f,$ and $\mathcal{N}_f$ is a power of the rational prime $p$ that measures the amplification. - In [\[1.13\]](#1.13){reference-type="eqref" reference="1.13"} the exponent of $\mathcal{N}_f$ is *linear* respect to $n,$ which is accountable to the subconvex bound $\delta\asymp n^{-4}$ in [\[1.4\]](#1.4){reference-type="eqref" reference="1.4"}. As a comparison, in the majorization of the geometric side $J_{\operatorname{Geo}}(f,\varphi')$ given in [@Nel21] --5.11, the exponent of $\mathcal{N}_f$ is a *quadratic* function of $n,$ which leads to $\delta\asymp n^{-5}$ in the setting of [\[1.\]](#1.){reference-type="eqref" reference="1."}. Moreover, noticing $\textbf{1}\boxplus\cdots\boxplus\textbf{1}$ in the continuous spectrum is tempered, we incorporate the Rankin-Selberg estimates into the amplification process towards the *$t$-aspect* **ScP** so that one can actually take $\mathcal{N}_f\asymp L$ rather than $\mathcal{N}_f\ll L^{n+1}$ in [@Nel21], where $L$ is the amplification scope (cf. [3.1](#3.1){reference-type="ref" reference="3.1"}). As a consequence, we obtain the stronger exponent $\delta\asymp n^{-3}$ in [\[1.\]](#1.){reference-type="eqref" reference="1."}. - The estimate [\[1.13\]](#1.13){reference-type="eqref" reference="1.13"} yields power savings in *both the eigenvalue and level* aspects. This is used in [11](#sec12){reference-type="ref" reference="sec12"} (cf. Theorem [Theorem 8](#G){reference-type="ref" reference="G"} in [11.3](#12.4){reference-type="ref" reference="12.4"}) to prove a quantitative form of simultaneous nonvanishing of central $L$-values for $\mathrm{GL}(n+1)\times\mathrm{GL}(n).$ *Remark 9*. We address that our choice of $f$ and $\phi'$ (via its Whittaker function) at the archimedean place is based on Nelson's construction in [@Nel21] (with some mild modification). In particular, $\phi'$ relies on $T,$ while in [@Yan22] the cusp form $\phi'$ is fixed. Moreover, one can also fix an arbitrary test function at the archimedean place, and investigate quantitative nonvanishing in the level aspect over number fields. ## Outline of the Paper ### The Relative Trace Formula with Amplification In [2](#sec2){reference-type="ref" reference="sec2"} we describe the relative trace formula (cf. Theorem [Theorem 4](#C){reference-type="ref" reference="C"}) which is developed in [@Yan22]. In [3](#11.2){reference-type="ref" reference="11.2"} we set up local and global data, including the choice of test functions, and parameters for the amplification (cf. Theorem [Theorem 5](#thmD'){reference-type="ref" reference="thmD'"}). In [4](#sec4.){reference-type="ref" reference="sec4."} we construct an appropriate cusp form $\phi'$ in $\pi'$ using its Kirillov model, closely following the methodology outlined in [@Nel21 Part 2]. Additionally, we establish several auxiliary estimates that will be utilized in the subsequent sections. ### The Spectral Side In [5](#sec5){reference-type="ref" reference="sec5"} we describe the amplified spectral side. Together with the local estimates developed in [4](#sec4.){reference-type="ref" reference="sec4."}, we prove a lower bound of the spectral side (cf. Proposition [Proposition 28](#thm6){reference-type="ref" reference="thm6"}) in terms of central $L$-values. ### The Geometric Side In [6](#sec5.1){reference-type="ref" reference="sec5.1"} we classify the support of the local test function in Iwasawa coordinates (cf. [6](#sec5.1){reference-type="ref" reference="sec5.1"}) at the place where we introduce the amplification. The classification will be used frequently in the following sections. In [7](#8.5.1){reference-type="ref" reference="8.5.1"}--[9](#sec10){reference-type="ref" reference="sec10"} we handle the geometric side $J_{\operatorname{Geo}}^{\operatorname{Reg},\heartsuit}(f,\mathbf{0}),$ which will be separated into 3 parts as follows. 1. the small cell orbital integral $J^{\operatorname{Reg}}_{\operatorname{Geo},\operatorname{Small}}(f,\mathbf{s})$, which, as one of the main terms, is handled by Proposition [Proposition 33](#prop54){reference-type="ref" reference="prop54"} in [7](#8.5.1){reference-type="ref" reference="8.5.1"}. 2. the dual orbital integral $J_{\operatorname{Geo},\operatorname{Dual}}^{\operatorname{Big}}(f,\textbf{s})$ is bounded by Proposition [\[8.1\]](#8.1){reference-type="ref" reference="8.1"} in [8](#8.5.2){reference-type="ref" reference="8.5.2"}. This integral is 'dual' to the small cell orbital integral via the Poisson summation, and contributes the other main term. 3. the regular orbital integrals, denoted as the convergent parts of $J_{\operatorname{Geo}}(f,\phi')$ in the pre-trace formula [\[39\]](#39){reference-type="eqref" reference="39"}, are analyzed in detail in Theorem [\[10.1\]](#10.1){reference-type="ref" reference="10.1"} of [9](#sec10){reference-type="ref" reference="sec10"}. While they are analogous to the integrals in [@Nel21 Part 4], our treatment differs significantly from Nelson's bilinear forms estimates. Our approach capitalizes on the orbital structure, allowing us to optimize the counting process for rational points, resulting in an improved bound for the amplification, and also addressing the level aspect. It is important to highlight that, akin to [@Nel21], we also make use of the "inverse transpose" trick, albeit in a different scenario. Specifically, we apply this technique to handle various types of orbital integrals (cf. Lemmas [Lemma 59](#lem9.14){reference-type="ref" reference="lem9.14"} and [Lemma 60](#lem9.15){reference-type="ref" reference="lem9.15"}). ### Proof of Main Results With the aforementioned preparations, we are able to prove the main results in the last two sections. In [10](#proof){reference-type="ref" reference="proof"} we specify the amplification data and put estimates from the spectral and geometric side all together, obtaining Theorem [Theorem 6](#E){reference-type="ref" reference="E"} in [10.4](#sec11.4){reference-type="ref" reference="sec11.4"} and Theorem [Theorem 7](#J){reference-type="ref" reference="J"} in [10.5](#sec12.5){reference-type="ref" reference="sec12.5"}, which leads to Theorems [Theorem 1](#A){reference-type="ref" reference="A"} and [Theorem 2](#B){reference-type="ref" reference="B"}. In [11](#sec12){reference-type="ref" reference="sec12"} we choose proper local and global data in the relative trace formula to study the nonvanishing problem. A lower bound of the geometric side $J_{\operatorname{Geo}}^{\operatorname{Reg},\heartsuit}(f,\textbf{0})$ is obtained in Proposition [Proposition 69](#proposition12.2){reference-type="ref" reference="proposition12.2"}. Gathering Theorem [Theorem 6](#E){reference-type="ref" reference="E"} with Proposition [Proposition 69](#proposition12.2){reference-type="ref" reference="proposition12.2"} we prove Theorem [Theorem 8](#G){reference-type="ref" reference="G"} (which implies Theorem [Theorem 3](#thmC){reference-type="ref" reference="thmC"}) in [11.3](#12.4){reference-type="ref" reference="12.4"}. ## Some Suggestions for Initial Reading During the initial reading, it is possible for the reader to disregard the ramification of $\pi'$ and the calculations specifically related to $p\mid M'$. These were introduced to simplify the relative trace formula and ensure the simultaneous vanishing of mixed orbital integrals and singular integrals. Additionally, [4](#sec4.){reference-type="ref" reference="sec4."} can be considered as a variant of [@Nel21 --] in the cuspidal case and can be skipped during the initial reading. Furthermore, [6](#sec5.1){reference-type="ref" reference="sec5.1"} can also be skipped as it is somewhat complex but elementary. The results presented therein can be revisited if necessary. It is recommended to focus on the case where $\pi$ is cuspidal in [5](#sec5){reference-type="ref" reference="sec5"}. The treatment of the non-cuspidal case, although a key aspect we would like to emphasize, is more technical and involved. One notable advancement and improvement, compared to [@Nel21 Part 4], can be found in [9](#sec10){reference-type="ref" reference="sec10"}. This particular section focuses on bounding the contribution from regular orbitals, providing different insights into the research. Following the above suggestions the reader may readily find the proof of $$L(1/2,\pi\times\pi')\ll T^{\frac{n(n+1)}{4}-\frac{1}{4\cdot (4n^2+2n-1)}+\varepsilon},\tag{\ref{b.}}$$ where $\pi$ is cuspidal. ## Further Discussions ### Relative Trace Formula v.s. Pre-trace Formula {#1.5.1} - In the pre-trace formula [\[39\]](#39){reference-type="eqref" reference="39"} the weighting $\varphi'$ is chosen to be a wave packet. As a consequence, Nelson [@Nel21] proves subconvexity for $L(1/2+it,\pi\times\sigma),$ where $\pi$ is a cuspidal representation of $\mathrm{GL}(n+1)/\mathbb{Q},$ $\sigma=\textbf{1}\boxplus \cdots\boxplus \textbf{1}$ is an Eisenstein series on $\mathrm{GL}(n)/\mathbb{Q}.$ We note that Nelson's approach should be generalized to pure isobaric representations $\sigma$ of $\mathrm{GL}(n)/\mathbb{Q},$ while $\pi$ is cuspidal since $J_{\operatorname{ER}}(f,\varphi')$ is dropped. - In the relative trace formula [Theorem 4](#C){reference-type="ref" reference="C"} the weighting $\phi'$ is a cusp form. We use it to prove a subconvex bound for $L(1/2+it,\pi\times\pi'),$ where $\pi$ is an isobraic representation of $\mathrm{GL}(n+1)/\mathbb{Q},$ $\pi'$ is a cuspidal representation of $\mathrm{GL}(n)/\mathbb{Q}.$ - Combining the above approaches, e.g., if $\phi'$ could be taken as an Eisenstein series, or if $J_{\operatorname{ER}}(f,\varphi')$ can be regularized and estimated properly, one should derive subconvexity for $L(1/2+it,\pi\times\sigma'),$ where $\pi$ (resp. $\sigma'$) is an pure isobaric representation of $\mathrm{GL}(n+1)/\mathbb{Q}$ (resp. $\mathrm{GL}(n)/\mathbb{Q}$), and $n\geq 1$. Let $(m,m')=1.$ Let $\Pi$ (resp. $\Pi'$) be a cuspidal representation of $\mathrm{GL}(m)/\mathbb{Q}$ (resp. $\mathrm{GL}(m')/\mathbb{Q}$). Let $l, l'\in\mathbb{Z}_{\geq 1}$ be such that $lm-l'm'=1$ or $lm-l'm'=-1.$ Let $n=\min\{lm,l'm'\},$ and $\pi=\Pi^{\boxplus l},$ $\sigma=\Pi'^{\boxplus l'}$ if $n=l'm',$ and $\pi=\Pi^{\boxplus l'},$ $\sigma=\Pi'^{\boxplus l}$ if $n=lm.$ Then the $t$-aspect subconvexity of $L(1/2+it,\Pi\times\Pi')$ follows from that of $L(1/2+it,\pi\times\sigma'),$ which would be available if one can gather the two above methods together. ### Over Number Fields We adopt adelic language throughout this paper, from which the interested readers may readily generalize our proofs at least to totally real fields. Over number fields with complex places, some extra local calculation (at these complex places) should be added, which seems not to bring significant obstacles. We also remark that one can directly generalize Theorem [Theorem 3](#thmC){reference-type="ref" reference="thmC"} to number fields by fixing an archimedean test function, and varying the level. As a consequence, one should deduce a quantitative (simultaneous) nonvanishing result on central $L$-values for $\mathrm{GL}(n+1)\times\mathrm{GL}(n)$ over number fieds, generalizing Theorem C in [@Yan22]. ### Bounding $L$-values for $\mathrm{GL}(n+1)\times\mathrm{GL}(n)$ in Other Aspects Our majorization of the geometric side $J_{\operatorname{Geo}}^{\operatorname{Reg},\heartsuit}(f,\mathbf{0})$ (cf. [7](#8.5.1){reference-type="ref" reference="8.5.1"}--[9](#sec10){reference-type="ref" reference="sec10"}) makes the relative trace formula Theorem [Theorem 4](#C){reference-type="ref" reference="C"} quite robust and flexible to study upper bound of $L$-values in multiple aspects. For instance, one may study the subconvexity problem in the depth aspect by fixing a prime $p$ and taking $f_p$ to be the function in [@Hu18] (3.23) on p.10. It is also possible to keep chasing the dependence of levels of $\pi$ and $\pi'$ to derive a hybrid subconvex result. In the $n=1$ case, this has been worked out by [@Yan23b] and [@Yan23c]. ## Notation Let $\mathbb{A}$ be the adele group of $\mathbb{Q}.$ For a rational prime $p,$ we denote by $\mathbb{Q}_p$ the corresponding local field and $\mathbb{Z}_p$ its ring of integers. Denote by $e_p(\cdot)$ the evaluation relative to $p$ normalized as $e_p(p)=1.$ Denote by $\widehat{\mathbb{Z}}=\prod_{p<\infty}\mathbb{Z}_{p}.$ Let $|\cdot|_p$ be the norm in $\mathbb{Q}_p.$ Let $|\cdot|_{\operatorname{fin}}=\prod_{v<\infty}|\cdot|_p.$ Let $|\cdot|_{\infty}$ be the norm in $\mathbb{R}.$ Let $|\cdot|_{\mathbb{A}}=|\cdot|_{\infty}\otimes|\cdot|_{\operatorname{fin}}$. We will simply write $|\cdot|$ for $|\cdot|_{\mathbb{A}}$ in calculation over $\mathbb{A}^{\times}$ or its quotient by $\mathbb{Q}^{\times}$. Let $\psi=\otimes_{p\leq \infty}\psi_p$ be the additive character on $\mathbb{Q}\backslash \mathbb{A}$ such that $\psi(t)=\exp(2\pi it),$ for $t\in \mathbb{R}\hookrightarrow\mathbb{A}.$ For $p\in \Sigma_{\mathbb{Q}},$ let $dt$ be the additive Haar measure on $\mathbb{Q}_p,$ self-dual relative to $\psi_p.$ Then $dt=\prod_{p\leq\infty}dt$ is the standard Tamagawa measure on $\mathbb{A}$. Let $d^{\times}t=\zeta_{p}(1)dt/|t|_p,$ where $\zeta_{p}(\cdot)$ is the local Riemann zeta factor. In particular, $\operatorname{Vol}(\mathbb{Z}_p^{\times},d^{\times}t)=\operatorname{Vol}(\mathbb{Z}_p,dt)=1$ for all finite place $p.$ Moreover, $\operatorname{Vol}(\mathbb{Q}\backslash\mathbb{A}; dt)=1$ and $\operatorname{Vol}(\mathbb{Q}\backslash\mathbb{A}^{(1)},d^{\times}t)=\underset{s=1}{\operatorname{Res}}\ \zeta(s)=1,$ where $\mathbb{A}^{(1)}$ is the subgroup of ideles $\mathbb{A}^{\times}$ with norm $1,$ and $\zeta(s)$ is the Riemann zeta function. More properties of $dt$ and $d^{\times}t$ can be found in [@Lan94], Ch. XIV. Let $G=\mathrm{GL}(n+1)$ and $G'=\mathrm{GL}(n).$ Denote by $Z$ (resp. $Z'$) the center of $G$ (resp. $G'$). Let $G'^0(\mathbb{A})$ be the subgroup of $G'(\mathbb{A})$ consisting of $g'\in G'(\mathbb{A})$ with $|\det g'|=1.$ Then the subgroup $Z'(\mathbb{A})G'^0(\mathbb{A})$ is open and has index $1$ (or $n-1$ if the base field is a function field) in $G'(\mathbb{A}).$ So we may write $G'(\mathbb{A})=Z'(\mathbb{A})G'^0(\mathbb{A}).$ Let $\overline{G}=Z\backslash G$ and $\overline{G'}=Z'\backslash G'.$ We will identify $\overline{G'}$ with $G'^0$ as the conventional notation from Rankin-Selberg theory. For $x\in G(\mathbb{A})$ or $G'(\mathbb{A}),$ we denote by $x^{\vee}$ the transpose inverse of $x.$ Fix the embedding from $G'$ to $G:$ $$\begin{aligned} \iota:\ G'\longrightarrow G,\quad \gamma\mapsto \begin{pmatrix} \gamma&\\ &1 \end{pmatrix}.\end{aligned}$$ For a matrix $g=(g_{i,j})\in G(\mathbb{A}),$ we denote by $E_{i,j}(g)=g_{i,j},$ the $(i,j)$-th entry of $g.$ For a vector $\mathbf{v}=(v_1, \cdots, v_m)$ denote by $E_i(\mathbf{v})=v_i,$ the $i$-th component of $\mathbf{v}.$ Let $m_1, m_2\in \mathbb{N}.$ We write $M_{m_1, m_2}$ for the group of $m_1\times m_2$ matrices. For an algebraic group $H$ over $\mathbb{Q}$, we will denote by $[H]:=H(\mathbb{Q})\backslash H(\mathbb{A}).$ We equip measures on $H(\mathbb{A})$ as follows: for each unipotent group $U$ of $H,$ we equip $U(\mathbb{A})$ with the Haar measure such that, $U(\mathbb{Q})$ being equipped with the counting measure and the measure of $[U]$ is $1.$ We equip the maximal compact subgroup $K$ of $H(\mathbb{A})$ with the Haar measure such that $K$ has total mass $1.$ When $H$ is split, we also equip the maximal split torus of $H$ with Tamagawa measure induced from that of $\mathbb{A}^{\times}.$ Let $\omega$ and $\omega'$ be unitary idele class characters on $\mathbb{A}^{\times}.$ Denote by $\mathcal{A}_0\left([G],\omega\right)$ (resp. $\mathcal{A}_0\left([G'],\omega'\right)$) the set of cuspidal representations on $G(\mathbb{A})$ (resp. $G'(\mathbb{A})$) with central character $\omega$ (resp. $\omega'$). Let $B$ (resp. $B'$) be the group of upper triangular matrices in $G$ (resp. $G'$). Let $T_B$ (resp. $T_{B'}$) be the diagonal subgroup of $B$ (resp. $B'$). Let $A=Z\backslash T_B$ and $A'=Z'\backslash T_{B'}.$ Let $N$ (resp. $N'$) be the unipotent radical of $B$ (resp. $B'$). Let $W_G$ be Weyl group of $G$ with respect to $(B,T_{B}).$ Let $\Delta=\{\alpha_{1,2},\alpha_{2,3},\cdots,\alpha_{n,n+1}\}$ be the set of simple roots, and for each simple root $\alpha_{k,k+1},$ $1\leq k\leq n,$ denote by $w_k$ the corresponding reflection. Explicitly, for each $1\leq k\leq n,$ $$\begin{aligned} w_k=\begin{pmatrix} I_{k-1} &\\ & S&\\ &&I_{n-k} \end{pmatrix},\ \text{where $S=\begin{pmatrix} &1\\ 1& \end{pmatrix}$}.\end{aligned}$$ For $1\leq k\leq n-1,$ denote by $w_k'$ the unique element in $G'$ such that $\iota(w_k')=w_k.$ Then $w_k$'s generate the Weyl group $W_{G'}$ of $G'$ with respect to $(B',T_{B'}).$ Let $\widetilde{w}_j'=w_j'w_{j+1}'\cdots w_{n-1}',$ $1\leq j\leq n-1.$ Denote by $N_j'=N'/(\widetilde{w}_j 'N' \widetilde{w}_j'^{-1}).$ Denote by $\widetilde{w}_n'=I_n$ and $N_n'=I_n.$ Define the generic character $\theta$ on $[N]$ by setting $\theta(u)=\prod_{j=1}^{n}\psi(u_{j,j+1})$ for $u=(u_{i,j})_{1\leq i, j\leq n+1}\in N(\mathbb{A}).$ Let $\theta'=\theta\mid_{[\iota(N')]}$ be the generic character on $[N'].$ Let $P$ (resp. $P'$) be the standard parabolic subgroup of $G$ (resp. $G'$) of type $(n,1)$ (resp. $(n-1,1)$). Let $P_0=Z\backslash P$ (resp. $P_0'=Z'\backslash P'$). We will denote by $Q$ a general parabolic subgroup of $G.$ Denote by $N_P$ (resp. $N_Q$) the unipotent radical of $P$ (resp. $Q$). For a function $h$ on $G(\mathbb{A}),$ we define $h^*$ by assigning $h^*(g)=\overline{h({g}^{-1})},$ $g\in G(\mathbb{A}).$ Let $F_1(s), F_2(s)$ be two meromorphic functions. Denote by $A\asymp B$ for $A, B\in\mathbb{R}$ if there are absolute constants $c_1$ and $c_2$ such that $c_1B\leq A\leq c_2B.$ Denote by $A\ll B$ for $A\in\mathbb{C}$ and $B\in\mathbb{R}_{>0}$ if there is an absolute constant $c$ such that $|A|\leq cB.$ Throughout, we follow the $\varepsilon$-convention: that is, $\varepsilon$ will always be positive number which can be taken as small as we like, but may differ from one occurrence to another. **Acknowledgements** I am deeply grateful to Paul Nelson for his helpful discussions. I would also like to express my gratitude to Valentin Blomer, Yongxiao Lin, Simon Marshall, Dinakar Ramakrishnan, and Peter Sarnak for their precise comments and valuable suggestions. # The Relative Trace Formula {#sec2} In this section we briefly summary the relative trace formula established in [@Yan22]. Let $G=\mathrm{GL}(n+1)$ and $G'=\mathrm{GL}(n).$ Let $\psi=\otimes_{p\leq\infty}\psi_p$ be the standard unramified additive character of $\mathbb{Q}\backslash\mathbb{A},$ i.e., $\psi_{\infty}(t)=\exp(2\pi it),$ for $t\in \mathbb{R}\hookrightarrow\mathbb{A}_{\mathbb{Q}}.$ Extend it naturally to the generic character $\theta'$ (resp. $\theta$) of $[N']$ (resp. $[N]$). Let $\pi_1', \pi_2'$ be fixed unitary cuspidal representations of $G'(\mathbb{A}).$ Let $\phi_i'\in\pi_i',$ $i=1, 2.$ Denote by $W_{\phi_i'}'(x):=\int_{[N']}\phi_i'(n'x)\theta'(n')dn'$ the Whittaker function of $\phi_i'$ relative to $\theta',$ $i=1, 2.$ Let $f$ be a continuous function on $G(\mathbb{A})$ with compact support modulo the center (via a unitary central character). Define two regions as follows: $$\begin{aligned} &\mathcal{R}^*:=\big\{\mathbf{s}=(s_1, s_2)\in\mathbb{C}^2:\ |\mathop{\mathrm{Re}}(s_1)|<1/(n+1),\ |\mathop{\mathrm{Re}}(s_2)|<1/(n+1)\big\},\\ &\mathcal{R}:=\big\{\mathbf{s}=(s_1,s_2)\in\mathbb{C}^2:\ \mathop{\mathrm{Re}}(s_1)>-1/(n+1),\ \mathop{\mathrm{Re}}(s_2)>-1/(n+1)\big\}.\end{aligned}$$ ## The Spectral Side $J_{\mathop{\mathrm{Spec}}}^{\operatorname{Reg},\heartsuit}(f,\mathbf{s})$ {#sec2.1.} The relative trace formula developed in [@Yan22] starts with the geometric and spectral expansions of $$\begin{aligned} J^{\operatorname{Reg}}(f,\textbf{s}):=\iint J_{\operatorname{Kuz}}\left(\begin{pmatrix} x&\\ &1 \end{pmatrix},\begin{pmatrix} y&\\ &1 \end{pmatrix}\right)\phi_1'(x)\overline{\phi_2'(y)}|\det x|^{s_1}|\det y|^{s_2}dxdy,\end{aligned}$$ where $\mathbf{s}=(s_1,s_2)\in\mathbb{C}^2$ is such that $\mathop{\mathrm{Re}}(s_1), \mathop{\mathrm{Re}}(s_2)\gg1,$ $x$ and $y$ ranges over $N'(\mathbb{Q})\backslash G'(\mathbb{A}),$ and for $g_1,$ $g_2\in G(\mathbb{A}),$ $$\begin{aligned} J_{\operatorname{Kuz}}(g_1,g_2):=\int_{[N]}\int_{[N]}\operatorname{K}(u_1g_2,u_2g_2)\theta(u_1)\overline{\theta}(u_2)du_1du_2.\end{aligned}$$ Here $\operatorname{K}=\operatorname{K}^f$ is the kernel function associated with $f.$ We suppress $\phi_1'$ and $\phi_2'$ from $J^{\operatorname{Reg}}(f,\textbf{s})$ to simplify the notation once no confusion arises. This abbreviation will be employed throughout the paper. Executing the spectral expansion (cf. in loc. cit.) of the kernel function $\operatorname{K}$ and swapping integrals, we obtain the spectral side of $J^{\operatorname{Reg}}(f,\textbf{s})$: $$\label{64} J_{\mathop{\mathrm{Spec}}}^{\operatorname{Reg}}(f,\mathbf{s})=\int_{\widehat{G(\mathbb{A})}_{\operatorname{gen}}}\sum_{\phi\in\mathfrak{B}_{\pi}}\Psi(s_1,\pi(f)W_{\phi},W_{\phi_1'}')\Psi(s_2,\widetilde{W}_{\phi},\widetilde{W}_{\phi_2'}')d\mu_{\pi},$$ where $\mathbf{s}=(s_1,s_2)\in\mathbb{C}^2$ is such that $\mathop{\mathrm{Re}}(s_1), \mathop{\mathrm{Re}}(s_2)\gg1,$ $\widehat{G(\mathbb{A})}_{\operatorname{gen}}$ refers to the generic automorphic representations (with a fixed central character) of $G(\mathbb{A}),$ and $\Psi(s,W_{\phi},W_{\phi_i'}')$ is the Rankin-Selberg period in the Whittaker form: $$\label{whittaker} \Psi(s,W_{\phi},W_{\phi_i'}')=\int_{N'(\mathbb{A})\backslash G'(\mathbb{A})}W_{\phi}\left(\begin{pmatrix} x\\ &1 \end{pmatrix}\right)W_{\phi_i'}'(x)|\det x|^sdx,\ \ \mathop{\mathrm{Re}}(s)\gg 1.$$ Here $W_{\phi}(g):=\int_{[N]}\phi(ng)\overline{\theta}(n)dn$ is the Whittaker function associated to $\phi,$ and $\mathfrak{B}_{\pi}$ is a set of basis according to the spectral decomposition (cf. [5.3.1](#sec5.2.1){reference-type="ref" reference="sec5.2.1"} in [5.3](#sec5.2){reference-type="ref" reference="sec5.2"}). The integral $J_{\mathop{\mathrm{Spec}}}^{\operatorname{Reg}}(f,\mathbf{s})$ converges absolutely in $\mathop{\mathrm{Re}}(s_1), \mathop{\mathrm{Re}}(s_2)\gg1$ (cf. [@Yan22]). By [@JPSS83] and [@Jac09] each $\Psi(s,W_{\phi},W_{\phi_i'}')$ admits a meromorphic continuation to $\mathbb{C},$ as a *holomorphic* multiple of the complete $L$-function $\Lambda(s+1/2,\pi\times\pi_i').$ Let $P(s,\phi,\phi_i'):=\Psi(s,W_{\phi},W_{\phi_i'}')/\Lambda(s+1/2,\pi\times\pi_i').$ Then $P(s,\phi,\phi_i')$ is entire. Denote by $\mathcal{P}(f,\textbf{s},\phi)=P(s_1,\pi(f)\phi,\phi_1')\overline{P(\overline{s_2},\phi,\phi_2')}.$ Define $$\begin{aligned} J_{\mathop{\mathrm{Spec}}}^{\operatorname{Reg},\heartsuit}(f,\mathbf{s})=\int_{\widehat{G(\mathbb{A})}_{\operatorname{gen}}}\sum_{\phi\in\mathfrak{B}_{\pi}} \Lambda(s_1+1/2,\pi\times\pi_1')\Lambda(s_2+1/2,\widetilde{\pi}\times\widetilde{\pi}_2')\mathcal{P}(f,\textbf{s},\phi) d\mu_{\pi}.\end{aligned}$$ Note that $J_{\mathop{\mathrm{Spec}}}^{\operatorname{Reg},\heartsuit}(f,\mathbf{s})$ is the same as $J_{\mathop{\mathrm{Spec}}}^{\operatorname{Reg}}(f,\mathbf{s})$ if we identify $\Psi(s,W_{\phi},W_{\phi_i'}')$ with its meromorphic continuation to $s\in\mathbb{C}.$ Shifting contour one can show (cf. [@Yan22]) that $J_{\mathop{\mathrm{Spec}}}^{\operatorname{Reg},\heartsuit}(f,\mathbf{s})$ admits a meromorphic continuation $\tilde{J}_{\mathop{\mathrm{Spec}}}^{\operatorname{Reg}}(f,\mathbf{s})$ to the region $\mathcal{R}.$ In particular, $$\label{59} \tilde{J}_{\mathop{\mathrm{Spec}}}^{\operatorname{Reg}}(f,\mathbf{s})=J_{\mathop{\mathrm{Spec}}}^{\operatorname{Reg},\heartsuit}(f,\mathbf{s})-\mathcal{G}_{\chi}(\textbf{s},\phi_1',\phi_2')+\widetilde{\mathcal{G}}_{\chi}(\textbf{s},\phi_1',\phi_2'),$$ where $\mathbf{s}=(s_1,s_2)\in \mathcal{R}^*\subset \mathcal{R},$ and $$\label{183} \mathcal{G}_{\chi}(\textbf{s},\phi_1',\phi_2'):=\sum_{\phi\in\mathfrak{B}_{Q,\chi}}\underset{\lambda=n(s_1-1/2)}{\operatorname{Res}}\Psi(s_1,\mathcal{I}(\lambda,f)W_{\phi_{\lambda}},W'_{\phi_1'})\overline{\Psi(\overline{s_2},W_{\phi_{\lambda}},W'_{\phi_2'})},$$ with $\phi_{\lambda}:=\phi\otimes|\cdot|^{\lambda}$, and similarly $$\label{198} \widetilde{\mathcal{G}}_{\chi}(\textbf{s},\phi_1',\phi_2'):=\sum_{\phi\in\mathfrak{B}_{Q,\chi}}\underset{\lambda=n(1/2-s_2)}{\operatorname{Res}}\Psi(s_1,\mathcal{I}(\lambda,f)W_{\phi_{\lambda}},W'_{\phi_1'})\overline{\Psi(\overline{s_2},W_{\phi_{\lambda}},W'_{\phi_2'})}.$$ By [@Yan22], the right hand side of [\[59\]](#59){reference-type="eqref" reference="59"} converges absolutely in $\mathcal{R}^*$. In particular, the integral $J_{\mathop{\mathrm{Spec}}}^{\operatorname{Reg}}(f,\mathbf{s})$ (cf. [\[64\]](#64){reference-type="eqref" reference="64"}), which is defined initially in the region $\mathop{\mathrm{Re}}(s_1), \mathop{\mathrm{Re}}(s_2)\gg1$, still converges absolutely in $|\mathop{\mathrm{Re}}(s_1)|, |\mathop{\mathrm{Re}}(s_2)|<1/(n+1)$ if we view the Rankin-Selberg periods $\Psi(\cdots)$ as meromorphic functions therein. The meromorphic continuation $\tilde{J}_{\mathop{\mathrm{Spec}}}^{\operatorname{Reg}}(f,\mathbf{s})$ coincide with $J_{\mathop{\mathrm{Spec}}}^{\operatorname{Reg}}(f,\mathbf{s})$ in $\mathop{\mathrm{Re}}(s_1), \mathop{\mathrm{Re}}(s_2)\gg1,$ but they are different (due to [\[59\]](#59){reference-type="eqref" reference="59"}) in the region $|\mathop{\mathrm{Re}}(s_1)|, |\mathop{\mathrm{Re}}(s_2)|<1/(n+1).$ In this paper we will take $s_1=s_2=0$ to prove the main subconvexity and nonvanishing results, which is the reason that we consider the region $\mathcal{R}^*\ni (0,0).$ To accomplish estimates of the relevant integrals, we need to describe [\[183\]](#183){reference-type="eqref" reference="183"} and [\[198\]](#198){reference-type="eqref" reference="198"} in a more geometric perspective as follows. Let $\eta=(0,\cdots,0,1)\in \mathbb{Q}^n.$ Define the Eisenstein series $E_P^{\dagger}(x,s;\widehat{f}_P,y)$ by $$\label{276} \sum_{\delta \in P'(\mathbb{Q})\backslash G'(\mathbb{Q})}\int_{Z'(\mathbb{A})}\int_{N_P(\mathbb{A})}\int_{\mathbb{A}^n}f\left(u(\mathbf{x})n\iota(y)\right)\psi(\eta \delta zx\mathfrak{u})d\mathbf{x}dn|\det zx|^{s}d^{\times}z,$$ where we write $n=\begin{pmatrix} I_n&\mathfrak{u}\\ &1 \end{pmatrix}\in N_P(\mathbb{A})$ and $u(\mathbf{x})=\begin{pmatrix} I_n&\\ \mathbf{x}&1 \end{pmatrix},$ $\mathbf{x}\in M_{1,n}(\mathbb{A})\simeq \mathbb{A}^n.$ Then $E_P^{\dagger}(x,s;\widehat{f}_P,y)$ converges absolutely when $\mathop{\mathrm{Re}}(s)>1,$ admits a functional equation and meromorphic continuation to $s\in\mathbb{C}.$ Likewise, the Eisenstein series $\widetilde{E}_P^{\dagger}(x,s;\widehat{f}_P,y)$ defined by $$\label{60} \sum_{\delta \in P'(\mathbb{Q})\backslash G'(\mathbb{Q})}\int_{Z'(\mathbb{A})}\int_{N_P(\mathbb{A})}\int_{\mathbb{A}^n}f\left(nu(\mathbf{x})\iota(y)\right)\psi(\eta \delta zx\mathfrak{u})d\mathbf{x}dn|\det zx|^{s}d^{\times}z$$ admits the same analytic properties. **Proposition 10** (Theorem 43 in op. cit.). *Let notation be as before. Then the function $\mathcal{G}_{\chi}(\textbf{s},\phi_1',\phi_2')$ is equal to $$\label{194} -\int_{G'(\mathbb{A})}\int_{[\overline{G'}]}\phi_1'(x)\overline{\phi_2'(xy)}E_P^{\dagger}(x,s_1+s_2;\widehat{f}_P,y)dx|\det y|^{s_2}dy.$$ Similarly, $\widetilde{\mathcal{G}}_{\chi}(\textbf{s},\phi_1',\phi_2')$ is equal to $$\begin{aligned} \int_{G'(\mathbb{A})}\int_{[\overline{G'}]}\phi_1'(x)\overline{\phi_2'(xy)}\widetilde{E}_P^{\dagger}(x,s_1+s_2;\widehat{f}_P,y)dx|\det y|^{s_2}dy. \end{aligned}$$ Moreover, $\mathcal{G}_{\chi}(\textbf{s},\phi_1',\phi_2')$ and $\widetilde{\mathcal{G}}_{\chi}(\textbf{s},\phi_1',\phi_2')$ admit meromorphic continuation to $\mathcal{R}.$* ## The Geometric Side $J^{\operatorname{Reg},\heartsuit}_{\operatorname{Geo}}(f,\textbf{s})$ One may also insert the expression $$\operatorname{K}(g_1,g_2)=\sum_{\gamma\in Z(\mathbb{Q})\backslash G(\mathbb{Q})}f(g_1^{-1}\gamma g_2),\ \ g_1,\ g_2\in G(\mathbb{A})$$ into $J^{\operatorname{Reg}}(f,\textbf{s})$ to obtain its elementary geometric expansion $J^{\operatorname{Reg}}_{\operatorname{Geo}}(f,\textbf{s}),$ which is defined in $\mathop{\mathrm{Re}}(s_1), \mathop{\mathrm{Re}}(s_2)\gg 1.$ This initial form of $J^{\operatorname{Reg}}_{\operatorname{Geo}}(f,\textbf{s})$ involves Kloosterman sums and is difficult to estimate or do meromorphic continuation. In [@Yan22] (cf. --) we obtain a decomposition of $J^{\operatorname{Reg}}_{\operatorname{Geo}}(f,\textbf{s})$ into a linear combination of period integrals, which admit meromorphic continuation $\tilde{J}^{\operatorname{Reg}}_{\operatorname{Geo}}(f,\textbf{s})$ to $\mathcal{R}.$ Hence, for $\textbf{s}\in\mathcal{R},$ we have the relative trace formula $$\label{rtf2.8} \tilde{J}^{\operatorname{Reg}}_{\mathop{\mathrm{Spec}}}(f,\textbf{s})=\tilde{J}^{\operatorname{Reg}}_{\operatorname{Geo}}(f,\textbf{s})$$ as an inequality of meromorphic functions. By [\[59\]](#59){reference-type="eqref" reference="59"} we define $$\begin{aligned} J^{\operatorname{Reg},\heartsuit}_{\operatorname{Geo}}(f,\textbf{s}):=\tilde{J}^{\operatorname{Reg}}_{\operatorname{Geo}}(f,\textbf{s})+\mathcal{G}_{\chi}(\textbf{s},\phi_1',\phi_2')-\widetilde{\mathcal{G}}_{\chi}(\textbf{s},\phi_1',\phi_2'),\ \ \textbf{s}\in\mathcal{R}.\end{aligned}$$ Although $\mathcal{G}_{\chi}(\textbf{s},\phi_1',\phi_2')$ and $\widetilde{\mathcal{G}}_{\chi}(\textbf{s},\phi_1',\phi_2')$ defined originally by spectral data, they can be written into geometric integrals according to Proposition [Proposition 10](#thm49){reference-type="ref" reference="thm49"}. By [@Yan22], we can decompose $J^{\operatorname{Reg},\heartsuit}_{\operatorname{Geo}}(f,\textbf{s})$ as $$\label{63} J^{\operatorname{Reg}}_{\operatorname{Geo},\operatorname{Small}}(f,\textbf{s})+J_{\operatorname{Geo},\operatorname{Dual}}^{\operatorname{Big}}(f,\textbf{s})-J^{\operatorname{Reg},\uppercase{\romannumeral 1\relax},\heartsuit}_{\operatorname{Geo},\operatorname{Big}}(f,\textbf{s})+J^{\operatorname{Reg},\uppercase{\romannumeral 2\relax}}_{\operatorname{Geo},\operatorname{Big}}(f,\textbf{s}),$$ where the summands will be described in [2.2.1](#2.2.1){reference-type="ref" reference="2.2.1"}--[2.2.3](#2.2.3){reference-type="ref" reference="2.2.3"} as follows. Consequently, $J^{\operatorname{Reg},\heartsuit}_{\operatorname{Geo}}(f,\textbf{s})$ continues to a meromorphic function in the region $\mathcal{R}^*,$ and [\[rtf2.8\]](#rtf2.8){reference-type="eqref" reference="rtf2.8"} becomes $J^{\operatorname{Reg},\heartsuit}_{\mathop{\mathrm{Spec}}}(f,\textbf{s})=J^{\operatorname{Reg},\heartsuit}_{\operatorname{Geo}}(f,\textbf{s}),$ $\mathbf{s}\in \mathcal{R}^*.$ See Theorem [Theorem 4](#C){reference-type="ref" reference="C"} below. ### The Geometric Side: $J^{\operatorname{Reg}}_{\operatorname{Geo},\operatorname{Small}}(f,\textbf{s})$ and $J_{\operatorname{Geo},\operatorname{Dual}}^{\operatorname{Big}}(f,\textbf{s})$ {#2.2.1} For $s\in\mathbb{C}$ with $\mathop{\mathrm{Re}}(s)>1,$ we define the Eisenstein series $$\begin{aligned} &E(s,x;\check{f}_P,y):= \sum_{\delta\in P_0'(\mathbb{Q})\backslash\overline{G'}(\mathbb{Q})}\int_{Z'(\mathbb{A})}\int_{\mathbb{A}^n}f\left(\begin{pmatrix} y&\mathfrak{u}\\ &1 \end{pmatrix}\right)\psi(\eta z_1\delta x\mathbf{u})|\det z_1x|^{s}d\mathbf{u}d^{\times}z_1,\\ &E(s,x;f,y):=\sum_{\delta\in P_0'(\mathbb{Q})\backslash \overline{G'}(\mathbb{Q})}\int_{Z'(\mathbb{A})}f(u(\eta z_1\delta x) \iota(y)) |\det z_1x|^{s}d^{\times}z_1,\end{aligned}$$ which converge absolutely in $\mathop{\mathrm{Re}}(s)>1,$ admit meromorphic continuation to $\mathbb{C}$ with a functional equation. Let $\mathop{\mathrm{Re}}(s_1), \mathop{\mathrm{Re}}(s_2)>1/2,$ define $$\begin{aligned} J^{\operatorname{Reg}}_{\operatorname{Geo},\operatorname{Small}}(f,\textbf{s}):=&\int_{G'(\mathbb{A})}\int_{[\overline{G'}]}\phi'(x)\overline{\phi'(xy)}E(1+s_1+s_2,x;\check{f}_P,y)dx|\det y|^{s_2}dy,\\ J_{\operatorname{Geo},\operatorname{Dual}}^{\operatorname{Big}}(f,\textbf{s}):=&\int_{G'(\mathbb{A})}\int_{[\overline{G'}]}\phi'(x)\overline{\phi'(xy)}E(s_1+s_2,x;f,y)dx|\det y|^{s_2}dy.\end{aligned}$$ We call $J^{\operatorname{Reg}}_{\operatorname{Geo},\operatorname{Small}}(f,\textbf{s})$ the *small cell orbital integral* since it comes from the small cell in the Bruhat decomposition; call $J_{\operatorname{Geo},\operatorname{Dual}}^{\operatorname{Big}}(f,\textbf{s})$ the *dual orbital integral*. According to the analytic behavior of the Eisenstein series, $J^{\operatorname{Reg}}_{\operatorname{Geo},\operatorname{Small}}(f,\textbf{s})$ (resp. $J^{\operatorname{Reg}}_{\operatorname{Geo},\operatorname{Dual}}(f,\textbf{s})$) admits a meromorphic continuation to $\mathbb{C}$, with possible (simple) poles at $s_1+s_2\in\{-1,0\}$ (resp. $s_1+s_2\in \{0,1\}$). In addition, $$\label{eq10.2} \underset{s=0}{\operatorname{Res}}\ J^{\operatorname{Reg}}_{\operatorname{Geo},\operatorname{Small}}(f,\textbf{s})+\underset{s=0}{\operatorname{Res}}\ J^{\operatorname{Reg}}_{\operatorname{Geo},\operatorname{Dual}}(f,\textbf{s})=0.$$ ### The Geometric Side: $J^{\operatorname{Reg},\uppercase{\romannumeral 1\relax},\heartsuit}_{\operatorname{Geo},\operatorname{Big}}(f,\textbf{s})$ {#2.2.2} For $s\in\mathbb{C}$ with $\mathop{\mathrm{Re}}(s)>1,$ we define the Eisenstein series $$\begin{aligned} &E^{\dagger}(x,s;f_P,y):= \sum_{\delta}\int_{Z'(\mathbb{A})}\int_{\mathbb{A}^n}f\left(\begin{pmatrix} I_n&\\ \eta \delta x&1 \end{pmatrix}\begin{pmatrix} y&u\\ &1 \end{pmatrix}\right)|\det zx|^{s}dud^{\times}z,\\ &\widetilde{E}_P^{\dagger}(x,s;f_P,y):=\sum_{\delta}\int_{Z'(\mathbb{A})}\int_{\mathbb{A}^n}f\left(\begin{pmatrix} I_n&u\\ &1 \end{pmatrix}\begin{pmatrix} y&\\ \eta \delta xy&1 \end{pmatrix}\right)|\det zx|^{s}dud^{\times}z,\end{aligned}$$ where $\delta$ ranges over $P_0'(\mathbb{Q})\backslash \overline{G'}(\mathbb{Q}).$ Eisenstein series $E^{\dagger}(x,s;f_P,y)$ and $\widetilde{E}_P^{\dagger}(x,s;f_P,y)$ converge absolutely in $\mathop{\mathrm{Re}}(s)>1,$ admit meromorphic continuation to $\mathbb{C}$ with a functional equation. We notice that, when replacing $f$ with $\ensuremath{\mskip 1mu\prescript{\smash{\mathrm t\mkern-3mu}}{}{\mathstrut f}}:$ $g\mapsto f(\ensuremath{\mskip 1mu\prescript{\smash{\mathrm t\mkern-3mu}}{}{\mathstrut g}}),$ $E^{\dagger}(x,s';f_P,y)$ (resp. $\widetilde{E}_P^{\dagger}(x,s;f_P,y)$) is the dual of $E_P^{\dagger}(x,s;\widehat{f}_P,y)$ (resp. $\widetilde{E}_P^{\dagger}(x,s_1+s_2;\widehat{f}_P,y)$) via the functional equation, which is defined by [\[276\]](#276){reference-type="eqref" reference="276"} (resp. [\[60\]](#60){reference-type="eqref" reference="60"}). Let $\mathop{\mathrm{Re}}(s_1), \mathop{\mathrm{Re}}(s_2)>0.$ Write $s'=s_1+s_2+1.$ Define $$\begin{aligned} &\mathcal{F}_{0,1}J^{\operatorname{Big}}_{\operatorname{Geo}}(f,\textbf{s})=\int_{[\overline{G'}]}\int_{{G'}(\mathbb{A})}\phi_1'(x)\overline{\phi_2'(xy)}E^{\dagger}(x,s';f_P,y)|\det x|^{s'}|\det y|^{s_2}dydx,\\ &\mathcal{F}_{1,0}J^{\operatorname{Big}}_{\operatorname{Geo}}(f,\textbf{s})=\int_{[\overline{G'}]}\int_{{G'}(\mathbb{A})}\phi_1'(x)\overline{\phi_2'(xy)}\widetilde{E}_P^{\dagger}(x,s';f_P,y)|\det x|^{s'}|\det y|^{s_2}dydx.\end{aligned}$$ Define the mixed orbital integral $J^{\operatorname{Reg},\uppercase{\romannumeral 1\relax},\heartsuit}_{\operatorname{Geo},\operatorname{Big}}(f,\textbf{s})$ by $$\label{62} J^{\operatorname{Reg},\uppercase{\romannumeral 1\relax},\heartsuit}_{\operatorname{Geo},\operatorname{Big}}(f,\textbf{s})=\mathcal{F}_{0,1}J^{\operatorname{Big}}_{\operatorname{Geo}}(f,\textbf{s})+\mathcal{F}_{1,0}J^{\operatorname{Big}}_{\operatorname{Geo}}(f,\textbf{s})-\mathcal{G}_{\chi}(\textbf{s},\phi_1',\phi_2')+\widetilde{\mathcal{G}}_{\chi}(\textbf{s},\phi_1',\phi_2').$$ By [@Yan22], functions $\mathcal{F}_{0,1}J^{\operatorname{Big}}_{\operatorname{Geo}}(f,\textbf{s})$ and $\mathcal{F}_{1,0}J^{\operatorname{Big}}_{\operatorname{Geo}}(f,\textbf{s})$ converge absolutely in $\mathop{\mathrm{Re}}(s_1+s_2)>0$ and they admit a meromorphic continuation to $\mathcal{R}.$ Along with Proposition [Proposition 10](#thm49){reference-type="ref" reference="thm49"} we conclude that $J^{\operatorname{Reg},\uppercase{\romannumeral 1\relax},\heartsuit}_{\operatorname{Geo},\operatorname{Big}}(f,\textbf{s})$ is meromorphic in $\mathcal{R}.$ ### The Geometric Side: $J^{\operatorname{Reg},\uppercase{\romannumeral 2\relax}}_{\operatorname{Geo},\operatorname{Big}}(f,\textbf{s})$ {#2.2.3} Let $\mathop{\mathrm{Re}}(s_1), \mathop{\mathrm{Re}}(s_2)>0,$ define $$\begin{aligned} J^{\operatorname{Reg},\uppercase{\romannumeral 2\relax}}_{\operatorname{Geo},\operatorname{Big}}(f,\textbf{s})=&\sum_{\substack{(t,\boldsymbol{\xi})\in \mathbb{Q}\oplus \mathbb{Q}^{n-1}\\ (t,\boldsymbol{\xi})\neq (0,\textbf{0})}}\int_{{G'}(\mathbb{A})}\int_{P_0'(\mathbb{Q})\backslash {G'}(\mathbb{A})}f\left(\iota(x)^{-1}\begin{pmatrix} I_{n-1}&&\boldsymbol{\xi}\\ &1&t\\ &1&1 \end{pmatrix}\iota(xy)\right)\\ &\qquad \qquad \qquad\qquad \phi'_1(x)\overline{\phi'_2(xy)}|\det x|^{s_1+s_2}|\det y|^{s_2}dxdy.\end{aligned}$$ We note that the excluded term $(t,\boldsymbol{\xi})=(0,\textbf{0})$ in the summand is exactly the orbital integral $J_{\operatorname{Geo},\operatorname{Dual}}^{\operatorname{Big}}(f,\textbf{s}).$ Unlike $J_{\operatorname{Geo},\operatorname{Dual}}^{\operatorname{Big}}(f,\textbf{s}),$ the integral $J^{\operatorname{Reg},\uppercase{\romannumeral 2\relax}}_{\operatorname{Geo},\operatorname{Big}}(f,\textbf{s})$ defines an entire function. **Proposition 11** ([@Yan22] Thm 5.6). *Let notation be as above. Then $J^{\operatorname{Reg},\uppercase{\romannumeral 2\relax}}_{\operatorname{Geo},\operatorname{Big}}(f,\textbf{s})$ converges absolutely everywhere in $\mathbb{C}^2.$* ## The Relative Trace Formula {#the-relative-trace-formula} With the above preparations, we can describe a consequence of the main results in [@Yan22] as the following identity. **Theorem 4**. *Let notation be as above. Let $\mathbf{s}=(s_1,s_2)\in\mathcal{R}^*.$ Then $$\label{67} J_{\mathop{\mathrm{Spec}}}^{\operatorname{Reg},\heartsuit}(f,\textbf{s})=\int_{\widehat{G(\mathbb{A})}_{\operatorname{gen}}}\sum_{\phi\in\mathfrak{B}_{\pi}}\Psi(s_1,\pi(f)W_{\phi},W_{\phi_1'}')\Psi(s_2,\widetilde{W}_{\phi},\widetilde{W}_{\phi_2'}')d\mu_{\pi},$$ which converges; and we have the identity of holomorphic functions $$\label{70} J_{\mathop{\mathrm{Spec}}}^{\operatorname{Reg},\heartsuit}(f,\textbf{s})=J_{\operatorname{Geo}}^{\operatorname{Reg},\heartsuit}(f,\textbf{s}),$$ where the geometric side $J_{\operatorname{Geo}}^{\operatorname{Reg},\heartsuit}(f,\textbf{s})$ is given by $$J^{\operatorname{Reg}}_{\operatorname{Geo},\operatorname{Small}}(f,\textbf{s})+J_{\operatorname{Geo},\operatorname{Dual}}^{\operatorname{Big}}(f,\textbf{s})-J^{\operatorname{Reg},\uppercase{\romannumeral 1\relax},\heartsuit}_{\operatorname{Geo},\operatorname{Big}}(f,\textbf{s})+J^{\operatorname{Reg},\uppercase{\romannumeral 2\relax}}_{\operatorname{Geo},\operatorname{Big}}(f,\textbf{s}).\tag{\ref{63}}$$* *Remark 12*. In [\[67\]](#67){reference-type="eqref" reference="67"} we identify $\Psi(s,W_{\phi},W_{\phi_i'}')$ with its meromorphic continuation $\Lambda(s+1/2,\pi\times\pi_i')P(s,\phi,\phi_i')$ (cf. [2.1](#sec2.1.){reference-type="ref" reference="sec2.1."}) to simply the notation. By setting $\textbf{s}=(0,0)\in\mathcal{R}^*$, we can investigate a linear combination of equation [\[70\]](#70){reference-type="eqref" reference="70"} using specific test functions $f$. This incorporation of test functions introduces an amplification structure to the relative trace formula. More detailed information will be given in [3.8](#sec3.7){reference-type="ref" reference="sec3.7"} in the following section. # Test Functions and Amplification {#11.2} ## Intrinsic Data {#3.1} In this section, we introduce some notation that will be extensively used throughout the rest of this paper. ### Local and Global Fields {#3.1.1.} Denote by $e_p(\cdot)$ the evaluation of $\mathbb{Q}_p$ normalized as $e_p(p)=1.$ Denote by $\widehat{\mathbb{Z}}=\prod_{p<\infty}\mathbb{Z}_{p}.$ Let $|\cdot|_p$ be the norm in $\mathbb{Q}_p.$ Let $|\cdot|_{\operatorname{fin}}=\prod_{v<\infty}|\cdot|_p.$ Let $|\cdot|_{\infty}$ be the norm in $\mathbb{R}.$ Let $|\cdot|_{\mathbb{A}}=|\cdot|_{\infty}\otimes|\cdot|_{\operatorname{fin}}$. We will simply write $|\cdot|$ for $|\cdot|_{\mathbb{A}}$ in calculation over $\mathbb{A}^{\times}$ or its quotient by $\mathbb{Q}^{\times}$. Let $\psi=\otimes_{p\leq \infty}\psi_p$ be the additive character on $\mathbb{Q}\backslash \mathbb{A}$ such that $\psi(t)=\exp(2\pi it),$ for $t\in \mathbb{R}\hookrightarrow\mathbb{A}.$ Let $\theta$ be the generic character induced by $\psi.$ ### The Automorphic Weight {#sec3.1.2} Let $\pi'=\otimes_{p\leq \infty}\pi_p'$ be a fixed unitary cuspidal representation of $G'(\mathbb{A})$ with central character $\omega'=\otimes_{p\leq\infty}\omega_p'$ and level $M'.$ Upon twisting a Dirichlet character (cf. [3.1.5](#seclevel){reference-type="ref" reference="seclevel"} below), we may assume that $\omega'\neq \textbf{1}.$ Let $M''>1$ be the arithmetic conductor of $\omega'.$ Then $M''\mid M'.$ The assumption $\omega'\neq \textbf{1}$ is introduced to facilitate the construction of a local test function at $p\mid M''$ that satisfies $J^{\operatorname{Reg},\uppercase{\romannumeral 1\relax},\heartsuit}_{\operatorname{Geo},\operatorname{Big}}(f,\textbf{s})\equiv 0$ for all $\textbf{s}\in\mathbb{C}^2.$ While it is true that the contribution from $J^{\operatorname{Reg},\uppercase{\romannumeral 1\relax},\heartsuit}_{\operatorname{Geo},\operatorname{Big}}(f,\textbf{s})$ is overshadowed by that of $J^{\operatorname{Reg},\uppercase{\romannumeral 2\relax},\heartsuit}_{\operatorname{Geo},\operatorname{Big}}(f,\textbf{s})$, employing this particular choice of test function simplifies the calculations without compromising the main results. ### Ramanujan Parameters {#sec3.1.3} Let $\vartheta_p$ is a Ramanujan bound towards the Satake parameters of $\pi_p'.$ By [@KS03] one has $\vartheta_p\leq \frac{1}{2}-\frac{2}{n(n+1)+2}$ for $p\leq \infty.$ ### Uniform Parameter Growth {#sec3.14} Let $\pi=\otimes_{p\leq \infty}\pi_p$ be a fixed unitary automorphic representation of $G(\mathbb{A})$ with central character $\omega$ and level $M.$ By classification theorem $\pi$ is the Langlands quotient of $\operatorname{Ind}_{Q}^{G}\pi_1\otimes\cdots\otimes\pi_m,$ where $Q$ is a parabolic subgroup subgroup of $G,$ $\pi_1\otimes\cdots\otimes\pi_m$ is a discrete automorphic representation of the Levi of $Q(\mathbb{A}).$ In particular, if $\pi$ is cuspidal, then $Q=G,$ $m=1$ and $\pi_1=\pi.$ Let $T\geq 1$ and $C_{\infty}>c_{\infty}>0.$ We assume that $\pi$ exhibits uniform parameter growth of size $(T;c_{\infty},C_{\infty})$ as defined in [1.2](#subc.){reference-type="ref" reference="subc."}, given by $$c_{\infty}T\leq |\lambda_{\pi_{\infty},j}|\leq C_{\infty}T,\ \ 1\leq j\leq n+1.\tag{\ref{1.6}}$$ ### Level Structure {#seclevel} For $p<\infty$ we define Hecke congruence subgroup by $$\label{equ3.4} K_p(M):=\Big\{\begin{pmatrix} A'&\mathfrak{b}\\ \mathfrak{c}& d \end{pmatrix}\in G(\mathbb{Z}_p):\ \mathfrak{c}\in p^{e_p(M)}M_{1,n}(\mathbb{Z}_p)\Big\},$$ where $p\mid M$ and $e_p(M)$ is the valuation of $M$ at $p.$ In particular, $K_p(M)=K_p:=G(\mathbb{Z}_p)$ for all $p\nmid M.$ Similarly, we define the congruence subgroup $$\label{3.4.} K_p'(M'):=\Big\{\begin{pmatrix} A'&\mathfrak{b}\\ \mathfrak{c}& d \end{pmatrix}\in G'(\mathbb{Z}_p):\ \mathfrak{c}\in p^{e_p(M')}M_{1,n-1}(\mathbb{Z}_p)\Big\}.$$ It should be noted that we do not assume $(M,M')=1$. In fact, when $\omega'=\textbf{1}$, we will select a prime $p'\equiv 2\pmod{n}$ that is coprime to $MM'$. Additionally, we choose a primitive Dirichlet character $\omega''$ modulo $p'$ and replace $\pi'$ with $\sigma':=\pi'\otimes\omega''$, and replace $\pi$ with $\sigma:=\pi\otimes\omega''^{-1}$. In this particular case, we have $L\left(s,\pi\times\pi'\right) = L\left(s,\sigma\times\sigma'\right).$ Since $p'\equiv 2\pmod{n}$, it follows that $(n-1)\nmid p'$. Hence, the character $\omega''^n$ is primitive with modulus $p'.$ Consequently, the level of $\sigma$ is $Mp'^{n+1}$, and the level of $\sigma'$ is $M'p'^{n}$, indicating that they share a nontrivial common factor. ### Deformation Parameters {#specpara} Define $\mu \in (i\mathbb{R})^{m-1}$ as follows: if $m=1,$ let $\mu=0;$ if $m\geq 2,$ let $\mu=(\mu_1,\cdots,\mu_m)$ with $\mu_j=2^{-j}\exp(-2\sqrt{\log T})i,$ $1\leq j\leq m-1,$ and $\mu_m=-\mu_1-\cdots-\mu_{m-1}.$ Let $\pi_{\mu}:=\pi_1|\cdot|^{\mu_1}\boxplus \cdots\boxplus \pi_m|\cdot|^{\mu_m}.$ We need this auxiliary parameter to handle the case that $\pi$ is an Eisenstein series, e.g., cf. Corollary [Corollary 1](#cor1.2){reference-type="ref" reference="cor1.2"} in [1.1](#subc){reference-type="ref" reference="subc"}. ### Amplification Parameters {#sec3.1.6} Let $L\gg 1$ be such that $\log L\asymp \log T.$ Let $\mathcal{L}$ be a subset of the set $\{\text{prime $p$}:\ L<p\leq 2L,\ p\nmid MM'\}.$ Let $\boldsymbol{\ell}=(l_p)_{p\in\mathcal{L}}$ be a sequence of integers such that $1\leq l_p\leq n+1$. Let $\boldsymbol{\alpha}=(\alpha_p)_{p\in\mathcal{L}}$ be a sequence of complex numbers. ### Other Notation {#3.1.6} For a function $h$ on $G(\mathbb{A})$ or $G(\mathbb{Q}_p),$ $p\leq \infty,$ define $h^*(g)=\overline{h(g^{-1})}$ and $(h*h^*)(g)=\int h(gg'^{-1})h^*(g')dg',$ where $g'$ ranges over the domain of $h.$ For a set $X$, we denote by $\textbf{1}_X$ the characteristic function of $X.$ ## Construction of Test Functions: the archimedean place {#11.1.1} We appeal to the test functions ${f}_{\infty}$ constructed in [@Nel20] (cf. .2 and on p.80) in the manner described in [@NV21], . See also [@Nel21], Part 2. ### Construction of ${f}_{\infty}$ {#3.2.1} Let $\mathfrak{g}$ (resp. $\mathfrak{g}'$) be the Lie algebras of $G(\mathbb{R})$ (resp. $G'(\mathbb{R})$), with imaginal dual $\hat{\mathfrak{g}}$ (resp. $\hat{\mathfrak{g}}'$). One can choose an element $\tau\in\hat{\mathfrak{g}}$ with the restriction $\tau'=\tau\mid_{G'}\in \hat{\mathfrak{g}}',$ so that $\tau$ (resp. $\tau'$) lies in the coadjoint orbit $\mathcal{O}_{\pi_{\infty}}$ of $\pi_{\infty}$ (resp. $\mathcal{O}_{\pi_{\infty}'}$ of $\pi_{\infty}'$). Let $\tilde{f}^{\wedge}_{\infty}:$ $\hat{\mathfrak{g}}\rightarrow\mathbb{C}$ be a smooth bump function concentrated on $\{\tau+(\xi,\xi^{\bot}):\ \xi\ll T^{\frac{1}{2}+\varepsilon},\ \xi^{\bot}\ll T^{\varepsilon}\},$ where $\xi$ lies in the tangent space of $\mathcal{O}_{\pi_{\infty}}$ at $\tau,$ and $\xi^{\bot}$ has the normal direction. Let $\tilde{f}_{\infty}^{\sharp}\in C_c^{\infty}(G(\mathbb{R}))$ be the pushforward of the Fourier transform of $\tilde{f}_{\infty}^{\wedge}.$ Denote by $\tilde{f}_{\infty}^{\dag}$ the truncation of $\tilde{f}_{\infty}^{\sharp}$ at the essentially support, namely, $$\label{245} \mathop{\mathrm{supp}}\tilde{f}_{\infty}^{\dagger}\subseteq \big\{g\in G(\mathbb{R}):\ g=I_{n+1}+O(T^{-\varepsilon}),\ \operatorname{Ad}^*(g)\tau=\tau+O(T^{-\frac{1}{2}+\varepsilon})\big\}.$$ Then, in the sense of [@NV21], , the operator $\pi_{\infty}(\tilde{f}_{\infty}^{\dagger})$ is approximately a rank one projector with range spanned by a unit vector microlocalized at $\tau.$ Let $$\label{eq3.2} \tilde{f}_{\infty}(g):=\int_{Z(\mathbb{R})}\tilde{f}_{\infty}^{\dagger}(zg)\omega_{\infty}(z)d^{\times}z,\ \ g\in G(\mathbb{R}).$$ Set the archimedean test function to be the convolution $f_{\infty}=\tilde{f}_{\infty}*\tilde{f}_{\infty}^*,$ where $\tilde{f}_{\infty}^*(g):=\overline{\tilde{f}_{\infty}(g^{-1})}$ (cf. [3.1.8](#3.1.6){reference-type="ref" reference="3.1.6"}). ### Application of Transversality By definition, one has (cf. (14.13) in [@Nel21]) $$\label{250} \|\tilde{f}_{\infty}\|_{\infty}\ll_{\varepsilon} T^{\frac{n(n+1)}{2}+\varepsilon},$$ where $\|\cdot \|_{\infty}$ is the sup-norm. For $g\in\overline{G}(\mathbb{R}),$ we may write $$\begin{aligned} g=\begin{pmatrix} A&\mathfrak{b}\\ \ensuremath{\mskip 1mu\prescript{\smash{\mathrm t\mkern-3mu}}{}{\mathstrut\mathfrak{c}}}&d \end{pmatrix},\ \ \ g^{-1}=\begin{pmatrix} A'&\mathfrak{b}'\\ \ensuremath{\mskip 1mu\prescript{\smash{\mathrm t\mkern-3mu}}{}{\mathstrut\mathfrak{c}}}'&d' \end{pmatrix},\end{aligned}$$ where $A, A'\in M_{n,n}(\mathbb{R}),$ $\mathfrak{b}, \mathfrak{b}', \mathfrak{c}, \mathfrak{c}'\in M_{n,1}(\mathbb{R}),$ and $d, d'\in \mathbb{R}.$ Define $$\label{dg} d_{G'}(g):=\begin{cases} \min\big\{1, \|d^{-1}\mathfrak{b}\|+ \|d^{-1}\mathfrak{c}\|+ \|d'^{-1}\mathfrak{b}'\|+\|d'^{-1}\mathfrak{c}'\| \big\},&\ \text{if $dd'\neq 0$}\\ 1,&\ \text{if $dd'=0$}. \end{cases}$$ **Proposition 13** (Theorem 15.1 of [@Nel20]). *Let notation be as above. Then there is a fixed neighborhood $\mathcal{Z}$ of the identity in $Z'(\mathbb{R})$ with the following property. Let $g$ be in a small neighborhood of $I_{n+1}$ in $\overline{G}(\mathbb{R}).$ Let $r>0$ be small. Then $$\begin{aligned} \operatorname{Vol}\left(\big\{z\in\mathcal{Z}:\ \operatorname{dist}(gz\tau, G'(\mathbb{R})\tau)\leq r\big\}\right)\ll \frac{r}{d_{G'}(g)}. \end{aligned}$$ Here $\operatorname{dist}(\cdots)$ denotes the infimum over $g'\in G'(\mathbb{R})$ of $\|gz\tau-g'\tau\|,$ where $\|\cdot\|$ is a fixed norm on $\hat{\mathfrak{g}}.$* Proposition [Proposition 13](#prop3.1){reference-type="ref" reference="prop3.1"} (with $r=T^{-1/2+\varepsilon}$) will be used to detect the restriction $\operatorname{Ad}^*(g)\tau=\tau+O(T^{-\frac{1}{2}+\varepsilon})$ in the support of $\tilde{f}_{\infty}.$ ## Construction of Test Functions: ramified places {#sec3.3.} ### Test Functions at $p\mid M,$ $p\nmid M'$ {#11.1.3} Recall the Hecke congruence subgroup $$K_p(M):=\Big\{\begin{pmatrix} A'&\mathfrak{b}\\ \mathfrak{c}& d \end{pmatrix}\in G(\mathbb{Z}_p):\ \mathfrak{c}\in p^{e_p(M)}M_{1,n}(\mathbb{Z}_p)\Big\}.\tag{\ref{equ3.4}}$$ We define the local test function $f_p$ on $G(\mathbb{Q}_p),$ supported on $Z(\mathbb{Q}_p)\backslash K_p(M),$ by $$f_p(zk)=\operatorname{Vol}(\overline{K_p(M)})^{-1}\omega_p(z)^{-1}\omega_p(E_{n+1,n+1}(k))^{-1},\ \ z\in Z(\mathbb{Q}_p),\ k\in K_p(M),$$ where $\overline{K_p(M)}$ is the image of $K_p(M)$ in $\overline{G}(\mathbb{Q}_p),$ and $E_{n+1,n+1}(k)$ is the $(n+1,n+1)$-th entry of $k\in K_p(M).$ ### Test Functions at $p\mid M'$ {#11.1.4} Let $p\mid M'$. Denote by $m':=e_p(M')$ and $m'':=e_p(M'').$ Recall that $\psi_p$ is the fixed standard local additive character (cf. [3.1.1](#3.1.1.){reference-type="ref" reference="3.1.1."}). Define the Gauss sum by $$\begin{aligned} G(\omega_p',\psi_p):=\sum_{\substack{\alpha_n\in (\mathbb{Z}/p^{m''}\mathbb{Z})^{\times}}}\psi_p(\alpha_np^{-m''})\omega_p'(\alpha_n).\end{aligned}$$ It is well known that $|G(\omega_p',\psi_p)|=p^{\frac{m''}{2}}.$ In particular, $|G(\omega_p',\psi_p)|=1$ if $m''=0.$ Let $\boldsymbol{\alpha}=(\alpha_1,\cdots,\alpha_{n-1},\alpha_n),$ where $\alpha_j\in \mathbb{Z}_p/p^{m'}\mathbb{Z}_p\simeq \mathbb{Z}/p^{m'}\mathbb{Z},$ and $\alpha_n\in (\mathbb{Z}_p/p^{m''}\mathbb{Z}_p)^{\times}\simeq (\mathbb{Z}/p^{m''}\mathbb{Z})^{\times}.$ Denote by $\textbf{u}_{\boldsymbol{\alpha}}=\ensuremath{\mskip 1mu\prescript{\smash{\mathrm t\mkern-3mu}}{}{\mathstrut(\alpha_1p^{-m'},\cdots, \alpha_{n-1}p^{-m'},\alpha_n p^{-m''})}},$ and $u_{\boldsymbol{\alpha}}:=\begin{pmatrix} I_n&\textbf{u}_{\boldsymbol{\alpha}}\\ &1 \end{pmatrix}.$ Define $$\label{3.4} f_p(g):=\tilde{f}_p(g)*\tilde{f}_p^*(g),\ \ \ g\in G(\mathbb{Q}_p),$$ where $\tilde{f}_p^*(g):=\overline{\tilde{f}_p(g^{-1})}$ (cf. [3.1.8](#3.1.6){reference-type="ref" reference="3.1.6"}), and $$\label{3.5} \tilde{f}_p(g):=v_p\sum_{\substack{\alpha_j\in \mathbb{Z}/p^{m'}\mathbb{Z}\\ 1\leq j<n}}\sum_{\alpha_n\in (\mathbb{Z}/p^{m''}\mathbb{Z})^{\times}}\omega_p'(\alpha_n)\int_{Z(\mathbb{Q}_p)}\textbf{1}_{u_{\boldsymbol{\alpha}}K_p(M)}(zg)\omega_p(z)d^{\times}z.$$ Here $v_p:=\operatorname{Vol}(K_p'(M'))^{-1}\operatorname{Vol}(K_p(M))^{-1}p^{-(n-1)m'}G(\omega_p',\psi_p)^{-1},$ with $K_p(M)$ and $K_p'(M')$ being defined by [\[equ3.4\]](#equ3.4){reference-type="eqref" reference="equ3.4"} and [\[3.4.\]](#3.4.){reference-type="eqref" reference="3.4."}, respectively. Note that $|v_p|\asymp p^{ne_p(M)-\frac{m''}{2}}.$ ## Construction of Test Functions: amplification {#11.1.5} Let $p\in\mathcal{L}.$ For $l\geq 0,$ let $T_{p^l}=p^{-\frac{nl}{2}}\textbf{1}_{K_p\operatorname{diag}(p^l,I_n)K_p}$, the Hecke operator at $p,$ where $K_p=G(\mathbb{Z}_p).$ Then $T^*_{p^l}=p^{-\frac{nl}{2}}\textbf{1}_{K_p\operatorname{diag}(I_n,p^{-l})K_p}$ is the adjoint operator. Recall that $\boldsymbol{\ell}=(l_p)_{p\in\mathcal{L}}$. Define $$\begin{aligned} f_p(g):=\int_{Z(\mathbb{Q}_p)}T_{p^{l_p}}(zg)\omega_p(z)d^{\times}z,\ \ g\in G(\mathbb{Q}_p).\end{aligned}$$ Then for $g\in G(\mathbb{Q}_p),$ we have $$\begin{aligned} f_p^*(g):=\int_{Z(\mathbb{Q}_p)}T_{p^{l_p}}^*(zg)\omega_p(z)d^{\times}z. \end{aligned}$$ For $p\in \mathcal{L}$ and $1\leq i\leq l_{p},$ we define $$\begin{aligned} f_{p,i}(g):=\int_{Z(\mathbb{Q}_p)}\textbf{1}_{K_{p}\operatorname{diag}(p^i,I_{n-1},p^{-i})K_{p}}(zg)\omega_p(z)d^{\times}z,\ \ g\in G(\mathbb{Q}_p).\end{aligned}$$ ## Construction of Test Functions: remaining places {#11.1.6} Let $p\notin\mathcal{L}$ be a prime such that $p\nmid MM'.$ Let $$\begin{aligned} f_p(g):=\int_{Z(\mathbb{Q}_p)}\textbf{1}_{K_p}(zg)\omega_p(z)d^{\times}z,\ \ g\in G(\mathbb{Q}_p).\end{aligned}$$ ## The Space of Test Functions {#testfunction} ### Decomposition of the test function {#sec3.6.1} Stick notations in [3.2](#11.1.1){reference-type="ref" reference="11.1.1"}--[3.5](#11.1.6){reference-type="ref" reference="11.1.6"}. Let $p_0\in\mathcal{L}.$ By [@BM15 Lemma 4.4] there exist constants $c_{p_0,i}\ll 1$ such that $$\label{3.66} T_{p_0^{l_{p_0}}}*T_{p_0^{l_{p_0}}}^*=\sum_{i=0}^{l_{p_0}}c_{p_0,i}{p_0}^{-ni}\textbf{1}_{K_{p_0}\operatorname{diag}(p_0^i,I_{n-1},p_0^{-i})K_{p_0}}.$$ For $p_0, p_1, p_2\in\mathcal{L}$ with $p_1\neq p_2$ and $0\leq i\leq l_{p_0}$ as in [\[3.66\]](#3.66){reference-type="eqref" reference="3.66"}, we define $$\begin{aligned} f(g;i,p_0):=&(({f}_{\infty}*{f}_{\infty}^*)\otimes f_{p_0,i}\otimes \otimes_{p<\infty,\ p\neq p_0}f_p)(g),\\ f(g;p_1,p_2):=&(({f}_{\infty}*{f}_{\infty}^*)\otimes{f}_{p_1}\otimes {f}_{p_2}^* \otimes\otimes_{p<\infty,\ p\nmid p_1p_2}f_p)(g).\end{aligned}$$ Ultimately the test function (involving an amplifier) will be taken as $$\label{73} \sum_{p_1\neq p_2\in\mathcal{L}}\alpha_{p_1}\overline{\alpha_{p_2}}f(g;p_1,p_2)+\sum_{p_0\in\mathcal{L}}\sum_{i=0}^{l_{p_0}}c_{p_0,i}|\alpha_{p_0}|^2f(g;i,p_0),\ \ g\in G(\mathbb{A}).$$ Since [\[73\]](#73){reference-type="eqref" reference="73"} is a linear combination of $f(g;p_1,p_2)$'s and $f(g;i,p_0)$'s, in [7](#8.5.1){reference-type="ref" reference="8.5.1"}--[9](#sec10){reference-type="ref" reference="sec10"} below we will take $f\in \big\{f(\cdot;i,p_0), f(\cdot;p_1,p_2)\big\}$ and estimate the associated orbital integrals. In [10](#proof){reference-type="ref" reference="proof"} we put the calculations in [7](#8.5.1){reference-type="ref" reference="8.5.1"}--[9](#sec10){reference-type="ref" reference="sec10"} together to derive an upper bound for the amplified geometric side of Theorem [Theorem 4](#C){reference-type="ref" reference="C"} under the test function given in [\[73\]](#73){reference-type="eqref" reference="73"}. ### Some auxiliary definitions {#3.6.2} Let $f\in \big\{f(\cdot;i,p_0), f(\cdot;p_1,p_2)\big\}.$ Denote by $$\label{61} \begin{cases} \nu(f):={p_1}{p_2},\ \mathcal{N}_f:=p_1^{l_{p_1}/2}p_2^{l_{p_2}/2}&\ \text{if $f=f(g;p_1,p_2)$},\\ \nu(f):={p_0},\ \mathcal{N}_f:=p_0^{i},&\ \text{if $f=f(g;i,p_0)$}.\\ \end{cases}$$ Note that the $f$ relies on the levels $M,$ $M',$ $M'',$ places $p_0$ or $p_1, p_2\in \mathcal{L},$ and the archimedean representation $\pi_{\infty}$. Recall that we defined $\tilde{f}_p$ for $p\mid M'$ or $p=\infty$ in [\[3.5\]](#3.5){reference-type="eqref" reference="3.5"} and [\[eq3.2\]](#eq3.2){reference-type="eqref" reference="eq3.2"}, respectively. We extend this definition to cover the remaining primes as follows: for $p\nmid \nu(f)$, we set $\tilde{f}_p=f_p$, and for $p\mid\nu(f)$, we define $$\begin{aligned} \tilde{f}_p(g)=\int_{Z(\mathbb{Q}_p)}\textbf{1}_{K_p}(zg)\omega_p(z)d^{\times}z,\ \ g\in G(\mathbb{Q}_p).\end{aligned}$$ Let $\tilde{f}:=\otimes_{p\leq\infty}\tilde{f}_p.$ It is worth noting that $\tilde{f}*\tilde{f}^*$ and $f$ have identical local components, except at primes $p\mid\nu(f)$. We can consider $\tilde{f}*\tilde{f}^*$ as the test function without amplification, while $f$ incorporates the amplification structure. The function $\tilde{f}$ will be used to prove Proposition [Proposition 28](#thm6){reference-type="ref" reference="thm6"} in [5.3](#sec5.2){reference-type="ref" reference="sec5.2"}. ## Vanishing of $J^{\operatorname{Reg},\uppercase{\romannumeral 1\relax},\heartsuit}_{\operatorname{Geo},\operatorname{Big}}(f,\textbf{s})$ **Lemma 14**. *Let notation be as in [2.2.2](#2.2.2){reference-type="ref" reference="2.2.2"}. Let $f$ be defined in [3.6](#testfunction){reference-type="ref" reference="testfunction"}. Then $$\label{3.8} J^{\operatorname{Reg},\uppercase{\romannumeral 1\relax},\heartsuit}_{\operatorname{Geo},\operatorname{Big}}(f,\textbf{s})\equiv 0,\ \ \textbf{s}\in\mathbb{C}^2.$$* *Proof.* Recall the definition in [2.2.2](#2.2.2){reference-type="ref" reference="2.2.2"}: $$J^{\operatorname{Reg},\uppercase{\romannumeral 1\relax},\heartsuit}_{\operatorname{Geo},\operatorname{Big}}(f,\textbf{s})=\mathcal{F}_{0,1}J^{\operatorname{Big}}_{\operatorname{Geo}}(f,\textbf{s})+\mathcal{F}_{1,0}J^{\operatorname{Big}}_{\operatorname{Geo}}(f,\textbf{s})-\mathcal{G}_{\chi}(\textbf{s},\phi_1',\phi_2')+\widetilde{\mathcal{G}}_{\chi}(\textbf{s},\phi_1',\phi_2').\tag{\ref{62}}$$ Here, $\mathcal{G}_{\chi}(\textbf{s},\phi_1',\phi_2')$ and $\widetilde{\mathcal{G}}_{\chi}(\textbf{s},\phi_1',\phi_2')$ are automorphic periods that involve the Eisenstein series $E_P^{\dagger}(x,s;\widehat{f}_P,y)$ (defined in [\[276\]](#276){reference-type="eqref" reference="276"}) and $\widetilde{E}_P^{\dagger}(x,s;\widehat{f}_P,y)$ (defined in [\[60\]](#60){reference-type="eqref" reference="60"}), respectively. Similarly, $\mathcal{F}_{0,1}J^{\operatorname{Big}}_{\operatorname{Geo}}(f,\textbf{s})$ and $\mathcal{F}_{1,0}J^{\operatorname{Big}}_{\operatorname{Geo}}(f,\textbf{s})$ are automorphic periods that involve the Eisenstein series $E^{\dagger}(x,s;f_P,y)$ and $\widetilde{E}_P^{\dagger}(x,s;f_P,y)$ (defined in [2.2.2](#2.2.2){reference-type="ref" reference="2.2.2"}), respectively. Let $p\mid M'.$ By definition [\[3.4\]](#3.4){reference-type="eqref" reference="3.4"} we have $$\label{3.13} f_p(g)=v_p^2\sum_{\boldsymbol{\alpha}}\sum_{\boldsymbol{\beta}}\omega_p'(\alpha_n)\omega_p'(\beta_n)\int_{Z(\mathbb{Q}_p)}\textbf{1}_{u_{\boldsymbol{\alpha}}K_pu_{\boldsymbol{\beta}}^{-1}}(zg)\omega_p(z)d^{\times}z,$$ where $\boldsymbol{\alpha}=(\alpha_1,\cdots,\alpha_{n-1},\alpha_n),$ with $\alpha_j\in \mathbb{Z}/p^{m'}\mathbb{Z},$ $1\leq j<n,$ and $\alpha_n\in (\mathbb{Z}/p^{m''}\mathbb{Z})^{\times},$ and $\boldsymbol{\beta}=(\beta_1,\cdots,\beta_{n-1},\beta_n),$ with $\beta_j\in \mathbb{Z}/p^{m'}\mathbb{Z},$ $1\leq j<n,$ and $\beta_n\in (\mathbb{Z}/p^{m''}\mathbb{Z})^{\times}.$ The nontriviality of $\omega'$ implies the orthogonality relation $$\sum_{\beta_n\in (\mathbb{Z}/p^{m''}\mathbb{Z})^{\times}}\omega_p'(\beta_n)=0.$$ As a consequence, for $\mathbf{x}\in M_{1,n}(\mathbb{Q}_p),$ $y_p\in G'(\mathbb{Q}_p),$ we have $$\label{eq3.9} \int_{N_P(\mathbb{Q}_p)}f_p\left(u(\mathbf{x})n\iota(y)\right)dn=\int_{N_P(\mathbb{Q}_p)}f_p\left(nu(\mathbf{x})\iota(y)\right)dn=0.$$ Let $\textbf{s}=(s_1,s_2)$ with $s_1> 2$ and $s_2>2.$ Then the aforementioned Eisenstein series converge absolutely. Inserting [\[eq3.9\]](#eq3.9){reference-type="eqref" reference="eq3.9"} into their series representations yields $$\begin{aligned} E_P^{\dagger}(x,s;\widehat{f}_P,y)= E^{\dagger}(x,s;f_P,y)=\widetilde{E}_P^{\dagger}(x,s;\widehat{f}_P,y)=\widetilde{E}_P^{\dagger}(x,s;f_P,y)\equiv 0.\end{aligned}$$ As a result, the RHS of [\[62\]](#62){reference-type="eqref" reference="62"} vanishes, implying that $J^{\operatorname{Reg},\uppercase{\romannumeral 1\relax},\heartsuit}_{\operatorname{Geo},\operatorname{Big}}(f,\textbf{s})$ is identically zero in the region $s_1>2$ and $s_2>2$. Recall that $J^{\operatorname{Reg},\uppercase{\romannumeral 1\relax},\heartsuit}_{\operatorname{Geo},\operatorname{Big}}(f,\textbf{s})$ is a meromorphic function. So it vanishes identically everywhere. Consequently, [\[3.8\]](#3.8){reference-type="eqref" reference="3.8"} holds. ◻ ## The Amplified Relative Trace Formula {#sec3.7} Let the local and global data be chosen as in [3.1](#3.1){reference-type="ref" reference="3.1"}--[3.6](#testfunction){reference-type="ref" reference="testfunction"}: $\mathcal{L},$ $\boldsymbol{\alpha},$ $\boldsymbol{\ell},$ and $f\in \big\{f(\cdot;i,p_0), f(\cdot;p_1,p_2)\big\}.$ ### The Spectral Side {#ampsp} Corresponding to [\[73\]](#73){reference-type="eqref" reference="73"} we define $\mathcal{J}_{\mathop{\mathrm{Spec}}}^{\heartsuit}(\boldsymbol{\alpha},\boldsymbol{\ell})$ by $$\begin{aligned} \sum_{p_1\neq p_2\in\mathcal{L}}\alpha_{p_1}\overline{\alpha_{p_2}}J_{\mathop{\mathrm{Spec}}}^{\operatorname{Reg},\heartsuit}(f(\cdot;p_1,p_2),\mathbf{0})+\sum_{p_0\in\mathcal{L}}\sum_{i=0}^{l_{p_0}}c_{p_0,i}|\alpha_{p_0}|^2J_{\mathop{\mathrm{Spec}}}^{\operatorname{Reg},\heartsuit}(f(\cdot;i,p_0),\mathbf{0}).\end{aligned}$$ ### The Geometric Side {#sec3.7.2} Let $\mathcal{J}_{\operatorname{Geo}}^{\heartsuit}(\boldsymbol{\alpha},\boldsymbol{\ell})$ be defined by $$\begin{aligned} \sum_{p_1\neq p_2\in\mathcal{L}}\alpha_{p_1}\overline{\alpha_{p_2}}J_{\operatorname{Geo}}^{\operatorname{Reg},\heartsuit}(f(\cdot;p_1,p_2),\mathbf{0})+\sum_{p_0\in\mathcal{L}}\sum_{i=0}^{l_{p_0}}c_{p_0,i}|\alpha_{p_0}|^2J_{\operatorname{Geo}}^{\operatorname{Reg},\heartsuit}(f(\cdot;i,p_0),\mathbf{0}),\end{aligned}$$ where $i$ and $c_{p_0,i}$ are defined by the Hecke relation [\[3.66\]](#3.66){reference-type="eqref" reference="3.66"}. ### The Amplified Relative Trace Formula {#3.7.3} As a consequence of Theorem [Theorem 4](#C){reference-type="ref" reference="C"} (with $\textbf{s}=(0,0)$) we have the following. **Theorem 5**. *Let notation be as before. Then $$\label{3.7} \mathcal{J}_{\mathop{\mathrm{Spec}}}^{\heartsuit}(\boldsymbol{\alpha},\boldsymbol{\ell})=\mathcal{J}_{\operatorname{Geo}}^{\heartsuit}(\boldsymbol{\alpha},\boldsymbol{\ell}).$$* *Remark 15*. The evaluation at $\textbf{s}=(0,0)$ is obligatory for the nonvanishing result as the spectral side gives the central $L$-values. # Construction of Test Vectors in $\pi'$ {#sec4.} Recall that $\pi'=\otimes_{p\leq \infty}\pi_p'$ is a fixed unitary cuspidal representation of $G'(\mathbb{A})$ with central character $\omega'$ and level $M'.$ The main goal of this section is to construct a suitable vector $\phi'\in \pi'$ which will be inserted into the relative trace formula Theorem [Theorem 4](#C){reference-type="ref" reference="C"} (cf. [\[70\]](#70){reference-type="eqref" reference="70"}). The vector $\phi'\in \pi'$ will be chosen as the cusp form whose Whittaker function is $\otimes_{p\leq \infty}W_p',$ where - at $p\mid M',$ the Whittaker function $W_p'$ is the new vector with $W_p'(I_n)=1$, - at a finite place $p$ with $p\nmid M',$ $W_p'$ is spherical vector with $W_p'(I_n)=1$, - at $p\mid\infty,$ $W_{\infty}'$ is in the Whittaker model of $\pi_{\infty}'$ constructed in [4.1](#spec){reference-type="ref" reference="spec"} below. ## Construction of the Test Whittaker Vector $W_{\infty}'$ {#spec} Our goal is to construct a normalized vector $W_{\infty}'$ in the Whittaker model of $\pi_{\infty}'$ such that $\mathop{\mathrm{Re}}(W_{\infty}'(x))$ admits a suitable lower bound for all $x\in \overline{B'}(\mathbb{R})$ with $x=I_n+O(T^{-1/2-\varepsilon}).$ In particular, when $\pi_{\infty}'$ is a fixed *unramified* and is *tempered*, i.e., satisfies the Ramanujan-Selberg conjecture, the construction of $W_{\infty}'$ follows readily from [@Nel21], , Part 2, with a slight modification. To handle the general $\pi_{\infty}',$ we adapt the following new inputs to Nelson's construction: Langlands classification theorem (cf. [@Lan89]), structure of generic representation (cf. [@Jac09]), and Casselman's subrepresentation theorem (cf. [@Cal75]) that for a real reductive Lie group, any irreducible admissible $(\mathfrak{g},K)$-module can be embedded as a *subrepresentation* of a parabolically induced representation. With the above ingredients, the construction of $W_{\infty}'$ can be reduced to [@Nel21]. ### Classification of $\pi_{\infty}'$ Let $\pi_{\infty}'$ be the archimedean component of $\pi'.$ Denote by $\mathrm{SL}_2^{\pm}(\mathbb{R})=\big\{g\in\mathrm{GL}_2(\mathbb{R}):\ \det g\in\{\pm 1\big\}.$ Let $l\geq 1.$ Denote by $D_l=\operatorname{Ind}_{\mathrm{SL}_2(\mathbb{R})}^{\mathrm{SL}_2^{\pm}(\mathbb{R})}D_l^+$ the sidcrete series, where $D_l^{+}$ acts in the space of analytic functions $h$ in the upper half-plane $\mathbb{H}$ with $\|h\|^2=\int_{\mathbb{H}}|h(z)|^2y^{l-1}dxdy<\infty,$ the action by $g=\begin{pmatrix} a&b\\ c&d \end{pmatrix}$ being $$D_l^+(g)h(z)=(cz+d)^{-l-1}h\left(\frac{az+b}{cz+d}\right).$$ By the classification theorem (cf. [@Lan89]) $\pi_{\infty}'$ is an irreducible quotient of an parabolic induction of the form $\operatorname{Ind}_{Q'(\mathbb{R})}^{G'(\mathbb{R})}(\sigma_1,\cdots, \sigma_r),$ for some parabolic subgroup $Q'$ of type $(n_1,\cdots,n_r),$ with $n_1+\cdots+n_r=n$ and $n_1,\cdots, n_r\in \{1,2\}.$ Here $\sigma_i$ is of the form $\textbf{1}\otimes |\cdot|_{\mathbb{R}}^{\lambda}$ or $\operatorname{sgn}\otimes |\cdot|_{\mathbb{R}}^{\lambda}$ if $n_i=1,$ and is of the form $D_l\otimes|\det(\cdot )|_{\mathbb{R}}^{\lambda}$ if $n_i=2,$ for some $\lambda\in\mathbb{C}.$ Write $\sigma_i=\sigma_{i}^{\circ}\otimes|\det (\cdot)|_{\mathbb{R}}^{\lambda_i},$ where $\sigma_{i}^{\circ}$ is of the form $\textbf{1}, \operatorname{sgn}$ if $n_i=1,$ and is $D_{l_i}$ for some $l_i\geq 1$ if $n_i=2.$ We may assume $n_1=\cdots =n_{r'}=1$ and $n_{r'+1}=\cdots=n_r=2.$ Note that $\pi_{\infty}'$ is generic. By [@Jac09], $\pi_{\infty}'$ is actually an induction representation. Hence we may write $\pi_{\infty}'=\operatorname{Ind}_{Q'(\mathbb{R})}^{G'(\mathbb{R})}(\sigma_1,\cdots, \sigma_r).$ Furthermore, by subrepresentation theorem, we can realize $D_{l_i}$ as a subrepresentation of the principal series $\operatorname{Ind}|\cdot|^{l_i}\otimes|\cdot|^{-l_i}$ of $\mathrm{GL}(2,\mathbb{R})$ (cf. [@Kna79]), $r'<i\leq r.$ Therefore, one can regard vectors in $\pi_{\infty}'$ as scalar functions on $B'(\mathbb{R})$ which are $N'(\mathbb{R})$-invariant. For $a=(a_1,\cdots, a_{r'}, a_{r'+1,1},a_{r'+1,2},\cdots, a_{r,1},a_{r,2})\in A'(\mathbb{R}),$ where $A'$ is the Levi of $B',$ we define the character $\chi$ associated to $\pi_{\infty}'$ by $$\begin{aligned} \chi(a)=\prod_{i=1}^{r'}|a_i|_{\mathbb{R}}^{\lambda_i}\cdot \prod_{j=r'+1}^r|a_{j,1}|_{\mathbb{R}}^{l_j+\lambda_j}|a_{j,2}|_{\mathbb{R}}^{-l_j+\lambda_j}.\end{aligned}$$ Then $\pi_{\infty}'$ can be realized as a subrepresentation of $\operatorname{Ind}_{B'(\mathbb{R})}^{G'(\mathbb{R})}\chi.$ **Definition 16**. We say that $\pi_{\infty}'$ is in the *stable range* relative to $\pi_{\infty}$ if $\lambda_i\ll T^{1-\varepsilon},$ $1\leq i\leq r',$ and $\pm l_i+\lambda_i\ll T^{1-\varepsilon},$ $r'+1\leq i\leq r.$ The concept of "stable range" here could be regarded as an archimedean counterpart of the phenomenon in the $p$-adic places, which was first studied by [@MR12]. ### Functions on $N'\backslash G'$ {#4.1.2} Let $\overline{N'}$ be the opposite of $N'.$ Fix $\beta\in C_c^{\infty}(A'(\mathbb{R}))$ such that $\int_{A'(\mathbb{R})} \beta(a^{-1})\chi(a)d^{\times}a=1.$ Define $$\begin{aligned} \mathcal{J}[\theta,\beta](g'):=\begin{cases} \delta_{B'}^{\frac{1}{2}}(a)\beta(a)\theta(u),&\ \text{if $g'=au\in A'(\mathbb{R})\times \overline{N'}(\mathbb{R})$}\\ 0, &\ \text{if $g'\notin A'(\mathbb{R})\times \overline{N'}(\mathbb{R})$}. \end{cases}\end{aligned}$$ Fix $h_0\in C_c^{\infty}(\mathfrak{g}')$ with $\int h_0=1.$ Recall that $\theta$ is the standard generic character of $N'(\mathbb{R}).$ Define $\theta_T$ by setting $\theta_T(\exp(x))=\theta(\exp(Tx))$ for $x\in\operatorname{Lie}(\overline{N'}(\mathbb{R})).$ Denote by $\theta^{\dagger}\in \operatorname{Lie}(G'(\mathbb{R}))$ the element whose restriction to $\operatorname{Lie}(\overline{N'}(\mathbb{R}))$ is the pullback $\exp^*\theta$ and whose restriction to $\operatorname{Lie}(\overline{B'}(\mathbb{R}))$ is trivial. In particular, $\theta(\exp(x))=\exp(\langle x,\theta^{\dagger}(x)\rangle)$ for $x\in \operatorname{Lie}(\overline{N'}(\mathbb{R})).$ Let $\gamma_0$ be the smooth measure on $\mathfrak{g}'$ characterized as follows: for any locally integrable function $\Phi$ on $G'(\mathbb{R}),$ $$\label{75} \int_{G'(\mathbb{R})}\gamma_0\cdot \Phi=\int_{\mathfrak{g}'}h_0(x)\Phi(\exp(xT^{-1/2}))\exp(-\langle T\theta^{\dagger}, xT^{-1/2}\rangle)dx.$$ Let $\gamma=\gamma_0^**\gamma_0.$ Define $$\label{65} \mathcal{J}[\theta_T,\beta,\gamma](g'):=\int_{G'(\mathbb{R})}\gamma(g'')\mathcal{J}[\theta_T,\beta](g'g'')dg''$$ and $\mathcal{J}_T[\theta,\beta,\gamma](g'):=\mathcal{J}[\theta_T,L(T^{\rho^{\vee}})\beta,\gamma](g')$ is defined similarly (by replacing $\beta$ with $L(T^{\rho^{\vee}})\beta$ in [\[65\]](#65){reference-type="eqref" reference="65"}), where $\rho$ is the half sum of the positive root relative to $(G',B')$ and $L(T^{\rho^{\vee}})\beta(a):=\delta_{B'}^{-\frac{1}{2}}(T^{\rho^{\vee}})\beta(T^{\rho^{\vee}}a),$ $a\in A'(\mathbb{R}).$ The Mellin transform $$\begin{aligned} \mathcal{M}_T[\theta,\beta,\gamma;\chi](g'):=\int_{A'(\mathbb{R})}\delta_{B'}^{\frac{1}{2}}(a)\mathcal{J}_T[\theta,\beta,\gamma](a^{-1}g')\chi(a)d^{\times}a,\end{aligned}$$ which is well defined, gives a scalar valued section in $\operatorname{Ind}_{B'(\mathbb{R})}^{G'(\mathbb{R})}\chi,$ which maps to $\operatorname{Ind}_{Q'(\mathbb{R})}^{G'(\mathbb{R})}(\sigma_1,\cdots, \sigma_r).$ Define the Whittaker function $$\begin{aligned} W_{\infty}'(g';\theta_T):=\int_{\overline{N'}(\mathbb{R})}\mathcal{M}_T[\theta,\beta,\gamma;\chi](ug')\overline{\theta}_T(u)du.\end{aligned}$$ Changing variables we obtain $\mathcal{M}_T[\theta,\beta,\gamma;\chi](g')=\chi(T^{\rho^{\vee}})\mathcal{M}[\theta_T,\beta,\gamma;\chi](g'),$ where $$\begin{aligned} \mathcal{M}[\theta_T,\beta,\gamma;\chi](g'):=\int_{A'(\mathbb{R})}\delta_{B'}^{\frac{1}{2}}(a)\mathcal{J}[\theta_T,\beta,\gamma](a^{-1}g')\chi(a)d^{\times}a.\end{aligned}$$ Since $\int_{A'(\mathbb{R})} \beta(a)\overline{\chi}(a)d^{\times}a=1,$ then by unfolding we have $$\begin{aligned} W_{\infty}'(g';\theta_T)=\chi(T^{\rho^{\vee}})\int_{\overline{N'}(\mathbb{R})}\mathcal{M}[\theta_T,\beta,\gamma_0;\chi](u)\overline{\mathcal{M}[\theta_T,\beta,\gamma_0*g'^{-1};\overline{\chi}^{-1}](u)}du.\end{aligned}$$ ### Construction of $W_{\infty}'$ and $\phi'$ {#sec4.1.3} **Definition 17**. Let $w_{n-1}$ be the long Weyl element in the Weyl group associated to $(G', B').$ For $g'\in G'(\mathbb{R}),$ let $$\begin{aligned} W_{\infty}'(g'):=\chi(T^{-\rho^{\vee}})T^{\frac{n(n-1)}{8}}\int_{N'(\mathbb{R})}\mathcal{M}_T[\theta,\beta,\gamma;\chi](w_{n-1}ug')\overline{\theta}(u)du.\end{aligned}$$ Let $\phi'\in \pi'$ be the cusp form whose Whittaker function is $\otimes_{p\leq \infty}W_p',$ where for $p<\infty ,$ $W_p'$ is the normalized local new Whittaker function. ## Miscellaneous Estimates In this section we prove various bounds for the Mellin transform $\mathcal{M}[\theta_T,\beta,\gamma_0;\chi]$ and the Whittaker function $W_{\infty}'(g';\theta_T).$ Recall $$\begin{aligned} \mathcal{M}[\theta_T,\beta,\gamma_0;\chi](u)=\int_{A'(\mathbb{R})}\delta_{B'}^{\frac{1}{2}}(a)\int_{G'(\mathbb{R})}\gamma_0(g)\mathcal{J}[\theta_T,\beta](a^{-1}ug)dg\chi(a)d^{\times}a.\end{aligned}$$ Since $\beta$ has the property that $\int_{A'(\mathbb{R})} \beta(a^{-1})\chi(a)d^{\times}a=1,$ then $$\label{74} \mathcal{M}[\theta_T,\beta,\gamma_0;\chi](u)=\int_{G'(\mathbb{R})}\gamma_0(g)\delta_{B'}^{\frac{1}{2}}((ug)_{A'})\theta_T((ug)_{\overline{N'}})\chi((ug)_{A'})dg.$$ **Lemma 18**. *Let notation be as before. Suppose that $\pi_{\infty}'$ is in the *stable range* relative to $\pi_{\infty}.$ Let $u\in \overline{N'}(\mathbb{R})$ with $u=I_n+O(T^{-1/2}).$ Then $$\label{78} \mathcal{M}[\theta_T,\beta,\gamma_0;\chi](u)=\theta_T(u)+O(T^{-\varepsilon}).$$* *Proof.* Substituting [\[75\]](#75){reference-type="eqref" reference="75"} into [\[74\]](#74){reference-type="eqref" reference="74"} we have $$\begin{aligned} \mathcal{M}[\theta_T,\beta,\gamma_0;\chi](u)=&\int_{\mathfrak{g}'}h_0(x)\delta_{B'}^{\frac{1}{2}}((u\exp(xT^{-1/2}))_{A'})\theta_T((u\exp(xT^{-1/2}))_{\overline{N'}})\\ &\qquad \exp(-\langle T\theta^{\dagger},xT^{-1/2}\rangle)\chi((u\exp(xT^{-1/2}))_{A'})dg.\end{aligned}$$ Write $u=\exp(y)$ for $y\in\operatorname{Lie}(\overline{N'}),$ with $\|y\|\ll T^{-1/2}.$ Then $$(u\exp(xT^{-1/2}))_{\overline{N'}}=\exp(y+x_{\overline{N'}}T^{-1/2}+O(T^{-1})),$$ where $x_{\overline{N'}}$ is the projection of $x$ into $\operatorname{Lie}(\overline{N'}).$ So $$\theta_T((u\exp(xT^{-1/2}))_{\overline{N'}})=\theta_T(u)\exp(\langle T\theta^{\dagger},xT^{-1/2}\rangle) +o(1).$$ Similarly, $(u\exp(xT^{-1/2}))_{A'}=I_n+O(T^{-1/2}).$ Therefore, $$\delta_{B'}^{\frac{1}{2}}((u\exp(xT^{-1/2}))_{A'})=1+O(T^{-1})$$ and $$\chi((u\exp(xT^{-1/2}))_{A'})=1+O(T^{-\varepsilon}),$$ if $\lambda_i\ll T^{1-\varepsilon},$ $1\leq i\leq r',$ and $\pm l_i+\lambda_i\ll T^{1-\varepsilon},$ $r'+1\leq i\leq r.$ Hence $$\begin{aligned} \mathcal{M}[\theta_T,\beta,\gamma_0;\chi](u)=&(\theta_T(u)+O(T^{-\varepsilon}))\int_{\mathfrak{g}'}h_0(x)dg=\theta_T(u)+O(T^{-\varepsilon}),\end{aligned}$$ as $\int h_0=1$ by out assumption. Then [\[78\]](#78){reference-type="eqref" reference="78"} holds, with the implied constant depending only on $\varepsilon.$ ◻ **Lemma 19**. *Let notation be as before. Suppose that $\pi_{\infty}'$ is in the *stable range* relative to $\pi_{\infty}.$ Let $g\in B'(\mathbb{R})$ with $g=I_n+O(T^{-1/2-\varepsilon}).$ Then $$\label{79} \int_{\overline{N'}(\mathbb{R})}\big|\mathcal{M}[\theta_T,\beta,\gamma_0*g'^{-1}-\gamma_0;\overline{\chi}^{-1}](u)\big|^2du\ll T^{-\frac{n(n-1)}{4}-2\varepsilon+o(1)},$$ where the implied constant depends at most on $\varepsilon.$* *Proof.* Denote by $\tilde{\gamma}_0=\gamma_0*g'^{-1}-\gamma_0.$ Then By fundamental calculus theorem (cf. Proposition 6.53 of [@Nel21]), we have $$\mathcal{J}_T[\theta,\beta,\tilde{\gamma}_0](u)=T^{-\varepsilon}\mathcal{J}_T[\theta,\beta^{\dagger},\gamma_0^{\dagger}](u)$$ for some $\beta^{\dagger}$ and $\gamma_0^{\dagger}$ of similar growth condition to $\beta$ and $\gamma_0,$ respectively. By Proposition 10.16 in loc. cit. one can decompose $\mathcal{J}_T[\theta,\beta^{\dagger},\gamma_0^{\dagger}](u)$ as $\mathcal{J}_T^{\sharp}[\theta,\beta^{\dagger},\gamma_0^{\dagger}](u)+\mathcal{J}_T^{\flat}[\theta,\beta^{\dagger},\gamma_0^{\dagger}](u),$ where - $\mathcal{J}_T^{\sharp}[\theta,\beta^{\dagger},\gamma_0^{\dagger}](T^{-\rho^{\vee}}a^{-1}u)\neq 0$ unless $\|a\|\ll T^{o(1)},$ $u=I_n+O(T^{-1/2+o(1)});$ - $\|\mathcal{J}_T^{\sharp}[\theta,\beta^{\dagger},\gamma_0^{\dagger}](T^{-\rho^{\vee}}a^{-1}u)\neq 0\|_{\infty}\ll \delta_{B'}^{1/2}(T^{-\rho^{\vee}})T^{o(1)};$ - $\kappa(\mathcal{J}_T^{\flat}[\theta,\beta^{\dagger},\gamma_0^{\dagger}](T^{-\rho^{\vee}}a^{-1}u)\neq 0)\ll T^{-\infty}$ for each fixed seminorm $\kappa.$ Let $\mathcal{M}^{\sharp}$ (resp. $\mathcal{M}^{\flat}$) the Mellin transform of $\mathcal{J}^{\sharp}[\theta_T,\beta^{\dagger},\gamma_0^{\dagger}]$ (resp. $\mathcal{J}^{\flat}[\theta_T,\beta^{\dagger},\gamma_0^{\dagger}]$) relative to $\overline{\chi}^{-1}.$ Using the relation $\mathcal{M}_T[\theta,\beta,\gamma;\chi](g')=\chi(T^{\rho^{\vee}})\mathcal{M}[\theta_T,\beta,\gamma;\chi](g'),$ we obtain, parallel to Lemma 10.20 in loc. cit., that $$\int_{\overline{N'}(\mathbb{R})}\big|\mathcal{M}^{\flat}[\theta_T,\beta^{\dagger},\gamma_0^{\dagger};\overline{\chi}^{-1}](u)\big|^2du\ll T^{-\infty},$$ and $$\begin{aligned} \int_{\overline{N'}(\mathbb{R})}\big|\mathcal{M}^{\sharp}[\theta_T,\beta^{\dagger},\gamma_0^{\dagger};\overline{\chi}^{-1}](u)\big|^2du\ll T^{-\frac{n(n-1)}{4}+o(1)},\end{aligned}$$ where the integral relative to $a,$ i.e., the Mellin transform, as it supports in $\|a\|\ll T^{o(1)}$ contributes $T^{o(1)};$ and the factor $T^{-\frac{n(n-1)}{4}}$ comes from the integral relative to $u\in \overline{N'}(\mathbb{R})$ (whose dimension is $n(n-1)/2$) as it supports in $I_n+O(T^{-1/2+o(1)}).$ Hence [\[79\]](#79){reference-type="eqref" reference="79"} follows. ◻ **Lemma 20**. *Let notation be as before. Let $|\lambda_i|=o(T),$ $1\leq i\leq n.$ Then for all $g'\in B'(\mathbb{R})$ with $g'=I_n+O(T^{-1/2-\varepsilon})$ for some fixed $\varepsilon>0,$ we have $$\label{81} \frac{W_{\infty}'(g';\theta_T)}{|\chi(T^{\rho^{\vee}})|^2}=\int_{\overline{N'}(\mathbb{R})}\big|\mathcal{M}[\theta_T,\beta,\gamma_0;\chi](u)\big|^2du+O\left(T^{-\frac{n(n-1)}{4}-2\varepsilon+o(1)}\right).$$ In particular, we have $$\label{80} T^{-\frac{n(n-1)}{4}}\cdot |\chi(T^{\rho^{\vee}})|^2\ll \mathop{\mathrm{Re}}(W_{\infty}'(g';\theta_T))\ll T^{-\frac{n(n-1)}{4}+\varepsilon}\cdot |\chi(T^{\rho^{\vee}})|^2.$$* *Proof.* Recall that $W_{\infty}'(g'):=\chi(T^{-\rho^{\vee}})W_{\infty}'(g';\theta_T)$ and $$\begin{aligned} W_{\infty}'(g';\theta_T)=\chi(T^{\rho^{\vee}})\int_{\overline{N'}(\mathbb{R})}\mathcal{M}[\theta_T,\beta,\gamma_0;\chi](u)\overline{\mathcal{M}[\theta_T,\beta,\gamma_0*g'^{-1};\overline{\chi}^{-1}](u)}du.\end{aligned}$$ Then [\[81\]](#81){reference-type="eqref" reference="81"} follows from Lemma [Lemma 19](#lem4.4){reference-type="ref" reference="lem4.4"}, and [\[80\]](#80){reference-type="eqref" reference="80"} is a consequence of Lemma [Lemma 18](#lem4.3){reference-type="ref" reference="lem4.3"}. ◻ ## Bounds of Norms {#sec4.5} $\langle \phi',\phi'\rangle :=\int_{[\overline{G'}]}\big|\phi'(x)\big|^2dx.$ Let $$\label{W_inf} \|W_{\infty}'\|_2:=\Bigg[\int_{N_H(\mathbb{R})\backslash H(\mathbb{R})}\big|W_{\infty}'(h)\big|^2dh\Bigg]^{1/2},$$ where $W_{\infty}'$ is defined by Definition [Definition 17](#def4.2){reference-type="ref" reference="def4.2"}. **Lemma 21**. *Let notation be as before. Let $H=\operatorname{diag}(\mathrm{GL}(n-1), 1).$ Denote by $N_H=N'\cap H.$ Then $$\label{83} \|W_{\infty}'\|_2\ll T^{o(1)},\ \ \langle \phi',\phi'\rangle \asymp_{M'} \|W_{\infty}'\|_2L(1,\pi',\operatorname{Ad}),$$ where the implied constant relies on $\pi_{\infty}'$ and $M',$ respectively.* *Proof.* Note that the measure on $G'(\mathbb{R})$ is a multiple of the measure pushed forward from the map $B'\times\overline{N'}\rightarrow G',$ $(b,u)\mapsto bu,$ whose image is dense in $G'$ by the Bruhat decomposition. Recall that $$\begin{aligned} W_{\infty}'(g')=T^{\frac{n(n-1)}{8}}\int_{N'(\mathbb{R})}\mathcal{M}[\theta_T,\beta,\gamma;\chi](w_{n-1}ug')\overline{\theta}(u)du,\end{aligned}$$ where $$\mathcal{M}[\theta_T,\beta,\gamma;\chi](g')=\int_{A'(\mathbb{R})}\delta_{B'}^{\frac{1}{2}}(c)\mathcal{J}[\theta_T,\beta,\gamma](c^{-1}g')\chi(c)d^{\times}c$$ is the Mellin transform of $\mathcal{J}[\theta_T,\beta,\gamma]$ as in [4.1.2](#4.1.2){reference-type="ref" reference="4.1.2"}. As a consequence, we have $$\label{4.3.93} \int_{N_H(\mathbb{R})\backslash H(\mathbb{R})}\big|W_{\infty}'(h)\big|^2dh\asymp T^{\frac{n(n-1)}{4}}\int_{K'_{\infty}}\big|\mathcal{M}[\theta_T,\beta,\gamma;\chi]\big|^2dk,$$ where the implied constant is absolute. By Lemma 10.20 in [@Nel21] we have $$\label{4.3.94} \int_{K'_{\infty}}\big|\mathcal{M}_T[\theta,\beta,\gamma;\chi]\big|^2dk\ll |\chi(T^{\rho^{\vee}})|^2\cdot T^{-\frac{n(n-1)}{4}}.$$ Then the upper bound in [\[83\]](#83){reference-type="eqref" reference="83"} follows from [\[4.3.93\]](#4.3.93){reference-type="eqref" reference="4.3.93"}, [\[4.3.94\]](#4.3.94){reference-type="ref" reference="4.3.94"} and the relation that $$\mathcal{M}_T[\theta,\beta,\gamma;\chi](g')=\chi(T^{\rho^{\vee}})\mathcal{M}[\theta_T,\beta,\gamma;\chi](g').$$ So the first inequality in [\[83\]](#83){reference-type="eqref" reference="83"} holds. The second inequality in [\[83\]](#83){reference-type="eqref" reference="83"} is a well known consequence of Rankin-Selberg convolution. ◻ ## Siegel Domain and Bounds of Cusp Forms {#sec4.4} Let $H=\operatorname{diag}(\mathrm{GL}(n-1),1).$ Denote by $A_H(\mathbb{R}):=\big\{\operatorname{diag}(a_1,\cdots,a_{n-1},1)\in A'(\mathbb{R}):\ a_1\geq\cdots\geq a_{n-1}>0\big\}.$ As a consequence of Siegel's theorem (cf. [@Bor62]) there exists a compact subset $\Omega$ of $P_0(\mathbb{R})$ such that for every nonnegative integrable function $h$ on $P_0'(\mathbb{Q})\backslash \overline{G'}(\mathbb{A})$ one has $$\label{87} \int_{P_0'(\mathbb{Q})\backslash \overline{G'}(\mathbb{A})}h(g)dg\leq \int_{A_H(\mathbb{R})}\int_{\Omega K_{\operatorname{fin}}'}h(ag)dg\delta_{B'}^{-1}(a)d^{\times}a,$$ Note that [\[87\]](#87){reference-type="eqref" reference="87"} is a variant of [@Nel21] Propositions 2.25 for parabolic subgroups. **Lemma 22**. *Let notation be as before. Then there exists an absolute constant $C_0$ such that $$\label{88} \int_{\Omega K_{\operatorname{fin}}'}h(ag)dg\ll \delta_{B'}(a)\int_{P_0'(\mathbb{Q})\backslash \overline{G'}(\mathbb{A})}h(g)\textbf{1}_{C_0|\det a|_{\infty}\leq |\det g|\leq 2C_0|\det a|_{\infty}}dg,$$ where the implied constant is absolute.* *Proof.* Following the proof of [@Nel21] Propositions 2.26 until the last step: $$\label{90} \int_{\Omega K_{\operatorname{fin}}'}h(ag)dg\ll \delta_{B'}(a)\int_{[N']}\int_{a\Omega_{A'}}\int_{K'}h(ua'k)dk\delta_{B'}^{-1}(a')d^{\times}a'du,$$ where $\Omega_{A'}\subset A_H'(\mathbb{R})$ is the set of semisimple components of elements in $\Omega$ in the Iwasawa decomposition. Unlike the bound in loc. cit. obtained by replacing $a\Omega_{A'}$ with $A_H'(\mathbb{R})$ according to $a\Omega_{A'}\subseteq A_H'(\mathbb{R}),$ we can bound the right hand side of [\[90\]](#90){reference-type="eqref" reference="90"} by $$\begin{aligned} \delta_{B'}(a)\int_{[N']}\int_{A_H'(\mathbb{R})}\int_{K'}h(ua'k)\textbf{1}_{C_0|\det a|_{\infty}\leq |\det g|\leq 2C_0|\det a|_{\infty}}dk\delta_{B'}^{-1}(a')d^{\times}a'du\end{aligned}$$ for some absolute constant $C_0$ determined by $\Omega.$ So [\[88\]](#88){reference-type="eqref" reference="88"} follows. ◻ For $\mathfrak{c}=(c_1,\cdots,c_n)\in\mathbb{A}^n,$ let $\|\mathfrak{c}_{\infty}\|_{\infty}:=\sqrt{|c_{1,\infty}|_{\infty}^2+\cdots+|c_{n,\infty}|_{\infty}^2}$ and $$\label{4.14} \|\mathfrak{c}\|:=\|\mathfrak{c}_{\infty}\|_{\infty}\cdot \prod_{p<\infty}\max\{|c_{1,p}|_p,\cdots,|c_{n,p}|_p\}.$$ Fix a function $h_{\infty}\in C_c^{\infty}(\mathbb{R})$ such that $0\leq h(t)\leq 1$ for all $t\in\mathbb{R},$ and $h_{\infty}(t)=1$ if $1\leq t\leq 2,$ and $h_{\infty}(t)=1$ if $t\leq 1/2$ or $t\geq 3.$ Let $h^{\dagger}(\mathfrak{c}):=h_{\infty}^{\dagger}(\mathfrak{c}_{\infty})\prod_{p<\infty}h_p^{\dagger}(\mathfrak{c}_p),$ where $h_{\infty}^{\dagger}(\mathfrak{c}_{\infty})=h_{\infty}(\|\mathfrak{c}_{\infty}\|_{\infty}),$ and $h_p^{\dagger}(\mathfrak{c}):=\mathbf{1}_{\mathbb{Z}_p}(c_{1,p})\cdots \mathbf{1}_{\mathbb{Z}_p}(c_{n,p})\textbf{1}_{\max\{|c_{1,p}|_p,\cdots,|c_{n,p}|_p\}=1}$ for $p<\infty$. Then $h^{\dagger}\in C_c(\mathbb{A}^n).$ In particular, $h^{\dagger}(\mathfrak{c})=0$ unless $1/2\leq \|\mathfrak{c}\|\leq 3.$ **Lemma 23**. *Let notation be as before. Let $C_0>0$ be an absolute constant. Define $$\label{4.14.} \mathcal{I}(a,y):=\int_{N'(\mathbb{A})\backslash G'(\mathbb{A})}\big|W'(gy)\big|^2\textbf{1}_{C_0|\det a|_{\infty}\leq |\det g|\leq 2^{n+1}C_0}h^{\dagger}(\eta g)dg,$$ where $\|\cdot \|$ is defined by [\[4.14\]](#4.14){reference-type="eqref" reference="4.14"}, and $\eta=(0,\cdots,0,1)\in \mathbb{Q}^n.$ Then $$\label{4.15.} I(a,y)\ll T^{\varepsilon}\cdot |\det a|_{\infty}^{-1}\cdot d(y)^2\cdot \langle\phi',\phi'\rangle,$$ where the implied constant relies at most on $\varepsilon,$ $n$ and $M',$ and $$\label{dy} d(y):=|\det y|_{\mathbb{A}}^{-1/2}\cdot \bigg[\int_{ K'}\|\eta ky^{-1}\|^{-n}dk\bigg]^{1/2}.$$* *Proof.* After a change of variable $g\mapsto gy^{-1},$ we obtain $$\mathcal{I}(a,y)=\int_{N'(\mathbb{A})\backslash G'(\mathbb{A})}\big|W'(g)\big|^2\textbf{1}_{C_0\leq \frac{|\det g|}{|\det a|_{\infty}|\det y|_{\mathbb{A}}}\leq 2^{n+1}C_0}h^{\dagger}(\eta gy^{-1})dg.$$ Write $g=zbk$ according to Iwasawa decomposition $G'(\mathbb{A})=Z'(\mathbb{A}) P_0'(\mathbb{A})K'.$ Then $$\mathcal{I}(a,y)\ll \int\int\big|W'(bk)\big|^2\textbf{1}_{\frac{|\det b|}{|\det a|_{\infty}|\det y|_{\mathbb{A}}\|\eta ky^{-1}\|^n}\asymp C_0}\int_{Z'(\mathbb{A})}h^{\dagger}(\eta zky^{-1})d^{\times}zdbdk,$$ where $b$ (resp. $k$) ranges over $N'(\mathbb{A})\backslash P_0'(\mathbb{A})$ (resp. $K'$). Let $\mathfrak{c}=\eta ky^{-1}\in M_{1,n}(\mathbb{A}).$ Then by identifying a central matrix with a scalar, $$\int_{Z'(\mathbb{A})}h^{\dagger}(\eta zky^{-1})d^{\times}z=\int_{\mathbb{R}^{\times}}h_{\infty}^{\dagger}(z_{\infty}\mathfrak{c}_{\infty})d^{\times}z_{\infty}\prod_{p<\infty}\int_{\mathbb{Q}_p^{\times}}h_{p}^{\dagger}(z_{p}\mathfrak{c}_p)d^{\times}z_{p}.$$ For $p<\infty,$ write $\mathfrak{c}_p=(p^{r_1}\beta_1,\cdots,p^{r_n}\beta_n),$ where $r_j\geq 0$ and $\beta_j\in\mathbb{Z}_p^{\times},$ $1\leq j\leq n.$ Write $z_p=p^{r}\beta,$ $r\in\mathbb{Z}$ and $\beta\in \mathbb{Z}_p^{\times}.$ By defintion, $h_{p}^{\dagger}(z_{p}\mathfrak{c}_p)=\textbf{1}_{r+\min\{r_1,\cdots,r_n\}=0}.$ So $$\int_{\mathbb{Q}_p^{\times}}h_{p}^{\dagger}(z_{p}\mathfrak{c}_p)d^{\times}z_{p}=1$$ for all $p<\infty.$ On the other hand, by a change of variable $z_{\infty}\mapsto (|c_{1,\infty}|_{\infty}^2+\cdots+|c_{n,\infty}|_{\infty}^2)^{-1/2}\cdot z_{\infty}$ we derive that $$\int_{\mathbb{R}^{\times}}h_{\infty}^{\dagger}(z_{\infty}\mathfrak{c}_{\infty})d^{\times}z_{\infty}=\int_{\mathbb{R}^{\times}}h_{\infty}(|z_{\infty}|)d^{\times}z_{\infty}=2\int_{\mathbb{R}}\frac{h_{\infty}(t)}{t}dt\asymp 1.$$ Therefore, $\int_{Z'(\mathbb{A})}h^{\dagger}(\eta zky^{-1})d^{\times}z=2\int_{\mathbb{R}}\frac{h_{\infty}(t)}{t}dt.$ Consequently, $$\begin{aligned} \mathcal{I}(a,y)\ll&\int_{K'}\frac{1}{|\det a|_{\infty}|\det y|_{\mathbb{A}}\|\eta ky^{-1}\|^n}\int_{N'(\mathbb{R})\backslash P_0'(\mathbb{R})}\big|W_{\infty}'(b_{\infty}k_{\infty})\big|^2|\det b_{\infty}|_{\infty}db_{\infty}\\ &\int_{N'(\mathbb{A}_{\operatorname{fin}})\backslash P_0'(\mathbb{A}_{\operatorname{fin}})}\big|W_p'(b_{\operatorname{fin}}k_{\operatorname{fin}})\big|^2|\det b_{\operatorname{fin}}|_p\textbf{1}_{|\det b_{\operatorname{fin}}|_{\operatorname{fin}}\asymp X}db_{\operatorname{fin}}dk,\end{aligned}$$ where $X:=|\det a|_{\infty}|\det y|_{\mathbb{A}}\|\eta ky^{-1}\|^nC_0|\det b_{\infty}|_{\infty}^{-1}.$ Denote by $$\begin{aligned} \mathcal{J}:=\int_{N'(\mathbb{A}_{\operatorname{fin}})\backslash P_0'(\mathbb{A}_{\operatorname{fin}})}\big|W_p'(b_{\operatorname{fin}}k_{\operatorname{fin}})\big|^2|\det b_{\operatorname{fin}}|_p\textbf{1}_{|\det b_{\operatorname{fin}}|_{\operatorname{fin}}\asymp X}db_{\operatorname{fin}}.\end{aligned}$$ By Macdonald formula's formula, $W_p'(b_{\operatorname{fin}}k_{\operatorname{fin}})=0$ unless $|\det b_p|_p\leq 1$ for all $p<\infty.$ Hence, $\mathcal{J}=0$ unless $X\geq 2^{-n-1}.$ Let $\lambda_{\pi'\times\pi'}(m)$ be the $m$-th Dirichlet coefficient of $L(s,\pi'\times\pi').$ By Rankin-Selberg convolution, we have $$\begin{aligned} \mathcal{J}\ll \sum_{m\asymp X}\frac{\lambda_{\pi'\times\pi'}(m)}{m}\ll L(1,\pi',\operatorname{Ad}),\end{aligned}$$ where the implied constants depend at most on $M'$. Let $\delta_{H\cap B'}$ be the modular character on $H\cap B'.$ Then $$\label{89} \delta_{B'}^{-1}(a')=|a_n'|^{n}|\det a'|^{-1}\delta_{H\cap B'}^{-1}(a'),\ a'=\operatorname{diag}(a_1',\cdots,a_n').$$ By the classification of irreducible unitary representations of $P_0'(\mathbb{R})$ and $P_0'(\mathbb{Q}_p)$ (cf. [@JS81] and [@JS83]), in conjunction with [\[89\]](#89){reference-type="eqref" reference="89"} and the local Rankin-Selberg theory at finite places, we have $$\begin{aligned} \mathcal{I}(a,y)\ll&\frac{L(1,\pi',\operatorname{Ad})}{|\det a|_{\infty}|\det y|_{\mathbb{A}}}\int_{K'}\frac{1}{\|\eta ky^{-1}\|^n}dk\int_{N_H(\mathbb{R})\backslash H(\mathbb{R})}\big|W_{\infty}'(h)\big|^2dh,\end{aligned}$$ where the implied constant depends at most on $M'$. Now [\[4.15.\]](#4.15.){reference-type="eqref" reference="4.15."} follows from Lemma [Lemma 21](#lem4.6){reference-type="ref" reference="lem4.6"}. ◻ **Lemma 24**. *Let notation be as before. Let $\phi'\in \pi'$ be defined in [4.1.3](#sec4.1.3){reference-type="ref" reference="sec4.1.3"}. Let $y\in G'(\mathbb{A}).$ Then $$\begin{aligned} \int_{\Omega K_{\operatorname{fin}}'}\big|\phi'(ag)\phi'(agy)\big|dg\ll T^{\varepsilon}\langle\phi',\phi'\rangle\delta_{B'}(a)\cdot \min\Bigg\{\frac{|\det a|_{\infty}d(\ensuremath{\mskip 1mu\prescript{\smash{\mathrm t\mkern-3mu}}{}{\mathstrut y}}^{-1})}{|a_1|_{\infty}^n},\frac{d(y)}{|\det a|_{\infty}}\Bigg\},\end{aligned}$$ where the implied constant depends at most on $\varepsilon,$ $n$ and $M',$ and for $g\in G'(\mathbb{A}),$ $d(g)$ is defined by [\[dy\]](#dy){reference-type="eqref" reference="dy"} in Lemma [Lemma 23](#lem4.8.){reference-type="ref" reference="lem4.8."}.* *Proof.* By replacing $\phi'$ with its contragredient, it suffices to show $$\label{91} \int_{\Omega K_{\operatorname{fin}}'}\big|\phi'(ag)\phi'(agy)\big|dg\ll \frac{\langle\phi',\phi'\rangle\delta_{B'}(a)}{|\det a|_{\infty}d(y)}\cdot T^{o(1)}.$$ Let $C_0>0$ be the constant in Lemma [Lemma 22](#lem4.7'){reference-type="ref" reference="lem4.7'"}. Then [\[91\]](#91){reference-type="eqref" reference="91"} follows from $$\label{94} \int_{P_0'(\mathbb{Q})\backslash \overline{G'}(\mathbb{A})}\big|\phi'(g)\phi'(gy)\big|\textbf{1}_{\substack{|\det g|\geq C_0|\det a|_{\infty}\\|\det g|\leq 2C_0|\det a|_{\infty}}}dg\ll \frac{T^{o(1)}\cdot d(y)}{|\det a|_{\infty}}\cdot \langle\phi',\phi'\rangle.$$ Hence we shall prove [\[94\]](#94){reference-type="eqref" reference="94"}. By Cauchy-Schwarz, the left hand side of [\[94\]](#94){reference-type="eqref" reference="94"} is $$\label{4.18} \ll \sqrt{\mathcal{J}(a,I_n)\mathcal{J}(a,y)}$$ where for $y\in G'(\mathbb{A}),$ we define $$\begin{aligned} \mathcal{J}(a,y):=\int_{P_0'(\mathbb{Q})\backslash \overline{G'}(\mathbb{A})}\big|\phi'(gy)\big|^2\textbf{1}_{\substack{|\det g|\geq C_0|\det a|_{\infty}\\|\det g|\leq 2C_0|\det a|_{\infty}}}dg.\end{aligned}$$ Unfolding $\phi'$ according to Fourier-Whittaker expansion, we obtain $$\begin{aligned} \mathcal{J}(a,y)=\int_{N'(\mathbb{A})\backslash \overline{G'}(\mathbb{A})}\big|W'(gy)\big|^2\textbf{1}_{C_0|\det a|_{\infty}\leq |\det g|\leq 2C_0|\det a|_{\infty}}dg.\end{aligned}$$ Let $h^{\dagger}$ be defined right before Lemma [Lemma 23](#lem4.8.){reference-type="ref" reference="lem4.8."}. For $z\in Z'(\mathbb{A}),$ we have $h^{\dagger}(\eta zk)=h_{\infty}(|\det z_{\infty}|_{\infty}^{1/n})\textbf{1}_{\substack{ z_{\operatorname{fin}}\in\widehat{\mathbb{Z}}^{\times}I_n}},$ uniformly for all $k\in K'.$ Then $\mathcal{J}(a,y)$ is $$\begin{aligned} \ll \int_{N'(\mathbb{A})\backslash G'(\mathbb{A})}\big|W'(gy)\big|^2\textbf{1}_{C_0|\det a|_{\infty}\leq |\det g|\leq 2^{n+1}C_0|\det a|_{\infty}}h^{\dagger}(\eta g)dg=\mathcal{I}(a,y),\end{aligned}$$ which is defined by [\[4.14.\]](#4.14.){reference-type="eqref" reference="4.14."}. Hence [\[94\]](#94){reference-type="eqref" reference="94"} follows from [\[4.18\]](#4.18){reference-type="eqref" reference="4.18"} and Lemma [Lemma 23](#lem4.8.){reference-type="ref" reference="lem4.8."}. So [\[91\]](#91){reference-type="eqref" reference="91"} holds. ◻ **Lemma 25**. *Let notation be as before. Let $\phi'\in \pi'$ be as above. Then for each $c_0\geq 0,$ there is a $c_1\geq 0$ such that for each fixed $c_2\in\mathbb{R},$ we have, for all $a\in A_H'(\mathbb{R}),$ and $g\in \Omega K_{\operatorname{fin}}',$ that $$\label{97} \big|\phi'(ag)\big|\ll \Bigg[\prod_{j=1}^{n-1}\max\{a_j,a_j^{-1}\}\Bigg]^{-c_0}T^{c_1}|\det a|^{c_2}.$$* *Proof.* Lemma [Lemma 25](#lem4.9){reference-type="ref" reference="lem4.9"} is the cuspidal analogue of Lemma 2.23 in [@Nel21]. The proof is similar. Note that there exists a fixed continuous seminorm $\nu$ such that $$\sup_{g\in [G']}|\det g|^{-c_2}\|g\|^{c_0}|\phi'(g)|\leq \nu(W'),$$ which is further bounded by a Soblev norm of $\mathcal{J}:=T^{\frac{n(n-1)}{4}}\mathcal{J}_T^{\sharp}[\theta,\beta,\gamma].$ By Lemma 21.27 in loc. cit., the norm is $\ll T^{O(1)}.$ ◻ # The Spectral Side {#sec5} Recall that in [3.1](#3.1){reference-type="ref" reference="3.1"} we consider a pure isobaric automotphic representation $\pi=\pi_{\infty}\otimes\pi_{\operatorname{fin}}\in\mathcal{A}([G],\omega),$ which is the Langlands quotient of $\operatorname{Ind}_{Q(\mathbb{A})}^{G(\mathbb{A})}\pi_1\otimes\cdots\otimes\pi_m.$ Here $Q$ is a standard parabolic subgroup of type $(n_1,\cdots,n_m)$ with $n_1+n_2+\cdots+n_m=n+1,$ and each $\pi_j$ is unitary cuspidal, $1\leq j\leq m$. In particular, $\pi$ is cuspidal if $m=1$. We assume that $\pi$ has *uniform parameter growth of size $(T;c_{\infty},C_{\infty})$* (cf. [\[1.6\]](#1.6){reference-type="eqref" reference="1.6"}). Recall the auxiliary parameter $\mu \in i\mathfrak{a}_{Q}^*/i\mathfrak{a}_G^*\simeq (i\mathbb{R})^{m-1}$ defined in [3.1](#3.1){reference-type="ref" reference="3.1"}: $$\label{5.1.} \mu=\begin{cases} 0,\ & \text{if $m=1$,}\\ (\mu_1,\cdots,\mu_m),\ & \text{if $m\geq 2$,} \end{cases}$$ where $\mu_j=2^{-j}\exp(-2\sqrt{\log T})i,$ $1\leq j< m,$ and $\mu_m=-\mu_1-\cdots-\mu_{m-1}.$ Define $$\label{mu} \boldsymbol{L}(\mu,\pi):=\prod_{j=1}^mL(1,\pi_j,\operatorname{Ad})\cdot\prod_{\substack{1\leq j_1<j_2\leq m}}\big|L(1+\mu_{j_1}-\mu_{j_2},\pi_{j_1}\otimes\widetilde{\pi}_{j_2})\big|^2.$$ Since $\mu_j\in i\mathbb{R}$ for $1\leq j\leq m,$ then $\boldsymbol{L}(\mu,\pi)\neq 0.$ For $\lambda=(\lambda_1,\cdots,\lambda_m)\in\mathbb{C}^m,$ we denote by $$\label{eq5.3} \pi_{\lambda}:=\pi_1|\cdot|^{\lambda_1}\boxplus \cdots\boxplus \pi_m|\cdot|^{\lambda_m}.$$ This is consistent with the definition of $\pi_{\mu}$ in [3.1.6](#specpara){reference-type="ref" reference="specpara"}. Recall that (cf. [3.8.1](#ampsp){reference-type="ref" reference="ampsp"}) the spectral side $\mathcal{J}_{\mathop{\mathrm{Spec}}}^{\heartsuit}(\boldsymbol{\alpha},\boldsymbol{\ell})$ is defined by $$\begin{aligned} \sum_{p_1\neq p_2\in\mathcal{L}}\alpha_{p_1}\overline{\alpha_{p_2}}J_{\mathop{\mathrm{Spec}}}^{\operatorname{Reg},\heartsuit}(f(\cdot;p_1,p_2),\mathbf{0})+\sum_{p_0\in\mathcal{L}}\sum_{i=0}^{l_{p_0}}c_{p_0,i}|\alpha_{p_0}|^2J_{\mathop{\mathrm{Spec}}}^{\operatorname{Reg},\heartsuit}(f(\cdot;i,p_0),\mathbf{0}).\end{aligned}$$ Our goal in this section is to bound $\mathcal{J}_{\mathop{\mathrm{Spec}}}^{\heartsuit}(\boldsymbol{\alpha},\boldsymbol{\ell})$ from below in terms of the central $L$-value $L(1/2,\pi\times\pi')$ as follows. thmthmf [\[prop11.2\]]{#prop11.2 label="prop11.2"} Let notation be as before. Let $\pi=\pi_1\boxplus\cdots\boxplus\pi_m$ be pure isobatic with *uniform parameter growth of size $(T;c_{\infty},C_{\infty})$*. Let $\pi_{\mu}:=\pi_1|\cdot|^{\mu_1}\boxplus \cdots\boxplus \pi_m|\cdot|^{\mu_m},$ where $\mu$ is defined as [\[5.1.\]](#5.1.){reference-type="eqref" reference="5.1."}. Suppose that $|L(1/2,\pi\times\pi')|\geq 2.$ Then $$\begin{aligned} \mathcal{J}_{\mathop{\mathrm{Spec}}}^{\heartsuit}(\boldsymbol{\alpha},\boldsymbol{\ell})\gg T^{-\frac{n^2}{2}-\varepsilon}\cdot \big|L(1/2,\pi\times\pi')\big|^2\cdot \bigg|\sum_{p\in \mathcal{L}}\alpha_p\lambda_{\pi_{\mu}}(p^{l_p})\bigg|^2\cdot \prod_{j=1}^m\frac{1}{L(1,\pi_j,\operatorname{Ad})},\end{aligned}$$ where the implied constant depends on $\varepsilon,$ parameters $c_{\infty}$ and $C_{\infty}$ defined in [3.1.4](#sec3.14){reference-type="ref" reference="sec3.14"}, and the conductor of $\pi_{\infty}'.$ ## Bounds of Archimedean Period Integrals **Lemma 26**. *Let notation be as before. Let $\mu\in (i\mathbb{R})^{m-1}$ be defined in [3.1](#3.1){reference-type="ref" reference="3.1"}. There exists a vector $W_{\mu,\infty}$ in the Whittaker model of $\pi_{\mu,\infty}$ such that $$\int_{N'(\mathbb{R})\backslash G'(\mathbb{R})}\Bigg|W_{\mu,\infty}\left(\begin{pmatrix} g'\\ &1 \end{pmatrix}\right)\Bigg|^2dg'\leq 1,$$ and $$\begin{aligned} T^{-\frac{n^2}{4}-\varepsilon}\ll \Bigg|\int_{N'(\mathbb{R})\backslash G'(\mathbb{R})}W_{\mu,\infty}\left(\begin{pmatrix} g'\\ &1 \end{pmatrix}\right)W_{\infty}'(g')dg'\Bigg|\ll T^{-\frac{n^2}{4}+\varepsilon}, \end{aligned}$$ where the implied constant relies on $\pi_{\infty}'$ and $\varepsilon.$* *Proof.* The vector $W_{\mu,\infty}$ can be chosen according to Lemma 8.6 of [@Nel21]. Then Lemma [Lemma 26](#prop4.11){reference-type="ref" reference="prop4.11"} follows from Lemma [Lemma 20](#lem4.5){reference-type="ref" reference="lem4.5"}. Here we also make use of the fact that $\mathop{\mathrm{Re}}(\mu_j)=0,$ and $|\mu_j|\leq 1,$ $1\leq j\leq m.$ ◻ ## Calculation of Period Integrals with Ramification Let $p\mid M''.$ Let $\tilde{f}_p$ be defined as in [3.3.2](#11.1.4){reference-type="ref" reference="11.1.4"}: $$\tilde{f}_p(g):=v_p\sum_{\boldsymbol{\alpha}}\omega_p'(\alpha_n)\int_{Z(\mathbb{Q}_p)}\textbf{1}_{u_{\boldsymbol{\alpha}}K_p}(zg)\omega_p(z)d^{\times}z,\tag{\ref{3.5}}$$ where $\boldsymbol{\alpha}=(\alpha_1,\cdots,\alpha_{n-1},\alpha_n),$ $\alpha_j\in \mathbb{Z}/p^{m'}\mathbb{Z},$ $1\leq j<n,$ $\alpha_n\in (\mathbb{Z}/p^{m''}\mathbb{Z})^{\times},$ and $u_{\boldsymbol{\alpha}}=\begin{pmatrix} I_n&\textbf{u}_{\boldsymbol{\alpha}}\\ &1 \end{pmatrix},$ with $\textbf{u}_{\boldsymbol{\alpha}}=\ensuremath{\mskip 1mu\prescript{\smash{\mathrm t\mkern-3mu}}{}{\mathstrut(\alpha_1p^{-m'},\cdots, \alpha_{n-1}p^{-m'},\alpha_n p^{-m''})}},$ and $$\begin{aligned} v_p=\operatorname{Vol}(K_p'(M'))^{-1}\operatorname{Vol}(K_p(M))^{-1}p^{-(n-1)m'}G(\omega_p',\psi_p)^{-1}.\end{aligned}$$ Let $W_p$ be a local new vector in the Whittaker model of $\pi_p.$ Let $W_p'$ be the normalized new vector in the Whittaker model of $\pi_p'$ as in [4](#sec4.){reference-type="ref" reference="sec4."}. Let $\mathop{\mathrm{Re}}(s)>1$ and $$\label{5.3..} \mathcal{P}(s,\pi_p(\tilde{f}_p)W_p,W_p'):=\int_{N'(\mathbb{Q}_p)\backslash G'(\mathbb{Q}_p)}\pi_p(\tilde{f}_p)W_p(\iota(x))W_p'(x)|\det x|_p^sdx.$$ **Lemma 27**. *Let notation be as above. Then $$\begin{aligned} \mathcal{P}(s,\pi_p(f)W_p,W_p')= W_p(I_{n+1})W_p'(I_n)L_p(s+1/2,\pi_p\times\pi_p'),\end{aligned}$$ where $L_p(s+1/2,\pi_p\times\pi_p')$ is the local Rankin-Selberg $L$-function.* *Proof.* Substituting the definition [\[3.5\]](#3.5){reference-type="eqref" reference="3.5"} into [\[5.3..\]](#5.3..){reference-type="eqref" reference="5.3.."} we obtain $$\begin{aligned} \mathcal{P}(s,\pi_p(f)W_p,W_p')=v_p\operatorname{Vol}(K_p(M))\int_{N'(\mathbb{Q}_p)\backslash G'(\mathbb{Q}_p)}W_p^{\dagger}(\iota(x))W_p'(x)|\det x|_p^sdx.\end{aligned}$$ where $$\begin{aligned} W_p^{\dagger}(\iota(x)):=\sum_{\boldsymbol{\alpha}}\omega_p'(\alpha_n)W_p\left(\begin{pmatrix} x&\\ &1 \end{pmatrix}\begin{pmatrix} I_n&\textbf{u}_{\boldsymbol{\alpha}}\\ &1 \end{pmatrix}\right).\end{aligned}$$ Here we make use of the fact that $W_p$ is right $K_p(M)$-invariant. Let $x=ak$ be the Iwasawa decomposition, where $a=\operatorname{diag}(p^{r_1},\cdots,p^{r_n}),$ $k\in G'(\mathbb{Z}_p).$ Write $(k_{n,1},\cdots, k_{n,n})$ for the last row of $k.$ Following [\[3.4.\]](#3.4.){reference-type="eqref" reference="3.4."} we define $$\begin{aligned} K_p'(p^{m'-r_n}):=\Big\{k\in K_p':\ k_{n,j}\in p^{m'-r_n}\mathbb{Z}_p,\ 1\leq j<n\Big\}.\end{aligned}$$ By the property of Whittaker function we have $$\begin{aligned} W_p\left(\begin{pmatrix} ak&ak\textbf{u}_{\boldsymbol{\alpha}}\\ &1 \end{pmatrix}\right)=\psi_p(\alpha_n k_{n,n}p^{r_n-m''})\prod_{j=1}^{n-1}\psi_p(\alpha_j k_{n,j}p^{r_n-m'}) W_p\left(\begin{pmatrix} a&\\ &1 \end{pmatrix}\right).\end{aligned}$$ Here we also utilize the fact that $W_p$ is right $\operatorname{diag}(k,1)$-invariant. Therefore, the sum $W_p^{\dagger}(\iota(x))$ is equal to $$\begin{aligned} W_p\left(\begin{pmatrix} a&\\ &1 \end{pmatrix}\right)\sum_{\substack{\alpha_n}}\omega_p'(\alpha_n)\psi_p(\alpha_n k_{n,n}p^{r_n-m''})\prod_{j=1}^{n-1}\Bigg[\sum_{\substack{\alpha_j\in \mathbb{Z}/p^{m'}\mathbb{Z}}}\psi_p(\alpha_j k_{n,j}p^{r_n-m'})\Bigg],\end{aligned}$$ where $\alpha_n\in (\mathbb{Z}/p^{m''}\mathbb{Z})^{\times}.$ Along with orthogonality we obtain that $$\begin{aligned} W_p^{\dagger}(\iota(x))=p^{(n-1)m'}\sum_{\substack{\alpha_n}}\omega_p'(\alpha_n)\psi_p(\alpha_n k_{n,n}p^{r_n-m''})\textbf{1}_{k\in K_p'(p^{m'-r_n})} W_p\left(\begin{pmatrix} a&\\ &1 \end{pmatrix}\right).\end{aligned}$$ Note that for $a=\operatorname{diag}(p^{r_1},\cdots,p^{r_n}),$ we have $W_p(\operatorname{diag}(a,1))=0$ unless $r_1\geq \cdots \geq r_n\geq 0.$ Therefore, the integral $\mathcal{P}(s,\pi_p(f)W_p,W_p')$ becomes $$\label{5.4.} v_p'\sum_{\substack{r_1\geq \cdots \geq r_n\geq 0}}W_p\left(\begin{pmatrix} a&\\ &1 \end{pmatrix}\right)\delta_{B'}^{-1}(a)|\det a|_p^s\int_{K_p'(p^{m'-r_n})}W_p'^{\dagger}(ak)dk,$$ where $v_p':=v_p\cdot \operatorname{Vol}(K_p(M))p^{(n-1)m'},$ $a=\operatorname{diag}(p^{r_1},\cdots,p^{r_n}),$ and $$\begin{aligned} W_p'^{\dagger}(ak):=\sum_{\substack{\alpha_n}}\omega_p'(\alpha_n)\psi_p(\alpha_n k_{n,n}p^{r_n-m''})W_p'(ak)=G(\omega_p',\psi_p)W_p'(ak)\overline{\omega_p'}(k_{n,n}).\end{aligned}$$ Since $W_p'$ is a local new vector of level $p^{m'},$ then $$\label{5.5} \int_{K_p'(p^{m'-r_n})}W_p'(ak)\overline{\omega_p'}(k_{n,n})dk=\textbf{1}_{r_n\leq 0}\cdot W_p'(a).$$ Substituting [\[5.5\]](#5.5){reference-type="eqref" reference="5.5"} into [\[5.4.\]](#5.4.){reference-type="eqref" reference="5.4."} the integral $\mathcal{P}(s,\pi_p(f)W_p,W_p')$ simplies to $$\begin{aligned} v_p'\cdot G(\omega_p',\psi_p)\sum_{\substack{r_1\geq \cdots \geq r_n=0\\ r_n\leq 0}}W_p\left(\begin{pmatrix} a&\\ &1 \end{pmatrix}\right)W_p'(a)\delta_{B'}^{-1}(a)|\det a|_p^s \operatorname{Vol}(K_p'(p^{m'})),\end{aligned}$$ which, by the definition of $v_p$ and $\operatorname{Vol}(K_p'(p^{m'}))=\operatorname{Vol}(K_p'(M')),$ amounts to $$\label{eq5.6} \mathcal{P}(s,\pi_p(f)W_p,W_p')=\sum_{\substack{r_1\geq \cdots \geq r_n=0\\ r_n=0}}W_p\left(\begin{pmatrix} a&\\ &1 \end{pmatrix}\right)W_p'(a)\delta_{B'}^{-1}(a)|\det a|_p^s.$$ Since $W_p'$ is a local new vector and $\pi_p'$ is ramified, then by [@Miy14 Theorem 4.1], $W_p'(\operatorname{diag}(p^{r_1},\cdots,p^{r_n}))=0$ unless $r_n=0.$ Hence, the constraint $r_n=0$ in [\[eq5.6\]](#eq5.6){reference-type="eqref" reference="eq5.6"} turns out to be redundant. Therefore, it follows form op.cit. and the Rankin-Selberg theory that $\mathcal{P}(s,\pi_p(f)W_p,W_p')=W_p(I_{n+1})W_p'(I_n)L_p(s+1/2,\pi_p\times\pi_p').$ ◻ ## Lower Bound of the Amplified Spectral Side {#sec5.2} ### Spectral Decomposition {#sec5.2.1} Let $X(G)_{\mathbb{Q}}$ be the space set of ${\mathbb{Q}}$-rational characters of $G.$ Denote by $\mathfrak{a}_G=\mathop{\mathrm{Hom}}_{\mathbb{Z}}(X(G)_{\mathbb{Q}},\mathbb{R}).$ Let $\mathfrak{a}_G^*=X(G)_{\mathbb{Q}}\otimes \mathbb{R}.$ Let $R$ be a standard parabolic subgroup of $G$ of type $(n_1',\cdots,n_l').$ Let $M_{R}$ (resp. $N_R$) be the Levi subgroup (resp. unipotent radical) of $R.$ Let $X(M_R)_{\mathbb{Q}}$ be the space set of ${\mathbb{Q}}$-rational characters of $M_R.$ Denote by $\mathfrak{a}_R=\mathfrak{a}_{M_R}=\mathop{\mathrm{Hom}}_{\mathbb{Z}}(X(M_R)_{\mathbb{Q}},\mathbb{R})$ and $\mathfrak{a}_R^*=\mathfrak{a}_{M_R}^*=X(M_R)_{\mathbb{Q}}\otimes \mathbb{R}.$ Write $\mathfrak{a}_0$ (resp. $\mathfrak{a}_0^*$) for $\mathfrak{a}_{B}$ (resp. $\mathfrak{a}_{B}^*$). The abelian group $X(M_R)_F$ has a canonical basis of rational characters $\nu_i: m\rightarrow \det m_i,$ for $,m=\begin{pmatrix} m_1\\ &\ddots\\ &&m_l \end{pmatrix}\in M_R.$ We then choose $\big\{n_i\nu_i^{-1}:\ 1\leq i\leq l\big\}$ to be a basis for $\mathfrak{a}_R^*,$ and choose a dual basis on $\mathfrak{a}_R$ to identified both $\mathfrak{a}_R$ and $\mathfrak{a}_{R}^*$ with $\mathbb{R}^l.$ Furthermore, define $$H_{R}(\mathbf{m})=\left(\frac{\log |\det m_1|}{n_1'},\cdots, \frac{\log |\det m_l|}{n_l'}\right)\in\mathfrak{a}_R,$$ where $\mathbf{m}=\operatorname{diag}(m_1,\cdots, m_l)\in M_R(\mathbb{A}).$ Denote by $\langle\cdot,\cdot\rangle$ the pair for $\mathfrak{a}_R$ and $\mathfrak{a}_R^*$ under the above chosen identification. Then by spectral theory (e.g., cf. p. 256 and p. 263 of [@Art79]), $$\label{49} L^2(R):=L^2\left(Z_{G}(\mathbb{A})N_R(\mathbb{A})M_R(F)\backslash G(\mathbb{A})\right)=\bigoplus_{\mathfrak{X}}L^2\left(R\right)_{\mathfrak{X}},$$ where $\mathfrak{X}$ ranges over cusp data, and $L^2\left(R\right)_{\mathfrak{X}}$ consists of functions $\phi\in L^2(R)$ such that for each standard parabolic subgroup $R_1\subset R,$ and almost all $x\in G(\mathbb{A}),$ the projection of the function $$m\mapsto x.\phi_{R_1}(m):=\int_{N_{R_1}(F)\backslash N_{R_1}(\mathbb{A})}\phi(nmx)dn$$ onto the space of cusp forms in $L^2\left(Z_{G}(\mathbb{A})M_{R_1}(F)\backslash M_{R_1}^1(\mathbb{A})\right)$ transforms under $M_{R_1}^1(\mathbb{A})$ as a sum of representations $\sigma,$ in which $(M_{R_1},\sigma)\in\mathfrak{X}.$ If there is no such pair in $\mathfrak{X},$ $x.\phi_{{R_1}}$ will be orthogonal to $\mathcal{A}_0\left(Z_{G}(\mathbb{A})M_{R_1}(F)\backslash M_{R_1}^1(\mathbb{A})\right).$ Denote by $\mathcal{H}_R$ the space of such $\phi$'s. Let $\mathcal{H}_{R,\mathfrak{X}}$ be the subspace of $\mathcal{H}_R$ such that for any $(M,\sigma)\notin \chi,$ with $M=M_{R_1}$ and $R_1\subset R,$ we have $$\int_{M(F)\backslash M(\mathbb{A})^1}\int_{N_{R_1}(F)\backslash N_{R_1}(\mathbb{A})}\varphi_0(m)\phi(nmx)dn=0,$$ for any $\varphi_0\in L^2_{0}\left(M(F)\backslash M(\mathbb{A})^1\right)_{\sigma},$ and almost all $x.$ This leads us to Langlands' decompostion $\mathcal{H}_R=\bigoplus_{\mathfrak{X}}\mathcal{H}_{R,\mathfrak{X}}.$ Let $\mathcal{B}_R$ be an orthonormal basis of $\mathcal{H}_R,$ then we can choose $\mathcal{B}_R=\bigcup_{\mathfrak{X}}\mathcal{B}_{R,\mathfrak{X}},$ where $\mathcal{B}_{R,\mathfrak{X}}$ is an orthonormal basis of the Hilbert space $\mathcal{H}_{R,\mathfrak{X}}.$ We may assume that vectors in each $\mathcal{B}_{R,\mathfrak{X}}$ are $K$-finite and are pure tensors. Given a spectral datum $\mathfrak{X}$ and $\lambda\in\mathfrak{a}_R^*\otimes\mathbb{C},$ we have the induced representation $\mathcal{I}_{R}(\cdot, \lambda):=\operatorname{Ind}_{R(\mathbb{A})}^{G(\mathbb{A})}\sigma\cdot e^{\langle \lambda, H_{M}(\cdot)\rangle}$ with the representation space being $\mathcal{H}_R.$ Explicitly, for $\phi\in \mathcal{H}_R,$ one has $$(\mathcal{I}_R(y,\lambda)\phi)(x)=\phi(xy)e^{\langle \lambda+\rho_R,H_{M_R}(xy)\rangle}e^{-\langle \lambda+\rho_R,H_{M_R}(x)\rangle},\ \ x, y\in G(\mathbb{A}),$$ where $\rho_R$ is the half sum of the positive roots of $(R, A_R).$ The associated Eisenstein series is defined by $$\begin{aligned} E(x,\phi,\lambda)=\sum_{\delta\in R(F)\backslash G(F)}\phi(\delta x)e^{\langle \lambda+\rho_Q,H_{M_Q}(\delta x)\rangle}.\end{aligned}$$ Let $W_{\phi,\lambda}(x):=\int_{[N]}E(ux,\phi,\lambda)\theta(ux)du$ be the Whittaker function of $E(x,\phi,\lambda).$ **Proposition 28**. *Let notation be as in [3](#11.2){reference-type="ref" reference="11.2"}.* 1. *Suppose $m=1,$ i.e., $\pi$ is cuspidal. Then $$\label{5.3'} \mathcal{J}_{\mathop{\mathrm{Spec}}}^{\heartsuit}(\boldsymbol{\alpha},\boldsymbol{\ell})\gg_{\varepsilon,\pi'} T^{-\frac{n^2}{2}-\varepsilon}\cdot \frac{\big|L(1/2,\pi\times\pi')\big|^2}{L(1,\pi,\operatorname{Ad})}\cdot \bigg|\sum_{p\in \mathcal{L}}\alpha_p\lambda_{\pi}(p^{l_p})\bigg|^2,$$ where the implied constant depends on $\varepsilon,$ $\pi',$ and $\pi_p,$ $p\mid M'.$* 2. *Suppose $m\geq 2.$ Let $\mu=(\mu_1,\cdots,\mu_m)\in i\mathfrak{a}_{Q}^*/i\mathfrak{a}_G^*\simeq (i\mathbb{R})^{m-1}$ with $\mu_j=2^{-j}\exp(-2\sqrt{\log T})i,$ $1\leq j\leq m-1$ and $\mu_m=-\mu_1-\cdots-\mu_{m-1}.$ Then $$\label{5.3''} \mathcal{J}_{\mathop{\mathrm{Spec}}}^{\heartsuit}(\boldsymbol{\alpha},\boldsymbol{\ell})\gg_{\varepsilon,\pi'} T^{-\frac{n^2}{2}-\varepsilon}\cdot \frac{\big|L(1/2,\pi_{\mu}\times\pi')\big|^2}{\boldsymbol{L}(\mu,\pi)}\cdot \bigg|\sum_{p\in \mathcal{L}}\alpha_p\lambda_{\pi_{\mu}}(p^{l_p})\bigg|^2,$$ where the implied constant depends on $\varepsilon$ and $\pi_p,$ $p\mid M'.$ Here $\boldsymbol{L}(\mu,\pi)$ is defined by [\[mu\]](#mu){reference-type="eqref" reference="mu"}.* *Proof.* Let $f\in \big\{f(\cdot;i,p_0), f(\cdot;p_1,p_2)\big\}$ and $\tilde{f}$ be constructed as described in [3.6.2](#3.6.2){reference-type="ref" reference="3.6.2"}. Note that $\tilde{f}*\tilde{f}^*$ can be thought of as the test function without amplification. By spectral expansion (cf. [5.3.1](#sec5.2.1){reference-type="ref" reference="sec5.2.1"}), we have $$\label{eqspec} J_{\mathop{\mathrm{Spec}}}^{\operatorname{Reg},\heartsuit}(f,\mathbf{0})=\sum_{\mathfrak{X}}\sum_{R}\frac{1}{n_R}\left(\frac{1}{2\pi i}\right)^{\dim A_R/A_G}\int_{i\mathfrak{a}_R^*/i\mathfrak{a}_G^*}\Psi^{\dagger}(\lambda)d\lambda,$$ where $$\begin{aligned} \Psi^{\dagger}(\lambda):=\sum_{\phi\in\mathfrak{B}_{R,\mathfrak{X}}}\Psi(0, \mathcal{I}_R(\lambda,f)W_{\phi,\lambda},W_{\phi'}')\overline{\Psi(0, W_{\phi,\lambda},W_{\phi'}')}.\end{aligned}$$ Here $\Psi(\cdots)$ is the Rankin-Selberg period (cf. [\[whittaker\]](#whittaker){reference-type="eqref" reference="whittaker"}) viewed as a meromorphic function. Since $\nu(f)$ is coprime to $M,$ then $$\begin{aligned} \mathcal{I}_R(\lambda,f)W_{\phi,\lambda}=\boldsymbol{c}_{\pi_{\lambda}}(f)\mathcal{I}_R(\lambda,\tilde{f}*\tilde{f}^*)W_{\phi,\lambda},\end{aligned}$$ where $\boldsymbol{c}_{\pi_{\lambda}}(f)$ is the scalar eigenvalue specific to $f$. Therefore, $$\label{eq5.10} \Psi^{\dagger}(\lambda)=\sum_{\phi\in\mathfrak{B}_{R,\mathfrak{X}}}\boldsymbol{c}_{\pi_{\lambda}}(f)\big|\Psi(0, \mathcal{I}_R(\lambda,\tilde{f})W_{\phi,\lambda},W_{\phi'}')\big|^2.$$ It follows from [\[3.66\]](#3.66){reference-type="eqref" reference="3.66"} that $$\label{5.9} \sum_{p_1\neq p_2\in\mathcal{L}}\alpha_{p_1}\overline{\alpha_{p_2}}\boldsymbol{c}_{\pi_{\lambda}}^{p_1,p_2}+\sum_{p_0\in\mathcal{L}}\sum_{i=0}^{l_{p_0}}c_{p_0,i}|\alpha_{p_0}|^2\boldsymbol{c}_{\pi_{\lambda}}^{i;p_0}=\bigg|\sum_{p\in \mathcal{L}}\alpha_p\lambda_{\pi_{\lambda}}(p^{l_p})\bigg|^2$$ where $\boldsymbol{c}_{\pi_{\lambda}}^{p_1,p_2}=\boldsymbol{c}_{\pi_{\lambda}}(f(\cdot;p_1,p_2))$ and $\boldsymbol{c}_{\pi_{\lambda}}^{i;p_0}:=\boldsymbol{c}_{\pi_{\lambda}}(f(\cdot;i,p_0)).$ Substituting [\[eq5.10\]](#eq5.10){reference-type="eqref" reference="eq5.10"} into [\[eqspec\]](#eqspec){reference-type="eqref" reference="eqspec"} and dropping all $R\neq Q,$ we obtain that $$\label{5.3.} J_{\mathop{\mathrm{Spec}}}^{\operatorname{Reg},\heartsuit}(f,\mathbf{0})\gg \int_{i\mathfrak{a}_{Q}^*/i\mathfrak{a}_G^*}\sum_{\phi\in\mathfrak{B}_{Q,\mathfrak{X}}}\boldsymbol{c}_{\pi_{\lambda}}(f)\big|\Psi(0,\mathcal{I}_{Q}(\lambda,\tilde{f})W_{\phi,\lambda},W_{\phi'}')\big|^2d\lambda,$$ where $\mathfrak{X}=\{(\pi_1\otimes\cdots\otimes\pi_m,M_Q)\}$ is the cuspidal datum corresponding to $\pi.$ 1. Suppose $m\geq 2.$ Recall that $W_p'(I_n)=1$ for all $p<\infty$ (cf. [4](#sec4.){reference-type="ref" reference="sec4."}). By Rankin-Selberg theory (cf. [@JPSS81] and [@JPSS83]) and Lemma [Lemma 27](#lem5.3){reference-type="ref" reference="lem5.3"}, $$\begin{aligned} \big|\Psi(0,\mathcal{I}_{Q}(\lambda,\tilde{f})W_{\phi,\lambda},W_{\phi'}')\big|^2=\big|\Psi_{\infty}(0,\mathcal{I}_{Q}(\lambda,\tilde{f})W_{\phi,\lambda},W_{\phi'}')\big|^2\cdot \frac{|L(1/2,\pi_{\lambda}\times\pi')|^2}{\boldsymbol{L}(\lambda,\pi)},\end{aligned}$$ where $\boldsymbol{L}(\lambda,\pi)$ is defined by [\[mu\]](#mu){reference-type="eqref" reference="mu"}. Along with [\[5.9\]](#5.9){reference-type="eqref" reference="5.9"} and Theorem 1.2.1 (and its corollary) in [@Ram95], the inequality [\[5.3.\]](#5.3.){reference-type="eqref" reference="5.3."} yields that $$\begin{aligned} \mathcal{J}_{\mathop{\mathrm{Spec}}}^{\heartsuit}(\boldsymbol{\alpha},\boldsymbol{\ell})\gg \mathcal{Q}_{\boldsymbol{\alpha},\boldsymbol{\ell}}(\pi,\pi';f,\mu)\sum_{\phi\in\mathfrak{B}_{Q,\mathfrak{X}}}\big|\Psi_{\infty}(0,\mathcal{I}_{Q}(\mu,\tilde{f})W_{\phi,\mu},W_{\phi'}')\big|^2,\end{aligned}$$ where $$\begin{aligned} \mathcal{Q}_{\boldsymbol{\alpha},\boldsymbol{\ell}}(\pi,\pi';f,\mu):=\frac{\big|L(1/2,\pi_{\mu}\times\pi')\big|^2}{\boldsymbol{L}(\mu,\pi)}\cdot \bigg|\sum_{p\in \mathcal{L}}\alpha_p\lambda_{\pi_{\mu}}(p^{l_p})\bigg|^2. \end{aligned}$$ So [\[5.3\'\'\]](#5.3''){reference-type="eqref" reference="5.3''"} follows from Lemma [Lemma 26](#prop4.11){reference-type="ref" reference="prop4.11"}, which implies that $$\sum_{\phi\in\mathfrak{B}_{Q,\mathfrak{X}}}\big|\Psi_{\infty}(0,\mathcal{I}_{Q}(\mu,\tilde{f})W_{\phi,\mu},W_{\phi'}')\big|^2\gg T^{-\frac{n^2}{2}-\varepsilon}.$$ 2. Suppose $m=1.$ Then $Q=G$ and thereby [\[5.3.\]](#5.3.){reference-type="eqref" reference="5.3."} becomes $$\begin{aligned} J_{\mathop{\mathrm{Spec}}}^{\operatorname{Reg},\heartsuit}(f,\mathbf{0})\gg \mathcal{Q}_{\boldsymbol{\alpha},\boldsymbol{\ell}}(\pi,\pi';f,\mu)\sum_{\phi\in\mathfrak{B}_{Q,\mathfrak{X}}}\big|\Psi_{\infty}(0,\pi(\tilde{f})W_{\phi},W_{\phi'}')\big|^2,\end{aligned}$$ where $\mu=0.$ So [\[5.3\'\]](#5.3'){reference-type="eqref" reference="5.3'"} follows from Lemma [Lemma 26](#prop4.11){reference-type="ref" reference="prop4.11"}. Therefore, Proposition [Proposition 28](#thm6){reference-type="ref" reference="thm6"} holds. ◻ ## Proof of Theorem [\[prop11.2\]](#prop11.2){reference-type="ref" reference="prop11.2"} Notice that Theorem [\[prop11.2\]](#prop11.2){reference-type="ref" reference="prop11.2"} follows from [\[5.3\'\]](#5.3'){reference-type="eqref" reference="5.3'"} if $m=1.$ To handle the general case, we make use of the following technical estimate, which is a generalization of [@Yan23b Lemma 2.3]. **Lemma 29**. *Let $\sigma$ be an automorphic representation of $\mathrm{GL}(m,\mathbb{A}),$ $m\geq 1.$ Let $t_0\in \mathbb{R}.$ Suppose $|L(1/2+it_0,\sigma\times\pi')|\geq 2.$ Let $\textbf{C}:=C(\sigma\times\pi'\otimes|\cdot|^{it_0})$ be the analytic conductor of $\sigma\boxtimes\pi'\otimes|\cdot|^{it_0}.$ Suppose that $\textbf{C}\geq 10^4.$ Then for all integer $l\geq 1,$ there exists some $d_l'\in [2^{-l}\exp(-3\sqrt{\log \textbf{C}}),2^{1-l}\exp(-3\sqrt{\log \textbf{C}})]$ such that for all $|t-t_0|=d_l',$ we have $$\label{11.1} |L(1/2+it_0,\sigma\times\pi')|\ll \exp(\log^{3/4}\textbf{C})\cdot |L(1/2+it,\sigma\times\pi')|,$$ and $$\label{5.8} |L(1/2+it,\sigma\times\pi')|\ll \exp(\log^{3/4}\textbf{C})\cdot |L(1/2+it_0,\sigma\times\pi')|,$$ where the implied constant depends at most on $l.$* *Proof.* Denote by $\mathcal{D}:=\big\{\rho\in\mathbb{C}:\ L(\rho,\sigma\times\pi')=0,\ |\rho-1/2-it_0|\leq \exp(-2\sqrt{\log \textbf{C}})\big\}.$ By Jensen's formula and the convexity bound for $L(s,\sigma\times\pi'),$ we have $$\begin{aligned} \#\mathcal{D}\cdot \log (\exp(\sqrt{\log \textbf{C}}))\leq \max_{|s-1/2-it_0|\leq \exp(-\sqrt{\log \textbf{C}})}\log \frac{|L(s,\sigma\times\pi')|}{|L(1/2+it_0,\sigma\times\pi')|}, \end{aligned}$$ which is $\ll \log \textbf{C},$ and the implied constant is absolute. Consequently, $$\#\mathcal{D}\ll \sqrt{\log \textbf{C}},$$ where the implied constant is absolute. By Pigeonhole principle there exists some $d_l\in [2^{-l}\exp(-2\sqrt{\log \textbf{C}}),2^{1-l}\exp(-2\sqrt{\log \textbf{C}})]$ such that $$\label{11.3.} |s-\rho|\geq \frac{\exp(-2\sqrt{\log \textbf{C}})}{3\cdot 2^l\cdot (1+\#\mathcal{D})},$$ for all $|s-1/2-it_0|=d_l$ and for all $\rho\in\mathcal{D}.$ Likewise, there exists some $d_l'\in [2^{-l}\exp(-3\sqrt{\log \textbf{C}}),2^{1-l}\exp(-3\sqrt{\log \textbf{C}})]$ such that for all $|s-1/2-it_0|=d_l'$ and for all $\rho\in\mathcal{D},$ we have $$\label{5.11} |s-\rho|\geq \frac{\exp(-3\sqrt{\log \textbf{C}})}{3\cdot 2^{l}\cdot (1+\#\mathcal{D})},$$ Let $\mathcal{B}_l:=\{s:\ |s-1/2-it_0|\leq d_l\}$ and $\mathcal{B}_l':=\{s:\ |s-1/2-it_0|\leq d_l'\}.$ Set $\mathbf{P}(s):=\prod_{\rho\in \mathcal{D}}(s-\rho).$ Then $\mathbf{P}$ is a polynomial of degree $\deg\mathbf{P}=\#\mathcal{D}.$ Define $$\mathbb{L}(s):=\log\frac{L(s,\sigma\times\pi')}{\mathbf{P}(s)},\ \ s\in\mathcal{B}_l.$$ Then $\mathbb{L}(s)$ is holomorphic in $\mathcal{B}_l.$ In particular, it is holomorphic in $\mathcal{B}_l'.$ Therefore, by Borel-Carathéodory theorem, we have, for all $s\in \mathcal{B}_l',$ that $$\begin{aligned} \big|\mathbb{L}(1/2+it_0)-\mathbb{L}(s)\big|\leq \frac{10\exp(-3\sqrt{\log \textbf{C}})\cdot \big[\max_{s'\in \mathcal{B}_l}\mathop{\mathrm{Re}}(\mathbb{L}(s'))-\mathop{\mathrm{Re}}(\mathbb{L}(1/2+it_0))\big]}{\exp(-2\sqrt{\log \textbf{C}})-\exp(-3\sqrt{\log \textbf{C}})}. \end{aligned}$$ Let $s_0\in\partial \mathcal{B}_l$ be such that $\mathop{\mathrm{Re}}(\mathbb{L}(s_0))=\max_{s'\in \mathcal{B}_l}\mathop{\mathrm{Re}}(\mathbb{L}(s')),$ where $\partial\mathcal{B}_l$ is the boundary of $\mathcal{B}_l$. Then $$\begin{aligned} \max_{s'\in \mathcal{B}_l}\mathop{\mathrm{Re}}(\mathbb{L}(s'))-\mathop{\mathrm{Re}}(\mathbb{L}(1/2+it_0))=\log\Big|\frac{L(s_0,\sigma\times\pi')}{L(1/2+it_0,\sigma\times\pi')}\Big|+\log\Big|\frac{\mathbf{P}(1/2+it_0)}{\mathbf{P}(s_0)}\Big|. \end{aligned}$$ Using the trivial bound $|\mathbf{P}(1/2+it_0)|\leq 1$ and [\[11.3.\]](#11.3.){reference-type="eqref" reference="11.3."}, $$\begin{aligned} \Big|\frac{\mathbf{P}(1/2+it_0)}{\mathbf{P}(s_0)}\Big|\ll \frac{1}{|\mathbf{P}(s_0)|}\ll \Big[3\cdot 2^l\cdot (1+\#\mathcal{D})\exp(2\sqrt{\log T})\Big]^{\deg \mathbf{P}}. \end{aligned}$$ Hence, we have, by $\#\mathcal{D}\ll \sqrt{\log \textbf{C}},$ that $$\log \Big|\frac{\mathbf{P}(1/2+it_0)}{\mathbf{P}(s_0)}\Big|\ll \deg \mathbf{P}\cdot \log \Big[3\cdot 2^l\cdot (1+\#\mathcal{D})\exp(2\sqrt{\log \textbf{C}})\Big]\ll \log \textbf{C},$$ where the implied constant depends on $l.$ Moreover, by convexity bound, $$\label{5.13} \log\Big|\frac{L(s_0,\sigma\times\pi')}{L(1/2+it_0,\sigma\times\pi')}\Big|\ll \log |L(s_0,\sigma\times\pi')|\ll \log \textbf{C}.$$ Therefore, uniformly for $s\in \mathcal{B}_l',$ we have $$\label{5.10} \big|\mathbb{L}(1/2+it_0)-\mathbb{L}(s)\big|\ll \exp(-\sqrt{\log \textbf{C}})\cdot \log \textbf{C}\ll 1.$$ Take $s=1/2+it\in \partial\mathcal{B}_l',$ then [\[5.10\]](#5.10){reference-type="eqref" reference="5.10"} leads to $$\begin{aligned} L(1/2+it_0,\sigma\times\pi')\ll \big|L(1/2+it,\sigma\times\pi')\big|\cdot \Big|\frac{\mathbf{P}(1/2+it_0)}{\mathbf{P}(1/2+it)}\Big|,\end{aligned}$$ where the implied constant is independent of $t.$ Moreover, by [\[5.11\]](#5.11){reference-type="eqref" reference="5.11"}, $\mathbf{P}(1/2+it)\neq 0.$ For $\rho\in\mathcal{D}$ and $s\in \partial \mathcal{B}_l',$ by [\[5.11\]](#5.11){reference-type="eqref" reference="5.11"} there exists some constant $C$ (which depends at most on $l$ and the analytic conductors of $\sigma$ and $\pi'$) such that for all $\rho\in\mathcal{D},$ $$\label{5.12} \frac{|1/2+it_0-\rho|}{|s-\rho|}\leq \begin{cases} 1,&\ \text{if $|s-1/2-it_0|\leq d_l'/2,$}\\ C,&\ \text{if $d_l'/2\leq |s-1/2-it_0|\leq 3d_l'/4,$}\\ C\cdot (1+\#\mathcal{D}),&\ \text{if $3d_l'/4\leq |s-1/2-it_0|\leq 5d_l'/4,$}\\ C,&\ \text{if $5d_l'/4\leq |s-1/2-it_0|\leq d_l.$} \end{cases}$$ Together with $\deg\mathbf{P}=\#\mathcal{D}\leq C_2\sqrt{\log \textbf{C}}$ for some absolute constant $C_2,$ we have $$\label{5.14} \Big|\frac{\mathbf{P}(1/2+it_0)}{\mathbf{P}(1/2+it)}\Big|\leq (C\sqrt{\log \textbf{C}})^{C_2\sqrt{\log \textbf{C}}}\ll \exp(\log^{3/4}\textbf{C}),$$ where the implied constant depends on $C,$ and $C_2.$ So [\[11.1\]](#11.1){reference-type="eqref" reference="11.1"} follows. The treatment of [\[5.8\]](#5.8){reference-type="eqref" reference="5.8"} is different since $|1/2+it_0-\rho|$ may be quite tiny. So we cannot bound ${|s-\rho|}/{|1/2+it_0-\rho|}$ as [\[5.12\]](#5.12){reference-type="eqref" reference="5.12"}. By Borel-Carathéodory theorem, $$\begin{aligned} \big|\mathbb{L}(1/2+it_0)-\mathbb{L}(s)\big|\leq 20\exp(-\sqrt{\log \textbf{C}})\log\Big|\frac{L(s_0,\sigma\times\pi')}{L(1/2+it_0,\sigma\times\pi')}\cdot\frac{\mathbf{P}(1/2+it_0)}{\mathbf{P}(s_0)}\Big|,\end{aligned}$$ where $s_0$ is defined as before. Employing [\[5.13\]](#5.13){reference-type="eqref" reference="5.13"} and the assumption that $\textbf{C}\geq 10^4$, $$\begin{aligned} \log \Big|\frac{L(1/2+it,\sigma\times\pi')}{L(1/2+it_0,\sigma\times\pi')}\cdot \frac{\mathbf{P}(1/2+it_0)}{\mathbf{P}(1/2+it)}\Big|\leq 20+\log\Big|\frac{\mathbf{P}(1/2+it_0)}{\mathbf{P}(s_0)}\Big|,\end{aligned}$$ where we use the fact that $\exp(-\sqrt{\log \textbf{C}})\log \textbf{C}\leq 1$ and $20\exp(-\sqrt{\log \textbf{C}})\leq 1.$ So $$\label{5.15} \log \Big|\frac{L(1/2+it,\sigma\times\pi')}{L(1/2+it_0,\sigma\times\pi')}\Big|\leq 20+\log\Big|\frac{\mathbf{P}(1/2+it)}{\mathbf{P}(s_0)}\Big|.$$ By [\[11.3.\]](#11.3.){reference-type="eqref" reference="11.3."}, a similar analysis of [\[5.12\]](#5.12){reference-type="eqref" reference="5.12"} works for $|1/2+it-\rho|/|s_0-\rho|,$ $\rho\in\mathcal{D}.$ As a consequence, we have an analogue of [\[5.14\]](#5.14){reference-type="eqref" reference="5.14"}: $$\label{5.16} \Big|\frac{\mathbf{P}(1/2+it)}{\mathbf{P}(s_0)}\Big|\leq (C'\sqrt{\log \textbf{C}})^{C_2\sqrt{\log \textbf{C}}}\ll \exp(\log^{3/4}\textbf{C}),$$ where $C'$ is some absolute constant, and thus the implied constant is absolute. Therefore, [\[5.8\]](#5.8){reference-type="eqref" reference="5.8"} follows from [\[5.15\]](#5.15){reference-type="eqref" reference="5.15"} and [\[5.16\]](#5.16){reference-type="eqref" reference="5.16"}. ◻ *Proof of Theorem [\[prop11.2\]](#prop11.2){reference-type="ref" reference="prop11.2"}.* Recall that Proposition [Proposition 28](#thm6){reference-type="ref" reference="thm6"} yields $$\label{11.4.} \mathcal{J}_{\mathop{\mathrm{Spec}}}^{\heartsuit}(\boldsymbol{\alpha},\boldsymbol{\ell})\gg_{\varepsilon,\pi'} T^{-\frac{n^2}{2}-\varepsilon}\cdot \frac{\big|L(1/2,\pi_{\mu}\times\pi')\big|^2}{\boldsymbol{L}(\mu,\pi)}\cdot \bigg|\sum_{p\in \mathcal{L}}\alpha_p\lambda_{\pi_{\mu}}(p^{l_p})\bigg|^2.$$ Note that for fixed $\pi',$ we have $$\boldsymbol{L}(\mu,\pi)\ll \prod_{1\leq j_1<j_2\leq m}|\mu_{j_1}-\mu_{j_2}|^{-2}\prod_{j=1}^mL(1,\pi_j,\operatorname{Ad})\ll T^{\varepsilon}\prod_{j=1}^mL(1,\pi_j,\operatorname{Ad}).$$ So [\[11.4.\]](#11.4.){reference-type="eqref" reference="11.4."} becomes $$\begin{aligned} \mathcal{J}_{\mathop{\mathrm{Spec}}}^{\heartsuit}(\boldsymbol{\alpha},\boldsymbol{\ell})\gg_{\varepsilon,\pi'} T^{-\frac{n^2}{2}-\varepsilon}\cdot \big|L(1/2,\pi_{\mu}\times\pi')\big|^2\cdot \bigg|\sum_{p\in \mathcal{L}}\alpha_p\lambda_{\pi_{\mu}}(p^{l_p})\bigg|^2.\end{aligned}$$ Therefore, Theorem [\[prop11.2\]](#prop11.2){reference-type="ref" reference="prop11.2"} follows from Lemma [Lemma 29](#lem11.1){reference-type="ref" reference="lem11.1"}. ◻ # Classification of the Amplification Support {#sec5.1} Recall that the orbital integrals (cf. [\[63\]](#63){reference-type="eqref" reference="63"}) are defined over the automorphic quotient by the mirabolic subgroup, while the local test function at $p\in\nu(f)$, as a Hecke operator, has support as a Cartan cell. In this section, our goal is to parameterize Cartan cells restricted to the mirabolic subgroup using Iwasawa coordinates. This parameterization will play a crucial role in effectively handling the amplified orbital integrals in the subsequent sections of this paper. ## Notation Recap Let $G'=\mathrm{GL}(n).$ Let $W'$ be the Weyl group of $G'.$ Let $B'$ be the standard Borel of $G'.$ Let $N'$ be the unipotent radical of $B'.$ Let $\delta_{B'}$ be the modular character of $B'.$ Recall the notation in [3.1.7](#sec3.1.6){reference-type="ref" reference="sec3.1.6"}: $\mathcal{L}$ is a subset of $\{\text{prime $p$}:\ L<p\leq 2L,\ p\nmid MM'\},$ and $\boldsymbol{\ell}=(l_p)_{p\in\mathcal{L}}$ is a sequence of integers such that $1\leq l_p\leq n+1$. Let $p_0, p_1, p_2\in\mathcal{L}$ with $p_1\neq p_2$ and $0\leq i\leq l_{p_0}$ as in [\[3.66\]](#3.66){reference-type="eqref" reference="3.66"} in [3.6.1](#sec3.6.1){reference-type="ref" reference="sec3.6.1"}. Given the independent and purely local nature of this section, we will omit the subscript $p$ to simplify the notation. For example, instead of writing $y_p\in G'(\mathbb{Q}_p)$, we will use the notation $y\in G'(\mathbb{Q}_p)$. ## The Amplification Support: Off-diagonal {#sec6.2} Let $y\in G'(\mathbb{Q}_p).$ By Cramer's rule there exists a Weyl element $w\in W'$ such that $wy=bk,$ where $b\in B'(\mathbb{Q}_p)$ and $k\in K_p'=G'(\mathbb{Z}_p),$ such that $e_p(E_{i,i}(b))\leq e_p(E_{i',j'}(b)),$ $1\leq i', j'\leq i,$ $1\leq i\leq n.$ Using the Levi decomposition, we can uniquely express this as: $$\label{dec} wy=tuk,\ t\in \operatorname{diag}(p^{\mathbb{Z}},\cdots,p^{\mathbb{Z}}),\ u\in N'(\mathbb{Q}_p),\ tu=b.$$ It is worth noting that $tut^{-1}\in K_p'$, and the expression [\[dec\]](#dec){reference-type="eqref" reference="dec"} actually corresponds to the Cartan decomposition $y=(w^{-1}tut^{-1})tk\in K_p'tK_p'.$ **Lemma 30**. *Let $p=p_2,$ $l=l_{p_2}.$ Let $y\in G'(\mathbb{Q}_p).$ Let $\mathfrak{u}\in M_{n,1}(\mathbb{Q}_p)\simeq\mathbb{Q}_p^n.$ Let $e, e'\geq 0$ such that $-e$ (resp. $-e'$) is the minimal evaluation of entries of $y$ (resp. $\mathfrak{u}$). Let $wy=tuk$ be the decomposition in [\[dec\]](#dec){reference-type="eqref" reference="dec"}. Suppose $f_p\left(\begin{pmatrix} y&\mathfrak{u}\\ &1 \end{pmatrix}\right)\neq 0.$ Then $\delta_{B'}^{-1}(t)\leq p^{(n-1)l}.$* 1. *Suppose $e\geq e'.$ Then $e=l$ and $$\begin{aligned} wy=\begin{pmatrix} I_{n-1}&\\ &p^{-l} \end{pmatrix}\begin{pmatrix} I_{n-1}&\mathfrak{u}''\\ &1 \end{pmatrix}k',\ \mathfrak{u}''\in p^{-l}\mathbb{Z}_p^{n-1},\ k\in K_p',\end{aligned}$$ and $w\mathfrak{u}=\begin{pmatrix} \mathfrak{u}'\\ \alpha\end{pmatrix}$ with $\mathfrak{u}'-\alpha p^l\mathfrak{u}''\in \mathbb{Z}_p^{n-1}$ and $\alpha\in p^{-l}\mathbb{Z}_p.$* 2. *Suppose $e<e'.$ Then $2e'-e=l,$ $w\mathfrak{u}=\begin{pmatrix} \mathfrak{u}'\\ \alpha\end{pmatrix}$ with $\mathfrak{u}'\in p^{-e'}\mathbb{Z}_p^{n-1},$ $\alpha\in p^{-e'}\mathbb{Z}_p^{\times},$ moreover, $$\begin{aligned} wy=\begin{pmatrix} p^{\frac{l-e}{2}}I_{n-1}&&\\ &p^{-e} \end{pmatrix}\begin{pmatrix} I_{n-1}&\mathfrak{u}''\\ &1 \end{pmatrix}k,\ \mathfrak{u}''-p^{-e'}\alpha^{-1}\mathfrak{u}'\in \mathbb{Z}_p^{n-1},\ k\in K_p'. \end{aligned}$$ In particular, $l>e'>e\geq 0.$* *Proof.* By Cramer's rule we can write $wy=tuk,$ with $k\in K_p',$ $u\in N'(\mathbb{Q}_p),$ and $t=\operatorname{diag}(p^{r_1-e},\cdots,p^{r_{n-1}-e},p^{-e}).$ By minimality of $-e,$ we must have $r_1\geq \cdots\geq r_{m}\geq 0.$ Write $u=\begin{pmatrix} u'&\mathfrak{u}''\\ &1 \end{pmatrix},$ with $\mathfrak{u}''\in\mathbb{Q}_p^{n-1}.$ Write $w\mathfrak{u}=\ensuremath{\mskip 1mu\prescript{\smash{\mathrm t\mkern-3mu}}{}{\mathstrut(\alpha_1,\cdots,\alpha_n)}}\in \mathbb{Q}_p^n.$ Let $t'=\operatorname{diag}(p^{r_1},\cdots,p^{r_{n-1}})$ and $\mathfrak{u}'=\ensuremath{\mskip 1mu\prescript{\smash{\mathrm t\mkern-3mu}}{}{\mathstrut(\alpha_1,\cdots,\alpha_{n-1})}}.$ Note that $f_p\left(\begin{pmatrix} y&\mathfrak{u}\\ &1 \end{pmatrix}\right)\neq 0$ amounts to $$\label{2} p^j\begin{pmatrix} tu&w\mathfrak{u}\\ &1 \end{pmatrix}\in K_p\begin{pmatrix} I_n\\ &p^{-l} \end{pmatrix}K_p$$ for some $j\in\mathbb{Z}.$ Then $-l\leq j\leq 0.$ 1. Suppose that $e\geq e'.$ So $j-e=-l.$ Then after an application of Cramer's rule, [\[2\]](#2){reference-type="eqref" reference="2"} becomes $$\begin{aligned} p^j\begin{pmatrix} p^{-e}t'u'&&\mathfrak{u}'-\alpha_n t'\mathfrak{u}''\\ &p^{-e}&\alpha_n\\ &&1 \end{pmatrix}\in K_p\begin{pmatrix} I_n\\ &p^{-l} \end{pmatrix}K_p.\end{aligned}$$ So $\alpha_n\in p^{-e}\mathbb{Z}_p,$ $p^j\begin{pmatrix} p^{-e}t'u'&\mathfrak{u}'-\alpha_n t'\mathfrak{u}''\\ &1 \end{pmatrix}\in K_p',$ leading to Case (A) and $\delta_{B'}^{-1}(t)\leq p^{(n-1)l}.$ 2. Suppose that $e< e'.$ Then $j-e'=-l.$ Let $m$ be the largest index such that $e_p(\alpha_m)=-e',$ $1\leq m\leq n.$ - Suppose $m=n.$ By Cramer's rule [\[2\]](#2){reference-type="eqref" reference="2"} becomes $$\begin{aligned} p^j\begin{pmatrix} p^{-e}t'u'&p^{-e}t'\mathfrak{u}''-p^{-e}\alpha_n^{-1}\mathfrak{u}'&\\ &p^{-e}&\alpha_n\\ &&1 \end{pmatrix}\in K_p\begin{pmatrix} I_n\\ &p^{-l} \end{pmatrix}K_p.\end{aligned}$$ So $2e'-e=l,$ $w\mathfrak{u}=\begin{pmatrix} \mathfrak{u}'\\ \alpha_n\end{pmatrix}$ with $\mathfrak{u}'\in p^{-e'}\mathbb{Z}_p^{n-1}$ and $\alpha_n\in p^{-e'}\mathbb{Z}_p^{\times},$ $$\begin{aligned} wy=\begin{pmatrix} p^{\frac{l-e}{2}}I_{n-1}&&\\ &p^{-e} \end{pmatrix}\begin{pmatrix} I_{n-1}&\mathfrak{u}''\\ &1 \end{pmatrix}k,\ \mathfrak{u}''-p^{j-e}\alpha_n^{-1}\mathfrak{u}'\in \mathbb{Z}_p^{n-1},\ k\in K_p'.\end{aligned}$$ By $2e'-e=l$ and $e<e'$ we have $e'<l.$ Hence, $\delta_{B'}^{-1}(t)=p^{\frac{(n-1)(l+e)}{2}}<p^{\frac{(n-1)(l+e')}{2}}<p^{(n-1)l}.$ - Suppose $m<n.$ By Cramer's rule [\[2\]](#2){reference-type="eqref" reference="2"} implies that $j-e=0,$ and $j-e+r_m+e'=0.$ So $r_m=-e'\leq -1,$ a contradiction! Therefore, when $e< e',$ Case (B) holds. In all, Lemma [Lemma 30](#lem7){reference-type="ref" reference="lem7"} follows. ◻ ## The Amplification Support: Diagonal {#sec6.2.} **Lemma 31**. *Let $p=p_0,$ $l=l_{p_0}.$ Let $y\in G'(\mathbb{Q}_p)$ and $\mathfrak{u}\in M_{n,1}(\mathbb{Q}_p)\simeq\mathbb{Q}_p^n.$ Let $e, e'\geq 0$ such that $-e$ (resp. $-e'$) is the minimal evaluation of entries of $y$ (resp. $\mathfrak{u}$). Let $wy=tuk$ be the decomposition in [\[dec\]](#dec){reference-type="eqref" reference="dec"}. Suppose $f_p\left(\begin{pmatrix} y&\mathfrak{u}\\ &1 \end{pmatrix}\right)\neq 0.$ Then $\delta_{B'}^{-1}(t)\leq p^{2(n-1)l}.$* 1. *Suppose $e\geq e'.$ Then we have the following two cases:* 1. *$e=l$ and $$\begin{aligned} wy=\begin{pmatrix} p^l\\ &I_{n-2}&\\ &&p^{-l} \end{pmatrix}\begin{pmatrix} 1&\mathfrak{c}&\mathfrak{u}_1''\\ &I_{n-2}&\mathfrak{u}_2''\\ &&1 \end{pmatrix}k,\ \begin{pmatrix} \mathfrak{u}_1''\\ \mathfrak{u}_2'' \end{pmatrix}\in \begin{pmatrix} p^{-2l}\mathbb{Z}_p\\ p^{-l}\mathbb{Z}_p^{n-2} \end{pmatrix},\end{aligned}$$ $\mathfrak{c}\in p^{-l}\mathbb{Z}_p^{n-2},$ $k\in K_p',$ and $w\mathfrak{u}=\begin{pmatrix} \mathfrak{u}'\\ \alpha\end{pmatrix}$ with $\mathfrak{u}'-\alpha p^l\begin{pmatrix} p^l\mathfrak{u}_{1}''\\ \mathfrak{u}_{2}'' \end{pmatrix}\in \mathbb{Z}_p^{n-1}$ and $\alpha \in p^{-l}\mathbb{Z}_p.$* 2. *$l<e\leq 2l.$ Let $r_1=3l-e.$ Write $w\mathfrak{u}=\begin{pmatrix} \mathfrak{u}'\\ \alpha\end{pmatrix}.$ Then $$\begin{aligned} wy=\begin{pmatrix} p^{-e+r_1}\\ &p^{l-e}I_{n-2}&\\ &&p^{-e} \end{pmatrix}\begin{pmatrix} 1&\mathfrak{c}&\mathfrak{u}_{1}''\\ &I_{n-2}&\mathfrak{u}_{2}''\\ &&1 \end{pmatrix}k,\ k\in K_p',\end{aligned}$$ $\mathfrak{u}_{2}''\in p^{-l}\mathbb{Z}_p^{n-1},$ $\mathfrak{u}_{1}''\in p^{-r_1}\mathbb{Z}_p,$ $\mathfrak{c}\in p^{l-r_1}\mathbb{Z}_p^{n-2},$ $\alpha\in p^{-l}\mathbb{Z}_p,$ and $\mathfrak{u}'-\alpha p^l\begin{pmatrix} \mathfrak{u}_{1}''\\ \mathfrak{u}_{2}'' \end{pmatrix}\in p^{l-e}\mathbb{Z}_p^{n-1}.$* 2. *Suppose $e<e'.$ Then we have the following three cases:* 1. *We have $2e'-e=l,$ and $w\mathfrak{u}=\begin{pmatrix} \mathfrak{u}'\\ \alpha\end{pmatrix},$ $\mathfrak{u}'\in p^{-e'}\mathbb{Z}_p^{n-1},$ $\alpha\in p^{-e'}\mathbb{Z}_p^{\times},$ $$\begin{aligned} wy=\begin{pmatrix} p^{l+e'-e}\\ &p^{e'-e}I_{n-2}&\\ &&p^{-e} \end{pmatrix}\begin{pmatrix} 1&\mathfrak{c}&\mathfrak{u}_{1}''\\ &I_{n-2}&\mathfrak{u}_2''\\ &&1 \end{pmatrix}k,\ \ k\in K_p',\end{aligned}$$ with $\mathfrak{c}\in p^{-l}\mathbb{Z}_p^{n-2},$ $\begin{pmatrix} \mathfrak{u}_1''\\ \mathfrak{u}_2'' \end{pmatrix}- \begin{pmatrix} p^{-l}\\ &I_{n-2} \end{pmatrix}p^{-e'}\alpha^{-1}\mathfrak{u}'\in \mathbb{Z}_p^{n-1}.$* 2. *We have $2e'-e=2l,$ and $w\mathfrak{u}=\begin{pmatrix} \mathfrak{u}'\\ \alpha\end{pmatrix},$ $\mathfrak{u}'\in p^{-e'}\mathbb{Z}_p^{n-1},$ $\alpha\in p^{-e'}\mathbb{Z}_p^{\times},$ $$\begin{aligned} wy=\begin{pmatrix} p^{e'-e-l}\\ &p^{e'-e-l}I_{n-2}&\\ &&p^{-e} \end{pmatrix}\begin{pmatrix} 1&&\mathfrak{u}_1''\\ &I_{n-2}&\mathfrak{u}_2''\\ &&1 \end{pmatrix}k,\end{aligned}$$ with $k\in K_p',$ and $\begin{pmatrix} \mathfrak{u}_1''\\ \mathfrak{u}_2'' \end{pmatrix}-p^{l-e'}\alpha^{-1}\mathfrak{u}'\in \mathbb{Z}_p^{n-1}.$* 3. *We have $l<2e'-e< 2l.$ Denote by $j=e'-l$ and $r_1=3l-3e'+2e.$ Then $1\leq j-e+r_1<l,$ $w\mathfrak{u}=\begin{pmatrix} \mathfrak{u}'\\ \alpha\end{pmatrix},$ with $\mathfrak{u}'=\begin{pmatrix} \mathfrak{u}_1'\\ \mathfrak{u}_2'\end{pmatrix},$ $\alpha\in p^{-e'}\mathbb{Z}_p^{\times},$ and $$\begin{aligned} wy=\begin{pmatrix} p^{3l-3e'+e}\\ &p^{l-e'}I_{n-2}&\\ &&p^{-e} \end{pmatrix}\begin{pmatrix} 1&\mathfrak{c}&\mathfrak{u}_1''\\ &I_{n-2}&\mathfrak{u}_2''\\ &&1 \end{pmatrix}k,\end{aligned}$$ $\mathfrak{c}\in p^{2e'-e-2l}\mathbb{Z}_p^{n-2},$ $k\in K_p',$ $\mathfrak{u}_2''\in p^{e'-e-l}\mathbb{Z}_p^{n-1},$ $\mathfrak{u}_1''\in p^{3e'-2e-3l}\mathbb{Z}_p,$ and $$\begin{pmatrix} \mathfrak{u}_1''\\ \mathfrak{u}_2'' \end{pmatrix}- \begin{pmatrix} p^{3e'-2e-3l}\\ &p^{e'-e-l}I_{n-2} \end{pmatrix}\alpha^{-1}\mathfrak{u}'\in \begin{pmatrix} p^{2e'-e-2l}\mathbb{Z}_p\\ \mathbb{Z}_p^{n-2} \end{pmatrix}.$$* *Proof.* Keep the notation in the proof of Lemma [Lemma 30](#lem7){reference-type="ref" reference="lem7"}: write $w\mathfrak{u}=\ensuremath{\mskip 1mu\prescript{\smash{\mathrm t\mkern-3mu}}{}{\mathstrut(\alpha_1,\cdots,\alpha_n)}}\in \mathbb{Q}_p^n,$ $t'=\operatorname{diag}(p^{r_1},\cdots,p^{r_{n-1}})$ and $\mathfrak{u}'=\ensuremath{\mskip 1mu\prescript{\smash{\mathrm t\mkern-3mu}}{}{\mathstrut(\alpha_1,\cdots,\alpha_{n-1})}}.$ Note that $f_p\left(\begin{pmatrix} y&\mathfrak{u}\\ &1 \end{pmatrix}\right)\neq 0$ amounts to the existence of $j\in\mathbb{Z}$ such that $$\label{2.} p^j\begin{pmatrix} tu&w\mathfrak{u}\\ &1 \end{pmatrix}\in K_p\begin{pmatrix} p^{l}\\ &I_{n-1}\\ &&p^{-l} \end{pmatrix}K_p.$$ Then $-l\leq j\leq l.$ We need an analogue of Lemma [Lemma 30](#lem7){reference-type="ref" reference="lem7"} in this case. 1. Suppose that $e\geq e'.$ So $j-e=-l.$ Then after an application of Cramer's rule, [\[2.\]](#2.){reference-type="eqref" reference="2."} becomes $$\begin{aligned} p^j\begin{pmatrix} p^{-e}t'u'&&\mathfrak{u}'-\alpha_nt'\mathfrak{u}''\\ &p^{-e}&\alpha_n\\ &&1 \end{pmatrix}\in K_p\begin{pmatrix} p^{l}\\ &I_{n-1}\\ &&p^{-l} \end{pmatrix}K_p.\end{aligned}$$ So $\alpha_n\in p^{-e}\mathbb{Z}_p,$ and $$\label{5} p^j\begin{pmatrix} p^{-e}t'u'&\mathfrak{u}'-\alpha_nt'\mathfrak{u}''\\ &1 \end{pmatrix}\in K_p'\begin{pmatrix} p^{l}\\ &I_{n-1} \end{pmatrix}K_p'.$$ - Suppose $j-e+r_1=l.$ Taking inverse of [\[5\]](#5){reference-type="eqref" reference="5"}, using $$\label{3} \begin{pmatrix} a&aC&b\\ &D&F\\ &&1 \end{pmatrix}^{-1}=\begin{pmatrix} a^{-1}&-CD^{-1}&-a^{-1}b+CD^{-1}F\\ &D^{-1}&-D^{-1}F\\ &&1 \end{pmatrix},$$ we then conclude that $j=0$ (by considering the $(n,n)$-th entry of [\[5\]](#5){reference-type="eqref" reference="5"}) and thereby the decomposition in the case (A.1). Also, $\delta_{B'}^{-1}(t)\leq p^{2(n-1)l}.$ - Suppose $j-e+r_1<l.$ Taking inverse of [\[5\]](#5){reference-type="eqref" reference="5"}, using [\[3\]](#3){reference-type="eqref" reference="3"}, $$\begin{aligned} wy=\begin{pmatrix} p^{-j-l+r_1}\\ &p^{-j}I_{n-2}&\\ &&p^{-j-l} \end{pmatrix}\begin{pmatrix} 1&\mathfrak{c}&\mathfrak{u}_1''\\ &I_{n-2}&\mathfrak{u}_2''\\ &&1 \end{pmatrix}k,\end{aligned}$$ $k\in K_p',$ $-j+e\leq r_1<l-j+e,$ $r_2=\cdots=r_{n-1}=l,$ $\mathfrak{u}_2''\in p^{-l}\mathbb{Z}_p^{n-1},$ $\mathfrak{u}_1''\in p^{-r_1}\mathbb{Z}_p,$ $\mathfrak{c}\in p^{-j+e-r_1}\mathbb{Z}_p^{n-2},$ and $w\mathfrak{u}=\begin{pmatrix} \mathfrak{u}'\\ \alpha_n\end{pmatrix}$ with $\mathfrak{u}'-\alpha_np^l\mathfrak{u}''\in p^{l-e}\mathbb{Z}_p^{n-1}$ and $\alpha_n\in p^{-l}\mathbb{Z}_p.$ Moreover, computing the determinant leads to that $3l=e+r_1.$ Since $r_1\geq r_2=l,$ $j-e=-l,$ and $-l\leq j\leq l,$ then $l\leq r_1<2l,$ and $e=j+l\leq 2l.$ So the case (A.2) holds. Since $l\leq r_1<2l,$ then $\delta_{B'}^{-1}(t)< p^{2(n-1)l}.$ 2. Suppose that $e< e'.$ Then $j-e'=-l.$ Then $e_p(\alpha_n)=-e'.$ By Cramer's rule [\[2.\]](#2.){reference-type="eqref" reference="2."} becomes $$\begin{aligned} p^j\begin{pmatrix} p^{-e}t'u'&p^{-e}t'\mathfrak{u}''-p^{-e}\alpha_n^{-1}\mathfrak{u}'&\\ &p^{-e}&\alpha_n\\ &&1 \end{pmatrix}\in K_p\begin{pmatrix} p^l\\ &I_{n-1}\\ &&p^{-l} \end{pmatrix}K_p.\end{aligned}$$ Consequently, $$\label{10} p^j\begin{pmatrix} p^{-e}t'u'&p^{-e}t'\mathfrak{u}''-p^{-e}\alpha_n^{-1}\mathfrak{u}'\\ &p^{e'-e} \end{pmatrix}\in K_p'\begin{pmatrix} p^{l}\\ &I_{n-1} \end{pmatrix}K_p'.$$ - Suppose $j-e+r_1=l.$ Taking inverse of [\[10\]](#10){reference-type="eqref" reference="10"}, we then conclude that $j+e'-e=0.$ So $r_2=\cdots=r_{n-1}=e'.$ Therefore, by [\[3\]](#3){reference-type="eqref" reference="3"}, $$\begin{aligned} wy=\begin{pmatrix} p^{l+e'-e}\\ &p^{e'-e}I_{n-2}&\\ &&p^{-e} \end{pmatrix}\begin{pmatrix} 1&\mathfrak{c}&\mathfrak{u}_1''\\ &I_{n-2}&\mathfrak{u}_2''\\ &&1 \end{pmatrix}k,\end{aligned}$$ $\mathfrak{c}\in p^{-l}\mathbb{Z}_p^{n-2},$ $k\in K_p',$ and $w\mathfrak{u}=\begin{pmatrix} \mathfrak{u}'\\ \alpha_n\end{pmatrix},$ $\mathfrak{u}'\in p^{-e'}\mathbb{Z}_p^{n-1},$ $\alpha_n\in p^{-e'}\mathbb{Z}_p^{\times},$ $\mathfrak{u}''-t'^{-1}\alpha_n^{-1}\mathfrak{u}'\in \mathbb{Z}_p^{n-1}.$ This verifies the case (B.1). In particular, $l>e'>e\geq 0,$ implying that $\delta_{B'}^{-1}(t)\leq p^{2(n-1)l}.$ - Suppose $j+e'-e=l.$ Then $2e'-e=2l.$ Taking inverse of [\[10\]](#10){reference-type="eqref" reference="10"}, using [\[3\]](#3){reference-type="eqref" reference="3"}, we obtain that $0\leq r_1=\cdots =r_{n-1}=e'-l,$ and thereby $$\begin{aligned} wy=\begin{pmatrix} p^{e'-e-l}\\ &p^{e'-e-l}I_{n-2}&\\ &&p^{-e} \end{pmatrix}\begin{pmatrix} 1&&\mathfrak{u}_1''\\ &I_{n-2}&\mathfrak{u}_2''\\ &&1 \end{pmatrix}k_p',\ k\in K_p',\end{aligned}$$ and $w\mathfrak{u}=\begin{pmatrix} \mathfrak{u}'\\ \alpha_n\end{pmatrix},$ $\mathfrak{u}'\in p^{-e'}\mathbb{Z}_p^{n-1},$ $\alpha_n\in p^{-e'}\mathbb{Z}_p^{\times},$ $\mathfrak{u}''-p^{l-e'}\alpha_n^{-1}\mathfrak{u}'\in \mathbb{Z}_p^{n-1}.$ Hence, the case (B.2) follows. From $2e'-e=2l$ and $e<e'$ we derive that $e'<2l,$ implying that $e'-l<e'<2l.$ Therefore, $\delta_{B'}^{-1}(t)=p^{(n-1)(e'-l)}< p^{2(n-1)l}.$ - Suppose $j-e+r_1<l$ and $j+e'-e<l.$ By the case (B) of Lemma [Lemma 30](#lem7){reference-type="ref" reference="lem7"} the $(1,n)$-th entry of the inverse of the left hand side of [\[10\]](#10){reference-type="eqref" reference="10"} has the least evaluation, which is $-l$. Note that [\[10\]](#10){reference-type="eqref" reference="10"} is equivalent to $$\begin{aligned} p^{j+e'-e}\begin{pmatrix} p^{-e'}t'u'&p^{-e'}t'\mathfrak{u}''-p^{-e'}\alpha_n^{-1}\mathfrak{u}'\\ &1 \end{pmatrix}\in K_p'\begin{pmatrix} p^{l}\\ &I_{n-1} \end{pmatrix}K_p',\end{aligned}$$ which is of form [\[5\]](#5){reference-type="eqref" reference="5"}. We then appeal to the argument in the case (A.2) above. Hence, $2j-2e+e'+r_1=l,$ and $j-e+r_m=0,$ $1<m<n.$ Since $j-e'=-l,$ then $3e'-2e+r_1=3l.$ So $-e+r_1=3l-3e'+e.$ Write $w\mathfrak{u}=\begin{pmatrix} \mathfrak{u}'\\ \alpha_n\end{pmatrix},$ where $\mathfrak{u}'=\begin{pmatrix} \mathfrak{u}_1'\\ \mathfrak{u}_2'\end{pmatrix},$ $\alpha_n\in p^{-e'}\mathbb{Z}_p^{\times}.$ Then $$\begin{aligned} wy=\begin{pmatrix} p^{-e+r_1}\\ &p^{-j}I_{n-2}&\\ &&p^{-e} \end{pmatrix}\begin{pmatrix} 1&\mathfrak{c}&\mathfrak{u}_1''\\ &I_{n-2}&\mathfrak{u}_2''\\ &&1 \end{pmatrix}k_p',\ k_p'\in K_p',\end{aligned}$$ $\mathfrak{c}\in p^{-j+e-r_1}\mathbb{Z}_p^{n-2},$ $\mathfrak{u}_2''\in p^{e'-e-l}\mathbb{Z}_p,$ $\mathfrak{u}_1''\in p^{3e'-2e-3l}\mathbb{Z}_p,$ and $$\mathfrak{u}''-\begin{pmatrix} p^{3e'-2e-3l}\\ &p^{e'-e-l}I_{n-2} \end{pmatrix}\alpha_n^{-1}\mathfrak{u}'\in \begin{pmatrix} p^{2e'-e-2l}\mathbb{Z}_p\\ \mathbb{Z}_p^{n-2} \end{pmatrix}.$$ This verifies the case (B.3). By assumption we have $j-e>j-e'=-l,$ and $j-e+r_1<l.$ So $\delta_{B'}^{-1}(t)=p^{(n-1)(j-e+r_1-j+e)}<p^{2(n-1)l}.$ In all, Lemma [Lemma 31](#lem7.){reference-type="ref" reference="lem7."} follows. ◻ *Remark 32*. Lemmas [Lemma 30](#lem7){reference-type="ref" reference="lem7"} and [Lemma 31](#lem7.){reference-type="ref" reference="lem7."} play an important role in the local estimate of all orbital integrals in the geometric side $\mathcal{J}_{\operatorname{Geo}}^{\heartsuit}(\boldsymbol{\alpha},\boldsymbol{\ell})$ (cf. [7.5](#sec5.4){reference-type="ref" reference="sec5.4"}, [8](#8.5.2){reference-type="ref" reference="8.5.2"}--[9](#sec10){reference-type="ref" reference="sec10"}). In fact, the explicit description of the support of $f_p$ accounts for $\delta\asymp n^{-3}$ in Corollary [Corollary 1](#cor1.2){reference-type="ref" reference="cor1.2"} parallel to the saving $\delta\asymp n^{-5}$ in [@Nel21]. # Geometric Side: The Small Cell Orbital Integral {#8.5.1} Let $\mathbf{s}=(s,0)\in\mathbb{C}^2.$ By definition in [2.2.1](#2.2.1){reference-type="ref" reference="2.2.1"}, the small cell orbital integral $J^{\operatorname{Reg}}_{\operatorname{Geo},\operatorname{Small}}(f,\textbf{s})$ converges absolutely in $\mathop{\mathrm{Re}}(s)>0,$ and admits a meromorphic continuation to $s\in\mathbb{C}$ with possible simple poles at $s\in\{0, -1\}.$ Define $$\label{7.1'} J_{\operatorname{Geo},\operatorname{Main}}^{\operatorname{Small}}(f,\textbf{s}):=J^{\operatorname{Reg}}_{\operatorname{Geo},\operatorname{Small}}(f,\textbf{s})-s^{-1}\underset{s=0}{\operatorname{Res}}\ J^{\operatorname{Reg}}_{\operatorname{Geo},\operatorname{Small}}(f,\textbf{s}).$$ Then $J_{\operatorname{Geo},\operatorname{Main}}^{\operatorname{Small}}(f,\textbf{s})$ is holomorphic in the region $\mathop{\mathrm{Re}}(s)>-1.$ The main result in this section is the following uniform upper bound of $J_{\operatorname{Geo},\operatorname{Main}}^{\operatorname{Small}}(f,\mathbf{0}).$ **Proposition 33**. *Let notation be as before. $f\in \big\{f(\cdot;i,p_0), f(\cdot;p_1,p_2)\big\}.$ Let $\varepsilon>0$ be a small constant. Then $$\begin{aligned} J_{\operatorname{Geo},\operatorname{Main}}^{\operatorname{Small}}(f,\mathbf{0}) \ll T^{\frac{n}{2}+\varepsilon}M'^{2n}M^{n+\varepsilon} \mathcal{N}_f^{-1+2\vartheta_p+\varepsilon}\langle\phi',\phi'\rangle\prod_{p\mid M'}p^{ne_p(M)},\end{aligned}$$ where the implied constant depends on $\varepsilon,$ parameters $c_{\infty}$ and $C_{\infty}$ defined in [3.1.4](#sec3.14){reference-type="ref" reference="sec3.14"}, and the conductor of $\pi_{\infty}'.$ Here $\vartheta_p$ is the Ramanujan parameter bound for $\pi'_p$ defined in [3.1.3](#sec3.1.3){reference-type="ref" reference="sec3.1.3"}, and $\mathcal{N}_f$ is the parameter defined by [\[61\]](#61){reference-type="eqref" reference="61"} in [3.6.2](#3.6.2){reference-type="ref" reference="3.6.2"}.* The Proof of Proposition [Proposition 33](#prop54){reference-type="ref" reference="prop54"} will be given in [7.5](#sec5.4){reference-type="ref" reference="sec5.4"}, based on the auxiliary results in [7.1](#6.2..){reference-type="ref" reference="6.2.."}--[7.4](#sec5.3){reference-type="ref" reference="sec5.3"}. ## Nonarchimedean Auxiliary Integrals () {#6.2..} Let $p< \infty$ be a prime. Let $\mathop{\mathrm{Re}}(s)>-1+2\vartheta_p.$ Define $$\label{eq7.2} \mathcal{I}_p(f,s)=\int_{N'(\mathbb{Q}_p)\backslash G'(\mathbb{Q}_p)}\int_{G'(\mathbb{Q}_p)}\kappa(x,y)W_p'(x)\overline{W_p'(xy)}|\det x|_p^{1+s}dxdy,$$ where $$\label{eq7.3} \kappa(x,y):=\int_{M_{n,1}(\mathbb{Q}_p)} f_p\left(\begin{pmatrix} y&\mathfrak{u}\\ &1 \end{pmatrix}\right)\psi_p(\eta x\mathfrak{u})d\mathfrak{u}.$$ Here $W_p',$ the vector in the Whittaker model of $\pi_p',$ is constructed in [4](#sec4.){reference-type="ref" reference="sec4."}. In particular, when $p\nmid M',$ $W_p'$ is the spherical vector normalized as $W_p'(I_n)=1.$ The main result in this section is an estimate of $\mathcal{I}_p(f,s)$ and its detivative at $s=0,$ $p\mid MM'\nu(f)$. We shall start with $\mathcal{I}_p(f,0),$ towards which one may make use of the Hecke structure. Write $y=w^{-1}tuk$ as in [\[dec\]](#dec){reference-type="eqref" reference="dec"}. At $p\mid\nu(f),$ $W_p'$ is spherical. Then $$\label{6.7} \int_{K_p'}\overline{W_p'(bk'y)}dk'=\gamma_{\pi_p'}(t)\overline{W_p'(b)},$$ where $\gamma_{\pi_p'}(t)$ is the normalized spherical function (cf. [@Mac71]). **Lemma 34**. *Let notation be as before. Let $p\nmid M'.$ Then $\mathcal{I}_p(f,0)$ is equal to $$\begin{aligned} L(1,\pi_p'\times\widetilde{\pi}_p')\int_{G'(\mathbb{Q}_p)}&\int_{\mathbb{Q}_p^n}\sum_{m\geq 0}p^{-mn}f_p\left(\begin{pmatrix} y&\mathfrak{u}\\ &1 \end{pmatrix}\right)\int_{\mathbb{Z}_p^{\times}}\psi_p(\eta p^m\beta \mathfrak{u})d^{\times}\beta d\mathfrak{u}\gamma_{\pi_p'}(t)dy.\end{aligned}$$* *Proof.* Let $H=\operatorname{diag}(\mathrm{GL}(n-1),1).$ By Iwasawa decomposition we can write $x=zbk',$ with $z=p^m\beta I_n,$ $m\in\mathbb{Z},$ $\beta\in \mathbb{Z}_p^{\times},$ $k'\in K_p'$ and $b\in N_H(\mathbb{Q}_p)\backslash H(\mathbb{Q}_p),$ where $N_H=N\cap H.$ Then $$\begin{aligned} \mathcal{I}_p(f,0)=\int_{K_p'}\int_{G'(\mathbb{Q}_p)}&\int_{\mathbb{Q}_p^n}\sum_{m\in\mathbb{Z}}p^{-mn}f_p\left(\begin{pmatrix} y&\mathfrak{u}\\ &1 \end{pmatrix}\right)\int_{\mathbb{Z}_p^{\times}}\psi_p(\eta p^m\beta k'\mathfrak{u})d^{\times}\beta d\mathfrak{u} \\ &\int_{N_H(\mathbb{Q}_p)\backslash H(\mathbb{Q}_p)}W_p'(b'k')\overline{W_p'(bk'y)}dbdydk'.\end{aligned}$$ By Kirillov model theory we have $$\begin{aligned} \mathcal{I}_p(f,0)=\int_{K_p'}\int_{G'(\mathbb{Q}_p)}&\int_{\mathbb{Q}_p^n}\sum_{m\in\mathbb{Z}}p^{-mn}f_p\left(\begin{pmatrix} y&\mathfrak{u}\\ &1 \end{pmatrix}\right)\int_{\mathbb{Z}_p^{\times}}\psi_p(\eta p^m\beta k'\mathfrak{u})d^{\times}\beta d\mathfrak{u} \\ &\int_{N_H(\mathbb{Q}_p)\backslash H(\mathbb{Q}_p)}W_p'(b)\overline{W_p'(by)}dbdydk'.\end{aligned}$$ As $f_p$ is left-$K_p(M)$-invariant, the above expression remains valid with $\mathfrak{u}$ replaced by $\mathfrak{u}+\mathfrak{v}$ for any $\mathfrak{v}\in M_{n,1}(\mathbb{Z}_p)$. Taking orthogonality into account, we observe that only $m\geq 0$ may contribute to $\mathcal{I}_p(f,0)$. Therefore, after a change of variables $\mathfrak{u}\mapsto k'^{-1}\mathfrak{u},$ $y\mapsto k'^{-1}y,$ and $k'\mapsto k'^{-1},$ we obtain that $$\begin{aligned} \mathcal{I}_p(f,0)=\int_{G'(\mathbb{Q}_p)}&\int_{\mathbb{Q}_p^n}\sum_{m\geq 0}p^{-mn}f_p\left(\begin{pmatrix} y&\mathfrak{u}\\ &1 \end{pmatrix}\right)\int_{\mathbb{Z}_p^{\times}}\psi_p(\eta p^m\beta \mathfrak{u})d^{\times}\beta d\mathfrak{u} \\ &\int_{N_H(\mathbb{Q}_p)\backslash H(\mathbb{Q}_p)}W_p'(b)\Big[\int_{K_p'}\overline{W_p'(bk'y)}dk'\Big]dbdy.\end{aligned}$$ Here we make use of the fact that $f_p$ is left $K_p(M)$-invariant, and $\iota(K_p')\subsetneq K_p(M).$ Therefore, Lemma [Lemma 34](#lem7.2){reference-type="ref" reference="lem7.2"} follows from the local Rankin-Selberg integral calculation in conjunction with [\[6.7\]](#6.7){reference-type="eqref" reference="6.7"}. Note that at $p\mid M'$ we make use of [@Miy14 Theorem 4.1] for the calculation of $W_p'.$ ◻ *Remark 35*. Note that we cannot extend the above calculation to $\mathcal{I}_p(f,s)$ when $s\neq 0$. This limitation arises from our inability to eliminate the dependence on $k$ in the integral: $$\begin{aligned} \int_{N_H(\mathbb{Q}_p)\backslash H(\mathbb{Q}_p)}W_p'(b'k')\overline{W_p'(bk'y)}|\det b|_p^sdb.\end{aligned}$$ Unless $s=0$, we are unable to eliminate the contribution from $k$. However, if $s=0$, we can exploit the Kirillov model to overcome this obstacle. **Lemma 36**. *Let notation be as before. Let $p$ be a prime with $p\nmid M'\nu(f).$ Then $$\begin{aligned} \mathcal{I}_p(f,s)=\operatorname{Vol}(\overline{K_p(M)})^{-1}\cdot L_p(1+s,\pi_p'\times\widetilde{\pi}_p'), \ \ \mathop{\mathrm{Re}}(s)>-1+2\vartheta_p.\end{aligned}$$* *Proof.* We notice that $\mathcal{I}_p(f,s)$ is equal to $$\begin{aligned} \sum_{m\geq 0}\frac{1}{p^{mn}}\int_{K_p'}\int_{G'(\mathbb{Q}_p)}&\int_{\mathbb{Q}_p^n}f_p\left(\begin{pmatrix} y&\mathfrak{u}\\ &1 \end{pmatrix}\right)\int_{\mathbb{Z}_p^{\times}}\theta_p(\eta p^m\beta k'\mathfrak{u})d^{\times}\beta d\mathfrak{u} \Phi(s)dydk',\end{aligned}$$ where $\Phi(s):=\Phi^r(s;m,y,k')$ is defined by $$\begin{aligned} \int_{N_H(\mathbb{Q}_p)\backslash H(\mathbb{Q}_p)}W_p'(b)\Big[\int_{K_p'}\overline{W_p'(bk'y)}dk'\Big]\Big[p^{-nms}|\det b|_p^s\Big]db_p.\end{aligned}$$ Since $p\nmid \nu(f),$ $f_p\left(\begin{pmatrix} y&\mathfrak{u}\\ &1 \end{pmatrix}\right)\neq 0$ unless $y\in K_p',$ and $\mathfrak{u}\in\mathbb{Z}_p^n.$ Hence, $$\begin{aligned} \mathcal{I}_p(f,s)=\frac{1}{\operatorname{Vol}(\overline{K_p(M)})}\sum_{m\geq 0}\frac{1}{p^{mn(1+s)}}\cdot \int_{N_H(\mathbb{Q}_p)\backslash H(\mathbb{Q}_p)}W_p'(b)\overline{W_p'(b)}|\det b|_p^sdb,\end{aligned}$$ which is equal to $\operatorname{Vol}(\overline{K_p(M)})^{-1}\cdot L_p(1+s,\pi_p'\times\widetilde{\pi}_p').$ ◻ **Lemma 37**. *Let notation be as before. Let $p\mid M'.$ Then for $\mathop{\mathrm{Re}}(s)>2\vartheta_p-1,$ $$\label{equ7.5} |\mathcal{I}_p(f,s)|\ll p^{2ne_p(M)+2ne_p(M')-e_p(M'')}L_p(1+\mathop{\mathrm{Re}}(s),\pi_p'\times\widetilde{\pi}_p'),$$ where the implied constant is absolute. In particular, $$\label{eq7.5.} |\mathcal{I}_p(f,0)|+\Big|\frac{d\mathcal{I}_p(f,s)}{ds}\mid_{s=0}\Big|\ll p^{2ne_p(M)+3ne_p(M')-e_p(M'')}L_p(1,\pi_p'\times\widetilde{\pi}_p'),$$ where the implied constant depends only on $n.$* *Proof.* Recall that $\mathcal{I}_p(f,s)$ is defined by [\[eq7.2\]](#eq7.2){reference-type="eqref" reference="eq7.2"}. Then Cauchy-Schwarz leads to $$\begin{aligned} |\mathcal{I}_p(f,s)|^2\leq \mathcal{I}_p^{(1)}(f,s)\cdot \mathcal{I}_p^{(2)}(f,s)\end{aligned}$$ where $$\begin{aligned} \mathcal{I}_p^{(1)}(f,s):=&\int_{N'(\mathbb{Q}_p)\backslash G'(\mathbb{Q}_p)}\int_{G'(\mathbb{Q}_p)}|\kappa(x,y)||W_p'(x)|^2|\det x|_p^{1+\mathop{\mathrm{Re}}(s)}dxdy,\\ \mathcal{I}_p^{(2)}(f,s):=&\int_{N'(\mathbb{Q}_p)\backslash G'(\mathbb{Q}_p)}\int_{G'(\mathbb{Q}_p)}|\kappa(x,y)||W_p'(xy)|^2|\det x|_p^{1+\mathop{\mathrm{Re}}(s)}dxdy.\end{aligned}$$ Here we recall that $$\begin{aligned} \kappa(x,y):=\int_{M_{n,1}(\mathbb{Q}_p)} f_p\left(\begin{pmatrix} y&\mathfrak{u}\\ &1 \end{pmatrix}\right)\psi_p(\eta x\mathfrak{u})d\mathfrak{u}.\tag{\ref{eq7.3}}\end{aligned}$$ By the change of variable $x\mapsto xy^{-1},$ the funntion $\mathcal{I}_p^{(2)}(f,s)$ becomes $$\begin{aligned} \int_{N'(\mathbb{Q}_p)\backslash G'(\mathbb{Q}_p)}\int_{G'(\mathbb{Q}_p)}|\kappa(xy^{-1},y)||\det y|_p^{-1-\mathop{\mathrm{Re}}(s)}|W_p'(x)|^2|\det x|_p^{1+\mathop{\mathrm{Re}}(s)}dxdy.\end{aligned}$$ Write $x=p^mak'\in N'(\mathbb{Q}_p)\backslash G'(\mathbb{Q}_p)$, where $a\in A'(\mathbb{Q}_p),$ and $k'\in K_p'.$ Then $$\begin{aligned} \kappa(x,y)=\int_{M_{n,1}(\mathbb{Q}_p)}f_p\left(\begin{pmatrix} y&\mathfrak{u}\\ &1 \end{pmatrix}\right)\psi_p(\eta p^mk'\mathfrak{u})d\mathfrak{u}.\end{aligned}$$ By the change of variable $\mathfrak{u}\mapsto y\mathfrak{u},$ we have $$\label{eq7.5} \frac{\kappa(xy^{-1},y)}{|\det y|_p^{1+\mathop{\mathrm{Re}}(s)}}=|\det y|_p^{-\mathop{\mathrm{Re}}(s)}\int_{M_{n,1}(\mathbb{Q}_p)} f_p\left(\begin{pmatrix} y&y\mathfrak{u}\\ &1 \end{pmatrix}\right)\psi_p(\eta x\mathfrak{u})d\mathfrak{u}.$$ Recall that $f_p$ is defined by [\[3.4\]](#3.4){reference-type="eqref" reference="3.4"} in [3.3.2](#11.1.4){reference-type="ref" reference="11.1.4"}. Explicitly, it is given by $$f_p(g)=v_p^2\sum_{\boldsymbol{\alpha}}\sum_{\boldsymbol{\beta}}\omega_p'(\alpha_n)\omega_p'(\beta_n)\int_{Z(\mathbb{Q}_p)}\textbf{1}_{u_{\boldsymbol{\alpha}}K_pu_{\boldsymbol{\beta}}^{-1}}(zg)\omega_p(z)d^{\times}z.\tag{\ref{3.13}}$$ Note that $f_p$ is bi-$\begin{pmatrix} I_n& M_{n,1}(\mathbb{Z}_p)\\ &1 \end{pmatrix}$-invariant. By orthogonality, $\kappa(x,y)=0$ or $\kappa(xy^{-1},y)=0$ unless $m\geq 0,$ which amounts to $\eta x\in M_{1,n}(\mathbb{Z}_p).$ So $$\begin{aligned} \mathcal{I}_p^{(1)}(f,s)=&\int\int_{G'(\mathbb{Q}_p)}|\kappa(x,y)|dy|W_p'(x)|^2|\det x|_p^{1+\mathop{\mathrm{Re}}(s)}\textbf{1}_{M_{1,n}(\mathbb{Z}_p)}(\eta x)dx,\\ \mathcal{I}_p^{(2)}(f,s)=&\int\int_{G'(\mathbb{Q}_p)}\frac{\kappa(xy^{-1},y)}{|\det y|_p^{1+\mathop{\mathrm{Re}}(s)}}dy|W_p'(x)|^2|\det x|_p^{1+\mathop{\mathrm{Re}}(s)}\textbf{1}_{M_{1,n}(\mathbb{Z}_p)}(\eta x)dx,\end{aligned}$$ where $x$ ranges through $N'(\mathbb{Q}_p)\backslash G'(\mathbb{Q}_p).$ Investigating the support of $f_p,$ we have $\kappa(xy^{-1},y)=0$ or $\kappa(x,y)=0$ unless $y\in K_p'$ and $\mathfrak{u}\in p^{-m'}M_{n,1}(\mathbb{Z}_p).$ Hence, by triangle inequality, $$\begin{aligned} \int_{G'(\mathbb{Q}_p)}|\kappa(x,y)|dy\leq \int_{K_P'}\int_{M_{n,1}(\mathbb{Q}_p)}f_p\left(\begin{pmatrix} y&\mathfrak{u}\\ &1 \end{pmatrix}\right)d\mathfrak{u}dy\leq v_p^2\operatorname{Vol}(\overline{\mathop{\mathrm{supp}}f_p})\leq v_p^2p^{2nm'},\end{aligned}$$ which is $\ll p^{2ne_p(M)+2nm'-m''}.$ Here the implied constant is absolute. Therefore, $$\begin{aligned} \mathcal{I}_p^{(1)}(f,s)\ll p^{2ne_p(M)+2nm'-m''}\int_{N'(\mathbb{Q}) \backslash G'(\mathbb{Q}_p)}|W_p'(x)|^2|\det x|_p^{1+\mathop{\mathrm{Re}}(s)}\textbf{1}_{M_{1,n}(\mathbb{Z}_p)}(\eta x)dx,\end{aligned}$$ which, by [@Miy14 Theorem 4.1], boils down to $$\begin{aligned} \mathcal{I}_p^{(1)}(f,s)\ll p^{2ne_p(M)+2nm'-m''}L_p(1+\mathop{\mathrm{Re}}(s),\pi_p'\times\widetilde{\pi}_p'). \end{aligned}$$ Likewise, by [\[eq7.5\]](#eq7.5){reference-type="eqref" reference="eq7.5"} and the change of variable $\mathfrak{u}\mapsto y^{-1}\mathfrak{u},$ we have $$\begin{aligned} \int_{G'(\mathbb{Q}_p)}\frac{\kappa(xy^{-1},y)}{|\det y|_p^{1+\mathop{\mathrm{Re}}(s)}}dy\leq \int_{K_P'}\int_{M_{n,1}(\mathbb{Q}_p)}f_p\left(\begin{pmatrix} y&\mathfrak{u}\\ &1 \end{pmatrix}\right)d\mathfrak{u}dy,\end{aligned}$$ which is $\ll p^{2ne_p(M)+2nm'-m''}.$ As a consequence, $$\begin{aligned} \mathcal{I}_p^{(2)}(f,s)\ll p^{2ne_p(M)+2nm'-m''}L_p(1+\mathop{\mathrm{Re}}(s),\pi_p'\times\widetilde{\pi}_p'). \end{aligned}$$ Therefore, [\[equ7.5\]](#equ7.5){reference-type="eqref" reference="equ7.5"} follows. Along with Cauchy's formula we obtain $$\begin{aligned} \big|\frac{d\mathcal{I}_p(f,s)}{ds}\mid_{s=0}\big|\leq \frac{1}{2\pi}\int_{|s|=n^{-100}}\frac{|\mathcal{I}_p(f,s)|}{|s|^2}|ds|\ll p^{2ne_p(M)+2nm'-m''},\end{aligned}$$ where the implied constant depends on $n.$ Here we also make use of the fact that $L_p(1+\mathop{\mathrm{Re}}(s),\pi_p'\times\widetilde{\pi}_p')\ll 1,$ uniformly for $|s|=n^{-100}.$ Therefor, [\[eq7.5.\]](#eq7.5.){reference-type="eqref" reference="eq7.5."} follows from the lower bound $L_p(1,\pi_p'\times\widetilde{\pi}_p')\gg 1.$ ◻ *Remark 38*. The bounds in [\[equ7.5\]](#equ7.5){reference-type="eqref" reference="equ7.5"} and [\[eq7.5.\]](#eq7.5.){reference-type="eqref" reference="eq7.5."} are not optimal; however, they serve our purpose in this paper, as we fix $\pi'$. ## Nonarchimedean Auxiliary Integrals () {#nonarchimedean-auxiliary-integrals} Recall that $$\mathcal{I}_p(f,s)=\int_{N'(\mathbb{Q}_p)\backslash G'(\mathbb{Q}_p)}\int_{G'(\mathbb{Q}_p)}\kappa(x,y)W_p'(x)\overline{W_p'(xy)}|\det x|_p^{1+s}dxdy,\tag{\ref{eq7.2}}$$ where $$\kappa(x,y):=\int_{M_{n,1}(\mathbb{Q}_p)} f_p\left(\begin{pmatrix} y&\mathfrak{u}\\ &1 \end{pmatrix}\right)\psi_p(\eta x\mathfrak{u})d\mathfrak{u}.\tag{\ref{eq7.3}}$$ In this section, our goal is to establish a precise bound for $\mathcal{I}_p(f,0)$ at $p\mid\nu(f)$, which has not been addressed in Lemmas [Lemma 36](#lem6.4.){reference-type="ref" reference="lem6.4."} and [Lemma 37](#lem11){reference-type="ref" reference="lem11"}. ### Notation Recap Let $W_H$ be the subgroup of the Weyl group $W_{G'}$ generalized by Weyl elements $w_1,\cdots, w_{n-2}.$ Identify $W_{G'}$ with $W_H\sqcup W_Hw_{n-1}W_H.$ Let $f\in \big\{f(\cdot;i,p_0), f(\cdot;p_1,p_2)\big\},$ and $p\mid\nu(f)$ (cf. [3.6.1](#sec3.6.1){reference-type="ref" reference="sec3.6.1"}--[3.6.2](#3.6.2){reference-type="ref" reference="3.6.2"}). Denote by $\mathcal{T}_p$ be the support of $t\in \operatorname{diag}(p^{\mathbb{Z}},\cdots, p^{\mathbb{Z}})$ such that $$f_p\left(\begin{pmatrix} w^{-1}tu&\mathfrak{u}\\ &1 \end{pmatrix}\right)\neq 0$$ for some $w\in W_{G'},$ $u\in N'(\mathbb{Q}_p),$ and $\mathfrak{u}\in M_{n,1}(\mathbb{Q}_p).$ By Lemmas [Lemma 30](#lem7){reference-type="ref" reference="lem7"} and [Lemma 31](#lem7.){reference-type="ref" reference="lem7."}, $\mathcal{T}_p$ is a finite set, with cardinality bounded by $O(1),$ where the implied constant depends at most on $n.$ Denote by $U_p(t,u,w)$ the set of $\mathfrak{u}\in M_{n,1}(\mathbb{Q}_p)$ such that $$f_p\left(\begin{pmatrix} w^{-1}tu& \mathfrak{u}\\ &1 \end{pmatrix}\right)\neq 0.$$ The set $U_p(t,u,w)$ has been described explicitly in Lemmas [Lemma 30](#lem7){reference-type="ref" reference="lem7"} and [Lemma 31](#lem7.){reference-type="ref" reference="lem7."}. Let $$\begin{aligned} \mathcal{J}_p^{\dagger}(t,u,w):=\sum_{m\geq 0}p^{-mn}\int_{\mathbb{Z}_p^{\times}}\int_{U_p(t,u,w)}\psi_p(\eta p^m\beta \mathfrak{u})d\mathfrak{u}d^{\times}\beta.\end{aligned}$$ Towards the inner integral relative to $\beta$ (i.e., Ramanujan sum), we have $$\label{8} p^{-m}p^{-\alpha}\int_{\mathbb{Z}_p^{\times}}\psi_p(p^{m}p^{\alpha}\beta)d^{\times}\beta=\begin{cases} (1-p^{-1})p^{-m-\alpha}& \text{if $m+\alpha\geq 0,$}\\ -(1-p^{-1})& \text{if $m+\alpha=-1,$}\\ 0& \text{if $m+\alpha\leq -2$}. \end{cases}$$ ### The case that $f=f(\cdot;p_1,p_2)$ We will follow the notation (e.g., $e,$ $e',$ $l,$ $\mathfrak{u},$ $\alpha,$ $\mathfrak{u}',$ etc) in Lemma [Lemma 30](#lem7){reference-type="ref" reference="lem7"} in [6.2](#sec6.2){reference-type="ref" reference="sec6.2"}. Write $y=w^{-1}tuk$ as in [\[dec\]](#dec){reference-type="eqref" reference="dec"}. Denote by $U_p(t)$ the range of $u=\begin{pmatrix} I_{n-1}&\mathfrak{u}''\\ &1 \end{pmatrix}$ according to Lemma [Lemma 30](#lem7){reference-type="ref" reference="lem7"}. **Lemma 39** (Off-diagonal Integrals). *Let notation be as before. Let $j\in \{1,2\}$ and $p=p_j\in\mathcal{L}.$ Write $l=l_p.$ Then $$\label{43} \mathcal{I}_p(f,0) \ll p^{(-1/2+\vartheta_p)l},$$ where the implied constant depends at most on $n.$* *Proof.* According to the definition in equation [\[eq7.2\]](#eq7.2){reference-type="eqref" reference="eq7.2"}, and after a change of variables $x\mapsto xy^{-1}$ and $\mathfrak{u}\mapsto y\mathfrak{u}$, it is sufficient to consider $p=p_2$ in order to prove [\[43\]](#43){reference-type="eqref" reference="43"}. If $w\in W_H,$ then by [\[8\]](#8){reference-type="eqref" reference="8"} we have $\mathcal{J}_p^{\dagger}(t,u,w)\ll 1,$ where the implied constant is absolute. Hence, by Lemmas [Lemma 30](#lem7){reference-type="ref" reference="lem7"} and [Lemma 34](#lem7.2){reference-type="ref" reference="lem7.2"}, $$\begin{aligned} \sum_{w\in W_H}\sum_{t}\int_{U_p(t)}|\gamma_{\pi_p'}(t)|\|f_p\|_{\infty}du\ll p^{\vartheta_p l-\frac{nl}{2}} \sum_{t}\delta_{B'}^{-\frac{1}{2}}(t)\int_{K_p'tK_p'}dy_p\ll p^{(-\frac{1}{2}+\vartheta_p)l},\end{aligned}$$ where $t\in\mathcal{T}_p.$ We also employ the fact that $\|f_p\|_{\infty}\ll p^{-nl/2},$ the bounds $\#\mathcal{T}_p\ll 1$ and $\delta_{B'}^{-1}(t)\leq p^{-(n-1)l}$ (cf. Lemma [Lemma 30](#lem7){reference-type="ref" reference="lem7"}) and Macdonald's formula to derive $$\label{6.10} |\gamma_{\pi_p'}(t)|\ll p^{l\vartheta_p}\cdot \delta_{B'}^{1/2}(t)$$ and to bound the volume of $K_p'tK_p'$ by $O(\delta_{B'}^{-1}(t)).$ Now we suppose $w\not\in W_H.$ Let $t\in \mathcal{T}_p$ be represented by $\begin{pmatrix} I_{n-1}\\ &p^{-a} \end{pmatrix}$ modulo $Z'(\mathbb{Q}_p),$ $0\leq a\leq l.$ Then by the calculation of Ramanujan sums [\[8\]](#8){reference-type="eqref" reference="8"}, and the relations between entries of $u$ and $\mathfrak{u}$ in Lemma [Lemma 30](#lem7){reference-type="ref" reference="lem7"}, we obtain that $$\label{6.10.} \int_{U(t)}\mathcal{J}_p^{\dagger}(t,u,w)du\ll p^a\cdot p^{(n-2)a}=p^{(n-1)a}.$$ Combining Lemmas [Lemma 30](#lem7){reference-type="ref" reference="lem7"} and [Lemma 36](#lem6.4.){reference-type="ref" reference="lem6.4."} with the estimate [\[6.10.\]](#6.10.){reference-type="eqref" reference="6.10."}, we then derive that $$\begin{aligned} \sum_{w\not\in W_H}\sum_{t}|\gamma_{\pi_p'}(t)|\|f_p\|_{\infty}\Big|\int_{U(t)}\mathcal{J}_p^{\dagger}(t,u,w)du\Big|\ll p^{\vartheta_p l-\frac{nl}{2}} \cdot p^{-\frac{(n-1)l}{2}}\cdot p^{(n-1)l}, \end{aligned}$$ which is $\ll p^{(-\frac{1}{2}+\vartheta_p)l}.$ Here we use [\[6.10\]](#6.10){reference-type="eqref" reference="6.10"} and the fact that $0\leq a\leq l\leq n+1.$ We thus conclude [\[43\]](#43){reference-type="eqref" reference="43"} form the above estimates. ◻ ### The case that $f=f(\cdot;i,p_0)$ We will adhere to the notation (e.g., $e,$ $e',$ $l,$ $\mathfrak{u},$ $\alpha,$ $\mathfrak{u}',$ $\mathfrak{u}_1'',$ $\mathfrak{u}_2'',$ $\mathfrak{c},$ etc) introduced in Lemma [Lemma 31](#lem7.){reference-type="ref" reference="lem7."} in [6.3](#sec6.2.){reference-type="ref" reference="sec6.2."}. Let us express $y$ as $y=w^{-1}tuk$ following equation [\[dec\]](#dec){reference-type="eqref" reference="dec"}. We will use $U_p(t)$ to represent the range of $u=\begin{pmatrix} 1&\mathfrak{c}&\mathfrak{u}_1''\\ &I_{n-2}&\mathfrak{u}_2''\\ &&1 \end{pmatrix}$ according to Lemma [Lemma 31](#lem7.){reference-type="ref" reference="lem7."}. **Lemma 40** (Diagonal Integrals). *Let notation be as before. Let $p=p_0\in\mathcal{L}.$ Write $l=i$ (cf. [3.6.2](#3.6.2){reference-type="ref" reference="3.6.2"}). Then $$\label{13} \mathcal{I}_p(f,0)\ll p^{(-1+2\vartheta_p)l},$$ where the implied constant depends at most on $n.$* *Proof.* First we assume that $w\in W_H.$ Then by [\[8\]](#8){reference-type="eqref" reference="8"} we have $\mathcal{J}_p^{\dagger}(t,u,w)\ll 1,$ where the implied constant is absolute. Hence, by Lemmas [Lemma 31](#lem7.){reference-type="ref" reference="lem7."} and [Lemma 34](#lem7.2){reference-type="ref" reference="lem7.2"}, $$\begin{aligned} \sum_{w\in W_H}\sum_{t}\int_{U_p(t)}|\gamma_{\pi_p'}(t)|\|f_p\|_{\infty}du\ll p^{2\vartheta_p i-ni} \sum_{t}\delta_{B'}^{-\frac{1}{2}}(t)\int_{K_p'tK_p'}dy\ll p^{(-1+2\vartheta_p)l},\end{aligned}$$ where we make use of Macdonald's formula to derive $$\label{7.7} |\gamma_{\pi_p'}(t)|\ll p^{2l\vartheta_p}\cdot \delta_{B'}^{1/2}(t)$$ and the fact that $\operatorname{Vol}(K_p'tK_p')\ll \delta_{B'}^{-1}(t).$ Therefore, the contribution from $w\in W_H$ to $\mathcal{I}_p(f,0)$ is $\ll p^{(-1+2\vartheta_p)l}.$ Henceforth we assume $w\not\in W_H.$ According to Lemma [Lemma 31](#lem7.){reference-type="ref" reference="lem7."}, we may write $\mathcal{T}_p\ni t\in \operatorname{diag}(p^b,I_{n-1},p^{-a})Z'(\mathbb{Q}_p),$ $0\leq a, b\leq l\leq n+1.$ The remaining estimates are similar to those in the proof of Lemma [Lemma 39](#lem8){reference-type="ref" reference="lem8"}. The major difference is that we need to investigate the cases (A.1), (A.2), (B.1), (B.2), (B.3) in Lemma [Lemma 31](#lem7.){reference-type="ref" reference="lem7."}. Denote by $\mathcal{I}_p(A.1),$ $\mathcal{I}_p(A.2),$ $\mathcal{I}_p(B.1),$ $\mathcal{I}_p(B.2)$, $\mathcal{I}_p(B.3)$ the contribution to $\mathcal{I}_p(f,0)$ from corresponding cases with $w\in W_{G'}-W_H,$ respectively. By Lemma [Lemma 34](#lem7.2){reference-type="ref" reference="lem7.2"}, $$\label{6.13} \mathcal{I}_p(*) \ll\sum_{w}\sum_{t}|\gamma_{\pi_p'}(t)|\|f_p\|_{\infty}\Big|\int_{U_p(t)}\mathcal{J}_p^{\dagger}(t,u,w)\textbf{1}_{*}(y)du\Big|,$$ where $*\in\{A.1, A.2, B.1, B.2, B.3\},$ and $\textbf{1}_{*}$ is the indicator function of the case $(*).$ Recall that $w\mathfrak{u}=\begin{pmatrix} \mathfrak{u}'\\ \alpha \end{pmatrix}.$ Write $\mathfrak{u}=\ensuremath{\mskip 1mu\prescript{\smash{\mathrm t\mkern-3mu}}{}{\mathstrut(\alpha_1',\cdots,\alpha_n')}}.$ Suppose $\alpha_n'$ is equal to $m'$-th entry of $\mathfrak{u}'$ for some $1\leq m'\leq n-1$ determined uniquely by $w.$ Let $p^{\mu_{m'}}\beta'$ be the $m'$-th entry of $\mathfrak{u}'':=\begin{pmatrix} \mathfrak{u}_1''\\ \mathfrak{u}_2'' \end{pmatrix},$ where $\beta'\in\mathbb{Z}_p^{\times}.$ 1. In the case (A.1) of Lemma [Lemma 31](#lem7.){reference-type="ref" reference="lem7."}, $\mathcal{T}_p\ni t\in \operatorname{diag}(p^l,I_{n-2},p^{-l})Z'(\mathbb{Q}_p).$ By orthogonality, we derive that $\mathcal{J}_p^{\dagger}(t,u,w)\ll p^{l},$ where the implied constant is absolute. In conjunction with [\[6.13\]](#6.13){reference-type="eqref" reference="6.13"} we obtain that $$\begin{aligned} \mathcal{I}_p(A.1)\ll \sum_{w}\sum_{t}|\gamma_{\pi_p'}(t)|\|f_p\|_{\infty}\int_{U_p(t)}du\cdot p^{l}\ll p^{2\vartheta_p l-l}.\end{aligned}$$ Here we use [\[7.7\]](#7.7){reference-type="eqref" reference="7.7"} and the facts that $\|f_p\|_{\infty}\ll p^{-nl}$ and $\operatorname{Vol}(U_p(t))\ll p^{(2n-3)l},$ which follows from the classification in Lemma [Lemma 31](#lem7.){reference-type="ref" reference="lem7."}. 2. In the case (A.2) of Lemma [Lemma 31](#lem7.){reference-type="ref" reference="lem7."}, $\mathcal{T}_p\ni t\in \operatorname{diag}(p^{2l-e},I_{n-2},p^{-l})Z'(\mathbb{Q}_p),$ where $l<e\leq 2l.$ Note that $\alpha\in p^{-l}\mathbb{Z}_p,$ and $\mathfrak{u}'-\alpha p^l\mathfrak{u}''\in p^{l-e}\mathbb{Z}_p^{n-1}.$ By [\[8\]](#8){reference-type="eqref" reference="8"} and the parametrization of $U_p(t)$ we have $$\label{6.14} \int_{U_p(t)}\mathcal{J}_p^{\dagger}(t,u,w)du\ll p^{l-e}\cdot p^l\cdot p^{(n-1)(2l-e)}\cdot p^{(n-2)l},$$ where the factor $p^l$ comes from the contribution from $\alpha,$ and $p^{l-e}$ is the contribution from $\alpha_1', \cdots,\alpha_{n-1}'$ and the exponential integral relative to the $m'$-th entry of $\mathfrak{u}',$ which ranges through $\alpha p^lp^{\mu_{m'}}\beta'+p^{l-e}\mathbb{Z}_p;$ and the factor $p^{(n-1)(2l-e)}$ is the contribution from $\mathfrak{u}'',$ and the factor $p^{(n-2)l}$ is the contribution from $\mathfrak{c}$ (cf. Lemma [Lemma 31](#lem7.){reference-type="ref" reference="lem7."}). Therefore, by [\[6.13\]](#6.13){reference-type="eqref" reference="6.13"} and the fact that $l<e\leq 2l,$ we derive $$\begin{aligned} \mathcal{I}_p(A.2) \ll& p^{2\vartheta_p l}p^{-nl}\sum_w\sum_{t}\delta_{B'}^{-1/2}(t)\cdot p^{l-e}\cdot p^l\cdot p^{(n-1)(2l-e)}\cdot p^{(n-2)l},\end{aligned}$$ which is $\ll p^{(-1+2\vartheta_p)l}.$ Here we use the uniform bound $\delta_{B'}(t)\leq p^{2(n-1)l}$ and the fact that the number of $t$ in the sum is $\leq 2(n+1).$ 3. In the case (B.1) of Lemma [Lemma 31](#lem7.){reference-type="ref" reference="lem7."}, $\mathcal{T}_p\ni t\in \operatorname{diag}(p^{l},I_{n-2},p^{-e'})Z'(\mathbb{Q}_p),$ where $e'<l.$ Note that $\alpha\in p^{-e'}\mathbb{Z}_p^{\times},$ and $\mathfrak{u}''-\begin{pmatrix} p^{-l}\\ &I_{n-2} \end{pmatrix}p^{-e'}\alpha^{-1}\mathfrak{u}'\in \mathbb{Z}_p^{n-1}.$ We need to separate the cases according to $m'.$ - Suppose $m'=1.$ Then the $m'$-th entry of $\mathfrak{u}'$ lies in $p^{\mu_{m'}}p^{e'+l}\alpha\beta'+p^{e'+l}\mathbb{Z}_p.$ By orthogonality, $$\begin{aligned} \mathcal{J}_p^{\dagger}(t,u,w)\ll p^{e'}\cdot p^{-l}\cdot\Big|\sum_{m\geq 0}p^{-mn}\int_{\mathbb{Z}_p^{\times}}\psi_p( p^{m+l+\mu_{m'}}\beta)d^{\times}\beta\Big|.\end{aligned}$$ where the factor $p^{e'}$ comes from the contribution from $\alpha,$ and $p^{-l}$ is the contribution from $\alpha_1',\cdots,\alpha_{n-1}'$. - Suppose $m'>1.$ Then the $m'$-th entry of $\mathfrak{u}'$ lies in $p^{\mu_{m'}}p^{e'}\alpha\beta'+p^{e'}\mathbb{Z}_p.$ By orthogonality, $$\begin{aligned} \mathcal{J}_p^{\dagger}(t,u,w)\ll p^{e'}\cdot \Big|\sum_{m\geq 0}p^{-mn}\int_{\mathbb{Z}_p^{\times}}\psi_p(p^{m+\mu_{m'}}\beta)d^{\times}\beta\Big|.\end{aligned}$$ where the factor $p^{e'}$ comes from the contribution from $\alpha.$ Using [\[8\]](#8){reference-type="eqref" reference="8"} to handle the above integrals relative to $\beta$, we then obtain $$\begin{aligned} \int_{U_p(t)}\mathcal{J}_p^{\dagger}(t,u,w)du\ll p^{e'}\cdot p^{(n-1)(l+e')-e'}. \end{aligned}$$ Together with [\[6.13\]](#6.13){reference-type="eqref" reference="6.13"} and the fact that $e'< l,$ we further derive that $$\begin{aligned} \mathcal{I}_p(B.1)\ll& p^{2\vartheta_p l}p^{-nl}\sum_{w}\sum_{t}\delta_{B'}^{-1/2}(t)\cdot p^{e'}\cdot p^{(n-1)(l+e')-e'}\ll p^{(-1+2\vartheta_p)l}.\end{aligned}$$ 4. In the case (B.2) of Lemma [Lemma 31](#lem7.){reference-type="ref" reference="lem7."}, $\mathcal{T}_p\ni t\in \operatorname{diag}(I_{n-1},p^{l-e'})Z'(\mathbb{Q}_p),$ where $l\leq e'<2l.$ Note that $\alpha\in p^{-e'}\mathbb{Z}_p^{\times},$ and $\mathfrak{u}''-p^{l-e'}\alpha^{-1}\mathfrak{u}'\in \mathbb{Z}_p^{n-1}.$ So $$\begin{aligned} \mathcal{J}_p^{\dagger}(t,u,w)\ll p^{e'}\cdot p^{-l}\cdot\Big|\sum_{m\geq l}p^{-mn}\int_{\mathbb{Z}_p^{\times}}\psi_p(p^{m-l+\mu_{m'}}\beta)d^{\times}\beta\Big|,\end{aligned}$$ where the analysis is similar to the above case (B.1) and $m'=1.$ Executing [\[8\]](#8){reference-type="eqref" reference="8"} to the integral relative to $\beta$, together with [\[6.13\]](#6.13){reference-type="eqref" reference="6.13"}, $$\begin{aligned} \mathcal{I}_p(B.2)\ll& p^{2\vartheta_p l}p^{-nl}\sum_w\sum_{t}\delta_{B'}^{-1/2}(t)\cdot p^{e'-l}\cdot p^{(n-1)(e'-l)-(e'-l)}\ll p^{(-1+2\vartheta_p)l}.\end{aligned}$$ Here we use the fact that $e'<2l$ and $\delta_{B'}^{-1}(t)\leq p^{(n-1)(e'-l)}.$ 5. In the case (B.3) of Lemma [Lemma 31](#lem7.){reference-type="ref" reference="lem7."}, $\mathcal{T}_p\ni t\in \operatorname{diag}(p^{2l-2e'+e},I_{n-2},p^{e'-e-l})Z'(\mathbb{Q}_p),$ where $l<2e'-e< 2l$ and $e'>e.$ Note that $\alpha\in p^{-e'}\mathbb{Z}_p^{\times},$ and $$\label{30} \mathfrak{u}''-\begin{pmatrix} p^{3e'-2e-3l}\\ &p^{e'-e-l}I_{n-2} \end{pmatrix}\alpha^{-1}\mathfrak{u}'\in \begin{pmatrix} p^{2e'-e-2l}\mathbb{Z}_p\\ \mathbb{Z}_p^{n-2} \end{pmatrix}.$$ We need to separate the cases according to $m'.$ The arguments here will be similar to the preceding case (B.1). - Suppose $m'=1.$ According to [\[30\]](#30){reference-type="eqref" reference="30"}, the $m'$-th entry of $\mathfrak{u}'$ ranges through $p^{\mu_{m'}}p^{2e+3l-3e'}\alpha\beta'+p^{e+l-2e'}\mathbb{Z}_p.$ By orthogonality, $$\begin{aligned} \mathcal{J}_p^{\dagger}(t,u,w)\ll p^{e'}\cdot p^{e+l-2e'}\cdot\Big|\sum_{m\geq 2e'-e-l}p^{-mn}\int_{\mathbb{Z}_p^{\times}}\psi_p(p^{m+2e+3l-4e'+\mu_{m'}}\beta)d^{\times}\beta\Big|,\end{aligned}$$ where the factor $p^{e'}$ comes from the contribution from $\alpha,$ and $p^{e+l-2e'}$ is the contribution from $\alpha_1',\cdots,\alpha_{n-1}'$. - Suppose $m'>1.$ According to [\[30\]](#30){reference-type="eqref" reference="30"}, the $m'$-th entry of $\mathfrak{u}'$ ranges through $p^{\mu_{m'}}p^{e+l-2e'}\beta'+p^{e+l-2e'}\mathbb{Z}_p.$ By orthogonality, $$\begin{aligned} \mathcal{J}_p^{\dagger}(t,u,w)\ll p^{e'}\cdot p^{e+l-2e'}\cdot\Big|\sum_{m\geq 2e'-e-l}p^{-mn}\int_{\mathbb{Z}_p^{\times}}\psi_p(p^{m+e+l-2e'+\mu_{m'}}\beta)d^{\times}\beta\Big|.\end{aligned}$$ Using [\[8\]](#8){reference-type="eqref" reference="8"} to handle the above integrals relative to $\beta$, we then obtain $$\begin{aligned} \int_{U_p(t)}\mathcal{J}_p^{\dagger}(t,u,w)du\ll p^{e'}\cdot p^{e+l-2e'}\cdot p^{(n-1)(2l-2e'+e)+e'-e-l}.\end{aligned}$$ Therefore, by [\[6.13\]](#6.13){reference-type="eqref" reference="6.13"} and the fact that $e'< l,$ we have $$\begin{aligned} \mathcal{I}_p(B.3)\ll& p^{2\vartheta_p l}p^{-nl}\sum_w\sum_{t}\delta_{B'}^{-1/2}(t)\cdot p^{e'}\cdot p^{e+l-2e'}\cdot p^{(n-1)(2l-2e'+e)+e'-e-l}.\end{aligned}$$ Since $\delta_{B'}^{-1}(t)=p^{(n-1)(2l-2e'+e)},$ and $2e+3l-3e'<e'+(e+l)+2l-3e'<e'+2e'+2l-3e'=2l,$ then $\mathcal{I}_p(B.3)\ll p^{(-1+2\vartheta_p)l}.$ We thus conclude [\[13\]](#13){reference-type="eqref" reference="13"} form the above estimates. ◻ ## Nonarchimedean Auxiliary Integrals () {#nonarchimedean-auxiliary-integrals-1} For $p\mid \nu(f)$, in this section, we aim to bound the derivative of $\mathcal{I}_p(f,s)$ at $s=0$. Notably, Lemmas [Lemma 39](#lem8){reference-type="ref" reference="lem8"} and [Lemma 40](#lem9){reference-type="ref" reference="lem9"} heavily rely on Lemma [Lemma 34](#lem7.2){reference-type="ref" reference="lem7.2"}, which only holds for $\mathcal{I}_p(f,s)$ at $s=0$. Consequently, we cannot employ it to establish a similar sharp bound for $\frac{d\mathcal{I}p(f,s)}{ds}\mid_{s=0}$. To overcome this, we will leverage techniques from complex analysis, as seen in the proof of Lemma [Lemma 29](#lem11.1){reference-type="ref" reference="lem11.1"}, to reduce the bound of $\frac{d\mathcal{I}p(f,s)}{ds}\mid_{s=0}$ to that of $\mathcal{I}_p(f,0)$. **Lemma 41**. *Let notation be as before. Let $p_0, p_1, p_2\in\mathcal{L}$ with $p_1\neq p_2.$ Let $l=l_p$ if $p\mid p_1p_2,$ and $l=i$ if $p=p_0$ (cf. [3.6.2](#3.6.2){reference-type="ref" reference="3.6.2"}). Then $$\label{7.11} \frac{d\mathcal{I}_p(f,s)}{ds}\mid_{s=0} \ll \begin{cases} p^{(-1/2+\vartheta_p)l+\varepsilon},\ & \text{if $p\mid \nu(f)=p_1p_2,$}\\ p^{(-1+2\vartheta_p)l+\varepsilon},\ & \text{if $p\mid \nu(f)=p_0,$} \end{cases}$$ where the implied constant depends at most on $n$ and $\varepsilon.$* *Proof.* Let $r=1$ if $\nu(f)=p_1p_2,$ and $r=2$ if $\nu(f)=p_0.$ By Lemmas [Lemma 39](#lem8){reference-type="ref" reference="lem8"} and [Lemma 40](#lem9){reference-type="ref" reference="lem9"} there exists an integer $m>0$ (depending on $n$) such that the function $$\begin{aligned} J(s):=\mathcal{I}_p(f,s)+mp^{(-1+2\vartheta_p)rl}+1,\ \ \mathop{\mathrm{Re}}(s)>-1+2\vartheta_p,\end{aligned}$$ satisfies that $J(0)\geq 1.$ By definition, $$\mathcal{I}_p(f,s)=\int_{N'(\mathbb{Q}_p)\backslash G'(\mathbb{Q}_p)}\int_{G'(\mathbb{Q}_p)}\kappa(x,y)W_p'(x)\overline{W_p'(xy)}|\det x|_p^{1+s}dxdy,\tag{\ref{eq7.2}}$$ where $$\kappa(x,y):=\int_{M_{n,1}(\mathbb{Q}_p)} f_p\left(\begin{pmatrix} y&\mathfrak{u}\\ &1 \end{pmatrix}\right)\psi_p(\eta x\mathfrak{u})d\mathfrak{u}.\tag{\ref{eq7.3}}$$ Following the proof of Lemma [Lemma 37](#lem11){reference-type="ref" reference="lem11"}, we have, by Cauchy-Schwarz, that $$\begin{aligned} |\mathcal{I}_p(f,s)|^2\leq \mathcal{I}_p^{(1)}(f,s)\cdot \mathcal{I}_p^{(2)}(f,s)\end{aligned}$$ where $$\begin{aligned} \mathcal{I}_p^{(1)}(f,s)=&\int\int_{G'(\mathbb{Q}_p)}|\kappa(x,y)|dy|W_p'(x)|^2|\det x|_p^{1+\mathop{\mathrm{Re}}(s)}\textbf{1}_{M_{1,n}(\mathbb{Z}_p)}(\eta x)dx,\\ \mathcal{I}_p^{(2)}(f,s)=&\int\int_{G'(\mathbb{Q}_p)}\frac{\kappa(xy^{-1},y)}{|\det y|_p^{1+\mathop{\mathrm{Re}}(s)}}dy|W_p'(x)|^2|\det x|_p^{1+\mathop{\mathrm{Re}}(s)}\textbf{1}_{M_{1,n}(\mathbb{Z}_p)}(\eta x)dx.\end{aligned}$$ Here $x$ ranges through $N'(\mathbb{Q}_p)\backslash G'(\mathbb{Q}_p).$ By Lemmas [Lemma 30](#lem7){reference-type="ref" reference="lem7"} and [Lemma 31](#lem7.){reference-type="ref" reference="lem7."} we have $f_p\left(\begin{pmatrix} y&\mathfrak{u}\\ &1 \end{pmatrix}\right)\neq 0$ unless $1\leq |\det y|_p\leq p^{(n+1)l}.$ Therefore, $$\begin{aligned} \int_{G'(\mathbb{Q}_p)}|\kappa(x,y)|dy\leq \int_{G'(\mathbb{Q}_p)}\int_{M_{n,1}(\mathbb{Q}_p)}f_p\left(\begin{pmatrix} y&y\mathfrak{u}\\ &1 \end{pmatrix}\right)|\det y|_pd\mathfrak{u}dy,\end{aligned}$$ which is $\leq p^{(n+1)l}\operatorname{Vol}(\mathop{\mathrm{supp}}f_p)\ll p^{(n+1)l}\cdot p^{(2n-1)l}=p^{3nl}.$ Similarly, $$\begin{aligned} \int_{G'(\mathbb{Q}_p)}\frac{\kappa(xy^{-1},y)}{|\det y|_p^{1+\mathop{\mathrm{Re}}(s)}}dy\leq \int_{G'(\mathbb{Q}_p)}\int_{M_{n,1}(\mathbb{Q}_p)}f_p\left(\begin{pmatrix} y&y\mathfrak{u}\\ &1 \end{pmatrix}\right)|\det y|_p^{-\mathop{\mathrm{Re}}(s)}d\mathfrak{u}dy,\end{aligned}$$ which is further bounded by $$\begin{aligned} \max\big\{1,p^{-(n+1)l\mathop{\mathrm{Re}}(s)}\big\}\cdot \operatorname{Vol}(\mathop{\mathrm{supp}}f_p)\ll p^{(2n-1)l+(n+1)l|\mathop{\mathrm{Re}}(s)|}.\end{aligned}$$ Therefore, $\mathcal{I}_p(f,s)\ll p^{3nl+(n+1)l|\mathop{\mathrm{Re}}(s)|}|L_p(1+\mathop{\mathrm{Re}}(s),\pi_p'\times\widetilde{\pi}_p')|,$ leading to $$\label{7.13} J(s)\ll p^{3nl+(n+1)l|\mathop{\mathrm{Re}}(s)|}|L_p(1+\mathop{\mathrm{Re}}(s),\pi_p'\times\widetilde{\pi}_p')|,$$ where the implied constant depends on $n.$ Now we can employ Jensen's formula and Borel-Carathéodory theorem as in the proof of Lemma [Lemma 29](#lem11.1){reference-type="ref" reference="lem11.1"} (and replace the convex bound therein by the above estimate [\[7.13\]](#7.13){reference-type="eqref" reference="7.13"}) to derive that $$\label{7.14} J(s)\ll \exp(\log^{3/4} p)J(0)$$ for all $|s|\leq 10^{-1}\exp(-2\sqrt{\log p}).$ By Cauchy formula we have $$\label{7.15} \frac{d\mathcal{I}_p(f,s)}{ds}\mid_{s=0}=\frac{dJ(s)}{ds}\mid_{s=0}=\frac{1}{2\pi i}\int_{|s|=100^{-1}\exp(-2\sqrt{\log p})}\frac{J(s)}{s^2}ds.$$ Therefore, the estimate [\[7.11\]](#7.11){reference-type="eqref" reference="7.11"} follows from [\[7.14\]](#7.14){reference-type="eqref" reference="7.14"},[\[7.15\]](#7.15){reference-type="eqref" reference="7.15"}, Lemmas [Lemma 39](#lem8){reference-type="ref" reference="lem8"} and [Lemma 40](#lem9){reference-type="ref" reference="lem9"}. ◻ ## Archimedean Auxiliary Integrals {#sec5.3} In this section we handle the archimedean integral, where the construction of $f_{\infty}$ (cf. [@Nel20 -14.6]) is the key ingredient. **Lemma 42**. *Let notation be as before. Then $$\begin{aligned} \mathcal{I}_{\infty}:=&\int_{K_{\infty}'}\int_{G'(\mathbb{R})}\int_{Z'(\mathbb{R})}\Bigg|\int_{\mathbb{R}^n}f_{\infty}\left(\begin{pmatrix} y&u\\ &1 \end{pmatrix}\right)\psi_{{\infty}}(\eta zku)du\Bigg||\det z|^{1+\varepsilon}_{\infty}d^{\times}z\\ &\int_{N'(\mathbb{R})\backslash P'_0(\mathbb{R})}\big|W_{\infty}'(pk)\overline{W_{\infty}'(pky)}\big||\det p|_{\infty}^{1+\varepsilon}d^{\times}pdydk\ll_{\varepsilon} T^{\frac{n}{2}+\varepsilon}\|W_{\infty}'\|_2^2,\end{aligned}$$ where the implied constant relies on $\varepsilon.$ Here $\|W_{\infty}'\|_2$ is defined by [\[W_inf\]](#W_inf){reference-type="eqref" reference="W_inf"} in [4.3](#sec4.5){reference-type="ref" reference="sec4.5"}.* *Proof.* For $k\in K_{\infty}',$ denote by $\eta k=(k_1,\cdots, k_n)$ the last row of $k.$ By the construction of $f_{\infty}$ (cf. [@Nel20 -14.6 ]) the integral $\mathcal{I}_{\infty}$ becomes $$\begin{aligned} \int_{K_{\infty}'}\int_{G'(\mathbb{R})}\int_{\mathbb{R}^{\times}}\Bigg|\int_{\mathbb{R}^n}\tilde{f}_{\infty}^{\sharp}\left(\begin{pmatrix} y&u\\ &1 \end{pmatrix}\right)\psi_{{\infty}}(\eta tku)du\Bigg|t^{n+n\varepsilon}d^{\times}t \mathcal{K}(k,y)dydk+O(T^{-\infty}),\end{aligned}$$ where $\tilde{f}_{\infty}^{\sharp}$ is defined in [3.2.1](#3.2.1){reference-type="ref" reference="3.2.1"} (and it is $\widetilde{Op}_h(a')$ in the sense of [@Nel20], i.e., without truncation at essential support), and $$\begin{aligned} \mathcal{K}(k,y):=\int_{N'(\mathbb{R})\backslash P'_0(\mathbb{R})}\big|W_{\infty}'(pk)\overline{W_{\infty}'(pky)}\big||\det p|_{\infty}^{1+\varepsilon }d^{\times}p.\end{aligned}$$ Using Cauchy-Schwarz and Lemma [Lemma 25](#lem4.9){reference-type="ref" reference="lem4.9"} we infer that $$\label{84} \int\big|W_{\infty}'(pk)\overline{W_{\infty}'(pky)}\big||\det p|^{1+\varepsilon}_{\infty}d^{\times}p\ll T^{O(\varepsilon)}\int\Bigg|W_{\infty}'\left(\begin{pmatrix} h\\ &1 \end{pmatrix}\right)\Bigg|^2dh,$$ where $p\in N'(\mathbb{R})\backslash P'_0(\mathbb{R}),$ and $h\in N_H(\mathbb{R})\backslash H(\mathbb{R}),$ with $H=\operatorname{diag}(\mathrm{GL}(n-1),1).$ Here the implied constant is independent of $k$ and $y.$ So Lemma [Lemma 21](#lem4.6){reference-type="ref" reference="lem4.6"} yields $$\mathcal{K}(k,y)\ll T^{o(1)},$$ uniformly for all $k$ and $y.$ Therefore, the integral $\mathcal{I}_{\infty}$ is $$\begin{aligned} \ll T^{o(1)}\int_{K_{\infty}'}\int_{G'(\mathbb{R})}\int_{\mathbb{R}^{\times}}\Bigg|\int_{\mathbb{R}^n}\tilde{f}_{\infty}^{\sharp}\left(\begin{pmatrix} y&u\\ &1 \end{pmatrix}\right)\psi_{{\infty}}(\eta tku)du\Bigg|t^{n+n\varepsilon}d^{\times}tdydk+O(T^{-\infty}).\end{aligned}$$ By definition, $$\int_{\mathbb{R}^n}\tilde{f}_{\infty}^{\sharp}\left(\begin{pmatrix} y&u\\ &1 \end{pmatrix}\right)\psi_{{\infty}}(\eta zku)du\neq 0$$ unless $tk_jT^{-1}-\tau_j\ll T^{-\frac{1}{2}+\varepsilon},$ where $\tau_j$ is determined by $\tau,$ $1\leq j\leq n.$ So $k_j-\tau_jTt^{-1}\ll T^{\frac{1}{2}+\varepsilon}t^{-1},$ $1\leq j\leq n.$ Since $k_1^2+\cdots+k_n^2=1,$ there must be some $|k_{j'}|\gg 1.$ Hence, $t\ll T^{1+\varepsilon}.$ Moreover, by decaying of Fourier transform of $\tilde{f}_{\infty}^{\sharp},$ $$\int_{\mathbb{R}^n}\tilde{f}_{\infty}^{\sharp}\left(\begin{pmatrix} y&u\\ &1 \end{pmatrix}\right)\psi_{{\infty}}(\eta zku)du=\int_{\mathcal{U}}\tilde{f}_{\infty}^{\sharp}\left(\begin{pmatrix} y&u\\ &1 \end{pmatrix}\right)\psi_{{\infty}}(\eta zku)du+O(T^{-\infty}),$$ where $\mathcal{U}:=\big\{\ensuremath{\mskip 1mu\prescript{\smash{\mathrm t\mkern-3mu}}{}{\mathstrut(u_1,\cdots,u_n)}}\in \mathbb{R}^n:\ u_j\ll T^{-1/2+\varepsilon},\ 1\leq j\leq n\big\}.$ Moreover, the Haar measure on $\mathrm{SO}(n)$ factors as the measures on $\prod_{j=1}^{n-1}\mathbb{S}^j,$ where $\mathbb{S}^j$ denotes the $j$-dimensional unit $j$-sphere in $\mathbb{R}^{j+1}.$ Therefore, $$\begin{aligned} \mathcal{I}_{\infty}\ll_{\varepsilon}&T^{\varepsilon}\int_{G'(\mathbb{R})}\int_{\mathcal{U}}\bigg|\tilde{f}_{\infty}^{\sharp}\left(\begin{pmatrix} y&u\\ &1 \end{pmatrix}\right)\bigg|dudy\int_{t\ll T^{1+\varepsilon}}\int_{\mathbb{S}_{T,\tau}^{n-1}(t)}d\mathbf{k}t^{n-1}dt+O(T^{-\infty}), \end{aligned}$$ where $\mathbb{S}_{T,\tau}^{n-1}(t)$ is the set defined by $$\label{98} \big\{(k_1,\cdots,k_n)\in\mathbb{R}^n:\ k_1^2+\cdots+k_n^2=1,\ tk_j-\tau_jT\ll T^{\frac{1}{2}+\varepsilon},\ 1\leq j\leq n\big\}.$$ Geometrically, $\mathbb{S}_{T,\tau}^{n-1}(t)$ is nonempty if and only if the $(n-1)$-sphere of radius $t$ intersects with the box centered at $(\tau_1,\cdots,\tau_{n})$ of side length $\ll_{\varepsilon} T^{\frac{1}{2}+\varepsilon}.$ Hence, $\mathbb{S}_{T,\tau}^{n-1}(t)$ is not empty unless $t$ ranges over an interval $I_{\tau}$ of length $\ll_{\varepsilon} T^{\frac{1}{2}+\varepsilon},$ determined by $\tau.$ Bounding the volume of $\mathbb{S}_{T,\tau}^{n-1}(t)$ by $O(T^{(n-1)/2+\varepsilon}t^{-(n-1)+\varepsilon})$ yields $$\label{7.23} \mathcal{I}_{\infty}\ll_{\varepsilon,\pi_{\infty}'}T^{\frac{n-1}{2}+\varepsilon}\int_{G'(\mathbb{R})}\int_{\mathcal{U}}\bigg|\tilde{f}_{\infty}^{\sharp}\left(\begin{pmatrix} y&u\\ &1 \end{pmatrix}\right)\bigg|dudy\int_{I_{\tau}}dt.$$ Recall that $\tilde{f}_{\infty}^{\dag}$ is the truncation of $\tilde{f}_{\infty}^{\sharp}$ at the essentially support, cf. [\[245\]](#245){reference-type="eqref" reference="245"} in [3.2.1](#3.2.1){reference-type="ref" reference="3.2.1"}. By [\[250\]](#250){reference-type="eqref" reference="250"} we have $\|\tilde{f}_{\infty}^{\sharp}\|_{\infty}\ll_{\varepsilon} T^{\frac{n(n+1)}{2}+\varepsilon}.$ It then follows from [\[7.23\]](#7.23){reference-type="eqref" reference="7.23"} that $$\begin{aligned} \mathcal{I}_{\infty}\ll_{\varepsilon,\pi_{\infty}'}&T^{\frac{n-1}{2}+\varepsilon}\cdot T^{\frac{n}{2}+\varepsilon}\cdot T^{-\frac{n}{2}+\varepsilon}\cdot T^{\frac{1}{2}+\varepsilon}, \end{aligned}$$ where the factor $T^{\frac{n}{2}+\varepsilon}$ is the contribution from $y$ and the $L^{\infty}$-norm of $\tilde{f}_{\infty}^{\sharp};$ the factor $T^{-\frac{n}{2}+\varepsilon}$ is the contribution from $u\in\mathcal{U};$ and the factor $T^{\frac{1}{2}+\varepsilon}$ comes from the length of $I_{\tau}.$ Hence Lemma [Lemma 42](#lem10){reference-type="ref" reference="lem10"} follows. ◻ *Remark 43*. Note that $f_{\infty}$ is not $\iota(K_{\infty}')$-invariant. So we cannot make use of Hecke structure as in the $p$-adic places (cf. [7.1](#6.2..){reference-type="ref" reference="6.2.."}). ## Proof of Proposition [Proposition 33](#prop54){reference-type="ref" reference="prop54"} {#sec5.4} Let $S=\{\infty\}\cup \{p:\ p\mid \nu(f)M'\}.$ Let $\mathbf{s}=(s,0).$ Then by the definition in [2.2.1](#2.2.1){reference-type="ref" reference="2.2.1"} and Lemma [Lemma 36](#lem6.4.){reference-type="ref" reference="lem6.4."} we have $$\label{7.1} J^{\operatorname{Reg}}_{\operatorname{Geo},\operatorname{Small}}(f,\textbf{s})=\prod_{\substack{p\mid M,\ p\nmid M'}}\operatorname{Vol}(\overline{K_p(M)})^{-1}\cdot L^{(S)}(1+s,\pi'\times\widetilde{\pi}')\cdot \mathcal{I}_S(f,s),$$ where $L^{(S)}(1+s,\pi'\times\widetilde{\pi}')$ is the partial $L$-function with local $L$-factors at $p\in S$ being removed, $\mathcal{I}_S(f,s)=\prod_{p\in S}\mathcal{I}_p(f,s),$ and for a place $p\leq \infty$, $$\begin{aligned} \mathcal{I}_p(f,s):=&\int_{G'(\mathbb{Q}_p)}\int_{M_{n,1}(\mathbb{Q}_p)}\int_{N'(\mathbb{Q}_p)\backslash G'(\mathbb{Q}_p)}f_p\left(\begin{pmatrix} y&\mathfrak{u}\\ &1 \end{pmatrix} \right)\theta_p(\eta x\mathfrak{u})\\ &\qquad \qquad W_p'(x)\overline{W_p'(xy)}|\det x|_p^{1+s}dxd\mathfrak{u}dy.\end{aligned}$$ Here we write $\mathbb{Q}_p=\mathbb{R}$ if $p=\infty.$ Observe that when $p<\infty,$ the above definition coincides with [\[eq7.2\]](#eq7.2){reference-type="eqref" reference="eq7.2"} in [7.1](#6.2..){reference-type="ref" reference="6.2.."}. Let $0<\varepsilon< \frac{4}{n(n+1)+2}.$ Note that [\[7.1\]](#7.1){reference-type="eqref" reference="7.1"} yields a meromorphic continuation of $J^{\operatorname{Reg}}_{\operatorname{Geo},\operatorname{Small}}(f,\textbf{s})$ to $\mathop{\mathrm{Re}}(s)>-\varepsilon.$ In addition, $\mathcal{I}_S(f,s)$ is holomorphic in $\mathop{\mathrm{Re}}(s)>-\varepsilon.$ Let $L^{(S)}(1+s,\pi'\times\widetilde{\pi}')=\frac{a_{-1}}{s}+a_0+a_1s+\cdots,$ and $\mathcal{I}_S(f,s)=b_0+b_1s+b_2s^2+\cdots$ be the Taylor expansions near $s=0,$ respectively. Then $$\begin{aligned} J^{\operatorname{Reg}}_{\operatorname{Geo},\operatorname{Small}}(f,\textbf{s})=\prod_{p\mid M,\ p\nmid M'}\operatorname{Vol}(\overline{K_p(M)})^{-1}\cdot\Big[\frac{a_{-1}b_0}{s}+a_0b_0+a_{-1}b_1+O(s)\Big].\end{aligned}$$ Consequently, we obtain that $$\begin{aligned} J_{\operatorname{Geo},\operatorname{Main}}^{\operatorname{Small}}(f,\textbf{0})=\lim_{s\rightarrow 0}\left(J^{\operatorname{Reg}}_{\operatorname{Geo},\operatorname{Small}}(f,\textbf{s})-\underset{s=0}{\operatorname{Res}}\ J^{\operatorname{Reg}}_{\operatorname{Geo},\operatorname{Small}}(f,\textbf{s})\right),\end{aligned}$$ which is equal to $\prod_{p\mid M,\ p\nmid M'}\operatorname{Vol}(\overline{K_p(M)})^{-1}\cdot(a_0b_0+a_{-1}b_1).$ It is well known that $|a_0|$ and $|a_1|$ are $\ll C(\pi')^{\varepsilon},$ where $C(\pi')$ is the analytic conductor of $\pi'$ and the implied constant depends only on $\varepsilon.$ It suffice to bound $|b_0|$ and $|b_1|.$ By definition, $b_0=\mathcal{I}_S(f,0)$ and $b_1=\frac{d\mathcal{I}_S(f,s)}{ds}\mid_{s=0}.$ Explicitly, we have $$\begin{aligned} b_0=&\int_{G'(\mathbb{A}_S)}\int_{M_{n,1}(\mathbb{A}_S)}\int_{N'(\mathbb{A}_S)\backslash G'(\mathbb{A}_S)}f_S\left(\begin{pmatrix} y_S&u_S\\ &1 \end{pmatrix} \right)\theta_S(\eta x_Su_S)\\ &\qquad \qquad W_S'(\iota(x_S))\overline{W_S'(x_Sy_S)}|\det x_S|_Sdx_Sdu_Sdy_S,\\ b_1=&\int_{G'(\mathbb{A}_S)}\int_{M_{n,1}(\mathbb{A}_S)}\int_{N'(\mathbb{A}_S)\backslash G'(\mathbb{A}_S)}f_S\left(\begin{pmatrix} y_S&u_S\\ &1 \end{pmatrix} \right)\theta_S(\eta x_Su_S)\\ &\qquad \qquad W_S'(\iota(x_S))\overline{W_S'(x_Sy_S)}|\det x_S|_S\log |\det x_S|_S dx_Sdu_Sdy_S.\end{aligned}$$ Gathering the estimates in Lemmas [Lemma 37](#lem11){reference-type="ref" reference="lem11"}, [Lemma 39](#lem8){reference-type="ref" reference="lem8"}, [Lemma 40](#lem9){reference-type="ref" reference="lem9"}, [Lemma 41](#lem7.7.){reference-type="ref" reference="lem7.7."}, and [Lemma 42](#lem10){reference-type="ref" reference="lem10"}, we obtain $$\begin{aligned} \prod_{p}\frac{(|a_0|+|a_1|)(|b_0|+|b_1|)}{\operatorname{Vol}(\overline{K_p(M)})}\ll_{\varepsilon}\frac{M'^{2n} T^{\frac{n}{2}+\varepsilon}M^{n+\varepsilon} \langle\phi',\phi'\rangle}{\mathcal{N}_f^{1-2\vartheta_p+\varepsilon}}\prod_{p\mid M'}p^{ne_p(M)},\end{aligned}$$ where the product is over $p\mid M$ but $p\nmid M',$ and $\mathcal{N}_f$ is defined by [\[61\]](#61){reference-type="eqref" reference="61"}. Here the implied constant depends on $\varepsilon$ and $M'.$ Therefore, Proposition [Proposition 33](#prop54){reference-type="ref" reference="prop54"} holds. # Geometric Side: Dual Orbital Integrals {#8.5.2} Let $\mathbf{s}=(s,0)\in\mathbb{C}^2.$ By definition, the dual orbital integral $$\begin{aligned} J_{\operatorname{Geo},\operatorname{Dual}}^{\operatorname{Big}}(f,\textbf{s}):=&\int_{G'(\mathbb{A})}\int_{[\overline{G'}]}\phi'(x)\overline{\phi'(xy)}E(s,x;f,y)dxdy,\end{aligned}$$ where $E(s,x;f,y)$ is the Eisenstein series defined in [2.2.1](#2.2.1){reference-type="ref" reference="2.2.1"}. So $J_{\operatorname{Geo},\operatorname{Dual}}^{\operatorname{Big}}(f,\textbf{s})$ converges absolutely in $\mathop{\mathrm{Re}}(s)>1,$ and admits a meromorphic continuation to $s\in\mathbb{C}$ with possible simple poles at $s\in\{0, 1\}.$ Define $$\label{8.1'} J_{\operatorname{Geo},\operatorname{Main}}^{\operatorname{Dual}}(f,\textbf{s}):=J^{\operatorname{Reg}}_{\operatorname{Geo},\operatorname{Dual}}(f,\textbf{s})-s^{-1}\underset{s=0}{\operatorname{Res}}\ J^{\operatorname{Reg}}_{\operatorname{Geo},\operatorname{Dual}}(f,\textbf{s}),$$ which is holomorphic in $\mathop{\mathrm{Re}}(s)<1.$ The main result is this section is the majorization of $J_{\operatorname{Geo},\operatorname{Main}}^{\operatorname{Dual}}(f,\mathbf{0})$ as follows. **Proposition 44**. *Let notation be as before. Let $\varepsilon>0$ be a small constant. Then $$\label{8.1} J_{\operatorname{Geo},\operatorname{Main}}^{\operatorname{Dual}}(f,\mathbf{0}) \ll T^{\frac{n}{2}+\varepsilon}M'^{2n}M^{n+\varepsilon} \mathcal{N}_f^{-1+2\vartheta_p+\varepsilon}\langle\phi',\phi'\rangle\prod_{p\mid M'}p^{ne_p(M)},$$ where $\mathcal{N}_f$ is defined by [\[61\]](#61){reference-type="eqref" reference="61"}. Here the implied constant depends on $\varepsilon,$ parameters $c_{\infty}$ and $C_{\infty}$ defined in [3.1.4](#sec3.14){reference-type="ref" reference="sec3.14"}, and the conductor of $\pi_{\infty}'.$* *Proof.* Let $\mathop{\mathrm{Re}}(s)<0.$ Employiong the functional equation of $E(s,x;f,y)$, the dual orbital integral $J^{\operatorname{Reg}}_{\operatorname{Geo},\operatorname{Dual}}(f,\textbf{s})$ can be expand as the convergent orbital $$\begin{aligned} \int\int_{G'(\mathbb{A})}\int_{M_{1,n}(\mathbb{A})} f\left(\begin{pmatrix} I_n&\\ \mathfrak{c}&1 \end{pmatrix}\begin{pmatrix} y&\\ &1 \end{pmatrix}\right)\psi(\eta \mathfrak{c}\ensuremath{\mskip 1mu\prescript{\smash{\mathrm t\mkern-3mu}}{}{\mathstrut x}})d\mathfrak{c} \overline{\phi'}(x)\phi'(x\ensuremath{\mskip 1mu\prescript{\smash{\mathrm t\mkern-3mu}}{}{\mathstrut y}}^{-1})|\det x|^{1-s}dydx,\end{aligned}$$ where $x\in P_0(\mathbb{Q})\backslash G'(\mathbb{A}).$ For a place $p\leq \infty$, we define $$\begin{aligned} \mathcal{J}_p(f,s):=&\int_{G'(\mathbb{Q}_p)}\int_{M_{n,1}(\mathbb{Q}_p)}\int_{N'(\mathbb{Q}_p)\backslash G'(\mathbb{Q}_p)}\ensuremath{\mskip 1mu\prescript{\smash{\mathrm t\mkern-3mu}}{}{\mathstrut f}}_p^{-1}\left(\begin{pmatrix} \ensuremath{\mskip 1mu\prescript{\smash{\mathrm t\mkern-3mu}}{}{\mathstrut y}}^{-1}&-\mathfrak{u}\\ &1 \end{pmatrix} \right)\theta_p(\eta \ensuremath{\mskip 1mu\prescript{\smash{\mathrm t\mkern-3mu}}{}{\mathstrut x}}\mathfrak{u})\\ &\qquad \qquad \overline{W_p'(x)}W_p'(x\ensuremath{\mskip 1mu\prescript{\smash{\mathrm t\mkern-3mu}}{}{\mathstrut y}}^{-1})|\det x|_p^{1-s}dxd\mathfrak{u}dy.\end{aligned}$$ Here $\mathbb{Q}_p=\mathbb{R}$ if $p=\infty,$ and the function $\ensuremath{\mskip 1mu\prescript{\smash{\mathrm t\mkern-3mu}}{}{\mathstrut f}}_p^{-1}$ is defined by $\ensuremath{\mskip 1mu\prescript{\smash{\mathrm t\mkern-3mu}}{}{\mathstrut f}}_p^{-1}(g):=f_p(\ensuremath{\mskip 1mu\prescript{\smash{\mathrm t\mkern-3mu}}{}{\mathstrut g}}^{-1})$ for all $g\in G(\mathbb{Q}_p).$ Then $$\begin{aligned} J^{\operatorname{Reg}}_{\operatorname{Geo},\operatorname{Dual}}(f,\textbf{s})=\prod_{p\leq \infty}\mathcal{J}_p(f,s),\ \ \mathop{\mathrm{Re}}(s)<0. \end{aligned}$$ Similarly to the local calculation in Lemma [Lemma 36](#lem6.4.){reference-type="ref" reference="lem6.4."}, we have $$\mathcal{J}_p(f,s)=\operatorname{Vol}(\overline{K_p(M)})^{-1}\cdot L_p(1-s,\pi_p'\times\widetilde{\pi}_p')$$ if $p\notin S:=\{\infty\}\cup \{p:\ p\mid \nu(f)M'\}.$ As a consequence, $$\begin{aligned} J_{\operatorname{Geo},\operatorname{Main}}^{\operatorname{Dual}}(f,\textbf{s})=L^{(S)}(1-s,\pi'\times\widetilde{\pi}')\cdot \mathcal{J}_S(f,s), \end{aligned}$$ where $L^{(S)}(1-s,\pi'\times\widetilde{\pi}'):=\prod_{s\notin S}L_p(1-s,\pi_p'\times\widetilde{\pi}_p') ,$ and $\mathcal{J}_S(f,s)=\prod_{p\in S}\mathcal{J}_p(f,s).$ Noticing the strong similarity between $\mathcal{J}_p(f,s)$ and $\mathcal{I}_p(f,s)$ (cf. [7.1](#6.2..){reference-type="ref" reference="6.2.."}), the proof of [\[8.1\]](#8.1){reference-type="eqref" reference="8.1"} then follows from that of Proposition [Proposition 33](#prop54){reference-type="ref" reference="prop54"}. ◻ # Geometric Side: Regular Orbital Integrals {#sec10} Recall that $J^{\operatorname{Reg},\uppercase{\romannumeral 2\relax}}_{\operatorname{Geo},\operatorname{Big}}(f,\textbf{0})$ is defined by $$\begin{aligned} \sum_{\substack{(\boldsymbol{\xi},t)\in \mathbb{Q}^{n}\\ (\boldsymbol{\xi},t)\neq \textbf{0}}}&\int_{P_0'(\mathbb{Q})\backslash {G'}(\mathbb{A})}\int_{{G'}(\mathbb{A})}f\left(\iota(x)^{-1}\begin{pmatrix} I_{n-1}&&\boldsymbol{\xi}\\ &1&t\\ &1&1 \end{pmatrix}\iota(xy)\right)\phi'(x)\overline{\phi'(xy)}dydx,\end{aligned}$$ which converges absolutely (cf. [@Yan22], Theorem 5.6). Quantitatively, the approach of estimating bilinear forms in Part 4 of [@Nel21] yields $$\label{10.1} J^{\operatorname{Reg},\uppercase{\romannumeral 2\relax}}_{\operatorname{Geo},\operatorname{Big}}(f,\textbf{0})\ll M^nT^{\frac{n}{2}-\frac{1}{4}+\varepsilon}\mathcal{N}_f^{3n^2+7n+4}\langle\phi',\phi'\rangle,$$ where $\mathcal{N}_f$ is defined in [\[61\]](#61){reference-type="eqref" reference="61"} in [3.6.2](#3.6.2){reference-type="ref" reference="3.6.2"}. The factor $\mathcal{N}_f^{3n^2+7n+4}$ has exponent $3n^2+7n+4,$ which is *quadratic* in $n.$ We will use a different strategy to improve the estimate [\[10.1\]](#10.1){reference-type="eqref" reference="10.1"} in multiple aspects. In particular, the exponent of $\mathcal{N}_f$ could be replaced by a *linear* function of $n$. Let $M$ and $M'$ be the levels defined in [3.1.2](#sec3.1.2){reference-type="ref" reference="sec3.1.2"}--[3.1.4](#sec3.14){reference-type="ref" reference="sec3.14"}. Define $$\label{equ9.4} M^{\dagger}:=\prod_{p\mid M,\ p\nmid M'}p^{e_p(M)}.$$ The main result in this section is as follows. **Theorem 45**. *Let notation be as before. Then $$\begin{aligned} \frac{J^{\operatorname{Reg},\uppercase{\romannumeral 2\relax}}_{\operatorname{Geo},\operatorname{Big}}(f,\textbf{0})}{\langle\phi',\phi'\rangle}\ll (TMM')^{\varepsilon}T^{\frac{n-1}{2}}M'^{2n}M^n\mathcal{N}_f^{2(n-1)}\prod_{p\mid M'}p^{ne_p(M)} \max_{1\leq l_0\leq n}\Bigg[\frac{M'^{2}\mathcal{N}_f^{2}}{M^{\dagger}}\Bigg]^{l_0},\end{aligned}$$ where the implied constant depends on $\varepsilon,$ parameters $c_{\infty}$ and $C_{\infty}$ defined in [3.1.4](#sec3.14){reference-type="ref" reference="sec3.14"}, and the conductor of $\pi_{\infty}'.$* *Remark 46*. The factor $M^{n-1}$ plays an important role in studying the nonvanishing problem quantitatively (cf. [11](#sec12){reference-type="ref" reference="sec12"}), since it allows amplification in the level aspect. Theorem [Theorem 45](#thm9.1){reference-type="ref" reference="thm9.1"} is a refinement of [@Yan22 Theorem 5.6] in the following sense: we execute the amplification here and explicate the dependence on the cusp form $\phi'$ which is allowed to vary, while in loc. cit., $\phi'$ is *fixed*. ## Preliminary Reduction By Cauchy-Schwarz, we have $$\label{9.1.} \big|J^{\operatorname{Reg},\uppercase{\romannumeral 2\relax}}_{\operatorname{Geo},\operatorname{Big}}(f,\textbf{0})\big|\leq \sqrt{\mathcal{J}_{\operatorname{Geo},\operatorname{Big}}^{\operatorname{Reg},\uppercase{\romannumeral 2\relax}}(f)\tilde{\mathcal{J}}_{\operatorname{Geo},\operatorname{Big}}^{\operatorname{Reg},\uppercase{\romannumeral 2\relax}}(f)},$$ where $\mathcal{J}_{\operatorname{Geo},\operatorname{Big}}^{\operatorname{Reg},\uppercase{\romannumeral 2\relax}}(f)$ is defined by $$\begin{aligned} \sum_{\substack{(\boldsymbol{\xi},t)\in \mathbb{Q}^{n}\\ (\boldsymbol{\xi},t)\neq \textbf{0}}}&\int_{{G'}(\mathbb{A})}\int_{P_0'(\mathbb{Q})\backslash {G'}(\mathbb{A})}\Bigg|f\left(\iota(x)^{-1}\begin{pmatrix} I_{n-1}&&\boldsymbol{\xi}\\ &1&t\\ &1&1 \end{pmatrix}\iota(xy)\right)\Bigg|\big|\phi'(x)\big|^2dxdy,\end{aligned}$$ and $\tilde{\mathcal{J}}_{\operatorname{Geo},\operatorname{Big}}^{\operatorname{Reg},\uppercase{\romannumeral 2\relax}}(f)$ is defined by $$\begin{aligned} \sum_{\substack{(\boldsymbol{\xi},t)\in \mathbb{Q}^{n}\\ (\boldsymbol{\xi},t)\neq \textbf{0}}}&\int_{{G'}(\mathbb{A})}\int_{P_0'(\mathbb{Q})\backslash {G'}(\mathbb{A})}\Bigg|f\left(\iota(x)^{-1}\begin{pmatrix} I_{n-1}&&\boldsymbol{\xi}\\ &1&t\\ &1&1 \end{pmatrix}\iota(xy)\right)\Bigg|\big|\phi'(xy)\big|^2dxdy.\end{aligned}$$ Here $\iota$ is the diagonal embedding of $G'$ into $G:$ $\iota(x)=\operatorname{diag}(x,1),$ $x\in G'(\mathbb{A}).$ **Lemma 47**. *Let notation be as before. Let $f^{-1}(g):=f(g^{-1}),$ $g\in G(\mathbb{A}).$ Then $$\tilde{\mathcal{J}}_{\operatorname{Geo},\operatorname{Big}}^{\operatorname{Reg},\uppercase{\romannumeral 2\relax}}(f)=\mathcal{J}_{\operatorname{Geo},\operatorname{Big}}^{\operatorname{Reg},\uppercase{\romannumeral 2\relax}}(f^{-1}).$$* *Proof.* Changing variable $x\mapsto xy^{-1}$ and $y\mapsto y^{-1},$ $\tilde{\mathcal{J}}_{\operatorname{Geo},\operatorname{Big}}^{\operatorname{Reg},\uppercase{\romannumeral 2\relax}}(f^{-1})$ becomes $$\begin{aligned} \sum_{\substack{(\boldsymbol{\xi},t)\in \mathbb{Q}^{n}\\ (\boldsymbol{\xi},t)\neq \textbf{0}}}&\int_{{G'}(\mathbb{A})}\int_{P_0'(\mathbb{Q})\backslash {G'}(\mathbb{A})}\Bigg|f\left(\iota(x)^{-1}\begin{pmatrix} I_{n-1}&&\boldsymbol{\xi}\\ &1&t\\ &1&1 \end{pmatrix}^{-1}\iota(xy)\right)\Bigg|\big|\phi'(x)\big|^2dxdy.\end{aligned}$$ A straightforward calculation shows that $$\begin{pmatrix} I_{n-1}&&\boldsymbol{\xi}\\ &1&t\\ &1&1 \end{pmatrix}^{-1}=\frac{1}{t-1}\begin{pmatrix} I_{n-1}&&-\boldsymbol{\xi}\\ &1&\\ &&1 \end{pmatrix}\begin{pmatrix} (t-1)I_{n-1}&&\\ &-1&t\\ &1&-1 \end{pmatrix}.$$ Swapping the integrals, after a series of changing variables $y\mapsto x^{-1}y,$ $y\mapsto \begin{pmatrix} -(t-1)^{-1}I_{n-1}\\ &1 \end{pmatrix}y,$ and $y\mapsto \begin{pmatrix} I_{n-1}&-\boldsymbol{\xi}\\ &1\\ &&1 \end{pmatrix}y,$ and $y\mapsto xy,$ $$\begin{aligned} \tilde{\mathcal{J}}_{\operatorname{Geo},\operatorname{Big}}^{\operatorname{Reg},\uppercase{\romannumeral 2\relax}}(f^{-1})=\sum_{\substack{(\boldsymbol{\xi},t)\in \mathbb{Q}^{n}\\ (\boldsymbol{\xi},t)\neq \textbf{0}}}&\int_{P_0'(\mathbb{Q})\backslash {G'}(\mathbb{A})}\int_{{G'}(\mathbb{A})}\big|f\left(\cdots\right)\big|\big|\phi'(x)\big|^2dydx,\end{aligned}$$ where the $``\cdots"$ in $f(\cdots)$ is $$\begin{aligned} \iota(x)^{-1}\begin{pmatrix} I_{n-1}&&-\boldsymbol{\xi}\\ &1&\\ &&1 \end{pmatrix}\begin{pmatrix} -I_{n-1}&&\\ &-1&t\\ &1&-1 \end{pmatrix}\begin{pmatrix} I_{n-1}&-\boldsymbol{\xi}\\ &1\\ &&1 \end{pmatrix}\iota(xy),\end{aligned}$$ which is equal to $\iota(x)^{-1}\begin{pmatrix} -I_{n-1}&&-\boldsymbol{\xi}\\ &-1&t\\ &1&-1 \end{pmatrix}\iota(xy) .$ Hence, swapping integrals and after a change of variable $x\mapsto \operatorname{diag}(I_{n-1},-1)x,$ we obtain $\tilde{\mathcal{J}}_{\operatorname{Geo},\operatorname{Big}}^{\operatorname{Reg},\uppercase{\romannumeral 2\relax}}(f^{-1})=\mathcal{J}_{\operatorname{Geo},\operatorname{Big}}^{\operatorname{Reg},\uppercase{\romannumeral 2\relax}}(f).$ ◻ By [\[9.1.\]](#9.1.){reference-type="eqref" reference="9.1."} and Lemma [Lemma 47](#lem9.2.){reference-type="ref" reference="lem9.2."}, Theorem [Theorem 45](#thm9.1){reference-type="ref" reference="thm9.1"} follows from the following proposition. **Proposition 48**. *Let notation be as before. Denote by $$\mathcal{J}:=\max\big\{\mathcal{J}_{\operatorname{Geo},\operatorname{Big}}^{\operatorname{Reg},\uppercase{\romannumeral 2\relax}}(f), \mathcal{J}_{\operatorname{Geo},\operatorname{Big}}^{\operatorname{Reg},\uppercase{\romannumeral 2\relax}}(f^{-1})\big\}.$$ Let $M^{\dagger}$ be defined by [\[equ9.4\]](#equ9.4){reference-type="eqref" reference="equ9.4"}. Then $$\begin{aligned} \mathcal{J}\ll (TMM')^{\varepsilon}T^{\frac{n-1}{2}}M'^{2n}M^n\mathcal{N}_f^{2(n-1)}\langle\phi',\phi'\rangle\prod_{p\mid M'}p^{ne_p(M)} \max_{1\leq l_0\leq n}\Bigg[\frac{M'^{2}\mathcal{N}_f^{2}}{M^{\dagger}}\Bigg]^{l_0},\end{aligned}$$ where the implied constant depends on $\varepsilon,$ parameters $c_{\infty}$ and $C_{\infty}$ defined in [3.1.4](#sec3.14){reference-type="ref" reference="sec3.14"}, and the conductor of $\pi_{\infty}'.$* After preparations in [9.3](#sec9.2){reference-type="ref" reference="sec9.2"}, [9.4](#sec9.3){reference-type="ref" reference="sec9.3"} and [9.5](#sec9.4){reference-type="ref" reference="sec9.4"}, a proof of Proposition [Proposition 48](#prop9.3.){reference-type="ref" reference="prop9.3."} is given in [9.6](#sec9.5){reference-type="ref" reference="sec9.5"}. ## Ad Hoc Notation In this section we introduce some notation that will be used extensively throughout this section to prove Proposition [Proposition 48](#prop9.3.){reference-type="ref" reference="prop9.3."}. ### Siegel Sets and Auxiliary Constants {#Sie} By Siegel's results on $[P_0']$ (cf. [4.4](#sec4.4){reference-type="ref" reference="sec4.4"}) every element may be represented in the form $b=ak^*\in A_H(\mathbb{R})\times\Omega^*,$ where $$A_H(\mathbb{R}):=\big\{\operatorname{diag}(a_1,\cdots,a_{n-1},1)\in A'(\mathbb{R}):\ a_1\geq\cdots\geq a_{n-1}>0\big\},$$ and $\Omega^*$ is a fixed compact set in $P_0'(\mathbb{A})$. Let $\mathcal{P}$ be the set of rational primes such that $k_p^*\in G'(\mathbb{Z}_p)$ whenever $p\notin \mathcal{P}.$ So $\mathcal{P}$ is fixed, relying only on $\Omega^*$. We may assume that $L$ is large enough so that $\mathcal{P}$ does not intersect with $\mathcal{L}.$ Write $\Omega^*=\otimes_{p\leq \infty}\Omega_p^*.$ For $p<\infty,$ define $$\label{equ9.3} e_{\min}(\Omega_p^*):=\min_{x\in \Omega_p^*}\{e_p(E_{i,j}(x)),\ e_p(E_{i,j}(x^{-1})):\ 1\leq i, j\leq n\},$$ where $E_{i,j}(x)$ (resp. $E_{i,j}(x^{-1})$) is the $(i,j)$-th entry of $x$ (resp. $x^{-1}$). Then $-1\ll e_{\min}(\Omega_p^*)\leq 0,$ and $e_{\min}(\Omega_p^*)=0$ if $\Omega_p^*=G(\mathbb{Z}_p).$ Define $C_{\Omega_p^*}:=p^{e_{\min}(\Omega_p^*)}\leq 1,$ and $C_{\Omega^*}=\prod_{p<\infty}C_{\Omega_p^*},$ which is a finite constant depending only on $\mathcal{P}.$ Let $M$ and $M'$ be the levels defined in [3.1.2](#sec3.1.2){reference-type="ref" reference="sec3.1.2"}--[3.1.4](#sec3.14){reference-type="ref" reference="sec3.14"}. Define $$\label{equ9.4} M^{\dagger}:=\prod_{p\mid M,\ p\nmid M'}p^{e_p(M)}.$$ ### Local Components {#sec9.2.2} Let $f\in \big\{f(\cdot;i,p_0), f(\cdot;p_1,p_2)\big\}$ be the test function defined in [3.6](#testfunction){reference-type="ref" reference="testfunction"}. Now we analyze $$f\left(\iota(x)^{-1}\begin{pmatrix} I_{n-1}&&(1-t)\boldsymbol{\xi}\\ &1&t\\ &1&1 \end{pmatrix}\iota(xy)\right)$$ to determine the support of $x$ and $y.$ By Iwasawa decomposition we may write $x=zbk,$ $z\in \mathbb{A}^{\times},$ $b=ak^*\in [P_0']\subset A_H(\mathbb{R})\times\Omega^*,$ and $k\in K',$ which is the maximal compact subgroup of $G'(\mathbb{A}).$ So $$f\left(\iota(x)^{-1}\begin{pmatrix} I_{n-1}&&(1-t)\boldsymbol{\xi}\\ &1&t\\ &1&1 \end{pmatrix}\iota(xy)\right)$$ factorizes as the product $\mathfrak{F}_{\infty}\cdot\mathfrak{F}_{\operatorname{fin}},$ where $$\label{55} \mathfrak{F}_{\infty}:=f_{\infty}\left(\iota(z_{\infty}ak_{\infty}^*k_{\infty})^{-1}\begin{pmatrix} I_{n-1}&&(1-t)\boldsymbol{\xi}\\ &1&t\\ &1&1 \end{pmatrix}\iota(z_{\infty}ak_{\infty}^*k_{\infty}y_{\infty})\right)$$ and $\mathfrak{F}_{\operatorname{fin}}:=\prod_{p< \infty}\mathfrak{F}_p,$ with $$\label{58} \mathfrak{F}_p:=f_p\left(\iota(z_pk_p^*k_p)^{-1}\begin{pmatrix} I_{n-1}&&(1-t)\boldsymbol{\xi}\\ &1&t\\ &1&1 \end{pmatrix}\iota(z_pk_p^*k_py_p)\right),\ \ p<\infty.$$ We will frequently use the notations $\mathfrak{F}_{\infty}$ and $\mathfrak{F}_{p}$ to simplify the expressions of local integrals in the subsequent subsections. ## Counting Rational Points {#sec9.2} In this section we shall classify the rational points $(\boldsymbol{\xi},t)\in \mathbb{Q}^n$ which contribute to $\mathcal{J}_{\operatorname{Geo},\operatorname{Big}}^{\operatorname{Reg},\uppercase{\romannumeral 2\relax}}(f)$ and $\mathcal{J}_{\operatorname{Geo},\operatorname{Big}}^{\operatorname{Reg},\uppercase{\romannumeral 2\relax}}(f^{-1}).$ ### The Nonarchimedean Constraint For $p<\infty,$ let $C_{\Omega_p^*}$ be the constant defined in [9.2.1](#Sie){reference-type="ref" reference="Sie"}. Recall that $M^{\dagger}:=\prod_{p\mid M,\ p\nmid M'}p^{e_p(M)}$ (cf. [\[equ9.4\]](#equ9.4){reference-type="eqref" reference="equ9.4"}), and $$\mathfrak{F}_p:=f_p\left(\iota(z_pk_p^*k_p)^{-1}\begin{pmatrix} I_{n-1}&&(1-t)\boldsymbol{\xi}\\ &1&t\\ &1&1 \end{pmatrix}\iota(z_pk_p^*k_py_p)\right),\ \ p<\infty. \tag{\ref{58}}$$ **Lemma 49**. *Let notation be as before. Let $p<\infty.$ Then $\prod_{p<\infty}\mathfrak{F}_p=0$ unless $(\boldsymbol{\xi},t)\in\mathfrak{X}(f),$ where $$\begin{aligned} \mathfrak{X}(f):=\Bigg\{(\xi_1,\cdots,\xi_{n-1},t)\in \mathbb{Q}^n:\ \xi_j\in \frac{C_{\Omega^*}^{2}M^{\dagger}}{M'^{2}\mathcal{N}_f^{2}}\mathbb{Z},\ \frac{t}{t-1}\in \frac{C_{\Omega^*}^2M^{\dagger}}{M'^{2}\mathcal{N}_f^{2}}\mathbb{Z},\ 1\leq j<n\Bigg\}.\end{aligned}$$* *Proof.* Write $\boldsymbol{\xi}=\ensuremath{\mskip 1mu\prescript{\smash{\mathrm t\mkern-3mu}}{}{\mathstrut(\xi_1,\cdots,\xi_{n-1})}}\in \mathbb{Q}^{n-1}.$ Let $p<\infty,$ write $z_p=p^rI_n.$ 1. Suppose $r\geq 0.$ By Cramer's rule there exists $k_{p}^{(1)}\in G(\mathbb{Z}_p)$ such that $$\begin{aligned} \iota(z_p)^{-1}\begin{pmatrix} I_{n-1}&&(1-t)\boldsymbol{\xi}\\ &1&t\\ &1&1 \end{pmatrix}\iota(z_p)=(1-t)k_{p}^{(1)}\begin{pmatrix} \frac{1}{1-t}I_{n-1}&&p^{-r}\boldsymbol{\xi}\\ &\frac{1}{1-t}&\frac{p^{-r}t}{1-t}\\ &&1 \end{pmatrix}. \end{aligned}$$ 2. Suppose $r< 0.$ By Cramer's rule there exists $k_{p}^{(2)}\in G(\mathbb{Z}_p)$ such that $$\begin{aligned} \iota(z_p)^{-1}\begin{pmatrix} I_{n-1}&&(1-t)\boldsymbol{\xi}\\ &1&t\\ &1&1 \end{pmatrix}\iota(z_p)=p^{-r}(1-t)k_{p}^{(2)}\begin{pmatrix} \frac{p^r}{1-t}I_{n-1}&&\boldsymbol{\xi}\\ &\frac{p^{2r}}{1-t}&\frac{p^{r}}{1-t}\\ &&1 \end{pmatrix}.\end{aligned}$$ Here $p^{-r}$ and $(1-t)$ are scalars identified with $p^{-r}I_{n+1}$ and $(1-t)I_{n+1},$ respectively. Let $t\in \mathbb{Q}-\{1\}.$ Consider the various cases as follows. 1. Let $p\nmid MM'\nu(f)$ and $p\notin\mathcal{P}.$ - Suppose $e_p(t-1)\geq 0$. From case (B) and the support of $f_p$, it follows that $r\geq 0$ by analyzing the $(n,n+1)$-th entry in case (B). So $$\begin{aligned} \mathfrak{F}_p=f_p\left( \begin{pmatrix} \frac{1}{1-t}I_{n-1}&&p^{-r}\boldsymbol{\xi}\\ &\frac{1}{1-t}&\frac{p^{-r}t}{1-t}\\ &&1 \end{pmatrix}\iota(k_p^*k_py_p) \right)\neq 0\end{aligned}$$ if and only if $r\geq 0$ and there exists some $\lambda_p\in \mathbb{Q}_p^{\times}$ such that $$\begin{aligned} \lambda_p\begin{pmatrix} \frac{1}{1-t}I_{n-1}&&p^{-r}\boldsymbol{\xi}\\ &\frac{1}{1-t}&\frac{p^{-r}t}{1-t}\\ &&1 \end{pmatrix}\iota(k_p^*k_py_p)\in G(\mathbb{Z}_p),\end{aligned}$$ which forces that $\lambda_p\in \mathbb{Z}_p^{\times}$ and $$\begin{aligned} \begin{cases} e_p(t)-e_p(t-1)\geq r\geq 0\\ e_p(\xi_j)\geq r\geq 0,\ 1\leq j\leq n-1. \end{cases}\end{aligned}$$ - Suppose $e_p(t-1)\leq -1.$ Then $e_p(t)=e_p(t-1).$ From the case (B) we obtain $e_p(t-1)\leq r\leq -1;$ in the case (A) we have $0\leq r\leq e_p(t)-e_p(t-1),$ implying $r=0.$ So $$\label{9.4..} \mathfrak{F}_p=\begin{cases} f_p\left( \begin{pmatrix} \frac{1}{1-t}I_{n-1}&&\boldsymbol{\xi}\\ &\frac{1}{1-t}&\frac{t}{1-t}\\ &&1 \end{pmatrix}\iota(k_p^*k_py_p) \right),&\text{if $r=0$}\\ f_p\left( \begin{pmatrix} \frac{p^r}{1-t}I_{n-1}&&\boldsymbol{\xi}\\ &\frac{p^{2r}}{1-t}&\frac{p^{r}}{1-t}\\ &&1 \end{pmatrix}\iota(k_p^*k_py_p) \right),& \text{if $r<0$}. \end{cases}$$ As a consequence, we have - Suppose $r=0.$ Then $\mathfrak{F}_p\neq 0$ unless $\boldsymbol{\xi}\in \mathbb{Z}_p^{n-1}$ and $e_p(t)-e_p(t-1)\geq 0.$ - Suppose $r<0.$ Then $\mathfrak{F}_p\neq 0$ unless $\boldsymbol{\xi}\in \mathbb{Z}_p^{n-1}$ and $r-e_p(t-1)\geq 0,$ which implies that $e_p(t-1)\leq r<0.$ Therefore, in the above cases we have $\mathfrak{F}_p\neq 0$ unless $$\begin{aligned} \begin{cases} e_p(t)-e_p(t-1)\geq 0\\ e_p(\xi_j)\geq 0,\ 1\leq j\leq n-1. \end{cases}\end{aligned}$$ 2. Let $p\mid M'$ or $p\in\mathcal{P}.$ Then by definition [\[3.13\]](#3.13){reference-type="eqref" reference="3.13"}, $\mathop{\mathrm{supp}}f_p=Z(\mathbb{Q}_p)D_p,$ where $$\begin{aligned} D_p:=\bigcup_{\substack{\alpha_j,\ \beta_j\in \mathbb{Z}/p^{m'}\mathbb{Z}\\ 1\leq j<n}}\bigcup_{\substack{\alpha_n \in (\mathbb{Z}/p^{m''}\mathbb{Z})^{\times}\\ \beta_n \in (\mathbb{Z}/p^{m''}\mathbb{Z})^{\times}}}\begin{pmatrix} I_n&\textbf{u}_{\boldsymbol{\alpha}}\\ &1 \end{pmatrix}K_p\begin{pmatrix} I_n&\textbf{u}_{\boldsymbol{\beta}}\\ &1 \end{pmatrix}.\end{aligned}$$ Here $\textbf{u}_{\boldsymbol{\alpha}}=\ensuremath{\mskip 1mu\prescript{\smash{\mathrm t\mkern-3mu}}{}{\mathstrut(\alpha_1p^{-m'},\cdots, \alpha_{n-1}p^{-m'},\alpha_n p^{-m''})}},$ and $\textbf{u}_{\boldsymbol{\beta}}$ is defined similarly. By the case (A) and case (B), there exists some $\lambda_p\in \mathbb{Q}_p^{\times}$ such that $$\label{eq9.4} \lambda_p\begin{pmatrix} \frac{1}{1-t}I_{n-1}&&p^{-r}\boldsymbol{\xi}\\ &\frac{1}{1-t}&\frac{p^{-r}t}{1-t}\\ &&1 \end{pmatrix}\iota(k_p^*k_py_p)\in \iota(k_p^*k_p)D_p$$ if $r\geq 0;$ and if $r<0,$ we have $$\label{eq9.5} \lambda_p\begin{pmatrix} \frac{p^r}{1-t}I_{n-1}&&\boldsymbol{\xi}\\ &\frac{p^{2r}}{1-t}&\frac{p^{r}}{1-t}\\ &&1 \end{pmatrix}\iota(k_p^*k_py_p)\in \iota(k_p^*k_p)D_p.$$ Recall that $m'=e_p(M').$ The constraint [\[eq9.4\]](#eq9.4){reference-type="eqref" reference="eq9.4"} implies that $$\label{eq9.9} \begin{cases} -m'+e_{\min}(\Omega_p^*)\leq e_p(\lambda_p)\leq m'-e_{\min}(\Omega_p^*),\ \ r\geq 0,\\ e_p(t)-e_p(t-1)-r+e_p(\lambda_p)\geq -m'+e_{\min}(\Omega_p^*)\\ e_p(\xi_j)-r+e_p(\lambda_p)\geq -m'+e_{\min}(\Omega_p^*),\ 1\leq j\leq n-1, \end{cases}$$ and [\[eq9.5\]](#eq9.5){reference-type="eqref" reference="eq9.5"} implies that $$\label{eq9.11} \begin{cases} -m'+e_{\min}(\Omega_p^*)\leq e_p(\lambda_p)\leq m'-e_{\min}(\Omega_p^*),\ \ r<0,\\ r-e_p(t-1)+e_p(\lambda_p)\geq -m'+e_{\min}(\Omega_p^*)\\ e_p(\xi_j)+e_p(\lambda_p)\geq -m'+e_{\min}(\Omega_p^*),\ 1\leq j\leq n-1, \end{cases}$$ where $e_{\min}(\Omega_p^*)$ was defined by [\[equ9.3\]](#equ9.3){reference-type="eqref" reference="equ9.3"}. Here the constraint $$e_p(\lambda_p)\leq m'-e_{\min}(\Omega_p^*)$$ follows by taking the inverse of [\[eq9.4\]](#eq9.4){reference-type="eqref" reference="eq9.4"} and [\[eq9.5\]](#eq9.5){reference-type="eqref" reference="eq9.5"} and considering the $(n+1,n+1)$-th entry. Consequently, we obtain from [\[eq9.9\]](#eq9.9){reference-type="eqref" reference="eq9.9"} and [\[eq9.11\]](#eq9.11){reference-type="eqref" reference="eq9.11"} that $$\begin{aligned} \begin{cases} e_p(t)-e_p(t-1)\geq -2m'+2e_{\min}(\Omega_p^*)\\ e_p(\xi_j)\geq -2m'+2e_{\min}(\Omega_p^*),\ 1\leq j\leq n-1. \end{cases}\end{aligned}$$ 3. Let $p\mid\nu(f).$ Then $p\notin\mathcal{P},$ namely, $\Omega_p^*=K_p.$ Recall definitions in [3.6.2](#3.6.2){reference-type="ref" reference="3.6.2"}, $\mathop{\mathrm{supp}}f_p=Z(\mathbb{Q}_p)D_p',$ where $$\begin{aligned} D_p':=\begin{cases} K_p\operatorname{diag}(p^{l_p},I_{n})K_p,\ \ & \text{if $f=f(g;p_1,p_2),$ and $p=p_1,$}\\ K_p\operatorname{diag}(I_{n},p^{-l_p})K_p,\ \ & \text{if $f=f(g;p_1,p_2),$ and $p=p_2,$}\\ K_p\operatorname{diag}(p^{i},I_{n-1},p^{-i})K_p,\ \ & \text{if $f=f(g;i,p_0),$ and $p=p_0$.} \end{cases}\end{aligned}$$ By the case (A) and case (B), there exists some $\lambda_p\in \mathbb{Q}_p^{\times}$ such that $$\begin{aligned} \lambda_p\begin{pmatrix} \frac{1}{1-t}I_{n-1}&&p^{-r}\boldsymbol{\xi}\\ &\frac{1}{1-t}&\frac{p^{-r}t}{1-t}\\ &&1 \end{pmatrix}\iota(k_p^*k_py_p)\in \iota(k_p^*k_p)D_p'=D_p'\end{aligned}$$ if $r\geq 0;$ and if $r<0,$ we have $$\begin{aligned} \lambda_p\begin{pmatrix} \frac{p^r}{1-t}I_{n-1}&&\boldsymbol{\xi}\\ &\frac{p^{2r}}{1-t}&\frac{p^{r}}{1-t}\\ &&1 \end{pmatrix}\iota(k_p^*k_py_p)\in \iota(k_p^*k_p)D_p'=D_p'.\end{aligned}$$ Analyzing the classifications given by Lemmas [Lemma 30](#lem7){reference-type="ref" reference="lem7"} and [Lemma 31](#lem7.){reference-type="ref" reference="lem7."} we then derive that $\mathfrak{F}_p\neq 0$ unless $$\label{9.6.3} \begin{cases} r\geq 0,\ e_p(t)-e_p(t-1)-r\geq -\delta_p\\ e_p(\xi_j)-r\geq -\delta_p,\ 1\leq j\leq n-1, \end{cases}$$ or $$\label{9.6.4} \begin{cases} r< 0,\ r-e_p(t-1)\geq -\delta_p\\ e_p(\xi_j)\geq -\delta_p,\ 1\leq j\leq n-1, \end{cases}$$ where $\delta_p=l_p$ if $f=f(\cdot;p_1,p_2)$ and $p\mid p_1p_2,$ and $\delta_p=2i$ if $f(\cdot;i,p_0)$ and $p=p_0.$ Both [\[9.6.3\]](#9.6.3){reference-type="eqref" reference="9.6.3"} and [\[9.6.4\]](#9.6.4){reference-type="eqref" reference="9.6.4"} yield that $$\begin{aligned} \begin{cases} e_p(t)-e_p(t-1)\geq -\delta_p\\ e_p(\xi_j)\geq -\delta_p,\ 1\leq j\leq n-1. \end{cases}\end{aligned}$$ 4. Finally we consider the case that $p\mid M,$ $p\nmid M'$ and $p\notin \mathcal{P}.$ By Iwasawa decomposition, we can write $k_p^*k_py_p=p^{r'}a_p'u_p'k_p'$ for some $r'\in\mathbb{Z},$ $a_p'\in A'(\mathbb{Q}_p),$ $u_p'\in N'(\mathbb{Q}_p)$ and $k_p'\in K_p'.$ Then by definition in [3.3.1](#11.1.3){reference-type="ref" reference="11.1.3"}, $$\begin{aligned} \mathfrak{F}_p=f_p\left(\iota(z_pk_p^*k_p)^{-1}\begin{pmatrix} I_{n-1}&&(1-t)\boldsymbol{\xi}\\ &1&t\\ &1&1 \end{pmatrix}\iota(z_pk_p^*k_py_p)\right)\neq 0\end{aligned}$$ unless $$\label{eq9.10} \lambda_p\iota(z_pk_p^*k_p)^{-1}\begin{pmatrix} I_{n-1}&&(1-t)\boldsymbol{\xi}\\ &1&t\\ &1&1 \end{pmatrix}\iota(z_pk_p^*k_py_p)\in K_p(M)$$ for some $\lambda_p\in \mathbb{Q}_p^{\times}.$ Observe that [\[eq9.10\]](#eq9.10){reference-type="eqref" reference="eq9.10"} amounts to $$\begin{aligned} \lambda_p\begin{pmatrix} *&*&p^{-r}(1-t)\boldsymbol{\xi}\\ &p^{r'}&p^{-r}t\\ &p^{r+r'}&1 \end{pmatrix}\in K_p(M),\end{aligned}$$ which leads to that $$\label{cons} \begin{cases} e_p(\lambda_p)+r'=0,\ e_p(\lambda_p)\geq 0,\\ e_p(\lambda_p)+r+r'\geq e_p(M),\\ e_p(\lambda_p)-r+e_p(t)\geq 0,\\ e_p(\lambda_p)-r+e_p(1-t)+e_p(\xi_j)\geq 0,\ 1\leq j\leq n-1,\\ 2e_p(\lambda_p)+r'+e_p(1-t)=0. \end{cases}$$ Notice that the last constraint in [\[cons\]](#cons){reference-type="eqref" reference="cons"} comes from the determinant of the lower right $2\times 2$-corner. So $r\geq e_p(M),$ $e_p(1-t)=-e_p(\lambda_p)\leq 0,$ and $e_p(\xi_j)\geq r\geq e_p(M),$ $1\leq j\leq n-1.$ Putting the above discussions together, then Lemma [Lemma 49](#lem9.2){reference-type="ref" reference="lem9.2"} follows. ◻ ### The Archimedean Constraint Recall that $$\mathfrak{F}_{\infty}:=f_{\infty}\left(\iota(z_{\infty}ak_{\infty}^*k_{\infty})^{-1}\begin{pmatrix} I_{n-1}&&(1-t)\boldsymbol{\xi}\\ &1&t\\ &1&1 \end{pmatrix}\iota(z_{\infty}ak_{\infty}^*k_{\infty}y_{\infty})\right).\tag{\ref{55}}$$ **Lemma 50**. *Let notation be as above. Then $\mathfrak{F}_{\infty}=0$ unless the vector $(\boldsymbol{\xi},t)=(\xi_1,\cdots,\xi_{n-1},t)\in \mathbb{Q}^n$ satisfies that $\xi_j\ll |z_{\infty}|a_j,$ $1\leq j\leq n-1,$ and $t\ll |z_{\infty}|,$ where the implied constants depend only on $c_{\infty}$ and $C_{\infty}$ defined in [3.1.4](#sec3.14){reference-type="ref" reference="sec3.14"}, and $\Omega_{\infty}^*$ in [9.2.1](#Sie){reference-type="ref" reference="Sie"}.* *Proof.* Recall the support of $f_{\infty}$ (cf. [\[245\]](#245){reference-type="eqref" reference="245"} in [3.2.1](#3.2.1){reference-type="ref" reference="3.2.1"}). We have $\mathfrak{F}_{\infty}=0$ unless $$\label{eq9.13} \iota(z_{\infty}ak_{\infty}^*k_{\infty})^{-1}\begin{pmatrix} I_{n-1}&&(1-t)\boldsymbol{\xi}\\ &1&t\\ &1&1 \end{pmatrix}\iota(z_{\infty}ak_{\infty}^*k_{\infty}y_{\infty})\in I_{n+1}+O(T^{-\varepsilon}).$$ Consider the left upper $n\times n$-corner we have $y_{\infty}\in I_n+O(T^{-\varepsilon}).$ Computing the determinant of [\[eq9.13\]](#eq9.13){reference-type="eqref" reference="eq9.13"} yields $|1-t|=1+O(T^{-\varepsilon}).$ Consider the $(j,n+1)$-th entry of [\[eq9.13\]](#eq9.13){reference-type="eqref" reference="eq9.13"}. We obtain $$\begin{aligned} \begin{pmatrix} I_n&&(1-t)\boldsymbol{\xi}\\ &1&t\\ &&1 \end{pmatrix}\in \iota(z_{\infty}ak_{\infty}^*k_{\infty})(I_{n+1}+O(T^{-\varepsilon})),\end{aligned}$$ which implies $$\begin{aligned} \begin{cases} (1-t)\xi_j\ll |z_{\infty}|a_j,\ \ 1\leq j\leq n-1,\\ t\ll |z_{\infty}|. \end{cases}\end{aligned}$$ Here the implied constant depends on $c_{\infty}$ and $C_{\infty}$ defined in [3.1.4](#sec3.14){reference-type="ref" reference="sec3.14"}, and $\Omega_{\infty}^*$ in [9.2.1](#Sie){reference-type="ref" reference="Sie"}. Then Lemma [Lemma 50](#lem9.1){reference-type="ref" reference="lem9.1"} follows. ◻ ### The Global Constraint **Corollary 51**. *Let notation be as before. Let $z_{\infty}\in\mathbb{R}^{\times}$ and $a\in A'(\mathbb{R}).$ Then there exist $z_{\operatorname{fin}}\in\mathbb{A}_{\operatorname{fin}}^{\times},$ $k^*\in \Omega^*,$ $k\in K',$ and $y\in G'(\mathbb{A})$ such that $$\label{9.6} f\left(\iota(z_{\infty}z_{\operatorname{fin}}ak^*k)^{-1}\begin{pmatrix} I_{n-1}&&(1-t)\boldsymbol{\xi}\\ &1&t\\ &1&1 \end{pmatrix}\iota(z_{\infty}z_{\operatorname{fin}}ak^*ky)\right)=0$$ unless $(\boldsymbol{\xi},t)\in \mathfrak{X}(f;a,z_{\infty}),$ where $\mathfrak{X}(f;a,z_{\infty})$ is by $$\begin{aligned} \big\{(\boldsymbol{\xi},t)=(\xi_1,\cdots,\xi_{n-1},t)\in \mathfrak{X}(f):\ \xi_j\ll |z_{\infty}|a_j,\ 1\leq j\leq n-1,\ t\ll |z_{\infty}|\ll 1\big\}.\end{aligned}$$ Here $\mathfrak{X}(f)$ is defined in Lemma [Lemma 49](#lem9.2){reference-type="ref" reference="lem9.2"}, and $\ll$ are defined in Lemma [Lemma 50](#lem9.1){reference-type="ref" reference="lem9.1"}.* **Definition 52**. Let notation be as before. Let $$\begin{aligned} \mathfrak{X}^*(f;a,z_{\infty}):=\big\{(\boldsymbol{\xi},t)\in \mathfrak{X}(f;a,z_{\infty}):\ (\boldsymbol{\xi},t)\neq \textbf{0}\big\},\end{aligned}$$ where $\mathfrak{X}(f;a,z_{\infty})$ is defined in Corollary [Corollary 51](#cor9.4){reference-type="ref" reference="cor9.4"}. A straightforward calculation yields that $$\label{9.7} \big|\mathfrak{X}^*(f;a,z_{\infty})\big|\ll \max\Bigg\{1,\frac{M'^2\mathcal{N}_f^2|z_{\infty}|}{C_{\Omega^*}^2M^{\dagger}}\Bigg\}\prod_{j=1}^{n-1}\max\Bigg\{1,\frac{M'^{2}\mathcal{N}_f^2|z_{\infty}|a_j}{C_{\Omega^*}^2M^{\dagger}}\Bigg\},$$ where the implied constant in [\[9.7\]](#9.7){reference-type="eqref" reference="9.7"} depends only on $c_{\infty}$ and $C_{\infty}$ defined in [3.1.4](#sec3.14){reference-type="ref" reference="sec3.14"}, and $\Omega_{\infty}^*$ in [9.2.1](#Sie){reference-type="ref" reference="Sie"}. ## Reduce to the Essential Range {#sec9.3} Recall that our goal is to prove Proposition [Proposition 48](#prop9.3.){reference-type="ref" reference="prop9.3."}, where $\mathcal{J}_{\operatorname{Geo},\operatorname{Big}}^{\operatorname{Reg},\uppercase{\romannumeral 2\relax}}(f)$ is defined by (after the change of variable $\boldsymbol{\xi}\mapsto (1-t)\boldsymbol{\xi}$) $$\begin{aligned} \int\int_{{G'}(\mathbb{A})}\sum_{\substack{(\boldsymbol{\xi},t)\in \mathfrak{X}^*(f;a,z_{\infty})}}\Bigg|f\left(\iota(x)^{-1}\begin{pmatrix} I_{n-1}&&(1-t)\boldsymbol{\xi}\\ &1&t\\ &1&1 \end{pmatrix}\iota(xy)\right)\Bigg|\big|\phi'(x)\big|^2dydx, \end{aligned}$$ where $x$ ranges through $P_0'(\mathbb{Q})\backslash {G'}(\mathbb{A}).$ The sum over $(\boldsymbol{\xi},t)$ is finite (cf. [\[9.7\]](#9.7){reference-type="eqref" reference="9.7"}). **Definition 53**. Let notation be as before. Let $c_0, c_1$ be constants in Lemma [Lemma 25](#lem4.9){reference-type="ref" reference="lem4.9"}. Take $c_0=11+n$. Then $c_1\geq 0$ is a constant depending at most on $n$. Let $c=2c_1+10n^2.$ Let $\varepsilon>0.$ Define $\mathfrak{A}_{c,\varepsilon}$ by $$\begin{aligned} \big\{a=(a_1,\cdots,a_{n-1},1)\in A_H(\mathbb{R}):\ \|a\|\leq T^{c},\ T^{-\varepsilon}\ll |\det a|_{\infty}\ll T^{\varepsilon}\big\}.\end{aligned}$$ We call $\mathfrak{A}_{c,\varepsilon}$ the essential range for $J^{\operatorname{Reg},\uppercase{\romannumeral 2\relax}}_{\operatorname{Geo},\operatorname{Big}}(f,\textbf{0}).$ **Lemma 54**. *Let notation be as before. Let $a\in A_H(\mathbb{R})-\mathfrak{A}_{c,\varepsilon}.$ Then for all $k^*k$ we have $$\label{9.10.} |\phi'(ak^*k)|\cdot \|\phi'\|_{\infty}\ll T^{-10n^2}\|a\|^{-n-10},$$ where the implied constant is absolute, and $\|a\|:=\prod_{j=1}^{n-1}\max\{a_j,a_j^{-1}\}$.* *Proof.* Let $a\in A_H(\mathbb{R})-\mathfrak{A}_{c,\varepsilon}.$ By Lemma [Lemma 25](#lem4.9){reference-type="ref" reference="lem4.9"}, $\|\phi'\|\ll T^{c_1}.$ 1. If $\|a\|>T^c,$ then by Lemma [Lemma 25](#lem4.9){reference-type="ref" reference="lem4.9"} (with $c_2=0$), for all $k^*k,$ $$|\phi'(ak^*k)|\cdot\|\phi'\|_{\infty}\ll \|a\|^{-11-n}T^{2c_1}\ll T^{-c}T^{2c_1}\|a\|^{-n-10} \ll T^{-10n^2}\|a\|^{-n-10}.$$ 2. If $\|a\|\leq T^c$ but $|\det a|_{\infty}\gg T^{\varepsilon}.$ Then taking $c_2=-(10n^2+2c_1+(10+n)c)/\varepsilon$ in Lemma [Lemma 25](#lem4.9){reference-type="ref" reference="lem4.9"} we then obtain, for all $k^*k,$ that $$|\phi'(ak^*k)|\cdot\|\phi'\|_{\infty}\ll T^{2c_1}T^{\varepsilon c_2}\ll T^{-10n^2}\|a\|^{-n-10}.$$ 3. If $\|a\|\leq T^c$ but $|\det a|_{\infty}\ll T^{-\varepsilon}.$ Then taking $c_2=(10n^2+2c_1+(10+n)c)/\varepsilon$ in Lemma [Lemma 25](#lem4.9){reference-type="ref" reference="lem4.9"} we then obtain, for all $k^*k,$ that $$|\phi'(ak^*k)|\cdot\|\phi'\|_{\infty}\ll T^{2c_1}T^{-\varepsilon c_2}\ll T^{-10n^2}\|a\|^{-n-10}.$$ In all, the estimate [\[9.10.\]](#9.10.){reference-type="eqref" reference="9.10."} holds. ◻ Write $x=zbk$ and $b=ak^*=ak_{\infty}^*k_{\operatorname{fin}}^*$ as before. By [\[87\]](#87){reference-type="eqref" reference="87"} and Corollary [Corollary 51](#cor9.4){reference-type="ref" reference="cor9.4"}, $$\begin{aligned} \mathcal{J}_{\operatorname{Geo},\operatorname{Big}}^{\operatorname{Reg},\uppercase{\romannumeral 2\relax}}(f)\ll \mathcal{J}_{\operatorname{Geo},\operatorname{Big}}^{\operatorname{Reg},\uppercase{\romannumeral 2\relax},\dagger}(f)+\mathcal{J}_{\operatorname{Geo},\operatorname{Big}}^{\operatorname{Reg},\uppercase{\romannumeral 2\relax},\heartsuit}(f),\end{aligned}$$ where $\mathcal{J}_{\operatorname{Geo},\operatorname{Big}}^{\operatorname{Reg},\uppercase{\romannumeral 2\relax},\dagger}(f)$ is defined by $$\begin{aligned} & \int_{Z'(\mathbb{A})}\int_{K'}\int_{G'(\mathbb{A})} \int_{\Omega^*}\int_{A_H(\mathbb{R})-\mathfrak{A}_{c,\varepsilon}}\sum_{\substack{(\boldsymbol{\xi},t)\in \mathfrak{X}^*(f;a,z_{\infty})}}\big|\phi'(ak^*k)\big|^2\\ &\Bigg|f_{\infty}\left(\iota(z_{\infty}ak_{\infty}^*k_{\infty})^{-1}\begin{pmatrix} I_{n-1}&&(1-t)\boldsymbol{\xi}\\ &1&t\\ &1&1 \end{pmatrix}\iota(z_{\infty}ak_{\infty}^*k_{\infty}y_{\infty})\right)\Bigg|\\ &\prod_{p<\infty}\Bigg|f_p\left(\iota(z_pk_p^*k_p)^{-1}\begin{pmatrix} I_{n-1}&&(1-t)\boldsymbol{\xi}\\ &1&t\\ &1&1 \end{pmatrix}\iota(z_pk_p^*k_py_p)\right)\Bigg|\delta_{B'}^{-1}(a)d^{\times}adk^*dydkd^{\times}z,\end{aligned}$$ and $\mathcal{J}_{\operatorname{Geo},\operatorname{Big}}^{\operatorname{Reg},\uppercase{\romannumeral 2\relax},\heartsuit}(f)$ is defined similarly except that $a$ ranges through $\mathfrak{A}_{c,\varepsilon}.$ In this section, we shall show that $\mathcal{J}_{\operatorname{Geo},\operatorname{Big}}^{\operatorname{Reg},\uppercase{\romannumeral 2\relax},\dagger}(f)$ is tiny, and thus the majorization of $\mathcal{J}^{\operatorname{Reg},\uppercase{\romannumeral 2\relax}}_{\operatorname{Geo},\operatorname{Big}}(f)$ is reduced to that of $\mathcal{J}_{\operatorname{Geo},\operatorname{Big}}^{\operatorname{Reg},\uppercase{\romannumeral 2\relax},\heartsuit}(f)$. ### Estimate of Nonarchimedean Local Integrals {#9.2.1} Define $\mathcal{J}_p^{\heartsuit}(\boldsymbol{\xi},t;k_{p}^*k_p)$ by $$\begin{aligned} \int_{Z'(\mathbb{Q}_p)}\int_{G'(\mathbb{Q}_{p})}\Bigg|f_p\left(\iota(z_pk_p^*k_p)^{-1}\begin{pmatrix} I_{n-1}&&(1-t)\boldsymbol{\xi}\\ &1&t\\ &1&1 \end{pmatrix}\iota(z_pk_p^*k_py_p)\right)\Bigg|dy_{p}d^{\times}z_p.\end{aligned}$$ **Lemma 55**. *Let notation be as before. Then $$\begin{aligned} \prod_{p<\infty}\mathcal{J}_p^{\heartsuit}(\boldsymbol{\xi},t;k_{p}^*k_p)\ll M'^{2n+\varepsilon}M^n\cdot \mathcal{N}_f^{2(n-1)}\cdot \prod_{p\mid M'}p^{ne_p(M)}\cdot \prod_{p<\infty}E_p(\boldsymbol{\xi},t),\end{aligned}$$ where $E_p(\boldsymbol{\xi},t):=\big|\min\{e_p(t)-e_p(t-1), e_p(\xi_j):\ 1\leq j\leq n-1\}\big|+\big|e_p(t-1)\big|+2e_p(M')-2e_{\min}(\Omega_p^*)+1,$ and the implied constant depends only on $\varepsilon.$* *Proof.* Consider various finite places as follows. We will make use of case (A) and case (B) in the proof of Lemma [Lemma 49](#lem9.2){reference-type="ref" reference="lem9.2"}. 1. Let $p\nmid MM'\nu(f)$ and $p\notin\mathcal{P}.$ Then $f_p$ is bi-invariant under $\Omega^*_pK_p'=K_p'.$ - Suppose $e_p(t-1)\geq 0.$ Then from the case (B) and the support of $f_p$ one must have $r\geq 0.$ Change variable $y_p\mapsto (k_p^*k_p)^{-1}y_p$ to obtain $$\begin{aligned} \mathcal{J}_p^{\heartsuit}(\boldsymbol{\xi},t;k_{p}^*k_p)=\sum_{r\geq 0}\int_{G'(\mathbb{Q}_{p})}f_p\left(\begin{pmatrix} I_{n-1}&&p^{-r}(1-t)\boldsymbol{\xi}\\ &1&p^{-r}t\\ &&1-t \end{pmatrix}\iota(y_p)\right)dy_{p}.\end{aligned}$$ By definition, $f_p\left(\begin{pmatrix} I_{n-1}&&p^{-r}(1-t)\boldsymbol{\xi}\\ &1&p^{-r}t\\ &&1-t \end{pmatrix}\iota(y_p)\right)\neq 0$ if and only if $$\begin{aligned} \begin{cases} y_p\in (1-t)K_p'\\ 0\leq r\leq \min\{e_p(t)-e_p(t-1),\ e_p(\xi_j):\ 1\leq j\leq n-1\}. \end{cases}\end{aligned}$$ Therefore, we have $$\label{56} \mathcal{J}_p^{\heartsuit}(\boldsymbol{\xi},t;k_{p}^*k_p)=\textbf{1}_{e_p(\xi_j)\geq e_p(t-1)\geq 0,\ 1\leq j\leq n-1}\sum_{\substack{0\leq r\leq e_p(t)-e_p(t-1)\\ r\leq e_p(\xi_j),\ 1\leq j\leq n-1}}1.$$ - Suppose $e_p(t-1)\leq -1.$ Then $e_p(t)=e_p(t-1).$ From the case (B) we obtain $e_p(t-1)\leq r\leq -1;$ in the case (A) we have $0\leq r\leq e_p(t)-e_p(t-1),$ implying $r=0.$ Changing variable $y_p\mapsto (k_p^*k_p)^{-1}y_p,$ $$\begin{aligned} \mathcal{J}_p^{\heartsuit}(\boldsymbol{\xi},t;k_{p}^*k_p)=&\int_{G'(\mathbb{Q}_{p})}f_p\left(\begin{pmatrix} I_{n-1}&&(1-t)\boldsymbol{\xi}\\ &1&t\\ &&1-t \end{pmatrix}\iota(y_p)\right)dy_{p}\\ &+\sum_{e_p(t-1)\leq r\leq -1}\int_{G'(\mathbb{Q}_{p})}f_p\left(\begin{pmatrix} I_{n-1}&&p^{-r}(1-t)\boldsymbol{\xi}\\ &p^{r}&1\\ &&p^{-r}(1-t) \end{pmatrix}\iota(y_p)\right)dy_{p}.\end{aligned}$$ By the support of $f_p,$ the first integral is nonzero unless $y_p\in (1-t)K_p',$ and in the second term, the integrand is nonzero unless $y_p\in (1-t)\operatorname{diag}(p^{-r}I_{n-1},p^{-2r})K_p'.$ Hence, $$\label{57} \mathcal{J}_p^{\heartsuit}(\boldsymbol{\xi},t;k_{p}^*k_p)=(1-e_p(t-1))\textbf{1}_{e_p(\xi_j)\geq 0,\ 1\leq j\leq n-1}\textbf{1}_{e_p(t-1)\leq -1}.$$ From the above discussions we see that $\mathcal{J}_p^{\heartsuit}(\boldsymbol{\xi},t;k_{p}^*k_p)=1$ if $e_p(t)=e_p(t-1)=0$ and $e_p(\xi_j)\geq 0,$ $1\leq j\leq n-1;$ and $y_p$ ranges over a compact set depending on $t.$ As a consequence, for fixed $\boldsymbol{\xi}$ and $t,$ $\mathcal{J}_p^{\heartsuit}(\boldsymbol{\xi},t;k_{p}^*k_p)=1$ for all but finitely many places $p.$ 2. Let $p\mid M'$ or $p\in \mathcal{P}.$ Suppose $$\begin{aligned} \mathfrak{F}_p:=f_p\left(\iota(z_pk_p^*k_p)^{-1}\begin{pmatrix} I_{n-1}&&(1-t)\boldsymbol{\xi}\\ &1&t\\ &1&1 \end{pmatrix}\iota(z_pk_p^*k_py_p)\right)\neq 0.\end{aligned}$$ By the case (A) and case (B), there exists some $\lambda_p\in p^{\mathbb{Z}}$ such that $$\lambda_p\begin{pmatrix} \frac{1}{1-t}I_{n-1}&&p^{-r}\boldsymbol{\xi}\\ &\frac{1}{1-t}&\frac{p^{-r}t}{1-t}\\ &&1 \end{pmatrix}\iota(k_p^*k_py_p)\in \iota(k_p^*k_p)D_p\tag{\ref{eq9.4}}$$ if $r\geq 0;$ and if $r<0,$ we have $$\lambda_p\begin{pmatrix} \frac{p^r}{1-t}I_{n-1}&&\boldsymbol{\xi}\\ &\frac{p^{2r}}{1-t}&\frac{p^{r}}{1-t}\\ &&1 \end{pmatrix}\iota(k_p^*k_py_p)\in \iota(k_p^*k_p)D_p.\tag{\ref{eq9.5}}$$ Let $D_p'\subset G'(\mathbb{Q}_p)$ be such that $\iota(D_p')=\operatorname{diag}(D_p',1)=\iota(G'(\mathbb{Q}_p))\cap D_p.$ Let $e_{\min}(\Omega_p^*)$ be defined by [\[equ9.3\]](#equ9.3){reference-type="eqref" reference="equ9.3"}. Recall that $m'=e_p(M').$ Then [\[eq9.4\]](#eq9.4){reference-type="eqref" reference="eq9.4"} implies $$\begin{aligned} \begin{cases} -m'+e_{\min}(\Omega_p^*)\leq e_p(\lambda_p)\leq m'-e_{\min}(\Omega_p^*),\ \ r\geq 0,\\ e_p(t)-e_p(t-1)-r+e_p(\lambda_p)\geq -m'+e_{\min}(\Omega_p^*)\\ e_p(\xi_j)-r+e_p(\lambda_p)\geq -m'+e_{\min}(\Omega_p^*),\ 1\leq j\leq n-1,\\ y_p\in \lambda_p^{-1}(1-t)D_p' \end{cases}\end{aligned}$$ and [\[eq9.5\]](#eq9.5){reference-type="eqref" reference="eq9.5"} implies that $$\begin{aligned} \begin{cases} -m'+e_{\min}(\Omega_p^*)\leq e_p(\lambda_p)\leq m'-e_{\min}(\Omega_p^*),\ \ r<0,\\ r-e_p(t-1)+e_p(\lambda_p)\geq -m'+e_{\min}(\Omega_p^*)\\ e_p(\xi_j)+e_p(\lambda_p)\geq -m'+e_{\min}(\Omega_p^*),\ 1\leq j\leq n-1,\\ y_p\in \lambda_p^{-1}p^{-r}(1-t)\iota(k_p^*k_p)^{-1}\operatorname{diag}(I_{n-1},p^{-r})\iota(k_p^*k_p)D_p. \end{cases}\end{aligned}$$ Here $m'=e_p(M').$ As a consequence, we have $$\begin{aligned} \begin{cases} 0\leq r\leq \min\{e_p(t)-e_p(t-1), e_p(\xi_j):\ 1\leq j\leq n-1\}+2m'-2e_{\min}(\Omega_p^*)\\ y_p\in \lambda_p^{-1}(1-t)D_p', \end{cases}\end{aligned}$$ in the case that $r\geq 0,$ and in the other case, i.e., $r<0,$ we obtain $$\begin{aligned} \begin{cases} e_p(t-1)-2m'+2e_{\min}(\Omega_p^*)\leq r<0\\ y_p\in \lambda_p^{-1}p^{-r}(1-t)\iota(k_p^*k_p)^{-1}\operatorname{diag}(I_{n-1},p^{-r})\iota(k_p^*k_p)D_p'. \end{cases}\end{aligned}$$ In conjunction with the invariance of Haar measure $dy_p,$ we have $$\begin{aligned} \mathcal{J}_p^{\heartsuit}(\boldsymbol{\xi},t;k_{p}^*k_p)&\ll \|f_p\|_{\infty}\sum_{\lambda_p}\sum_{0\leq r\leq d_1}\int_{\lambda_p^{-1}(1-t)D_p'}dy_p\\ &+\|f_p\|_{\infty}\sum_{\lambda_p}\sum_{d_2\leq r< 0}\int_{\lambda_p^{-1}p^{-r}(1-t)\iota(k_p^*k_p)^{-1}\operatorname{diag}(I_{n-1},p^{-r})\iota(k_p^*k_p)D_p'}dy_p,\end{aligned}$$ where $\lambda_p\in p^{\mathbb{Z}}$ with $|e_p(\lambda_p)|\leq m'-e_{\min}(\Omega_p^*),$ $d_1:=\min\{e_p(t)-e_p(t-1), e_p(\xi_j):\ 1\leq j\leq n-1\}+2m'-2e_{\min}(\Omega_p^*),$ $d_2:=e_p(t-1)-2m'+2e_{\min}(\Omega_p^*),$ and $\|f_p\|_{\infty}\ll p^{2ne_p(M)-m''}$ (cf. [3.3.2](#11.1.4){reference-type="ref" reference="11.1.4"}). Therefore, $$\begin{aligned} \mathcal{J}_p^{\heartsuit}(\boldsymbol{\xi},t;k_{p}^*k_p)\ll (m'-e_{\min}(\Omega_p^*)+1)E_p(\boldsymbol{\xi},t)\operatorname{Vol}(D_p')\ll p^{2nm'+2ne_p(M)+\varepsilon}E_p(\boldsymbol{\xi},t). \end{aligned}$$ 3. Let $p\mid M^{\dagger}:=\prod_{p\mid M,\ p\nmid M'}p^{e_p(M)}$ and $p\notin \mathcal{P}.$ By a similar argument, $$\begin{aligned} \mathcal{J}_p^{\heartsuit}(\boldsymbol{\xi},t;k_{p}^*k_p)\ll p^{ne_p(M^{\dagger})}\cdot E_p(\boldsymbol{\xi},t).\end{aligned}$$ 4. Let $p\mid\nu(f).$ Assume $\mathfrak{F}_p\neq 0.$ Let $$\begin{aligned} M_{n,1}(\mathbb{Q}_p)\ni \mathfrak{u}:=\begin{cases} \ensuremath{\mskip 1mu\prescript{\smash{\mathrm t\mkern-3mu}}{}{\mathstrut(\xi_1,\cdots, \xi_{n-1},\frac{p^{-r}t}{1-t})}},\ &\text{if $r\geq 0$,}\\ \ensuremath{\mskip 1mu\prescript{\smash{\mathrm t\mkern-3mu}}{}{\mathstrut(\xi_1,\cdots, \xi_{n-1},\frac{p^{r}}{1-t})}},\ &\text{if $r<0$.} \end{cases}\end{aligned}$$ Let $\mathcal{D}_p(\mathfrak{u})$ be the set of $y_p$ such that $f_p\left(\begin{pmatrix} y_p&\mathfrak{u}\\ &1 \end{pmatrix}\right)\neq 0.$ The structure of $\mathcal{D}_p(\mathfrak{u})$ has been classified by Lemmas [Lemma 30](#lem7){reference-type="ref" reference="lem7"} and [Lemma 31](#lem7.){reference-type="ref" reference="lem7."}. Set $$\begin{aligned} \delta_p:=\begin{cases} l_p, &\text{if $f=f(\cdot;p_1,p_2)$ and $p\mid p_1p_2,$}\\ 2i, &\text{ if $f(\cdot;i,p_0)$ and $p=p_0.$} \end{cases}\end{aligned}$$ By [\[9.6.3\]](#9.6.3){reference-type="eqref" reference="9.6.3"} and [\[9.6.4\]](#9.6.4){reference-type="eqref" reference="9.6.4"}, $\mathfrak{F}_p\neq 0$ implies that $$\begin{aligned} \begin{cases} r\geq \min\{e_p(t-1),0\}-\delta_p,\\ r\leq \min\{e_p(t)-e_p(t-1), e_p(\xi_j):\ 1\leq j\leq n-1\}+\delta_p,\\ y_p\in (1-t)\mathcal{D}_p(\mathfrak{u}),\ \text{if $r\geq 0,$}\\ y_p\in (1-t)\operatorname{diag}(p^{-r}I_{n-1},p^{-2r})\mathcal{D}_p(\mathfrak{u}),\ \text{if $r<0.$} \end{cases}\end{aligned}$$ Therefore, $$\begin{aligned} \mathcal{J}_p^{\heartsuit}(\boldsymbol{\xi},t;k_{p}^*k_p)\ll \sum_{0\leq r\leq d_3}\int_{(1-t)\mathcal{D}_p(\mathfrak{u})}dy_p+\sum_{d_3\leq r< 0}\int_{(1-t)\operatorname{diag}(p^{-r}I_{n-1},p^{-2r})\mathcal{D}_p(\mathfrak{u})}dy_p,\end{aligned}$$ where $d_3:=\min\{e_p(t)-e_p(t-1), e_p(\xi_j):\ 1\leq j\leq n-1\}+\delta_p$ and $d_4:=\min\{e_p(t-1),0\}-\delta_p.$ Therefore, and the fact that $\delta_p\leq n+1,$ $$\begin{aligned} \mathcal{J}_p^{\heartsuit}(\boldsymbol{\xi},t;k_{p}^*k_p)\ll E_p(\boldsymbol{\xi},t)\cdot \max_{\mathfrak{u}}\operatorname{Vol}(\mathcal{D}_p(\mathfrak{u})).\end{aligned}$$ The volume $\operatorname{Vol}(\mathcal{D}_p(\mathfrak{u}))$ can be computed by lemmas [Lemma 30](#lem7){reference-type="ref" reference="lem7"} and [Lemma 31](#lem7.){reference-type="ref" reference="lem7."}. - Suppose $\nu(f)=p_1p_2$ and $p\mid \nu(f),$ where $\nu(f)$ is defined by [\[61\]](#61){reference-type="eqref" reference="61"}. Investigating cases (A) and (B) in lemma [Lemma 30](#lem7){reference-type="ref" reference="lem7"}, $$\begin{aligned} \max_{\mathfrak{u}}\operatorname{Vol}(\mathcal{D}_p(\mathfrak{u}))\ll p^{(n-1)l_p},\end{aligned}$$ where the implied constant is absolute. - Suppose $\nu(f)=p_0$ and $p\mid\nu(f).$ Investigating cases (A.1), (A.2), (B.1), (B.2), and (B.3) in Lemma [Lemma 31](#lem7.){reference-type="ref" reference="lem7."}, $$\begin{aligned} \max_{\mathfrak{u}}\operatorname{Vol}(\mathcal{D}_p(\mathfrak{u}))\ll p^{2(n-1)i},\end{aligned}$$ where the implied constant is absolute, and the case (A.2) contributes $$\max_{\mathfrak{u}}\operatorname{Vol}(\mathcal{D}_p(\mathfrak{u}))\ll p^{(n-2)(r_1-i)+(n-1)(e-i)}\ll p^{2(n-1)i}.$$ Here $r_1$ and $e$ are defined in Lemma [Lemma 31](#lem7.){reference-type="ref" reference="lem7."}, case (A.2). Note that in the above estimate of $\max \operatorname{Vol}(\mathcal{D}_p(\mathfrak{u}))$ we make use of the constraint between $\mathfrak{u}$ and the uniponent part of $y_p$ in the Iwasawa decomposition (i.e., $\mathfrak{u}''$ in the notation of lemmas [Lemma 30](#lem7){reference-type="ref" reference="lem7"} and [Lemma 31](#lem7.){reference-type="ref" reference="lem7."}). Putting the above discussions together we then derive Lemma [Lemma 55](#lem9.7){reference-type="ref" reference="lem9.7"}. ◻ ### Separate Out the Automorphic Weight By Lemma [Lemma 20](#lem4.5){reference-type="ref" reference="lem4.5"} one has $\langle\phi',\phi'\rangle\gg\|W_{\infty}'\|_2\gg T^{-n^2}.$ In conjunction with Lemma [Lemma 54](#lem9.7.){reference-type="ref" reference="lem9.7."} we have, for $a\notin \mathfrak{A}_{c,\varepsilon},$ that $$\label{9.13} \big|\phi'(ak^*k)\big|^2\leq |\phi'(ak^*k)|\cdot \|\phi'\|_{\infty}\ll T^{-10n^2}\|a\|^{-n-10}\ll \frac{\langle\phi',\phi'\rangle}{T^{9n^2}\|a\|^{n+10}}.$$ We shall substitute [\[9.13\]](#9.13){reference-type="eqref" reference="9.13"} into $\mathcal{J}_{\operatorname{Geo},\operatorname{Big}}^{\operatorname{Reg},\uppercase{\romannumeral 2\relax},\dagger}(f)$ to handle the integral relative to $\phi',$ which is initially not factorizable. ### Bounding $\mathcal{J}_{\operatorname{Geo},\operatorname{Big}}^{\operatorname{Reg},\uppercase{\romannumeral 2\relax},\dagger}(f)$ Recall that (cf. [9.4](#sec9.3){reference-type="ref" reference="sec9.3"}, right before [9.4.1](#9.2.1){reference-type="ref" reference="9.2.1"}) $$\begin{aligned} \mathcal{J}_{\operatorname{Geo},\operatorname{Big}}^{\operatorname{Reg},\uppercase{\romannumeral 2\relax},\dagger}(f):=& \int_{Z'(\mathbb{R})}\int_{K'}\int_{G'(\mathbb{R})} \int_{\Omega^*}\int_{A_H(\mathbb{R})-\mathfrak{A}_{c,\varepsilon}}\sum_{\substack{(\boldsymbol{\xi},t)\in \mathfrak{X}^*(f;a,z_{\infty})}}\big|\phi'(ak^*k)\big|^2\\ &\Bigg|f_{\infty}\left(\iota(z_{\infty}ak_{\infty}^*k_{\infty})^{-1}\begin{pmatrix} I_{n-1}&&(1-t)\boldsymbol{\xi}\\ &1&t\\ &1&1 \end{pmatrix}\iota(z_{\infty}ak_{\infty}^*k_{\infty}y_{\infty})\right)\Bigg|\\ &\prod_{p<\infty}\mathcal{J}_p^{\heartsuit}(\boldsymbol{\xi},t;k_{p}^*k_p)\delta_{B'}^{-1}(a)d^{\times}adk^*dy_{\infty}dkd^{\times}z_{\infty},\end{aligned}$$ where $\mathcal{J}_p^{\heartsuit}(\boldsymbol{\xi},t;k_{p}^*k_p)$ was defined in [9.4.1](#9.2.1){reference-type="ref" reference="9.2.1"}. **Proposition 56**. *Let notation be as before. Then $$\label{eq9.22} \frac{\mathcal{J}_{\operatorname{Geo},\operatorname{Big}}^{\operatorname{Reg},\uppercase{\romannumeral 2\relax},\dagger}(f)}{\langle\phi',\phi'\rangle}\ll \frac{(TMM')^{\varepsilon}M'^{2n}M^n}{T^{8n^2}\mathcal{N}_f^{2(1-n)}}\prod_{p\mid M'}p^{ne_p(M)}\max_{1\leq l_0\leq n}\Bigg[\frac{M'^{2}\mathcal{N}_f^{2}}{M^{\dagger}}\Bigg]^{l_0},$$ where the implied constant depends on $\varepsilon,$ parameters $c_{\infty}$ and $C_{\infty}$ defined in [3.1.4](#sec3.14){reference-type="ref" reference="sec3.14"}, and the conductor of $\pi_{\infty}'.$* *Proof.* Substitute [\[9.13\]](#9.13){reference-type="eqref" reference="9.13"} into $\mathcal{J}_{\operatorname{Geo},\operatorname{Big}}^{\operatorname{Reg},\uppercase{\romannumeral 2\relax},\dagger}(f)/\langle\phi',\phi'\rangle$ to majorized it by $$\begin{aligned} &T^{-9n^2}\int_{K'} \int_{\Omega^*}\int_{A_H(\mathbb{R})-\mathfrak{A}_{c,\varepsilon}}\|a\|^{-n-10} \int_{Z'(\mathbb{R})}\int_{G'(\mathbb{R})}\sum_{\substack{(\boldsymbol{\xi},t)\in \mathfrak{X}^*(f;a,z_{\infty})}}\prod_{p<\infty}\mathcal{J}_p^{\heartsuit}(\boldsymbol{\xi},t;k_{p}^*k_p)\\ &\Bigg|f_{\infty}\left(\iota(z_{\infty}ak_{\infty}^*k_{\infty})^{-1}\begin{pmatrix} I_{n-1}&&\boldsymbol{\xi}\\ &1&t\\ &1&1 \end{pmatrix}\iota(z_{\infty}ak_{\infty}^*k_{\infty}y_{\infty})\right)\Bigg|dy_{\infty}d^{\times}z_{\infty}\frac{d^{\times}a}{\delta_{B'}(a)}dk^*dk.\end{aligned}$$ By Lemma [Lemma 55](#lem9.7){reference-type="ref" reference="lem9.7"} and the fact that $\delta_{B'}(a)\geq \|a\|^{-n},$ the above integral is further bounded by $$\begin{aligned} &C^{\dag}\int_{K'_{\infty}} \int_{\Omega^*_{\infty}}\int_{A_H(\mathbb{R})-\mathfrak{A}_{c,\varepsilon}}\|a\|^{-10} \int_{Z'(\mathbb{R})}\sum_{\substack{(\boldsymbol{\xi},t)\in \mathfrak{X}^*(f;a,z_{\infty})}}\int_{G'(\mathbb{R})} \prod_{p<\infty}E_p(\boldsymbol{\xi},t)\\ &\Bigg|f_{\infty}\left(\iota(z_{\infty}ak_{\infty}^*k_{\infty})^{-1}\begin{pmatrix} I_{n-1}&&\boldsymbol{\xi}\\ &1&t\\ &1&1 \end{pmatrix}\iota(z_{\infty}ak_{\infty}^*k_{\infty}y_{\infty})\right)\Bigg|dy_{\infty}d^{\times}z_{\infty}d^{\times}adk_{\infty}^*dk_{\infty},\end{aligned}$$ where $C^{\dag}:=T^{-9n^2}M'^{2n+\varepsilon}M^n\cdot \mathcal{N}_f^{2(n-1)}\cdot \prod_{p\mid M'}p^{ne_p(M)}.$ By [\[9.7\]](#9.7){reference-type="eqref" reference="9.7"} and the condition that $(\boldsymbol{\xi},t)\neq \textbf{0}$ we have $$\big|\mathfrak{X}^*(f;a,z_{\infty})\big|\ll \max\Bigg\{1,\frac{M'^2\mathcal{N}_f^2|z_{\infty}|}{C_{\Omega^*}^2M^{\dagger}}\Bigg\}\prod_{j=1}^{n-1}\max\Bigg\{1,\frac{M'^{2}\mathcal{N}_f^2|z_{\infty}|a_j}{C_{\Omega^*}^2M^{\dagger}}\Bigg\},\tag{\ref{9.7}}$$ and either $M'^2\mathcal{N}_f^2|z_{\infty}|\gg C_{\Omega^*}^2M^{\dagger}$ or $M'^{2}\mathcal{N}_f^2|z_{\infty}|a_1\gg C_{\Omega^*}^2M^{\dagger},$ depending on whether $t=0$ or not. Hence, in conjunction with the restriction that $|z_{\infty}|\ll 1,$ we have $$\label{10.17} C_{\Omega^*}^2M^{\dagger}M'^{-2}\mathcal{N}_f^{-2}\textbf{1}_{t\neq 0}+C_{\Omega^*}^2M^{\dagger}M'^{-2}\mathcal{N}_f^{-2}a_1^{-1}\textbf{1}_{t=0}\ll |z_{\infty}|\ll 1.$$ By definition of $\mathfrak{X}^*(f;a,z_{\infty}),$ we have, for $(\boldsymbol{\xi},t)\in \mathfrak{X}^*(f;a,z_{\infty}),$ that $$\label{eq9.24.} \prod_{p<\infty}E_p(\boldsymbol{\xi},t)\ll (TMM')^{\varepsilon}.$$ - Suppose $M'^2\mathcal{N}_f^2|z_{\infty}|\gg C_{\Omega^*}^2M^{\dagger}.$ Then $$\label{eq9.23} \big|\mathfrak{X}^*(f;a,z_{\infty})\big|\ll \frac{M'^{2n}\mathcal{N}_f^{2n}|z_{\infty}|}{(C_{\Omega^*}^2M^{\dagger})^n}\prod_{j=1}^{n-1}\max\{a_j,a_j^{-1}\}=\frac{M'^{2n}\mathcal{N}_f^{2n}|z_{\infty}|}{(C_{\Omega^*}^2M^{\dagger})^n}\|a\|.$$ - Suppose $M'^2\mathcal{N}_f^2|z_{\infty}|=o(C_{\Omega^*}^2M^{\dagger}).$ Then by [\[10.17\]](#10.17){reference-type="eqref" reference="10.17"} we have $t=0,$ and thus $$M'^2\mathcal{N}_f^2|z_{\infty}|a_1\geq c' C_{\Omega^*}^2M^{\dagger}$$ for some constant $c'>0.$ Here $c'$ depends only on $c_{\infty}$ and $C_{\infty}$ defined in [3.1.4](#sec3.14){reference-type="ref" reference="sec3.14"}. Recall that $a_1\geq a_2\geq \cdots \geq a_{n-1}>0.$ Let $1\leq l_0\leq n-1$ be the largest integer such that $$M'^2\mathcal{N}_f^2|z_{\infty}|a_{l_0}\geq 10^{-1}c' C_{\Omega^*}^2M^{\dagger}.$$ Then $\big|\mathfrak{X}^*(f;a,z_{\infty})\big|$ is $$\begin{aligned} \ll \prod_{j=1}^{n-1}\max\Bigg\{1,\frac{M'^{2}\mathcal{N}_f^2|z_{\infty}|a_j}{C_{\Omega^*}^2M^{\dagger}}\Bigg\}\ll \Bigg[\frac{M'^{2}\mathcal{N}_f^{2}|z_{\infty}|}{C_{\Omega^*}^2M^{\dagger}}\Bigg]^{l_0}\cdot \prod_{j=1}^{l_0}\max\{a_j,a_j^{-1}\}.\end{aligned}$$ Since $|z_{\infty}|\ll 1$ and $\max\{a_j,a_j^{-1}\}\geq 1$ for all $1\leq j\leq n-1,$ then $$\label{eq9.24} \big|\mathfrak{X}^*(f;a,z_{\infty})\big|\ll \Bigg[\frac{M'^{2}\mathcal{N}_f^{2}}{C_{\Omega^*}^2M^{\dagger}}\Bigg]^{l_0}\cdot |z_{\infty}|\cdot \|a\|.$$ Combining [\[eq9.24.\]](#eq9.24.){reference-type="eqref" reference="eq9.24."} with [\[eq9.23\]](#eq9.23){reference-type="eqref" reference="eq9.23"} and [\[eq9.24\]](#eq9.24){reference-type="eqref" reference="eq9.24"} we derive that $$\begin{aligned} \frac{\mathcal{J}_{\operatorname{Geo},\operatorname{Big}}^{\operatorname{Reg},\uppercase{\romannumeral 2\relax},\dagger}(f)}{\langle\phi',\phi'\rangle}\ll C^{\dag}T^{\frac{n(n+1)}{2}}(TMM')^{\varepsilon}\max_{1\leq l_0\leq n}\Bigg[\frac{M'^{2}\mathcal{N}_f^{2}}{C_{\Omega^*}^2M^{\dagger}}\Bigg]^{l_0}\int \frac{1}{\|a\|^{9}}\int |z_{\infty}|d^{\times}z_{\infty}d^{\times}a,\end{aligned}$$ where $a\in A_H(\mathbb{R})-\mathfrak{A}_{c,\varepsilon},$ and $C_{\Omega^*}^2M^{\dagger}M'^{-2}\mathcal{N}_f^{-2}a_1^{-1}\ll |z_{\infty}|\ll 1.$ Since $|z_{\infty}|d^{\times}=dz_{\infty}$ is the additive Haar measure, then $$\begin{aligned} \int |z_{\infty}|d^{\times}z_{\infty}\leq \int_0^{O(1)}dz_{\infty}\ll 1.\end{aligned}$$ In conjunction with $\int \|a\|^{-9}d^{\times}a\ll 1$ we obtain $$\begin{aligned} \frac{\mathcal{J}_{\operatorname{Geo},\operatorname{Big}}^{\operatorname{Reg},\uppercase{\romannumeral 2\relax},\dagger}(f)}{\langle\phi',\phi'\rangle}\ll (TMM')^{\varepsilon}T^{-8n^2}M'^{2n}M^n\mathcal{N}_f^{2(n-1)}\prod_{p\mid M'}p^{ne_p(M)}\max_{1\leq l_0\leq n}\Bigg[\frac{M'^{2}\mathcal{N}_f^{2}}{C_{\Omega^*}^2M^{\dagger}}\Bigg]^{l_0}.\end{aligned}$$ Therefore, [\[eq9.22\]](#eq9.22){reference-type="eqref" reference="eq9.22"} follows. ◻ ## Estimates in the Essential Range {#sec9.4} Recall $\mathcal{J}_{\operatorname{Geo},\operatorname{Big}}^{\operatorname{Reg},\uppercase{\romannumeral 2\relax},\heartsuit}(f)$ is defined by $$\begin{aligned} &\int_{\mathfrak{A}_{c,\varepsilon}} \int_{K'} \int_{\Omega^*}\big|\phi'(ak^*k)\big|^2\int_{G'(\mathbb{A})} \int_{Z'(\mathbb{A})}\sum_{\substack{(\boldsymbol{\xi},t)\in \mathfrak{X}^*(f;a,z_{\infty})}}\big|\mathfrak{F}_{\infty}\cdot\mathfrak{F}_{\operatorname{fin}}\big|d^{\times}zdydk^*dk\frac{d^{\times}a}{\delta_{B'}(a)},\end{aligned}$$ where $\mathfrak{F}_{\infty}$ and $\mathfrak{F}_{\operatorname{fin}}$ were defined in [9.2.2](#sec9.2.2){reference-type="ref" reference="sec9.2.2"}: $$\mathfrak{F}_{\infty}:=f_{\infty}\left(\iota(z_{\infty}ak_{\infty}^*k_{\infty})^{-1}\begin{pmatrix} I_{n-1}&&(1-t)\boldsymbol{\xi}\\ &1&t\\ &1&1 \end{pmatrix}\iota(z_{\infty}ak_{\infty}^*k_{\infty}y_{\infty})\right)\tag{\ref{55}}$$ and $\mathfrak{F}_{\operatorname{fin}}:=\prod_{p< \infty}\mathfrak{F}_p,$ with $$\mathfrak{F}_p:=f_p\left(\iota(z_pk_p^*k_p)^{-1}\begin{pmatrix} I_{n-1}&&(1-t)\boldsymbol{\xi}\\ &1&t\\ &1&1 \end{pmatrix}\iota(z_pk_p^*k_py_p)\right),\ \ p<\infty. \tag{\ref{58}}$$ The goal in this section is to estimate $\mathcal{J}_{\operatorname{Geo},\operatorname{Big}}^{\operatorname{Reg},\uppercase{\romannumeral 2\relax},\heartsuit}(f)$ as follows. **Proposition 57**. *Let notation be as before. Then $$\begin{aligned} \frac{\mathcal{J}_{\operatorname{Geo},\operatorname{Big}}^{\operatorname{Reg},\uppercase{\romannumeral 2\relax},\heartsuit}(f)}{\langle\phi',\phi'\rangle}\ll (TMM')^{\varepsilon}T^{\frac{n-1}{2}}M'^{2n}M^n\mathcal{N}_f^{2(n-1)}\prod_{p\mid M'}p^{ne_p(M)} \max_{1\leq l_0\leq n}\Bigg[\frac{M'^{2}\mathcal{N}_f^{2}}{M^{\dagger}}\Bigg]^{l_0}, \end{aligned}$$ where the implied constant depends on $\varepsilon,$ parameters $c_{\infty}$ and $C_{\infty}$ defined in [3.1.4](#sec3.14){reference-type="ref" reference="sec3.14"}, and the conductor of $\pi_{\infty}'.$* ### Separate Out the Automorphic Weight By Lemma [Lemma 24](#norm){reference-type="ref" reference="norm"} we have $$\label{9.14} \int_{K'}\int_{\Omega^*}\frac{\big|\phi'(ak^*k)\big|^2}{\langle\phi',\phi'\rangle}dk^*dk\ll T^{o(1)}\delta_{B'}(a)\min\Big\{\frac{|\det a|_{\infty}}{|a_1|_{\infty}^n},\frac{1}{|\det a|_{\infty}}\Big\}.$$ ### Further Reductions Swapping the integrals we obtain $$\begin{aligned} \mathcal{J}_{\operatorname{Geo},\operatorname{Big}}^{\operatorname{Reg},\uppercase{\romannumeral 2\relax},\heartsuit}(f)\ll \int_{\mathfrak{A}_{c,\varepsilon}}I(a)\Bigg[\max_{\substack{k^*\in\Omega^*\\ k\in K'}}\int_{G'(\mathbb{A})}\int_{Z'(\mathbb{A})}\sum_{\substack{(\boldsymbol{\xi},t)}}\big|\mathfrak{F}_{\infty}\cdot \mathfrak{F}_{\operatorname{fin}}\big|d^{\times}zdy\Bigg]\delta_{B'}^{-1}(a)d^{\times}a,\end{aligned}$$ where $\mathfrak{F}_{\infty}$ and $\mathfrak{F}_{\operatorname{fin}}$ are defined by [\[55\]](#55){reference-type="eqref" reference="55"} and [\[58\]](#58){reference-type="eqref" reference="58"}, $(\boldsymbol{\xi},t)\in \mathfrak{X}^*(f;a,z_{\infty})$, and $$I(a):=\int_{K'}\int_{\Omega^*}\big|\phi'(ak^*k)\big|^2dk^*dk.$$ Using [\[9.14\]](#9.14){reference-type="eqref" reference="9.14"} to bound $I(a),$ then $\mathcal{J}_{\operatorname{Geo},\operatorname{Big}}^{\operatorname{Reg},\uppercase{\romannumeral 2\relax},\heartsuit}(f)/\langle\phi',\phi'\rangle$ is $$\begin{aligned} \ll & T^{\varepsilon}\int_{\mathfrak{A}_{c,\varepsilon}}\Bigg[ \max_{\substack{k^*\in\Omega^*\\ k\in K'}}\int_{G'(\mathbb{R})}\int_{Z'(\mathbb{R})}\sum_{\substack{(\boldsymbol{\xi},t)}}|\mathfrak{F}_{\infty}|\prod_{p<\infty}\mathcal{J}_p^{\heartsuit}(\boldsymbol{\xi},t;k_{p}^*k_p)d^{\times}z_{\infty}dy_{\infty}\Bigg]h(a)d^{\times}a,\end{aligned}$$ where $(\boldsymbol{\xi},t)\in \mathfrak{X}^*(f;a,z_{\infty}),$ $\mathcal{J}_p^{\heartsuit}(\boldsymbol{\xi},t;k_{p}^*k_p)$ was defined in [9.4.1](#9.2.1){reference-type="ref" reference="9.2.1"}, and $$h(a):=\min\Big\{\frac{|\det a|_{\infty}}{|a_1|_{\infty}^n},\frac{1}{|\det a|_{\infty}}\Big\}.$$ Bounding $\mathcal{J}_p^{\heartsuit}(\boldsymbol{\xi},t;k_{p}^*k_p)$ by Lemma [Lemma 55](#lem9.7){reference-type="ref" reference="lem9.7"}, we then obtain that $$\begin{aligned} \mathcal{J}_{\operatorname{Geo},\operatorname{Big}}^{\operatorname{Reg},\uppercase{\romannumeral 2\relax},\heartsuit}(f)\ll &T^{\varepsilon}M'^{2n+\varepsilon}M^n\mathcal{N}_f^{2(n-1)}\langle\phi',\phi'\rangle\prod_{p\mid M'}p^{ne_p(M)} \int_{\mathfrak{A}_{c,\varepsilon}}\max_{\substack{k_{\infty}^*\in\Omega_{\infty}^*\\ k_{\infty}\in K_{\infty}'}}\int_{Z'(\mathbb{R})} \\ &\sum_{\substack{(\boldsymbol{\xi},t)\in \mathfrak{X}^*(f;a,z_{\infty})}}\Bigg[\int_{G'(\mathbb{R})}\big|\mathfrak{F}_{\infty}\big|dy_{\infty}\Bigg] h(a)\prod_{p<\infty}E_p(\boldsymbol{\xi},t)d^{\times}z_{\infty}d^{\times}a.\end{aligned}$$ ### Bounding the Inner Integral In this section we bound the inner integral $\int_{G'(\mathbb{R})}\big|\mathfrak{F}_{\infty}\big|dy_{\infty}$ as a function of $(\boldsymbol{\xi},t),$ $z_{\infty},$ $k_{\infty}^*,$ $k_{\infty}$ and $a,$ where $$\mathfrak{F}_{\infty}:=f_{\infty}\left(\iota(z_{\infty}ak_{\infty}^*k_{\infty})^{-1}\begin{pmatrix} I_{n-1}&&(1-t)\boldsymbol{\xi}\\ &1&t\\ &1&1 \end{pmatrix}\iota(z_{\infty}ak_{\infty}^*k_{\infty}y_{\infty})\right).\tag{\ref{55}}$$ Denote by $$\label{9.19.} g=\iota(z_{\infty}ak_{\infty}^*k_{\infty})^{-1}\begin{pmatrix} I_{n-1}&&(1-t)\boldsymbol{\xi}\\ &1&t\\ &1&1 \end{pmatrix}\iota(z_{\infty}ak_{\infty}^*k_{\infty})\in G(\mathbb{R}).$$ **Lemma 58**. *Let notation be as before. Then $$\begin{aligned} \int_{G'(\mathbb{R})}\big|\mathfrak{F}_{\infty}\big|dy_{\infty}\ll T^{\frac{n}{2}+\varepsilon}\cdot\min\Bigg\{1,\frac{T^{-1/2}}{d_{G'}(g)}\Bigg\},\end{aligned}$$ where $d_{G'}$ is defined by [\[dg\]](#dg){reference-type="eqref" reference="dg"}, $g$ is defined by [\[9.19.\]](#9.19.){reference-type="eqref" reference="9.19."}, and the implied constant depends on $\varepsilon,$ parameters $c_{\infty}$ and $C_{\infty}$ defined in [3.1.4](#sec3.14){reference-type="ref" reference="sec3.14"}, and the conductor of $\pi_{\infty}'.$* *Proof.* According to the support of $f_{\infty}$ (cf. [\[245\]](#245){reference-type="eqref" reference="245"}), $\mathfrak{F}_{\infty}=f_{\infty}(g\iota(y_{\infty}))=0$ unless $gy_{\infty}=I_{n+1}+o(1)$ (which implies that $y_{\infty}=I_n+o(1)$) and $\operatorname{dist}(gy_{\infty}\tau, \tau)\ll T^{-1/2+\varepsilon},$ where $\operatorname{dist}(\cdot)$ is defined in Proposition [Proposition 13](#prop3.1){reference-type="ref" reference="prop3.1"}. So $$\label{9.18.} \int_{G'(\mathbb{R})}\big|\mathfrak{F}_{\infty}\big|dy_{\infty}\ll \|f_{\infty}\|_{\infty}\int_{I_n+o(1)}\textbf{1}_{\operatorname{dist}(gy_{\infty}\tau, \tau)\ll T^{-1/2}}dy_{\infty}.$$ Write $y_{\infty}=\exp(\mathfrak{y})$ with $\mathfrak{y}$ ranges over a ball whose center relies on $g$ and whose radius is $O(T^{-1/2+\varepsilon}),$ with implied constant depending at most on $\varepsilon$. Therefore, $$\label{9.18..} \int_{G'(\mathbb{R})}\textbf{1}_{\operatorname{dist}(gy_{\infty}\tau, \tau)\ll T^{-1/2}}dy_{\infty}\ll \int d\mathfrak{y}\ll T^{-n^2/2+\varepsilon},$$ where the integral relative to $\mathfrak{y}$ is over a ball of radius $O(T^{-1/2+\varepsilon})$ and the bound $T^{-n^2/2+\varepsilon}$ follows from the fact that $\dim\operatorname{Lie}(G')=n^2.$ Let $\mathcal{Z}$ be the set defined in Proposition [Proposition 13](#prop3.1){reference-type="ref" reference="prop3.1"}. Changing variable $y_{\infty}\mapsto zy_{\infty}$ and swapping integrals we obtain $$\begin{aligned} \int\textbf{1}_{\operatorname{dist}(gy_{\infty}\tau, \tau)\ll T^{-1/2}}dy_{\infty}=\int \frac{1}{\operatorname{Vol}(\mathcal{Z})}\int_{\mathcal{Z}}\textbf{1}_{\operatorname{dist}(gy_{\infty}z\tau, \tau)\ll T^{-1/2}}dzdy_{\infty},\end{aligned}$$ where $y_{\infty}$ ranges over $I_n+o(1)\subset G'(\mathbb{R}).$ By mean value theorem, $$\begin{aligned} \int_{\mathcal{Z}}\textbf{1}_{\operatorname{dist}(gy_{\infty}z\tau, \tau)\ll T^{-1/2}}dz=\textbf{1}_{\operatorname{dist}(gy_{\infty}z_0\tau, \tau)\ll T^{-1/2}}\cdot \operatorname{Vol}(\mathcal{Z}_{gy_{\infty}})\end{aligned}$$ for some $z_0\in\mathcal{Z},$ where $\mathcal{Z}_{gy_{\infty}}:=\big\{z\in\mathcal{Z}:\ \operatorname{dist}(gz\tau, \tau)\ll T^{-1/2+\varepsilon}\big\}.$ Using Proposition [Proposition 13](#prop3.1){reference-type="ref" reference="prop3.1"} to bound $\operatorname{Vol}(\mathcal{Z}_{gy_{\infty}}),$ together with the change of variable $y_{\infty}\mapsto z_0^{-1}y_{\infty}$ we then derive that $$\begin{aligned} \int_{G'(\mathbb{R})}\textbf{1}_{\operatorname{dist}(gy_{\infty}\tau, \tau)\ll T^{-1/2}}dy_{\infty}\ll \frac{T^{-1/2}}{d_{G'}(g)}\int_{G'(\mathbb{R})}\textbf{1}_{\operatorname{dist}(gy_{\infty}\tau, \tau)\ll T^{-1/2}}dy_{\infty}.\end{aligned}$$ In conjunction with [\[9.18.\]](#9.18.){reference-type="eqref" reference="9.18."} and [\[9.18..\]](#9.18..){reference-type="eqref" reference="9.18.."} we then obtain that $$\begin{aligned} \int_{G'(\mathbb{R})}\big|\mathfrak{F}_{\infty}\big|dy_{\infty}\ll T^{\frac{n(n+1)}{2}+\varepsilon}\int\textbf{1}_{\operatorname{dist}(gy_{\infty}\tau, \tau)\ll T^{-1/2}}dy_{\infty}\ll T^{\frac{n}{2}+\varepsilon}\min\Bigg\{1,\frac{T^{-1/2}}{d_{G'}(g)}\Bigg\},\end{aligned}$$ where we also use the fact that $\|f_{\infty}\|_{\infty}\ll T^{n(n+1)/2+\varepsilon}$ (cf. [\[250\]](#250){reference-type="eqref" reference="250"}). ◻ ### Bounding the Outer Integrals Here we recall some arguments in the proof of Proposition [Proposition 56](#prop9.11.){reference-type="ref" reference="prop9.11."}. By [\[9.7\]](#9.7){reference-type="eqref" reference="9.7"} and the condition that $(\boldsymbol{\xi},t)\neq \textbf{0}$ we have $$\big|\mathfrak{X}^*(f;a,z_{\infty})\big|\ll \max\Bigg\{1,\frac{M'^2\mathcal{N}_f^2|z_{\infty}|}{C_{\Omega^*}^2M^{\dagger}}\Bigg\}\prod_{j=1}^{n-1}\max\Bigg\{1,\frac{M'^{2}\mathcal{N}_f^2|z_{\infty}|a_j}{C_{\Omega^*}^2M^{\dagger}}\Bigg\},\tag{\ref{9.7}}$$ and either $M'^2\mathcal{N}_f^2|z_{\infty}|\gg C_{\Omega^*}^2M^{\dagger}$ or $M'^{2}\mathcal{N}_f^2|z_{\infty}|a_1\gg C_{\Omega^*}^2M^{\dagger},$ depending on whether $t=0$ or not. Hence, in conjunction with the restriction that $|z_{\infty}|\ll 1,$ we have $$C_{\Omega^*}^2M^{\dagger}M'^{-2}\mathcal{N}_f^{-2}\textbf{1}_{t\neq 0}+C_{\Omega^*}^2M^{\dagger}M'^{-2}\mathcal{N}_f^{-2}a_1^{-1}\textbf{1}_{t=0}\ll |z_{\infty}|\ll 1.\tag{\ref{10.17}}$$ By definition of $\mathfrak{X}^*(f;a,z_{\infty}),$ we have, for $(\boldsymbol{\xi},t)\in \mathfrak{X}^*(f;a,z_{\infty}),$ that $$\begin{aligned} \prod_{p<\infty}E_p(\boldsymbol{\xi},t)\ll (TMM')^{\varepsilon}.\tag{\ref{eq9.24.}}\end{aligned}$$ The main results in this section are the following two lemmas. **Lemma 59**. *Let notation be as before. Then $$\label{lem9.14.} \int_{Z'(\mathbb{R})}\frac{T^{-1/2}}{d_{G'}(g)}\sum_{\substack{(\boldsymbol{\xi},t)\in \mathfrak{X}^*(f;a,z_{\infty})\\ \boldsymbol{\xi}=\textbf{0}}}h(a)\prod_{p<\infty}E_p(\boldsymbol{\xi},t)d^{\times}z_{\infty}\ll \frac{(TMM')^{\varepsilon}M'^2\mathcal{N}_f^2}{C_{\Omega^*}^2M^{\dagger}T^{1/2}},$$ where $g$ is defined by [\[9.19.\]](#9.19.){reference-type="eqref" reference="9.19."}, and the implied constant depends on $\varepsilon,$ parameters $c_{\infty}$ and $C_{\infty}$ defined in [3.1.4](#sec3.14){reference-type="ref" reference="sec3.14"}, and the conductor of $\pi_{\infty}'.$* *Proof.* Suppose $(\boldsymbol{\xi},t)\in \mathfrak{X}^*(f;a,z_{\infty})$ and $\boldsymbol{\xi}=\textbf{0}.$ Since $(\boldsymbol{\xi},t)\neq \textbf{0},$ then $t\neq 0,$ which implies that $|z_{\infty}|\gg C_{\Omega^*}^2M^{\dagger}M'^{-2}\mathcal{N}_f^{-2}.$ Moreover, by definition we have $$\label{9.22e} \sum_{\substack{(\boldsymbol{\xi},t)\in \mathfrak{X}^*(f;a,z_{\infty})\\ \boldsymbol{\xi}=\textbf{0}}}1\ll \max\Bigg\{1,\frac{M'^2\mathcal{N}_f^2|z_{\infty}|}{C_{\Omega^*}^2M^{\dagger}}\Bigg\}\ll \frac{M'^2\mathcal{N}_f^2|z_{\infty}|}{C_{\Omega^*}^2M^{\dagger}}.$$ Note that $h(a)\leq |\det a|_{\infty}^{-1}\ll T^{\varepsilon}$ for all $a\in \mathfrak{A}_{c,\varepsilon}.$ Combining [\[eq9.24.\]](#eq9.24.){reference-type="eqref" reference="eq9.24."} with [\[9.22e\]](#9.22e){reference-type="eqref" reference="9.22e"} we have $$\begin{aligned} \sum_{\substack{(\boldsymbol{\xi},t)\in \mathfrak{X}^*(f;a,z_{\infty})\\ \boldsymbol{\xi}=\textbf{0}}}h(a)\prod_{p<\infty}E_p(\boldsymbol{\xi},t) \ll \frac{M'^2\mathcal{N}_f^2|z_{\infty}|}{C_{\Omega^*}^2M^{\dagger}}\cdot (TMM')^{\varepsilon}.\end{aligned}$$ By definition (cf. [\[dg\]](#dg){reference-type="eqref" reference="dg"} and [\[9.19.\]](#9.19.){reference-type="eqref" reference="9.19."}) and the fact that $|z_{\infty}|\ll 1$, one has $$\begin{aligned} d_{G'}(g)\gg\min\big\{1,|z_{\infty}|\big\}\gg |z_{\infty}|. \end{aligned}$$ So the left hand side of [\[lem9.14.\]](#lem9.14.){reference-type="eqref" reference="lem9.14."} is $$\begin{aligned} \ll \frac{M'^2\mathcal{N}_f^2}{C_{\Omega^*}^2M^{\dagger}}\cdot (TMM')^{\varepsilon}\cdot \int \frac{T^{-1/2}}{z_{\infty}}dz_{\infty}\ll \frac{M'^2\mathcal{N}_f^2T^{-1/2}}{C_{\Omega^*}^2M^{\dagger}}\cdot (TMM')^{\varepsilon}(1+\log \nu(f)),\end{aligned}$$ where the range of $z_{\infty}$ is $C_{\Omega^*}^2M^{\dagger}M'^{-2}\mathcal{N}_f^{-2}\ll |z_{\infty}|\ll 1.$ Hence [\[lem9.14.\]](#lem9.14.){reference-type="eqref" reference="lem9.14."} follows from the assumption that $\log \nu(f)\asymp \log L\asymp \log T.$ ◻ **Lemma 60**. *Let notation be as before. Then $$\label{lem9.15.} \int \frac{T^{-\frac{1}{2}}}{d_{G'}(g)}\sum_{\substack{(\boldsymbol{\xi},t)\\ \boldsymbol{\xi}\neq \textbf{0}}}h(a)\prod_{p<\infty}E_p(\boldsymbol{\xi},t)d^{\times}z_{\infty}\ll \frac{(TMM')^{\varepsilon}(C_{\Omega^*}^2M^{\dagger}+M'^{2}\mathcal{N}_f^2)}{T^{\frac{1}{2}}(C_{\Omega^*}^2M^{\dagger})^{n}(M'^{2}\mathcal{N}_f^2)^{1-n}},$$ where $z_{\infty}\in Z'(\mathbb{R}),$ $(\boldsymbol{\xi},t)\in \mathfrak{X}^*(f;a,z_{\infty}),$ $g$ is defined by [\[9.19.\]](#9.19.){reference-type="eqref" reference="9.19."}, and the implied constant depends on $\varepsilon,$ parameters $c_{\infty}$ and $C_{\infty}$ defined in [3.1.4](#sec3.14){reference-type="ref" reference="sec3.14"}, and the conductor of $\pi_{\infty}'.$* *Proof.* Suppose $\boldsymbol{\xi}\neq \textbf{0}.$ Then $M'^{2}\mathcal{N}_f^2|z_{\infty}|a_1\gg C_{\Omega^*}^2M^{\dagger}.$ By Definition [Definition 53](#defn9.9){reference-type="ref" reference="defn9.9"} in [9.4](#sec9.3){reference-type="ref" reference="sec9.3"}, we have $T^{-\varepsilon}\ll |\det a|_{\infty}\ll T^{\varepsilon}$ and $\|a\|\ll T^c$ for $a\in \mathfrak{A}_{c,\varepsilon}.$ Recall that (cf. [9.2.1](#Sie){reference-type="ref" reference="Sie"}) $a_1\geq a_2\geq \cdots\geq a_{n-1}>0.$ Then $a_1^{n-1}\geq |\det a|_{\infty}\gg T^{-\varepsilon}.$ So $a_1^{-1}\ll T^{\varepsilon}.$ By $\|a\|\leq T^c$ we obtain $a_1\leq T^c.$ Hence, $$\label{eq9.35} T^{-c}\leq a_1^{-1}\ll T^{\varepsilon},$$ where $c=2c_1+10n^2$ and $c_1$ is the constant in Lemma [Lemma 25](#lem4.9){reference-type="ref" reference="lem4.9"}. Note that $c_1$ (and thus $c$) depends only on $n.$ Substituting $M'^{2}\mathcal{N}_f^2|z_{\infty}|a_1\gg C_{\Omega^*}^2M^{\dagger}$ into [\[9.7\]](#9.7){reference-type="eqref" reference="9.7"} we deduce that $$\begin{aligned} \sum_{\substack{(\boldsymbol{\xi},t)\in \mathfrak{X}^*(f;a,z_{\infty})\\ \boldsymbol{\xi}\neq \textbf{0}}}1\ll \max\Bigg\{1,\frac{M'^2\mathcal{N}_f^2|z_{\infty}|}{C_{\Omega^*}^2M^{\dagger}}\Bigg\}\cdot\Bigg[\frac{M'^{2}\mathcal{N}_f^2|z_{\infty}|}{C_{\Omega^*}^2M^{\dagger}}\Bigg]^{n-1}\cdot a_1^{n-1}.\end{aligned}$$ It then follows from the fact that $|z_{\infty}|\ll 1$ and $n-1\geq 1$ that $$\label{9.36} \sum_{\substack{(\boldsymbol{\xi},t)\in \mathfrak{X}^*(f;a,z_{\infty})\\ \boldsymbol{\xi}\neq \textbf{0}}}h(a)\prod_{p<\infty}E_p(\boldsymbol{\xi},t)\ll C\cdot |z_{\infty}|a_1^{n-1}\cdot \frac{|\det a|_{\infty}}{a_1^n}\ll CT^{\varepsilon}\cdot |z_{\infty}|,$$ where $C:=(TMM')^{\varepsilon}\cdot \max\big\{1,M'^2\mathcal{N}_f^2(C_{\Omega^*}^2M^{\dagger})^{-1}\big\}\cdot\big[M'^{2}\mathcal{N}_f^2(C_{\Omega^*}^2M^{\dagger})^{-1}\big]^{n-1}.$ By $M'^{2}\mathcal{N}_f^2|z_{\infty}|a_1\gg C_{\Omega^*}^2M^{\dagger}$ one has $|z_{\infty}|\gg C_{\Omega^*}^2M^{\dagger}(M'^{2}\mathcal{N}_f^2)^{-1}a_1^{-1}.$ Hence, $$\label{9.35} C_{\Omega^*}^2M^{\dagger}(M'^{2}\mathcal{N}_f^2)^{-1}a_1^{-1}\ll |z_{\infty}|\ll 1.$$ Note also that $d_{G'}(g)\gg\min\big\{1,|z_{\infty}|\big\}\gg |z_{\infty}|.$ Therefore, by [\[eq9.35\]](#eq9.35){reference-type="eqref" reference="eq9.35"}, [\[9.36\]](#9.36){reference-type="eqref" reference="9.36"}, and [\[9.35\]](#9.35){reference-type="eqref" reference="9.35"}, the left hand side of [\[lem9.15.\]](#lem9.15.){reference-type="eqref" reference="lem9.15."} is $$\begin{aligned} \ll CT^{\varepsilon}\int\frac{T^{-\frac{1}{2}}}{z_{\infty}}dz_{\infty}\ll (TMM')^{\varepsilon}T^{-\frac{1}{2}}\max\Bigg\{1,\frac{M'^2\mathcal{N}_f^2}{C_{\Omega^*}^2M^{\dagger}}\Bigg\}\cdot\Bigg[\frac{M'^{2}\mathcal{N}_f^2}{C_{\Omega^*}^2M^{\dagger}}\Bigg]^{n-1},\end{aligned}$$ where the range of $z_{\infty}$ is $C_{\Omega^*}^2M^{\dagger}(M'^{2}\mathcal{N}_f^2)^{-1}T^{-c}\ll |z_{\infty}|\ll 1.$ ◻ ### Bounding $\mathcal{J}_{\operatorname{Geo},\operatorname{Big}}^{\operatorname{Reg},\uppercase{\romannumeral 2\relax},\heartsuit}(f)$ By definition we have $\int_{\mathfrak{A}_{c,\varepsilon}} d^{\times}a\ll T^{\varepsilon},$ and $$\begin{aligned} \frac{(C_{\Omega^*}^2M^{\dagger}+M'^{2}\mathcal{N}_f^2)}{(C_{\Omega^*}^2M^{\dagger})^{n}(M'^{2}\mathcal{N}_f^2)^{1-n}}\ll \frac{M'^2\mathcal{N}_f^2}{C_{\Omega^*}^2M^{\dagger}}+\frac{(M'^{2}\mathcal{N}_f^2)^n}{(C_{\Omega^*}^2M^{\dagger})^{n}}\ll \max_{1\leq l_0\leq n}\Bigg[\frac{M'^{2}\mathcal{N}_f^{2}}{M^{\dagger}}\Bigg]^{l_0}.\end{aligned}$$ By Lemmas [Lemma 58](#lem9.13.){reference-type="ref" reference="lem9.13."}, [Lemma 59](#lem9.14){reference-type="ref" reference="lem9.14"}, and [Lemma 60](#lem9.15){reference-type="ref" reference="lem9.15"}, noting that the implied constants therein are independent of $k_{\infty}^*,$ $k_{\infty},$ or $a,$ we obtain that $$\begin{aligned} \frac{\mathcal{J}_{\operatorname{Geo},\operatorname{Big}}^{\operatorname{Reg},\uppercase{\romannumeral 2\relax},\heartsuit}(f)}{\langle\phi',\phi'\rangle}\ll (TMM')^{\varepsilon}T^{\frac{n-1}{2}}M'^{2n}M^n\mathcal{N}_f^{2(n-1)}\prod_{p\mid M'}p^{ne_p(M)} \max_{1\leq l_0\leq n}\Bigg[\frac{M'^{2}\mathcal{N}_f^{2}}{M^{\dagger}}\Bigg]^{l_0},\end{aligned}$$ where the implied constant depends on $\varepsilon,$ parameters $c_{\infty}$ and $C_{\infty}$ defined in [3.1.4](#sec3.14){reference-type="ref" reference="sec3.14"}, and the conductor of $\pi_{\infty}'.$ Hence Proposition [Proposition 57](#prop9.12){reference-type="ref" reference="prop9.12"} follows. ## Proof of Proposition [Proposition 48](#prop9.3.){reference-type="ref" reference="prop9.3."} {#sec9.5} Combining the estimate $$\mathcal{J}_{\operatorname{Geo},\operatorname{Big}}^{\operatorname{Reg},\uppercase{\romannumeral 2\relax}}(f)\ll \mathcal{J}_{\operatorname{Geo},\operatorname{Big}}^{\operatorname{Reg},\uppercase{\romannumeral 2\relax},\dagger}(f)+\mathcal{J}_{\operatorname{Geo},\operatorname{Big}}^{\operatorname{Reg},\uppercase{\romannumeral 2\relax},\heartsuit}(f)$$ with Propositions [Proposition 56](#prop9.11.){reference-type="ref" reference="prop9.11."} and [Proposition 57](#prop9.12){reference-type="ref" reference="prop9.12"} we then derive that $$\begin{aligned} \frac{\mathcal{J}_{\operatorname{Geo},\operatorname{Big}}^{\operatorname{Reg},\uppercase{\romannumeral 2\relax}}(f)}{\langle\phi',\phi'\rangle}\ll (TMM')^{\varepsilon}T^{\frac{n-1}{2}}M'^{2n}M^n\mathcal{N}_f^{2(n-1)}\prod_{p\mid M'}p^{ne_p(M)} \max_{1\leq l_0\leq n}\Bigg[\frac{M'^{2}\mathcal{N}_f^{2}}{M^{\dagger}}\Bigg]^{l_0},\end{aligned}$$ where the implied constant depends on $\varepsilon,$ parameters $c_{\infty}$ and $C_{\infty}$ defined in [3.1.4](#sec3.14){reference-type="ref" reference="sec3.14"}, and the conductor of $\pi_{\infty}'.$ Note that $f_{p}^{-1}=f_p$ for all $p\nmid M'\nu(f)$ or $p=p_0.$ Also, $f_{p_1}^{-1}=f_{p_2},$ $f_{p_2}^{-1}=f_{p_1},$ and $f_{\infty}^{-1}$ has a similar support and sup-norm as that of $f.$ Therefore, we may repeat the arguments in [9.3](#sec9.2){reference-type="ref" reference="sec9.2"}, [9.4](#sec9.3){reference-type="ref" reference="sec9.3"} and [9.5](#sec9.4){reference-type="ref" reference="sec9.4"} with the test function $f$ replaced by $f^{-1}$ to obtain similarly that $$\begin{aligned} \frac{\mathcal{J}_{\operatorname{Geo},\operatorname{Big}}^{\operatorname{Reg},\uppercase{\romannumeral 2\relax}}(f^{-1})}{\langle\phi',\phi'\rangle}\ll (TMM')^{\varepsilon}T^{\frac{n-1}{2}}M'^{2n}M^n\mathcal{N}_f^{2(n-1)}\prod_{p\mid M'}p^{ne_p(M)} \max_{1\leq l_0\leq n}\Bigg[\frac{M'^{2}\mathcal{N}_f^{2}}{M^{\dagger}}\Bigg]^{l_0}.\end{aligned}$$ Here we notice that $\mathcal{N}_{f^{-1}}=\mathcal{N}_f.$ Then Proposition [Proposition 48](#prop9.3.){reference-type="ref" reference="prop9.3."} follows. # Applications to the Subconvexity Problem {#proof} ## The Spectral Side {#the-spectral-side-1} The spectral side $\mathcal{J}_{\mathop{\mathrm{Spec}}}^{\heartsuit}(\boldsymbol{\alpha},\boldsymbol{\ell})$ has been handled in [5](#sec5){reference-type="ref" reference="sec5"}. Here we recall the lower bound of $\mathcal{J}_{\mathop{\mathrm{Spec}}}^{\heartsuit}(\boldsymbol{\alpha},\boldsymbol{\ell})$ as follows. ## The Geometric Side {#sec11.2} Recall that (cf. [3.8.2](#sec3.7.2){reference-type="ref" reference="sec3.7.2"}) the geometric side of the amplified relative trace formula $\mathcal{J}_{\operatorname{Geo}}^{\heartsuit}(\boldsymbol{\alpha},\boldsymbol{\ell})$ is defined by $$\begin{aligned} \sum_{p_1\neq p_2\in\mathcal{L}}\alpha_{p_1}\overline{\alpha_{p_2}}J_{\operatorname{Geo}}^{\operatorname{Reg},\heartsuit}(f(\cdot;p_1,p_2),\mathbf{0})+\sum_{p_0\in\mathcal{L}}\sum_{i=0}^{l_{p_0}}c_{p_0,i}|\alpha_{p_0}|^2J_{\operatorname{Geo}}^{\operatorname{Reg},\heartsuit}(f(\cdot;i,p_0),\mathbf{0}),\end{aligned}$$ where for any test function $f$ the geometric side (cf. [\[63\]](#63){reference-type="eqref" reference="63"}) $$\begin{aligned} J^{\operatorname{Reg},\heartsuit}_{\operatorname{Geo}}(f,\textbf{s}):=J^{\operatorname{Reg}}_{\operatorname{Geo},\operatorname{Small}}(f,\textbf{s})+J_{\operatorname{Geo},\operatorname{Dual}}^{\operatorname{Big}}(f,\textbf{s})-J^{\operatorname{Reg},\uppercase{\romannumeral 1\relax},\heartsuit}_{\operatorname{Geo},\operatorname{Big}}(f,\textbf{s})+J^{\operatorname{Reg},\uppercase{\romannumeral 2\relax}}_{\operatorname{Geo},\operatorname{Big}}(f,\textbf{s})\end{aligned}$$ is an entire function (though each individual term might be just meromorphic). **Proposition 61**. *Let notation be as before. Then $$\label{eq10.1} J_{\operatorname{Geo}}^{\operatorname{Reg},\heartsuit}(f,\textbf{s})=J_{\operatorname{Geo},\operatorname{Main}}^{\operatorname{Small}}(f,\textbf{s})+J_{\operatorname{Geo},\operatorname{Main}}^{\operatorname{Dual}}(f,\textbf{s})+J^{\operatorname{Reg},\uppercase{\romannumeral 2\relax}}_{\operatorname{Geo},\operatorname{Big}}(f,\textbf{s}),$$ where each term on the RHS is holomorphic near $\mathbf{s}=(0,0).$ Here $J_{\operatorname{Geo},\operatorname{Main}}^{\operatorname{Small}}(f,\textbf{s})$ is defined by [\[7.1\'\]](#7.1'){reference-type="eqref" reference="7.1'"} in [7](#8.5.1){reference-type="ref" reference="8.5.1"}, $J_{\operatorname{Geo},\operatorname{Main}}^{\operatorname{Dual}}(f,\textbf{s})$ is defined by [\[8.1\'\]](#8.1'){reference-type="eqref" reference="8.1'"} in [8](#8.5.2){reference-type="ref" reference="8.5.2"}, and $J^{\operatorname{Reg},\uppercase{\romannumeral 2\relax}}_{\operatorname{Geo},\operatorname{Big}}(f,\textbf{s})$ is defined in [2.2.3](#2.2.3){reference-type="ref" reference="2.2.3"}.* *Proof.* We follow the notation in [2.2.1](#2.2.1){reference-type="ref" reference="2.2.1"}. Recall that $J_{\operatorname{Geo}}^{\operatorname{Reg},\heartsuit}(f,\textbf{s})$ is equal to $$J^{\operatorname{Reg}}_{\operatorname{Geo},\operatorname{Small}}(f,\textbf{s})+J_{\operatorname{Geo},\operatorname{Dual}}^{\operatorname{Big}}(f,\textbf{s})-J^{\operatorname{Reg},\uppercase{\romannumeral 1\relax},\heartsuit}_{\operatorname{Geo},\operatorname{Big}}(f,\textbf{s})+J^{\operatorname{Reg},\uppercase{\romannumeral 2\relax}}_{\operatorname{Geo},\operatorname{Big}}(f,\textbf{s})\tag{\ref{63}}.$$ By Lemma [Lemma 14](#lem3.2){reference-type="ref" reference="lem3.2"} we have $J^{\operatorname{Reg},\uppercase{\romannumeral 1\relax},\heartsuit}_{\operatorname{Geo},\operatorname{Big}}(f,\textbf{s})\equiv 0.$ Moreover, $$\underset{s=0}{\operatorname{Res}}\ J^{\operatorname{Reg}}_{\operatorname{Geo},\operatorname{Small}}(f,\textbf{s})+\underset{s=0}{\operatorname{Res}}\ J^{\operatorname{Reg}}_{\operatorname{Geo},\operatorname{Dual}}(f,\textbf{s})=0.\tag{\ref{eq10.2}}$$ Therefore, [\[eq10.1\]](#eq10.1){reference-type="eqref" reference="eq10.1"} follows from [\[63\]](#63){reference-type="eqref" reference="63"} and [\[eq10.2\]](#eq10.2){reference-type="eqref" reference="eq10.2"}. ◻ Combining Propositions [Proposition 33](#prop54){reference-type="ref" reference="prop54"}, [\[8.1\]](#8.1){reference-type="ref" reference="8.1"}, [Proposition 61](#prop12.1){reference-type="ref" reference="prop12.1"}, and Theorem [Theorem 45](#thm9.1){reference-type="ref" reference="thm9.1"}, we obtain the following corollary. **Corollary 62**. *Let notation be as before. Let $f=\big\{f(\cdot ;i,p_0),f(\cdot;p_1,p_2)\big\}$ (cf. [3.6](#testfunction){reference-type="ref" reference="testfunction"}). Then $$\begin{aligned} \frac{J_{\operatorname{Geo}}^{\operatorname{Reg},\heartsuit}(f,\mathbf{0})}{\langle\phi',\phi'\rangle}&\ll T^{\frac{n}{2}+\varepsilon}M'^{2n}M^{n+\varepsilon} \mathcal{N}_f^{-1+2\vartheta_p+\varepsilon}\prod_{p\mid M'}p^{ne_p(M)}\\ &+(TMM')^{\varepsilon}T^{\frac{n-1}{2}}M'^{2n}M^n\mathcal{N}_f^{2(n-1)}\prod_{p\mid M'}p^{ne_p(M)} \max_{1\leq l_0\leq n}\Bigg[\frac{M'^{2}\mathcal{N}_f^{2}}{M^{\dagger}}\Bigg]^{l_0},\end{aligned}$$ where the implied constant depends on $\varepsilon,$ parameters $c_{\infty}$ and $C_{\infty}$ defined in [3.1.4](#sec3.14){reference-type="ref" reference="sec3.14"}, and the conductor of $\pi_{\infty}'.$* ## An Upper Bound of the Geometric Side: the General Case ### Choice of the set $\mathcal{L}$ {#sec11.3.1} Let $0<\beta<1.$ Recall that (cf. [3.1](#3.1){reference-type="ref" reference="3.1"}) we have $\log L\asymp \log T.$ By Rankin-Selberg theory we have $$\begin{aligned} \frac{L}{\log L}\gg\sum_{\substack{L<p\leq 2L}}|\lambda_{\pi'}(p)|^2\geq\sum_{\substack{L<p\leq 2L,\ p\nmid MM'\\ |\lambda_{\pi'}(p)|\geq p^{\beta}}}|\lambda_{\pi'}(p)|^2\geq p^{2\beta} \sum_{\substack{L<p\leq 2L,\ p\nmid MM'\\ |\lambda_{\pi'}(p)|\geq p^{\beta}}}1,\end{aligned}$$ where $p$ ranges over primes. As a consequence, we have $$\label{11.4} \sum_{\substack{L<p\leq 2L,\ p\nmid MM',\ |\lambda_{\pi'}(p)|\geq p^{\beta}}}1\ll \frac{L^{1-2\beta}}{\log L},$$ where the implied constant depends on the analytic conductor of $\pi'.$ Define $$\label{L} \mathcal{L}:=\big\{\text{prime $p$}:\ L<p\leq 2L,\ p\nmid MM',\ |\lambda_{\pi'}(p)|\leq \exp(\sqrt{\log L})\big\}.$$ By prime number theorem and [\[11.4\]](#11.4){reference-type="eqref" reference="11.4"} (with $\beta=1/\sqrt{\log L}$) we have $\#\mathcal{L}\asymp L/\log L.$ Moreover, for $p\in\mathcal{L},$ one has $\vartheta_p\leq 1/\sqrt{\log L}.$ ### Choice of $\boldsymbol{\ell}$ and $\boldsymbol{\alpha}$ {#alpha} Let $\mu \in i\mathfrak{a}_{Q}^*/i\mathfrak{a}_G^*$ be defined in Theorem [\[prop11.2\]](#prop11.2){reference-type="ref" reference="prop11.2"}. Recall that $\lambda_{\pi_{\mu}}(p^l)$ is the eigenvalue of $T_{p^l}$ acting on $\pi_{1,p}|\cdot|_p^{\mu_1}\boxplus\cdots\boxplus\pi_{m,p}|\cdot|_p^{\mu_m},$ $p\in\mathcal{L},$ $l\geq 0.$ By Hecke relations (cf. [@BM15], Cor 4.3) one has $$\begin{aligned} \sum_{j=1}^{n+1}|\lambda_{\pi_{\mu}}(p^l)|\gg 1,\ \ p\nmid M.\end{aligned}$$ By the pigeonhole principle, there exists, for each $p\in\mathcal{L},$ $l_p\in\{1,\cdots,n+1\}$ such that $|\lambda_{\pi_{\mu}}(p^l)|\gg 1.$ We henceforth fix a choice of $l_p$ for each $p\in\mathcal{L}.$ Note that $l_p$ depends on $\pi_p$. Let $\alpha_p:=\overline{\lambda_{\pi_{\mu}}(p^{l_p})}/|\overline{\lambda_{\pi_{\mu}}(p^{l_p})}|$ be the sign of $\overline{\lambda_{\pi_{\mu}}(p^{l_p})},$ $p\in\mathcal{L}.$ ### Majorization of $\mathcal{J}_{\operatorname{Geo}}^{\heartsuit}(\boldsymbol{\alpha},\boldsymbol{\ell})$ **Proposition 63**. *Let notation be as before. Let $\mathcal{L}$ be defined by [\[L\]](#L){reference-type="eqref" reference="L"} and the sequences $\boldsymbol{\ell}$ and $\boldsymbol{\alpha}$ be defined in [10.3.2](#alpha){reference-type="ref" reference="alpha"}. Let $\phi'\in \pi'$ be defined by Definition [Definition 17](#def4.2){reference-type="ref" reference="def4.2"}. Suppose $L=T^{\delta}$ with $\delta\in (0, 2).$ Then $$\begin{aligned} \frac{\mathcal{J}_{\operatorname{Geo}}^{\heartsuit}(\boldsymbol{\alpha},\boldsymbol{\ell})}{\langle\phi',\phi'\rangle T^{\frac{n}{2}}}\ll (T M)^{\varepsilon}(M^n L+ T^{-\frac{1}{2}} M^{n-1}L^{2n(n+1)+2}+T^{-\frac{1}{2}} L^{(4n-2)(n+1)+2}), \end{aligned}$$ where the implied constant depends on $\varepsilon,$ parameters $c_{\infty}$ and $C_{\infty}$ defined in [3.1.4](#sec3.14){reference-type="ref" reference="sec3.14"}, and the conductor of $\pi'.$* *Proof.* For $f\in \big\{f(\cdot;i,p_0), f(\cdot;p_1,p_2)\big\}$ (cf. [3.6.2](#3.6.2){reference-type="ref" reference="3.6.2"}), Corollary [Corollary 62](#cor11.3){reference-type="ref" reference="cor11.3"} yields that $$\label{10.4} \frac{J_{\operatorname{Geo}}^{\operatorname{Reg},\heartsuit}(f,\mathbf{0})}{\langle\phi',\phi'\rangle}\ll T^{\frac{n}{2}+\varepsilon}M^{n+\varepsilon} \mathcal{N}_f^{-1+\varepsilon}+(TM)^{\varepsilon}T^{\frac{n-1}{2}}(M^{n-1}\mathcal{N}_f^{2n}+\mathcal{N}_f^{4n-2}),$$ where the implied constant depends on $\varepsilon,$ $c_{\infty},$ $C_{\infty}$ and the conductor of $\pi'.$ Here we make use of the construction in [10.3.1](#sec11.3.1){reference-type="ref" reference="sec11.3.1"}, namely, for $p\in\mathcal{L},$ $\vartheta_p\leq 1/\sqrt{\log L}.$ Substituting the estimate [\[10.4\]](#10.4){reference-type="eqref" reference="10.4"} into the definition of $\mathcal{J}_{\operatorname{Geo}}^{\heartsuit}(\boldsymbol{\alpha},\boldsymbol{\ell}),$ i.e., $$\begin{aligned} \sum_{p_1\neq p_2\in\mathcal{L}}\alpha_{p_1}\overline{\alpha_{p_2}}J_{\operatorname{Geo}}^{\operatorname{Reg},\heartsuit}(f(\cdot;p_1,p_2),\mathbf{0})+\sum_{p_0\in\mathcal{L}}\sum_{i=0}^{l_{p_0}}c_{p_0,i}|\alpha_{p_0}|^2J_{\operatorname{Geo}}^{\operatorname{Reg},\heartsuit}(f(\cdot;i,p_0),\mathbf{0}),\end{aligned}$$ noting that $\mathcal{N}_f\ll L^{n+1},$ then Proposition [Proposition 63](#cor11.4){reference-type="ref" reference="cor11.4"} follows. ◻ ## Put It All Together {#sec11.4} In this section we prove the following main result and establish Theorem [Theorem 2](#B){reference-type="ref" reference="B"} as a consequence. **Theorem 6**. *Let $n\geq 2.$ Let $\pi=\pi_1\boxplus\cdots\boxplus\pi_m$ be a unitary pure isobaric automorphic representation of $\mathrm{GL}(n+1)/\mathbb{Q}.$ Suppose that $\pi$ has uniform parameter growth of size $(T;c_{\infty},C_{\infty}).$ Let $\pi'$ be a unitary cuspidal representation of $\mathrm{GL}(n)/\mathbb{Q}.$ Let $$\begin{aligned} \mathbf{L}:= \begin{cases} M^{\frac{n}{2\cdot (4n^2+2n-1)}}T^{\frac{1}{4\cdot (4n^2+2n-1)}},\ \ & \text{if $M\leq T^{\frac{n+1}{2n^2-1}}$,}\\ M^{\frac{1}{4n^2+4n+2}}T^{\frac{1}{4\cdot (2n^2+2n+1)}},\ \ & \text{if $M>T^{\frac{n+1}{2n^2-1}}.$} \end{cases}\end{aligned}$$ Then we have $$\label{nb} \frac{L(1/2,\pi\times\pi')}{\sqrt{\langle\phi',\phi'\rangle }\prod_{j=1}^m\sqrt{L(1,\pi_j,\operatorname{Ad})}}\ll T^{\frac{n(n+1)}{4}+\varepsilon}M^{\frac{n}{2}+\varepsilon}\mathbf{L}^{-1},$$ where the implied constant depends on $\varepsilon,$ parameters $c_{\infty}$ and $C_{\infty}$ defined in [3.1.4](#sec3.14){reference-type="ref" reference="sec3.14"}, and the conductor of $\pi'.$* *Proof.* From the choice of $\boldsymbol{\ell}$ and $\boldsymbol{\alpha}$ (cf. [10.3.2](#alpha){reference-type="ref" reference="alpha"}) we have $$\begin{aligned} \sum_{p\in \mathcal{L}}\alpha_p\lambda_{\pi_{\mu}}(p^{l_p})=\sum_{p\in \mathcal{L}}\big|\lambda_{\pi_{\mu}}(p^{l_p})\big|\gg \#\mathcal{L}\gg \frac{L}{\log L}.\end{aligned}$$ Hence it follows from Theorem [\[prop11.2\]](#prop11.2){reference-type="ref" reference="prop11.2"} and Proposition [Proposition 63](#cor11.4){reference-type="ref" reference="cor11.4"} that $$\begin{aligned} \frac{\big|L(1/2,\pi\times\pi')\big|^2}{\langle\phi',\phi'\rangle T^{\frac{n(n+1)}{2}+\varepsilon}}\prod_{j=1}^m\frac{1}{L(1,\pi_j,\operatorname{Ad})} \ll\frac{M^{n+\varepsilon}}{L^{1+\varepsilon}}+ \frac{L^{4n^2+2n-2+\varepsilon}+ M^{n-1}L^{2n(n+1)+\varepsilon}}{T^{\frac{1}{2}}}.\end{aligned}$$ Then [\[nb\]](#nb){reference-type="eqref" reference="nb"} follows from optimizing the parameter $L$. ◻ By Lemma [Lemma 21](#lem4.6){reference-type="ref" reference="lem4.6"} and Theorem 2 in [@Li10], $\langle\phi',\phi'\rangle=O(T^{\varepsilon}).$ Hence Theorem [Theorem 2](#B){reference-type="ref" reference="B"} holds. *Remark 64*. It is observable from Corollary [Corollary 62](#cor11.3){reference-type="ref" reference="cor11.3"} that the implied constant in Theorem [Theorem 2](#B){reference-type="ref" reference="B"} have polynomial dependence in the conductor of $\pi'$. ## An Upper Bound of the Geometric Side: the $t$-aspect {#sec12.5} ### Choice of $\mathcal{L}$, $\boldsymbol{\alpha}$ and $\boldsymbol{\ell}$ {#12.4.1} Construct the set $\mathcal{L}$ and the sequence $\boldsymbol{\alpha}$ as in [10.3.1](#sec11.3.1){reference-type="ref" reference="sec11.3.1"} and [10.3.2](#alpha){reference-type="ref" reference="alpha"}, respectively. Let $l_p=1$ for all $p\in\mathcal{L},$ which is the only difference from the construction in [10.3.2](#alpha){reference-type="ref" reference="alpha"}. ### Majorization of $\mathcal{J}_{\operatorname{Geo}}^{\heartsuit}(\boldsymbol{\alpha},\boldsymbol{\ell})$ Substituting the estimate $$\frac{J_{\operatorname{Geo}}^{\operatorname{Reg},\heartsuit}(f,\mathbf{0})}{\langle\phi',\phi'\rangle}\ll T^{\frac{n}{2}+\varepsilon}M^{n+\varepsilon} \mathcal{N}_f^{-1+\varepsilon}+(TM)^{\varepsilon}T^{\frac{n-1}{2}}(M^{n-1}\mathcal{N}_f^{2n}+\mathcal{N}_f^{4n-2})\tag{\ref{10.4}}$$ into the definition of $\mathcal{J}_{\operatorname{Geo}}^{\heartsuit}(\boldsymbol{\alpha},\boldsymbol{\ell}),$ we the obtain the following analogue of Proposition [Proposition 63](#cor11.4){reference-type="ref" reference="cor11.4"}. **Proposition 65**. *Let notation be as before. Let $\mathcal{L},$ $\boldsymbol{\ell}$ and $\boldsymbol{\alpha}$ be defined in [10.5.1](#12.4.1){reference-type="ref" reference="12.4.1"}. Let $\phi'\in \pi'$ be defined by Definition [Definition 17](#def4.2){reference-type="ref" reference="def4.2"}. Suppose $L=T^{\delta}$ with $\delta\in (0, 2).$ Then $$\begin{aligned} \frac{\mathcal{J}_{\operatorname{Geo}}^{\heartsuit}(\boldsymbol{\alpha},\boldsymbol{\ell})}{\langle\phi',\phi'\rangle T^{\frac{n}{2}}}\ll &T^{\varepsilon} M^{n+\varepsilon} L+ T^{-\frac{1}{2}+\varepsilon} M^{n-1}L^{2n+2}+T^{-\frac{1}{2}+\varepsilon} L^{4n}, \end{aligned}$$ where the implied constant depends on $\varepsilon,$ parameters $c_{\infty}$ and $C_{\infty}$ defined in [3.1.4](#sec3.14){reference-type="ref" reference="sec3.14"}, and the conductor of $\pi'.$* ### Proof of Theorem [Theorem 1](#A){reference-type="ref" reference="A"}: Tempered Case {#proof-of-theorem-a-tempered-case} The estimate [\[1.4\*\]](#1.4*){reference-type="eqref" reference="1.4*"} in Theorem [Theorem 1](#A){reference-type="ref" reference="A"} follows readily from the following. **Theorem 7**. *Let notation be as before. Suppose that $\pi$ is tempered. Then $$\label{nb.} L(1/2+it,\pi\times\pi')\ll_{\varepsilon,\pi,\pi'} (1+|t|)^{\frac{n(n+1)}{4}-\frac{1}{4\cdot(4n-1)}+\varepsilon},$$ where the implied constant relies on $\varepsilon,$ and the conductors of $\pi$ and $\pi'.$* *Proof.* Let $\sigma=\pi|\cdot|^{it}.$ Let $T=1+|t|.$ Suppose $L>M.$ Let $\mathcal{L}$ be defined in [10.3.1](#sec11.3.1){reference-type="ref" reference="sec11.3.1"}. Since $\pi$ is tempered, then $|\lambda_{\pi}(p)|\ll 1$ if $p\nmid M.$ Hence $|\lambda_{\sigma_{\mu}}(p)|=|\lambda_{\pi_{\mu}}(p)\cdot p^{it}|\ll 1$ for all rational primes $p\nmid M,$ where $\sigma_{\mu}$ was defined by [\[5.1.\]](#5.1.){reference-type="eqref" reference="5.1."} and [\[eq5.3\]](#eq5.3){reference-type="eqref" reference="eq5.3"}. By Cauchy-Schwarz inequality we have $$\label{12.6} \Big[\sum_{p\in \mathcal{L}}\big|\lambda_{\sigma_{\mu}}(p)\big|^2\Big]^2\leq \Big[\sum_{p\in \mathcal{L}}\big|\lambda_{\sigma_{\mu}}(p)\big|\Big]\cdot \Big[\sum_{p\in \mathcal{L}}\big|\lambda_{\sigma_{\mu}}(p)\big|^3\Big].$$ Note that $|\lambda_{\sigma_{\mu}}(p)|=|\lambda_{\pi_{\mu}}(p)|.$ Then by Rankin-Selberg convolution, $$\begin{aligned} \sum_{p\sim L}\big|\lambda_{\sigma_{\mu}}(p)\big|^2=\sum_{p\sim L}\big|\lambda_{\pi_{\mu}}(p)\big|^2\asymp \frac{L}{\log L},\end{aligned}$$ where the implied constant depends on the conductor of $\pi.$ Along with the estimate [\[11.4\]](#11.4){reference-type="eqref" reference="11.4"} in [10.3.1](#sec11.3.1){reference-type="ref" reference="sec11.3.1"}, and $|\lambda_{\sigma_{\mu}}(p)|\ll 1,$ we obtain $$\begin{aligned} \sum_{p\in \mathcal{L}}\big|\lambda_{\sigma_{\mu}}(p)\big|^2=\sum_{p\sim L}\big|\lambda_{\sigma_{\mu}}(p)\big|^2-\sum_{p\not\in \mathcal{L}}\big|\lambda_{\sigma_{\mu}}(p)\big|^2\asymp \frac{L}{\log L}\cdot (1+O(\exp(-2\sqrt{\log L}))).\end{aligned}$$ Hence, by [\[12.6\]](#12.6){reference-type="eqref" reference="12.6"}, and the estimate that $$\begin{aligned} \sum_{p\in \mathcal{L}}\big|\lambda_{\sigma_{\mu}}(p)\big|^3\ll \sum_{p\in \mathcal{L}}1\ll \sum_{p\sim L}1\asymp \frac{L}{\log L},\end{aligned}$$ we have $$\begin{aligned} \sum_{p\in \mathcal{L}}\big|\lambda_{\sigma_{\mu}}(p)\big|\gg \frac{L}{\log L}. \end{aligned}$$ It then follows from Theorem [\[prop11.2\]](#prop11.2){reference-type="ref" reference="prop11.2"} and Proposition [Proposition 65](#prop12.5){reference-type="ref" reference="prop12.5"} that $$\begin{aligned} \frac{\big|L(1/2,\sigma\times\pi')\big|^2}{\langle\phi',\phi'\rangle T^{\frac{n(n+1)}{2}+\varepsilon}}\prod_{j=1}^m\frac{1}{L(1,\pi_j,\operatorname{Ad})} \ll & M^{n+\varepsilon} L^{-1+\varepsilon}+T^{-\frac{1}{2}} (L^{4n-2}+ M^{n-1}L^{2n})L^{\varepsilon},\end{aligned}$$ which is $\ll L^{-1}+T^{-\frac{1}{2}} L^{4n-2}.$ Here the implied constant depends on $\varepsilon,$ and conductors of $\pi$ and $\pi'.$ We also use the fact that $L(1,\pi_j|\cdot|^{it},\operatorname{Ad})=L(1,\pi_j,\operatorname{Ad})$ for each component $\pi_j$ of $\pi.$ So [\[nb.\]](#nb.){reference-type="eqref" reference="nb."} follows from setting $L=T^{\frac{1}{2(4n-1)}}.$ ◻ # Quantitative Nonvanishing for $\mathrm{GL}(n+1)\times\mathrm{GL}(n)$ {#sec12} In this section, we shall make use of the relative trace formula Theorem [Theorem 4](#C){reference-type="ref" reference="C"} in conjunction with Theorem [Theorem 6](#E){reference-type="ref" reference="E"} to prove Theorem [Theorem 3](#thmC){reference-type="ref" reference="thmC"} in [1.3](#nonv){reference-type="ref" reference="nonv"}. ## Choice of Local and Global Data {#sec12.1} ### Construction of Automorphic Weights $\phi_j'\in\pi_j'$ Let $M>1$ be an integer. Let $p_*\nmid M$ be a fixed prime. Let $\pi_j'=\otimes_{p\leq \infty}\pi_{i,p}'$ be unitary cuspidal representations of $G'(\mathbb{A})$ with central character $\omega_j=\otimes_{p\leq \infty}\omega_{j,p},$ and arithmetic conductor $M_j',$ $1\leq j\leq 2.$ Suppose that $\pi_{1,\infty}'\simeq \pi_{2,\infty}',$ $\pi_{1,p_*}'\simeq \pi_{2,p_*}'.$ Suppose that $M_1'M_2'>1,$ $(M_1',M_2')=1$ and $(p_*M, M_1'M_2')=1.$ For $p\mid M_1'M_2',$ by [@Kim10], one has $$\label{12.3} \int_{N'(\mathbb{Q}_p) \backslash \overline{G'}(\mathbb{Q}_p)}W_{1,p}'(x_p)\overline{W_{2,p}'(x_p)}|\det x_p|_pdx_p=L_p(1,\pi_{1,p}'\times\widetilde{\pi}_{2,p}').$$ where $W_{j,p}'$ is the normalized new vector in the Whittaker model of $\pi_{j,p}',$ $1\leq j\leq 2.$ Let $W_{\infty}'$ be defined by Definition [Definition 17](#def4.2){reference-type="ref" reference="def4.2"} in [4.1.3](#sec4.1.3){reference-type="ref" reference="sec4.1.3"}. For $p<\infty,$ let $W_{j,p}'$ be the normalized new vector in the Whittaker model of $\pi_{j,p}',$ i.e., $W_{j,p}'(I_n)=1;$ let $\phi_j'\in \pi_j'$ be the cusp form corresponding to the Whittaker vector $W_{\infty}'\otimes\otimes_{\substack{p<\infty}}W_{j,p}',$ $j=1, 2.$ We note that $\pi_1'\not\simeq\pi_2'$ as a consequence of $(M_1',M_2')=1$ and $M_1'M_2'>1.$ ### Space of Cuspidal Representations $\mathcal{A}_0(T,M;\pi_{\infty},\pi_{p_*},\pi_1',\pi_2')$ {#12.1.3} Recall that $M>1$ is an integer, and $p_*\nmid M$ is a fixed prime. Let $\pi_{p_*}$ be a *fixed* supercuspidal representation of $G(\mathbb{Q}_{p_*})$. Let $c_{\infty},$ $C_{\infty}>0$ be fixed constants. Let $\pi_{\infty}$ be an unitary irreducible admissible representation of $\mathrm{PGL}_{n+1}(\mathbb{R}),$ with the property that $$c_{\infty} T\leq |\lambda_{\pi_{\infty},j}|\leq C_{\infty} T,\ \ 1\leq j\leq n+1,$$ where $\boldsymbol{\lambda}_{\pi_{\infty}}=(\lambda_{\pi_{\infty},1},\cdots,\lambda_{\pi_{\infty},n+1})$ is the Langlands parameter of $\pi_{\infty}.$ **Definition 66**. Let $\mathcal{A}_0(T,M;\pi_{\infty},\pi_{p_*},\pi_1',\pi_2')$ be the set of equivalent classes of cuspidal representations $\sigma=\otimes_{p\leq \infty}\sigma_p\in\mathcal{A}_0([G];\omega)$ such that - $\sigma_{\infty}\simeq\pi_{\infty},$ $\sigma_{p_*}\simeq \pi_{p_*},$ - $\sigma_p$ is right $K_p(MM_1')$-invariant at $p\nmid p_*M_2',$ - $\sigma_p$ is right $I_p(M_2')$-invariant at $p\mid M_2'.$ By Weyl law we have $$\label{12.2.2} |\mathcal{A}_0(T,M;\pi_{\infty},\pi_{p_*},\pi_1',\pi_2')|\asymp T^{\frac{n(n+1)}{2}+o(1)}M^n,$$ where the implied constant depends on $M_1',$ $M_2',$ and $p_*.$ Since $\pi_{p_*}$ is tempered, by the bound towards the Satake parameter of $\pi_{p_*}'$ (cf. [@KS03]), $L_{p_*}(1/2,\pi_{p_*}\times\pi_{p_*}')$ is finite. Let $W_{p_*}$ (resp. $W_{p_*}'$) be the normalized new vector in the Whittaker model of $\pi_{p_*}$ (resp. $\pi_{p_*}'$). As $(p_*,M_1'M_2')=1,$ $\pi_{p_*}'$ is unramified. Hence, [@JPSS81] yields that $$\label{12.1} \int_{N'(\mathbb{Q}_{p_*})\backslash G'(\mathbb{Q}_{p_*})}W_{p_*}(\iota(x_{p_*}))W_{p_*}'(x_{p_*})dx_{p_*}=L_{p_*}(1/2,\pi_{p_*}\times\pi_{p_*}')\neq 0.$$ ### Construction of the test function $f$ Define $f=\otimes_{p\leq \infty}f_p$ as follows: - $f_{\infty}$ is defined as in [3.2](#11.1.1){reference-type="ref" reference="11.1.1"}; - $p\mid M,$ $f_p$ is defined in [3.3](#sec3.3.){reference-type="ref" reference="sec3.3."}; - $p=p_*,$ $f_p(g):=\langle \pi_{p_*}(g)W_{p_*},W_{p_*}\rangle,$ $g\in G(\mathbb{Q}_{p_*});$ - $p\nmid p_*MM_1'M_2'$ and $p<\infty,$ $f_p$ is defined in [3.5](#11.1.6){reference-type="ref" reference="11.1.6"}; - $p\mid M_1'.$ Define the function $$\begin{aligned} \widehat{\Phi}_p(b_1,\cdots, b_n)=\begin{cases} \overline{\omega}_{1,p}(b_n)\omega_{2,p}(b_n),\ &\text{if $b_1,\cdots, b_{n-1}\in M_1'\mathbb{Z}_p,$ $b_n\in\mathbb{Z}_p^{\times},$}\\ 0,\ & \text{otherwise}. \end{cases}\end{aligned}$$ Let $\psi_p$ be the standard unramified additive character of $\mathbb{Q}_p$ (cf. [1.8](#notation){reference-type="ref" reference="notation"}). Let $\Phi_p$ be the Fourier inversion of $\widehat{\Phi}_p$ relative to $\psi_p.$ Define $f_p(g_p):=\int_{Z(\mathbb{Q}_p)}\tilde{f}_p(z_pg_p)\omega_p(z_p)d^{\times}z_p,$ $g_p\in G(\mathbb{Q}_p),$ where $$\begin{aligned} \tilde{f}_p(g_p)=\frac{\textbf{1}_{M_{n,n}(\mathbb{Z}_p)}(A)\Phi_p(\mathfrak{b})\textbf{1}_{M_{1,n}(\mathbb{Z}_p)}(\mathfrak{c})\textbf{1}_{\mathbb{Z}_p}(d)\textbf{1}_{G(\mathbb{Z}_p)}(g_p)}{\operatorname{Vol}(K_p'(M_1'))}\end{aligned}$$ for all $g_p=\begin{pmatrix} A&\mathfrak{b}\\ \mathfrak{c}&d \end{pmatrix}\in G(\mathbb{Q}_p).$ Here $K_p'(M_1')$ is the Hecke congruence subgroup of $K_p'$ of level $e_p(M_1').$ Recall [\[3.4.\]](#3.4.){reference-type="eqref" reference="3.4."} for its definition. In particular, $f_p$ is right $K_p(M_1')$-invariant, where $K_p(M')$ is defined in the manner of [\[equ3.4\]](#equ3.4){reference-type="eqref" reference="equ3.4"}. - $p\mid M_2'.$ Define $f_p(g_p):=\int_{Z(\mathbb{Q}_p)}\tilde{f}_p(z_pg_p)\omega_p(z_p)d^{\times}z_p,$ $g_p\in G(\mathbb{Q}_p),$ where $$\begin{aligned} \tilde{f}_p(g_p)=\frac{\textbf{1}_{K_p'(M_2')}(A)\Phi_p(\mathfrak{b})\textbf{1}_{M_{1,n}(\mathbb{Z}_p)}(\mathfrak{c})\textbf{1}_{\mathbb{Z}_p}(d)\textbf{1}_{G(\mathbb{Z}_p)}(g_p)}{\operatorname{Vol}(K_p'(M_2')}\end{aligned}$$ for all $g_p=\begin{pmatrix} A&\mathfrak{b}\\ \mathfrak{c}&d \end{pmatrix}\in G(\mathbb{Q}_p).$ In particular, $f_p$ is right $I_p(M_2')$-invariant, $p\mid M_2',$ where $I_p(M_2'):=\big\{g_p=(g_{i,j})_{1\leq i,j \leq n}\in G(\mathbb{Z}_p):\ g_{i,j}\in M_2'\mathbb{Z}_p,\ i>j\big\}$ is the Iwahori subgroup of level $e_p(M_2').$ ## The Relative Trace Formula {#the-relative-trace-formula-1} We will take the above data ($f,$ $\phi_1',$ and $\phi_2'$) into Theorem [Theorem 4](#C){reference-type="ref" reference="C"} (cf. [2](#sec2){reference-type="ref" reference="sec2"}) to obtain $J_{\mathop{\mathrm{Spec}}}^{\operatorname{Reg},\heartsuit}(f,\textbf{0})=J_{\operatorname{Geo}}^{\operatorname{Reg},\heartsuit}(f,\textbf{0}).$ ### The Spectral Side {#the-spectral-side-2} **Lemma 67**. *Let notation be as before. Then $$\begin{aligned} J_{\mathop{\mathrm{Spec}}}^{\operatorname{Reg},\heartsuit}(f,\mathbf{0})\ll \sum_{\sigma\in \mathcal{A}_0(T,M;\pi_{\infty},\pi_{p_*},\pi_1',\pi_2')}\frac{T^{-\frac{n^2}{2}+\varepsilon}\big|L(1/2,\sigma\times\pi_1')L(1/2,\sigma\times\pi_2')\big|}{L(1,\sigma,\operatorname{Ad})},\end{aligned}$$ where the implied constant relies on $\pi_{\infty},$ $\pi_{p_*},$ $\phi_1'$ and $\phi_2'.$* *Proof.* Using the notation in [5.3.1](#sec5.2.1){reference-type="ref" reference="sec5.2.1"} and the proof of Proposition [Proposition 28](#thm6){reference-type="ref" reference="thm6"}, the spectral side $J_{\mathop{\mathrm{Spec}}}^{\operatorname{Reg},\heartsuit}(f,\mathbf{0})$ is equal to $$\begin{aligned} \sum_{\mathfrak{X}}\sum_{R}\frac{1}{n_R}\left(\frac{1}{2\pi i}\right)^{\dim A_R/A_G}\int_{i\mathfrak{a}_R^*/i\mathfrak{a}_G^*}\sum_{\phi\in\mathfrak{B}_{R,\mathfrak{X}}}\big|\Psi(0, \mathcal{I}_R(\lambda,\tilde{f})W_{\phi,\lambda},W_{\phi'}')\big|^2d\lambda.\end{aligned}$$ Since $f_{p_*}$ is the matrix coefficient of a supercuspidal representation of $G(\mathbb{Q}_{p_*}),$ only cuspidal spectrum contributes in the above decomposition. Hence, $$\begin{aligned} J_{\mathop{\mathrm{Spec}}}^{\operatorname{Reg},\heartsuit}(f,\mathbf{0})=\sum_{\sigma\in \mathcal{A}_0(T,M,S;\pi_{p_*})}\sum_{\phi\in\mathfrak{B}_{\sigma}}\big|\Psi(0, \sigma(\tilde{f})W_{\phi},W_{\phi'}')\big|^2,\end{aligned}$$ where $\mathfrak{B}_{\sigma}$ is an orthonormal basis of $\sigma.$ Then Lemma [Lemma 67](#lemma12.1){reference-type="ref" reference="lemma12.1"} follows from Lemma [Lemma 26](#prop4.11){reference-type="ref" reference="prop4.11"} and [\[12.1\]](#12.1){reference-type="eqref" reference="12.1"}. ◻ ### The Geometric Side {#the-geometric-side-1} As a consequence of the assumption that $M_1'M_2'>1,$ we have the following. **Lemma 68**. *Let notation be as before. Then $J_{\operatorname{Geo},\operatorname{Dual}}^{\operatorname{Big}}(f,\textbf{s})\equiv 0.$* *Proof.* Let $\mathbf{s}=(s,0)$ be such that $\mathop{\mathrm{Re}}(s)>1.$ Then $J_{\operatorname{Geo},\operatorname{Dual}}^{\operatorname{Big}}(f,\textbf{s})$ converges absolutely. So $J_{\operatorname{Geo},\operatorname{Dual}}^{\operatorname{Big}}(f,\textbf{s})$ factors through $$\begin{aligned} \mathcal{I}_p(y_p):=\int_{N'(\mathbb{Q}_p)\backslash G'(\mathbb{Q}_p)}f_p\left(\begin{pmatrix} y_p&\\ \eta x_p y_p&1 \end{pmatrix}\right)W_{1,p}'(x_p)\overline{W_{2,p}'(x_py_p)}|\det x_p|^sdx_p\end{aligned}$$ for $p\mid M_1'M_2'.$ We now consider the following cases. 1. Suppose $p\mid M_1'.$ Then $W_{2,p}'$ is spherical and $W_{1,p}'$ is a new vector which is *not* spherical. By definition, $f_p\left(\begin{pmatrix} y_p&\\ \eta x_p y_p&1 \end{pmatrix}\right)=0$ unless $y_p\in K_p'.$ So $$\begin{aligned} \mathcal{I}_p(y_p)=\textbf{1}_{K_p'}(y_p)\int_{N'(\mathbb{Q}_p)\backslash G'(\mathbb{Q}_p)}\textbf{1}_{M_{1,n}}(\eta x_p)W_{1,p}'(x_p)\overline{W_{2,p}'(x_p)}|\det x_p|^sdx_p=0\end{aligned}$$ as a consequence of the assumption that $\pi_1'$ is ramified and $W_{2,p}'$ is spherical. 2. Suppose $p\mid M_2'.$ Then $W_{1,p}'$ is spherical and $W_{2,p}'$ is a new vector but is not spherical. By definition of $f_p,$ we have $$\begin{aligned} \mathcal{I}_p(y_p)=\textbf{1}_{K_p'(M_2')}(y_p)\int_{N'(\mathbb{Q}_p)\backslash G'(\mathbb{Q}_p)}\textbf{1}_{M_{1,n}}(\eta x_p)W_{1,p}'(x_p)\overline{W_{2,p}'(x_p)}|\det x_p|^sdx_p,\end{aligned}$$ which is vanishing since $\pi_{2,p}'$ is ramified while $W_{1,p}'$ is spherical. Since we assume that $M_1'M_2'>1,$ then $\prod_{p\mid M_1'M_2'}\mathcal{I}_p(y_p)\equiv 0.$ So $J_{\operatorname{Geo},\operatorname{Dual}}^{\operatorname{Big}}(f,\textbf{s})\equiv 0$ in the range $\mathop{\mathrm{Re}}(s)>1.$ By meromorphic continuation, $J_{\operatorname{Geo},\operatorname{Dual}}^{\operatorname{Big}}(f,\textbf{s})\equiv 0$ everywhere. Hence Lemma [Lemma 68](#lem12.2){reference-type="ref" reference="lem12.2"} follows. ◻ **Proposition 69**. *Let notation be as before. Then $$\label{12.44} J_{\operatorname{Geo}}^{\operatorname{Reg},\heartsuit}(f,\textbf{0})\gg_{\varepsilon,\pi_1',\pi_2',\pi_{p_*}} T^{\frac{n}{2}-\varepsilon}M^n\|W_{\infty}'\|_{L^2}^2,$$ where the implied constant relies on $\varepsilon,$ $\pi_1',$ $\pi_2',$ and $\pi_{p_*},$ and $\|\cdot \|_{L^2}$ is the $L^2$-norm.* *Proof.* Recall that geometric side (cf. [\[63\]](#63){reference-type="eqref" reference="63"}), $$\begin{aligned} J^{\operatorname{Reg},\heartsuit}_{\operatorname{Geo}}(f,\textbf{s})=J^{\operatorname{Reg}}_{\operatorname{Geo},\operatorname{Small}}(f,\textbf{s})+J_{\operatorname{Geo},\operatorname{Dual}}^{\operatorname{Big}}(f,\textbf{s})-J^{\operatorname{Reg},\uppercase{\romannumeral 1\relax},\heartsuit}_{\operatorname{Geo},\operatorname{Big}}(f,\textbf{s})+J^{\operatorname{Reg},\uppercase{\romannumeral 2\relax}}_{\operatorname{Geo},\operatorname{Big}}(f,\textbf{s})\end{aligned}$$ is an entire function (though each individual term might be just meromorphic). Since $f_{p_*}$ is the matrix coefficient of a supercuspidal representation, it has no constant in the Fourier expansion. Hence, by Proposition [Proposition 10](#thm49){reference-type="ref" reference="thm49"} and the definition in [2.2.2](#2.2.2){reference-type="ref" reference="2.2.2"} we have that $J^{\operatorname{Reg},\uppercase{\romannumeral 1\relax},\heartsuit}_{\operatorname{Geo},\operatorname{Big}}(f,\textbf{s})\equiv 0.$ By Lemma [Lemma 68](#lem12.2){reference-type="ref" reference="lem12.2"}, we have $J_{\operatorname{Geo},\operatorname{Dual}}^{\operatorname{Big}}(f,\textbf{s})\equiv 0.$ Moreover, using Cauchy-Schwarz to separate $\phi_1'$ and $\phi_2',$ it follows from the proof of Theorem [Theorem 45](#thm9.1){reference-type="ref" reference="thm9.1"} that $$\begin{aligned} \frac{J^{\operatorname{Reg},\uppercase{\romannumeral 2\relax}}_{\operatorname{Geo},\operatorname{Big}}(f,\textbf{0})}{\|W_{\infty}'\|_2^2}\ll_{\varepsilon,\pi_1',\pi_2',\pi_{p_*}}M^{n-1}T^{\frac{n}{2}-\frac{1}{2}+\varepsilon},\end{aligned}$$ where the implied constant relies on $\varepsilon,$ $\pi_1',$ $\pi_2',$ and $\pi_{p_*}.$ Now we handle $J^{\operatorname{Reg}}_{\operatorname{Geo},\operatorname{Small}}(f,\textbf{0}).$ Let $S=\{\text{$p$ prime}:\ p\mid p_*M_1'M_2'\}.$ Let $\mathbf{s}=(s,0).$ When $\mathop{\mathrm{Re}}(s)>0$ we have $$\begin{aligned} J^{\operatorname{Reg}}_{\operatorname{Geo},\operatorname{Small}}(f,\textbf{s})=\frac{L^{(S)}(1+s,\pi_1'\times\widetilde{\pi}_2')}{\operatorname{Vol}(K_0(M))}\cdot \mathcal{I}_S(f,s),\end{aligned}$$ where $\mathcal{I}_S(f,s)=\prod_{p\in S}\mathcal{I}_p(f,s),$ and for a place $p$ of $\mathbb{A},$ $$\begin{aligned} \mathcal{I}_v(f,s):=&\int_{G'(\mathbb{Q}_p)}\int_{M_{1,n}(\mathbb{Q}_p)}\int_{N'(\mathbb{Q}_p)\backslash G'(\mathbb{Q}_p)}f_p\left(\begin{pmatrix} y&\mathfrak{u}\\ &1 \end{pmatrix} \right)\psi_p(\eta x\mathfrak{u})\\ &\qquad \qquad W_p'(\iota(x))\overline{W_p'(xy)}|\det x|_p^{1+s}dxd\mathfrak{u}dy.\end{aligned}$$ Note that $\mathcal{I}_p(f,s)$ converges absolutely when $s=0.$ Since $\pi_1'\not\simeq \pi_2',$ the function $L^{(S)}(1+s,\pi_1'\times\widetilde{\pi}_2')$ is holomorphic at $s=0.$ Therefore, $$\begin{aligned} J^{\operatorname{Reg}}_{\operatorname{Geo},\operatorname{Small}}(f,\textbf{0})=\frac{L^{(S)}(1,\pi_1'\times\widetilde{\pi}_2')}{\operatorname{Vol}(K_0(M))}\cdot \mathcal{I}(f,0).\end{aligned}$$ - Suppose $p\mid M_1'M_2'.$ By the definition of $f_p$ and [@Kim10 Theorem 2.1.1], $$\begin{aligned} \mathcal{I}_p(f,0)=\int_{N'(\mathbb{Q}_p)\backslash G'(\mathbb{Q}_p)}W_{1,p}'(x_p)\overline{W_{2,p}'(x_p)}\widehat{\Phi}_p(\eta x_p)|\det x_p|_pdx_p=L_p(1,\pi_{1,p}'\times\widetilde{\pi}_{2,p}').\end{aligned}$$ - Suppose $p=p_*.$ By Iwasawa decomposition we have $$\begin{aligned} \mathcal{I}_p(f,0)=&\int_{K_p'}\int_{G'(\mathbb{Q}_p)}\int_{M_{1,n}(\mathbb{Q}_p)}\int_{Z'(\mathbb{Q}_p)}f_p\left(\begin{pmatrix} y_p&u_p\\ &1 \end{pmatrix} \right)\psi_p(\eta z_pk_pu_p)|z_p|_p^{n}d^{\times}z_p\\ &\qquad \int_{N'(\mathbb{Q}_p)\backslash P_0'(\mathbb{Q}_p)}W_p'(b_pk_p)\overline{W_p'(b_pk_py_p)}|\det b_p|_pdb_pdu_pdy_pdk_p.\end{aligned}$$ Let $H=\operatorname{diag}(\mathrm{GL}(n-1),1)$ and $N_H=N'\cap H.$ Then $$\begin{aligned} \int W_p'(b_pk_p)\overline{W_p'(b_pk_py_p)}|\det b_p|_pdb_p=\int_{N_H(\mathbb{Q}_p)\backslash H(\mathbb{Q}_p)}W_p'(h_pk_p)\overline{W_p'(h_pk_py_p)}dh_p,\end{aligned}$$ where $x_p\in N'(\mathbb{Q}_p)\backslash P_0'(\mathbb{Q}_p).$ By Kirillov model theory, $$\begin{aligned} \int_{N_H(\mathbb{Q}_p)\backslash H(\mathbb{Q}_p)}W_p'(h_pk_p)\overline{W_p'(h_pk_py_p)}dh_p=\int_{N_H(\mathbb{Q}_p)\backslash H(\mathbb{Q}_p)}W_p'(h_p)\overline{W_p'(h_py_p)}dh_p.\end{aligned}$$ On the other hand, by Fourier inversion and polar coordinates, $$\begin{aligned} \int_{K_p'}\int_{Z'(\mathbb{Q}_p)}\int_{M_{1,n}(\mathbb{Q}_p)}f_p\left(\begin{pmatrix} y_p&u_p\\ &1 \end{pmatrix} \right)\psi_p(\eta z_pk_pu_p)du_p|z_p|_p^{n}d^{\times}z_pdk_p\end{aligned}$$ is equal to $c_{n,p}f_p\left(\begin{pmatrix} y_p\\ &1 \end{pmatrix} \right)$ for some positive constant $c_{n,p}.$ Here we factor $dk_p$ through the measure on $\eta K_p'=\big\{(k_{p,1},\cdots,k_{p,n})\in M_{n,1}(\mathbb{Z}_p):\ \min_{1\leq j\leq n}e_p(k_{p,j})=0\big\}$ and the multiplier $c_{n,p}$ comes from this decomposition of $dk_p.$ Therefore, $$\begin{aligned} \mathcal{I}_p(f,0)=&c_{n,p}\int_{G'(\mathbb{Q}_p)}\langle \pi_{p}(\iota(y_p))W_{p},W_{p}\rangle\overline{\langle\pi_{p}'(y_p)W_p'(y_p),W_{p}'\rangle}dy_p,\end{aligned}$$ which is equal to $c_{n,p}d_{\pi_{p}'}^{-1}\langle W_p,W_p\rangle \overline{\langle W_p',W_p'\rangle}\textbf{1}_{\pi_{p}\mid_{G'(\mathbb{Q}_{p})}\supseteq \pi_{p}'},$ with $d_{\pi_p'}$ being the formal degree of $\pi_p'.$ By [\[12.1\]](#12.1){reference-type="eqref" reference="12.1"} we have $\dim\mathop{\mathrm{Hom}}_{G'(\mathbb{Q}_{p})}(\pi_{p}\mid_{G'(\mathbb{Q}_{p})},\pi_{p}')\geq 1,$ which implies that $\textbf{1}_{\pi_{p}\mid_{G'(\mathbb{Q}_{p})}\supseteq \pi_{p}'}\equiv 1.$ Therefore, $\mathcal{I}_p(f,0)\gg 1\gg L_p(1,\pi_1'\times\widetilde{\pi}_{2}').$ - Suppose $p=\infty.$ Following the above arguments we obtain $$\begin{aligned} \mathcal{I}_{\infty}(f,0)=&c_{n,\infty}\int_{G'(\mathbb{R})}f_{\infty}(\iota(y_{\infty}))\overline{\langle \pi_{\infty}'(y_{\infty})W_{\infty}',W_{\infty}'\rangle} dy_{\infty},\end{aligned}$$ where $c_{n,\infty}$ is a positive constant depending on $n.$ By construction of $f_{\infty}$ and $W_{\infty}',$ we have $$\begin{aligned} \mathcal{I}_{\infty}(f,0)\gg \langle W_{\infty}',W_{\infty}'\rangle\Big|\int_{G'(\mathbb{R})}f_{\infty}(\iota(y_{\infty}))dy_{\infty}\Big|\gg T^{\frac{n}{2}-\varepsilon}\|W_{\infty}\|_{\infty}^2.\end{aligned}$$ Gathering the above estimates we obtain $$\begin{aligned} J_{\operatorname{Geo}}^{\operatorname{Reg},\heartsuit}(f,\textbf{0})\gg T^{\frac{n}{2}-\varepsilon}M^n\|W_{\infty}'\|_2^2L(1,\pi_1'\times\widetilde{\pi}_2')+O(M^{n-1}T^{\frac{n}{2}-\frac{1}{2}+\varepsilon}\|W_{\infty}'\|_2^2),\end{aligned}$$ from which the inequality [\[12.44\]](#12.44){reference-type="eqref" reference="12.44"} follows. ◻ ## Nonvanishing Problems {#12.4} Theorem [Theorem 3](#thmC){reference-type="ref" reference="thmC"} follows readily from the following. **Theorem 8**. *Let notation be as in [11.1](#sec12.1){reference-type="ref" reference="sec12.1"}. Then $$\label{12.5.} \sum_{\substack{\pi\in \mathcal{A}_0(T,M;\pi_{\infty},\pi_{p_*},\pi_1',\pi_2')\\ L(1/2,\pi\times\pi_1')L(1/2,\pi\times\pi_2')\neq 0}}1\gg \begin{cases} M^{\frac{n}{(4n^2+2n-1)}}T^{\frac{1}{2\cdot (4n^2+2n-1)}}, & \text{if $M\leq T^{\frac{n+1}{2n^2-1}}$,}\\ M^{\frac{1}{2n^2+2n+1}}T^{\frac{1}{2\cdot (2n^2+2n+1)}}, & \text{if $M>T^{\frac{n+1}{2n^2-1}}.$} \end{cases}$$ where the implied constant relies on $\varepsilon,$ $c_{\infty},$ $C_{\infty},$ $\pi_1',$ $\pi_2',$ and $\pi_{p_*}.$ In particular, $$\label{12.5} \sum_{\substack{\pi\in \mathcal{A}_0(T,M;\pi_{\infty},\pi_{p_*},\pi_1',\pi_2')\\ L(1/2,\pi\times\pi_1')L(1/2,\pi\times\pi_2')\neq 0}}1\gg |\mathcal{A}_0(T,M;\pi_{\infty},\pi_{p_*},\pi_1',\pi_2')|^{\frac{1}{n(n+1)(4n^2+2n-1)}-\varepsilon},$$ where the implied constant relies on $\varepsilon,$ $c_{\infty},$ $C_{\infty},$ $\pi_1',$ $\pi_2',$ and $\pi_{p_*}.$* *Proof.* By Lemma [Lemma 67](#lemma12.1){reference-type="ref" reference="lemma12.1"} and Proposition [Proposition 69](#proposition12.2){reference-type="ref" reference="proposition12.2"} we obtain $$\begin{aligned} \sum_{\pi\in \mathcal{A}_0(T,M;\pi_{\infty},\pi_{p_*},\pi_1',\pi_2')}\frac{T^{-\frac{n^2}{2}+\varepsilon}\big|L(1/2,\pi\times\pi_1')L(1/2,\pi\times\pi_2')\big|}{L(1,\pi,\operatorname{Ad})\|W_{\infty}'\|_2^2}\gg T^{\frac{n}{2}-\varepsilon}M^n.\end{aligned}$$ By [\[nb\]](#nb){reference-type="eqref" reference="nb"} in Theorem [Theorem 6](#E){reference-type="ref" reference="E"} (cf. [10.4](#sec11.4){reference-type="ref" reference="sec11.4"}) and $\langle\phi_j',\phi_j'\rangle\asymp \|W_{\infty}'\|_2^2,$ $j=1,2,$ we obtain $$\begin{aligned} T^{\frac{n(n+1)}{2}+\varepsilon}M^{n+\varepsilon}\mathbf{L}^{-2}\sum_{\substack{\pi\in \mathcal{A}_0(T,M;\pi_{\infty},\pi_{p_*},\pi_1',\pi_2')\\ L(1/2,\pi\times\pi_1')L(1/2,\pi\times\pi_2')\neq 0}}1\gg T^{\frac{n(n+1)}{2}-\varepsilon}M^n,\end{aligned}$$ where the implied constant relies on $\varepsilon,$ $\pi_1',$ $\pi_2',$ and $\pi_{p_*}.$ Hence [\[12.5.\]](#12.5.){reference-type="eqref" reference="12.5."} follows. Therefore, [\[12.5\]](#12.5){reference-type="eqref" reference="12.5"} follows from [\[12.5.\]](#12.5.){reference-type="eqref" reference="12.5."} and the estimate $$|\mathcal{A}_0(T,M;\pi_{\infty},\pi_{p_*},\pi_1',\pi_2')|\asymp T^{\frac{n(n+1)}{2}+o(1)}M^n,\tag{\ref{12.2.2}}$$ where the implied constant depends on $M_1',$ $M_2',$ and $p_*.$ ◻ *Proof of Corollary [Corollary 7](#1.10){reference-type="ref" reference="1.10"}.* Take $p_*=2.$ Let $\pi_{p_*}$ be a supercuspidal representation of $G(\mathbb{Q}_{p_*})$ of depth zero. So the conductor of $\pi_{p_*}$ is $2^{n}.$ If $M'>1,$ then Corollary [Corollary 7](#1.10){reference-type="ref" reference="1.10"} follows from Theorem [Theorem 3](#thmC){reference-type="ref" reference="thmC"} with $\pi_1'=\pi',$ $M_2'=1.$ If $M'=1,$ then Corollary [Corollary 7](#1.10){reference-type="ref" reference="1.10"} follows from Theorem [Theorem 3](#thmC){reference-type="ref" reference="thmC"} with $\pi_1'=\pi',$ $M_2'=3.$ ◻

arxiv_math

--- abstract: | We present an optimal transport approach for mesh adaptivity and shock capturing of compressible flows. Shock capturing is based on a viscosity regularization of the governing equations by introducing an artificial viscosity field as solution of the Helmholtz equation. Mesh adaptation is based on the optimal transport theory by formulating a mesh mapping as solution of Monge-Ampère equation. The marriage of optimal transport and viscosity regularization for compressible flows leads to a coupled system of the compressible Euler/Navier-Stokes equations, the Helmholtz equation, and the Monge-Ampère equation. We propose an iterative procedure to solve the coupled system in a sequential fashion using homotopy continuation to minimize the amount of artificial viscosity while enforcing positivity-preserving and smoothness constraints on the numerical solution. We explore various mesh monitor functions for computing r-adaptive meshes in order to reduce the amount of artificial dissipation and improve the accuracy of the numerical solution. The hybridizable discontinuous Galerkin method is used for the spatial discretization of the governing equations to obtain high-order accurate solutions. Extensive numerical results are presented to demonstrate the optimal transport approach on transonic, supersonic, hypersonic flows in two dimensions. The approach is found to yield accurate, sharp yet smooth solutions within a few mesh adaptation iterations. author: - Ngoc Cuong Nguyen - R. Loek Van Heyningen - Jordi Vila-Pérez - Jaime Peraire bibliography: - library.bib - extrarefs.bib title: Optimal transport for mesh adaptivity and shock capturing of compressible flows --- optimal transport ,compressible flows ,shock capturing ,mesh adaptation ,artificial viscosity ,discontinuous Galerkin methods ,finite element methods # Introduction {#sec:intro} Compressible flows at high Mach number lead to shock waves which pose one of the most challenging problems for numerical methods. For high-order numerical methods, insufficient resolution or an inadequate treatment of shocks can result in Gibbs oscillations, which grow rapidly and contribute to numerical instabilities. Effective treatment of shock waves requires both shock capturing and mesh adaptation. Shock capturing methods lie within one of the following two categories: limiters and artificial viscosity. Limiters, in the form of flux limiters [@Burbeau2001; @Cockburn1989; @Krivodonova2007], slope limiters [@Cockburn1998a; @MR2056921; @Lv2015; @Sonntag2017], and WENO-type schemes [@Luo2007; @Qiu2005; @Zhu2008; @Zhu2013] pose implementation difficulties for implicit time integration schemes and high-order methods on complex geometries. As for artificial viscosity methods, Laplacian-based [@Barter2010; @Hartmann2013; @Lv2016; @Moro2016; @Nguyen2011a; @persson06:_shock_capturing; @Persson2013] and physics-based [@Abbassi2014; @Chaudhuri2017; @Cook2004; @Cook2005; @Fernandez2018; @Fiorina2007; @Kawai2008; @Kawai2010; @Mani2009; @Olson2013; @persson06:_shock_capturing] approaches have been proposed. When the amount of viscosity is properly added in a neighborhood of shocks, the solution can converge uniformly except in the region around shocks, where it is smoothed and spread out over some length scale. Artificial viscosity has been widely used in finite volume methods [@Jameson1995], streamline upwind Petrov-Galerkin (SUPG) methods [@HughesMalletMisukamiII86], spectral methods [@madtad93; @Tadmor89], as well as DG methods [@Barter2010; @Ching2019; @HartmannHoustonCompressible02; @persson06:_shock_capturing; @Bai2022a; @Vila-Perez2021]. Both Laplacian-based [@Hartmann2013; @Lv2016; @Nguyen2011a; @Moro2016; @persson06:_shock_capturing; @Persson2013] and physics-based [@Abbassi2014; @Bhagatwala2009; @Mani2009; @Cook2005; @Cook2007; @Fiorina2007; @Kawai2008; @Kawai2010; @Olson2013; @Premasuthan2013] artificial viscosity methods have been used for shock capturing. Recent advances lead to shock-fitting methods that do not require limiter and artificial viscosity to stabilize shocks. The recent work [@Zahr2018; @Zahr2020; @Shi2022] introduces a high-order implicit shock tracking (HOIST) method for resolving discontinuous solutions of conservation laws with high-order numerical discretizations that support inter-element solution discontinuities, such as discontinuous Galerkin or finite volume methods [@Zahr2020]. The method aims to align mesh elements with shock waves by deforming the computational mesh in order to obtain accurate high-order solutions. It requires solution of a PDE-constrained optimization problem for both the computational mesh and the numerical solution using sequential quadratic programming solver. Recently, a moving discontinuous Galerkin finite element method with interface condition enforcement (MDG-ICE) [@Corrigan2019; @Kercher2021; @Kercher2021a] is formulated for shock flows by enforcing the interface condition separately from the conservation laws. In the MDG-ICE method, the grid coordinates are treated as additional unknowns to detect interfaces and satisfy the interface condition, thereby directly fitting shocks and preserving high-order accurate solutions. The Levenberg-Marquardt method is used to solve the coupled system of the conservation laws and the interface condition to obtain the numerical solution and the shock-fitted mesh. In a recent work [@Nguyen2023c], we introduce an adaptive viscosity regularization scheme for the numerical solution of nonlinear conservation laws with shock waves. The scheme solves a sequence of viscosity-regularized problems by using homotopy continuation to minimize the amount of viscosity subject to relevant physics and smoothness constraints on the numerical solution. The scheme is coupled to mesh adaptation algorithms that identify the shock location and generate shock-aligned meshes in order to further reduce the amount of artificial dissipation. In particular, shocks curves are constructed by determining shock-containing elements and finding a collection of points at which the artificial viscosity reaches its maximum value along streamline directions. A shock-aligned grid is generated by replicating shock curves along streamline directions. While the mesh alignment procedure is simple, it is not practical for complex geometries and shock patterns. In this paper, we present an optimal transport approach for mesh adaptivity and shock capturing of compressible flows. Shock capturing is based on the viscosity regularization method introduced in [@Nguyen2023c]. Mesh adaptation is based on the optimal transport theory by formulating a mesh mapping as solution of Monge-Ampère equation [@nguyen2023hybridizable]. The marriage of optimal transport and viscosity regularization for compressible flows leads to a coupled system of the compressible Euler/Navier-Stokes equations, the Helmholtz equation, and the Monge-Ampère equation. We propose an iterative solution procedure to solve the coupled system in a sequential manner. We explore various mesh monitor functions for computing r-adaptive meshes in order to reduce the amount of artificial dissipation and improve the accuracy of the numerical solution. The hybridizable discontinuous Galerkin (HDG) method is used for the spatial discretization of the governing equations owing to its efficiency and high-order accuracy [@Nguyen2012; @Fernandez2018a; @Moro2011a; @Peraire2010; @Vila-Perez2021; @Woopen2014c; @Fidkowski2016; @Fernandez2017a; @williams2018entropy]. Extensive numerical results are presented to demonstrate the proposed approach on a wide variety of transonic, supersonic, hypersonic flows in two dimensions. The approach is found to yield accurate, sharp yet smooth solutions within a few mesh adaptation iterations. It is capable of moving mesh points to resolve complex shock patterns without creating new mesh points or modifying the connectivity of the initial mesh. The generated r-adaptive mesh can significantly improve the accuracy of the numerical solution relative to the initial mesh. Accurate prediction of drag forces and heat transfer rates for viscous shock flows requires meshes to resolve both shocks and boundary layers. We show that the approach can generate r-adaptive meshes that resolve not only shocks but also boundary layers for viscous shock flows. The approach predicts heat transfer coefficient accurately by adapting the initial mesh to resolve bow shock and boundary layer when it is applied to viscous hypersonic flows past a circular cylinder. The paper is organized as follows. We describe the adaptive viscosity regularization method in Section [2](#sec:viscosity){reference-type="ref" reference="sec:viscosity"} and the optimal transport approach in Section 3. In Section [4](#sec:results){reference-type="ref" reference="sec:results"}, we present numerical results to assess the performance of the proposed approach on a wide variety of transonic, supersonic, and hypersonic flows. Finally, in Section [5](#sec:conclusions){reference-type="ref" reference="sec:conclusions"}, we conclude the paper with some remarks and future work. # Adaptive Viscosity Regularization Method {#sec:viscosity} ## Governing equations We consider the steady-state conservation laws of $m$ state variables, defined on a physical domain $\Omega \in \mathbb{R}^d$ and subject to appropriate boundary conditions, as follows $$\label{eq1} \nabla \cdot \bm F(\bm u, \nabla \bm u) = 0 \quad \mbox{in }\Omega,$$ where $\bm u(\bm x) \in \mathbb{R}^m$ is the solution of the system of conservation laws at $\bm x \in \Omega$ and the physical fluxes $\bm F = (\bm f_1(\bm u, \nabla \bm u), \ldots, \bm f_d(\bm u, \nabla \bm u)) \in \mathbb{R}^{m \times d}$ include $d$ vector-valued functions of the solution. This paper focuses on the compressible Euler and Navier-Stokes equations. For the compressible Euler equations, the state vector and physical fluxes are given by $$\bm u = \begin{pmatrix} \rho \\ \rho v_i \\ \rho E \\ \end{pmatrix}, \qquad \bm F(\bm u) = \begin{pmatrix} \rho v_j \\ \rho v_i v_j + \delta_{ij} p \\ \rho v_j H \end{pmatrix} \label{eq:euler}$$ with density $\rho$, velocity $\bm v$, total energy $E$, total specific enthalpy $H = E + p/\rho$ and pressure $p$ given by the ideal gas law $p = (\gamma - 1) \rho ( E-\frac{1}{2}v_i \ v_i)$. Let $\Gamma_{\rm wall} \subset \partial \Omega$ be the wall boundary. The boundary condition at the wall boundary $\Gamma_{\rm wall}$ is $\bm v \cdot \bm n = 0$, where $\bm v$ is the velocity field and $\bm n$ is the unit normal vector outward the boundary. For supersonic and hypersonic flows, supersonic inflow and outflow conditions are imposed on the inflow and outflow boundaries, respectively. For transonic flows, a freastream boundary condition is imposed at the far field boundary by using the freestream state $\bm u_\infty$. The freestream Mach number $M_\infty$ enters through the non-dimensional freestream pressure $p_{\infty} = 1 / (\gamma M^2_{\infty})$. Here $\gamma$ denotes the specific heat ratio. For the compressible Navier-Stokes equations, the fluxes are given by $$\label{flux} \bm{F}(\bm{u},\nabla \bm u) = \left( \begin{array}{c} \rho v_j \\ \rho v_i v_j + \delta_{ij} p \\ \rho v_j H \end{array} \right) - \left( \begin{array}{c} 0 \\ \tau_{ij} \\ v_i \tau_{ij} + f_j \end{array} \right) .$$ For a Newtonian, calorically perfect gas in thermodynamic equilibrium, the non-dimensional viscous stress tensor and heat flux are given by $$\label{closures} \tau_{ij} = \frac{1}{Re} \bigg[ \Big( \frac{\partial v_i}{\partial x_j}+\frac{\partial v_j}{\partial x_i} \Big) -\frac{2}{3}\frac{\partial v_k}{\partial x_k}\delta_{ij} \bigg] , \qquad f_j = - \frac{\gamma}{Re \ Pr} \ \frac{\partial T}{\partial x_j} ,$$ respectively. Here $Re$ denotes the Reynolds number, and $Pr$ the Prandtl number. For high Mach number flows, Sutherland's law is used to obtain the dynamic viscosity, thereby rendering the Reynolds number dependent on the temperature. The boundary conditions at the wall are zero velocity and either isothermal or adiabatic temperature. Other boundary conditions are similar to those of the compressible Euler equations. ## Viscosity regularization of compressible flows Shock waves have always been a considerable source of difficulties toward a rigorous numerical solution of compressible flows. In order to treat shock waves, we follow the recent work [@Barter2010; @Ching2019] by considering the viscosity regularization of the conservation laws ([\[eq1\]](#eq1){reference-type="ref" reference="eq1"}) as follows [\[eq3\]]{#eq3 label="eq3"} $$\begin{aligned} {2} \nabla \cdot \bm F(\bm u, \nabla \bm u) - \lambda_1 \nabla \cdot \bm G(\bm u, \nabla \bm u, \eta) = 0 \quad \mbox{in }\Omega, \\ \eta - \lambda_2^2 \nabla \cdot \left( \ell^2 \nabla \eta \right)- s(\bm u, \nabla \bm u) = 0 \quad \mbox{in }\Omega ,\end{aligned}$$ where $\eta(\bm x)$ is the solution of the Helmholtz equation ([\[eq3\]](#eq3){reference-type="ref" reference="eq3"}b) with homogeneous Neumann boundary conditions $$\eta = 0 \quad \mbox{on } \Gamma_{\rm wall}, \qquad \ell^2 \nabla \eta \cdot \bm n = 0 \quad \mbox{on }\partial \Omega \backslash \Gamma_{\rm wall} \ .$$ The artificial fluxes $\bm G$ provide a viscosity regularization to smooth out discontinuities in the shock region. There are a number of different options for the artificial fluxes $\bm G$. In this paper, we use the Laplacian fluxes of the form $$\bm G(\bm u, \nabla \bm u, \eta) = \mu(\eta) \nabla \bm u ,$$ where $$\mu(\eta) = (\bar{\eta}-\bar{\eta}_{\rm T})\left(\frac{\arctan(100(\bar{\eta} -\bar{\eta}_{\rm T}))}{\pi} + \frac{1}{2} \right) - \frac{\arctan(100)}{\pi} + \frac{1}{2}$$ is a smooth approximation of a ramp function. Here $\bar{\eta} = \eta/\|\eta\|_\infty$ is the normalized function with $\|\eta\|_{\infty} = \max_{\bm x \in \Omega} |\eta(\bm x)|$ being the $L_\infty$ norm. Note that $\bar{\eta}_{\rm T}$ is the artificial viscosity threshold that makes $\mu(\eta)$ vanish to zero when $\bar{\eta} \le \bar{\eta}_{\rm T}$. In other words, artificial viscosity is only added to the shock region where $\bar{\eta}$ exceeds $\bar{\eta}_{\rm T}$. Therefore, the threshold $\bar{\eta}_{\rm T}$ will help remove excessive artificial viscosity. Since $\|\bar{\eta}\|_\infty = 1$, $\bar{\eta}_{\rm T} = 0.2$ is a sensible choice. Note that the artificial viscosity field is equal to $\lambda_1 \mu(\bm x)$, where $\mu(\bm x)$ is bounded by $\mu(\bm x) \in [0, 1 - \bar{\eta}_{\rm T}]$ for any $\bm x \in \Omega$. We can also consider a more general form $\bm G = \mu(\eta) \nabla \bm u^*$ [@Barter2010; @Nguyen2011a], where $\bm u^*$ is a modified state vector. Another option is physics-based artificial viscosity by taking $\bm G$ to be the viscous stress tensor and the heat flux of the Navier-Sokes equation and adding the artificial viscosity to the physical viscosities and thermal conductivity [@CuongNguyen2022; @Nguyen2023a]. The source term $s$ in ([\[eq3\]](#eq3){reference-type="ref" reference="eq3"}b) is required to determine $\eta$ and defined as follows $$\label{avsource} s(\bm u, \nabla \bm u) = {g}(S(\bm u, \nabla \bm u)) % S\left(\frac{\arctan(\alpha S)}{\pi} + \frac{1}{2} \right) - \frac{\arctan(\alpha)}{\pi} + \frac{1}{2}$$ where $g(S)$ is a smooth approximation of the following step function $$\label{eq8g} \tilde{g}(S) = \left\{ \begin{array}{cl} 0 & \mbox{if } S < 0, \\ S & \mbox{if } 0 \le S \le s_{\rm max}, \\ s_{\rm max} & \mbox{if } S > s_{\rm max} . \end{array} \right.$$ The quantity $S(\bm u, \nabla \bm u)$ is a measure of the shock strength which is given by $$S(\bm u, \nabla \bm u) = -\nabla \cdot \bm v \ ,$$ where $\bm v$ is the non-dimensional velocity field that is determined from the state vector $\bm u$. The use of the velocity divergence as shock strength for defining an artificial viscosity field follows from [@Fernandez2018; @Moro2016; @Nguyen2011a]. The parameter $s_{\max}$ is used to put an upper bound on the source term when the divergence of the velocity becomes too negatively large. Herein we choose $s_{\max} = 0.5 \|S\|_\infty$, where $\|S\|_{\infty} = \max_{\bm x \in \Omega} |S(\bm x)|$ is the $L_\infty$ norm. It remains to determine $\lambda_1$ and $\lambda_2$ in order to close the system ([\[eq3\]](#eq3){reference-type="ref" reference="eq3"}). We propose to solve the following minimization problem [\[eq5\]]{#eq5 label="eq5"} $$\begin{aligned} {2} \min_{\lambda_1 \in \mathbb{R}^+, \lambda_2 \ge 1, \bm u, \eta} & \quad \lambda_1 \lambda_2 \\ \mbox{s.t.} & \quad \mathcal{L}(\bm u, \eta, \bm \lambda) = 0 \\ & \quad \bm u \in \mathcal{C} .\end{aligned}$$ Here $\mathcal{L}$ represents the spatial discretization of the coupled system ([\[eq3\]](#eq3){reference-type="ref" reference="eq3"}) by a numerical method and $\mathcal{C}$ represents a set of constraints on the numerical solution. The objective function is to minimize the amount of artificial viscosity which is proportional to $\lambda_1 \lambda_2$. The constraints are specified to rule out unwanted solutions of the discrete system ([\[eq5\]](#eq5){reference-type="ref" reference="eq5"}b) and play an important role in yielding a high-quality numerical solution. Hence, the optimization problem ([\[eq5\]](#eq5){reference-type="ref" reference="eq5"}) is to minimize the amount of artificial viscosity while ensuring the smoothness of the numerical solution. ## Solution constraints We introduce the constraints to ensure the quality of the numerical solution. The physical constraints are that pressure and density must be positive. In order to establish a smoothness constraint on the numerical solution, we express an approximate scalar variable $\xi$ of degree $k$ within each element in terms of an orthogonal basis and its truncated expansion of degree $k-1$ as $$\xi = \sum_{i=1}^{N(k)} \xi_i \psi_i, \qquad \xi^* = \sum_{i=1}^{N(k-1)} \xi_i \psi_i$$ where $N(k)$ is the total number of terms in the $k$-degree expansion and $\psi_i$ are the basis functions [@persson06:_shock_capturing]. Here $\xi$ is chosen to be either density, pressure, or local Mach number. We introduce the following quantity $$\label{eq10} \sigma(\bm \lambda) = \max_{K \in \mathcal{T}_h^{\rm shock}} \sigma_K(\bm \lambda), \qquad \sigma_K(\bm \lambda) \equiv \frac{\int_K |\xi/\xi^* - 1| d \bm x}{\int_K d \bm x} ,$$ where $\mathcal{T}_h^{\rm shock}$ is the set of elements defining the shock region $$\mathcal{T}_h^{\rm shock} = \{K \in \mathcal{T}_h \ : \ \int_K \bar{\eta} d \bm x \ge \bar{\eta}_{\rm T} |K| \}$$ and $\mathcal{T}_h$ is a collection of high-order elements on the physical domain $\Omega$ $$\label{homesh} \mathcal{T}_h = \{K_n \in \Omega \ : \cup_{n=1}^{N_e} \bar{K}_n = \bar{\Omega}, \bm x|_{K_n} \in [\mathcal{P}_k(K_{\rm ref})]^d, 1 \le n \le N_{\rm e}\} .$$ Here $N_e$ is the number of elements, $K_{\rm ref}$ is the master element, and $\mathcal{P}_k(K_{\rm ref})$ is the space of polynomials of degree $k$ on $K_{\rm ref}$. The constraint set $\mathcal{C}$ in ([\[eq5\]](#eq5){reference-type="ref" reference="eq5"}) consists of the following contraints $$\rho(\bm x) > 0, \quad p(\bm x) > 0, \quad \sigma(\bm \lambda) \le C_0 \, \sigma(\bm \lambda_0) ,$$ where $\bm \lambda_0$ is an initial value and the constant $C_0$ is set to 5. The first two constraints enforce the positivity of density and pressure, while the last constraint guarantees the smoothness of the numerical solution. The smoothness constraint imposes a degree of regularity on the numerical solution and plays a vital role in yielding sharp and smooth solutions. ## Homotopy continuation of the regularization parameters The pair of regularization parameters $\bm \lambda = (\lambda_1, \lambda_2)$ controls the magnitude and thickness of the artificial viscosity in order to obtain accurate solutions. On the one hand, if $\bm \lambda$ is too small then the numerical solution can develop oscillations across the shock waves. On the other hand, if $\bm \lambda$ is too large the solution becomes less accurate in the shock region, which in turn affects the accuracy of the solution in the remaining region. Therefore, we propose a homotopy continuation method to determine $\bm \lambda$. The key idea is to solve the regularized system with a large value of $\bm \lambda$ first and then gradually decrease $\bm \lambda$ until any of the physics or smoothness constraints on the numerical solution are violated. At this point, we take the value of $\bm \lambda$ from the previous iteration where the numerical solution still satisfies all of the physics and smoothness constraints. This procedure is summarized in the following algorithm: ## Finite element approximations {#sec:FEmethods} The homotopy continuation solves the Helmholtz equation ([\[eq3\]](#eq3){reference-type="ref" reference="eq3"}b) separately from the regularized system ([\[eq3\]](#eq3){reference-type="ref" reference="eq3"}a). Hence, different numerical methods can be used to solve ([\[eq3\]](#eq3){reference-type="ref" reference="eq3"}a) and ([\[eq3\]](#eq3){reference-type="ref" reference="eq3"}b) separately. In this paper, we employ the hybridizable discontinuous Galerkin (HDG) method to solve the former and the continuous Galerkin (CG) method to solve the latter. We use the CG method since it allows us to obtain a continuous artificial viscosity field. The HDG method [@Nguyen2012; @Fernandez2018a; @Moro2011a; @Peraire2010; @Vila-Perez2021; @Woopen2014c; @Fidkowski2016; @Fernandez2017a; @williams2018entropy] is suitable for solving the regularized conservation laws because of its efficiency and high-order accuracy. # Mesh adaptation via optimal transport {#sec:ot} ## Optimal transport theory The optimal transport (OT) problem is described as follows. Suppose we are given two probability densities: $\varrho(\bm x)$ supported on $\Omega \in \mathbb{R}^d$ and $\varrho'(\bm x')$ supported on $\Omega' \in \mathbb{R}^d$. The source density $\varrho(\bm x)$ may be discontinuous and even vanish. The target density $\varrho'(\bm x')$ must be strictly positive and Lipschitz continuous. The OT problem is to find a map $\bm \phi : \Omega \to \Omega'$ such that it minimizes the following functional $$\inf_{\bm \phi \in \mathcal{M}} \int_{\Omega} \|\bm x - \bm \phi(\bm x)\|^2 \varrho(\bm x) d \bm x ,$$ where $$\label{otset} \mathcal{M} = \{\bm \phi : \Omega \to \Omega', \ \varrho'(\bm \phi(\bm x)) \det (\nabla \bm \phi(\bm x)) = \varrho(\bm x), \ \forall \bm x \in \Omega \} ,$$ is the set of mappings which map the source density $\varrho(\bm x)$ onto the target density $\varrho'(\bm x')$. Here $\det$ denotes determinants for $d \times d$ matrices. Whenever the infimum is achieved by some map $\bm \phi$, we say that $\bm \phi$ is an optimal map. In [@Brenier1991], Brenier gave the proof of the existence and uniqueness of the solution of the OT problem. Furthermore, the optimal map $\bm \phi$ can be written as the gradient of a unique (up to a constant) convex potential $u$, so that $\bm \phi(\bm x) = \nabla u(\bm x)$, $\Delta u(\bm x) > 0$. Substituting $\bm \phi(\bm x) = \nabla u(\bm x)$ into ([\[otset\]](#otset){reference-type="ref" reference="otset"}) results in the Monge--Ampère equation $$\label{mae} \varrho'(\nabla u(\bm x)) \det (D^2 u(\bm x)) = \varrho(\bm x) \quad \mbox{in } \Omega,$$ along with the restriction that $u$ is convex. This equation lacks standard boundary conditions. However, it is geometrically constrained by the fact that the gradient map takes $\partial \Omega$ to $\partial \Omega'$: $$\label{maebc} \nabla u(\bm x) \in \partial \Omega', \quad \forall \bm x \in \Omega .$$ This constraint is referred to as the second boundary value problem for the Monge--Ampère equation. If the boundary $\partial \Omega'$ can be expressed by $$\partial \Omega' = \{\bm x' \in \Omega' : c(\bm x') = 0\},$$ then the boundary constraint ([\[maebc\]](#maebc){reference-type="ref" reference="maebc"}) becomes the following Neumann boundary condition $$\label{maenm} c(\nabla u(\bm x)) = 0, \quad \forall \bm x \in \partial \Omega .$$ The scalar potential $u$ is required to satisfy $\int_{\Omega} u(\bm x) d \bm x = 0$ for uniqueness. For problems where densities are periodic, it is natural and convenient to use periodic boundary conditions instead. ## Equidistribution principle Mesh adaptation is based on the equidistribution principle that equidistributes the target density function $\varrho'$ so that the source density $\varrho$ is uniform on $\Omega$ [@Delzanno2008; @Chacon2011]. The equidistribution principle leads to a constant source density $\varrho(\bm x) = \theta$, where $\theta = \int_{\Omega'} \varrho'(\bm x') d \bm x' / \int_{\Omega} d \bm x$. Using the optimal transport theory, the optimal map is sought by solving the Monge--Ampère equation: $$\label{maem} \begin{split} \varrho'(\nabla u(\bm x)) \det (D^2 u(\bm x)) & = \theta, \quad \mbox{in } \Omega, \\ c(\nabla u(\bm x)) & = 0, \quad \mbox{on } \partial \Omega , \end{split}$$ with the constraint $\int_{\Omega} u(\bm x) d \bm x = 0$. In the context of mesh adaptation, the target boundary $\partial \Omega'$ coincides with $\partial \Omega$. Hence, the root of the equation $c(\bm x) = 0$ defines $\partial \Omega$. It means that $c(\nabla u(\bm x)) = c(\bm x) = 0, \forall \bm x \in \partial \Omega$. ## Mesh density function In the context of mesh adaptation, $\varrho'(\bm x')$ is the mesh density function and $\mathcal{T}_h$ is the initial mesh. The optimal map $\bm \phi(\bm x) = \nabla u(\bm x)$ drives the coordinates of the initial mesh to concentrate around a region where the mesh density function is high. Therefore, we need to make $\varrho'(\bm x')$ large in the shock region and small in the smooth region. It is also necessary for $\varrho'(\bm x')$ to be sufficiently smooth, so that the numerical approximation of the Monge--Ampère equation ([\[maem\]](#maem){reference-type="ref" reference="maem"}) is convergent. To this end, we compute $\varrho'(\bm x')$ as solution of the Helmholtz equation $$\label{hmmd} \varrho'(\bm x') - \nabla \cdot \left( \ell^2 \nabla \varrho'(\bm x') \right) = b(\bm x') \quad \mbox{in }\Omega,$$ with homogeneous Neumann boundary condition. Here $b$ is a resolution indicator function that is large in the shock region and small elsewhere. We explore two different options for the indicator function. The first option is to define it as a function of the velocity divergence as $$b(\bm x') = \sqrt{1 + \beta s(\bm u, \nabla \bm u)}$$ where $s$ is given by ([\[avsource\]](#avsource){reference-type="ref" reference="avsource"}) and $\beta$ is a specified constant. The second option is to define it as a function of the density gradient as $$\label{mdf2} b(\bm x') = \sqrt{1 + \beta g(|\nabla \rho(\bm x')|) }$$ where $g(\cdot)$ is given by ([\[eq8g\]](#eq8g){reference-type="ref" reference="eq8g"}). Other indicator functions are possible, such as those based on some combination of physics-based sensors that can distinguish between shocks, large temperature gradients, and other sharp features. The Helmholtz equation ([\[hmmd\]](#hmmd){reference-type="ref" reference="hmmd"}) is numerically solved by using the CG method in which the same polynomial spaces are used to represent both the numerical solution and the geometry. In this case, the value of the mesh density function at any given point $\bm x' \in K_i \subset \mathcal{T}_h$ is calculated as $$\label{mdp1} \varrho'(\bm x')|_{K_i} = \sum_{j=1}^{N_p} \rho_{ij} \varphi_j(\bm \xi(\bm x'))$$ where $\bm \xi(\bm x')$ is found by solving the following system $$\label{mdp2} \sum_{j=1}^{N_p} \bm x_{ij} \varphi_j(\bm \xi) = \bm x' .$$ Here $N_p$ is the number of polynomials per element, $\bm x_{ij}$ are the mesh nodes on element $K_n$, $\rho_{ij}$ are the degrees of freedom of the function $\rho'$ on $K_n$, and $\varphi_j(\bm \xi), 1 \le j \le N_p,$ are polynomials of degree $k$ defined on the master element $K_{\rm ref}$. We note that the system ([\[mdp2\]](#mdp2){reference-type="ref" reference="mdp2"}) is linear for $k=1$ and nonlinear for $k > 1$. The mesh density function is the numerical solution of the Helmholtz equation ([\[hmmd\]](#hmmd){reference-type="ref" reference="hmmd"}) whose source term depends on the flow state $\bm u$. In practice, we compute the approximate solution of the flow state by using the adaptive viscosity regularization method to solve the problem ([\[eq5\]](#eq5){reference-type="ref" reference="eq5"}) on the initial mesh $\mathcal{T}_h$ or on the previous adaptive mesh during the mesh adaptation procedure described in subsection 3.5. ## Numerical solution of the Monge--Ampère equation In a recent paper [@nguyen2023hybridizable], we introduce HDG methods for numerically solving the Monge--Ampère equation in which the mesh density function is an analytical function. In order to solve the Monge--Ampère equation in which the mesh density function is approximated by local spaces of polynomials in ([\[mdp1\]](#mdp1){reference-type="ref" reference="mdp1"})-([\[mdp2\]](#mdp2){reference-type="ref" reference="mdp2"}), we propose to extend the HDG methods introduced in [@nguyen2023hybridizable]. In two dimensions, the Monge-Ampère equation ([\[maem\]](#maem){reference-type="ref" reference="maem"}) can be rewritten as a first-order system of equations $$\begin{array}{rcll} \bm{H} - \nabla \bm q & = & 0, \quad & \mbox{in } \Omega, \\ \bm{q} - \nabla u & = & 0, \quad & \mbox{in } \Omega, \\ f(\bm H, \bm q) - \nabla \cdot \bm q & = & 0 , \quad & \mbox{in } \Omega, \\ c(\bm q) & = & 0, \quad & \mbox{on } \partial \Omega , \\ \int_{\Omega} u(\bm x) d \bm x &= & 0, \end{array} \label{maem3}$$ where $f(\bm H,\bm q) = \sqrt{H_{11}^2 + H_{22}^2 + H_{12}^2 + H_{21}^2 + 2 \theta/\varrho'(\bm q)}$. The HDG discretization of the system ([\[maem3\]](#maem3){reference-type="ref" reference="maem3"}) is to find $(\bm{H}_h, \bm{q}_h,u_h,\widehat{u}_h) \in \bm{W}_{h}^p \times \bm{V}_{h}^p \times U_{h}^p \times M_h^p$ such that $$\label{eq82} \begin{array}{rcl} \left(\bm{H}_h, \bm{G}\right)_{\mathcal{T}_h} + \left(\bm q_h, \nabla \cdot \bm{G}\right)_{\mathcal{T}_h} - \left\langle \widehat{\bm q}_h, \bm{G} \cdot \bm{n} \right\rangle_{\partial \mathcal{T}_h} & = & 0, \\ \left(\bm{q}_h, \bm{v}\right)_{\mathcal{T}_h} + \left(u_h, \nabla \cdot \bm{v}\right)_{\mathcal{T}_h} - \left\langle \widehat{u}_h, \bm{v} \cdot \bm{n} \right\rangle_{\partial \mathcal{T}_h} & = & 0, \\ \left(\bm{q}_h, \nabla w\right)_{\mathcal{T}_h} - \left\langle \widehat{\bm{q}}_h \cdot \bm{n}, w \right\rangle_{\partial \mathcal{T}_h} + (f(\bm H_h, \bm q_h),w)_{\mathcal{T}_h} & = & 0, \\ \left\langle \widehat{\bm{q}}_h \cdot \bm{n} , \mu \right\rangle_{\partial \mathcal{T}_h \backslash \partial \Omega} + \left\langle c(\bm q_h) + \tau (\widehat{u}_h - u_h), \mu \right\rangle_{\partial \Omega} & = & 0, \\ (u_h,1)_{\mathcal{T}_h} & = & 0, \end{array}$$ for all $(\bm G, \bm{v}, w, \mu) \in \bm{W}_h^p \times \bm{V}_h^p \times U_h^p \times M_h^p$, where $$\widehat{\bm{q}}_h = {\bm{q}_h} - \tau (u_h - \widehat{u}_h) \bm{n}, \quad \mbox{on } \mathcal{E}_h. \label{fluxdef2}$$ We are going to use the fixed point method to solve this nonlinear system of equations. To deal with the nonlinear boundary condition $c(\bm q) = 0$, we linearize it around the previous solution $\bm q^{\ell -1}$ to obtain $$c(\bm q^{l-1}) + \partial c_{\bm q}(\bm q^{l-1}) \cdot \left( \bm q^{l} - \bm q^{l-1} \right) = 0 ,$$ where $\partial c_{\bm q}$ denotes the partial derivative of $c$ with respect to $\bm q$. Starting from an initial guess $(\bm H^0_h, \bm q_h^0, u_h^0)$ we find $(\bm{q}_h^l,u_h^l,\widehat{u}_h^l) \in \bm{V}_{h}^k \times U_{h}^k \times M_h^k$ such that $$\label{nfpHDG} \begin{array}{rcl} \left(\bm{q}_h^l, \bm{v}\right)_{\mathcal{T}_h} + \left(u_h^l, \nabla \cdot \bm{v}\right)_{\mathcal{T}_h} - \left\langle \widehat{u}_h^l, \bm{v} \cdot \bm{n} \right\rangle_{\partial \mathcal{T}_h} & = & 0, \\ \left(\bm{q}_h^l, \nabla w\right)_{\mathcal{T}_h} - \left\langle \widehat{\bm{q}}_h^l \cdot \bm{n}, w \right\rangle_{\partial \mathcal{T}_h} & = & - (f(\bm H_h^{l-1}, \bm q_h^{l-1}),w)_{\mathcal{T}_h}, \\ \left\langle \widehat{\bm{q}}_h^l \cdot \bm{n} , \mu \right\rangle_{\partial \mathcal{T}_h \backslash \partial \Omega} + \left\langle \partial c_{\bm q}(\bm q_h^{l-1}) \cdot \bm q_h^l + \tau ( \widehat{u}_h^l - u_h^l) , \mu \right\rangle_{\partial \Omega} & = & -\left\langle a(\bm q^{l-1}) , \mu \right\rangle_{\partial \Omega}, \\ (u_h^l,1)_{\mathcal{T}_h} & = & 0, \end{array}$$ for all $(\bm{v}, w, \mu) \in \bm{V}_h^k \times U_h^k \times M_h^k$, and then compute $\bm{H}_h^l\in \bm{W}_{h}^k$ such that $$\label{neqH} \left(\bm{H}_h^l, \bm{G}\right)_{\mathcal{T}_h} = - \left(\bm q_h^l, \nabla \cdot \bm{G}\right)_{\mathcal{T}_h} + \left\langle \widehat{\bm q}_h^l, \bm{G} \cdot \bm{n} \right\rangle_{\partial \mathcal{T}_h}, \quad \forall \bm G \in \bm{W}_h^k .$$ Note here that $a(\bm q_h^{l-1}) = c(\bm q_h^{l-1}) - \partial c_{\bm q}(\bm q_h^{l-1}) \cdot \bm q_h^{l-1}$, and that the numerical flux $\widehat{\bm q}_h^l$ is defined by ([\[fluxdef2\]](#fluxdef2){reference-type="ref" reference="fluxdef2"}). We refer to [@nguyen2023hybridizable] for the definition of the finite element spaces associated with the fixed-point HDG formulation ([\[nfpHDG\]](#nfpHDG){reference-type="ref" reference="nfpHDG"})-([\[neqH\]](#neqH){reference-type="ref" reference="neqH"}) and the detailed implementation. At each iteration of the fixed-point HDG method, the weak formulation ([\[nfpHDG\]](#nfpHDG){reference-type="ref" reference="nfpHDG"}) yields a matrix system which can be solved efficiently by locally eliminating the degrees of freedom of $(\bm{q}_h^l,u_h^l)$ to obtain a global linear system in terms of the degrees of freedom of $\widehat{u}_h^l$. While it is straightforward to form the matrix, computing the right-hand side vector is more complicated because we need to evaluate $f(\bm H_h^{l-1}, \bm q_h^{l-1})$. Henceforth, we must compute $\varrho'(\bm q_h^{l-1})$ by replacing $\bm x'$ with $\bm q_h^{l-1}$ in ([\[mdp1\]](#mdp1){reference-type="ref" reference="mdp1"}) and solve the resulting system ([\[mdp2\]](#mdp2){reference-type="ref" reference="mdp2"}) by using Newton's method for all quadrature points. ## Mesh adaptation procedure We start mesh adaptation with an initial mesh $\mathcal{T}_h$ and compute the initial solution $\bm u_h$. Next, we compute a mesh density function based on $\bm u_h$ and solve the Monge-Ampère equation to obtain an adaptive mesh $\mathcal{T}_h^*$. Finally, we interpolate $\bm u_h$ onto $\mathcal{T}_h^*$ and use it as an initial guess to solve for the final solution $\bm u_h^*$ on the adaptive mesh. The mesh adaptation procedure is described in Algorithm 1. The adaptation procedure can be repeated by using the adaptive mesh as an initial mesh in the next iteration until $\|\bm u_h^* - \bm u_h\|_{\Omega}$ is less than a specified tolerance. It should be pointed out that we do not perform the homotopy continuation at every mesh adaptation iterations. We perform the homotopy continuation to compute the numerical solution on the final adaptive mesh only. This will considerably reduce the number of times we solve the compressible Euler/Navier-Stokes equations. We demonstrate the action of Algorithm 1 by applying it to an inviscid supersonic flow in a channel with a 4% thick circular bump [@Nguyen2011d]. The length and height of the channel are 3 and 1, respectively. The inlet Mach number is $M_\infty = 1.4$. Supersonic inlet/outlet conditions are prescribed at the left/right boundaries, while inviscid wall boundary condition is used on the top and bottom sides. Isoparametric elements with the polynomials of degree $k = 4$ are used to represent both the numerical solution and geometry. Representative inputs and outputs are shown in Figure 1. {width="\\textwidth"} {width="\\textwidth"} \ {width="\\textwidth"} {width="\\textwidth"} \ {width="\\textwidth"} {width="\\textwidth"} # Numerical Results {#sec:results} In this section, we present numerical results for a number of well-known test cases to demonstrate the proposed approach. Unless otherwise specified, polynomial degree $k = 4$ is used to represent both the numerical solution and the geometry. Although the polynomial degree $k=4$ is relatively high for shock flows, our approach can compute the numerical solution without using the solutions computed with lower polynomial degrees. ## Inviscid transonic flow past NACA 0012 airfoil The first test case is an inviscid transonic flow past a NACA 0012 airfoil at angle of attack $\alpha = 1.5^{\rm o}$ and freestream Mach number $M_\infty = 0.8$ [@Nguyen2011a]. Slip velocity boundary condition is imposed on the airfoil, while far-field boundary condition is imposed on the rest of the boundary. A shock is formed on the upper surface, while another weaker shock is formed under the lower surface. Figure [\[fignaca2\]](#fignaca2){reference-type="ref" reference="fignaca2"} depicts the initial mesh and three consecutive adaptive meshes near the airfoil surface. Figure [\[fignaca3\]](#fignaca3){reference-type="ref" reference="fignaca3"} shows the Mach number computed on the initial mesh and the r-adaptive meshes. {width="\\textwidth"} {width="\\textwidth"} {width="\\textwidth"} {width="\\textwidth"} {width="\\textwidth"} {width="\\textwidth"} {width="\\textwidth"} {width="\\textwidth"} It is interesting to see how the elements of the initial mesh are moved to create new meshes that align well with the shocks. The results also show how the numerical solution is improved and how the shocks are better resolved over each iteration of the mesh adaptation procedure. We observe that the shocks are well resolved on the final adaptive mesh and that the solution on the final mesh is accurate. This can be clearly seen from the profiles of the computed pressure coefficient shown in Figure [1](#fignaca4){reference-type="ref" reference="fignaca4"}. We see that the pressure coefficient profiles converge rapidly and that the profile computed on the second adaptive mesh is very similar to that computed on the third adaptive mesh. We emphasize that the profile on the third adaptive mesh is very sharp at the shocks, yet there is no oscillation and overshoot. {#fignaca4 width="65%"} ## Inviscid supersonic flow over a double ramp This test case is used in [@carnes2019code] as a building block towards more complicated double wedge and cone flows. The geometry is a double-ramp with a $25^{\circ}$ incline for the first ramp and $37^{\circ}$ incline for the second. Note that the second angle is shallower than typical hypersonic double wedge or cone flows [@olejniczak1997numerical]. We consider supersonic flow at a free-stream Mach number of 3.6, for which the resulting flow-field is relatively simple. Two shocks are expected to emanate from the corners and intersect to form a third shock. The purpose of this example is to examine the ability of the Monge-Ampère solver to refine the mesh on polygonal domains with flow over corners of the domain. The boundary consists of six line segments $\{\bm \Gamma_i\}_{i=1}^6$ defined as $c_i(\bm x) := \bm A_i \bm x + \bm b_i = 0$. Enforcing that each $\bm q_h$ on $\Omega_i$ must satisfy $c_i(\bm q_h) = 0$ led to meshes that would detach at the corners, hampering convergence. This is demonstrated in Figure [\[fig:cornersep\]](#fig:cornersep){reference-type="ref" reference="fig:cornersep"} with an artificial target density. Whether this phenonema is a result of the HDG discretization or the formulation of Monge-Amper̀e on this domain remains to be determined. {width="80%"} {width="80%"} {width="80%"} This issue is addressed by changing the Neumann boundary condition to obey a global description of the geometry; instead of enforcing that $\bm q_h$ at $\bm \Gamma_i$ must satisfy $c_i(\bm q_h) = 0$, it is allowed to transition onto adjacent faces if the value of $\bm q_h$ leaves the bounds of $\bm \Gamma_i$. In this way, boundary nodes are allowed to slide along the boundary and move from one face to another. Other $r$-adaptive methods have found it advantageous to fix nodes at boundaries rather than let them transition from one boundary to another. Since the domain mapping is determined as the gradient of a scalar potential, we cannot explicitly fix the location of certain nodes. Instead, after the adaptive mesh is formed, the element that crosses a corner is identified and its closest vertex is moved to that same corner, in order to not change the definition of the geometry. This procedure is illustrated in Figure [\[fig:corners\]](#fig:corners){reference-type="ref" reference="fig:corners"}. {width="80%"} {width="80%"} The starting grid consists of 909 elements and polynomial order $k=3$. The results on the initial mesh are shown in Figure [\[fig:wedge\]](#fig:wedge){reference-type="ref" reference="fig:wedge"}. We use the sensor based off the gradient of the physical density [\[mdf2\]](#mdf2){reference-type="eqref" reference="mdf2"} with $\beta = 1$ in order to get some refinement along the contact discontinuity, which would be missed with the sensor based off the divergence of the velocity. While the starting mesh is fine enough to capture the density and pressure well, visible oscillations are present in the Mach number field. These oscillations are not visible with mesh adaptation and the primary shocks and contact discontinuities are sharper than on the starting mesh. See Figure [\[fig:wedgeadapt\]](#fig:wedgeadapt){reference-type="ref" reference="fig:wedgeadapt"}. {width="80%"} {width="80%"} {width="80%"} {width="80%"} {width="80%"} {width="80%"} ## Inviscid hypersonic flow past unit circular cylinder This test case involves hypersonic flow past a unit circular cylinder at $M_\infty = 7$ and serves to demonstrate the effectiveness of our approach for strong bow shocks in the hypersonic regime . The cylinder wall is modeled with slip wall boundary condition. Supersonic outflow condition is used at the outlet, while supersonic inflow condition is imposed at the inlet. Figure [\[cyl7a\]](#cyl7a){reference-type="ref" reference="cyl7a"} shows the mesh density functions used in the numerical solution of the Monge-Ampère equation to generate the three adaptive meshes shown in Figure [\[cyl70\]](#cyl70){reference-type="ref" reference="cyl70"}. These mesh density functions are computed from the mesh indicator ([\[mdf2\]](#mdf2){reference-type="ref" reference="mdf2"}) with using the numerical solutions on the initial mesh, the 1st adaptive mesh, and the 2nd adaptive mesh. We notice that the amplitude of the mesh density function increases with the mesh adaptation iteration because the numerical solution becomes sharper due to better resolution of the bow shock. This is because the optimal transport moves the elements toward the shock region and aligns them along the shock curves according to the mesh density function. {width="60%"} {width="66%"} {width="65%"} {width="70%"} {width="70%"} {width="70%"} {width="70%"} Figure [9](#cyl71){reference-type="ref" reference="cyl71"} depicts the numerical solution computed on the initial and adaptive meshes. We see how the artificial viscosity fields are reduced in amplitude and width as the mesh adaptation procedure iterates. The numerical solution computed on the final adaptive mesh is clearly more accurate than those computed on the previous adaptive meshes. This can also be seen in Figure [11](#cyl7b){reference-type="ref" reference="cyl7b"} which shows the profiles of pressure and Mach number along the line $y=0$. We see that these profiles converge rapidly with the adaptation iteration. The profiles computed on the second adaptive mesh are close to those computed on the third adaptive mesh, which are sharp and smooth. There is no oscillation and overshoot in the numerical solution on the final adaptive mesh. These results demonstrate the robustness of the proposed approach for strong bow shocks. {#cyl71 width="80%"} {#cyl71 width="80%"} {#cyl71 width="80%"} {#cyl71 width="80%"} \ {#cyl71 width="70%"} {#cyl71 width="70%"} {#cyl71 width="70%"} {#cyl71 width="70%"} \ {width="77%"} {width="77%"} {width="77%"} {width="77%"} {#cyl7b width="\\textwidth"} {#cyl7b width="\\textwidth"} ## Inviscid type IV shock-shock interaction Type IV Shock-shock interaction results in a very complex flow field with high pressure and heat flux peak in localized region. It occurs when the incident shock impinges on a bow shock and results in the formation of a supersonic impinging jet, a series of shock waves, expansion waves, and shear layers in a local area of interaction. The supersonic impinging jet, which is bounded by two shear layers separating the jet from the upper and lower subsonic regions, impinges on the body surface, and is terminated by a jet bow shock just ahead of the surface. This impinging jet bow shock wave creates a small stagnation region of high pressure and heating rates. Meanwhile, shear layers are formed to separate the supersonic jet from the lower and upper subsonic regions. Type IV hypersonic flows were experimentally studied by Wieting and Holden [@WIETING1989]. Over the years, many numerical methods have been used in the study of type IV shock-shock interaction [@Hsu1996; @Nguyen2020gpu; @Thareja1989; @Yamamoto1998; @Xu2005; @Zhong1994a]. In the present work, we consider an inviscid type IV interaction with freestream Mach number $M_\infty = 8.03$. Based on the experimental measurement and the numerical calculations, Thareja et al. [@Thareja1989] summarized that the position of incident impinging shock on the cylinder can be approximated by the curve $y = 0.3271x + 0.4147$ for the experiment (Run 21) [@WIETING1989]. Boundary conditions are the same as those for the test case presented in Subsection 4.2, where the freemstream state $\bm u_\infty$ is represented by a hyperbolic tangent function to account for the incident impinging shock. Figure [\[typeiva\]](#typeiva){reference-type="ref" reference="typeiva"} shows the initial and adaptive meshes as well as the mesh density functions used to obtain the adaptive meshes. The mesh density functions are computed from the mesh indicator ([\[mdf2\]](#mdf2){reference-type="ref" reference="mdf2"}) with using the numerical solutions on the initial mesh and the 1st adaptive mesh. The optimal transport moves the elements toward the shock region and aligns them along the shock curves. Furthermore, it also distributes elements around supersonic impinging jet, jet bow shock, expansion waves, and shear layers according to the mesh density function. As a result, the optimal transport can adapt meshes to capture complicated flow features without increasing the number of elements and modifying data structure. {width="\\textwidth"} {width="\\textwidth"} {width="85%"} {width="\\textwidth"} {width="\\textwidth"} We present the numerical solution computed on the initial mesh in Figure [\[typeivc\]](#typeivc){reference-type="ref" reference="typeivc"} and on the second adaptive mesh in Figure [\[typeivd\]](#typeivd){reference-type="ref" reference="typeivd"}. We notice that the numerical solution on the second adaptive mesh reveals supersonic impinging jet, jet bow shock, expansion waves, and shear layers of the flow, whereas the solution on the initial mesh does not possess some of these features. This is because the initial mesh does not have enough grid points to resolve those features even though it has the same number of elements as the second adaptive mesh. By redistributing the elements of the initial mesh to resolve shocks, impinging jet, jet bow shock, expansion waves, and shear layers, the optimal transport considerably improves the numerical solution. This test case shows the ability of the optimal transport for dealing with complex shock flows. {width="\\textwidth"} {width="\\textwidth"} {width="\\textwidth"} {width="\\textwidth"} {width="\\textwidth"} {width="\\textwidth"} {width="\\textwidth"} {width="\\textwidth"} Finally, we present in Figure [12](#typeive){reference-type="ref" reference="typeive"} the profiles of the computed pressure along the cylindrical surface, where the symbols $\circ$ are the experimental data [@WIETING1989]. We see that the pressure profile computed on the second adaptive mesh has larger peak than those on the initial mesh and the first adaptive mesh. This is because the second adaptive mesh has a lot more elements in the supersonic jet region than the initial mesh and the first adaptive mesh. As a result, the computed pressure on the second adaptive mesh agrees with the experimental measurement better than those on the other meshes. ![Profiles of the computed pressure ratio $p/p_0$ along the cylindrical surface for the type IV case. Here $p_0$ is the pressure at the stagnation point for inviscid hypersonic flow past the cylinder at $M_\infty = 8.03$. The symbols $\circ$ are the experimental data [@WIETING1989].](figures/typeiv_surfacepressure.png){#typeive width="70%"} ## Viscous hypersonic flow past unit circular cylinder The last test case involves viscous hypersonic flow past unit circular cylinder at $M_\infty = 17.6$ and $Re=376,000$. The freestream temperature is $T_\infty = 200^{\rm o}$ K. The cylinder surface is isothermal with wall temperature $T_{\rm wall} = 500^{\rm o}$ K. Supersonic inflow and outflow boundary conditions are imposed at the inlet and outlet, respectively. This test case serves to demonstrate the ability of the optimal transport approach to deal with very strong bow shocks and extremely thin boundary layers. This problem was studied by Gnoffo and White [@Gnoffo2004] comparing the structured code LAURA and the unstructured code FUN3D. The simple geometry and strong shock make it a common benchmark case for assessing the performance of numerical methods and solution algorithms in hypersonic flow predictions [@Barter2010; @Ching2019; @Gnoffo2004a; @Kitamura2013; @Nguyen2020gpu]. This test case will demonstrate the ability of the optimal transport for dealing with very strong bow shock and extremely thin boundary layer. Figure [\[cyl18a\]](#cyl18a){reference-type="ref" reference="cyl18a"} shows the initial and adaptive meshes as well as the mesh density function used to obtain the adaptive mesh. The mesh density function is computed from the mesh indicator ([\[mdf2\]](#mdf2){reference-type="ref" reference="mdf2"}) with using the numerical solutions on the initial mesh. The optimal transport moves the elements of the initial mesh toward the shock and the boundary layer regions because the mesh density function is high in those regions. As a result, the optimal transport can adapt meshes to capture shocks and resolve boundary layers. To see this feature more clearly, in Figure [14](#cyl18b){reference-type="ref" reference="cyl18b"}, we plot $\log_{10}(h_n)$ as a function of $n$ for both the initial mesh and the adaptive mesh, where $h_n$ denotes the element size of an $n$th element starting from the cylinder wall along the horizontal line $y=0$. We see that the adaptive mesh has smaller element sizes than the initial mesh near the wall and in the shock region. As a result, the adaptive mesh should be able to resolve the boundary layer and shock better than the initial mesh. {width="68%"} {width="90%"} {width="90%"} {width="68%"} {#cyl18b width="\\textwidth"} {#cyl18b width="\\textwidth"} We present the numerical solution computed on the initial mesh in Figure [\[cyl18c\]](#cyl18c){reference-type="ref" reference="cyl18c"} and on the adaptive mesh in Figure [\[cyl18d\]](#cyl18d){reference-type="ref" reference="cyl18d"}. We observe that pressure and temperature rise rapidly behind the bow shock, which create very strong pressure and high temperature environments surrounding the cylinder. In addition, Figure [18](#cyl18e){reference-type="ref" reference="cyl18e"} shows profiles of the numerical solution along the horizontal line $y=0$. We notice that the numerical solution on the adaptive mesh has higher gradient than that on the initial mesh in the shock region and boundary layer. This is because the adaptive mesh has more grid points to resolve those features than the initial mesh. By redistributing the elements of the initial mesh to resolve the bow shock and boundary layer, the optimal transport can considerably improve the prediction of heating rate as shown in Figure [20](#cyl18f){reference-type="ref" reference="cyl18f"}. We see that while the pressure coefficient on the initial mesh is very similar to that on the adaptive mesh, the heat transfer coefficient on the initial mesh is lower than that on the adaptive mesh. The heat transfer coefficient on the adaptive mesh agrees very well with the prediction by Gnoffo and White [@Gnoffo2004]. {width="\\textwidth"} {width="\\textwidth"} {width="\\textwidth"} {width="\\textwidth"} {width="\\textwidth"} {width="\\textwidth"} {width="\\textwidth"} {width="\\textwidth"} {#cyl18e width="\\textwidth"} {#cyl18e width="\\textwidth"} \ {#cyl18e width="\\textwidth"} {#cyl18e width="\\textwidth"} ![Pressure coefficient (left) and heat transfer coefficient (right) along the cylinder surface for the viscous hypersonic flow past a circular cylinder. The lines in the third legend correspond to the results obtained using LAURA code by Gnoffo and White [@Gnoffo2004].](figures/nsmahcyl18_pressurecoefficient.png){#cyl18f width="\\textwidth"} ![Pressure coefficient (left) and heat transfer coefficient (right) along the cylinder surface for the viscous hypersonic flow past a circular cylinder. The lines in the third legend correspond to the results obtained using LAURA code by Gnoffo and White [@Gnoffo2004].](figures/nsmahcyl18_stanton.png){#cyl18f width="\\textwidth"} # Concluding remarks {#sec:conclusions} We have presented an optimal transport approach for the numerical solution of compressible flows with shock waves. The approach couples an adaptive viscosity regularization method and optimal transport theory in order to capture shocks and adapt meshes. The marriage of optimal transport and viscosity regularization for compressible flows leads to a coupled system of the compressible Euler/Navier-Stokes equations, the Helmholtz equation, and the Monge-Ampère equation. The hybridizable discontinuous Galerkin method is used for the spatial discretization of the governing equations to obtain high-order accurate solutions. We devise a mesh adaptation procedure to solve the coupled system in an iterative and sequential fashion. The approach is found to yield accurate, sharp yet smooth solutions within a few mesh adaptation iterations. We explore two different options to define the mesh indicator function for computing adaptive meshes. The option based on density gradient is more effective than that based on velocity divergence for dealing with shock flows that have more complex structures such as boundary layers, shear layers, and expansion waves. We have presented a wide variety of transonic, supersonic and supersonic flows in two dimensions in order to demonstrate the performance of the proposed approach. The approach is capable of moving mesh points to resolve complex shock patterns without creating new mesh points or modifying the connectivity of the initial mesh. The generated r-adaptive meshes can significantly improve the accuracy of the numerical solution relative to the initial mesh. Accurate prediction of aerodynamic forces and heat transfer rates for viscous shock flows requires meshes to resolve both shocks and boundary layers. Numerical results show that the approach can generate r-adaptive meshes that resolve not only shocks but also boundary layers for viscous shock flows. It yields accurate predictions of pressure and heat transfer coefficients by adapting the initial mesh to resolve shocks and boundary layers. Moreover, the approach can also adapt the initial mesh to resolve other flow structures such as shear layers and expansion waves. The approach presented herein can be extended to chemically reacting hypersonic flows without loss of generality. To this end, different variants of the regularized viscosity can be devised, including physics-based artificial viscosity terms that augment the molecular viscous components. The approach can also be extended to compressible flows in three dimensions. We are going to pursue these extensions in future work. Another interesting application of the optimal transport approach is model reduction of compressible flows with shock waves. We show in a recent paper [@Heyningen2023] that the optimal transport provides an effective treatment of shock waves for model reduction because it can generate snapshots that are aligned well with the shocks. Hence, it results in stable, robust and accurate reduced order models of parametrized compressible flows. In future reserach, we would like to couple the optimal transport theory with the first-order empirical interpolation method [@Nguyen2023d] to develop an efficient intrusive reduced order modeling for compressible flows. # Acknowledgements {#acknowledgements .unnumbered} We gratefully acknowledge the United States Department of Energy under contract DE-NA0003965, the National Science Foundation for supporting this work (under grant number NSF-PHY-2028125), and the Air Force Office of Scientific Research under Grant No. FA9550-22-1-0356 for supporting this work.

arxiv_math

--- abstract: | This paper investigates the problem of estimating the larger location parameter of two general location families from a decision-theoretic perspective. In this estimation problem, we use the criteria of minimizing the risk function and the Pitman closeness under a general bowl-shaped loss function. Inadmissibility of a general location and equivariant estimators is provided. We prove that a natural estimator (analogue of the BLEE of unordered location parameters) is inadmissible, under certain conditions on underlying densities, and propose a dominating estimator. We also derive a class of improved estimators using the Kubokawa's IERD approach and observe that the boundary estimator of this class is the Brewster-Zidek type estimator. Additionally, under the generalized Pitman criterion, we show that the natural estimator is inadmissible and obtain improved estimators. The results are implemented for different loss functions, and explicit expressions for the dominating estimators are provided. We explore the applications of these results to for exponential and normal distribution under specified loss functions. A simulation is also conducted to compare the risk performance of the proposed estimators. Finally, we present a real-life data analysis to illustrate the practical applications of the paper's findings.\ **Keywords**: Decision theory, location family, Improved estimator, Brewster-Zidek type estimator, IERD approach, Pitman nearness. author: - | Naresh Garg$^a$[^1], Lakshmi Kanta Patra$^b$ and Neeraj Misra$^a$\ $^a$Department of Mathematics and Statistics, Indian Institute of Technology Kanpur\ $^b$Department of Mathematics , Indian Institute of Technology Bhilai bibliography: - references.bib title: On improved estimation of the larger location parameter --- # Introduction The problem of estimating ordered parameters, whether the correct ordering between the parameters apriori, is known, has been extensively discussed in the literature. When the ordering among the parameters is known apriori, numerous studies have focused on estimating the smallest and the largest parameters, with contributions from [@blumenthal1968estimation], [@MR1370413], [@kumar2005james], [@chang2015] [@patra], [@chang2017estimation] and [@garg2021componentwise]. For a detailed review on estimation of restricted parameter we refer to [@MR0326887], [@MR961262] and [@MR2265239]. However, limited attention has been given to the estimation of the smallest and the largest parameters when the correct ordering between the parameters is unknown. This problem can be viewed as an estimation counterpart to the well-known \"Ranking and Selection\" problems, where a basic goal is to select the population associated with the largest (or smallest) parameter while lacking knowledge of the correct ordering among parameters (refer to [@dudewicz1982complete], for an extensive bibliography on ranking and selection problems). In many real-world scenarios, such as in environmental studies, finance, or risk management, estimating the largest (or smallest) location/scale parameter is essential for assessing extreme events, outliers, or rare occurrences. For example, in environmental studies, estimating the largest (or smallest) location parameter of pollutant concentrations helps in determining critical levels at which adverse effects may occur. In finance, estimating the largest (or smallest) location parameter of stock returns allows investors to understand the potential for extreme losses or gains. In early work in this area, [@blumenthal1968estimation] considered estimation of the larger mean of two normal distributions having a common known variance. They proposed various estimators for the larger mean and compared their performances under the squared error loss function. [@dhariyal1982estimation] extended the class of estimators proposed by [@blumenthal1968estimation] by introducing two new estimators for estimating the larger mean of two normal distributions. [@elfessi1992estimation] focused on the estimation of the smaller and larger scale parameters of two uniform distributions. They proposed improved estimators that outperformed the usual estimators based on the mean squared error criterion and the Pitman nearness criterion. [@misra1997estimation] consider two exponential distributions with unknown scale parameters. They dealt with estimation of the smaller and the larger scale parameters and obtained the MLEs. They showed that the MLEs are inadmissible and better estimators are derived under the squared error loss function. Most of the studies related to this problem have focused on specific distributions with independent marginals and specific loss functions. Some other contributions on the estimation of the larger and the smaller location/scale parameters can be found in [@kumar1993unbiased], [@misra1994estimation], [@mitra1994estimating], and [@misra2002natural]. For a general framework, [@Misra2003] dealt with estimation of the largest scale parameter of $k\; (\geq 2)$ independent and absolutely continuous scale parameter distributions (general probability scale models). Under the assumption of a monotone likelihood ratio on the probability models and the squared error loss function, they established that a natural estimator is inadmissible and obtained a dominating estimator. They also provided applications of these results to some specific probability models. In this paper, we make an attempt to unify/extend various studies by considering estimation of the larger location parameters of two general probability models under a general loss function.\ In most of the above studies, criterion of minimizing the risk function is used to obtain estimators outperforming usual estimators (such as those based on the component-wise best location/scale equivariant or the maximum likelihood estimators) under the squared error loss function. A popular alternative criterion to compare different estimators is the Pitman nearness (PN) criterion, due to [@pitman1937]. It compares two estimators based on the probability of one estimator being closer to the estimand than the other estimator under the absolute error loss function. [@rao1981] has pointed out some advantages of the Pitman nearness (PN) criterion over the mean squared error criterion. Keating ([-@keating1985]) further supported Rao's observations through certain estimation problems, and [@keating1985m] provided some practical examples where the PN criterion is more relevant than minimizing the risk function. Additionally, [@peddada1985] and [@MR860477] extended the PN criterion to the generalized Pitman Criterion (GPN) by considering the general loss function instead of the absolute error loss function. A detailed description of the PN criterion and the relevant literature can be found in the monograph by [@keating1993]. The PN criterion has been extensively used in the literature for comparing various estimators in different estimation problems. However, there are only limited number in studies on the use of the PN criterion following Stein (1964) approach to obtain improvements over the usual estimators ([@nayak1990] and [@kubokawa1991]). Moreover, all these studies are centred around specific probability distributions (mostly, normal and gamma) and absolute error loss in the PN criterion. In this paper, we consider the problem of estimation of the larger location parameter of two general location models under the GPN criterion. We develop a result that is useful in finding improvements over location equivariant estimators in certain situations. Throughout, $\Re$ will denote the real line and $\Re^2=\Re \times \Re$ will denote the two dimensional Euclidean space. for any two real numbers $x_1$ and $x_2$, $x_{(1)}=\min\{x_1,x_2\}$ and $x_{(2)}=\max\{x_1,x_2\}$ also denote the smaller and larger of them respectively. Let $X_1$ and $X_2$ be two independently distributed random variable with densities $f(x_1-\theta_1),\; x_1\in \Re,\; \theta_1\in \Re,$ and $f(x_2-\theta_2),\; x_2\in \Re,\; \theta_2\in \Re,$ respectively. Our aim is to estimate the larger location parameter $\theta_{(2)}=\max\{\theta_1,\theta_2\}$ under a non-negative loss function $L((\theta_1,\theta_2),a)$, $(\theta_1,\theta_2) \in \Theta=\Re^2$ and $a\in \mathcal{A}=\Re$, here $\Theta$ and $\mathcal{A}$ denotes the parametric space and the action space, respectively. At first we will invoke the principle of invariance under a suitable group of transformation. For this purpose, we consider the group $\mathcal{G}$ of transformation, where $$\begin{aligned} \mathcal{G}=\{g_c:c\in \mathbb{R}\}\cup \{g^*_1,g^*_2\},\end{aligned}$$ $$\begin{aligned} g_c(x_1,x_2)=(x_1+c,x_2+c), ~~g^*_1(x_1,x_2)=(x_1,x_2)~\mbox{ and }~g^*_2(x_1,x_2)=(x_2,x_1),\;\; x_1\in \Re,\; x_2\in \Re,\; c\in \Re.\end{aligned}$$ It can be easily seen that, under the group of transformation the family of distributions under consideration is invariant and induced group of transformation of the parametric space $\Theta$ and the action space $\mathcal{A}$ as $$\bar{\mathcal{G}}=\{\bar{g}_c~:~ c\in \Re\}~~\mbox{ and }~~\tilde{\mathcal{G}}=\{\tilde{g}_c~:~c\in \Re\}$$ respectively, where for every $(x_1,x_2) \in \Theta$ and $c\in \Re$ $$\begin{aligned} \overline{g}_c(x_1,x_2)=(x_1+c,x_2+c), ~~\overline{g}^*_1(x_1,x_2)=(x_1,x_2)~\mbox{ and }~\overline{g}^*_2(x_1,x_2)=(x_2,x_1)\end{aligned}$$ and for any $a \in \mathcal{A}$ and $c \in \Re$ $$\begin{aligned} \widetilde{g}_c(a)=a+c, ~~\widetilde{g}_1^*(a)=a~~\mbox{ and }~~\widetilde{g}_2^*(a)=a\end{aligned}$$ Now the loss function $L((\theta_1,\theta_2),a)$ is invariant under the group $\mathcal{G}$ if and only is for any $(x_1,x_2) \in \Theta$, $a \in \mathcal{A}$ and $c\in \Re$ $$\begin{aligned} \label{inv1} L(\overline{g}_c(x_1,x_2),\widetilde{g}_c(a))=L((x_1,x_2),a), ~\mbox{that is}~ L((x_1+c,x_2+c),a+c)=L((x_1,x_2),a),\end{aligned}$$ and $$\begin{aligned} \label{inv2} L(\overline{g}^*_2(x_1,x_2),\widetilde{g}_2^*(a))=L((x_1,x_2),a), ~\mbox{that is}~ L((x_2,x_1),a)=L((x_1,x_2),a).\end{aligned}$$ So combining conditions ([\[inv1\]](#inv1){reference-type="ref" reference="inv1"}) and ([\[inv2\]](#inv2){reference-type="ref" reference="inv2"}) we have, for any function $V:\Re^2\rightarrow [0,\infty)$ and for all $(x_1, x_2)\in \Theta$, $a \in \mathcal{A}$ and $c\in \Re$ $$\begin{aligned} L((x_1,x_2),a)=V((x_{(1)},x_{(2)}),a)~\mbox{and}~ V((x_{(1)},x_{(2)}),a)= V((x_{(1)}+c,x_{(2)}+c),a+c).\end{aligned}$$ This suggests us to consider the loss function as $$\begin{aligned} \label{loss1} L((\theta_1,\theta_2),a)=W(a-\theta_{(2)}),\; \;(\theta_1,\theta_2)\in \Theta,\; a\in \Re,\end{aligned}$$ where $W:\Re\rightarrow [0,\infty)$ is given function. Now onwards we denote $\boldsymbol{\theta}=(\theta_1,\theta_2)$. Further we will make the following assumptions on the function $W(.)$: - $W(0)=0$, $W(t)$ is strictly decreasing on $(-\infty,0)$ and is strictly increasing on $(0,\infty)$, that is, $W(t)$ is strictly bowl shaped function in $t\in (-\infty,\infty)$; - $W'(t)$ is increasing, almost everywhere; - Integrals involving $W(t)$ are finite and differentiation under the integral sign is permissible. An estimator $\delta(x_1,x_2)$ is invariant under the group of transformation $\mathcal{G}$ if, and only if for any $(x_1,x_2)\in \Re$ and $c\in \Re$ $$\begin{aligned} \label{inavriance} \delta(x_1+c,x_2+c)=\delta(x_1,x_2)+c~~\mbox{ and }~~\delta(x_1,x_2)=\delta(x_2,x_1).\end{aligned}$$ From the second condition of ([\[inavriance\]](#inavriance){reference-type="ref" reference="inavriance"}), we have $\delta(x_1,x_2)=\delta^*(x_{(1)},x_{(2)})$, for some function $\delta^*$, and from the first condition of ([\[inavriance\]](#inavriance){reference-type="ref" reference="inavriance"}) we have $\delta^*(x_{(1)}+c,x_{(2)}+c)=\delta^*(x_{(1)},x_{(2)})+c$. This suggests that any invariant estimator has the form $\delta_{\phi}(X_1,X_2)$, where $$\label{investi} \delta_{\phi}(X_{(1)},X_{(2)})=X_{(2)}-\phi(X_{(2)}-X_{(1)})=X_{(2)}-\phi(U)$$ where $U=X_{(2)}-X_{(1)}$ and $\phi:[0,\infty)\rightarrow \Re$ is real valued function.\ For the component problem of estimating $\theta_i$ usual estimator that is the best location equivariant estimator (BLEE) of $\theta_i$ with respect the loss function $L_i(\theta_i,\delta)=W(\delta-\theta_i)$ is obtained as $$\begin{aligned} \delta^i_{c_0}(\bold{X})=X_i-c_0,\;\;i=1,2, \end{aligned}$$ where $c_0$ is the unique solution of equation $$\begin{aligned} \int_{-\infty}^{\infty}W^{\prime}(x-c_0)f(x)dx=0.\end{aligned}$$ So, we consider a natural estimator of $\theta_{(2)}$ as $$\label{usual estimator} \delta_{c_0}(\bold{X})=X_{(2)}-c_0.$$ Our aim is to find estimators which improve upon the natural estimator $\delta_{c_0}$, for estimating $\theta_{(2)}$ under the loss function $L((\theta_1,\theta_2),a)$ defined by ([\[loss1\]](#loss1){reference-type="ref" reference="loss1"}).\ The rest of the paper is organized as follows. Inadmissibility of the usual estimator $\delta_{c_0}$ has been proved in Section [2](#sec2){reference-type="ref" reference="sec2"}. We have obtained [@stein1964]-type dominating estimator to demonstrate the inadmissibility. In Section [2.1](#sec3){reference-type="ref" reference="sec3"}, we consider a class of natural estimators for estimating $\theta_{(2)}$ as $\mathcal{D}=\{\delta_{b}=X_{(2)}-b: \;b\in \Re\}$. We obtain admissible estimators within the class $\mathcal{D}$. It is seen that under the condition $\lim_{t\rightarrow \infty} f(t)=0$, one of the boundary estimators of this admissible class is $\delta_{c_0}$. Furthermore, we derive a class of improved estimators over a boundary estimator of the class of admissible estimators using the IERD approach proposed by [@kubokawa1994unified]. Additionally, we obtain a [@brewster1974improving]-type improved estimator, which improves upon the boundary estimator of the admissible class of estimators. In Section [3](#sec4){reference-type="ref" reference="sec4"}, we have obtained improved estimators for the special loss function: the squared error loss, the linex loss, and the absolute error loss.\ In section [4](#sec5){reference-type="ref" reference="sec5"}, we consider the estimation of $\theta_{(2)}$ with respect to the Pitman closeness criterion. Under the Pitman closeness, we obtain the [@stein1964]-type estimator, which dominates the usual estimator of $\theta_{(2)}$. As an application, in section [5](#sec6){reference-type="ref" reference="sec6"}, we derive improved estimators under the squared error loss, the linex loss, and the absolute error loss functions for the normal distribution and exponential distribution. We observe that the [@stein1964]-type improved estimator, and the usual estimator for the normal distribution with respect to the squared error loss and the absolute error loss are the same due to the symmetric nature of these loss functions. In Section [6](#sec7){reference-type="ref" reference="sec7"}, a simulation is carried out to compare the risk performance of the proposed estimators. Section [7](#sec8){reference-type="ref" reference="sec8"} presents a real-life data analysis, showcasing the practical applications of our paper's findings. Lastly, in Section [8](#sec9){reference-type="ref" reference="sec9"}, we offer our concluding remarks on the paper's contributions. # Inadmissibility of usual estimator {#sec2} In this section, we will prove inadmissibility of the natural estimator $\delta_{c_0}=X_{(2)}-c_0$ by deriving dominating estimators under the bowl shaped $\mathcal{G}$-invariant loss function [\[loss1\]](#loss1){reference-type="eqref" reference="loss1"}, where $W(\cdot)$ satisfies assumptions (C1)-(C3). To prove the dominance result we require the following assumptions: - The family $\{f(x-\eta):\eta\in \Re\}$ of p.d.f.s holds the MLR property, i.e., for $-\infty<x_1<x_2<\infty$ and $-\infty<\eta_1<\eta_2<\infty$, $f(x_1-\eta_1)f(x_2-\eta_2)\geq f(x_1-\eta_2)f(x_2-\eta_1)$. ```{=html} <!-- --> ``` - For every fixed $u>0$ and $\theta\geq 0$, let $c\equiv c(\theta,u)$ be the unique minimizer of the following function $$\frac{\int_{-\infty}^{\infty}W(z-c)[f(z-u+\theta)f(z)+f(z-u)f(z+\theta)]dz} {\int_{-\infty}^{\infty}[f(z-u+\theta)f(z)+f(z-u)f(z+\theta)]dz}.$$ That implies, for every fixed $u>0$ and $\theta\geq 0$, $c\equiv c(\theta,u)$ is the unique solution of the equation $$\begin{aligned} \int_{-\infty}^{\infty}W^{'}(z-c(\theta,u))[f(z-u+\theta)f(z)+f(z-u)f(z+\theta)]dz =0. \end{aligned}$$ **Theorem 1**. *Let assumptions (A), (B), and (C1)-(C3) hold. For any fixed $u>0$, let $c \equiv c(0,u)$ be the unique solution of the equation $$\begin{aligned} \label{eq: 2.1} \int_{-\infty}^{\infty}W^{'}(z-c)f(z-u)f(z)dz=0. \end{aligned}$$ Then the estimator, $$\begin{aligned} \label{deltast1} \delta_{\phi_0}(\bold{X})= X_{(2)}-\min\{\phi(U),c(0,U)\} \end{aligned}$$ improves upon the equivariant estimator $\delta_{\phi}(\bold{X})=X_{(2)}-\phi(U)$ under the loss ([\[loss1\]](#loss1){reference-type="ref" reference="loss1"}), provided $P_{\boldsymbol{\theta}}\left(\phi(U) > c(0,U)\right) >0,$ at least for some $\boldsymbol{\theta}\in \Theta$.* ***Proof:*** The risk function of any equivariant estimator $\delta_{\phi}(\bold{X})=X_{(2)}-\phi(U)$ is $$\begin{aligned} R(\boldsymbol{\theta},\delta_{\phi})&=&E_{\boldsymbol{\theta}}\left[W(X_{(2)}-\phi(U)-\theta_{(2)})\right]\\ &=&E_{\boldsymbol{\theta}}^U\left[E_{\boldsymbol{\theta}}^{X/U}\left[W\left(X_{(2)}-\phi(U)-\theta_{(2)}\right)|U\right]\right], \;\;\boldsymbol{\theta}\in \Theta.\end{aligned}$$ For any fixed $\boldsymbol{\theta}\in \Theta$ and $u>0$, consider $$\begin{aligned} R_{\boldsymbol{\theta},u}(c)&=&E_{\boldsymbol{\theta}}^{X/U}\left[W\left(X_{(2)}-c-\theta_{(2)}\right)|U=u\right]\\ &=& \frac{\int_{-\infty}^{\infty}W(y-\theta_{(2)}-c)[f(y-u-\theta_{(1)})f(y-\theta_{(2)}) +f(y-u-\theta_{(2)})f(y-\theta_{(1)})]dy}{\int_{-\infty}^{\infty}[f(y-u-\theta_{(1)})f(y-\theta_{(2)}) +f(y-u-\theta_{(2)})f(y-\theta_{(1)})]dy}\\ &=& \frac{\int_{-\infty}^{\infty}W(z-c)[f(z-u+\theta)f(z)+f(z-u)f(z+\theta)]dz} {\int_{-\infty}^{\infty}[f(z-u+\theta)f(z)+f(z-u)f(z+\theta)]dz},\;\; -\infty<c<\infty,\end{aligned}$$ here $\theta=\theta_{(2)}-\theta_{(1)}\in [0,\infty)$. Using the assumption (B), for every fixed $\theta \geq 0$ and $u>0$, there exists a unique minimizer of $R_{\boldsymbol{\theta},u}(c)$ say $c\equiv c(\theta,u)$ which is the unique solution of the equation $R^{\prime}_{\boldsymbol{\theta},u}(c)=0$, i.e., $$\begin{aligned} \frac{\int_{-\infty}^{\infty}W^{'}(z-c(\theta,u))[f(z-u+\theta)f(z)+f(z-u)f(z+\theta)]dz} {\int_{-\infty}^{\infty}[f(z-u+\theta)f(z)+f(z-u)f(z+\theta)]dz}=0.\end{aligned}$$ For any fixed $\theta\ge 0$ and $u>0$, let $Z_{\theta,u}$ be a random variable having the density $$\Pi_{\theta,u}(z) = \frac{f(z-u+\theta)f(z)+f(z-u)f(z+\theta)}{\int_{-\infty}^{\infty}f(t-u+\theta)f(t)+f(t-u)f(t+\theta)dt},\;-\infty<z<\infty,$$ so that $E[W^{'}(Z_{\theta,u}-c(\theta,u))]=0.$ Then, for any fixed $\theta\geq 0$ and $u>0$, $$\begin{aligned} \frac{\Pi_{\theta,u}(z)}{\Pi_{0,u}(z)}&=&d(\theta,u)\frac{f(z-u+\theta)f(z)+f(z-u)f(z+\theta)} {2f(z-u)f(z)}\\ &=&\frac{1}{2}\left[\frac{f(z-u+\theta)}{f(z-u)}+\frac{f(z+\theta)}{f(z)}\right],\; -\infty<z<\infty,\end{aligned}$$ where $d(\theta,u)$ is a positive constant.\ By the assumption (A), we have $\displaystyle\frac{\Pi_{\theta,u}(z)}{\Pi_{0,u}(z)}$ decreasing in $z$, for any fixed $\theta\geq 0$ and $u>0$. Since, for any constant $c$, $W^{'}(z-c)$ is an almost everywhere increasing function of $z$, we conclude that, for any $u>0$, $$\label{eq: 2.3} E[W^{'}(Z_{\theta,u}-c)] \le E[W^{'}(Z_{0,u}-c)],\;\forall \; \theta\geq 0,\; c\in \Re.$$ Taking $c=c(\theta,u)$ in [\[eq: 2.3\]](#eq: 2.3){reference-type="eqref" reference="eq: 2.3"}, we have, for any $\theta \geq 0$ and $u>0$, $$\begin{aligned} 0=&E[W^{'}(Z_{\theta,u}-c(\theta,u))] \le E[W^{'}(Z_{0,u}-c(\theta,u))]\\ \implies\qquad 0=&E[W^{'}(Z_{0,u}-c(0,u))] \le E[W^{'}(Z_{0,u}-c(\theta,u))]\end{aligned}$$ Since, for any fixed $t$, $W^{'}(t-c)$ is a decreasing function of $c\in \Re$, we get $$\begin{aligned} c(\theta,u)\le c(0,u),\;\;\forall\;u>0,\; \theta\geq 0.\end{aligned}$$ Now consider the function $\phi_0(u)=\min\{\phi(u),c(0,u)\},\; u>0$. Then, for any fixed $\theta \geq 0$ and $u>0$, we have $c(\theta,u)\le \phi_0(u) < \phi(u)$, provided $\phi(u)>c(0,u)$. Using condition (C1), for any fixed $\theta \geq 0$ and $u>0$, $R_{\boldsymbol{\theta},u}(c)$ is increasing in $c\in [c(0,u),\infty)$. Consequently we get $$\begin{aligned} E_{\boldsymbol{\theta}}^{X/U}\left[W\left(\delta_{\phi_0}-\theta_{(2)}\right)|U=u\right]\le E_{\boldsymbol{\theta}}^{X/U}\left[W\left(\delta_{\phi}-\theta_{(2)}\right)|U=u\right]\end{aligned}$$ for all $\boldsymbol{\theta}\in\Theta$ and $u>0$ and strict inequity holds for some $u>0$. Hence we have $R(\boldsymbol{\theta},\delta_{\phi_0}) \le R(\boldsymbol{\theta},\delta_{\phi})$. This proves the theorem. $\blacksquare$ **Corollary 2**. *Let the assumption (A) and assumptions (C1)-(C3) hold. Then the estimator, $$\begin{aligned} \label{deltast1} \delta_{\phi_0}(\bold{X})= X_{(2)}-\min\{c_0,c(0,U)\} \end{aligned}$$ improves upon the natural estimator $\delta_{c_0}(\bold{X})=X_{(2)}-c_0$ under the loss ([\[loss1\]](#loss1){reference-type="ref" reference="loss1"}) provided $P_{\boldsymbol{\theta}}(c_0>c(0,U))> 0$, for some $\boldsymbol{\theta}\in \Theta$.* **Remark 1**. *If $f(x)$ is decreasing in $x$ then $P_{\boldsymbol{\theta}}(c_0>c(0,U))>0$ for some $\boldsymbol{\theta}\in \Theta$.* ## A Class of improved estimators {#sec3} A natural class of estimators for estimating $\theta_{(2)}$ is $\mathcal{D}=\{\delta_b=X_{(2)}-b: \;b\in \Re\}$. Firstly, we find admissible estimators within the class of estimators $\mathcal{D}$. The risk function of an estimator $\delta_b$ is $$\begin{aligned} \nonumber R(\boldsymbol{\theta},\delta_b) &=E_{\boldsymbol{\theta}}[W(X_{(2)}-\theta_{(2)}-b)]\\ \nonumber &=\iint \limits _{-\infty<x_1\leq x_2<\infty} W(x_2-\theta_{(2)}-b)\left[f(x_1-\theta_{(1)})f(x_2-\theta_{(2)})+f(x_1-\theta_{(2)})f(x_2-\theta_{(1)})\right] dx_1\,dx_2 \\ \nonumber &=\int_{-\infty}^{\infty}\int_{-\infty}^{z}W(z-b) \left[f(x+\theta) f(z)+f(x)f(z+\theta)\right] dx\,dz\\ &=E_{\theta}[W(Z-b)],\;\;\theta\geq 0, \label{risk function}\end{aligned}$$ where $Z$ is a r.v. with the density $g_{\theta}(z)=F(z+\theta)f(z)+F(z)f(z+\theta),\;z\in \Re,\; \theta\geq 0$, and $F(z)=\int_{-\infty}^{z} f(t)\,dt,\;z\in \Re$. Using the assumption (A), it is easy to verify that, for every $\theta\geq 0$, $g_{\theta}(z)/g_{0}(z)$ is decreasing in $z$. Let $b_{\theta}$ be the continues function and be the unique solution of the equation $E_{\theta}[W^{'}(Z-b)]=\int_{-\infty}^{\infty} W^{'}(x-b) g_{\theta}(x) dx=0$. Since, for every $\theta\geq 0$, $g_{\theta}(x)/g_{0}(x)$ is decreasing in $x$ and $W^{'}(x-b)$ is decreasing in $b$, and under the assumption $\lim_{t\to \infty}f(t)=0$, it can easy to see that $$\inf_{\theta \geq 0} b_{\theta}=b_{\infty}=c_0\,\leq \,b_{\theta}\,\leq \, b_{0},\; \; \forall\; \theta\geq 0.$$ **Theorem 3**. *Suppose that the assumption (A) holds and $\lim_{t\to \infty}f(t)=0$. Then the estimators that are admissible within the class $\mathcal{D}$ are $\{X_{(2)}-b:\; b_{\infty}\,\leq \,b_{\theta}\,\leq \, b_{0}\}$.* ***Proof:*** Note that, for any fixed $\boldsymbol{\theta}\in\Theta$ (or fixed $\theta\geq 0$), the risk function $R(\boldsymbol{\theta},\delta)$, given by [\[risk function\]](#risk function){reference-type="eqref" reference="risk function"}, is uniquely minimized at $b=b_{\theta}$, it is a strictly decreasing function of $b$ on $(-\infty,b_{\theta})$ and strictly increasing function of $b$ on $(b_{\theta},\infty)$. Since, for any $\theta \geq 0$, $b_{\theta}$ is a continuous function of $\theta\in[0,\infty)$, it assumes all values between $\inf_{\theta\geq 0} b_{\theta}=b_{\infty}=c_0$ and $\sup_{\theta\geq 0}b_{\theta}=b_{0}$, as $\theta$ varies on $[0,\infty)$. It follows that, each $b\in[b_{\infty},b_{0}]$ uniquely minimizes the risk function $R(\boldsymbol{\theta},\delta)$ at some $\boldsymbol{\theta}\in\Theta$ (or at some $\theta\geq 0$). This proves that the estimators $\{X_{(2)}-b:\; b_{\infty}\,\leq \,b_{\theta}\,\leq \, b_{0}\}$ are admissible among the estimators in the class $\mathcal{D}$. $\blacksquare$ Hence the subclass of estimators $\mathcal{D}_0=\{X_{(2)}-b:\; b_{\infty}=c_0\,\leq \,b_{\theta}\,\leq \, b_{0}\}$ is admissible within the class $\mathcal{D}$. Now, one can also consider whether improvements can be made to the estimators within the class $\mathcal{D}_0$, but it may not be possible to obtain improvements over all estimators in $\mathcal{D}_0$. Therefore, in this section, we aim to find improvements specifically for the boundary estimator $\delta_{b_0}(\bold{X}) = X_{(2)}-b_0$ within the class $\mathcal{D}_0$, where $b_0$ is the unique solution of equation $$\begin{aligned} \label{eq:2.5} \int_{-\infty}^{\infty}W^{\prime}(x-b_0)f(x)\,F(x)\,dx=0,\end{aligned}$$ where $F(x)=\int_{-\infty}^{x} f(y)\,dy,\;x\in \Re$. Now, we use the IERD approach of Kubokawa (1994) to propose a class of estimators dominating over the estimator $\delta_{b_0}$. Further, we will obtain the Brewster-Zidek (1974) type estimator improving over $\delta_{b_0}$. Consider estimation of $\theta_{(2)}$ under the loss function (1.2). Assume that the function $W(\cdot)$ is absolute continuous and satisfies the assumptions (C1), (C2) and (C3). The following two lemmas will be useful in proving the next result. The lemma stated in the following lemma follows from relationship between the likelihood ratio order and the revised failure rate order in the theory of stochastic orders (see [@MR2265633]). The proof of the following lemma, being straightforward, is also omitted. **Lemma 4**. *Let $s_0\in \Re$ and let $M:\Re\rightarrow\Re$ be such that $M(s)\leq 0,\; \forall \; s<s_0,$ and $M(s)\geq 0,\; \forall \; s> s_0$. Let $M_i:\Re\rightarrow [0,\infty), \; i=1,2,$ be non-negative functions such that $M_1(s) M_2(s_0) \geq (\leq)\, M_1(s_0) M_2(s),\; \forall \; s<s_0, \text{ and } M_1(s) M_2(s_0) \leq\,(\geq)\; M_1(s_0) M_2(s),\; \forall \; s$ $>s_0.$ Then, $$M_2(s_0) \int\limits_{-\infty}^{\infty} M(s) \, M_1(s) ds\leq\;(\geq)\; M_1(s_0) \int\limits_{-\infty}^{\infty} M(s) \, M_2(s) ds.$$* In the following theorem, we provide a class of estimators that improve upon the natural estimator $\delta_{b_0}$. **Theorem 5**. *Suppose that the assumption (A) holds. Additionally, assume that $W(\cdot)$ is absolutely continuous and satisfies (C1), (C2) and (C3). Let $\delta_{\phi}(\bold{X})=X_{(2)} - \phi(U)$ be a location equivariant estimator of $\theta_{(2)}$ such that* - *$\phi(t)$ is increasing in $t\in [0,\infty)$,* - *$\lim_{t\to\infty} \phi(t)=b_0$* - *$\int_{-\infty}^{\infty} W^{'}(z-\phi(t))\;[F(z)-F(z-t)]f(z)\,dz \, \leq \,0,\; \forall\; t\in [0,\infty).$* *Then, the estimator $\delta_{\phi}(\bold{X})=X_{(2)} - \phi(U)$ is an improvement over the estimator $\delta_{b_0}(\bold{X})=X_{(2)}-b_0$.* **Proof:** Let us fix $\boldsymbol{\theta}\in\Theta$ and let $\theta=\theta_{(2)}-\theta_{(1)}$, so that $\theta \geq 0$. Consider the risk difference\  \ $\Delta(\boldsymbol{\theta})$ $$\begin{aligned} &= R(\boldsymbol{\theta},\delta_{b_0})-R(\boldsymbol{\theta},\delta_{\phi})\\ & = E_{\boldsymbol{\theta}}[W(X_{(2)}-\theta_{(2)}-b_0)- W(X_{(2)}-\theta_{(2)}-\phi(U))] \\ &= E_{\boldsymbol{\theta}}\left[\int_{U}^{\infty}\Big\{ \frac{d}{dt} W(X_{(2)}-\theta_{(2)}-\phi(t))\Big\}\; dt\right],\\ &= \int_{-\infty}^{\infty}\int_{u=0}^{\infty}\left[\int_{u}^{\infty}\Big\{ \frac{d}{dt} W(y-\theta_{(2)}-\phi(t))\Big\}\; dt\right][f(y-u-\theta_{(1)})f(y-\theta_{(2)})+f(y-u-\theta_{(2)})f(y-\theta_{(1)})]\,du\,dy,\end{aligned}$$ After changing the order of integration we have\  \ $\Delta(\boldsymbol{\theta})$ $$\begin{aligned} &= \int_{t=0}^{\infty}\int_{-\infty}^{\infty}\int_{u=0}^{t}\Big\{ \frac{d}{dt} W(y-\theta_{(2)}-\phi(t))\Big\}\; [f(y-u-\theta_{(1)})f(y-\theta_{(2)})+f(y-u-\theta_{(2)})f(y-\theta_{(1)})]dudydt,\\ &=- \!\int_{t=0}^{\infty}\phi^{\prime}(t)\!\left[\int_{-\infty}^{\infty}\int_{u=0}^{t}\!\!\Big\{ W^{\prime}(y-\theta_{(2)}-\phi(t))\Big\}\! [f(y-u-\theta_{(1)})f(y-\theta_{(2)})\!+\!f(y-u-\theta_{(2)})f(y-\theta_{(1)})]dudy\right]\!dt.\end{aligned}$$ Since $\phi(t)$ is a increasing function of $t$, it suffices to show that, for every $t>0$, $$\begin{aligned} \label{eq:2.10} &\int_{-\infty}^{\infty}\int_{u=0}^{t}\Big\{ W^{\prime}(y-\theta_{(2)}-\phi(t))\Big\}\; [f(y-u-\theta_{(1)})f(y-\theta_{(2)})+f(y-u-\theta_{(2)})f(y-\theta_{(1)})]dudy\leq \,0 \nonumber\\ \iff \qquad & \int_{u=0}^{t}\int_{-\infty}^{\infty}\Big\{ W^{\prime}(z-\phi(t))\Big\}\; [f(z-u+\theta)f(z)+f(z-u)f(z+\theta)]dzdu\leq \,0.\end{aligned}$$ Now, since $W^{'}(t)$ is increasing function of $t$ and, for every fixed $\theta\geq 0$ and $u\in \Re$, $\frac{f(z-u+\theta)f(z)+f(z-u)f(z+\theta)}{2 f(z-u)f(z)}$ is decreasing in $z$, then, for $\theta\geq 0$, we have $$\begin{aligned} &\int_{u=0}^{t}\left[\int_{-\infty}^{\infty} W^{'}(z-\phi(t))\left[f(z-u+\theta)f(z)+f(z-u)f(z+\theta)\right]dz\right]du\\ &\leq \,\int_{u=0}^{t}\left[\int_{-\infty}^{\infty} W^{'}(z-\phi(t))\;[f(z-u)f(z)+f(z-u)f(z)]dz\right] du\\ &= \, \,2\int_{-\infty}^{\infty} W^{'}(z-\phi(t))\;[F(z)-F(z-t)]f(z)\,dz\end{aligned}$$ Now, using hypothesis (iii), we obtain [\[eq:2.10\]](#eq:2.10){reference-type="eqref" reference="eq:2.10"}. This completes proof of the theorem. $\blacksquare$\ In the following we will prove a corollary which will provide [@brewster1974improving] type improved estimators. **Corollary 6**. *Suppose that assumptions (A), (C1), (C2) and (C3) hold. Additionally suppose that, for every fixed $t$, the equation $$k_1(c\vert t)=\int_{-\infty}^{\infty} \; W^{'}(z-c)\; [F(z)-F(z-t)] f(z)\,dz =0$$ has the unique solution $c\equiv \phi_{0}(t)$. Then $$R(\boldsymbol{\theta},\delta_{\phi_{0}})\leq R(\boldsymbol{\theta},\delta_{0}), \;\;\; \forall \; \; \boldsymbol{\theta} \in \Theta,$$ where $\delta_{\phi_{0}}(\bold{X})=X_{(2)}-\phi_{0}(U)$.* **Proof:** It is suffices to show that $\phi_{0}(t)$ satisfies conditions of Theorem [Theorem 5](#kubothm){reference-type="ref" reference="kubothm"}. Note that a hypothesis of the corollary ensures that $\lim_{t\to\infty} \phi_{0}(t)=b_0$. To show that $\phi_{0}(t)$ is an increasing function of $t$, suppose that, there exist numbers $t_1$ and $t_2$ such that $0<t_1<t_2$ and $\phi_{0}(t_1)\neq \phi_{0}(t_2).$ Under the hypotheses of the corollary, we have $k_1(\phi_{0}(t_1)\vert t_1)=0$, $\phi_{0}(t_2)$ is the unique solution of $k_1(c\vert t_2)=0$ and $k_1(c\vert t_2)$ is a decreasing function of $c$. Let $s_0 = \phi_{0}(t_1), \; M(s)=W^{'}(s-s_0)f(s),\; M_1(s)=\int_{0}^{t_2}f(s-u)du$ and $M_2(s)=\int_{0}^{t_1}f(s-u)du$. Then, using Lemma [Lemma 4](#garglemma){reference-type="ref" reference="garglemma"}, we get $$\int_{0}^{t_1}f(\phi_{0,1}(t_1)-w)\,du\; \left(\int_{-\infty}^{\infty}\; W^{'}(z-\phi_{0}(t_1))\,f(z)\;\int_{0}^{t_2} f(z-u)\,du \;dz\right) \qquad \qquad \qquad \qquad \qquad \qquad \quad$$ $$\qquad \qquad \qquad \qquad \geq\; \int_{0}^{t_2}f(\phi_{0}(t_1)-w)du\; \left( \int_{-\infty}^{\infty}\; W^{'}(z-\phi_{0}(t_1))\,f(z)\;\int_{0}^{t_1} f(z-u)\,du \;dz \right)=0.$$ This implies that $$k_1(\phi_{0}(t_1)\vert t_2)=\int_{-\infty}^{\infty}\int_{0}^{t_2}\; W^{'}(z-\phi_{0,1}(t_1))f(z-u)f(z)\,du\, dz\geq \; 0.$$ So we have $k_1(\phi_{0}(t_1)\vert t_2)\,> \, 0$, as $k_1(c\vert t_2)=0$ has the unique solution $c\equiv \phi_{0}(t_2)$ and $\phi_{0}(t_1)\neq\phi_{0}(t_2)$. Since $k_1(c\vert t_2)$ is a decreasing function of $c$, $k_1(\phi_{0}(t_2)\vert t_2)$ $=0$ and $k_1(\phi_{0}(t_1)\vert t_2)\,> \, 0$, it follows that $\phi_{0}(t_1)<\, \phi_{0}(t_2)$. Hence the result follows. $\blacksquare$ # Dominance result for special loss functions {#sec4} In this section, we have obtained the improved estimators for three special loss functions namely squared error loss $L_1:W(t)=t^2,\; t\in \Re$, linex loss $L_2:W(t)=e^{at}-at-1,,\; t\in \Re,\; a\ne 0$ and absolute error loss $L_3:W(t)=|t|,\; t\in \Re$. **Theorem 7**. *Suppose that the assumption (A) holds. Then for estimating $\theta_{(2)}$ with respect to loss function $L_1$ the estimator $$\begin{aligned} \label{deltastl1} \delta_{ST}(\bold{X})= X_{(2)}-\min\{c_0,c(0,u)\} \end{aligned}$$ improves upon the estimator $\delta_{0}(\bold{X})$, provided $P_{\boldsymbol{\theta}}(c_0 > c(0,U)) \ne0$, for some $\boldsymbol{\theta}\in \Theta$, where $$\begin{aligned} c(0,u)=\int_{-\infty}^{\infty}zf(z-u)f(z)dz\bigg/\int_{-\infty}^{\infty}f(z-u)f(z)dz. \end{aligned}$$ and $c_0=\int_{-\infty}^{\infty}xf(x)dx.$* **Theorem 8**. *The estimator $$\begin{aligned} \label{deltastl2} \delta_{ST}(\bold{X})= X_{(2)}-\min\{c_0,c(0,U)\}, \end{aligned}$$ improves upon the estimator $\delta_{c_0}(\bold{X})$ with respect to the loss $L_2$, where $c_0=\frac{1}{a}\ln \int_{\infty}^{\infty}e^{ax}f(x)dx$ and $c(0,U)=\frac{1}{a}\ln H(U)$ with $$\begin{aligned} H(u)=\int_{-\infty}^{\infty}e^{az}f(z-u)f(z)dz\bigg/\int_{-\infty}^{\infty}f(z-u)f(z)dz\end{aligned}$$ provided the assumption (A) holds and $P_{\boldsymbol{\theta}}(c_0> \frac{1}{a}\ln H(u))\ne 0$, for some $\boldsymbol{\theta}\in \Theta$.* **Theorem 9**. *The estimator $$\begin{aligned} \label{deltastl2} \delta_{ST}(\bold{X})= X_{(2)}-\min\{c_0,c(0,U)\}, \end{aligned}$$ improves upon the estimator $\delta_{c_0}(\bold{X})$ with respect to the loss $L_3$, where $c_0$ and $c(0,u)$ are such that $$\begin{aligned} \int_{-\infty}^{c_0}f(z)dz=\frac{1}{2} \; \text{ and } \; \int_{-\infty}^{c(0,u)}f(z-u)f(z)dz\bigg/\int_{-\infty}^{\infty}f(z-u)f(z)dz=\frac{1}{2}, \end{aligned}$$ respectively, provided the assumption (A) holds and $P_{\boldsymbol{\theta}}(c_0> c(0,U))\ne 0$, for some $\boldsymbol{\theta}\in \Theta$.* Now we will apply Theorem [Theorem 5](#kubothm){reference-type="ref" reference="kubothm"} to particular loss functions and provide a class of improved estimators over the estimator $\delta_{b_0}(\bold{X}) = X_{(2)}-b_0,$ where $b_0$ be as defined by the equation [\[eq:2.5\]](#eq:2.5){reference-type="eqref" reference="eq:2.5"}. **Theorem 10**. *Suppose that the assumption (A) holds. Let $\delta_{\phi}(\bold{X})=X_{(2)} - \phi(U)$ be a location equivariant estimator of $\theta_{(2)}$ such that* - *$\phi(t)$ is increasing in $t$,* - *$\lim_{t\to\infty} \phi(t)=2\int_{-\infty}^{\infty}xf(x)F(x)dx=b_0,$* - *$\phi(t) \ge \frac{\int_{-\infty}^{\infty}\int_{0}^{t}zf(z-u)f(z)\,du\,dz}{\int_{-\infty}^{\infty}\int_{0}^{t}f(z-u)f(z)\,du\,dz}$.* *Then the estimator $\delta_{\phi}(\bold{X})=X_{(2)} - \phi(U)$ improves upon the estimator $\delta_{b_0}(\bold{X})=X_{(2)}-b_0$ with respect to the $L_1$.* **Remark 2**. *The boundary estimator of the class estimators given by Theorem [Theorem 10](#kubothml1){reference-type="ref" reference="kubothml1"} is the Brewster-Zidek type estimator. So the Brewster-Zidek type estimator is obtained as $$\delta_{BZ}(\bold{X})=X_{(2)} - \phi_{BZ}(U)$$ with $$\phi_{BZ}(t)=\frac{\int_{-\infty}^{\infty}\int_{0}^{t}zf(z-u)f(z)\,du\,dz}{\int_{-\infty}^{\infty}\int_{0}^{t}f(z-u)f(z)\,du\,dz}.$$* **Theorem 11**. *Suppose that the assumption (A) holds. Let $\delta_{\phi}(\bold{X})=X_{(2)} - \phi(U)$ be a location equivariant estimator of $\theta_{(2)}$ such that* - *$\phi(t)$ is increasing in $t$,* - *$\lim_{t\to\infty} \phi(t)=\frac{1}{a}\ln\left( 2\int_{-\infty}^{\infty}e^{ax}f(x)F(x)dx\right)=b_0,$* - *$\phi(t) \le\frac{1}{a}\ln \left( \frac{\int_{-\infty}^{\infty}\int_{0}^{t}e^{az}f(z-u)f(z)\,du\,dz}{\int_{-\infty}^{\infty}\int_{0}^{t}f(z-u)f(z)\,du\,dz}\right)$.* *Then the estimator $\delta_{\phi}(\bold{X})=X_{(2)} - \phi(U)$ improves upon the estimator $\delta_{b_0}(\bold{X})=X_{(2)}-b_0$ with respect to the $L_2$.* **Remark 3**. *The boundary estimator of the class estimators given by Theorem [Theorem 11](#kubothml2){reference-type="ref" reference="kubothml2"} is the Brewster-Zidek type estimator. So the Brewster-Zidek type estimator is obtained as $$\delta_{BZ}(\bold{X})=X_{(2)} - \phi_{BZ}(U)$$ with $$\phi_{BZ}(t)=\frac{1}{a}\ln \left( \frac{\int_{-\infty}^{\infty}\int_{0}^{t}e^{az}f(z-u)f(z)\,du\,dz}{\int_{-\infty}^{\infty}\int_{0}^{t}f(z-u)f(z)\,du\,dz}\right),\;\;t>0.$$* **Theorem 12**. *Suppose that the assumption (A) holds and $c_0$ be as in Theorem [Theorem 9](#thl3){reference-type="ref" reference="thl3"}. Let $\delta_{\phi}(\bold{X})=X_{(2)} - \phi(U)$ be a location equivariant estimator of $\theta_{(2)}$ such that* - *$\phi(t)$ is increasing in $t$,* - *$\lim_{t\to\infty} \phi(t)=b_0$, where $b_0$ is the solution of the equation $2\int_{-\infty}^{b_0} f(x) F(x) dx=\frac{1}{2}$,* - *$\phi(t)$ be such that it satisfies $$\int_{-\infty}^{\phi(t)}\int_{0}^{t}f(z-u)f(z)\,du\,dz \ge \frac{1}{2} \int_{-\infty}^{\infty}\int_{0}^{t}f(z-u)f(z)\,du\,dz.$$* *Then the estimator $\delta_{\phi}(\bold{X})=X_{(2)} - \phi(U)$ improves upon the estimator $\delta_{b_0}(\bold{X})=X_{(2)}-b_0$ with respect to the $L_3$.* **Remark 4**. *The boundary estimator of the class estimators given by Theorem [Theorem 12](#kubothml3){reference-type="ref" reference="kubothml3"} is the Brewster-Zidek type estimator. So the Brewster-Zidek type estimator is obtained as $$\delta_{BZ}(\bold{X})=X_{(2)} - C,$$ where $C$ is the unique solution of the equation $$\int_{-\infty}^{C}\int_{0}^{t}f(z-u)f(z)\,du\,dz = \frac{1}{2} \int_{-\infty}^{\infty}\int_{0}^{t}f(z-u)f(z)\,du\,dz,\;\;\;t>0.$$* # Improved estimator under generalized Pitman closeness criterion {#sec5} In this section we consider the estimation problem under the generalized Pitman nearness (GPN) criterion. A brief discussion on the Pitman nearness criterion is given in [@garg2022]. For completeness in our presentation, we are again discussing it here. The notion of the Pitman nearness criterion was first introduced by Pitman ([-@pitman1937]), as defined below. **Definition 1**. *Let $\bold{X}$ be a random vector having a probability distribution involving an unknown parameter $\boldsymbol{\theta}\in \Theta$ ($\boldsymbol{\theta}$ may be vector valued). Let $\delta_1$ and $\delta_2$ be two estimators of a real-valued estimand $\tau(\boldsymbol{\theta})$. Then, the Pitman nearness (PN) of $\delta_1$ relative to $\delta_2$ is defined by $$PN(\delta_1,\delta_2;\boldsymbol{\theta})=P_{\boldsymbol{\theta}}[\vert \delta_1-\tau(\boldsymbol{\theta})\vert <\vert \delta_2-\tau(\boldsymbol{\theta})\vert], \; \;\boldsymbol{\theta}\in\Theta,$$ and the estimator $\delta_1$ is said to be nearer to $\tau(\boldsymbol{\theta})$ than $\delta_2$ if $PN(\delta_1,\delta_2;\boldsymbol{\theta})\geq \frac{1}{2},\;\forall\; \boldsymbol{\theta}\in\Theta$, with strict inequality for some $\boldsymbol{\theta}\in\Theta$.* Nayak ([-@nayak1990]) and Kubokawa ([-@kubokawa1991]) modified the Pitman ([-@pitman1937]) nearness criterion and defined the generalized Pitman nearness (GPN) criterion based on general loss function $L(\boldsymbol{\theta},\delta).$ **Definition 2**. *Let $\bold{X}$ be a random vector having a distribution involving an unknown parameter $\boldsymbol{\theta}\in \Theta$ and let $\tau(\boldsymbol{\theta})$ be a real-valued estimand. Let $\delta_1$ and $\delta_2$ be two estimators of the estimand $\tau(\boldsymbol{\theta})$. Also, let $L(\boldsymbol{\theta},a)$ be a specified loss function for estimating $\tau(\boldsymbol{\theta})$. Then, the generalized Pitman nearness (GPN) of $\delta_1$ relative to $\delta_2$ is defined by $$GPN(\delta_1,\delta_2;\boldsymbol{\theta})=P_{\boldsymbol{\theta}}[L(\boldsymbol{\theta},\delta_1) <L(\boldsymbol{\theta},\delta_2)]+\frac{1}{2} P_{\boldsymbol{\theta}}[L(\boldsymbol{\theta},\delta_1) =L(\boldsymbol{\theta},\delta_2)], \; \; \boldsymbol{\theta}\in\Theta.$$ The estimator $\delta_1$ is said to be nearer to $\tau(\boldsymbol{\theta})$ than $\delta_2$, under the GPN criterion, if $GPN(\delta_1,\delta_2;\boldsymbol{\theta})\geq \frac{1}{2},\;\forall\; \boldsymbol{\theta}\in\Theta$, with strict inequality for some $\boldsymbol{\theta}\in\Theta$.* The following result, popularly known as Chebyshev's inequality, will be used in our study (see [@MR2363282]). **Proposition 13**. *Let $S$ be random variable and let $k_1(\cdot)$ and $k_2(\cdot)$ be real-valued monotonic functions defined on the distributional support of the r.v. $S$. If $k_1(\cdot)$ and $k_2(\cdot)$ are monotonic functions of the same (opposite) type, then $$E[k_1(S)k_2(S)]\geq (\leq ) E[k_1(S)] E[k_2(S)],$$ provided the above expectations exist.* The following lemma, taken from [@garg2022], will be useful in proving the main results of this section (also see [@nayak1990]) and [@zhou2012]). **Lemma 14** ([@garg2022]). *Let $Y$ be a random variable having the Lebesgue probability density function and let $m_Y$ be the median of $Y$. Let $W:\Re\rightarrow [0,\infty)$ be a function such that $W(0)=0$, $W(t)$ is strictly decreasing on $(-\infty,0)$ and strictly increasing on $(0,\infty)$. Then, for $-\infty< c_1<c_2\leq m_Y$ or $-\infty<m_Y\leq c_2<c_1$, $GPN= P[W(Y-c_2)<W(Y-c_1)]+\frac{1}{2} P[W(Y-c_2)=W(Y-c_1)]>\frac{1}{2}$.* **Lemma 15**. *Suppose that assumptions (A) and (C1)-(C3) hold. For $u>0$ and $\boldsymbol{\theta}\in \Theta$, let $m(\boldsymbol{\theta},u)$ denote the median of the conditional distribution of $X_{(2)}-\theta_{(2)}$, given $U=u$. Then $$m(\boldsymbol{\theta},u)\leq m(\boldsymbol{0},u),\; \; \forall \; u>0.$$* ***Proof:*** The joint distribution of $(Y,U)=(X_{(2)}-\theta_{(2)},X_{(2)}-X_{(1)})$ is $$\begin{aligned} h_{\theta}(y,u)=f(y-u+\theta)f(y) +f(y-u)f(y+\theta),\;\; -\infty<y<\infty,\; u>0,\end{aligned}$$ where $\theta=\theta_{(2)}-\theta_{(1)}\geq 0$. The conditional p.d.f. of $Y$ given $U=u$ is $\Pi_{\theta}(z\vert u) = \frac{1}{c_{\theta}}f(z-u+\theta)f(z)+f(z-u)f(z+\theta),\,z\in \Re$, where $c_{\theta}=\int_{-\infty}^{\infty}\left[f(t-u+\theta)f(t)+f(t-u)f(t+\theta)\right] dt$.\ Now, for any fixed $u>0$ and $\theta\geq 0$, we have $$\begin{aligned} \frac{\Pi_{\theta}(z\vert u)}{\Pi_{0}(z\vert u)}&=&\frac{c_0}{c_{\theta}}\frac{f(z-u+\theta)f(z)+f(z-u)f(z+\theta)} {2f(z-u)f(z)}\\ &=&\frac{c_0}{2\,c_{\theta}}\left[\frac{f(z-u+\theta)}{f(z-u)}+\frac{f(z+\theta)}{f(z)}\right].\end{aligned}$$ By the assumption (A), for $\theta>0$, $\frac{\Pi_{\theta}(z\vert u)}{\Pi_{0}(z\vert u)}$ is decreasing in $z$. For every fixed $u>0$ and $\theta\geq 0$, take $k_1(s)=I_{(-\infty,m(\boldsymbol{\theta},u))}(s),\;s\in \Re,$ and $k_2(s)= \frac{\Pi_{\theta}(s\vert u)}{\Pi_{0}(s\vert u)},\; \in \Re$, where $I_A(\cdot)$ denotes the indicator function of set $A \subseteq \Re$. Here $k_1(s)$ and $k_2(s)$ are decreasing functions of $s$. Using Proposition 4.1, for any $u>$ and $\theta\geq 0$, we get $$\begin{aligned} \frac{1}{2}=&\int_{-\infty}^{\infty} k_1(s) k_2(s) \Pi_{0}(s\vert u)ds \geq \left( \int_{-\infty}^{\infty} k_1(s) \Pi_{0}(s\vert u)ds \right) \left( \int_{-\infty}^{\infty} k_2(s) \Pi_{0}(s\vert u)ds\right)\\ \implies &\int_{-\infty}^{m(\boldsymbol{\theta},u)} \Pi_{\theta}(s\vert u)ds=\int_{-\infty}^{m(\boldsymbol{0},u)} \Pi_{0}(s\vert u)ds=\frac{1}{2} \geq \int_{-\infty}^{m(\boldsymbol{\theta},u)} \Pi_{0}(s\vert u)ds\\ \implies & m(\boldsymbol{0},u)\geq \, m(\boldsymbol{\theta},u),\;\; \forall \; u>0, \end{aligned}$$ establishing the assertion. $\blacksquare$ **Theorem 16**. *Suppose that assumptions (A) and (C1)-(C3) hold. Let $\delta_{\phi}=X_{(2)}-\phi(U)$ be an estimator of $\theta_{(2)}$ such that $P_{\boldsymbol{\theta}}(\phi(U)>m(\boldsymbol{0},U)) \ne 0$, for some $\boldsymbol{\theta}\in \Theta$. Then the estimator $$\begin{aligned} \label{deltast} \delta_{\phi_0}(\bold{X})=X_{(2)}-\min\{\phi(U),m(\boldsymbol{0},U)\} \end{aligned}$$ improves over the estimator $\delta_{\phi}(\bold{X})=X_{(2)}-\phi(U)$ in terms of the GPN criterion with a general loss ( [\[loss1\]](#loss1){reference-type="ref" reference="loss1"}).* ***Proof:*** Let $\phi_0(t)=\min\{\phi(t),m(\boldsymbol{0},t)\},\; t\geq 0.$ The GPN of $\delta_{\phi_0}(\bold{X})=X_{(2)}-\phi_0(U)$ relative to $\delta_{\phi}(\bold{X})$ is given by $$\begin{aligned} GPN(\delta_{\phi_0},\delta_{\phi};\boldsymbol{\theta})&=P_{\boldsymbol{\theta}}[W(X_{(2)}-\theta_{(2)}-\phi_0(U)) <W(X_{(2)}-\theta_{(2)}-\phi(U))] \\&\quad +\frac{1}{2} P_{\boldsymbol{\theta}}[W(X_{(2)}-\theta_{(2)}-\phi_0(U)) =W(X_{(2)}-\theta_{(2)}-\phi(U))]\\ &=\int_{-\infty}^{\infty} P_{\boldsymbol{\theta}}[W(X_{(2)}-\theta_{(2)}-\phi_0(u)) <W(X_{(2)}-\theta_{(2)}-\phi(u)) \vert U=u]\,f_U(u) \,du \\&\quad +\frac{1}{2}\, \int_{-\infty}^{\infty}P_{\boldsymbol{\theta}}[W(X_{(2)}-\theta_{(2)}-\phi_0(u)) =W(X_{(2)}-\theta_{(2)}-\phi(u)) \vert U=u] \,f_U(u) \,du,\end{aligned}$$ where $f_U(u)=\int_{-\infty}^{\infty}\left[f(t-u+\theta)f(t)+f(t-u)f(t+\theta)\right] dt,\;u>0,\;\theta=\theta_{(2)}-\theta_{(1)}\geq 0,$ is p.d.f. of r.v. $U$. Now, define $A=\{u>0: \phi(u)\leq m(\bold{0},u)\}$ and $B=\{u>0: \phi(u)> m(\bold{0},u)\}$, we get $$\begin{aligned} GPN(\delta_{\phi_0},\delta_{\phi};\boldsymbol{\theta})&=\int_{A} P_{\boldsymbol{\theta}}[W(X_{(2)}-\theta_{(2)}-\phi(u)) <W(X_{(2)}-\theta_{(2)}-\phi(u)) \vert U=u]\,f_U(u) \,du \\&\quad +\frac{1}{2}\, \int_{A}P_{\boldsymbol{\theta}}[W(X_{(2)}-\theta_{(2)}-\phi(u)) =W(X_{(2)}-\theta_{(2)}-\phi(u)) \vert U=u] \,f_U(u) \,du \\ &\quad + \int_{B} P_{\boldsymbol{\theta}}[W(X_{(2)}-\theta_{(2)}-m(\bold{0},u)) <W(X_{(2)}-\theta_{(2)}-\phi(u)) \vert U=u]\,f_U(u) \,du \\&\quad +\frac{1}{2}\, \int_{B}P_{\boldsymbol{\theta}}[W(X_{(2)}-\theta_{(2)}-m(\bold{0},u)) =W(X_{(2)}-\theta_{(2)}-\phi(u)) \vert U=u] \,f_U(u) \,du\\ &=\frac{1}{2}\, \int_{A}\,f_U(u) \,du \\ &\quad + \int_{B} P_{\boldsymbol{\theta}}[W(X_{(2)}-\theta_{(2)}-m(\bold{0},u)) <W(X_{(2)}-\theta_{(2)}-\phi(u)) \vert U=u]\,f_U(u) \,du.\end{aligned}$$ From Lemma [Lemma 15](#pitlemma2){reference-type="ref" reference="pitlemma2"}, we have $$\begin{aligned} m(\boldsymbol{\theta},u)\le m(\boldsymbol{0},u),\;\;\forall\;u>0.\end{aligned}$$ Then, for every $\boldsymbol{\theta}\in \Theta$, we have $-\infty<m(\boldsymbol{\theta},u)\le m(\boldsymbol{0},u) < \phi(u)$,$\; \forall\; u\in B$. Since, for $u\in B$, $m(\boldsymbol{\theta},u)$ is the median of the conditional distribution of $X_{(2)}-\theta_{(2)}$ given $U=u$ and using Lemma [Lemma 14](#garg1){reference-type="ref" reference="garg1"}, we have $P_{\boldsymbol{\theta}}[W(X_{(2)}-\theta_{(2)}-m(\bold{0},u)) <W(X_{(2)}-\theta_{(2)}-\phi(u)) \vert U=u]\geq \frac{1}{2},\;\forall\; u\in B, \; \boldsymbol{\theta}\in \Theta.$ Hence we get $$\begin{aligned} GPN(\delta_{\phi_0},\delta_{\phi};\boldsymbol{\theta})\, \geq \,\frac{1}{2},\;\forall \; \boldsymbol{\theta}\in\Theta, \;u>0,\end{aligned}$$ and strict inequity holds for some $u>0$. This proves the theorem. $\blacksquare$ An immediate consequence of Lemma [Lemma 14](#garg1){reference-type="ref" reference="garg1"}, the Pitman nearest (PN) equivariant estimator of $\theta_{(2)}$ within the class $\mathcal{D}$, under the GPN criterion, is obtained as $$\delta_{PN}(\bold{X})=X_{(2)}-m_{0},$$ where $m_{0}$ is such that $\int_{-\infty}^{m_0} f(x)\,dx=\frac{1}{2}$. **Corollary 17**. *Suppose that assumptions (A) and (C1)-(C3) hold. Then for estimating $\theta_{(2)}$ under the GPN criterion, the estimator $$\begin{aligned} \label{deltast1} \delta^*_{PN}(\bold{X})=X_{(2)}-\min\{m_0,m(\boldsymbol{0},U)\} \end{aligned}$$ is Pitman nearer to $\theta_{(2)}$ than the estimator $\delta_{PN}(\bold{X})=X_{(2)}-m_{0}$, provided $P_{\boldsymbol{\theta}}(m(\boldsymbol{0},U)<m_0) \ne0$, for some $\boldsymbol{\theta}\in \Theta$.* The following corollary provides an improvement over the usual estimator $\delta_{0}(\bold{X})=X_{(2)}-c_{0}$ (as defined by [\[usual estimator\]](#usual estimator){reference-type="eqref" reference="usual estimator"}). **Corollary 18**. *Suppose that assumptions (A) and (C1)-(C3) hold. Then for estimating $\theta_{(2)}$ under the GPN criterion, the estimator $$\begin{aligned} \label{deltast1} \delta^*_{PN}(\bold{X})=X_{(2)}-\min\{c_0,m(\boldsymbol{0},U)\} \end{aligned}$$ is Pitman nearer to $\theta_{(2)}$ than the estimator $\delta_{0}(\bold{X})=X_{(2)}-c_{0}$, provided $P_{\boldsymbol{\theta}}(m(\boldsymbol{0},U)<c_0) \ne 0$, for some $\boldsymbol{\theta}\in \Theta$.* # Applications {#sec6} **Example 1**. *Let $X_1\sim N(\theta_1,\sigma^2)$ and $X_2\sim N(\theta_2,\sigma^2)$, where $(\theta_1,\theta_2)\in \Theta_0$ is vector of unknown means and $\sigma^2>0$ is known common variance. The pdf of $X_i$ is $f(x-\theta_i)=\frac{1}{\sqrt{2 \pi} \sigma} e^{-\frac{1}{2\sigma^2}(x-\theta_i)^2},\;x\in \Re,\;i=1,2.$ Here it is easy to see that $f(\cdot)$ holds the assumption (A).* *Consider the estimation of parameter $\theta_{(2)}=\max\{\theta_1,\theta_2\}$ under the loss function $$\begin{aligned} \label{eq:4.1} L((\theta_1,\theta_2),a)=W(a-\theta_{(2)}), \; (\theta_1,\theta_2)\in \Re^2,\; a\in \Re.\end{aligned}$$ For the squared error loss (i.e., $W(t)=t^2,\;t\in \Re$), the usual estimator of $\theta_{(2)}$ is $\delta_{c_0}=X_{(2)}$. In this case, using Theorem 3.1, there is no improvement over the usual estimator $\delta_{c_0}$ and the improved estimator $\delta_{ST}(\bold{X})=X_{(2)}-\min\bigg\{0,\frac{U}{2}\bigg\}=X_{(2)}$ is same. Also, using Theorem 3.4, the estimator $\delta_{b_0}(\bold{X})=X_{(2)}-\frac{\sigma}{\sqrt{\pi}}$ is dominated by the estimator $\delta_{BZ}(\bold{X})$, where $$\begin{aligned} \delta_{BZ}(\bold{X})&=X_{(2)}-\phi_{BZ}(U)=X_{(2)}-\frac{\sigma}{\sqrt{2}} \frac{\left(\frac{1}{\sqrt{2\pi}}-\phi\left(\frac{U}{\sqrt{2} \sigma}\right)\right)}{\left(\Phi\left(\frac{U}{\sqrt{2} \sigma}\right)-0.5\right)},\end{aligned}$$ $U=X_{(2)}-X_{(1)}$, $\phi(\cdot)$ is the p.d.f. of standard normal distribution and $\Phi(\cdot)$ is the d.f. of standard normal distribution.* *For $W(t)=e^{at}-at-1,\;t\in \Re,\; a\neq 0$, (i.e., the Linex loss) the usual estimator of $\theta_{(2)}$ is $\delta_{c_0}=X_{(2)}-\frac{a \sigma^2}{2}$. In this case, using Theorem 3.2, the estimator $\delta_{c_0}$ is dominated by the estimator $\delta_{ST}(\bold{X})$, where $$\begin{aligned} \delta_{ST}(\bold{X})&=X_{(2)}-\min\bigg\{\frac{a \sigma^2}{2},\frac{U}{2}+\frac{a \sigma^2}{4}\bigg\}=\max\bigg\{X_{(2)}-\frac{a \sigma^2}{2},\frac{X_1+X_2}{2}-\frac{a \sigma^2}{4}\bigg\}.\end{aligned}$$ Also, using Theorem 3.5, the estimator $\delta_{b_0}(\bold{X})=X_{(2)}-\frac{1}{a}\left[\ln2+\frac{a^2\sigma^2}{2} +\ln\left(\Phi\left(\frac{a\sigma}{\sqrt{2}}\right)\right)\right]$ is dominated by the estimator $\delta_{BZ}(\bold{X})$, where $$\delta_{BZ}(\bold{X})=X_{(2)}-\frac{1}{a} \left[\frac{a^2\sigma^2}{2} +\ln\left(\Phi\left(\frac{a\sigma}{\sqrt{2}}\right)-\Phi\left(\frac{-U+a\sigma^2}{\sqrt{2} \sigma}\right)\right) -\ln\left(\frac{1}{2}-\Phi\left(\frac{-U}{\sqrt{2} \sigma}\right)\right) \right].$$* *Now we consider $W(t)=\vert t\vert,\;t\in \Re,$ (i.e., the absolute error loss). Under this loss function the usual estimator of $\theta_{(2)}$ is $\delta_{c_0}=X_{(2)}$. In this case, using Theorem 3.3, the estimator $\delta_{c_0}$ is dominated by the estimator $$\begin{aligned} \delta_{ST}(\bold{X})&=X_{(2)}-\min\bigg\{0,\frac{U}{2}\bigg\}=X_{(2)}.\end{aligned}$$ Also, using Theorem 3.6, the estimator $\delta_{b_0}(\bold{X})=X_{(2)}-b_0$, where $b_0$ is the solution of the equation $\int_{-\infty}^{C} \Phi\left(\frac{s}{\sigma}\right) \phi\left(\frac{s}{\sigma}\right)ds=\frac{\sigma}{4}$, is dominated by the estimator $\delta_{BZ}(\bold{X})=X_{(2)}-C,$ where $C$ is the unique solution of the following equation $$\int_{-\infty}^{C}\left[ \Phi\left(\frac{s}{\sigma}\right)- \Phi\left(\frac{s-U}{\sigma}\right)\right] \phi\left(\frac{s}{\sigma}\right)ds=\frac{\sigma}{4}-\frac{\sigma}{2}\Phi\left(\frac{-U}{\sqrt{2} \sigma}\right).$$* *Now, we will illustrate an application of Theorem [Theorem 16](#th1){reference-type="ref" reference="th1"} and Corollary [Corollary 17](#pitman corr){reference-type="ref" reference="pitman corr"}. Consider the estimation of parameter $\theta_{(2)}$ under the GPN criterion with the general loss function ($W(\cdot)$ satisfy (C1), (C2) and (C3)) $$\begin{aligned} \label{eq:3.1} L(\boldsymbol{\theta},a)=W(a-\theta_{(2)}), \; \boldsymbol{\theta}\in \Re^2,\; a\in \Re.\end{aligned}$$ Under the GPN criterion, the Pitman nearest (PN) estimator of $\theta_{(2)}$ is $X_{(2)}$. In this case, using Theorems [Theorem 16](#th1){reference-type="ref" reference="th1"}, the improved estimator $\delta_{\phi^*}(\bold{X})=X_{(2)}-\min\big\{0,\frac{U}{2}\big\}=X_{(2)}$ is same as PN estimator. Hence, there is no improvement over the estimator $X_{(2)}$ using our results.* **Example 2**. *Let $X_1$ and $X_2$ be independent exponential random variables with $X_i$ having the p.d.f. $f(x-\theta_i),\;i=1,2,$ where $$f(z)=\begin{cases} \frac{1}{\sigma}\, e^{-\frac{z}{\sigma}}\, ,&\text{if }\; z\geq 0\\ 0,&\text{if }\; z<0 \end{cases},$$ $\sigma>0$ is known positive constant and $\boldsymbol{\theta} \in \Re^2$ is the vector of unknown location parameters. Consider estimation of $\theta_{(2)}$ under the loss function $$\begin{aligned} \label{eq:4.4} L(\boldsymbol{\theta},a)=W(a-\theta_{(2)}), \; \boldsymbol{\theta}\in \Re^2,\; a\in \Re.\end{aligned}$$ Under the squared error loss (i.e., $W(t)=t^2,\;t\in \Re$), the usual estimator of $\theta_{(2)}$ is $X_{(2)}-\sigma$. In this case, using Theorem 3.1, the BLEE $X_{(2)}-\sigma$ is dominated by the estimator $\delta_{ST}(\bold{X})$, where $$\begin{aligned} \delta_{ST}(\bold{X})&=X_{(2)}-\min\bigg\{\sigma,\frac{\sigma}{2}+U\bigg\}=\max\bigg\{X_{(2)}-\sigma,X_{(1)}-\frac{\sigma}{2}\bigg\}.\end{aligned}$$ Also, using Theorem 3.4, the estimator $\delta_{b_0}(\bold{X})=X_{(2)}-\frac{3\sigma}{2}$ is dominated by the estimator $\delta_{BZ}(\bold{X})$, where $$\begin{aligned} \delta_{BZ}(\bold{X})&=X_{(2)}-\phi_{BZ}(U)=X_{(2)}-\frac{3\sigma-(2U+3\sigma)e^{-\frac{U}{\sigma}}}{2(1-e^{-\frac{U}{\sigma}})},\end{aligned}$$ and $U=X_{(2)}-X_{(1)}$.* *Under the Linex loss function $W(t)=e^{at}-at-1,\;t\in \Re,\; a\neq 0$, the usual estimator of $\theta_{(2)}$ is $\delta_{c_0}=X_{(2)}+\frac{1}{a} \ln(1-a\sigma)$, whenever $a\sigma<1$. In this case, using Theorem 3.2, the estimator $X_{(2)}$ is dominated by the estimator $\delta_{ST}(\bold{X})$, where $$\begin{aligned} \delta_{ST}(\bold{X})&=X_{(2)}-\min\bigg\{-\frac{1}{a} \ln(1-a\sigma),U+\frac{\ln(2)}{a}-\frac{1}{a} \ln(2-a\sigma)\bigg\}\\ &=\max\bigg\{X_{(2)}+\frac{1}{a} \ln(1-a\sigma),X_{(1)}+\frac{1}{a} \ln(2-a\sigma)-\frac{\ln(2)}{a}\bigg\}.\end{aligned}$$ Also, using Theorem 3.5 the estimator $\delta_{b_0}(\bold{X})=X_{(2)}-\frac{1}{a}[\ln2 -\ln(1-a\sigma)-\ln(2-a\sigma)]$, whenever $a\sigma<1$, is dominated by the estimator $\delta_{BZ}(\bold{X})$, where $$\delta_{BZ}(\bold{X})=X_{(2)}-\phi_{BZ}(U)=X_{(2)}-\frac{1}{a}\left[\ln2 -\ln(1-a\sigma)-\ln(2-a\sigma)+\ln\left(1-e^{-U\left(\frac{1}{\sigma}-a\right)}\right)-\ln\left(1-e^{-\frac{U}{\sigma}}\right)\right].$$* *Under the absolute error loss $W(t)=\vert t\vert,\;t\in \Re,$ the usual estimator of $\theta_{(2)}$ is $\delta_{c_0}=X_{(2)}-\sigma \ln(2)$. In this case, using Theorem 3.3, the estimator $X_{(2)}-\sigma \ln(2)$ is dominated by the estimator $$\begin{aligned} \delta_{ST}(\bold{X})&=X_{(2)}-\min\bigg\{\sigma \ln(2),U+\frac{\sigma}{2}\ln(2)\bigg\}=\max\bigg\{X_{(2)}-\sigma \ln(2),X_{(1)}-\frac{\sigma}{2}\ln(2)\bigg\}.\end{aligned}$$ Also, using Theorem 3.6, the estimator $\delta_{b_0}(\bold{X})=X_{(2)}+\sigma\ln\left(1-\frac{1}{\sqrt{2}}\right)$ is dominated by the estimator $\delta_{BZ}(\bold{X})=X_{(2)}-C,$ where $C>0$ is the unique solution of the following equation $$\begin{aligned} \int_{0}^{C}\int_{0}^{\min\{U,s\}} e^{-\frac{2s}{\sigma}}e^{\frac{y}{\sigma}}dy\,ds=\frac{\sigma^2}{4}\left(1-e^{-\frac{U}{\sigma}}\right).\end{aligned}$$* *Now, we will illustrate an application of Theorem [Theorem 16](#th1){reference-type="ref" reference="th1"} and Corollary [Corollary 17](#pitman corr){reference-type="ref" reference="pitman corr"}. Consider the estimation of parameter $\theta_{(2)}$ under the GPN criterion with the general loss function ($W(\cdot)$ satisfy (C1), (C2) and (C3)) $$\begin{aligned} \label{eq:3.1} L(\boldsymbol{\theta},a)=W(a-\theta_{(2)}), \; \boldsymbol{\theta}\in \Re^2,\; a\in \Re.\end{aligned}$$ Under the GPN criterion, the PN estimator of $\theta_{(2)}$ is $X_{(2)}-\sigma \ln(2)$. In this case, using Theorems [Theorem 16](#th1){reference-type="ref" reference="th1"}, the estimator $$\delta_{\phi_0}(\bold{X})=\max\bigg\{X_{(2)}-\sigma \ln(2),X_{(1)}-\frac{\sigma}{2}\ln(2)\bigg\}$$ is the Pitman nearer to $\theta_{(2)}$ than the estimator $X_{(2)}-\sigma \ln(2)$, under the GPN criterion.* # Simulation study {#sec7} In Example 4.1, we considered the estimation of the larger mean $\theta_{(2)}$ of two independent normal distributions with a known common variance ($\sigma^2$). We derived estimators that improved upon the usual estimator $\delta_{c_0}(\boldsymbol{X}) = X_{(2)}$ and the estimator $\delta_{1}(\boldsymbol{X}) = X_{(2)} - b_0$ under three different loss functions. To evaluate the risk performance of these estimators of $\theta_{(2)}$ under these loss functions, we conducted Monte Carlo simulations to compare the risk performances of the usual estimator $X_{(2)}$, the Stein-type estimator $\delta_{ST}$, the estimator $\delta_{1}(\boldsymbol{X})$, and the Brewster-Zidek type estimator $\delta_{BZ}$. We computed the simulated risks based on 50000 simulations from relevant distributions, and the resulting risk function of the different estimators were plotted in Figures [\[fig1\]](#fig1){reference-type="ref" reference="fig1"}-[\[fig3\]](#fig3){reference-type="ref" reference="fig3"}. The following observations are evident from Figures [\[fig1\]](#fig1){reference-type="ref" reference="fig1"}-[\[fig3\]](#fig3){reference-type="ref" reference="fig3"}: - The risk of the Stein type estimator $\delta_{ST}$ has smaller risk than the usual estimator $\delta_{c_0}$. For smaller $\sigma$, it can be seen that the risk values of both the estimators are almost equal when $\theta_{(2)}-\theta_{(1)}$ takes values close to zero and approximately more than 3. For higher values of $\sigma$, we observe that $\delta_{ST}$ performs significantly better than $\delta_{c_0}$. In this case, the region on improvement is better than the previous (small $\sigma$) case. - The risk performance of $\delta_{BZ}$ is better then $\delta_{1}$ for all $\theta_{(2)}-\theta_{(1)}$. It is observed that for larger values of $\sigma$, the improvement interval is bigger compared to the smaller values of $\sigma$. - We also found that there was no clear winner among the various estimators, as the estimator $\delta_{1}$ and the $\delta_{BZ}$ performed better than the usual estimator and $\delta_{ST}$ for small and moderate values of $\theta_{(2)}-\theta_{(1)}$, whereas the estimator $\delta_{ST}$ dominated the other two estimators for large values of $\theta_{(2)}-\theta_{(1)}$. {width="80mm"} {width="80mm"} \ {width="80mm"} {width="80mm"} {width="70mm"} {width="70mm"} \ {width="70mm"} {width="70mm"} \ {width="70mm"} {width="70mm"} {width="80mm"} {width="80mm"} \ {width="80mm"} {width="80mm"} # **Real-life Data Analysis** {#sec8} For real life data analysis, we have considered the "Jute fiber breaking strength data\", discussed by [@xia2009study] and presented in Table [1](#table:1){reference-type="ref" reference="table:1"}. This data represents the breaking strength of jute fibre of two different gauge lengths. Jute is a versatile natural fiber widely employed in various products, including textiles, ropes, sacks, and geotextiles. The breaking strength of jute fibers plays a pivotal role in determining the quality and durability of these products. By estimating the maximum breaking strength with different gauge lengths, manufacturers can ensure that their products conform to the required standards and can withstand a range of stresses and loads. Therefore, the estimation of the maximum breaking strength of jute fibers using two different gauge lengths holds significant importance in the realm of material science and engineering for several compelling reasons. For data analysis, we initially examined whether these datasets follow two-parameter exponential distributions. We used the one-sample Kolmogorov-Smirnov test to see if the data with gauge lengths 10 mm and 15 mm may be following a two parameter exponential distributions and found p-values of $0.755$ and $0.306$, respectively, indicating that the data with gauge lengths 10 mm and 15 mm follow a two parameter exponential distributions with a common scale parameter value of $322$. No. gauge length 10 mm gauge length 15 mm ----- -------------------- -------------------- 1 43.93 594.4 2 50.16 202.75 3 101.15 168.37 4 108.94 574.86 5 123.06 225.65 6 141.38 76.38 7 151.48 156.67 8 163.4 127.81 9 177.25 813.87 10 183.16 562.39 11 212.13 468.47 12 257.44 135.09 13 262.9 72.24 14 291.27 497.94 15 303.9 355.56 16 323.83 569.07 17 353.24 640.48 18 376.42 200.76 19 383.43 550.42 20 422.11 748.75 21 506.6 489.66 22 530.55 678.06 23 590.48 457.71 24 637.66 106.73 25 671.49 716.3 26 693.73 42.66 27 700.74 80.4 28 704.66 339.22 29 727.23 70.09 30 778.17 193.42 : The breaking strength of jute fibre [\[table:1\]]{#table:1 label="table:1"} Let $X_1$ and $X_2$ be the two independent random variables representing the breaking strength of jute fibre corresponding to gauge length 10 mm and 15 mm, respectively. Therefore $X_1\sim Exp(\theta_1,\sigma)$ and $X_2\sim Exp(\theta_2,\sigma)$, where $\theta_1$ and $\theta_2$ are location parameters, and common known scale parameter $\sigma= 322/30 = 10.73$. Using our finding of this paper, we obtain estimates that are better than the natural estimates of parameter $\max\{\theta_1,\theta_2\}$. In Example 5.2, we have presented various estimators under the squared error loss, linex loss and absolute error loss functions, which are given in Table [2](#table:2){reference-type="ref" reference="table:2"}, Table [3](#table:3){reference-type="ref" reference="table:3"} and Table [4](#table:4){reference-type="ref" reference="table:4"}, respectively. From theoretical results (in Example 5.2), we infer that the Stein type estimated value $\delta_{ST}(\bold{x})$ is better than estimated value of $\delta_{c_0}(\bold{x})$ and the Brewster-Zidek type estimated value $\delta_{BZ}(\bold{x})$ is better than estimated value of $\delta_{b_0}(\bold{x})$ for parameter $\max\{\theta_1,\theta_2\}$. $\;\;\delta_{c_0}(\bold{x})\;\;$ $\quad\delta_{ST}(\bold{x}) \quad$ $\quad\delta_{b_0}(\bold{x}) \quad$ $\quad \delta_{BZ}(\bold{x}) \quad$ ---------------------------------- ------------------------------------ ------------------------------------- ------------------------------------- 33.2 37.3 27.835 37.94 : Various estimated values of parameter $\max\{\theta_1,\theta_2\}$:\ under the squared error loss [\[table:2\]]{#table:2 label="table:2"} $\;\;\delta_{c_0}(\bold{x})\;\;$ $\quad\delta_{ST}(\bold{x}) \quad$ $\quad\delta_{b_0}(\bold{x}) \quad$ $\quad \delta_{BZ}(\bold{x}) \quad$ ---------------------------------- ------------------------------------ ------------------------------------- ------------------------------------- 41.47 41.47 39.62 41.52 : Various estimated values of parameter $\max\{\theta_1,\theta_2\}$:\ under the Linex loss with $a=-1$ [\[table:3\]]{#table:3 label="table:3"} $\;\;\delta_{c_0}(\bold{x})\;\;$ $\quad\delta_{ST}(\bold{x}) \quad$ $\quad\delta_{b_0}(\bold{x}) \quad$ $\quad \delta_{BZ}(\bold{x}) \quad$ ---------------------------------- ------------------------------------ ------------------------------------- ------------------------------------- 36.5 38.94 30.75 45.65 : Various estimated values of parameter $\max\{\theta_1,\theta_2\}$:\ under the absolute error loss [\[table:4\]]{#table:4 label="table:4"} # Conclusions {#sec9} In the present article, we have considered the estimation, from a decision-theoretic point of view, of the larger location parameter of two general location family of distributions under a general location invariant loss function. We have proposed a Stein-type estimator, which improves upon the usual estimator. Next, we have proposed an estimator, $\delta_{b_0}$, which is similar to the usual estimator. Using the IERD approach of Kubokawa, a class of estimators has been derived that dominates $\delta_{b_0}$. It is seen that the boundary estimator of this class is the Brewster-Zidek type estimator. As an application, the improved estimators are derived for particular loss functions. We have also considered the same estimation problem with respect to the generalized Pitman nearness criterion. We have proposed an estimator that is nearer than the usual estimator. As an application, explicit expressions of all the improved estimators are obtained for the normal and exponential models. Further, we have compared the risk performance of the proposed estimators using the Monte Carlo simulation. Finally, we present a real-life application that demonstrates the practical significance of the findings presented in this paper. [^1]:  garg.naresh22\@gmail.com

arxiv_math

--- abstract: | We let $\Omega$ be a bounded domain of $\mathbb{R}^3$ and $\Gamma$ be a closed curve contained in $\Omega$. We study existence of positive solutions $u \in H^1_0\left(\Omega\right)$ to the equation $$-\Delta u+hu=\lambda \rho^{-s_1}_\Gamma u^{5-2s_1}+\rho^{-s_2}_\Gamma u^{5-2s_2} \qquad \textrm{ in } \Omega$$ where $h$ is a continuous function and $\rho_\Gamma$ is the distance function to $\Gamma$. We prove existence of solutions depending on the regular part of the Green function of linear operator. We prove the existence of positive mountain pass solutions for this Euler-Lagrange equation depending on the mass which is the regular part of the Green function of the linear operator $-\Delta +h$. address: - "H. E. A. T. : Université Iba Der Thiam de Thies, UFR des Sciences et Techniques, département de mathématiques, Thies." - "H. E. A. T. : Université Iba Der Thiam de Thies, UFR des Sciences et Techniques, département de mathématiques, Thies." author: - El Hadji Abdoulaye THIAM title: Mass effect on an elliptic PDE involving two Hardy-Sobolev critical exponents --- **Key Words**: Two Hardy-Sobolev critical exponents; Green function; Positive mass; Mountain Pass solution; Curve singularity. # Introduction In this paper, we are concerned with the mass effect on the existence of mountain pass solutions of the following nonlinear partial differential equation involving two Hardy-Sobolev critical exponents in $\mathbb{R}^3$. More precisely, letting $h$ be a continuous function and $\lambda$ be a real parameter, we consider $$\label{Euler-Lagrange11} \begin{cases} \displaystyle-\Delta u(x)+ h u(x)=\lambda \frac{u^{5-2s_1}(x)}{\rho_\Gamma^{s_1}(x)}+\frac{u^{5-2s_2}(x)}{\rho_\Gamma^{s_2}(x)} \qquad & \textrm{ in $\Omega $}\\\\ u(x)>0 \qquad \textrm{ and } \qquad u(x)=0 &\textrm{ on $\partial\Omega$}, \end{cases}$$ where $\rho_\Gamma(x):=\inf_{y \in \Gamma}|y-x|$ is the distance function to the curve $\Gamma$ and for $0< s_2 <s_1<2$, $2^*_{s_1}:=6-2s_1$ and $2^*_{s_2}:=6-2s_2$ are two critical Hardy-Sobolev exponents. To study the equation [\[Euler-Lagrange11\]](#Euler-Lagrange11){reference-type="eqref" reference="Euler-Lagrange11"}, we consider the following non-linear functional $\Psi: H^1_0(\Omega ) \to \mathbb{R}$ defined by: $$\label{Functional} \Psi(u):=\frac{1}{2} \int_\Omega |\nabla u|^2 dx+\frac{1}{2} \int_\Omega h(x) u^2 dx- \frac{\lambda }{2^*_{s_1}} \int_\Omega \rho_\Gamma^{-s_1}(x) |u|^{2^*_{s_1}} dx-\frac{1}{2^*_{s_2}} \int_\Omega \rho_\Gamma^{-s_2}(x) |u|^{2^*_{s_2}} dx.$$ Then there exists a positive constant $r>0$ and $u_0 \in H^1_0(\Omega)$ such that $\|u_0\|_{H^1_0(\Omega )}>r$ and $$\inf_{\|u\|_{H^1_0(\Omega )}=r} \Psi(u) >\Psi(0) \geq \Phi(u_0),$$ see for instance the paper of the author \[[@Thiam1], Lemma 4.5\]. Then the point $(0, \Psi(0))$ is separated from the point $(u_0, \Psi(u_0))$ by a ring of mountains. Set $$\label{Heat} c^*:=\inf_{P \in \mathcal{P}} \max_{v \in P} \Psi(v),$$ where $\mathcal{P}$ is the class of continuous paths in $H^1_0(\Omega )$ connecting $0$ to $u_0$. Since $2^*_{s_2}>2^*_{s_1}$, the function $t \longmapsto \Psi(tv)$ has the unique maximum for $t \geq 0$. Furthermore, we have $$c^*:=\inf_{u \in H^1_0(\Omega ), u \geq 0, u \neq 0} \max_{t \geq 0} \Psi(tu).$$ Due to the fact that the embedding of $H^1_0(\Omega )$ into the weighted Lebesgue spaces $L^{2^*_{si}}(\rho_\Gamma^{-si} dx)$ is not compact, the functional $\Psi$ does not satisfy the Palais-Smale condition. Therefore, in general $c^*$ might not be a critical value for $\Psi$. To recover compactness, we study the following non-linear problem: let $x=(y, z) \in \mathbb{R}\times \mathbb{R}^{2}$ and consider $$\label{A1A1} \begin{cases} \displaystyle-\Delta u=\lambda \frac{u^{2^*_{s_1}-1}(x)}{|z|^{s_1}}+\frac{u^{2^*_{s_2}-1}}{|z|^{s_2}} \qquad & \textrm{ in $\mathbb{R}^3$}\\\ u(x)>0 & \textrm{ in $\mathbb{R}^3$}. \end{cases}$$ To obtain solutions of [\[A1A1\]](#A1A1){reference-type="eqref" reference="A1A1"}, we consider the functional $\Phi: \mathcal{D}^{1,2}(\mathbb{R}^N)$ defined by $$\label{Functional1} \Phi(u):=\frac{1}{2} \int_{\mathbb{R}^3} |\nabla u|^2 dx- \frac{\lambda }{2^*_{s_1}} \int_{\mathbb{R}^3} |z|^{-s_1}|u|^{2^*_{s_1}} dx-\frac{1}{2^*_{s_2}} \int_{\mathbb{R}^3} |z|^{-s_2}|u|^{2^*_{s_2}} dx.$$ Next, we define $$\beta^*:=\inf_{u \in D^{1, 2}(\mathbb{R}^3), u \geq 0, u \neq 0} \max_{t \geq 0} \Phi(tu).$$ Then we get compactness provided $$c^*<\beta^*,$$ see Proposition 4.3 in [@Thiam1]. Therefore the existence, symmetry and decay estimates of non-trivial solution $w\in \mathcal{D}^{1,2}(\mathbb{R}^3)$ of [\[A1A1\]](#A1A1){reference-type="eqref" reference="A1A1"} play an important role in problem [\[Euler-Lagrange11\]](#Euler-Lagrange11){reference-type="eqref" reference="Euler-Lagrange11"}. Then we have the following results. **Proposition 1**. *Let $0 \leq s_2<s_1<2$, $\lambda \in \mathbb{R}$. Then equation $$\label{Euler-Lagrange1} \begin{cases} \displaystyle-\Delta u=\lambda \frac{u^{2^*_{s_1}-1}(x)}{|z|^{s_1}}+\frac{u^{2^*_{s_2}-1}}{|z|^{s_2}} \qquad & \textrm{ in $\mathbb{R}^3$}\\\ u(x)>0 & \textrm{ in $\mathbb{R}^3$} \end{cases}$$ has a positive ground state solution $w \in \mathcal{D}^{1,2}(\mathbb{R}^3)$ depending only on $|y|$ and $|z|$. Moreover $$\label{eq:up-low-bound-w-3D} \frac{C_1}{1+|x|} \leq w(x) \leq \frac{C_2}{1+|x|} \qquad \textrm{ in $\mathbb{R}^3$}.$$ Moreover, for $|x|= |(t,z)|\leq 1$, we have $$\label{eq:up-low-bound-w-3D1} |\nabla w (x)|+ |x| |D^2 w (x)|\leq C_2 |z|^{1-s_1}$$ and if $|x|= |(t,z)|\geq 1$, we have $$\label{eq:up-low-bound-w-3D2} |\nabla w(x)|+ |x| |D^2 w(x)|\leq C_2 \max(1, |z|^{-s_1})|x|^{1 -N}.$$* Next, we let $G(x,y)$ be the Dirichlet Green function of the operator $-\Delta +h$, with zero Dirichlet data. It satisfies $$\label{eq:Green-expan-introduction} \begin{cases} -\Delta _x G(x,y)+h(x) G(x,y)=0& \qquad\textrm{ for every $x\in \Omega \setminus\{y\}$}\\ % G(x,y)=0 & \qquad\textrm{ for every $x\in\partial\Omega $.} \end{cases}$$ In addition there exists a continuous function $\textbf{m}:\Omega \to \mathbb{R}$ and a positive constant $c>0$ such that $$\label{eq:expans-Green} G (x,y)=\frac{c}{ |x- y|}+ c\, \textbf{m}(y)+o(1) \qquad \textrm{ as $x \to y.$}$$ We call the function $\textbf{m}:\Omega \to \mathbb{R}$ the *mass* of $-\Delta +h$ in $\Omega$. We note that $-\textbf{m}$ is occasionally called the *Robin function* of $-\Delta +h$ in the literature. Then our main result is the following. Then we have **Theorem 2**. *Let $0\leq s_2 <s_1<2$ and $\Omega$ be a bounded domain of $\mathbb{R}^3$. Consider $\Gamma$ a smooth closed curve contained in $\Omega$. Let $h$ be a continuous function such that the linear operator $-\Delta +h$ is coercive. We assume that there exists $y_0 \in \Gamma$ such that $$\label{ExistenceAssumption} m(y_0)>0.$$ Moreover there exists $u \in H^1_0(\Omega)\setminus \lbrace 0\rbrace$ non-negative solution of $$-\Delta u(x)+ h u(x)=\lambda \frac{u^{5-2s_1}(x)}{\rho_\Gamma^{s_1}(x)}+\frac{u^{5-2s_2}(x)}{\rho_\Gamma^{s_2}(x)} \qquad \textrm{ in $\Omega $}.$$* In contrast to the case $N \geq 4$ (see [@Thiam1] for more details), the existence of solution does not depend on the local geometry of the singularity but on the location of the curve $\Gamma$. Besides in the study of Hardy-Sobolev equations in domains with interior singularity for the Three dimensional case, the effect of the mass plays an important role in the existence of positive solutions. For Hardy-Sobolev inequality on Riemannian manifolds with singularity a point, Jaber [@Jaber1] proved the existence of positive solutions when the mass is positive. We refer also to [@Jaber2] for existence of mountain pass solution to a Hardy-Sobolev equation with an additional perturbation term. For the Hardy-Sobolev equations on domains with singularity a curve, we refer to the papers of the author and Fall [@Fall-Thiam] and the author and Ijaodoro [@Esther]. We also suggest to the interested readers the nice work of Schoen-Yau [@Schoen-Yau] and [@Schoen-Yau1] for more details related to the positive mass theorem. We also mention that this paper is the 3-dimensional version of the work of thye author [@Thiam1].\ The proof of Theorem [Theorem 2](#th:main1){reference-type="ref" reference="th:main1"} relies on test function methods. Namely we build appropriate test functions allowing to compare $c^*$ and $\beta ^*$. Near the concentration point $y_0 \in \Gamma$, the test function is similar to the test function in the case $N \geq 4$ but away from it is replaced with the regular part of the Green function which makes apear the mass, see Section [3](#ss:proof-of-th-3D){reference-type="ref" reference="ss:proof-of-th-3D"}. # Tool Box {#s:3D-case} We consider the function $${\mathcal R}:\mathbb{R}^3\setminus\{0\}\to \mathbb{R}, \qquad x\mapsto {\mathcal R}(x)=\frac{1}{|x|}$$ which satisfies $$\label{eq:Green-R3-3D} -\Delta {\mathcal R}=0 \qquad \textrm{ in $\mathbb{R}^3\setminus\{0\}$. }$$ We denote by $G$ the solution to the equation $$\label{eq:Green-3D} \begin{cases} -\Delta _x G(y, \cdot)+h G(y,\cdot)=0& \qquad \textrm{ in $\Omega \setminus \{y\}$. } \\ % G(y,\cdot )=0& \qquad \textrm{ on $\partial\Omega $, } \end{cases}$$ and satisfying $$\label{eq:expand-Green-trace} G(x,y)= {\mathcal R}(x-y)+O(1)\qquad\textrm{ for $x, y\in \Omega $ and $x\not= y$.}$$ We note that $G$ is proportional to the Green function of $-\Delta +h$ with zero Dirichlet data.\ We let $\chi\in C^\infty_c(-2,2)$ with $\chi \equiv 1$ on $(-1,1)$ and $0\leq \chi<1$. For $r>0$, we consider the cylindrical symmetric cut-off function $$\label{eq:def-cut-off-cylind} \eta_r(t,z)=\chi\left(\frac{|t|+|z|}{r} \right) \qquad \qquad\textrm{ for every $(t,z)\in \mathbb{R}\times \mathbb{R}^2$}.$$ It is clear that $$\eta_r\equiv 1\quad \textrm{ in ${Q}_r$},\qquad \eta_r\in H^1_0({Q}_{2r}),\qquad |\nabla \eta_r|\leq \frac{C}{r} \quad\textrm{ in $\mathbb{R}^3$}.$$ For $y_0\in \Omega$, we let $r_0\in (0,1)$ such that $$\label{eq:def-r0} y_0+ Q_{2r_0}\subset\Omega .$$ We define the function $M_{y_0}: Q_{2r_0}\to \mathbb{R}$ given by $$\label{C9} M_{y_0}(x):= G (y_0,x+y_0)-{\eta_r}(x)\frac{1}{|x|} \qquad \textrm{ for every $x\in Q_{2r_0}$}.$$ It follows from [\[eq:expand-Green-trace\]](#eq:expand-Green-trace){reference-type="eqref" reference="eq:expand-Green-trace"} that $M_{y_0}\in L^\infty(Q_{r_0})$. By [\[eq:Green-3D\]](#eq:Green-3D){reference-type="eqref" reference="eq:Green-3D"} and [\[eq:Green-R3-3D\]](#eq:Green-R3-3D){reference-type="eqref" reference="eq:Green-R3-3D"}, $$|-\Delta {M}_{y_0}(x)+h(x) {M}_{y_0}(x)|\leq \frac{C}{|x|}= C {\mathcal R}(x) \qquad \textrm{ for every $x\in Q_{r_0}$},$$ whereas ${\mathcal R}\in L^p( Q_{r_0})$ for every $p\in (1,3)$. Hence by elliptic regularity theory, $M_{y_0}\in W^{2,p}(Q_{r_0/2})$ for every $p\in (1,3)$. Therefore by Morrey's embdding theorem, we deduce that $$\label{eq:regul-beta} \|M_{y_0}\|_ {C^{1,\varrho}(Q_{r_0/2})}\leq C \qquad \textrm{ for every $\varrho\in (0,1)$.}$$ In view of [\[eq:expans-Green\]](#eq:expans-Green){reference-type="eqref" reference="eq:expans-Green"}, the mass of the operator $-\Delta +h$ in $\Omega$ at the point $y_0\in \Omega$ is given by $$\label{eq:def-mass} \textbf{m}(y_0)={M}_{y_0}(0).$$ Next, we have the following result which will be important in the sequel. **Lemma 3**. *Consider the function $v_\varepsilon : \mathbb{R}^3\setminus\{0\}\to \mathbb{R}$ given by $$v_\varepsilon (x)= \varepsilon ^{-1} w\left(\frac{x}{\varepsilon }\right).$$ Then there exists a constant $\textbf{c}>0$ and a sequence $(\varepsilon _n)_{n\in \mathbb{N}}$ (still denoted by $\varepsilon$) such that $$v_\varepsilon (x) \to \frac{\textbf{c}}{|x|} \qquad \textrm{ and } \qquad \nabla v_\varepsilon (x) \to -\textbf{c} \frac{x}{|x|^3} \qquad \textrm{ for all most every $x\in \mathbb{R}^3 $ }$$ and $$\label{eq:nv-eps-to-nv-C1} v_\varepsilon (x) \to \frac{\textbf{c}}{|x|} \qquad \textrm{ and } \qquad \nabla v_\varepsilon (x) \to -\textbf{c} \frac{x}{|x|^3} \qquad \textrm{ for every $x\in \mathbb{R}^3\setminus\{z=0\}$. }$$* *Proof.* By Proposition [Proposition 1](#TheoremA){reference-type="ref" reference="TheoremA"}, we have that $(v_\varepsilon )$ is bounded in $C^2_{loc}(\mathbb{R}^3\setminus\{z=0\})$. Therefore by Arzelá-Ascolli's theorem $v_\varepsilon$ converges to $v$ in $C^1_{loc}(\mathbb{R}^3\setminus\{z=0\})$. In particular, $$v_\varepsilon \to v \qquad \textrm{ and } \qquad \nabla v_\varepsilon \to \nabla v \qquad\textrm{ almost every where on $\mathbb{R}^3$.}$$ It is plain, from [\[eq:up-low-bound-w-3D\]](#eq:up-low-bound-w-3D){reference-type="eqref" reference="eq:up-low-bound-w-3D"}, that $$\label{eq:up-low-bound-v-eps-3D} 0<\frac{C_1}{\varepsilon +|x|} \leq v_\varepsilon (x) \leq \frac{C_2}{\varepsilon + |x|} \qquad \textrm{ for almost every $x\in \mathbb{R}^3 $.}$$ By [\[A1A1\]](#A1A1){reference-type="eqref" reference="A1A1"}, we have $$\label{eq:eq-for-v-eps} -\Delta v_{\varepsilon }(x)=\lambda {\varepsilon }^{2-s_1} \frac{v_\varepsilon ^{5-2s_1}(x)}{|z|^{s_1}}+{\varepsilon }^{2-s_2} \frac{v_\varepsilon ^{5-2s_2}(x)}{|z|^{s_2}} \qquad \textrm{ in } \mathbb{R}^3.$$ Newt, we let $\varphi \in C^\infty_c\left(\mathbb{R}^3\setminus \lbrace 0\rbrace\right)$. We multiply [\[eq:eq-for-v-eps\]](#eq:eq-for-v-eps){reference-type="eqref" reference="eq:eq-for-v-eps"} by $\varphi$ and integrate by parts to get $$-\int_{\mathbb{R}^3} v_{\varepsilon } \Delta \varphi dx= \lambda {\varepsilon }^{2-s_1} \int_{\mathbb{R}^3} \frac{v_\varepsilon ^{5-2s_1}(x)}{|z|^{s_1}} \varphi(x) dx+{\varepsilon }^{2-s_2} \int_{\mathbb{R}^3} \frac{v_\varepsilon ^{5-2s_2}(x)}{|z|^{s_2}} \varphi(x) dx.$$ By [\[eq:up-low-bound-v-eps-3D\]](#eq:up-low-bound-v-eps-3D){reference-type="eqref" reference="eq:up-low-bound-v-eps-3D"} and the dominated convergence theorem, we can pass to the limit in the above identity and deduce that $$\Delta v=0 \qquad \quad\textrm{ in } \mathcal{D}^\prime\left(\mathbb{R}^3\setminus \lbrace 0\rbrace\right).$$ In particular $v$ is equivalent to a function of class $C^\infty\left(\mathbb{R}^3 \setminus \lbrace0\rbrace\right)$ which is still denoted by $v$. Thanks to [\[eq:up-low-bound-v-eps-3D\]](#eq:up-low-bound-v-eps-3D){reference-type="eqref" reference="eq:up-low-bound-v-eps-3D"}, by Bôcher's theorem, there exists a constant $\textbf{c}>0$ such that $v(x)=\frac{\textbf{c}}{|x|}.$ The proof of the lemma is thus finished. ◻ We finish this section by the following estimates. Thanks to the decay estimates in Proposition [Proposition 1](#TheoremA){reference-type="ref" reference="TheoremA"}, we have **Lemma 4**. *There exists a constant $C>0$ such that for every $\varepsilon ,r\in (0,r_0/2)$ and for $s \in (0,2)$, we have $$\label{eq:est-nw-sq-3D} \int_{ {Q}_{r/\varepsilon }} |\nabla w |^2 dx\leq C\max\left(1, \frac{\varepsilon }{r}\right),\qquad \int_{ {Q}_{r/\varepsilon }} | w |^2 dx\leq C \max\left(1 , \frac{r}{\varepsilon } \right),$$ $$\label{eq:est-wnw-sq-3D} \int_{ {Q}_{r/\varepsilon }} w |\nabla w| dx\leq C \max\left( 1, \log\frac{r}{\varepsilon }\right),$$ $$\label{eq:est-nw-3D} \int_{ {Q}_{r/\varepsilon }} |\nabla w| dx \leq C \max\left(1 , \frac{r}{\varepsilon } \right), \qquad \int_{ {Q}_{r/\varepsilon }} | w| dx \leq C \max\left(1 , \frac{r^2}{\varepsilon ^2} \right)$$ and $$\label{eq:est-L2-star-3D} \varepsilon ^2 \int_{{Q}_{r/\varepsilon }} |z|^{-s}|x|^2 w ^{2^*_s} dx+\varepsilon \int_{{Q}_{4r/\varepsilon }\setminus {Q}_{r/\varepsilon }} |z|^{-s} w ^{2^*_s-1} dx+ \int_{\mathbb{R}^3\setminus {Q}_{r/\varepsilon }} |z|^{-s} w ^{2^*_s} dx=o(\varepsilon ).$$* # Proof of the main result {#ss:proof-of-th-3D} Given $y_0\in \Gamma\subset\Omega \subset \mathbb{R}^3$, we let $r_0$ as defined in [\[eq:def-r0\]](#eq:def-r0){reference-type="eqref" reference="eq:def-r0"}. For $r\in (0, r_0/2)$, we consider $F_{y_0}: Q_r\to \Omega$ parameterizing a neighborhood of $y_0$ in $\Omega$, with the property that $F_{y_0}(0)=y_0$, $$\label{Sym-dist} \rho_\Gamma(F_{y_0}(x))= |z|, \qquad \textrm{ for all $x=(y, z)\in Q_r$}.$$ Moreover in these local coordinates, we have $$\label{Metric} g_{ij}(x)= \delta _{ij}+O(|x|)$$ and $$\label{DeterminantMetric} \sqrt{|g|}(x)=1+\langle A, z\rangle+O\left(|x|^2\right),$$ where $A\in \mathbb{R}^2$ is the vector curvature of $\Gamma$ and $|g|$ stands for the determinant of $g$, see [@Fall-Thiam] for more details related to this parametrization.\ Next, for $\varepsilon >0$, we consider $u_\varepsilon : \Omega \to \mathbb{R}$ given by $$%\label{eq:TestFunction-u-eps-3D} u_\varepsilon (y):=\varepsilon ^{-1/2} \eta_r(F^{-1}_{y_0}(y)) w \left(\frac{F^{-1}_{y_0}(y)}{\varepsilon } \right).$$ We can now define the test function $\Psi_\varepsilon :\Omega \to \mathbb{R}$ by $$\label{eq:TestFunction-Om-3D} \Psi_\varepsilon \left(y\right)=u_\varepsilon (y)+\varepsilon ^{1/2} \textbf{c}\, \eta_{2r}(F^{-1}_{y_0}(y) ){M}_{y_0}(F^{-1}_{y_0}(y) ). %$$ It is plain that $\Psi_\varepsilon \in H^1_0(\Omega )$ and $$%\label{eq:TestFunction-flat-3D} \Psi_\varepsilon \left(F_{y_0}(x)\right)=\varepsilon ^{-1/2} \eta_r(x) w \left(\frac{x}{\varepsilon } \right)+\varepsilon ^{1/2} \textbf{c} \, \eta_{2r}(x) {M}_{y_0}(x) \qquad\textrm{ for every $x\in \mathbb{R}^N$.} %$$ To alleviate the notations, we will write $\varepsilon$ instead of $\varepsilon _n$ and we will remove the subscript $y_0$, by writing $M$ and $F$ in the place of ${M}_{y_0}$ and $F_{y_0}$ respectively. We define $$%\label{eq:def-tilde-change-var-3D} \widetilde\eta_r(y):=\eta_r(F^{-1}(y)),\qquad V_\varepsilon (y):=v_\varepsilon (F^{-1}(y)) \qquad\textrm{ and } \qquad \widetilde M_{2r}(y):=\eta_{2r}(F^{-1}(y) ) M ( F^{-1}(y)) ,$$ where $v_\varepsilon (x)=\varepsilon ^{-1} w\left(\frac{x}{\varepsilon } \right).$ With these notations, [\[eq:TestFunction-Om-3D\]](#eq:TestFunction-Om-3D){reference-type="eqref" reference="eq:TestFunction-Om-3D"} becomes $$\label{eq:def-W-eps-3D} \Psi_\varepsilon (y) = u_\varepsilon (y)+ \varepsilon ^{\frac{1}{2}} \textbf{c}\, \widetilde M_{2r}(y)= \varepsilon ^{\frac{1}{2}} V_\varepsilon (y)+ \varepsilon ^{\frac{1}{2}} \textbf{c}\, \widetilde M_{2r}(y) .$$ In the sequel we define $\mathcal{O}_{r, \varepsilon }$ as $$\lim_{r \to 0}\frac{\mathcal{O}_{r, \varepsilon }}{\varepsilon }=0.$$ Then we have the following. **Lemma 5**. *We have $$\begin{aligned} \label{eq:Dirichlet-Psi-eps000} \int_\Omega |\nabla \Psi_\varepsilon |^2 dy+ \int_{\Omega } h |\Psi_\varepsilon |^2 dy=&\int_{\mathbb{R}^3} |\nabla w|^2 dx +\pi \varepsilon \textbf{m}(y_0) \textbf{c}^2+\mathcal{O}_r(\varepsilon ),\end{aligned}$$ as $\varepsilon \to 0$.* *Proof.* Recalling [\[eq:def-W-eps-3D\]](#eq:def-W-eps-3D){reference-type="eqref" reference="eq:def-W-eps-3D"}, direct computations give $$\begin{aligned} \label{eq:nab-Psi} \int_{F({Q}_{2r})\setminus F\left({Q}_r\right)} |\nabla \Psi_\varepsilon |^2 dy&= \int_{F({Q}_{2r}) \setminus F\left({Q}_r\right)} |\nabla \left( \widetilde\eta_r u_\varepsilon \right)|^2 dy+\varepsilon \textbf{c} ^2 \int_{F({Q}_{2r})\setminus F\left({Q}_r\right)} |\nabla \widetilde M_{2r} |^2 dy \nonumber\\ % &+2 \varepsilon ^{1/2} \textbf{c} \int_{F({Q}_{2r}) \setminus F\left({Q}_r\right)} \nabla \left( \widetilde\eta_r u_\varepsilon \right) \cdot \nabla \widetilde M_{2r} dy \nonumber\\ % &= \varepsilon \int_{F({Q}_{2r}) \setminus F\left({Q}_r\right)} |\nabla \left( \widetilde\eta_r V_\varepsilon \right)|^2 dy+\varepsilon \textbf{c}^2 \int_{F({Q}_{2r}) \setminus F\left({Q}_r\right)} |\nabla \widetilde M_{2r} |^2 dy \nonumber\\ % &+2 \varepsilon \textbf{c} \int_{F({Q}_{2r})\setminus F\left({Q}_r\right)} \nabla \left( \widetilde\eta_r V_\varepsilon \right) \cdot \nabla \widetilde M_{2r} dy. %\end{aligned}$$ By [\[eq:def-cut-off-cylind\]](#eq:def-cut-off-cylind){reference-type="eqref" reference="eq:def-cut-off-cylind"}, $\eta_r v_\varepsilon = \eta_r \varepsilon ^{-1}w(\cdot/\varepsilon )$ is cylindrically symmetric. Therefore by the change variable $y=F(x)$ and using [\[Metric\]](#Metric){reference-type="eqref" reference="Metric"}, we get $$\begin{aligned} \label{eq:enev} \varepsilon \int_{F({Q}_{2r}) \setminus F\left({Q}_r\right)} |\nabla \left( \widetilde\eta_r V_\varepsilon \right)|^2 dy&= \varepsilon \int_{{Q}_{2r} \setminus {Q}_r } |\nabla \left( \eta_r {v}_\varepsilon \right)|^2_g \sqrt{g} dx \nonumber\\ % &= \varepsilon \int_{{Q}_{2r} \setminus {Q}_r} |\nabla \left( \eta_r {v}_\varepsilon \right)|^2 dx+O\left(\varepsilon r^2 \int_{{Q}_{2r} \setminus {Q}_r} |\nabla \left( \eta_r {v}_\varepsilon \right)|^2 dx \right).\end{aligned}$$ By computing, we find that $$\begin{aligned} \varepsilon \int_{{Q}_{2r} \setminus {Q}_r} |\nabla \left( \eta_r {v}_\varepsilon \right)|^2 dx& \leq \varepsilon \int_{{Q}_{2r} \setminus {Q}_r} |\nabla {v}_\varepsilon |^2 dx+\varepsilon \int_{{Q}_{2r} \setminus {Q}_r} v_\varepsilon ^2 |\nabla \eta_r |^2 dx + 2 \varepsilon \int_{{Q}_{2r} \setminus {Q}_r} v_\varepsilon |\nabla v_\varepsilon || \nabla \eta_r| dx \nonumber\\ % & \leq \varepsilon \int_{{Q}_{2r} \setminus {Q}_r} |\nabla {v}_\varepsilon |^2 dx+ \frac{C}{r^2} \varepsilon \int_{{Q}_{2r} \setminus {Q}_r} v_\varepsilon ^2 dx + \frac{C}{r} \varepsilon \int_{{Q}_{2r} \setminus {Q}_r} v_\varepsilon |\nabla v_\varepsilon | dx \nonumber\\ % &=\int_{{Q}_{2r/\varepsilon } \setminus {Q}_{r/\varepsilon }} |\nabla w |^2 dx+C \frac{\varepsilon }{r^2} \int_{{Q}_{2r/\varepsilon } \setminus {Q}_{r/\varepsilon }}w^2 dx + \frac{C}{r} \varepsilon \int_{{Q}_{2r/\varepsilon } \setminus {Q}_{r/\varepsilon }} w |\nabla w| dx.\end{aligned}$$ From this and [\[eq:est-nw-sq-3D\]](#eq:est-nw-sq-3D){reference-type="eqref" reference="eq:est-nw-sq-3D"} and [\[eq:est-wnw-sq-3D\]](#eq:est-wnw-sq-3D){reference-type="eqref" reference="eq:est-wnw-sq-3D"}, we get $$O\left(\varepsilon r^2 \int_{{Q}_{2r} \setminus {Q}_r} |\nabla \left( \eta_r {v}_\varepsilon \right)|^2 dx \right)= \mathcal{O}_r(\varepsilon ).$$ We replace this in [\[eq:enev\]](#eq:enev){reference-type="eqref" reference="eq:enev"} to have $$\begin{aligned} \label{eq:int-nv-et-v} \varepsilon \int_{F({Q}_{2r}) \setminus F\left({Q}_r\right)} |\nabla \left( \widetilde\eta_r V_\varepsilon \right)|^2 dy&= \varepsilon \int_{{Q}_{2r} \setminus {Q}_r} |\nabla (\eta_r {v}_\varepsilon ) |^2 dx + \mathcal{O}_r(\varepsilon ).\end{aligned}$$ We have the following estimates $$\label{eq:est-nv-eps} 0\leq v_\varepsilon \leq C |x|^{-1}\quad \textrm{ for $x\in \mathbb{R}^3\setminus\{0\}$ }\qquad \textrm{ and }\qquad |\nabla v_\varepsilon (x)| \leq C |x|^{-2} \quad\textrm{ for $|x|\geq \varepsilon $, }$$ which easily follows from [\[eq:up-low-bound-w-3D\]](#eq:up-low-bound-w-3D){reference-type="eqref" reference="eq:up-low-bound-w-3D"}, [\[Metric\]](#Metric){reference-type="eqref" reference="Metric"} and [\[eq:Green-R3-3D\]](#eq:Green-R3-3D){reference-type="eqref" reference="eq:Green-R3-3D"}. By these estimates, [\[Metric\]](#Metric){reference-type="eqref" reference="Metric"}, [\[DeterminantMetric\]](#DeterminantMetric){reference-type="eqref" reference="DeterminantMetric"} and [\[eq:regul-beta\]](#eq:regul-beta){reference-type="eqref" reference="eq:regul-beta"} together with the change of variable $y=F(x)$, we have $$\begin{aligned} \varepsilon \int_{F({Q}_{2r})\setminus F\left({Q}_r\right)} \nabla \left( \widetilde\eta_r V_\varepsilon \right) \cdot \nabla \widetilde M_{2r} dy=& \varepsilon \int_{ {Q}_{2r}\setminus {Q}_{r} } \nabla \left( \eta_{ {r} } v_\varepsilon \right) \cdot \nabla M dx\\ % &+ O\left( \varepsilon \int_{ {{Q}}_{2r}\setminus {Q}_{r} }|\nabla v_\varepsilon | dx + \frac{\varepsilon }{r} \int_{ {{Q}}_{2r}\setminus {Q}_{r} } v_\varepsilon dx \right)\nonumber\\ % % = &\varepsilon \int_{ {{Q}}_{2r}\setminus {Q}_{r} } \nabla \left( \eta_{ {r} } v_\varepsilon \right) \cdot \nabla M dx + \mathcal{O}_r(\varepsilon ).\end{aligned}$$ This with [\[eq:int-nv-et-v\]](#eq:int-nv-et-v){reference-type="eqref" reference="eq:int-nv-et-v"}, [\[eq:regul-beta\]](#eq:regul-beta){reference-type="eqref" reference="eq:regul-beta"} and [\[eq:nab-Psi\]](#eq:nab-Psi){reference-type="eqref" reference="eq:nab-Psi"} give $$\begin{aligned} \int_{F({Q}_{2r})\setminus F\left({Q}_r\right)} |\nabla \Psi_\varepsilon |^2 dy&= \varepsilon \int_{ {Q}_{2r} \setminus {Q}_r } |\nabla \left( \eta_r {v}_\varepsilon \right)|^2 dx+\varepsilon \textbf{c}^2 \int_{ {Q}_{2r} \setminus {Q}_r } |\nabla (\eta_{2r} M )|^2 dx\\ % &+2 \varepsilon \textbf{c} \int_{{Q}_{2r}\setminus {Q}_r } \nabla \left( \eta_r {v}_\varepsilon \right) \cdot \nabla M dx+ \mathcal{O}_r(\varepsilon ). \end{aligned}$$ Thanks to Lemma [Lemma 3](#lem:v-to-cR){reference-type="ref" reference="lem:v-to-cR"} and [\[eq:est-nv-eps\]](#eq:est-nv-eps){reference-type="eqref" reference="eq:est-nv-eps"}, we can thus use the dominated convergence theorem to deduce that, as $\varepsilon \to 0$, $$\label{eq:claim1-est-num} \int_{ {Q}_{2r} \setminus {Q}_r } |\nabla \left( \eta_r {v}_\varepsilon \right)|^2 dx= \textbf{c}^2\int_{ {Q}_{2r} \setminus {Q}_r } |\nabla \left( \eta_r {\mathcal R}\right)|^2 dx+o(1).$$ Similarly, we easily see that $$\int_{{Q}_{2r}\setminus {Q}_r } \nabla \left( \eta_r {v}_\varepsilon \right) \cdot \nabla M dx= \textbf{c} \int_{{Q}_{2r}\setminus {Q}_r } \nabla \left( \eta_r {\mathcal R}\right) \cdot \nabla M dx+o(1)\qquad\textrm{ as $\varepsilon \to 0$.}$$ This and [\[eq:claim1-est-num\]](#eq:claim1-est-num){reference-type="eqref" reference="eq:claim1-est-num"}, then give $$\begin{aligned} \int_{F({Q}_{2r})\setminus F\left({Q}_r\right)} |\nabla \Psi_\varepsilon |^2 dy &= \varepsilon \textbf{c}^2 \int_{ {Q}_{2r} \setminus {Q}_r } |\nabla \left( \eta_r {\mathcal R}\right)|^2 dx+\varepsilon \textbf{c}^2 \int_{ {Q}_{2r} \setminus {Q}_r } |\nabla M |^2 dx \nonumber\\ % &+2 \varepsilon \textbf{c}^2 \int_{{Q}_{2r}\setminus {Q}_r } \nabla \left( \eta_r {\mathcal R}\right) \cdot \nabla M dx+ \mathcal{O}_r(\varepsilon ) \nonumber\\ % &=\varepsilon \textbf{c}^2 \int_{{Q}_{2r}\setminus {Q}_r } | \nabla ( \eta_r {\mathcal R}+ M )|^2 dx+ \mathcal{O}_r(\varepsilon ). \label{eq:nPs-B2r-Br} %\end{aligned}$$ Since the support of $\Psi_\varepsilon$ is contained in ${Q}_{4r}$ while the one of $\eta_r$ is in ${Q}_{2r}$, it is easy to deduce from [\[eq:regul-beta\]](#eq:regul-beta){reference-type="eqref" reference="eq:regul-beta"} that $$\begin{aligned} \int_{\Omega \setminus F\left({Q}_{2r}\right)} |\nabla \Psi_\varepsilon |^2 dy&= \varepsilon \textbf{c} ^2 \int_{F({Q}_{4r})\setminus F\left({Q}_{2r}\right)} |\nabla \widetilde M_{2r} |^2 dy= \mathcal{O}_r(\varepsilon )\end{aligned}$$ and from Lemma [Lemma 4](#lem:estimates-for-3D){reference-type="ref" reference="lem:estimates-for-3D"}, that $$\int_{\Omega \setminus F\left({Q}_r\right)}h | \Psi_\varepsilon |^2 dy=\varepsilon \textbf{c} ^2 \int_{F({Q}_{4r})\setminus F\left({Q}_{r}\right)} h | \eta_r V_\varepsilon +\widetilde M_{2r} |^2 dy= \mathcal{O}_r(\varepsilon ).$$ Therefore by [\[eq:nPs-B2r-Br\]](#eq:nPs-B2r-Br){reference-type="eqref" reference="eq:nPs-B2r-Br"}, we conclude that $$\begin{aligned} \int_{\Omega \setminus F\left({Q}_r\right)} |\nabla \Psi_\varepsilon |^2 dy&+\int_{\Omega \setminus F\left({Q}_r\right)}h | \Psi_\varepsilon |^2 dy \\ % &\qquad=\varepsilon \textbf{c}^2 \int_{{Q}_{2r}\setminus {Q}_r } | \nabla ( \eta_r {\mathcal R}+ M )|^2 dx+\varepsilon \textbf{c}^2 \int_{{Q}_{2r}\setminus {Q}_r }h(\cdot+y_0) | \eta_r {\mathcal R}+ M |^2 dx+ \mathcal{O}_r(\varepsilon ). %\end{aligned}$$ Recall that $G(x+y_0,y_0)= \eta_r(x) {\mathcal R}(x)+ M(x )$ for ever $x\in {Q}_{2r}$ and that by [\[eq:Green-3D\]](#eq:Green-3D){reference-type="eqref" reference="eq:Green-3D"}, $$-\Delta _x G(x+y_0,y_0)+h(x+y_0) G(x+y_0,y_0)=0 \qquad \textrm{ for every $x \in Q_{2r}\setminus Q_r $. }$$ Therefore, by integration by parts, we find that $$\begin{aligned} %\label{eq:est-int-J-3} \int_{\Omega \setminus F\left({Q}_r\right)} |\nabla \Psi_\varepsilon |^2 dy+&\int_{\Omega \setminus F\left({Q}_r\right)}h | \Psi_\varepsilon |^2 dy=\textbf{c}^2\int_{\partial({{Q}_{2r}\setminus {Q}_r )}} ( \eta_r {\mathcal R}+ M ) \frac{\partial( \eta_r {\mathcal R}+ M )}{\partial\overline\nu}\sigma (x)+\mathcal{O}_r(\varepsilon ),\end{aligned}$$ where $\overline\nu$ is the exterior normal vectorfield to ${Q}_{2r}\setminus {Q}_r$. Thanks to [\[eq:regul-beta\]](#eq:regul-beta){reference-type="eqref" reference="eq:regul-beta"}, we finally get $$\begin{aligned} \label{eq:nPsi-Ome-Br} %\label{eq:est-int-J-3} \int_{\Omega \setminus F\left({Q}_r\right)} |\nabla \Psi_\varepsilon |^2 dy+&\int_{\Omega \setminus F\left({Q}_r\right)}h | \Psi_\varepsilon |^2 dy &=-\varepsilon \textbf{c}^2\int_{\partial{Q}_r } {{\mathcal R}} \frac{\partial{{\mathcal R}}}{\partial\nu} d\sigma (x)- \varepsilon \textbf{c}^2 \int_{\partial{Q}_r } M \frac{\partial{{\mathcal R}}}{\partial\nu} d\sigma (x) + \mathcal{O}_r(\varepsilon ),\end{aligned}$$ where $\nu$ is the exterior normal vectorfield to ${Q}_r$.\ Next we make the expansion of $\int_{F\left({Q}_r\right)} |\nabla \Psi_\varepsilon |^2 dy$ for $r$ and $\varepsilon$ small. First, we observe that, by Lemma [Lemma 4](#lem:estimates-for-3D){reference-type="ref" reference="lem:estimates-for-3D"} and [\[eq:regul-beta\]](#eq:regul-beta){reference-type="eqref" reference="eq:regul-beta"}, we have $$\begin{aligned} \int_{F\left({Q}_r\right)}& |\nabla \Psi_\varepsilon |^2 dy =\int_{F\left({Q}_r\right)} |\nabla u_\varepsilon |^2 dy+\varepsilon \textbf{c}^2 \int_{F\left({Q}_r\right)} |\nabla M |^2 dy+2\varepsilon ^{1/2}\textbf{c} \int_{F\left({Q}_r\right)} \nabla u_\varepsilon \cdot \nabla \widetilde M_{2r} dy\\ &= \int_{ {Q}_{r/\varepsilon }} |\nabla w|^2 dx +O\left( \varepsilon ^2 \int_{ {Q}_{r/\varepsilon }} |x|^2 |\nabla w|^2 dx +\varepsilon ^{2} \int_{ {Q}_{r/\varepsilon } } |\nabla w | dx \right) +\mathcal{O}_r(\varepsilon ) = \int_{ {Q}_{r/\varepsilon }} |\nabla w|^2 dx+ \mathcal{O}_r(\varepsilon ). % \end{aligned}$$ By integration by parts and using [\[eq:est-L2-star-3D\]](#eq:est-L2-star-3D){reference-type="eqref" reference="eq:est-L2-star-3D"}, we deduce that $$\begin{aligned} \label{eq:expans-num-FQr} \int_{F\left({Q}_r\right)} |\nabla \Psi_\varepsilon |^2 dy % &=\int_{\mathbb{R}^3} |\nabla w|^2 dx+ \int_{\partial{Q}_{r/\varepsilon }} w\frac{\partial w }{\partial\nu } d\sigma (x)+\mathcal{O}_r(\varepsilon ) \nonumber\\ % % &=\int_{\mathbb{R}^3} |\nabla w|^2 dx + \varepsilon \int_{\partial{Q}_{r}} v_\varepsilon \frac{\partial v_\varepsilon }{\partial\nu } d\sigma (x) + \mathcal{O}_r(\varepsilon ). \end{aligned}$$ Now [\[eq:est-nv-eps\]](#eq:est-nv-eps){reference-type="eqref" reference="eq:est-nv-eps"}, [\[eq:nv-eps-to-nv-C1\]](#eq:nv-eps-to-nv-C1){reference-type="eqref" reference="eq:nv-eps-to-nv-C1"} and the dominated convergence theorem yield, for fixed $r>0$ and $\varepsilon \to 0$, $$\begin{aligned} \label{eq:d-eps-nv-eps} \int_{\partial{Q}_{r}} v_\varepsilon \frac{\partial v_\varepsilon }{\partial\nu } d\sigma (x)&=\int_{\partial B^2_{\mathbb{R}^2}(0,r)}\int_{-r}^r v_\varepsilon (t,z)\nabla v_\varepsilon (t,z)\cdot\frac{z}{|z|} d\sigma (z) dt+2\int_{B^2_{\mathbb{R}^2}} v_\varepsilon (r,z) \partial_tv_\varepsilon (r,z) dz \nonumber\\ % &= \textbf{c}^2\int_{\partial B^2_{\mathbb{R}^2}(0,r)}\int_{-r}^r {\mathcal R}(t,z)\nabla {\mathcal R}(t,z)\cdot\frac{z}{|z|} d\sigma (z) dt+2 \textbf{c}^2\int_{B^2_{\mathbb{R}^2}} {\mathcal R}(r,z) \partial_t {\mathcal R}(r,z) dz+o(1)\nonumber\\ % & = \textbf{c}^2\int_{\partial{Q}_r } {{\mathcal R}} \frac{\partial{{\mathcal R}}}{\partial\nu} d\sigma (x) + o(1).\end{aligned}$$ Moreover [\[eq:est-nw-3D\]](#eq:est-nw-3D){reference-type="eqref" reference="eq:est-nw-3D"} implies that $$\begin{aligned} \int_{F({Q}_{r})} h \Psi_\varepsilon ^2 dy = \mathcal{O}_r(\varepsilon ).\end{aligned}$$ From this together with [\[eq:expans-num-FQr\]](#eq:expans-num-FQr){reference-type="eqref" reference="eq:expans-num-FQr"} and [\[eq:d-eps-nv-eps\]](#eq:d-eps-nv-eps){reference-type="eqref" reference="eq:d-eps-nv-eps"}, we obtain $$\begin{aligned} \int_{F\left({Q}_r\right)} |\nabla \Psi_\varepsilon |^2 dy + \int_{F({Q}_{r})} h \Psi_\varepsilon ^2 dy &= \int_{\mathbb{R}^3} |\nabla w|^2 dx + \textbf{c}^2\varepsilon \int_{\partial{Q}_r } {{\mathcal R}} \frac{\partial{{\mathcal R}}}{\partial\nu} d\sigma (x) +\mathcal{O}_r(\varepsilon ). \end{aligned}$$ Combining this with [\[eq:nPsi-Ome-Br\]](#eq:nPsi-Ome-Br){reference-type="eqref" reference="eq:nPsi-Ome-Br"}, we then have $$\begin{aligned} \label{eq:Dirichlet-Psi-eps} \int_\Omega |\nabla \Psi_\varepsilon |^2 dy+ \int_{\Omega } h \Psi_\varepsilon ^2 dy=&\int_{\mathbb{R}^3} |\nabla w|^2 dx - \varepsilon \textbf{c}^2\int_{\partial{Q}_r } M \frac{\partial{{\mathcal R}}}{\partial\nu} d\sigma (x)+\mathcal{O}_r(\varepsilon ) +o\left(\varepsilon \right).\end{aligned}$$ Recalling that ${\mathcal R}(x)=\frac{1}{|x|}$, we have $$\begin{aligned} \int_{\partial{Q}_r}\frac{\partial{\mathcal R}}{\partial\nu}\, d\sigma (x)&=- \int_{\partial{Q}_r}\frac{x\cdot \nu (x)}{|x|^3}\, d\sigma (x)= \int_{ B_{\mathbb{R}^2}(0, r)}\frac{- 2 r }{r^2+|z|^2}\, dz - 2\pi\int_{-r}^r\frac{r^3}{r^2+t^2}dt=- \pi^2 (1+ r^2) .\end{aligned}$$ Since (recalling [\[eq:def-mass\]](#eq:def-mass){reference-type="eqref" reference="eq:def-mass"}) $M(y)=M(0)+O(r)=\textbf{m}(y_0)+O(r)$ in ${Q}_{2r}$, we get [\[eq:Dirichlet-Psi-eps000\]](#eq:Dirichlet-Psi-eps000){reference-type="eqref" reference="eq:Dirichlet-Psi-eps000"}. This then ends the proof. ◻ We finish by the following expansion **Lemma 6**. *$$\begin{aligned} \frac{\lambda }{2^*_{s_1}}\int_{\Omega } \rho^{-s_1}_\Gamma|\Psi_{\varepsilon }|^{2^*_{s_1}} dy&+\frac{1}{2^*_{s_2}}\int_{\Omega } \rho^{-s_2}_\Gamma|\Psi_{\varepsilon }|^{2^*_{s_2}} dy= \frac{\lambda }{2^*_{s_1}}\int_{\mathbb{R}^3} |z|^{-s_1} |w|^{2^*_{s_1}} dx\\\\ &+\frac{1}{2^*_{s_2}}\int_{\mathbb{R}^3} |z|^{-s_2} |w|^{2^*_{s_2}} dx+{\varepsilon } \pi^2 \textbf{c}^2 \textbf{m}(y_0)+\mathcal{O}_r(\varepsilon ).\end{aligned}$$* *Proof.* Let $p>2$. Then there exists a positive constant $C(p)$ such that $$||a+b|^p-|a|^{p}-p ab |a|^{p-2}| \leq C(p) \left(|a|^{p-2} b^2+|b|^{p}\right)\qquad\textrm{ for all $a,b \in \mathbb{R}$.}$$ As a consequence, we obtain, for $s \in (0,2)$, that $$\begin{aligned} \label{eq:expan-L-2-star-1} % \displaystyle\int_{\Omega } &\rho^{-s}_\Gamma|\Psi_{\varepsilon }|^{2^*_s} dy= \displaystyle \int_{F({Q}_{r})} \rho^{-s}_\Gamma|u_{\varepsilon }+ \varepsilon ^{\frac{1}{2}} \widetilde{M}_{2r}|^{2^*_s} dy+ \int_{F({Q}_{4r}) \setminus F({Q}_{r})} \rho^{-s}_\Gamma|W_{\varepsilon }+ \varepsilon ^{\frac{1}{2}} \widetilde{M}_{2r}|^{2^*_s} dy \nonumber\\ % &=\displaystyle \int_{F({Q}_{r})} \rho^{-s}_\Gamma| u_{\varepsilon }|^{2^*_s} dy + 2^*_s \textbf{c} {\varepsilon }^{1/2} \int_{F({Q}_{r})} \rho^{-s}_\Gamma| u_{\varepsilon }|^{2^*_s-1} \widetilde M_{2r} dy \nonumber\\ % &\quad \displaystyle + O\left(\int_{F\left({Q}_{ 4r}\right)} \rho^{-s}_\Gamma|\eta_r u_{\varepsilon }|^{2^*_s-2} \left({\varepsilon }^{1/2}\widetilde{M}_{2r} \right)^2 dy + \int_{F\left({Q}_{ 4r}\right)} \rho^{-s}_\Gamma|{\varepsilon }^{1/2} \widetilde{M}_{2r} |^{2^*_s} dy\right) \nonumber\\ % &\quad \displaystyle+O \left( \int_{F({Q}_{4r}) \setminus F({Q}_{r})} \rho^{-s}_\Gamma| u_{\varepsilon }|^{2^*_s} dy + 2^*_s \textbf{c} {\varepsilon }^{1/2} \int_{F({Q}_{4r}) \setminus F({Q}_{r})} \rho^{-s}_\Gamma| u_{\varepsilon }|^{2^*_s-1} \widetilde{M}_{2r} dy \right).\end{aligned}$$ By Hölder's inequality and [\[DeterminantMetric\]](#DeterminantMetric){reference-type="eqref" reference="DeterminantMetric"}, we have $$\begin{aligned} \label{eq:expan-L-2-star-2} \int_{F\left({Q}_{ 4r}\right)} \rho^{-s}_\Gamma|\eta u_{\varepsilon }|^{2^*_s-2} \left({\varepsilon }^{1/2}\widetilde\beta _r \right)^2 dy& \leq \varepsilon \|u_{\varepsilon }\|_{L^{2^*_s}(F({Q}_{4 r} );\rho^{-s})}^{{2^*_s-2}}\|\widetilde{M}_{2r}\|_{L^{2^*_s}(F({Q}_{4 r} );\rho^{-s}_\Gamma)}^{{2} }\nonumber\\ % &=\varepsilon \|w\|_{L^{2^*_s}( {Q}_{ 4r};|z|^{-s}\sqrt{|g|} )}^{{2^*_s-2}} \|\widetilde{M}_{2r}\|_{L^{2^*_s}(F({Q}_{4 r} );\rho^{-s}_\Gamma)}^{{2} }\nonumber\\ % &\leq \varepsilon (1+ Cr)\|\widetilde{M}_{2r}\|_{L^{2^*_s}(F({Q}_{ 4r} );\rho^{-s}_\Gamma)}^{{2} }=\mathcal{O}_r(\varepsilon ).\end{aligned}$$ Furthermore, since $2^*_s>2$, by [\[eq:regul-beta\]](#eq:regul-beta){reference-type="eqref" reference="eq:regul-beta"}, we easily get $$\begin{aligned} \label{eq:expan-L-2-star-2-00} \int_{F\left({Q}_{4 r}\right)} \rho^{-s}_\Gamma|{\varepsilon }^{1/2} \widetilde{M}_{2r} |^{2^*_s} dy=o(\varepsilon ).\end{aligned}$$ Moreover by change of variables and [\[eq:est-L2-star-3D\]](#eq:est-L2-star-3D){reference-type="eqref" reference="eq:est-L2-star-3D"}, we also have $$\begin{aligned} \int_{F({Q}_{4r}) \setminus F({Q}_{r})} \rho^{-s}_\Gamma| u_{\varepsilon }|^{2^*_s} dy + 2^*_s \textbf{c} {\varepsilon }^{1/2} &\int_{F({Q}_{4r}) \setminus F({Q}_{r})} \rho^{-s}_\Gamma| u_{\varepsilon }|^{2^*_s-1} \widetilde M_{2r} dy\\ % \leq C& \int_{{Q}_{4r/\varepsilon } \setminus {Q}_{r/\varepsilon }} |z|^{-s} | w|^{2^*_s} dx + C {\varepsilon } \int_{{Q}_{4r/\varepsilon } \setminus {Q}_{r/\varepsilon } } |z|^{-s} | w|^{2^*_s-1} dx =o(\varepsilon ).\end{aligned}$$ By this, [\[eq:expan-L-2-star-1\]](#eq:expan-L-2-star-1){reference-type="eqref" reference="eq:expan-L-2-star-1"}, [\[eq:expan-L-2-star-2-00\]](#eq:expan-L-2-star-2-00){reference-type="eqref" reference="eq:expan-L-2-star-2-00"} and [\[eq:expan-L-2-star-2\]](#eq:expan-L-2-star-2){reference-type="eqref" reference="eq:expan-L-2-star-2"}, it results $$\begin{aligned} % \displaystyle\int_{\Omega } \rho^{-s}_\Gamma|\Psi_{\varepsilon }|^{2^*_s} dy&= \displaystyle \int_{F({Q}_{r})} \rho^{-s}_\Gamma| u_{\varepsilon }|^{2^*_s} dy + 2^*_s \textbf{c} {\varepsilon }^{1/2} \int_{F({Q}_{r})} \rho^{-\sigma }_\Gamma| u_{\varepsilon }|^{2^*_\sigma -1} \widetilde M_{2r} dy +\mathcal{O}_r(\varepsilon ). %\end{aligned}$$ We define $B_\varepsilon (x):=M(\varepsilon x) \sqrt{|g_\varepsilon |}(x) =M(\varepsilon x) \sqrt{|g|}(\varepsilon x)$. Then by the change of variable $y=\frac{F(x)}{\varepsilon }$ in the above identity and recalling [\[DeterminantMetric\]](#DeterminantMetric){reference-type="eqref" reference="DeterminantMetric"}, then by oddness, we have $$\begin{aligned} \displaystyle\int_{\Omega } \rho^{-s}_\Gamma|\Psi_{\varepsilon }|^{2^*_s} dy &= \displaystyle\int_{{Q}_{r/\varepsilon }} |z|^{-s} w ^{2^*_s} \sqrt{|g_\varepsilon |}dx + 2^*_s {\varepsilon } \textbf{c} \int_{{Q}_{r/\varepsilon }}|z|^{-s} | w |^{2^*_s-1} B_\varepsilon dx+ \mathcal{O}_r(\varepsilon )\\ % &= \displaystyle\int_{{Q}_{r/\varepsilon }} |z|^{-s} w ^{2^*_s} dx + 2^*_s {\varepsilon } \textbf{c} \int_{{Q}_{r/\varepsilon }}|z|^{-s} | w |^{2^*_s-1} B_\varepsilon dx+ \mathcal{O}_r(\varepsilon )\\ % &\quad \displaystyle+ O\left( \varepsilon ^2\int_{{Q}_{r/\varepsilon }} |z|^{-s} |x|^2 w ^{2^*_s} dx\right)\\ % &=\displaystyle \int_{\mathbb{R}^3} |z|^{-s} |w|^{2^*_s} dx + 2^*_s {\varepsilon } \textbf{c} \int_{{Q}_{r/\varepsilon }}|z|^{-s} | w |^{2^*_s-1} B_\varepsilon dx\\ % &\quad \displaystyle+O\left( \int_{\mathbb{R}^3\setminus {Q}_{r/\varepsilon }} |z|^{-s} w ^{2^*_s} dx+ \varepsilon ^2\int_{{Q}_{r/\varepsilon }} |z|^{-s} |x|^2 w ^{2^*_s} dx\right)+ \mathcal{O}_r(\varepsilon ). %\end{aligned}$$ By [\[eq:est-L2-star-3D\]](#eq:est-L2-star-3D){reference-type="eqref" reference="eq:est-L2-star-3D"} we then have $$\begin{aligned} \label{eq:est-L-2star-Psi-not-ok} \int_{\Omega } \rho^{-s}_\Gamma|\Psi_{\varepsilon }|^{2^*_s} dy = \displaystyle \int_{\mathbb{R}^3} |z|^{-s} |w|^{2^*_s} dx + 2^*_s {\varepsilon } \textbf{c} \int_{{Q}_{r/\varepsilon }}|z|^{-s} | w |^{2^*_s-1} B_\varepsilon (x) dx +\mathcal{O}_r(\varepsilon ).\end{aligned}$$ Therefore for $0<s_2<s_1<2$, we have $$\begin{aligned} \frac{\lambda }{2^*_{s_1}}\int_{\Omega } \rho^{-s_1}_\Gamma|\Psi_{\varepsilon }|^{2^*_{s_1}} dy&+\frac{1}{2^*_{s_2}}\int_{\Omega } \rho^{-s_2}_\Gamma|\Psi_{\varepsilon }|^{2^*_{s_2}} dy= \frac{\lambda }{2^*_{s_1}}\int_{\mathbb{R}^3} |z|^{-s_1} |w|^{2^*_{s_1}} dx+\frac{1}{2^*_{s_2}}\int_{\mathbb{R}^3} |z|^{-s_2} |w|^{2^*_{s_2}} dx\\\\ &+{\varepsilon } \textbf{c} \lambda \int_{{Q}_{r/\varepsilon }}|z|^{-s_1} | w |^{2^*_{s_1}-1} B_\varepsilon (x) dx+{\varepsilon } \textbf{c} \int_{{Q}_{r/\varepsilon }}|z|^{-s_2} | w |^{2^*_{s_2}-1} B_\varepsilon (x) dx +\mathcal{O}_r(\varepsilon ).\end{aligned}$$ We multiply [\[A1A1\]](#A1A1){reference-type="eqref" reference="A1A1"} by $B_\varepsilon \in {\mathcal C}^1(\overline{Q_r})$ and we integrate by parts to get $$\begin{aligned} \lambda \int_{{Q}_{r/\varepsilon }}|z|^{-s_1} | w |^{2^*_{s_1}-1} B_\varepsilon dx+\int_{{Q}_{r/\varepsilon }}|z|^{-s_2} | w |^{2^*_{s_2}-1} B_\varepsilon dx& = \int_{ {Q}_{r/\varepsilon } } \nabla w \cdot \nabla B_\varepsilon dx -\int_{ \partial{Q}_{r/\varepsilon } }B_\varepsilon \frac{ \partial w}{\partial\nu} d\sigma (x)\\ % &= \int_{ {Q}_{r/\varepsilon } } \nabla w \cdot \nabla B_\varepsilon dx-\int_{ \partial{Q}_{r} }B_1 \frac{ \partial v_\varepsilon }{\partial\nu} d\sigma (x).\end{aligned}$$ Since $|\nabla B_\varepsilon |\leq C \varepsilon$, by Lemma [Lemma 3](#lem:v-to-cR){reference-type="ref" reference="lem:v-to-cR"} and [\[eq:regul-beta\]](#eq:regul-beta){reference-type="eqref" reference="eq:regul-beta"}, we then have $$\varepsilon \int_{ {Q}_{r/\varepsilon } } \nabla w \cdot \nabla B_\varepsilon dx =O\left( \varepsilon ^2 \int_{ {Q}_{r/\varepsilon } } |\nabla w| dx \right)= \mathcal{O}_r(\varepsilon ).$$ Consequently, on the one hand, $$\begin{aligned} \lambda \varepsilon \int_{{Q}_{r/\varepsilon }}|z|^{-s_1} | w |^{2^*_{s_1}-1} B_\varepsilon dx+\varepsilon \int_{{Q}_{r/\varepsilon }}|z|^{-s_2} | w |^{2^*_{s_2}-1} B_\varepsilon dx % &= -\varepsilon \int_{ \partial{Q}_{r} }B_1 \frac{ \partial v_\varepsilon }{\partial\nu} d\sigma (x)+ \mathcal{O}_r(\varepsilon ).\end{aligned}$$ On the other hand by Lemma [Lemma 3](#lem:v-to-cR){reference-type="ref" reference="lem:v-to-cR"}, [\[eq:regul-beta\]](#eq:regul-beta){reference-type="eqref" reference="eq:regul-beta"} and the dominated convergence theorem, we get $$\begin{aligned} \int_{ \partial{Q}_{r} } B_1 \frac{\partial v_\varepsilon }{\partial\nu} d\sigma (x)= \textbf{c} \int_{ \partial{Q}_{r} } B_1 \frac{\partial{\mathcal R}}{\partial\nu} d\sigma (x)+o(1)= \textbf{c} {M}(0) \int_{ \partial{Q}_{r} } \frac{\partial{\mathcal R}}{\partial\nu} d\sigma (x)+O(r)+o(1),\end{aligned}$$ so that $$\begin{aligned} \lambda \varepsilon c\int_{{Q}_{r/\varepsilon }}|z|^{-s_1} | w |^{2^*_{s_1}-1} B_\varepsilon dx+\varepsilon c\int_{{Q}_{r/\varepsilon }}|z|^{-s_2} | w |^{2^*_{s_2}-1} B_\varepsilon dx % &= -\varepsilon \textbf{c}^2 M(0)\int_{ \partial{Q}_{r} } \frac{ \partial{\mathcal R}}{\partial\nu} d\sigma (x)+ \mathcal{O}_r(\varepsilon ).\end{aligned}$$ It then follows from [\[eq:est-L-2star-Psi-not-ok\]](#eq:est-L-2star-Psi-not-ok){reference-type="eqref" reference="eq:est-L-2star-Psi-not-ok"} that $$\begin{aligned} \frac{\lambda }{2^*_{s_1}}\int_{\Omega } \rho^{-s_1}_\Gamma|\Psi_{\varepsilon }|^{2^*_{s_1}} dy&+\frac{1}{2^*_{s_2}}\int_{\Omega } \rho^{-s_2}_\Gamma|\Psi_{\varepsilon }|^{2^*_{s_2}} dy= \frac{\lambda }{2^*_{s_1}}\int_{\mathbb{R}^3} |z|^{-s_1} |w|^{2^*_{s_1}} dx\\\\ &+\frac{1}{2^*_{s_2}}\int_{\mathbb{R}^3} |z|^{-s_2} |w|^{2^*_{s_2}} dx-{\varepsilon } \textbf{c}^2 {M}(0) \int_{ \partial{Q}_{r} } \frac{\partial{\mathcal R}}{\partial\nu} d\sigma (x) +\mathcal{O}_r(\varepsilon ).\end{aligned}$$ Finally, recalling that ${\mathcal R}(x)=\frac{1}{|x|}$, we have $$\begin{aligned} \int_{\partial{Q}_r}\frac{\partial{\mathcal R}}{\partial\nu}\, d\sigma (x)&=- \int_{\partial{Q}_r}\frac{x\cdot \nu (x)}{|x|^3}\, d\sigma (x)= \int_{ B_{\mathbb{R}^2}(0, r)}\frac{- 2 r }{r^2+|z|^2}\, dz - 2\pi\int_{-r}^r\frac{r^3}{r^2+t^2}dt=- \pi^2 (1+ r^2) .\end{aligned}$$ Since ${M} (0)=\textbf{m}(y_0)$, see [\[eq:def-mass\]](#eq:def-mass){reference-type="eqref" reference="eq:def-mass"}, the proof of the lemma is thus finished. ◻ Now we are in position to complete the proof of our main result. *Proof.* **of Theorem [Theorem 2](#th:main1){reference-type="ref" reference="th:main1"}**\ Combining Lemma [Lemma 5](#lem:expans-num-3D){reference-type="ref" reference="lem:expans-num-3D"} and Lemma [Lemma 6](#lem:expans-denom-3D){reference-type="ref" reference="lem:expans-denom-3D"} and recalling [\[Functional\]](#Functional){reference-type="eqref" reference="Functional"} and [\[Functional1\]](#Functional1){reference-type="eqref" reference="Functional1"}, we have $$\begin{aligned} J\left(t u_\varepsilon \right)=\Psi(t w)+\mathcal{M}_{r, \varepsilon }(t w),\end{aligned}$$ for some function $\mathcal{M}: \mathcal{D}^{1, 2}(\mathbb{R}^N) \to \mathbb{R}$ satisfying $$\mathcal{M}_{r, \varepsilon }(w)=-\frac{\varepsilon }{2} c^2 \pi^2 m(y_0)+\mathcal{O}_{r, \varepsilon }.$$ Since $2^*_{s_2}> 2^*_{s_1}$, $\Psi(tu_\varepsilon )$ has a unique maximum, we have $$\max_{t\geq 0} \Psi(tw)=\Psi(w)=\beta ^*.$$ Therefore, the maximum of $J(t u_\varepsilon )$ occurs at $t_\varepsilon :=1+o_\varepsilon (1)$. Thanks to assumption [\[ExistenceAssumption\]](#ExistenceAssumption){reference-type="eqref" reference="ExistenceAssumption"}, we have $$\mathcal{M}_{r, \varepsilon }(w)<0.$$ Therefore $$\max_{t \geq 0} J(t u_\varepsilon ):= J(t_\varepsilon u_\varepsilon )\leq \Psi(t_\varepsilon w)+\varepsilon ^2 \mathcal{G}(t_\varepsilon w) \leq \Psi(t_\varepsilon w) < \Psi(w)=\beta ^*.$$ We thus get the desired result. ◻ M. M. Fall and E. H. A. Thiam, *Hardy-Sobolev inequality with singularity a curve*, (2018): 151-181.\ I. E. Ijaodoro and E. H. A. Thiam, *Influence of an lp--perturbation on Hardy-Sobolev inequality with singularity a curve*, Opuscula Mathematica 41.2 (2021): 187-204.\ H. Jaber, *Hardy-Sobolev equations on compact Riemannian manifolds*, Nonlinear Anal. 103(2014), 39-54.\ H. Jaber, Mountain pass solutions for perturbed Hardy--Sobolev equations on compact manifolds Analysis, 2016, vol. 36, no 4, p. 287-296.\ R. Schoen and S. Yau, *On the proof of the positive mass conjecture in general relativity*, Comm. Math. Phys. 65 (1979), no. 1, 45--76.\ R. Schoen and S. Yau, *Proof of the positive action-conjecture in quantum relativity*, Phys. Rev. Lett. 42 (1979), no. 9, 547--548.\ E. H. A. Thiam, *A nonlinear Elliptic PDE involving two Hardy-Sobolev critical exponents in domains with curve singularity*, Preprint.

arxiv_math

--- abstract: | In this work we determine the wavefront set of certain eigendistributions of the Laplace-Beltrami operator on the de Sitter space. Let $G = \mathrm{SO}_{1,n}(\mathbb{R}^{})_e$ be the connected component of identity of Lorentz group and let $H = \mathrm{SO}_{1,n-1}(\mathbb{R}^{})_e \subset G$. The de Sitter space $\mathrm{dS}^n$, is the one-sheeted hyperboloid in $\mathbb{R}^{1,n}$ isomorphic to $G/H$. A *spherical distribution*, is an $H$-invariant, eigendistribution of the Laplace-Beltrami operator on $\mathrm{dS}^n$. The space of spherical distributions with eigenvalue $\lambda$, denoted by $\mathcal{D}'_{\lambda}(\mathrm{dS}^n)$, has dimension 2. In this article we construct a basis for the space of positive-definite spherical distributions as boundary value of sesquiholomorphic kernels on the crown domains, an open $G$-invariant domain in $\mathrm{dS}^n_{\mathbb{C}^{}}$. It contains $\mathrm{dS}^n$ as a $G$-orbit on the boundary. We characterize the analytic wavefront set for such distributions. Moreover, if a spherical distribution $\Theta \in \mathcal{D}'_{\lambda}(\mathrm{dS}^n)$ has the wavefront set same as one of the basis element, then it must be a constant multiple of that basis element. Using the analytic wavefront sets we show that the basis elements of $\mathcal{D}'_{\lambda}(\mathrm{dS}^n)$ can not vanish in any open region. address: - Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, USA - Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, USA author: - Gestur Ólafsson - Iswarya Sitiraju title: Analytic wavefront sets of spherical distributions on de Sitter space --- # Introduction For a distribution $\Theta$ on a smooth manifold, the *wavefront set* $WF(\Theta)$ characterizes the singular support and singular directions of $\Theta$. Thus, it describes the set of points where $\Theta$ is not given by a smooth function and the direction in which the singularity occurs. The singularities of a distribution $\Theta$ can also be studied in terms of analytic wavefront set denoted as $WF_A(\Theta)$. These are the set of points having no neighbourhood where $\Theta$ is real analytic and the direction in which the singularity occurs. The term wavefront set was first introduced by Lars Hörmander in [@H70] to study the propagation of singularities of pseudo-differential operators. A full account on wavefront set is given in [@H63 Chap. 8]. The wavefront set is a crucial concept in quantum field theory(QFT). One of the initial papers using wavefront sets in QFT was [@Di79]. The author constructs a scattering operator associated with a unique field operator satisfying the Klein-Gordon equations in $\mathbb{R}^{4}$ with certain class of metrics. To do that he uses the estimates of the fundamental solutions of the Klein-Gordon equation away from its wavefront set. Later the wavefront set was brought into the context of Hadarmard distributions in [@RM]. It was shown that the Hadamard condition of a two point distribution of a quasi-free quantum field is equivalent to a condition its wavefront set. In algebraic quantum field theory, the condition on the wavefront set of the states of quantum fields is related to Reeh-Schleider property (see [@SVW02; @V99]). The wavefront set was conceptualized in the context of unitary representations of Lie groups, a research direction initiated by the fundamental paper [@HR]. One of the direction was to understand the wavefront set of induced representations which was conjectured in that paper. The results for compact Lie group and parabolic induction on connected semisimple Lie group with finite center were obtained. A more general result was established in [@HHO]. These are a few studies, among others, on wavefront sets. For $n \geq 2$, let $G = \mathrm{SO}_{1,n} (\mathbb{R}^{})_e$ be the connected component of identity of Lorentz group and $H = \mathrm{SO}_{1,n-1}(\mathbb{R}^{})_e \subset G$, the stabilizer of $e_n$. The de Sitter space is the one-sheeted hyperboloid defined as $$\mathrm{dS}^n= \{x \in \mathbb{R}^{1+n}: -x_0^2 +x_1^2 +...+x_n^2 = 1 \}.$$ Moreover, for $e_n = (0,0,...,0,1)$ $$\mathrm{dS}^n= G/H = g \cdot e_n.$$ The de Sitter space is a homogenous Lorentzian manifold and one of the simple models of the universe in special theory of relativity. Let $[.,.]$ be the Lorentzian bilinear form $$[x,y] = -x_0y_0 +x_1y_1+...+x_ny_n.$$ In [@NO18], the authors showed that the de Sitter space lies on the boundary of the open complex domains $\Xi$ and $\widetilde{\Xi}$, where $$\Xi = G\cdot \mathbb{S}^n_{+}, \qquad \widetilde{\Xi} = G \cdot \mathbb{S}^n_{-},$$ where $\mathbb{S}^n= \{x : ||x|| = 1\}$ is the unit sphere and $\mathbb{S}^n_{\pm} = \{v \in \mathbb{S}^n: \pm v_0>0\}$ is the upper respectively lower hemisphere. In that paper the authors also showed that the cone of $G$-invariant sesquiholomorhpic kernels on $\Xi$ are parametrized by $i[0,\infty) \cup [0,\rho)$ where $\rho = (n-1)/2$. A set of representatives $\Psi_\lambda$ was given in terms of hypergeometric functions (see bellow). The same idea works for $\widetilde{\Xi}$. For $\lambda \in i[0,\infty) \cup [0,\rho)$ we define the sesquiholomorphic kernels $\Psi_{\lambda}$ and $\widetilde{\Psi}_{\lambda}$ on $\Xi \times \Xi$ and $\widetilde{\Xi}\times \widetilde{\Xi}$, respectively. In this paper we will show that for each $x \in \mathrm{dS}^n$, the boundary values of these kernels $\Psi_x^{\lambda}, \widetilde{\Psi}_x^{\lambda}$ defines distributions on $\mathrm{dS}^n$. These distributions are eigendistributions of the Laplace Beltrami operator $\Delta$. Moreover, $\Psi_{e_n}^{\lambda}$ and $\widetilde{\Psi}_{e_n}^{\lambda}$ are $H-$invariant distributions and span the space of H-invariant eigendistributions called spherical distributions with eigenvalue $\rho^2-\lambda^2$. A similar work was done previously in [@BM96; @BM04; @BV96; @BV97]. The crown $\Xi$ is holomorphically equivalent to the Lorentzian tuboid $\mathcal{T}^{+} = (\mathbb{R}^{1,n} + i\Omega)\cap \mathrm{dS}^n_{\mathbb{C}^{}}$, where $\Omega = \{v \in \mathbb{R}^{1,n}: [v,v]<0,v_0>0\}$ and $\mathrm{dS}^n_{\mathbb{C}^{}}$ is the complexification of $\mathrm{dS}^n$. Similarily, $\widetilde{\Xi}$ is equivalent to the tuboid $(\mathbb{R}^{1,n} - i\Omega)\cap \mathrm{dS}^n_{\mathbb{C}^{}}$. The Perikernels, defined as the distributional solutions of $(\Delta - (\rho^2-\lambda^2))\phi = 0$ on the de Sitter space, corresponds to the kernel $\Psi_{\lambda}$. They show that the Perikernels can be extended holomorphically to the domain $\mathcal{T}^\pm$. In [@BV97], the authors discuss how the perikernels are holomorphic in the cut domain of $\mathrm{dS}^n\times \mathrm{dS}^n$ of the form $\mathrm{dS}^n\times \mathrm{dS}^n\setminus \Sigma$ where, $\Sigma$ is the set of tuples $(x,y)$ with $[x-y,x-y] \leq 0$. In this paper we will show something more that the distributions $\Psi_x^{\lambda}$ and $\widetilde{\Psi}_x^{\lambda}$ are real analytic on the cut domain $\mathrm{dS}^n\times \mathrm{dS}^n\setminus \{(x,y): [x-y,x-y] =0 \}$ and have jump discontinuities along the cut. The cut is where the distributions have singularities and we will study these singularities in terms of analytic wavefront sets. In this paper, we will characterize the wavefront sets of positive definite spherical distributions. In [@NO18], the authors have introduced the kernel $\Psi_\lambda$ upto a constant and showed that $\Psi_{\lambda}$ was represented as a hypergeometric function. The kernels $\Psi_{\lambda}$ and $\widetilde{\Psi}_{\lambda}$ are given as follows: $$\Psi_{\lambda}(z,w) = {}_2F_1\Big(\rho+\lambda,\rho-\lambda; \frac{n}{2}; \mbox{\Large$\frac{1+[z,\Bar{w}]}{2}$}\Big) \quad z,w \in \Xi;$$[\[eq : pm1\]]{#eq : pm1 label="eq : pm1"} and, $$\widetilde{\Psi}_{\lambda}({z},{w}) = {}_2F_1\Big(\rho+\lambda,\rho-\lambda; \frac{n}{2}; \mbox{\Large$\frac{1+[{z},\Bar{w}]}{2}$}\Big) \quad {z},{w} \in \widetilde{\Xi}.$$[\[eq:pm2\]]{#eq:pm2 label="eq:pm2"} These are well-defined sesquiholomorphic, positive-definite, G-invariant kernels. The boundary value from $\Xi$ and $\widetilde{\Xi}$ on $\mathrm{dS}^n$ is taken in the sense of distributions as follows: for $x \in \mathrm{dS}^n$ $$\begin{split} \Psi_x^{\lambda} &= \underset{z \rightarrow x}{\mathrm{lim}}\Psi_\lambda(z, .),\quad z\in \Xi; \\ \widetilde{\Psi}_x^{\lambda} &= \underset{{z} \rightarrow x}{\mathrm{lim}}\widetilde{\Psi}_\lambda({z},.), \quad z \in \widetilde{\Xi}. \end{split}$$ The main theorem of this paper is **Theorem 1**. *The distributions $\Psi_{e_n}^{\lambda}$ and $\widetilde{\Psi}_{e_n}^{\lambda}$ are spherical distributions and span $\mathcal{D}_{m^2}^{H}(\mathrm{dS}^n)$, where $m^2 = \rho^2 - \lambda^2$ and, $\lambda \in i[0,\infty) \cup [0,\rho)$. Moreover, the following holds for a non-zero spherical distribution $\Theta$ on $\mathrm{dS}^n$:* 1. *$WF_A(\Theta)\subset WF_A(\Psi_{e_n}^{\lambda}) \cup WF_A(\widetilde{\Psi}_{e_n}^{\lambda})$.* 2. *If $WF_A(\Theta) = WF_A(\Psi_{e_n}^{\lambda})$ then there exists a nonzero constant $c$ such that $\Theta=c\Psi_{e_n}^{\lambda}$.* 3. *If $WF_A(\Theta)=WF_A(\widetilde{\Psi}_{e_n}^{\lambda})$ then there exists a nonzero constant $c$ such that $\Theta=c\widetilde{\Psi}_{e_n}^{\lambda}$.* We also obtain from [Theorem 47](#thm:supp){reference-type="ref" reference="thm:supp"} that the support of the distributions $\Psi_x^{\lambda}$ and, $\widetilde{\Psi}_x^{\lambda}$ is the entire space $\mathrm{dS}^n$. This paper is setup in the following way. The geometrical setting of $\mathrm{dS}^n$ and the crown domains $\Xi$ and $\widetilde{\Xi}$ is given in [2](#sec:ds){reference-type="ref" reference="sec:ds"}. In [3](#sec:ker){reference-type="ref" reference="sec:ker"}, we will study the kernels $\Psi_\lambda$ and $\widetilde{\Psi}_\lambda$. We will prove that the boundary values $\Psi_x^{\lambda}$ and $\widetilde{\Psi}_x^{\lambda}$ are eigen-distributions of $\Delta$ on the de Sitter space in [4](#sec:bv){reference-type="ref" reference="sec:bv"}. We will then review the concept of wavefront sets with some examples in [5](#sec: wf){reference-type="ref" reference="sec: wf"}. In [6](#sec:ws){reference-type="ref" reference="sec:ws"} and [7](#sec:proof){reference-type="ref" reference="sec:proof"}, we will calculate the wavefront set of these distributions and prove [Theorem 1](#thm: sd1){reference-type="ref" reference="thm: sd1"}. The theory discussed in this article is closely related to the representation of the group $G$ and is hidden in the background. We discuss this briefly in the beginning of Section [3.2](#seBoundVal){reference-type="ref" reference="seBoundVal"}. The motivation for [@NO18] was the reflection positivity on the sphere related to the resolvent of the Laplacian. This lead naturally to the kernels $\Psi_\lambda$ and the author showed that the functions $x\mapsto \Psi_\lambda (x,i e_0)$ are the spherical functions on $G/K=\mathrm{H}$ and that all positive definite spherical functions on $\mathrm{H}$ are obtained in this way. This related the analysis to the spherical principal series representations of $G$, their distribution vectors and to the general theory of the crown as presented in [@AG90; @vdBD88; @BD92; @GK02a; @GK02b; @GKO04; @KSt04]. Our results are independent of the representation theory, but we refer to [@FNO23] were the analytic continuation and the construction of the spherical distribution vectors are discussed from the point of view of representation theory. ## Notations: {#notations .unnumbered} We use the following notations throughout the article: - $\mathrm{G} = \mathrm{SO}_{1,n}(\mathbb{R}^{})_e$, - $\mathrm{H} = \mathrm{SO}_{1,n-1}(\mathbb{R}^{})_e\subset G$, - $\mathrm{K} = \mathrm{SO}_{n}(\mathbb{R}^{})\subset G$, - For $z,w \in \mathbb{C}^{1+n}$, $[z,w] = -z_0w_0 +z_1w_1 +...+ z_nw_n$, - $\mathbb{R}^{1,n} = (\mathbb{R}^{1+n},[\, , \,])$, - $\mathrm{dS}^n=\{x\in \mathbb{R}^{1+n} : [x,x]=1\} = G/H$, - $\mathrm{dS}^n_{\mathbb{C}}= \{z\in \mathbb{C}^{1+n}: [z,z]=1\}$, - $\mathbb{H}^n = \{ix\in i\mathbb{R}^{1+n} : x_0 > 0, -x_0^2 + \mathbf{x}^2 = -1\} \simeq G/K\subset \mathrm{dS}^n_{\mathbb{C}}$, - $\overline{{\mathbb H}}^n= \{ix\in i\mathbb{R}^{1+n} :x_0 < 0, -x_0^2 + \mathbf{x}^2 = -1\} \simeq G/K$, - $\mathbb{S}^n = \{(ix_0,\mathbf{x}) : x_0^2 + \mathbf{x}^2 = 1\}$, - $\mathbb{S}^n_{\pm} = \{(ix_0,\mathbf{x}) \in \mathbb{S}^n: \pm x_0>0\}$, - $\Gamma^{\pm}(x) = \{ y \in \mathrm{dS}^n: \text{for}\, x \in \mathrm{dS}^n, [y-x,y-x] < 0, \pm y_0 > x_0 \}$, - $\Gamma(x) = \Gamma^{+}\cup \Gamma^{-}$, - $\mathbb{L}_n = \{ v \in \mathbb{R}^{1,n}: [v,v]=0\}$, - $\Omega = \{v \in \mathbb{R}^{1,n}: [v,v]<0, v_0 >0\}$, - $T_\Omega = \mathbb{R}^{1+n} + i\Omega$, - $\rho = (n-1)/2$ for $n \geq 2$. # The Hyperboloid, the De Sitter Space and the crown {#sec:ds} In this section we recall some basic geometric facts about the hyperboloid $\mathbb{H}^n$ and the de Sitter space $\mathrm{dS}^n$, the two main spaces that we will discuss in this article. The material is well known. Our main reference is [@NO20]. We write elements in $\mathbb{C}^{n+1}$ often as $z = (z_0,\mathbf{z})$ with $z_0\in \mathbb{C}$ and $\mathbf{z}\in\mathbb{C}^n$. We write $\mathbf{z}\cdot \mathbf{w}= \sum_{j=1}^n z_jw_j$ and $\mathbf{z}^2 = \mathbf{z}\cdot \mathbf{z}$. Denote by $e_j$ the standard basis of $\mathbb{C}^{1+n}$ and by $\beta_{1+n}=[\,\, ,\,\,]$, the bilinear form on $\mathbb{C}^{1+n}$ given by $$[z,w] = -z_0w_0 + \sum_{j=1}^n z_j w_j = -z_0w_0 + \mathbf{z}\cdot \mathbf{w}.$$ We denote by $\mathbb{R}^{1,n}$ the space $\mathbb{R}^{1+n}$ viewed as a Lorentzian space with Lorentzian form $\beta_{n+1}$. We say that a vector $v \in \mathbb{R}^{1,n}$ is time-like if $[v,v]<0$. ## The hyperboloid and the De Sitter space Define the *hyperbolic space* $\mathbb{H}^n$ and the *de Sitter space $\mathrm{dS}^n$* by $$\mathbb{H}^n=\{x\in i\mathbb{R}^{1+n}\mid [x,x]=1, x_0>0\}\quad\text{and}\quad \mathrm{dS}^n=\{w\in \mathbb{R}^{1+n}\mid [w,w]=1\}.$$ Both spaces are closed submanifold of the complex manifold $$\mathrm{dS}^n_\mathbb{C}=\{z\in \mathbb{C}^{1+n}\mid [z,z]=1\}.$$ We set $$\overline{{\mathbb H}}^n=\{x\in i\mathbb{R}^{1+n}\mid [x,x]=1, x_0<0\} = \sigma (\mathbb{H}^n),$$ where $\sigma$ is the complex conjugation $\sigma(z) =\overline{z}$. We also write $V= i\mathbb{R}^{1+n}$ and $\sigma_V = -\sigma$, the conjugation w.r.t. $V$. We are mostly interested in the de Sitter space so we restrict our discussion to that case. Let $x \in \mathrm{dS}^n$, then we denote by the future(past) cone of $x$ as $\Gamma^{+}(x)(\Gamma^-(x))$ where $$\Gamma^{\pm}(x) := \{y\in \mathrm{dS}^n: [y-x,y-x] <0, \pm y_0 >0\}.$$ For $x\in \mathrm{dS}^n$ the set $\{y \in \mathrm{dS}^n: [y-x,y-x]=0\}$ is called the light cone of $x$ in $\mathrm{dS}^n$. For $x\in \mathrm{dS}^n$ we have $$\mathrm{T}_x(\mathrm{dS}^n) =\{y\in \mathbb{R}^{1+n}\mid [x,y]=0\}\cong \mathbb{R}^{1,n-1} .$$ In particular we have $$\mathrm{T}_{e_n}(\mathrm{dS}^n) = \{y\in \mathbb{R}^{1+n}\mid y_n =0\}.$$ The tangent bundle is then given by $$\mathrm{T}(\mathrm{dS}^n) = \{(x,v)\in \mathbb{R}^{1+n}\times \mathbb{R}^{1+n}\mid x\in \mathrm{dS}^n\text{ and } [x,v]=0\}.$$ Let $\tau : G \rightarrow G$ be the involution given by $\tau(g) = JgJ$, where $J$ is the orthogonal reflection in the hyperplane $x_n = 0$. Furthermore, $$\mathfrak{g} = \mathfrak{h} \oplus \mathfrak{q}$$ with $\mathfrak{h} = \mathrm{ker}(\tau -1)$ and $\mathfrak{q} = \mathrm{ker}(\tau + 1)$. Then we have that $$\mathrm{T}_{e_n} \mathrm{dS}^n\cong \mathfrak{q} \cong \mathbb{R}^{1,n-1}.$$ The $G=\mathrm{SO}(1,n)_e$ be the connected component of identity of the isometry group of $\beta_{1+n}$. We denote by $K=\mathrm{SO}(n)$ the maximal compact subgroup $$\begin{aligned} K& =\{k\in G\mid g\cdot e_0= e_0\} = \left\{\left. \begin{pmatrix} 1 & 0 \\ 0 & a\end{pmatrix}\, \right|\, a\in \mathrm{SO}(n)\right\},\\[2mm] A& = \Bigg \{a_t = \begin{pmatrix} \cosh t & 0 & \sinh t\\ 0& I_{n-1}&0\\ \sinh t & 0 & \cosh t \end{pmatrix} : t \in \mathbb{R}^{}\Bigg \},\end{aligned}$$ and by $$\begin{aligned} H &= \{h\in G\mid h\cdot e_n = e_n\} = \left\{\left. \begin{pmatrix} h & 0 \\ 0 & 1\end{pmatrix}\, \right|\, h\in \mathrm{SO}(1,n-1)_e\right\}\\ &= \mathrm{SO}(1,n-1)_e .\end{aligned}$$ The group $G$ acts transitively on $\mathbb{H}^n$ and $\mathrm{dS}^n$ and, $$\mathbb{H}^n= G\cdot ie_0 \simeq G/K\simeq \overline{{\mathbb H}}^n= G\cdot (-ie_0) \quad\text{and} \quad \mathrm{dS}^n= G\cdot e_n \simeq G/H .$$ We write $x_0=ie_0$. We will also write $\ell_g$ for the diffeomorphism $\ell_gx = gx$. The group $G$ acts on the tangent bundle by $$g\cdot (x,v) =(d\ell_g)_x (v) = (gx,gv),$$ where the action on the right is the natural linear action. It is well know that if $(x,v), (y,w)\in T(\mathrm{dS}^n)$ with $[v,v]=[w,w]$ then there exists a $g\in G$ such that $g\cdot (x,v) = (y,w)$. The exponential function can be written using analytic functions $\mathrm{C},\mathrm{S}:\mathbb{C}\rightarrow \mathbb{C}$ defined by $$\mathrm{C}(z):= \sum_{k=0}^\infty \frac{(-1)^k}{(2k)!}z^k \quad \text{and} \quad \mathrm{S}(z):=\sum_{k=0}^\infty \frac{(-1)^k}{(2k+1)!}z^k .$$ Thus $\mathrm{C}(z) = \cos \sqrt{z}$ and $\mathrm{S}(z) =\frac{ \sin \sqrt{z}}{\sqrt{z}}$. Note that this is well defined as the functions $y \mapsto \cos (y), \sin (y)/y$ are both even. With this notation we have [@NO20 p. 15] **Lemma 2**. *The exponential function $\mathrm{Exp}_x : \mathrm{T}_x(\mathrm{dS}^n)\to \mathrm{dS}^n$ is given by $$\mathrm{Exp}_x(v) = C([v,v])x + S([v,v])v, \quad v \in \mathrm{T}_x (\mathrm{dS}^n)$$ and satisfies $$\ell_g \circ \mathrm{Exp}_x = \mathrm{Exp}_{g\cdot x} \circ (d\ell_g)_x.$$* For $x\in \mathrm{dS}^n$, let $U_{x} =\{v\in \mathrm{T}_{x}(\mathrm{dS}^n) \mid [v,v]<\pi/2\}$ and note that if $x=g\cdot e_n$ then $U_x = g\cdot U_{e_n}$. Let $V_x = \mathrm{Exp}_x U_x \subset \mathrm{dS}^n$. Then the following holds **Lemma 3**. *$V_x$ is open and $\mathrm{Exp}_x : U_x \to V_x$ is a diffeomorphism.* *Proof.* Clearly, the map is analytic. It is enough to prove this for $x = e_n$. Let the map $\alpha$ be given by $u \in \mathrm{T}_{e_n}\mathrm{dS}^n\rightarrow X_u \in \mathfrak{q}$ where, $$\alpha(u) := X_u := \left(\begin{array}{c|c} \mbox{\normalfont\Large\bfseries 0} & u \\ \hline\\[-2ex]%<-- -u^T & 0\\ \end{array}\right)$$ The map $\alpha$ is an isomorphism. Consider the map from $\mathrm{T}_{e_n}\mathrm{dS}^n$ into $G$ given by $$u \mapsto \mathrm{exp}(\alpha(u)) = \left( \begin{array}{c|c} \begin{array}{cc} C[u,u] & 0 \\ 0 & \mathrm{Id}_{n-1} \end{array} & S[u,u]u \\ \hline \\[-2ex] -S[u,u] u^T & C[u,u] \end{array} \right) .$$ We claim that the restriction of this map to the set $U_{e_n}$ is injective. Suppose, $\mathrm{exp}(X_u) = \mathrm{Id}_{1+n}$. It follows that $C[u,u] = 1$ and $S[u,u]u=0$. This is true only if either $u = 0$ or $[u,u] = 4m^2\pi^2$, for $m \in \mathbb{Z} \setminus 0$. Thus, the claim follows for the restriction to $U_{e_n}$. Observe that $$\label{eq: exp} \mathrm{exp}(X_u)e_n = \mathrm{Exp}_{e_n}(u).$$ Since $u \rightarrow \mathrm{exp}(X_u)$ is injective, the lemma is proved. ◻ The following lemma has been proved in [@NO20 Lemma 6.3]. **Lemma 4**. *$G = HAK = KAH$ and, $$G/H = KA.e_n = \mathrm{dS}^n.$$* There exists a unique upto a constant $G$-invariant measure on $\mathrm{dS}^n$. For more discussions see [@D09 p. 159] ## Invariant differential operator Let $L$ be a Lie group and assume that $L$ acts on the manifold $X$ by $g\cdot x = \ell_g(x)$. Then a differential operator $D: C_c^\infty (X) \to C_c^\infty (X)$ is *invariant* if for all $\Theta\in C_c^\infty (X)$ and all $g\in L$, we have $D(\Theta\circ \ell_g) = (Df)\circ \ell_g$. We denote by $\mathbb{D}(X)$ the algebra of invariant differential operators. It is known [@F79] that $\mathbb{D}(\mathrm{dS}^n) = \mathbb{C}^{} [\Delta]$, the algebra of polynomials in the Laplacian which we define in two equivalent ways. First let $$\square_{n+1} = -\dfrac{\partial^2}{\partial x_0^2} + \sum_{j=1}^n \dfrac{\partial^2}{\partial x_j^2}$$ in $\mathbb{R}^{1,n}$. Let $\varphi \in C_c^\infty (\mathbb{R}^{})$, $\varphi = 1$ in a neighborhood of $1$ and, $\varphi (t) =0$ for $|t-1|> 1/2$. For $\Theta\in C_c^\infty (\mathrm{dS}^n)$ define $$\tilde \Theta (x) = \varphi ([x,x])\Theta(x/| [x,x]|^{1/2}),\quad x\in \mathbb{R}^{1,n}.$$ Then $\widetilde{\Theta}\in C^\infty_c(\mathbb{R}^{1,n})$ and we define $$\widetilde{\Delta}\Theta := (\square_{n+1}\widetilde{\Theta})|_{\mathrm{dS}^n}.$$ It is a well defined $G$-invariant differential operator on $\mathrm{dS}^n$, see [@D09 p. 110,160]. We can also define $\Delta$ using the tangent space and the exponential map. Note that $\square_n$ is a well defined $H=\mathrm{SO}(1,n-1)$-invariant differential operator on $T_{e_n}(\mathrm{dS}^n) \simeq \mathbb{R}^{1,n-1}$. Define $$(\Delta \Theta)\circ \mathrm{Exp}_{e_n} := \square_{e_n}(\Theta\circ \mathrm{Exp}_{e_n}),\quad \Theta\in C_c^\infty (\mathrm{dS}^n).$$ As $\Delta_{e_n}$ is $H$-invariant we have a well defined $G$-invariant differential operator $\Delta$ on $\mathrm{dS}^n$ given by $$\Delta \Theta(g\cdot e_n) = \Delta (\Theta\circ \ell_g)(e_n) .$$ As both $\widetilde{\Delta}$ and $\Delta$ are second order invariant differential operators annihilating the constants it follows that there exists a $c>0$ such that $\Delta = c\widetilde{\Delta}$. ## The Crown $\Xi$, $\widetilde{\Xi}$ The complex crown $\Xi$ of a Riemannian symmetric space $G/K$ is a natural complex open domain in the complexification $G_\mathbb{C}/K_\mathbb{C}$ with the property that eigenfunctions of the algebra of invariant differential operators extends to $\Xi$. It was introduced in [@AG90]. It was studied by several authors but for us the articles [@GK02a; @GK02b; @KSt04] are of most importance, in particular the the articles [@GK02a; @KSt04] finished the description of the crown and [@GK02b] showed that the non-compactly causal symmetric spaces [@HO97], including the de Sitter space, can be realized as open orbit in the boundary of the crown. The crown showed up in a natural way in [@NO20] in relation to reflection positivity and we will collect those results here. Let $h =E_{0n} + E_{n0}\in \mathfrak{so}(1,n)$ be the operator $$h(x_0,x_1, \ldots , x_{n-1}, x_n) = (x_n,0, \ldots ,0,x_1).$$ Then $\mathrm{ad}h$ has the eigenvalues $0, 1,-1$. Thus, we have the eigen space decomposition $$\mathfrak{g}=\mathfrak{g}_{-1}\oplus \mathfrak{g}_0 \oplus \mathfrak{g}_{+1}$$ and the space $\mathfrak{g}_{\pm1}$ are $\mathfrak{g}_0$-invariant. The crown of $\mathbb{H}^n$ is defined to be $$\Xi = G\exp (i(-\frac{\pi}{2},\frac{\pi}{2}) h )\cdot x_0 = G\cdot \{(i\cos t, 0,\ldots , -\sin t)\mid |t|<\pi /2\}$$ and similarly for $\overline{{\mathbb H}}^n$ $$\overline{\Xi}= G\exp i(-\pi/2, \pi /2)h \cdot \overline{x}_0 = \sigma (\Xi)\quad\text{with} \quad \overline{x}_0=-ie_0 .$$ The crown domains $\Xi$ and, $\overline{\Xi}$ are subsets of $G_{\mathbb{C}^{}}/K_{\mathbb{C}^{}}$. We now recall the description of $\Xi$ and its properties, see [@NO20]. The corresponding statements for $\overline{\Xi}$ follows by taking the complex conjugation $\sigma$. **Remark 5**. Recall that an element $h\in \mathfrak{g}$, $h\not = 0$, is called an Euler element if $\mathrm{ad}h$ has eigenvalues $0, 1, -1$. The crown does only depend on $G/K$ and is independent of the choice of Euler element. Consider the open future light cone $\Omega$ given by $$\Omega = \{x \in \mathbb{R}^{1,n}: [x,x]<0, x_0>0\}.$$ The corresponding future tube is given by $$T_{\Omega} = \mathbb{R}^{1,n} + i\Omega.$$ Similarily, the past tube is $$T_{-\Omega} = \mathbb{R}^{1,n} - i\Omega .$$ We realize the unite sphere in $i\mathbb{R}^{}e_0 + \mathbb{R}^{n}$ by $\mathbb{S}^n=\{x\in V\mid [x,x]=1\}$. Set $\mathbb{S}^n_+ = \{ x\in \mathbb{S}^n: x_0>0\}$ and $\mathbb{S}^n_- = \{ x\in \mathbb{S}^n: x_0 < 0\} = \sigma (\mathbb{S}^n_+).$ **Lemma 6** (NÓ 2020). *The crowns can be described as $$\begin{aligned} \Xi & = G \mathbb{S}^n_+ = T_\Omega \cap \mathbb{S}^n_{+,\mathbb{C}^{}} = T_\Omega \cap \mathrm{dS}^n_{\mathbb{C}^{}}\\ &=\{ u + iv : [u,u]-[v,v]=1, [u,v]=0,[v,v]<0, v_0>0\};\\ \overline{\Xi}&= G (\mathbb{S}^n_-) = T_{-\Omega} \cap \mathbb{S}^n_{-,\mathbb{C}^{}} = T_{-\Omega} \cap \mathrm{dS}^n_{\mathbb{C}^{}}\\ &= \{ u - iv :[u,u]-[v,v]=1, [u,v]=0,[v,v]<0, v_0 > 0\}. \end{aligned}$$* *Proof.* The first part is [@NO20 Lem. 3.1] and [@NO20 Prop. 3.2]. The second part follows by applying $\sigma$ to $\Xi$. ◻ The following proposition is the key for the kernels $\Psi_{\lambda}$ and $\widetilde{\Psi}_{\lambda}$, which we will see in [3](#sec:ker){reference-type="ref" reference="sec:ker"}, to be well defined on the crown domains $\Xi$ and $\widetilde{\Xi}$ respectively. **Proposition 7**. *We have $$\{[z,\sigma(w)]\mid z,w \in \Xi \} = \mathbb{C}^{} \setminus [1,\infty) =\{[z,\sigma (w ) ] \mid z,w \in \overline{\Xi}\} .$$* *Proof.* The crown is invariant under the conjugation $z\mapsto -\sigma (z)$. The claim therefore follows from [@NO20 Lem. 3.5] using that the Lorentz form in [@NO20] is the negative of the form considered here. The claim for $\overline{\Xi}$ follows from the first part using that $\overline{\Xi}= \sigma (\Xi)$. ◻ For $U\subset \mathbb{C}^{n+1}$ denote by $\mathrm{cl}\, (U)$ the closure of $U$ in $\mathbb{C}^n$. The boundary $\partial U$ is then $\partial U = \mathrm{cl}\, (U) \setminus U$. **Lemma 8**. *The boundary of $\Xi$, respectively $\overline{\Xi}$, in $\mathbb{S}^n_{\mathbb{C}}$ is given by $$\begin{aligned} \partial\,\Xi &= \{x+iy: x,y \in \mathbb{R}^{1,n}, \,[x,x]=1, [y,y]=0,y_0 \geq 0, [x,y] =0\}; \\ \partial\, \overline{\Xi}&= \{x-iy: x,y \in \mathbb{R}^{1,n}, \,[x,x]=1, [y,y]=0,y_0 \geq 0, [x,y] =0\}\\ &=\sigma (\partial \Xi). \end{aligned}$$* *Proof.* The first part is [@NO20 Lem. 3.7] and the second claim then follows from $\overline{\Xi}= \sigma (\Xi)$. ◻ **Corollary 9**. *$\mathrm{dS}^n=\partial \, \Xi \cap \partial\, \overline{\Xi}$.* *Proof.* The above description of the boundary implies that $\mathrm{dS}^n\subset \partial\, \Xi \cap \partial\, \overline{\Xi}$. If $z= x+iy\in \partial\, \Xi \cap \partial\, \overline{\Xi}$ then $y_0 \geq 0$. Hence $0 = [y,y] = \mathbf{y}^2$ which happens if and only if $\mathbf{y}=0$. Hence $y=0$ and, $[x,x]=1$ implies that $x\in \mathrm{dS}^n$. ◻ The next proposition shows that around each point $x \in \mathrm{dS}^n$, the crown can be represented locally as a tuboid of the form $U + i\Omega'$ where, $U$ is an open set and $\Omega$ is a pointed cone in the tangent space of $x$. Let $U_x$ be the coordinate chart around $x = g\cdot e_n \in \mathrm{dS}^n$. Let $\Omega_{e_n}' = \{v \in \mathrm{T}_{e_n}(\mathrm{dS}^n) : [v,v] < 0, v_0>0\}$ be the open future $H$-invariant cone in $\mathbb{R}^{1,n-1}$. Write $\Omega'_x = g \cdot \Omega_{e_n}' \subset \mathrm{T}_x(\mathrm{dS}^n)$. **Proposition 10**. *Let $g= ka_th$ and, $x=g \cdot e_n \in \mathrm{dS}^n$.* 1. *The map $\kappa_x : U_{x} + i\Omega'_x \rightarrow \Xi$ where, $$\label{eq:kappa} \kappa_x : u + iv \mapsto (\sqrt{1+[v,v]})\mathrm{Exp}_{x}(u) + i \,\mathrm{exp}(X_{{(ka_t)}^{-1}\cdot u})\cdot v$$ is well-defined and biholomorphic onto its image.\ * 2. *The map $\widetilde{\kappa}_x : U_{x} - i\Omega'_x \rightarrow \widetilde{\Xi}$ where, $$\label{eq:kappa2} \widetilde{\kappa}_x : u - iv \mapsto (\sqrt{1+[v,v]})\mathrm{Exp}_{x}(u) - i \,\mathrm{exp}(X_{{(ka_t)}^{-1}\cdot u})\cdot v$$ is well-defined and bi-antiholomorphic onto its image.* *Proof.* Note that if $[v,v] < -1$ for $v \in \Omega'$, then $k_{e_n}(u+iv)\in \mathbb{H}^n$. That is because $k_{e_n}(u+iv) = i \mathrm{exp}(X_u)\cdot (\sqrt{-[v,v]-1}e_n + v)$ and the group $G$ preserves the direction of time-like vector with $v_0 \neq 0$. Following the proof of [Lemma 3](#lemma: exp_x){reference-type="ref" reference="lemma: exp_x"} and [Lemma 4](#lemma:KAH){reference-type="ref" reference="lemma:KAH"}, the maps $\kappa$ and $\widetilde{\kappa}$ are well defined and biholomorphic and bi-antiholomorphic, respectively. ◻ For $u \in U_{e_n}$, $v \in \Omega'_{e_n}$, $h\in H$ and, using the fact tha $\mathrm{exp}(X_u) = \mathrm{exp}(X_{h\cdot u})$, we obtain $$\label{eq:kappa3} \begin{split} \kappa_x \circ (dl_g)_{e_n}(u+iv) &= g\cdot \kappa_{e_n}(u+iv);\\ \widetilde{\kappa}_x \circ (dl_g)_{e_n}(u-iv) &= g\cdot \widetilde{\kappa}_{e_n}(u-iv). \end{split}$$ **Corollary 11**. *We have that $$\Xi = G\cdot \kappa_{e_n}(U_{e_n} + i\Omega_{e_n}') = \bigcup_{x \in \mathrm{dS}^n}\kappa_x(U_{x} + i\Omega'_x)$$ and, $$\widetilde{\Xi}= G\cdot \widetilde{\kappa}_{e_n}(U_{e_n} -i\Omega_{e_n}') = \bigcup_{x \in \mathrm{dS}^n}\widetilde{\kappa}_x(U_{x} - i\Omega'_x).$$* *Proof.* Because of [\[eq:kappa3\]](#eq:kappa3){reference-type="ref" reference="eq:kappa3"}, it is enough to prove the first equality. From the above proposition we clearly have that $G\cdot \kappa_{e_n}(U_{e_n} + i\Omega') \subseteq \Xi$. Now, let $z \in \Xi$. As $\Xi = G \cdot \mathbb{S}^n_+$ it follows that $z = g\cdot (i\cos (t)e_0 + \sin (t) e_n)$ for $t \in (-\pi/2,\pi/2)$. Clearly, $i\cos (t)e_0 + \sin (t) e_n = \kappa_{e_n}(i\cos (t)e_0)$. Thus, the first equality holds. We follow the same arguments for $\widetilde{\Xi}$. ◻ # The Kernels $\Psi_{\lambda}$ and $\widetilde{\Psi}_{\lambda}$ {#sec:ker} In this section we will study the sesquiholomorphic kernels $\Psi_{\lambda}$ and $\widetilde{\Psi}_\lambda$ on the crown domains $\Xi$ and, $\widetilde{\Xi}$, respectively. These kernels are positive-definite for specific parameter $\lambda$ which are related to the spherical representations of $G$ on the Reimannian symmetric space of hyperboloids $\mathbb{H}^n$ and, $\overline{\mathbb{H}}^n$, respectively (see [@D09 Chap. 7], [@NO20]). We will study the boundary values of these kernels on the de-Sitter space. ## The positive definite kernels $\Psi_\lambda$ and $\widetilde{\Psi}_\lambda$ We recall that reflection positivity on the sphere [@NO20], see also [@NO22], lead to a positive definite kernel $\Psi_\lambda$ (also denoted by $\Psi_m$ and with a different normalization in [@NO20 Thm. 4.12, Thm. 4.19]) given for $\lambda \in i[0,\infty) \cup [0,\frac{n-1}{2})$ by $$\label{def:PsiLambda} \Psi_\lambda (z,w) = {}_2F_1\left( \rho + \lambda, \rho-\lambda ; \frac{n}{2}; \frac{1+[z,\sigma (w)]}{2}\right), \quad \rho = \frac{n-1}{2}$$ As both $n$ and $\lambda$ are fixed most of the time we simplify our notation and write $${}_2F_1(z) = {}_2F_1(n,\lambda ;z) = {}_2F_1\left( \frac{n-1}{2}+ \lambda, \frac{n-1}{2}-\lambda ; \frac{n}{2}; z\right).$$ Here ${}_2F_1(a,b;c;z)$ denotes the Gauss hypergeometric function $${}_2F_1(a,b;c;z) = \sum_{n=0}^\infty \frac{(a)_n(b)_n}{(c)_n}\frac{z^n}{n!}$$ where $(d)_n = d(d+1)\cdots (d+n-1)$, $c\not\in -\mathbb{N}_0$ and $|z|<1$. Recall the following facts about the hypergeometric function: **Theorem 12**. *The hypergeometric function ${}_2F_1$ extends to a holomorphic function on $\mathbb{C}\setminus [1,\infty)$. Furthermore for a fixed $z\in \mathbb{C}\setminus [1,\infty)$ the function $$(a,b,c) \mapsto \frac{{}_2F_1(a,b;c;z)}{\Gamma (c)}$$ is an entire function of $a$, $b$, and $c$.* *Proof.* See [@LS p.241,245--246]. ◻ We note that $\Psi_\lambda$ is uniquely determined by $$\phi_\lambda (x) = \Psi_\lambda (x,ie_0)= {}_2F_1 \left( \rho + \lambda, \rho -\lambda ; \frac{n}{2}; \frac{1+ix_0}{2}\right),\quad ix\in\mathbb{H}^n$$ which is the positive definite spherical function on $\mathbb{H}^n$ with spectral parameter $\lambda$. Here, as usually, $\rho =(n-1)/2$. We have, using that $-\sinh^2 (t/2) = \frac{1-\cosh (t)}{2}$: $$\label{eq:sphf} \phi_\lambda( \exp (th)\cdot ie_0) = {}_2F_1 \left( \rho + \lambda, \rho-\lambda ; \frac{n}{2}; -\sinh^2(t/2) \right).$$ Similarly we have that the kernel $\widetilde{\Psi}_\lambda$ on $\overline{\Xi}\times \overline{\Xi}$ uniquely determined by $\widetilde{\Psi}_\lambda(\cdot , -ie_0)|_{\overline{{\mathbb H}}^n}$ which is the spherical function on $\overline{{\mathbb H}}^n$ with spectral parameter $\lambda$. We collect the main facts together in the following Lemma. Part (1) is in [@NO20; @NO22] using that $\sigma_V= -\sigma$: **Theorem 13**. *Let $\rho = \frac{n-1}{2}$. Then* - *The kernel $\Psi_{\lambda}(z,w)$ is a positive definite $G$ invariant kernel on $\Xi \times \Xi$ which is holomorphic in first variable and anti-holomorphic in the second variable. It is given by $$\Psi_{\lambda}(z,w) = \, {}_2F_1\left(\rho+\lambda,\rho-\lambda; \frac{n}{2}; \mbox{\Large$\frac{1+[z, \sigma( w)]}{2}$}\right), \quad z,w \in \Xi;$$* - *Let $z,w\in\Xi$. Then $$\overline{\Psi_\lambda (z,w)} = \Psi_\lambda( \sigma(z),\sigma (w))$$* - *The kernel $\widetilde{\Psi}_{\lambda}(z,w)$ is a $G$-invariant positive definite kernel on $\overline{\Xi}\times \overline{\Xi}$ holomorphic in the first variable and anti-holomorphic in the second variable given by: $$\widetilde{\Psi}_{\lambda}(z,w) = \, {}_2F_1\Big(\rho+\lambda,\rho-\lambda; \frac{n}{2}; \mbox{\Large$\frac{1+[{z}, \sigma(w)]}{2}$}\Big) \quad z,w \in \overline{\Xi}.$$* *We also have that ${\Psi}_{\lambda}(z,w) = \overline{\widetilde{\Psi}_{\lambda}(\bar{z},\bar{w})} = \widetilde{\Psi}_\lambda(\Bar{w},\Bar{z})$.* *Proof.* We use the simplified notation ${}_2F_1(u)= {}_2F_1(\rho + \lambda, \rho -\lambda ;n/2; u)$. (1) is [@NO20 Thm. 4.12] and (2) follows from (1) and the fact that for $\lambda \in i\mathbb{R}\cup \mathbb{R}$ we have for $z\in \mathbb{C}^{}\setminus [1,\infty)$: $$\overline{{}_2F_1(u)} = {}_2F_1(\bar z)$$ as ${}_2F_1(a,b;c;z) = {}_2F_1 (b,a; c; z)$. For (3) let $\widetilde{\phi}_\lambda (z) = \widetilde{\Psi}_\lambda (z, -ie_0)$, $z\in \overline{{\mathbb H}}^n$, be the spherical function on $\overline{{\mathbb H}}^n$. Then, using that $\exp (th) \overline{x}_0 =-i (\cosh (t)e_0 + \sinh (t)e_n)$, we get $$\begin{aligned} \widetilde{\phi} (\exp th \overline{x}_0) & ={}_2F_1 \left(\frac{1+\cosh t}{2}\right)\\ &={}_2F_1 \left(\frac{1+[\exp(th)\overline{x}_0,\sigma (\overline{x}_0)]}{2}\right).\end{aligned}$$ From this it follows that for all $z\in \overline{{\mathbb H}}^n$ we have $$\widetilde{\phi} (z) = {}_2F_1 \left(\frac{1+[z,\sigma (\overline{x}_0)]}{2}\right)$$ because $\overline{{\mathbb H}}^n=K\exp\mathbb{R}^{}h \cdot (-ie_0)$ and $K$ fixes $\pm ie_0$. As $\widetilde{\Psi}(\cdot , \overline{x}_0)$ is holomorphic on $\overline{\Xi}$ it follows that $$\widetilde{\Psi}(z, \overline{x}_0) = {}_2 F_1 \left(\frac{1+[z,\sigma (\overline{x}_0)]}{2}\right),\quad z\in \overline{\Xi}.$$ Using that $\widetilde{\Psi}$ is $G$-invariant it follows that $$\widetilde{\Psi}(\cdot , w) = {}_2 F_1 \left(\frac{1+[z,\sigma (w)]}{2}\right), \quad\text{ for all }z\in \overline{\Xi}\text{ and } w\in \overline{{\mathbb H}}^n.$$ The claim now follows using that $w\mapsto \widetilde{\Psi}_\lambda (z,w)$ is antiholomorphic and hence determined by the restriction to $\overline{{\mathbb H}}^n$. The last claim follows by $\overline{\Xi}= \sigma (\Xi)$ and ${\Psi}_{\lambda}(z,w) = \widetilde{\Psi}(\overline{w},\overline{z})$. ◻ ## The boundary value distribution {#seBoundVal} It was already proved in [@GKO04] that $\lim_{\pi/2 > t\to \pi/2} \Psi_\lambda (z, \exp (it h)\cdot x_0)$ exists as a distribution on $\mathrm{dS}^n$. It was proved for all ncc symmetric spaces using the *Automatic Continuation Theorem* of van den Ban, Brylinski and Delorme, see [@vdBD88 Thm. 2.1] and [@BD92 Thm. 1] and Hardy space approximation and restated in [@NO18] for the specific case of $\mathrm{dS}^n$. A different and less abstract proof was given in [@FNO23]. We give here a third proof of this fact. As a motivation let us recall some facts from representations theory and from the above mentioned references. Let $\mathfrak{a}=\mathbb{R}^{\,} h$, $\mathfrak{n}= \mathfrak{g}_1$ and $\mathfrak{m}= \mathfrak{z}_\mathfrak{k}(\mathfrak{a})$. Then $\mathfrak{p}= \mathfrak{m}\oplus \mathfrak{a}\oplus \mathfrak{n}$ is a minimal parabolic subalgebra. The corresponding minimal parabolic subgroup is $P = N_G (\mathfrak{p}) = MAN$ where $M = Z_K (\mathfrak{a})$, $A = \exp \mathbb{R}^{} h$ and $N =\exp \mathfrak{a}$. Note that $N$ is abelian in this case. Furthermore $G = KAN \simeq K \times A\times N$. We write accordingly $g = k(g) a(g) n(g)$ . We have $G/P = K/M = \mathrm{S}^{n-1}$ and $G$ acts on $\mathrm{S}^{n-1}$ by $g\cdot v= k(g)v$. Denote by $\pi_\lambda$ the principal series representation with spectral parameter $\lambda$. It acts on the Hilbert space $\mathrm{H}_\lambda=L^2(\mathrm{S}^{n-1})$ (with the $K$-invariant probability measure) by $$\pi_{\lambda }(x) f(v) = a(x^{-1}k)^{-\lambda - \rho} f(x^{-1}\cdot v)$$ where $v= ke_1\in\mathrm{S}^{n-1}$, $x\in G$ and $f\in L^2(\mathrm{S}^{n-1})$. The constant function $e_\lambda (v) = 1$ is $K$-invariant with norm $1$ and the associated spherical function is $$\phi_\lambda (x) = \langle \pi_\lambda (x)e_\lambda,e_\lambda\rangle = \int_{\mathrm{S}^n} a(x^{-1}v)^{-\lambda - \rho}dv .$$ We note that $x\mapsto \pi_\lambda (x)e_\lambda$ is right $K$-invariant, hence $\pi_\lambda (z)e_\lambda$ is well defined and the kernel $\Psi_\lambda$ is given by $$\Psi_\lambda (z,w) = \langle \pi_\lambda (z)e_\lambda,\pi_\lambda (w)e_\lambda\rangle.$$ For $g \in G$ we have $g\mathrm{exp}(-ith)e_0 \in \Xi$ and, the orbit map $$(-\pi/2,\pi/2)\mapsto \pi_\lambda (g\exp (-ith ))e_\lambda$$ is analytic and $$\label{eq:eH} e_\lambda^H = \lim_{t\to \pi/2} \pi_\lambda (\exp (-ith ))e_\lambda$$ exists in $\mathrm{H}_\lambda^{-\infty}$ and defines a $H$-invariant distribution vector [@FNO23 Sec. 5]. Furthermore $\pi^{-\infty}_\lambda (\varphi)e_\lambda^H\in \mathrm{H}^\infty_\lambda$ for $\varphi \in C_c^\infty (G/H)$, see [@NO18 Chap. 7] Hence $$\Theta_\lambda (\varphi ) = \langle e_\lambda,\pi_\lambda^{-\infty}(\varphi)e_\lambda^H\rangle$$ defines a $H$-invariant distribution. Furthermore $$\Delta \Theta_\lambda = ( \rho^2 - \lambda^2)\Theta_\lambda .$$ This can be reformulated in terms of the kernel $\Psi_\lambda$. For that let $z\in \Xi$ and $g\in G$. Then $$t\mapsto \Psi_\lambda (z,g\exp (- ith)x_0)=\Psi_\lambda (z,g(i\cos t e_0 + \sin t e_n))$$ is analytic on an open interval containing $(-\pi/2,\pi/2)$ with limit $${}_2F_1\left(\rho + \lambda , \rho-\lambda ; \frac{n}{2}; \frac{1 +[z,ge_n]}{2}\right) = \Psi_\lambda (z,ge_n).$$ is analytic and extend to a continuous map to an open interval containing $\pi/2$, see more detailed discussion in a moment. We then get a distribution on $\mathrm{dS}^n$ by $$\Theta_\lambda(z; \varphi ) = \int_{\mathrm{dS}^n} \overline{\varphi (y)} \Psi_{\lambda}(z,y)d\mu_{\mathrm{dS}^n}(y) = \langle \pi(z)e_\lambda,\pi^{-\infty}_\lambda (\varphi)e_\lambda^H\rangle$$ where $\mu_{\mathrm{dS}^n}$ is a $G$-invariant measure on $\mathrm{dS}^n$. Taking the limit $z\to e_n$ leads then to the eigen-distribution $\Theta_\lambda$: $$\Theta_\lambda (\varphi) = \lim_{t \to \pi^-/2} \int_{G/H} \overline{\varphi (ge_n)} \Psi_{\lambda}(\exp (-ith)e_n,y)d\mu_{\mathrm{dS}^n}(y)$$ or $$\Theta_\lambda = \lim_{t\to \pi^-/2} \Psi_\lambda (\exp (-ith ) e_n, \cdot ) .$$ We will discuss this in more detail in a moment without using the existence of the $H$-invariant distribution vector $e^H_\lambda$ in [\[eq:eH\]](#eq:eH){reference-type="eqref" reference="eq:eH"}. Similar discussion holds for $\overline{\Xi}$. The following lemma has been proved in [@NO20 Lem. 6.4]. **Lemma 14**. *We have $$[\mathrm{dS}^n,\Xi]\cap \mathbb{R}= [\mathrm{dS}^n, \overline{\Xi}] \cap \mathbb{R}= (-1,1).$$* From this and the properties of the hypergeometric function we get: **Proposition 15**. *The kernel $\Psi_{\lambda}$ can be extended continuously to $\Xi \times (\mathrm{dS}^n\cup\Xi)$ and the kernel $\widetilde{\Psi}_{\lambda}$ can be extended continuously to $\overline{\Xi}\times( \mathrm{dS}^n\cup\overline{\Xi})$.* This implies that for all $z\in\Xi, w \in \overline{\Xi}$ and $y\in \mathrm{dS}^n$ we have $(1+[y,z])/2, (1+[y,w])/2\not\in [1,\infty)$ and hence $$(z,y)\mapsto \Psi_\lambda (y,z), \quad (w,y)\mapsto \widetilde{\Psi}_\lambda (y,w)$$ are analytic maps. For $y = e_n$ we have $[z,y] =z_n$ so that $(1+[y,z])/2 \not\in [1,\infty)$ is equivalent to $z\in \mathbb{C}^{} \setminus [1,\infty)$. In particular for $z \in\mathrm{dS}^n$ we have that $z\mapsto \Psi_\lambda (y,z)$ is analytic on $\{z\in \mathrm{dS}^n\, : \, z_n< 1\}$. The same holds for the functions $z\mapsto \Psi_\lambda (y,z), \widetilde{\Psi}_\lambda (y,z),\widetilde{\Psi}(z,y)$. Recall that $\Gamma(x) = \{y\in \mathrm{dS}^n\, : \, [y-x,y-x]<0\}$. We claim that **Proposition 16**. *For $y\in \mathrm{dS}^n$, $y_n \geq 1 \Leftrightarrow y \in \overline{\Gamma(e_n)}$, where $\Gamma(e_n)$ is the open light cone of $e_n$.* *Proof.* We have $$\begin{aligned} &= [y,y] -2[y,e_n] +[e_n,e_n]\\ &= 1 -2y_n +1 = 2(1-y_n) \end{aligned}$$ and $2(1-y_n)\le 0$ if and only if $y_n\ge 1$. ◻ **Lemma 17**. *Let $x\in \mathrm{dS}^n$ and $g\in G$. Then $\Gamma (gx) = g\Gamma (x)$.* *Proof.* We have $[y - gx , y -gx] = [g^{-1} y-x,g^{-1}y-x]$ because of the invariance of $[\cdot , \cdot ]$. This implies that $y \in \Gamma (g x)$ if and only if $g^{-1}y\in \Gamma (x)$ which is equivalent to $y \in g\Gamma (x)$. ◻ **Corollary 18**. *Let $x\in \mathrm{dS}^n$. Then $\mbox{\Large$\frac{1+[y,x]}{2}$} \in [1,\infty)$ if and only if $y \in \overline{\Gamma(x)}$.* *Proof.* Write $x =ge_n$. Then $[y,x] = [y,ge_n] = [g^{-1}y,e_n]$. The claim now follows from Proposition [Proposition 16](#pro:Gamma){reference-type="ref" reference="pro:Gamma"} and Lemma [Lemma 17](#lem:gGamma){reference-type="ref" reference="lem:gGamma"}. ◻ # Distributions as Boundary value of holomorphic functions {#sec:bv} In the previous section we saw that the kernels $\Psi_\lambda(z,y)$ and $\widetilde{\Psi}_\lambda({z},y)$ are analytic for $z$ in their respective crown domains and $y \in \mathrm{dS}^n$. In this section we prove that the boundary value of the kernels $\Psi_\lambda(z,.)$ and $\widetilde{\Psi}_\lambda(\bar{z},.)$ are distributions as $z$ and $\bar{z}$ tends to an element in $\mathrm{dS}^n$. As a motivation we start with simpler kernel $\Phi_{\lambda}$ and $\widetilde{\Phi}_{\lambda}$. We use the usual notation $\mathcal{D}(\mathrm{dS}^n) = C_c^{\infty}(\mathrm{dS}^n)$, $\mathcal{E}(\mathrm{dS}^n)= C^\infty(\mathrm{dS}^n)$ with the standard topology, $\mathcal{E}'(\mathrm{dS}^n)$ the space of distributions with compact support and, $\mathcal{D}'(\mathrm{dS}^n)$ the space of distributions on the de Sitter space. From this section onwards we will denote the elements in $\widetilde{\Xi}$ as $\bar{z}$, since $\sigma(\bar{z})=z$ lies in $\Xi$. ## The kernels $\Phi_{\lambda}$ and $\widetilde{\Phi}_{\lambda}$ {#subsec:Q} Before discussing the distributional limits of the hypergeometric functions let us start with a simple example. For $\lambda \in \mathbb{C}$ and $z,w\in \mathbb{C}^{n+1}$ with $[z,w]\not\in [1,\infty)$ let $$\Phi(z,w) = \frac{1-[z,\Bar{w}]}{2}$$ and for $\lambda \in \mathbb{C}$ $$\Phi_\lambda (z,w) = \left(\frac{1-[z,\Bar{w}]}{2}\right)^\lambda$$ where ever defined. Note that $\Phi_\lambda$ is well defined for $z,w\in\Xi$ and ${z},{w}\in \overline{\Xi}$ or if one of the points $z$ or $w$ is in $\Xi$, respectively $\overline{\Xi}$ and the other is from $\mathrm{dS}^n$. In the case of $\overline{\Xi}$ we sometimes write $\widetilde{\Phi}$ to indicate the domain that we are looking at. We note that the kernels $\Psi_\lambda$ and $\widetilde{\Psi}_\lambda$ behave approximately as a constant multiple $\Phi_{\frac{2-n}{2}}$ and $\widetilde{\Phi}_{\frac{2-n}{2}}$ respectively near $[z,\Bar{w}] = 1 = [\bar{z}, w]$, where the constant depends on $\lambda$ and $n$. Fix $z\in \Xi$ and $\overline{z}\in \overline{\Xi}$. Then the the functions $\Phi_\lambda(z,\cdot )$ and $\widetilde{\Phi}_\lambda(\bar{z},\cdot )$ extends to analytic functions on $\mathrm{dS}^n$ and hence defines distributions $\Phi_z ^\lambda$ and $\widetilde{\Phi}_{\bar{z}}^\lambda$ on $\mathrm{dS}^n$. For $y \in \mathrm{dS}^n$, we want to prove that $\underset{z \rightarrow x}{\mathrm{lim}} \Phi_\lambda(z,y)$ is a distribution for any $x \in \mathrm{dS}^n$. For simplicity we start by taking $x = e_n$. Let $U_{e_n}$ be the local co-ordinate chart around $e_n$ and $\Omega_{e_n}' = \{v \in \mathrm{T}_{e_n}(\mathrm{dS}^n) : [v,v] < 0, v_0>0\}$. Let $\kappa = \kappa_{e_n}$ be the map defined in [\[eq:kappa\]](#eq:kappa){reference-type="ref" reference="eq:kappa"}. Consider the limit $z = \sqrt{1-[v,v]}e_n +iv \rightarrow e_n$ as $v \rightarrow 0$ in $\Omega_{e_n}'$. In $U_{e_n}$, $$\begin{aligned} \Phi_{e_n}^{\lambda}(y) &= \underset{z \rightarrow e_n}{\mathrm{lim}}\Phi_z^{\lambda}(y) =\underset{z \rightarrow e_n}{\mathrm{lim}} \Phi_\lambda(z,y) \\ &= \underset{v \rightarrow 0}{\mathrm{lim}} \Phi_\lambda(\kappa(e_n + iv),y) \\ &= \underset{v \rightarrow 0}{\mathrm{lim}} \left(\frac{[\kappa(e_n+iv) - \mathrm{Exp}_{e_n}(y),\kappa(e_n + iv) - \mathrm{Exp}_{e_n}(y) ]}{2}\right)^{\lambda}.\end{aligned}$$ For $\varphi \in C_c(\mathrm{dS}^n)$ with $\mathrm{supp}(\varphi) \subset U_x$, where $U_x$ is the local chart around $x = g \cdot e_n$. We have $$\underset{z \rightarrow e_n}{\mathrm{lim}} \int_{\mathrm{dS}^n} \Phi_{\lambda}(z,y)\varphi(y) dy = \underset{z \rightarrow e_n}{\mathrm{lim}} \int_{\mathrm{dS}^n} \Phi_{\lambda}(z,g^{-1}y)\varphi(g^{-1}y) dy; \quad$$ exists as $U_{e_n} = g^{-1}U_x$. Thus for any $\varphi \in C_c(\mathrm{dS}^n)$, by using partition of unity, the limit of $\underset{z \rightarrow e_n}{\mathrm{lim}} \Phi_n(z,y)$ exists in distributions. Similarly for $\bar{z}\in \widetilde{\Xi}$, using the definition of $\widetilde{\kappa} = \widetilde{\kappa}_{e_n}$ as in [\[eq:kappa2\]](#eq:kappa2){reference-type="ref" reference="eq:kappa2"} and, same argument as above it is enough to prove that $$\widetilde{\Phi}_{\bar{z}}^\lambda(y) = \underset{\bar{z} \rightarrow e_n}{\mathrm{lim}}\widetilde{\Phi}_{\bar{z}}^{\lambda}(y) =\underset{\bar{z} \rightarrow e_n}{\mathrm{lim}}{\Phi}_{\lambda}(\bar{z},y) = \underset{v \rightarrow 0}{\mathrm{lim}}\, \Phi_\lambda(\widetilde{\kappa}(e_n - iv),y).$$ is a distribution on $U_{e_n}$. By the discussions so far it is enough to calculate the boundary value for $z = z_t = i\cos(t)e_0 + \sin(t)e_n= \exp (-i t h)$ and $\bar{z} = \bar{z}_t = -i\cos(t)e_0 + \sin(t)e_n= \exp (i t h)$ as $t \rightarrow \frac{\pi}{2}^{-}$. If $z\in \Xi$ then, as mentioned earlier, there exists $t \in (-\pi/2,\pi/2)$ and $g\in G$ such that $z (g,t) =g.z_t= g \exp (-i t h)x_0$ and we have $z (g,t) \to ge_n\in \mathrm{dS}^n$ as $t \to \pi/2$. Similary for $\bar{z} \in\overline{\Xi}$, there exists a $t \in (-\pi/2,\pi/2)$ such that $\bar{z}(g,t) = g.\bar{z}_t$. In particular, if we take $g= Id$ then $z(Id,t )= z_t \to e_n$. For $\mathrm{Re}(\lambda) > 0$, the limit is well defined in distributions. We will use analytic continuation to extend the definition to $\mathrm{Re}(\lambda)<0$. Using the local co-ordinates and after some calculations we arrive at the following: **Lemma 19**. *Let $L_{\lambda} = \Delta + \lambda(\lambda-1 + n)$, where $\Delta$ is the Laplace-Beltrami operator on $\mathrm{dS}^n$. Then the distributions $\Phi_z^{\lambda}$ and $\widetilde{\Phi}_{\bar{z}}^{\lambda}$ satisfy $$L_{\lambda + 1} \Phi_z^{\lambda + 1} = (\lambda + 1)\left(\lambda + \frac{n}{2}\right)\Phi_z^{\lambda}$$ and, $$L_{\lambda + 1} \widetilde{\Phi}_{\bar{z}}^{\lambda + 1} = (\lambda + 1)\left(\lambda + \frac{n}{2}\right)\widetilde{\Phi}_{\bar{z}}^{\lambda}.$$* It follows from the above lemma that $$\begin{split} \Phi_{z}^{\lambda} &= {L_{\lambda + 1}\ldots L_{\lambda + k} \,\Phi_{z}^{\lambda + k}} \\ \widetilde{\Phi}_{\bar{z}}^{\lambda} &= {L_{\lambda + 1}\ldots L_{\lambda + k} \,\widetilde{\Phi}_{\bar{z}}^{\lambda + k}}. \end{split}$$ For $\lambda \neq -1,-2,...,-n/2, -n/2-1,...$, we can thus define the analytic continuation of $\Phi_z^{\lambda}$ and $\widetilde{\Phi}_{\bar{z}}^{\lambda}$. For the residue at the singular points we refer to [@GS64 Sec III.2]. Therefore we obtain that the limits $$\underset{z \rightarrow x}{\mathrm{lim}} \, \Phi_z^{\lambda}, \quad \underset{\bar{z} \rightarrow x}{\mathrm{lim}} \,\widetilde{\Phi}_{\bar{z}}^{\lambda}$$ are distributions on $\mathrm{dS}^n$. **Corollary 20**. *For $\lambda = (2-n)/2$ the distributions $\Phi_z^{\lambda}$ and $\widetilde{\Phi}_{\bar{z}}^{\lambda}$ and their limits $\Phi_x^{\lambda}$, $\widetilde{\Phi}_x^{\lambda}$ are eigendistributions of the Laplace-Beltrami operator $\Delta$ with eigenvalue $\frac{n}{2}(\frac{n-2}{2})$.* *Proof.* For $\lambda = 1-n/2$, it follows from [Lemma 19](#lem: L){reference-type="ref" reference="lem: L"} that, $$\Delta \Phi_z^{\frac{2-n}{2}} = \frac{n}{2}\Big(\frac{n-2}{2}\Big)\Phi_z^{\frac{2-n}{2}}.$$ and same for $\widetilde{\Phi}_z^{\frac{2-n}{2}}$. Since differentiation is continuous on the space of distributions and, $\Delta$ is invariant under the group $G$ we have that $$\label{eq:dq} \Delta \Phi_x^{\frac{2-n}{2}} = \frac{n}{2}\Big(\frac{n-2}{2}\Big)\Phi_x^{\frac{2-n}{2}}, \qquad \Delta \widetilde{\Phi}_x^{\frac{2-n}{2}} = \frac{n}{2}\Big(\frac{n-2}{2}\Big) \widetilde{\Phi}_x^{\frac{2-n}{2}}.$$ ◻ We start with the special case $g=Id$ and write $y' = (1-y_n)/2$. From now on we denote $\Phi_x = \Phi_x^{\frac{2-n}{2}}$ and $\widetilde{\Phi}_x = \widetilde{\Phi}_x^{\frac{2-n}{2}}$. From [9](#appendixa){reference-type="ref" reference="appendixa"} we obtain that $$\Phi_{e_n} (y)=\lim_{t\to \frac{\pi}{2}^-}\Phi_{z_t} = (y' \pm i0)^{\frac{2-n}{2}} \quad \mathrm{for}\; \pm y_0 >0,$$ and, $$\widetilde{\Phi}_{e_n}(y) = \lim_{t\to \frac{\pi}{2}^-}\widetilde{\Phi}_{\bar{z}_t} = (y' \mp i0)^{\frac{2-n}{2}}\quad \mathrm{for}\; \pm y_0 >0$$ because $$\frac{1- [z_t, y]}{2} = \frac{1-\sin(t)y_n}{2} + i\frac{\cos (t)y_0}{2}.$$ For $n$ even we have by [\[eq:AppnEven\]](#eq:AppnEven){reference-type="eqref" reference="eq:AppnEven"}: $$\Phi_{e_n} (y) = (y')_+^{\frac{2-n}{2}} + (-1)^{\frac{n-2}{2}} (y')_-^{\frac{2-n}{2}} - (-1)^{\frac{n-2}{2}} \mathrm{sgn}(y_0)\mbox{\Large$\frac{ i \pi}{(n/2 -2)!}$}\delta^{\frac{n-2}{2}}(y'),$$ and if $n$ is odd then [\[eq:AppnOdd\]](#eq:AppnOdd){reference-type="eqref" reference="eq:AppnOdd"} leads to $$\Phi_{e_n}(y) = (y')_+^{\frac{2-n}{2}} + (-i \,\mathrm{sgn}( y_0))^{n-2} (y')_-^{\frac{2-n}{2}} .$$ Correspondingly, when n is even we have $$\begin{aligned} \widetilde{\Phi}_{e_n}(y) &= (y' \mp \mathrm{sgn}( y_0)\, i0)^{\frac{2-n}{2}} \\ & = (y')_+^{\frac{2-n}{2}} + (-1)^{\frac{n-2}{2}} (y')_-^{\frac{2-n}{2}} + (-1)^{\frac{n-2}{2}} \mathrm{sgn}(y_0)\mbox{\Large$\frac{ i \pi}{(n/2 -2)!}$}\delta^{\frac{n-2}{2}}(y'), \end{aligned}$$ and when n is odd $$\widetilde{\Phi}_{e_n}(y) = (y' \mp \mathrm{sgn}( y_0)\, i0)^{\frac{2-n}{2}} = (y')_+^{\frac{2-n}{2}} + (i\, \mathrm{sgn}(y_0))^{n-2} (y')_-^{\frac{2-n}{2}} .$$ Observe that $\widetilde{\Phi}_x = \overline{\Phi_x}$. Thus, we obtain the following theorem : **Theorem 21**. *The limits $$\lim_{t \rightarrow \pi/2^-} \Phi (g.z_t,\cdot ) =\Phi_x \quad\text{and}\quad \lim_{t \rightarrow \pi/2^-}\widetilde{\Phi} (g.\bar{z}_t,\cdot )=\widetilde{\Phi}_x ,\quad x=ge_n$$ exist in $\mathcal{D}^\prime (\mathrm{dS}^n)$. The distributions $\Phi_x$ and $\widetilde{\Phi}_x$ satisfy [\[eq:dq\]](#eq:dq){reference-type="ref" reference="eq:dq"}. Finally we have $$\Phi_{x}(y)= \begin{cases} \Big(\mbox{\Large$\frac{1-[x,y]}{2}$} \Big)_+^{\frac{2-n}{2}} + (-1)^{\frac{n-2}{2}} \Big(\mbox{\Large$\frac{1-[x,y]}{2}$} \Big)_-^{\frac{2-n}{2}} \\[3mm] -(-1)^{\frac{n-2}{2}} \mathrm{sgn}(y_0-x_0)\mbox{\Large$\frac{ i \pi}{(n/2 -2)!}$}\delta^{\frac{n-2}{2}}\Big(\frac{1-[x,y]}{2}\Big) & \text{if n even};\\ \Big(\mbox{\Large$\frac{1-[x,y]}{2}$} \Big)_+^{\frac{2-n}{2}} +(-i \, \mathrm{sgn}(y_0-x_0))^{n-2} \Big(\mbox{\Large$\frac{1-[x,y]}{2}$} \Big)_-^{\frac{2-n}{2}} & \text{if n is odd}. \end{cases}$$ and, $$\widetilde{\Phi}_{x}(y)= \begin{cases} \Big(\mbox{\Large$\frac{1-[x,y]}{2}$} \Big)_+^{\frac{2-n}{2}} + (-1)^{\frac{n-2}{2}} \Big(\mbox{\Large$\frac{1-[x,y]}{2}$} \Big)_-^{\frac{2-n}{2}} \\[3mm] +(-1)^{\frac{n-2}{2}} \mathrm{sgn}(y_0-x_0)\mbox{\Large$\frac{ i \pi}{(n/2 -2)!}$}\delta^{\frac{n-2}{2}}\Big(\frac{1-[x,y]}{2}\Big) & \text{if n even};\\ \Big(\mbox{\Large$\frac{1-[x,y]}{2}$} \Big)_+^{\frac{2-n}{2}} + (i\,\mathrm{sgn}(y_0-x_0))^{n-2} \Big(\mbox{\Large$\frac{1-[x,y]}{2}$} \Big)_-^{\frac{2-n}{2}} & \text{if n is odd}. \end{cases}$$* *Proof.* Clearly, the limits are well-defined . The rest follows from the above discussion and the fact that $\Phi_{\frac{2-n}{2}}(g.z_t,y) = \Phi_{\frac{2-n}{2}}(z_t, g^{-1}.y)$ and, $\widetilde{\Phi}_{\frac{2-n}{2}}(g.\bar{z}_t,y) = \widetilde{\Phi}_{\frac{2-n}{2}}(\bar{z}_t, g^{-1}.y)$. ◻ Immediately, we obtain the following corollary: **Corollary 22**. *The distributions $\Phi_{e_n}$ and, $\widetilde{\Phi}_{e_n}$ are $H$-invariant distributions.* ## The kernels $\Psi_\lambda$ and $\widetilde{\Psi}_\lambda$ {#the-kernels-psi_lambda-and-widetildepsi_lambda} For $\lambda \in i[0,\infty) \cup [0,\frac{n-1}{2})$, $a= \rho + \lambda$, $b =\rho-\lambda$ and, $c= n/2$ we will denote ${}_2F_1(z)={}_2F_1(a,b;c;z)$. From the [\[eq:kappa\]](#eq:kappa){reference-type="ref" reference="eq:kappa"}, [\[eq:kappa2\]](#eq:kappa2){reference-type="ref" reference="eq:kappa2"} and following previous subsection it is enough to prove that the limits $$\underset{t \rightarrow \pi /2^-}{\mathrm{lim}} \Psi_\lambda(g.z_t, y) \quad \text{and} \quad \underset{t \rightarrow \pi /2^-}{\mathrm{lim}} \widetilde{\Psi}_\lambda(g.\bar{z}_t, y)$$ are distributions. We drop the dependence on $\lambda$ for the limit distributions as it will be clear from the context. From [Theorem 54](#thm:hgf){reference-type="ref" reference="thm:hgf"} the point-wise limit is the following: $$\begin{aligned} \Psi_{e_n}^{\lambda}(y) &= \underset{t \rightarrow \pi /2^-}{\mathrm{lim}} {}_2F_1\Big( \frac{1+[z_t,y]}{2}\Big)\\ &= \underset{t \rightarrow \pi /2^-}{\mathrm{lim}} {}_2F_1\Big( \frac{1+\sin (t) y_n - i\cos (t)y_0}{2}\Big) \\[2mm] &= \begin{cases} {}_2F_1\Big(\frac{1+y_n}{2}\Big) &\text{if $y_n < 1$},\\ {}_2F_1\Big(\frac{1+y_n}{2} - i0\Big) &\text{if $y_n > 1,y_0 > 0$} ,\\ {}_2F_1\Big(\frac{1+y_n}{2} + i0\Big) &\text{if $y_n>1, y_0 < 0$}; \end{cases}\end{aligned}$$ where for $x>1$, ${}_2F_1( x \pm i0 )$ has been calculated in [8](#sec:hgf){reference-type="ref" reference="sec:hgf"}. For the other kernel we get $$\begin{aligned} \widetilde{\Psi}_{e_n}^{\lambda}(y) &= \underset{t \rightarrow \pi /2^-}{\mathrm{lim}} {}_2F_1\Big( \frac{1+\sin (t) y_n + i\cos (t)y_0}{2}\Big) \\[2mm] &= \begin{cases} {}_2F_1\Big(\frac{1+y_n}{2}\Big) &\text{if $y_n < 1$},\\ {}_2F_1\Big(\frac{1+y_n}{2} + i0\Big) &\text{if $y_n > 1,y_0 > 0$}, \\ {}_2F_1\Big(\frac{1+y_n}{2} - i0\Big) &\text{if $y_n>1, y_0 < 0$}. \end{cases}\end{aligned}$$ From the [Theorem 54](#thm:hgf){reference-type="ref" reference="thm:hgf"} we have that in each of the disjoint region the limit is uniform on compact sets. Next step is to prove that the limit actually converges to a distribution. Let $n \geq 2$ and $\varphi$ be such that $\mathrm{supp}(\varphi) \cap \{y_n =1\} = \emptyset$. Since $\Psi_\lambda(z_t,y)$ and $\widetilde{\Psi}_\lambda(\bar{z}_t,y)$ converges to $\Psi_{e_n}^{\lambda}(y)$ and $\widetilde{\Psi}_{e_n}^{\lambda}(y)$ uniformly on compact sets in the region $\mathrm{dS}^n\setminus \{y_n =1\}$, we have that $$\underset{t \rightarrow \pi/2^-}{\mathrm{lim}}\;\int_{\mathrm{dS}^n} \Psi_\lambda(z_t,y) \varphi(y) dy \longrightarrow \int_{\mathrm{dS}^n} \Psi_{e_n}^{\lambda}(y) \varphi(y) dy$$ and, $$\underset{t \rightarrow \pi/2^-}{\mathrm{lim}}\;\int_{\mathrm{dS}^n} \widetilde{\Psi}_\lambda(\bar{z}_t,y) \varphi(y) dy \longrightarrow \int_{\mathrm{dS}^n} \widetilde{\Psi}_{e_n}^{\lambda}(y) \varphi(y) dy.$$ ## Case: dimension 2 {#case-dimension-2 .unnumbered} On the other hand if $\mathrm{supp}(\varphi) \cap \{y_2 =1\} \neq \{ \emptyset \}$ for $\varphi \in \mathcal{D}(X)$, without loss of generality we can take $\varphi$ such that in any local co-ordinates, ${\mathrm{max}}\,[\mathrm{d}(y, \{y_2 =1\})] <\epsilon$, for $y \in \mathrm{supp}(\varphi)$ and very small $\epsilon >0$. We know that, close to the set $\{y_2 =1\}$, $$\Psi_\lambda(z_t,y) \approx -\mbox{\Large$\frac{\Gamma(1)}{\Gamma(\frac{1}{2} + \lambda)\Gamma(\frac{1}{2} - \lambda)}$} \ln\left(\frac{1-[z_t,y]}{2}\right).$$ and, $$\widetilde{\Psi}_\lambda(\bar{z}_t,y) \approx -\mbox{\Large$\frac{\Gamma(1)}{\Gamma(\frac{1}{2} + \lambda)\Gamma(\frac{1}{2} - \lambda)}$}\ln\left(\frac{1-[\bar{z}_t,y]}{2}\right).$$ Since logarithm is locally integrable function and by appendix [9](#appendixa){reference-type="ref" reference="appendixa"} and [@GS64 Sec 2.4, Example 4] we see that the limit convergences in distribution. ## Case: $n \geq 3$ {#case-n-geq-3 .unnumbered} For $n \geq 3$, we have that $\mathrm{Re}(c-a-b) = \mbox{\Large$\frac{2-n}{2}$} < 0$. Without loss of generality we choose $\varphi$ as we did in the 2-dimensional case. Close to $y_n = 1$, the kernels behave as: $$\Psi_\lambda(z_t,y) \approx \mbox{\Large$\frac{\Gamma(n/2)\Gamma((n-2)/2)}{\Gamma(\rho +\lambda)\Gamma(\rho-\lambda)}$} \Bigg(\mbox{\Large$\frac{1-[z_t,y]}{2}$}\Bigg)^{\frac{2-n}{2}};$$ $$\widetilde{\Psi}_\lambda(\bar{z}_t,y) \approx \mbox{\Large$\frac{\Gamma(n/2)\Gamma((n-2)/2)}{\Gamma(\rho +\lambda)\Gamma(\rho-\lambda)}$} \Bigg(\mbox{\Large$\frac{1-[\bar{z}_t,y]}{2}$}\Bigg)^{\frac{2-n}{2}}.$$ Therefore by [Theorem 21](#thm : 4){reference-type="ref" reference="thm : 4"}, appendix [9](#appendixa){reference-type="ref" reference="appendixa"} and [@GS64 Sec 3.6] as we take $t \rightarrow \pi/2^-$, the kernels $\Psi_\lambda(z_t,y)$ and $\widetilde{\Psi}_\lambda(\bar{z}_t,y)$ converge to corresponding distributions $\Psi_{e_n}^{\lambda}$ and $\widetilde{\Psi}_{e_n}^{\lambda}$ . These limits are well defined as $\Psi_{e_n}^{\lambda}$ and $\widetilde{\Psi}_{e_n}^{\lambda}$ are $H$-invariant. To see that let $h \in H$ for which $h.e_n = e_n$. Let $\varphi \in \mathcal{D}(\mathrm{dS}^n)$. If $\mathrm{supp}(\varphi) \cap \{y_n = 1\} = \emptyset$ then clearly $$\underset{t \rightarrow \pi/2^-}{\mathrm{lim}}\;\int_{\mathrm{dS}^n} \Psi_\lambda(h.z_t,y) \varphi(y) dy =\underset{t \rightarrow \pi/2^-}{\mathrm{lim}}\;\int_{\mathrm{dS}^n} \Psi_\lambda(z_t,y) \varphi(y) dy$$ and, $$\underset{t \rightarrow \pi/2^-}{\mathrm{lim}}\;\int_{\mathrm{dS}^n} \widetilde{\Psi}_\lambda(h.\bar{z}_t,y) \varphi(y) dy = \underset{t \rightarrow \pi/2^-}{\mathrm{lim}}\;\int_{\mathrm{dS}^n} \widetilde{\Psi}_\lambda(\bar{z}_t,y) \varphi(y) dy.$$ If $\mathrm{supp}(\varphi) \cap \{y_n = 1\} \neq \emptyset$ as in previous steps. For $n\geq 3$ and, some constant $c$, we obtain that $$\begin{aligned} | \langle \underset{t \rightarrow \pi/2^-}{\mathrm{lim}}\;(\Psi_\lambda(h.z_t,.) - \Psi_\lambda(z_t,.)),\varphi \rangle| &\leq c \underset{t \rightarrow \pi/2^-}{\mathrm{lim}}|\langle \Phi_{\frac{2-n}{2}}(h.z_t, .)-\Phi_{\frac{2-n}{2}}(z_t,.), \varphi\rangle|\\ & = 0.\end{aligned}$$ The last equality is due to [Corollary 22](#cor:eta){reference-type="ref" reference="cor:eta"}. The same steps can be followed for $n=2$ and also for $\widetilde{\Psi}_{e_n}^{\lambda}$. Thus proving that the limits are well-defined. For $g \in G$ and, $x=g \cdot e_n$ $$\Psi_x^{\lambda}(y) = \underset{t \rightarrow \pi/2^-}{\mathrm{lim}}\;\Psi_\lambda(g.z_t,y) = \underset{t \rightarrow \pi/2^-}{\mathrm{lim}}\;\Psi_\lambda(z_t,g^{-1}.y)$$ and, $$\widetilde{\Psi}_x^{\lambda}(y) = \underset{t \rightarrow \pi/2^-}{\mathrm{lim}}\;\widetilde{\Psi}_\lambda(g.\bar{z}_t,y) = \underset{t \rightarrow \pi/2^-}{\mathrm{lim}}\;\widetilde{\Psi}_\lambda(\bar{z}_t,g^{-1}.y)$$ are also distributions. And, the distributions $\Psi_x^{\lambda}$ and $\widetilde{\Psi}_x^{\lambda}$ are $G$-invariant.\ Now, we claim that for $m^2 = \rho^2 - \lambda^2$, $(\Delta - m^2)\Psi_{e_n}^{\lambda} = 0$. Using fact that differentiation is a continuous linear map on space of distributions, we obtain $$\underset{t \rightarrow \pi /2^-}{\mathrm{\mathrm{lim}}}(\Delta - m^2)\Psi_\lambda(z_t, y) = (\Delta - m^2)\Psi_{e_n}^{\lambda}.$$ Now, let $a = \rho +\lambda$, $b=\rho-\lambda$, $c = n/2$ and $w_t = \frac{1+[z_t,y]}{2}$. Using local co-ordinates we arrive at $$\begin{aligned} (\Delta - m^2)\Psi_\lambda(z_t,y) & = \mbox{\Large$\frac{ab}{c}$}[w_t(1-w_t)\frac{(a+1)(b+1)}{(c+1)}{}_2F_1(a+2,b+2,c+2,w_t) \\ &+(\frac{n}{2}-nw_t){}_2F_1(a+1,b+1,c+1,w_t)- c \,{}_2F_1(a,b,c,w_t)] \\ &=0. \end{aligned}$$ using the properties of hypergeometric function. Therefore we have that as distributions $(\Delta -m^2)\Psi_{e_n}^{\lambda} = 0$. Following the same steps we obtain $(\Delta - m^2)\widetilde{\Psi}_{e_n}^{\lambda} = 0$. As $(\Delta - m^2)$ is a $G$ invariant operator, we have that $(\Delta - m^2)\Psi_x^{\lambda} = 0 = (\Delta - m^2)\widetilde{\Psi}_x^{\lambda}$. Therefore we have proved that : **Theorem 23**. *For $n \geq 2$, $\underset{t \rightarrow \pi/2^-}{\mathrm{lim}}\Psi_\lambda(g.z_t,y)$ and $\underset{t \rightarrow \pi/2^-}{\mathrm{lim}}\widetilde{\Psi}_\lambda(g.\bar{z}_t,y)$ converge to distributions $\Psi_x^{\lambda}$ and $\widetilde{\Psi}_x^{\lambda}$ respectively, on $\mathrm{dS}^n$ with $x= g \cdot e_n$ and, $(\Delta - m^2)\Psi_x^{\lambda} = 0 = (\Delta - m^2)\widetilde{\Psi}_x^{\lambda}$. Also, $\Psi_x^{\lambda}$ and $\widetilde{\Psi}_x^{\lambda}$ can be represented as analytic functions in the following regions: $$\begin{aligned} \Psi_x^{\lambda}(y) = \begin{cases} {}_2F_1\Big(\frac{1+[x,y]}{2}\Big) &\text{if $y \notin \overline{\Gamma(x)}$},\\ {}_2F_1\Big(\frac{1+[x,y]}{2} - i0\Big) &\text{if $y \in \Gamma^+(x)$}, \\ {}_2F_1\Big(\frac{1+[x,y]}{2} + i0\Big) &\text{if $y \in \Gamma^-(x)$}; \end{cases}\\[2mm] \widetilde{\Psi}_x^{\lambda}(y) = \begin{cases} {}_2F_1\Big(\frac{1+[x,y]}{2}\Big) &\text{if $y \notin \overline{\Gamma(x)}$},\\ {}_2F_1\Big(\frac{1+[x,y]}{2} + i0\Big) &\text{if $y \in \Gamma^+(x)$}, \\ {}_2F_1\Big(\frac{1+[x,y]}{2} - i0\Big) &\text{if $y \in \Gamma^-(x)$}. \end{cases} \end{aligned}$$* As a conclusion it implies that $\Psi_x^{\lambda} = \overline{\widetilde{\Psi}_x^{\lambda}}$ where it is defined pointwise as a function. # Wavefront Sets {#sec: wf} The wavefront set of a distribution was introduced by L. Hörmander in 1970. It gives more information about singularities. In particular, it gives the singular support of a distribution and the direction where the distribution is not smooth or analytic. We apply this notion to the distributions $\Psi_x^{\lambda}$ and $\widetilde{\Psi}_x^{\lambda}$. As we have seen in previous section the distributions $\Psi_x^{\lambda}$ and $\widetilde{\Psi}_x^{\lambda}$ can be written as analytic functions everywhere on de-Sitter except at the boundary of the light cone of $x$. That is where the distributions are singular. We will now recall the wavefront set of distributions. Let $X \subset \mathbb{R}^{1,n}$ be an open subset. Suppose, $\Theta \in \mathcal{E}'(X)$ is a distribution with compact support then we can define Fourier transform of $\Theta$ as following: for $\xi\in (\mathbb{R}^{1,n}\setminus 0)$: $$\widehat{\Theta}(\xi) = \Theta(e^{-2\pi i [x,\xi]}).$$ where $[x,\xi] = -x_0\xi_0 + x_1\xi_1...+x_{n}\xi_{n}$. **Definition 24**. Let $\Theta$ in $\mathcal{D}'(X)$ be a distribution. We say $(x_0,\xi_0) \in \mathrm{T}^*(X)\setminus \{0\}$ is a regular directed point if there exist an open neighbourhood U of $x_0$, a conical neighbourhood $V$ of $\xi_0$ and $\varphi \in C_c^{\infty}(U)$ with $\varphi(x_0) \neq 0$ such that for all $N \in \mathbb{N}$: $$\label{ft} |\widehat{\varphi \Theta}(\tau \xi)| \leq C_{N,\varphi}(1+|\tau|)^{-N}, \quad \forall \xi \in V.$$ The wavefront set $WF(\Theta) \in \mathrm{T}^*(X)\setminus \{0\}$ is the complement of the regular directed set. **Definition 25**. Let $\Theta \in \mathcal{D}'(X)$. The *singular support* of $\Theta$ is set of all points $x$ such that there is no neighbourhood of x to which the restriction of $\Theta$ is a $C^{\infty}$ function . **Lemma 26**. *If $\Theta \in \mathcal{D}'(X)$, then the projection of $WF(\Theta)$ onto $X$ is the singular support of $\Theta$.* ***Remark** 27*. The $WF(\Theta)$ is a conic set, that is if $(x,\xi) \in WF(\Theta)$, then for $\tau > 0$, $(x,\tau \xi) \in WF(\Theta)$. Here are some examples. **Example 28**. We will consider the basic distribution, the Dirac-delta distribution in $\mathbb{R}^{1,n}$. Then the $\mathrm{supp}(\delta_0) = \{0\}$. Let $\varphi \in C_c^{\infty}(\mathbb{R}^{1,n})$ with $\varphi(0) = c \neq 0$. Now, choose any $\xi \in \mathbb{R}^{1,n}\setminus \{0\}$, we see that because $$\begin{aligned} \widehat{\varphi \delta_0}(\xi) = \delta_0(\varphi(x)e^{-2\pi i [x, \xi]}) = \varphi(0) \neq 0, \end{aligned}$$ the Fourier transform is not rapidly decreasing in $\xi$ for any $\xi \in \mathbb{R}^{1,n} \setminus 0$. Hence $$\pushQED{\qed} \mathrm{WF}(\delta_0)= \{(0,\xi) : \xi \in \mathbb{R}^{1,n} \setminus 0\}. \qedhere$$ **Example 29**. Consider the Heaviside function as distribution. That is, $$H(x) = \begin{cases} 1 & x >0 \\ 0 & x\leq 0. \end{cases}$$ Clearly it is smooth function away from zero. Let $\varphi \in C_c^{\infty}(\mathbb{R}^{})$ with $\varphi(0)\neq 0$. Then using integration by parts we obtain: $$\begin{aligned} \widehat{\varphi H}(\xi) & = \int_0^\infty \varphi(x) e^{-2 \pi i x\xi} dx. \\ &= \frac{\varphi(0)}{2 \pi i \xi} + \int_0^\infty \varphi'(x)e^{-2\pi i x\xi} dx\\ &= \frac{\varphi(0)}{2 \pi i \xi} + \frac{\varphi'(0)}{(2\pi i \xi)^2} + \frac{1}{(2\pi i \xi)^2}\int_0^\infty \varphi''(x)e^{-2\pi i x\xi} dx.\\ \end{aligned}$$ The first term is of order 1 and the rest are atleast of order 2. Thus, the Fourier transform does not decay rapidly enough for any $\xi \neq 0$ in $\mathbb{R}^{}$. Hence, $$\pushQED{\qed} WF(H) = \{0\} \times (\mathbb{R}^{}\setminus 0). \qedhere$$ We will now introduce analytic wavefront sets. We follow the definition from [@H63 def. 8.4.3]. Since by multiplying the distribution with smooth function will only increase the analytic wavefront set and there is no non-zero real analytic function with compact support. To circumvent this problem the following proposition (see [@H63 Proposition 8.4.2]) is the basis for the definition of analytic wavefront set. **Proposition 30**. *Let $X$ be an open subset of $\mathbb{R}^{1,n}$ and $\Theta \in \mathcal{D}'(X)$. Then $\Theta$ is real analytic in a neighbourhood $U$ of $x_0$ if and only if there is a bounded sequence $\Theta_N$ of distributions with compact support which is equal to $\Theta$ in $U$ satisfying, $$|\widehat{\Theta}_N(\xi)|\leq C^{N+1}(N/|\xi|)^{N}, \quad N=1,2,...$$ for $C>0$.* **Definition 31**. If $X$ is an open subset of $\mathbb{R}^{1,n}$ and $\Theta \in \mathcal{D}'(X)$, we denote $WF_A(\Theta)$ to be the complement in $X \times (\mathbb{R}^{1,n}\setminus 0)$ of the set $(x_0,\xi_0)$ such that there is an open neighbourhood $U \subset X$ of $x_0$, a conic neighbourhood $\Gamma$ of $\xi_0$ and a bounded sequence of $\Theta_N \in \mathcal{E}'(X)$ which is equal to $\Theta$ in $U$ and satisfies $$|\widehat{\Theta_N}(\xi)| \leq C^{N+1}(N/|\xi|)^N \quad N=1,2,...$$ when $\xi \in \Gamma$ and for some $C>0$. The following lemma shows that $\Theta_N$ can always be chosen as a product of $\Theta$ with some suitable functions. **Lemma 32**. *Let $\Theta \in \mathcal{D}'(X)$. Let $\Gamma$ and $U$ be as in the definition above. We have that $(x_0,\xi_0) \notin WF_A(\Theta)$ if and only if for $K$ a compact neighbourhood of $x_0$ in U, $\Theta$ a closed conic neighbourhood of $\xi_0$ in $\Gamma$, there exists functions $\chi_N \in C_c^\infty(U)$ such that $\chi_N = 1$ on $K$ with $$|D^{t +\beta}\chi_N| \leq C_t^{N+1}N^{|\beta|}, \quad |\beta| \leq N,$$ then, it follows that the sequence $\chi_N\Theta$ is bounded in $\mathcal{E}'$ and satisfies the following: $$\label{un} |\widehat{\chi_N \Theta}(\xi)| \leq C(C(N+1)/|\xi|)^N.$$* The proof of the above lemma can be found in [@H63 Chap 8]. **Example 33**. Let $u=\delta_0$ in $\mathbb{R}^{1,n}$. We can see that $WF_A(\delta_0) \subset \{0\}\times (\mathbb{R}^{1,n}\setminus 0)$. Let $\chi_N$ be a sequence of functions as in the above lemma. Then for $\xi \neq 0$ $$\begin{aligned} \widehat{\chi_N\delta_0}(\xi) = \chi_N(0) = 1, \end{aligned}$$ which does not decay at infinity. Therefore, $WF_A(\delta_0) = \{0\}\times (\mathbb{R}^{1,n}\setminus 0).$ 0◻ The following lemma tells us the relation between wavefront sets and analytic wavefront set. **Lemma 34**. *Let $\Theta \in \mathcal{D}'(X)$, we have that $WF(\Theta) \subset WF_A(\Theta)$.* *Proof.* Suppose $(x_0,\xi_0) \notin WF_A(\Theta)$ then there exist an open neighbourhood $U \ni x_0$, an open cone $\Gamma \ni \xi_0$ and a bounded sequence of $\Theta_N$ with compact support such that $\Theta_N = \Theta$ in U and $$|\widehat{\Theta}_N(\xi)| \leq C^{N+1}(N/|\xi|)^{N}, \quad \xi \in \Gamma.$$ Then for $x \in U$, $$D^\alpha \Theta(x) = D^\alpha \Theta_N(x) = \int \xi^\alpha \widehat{\Theta}_N(\xi) e^{2\pi i <x,\xi>} d\xi.$$ It follows since $\xi^\alpha \widehat{\Theta}_N(\xi)$ is integrable for $N = |\alpha| + n + 1$, as $1/|\xi|^{1+n}$ is integrable outside unit ball and $|\widehat{\Theta}_N(\xi)| \leq C(1+|\xi|)^M$. Hence $\Theta$ is smooth in U. ◻ We will now show an examples of a distribution whose analytic wavefront set is strictly bigger than wavefront set. Before that let us look at a characterization of real analytic function. A smooth function $\Theta$ is real analytic if and only if for every compact set $K \subset \mathbb{R}^{}$ there is a constant $C_K$ with $$|D^N \Theta(x)| \leq C_K^{N+1} (N)^N, \quad x \in K,$$ for all $N \geq 0$. Indeed, by Taylor's theorem $$\Theta(x) = \sum_{i=0}^n \Theta^{(i)}(x_0)\mbox{\Large$\frac{(x-x_0)^i}{i!}$} + \frac{1}{n!}\int_{x_0}^x \Theta ^{(n+1)}(t)(x-t)^n dt.$$ We have that, for $|x-x_0|<\delta < 1/(3C_K)$ and $N^N \leq 3^N N!$, $$\begin{aligned} \Big|\frac{1}{n!}\int_{x_0}^x \Theta ^{(n+1)}(t)(x-t)^n dt\Big| &\leq \frac{C_K^{N+1} (N)^N}{N!} |\int_{x_0}^x (x-t)^N dt| \\ & = \frac{C_K^{N+1} (N)^N}{(N+1)!} |x-x_0|^{N+1}\\ & \leq (3C_K\delta)^{N+1} \rightarrow 0, \; \text{as}\; N \rightarrow \infty. \end{aligned}$$ Hence $\Theta$ is real analytic function. On the other hand we get that $\Theta$ satisfies the above conditions if it is real analytic by Cauchy's inequalities. **Example 35**. We know that the function $$\begin{aligned} \Theta(x) = \begin{cases} e^{-1/x} & \text{if $x > 0$}\\ 0 & \text{if $x\leq 0$} \end{cases} \end{aligned}$$ is smooth everywhere but not real analytic at origin. It is obvious that $\Theta$ is a distribution. Let $\varphi$ be a smooth function with compact support in a small neighbourhood of 0 with $\varphi(0)=1$. Then, $$\begin{aligned} |\xi^N\widehat{\varphi \Theta}(\xi)| &= |\int_0^\infty D^N (\varphi e^{-1/x}) e^{-2\pi i x \xi} dx|\\ & \leq \int_0^\infty| D^N (\varphi e^{-1/x})| dx \\ &\leq C_N, \end{aligned}$$ where the last inequality is because all the derivatives of $e^{-1/x}$ are bounded and $\varphi$ is smooth with compact support. Therefore, $D^N(\varphi \Theta)$ is integrable for all $N$. Hence $$WF(\Theta) = \emptyset.$$ Now, let $K = [-\epsilon,\epsilon]$, for $\epsilon$ very small. We have that $$D^N(e^{-1/x}) = \mbox{\Large$\frac{e^{-1/x}p_N(x)}{x^{2N}}$},$$ where $p_N(x)$ is a polynomial of degree N with constant coefficient 1. Thus for x in K, $D^N(e^{-1/x}) \approx \mbox{\Large$\frac{e^{-1/x}}{x^{2N}}$}$. Hence the maximum is approximately at $x= 1/2N$ and the maximum value is $e^{-2N}(2N)^{2N}$. For sufficiently large N, $$\underset{x \in K}{\rm{max}}|D^N(e^{-1/x}) | \approx e^{-2N}(2N)^{2N} > N^N.$$ We see that the derivatives of $\Theta$ do not have the desired growth near zero. Hence $\Theta$ is not real analytic at 0 and $\emptyset \neq WF_A(\Theta) \subset \{0\} \times (\mathbb{R}^{} \setminus 0)$. From [Theorem 47](#thm:supp){reference-type="ref" reference="thm:supp"}, we obtain that if $WF_A(\Theta) \cap -WF_A(\Theta) = \emptyset$, then $\Theta$ can not vanish on any open set of $\mathbb{R}^{}$. This implies that $$\pushQED{\qed} WF_A(\Theta) = \{0\} \times (\mathbb{R}^{} \setminus 0). \qedhere$$ Generally, the pull back of a distribution under a map is not continuous. For example, consider the map $\iota : \mathbb{R}^{} \rightarrow \mathbb{R}^{2}$ by $\iota(x) = (x,0)$. Then the pull back must be defined such that $\iota^*(\Theta) = \Theta\circ \iota$ for $\Theta$ a smooth map. For $\Theta \geq 0$, smooth with $\rm{supp}(\Theta) \subseteq \overline{B(0,1)}$, $f_k = k^2f(kx)$, we have $f_k \rightarrow \delta_{(0,0)}$. Let $\varphi \in C_c^\infty(\mathbb{R}^{})$ and $\varphi \geq 0$, $$\begin{aligned} <\iota^*(f_k),\varphi > &= \int_{\mathbb{R}^{}} (f_k\circ \iota)(x) \varphi(x) dx\\ & = k\int_{-1}^1 \Theta(x,0)\varphi (x)dx \rightarrow \infty \quad \text{as} \; k \rightarrow \infty. \end{aligned}$$ Therefore the pull back is not continuous. Define the normal set of the map $\iota$ by $$\begin{aligned} N_\iota &= \{(\iota(x),\Phi) \in \mathbb{R}^{2} \times \mathbb{R}^{2} : {}^td\iota_x(\Phi) =0 \};\\ & = \{((x,0); (0,\Phi_2)) : x, \Phi_2 \in \mathbb{R}^{}\}. \end{aligned}$$ where ${}^td\iota_x = [1,0]$.\ We have that $WF_A(\delta_{(0,0)}) = \{((0,0); (\Phi_1,\Phi_2)\}$.\ Observe that $WF_A(\delta_{(0,0)}) \cap N_{\iota} \neq \emptyset$. We will now see the relation between the set of normals, wavefront set and pullback of distribution. The following theorem says under what condition we can define a pull back of a distribution. The proof can be found in [@H63 Theorem 8.2.4, Theorem 8.5.1] **Theorem 36**. *Let $X$ and $Y$ be open subsets of $\mathbb{R}^{n}$ and $\mathbb{R}^{m}$ respectively and let $\iota :X \rightarrow Y$ be a real analytic map. Denote the normal set of the map by $$N_{\iota} = \{(\iota(x),\Phi) \in Y\times \mathbb{R}^{n} : {}^td\iota_x(\Phi) =0 \}.$$ Then the pull back $\iota^*\Theta$ can be defined in one and only one way for all $\Theta\in \mathcal{D}'(Y)$ with $$N_\iota \cap WF_A(\Theta) = \emptyset$$ so that $\iota^*(\Theta) = \Theta \circ \iota$ when $\Theta \in C^\infty$ and for any closed conic subset $\Gamma$ of $Y \times (\mathbb{R}^{n} \setminus 0)$ with $\Gamma \cap N_{\iota} = \emptyset$ we have $$\iota^*(\Gamma)= \{(x,{}^td\iota_x(\Phi)): (\iota(x),\Phi) \in \Gamma \}.$$ In particular, if $\Theta \in \mathcal{D}'(Y)$ with $N_\iota \cap WF_A(\Theta) = \emptyset$ then $$WF_A(\iota^*\Theta) \subset \iota^*WF_A(\Theta).$$* The above theorem lets us define the analytic wavefront set if $X$ is a real analytic manifold. **Definition 37**. If $X$ is a real analytic manifold, and $(U_k,k)$ be the analytic local co-ordinates on X. We define $WF_A(\Theta) \subset T^*(X)\setminus 0$ to be the set $$k^*WF((k^{-1})^*\Theta) := \{(x, {}^tdk^{-1}_x(\eta)); (k^{-1}(x),\eta) \in WF((k^{-1})^*\Theta),$$ where $(k^{-1})^*\Theta (\varphi)= \Theta(\varphi\circ k^{-1})$ for $\varphi \in C_c^\infty (U_k)$. The [Theorem 36](#cw){reference-type="ref" reference="cw"} tells us that the above definition is invariant under co-ordinate change. The next theorem describes the analytic wavefront sets of distributions which are boundary value of analytic functions. Let $\Gamma$ be an open convex cone, then the *dual cone* $\Gamma^{\circ}$ is defined as $$\Gamma^{\circ} = \{ \eta \in \mathbb{R}^{1+n} : \eta_0\xi_0 + ...+ \eta_n\xi_n \geq 0, \;\forall \xi \in \Gamma\}.$$ **Theorem 38**. *Let $X \subset \mathbb{R}^{1,n}$ be an open set and $\Gamma$ an open convex cone in $\mathbb{R}^{}$ and for some $\gamma > 0$, $$Z = \{ z \in \mathbb{C}^{n} : \rm{Re}\, z \in X, \rm{Im}\, z \in \Gamma, |\rm{Im} z| < \gamma\}.$$ If $\Theta$ is an analytic function in $Z$ such that $$|\Theta(z)| \leq C |\rm{Im} \,z|^{-N}$$ for some $N$ and some constant $C >0$, the $\underset{y\searrow 0}{\rm{lim}}\Theta(.+iy) = \Theta_0$ exists in terms of distribution and is of order $N$. We also have that $$WF_A (\Theta_0) \subset X \times (\Gamma^{\circ} \setminus 0).$$* *Proof.* See theorem 3.1.15 and theorem 8.4.8 in [@H63]. ◻ **Example 39**. Consider the distribution on $\mathbb{R}^{}$, $\Theta = (x+i0)^{\frac{2-n}{2}}$. It is the limit of the analytic function $(x+iy)^{\frac{2-n}{2}}$ for $x \in \mathbb{R}^{}$ and $y \in \Gamma = \mathbb{R}^{}_+$. Then its dual cone is $\Gamma^{\circ} = \mathbb{R}^{}_{\geq 0 }$. By [Theorem 38](#bd){reference-type="ref" reference="bd"}, $WF_A(\Theta) \subset \mathbb{R}^{} \times \mathbb{R}^{}_+$. It is obvious that the distribution has singularities only at $x =0$. Therefore, $$WF_A((x+i0)^{\frac{2-n}{2}}) = \{(0, \tau): \tau > 0\}.$$ Similarily, $$\pushQED{\qed} WF_A((x-i0)^{\frac{2-n}{2}}) = \{(0, \tau): \tau < 0\}. \qedhere$$ **Example 40**. Let $\Theta = \ln({x+i0})$ which is a boundary value of holomorphic function $\ln({x +iy})$ for $y >0$. Since logarithm grows slower than any negative power of $|y|$, the limit $y \rightarrow 0$ is a distribution on $\mathbb{R}^{}$. It follows from [Theorem 38](#bd){reference-type="ref" reference="bd"} that $$WF_A(\ln({x+i0})) = \{(0,\tau):\tau>0\}.$$ Likewise we have that, $$\pushQED{\qed} WF_A(\ln({x-i0})) = \{(0,\tau): \tau<0\}. \qedhere$$ **Example 41**. Let $\Theta = {}_2F_1(x + i0)$, the boundary value of the holomorphic function ${}_2F_1(x +iy)$ for $y>0$. We have proved in [8](#sec:hgf){reference-type="ref" reference="sec:hgf"} that it is a distribution which has analytic singularity at $x=1$. As a result of [Theorem 38](#bd){reference-type="ref" reference="bd"}, the analytic wavefront set is $$WF_A({}_2F_1(x + i0)) = \{(1, \tau):\tau > 0\}$$ and it also follows that $$\pushQED{\qed} WF_A({}_2F_1(x - i0))= \{(1, \tau):\tau < 0\}.\qedhere$$ Let $P(x,D) = \underset{|t| \leq m}{\Sigma} a_t(x) D^\alpha$ be a differential operator on $X$ with analytic coefficients. Then we have that $$WF_A(P(x,D)\Theta) \subset WF_A(\Theta).$$ The following theorem is a converse to the above statement which can be found in [@H63]. **Theorem 42**. *If P(x,D) is a differential operator of order m with real analytic coefficients in X, then $$WF_A(\Theta) \subset WF_A(Pf) \cup \mathrm{Char} (P),$$ where the characteristic set of P is defined by $$\mathrm{Char} P = \{ (x,\xi) \in T ^*(X)\setminus 0 : P_s(x,\xi) := \underset{|\alpha| = s}{\Sigma}a_\alpha \xi^\alpha=0\}.$$* Consider the differential operator $P(x,D)$ in a manifold $X$ with real analytic co-eficients,. In local coordinates, the principle symbol is $P_s = \underset{|\alpha| = s}{\Sigma}a_\alpha \xi^t$. We say that the curve $(x(t),\xi(t))$ in $T^*(\mathrm{dS}^n)$ is a bicharacteristic strip if $P_s(x(t), \xi(t))=0$ for all with initial data $(x_0,\xi_0) \in \rm{Char} P_s$ and, satisfies Hamiltonian equations defined as: $$\frac{dx}{dt} = \frac{\partial P_s(x,\xi)}{\partial \xi}, \qquad \frac{d\xi}{dt} =- \frac{\partial P_s(x,\xi)}{\partial x}.$$ Let $S$ be a closed conic set in $T^*(X)$. We say that it is *invariant under the Hamiltonian vector field of $P_s$* if $S \subseteq \rm{Char} P_s$ and for a bicharacteristic strip $(x(t),\xi(t))$ passing through $(x_0,\xi_0) \in S$, then $(x(t),\xi(t))$ must lie in $S$ for all $t$.\ The following result can be found in [@H71]. **Theorem 43** (Propagation of Singularities). *Let $P$ be a differential operator with analytic coefficients and $P_s$ be its real principle symbol. If $\Theta \in \mathcal{D}'(X)$ and $P\Theta = f$, it follows that $WF_A(\Theta) \setminus WF_A(f)$ is invariant under the Hamiltonian vector field of $P_s$ when $\partial P_s(x,\xi)/\partial \xi \neq 0$.* We say that a curve $(x(t), \xi(t))$ is a *null geodesic strip* if $[\dot{x(t)},\dot{x(t)}] = 0$ and $\xi(t)$ is the dual of $\dot{x}(t)$. The following proposition is a well known fact. **Proposition 44**. *On $\mathrm{dS}^n$, the bicharacteristics strip for $\Delta - m^2$ are exactly the null geodesic strip for $\mathrm{dS}^n$.* # Wavefront set of spherical distributions on $\mathrm{dS}^n$ {#sec:ws} In this section we will state the main theorem and its implications. The proofs of the [Theorem 45](#thm:wf){reference-type="ref" reference="thm:wf"} and [Theorem 46](#thm:wfq){reference-type="ref" reference="thm:wfq"} are postponed to the next section. **Theorem 45**. *Let $\Psi_x^{\lambda} = \underset{t \rightarrow \pi/2^-}{\mathrm{lim}}\Psi_\lambda(g.z_t,y)$ and $\widetilde{\Psi}_x^{\lambda} = \underset{t \rightarrow \pi/2^-}{\mathrm{lim}}\Psi_\lambda(g.\bar{z}_t,y)$, where $x = g \cdot e_n$. Then analytic wavefront sets of these distributions are given by $$\begin{aligned} WF_A(\Psi_x^{\lambda}) &= \{(x +v, \tau (-v_0, v_1,...,v_{n-1}) , v_0>0\}\cup \\ &\{(x+v,\tau (v_0, -v_1,...,-v_{n-1})), v_0 <0\} \cup \{(x,v): v_0 < 0 \}; \\ WF_A(\widetilde{\Psi}_x^{\lambda}) &= \{(x +v, \tau (v_0, -v_1,...,-v_{n-1}) , v_0>0\}\cup \\ &\{(x+v,\tau (-v_0, v_1,...,v_{n-1})), v_0 <0\} \cup \{(x,v): v_0 > 0 \}, \end{aligned}$$ for $v \in \mathbb{L}_{n-1}$.* In [\[fig:wavefront set\]](#fig:wavefront set){reference-type="ref" reference="fig:wavefront set"} we can see the analytic wavefront set in $T_x\mathrm{dS}^n$. **Theorem 46**. *For the distributions $\Phi_x$ and $\widetilde{\Phi}_x$, the analytic wavefront set is given by $$\begin{aligned} WF_A(\Phi_x) &= \{(x +v, \tau (-v_0, v_1,...,v_{n-1}) , v_0>0\}\cup \\ &\{(x+v,\tau (v_0, -v_1,...,-v_{n-1})), v_0 <0\} \cup \{(x,v): v_0 < 0 \}; \\ WF_A(\widetilde{\Phi}_x) &= \{(x +v, \tau (v_0, -v_1,...,-v_{n-1}) , v_0>0\}\cup \\ &\{(x+v,\tau (-v_0, v_1,...,v_{n-1})), v_0 <0\} \cup \{(x,v): v_0 > 0 \}, \end{aligned}$$ for $v \in \mathbb{L}_{n-1}$.* Using the analytic wavefront sets, we can prove that the distributions can not vanish on any non-empty open set $O$ of $\mathrm{dS}^n$. The following theorem is due to Strohmaier, Verch and, Wollenberg, see [@SVW02 Proposition 5.3]. **Theorem 47**. *Let X be a real analytic manifold and $\Theta \in \mathcal{D}'(X)$. If $WF_A(\Theta) \cap -WF_A(\Theta) = \emptyset$ then for an open region $O$ in $X$ $$\Theta|_O \Rightarrow \Theta=0 ,$$ where $-WF_A(\Theta) = \{(x,\xi) : (x, -\xi) \in WF_A(\Theta)\}$.* This theorem is not true in the case of smooth wavefront set. Consider the distribution $\Theta$ from [Example 35](#eg:e){reference-type="ref" reference="eg:e"}. The wavefront set of $\Theta$ satisfies the condition that $WF(\Theta) \cap -WF(\Theta) = \emptyset$. Obviously, $\Theta$ is not a zero distribution, however it is zero in the open region $(-\infty,0)$. Since the wavefront sets of the distributions $\Psi_x^{\lambda}$ is such that $WF_A(\Psi_x^{\lambda}) \cap -WF_A(\Psi_x^{\lambda}) = \emptyset$ for all x, which is same for $\widetilde{\Psi}_x^{\lambda}$, immediately as a corollary we obtain that, **Corollary 48**. *The distributions $\Psi_x^{\lambda}$, $\widetilde{\Psi}_x^{\lambda}$, $\Phi_x$ and, $\widetilde{\Phi}_x$ can not vanish on any open regions of $\mathrm{dS}^n$.* Let $\Theta$ be a distribution on $\mathrm{dS}^n$. Then $G$ acts on $\Theta$ by $$\begin{aligned} \pi_{-\infty}(g)\Theta (\varphi) = \Theta(\pi_{\infty}(g^{-1})\varphi) \quad \varphi \in \mathcal{D}(\mathrm{dS}^n)\end{aligned}$$ where $\pi_{\infty}(g)\varphi(x) = \varphi(g^{-1}\cdot x)$. **Definition 49**. Let $H$ be a closed subgroup of $G$. We say that a distribution $\Theta$ is $H$-invariant if $\pi_{-\infty}(h)\Theta = \Theta$ for all $h \in H = G_{e_n}$. **Definition 50**. A distribution $\Theta$ is said to be a spherical distribution if it is H-invariant eigendistribution of the Laplace-Beltrami operator $\Delta$. Let $\mathcal{D}_{\lambda}^{H}(\mathrm{dS}^n)$ be the space of spherical distributions on the de Sitter space with $\Delta (\Theta) = \lambda \Theta$. Then according to [@D09 Theorem 9.2.5] **Theorem 51**. *The dimension of $\mathcal{D}_{\lambda}^{H}(\mathrm{dS}^n)$ is 2.* We will now restate the main theorem of this paper. **Theorem 1.1 1**. The distributions $\Psi_{e_n}^{\lambda}$ and $\widetilde{\Psi}_{e_n}^{\lambda}$ are spherical distributions and span $\mathcal{D}_{m^2}^{H}(\mathrm{dS}^n)$, where $m^2 = \rho^2 - \lambda^2$ and, $\lambda \in i[0,\infty) \cup [0,\rho)$. Moreover, the following holds for a non-zero spherical distribution $\Theta$ on $\mathrm{dS}^n$: 1. $WF_A(\Theta)\subset WF_A(\Psi_{e_n}^{\lambda}) \cup WF_A(\widetilde{\Psi}_{e_n}^{\lambda})$. 2. If $WF_A(\Theta) = WF_A(\Psi_{e_n}^{\lambda})$ then there exists a nonzero constant $c$ such that $\Theta=c\Psi_{e_n}^{\lambda}$. 3. If $WF_A(\Theta)=WF_A(\widetilde{\Psi}_{e_n}^{\lambda})$ then there exists a nonzero constant $c$ such that $\Theta=c\widetilde{\Psi}_{e_n}^{\lambda}$. *Proof.* Clearly, it follows from [Theorem 23](#thm:psi){reference-type="ref" reference="thm:psi"} that $\Psi_{e_n}^{\lambda}$ and $\widetilde{\Psi}_{e_n}^{\lambda}$ are linearly independent H-invariant distribution with eigenvalue $\rho^2 - \lambda^2$ for the operator $\Delta$. Hence, $\Psi_{e_n}^{\lambda}$ and $\widetilde{\Psi}_{e_n}^{\lambda}$ span $\mathcal{D}_{m^2}^{H}(\mathrm{dS}^n)$. Now, suppose that $WF_A(\Theta) = WF_A(\Psi_{e_n}^{\lambda})$ and $\Theta = c_1\Psi_{e_n}^{\lambda} + c_2\widetilde{\Psi}_{e_n}^{\lambda}$ with $c_2 \neq 0$. We have $\widetilde{\Psi}_{e_n}^{\lambda} =\frac{1}{c_2}( \Theta - c_1\Psi_{e_n}^{\lambda})$, with $WF_A(\frac{1}{c_2}( \Theta - c_1\Psi_{e_n}^{\lambda})) \subseteq WF_A(\Psi_{e_n}^{\lambda})$. Since $WF_A(\Psi_{e_n}^{\lambda}) \cap WF_A(\widetilde{\Psi}_{e_n}^{\lambda}) = \emptyset$, we arrive at a contradiction. Thus, $\Theta = c_1\Psi_{e_n}^{\lambda}$. ◻ ***Remark** 52*. Observe that for a spherical distribution which is linear combination of $\Psi_{e_n}^{\lambda}$ and $\widetilde{\Psi}_{e_n}^{\lambda}$ with both the the coefficients not zero, the analytic wavefront set must be $WF_A(\Psi_{e_n}^\lambda) \cup WF_A(\widetilde{\Psi}_{e_n}^{\lambda})$. From the above theorem and [Theorem 23](#thm:psi){reference-type="ref" reference="thm:psi"} we obtain the following: **Corollary 53**. *The wavefront set of the distribution $\Psi_x^{\lambda} - \widetilde{\Psi}_x^{\lambda}$ is* *$$WF_A(\Psi_x^{\lambda} - \widetilde{\Psi}_x^{\lambda}) = WF_A(\Psi_x^{\lambda})\cup WF_A(\widetilde{\Psi}_x^{\lambda}).$$ Moreover,* 1. *For odd $n$ and $c_1 = (-1)^{\frac{n-1}{2}}\frac{i\Gamma(n/2)\Gamma((n-2)/2)}{\Gamma(\rho + \lambda)\Gamma(\rho-\lambda)}$, the distribution is given by: $(\Psi_x^{\lambda} - \widetilde{\Psi}_x^{\lambda})(y)=$* *$$c_1\begin{cases} 0 &\text{if $y \notin \overline{\Gamma(x)}$}\\ \left(\frac{[x,y]-1}{2}\right)^{\frac{2-n}{2}} {}_2F_1(1/2-\lambda,1/2+\lambda;\frac{4-n}{2};\frac{1-[x,y]}{2}) &\text{if $y \in \Gamma^+(x)$} \\ -\left(\frac{[x,y]-1}{2}\right)^{\frac{2-n}{2}} {}_2F_1(1/2-\lambda,1/2+\lambda;\frac{4-n}{2};\frac{1-[x,y]}{2}) &\text{if $y \in \Gamma^-(x)$}. \end{cases}$$* 2. *For even $n$ and $c_2 = (-1)^{\frac{n-2}{2}} \frac{i\pi\Gamma(n/2)}{\Gamma(1/2 + \lambda)\Gamma(1/2-\lambda)}$, the distribution is given by: $(\Psi_x^{\lambda} - \widetilde{\Psi}_x^{\lambda})(y)=$ $$c_2\begin{cases} 0 &\text{if $y \notin \overline{\Gamma(x)}$} \\ \sum_{k=0}^{\infty}\frac{(\rho+\lambda)_k(\rho - \lambda)_k}{k!(n/2 - 1 +k)!}\left(\frac{[x,y]-1}{2}\right)^k & \text{if $y \in \Gamma^+(x)$} \\[0.5em] -\sum_{k=0}^{\infty}\frac{(\rho+\lambda)_k(\rho - \lambda)_k}{k!(n/2 - 1 +k)!}\left(\frac{[x,y]-1}{2}\right)^k & \text{if $y \in \Gamma^-(x)$}. \end{cases}$$* # Proof of [Theorem 45](#thm:wf){reference-type="ref" reference="thm:wf"} {#sec:proof} We are now ready to look at the proof of [Theorem 45](#thm:wf){reference-type="ref" reference="thm:wf"} and [Theorem 46](#thm:wfq){reference-type="ref" reference="thm:wfq"}. ***Proof of [Theorem 45](#thm:wf){reference-type="ref" reference="thm:wf"}**.* First step is to find the analytic wavefront sets of $\Psi_{e_n}^{\lambda}$ and $\widetilde{\Psi}_{e_n}^{\lambda}$. Since the distributions are solutions of $P= \Delta_y - m^2$, therefore, we have that $$WF_A(\Psi_x^{\lambda}), WF_A(\widetilde{\Psi}_x^{\lambda}) \subset \mathrm{Char} P$$ where $$\mathrm{Char} P = \{(x,\xi) \in T^* (X)\setminus 0, P_n(x,\xi) =0\},$$ and $P_n$ is the principle symbol of the differential operator $P$. \(1\) Let $U = U_{e_n}$ be the local chart around $e_n$ and the co-ordinate map be the exponential map: $$y = \mathrm{Exp}_{e_n}(v) = C([v,v])e_n + S([v,v])v.$$ As we know that the singularities of $\Psi_{e_n}^{\lambda}$ and $\widetilde{\Psi}_{e_n}^{\lambda}$ lie on the boundary of the light cone of $e_n$, it is enough to calculate the wavefront set in $U$. Now consider the map $f:U \rightarrow \mathbb{R}^{}$ defined by $$f(v) = \mbox{\Large$\frac{1+C[v,v]}{2}$}.$$ The distribution $\Psi_{e_n}^{\lambda}$ is the distribution ${}_2F_1(f(v) - i0)$ in the open set $U|_{v_0>0}$ and is ${}_2F_1(f(v) + i0)$ in $U|_{v_0<0}$. The differential operator $P$ in $U$ is given by $$\Delta - m^2 = -\frac{\partial^2}{\partial v_0^2} + \sum_{i=1}^{n-1} \frac{\partial^2}{\partial v_i^2} -m^2.$$ Moreover, the principle symbol of the differential operator $P$ is $P_n = \xi_0^2 - \sum_{i=1}^{n-1} \xi_i^2$. Therefore the analytic wavefront set lies in the set $\{(v,\xi) : v \in \mathbb{L}_{n-1}, \xi \in \mathbb{L}_{n-1}^*\}$, where $\mathbb{L}_{n-1}$ is the light cone in $U \subset \mathbb{R}^{1,n-1}$ and $\mathbb{L}_{n-1}^*$ is the dual of the light cone. \(2\) The tangent map of $f$ is $$df_v = \frac{-S[v,v]}{4} \begin{bmatrix}-2v_0 &2v_1 & ...& 2v_n \end{bmatrix}, \quad v \in U.$$ For $\eta \in \mathbb{R}^{}$,$v \in U$, if ${}^tdf_v(\eta) = 0$ then either $\eta = 0$ or $S[v,v]=0$. Now, using the relation that $S(z^2) = \sin{z}/z$ when $z \neq 0$, we obtain that $S[v,v] =0$ when $[v,v] = m^2\pi^2$ for $m \in \mathbb{Z} \setminus 0$. Such a $v$ does not belong to the set $U$. Thus the set of normals of $f$ is $N_f = \{(f(v), 0): v \in U\}$. \(3\) In this step we will calculate the singularities when $v \neq 0$. We will write $v = (v_0, \mathbf{v})$. Consider the distributions ${}_2F_1(x+i0)$ and ${}_2F_1(x -i0)$. Then from [Example 41](#eg:hgf){reference-type="ref" reference="eg:hgf"}, it follows $N_f \cap WF_A({}_2F_1(x - i0)) = \emptyset$ and $N_f \cap WF_A({}_2F_1(x + i0)) = \emptyset$. As a result of [Theorem 36](#cw){reference-type="ref" reference="cw"}, the distribution $\Psi_{e_n}^{\lambda}$ is the pullback of the distributions ${}_2F_1(x -i0)$ and ${}_2F_1(x+i0)$ under the restriction of $f$ at $U|_{v_0 > 0}$ and $U|_{v_0< 0}$, respectively. Consequently, in $U_{v_0 >0}$ $$WF_A({}_2F_1(f(v) - i0)) \subseteq \{(v, {}^tdf_v(\Phi)): v_0 >0, (f(v), \Phi) \in WF_A({}_2F_1(x - i0)));$$ and in $U|{v_0 <0}$, $$WF_A({}_2F_1(f(v) + i0)) \subseteq \{(v, {}^tdf_v(\Phi)): v_0<0, (f(v), \Phi) \in WF_A({}_2F_1(x + i0))\}.$$ That is , $$WF_A({}_2F_1(f(v) - i0)) = \{ (v, \xi) : [v,v]=0, \xi = \tau(-v_0,\mathbf{v}), \tau>0, v_0 >0\};$$ and, $$WF_A({}_2F_1(f(v) + i0)) = \{ (v, \xi) :[v,v] =0, \xi = \tau(v_0,-\textbf{v}), \tau>0, v_0 <0\}.$$ We get the equality since the analytic wavefront set cannot be empty as the points $[v,v] =0$ lies in analytic singular support of ${}_2F_1(f(v) - i0)$ and ${}_2F_1(f(v) + i0)$ in their respective domains. \(4\) Now that we have calculated wavefront set of $\Psi_{e_n}^{\lambda}$ when $v \neq 0$. The next step is to calculate at $v=0$. For that we will use Propagation of Singularity theorem, which says that analytic wavefront set is invariant under Hamiltonian $P_n$ when $\frac{\partial P_n}{\partial \xi} \neq 0$. We have that $\frac{\partial P_n}{\partial \xi} = 0$ only if $\bf{\xi} =0$. Hence we can apply [Theorem 43](#PS){reference-type="ref" reference="PS"}. Now the Hamiltonian equations in the local coordinates are $$\frac{\partial v}{\partial t} = \frac{\partial P_n}{\partial \xi}, \qquad \frac{\partial \xi}{\partial t} = -\frac{\partial P_n}{\partial v}.$$ That is, for $\xi \in \mathbb{L}_{n-1}^*$ $$\begin{aligned} \dot{v_0} = 2\xi_0 &, \qquad \dot{\xi_0}=0 \\ \dot{v_i} = -2\xi_i&, \qquad \dot{\xi_i} = 0, \quad \text{for}\; i=1,...,n.\end{aligned}$$ which gives us, $$v_0(t) = 2\xi_0t, \; v_i(t) = -2\xi_it; \quad \xi(t) = const.$$ with $v(0) = 0$. That is, $v(t)$ lies on the light cone of $0$. From what we have calculated in step (3), choose $\tau = 1$, then $\xi_0 = -v_0 < 0$, $\xi_i = v_i$ when $v_0 >0$ and $\xi_0 = v_0<0$ and $\xi_i = -v_i$ when $v_0<0$. This says that $\xi_0 < 0$ and thus the null geodesic $v(t)$ is the past directed curve. At $t=0$, $(0,(\dot{v_0}, -\dot{\textbf{v}}))$ must be in the wavefront set for all null geodesics $v(t)$ satisfying the Hamiltonian equations and fitting in what we have calculated in step (3). Thus so far what we have calculated is $$\begin{aligned} &WF_A((\mathrm{Exp}_{e_n}^{-1})^{*}\Psi_{e_n}^{\lambda}) = WF_A \left(\underset{t \rightarrow \pi/2}{\mathrm{lim}}\, {}_2F_1\left(\frac{1+[z_t, \mathrm{Exp}_{e_n}(v)]}{2} \right)\right) \\[1em] &= \{(0,\tau v): v_0 <0\} \cup \{(v, \tau(-v_0,\textbf{v}), v_0>0\} ) \cup \{(v,\tau(v_0,-\textbf{v}), v_0 < 0 \};\end{aligned}$$ for $v \in \mathbb{L}_{n-1}$ and $\tau > 0$. \(5\) Now that the wavefront set has been calculated in local coordinates, we pull the wavefront set back to the de Sitter space. If $v \in \mathbb{L}_{n-1}$, then $y= C[v,v]e_n + S[v,v]v = e_n + v$ and $[y-e_n,y-e_n]=0$ which implies that $y$ lies on the light cone of $e_n$. We now conclude that the wavefront of $\Psi_{e_n}^{\lambda}$ is given by $$WF_A(\Psi_{e_n}^{\lambda}) = \mathrm{(Exp_{e_n})}^*WF_A((\mathrm{Exp_{e_n}}^{-1})^*\Psi_{e_n}^{\lambda}).$$ That is, for all $v \in \mathbb{L}_{n-1}$ and $\tau >0$ $$\begin{aligned} WF_A(\Psi_{e_n}^{\lambda}) &= \{(e_n,v): v_0>0 \} \cup \{(e_n +v, \tau (-v_0, v_1,...,v_{n-1}) , v_0>0\}\cup \\ &\{(e_n+v,\tau (v_0, -v_1,...,-v_{n-1})), v_0 <0\}.\end{aligned}$$ (6) Lastly, consider $\Psi_x^{\lambda} = \underset{t \rightarrow \pi/2^-}{\mathrm{lim}}\Psi_\lambda(g\cdot z_t,y)$. For $x = g \cdot e_n$, define a map $l_g : \mathrm{dS}^n\rightarrow \mathrm{dS}^n$ by $l_g(y) = g^{-1}\cdot y$. Since $l_g$ is an analytic diffeomorphism, we have that $dl_g$ is an isomorphism of tangent spaces. Therefore, $N_{l_g} = \{(y, 0): y \in \mathrm{dS}^n\}$ and the pull back of the distribution $\Psi_{e_n}^{\lambda}$ under the map $l_g$ is $\Psi_{x}^{\lambda}$. Thus, $WF_A(\Psi_x^{\lambda}) = l_g^* WF_A(\Psi_{e_n}^{\lambda})$. That is, $(y,\xi) \in WF_A(\Psi_x^{\lambda})$ if $(l_g(y), {}^tdl_{g^{-1}}\xi) \in WF_A(\Psi_{e_n}^{\lambda})$. This implies that $y = x + v$ for $v \in \mathbb{L}_{n-1}$ as the $G$ acts transitively on light cone and $({}^tdl_{g^{-1}}\xi) = g\cdot \xi$ is also on the dual light cone, as a result $$\begin{aligned} WF_A(\Psi_x^{\lambda}) &= \{(x,v): v_0 < 0 \} \cup \{(x +v, \tau (-v_0, v_1,...,v_{n-1}) , v_0>0\}\cup \\ &\{(x+v,\tau (v_0, -v_1,...,-v_{n-1})), v_0 <0\}.\end{aligned}$$ for $v \in \mathbb{L}_{n-1}$. \(7\) Finally, in local coordinates $\widetilde{\Psi}_{e_n}^{\lambda}$ is the distribution ${}_2F_1(f(v) + i0)$ in the open set $U|_{v_0>0}$ and as ${}_2F_1(f(v) - i0)$ in $U|_{v_0<0}$. Following all the steps above we obtain for $v \in \mathbb{L}_{n-1}$, $$\begin{aligned} WF_A(\widetilde{\Psi}_{e_n}^{\lambda}) &= \{(e_n,v): v_0 > 0 \} \cup \{(e_n +v, \tau (v_0, -v_1,...,-v_{n-1}) , v_0>0\}\cup \\ &\{(e_n+v,\tau (-v_0, v_1,...,v_{n-1})), v_0 <0\},\end{aligned}$$ and $$\begin{aligned} WF_A(\widetilde{\Psi}_x^{\lambda}) &= \{(x,v): v_0 > 0 \} \cup \{(x +v, \tau (v_0, -v_1,...,-v_{n-1}) , v_0>0\}\cup \\ &\{(x+v,\tau (-v_0, v_1,...,v_{n-1})), v_0 <0\}. \end{aligned}$$ Thus we have proved the theorem. ◻ ***Proof of[Theorem 46](#thm:wfq){reference-type="ref" reference="thm:wfq"}**.* The proof of [Theorem 46](#thm:wfq){reference-type="ref" reference="thm:wfq"} is similar to the above proof. In the local co-ordinates $(U_{e_n},v)$ around $e_n$, consider the map $$f: U_{e_n} \rightarrow \mathbb{R}^{}, \quad v \mapsto \Big(\mbox{\Large$\frac{1 -C[v,v]}{2}$}\Big)^{\frac{2-n}{2}},$$ where, $N_f = \{(g(v),0):v\in U_{e_n}\}$. The distribution $\Phi_{e_n}$ is the pull back of $(x + i0)^{\frac{2-n}{2}}$ and $(x - i0)^{\frac{2-n}{2}}$ under the restriction of $f$ at $U_{e_n}|{v_0 >0}$ and $U_{e_n}|{v_0 < 0}$. We follow the similar steps as (1)-(7) of the proof of [Theorem 45](#thm:wf){reference-type="ref" reference="thm:wf"} and obtain that $WF_A(\Phi_x) = WF_A(\Psi_x^{\lambda})$ and $WF_A(\widetilde{\Phi}_x) = WF_A(\widetilde{\Psi}_x^{\lambda})$, thus proving the [Theorem 46](#thm:wfq){reference-type="ref" reference="thm:wfq"}. ◻ # Boundary value of Hypergeometric function {#sec:hgf} For simplicity we will write ${}_2F_1(a,b;c,z) = {}_2F_1(z)$. In this section we will show that ${}_2F_1(x+i0) := \underset{y \rightarrow 0}{\mathrm{lim}}{}_2F_1(x+iy)$ for $y>0$ and ${}_2F_1(x-i0) := \underset{y \rightarrow 0}{\mathrm{lim}}{}_2F_1(x-iy)$ for $y>0$, are distributions for $a = \rho + \lambda$, $b = \rho - \lambda$ and $c= n/2$. It is a fact that ${}_2F_1( z )$ has a branch cut on $[1,\infty)$. Hence, the convergence for $x < 1$ is uniform on compact sets. The case when $x > 1$ and the growth near $z=1$ will determine whether it will be a distribution or not. **Theorem 54**. *The limit $\underset{y \rightarrow 0}{\mathrm{lim}} \; {}_2F_1(\rho + \lambda, \rho - \lambda, n/2, x \pm iy)$ for $y >0$ exists in the sense of distributions where for $\rm{Re}(z) > 1$ the limit converges uniformly on compact sets. For $1<x<2$, if $n$ is odd $$\label{eq:2} \begin{split} &{}_2F_1( x\pm i0 ) = \frac{\Gamma(n/2)\Gamma((2-n)/2)}{\Gamma(1/2 + \lambda)\Gamma(1/2-\lambda)} {}_2F_1(\rho + \lambda,\rho-\lambda; \frac{n}{2};1-x) \\ &+ e^{\mp i\pi(\frac{2-n}{2})}(x-1)^{\frac{2-n}{2}} \mbox{\Large$\frac{\Gamma(n/2)\Gamma((n-2)/2)}{\Gamma(\rho + \lambda)\Gamma(\rho-\lambda)}$} {}_2F_1(1/2- \lambda,1/2+\lambda; \frac{4-n}{2};1-x). \end{split}$$ and if $n$ is even ${}_2F_1( x \pm i0 ) =$* *$$\label{eq:3} \begin{split} (-1)^{\frac{2-n}{2}}& \Bigg( \mbox{\Large$\frac{\Gamma(n/2)}{\Gamma(\rho + \lambda)\Gamma(\rho-\lambda)}$}\sum_{k=0}^{n/2-2}\mbox{\Large$\frac{(n/2-k-2)!(1/2+\lambda)_k(1/2-\lambda)_k}{k!}$}(x-1)^{k+1-\frac{n}{2} }\\ &+ \mbox{\Large$\frac{\Gamma(n/2)}{\Gamma(1/2 + \lambda)\Gamma(1/2-\lambda)}$}\sum_{k=0}^{\infty}\mbox{\Large$\frac{(\rho+\lambda)_k(\rho-\lambda)_k}{k!(n/2 - 1 + k)!}$}[\psi(k+1) + \psi(n/2 + k) \\ & - \psi(\rho+\lambda +k) - \psi(\rho-\lambda +k) - \ln (x-1) \pm i\pi](-1)^k(x-1)^{k} \Bigg), \end{split}$$* *where $\psi(z) = \Gamma'(z)/\Gamma(z)$ Furthermore, the behaviour of the hypergeometric function near $z=1$ as distributions is given as follows: for $n = 2$, $${}_2F_1(z) \approx \mbox{\Large$\frac{1}{\Gamma(\rho+\lambda)\Gamma(\rho-\lambda)}$} (-\ln{(1-z}))$$ and for $n \geq 3$, $${}_2F_1( z ) \approx \frac{\Gamma(n/2)\Gamma((n-2)/2)}{\Gamma(\rho+\lambda)\Gamma(\rho-\lambda)}(1-z)^{\frac{2-n}{2}}. \label{4}$$* *Proof.* Let $n \geq 2$. Suppose that n is odd. Then $c-a-b = \frac{2-n}{2}$ is not an integer. Therefore, for $|z-1|<1$ and $|\mathrm{arg}(1-z)|<\pi$ we can use the following transformation $$\label{eq: 1} \begin{split} &{}_2F_1( z ) = \mbox{\Large$\frac{\Gamma(n/2)\Gamma((2-n)/2)}{\Gamma(1/2 + \lambda)\Gamma(1/2-\lambda)}$} {}_2F_1(\rho + \lambda,\rho-\lambda; \frac{n}{2};1-z) \\ &+ (1-z)^{\frac{2-n}{2}} \mbox{\Large$\frac{\Gamma(n/2)\Gamma((n-2)/2)}{\Gamma(\rho + \lambda)\Gamma(\rho-\lambda)}$} {}_2F_1(1/2- \lambda,1/2+\lambda; \frac{4-n}{2};1-z). \end{split}$$ Suppose that $1<x<2$ then, $$\label{eq:2} \begin{split} &{}_2F_1( x\pm i0 ) = \mbox{\Large$\frac{\Gamma(n/2)\Gamma((2-n)/2)}{\Gamma(1/2 + \lambda)\Gamma(1/2-\lambda)}$} {}_2F_1(\rho + \lambda,\rho-\lambda; \frac{n}{2};1-x) \\ &+ e^{\mp i\pi(\frac{2-n}{2})}(x-1)^{\frac{2-n}{2}} \mbox{\Large$\frac{\Gamma(n/2)\Gamma((n-2)/2)}{\Gamma(\rho + \lambda)\Gamma(\rho-\lambda)}$} {}_2F_1(1/2- \lambda,1/2+\lambda; \frac{4-n}{2};1-x). \end{split}$$ For $x\geq 2$ we can use linear transformations of hypergeometric functions to extend ${}_2F_1( x\pm i0 )$ analytically. If $n$ is even, we obtain [\[eq:3\]](#eq:3){reference-type="ref" reference="eq:3"} for $1 < x <2$ from [@GS64 Eq 9.7.5, 9.7.6]. Similarly, we can extend ${}_2F_1( x \pm i0 )$ for $x > 2$ using the formulae from [@GS64 Sec 9.7] Now let us calculate the behaviour of the hypergeometric function near $x=1$. Let $n\geq 3$. We have that for $\mathrm{Re}(c-a-b) = 1-n/2 <0$ and $x <1$, $$\underset{x \rightarrow 1^-}{\mathrm{lim}} \frac{{}_2F_1( x )}{(1-x)^{\frac{2-n}{2}}} = \frac{\Gamma(n/2)\Gamma((n-2)/2)}{\Gamma(\rho + \lambda)\Gamma(\rho-\lambda)}.\label{eq:4}$$ From [\[eq:2\]](#eq:2){reference-type="ref" reference="eq:2"} and [\[eq:3\]](#eq:3){reference-type="ref" reference="eq:3"} we obtain that $$\underset{x \rightarrow 1^+}{\mathrm{lim}} \frac{{}_2F_1( x\pm i0 )}{(x-1)^{c-a-b}} = e^{\mp i \pi (\frac{2-n}{2})}\frac{\Gamma(n/2)\Gamma((n-2)/2)}{\Gamma(\rho +\lambda)\Gamma(\rho-\lambda)}.$$ From this we can say that around 1 (see [9](#appendixa){reference-type="ref" reference="appendixa"}), $${}_2F_1( z ) \approx \frac{\Gamma(n/2)\Gamma((n-2)/2)}{\Gamma(\rho +\lambda)\Gamma(\rho-\lambda)}{(1-z)^{\frac{2-n}{2}}}. \label{4}$$ The growth of ${}_2F_1( z )$ near $z=1$ is $$|{}_2F_1( z )| \approx \Big|\mbox{\Large$\frac{\Gamma(n/2)\Gamma((n-2)/2)}{\Gamma(\rho + \lambda)\Gamma(\rho - \lambda)}$}\Big|(|1-z|^{\frac{2-n}{2}}) \leq \rm{const.}\; |y|^{\frac{2-n}{2}}.$$ Hence, it follows from [@H63 Theorem 3.1.11] that the limit converges to a distribution. If $n=2$, then $c = a+ b$ and $$\underset{x \rightarrow 1^-}{\mathrm{lim}} \frac{{}_2F_1( x )}{- \ln (1-x)} = \frac{1}{\Gamma(\rho + \lambda)\Gamma(\rho-\lambda)}.\label{eq:5}$$ For $n=2$, the first summation in [\[eq:3\]](#eq:3){reference-type="ref" reference="eq:3"} does not appear. Thus, we obtain that $$\underset{x \rightarrow 1^-}{\mathrm{lim}} \frac{{}_2F_1( x \pm i0 )}{- \ln (x-1) \pm i \pi} = \frac{1}{\Gamma(\rho + \lambda)\Gamma(\rho-\lambda)}.\label{eq:6}$$ Therefore, around $z=1$ $${}_2F_1( z ) \approx \frac{1}{\Gamma(\rho + \lambda)\Gamma(\rho-\lambda)} (- \ln (1-z)).$$ Since logarithm is an integrable function on compact sets we have that for $n=2$, ${}_2F_1( z )$ is a distribution. ◻ # Distributions: $(x+i0)^\frac{2-n}{2}$ and $\log (x+i0)$ {#appendixa} Here, we will recall the distributions $(x\pm i0)^{\frac{2-n}{2}}$ and $\ln (x \pm i0)$ (see [@GS64]). First for $n$ an odd number, we look at the distributions $x^{\frac{2-n}{2}}_+$ and $x^{\frac{2-n}{2}}_-$. Let $\varphi \in C_c^\infty(\mathbb{R}^{})$. We will look at the case when $n$ is odd dimension. For $n=3$ $$(x^{-\frac{1}{2}}_+, \varphi) = \int_0^\infty x^{-\frac{1}{2}}\varphi(x) dx,$$ is the regular distribution. However, for $n \geq 5$, $m= (n-5)/2$ we have $$(x^{\frac{2-n}{2}}_+, \varphi) = \int_0^\infty x^{\frac{2-n}{2}}\Big[\varphi(x) - \varphi(0)-x\varphi'(0)-...-\mbox{\Large$\frac{x^m}{(m)!}$}\varphi^m(0) \Big] dx.$$ The distribution $x^{\frac{2-n}{2}}_-$ is defined as follows: $$(x^{\frac{2-n}{2}}_-, \varphi(x)) = (x^{\frac{2-n}{2}}_+, \varphi(-x)).$$ Now we will look at the case when $n$ is even dimension: For $k = (n-2)/2$ and $k$ is even, $$\begin{aligned} (x^{-k}, \varphi) &= \int_0^\infty x^{-k} \Big( \varphi(x) + \varphi(-x) \\ -2\Big[\varphi(0) &+ \frac{x^2}{2!}\varphi''(0)+...+\frac{x^{k-2}}{(k-2)!}\varphi^{k-2}(0) \Big]\Big) dx. \end{aligned}$$ For $k = (n-2)/2$ and $k$ an odd number: $$\label{eq:nEvenkOdd} \begin{aligned} (x^{-k}, \varphi) &= \int_0^\infty x^{-k} \Big( \varphi(x) - \varphi(-x) \\ -2\Big[x\varphi'(0) &+ \frac{x^3}{3!}\varphi'''(0)+...+\frac{x^{k-2}}{(k-2)!}\varphi^{k-2}(0) \Big]\Big) dx. \end{aligned}$$ Let us consider the distributions given as follows: $$(x \pm i0)^{\frac{2-n}{2}} = \underset{y \rightarrow 0^+}{\mathrm{lim}} (x \pm iy)^{\frac{2-n}{2}}.$$ When $n$ is odd, $$\begin{aligned} \label{eq:AppnOdd} (x + i0)^{\frac{2-n}{2}} &= x^{\frac{2-n}{2}}_+ + e^{i \pi \frac{2-n}{2}} x^{\frac{2-n}{2}}_-,\\ (x - i0)^{\frac{2-n}{2}} &= x^{\frac{2-n}{2}}_+ + e^{-i \pi \frac{2-n}{2}} x^{\frac{2-n}{2}}_-.\end{aligned}$$ When $n$ is even, $k = (n-2)/2$ we have $$\begin{aligned} \label{eq:AppnEven} (x + i0)^{-k} &= x^{-k} - \mbox{\Large$\frac{i \pi (-1)^{k-1}}{(k-1)!}$}\delta^{k-1}(x),\\ (x - i0)^{-k} &= x^{-k} + \mbox{\Large$\frac{i \pi (-1)^{k-1}}{(k-1)!}$}\delta^{k-1}(x).\end{aligned}$$ Finally we have the distribution $$\ln (x \pm i0) = \underset{y \rightarrow 0}{\mathrm{lim}} \ln (x \pm iy),$$ where $$\ln (x \pm i0) = \begin{cases} \ln |x| \pm i\pi & \mathrm{for} \; x< 0,\\ \ln x & \mathrm{for}\; x > 0. \end{cases}$$ AAAA D. 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--- abstract: | We deal, for the classical $N$-body problem, with the existence of action minimizing half entire expansive solutions with prescribed asymptotic direction and initial configuration of the bodies. We tackle the cases of hyperbolic, hyperbolic-parabolic and parabolic arcs in a unitary manner. Our approach is based on the minimization of a renormalized Lagrangian action, on a suitable functional space. With this new strategy, we are able to confirm the already-known results of the existence of both hyperbolic and parabolic solutions, and we prove for the first time the existence of hyperbolic-parabolic solutions for any prescribed asymptotic expansion in a suitable class. Associated with each element of this class we find a viscosity solution of the Hamilton-Jacobi equation as a linear correction of the value function. Besides, we also manage to give a better description of the growth of parabolic and hyperbolic-parabolic solutions. author: - Davide Polimeni and Susanna Terracini title: "**On the existence of minimal expansive solutions to the $N$-body problem**" --- # Introduction and main results {#sec_intro} In this paper, we deal with half entire solutions to the $N$-body problem of Celestial Mechanics in the Euclidean space $\mathbb{R}^d$ of hyperbolic, parabolic or mixed hyperbolic-parabolic type. We first investigate the existence of trajectories to the gravitational $N$-body problem having prescribed growth at infinity. This classical line of research has recently been re-energized by the injection of new methods of analysis, of perturbative, variational, geometric and/or analytic functional nature. Indeed, in addition to the classical literature on the subject [@Alekseev_FinalMotions3Body; @Chazy; @MarchalSaari_FinalEvolution; @Saari_ExpandingGravitational; @Saari_ManifoldStructure], we quote the recent results about existence of hyperbolic solutions [@MR4121133; @MR3868425; @MR4600219; @MadernaVenturelli_GloballyMinimizingParabolic], parabolic ones [@BTV2; @BTV1; @BDFT; @LuzMaderna_FreeTimeMinimizers; @MadernaVenturelli_HyperbolicMotions; @SaariHulkower_TotalCollapse] and hyperbolic-parabolic ones [@Burgos_PartiallyHyperbolic], without neglecting those ending in an oscillatory manner [@MR3455155; @MR4265664; @paradela2022oscillatory] and references therein. To start with, let us consider $N$ point masses $m_1,...,m_N > 0$ moving under the action of the mutual attraction, with the inverse-square law of universal gravitation. We denote the components of the configuration vector $x = (r_1,...,r_N) \in \mathbb{R}^{dN}$ of the positions of the bodies and by $|r_i - r_j|$ the Euclidean distance between two bodies $i$ and $j$. Newton's equation of motion for the $i$-th body of the $N$-body problem reads as $$m_i\ddot{r_i} = -\sum_{j=1,...,N,\ j\neq i}^{N} m_im_j \frac{r_i - r_j}{|r_i - r_j|^3}.$$ Since these equations are invariant by translation, we can fix the origin of our inertial frame at the center of mass of the system. We can thus define the configuration space of the system as $$\mathcal{X} = \biggl\{x=(r_1,...,r_N)\in\mathbb{R}^{dN},\ \sum_{i=1}^{N} m_i r_i = 0\biggl\}$$ and denote by $\Omega = \{x\in \mathcal{X} \ | \ r_i\neq r_j\ \forall \ i \neq j\} \subset \mathcal{X}$ the set of configurations without collisions, which is open and dense in $\mathcal{X}$, and with $\Delta$ its complement, that is the collision set. Now we can write the equations of motion as $$\label{eq_newton} \mathcal{M}\ddot{x} = \nabla U(x),$$ where $\mathcal{M}=\text{diag}(m_1 I_d,...,m_N I_d)$ is the matrix of the masses and the function $U:\Omega \rightarrow \mathbb{R} \cup \{+\infty\}$ is the Newtonian potential $$\label{eq:potential} U(x) = \sum_{i<j} \frac{m_i m_j}{|r_i-r_j|}.$$ Newton's equations define an analytic local flow on $\Omega \times \mathbb{R}^{dN}$ with a first integral given by the mechanical energy: $$h = \frac{1}{2}\|\dot{x}\|_\mathcal{M}^2 - U(x).$$ We will use $\|\cdot\|_\mathcal{M}$ to denote the norm induced by the mass scalar product $$\langle x,y\rangle_\mathcal{M} = \sum_{i=1}^{N} m_i \langle r_i,s_i\rangle,\qquad \text{for any }x=(r_1,...,r_N),\ y=(s_1,...,s_N)\in\mathcal{X},$$ where, with a little abuse, $\langle\cdot,\cdot\rangle$ denoes the standard scalar product in $\mathbb{R}^d$ and also in $\mathcal{X}$. In this paper we will be concerned with the class of expansive motions, which is defined in the following way. **Definition 1**. A motion $x:[0,+\infty) \rightarrow \Omega$ is said to be expansive when all the mutual distances diverge, that is, when $|r_i(t)-r_j(t)| \rightarrow +\infty$ as $t\rightarrow+\infty$ for all $i < j$. Equivalently, the motion is expansive if $U(x(t)) \rightarrow 0$ as $t\rightarrow+\infty$. From the conservation of the energy, we observe that, since $U(x(t)) \rightarrow 0$ implies $\|\dot{x}(t)\|_\mathcal{M}^2\rightarrow 2h$ as $t\rightarrow+\infty$, expansive motions can only occur at nonnegative energies. For a given motion, we introduce the minimum and the maximum separation between the bodies at time $t$ as the two functions $$r(t) = \min_{i<j} |r_i(t) - r_j(t)| \quad \text{and} \quad R(t) = \max_{i<j} |r_i(t) - r_j(t)|,$$ where we write $|\cdot|$ to denote the standard Euclidean norm in $\mathbb{R}^d$. The next fundamental theorems give us a more accurate description of the system's expansion. **Theorem 2** (Pollard, 1967 [@Pollard_BehaviorOfGravitationalSystems]). *Let $x$ be a motion defined for all $t>t_0$. If $r$ is bounded away from zero, then we have that $R=O(t)$ as $t\rightarrow+\infty$. In addition, $R(t)/t\rightarrow+\infty$ if and only if $r(t)\rightarrow0$.* **Theorem 3** (Marchal-Saari, 1976 [@MarchalSaari_FinalEvolution]). *Let $x$ be a motion defined for all $t > t_0$. Then either $R(t)/t \rightarrow +\infty$ and $r(t) \rightarrow 0$, or there is a configuration $a \in \mathcal{X}$ such that $x(t) = at + O(t^{2/3})$. In particular, for superhyperbolic motions (i.e. motions such that $\limsup_{t \rightarrow +\infty} R(t)/t = +\infty$) the quotient $R(t)/t$ diverges.* **Theorem 4** (Marchal-Saari, 1976 [@MarchalSaari_FinalEvolution]). *Suppose that $x(t) = at + O(t^{2/3})$ for some $a \in \mathcal{X}$ and that the motion is expansive. Then, for each pair $i < j$ such that $a_i = a_j$, we have $|r_i(t)-r_j(t)| \approx\footnote{ Given positive functions $f$ and $g$, we write $f \approx g$ when there exist two positive constants $\alpha$ and $\beta$ such that $\alpha \leq \frac{f}{g} \leq \beta$. } t^{2/3}$.* Next, let us recall the well known Chazy classification of the expansive motions for the three-body problem (cfr. [@Chazy]), based on the asymptotic order of growth of the distances between the bodies. This prevents an expansive motion to be superhyperbolic, so we can assume that it is of the form $x(t) = at + O(t^{2/3})$ for some limit $a \in \mathcal{X}$. Assuming that the center of mass of the system is at rest, Chazy classified these motions as follows: - *Hyperbolic*: $a \in \Omega$ and $|r_i(t)-r_j(t)| \approx t$ for all $i < j$; - *Hyperbolic-parabolic*: $a \in \Delta$ but $a \neq 0$; - *Completely parabolic*: $a = 0$ and $|r_i(t)-r_j(t)| \approx t^{2/3}$ for all $i < j$. The following definition is in order. **Definition 5**. A motion $x(t)$ is said to have limit shape when there is a time dependent similarity $S(t)$ of the space $\mathbb{R}^d$ such that $S(t)x(t)$ converges to some configuration $a \neq 0$. In our case, there is a diagonal action of $S(t)$, which means that $S(t)x=(S(t)r_1,...,S(t)r_N)$ for $x=(r_1,...,r_N)\in \mathcal{X}$. In particular, for the case of (half) hyperbolic motions, we can say that the limit shape of such a motion is its asymptotic velocity $a = \lim_{t\rightarrow +\infty} \frac{x(t)}{t}$. Similarily, (half) parabolic motions also possess a limit shape, which is now bound to be a central configuration, that is, a critical point of the potential $U$ constrained on the intertia ellipsoid $\mathcal E=\{x\in\mathcal X\,:\, \|x\|_\mathcal{M}^2=1\}$.\ In this paper, we are going to tackle the existence of half entire expansive solutions for the Newtonian $N$-body problem from a unitary perspective by a global variational approach, using a suitable renormalized action functional, as the Lagrangian is not expected to be integrable on the half line. In particular, referring to Chazy's classification, we will show a proof of existence of motions for each one of the previous three classes of motions. As a first step, we shall revisit recent works by E. Maderna and A. Venturelli about the existence of half hyperbolic and parabolic trajectories from this new angle. **Theorem 6** (Maderna and Venturelli 2020, [@MadernaVenturelli_HyperbolicMotions]). *Given $d\in\mathbb{N}$, $d\geq2$, for the Newtonian $N$-body problem in $\mathbb{R}^{d}$ there is a hyperbolic motion $x:[1,+\infty)\rightarrow\mathcal{X}$ of the form $$x(t) = at- \log (t) \nabla U(a) + o(1)\quad\text{as }t\rightarrow+\infty,$$ for any initial configuration $x^0=x(1)\in\mathcal{X}$ and for any collisionless configuration $a\in\Omega$.* As far as the parabolic case is concerned, in addition to providing an alternative proof, we will be able to extend the result of Maderna and Venturelli [@MadernaVenturelli_GloballyMinimizingParabolic] by improving the estimate of the remainder as follows. **Theorem 7**. *Given $d\in\mathbb{N}$, $d\geq2$, for the Newtonian $N$-body problem in $\mathbb{R}^{d}$ there is a parabolic solution $x:[1,+\infty)\rightarrow\mathcal{X}$ of the form $$\label{eq:parabolic} x(t) = \beta b_m t^{2/3}+o(t^{1/3^+})\quad\text{as }t\rightarrow+\infty,$$ for any initial configuration $x^0=x(1)\in\mathcal{X}$, for any minimal normalized central configuration $b_m$ and for $\beta = \sqrt[3]{\frac{9}{2}U(b_m)}$.* Here, a minimal central configuration is a minimizer of the potential $U$ constrained to the intertia ellipsoid $\mathcal E=\{x\in\mathcal X\,:\, \|x\|_\mathcal{M}^2=1\}$. As said, the existence of hyperbolic and parabolic solutions for the Newtonian $N$-body problem has already been proved by Maderna and Venturelli in 2020 and 2009, respectively. In [@MadernaVenturelli_HyperbolicMotions], the authors proved the existence of hyperbolic motions for any prescribed limit shape, any initial configuration of the bodies and any positive value of the energy. These solutions, whose actions are infinite, were found as the limits of locally converging subsequences in families of minimizing motions, where the existence of the approximate solutions are minimal geodesics of the Maupertuis'-Jacobi metric. More specifically, these solutions were obtained as the limits of solutions of sequences of approximating two-point boundary value problems. To exclude collisions, both proofs in [@MadernaVenturelli_HyperbolicMotions] and [@MadernaVenturelli_GloballyMinimizingParabolic] invoke Marchal's Principle ensuring the absence of collisions for action-minimizing paths (Theorem [Theorem 10](#thm_marchal){reference-type="ref" reference="thm_marchal"}). There tarjectories are characteristic curves of a global viscosity solutions for the Hamilton-Jacobi equation $H(x, \nabla u) = h$. In such, these solutions are fixed points of the associated Lax-Oleinik semigroup. In [@MadernaVenturelli_GloballyMinimizingParabolic], for any starting configuration they proved the existence of parabolic arcs asymptotic to any prescribed normalized minimal central configuration. Compared to Maderna and Venturelli's articles, in this paper we show alternative and simpler proofs for the existence of hyperbolic and parabolic solutions in a unitary framework, which is based on a straightforward application of the Direct Method of the Calculus of Variations to minimize the renormalized Lagrangian actions associated to the problem. This approach has the advantage of allow us to complement the existence of parabolic arcs with their (almost exact) expansion [\[eq:parabolic\]](#eq:parabolic){reference-type="eqref" reference="eq:parabolic"}. Finally, after proving Theorems [Theorem 6](#thm_hyperbolic){reference-type="ref" reference="thm_hyperbolic"} and [Theorem 7](#thm_parabolic){reference-type="ref" reference="thm_parabolic"}, we will extend our approach to similarly prove the existence of hyperbolic-parabolic solutions for the $N$-body problem. In order to state our main result we need to introduce the *$a$-cluster partition* associated with a collision asymptotic velocity $a\in\Delta\setminus\{0\}$, where clusters are the equivalence classes of the relation $i\sim j \Longleftrightarrow a_i-a_j=0$. Given a cluster $K$, we consider the associated partial potential $U_K$, where the sum in [\[eq:potential\]](#eq:potential){reference-type="eqref" reference="eq:potential"} is restricted to the cluster $K$. The $a$-clustered potential $U_a$ is the sum of all the cluster potentials of the partition. Now we can state our main theorem: **Theorem 8**. *Given $d\in\mathbb{N}$, $d\geq2$, for the Newtonian $N$-body problem in $\mathbb{R}^{d}$ there is a hyperbolic-parabolic motion $x:[1,+\infty)\rightarrow\mathcal{X}$ of the form $$x(t) = at+\beta b_m t^{2/3}+o(t^{1/3^+})\quad\text{as }t\rightarrow+\infty,$$ for any initial configuration $x^0=x(1)\in\mathcal{X}$, for any collision configuration $a\in\Delta$, for any normalized minimal central configuration[^1] $b_m\in\mathcal{X}$ of the $a$-clustered potential and for any choice of the energy constant $h>0$.* Intuitively, hyperbolic-parabolic motions are those expansive motions of the form $x(t)=at+o(t)$, as $t\rightarrow+\infty$, when their limit shapes have collisions, that is, $a\in \Delta\setminus\{0\}$. This means that hyperbolic-parabolic motions can be viewed as clusters of bodies moving asymptotically with a linear growth, while the distances of the bodies inside each cluster grow with a rate of order $t^{2/3}$ and, referred to its center of mass, the cluster has a limit shape which is a prescribed minimal configuration of the cluster potential $U_K$. For the Newtonian $N$-body problem, the existence of hyperbolic-parabolic solutions for any prescribed positive energy and any given initial configuration of the bodies has been tackled by Burgos in [@Burgos_PartiallyHyperbolic], where his proof follows from and application of Maderna and Venturelli's Theorem on the existence of hyperbolic motions and a limiting procedure as the limit shape approaches the collision set. With respect to Burgos' result, we can provide a a much wider class of such hyperbolic-parabolic trajectories. Moreover, our approach provides a much more detailed information about the asymptotic behaviour of the solution and a better description of the motion of the bodies. Indeed, to prove Theorem [Theorem 8](#thm_partially_hyperbolic){reference-type="ref" reference="thm_partially_hyperbolic"}, we partition the set of bodies following the natural cluster partition that was presented by Burgos and Maderna in [@BurgosMaderna_GeodesicRays] and is defined as follows: if $x(t)=(r_1(t),...,r_N(t))$ and $a=(a_1,...,a_N)$, then $a_i=a_j$ if and only if $|r_i(t)-r_j(t)|=O(t^{2/3})$, and the partition of the set of bodies is defined by this equivalence relation. Using this particular partition, we are able to decompose the Lagrangian action into two terms: the first is related to the hyperbolic motion of the clusters and the second is related to the parabolic motion of the bodies inside the clusters. Through similar proofs to the ones in Theorems [Theorem 6](#thm_hyperbolic){reference-type="ref" reference="thm_hyperbolic"} and [Theorem 7](#thm_parabolic){reference-type="ref" reference="thm_parabolic"}, we can thus apply the Direct Method of the Calculus of Variation and Marchal's Theorem also to the case of hyperbolic-parabolic motions. **Corollary 9**. *The motions $x(t)$ given by Theorems [Theorem 6](#thm_hyperbolic){reference-type="ref" reference="thm_hyperbolic"}, [Theorem 7](#thm_parabolic){reference-type="ref" reference="thm_parabolic"} and [Theorem 8](#thm_partially_hyperbolic){reference-type="ref" reference="thm_partially_hyperbolic"} are continuous at $t=1$ and collisionless for $t>1$. Moreover they are free time action minimizers at their energy level.* As already pointed out by Maderna and Venturelli, a family of hyperbolic trajectories that are minimal in free time is associated, via the Busemann function, with a solution of the time-independent Hamilton-Jacobi equation. A further advantage of the approach through the direct minimization of a renormalized action functional is that a value function, dependent on the initial point, is directly defined. As we shall outline in Section [7](#sec:HJ){reference-type="ref" reference="sec:HJ"}, a linear correction to the value function is, as expected from theory, a solution of the Hamilton-Jacobi equation. Our general strategy in the proofs of Theorems [Theorem 6](#thm_hyperbolic){reference-type="ref" reference="thm_hyperbolic"}, [Theorem 7](#thm_parabolic){reference-type="ref" reference="thm_parabolic"} and [Theorem 8](#thm_partially_hyperbolic){reference-type="ref" reference="thm_partially_hyperbolic"} is to seek solutions to [\[eq_newton\]](#eq_newton){reference-type="eqref" reference="eq_newton"} which are lower order perturbations of a given path: $$x(t) = r_0(t) + \varphi(t)+\Tilde x_0\,, \qquad \Tilde x_0=x^0-t_0(1).$$ Here $\varphi(t)$ is the lower order term and the reference path $r_0$ is linear in the hyperbolic case, is a parabolic self-similar solution in the parabolic one and mixes the two types in the hyperbolic-parabolic case and $\Tilde x_0=x_0-r_0(1)$. In particular, we will consider functions $\varphi$ belonging to the functional space of continuous functions on $[1,+\infty)$ which vanish at $t = 1$ and can be written as primitives of functions in $L^2(1,+\infty)$. With this choice of space, which is denoted by $\mathcal{D}_0^{1,2}(1,+\infty)$, we will be able to give the problem a global variational structure, so that we can prove the existence of solutions of the $N$-body problem through the minimization of a Lagrangian action on the space $\mathcal{D}_0^{1,2}(1,+\infty)$. The crucial idea will be to minimize the action after a necessary proper renormalization (cfr. Definition [Definition 14](#def:renormalized_action){reference-type="ref" reference="def:renormalized_action"}), since the Lagrangian is never integrable at infinity. # The variational setting For the $N$-body problem, the Hamiltonian $H$ is defined over $\Omega\times\mathbb{R}^{dN}$ as $$\label{eq:hamiltonian} H(x,p) = \frac{1}{2}\|p\|_{\mathcal{M}^{-1}}^2 - U(x),$$ while the Lagrangian is defined over $\Omega\times\mathbb{R}^{dN}$ as $$L(x,v) = \frac{1}{2}\|v\|_\mathcal{M}^2 + U(x).$$ This means, in particular, that $L$ and $H$ become infinite when $x$ has collisions. Given two configurations $x,y\in\mathcal{X}$ and $T>0$, we denote by $\mathcal{C}(x,y,T)$ the set of absolutely continuous curves $\gamma:[a,b]\rightarrow\mathcal{X}$ going from $x$ to $y$ in time $T=b-a$ and we write $\mathcal{C}(x,y)=\bigcup_{T>0}\mathcal{C}(x,y,T)$. We define the Lagrangian action of a curve $\gamma\in\mathcal{C}(x,y,T)$ as the functional $$\mathcal{A}_L(\gamma) = \int_{a}^{b} L(\gamma,\dot{\gamma})\ \mathrm{d}t = \int_{a}^{b} \frac{1}{2}\|\dot{\gamma}\|_\mathcal{M}^2 + U(\gamma)\ \mathrm{d}t.$$ Hamilton's principle of least action implies that if a curve $\gamma$ is a minimizer of the Lagrangian action in $\mathcal{C}(x,y,T)$, then $\gamma$ satisfies Newton's equations at every time $t\in[a,b]$ in which $\gamma(t)$ has no collisions. However, as Poincaré already noticed in [@Poincare_SolutionsPeriodiques], there are curves with isolated collisions and finite action, which means that minimizing orbits may not always be true motions. The following theorem represents a big step forward in this theory, since it enabled the application of variational techniques to study the Newtonian $N$-body problem. The main idea to prove the theorem was given by Marchal in [@Marchal_MethodOfMinimization], while more complete proofs are due to Chenciner in [@Chenciner] and Ferrario and Terracini in [@FerrarioTerracini]. **Theorem 10** (Marchal [@Marchal_MethodOfMinimization], Chenciner [@Chenciner], Ferrario and Terracini [@FerrarioTerracini]). *Given $x,y\in\mathcal{X}$, if $\gamma\in\mathcal{C}(x,y)$ is defined on some interval $[a,b]$ and satisfies $$\mathcal{A}_L(\gamma) = \min\{\mathcal{A}(\sigma)\ |\ \sigma\in\mathcal{C}(x,y,b-a)\},$$ then $\gamma(t) \in \Omega$ for all $t \in (a,b)$.* Marchal's Theorem will be fundamental in our proofs, since it will guarantee that the minimizers of the action (whose existence is the object of our proofs) are in fact true motions of the $N$-body problem free of collisions. The Principle of Least Action, jointly with Theorem [Theorem 10](#thm_marchal){reference-type="ref" reference="thm_marchal"}, has been widely applied in the search for collisionless periodic solutions to the N-body problem (cfr. e.g. [@MR1610784; @FerrarioTerracini]). However, we must now build a suitable variational framework for the search of expansive solutions. Our minimization will take place on the functional space $$\mathcal{D}_0^{1,2}([1,+\infty),\mathcal{X}) = \{ \varphi\in H_{loc}^1([1,+\infty),\mathcal{X})\ :\ \varphi(1)=0\ \text{and}\ \int_{1}^{+\infty} \|\dot{\varphi}(t)\|_\mathcal{M}^2\ \mathrm{d}t<+\infty \},$$ which is endowed with the norm $$\|\varphi\|_{\mathcal{D}} = \bigg( \int_{1}^{+\infty} \|\dot{\varphi}(t)\|_\mathcal{M}^2\ \mathrm{d}t \bigg)^{1/2}.$$ **Remark 11**. Given a configuration $\varphi=(\varphi_1,...,\varphi_n)\in\mathcal{D}_0^{1,2}([1,+\infty),\mathcal{X})$, we will say that its components belong to the space $\mathcal{D}_0^{1,2}([1,+\infty),\mathbb{R}^{d})$ and the $\mathcal{D}_0^{1,2}$-norm of each component is $$\|\varphi_i\|_\mathcal{D} = \bigg( \int_{1}^{+\infty} |\dot{\varphi}_i(t)|^2\ \mathrm{d}t \bigg)^{1/2},$$ for $i=1,...,N$. We will write $\mathcal{D}_0^{1,2}(1,+\infty)$ to denote both the spaces $\mathcal{D}_0^{1,2}([1,+\infty),\mathcal{X})$ and $\mathcal{D}_0^{1,2}([1,+\infty),\mathbb{R}^{d})$, since it will be trivial to distinguish them. **Proposition 12** (Cfr. Boscaggin-Dambrosio-Feltrin-Terracini, 2021 [@BDFT]). *The space $\mathcal{D}_0^{1,2}(1,+\infty)$ is a Hilbert space containing the set $C_c^\infty(1,+\infty)$ as a dense subspace.* We recall here the following paramount Hardy-type inequality, which will be used several times in the paper. It states that the space $\mathcal{D}_0^{1,2}(1,+\infty)$ is continuously embedded in a weighted $L^2$-space with measure $\mathrm{d}t/t^2$. **Proposition 13** (*Hardy inequality*, Cfr. Boscaggin-Dambrosio-Feltrin-Terracini, 2021 [@BDFT]). *For every $\varphi\in\mathcal{D}_0^{1,2}(1,+\infty)$, it holds that $$\int_{1}^{+\infty} \frac{\|\varphi(t)\|_\mathcal{M}^2}{t^2}\ \mathrm{d}t \leq 4 \int_{1}^{+\infty}\|\dot{\varphi}(t)\|_\mathcal{M}^2\ \mathrm{d}t,$$ and moreover $$\label{dis_space_D012} \sup_{t\in[1,+\infty)}\frac{\|\varphi(t)\|_\mathcal{M}^2}{t-1} \leq \int_{1}^{+\infty}\|\dot{\varphi}(t)\|_\mathcal{M}^2\ \mathrm{d}t.$$* In order to prove the existence of minima for the functional $\mathcal{A}$ on $\mathcal{D}_0^{1,2}(1,+\infty)$, we will properly renormalize the Lagrangian action and, after proving its coercivity and weak lower semicontinuity, we will apply the Direct Method of the Calculus of Variations. In particular, we will use the following renormalization. **Definition 14** (Renormalized Lagrangian action). Given a motion $x(t)\in\Omega$ of the form $x(t) = \varphi(t) + r_0(t) + \Tilde{x}^0$, where $\varphi\in\mathcal{D}_0^{1,2}(1,+\infty)$, $r_0(t) = at + \beta bt^{2/3}$ for proper $a,\beta b\in\mathcal{X}$, and $\Tilde{x}^0\in\mathcal{X}$, then we can define the renormalized Lagrangian action $$\label{eq:renorm_action} \mathcal{A}^{ren}(\varphi) = \int_{1}^{+\infty} \frac{1}{2}\|\dot{\varphi}(t)\|_\mathcal{M}^2 + U(\varphi(t) + r_0(t) + \tilde{x}^0) - U(r_0(t)) - \langle \mathcal{M}\ddot{r}_0(t),\varphi(t)\rangle\ \mathrm{d}t.$$ In order to shorten the notation, throughout the paper we will usually write $\mathcal{A}$ instead of $\mathcal{A}^{ren}$. To describe the asymptotic expansion of our motions, we will use the following theorem and lemma. The former, applies to the cases of hyperbolic and hyperbolic-parabolic motions, while the latter, which is typically known as *Chazy's Lemma*, states that the set of initial conditions in the phase space that generate hyperbolic motions is an open set and that the map defined on this set that gives the asymptotic velocity in the future is continuous. **Theorem 15** (Chazy, 1922 [@Chazy]). *Let $x(t)$ be a motion with energy constant $h > 0$ and defined for all $t > t_0$.* 1. *The limit $$\lim_{t \rightarrow +\infty} R(t)r(t)^{-1} = L \in [1,+\infty]$$ always exists.* 2. *If $L < +\infty$, then there are a configuration $a \in \Omega$ and some function $P$, which is analytic in a neighbrhood of $(0,0)$, such that for every $t$ large enough, we have $$x(t) = at - \log (t) \nabla U(a) + P(u,v),$$ where $u = 1/t$ and $v = \log (t) /t$.* **Lemma 16** (Maderna-Venturelli, 2020 [@MadernaVenturelli_HyperbolicMotions]). *Working on an Euclidean space $E$, which is endowed with an Euclidean norm $\|\cdot\|$, let $U:E^N\rightarrow\mathbb{R}\cup\{+\infty\}$ be a homogeneous potential of degree -1 of class $C^2$ on the open set $\Omega = \{ x\in E^N\ |\ U(x)<+\infty \}$. Let $x:[0,+\infty)\rightarrow\Omega$ be a given solution of $\ddot{x}=\nabla U(x)$ satisfying $x(t)=at+o(t)$ as $t\rightarrow+\infty$ with $a\in\Omega$. Then we have the following:* 1. *The solution $x$ has asymptotic velocity $a$, meaning that $$\lim_{t\rightarrow+\infty} \dot{x}(t)=a.$$* 2. *(Chazy's continuity of the limit shape). Given $\varepsilon >0$, there are constants $t_1 > 0$ and $\delta > 0$ such that, for any maximal solution $y:[0,T)\rightarrow \Omega$ satisfying $\|y(0)-x(0)\| < \delta$ and $\|\dot{y}(0)-\dot{x}(0)\|<\delta$, we have* - *$T=+\infty$, $\|y(t)-at\|<t\varepsilon$ for all $t>t_1$;* - *there is $b\in\Omega$ with $\|b-a\| < \varepsilon$ for which $y(t)=bt+o(t).$* # Existence of minimal half hyperbolic motions This section is devoted to the proof of Theorem [Theorem 6](#thm_hyperbolic){reference-type="ref" reference="thm_hyperbolic"}. The class of hyperbolic motions has the following equivalent definition, also due to Chazy (see [@Chazy]). **Definition 17**. Hyperbolic motions are those motions such that each body has a different limit velocity vector, that is, $\dot{r}_i(t) \rightarrow a_i \in \mathbb{R}^d$, as $t \rightarrow +\infty$, and $a_i \neq a_j$ whenever $i \neq j$. We consider the differential system $$\label{newton_eq} \begin{cases} \mathcal{M}\ddot{x}=\nabla U(x)\\ x(1)=x^0\\ \lim_{t\rightarrow+\infty}\dot{x}(t)=a \end{cases},$$ where $x^0\in\mathcal{X}$ and $a\in\Omega$. To prove the existence of hyperbolic motions to Newton's equations [\[newton_eq\]](#newton_eq){reference-type="eqref" reference="newton_eq"}, we will look for solutions having the form $x(t)=\varphi(t)+at+x^0-a$, where $\varphi:[1,+\infty)\rightarrow\mathcal{X}$ belongs to the space $\mathcal{D}_0^{1,2}(1,+\infty)$. We can thus equivalently study the system $$\label{newton_eq_phi} \begin{cases} \mathcal{M}\ddot{\varphi}=\nabla U(\varphi + x^0 - a + at)\\ \varphi(1)=0\\ \lim_{t\rightarrow+\infty}\dot{\varphi}(t)=0 \end{cases}.$$ Taking advantage of the problem's variational structure, we would be tempted to prove the existence of hyperbolic motions through the minimization of the Lagrangian action associated to the system [\[newton_eq_phi\]](#newton_eq_phi){reference-type="eqref" reference="newton_eq_phi"}, that is, the functional $$\label{lag_ac_not_renorm} \int_{1}^{+\infty} \frac{1}{2}\|\dot{\varphi}(t)\|_\mathcal{M}^2 + U(\varphi(t) + x^0 - a +at)\ \mathrm{d}t,$$ where $$U(\varphi(t) + x^0 - a +at) = \sum_{i<j} \frac{m_i m_j}{|(\varphi_i(t) + x^0_i - a_i + a_i t) - (\varphi_j(t) + x^0_j - a_j + a_j t)|}.$$ In attempting to work with the action functional as above, the major problem we encounter is that $U(\varphi(t) + x^0 - a +at)$ needs not to be integrable at infinity. Indeed, when $\varphi\in C_0^\infty([1,+\infty))$, $U(\varphi(t) + x^0 - a +at)$ decays as $\frac{1}{t}$ for $t\rightarrow+\infty$. To overcome this problem, as we can add arbitrary functions to the Lagrangian without changing the associated Euler-Lagrange equations, we can renormalize the action functional in order to have a finite integral in the following way: $$\mathcal{A}(\varphi) = \mathcal{A}^{ren}(\varphi) = \int_{1}^{+\infty} \frac{1}{2}\|\dot{\varphi}(t)\|_\mathcal{M}^2 + U(\varphi(t) + x^0 - a +at) - U(at)\ \mathrm{d}t.$$ ## Coercivity In order to apply the Direct Method of the Calculus of Variations, we start by proving the coercivity of the functional, that is to say, that $\mathcal{A}(\varphi) \rightarrow+\infty$ as $\|\varphi\|_\mathcal{D}\rightarrow+\infty$. From now on, we will use the notations $\varphi_{ij}=\varphi_i - \varphi_j$, $x^0_{ij}=x^0_i - x^0_j$ and $a_{ij}=a_i-a_j$. We observe that the action can be equivalently written as $$\mathcal{A}(\varphi) = \int_{1}^{+\infty} \frac{1}{2}\sum_{i=1}^{N} m_i|\dot{\varphi}_i(t)|^2 + U(\varphi(t) + x^0 - a +at) - U(at)\ \mathrm{d}t,$$ where $$\begin{split} U(\varphi(t) + x^0 - a +at) - U(at) &= \sum_{i<j} \bigg(\frac{m_i m_j}{|(\varphi_i(t) + x^0_i - a_i + a_i t) - (\varphi_j(t) + x^0_j - a_j + a_j t)|} - \frac{m_i m_j}{|a_i - a_j|t}\bigg) \\ & = \sum_{i<j} \bigg(\frac{m_i m_j}{|\varphi_{ij}(t) + x^0_{ij} - a_{ij} + a_{ij} t|} - \frac{m_i m_j}{|a_{ij}|t}\bigg). \end{split}$$ Since we are working in the space of configurations whose center of mass is null at every time, it can easily be proved that $$\label{alternative_proof_kinetic_term} \sum_{i=1}^{N} m_i |\dot{\varphi}_i(t)|^2 = \frac{1}{M}\sum_{i<j} m_i m_j |\dot{\varphi}_i(t) - \dot{\varphi}_j(t)|^2,$$ where $M=\sum_{i=1}^{N}m_i$. Indeed, we have $$\begin{split} \sum_{i<j} m_i m_j |\dot{\varphi}_i(t) - \dot{\varphi}_j(t)|^2 &= \frac{1}{2} \sum_{i,j} m_i m_j (|\dot{\varphi}_i(t)|^2 +|\dot{\varphi}_j(t)|^2 - 2\langle\dot{\varphi}_i(t),\dot{\varphi}_j(t)\rangle)\\ & = \frac{1}{2}\bigg( M\sum_{i=1}^{N}m_i |\dot{\varphi}_i(t)|^2 + M\sum_{j=1}^{N}m_j |\dot{\varphi}_j(t)|^2 - 2 \langle\sum_{i=1}^{N}m_i \dot{\varphi}_i(t) , \sum_{j=1}^{N}m_j \dot{\varphi}_j(t) \rangle \bigg) \\ & = \frac{1}{2}\bigg( M\sum_{i=1}^{N}m_i |\dot{\varphi}_i(t)|^2 + M\sum_{j=1}^{N}m_j |\dot{\varphi}_j(t)|^2 \bigg) \\ & = M \sum_{i=1}^{N}m_i |\dot{\varphi}_i(t)|^2. \end{split}$$ Using [\[alternative_proof_kinetic_term\]](#alternative_proof_kinetic_term){reference-type="eqref" reference="alternative_proof_kinetic_term"}, we can then write the Lagrangian action as $$\mathcal{A}(\varphi) = \int_{1}^{+\infty} \sum_{i<j} m_i m_j \bigg( \frac{|\dot{\varphi}_{ij}(t)|^2}{2M} + \frac{1}{|\varphi_{ij}(t) + x^0_{ij} - a_{ij} + a_{ij} t|} - \frac{1}{|a_{ij}|t} \bigg)\ \mathrm{d}t.$$ Since $\|\dot{\varphi}\|_{L^2}\rightarrow+\infty$ if and only if there is $i<j$ such that $\|\dot{\varphi}_i - \dot{\varphi}_j\|_{L^2}\rightarrow+\infty$, we can prove the coercivity of the action by proving the coercivity of each term $\mathcal{A}_{ij}$, where $$\mathcal{A}(\varphi) = \sum_{i<j} \mathcal{A}_{ij}(\varphi)$$ and $$\mathcal{A}_{ij}(\varphi) = \int_{1}^{+\infty} m_i m_j \bigg( \frac{|\dot{\varphi}_{ij}(t) |^2}{2M} + \frac{1}{|\varphi_{ij}(t) + x^0_{ij} - a_{ij} + a_{ij} t|} - \frac{1}{|a_{ij}|t} \bigg)\ \mathrm{d}t.$$ Using the inequality $$\label{2.3} |\varphi_i(t)|\leq\|\varphi_i\|_\mathcal{D}\sqrt{t},\qquad \text{for every }i=1,...,N,\ t\geq1\text{ and }\varphi_i\in\mathcal{D}_0^{1,2},$$ which follows from [\[dis_space_D012\]](#dis_space_D012){reference-type="eqref" reference="dis_space_D012"}, we have $$U(\varphi(t) + x^0 - a +at) - U(at) \geq \sum_{i<j} \bigg(\frac{m_i m_j}{\|\varphi_{ij}\|_{\mathcal{D}}\sqrt{t} + |x^0_{ij} - a_{ij}| + |a_{ij}| t} - \frac{m_i m_j}{|a_{ij}|t}\bigg);$$ We can then look for an upper bound for the integral $$\int_{1}^{+\infty} \bigg( \frac{1}{|a_{ij}|t} - \frac{1}{|a_{ij}|t + \|\varphi_{ij}\|_{\mathcal{D}}\sqrt{t} + |x^0_{ij}|} \bigg)\ \mathrm{d}t.$$ Using the change of variables $t=s^2$, we obtain $$\label{int_1_hyp} \frac{2}{|a_{ij}|} \int_{1}^{+\infty} \bigg( \frac{1}{s^2} - \frac{1}{s^2 + \frac{\|\varphi_{ij}\|_{\mathcal{D}}}{|a_{ij}|}s + \frac{|x^0_{ij}-a_{ij}|}{|a_{ij}|}}\bigg)s\ \mathrm{d}s.$$ Since $$\begin{split} s^2 + \frac{\|\varphi_{ij}\|_{\mathcal{D}}}{|a_{ij}|}s + \frac{|x^0_{ij}-a_{ij}|}{|a_{ij}|} & = \bigg( s + \frac{\|\varphi_{ij}\|_{\mathcal{D}}}{2|a_{ij}|} \bigg)^2 - \frac{\|\varphi_{ij}\|^2_{\mathcal{D}}}{4|a_{ij}|^2} + \frac{|x^0_{ij}-a_{ij}|}{|a_{ij}|}\\ & = \frac{\|\varphi_{ij}\|^2_{\mathcal{D}}}{4|a_{ij}|^2} \bigg[ \bigg( \frac{2|a_{ij}|s}{\|\varphi_{ij}\|_{\mathcal{D}}} + 1 \bigg)^2 -1 +\frac{4|x^0_{ij}-a_{ij}| |a_{ij}|}{\|\varphi_{ij}\|_{\mathcal{D}}^2} \bigg], \end{split}$$ [\[int_1\_hyp\]](#int_1_hyp){reference-type="eqref" reference="int_1_hyp"} is equal to $$\label{int_2_hyp} \frac{2}{|a_{ij}|}\frac{4|a_{ij}|^2}{\|\varphi_{ij}\|^2_{\mathcal{D}}} \int_{1}^{+\infty} \bigg[ \frac{1}{\bigg( \frac{2|a_{ij}|s}{\|\varphi_{ij}\|_{\mathcal{D}}} \bigg)^2} - \frac{1}{\bigg( \frac{2|a_{ij}|s}{\|\varphi_{ij}\|_{\mathcal{D}}} +1 \bigg)^2 - 1 + \frac{4|x^0_{ij}-a_{ij}| |a_{ij}|}{\|\varphi_{ij}\|^2_{\mathcal{D}}}} \bigg] s\ \mathrm{d}s.$$ Changing variables again with $\tau = \frac{2|a_{ij}|s}{\|\varphi_{ij}\|_\mathcal{D}}$, we obtain that [\[int_2\_hyp\]](#int_2_hyp){reference-type="eqref" reference="int_2_hyp"} is equal to $$\frac{2}{|a_{ij}|} \int_{\frac{2|a_{ij}|}{\|\varphi_{ij}\|_\mathcal{D}}}^{+\infty} \bigg[ \frac{1}{\tau^2} - \frac{1}{(\tau+1)^2 - 1 + \frac{4|x^0_{ij}-a_{ij}| |a_{ij}|}{\|\varphi_{ij}\|^2_\mathcal{D}}} \bigg] \tau\ \mathrm{d}\tau.$$ Since we are interested in large values of $\|\varphi_{ij}\|_\mathcal{D}$, we can suppose that there is some $\lambda<1$ such that $\frac{4|x^0_{ij}-a_{ij}| |a_{ij}|}{\|\varphi_{ij}\|^2_\mathcal{D}} \leq \lambda$. We then have $$\label{alternative_proof1} \frac{2}{|a_{ij}|} \int_{\frac{2|a_{ij}|}{\|\varphi_{ij}\|_\mathcal{D}}}^{+\infty} \bigg[ \frac{1}{\tau^2} - \frac{1}{(\tau+1)^2 - 1 + \frac{4|x^0_{ij}-a_{ij}| |a_{ij}|}{\|\varphi_{ij}\|^2_\mathcal{D}}} \bigg] \tau\ \mathrm{d}\tau \leq \frac{2}{|a_{ij}|} \int_{\frac{2|a_{ij}|}{\|\varphi_{ij}\|_\mathcal{D}}}^{+\infty} \bigg[ \frac{1}{\tau^2} - \frac{1}{(\tau+1)^2 - 1 + \lambda} \bigg] \tau\ \mathrm{d}\tau.$$ The integrand of the last integral is a positive function. We observe that it is asymptotic to $\frac{1}{\tau}$ as $\tau\rightarrow0$ and to $\frac{1}{\tau^2}$ as $\tau\rightarrow+\infty$. In particular, the integral exists at infinity, uniformly in $\lambda$. Taking $\|\varphi_{ij}\|_\mathcal{D}$ large enough, we can equivalently study the integral $$\int_{\varepsilon}^{+\infty} \bigg[ \frac{1}{\tau^2} - \frac{1}{(\tau+1)^2 - 1 + \lambda} \bigg] \tau\ \mathrm{d}\tau,$$ where $\varepsilon = \frac{2|a_{ij}|}{\|\varphi_{ij}\|_\mathcal{D}} < 1$. Since the integrand is asymptotic to $\frac{1}{\tau}$ as $\tau\rightarrow0$, it is equivalent to consider the sum of integrals $$\int_{\varepsilon}^{1} \frac{1}{\tau}\ \mathrm{d}\tau + \int_{1}^{+\infty} \bigg[ \frac{1}{\tau^2} - \frac{1}{(\tau+1)^2 - 1 + \lambda} \bigg] \tau\ \mathrm{d}\tau,$$ where the second integral is constant (we will call it $C_1$) and does not depend on $\varepsilon$. We have $$\int_{\varepsilon}^{1} \frac{1}{\tau}\ \mathrm{d}\tau + \int_{1}^{+\infty} \bigg[ \frac{1}{\tau^2} - \frac{1}{(\tau+1)^2 - 1 + \lambda} \bigg] \tau\ \mathrm{d}\tau = \log \tau\bigg|_\varepsilon^1 + C_1 = -\log \varepsilon + C_1.$$ Then, as $\|\varphi_{ij}\|_\mathcal{D}\rightarrow+\infty$, we know that the integral on the right-hand side of [\[alternative_proof1\]](#alternative_proof1){reference-type="eqref" reference="alternative_proof1"} behaves like $$\frac{2}{|a_{ij}|}\bigg(-\log \frac{2|a_{ij}|}{\|\varphi_{ij}\|_\mathcal{D}} + C_1 \bigg) = \frac{2}{|a_{ij}|}\bigg(\log \|\varphi_{ij}\|_\mathcal{D} + C_1 - \log 2|a_{ij}| \bigg) = \frac{2}{|a_{ij}|}(\log \|\varphi_{ij}\|_\mathcal{D} + C_2 ),$$ where $C_2 = C_1 -\log 2|a_{ij}|$. We have thus proved that $$\int_{1}^{+\infty} \bigg( \frac{1}{|a_{ij}|t} - \frac{1}{|a_{ij}|t + \|\varphi_{ij}\|_{\mathcal{D}}\sqrt{t} + |x^0_{ij}-a_{ij}|}\bigg)\ \mathrm{d}t \leq \frac{2}{|a_{ij}|}(\log \|\varphi_{ij}\|_\mathcal{D} + C_2 ).$$ This means that given $R>0$, when $\|\varphi_{ij}\|_\mathcal{D} \geq R$ for $R$ large enough, we have $$\mathcal{A}_{ij}(\varphi) \geq m_i m_j \bigg[ \frac{\|\varphi_{ij}\|^2_\mathcal{D}}{2M} - \frac{2}{|a_{ij}|} (\log\|\varphi_{ij}\|_\mathcal{D} + C_2 )\bigg]$$ and we can conclude that $\mathcal{A}_{ij}(\varphi) \rightarrow+\infty$ as $\|\varphi_{ij}\|_\mathcal{D}\rightarrow+\infty$. ## Weak lower semicontinuity Now, we prove that the functional $\mathcal{A}$ is weakly lower semicontinuous. Since the kinetic term $\frac{1}{2}\|\dot{\varphi}(t)\|_\mathcal{M}$ is convex, it is straightforward that the term $\int_{1}^{+\infty} \frac{1}{2}\|\dot{\varphi}(t)\|_\mathcal{M}^2\ \mathrm{d}t$ is weakly lower semicontinuous. However it is worthwhile noticing that Fatou's Lemma cannot be applied to the term $\int_{1}^{+\infty} U(\varphi(t) + x^0 - a + at) - U(at)\ \mathrm{d}t$, since the integrand is not a positive function, and we must proceed in a different way. We know that there is at least a sequence of functions in $\mathcal{D}_0^{1,2}(1,+\infty)$ that converges uniformly on the compact subsets of $[1,+\infty)$. To show this, consider a bounded sequence $(\varphi^n)_n$ in $\mathcal{D}_0^{1,2}(1,+\infty)$. We also know, by the definition of this space, that $\|\dot{\varphi}^n\|_{L^2([1,+\infty))}<+\infty$ and that $\varphi^n(1) = 0$, for every $n$. From the inequality $$\|\varphi(t)\|_\mathcal{M}\leq \|\dot{\varphi}\|_{L^2}\sqrt{t-1}\leq\|\dot{\varphi}\|_{L^2}\sqrt{t}\qquad\text{ for every }t\geq1,$$ we have $\|\varphi^n(t)\|_\mathcal{M} \leq \|\dot{\varphi}^n\|_{L^2}\sqrt{t}$ for every $t\geq1$ and for every $n$, which means that the $L^{\infty}$-norm in $[1,T]$ of $\varphi^n$ is bounded, for every fixed $T\geq1$ and for every $n$. On the other hand, we have $$\|\varphi^n(t_1)-\varphi^n(t_2)\|_\mathcal{M} \leq \|\dot{\varphi}^n\|_{L^2}\sqrt{t_1 - t_2},$$ for every $t_1,t_2\in[1,+\infty)$ and for every $n$, which implies that the sequence $(\varphi^n)_n$ is equicontinuous on each interval $[1,T]$, for $T$ fixed. Then, by Ascoli-Arzelà's Theorem, we can say that for every fixed $T\geq1$ there is a subsequence $(\varphi^{n_k})_k$ that converges uniformly on $[1,T]$ (and, consequently, it converges pointwise on each compact). Besides, it can also be proved, through a diagonal procedure, that there is a subsequence converging pointwise in $[1,+\infty)$. Consider now a sequence $(\varphi^n)_n$ in $\mathcal{D}_0^{1,2}(1,+\infty)$ converging weakly to some limit $\varphi\in\mathcal{D}_0^{1,2}(1,+\infty)$. By the properties of weak convergence we know that the sequence is bounded on $\mathcal{D}_0^{1,2}(1,+\infty)$ and, from the previous considerations, there is a subsequence $(\varphi^{n_k})_k$ converging uniformly on compact subsets of $[1,+\infty)$ (and hence pointwise in $[1,+\infty)$). We write $$\label{alternative_proof_equality_with_s} \frac{1}{|x^0_{ij} - a_{ij} + a_{ij}t + \varphi^n_{ij}(t)|} - \frac{1}{|a_{ij}t|} = \int_{0}^{1} \frac{\mathrm{d}}{\mathrm{d}s}\bigg[\frac{1}{|a_{ij}t + s(x^0_{ij} - a_{ij} +\varphi^n_{ij}(t))|}\bigg]\ \mathrm{d}s.$$ However, this inequality holds only when the denominator of the integrand is not zero, which happens for $t$ sufficiently small. In particular, for all $s\in(0,1)$ we have $$\begin{split} |a_{ij}t + s(x^0_{ij} - a_{ij} + \varphi^n_{ij}(t))| & \geq |a_{ij}|t - s(|x^0_{ij} - a_{ij}| + \|\varphi^n_{ij}\|_\mathcal{D}\sqrt{t}) \\ & > |a_{ij}|t - (|x^0_{ij} - a_{ij}| + \|\varphi^n_{ij}\|_\mathcal{D}\sqrt{t}), \end{split}$$ and, since $|\varphi^n_{ij}(t)| \leq k\sqrt{t}$ for $k\in\mathbb{R}^+$ large enough, we have $$|a_{ij}t + s(x^0_{ij} - a_{ij} + \varphi^n_{ij}(t))| > |a_{ij}|t - (|x^0_{ij} - a_{ij}| + k\sqrt{t}),$$ where the last term is larger then zero if $t$ is larger than some $\bar{T}=\bar{T}(k)$; it is easy to compute $\bar{T}$ by studying the function $g(t)=|a_{ij}|t - [|x^0_{ij} - a_{ij}| + k\sqrt{t}]$. For these reasons, it is better to study the potential term separately on the two intervals $[1,\bar{T}]$ and $[\bar{T},+\infty)$. We observe that $U(x^0 - a + at + \varphi)\in L^1([1,\bar{T}])$, since $$\frac{1}{|x^0_{ij} - a_{ij} + a_{ij}t + \varphi^n_{ij}(t)|} \leq \frac{1}{|x^0_{ij} - a_{ij}| - |a_{ij}|t - \|\varphi^n_{ij}\|_\mathcal{D}\sqrt{t}}.$$ Besides, since $U$ is a positive function, we can use the pointwise convergence of the sequence and Fatou's Lemma to state that $$\int_{1}^{\bar{T}} \frac{1}{|x^0_{ij} - a_{ij} + a_{ij}t + \varphi_{ij}(t)|}\ \mathrm{d}t \leq \liminf_{n\rightarrow+\infty} \int_{1}^{\bar{T}} \frac{1}{|x^0_{ij} - a_{ij} + a_{ij}t + \varphi^n_{ij}(t)|}\ \mathrm{d}t.$$ Now, knowing that the sequence $(\varphi^n)_n$ is bounded, we wish to prove that the term $U(\varphi^n(t) + x^0 - a + at) - U(at)$ converges in $L^1([\bar{T},+\infty))$. By using [\[alternative_proof_equality_with_s\]](#alternative_proof_equality_with_s){reference-type="eqref" reference="alternative_proof_equality_with_s"}, we can write $$\begin{split} \int_{\bar{T}}^{+\infty}& \frac{1}{|x^0_{ij} - a_{ij} + a_{ij}t + \varphi^n_{ij}(t)|} - \frac{1}{|a_{ij}t|}\ \mathrm{d}t\\ & = \int_{\bar{T}}^{+\infty} \bigg( \int_{0}^{1} -\frac{[a_{ij}t + s(x^0_{ij} - a_{ij} +\varphi^n_{ij}(t))](x^0_{ij}-a_{ij}+\varphi^n_{ij}(t))}{|a_{ij}t + s(x^0_{ij} - a_{ij} +\varphi^n_{ij}(t))|^3}\ \mathrm{d}s \bigg)\ \mathrm{d}t. \end{split}$$ Our goal is to find an upper bound for the term $$\int_{\bar{T}}^{+\infty} \bigg|\frac{1}{|x^0_{ij} - a_{ij} + a_{ij}t + \varphi^n_{ij}(t)|} - \frac{1}{|a_{ij}t|}\bigg|\ \mathrm{d}t.$$ To find the upper bound, we will need the inequality $$\label{alternative_proof_inequality} \frac{|b+c|^2}{|b|^2 - |c|^2} \geq \frac{1}{3}, \qquad \text{for each }b,c\in\mathbb{R}^d \text{ such that }|b|\geq2|c|,$$ which can easily be proved by elementary calculus. By [\[alternative_proof_inequality\]](#alternative_proof_inequality){reference-type="eqref" reference="alternative_proof_inequality"} and using the fact that $|x^0_{ij} - a_{ij}|+\|\varphi^n_{ij}\|_\mathcal{D}\sqrt{t} \leq k'\sqrt{t}$ for $k'\in\mathbb{R}^+$ large enough, we thus have $$\begin{split} \int_{\bar{T}}^{+\infty} \bigg| \int_{0}^{1} & -\frac{[a_{ij}t + s(x^0_{ij} -a_{ij} +\varphi^n_{ij}(t))](x^0_{ij}-a_{ij}+\varphi^n_{ij}(t))}{|a_{ij}t + s(x^0_{ij} -a_{ij} +\varphi^n_{ij}(t))|^3}\ \mathrm{d}s \bigg|\ \mathrm{d}t \\ & \leq \int_{\bar{T}}^{+\infty} \bigg( \int_{0}^{1} \frac{|x^0_{ij}-a_{ij}+\varphi^n_{ij}(t)|}{|a_{ij}t + s(x^0_{ij} -a_{ij} +\varphi^n_{ij}(t))|^2}\ \mathrm{d}s \bigg)\ \mathrm{d}t \\ & \leq \int_{\bar{T}}^{+\infty} \bigg( \int_{0}^{1} 3\frac{|x^0_{ij}-a_{ij}|+\|\varphi^n_{ij}\|_\mathcal{D}\sqrt{t}}{|a_{ij}t|^2 - s|x^0_{ij} - a_{ij} +\|\varphi^n_{ij}\|_\mathcal{D}\sqrt{t}|^2}\ \mathrm{d}s \bigg)\ \mathrm{d}t \\ & \leq \int_{\bar{T}}^{+\infty} \bigg( \int_{0}^{1} \frac{3k'\sqrt{t}}{|a_{ij}|^2 t^2 - sk't}\ \mathrm{d}s \bigg)\ \mathrm{d}t. \end{split}$$ By choosing $\bar{T}(k)\gg k'/|a_{ij}|^2$ so that $|a_{ij}|^2 t > sk'$ for all $s\in(0,1)$ and for all $t\in[\bar{T},+\infty)$ (take $k$ large enough), we have that the last integral is finite and we have thus proved that there is a $\hat{T}$ such that, for all $\bar{T}\geq\hat{T}$, $$\int_{\bar{T}}^{+\infty} \bigg|\frac{1}{|x^0_{ij} - a_{ij} + a_{ij}t + \varphi^n_{ij}(t)|} - \frac{1}{|a_{ij}t|}\bigg|\ \mathrm{d}t < +\infty.$$ From this result, the $L^1$ convergence of the term $U(\varphi^n(t) + x^0 - a + at) - U(at)$ follows: by the dominated convergence Theorem we have, in particular, $$\lim_{n\rightarrow+\infty} \int_{\bar{T}}^{+\infty} U(\varphi^n(t) + x^0 - a + at) - U(at)\ \mathrm{d}t = \int_{\bar{T}}^{+\infty} U(\varphi(t) + x^0 - a + at) - U(at)\ \mathrm{d}t.$$ Thus, if we consider any sequence $(\varphi^n)_n$ in $\mathcal{D}_0^{1,2}(1,+\infty)$ converging weakly to some $\varphi\in\mathcal{D}_0^{1,2}(1,+\infty)$, we have $$\mathcal{A}(\varphi) \leq \liminf_{n\rightarrow+\infty} \int_{1}^{+\infty} \frac{1}{2} \|\dot{\varphi}^n(t)\|_\mathcal{M}^2 + U(\varphi^n(t) + x^0 - a + at) - U(at)\ \mathrm{d}t,$$ which proves the weak lower semicontinuity of the renormalized Lagrangian action in the space $\mathcal{D}_0^{1,2}(1,+\infty)$. **Remark 18**. The same reasoning leads to the continuity of the renormalised action with respect to the strong topology, in all elements $\varphi$ that do not give rise to collisions. ## Absence of collisions and hyperbolicity of the motion Now we can apply the Direct Method of the Calculus of Variations, obtaining a minimizer $\varphi$ on $\mathcal{D}_0^{1,2}(1,+\infty)$ of the renormalized action $\mathcal{A}$ and, by Marchal's Principle applied to $x(t) = \varphi(t) + x^0 - a + at$, we have that $x(t)\in\Omega$ for all $t\in(1,+\infty)$. Indeed, in each finite time interval, the full path $x$ minimizes the Lagrangian action among all paths joining the two ends. Being free of collisions, it solves the associated Euler-Lagrange equations. It remains to prove that $\lim_{t\rightarrow+\infty}\dot{\varphi}(t)=0$. We already know that $\dot{\varphi}\in L^2$ and that there is some $k\in\mathbb{R}^+$ such that $\|\varphi(t)\|_\mathcal{M} \leq k\sqrt{t}$. By this last inequality, we have that $$\sum_{i<j} m_i m_j \frac{1}{|a_{ij}t + x^0_{ij} - a_{ij} + \varphi_{ij}(t)|} \leq \sum_{i<j} m_i m_j \frac{1}{|a_{ij}|t - |x^0_{ij}-a_{ij}| - k\sqrt{t}}$$ and since $|a_{ij}|t - |x^0_{ij}-a_{ij}| - k\sqrt{t}\rightarrow+\infty$ as $t\rightarrow+\infty$ for all $i,j=1,...,N$, we obtain that $\lim_{t\rightarrow+\infty}U(x(t))=0$. Besides, since $\int_{1}^{+\infty} |\dot{\varphi}_{ij}(t)|^2\ \mathrm{d}t < +\infty$, we have that $$\label{alternative_proof_liminf} \liminf_{t\rightarrow+\infty} |\dot{\varphi}_{ij}(t)|=0.$$ **Remark 19**. A solution $x(t)=\varphi(t)+at+x^0-a$ of the equation $\mathcal{M}\ddot{x}=\nabla U(x)$ has positive energy. Indeed, $$\frac{1}{2}\|\dot{x}(t)\|^2_\mathcal{M} - U(x(t)) = \frac{1}{2}\sum_{i=1}^{N} m_i |\dot{\varphi}_i(t) + a_i|^2 - U(x(t)) = h,$$ and since by [\[alternative_proof_liminf\]](#alternative_proof_liminf){reference-type="eqref" reference="alternative_proof_liminf"} there is some $t_k\rightarrow+\infty$ such that $\lim_{t_k\rightarrow+\infty} \dot{\varphi}_i(t_k) = 0$, we have $h = \frac{1}{2}\|a\|_\mathcal{M}$. By Remark [Remark 19](#rem_lim_hyp){reference-type="ref" reference="rem_lim_hyp"}, we can apply Chazy's Lemma (Lemma [Lemma 16](#art2_lem4.1){reference-type="ref" reference="art2_lem4.1"}), which implies that the limit of $\dot{x}(t)$ exists for $t\rightarrow+\infty$. Since, by [\[alternative_proof_liminf\]](#alternative_proof_liminf){reference-type="eqref" reference="alternative_proof_liminf"}, there is at least a sequence $(t_k)_k$ such that $\dot{x}(t_k)\rightarrow a$ as $t_k\rightarrow+\infty$, we can conclude that $$\lim_{t\rightarrow+\infty}\dot{x}(t)=a.$$ Besides, we can apply Chazy's Theorem (Theorem [Theorem 15](#art2_thm_chazy){reference-type="ref" reference="art2_thm_chazy"}) to state that the minimizing motion $x$ has the asymptotic expansion $$x(t) = at- \log (t) \nabla U(a) + o(1)\quad\text{as }t\rightarrow+\infty.$$ We have thus proved that $x$ is a solution of the system $$\begin{cases} \mathcal{M}\ddot{x} = \nabla U(x)\\ x(1) = x^0\\ \lim_{t\rightarrow+\infty}\dot{x}(t) = a \end{cases},$$ which means that there is a hyperbolic motion for the $N$-body problem, starting at any initial configuration $x^0$ and having prescribed asymptotic velocity $a$ without collisions. # Existence of minimal half completely parabolic motions {#sec_parabolic} We now focus on the class of completely parabolic motions, that is, those motions that have the form $x(t) = at + O(t^{2/3})$ for $t\rightarrow+\infty$, with $a=0$ and $|r_i(t)-r_j(t)|\approx t^{2/3}$ for $i<j$. Equivalently, we have the following definition. **Definition 20**. An expansive solution $x$ of the $N$-body problem is said to be parabolic if every body approaches infinity with zero velocity. In this section we will prove Theorem [Theorem 7](#thm_parabolic){reference-type="ref" reference="thm_parabolic"}. More specifically, we will prove, for the $N$-body problem, the existence of orbits having the form $$x(t) = \beta b t^{2/3} + o(t^{1/3^+}), \quad \text{as }t\rightarrow+\infty,$$ where $\beta\in\mathbb{R}$ is a proper value and $b$ is a minimal central configuration. The remainder is $o(t^{1/3^+})$ in the sense that it grows less than order $\gamma$ for every $\gamma>1/3$. **Definition 21**. We say that $b\in\mathcal{X}$ is a central configuration if it is a critical point of $U$ when restricted to the inertial ellipsoid $$\mathcal{E} = \{x\in\mathcal{X}\ :\ \langle \mathcal{M} x,x\rangle=1\}.$$ A central configuration $b_m\in \mathcal{E}$ is said to be minimal if $$U(b_m)=\min_{b\in\mathcal{E}} U(b).$$ More precisely, we will work with normalized central configurations, that is, central configurations $b$ such that $\langle \mathcal{M}b,b\rangle=1$. **Remark 22**. Obviously, as $U$ is infinite on collisions, minimal central configuration $b_m$ are non collision, i.e. $b_m\in\Omega$. Given a Kepler potential $U$, we observe that from the definition of central configurations, it follows $$\nabla U(b) = \lambda \mathcal{M}b,$$ where $\lambda$ is a Lagrange multiplier. Besides, we have the equality $$\label{parabolic_lambda} \lambda = \lambda\langle \mathcal{M}b,b\rangle = \langle\nabla U(b),b\rangle = -U(b).$$ We first recall that there are self-similar solutions to Newton's equations $\mathcal{M}\ddot{x} = \nabla U(x)$ having the form $$x(t) = \beta b t^{2/3},$$ for a proper constant $\beta$ and a central configuration $b$. Indeed $$\mathcal{M}\ddot{x} = -\frac{2}{9}\mathcal{M}\beta b t^{-4/3} = \nabla U(x) = \nabla U(\beta b t^{2/3}) = \frac{1}{\beta^2}t^{-4/3}\nabla U(b) = \frac{1}{\beta^2}t^{-4/3}\lambda \mathcal{M} b$$ and, by [\[parabolic_lambda\]](#parabolic_lambda){reference-type="eqref" reference="parabolic_lambda"}, we also have $$\beta^3 = \frac{9}{2}U(b).$$ This means that for $\beta = \sqrt[3]{\frac{9}{2}U(b)}$, the orbit $x(t)=\beta b t^{2/3}$ can be a homothetic solution of Newton's equations. Now, let us define $$r_0(t) = \beta b_m t^{2/3},$$ where $b_m\in\Omega$ is a normalized minimal central configuration. We wish to prove the existence of solutions of the system $$\begin{cases} \mathcal{M}\ddot{x} = \nabla U(x)\\ x(1)=x^0\\ \lim_{t\rightarrow+\infty}\dot{x}(t)=0 \end{cases},$$ given $x^0\in\mathcal{X}$. We seek solutions having the form $$\label{expr_parabolic_motion} x(t) = r_0(t) + \varphi(t) - r_0(1) - x^0 = r_0(t) + \varphi(t) + \Tilde{x}^0,$$ where $\varphi\in\mathcal{D}_0^{1,2}(1,+\infty)$. In this case, we have $$\nabla U(x(t)) = \mathcal{M}\ddot{x}(t) = \mathcal{M}\ddot{r_0}(t) + \mathcal{M}\ddot{\varphi}(t) = \nabla U(r_0(t)) + \mathcal{M}\ddot{\varphi}(t),$$ which means that $$\mathcal{M}\ddot{\varphi}(t) = \nabla U(r_0(t)+\varphi(t)+\Tilde{x}^0)-\nabla U(r_0(t)).$$ We can thus define the renormalized Lagrangian action as $$\label{parabolic_action} \mathcal{A}(\varphi) = \int_{1}^{+\infty} \frac{1}{2}\langle \mathcal{M}\dot{\varphi}(t),\dot{\varphi}(t)\rangle + U(r_0(t)+\varphi(t)+\Tilde{x}^0) - U(r_0(t)) - \langle \nabla U(r_0(t)),\varphi(t)\rangle\ \mathrm{d}t.$$ Besides the coercivity and weak lower semicontinuity of the Lagrangian action, we have to verify that: - $\forall\ \varphi\in\mathcal{D}_0^{1,2}(1,+\infty)$ such that $r_0(t)+\varphi(t)+\Tilde{x}^0(t)\neq0$ for all $t\geq1$, $\mathcal{A}(\varphi)<+\infty$; - the action is continuous and $C^1$ on $\mathcal{D}_0^{1,2}\setminus\{\varphi\in\mathcal{D}_0^{1,2}\ :\ \exists\ t\text{ such that }r_0(t)+\varphi(t)+\Tilde{x}^0(t)=0\}$. ## Coercivity To minimize the action on the set $\mathcal{D}_0^{1,2}(1,+\infty)$, we start by proving its coercivity. We do this by reconducting the problem to a Kepler problem, where we denote $U_{min}=\min_{b\in\mathcal{E}}U(b)$. We notice that, for any orbit $x$, $$U(x) \geq \frac{U_{min}}{\|x\|},$$ where $\|\cdot\|$ represents the Euclidean norm on $\mathbb{R}^{dN}$. Indeed, because of the homogeneity of the potential, $$\label{parabolic_inequality(a)} U(x) = U\bigg(\|x\| \frac{x}{\|x\|}\bigg) = \frac{1}{\|x\|}U\bigg(\frac{x}{\|x\|}\bigg) \geq \frac{1}{\|x\|}U_{min}.$$ Besides, $$\label{parabolic_equality(b)} \nabla U(r_0) = \nabla U(\beta b_m t^{2/3}) = \frac{1}{\beta^2 t^{4/3}}\nabla U(b_m) = \frac{1}{\beta^2 t^{4/3}}\lambda \mathcal{M} b_m = -\frac{U_{min}}{\beta^2 t^{4/3}}\mathcal{M} b_m.$$ Using [\[parabolic_inequality(a)\]](#parabolic_inequality(a)){reference-type="eqref" reference="parabolic_inequality(a)"} and [\[parabolic_equality(b)\]](#parabolic_equality(b)){reference-type="eqref" reference="parabolic_equality(b)"}, we can then write $$\begin{split} \mathcal{A}(\varphi) &\geq \int_{1}^{+\infty} \frac{1}{2}\langle \mathcal{M}\dot{\varphi}(t),\dot{\varphi}(t)\rangle + \frac{U_{min}}{\|r_0(t)+\varphi(t)+\Tilde{x}^0\|}-\frac{U_{min}}{\|r_0(t)\|} + \frac{1}{\beta^2 t^{4/3}}\langle U_{min}\mathcal{M} b,\varphi(t)\rangle\ \mathrm{d}t \\ & = \int_{1}^{+\infty} \frac{1}{2}\langle \mathcal{M}\dot{\varphi}(t),\dot{\varphi}(t)\rangle + \frac{U_{min}}{\|r_0(t)+\varphi(t)+\Tilde{x}^0\|}-\frac{U_{min}}{\|r_0(t)\|} + \frac{\langle U_{min}\mathcal{M}r_0(t),\varphi(t)\rangle}{\|r_0(t)\|^3}\ \mathrm{d}t. \end{split}$$ We have $$\|r_0(t)+\varphi(t)+\Tilde{x}^0\|^2 = \|r_0(t)\|^2 + 2\langle \mathcal{M}r_0(t),\varphi(t)\rangle + 2\langle \mathcal{M}\varphi(t),x^0\rangle + 2\langle \mathcal{M} r_0(t),x^0\rangle + \|\varphi(t)\|^2 + \|x^0\|^2 = u + v,$$ where we define $$\begin{split} & u:= \|r_0(t)\|^2\\ & v:= 2\langle \mathcal{M}r_0(t),\varphi(t)\rangle + 2\langle \mathcal{M}\varphi(t),x^0\rangle + 2\langle \mathcal{M} r_0(t),x^0\rangle + \|\varphi(t)\|^2 + \|x^0\|^2. \end{split}$$ **Remark 23**. The following equalities hold true: $$\begin{split} & U(b+s)-U(b) = \int_{0}^{1} \frac{\mathrm{d}}{\mathrm{d}t} U(b+st)\ \mathrm{d}t = \int_{0}^{1} \nabla U(b+st)\ \mathrm{d}t,\\ & U(b+s)-U(b)-\nabla U(b)s = \int_{0}^{1}\int_{0}^{1} \langle\nabla^2 U(b+st_1 t_2)s,s\rangle t_2\ \mathrm{d}t_1\ \mathrm{d}t_2. \end{split}$$ Using Remark [Remark 23](#rem_taylor){reference-type="ref" reference="rem_taylor"}, we then have $$\|r_0(t)+\varphi(t)+\Tilde{x}^0\|^{-1} = (u+v)^{-1/2} = u^{-1/2} -\frac{1}{2}u^{-3/2}v + \frac{3}{4}\int_{0}^{1}\int_{0}^{1}\langle(u+stv)^{-5/2}v,v\rangle s\ \mathrm{d}s\ \mathrm{d}t.$$ Since the integral in the last expression is positive, it follows $$\label{parabolic_inequality3} \begin{split} \|r_0(t)+\varphi(t)+\Tilde{x}^0\|^{-1} &= (u+v)^{-1/2}\\ &\geq u^{-1/2} -\frac{1}{2}u^{-3/2}v \\ &= \|r_0(t)\|^{-1} -\frac{1}{2\|r_0(t)\|^3}[2\langle \mathcal{M}r_0(t),\varphi(t)\rangle + 2\langle \mathcal{M}\varphi(t),x^0\rangle + 2\langle \mathcal{M} r_0(t),x^0\rangle + \|\varphi(t)\|^2 + \|x^0\|^2]\\ & = \|r_0(t)\|^{-1} -\frac{\langle \mathcal{M}r_0,\varphi(t)\rangle}{\|r_0(t)\|^3} - \frac{\langle \mathcal{M}\varphi(t),x^0\rangle}{\|r_0(t)\|^3} - \frac{\langle \mathcal{M} r_0(t),x^0\rangle}{\|r_0(t)\|^3} -\frac{1}{2}\frac{\|\varphi(t)\|^2}{\|r_0(t)\|^3} - \frac{1}{2}\frac{\|x^0\|^2}{\|r_0(t)\|^3}. \end{split}$$ At this point we can use [\[parabolic_inequality3\]](#parabolic_inequality3){reference-type="eqref" reference="parabolic_inequality3"} to obtain $$\begin{split} \mathcal{A}(\varphi) &\geq \int_{1}^{+\infty} \frac{1}{2}\langle \mathcal{M}\dot{\varphi}(t),\dot{\varphi}(t)\rangle + \frac{U_{min}}{\|r_0(t)+\varphi+\Tilde{x}^0\|}-\frac{U_{min}}{\|r_0(t)\|} + \frac{\langle U_{min}\mathcal{M} r_0(t),\varphi(t)\rangle}{\|r_0(t)\|^3}\ \mathrm{d}t \\ & \geq \int_{1}^{+\infty} \frac{1}{2}\langle \mathcal{M}\dot{\varphi}(t),\dot{\varphi}(t)\rangle - \frac{U_{min}}{2}\frac{\|\varphi(t)\|^2}{\|r_0(t)\|^3}-\frac{\langle U_{min}\mathcal{M} \varphi(t),x^0\rangle}{\|r_0(t)\|^3}\ \mathrm{d}t + C_3, \end{split}$$ where $C_3$ is a constant. By Hardy inequality [\[dis_hardy\]](#dis_hardy){reference-type="eqref" reference="dis_hardy"} and the fact that, for $\beta=\sqrt[3]{\frac{9}{2}U(b)}$, $$\label{parabolic_equality4} \frac{U_{min}}{\|r_0(t)\|^3} = \frac{U_{min}}{\|\beta b_m t^{2/3}\|^3}=\frac{U_{min}}{\beta^3 t^2\| b_m\|^3}=\frac{2}{9}\frac{1}{t^2},$$ we have $$\begin{split} \mathcal{A}(\varphi) &\geq \int_{1}^{+\infty} \frac{1}{2}\bigg[\langle \mathcal{M}\dot{\varphi}(t),\dot{\varphi}(t)\rangle -\frac{8}{9}\langle \mathcal{M}\dot{\varphi}(t),\dot{\varphi}(t)\rangle \bigg] - \frac{U_{min}\langle \mathcal{M}\varphi(t),x^0\rangle}{\|r_0(t)\|^3}\ \mathrm{d}t\\ &=\int_{1}^{+\infty}\frac{1}{18} \langle \mathcal{M}\dot{\varphi}(t),\dot{\varphi}(t)\rangle - \frac{U_{min}\langle \mathcal{M}\varphi(t),x^0\rangle}{\|r_0(t)\|^3}\ \mathrm{d}t. \end{split}$$ Using again [\[parabolic_equality4\]](#parabolic_equality4){reference-type="eqref" reference="parabolic_equality4"}, we observe that $$\frac{U_{min}\langle \mathcal{M}\varphi(t),x^0\rangle}{\|r_0(t)\|^3} = \frac{2}{9}\frac{\langle \mathcal{M}\varphi(t),x^0\rangle}{t^2}.$$ By Cauchy-Scwartz and Hardy inequalities, it follows $$\begin{split} \int_{1}^{+\infty} - \frac{U_{min}\langle \mathcal{M}\varphi(t),x^0\rangle}{\|r_0(t)\|^3}\ \mathrm{d}t & \geq - \int_{1}^{+\infty} \frac{2}{9}\frac{|\langle \mathcal{M}\varphi(t),x^0\rangle|}{t^2}\ \mathrm{d}t \geq - \int_{1}^{+\infty} \frac{2}{9}\frac{\|\varphi(t)\|_\mathcal{M}}{t}\frac{\|x^0\|_\mathcal{M}}{t}\ \mathrm{d}t \\ & \geq -\frac{2}{9}\bigg(\int_{1}^{+\infty}\frac{\|\varphi(t)\|_\mathcal{M}^2}{t^2}\ \mathrm{d}t\bigg)^{1/2}\bigg(\int_{1}^{+\infty}\frac{\|x^0\|_\mathcal{M}^2}{t^2}\ \mathrm{d}t\bigg)^{1/2}\ \mathrm{d}t \\ &\geq -\frac{4}{9}C_4\|\varphi\|_\mathcal{D}, \end{split}$$ where $C_4$ is constant. This means that $$\mathcal{A}(\varphi) \geq \frac{1}{18}\|\varphi\|_\mathcal{D}^2-\frac{4}{9}C_4\|\varphi\|_\mathcal{D},$$ which proves the coercivity of the action. ## Weak-lower semicontinuity Now, we can focus on the proof of the weak lower semicontinuity of the action. Consider a sequence of functions $(\varphi^n)_n\subset\mathcal{D}_0^{1,2}(1,+\infty)$ converging weakly in $\mathcal{D}_0^{1,2}(1,+\infty)$ to some $\varphi$, for $n\rightarrow+\infty$. It trivially follows that, for every $n$, $\|\varphi\|_\mathcal{D}<+\infty$ and $\|\varphi^n\|_\mathcal{D}<+\infty$. Let us divide the action in two parts: $$\mathcal{A}(\varphi) = \mathcal{A}_{[1,\overline{T})}(\varphi) + \mathcal{A}_{[\overline{T},+\infty)}(\varphi),$$ where $$\begin{split} \mathcal{A}_{[1,\overline{T})}(\varphi)& = \int_{1}^{\overline{T}} \frac{1}{2}\|\dot{\varphi}(t)\|_\mathcal{M}^2 + U(r_0(t)+\varphi(t)+\Tilde{x}^0) - U(r_0(t)) - \langle \nabla U(r_0(t)),\varphi(t)\rangle \mathrm{d}t,\\ \mathcal{A}_{[\overline{T},+\infty)}(\varphi) &= \int_{\overline{T}}^{+\infty} \frac{1}{2}\|\dot{\varphi}(t)\|_\mathcal{M}^2 + U(r_0(t)+\varphi(t)+\Tilde{x}^0) - U(r_0(t)) - \langle \nabla U(r_0(t)),\varphi(t)\rangle\ \mathrm{d}t \end{split}$$ for some $\overline{T}\in(1,+\infty)$. Using Ascoli-Arzelà's Theorem, we can say that $\varphi^n\rightarrow\varphi$ uniformly on compact sets, which implies that $\langle \nabla U(r_0),\varphi^n\rangle \rightarrow \langle \nabla U(r_0),\varphi\rangle$ uniformly in $[1,\overline{T}]$, as $n\rightarrow+\infty$, for every $\overline{T}<+\infty$. Then, using Fatou's Lemma, it easily follows that the term $\mathcal{A}_{[1,\overline{T})}(\varphi)$ is weak lower semicontinuous. Concerning the term $\mathcal{A}_{[\overline{T},+\infty)}(\varphi)$, we can write: $$\begin{split} \mathcal{A}_{[\overline{T},+\infty)}(\varphi) = \int_{\overline{T}}^{+\infty} &\frac{1}{2}\|\dot{\varphi}(t)\|_\mathcal{M}^2 + \frac{1}{2}\langle \nabla^2 U(r_0(t))\varphi(t),\varphi(t)\rangle\\ &+ U(r_0(t)+\varphi(t)+\Tilde{x}^0) - U(r_0(t)) - \langle \nabla U(r_0(t)),\varphi(t)\rangle - \frac{1}{2}\langle \nabla^2 U(r_0(t))\varphi(t),\varphi(t)\rangle\ \mathrm{d}t. \end{split}$$ **Claim**: The map $\varphi(t) \mapsto \bigg( \int_{1}^{+\infty} \frac{1}{2}\|\dot{\varphi}(t)\|_\mathcal{M}^2 + \frac{1}{2}\langle \nabla^2 U(r_0(t))\varphi(t),\varphi(t)\rangle\ \mathrm{d}t \bigg)^{1/2}$ is an equivalent norm to $\|\cdot\|_\mathcal{D}$. Indeed: - Since $U(x)\geq \frac{U_{min}}{\|x\|}$ for each $x\neq0$, it follows that $\nabla^2 U(x) \geq -U_{min}\frac{Id}{\|x\|^3}$, which implies $\langle \nabla^2 U(r_0(t))\varphi(t),\varphi(t)\rangle \geq -\frac{2}{9}\frac{\|\varphi(t)\|_\mathcal{M}^2}{t^2}$ for each $t\in[1,+\infty)$. Then, by Hardy inequality, we have $$\int_{1}^{+\infty} \frac{1}{2}\|\dot{\varphi}(t)\|_\mathcal{M}^2 + \frac{1}{2}\langle \nabla^2 U(r_0(t))\varphi(t),\varphi(t)\rangle\ \mathrm{d}t \geq \bigg( 1-\frac{8}{9} \bigg)\|\varphi\|^2_\mathcal{D} = \frac{1}{9}\|\varphi\|^2_\mathcal{D}.$$ - Using the fact that, for some constant $C_5>0$, $$\langle \nabla^2 U(r_0(t))\varphi(t),\varphi(t)\rangle \leq C_5\frac{\|\varphi(t)\|_\mathcal{M}}{t^2}$$ and Hardy inequality, we have $$\int_{1}^{+\infty} \frac{1}{2}\|\dot{\varphi}(t)\|_\mathcal{M}^2 + \frac{1}{2}\langle \nabla^2 U(r_0(t))\varphi(t),\varphi(t)\rangle\ \mathrm{d}t \leq C_6\|\varphi\|_\mathcal{D}^2,$$ for some constant $C_6>0$. From the equivalence between the two norms, we have that the term $\int_{\overline{T}}^{+\infty} \frac{1}{2}\|\dot{\varphi}(t)\|_\mathcal{M}^2 + \frac{1}{2}\langle \nabla^2 U(r_0(t))\varphi(t),\varphi(t)\rangle\ \mathrm{d}t$ is weak lower semicontinuous. Using Taylor's series expansion, we can write $$\begin{split} \int_{\overline{T}}^{+\infty} & U(r_0(t)+\varphi(t)+\Tilde{x}^0) - U(r_0(t)) - \langle \nabla U(r_0(t)),\varphi(t)\rangle - \frac{1}{2}\langle \nabla^2 U(r_0(t)),\varphi(t),\varphi(t)\rangle\ \mathrm{d}t \\ & = \int_{\overline{T}}^{+\infty}\int_{0}^{1}\int_{0}^{1}\int_{0}^{1} \langle \nabla^3 U(r_0(t) + \tau_1\tau_2\tau_3 (\varphi^n(t)+\Tilde{x}^0))(\varphi^n(t)+\Tilde{x}^0),\varphi^n(t)+\Tilde{x}^0,\varphi^n(t)+\Tilde{x}^0\rangle \tau_1 \tau_2^2\ \mathrm{d}\tau_1\ \mathrm{d}\tau_2\ \mathrm{d}\tau_3\ \mathrm{d}t. \end{split}$$ Obviously there is a $\Tilde{t}>1$ such that $$\|r_0(t) + \tau_1\tau_2\tau_3 (\varphi^n(t)+\Tilde{x}^0)\|_\mathcal{M}>0$$ for every $t\geq\Tilde{t}$. We can then choose $\overline{T}\geq\Tilde{t}$ and we have $$\langle \nabla^3 U(r_0(t) + \tau_1\tau_2\tau_3 (\varphi^n(t)+\Tilde{x}^0))(\varphi^n(t)+\Tilde{x}^0),\varphi^n(t)+\Tilde{x}^0,\varphi^n(t)+\Tilde{x}^0\rangle \leq C_7\frac{\|\varphi^n(t) + \Tilde{x}^0\|_\mathcal{M}^3}{t^{8/3}} \leq C_8\frac{\|\varphi^n\|_\mathcal{D}^3 t^{3/2}}{t^{8/3}} \leq \frac{C_9}{t^{7/6}},$$ for every $t\geq\overline{T}$ and for proper constants $C_7,C_8,C_9>0$. This means that the term $\langle \nabla^3 U(r_0(t) + \tau_1\tau_2\tau_3 (\varphi^n(t)+\Tilde{x}^0))(\varphi^n(t)+\Tilde{x}^0),\varphi^n(t)+\Tilde{x}^0,\varphi^n(t)+\Tilde{x}^0\rangle \tau_1 \tau_2^2$ is $L^1$-dominated and the weak lower semicontinuity of $\mathcal{A}_{[\overline{T},+\infty)}$ follows from the Dominated Convergence Theorem. ## The renormalized action is of class $C^1$ over non-collision sets Now, we prove that the action $\mathcal{A}$ is $C^1$ over the set $\mathcal{D}_0^{1,2}([1,+\infty))\setminus\{\varphi\in\mathcal{D}_0^{1,2}\ : \exists\ t\text{ such that }r_0(t)+\varphi(t)+\Tilde{x}^0=0 \}$. The term $\int_{1}^{+\infty}\frac{1}{2}\langle \mathcal{M}\dot{\varphi}(t),\dot{\varphi}(t)\rangle\ \mathrm{d}t = \frac{1}{2}\|\varphi\|^2_\mathcal{D}$ is of course a smooth functional, so we focus on the term $$\mathcal{A}^2(\varphi) := \int_{1}^{+\infty} K(t,\varphi(t))\ \mathrm{d}t,$$ where $$K(t,\varphi(t)):=U(r_0(t)+\varphi(t)+\Tilde{x}^0) - U(r_0(t)) - \langle \nabla U(r_0(t)),\varphi(t)\rangle.$$ We have $$\mathrm{d}\mathcal{A}^2(\varphi)[\psi] = \int_{1}^{+\infty} \langle\nabla K(t,\varphi(t)),\psi(t)\rangle\ \mathrm{d}t = \int_{1}^{+\infty} \langle\nabla U(r_0(t)+\varphi(t)+\Tilde{x}^0)-\nabla U(r_0(t)),\psi(t)\rangle\ \mathrm{d}t$$ for every $\psi\in\mathcal{D}_0^{1,2}(1,+\infty)$. Given a sequence $(\varphi^n)_n\subset\mathcal{D}_0^{1,2}(1,+\infty)$ we have to prove that if $\varphi^n\rightarrow\varphi$ in $\mathcal{D}_0^{1,2}(1,+\infty)$, then $$\sup_{\|\psi\|_\mathcal{D}\leq 1} \bigg|\int_{1}^{+\infty} \langle \nabla K(t,\varphi^n(t))-\nabla K(t,\varphi(t)),\psi(t)\rangle\ \mathrm{d}t\bigg|\rightarrow0.$$ Since $$\nabla K(t,\varphi(t)) = \nabla U(r_0(t)+\varphi(t)+\Tilde{x}^0)-\nabla U(r_0(t)) = \int_{0}^{1} \nabla^2 K(t,s\varphi(t))\varphi(t)\ \mathrm{d}s,$$ we can estimate $$\label{ex_par_inequality_K} \|\nabla K(t,\varphi(t))\|_\mathcal{M} \leq \int_{0}^{1} \|\nabla^2 K(t,s\varphi(t))\|_\mathcal{M} \|\varphi(t)\|_\mathcal{M}\ \mathrm{d}s \leq C_{10}\frac{\|\varphi(t)\|_\mathcal{M}}{t^2},$$ where $C_{10}>0$ is a proper constant. Using the Cauchy-Schwartz inequality we can then compute $$\begin{split} &\sup_{\|\psi\|_\mathcal{D}\leq 1} \bigg|\int_{1}^{+\infty} \langle \nabla K(t,\varphi^n(t))-\nabla K(t,\varphi(t)),\psi(t)\rangle\ \mathrm{d}t\bigg| \\ &\leq \sup_{\|\psi\|_\mathcal{D}\leq 1} \int_{1}^{+\infty} t\| \nabla K(t,\varphi^n(t))-\nabla K(t,\varphi(t))\|_\mathcal{M}\frac{\|\psi(t)\|_\mathcal{M}}{t}\ \mathrm{d}t\\ &\leq \sup_{\|\psi\|_\mathcal{D}\leq 1} \bigg( \int_{1}^{+\infty} \frac{\|\psi(t)\|_\mathcal{M}^2}{t^2}\ \mathrm{d}t\bigg)^{1/2} \bigg( \int_{1}^{+\infty} t^2 \| \nabla K(t,\varphi^n(t))-\nabla K(t,\varphi(t))\|_\mathcal{M}^2\ \mathrm{d}t\bigg)^{1/2} \\ & \leq 2 \bigg( \int_{1}^{+\infty} t^2 \| \nabla K(t,\varphi^n(t))-\nabla K(t,\varphi(t))\|_\mathcal{M}^2\ \mathrm{d}t\bigg)^{1/2}. \end{split}$$ Now, using [\[ex_par_inequality_K\]](#ex_par_inequality_K){reference-type="eqref" reference="ex_par_inequality_K"} $$\begin{split} \| \nabla K(t,\varphi^n(t))-\nabla K(t,\varphi(t))\|_\mathcal{M}^2 &= \bigg| \int_{0}^{1} \nabla^2 K(t,\varphi(t)+\sigma(\varphi^n(t)-\varphi))(\varphi^n(t)-\varphi(t))\ \mathrm{d}\sigma \bigg|^2\\ & \leq \bigg( \int_{0}^{1} \|\nabla^2 K(t,\varphi(t)+\sigma(\varphi^n(t)-\varphi(t)))(\varphi^n(t)-\varphi(t))\|_\mathcal{M}\ \mathrm{d}\sigma \bigg)^2\\ & \leq \bigg( \int_{0}^{1} \frac{\|\varphi^n(t)-\varphi(t)\|_\mathcal{M}}{t^2}\ \mathrm{d}\sigma \bigg)^2 \\ & = \frac{\|\varphi^n(t)-\varphi(t)\|_\mathcal{M}^2}{t^4}. \end{split}$$ From this last computation, it follows that $$\begin{split} \bigg( \int_{1}^{+\infty} t^2 \| \nabla K(t,\varphi^n(t))-\nabla K(t,\varphi(t))\|_\mathcal{M}^2\ \mathrm{d}t\bigg)^{1/2} & \leq \bigg( \int_{1}^{+\infty} \frac{\|\varphi^n(t)-\varphi(t)\|_\mathcal{M}^2}{t^2}\ \mathrm{d}t\bigg)^{1/2}\\ & \leq 2 \bigg( \int_{1}^{+\infty} \|\dot{\varphi}^n(t)-\dot{\varphi}(t)\|_\mathcal{M}^2\ \mathrm{d}t\bigg)^{1/2}\\ & = 2\|\varphi^n - \varphi\|_\mathcal{D} \end{split}$$ and since $\|\varphi^n - \varphi\|_\mathcal{D}\rightarrow0$ as $n\rightarrow+\infty$, this proves our thesis. ## Absence of collisions and parabolicity of the motion Given a minimizer of the Lagrangian action $\varphi\in\mathcal{D}_0^{1,2}(1,+\infty)$, we apply Marchal's Theorem to state that $\varphi$ has no collisions. To conclude, we observe that given $$x(t) = \varphi(t) + \beta b_m t^{2/3} + \Tilde{x}^0,$$ we have $$\dot{x}(t) = \dot{\varphi}(t) + \frac{2}{3}\beta b_m t^{-1/3}.$$ To prove that the motion $x$ is indeed parabolic, we still have to prove that $$\lim_{t\rightarrow+\infty}\dot{x}(t) = \lim_{t\rightarrow+\infty}\dot{\varphi}(t) = 0.$$ Since $\int_{1}^{+\infty} |\dot{\varphi}_{ij}(t)|^2\ \mathrm{d}t < +\infty$, we have $$\liminf_{t\rightarrow+\infty} |\dot{\varphi}_{ij}(t)| = 0.$$ Because of the conservation of the energy along the motion, we have $$\frac{1}{2}\|\dot{x}(t)\|^2_\mathcal{M} - U(x(t)) = \frac{1}{2}\sum_{i=1}^{N} m_i \bigg|\dot{\varphi}_i(t) + \frac{2}{3}\beta b_{m_i} t^{-1/3} \bigg|^2 - U(x(t)) = h.$$ Since there is at least a subsequence $(t_k)_k$, with $t_k\rightarrow+\infty$, such that $\lim_{t_k\rightarrow+\infty} \dot{\varphi}_i(t_k)=0$, it follows that $h=0$ and, consequently, $$\frac{1}{2}\|\dot{x}(t)\|^2_\mathcal{M} - U(x(t)) = 0.$$ From this, we have that $\lim_{t\rightarrow+\infty} \dot{x}(t) = 0$. ## Asymptotic estimates for half parabolic motions In order to give a better description of the asymtptoic expansion of parabolic motions, we can improve inequality [\[2.3\]](#2.3){reference-type="eqref" reference="2.3"}. In particular, we can show that for any $\varphi\in\mathcal{D}_0^{1,2}(1,+\infty)$, it holds $$\label{new_estimate_parabolic} \|\varphi(t)\|_\mathcal{M} \leq c t^{\frac{1}{3}+\varepsilon},\quad \forall\varepsilon>0,$$ for a proper constant $c\in\mathbb{R}$. This section is devoted to the proof of this estimate. Let us consider a half parabolic motion $x(t)$ having the form [\[expr_parabolic_motion\]](#expr_parabolic_motion){reference-type="eqref" reference="expr_parabolic_motion"}, where $\varphi\in\mathcal{D}_0^{1,2}(1,+\infty)$ is a solution of the equations of motion $\mathcal{M}\ddot{\varphi}(t) = \nabla U(r_0(t) + \varphi(t) + \tilde{x}^0) - \nabla U(r_0(t))$. We can write: $$\begin{split} \mathcal{M}\ddot{\varphi}(t) &= \frac{1}{\beta^2 t^{4/3}} \bigg[ \nabla U\bigg(\frac{x(t)}{\beta t^{2/3}}\bigg) - \nabla U\bigg(\frac{r_0(t)}{\beta t^{2/3}}\bigg) \bigg] \\ & = \frac{1}{\beta^2 t^{4/3}} \bigg[ \nabla U\bigg(b_m + \frac{\varphi(t)}{\beta t^{2/3}} + \frac{\Tilde{x}^0}{\beta t^{2/3}}\bigg) - \nabla U(b_m) \bigg]\\ & = \frac{1}{\beta^3 t^2}\int_{0}^{1} \nabla^2 U\bigg(b_m + \theta\frac{(\varphi(t)+\tilde{x}^0)}{\beta t^{2/3}}\bigg)(\varphi(t)+\tilde{x}^0)\ \mathrm{d}\theta\\ & = \frac{1}{\beta^3 t^2}\bigg[\int_{0}^{1} \nabla^2 U\bigg(b_m + \theta\frac{(\varphi(t)+\tilde{x}^0)}{\beta t^{2/3}}\bigg)\ \mathrm{d}\theta\bigg](\varphi(t)+\tilde{x}^0), \end{split}$$ where we can view the integral term as a matrix. Fixing a real constant $\delta\in(1,2)$ and a sufficiently big constant $k\in\mathbb{R}$, we define a test function $\psi_k:\mathbb{R}\rightarrow\mathcal{X}$ as $$\psi_k(t)=\eta^2\min\{k,\|\varphi(t)\|_\mathcal{M}^{\delta-1}\}\varphi(t)$$ where $\eta:\mathbb{R}\rightarrow\mathbb{R}$ is a $C^\infty$-class cut-off function having the form $$\eta(t) = \begin{cases} 0, \quad t\in[1,R] \\ 1, \quad t\in[2R,+\infty) \end{cases},$$ for $R$ big enough, with $0<\eta(t)<1,\ \forall t\in(R,2R)$ . We point out that $k$ can be chosen such that $\eta\equiv1$ when $\|\varphi(t)\|_\mathcal{M}^{\delta-1}>k$, so that we have $$\dot{\psi}_k(t) = \begin{cases} 2\eta\dot{\eta}\|\varphi(t)\|_\mathcal{M}^{\delta-1}\varphi(t) + \eta^2\delta\|\varphi(t)\|_\mathcal{M}^{\delta-2}\langle\varphi(t),\dot{\varphi}(t)\rangle_\mathcal{M},\qquad & t\in I_k\\ k\dot{\varphi}(t),\qquad&t\in\hat{I}_k \end{cases},$$ where $I_k = \{t\in[1,+\infty) : \|\varphi(t)\|_\mathcal{M}^{\delta-1}\leq k\}$ and $\hat{I}_k = [1,+\infty)\setminus I_k = \{t\in[1,+\infty) : \|\varphi(t)\|_\mathcal{M}^{\delta-1}> k\}$. Multiplying the equations of motion for $\psi_k(t)$ and integrating, we obtain $$\begin{split} &\int_{R}^{+\infty} -\langle\ddot{\varphi}(t),\psi_k(t)\rangle_\mathcal{M} + \bigg\langle\frac{1}{\beta^3 t^2} \bigg[\int_{0}^{1} \nabla^2 U\bigg(b_m + \theta\frac{(\varphi(t)+\tilde{x}^0)}{\beta t^{2/3}}\bigg)\ \mathrm{d}\theta\bigg](\varphi(t)+\tilde{x}^0),\psi_k\bigg\rangle\ \mathrm{d}t\\ & = \int_{R}^{+\infty}\langle\dot{\varphi}(t),\dot{\psi}_k(t)\rangle_\mathcal{M} + \bigg\langle\frac{1}{\beta^3 t^2} \bigg[\int_{0}^{1} \nabla^2 U\bigg(b_m + \theta\frac{(\varphi(t)+\tilde{x}^0)}{\beta t^{2/3}}\bigg)\ \mathrm{d}\theta\bigg](\varphi(t)+\tilde{x}^0),\psi_k\bigg\rangle\ \mathrm{d}t. \end{split}$$ Recalling that $\|\nabla^2 U(r_0 + \theta(\varphi(t)+\tilde{x}^0))\|_\mathcal{M} \leq \frac{C_{11}}{t^2}$ for a proper constant $C_{11}$, for every $t>1$ and for every $\theta\in[0,1]$, we can use Hölder's and Hardy's inequalities to estimate $$\begin{split} & \int_{R}^{+\infty}\langle\dot{\varphi}(t),\dot{\psi}_k(t)\rangle_\mathcal{M} + \bigg\langle \bigg[\int_{0}^{1} \nabla^2 U(r_0(t) + \theta(\varphi(t)+\tilde{x}^0))\ \mathrm{d}\theta\bigg]\varphi(t),\psi_k(t)\bigg\rangle\ \mathrm{d}t \\ & = -\int_{R}^{+\infty}\bigg\langle \bigg[\int_{0}^{1} \nabla^2 U(r_0(t) + \theta(\varphi(t)+\tilde{x}^0))\ \mathrm{d}\theta\bigg]\tilde{x}^0,\psi_k(t)\bigg\rangle\ \mathrm{d}t \\ & \leq C_{11}\int_{R}^{+\infty}\frac{\|\psi_k(t)\|_\mathcal{M}}{t^2}\ \mathrm{d}t \\ & \leq C_{11} \int_{R}^{+\infty}\frac{\|\varphi(t)\|_\mathcal{M}^\delta}{t^2}\ \mathrm{d}t \\ & = C_{11} \int_{R}^{+\infty}\frac{1}{t^{2-\delta}}\frac{\|\varphi(t)\|_\mathcal{M}^\delta}{t^\delta}\ \mathrm{d}t \\ & \leq C_{11} \bigg( \int_{R}^{+\infty}\frac{1}{t^2}\ \mathrm{d}t\bigg)^{(2-\delta)/2}\bigg( \int_{R}^{+\infty}\frac{\|\varphi(t)\|_\mathcal{M}^2}{t^2}\ \mathrm{d}t \bigg)^{\delta/2} \\ & \leq C_{12} \|\varphi\|_\mathcal{D}^\delta, \end{split}$$ where $C_{12}$ is a proper constant. **Remark 24**. We recall that the Keplerian potential $U$ is homogeneous of degree -1. For any configuration $x\in\mathcal{X}$, denoting $s=\frac{x}{\|x\|_\mathcal{M}}$, $$U(x) = U\bigg(\|x\|_\mathcal{M} \frac{x}{\|x\|_\mathcal{M}}\bigg) = \frac{U(s)}{\|x\|_\mathcal{M}}.$$ The Hessian matrix of $U$ with respect to $x$ is $$\nabla^2 U(x) = -\frac{U(s)\mathcal{M}}{\|x\|_\mathcal{M}^3} + 3\frac{U(s)}{\|x\|_\mathcal{M}^5}\mathcal{M}x\otimes \mathcal{M}x - 2\frac{\nabla_s U(s)\otimes \mathcal{M}x}{\|x\|_\mathcal{M}^4} + \frac{\nabla_s^2 U(s)}{\|x\|_\mathcal{M}^3},$$ where $x \otimes x$ denotes the symmetric square matrix with components $(x \otimes x)_{ij} = x_i x_j$ for $i,j \in {1,...,N}$, and $\nabla_s U(s)$ and $\nabla_s^2 U(s)$ represent the gradient and the Hessian matrix of $U$ with respect to $s$, respectively. Choosing $s=b_m$, since $b_m$ is the minimum of the restricted potential, we have $\frac{\nabla_s U(s)\otimes \mathcal{M}x}{\|x\|_\mathcal{M}^4} = 0$. Besides, since $\mathcal{M}x\otimes \mathcal{M}x$ and $\nabla_s^2 U(s)$ are positive semidefinite quadratic forms, it holds $$\label{diseq_hessian_kepler_potential} \nabla^2 U(x) \geq -\frac{U(b_m)\mathcal{M}}{\|x\|_\mathcal{M}^3}.$$ Using a continuity argument and [\[diseq_hessian_kepler_potential\]](#diseq_hessian_kepler_potential){reference-type="eqref" reference="diseq_hessian_kepler_potential"}, we can also say that for every $\mu>0$ there is a $\bar{T}>0$ such that, for every $t>\bar{T}$, $$\frac{1}{\beta^3 t^2} \nabla^2 U\bigg(b_m + \theta\frac{(\varphi(t)+\tilde{x}^0)}{\beta t^{2/3}}\bigg) \geq -\frac{2}{9}(1+\mu)\frac{\mathcal{M}}{t^2}$$ in the sense of quadratic forms. It follows $$\begin{split} & \int_{R}^{+\infty}\langle\dot{\varphi}(t),\dot{\psi}_k(t)\rangle_\mathcal{M} + \bigg\langle\frac{1}{\beta^3 t^2} \bigg[\int_{0}^{1} \nabla^2 U\bigg(b_m + \theta\frac{(\varphi(t)+\tilde{x}^0)}{\beta t^{2/3}}\bigg)\ \mathrm{d}\theta\bigg]\varphi(t),\psi_k(t)\bigg\rangle\ \mathrm{d}t \\ & \geq \int_{R}^{+\infty}\langle\dot{\varphi}(t),\dot{\psi}_k(t)\rangle_\mathcal{M} -\frac{2}{9}(1+\mu)\bigg\langle\frac{\varphi(t)}{t^2},\psi_k(t)\bigg\rangle_\mathcal{M}\ \mathrm{d}t. \end{split}$$ To estimate the right-hand side of the last inequality, we study the integral separately on the two complementary sets $I_k$ and $\hat{I}_k$. In $I_k$, we have $$\begin{split} & \int_{I_k}\langle\dot{\varphi}(t),\dot{\psi}_k(t)\rangle_\mathcal{M} -\frac{2}{9}(1+\mu)\bigg\langle\frac{\varphi(t)}{t^2},\psi_k(t)\bigg\rangle_\mathcal{M}\ \mathrm{d}t \\ & = \int_{I_k} 2\eta\dot{\eta}\|\varphi(t)\|_\mathcal{M}^{\delta-1}\langle\dot{\varphi}(t),\varphi(t)\rangle_\mathcal{M} + \eta^2\delta\|\varphi(t)\|_\mathcal{M}^{\delta-1}\|\dot{\varphi(t)}\|_\mathcal{M} -\frac{2}{9}(1+\mu)\eta^2\frac{\|\varphi(t)\|_\mathcal{M}^{\delta+1}}{t^2}\ \mathrm{d}t, \end{split}$$ which implies $$\int_{I_k} \eta^2\delta\|\varphi(t)\|_\mathcal{M}^{\delta-1}\|\dot{\varphi}(t)\|_\mathcal{M} -\frac{2}{9}(1+\mu)\eta^2\frac{\|\varphi(t)\|_\mathcal{M}^{\delta+1}}{t^2}\ \mathrm{d}t \leq \int_{I_k} 2\eta\dot{\eta}\|\varphi(t)\|_\mathcal{M}^{\delta}\|\dot{\varphi}(t)\|_\mathcal{M}\ \mathrm{d}t + C_{12}\|\varphi\|_\mathcal{D}^\delta. %\\ %& \leq C''\int_{I_k} \|\varphi\|_\mathcal{M}^{\delta}\|\dot{\varphi}\|_\mathcal{M}\ \ud t + C'\|\varphi\|_\mathcal{D}^\delta \\ %& \leq C'''\|\varphi\|_\mathcal{D} + C'\|\varphi\|_\mathcal{D}^\delta,$$ where the cut-off function makes sure that the last integral is finite. Besides, we also have $$\begin{split} & \int_{I_k} \eta^2\delta\|\varphi(t)\|_\mathcal{M}^{\delta-1}\|\dot{\varphi}(t)\|_\mathcal{M} -\frac{2}{9}(1+\mu)\eta^2\frac{\|\varphi(t)\|_\mathcal{M}^{\delta+1}}{t^2}\ \mathrm{d}t \\ & = \int_{I_k} \frac{4\delta}{(\delta+1)^2}\bigg(\eta\frac{\mathrm{d}}{\mathrm{d}t}\|\varphi(t)\|_\mathcal{M}^{\frac{\delta+1}{2}}\bigg)^2 -\frac{2}{9}(1+\mu)\eta^2\frac{\|\varphi(t)\|_\mathcal{M}^{\delta+1}}{t^2}\ \mathrm{d}t. \end{split}$$ On the other hand, working on the interval $\hat{I}_k$ we obtain $$\begin{split} & \int_{\hat{I}_k}\langle\dot{\varphi}(t),\dot{\psi}_k(t)\rangle_\mathcal{M} -\frac{2}{9}(1+\mu)\bigg\langle\frac{\varphi(t)}{t^2},\psi_k(t)\bigg\rangle_\mathcal{M}\ \mathrm{d}t \\ & = \int_{\hat{I}_k} k\|\dot{\varphi}(t)\|_\mathcal{M}^2 - \frac{2}{9}(1+\mu)k\frac{\|\varphi(t)\|_\mathcal{M}^2}{t^2}\ \mathrm{d}t \\ & \geq \int_{\hat{I}_k} \frac{4\delta}{(\delta+1)^2} k\|\dot{\varphi}(t)\|_\mathcal{M}^2 - \frac{2}{9}(1+\mu)k\frac{\|\varphi(t)\|_\mathcal{M}^2}{t^2}\ \mathrm{d}t, \end{split}$$ where we used the fact that $\frac{4\delta}{(\delta+1)^2} < 1$ for every $\delta\in(1,2)$. Now, we define a function $u_k:\mathbb{R}\rightarrow\mathbb{R}$ as $$u_k(t) = \min\{\eta\|\varphi(t)\|_\mathcal{M}^{\frac{\delta-1}{2}},k^{1/2}\}\|\varphi(t)\|_\mathcal{M}.$$ Putting everything together, we can use Hardy's inequality to say that $$\begin{split} &\int_{I_k} \frac{4\delta}{(\delta+1)^2}\bigg(\eta\frac{\mathrm{d}}{\mathrm{d}t}\|\varphi(t)\|_\mathcal{M}^{\frac{\delta+1}{2}}\bigg)^2 -\frac{2}{9}(1+\mu)\eta^2\frac{\|\varphi(t)\|_\mathcal{M}^{\delta+1}}{t^2}\ \mathrm{d}t + \int_{\hat{I}_k} \frac{4\delta}{(\delta+1)^2} k\|\dot{\varphi}(t)\|_\mathcal{M}^2 - \frac{2}{9}(1+\mu)k\frac{\|\varphi(t)\|_\mathcal{M}^2}{t^2}\ \mathrm{d}t \\ & = \int_{1}^{+\infty} \frac{4\delta}{(\delta+1)^2} \|\dot{u}_k(t)\|_\mathcal{M}^2 - \frac{2}{9}(1+\mu)\frac{\|u_k(t)\|_\mathcal{M}^2}{t^2}\ \mathrm{d}t \\ & \geq \int_{1}^{+\infty} \bigg(\frac{4\delta}{(\delta+1)^2} - \frac{8}{9}(1+\mu)\bigg)\|\dot{u}_k(t)\|_\mathcal{M}^2\ \mathrm{d}t. \end{split}$$ In particular, we can choose $\mu$ such that $\frac{4\delta}{(\delta+1)^2} - \frac{8}{9}(1+\mu) > 0$, which proves that $u_k\in\mathcal{D}_0^{1,2}(1,+\infty)$. Since the estimates we obtained do not depend on $k$, we can take $k\rightarrow+\infty$ so that [\[2.3\]](#2.3){reference-type="eqref" reference="2.3"} leads us to the conclusion of our proof. We have thus shown that for any $\varphi\in\mathcal{D}_0^{1,2}(1,+\infty)$ and for any $\delta\in(1,2)$ there is a constant $c$, which depends on $\delta$ and $\|\varphi\|_\mathcal{D}$, such that $$\|\varphi(t)\|_\mathcal{M} \leq c t^{\frac{1}{\delta+1}},\qquad\forall t\geq1.$$ # Existence of minimal half hyperbolic-parabolic motions {#sec_hyperbolic_parabolic} This last section is devoted to the proof of Theorem [Theorem 8](#thm_partially_hyperbolic){reference-type="ref" reference="thm_partially_hyperbolic"}. To prove the existence of hyperbolic-parabolic solutions in the $N$-body problem, we will use the cluster decomposition that we briefly introduced in Section [1](#sec_intro){reference-type="ref" reference="sec_intro"} to decompose the Lagrangian action, so that we will finally be able to minimize the renormalized action over the set $\mathcal{D}_0^{1,2}(1,+\infty)$. **Definition 25**. Given a configuration $a\in\mathcal{X}$ and a motion $x(t)=at+O(t^{2/3})$ as $t\rightarrow+\infty$, its corresponding natural partition ($a$-partition) of the index set $\mathcal{N}=\{1,...,N\}$ is the one for which $i,j\in\mathcal{N}$ belong to the same class if and only if the mutual distance $|r_i(t)-r_j(t)|$ grows as $O(t^{2/3})$. Equivalently, if $a = (a_1,...,a_N)$, then the natural partition is defined by the relation $i \sim j$ if and only if $a_i = a_j$. The partition classes will be called clusters. We give now some definitions and basic notations related to a given partition $\mathcal{P}$ of the set $\mathcal{N}=\{1,...,N\}$. **Definition 26**. Let $\mathcal{P}$ be a given partition of $\mathcal{N}$ and consider a configuration $x=(r_1,...,r_N)\in \mathcal{X}$. For each cluster $K\in\mathcal{P}$ we define the mass of the cluster as $$M_K=\sum_{i\in K}m_i.$$ Besides, for any couple of clusters $K_1,K_2\in\mathcal{P}$, $K_1\neq K_2$, we define the mass of the two clusters as $$M_{K_{1,2}}=\sum_{i\in K_1 \cup K_2}m_i.$$ **Definition 27**. Let $\mathcal{P}$ be any given partition of $\mathcal{N}$. Then, for every given curve $x(t)=(r_1(t),...,r_N(t))$ in $\mathcal{X}$ and for each cluster $K\in\mathcal{P}$ we define the function $$U_K(t) = \sum_{i,j\in K,\ i<j}\frac{m_i m_j}{|r_i(t)-r_j(t)|},$$ which represents the restriction of the potential $U$ to the cluster $K$. The system we are studying here is $$\begin{cases} \mathcal{M}\ddot{x}=\nabla U(x)\\ x(1)=x^0\\ \lim_{t\rightarrow+\infty}\dot{x}(t)=a \end{cases},$$ where $x^0\in\mathcal{X}$ and $a$ is a configuration with collisions. We will look for solutions of the form $x(t) = \varphi(t)+\gamma_0(t)$, for any $\varphi\in\mathcal{D}_0^{1,2}(1,+\infty)$, where $\gamma_0$ is a proper function, so that our problem equivalently reads $$\begin{cases} \mathcal{M}\ddot{\varphi}=\nabla U(\varphi+\gamma_0)-\ddot{\gamma}_0\\ \varphi(0)=0\\ \lim_{t\rightarrow+\infty}\dot{\varphi}(t)=0 \end{cases}.$$ We can thus prove the existence of solutions to the last system by minimizing the associated renormalized Lagrangian action. We partition the indexes according to the natural cluster partition, so that we obtain a partition of $\mathcal{N}$ of the form $$K_1:=\{1,...,k_1\},\ K_2:=\{k_1+1,...,k_2\},\ K_3:=\{k_2+1,...,k_3\}...$$ For every $K_i$, we can choose a central configuration $b^{K_i}$ which is minimal for that particular cluster and we can define the configuration $$b=(b^{K_1},b^{K_2},...)\in \mathcal{X}.$$ Using this particular definition of $b$, we can then look for solutions of the form $$\label{explicit_hyperbolic-parabolic} x(t) = \varphi(t) + at +\beta b t^{2/3} -a -b +x^0 = \varphi(t) + at +\beta b t^{2/3} + \Tilde{x}^0.$$ Here, $\beta$ is a real vector with as many components as the number of clusters. Precisely, we have $$\beta=(\beta_{K_1},\beta_{K_2},...),$$ with $$\beta_{K_1}=\sqrt[3]{\frac{9}{2}U_{min}^{K_1}}$$and $U_{min}^{K_1}$ denotes the minimum of the potential $U$ restricted to the first cluster; $$\beta_{K_2}=\sqrt[3]{\frac{9}{2}U_{min}^{K_2}}$$ and $U_{min}^{K_2}$ denotes the minimum of the potential $U$ restricted to the second cluster, and so on. With an abuse of notation, in this section we write $\beta b$ to denote the configuration $(\beta_{K_1}b^{K_1}, \beta_{K_2}b^{K_2},...)\in\mathcal{X}$. Using the aforementioned partition of the bodies, it is possible to decompose the Lagrangian action of a curve: for every $\varphi\in\mathcal{D}_0^{1,2}(1,+\infty)$, we define $$\label{action_partially_hyperbolic} \begin{split} \mathcal{A}(\varphi) & := \sum_{K\in\mathcal{P}} \mathcal{A}_{K}(\varphi) + \sum_{K_1,K_2\in\mathcal{P},\ K_1\neq K_2} \mathcal{A}_{K_1,K_2}(\varphi)\\ & = \sum_{K\in\mathcal{P}}\bigg(\sum_{i,j\in K,\ i<j} \mathcal{A}_{K}^{ij}(\varphi)\bigg) + \frac{1}{2}\sum_{K_1,K_2\in\mathcal{P},\ K_1\neq K_2}\bigg(\sum_{i\in K_1,\ j\in K_2} \mathcal{A}_{K_1,K_2}^{ij}(\varphi)\bigg), \end{split}$$ where $$\label{action_partially_hyperbolic_inside} \begin{split} \mathcal{A}_{K}^{ij}(\varphi) := & \int_{1}^{+\infty} \frac{1}{2M_K} m_i m_j|\dot{\varphi}_i(t)-\dot{\varphi}_j(t)|^2 + \frac{m_i m_j}{|\varphi_{ij}(t)+a_{ij}t+\beta_K b^K_{ij}t^{2/3}+\Tilde{x}^0_{ij}|} - \frac{m_i m_j}{|\beta_K b^K_{ij}t^{2/3}|} \\ &+ \frac{2}{9}\frac{\beta_K}{M_K}m_i m_j \frac{\langle b^K_{ij},\varphi_{ij}(t)\rangle}{t^{4/3}}\ \mathrm{d}t, \end{split}$$ $$\label{action_partially_hyperbolic_outside} \begin{split} \mathcal{A}_{K_1,K_2}^{ij}(\varphi) := &\int_{1}^{+\infty}\frac{1}{2M_{K_{1,2}}} m_i m_j|\dot{\varphi}_i(t)-\dot{\varphi}_j(t)|^2+ \frac{m_i m_j}{|\varphi_{ij}(t)+a_{ij}t+\beta_{K_{1,2}} b^{K_{1,2}}_{ij}t^{2/3}+\Tilde{x}^0_{ij}|} - \frac{m_i m_j}{|a_{ij}t|}\ \mathrm{d}t. \end{split}$$ Here, we used the notations: $$\begin{split} &b^{K_{1,2}}=(b^{K_1},b^{K_2})\\ &\beta_{K_{1,2}}b^{K_{1,2}}=(\beta_{K_1}b^{K_1},\beta_{K_2}b^{K_2}) \end{split}$$ We point out that the term [\[action_partially_hyperbolic_inside\]](#action_partially_hyperbolic_inside){reference-type="eqref" reference="action_partially_hyperbolic_inside"} is the part of the Lagrangian action that refer to the (parabolic) motion of the bodies inside each cluster, while the term [\[action_partially_hyperbolic_outside\]](#action_partially_hyperbolic_outside){reference-type="eqref" reference="action_partially_hyperbolic_outside"} refers to the (linear) motion of the cluster. In the following sections, we will study the two terms separately, in order to apply the Direct Method. ## Coercivity of $\mathcal{A}(\varphi)$ We start with the proof of the coercivity of the Lagrangian action when restricted to a general cluster, where we denote by $K$ the set of indexes related to this cluster. Because of the natural cluster partition of the bodies, we have $a_i = a_j$ for any $i,j\in K$. This means that for any $\varphi\in\mathcal{D}_0^{1,2}(1,+\infty)$, $$\begin{split} \mathcal{A}_{K}(\varphi) = \sum_{i,j\in K,\ i<j} &\int_{1}^{+\infty} \frac{1}{2M_K} m_i m_j|\dot{\varphi}_i(t)-\dot{\varphi}_j(t)|^2 + \frac{m_i m_j}{|\varphi_{ij}(t)+\beta_K b^K_{ij}t^{2/3}+\Tilde{x}^0_{ij}|} - \frac{m_i m_j}{|\beta_K b^K_{ij}t^{2/3}|} \\ &+ \frac{2}{9}\frac{\beta_K}{M_K}m_i m_j \frac{\langle b^K_{ij},\varphi_{ij}(t)\rangle}{t^{4/3}}\ \mathrm{d}t. \end{split}$$ Using the homogeneity of the potential and denoting by $U_K$ the potential $U$ when restricted to the cluster $K$, we apply the inequality $$U_K(x) \geq \frac{U_K(b^K)}{\|x\|_\mathcal{M}} = \frac{U_{min}}{\|x\|_\mathcal{M}}$$ to every configuration $x$ restricted to the cluster $K$. It follows $$\begin{split} \mathcal{A}_K(\varphi) \geq &\int_{1}^{+\infty} \sum_{i,j\in K,\ i<j} \bigg(\frac{1}{2M_K} m_i m_j|\dot{\varphi}_i(t)-\dot{\varphi}_j(t)|^2\bigg) + \frac{U_{min}}{\|\varphi(t) + \beta_K b^K t^{2/3}+\Tilde{x}^0\|_\mathcal{M}} - \frac{U_{min}}{\|\beta_K b^K t^{2/3}\|_\mathcal{M}}\\ &+ \frac{2}{9}\frac{\beta_K}{M_K}\langle \mathcal{M}_K b^K,\varphi(t)\rangle\ \mathrm{d}t, \end{split}$$ where $\mathcal{M}_K$ denotes the matrix of the masses of the cluster $K$. Using the inequality $$\begin{split} \frac{1}{\|\varphi(t) + \beta_K b^K t^{2/3}+\Tilde{x}^0\|_\mathcal{M}} \geq &\frac{1}{\|\beta_K b^K t^{2/3}\|_\mathcal{M}} - \frac{1}{2\|\beta_K b^K\|_\mathcal{M}^3 t^2}(2t^{2/3}\beta_K\langle \mathcal{M}_K b^K,\varphi(t)\rangle \\ & + 2\langle \mathcal{M}_K\varphi(t),\Tilde{x}^0\rangle + 2t^{2/3}\beta_K\langle\mathcal{M}_K b^K,\Tilde{x}^0\rangle + \|\varphi(t)\|_\mathcal{M}^2+\|\Tilde{x}^0\|_\mathcal{M}^2), \end{split}$$ which holds because of the convexity of the norm, we obtain $$\begin{split} \mathcal{A}_K(\varphi) &\geq \int_{1}^{+\infty} \sum_{i,j\in K,\ i<j} \frac{1}{2M_K} m_i m_j|\dot{\varphi}_{ij}(t)|^2 + \frac{2}{9}\frac{\beta_K}{M_K}\langle \mathcal{M}_K b^K,\varphi(t)\rangle \\ & \hspace{4mm}- \frac{U_{min}}{2\beta_K^3\|b^K\|_\mathcal{M}^3 t^2}(2t^{2/3}\beta_K\langle \mathcal{M}_K b^K,\varphi(t)\rangle + 2\langle \mathcal{M}_K\varphi(t),\Tilde{x}^0\rangle + 2t^{2/3}\beta_K\langle\mathcal{M}_K b^K,\Tilde{x}^0\rangle + \|\varphi(t)\|_\mathcal{M}^2+\|\Tilde{x}^0\|_\mathcal{M}^2)\ \mathrm{d}t\\ & = \int_{1}^{+\infty} \frac{1}{2}\|\dot{\varphi}(t)\|_\mathcal{M}^2 \\ & \hspace{4mm} - \frac{U_{min}}{2\beta_K^3\|b^K\|_\mathcal{M}^3 t^2}(2\langle \mathcal{M}_K\varphi(t),\Tilde{x}^0\rangle + 2t^{2/3}\beta_K\langle\mathcal{M}_K b^K,\Tilde{x}^0\rangle + \|\varphi(t)\|_\mathcal{M}^2+\|\Tilde{x}^0\|_\mathcal{M}^2)\ \mathrm{d}t \end{split}$$ We notice that the term $$C_{13}:=\int_{1}^{+\infty} - \frac{U_{min}}{2\beta_K^3\|b^K\|_\mathcal{M}^3 t^2}( 2t^{2/3}\beta_K\langle\mathcal{M}_K b^K,\Tilde{x}^0\rangle + \|\Tilde{x}^0\|_\mathcal{M}^2)\ \mathrm{d}t$$ is constant and finite. Using Hardy and Cauchy-Schwartz inequalities we also have $$\begin{split} -\int_{1}^{+\infty} \frac{U_{min}}{\beta_K^3\|b^K\|_\mathcal{M}^3 t^2} \langle \mathcal{M}_K\varphi(t),\Tilde{x}^0\rangle\ \mathrm{d}t &= -\frac{2}{9} \int_{1}^{+\infty} \frac{1}{t^2} \langle \mathcal{M}_K\varphi(t),\Tilde{x}^0\rangle\ \mathrm{d}t\\ & \geq - \frac{2}{9} \bigg(\int_{1}^{+\infty} \frac{\|\varphi(t)\|_\mathcal{M}^2}{t^2}\ \mathrm{d}t\bigg)^{1/2}\bigg( \int_{1}^{+\infty} \frac{\|\Tilde{x}^0\|_\mathcal{M}^2}{t^2}\ \mathrm{d}t\bigg)^{1/2} \\ & \geq -C_{14} \|\varphi\|_\mathcal{D}, \end{split}$$ where $C_{14}:= \frac{8}{9}\big(\int_{1}^{+\infty} \frac{\|\Tilde{x}^0\|_\mathcal{M}^2}{t^2}\ \mathrm{d}t\big)^{1/2} < +\infty$. Again by Hardy inequality, we obtain $$\mathcal{A}_K(\varphi) \geq \frac{1}{18}\|\varphi\|_\mathcal{D}^2- C_{14}\|\varphi\|_\mathcal{D}+C_{13},$$ which implies that the functional $\mathcal{A}_K$ is coercive.\ We now focus on studying the terms $$\begin{split} \mathcal{A}_{K_1,K_2}(\varphi) := \sum_{i\in K_1,\ j\in K_2} &\int_{1}^{+\infty}\frac{1}{2M_{K_{1,2}}} m_i m_j|\dot{\varphi}_i(t)-\dot{\varphi}_j(t)|^2+ \frac{m_i m_j}{|\varphi_{ij}(t)+a_{ij}t+\beta_{K_{1,2}} b^{K_{1,2}}_{ij}t^{2/3}+\Tilde{x}^0_{ij}|} - \frac{m_i m_j}{|a_{ij}t|}\ \mathrm{d}t. \end{split}$$ **Remark 28**. We notice that if two bodies of the configuration $b^{K_{1,2}}$ belong to different clusters and have collisions, that is, if there are $i\in K_1$ and $j\in K_2$ such that $b^{K_{1,2}}_i=b^{K_{1,2}}_j$, then the functional reads $$\mathcal{A}_{K_1,K_2}(\varphi) = \sum_{i\in K_1,\ j\in K_2} \int_{1}^{+\infty} \frac{1}{2M_{K_{1,2}}} m_i m_j|\dot{\varphi}_i(t)-\dot{\varphi}_j(t)|^2+\frac{m_i m_j}{|\varphi_{ij}(t)+a_{ij}t+\Tilde{x}^0_{ij}|} - \frac{m_i m_j}{|a_{ij}t|}\ \mathrm{d}t.$$ Since $a_i\neq a_j$ when $i\in K_1$, $j\in K_2$ and $K_1\neq K_2$, we have already proved that in this case the action functional $\mathcal{A}$ is coercive. Assuming $b^{K_{1,2}}$ without collisions, we proceed in the following way. By the triangular inequality, we have $$\begin{split} & \int_{1}^{+\infty} \frac{1}{|\varphi_{ij}(t)+a_{ij}t+\beta_{K_{1,2}}b^{K_{1,2}}_{ij}t^{2/3}+\Tilde{x}^0_{ij}|} - \frac{1}{|a_{ij}t|}\ \mathrm{d}t \\ &\geq \int_{1}^{+\infty} \frac{1}{\|\varphi_{ij}\|_\mathcal{D} t^{1/2}+|a_{ij}|t+\beta_{K_{1,2}}|b^{K_{1,2}}_{ij}|t^{2/3} + |\Tilde{x}^0_{ij}|} - \frac{1}{|a_{ij}|t}\ \mathrm{d}t. \end{split}$$ Using the changes of variables $s=\|\varphi\|_\mathcal{D}u$, we obtain $$\begin{split} &\int_{1}^{+\infty} \frac{1}{\|\varphi_{ij}\|_\mathcal{D} t^{1/2}+|a_{ij}|t+\beta_{K_{1,2}}|b^{K_{1,2}}_{ij}|t^{2/3} + |\Tilde{x}^0_{ij}|} - \frac{1}{|a_{ij}|t}\ \mathrm{d}t\\ & = 2 \int_{1}^{+\infty} \bigg(\frac{1}{\|\varphi_{ij}\|_\mathcal{D} s+|a_{ij}|s^2+\beta_{K_{1,2}}|b^{K_{1,2}}_{ij}|s^{4/3} + |\Tilde{x}^0_{ij}|} - \frac{1}{|a_{ij}|s^2}\bigg)s\ \mathrm{d}s\\ & = \frac{2}{\|\varphi\|_\mathcal{D}|a_{ij}|} \int_{1}^{+\infty} \bigg(\frac{1}{s^2+\frac{\beta_{K_{1,2}}|b^{K_{1,2}}_{ij}|}{|a_{ij}|}\frac{s^{4/3}}{\|\varphi\|_\mathcal{D}^{2/3}}+\frac{s}{|a_{ij}|\|\varphi\|_\mathcal{D}}+\frac{|\Tilde{x}^0_{ij}|}{|a_{ij}|\|\varphi\|_\mathcal{D}}} -\frac{1}{\frac{s^2}{\|\varphi\|_\mathcal{D}^2}}\bigg)s\ \mathrm{d}s\\ & = \frac{2}{|a_{ij}|} \int_{1/\|\varphi\|_\mathcal{D}}^{+\infty}\bigg( \frac{1}{u^2+\frac{\beta_{K_{1,2}}|b^{K_{1,2}}_{ij}|}{|a_{ij}|}\frac{u^{4/3}}{\|\varphi\|_\mathcal{D}^{2/3}}+\frac{u}{|a_{ij}|}+\frac{|\Tilde{x}^0_{ij}|}{|a_{ij}|\|\varphi\|_\mathcal{D}}} -\frac{1}{u^2}\bigg)u\ \mathrm{d}u. \end{split}$$ We can observe that for $\|\varphi\|_\mathcal{D}\rightarrow+\infty$, we have $\frac{\beta_{K_{1,2}}|b^{K_{1,2}}_{ij}|}{|a_{ij}|\|\varphi\|_\mathcal{D}^{2/3}}\leq 1$ and $\frac{|\Tilde{x}^0_{ij}|}{|a_{ij}|\|\varphi\|_\mathcal{D}} \leq 1$. So, for $\|\varphi\|_\mathcal{D}\rightarrow+\infty$, it follows $$\begin{split} & \frac{2}{|a_{ij}|} \int_{1/\|\varphi\|_\mathcal{D}}^{+\infty}\bigg( \frac{1}{u^2+\frac{\beta_{K_{1,2}}|b^{K_{1,2}}_{ij}|}{|a_{ij}|}\frac{u^{4/3}}{\|\varphi\|_\mathcal{D}^{2/3}}+\frac{u}{|a_{ij}|}+\frac{|\Tilde{x}^0_{ij}|}{|a_{ij}|\|\varphi\|_\mathcal{D}}} -\frac{1}{u^2}\bigg)u\ \mathrm{d}u \\ & \geq \frac{2}{|a_{ij}|} \int_{1/\|\varphi\|_\mathcal{D}}^{+\infty}\bigg( \frac{1}{u^2+u^{4/3}+\frac{u}{|a_{ij}|}+1} -\frac{1}{u^2}\bigg)u\ \mathrm{d}u\\ & = \frac{2}{|a_{ij}|} \int_{1/\|\varphi\|_\mathcal{D}}^{+\infty} \frac{1}{u} \bigg( \frac{1}{1+u^{-2/3}+\frac{u^{-1}}{|a_{ij}|}+u^{-1}} -1 \bigg)\ \mathrm{d}u. \end{split}$$ Since $1/\|\varphi\|_\mathcal{D}\leq1$ when $\|\varphi\|_\mathcal{D}\rightarrow+\infty$, we can study the integral separately on the intervals $[1/\|\varphi\|_\mathcal{D},1]$ and $[1,+\infty)$. On the second interval, the integral is constant (let us say that it is equal to a constant $C_{15})$. On the other interval, we have $$\begin{split} & \frac{2}{|a_{ij}|} \int_{1/\|\varphi\|_\mathcal{D}}^{1} \frac{1}{u} \bigg( \frac{1}{1+u^{-2/3}+\frac{u^{-1}}{|a_{ij}|}+u^{-1}} -1 \bigg)\ \mathrm{d}u \geq \frac{2}{|a_{ij}|} \int_{1/\|\varphi\|_\mathcal{D}}^{1} -\frac{\mathrm{d}u}{u}. \end{split}$$ We have thus demonstrated that $$\begin{split} \int_{1}^{+\infty} \frac{1}{|\varphi_{ij}(t)+a_{ij}t+\beta_{K_{1,2}}b^{K_{1,2}}_{ij}t^{2/3}+\Tilde{x}^0_{ij}|} - \frac{1}{|a_{ij}t|}\ \mathrm{d}t \geq \frac{2}{|a_{ij}|} \log{\frac{1}{\|\varphi\|_\mathcal{D}}}+C_{15} = -\frac{2}{|a_{ij}|}\log{\|\varphi\|_\mathcal{D}}+C_{15}, \end{split}$$ which concludes the proof of the coercivity of the Lagrangian action. ## Weak lower semicontinuity of $\mathcal{A}(\varphi)$ In order to prove the weak lower semicontinuity of the Lagrangian action, we can use the decomposition [\[action_partially_hyperbolic\]](#action_partially_hyperbolic){reference-type="eqref" reference="action_partially_hyperbolic"} and study the weak lower semicontinuity of the terms $\mathcal{A}_K$ and $\mathcal{A}_{K_1,K_2}$ separately, given arbitrary clusters $K,K_1,K_2\in\mathcal{P}$. Concerning the term $\mathcal{A}_K$, we can refer to Section [4](#sec_parabolic){reference-type="ref" reference="sec_parabolic"}, since our choice of $\beta_K b^K$ leads us to the same computations. For the proof of the weak lower semicontinuity of the terms $\mathcal{A}_{K_1,K_2}$, let us consider a sequence $(\varphi^n)_n\subset\mathcal{D}_0^{1,2}(1,+\infty)$ converging weakly in $\mathcal{D}_0^{1,2}(1,+\infty)$ to some $\varphi$, as $n\rightarrow+\infty$. It follows that there is a constant $k\in\mathbb{R}$ such that $\|\varphi^n\|_\mathcal{D}\leq k$ and $\|\varphi\|_\mathcal{D}\leq k$ for every $n\in\mathbb{N}$. We would like to use the inequality $$\label{alternative_proof_second_equality_with_s} \frac{1}{|\varphi^n_{ij}(t) + a_{ij}t + \beta_{K_{1,2}}b^{K_{1,2}}_{ij}t^{2/3} + \Tilde{x}^0_{ij}|} - \frac{1}{|a_{ij}t|} = \int_{0}^{1} \frac{\mathrm{d}}{\mathrm{d}s}\bigg[\frac{1}{|a_{ij}t + s(\varphi^n_{ij}(t) + \beta_{K_{1,2}}b^{K_{1,2}}_{ij}t^{2/3} + \Tilde{x}^0_{ij})|}\bigg]\ \mathrm{d}s,$$ which holds true when the denominator of the integrand is not zero. For all $s\in(0,1)$ we have $$\begin{split} |a_{ij}t + s(\varphi^n_{ij}(t) + \beta_{K_{1,2}}b^{K_{1,2}}_{ij}t^{2/3} + \Tilde{x}^0_{ij})| & \geq |a_{ij}|t - s(\|\varphi^n_{ij}\|_\mathcal{D}t^{1/2} + |\beta_{K_{1,2}}b^{K_{1,2}}_{ij}|t^{2/3} + |\Tilde{x}^0_{ij}|) \\ & > |a_{ij}|t - (\|\varphi^n_{ij}\|_\mathcal{D}t^{1/2} + |\beta_{K_{1,2}}b^{K_{1,2}}_{ij}|t^{2/3} + |\Tilde{x}^0_{ij}|), \end{split}$$ and since $\|\varphi^n_{ij}\|_\mathcal{D}\leq k$, we have $$|a_{ij}t + s(\varphi^n_{ij}(t) + \beta_{K_{1,2}}b^{K_{1,2}}_{ij}t^{2/3} + \Tilde{x}^0_{ij})| > |a_{ij}|t - (k t^{1/2} + |\beta_{K_{1,2}}b^{K_{1,2}}_{ij}|t^{2/3} + |\Tilde{x}^0_{ij}|),$$ where the last term is larger then zero if $t\geq\bar{T}=\bar{T}(k)$, for a proper $\bar{T}$. We can thus study the weak lower semicontinuity of the potential term separately on the two intervals $[1,\bar{T}]$ and $[\bar{T},+\infty)$. On $[1,\bar{T}]$, the weak lower semicontinuity easily follows from Fatou's Lemma. On $[\bar{T},+\infty)$, we can use [\[alternative_proof_second_equality_with_s\]](#alternative_proof_second_equality_with_s){reference-type="eqref" reference="alternative_proof_second_equality_with_s"}: $$\begin{split} &\int_{\bar{T}}^{+\infty} \frac{1}{|\varphi^n_{ij}(t) + a_{ij}t + \beta_{K_{1,2}}b^{K_{1,2}}_{ij}t^{2/3} + \Tilde{x}^0_{ij}|} - \frac{1}{|a_{ij}t|}\ \mathrm{d}t \\ & = \int_{\bar{T}}^{+\infty} \bigg( \int_{0}^{1} -\frac{[a_{ij}t + s(\varphi^n_{ij}(t) + \beta_{K_{1,2}}b^{K_{1,2}}_{ij}t^{2/3} + \Tilde{x}^0_{ij})](\varphi^n_{ij}(t) + \beta_{K_{1,2}}b^{K_{1,2}}_{ij}t^{2/3} + \Tilde{x}^0_{ij})}{|a_{ij}t + s(\varphi^n_{ij}(t) + \beta_{K_{1,2}}b^{K_{1,2}}_{ij}t^{2/3} + \Tilde{x}^0_{ij})|^3}\ \mathrm{d}s \bigg)\ \mathrm{d}t. \end{split}$$ Using [\[alternative_proof_inequality\]](#alternative_proof_inequality){reference-type="eqref" reference="alternative_proof_inequality"}, we then have $$\begin{split} &\int_{\bar{T}}^{+\infty}\bigg| \frac{1}{|\varphi^n_{ij}(t) + a_{ij}t + \beta_{K_{1,2}}b^{K_{1,2}}_{ij}t^{2/3} + \Tilde{x}^0_{ij}|} - \frac{1}{|a_{ij}t|}\bigg|\ \mathrm{d}t \\ & \leq \int_{\bar{T}}^{+\infty} \bigg( \int_{0}^{1} \frac{|\varphi^n_{ij}(t) + \beta_{K_{1,2}}b^{K_{1,2}}_{ij}t^{2/3} + \Tilde{x}^0_{ij}|}{|a_{ij}t + s(\varphi^n_{ij}(t) + \beta_{K_{1,2}}b^{K_{1,2}}_{ij}t^{2/3} + \Tilde{x}^0_{ij})|^2}\ \mathrm{d}s\bigg)\ \mathrm{d}t\\ & \leq \int_{\bar{T}}^{+\infty} \bigg( \int_{0}^{1} \frac{3(|kt^{1/2} + \beta_{K_{1,2}}b^{K_{1,2}}_{ij}t^{2/3}| + |\Tilde{x}^0_{ij}|)}{|a_{ij}t|^2 - s|kt^{1/2} + \beta_{K_{1,2}}b^{K_{1,2}}_{ij}t^{2/3} + \Tilde{x}^0_{ij}|^2}\ \mathrm{d}s\bigg)\ \mathrm{d}t\\ & \leq \int_{\bar{T}}^{+\infty} \bigg( \int_{0}^{1} \frac{3k't^{2/3}}{|a_{ij}|^2 t^2 - sk't^{4/3}}\ \mathrm{d}s\bigg)\ \mathrm{d}t, \end{split}$$ where $k'\in\mathbb{R}$ is big enough so that $|kt^{1/2}| + |\beta_{K_{1,2}}b^{K_{1,2}}_{ij}t^{2/3} + \Tilde{x}^0_{ij}|\leq \sqrt{k'}t^{2/3}$. The denominator of the last integral is positive when $$t>\bigg(\frac{|a_{ij}|^2}{k'}\bigg)^{2/3}=:\hat{T}.$$ If we choose $\bar{T}(k)\gg\hat{T}$, the last integral is finite, which means that $$\int_{\bar{T}}^{+\infty}\bigg| \frac{1}{|\varphi^n_{ij}(t) + a_{ij}t + \beta_{K_{1,2}}b^{K_{1,2}}_{ij}t^{2/3} + \Tilde{x}^0_{ij}|} - \frac{1}{|a_{ij}t|}\bigg|\ \mathrm{d}t < +\infty.$$ This implies the $L^1$-convergence of the potential term, which proves its weak lower semicontinuity. ## The action is of class $C^1$ over non-collision sets The last thing we have to prove is that the action is of class $C^1$ over sets of motions that don't undergo collisions. We have already proved this result for the terms $\mathcal{A}_K$, so we can only focus on the terms $\mathcal{A}_{K_1,K_2}$. In particular, denoting by $\mathcal{A}_{K_1,K_2}^2$ the potential term, we wish to prove that the differential $$\mathrm{d}\mathcal{A}_{K_1,K_2}(\varphi)[\psi] = \int_{1}^{+\infty} \langle \nabla U(\varphi(t)+at+\beta_{K_{1,2}} b^{K_{1,2}}t^{2/3}+\Tilde{x}^0), \psi(t)\rangle\ \mathrm{d}t$$ is continuous, for every $\varphi,\psi\in\mathcal{D}_0^{1,2}(1,+\infty)$, over the set of non-collisional configurations when the potential $U$ is restricted to the clusters $K_1$ and $K_2$. First of all, we have $$\|\nabla U(\varphi(t)+at+\beta_{K_{1,2}} b^{K_{1,2}}t^{2/3}+\Tilde{x}^0)\|_\mathcal{M} \leq C_{16}\sum_{i\in K_1,\ j\in K_2}\frac{1}{|\varphi^n_{ij}(t) + a_{ij}t + \beta_{K_{1,2}}b^{K_{1,2}}_{ij}t^{2/3} + \Tilde{x}^0_{ij}|^2}$$ for a proper constant $C_{16}$, where the right-hand side term behaves like $1/t^2$ when $t\rightarrow+\infty$. This, together with the Cauchy-Schwartz inequality, proves that the differential is well-defined. Now, given $(\varphi^n)_n\subset\mathcal{D}_0^{1,2}(1,+\infty)$ such that $\varphi^n\rightarrow\varphi$ in $\mathcal{D}_0^{1,2}(1,+\infty)$ for some $\varphi$, we wish to prove that $$\sup_{\|\psi\|_\mathcal{D}\leq 1} \bigg|\int_{1}^{+\infty} \langle \nabla U(t,\varphi^n(t))-\nabla U(t,\varphi(t)),\psi(t)\rangle\ \mathrm{d}t\bigg| \rightarrow0,\qquad\text{as }n\rightarrow+\infty,$$ where we write $U(t,\varphi(t)):=U(\varphi(t)+at+\beta_{K_{1,2}} b^{K_{1,2}}t^{2/3}+\Tilde{x}^0)$ to lighten the notation. Using Cauchy-Schwartz and Hardy inequalities, we have $$\begin{split} &\sup_{\|\psi\|_\mathcal{D}\leq 1} \bigg|\int_{1}^{+\infty} \langle \nabla U(t,\varphi^n(t))-\nabla U(t,\varphi(t)),\psi(t)\rangle\ \mathrm{d}t\bigg| \\ &\leq \sup_{\|\psi\|_\mathcal{D}\leq 1} \int_{1}^{+\infty} t\| \nabla U(t,\varphi^n(t))-\nabla U(t,\varphi(t))\|_\mathcal{M} \frac{\|\psi(t)\|_\mathcal{M}}{t}\ \mathrm{d}t\\ &\leq \sup_{\|\psi\|_\mathcal{D}\leq 1} \bigg( \int_{1}^{+\infty} \frac{\|\psi(t)\|_\mathcal{M}^2}{t^2}\ \mathrm{d}t\bigg)^{1/2} \bigg( \int_{1}^{+\infty} t^2 \| \nabla U(t,\varphi^n(t))-\nabla U(t,\varphi(t))\|_\mathcal{M}^2\ \mathrm{d}t\bigg)^{1/2} \\ & \leq 2 \bigg( \int_{1}^{+\infty} t^2 \| \nabla U(t,\varphi^n(t))-\nabla U(t,\varphi(t))\|_\mathcal{M}^2\ \mathrm{d}t\bigg)^{1/2}. \end{split}$$ Now, we can write $$\begin{split} &\int_{1}^{+\infty} t^2 \| \nabla U(t,\varphi^n(t))-\nabla U(t,\varphi(t))\|_\mathcal{M}^2\ \mathrm{d}t \\ & = \int_{1}^{+\infty} t^2 \bigg| \int_{0}^{1} \nabla^2 U(\varphi(t)+at+\beta_{K_{1,2}} b^{K_{1,2}}t^{2/3}+\Tilde{x}^0+s(\varphi^n(t)-\varphi(t)))(\varphi^n(t)-\varphi(t))\ \mathrm{d}s\ \bigg|^2\ \mathrm{d}t \\ & \leq \int_{1}^{+\infty} t^2 \bigg( \int_{0}^{1} C_{16} \sum_{i\in K_1,\ j\in K_2}\frac{1}{|\varphi_{ij}(t) + a_{ij}t + \beta_{K_{1,2}}b^{K_{1,2}}_{ij}t^{2/3} + \Tilde{x}^0_{ij} + s(\varphi^n_{ij}(t)-\varphi_{ij}(t))|^3}\|\varphi^n(t)-\varphi(t)\|_\mathcal{M}\ \mathrm{d}s\bigg)^2\ \mathrm{d}t \\ & \leq \int_{1}^{+\infty} \bigg( \int_{0}^{1} C_{16} \sum_{i\in K_1,\ j\in K_2}\frac{1}{|\varphi_{ij}(t) + a_{ij}t + \beta_{K_{1,2}}b^{K_{1,2}}_{ij}t^{2/3} + \Tilde{x}^0_{ij} + s(\varphi^n_{ij}(t)-\varphi_{ij}(t))|^3}\|\varphi^n-\varphi\|_\mathcal{D} t^{3/2}\ \mathrm{d}s\bigg)^2\ \mathrm{d}t \\ & \leq C_{17}\|\varphi^n-\varphi\|_\mathcal{D} \end{split}$$ for a proper constant $C_{17}\in\mathbb{R}$, where the last term goes to zero as $n\rightarrow+\infty$. This concludes the proof. ## Absence of collisions and partial hyperbolicity of the motion Again, Marchal's Theorem implies that the motion we are considering has no collisions. Given $$x(t) = \varphi(t) + at + \beta b t^{2/3} + \Tilde{x}^0,$$ we have $$\dot{x}(t) = \dot{\varphi}(t) + a + \frac{2}{3}\beta b t^{-1/3}.$$ In this case, the conservation of the energy implies that the energy of the motion is positive. **Remark 29**. We observe that Chazy's Theorem can be applied to the cases of hyperbolic and hyperbolic-parabolic motions, because for completely parabolic motions the energy constant of the internal motion is null. In such cases, the limit shape of $x(t)$ is the shape of the configuration $a$ and, moreover, $L = \lim_{t\rightarrow+\infty}\frac{\max_{i<j}|x_{ij}(t)|}{\min_{i<j}|x_{ij}(t)|} < +\infty$ if and only if $x$ is hyperbolic. If the energy $h > 0$ and $L = +\infty$, then either the motion is hyperbolic-parabolic or it is not expansive. In our case, it is trivial to prove that $L=+\infty$, which implies that the motion is hyperbolic-parabolic. **Remark 30**. We can observe that if the indexes $i,j$ belong to the same cluster, we have $\dot{x}_{ij}(t)\rightarrow0$ when $t\rightarrow+\infty$, while if $i,j$ belong to different clusters, we have $\dot{x}_{ij}(t)\rightarrow a_{ij}$ when $t\rightarrow+\infty$. ## Hyperbolic-parabolic motions' asymtptotic expansion We have seen that a hyperbolic-parabolic motion $x$ can be written in the form $x(t) = at + \beta bt^{2/3} + \varphi(t) + \Tilde{x}^0$, as shown in [\[explicit_hyperbolic-parabolic\]](#explicit_hyperbolic-parabolic){reference-type="eqref" reference="explicit_hyperbolic-parabolic"}, and that the bodies can be divided into subgroups following the natural cluster partition introduced in Definition [Definition 25](#def_cluster_partition){reference-type="ref" reference="def_cluster_partition"}. In this section, we will prove that the centers of mass of the clusters follow hyperbolic orbits. Besides, we will show that inside each cluster, the bodies move with respect to the center of mass of the cluster following a parabolic path. We start with proving that the centers of mass of each cluster have a hyperbolic expansion. For a cluster $K$, denoting the center of mass of $K$ as $$c^K(t) = \frac{1}{M_K}\sum_{i\in K}m_i x_i(t),$$ we can compute the equations of motion of the center of mass as $$\begin{split} M_K \ddot{c}^K(t) & = \sum_{i\in K} m_i \ddot{x}_i(t) \\ & = -\sum_{i\in K}\sum_{j\neq i} m_i m_j \frac{x_i(t) - x_j(t)}{|x_i(t) - x_j(t)|^3} \\ & = -\sum_{i\in K}\sum_{j\notin K} m_i m_j \frac{x_i(t) - x_j(t)}{|x_i(t) - x_j(t)|^3}. \end{split}$$ It is easy to see that the right-hand side of the equation is a $O\big(\frac{1}{t^2}\big)$-term for $t\rightarrow+\infty$. We also notice that $$-\sum_{i\in K}\sum_{j\notin K} m_i m_j \frac{x_i(t) - x_j(t)}{|x_i(t) - x_j(t)|^3} \simeq -\frac{1}{t^2}\sum_{i\in K}\sum_{j\notin K} m_i m_j \frac{a_i - a_j}{|a_i - a_j|^3} + O\bigg(\frac{1}{t^3}\bigg),$$ for $t\rightarrow+\infty$. We can define $$\Tilde{\nabla}U(a^K) = -\sum_{i\in K}\sum_{j\notin K} m_i m_j \frac{a_i - a_j}{|a_i - a_j|^3},$$ which can be seen as a restriction of $\nabla U(a^K)$. Denoting with $a^K$ the restriction of the configuration $a$ to the cluster $K$, we can thus compute $$\lim_{t\rightarrow+\infty} \frac{M_K c^K(t)}{\log t} = \lim_{t\rightarrow+\infty} \frac{M_K \dot{c}^K(t)}{\frac{1}{t}} = -\lim_{t\rightarrow+\infty} \frac{M_K \ddot{c}^K(t)}{\frac{1}{t^2}} = -\Tilde{\nabla}U(a^K).$$ This implies that the center of mass of the cluster $K$ has the hyperbolic asymptotic expansion $$c^K(t) = a^K t - \Tilde{\nabla}U(a^K)\log t + o(\log t),$$ for $t\rightarrow +\infty$. Now, considering an index $i\in K$, we denote the motion of a body $x_i$ with respect to the center of mass of its cluster as $$y_i(t) = x_i(t) - c^K_i(t).$$ We are going to show that its asymptotic expansion is a parabolic one. If the cluster only has one element, we obviously have $y_i\equiv0$, so we consider the case where $K$ has two or more elements. The equation of motion reads $$\begin{split} m_i \Ddot{y}_i(t) & = m_i \Ddot{x}_i(t) - m_i\Ddot{c}^K_i(t) \\ & = -\sum_{j\in K} m_i m_j \frac{x_i(t) - x_j(t)}{|x_i(t) - x_j(t)|^3} - \sum_{j\notin K} m_i m_j \frac{x_i(t) - x_j(t)}{|x_i(t) - x_j(t)|^3} - m_i \Ddot{c}^K_i(t). \end{split}$$ Since we already know that $-\sum_{j\notin K} m_i m_j \frac{x_i(t) - x_j(t)}{|x_i(t) - x_j(t)|^3} - m_i \Ddot{c}^K_i(t) = O\big(\frac{1}{t^2}\big)$ for $t\rightarrow+\infty$, we can then say that $$m_i \Ddot{y}_i(t) = -\sum_{j\in K} m_i m_j \frac{y_i(t) - y_j(t)}{|y_i(t) - y_j(t)|^3} + O\big(\frac{1}{t^2}\big).$$ Using the definition of $x(t)$ and the asymptotic expansion of $c^K(t)$ we found above, we can easily see that $$y_i(t) = \beta_K b^K_i t^{2/3} + \varphi_i(t) - \log t\sum_{j\notin K} m_i m_j \frac{a_i - a_j}{|a_i - a_j|^3} + o(\log t),$$ for $t\rightarrow+\infty$, where $\beta_K = \sqrt[3]{\frac{9}{2}\min_{K} U}$. Defining $\psi_i(t):=\varphi_i(t) - \sum_{j\notin K} m_i m_j \frac{a_i - a_j}{|a_i - a_j|^3} + o(\log t)$, it is easy to prove that $\psi_i\in\mathcal{D}^{1,2}(1,+\infty)$. We can then apply the estimate [\[new_estimate_parabolic\]](#new_estimate_parabolic){reference-type="eqref" reference="new_estimate_parabolic"} to say that $$y_i(t) = \beta_K b_i^K t^{2/3} + o(t^{\frac{1}{3}+}),$$ for $t\rightarrow+\infty$. # Free-time minimization property "Jacobi's principle brings out vividly the intimate relationship which exists between the motions of conservative holonomic systems and the geometry of curved space" (C. Lanczos, [@Lanczos page 138]). Accordingly, trajectories of the $N$-body problem at energy $h$ are geodesics of the Jacobi-Maupertuis' metric of level $h$ in the configuration space, i.e., $$\mathrm{d}\sigma^2=(U+h)\mathrm{d}s_\mathcal{M}^2,$$ being $\mathrm{d}s_\mathcal{M}^2$ the mass Euclidean metric in the configuration space. **Definition 31**. A curve $x:[1, +\infty) \rightarrow E^N$ is said to be a geodesic ray from $p \in E^N$ if $x(1) = p$ and each restriction to a compact interval is a minimizing geodesic. In [@MadernaVenturelli_HyperbolicMotions], Maderna and Venturelli also proved the following theorem. **Theorem 32** (Maderna-Venturelli, 2020 [@MadernaVenturelli_HyperbolicMotions]). *Let $E$ be an Euclidean space. For any $h>0$, $p \in E^N$ and $a \in \Omega$, there is geodesic ray of the Jacobi-Maupertuis' metric of level $h$ with asymptotic direction $a$ and starting at $p$.* In order to relate geodesics of the Jacobi-Maupertuis' metric to the action minimizing trajectories of our Lagrangian systems we need the following definition. **Definition 33**. A curve $x:I\rightarrow\mathcal{X}$ is a free-time minimizer for the Lagrangian action at energy $h$ if $\forall\ [a,b],[a',b']\subset I$ and $\forall\ \sigma:[a',b']\rightarrow\mathcal{X}$ such that $\gamma(a)=\sigma(a')$ and $\gamma(b)=\sigma(b')$, it holds $$\int_{a}^{b} L(\gamma,\dot{\gamma})\ \mathrm{d}t+h(b-a) \leq \int_{a'}^{b'} L(\sigma,\dot{\sigma})\ \mathrm{d}t+h(b'-a').$$ In light of the equivalence between the variational property of being an unbouded free-time minimizer of the Lagrangian action at energy $h$ and the geometrical property of being a geodesic ray for the Jacobi-Mapertuis metric at the same energy level (cfr. [@Lanczos; @MR2269239]), we show here that our existence results of expansive motions through the minimization of the renormalized action do indeed agree with Theorem [Theorem 32](#thm_MV_geodesicrays){reference-type="ref" reference="thm_MV_geodesicrays"}. More precisely, we prove the following corollary. **Corollary 34**. *Consider an expansive motion $x:[1,+\infty)\rightarrow\mathcal{X}$ of the Newtonian $N$-body problem of the form $$x(t)=r_0(t) + \varphi(t)+\Tilde x_0,$$ where $\varphi\in\mathcal{D}_0^{1,2}(1,+\infty)$ minimizes the renormalized action in [\[eq:renorm_action\]](#eq:renorm_action){reference-type="eqref" reference="eq:renorm_action"} in any of the settings of Theorems [Theorem 6](#thm_hyperbolic){reference-type="ref" reference="thm_hyperbolic"}, [Theorem 7](#thm_parabolic){reference-type="ref" reference="thm_parabolic"} and [Theorem 8](#thm_partially_hyperbolic){reference-type="ref" reference="thm_partially_hyperbolic"}. Then $x$ is actually a free-time minimizer at its energy level. Therefore it is a geodesic ray for the Jacobi-Maupertuis' metric.* *Proof.* We consider a curve $\gamma:[1,+\infty)\rightarrow\mathcal{X}$ of the form $\gamma(t) = r_0(t) + \varphi(t) + \Tilde{x}^0$ such that $\varphi$ minimizes the renormalized Lagrangian action on $\mathcal{D}_0^{1,2}(1,+\infty)$. By contradiction, we suppose that there are $T$ and $\bar{T}$, $\varepsilon>0$ and there is some curve $\bar{\sigma}:[1,\bar{T}]\rightarrow\mathcal{X}$ with $\gamma(T)=\bar{\sigma}(\bar{T})$ such that $$\label{eq_freetime_1} \int_{1}^{T} L(\gamma,\dot{\gamma})\ \mathrm{d}t +hT> \int_{1}^{\bar{T}} L(\bar{\sigma},\dot{\bar{\sigma}})\ \mathrm{d}t+h\bar T + \varepsilon.$$ By a density and continuity argument, we can then define a compactly supported function $\Tilde{\varphi}$ such that $\Tilde{\varphi}(t)=\varphi(t)$ on $[1,\hat{T}]$, where $\hat{T}\gg \max\{T,\bar{T}\}$, and $\Tilde{\varphi}$ is close enough to $\varphi$ in the $\mathcal{D}_0^{1,2}$-norm to have $$\mathcal{A}(\Tilde{\varphi}) \leq \mathcal{A}(\varphi) + \varepsilon,$$ where $\mathcal{A}$ is the renormalized Lagrangian action. By the minimizing property of $\varphi$ we infer $$\label{eq:minepsilon} \mathcal{A}(\Tilde{\varphi}) \leq \mathcal{A}(\psi) + \varepsilon,\quad\forall \psi\in\mathcal D^{1,2}_0([1,+\infty)).$$ Now, denoting $\Tilde{\gamma}(t) = r_0(t) + \Tilde{\varphi}(t) + \Tilde{x}^0$, we build a curve $\tilde{\sigma}:[1,+\infty)\rightarrow\mathcal{X}$ such that $$\tilde{\sigma}(t) = \begin{cases} \bar{\sigma}(t),\quad t\in[1,\bar{T}]\\ \tilde{\gamma}(t-\bar{T}+T),\quad t\in[\bar{T},+\infty) \end{cases}.$$ Since we supposed that $\gamma(T)=\bar{\sigma}(\bar{T})$, we know for sure that $\tilde{\sigma}$ is continuous. Moreover we define $\bar\varphi(t)=\bar\sigma(t)-r_0(t)-\tilde x_0$, so that $\bar\varphi\in\mathcal D^{1,2}_0(1,+\infty)$ and, by its definition, we have $$\label{eq:barphi} \bar\varphi(t)\equiv r_0(t-\bar T+T)-r_0(t)=a(T-\bar T)+o(1), \quad\forall t\gg\max\{T,\bar T\},$$ as $\Tilde{\varphi}$ is compactly supported. We notice that we can write $$\mathcal{A}(\tilde{\varphi}) = \int_{1}^{+\infty} L(\Tilde{\gamma},\dot{\Tilde{\gamma}}) - L_0(t)\ \mathrm{d}t,$$ which easily follows from the fact that also $L(\Tilde{\gamma},\dot{\tilde{\gamma}}) - L_0(t)\in L^1[1,+\infty)$ and furthermore $$\int_{1}^{+\infty} -\langle \mathcal{M}\ddot{r_0},\tilde\varphi\rangle\ \mathrm{d}t = -\langle \mathcal{M}\dot{r_0},\tilde\varphi\rangle\bigg|_1^{+\infty} + \int_{1}^{+\infty} \langle \mathcal{M}\dot{r_0},\dot{\tilde\varphi}\rangle\ \mathrm{d}t = \int_{1}^{+\infty} \langle \mathcal{M}\dot{r_0},\dot{\tilde\varphi}\rangle\ \mathrm{d}t.$$ On the other hand, from [\[eq:barphi\]](#eq:barphi){reference-type="eqref" reference="eq:barphi"}, using $\dot r_0\simeq t^{-1/3}$, it follows that $$\begin{split} \int_{1}^{+\infty} -\langle \mathcal{M}\ddot{r_0},\bar\varphi\rangle\ \mathrm{d}t = -\langle \mathcal{M}\dot{r_0},\bar\varphi\rangle\bigg|_1^{+\infty} + \int_{1}^{+\infty} \langle \mathcal{M}\dot{r_0},\dot{\bar\varphi}\rangle\ \mathrm{d}t= \langle \mathcal{M}a,a \rangle (\bar T-T)+\int_{1}^{+\infty} \langle \mathcal{M}\dot{r_0},\dot{\bar\varphi}\rangle\ \mathrm{d}t \\= 2h (\bar T-T)+\int_{1}^{+\infty} \langle \mathcal{M}\dot{r_0},\dot{\bar\varphi}\rangle\ \mathrm{d}t, \end{split}$$ where $h=H(r_0,\dot{r}_0)$ is the energy of $r_0$, which is positive equal to $\Vert a\Vert^2_\mathcal{M}/2$ in the hyperbolic and hyperbolic-parabolic case and zero in the completely parabolic case. Consequently we have $$\mathcal{A}(\bar{\varphi}) = 2h (\bar T-T)+\int_{1}^{+\infty} L(\bar{\sigma},\dot{\bar{\sigma}}) - L_0(t)\ \mathrm{d}t.$$ Let us denote $L^h=L-h$ and $L_0^h(t):=L(r_0(t))-h$. By [\[eq_freetime_1\]](#eq_freetime_1){reference-type="eqref" reference="eq_freetime_1"}, we can say that $$\label{eq_freetime_2} \begin{split} & \int_{1}^{T} L^h(\gamma,\dot{\gamma})\ \mathrm{d}t + \int_{\bar{T}}^{+\infty} L^h(\tilde{\sigma},\dot{\tilde{\sigma}})-L^h_0(t-\bar{T}+T)\ \mathrm{d}t \\ & > \int_{1}^{\bar{T}} L^h(\bar{\sigma},\dot{\bar{\sigma}})\ \mathrm{d}t + \int_{\bar{T}}^{+\infty} L^h(\tilde{\sigma},\dot{\tilde{\sigma}})-L^h_0(t-\bar{T}+T)\ \mathrm{d}t + \varepsilon+2h(\bar T-T). \end{split}$$ Working on left-hand side of equation [\[eq_freetime_2\]](#eq_freetime_2){reference-type="eqref" reference="eq_freetime_2"}, we obtain $$\begin{split} & \int_{1}^{T} L^h(\gamma,\dot{\gamma})\ \mathrm{d}t + \int_{\bar{T}}^{+\infty} L^h(\tilde{\sigma},\dot{\tilde{\sigma}})-L^h_0(t-\bar{T}+T)\ \mathrm{d}t \\ & = \int_{1}^{T} L^h(\gamma,\dot{\gamma})\ \mathrm{d}t + \int_{\bar{T}}^{+\infty} L^h(\Tilde{\gamma}(t-\bar{T}+T),\dot{\Tilde{\gamma}}(t-\bar{T}+T))-L^h_0(t-\bar{T}+T)\ \mathrm{d}t\\ & = \int_{1}^{T} L^h(\gamma,\dot{\gamma}) - L^h_0(t)\ \mathrm{d}t + \int_{T}^{+\infty} L^h(\Tilde{\gamma},\dot{\Tilde{\gamma}}) - L^h_0(t)\ \mathrm{d}t + \int_{1}^{T}L^h_0(t)\ \mathrm{d}t\\ & = \int_{1}^{+\infty} L^h(\Tilde{\gamma},\dot{\Tilde{\gamma}}) - L^h_0(t)\ \mathrm{d}t + \int_{1}^{T}L^h_0(t)\ \mathrm{d}t. \end{split}$$ On the other hand, working on right-hand side of [\[eq_freetime_2\]](#eq_freetime_2){reference-type="eqref" reference="eq_freetime_2"}, we have $$\begin{split} & \int_{1}^{\bar{T}} L^h(\bar{\sigma},\dot{\bar{\sigma}})\ \mathrm{d}t + \int_{\bar{T}}^{+\infty} L^h(\tilde{\sigma},\dot{\tilde{\sigma}})-L^h_0(t-\bar{T}+T)\ \mathrm{d}t +2h(\bar T-T)+ \varepsilon \\ & = \int_{1}^{\bar{T}} L^h(\bar{\sigma},\dot{\bar{\sigma}}) - L^h_0(t)\ \mathrm{d}t + \int_{\bar{T}}^{+\infty} L^h(\tilde{\sigma},\dot{\tilde{\sigma}})-L^h_0(t-\bar{T}+T) + L^h_0(t) - L^h_0(t)\ \mathrm{d}t + \\ &\hspace{9cm} +\int_{1}^{\bar{T}} L^h_0(t)\ \mathrm{d}t +2h(\bar T-T)+ \varepsilon \\ & = \int_{1}^{\bar{T}} L^h(\bar{\sigma},\dot{\bar{\sigma}}) - L^h_0(t)\ \mathrm{d}t + \int_{\bar{T}}^{+\infty} L^h(\tilde{\sigma},\dot{\tilde{\sigma}})-L^h_0(t)\ \mathrm{d}t+ \\ &\hspace{4cm} + \int_{1}^{\bar{T}} L^h_0(t)\ \mathrm{d}t + \int_{\bar{T}}^{+\infty} L^h_0(t) - L^h_0(t-\bar{T}+T)\ \mathrm{d}t +2h(\bar T-T)+ \varepsilon \\ & = \int_{1}^{+\infty} L^h(\tilde{\sigma},\dot{\tilde{\sigma}})-L^h_0(t)\ \mathrm{d}t + \int_{1}^{\bar{T}} L^h_0(t)\ \mathrm{d}t + \int_{\bar{T}}^{+\infty} L^h_0(t) - L^h_0(t-\bar{T}+T)\ \mathrm{d}t +2h(\bar T-T)+ \varepsilon. \end{split}$$ It thus follows that $$\begin{split} & \int_{1}^{+\infty} L^h(\Tilde{\gamma},\dot{\Tilde{\gamma}}) - L^h_0(t)\ \mathrm{d}t \\ &> \int_{1}^{+\infty} L^h(\tilde{\sigma},\dot{\tilde{\sigma}})-L^h_0(t)\ \mathrm{d}t + \int_{T}^{\bar{T}} L^h_0(t)\ \mathrm{d}t + \int_{\bar{T}}^{+\infty} L^h_0(t) - L^h_0(t-\bar{T}+T)\ \mathrm{d}t + 2h(\bar T-T)+\varepsilon. \end{split}$$ We recall the following property, which can be demonstrated as a simple exercise. **Proposition 35**. *Given a function $f\in L^1_{loc}(\mathcal{X})$ such that $f(t)\rightarrow0$ as $t\rightarrow\pm\infty$ and such that $f(t)-f(t-\tau)\in L^1$ for some $\tau\in\mathbb{R}$, then $$\int_{-\infty}^{+\infty} f(t) - f(t-\tau)\ \mathrm{d}t = 0.$$* Since $$\int_{T}^{\bar{T}} L^h_0(t)\ \mathrm{d}t + \int_{\bar{T}}^{+\infty} L^h_0(t) - L^h_0(t-\bar{T}+T)\ \mathrm{d}t = \int_{-\infty}^{+\infty} L^h_0(t)\mathcal{X}_{\{t>T\}} - L^h_0(t-\bar{T}+T)\mathcal{X}_{\{t>\bar{T}\}}\ \mathrm{d}t,$$ we can apply the Proposition [Proposition 35](#lem_freetime){reference-type="ref" reference="lem_freetime"} to the function $L^h_0(t)\mathcal{X}_{\{t>T\}}$. This eventually yields $$\int_{1}^{+\infty} L^h(\Tilde{\gamma},\dot{\Tilde{\gamma}}) - L^h_0(t)\ \mathrm{d}t \\ > \int_{1}^{+\infty} L^h(\tilde{\sigma},\dot{\tilde{\sigma}})-L^h_0(t)\ \mathrm{d}t + 2h(\bar T-T)+\varepsilon,$$ and finally $$\mathcal A(\Tilde\varphi)>\mathcal A(\bar\varphi)+\varepsilon,$$ in clear contradiction with [\[eq:minepsilon\]](#eq:minepsilon){reference-type="eqref" reference="eq:minepsilon"}. ◻ # Hamilton-Jacobi equations {#sec:HJ} We now emphasize the dependence on the initial point $x^0$ and define the function $$\begin{split}\label{eq:sol_HJ} v(x_0)&=\min_{\varphi \in \mathcal D^{1,2}_0(1,+\infty)} \left\{ \int_{1}^{+\infty} \frac{1}{2}\|\dot{\varphi}(t)\|_\mathcal{M}^2 \ \mathrm{d}t\right.+ \\&+ \left. \int_{1}^{+\infty} \ U(\varphi(t) + r_0(t) + x^0-r_0(1)) - U(r_0(t)) - \langle \ddot{r}_0(t),\varphi(t)\rangle_\mathcal{M}\ \mathrm{d}t\right\}- \langle a, x^0\rangle_\mathcal{M}. \end{split}$$ We claim that $u$ solves the Hamilton-Jacobi equation $$\label{eq:HJ} H(x,\nabla v(x))=h$$ in the viscosity sense. This can be easily seen by taking a point $x^0$ of differentiability, and formally differentiate [\[eq:sol_HJ\]](#eq:sol_HJ){reference-type="eqref" reference="eq:sol_HJ"} with respect to $x^0$, finding $$\nabla v(x^0)=-\mathcal{M}\dot x(1)$$ where $x(t)=r_0(t)+\varphi(t)+x^0-r_0(1)$ and $\varphi$ is the minimizer of the renormalized action associated with $x^0$. Therefore $\mathcal{M}^{-1/2}\nabla v(x^0)=-\mathcal{M}^{1/2}\dot x(1)$ and we easily obtain [\[eq:HJ\]](#eq:HJ){reference-type="eqref" reference="eq:HJ"} from the expression of the Hamiltonian [\[eq:hamiltonian\]](#eq:hamiltonian){reference-type="eqref" reference="eq:hamiltonian"}. Making this argument fully rigorous goes beyond the scope of this paper. The interested reader can retrace step by step the method explained in [@MadernaVenturelli_HyperbolicMotions], also taking into account that it is known that the singular set is contained in a locally countable union of smooth hypersurfaces of codimension at least one (cfr. [@MR2041617]). Fixing $x^0$ and $T>0$, we now consider the boundary value problem $$\begin{cases} \mathcal{M}\ddot{x} = \nabla U(x)\\ x(1)=x^0\\ \dot{x}(T)=\dot{r}_0(T) \end{cases}$$ and introduce the associated value function $$u(T,x^0) = \min_{\gamma\in H^1([1,T]),\ \gamma(1)=x^0} \int_{1}^{T} \frac{1}{2}\|\dot{\gamma}(t)\|_\mathcal{M}^2 + U(\gamma(t))\ \mathrm{d}t - \langle\dot{r}_0(T),\gamma(T)\rangle_\mathcal{M}.$$ It is a standard result of the theory of Hamilton-Jacobi equations (cfr. [@MR2041617]) that $u$ is a viscosity solution of $$-\frac{\partial u}{\partial T} = \frac{1}{2}\|\nabla u\|^2_{\mathcal{M}^{-1}} - U(x),$$ where the gradient is taken with respect to the second variable. **Remark 36**. Notice that, compared with [@MR2041617], we have reversed time orientation. Now, we define $$v(T,x) = u(T,x) + \int_{1}^{T}\frac{1}{2}\|\dot{r}_0(t)\|_\mathcal{M}^2 - U(r_0(t))\ \mathrm{d}t = u(T,x) + \int_{1}^{T} H(r_0(t),\dot{r}_0(t))\ \mathrm{d}t$$ and observe that $$-\frac{\partial v}{\partial T} = \frac{1}{2}\|\nabla v\|_{\mathcal{M}^{-1}} - U(x) - H(r_0,\dot{r}_0).$$ Assume that $v(T,x)$ converges uniformly to some $v(x)$ as $T\rightarrow+\infty$. Then, $v$ is a stationary viscosity solution to the stationary Hamilton-Jacobi equation $$\frac{1}{2}\|\nabla v\|_{\mathcal{M}^{-1}} - U(x) = \lim_{T\rightarrow+\infty} H(r_0,\dot{r}_0) = \frac{1}{2}\|a\|_\mathcal{M}^2.$$ To relate the modified value function $v$ with the minimum of our renormalized action, let us write $$\gamma(t) = r_0(t) + \varphi(t) + \Tilde{x}^0,$$ with $\Tilde{x}^0=x^0-r_0(1)$, and compute $$\begin{split} &\int_{1}^{T} \frac{1}{2}\|\dot{r}_0(t) + \dot{\varphi}(t)\|_\mathcal{M}^2 + U(r_0(t)+\varphi(t)+\Tilde{x}^0) + \frac{1}{2}\|\dot{r}_0(t)\|_\mathcal{M}^2 - U(r_0(t))\ \mathrm{d}t - \langle\dot{r}_0(T),r_0(T)+\varphi(T)+x^0-r_0(1)\rangle_\mathcal{M}\\ & = \int_{1}^{T} \frac{1}{2}\|\dot{\varphi}(t)\|_\mathcal{M}^2 + U(r_0(t)+\varphi(t)+\Tilde{x}^0) - U(r_0(t)) - \langle\ddot{r}_0(t),\varphi(t)\rangle_\mathcal{M}\ \mathrm{d}t - \langle\dot{r}_0(T), x^0\rangle_\mathcal{M}, \end{split}$$ which follows from some integration by parts. Therefore, we have $$v(T,x^0) = \min_{\varphi\in H^1([1,T]),\ \varphi(1)=0} \mathcal{A}_{[1,T]}^{ren}(\varphi) - \langle\dot{r}_0(T), x^0\rangle_\mathcal{M},$$ where we denoted $$\mathcal{A}_{[1,T]}^{ren}(\varphi) = \int_{1}^{T} \frac{1}{2}\|\dot{\varphi}(t)\|_\mathcal{M}^2 + U(r_0(t)+\varphi(t)+\Tilde{x}^0) - U(r_0(t)) - \langle\ddot{r}_0(t),\varphi(t)\rangle_\mathcal{M}\ \mathrm{d}t.$$ Then, it becomes natural to let $T\to +\infty$ and define $$v(x^0) = \min_{\varphi\in\mathcal{D}_0^{1,2}(1,+\infty)}\int_{1}^{+\infty} \frac{1}{2}\|\dot{\varphi}(t)\|_\mathcal{M}^2 + U(r_0(t) + \varphi(t) + x^0 - r_0(1)) - U(r_0(t)) - \langle\ddot{r}_0(t),\varphi(t)\rangle_\mathcal{M}\ \mathrm{d}t - \langle a, x^0\rangle_\mathcal{M}.$$ We will prove in a forthcoming paper that $$v(x) = \lim_{T\rightarrow+\infty} v(T,x)$$ uniformly on compact sets of $\mathbb{R}^{dN}$ (actually, in the Hölder norms), so that $v$ solves $$\frac{1}{2}\|\nabla v\|_{\mathcal{M}^{-1}} - U(x) = \frac{1}{2}\|a\|_\mathcal{M}^2$$ in the viscosity sense. 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E. Maderna and A. Venturelli, *Globally minimizing parabolic motions in the Newtonian $n$-body problem*, Archive for Rational Mechanics and Analysis, 194 (2009), pp. 283--313. height 2pt depth -1.6pt width 23pt, *Viscosity solutions and hyperbolic motions: a new pde method for the $n$-body problem*, Annals of Mathematics, 192 (2020), pp. 499--550. height 2pt depth -1.6pt width 23pt, *Uniqueness of hyperbolic Busemann functions for the Newtonian n-body problem*, (in preparation). C. Marchal, *How the method of minimization of action avoids singularities*, Celestial Mechanics and Dynamical Astronomy, 83 (2002), pp. 325--353. C. Marchal and D. G. Saari, *On the final evolution of the $n$-body problem*, Journal of Differential Equations, 20 (1976), pp. 150--186. R. Montgomery, *The $N$-body problem, the braid group, and action-minimizing periodic solutions*, Nonlinearity, 11 (1998), pp. 363--376. J. Paradela and S. Terracini, *Oscillatory motions in the restricted 3-body problem: A functional analytic approach*, 2022. H. Poincaré, *Sur les solutions périodiques et le principe de moindre action*, Comptes rendus hebdomadaires des séances de l'Académie des sciences de Paris, 123 (1896), pp. 915--918. H. Pollard, *The behavior of gravitational systems*, Journal of Mathematics and Mechanics, 17 (1967), pp. 601--611. D. G. Saari, *Expanding gravitational systems*, Transactions Of The American Mathematical Society, 156 (1971), pp. 219--240. height 2pt depth -1.6pt width 23pt, *The manifold structure for collision and for hyperbolic-parabolic orbits in the $N$-body problem*, Journal of Differential Equations, 5 (1984), pp. 300--329. D. G. Saari and N. D. Hulkower, *On the manifolds of total collapse orbits and of completely parabolic orbits for the n-body problem*, Journal of Differential Equations, 41 (1981), pp. 27--43. [^1]: *See Section [5](#sec_hyperbolic_parabolic){reference-type="ref" reference="sec_hyperbolic_parabolic"} for the exact definition of $\beta$ and $b_m$.*

arxiv_math

--- abstract: - | Chance constraints are frequently used to limit the probability of constraint violations in real-world optimization problems where the constraints involve stochastic components. We study chance-constrained submodular optimization problems, which capture a wide range of optimization problems with stochastic constraints. Previous studies considered submodular problems with stochastic knapsack constraints in the case where uncertainties are the same for each item that can be selected. However, uncertainty levels are usually variable with respect to the different stochastic components in real-world scenarios, and rigorous analysis for this setting is missing in the context of submodular optimization. This paper provides the first such analysis for this case, where the weights of items have the same expectation but different dispersion. We present greedy algorithms that can obtain a high-quality solution, i.e., a constant approximation ratio to the given optimal solution from the deterministic setting. In the experiments, we demonstrate that the algorithms perform effectively on several chance-constrained instances of the maximum coverage problem and the influence maximization problem. - | Chance constraints are frequently used to limit the probability of constraint violations in real-world optimization problems where the constraints involve stochastic components. We study chance-constrained submodular optimization problems, which capture a wide range of optimization problems with stochastic constraints. Previous studies considered submodular problems with stochastic knapsack constraints in the case where uncertainties are the same for each item that can be selected. However, uncertainty levels are usually variable with respect to the different stochastic components in real-world scenarios, and rigorous analysis for this setting is missing in the context of submodular optimization. This paper provides the first such analysis for this case, where the weights of items have the same expectation but different dispersion. We present greedy algorithms that can obtain a high-quality solution, i.e., a constant approximation ratio to the given optimal solution from the deterministic setting. In the experiments, we demonstrate that the algorithms perform effectively on several chance-constrained instances of the maximum coverage problem and the influence maximization problem. address: - Optimisation and Logistics, University of Adelaide, Adelaide, Australia - School of Computer Science and Engineering, Central South University, Changsha, P.R. China - Xiangjiang Laboratory, Changsha, P.R. China - University of Birmingham, Birmingham, UK author: - " [^1]" -   -   -   -   bibliography: - mainRef.bib title: | Optimizing Chance-Constrained Submodular Problems\ with Variable Uncertainties --- # Introduction Stochastic components can significantly affect the quality of solutions for a given stochastic optimization problem. Reducing the uncertain effect of stochastic components is vital to avoid potentially disruptive incidents in the complex and expensive system. *Chance constraints* can be applied to optimization tasks, which limit the probability of incidental constraint violations [@charnes1959chance; @iwamura1996genetic; @miller1965chance; @poojari2008genetic]. A chance-constrained optimization problem can be described as finding an optimal solution subject to the condition that the constraints are only violated with a given small probability. Recently, the problem has been investigated widely [@doerr2020optimization; @neumann2020optimising; @neumann2019runtime; @neumann2022runtime; @shi2022runtime; @xie2019evolutionary; @xie2020specific; @xie2021runtime]. A typical technique for taking chance constraints into account for a given optimization problem is to convert the stochastic constraints to their respective deterministic equivalents for a given confidence level, which is possible when considering normally distributed stochastic components. *Submodular functions* [@nemhauser1978analysis] capture problems of diminishing returns which frequently appear in real-world scenarios. They constitute a significant category of optimization challenges. In the artificial intelligence literature, greedy algorithms[@das2011submodular; @doerr2020optimization; @friedrich2019greedy; @zhang2016submodular] and Pareto optimization approaches[@neumann2020optimising; @roostapour2022pareto; @qian2017subset; @qian2017subsets] based on evolutionary multi-objective algorithms have been widely examined for submodular optimization problems. The goal for a submodular optimization problem with a given knapsack constraint is to find a set of elements with the maximal value of the submodular function whose total weight does not exceed the budget of the given knapsack. There are many analyses on the deterministic version of this submodular optimization problem [@khuller1999budgeted; @krause2014submodular; @nemhauser1978analysis; @qian2017subset]. Often the weights of elements might be stochastic and sampled from a probability distribution. The *Chance-constrained Submodular Problem*  [@chen2019chance] has been proposed to model this case. Here, the goal of the problem is to maximize a given monotone submodular function subject to the constraint that the probability of violating the knapsack constraint is no more than a small threshold value. For this problem, Chen and Maehara [@chen2019chance] reduced the chance constraint of the problem into multiple deterministic constraints by guessing the parameters and relaxed the knapsack budget and threshold. They rigorously analyzed an algorithm that enumerates all parameters for the abstracted problem with random weights sampled from arbitrary known distributions, which meets the relaxed constraint. Doerr et al., [@doerr2020optimization] investigated a specific variant of the problem where the weights are sampled from a uniform distribution with an identical dispersion. They applied the one-sided Chebyshev's inequality and a Chernoff bound separately to construct the surrogates that helps to estimate the probability of constraint violation. In addition, they empirically showed that using the greedy algorithms based on such surrogates gives high-quality solutions in stochastic scenarios. Furthermore, multi-objective evolutionary algorithms have also been employed to tackle this problem, e.g., the GSEMO algorithm [@neumann2020optimising]. It has been theoretically analyzed and found to achieve comparable performance to the greedy algorithms in the worse case within polynomial time. However, uncertainties of items usually vary between the items and greedy algorithms with theoretical performance guarantees missing in the literature. Such an analysis is supposed to be more challenging than the one carried out in [@doerr2020optimization] since variable uncertainties of the weights lead to more intricate effects than identical uncertainties, which is reflected in the surrogate based on one-sided Chebyshev's inequality. In this paper, we focus on a general setting of the problem studied in [@doerr2020optimization; @neumann2020optimising], i.e., the weights of the elements are sampled from uniform distributions with the same expectation but different dispersion values, instead of from an identical uniform distribution. We remark the item's dispersion as its uncertainty level, such that the uncertainty level varies from item to item in this setting. The one-sided Chebyshev's inequality is used to construct a surrogate of the chance constraint. In addition to the greedy algorithm (GA) and the generalized greedy algorithm (GGA), our analysis also encompasses another studied greedy algorithm, the generalized greedy$+$Max algorithm (GGMA) [@yaroslavtsev2020bring]. Our rigorous analysis demonstrates that the GA struggles to effectively obtain an acceptable solution in the worse case due to a heavy impact arising from the variable uncertainties. For the GGA and the GGMA, we first use a simple strategy for element selection, which only considers the sum of the dispersion values. The algorithms cannot guarantee a high-quality solution in some linear instances. Instead of using this simple strategy, simultaneously considering the expectation and the dispersion is promising to fill this gap. We adopt an improved strategy that applies the surrogate of the chance constraints in selecting elements. Using this strategy, the GGA and the GGMA can obtain a $(1/2-o(1))(1-1/e)-$approximation and $(1/2-o(1))-$approximation, respectively. Finally, we empirically analyze the performance of the algorithms on twelve chance-constrained instances of the maximum coverage problem and the influence maximization problem. The empirical results show that the GGA and the GGMA beat the GA in most instances. Furthermore, the GGMA obtains a solution of similar quality as the GGA, which verifies and supplements our theoretical results. The paper is structured as follows. Sections [2](#sec:pl){reference-type="ref" reference="sec:pl"}-[3](#sec:algs){reference-type="ref" reference="sec:algs"} introduce the studied problem and the algorithms. Our theoretical results of the investigated algorithms are shown in Sections [4](#sec:pga){reference-type="ref" reference="sec:pga"}-[6](#sec:pggma){reference-type="ref" reference="sec:pggma"}. We present our experimental results in Section [7](#sec:exp){reference-type="ref" reference="sec:exp"} and finish with some conclusions in Section [8](#sec:con){reference-type="ref" reference="sec:con"}. # Preliminaries {#sec:pl} ## Problem Definition Consider a set $V = \{1,...,n\}$, in which each element $i \in V$ has a weight $w_i$, and a function $f:V'\to \mathbb{R}_{\geq 0}$ defined on the subsets $V' \subseteq V$. The function $f$ is *monotone* iff for any two subsets $S, T\subseteq V$ with $S \subseteq T$, $f(S) \le f(T)$ holds. Besides, the function $f$ is *submodular* iff for any two subsets $S$, $T\subseteq V$ with $S\subseteq T$ and any element $e \notin T$, $$\label{equ:1} \centering f(S\cup \{e\})-f(S) \ge f(T\cup \{e\}) - f(T).$$ Given a *monotone* submodular function $f$ defined on the subsets of $V$ and a budget $B$, the problem named *the submodular problem with respect to $V$ and $B$* is to look for a subset $S \subseteq V$ such that $f(S)$ is maximized and $W(S) \le B$, where $W(S) = \sum_{i \in S} w_i$. Within the investigation, we focus on a *chance-constrained* version of the submodular problem, in which the weight $w_i$ of each element $i \in V$ is random (not deterministic) and has expected value $E[w_i] = a_i$ and variance $\sigma^2_i\geq 0$. The aim is find a subset $S \subseteq V$ such that $f(S)$ is maximized and subject to the constraint that $Pr[W(S)>B] \le \alpha$, where the threshold $0 \le \alpha \le 1$ is given which upper bounds the probability of a constraint violation. As mentioned above, given an instance of the chance-constrained submodular problem, the weight $W(S) = \sum_{i\in S} w_i$ of a solution $S$ to it is random but has an expectation $E[W(S)] = \sum_{i\in S} a_i$, and variance $Var[W(S)] = \sum_{i\in S}\sigma^2_i$. Within the paper, we consider a specific setting for the chance-constrained submodular problem, in which the random weight $w_i$ of the element $i\in V$ is independently uniformly sampled from the interval $[a_i-\delta_i, a_i+\delta_i]$ at random. Besides, we consider $a_i =1$ and $0\leq\delta_i\leq 1$. Note that, for a uniform distribution, the expectation and variance can be calculated by the given interval bounds [@xie2019evolutionary]. Therefore the expected weights of all elements are $1$, and the variance of each element $i$ is $Var[W(i)] = \delta_i^2/3$. Furthermore, w.l.o.g., assume every single element is feasible with respect to the budget $B$. Observe that the case that $0 \le B \le 1$ is meaningless, thus we assume $B>1$ such that at least one item is in the solution throughout the paper. ## Surrogate of the Chance constraint For the probability $Pr[W(S)>B]$, as the work given in [@xie2019evolutionary], we consider the one-sided Chebyshev's inequality to construct a usable surrogate of the chance constraint, whose formulation is given below. **Theorem 1** (*One-sided Chebyshev's inequality*). *For any random variable $X$ and $\lambda \geq 0$, $Pr[X>E[X]+\lambda]\leq \frac{Var[X]}{Var[X]+\lambda^2}.$* For a solution $S$ to the chance-constrained submodular problem, if $Pr[W(S) > B] \le \alpha$, then it is *feasible*; otherwise, *infeasible*. By the one-sided Chebyshev's inequality, we have the following observation directly, which considers the feasibility of a given solution. **Observation 1**. *Given a solution $S$ to the chance-constrained submodular problem, if $E[W(S)] + \sqrt{\frac{(1-\alpha)Var[W(S)]}{\alpha}}$ $\leq B$ then the solution $S$ is feasible.* By the above observation, the *surrogate weight* of a solution can be defined as $\Gamma(S) := E[W(S)] + \kappa_{\alpha}\sqrt{ Var[W(S)]},$ where $\kappa_{\alpha} = \sqrt{\frac{1-\alpha}{\alpha}}$. # Algorithms {#sec:algs} The first algorithm studied is the greedy algorithm (GA, see Algorithm [\[alg:ga\]](#alg:ga){reference-type="ref" reference="alg:ga"}), which was analyzed in [@doerr2020optimization] for the chance-constrained submodular problem with all elements having iid weights. The GA starts with an empty set and picks the element with the largest marginal gain that meets the constraint in each iteration. It stops when no more elements can be accepted without violating the constraint. Considering that the elements may have different weights, the generalized greedy algorithm (GGA, see Algorithm [\[alg:gga\]](#alg:gga){reference-type="ref" reference="alg:gga"}) is studied. Similar to the mechanism of the GA, the GGA starts with an empty set and stops when no more elements can be added due to the chance constraint. However, the GGA selects the element that satisfies the chance constraint and maximizes the ratio between the additional gain in the objective function $f$ and that in a non-decreasing function $h$. As the variances and surrogate weights of the solutions to the problem are non-decreasing, two strategies based on two non-decreasing functions $h$ are respectively studied: **Strategy $h(S) := \sum_{i\in S}\delta_i^2$**, and **Strategy $h(S) :=\Gamma(S)$**. Uncertainties of the solution are only considered in Strategy and the surrogate weight based on the one-sided Chebyshev's inequality is studied in Strategy . Furthermore, Lines [\[gga:line 10\]](#gga:line 10){reference-type="ref" reference="gga:line 10"}-[\[gga:line 11\]](#gga:line 11){reference-type="ref" reference="gga:line 11"} of Algorithm [\[alg:gga\]](#alg:gga){reference-type="ref" reference="alg:gga"} are required when there exists an element with an extremely high objective value, see [@khuller1999budgeted; @leskovec2007cost] for more details. Additionally, the generalized greedy$+$Max algorithm (GGMA, see Algorithm [\[alg:ggma\]](#alg:ggma){reference-type="ref" reference="alg:ggma"}) is studied. The GGMA adopts the same greedy strategies as the GGA, but it uses the feasible item having the largest marginal gain to augment every partial greedy solution. More specifically, the augmenting item is selected among the remaining items that still meet the constraint in each iteration. Until no element can fit into the solution, the GGMA stops and outputs the best-augmented solution. Note that the surrogate is applied to the algorithms instead of calculating the probability $Pr[W(S)>B]$. We only are using the exact calculation for $Pr[W(v)>B]$ when considering a single element at line [\[gga:line 10\]](#gga:line 10){reference-type="ref" reference="gga:line 10"} in the GGA. **Input**:\ **Output**:$S$ $S \gets \emptyset,V'\gets V$[\[line:GA_feasible_options\]]{#line:GA_feasible_options label="line:GA_feasible_options"} $v^* \gets \arg\max_{v\in V'}f(S\cup\{v\})-f(S)$[\[line:GA_ratio\]]{#line:GA_ratio label="line:GA_ratio"} $S\gets S\cup \{v^*\}$ $V'\gets V'\setminus\{v^*\}$ **Input**:\ **Output**: $S$ $S \gets \emptyset,V'\gets V$[\[line:GGA_feasible_options\]]{#line:GGA_feasible_options label="line:GGA_feasible_options"} $v^* \gets \arg\max_{v\in V'}\frac{f(S\cup\{v\})-f(S)}{h(S\cup \{v\})-h(S)}$[\[line:GGA_ratio\]]{#line:GGA_ratio label="line:GGA_ratio"} $S\gets S\cup \{v^*\}$ $V'\gets V'\setminus\{v^*\}$ $v^*\gets \arg\max_{\{v\in V; Pr[W(v)>B]\le \alpha\}} f(v)$ [\[gga:line 10\]]{#gga:line 10 label="gga:line 10"} $S\gets\arg\max_{Y\in\{S,\{v^*\}\}}f(Y)$[\[gga:line 11\]]{#gga:line 11 label="gga:line 11"} **Input**:\ **Output**:$T$ $T \gets \emptyset,S \gets \emptyset,V'\gets V$ $V'\gets \left\{v\in V'\setminus S \mid \Gamma(S\cup \{v\}) \leq B \right\}$[\[ggma:us\]]{#ggma:us label="ggma:us"} $v' \gets \arg\max_{v\in V'}{f(S\cup\{v\})}$[\[line:GGMA_ratio\]]{#line:GGMA_ratio label="line:GGMA_ratio"} $T\gets S\cup \{v'\}$ $v^* \gets \arg\max_{v\in V'}\frac{f(S\cup\{v\})-f(S)}{h(S\cup \{v\})-h(S)}$ $S\gets S\cup \{v^*\}$ Update $V'$ as Line [\[ggma:us\]](#ggma:us){reference-type="ref" reference="ggma:us"} # Performance of the GA {#sec:pga} According to the previous work [@doerr2020optimization], the GA is theoretically proven that works well in chance-constrained submodular problems with identical weight and uncertainty. However, since the uncertainties become variable, the GA is hard to obtain a high-quality solution, which is proved in the below. Let $S_{cc}$ be the solution obtained by the GA. From Theorem [Theorem 2](#thm:gap){reference-type="ref" reference="thm:gap"}, we find that the GA performs badly on some linear instances. Before the statement, we define such a linear instance $I_1$, in which let $V$ have at least $B+1$ elements, $f(S)=|S|$, $\gamma\in(0,1]$, $\alpha\in\left(\frac{3\gamma}{(B-1)^2+3\gamma},\frac{3\gamma}{(B-2)^2+3\gamma}\right)$, $\delta_1=\sqrt{\gamma}$, and $\delta_i=0$ for all $i\geq2$. **Theorem 2**. *There exists a linear instance $I_1$ such that the GA fails to guarantee better than $(1/B)$-approximation.* *Proof.* Considering the instance $I_1$, we have $\Gamma(\{1\})\in(B-1,B)$ and the GA on $I_1$ can pick element $1$ in the first iteration, preventing it from continuing. Thus $f(S_{cc})=1$, and the solution is $(1/B)-$ approximation while $Y=\{2,\ldots,B+1\}$, $f(Y)=\Gamma(Y)=B$. The claim is proved. ◻ The proof reveals that the GA rapidly exhausts the budget (i.e., selecting only one element) due to the significant influence of dispersion in the surrogate. This is the primary factor leading to the suboptimal performance of the GA. # Performance of the GGA {#sec:pgga} ## Analysis of Using Strategy {#ssec: gga1} For Strategy : $h(S) := \sum_{i\in S}\delta_i^2$, we also find that there exists a collection of linear instances of the problem, for which the GGA is hard to obtain a high-quality solution. To facilitate the construction of these instances, a solution $S$ is encoded as a decision vector $X = x_1 x_2...x_n$ with length $n$, where $x_i=1$ means that the element $i \in V$ is selected into the solution $S$. Then we define such an instance $I_2$ with a linear function $f$, in which let $V = 1,\ldots,n$, $0 < \alpha <0.5$, and $B= \varepsilon+1$ where $n \ge 2\varepsilon$ and $\varepsilon\geq 1$. The function $f$ represented by the decision vector $X$ is given as: $$\label{equ:4} f(X) = \sum_{i=1}^{\varepsilon}x_i+\varepsilon\sum_{i = \varepsilon+1}^{n}x_i.$$ Besides the dispersion of each element in $I_2$ is considered as $\delta_i=\sqrt{\frac{\gamma}{\varepsilon}}$ for $i \in [1,\varepsilon]$, and $\delta_j=\sqrt{\frac{\varepsilon\gamma+\beta}{\varepsilon}}$ for $j \in [\varepsilon+1,n]$ subjected to $0<\gamma$, $0<\beta$ and $\varepsilon\gamma+\beta\leq 3\alpha/(1-\alpha)$, which indicated by Theorem [Theorem 3](#thm:2){reference-type="ref" reference="thm:2"}. **Theorem 3**. *Given $\varepsilon\geq 1$, there exists a linear instance $I_2$ such that the generalized greedy algorithm GGA applying $h := \sum_{i\in S}\delta^2_i$ fails to guarantee better than $(1/\varepsilon)-$approximation.* ## Analysis of Using Strategy {#analysis-of-using-strategy} Since $h(S) :=\Gamma(S)$ in Strategy , we know that $h$ is a non-linear function therefore the surrogate weight of each element is changed as the size of the solution grows. For the analysis (Theoreom [Theorem 4](#the:3){reference-type="ref" reference="the:3"}), some useful notations and definitions are introduced first. Let $S_{cc}$ be the greedy solution generated by the GGA, $v_{i}$ be the $i$-th element added to the solution $S_{cc}$, and $S_i = \{v_1,\ldots v_i\} \subseteq S_{cc}$ ($1 \le i \le |S_{cc}|$) be the set containing the first $i$ elements. Then we define a set $A_i$ to collect all abandoned elements due to the constraint violation before the GGA adds $v_i$ into $S_{i-1}$. Note that $A_{i-1}\subseteq A_i$. Besides, the surrogate weight of the element $v_i$ is denoted by $c_i$, where $c_{i} = \Gamma(S_{i}) - \Gamma(S_{i-1})$. Moreover, given any two sets $S, T\subset V$, let $f(S\mid T) := f(S\cup T) - f(T)$. Let $OPT_d$ be the optimal solution of the deterministic instance of the problem. Given a partial greedy solution $S_k$ generated by the GGA, the relation between $S_k$ and $OPT_d$ is first investigated in Lemma [Lemma 1](#lemma:2){reference-type="ref" reference="lemma:2"}. Observe that $|OPT_d| = \lfloor B \rfloor$ as the expected weight is exactly one. **Lemma 1**. *Let $\zeta = \kappa_{\alpha}\sum_{j\in OPT_d}\sqrt{\delta_j^2/3}$. Given a partial greedy solution $S_{k}$, if $A_k\cap OPT_d = \emptyset$, then $$f(S_{k+1})-f(S_{k})\geq \frac{c_{k+1}}{\lfloor B \rfloor+\zeta }\cdot (f(OPT_d)-f(S_{k})).$$* After that, we can get a relation between $OPT_d$ and $S_{cc}$ by using Lemma [Lemma 1](#lemma:2){reference-type="ref" reference="lemma:2"}. **Theorem 4**. *The solution obtained by the GGA applying $h := \Gamma(S)$ is a $(1/2-o(1))(1-1/e)-$approximation.* *Proof.* Consider the upper bound of $k$ that is denoted by $k^*$. It has the set $S_{k^*}$ such that the element from $OPT_d$ is first abandoned due to the constraint when the GGA attempts to add it into the set. We denote the abandoned element by $z$ and derive a relation between $S_{k^*}$ and $OPT_d$. Note that $A_{i}\cap OPT_d = \emptyset$ for $1\leq i\leq k^*$. Following Lemma [Lemma 1](#lemma:2){reference-type="ref" reference="lemma:2"}, it gives $$\label{equ:rskopt} % f(S_{k^*}) -f(S_{k^*-1}) \geq \frac{c_{k+1}}{\lfloor B\rfloor+\zeta }\cdot(f(OPT_d)-f(S_{k^*-1})). f(S_{k^*+1}) -f(S_{k^*}) \geq \frac{c_{k+1}}{\lfloor B\rfloor+\zeta }\cdot(f(OPT_d)-f(S_{k^*})).$$ As we know that $(1-x)\leq e^{-x}$, then rearranging ([\[equ:rskopt\]](#equ:rskopt){reference-type="ref" reference="equ:rskopt"}) gives $$\begin{aligned} f(OPT_d)&-f(S_{k^*+1}) \\ &\leq \left(1-\frac{c_{k^*+1}}{\lfloor B\rfloor +\zeta }\right)\cdot(f(OPT_d)-f(S_{k^*})) \\ & \leq e^{-\frac{c_{k^*+1}}{\lfloor B\rfloor +\zeta }} \cdot (f(OPT_d)-f(S_{k^*})). \end{aligned}$$ Recursively, $$\begin{aligned} f&(OPT_d) - f(S_{k^*+1}) \\ &\leq e^{-\frac{c_{k^*+1}}{\lfloor B\rfloor+\zeta}} \cdot (f(OPT_d)-f(S_{k^*})) \\ &\leq e^{-\frac{c_{k^*+1}+c_{k^*}}{\lfloor B\rfloor+\zeta }} \cdot (f(OPT_d)-f(S_{k^*-1})) \\ &\leq \ldots \leq e^{-\frac{\sum_{i=1}^{k^*+1} c_i}{\lfloor B\rfloor+\zeta }} \cdot f(OPT_d) \\ &= e^{-\frac{\Gamma(S_{k^*+1})}{\lfloor B\rfloor+\zeta }} \cdot f(OPT_d), \end{aligned}$$ Consequently, we get the relation between $S_{k^*}$ and $OPT_d$ as $$\label{equ:rso} % f(S_{k^*}) \geq \left(1-e^{-\Gamma(S_{k^*})/(\lfloor B\rfloor+\zeta)}\right)\cdot f(OPT_d). f(S_{k^*+1}) \geq \left(1-e^{-\Gamma(S_{k^*+1})/(\lfloor B\rfloor+\zeta)}\right)\cdot f(OPT_d).$$ Then we investigate the approximation by using the relation and the abandoned element $z$. By Observation [Observation 1](#obs:feasibility){reference-type="ref" reference="obs:feasibility"} and definitions, it observes that $\Gamma(S_{k^*}\cup \{z\}) = \Gamma(S_{k^*}) + c' > \lfloor B\rfloor$, where $c' = \Gamma(z\mid S_{k^*})$. Putting it with ([\[equ:rso\]](#equ:rso){reference-type="ref" reference="equ:rso"}) together gives $$\begin{aligned} &f(S_{k^*+1}) \geq \left(1-e^{-\frac{\Gamma(S_{k^*+1})}{\Gamma(S_{k^*})+c'+\zeta}}\right)\cdot f(OPT_d) \\ &= \left(1-e^{-1}exp\left({\frac{\zeta+c'-c_{k^*+1}}{\Gamma(S_{k^*})+c'+\zeta}}\right)\right)\cdot f(OPT_d). \end{aligned}$$ As $S_{k^*+1}$ at least include one element, the expression of $exp\left(\cdot\right)$ is $(1+o(1))$. Moreover, let $v^*\in V\setminus S_{k^*}$ be the element that has the largest function value. Observe that $f(v^*) \geq f(v_{k^*+1})$ and $f(S_{cc})>f(S_{k^*})$. It gets $f(S_{cc}) + f(v^*) \geq f(S_{k^*}) + f(v^*) \geq f(S_{k^*+1})$. Putting them together gets $f(S_{cc}) + f(v^*)\geq (1-o(1))(1-1/e)\cdot f(OPT_d)$, and therefore $\max_{Y\in\{S_{cc},\{v^*\}\}}f(Y) \geq (1/2-o(1))(1-1/e)\cdot f(OPT_d).$ ◻ # Performance of the GGMA {#sec:pggma} In this section, we analyze the approximation behavior of the GGMA. The performance of the algorithm applying two different strategies is investigated separately. ## Analysis of Using Strategy {#analysis-of-using-strategy-1} Theorem [Theorem 5](#thm: pgga2){reference-type="ref" reference="thm: pgga2"} implies that using Strategy : $h(S):=\sum_{i\in S}\delta_i^2$, the GGMA also performs badly in the instances $I_2$, which was presented in the Section [5.1](#ssec: gga1){reference-type="ref" reference="ssec: gga1"}. **Theorem 5**. *Given $\varepsilon\geq 1$, there exists an instance $I_2$ such that the GGMA applying $h := \sum_{i\in S}\delta^2_i$ fails to guarantee better than $(2/\varepsilon - 1/\varepsilon^2)-$approximation.* ## Analysis of Using Strategy {#analysis-of-using-strategy-2} For Strategy : $h(S):=\Gamma(S)$, let $S_{cc}$ be the greedy solution constructed by the greedy strategy for the instance in the chance-constrained setting, and $OPT_d$ be the optimal solution for the corresponding deterministic instance. In the $i$-th generation, the element $v_i$ is selected by the algorithm, and its surrogate weight is denoted by $c_i$. Besides, the partial solution containing the first $i$ item is denoted by $S_i\subseteq S_{cc}$. Then some useful greedy performance functions are defined to track the performance of the algorithm for the chance-constrained setting. For a fixed $x \in [0, B]$, let $i$ be the smallest greedy index so that $\Gamma(S_i)> x$. Then to track the performance of the greedy strategy, a continuous and monotone piecewise-linear function $g(x)$ is defined as $g(x) = f(S_{i-1})+(x-\Gamma(S_{i-1}))\frac{f(v_i|S_{i-1})}{c_i}$, and $g(0):= 0$. Additionally, $g'$ denotes the right derivative for $g$ on the interval $[0, \Gamma(S_{cc}))$. Observe that $g'$ is always non-negative as the objective value of the greedy solution does not decrease after including a new item. Besides, to track the performance of the greedy$+$Max when the greedy solution collects a set of cost $x$, we define the function $g_+(x) = g(x) + f(v\mid S_{i-1})$, where $v = \mathop{\mathrm{argmax}}_{j\in V\setminus S_{i-1}:\Gamma(\{j\}\cup S_{i-1})\leq B} f(j\mid S_{i-1})$. After that, we consider the lower bound of the function $g_+$ in the specific interval. Following the definition of the fixed greedy index $i$, $z_{max}$ denotes the element that has the largest dispersion in $OPT_d\setminus S_{i-1}$. Note that $z_{max}$ is the first abounded element from $OPT_d$ due to the chance constraint. Denote by $S_{k^*}$ the partial solution. If $S_{k^*}$ is obtained then $z_{max}$ is removed. That implies $z_{max}$ can be selected by the algorithm as the augmenting item for the set $S_i$ where $0\leq i\leq k^*-1$. Therefore for $x \in [0, \Gamma(S_{k^*-1})]$, we define the greedy$+$Max performance lower bound as $g_1(x) = g(x) + f(z_{max}\mid S_{i-1})$, so that $g_1\leq g_+$ for $x \in [0, \Gamma(S_{k^*-1})]$. Now we investigate the relation between $OPT_d$, $g_1(x)$ and $g'(x)$ while $x \in [0, \Gamma(S_{k^*-1})]$ in Lemma [Lemma 2](#lemma:3){reference-type="ref" reference="lemma:3"} (proof in Appendix [9.2](#sect_app_proof_lemma3){reference-type="ref" reference="sect_app_proof_lemma3"}). **Lemma 2**. *For any $x \in [0, \Gamma(S_{k^*-1})]$, let $i$ be the smallest greedy index so that $\Gamma(S_i)> x$. It holds that $$f(OPT_d) \leq g_1(x) + g'(x)\sum_{j\in OPT_d \setminus z_{max}} \Gamma(j|S_{i-1}).$$* Then we focus on the point $x = \Gamma(S_{k^*-1})$ and analyze the approximation behavior of the GGMA (Thereom [Theorem 6](#the:5){reference-type="ref" reference="the:5"}) via Lemma [Lemma 2](#lemma:3){reference-type="ref" reference="lemma:3"}. **Theorem 6**. *The solution obtained by the GGMA applying $h := \Gamma(S)$ is a $(1/2-o(1))-$approximation.* *Proof.* Following Lemma [Lemma 2](#lemma:3){reference-type="ref" reference="lemma:3"} and applying $x = \Gamma(S_{k^*-1})$, it gives $$\begin{aligned} f(OPT_d) &\leq g_1(\Gamma(S_{k^*-1}))\\ &+ g'(\Gamma(S_{k^*-1}))\sum_{j\in OPT_d \setminus Z_{max}} \Gamma(j|S_{k^*-1}). \end{aligned}$$ Here the upper bound of the last term is given below. Recall that $|OPT_d| = \lfloor B\rfloor$. Let $\psi := \sqrt{Var[W(S_{k^*-1})]}$. It holds that $$\begin{aligned} &\sum_{j\in OPT_d \setminus z_{max}} \Gamma(j|S_{k^*-1})\\ &=\sum_{j\in OPT_d} \Gamma(j|S_{k^*-1}) - \Gamma(z_{max}|S_{k^*-1}) \\ & \leq \sum_{j\in OPT_d} \Gamma(j|S_{k^*-1}) - \Gamma(z_{max}|S_{k^*}) \\ &=\lfloor B\rfloor + \eta - c'_{max}, \end{aligned}$$ where $\eta = \kappa_{\alpha}\sum_{j\in OPT_d} \left(\sqrt{Var[W(S_{k^*-1}\cup \{j\})]}- \psi\right)$ and $c'_{max} = \Gamma(z_{max}|S_{k^*})$. Consequently, we have $$\label{equ:10} \begin{aligned} f(OPT_d) \leq g_1&(\Gamma(S_{k^*-1})) \\ &+ g'(\Gamma(S_{k^*-1}))( \lfloor B\rfloor + \eta -c'_{max}). \end{aligned}$$ After that, we consider the value of $g_1(\Gamma(S_{k^*-1}))$ in ([\[equ:10\]](#equ:10){reference-type="ref" reference="equ:10"}). Recall that the surrogate weight of $v_{k^*}$ is $c_{k^*}$. Then let $c^* := 1+\kappa_{\alpha}\sqrt{\delta_{k^*}^2/3}$ and $\phi := \frac{c_{k^*}\cdot(\Gamma(S_{k^*-1})+c^*)}{\Gamma(S_{k^*})+\eta}$. Two possible cases for it are listed as follows. **Case 1**. $g_1(\Gamma(S_{k^*-1}))\geq \frac{\phi}{c^*+\phi}\cdot f( OPT_d)$. Since $g_1(x)\leq g_+(x)$ for $x\in [0, \Gamma(S_{k^*-1})]$, it directly holds $g_+(\Gamma(S_{k^*-1}))\geq \frac{\phi}{c^*+\phi}\cdot f(OPT_d)$. **Case 2**. $g_1(\Gamma(S_{k^*-1}))< \frac{\phi}{c^*+\phi}\cdot f( OPT_d)$. For this case, we can prove that $g(\Gamma(S_{k^*}))\geq \frac{\phi}{c^*+\phi}\cdot f(OPT_d)$ as following. First of all, rearranging ([\[equ:10\]](#equ:10){reference-type="ref" reference="equ:10"}) gets $$\label{equ:gphi} \begin{aligned} g'(\Gamma(S_{k^*-1})) &\geq \frac{f( OPT_d)-g_1(\Gamma(S_{k^*-1}))}{\lfloor B\rfloor+ \eta - c'_{max} }\\ &\geq \frac{c^*\cdot f( OPT_d)}{(c^*+\phi)(\lfloor B\rfloor+ \eta - c'_{max})}. \end{aligned}$$ Besides, let $g'_{min} := \arg\min_{i\in[1,k^*]} g'(\Gamma(S_{i-1}))$. Recall $g'$ is non-negative and $g(0)=0$, thus it holds that $$g(x)\geq x \cdot g'_{min},$$ for any $x\in [0,\Gamma(S_{k^*-1})]$. Now we show that $g'_{min} \geq \frac{f(v_{k^*}\mid S_{k^*-1})}{c^*}$. Considering the greedy strategy of the GGMA, it holds that $g'(\Gamma(S_{i-1})) = \frac{f(v_i|S_{i-1})}{c_{i}}\geq \frac{f(v_{k^*}|S_{i-1})}{\Gamma({v_{k^*}|S_{i-1}})}$ in the $i$-th generation for $1\leq i \leq k^*$. Observe $\Gamma({v_{k^*}|S_{i-1}}) \leq c^*$ and $f(v_{k^*}|S_{i-1})\geq f(v_{k^*}|S_{k^*-1})$ for $i \leq k^*$ as $S_{i-1}\subseteq S_{{k^*-1}}$ (recall ([\[equ:1\]](#equ:1){reference-type="ref" reference="equ:1"})). Therefore we have $\frac{f(v_{k^*}|S_{i-1})}{\Gamma({v_{k^*}|S_{i-1}})}\geq \frac{f(v_{k^*}|S_{k^*-1})}{c^*}.$ Putting them together yields $g'_{min} \geq \frac{f(v_{k^*}\mid S_{k^*-1})}{c^*}$. For any $x \in [0,\Gamma(S_{k^*-1})]$, therefore it holds that $$\label{equ:11} g(x) \geq x\cdot \frac{f(v_{k^*}|S_{k^*-1})}{c^*}.$$ Then applying $x = \Gamma(S_{k^*-1})$ to ([\[equ:11\]](#equ:11){reference-type="ref" reference="equ:11"}), it gets $$\begin{aligned} g(\Gamma(S_{k^*-1})) &\geq \Gamma(S_{k^*-1})\cdot \frac{f(v_{k^*}\mid S_{k^*-1})}{c^*} \end{aligned}$$ Besides, since $g'(\Gamma(S_{k^*-1})) = \frac{f(v_{k^*}\mid S_{k^*-1})}{c_{k^*}}$, rearranging ([\[equ:gphi\]](#equ:gphi){reference-type="ref" reference="equ:gphi"}) gets $\frac{f(v_{k^*}\mid S_{k^*-1})}{c^*}\geq \frac{c_{k^*}\cdot f( OPT_d) }{(c^*+\phi)(\lfloor B\rfloor+ \eta - c'_{max})}.$ Putting them together, we have $g(\Gamma(S_{k^*-1}))\geq \frac{f( OPT_d)\cdot\Gamma(S_{k^*-1})\cdot c_{k^*}}{(c^*+\phi)\left(\lfloor B\rfloor+ \eta - c'_{max}\right)}.$ Now we can derive a lower bound for the objective value of the set $S_{k^*}$. Recall that $v_{k^*}$ is the next added element for the solution $S_{k^*-1}$. Thus $f(S_{k^*})$ is at least $$\begin{aligned} g&(\Gamma(S_{k^*}))= g(\Gamma(S_{k^*-1})) + c_{k^*} g'(\Gamma(S_{k^*-1}))\\ % &\geq \frac{f( OPT_d)\cdot\Gamma(S_{k^*-1})\cdot c_{k^*}}{(c^*+\phi)(\lfloor B\rfloor+ \eta - c'_{max} )} + \frac{f( OPT_d)\cdot c^* \cdot c_{k^*}}{(c^*+\phi)(\lfloor B\rfloor+ \eta - c'_{max} )} \\ & \geq \frac{ c_{k^*}\cdot(\Gamma(S_{k^*-1}) + c^*)}{(c^*+\phi)(\lfloor B\rfloor+ \eta - c'_{max} )}\cdot f( OPT_d). \end{aligned}$$ Furthermore, by Observation [Observation 1](#obs:feasibility){reference-type="ref" reference="obs:feasibility"} it yields that $\Gamma(S_{k^*-1}\cup\{v_{k^*}, z_{max}\}) = \Gamma(S_{k^*}) + c'_{max} \geq \lfloor B\rfloor.$ Put them together gets $$\begin{aligned} g(\Gamma(S_{k^*}))&\geq \frac{ c_{k^*}\cdot(\Gamma(S_{k^*-1}) + c^*)}{(c^*+\phi)(\lfloor B\rfloor+ \eta - c'_{max} )}\cdot f(OPT_d)\\ &\geq\frac{c_{k^*}\cdot(\Gamma(S_{k^*-1})+c^* )}{(c^*+\phi)(\Gamma(S_{k^*}) + \eta)}\cdot f( OPT_d) \\ &= \frac{\phi}{c^*+\phi}\cdot f( OPT_d). \end{aligned}$$ Therefore the GGMA achieves a $\frac{\phi}{c^*+\phi}$-approximation. Since at least one element is included in the greedy solution, the expression of $\frac{\phi}{c^*+\phi}$ is $(1/2-o(1))$. Additionally, as the objective value of the augmented output solution $T$ is no worse than $g_+(\Gamma(S_{k^*}))$, it holds $f(T)\geq (1/2-o(1))\cdot f(OPT_d)$. ◻ # Experiments {#sec:exp} {width="\\textwidth"} [\[fig:1\]]{#fig:1 label="fig:1"} {width="99%"} [\[fig:2\]]{#fig:2 label="fig:2"} {width="90%"} {width="\\textwidth"} In this section, we regard the GA as the baseline algorithm and evaluate the experimental performance of other algorithms (namely the GGA and the GGMA) on two significant submodular optimization problems such as the maximum coverage problem (MCP) and the influence maximum problem (IMP) with chance constraint. Following the specific setting described in Section [2](#sec:pl){reference-type="ref" reference="sec:pl"}, the expectation of each element's weight is set as $1$, and the dispersion value of each item is different. ## The Maximum Coverage Problem The first submodular problem is the maximum coverage problem [@feige1998threshold; @khuller1999budgeted]. We consider a chance-constrained version of the MCP based on the graph. Given an undirected graph $G = (V,E)$, we denote the degree of the node $v_i$ by $D(v_i)$, and the number of all nodes of $V'\subseteq V$ and their neighbors in $G$ by the objective function $N(V')$. The MCP aims to find a subset $V'$ so that $N(V')$ is maximized under the constraint. Moreover, given a linear cost function $c: V\to \mathbb{R}^{+}$, the problem under chance constraint is formulated as $$\mathop{\mathrm{argmax}}_{V'\subseteq V}~N(V')~s.t.~Pr[c(V')>B]\leq \alpha.$$ The graph used in the instances are frb30-15-01 (450 nodes and 17827 edges) and frb35-17-01 (595 nodes and 27 856 edges) [@nr]. In terms of settings, for each node $v_i$, the cost $a_i$ is 1 but the value of the dispersion $\delta_i$ is set by two different methods. The first method is that $\delta_i$ is independently uniformly at random sampled from $[0,1]$. To analyze our experiments more rigorously, we independently randomly sample the value of dispersion five times for each graph. Moreover, we also consider $\delta_i$ is associated with the degree of the node, which can be expressed as $\delta_i = D(v_i)/\sum_{v\in V}D(v)$. Furthermore, we investigate all combinations of $\alpha \in \{10^{-4},10^{-5}, 10^{-6}\}$ and $B \in \{10, 15, 20, 25\}$ for the experimental investigation of the algorithms. The performance of the algorithms is measured in terms of the function value $N(V')$. The experimental results are shown in figures [\[fig:1\]](#fig:1){reference-type="ref" reference="fig:1"}, [\[fig:2\]](#fig:2){reference-type="ref" reference="fig:2"} and [\[fig:3\]](#fig:3){reference-type="ref" reference="fig:3"}, which indicate that for the instances with the same budget, both the function value and the number of nodes of the solution by the algorithms GGA and GGMA decline as $\alpha$ increasing. They also show that the performance of the GA is worse than GGA and GGMA using strategy among most instances. It also can be found that GA collects fewer items before reaching the budget, which matches our theoretical analysis. Besides, for the same strategy applied to the different algorithms, the GGMA slightly outperforms the GGA while applying strategy to the instances with a lower budget. Additionally, the performance of the GGMA using strategy is comparable to the GGA. In terms of strategies, we observe from figures that using strategy can improve the performance of all algorithms by strategy . More precisely, the algorithms with strategy can output a solution that includes more nodes but is with a lower function value, among most instances. It is noticeable that the GGMA with strategy can obtain high-quality solutions for these instances. ## The Influence Maximization Problem We now study the influence maximization problem [@zhang2016submodular; @qian2017subset; @leskovec2007cost]. The IMP aims to identify the set of users, who are the most influential in a large-scale social network. The goal of the IMP is to maximize the spread of influence over a given social network, i.e., a graph of interactions within a group of users [@kempe2003maximizing]. This section presents the experimental analysis of some chance-constrained IMP instances. Let a directed graph be $G = (V,E)$ to represent a social network, in which each node $v_i\in V$ corresponds to a user, and the probability $p_{i,j}$ of the edge in $E$ represents the strength of the influence between a pair of users $v_i$ and $v_j$. The IMP aims to find a subset $X \subseteq V$ such that the expected number of nodes $E[I(X)]$ (the objective function of the problem) activated by propagating from $X$ is maximized subject to the constraints. Given a linear cost function $c: V\to \mathbb{R}^{+}$ and a budget $B$, the chance constraint version of the IMP is formulated as $$\mathop{\mathrm{argmax}}_{X\subseteq V}~E[I(X)]~s.t.~ Pr[c(X)> B]\leq \alpha.$$ The dataset *Social circles: Facebook* consists of friends lists collected from a social networking service, which includes 4,039 nodes and 176,468 edges  [@leskovec2012learning]. We transform the instances in this dataset to chance-constrained IMP instances. For each node $v_i$, its expected cost is set as 1, and the dispersion $\delta_i$ is independently and uniformly sampled from $[0,1]$. The algorithms are evaluated for all pairs of budgets $B\in \{20,50,100,150\}$ and tolerate probabilities $\alpha \in \{10^{-3},10^{-4},10^{-5}\}$. We use the function value $E[I(X)]$ to evaluate the performance of algorithms, and independently sample the value of dispersion three times to analyze our experiments more rigorously. Figure [\[fig:4\]](#fig:4){reference-type="ref" reference="fig:4"} clearly shows that the GA collects fewer nodes than other algorithms and performs unwell when $\alpha$ is small. For the same budget instances, the function value obtained sharply decreases with the increasing value of $\alpha$. This phenomenon is common among those algorithms. Moreover, we observe that the GGMA includes fewer nodes than GGA but obtains higher function values in most instances. In terms of strategies, the results demonstrate that the algorithms with strategy is significantly worse in some instances than them with strategy , even worse than the GA although it collects more elements. On the other hand, the algorithms applying strategy can fix it in most instances, which coincides with our theoretical analysis. In addition, the GGMA beats the GGA in terms of the quality of the output solution in most instances. # Conclusion {#sec:con} The paper studied a chance-constrained submodular optimization problem with variable uncertainties and investigated the performance of the GA, the GGA, and the GGMA on it. In the setting, the weights of elements are sampled from distributions with the same expectation but varied dispersion values. We found that the GA does not perform well even in some linear instances. Besides the GGA and the GGMA respectively can achieve guarantee a $(1/2-o(1))(1-1/e)-$approximation and a $(1/2-o(1))-$approximation of the optimal solution for a deterministic setting. Additionally, the experimental results showed that the GGMA using the surrogate weight based on the one-sided Chebyshev's inequality beats other algorithms in some instances of the MCP and the IMP which are typical submodular problems. The future work is to broaden the exploration of chance-constrained submodular problems with variable weights and uncertainties, and potentially different distributions. These subsequent studies will be both challenging and engaging, with the aim of yielding more meaningful insights to enhance our comprehension of the problem. This work has been supported by the Australian Research Council (ARC) through grant FT200100536, the Hunan Provincial Natural Science Foundation of China through grant 2021JJ40791, and the Open Project of Xiangjiang Laboratory (No.22XJ03005). # Proofs ## Proof of Lemma [Lemma 1](#lemma:2){reference-type="ref" reference="lemma:2"} {#sect_app_proof_lemma2} *Proof.* By the monotonicity and submodularity, it has $$\label{equ:suq} \begin{aligned} f(OPT_d)&\leq f(OPT_d\cup S_k) \\ &\leq f(S_k) + f(OPT_d\setminus S_k|S_k)\\ &\leq f(S_k) +\sum_{{j}\in OPT_d\setminus S_k} \Gamma(j|S_k)\cdot\frac{f(j\mid S_k)}{\Gamma(j|S_k)}, \end{aligned}$$ where $\Gamma(j|S_k) = \Gamma(S_k\cup \{j\}) - \Gamma(S_k)$. Then according to the greedy strategy, for any element $j\in OPT_d\setminus S_k$, it gets $\frac{f({v_{k+1}}\mid S_k)}{c_{k+1}} \geq \frac{f(j\mid S_k)}{\Gamma(j|S_k)}$, since $OPT_d\cap A_k = \emptyset$. Consequently putting it with ([\[equ:suq\]](#equ:suq){reference-type="ref" reference="equ:suq"}) together gives that $$\label{equ:ggap} \begin{aligned} f(OPT_d) \leq f&(S_k)\\ &+ \frac{f(v_{k+1}|S_k)}{c_{k+1}} \sum_{j\in OPT_d\setminus S_k}&\Gamma(j|S_k) . \end{aligned}$$ There is an upper bound for $\sum_{j\in OPT_d\setminus S_k}\Gamma(j|S_k)$. As it has $\Gamma(j|S_k) \leq 1+\kappa_{\alpha} \sqrt{\delta_j^2/3}$ for any $j\in OPT_d\setminus S_k$, it holds that $\sum_{j\in OPT_d\setminus S_k}\Gamma(j|S_k) \leq \lfloor B\rfloor + \zeta$, where $\zeta = \kappa_{\alpha} \sum_{j\in OPT_d}\sqrt{\delta_j^2/3}$. Substituting it into ([\[equ:ggap\]](#equ:ggap){reference-type="ref" reference="equ:ggap"}) completes the proof. ◻ ## Proof of Lemma [Lemma 2](#lemma:3){reference-type="ref" reference="lemma:3"} {#sect_app_proof_lemma3} *Proof.* It suffices to show the statement only for the points $x = \Gamma(S_{i-1})$ where $1\leq i \leq k^*$. Let $S'_{i-1}:=S_{i-1}\cup z_{max}$. Obverse that $g_1(x) = g(\Gamma(S_{i-1})) + f(z_{max}\mid S_{i-1})= f(S_{i-1}\cup z_{max}) = f(S'_{i-1}).$ By monotonicity and submodularity, it gives that $$\label{equ:7} \begin{aligned} f(OPT_d) & \leq f(S_{i-1}\cup OPT_d) \\ &= f(S'_{i-1}) + f(OPT_d\setminus (S'_{i-1})\mid S'_{i-1}\}) \\ & \leq g_1(x) + \sum_{j\in OPT_d\setminus S'_{i-1}} f(j\mid S'_{i-1}) \end{aligned}$$ Since $S_{i-1}\subset S'_{i-1}$, it holds $f(j\mid S'_{i-1}) \leq f(j\mid S_{i-1})$ for any element $j\in OPT_d\setminus S'_{i-1}$ by submodularity (recall the inequality ([\[equ:1\]](#equ:1){reference-type="ref" reference="equ:1"})). Furthermore, recall that $z_{max}$ has the largest dispersion in $OPT_d\setminus S_{i-1}$ and it still can be added into $S_i$, thus any elements from $OPT_d\setminus S'_{i-1}$ also can be added into $S_i$ without violating the constraint. According to the greedy strategy, it gives that $\frac{f(j\mid S_{i-1})}{\Gamma(j|S_{i-1}) } \leq \frac{f(v_i\mid S_{i-1})}{c_i} = g'(x),$ for any element $j\in OPT_d\setminus S'_{i-1}$. Consequently, putting them with ([\[equ:7\]](#equ:7){reference-type="ref" reference="equ:7"}), it gets that $$\label{equ:8} \begin{aligned} &f(OPT_d) \\ & \leq g_1(x) + \sum_{j\in OPT_d\setminus S'_{i-1}} f(j\mid S'_{i-1}) \\ &\leq g_1(x) + \sum_{j\in OPT_d\setminus S'_{i-1}} \Gamma(j|S_{i-1}) \frac{f(j\mid S_{i-1})}{\Gamma(j|S_{i-1})} \\ &\leq g_1(x) + \sum_{j\in OPT_d\setminus S'_{i-1}} \Gamma(j|S_{i-1}) \cdot g'(x) \\ &\leq g_1(x) + g'(x) \sum_{j\in OPT_d\setminus z_{max}} \Gamma(j|S_{i-1}).\\ \end{aligned}$$ The proof is completed. ◻ ## Proof of Theorem [Theorem 3](#thm:2){reference-type="ref" reference="thm:2"} {#sect_app_proof_thm:2} *Proof.* We consider the instance $I_2$. Given a $\varepsilon-$size solution $X$, it can be found that the largest surrogate weight of $X$ is $\varepsilon+\sqrt{\frac{1-\alpha}{3\alpha}(\varepsilon\gamma+\beta)}\leq \varepsilon+1$. Thus every $\varepsilon-$size solution is feasible for $I_2$. Observe that any $(\varepsilon+1)-$size solution cannot be accepted. Then we analyze the solution returned by the GGA for $I_2$. For the first $\varepsilon$ elements and the remaining, the marginal ratios between $f$ and $h$ are $\frac{\varepsilon}{\gamma}$ and $\frac{\varepsilon^2}{\varepsilon\gamma+\beta}$, respectively. By the greedy strategy of the GGA, the algorithm always selects the items from the first $\varepsilon$ of them since $\frac{\varepsilon}{\gamma}> \frac{\varepsilon^2}{\varepsilon\gamma+\beta}$ until no more element can be added. Consequently, the algorithm returns the solution $X$ that contains all the first $\varepsilon$ elements, and $f(X)=\varepsilon$. Moreover, considering line [\[gga:line 10\]](#gga:line 10){reference-type="ref" reference="gga:line 10"} in the GGA, the algorithm gets $f(v^*) = \varepsilon.$ Finally, GGA outputs the solution $S_{cc}$ with the value $f(S_{cc}) = \varepsilon$. Besides, we define a feasible solution $Y\subseteq \{\varepsilon+1,\ldots, n\}$ that has $\varepsilon$ elements. It has $f(Y) = \varepsilon^2$ by Equation ([\[equ:4\]](#equ:4){reference-type="ref" reference="equ:4"}). Denote the optimal solution of $I_2$ for the deterministic setting by $OPT_d$. Obviously, $f(OPT_{d}) \geq f(Y)$ and $f(S_{cc})/f(OPT_{d}) \leq f(S_{cc})/f(Y)=1/\varepsilon$. The claim is proved. ◻ ## Proof of Theorem [Theorem 5](#thm: pgga2){reference-type="ref" reference="thm: pgga2"} {#sect_app_proof_pgga2} *Proof.* Considering the given linear instance $I_2$, recall that any $\varepsilon-$size solution is feasible. Because the marginal ratio between the additional gain in $f$ and $h$ are $\frac{\varepsilon}{\gamma}$ and $\frac{\varepsilon^2}{\varepsilon\gamma+\beta}$ for the first $\varepsilon$ elements and rest of them respectively, the GGMA constructs a $(\varepsilon-1)-$size partial solution $X\subseteq \{1,\ldots,\varepsilon\}$ and selects one augmenting item $j\in \{\varepsilon+1,\ldots n\}$ with objective value $\varepsilon$. Thus the output solution is $S_{cc} = X\cup \{j\}$ with $f(S_{cc}) = 2\varepsilon - 1$. Besides it has $f(OPT_{d})\geq \varepsilon^2$. Observe that $f(S_{cc})/f(OPT_{d}) \leq 2/\varepsilon - 1/\varepsilon^2$. The proof is completed. ◻ [^1]: Corresponding Author. Email: xiankun.yan\@adelaide.edu.au.

arxiv_math

--- abstract: | The goal of this paper is to show that a wide class of Harish-Chandra $(\mathfrak{g},K)$-modules including all irreducible ones come with a certain canonical filtration. author: - Ivan Losev title: Canonical filtrations on Harish-Chandra modules --- # Introduction Let $G$ be a semisimple algebraic group over $\mathbb{C}$ and $\mathfrak{g}$ be its Lie algebra. Let $\mathcal{U}$ denote the universal enveloping algebra $U(\mathfrak{g})$. Fix an involution $\sigma$ of $\mathfrak{g}$ and let $\mathfrak{k}$ be the fixed point locus $\mathfrak{g}^\sigma$. Let $K$ denote the corresponding connected algebraic subgroup of $G$ (one can deal with a more general choice of $K$ but for the purposes of the introduction we are not going to do this). Then we can talk about Harish-Chandra (shortly, HC) $(\mathfrak{g},K)$-modules, i.e., finitely generated $\mathcal{U}$-modules, where the $\mathfrak{k}$-action integrates to $K$. Below we will only consider those HC modules that have finite length. The category of such modules will be denoted by $\operatorname{HC}(\mathcal{U},K)$. Let $M$ be HC $(\mathfrak{g},K)$-module. An important tool to study $M$ are *good filtrations* on $M$, i.e., $K$-stable $\mathcal{U}$-module filtrations such that $\operatorname{gr}M$ is finitely generated over $S(\mathfrak{g})$ (in fact, $\operatorname{gr}M$ is then an $S(\mathfrak{g}/\mathfrak{k})$-module). In general, there is no canonical choice of such a filtration. The goal of this paper is to produce one under certain additional condition on $M$ that, in particular, is satisfied by all irreducible modules. In order to state the definition, we need to recall the notion of the associated variety of a HC module. The support of $\operatorname{gr}M$ in $(\mathfrak{g}/\mathfrak{k})^*$ is independent of the choice of a good filtration on $M$. It is called the *associated variety* of $M$ and is denoted by $\operatorname{V}(M)$. Since we assume that $M$ has finite length, the associated variety $\operatorname{V}(M)$ is contained in the nilpotent cone $\mathcal{N}_K\subset (\mathfrak{g}/\mathfrak{k})^*$ and so is the union of finitely many $K$-orbits. If $M$ is irreducible, then $\overline{G \operatorname{V}(M)}$ is the closure of a single nilpotent orbit, it is the set of zeroes of the ideal $\operatorname{gr}\operatorname{Ann}_{\mathcal{U}}(M)$. For a $K$-stable closed subvariety $Y\subset \mathcal{N}_K$, we write $\operatorname{HC}_Y(\mathcal{U},K)$ for the full subcategory of $\operatorname{HC}(\mathcal{U})$ consisting of all $M$ such that $\operatorname{V}(M)\subset Y$. If $Y=X\cap (\mathfrak{g}/\mathfrak{k})^*$, where $X$ is a closed $G$-stable subset in the nilpotent cone of $\mathfrak{g}^*$, then we write $\operatorname{HC}_X(\mathcal{U},K)$ for $\operatorname{HC}_Y(\mathcal{U},K)$. Now fix a nilpotent orbit $\mathbb{O}\subset \mathfrak{g}^*$. Consider the full (but not abelian) subcategory $\operatorname{HC}^\partial_\mathbb{O}(\mathcal{U},K)\subset \operatorname{HC}_{\overline{\mathbb{O}}}(\mathcal{U},K)$ consisting of all objects that do not have nonzero submodules with associated variety contained in $\overline{\mathbb{O}}\setminus \mathbb{O}$. For example, every simple HC module $M$ belongs to exactly one category $\operatorname{HC}^\partial_{\mathbb{O}}(\mathcal{U},K)$, here $\overline{\mathbb{O}}=\overline{G\operatorname{V}(M)}$. Our goal is to equip $M$ with a canonical good filtration. The characterization only involves the $S(\mathfrak{g})$-module $\operatorname{gr}M$ and has three conditions. Here are the first two conditions: - The annihilator $I$ of $\operatorname{gr}M$ in $S(\mathfrak{g})$ is radical. - $\operatorname{gr}M$ has *depth at least 1* meaning that every element in $S(\mathfrak{g})/I$ that is not a zero divisor in the algebra is not a zero divisor in $\operatorname{gr}M$ either. The third condition involves a certain normalization. Take a $K$-equivariant graded finitely generated $S(\mathfrak{g}/\mathfrak{k})$-module $N$ (an example is provided by $\operatorname{gr}M$). Take a point $\chi$ in an open $K$-orbit in $\operatorname{V}(M)$. Then the fiber $N_\chi$ is a finite dimensional module over the stabilizer $K_\chi$ of $\chi$ in $K$. It turns out it comes also with a preferred choice of a grading. Namely, pick a nondegenerate invariant symmetric form $(\cdot,\cdot)$ on $\mathfrak{g}$ and use it to identify $\mathfrak{g}$ with $\mathfrak{g}^*$. Let $e\in \mathfrak{g}^*$ be the element of $\mathfrak{g}$ corresponding to $\chi$. We include it into an $\mathfrak{sl}_2$-triple $(e,h,f)$ with $h\in \mathfrak{k}$ and $f\in \mathfrak{k}^\perp$. We note that $h$ is uniquely determined up to $K_\chi$-conjugacy. Let $\gamma:\mathbb{C}^\times\rightarrow G$ be the one-parameter subgroup with $d_1\gamma(1)=h$. In particular, the image lies in $K$. Then the $\mathbb{C}^\times$-action on $\mathfrak{g}^*$ given by $t.\alpha=t^{-2}\gamma(t)\alpha$ fixes $\chi$ and so gives a $\mathbb{C}^\times$-action (equivalently, a grading) on $N_\chi$ to be denoted by $N_\chi=\bigoplus_{i} N_{\chi,i}$. We impose the following normalization condition: - We have $(\operatorname{gr}M)_{\chi,i}=\{0\}$ for $i\neq 0,1$. We note that this condition is independent of the choice of $h$ (different choices are conjugate by $K_\chi$ and so the resulting gradings are conjugate) and of the choice of $\chi$ up to the $K$-conjugacy. So, we have one condition (c) for each $K$-orbit in $\operatorname{V}(M)\cap \mathbb{O}$. Here is the main result of this paper. **Theorem 1**. *Let $M\in \operatorname{HC}^\partial_\mathbb{O}(\mathcal{U},K)$. There is a unique good filtration on $M$ (to be called canonical) such that (a) and (b) satisfied, while (c) is satisfied for all $K$-orbits in $\operatorname{V}(M)\cap \mathbb{O}$.* We will also see that the canonical filtrations are functorial. **Example 2**. Let $\mathfrak{g}=\mathfrak{sl}_2$ and $\mathfrak{k}$ be its Cartan subalgebra. Pick numbers $z\in \mathbb{C}, a\in \mathbb{Z}$, and consider the HC module $M:=\mathcal{U}/\mathcal{U}(C-z,h-a)$, where $C$ is the Casimir element. It comes with the filtration induced from the usual filtration on $\mathcal{U}$. We claim that this filtration is canonical. The associated graded is $\operatorname{gr}M=\mathbb{C}[e,f]/(ef)$, it satisfies conditions (a) and (b). Now we check (c). There are two $K=\mathbb{C}^\times$-orbits in $\mathbb{O}\cap (\mathfrak{g}/\mathfrak{k})^*$: they are $(e\neq 0, f=0)$ and $(e=0, f\neq 0)$. The grading on $\operatorname{gr}M$ coming from the Kazhdan action associated to the former has $\deg e=0, \deg f=4$. So the fiber of $(\operatorname{gr}M)$ at $e=1,f=0$ is a one-dimensional space in degree $0$. The same conclusion holds for the other $K$-orbit. This establishes (c) for all orbits. So this filtration is canonical. Using the canonical filtration (or any good filtration satisfying (b)) we will prove the following result conjectured by Vogan. Let $\operatorname{V}(M)^{\geqslant 2}$ denote the complement in $\operatorname{V}(M)$ to the union of $K$-orbits of dimension at most $\dim \operatorname{V}(M)-2$. **Theorem 3**. *Let $M$ be an irreducible HC $(\mathfrak{g},\mathfrak{k})$-module, i.e., an irreducible $\mathfrak{g}$-module, where $\mathfrak{k}$ acts locally finitely. Then the variety $\operatorname{V}(M)^{\geqslant 2}$ is connected.* The paper is organized as follows. The construction of the canonical filtration is crucially based on the restriction functors for HC modules sketched in [@Wdim Section 6.1] and elaborated in [@LY Section 5]. These functors will be recalled in Section [2](#S_res_fun){reference-type="ref" reference="S_res_fun"}. Section [3](#S_canon_filtr){reference-type="ref" reference="S_canon_filtr"} is the main part of the paper, there we construct canonical filtrations and establish their properties. **Acknowledgements**: I am grateful to Pavel Etingof and Victor Ginzburg for stimulating discussions. My work has been partially supported by the NSF under grant DMS-2001139. # Restriction functors {#S_res_fun} In this section we will briefly recall the construction of the restriction functor for HC modules following [@Wdim Section 6.1]. This functor goes from a suitable category of HC modules to the category of modules over the W-algebra $\mathcal{W}$. So, we first recall the W-algebras, Section [2.1](#SS_W_alg){reference-type="ref" reference="SS_W_alg"}, and then discuss a general setting for HC modules, Section [2.2](#SS_wHC){reference-type="ref" reference="SS_wHC"}. After that we recall the restriction functor and its properties, Section [2.3](#SS_res_fun_constr){reference-type="ref" reference="SS_res_fun_constr"}. ## W-algebras {#SS_W_alg} Let $\sigma$ be an involution of $\mathfrak{g}$. Set $\mathfrak{k}:=\mathfrak{g}^\sigma$. Take a nilpotent $G$-orbit $\mathbb{O}\subset \mathfrak{g}^*$ whose intersection with $(\mathfrak{g}/\mathfrak{k})^*$ is non-empty and a $(G^\sigma)^\circ$-orbit $\mathbb{O}_K\subset \mathbb{O}\cap (\mathfrak{g}/\mathfrak{k})^*$. Pick $\chi\in \mathbb{O}_K$ and consider the transverse Slodowy slice $S\subset \mathfrak{g}^*$ through $\chi$. We can and will assume that $S$ is stabilized by $\theta:=-\sigma:\mathfrak{g}^*\rightarrow \mathfrak{g}^*$. The slice $S$ is stable with respect to the $\mathbb{C}^\times$-action recalled before Theorem [Theorem 1](#Thm:main){reference-type="ref" reference="Thm:main"}. The grading on $\mathbb{C}[S]$ induced by this action is positive. Also it is easy to see that $$\label{eq:Slodowy_vanish1} \mathbb{C}[S]_1=\{0\}.$$ There is a natural Poisson bracket of degree $-2$ on $\mathbb{C}[S]$. So $\mathbb{C}[S]$ is a graded Poisson algebra. It admits a filtered quantization $\mathcal{W}$. This algebra was introduced by Premet in [@Premet]. The definition we are using originates from [@Walg], a more technical version is given in [@W_prim Section 2.2]. Let $\mathcal{U}_\hbar$ denote the Rees algebra of $\mathcal{U}$ for the PBW filtration. We rescale the grading by $2$ so that $\mathfrak{g}$ and $\hbar$ are now in degree $2$. The anti-involution $\theta=-\sigma$ of $\mathfrak{g}$ extends to a $\mathbb{C}[\hbar]$-linear anti-involution of $\mathcal{U}_\hbar$ again denoted by $\theta$. We can view $\chi$ as a homomorphism $\mathcal{U}_\hbar\rightarrow \mathbb{C}$ and consider the completion $\mathcal{U}_\hbar^{\wedge_\chi}$ at the kernel of $\chi$. A standard argument shows that it is a flat $\mathcal{U}_\hbar$-module. It inherits the anti-involution $\theta$ from $\mathcal{U}_\hbar$. Set $V:=T_\chi\mathbb{O}$, this is a symplectic vector space with form $\omega$. It is $\mathbb{C}^\times$-stable and also stable under the anti-involution $\theta$ of $\mathfrak{g}$. Consider its Weyl algebra $\mathsf{A}$ and its homogenized version $\mathsf{A}_\hbar=T(V)[\hbar]/([u,v]-\hbar \omega(u,v))$. Thanks to the decomposition $\mathfrak{g}^*=T_\chi\mathbb{O}\oplus T_\chi S$, we can view $V$ as a subspace in $(T_\chi \mathfrak{g}^*)^*$. Then we can $\mathbb{C}^\times$- and $\theta$-equivariantly lift this embedding to an embedding $V\hookrightarrow \mathcal{U}_\hbar^{\wedge_\chi}$ subject to the following condition: - it lifts to an algebra homomorphism $\mathsf{A}_\hbar^{\wedge_0}\hookrightarrow \mathcal{U}_\hbar^{\wedge_\chi}$, where $\bullet^{\wedge_0}$ stands for the completion at the maximal ideal of $0$. Let $\mathcal{W}_\hbar'$ denote the centralizer of the image of $V$ in $\mathcal{U}_\hbar^{\wedge_\chi}$. Note that we have a decomposition $$\label{eq:algebra_decomposition} \mathcal{U}_\hbar^{\wedge_\chi}=\mathsf{A}_\hbar^{\wedge_0}\widehat{\otimes}_{\mathbb{C}[[\hbar]]}\mathcal{W}_\hbar',$$ where $\widehat{\otimes}$ is the completed tensor product. The algebra $\mathcal{W}_\hbar'$ comes with a $\mathbb{C}^\times$-action. Let $\mathcal{W}_\hbar$ denote the $\mathbb{C}^\times$-finite part. Finally, set $\mathcal{W}:=\mathcal{W}_\hbar/(\hbar-1)\mathcal{W}_\hbar$. The algebra $\mathcal{W}$ is called the *finite W-algebra*, it quantizes $\mathbb{C}[S]$ in the following sense: the grading on $\mathcal{W}_\hbar$ induces an ascending filtration $\mathcal{W}=\bigcup_{i\geqslant 0}\mathcal{W}_{\leqslant i}$, and we have a graded Poisson algebra isomorphism $\operatorname{gr}\mathcal{W}\xrightarrow{\sim}\mathbb{C}[S]$. We note that by the construction, $\mathcal{W}$ comes with a *parity involution*, to be denoted by $\varsigma$, that preserves the filtration and acts on $\mathbb{C}[S]=\operatorname{gr}\mathcal{W}$ as $-1\in \mathbb{C}^\times$. This is because $\hbar$ has degree $2$ for the $\mathbb{C}^\times$-action. Also note that the algebra $\mathcal{W}$ (as a filtered algebra with a parity involution) only depends on $\mathbb{O}$ up to an isomorphism preserving these structures (the algebras for points $\chi,g\chi$ are isomorphic via the action of $g$). Also, $\mathcal{W}$ inherits the anti-involution $\theta$, it preserves the filtration and commutes with $\varsigma$. This structure depends on $\mathbb{O}_K$ and not just on $\mathbb{O}$. ## Weakly HC modules {#SS_wHC} Now we introduce a category of weakly HC $\mathcal{U}_\hbar$-modules from [@LY Section 2]. More generally, let $\mathcal{A}_\hbar$ be a $\mathbb{C}[\hbar]$-algebra that is equipped with a $\mathbb{C}[\hbar]$-linear anti-involution $\theta$. We assume that $A:=\mathcal{A}_\hbar/(\hbar)$ is commutative. Let $A^{-\theta}$ denote the $-1$-eigenspace for $\theta$. Consider the two-sided ideal $\mathcal{J}_\hbar\subset \mathcal{A}_\hbar$ defined as the preimage of $AA^{-\theta}$. This is the Lie subalgebra in $\mathcal{U}_\hbar$ with respect to the bracket given by $[a,b]_\hbar:=\hbar^{-1}(ab-ba)$. Following [@LY Section 2.3], by a *weakly HC* $\mathcal{A}_\hbar$-module we mean a finitely generated $\mathcal{A}_\hbar$-module $M_\hbar$ equipped, in addition, with a Lie algebra action of $\mathcal{J}_\hbar$, $(a,m)\mapsto a.m$, subject to the following conditions: 1. $am=\hbar(a.m)$, $\forall\, a \in \mathcal{J}_\hbar$. 2. $(ba).m = b(a.m)$, $\forall\, a \in \mathcal{J}_\hbar, b \in \mathcal{A}_\hbar$. 3. $a.(bm)=[a,b]_\hbar m+b(a.m), \forall a\in \mathcal{J}_\hbar, b\in \mathcal{A}_\hbar$. By a homomorphism of weakly HC modules we mean an $\mathcal{A}_\hbar$-linear map that intertwines the actions of $\mathcal{J}_\hbar$. The resulting category will be denoted by $\operatorname{wHC}(\mathcal{A}_\hbar)$. If $\mathcal{A}_\hbar$ is graded in such a way that $\deg \hbar=d$ and $\theta$ preserves the grading, it makes sense to speak about graded weakly HC modules: the action of $\mathcal{J}_\hbar$ is of degree $-d$. The category of graded weakly HC modules will be denoted by $\operatorname{wHC}^{gr}(\mathcal{A}_\hbar)$. Here is the most important special case. The anti-involution $\theta$ of $U(\mathfrak{g})$ lifts to a graded $\mathbb{C}[\hbar]$-linear involution of $\mathcal{U}_\hbar$. Then $A^{-\theta}=S(\mathfrak{g})\mathfrak{k}$ and $\mathcal{J}_\hbar=\mathcal{U}_\hbar\mathfrak{k}+\hbar \mathcal{U}_\hbar$. Let $M$ be a HC $(\mathfrak{g},\mathfrak{k})$-module, meaning a finitely generated $U(\mathfrak{g})$-module with locally finite action of $\mathfrak{k}$. Equip it with a good $\mathfrak{k}$-stable filtration. Then its Rees module $R_\hbar(M)$ is weakly HC: note that for every $a\in \mathcal{J}_\hbar$ and every $m\in R_\hbar(M)$ we have $am\in \hbar M_\hbar$ and we define $a.m$ by $\hbar^{-1}(am)$: since $R_\hbar(M)$ is flat over $\mathbb{C}[\hbar]$, the division makes sense. Conversely, if $M_\hbar$ is a graded weakly HC module over $\mathcal{U}_\hbar$, then $M_\hbar/(\hbar-1)M_\hbar$ is a HC $(\mathfrak{g},\mathfrak{k})$-module. The grading on $M_\hbar$ induces a good filtration on $M_\hbar/(\hbar-1)M_\hbar$. We can apply this construction to the Lie algebra $\mathfrak{g}\times \mathfrak{g}^{opp}$ and the anti-involution given by $(x,y)\mapsto (y,x)$. The resulting weakly HC modules will be referred to as weakly HC $\mathcal{U}_\hbar$-bimodules. Note that they form a monoidal category, while $\operatorname{wHC}^{gr}(\mathcal{U}_\hbar)$ is its module category. Another special case of $\mathcal{A}_\hbar$ is $\mathcal{W}_\hbar$ with its anti-involution $\theta$. ## Construction and properties {#SS_res_fun_constr} Let $K$ denote a connected algebraic group with Lie algebra $\mathfrak{k}$ and a homomorphism to $G$ that induces the identity automorphism of $\mathfrak{k}$. Fix a character $\kappa$ of $K$ Let $\operatorname{wHC}^{gr}(\mathcal{U}_\hbar)^{K,\kappa}$ denote the category of *strongly $K$-equivariant* objects in $\operatorname{wHC}^{gr}(\mathcal{U}_\hbar)$. By definition, this means that, for every $\xi\in \mathfrak{k}$, the operator $m\mapsto \xi m-\langle\kappa,\xi\rangle m$ coincides with the image of $\xi$ under the differential of the $K$-action. Similarly, we can talk about HC $(\mathfrak{g},K,\kappa)$-modules, their category will be denoted by $\operatorname{HC}(\mathfrak{g},K,\kappa)$. Note that in Introduction we considered the situation when $K\hookrightarrow G$ and $\kappa=0$. Now we proceed to the *restriction functor* $\operatorname{wHC}^{gr}(\mathcal{U}_\hbar)^{K,\kappa}\rightarrow \operatorname{wHC}^{gr}(\mathcal{W}_\hbar)$ to be denoted by $\bullet_{\dagger,\chi}$ (first sketched in [@Wdim Section 6.1], see also [@LY Section 5.4]). It is constructed as follows, compare to [@LY Section 5.4]. Take $M_\hbar\in \operatorname{wHC}^{gr}(\mathcal{U}_\hbar)^{K,\kappa}$. Consider its completion $$M_\hbar^{\wedge_\chi}\cong \mathcal{U}_\hbar^{\wedge_\chi}\otimes_{\mathcal{U}_\hbar}R_\hbar(M).$$ We view $M_\hbar^{\wedge_\chi}$ as a module over the right hand side of ([\[eq:algebra_decomposition\]](#eq:algebra_decomposition){reference-type="ref" reference="eq:algebra_decomposition"}). One can equip $M_\hbar^{\wedge_\chi}$ with a $\mathbb{C}^\times$-action as follows. Let $\iota$ denote the homomorphism $K\rightarrow G$. Set $d:=2|\ker\iota|$. Then $dh/2\in \mathfrak{k}$ is the differential at $1$ of a one-parameter subgroup $\mathbb{C}^\times\rightarrow K$ to be denoted by $\gamma$ (one can take $d=2$ if $K\hookrightarrow G$). Then we can define an action of $\mathbb{C}^\times$ on $M_\hbar$ by $t.m:=t^{di}\gamma(t)^{-1}m$ for $m\in M_{\hbar}$ of degree $i$. Note that the corresponding action of $\mathbb{C}^\times$ on $\mathfrak{g}^*$ preserves $\chi$, and so extends to $M_\hbar^{\wedge_\chi}$. The latter becomes a $\mathbb{C}^\times$-equivariant $\mathcal{U}_\hbar^{\wedge_\chi}$-module, where the $\mathbb{C}^\times$-action on $\mathcal{U}_\hbar^{\wedge_\chi}$ is rescaled $d/2$ times (so that $\hbar$ is in degree $d$). One can show, [@LY Lemma 5.2], that $M_\hbar^{\wedge_\chi}$ decomposes as $\mathbb{C}[[L,\hbar]]\widehat{\otimes}_{\mathbb{C}[[\hbar]]}N'_\hbar$, where $L$ is the $-1$-eigenspace for $\theta$ in $T_\chi G\chi$, a lagrangian subspace, and $N'_\hbar$ is the annihilator of the Lie algebra action of $L$ on $M_\hbar^{\wedge_\chi}$. Taking the locally finite elements for the action of $\mathbb{C}^\times$ in $N'_\hbar$ we get an object in $\operatorname{wHC}^{gr}(\mathcal{W}_\hbar)$ (where the grading on $\mathcal{W}_\hbar$ is rescaled $d/2$ times) to be denoted by $(M_\hbar)_{\dagger,\chi}$, giving us the required functor. Note that if $M_\hbar$ is flat over $\mathbb{C}[\hbar]$, then so is $N_\hbar$, as all operations above preserve the flatness. Now we can take an object $M\in \operatorname{HC}(\mathcal{U})^{K,\kappa}$. Equip it with a good filtration, and take the Rees module $M_\hbar$. Rescale the grading $d$ times. We get an object $M_{\hbar,\dagger,\chi}\in \operatorname{wHC}^{gr}(\mathcal{W}_\hbar)$. Consider the quotient $M_{\hbar,\dagger,\chi}/(\hbar-1)M_{\hbar,\dagger,\chi}$. Since $\hbar$ has degree $d$, we get a $\mathbb{Z}/d\mathbb{Z}$-grading on the quotient. Rescale the $\mathbb{Z}/2\mathbb{Z}$-grading on $\mathcal{W}$ $d/2$ times so that only the components of degrees $0$ and $d/2$. Denote the category of $\mathbb{Z}/d\mathbb{Z}$-graded modules over $\mathcal{W}$ by $\mathcal{W}\operatorname{-mod}^{\mathbb{Z}/d\mathbb{Z}}$. Similarly to [@HC Section 3.4], $\operatorname{wHC}^{gr}(\mathcal{U}_\hbar)^{K,\kappa}\rightarrow \operatorname{wHC}^{gr}(\mathcal{W}_\hbar)$ descends to a functor $\operatorname{HC}(\mathcal{U})^{K,\kappa}\rightarrow\mathcal{W}\operatorname{-mod}^{\mathbb{Z}/d\mathbb{Z}}$. Now we summarize some properties of the restriction functor. First, note that all intermediate functors in the construction of $$\bullet_{\dagger,\chi}: \operatorname{wHC}^{gr}(\mathcal{U}_\hbar)^{K,\kappa}\rightarrow \operatorname{wHC}^{gr}(\mathcal{W}_\hbar).$$ (the completion, the pushforward under an isomorphism and taking the locally finite part for the $\mathbb{C}^\times$-action) are exact. So $\bullet_{\dagger,\chi}$ is exact. Further, take $M_\hbar\in \operatorname{wHC}(\mathcal{U}_\hbar)$ and write $\operatorname{Supp}(M_\hbar)$ for its support in $(\mathfrak{g}/\mathfrak{k})^*$ (so that for $M\in \operatorname{HC}(\mathcal{U},\mathfrak{k})$, we have $\operatorname{Supp}(R_\hbar(M))=\operatorname{V}(M)$). Similarly, we can talk about the supports of modules over the completed algebras involved in the construction of $\bullet_{\dagger,\chi}$. The support of the $\mathsf{A}_\hbar^{\wedge_0}$-module $\mathbb{C}[[L,\hbar]]$ is $(\mathfrak{g}/\mathfrak{k})^*\cap T_\chi \mathbb{O}$. It follows that $$\label{eq:support_restriction} \operatorname{Supp}((M_\hbar)_{\dagger,\chi})=\operatorname{Supp}(M_\hbar)\cap S.$$ Next, we will need the compatibility between $\bullet_{\dagger,\chi}$ and Hom's from HC bimodules. Let $\mathcal{B}_\hbar$ denote a weakly HC $\mathcal{U}_\hbar$-bimodule and $M_\hbar$ a weakly $(K,\kappa)$-equivariant HC $\mathcal{U}_\hbar$-module. It is easy to see that $\operatorname{Hom}_{\mathcal{U}_\hbar}(\mathcal{B}_\hbar,M_\hbar)$ is a $(K,\kappa)$-equivariant weakly HC $\mathcal{U}_\hbar$-module. The following claim follows from [@LY Remark 5.7]. **Lemma 4**. *There is a bi-functorial isomorphism $$\operatorname{Hom}_{\mathcal{U}_\hbar}(\mathcal{B}_\hbar,M_\hbar)_{\dagger,\chi}\xrightarrow{\sim} \operatorname{Hom}_{\mathcal{W}_\hbar}((\mathcal{B}_\hbar)_{\dagger,\chi},(M_\hbar)_{\dagger,\chi}).$$* ## Generalization to Dixmier algebras {#SS_Dixmier_generalization} Recall that by a Dixmier algebra one means an associative algebra $\mathcal{A}$ equipped with a rational action of $G$ that comes with a quantum comoment map $\mathcal{U}\rightarrow \mathcal{A}$ (a $G$-equivariant map such that the resulting adjoint action of $\mathfrak{g}$ on $\mathcal{A}$ coincides with the differential of the $G$-action) such that $\mathcal{A}$ is finitely generated as a left module over $U(\mathfrak{g})$. So $\mathcal{A}$ becomes a HC bimodule over $\mathcal{U}$. One can show, [@LY Lemma 5.1], that there is a $G$-stable algebra filtration on $\mathcal{A}$ that is also a good bimodule filtration. Consider the Rees algebra $\mathcal{A}_\hbar$ of $\mathcal{A}$ with respect to such a filtration and its completion $\mathcal{A}_\hbar^{\wedge_\chi}$. Note that there is a natural homomorphism $\mathcal{U}_\hbar\rightarrow \mathcal{A}_\hbar$ and hence $\mathcal{U}_\hbar^{\wedge_\chi}\rightarrow \mathcal{A}_\hbar^{\wedge_\chi}$. Then $\mathcal{A}_{\hbar}^{\wedge_\chi}$ decomposes as $\mathsf{A}_\hbar^{\wedge_0}\widehat{\otimes}_{\mathbb{C}[[\hbar]]}\mathcal{A}_\hbar'$, where $\mathcal{A}_\hbar'$ is the centralizer of the image of $V$. Let $\mathcal{A}_{\hbar,\dagger}$ denote the $\mathbb{C}^\times$-locally finite part of $\mathcal{A}_\hbar'$, and $\mathcal{A}_{\dagger}$ be its specialization at at $\hbar=1$. This is a filtered algebra equipped with a parity involution $\varsigma$ and a filtered algebra homomorphism $\mathcal{W}\rightarrow \mathcal{A}_{\dagger}$ compatible with parity involutions. By a weakly HC $\mathcal{A}_\hbar$-module we mean an $\mathcal{A}_\hbar$-module that becomes weakly HC after pulling back to $\mathcal{U}_\hbar$. Similarly, we can talk about graded weakly HC modules. The corresponding category will be denoted by $\operatorname{wHC}^{gr}(\mathcal{A}_\hbar)$. And similarly, we can talk about HC $\mathcal{A}$-modules, weakly HC $\mathcal{A}_{\hbar,\dagger}$-modules, etc. Then we get the functor $\bullet_{\dagger,\chi}:\operatorname{wHC}^{gr}(\mathcal{A}_\hbar)\rightarrow \operatorname{wHC}^{gr}(\mathcal{A}_{\hbar,\dagger})$ lifting the functor from Section [2.3](#SS_res_fun_constr){reference-type="ref" reference="SS_res_fun_constr"}. We finish this section with a discussion of an interesting class of Dixmier algebras. Let $\tilde{\mathbb{O}}$ be a $G$-equivariant cover of a nilpotent orbit in $\mathfrak{g}^*$, let $\mu$ denote the corresponding map $\tilde{\mathbb{O}}\rightarrow \mathfrak{g}^*$. The algebra $\mathbb{C}[\tilde{\mathbb{O}}]$ is Poisson and graded. In many cases (for example, when $\tilde{\mathbb{O}}\hookrightarrow \mathfrak{g}^*$) one can assume that the degree of the Poisson bracket is $-1$, while in general one has degree $-2$. One can talk about filtered quantizations of $\mathbb{C}[\tilde{\mathbb{O}}]$. The algebra $\mathbb{C}[\tilde{\mathbb{O}}]$ is finitely generated, the corresponding variety $X$ is singular symplectic. Using this, one can give a classification of quantizations of $\mathbb{C}[\tilde{\mathbb{O}}]$, [@orbit Theorem 3.4]. These quantizations are Dixmier algebras, [@orbit Section 5.2], and in the case when the degree of the bracket is $-1$ (which is what we are going to assume for now), the corresponding filtration on a quantization $\mathcal{A}$ is a good algebra filtration. The algebra $\mathcal{A}_{\dagger}$ is isomorphic to $\mathbb{C}[\mu^{-1}(\chi)]$ and the filtration is trivial meaning that $\mathcal{A}_{\dagger,\leqslant 0}=\{0\},\mathcal{A}_{\dagger,\leqslant 0}=\mathcal{A}_{\dagger}$, [@orbit Section 5.2]. **Remark 5**. Now consider the case when the degree of the bracket is $-2$. In this case the default filtration on $\mathcal{A}$ is not compatible with the PBW filtration on $\mathcal{U}$, however, it is compatible with doubled PBW filtration on $\mathcal{U}$. And we have the parity involution on a quantization $\mathcal{A}$ and can talk about $\mathbb{Z}/2\mathbb{Z}$-graded HC modules. Their category will be denoted by $\operatorname{HC}(\mathcal{A})^{K,\kappa,\mathbb{Z}/2\mathbb{Z}}$. The constructions of the algebra $\mathcal{A}_\hbar$ and the functor $\bullet_{\dagger,\chi}$ generalize to the present setting by replacing the Rees construction with its modification. Namely, we consider the modified Rees construction for $\mathbb{Z}/d\mathbb{Z}$-graded spaces, compare to [@LY Section 2.3]. Let $V=\bigoplus_{a\in \mathbb{Z}/d\mathbb{Z}}V_a$ be such a space. Choose a filtration $V=\bigcup V_{\leqslant i}$ that is *compatible* with the $\mathbb{Z}/d\mathbb{Z}$-grading in the following sense: each $V_{\leqslant i}$ is graded, and the resulting $\mathbb{Z}/d\mathbb{Z}$-grading on $\operatorname{gr}V$ comes from the $\mathbb{Z}$-grading meaning that $(\operatorname{gr}V)_i\subset \operatorname{gr}(V_{i\operatorname{mod} d})$. By the *modified Rees module* we mean $R^{\mathbb{Z}/d\mathbb{Z}}_\hbar(V):=\bigoplus_{i} (V_{\leqslant i}\cap V_{i\mod d})\hbar^{i/d}\subset V[\hbar^{\pm 1/d}]$, this is a graded $\mathbb{C}[\hbar]$-submodule with $\hbar$ of degree $d$. Now we can form $\mathcal{A}_\hbar:=R^{\mathbb{Z}/2\mathbb{Z}}_\hbar(\mathcal{A})$. With this, the construction in Section [2.3](#SS_res_fun_constr){reference-type="ref" reference="SS_res_fun_constr"} goes through verbatim. # Canonical filtration {#S_canon_filtr} Recall that $\mathbb{O}\subset \mathfrak{g}^*$ is a nilpotent orbit. Let $\mathcal{A}$ be a Dixmier algebra that we equip with a good filtration. The goal of this section is to construct a distinguished good filtration on $M\in \operatorname{HC}_\mathbb{O}^\partial(\mathcal{A})^{K,\kappa}$. For $\mathcal{A}=\mathcal{U}$ (with the PBW filtration) $K\subset G$ and $\kappa=0$ we recover the situation considered in the introduction. ## Comparable lattices In what follows we will use the correspondence between good filtrations on $M$ (here we do not require them to be $K$-stable) and graded lattices in $M[\hbar^{\pm 1}]$. Namely, by a *lattice* in $M[\hbar^{\pm 1}]$ we mean a finitely generated $\mathcal{A}_\hbar$-submodule $M_\hbar$ such that $M_\hbar[\hbar^{-1}]\xrightarrow{\sim} M[\hbar^{\pm 1}]$. There is a one-to-one correspondence between graded lattices in $M[\hbar^{\pm 1}]$ and good $\mathcal{A}$-module filtrations on $M$: we send a good filtration to the corresponding Rees module, and we send a lattice $M_\hbar$ to the filtered module $M_\hbar/(\hbar-1)$ (naturally identified with $M$). The condition that a good filtration is $K$-stable translates to the condition that the lattice is $K$-stable (where $\mathfrak{k}$-action comes from the action of the Lie algebra $\mathcal{J}_\hbar:=\mathcal{U}_\hbar\mathfrak{k}+ \mathcal{U}_\hbar\hbar\subset \mathcal{U}_\hbar$). A lattice is $K$-stable if and only if it is $\mathcal{J}_\hbar$-stable. Another easy remark: if $M^1_\hbar,M^2_\hbar$ are two lattices in $M[\hbar^{\pm 1}]$, then so are their sum and intersection. In what follows all lattices in $M[\hbar^{\pm 1}]$ (for $M\in \operatorname{HC}(\mathcal{A})$) we consider are going to be graded $K$-stable lattices. Similarly, there is a one-to-one correspondence between compatible (in the sense of Remark [Remark 5](#Rem:doubled_grading){reference-type="ref" reference="Rem:doubled_grading"}) good filtrations on objects $N\in \mathcal{W}\operatorname{-mod}^{\mathbb{Z}/d\mathbb{Z}}$ and graded $\mathcal{W}_\hbar$-lattices in $N[\hbar^{\pm 1}]$. To get from a filtration to a lattice, we take the modified Rees module $R_\hbar^{\mathbb{Z}/d\mathbb{Z}}(N)$, compare to Remark [Remark 5](#Rem:doubled_grading){reference-type="ref" reference="Rem:doubled_grading"}. **Definition 6**. Let $M\in \operatorname{HC}(\mathcal{A})$, and $M^1_\hbar,M^2_\hbar$ be two (graded $K$-stable) lattices in $M[\hbar^{\pm 1}]$. We say that $M^1_\hbar,M^2_\hbar$ are *comparable* if the dimension of the support of $M^i_\hbar/(M^1_\hbar\cap M^2_\hbar)$ for both $i=1,2$ is strictly less than $\dim \operatorname{V}(M)$. The following lemma describes basic properties of comparable lattices. The proof is easy and is left as an exercise. **Lemma 7**. *The following claims are true:* 1. *Let $M_\hbar, M'_\hbar$ be lattices such that $M'_\hbar$ is contained a lattice comparable with $M_\hbar$. Then $M_\hbar+ M'_\hbar$ is comparable with $M_\hbar$.* 2. *Let $M_\hbar, M'_\hbar$ be lattices such that $M'_\hbar$ contains a lattice comparable with $M_\hbar$. Then $M_\hbar\cap M'_\hbar$ is comparable with $M_\hbar$.* 3. *The comparability is an equivalence relation.* Here is our main result about comparable lattices. **Theorem 8**. *Suppose that $M\in \operatorname{HC}^\partial_\mathbb{O}(\mathcal{A})^{K,\kappa}$. In each comparability class, there is the unique [maximal]{.ul} (w.r.t inclusion) lattice. Moreover, for a graded $K$-stable lattice $M_\hbar\subset M[\hbar^{\pm 1}]$, the following two conditions are equivalent:* 1. *$M_\hbar$ is maximal in its comparability class.* 2. *The dimension of the support of every nonzero $\operatorname{gr}\mathcal{A}$-submodule in $\operatorname{gr}M$ is equal to $\dim \operatorname{V}(M)$.* The filtration mentioned in Theorem [Theorem 1](#Thm:main){reference-type="ref" reference="Thm:main"} is the maximal filtration in a certain comparability class. Theorem [Theorem 8](#Thm:max_filtration){reference-type="ref" reference="Thm:max_filtration"} will be proved in Section [3.4](#SS_thm_normalized_proof){reference-type="ref" reference="SS_thm_normalized_proof"} after some preparation. **Remark 9**. We remark that if $M\in \operatorname{HC}_{\overline{\mathbb{O}}}(\mathcal{A})$ has a submodule supported on $\partial \mathbb{O}$, then it cannot have a filtration satisfying (2). ## Classification In this section, we give a classification of lattices in $M[\hbar^{\pm 1}]$ up to comparability. We start with some terminology. Let $V$ be a finite dimensional module in $\mathcal{A}_{\dagger}\operatorname{-mod}^{\pm 1}$. An $\mathcal{A}_{\dagger}$-module filtration on $V$ is called Harish-Chandra if for all $a\in \mathcal{W}^{-\theta}_{\leqslant i}$ (the superscript means the $-1$ eigenspace for $\theta$) and $v\in V_{\leqslant j}$, we have $av\in V_{\leqslant i+j-2}$. Equivalently, the corresponding modified Rees module $R_\hbar(V)\subset V[\hbar^{\pm 1}]$ is stable under the Lie algebra action of the ideal $\mathcal{J}_\hbar\subset\mathcal{W}_\hbar$. We say that graded $\mathcal{A}_\hbar$-lattices in $V[\hbar^{\pm 1}]$ stable under the $\mathcal{J}_\hbar$-action are *HC lattices*. Let $\mathbb{O}^1_K,\ldots, \mathbb{O}^\ell_K$ be all open $K$-orbits in $\operatorname{V}(M)$ (they all have dimension $\dim \operatorname{V}(M)=\frac{1}{2}\dim \mathbb{O}$). Pick $\chi^i\in \mathbb{O}^i_K$. In particular, by ([\[eq:support_restriction\]](#eq:support_restriction){reference-type="ref" reference="eq:support_restriction"}), we have $\dim M_{\dagger,\chi^i}<\infty$ for all $i$. By the construction in Section [2.3](#SS_res_fun_constr){reference-type="ref" reference="SS_res_fun_constr"}, a choice of a good filtration on $M$ gives rise to a choice of a HC filtration on $M_{\dagger,\chi^i}$ (compatible with the $\mathbb{Z}/d\mathbb{Z}$-grading). **Proposition 10**. *The map that sends a lattice in $M[\hbar^{\pm 1}]$ to the collection of induced graded HC lattices in $M_{\dagger,\chi^i}[\hbar^{\pm 1}]$ defines a bijection between the sets of* - *The comparability classes of graded $K$-stable $\mathcal{A}_\hbar$-lattices in $M[\hbar^{\pm 1}]$.* - *The collections of graded HC $\mathcal{A}_{\hbar,\dagger}$-lattices in $M_{\dagger,\chi^i}[\hbar^{\pm 1}]$ for $i=1,\ldots,k$.* *Proof.* First, we prove that the map is constant on comparability classes. The functor $\bullet_{\dagger,\chi}$ kills the weakly HC modules whose support does not intersect $\mathbb{O}_K$. It follows that if $M^1_\hbar,M^2_\hbar$ are compatible, then $(M^j_\hbar/(M^1_\hbar\cap M^2_\hbar))_{\dagger,\chi^i}=0, j=1,2, i=1,\ldots,k$. Since $\bullet_{\dagger,\chi^i}$ is an exact functor, we see that $(M^1_\hbar)_{\dagger,\chi^i}=(M^2_\hbar)_{\dagger,\chi^i}$ for all $i$. Hence, we indeed get a map from (a) to (b). Now we show that the map is injective. Suppose $M^1_\hbar,M^2_\hbar$ are two lattices such that $(M^1_\hbar)_{\dagger,\chi^i}=(M^2_\hbar)_{\dagger,\chi^i}$, an equality of lattices in $M_{\dagger,\chi^i}[\hbar^{\pm 1}]$ for all $i=1,\ldots,k$. Since $\bullet_{\dagger,\chi^i}$ is exact, it sends $(M^1_\hbar\cap M^2_\hbar)_{\dagger,\chi^i}$ to $(M^1_\hbar)_{\dagger,\chi^i}\cap (M^2_\hbar)_{\dagger,\chi^i}=M^j_{\hbar,\dagger,\chi^i}$ for $j=1,2$. It follows that $\bullet_{\dagger,\chi^i}$ kills $M^j_\hbar/(M^1_\hbar\cap M^2_\hbar)$. So $M^1_\hbar, M^2_\hbar$ are comparable. Now we prove the surjectivity. Choose graded HC $\mathcal{A}_{\hbar,\dagger}$-lattices $\underline{M}'^i_\hbar\subset M_{\dagger,\chi^i}[\hbar^{\pm 1}]$. We need to prove that there is a graded $K$-stable $\mathcal{A}_\hbar$-lattice $M'_\hbar\subset M[\hbar^{\pm 1}]$ with $M'_{\hbar,\dagger,\chi^i}=\underline{M}'^i_\hbar$. Pick some lattice $M_\hbar \subset M[\hbar^{\pm 1}]$. And set $\underline{M}^i_\hbar:=M_{\hbar,\dagger,\chi^i}$. Let $\varphi_i$ denote the natural map $M_\hbar[\hbar^{-1}]\rightarrow M_\hbar^{\wedge_{\chi_i}}[\hbar^{-1}]$. Let $L_i:=T_{\chi^i}\mathbb{O}^i_K$, so that $M_\hbar^{\wedge_{\chi_i}}$ decomposes as $\mathbb{C}[[L_i,\hbar]]\otimes_{\mathbb{C}[[\hbar]]}\underline{M}^{i,\wedge_0}_{\hbar}$, where $\underline{M}^{i,\wedge_0}_\hbar$ stands for the $\hbar$-adic completion of $\underline{M}^i_\hbar$. Set $M_\hbar'^{i}:=\mathbb{C}[[L_i,\hbar]]\otimes_{\mathbb{C}[[\hbar]]}\underline{M}^{i,\wedge_0}_\hbar$. This is an $\mathcal{A}_\hbar^{\wedge_{\chi^i}}$-lattice in $M_\hbar^{\wedge_{\chi_i}}[\hbar^{\pm 1}]$. It is $\mathbb{C}^\times$-stable by the construction. We claim that it is HC in the sense to be explained now. Let $\mathcal{J}^U_\hbar, \mathcal{J}^\mathsf{A}_\hbar, \mathcal{J}^{\mathcal{W}}_\hbar$ denote the analogs of $\mathcal{J}_\hbar\subset \mathcal{U}_\hbar$ in $\mathcal{U}_\hbar^{\wedge_{\chi^i}}, \mathsf{A}_\hbar^{\wedge_0}, \mathcal{W}_\hbar^{\wedge_0}$. Then $$\label{eq:ideal_decomp} \mathcal{J}_\hbar^\mathcal{U}=\mathsf{A}_\hbar^{\wedge_0}\widehat{\otimes}_{\mathbb{C}[[\hbar]]}\mathcal{J}^{\mathcal{W}}_\hbar+ \mathcal{J}^\mathsf{A}_\hbar\widehat{\otimes}_{\mathbb{C}[[\hbar]]}\mathcal{W}_\hbar^{\wedge_0}.$$ When we say that a lattice in $M_\hbar^{\wedge_{\chi_i}}[\hbar^{-1}]$ is HC, we mean that it is stable under the Lie algebra action of $\mathcal{J}^U_\hbar$. The lattice $\mathbb{C}[[L_i,\hbar]]$ in its localization is stable under $\mathcal{J}^\mathsf{A}_\hbar$. Thanks to ([\[eq:ideal_decomp\]](#eq:ideal_decomp){reference-type="ref" reference="eq:ideal_decomp"}), $M_\hbar^{\wedge_{\chi_i}}\subset M_\hbar^{\wedge_{\chi_i}}[\hbar^{-1}]$ is stable under $\mathcal{J}^\mathcal{U}_\hbar$ if and only if $\underline{M}'^i_\hbar\subset \underline{M}'^i_\hbar[\hbar^{-1}]$ is stable under $\mathcal{J}^\mathcal{W}_\hbar$. The latter holds by assumption, establishing the claim in the beginning of the paragraph. Note that there is $f>0$ such that $\hbar^{f}M_\hbar^{\wedge_{\chi_i}}\subset M_\hbar'^i\subset \hbar^{-f}M_\hbar^{\wedge_{\chi_i}}$ for all $i$ (as both $M_\hbar^{\wedge_{\chi_i}}, M_\hbar'^i$ are lattices). Set $$\label{eq:interm_lattice} \tilde{M}_\hbar^i:=\hbar^{-f}M_\hbar\cap \varphi^{-1}_i(M_\hbar'^i).$$ This is a $\mathcal{J}_\hbar$- (hence $\mathfrak{k}$-) stable and $\mathbb{C}^\times$-stable $\mathcal{A}_\hbar$-lattice in $M_\hbar[\hbar^{-1}]$. Note that, since the completion functor intertwines the intersections, we have $$\label{eq:interm_lattice4} (\tilde{M}_\hbar^i)^{\wedge_{\chi_i}}=M'^i_\hbar,$$ Also, $$\label{eq:interm_lattice1}(\tilde{M}_\hbar^i)^{\wedge_{\chi_j}}\subset \hbar^{-e}M'^j_\hbar$$ for some fixed integer $e$ and all $j\neq i$. Now we construct a lattice $M'_\hbar\subset M_\hbar[\hbar^{-1}]$ from the lattices $\tilde{M}_\hbar^i$. Let $J_i$ denote the ideal of all elements in $S(\mathfrak{g})=\mathcal{U}_\hbar/(\hbar)$ that vanish on $\bigcup_{j\neq i}\mathbb{O}_K^j$. Note that $\hat{J}_i$ is $K$- and $\mathbb{C}^\times$-stable two-sided ideal. Also note $$\label{eq:interm_lattice3} \hat{J}_iM_\hbar'^i=M_\hbar'^i$$ because we can find an element of $J_i$ equal to $1$ at $\chi^i$. On the other hand, there is $r>0$ such that $$\label{eq:interm_lattice2} \hat{J}_i^r M_\hbar'^j\subset \hbar M_\hbar'^j$$ for all $j\neq i$. Let $\hat{I}_i$ denote the left ideal in $\mathcal{A}_\hbar$ generated by $\hat{J}_i^{er}$, where $e$ is as in ([\[eq:interm_lattice1\]](#eq:interm_lattice1){reference-type="ref" reference="eq:interm_lattice1"}), $r$ is as in ([\[eq:interm_lattice2\]](#eq:interm_lattice2){reference-type="ref" reference="eq:interm_lattice2"}). ([\[eq:interm_lattice3\]](#eq:interm_lattice3){reference-type="ref" reference="eq:interm_lattice3"}) gives $$\label{eq:interm_lattice3a} \hat{I}_iM_\hbar'^i=M_\hbar'^i,$$ while from ([\[eq:interm_lattice2\]](#eq:interm_lattice2){reference-type="ref" reference="eq:interm_lattice2"}) we get $$\label{eq:interm_lattice2a} \hat{I}_i M_\hbar'^j\subset \hbar^e M_\hbar'^j$$ We set $$M'_\hbar:=\hbar^{f} M_\hbar+\sum_{i=1}^\ell \hat{I}_i\tilde{M}_\hbar^i,$$ where $f$ is as in ([\[eq:interm_lattice\]](#eq:interm_lattice){reference-type="ref" reference="eq:interm_lattice"}). This $\mathcal{A}_\hbar$-submodule is $K$-stable and graded by the construction. It generates $M_\hbar[\hbar^{-1}]$ as an $\mathcal{A}_\hbar[\hbar^{-1}]$-module because it contains $\hbar^f M_\hbar$. It is a lattice because $M_\hbar$ and all $\tilde{M}_\hbar^i$ are lattices. We have $\tilde{M}_\hbar^{\wedge_{\chi_i}}=M'^i_\hbar$ for all $i$ because - the completion functor is exact and so intertwines the sums, - $(\hat{I}_i\tilde{M}_\hbar^i)^{\wedge_{\chi_i}}=M_\hbar'^i$ by ([\[eq:interm_lattice4\]](#eq:interm_lattice4){reference-type="ref" reference="eq:interm_lattice4"}) combined with ([\[eq:interm_lattice3a\]](#eq:interm_lattice3a){reference-type="ref" reference="eq:interm_lattice3a"}), - $(\hbar^d M_\hbar)^{\wedge_{\chi^i}}\subset M_\hbar'^i$ by ([\[eq:interm_lattice\]](#eq:interm_lattice){reference-type="ref" reference="eq:interm_lattice"}), - $(\hat{I}_j\tilde{M}_\hbar^i)^{\wedge_{\chi_i}}\subset M_\hbar'^i$ by ([\[eq:interm_lattice2a\]](#eq:interm_lattice2a){reference-type="ref" reference="eq:interm_lattice2a"}) combined with ([\[eq:interm_lattice1\]](#eq:interm_lattice1){reference-type="ref" reference="eq:interm_lattice1"}). So we have constructed a lattice $M'_\hbar\subset M_\hbar[\hbar^{-1}]$ with required properties. This finishes the proof of the surjectivity, and hence of the proposition. ◻ ## Fd-comparable filtrations In this section we consider the following situation. Let $\mathbb{O}_K$ be a nilpotent $K$-orbit in $(\mathfrak{g}/\mathfrak{k})^*$. Pick $\chi\in \mathbb{O}_K$. Let $S$ denote a $\theta$-stable Slodowy slice at $\chi$. Consider the algebra $\mathcal{A}_{\dagger}$. Take $N\in \mathcal{A}_\dagger\operatorname{-mod}^{\mathbb{Z}/d\mathbb{Z}}$. Suppose that $N$ does not have nonzero finite dimensional $\mathcal{A}_\dagger$-submodules. **Definition 11**. Two graded $\mathcal{A}_{\hbar,\dagger}$-lattices $N_\hbar^{1}$ and $N_\hbar^{2}$ are said to be *fd-comparable* if $N_\hbar^1/(N_\hbar^1\cap N_\hbar^2), N_\hbar^2/(N_\hbar^1\cap N_\hbar^2)$ are finite dimensional. Note that being fd-comparable is an equivalence relation, compare to Lemma [Lemma 7](#Lem:compar_properties){reference-type="ref" reference="Lem:compar_properties"}, so we can talk about the fd-comparability classes of lattices. The goal of this section is to prove the following result that will be used in the proof of Theorem [Theorem 8](#Thm:max_filtration){reference-type="ref" reference="Thm:max_filtration"}. **Proposition 12**. *Recall that we assume that $N$ does not have finite dimensional submodules. In each fd-comparability class, there is the unique maximal (w.r.t. inclusion) element.* *Proof.* Pick a graded $\mathcal{A}_{\hbar,\dagger}$-lattice $N_\hbar$. For each $k\geqslant 0$, we have the unique lattice fd-comparable with $N_\hbar$ and maximal among all fd-comparable lattices contained in $\hbar^{-k}N_\hbar$: the sum of all fd-comparable lattices, which coincides with the sum of finitely many of them because $\mathcal{A}_{\hbar,\dagger}$ is a Noetherian algebra (it is a finitely generated module over the Noetherian algebra $\mathcal{W}_\hbar$). Denote this maximal lattice by $N_\hbar^k$. So we have the increasing chain of fd-comparable lattices $$N_\hbar=N^0_\hbar\subset N^1_\hbar\subset\ldots\subset N^k_\hbar\subset\ldots$$ and the claim of the proposition amounts to showing that this chain terminates. The proof of this is in several steps. *Step 1*. We claim that for each $i$ we have $$\begin{aligned} \label{eq:containment1} &\hbar N_\hbar^{i+1}\subset N_\hbar^i,\\\label{eq:containment2} &N_\hbar^i=N_\hbar^{i+1}\cap \hbar^{-i}N_{\hbar}.\end{aligned}$$ To show ([\[eq:containment1\]](#eq:containment1){reference-type="ref" reference="eq:containment1"}) observe that $\hbar N_{\hbar}^{i+1}+N_\hbar$ is fd-comparable to $N_\hbar$ by the direct analog of (1) of Lemma [Lemma 7](#Lem:compar_properties){reference-type="ref" reference="Lem:compar_properties"} and is contained in $\hbar^{-i}N_\hbar$. ([\[eq:containment2\]](#eq:containment2){reference-type="ref" reference="eq:containment2"}) is similar: we use the direct analog of (2) of Lemma [Lemma 7](#Lem:compar_properties){reference-type="ref" reference="Lem:compar_properties"}. *Step 2*. We claim that the inclusions $\hbar N_{\hbar}^{i+1}\subset N_\hbar^i, \hbar N_{\hbar}^{i}\subset N_\hbar^{i-1}$ induce an embedding $$\label{eq:containment3} \hbar N_\hbar^{i+1}/\hbar N_{\hbar}^i\hookrightarrow N_{\hbar}^i/N_{\hbar}^{i-1}, \forall i\geqslant 1.$$ Indeed, thanks to ([\[eq:containment1\]](#eq:containment1){reference-type="ref" reference="eq:containment1"}), we have $$N_\hbar^{i+1}/N_\hbar^i=N_\hbar^{i+1}/(N_\hbar^{i+1}\cap \hbar^{-i}N_{\hbar}) \hookrightarrow (N_{\hbar}^{i+1}+\hbar^{-i}N_\hbar)/(\hbar^{-i}N_\hbar).$$ The multiplication by $\hbar$ identifies the latter space with $(\hbar N_{\hbar}^{i+1}+\hbar^{1-i}N_\hbar)/(\hbar^{1-i}N_\hbar)$. Thanks to ([\[eq:containment1\]](#eq:containment1){reference-type="ref" reference="eq:containment1"}), this space includes into $$(N_{\hbar}^{i}+\hbar^{1-i}N_\hbar)/(\hbar^{1-i}N_\hbar)=N_\hbar^{i}/N_{\hbar}^{i-1},$$ where the equality is thanks to ([\[eq:containment2\]](#eq:containment2){reference-type="ref" reference="eq:containment2"}) (with $i$ replaced by $i-1$). *Step 3*. Note that all spaces $N_\hbar^{i+1}/N_{\hbar}^i$ are finite dimensional by the definition of fd-comparable lattices. Thanks to ([\[eq:containment3\]](#eq:containment3){reference-type="ref" reference="eq:containment3"}), the sequence $N_{\hbar}^{i+1}/N_{\hbar}^i$ stabilizes. We need to show that it stabilizes to $0$. Assume the contrary. Note that all spaces $N_{\hbar}^{i+1}/N_\hbar^i$ are graded, and ([\[eq:containment3\]](#eq:containment3){reference-type="ref" reference="eq:containment3"}) is homogeneous. Consider the sequence $\hbar^{i}N_\hbar^{i}\subset N_\hbar$, it is decreasing by ([\[eq:containment1\]](#eq:containment1){reference-type="ref" reference="eq:containment1"}). Let $K_\hbar$ denote the intersection $\bigcap_{i\geqslant 0}\hbar^{i}N_\hbar^{i}$. We claim that $$\label{eq:containment4} K_\hbar/\hbar K_\hbar\xrightarrow{\sim} \varprojlim N_\hbar^{i+1}/N_{\hbar}^i.$$ Indeed, take $j$ such that the maps ([\[eq:containment3\]](#eq:containment3){reference-type="ref" reference="eq:containment3"}) are isomorphisms for $i>j$. This is equivalent to $N^{i}_\hbar=N^{i-1}_\hbar+\hbar N^{i+1}_\hbar$ for all $i>j$, equivalently $$\label{eq:containment5} \hbar^i N^i_\hbar=\hbar^{i-j}(\hbar^jN^{j}_\hbar)+ \hbar^{i+1}N^{i+1}_\hbar, \forall i>j.$$ Now take a homogeneous element $n\in N_\hbar^{j+1}/N_{\hbar}^j$ of degree $f$, and let $f'$ be the minimal degree in $N^j_\hbar$ (that exists because $N^j_\hbar$ is a finitely generated module over the positively graded algebra $\mathcal{W}_\hbar$). Take the component of degree $i$ in ([\[eq:containment5\]](#eq:containment5){reference-type="ref" reference="eq:containment5"}) for $i-j>f-f'$, and get $(\hbar^i N^i_\hbar)_{f}= (\hbar^{i+1}N^{i+1}_\hbar)_{f}$. This implies that ([\[eq:containment4\]](#eq:containment4){reference-type="ref" reference="eq:containment4"}) is an isomorphism on the elements of degree $f$. So, it is an isomorphism. *Step 4*. In particular, $K_\hbar$ is a nonzero graded $\mathcal{A}_{\hbar,\dagger}$-submodule in $N_\hbar$ that is supported at the point $\chi\in S\cap (\mathfrak{g}/\mathfrak{k})^*$. Then $K_\hbar/(\hbar-1)K_\hbar$ is a nonzero finite dimensional $\mathcal{A}_{\dagger}$-submodule of $N$. This contradiction shows that ([\[eq:containment3\]](#eq:containment3){reference-type="ref" reference="eq:containment3"}) stabilizes at $0$ and finishes the proof. ◻ ## Proof of Theorem [Theorem 8](#Thm:max_filtration){reference-type="ref" reference="Thm:max_filtration"} {#SS_thm_normalized_proof} We start the following result that will be used in the proof. **Lemma 13**. *Let $M\in \operatorname{HC}_{\mathbb{O}}^\partial(\mathcal{A})^{K,\kappa}$. Pick a point $\chi\in \operatorname{V}(M)\setminus \mathbb{O}$. Then $M_{\dagger,\chi}$ has no nonzero finite dimensional $\mathcal{A}_{\dagger,\chi}$-submodules.* *Proof.* It is enough to prove this claim for $\mathcal{A}=\mathcal{U}$. Assume the contrary, let $N$ be a nonzero finite dimensional $\mathcal{W}$-submodule of $M_{\dagger,\chi}$. Let $\mathbb{O}':=G\chi$. We can assume that $N$ is irreducible. Using [@HC Theorem 1.2.2], we see that there is an ideal $\mathcal{J}\subset \mathcal{U}$ with $\operatorname{V}(\mathcal{U}/\mathcal{J})=\overline{\mathbb{O}}'$ such that $\mathcal{J}_{\dagger}$ annihilates $N$. It follows that $\operatorname{Hom}_{\mathcal{W}}((\mathcal{U}/\mathcal{J})_{\dagger,\chi}, M_{\dagger,\chi})\neq \{0\}$. By Lemma [Lemma 4](#Lem:HC_hom_restr_compat){reference-type="ref" reference="Lem:HC_hom_restr_compat"} (applied to HC bimodules/modules rather than weakly HC ones), we have $\operatorname{Hom}_{\mathcal{U}}(\mathcal{U}/\mathcal{J},M)\neq 0$, equivalently, the annihilator of $\mathcal{J}$ in $M$ is nonzero. This annihilator must be supported on $\overline{\mathbb{O}}'\cap (\mathfrak{g}/\mathfrak{k})^*$, which is impossible because $M\in \operatorname{HC}^\partial_{\mathbb{O}}(\mathfrak{g},K)$. ◻ *Proof of Theorem [Theorem 8](#Thm:max_filtration){reference-type="ref" reference="Thm:max_filtration"}.* First, we prove that a maximal graded $K$-stable lattice in the comparability class of $M_\hbar$ exists (the uniqueness is clear). As in the proof of Proposition [Proposition 12](#Prop:max_filtr_dim1){reference-type="ref" reference="Prop:max_filtr_dim1"}, we have the maximal such lattice contained in $\hbar^{-i}M_\hbar$, denote it by $M_\hbar^i$. We need to show that the sequence $(M_\hbar^i)$ stabilizes. Let $\chi\in \operatorname{V}(M)\setminus \mathbb{O}$. We prove that $(M_\hbar^i)_{\dagger,\chi}$ terminates by induction on $\dim \operatorname{V}(M)-\dim K\chi$. The base case is when $\dim \operatorname{V}(M)-\dim K\chi=1$. Here the lattices $M^i_{\hbar,\dagger,\chi}\subset M_{\dagger,\chi}[\hbar^{\pm 1}]$ are fd-comparable. Thanks to Lemma [Lemma 13](#Lem:no_fin_dim_subs){reference-type="ref" reference="Lem:no_fin_dim_subs"}, $M_{\dagger,\chi}$ satisfies the assumptions of Proposition [Proposition 12](#Prop:max_filtr_dim1){reference-type="ref" reference="Prop:max_filtr_dim1"} and we are done by this proposition. Now suppose that $M^i_{\hbar,\dagger,\chi}$ stabilizes for all $\chi$ with $$\label{eq:dim_induction}\dim \operatorname{V}(M)-\dim K\chi<\ell$$ (with some $\ell$). As we only have finitely many orbits $K\chi$, we can assume that there is $j$ with $M^i_{\hbar,\dagger,\chi}=M^j_{\hbar,\dagger,\chi}$ for all $i>j$ and all $\chi$ satisfying ([\[eq:dim_induction\]](#eq:dim_induction){reference-type="ref" reference="eq:dim_induction"}). It follows that the support of $M^i_{\hbar}/M^j_\hbar$ has dimension at most $\dim \operatorname{V}(M)-\ell$. So, for $\chi$ with $\dim \operatorname{V}(M)-\dim K\chi=\ell$, the lattices $M^i_{\hbar,\dagger,\chi}\subset M_{\dagger,\chi}[\hbar^{\pm 1}]$ with $i\geqslant j$ are fd-comparable. Similarly to the previous paragraph, this sequence stabilizes. This establishes the induction step and finishes the proof of the existence of a maximal lattice in a given comparability class. Now we prove the equivalence of (1) and (2). First, suppose that $M_\hbar$ is maximal in its comparability class. Let $N\subset \operatorname{gr}M$ be a $\mathbb{C}[\mathfrak{g}^*]$-submodule such that $\dim \operatorname{Supp}(N)<\dim \operatorname{V}(M)$. We can choose it to be maximal of all submodules with this property. Then it must be $\mathbb{C}^\times$-stable as the action of $\mathbb{C}^\times$ maps one submodule with the required property into another. It also must be $K$-stable for a similar reason. The preimage of $N$ in $\hbar^{-1}M_\hbar$ under the projection to $\hbar^{-1}M_\hbar/M_\hbar\cong \operatorname{gr}M$ is comparable with $M_\hbar$, a contradiction. Now suppose that $\operatorname{gr}M$ has no submodules with dimension of support strictly less than $\dim \operatorname{V}(M)$. Suppose $M_\hbar$ is not maximal. Then we can find a comparable graded $K$-stable lattice $\tilde{M}_\hbar$ between $M_\hbar$ and $\hbar^{-1}M_\hbar$. Then $\tilde{M}_\hbar/M_\hbar$ is a submodule in $\hbar^{-1}M_\hbar/M_{\hbar}\cong \operatorname{gr}M$ whose support has dimension strictly less than $\dim \operatorname{V}(M)$. A contradiction. ◻ ## A question of Vogan The goal of this section is to use Theorem [Theorem 8](#Thm:max_filtration){reference-type="ref" reference="Thm:max_filtration"} to prove Theorem [Theorem 3](#Thm:codim1_connected){reference-type="ref" reference="Thm:codim1_connected"} from Introduction. *Proof of Theorem [Theorem 3](#Thm:codim1_connected){reference-type="ref" reference="Thm:codim1_connected"}.* Assume the contrary, $X:=\operatorname{V}(M)\setminus \operatorname{V}(M)^{\geqslant 2}$ splits as a disjoint union of two closed nonempty subvarieties $X_1$ and $X_2$. Thanks to [@LY Lemma 2.5], we can find - a connected reductive group $K$ with a homomorphism to $G$ that identifies the Lie algebra of $K$ with $\mathfrak{k}$, - and a character $\kappa$ of $\mathfrak{k}$ so that $M$ becomes a strongly $(K,\kappa)$-equivariant $\mathcal{U}$-module. Equip $M$ with a $K$-stable good filtration that is maximal in its comparability class. Let $I$ denote the annihilator of $\operatorname{gr}M$ in $S(\mathfrak{g})$ and $A:=S(\mathfrak{g})$. By condition (2) of Theorem [Theorem 8](#Thm:max_filtration){reference-type="ref" reference="Thm:max_filtration"}, a non-zero divisor of $A$ is also a non-zero divisor in $\operatorname{gr}M$. Let $(\operatorname{gr}M)^0$ denote the restriction of $\operatorname{gr}M$ to $X_1$: formally it is the localization of $A$-module $\operatorname{gr}M$ to the open subset $\operatorname{Spec}(A)\setminus [\operatorname{V}(M)^{\geqslant 2}\sqcup X_2]$. The key observation is that $\Gamma((\operatorname{gr}M)^0)$ is finitely generated. Indeed, the locus in $\operatorname{Spec}(A)$, where $\operatorname{gr}M$ fails to be Cohen-Macaulay has codimension greater than $1$. This locus has to be $K$-stable because of the $K$-action on $\operatorname{gr}M$. So it is contained in $\operatorname{V}(M)^{\geqslant 2}$. Hence $(\operatorname{gr}M)^0$ is Cohen-Macaulay. The push-forward of a maximal Cohen-Macaulay coherent sheaf from an open subset whose complement has codimension at least $2$ is coherent. Moreover, the support of $\Gamma((\operatorname{gr}M)^0)$ is contained in $\overline{X}_1$. Then the argument proceeds as in the proof of [@Vogan Theorem 4.6]. We can microlocalize $M$ to a sheaf on $\operatorname{V}(M)$, in particular, consider the restriction $M^0$ of this sheaf to $X_1$. This sheaf carries a complete and separated $K$-stable filtration whose associated graded is $(\operatorname{gr}M)^0$. It follows that $\operatorname{gr}\Gamma(M^0)\hookrightarrow \Gamma((\operatorname{gr}M)^0)$, hence is a finitely generated $S(\mathfrak{g}^*)$-module. So $\Gamma(M^0)$ carries a good filtration, hence is a HC $(\mathfrak{g},K,\kappa)$-module. By the construction, its associated variety is contained in the support of $\Gamma((\operatorname{gr}M)^0)$, which, in its turn, is contained in $\overline{X}_1$. On the other hand, by the construction, we have a $\mathcal{U}$-module homomorphism $M\rightarrow M^0$, which is nonzero because $X_1\subset \operatorname{V}(M)$. So we get a nonzero homomorphism $M\rightarrow \Gamma(M^0)$. Since $M$ is simple, this homomorphism has to be injective. But this leads to a contradiction: as we have seen above in this proof, $\operatorname{V}(M)\not\subset X_1$. We have proved that $\operatorname{V}(M)\setminus \operatorname{V}(M)^{\geqslant 2}$ is connected. ◻ ## Normalized and canonical filtrations Recall the integer $d:=2|\ker\iota|$, where $\iota$ is the homomorphism $K\rightarrow G$. Let $\mathcal{A}$ be a Dixmier algebra. We assume that it is equipped with a good filtration subject to certain conditions. Namely, fix a nilpotent orbit $\mathbb{O}\subset \mathfrak{g}^*$, take a point $\chi\in \mathbb{O}$ and let $S$ be a Slodowy slice through $\chi$. The restriction $(\operatorname{gr}\mathcal{A})|_S$ has a natural grading, it is induced by the action $t\mapsto t^{-d}\gamma(t)$. Here are the two conditions on $\mathcal{A}$ that we need: - $\operatorname{gr}\mathcal{A}$ is commutative, - $(\operatorname{gr}\mathcal{A})|_S$ is positively graded (meaning that $(\operatorname{gr}\mathcal{A})|_{S,i}$ is zero for $i<0$ and is 1-dimensional for $i=0$). Moreover, $(\operatorname{gr}\mathcal{A})_{S,i}=\{0\}$ for $0<i<d$. For example, $\mathcal{A}=\mathcal{U}$ satisfies both conditions. Another example is when $\mathcal{A}$ is a quantization of $\mathbb{C}[\mathbb{O}]$ (or of $\mathbb{C}[\tilde{\mathbb{O}}]$, where $\tilde{\mathbb{O}}$ is an equivariant cover of $\mathbb{O}$ such that the scaling action of $\mathbb{C}^\times$ on $\mathbb{O}$ lifts to $\tilde{\mathbb{O}}$). As was noted in Section [2.4](#SS_Dixmier_generalization){reference-type="ref" reference="SS_Dixmier_generalization"}, $(\operatorname{gr}\mathcal{A})_S$ is concentrated in degree $0$. **Definition 14**. Let $N$ be a finite dimensional $\mathbb{Z}/d\mathbb{Z}$-graded $\mathcal{A}_{\dagger}$-module. By the *trivial filtration* on $N$ we mean the filtration given by $N_{\leqslant -1}=\{0\}, N_{\leqslant i}=\sum_{j=0}^i N_{i\operatorname{mod} d}$ for $i=0,\ldots,d-1$. The following lemma describes easy properties of this filtration. **Lemma 15**. *The following claims are true:* - *The trivial filtration is a compatible HC filtration.* - *The associated graded of the trivial filtration is annihilated by $(\operatorname{gr}\mathcal{A}_\dagger)_{>0}$.* - *Suppose that $N$ is equipped with a compatible filtration such that the only nonzero components in $\operatorname{gr}N$ are in degrees $0,\ldots,d-1$. Then the filtration is trivial.* *Proof.* (1) and (2) easily follow from (b). (3) is an easy exercise. ◻ Now we can define the canonical filtration on an object $M\in \operatorname{HC}^\partial_{\mathbb{O}}(\mathcal{A},K,\kappa)$. Recall that each comparability class of good $K$-stable $\mathcal{A}$-module filtrations on $M$ has the unique maximal element, Theorem [Theorem 8](#Thm:max_filtration){reference-type="ref" reference="Thm:max_filtration"}. Further, if $\mathbb{O}^1_K,\ldots,\mathbb{O}^\ell_K$ be all $K$-orbits of dimension $\frac{1}{2}\dim \mathbb{O}$ in $\operatorname{V}(M)$, then there is a bijection between the comparability classes of good filtrations on $M$ and the $\ell$-tuples of HC filtrations compatible with the $\mathbb{Z}/d\mathbb{Z}$-gradings on $M_{\dagger,\chi^i}$ for $\chi^i\in \mathbb{O}^i_K$, Proposition [Proposition 10](#Prop:lattices_classification){reference-type="ref" reference="Prop:lattices_classification"}. **Definition 16**. By a *normalized filtration* on $M$ we mean a good $K$-stable $\mathcal{A}$-module filtration such that the corresponding filtrations on the modules $M_{\dagger,\chi^i}$ are trivial. By the *canonical filtration* on $M$ we mean the unique normalized filtration such that the corresponding lattice on $M[\hbar^{\pm 1}]$ is maximal. Our next goal is to characterize the canonical filtration in elementary terms, i.e., without using restriction functors, proving a direct generalization of Theorem [Theorem 1](#Thm:main){reference-type="ref" reference="Thm:main"}. **Theorem 17**. *For a good $K$-stable $\mathcal{A}$-module filtration on $M\in \operatorname{HC}^\partial_{\mathbb{O}}(\mathcal{A},K,\kappa)$, the following two conditions are equivalent:* 1. *The filtration is canonical.* 2. - *The annihilator $I$ of $\operatorname{gr}M$ in $\operatorname{gr}\mathcal{A}$ is radical.* - *Every nonzero divisor in $(\operatorname{gr}\mathcal{A})/I$ is also a nonzero divisor in $\operatorname{gr}M$.* - *For each $i=1,\ldots,\ell$, the graded module $(\operatorname{gr}M)_{\chi^i}$ is concentrated in degrees $0,\ldots,d-1$.* *Proof.* We start with (1)$\Rightarrow$(2). We claim that - The $S(\mathfrak{g}/\mathfrak{k})$-module $\sqrt{I}/I$ is supported away from $\bigsqcup_{i=1}^\ell \mathbb{O}^i_K$. We write $\chi$ for $\chi^i$ and $L$ for the lagrangian subspace $L_i$. By Lemma [Lemma 15](#Lem:triv_filtr_properties){reference-type="ref" reference="Lem:triv_filtr_properties"}, $\operatorname{gr}(M_{\dagger,\chi})$ is killed by $(\operatorname{gr}\mathcal{A}_\dagger)_{>0}$. Note that $\operatorname{gr}M$ is an object in $\operatorname{wHC}^{gr}(\mathcal{A}_\hbar)$, and $(\operatorname{gr}M_{\dagger,\chi})\cong \operatorname{gr}(M_{\dagger,\chi})$. In particular, $(\operatorname{gr}M)^{\wedge_{\chi}}\cong \mathbb{C}[[L]]\otimes \operatorname{gr}(M_{\dagger,\chi})$. The annihilator of this module in $(\operatorname{gr}\mathcal{A})^{\wedge_\chi}\cong \mathbb{C}[[T_{\chi}\mathbb{O}]]\widehat{\otimes}(\operatorname{gr}\mathcal{A}_\dagger)^{\wedge_\chi}$ is the vanishing ideal of $\mathbb{C}[[T_{\chi}\mathbb{O}]]\widehat{\otimes}(\operatorname{gr}\mathcal{A}_\dagger)^{\wedge_\chi}_{>0}$. So it is radical. We note that the annihilator of the completion is the completion of the annihilator. Also the completion of the radical is the radical of the completion. (\*) follows. Now note that modulo (\*), (a) is equivalent to (2) of Theorem [Theorem 8](#Thm:max_filtration){reference-type="ref" reference="Thm:max_filtration"}. (b) is equivalent to (2) of Theorem [Theorem 8](#Thm:max_filtration){reference-type="ref" reference="Thm:max_filtration"} unconditionally. And (c) is satisfied because the filtration is normalized. To prove (2)$\Rightarrow$(1), note that (a) and (b) together imply (2) of Theorem [Theorem 8](#Thm:max_filtration){reference-type="ref" reference="Thm:max_filtration"}. Hence the filtration on $M$ is maximal in its comparability class. Thanks to (c) and (3) of Lemma [Lemma 15](#Lem:triv_filtr_properties){reference-type="ref" reference="Lem:triv_filtr_properties"}, the filtration is normalized. So, it is canonical. ◻ To finish the section we prove the following functoriality result. **Proposition 18**. *Let $M_1,M_2\in \operatorname{HC}^\partial_\mathbb{O}(\mathcal{A},K,\kappa)$ and $\varphi:M_1\rightarrow M_2$ be a homomorphism. Then $\varphi$ is filtration preserving with respect to the canonical filtrations.* *Proof.* The proof is in two steps. *Step 1*. First, assume $M_i=M, i=1,2,$ and $\varphi$ is an automorphism. Note that $\varphi$ (extended to $M[\hbar^{\pm 1}]$ in an obvious way) sends comparable lattices to comparable lattices, preserving inclusions. So, one just needs to check that a normalized filtration is sent to a normalized filtration. This is because any morphism in the category of finite dimensional objects in $\mathcal{A}_{\dagger}\operatorname{-mod}^{\mathbb{Z}/d\mathbb{Z}}$ is compatible with the trivial filtrations. *Step 2*. We deduce the general case from Step 1. Note that $M_1\oplus M_2$ carries the direct sum filtration. It is canonical, this follows for example from Theorem [Theorem 17](#Thm:canonical_charact){reference-type="ref" reference="Thm:canonical_charact"}. Now consider the automorphism of $M_1\oplus M_2$ sending $(m_1,m_2)$ to $(m_1,m_2+\varphi(m_1))$. Applying the claim of Step 1 finishes the proof. ◻ 99 I. Losev, *Quantized symplectic actions and W-algebras*. J. Amer. Math. Soc. 23 (2010), no. 1, 35-59. I. Losev, *Finite dimensional representations of W-algebras*. Duke Math J. 159(2011), n.1, 99-143. I. Losev, *Dimensions of irreducible modules over W-algebras and Goldie ranks*. Invent. Math. 200 (2015), N3, 849-923. I. Losev, *Primitive ideals in W-algebras of type A*. J. Algebra, 359 (2012), 80-88. I. Losev, *Bernstein inequality and holonomic modules*. Adv. Math. 308 (2017), 941-963. I. Losev, *Deformations of symplectic singularities and Orbit method for semisimple Lie algebras*. Selecta Math, 28 (2022), N2, paper N30, 52 pages. I. Losev, S. Yu, *On Harish-Chandra modules over quantizations of nilpotent orbits*. arXiv:2309.11191. A. Premet, *Special transverse slices and their enveloping algebras*. With an appendix by Serge Skryabin. Adv. Math. 170 (2002), no. 1, 1-55. D. Vogan, *Associated varieties and unipotent representations*. Harmonic analysis on reductive groups (Brunswick, ME, 1989), 315-388, Progr. Math., 101, Birkhäuser Boston, Boston, MA, 1991.

arxiv_math

--- abstract: | For $L$-functions attached to automorphic representations of unitary groups $U_{n+1}\times U_n$, we establish a subconvex bound valid in certain horizontal aspects, where the set of ramified places is allowed to vary. address: - YMSC, Tsinghua University - Aarhus University author: - Yueke Hu - Paul D. Nelson title: Subconvex bounds for $U_{n+1}\times U_n$ in horizontal aspects --- # Introduction ## Overview This paper concerns the quantitative analysis of automorphic $L$-functions on higher rank classical groups. Recent works on this topic include - an approach to the analytic test vector problem based on the orbit method, applied to the asymptotic evaluations of moments (see [@nelson-venkatesh-1]), and - subconvex bounds on $\mathop{\mathrm{U}}_{n+1} \times \mathop{\mathrm{U}}_n$ and $\mathop{\mathrm{GL}}_n$ (see [@2020arXiv201202187N; @2021arXiv210915230N] and announcement talks of Marshall). These works study sequences of automorphic forms whose ramification increases inside some *fixed* finite set of places, a setup known as the *depth* aspect. By a *horizontal* aspect, we mean one where the set of ramified places is itself allowed to vary. For example, one can study Dirichlet characters of conductor $p^m$ for fixed $p$ as $m \rightarrow \infty$ (depth) and for fixed $m$ as $p \rightarrow \infty$ (horizontal). This paper establishes subconvex bounds on $U_{n +1} \times U_n$ in certain horizontal aspects. Our results are uniform enough to apply also in the depth aspect (indeed, as "$p^m \rightarrow \infty$"), but this is not the main novelty. The depth aspect often has a Lie-algebraic flavor, involving tools such as stationary phase analysis and Taylor approximation. For instance, the works noted above make heavy use of the exponential map and Lie algebra for groups such as $\mathop{\mathrm{GL}}_n(F)$, with $F$ a fixed local field. These techniques are applied to spectral problems (construction and analysis of test vectors) and geometric problems (the "volume bound"). In horizontal aspects, Lie-algebraic techniques are often less relevant. For this paper, we restrict to "even depth" cases ("$p^{2 m}$") where the spectral analysis can still be carried out using Lie-algebraic techniques, but the geometric analysis requires new arguments, of algebro-geometric rather than Lie-algebraic flavor. We now describe our main result (Theorem [Theorem 1](#theorem:cj3ngw7u2s){reference-type="ref" reference="theorem:cj3ngw7u2s"}) and the main ideas of its proof, emphasizing new features encountered in horizontal aspects. ## The refined Gan--Gross--Prasad conjectures {#sec:cj3siudn3c} These conjectures, now known in many cases, provide a link between values of $L$-functions and integrals of automorphic forms. The body of this paper addresses the local problems that arise in estimating such integrals. The link described here is not otherwise applied in this paper, but provides context for interpreting our results. Let $F$ be a number field with adele ring $\mathbb{A}$, let $E/F$ be a quadratic extension, let $V$ be an $(n+1)$-dimensional hermitian space over $E$, and let $W$ be an $n$-dimensional nondegenerate subspace of $V$. Define the pair of unitary groups $(G,H) := (\mathop{\mathrm{U}}(V), \mathop{\mathrm{U}}(W))$ over $F$. Given a pair of cuspidal automorphic representations $\pi$ and $\sigma$ of $G$ and $H$ that are locally distinguished, one may attach a branching coefficient $\mathcal{L}(\pi,\sigma)$ quantifying how automorphic forms in $\pi$ correlate against those in $\sigma$. The definition depends upon the choice of a finite set $S$ of places of $F$, taken large enough to contain every place that is archimedean or at which $\pi$ or $\sigma$ is ramified, and requires $\pi$ and $\sigma$ to be (nearly) tempered inside $S$. It is characterized by the following family of identities: for all $v \in \pi$ and $u \in \sigma$ that are unramified outside $S$, and with suitable normalization of measure, $$\label{eq:cj3tfrrhlc} \left\lvert \int_{H(F) \backslash H(\mathbb{A})} v \bar{u} \right\rvert^2 = \mathcal{L}(\pi, \sigma) \int_{H(F_S)} \langle h v, v \rangle \langle u, h u \rangle \, d h.$$ We refer to [@2020arXiv201202187N §3.7] for further details. It has been conjectured by Ichino--Ikeda [@MR2585578] and N. Harris [@MR3159075] that if $S$ is large enough in the sense recorded in [@MR2585578 §1], then $\mathcal{L}(\pi,\sigma)$ is given by a ratio of $L$-values, namely, with notation as in [@MR4426741], $$\label{eqn:20230517061322} \mathcal{L}(\pi,\sigma) = 2^{- \beta} \frac{L^{(S)}(\pi_E \otimes \sigma_E^{\vee},1/2)}{L^{(S)}(\mathop{\mathrm{Ad}}, \pi \boxtimes {\sigma}^{\vee},1)} \Delta_G^{(S)}.$$ This expectation has been proved at least when $\pi$ and $\sigma$ are tempered at all places [@MR4426741]. ## Main result {#sec:cj4vjf3x2x} We establish a subconvex bound on $\mathop{\mathrm{U}}_{n+1} \times \mathop{\mathrm{U}}_n$, with $\mathop{\mathrm{U}}_n$ anisotropic, for pairs $(\pi,\sigma)$ whose ramification concentrates at some (varying) finite place $\mathfrak{p}$, provided that the conductor does not drop, under some local assumptions (e.g., principal series of even depth). In more detail: **Theorem 1**. *Fix $F, E, G, H$ as in §[1.2](#sec:cj3siudn3c){reference-type="ref" reference="sec:cj3siudn3c"}, as well as a finite set $S$ of of places of $F$, large enough in the sense specified in [@2020arXiv201202187N §3.6]. Let $\mathcal{F}$ be a family of tuples $(\pi,\sigma, \mathfrak{p}, \mathfrak{q})$ satisfying the following conditions:* (a) *$\pi$ and $\sigma$ are cuspidal automorphic representations of $G$ and $H$, respectively, having unitary central characters.* (b) *$\mathfrak{p} \notin S$ is a non-archimedean place (or prime ideal) of $F$ at which $E/F$ splits, so that (see [@2020arXiv201202187N §3.3]) $$(G(F_\mathfrak{p}), H(F_\mathfrak{p})) \cong (\mathop{\mathrm{GL}}_{n+1}(F_\mathfrak{p}), \mathop{\mathrm{GL}}_n(F_\mathfrak{p})).$$* (c) *$(\pi,\sigma)$ is locally distinguished: there is a nonzero $H(\mathbb{A})$-invariant functional $\pi \rightarrow \sigma$.* (d) *Inside $S$, the representations $\pi$ and $\sigma$ are tempered, and their depth is uniformly bounded as $(\pi,\sigma)$ traverses $\mathcal{F}$ (see [@2020arXiv201202187N §1.3]).* (e) *[\[itemize:cj3ubog35g\]]{#itemize:cj3ubog35g label="itemize:cj3ubog35g"} Outside $S \cup \{\mathfrak{p} \}$, the representations $\pi$ and $\sigma$ are unramified, and $\sigma$ satisfies a uniform bound towards Ramanujan at places where $E/F$ splits: it is $\vartheta$-tempered at such places for some fixed $0 \leq \vartheta < 1/2$ (see [@2020arXiv201202187N §5.2.1]).* (f) *$\mathfrak{q}$ is a positive power of the prime ideal $\mathfrak{p}$.* (g) *At $\mathfrak{p}$, the representations $\pi$ and $\sigma$ belong to the principal series and are induced by characters of conductor dividing $\mathfrak{q}^2$ for which the local analytic conductor for $\mathcal{L}(\pi,\sigma)$ at $\mathfrak{p}$ is as large as possible (see [\[eqn:cj2dzunqa6\]](#eqn:cj2dzunqa6){reference-type="eqref" reference="eqn:cj2dzunqa6"} for details, and Remark [Remark 6](#remark:cj3u9047n5){reference-type="ref" reference="remark:cj3u9047n5"} for weaker allowable hypotheses).* *Assume that $V$ (hence also $W$) is positive-definite. In particular, $H$ is anisotropic, and at each archimedean place of $F$, the groups $G$ and $H$ are compact.* *Then the branching coefficients $\mathcal{L}(\pi,\sigma)$, defined relative to the set $S \sqcup \{\mathfrak{p} \}$, satisfy a subconvex bound (see [\[eq:cj3tfrulcn\]](#eq:cj3tfrulcn){reference-type="eqref" reference="eq:cj3tfrulcn"} and [\[eq:cj3ubs6lvw\]](#eq:cj3ubs6lvw){reference-type="eqref" reference="eq:cj3ubs6lvw"}) as $(\pi,\sigma)$ varies over the family $\mathcal{F}$.* ## Some details {#sec:cj3v663hcw} Here we clarify the statement of Theorem [Theorem 1](#theorem:cj3ngw7u2s){reference-type="ref" reference="theorem:cj3ngw7u2s"}. We may write the local components at the "interesting" place $\mathfrak{p}$ as the normalized inductions $$\label{eqn:cj2dzujg9c} \pi_\mathfrak{p}= \chi_1 \boxplus \dotsb \boxplus \chi_{n+1}, \qquad \sigma_\mathfrak{p}= \eta_1 \boxplus \dotsb \boxplus \eta_n$$ for some characters $\chi_i$ and $\eta_j$ of $F_\mathfrak{p}^\times$. For a character $\omega$ of $F_\mathfrak{p}^\times$, we denote by $C(\omega)$ the analytic conductor, i.e., the absolute norm of the largest integral ideal $\mathfrak{a}$ in $F_\mathfrak{p}$ such that $\omega(x) = 1$ whenever $x - 1 \in \mathfrak{a}$. We set $$\label{eq:cj3ubsuj6v} T := \text{the absolute norm of } \mathfrak{q}^2.$$ With this notation, the "depth $\mathfrak{q}^2$" assumption means $$\label{eqn:20230516205128} C(\chi_i) \leq T, \qquad C(\eta_j) \leq T,$$ while "conductor as large as possible" means $$\label{eqn:cj2dzunqa6} \prod_{i, j} {C(\chi_i/\eta_j)}^2 = T^{2 n(n+1)},$$ which is equivalent to requiring that for all $i$ and $j$, $$\label{eqn:cj3ngvwrm4} C(\chi_i / \eta_j) = T.$$ We note that this forces $T$ to be the maximum of the $C(\chi_i)$ and $C(\eta_j)$. "Subconvex bound" means that $$\label{eq:cj3tfrulcn} \mathcal{L}(\pi,\sigma) \leq c_{\mathcal{F}} T^{2 n (n + 1)(1/4 - \delta_n)}$$ for some $c _{\mathcal{F}} \geq 0$ (resp. $\delta_n > 0$) depending only upon $\mathcal{F}$ (resp. $n$). In cases where $\mathcal{L}(\pi,\sigma)$ is known to coincide with an $L$-value (e.g., the case that $\pi$ and $\sigma$ are everywhere tempered), the estimate [\[eq:cj3tfrulcn\]](#eq:cj3tfrulcn){reference-type="eqref" reference="eq:cj3tfrulcn"} improves upon the convexity bound, which would assert the same for some $\delta_n \leq 0$ (see [@2020arXiv201202187N Proof of Cor 1.2] for details). In the case $n+1 = 2$, such an estimate has been known for a while [@michel-2009]. When $n+1 \geq 3$, such an estimate is new; we will show then that [\[eq:cj3tfrulcn\]](#eq:cj3tfrulcn){reference-type="eqref" reference="eq:cj3tfrulcn"} holds for any $$\label{eq:cj3ubs6lvw} \delta_n < \frac{1 - 2 \vartheta }{4 n(n+1) (A + 1 - 2 \vartheta )}, \quad A := (2 {(n + 1 )}^2 - n) (n + 1),$$ with $\vartheta$ as in assumption [\[itemize:cj3ubog35g\]](#itemize:cj3ubog35g){reference-type="eqref" reference="itemize:cj3ubog35g"} above (and $c_\mathcal{F}$ allowed to depend upon $\delta_n$). **Remark 2**. We do not address the interesting challenge of improving the numerical strength of the exponent [\[eq:cj3ubs6lvw\]](#eq:cj3ubs6lvw){reference-type="eqref" reference="eq:cj3ubs6lvw"}. Several avenues for doing so were mentioned in [@2020arXiv201202187N Rmk 1.4], many of which could be pursued in the present context. **Remark 3**. The "large conductor" assumption [\[eqn:cj2dzunqa6\]](#eqn:cj2dzunqa6){reference-type="eqref" reference="eqn:cj2dzunqa6"} is the most serious one --- it is an open problem to give any genuine subconvex bound for $\mathop{\mathrm{GL}}_3$ or higher in any case where the conductor drops (see the final paragraph of [@2021arXiv210915230N §1.4] and references). ## Related results {#sec:cj4vjryk3u} In talks starting in March 2018, Simon Marshall tentatively announced results in the direction of Theorem [Theorem 1](#theorem:cj3ngw7u2s){reference-type="ref" reference="theorem:cj3ngw7u2s"}, introducing important and fundamental ideas, under some additional assumptions: (i) [\[enumerate:20230516210021\]]{#enumerate:20230516210021 label="enumerate:20230516210021"} (Wall-avoidance) For all $i \neq j$, one has $$C(\chi_i / \chi_j) = T, \qquad C(\eta_i / \eta_j) = T.$$ (ii) [\[enumerate:20230516210903\]]{#enumerate:20230516210903 label="enumerate:20230516210903"} (Depth aspect) $\mathfrak{p}$ is a *fixed* (i.e., independent of $\mathcal{F}$) non-archimedean place. An important input to his arguments is a certain *volume bound* (see §[1.7](#sec:cj3ubuyr8u){reference-type="ref" reference="sec:cj3ubuyr8u"} and [@2020arXiv201202187N §1.5]), which Marshall announces he can establish assuming [\[enumerate:20230516210021\]](#enumerate:20230516210021){reference-type="eqref" reference="enumerate:20230516210021"} and [\[enumerate:20230516210903\]](#enumerate:20230516210903){reference-type="eqref" reference="enumerate:20230516210903"}. The paper [@2020arXiv201202187N] gave analogues of Theorem [Theorem 1](#theorem:cj3ngw7u2s){reference-type="ref" reference="theorem:cj3ngw7u2s"} at an archimedean place $\mathfrak{p}$ (without assuming "principal series" or "$E/F$ split at $\mathfrak{p}$"). Parts of the proof apply to the non-archimedean "$\mathfrak{p}$ fixed" case of Theorem [Theorem 1](#theorem:cj3ngw7u2s){reference-type="ref" reference="theorem:cj3ngw7u2s"}; for instance, the volume bound [@2020arXiv201202187N Thm 15.2] was established over any fixed local field, without assuming wall-avoidance. In summary, the "$\mathfrak{p}$ fixed" case of Theorem [Theorem 1](#theorem:cj3ngw7u2s){reference-type="ref" reference="theorem:cj3ngw7u2s"} --- an estimate like [\[eq:cj3tfrulcn\]](#eq:cj3tfrulcn){reference-type="eqref" reference="eq:cj3tfrulcn"}, but with $c_{\mathcal{F}}$ allowed to depend also upon $\mathfrak{p}$ --- is closely related to existing results. The main novelty here is to allow $\mathfrak{p}$ to vary. **Remark 4**. Strictly speaking, one could quantify the available arguments to obtain *some* uniformity, e.g., [\[eq:cj3tfrulcn\]](#eq:cj3tfrulcn){reference-type="eqref" reference="eq:cj3tfrulcn"} with $c_{\mathcal{F}}$ depending polynomially upon $\mathfrak{p}$. This yields subconvexity under a "sufficient depth" restriction, namely, that $\mathfrak{q} = \mathfrak{p}^m$ with $m$ large enough in terms of the rank $n$. Our main novelty is thus to allow $\mathfrak{p}$ to vary while $\mathfrak{q}$ is a small power of $\mathfrak{p}$, e.g., $\mathfrak{q} = \mathfrak{p}$. Such "horizontal" cases do not follow from direct quantification. ## Division of the proof {#sec:cj4vjf3iqs} Like in previous works, the basic object of study is the integral $$\label{eq:cj3v66d7v4} \int_{[H]} \pi(\omega) v \cdot \overline{u}$$ for suitable vectors $v \in \pi$ and $u \in \sigma$ and a "convolution kernel" or "amplifier" $\omega \in C_c^\infty(G(\mathbb{A}))$, chosen so that $\pi(\omega) v$ approximates $v$. One seeks to bound the integrals [\[eq:cj3v66d7v4\]](#eq:cj3v66d7v4){reference-type="eqref" reference="eq:cj3v66d7v4"} simultaneously - from below, using [\[eq:cj3tfrrhlc\]](#eq:cj3tfrrhlc){reference-type="eqref" reference="eq:cj3tfrrhlc"}, in terms of $\mathcal{L}(\pi,\sigma)$, and - from above, by their second moment over "all" $\pi$ and $v$, using its "relative trace formula" expansion $$\label{eq:cj3tv4jpta} \int_{x, y \in [H]} \bar{u}(x) u(y) \sum_{\gamma \in G(F)} \omega (x^{-1} \gamma y) \, d x \, d y.$$ We work with factorizable vectors $v$ and $u$ and a factorizable test function $\omega$. Their local components at the "uninteresting" places are chosen in a soft and general way [@2020arXiv201202187N §5]. The key point is to choose the components $v_\mathfrak{p}, u_\mathfrak{p}$ and $\omega_\mathfrak{p}$ at the "interesting" place and then to establish the required estimates. The proof may be divided into roughly the following steps (see [@2020arXiv201202187N §2] for a more leisurely overview). 1. [\[enumerate:cj3ubymeey\]]{#enumerate:cj3ubymeey label="enumerate:cj3ubymeey"} For an individual representation $\pi$ of $G$, a construction of vectors $v \in \pi$ that are *microlocalized* with respect to some parameter $\tau$ [@nelson-venkatesh-1 §1.7]. 2. [\[enumerate:cj3ubymf5o\]]{#enumerate:cj3ubymf5o label="enumerate:cj3ubymf5o"} For a pair $(\pi,\sigma)$ as above, a study of the pairs of parameters $(\tau, \tau_H)$, at which microlocalized vectors $v \in \pi$ and $u \in \sigma$ exist, that are *compatible* in the sense that $\tau$ restricts to $\tau_H$ [@nelson-venkatesh-1 §13-14]. 3. [\[enumerate:cj3ubymh5v\]]{#enumerate:cj3ubymh5v label="enumerate:cj3ubymh5v"} Spectral estimates: the estimation (in particular, lower bound) of integrals of matrix coefficients of microlocalized vectors [@nelson-venkatesh-1 §18]. 4. [\[enumerate:cj3ubymi4d\]]{#enumerate:cj3ubymi4d label="enumerate:cj3ubymi4d"} Geometric estimates, namely the "volume bound", which is the key technical problem that arises when attempting to estimate [\[eq:cj3tv4jpta\]](#eq:cj3tv4jpta){reference-type="eqref" reference="eq:cj3tv4jpta"}. The first three of these steps adapt readily to the setting of this paper: 1. We are content to impose local conditions that make the construction of microlocalized vectors particularly concrete (§[8](#sec:part-body-paper){reference-type="ref" reference="sec:part-body-paper"}). 2. The study of compatible parameters was addressed in [@nelson-venkatesh-1 §13-14] over any base field of characteristic zero; we have found it convenient here to extend that study to a general base ring (§[4](#sec:20230514080526){reference-type="ref" reference="sec:20230514080526"}), using arguments quite similar to those in *loc. cit.* 3. The spectral estimates proceed in our context as in [@nelson-venkatesh-1 §18], with simplification. The focus of this paper is thus on the geometric estimates, namely, in establishing a volume bound that is uniform in $\mathfrak{p}$. We discuss this further starting in §[1.7](#sec:cj3ubuyr8u){reference-type="ref" reference="sec:cj3ubuyr8u"}. **Remark 5**. The method and ideas employed here have a long history. We refer to [@2020arXiv201202187N §2.6] for an overview, but emphasize the use of amplification following Duke--Friedlander--Iwaniec [@MR1942691; @DFI94; @MR1923476], the systematic study of period integrals and test vectors following Bernstein--Reznikov [@MR2726097], Venkatesh [@venkatesh-2005] and Michel--Venkatesh [@michel-2009], and the influential papers of Sarnak [@MR780071] and Iwaniec--Sarnak [@iwan-sar]. **Remark 6**. To simplify the statement of Theorem [Theorem 1](#theorem:cj3ngw7u2s){reference-type="ref" reference="theorem:cj3ngw7u2s"} and focus on the primary "horizontal" novelty, we assumed that the local components $\pi_\mathfrak{p}$ and $\sigma_\mathfrak{p}$ belong to the principal series. We will actually establish a more general result, where that assumption is weakened to a more complicated condition: that $\pi_\mathfrak{p}$ and $\sigma_\mathfrak{p}$ are "regular at depth $\mathfrak{q}^2$" (Definition [Definition 60](#definition:we-say-that-pi-emphr-at-depth-mathfr-if-there-cycl-regular-depth-parameter-polynomial){reference-type="ref" reference="definition:we-say-that-pi-emphr-at-depth-mathfr-if-there-cycl-regular-depth-parameter-polynomial"}), with the analogue of the "large conductor" assumption [\[eqn:cj3ngvwrm4\]](#eqn:cj3ngvwrm4){reference-type="eqref" reference="eqn:cj3ngvwrm4"} being that the pair $(\pi_\mathfrak{p}, \sigma_\mathfrak{p})$ is "stable at depth $\mathfrak{q}^2$" (Definition [Definition 65](#definition:let-pi-sigma-be-repr-pair-gener-line-groups-over-f){reference-type="ref" reference="definition:let-pi-sigma-be-repr-pair-gener-line-groups-over-f"}). We formulate these definitions in terms of what is needed by our argument (the existence of suitable vectors), and verify (§[8](#sec:part-body-paper){reference-type="ref" reference="sec:part-body-paper"}) that they are closed under parabolic induction and apply to certain supercuspidals (e.g., characters). We do not exhaustively classify the representations to which they apply. Giving such a classification is an interesting problem in the representation theory of supercuspidals that belongs to step [\[enumerate:cj3ubymeey\]](#enumerate:cj3ubymeey){reference-type="eqref" reference="enumerate:cj3ubymeey"} in the proof strategy outlined above. For the other steps [\[enumerate:cj3ubymf5o\]](#enumerate:cj3ubymf5o){reference-type="eqref" reference="enumerate:cj3ubymf5o"}, [\[enumerate:cj3ubymh5v\]](#enumerate:cj3ubymh5v){reference-type="eqref" reference="enumerate:cj3ubymh5v"} and [\[enumerate:cj3ubymi4d\]](#enumerate:cj3ubymi4d){reference-type="eqref" reference="enumerate:cj3ubymi4d"}, our treatment is general. **Remark 7**. It would be desirable to generalize Theorem [Theorem 1](#theorem:cj3ngw7u2s){reference-type="ref" reference="theorem:cj3ngw7u2s"} by removing the parity condition on the conductor exponent, i.e., by replacing "$\mathfrak{q}^2$" with $\mathfrak{q}$ in the statement, or equivalently, by allowing $T$ to be the norm of any ideal (rather than the square an ideal) in the discussion following Theorem [Theorem 1](#theorem:cj3ngw7u2s){reference-type="ref" reference="theorem:cj3ngw7u2s"}. Such a generalization seems accessible in the depth aspect, but would require new ideas in horizontal aspects. The issues touch upon all steps [\[enumerate:cj3ubymeey\]](#enumerate:cj3ubymeey){reference-type="eqref" reference="enumerate:cj3ubymeey"}, [\[enumerate:cj3ubymf5o\]](#enumerate:cj3ubymf5o){reference-type="eqref" reference="enumerate:cj3ubymf5o"}, [\[enumerate:cj3ubymh5v\]](#enumerate:cj3ubymh5v){reference-type="eqref" reference="enumerate:cj3ubymh5v"} and [\[enumerate:cj3ubymi4d\]](#enumerate:cj3ubymi4d){reference-type="eqref" reference="enumerate:cj3ubymi4d"} in the proof strategy outlined above; for instance, in place of the volume bound, one might need to estimate character sums (see the third paragraph of [@2021arXiv210915230N §1.4] and references). This seems to us an interesting direction for future work. **Remark 8**. Using the methods of [@2021arXiv210915230N], it should be possible to extend Theorem [Theorem 1](#theorem:cj3ngw7u2s){reference-type="ref" reference="theorem:cj3ngw7u2s"} to the split case $E = F \times F$ and to Eisenstein series. This would yield subconvex bounds in horizontal aspects for standard $L$-functions away from conductor dropping. Such bounds should apply to character twists $L(\pi \otimes \chi, \tfrac{1}{2} + i t)$, with $\pi$ fixed and $\chi$ of *square* conductor. More generally, it should be possible to establish subconvex bounds for standard $L$-functions $L(\pi,\tfrac{1}{2})$ when $\pi$ is induced at each finite place from characters all having the same square conductor. To remove the squareness assumption would require extending our methods as in Remark [Remark 7](#remark:cj3ubzzdb9){reference-type="ref" reference="remark:cj3ubzzdb9"}. **Remark 9**. One motivation for studying horizontal aspects comes from conjectures and results of Lapid--Mao [@MR3267120; @MR3619910; @MR3649366], which relate quadratic twists of self-dual standard $L$-functions on $\mathop{\mathrm{GL}}_n$ to Fourier coefficients of automorphic forms on $\mathop{\mathrm{Mp}}_{2n}$. Estimates for such coefficients could have applications to representation problems concerning quadratic forms. The present work does not apply to such twists due to the squareness restriction in Remark [Remark 8](#remark:cj3ubsydvn){reference-type="ref" reference="remark:cj3ubsydvn"}, but may be understood as a step in that direction. ## Geometric estimates: the uniform volume bound {#sec:cj3ubuyr8u} Continuing the discussion of §[1.6](#sec:cj4vjf3iqs){reference-type="ref" reference="sec:cj4vjf3iqs"}, we formulate the volume bound below as Problem [Problem 10](#Probleminformal){reference-type="ref" reference="Probleminformal"}. The asymptotic determination of [\[eq:cj3tv4jpta\]](#eq:cj3tv4jpta){reference-type="eqref" reference="eq:cj3tv4jpta"} reduces to the volume bound as in previous work (see [@2020arXiv201202187N §1.5.3] or §[7](#Sec:bilinear){reference-type="ref" reference="Sec:bilinear"}). Cases of the volume bound sufficient for the depth aspect were established in [@2020arXiv201202187N], using reductions specific to that aspect (see §[1.9](#sec:cj4t64wm4h){reference-type="ref" reference="sec:cj4t64wm4h"} for details). We develop a different approach (§[1.8](#sec:cj4t647ljn){reference-type="ref" reference="sec:cj4t647ljn"}) that gives the volume bound in general, uniformly in $\mathfrak{p}$. The volume bound is a local assertion concerning the "interesting" place $\mathfrak{p}$. To simplify notation, we write simply $G, H$, etc., for points over $F_\mathfrak{p}$, and drop the subscripts $\mathfrak{p}$. Thus, $F$ now denotes a non-archimedean local field, arising as the local component "$F_\mathfrak{p}$" of the global field considered above. We write simply $\mathfrak{p}$ for its maximal ideal, $\mathfrak{o}$ for its maximal order, and $q := \left\lvert \mathfrak{o} / \mathfrak{p} \right\rvert$ for the residue field cardinality. We set $$(G,H,M, M_H) = (\mathop{\mathrm{GL}}_{n+1}(F), \mathop{\mathrm{GL}}_n(F), \mathop{\mathrm{Mat}}_{n+1}(F), \mathop{\mathrm{Mat}}_n(F)),$$ where $\mathop{\mathrm{Mat}}_n$ means "$n \times n$ matrices". Let $K = \mathop{\mathrm{GL}}_{n+1}(\mathfrak{o}) < G$ denote the standard maximal compact subgroup. Recall from Theorem [Theorem 1](#theorem:cj3ngw7u2s){reference-type="ref" reference="theorem:cj3ngw7u2s"} the positive power $\mathfrak{q}$ of $\mathfrak{p}$, with square $\mathfrak{q}^2$ of norm $T$, that controls the depths of the representations $\pi$ of $G$ and $\sigma$ of $H$ that we consider. Let $\tau \in M$, with upper-left block $\tau_H \in M_H$. Let $G_\tau$ and $H_{\tau_H}$ denote the respective centralizers of $\tau$ in $G$ and of $\tau_H$ in $H$. We say that $\tau$ is *stable* if it has no eigenvalues in common with $\tau_H$ over an algebraic closure, or equivalently, if the characteristic polynomials of $\tau$ and $\tau_H$ generate the unit ideal. This is an avatar for the "large conductor" assumption [\[eqn:cj3ngvwrm4\]](#eqn:cj3ngvwrm4){reference-type="eqref" reference="eqn:cj3ngvwrm4"} (see [@nelson-venkatesh-1 §15]) and is equivalent to the geometric invariant theory notion of stability (see [@nelson-venkatesh-1 §14]). **Problem 10**. Fix a stable element $\tau \in M$. Fix $a \in G - H Z$, where $Z < G$ denotes the center. Give a nontrivial bound for the volume of the set of all $y \in H_{\tau_H} \cap K$ for which $a y$ is congruent modulo $\mathfrak{q}$ to an element of $H G_\tau$. Here "nontrivial" means a power saving in $T$ over the trivial bound, $\mathop{\mathrm{vol}}(H_{\tau_H} \cap K)$. Strictly speaking, we need mild refinements of the problem statement depending quantitatively upon $a$. A key result of this paper is a solution to this problem (Theorem [Theorem 57](#theorem:volume-bound){reference-type="ref" reference="theorem:volume-bound"}) that is uniform with respect to variation of the underlying local field $F$. In §[1.8](#sec:cj4t647ljn){reference-type="ref" reference="sec:cj4t647ljn"}, we summarize the proof. In §[1.9](#sec:cj4t64wm4h){reference-type="ref" reference="sec:cj4t64wm4h"}, we explain why the depth aspect treatment given in [@2020arXiv201202187N] is not uniform. ## The approach of this paper {#sec:cj4t647ljn} The volume bound controls elements of $H_{\tau_H}$ lying close to a certain subvariety, with "close" quantified by the ideal $\mathfrak{q}$. We deduce it from a purely algebraic statement, Theorem [Theorem 11](#theorem:cj2dzotgu8){reference-type="ref" reference="theorem:cj2dzotgu8"}, whose informal content is that, among the equations defining that subvariety, there is at least one whose coefficients are at least as large as the distance from $a$ to $H Z$. To formulate that algebraic statement, we denote now by $(\mathbf{G}, \mathbf{H}, \mathbf{M}, \mathbf{M}_H)$ the group schemes $(\mathop{\mathrm{GL}}_{n+1}, \mathop{\mathrm{GL}}_n, \mathop{\mathrm{Mat}}_{n+1}, \mathop{\mathrm{Mat}}_n)$ of invertible and all square matrices of the indicated dimensions, and write $\mathbf{Z}$ for the center of $\mathbf{G}$. Let $R$ be a ring, and let $\tau \in \mathbf{M}(R)$ satisfy the following "stability" hypothesis (§[4](#sec:20230514080526){reference-type="ref" reference="sec:20230514080526"}): the characteristic polynomials of $\tau$ and its upper-left block $\tau_H$ generate the unit ideal. Let $\mathbf{G}_\tau$ and $\mathbf{H}_{\tau_H}$ denote the centralizers of $\tau$ and $\tau_H$, regarded as subgroup schemes of $\mathbf{G}$ and $\mathbf{H}$ defined over $R$, and let $a \in \mathbf{G}(R)$. For each $R$-algebra $R'$, we define the following set (compare with Problem [Problem 10](#Probleminformal){reference-type="ref" reference="Probleminformal"}): $$\mathbf{X}_{\tau,a}(R') := \left\{ y \in \mathbf{H}_{\tau_H}(R') : a y \in \mathbf{H}(R') \mathbf{G}_{\tau}(R') \right\}.$$ The stability hypothesis on $\tau$ turns out to imply that $\mathbf{X}_{\tau,a}$ is naturally a closed subscheme of $\mathbf{H}_{\tau_H}$ over $R$, defined by finitely many polynomials equations (see Lemma [Lemma 37](#lemma:each-ring-extens-r-r-we-have-begin-mathbfhr-m){reference-type="ref" reference="lemma:each-ring-extens-r-r-we-have-begin-mathbfhr-m"}). It is easy to see that if $a$ lies in $\mathbf{H}(R) \mathbf{Z}(R)$, then $\mathbf{X}_{\tau,a} = \mathbf{H}_{\tau_H}$ as schemes over $R$ (i.e., their point sets coincide for all $R'$). We establish a converse: **Theorem 11** (Theorem [Theorem 30](#theorem:main-transversality-general-ring){reference-type="ref" reference="theorem:main-transversality-general-ring"}). *Assume that $n+1 \geq 3$, that $2$ is a unit in $R$, and that $\mathbf{X}_{\tau,a} = \mathbf{H}_{\tau_H}$ as schemes over $R$. Then $a \in \mathbf{H}(R) \mathbf{Z}(R)$.* We refer to §[5.1](#sec:transversality-statement-results){reference-type="ref" reference="sec:transversality-statement-results"} for refinements and discussion of why the hypotheses are necessary. Informally, the condition $\mathbf{X}_{\tau,a} = \mathbf{H}_{\tau_H}$ says that the defining equations for $\mathbf{X}_{\tau,a}$ inside $\mathbf{H}_{\tau_H}$ are tautological, i.e., their coefficients are all zero. The informal content of Theorem [Theorem 11](#theorem:cj2dzotgu8){reference-type="ref" reference="theorem:cj2dzotgu8"} is thus that if $a \notin \mathbf{H}(R) \mathbf{Z}(R)$, then we can find a nonzero (bounded degree) polynomial on $\mathbf{H}_{\tau_H}$ over $R$ whose zero locus contains $\mathbf{X}_{\tau,a}$. This result, applied in the context of Problem [Problem 10](#Probleminformal){reference-type="ref" reference="Probleminformal"} with $R$ a suitable quotient of $\mathfrak{o}/\mathfrak{q}$, implies that the set whose volume we must bound is contained in the locus of a polynomial whose coefficients are not all too small. The required uniform volume bound (Theorem [Theorem 57](#theorem:volume-bound){reference-type="ref" reference="theorem:volume-bound"}) then follows via general bounds for solutions to polynomial congruences (§[6.1](#sec:cj3v60uw7y){reference-type="ref" reference="sec:cj3v60uw7y"}). To prove Theorem [Theorem 11](#theorem:cj2dzotgu8){reference-type="ref" reference="theorem:cj2dzotgu8"}, we apply the assumed equality between $\mathbf{X}_{\tau,a}$ and $\mathbf{H}_{\tau_H}$ first over the ring of dual numbers $R' = R[\varepsilon]/(\varepsilon^2)$, then over $R'' = R[\varepsilon_1,\varepsilon_2]/(\varepsilon_1,\varepsilon_2)$. We refer to these steps as linear and quadratic analysis, respectively. They may be understood as studying the consequences of the vanishing of linear and quadratic coefficients of the defining equations for $\mathbf{X}_{\tau,a}$ inside $\mathbf{H}_{\tau_H}$. The arguments sketched below were found after extensive numerical study of these coefficients,[^1] and represent a main novelty of this paper. The key case is when $a \in \mathbf{G}_\tau(R)$; we must show then that $a \in \mathbf{Z}(R)$. From the linear analysis, we deduce that $a^2 \in \mathbf{Z}(R)$. To do so, we determine the linear coefficients with respect to the basis for $\mathop{\mathrm{Lie}}(\mathbf{H}_{\tau_H})$ given by powers of $\tau_H$, and observe that they may be related via an upper-triangular substitution to some invariants of $\tau$ and $a$ whose vanishing forces $a^2$ to be central. We refer to §[5.4](#sec:analysis-first-derivatives){reference-type="ref" reference="sec:analysis-first-derivatives"} for details. This step of the argument generalizes the approach of [@2020arXiv201202187N], recalled below in §[1.9](#sec:cj4t64wm4h){reference-type="ref" reference="sec:cj4t64wm4h"}, which amounts to studying just the linear coefficient for the central direction in $\mathop{\mathrm{Lie}}(\mathbf{H}_{\tau_H})$. From the quadratic analysis, we construct (Lemma [Lemma 46](#lemma:second-derivatives-yield-homomorphism-property){reference-type="ref" reference="lemma:second-derivatives-yield-homomorphism-property"}) a multiplicative linear map $\mathop{\mathrm{Lie}}(\mathbf{H}_{\tau_H}) \rightarrow \mathop{\mathrm{Lie}}(\mathbf{G}_{\tau})$. Our argument then divides according to whether this map preserves unit elements (residually). If it does not, then we may apply multiplicativity in the central direction to see that $a$ is central. If it does, then we show by two applications of the linear analysis --- first for $a$ over $R$, then for suitable "nearby" $b \in \mathbf{G}_{\tau}(R')$ over $R' := R[\varepsilon_2]/(\varepsilon_2^2)$, with $a(1+\varepsilon_1\tau_H) \in \mathbf{H}(R') b$ --- that $\tau$ satisfies a monic polynomial of degree $2$, contrary our assumption $n+1\geq 3$. We refer to §[5.5](#sec:analysis-second-derivatives){reference-type="ref" reference="sec:analysis-second-derivatives"} for details. **Remark 12**. In the depth aspect, we may assume via the "near identity" reduction (§[1.9.3](#sec:cj4t69an73){reference-type="ref" reference="sec:cj4t69an73"}) that $a$ is close to the identity. Then $a^2 \in \mathbf{Z}(R) \implies a \in \mathbf{Z}(R)$, so the "linear" part of above argument suffices, yielding a simpler proof than in [@2020arXiv201202187N] of the volume bound required in the depth aspect (see Remark [Remark 43](#Remark:recoverLiealgproof){reference-type="ref" reference="Remark:recoverLiealgproof"}). ## Available treatment in the depth aspect {#sec:cj4t64wm4h} We indicate here why the approach to the volume bound given in [@2020arXiv201202187N] does not apply in horizontal aspects, motivating the approach described above and pursued in this paper. ### The short spectral projector $\omega$ We construct in §[9.3](#sec:cj4t6ynizh){reference-type="ref" reference="sec:cj4t6ynizh"} a convolution kernel $\omega \in C_c^\infty(K)$, the local component of the global amplifier discussed after [\[eq:cj3v66d7v4\]](#eq:cj3v66d7v4){reference-type="eqref" reference="eq:cj3v66d7v4"}. It is supported on a certain open subgroup $J_\tau < K$, and given there by a multiple of a certain character of $J_\tau$, depending upon $\tau$ and some additional data. Roughly, $J_\tau$ is the inverse image modulo $\mathfrak{q}$ of the image of $G_\tau \cap K$. For example, when $\tau$ is a (regular) diagonal matrix, $J_\tau$ consists of elements of $K$ that are congruent modulo $\mathfrak{q}$ to the diagonal. ### Reductions The depth aspect case of the volume bound (i.e., $\mathfrak{p}$ fixed) was addressed in [@2020arXiv201202187N] following a series of reductions: (i) **Reduction to the near-identity case**. By [@2020arXiv201202187N §15], it is not necessary to solve Problem [Problem 10](#Probleminformal){reference-type="ref" reference="Probleminformal"} in general --- it suffices to treat the "near-identity" case, where $a$ lies in an arbitrarily small (but fixed) neighborhood of the identity (see §[1.9.3](#sec:cj4t69an73){reference-type="ref" reference="sec:cj4t69an73"}). (ii) **Reduction to the Lie algebra**. The near-identity case reduces further, via Lie-theoretic arguments [@2020arXiv201202187N §16], to a Lie algebra problem, where $\mathop{\mathrm{Lie}}(H_{\tau_H})$ plays the role of $H_{\tau_H}$. (iii) **Reduction to central directions**. The Lie algebra problem concerns certain subspaces of $\mathop{\mathrm{Lie}}(H_{\tau_H})$, whose definition we omit here. The problem is to show that the subspaces are not the entire space. This problem was addressed in [@2020arXiv201202187N §17] by showing that the subspaces do not contain the one-dimensional central subspace; in effect, this shows that central cosets in $H_{\tau_H} \cap K$ suffice to solve Problem [Problem 10](#Probleminformal){reference-type="ref" reference="Probleminformal"}. These reductions fail in horizontal aspects, for independent reasons: (i) The near-identity reduction in the depth aspect comes from the freedom to take the convolution kernel $\omega$ supported close to the identity, which is unavailable in horizontal aspects (see §[1.9.3](#sec:cj4t69an73){reference-type="ref" reference="sec:cj4t69an73"}). (ii) The reduction to the Lie algebra may be adapted beyond the near-identity case, but the resulting Lie algebra problem is unsolvable: there are natural families of counterexamples (see Theorem [Theorem 34](#theorem:characterization-of-tangential-points){reference-type="ref" reference="theorem:characterization-of-tangential-points"} and Remark [Remark 45](#remark:cj4u2pnri2){reference-type="ref" reference="remark:cj4u2pnri2"}). (iii) Central directions do not suffice to solve the general case of Problem [Problem 10](#Probleminformal){reference-type="ref" reference="Probleminformal"}; we have again identified counterexamples (see Remark [Remark 49](#remark:gl6-example){reference-type="ref" reference="remark:gl6-example"}). ### The near-identity reduction {#sec:cj4t69an73} We conclude §[1.9](#sec:cj4t64wm4h){reference-type="ref" reference="sec:cj4t64wm4h"} by explaining the "near-identity" reduction mentioned above, whose failure in horizontal aspects motivates why the geometric estimates in this paper are of "algebro-geometric" rather than "Lie-algebraic" flavor compared to those in [@2020arXiv201202187N]. Fix a small natural number $d$. Let $K[d]$ denote the $d$th principal congruence subgroup of $K$. The convolution kernel $\omega$ is supported in $K$, but not in $K[1]$. It projects onto a "short" family: the integral operator $\pi(\omega)$ vanishes unless the irreducible representation $\pi$ lies in such a family. The normalized restriction $\omega^{[d]}$ of $\omega$ to $K[d]$ projects onto a larger family; in global settings, it is larger by a factor of roughly $$\label{eq:cj4t64h1mo} q^{{(n+1)}^2 d}.$$ **Example 13**. If $\tau$ is diagonal, then the operator $\pi(\omega)$ (resp. $\pi(\omega^{[d]})$) vanishes unless $\pi$ is a principal series representation $\chi_1 \boxplus \dotsb \boxplus \chi_{n+1}$ induced by characters $\chi_j$ of $F^\times$ having prescribed restrictions to $\mathfrak{o}^\times$ (resp. to $1 + \mathfrak{p}^d$). The near-identity reduction arises from the freedom to replace $\omega$, supported on $G_\tau \cap K$, with $\omega^{(d)}$, supported on the smaller group $G_\tau \cap K[d]$. We can do so in the depth aspect, where $q$ is fixed, because the factors [\[eq:cj4t64h1mo\]](#eq:cj4t64h1mo){reference-type="eqref" reference="eq:cj4t64h1mo"} are then harmless. In horizontal aspects --- where $q$ is large --- such factors ruin the near-identity reduction: any small power of $q$ saved by amplification is swamped by the large power of $q$ in [\[eq:cj4t64h1mo\]](#eq:cj4t64h1mo){reference-type="eqref" reference="eq:cj4t64h1mo"}, yielding estimates that fail even to recover convexity. We must thus work with projectors $\omega$ supported on the full maximal compact subgroup $K$. ## Organization of this paper This paper consists almost exclusively of local analysis at the "interesting" place $\mathfrak{p}$. It culminates in our main local result, Theorem [Theorem 89](#theorem:main-local-result){reference-type="ref" reference="theorem:main-local-result"}, which is a direct analogue of its archimedean counterpart, [@2020arXiv201202187N Theorem 4.2]. The auxiliary arguments of [@2020arXiv201202187N §4--6] (concerning "uninteresting" places, amplification, counting, etc.) combine with our main local result to yield a subconvex bound, exactly as in [@2020arXiv201202187N]. While our local analysis is logically self-contained, its global motivation might be clarified by skimming [@2020arXiv201202187N §6]. We conclude with the detailed breakdown. §[2](#sec:notation){reference-type="ref" reference="sec:notation"}--§[3](#sec:cyclic-matrices){reference-type="ref" reference="sec:cyclic-matrices"} contain preliminaries. §[4](#sec:20230514080526){reference-type="ref" reference="sec:20230514080526"} treats "stability" for $(\mathop{\mathrm{GL}}_{n+1}, \mathop{\mathrm{GL}}_n)$ over a general ring. §[5](#sec:transversality){reference-type="ref" reference="sec:transversality"} establishes our key algebraic result (Theorem [Theorem 30](#theorem:main-transversality-general-ring){reference-type="ref" reference="theorem:main-transversality-general-ring"}, or Theorem [Theorem 11](#theorem:cj2dzotgu8){reference-type="ref" reference="theorem:cj2dzotgu8"} above). §[6](#sec:volumebound){reference-type="ref" reference="sec:volumebound"} applies that result to derive the uniform volume bound (Theorem [Theorem 57](#theorem:volume-bound){reference-type="ref" reference="theorem:volume-bound"}), which we further apply in §[7](#Sec:bilinear){reference-type="ref" reference="Sec:bilinear"} to estimate bilinear forms relevant for [\[eq:cj3tv4jpta\]](#eq:cj3tv4jpta){reference-type="eqref" reference="eq:cj3tv4jpta"}. §[8](#sec:part-body-paper){reference-type="ref" reference="sec:part-body-paper"} concerns "microlocal analysis" involving representations $\pi,\sigma$ and parameters $\tau,\tau_H$. §[9](#sec:mainlocal){reference-type="ref" reference="sec:mainlocal"} establishes our main local result (Theorem [Theorem 89](#theorem:main-local-result){reference-type="ref" reference="theorem:main-local-result"}), a uniform non-archimedean analogue of [@2020arXiv201202187N Theorem 4.2]. §[10](#Sec:final){reference-type="ref" reference="Sec:final"} combines our main local result with the auxiliary arguments of [@2020arXiv201202187N] to complete the proof of Theorem [Theorem 1](#theorem:cj3ngw7u2s){reference-type="ref" reference="theorem:cj3ngw7u2s"}. ## Acknowledgment Y.H. is supported by the National Key Research and Development Program of China (No. 2021YFA1000700). P.N. worked on this project while he was a von Neumann Fellow at the Institute for Advanced Study, supported by the National Science Foundation under Grant No. DMS-1926686. The paper was completed while P.N. was a Villum Investigator based at Aarhus University, supported by a research grant (VIL54509) from VILLUM FONDEN. # Notation {#sec:notation} We record general notation and conventions, used throughout the paper. ## Vectors and matrices {#sec:vectors-matrices} Let $\mathbf{V}$ be a finite free $\mathbb{Z}$-module, thus $\mathbf{V} \cong \mathbb{Z}^n$ for some $n$, the *rank* of $\mathbf{V}$. It will occasionally be convenient to denote the rank instead by $n+1$, so that $n$ refers instead to the rank of a codimension one submodule (see §[2.2](#sec:general-linear-ggp-pairs){reference-type="ref" reference="sec:general-linear-ggp-pairs"} below). We denote by $\mathbf{V}^* = \mathop{\mathrm{Hom}}(\mathbf{V},\mathbb{Z}) \cong \mathbb{Z}^n$ the dual module and by $\mathbf{M} := \mathop{\mathrm{End}}(\mathbf{V}) \cong \mathop{\mathrm{Mat}}_n(\mathbb{Z})$ the endomorphism ring. We denote by juxtaposition the natural pairing $$\mathbf{V}^* \otimes \mathbf{V} \rightarrow \mathbb{Z}, \quad \ell \otimes v \mapsto \ell v := \ell(v),$$ as well as the left action, right action and outer product, respectively: $$\begin{aligned} &\mathbf{M} \otimes \mathbf{V} \rightarrow \mathbf{V}, \quad &&a \otimes v \mapsto a v := a(v), \\ &\mathbf{V}^* \otimes \mathbf{M} \rightarrow \mathbf{V}^*, \quad &&\ell \otimes a \mapsto \ell a := [u \mapsto \ell(a(u))] \\ &\mathbf{V} \otimes \mathbf{V}^* \rightarrow \mathbf{M}, \quad &&v \otimes \ell \mapsto v \ell := [u \mapsto \ell(u) v]. \end{aligned}$$ The motivation for the notation is that, by choosing a basis, we may regard $\mathbf{V}$ (resp. $\mathbf{V}^*$) as spaces of column (resp. row) vectors and $\mathbf{M}$ as a space of matrices, in which case the above pairings are given by matrix multiplication. We use the same notation for the extensions of these pairings to more general rings given below. For us, a *ring* $R$ is a commutative ring with identity, an *algebra* or *ring extension* $R'$ of $R$ is a ring map $R \rightarrow R'$, and an *affine scheme* $\mathbf{X}$ is a functor that assigns to each ring $R$ a set $\mathbf{X}(R)$ that functorially identifies with the set of ring maps $\mathbb{Z}[\mathbf{X}] \rightarrow R$ for some ring $\mathbb{Z}[\mathbf{X}]$, the *coordinate ring* of $\mathbf{X}$. Affine schemes over $R$ are defined similarly, but restricting to $R$-algebras; the above discussion then applies upon replacing $\mathbb{Z}$ with $R$. By abuse of notation, we denote also by $\mathbf{V}, \mathbf{V}^*$ and $\mathbf{M}$ the affine group schemes over $\mathbb{Z}$ given by $$\begin{aligned} \mathbf{V}(R) &:= \mathbf{V} \otimes R \cong R^n, \\ \mathbf{V}^*(R) &:= \mathbf{V}^* \otimes R \cong R^n, \\ \mathbf{M}(R) &:= \mathop{\mathrm{End}}_R(\mathbf{V}(R)) \cong \mathop{\mathrm{Mat}}_n(R).\end{aligned}$$ The coordinate rings of $\mathbf{V}, \mathbf{V}^*$ and $\mathbf{M}$ identify with the polynomial rings over $\mathbb{Z}$ in $n, n$ and $n^2$ variables, respectively, where $n$ denotes the rank of $\mathbf{V}$. We denote by $\mathbf{G}$ the affine group scheme $$\mathbf{G}(R) := \mathop{\mathrm{Aut}}_R(\mathbf{V}(R)) \cong \mathop{\mathrm{GL}}_n(R),$$ whose coordinate ring is obtained from that of $\mathbf{M}$ by inverting the determinant. We denote by $\mathbf{Z}$ the center of $\mathbf{G}$, consisting of scalar matrices: $$\mathbf{Z}(R) = R^\times \hookrightarrow \mathbf{G}(R).$$ For $g \in \mathbf{G}(R)$, we denote by $\mathop{\mathrm{Ad}}(g) : \mathbf{M}(R) \rightarrow \mathbf{M}(R)$ the conjugation map $$\mathop{\mathrm{Ad}}(g) x := g x g ^{-1}.$$ ## General linear GGP pairs {#sec:general-linear-ggp-pairs} In some parts of this paper, we consider just one general linear group $\mathbf{G}$. In others, we work with inclusions $\mathbf{H} \leq \mathbf{G}$ of general linear groups of neighboring rank. We then use the following notation. Suppose given $e \in \mathbf{V}$ and $e^* \in \mathbf{V}^*$ such that $e^* e= 1$. We then define the submodule $$\mathbf{V}_H := \left\{ v \in \mathbf{V} : e^* v = 0 \right\},$$ which participates in the direct sum decomposition $$\label{eqn:lambda-=-lambda_h-oplus-mathbbz-e-} \mathbf{V} = \mathbf{V}_H \oplus \mathbb{Z} e,$$ $$v = (1 - e e^*) v + (e^* v) e.$$ This extends to any ring. We define affine group schemes $$\mathbf{V}_H, \quad \mathbf{V}_H^*, \quad \mathbf{M}_H, \quad \mathbf{H}$$ in terms of $\mathbf{V}_H$, by analogy to the definitions of §[2.1](#sec:vectors-matrices){reference-type="ref" reference="sec:vectors-matrices"}. These define closed subgroup schemes of $\mathbf{V}, \mathbf{V}^*, \mathbf{M}, \mathbf{G}$, respectively. For any ring $R$, we denote by $1_H \in \mathbf{M}_H(R)$ the identity operator. It is given in terms of the identity operator $1 \in \mathbf{M}(R)$ by $$1_H = 1 - e e^*.$$ Given $\tau \in \mathbf{H}(R)$, we denote by $$\tau_H := 1_H \tau 1_H \in \mathbf{M}_H(R)$$ the element induced by the decomposition [\[eqn:lambda-=-lambda_h-oplus-mathbbz-e-\]](#eqn:lambda-=-lambda_h-oplus-mathbbz-e-){reference-type="eqref" reference="eqn:lambda-=-lambda_h-oplus-mathbbz-e-"}. **Example 14**. Suppose $\mathbf{V}$ has rank three. Let $e_1, e_2$ be a basis for $\mathbf{V}_H$. Then $e_1,e_2,e$ is a basis for $\mathbf{V}$. Using this basis to identify $\mathbf{M}$ with the space of $3 \times 3$ matrices, we have $$\mathbf{M}_H = \begin{pmatrix} \ast & \ast & 0 \\ \ast & \ast & 0 \\ 0 & 0 & 0 \\ \end{pmatrix}, \quad \mathbf{H} = \begin{pmatrix} \ast & \ast & 0 \\ \ast & \ast & 0 \\ 0 & 0 & 1 \\ \end{pmatrix}, \quad 1_H = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \\ \end{pmatrix},$$ $$\tau = \begin{pmatrix} \tau _{11} & \tau _{12} & \tau _{13} \\ \tau _{21} & \tau _{22} & \tau _{23} \\ \tau _{31} & \tau _{32} & \tau _{33} \\ \end{pmatrix} \implies \tau_H = \begin{pmatrix} \tau _{11} & \tau _{12} & 0 \\ \tau _{21} & \tau _{22} & 0 \\ 0 & 0 & 0 \\ \end{pmatrix}.$$ ## Centralizers For $\tau \in \mathbf{M}(R)$ or $\sigma \in \mathbf{M}_{H}(R)$, we denote by $$\mathbf{G}_\tau \leq \mathbf{G}, \quad \mathbf{M}_{\tau} \leq \mathbf{M}, \quad \mathbf{H}_{\sigma} \leq \mathbf{H}, \quad \mathbf{M}_{H,\sigma} \leq \mathbf{M}_H$$ the centralizers, regarded as affine group schemes over $R$. ## Local fields and congruence subgroups {#sec:local-fields-congruence-subgroups} Let $F$ be a non-archimedean local field. We denote then by $$\mathfrak{o}, \qquad \mathfrak{p}, \qquad \varpi \in \mathfrak{p} ,\qquad q = \lvert \mathfrak{o} / \mathfrak{p} \rvert$$ its ring of integers, its maximal ideal, a uniformizer, and the cardinality of the residue field. We denote by $|.|_F$ the normalized valuation. For each ideal $\mathfrak{a} \subseteq \mathfrak{o}$, we set $$\label{eqn:cj3m0de4ie} K(\mathfrak{a}) := \ker(\mathbf{G}(\mathfrak{o}) \rightarrow \mathbf{G}(\mathfrak{o}/\mathfrak{a})), \quad K_H(\mathfrak{a}) := \ker(\mathbf{H}(\mathfrak{o}) \rightarrow \mathbf{H}(\mathfrak{o}/\mathfrak{a})),$$ We sometimes write simply $$K := K(\mathfrak{o}) = \mathbf{G}(\mathfrak{o}), \quad K_H := K_H(\mathfrak{o}) = \mathbf{H}(\mathfrak{o}), \quad K_Z := \mathbf{Z}(\mathfrak{o})$$ for the standard maximal compact subgroups of $\mathbf{G}(F)$, $\mathbf{H}(F)$ and $\mathbf{Z}(F)$, respectively. We equip the latter groups with the Haar measure that assigns volume one to their maximal compact subgroups $K _G, K _H$ and $K _Z$. # Cyclic matrices {#sec:cyclic-matrices} Let $R$ be a ring. In this section, we write $V, M, H$, etc., for the set of $R$-points of the corresponding bold-faced group schemes. Let $\tau \in M$. We denote by $$P_\tau(X) = \det(X - \tau) \in R[X]$$ its characteristic polynomial. It is a monic polynomial whose degree is the rank of $\mathbf{V}$. We recall the Cayley--Hamilton theorem (see [@MR1322960 Theorem 4.3]): $P_\tau(\tau) = 0$. **Definition 15**. Let $\tau \in M$. We say that $v \in V$ is *$\tau$-cyclic* if $R[\tau] v = V$. We say that *$V$ is $\tau$-cyclic* (or simply that *$\tau$ is cyclic* when $V$ is understood, or that *$V$ is cyclic* when $\tau$ is understood) if there exists a $\tau$-cyclic vector $v \in V$. **Example 16**. It is clear that any Jordan block is cyclic. **Example 17**. It follows from the Vandermonde determinant calculation that a diagonal matrix is cyclic when the pairwise differences between its diagonal entries are invertible. **Example 18**. If $v$ is $\tau$-cyclic, then it remains so upon passing to any ring extension $R'$ of $R$. In particular, the cyclicity of $\tau$ is preserved under base extension. **Example 19**. Let $R$ be a local ring. By Nakayama's lemma, we can test whether $\tau$ is cyclic over the residue field. When $R$ is a field, the structure theorem for modules over the principal ideal domain $R[X]$ says that $V$ is isomorphic to a direct sum $\oplus_i R / (p_i)$, where $p_i$ are monic primary polynomials (i.e., powers of irreducible polynomials). Then $\tau$ is cyclic if and only if the $p_i$ are pairwise relatively prime. For instance, when $R$ is algebraically closed, this says that no two Jordan blocks for $\tau$ have the same eigenvalue. **Lemma 20**. Write $n = \mathop{\mathrm{rank}}(\mathbf{V})$. Then $v$ is $\tau$-cyclic if and only if the map of modules $$\label{eqn:rn-rightarrow-v} R^n \rightarrow V$$ $$\label{eqn:let-tau-in-c0-through-cn1-sum-cj-tau} (c_0,\dotsc,c_{n-1}) \mapsto \sum_{j < n} c_j \tau^j v$$ is surjective, in which case it is an isomorphism. *Proof.* The first statement follows from the Cayley--Hamilton theorem, the second from the fact that any surjective map of free modules of the same finite rank is an isomorphism [@MR1322960 Corollary 4.4a]. ◻ In particular, each cyclic element $\tau$ admits a *cyclic basis* $e_1, \dotsc, e_n$ such that $$\label{eqn:beginpmatrix-0--0--ast-} \tau e_j = e_{j+1} \text{ for } j < n, \quad \text{e.g., } \tau = \begin{pmatrix} 0 & 0 & \ast \\ 1 & 0 & \ast \\ 0 & 1 & \ast \\ \end{pmatrix}.$$ **Lemma 21**. Let $\tau \in M$, and suppose that $v \in V$ is $\tau$-cyclic. Then the map $$M_\tau \rightarrow V, \qquad x \mapsto x v$$ is an isomorphism of $R$-modules. Moreover, $$M_\tau = R[\tau] = \left\{ \sum_{j=0}^{n-1} c_j \tau^j : (c_0,\dotsc,c_{n-1}) \in R^{n} \right\}.$$ In particular, $M_\tau$ is a commutative $R$-algebra, and $G_\tau$ is an abelian group. *Proof.* Since $V$ is spanned by the $\tau^j v$ and $x \tau^j v = \tau^j x v$, we see that $x$ is determined by $x v$. The map $M_\tau \rightarrow V$ is thus injective. We next verify that $M_\tau = R[\tau]$. Let $x \in M_\tau$. By Lemma [Lemma 20](#lemma:v-tau-cycl-if-only-if-map-modul-begin-rn-right-v-){reference-type="ref" reference="lemma:v-tau-cycl-if-only-if-map-modul-begin-rn-right-v-"}, we may choose $(c_0,\dotsc,c_{n-1}) \in R^n$ so that $x v = \sum _{j=0}^{n-1} c_j \tau^j v$. By the noted injectivity of $M_\tau \rightarrow V$, it follows that $x = \sum_{j=0}^{n-1} c_j \tau^j$. Thus $M_\tau \subseteq R[\tau]$. The reverse containment is evident. The surjectivity of $M_\tau \rightarrow V$ follows from the identity $M_\tau = R[\tau]$ and Lemma [Lemma 20](#lemma:v-tau-cycl-if-only-if-map-modul-begin-rn-right-v-){reference-type="ref" reference="lemma:v-tau-cycl-if-only-if-map-modul-begin-rn-right-v-"}. ◻ **Lemma 22**. Suppose that $\tau_1, \tau_2 \in M$ are cyclic elements with the same characteristic polynomial. Then they are conjugate under $G$. *Proof.* Let $e_1^{(j)},\dotsc,e_n^{(j)}$ be a cyclic basis for $\tau_1$ (resp. $\tau_2$). By Cayley--Hamilton, we have $\tau_j e_n^{(j)} = \sum_{i=1}^{n} c_i e_i^{(j)}$, where the coefficients $c_i \in R$ depend only upon the characteristic polynomial, hence are the same for $\tau_1$ and $\tau_2$. Let $g \in G$ be the unique map that sends $e_i^{(1)}$ to $e_i^{(2)}$. Then $g \tau_1 = \tau_2 g$, so $\tau_1$ and $\tau_2$ are conjugate. ◻ # Stability {#sec:20230514080526} **Definition 23**. Let $R$ be a ring and $\tau \in \mathbf{M}(R)$. We say that $\tau$ is *stable* if the vectors $e$ and $e^*$ are both $\tau$-cyclic. We denote by $\mathbf{M}_{\mathop{\mathrm{stab}}}(R)$ the subset of stable elements. **Remark 24**. This condition was studied by Rallis--Schiffmann [@2007arXiv0705.2168R] under the name "regular", and also in [@nelson-venkatesh-1 §14], where it was related (for $R$ a field of characteristic zero) to the geometric invariant theory notion of stability. See especially [@2007arXiv0705.2168R Thm 6.1] and [@nelson-venkatesh-1 Lem 14.8]. Many of the results of this section were established in [@nelson-venkatesh-1 §13-14] when $R$ is a field of characteristic zero, and in some respects, more systematically and generally (e.g., also for orthogonal GGP pairs). We have chosen to give short proofs of what we require, repeating or adapting the arguments of the cited reference. The adapted arguments could likely be extended to the orthogonal case along the same lines as in [@nelson-venkatesh-1 §14]. **Example 25**. Suppose $R$ is a local ring, with maximal ideal $\mathfrak{p}$. Let $\tau \in \mathbf{M}(R)$, with image $\bar{\tau} \in \mathbf{M}(R/\mathfrak{p})$. Nakayama's lemma implies that $\tau$ is stable if and only if $\bar{\tau}$ is stable. We denote by $n+1$ the rank of $\mathbf{V}$, so that $n$ is the rank of $\mathbf{V}_H$. For a ring $R$, we define $$\Delta : \mathbf{M}(R) \rightarrow R$$ $$\Delta(\tau) := \det (e^*, e^* \tau, \dotsc, e^* \tau ^n ) \det (e, \tau e, \dotsc, \tau ^n e),$$ where $\det(\dotsb)$ denotes the determinant of the matrix having the indicated row vectors $e^* \tau^i \in \mathbf{V}^*(R)$ or column vectors $\tau^j e \in \mathbf{V}(R)$, with such vectors realized as $(n+1)$-tuples using some chosen basis for the free $\mathbb{Z}$-module $\mathbf{V}$. Choosing a different basis has the effect of multiplying the two determinants by mutually inverse factors, hence has no effect on $\Delta(\tau)$. This family of maps $\Delta$ defines a regular function on the scheme $\mathbf{M}$. It follows from Lemma [Lemma 20](#lemma:v-tau-cycl-if-only-if-map-modul-begin-rn-right-v-){reference-type="ref" reference="lemma:v-tau-cycl-if-only-if-map-modul-begin-rn-right-v-"} that $\tau$ is stable if and only if $\Delta(\tau)$ is a unit in $R$. In particular, $\mathbf{M} _{\mathop{\mathrm{stab}}}$ defines an open affine subscheme of $\mathbf{M}$. In the remainder of this section, we focus on an individual (arbitrary) ring $R$ and abbreviate $V := \mathbf{V}(R), G := \mathbf{G}(R)$, etc. For $\tau \in M$, we may form the characteristic polynomials $$P_\tau \in R[X] \quad \text{and} \quad P_{\tau_H} \in R[X].$$ These are monic polynomials of degrees $n+1$ and $n$, respectively. **Lemma 26**. Let $P, P_H \in R[X]$ be monic polynomials of degrees $n+1$ and $n$, respectively. There exists $\tau \in M$ such that $P_{\tau} = P$ and $P_{\tau_H} = P_H$. *Proof.* One can deduce the existence of $\tau$ directly from the spherical property of $H \hookrightarrow G \times H$ (see [@nelson-venkatesh-1 §14.6]). For variety of exposition, we record an "explicit" construction, following [@MR3164988 §6.2]. Supposing for instance that $H = \mathop{\mathrm{GL}}_2(R) \hookrightarrow G = \mathop{\mathrm{GL}}_3(R)$, embedded as the upper-left block, we take $$\tau = \begin{pmatrix} a _1 & 1 & 0 \\ a _2 & 0 & 1 \\ b _2 & b _1 & b _0 \\ \end{pmatrix}.$$ The coefficients of ${\operatorname{charpoly}} {\paren*{\tau_H}}$ are then $\pm a_j$, while the coefficients of ${\operatorname{charpoly}} {\paren*{\tau}}$ are of the form $\pm (b_j + c_j)$, where $c_j$ depends upon the $a_k$ and $b_l$ for $l<j$. By choosing suitable $a_j$ and $b_j$ inductively, we may thus arrange that $\tau$ and $\tau_H$ have prescribed characteristic polynomials. ◻ **Lemma 27**. For $\tau \in M$, the following are equivalent. (i) [\[enumerate:p_tau-p_tau_h-generate-unit-ideal.-\]]{#enumerate:p_tau-p_tau_h-generate-unit-ideal.- label="enumerate:p_tau-p_tau_h-generate-unit-ideal.-"} $P_\tau$ and $P_{\tau_H}$ generate the unit ideal. (ii) [\[enumerate:there-are-no-nontr-tau-invar-subsp-v_h-or-v_h.-\]]{#enumerate:there-are-no-nontr-tau-invar-subsp-v_h-or-v_h.- label="enumerate:there-are-no-nontr-tau-invar-subsp-v_h-or-v_h.-"} There are no nonzero $\tau$-invariant submodules of $V_H$ or of $V_H^*$. (iii) [\[enumerate:vectors-e-f-are-tau-cyclic.-\]]{#enumerate:vectors-e-f-are-tau-cyclic.- label="enumerate:vectors-e-f-are-tau-cyclic.-"} $\tau \in M_{\mathop{\mathrm{stab}}}$. *Proof.* We show first that [\[enumerate:p_tau-p_tau_h-generate-unit-ideal.-\]](#enumerate:p_tau-p_tau_h-generate-unit-ideal.-){reference-type="eqref" reference="enumerate:p_tau-p_tau_h-generate-unit-ideal.-"} implies [\[enumerate:there-are-no-nontr-tau-invar-subsp-v_h-or-v_h.-\]](#enumerate:there-are-no-nontr-tau-invar-subsp-v_h-or-v_h.-){reference-type="eqref" reference="enumerate:there-are-no-nontr-tau-invar-subsp-v_h-or-v_h.-"}. By assumption, we may write $a P_\tau + b P_{\tau_H} = 1$ for some $a,b \in R[X]$. Since $P_\tau(\tau) = 0$, it follows that $b(\tau) P_{\tau_H}(\tau) = 1$. On the other hand, if $U$ is a $\tau$-invariant submodule of $V_H$ and $u \in U$, then $f(\tau) u = f(\tau_H) u$ for each $f \in R[x]$, hence $P_{\tau_H}(\tau) u = P_{\tau_H}(\tau_H) u = 0$. Therefore $u =0$, as required. We show next that [\[enumerate:there-are-no-nontr-tau-invar-subsp-v_h-or-v_h.-\]](#enumerate:there-are-no-nontr-tau-invar-subsp-v_h-or-v_h.-){reference-type="eqref" reference="enumerate:there-are-no-nontr-tau-invar-subsp-v_h-or-v_h.-"} implies [\[enumerate:vectors-e-f-are-tau-cyclic.-\]](#enumerate:vectors-e-f-are-tau-cyclic.-){reference-type="eqref" reference="enumerate:vectors-e-f-are-tau-cyclic.-"}. We must check that $e$ and $e^*$ are $\tau$-cyclic. Suppose otherwise that $R[\tau] e \neq V$ or that $e^* R[\tau] \neq V^*$. In the first case, we set $U := {(R[\tau] e)}^\perp \subseteq V^*$; in the second, $U := {(e^* R [\tau ])}^\perp \subseteq V$. In either case, we see that $U$ is a non-zero $\tau$-invariant submodule of $V_H^*$ or $V_H$, contrary to [\[enumerate:there-are-no-nontr-tau-invar-subsp-v_h-or-v_h.-\]](#enumerate:there-are-no-nontr-tau-invar-subsp-v_h-or-v_h.-){reference-type="eqref" reference="enumerate:there-are-no-nontr-tau-invar-subsp-v_h-or-v_h.-"}. We show finally that [\[enumerate:vectors-e-f-are-tau-cyclic.-\]](#enumerate:vectors-e-f-are-tau-cyclic.-){reference-type="eqref" reference="enumerate:vectors-e-f-are-tau-cyclic.-"} implies [\[enumerate:p_tau-p_tau_h-generate-unit-ideal.-\]](#enumerate:p_tau-p_tau_h-generate-unit-ideal.-){reference-type="eqref" reference="enumerate:p_tau-p_tau_h-generate-unit-ideal.-"}. Suppose that $\tau$ is stable, but that the ideal $\mathfrak{a} \subseteq R[X]$ generated by $P_\tau$ and $P_{\tau_H}$ is not the unit ideal (1). Let $\mathfrak{m}$ be a maximal ideal that contains $\mathfrak{a}$, and set $\mathfrak{p} := R \cap \mathfrak{m}$. The stability of $\tau$ and the fact that $\mathfrak{a} \neq (1)$ are unaffected by modding out by $\mathfrak{p}$, passing to the field of fractions of $R / \mathfrak{p}$, and then passing to the algebraic closure of that field. We thereby reduce to the case that $R$ is an algebraically closed field. Then $P_\tau$ and $P_{\tau_H}$ share a common root $c$. The remainder of the proof closely follows that of [@nelson-venkatesh-1 Lemma 14.4]. We may find - an eigenvector $v \in V_H$ for $\tau_H$ with eigenvalue $c$, and also - an eigenvector $\ell \in V^*$ for $\tau$ with eigenvalue $c$. Then $$\label{eqn:ell-cdot-tau-c-cdot-v-=-0.-} \ell \cdot (\tau - c) \cdot v = 0.$$ We consider two cases. - $(\tau - c) v = 0$, so that $v \in V_H$ is an eigenvector for $\tau$. - $(\tau - c) v \neq 0$. Since $(\tau_H - c) v = 0$, it follows that $e$ is a multiple of $(\tau - c) v$, hence by [\[eqn:ell-cdot-tau-c-cdot-v-=-0.-\]](#eqn:ell-cdot-tau-c-cdot-v-=-0.-){reference-type="eqref" reference="eqn:ell-cdot-tau-c-cdot-v-=-0.-"} that $\ell e = 0$, so that $\ell \in V_H^*$ is an eigenvector for $\tau$. In either case, we have produced an eigenvector for $\tau$ in $V_H$ or in $V_H^*$. Since $R$ is a field, the orthogonal complement of that eigenvector defines a proper $\tau$-invariant subspace of $V^*$ (resp. $V$) that contains the vector $e^*$ (resp. $e$), contrary to our assumption that that vector is cyclic. ◻ **Lemma 28**. Let $\tau \in M_{\mathop{\mathrm{stab}}}$. Then $H \cap G_\tau = \{1\}$. *Proof.* This follows from the fact that $e$ is $\tau$-cyclic and fixed by $H$, which shows that if $h \in H$ commutes with $\tau$, then it fixes all of $V$, hence $h = 1$. ◻ **Lemma 29**. Let $\tau \in M_{\mathop{\mathrm{stab}}}$. Then $\tau_H$ is cyclic, that is to say, $V_H$ and, equivalently, $V_H^*$, are $\tau_H$-cyclic. *Proof.* We will show that $V_H^*$ is cyclic; a similar argument applies to $V_H$. We will verify more precisely that $$\ell : V_H \rightarrow R, \qquad \ell(v) := e^* \tau v$$ defines a $\tau_H$-cyclic vector $\ell \in V_H^*$, that is to say, $$\label{eqn:ell-rtau_h-=-v_h.-} \ell R[\tau_H] = V_H^*.$$ In verifying this, we may assume that $R$ is a local ring, and then (by Nakayama's lemma) that $R$ is a field. In that case, if [\[eqn:ell-rtau_h-=-v_h.-\]](#eqn:ell-rtau_h-=-v_h.-){reference-type="eqref" reference="eqn:ell-rtau_h-=-v_h.-"} fails, then the orthogonal complement of $\ell R[\tau_H]$ is a nonzero $\tau_H$-invariant subspace $U$ of $V_H$. By construction, $0 = \ell(U) = e^* \tau U$, so $\tau|_{U} = \tau_H|_{U}$. Thus $U$ is in fact a $\tau$-invariant subspace of $V_H$, which by part [\[enumerate:there-are-no-nontr-tau-invar-subsp-v_h-or-v_h.-\]](#enumerate:there-are-no-nontr-tau-invar-subsp-v_h-or-v_h.-){reference-type="eqref" reference="enumerate:there-are-no-nontr-tau-invar-subsp-v_h-or-v_h.-"} of Lemma [Lemma 27](#lemma:stability-equivalences){reference-type="ref" reference="lemma:stability-equivalences"} yields $U = 0$, giving the required contradiction. ◻ # Transversality {#sec:transversality} ## Statement of results {#sec:transversality-statement-results} For $\tau \in \mathbf{M}_{\mathop{\mathrm{stab}}}(R)$, $a \in \mathbf{G}(R)$, we define a subfunctor $\mathbf{X}_{\tau,a}$ of $\mathbf{H}_{\tau_H}$ over $R$, as follows: for each ring extension $R'$ of $R$, $$\mathbf{X}_{\tau,a}(R') := \left\{ y \in \mathbf{H}_{\tau_H}(R') : a y \in \mathbf{H}(R') \mathbf{G}_{\tau}(R') \right\}.$$ As we explain below (Lemma [Lemma 37](#lemma:each-ring-extens-r-r-we-have-begin-mathbfhr-m){reference-type="ref" reference="lemma:each-ring-extens-r-r-we-have-begin-mathbfhr-m"}), it defines a closed subscheme. We observe that if $a$ lies in $\mathbf{H}(R) \mathbf{Z}(R)$, then $\mathbf{X}_{\tau,a} = \mathbf{H}_{\tau_H}$, that is, their point sets coincide for all ring extensions $R'$ of $R$. Indeed, if $a = h z$ with $(h,z) \in \mathbf{H}(R) \mathbf{Z}(R)$, then for each $y \in \mathbf{H}_{\tau_H}(R')$, we have $$a y = h z y = (h y) z \in \mathbf{H}(R') \mathbf{G}_{\tau}(R').$$ The main result of this section is the following converse: **Theorem 30**. *Assume that $\mathop{\mathrm{rank}}(\mathbf{V}) \geq 3$. Let $R$ be a ring in which $2$ is a unit. Let $(\tau,a) \in \mathbf{M}_{\mathop{\mathrm{stab}}}(R) \times \mathbf{G}(R)$. Suppose that $\mathbf{X}_{\tau,a} = \mathbf{H}_{\tau_H}$ (as schemes over $R$). Then $a \in \mathbf{H}(R) \mathbf{Z}(R)$.* **Remark 31** (Sharpness with respect to the rank). The conclusion of Theorem [Theorem 30](#theorem:main-transversality-general-ring){reference-type="ref" reference="theorem:main-transversality-general-ring"} clearly fails when $\mathop{\mathrm{rank}}(\mathbf{V}) = 1$, since in that case, $\mathbf{H}_{\tau_H}$ is trivial. The conclusion may fail also when $\mathop{\mathrm{rank}}(\mathbf{V}) = 2$. For example, take $\mathbf{V} = \mathbb{Z}^2$ with standard basis $e_1,e_2$. Let $e_1^*,e_2^*$ denote the corresponding dual basis of $\mathbf{V}^*$. Take $e := e_2$ and $e^* := e_2^*$, so that $$\mathbf{H} = \begin{pmatrix} \ast & 0 \\ 0 & 1 \\ \end{pmatrix} \leq \mathbf{G}.$$ Take $$\tau := a := \begin{pmatrix} 0 & 1 \\ 1 & 0 \\ \end{pmatrix}.$$ Then for any ring $R$, we see that $a$ does not lie in $\mathbf{H}(R) \mathbf{Z}(R)$ (although it does normalize it). On the other hand, for any $$y = \begin{pmatrix} y_1 & 0 \\ 0 & 1 \\ \end{pmatrix} \in \mathbf{H}_{\tau_H}(R) = \mathbf{H}(R),$$ we have $$a y = h b \quad \text{ with } h := \begin{pmatrix} y_1^{-1} & 0 \\ 0 & 1 \\ \end{pmatrix} \in \mathbf{H}(R), \quad b := \begin{pmatrix} 0 & y_1 \\ y_1 & 0 \\ \end{pmatrix} \in \mathbf{G}_\tau(R).$$ Thus $\mathbf{X}_{\tau,a} = \mathbf{H}_{\tau_H}$. The proof of Theorem [Theorem 30](#theorem:main-transversality-general-ring){reference-type="ref" reference="theorem:main-transversality-general-ring"} can be adapted to show that these are essentially the only counterexamples when $\mathop{\mathrm{rank}}(\mathbf{V}) = 2$. This discussion suggests that Theorem [Theorem 30](#theorem:main-transversality-general-ring){reference-type="ref" reference="theorem:main-transversality-general-ring"} should be related to the fact that $\mathbf{H} \mathbf{Z}$ has trivial normalizer in $\mathbf{G}$ when $\mathop{\mathrm{rank}}(\mathbf{V}) \geq 3$, although we did not spot a direct way to relate the two conditions. **Remark 32** (Sharpness with respect to the ring). We have seen by computer calculation with Gröbner bases that when $\mathop{\mathrm{rank}}(\mathbf{V}) \in \{3,4\}$, there exist fields $R$ of characteristic $2$ over which there are counterexamples to the conclusion of Theorem [Theorem 30](#theorem:main-transversality-general-ring){reference-type="ref" reference="theorem:main-transversality-general-ring"}. We see no reason why such examples should not exist in any rank. Theorem [Theorem 30](#theorem:main-transversality-general-ring){reference-type="ref" reference="theorem:main-transversality-general-ring"} is a consequence of some more precise assertions that we now formulate. **Definition 33**. Let $R$ be a ring. We define the extension rings $$R' := R[\varepsilon]/(\varepsilon^2), \quad R'' := R[\varepsilon_1,\varepsilon_2]/(\varepsilon_1^2,\varepsilon_2^2),$$ which come with natural maps $R' \rightarrow R$ and $R'' \rightarrow R$ obtained by sending $\varepsilon, \varepsilon_1, \varepsilon_2$ to zero. For a scheme $\mathbf{X}$, we say that $y' \in \mathbf{X}(R')$ lies over $y \in \mathbf{X}(R)$ if $y' \mapsto y$ under the induced map $\mathbf{X}(R') \rightarrow \mathbf{X}(R)$, and similarly for $R''$. Let $\tau \in \mathbf{M}_{\mathop{\mathrm{stab}}}(R)$, $a \in \mathbf{G}(R)$ and $y \in \mathbf{X}_{\tau,a}(R)$. 1. We say that $\mathbf{X}_{\tau,a}$ is *tangential at $y$ over $R$* if for all $y' \in \mathbf{H}_{\tau_H}(R')$ lying over $y$, we have $y' \in \mathbf{X}_{\tau,a}(R')$. 2. We say that $\mathbf{X}_{\tau,a}$ is *doubly-tangential at $y$ over $R$* if for all $y'' \in \mathbf{H}_{\tau_H}(R'')$ lying over $y$, we have $y'' \in \mathbf{X}_{\tau,a}(R'')$. The informal content is that $\mathbf{X}_{\tau,a}$ is tangential (resp. doubly-tangential) at $y$ over $R$ if the polynomials whose vanishing defines $\mathbf{X}_{\tau,a}$ over $R$ have the property that their Taylor series at $y$ vanish to order at least $2$ (resp. $3$). **Theorem 34**. *Retain the setting of Definition [Definition 33](#definition:let-r-be-ring.-let-a-in-mathbfgr-y-in-mathbfx_t-ar){reference-type="ref" reference="definition:let-r-be-ring.-let-a-in-mathbfgr-y-in-mathbfx_t-ar"}. The following are equivalent:* (i) *$\mathbf{X}_{\tau,a}$ is tangential at $y$ over $R$.* (ii) *For the (unique) elements $(h,b) \in \mathbf{H}(R) \times \mathbf{G}_\tau(R)$ defined by writing $a y = h b$, we have $b^2 \in \mathbf{Z}(R)$.* **Theorem 35**. *Retain the setting of Definition [Definition 33](#definition:let-r-be-ring.-let-a-in-mathbfgr-y-in-mathbfx_t-ar){reference-type="ref" reference="definition:let-r-be-ring.-let-a-in-mathbfgr-y-in-mathbfx_t-ar"}. Assume that* - *$2$ is a unit in $R$, and* - *$\mathop{\mathrm{rank}}(\mathbf{V}) \geq 3$.* *Then the following are equivalent:* (i) *$\mathbf{X}_{\tau,a}$ is doubly-tangential at $y$ over $R$.* (ii) *We have $a \in \mathbf{H}(R) \mathbf{Z}(R)$.* Theorem [Theorem 35](#theorem:reta-sett-defin-refd-r-be-ring.-let-mathbfgr-y-mat-doubly-tangential){reference-type="ref" reference="theorem:reta-sett-defin-refd-r-be-ring.-let-mathbfgr-y-mat-doubly-tangential"} implies in particular that, under the hypotheses of Theorem [Theorem 30](#theorem:main-transversality-general-ring){reference-type="ref" reference="theorem:main-transversality-general-ring"}, if $a \notin \mathbf{H}(R) \mathbf{Z}(R)$, then $\mathbf{X}_{\tau,a}$ fails to be doubly-tangential at every point $y \in \mathbf{X}_{\tau,a}(R)$. In particular, $\mathbf{X}_{\tau,a} \neq \mathbf{H}_{\tau_H}$, so the conclusion of Theorem [Theorem 30](#theorem:main-transversality-general-ring){reference-type="ref" reference="theorem:main-transversality-general-ring"} holds. The remainder of this section is devoted to the proofs of Theorems [Theorem 34](#theorem:characterization-of-tangential-points){reference-type="ref" reference="theorem:characterization-of-tangential-points"} and [Theorem 35](#theorem:reta-sett-defin-refd-r-be-ring.-let-mathbfgr-y-mat-doubly-tangential){reference-type="ref" reference="theorem:reta-sett-defin-refd-r-be-ring.-let-mathbfgr-y-mat-doubly-tangential"}. ## Defining $\mathbf{X}_{\tau,a}$ by polynomial equations {#Sec:polyEqn} Let $R$ be a ring, and let $\tau \in \mathbf{M}_{\mathop{\mathrm{stab}}}(R)$. By Lemma [Lemma 21](#lemma:centralizer-description){reference-type="ref" reference="lemma:centralizer-description"}, the maps $$\begin{aligned} &\bullet e : \mathbf{M}_{\tau}(R) \rightarrow \mathbf{V}(R), \quad &&x \mapsto x e \\ &e^* \bullet : \mathbf{M}_{\tau}(R) \rightarrow \mathbf{V}^*(R), \quad &&x \mapsto e^* x \end{aligned}$$ are linear isomorphisms (i.e., isomorphisms of $R$-modules). We denote by $$\begin{aligned} &{[\bullet e]}^{-1} : \mathbf{V}(R) \rightarrow \mathbf{M}_\tau(R), \quad \\ &{[e^* \bullet]}^{-1} : \mathbf{V}^*(R) \rightarrow \mathbf{M}_\tau(R) \end{aligned}$$ their inverses. These extend to linear isomorphisms over any ring extension $R'$ of $R$. We note the following equivariance property: **Lemma 36**. For any $v \in \mathbf{V}(R)$ (resp. $v^* \in \mathbf{V}^*(R)$) and $a \in \mathbf{G}_{\tau}(R)$, we have $${[\bullet e]}^{-1}(a v) = a {[\bullet e]}^{-1}(v), \qquad {[e^* \bullet ]}^{-1}(v^* a) = {[e^* \bullet]}^{-1}(v^*) a.$$ *Proof.* By definition, ${[\bullet e]}^{-1}(v)$ is the unique element of $\mathbf{M}_\tau(R)$ with ${[\bullet e]}^{-1}(v) e = v$, while ${[\bullet e]}^{-1}(a v)$ is the unique element with ${[\bullet e]}^{-1}(a v) e = a v$. It is clear then that $a {[\bullet e]}^{-1}(v) = {[\bullet e]}^{-1}(a v)$. A similar argument gives the second identity. ◻ **Lemma 37**. For each ring extension $R'$ of $R$, we have $$\label{eqn:mathbfhr-mathbfg_t-=-left-g-in-mathbfgr-:-bull-e-1} \mathbf{H}(R') \mathbf{G}_\tau(R') = \left\{ g \in \mathbf{G}(R') : {[e^* \bullet]}^{-1}( e^* g) {[\bullet e]}^{-1}(g^{-1} e) = 1 \right\}.$$ In particular, $R' \mapsto \mathbf{H}(R') \mathbf{G}_\tau(R')$ defines a closed subscheme of $\mathbf{G}$ over $R$. *Proof.* Suppose $g = h b$ with $(h,b) \in \mathbf{H}(R') \times \mathbf{G}_\tau(R')$. Then $g^{-1} e = b^{-1} e$ and $e^* g = e^* b$, so $${[e^* \bullet]}^{-1}( e^* g) {[\bullet e]}^{-1}(g^{-1} e) = b b^{-1} = 1.$$ Conversely, suppose $g$ belongs to the right hand side of [\[eqn:mathbfhr-mathbfg_t-=-left-g-in-mathbfgr-:-bull-e-1\]](#eqn:mathbfhr-mathbfg_t-=-left-g-in-mathbfgr-:-bull-e-1){reference-type="eqref" reference="eqn:mathbfhr-mathbfg_t-=-left-g-in-mathbfgr-:-bull-e-1"}, Then, defining $b, b' \in \mathbf{M}_{\tau}(R')$ by $b' := {[\bullet e]}^{-1}(g^{-1} e)$ and $b := {[e^* \bullet]}^{-1}(e^* g)$, we have $b' b = 1$. It follows that $b \in \mathbf{G}_\tau(R')$. Moreover, $h := g b^{-1}$ satisfies $h e = e$ and $e^* h = e^*$, hence $h \in \mathbf{H}(R')$. ◻ **Corollary 38**. *Let $$f : \mathbf{H}_{\tau_H} \rightarrow \mathbf{M}_\tau$$ denote the map of schemes over $R$ defined by $$f(y) := {[e^* \bullet]}^{-1}( e^* a y) {[\bullet e]}^{-1}({(a y)}^{-1} e).$$ Then $$\mathbf{X}_{\tau, a}(R') = \left\{ y \in \mathbf{H}_{\tau_H}(R') : f(y) = 1 \right\}.$$ In particular, $\mathbf{X}_{\tau,a}$ defines a closed subscheme of $\mathbf{H}_{\tau_H}$ over $R$.* **Remark 39**. We may rewrite the defining equation $f(y) = 1$ for $\mathbf{X}_{\tau,a}$ using Cramer's rule for $y^{-1}$, as follows: $${[e^* \bullet]}^{-1}( e^* a y) {[\bullet e]}^{-1}(y^{\mathrm{adj}} a^{-1} e) = \det(y).$$ Here $y^{\mathrm{adj}}$ denotes the adjucate or cofactor matrix, characterized by the identity $y y^{\mathrm{adj}} = \det(y)$. Writing $n+1 = \mathop{\mathrm{rank}}(\mathbf{V})$, so that $\mathbf{M}_{\tau}$ has rank $n+1$, one can see that $\mathbf{X}_{\tau,a}$ is defined by $n+1$ polynomial equations of degree at most $n$ in the entries of $y$. Indeed, one can parameterize $\mathbf{H}_{\tau_H}$ using Lemma [Lemma 21](#lemma:centralizer-description){reference-type="ref" reference="lemma:centralizer-description"} with $n$ variables, say, $y_1,\cdots y_n$. Then - each entry of the row vector $e^* a y$ is an affine function of the $y_i$ (i.e., a linear combination of the $y_i$ and $1$), - the coefficients of ${[e^* \bullet]}^{-1}( e^* a y)\in \mathbf{M}_\tau$ are likewise affine functions of the $y_i$, - the matrices $y^{\mathrm{adj}}, y^{\mathrm{adj}} a^{-1} e$ and ${[\bullet e]}^{-1}(y^{\mathrm{adj}} a^{-1} e)$ have entries given by polynomials in the $y_i$ of total degree at most $n-1$, and - $\det(y)$ is a polynomial in the $y_i$ of total degree $n$. Finally, the condition $${[e^* \bullet]}^{-1}( e^* a y) {[\bullet e]}^{-1}(y^{\mathrm{adj}} a^{-1} e) = \det(y)$$ is an equality of elements of $\mathbf{G}_\tau$, whose $n+1$ coefficients give rise to $n+1$ polynomial equations of degree at most $n$ in the $y_i$. ## Reduction to the centralizer {#sec:reduction-centralizer} The set $\mathbf{X}_{\tau,a}(R)$ is empty unless we can find $h \in \mathbf{H}(R)$ and $y \in \mathbf{H}_{\tau_H}(R)$ so that $b := h a y \in \mathbf{G}_\tau(R)$. Then for all ring extensions $R'$ of $R$, we have $$\mathbf{X}_{\tau,a}(R') = y \mathbf{X}_{\tau,b}(R').$$ Furthermore, for each point $z\in \mathbf{X}_{\tau, b}(R)$, the following are equivalent: 1. $\mathbf{X}_{\tau,a}$ is tangential (resp. doubly-tangential) at $yz$. 2. $\mathbf{X}_{\tau,b}$ is tangential (resp. doubly-tangential) at $z$. This last equivalence holds in particular for $z = 1$, which lies in $\mathbf{X}_{\tau,b}(R)$ because $b$ lies in $\mathbf{G}_\tau(R)$. In this way, many questions concerning the $\mathbf{X}_{\tau,a}$ may be reduced to the case $a \in \mathbf{G}_\tau(R)$ and $y = 1$. **Definition 40**. Given $a \in \mathbf{G}_\tau(R)$, we denote by $$\mu, \nu : \mathbf{M}_{H,\tau_H}(R) \rightarrow \mathbf{M}_{\tau}(R)$$ the linear maps given by $$\mu(u) := {[e^* \bullet]}^{-1}( e^* a u a^{-1} ),$$ $$\nu(u) := {[\bullet e]}^{-1}(a u a^{-1} e).$$ We may use the equivariance property noted in Lemma [Lemma 36](#lemma:equivariance-property-of-bullet-e-e-star-inverse-maps){reference-type="ref" reference="lemma:equivariance-property-of-bullet-e-e-star-inverse-maps"} to rewrite the map $f$ from Corollary [Corollary 38](#corollary:let-begin-f-:-mathbfh_t-right-mathbfm_t-){reference-type="ref" reference="corollary:let-begin-f-:-mathbfh_t-right-mathbfm_t-"} as $$f(y) = \mu(y) \nu(y^{-1}).$$ ## Linear analysis {#sec:analysis-first-derivatives} In this section, we prove Theorem [Theorem 34](#theorem:characterization-of-tangential-points){reference-type="ref" reference="theorem:characterization-of-tangential-points"}. By replacing $a$ with $a y$, we may reduce to the case $y = 1$. By then replacing $a$ with $h^{-1} a$, we may reduce further to the case that $a = b \in \mathbf{G}_\tau(R)$. The required equivalence then follows from that between [\[itemize:mathbfx_tau-a-has-first-order-tangency-at-1.-\]](#itemize:mathbfx_tau-a-has-first-order-tangency-at-1.-){reference-type="eqref" reference="itemize:mathbfx_tau-a-has-first-order-tangency-at-1.-"} and [\[itemize:we-have-a2-in-mathbfzr.-\]](#itemize:we-have-a2-in-mathbfzr.-){reference-type="eqref" reference="itemize:we-have-a2-in-mathbfzr.-"} in Proposition [Proposition 41](#lemma:let-r-be-ring-tau-in-mathbfm_st-a-in-mathbfg_t-fol){reference-type="ref" reference="lemma:let-r-be-ring-tau-in-mathbfm_st-a-in-mathbfg_t-fol"}, below. **Proposition 41**. *Let $R$ be a ring, $\tau \in \mathbf{M}_{\mathop{\mathrm{stab}}}(R)$, $a \in \mathbf{G}_\tau(R)$. The following are equivalent:* (i) *[\[itemize:mathbfx_tau-a-has-first-order-tangency-at-1.-\]]{#itemize:mathbfx_tau-a-has-first-order-tangency-at-1.- label="itemize:mathbfx_tau-a-has-first-order-tangency-at-1.-"} $\mathbf{X}_{\tau,a}$ is tangential at $1$ over $R$.* (ii) *[\[itemize:line-isom-begin-mu-nu-:-mathbfm_h-tau_hr-right-mat\]]{#itemize:line-isom-begin-mu-nu-:-mathbfm_h-tau_hr-right-mat label="itemize:line-isom-begin-mu-nu-:-mathbfm_h-tau_hr-right-mat"} The maps $\mu$ and $\nu$ attached to $a$ as in §[5.3](#sec:reduction-centralizer){reference-type="ref" reference="sec:reduction-centralizer"} satisfy $$\label{eqn:mu-=-nu.-} \mu = \nu.$$* (iii) *[\[itemize:j-geq-0-define-begin-a_j-:=-e-tauj-e-quad-b_j-:=-e\]]{#itemize:j-geq-0-define-begin-a_j-:=-e-tauj-e-quad-b_j-:=-e label="itemize:j-geq-0-define-begin-a_j-:=-e-tauj-e-quad-b_j-:=-e"} For $j \geq 0$, define $$A_j := e^* a \tau^j e, \quad B_j := e^* a^{-1} \tau^j e.$$ Then $$\label{eqn:a_j-a-1-=-b_j-} A_j a^{-1} = B_j a.$$* (iv) *[\[itemize:we-have-a2-in-mathbfzr.-\]]{#itemize:we-have-a2-in-mathbfzr.- label="itemize:we-have-a2-in-mathbfzr.-"} We have $a^2 \in \mathbf{Z}(R)$.* The proof uses the following key lemma. **Lemma 42**. There exist $c_{ij} \in M_\tau$ ($0 \leq i \leq j$), with $c_{i i} = 1$, so that for each $j \geq 0$, $$\label{eqn:mu-tau_hj--nu-tau_hj--=-sum_i-leq-j-c_i-j--left-a} \mu (\tau_H^j ) - \nu (\tau_H^j ) = \sum_{i \leq j} c_{i j } \left( B_i a - A_i a^{-1} \right).$$ Here and henceforth, we adopt the convention $$\tau_H^0 := 1_H.$$ *Proof.* We induct on $j$. We consider first the case $j = 0$. By definition, $\nu(1_H)$ is the element of $\mathbf{M}_\tau(R)$ such that $$\nu(1_H) e = a 1_H a ^{-1} e.$$ Expanding out $1_H = 1 - e e^*$ gives $$a 1_H a^{-1} e = e - a e e^* a ^{-1} e.$$ Recognizing $e^* a^{-1} e$ as the scalar $B_0$, we deduce that $$\nu(1_H) e = (1 - B_0 a) e.$$ Since $1 - B_0 a \in \mathbf{M}_{\tau}(R)$, it follows from the definition of $\nu$ that $$\nu(1_H) = 1 - B_0 a.$$ A similar argument gives $$\label{eqn:nu1_h-=-1-a_0-a-1.-} \mu(1_H) = 1 - A_0 a^{-1}.$$ Thus the required identity [\[eqn:mu-tau_hj\--nu-tau_hj\--=-sum_i-leq-j-c_i-j\--left-a\]](#eqn:mu-tau_hj--nu-tau_hj--=-sum_i-leq-j-c_i-j--left-a){reference-type="eqref" reference="eqn:mu-tau_hj--nu-tau_hj--=-sum_i-leq-j-c_i-j--left-a"} holds for $j = 0$ (with $c_{0 0} = 1$). We now aim to verify [\[eqn:mu-tau_hj\--nu-tau_hj\--=-sum_i-leq-j-c_i-j\--left-a\]](#eqn:mu-tau_hj--nu-tau_hj--=-sum_i-leq-j-c_i-j--left-a){reference-type="eqref" reference="eqn:mu-tau_hj--nu-tau_hj--=-sum_i-leq-j-c_i-j--left-a"} for given $j \geq 1$, assuming that it holds for all smaller values of $j$. We expand $$\begin{aligned} \tau_H^j &= (1 - e e^*) \tau (1 - e e^\ast) \tau \dotsb (1 - e e^*) \tau (1 - e e^*) \\ &= \tau^j - e e^* \tau^j - \tau^j e e^* + \dotsb, \end{aligned}$$ where $\dotsb$ denotes a linear combination of terms of the form $$\label{eqn:tauj_1-e-e-tauj_2-quad-0-leq-j_1-j_2--j-} \tau^{j_1} e e^* \tau^{j_2} \quad (0 \leq j_1, j_2 < j)$$ that is symmetric with respect to $j_1$ and $j_2$. For instance, when $j = 1$, we have $$\dotsb = e e^\ast \tau e e^\ast = e (e^\ast \tau e) e^\ast = (e^* \tau e) e e^*,$$ which is a multiple of the vector $e e^\ast$ (i.e., [\[eqn:tauj_1-e-e-tauj_2-quad-0-leq-j_1-j_2\--j-\]](#eqn:tauj_1-e-e-tauj_2-quad-0-leq-j_1-j_2--j-){reference-type="eqref" reference="eqn:tauj_1-e-e-tauj_2-quad-0-leq-j_1-j_2--j-"} with $j_1 = j_2 = 0$). Arguing like in the $j=0$ case, we obtain $$\begin{aligned} \label{eqn:20230516005320} \mu(\tau_H^j) &= \tau^j - A_j a^{-1} - A_0 a^{-1} \tau ^j + \dotsb, \\ \nonumber \nu(\tau_H^j) &= \tau^j - B_j a - B_0 a \tau ^j + \dotsb, \end{aligned}$$ where $\dotsb$ denotes a linear combination of terms given in the first case by $A_{j_1} \tau^{j_2} a^{-1}$ and in the second by $B_{j_1} a \tau^{j_2}$. Taking differences, we obtain $$\mu(\tau_H^j) - \nu(\tau_H^j) = (B_j a - A_j a^{-1} ) + \tau^j (B_0 a - A_0 a^{-1}) + \dotsb,$$ where $\dotsb$ denotes a linear combination of differences $$\tau^{j_2} (B_{j_1} a - A_{j_1} a^{-1}).$$ with $j_1,j_2 < j$. ◻ *Proof of Proposition [Proposition 41](#lemma:let-r-be-ring-tau-in-mathbfm_st-a-in-mathbfg_t-fol){reference-type="ref" reference="lemma:let-r-be-ring-tau-in-mathbfm_st-a-in-mathbfg_t-fol"}.* We check first that [\[itemize:mathbfx_tau-a-has-first-order-tangency-at-1.-\]](#itemize:mathbfx_tau-a-has-first-order-tangency-at-1.-){reference-type="eqref" reference="itemize:mathbfx_tau-a-has-first-order-tangency-at-1.-"} is equivalent to [\[itemize:line-isom-begin-mu-nu-:-mathbfm_h-tau_hr-right-mat\]](#itemize:line-isom-begin-mu-nu-:-mathbfm_h-tau_hr-right-mat){reference-type="eqref" reference="itemize:line-isom-begin-mu-nu-:-mathbfm_h-tau_hr-right-mat"}. For $u \in \mathbf{M}_{H,\tau_H}(R)$, set $$y := 1 + \varepsilon u \in \mathbf{H}_{\tau_H}(R').$$ Then $y^{-1} = 1 - \varepsilon u$, so $${[e^* \bullet]}^{-1}(e^* a y) = a + \varepsilon{[e^* \bullet]}^{-1}(e^* a u),$$ $${[\bullet e]}^{-1} ( {(a y ) }^{-1} e) = a^{-1} - \varepsilon{[\bullet e]}^{-1} ( u a^{-1} e).$$ Multiplying these together and applying the equivariance properties noted in Lemma [Lemma 36](#lemma:equivariance-property-of-bullet-e-e-star-inverse-maps){reference-type="ref" reference="lemma:equivariance-property-of-bullet-e-e-star-inverse-maps"}, we obtain $$\begin{aligned} f(y) &= 1 + \varepsilon\left( {[e^* \bullet]}^{-1}(e^* a u a^{-1}) - {[\bullet e]}^{-1}(a u a ^{-1} e) \right) \\ &= 1 + \varepsilon\left( \mu(u) - \nu(u) \right). \end{aligned}$$ Thus $f(y) = 1$ for all $u$ if and only if the identity [\[eqn:mu-=-nu.-\]](#eqn:mu-=-nu.-){reference-type="eqref" reference="eqn:mu-=-nu.-"} holds. To verify the equivalence between [\[itemize:line-isom-begin-mu-nu-:-mathbfm_h-tau_hr-right-mat\]](#itemize:line-isom-begin-mu-nu-:-mathbfm_h-tau_hr-right-mat){reference-type="eqref" reference="itemize:line-isom-begin-mu-nu-:-mathbfm_h-tau_hr-right-mat"} and [\[itemize:j-geq-0-define-begin-a_j-:=-e-tauj-e-quad-b_j-:=-e\]](#itemize:j-geq-0-define-begin-a_j-:=-e-tauj-e-quad-b_j-:=-e){reference-type="eqref" reference="itemize:j-geq-0-define-begin-a_j-:=-e-tauj-e-quad-b_j-:=-e"}, observe first that powers $\tau_H^j$ for $j \geq 0$ span $\mathbf{M}_{H,\tau_H}(R)$, so the identity [\[eqn:mu-=-nu.-\]](#eqn:mu-=-nu.-){reference-type="eqref" reference="eqn:mu-=-nu.-"} is equivalent to the identity $$\label{eqn:mutau_hj-=-nutau_hj-quad-text-all--j-geq-0.-} \mu(\tau_H^j) = \nu(\tau_H^j)$$ holding for all $j \geq 0$. By inductive application of Lemma [Lemma 42](#lemma:there-exist-c_ij-in-rtau-0-leq-i-leq-j-with-c_i-i-){reference-type="ref" reference="lemma:there-exist-c_ij-in-rtau-0-leq-i-leq-j-with-c_i-i-"}, we see that [\[eqn:mutau_hj-=-nutau_hj-quad-text-all\--j-geq-0.-\]](#eqn:mutau_hj-=-nutau_hj-quad-text-all--j-geq-0.-){reference-type="eqref" reference="eqn:mutau_hj-=-nutau_hj-quad-text-all--j-geq-0.-"} holds for all $j$ if and only if [\[eqn:a_j-a-1-=-b_j-\]](#eqn:a_j-a-1-=-b_j-){reference-type="eqref" reference="eqn:a_j-a-1-=-b_j-"} does. The required equivalence follows. We verify finally that [\[itemize:j-geq-0-define-begin-a_j-:=-e-tauj-e-quad-b_j-:=-e\]](#itemize:j-geq-0-define-begin-a_j-:=-e-tauj-e-quad-b_j-:=-e){reference-type="eqref" reference="itemize:j-geq-0-define-begin-a_j-:=-e-tauj-e-quad-b_j-:=-e"} and [\[itemize:we-have-a2-in-mathbfzr.-\]](#itemize:we-have-a2-in-mathbfzr.-){reference-type="eqref" reference="itemize:we-have-a2-in-mathbfzr.-"} are equivalent. The relation [\[eqn:a_j-a-1-=-b_j-\]](#eqn:a_j-a-1-=-b_j-){reference-type="eqref" reference="eqn:a_j-a-1-=-b_j-"} maybe be rewritten $$B_j a^2 = A_j 1_G.$$ For notational clarity in what follows, let us choose a basis of $\mathbf{V}(R)$, so that we may describe $a^2$ and $1_G$ by their matrix entries $a_{ij}$ and $\delta_{i j}$. Let us also define the row vectors $A = (A _0, \dotsc, A _n )$ and $B = (B _0, \dotsc, B _n)$. Then we further rewrite $$\label{eqn:delta_i-j-=-b-a2_ij-.-} {(a^2)}_{ij } B = \delta_{i j} A.$$ Let $P$ denote the matrix with columns $e, \tau e, \dotsc, \tau^n e$. Since $\tau \in \mathbf{M}_{\mathop{\mathrm{stab}}}(R)$, the matrix $P$ is invertible. We compute $$A = e^* a P, \quad B = e^* a^{-1} P,$$ $$B P^{-1} a e = e^* e = 1, \quad$$ $$A P^{-1} a e = e^* a^2 e.$$ Multiplying the relation [\[eqn:delta_i-j-=-b-a2_ij-.-\]](#eqn:delta_i-j-=-b-a2_ij-.-){reference-type="eqref" reference="eqn:delta_i-j-=-b-a2_ij-.-"} on the right by $P^{-1} a e$ gives $${(a^2)}_{i j} = \delta_{i j} e^* a^2 e,$$ hence $$a^2 = (e^* a^2 e) 1_G \in \mathbf{Z}(R).$$ This shows that [\[itemize:j-geq-0-define-begin-a_j-:=-e-tauj-e-quad-b_j-:=-e\]](#itemize:j-geq-0-define-begin-a_j-:=-e-tauj-e-quad-b_j-:=-e){reference-type="eqref" reference="itemize:j-geq-0-define-begin-a_j-:=-e-tauj-e-quad-b_j-:=-e"} implies [\[itemize:we-have-a2-in-mathbfzr.-\]](#itemize:we-have-a2-in-mathbfzr.-){reference-type="eqref" reference="itemize:we-have-a2-in-mathbfzr.-"}. The converse may be verified by reversing the above steps, or directly from the definitions. ◻ **Remark 43**. The arguments used to establish Theorem [Theorem 34](#theorem:characterization-of-tangential-points){reference-type="ref" reference="theorem:characterization-of-tangential-points"} lead to a much simpler proof of the following generalizations of Theorem [@2020arXiv201202187N Theorem 16.3]: *for each ring* $R$ *with* $2 \in R^\times$*, each* $\tau \in \mathbf{M}_{\mathop{\mathrm{stab}}}(R)$ *and* $x \in \mathbf{M}_\tau(R) - R$*, we have* $$\label{eqn:x-1_h-notin-mathbfm_hr-+-mathbfm_taur.-} [x,1_H] \notin \mathbf{M}_{H}(R) + \mathbf{M}_{\tau}(R).$$ (The quoted result asserts the same conclusion when $R$ is a field of characteristic zero.) To see this, suppose otherwise that $[x,1_H] \in t + \mathbf{M}_H(R)$ for some $t \in \mathbf{M}_\tau(R)$. Then $$\label{eqn:e-t-=-ex-1_h-quad-t-e-=-x-1_h-e.-} e^* t = e^*[x,1_H], \quad t e = [x,1_H] e.$$ Using that $1_H = 1 - e e^*$ and identities like $e^* x e e^* = e^* (e^* x e)$, we compute that $$\label{eqn:e-x-1_h-=-e-x-e-x-e-quad-x-1_h-e-=-x} e^* [x,1_H] = e^* (x - e^* x e), \quad [x,1_H] e = - (x - e^* x e) e.$$ The stability of $\tau$ implies that $t \in \mathbf{M}_{\tau}(R)$ is determined by either of the identities in [\[eqn:e-t-=-ex-1_h-quad-t-e-=-x-1_h-e.-\]](#eqn:e-t-=-ex-1_h-quad-t-e-=-x-1_h-e.-){reference-type="eqref" reference="eqn:e-t-=-ex-1_h-quad-t-e-=-x-1_h-e.-"}. Since both factors in [\[eqn:e-x-1_h-=-e-x-e-x-e-quad-x-1_h-e-=-x\]](#eqn:e-x-1_h-=-e-x-e-x-e-quad-x-1_h-e-=-x){reference-type="eqref" reference="eqn:e-x-1_h-=-e-x-e-x-e-quad-x-1_h-e-=-x"} lie in $\mathbf{M}_{\tau}(R)$, it follows that $$x - e^* x e = t = - x + e^* x e,$$ which simplifies to $2 x = 2 e^* x e$. Since $2 \in R^\times$, we deduce that $x = e^* x e \in R$, contrary to assumption. **Remark 44**. There are many examples of $\tau$ for which we have the implication $$a \in \mathbf{G}_\tau(R), \, a^2 \in \mathbf{Z}(R) \implies a \in \mathbf{Z}(R).$$ For instance, this implication holds if $R$ is a field of characteristic $\neq 2$ and $a$ is a cyclic element that is not semisimple. Theorem [Theorem 34](#theorem:characterization-of-tangential-points){reference-type="ref" reference="theorem:characterization-of-tangential-points"} implies that the conclusion of Theorem [Theorem 30](#theorem:main-transversality-general-ring){reference-type="ref" reference="theorem:main-transversality-general-ring"} holds for such $\tau$. **Remark 45**. The basic cases where Theorem [Theorem 34](#theorem:characterization-of-tangential-points){reference-type="ref" reference="theorem:characterization-of-tangential-points"} on its own does not suffice to prove Theorem [Theorem 30](#theorem:main-transversality-general-ring){reference-type="ref" reference="theorem:main-transversality-general-ring"} are when, with respect to some basis, $\tau$ and $a$ are both diagonal, the entries of $a$ equal to $\pm 1$ but not all the same (cf. Remark [Remark 49](#remark:gl6-example){reference-type="ref" reference="remark:gl6-example"}). When $R$ is a field, the stability assumption then says that $e$ and $e^*$ are vectors (with inner product $e^* e = 1$) all of whose entries are nonzero. ## Quadratic analysis {#sec:analysis-second-derivatives} Here we prove Theorem [Theorem 35](#theorem:reta-sett-defin-refd-r-be-ring.-let-mathbfgr-y-mat-doubly-tangential){reference-type="ref" reference="theorem:reta-sett-defin-refd-r-be-ring.-let-mathbfgr-y-mat-doubly-tangential"}, following several preparatory lemmas. **Lemma 46**. Let $R$ be ring, $\tau \in \mathbf{M}_{\mathop{\mathrm{stab}}}(R)$ and $a \in \mathbf{G}_{\tau}(R)$. Assume that $\mathbf{X}_{\tau,a}$ is doubly-tangential at $1$ over $R$; in particular, it is tangential, so by Proposition [Proposition 41](#lemma:let-r-be-ring-tau-in-mathbfm_st-a-in-mathbfg_t-fol){reference-type="ref" reference="lemma:let-r-be-ring-tau-in-mathbfm_st-a-in-mathbfg_t-fol"}, the linear maps $\mu$ and $\nu$ from §[5.3](#sec:reduction-centralizer){reference-type="ref" reference="sec:reduction-centralizer"} coincide. Then $$\label{eqn:cool-homomorphism-property-u1-u2} 2 \left( \mu (u _1 u _2 ) - \mu (u _1 ) \mu (u _2 ) \right) = 0 \quad \text{ for all } u_1, u_2 \in \mathbf{M}_{H,\tau_H}(R).$$ *Proof.* With $y := 1 + \varepsilon_1 u_1 + \varepsilon_2 u_2$, we see by a short calculation that $$y^{-1} = 1 - \varepsilon_1 u_1 - \varepsilon_2 u_2 + 2 \varepsilon_1 \varepsilon_2 u_1 u_2,$$ hence $$\mu (y) = 1 + \varepsilon_1 \mu (u _1 ) + \varepsilon_2 \mu (u _2 ).$$ $$\nu(y^{-1}) = 1 - \varepsilon_1 \nu(u _1 ) - \varepsilon_2 \nu (u _2 ) + 2 \varepsilon_1 \varepsilon_2 \nu (u _1 u _2 ),$$ By our doubly-tangential hypothesis, we have the relation $1 = f(y) = \mu(y) \nu(y^{-1})$. Expanding this out and using that $\mu = \nu$ gives $$2 \mu (u _1 u _2 ) = \mu (u _1 ) \mu (u _2 ) + \mu (u _2 ) \mu (u _1 ).$$ The claimed identity follows now from the commutativity of $\mathbf{M}_\tau(R)$ (Lemma [Lemma 21](#lemma:centralizer-description){reference-type="ref" reference="lemma:centralizer-description"}). ◻ **Lemma 47**. Under the same hypotheses as Lemma [Lemma 46](#lemma:second-derivatives-yield-homomorphism-property){reference-type="ref" reference="lemma:second-derivatives-yield-homomorphism-property"}, and with $A_0 := e^* a e$ as in Proposition [Proposition 41](#lemma:let-r-be-ring-tau-in-mathbfm_st-a-in-mathbfg_t-fol){reference-type="ref" reference="lemma:let-r-be-ring-tau-in-mathbfm_st-a-in-mathbfg_t-fol"}, we have $$\label{eqn:2-a_0--a_0-=-0.-} 2 A_0 ( a - A_0) = 0.$$ *Proof.* We specialize [\[eqn:cool-homomorphism-property-u1-u2\]](#eqn:cool-homomorphism-property-u1-u2){reference-type="eqref" reference="eqn:cool-homomorphism-property-u1-u2"} to $u_1 = u_2 = 1_H$. The identity [\[eqn:nu1_h-=-1-a_0-a-1.-\]](#eqn:nu1_h-=-1-a_0-a-1.-){reference-type="eqref" reference="eqn:nu1_h-=-1-a_0-a-1.-"} for $\mu = \nu$ gives $$\mu(1_H) = 1 - A_0 a^{-1}.$$ A short calculation then gives $$A_0(a - A_0) = a^2 (\mu(1_H) - {\mu(1_H)}^2).$$ The conclusion now follows from [\[eqn:cool-homomorphism-property-u1-u2\]](#eqn:cool-homomorphism-property-u1-u2){reference-type="eqref" reference="eqn:cool-homomorphism-property-u1-u2"}. ◻ **Remark 48**. We pause to sketch a proof of Theorem [Theorem 30](#theorem:main-transversality-general-ring){reference-type="ref" reference="theorem:main-transversality-general-ring"} in the special case $$\label{eq:cj3szcth5l} A_0 = A_1 = 0$$ by an argument that seems more illuminating than the general argument given below. (We note that the hypothesis [\[eq:cj3szcth5l\]](#eq:cj3szcth5l){reference-type="eqref" reference="eq:cj3szcth5l"} excludes the case $a \in \mathbf{H}(R) \mathbf{Z}(R)$, since then $A_0$ is a unit.) Since $A_0 = 0$, we have $$\mu(1_H) = 1.$$ Since $A_1 = 0$, we see from the formula [\[eqn:20230516005320\]](#eqn:20230516005320){reference-type="eqref" reference="eqn:20230516005320"} (specialized to $j=1$) that $$\mu(\tau_H) = \tau.$$ Since $2 \notin R^\times$, we deduce from [\[eqn:cool-homomorphism-property-u1-u2\]](#eqn:cool-homomorphism-property-u1-u2){reference-type="eqref" reference="eqn:cool-homomorphism-property-u1-u2"} the homomorphism property $$\mu(u_1) \mu(u_2) = \mu(u_1 u_2).$$ By iterating this and applying the Cayley--Hamilton theorem to $\tau_H$, we obtain $$0 = \mu(P_{\tau_H}(\tau_H)) = P_{\tau_H}(\mu(\tau_H)) = P_{\tau_H}(\tau).$$ Since $\tau$ is stable, the characteristic polynomials $P_\tau$ and $P_{\tau_H}$ generate the unit ideal (Lemma [Lemma 27](#lemma:stability-equivalences){reference-type="ref" reference="lemma:stability-equivalences"}), contradicting the Cayley--Hamilton theorem for $\tau$. **Remark 49**. The argument sketched in [Remark 48](#remark:sketch-proof-when-A0-A1-vanish){reference-type="ref" reference="remark:sketch-proof-when-A0-A1-vanish"}, while aesthetically pleasing, does not suffice to establish the general case of Theorem [Theorem 30](#theorem:main-transversality-general-ring){reference-type="ref" reference="theorem:main-transversality-general-ring"}. We record a counterexample with $\mathop{\mathrm{rank}}(\mathbf{V}) = 6$. Take $R = \mathbb{Q}(\alpha)$ with $\alpha^2 = 2$, $$\tau = \mathop{\mathrm{diag}}(0,1,2,2 \alpha,1 + 2 \alpha,2 + 2 \alpha),$$ $$e^* = (1,1,1,1,1,1), \quad e = \tfrac{1}{6} {(1,1,1,1,1,1)}^t,$$ $$a = \mathop{\mathrm{diag}}(1,1,1,-1,-1,-1).$$ One can verify that $A_0 =B_0= 0$, $A_1 = -\alpha$, $$\mu(1_H) = 1, \quad \mu(\tau_H) = \mathop{\mathrm{diag}}\left( \alpha, \alpha + 1, \alpha + 2, \alpha, \alpha + 1, \alpha + 2 \right),$$ and that $P_{\tau_H}(\mu(\tau_H)) = 0$. We have checked by computer calculation with Gröbner bases that no similar examples exist when $\mathop{\mathrm{rank}}(\mathbf{V}) \in \{3,4\}$. One can also check in this example that $\mathbf{X}_{\tau,a}$ contains the center of $\mathbf{H}$. (Of course, once we have proved Theorem [Theorem 30](#theorem:main-transversality-general-ring){reference-type="ref" reference="theorem:main-transversality-general-ring"}, we will know that $\mathbf{X}_{\tau,a}$ does not contain the full centralizer $\mathbf{H}_{\tau_H}$.) The following lemma contains the final main idea for the proof of Theorem [Theorem 35](#theorem:reta-sett-defin-refd-r-be-ring.-let-mathbfgr-y-mat-doubly-tangential){reference-type="ref" reference="theorem:reta-sett-defin-refd-r-be-ring.-let-mathbfgr-y-mat-doubly-tangential"}. **Lemma 50**. Let $(R,\mathfrak{m})$ be a local ring with $2 \in R^\times$. Let $\tau \in \mathbf{M}_{\mathop{\mathrm{stab}}}(R)$, $a \in \mathbf{G}_{\tau}(R)$. Assume that $\mathbf{X}_{\tau,a}$ is doubly-tangential at $1$ over $R$. Then one of the following is true: (i) $a \in \mathbf{Z}(R)$. (ii) $a \notin \mathbf{Z}(R)$, $a^2 \in \mathbf{Z}(R)$ and $$\tau \in R a + R \subseteq \mathbf{M}(R).$$ *Proof.* The hypotheses of Lemma [Lemma 47](#lemma:A0-times-a-minus-A0-vanishes){reference-type="ref" reference="lemma:A0-times-a-minus-A0-vanishes"} apply, so the identity [\[eqn:2-a_0\--a_0-=-0.-\]](#eqn:2-a_0--a_0-=-0.-){reference-type="eqref" reference="eqn:2-a_0--a_0-=-0.-"} holds. The argument divides according to whether $A_0$ lies in $\mathfrak{m}$ or not. If $A_0 \notin \mathfrak{m}$, so that $A_0$ is a unit, then from [\[eqn:2-a_0\--a_0-=-0.-\]](#eqn:2-a_0--a_0-=-0.-){reference-type="eqref" reference="eqn:2-a_0--a_0-=-0.-"} and our assumption $2 \in R^\times$, we see that $a \in \mathbf{Z}(R)$. The remainder of the proof concerns the case $A_0 \in \mathfrak{m}$. By Theorem [Theorem 34](#theorem:characterization-of-tangential-points){reference-type="ref" reference="theorem:characterization-of-tangential-points"} and our hypothesis that $\mathbf{X}_{\tau,a}$ is (doubly-)tangential at $1$ over $R$, we have $$a^2 \in \mathbf{Z}(R).$$ This hypothesis further implies that for $$R' := R[\varepsilon] / (\varepsilon^2), \quad y := 1 + \varepsilon\tau _H \in \mathbf{H}_{\tau_H}(R'),$$ there exists $(h,b) \in \mathbf{H}(R') \times \mathbf{G}_\tau(R')$ such that $a y = h b$. It implies finally that $\mathbf{X}_{\tau,a}$ is tangential at $y$ over $R'$, hence by Theorem [Theorem 34](#theorem:characterization-of-tangential-points){reference-type="ref" reference="theorem:characterization-of-tangential-points"} that $$b^2 \in \mathbf{Z}(R').$$ We retain the notation $A_j, B_j$ from Proposition [Proposition 41](#lemma:let-r-be-ring-tau-in-mathbfm_st-a-in-mathbfg_t-fol){reference-type="ref" reference="lemma:let-r-be-ring-tau-in-mathbfm_st-a-in-mathbfg_t-fol"}. From the calculation $$e^* a \tau_H = e^* a (1 - e e^*) \tau ( 1- e e^*) = e^* \left( a \tau - A_0 \tau - A_1 + A_0 (e^* \tau e) \right),$$ we see that $$b = a + \varepsilon\left(a \tau - A _0 \tau - A _1 + A _0 (e^* \tau e)\right).$$ Squaring this relation gives $$\label{eqn:b2-=-a2-+-2-eps-lefta-tau-_0-tau-_1-+-_0-e-tau} b^2 = a^2 + 2 \varepsilon a \left(a \tau - A _0 \tau - A _1 + A _0 (e^* \tau e)\right),$$ using here that $a$ and $\tau$ commute. Using now that both $a^2$ and $b^2$ lie in $\mathbf{Z}(R')$, together with our assumption $2 \in R^\times$, we deduce that $$a (a - A_0) \tau \in R a + R \subseteq \mathbf{M}(R).$$ Using next our assumption $A_0 \in \mathfrak{m}$, we see that $a(a-A_0)$ is invertible. Using again that $a^2 \in \mathbf{Z}(R)$, we conclude that $\tau \in R a + R$. ◻ *Proof of Theorem [Theorem 35](#theorem:reta-sett-defin-refd-r-be-ring.-let-mathbfgr-y-mat-doubly-tangential){reference-type="ref" reference="theorem:reta-sett-defin-refd-r-be-ring.-let-mathbfgr-y-mat-doubly-tangential"}.* We have already noted, at the start of §[5.1](#sec:transversality-statement-results){reference-type="ref" reference="sec:transversality-statement-results"}, that if $a \in \mathbf{H}(R) \mathbf{Z}(R)$, then $\mathbf{X}_{\tau,a} = \mathbf{H}_{\tau_H}$; in particular, $\mathbf{X}_{\tau,a}$ is doubly-tangential at $y$ over $R$. The converse is the interesting direction. Suppose, thus, that $2$ is a unit in $R$, $\mathop{\mathrm{rank}}(\mathbf{V}) \geq 3$, $\tau \in \mathbf{M}_{\mathop{\mathrm{stab}}}$ and $a \notin\mathbf{H}(R) \mathbf{Z}(R)$, but $\mathbf{X}_{\tau,a}$ is doubly-tangential at some $y \in \mathbf{X}_{\tau,a}(R)$ over $R$. We aim to derive a contradiction. By passing to the localization of $R$ at a prime $\mathfrak{p}$ for which the image of $a$ does not lie in $\mathbf{H}(R_\mathfrak{p}) \mathbf{Z}(R_\mathfrak{p})$, we may assume that $R$ is a local ring. We may reduce further to the case $a \in \mathbf{G}_\tau(R) - \mathbf{Z}(R)$ and $y=1$, for the same reasons as noted at the beginning of §[5.4](#sec:analysis-first-derivatives){reference-type="ref" reference="sec:analysis-first-derivatives"}. Since $\mathbf{X}_{\tau,a}$ is doubly-tangential at $1$ over $R$, we know from Theorem [Theorem 34](#theorem:characterization-of-tangential-points){reference-type="ref" reference="theorem:characterization-of-tangential-points"} and Lemma [Lemma 50](#lemma:let-r-mathfr-be-local-ring-with-2-in-rtim-let-tau-a-in-Z-or-tau-in-rank-two){reference-type="ref" reference="lemma:let-r-mathfr-be-local-ring-with-2-in-rtim-let-tau-a-in-Z-or-tau-in-rank-two"} that $a^2 \in \mathbf{Z}(R)$ and $\tau \in R a + R$. In particular, we may find $c_0,c_1,c_2 \in R$ so that $\tau = c_0 + c_1 a$ and $a^2 = c_2$. Then $\tau ^2 = c _0 ^2 + c _1^2 c _2 + 2 c _0 c _1 a$, hence $\tau ^2 - 2 c _0 \tau + c _0 ^2 - c _1 ^2 c _2 =0$. Since $\tau$ is stable, the vector $e$ is $\tau$-cyclic, so $$\mathbf{V}(R) = R [\tau] e = R e + R \tau e.$$ Since the rank of $\mathbf{V}$ is $\geq 3$, we obtain a contradiction. ◻ # Volume bounds {#sec:volumebound} We now apply the results of §[5](#sec:transversality){reference-type="ref" reference="sec:transversality"} to deduce a uniform solution to Problem [Problem 10](#Probleminformal){reference-type="ref" reference="Probleminformal"}. ## Bounds for polynomial congruences {#sec:cj3v60uw7y} Let $n$ and $d$ be natural numbers. In §[6.1](#sec:cj3v60uw7y){reference-type="ref" reference="sec:cj3v60uw7y"}, we use the equivalent notations $A \ll B$ and $A = \operatorname{O}(B)$ to denote that $|A| \leq C |B|$, where $C$ depends at most upon $n$ and $d$. **Lemma 51**. Let $F$ be a finite field, of cardinality $q$. Let $P \in F[x_1,\dotsc,x_n]$ be a polynomial of degree at most $d$ whose coefficients are not all zero. Then $$\lvert \{x \in F^n : P(x) = 0\} \rvert \ll q^{n-1}.$$ *Proof.* By the fundamental theorem of algebra, we know that for all but $\operatorname{O}(1)$ many $x_1$, the polynomial that we get by specializing to that value is nonzero. By iterating this observation for $x_2, x_3$, and so on, we obtain the required estimate. ◻ **Lemma 52**. Let $m$ be a natural number. Suppose given a subset $\mathcal{D} \subseteq {(\mathfrak{o}/\mathfrak{p})}^n$ and a polynomial $P \in (\mathfrak{o}/\mathfrak{p}^m)[X_1,\dotsc,X_n]$ of degree $\leq d$. Let $\mathcal{D}_m \subseteq {(\mathfrak{o}/\mathfrak{p}^m)}^n$ denote the inverse image of $\mathcal{D}$. (i) [\[enumerate:assume-that-each-y-in-mathc-one-line-tayl-coeff-p-\]]{#enumerate:assume-that-each-y-in-mathc-one-line-tayl-coeff-p- label="enumerate:assume-that-each-y-in-mathc-one-line-tayl-coeff-p-"} Assume that for each $y \in \mathcal{D}_m$, at least one of the linear Taylor coefficients for $P$ at $y$ is a unit. Then $$\lvert \left\{ y \in \mathcal{D}_m : P(y) \in \mathfrak{p}^m \right\} \rvert \ll q^{m n - m}.$$ (ii) [\[enumerate:assume-that-each-y-in-mathc-one-line-or-quadr-tayl\]]{#enumerate:assume-that-each-y-in-mathc-one-line-or-quadr-tayl label="enumerate:assume-that-each-y-in-mathc-one-line-or-quadr-tayl"} Assume that for each $y \in \mathcal{D}_m$, at least one of the linear or quadratic Taylor coefficients for $P$ at $y$ is a unit. Then $$\lvert \left\{ y \in \mathcal{D}_m : P(y) \in \mathfrak{p}^m \right\} \rvert \ll q^{m n - \lceil m/2 \rceil}.$$ *Proof.* In the case $m=1$, either assertion reduces to the hypersurface bound (Lemma [Lemma 51](#lemma:hypersurface-count){reference-type="ref" reference="lemma:hypersurface-count"}). The case $m \geq 2$ of part [\[enumerate:assume-that-each-y-in-mathc-one-line-tayl-coeff-p-\]](#enumerate:assume-that-each-y-in-mathc-one-line-tayl-coeff-p-){reference-type="eqref" reference="enumerate:assume-that-each-y-in-mathc-one-line-tayl-coeff-p-"} reduces via Hensel's lemma to the case $m = 1$. Indeed, under the stated assumptions, we have more precisely that each fiber of $\mathcal{D}_m \rightarrow \mathcal{D}$ contains at most $q^{(m-1)(n-1)}$ elements $y$ for which $P(y) \in \mathfrak{p}^m$. To see this, suppose for instance that the coefficient of $X_1$ in the Taylor expansion of $P$ at $y$ is a unit. We may then use Hensel's lemma and the condition $P(y) \in \mathfrak{p}^m$ to determine $y_1 \in \mathfrak{o}/\mathfrak{p}^m$ in terms of the other coefficients. The number of possibilities for each of $y_2,\dotsc,y_n$ is at most $q^{m-1}$, giving the required estimate. We turn to part [\[enumerate:assume-that-each-y-in-mathc-one-line-or-quadr-tayl\]](#enumerate:assume-that-each-y-in-mathc-one-line-or-quadr-tayl){reference-type="eqref" reference="enumerate:assume-that-each-y-in-mathc-one-line-or-quadr-tayl"}. The case $m=2$ follows formally from the case $m=1$, using that solutions modulo $\mathfrak{p}^2$ map to solutions modulo $\mathfrak{p}$. We address the cases $m \geq 3$ by induction. We first observe that if $P(y) \in \mathfrak{p}^m$, then certainly $P(y) \in \mathfrak{p}$, and the latter condition depends only upon the class of $y$ modulo $\mathfrak{p}$. By the case $m=1$ that we have already addressed, the number of such classes is $\ll q^{n-1}$. By replacing $P$ with a translate, we reduce to showing that when $P(0) \in \mathfrak{p}^m$, we have $$\lvert \left\{ y \in {(\mathfrak{p}/\mathfrak{p}^m)}^n : P(y) \in \mathfrak{p}^m \right\} \rvert \ll q^{m n - \lceil m/2 \rceil - (n-1)}.$$ We now consider two cases separately: The first is when some linear Taylor coefficient of $P$ is a unit, say the coefficient of $X_1$. Then, arguing as in part [\[enumerate:assume-that-each-y-in-mathc-one-line-tayl-coeff-p-\]](#enumerate:assume-that-each-y-in-mathc-one-line-tayl-coeff-p-){reference-type="eqref" reference="enumerate:assume-that-each-y-in-mathc-one-line-tayl-coeff-p-"}, we see that the cardinality in question is at most $q^{(m-1) (n-1)}$, which is better than the required estimate because $m - 1 \geq \lceil m/2 \rceil$ for $m \geq 3$. The second is when all linear Taylor coefficients of $P$ lie in $\mathfrak{p}$. We then form the polynomial $$Q(y) := \varpi^{-2} P(\varpi y) \in (\mathfrak{o} / \mathfrak{p}^{m-2})[X_1,\dotsc,X_n]$$ and we see that $$\lvert \left\{ y \in {(\mathfrak{p}/\mathfrak{p}^m)}^n : P(y) \in \mathfrak{p}^m \right\} \rvert = q^n \lvert \left\{ y \in {(\mathfrak{o}/\mathfrak{p}^{m-2})}^n : Q(y) \in \mathfrak{p}^{m-2} \right\} \rvert.$$ By hypothesis, some quadratic Taylor coefficient of $P$ is a unit, hence the same holds for $Q$. We may thus apply our inductive hypothesis to bound the cardinality on the right hand side, giving that the left hand side satisfies the estimate $$\ll q^n q ^{(m- 2) n - \lceil (m-2)/2 \rceil} = q ^{m n - \lceil m/2 \rceil - (n-1)},$$ as required. ◻ ## Norms and distance functions {#sec:norms-distance-functions} For each $m \geq 0$, we equip the vector space $F^m$ with the norm $$\left\lvert (v_1,\dotsc,v_m) \right\rvert_F := \max_{1 \leq j \leq m} \lvert v _j \rvert_F.$$ We apply this notation more generally to any vector space that comes with a natural basis, e.g., to the space of $m_1 \times m_2$ matrices over $F$. We choose a basis $\{e_1,\dotsc,e_n\}$ of $\mathbf{V}_H(\mathfrak{o})$, so that $\{e_1,\dotsc,e_n,e\}$ is a basis of $\mathbf{V}(\mathfrak{o})$, and use these bases to identify $\mathbf{V}(F), \mathbf{V}^*(F)$ and $\mathbf{G}(F)$ with matrices over $F$. The above notation then applies. For $h \in \mathbf{G}(F)$, we write $$h = \begin{pmatrix} a & b \\ c & d \\ \end{pmatrix}, \quad h^{-1} = \begin{pmatrix} a' & b' \\ c' & d' \\ \end{pmatrix},$$ where $a$ and $a'$ are $n \times n$ block matrices, and set $$d_H(g) := \min\left(1, \max \left\{ \lvert b/d \rvert_F, \lvert b'/d' \rvert_F, \lvert c/d \rvert_F, \lvert c'/d' \rvert_F \right\} \right).$$ This definition differs mildly from that in [@2020arXiv201202187N §4.2], but agrees up to a constant factor, for reasons explained there. **Lemma 53**. Let ${\ell} \geq 0$ and $g \in K$ be such that $$\label{eqn:g-in-k_h-k_z-k_gmathfrakpd.-} g \in K_H K_Z K(\mathfrak{p}^\ell).$$ Then $d_H(g) \leq q^{-\ell}$, with equality if $g \notin K_H K_Z K(\mathfrak{p}^{\ell+1})$. *Proof.* This follows readily from the following observation: for $\ell \geq 1$, the membership [\[eqn:g-in-k_h-k_z-k_gmathfrakpd.-\]](#eqn:g-in-k_h-k_z-k_gmathfrakpd.-){reference-type="eqref" reference="eqn:g-in-k_h-k_z-k_gmathfrakpd.-"} says that $d, d' \in \mathfrak{o}^\times$ and that at least one $b$ or $c$ (and $b'$ or $c'$) lies in $\mathfrak{p}^{\ell}$. ◻ In other words, for $g \in K = \mathbf{G}(\mathfrak{o})$, the quantity $d_H(g)$ is the infimum of $q^{-\ell}$ taken over all $\ell \geq 0$ for which the image of $g$ in $\mathbf{G}(\mathfrak{o}/\mathfrak{p}^{\ell})$ lies in $\mathbf{H}(\mathfrak{o}/\mathfrak{p}^{\ell})\mathbf{Z}(\mathfrak{o}/\mathfrak{p}^{\ell})$. For example, if the image of $g$ does not lie in $\mathbf{H} (\mathfrak{o}/\mathfrak{p})$, then $d_H(g) = 1$, while if $g$ lies in $\mathbf{Z}(\mathfrak{o}) \mathbf{H}(\mathfrak{o})$, then $d_H(g) = 0$. We abbreviate $${d_H(g)}^\infty := \begin{cases} 1 & \text{ if } d_H(a) = 1, \\ 0 & \text{ if } d_H(a) < 1. \end{cases}$$ We note that for a nonzero ideal $\mathfrak{q}\subseteq \mathfrak{p}$ and $a \in \mathbf{G}(\mathfrak{o}/\mathfrak{q}) - \mathbf{H}(\mathfrak{o}/\mathfrak{q}) \mathbf{Z}(\mathfrak{o}/\mathfrak{q})$, the quantity $d_H(a)$ is well-defined. ## Miscellaneous lemmas Let $(F,\mathfrak{o},\mathfrak{p},q)$ be a non-archimedean local field, and let $\mathfrak{q}\subseteq \mathfrak{p}$ be a nonzero $\mathfrak{o}$-ideal. **Lemma 54**. Assume that $q$ is odd. Let $a \in \mathbf{G}(\mathfrak{o}/\mathfrak{q})$ such that (i) the image of $a$ in $\mathbf{G}(\mathfrak{o}/\mathfrak{p})$ lies in $\mathbf{Z}(\mathfrak{o}/\mathfrak{p})$, and (ii) $a^2 \in \mathbf{Z}(\mathfrak{o}/\mathfrak{q})$. Then $a \in \mathbf{Z}(\mathfrak{o}/\mathfrak{q})$. *Proof.* Write $\mathfrak{q}= \mathfrak{p}^m$. If the conclusion fails, then we may write $a = \lambda (1 + \varpi^{\ell} x)$, where $\lambda \in {(\mathfrak{o}/\mathfrak{q})}^\times$, $1 \leq \ell < m$ and $x \in \mathbf{M}(\mathfrak{o}/\mathfrak{q})$ has the property that its image $\bar{x}$ in $\mathbf{M}(\mathfrak{o}/\mathfrak{p})$ does not lie in $\mathbf{Z}(\mathfrak{o}/\mathfrak{p})$. Then $$\mathbf{Z}(\mathfrak{o}/\mathfrak{q}) \ni a^2 / \lambda^2 = 1 + 2 \varpi^{\ell} x + \varpi^{2 \ell} x^2.$$ Reducing this identity modulo $\mathfrak{p}^{\ell+1}$ and using that $2$ is a unit, we deduce that $\bar{x}$ lies in $\mathbf{Z}(\mathfrak{o}/\mathfrak{p})$, giving the required contradiction. ◻ Recall from Definition [Definition 33](#definition:let-r-be-ring.-let-a-in-mathbfgr-y-in-mathbfx_t-ar){reference-type="ref" reference="definition:let-r-be-ring.-let-a-in-mathbfgr-y-in-mathbfx_t-ar"} the meaning of "tangential". **Lemma 55**. Assume that $q$ is odd. Let $a \in \mathbf{G}(\mathfrak{o}/\mathfrak{q}) - \mathbf{H}(\mathfrak{o}/\mathfrak{q}) \mathbf{Z}(\mathfrak{o}/\mathfrak{q})$ with $d_H(a) < 1$. Then for each $y \in \mathbf{X}_{\tau,a}(\mathfrak{o}/\mathfrak{q})$, we have that $\mathbf{X}_{\tau,a}$ is not tangential at $y$ over $\mathfrak{o}/\mathfrak{q}$. *Proof.* Suppose otherwise that $\mathbf{X}_{\tau,a}$ is tangential at $y$ over $\mathfrak{o}/\mathfrak{q}$. Writing $a y = h b$ with $(h,b) \in \mathbf{H}(\mathfrak{o}/\mathfrak{q}) \times \mathbf{G}_\tau(\mathfrak{o}/\mathfrak{q})$, it follows then by Theorem [Theorem 34](#theorem:characterization-of-tangential-points){reference-type="ref" reference="theorem:characterization-of-tangential-points"} that $b^2 \in \mathbf{Z}(\mathfrak{o}/\mathfrak{q})$. On the other hand, since $d_H(a) < 1$, the image $\bar{a}$ of $a$ modulo $\mathfrak{p}$ lies in $\mathbf{H}(\mathfrak{o}/\mathfrak{p}) \mathbf{Z}(\mathfrak{o}/\mathfrak{p})$. By the uniqueness of the decomposition $\bar{a} \bar{y} = \bar{h} \bar{b}$, it follows that $\bar{b}$ lies in $\mathbf{Z}(\mathfrak{o}/\mathfrak{p})$. By Lemma [Lemma 54](#lemma:supp-char-mathfr-odd.-let-a-in-mathbfgm){reference-type="ref" reference="lemma:supp-char-mathfr-odd.-let-a-in-mathbfgm"}, it follows that $b \in \mathbf{Z}(\mathfrak{o}/\mathfrak{q})$. But then $a \in \mathbf{H}(\mathfrak{o}/\mathfrak{q}) \mathbf{Z}(\mathfrak{o}/\mathfrak{q})$, contrary to assumption. ◻ **Lemma 56**. Let $\tau \in \mathbf{M}(\mathfrak{o}/\mathfrak{q})$ be cyclic. Write $n$ for the rank of $\mathbf{V}$. Then the group $\mathbf{G}_{\tau}(\mathfrak{o}/\mathfrak{q})$ has cardinality at least ${(1 - q^{-1})}^n {[\mathfrak{o}:\mathfrak{q}]}^n$. *Proof.* Since $\mathbf{G}_\tau(\mathfrak{o}/\mathfrak{q}) = {\left( \mathbf{M}_\tau(\mathfrak{o}/\mathfrak{q}) \twoheadrightarrow \mathbf{M}_\tau(\mathfrak{o}/\mathfrak{p}) \right)}^{-1}(\mathbf{G}_\tau(\mathfrak{o}/\mathfrak{p}))$, we may reduce to the case $\mathfrak{q}= \mathfrak{p}$. Since $k := \mathfrak{o}/\mathfrak{p}$ is a field, we may appeal to Lemma [Lemma 21](#lemma:centralizer-description){reference-type="ref" reference="lemma:centralizer-description"} and the structure theorem for modules over the PID to see that the ring $\mathbf{M}_\tau(k)$ is isomorphic to a product of rings $k[X]/(p_i)$, where the $p_i$ are primary polynomials whose degrees sum to $n$. Let $d_i$ denote the degree of the irreducible polyomial of which $p_i$ is a power. Taking unit groups, we obtain $\lvert \mathbf{G}_\tau(k) \rvert = q^n \prod_i (1 - q^{-d_i})$, and the claim follows from the fact that $\sum_i d_i \leq n$. ◻ ## Main result {#sec:cj3v60wplb} **Theorem 57**. *Let $(F,\mathfrak{o},\mathfrak{p},q)$ be a non-archimedean local field with $q$ odd. Let $\mathfrak{q}\subseteq \mathfrak{p}$ be a nonzero $\mathfrak{o}$-ideal. Denote by $Q := [\mathfrak{o}:\mathfrak{q}]$ its absolute norm. Set $$Q^* := q^{\lceil m/2 \rceil} \text{ if } Q = q^m.$$ Let $\tau \in \mathbf{M}_{\mathop{\mathrm{stab}}}(\mathfrak{o}/\mathfrak{q})$ and $a \in \mathbf{G}(\mathfrak{o}/\mathfrak{q}) - \mathbf{H}(\mathfrak{o}/\mathfrak{q}) \mathbf{Z}(\mathfrak{o}/\mathfrak{q})$. Then $$\left\lvert \mathbf{X}_{\tau,a}(\mathfrak{o}/\mathfrak{q}) \right\rvert \leq C \left( \frac{1}{1 + Q d_H(a)} + \frac{{d_H(a)}^\infty }{Q^{*}} \right) \lvert \mathbf{H}_{\tau_H}(\mathfrak{o}/\mathfrak{q}) \rvert,$$ where $C \geq 0$ depends at most upon the rank of $\mathbf{V}$.* *Proof.* By Lemma [Lemma 56](#lemma:cardinality-centralizer){reference-type="ref" reference="lemma:cardinality-centralizer"}, the cardinality of $\mathbf{H}_{\tau_H}(\mathfrak{o}/\mathfrak{q})$ is comparable to $Q^{n}$, where we write $$\mathop{\mathrm{rank}}(\mathbf{V}) = n+1.$$ For this reason, it suffices to establish the modified estimate obtained by replacing $\lvert \mathbf{H}_{\tau_H}(\mathfrak{o}/\mathfrak{q}) \rvert$ with $Q^n$. Moreover, in view of the trivial bound $\lvert \mathbf{X}_{\tau,a}(\mathfrak{o}/\mathfrak{q}) \rvert \leq \lvert \mathbf{H}_{\tau_H}(\mathfrak{o}/\mathfrak{q}) \rvert$, it suffices to show that $$\left\lvert \mathbf{X}_{\tau,a}(\mathfrak{o}/\mathfrak{q}) \right\rvert \ll \left( \frac{1}{Q d_H(a)} + \frac{{d_H(a)}^\infty }{Q^{*}} \right) Q^n,$$ where $\ll$ means "bounded in magnitude by a scalar depending at most upon the rank of $\mathbf{V}$". We will establish this separately when $d_H(a) < 1$ and when $d_H(a) = 1$. Recall from Remark [Remark 39](#remark:rewrite-defining-equations-for-X-as-polynomials){reference-type="ref" reference="remark:rewrite-defining-equations-for-X-as-polynomials"} that $\mathbf{X}_{\tau,a}$ is defined by a system of $n+1$ polynomial equations in the entries of $y \in \mathbf{H}_{\tau_H}(\mathfrak{o}/\mathfrak{q})$, each of degree at most $n$. Consider first the case $d_H(a) < 1$. Write $Q = q^m$ and $d_H(a) = q^{-\ell}$, so that $1 \leq \ell < m$. The image of $a$ lies in $\mathbf{H}(\mathfrak{o}/\mathfrak{p}^{\ell}) \mathbf{Z}(\mathfrak{o}/\mathfrak{p}^{\ell})$, thus the defining polynomial equations are trivial mod $\mathfrak{p}^{\ell}$. The image of $a$ does not lie in $\mathbf{H}(\mathfrak{o}/\mathfrak{p}^{\ell+1}) \mathbf{Z}(\mathfrak{o}/\mathfrak{p}^{\ell+1})$, so by Lemma [Lemma 55](#lemma:supp-char-mathfr-odd.-let-a-in-mathbfgm-not-tangential){reference-type="ref" reference="lemma:supp-char-mathfr-odd.-let-a-in-mathbfgm-not-tangential"}, we see that $\mathbf{X}_{\tau,a}$ is not tangential over $\mathfrak{o} / \mathfrak{p}^{\ell+1}$ at any $y \in \mathbf{X}_{\tau,a}(\mathfrak{o}/\mathfrak{p}^{\ell+1})$. This implies that at least one of the polynomial equations in our system has the property that at least one of the linear Taylor coefficients at $y$ lies in $\mathfrak{p}^{\ell}-\mathfrak{p}^{\ell+1}$. Let us divide that polynomial congruence by $\varpi^{\ell}$ and view it now as a polynomial congruence taken modulo $\mathfrak{p}^{m-\ell}$. Each solution of the new congruence corresponds to exactly $q^{\ell n}$ solutions of the old congruence, but at least one linear Taylor coefficient for the new congruence is a unit, so by part [\[enumerate:assume-that-each-y-in-mathc-one-line-tayl-coeff-p-\]](#enumerate:assume-that-each-y-in-mathc-one-line-tayl-coeff-p-){reference-type="eqref" reference="enumerate:assume-that-each-y-in-mathc-one-line-tayl-coeff-p-"} of Lemma [Lemma 52](#lemma:let-d-geq-0.-supp-given-subs-mathc-subs-mathfr-p){reference-type="ref" reference="lemma:let-d-geq-0.-supp-given-subs-mathc-subs-mathfr-p"} (applied with $\mathcal{D} = \mathbf{H}_{\tau_H}(\mathfrak{o}/\mathfrak{p})$), we obtain $$\lvert \mathbf{X}_{\tau,a}(\mathfrak{o}/\mathfrak{q}) \rvert \ll q^{\ell n} q^{(m-\ell) n - (m-\ell)} = Q^{n-1} q^{\ell} = \frac{1}{Q d_H(a)} Q^n,$$ as required in this case. It remains to consider the case $d_H(a) = 1$, where our goal bound is now $$\lvert \mathbf{X}_{\tau,a}(\mathfrak{o}/\mathfrak{q}) \rvert \ll \frac{1}{Q^*} Q^n.$$ We now have that the image of $a$ does not lie in $\mathbf{H}(\mathfrak{o}/\mathfrak{p}) \mathbf{Z}(\mathfrak{o}/\mathfrak{p})$. We may thus apply Theorem [Theorem 35](#theorem:reta-sett-defin-refd-r-be-ring.-let-mathbfgr-y-mat-doubly-tangential){reference-type="ref" reference="theorem:reta-sett-defin-refd-r-be-ring.-let-mathbfgr-y-mat-doubly-tangential"} to see that for each $y \in \mathbf{X}_{\tau,a}(\mathfrak{o}/\mathfrak{p})$, at least one of the linear or quadratic Taylor coefficients for one of the defining polynomials for $\mathbf{X}_{\tau,a}$ is a unit. By part [\[enumerate:assume-that-each-y-in-mathc-one-line-or-quadr-tayl\]](#enumerate:assume-that-each-y-in-mathc-one-line-or-quadr-tayl){reference-type="eqref" reference="enumerate:assume-that-each-y-in-mathc-one-line-or-quadr-tayl"} of Lemma [Lemma 52](#lemma:let-d-geq-0.-supp-given-subs-mathc-subs-mathfr-p){reference-type="ref" reference="lemma:let-d-geq-0.-supp-given-subs-mathc-subs-mathfr-p"}, we deduce the required bound. ◻ **Remark 58**. For applications to subconvexity involving a *fixed* local field of characteristic zero, the "near-identity reduction" (see §[1.9.3](#sec:cj4t69an73){reference-type="ref" reference="sec:cj4t69an73"} and [@2020arXiv201202187N §15.6]) reduces our task, as far as volume bounds like in Theorem [Theorem 57](#theorem:volume-bound){reference-type="ref" reference="theorem:volume-bound"} are concerned, to the case that $a$ is close to the identity element. In that case, an adequate volume bound follows from [@2020arXiv201202187N Theorem 15.2]. The novelty of Theorem [Theorem 57](#theorem:volume-bound){reference-type="ref" reference="theorem:volume-bound"} relative to the cited result is its validity for all $a$ in the maximal compact subgroup, uniformly in the local field. The restriction to characteristic $\neq 2$ in Theorem [Theorem 57](#theorem:volume-bound){reference-type="ref" reference="theorem:volume-bound"} is thus harmless in applications. With additional work, this restriction could be eliminated in the case of a non-archimedean local field of characteristic zero and residue characteristic $2$, basically because the proof of Theorem [Theorem 30](#theorem:main-transversality-general-ring){reference-type="ref" reference="theorem:main-transversality-general-ring"} requires only a bounded number of divisions by $2$. There remains the case of a local field of characteristic $2$, where the conclusion of Theorem [Theorem 57](#theorem:volume-bound){reference-type="ref" reference="theorem:volume-bound"} fails for the reasons indicated in Remark [Remark 32](#remark:we-have-seen-comp-calc-with-grobn-bases-that-when-){reference-type="ref" reference="remark:we-have-seen-comp-calc-with-grobn-bases-that-when-"}. That case is not relevant for the subconvexity problem over number fields, but would be relevant for studying moments of $L$-functions in horizontal aspects over a function field of characteristic $2$. Estimating such moments thus presents an interesting challenge to which our results do not apply. # Bilinear forms estimates {#Sec:bilinear} We now apply the volume bound to estimate bilinear forms relevant for [\[eq:cj3tv4jpta\]](#eq:cj3tv4jpta){reference-type="eqref" reference="eq:cj3tv4jpta"}. Let $(F,\mathfrak{o},\mathfrak{p},\varpi,q)$ be a non-archimedean local field of odd residue characteristic. Let $\mathfrak{q}\subseteq \mathfrak{p}$ be a nonzero $\mathfrak{o}$-ideal. In this section, we abbreviate $$G := \mathbf{G}(\mathfrak{o}/\mathfrak{q}), \qquad H := \mathbf{H}(\mathfrak{o}/\mathfrak{q}),$$ etc. Write $Q := [\mathfrak{o}:\mathfrak{q}]$, and define $Q^*$ as in the statement of Theorem [Theorem 57](#theorem:volume-bound){reference-type="ref" reference="theorem:volume-bound"}. Recall from §[6.2](#sec:norms-distance-functions){reference-type="ref" reference="sec:norms-distance-functions"} the "distance from $H Z$" function $d_H$. **Theorem 59**. *Let $\tau \in M_{\mathop{\mathrm{stab}}}$. Let $u_1, u_2 : H \rightarrow \mathbb{R}_{\geq 0}$ be nonnegative functions that are right-invariant under $H_{\tau_H}$. Let $\gamma \in G$. Then the quantity $$\label{eqn:i-:=-frac1lv-h-rvert2-sum-_-subst-x-y-in-h-:-} I := \frac{1}{\lvert H \rvert^2} \sum _{ \substack{ x, y \in H : \\ x ^{-1} \gamma y \in G_\tau } } u_1(x) u_2(y)$$ satisfies the trivial bound $$\label{eqn:volume-bound-trivial} I \leq \frac{1}{|H|} \lVert u_1 \rVert_{L^2} \lVert u_2 \rVert_{L^2}$$ and, for $\gamma \notin H Z$, the refined bound $$\label{eqn:volume-bound-refined} I \leq \frac{C}{|H|} \left( \frac{1}{1 + Q d_H(\gamma)} + \frac{{d_H(\gamma)}^\infty }{Q^*} \right) \lVert u_1 \rVert_{L^2} \lVert u_2 \rVert_{L^2}.$$ Here $C$ depends at most upon $\mathop{\mathrm{rank}}(\mathbf{V})$, and may be taken to be the same constant as in Theorem [Theorem 57](#theorem:volume-bound){reference-type="ref" reference="theorem:volume-bound"}. The $L^2$-norms are defined using the invariant probability measures.* *Proof.* We closely follow the proof of [@2020arXiv201202187N Thm 15.1, Lem 15.3]. By Cauchy--Schwarz, we have $I \leq \sqrt{I_1 I_2}$, where $$I_1 :=\frac{1}{\lvert H \rvert^2} \sum _{ \substack{ x, y \in H : \\ x ^{-1} \gamma y \in G_{\tau} } } {u_1(x)}^2,$$ $$I_2 :=\frac{1}{\lvert H \rvert^2} \sum _{ \substack{ x, y \in H : \\ x ^{-1} \gamma y \in G_{\tau} } } {u_2(y)}^2.$$ The set $G_{\tau}$, the condition $\gamma \notin H Z$ and the quantity $d_H(\gamma)$ are invariant under inversion, so it will suffice to obtain a suitable estimate for $I_1$, as the same argument then applies to $I_2$. By definition, $$I_1 = \frac{1}{|H|^2} \sum _{x \in H} {u_1(x)}^2 \nu(x), \quad \nu(x) := \left\lvert \left\{ y \in H : x ^{-1} \gamma y \in G_{\tau} \right\} \right\rvert.$$ By Lemma [Lemma 28](#lemma:stable-implies-trivial-stabilizer){reference-type="ref" reference="lemma:stable-implies-trivial-stabilizer"}, we have $H \cap G_{\tau} = \{1\}$, and so $\nu(x) \in \{0, 1\}$. Thus $$I_1 \leq \frac{1}{|H|} I_1', \quad I_1' := \frac{1}{|H|} \sum _{ \substack{ x \in H : \\ \gamma ^{-1} x \in H G_{\tau} } } {u_1(x)}^2.$$ The trivial bound $I_1' \leq \lVert u_1 \rVert_{L^2}^2$, obtained by dropping the summation condition, yields the first required estimate [\[eqn:volume-bound-trivial\]](#eqn:volume-bound-trivial){reference-type="eqref" reference="eqn:volume-bound-trivial"}. For the second required estimate, we appeal to the right $H_{\tau_H}$-invariance of $u_1$ to write $$I_1' := \frac{1}{|H|} \sum _{ x \in H } {u_1(x)}^2 \mu(x), \quad \mu(x) := \frac{ \left\lvert \left\{ u \in H_{\tau_H} : \gamma ^{-1} x u \in H G_{\tau} \right\} \right\rvert }{ \left\lvert H_{\tau_H} \right\rvert }.$$ If $\gamma \notin H Z$, then we have $\gamma ^{-1} x \notin H Z$ and $d_H(\gamma^{-1} x) = d_H(\gamma)$ for all $x \in H$. Estimating $\mu(x)$ via Theorem [Theorem 57](#theorem:volume-bound){reference-type="ref" reference="theorem:volume-bound"} yields the second estimate [\[eqn:volume-bound-refined\]](#eqn:volume-bound-refined){reference-type="eqref" reference="eqn:volume-bound-refined"}. ◻ # Construction of test vectors {#sec:part-body-paper} The primary aim of this section is to make Theorem [Theorem 1](#theorem:cj3ngw7u2s){reference-type="ref" reference="theorem:cj3ngw7u2s"} precise by specifying the local conditions required at the distinguished place (Remark [Remark 6](#remark:cj3u9047n5){reference-type="ref" reference="remark:cj3u9047n5"}). To that end, we define "stable" pairs of representations of general linear groups (Definition [Definition 65](#definition:let-pi-sigma-be-repr-pair-gener-line-groups-over-f){reference-type="ref" reference="definition:let-pi-sigma-be-repr-pair-gener-line-groups-over-f"}); informally, such representations contain "microlocalized" test vectors whose localization parameters have no matching eigenvalues. To apply this definition, we need to check when it holds. Since representation-theoretic issues are orthogonal to the main novelty of this paper, we are content here to check that our definition is preserved under parabolic induction (Example [Example 66](#example:cj3m0jsjik){reference-type="ref" reference="example:cj3m0jsjik"}) and to investigate fully the case of principal series (Example [Example 67](#example:20230516204714){reference-type="ref" reference="example:20230516204714"}), leaving open the case of general supercuspidals (see Example [Example 68](#example:cj3m0jzefl){reference-type="ref" reference="example:cj3m0jzefl"} for some special cases). We note that many of the ideas and results recorded in this section are well-known in the type theory literature, going back to work of Howe from the 1970s (see, e.g., [@MR0579176; @MR0327982] and [@MR2016587 §5]). The cited works would be most relevant to the special cases of our results where $\tau$ and $\tau_H$ are diagonalizable, a condition that corresponds to the "wall-avoidance" hypothesis [\[enumerate:20230516210021\]](#enumerate:20230516210021){reference-type="eqref" reference="enumerate:20230516210021"} mentioned in §[1.5](#sec:cj4vjryk3u){reference-type="ref" reference="sec:cj4vjryk3u"}. Since we do not wish to impose such hypotheses, we give short proofs of what we need here, without claiming particular novelty. ## Overview {#sec:20230516194628} In §[8.1](#sec:20230516194628){reference-type="ref" reference="sec:20230516194628"}, we summarize the main definitions and results of §[8](#sec:part-body-paper){reference-type="ref" reference="sec:part-body-paper"}; the remaining subsections are then devoted to the proofs of these results. Let $F$ be a non-archimedean local field, with ring of integers $\mathfrak{o}$ and maximal ideal $\mathfrak{p}$. Let $\mathfrak{q} \subseteq \mathfrak{p}$ be an $\mathfrak{o}$-ideal. Let $\psi$ be a unitary character of $\mathfrak{q}$ that is trivial on $\mathfrak{q}^2$, but not on $\mathfrak{p}^{-1}\mathfrak{q}^2$. For $\tau \in \mathbf{M}(\mathfrak{o}/\mathfrak{q})$, we denote by $\chi_\tau$ the character of the group $K(\mathfrak{q})/K(\mathfrak{q}^2)$ given by $$\label{eqn:definition-of-chi-tau-as-psi-of-trace-x-tau} \chi _\tau (1 + x) = \psi (\mathop{\mathrm{trace}}(x \tau )).$$ Every character of that group arises in this way. In what follows, *representation* always means "complex representation". We introduce some terminology concerning representations $\pi$ of $\mathbf{G}(\mathfrak{o})$. We will apply this terminology also to representations of $\mathbf{G}(F)$, through their restrictions. **Definition 60**. Let $\pi$ be a representation of $\mathbf{G}(\mathfrak{o})$. We say that $\pi$ is *regular at depth $\mathfrak{q}^2$* if there is a cyclic element $\tau \in \mathbf{M}(\mathfrak{o}/\mathfrak{q})$ (see §[3](#sec:cyclic-matrices){reference-type="ref" reference="sec:cyclic-matrices"}) such that $\pi$ contains a nonzero vector $v$ that transforms under $K (\mathfrak{q})$ via the character $\chi_\tau$, i.e., $$\label{eqn:g-v-=-chi_taug-v-quad-text-all--g-in-kmathfrakq.-} g v = \chi_\tau(g) v \quad \text{ for all } g \in K(\mathfrak{q}).$$ In that case, we refer to $\tau$ as a *regular parameter for* $\pi$ *at depth* $\mathfrak{q}^2$, to its characteristic polynomial $P \in (\mathfrak{o}/\mathfrak{q})[X]$ as a *polynomial for* $\pi$ *at depth* $\mathfrak{q}^2$, and to the dimension of the space of $v$ satisfying [\[eqn:g-v-=-chi_taug-v-quad-text-all\--g-in-kmathfrakq.-\]](#eqn:g-v-=-chi_taug-v-quad-text-all--g-in-kmathfrakq.-){reference-type="eqref" reference="eqn:g-v-=-chi_taug-v-quad-text-all--g-in-kmathfrakq.-"} as the *multiplicity* of $\tau$ or of $P$ in $\pi$. (These notions depend, of course, upon the choice of $\psi$.) **Example 61**. Let $\chi$ be a character of $\mathop{\mathrm{GL}}_1(\mathfrak{o}) = \mathfrak{o}^\times$, regarded also as a one-dimensional representation. We denote by $$c(\chi) \subseteq \mathfrak{o}$$ the largest ideal $\mathfrak{a}$ of $\mathfrak{o}$ for which $\chi$ has trivial restriction to $\mathfrak{o}^\times \cap (1 + \mathfrak{a})$. It is clear then that $$c(\chi) \supseteq \mathfrak{q}^2 \iff \text{$\chi$ is regular at depth $\mathfrak{q}^2$}.$$ **Example 62**. We verify in Proposition [Proposition 78](#proposition:regular-stable-closed-under-parabolic-induction){reference-type="ref" reference="proposition:regular-stable-closed-under-parabolic-induction"} that being regular at depth $\mathfrak{q}^2$ is closed under parabolic induction. That is to say, suppose given a partition $n = m_1 + \dotsb + m_k$, corresponding to a standard parabolic subgroup of $\mathop{\mathrm{GL}}_n(\mathfrak{o})$. For each $j \in \{1, \dotsc, k\}$, let $\pi_j$ be a representation of $\mathop{\mathrm{GL}}_{m_j}(\mathfrak{o})$ that is regular at depth $\mathfrak{q}^2$ with polynomial $P_j$. Then the same holds for the parabolic induction of $\pi_1 \otimes \dotsb \otimes \pi_k$ to $\mathop{\mathrm{GL}}_n(\mathfrak{o})$, with polynomial at depth $\mathfrak{q}^2$ given by $P=\prod\limits_{i} P_i$ (unique if the $P_i$ are). **Example 63**. By combining Examples [Example 61](#example:cj2i1fuxey){reference-type="ref" reference="example:cj2i1fuxey"} and [Example 62](#example:cj2i1fu0az){reference-type="ref" reference="example:cj2i1fu0az"}, we deduce that a principal series representation induced by characters $\chi_i$ with $c(\chi_i) \supseteq \mathfrak{q}^2$ is regular at depth $\mathfrak{q}^2$. Its unique polynomial at depth $\mathfrak{q}^2$ is the product of linear factors $\prod_{i} (X - \xi_i)$, where $\xi_i$ is characterized by the identity $\chi_i(1+y) = \psi(y \xi_i)$ for all $y \in \mathfrak{q}$. **Example 64**. We verify in §[8.8](#sec:supercuspidals){reference-type="ref" reference="sec:supercuspidals"} that certain supercuspidal representations are regular at depth $\mathfrak{q}^2$. We do not determine precisely which supercuspidal representations are regular at depth $\mathfrak{q}^2$, but it may be possible to do so using known classifications of the latter (see Remark [Remark 86](#Rem:supercuspidalclassification){reference-type="ref" reference="Rem:supercuspidalclassification"}). **Definition 65**. Let $m$ and $n$ be natural numbers. Let $\pi$ and $\sigma$ be representations of $\mathop{\mathrm{GL}}_{n}(\mathfrak{o})$ and $\mathop{\mathrm{GL}}_m(\mathfrak{o})$. We say that $(\pi,\sigma)$ is *stable at depth* $\mathfrak{q}^2$ if - both $\pi$ and $\sigma$ are regular at depth $\mathfrak{q}^2$, and - there are polynomials $P_\pi$ (resp. $P_\sigma$) for $\pi$ (resp. $\sigma$) at depth $\mathfrak{q}^2$ that generate the unit ideal of $(\mathfrak{o}/\mathfrak{q})[X]$. **Example 66**. Being stable at depth $\mathfrak{q}^2$ is closed under parabolic induction (Proposition [Proposition 80](#proposition:stable-pairs-closed-under-parabolic-induction){reference-type="ref" reference="proposition:stable-pairs-closed-under-parabolic-induction"}): if $\pi$ (resp. $\sigma$) is induced by some collection of representations $\pi_i$ (resp. $\sigma_j$) of $\mathop{\mathrm{GL}}_{n_i}(\mathfrak{o})$ (resp. $\mathop{\mathrm{GL}}_{m_j}(\mathfrak{o})$) such that each pair $(\pi_i,\sigma_j)$ is stable at depth $\mathfrak{q}^2$, then the same holds for $(\pi,\sigma)$. **Example 67**. If $\pi$ (resp. $\sigma$) is a principal series representation induced by characters $\chi_i$ (resp. $\eta_j$), then the pair $(\pi,\sigma)$ is stable at depth $\mathfrak{q}^2$ provided that for all relevant indices, $$\label{eqn:20230516200125} c(\chi_i) \supseteq \mathfrak{q}^2, \qquad c(\eta_j) \supseteq \mathfrak{q}^2, \qquad c (\chi_i / \eta_j) = \mathfrak{q}^2.$$ Assuming the first two conditions in [\[eqn:20230516200125\]](#eqn:20230516200125){reference-type="eqref" reference="eqn:20230516200125"}, the third condition is equivalent to $$\label{eqn:20230516200301} \prod_{i,j} [\mathfrak{o}:c(\chi_i/\eta_j)] = {[\mathfrak{o}:\mathfrak{q}^2]}^{n(n+1)}.$$ The left hand side may be understood as the conductor of the Rankin--Selberg convolution $\pi \times \sigma^\vee$, and the equality [\[eqn:20230516200301\]](#eqn:20230516200301){reference-type="eqref" reference="eqn:20230516200301"} may be understood as an analogue of the "no conductor dropping" or "uniform growth" conditions considered in [@2020arXiv201202187N] and [@2021arXiv210915230N]. **Example 68**. When $(\pi,\sigma)$ is a pair of regular supercuspidal representations of $(\mathop{\mathrm{GL}}_{m},\mathop{\mathrm{GL}}_{n})$ as discussed in Example [Example 64](#Example:supercuspidaloverview){reference-type="ref" reference="Example:supercuspidaloverview"} or §[8.8](#sec:supercuspidals){reference-type="ref" reference="sec:supercuspidals"}, it is automatically stable when $m\neq n$; when $m=n$, one needs to impose additional conditions similar to the case of principal series representations (see Lemma [Lemma 87](#Lem:supercuspidalstablepair){reference-type="ref" reference="Lem:supercuspidalstablepair"}). ## Characters {#sec:characters-in-construction-test-vectors} Let $\mathfrak{q}\subseteq \mathfrak{p}$ be an $\mathfrak{o}$-ideal. Let $\psi : \mathfrak{q}/ \mathfrak{q}^2 \rightarrow \mathbb{C}^\times$ be a character that is nontrivial on $\mathfrak{p} ^{-1} \mathfrak{q}^2 / \mathfrak{q}^2$. The character $\psi$ induces an identification $$\mathfrak{o} / \mathfrak{q}\xrightarrow{\cong } \left\{ \text{characters } \mathfrak{q}/ \mathfrak{q}^2 \rightarrow \mathbb{C} ^\times \right\},$$ $$\xi \mapsto \psi(\xi \bullet).$$ For a character $\chi$ of $1 + \mathfrak{q}$ that is trivial on $1 + \mathfrak{q}^2$, we denote by $\xi_\chi \in \mathfrak{o}/\mathfrak{q}$ the unique element with $$\chi(1 + x) = \psi(\xi_\chi x) \quad \text{ for } x \in \mathfrak{q}.$$ ## Centralizers of characters {#sec:centralizers-characters-J-tau} Given $\tau \in \mathbf{M}(\mathfrak{o}/\mathfrak{q})$, we denote by $$J_\tau \leq \mathbf{G}(\mathfrak{o})$$ the inverse image of the mod-$\mathfrak{q}$ centralizer $\mathbf{G}_\tau(\mathfrak{o}/\mathfrak{q})$. It is a subgroup that contains $K(\mathfrak{q})$. ## Preliminary lemmas regarding regular parameters {#sec:regular-uniform-representations} **Lemma 69**. Let $P$ be a polynomial for $\pi$ at depth $\mathfrak{q}^2$. Then any cyclic element $\tau \in \mathbf{M}(\mathfrak{o}/\mathfrak{q})$ whose characteristic polynomial is $P$ is a regular parameter for $\pi$ at depth $\mathfrak{q}^2$. *Proof.* We observe first that the set of regular parameters for $\pi$ at depth $\mathfrak{q}^2$ is invariant under conjugation by $\mathbf{G}(\mathfrak{o}/\mathfrak{q})$. The conclusion of the lemma then follows from the fact (Lemma [Lemma 22](#lemma:regular-elements-same-characteristic-polynomial-are-conjugate){reference-type="ref" reference="lemma:regular-elements-same-characteristic-polynomial-are-conjugate"}) that any two cyclic elements having the same characteristic polynomial are conjugate. ◻ **Lemma 70**. Let $\tau \in \mathbf{M}(\mathfrak{o}/\mathfrak{q})$ be cyclic. Then $\ker(\chi_\tau)$ is a normal subgroup of $J_\tau$, and the quotient group $J_\tau / \ker(\chi_\tau)$ is abelian. *Proof.* We verify first that the kernel is normal. Let $x \in J_\tau$ and $y \in \ker(\chi_\tau)$. Then $x y x^{-1}$ lies in $K(\mathfrak{q})$ (because $y$ lies in $K(\mathfrak{q})$, which is normal in $\mathbf{G}(\mathfrak{o})$) and also in $\ker(\chi_\tau)$ (because $x$ commutes with $\tau$ modulo $\mathfrak{q}$). We verify next that the quotient is abelian. Choose an arbitrary lift $\tilde{\tau} \in \mathbf{G}(\mathfrak{o})$ of $\tau$. Then, since $\mathbf{M}_\tau(\mathfrak{o}/\mathfrak{q})$ consists of polynomials in $\tau$ (Lemma [Lemma 21](#lemma:centralizer-description){reference-type="ref" reference="lemma:centralizer-description"}), we see that any element of $J_\tau$ may be written as a product $a b$, where $a$ is a polynomial in $\tilde{\tau}$ and $b$ lies in $K(\mathfrak{q})$. The images of such elements generate the quotient group $J_\tau / \ker(\chi_\tau)$, so to verify that the latter is abelian, we reduce to checking that any commutator $(a_1,a_2)$ or $(a_1,b_1)$ or $(b_1,b_2)$ lies in $\ker(\chi_\tau)$, where the $a_i$ and $b_j$ are as before. Indeed, we have $(a_1,a_2) = 1$ (because any two polynomials in $\tilde{\tau}$ commute with one another) and $\chi_\tau((a_1,b_1)) = 1$ (because $a_1$ commutes with $\tau$ modulo $\mathfrak{q}$) and $\chi_\tau((b_1,b_2)) = 1$ (because $\chi_\tau$ is a character of $K(\mathfrak{q})$). ◻ **Lemma 71**. Let $\tau \in \mathbf{M} (\mathfrak{o} / \mathfrak{q})$ be a regular parameter for $\pi$ at depth $\mathfrak{q}^2$. Then there is an extension of $\chi_\tau$ to a character $\tilde{\chi}_\tau$ of $J_\tau$ and a nonzero vector $v \in \pi$ that transforms under $J_\tau$ via $\tilde{\chi }_\tau$. If $\tau$ occurs with multiplicity one, then $v$ is unique up to scalar multiple. *Proof.* Let $v_0$ be a nonzero vector in $\pi$ that transforms under $K(\mathfrak{q})$ via $\chi_\tau$. Let $V$ denote the span of $v_0$ under $J_\tau$. This is a vector space of dimension at most $[J_\tau:K(\mathfrak{q})]$, on which $K(\mathfrak{q}^2)$ acts trivially. By Lemma [Lemma 70](#lemma:cj3m0fjios){reference-type="ref" reference="lemma:cj3m0fjios"}, the action of $J_\tau$ on $V$ factors through that of the quotient group $J_\tau / \ker(\chi_\tau)$. The latter is a finite abelian group, so its action on $V$ decomposes as a direct sum of one-dimensional subspaces. Take $v$ to lie in one of these subspaces, and $\tilde{\chi}_\tau$ to be the character describing the eigenvalues for $J_\tau$ acting on $v$. The final assertion is clear. ◻ **Example 72**. Let $\tau \in \mathbf{M}(\mathfrak{o}/\mathfrak{q})$ be cyclic. It defines a character $\chi_\tau$ of the subgroup $K(\mathfrak{q})$ of $J_\tau$. Suppose given an extension $\tilde{\chi}_\tau$ of $\chi_\tau$ to a character of $J_\tau$. Let $\pi$ denote the representation of $\mathbf{G}(\mathfrak{o})$ induced from $\tilde{\chi}_\tau$. Then $\pi$ is regular at depth $\mathfrak{q}^2$, with regular parameter $\tau$. These elementary observations imply that certain supercuspidal representations satisfy the above definitions (specifically, those of "odd depth" and "generic inducing data" --- see §[8.8](#sec:supercuspidals){reference-type="ref" reference="sec:supercuspidals"}). ## Some linear algebra {#sec:cj3m0c59nb} We record some general lemmas to be applied in the following subsection. Here we focus on an individual ring $R$ and abbreviate $V := \mathbf{V}(R)$, $M := \mathbf{M}(R)$, etc. **Lemma 73**. Let $\tau \in M$. Then $V$ is $\tau$-cyclic if and only if $V^*$ is $\tau$-cyclic. More precisely, if the vector $v \in V$ is $\tau$-cyclic, then functional $v^* \in V^*$ given by $\sum c _j \tau ^j v \mapsto c_{n-1}$ is $\tau$-cyclic. *Proof.* This is the observation that the transpose of a matrix like [\[eqn:beginpmatrix-0\--0\--ast-\]](#eqn:beginpmatrix-0--0--ast-){reference-type="eqref" reference="eqn:beginpmatrix-0--0--ast-"} is cyclic; we leave it to the reader. ◻ In what follows, by a *flag* $V_0 \subset \dotsb \subset V_k$ in $V$, we mean a sequence of free submodules, with $V_0 = \{0\}$ and $V_k = V$, such that each quotient $V_j / V_{j-1}$ is also free. We say that $\tau \in M$ preserves this flag if $\tau V_j \subseteq V_j$ for each $j \in \{1, \dotsc, k\}$. **Lemma 74**. Let $\tau \in M$ be cyclic. Let $V_0 \subset \dotsb \subset V_k$ be a flag. Suppose that $\tau$ preserves the flag, so that $\tau$ induces endomorphisms $\tau_j \in \mathop{\mathrm{End}}(V_j/ V_{j-1})$. Then each $\tau_j$ is cyclic. *Proof.* By induction, we may reduce to the case $k = 2$. It is clear that $\tau_2$ is cyclic, because the image in $V/V_1$ of any cyclic vector for $\tau$ is a cyclic vector for $\tau_2$. It remains to show that $\tau_1$ is cyclic. By Lemma [Lemma 73](#lemma:v-tau-cycl-if-only-if-v-tau-cycl-more-prec-if-vect){reference-type="ref" reference="lemma:v-tau-cycl-if-only-if-v-tau-cycl-more-prec-if-vect"}, we know that $\tau^* \in \mathop{\mathrm{End}}(V^*)$ is cyclic, and it suffices to show that $\tau_1^* \in \mathop{\mathrm{End}}(V_1^*)$ is cyclic. By identifying $V_1^*$ with the quotient $V^* / V_1^\perp$, we reduce to the previous observation. ◻ **Lemma 75**. If $\tau$ is cyclic, then its characteristic polynomial $P_\tau$ generates the annihilator ideal $\{f \in R[X] : f(\tau) = 0\}$. *Proof.* Otherwise, let $f$ be polynomial of minimal degree that is not divisible by $P_\tau$ and yet for which $f(\tau) = 0$. If the degree of $f$ is less than the rank of $\mathbf{V}$, then we obtain a contradiction from Lemma [Lemma 20](#lemma:v-tau-cycl-if-only-if-map-modul-begin-rn-right-v-){reference-type="ref" reference="lemma:v-tau-cycl-if-only-if-map-modul-begin-rn-right-v-"}. Otherwise, we can apply division with remainder using the monic polynomial $P_\tau$ to contradict the minimality of $f$. ◻ **Lemma 76**. Let $V_1 \subset \dotsb \subset V_k$ be a flag, and let $\tau_j \in \mathop{\mathrm{End}}(V_j / V_{j-1})$ be cyclic elements. Then there is a cyclic element $\tau \in \mathop{\mathrm{End}}(V)$ that preserves the flag and for which the induced action on $V_j / V_{j-1}$ is $\tau_j$. *Proof.* By induction, we may reduce to the case $k = 2$ and $V_k = V$. We may find a basis $e_1,\dotsc,e_m$ for $V_1$ with respect to which $\tau_1$ is cyclic and a basis $\bar{f}_1,\dotsc,\bar{f}_n$ for $V/V_1$ with respect to which $\tau_2$ is cyclic, i.e., $$\tau_1 e_i = e_{i+1} \quad (i < m), \qquad \tau_2 \bar{f}_j = \bar{f}_{j+1} \quad (j < n).$$ By choosing a splitting $V = V_1 \oplus V_2$, we identify $\tau_2$ with an endomorphism of $V_2$, extended by zero to $V$. We then define $\tau \in \mathop{\mathrm{End}}(V)$ by taking $$\tau e_i = \tau_1 e_{i}, \qquad \tau f_j = \tau_2 f_j + \begin{cases} e_1 & \text{ if } j = n, \\ 0 & \text{ otherwise.} \end{cases}$$ We claim then that $f_1$ is a cyclic vector for $\tau$. Indeed, the set $\{f_1, \tau f_1, \dotsc, \tau^{n-1} f_1\} = \{f_1, f_2, \dotsc, f_{n}\}$ spans $V_2$, while the set $\{\tau^n f_1, \dotsc, \tau^{n+m-1} f_1\}$ is congruent modulo $V_2$ to the spanning set $\{e_1, \dotsc, e_m\}$ for $V_1$. ◻ **Lemma 77**. Let $V_1 \subset \dotsb \subset V_k$ be a flag. Let $\tau \in M$ be a cyclic element that preserves this flag. Suppose that $g \in G$ has the following properties: (i) $\mathop{\mathrm{Ad}}(g) \tau$ preserves the flag. (ii) For each $j$, the induced actions of $\tau$ and $\mathop{\mathrm{Ad}}(g) \tau$ on $V_j / V_{j-1}$ have the same characteristic polynomial. Then $g$ preserves the flag. *Proof.* By induction, we may reduce to the case $k = 2$. Let $v \in V$ be $\tau$-cyclic. Then every element of $V$ may be written $f(\tau) v$ for some polynomial $f \in R[X]$, so it suffices to show that if $f(\tau) v \in V_1$, then $g f(\tau) v \in V_1$. Suppose, thus, that $f(\tau) v \in V_1$. Let $v_2 \in V/ V_1$ denote the image of $v$, and write $\tau_2 \in \mathop{\mathrm{End}}(V/V_1)$ for the induced action of $\tau$ on the quotient. Then $v_2$ is $\tau_2$-cyclic and $f(\tau_2) v_2 = 0$, hence $f(\tau_2) = 0$. It follows that $f$ is divisible by the characteristic polynomial of $\tau_2$ (Lemma [Lemma 75](#lemma:if-tau-cycl-then-char-polyn-p_tau-gener-annih-idea){reference-type="ref" reference="lemma:if-tau-cycl-then-char-polyn-p_tau-gener-annih-idea"}). Since the induced action of $\mathop{\mathrm{Ad}}(g) \tau$ on $V / V_1$ has the same characteristic polynomial as $\tau_2$, it follows from the Cayley--Hamilton theorem that $f(\mathop{\mathrm{Ad}}(g) \tau) V \subseteq V_1$. But then $g f(\tau) v = f(\mathop{\mathrm{Ad}}(g) \tau) g v \in V_1$, as required. ◻ ## Parabolic induction {#sec:cj3m0c6a7b} The following proposition shows that the existence and uniqueness of localized vectors is preserved by induction. (The "uniqueness" assertions will not be used here, but might be useful for certain extensions of the work of this paper.) **Proposition 78**. *The class of representations of general linear groups over $\mathfrak{o}$ that are regular at depth $\mathfrak{q}^2$ is closed under parabolic induction. That is to say, writing $n = \mathop{\mathrm{rank}}(\mathbf{V})$, suppose given a partition $n = m_1 + \dotsb + m_k$, corresponding to a parabolic subgroup $\mathbf{P}$ of $\mathbf{G}$, together with a collection of representations $\pi_j$ ($j = 1, \dotsc, k$) of $\mathop{\mathrm{GL}}_{m_j}(\mathfrak{o})$ each of which is regular at depth $\mathfrak{q}^2$. Then the same holds for the parabolic induction $$\pi := \mathop{\mathrm{Ind}}_{\mathbf{P}(\mathfrak{o})}^{\mathbf{G}(\mathfrak{o})}(\pi_1 \otimes \dotsb \otimes \pi_k).$$ Moreover:* (i) *Every polynomial for $\pi$ at depth $\mathfrak{q}^2$ is a product of polynomials for $\pi_1,\dotsc,\pi_k$ at depth $\mathfrak{q}^2$.* (ii) *For any polynomials $P_1,\dotsc,P_k$ for $\pi_1,\dotsc,\pi_k$ at depth $\mathfrak{q}^2$, the product $P_1 \dotsm P_k$ is a polynomial for $\pi$ at depth $\mathfrak{q}^2$.* (iii) *If $\pi_i$ admits a unique polynomial $P_i$ at depth $\mathfrak{q}^2$, and if $P_i$ occurs with multiplicity one, then $P_1 \dotsb P_k$ is likewise the unique polynomial for $\pi$ at depth $\mathfrak{q}^2$, and occurs with multiplicity one.* *Proof.* For $\tau \in \mathbf{M}(\mathfrak{o}/\mathfrak{q})$, let us temporarily write $\pi^{\tau}$ for the subspace of $\pi$ on which $K(\mathfrak{q})$ acts via $\chi_\tau$. Similarly, for an $m_i \times m_i$ matrix $\tau_i$ over $\mathfrak{o}/\mathfrak{q}$, write $\pi_i^{\tau_i}$ for the associated subspace of $\pi_i$. Let us take for $\mathbf{P}$ the standard upper-triangular parabolic subgroup associated to the given partition. Let $\mathbf{U} < \mathbf{P}$ denote the unipotent radical and $\mathbf{L}$ the block-diagonal Levi factor, thus $\mathbf{L} \cong \prod_{i=1}^k \mathbf{G}_i$, where each $\mathbf{G}_i$ is a general linear group of rank $m_i$. We may regard $$\Pi := \otimes_i \pi_i$$ as a representation of $\mathbf{L}(\mathfrak{o})$. We extend it to a representation of $\mathbf{P}(\mathfrak{o})$ by letting $\mathbf{U}(\mathfrak{o})$ act trivially, and realize $\pi$ as the space of functions $v : \mathbf{G}(\mathfrak{o}) \rightarrow \Pi$ such that for all $p \in \mathbf{P}(\mathfrak{o})$, $$v(p g) = \Pi(p) v(g),$$ with the action given by right translation. By Mackey theory, a basis for $\pi^\tau$ is indexed by representatives $g$ for the double quotient $$\label{eqn:mathbfp-mathfr-backsl-mathbfgm--kmathfr-cong-mathb} \mathbf{P} (\mathfrak{o}) \backslash \mathbf{G}(\mathfrak{o}) / K(\mathfrak{q}) \cong \mathbf{P}(\mathfrak{o}/\mathfrak{q}) \backslash \mathbf{G}(\mathfrak{o}/\mathfrak{q})$$ together with a basis for the space of vectors $t \in \Pi$ such that the formula $$v_g (p g h) = \begin{cases} \chi_\tau(h) \Pi(p) t & \text{ if } (p,h) \in \mathbf{P}(\mathfrak{o}) \times K(\mathfrak{q}), \\ 0 & \text{ otherwise} \end{cases}$$ is well-defined, that is, for which $p g h = g \implies \chi_\tau(h) \Pi(p) t = t$, or equivalently, $$\label{eqn:p-in-mathbfpm-times-g-kmathfr-g-1-impl-chi_-tau-p-} p \in \mathbf{P}(\mathfrak{o}) \cap g K(\mathfrak{q}) g^{-1} \implies \chi_{\mathop{\mathrm{Ad}}(g) \tau }(p) t = \chi_\tau(g^{-1} p g) t = \Pi(p) t.$$ Since $K(\mathfrak{q})$ is normal in $\mathbf{G}(\mathfrak{o})$, we have $$\mathbf{P}(\mathfrak{o}) \cap g K(\mathfrak{q}) g^{-1} = K_P(\mathfrak{q}) := \ker ( \mathbf{P}(\mathfrak{o}) \rightarrow \mathbf{P}(\mathfrak{o}/\mathfrak{q}) ).$$ Any element $p$ of the latter may be written uniquely as $p = m u$, where $(m, u)$ lies in $K_L(\mathfrak{q}) \times K_U(\mathfrak{q})$ (with each factor defined like $K_P(\mathfrak{q})$). Assuming that the vector $t$ is nonzero, the condition [\[eqn:p-in-mathbfpm-times-g-kmathfr-g-1-impl-chi\_-tau-p-\]](#eqn:p-in-mathbfpm-times-g-kmathfr-g-1-impl-chi_-tau-p-){reference-type="eqref" reference="eqn:p-in-mathbfpm-times-g-kmathfr-g-1-impl-chi_-tau-p-"} is then equivalent to the following pair of assertions. - $\chi_{\mathop{\mathrm{Ad}}(g) \tau}(u) = 1$ for all $u \in K_U(\mathfrak{q})$. Translating this assertion via the trace pairing, it says that the element $\mathop{\mathrm{Ad}}(g) \tau$ of $\mathbf{M}(\mathfrak{o}/\mathfrak{q})$ preserves the flag $\mathcal{F}$ in $\mathbf{M}(\mathfrak{o}/\mathfrak{q})$ of which $\mathbf{P}(\mathfrak{o}/\mathfrak{q})$ is the stabilizer. - $\Pi(m) t = \chi_{\mathop{\mathrm{Ad}}(g) \tau}(m) t$ for all $m \in K_L(\mathfrak{q})$. Writing ${(\mathop{\mathrm{Ad}}(g) \tau )}_{ii}$ for the $i$th block diagonal $m_i \times m_i$ matrix, this is equivalent to the condition $t \in \otimes_{i=1}^k \pi_i^{{(\mathop{\mathrm{Ad}}(g) \tau)}_{i i}}$. In summary, $$\label{eqn:pitau--=-oplus_-subst-g-in-mathbfpm-backsl-mathbfg} \pi^{\tau } = \oplus_{ \substack{ g \in \mathbf{P}(\mathfrak{o}) \backslash \mathbf{G}(\mathfrak{o}) / K(\mathfrak{q}) : \\ \mathop{\mathrm{Ad}}(g) \tau \text{ preserves } \mathcal{F} } } \otimes_{i=1}^k \pi_i^{{(\mathop{\mathrm{Ad}}(g) \tau)}_{i i}}.$$ The matrices ${(\mathop{\mathrm{Ad}}(g) \tau)}_{i i}$ are cyclic, as their images in $\mathbf{M}_{m_i}(\mathfrak{o}/\mathfrak{q})$ are cyclic by Lemma [Lemma 74](#lemma:let-v_0-be-free-tau-invar-subm-v.-assume-tau-regul){reference-type="ref" reference="lemma:let-v_0-be-free-tau-invar-subm-v.-assume-tau-regul"}. If the spaces $\pi_i^{{(\mathop{\mathrm{Ad}}(g) \tau )}_{i i}}$ are nonzero, then the characteristic polynomial of ${(\mathop{\mathrm{Ad}}(g) \tau )}_{i i}$ must be a polynomial for $\pi_j$ at depth $\mathfrak{q}^2$. Thus every polynomial for $\pi$ is a product of polynomials for the $\pi_j$ inside the ring $(\mathfrak{o}/\mathfrak{q})[X]$. Conversely, suppose given polynomials $P_1, \dotsc, P_k$ for $\pi_1,\dotsc,\pi_k$ at depth $\mathfrak{q}^2$. By Lemma [Lemma 76](#lemma:let-v_1-subs-dotsb-subs-v_k-be-flag-let-tau_j-in-e){reference-type="ref" reference="lemma:let-v_1-subs-dotsb-subs-v_k-be-flag-let-tau_j-in-e"}, there is a cyclic element $\tau$ which preserves the flag and for which each $\tau_{ii}$ has characteristic polynomial $P_i$. Then the characteristic polynomial of $\tau$ is $P_1 \dotsb P_k$. On the other hand, the contribution to [\[eqn:pitau\--=-oplus\_-subst-g-in-mathbfpm-backsl-mathbfg\]](#eqn:pitau--=-oplus_-subst-g-in-mathbfpm-backsl-mathbfg){reference-type="eqref" reference="eqn:pitau--=-oplus_-subst-g-in-mathbfpm-backsl-mathbfg"} from $g = 1$ is positive-dimensional, so $\pi^\tau$ is nonzero. Thus $P_1 \dotsb P_k$ is a polynomial for $\pi$. It remains to verify the final assertion concerning uniqueness and multiplicity one. Suppose that $\pi^\tau$ is nonzero for some cyclic $\tau \in M$. By [\[eqn:pitau\--=-oplus\_-subst-g-in-mathbfpm-backsl-mathbfg\]](#eqn:pitau--=-oplus_-subst-g-in-mathbfpm-backsl-mathbfg){reference-type="eqref" reference="eqn:pitau--=-oplus_-subst-g-in-mathbfpm-backsl-mathbfg"}, we may assume (after conjugating $\tau$ if necessary) that $\tau$ preserves $\mathcal{F}$ and that each $\pi_i^{\tau_{ii}}$ is nonzero. By hypothesis, the characteristic polynomial of $\tau_{ii}$ is $P_i$ and $\pi_i^{\tau_{ii}}$ is one-dimensional. Moreover, by Lemma [Lemma 77](#lemma:let-r-be-ring.-write-v-:=-mathbfvr-m-:=-mathbfmr-e){reference-type="ref" reference="lemma:let-r-be-ring.-write-v-:=-mathbfvr-m-:=-mathbfmr-e"} (applied over $R = \mathfrak{o} / \mathfrak{q}$), the only $g \in \mathbf{G}(\mathfrak{o})$ for which $\mathop{\mathrm{Ad}}(g) \tau$ preserves $\mathcal{F}$ and ${(\mathop{\mathrm{Ad}}(g) \tau)}_{ii}$ has characteristic polynomial $P_i$ lie in the trivial double coset. Thus [\[eqn:pitau\--=-oplus\_-subst-g-in-mathbfpm-backsl-mathbfg\]](#eqn:pitau--=-oplus_-subst-g-in-mathbfpm-backsl-mathbfg){reference-type="eqref" reference="eqn:pitau--=-oplus_-subst-g-in-mathbfpm-backsl-mathbfg"} reduces to the one-dimensional summand $\otimes_{i=1}^k \pi_i^{\tau_{ii}}$. This proves that $P_1 \dotsb P_k$ is the unique polynomial for $\pi$ and occurs with multiplicity one. ◻ ## Stable pairs of representations {#sec:cj3m0c6cif} Recall that in Definition [Definition 65](#definition:let-pi-sigma-be-repr-pair-gener-line-groups-over-f){reference-type="ref" reference="definition:let-pi-sigma-be-repr-pair-gener-line-groups-over-f"}, we defined when a pair of representations $(\pi,\sigma)$ is stable at depth $\mathfrak{q}^2$. **Example 79**. Let $\chi$ and $\eta$ be characters of $\mathfrak{o}^\times = \mathop{\mathrm{GL}}_1(\mathfrak{o})$. Then the pair $(\chi,\eta)$ is stable at depth $\mathfrak{q}^2$ precisely when $\chi$ and $\eta$ have conductor dividing $\mathfrak{q}^2$ and the ratio $\chi/\eta$ has conductor exactly $\mathfrak{q}^2$. In verifying this, we observe that $\chi(x) / \eta (x) = \psi (x (\xi_\chi - \xi_\eta ))$, which is nontrivial for some $x \in \mathfrak{p}^{-1} \mathfrak{q}^2$ precisely when $\xi_\chi - \xi_\eta$ is a unit, or equivalently, when the polynomials $X - \xi_\chi$ and $X - \xi_\eta$ generate the unit ideal in $(\mathfrak{o} / \mathfrak{q})[X]$. **Proposition 80**. *The class of pairs of representations $(\pi,\sigma)$ that are stable at depth $\mathfrak{q}^2$ is closed under parabolic induction, in the following sense. Suppose given a finite collection of representations $\pi_i$ and $\sigma_j$, where each pair $(\pi_i,\sigma_j)$ is stable at depth $\mathfrak{q}^2$. Let $\pi$ (resp. $\sigma$) denote the parabolic induction of the $\pi_i$ (resp. $\sigma_j$), as in Proposition [Proposition 78](#proposition:regular-stable-closed-under-parabolic-induction){reference-type="ref" reference="proposition:regular-stable-closed-under-parabolic-induction"}. Then the pair $(\pi,\sigma)$ is likewise stable at depth $\mathfrak{q}^2$.* *Proof.* This reduces to Proposition [Proposition 78](#proposition:regular-stable-closed-under-parabolic-induction){reference-type="ref" reference="proposition:regular-stable-closed-under-parabolic-induction"} and the following observation: given finite collections of polynomials $\{P_i\}$ and $\{Q_j\}$ in $(\mathfrak{o} / \mathfrak{q})[X]$ such that each pair $(P_i,Q_j)$ generates the unit ideal, the same holds for the pair $(P,Q)$ given by $P := \prod_i P_i$ and $Q := \prod_j Q_j$. (To see this, write $1 = a_i P_i + b_j Q_j$ and observe that $\prod_{i,j} (a_i P_i + b_j Q_j)$ lies in the ideal generated by $P$ and $Q$.) ◻ **Example 81**. Let $\pi$ (resp. $\sigma$) be a principal series representation induced by characters $\chi_i$ (resp. $\eta_j$), each of conductor dividing $\mathfrak{q}^2$. Suppose that each ratio $\chi_i / \eta_j$ has conductor $\mathfrak{q}^2$. Then $(\pi,\sigma)$ is stable at depth $\mathfrak{q}^2$. In verifying this, we may reduce via Proposition [Proposition 80](#proposition:stable-pairs-closed-under-parabolic-induction){reference-type="ref" reference="proposition:stable-pairs-closed-under-parabolic-induction"} to the case of a pair of characters $(\chi,\eta)$ discussed in Example [Example 79](#example:stable-pairs-of-characters){reference-type="ref" reference="example:stable-pairs-of-characters"}. ## Supercuspidals {#sec:supercuspidals} We verify that the above terminology applies to certain supercuspidal representations of $\mathbf{G}(F)$. We do not aim here for an exhaustive treatment, as representation-theoretic issues are not the focus of this paper. Write $n := \dim(V)$. Let $\tau \in \mathbf{M}(\mathfrak{o})$ be such that the $F$-subalgebra $$E := F[\tau] \subseteq \mathbf{M}(F)$$ is a field extension of $F$ having the largest possibly degree, namely $n$. Let $$\eta : E^\times \rightarrow \mathbb{C}^\times$$ be a character. We consider the following special case: **Assumption 82**. The element $\tau$ satisfies the following conditions: 1. [\[enumerate:d1bc2d5fbd54\]]{#enumerate:d1bc2d5fbd54 label="enumerate:d1bc2d5fbd54"} The image $\overline{\tau}$ of $\tau$ in $\mathbf{M}(\mathfrak{o}/\mathfrak{p})$ also generates a degree $n$ extension of the residue field. In particular, $\det(\tau)\in \mathfrak{o}^\times$. 2. For each $e \in E^\times \cap K (\mathfrak{q})$, we have the compatibility condition $$\eta(e) = \psi(\mathop{\mathrm{trace}}(\tau (e-1))).$$ Recall that $\psi : \mathfrak{q}/ \mathfrak{q}^2 \rightarrow \mathbb{C}^\times$ is a character nontrivial on $\mathfrak{p} ^{-1} \mathfrak{q}^2 / \mathfrak{q}^2$. As $\tau$ normalizes $K(\mathfrak{q})$, so does $E^\times$. The subgroup $$J_E := E^\times K(\mathfrak{q})$$ is well-defined. It admits the character $\chi : J_E \rightarrow \mathbb{C}^\times$ defined by the formula $$\label{Eq:chi} \chi(e g) := \eta(e) \psi(\mathop{\mathrm{trace}}(\tau (g-1))) \quad \text{ for } (e,g) \in E^\times \times K(\mathfrak{q}).$$ **Remark 83**. The assumption [\[enumerate:d1bc2d5fbd54\]](#enumerate:d1bc2d5fbd54){reference-type="eqref" reference="enumerate:d1bc2d5fbd54"} implies that $E$ is an unramified extension of $F$. It also implies that any nonzero element in $\mathbf{V}(\mathfrak{o}/\mathfrak{p})$ is cyclic with respect to the image $\overline{\tau}$ of $\tau$ in $\mathbf{M}(\mathfrak{o}/\mathfrak{p})$, as $\mathbf{V}(\mathfrak{o}/\mathfrak{p})$ is then a one-dimensional vector space over the field $(\mathfrak{o}/\mathfrak{p})[\overline{\tau}] \subseteq \mathbf{M}(\mathfrak{o}/\mathfrak{p})$. **Lemma 84**. The compact induction $$\pi :=\mathop{\mathrm{ind}}_{J_E}^{\mathbf{G}(F)} \chi$$ is an irreducible supercuspidal representation of $\mathbf{G}(F)$. *Proof.* For the notations as above, the pair $(J_E, \chi)$ is a maximal type in the sense of [@BK (6.2)]. Thus by [@BK (6.2.2)], the representation $\pi$ constructed above is a supercuspidal representation. ◻ **Lemma 85**. The representation $\pi$ constructed in Lemma [Lemma 84](#lemma:d1bfac67927b){reference-type="ref" reference="lemma:d1bfac67927b"} is regular at depth $\mathfrak{q}^2$, with regular parameter given by $\tau$ modulo $\mathfrak{q}$ (see §[8.1](#sec:20230516194628){reference-type="ref" reference="sec:20230516194628"} for definitions). *Proof.* By definition of compact induction, there exists an element $v\in \pi$ which identifies with $\chi$ on $J_E$ and vanishes elsewhere. In particular $J_E$ acts on $v$ by $\chi$, and so $K(\mathfrak{q})\subset J_E$ acts on $v$ by $\chi_\tau$ as in Definition [Definition 60](#definition:we-say-that-pi-emphr-at-depth-mathfr-if-there-cycl-regular-depth-parameter-polynomial){reference-type="ref" reference="definition:we-say-that-pi-emphr-at-depth-mathfr-if-there-cycl-regular-depth-parameter-polynomial"}. The element $\tau$ is also cyclic in the sense of Definition [Definition 15](#definition:cyclic){reference-type="ref" reference="definition:cyclic"}, by Remark [Remark 83](#Remark:assumption1){reference-type="ref" reference="Remark:assumption1"} above. Thus $\tau$ is a regular parameter for $\pi$ at depth $\mathfrak{q}^2$. ◻ **Remark 86**. We expect that one could verify that every supercuspidal representation with "odd integral depth" and "generic induction datum" arises via Lemma [Lemma 84](#lemma:d1bfac67927b){reference-type="ref" reference="lemma:d1bfac67927b"}. Here "depth" should be understood according to the conventions of [@moyUnrefinedMinimalKtypes1994 §5.1], while by "generic induction datum", we mean that the parameter $\beta$ associated to a simple type contained in $\pi$ (see [@BK (3.2.1)]) generates a field extension of $F$ of maximal degree, and that $\beta$ is minimal in the sense of [@BK (1.4.14)]. For the above construction, if $\mathfrak{q}= \mathfrak{p}^k$, then the "depth" is $2k - 1$, while we may take $\beta$ to be $\varpi^{- 2k + 1} \tau$. **Lemma 87**. Let $m,n$ be positive integers, and let $\pi, \sigma$ be regular supercuspidal representations of $\mathop{\mathrm{GL}}_{m},\mathop{\mathrm{GL}}_{n}$ at depth $\mathfrak{q}^2$, constructed above in terms of parameters $\tau_\pi$ and $\tau_\sigma$. We consider the following two cases: - Case $m\neq n$. - Case $m=n$. Then the degree $n$ inert field extensions generated by $\tau_\pi$ and $\tau_\sigma$ can be identified, uniquely up to the action of $\mathop{\mathrm{Gal}}(E/F) \cong \mathop{\mathrm{Gal}}(k_E/k_F)$. We suppose that the images of $\tau_\pi$ and $\tau_\sigma$ in the residue field $k_E$ of $E$ satisfy $$\overline{\tau}_\sigma-\overline{\tau}_\pi^\iota\neq 0$$ for all $\iota \in \mathop{\mathrm{Gal}}(k_E/k_F)$. Then, in either case, the pair $(\pi,\sigma)$ is stable at depth $\mathfrak{q}^2$. Furthermore, the local analytic conductor of the Rankin--Selberg pair $\pi\times \sigma$ is $$\label{eq:cj3u9x1ked} C (\pi\times \sigma)={[\mathfrak{o}:\mathfrak{q}]}^{2mn}.$$ Note that [\[eq:cj3u9x1ked\]](#eq:cj3u9x1ked){reference-type="eqref" reference="eq:cj3u9x1ked"} generalizes the conductor formula given in Example [Example 79](#example:stable-pairs-of-characters){reference-type="ref" reference="example:stable-pairs-of-characters"}, which concerns the special case of characters (i.e., $m = n = 1$). *Proof.* Let $P_\pi, P_\sigma$ be the polynomials for $\pi$ and $\sigma$ at depth $\mathfrak{q}^2$. By Assumption [Assumption 82](#Assumption:supercuspidal){reference-type="ref" reference="Assumption:supercuspidal"}, part [\[enumerate:d1bc2d5fbd54\]](#enumerate:d1bc2d5fbd54){reference-type="eqref" reference="enumerate:d1bc2d5fbd54"}, they are irreducible polynomials of degrees $m$ and $n$. When $m\neq n$, they necessarily generate the unit ideal. When $m=n$, the assumption on $\overline\tau_\pi,\overline\tau_\sigma$ implies that the images $\overline{P}_\pi, \overline{P}_\sigma$ of $P_\pi,P_\sigma$ mod $\mathfrak{p}$ satisfy $(\overline{P}_\pi, \overline{P}_\sigma)=1$. By Nakayama's lemma, we then have $(P_\pi,P_\sigma)=1$. By Definition [Definition 65](#definition:let-pi-sigma-be-repr-pair-gener-line-groups-over-f){reference-type="ref" reference="definition:let-pi-sigma-be-repr-pair-gener-line-groups-over-f"} and Lemma [Lemma 85](#Lem:supercuspidalregular){reference-type="ref" reference="Lem:supercuspidalregular"}, we deduce that the pair $(\pi,\sigma)$ is stable at depth $\mathfrak{q}^2$. The conductor formula follows readily from [@MR1606410 Theorem 6.5(ii)]. Indeed, observe first that $\pi,\sigma$ are "completely distinct" in the sense of [@MR1606410 §6.2], so the cited result is applicable. The ramification indices $e_i$ there are all $1$ in our case. The claimed conductor formula follows by writing $\mathfrak{q}=\mathfrak{p}^k$ and taking $m=2k-1$ in [@MR1606410 Theorem 6.5(ii)]. ◻ **Remark 88**. Note that the additional condition in the $m=n$ case is necessary, as the conductor would otherwise be smaller. It is the analogue of the condition we imposed for a pair of characters in Example [Example 79](#example:stable-pairs-of-characters){reference-type="ref" reference="example:stable-pairs-of-characters"}. # Main local result {#sec:mainlocal} In this section, we retain the general notation and conventions of §[2](#sec:notation){reference-type="ref" reference="sec:notation"}, and denote by $n+1$ the rank of $\mathbf{V}$. We use the notation $A \ll B$ to denote that $\lvert A \rvert \leq C \lvert B \rvert$, where $C$ depends at most upon $n$. We write $A \asymp B$ to denote that $A \ll B \ll A$. Let $(F,\mathfrak{o},\mathfrak{p},\varpi,q)$ be a non-archimedean local field, with associated data as in §[2.4](#sec:local-fields-congruence-subgroups){reference-type="ref" reference="sec:local-fields-congruence-subgroups"}. ## Statement of result The following may be understood as a non-archimedean analogue, uniform with respect to variation of the local field, of the archimedean result [@2020arXiv201202187N Thm 4.2]. It encapsulates the local results needed to prove Theorem [Theorem 1](#theorem:cj3ngw7u2s){reference-type="ref" reference="theorem:cj3ngw7u2s"}. **Theorem 89**. *Let $\mathfrak{q}\subseteq \mathfrak{p}$ be a nonzero $\mathfrak{o}$-ideal. Denote by $Q := [\mathfrak{o}:\mathfrak{q}]$ its absolute norm.* *Let $\pi$ and $\sigma$ be representations of $\mathbf{G}(F)$ and $\mathbf{H}(F)$ equipped with inner products $\langle , \rangle$ invariant by $K(\mathfrak{q})$ and $K_H(\mathfrak{q})$, respectively. Assume that the pair $(\pi,\sigma)$ is stable at depth $\mathfrak{q}^2$ (Definition [Definition 65](#definition:let-pi-sigma-be-repr-pair-gener-line-groups-over-f){reference-type="ref" reference="definition:let-pi-sigma-be-repr-pair-gener-line-groups-over-f"}). Then there exists* - *a unit vector $v \in \pi$,* - *a self-adjoint idempotent $\omega \in C_c^\infty(K)$ with $\pi(\omega) v = v$,* - *a compact open subgroup $J_H$ of $K_H$, and* - *a unit vector $u \in \sigma$ that is a $J_H$-eigenvector,* *with the following properties.* *Let $\pi|_{Z} : \mathbf{Z}(F) \rightarrow \mathop{\mathrm{U}}(1)$ denote the central character of $\pi$. Define $$\omega ^\sharp (g) := \int _{z \in \mathbf{Z}(F) } \pi|_{Z} (z) \omega (z g ) \, d z.$$* (i) *[\[enumerate:20230517151810\]]{#enumerate:20230517151810 label="enumerate:20230517151810"} We have $$\label{eqn:sum-_v-in-mathc-mathc-pif-v-otim-u-gg-q-n2-} \int _{h \in \mathbf{H}(F)} \langle h v, v \rangle \langle u, h u \rangle \, d h \asymp Q^{-n^2},$$ where the integrand is compactly-supported, hence converges absolutely.* (ii) *[\[enumerate:20230517151812\]]{#enumerate:20230517151812 label="enumerate:20230517151812"} We have $$\label{eqn:int-_mathbfhf-lvert-f-sharp-rvert-ll-q-n.-} \int _{\mathbf{H}(F)} \lvert \omega ^\sharp \rvert \ll Q ^{n}.$$* (iii) *[\[enumerate:20230517165012\]]{#enumerate:20230517165012 label="enumerate:20230517165012"} Let $\Psi_1, \Psi_2 : K_H \rightarrow \mathbb{C}$ be functions satisfying $$\lvert \Psi_j(g z) \rvert = \left\lvert \Psi_j(g) \right\rvert \text{ for all } (g,z) \in K_H \times J_H.$$ Let $$\gamma \in \mathbf{G}(F) - \mathbf{H}(F) \mathbf{Z}(F).$$ Then $$\label{eqn:int-_x-y-in-k_h-leftlv-psi_1x-psi_2y-omega-sharp-x} \int _{x, y \in K_H} \left\lvert \Psi_1(x) \Psi_2(y) \omega ^\sharp (x ^{-1} \gamma y) \right\rvert \ll Q^n \left( \frac{1}{1 + Q d_H(\gamma)} + \frac{{d_H(\gamma)}^{\infty}}{Q^*} \right) \lVert \Psi_1 \rVert_{L^2} \lVert \Psi_2 \rVert_{L^2}.$$ Here the notation $d_H(\gamma)$ and $d_H(\gamma)^\infty$ is as in §[6.2](#sec:norms-distance-functions){reference-type="ref" reference="sec:norms-distance-functions"}, while $Q^*$ is as in the statement of Theorem [Theorem 57](#theorem:volume-bound){reference-type="ref" reference="theorem:volume-bound"}.* The proof is given in §[9.4](#sec:proof-main-local-compact){reference-type="ref" reference="sec:proof-main-local-compact"}, following some preliminaries. ## Stability and matrix coefficients {#sec:20230517161735} Here we adapt some arguments from [@nelson-venkatesh-1 §19] to the non-archimedean setting, where they simplify considerably. **Lemma 90**. Let $\tau \in \mathbf{M}_{\mathop{\mathrm{stab}}}(\mathfrak{o}/\mathfrak{q})$. (i) [\[enumerate:cj3twlzoes\]]{#enumerate:cj3twlzoes label="enumerate:cj3twlzoes"} Let $\pi$ be a representation of $\mathbf{G}(\mathfrak{o})$, equipped with a $K(\mathfrak{q})$-invariant inner product $\langle , \rangle$, and let $v \in \pi$ be a vector that transforms under $K(\mathfrak{q})$ according to $\chi_\tau$. Then for $h \in \mathbf{H}(\mathfrak{o})$, we have $$\label{eqn:20230516004612} \langle h v, v \rangle = 0 \quad \text{unless } h \in K_H(\mathfrak{q}).$$ (ii) [\[enumerate:cj3twlzxgy\]]{#enumerate:cj3twlzxgy label="enumerate:cj3twlzxgy"} Let $\pi$ be a representation of $\mathbf{G}(F)$ satisfying the same hypotheses as above. Then [\[eqn:20230516004612\]](#eqn:20230516004612){reference-type="eqref" reference="eqn:20230516004612"} holds for $h \in \mathbf{H}(F)$. *Proof.* The group $$\label{eqn:kmathfrakq-cap-h-kmathfrakq-h-1-subseteq-k_g-intersection-where-u-goes} K(\mathfrak{q}) \cap h K(\mathfrak{q}) h^{-1} \subseteq K$$ acts on $h v$ by the character $\chi_\tau(h^{-1} \bullet h)$ and on $v$ by $\chi_\tau$, so by orthogonality of characters, it suffices to show that whenever $h \notin K_H(\mathfrak{q})$, there exists $u$ in the intersection [\[eqn:kmathfrakq-cap-h-kmathfrakq-h-1-subseteq-k_g-intersection-where-u-goes\]](#eqn:kmathfrakq-cap-h-kmathfrakq-h-1-subseteq-k_g-intersection-where-u-goes){reference-type="eqref" reference="eqn:kmathfrakq-cap-h-kmathfrakq-h-1-subseteq-k_g-intersection-where-u-goes"} such that $$\chi_\tau (h ^{-1} u h) \neq \chi _\tau (u).$$ We consider two cases. (i) If $h \in \mathbf{H}(\mathfrak{o})$, then $\chi_\tau(h ^{-1} u h) = \chi_{\mathop{\mathrm{Ad}}(h) \tau}(u)$, and $K(\mathfrak{q}) = h K(\mathfrak{q}) h^{-1}$. For this reason, it suffices to show that $\mathop{\mathrm{Ad}}(h) \tau \equiv \tau \mod\mathfrak{q}$ if and only if $h \in K_H(\mathfrak{q})$. This is the content of Lemma [Lemma 28](#lemma:stable-implies-trivial-stabilizer){reference-type="ref" reference="lemma:stable-implies-trivial-stabilizer"}. (ii) The case $h \in \mathbf{H}(F) - \mathbf{H}(\mathfrak{o})$ is addressed by Lemma [Lemma 91](#lemma:let-tau-in-barm_st-h-in-mathbfhf-mathbfhm-then-the){reference-type="ref" reference="lemma:let-tau-in-barm_st-h-in-mathbfhf-mathbfhm-then-the"}, below.  ◻ **Lemma 91**. Let $\tau \in \mathbf{M}_{\mathop{\mathrm{stab}}}(\mathfrak{o}/\mathfrak{q})$ and $h \in \mathbf{H}(F) - \mathbf{H}(\mathfrak{o})$. Then there exists $u \in K(\mathfrak{q}) \cap h K(\mathfrak{q}) h^{-1}$ such that $$\label{eqn:k1-cap-h-k1-h-1-} \chi_\tau(h ^{-1} u h) \neq \chi _\tau (u).$$ *Proof.* The proof is an adaptation of [@nelson-venkatesh-1 Lemma 19.7]. We will show more precisely that there exists $$u \in K(\mathfrak{q}^2) \cap h K(\mathfrak{q}) h ^{-1}$$ such that $\chi _\tau (h ^{-1} u h) \neq 1$. This suffices in view of the fact that $\chi_\tau(u) = 1$. We choose a basis $e_1,\dotsc,e_n$ for $\mathbf{V}_H(\mathfrak{o})$. We denote by $A_H$ the subgroup of $\mathbf{H}(F)$ diagonalized by this basis. Let $A_H^+$ denote the subgroup consisting of diagonal matrices whose entries are integral powers of the uniformizer $\varpi$ of $\mathfrak{p}$. We first reduce to the special case that $h \in A_H^+ - K_H$. To that end, we apply the Cartan decomposition to write $$h = k_1 a k_2,$$ where $k_1, k_2 \in K_H$ and $a \in A_H^+ - K_H$. Then $$\chi_\tau(h^{-1} u h) = \chi_{\mathop{\mathrm{Ad}}(k_2) \tau}(a^{-1} k_1^{-1} u k_1 a),$$ $$K(\mathfrak{q}^2) \cap h K(\mathfrak{q}) h^{-1} = K(\mathfrak{q}^2) \cap k_1 a K(\mathfrak{q}) a^{-1}k_1^{-1}.$$ We observe that $\mathop{\mathrm{Ad}}(k_2) \tau$ satisfies the same hypotheses as $\tau$. Suppose we can find some $u \in K(\mathfrak{q}^2) \cap a K(\mathfrak{q}) a^{-1}$ so that $$\chi_{\mathop{\mathrm{Ad}}(k_2) \tau}(a^{-1} u a) \neq 1.$$ Then, setting $$u' := k_1 u k_1^{-1} \in K(\mathfrak{q}^2) \cap h K(\mathfrak{q}) h^{-1},$$ we obtain $$\chi_\tau(h^{-1} u' h) = \chi_{\mathop{\mathrm{Ad}}(k_2) \tau}(a^{-1} u a ) \neq 1,$$ as required. Thus, let $a \in A_H^+ - K_H$. Consider the adjoint action of $a$ on $\mathbf{M}(F)$. Since $a \notin K_H$, there are nontrivial weights for this action. Suppose for instance that there are positive weights (an identical argument will apply if there are negative weights). By conjugating $a$ by a permutation matrix (as we may, by the preceding argument), we may assume that the largest diagonal coordinate of $a$, say $\varpi^{-\ell}$ with $\ell \geq 1$, appears in components $1, \dotsc, m$, where $m \geq 1$. If we use the partition $$n+1 = m + (n -m) + 1$$ to describe $\mathbf{M}(F)$ as a space of $3 \times 3$ block matrices, then we may describe the weight spaces for $a$ as follows: $$\begin{pmatrix} 0 & + & + \\ - & \ast & \ast \\ - & \ast & \ast \\ \end{pmatrix}$$ Here the symbol $0$ indicates where $a$ acts trivially, while $+$ (resp. $-$) describe $\mathfrak{o}$-submodules $$\mathbf{M}_{\pm} \subseteq \mathbf{M}$$ such that for $x \in \mathbf{M}_{\pm}(F)$, we have $$a^{-1} x a = \varpi^{\pm \ell } x;$$ asterisks denote some unspecified combination of trivial, positive and negative weights. We obtain in particular subspaces $$\bar{M}_{\pm } := \mathbf{M}_{\pm}(\mathfrak{o}/\mathfrak{q}) \subseteq \bar{M} = \mathbf{M}(\mathfrak{o}/\mathfrak{q}).$$ Let $\tau_+ \in \bar{M}_+$ denote the component of $\tau \in \bar{M}$. We claim that $\tau_+$ is nonzero modulo $\mathfrak{p}$, i.e., that $\tau$ modulo $\mathfrak{p}$ is not of the form $$\begin{pmatrix} \ast & 0 & 0 \\ \ast & \ast & \ast \\ \ast & \ast & \ast \\ \end{pmatrix}.$$ Indeed, if it were, then $\tau$ modulo $\mathfrak{p}$ would stabilize an $m$-dimensional subspace of $\bar{V}_H^*$ modulo $\mathfrak{p}$. This contradicts the characterization [\[enumerate:there-are-no-nontr-tau-invar-subsp-v_h-or-v_h.-\]](#enumerate:there-are-no-nontr-tau-invar-subsp-v_h-or-v_h.-){reference-type="eqref" reference="enumerate:there-are-no-nontr-tau-invar-subsp-v_h-or-v_h.-"} of stability recorded in Lemma [Lemma 27](#lemma:stability-equivalences){reference-type="ref" reference="lemma:stability-equivalences"}, noting that $\tau$ modulo $\mathfrak{p}$ is likewise stable (Example [Example 25](#example:cj3twmtcpz){reference-type="ref" reference="example:cj3twmtcpz"}). The trace pairing puts $M_+$ and $M_-$ in duality, so that for each $x \in M_-$, we have $$\label{eq:cj3twm8wr8} \mathop{\mathrm{trace}}(x \tau) = \mathop{\mathrm{trace}}(x \tau_+).$$ Since $\tau_+$ is nonzero modulo $\mathfrak{p}$, we may find such an $x$ for which $$\label{eqn:tracex-tau-in-o-minus-p} \mathop{\mathrm{trace}}(x \tau_+) \in \mathfrak{o} - \mathfrak{p}.$$ Let $t \in \mathfrak{p} ^{-1} \mathfrak{q}^2 - \mathfrak{q}^2$, to be determined later, and take $$u := 1 + \varpi^{\ell} t x.$$ We note that, since $\ell \geq 1$, we have $$\label{eqn:u-in-kmathfrakq2} u \in K(\mathfrak{q}^2).$$ On the other hand, $$\label{eqn:a-1-u} a^{-1} u a = 1 + t x \in K(\mathfrak{p}^{-1} \mathfrak{q}^2) \subseteq K(\mathfrak{q}),$$ while $$\chi_\tau(a^{-1} u a) = \chi_\tau(1 + t x) = \psi(\mathop{\mathrm{trace}}(t x \tau )).$$ By [\[eq:cj3twm8wr8\]](#eq:cj3twm8wr8){reference-type="eqref" reference="eq:cj3twm8wr8"} and [\[eqn:tracex-tau-in-o-minus-p\]](#eqn:tracex-tau-in-o-minus-p){reference-type="eqref" reference="eqn:tracex-tau-in-o-minus-p"}, we have $t \mathop{\mathrm{trace}}( x \tau) \in \mathfrak{p}^{-1} \mathfrak{q}^2 - \mathfrak{q}^2$. By choosing $t$ suitably, we may thus arrange that $$\label{eqn:chi_taua-1-u} \chi_\tau(a^{-1} u a) = \psi(t \mathop{\mathrm{trace}}(x \tau)) \neq 1.$$ The required conclusion is then immediate from [\[eqn:u-in-kmathfrakq2\]](#eqn:u-in-kmathfrakq2){reference-type="eqref" reference="eqn:u-in-kmathfrakq2"}, [\[eqn:a-1-u\]](#eqn:a-1-u){reference-type="eqref" reference="eqn:a-1-u"} and [\[eqn:chi_taua-1-u\]](#eqn:chi_taua-1-u){reference-type="eqref" reference="eqn:chi_taua-1-u"}. ◻ **Lemma 92**. Let $\pi$ and $\sigma$ be representations of $\mathbf{G}(\mathfrak{o})$ and $\mathbf{H}(\mathfrak{o})$, respectively, such that the pair $(\pi,\sigma)$ is stable at depth $\mathfrak{q}^2$. (i) There is a stable element $\tau \in \mathbf{M}(\mathfrak{o}/\mathfrak{q})$ such that $\tau$ (resp. $\tau_H$) is a regular parameter for $\pi$ (resp. $\sigma$) at depth $\mathfrak{q}^2$. (ii) [\[enumerate:cj3twlwnkr\]]{#enumerate:cj3twlwnkr label="enumerate:cj3twlwnkr"} Suppose $\pi$ (resp. $\sigma$) is equipped with an inner product $\langle , \rangle$ invariant by $K(\mathfrak{q})$ (resp. $K_H(\mathfrak{q})$). Let $\tau$ be any element as above. Let $v$ (resp. $u$) be unit vectors that transform under $K(\mathfrak{q})$ (resp. $K_H(\mathfrak{q})$) according to $\chi_\tau$ (resp. $\chi_{\tau_H}$). Then $$\label{eq:cj3twlxge3} \int_{h \in \mathbf{H}(\mathfrak{o})} \langle h v, v \rangle \langle u, h u \rangle \, d h = \mathop{\mathrm{vol}}(K_H(\mathfrak{q})).$$ (iii) [\[enumerate:cj3twly3hj\]]{#enumerate:cj3twly3hj label="enumerate:cj3twly3hj"} Retaining the hypotheses of [\[enumerate:cj3twlwnkr\]](#enumerate:cj3twlwnkr){reference-type="eqref" reference="enumerate:cj3twlwnkr"}, if $\pi$ and $\sigma$ arise as restrictions of representations of $\mathbf{G}(F)$ and $\mathbf{H}(F)$, then the formula [\[eq:cj3twlxge3\]](#eq:cj3twlxge3){reference-type="eqref" reference="eq:cj3twlxge3"} remains valid after extending the integral to $\mathbf{H}(F)$. *Proof.* Let $P_\pi$ and $P_\sigma$ be polynomials for $\pi$ and $\sigma$ at depth $\mathfrak{q}^2$ that generate the unit ideal. By Lemma [Lemma 26](#lemma:let-p-p_h-in-rx-be-monic-polyn-degr-n+1-n-resp-the){reference-type="ref" reference="lemma:let-p-p_h-in-rx-be-monic-polyn-degr-n+1-n-resp-the"}, there exists $\tau \in \mathbf{M}(\mathfrak{o}/\mathfrak{q})$ such that $P_{\tau} = P_{\pi}$ and $P_{\tau_H} = P_{\sigma}$. By Lemma [Lemma 27](#lemma:stability-equivalences){reference-type="ref" reference="lemma:stability-equivalences"}, $\tau$ is stable, and in particular, cyclic. By Lemma [Lemma 29](#lemma:tau-stable-implies-tauH-cyclic){reference-type="ref" reference="lemma:tau-stable-implies-tauH-cyclic"}, $\tau_H \in \mathbf{M}_H(\mathfrak{o}/\mathfrak{q})$ is cyclic. By Lemma [Lemma 69](#lemma:let-p-be-polyn-pi-at-depth-mathfr-let-tau-in-mathb){reference-type="ref" reference="lemma:let-p-be-polyn-pi-at-depth-mathfr-let-tau-in-mathb"}, it follows that $\tau$ (resp. $\tau_H$) is a regular parameter for $\pi$ (resp. $\sigma$) at depth $\mathfrak{q}^2$. For the second assertion [\[enumerate:cj3twlwnkr\]](#enumerate:cj3twlwnkr){reference-type="eqref" reference="enumerate:cj3twlwnkr"} concerning the matrix coefficient integral, we apply part [\[enumerate:cj3twlzoes\]](#enumerate:cj3twlzoes){reference-type="eqref" reference="enumerate:cj3twlzoes"} of Lemma [Lemma 90](#lemma:matrix-coefficients-and-stability-one-rep){reference-type="ref" reference="lemma:matrix-coefficients-and-stability-one-rep"} to truncate that integral to $h \in K_H(\mathfrak{q})$, where the integrand evaluates to $$\chi_\tau(h) {\chi_{\tau_H}(h)}^{-1} = 1.$$ For the final assertion [\[enumerate:cj3twly3hj\]](#enumerate:cj3twly3hj){reference-type="eqref" reference="enumerate:cj3twly3hj"}, we argue similarly using part [\[enumerate:cj3twlzxgy\]](#enumerate:cj3twlzxgy){reference-type="eqref" reference="enumerate:cj3twlzxgy"} of Lemma [Lemma 90](#lemma:matrix-coefficients-and-stability-one-rep){reference-type="ref" reference="lemma:matrix-coefficients-and-stability-one-rep"}. ◻ ## Setup for the proof {#sec:cj4t6ynizh} We retain the hypotheses of Theorem [Theorem 89](#theorem:main-local-result){reference-type="ref" reference="theorem:main-local-result"}. By Lemma [Lemma 92](#lemma:let-pi-sigma-be-pair-repr-mathbfgm-mathbfhm-test-vectors-stable-pairs){reference-type="ref" reference="lemma:let-pi-sigma-be-pair-repr-mathbfgm-mathbfhm-test-vectors-stable-pairs"}, we obtain a stable element $\tau \in \mathbf{M}(\mathfrak{o}/\mathfrak{q})$ such that $\tau$ (resp. $\tau_H$) is a regular parameter for $\pi$ (resp. $\sigma$) at depth $\mathfrak{q}^2$. By Lemma [Lemma 71](#lemma:let-tau-in-mathbfm-mathfr--mathfr-be-regul-param-p-extension-chi-tau-to-J-tau){reference-type="ref" reference="lemma:let-tau-in-mathbfm-mathfr--mathfr-be-regul-param-p-extension-chi-tau-to-J-tau"}, we may find unit vectors $v \in \pi$ and $u \in \sigma$ that transform under $J_\tau$ and $J_{\tau_H}$ by characters $\tilde{\chi}_\tau$ and $\tilde{\chi}_{\tau_H}$ that extend $\chi_\tau$ and $\chi_{\tau_H}$, respectively. We set $(J_G, J_H, \chi_G, \chi_H) := (J_\tau, J_{\tau_H}, \tilde{\chi}_\tau, \tilde{\chi}_{\tau_H})$ and $$\omega := {\mathop{\mathrm{vol}}(J_G)}^{-1} \chi_G^{-1} \in C_c^\infty(K),$$ so that $\pi(\omega)$ is a self-adjoint idempotent with $\pi(\omega) v = v$. We pause to clarify the shape of the function $\omega ^\sharp$. **Lemma 93**. For $g \in \mathbf{G}(F)$, we have $$\lvert \omega ^\sharp (g) \rvert \ll {\mathop{\mathrm{vol}}(J_G)}^{-1} 1 _{g \in \mathbf{Z}(F) J_G}.$$ *Proof.* By definition, $$\omega ^\sharp (g) = \int _{z \in \mathbf{Z}(F)} \pi|_{Z}(z) \omega(z g) \, d z.$$ The function $\omega$ is supported on $J_G$ and has $L^\infty$-norm ${\mathop{\mathrm{vol}}(J_G)}^{-1}$. It follows that if $g \notin \mathbf{Z}(F) J_G$, then $\omega ^\sharp (g)$ vanishes. We have $Z(\mathbf{F}) \cap J_G = K_Z$, so the set of $z$ for which $\omega(z g) \neq 0$ is a $K_Z$-coset, and so has volume $\ll 1$. We have assumed that $\pi$ is unitary, so $\lvert \pi|_{Z} (z) \rvert = 1$. The stated estimate follows from these observations and the triangle inequality. ◻ ## The proof {#sec:proof-main-local-compact} We now verify each numbered assertion in turn. (i) We see from Lemmas [Lemma 90](#lemma:matrix-coefficients-and-stability-one-rep){reference-type="ref" reference="lemma:matrix-coefficients-and-stability-one-rep"} and [Lemma 92](#lemma:let-pi-sigma-be-pair-repr-mathbfgm-mathbfhm-test-vectors-stable-pairs){reference-type="ref" reference="lemma:let-pi-sigma-be-pair-repr-mathbfgm-mathbfhm-test-vectors-stable-pairs"} that the integrand in [\[eqn:sum-\_v-in-mathc-mathc-pif-v-otim-u-gg-q-n2-\]](#eqn:sum-_v-in-mathc-mathc-pif-v-otim-u-gg-q-n2-){reference-type="eqref" reference="eqn:sum-_v-in-mathc-mathc-pif-v-otim-u-gg-q-n2-"} is supported on $h \in K_H(\mathfrak{q})$ and the integral evaluates to $\mathop{\mathrm{vol}}(K_H(\mathfrak{q})) \asymp Q^{-n^2}$, as required. (ii) By Lemma [Lemma 93](#lemma:g-in-mathbfgf-we-have-begin-lvert-f-sharp-g-rvert-clarify-f-sharp){reference-type="ref" reference="lemma:g-in-mathbfgf-we-have-begin-lvert-f-sharp-g-rvert-clarify-f-sharp"}, we have $$\label{eqn:int-_mathbfhf-lvert-f-sharp-rvert-ll-frac1v-int-_h} \int _{\mathbf{H}(F)} \lvert \omega ^\sharp \rvert \ll \frac{1}{\mathop{\mathrm{vol}}(J_G)} \int _{h \in \mathbf{H}(F)} 1 _{\mathbf{Z}(F) J_G }(h) \, d h.$$ We must verify that the above is $\ll Q^n$. Suppose $h \in \mathbf{H}(F)$ may be written $h = z g$ with $(z, g ) \in \mathbf{Z} (F) \times J _G$. Then $$z^{-1} h \in J_G \cap \mathbf{H}(F) \mathbf{Z}(F) = J_G \cap K_H K_Z = (J_G \cap K_H)K_Z,$$ since $K_Z \subseteq J_G$. By Lemma [Lemma 28](#lemma:stable-implies-trivial-stabilizer){reference-type="ref" reference="lemma:stable-implies-trivial-stabilizer"}, we have $$J_G \cap K_H = K_H(\mathfrak{q}).$$ We have $$\label{eqn:volj_g-=-k_g-:-j_g-1-asymp-qn+12-n+1-=-qnn+1.-vol-J-G} {\mathop{\mathrm{vol}}(J_G)}^{-1} \mathop{\mathrm{vol}}(K_H(\mathfrak{q})) = \frac{[K : J_G]}{[K_H : K_H(\mathfrak{q})]} \asymp \frac{Q^{{(n+1)}^2 - (n+1)}}{Q^{n^2}} = Q^{n}.$$ The required estimate follows. (iii) We are given $\Psi_1, \Psi_2 : K_H \rightarrow \mathbb{C}$ whose magnitudes are right-invariant by $J_{\tau_H}$, and $\gamma \in \mathbf{G}(F) - \mathbf{H}(F) \mathbf{Z}(F)$. We must verify that $$\label{eqn:i_1-:=-int-_x-y-in-k_h-leftlv-psi_1x-psi_2y-f-shar-task} I_1 := \int _{x, y \in K_H} \left\lvert \Psi_1(x) \Psi_2(y) \omega ^\sharp (x ^{-1} \gamma y) \right\rvert \ll \Delta Q^n \lVert \Psi_1 \rVert_{L^2} \lVert \Psi_2 \rVert_{L^2},$$ where $\Delta$ is the parenthetical quantity on the right hand side of [\[eqn:int-\_x-y-in-k_h-leftlv-psi_1x-psi_2y-omega-sharp-x\]](#eqn:int-_x-y-in-k_h-leftlv-psi_1x-psi_2y-omega-sharp-x){reference-type="eqref" reference="eqn:int-_x-y-in-k_h-leftlv-psi_1x-psi_2y-omega-sharp-x"}. By the bound for $\omega ^\sharp$ recorded in Lemma [Lemma 93](#lemma:g-in-mathbfgf-we-have-begin-lvert-f-sharp-g-rvert-clarify-f-sharp){reference-type="ref" reference="lemma:g-in-mathbfgf-we-have-begin-lvert-f-sharp-g-rvert-clarify-f-sharp"}, we see that the left hand side of [\[eqn:i_1-:=-int-\_x-y-in-k_h-leftlv-psi_1x-psi_2y-f-shar-task\]](#eqn:i_1-:=-int-_x-y-in-k_h-leftlv-psi_1x-psi_2y-f-shar-task){reference-type="eqref" reference="eqn:i_1-:=-int-_x-y-in-k_h-leftlv-psi_1x-psi_2y-f-shar-task"} vanishes unless there exists $z \in \mathbf{Z}(F)$ so that $$z \gamma \in K_H J_G K_H \subseteq K.$$ Replacing $\gamma$ by $z^{-1} \gamma$ has no effect on either side of the desired estimate, so we may suppose that $\gamma \in K$. Then, since $x,y \in K_H \subseteq K$ and $\mathbf{Z}(F) J_G \cap K = J_G$, we have $$\lvert \omega ^\sharp (x ^{-1} \gamma y) \rvert \ll {\mathop{\mathrm{vol}}(J_G)}^{-1} 1 _{x ^{-1} \gamma y \in J_G}.$$ The integral $I_1$ descends to the quotient $$\bar{H} := K_H / K_H(\mathfrak{q}).$$ For $j=1,2$, let $u_j : \bar{H} \rightarrow \mathbb{R}_{\geq 0}$ denote the function induced by $\lvert \Psi_j \rvert$. By definition, the image of $J_G$ in $K / K(\mathfrak{q})$ is the centralizer of $\tau$. The integral $I_1$ thus satisfies $$I_1 \ll {\mathop{\mathrm{vol}}(J_G)}^{-1} I,$$ where $I$ is the integral [\[eqn:i-:=-frac1lv-h-rvert2-sum-\_-subst-x-y-in-h-:-\]](#eqn:i-:=-frac1lv-h-rvert2-sum-_-subst-x-y-in-h-:-){reference-type="eqref" reference="eqn:i-:=-frac1lv-h-rvert2-sum-_-subst-x-y-in-h-:-"} defined in the statement of Theorem [Theorem 59](#theorem:bilinear-forms-estimate){reference-type="ref" reference="theorem:bilinear-forms-estimate"}, which gives $$I \ll \Delta \lvert \bar{H} \rvert^{-1} \lVert u_1 \rVert _{L^2} \lVert u_2 \rVert_{L^2}.$$ The required estimate for $I_1$ follows now from [\[eqn:volj_g-=-k_g-:-j_g-1-asymp-qn+12-n+1-=-qnn+1.-vol-J-G\]](#eqn:volj_g-=-k_g-:-j_g-1-asymp-qn+12-n+1-=-qnn+1.-vol-J-G){reference-type="eqref" reference="eqn:volj_g-=-k_g-:-j_g-1-asymp-qn+12-n+1-=-qnn+1.-vol-J-G"}. # Completion of the proof {#Sec:final} We now deduce our main result Theorem [Theorem 1](#theorem:cj3ngw7u2s){reference-type="ref" reference="theorem:cj3ngw7u2s"}, from our main local result, Theorem [Theorem 89](#theorem:main-local-result){reference-type="ref" reference="theorem:main-local-result"}. The deduction is exactly as in [@2020arXiv201202187N §6], so we will be brief. We denote in what follows by $\mathbb{Z}_{\mathfrak{l}} \leq F_{\mathfrak{l}}$ the ring of integers in the completion of $F$ at a finite place $\mathfrak{l}$. **Proposition 94**. *Fix $\alpha > 0$ and $\varepsilon> 0$. Retain the setting of Theorem [Theorem 1](#theorem:cj3ngw7u2s){reference-type="ref" reference="theorem:cj3ngw7u2s"}; in particular, $F$ is a number field, $(G,H) = (\mathop{\mathrm{U}}(V), \mathop{\mathrm{U}}(W))$ is a pair of unitary groups attached to a nondegenerate codimension one inclusion $W \hookrightarrow V$ of positive-definite hermitian spaces over $F$, $S$ is a large enough finite set of places, $(\pi,\sigma,\mathfrak{p},\mathfrak{q}) \in \mathcal{F}$, $T$ is the absolute norm of $\mathfrak{q}^2$, and $\vartheta \in [0,1/2)$ is such that $\sigma$ is $\vartheta$-tempered at every finite place $\mathfrak{l}\notin S \cup \{\mathfrak{p}\}$ that splits in $E$. Set $L := T^{\alpha}$. Let $\omega \in C_c^\infty(G(\mathbb{Z}_\mathfrak{p}))$ and $J_H \leq H(\mathbb{Z}_\mathfrak{p})$ be as in Theorem [Theorem 89](#theorem:main-local-result){reference-type="ref" reference="theorem:main-local-result"} applied to $(\pi_\mathfrak{p},\sigma_\mathfrak{p})$. There exists $c > 0$, depending only upon $\alpha$ and $(G,H,S)$, with the following property. For $j=0,\dotsc,n+1$, let $\Delta_j$ denote the infimum of all nonnegative quantities with the following property: for all $u_1, u_2 \in L^2(H(\mathbb{Z}_\mathfrak{p}))$ of unit norm that transform on the right under $J_H$ via some unitary character, and all $\gamma \in G(F_\mathfrak{p})$ with $d_{H_\mathfrak{p}}(\gamma) \geq c L^{-j}$, we have $$\label{eqn:20230517152904} \int_{x, y \in H(\mathbb{Z}_\mathfrak{p})} \left\lvert u_1(x) u_2(y) \omega^\sharp (x^{-1} \gamma y ) \right\rvert \, d x \, d y \leq T^{n / 2} \Delta_j.$$ Then $$\begin{aligned} \frac{\mathcal{L} (\pi,\sigma)}{T^{n(n+1)/2 + \varepsilon}} &\ll \sum _{j=1}^{n+1} \left( L^{-(1-2 \vartheta) j} + L^{(2 {(n+1)}^2 - n)j} \Delta_j \right)\\ &\quad + L^{-1} \sum _{j=0}^{n+1} \left( L^{-(1-2 \vartheta) j} + L^{(2 {(n+1)}^2 - n )j} \Delta_j \right), \end{aligned}$$ where the meaning of $\ll$ is "bounded in magnitude up to a factor depending only upon $(\mathcal{F},\alpha,\varepsilon, \vartheta)$".* *Proof.* This follows from the arguments of [@2020arXiv201202187N §6], applied with $Q=T^{1/2}$. The positive-definiteness assumption is used in two ways: so that the quotient $H(F) \backslash H(\mathbb{A})$ is compact, and so that each of the groups $G$ and $H$ are compact at infinity. This last property is used in applying [@2020arXiv201202187N Lem 5.1]. In more detail, we repeat [@2020arXiv201202187N §6.1-6.4] verbatim (omitting the adjective "archimedean" in a couple places --- this property is never used). In the first line of [@2020arXiv201202187N §6.5.1], we observe that the integrand in [@2020arXiv201202187N (6.5)] is now invariant merely under $\prod_{\mathfrak{l}\notin S \cup \{\mathfrak{p}\}} H(\mathbb{Z}_\mathfrak{l})$, rather than the larger group $\prod_{\mathfrak{l}\notin S} H(\mathbb{Z}_\mathfrak{l})$. (This is because of a difference in conventions: in [@2020arXiv201202187N], $S$ contains the interesting place, while here, it does not.) In the subsequent estimates, we accordingly replace the factorizable neighborhood $\Theta_S$ with $$\Theta_{S \cup \{\mathfrak{p}\}} := \Theta_S H(\mathbb{Z}_\mathfrak{p}).$$ The remainder of [@2020arXiv201202187N §6.5.1-6.5.2] is unchanged. The integral on the left hand side of [\[eqn:20230517152904\]](#eqn:20230517152904){reference-type="eqref" reference="eqn:20230517152904"} arises naturally following the proof[^2] of [@2020arXiv201202187N Lem 6.7]. We feed the resulting estimate into the summary of [@2020arXiv201202187N §6.6] to obtain the stated bound. ◻ We now apply Theorem [Theorem 89](#theorem:main-local-result){reference-type="ref" reference="theorem:main-local-result"}, part [\[enumerate:20230517165012\]](#enumerate:20230517165012){reference-type="eqref" reference="enumerate:20230517165012"} to see that the quantities $\Delta_j$ as in Proposition [Proposition 94](#proposition:20230517165122){reference-type="ref" reference="proposition:20230517165122"} satisfy $$\label{eqn:20230517165322} \Delta_j \ll \frac{L^j}{T^{1/2}} + \frac{1}{R },$$ where $R$ is given by $$T = q^{2 m} \implies R = q^{\lceil m/2 \rceil}.$$ In particular, $R$ satisfies the slightly wasteful estimate $R \geq T^{1/4}$. Substituting this estimate into the right hand side of the bound stated in Proposition [Proposition 94](#proposition:20230517165122){reference-type="ref" reference="proposition:20230517165122"}, we see that two terms dominate, giving $$\label{eqn:20230517170454} \frac{\mathcal{L}(\pi,\sigma)}{ T^{n (n + 1 )/2 + \varepsilon}} \ll L^{-(1 - 2 \vartheta )} + L^{(2 {(n + 1 )}^2 - n ) (n+1)} \left( \frac{L^{n+1}}{ T^{1/2} } + \frac{1}{T^{1/4} }\right).$$ Expanding the right hand side as a sum of three terms, the first and third terms are equal if $$L^{- (1 - 2 \vartheta )} = L^{(2 {(n + 1 )}^2 - n ) (n+1)} \frac{1}{T^{1/4} },$$ or equivalently, $$\alpha = \frac{1}{4 ( A + 1 - 2 \vartheta)}, \quad A := (2 {(n + 1 )}^2 - n ) (n + 1).$$ We then have $A \geq n+1$, hence $(n+1) \alpha \leq 1/4$, so the second term on the right hand side of [\[eqn:20230517170454\]](#eqn:20230517170454){reference-type="eqref" reference="eqn:20230517170454"} is dominated by the third term. The estimate [\[eqn:20230517170454\]](#eqn:20230517170454){reference-type="eqref" reference="eqn:20230517170454"} thus simplifies to the required bound $$\frac{\mathcal{L}(\pi,\sigma)}{ T^{n (n + 1 )/2 + \varepsilon}} \ll T^{-\delta}, \quad \delta := \frac{1 - 2 \vartheta }{4 (A + 1 - 2 \vartheta )}.$$ 10 Ehud Moshe Baruch. Bessel functions for GL(3) over a $p$-adic field. , 211(1):1--33, 2003. Joseph Bernstein and Andre Reznikov. Subconvexity bounds for triple $L$-functions and representation theory. , 172(3):1679--1718, 2010. Raphaël Beuzart-Plessis, Pierre-Henri Chaudouard, and MichałZydor. The global Gan-Gross-Prasad conjecture for unitary groups: the endoscopic case. , 135:183--336, 2022. Colin J. Bushnell, Guy M. Henniart, and Philip C. Kutzko. Local Rankin-Selberg convolutions for ${\rm GL}_n$: explicit conductor formula. , 11(3):703--730, 1998. Colin J. Bushnell and Philip C. Kutzko. , volume 129 of *Annals of Mathematics Studies*. 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arxiv_math

--- abstract: | In this paper, the dispersive revival and fractalization phenomena for bidirectional dispersive equations on a bounded interval subject to periodic boundary conditions and discontinuous initial profiles are investigated. Firstly, we study the periodic initial-boundary value problem of the linear beam equation with step function initial data, and analyze the manifestation of the revival phenomenon for the corresponding solution at rational times. Next, we extend the investigation to periodic initial-boundary value problems of more general bidirectional dispersive equations. We prove that, if the initial functions are of bounded variation, the dynamical evolution of such periodic problems depend essentially upon the large wave number asymptotics of the associated dispersion relations. Integral polynomial or asymptotically integral polynomial dispersion relations produce dispersive revival/fractalization rational/irrational dichotomies, whereas those with non-polynomial growth result in fractal profiles at all times. Finally, numerical experiments, in the concrete case of the nonlinear beam equation, are used to demonstrate how such effects persist into the nonlinear regime. address: - George Farmakis Heriot-Watt University & Maxwell Institute for Mathematical Sciences, Edinburgh, UK - Jing Kang Center for Nonlinear Studies and School of Mathematics, Northwest University, Xi'an 710069, P.R. China - Peter J. Olver School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA - Changzheng Qu School of Mathematics and Statistics, Ningbo University, Ningbo 315211, P.R. China - Zihan Yin School of Mathematics, Northwest University, Xi'an 710069, P.R. China author: - George Farmakis - Jing Kang - Peter J. Olver - Changzheng Qu - Zihan Yin title: New revival phenomena for bidirectional dispersive hyperbolic equations --- *Key words and phrases:* beam equation; revival; fractalization; Talbot effect; dispersive equation. (2020): 37K55, 35Q51\ # Introduction This paper is devoted to the study of the periodic initial-boundary value problem for bidirectional dispersive partial differential equations. We prove that, for linear equations, if the initial condition at time zero is a step function or, more generally, a function of bounded variation, the time evolution of the bidirectional dispersive equations subject to periodic boundary conditions will exhibit new revival phenomena at rational times, of a different form from that previously observed in unidirectional dispersive evolution equations, whereas at irrational times the solution exhibits a continuous, but non-differentiable fractal profile. The term "revival" is based on the experimentally observed phenomenon of quantum revival [@BMS; @VVS], in which an electron that is initially concentrated near a single location of its orbital shell is re-concentrated near a finite number of orbital locations at certain times. A precursor of the revival phenomenon was observed as far back as 1834 in a striking optical experiment, [@Tal], conducted in 1836 by William Henry Fox Talbot. This motivated the pioneering work of Berry and his collaborators, [@Ber; @BK; @BMS], on what they called the Talbot effect in the context of the linear free space Schrödinger equation. Rigorous analytical results and estimates justifying the Talbot effect can be found in the work of Kapitanski and Rodnianski, [@KR; @Rod], Oskolkov, [@Osk98; @Osk92], and Taylor, [@Tay]. The Talbot effect governs, in the quantum mechanical setting, the behavior of rough solutions subject to periodic boundary conditions. The evolution of the rough initial profile, for instance, a step function, also known as the Riemann problem [@Whi], "quantizes" into a dispersive revival profile at rational times, but "fractalizes" into a continuous but nowhere differentiable profile having a specific fractal dimension at irrational times. In [@CO12; @Olv10], the same Talbot effect, which the authors called dispersive quantization and fractalization, was shown to appear in general periodic linear dispersive equations possessing an "integral polynomial" (a polynomial with integer coefficients) dispersion relation, which included the prototypical linearized Schrödinger and Korteweg-de Vries (KdV) equations. Based on these investigations, one learns that a linear dispersive equation admitting a polynomial dispersion relation and subject to periodic boundary conditions will exhibit the revival phenomenon at each rational time, which means that the fundamental solution, i.e., that induced by a delta function initial condition, localizes into a finite linear combination of delta functions. This has the remarkable consequence that the solution, to any initial value problem, at rational times is a finite linear combination of translates of the initial data and hence its value at any point on the periodic domain depends only upon finitely many of the initial values. In [@OSS], the revival phenomenon for the linear free space Schrödinger equation subject to pseudo-periodic boundary conditions was investigated, see also [@BFP] for the same model and for the quasi-periodic linear KdV equation. In [@BOPS], a more general revival phenomenon, that produces dispersively quantized cusped solutions of the periodic Riemann problem for three linear integro-differential equations, including the Benjamin-Ono equation, the Intermediate Long Wave equation and the Smith equation were studied. More recently, these phenomena were shown to extend to multi-component dispersive equations, see [@YKQ]. For a class of two-component linear systems of dispersive evolution equations, the dispersive quantization conditions, which may yield quantized structures for step-function initial value at rational times, are provided. Inspired by these linear results, the phenomena of dispersive quantization and fractalization for the periodic Riemann problem for nonlinear dispersive evolution equations on periodic domains, including the integrable nonlinear Schrödinger (NLS), KdV and modified KdV (mKdV) equations as well as non-integrable versions with higher-order nonlinearities were studied numerically in [@CO14]. Erdo$\mathrm{\breve{g}}$an, Tzirakis and their collaborators established rigorous results on the fractalization for the nonlinear equations at a dense set of times. Quantifying the irrational time fractalization in terms of the estimate on the fractal dimension, their results, on the one hand extend the results of Oskolkov and Rodnianski to a class of nonlinear integer polynomial dispersive equations subject to initial data of bounded variation, and, on the other hand, confirm the numerical observations of fractalization in [@CO14]. Erdoǵan and Tzirakis studied the cubic NLS and KdV equations on a periodic domain with initial data of bounded variation in [@ET13-nls] and [@ET13-kdv], respectively. Subsequently, together with Chousionis, they obtained some results on the Minkowski dimension of the fractalization profiles for dispersive linear partial differential equations with monomial dispersion relation [@CET]. We refer the reader to the survey texts [@ES19; @ET16] for irrational time fractalization results. See also the recent survey [@Smi]. To date, investigations have almost all concentrated on unidirectional dispersive systems. In the present paper, we will show that the dispersive revival/fractalization rational/irrational dichotomy extends to bidirectional dispersive equations of the form $$\label{bi-eq-in} u_{tt}=L[u],$$ where $L$ is a scalar differential operator with constant coefficients. Obviously, equation [\[bi-eq-in\]](#bi-eq-in){reference-type="eqref" reference="bi-eq-in"} is equivalent to the following two-component evolutionary system $$\label{2comp-ev} u_{t}=v,\qquad v_t=L[u],$$ which, however, does not satisfy the dispersive quantization conditions given in [@YKQ]. As we describe below, in the bidirectional setting, if we set the initial conditions equal to the same step function, then the solution of the corresponding periodic Riemann problem will exhibit qualitative dispersive quantization behaviour, of a different form than the standard piecewise constant solutions admitted by the unidirectional systems, such as the linear KdV and Schrödinger equations and their associated multi-component generalizations. Interestingly, in the concrete case of the linear beam equation $$\label{beam-eq} u_{tt}+u_{xxxx}=0,$$ these solutions at rational times $t^\ast=\pi p/q$ with $q> 2$ appear to be piecewise parabolic, non-constant between jump discontinuities, whereas at the rational times $t_k^0=\pi(2k-1)/2$, the solution becomes a continuously differentiable curve, with analytical expression [\[sol-pi2\]](#sol-pi2){reference-type="eqref" reference="sol-pi2"}. Similar studies were initiated in one of the authors' recent Ph.D. thesis [@Fa], which studies the revival property in bidirectional dispersive equations [\[bi-eq-in\]](#bi-eq-in){reference-type="eqref" reference="bi-eq-in"} where the operator $L$ is an even-order poly-Laplacian, which includes the linear wave equation and the beam equation [\[beam-eq\]](#beam-eq){reference-type="eqref" reference="beam-eq"}, subject to periodic and quasi-periodic boundary conditions. We should further mention that from a general perspective, the form of the revival effect in the periodic bidirectional problems considered here resembles this of the revival effect in the free linear Schrödinger equation with Robin boundary conditions $bu(t,0) = (1-b)u_{x}(t,\pi)$, where $b\in (0,1)$ is a parameter, see [@BFP]. Indeed, in both cases the solution at rational times is given as the sum of the revival of the initial condition and a more regular function, which can be considered as a weak type of revival. Other models in the literature that exhibit such weak revivals include the periodic cubic NLS and KdV equations [@ET13-nls; @ET13-kdv], the periodic linear Schrödinger equation with periodic potential [@Rod-potential; @CKKKS] and the linear Schrödinger equation subject to Dirichlet boundary conditions [@BFP-Pot]. The linear beam equation, which is studied in Section 2, is a typical example of a model with the dispersion relation of the form $\omega(k)=\pm k^N,\;2\leq N\leq \mathbb{Z}^+$, namely when $N=2$. Although it is a special case, it motivates the study of the more general case and illustrates the idea and the method in the proof. More importantly, through the analysis and derivation of its periodic initial-boundary value problem, we arrive at some classical results for the Riemann zeta function. It implies that the periodic initial-boundary value problems for such systems can provide an alternative mechanism for establishing such classical identities. The three concrete goals of the present paper are as follows. The first is to investigate the new phenomenon of dispersive revival in greater detail, by examining the periodic initial-boundary value problems for bidirectional dispersive equations. We will provide an explicit characterization of the solution profiles of the periodic Riemann problem for the linear beam equation, leading to the general form of dispersive revival for bidirectional periodic initial-boundary value problems with various dispersion relations, including integral polynomial and non-polynomial. Our main results contain the analytic description of the new phenomena of the dispersive revival, which can be found in Section 2 for the linear beam equation, and Section 3 for general bidirectional equations, respectively. In the particular case of monomial dispersion relations, we present an alternative approach. Secondly, with the aim to show that such effects can persist into the nonlinear regime, we present numerical simulations, based on the Fourier spectral method, of the periodic Riemann problem for the nonlinear beam equation in Section 4. Numerical approximation supplies strong evidence that the dispersive revival/fractalization rational/irrational dichotomy persists into the nonlinear regime, whereas, when compared with the unidirectional systems, the nonlinear terms induce greater variations of the curve profiles, including their convexities. Finally, in the course of our analysis, we find that the solutions at rational times of the periodic Riemann problem for bidirectional dispersive equations with integral polynomial dispersion relations are closely related to identities for the Riemann zeta function, which is of great importance in analytic number theory. In summary, these new revival phenomena warrant further investigation, both mathematically, and to develop its profound applications. # Revival for the linear beam equation The starting point is the periodic initial-boundary value problem for the linear beam equation on the interval $0\leq x \leq 2\pi$: $$\label{ibv-beam} \begin{cases} u_{tt}+u_{xxxx}=0,\\ u(0, x)=f(x), \qquad \qquad u_t(0, x)=g(x),\\ {\displaystyle\partial_{x}^{j}u(t,0) = \partial_{x}^{j}u(t,2\pi)\atop \displaystyle\partial_{x}^{j}\partial _tu(t,0) = \partial_{x}^{j}\partial _tu(t,2\pi),} \qquad j =0, \ 1, \ 2, \ 3. %\displaystyle\frac{\partial^ku}{\partial_{t}^i\partial_{x}^j}\Big |_{x=0}=\frac{\partial^ku}{\partial_{t}^i\partial_{x}^j}\Big |_{x=2\pi},\quad i+j=k,\;0\leq i\leq 1,\; 0\leq k\leq 3. \end{cases}$$ The linear beam equation is a bidirectional dispersive equation modeling small vibrations of a thin elastic beam, with quadratic dispersion relation $\omega(k)=\pm k^2$. We focus our attention on the initial data given by a step function: $$\label{iv-s} f(x)=g(x)=\sigma(x)= \begin{cases} -1 ,\qquad & 0\leq x<\pi,\\ 1 ,\qquad & \pi\leq x<2\pi, \end{cases}$$known as the *Riemann problem*. Without further mention, here and elsewhere below, we assume that functions and distributions defined on $[\,0, \,2\pi \,]$ are extended $2\pi$-periodically to $\mathbb{R}$ in the usual way, when required. For the solution of [\[ibv-beam\]](#ibv-beam){reference-type="eqref" reference="ibv-beam"} with initial data [\[iv-s\]](#iv-s){reference-type="eqref" reference="iv-s"}, we have the following result. **Lemma 1**. *The periodic initial-boundary value problem [\[ibv-beam\]](#ibv-beam){reference-type="eqref" reference="ibv-beam"}-[\[iv-s\]](#iv-s){reference-type="eqref" reference="iv-s"} has the following solution $$\label{sol1-beam} u(t, x)=-\frac{4}{\pi}\left(\sum_{n=0}^{+\infty}\frac{\cos((2n+1)^2 t)\sin((2n+1) x)}{2n+1}+\sum_{n=0}^{+\infty}\frac{\sin((2n+1)^2 t)\sin((2 n+1)x)}{(2n+1)^3}\right).$$* Throughout the paper, a time $t>0$ will be designated as *rational* if $t/\pi \in \mathbb{Q}$, i.e., $t=t^\ast=\pi p/q$, with $p$ and $0\neq q\in \mathbb{Z}^+$ having no common factors. Otherwise, if $t/\pi \notin \mathbb{Q}$, the time is called *irrational*. To analyze the qualitative behavior of solution [\[sol1-beam\]](#sol1-beam){reference-type="eqref" reference="sol1-beam"} at the rational times, we invoke the following Lemma. **Lemma 2**. *Given $j,q\in \mathbb{Z^+}$ with $q \ne 0$, let $\sigma^{j, q}(x)$ be the box function defined as $$\label{box} \sigma^{j, q}(x)= \begin{cases} \,1,\qquad &\displaystyle\frac{\pi j}{q} \leq x < \frac{\pi(j+1)}{q},\quad 0\leq j\leq 2q-1,\\ \,0,\qquad &otherwise. \end{cases}$$ Let $N,\,p,\,q\in \mathbb{Z^+},\,N\geq 2,\,q\neq 0$. Then, the following formulae hold.* *(*i*) $$\label{eq1-l1}-\frac{4}{\pi}\sum_{n=0}^{+\infty}\frac{\displaystyle\cos\left((2n+1)^N \frac{\pi p}{q}\right)}{2n+1}\sin((2n+1) x)=\sum_{j=0}^{2q-1}a_j\left(\frac{p}{q}\right)\sigma^{j, q}(x),$$* *(*ii*) For each even $N$, $$\label{eq2-l1}-\frac{4}{\pi}\sum_{n=0}^{+\infty}\frac{\displaystyle\sin\left((2n+1)^N \frac{\pi p}{q}\right)}{2n+1}\sin((2n+1) x)=\sum_{j=0}^{2q-1}b_j\left(\frac{p}{q}\right)\sigma^{j, q}(x),$$* *(*iii*) For each odd $N$, $$\label{eq3-l1}-\frac{4}{\pi}\sum_{n=0}^{+\infty}\frac{\displaystyle\sin\left(((2n+1)^N \frac{\pi p}{q}\right)}{2n+1}\cos((2n+1) x)=\sum_{j=0}^{2q-1}\tilde{b}_j\left(\frac{p}{q}\right)\sigma^{j, q}(x),$$ where $a_j, b_j, \tilde{b}_j \in \mathbb{R}$, $j=0,\ldots, 2q-1$, are certain constants which depend on $p$ and $q$.* To prove Lemma [Lemma 2](#beam-le1){reference-type="ref" reference="beam-le1"}, we need the following theorem, which is based on Theorem 3.2 and Corollary 3.4 in [@CO12], and underlies the dispersive quantization effect for equations with "integral polynomial" dispersion relation. **Definition 3**. *A polynomial $P(k)=c_0+c_1k+\cdots+c_Nk^N$ is called an integral polynomial if its coefficients are integers: $c_i\in \mathbb{Z}, i=0, \ldots, N$.* **Theorem 4**. *Suppose that the dispersion relation of the evolution equation $u_t=L[u]$ is an integral polynomial: $$\omega(k)= P(k).$$ Then at every rational time $t^\ast=\pi p/q$, with $p$ and $0\neq q\in \mathbb{Z}$, the fundamental solution $$F(t, x)\sim \frac{1}{2\pi}\sum_{k=-\infty}^{\infty}e^{\,{\rm i}\,(k x+\omega(k)t)}$$ is a linear combination of $q$ periodically extended delta functions concentrated at the rational nodes $x_j=\pi j/q$ for $j\in \mathbb{Z}$. Moreover, at $t^\ast=\pi p/q$, the solution profile to the periodic initial-boundary value problem for $u_t=L[u]$ is a linear combination of $\leq 2q$ translates of its initial data $u(0, x)=f(x)$, i.e., $$u\left(\frac{\pi p}{q}, x\right)=\sum_{j=0}^{2q-1}a_j\left(\frac{p}{q}\right)f\left(x-\frac{\pi j}{q}\right).$$* **Remark 5**. *Slightly more generally, the dispersion relation can be a nonzero multiple of an integral polynomial. By suitably rescaling time, the stated result holds, the only difference being which times are designated as rational or irrational. A similar remark holds if one rescales space to consider the equation on a different spatial interval.* ***Proof of Lemma [Lemma 2](#beam-le1){reference-type="ref" reference="beam-le1"}**.* According to Theorem [Theorem 4](#thm-olv1){reference-type="ref" reference="thm-olv1"}, if the underlying equation admits a dispersion relation $\omega(k)=\pm k^N$, and the initial data is the unit step function $$u(0, x)= \begin{cases} 0 ,\qquad & 0\leq x<\pi,\\ 1 ,\qquad & \pi\leq x<2\pi, \end{cases}$$ with Fourier coefficients $$\begin{aligned} c_k= \begin{cases} \,\displaystyle\frac{1}{2},\qquad &k=0,\\ \,0,\qquad &k\neq0\; \mathrm{even},\\ \, \displaystyle\frac{\,{\rm i}\,}{\pi k},\qquad &k\; \mathrm{odd}. \end{cases}\end{aligned}$$ Then, at a rational time $t^\ast=\pi p/q$, the corresponding solution has the Fourier series form $$u^\pm(t, x)=\sum_{k=-\infty}^{+\infty}c_ke^{\,{\rm i}\,(kx\pm k^N t)}$$ and hence is constant on every subinterval $\pi j/q< x<\pi(j+1)/q$, for $j=0, \ldots, 2q-1$, namely $$\label{sol-l1} u^\pm(t^\ast, x)=\sum_{j=0}^{2q-1}\gamma_j^\pm\left(\frac{p}{q}\right)\sigma^{j, q}(x),$$ for certain $\gamma_j^\pm\in \mathbb{C}$, $j=0, \ldots, 2q-1$, dependent on $p$ and $q$. We thus need to distinguish two cases: **Case 1.**$N$ is even. It is easy to see that $$\begin{aligned} \begin{aligned} u^+(t^\ast, x)+u^-(t^\ast, x)&=\sum_{k=-\infty}^{+\infty}c_k\left(e^{\,{\rm i}\,(kx+ k^N t^\ast)}+e^{\,{\rm i}\,(kx- k^N t^\ast)}\right)\\ &=1-\frac{4}{\pi}\sum_{n=0}^{+\infty}\frac{\cos\left((2n+1)^N t^\ast\right)}{2n+1}\sin((2n+1) x),\\ u^+(t^\ast, x)-u^-(t^\ast, x)&=\sum_{k=-\infty}^{+\infty}c_k\left(e^{\,{\rm i}\,(kx+ k^N t^\ast)}-e^{\,{\rm i}\,(kx- k^N t^\ast)}\right)\\ &=-\frac{4\,{\rm i}\,}{\pi}\sum_{n=0}^{+\infty}\frac{\sin\left((2n+1)^N t^\ast\right)}{2n+1}\sin((2n+1) x). \end{aligned}\end{aligned}$$ **Case 2.**$N$ is odd. Similar to the above, we have $$\begin{aligned} \begin{aligned} u^+(t^\ast, x)+u^-(t^\ast, x)&=\sum_{k=-\infty}^{+\infty}c_k\left(e^{\,{\rm i}\,(kx+ k^N t^\ast)}+e^{\,{\rm i}\,(kx- k^N t^\ast)}\right)\\ &=1-\frac{4}{\pi}\sum_{n=0}^{+\infty}\frac{\cos\left((2n+1)^N t^\ast\right)}{2n+1}\sin((2n+1) x),\\ u^+(t^\ast, x)-u^-(t^\ast, x)&=\sum_{k=-\infty}^{+\infty}c_k\left(e^{\,{\rm i}\,(kx+ k^N t^\ast)}-e^{\,{\rm i}\,(kx- k^N t^\ast)}\right)\\ &=-\frac{4}{\pi}\sum_{n=0}^{+\infty}\frac{\sin\left((2n+1)^N t^\ast\right)}{2n+1}\cos((2n+1) x). \end{aligned}\end{aligned}$$ Finally, these formulae, together with [\[sol-l1\]](#sol-l1){reference-type="eqref" reference="sol-l1"} yield [\[eq1-l1\]](#eq1-l1){reference-type="eqref" reference="eq1-l1"}-[\[eq3-l1\]](#eq3-l1){reference-type="eqref" reference="eq3-l1"}, respectively, proving the lemma. ◻ Denote the solution [\[sol1-beam\]](#sol1-beam){reference-type="eqref" reference="sol1-beam"} as $$\begin{aligned} \label{beam-solut-1} u(t,x):=\mathrm{I}(t, x)+\mathrm{II}(t, x).\end{aligned}$$ Note that [\[eq1-l1\]](#eq1-l1){reference-type="eqref" reference="eq1-l1"} implies that the first summation in [\[beam-solut-1\]](#beam-solut-1){reference-type="eqref" reference="beam-solut-1"} evaluated at rational times $t^\ast$ has the representation $$\label{eq-s1-l1} \mathrm{I}(t^\ast, x)=\sum_{j=0}^{2q-1}a_j\left(\frac{p}{q}\right)\sigma^{j, q}(x),$$ for certain constants $a_0, \ldots, a_{2q-1}$ determined by [\[eq1-l1\]](#eq1-l1){reference-type="eqref" reference="eq1-l1"}. In particular, if $q=2$, at the corresponding specific rational time $t^0_k=\pi(2k-1)/2,\,k\in \mathbb{Z^+}$, it vanishes identically. When it comes to $\mathrm{II}(t^\ast, x)$, firstly, a direct computation shows that for any $t>0$ we have that $$\begin{aligned} \begin{aligned}\label{eq-s2-l1} &\frac{\sin((2n+1)^2 t)\sin((2 n+1)x)}{(2n+1)^3}\\ &\hskip.7in =-\int_{0}^{x}\int_{0}^{y}\frac{\sin((2n+1)^2t)\sin((2n+1)z)}{2n+1}\;\mathrm{d}z\mathrm{d}y+\frac{\sin((2n+1)^2t)}{(2n+1)^2} x. \end{aligned}\end{aligned}$$ Next, again thanks to Lemma 2.1, we find that the first of the two components in $\mathrm{II}(t, x)$ satisfies, $$\begin{aligned} \begin{aligned}\label{II(1)-t1} \mathrm{II}^{(1)}(t^\ast, x)&=\frac{4}{\pi}\int_{0}^{x}\int_{0}^{y}\sum_{n=0}^{+\infty}\frac{\sin((2n+1)^2 t^\ast)\sin((2n+1)z)}{2n+1}\;\mathrm{d}z\mathrm{d}y\\ &=-\int_{0}^{x}\int_{0}^{y}\sum_{j=0}^{2q-1}b_j\left(\frac{p}{q}\right)\sigma^{j, q}(z)\;\mathrm{d}z\mathrm{d}y, \end{aligned}\end{aligned}$$ for certain constants $b_j,\,j=0,\ldots, 2q-1$ determined by [\[eq2-l1\]](#eq2-l1){reference-type="eqref" reference="eq2-l1"}. We denote $$F(y)=\int_{0}^{y}\sum_{j=0}^{2q-1}b_j\sigma^{j, q}(z)\;\mathrm{d}z,\quad\mathrm{for}\quad 0\leq y\leq x,$$ and set $$H(x)=\int_{0}^{x}F(y)\mathrm{d}y.$$ It is easy to verify that $$F(y)= \begin{cases} b_0 y, \quad \quad &0\leq y\leq \frac{\pi }{q},\\ \displaystyle b_j y+\frac{\pi}{q}\sum\limits_{m=0}^{j-1}b_m-\frac{\pi}{q}j b_j, \quad & \displaystyle\frac{\pi}{q}j\leq y \leq \frac{\pi}{q}(j+1),\quad j=1,\ldots, 2q-1, \end{cases}$$ and hence $$\label{Hx} H(x) = \begin{cases} \frac{1}{2}b_0 x^2, \quad \quad &\displaystyle 0\leq x\leq \frac{\pi }{q},\hbox{\vrule height 0pt depth 13pt width 0pt}\\ \frac{1}{2}b_1 x^2+\displaystyle\frac{\pi}{q}(b_0-b_1)x+\frac{\pi^2}{2q^2}(b_1-b_0),\quad &\displaystyle\frac{\pi }{q}\leq x\leq \frac{2\pi }{q}, \hbox{\vrule height 0pt depth 15pt width 0pt}\\ \frac{1}{2}b_j x^2+h_1x+h_0,\quad &\displaystyle\frac{\pi }{q}j\leq x\leq \frac{\pi }{q}(j+1), \quad j=1,\ldots, 2q-1, \end{cases}$$ where $$h_1=\frac{\pi}{q}\left(\sum_{m=0}^{j-1}b_m-mb_m\right)\quad \mathrm{and}\quad h_0=\frac{\pi^2}{q^2}\left[\>\sum_{m=1}^{j-1}\left(\sum\limits_{i=0}^{m-1}b_i+\frac{b_m}{2}\right)+\frac{j^2}{2}b_j-j\sum_{m=0}^{j-1}b_m+\frac{b_0}{2}\>\right].$$ This, when combined with [\[eq-s1-l1\]](#eq-s1-l1){reference-type="eqref" reference="eq-s1-l1"} and [\[eq-s2-l1\]](#eq-s2-l1){reference-type="eqref" reference="eq-s2-l1"}, allows us to arrive at the exact result for the solution [\[beam-solut\]](#beam-solut){reference-type="eqref" reference="beam-solut"} at the rational times, which is summarized in the following theorem. **Theorem 6**. *At a rational time $t^\ast=\pi p/q$, the solution to the periodic initial-boundary value problem [\[ibv-beam\]](#ibv-beam){reference-type="eqref" reference="ibv-beam"} for the linear beam equation with the step function initial datum [\[iv-s\]](#iv-s){reference-type="eqref" reference="iv-s"} takes the form $$\label{sol-ra-beam} u(t^\ast, x)=\sum_{j=0}^{2q-1}a_j\left(\frac{p}{q}\right)\sigma^{j, q}(x)-H(x)+C(t^\ast)x,$$ where $$\label{C} C(t^\ast)=-\frac{4}{\pi}\sum_{n=0}^{+\infty}\frac{\sin((2n+1)^2t^\ast)}{(2n+1)^2},$$ $\sigma^{j, q}(x)$ is the box function defined in [\[box\]](#box){reference-type="eqref" reference="box"}, $H(x)$ is the piecewise quadratic function defined in [\[Hx\]](#Hx){reference-type="eqref" reference="Hx"}, and $a_j$, $j=0,\ldots, 2q-1$, are certain constants determined by equation [\[eq1-l1\]](#eq1-l1){reference-type="eqref" reference="eq1-l1"}.* With the explicit expression [\[sol1-beam\]](#sol1-beam){reference-type="eqref" reference="sol1-beam"} of the solution in hand, we now analyze its qualitative behaviour. First of all, as a direct corollary of the estimate on the solution of the linearized dispersion equation given by Oskolkov [@Osk92] and Rodnianski [@Rod] --- see also [@ET16] --- one has, for almost all irrational $t/(2\pi)$, the first summation $\mathrm{I}\in\bigcap_{\epsilon>0}C^{\frac{1}{2}-\epsilon}$, which in turn indicates that the second one $\mathrm{II}\in\bigcap_{\epsilon>0}C^{\frac{5}{2}-\epsilon}$. We thus conclude that, at the irrational times, the profile of the solution [\[beam-solut\]](#beam-solut){reference-type="eqref" reference="beam-solut"} is a continuous fractal, with fractal dimension $D=3/2$. When it comes to the rational times, note that in the expression of the solution [\[sol-ra-beam\]](#sol-ra-beam){reference-type="eqref" reference="sol-ra-beam"}, on the one hand, $H(x)\in C^1$, on the other hand, $\mathrm{I}(t^\ast, x)$ is a piecewise constant with $\leq 2q$ discontinuities, apart from the specific times $t^0_k=\pi(2k-1)/2, \,k\in\mathbb{Z^+}$, which will result in $\mathrm{I}(t^0_k, x)\equiv0$. Therefore, we need to distinguish two cases: **Case 1.** $q=2$. At times $t^0_k=\pi(2k-1)/2, \,k\in\mathbb{Z^+}$, $$u(t^0_k, x)=-H(x)+(-1)^k\frac{4}{\pi}\sum_{n=0}^{+\infty}\frac{1}{(2n+1)^2}x\in C^1.$$ Referring back to the relation [\[eq2-l1\]](#eq2-l1){reference-type="eqref" reference="eq2-l1"} corresponding to $q=2, p=2k-1$, and $N=2$, and solving for $b_j$ by making use of the inverse discrete Fourier transform (IDFT) gives rise to $$\label{b-2-pi2} b_0=b_1=(-1)^{k},\quad b_2=b_3=(-1)^{k-1}.$$ Then, [\[Hx\]](#Hx){reference-type="eqref" reference="Hx"} readily leads to $$\begin{aligned} \begin{aligned}\label{Hx-ex1} H_k(x) &=(-1)^{k-1} \begin{cases} -\frac{1}{2} x^2, \quad \quad &0\leq x \leq \pi,\\ \frac{1}{2}x^2-2\pi x+\pi^2,\quad &\pi \leq x \leq 2\pi, \end{cases} \end{aligned}\end{aligned}$$ which, together with the fact that the solution [\[sol1-beam\]](#sol1-beam){reference-type="eqref" reference="sol1-beam"} is $2\pi$-periodic, i.e., $u(t^0_k, 0)=u(t^0_k, 2\pi)$, yields $$\label{C-pi2} \sum_{n=0}^{+\infty}\frac{1}{(2n+1)^2}=\frac{\pi^2}{8}.$$ It follows that, evaluated at each $t^0_k=(2k-1)\pi/2, \,k\in\mathbb{Z^+}$, the solution can be written as $$\begin{aligned} \begin{aligned}\label{sol-pi2} u(t^0_k, x) =(-1)^{k-1} \begin{cases} \frac{1}{2} (x^2-\pi x), \quad \quad &0\leq x \leq\pi,\\ -\frac{1}{2}(x^2-3\pi x+2\pi^2),\quad &\pi \leq x \leq 2\pi. \end{cases} \end{aligned}\end{aligned}$$ More interestingly, we find that the conclusion [\[C-pi2\]](#C-pi2){reference-type="eqref" reference="C-pi2"} agrees with the classical result $$\label{zeta-2} \zeta (2) = \sum_{n=0}^{+\infty}\frac{1}{n^2}=\frac{\pi^2}{6},$$ where $$\label{zeta} \zeta(s)=\sum_{n=0}^{+\infty}\frac{1}{n^s},\quad s>1$$ is the Riemann zeta function. The above procedure provides an alternative mechanism for establishing such classical identities, and the behavior of these solutions at rational times has intriguing connections with such number-theoretic exponential sums. **Case 2.** $q\neq 2$. According to [\[sol-ra-beam\]](#sol-ra-beam){reference-type="eqref" reference="sol-ra-beam"}, at each rational time $t^\ast=\pi p/q, \,q\neq 2$, the solution consists of a piecewise constant function, which is constant on the intervals $\pi j/q< x< \pi(j+1)/q$ for $j=0, \ldots, 2q-1$, combined with a continuously differentiable function $-H(x)+C(t^\ast)x$, being composed of different parabolas defined on the intervals $\pi j/q\leq x< \pi(j+1)/q, \ j=0, \ldots, 2q-1$. Therefore, in the present case, the solution profile is a discontinuous, piecewise parabolic curve. For instance, let us take $t^\ast=\pi/3$ as an example. On the one hand, $$\mathrm{I}\left(\frac{\pi}{3}, x\right)=-\frac{4}{\pi}\sum_{n=0}^{+\infty}\frac{\cos((2n+1)^2 \frac{\pi}{3})}{2n+1}\sin((2n+1) x)=\sum_{j=0}^{5}a_j\left(\frac{1}{3}\right)\sigma^{j, q}(x),$$ where, a direct computation through IDFT shows that $$a_0=a_2=a_3=a_5=0,\quad a_1=-1,\quad a_4=1.$$ On the other hand, $b_j\,( 0\leq j\leq 5)$ in $H(x)$ are obtained from the relation $$-\frac{4}{\pi}\sum_{n=0}^{+\infty}\frac{\sin((2n+1)^2 \frac{\pi}{3})}{2n+1}\sin((2n+1) x)=\sum_{j=0}^{5}b_j\left(\frac{1}{3}\right)\sigma^{j, q}(x).$$ We have $$b_0=b_2=-b_3=-b_5=-\frac{\sqrt{3}}{3},\quad b_1=-b_4=-\frac{2\sqrt{3}}{3}.$$ Again, owing to the periodicity of the solution, one has $$\label{R-series} \sum_{n=0}^{+\infty}\frac{\sin((2n+1)^2 \frac{\pi}{3})}{(2n+1)^2}=-\frac{\pi}{4}C\left(\frac{\pi}{3}\right)=\frac{\sqrt{3}\,\pi^2}{18}.$$ Finally, inserting them into [\[Hx\]](#Hx){reference-type="eqref" reference="Hx"} and then [\[sol-ra-beam\]](#sol-ra-beam){reference-type="eqref" reference="sol-ra-beam"} gives rise to the explicit solution at time $t^\ast=\pi/3$, namely $$\begin{aligned} \begin{aligned}\label{sol-pi3-1} u\left(\frac{\pi}{3}, x\right) = \begin{cases} \frac{\sqrt{3}}{6} \left(x^2-\frac{4\pi}{3} x\right), \quad \quad &0\leq x\leq \frac{\pi}{3},\\ \frac{\sqrt{3}}{3} \left(x^2-\pi x+\frac{\pi^2}{18}\right)-1, \quad \quad &\frac{\pi}{3}\leq x\leq\frac{2\pi}{3},\\ \frac{\sqrt{3}}{6} \left(x^2-\frac{2\pi}{3} x-\frac{\pi^2}{3}\right), \quad \quad &\frac{2\pi}{3}\leq x\leq\pi,\\ -\frac{\sqrt{3}}{6} \left(x^2-\frac{10\pi}{3} x+\frac{7\pi^2}{3}\right), \quad \quad &\pi\leq x\leq\frac{4\pi}{3},\\ -\frac{\sqrt{3}}{3} \left(x^2-3\pi x+\frac{37\pi^2}{18}\right)+1, \quad \quad &\frac{4\pi}{3}\leq x\leq\frac{5\pi}{3},\\ -\frac{\sqrt{3}}{6} \left(x^2-\frac{8\pi}{3} x+\frac{4\pi^2}{3}\right), \quad \quad &\frac{5\pi}{3}\leq x\leq 2\pi. \end{cases} \end{aligned}\end{aligned}$$ **Remark 7**. *Note that, at the rational points, the series [\[R-series\]](#R-series){reference-type="eqref" reference="R-series"} contains the odd terms of Riemann's non-differentiable function, which was introduced by Riemann in 1872; see [@Dui] for details. The connection between the Riemann's non-differentiable function and solutions of the vortex filament equation with polygonal initial data was recently established in [@HKV1; @HKV2].* All in all, in the context of the linear beam equation, the evolution of the periodic step function initial datum will take on three different qualitative behaviors. At irrational times, it evolves into continuous but non-differentiable fractal-like profile. At rational times $t^\ast=\pi p/q\,(q\neq 2)$, the solution takes on a discontinuous, piecewise parabola behavior. On the other hand, at each specific rational time $t^0_k=\pi(2k-1)/2, \,k\in\mathbb{Z^+}$, the quantization effect disappears entirely, and the solution instantly becomes a continuously differentiable function, emerging at regular $\pi$-periodic intervals. We conclude that the revival phenomena exhibited by the periodic evolution of the linear beam equation differs from that arising in the linear KdV, the linear Schrödinger, and other unidirectional linear dispersive evolution equations studied previously. In Figures [\[lin-beam-ra\]](#lin-beam-ra){reference-type="ref" reference="lin-beam-ra"} and [\[lin-beam-irra\]](#lin-beam-irra){reference-type="ref" reference="lin-beam-irra"}, we display the graphs of the solution at some representative rational and irrational times, respectively. These figures are plotted by straightforwardly applying the Fourier series representation of $u(t, x)$ given by [\[sol1-beam\]](#sol1-beam){reference-type="eqref" reference="sol1-beam"}. We sum over 1001 terms[^1] to obtain the numerical approximation of the solution. As illustrated in Figure [\[lin-beam-ra\]](#lin-beam-ra){reference-type="ref" reference="lin-beam-ra"}(a), the solution is continuous at $\pi/2$. While, referring to Figure [\[lin-beam-ra\]](#lin-beam-ra){reference-type="ref" reference="lin-beam-ra"}(b) and Figure [\[lin-beam-ra\]](#lin-beam-ra){reference-type="ref" reference="lin-beam-ra"}(c), it appears that, at $\pi/3$ and $\pi/5$, there exist a finite number of jump discontinuities, and between which parabolic curves of different form arise. Obviously, the plots in Figure [\[lin-beam-ra\]](#lin-beam-ra){reference-type="ref" reference="lin-beam-ra"}, which obtained by simply truncating the Fourier series [\[sol1-beam\]](#sol1-beam){reference-type="eqref" reference="sol1-beam"}, are entirely consistent with the explicit expressions given by [\[sol-pi2\]](#sol-pi2){reference-type="eqref" reference="sol-pi2"} and [\[sol-pi3-1\]](#sol-pi3-1){reference-type="eqref" reference="sol-pi3-1"}. On the other hand, Figure [\[lin-beam-irra\]](#lin-beam-irra){reference-type="ref" reference="lin-beam-irra"} shows that, at irrational times, the solution displays continuous, but nowhere differentiable fractal-like profiles, as claimed above. **Remark 8**. *It is worth mentioning that, in view of the series [\[eq-s1-l1\]](#eq-s1-l1){reference-type="eqref" reference="eq-s1-l1"} and that in $\mathrm{II}^{(1)}(t^\ast, x)$, the distribution of the discontinuity points in $H(x)$ depends on the value of $q$, especially on its parity. As studied in [@OT18], general speaking, the piecewise subintervals for these series are $\pi j/q\leq x< \pi(j+1)/q, j=0, \ldots, 2q-1$. However, if $q$ is even (for instance $q=2$, whose corresponding solution is a representative example which can manifest such characteristic), the solutions sometimes assume identical values on adjacent subintervals, and so exhibits larger regions of constancy. See also [@OT18] for a number-theoretic characterization of these occurrences.* \ \ \ Formula [\[sol1-beam\]](#sol1-beam){reference-type="eqref" reference="sol1-beam"} provides the solution of [\[ibv-beam\]](#ibv-beam){reference-type="eqref" reference="ibv-beam"} with the same step function $f=g=\sigma(x)$ [\[iv-s\]](#iv-s){reference-type="eqref" reference="iv-s"} as the initial data. Indeed, in view of Theorem [Theorem 9](#thm-ge-eq){reference-type="ref" reference="thm-ge-eq"} below, we find that the first and second terms in solution [\[sol1-beam\]](#sol1-beam){reference-type="eqref" reference="sol1-beam"} are induced by the initial data $u|_{t=0}=f(x)$ and $u_{t}|_{t=0}=g(x)$, respectively. It is noticed that the emergence of such a dichotomy phenomenon is not only for the case $f=g$. For instance, if we take the initial $g(x)=\sigma(x)$, while $$f(x)=\tilde{\sigma}(x)= \begin{cases} 0 ,\qquad & 0\leq x<\pi,\\ 1 ,\qquad & \pi\leq x<2\pi. \end{cases}$$ Figure [\[lin-beam-f01\]](#lin-beam-f01){reference-type="ref" reference="lin-beam-f01"} suggests that the different steps functions will also evolve into three different qualitative behaviors. Furthermore, in Figure [\[lin-beam-f0\]](#lin-beam-f0){reference-type="ref" reference="lin-beam-f0"} and Figure [\[lin-beam-g0\]](#lin-beam-g0){reference-type="ref" reference="lin-beam-g0"}, we display the graphs of solutions corresponding to $f(x)=0,\,g(x)=\sigma(x)$, and $f(x)=\sigma(x),\,g(x)=0$, respectively. As demonstrated in Figure [\[lin-beam-f0\]](#lin-beam-f0){reference-type="ref" reference="lin-beam-f0"}, if $f(x)=0$, the dispersive quantization induced by $f(x)$ dispears entirely, and then the solution will retain a $C^1$ profile all the time. On the other hand, if $f(x)=\sigma(x),\,g(x)=0$, referring to Figure [\[lin-beam-g0\]](#lin-beam-g0){reference-type="ref" reference="lin-beam-g0"}(b) and Figure [\[lin-beam-g0\]](#lin-beam-g0){reference-type="ref" reference="lin-beam-g0"}(c), the solutions take on dispersive quantization at the rational times. While, as shown in Figure [\[lin-beam-g0\]](#lin-beam-g0){reference-type="ref" reference="lin-beam-g0"}(a), the solution will vanish at $\pi/2$, since $\mathrm{I}(\pi/2, x)\equiv0$ as claimed above. # Revival for bidirectional dispersive equations In the preceding section, we concentrated on the periodic initial-boundary problem for the linear beam equation. In this section, we will seek to extend our analysis to bidirectional dispersive equations possessing more general dispersion relations, as before, subject to general initial conditions and periodic boundary conditions. Let $L$ be a scalar, constant coefficient integro-differential operator with real-valued Fourier transform $\widehat{L}(k)=-\varphi(k)$, where $\varphi(k)>0$ is a real-valued function of $k$. The associated bidirectional scalar equation $$\label{ge-eq} u_{tt}=L[u]$$ has real dispersion relation $\omega(k)=\pm \sqrt{\varphi(k)}$. We subject [\[ge-eq\]](#ge-eq){reference-type="eqref" reference="ge-eq"} to periodic boundary conditions posed on the interval $0\leq x \leq 2\pi$, and initial conditions $$\label{ge-ic} u(0, x)=f(x),\quad u_t(0, x)=g(x),$$ where $f(x)$ and $g(x)$ are of bounded variation, and $g(x)$ is required to satisfy $\int_0^{2\pi} g(x)\;\mathrm{d}x=0$. As usual, the first step is to construct the (formal) solution as a Fourier series $$u(t,x) \sim \sum_{k=-\infty}^{+\infty}a_k(t)e^{\,{\rm i}\,kx}.$$ To this end, we first expand the initial data $f(x)$ and $g(x)$ in Fourier series $$\label{f-fs1} f(x)\sim \sum_{k=-\infty}^{+\infty}c_ke^{\,{\rm i}\,k x},\quad \mathrm{where}\quad c_k=\widehat{f}(k)=\frac{1}{2\pi}\int_0^{2\pi} \, f(x)e^{-\,{\rm i}\,k x}\,\mathrm{d}x$$ and $$\label{f-fs2} g(x)\sim \sum_{k=-\infty}^{+\infty}d_ke^{\,{\rm i}\,k x},\quad \mathrm{where}\quad d_k=\widehat{g}(k)=\frac{1}{2\pi}\int_0^{2\pi} \, g(x)e^{-\,{\rm i}\,k x}\,\mathrm{d}x.$$ Next, an analogous analysis as used for the linear beam equation with step function initial conditions, implies that the corresponding coefficients $a_k(t)$ satisfy the following linear ODE $$\label{ak-ode-ge} a_k''(t)+\varphi(k)a_k(t)=0.$$ Solving it yields $$a_k(t)=A_ke^{\,{\rm i}\,\sqrt{\varphi(k)}t}+B_ke^{-\,{\rm i}\,\sqrt{\varphi(k)} t}.$$ Finally, using the initial data again, we find that the solution to the periodic initial-boundary value problem [\[ge-eq\]](#ge-eq){reference-type="eqref" reference="ge-eq"}-[\[ge-ic\]](#ge-ic){reference-type="eqref" reference="ge-ic"} is given by $$\begin{aligned} \begin{aligned}\label{sol-ge-eq} u(t, x)=\sum_{k}\widehat{f}(k)\cos\left(\sqrt{\varphi(k)}\> t\right)e^{\,{\rm i}\,kx}+\sum_{k\neq 0}\frac{\widehat{g}(k)}{\sqrt{\varphi(k)}}\sin\left(\sqrt{\varphi(k)}\>t\right)e^{\,{\rm i}\,kx}. \end{aligned}\end{aligned}$$ With the Fourier series representation [\[sol-ge-eq\]](#sol-ge-eq){reference-type="eqref" reference="sol-ge-eq"} in hand, we are now able to analyze the qualitative behavior of the solution at rational times. We will show that the dynamical evolution of equation [\[ge-eq\]](#ge-eq){reference-type="eqref" reference="ge-eq"} on periodic domains with initial profiles [\[ge-ic\]](#ge-ic){reference-type="eqref" reference="ge-ic"} depends dramatically upon the asymptotics of the dispersion relation at large wave number. In all cases considered here, the large wave number asymptotics of the dispersion relation is given by a positive power of the wave number: $$\sqrt{\varphi(k)}\sim \vert k\vert ^\alpha,\quad 2\leq \alpha\in\mathbb{R}, \quad\mathrm{as}\quad \vert k\vert\rightarrow \infty.$$ ## Monomial dispersion relation: As the first step, we will study the special case of monomial dispersion relation given by $$\label{monomial} \omega(k)=\pm k^N,\quad 2\le N\in\mathbb{Z}^+.$$ The main results for the corresponding solutions are summarized in Theorem [Theorem 9](#thm-ge-eq){reference-type="ref" reference="thm-ge-eq"} below. Hereafter, we define the operator $\partial_x^{-1}$ by the formula $$\label{minus1} \partial_x^{-1}\,P(x)=\int_{2(k-1)\pi}^x\,P(y)\, \mathrm{d}y,\qquad x\in [ \,2(k-1)\pi, \,2k\pi \,].$$ We further define its $M$-th order power $\partial_x^{-M}$ via the recursive relation $\partial_x^{-M}=\partial_x^{-1}\partial_x^{-(M-1)}$, for $M\geq 1$. **Theorem 9**. *Suppose that equation [\[ge-eq\]](#ge-eq){reference-type="eqref" reference="ge-eq"} has the monomial dispersion relation [\[monomial\]](#monomial){reference-type="eqref" reference="monomial"}, the initial data $f(x)$ and $g(x)$ in [\[ge-ic\]](#ge-ic){reference-type="eqref" reference="ge-ic"} are of bounded variation, and $g(x)$ satisfies $\int_0^{2\pi} g(x)\;\mathrm{d}x=0$. Let $G(x)=\partial_x^{-N}g(x)$. Then at each rational time $t^\ast=\pi p/q$, the solution to the periodic initial-boundary value problem [\[ge-eq\]](#ge-eq){reference-type="eqref" reference="ge-eq"}-[\[ge-ic\]](#ge-ic){reference-type="eqref" reference="ge-ic"} takes the form $$\label{sol-ra-ge-eq} u(t^\ast, x)=\sum_{j=0}^{2q-1}a_j\left(\frac{p}{q}\right)\,f\left(x-\frac{\pi j}{q}\right)+\,{\rm i}\,^N \sum_{j=0}^{2q-1}b_j\left(\frac{p}{q}\right)\,G\left(x-\frac{\pi j}{q}\right)+\sum_{j=0}^{N-1}C_jx^j,$$ $$\label{Cj} C_j=\frac{\,{\rm i}\,^j}{j\,!}\sum_{k\neq 0}\frac{\widehat{g}(k)\sin(k^Nt^\ast)}{k^{N-j}},$$ where the coefficients $a_j, \,b_j\in \mathbb{C}$, $j=0,\ldots, 2q-1$, are constants depending on $p$ and $q$.* The proof of the theorem relies on the following lemma, which is a direct corollary of Theorem 3.2 established in [@CO12] and is a special case of Lemmas 7.5 and 7.6 in [@Fa]. Thus, we omit the proof. Moreover, we remark that the expression [\[sol-ra-ge-eq\]](#sol-ra-ge-eq){reference-type="eqref" reference="sol-ra-ge-eq"} for the exact solution is equivalent to [\[sol-ra-ge-eq 2\]](#sol-ra-ge-eq 2){reference-type="eqref" reference="sol-ra-ge-eq 2"} in the next subsection, although not exactly the same in form. **Lemma 10**. *Let $P(k)$ be an integral polynomial. Assume that $f(x)$ is of bounded variation, and let $\widehat{f}(k)$ be the Fourier coefficient of $f(x)$, i.e., $$\widehat{f}(k)=\frac{1}{2\pi}\int_0^{2\pi} \, f(x)e^{-\,{\rm i}\,k x}\,\mathrm{d}x.$$ Given $t^\ast=\pi p/q$, with $p$ and $0\neq q\in \mathbb{Z}^+$, there exist constants $a_j^1, a_j^2\in \mathbb{C}$, $j=0,\ldots, 2q-1$, depending on $p$ and $q$, such the following two formulae hold:\ $$\label{eq1-l2}\sum_{k=-\infty}^{\infty}\widehat{f}(k)\cos\left(P(k) t^\ast\right) e^{\,{\rm i}\,kx}=\sum_{j=0}^{2q-1}a_j^1\left(\frac{p}{q}\right)f\left(x-\frac{\pi j}{q}\right),$$ $$\label{eq2-l2}\sum_{k=-\infty}^{\infty}\widehat{f}(k)\sin\left(P(k) t^\ast\right) e^{\,{\rm i}\,kx}=\sum_{j=0}^{2q-1}a_j^2\left(\frac{p}{q}\right)f\left(x-\frac{\pi j}{q}\right).$$* ***Proof of Theorem [Theorem 9](#thm-ge-eq){reference-type="ref" reference="thm-ge-eq"}**.* First of all, according to [\[sol-ge-eq\]](#sol-ge-eq){reference-type="eqref" reference="sol-ge-eq"}, under the assumption of the theorem, the solution to the corresponding periodic initial-boundary problem has the form $$\begin{aligned} \begin{aligned}\label{sol-ge-eq-N} u(t, x)&=\sum_{k}\widehat{f}(k)\cos(k^N t)e^{\,{\rm i}\,kx}+\sum_{k\neq 0}\frac{\widehat{g}(k)\sin(k^N t)}{k^N}e^{\,{\rm i}\,kx}:=\mathrm{I}(t, x)+\mathrm{II}(t, x). \end{aligned}\end{aligned}$$ Furthermore, since $f(x)$ and $g(x)$ are of bounded variation and $N\geq 2$, the first summation in expression [\[sol-ge-eq-N\]](#sol-ge-eq-N){reference-type="eqref" reference="sol-ge-eq-N"} is conditionally convergent, and the second one is absolutely convergent. At the rational times $t^\ast=\pi p/q$, by Lemma 3.2, the first summation is a linear combination of translates of $f(x)$, i.e., $$\mathrm{I}(t^\ast, x)=\sum_{j=0}^{2q-1}a^1_j\left(\frac{p}{q}\right)\,f\left(x-\frac{\pi j}{q}\right),$$ for certain $a_0^1, \ldots, a_{2q-1}^1\in \mathbb{C}$ determined by [\[eq1-l2\]](#eq1-l2){reference-type="eqref" reference="eq1-l2"} with $P(k)=k^N$. Note that $$\label{parJ} \partial_x^{-M}e^{\,{\rm i}\,kx}=\frac{1}{(\,{\rm i}\,k)^M}e^{\,{\rm i}\,kx}-\sum_{j=0}^{M-1}\frac{1}{j !\>(\,{\rm i}\,k)^{M-j}}x^j, \quad {\rm for} \quad 0 \leq x \leq 2 \pi.$$ It follows that, at the rational times $t^\ast=\pi p/q$, the second summation satisfies $$\begin{aligned} \begin{aligned} \mathrm{II}(t^\ast, x)&=\sum_{k}\,{\rm i}\,^N\,\widehat{g}(k)\sin(k^N t^\ast)\partial_x^{-N}e^{\,{\rm i}\,kx}+\sum_{k\neq 0} \widehat{g}(k)\sin(k^N t^\ast) \sum_{j=0}^{N-1}\frac{\,{\rm i}\,^j}{j! \>k^{N-j}}x^j\\ &:=\mathrm{II}^{(1)}(x)+\mathrm{II}^{(2)}(x). \end{aligned}\end{aligned}$$ Since $g(x)$ is of bounded variation, the series $C_j$ given in [\[Cj\]](#Cj){reference-type="eqref" reference="Cj"} is convergent for each $j=0, \ldots, N-1$, then the second component $\mathrm{II}^{(2)}(x)$ readily leads to the last term in [\[sol-ra-ge-eq\]](#sol-ra-ge-eq){reference-type="eqref" reference="sol-ra-ge-eq"}. On the other hand, in the case of $P(k)=k^N$, applying equation [\[eq2-l2\]](#eq2-l2){reference-type="eqref" reference="eq2-l2"} to the delta function $\delta(x)$ yields $$\frac{1}{2\pi}\sum_{k=-\infty}^{\infty}\sin\left(k^N t^\ast\right) e^{\,{\rm i}\,kx}=\sum_{j=0}^{2q-1}b_j\left(\frac{p}{q}\right)\delta\left(x-\frac{\pi j}{q}\right),$$ for some constants $b_j\in \mathbb{C},\; j=0, \ldots, 2q-1.$ We thus deduce $\mathrm{II}^{(1)}(x)$ as follows: $$\begin{aligned} \begin{aligned} \mathrm{II}^{(1)}(x)&=\frac{\,{\rm i}\,^N}{2\pi}\sum_{k}\sin(k^N t^\ast)\int_0^{2\pi}g(y)e^{-\,{\rm i}\,ky}\partial_x^{-N}e^{\,{\rm i}\,kx}\,\mathrm{d}y\\ &=\frac{\,{\rm i}\,^N}{2\pi}\sum_{k}\sin(k^N t^\ast)\int_0^{2\pi}e^{\,{\rm i}\,ky}\partial_x^{-N}g(x-y)\,\mathrm{d}y\\ &=\,{\rm i}\,^N\int_0^{2\pi}G(x-y)\sum_{j=0}^{2q-1}b_j\delta\left(y-\frac{\pi j}{q}\right)\,\mathrm{d}y =\,{\rm i}\,^N\sum_{j=0}^{2q-1}b_jG\left(x-\frac{\pi j}{q}\right). \end{aligned}\end{aligned}$$ Summing $\mathrm{II}^{(1)}(x)$, $\mathrm{II}^{(2)}(x)$ and $\mathrm{I}(t^\ast, x)$ gives [\[sol-ra-ge-eq\]](#sol-ra-ge-eq){reference-type="eqref" reference="sol-ra-ge-eq"}, which justifies the statement of the theorem. ◻ In particular, if the initial data $f(x)$ and $g(x)$ are the step function $\sigma(x)$ given in [\[iv-s\]](#iv-s){reference-type="eqref" reference="iv-s"}, the following corollary holds. **Corollary 11**. *Let $\sigma^{j, q}(x)$ be the box function defined in [\[box\]](#box){reference-type="eqref" reference="box"}. At a rational time $t^\ast=\pi p/q$, the solution to the periodic initial-boundary value problem [\[ge-eq\]](#ge-eq){reference-type="eqref" reference="ge-eq"}-[\[ge-ic\]](#ge-ic){reference-type="eqref" reference="ge-ic"} on the interval $0\leq x\leq 2\pi$, with initial data $f(x)=g(x)=\sigma(x)$ given in [\[iv-s\]](#iv-s){reference-type="eqref" reference="iv-s"} takes the form $$\label{sol-ra-geeq-sic} u(t^\ast, x)=\sum_{j=0}^{2q-1}a_j\left(\frac{p}{q}\right)\sigma^{j, q}(x)+(-1)^{\left[\frac{N}{2}\right]}\partial_x^{-N}\sum_{j=0}^{2q-1}b_j\left(\frac{p}{q}\right)\sigma^{j, q}(x)+\sum_{j=0}^{\left[\frac{N}{2}\right]-1}D_j x^{2j+1},$$ where $$\label{Dj} D_j=\frac{(-1)^{j+1}4}{\pi(2j+1)!}\sum_{n=0}^{+\infty}\frac{\sin((2n+1)^Nt^\ast)}{(2n+1)^{N-2j}}, \qquad j=0, \ldots, \left[\frac{N}{2}\right]-1,$$ and the coefficients $a_j, j=0,\ldots, 2q-1$ are determined by formula [\[eq1-l1\]](#eq1-l1){reference-type="eqref" reference="eq1-l1"} in Lemma 2.1, $b_j, j=0,\ldots, 2q-1$ satisfy [\[eq2-l1\]](#eq2-l1){reference-type="eqref" reference="eq2-l1"} for even $N$, and [\[eq3-l1\]](#eq3-l1){reference-type="eqref" reference="eq3-l1"} for odd $N$, respectively.* *Proof.* If $f(x)=g(x)=\sigma(x)$, the corresponding solution [\[sol-ge-eq\]](#sol-ge-eq){reference-type="eqref" reference="sol-ge-eq"} reduces to $$\begin{aligned} \begin{aligned}\label{sol-beam} u(t^\ast, x)&=-\frac{4}{\pi}\left[\>\sum_{n=0}^{+\infty}\frac{\cos((2n+1)^N t^\ast)\sin((2n+1) x)}{2n+1}+\sum_{n=0}^{+\infty}\frac{\sin((2n+1)^N t^\ast)\sin((2 n+1)x)}{(2n+1)^{N+1}}\>\right]. \end{aligned}\end{aligned}$$ Obviously, the first summation is exactly the first term in [\[sol-ra-geeq-sic\]](#sol-ra-geeq-sic){reference-type="eqref" reference="sol-ra-geeq-sic"}. As for the second summation, a direct induction procedure shows that, if $N$ is even, $$\frac{\sin((2 n+1)x)}{(2n+1)^{N+1}}=(-1)^{\frac{N}{2}}\partial_x^{-N}\frac{\sin((2 n+1)x)}{2n+1}+\sum_{j=0}^{\frac{N}{2}-1}\frac{(-1)^{j}}{(2j+1)!(2n+1)^{N-2j}}x^{2j+1},$$ whereas, if $N$ is odd, $$\frac{\sin((2 n+1)x)}{(2n+1)^{N+1}}=(-1)^{\frac{N-1}{2}}\partial_x^{-N}\frac{\cos((2 n+1)x)}{2n+1}+\sum_{j=0}^{\frac{N-1}{2}-1}\frac{(-1)^{j}}{(2j+1)!(2n+1)^{N-2j}}x^{2j+1}.$$ Substituting into the second summation, and making use of formulae [\[eq2-l1\]](#eq2-l1){reference-type="eqref" reference="eq2-l1"} and [\[eq3-l1\]](#eq3-l1){reference-type="eqref" reference="eq3-l1"} for even and odd $N$, respectively, verifies [\[sol-ra-geeq-sic\]](#sol-ra-geeq-sic){reference-type="eqref" reference="sol-ra-geeq-sic"}, proving the corollary. ◻ More specifically, if the underlying equation is exactly the linear beam equation in [\[ibv-beam\]](#ibv-beam){reference-type="eqref" reference="ibv-beam"} with dispersion relation $\omega(k)=\pm k^2$. It follows that the second term in [\[sol-ra-geeq-sic\]](#sol-ra-geeq-sic){reference-type="eqref" reference="sol-ra-geeq-sic"} reduces to [\[II(1)-t1\]](#II(1)-t1){reference-type="eqref" reference="II(1)-t1"}, which is nothing but $-H(x)$ in [\[sol-ra-beam\]](#sol-ra-beam){reference-type="eqref" reference="sol-ra-beam"}. Meanwhile, the third term is identical to $C(t^\ast)x$ in [\[sol-ra-beam\]](#sol-ra-beam){reference-type="eqref" reference="sol-ra-beam"}. This indicates that in this particular case, Corollary [Corollary 11](#sol-ge-eqN-ivs){reference-type="ref" reference="sol-ge-eqN-ivs"} is in accordance with Theorem [Theorem 6](#thm-beam){reference-type="ref" reference="thm-beam"}. As in Section 2, let us now illustrate how, by Corollary [Corollary 11](#sol-ge-eqN-ivs){reference-type="ref" reference="sol-ge-eqN-ivs"}, we can calculate the value of the Riemann zeta function at $s=4$. We define $$H_{p, q}^N(x)=\partial_x^{-N}\sum_{j=0}^{2q-1}b_j\left(\frac{p}{q}\right)\sigma^{j, q}(x),$$ where $b_j$ are determined by the formulae [\[eq2-l1\]](#eq2-l1){reference-type="eqref" reference="eq2-l1"} for even $N$, or [\[eq3-l1\]](#eq3-l1){reference-type="eqref" reference="eq3-l1"} for odd $N$, respectively. Denote $$\label{series} S_{l}^N(t)=\sum_{n=0}^{+\infty}\frac{\sin((2n+1)^N t)}{(2n+1)^l},\quad\quad \mathrm{for} \quad l\in\mathbb{Z^+}, \quad\mathrm{with}\quad l\geq 2,$$ and let $$\label{gamma} \Gamma_N = \begin{cases} \displaystyle\sum\limits_{k=1}^{\frac{N}{2}}\frac{(-1)^{k}(2\pi)^{N-2k+1}}{(N-2k+1)!}S_{2k}^N(t^\ast), \quad \quad & \mathrm{if} \;N\; \mathrm{even},\\ \displaystyle\sum\limits_{k=1}^{\frac{N-1}{2}}\frac{(-1)^{k}(2\pi)^{N-2k}}{(N-2k)!}S_{2k+1}^N(t^\ast), \quad \quad & \mathrm{if} \;N\geq 3\; \mathrm{odd}. \end{cases}$$ According to [\[sol-ra-geeq-sic\]](#sol-ra-geeq-sic){reference-type="eqref" reference="sol-ra-geeq-sic"}, we find a formula involving the sum $\Gamma_N$, which along with the periodicity produces $$\label{SN} \Gamma_N=\frac{\pi}{4}H_{p, q}^N(2\pi).$$ Note that, if $N$ is even, at the special rational times $t^\ast_l=(2l-1)\pi/2,\,l\in \mathbb{Z^+}$, $$S_N^N\left(t^\ast_l\right)=(-1)^{l-1}\sum_{n=0}^{+\infty}\frac{1}{(2n+1)^N},$$ while, if $N$ is odd, $S_N^N\left(t^\ast_l\right)$ is a alternating series, namely, $$S_N^N\left(t^\ast_l\right)=(-1)^{l-1}\sum_{n=0}^{+\infty}\frac{(-1)^n}{(2n+1)^N}.$$ Hereafter, we denote $$\sigma (N)=\sum_{n=0}^{+\infty}\frac{1}{(2n+1)^{N}},\quad \mathrm{for}\; \mathrm{even} \;N,\qquad \tau(N)=\sum_{n=0}^{+\infty}\frac{(-1)^n}{(2n+1)^{N}},\quad \mathrm{for}\; \mathrm{odd} \;N,$$ respectively. Therefore, in the special rational times $t^\ast_l$ setting, [\[SN\]](#SN){reference-type="eqref" reference="SN"} establishes the recursion formulae for $\sigma(2k)$ and $\tau(2k+1)$ for each $k\in \mathbb{Z^+}$. More precisely, for even $N$ $$\sigma(N)=\frac{(-1)^{\frac{N}{2}}}{8}\left(H_{1, 2}^N(2\pi)-\frac{4}{\pi}\sum_{k=0}^{\frac{N}{2}-1}\frac{(-1)^{k}(2\pi)^{N-2k+1}}{(N-2k+1)!}\sigma(2k)\right),$$ or, for odd $N$, $$\tau(N)=\frac{(-1)^{\frac{N-1}{2}}}{8}\left(H_{1, 2}^N(2\pi)-\frac{4}{\pi}\sum_{k=0}^{\frac{N-1}{2}-1}\frac{(-1)^{k}(2\pi)^{N-2k}}{(N-2k)!}\tau(2k+1)\right),$$ which are initiated by the series $\sigma(2)$ [\[C-pi2\]](#C-pi2){reference-type="eqref" reference="C-pi2"} for even $N$, or $\tau(3)$ for odd $N$, respectively. As far as $\tau(3)$ is concerned, one can verify from [\[eq3-l1\]](#eq3-l1){reference-type="eqref" reference="eq3-l1"} for $N=3$ that $$\tilde{b}_0=-1,\quad \tilde{b}_1=\tilde{b}_2=1,\quad \tilde{b}_3=-1,$$ which immediately yields $$\begin{aligned} \begin{aligned}\label{sol-pi3} H_{1, 2}^3(x) = \begin{cases} -\frac{1}{6} x^3, \quad \quad &0\leq x\leq \frac{\pi}{2},\\ \frac{1}{6} \left(x^3-3\pi x^2+\frac{3\pi^2}{2}x-\frac{\pi^3}{4}\right), \quad \quad &\frac{\pi}{2}\leq x\leq \frac{3\pi}{2},\\ -\frac{1}{6} \left(x^3-6\pi x^2+12\pi^2x-\frac{13\pi^3}{2}\right), \quad \quad &\frac{3\pi}{2}\leq x\leq 2\pi. \end{cases} \end{aligned}\end{aligned}$$ We thus arrive at $$\tau(3)=\sum_{n=0}^{+\infty}\frac{(-1)^n}{(2n+1)^3}=\frac{\pi^3}{32}.$$ When it comes to $\zeta(4)$, we calculate from [\[b-2-pi2\]](#b-2-pi2){reference-type="eqref" reference="b-2-pi2"} that $$\begin{aligned} \begin{aligned}\label{sol-zeta4} H_{1, 2}^4(x) = \begin{cases} -\frac{1}{24} x^4, \quad \quad &0\leq x\leq \pi,\\ \frac{1}{24} \left(x^4-8\pi x^3+12\pi^2x^2-8\pi^3x+2\pi^4\right), \quad \quad &\pi \leq x\leq 2\pi. \end{cases} \end{aligned}\end{aligned}$$ Consequently, $$\sigma(4)=\sum_{n=0}^{+\infty}\frac{1}{(2n+1)^4}=\frac{1}{8}\left(H_{1, 2}^4(2\pi)+\frac{4\pi^2}{3!}\bar{\zeta}(2)\right)=\frac{\pi^4}{96},$$ which further yields the following classical result for the Riemann zeta function at $s=4$: $$\zeta(4)=\sum_{n=0}^{+\infty}\frac{1}{n^4}=\frac{\pi^4}{90}.$$ ## Monomial dispersion relation --- second approach We now briefly consider a different approach in the monomial case, which is based on [@Fa Chapter 7] and derive an alternative representation of the solution at rational times. Hence, the dispersion relation assumes the form [\[monomial\]](#monomial){reference-type="eqref" reference="monomial"}. Moreover, only for this subsection, we relax the condition on $g$ and allow it to have non-zero mean over $[0,2\pi]$. The solution to the periodic initial-boundary value problem [\[ge-eq\]](#ge-eq){reference-type="eqref" reference="ge-eq"}-[\[ge-ic\]](#ge-ic){reference-type="eqref" reference="ge-ic"} is given by $$\begin{aligned} \begin{aligned}\label{sol-ge-eq-N1} u(t, x)&=\sum_{k}\widehat{f}(k)\cos(k^N t)e^{\,{\rm i}\,kx}+\frac{1}{2\pi}\int_{0}^{2\pi}g(y)\mathrm{d}y \ t + \sum_{k\neq 0}\frac{\widehat{g}(k)\sin(k^N t)}{k^N}e^{\,{\rm i}\,kx}\\ &:=\mathrm{I}(t,x)+ \langle g \rangle \ t + \mathrm{II}(t,x), \end{aligned}\end{aligned}$$ where $\langle g \rangle$ is the mean of $g$ and $\mathrm{I}(t,x)$, $\mathrm{II}(t,x)$ correspond to the two Fourier series representations respectively. In the following, we derive an alternative representation of the term $\mathrm{II}(t, x)$. In particular, we will show that $\mathrm{II}(t,x)$ can be expressed as the time-evolution of the periodic convolution of the function $g - \langle g \rangle$ with a polynomial of degree $N\geq 2$. As it is known, see for example [@Iorio Proposition 2.76], the convolution gains the regularity of the most regular function between the two involved. Consequently, we may deduce that at any time $t>0$, either rational or irrational, the Fourier series representation of $\mathrm{II}(t,x)$ defines a $2\pi$-periodic function of class $C^{N}(\mathbb{R})$. Let us make all the above precise and define first the following family of polynomials on $[0,2\pi]$. **Definition 12**. *Let $N\geq 1$ be an integer. We denote by $Q_{N}:[0,2\pi]\rightarrow \mathbb{C}$ the polynomial of degree $N$, defined inductively by the formula $$\label{QN-Poly} Q_{N}(x) = \frac{(-\,{\rm i}\,)^{N} x^{N}}{(-1)^{N-1} N!} - \sum_{\ell=1}^{N-1} \frac{(-1)^{\ell - N}}{(-\,{\rm i}\,)^{\ell - N}} \frac{(2\pi)^{N-\ell}}{(N-\ell +1)!} Q_{\ell}(x).$$* The crucial feature of the polynomial $Q_{N}$ is the form of its Fourier coefficients, which are equal to $k^{-N}$. As we shall shortly see, this will allow us to invoke the operation of the periodic convolution in the representation of $\mathrm{II}(t,x)$. **Lemma 13**. *Fix an integer $N\geq 1$ and consider the polynomial $Q_{N}:[0,2\pi]\rightarrow \mathbb{C}$. Then, for $k\not=0$, $\widehat{Q_{N}}(k) = k^{-N}$.* *Proof.* The proof follows by induction on $N$. It is easy to show that the statement holds for $N=1$ and $N=2$. We assume that $\widehat{Q_{\ell}}(k) = k^{-\ell}$ for $\ell=1,2,\dots, N$, with $N\geq3$, and calculate the Fourier coefficients of $Q_{N+1}$. Let $k\not=0$. Then, we have that $$\begin{aligned} \widehat{Q_{N+1}}(k) &= \frac{1}{2\pi}\int_{0}^{2\pi} Q_{N+1}(y) e^{-\,{\rm i}\,k y}dy \\ &= \frac{(-i)^{N+1}}{2\pi (-1)^{N} (N+1)!}\int_{0}^{2\pi} y^{N+1}e^{-\,{\rm i}\,k y}dy - \sum_{\ell=1}^{N} \frac{(-1)^{\ell - N-1}}{(-\,{\rm i}\,)^{\ell - N-1}} \frac{(2\pi)^{N+1-\ell}}{(N-\ell)!} \frac{1}{k^{\ell}}. \end{aligned}$$ However, a direct calculation shows that $$\int_{0}^{2\pi} y^{N+1}e^{-\,{\rm i}\,k y}dy = \frac{2 \pi (-1)^{N} (N+1)!}{(-\,{\rm i}\,)^{N+1} k^{N+1}} + (N+1)! \sum_{\ell = 1}^{N} \frac{(-1)^{\ell -1}}{(-\,{\rm i}\,)^{\ell}} \frac{(2\pi)^{N + 2-\ell}}{(N-\ell)!} \frac{1}{k^{\ell}}.$$ Substituting back for $\widehat{Q_{N+1}}(k)$ we find that $$\begin{aligned} \widehat{Q_{N+1}}(k) = \frac{1}{k^{N+1}}, \end{aligned}$$ which concludes the proof. ◻ We now turn our attention to the second ingredient needed for the alternative representation of $\mathrm{II}(t,x)$. Thus, we recall the definition of the periodic convolution, see [@Stein2011]. **Definition 14**. *Let $f$ and $g$ be $2\pi$-periodic on $\mathbb{R}$ and such that $f$, $g$ $\in L^{1}(0,2\pi)$. Then the *$2\pi$-periodic convolution* of $f$ and $g$ is defined by $$\label{Periodic Conv} f\ast g (x) = \frac{1}{2\pi}\int_{0}^{2\pi} f(x - y) g(y) dy, \quad x\in[0,2\pi].$$* From [@Stein2011 Proposition 3.1], we know that $f\ast g$ defines a $2\pi$-periodic continuous function whose Fourier coefficients are given by $\widehat{f \ast g}(k) = \widehat{f}(k) \widehat{g}(k)$. Summarizing the above, we arrive at the following lemma which identifies $\mathrm{II}(t,x)$ based on the convolution of $g - \langle g \rangle$ with $Q_{N}$. **Lemma 15**. *Assume that $g$ is of bounded variation over $[0,2\pi]$. Fix integer $N\geq 2$ and consider the function $$\label{v function} v(x) = (g-\langle g \rangle)\ast Q_{N} (x), \quad x\in [0,2\pi].$$ Then, at any fixed time $t\geq 0$, we have that $$\label{II representation} \mathrm{II}(t,x) = \sum_{k=-\infty}^{\infty} \widehat{v}(k)\sin(k^{N} t) e^{\,{\rm i}\,kx}.$$* *Proof.* Let $\tilde{g} = g - \langle g \rangle$. Then, the Fourier coefficients of $\tilde{g}$ are given by $$\widehat{\tilde{g}}(k) = \widehat{g}(k), \quad k\not=0, \quad \widehat{\tilde{g}}(0) = 0.$$ Moreover, from Lemma [Lemma 13](#QN-Poly-FC){reference-type="ref" reference="QN-Poly-FC"}, we know that for $k\not=0$, $\widehat{Q_{N}}(k) = k^{-N}$. Hence, $$\widehat{v}(k) = \widehat{\tilde{g}}(k)\, \widehat{Q_{N}}(k) = \begin{cases} 0, \quad k = 0, \\ \frac{\widehat{g}(k)}{k^{N}}, \quad k\not=0, \end{cases}$$ which implies that for any $t>0$, $$\sum_{k=-\infty}^{\infty} \widehat{v}(k)\sin(k^{N} t) e^{\,{\rm i}\,kx} = \sum_{k\not=0} \frac{\widehat{g}(k)}{k^{N}}\sin(k^{N} t) e^{\,{\rm i}\,kx} = \mathrm{II}(t,x).$$ ◻ The validity of the revival effect at rational times $t^\ast=\pi p/q$ follows again by Lemma [Lemma 10](#Revival Representations){reference-type="ref" reference="Revival Representations"} applied directly on $\mathrm{I}(t^\ast,x)$ and $\mathrm{II}(t^\ast,x)$ in conjunction with Lemma [Lemma 15](#II representation lemma){reference-type="ref" reference="II representation lemma"}. This is the context of the next theorem. **Theorem 16**. *Suppose that equation [\[ge-eq\]](#ge-eq){reference-type="eqref" reference="ge-eq"} admits the monomial dispersion relation [\[monomial\]](#monomial){reference-type="eqref" reference="monomial"}, the initial data $f(x)$ and $g(x)$ in [\[ge-ic\]](#ge-ic){reference-type="eqref" reference="ge-ic"} are of bounded variation. Then, at each rational time $t^\ast=\pi p/q$, the solutions to the periodic initial-boundary value problem [\[ge-eq\]](#ge-eq){reference-type="eqref" reference="ge-eq"}-[\[ge-ic\]](#ge-ic){reference-type="eqref" reference="ge-ic"} take the form $$\label{sol-ra-ge-eq 2} u(t^\ast, x)=\sum_{j=0}^{2q-1}a_j\left(\frac{p}{q}\right)\,f\left(x-\frac{\pi j}{q}\right)+ \langle g \rangle \,t^{\ast} + \sum_{j=0}^{2q-1}d_j\left(\frac{p}{q}\right)\,v\left(x-\frac{\pi j}{q}\right),$$ where $v(x) = ( g - \langle g \rangle ) \ast Q_{N} (x)$ and the coefficients $a_j, \,d_j\in \mathbb{C}$, $j=0,\ldots, 2q-1$, are certain constants depending on $p,q$.* As a consequence of Theorem [Theorem 16](#thm-ge-eq 2){reference-type="ref" reference="thm-ge-eq 2"}, equivalently of Theorem [Theorem 9](#thm-ge-eq){reference-type="ref" reference="thm-ge-eq"}, the solution at rational times is, at least, piecewise continuous, given that $f(x)$ has finitely many jump discontinuities. More specifically, the first term in [\[sol-ra-ge-eq 2\]](#sol-ra-ge-eq 2){reference-type="eqref" reference="sol-ra-ge-eq 2"}, $\mathrm{I}(t^\ast,x)$, corresponds to the revival of the initial function $f(x)$, whereas the third term, $\mathrm{II}(t^\ast)$, is the revival of the $2\pi$-periodic, $C^{N}(\mathbb{R})$ function $v(x) = ( g - \langle g \rangle ) \ast Q_{N} (x)$ and thus, together with the constant term, $\langle g \rangle t^{\ast}$, a $2\pi$-periodic, $C^{N}(\mathbb{R})$ function. Therefore, the solution is given as the sum of the revival of the initial condition $f(x)$ and a more regular function, which ensures the revival of the initial jump discontinuities of $f(x)$. ## Integral polynomial dispersion relation This subsection is concerned with the case that $\sqrt{\phi(k)}$ is an integral polynomial $P(k)$. The corresponding solution takes the form $$\begin{aligned} \begin{aligned}\label{sol-ge-eq-Pk} u(t, x)&=\sum_{k}\widehat{f}(k)\cos(P(k) t)e^{\,{\rm i}\,kx}+\sum_{k\neq 0}\frac{\widehat{g}(k)\sin(P(k) t)}{P(k)}e^{\,{\rm i}\,kx}:=\mathrm{I}(t, x)+\mathrm{II}(t, x). \end{aligned}\end{aligned}$$ Firstly, using [\[eq1-l2\]](#eq1-l2){reference-type="eqref" reference="eq1-l2"} again, we obtain that at each rational time $t^\ast=\pi p/q$, the first term in [\[sol-ge-eq-Pk\]](#sol-ge-eq-Pk){reference-type="eqref" reference="sol-ge-eq-Pk"} satisfies $$\mathrm{I}(t^\ast, x)=\sum_{j=0}^{2q-1}a_j\left(\frac{p}{q}\right)\,f\left(x-\frac{\pi j}{q}\right).$$ Next, notice that $$\begin{aligned} \begin{aligned} \left| \>\mathrm{II}(t, x)-\sum_{k\neq 0}\frac{\widehat{g}(k)\sin(P(k) t)}{c_Nk^N}e^{\,{\rm i}\,kx}\>\right|&\leq \sum_{k\neq 0}\frac{\vert c_{N-1}k^{N-1}+\cdots+c_{1}k+c_0 \vert }{\vert c_N P(k)k^N \vert}\vert \widehat{g}(k)\vert\\ &\lesssim \sum_{k\neq 0}\frac{1}{k^{N+2}}, \end{aligned}\end{aligned}$$ where the fact that $g(x)$ is of bounded variation has been used in the last inequality and $c_{N}$ denotes the coefficient of the highest power of $P(k)$. Since $N\geq 2$, the final series is absolutely convergent, whose sum is a constant. Thus, the above estimate implies that the qualitative behavior of the second term relies crucially on that of the series $$\label{se-II} \sum_{k\neq 0}\frac{\widehat{g}(k)\sin(P(k) t)}{c_Nk^N}e^{\,{\rm i}\,kx}.$$ While, as for [\[se-II\]](#se-II){reference-type="eqref" reference="se-II"}, a direct generalization of the proof of Theorem [Theorem 9](#thm-ge-eq){reference-type="ref" reference="thm-ge-eq"} shows that, at each rational time $t^\ast=\pi p/q$, $$\begin{aligned} \begin{aligned} \sum_{k\neq 0}\frac{\widehat{g}(k)\sin(P(k) t^\ast)}{c_Nk^N}e^{\,{\rm i}\,kx}=\,{\rm i}\,^N\sum_{j=0}^{2q-1}\bar{b}_j\left(\frac{p}{q}\right)\,G\left(x-\frac{\pi j}{q}\right)+\sum_{j=0}^{N-1}\bar{C}_jx^j, \end{aligned}\end{aligned}$$ where the coefficients $\bar{b}_0, \ldots, \bar{b}_{2q-1}$ are determined by $$\sum_{k}\sin\left(P(k) t^\ast\right) e^{\,{\rm i}\,kx}=\sum_{j=0}^{2q-1}\bar{b}_j\left(\frac{p}{q}\right)\delta\left(x-\frac{\pi j}{q}\right),\quad \hbox{and} \quad \bar{C}_j=\sum_{k\neq 0}\frac{\widehat{g}(k)\sin(P(k)t^\ast)}{c_Nj! k^{N-j}}.$$ We thus conclude that, at each rational time, the series [\[se-II\]](#se-II){reference-type="eqref" reference="se-II"} admits the same discontinuities and revival structure as the second summation in solution [\[sol-ge-eq-N\]](#sol-ge-eq-N){reference-type="eqref" reference="sol-ge-eq-N"}. All in all, we may safely draw the conclusion that, in the present case, the discontinuities of the solution will be determined by the initial data. For instance, if the initial data are the step function $\sigma(x)$, as in [\[iv-s\]](#iv-s){reference-type="eqref" reference="iv-s"}, by Corollary [Corollary 11](#sol-ge-eqN-ivs){reference-type="ref" reference="sol-ge-eqN-ivs"}, $u(t, x)$ will be a $C^{N-1}$ curve at each $t_k^0=(2k-1)\pi/2,\;k\in \mathbb{Z}^+$, and exhibit jump discontinuities and revival profile at other rational times. ## Non-polynomial dispersion relation If $\sqrt{\varphi(k)}$ is not a polynomial, we distinguish two cases. The first one assumes that, for large wave numbers, the dispersion relation is asymptotically close to an integral polynomial $P(k)$. Hence, suppose $$\sqrt{\varphi(k)} \sim P(k)+O(k^{-1}),\quad \mathrm{as}\quad \vert k\vert\ \rightarrow \infty.$$ Firstly, under the assumption that $f(x)$ is of bounded variation, the first summation in [\[sol-ge-eq\]](#sol-ge-eq){reference-type="eqref" reference="sol-ge-eq"} satisfies $$\begin{aligned} \begin{aligned}\label{non-poly-1} &\left|\ \sum_{k}\widehat{f}(k)\cos\left(\sqrt{\varphi(k)}\> t\right)e^{\,{\rm i}\,kx}-\sum_{k}\widehat{f}(k)\cos(P(k) t)e^{\,{\rm i}\,kx}\ \right|\\ &\hskip1.5in\leq \sum_{k} \vert \widehat{f}(k)\vert\,\vert \cos\left(\sqrt{\varphi(k)}\> t\right)-\cos(P(k) t)\vert\lesssim \sum_{k\neq 0}\frac{1}{k^{2}}, \end{aligned}\end{aligned}$$ where the mean-value theorem has been used in the last inequality. Next, for the second summation, one has $$\begin{aligned} \begin{aligned} &\left|\ \sum_{k\neq 0}\frac{\widehat{g}(k)}{\sqrt{\varphi(k)}}\sin\left(\sqrt{\varphi(k)}\> t\right)e^{\,{\rm i}\,kx}-\sum_{k\neq 0}\frac{\widehat{g}(k)}{k^N}\sin(P(k) t)e^{\,{\rm i}\,kx}\ \right|\\ &\hskip1in\leq \left|\ \sum_{k\neq 0}\frac{\widehat{g}(k)}{\sqrt{\varphi(k)}}\sin\left(\sqrt{\varphi(k)}\> t\right)e^{\,{\rm i}\,kx}-\sum_{k\neq 0}\frac{\widehat{g}(k)}{P(k)}\sin\left(\sqrt{\varphi(k)}\> t\right)e^{\,{\rm i}\,kx}\ \right|\\ &\hskip1.5in{}+\left|\ \sum_{k\neq 0}\frac{\widehat{g}(k)}{P(k)}\sin\left(\sqrt{\varphi(k)}\> t\right)e^{\,{\rm i}\,kx}-\sum_{k\neq 0}\frac{\widehat{g}(k)}{P(k)}\sin(P(k) t)e^{\,{\rm i}\,kx}\ \right|\\ &\hskip1.5in{}+\left|\ \sum_{k\neq 0}\frac{\widehat{g}(k)}{P(k)}\sin(P(k) t)e^{\,{\rm i}\,kx}-\sum_{k\neq 0}\frac{\widehat{g}(k)}{k^N}\sin(P(k) t)e^{\,{\rm i}\,kx} \right|\\ &\hskip1in:=\mathrm{II}^{(1)}+\mathrm{II}^{(2)}+\mathrm{II}^{(3)}. \end{aligned}\end{aligned}$$ We directly estimate the above three terms as follows: $$\begin{aligned} \begin{aligned} \mathrm{II}^{(1)}&\leq \sum_{k\neq 0}\frac{\vert O(k^{-1})\widehat{g}(k)\vert}{\vert P(k)\varphi(k)\vert}\lesssim \sum_{k\neq 0}\frac{1}{k^{2N+2}} , \\ \mathrm{II}^{(2)}&\leq \sum_{k\neq 0}\frac{\vert \sin(\sqrt{\varphi(k)} t)-\sin(P(k) t)\vert \vert \widehat{g}(k)\vert}{\vert P(k)\vert}\lesssim \sum_{k\neq 0}\frac{1}{k^{N+2}} , \\ \mathrm{II}^{(3)}&\leq \sum_{k\neq 0}\frac{\vert c_{N-1}k^{N-1}+\ldots+c_0\vert \vert \widehat{g}(k)\vert}{\vert P(k)k^N\vert}\lesssim \sum_{k\neq 0}\frac{1}{k^{N+2}} . \end{aligned}\end{aligned}$$ We thus conclude that $$\begin{aligned} \begin{aligned} \left|\; \sum_{k\neq 0}\frac{\widehat{g}(k)}{\sqrt{\varphi(k)}}\sin(\sqrt{\varphi(k)} t)e^{\,{\rm i}\,kx}-\sum_{k\neq 0}\frac{\widehat{g}(k)}{k^N}\sin(P(k) t)e^{\,{\rm i}\,kx} \;\right|\lesssim \sum_{k\neq 0}\frac{1}{k^{N+2}}, \end{aligned}\end{aligned}$$ which, together with the estimate [\[non-poly-1\]](#non-poly-1){reference-type="eqref" reference="non-poly-1"} imply that, in the present case, the solution $u(t, x)$ will exhibit the same asymptotic behavior as the polynomial case. The times at which the solution (approximately) exhibits revivals are densely embedded in the times at which it has a continuous, fractal profile. For example, the Boussinesq equation $$\label{bs} u_{tt}+\frac{1}{3}u_{xxxx}-u_{xx}+\frac{3}{2}\alpha(u^2)_{xx}=0,$$ has the linear dispersion relation $\omega(k)=\pm k\sqrt{\frac{1}{3}k^2+1}$, and its leading order asymptotics is $\pm \frac{1}{\sqrt{3}}k^2$. The solution of the periodic initial-boundary value problem for the linearization of equation [\[bs\]](#bs){reference-type="eqref" reference="bs"} subject to the step function initial data [\[iv-s\]](#iv-s){reference-type="eqref" reference="iv-s"} at several representative rational and irrational times are plotted in Figure [\[lin-bs\]](#lin-bs){reference-type="ref" reference="lin-bs"}. As illustrated in these figures, the solutions exhibit the (approximately) revival profile at rational times, and the overall jump discontinuities and revival structure is very similar to that of the linear beam equation. \ On the other hand, if the equation admits a non-polynomial dispersion relation with an non-integral asymptotic exponent, i.e., $$\sqrt{\varphi(k)}\sim \vert k\vert ^\alpha,\quad 2\leq \alpha\notin \mathrm{Z}, \;\mathrm{as}\quad \vert k\vert\rightarrow \infty,$$ we still estimate the two summations in [\[sol-ge-eq\]](#sol-ge-eq){reference-type="eqref" reference="sol-ge-eq"} separately. As far as the first one is concerned, as studied in [@CO12], its overall qualitative behavior is entirely determined by the asymptotic exponent $\alpha$. In particular, when $2\leq \alpha\notin \mathrm{Z}$ is not an integer, only fractal solution profiles will be observed at every time. While, as for the second term, observe that in the present situation, $$\begin{aligned} \begin{aligned} &\left|\ \sum_{k\neq 0}\frac{\widehat{g}(k)}{\sqrt{\varphi(k)}}\sin(\sqrt{\varphi(k)} t)e^{\,{\rm i}\,kx}-\sum_{k\neq 0}\frac{\widehat{g}(k)}{\vert k\vert ^{[\alpha]+1} }\sin(\vert k\vert ^\alpha t)e^{\,{\rm i}\,kx}\ \right|\\ &\hskip1.5in\leq \sum_{k\neq 0}\frac{\vert k\vert ^{[\alpha]+1}-\vert k\vert ^{\alpha}}{\vert k\vert ^{\alpha}\vert k\vert ^{[\alpha]+1}}\vert \widehat{g}(k)\vert \lesssim \sum_{k\neq 0}\frac{\vert k\vert ^{\alpha'}\ln \vert k\vert}{\vert k\vert ^{\alpha}\vert k\vert ^{[\alpha]+1}}\vert \widehat{g}(k)\vert, \end{aligned}\end{aligned}$$ for some $\alpha <\alpha'<[\alpha]+1$. Note that if $g(x)$ is of bounded variation, the estimate can only be obtained by using $\sum_{k\neq 0}{\vert k\vert ^{-[\alpha]-1}}$. In view of this situation, we need to further require that $g(x)$ satisfies $$\int\,e^{\,{\rm i}\,kx}\;\mathrm{d}g\sim O(k^{(\alpha-\alpha')-\delta }) \quad{\rm for\ all} \quad\delta > 0.$$ Under this hypothesis, the above estimate is bounded by $\sum_{k\neq 0}{\vert k\vert ^{-[\alpha]-2}}$, and hence the second term is completely determined by the series $$\sum_{k\neq 0}\frac{\widehat{g}(k)}{\vert k\vert ^{[\alpha]+1}}\sin(\vert k\vert ^\alpha t)e^{\,{\rm i}\,kx},$$ which, compared with the first term, will admit better regularity. \ We conclude that, in the present case, the solutions will retain a fractal profile at all times. Results confirming this are displayed in Figure [\[bi-BO\]](#bi-BO){reference-type="ref" reference="bi-BO"}, which are the plots of the solutions to the periodic Riemann problem for the case of three-halves dispersion relation $\omega(k)=\pm \vert k\vert^{\frac{3}{2}}$ corresponding to the equation $$\label{th-eq} u_{tt}=\mathcal{H}[u_{xxx}],$$ where $\mathcal{H}$ denotes the periodic Hilbert transform, $$\mathcal{H}[f](x)=\frac{1}{\pi}\sum_{-\infty}^{+\infty}\,\int_{-\pi}^{\pi}\hskip-19pt \hbox{---} \hskip10pt \frac{f(y)}{x-y+2\pi k}\,\mathrm{d}y =\frac1{2\pi}\, \int_{-\pi}^{\pi}{\hskip-19pt \hbox{---} \hskip10pt \cot\frac{x-y}2 f(y)}\,\mathrm{d}y .$$ # Numerical simulation of dispersive revival for nonlinear equations In this section, we will explore the effect of periodicity on rough initial data for nonlinear equations in the context of the nonlinear defocusing cubic beam equation of the form $$\label{non-beam} u_{tt}+u_{xxxx}+\mu \,u+ \varepsilon \,\vert u\vert^2u=0,$$ which is motivated by the nonlinear Boussinesq equation, see [@MMS] for details. We will numerically approximate the solutions to the periodic initial-boundary value problem for the beam equation [\[non-beam\]](#non-beam){reference-type="eqref" reference="non-beam"} subject to periodic boundary conditions on $[-\pi,\,\pi]$, with the same step function [\[iv-s\]](#iv-s){reference-type="eqref" reference="iv-s"} as initial data. The goal of this section is to investigate to what extent revival and fractalization phenomena persist into the nonlinear regime. A basic numerical technique, the Fourier spectral method, will be employed to approximate the solution to this initial-boundary value problem. As we will see, our numerical studies strongly indicate that the dispersive revival phenomenon admitted by the associated linearized equation will persist into the nonlinear regime. However, some of the qualitative details --- for instance, the convexity of the curves between the jump discontinuities --- will be affected by the nonlinearity, in contrast to what was observed in the unidirectional case. ## The Fourier spectral method Let us first summarize the basic ideas behind the Fourier spectral method for approximating the solutions to nonlinear equations. One can refer to [@GO; @Tre] for details of the method. Formally, consider the initial value problem for a nonlinear evolution equation $$\label{ns-eq} u_t=K[u],\qquad u(0, x)=u_0(x),$$ where $K$ is a differential operator in the spatial variable with no explicit time dependence. Suppose $K$ can be written as $K=L+N$, in which $L$ is a linear operator characterized by its Fourier transform $\widehat{Lu}(k)=\omega(k)\widehat{u}(k)$, while $N$ is a nonlinear operator. We use $\mathcal{F}[\cdot]$ and $\mathcal{F}^{-1}[\cdot]$ to denote the Fourier transform and the inverse Fourier transform of the indicated function, respectively, so that the Fourier transform for equation in [\[ns-eq\]](#ns-eq){reference-type="eqref" reference="ns-eq"} takes the form $$\widehat{u}_t=\omega(k)\widehat{u}+\mathcal{F}[\,N(\mathcal{F}^{-1}[\widehat{u}])\,].$$ Firstly, periodicity and discretization of the spatial variable enables us to apply the fast Fourier transform (FFT) based on, for instance, 512 space nodes, and arrive at a system of ordinary differential equations (ODEs), which we solve numerically. For simplicity, we adopt a uniform time step $0<\Delta t\ll1$, and seek to approximate the solution $\hat{u}(t_n)$ at the successive times $t_n=n\Delta t$ for $n=0, 1, \ldots$. The classic fourth-order Runge-Kutta method, which has a local truncation error of $O((\Delta t)^5)$, is adopted, and its iterative scheme is given by $$\widehat{u}(t_{n+1})=\widehat{u}(t_n)+\frac{1}{6}(f_{k_1}+2f_{k_2}+2f_{k_3}+f_{k_4}),\quad n=0, 1, \ldots, \quad \widehat{u}(t_0)=\widehat{u}_0(k),$$ where $$\begin{aligned} \begin{aligned} f_{k_1}&=f(t_n, \widehat{u}(t_n)),& f_{k_2}&= f(t_n+\Delta t/2,\widehat{u}(t_n)+\Delta t f_{k_1}/2),\\ f_{k_3}&= f(t_n+\Delta t/2, \widehat{u}(t_n)+\Delta t f_{k_2}/2),&\qquad f_{k_4}&= f(t_n+\Delta t, \widehat{u}(t_n)+\Delta t f_{k_3}) \end{aligned}\end{aligned}$$ with $$\begin{aligned} f(t, \widehat{u})=\omega(k)\widehat{u}+\mathcal{F}[\,N(\mathcal{F}^{-1}[\widehat{u}])\,].\end{aligned}$$ Accordingly, the approximate solution $u(t, x)$ can be obtained through the inverse discrete Fourier transform. Since the Runge-Kutta method is designed for first order systems of ordinary differential equations, we convert our bidirectional second order in time system [\[non-beam\]](#non-beam){reference-type="eqref" reference="non-beam"} into a first order system by setting $$v=u_t,$$ an hence the beam equation [\[non-beam\]](#non-beam){reference-type="eqref" reference="non-beam"} is mapped to the following evolutionary system $$\label{non-beam-sys} u_{t}=v,\qquad v_t=-u_{xxxx}-\mu \,u-\varepsilon \,\vert u\vert^2u.$$ The Fourier transform for [\[non-beam-sys\]](#non-beam-sys){reference-type="eqref" reference="non-beam-sys"} takes the form $$\label{non-beam-sys-ft} \widehat{u}_{t}=\widehat{v},\qquad \widehat{v}_t=-( k)^4\widehat{u}-\mu \,\widehat{u}-\varepsilon \,\mathcal{F}[\,\vert \mathcal{F}^{-1}[\widehat{u}] \vert^2\mathcal{F}^{-1}[\widehat{u}]\,].$$ Using the classic fourth-order Runge-Kutta method to solve the resulting system [\[non-beam-sys-ft\]](#non-beam-sys-ft){reference-type="eqref" reference="non-beam-sys-ft"}, and then taking the inverse discrete Fourier transform, one can obtain the numerical solution to the periodic initial-boundary value problem for the nonlinear beam equation [\[non-beam\]](#non-beam){reference-type="eqref" reference="non-beam"}. ## Numerical Results Figure [\[non-beam-ra\]](#non-beam-ra){reference-type="ref" reference="non-beam-ra"} and Figure [\[non-beam-irra\]](#non-beam-irra){reference-type="ref" reference="non-beam-irra"} display some results from our numerical approximations of the solutions to the nonlinear beam equation [\[non-beam\]](#non-beam){reference-type="eqref" reference="non-beam"} with periodic boundary conditions and initial conditions [\[iv-s\]](#iv-s){reference-type="eqref" reference="iv-s"} at some representative rational and irrational times. Comparing the graphs in these two figures with the graphs corresponding to the same times in Figure 1 and Figure 2, we find that, at each irrational time, all sets of plots are fairly similar to those from the associated linear beam equation, and the solution still takes a continuous, non-differentiable profile. When it comes to the rational times, the same jump discontinuities consistency for nonlinear and linear equations emerges as well. Meanwhile, closer inspection will reveal some differences. The most noticeable is that, the shape of the curves between jump discontinuities will change with time evolution. More precisely, the graphs corresponding to $t=\pi/5$ show that the differences of the solution profile between the linear and nonlinear equations is slight, except that, in the nonlinear case, the curves between the jumps become closer to constants. Further, as the power $p$ decreases with increasing time, the variation in the shape of the curves from linear to nonlinear becomes greater and greater. As illustrated in the graphs corresponding to $t=\pi/3$ and $t=\pi/2$, the convexity of the curves has completely changed. These differences of the qualitative behavior of the solutions exhibit the effect of the nonlinearity. Furthermore, in order to better understand the effect of the nonlinearity, we perform further numerical experiments for smaller values of coefficients $\varepsilon$ and $\mu$ in equation [\[non-beam\]](#non-beam){reference-type="eqref" reference="non-beam"}. Referring to Figure [\[non-beam-pi3\]](#non-beam-pi3){reference-type="ref" reference="non-beam-pi3"}, it appears that solution at $t=\pi/3$ tends to the linear profile as $\varepsilon$ tends to zero. Meanwhile, the shape of the curves between jump discontinuities will change as $\varepsilon$ increases, the most noticeable variation being the changes in convexity. More unexpected phenomena appear when $t=\pi/2$. We find the variation of the profile of the solution will be affected not only by the nonlinear term but also by the linear term involving $u$. The plots displayed in Figure [\[non-beam-pi2\]](#non-beam-pi2){reference-type="ref" reference="non-beam-pi2"}, corresponding to some representative coefficients $\varepsilon$ and $\mu$, suggest that the solution profile, including its convexity and the values of its peak and trough will be affected by the combination of both coefficients $\varepsilon$ and $\mu$. -.3in -.3in Recall that the numerical experiments to the periodic initial-boundary value problem for the KdV equation, the NLS equation and the multi-component KdV system have been previously analyzed in [@CO14; @YKQ], which show that, in the unidirectional regime, the effect of the nonlinear flow can be regarded as a perturbation of the linearized flow. When it comes to the bidirectional dispersive equations, our numerical simulation strongly indicates that, the dichotomy of revival/fractalization at rational/irrational times in linearization will persist into the nonlinear regime, and the finite "revival" nature of the solutions at rational times is not affected by the nonlinearity, however, the influence of the nonlinearity on the qualitative behavior of the solutions is much greater than in the unidirectional setting. Motivated by this observation, formulation of theorems and rigorous proofs concerning this novel revival phenomenon in the nonlinear bidirectional regime, specially for the nonlinear beam and Boussinesq equations, is eminently worth further study. Part of Farmakis' research was conducted during his Ph.D studies which were supported by the Maxwell Institute Graduate School in Analysis and its Applications, a Centre for Doctoral Training funded by EPSRC (grant EP/L016508/01), the Scottish Funding Council, Heriot-Watt University and the University of Edinburgh. Kang's research was supported by NSFC (Grants-11631007 and 11871395 and Basic Science Program of Shaanxi Province (Grant-2019JC-28). Qu's research was supported by NSFC (Grants-11971251, 11631007 and 12111530003). Yin's research was supported by the NSFC (Grant-11631007). $\,$ 50 [M.V. Berry,]{.smallcaps} Quantum fractals in boxes, *J. Phys. A* **29** (1996), 6617-6629. [G. Chen and P.J. Olver,]{.smallcaps} Dispersion of discontinuous periodic waves, *Proc. Roy. Soc. A* **469** (2012), 20120407. [G. Chen and P.J. Olver,]{.smallcaps} Numerical simulation of nonlinear dispersive quantization, *Discrete Contin. Dyn. Syst.* **34** (2014), 991-1008. 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[I. Rodnianski,]{.smallcaps} Continued Fractions and Schrödinger Evolution, *Contemp. Math.* **236** (1999), 311-321. [M.M. Shakiryanov,]{.smallcaps} On the asymptotic reduction to the multidimensional nonlinear Schrödinger equation, arXiv:math-ph/9907003. [D.A. Smith,]{.smallcaps} Revivals and Fractalization, *Dyn. Sys. Web* **2020** (2020), 1-8. [E.M. Stein, R.Shakarchi,]{.smallcaps} Princeton Lectures in Analysis I. Fourier Analysis: An Introduction, Princeton University Press, 2011. [H.F. Talbot,]{.smallcaps} Facts related to optical science, No. IV, *Philos. Mag.* **9** (1836), 401-407. [M.Taylor,]{.smallcaps}  The Schrödinger equation on spheres, *Pacific J. Math.* **209** (2003), 145-155. [L.N. Trefethen,]{.smallcaps} Spectral Methods in Matlab, Society for Industrial and Applied Mathematics, SIAM, 2001. [M.J.J. Vrakking, D.M. Villeneuve and A. Stolow,]{.smallcaps}  Observation of fractional revivals of a molecular wavepacket, *Phys. Rev. A* **54** (1996), R37-R40. 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arxiv_math

--- abstract: | Gilbert--Steiner problem is a generalization of the Steiner tree problem on a specific optimal mass transportation. We show that every branching point in a solution of the planar Gilbert--Steiner problem has degree 3. author: - Danila Cherkashin - Fedor Petrov bibliography: - main.bib title: Branching points in the planar Gilbert--Steiner problem have degree 3 --- # Introduction One of the first models for branched transport was introduced by Gilbert [@gilbert1967minimum]. **Definition 1**. *Let $\mu^+,\mu^-$ be two finite measures on a metric space $(X,|\cdot\cdot|)$ with finite supports such that total masses $\mu^+(X)=\mu^-(X)$ are equal. Let $V\subset X$ be a finite set containing the support of the signed measure $\mu^+-\mu^-$, the elements of $V$ are called *vertices*. Further, let $E$ be a finite collection of unordered pairs $\{x,y\}\subset V$ which we call *edges*. So, $(V,E)$ is a simple undirected finite graph. Assume that for every $\{x,y\}\in E$ two non-zero real numbers $m(xy)$ and $m(yx)$ are defined so that $m(xy)+m(yx)=0$. This data set is called a *$(\mu^+,\mu^-)$-flow* if $$\mu^- - \mu^+=\sum_{\{x,y\}\in E} m(xy)\cdot (\delta_y-\delta_x)$$ where $\delta_x$ denotes a delta-measure at $x$ (note that the summand $m(xy)\cdot (\delta_y-\delta_x)$ is well-defined in the sense that it does not depend on the order of $x$ and $y$.)* Let $C\colon (0,\infty)\to (0,\infty)$ be a *cost function*. The expression $$\sum_{\{x,y\}\in E} C(|m(xy)|) \cdot |xy|$$ is called the *Gilbert functional* of the $(\mu^+,\mu^-)$-flow. The *Gilbert--Steiner problem* is to find the flow which minimizes the Gilbert functional with cost function $C(x) = x^p$, for a fixed $p \in (0,1)$; we call a solution *minimal flow*. Vertices from $\mathop{\mathrm{supp}}(\mu^+) \setminus \mathop{\mathrm{supp}}(\mu^-)$ are called *terminals*. A vertex from $V \setminus \mathop{\mathrm{supp}}(\mu^+) \setminus \mathop{\mathrm{supp}}(\mu^-)$ is called a *branching point*. Formally, we allow a branching point to have degree 2, but clearly it never happens in a minimal flow. Local structure in the Gilbert--Steiner problem was discussed in [@bernot2008optimal], and the paper [@lippmann2022theory] deals with planar case. A local picture around a branching point $b$ of degree 3 is clear due to the initial paper of Gilbert. Similarly to the finding of the Fermat--Torricelli point in the celebrated Steiner problem one can determine the angles around $b$ in terms of masses (see Lemma [\[main_l\]](#main_l){reference-type="ref" reference="main_l"}). **Theorem 1** (Lippmann--Sanmartı́n--Hamprecht [@lippmann2022theory], 2022). *A solution of the planar Gilbert--Steiner problem has no branching point of degree at least 5.* The goal of this paper is to give some conditions on a cost function under which all branching points in a planar solution have degree 3. In particular, this covers the case of the standard cost function $x^p$, $0<p<1$. The following main theorem is the part of a more general Theorem [Theorem 3](#theorem:general){reference-type="ref" reference="theorem:general"}. **Theorem 2**. *A solution of the planar Gilbert--Steiner problem has no branching point of degree at least 4.* # Preliminaries We need the following lemmas. **Lemma 1** (Folklore). *Let $PQR$ be a triangle and $w_1$, $w_2$, $w_3$ be non-negative reals. For every point $X \in \mathbb{R}^2$ consider the value $$L(X) := w_1 \cdot |PX| + w_2 \cdot |QX| + w_3 \cdot |RX|.$$ Then* - *a minimum of $L(X)$ is achieved at a unique point $X_{min}$;* - *if $X_{min} = P$ then $w_1 \geq w_2 + w_3$ or there is a triangle $\Delta$ with sides $w_1$, $w_2$, $w_3$ and $\angle P$ is at least the outer angle between $w_2$ and $w_3$ in $\Delta$.* *[\[main_l\]]{#main_l label="main_l"}* Hereafter the metric space is the Euclidean plane $\mathbb{R}^2$. **Definition 2**. *Let $\lambda$ be a Borel measure on $\mathbb{R}$ for which $\int \min(x^2,1)d\lambda(x)<\infty$. Assume additionally that the support of $\lambda$ is uncountable. A function $f\colon \mathbb{R}_{\geqslant 0}\to \mathbb{R}_{\geqslant 0}$ of the form $$f(t)=\sqrt{\int \sin^2 tx d\lambda(x)}$$ is called *admissible*.* Further we are going to consider only admissible cost functions. Note that admissibility implies some properties one may expect from a cost function, in particular subadditivity and $f(0) = 0$. On the other hand it does not imply monotonicity (for instance for $\mathop{\mathrm{supp}}\lambda \subset [0.9, 1.1]$ we have $f(\pi) < f(\pi/2)$). Hereafter $L^2(\lambda)$ for a measure $\lambda$ on $\mathbb{R}$ is understood as a *real* Hilbert space of complex-valued square summable w.r.t. $\lambda$ functions (strictly speaking, of classes of equivalences of such functions modulo coincidence $\lambda$-almost everywhere). **Proposition 1**. *If $\lambda$ is a Borel measure on $\mathbb{R}$ with uncountable support such that $$\int \min(x^2,1)d\lambda(x)<\infty,$$ then any finite collection of functions of the form $e^{iax}-1$, $a\in \mathbb{R}$, is affinely independent in $L^2(\lambda)$.* *Proof.* Assume the contrary. Then there exist distinct real numbers $a_1,\ldots,a_n$ and non-zero real coefficients $t_1,\ldots,t_n$ such that $\sum t_j=0$ and $\sum t_j(e^{ia_jx}-1)=0$ $\lambda$-almost everywhere. But the analytic function $\sum t_j(e^{ia_jx}-1)$ is either identically zero, or has at most countably many (and separated) zeroes. In the latter case, it is not zero $\lambda$-almost everywhere, since the support of $\lambda$ is uncountable. The former case is not possible: indeed, if $\sum t_je^{ia_jx}\equiv 0$, then taking the Taylor expansion at 0 we get $\sum t_j a_j^k=0$ for all $k=0,1,2,\ldots$. Therefore $\sum t_j W(a_j)=0$ for any polynomial $W$. Choosing $W(t)=\prod_{j=2}^n (t-a_j)$ we get $t_1=0$, a contradiction. ◻ **Lemma 2**. *Let $C$ be an admissible cost function. Define $h(m_1,m_2)$ as the value of the outer angle in the triangle with sides $C(|m_1|)$, $C(|m_2|)$, $C(|m_1+m_2|)$ (it exists by Proposition [Proposition 1](#independence){reference-type="ref" reference="independence"}) for real $m_1,m_2$. Suppose that $OV_1$, $OV_2$ are edges in a minimal flow with masses $m_1$ and $m_2$. Then the angle between $OV_1$ and $OV_2$ is at least $h(m_1,m_2)$.* *Proof.* Assume the contrary, then by Lemma [\[main_l\]](#main_l){reference-type="ref" reference="main_l"} with $P = O$, $Q = V_1$, $R = V_2$, $w_1 = C(|m_1|)$, $w_2 = C(|m_2|)$, $w_3 = C(|m_1+m_2|)$ we have $X_{min} \neq O$. Then we can replace $[OV_1] \cup [OV_2]$ with $[X_{min}O] \cup [X_{min}V_1] \cup [X_{min}V_2]$ with the corresponding masses in our flow; this contradicts the minimality of the flow. ◻ **Lemma 3**. *For $0<p<1$, the function $f(x)=x^p$ is admissible.* *Proof.* Consider the measure $d\lambda=x^{-2p-1}dx$ on $[0,\infty)$. Then $\int_0^\infty \min(x^2,1)d\lambda<\infty$ and for $t>0$ we have $$\int_0^\infty \sin^2txd\lambda(x)= \int_0^\infty \sin^2tx \, x^{-2p-1}dx= t^{2p}\int_0^\infty \sin^2 y \, y^{-2p-1}dy,$$ thus the measure $\lambda$ multiplied by an appropriate positive constant proves the result. ◻ The following lemma is essentially well-known, but for the sake of completeness and for covering degeneracies and the equality cases we provide a proof. **Lemma 4**. *Let $X$ be a finite-dimensional Euclidean space, let the points $A_0,A_1,A_2,\ldots,A_{n-1},A_n=A_0,A_{n+1}=A_1$ in $X$ be chosen so that $A_i\ne A_{i+1}$ for all $i=1,2,\ldots,n$. Denote $\varphi_i:=\pi-\angle A_{i-1}A_iA_{i+1}$ for $i=1,2,\ldots,n$. Then $\sum \varphi_i\geqslant 2\pi$, and if the equality holds then the points $A_1,\ldots,A_n$ belong to the same two-dimensional affine plane.* *Proof.* Let $u$ be a randomly chosen unit vector in $X$ (with respect to a uniform distribution on the sphere). For $j=1,2,\ldots,n$ denote by $U(j)$ the following event: $\langle u,A_j\rangle =\max_{1\leqslant i\leqslant n} \langle u,A_i\rangle$, where $\langle \cdot,\cdot\rangle$ denotes the inner product in $X$; and by $V(j)$ the event $\langle u,A_j\rangle =\max_{j-1\leqslant i\leqslant j+1} \langle u,A_i\rangle$. Obviously, $\mathop{\mathrm{prob}}U(j)\leqslant \mathop{\mathrm{prob}}V(j)$. Also, $\mathop{\mathrm{prob}}V(j)=\frac{\varphi_j}{2\pi}$, since the set of directions of $u$ for which $V(j)$ holds is the dihedral angle of measure $\varphi_j$. Thus, since always at least one event $U(j)$ holds, we get $$1\leqslant \sum_{j=1}^n \mathop{\mathrm{prob}}U(j)\leqslant \sum_{j=1}^n \mathop{\mathrm{prob}}V(j)=\frac1{2\pi}\sum_{j=1}^n \varphi_j.$$ This proves the inequality. It remains to prove that it is strict assuming that not all the points belong to a two-dimensional plane. Note that if every three consecutive points $A_{j-1},A_j,A_{j+1}$ are collinear, then all the points $A_1,\ldots,A_n$ are collinear that contradicts to our assumption. If $A_{j-1},A_j,A_{j+1}$ are not collinear, denote by $\alpha$ the two-dimensional plane they belong to. There exists $i$ for which $A_i\notin \alpha$. Then $\mathop{\mathrm{prob}}U(j)<\mathop{\mathrm{prob}}V(j)$, since there exist planes passing through $A_j$ which separate the triangle $A_{j-1}A_jA_{j+1}$ and the point $A_i$, and the measure of directions of such planes is strictly positive. Therefore, our inequality is strict. ◻ # Main result **Theorem 3**. *Let $\mu^+,\mu^-$ be two measures with finite support on the Euclidean plane $\mathbb{R}^2$, and assume that the cost function $C$ is admissible. Then if a $(\mu^+,\mu^-)$-flow has a branching point of degree at least 4, then there exists a $(\mu^+,\mu^-)$-flow with strictly smaller value of Gilbert functional.* *Proof.* Assume the contrary. Let $O$ be a branching point, $OV_1,OV_2,\ldots,OV_k$, $k\geqslant 4$, be the edges incident to $O$, enumerated counterclockwise. Further the indices of $V_i$'s are taken modulo $k$, so that $V_1=V_{k+1}$ etc. Denote $m_i=m(OV_i)$, then by the definition of flow we get $\sum m_i=0$. By Lemma [Lemma 2](#d1){reference-type="ref" reference="d1"}, $\angle V_iOV_{i+1}\geqslant h(m_i,m_{i+1})$. Consider the functions $A_j(x):=e^{i(m_1+\ldots+m_j)x}-1$ for $j=1,2,\ldots$ (here $i$ is the imaginary unit). Then $\sum m_j=0$ yields that $A_{j+k}\equiv A_j$ for all $j>0$. Since the cost function $C(t)$ is admissible, there exists a Borel measure $\lambda$ on $\mathbb{R}$ with uncountable support such that $\int \min(x^2,1)d\lambda(x)<\infty$ and $$C(t)=\sqrt{\int 4\sin^2 \frac{tx}2 d\lambda(x)}.$$ Using the identity $|e^{ia}-e^{ib}|^2=4\sin^2\frac{a-b}2$ for real $a,b$ we note that for $j,s>0$ in the Hilbert space $L^2(\lambda)$ we have $$\|A_{j+s}-A_j\|^2 = C(m_{j+1}+\ldots+m_{j+s})^2.$$ In particular, the lengths of the sides of the triangle $A_{j-1}A_jA_{j+1}$ are equal to $C(m_j),C(m_{j+1}),C(m_{j}+m_{j+1})$. Therefore $\varphi_j:=\pi-\angle A_{j-1}A_jA_{j+1} = h(m_j,m_{j+1})$. By Lemma [Lemma 4](#f1){reference-type="ref" reference="f1"} we get $\sum \varphi_j\geqslant 2\pi$. By Lemma [\[main_l\]](#main_l){reference-type="ref" reference="main_l"}, this yields $2\pi=\sum_{j=1}^k \angle V_jOV_{j+1}\geqslant \sum \varphi_j \geqslant 2\pi$. Therefore, the equality must take place. Again by Lemma [Lemma 4](#f1){reference-type="ref" reference="f1"} it follows that the points $A_j$ belong to the same 2-dimensional subspace. But by Proposition [Proposition 1](#independence){reference-type="ref" reference="independence"}, distinct points between $A_j$'s are affinely independent. Therefore, there exist at most three distinct $A_j$'s, and if exactly three, they are not collinear. It is easy to see that the equality $\sum \varphi_j=2\pi$ under these conditions does not hold when $k>3$. A contradiction. ◻ # Open questions Our definition of admissible functions is motivated by the famous Bochner characterization of positive definite kernels. It would be interesting to describe all cost functions for which the conclusion of Theorem [Theorem 3](#theorem:general){reference-type="ref" reference="theorem:general"} holds. Now let us focus on the cost function $C(x) = x^p$. Having a knowledge that every branching point has degree 3 one can adapt Melzak algorithm [@melzak1961problem] from Steiner trees to Gilbert--Steiner problem. The idea of the algorithm is that after fixing the combinatorial structure one can find two terminals $t_1$, $t_2$ connected with the same branching point $b$. Then one may reconstruct the solution for $V$ from the solution for $V \setminus \{t_1,t_2\} \cup \{t'\}$ for a proper $t'$ which depend only on $t_1,t_2$. When the underlying graph is a matching we finish in an obvious way. Application of this procedure for all possible combinatorial structures gives a slow but mathematically exhaustive algorithm in the planar case. However there is no known algorithm in $\mathbb{R}^d$ for $d > 2$ (see Problem 15.12 in [@bernot2008optimal]). Recall that we have to consider a high-degree branching. A naturally related problem is to evaluate the maximal possible degree of a branching point in the $d$-dimensional Euclidean space for every $d$. Note that the dependence on the cost function may be very complicated. Some other questions are collected in Section 15 of [@bernot2008optimal] (note that some of them may be solved, in particular Problem 15.1 is solved in [@colombo2021well]). #### Acknowledgements. The research is supported by RSF grant 22-11-00131. The authors are grateful to Guy David and Yana Teplitskaya for sharing the problem.

arxiv_math

--- abstract: | We present two geometric applications of heat flow methods on the discrete hypercube $\{-1,1\}^n$. First, we prove that if $X$ is a finite-dimensional normed space, then the bi-Lipschitz distortion required to embed $\{-1,1\}^n$ equipped with the Hamming metric into $X$ satisfies $$\mathsf{c}_X\big(\{-1,1\}^n\big) \gtrsim \sup_{p\in[1,2]} \frac{n}{\mathsf{T}_p(X) \min\{n,\mathrm{dim}(X)\}^{1/p}},$$ where $\mathsf{T}_p(X)$ is the Rademacher type $p$ constant of $X$. This estimate yields a mutual refinement of distortion lower bounds which follow from works of Oleszkiewicz (1996) and Ivanisvili, van Handel and Volberg (2020) for low-dimensional spaces $X$. The proof relies on an extension of an important inequality of Pisier (1986) on the biased hypercube combined with an application of the Borsuk--Ulam theorem from algebraic topology. Secondly, we introduce a new metric invariant called metric stable type as a functional inequality on the discrete hypercube and prove that it coincides with the classical linear notion of stable type for normed spaces. We also show that metric stable type yields bi-Lipschitz nonembeddability estimates for weighted hypercubes. address: CNRS, Institut de Mathématiques de Jussieu, Sorbonne Université, France and Trinity College, University of Cambridge, UK. author: - Alexandros Eskenazis bibliography: - flow-applications.bib title: Some geometric applications of the discrete heat flow --- Primary: 46B85; Secondary: 30L15, 42C10, 46B07. Hamming cube, Rademacher type, metric embeddings, Borsuk--Ulam theorem, stable type. # Introduction Let $\{-1,1\}^n$ be the $n$-dimensional discrete hypercube equipped with the Hamming metric $$\forall\ x,y\in\{-1,1\}^n,\qquad \rho(x,y) = \frac{1}{2} \sum_{i=1}^n |x(i)-y(i)|,$$ where $x=(x(1),\ldots,x(n))$ and $y=(y(1),\ldots,y(n))$. The purpose of the present paper is to investigate certain metric properties of the Hamming cube via heat flow methods. ## Dimensionality of Hamming metrics If $(\mathsf{M},d_\mathsf{M})$ is a metric space and $(Y,\|\cdot\|_Y)$ is a normed space, we say that $\mathsf{M}$ embeds into $Y$ with bi-Lipschitz distortion at most $D\in[1,\infty)$, if there exists a mapping $f:\mathsf{M}\to Y$ satisfying $$\label{eq:bi-Lip} \forall \ p,q\in\mathsf{M},\qquad d_\mathsf{M}(p,q) \leq \|f(p)-f(q)\|_Y \leq D d_\mathsf{M}(p,q).$$ The least $D\geq1$ for which such an embedding exists will be denoted by $\mathsf{c}_Y(\mathsf{M})$. The rapidly growing field of metric dimension reduction aims to uncover conditions under which given families of metric spaces admit (or do not admit) embeddings into low-dimensional normed spaces with prescribed properties. Without attempting to survey this vast area, we note that important contributions have been made on low-dimensional embeddings of finite subsets of Hilbert space [@JL84], arbitrary finite metric spaces [@JLS87; @AdRRP92; @Mat92; @Mat96], discrete hypercubes [@Ole96; @LMN05], diamond graphs [@BC05; @LN04; @NPS20], Laakso graphs [@GKL03; @LMN05], ultrametric spaces [@BM04], series-parallel graphs [@BKL07], recursive cycle graphs [@ACNN11], Heisenberg-type metrics [@LN14; @NY22], $\ell_p$ variants of thin Laakso structures [@BGN15; @BSS21] and expander graphs [@Nao17; @Nao21]. We refer to the survey [@Nao18] for more bibliographic information and to [@Ind01; @Lin02; @Vem05; @AIR18] for a sample of algorithmic applications. The first to study the bi-Lipschitz embeddability of hypercubes into normed spaces was Enflo. In the seminal work [@Enf69], he introduced the notion of *roundness* of a metric space and used it to show that any embedding of the Hamming cube $\{-1,1\}^n$ into an $L_p(\mu)$ space, where $p\in[1, 2]$, incurs bi-Lipschitz distortion at least $n^{1-1/p}$ (see also [@Enf70; @Enf78] for additional early results along these lines). More specifically, Enflo proved that if $p\in[1,2]$, then any mapping of the form $f:\{-1,1\}^n\to L_p(\mu)$ satisfies the estimate $$\label{eq:enflo} \int_{\{-1,1\}^n} \big\| f(x)-f(-x)\big\|_{L_p(\mu)}^p \,\mathop{}\!\mathrm{d}\sigma_n(x) \leq \sum_{i=1}^n \int_{\{-1,1\}^n} \big\|f(x)-f\big(x(1),\ldots,-x(i),\ldots,x(n)\big)\big\|_{L_p(\mu)}^p\,\mathop{}\!\mathrm{d}\sigma_n(x),$$ where $\sigma_n$ is the uniform probability on $\{-1,1\}^n$. This readily implies that if $f$ has bi-Lipschitz distortion $D$, then $D\geq n^{1-1/p}$. In the follow-up work [@Enf78], he raised an influential problem by asking for which normed spaces $(X,\|\cdot\|_X)$, inequality [\[eq:enflo\]](#eq:enflo){reference-type="eqref" reference="eq:enflo"} is satisfied for $X$-valued functions $f$ up to a multiplicative constant $T$, independent of the choice of $f$ or the dimension $n$. Restricting this requirement to linear functions $f(x) = \sum_{i=1}^n x_i v_i$, one recovers the necessary condition $$\label{eq:type} \int_{\{-1,1\}^n} \Big\| \sum_{i=1}^n x_i v_i\Big\|_X^p\,\mathop{}\!\mathrm{d}\sigma_n(x) \leq T^p \sum_{i=1}^n \|v_i\|_X^p,$$ which must be satisfied for every $n\in\mathbb{N}$ and vectors $v_1,\ldots,v_n\in X$. If a normed space $X$ satisfies [\[eq:type\]](#eq:type){reference-type="eqref" reference="eq:type"}, we say that $X$ has *Rademacher type* $p$ and the least constant $T$ is denoted by $\mathsf{T}_p(X)$. After decades of substantial efforts (see [@BMW86; @Pis86; @NS02; @HN13; @Esk21]), Ivanisvili, van Handel and Volberg resolved Enflo's problem in the breakthrough work [@IVV20] by proving the sufficiency of this condition, namely that any normed space of Rademacher type $p$ also has Enflo's nonlinear type $p$. Consequently, any bi-Lipschitz embedding of $\{-1,1\}^n$ into a normed space $X$ of Rademacher type $p$ incurs distortion at least a constant multiple of $\mathsf{T}_p(X)^{-1}n^{1-1/p}$. We note in passing that, conversely, a classical theorem of Pisier [@Pis73] implies that if $X$ does not have type $p$ for any $p>1$, then $\{-1,1\}^n$ embeds into $X$ with bi-Lipschitz distortion at most $1+\varepsilon$, for any $\varepsilon>0$. Independently of this line of research, the beautiful (but perhaps overlooked) work [@Ole96] of Oleszkiewicz established a nonembeddability result for discrete hypercubes in the context of dimensionality reduction. Following Ball, Carlen and Lieb [@BCL94], we say that a normed space is $p$-*uniformly smooth*, where $p\in[1,2]$, if there exists a constant $S>0$ such that $$\label{eq:smooth} \forall \ x,y\in X,\qquad \frac{\|x\|_X^p+\|y\|_X^p}{2}\leq \Big\|\frac{x+y}{2}\Big\|_X^p +S^p \Big\|\frac{x-y}{2}\Big\|_X^p;$$ the least such constant $S$ is denoted by $\mathsf{S}_p(X)$. A well-known tensorization argument due to Pisier [@Pis75] shows that $\mathsf{T}_p(X) \leq \mathsf{S}_p(X)$, yet there exist examples of normed spaces $X$ for which $\mathsf{T}_p(X)<\infty$ whereas $\mathsf{S}_p(X)=\infty$ for $p\in(1,2]$, see [@Pis75b; @Jam78; @PX87]. The main result of Oleszkiewicz's paper [@Ole96] asserts that the distortion required to embed $\{-1,1\}^n$ into a finite-dimensional normed space $X$ satisfies $$\label{eq:ole} \mathsf{c}_X\big(\{-1,1\}^n\big) \geq \sup_{p\in[1,2]} \frac{n}{\mathsf{S}_p(X) \min\{n,\dim(X)\}^{1/p}}$$ (see also [@BG81] for a precursor of this result for linear embeddings). This substantially improves the bound $\mathsf{c}_X(\{-1,1\}^n)\gtrsim \sup_{p\in[1,2]} \mathsf{T}_p(X)^{-1}n^{1-1/p}$ which follows from [@IVV20], at least for spaces $X$ with $p$-smoothness constant $\mathsf{S}_p(X)\asymp 1$ and dimension $\dim(X)<\!\!\!< n$. The first goal of the present paper is to revisit the technique used for Oleszkiewicz's nonembeddability theorem [@Ole96], in particular proving the following mutual refinement of his result and of the recent work of Ivanisvili, van Handel and Volberg [@IVV20]. **Theorem 1**. *Let $(X,\|\cdot\|_X)$ be a finite-dimensional normed space. Then, for any $n\geq1$, we have $$\label{eq:main} \mathsf{c}_X\big(\{-1,1\}^n\big) \gtrsim \sup_{p\in[1,2]} \frac{n}{\mathsf{T}_p(X) \min\{n,\dim(X)\}^{1/p}}$$* We emphasize that, in contrast to Oleszkiewicz's bound [\[eq:ole\]](#eq:ole){reference-type="eqref" reference="eq:ole"}, Theorem [Theorem 1](#thm:main){reference-type="ref" reference="thm:main"} captures more accurately the nonembeddability of the hypercube into (finite-dimensional subspaces of) normed spaces which have Rademacher type $p$ but are not $r$-smooth for any $r\in(1,2]$, see [@Jam78; @PX87]. ### About the proof {#about-the-proof .unnumbered} Theorem [Theorem 1](#thm:main){reference-type="ref" reference="thm:main"} is proven by a combination of semigroup tools with a clever topological trick of [@Ole96]. More specifically, let $f:\{-1,1\}^n\to X$ be a function, where $X$ is a $d$-dimensional normed space and $d<n$. An application of the Borsuk--Ulam theorem [@Mat03] for the unique multilinear extension of $f$ implies that there exists a subset $\sigma\subseteq\{1,\ldots,n\}$ with $|\sigma|=d$, a *product* measure $\nu$ on $\{-1,1\}^{\sigma}$ and a point $w\in\{-1,1\}^{\sigma^c}$ such that $$\label{eq:top-info} \int_{\{-1,1\}^\sigma} f(x,w) \,\mathop{}\!\mathrm{d}\nu(x) = \int_{\{-1,1\}^\sigma} f(-x,-w) \,\mathop{}\!\mathrm{d}\nu(x).$$ Then, a Poincaré inequality à la Enflo for the product measure $\nu$ (instead of the uniform measure $\sigma_d$) on the $d$-dimensional subcubes $\{-1,1\}^\sigma\times\{w\}$ and $\{-1,1\}^\sigma\times\{-w\}$ yields the distortion bounds of Theorem [Theorem 1](#thm:main){reference-type="ref" reference="thm:main"} (see Theorem [Theorem 14](#thm:poincare){reference-type="ref" reference="thm:poincare"} and also equation [\[eq:real-conclusion\]](#eq:real-conclusion){reference-type="eqref" reference="eq:real-conclusion"} below). In the case of $p$-uniformly smooth spaces, Oleszkiewicz [@Ole96] used [\[eq:top-info\]](#eq:top-info){reference-type="eqref" reference="eq:top-info"} and a bootstrap argument for the Lipschitz constant of $f$, based on the two-point inequality [\[eq:smooth\]](#eq:smooth){reference-type="eqref" reference="eq:smooth"}, to obtain [\[eq:ole\]](#eq:ole){reference-type="eqref" reference="eq:ole"}. In our case, the biased Poincaré inequality which will yield [\[eq:main\]](#eq:main){reference-type="eqref" reference="eq:main"} is an extension of an inequality for the uniform measure that was proven in [@IVV20]. The key technical contribution of [@IVV20] was a novel representation of the time derivative of the heat flow on $\{-1,1\}^n$. Instead, we consider a Markov process having the product measure $\nu$ as stationary measure (see Section [2](#sec:prel){reference-type="ref" reference="sec:prel"}) and prove a formula for the time derivative of the corresponding semigroup (see Proposition [Proposition 13](#prop:ident){reference-type="ref" reference="prop:ident"}) which extends the formula of [@IVV20] (see also [\[eq:ivv\]](#eq:ivv){reference-type="eqref" reference="eq:ivv"} below). Due to the fact that our product measure $\nu$ is no longer the stationary measure of the random walk on a group, the resulting identity lacks some homogeneity properties that were used in [@IVV20], but nevertheless it is sufficient for the proof of the biased Poincaré inequality which is needed for our geometric application. ## A Pisier--Talagrand inequality on the biased cube For $\alpha\in(0,1)$ we denote by $\mu_\alpha$ the $\alpha$-biased probability measure on $\{-1,1\}$ with $\mu_\alpha\{1\}=\alpha$ and $\mu_\alpha\{-1\}=1-\alpha$. The proof of Theorem [Theorem 1](#thm:main){reference-type="ref" reference="thm:main"} yields as a consequence the following extension and refinement of Pisier's inequality [@Pis86]. **Theorem 2**. *For every $p\in[1,\infty)$ and $\alpha\in(0,1)$, there exist $\mathsf{K}_{p,\alpha}, \mathsf{C}_\alpha\in(0,\infty)$ such that the following holds. For any normed space $(X,\|\cdot\|_X)$ and any $n\in\mathbb{N}$, every function $f:\{-1,1\}^n\to X$ satisfies $$\label{eq:p1} \begin{split} \bigg\| f - \int_{\{-1,1\}^n} f\,\mathop{}\!\mathrm{d}\mu_\alpha^n\bigg\|_{L_p(\log L)^{p/2}(\mu_\alpha^n;X)} \leq \mathsf{K}_{p,\alpha} & \left( \int_{\{-1,1\}^n} \Big\| \sum_{i=1}^n \delta_i \partial_i^\alpha f\Big\|_{L_p(\mu_\alpha^n;X)}^p \,\mathop{}\!\mathrm{d}\sigma_n(\delta)\right)^{1/p} \\ & + \mathsf{C}_{\alpha}(\log n+1) \int_{\{-1,1\}^n} \Big\|\sum_{i=1}^n\delta_i \partial_i^\alpha f\Big\|_{L_1(\mu_\alpha^n;X)}\,\mathop{}\!\mathrm{d}\sigma_n(\delta). \end{split}$$ If additionally $X$ is assumed to be of finite cotype, then there exists $\mathsf{K}_{p,\alpha}(X)\in(0,\infty)$ such that $$\label{eq:p2} \bigg\| f - \int_{\{-1,1\}^n} f\,\mathop{}\!\mathrm{d}\mu_\alpha^n\bigg\|_{L_p(\log L)^{p/2}(\mu_\alpha^n;X)} \leq \mathsf{K}_{p,\alpha}(X) \left( \int_{\{-1,1\}^n} \Big\| \sum_{i=1}^n \delta_i \partial_i^\alpha f\Big\|_{L_p(\mu_\alpha^n;X)}^p \,\mathop{}\!\mathrm{d}\sigma_n(\delta)\right)^{1/p}.$$* The (standard) definitions of Orlicz norms, cotype and discrete derivatives will be given at the main part of the article. Theorem [Theorem 2](#thm:pisier){reference-type="ref" reference="thm:pisier"} is an optimal vector-valued version of a deep logarithmic Sobolev inequality of Talagrand [@Tal93]. In the case of the uniform measure which corresponds to $\alpha=\tfrac{1}{2}$, Theorem [Theorem 2](#thm:pisier){reference-type="ref" reference="thm:pisier"} was proven recently in [@CE24]. ## Metric stable type Besides its interest in the context of embedding theory, the work [@IVV20] of Ivanisvili, van Handel and Volberg was a major contribution to a long-standing research program in nonlinear functional analysis called the *Ribe program*. Put simply, the foundational idea of the program (as put forth by Bourgain [@Bou86], who was inspired by a landmark rigidity theorem of Ribe [@Rib76]) is that isomorphic local properties of normed spaces can be equivalently reformulated using only distances between pairs of points, with no reference to the spaces' linear structure (see also [@Nao12; @Bal13]). In this sense, the main result of [@IVV20] completes the Ribe program for Rademacher type by proving that it is equivalent to nonlinear Enflo type. Among the various isomorphic properties studied in the local theory of normed spaces, *stable type* (see [@Pis74a; @Pis74b]) plays a prominent role, in particular due to its relation to the problem of identifying $\ell_p^n$ subspaces of Banach spaces [@Pis83]. Recall that a symmetric random variable $\theta$ is distributed according to the standard $p$-stable law, where $p\in(0,2]$, if it satisfies $\mathbb{E} \exp(it\theta) = \exp(-|t|^p)$ for every $t\in\mathbb{R}$. A Banach space $(X,\|\cdot\|_X)$ is said to have stable type $p$ if for every $r<p$, every $n\in\mathbb{N}$ and every vectors $v_1,\ldots,v_n\in X$, we have $$\Big( \mathbb{E} \Big\| \sum_{i=1}^n \theta_i v_i\Big\|_X^r \Big)^{1/r} \leq \mathsf{C}_{p,r}(X) \Big( \sum_{i=1}^n \|v_i\|_X^p\Big)^{1/p},$$ where the constant $\mathsf{C}_{p,r}(X)$ depends only on $p, r$ and $X$, but not on $n$ or the choice of $v_1,\ldots,v_n$, and $\theta_1,\theta_2,\ldots$ are independent standard symmetric $p$-stable random variables. In the special case $p=2$, $p$-stable random variables are Gaussians and thus stable type 2 coincides with Rademacher type 2. Moreover, any space with stable type $p\in[1,2)$ also has Rademacher type $p$ but the converse does not necessarily hold. In fact, early results of Maurey and Pisier in the theory of type (see [@Pis74a Proposition 3] and [@Pis74b Théorème 1]), show that a normed space $X$ has stable type $p\in[1,2)$ if and only if $X$ has Rademacher type $p+\varepsilon$ for some $\varepsilon>0$. Therefore, formally, the Ribe program for stable type $p\in[1,2)$ has been completed by [@IVV20]: a normed space $X$ has stable type $p$ if and only if $X$ has Enflo type $p+\varepsilon$ for some $\varepsilon>0$. In other words, stable type $p$ of a normed space $X$ can be characterized by the validity of some, a priori unknown, $X$-valued Poincaré-type inequality from a given family. To amend this, we shall introduce a new nonlinear invariant called *metric stable type* which is again a Poincaré-type inequality for functions defined on the discrete hypercube and will serve as a metric characterization of stable type for normed spaces. If $(\mathsf{M},d_\mathsf{M})$ is a metric space and $f:\{-1,1\}^n\to \mathsf{M}$ is an $\mathsf{M}$-valued function, we denote $$\forall \ x\in\{-1,1\}^n, \qquad \mathfrak{d}_if(x) \stackrel{\mathrm{def}}{=}\tfrac{1}{2}d_\mathsf{M}\big(f(x), f\big(x(1),\ldots,-x(i),\ldots,x(n)\big)\big).$$ Moreover, we shall denote the weak $\ell_p$ norm on $\mathbb{R}^n$ by $$\forall \ w\in\mathbb{R}^n,\qquad \|w\|_{\ell_{p,\infty}^n} \stackrel{\mathrm{def}}{=}\sup_{r\geq0} \big\{ r\cdot \# \{i: \ |w_i|\geq r\}^{1/p}\big\},$$ where $\# S$ denotes the cardinality of a finite set $S$. **Definition 3**. *A metric space $(\mathsf{M},d_\mathsf{M})$ has metric stable type $p\in(0,2)$ with constant $S\in(0,\infty)$ if for any $n\in\mathbb{N}$, every function $f:\{-1,1\}^n\to\mathsf{M}$ satisfies $$\label{eq:st} \int_{\{-1,1\}^n} d_\mathsf{M}\big( f(x),f(-x)\big)^p \,\mathop{}\!\mathrm{d}\sigma_n(x) \leq S^p \int_{\{-1,1\}^n} \big\| \big(\mathfrak{d}_1f(x),\ldots,\mathfrak{d}_nf(x)\big)\big\|^p_{\ell_{p,\infty}^n} \,\mathop{}\!\mathrm{d}\sigma_n(x).$$* Since $\|\cdot\|_{\ell_{p,\infty}^n} \leq\|\cdot\|_{\ell_p^n}$ by Markov's inequality, any space with metric stable type $p$ also has Enflo type $p$ with the same constant. We shall prove the following metric characterization. **Theorem 4**. *A normed space $X$ has metric stable type $p\in[1,2)$ if and only if $X$ has stable type $p$.* Pisier's K-convexity theorem [@Pis82], asserts that a Banach space $X$ is K-convex if and only if it has Rademacher type $p$ for some $p>1$. In view of the aforementioned characterization of stable type [@Pis74a; @Pis74b], this is equivalent to $X$ having stable type 1 and thus we derive the following metric characterization of $K$-convexity by a Poincaré inequality. **Corollary 5**. *A normed space $X$ is K-convex if and only if $X$ has metric stable type $1$.* ### Embeddings {#embeddings .unnumbered} The fact that stable type of Banach spaces is a refinement of Rademacher type can also be seen at the level of (linear) embeddings. Indeed, if $X$ is a space of Rademacher type $p$, then standard considerations show that the Banach--Mazur distance between $\ell_1^n$ and any $n$-dimensional subspace of $X$ is at least a constant multiple of $n^{1-1/p}$. This estimate is sharp for $X=\ell_p$. On the other hand, if $X$ has stable type $p$, then the characterization of [@Pis74a; @Pis74b] implies that $X$ has Rademacher type $r$ for some $r>p$. Hence, there exists $\varepsilon(X)>0$ such that the Banach--Mazur distance between $\ell_1^n$ and any $n$-dimensional subspace of $X$ is at least $n^{1-1/p+\varepsilon(X)}$. In the metric setting, Enflo type gives distortion lower bounds for embeddings of $\{-1,1\}^n$. More generally, given a vector ${\bf w}=(w_1,\ldots,w_n) \in\mathbb{R}_+^n$, consider the metric space $\{-1,1\}^n_{{\bf w}}$ which is the hypercube $\{-1,1\}^n$ equipped with the *weighted* Hamming metric $$\forall \ x,y\in\{-1,1\}^n,\qquad \rho_{{\bf w}} (x,y) = \frac{1}{2} \sum_{i=1}^n w_i |x(i)-y(i)|.$$ It follows readily from the definitions that if $\mathsf{M}$ has Enflo type $p$, then any embedding of $\{-1,1\}^n_{{\bf w}}$ into $\mathsf{M}$ incurs distortion at least a constant multiple of $\|{\bf w}\|_{\ell_1^n}/\|{\bf w}\|_{\ell_{p}^n}$. For spaces of metric stable type $p$, we have the following improvement for the distortion of weighted hypercubes. **Proposition 6**. *If a metric space $(\mathsf{M},d_\mathsf{M})$ has metric stable type $p$ with constant $S$, then $$\label{eq:dist} \forall \ {\bf w}\in\mathbb{R}_+^n,\qquad \mathsf{c}_\mathsf{M}\big(\{-1,1\}^n_{{\bf w}} \big) \geq \frac{\|{\bf w}\|_{\ell_1^n}}{S\|{\bf w}\|_{\ell_{p,\infty}^n}}.$$* Concretely, choosing the weight vector ${\bf w}$ with $w_i = i^{-1/p}$, we have that $\|{\bf w}\|_{\ell_{p,\infty}^n} \asymp 1$, whereas $\|{\bf w}\|_{\ell_{p}^n} \asymp (\log n)^{1/p}$, therefore any embedding of $\{-1,1\}^n_{{\bf w}}$ into a metric space of Enflo type $p$ incurs distortion at least $n^{1-1/p}/(\log n)^{1/p}$ instead of the asymptotically stronger lower bound $n^{1-1/p}$ which one gets for target spaces of metric stable type $p$. ### On the nonlinear Maurey--Pisier problem for type {#on-the-nonlinear-maureypisier-problem-for-type .unnumbered} A landmark theorem of Maurey and Pisier [@MP76] asserts that if $p_X\in[1,2]$ is the supremal Rademacher type of an infinite-dimensional Banach space $X$, then $X$ containts the spaces $\ell_{p_X}^n$ uniformly. In [@Pis83], Pisier used stable type to give a much simpler proof of this important theorem, while simultaneously obtaining the following quantitative refinement: for any $\varepsilon>0$ and $p\in[1,2)$, every normed space $X$ containts a subspace which is $(1+\varepsilon)$-isomorphic to $\ell_p^d$ provided that $d$ is smaller than some explicit (unbounded) function of $\varepsilon$ and the stable type $p$ constant of $X$. When $X$ is infinite-dimensional, this yields the result of [@MP76] since $X$ does not have stable type $p_X$ by the results of [@Pis74a; @Pis74b]. In [@BMW86], Bourgain, Milman and Wolfson introduced a notion of nonlinear type, which is now referred to as BMW type, and showed that if a metric space $\mathsf{M}$ does not have BMW type $p$ for any $p>1$, then $\mathsf{c}_\mathsf{M}(\{-1,1\}^n) = 1$ for any $n\in\mathbb{N}$. Finding a satisfactory nonlinear Maurey--Pisier theorem for metric spaces of supremal nonlinear type $p>1$ remains an open problem (see also [@Nao14 Section 6]). It would be interesting to understand whether the newly introduced notion of metric stable type can lead to a nonlinear version of the result of [@Pis83]. We refer to Section [6](#sec:disc){reference-type="ref" reference="sec:disc"} for further remarks and open problems which naturally arise from this work. ## Acknowledgements {#acknowledgements .unnumbered} I wish to thank Florent Baudier, Paata Ivanisvili and Assaf Naor for their constructive feedback on this work. # Preliminaries {#sec:prel} ## Probability In this section, we outline the basics of analysis on the biased hypercube, with an emphasis on the underlying semigroup structure. ### The biased measure {#the-biased-measure .unnumbered} Recall that, for $\alpha\in(0,1)$, the $\alpha$-biased probability measure $\mu_\alpha$ on $\{-1,1\}$ is given by $\mu_\alpha\{1\}=\alpha$ and $\mu_\alpha\{-1\}=1-\alpha$. Moreover, if $\pmb{\alpha} = (\alpha_1,\ldots,\alpha_n)\in(0,1)^n$, then we shall denote by $\mu_{\pmb{\alpha}}$ the product measure $\mu_{\alpha_1}\otimes\cdots\otimes\mu_{\alpha_n}$ on the hypercube $\{-1,1\}^n$. ### The Markov process {#the-markov-process .unnumbered} For $\alpha\in(0,1)$, consider the transition matrices $\{p_t^\alpha\}_{t\geq0}$ on $\{-1,1\}$ given by $$\label{eq:transition} \forall \ t\geq0,\qquad \begin{pmatrix} p_t^\alpha(1,1) & p_t^\alpha(1,-1) \\ p_t^\alpha(-1,1) & p_t^\alpha(-1,-1) \end{pmatrix} = \begin{pmatrix} 1-(1-e^{-t})(1-\alpha) & (1-e^{-t})(1-\alpha) \\ (1-e^{-t})\alpha & 1-(1-e^{-t})\alpha \end{pmatrix}$$ Moreover, for $\pmb{\alpha} = (\alpha_1,\ldots,\alpha_n)\in(0,1)^n$ consider the corresponding tensor products $\{p_t^{\pmb{\alpha}}\}_{t\geq0}$ on $\{-1,1\}^n$ given by $$\label{eq:transition2} \forall \ x,y\in\{-1,1\}^n, \qquad p_t^{\pmb{\alpha}}(x,y) = \prod_{i=1}^n p_t^{\alpha_i}\big(x(i),y(i)\big).$$ As each $p_t^{\alpha_i}$ is a row-stochastic $2\times2$ matrix with nonnegative entries, the same holds also for the $2^n\times2^n$ matrices $p_t^{\pmb{\alpha}}$. Therefore, $\{p_t^{\pmb{\alpha}}\}_{t\geq0}$ is the transition kernel of a time-homogeneous Markov chain $\{X_t^{\pmb{\alpha}}\}_{t\geq0}$ on $\{-1,1\}^n$, that is $$\forall \ t, s\geq0,\qquad \mathbb{P}\big\{X_{t+s}^{\pmb{\alpha}} = y \ \big| \ X_s^{\pmb{\alpha}}=x \big\} = p_t^{\pmb{\alpha}}(x,y),$$ where $x,y\in\{-1,1\}^n$. We shall need the following simple facts for this process. **Lemma 7**. *Fix $\pmb{\alpha}\in(0,1)^n$ and let $\{X_t^{\pmb{\alpha}}\}_{t\geq0}$ be a Markov process on $\{-1,1\}^n$ with transition kernels $\{p_t^{\pmb{\alpha}}\}_{t\geq0}$. Then, $\{X_t^{\pmb{\alpha}}\}_{t\geq0}$ is stationary and reversible with respect to $\mu_{\pmb{\alpha}}$.* *Proof.* Due to the product structure of the Markov chain, it suffices to consider the case $n=1$, that is, to prove that for $\alpha\in(0,1)$, $$\forall \ x,y\in\{-1,1\},\qquad \mu_\alpha(x) p_t^\alpha(x,y) = \mu_\alpha(y) p_t^\alpha(y,x).$$ This follows automatically by the expression [\[eq:transition\]](#eq:transition){reference-type="eqref" reference="eq:transition"} for the transition matrix. The simple fact that reversibility implies stationarity is well-known [@LLP17 Proposition 1.20]. ◻ The stationary Markov process $\{X_t^{\pmb{\alpha}}\}_{t\geq0}$ has a simple probabilistic interpretation which we shall now describe. For $i=1,\ldots,n$, let $\{N_t(i)\}_{t\geq0}$ be $n$ independent Poisson processes of unit rate and suppose that $X_0^{\pmb{\alpha}}$ is sampled from $\mu_{\pmb{\alpha}}$ independently of $\{N_t\}_{t\geq0}$. Then, at any time $t>0$ for which the process $N_t(i)$ jumps for some $i\in\{1,\ldots,n\}$, the corresponding value $X_t^{\pmb{\alpha}}(i)$ is updated independently from $\mu_{\alpha_i}$. An explicit calculation shows that this probabilistic construction gives rise exactly to the transition kernel of [\[eq:transition\]](#eq:transition){reference-type="eqref" reference="eq:transition"} and [\[eq:transition2\]](#eq:transition2){reference-type="eqref" reference="eq:transition2"}. ### The corresponding semigroup {#the-corresponding-semigroup .unnumbered} Fix $\pmb{\alpha}\in(0,1)^n$ and let $\{P_t^{\pmb{\alpha}}\}_{t\geq0}$ be the Markov semigroup associated to the process $\{X_t^{\pmb{\alpha}}\}_{t\geq0}$. Concretely, if $X$ is a vector space, then for every function $f:\{-1,1\}^n\to X$ and $t\geq0$, we denote by $$\label{eq:sgp} \forall \ x\in\{-1,1\}^n,\qquad P_t^{\pmb{\alpha}}f(x) = \mathbb{E}\big[ f\big(X_t^{\pmb{\alpha}}\big) \ \big| \ X_0^{\pmb{\alpha}} = x\big].$$ In view of the above interpretation of $\{X_t^{\pmb{\alpha}}\}_{t\geq0}$ by means of a Poisson process, the action of the semigroup $\{P_t^{\pmb{\alpha}}\}_{t\geq0}$ can be computed via the identity $$\label{eq:sgp2} \begin{split} P_t^{\pmb{\alpha}}f & = \sum_{S\subseteq\{1,\ldots,n\}} \mathbb{P}\big\{N_t(i)>0 \ \mbox{for } i\in S \mbox{ and } N_t(i)=0 \mbox{ for } i\notin S\big\} \int_{\{-1,1\}^S} f \, \mathop{}\!\mathrm{d}\prod_{i\in S} \mu_{\alpha_i} \\ & = \sum_{S\subseteq\{1,\ldots,n\}} (1-e^{-t})^{|S|} e^{-t(n-|S|)} \int_{\{-1,1\}^S} f \, \mathop{}\!\mathrm{d}\prod_{i\in S} \mu_{\alpha_i}. \end{split}$$ **Lemma 8**. *Fix $\pmb{\alpha}=(\alpha_1,\ldots,\alpha_n)\in(0,1)^n$ and let $\{P_t^{\pmb{\alpha}}\}_{t\geq0}$ be the semigroup [\[eq:sgp\]](#eq:sgp){reference-type="eqref" reference="eq:sgp"}. Then, the action of its generator $\mathscr{L}_{\pmb{\alpha}}$ on a function $f:\{-1,1\}^n\to X$, where $X$ is a vector space, is given by $$\label{eq:gen} \forall \ x\in\{-1,1\}^n,\qquad \mathscr{L}_{\pmb{\alpha}} f(x) = -\sum_{i=1}^n \partial_i^{\alpha_i} f(x)$$ where $\partial_i^\beta f(x) = f(x) - \int_{\{-1,1\}} f(x_1,\ldots,x_{i-1},y,x_{i+1},\ldots,x_n) \, \mathop{}\!\mathrm{d}\mu_\beta(y)$ for $\beta\in(0,1)$.* *Proof.* The claim follows from the expression [\[eq:sgp2\]](#eq:sgp2){reference-type="eqref" reference="eq:sgp2"} of the semigroup and the definition $$\forall \ x\in\{-1,1\}^n,\qquad \mathscr{L}_{\pmb{\alpha}} f(x) = \frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}t}\Big|_{t=0} P_t^{\pmb{\alpha}}f(x). \qedhere$$ ◻ ## Topology Apart from the probabilistic elements from analysis on biased hypercubes, the proof of Theorem [Theorem 1](#thm:main){reference-type="ref" reference="thm:main"} also has a crucial topological component, following an idea of [@Ole96]. ### The Borsuk--Ulam theorem {#the-borsukulam-theorem .unnumbered} While the Poincaré-type inequality of Enflo for $X$-valued functions on $\{-1,1\}^n$ cannot capture the dimension of the target space $X$, a key part of the argument towards Theorem [Theorem 1](#thm:main){reference-type="ref" reference="thm:main"} is to show that there exists a $\dim(X)$-dimensional subcube of $\{-1,1\}^n$ along with a bias vector $\pmb{\alpha}$ for which the $\pmb{\alpha}$-biased Poincaré inequalities (see Theorem [Theorem 14](#thm:poincare){reference-type="ref" reference="thm:poincare"} below) on this subcube and its antipodal yield much better distortion lower bounds. This will be proven using the Borsuk--Ulam theorem from algebraic topology, see [@Mat03]. **Theorem 9** (Borsuk--Ulam). *For every continuous function $g:\mathbb{S}^d\to\mathbb{R}^d$, where $d\in\mathbb{N}$, there exists a point $w\in\mathbb{S}^d$ such that $g(w)=g(-w)$.* ### Multilinear extension and low-dimensional faces of the cube {#multilinear-extension-and-low-dimensional-faces-of-the-cube .unnumbered} Every function $f:\{-1,1\}^n\to X$ admits a unique multilinear extension on the solid cube $[-1,1]^n$, given by $$\label{eq:extension} \forall \ y\in[-1,1]^n, \qquad F(y) \stackrel{\mathrm{def}}{=}\sum_{S\subseteq \{-1,1\}^n} \Big( \frac{1}{2^n} \sum_{x\in\{-1,1\}^n} f(x) w_S(x) \Big) w_S(y),$$ where $w_S(a) = \prod_{i\in S} a_i$, which is usually referred to as the Fourier--Walsh expansion of $f$. Extending $f$ to the continuous cube allows for the use of topological methods. In what follows, we will exploit the fact that the cube $[-1,1]^n$ is equipped with a canonical CW complex structure. Concretely, for $d\in\{1,\ldots,n\}$, consider the subsets $$\mathscr{C}_d^n = \big\{ x \in[-1,1]^n: \ \mbox{there exists } \sigma\subseteq\{1,\ldots,n\} \mbox{ with } |\sigma|\geq n-d \mbox{ and } |x(i)|=1, \ \forall \ i\in\sigma\big\}$$ consisting of all $\ell$-dimensional faces of $[-1,1]^n$ for $\ell\leq d$, so that $\mathscr{C}_n^n=[-1,1]^n$ and $\mathscr{C}_0^n=\{-1,1\}^n$. We shall use the following elementary topological fact (see [@Ole96 Lemma 1]). **Lemma 10**. *If $d<n$, there exists a continuous map $h_d:\mathbb{S}^d\to \mathscr{C}_d^n$ with $h_d(-x)=-h_d(x)$, $\forall \ x\in\mathbb{S}^d$.* Combining this and the Borsuk--Ulam theorem, we deduce the following useful lemma. **Lemma 11**. *If $n,d\in\mathbb{N}$ with $d<n$, then for every continuous function $F:\mathscr{C}_d^n\to \mathbb{R}^d$ there exists a point $z\in\mathscr{C}_d^n$ such that F(z)=F(-z).* *Proof.* Consider the function $g\stackrel{\mathrm{def}}{=}F\circ h_d : \mathbb{S}^d\to\mathbb{R}^d$, where $h_d$ is the function of Lemma [Lemma 10](#lem:ole){reference-type="ref" reference="lem:ole"}. By the Borsuk--Ulam theorem and the oddness of $h_d$, there exists a point $w\in\mathbb{S}^d$ such that $$F(h_d(w)) = g(w) = g(-w) = F(h_d(-w)) = F(-h_d(w))$$ and the conclusion follows by choosing $z=h_d(w)\in\mathscr{C}_d^n$. ◻ # Proof of Theorem [Theorem 1](#thm:main){reference-type="ref" reference="thm:main"} {#proof-of-theorem-thmmain} We are now ready to proceed to the main part of the proof. The main analytic component is a biased version of the key formula of [@IVV20] for the time derivative of the heat flow on $\{-1,1\}^n$. Given $t\geq0$, $\alpha\in(0,1)$ and an auxiliary parameter $\theta\in\mathbb{R}$, consider the matrix $\eta_t^\alpha(\cdot,\cdot;\theta)$ given by $$\label{eq:eta} \forall \ t\geq0,\qquad \begin{pmatrix} \eta_t^\alpha(1,1;\theta) & \eta_t^\alpha(1,-1;\theta) \\ \eta_t^\alpha(-1,1;\theta) & \eta_t^\alpha(-1,-1;\theta) \end{pmatrix} = \begin{pmatrix} \frac{e^{-t}-\theta}{p_t^\alpha(1,1)} & \frac{-\theta}{p_t^\alpha(-1,1)} \\ \frac{\theta-e^{-t}}{p_t^\alpha(1,-1)} & \frac{\theta}{p_t^\alpha(-1,-1)} \end{pmatrix}.$$ For future reference, we record the following straightforward properties of $\eta_t^\alpha(\cdot,\cdot;\theta)$. **Lemma 12**. *Fix $t\geq0$ and $\alpha\in(0,1)$. Then, $$\label{eq:center} \forall \ x\in\{-1,1\}, \ \theta\in \mathbb{R}, \qquad p_t^\alpha(x,1) \eta_t^\alpha(1,x;\theta) + p_t^\alpha(x,-1) \eta_t^\alpha(-1,x;\theta) = 0$$ and $$\label{eq:second-mom} \begin{split} \min_{\theta\in\mathbb{R}} \max_{x\in\{-1,1\}} \Big\{ p_t^\alpha(x,1) \eta_t^\alpha(1,x;\theta)^2 & + p_t^\alpha(x,-1) \eta_t^\alpha(-1,x;\theta)^2 \Big\} \\ & = \frac{e^{-t}}{(e^t-1)(\sqrt{\alpha p_t^\alpha(-1,-1)}+\sqrt{(1-\alpha)p_t^\alpha(1,1)})^2} \leq \frac{1}{e^t-1}. \end{split}$$* *Proof.* The centering condition [\[eq:center\]](#eq:center){reference-type="eqref" reference="eq:center"} can be checked easily using the explicit formulas [\[eq:transition\]](#eq:transition){reference-type="eqref" reference="eq:transition"} and [\[eq:eta\]](#eq:eta){reference-type="eqref" reference="eq:eta"} of the matrices. For [\[eq:second-mom\]](#eq:second-mom){reference-type="eqref" reference="eq:second-mom"}, we compute that for any $\theta\in\mathbb{R}$, $$\begin{split} \max_{x\in\{-1,1\}} \Big\{ p_t^\alpha(x,1) \eta_t^\alpha(1,x;\theta)^2 & + p_t^\alpha(x,-1) \eta_t^\alpha(-1,x;\theta)^2 \Big\} \\ & = \max\Big\{ \frac{(e^{-t}-\theta)^2}{p_t^\alpha(1,1) p_t^\alpha(1,-1)}, \frac{\theta^2}{p_t^\alpha(-1,1) p_t^\alpha(-1,-1)},\Big\}. \end{split}$$ As this is the maximum of two quadratic functions in $\theta$, its minimum is attained at the point $\theta^\ast$ where they intersect in the interval $(0,e^{-t})$, namely at $$\theta^\ast = \frac{e^{-t}\sqrt{\alpha p_t^\alpha(-1,-1)}}{\sqrt{\alpha p_t^\alpha(-1,-1)}+\sqrt{(1-\alpha) p_t^\alpha(1,1)}}.$$ The first equality in [\[eq:second-mom\]](#eq:second-mom){reference-type="eqref" reference="eq:second-mom"} is immediate, whereas for the inequality we compute $$\begin{split} \frac{e^{-t}}{(e^t-1)(\sqrt{\alpha p_t^\alpha(-1,-1)}+\sqrt{(1-\alpha)p_t^\alpha(1,1)})^2} & \leq \frac{e^{-t}}{(e^t-1)(\alpha p_t^\alpha(-1,-1)+(1-\alpha)p_t^\alpha(1,1))} \\ & = \frac{e^{-t}}{(e^t-1)\big(1-(1-e^{-t})(\alpha^2+(1-\alpha)^2)\big)} \leq \frac{1}{e^t-1}, \end{split}$$ where both inequalities follow from the convexity of $x\mapsto x^2$. ◻ The key technical ingredient in the proof of Theorem [Theorem 1](#thm:main){reference-type="ref" reference="thm:main"} is the following identity. **Proposition 13**. *Fix $n\in\mathbb{N}$, $\pmb{\alpha}=(\alpha_1,\ldots,\alpha_n)\in(0,1)^n$, $t\geq0$ and $\theta_1,\ldots,\theta_n\in\mathbb{R}$. Then, for every function $f:\{-1,1\}^n\to X$, where $X$ is a vector space, we have $$\label{eq:identi} \forall \ x\in\{-1,1\}^n,\qquad \mathscr{L}_{\pmb{\alpha}} P_t^{\pmb{\alpha}} f(x) = - \mathbb{E}\Big[ \sum_{i=1}^n \eta_t^{\alpha_i}\big(x(i),X_t^{\pmb{\alpha}}(i);\theta_i\big) \partial_i^{\alpha_i} f(X_t^{\pmb{\alpha}}) \ \Big| \ X_0^{\pmb{\alpha}}=x\Big].$$* *Proof.* In view of [\[eq:gen\]](#eq:gen){reference-type="eqref" reference="eq:gen"} and the product structure of the process $\{X_t^{\pmb{\alpha}}\}_{t\geq0}$, it suffices to check the claim for $n=1$, namely that for every $\beta\in(0,1)$, $\theta\in\mathbb{R}$ and $f:\{-1,1\}\to X$, $$\forall \ x\in\{-1,1\},\qquad e^{-t} \partial^\beta f(x) = P_t^\beta \partial^\beta f(x) = \mathbb{E}\big[ \eta_t^\beta(x,X_t^\beta;\theta) \partial^\beta f(X_t^\beta) \ \big| \ X_0^\beta=x\big],$$ where the first equality follows from the probabilistic representation [\[eq:sgp2\]](#eq:sgp2){reference-type="eqref" reference="eq:sgp2"}. Taking into account that $\beta \partial^\beta f(1) + (1-\beta) \partial^\beta f(-1) = 0$, this amounts to the system of equations $$\begin{cases} e^{-t} = p_t^\beta(1,1) \eta_t^\beta(1,1;\theta) - \frac{\beta}{1-\beta} p_t^\beta(1,-1) \eta_t^\beta(1,-1;\theta) \\ e^{-t} = -\frac{1-\beta}{\beta} p_t^\beta(-1,1) \eta_t^\beta(-1,1;\theta) + p_t^\beta(-1,-1) \eta_t^\beta(-1,-1;\theta) \end{cases}$$ which can be easily verified by direct computation. ◻ **Theorem 14**. *Fix $p\in[1,2]$ and let $(X,\|\cdot\|_X)$ be a normed space of Rademacher type $p$. Then, for any $n\in\mathbb{N}$ and $\pmb{\alpha}=(\alpha_1,\ldots, \alpha_n)\in(0,1)^n$, every function $f:\{-1,1\}^n\to X$ satisfies $$\int_{\{-1,1\}^n}\Big\| f(x) - \int_{\{-1,1\}^n} f \,\mathop{}\!\mathrm{d}\mu_{\pmb{\alpha}} \Big\|_X^p \,\mathop{}\!\mathrm{d}\mu_{\pmb{\alpha}}(x) \leq \big(2\pi \mathsf{T}_p(X)\big)^p \sum_{i=1}^n \int_{\{-1,1\}^n} \big\| \partial_i^{\alpha_i} f(x)\big\|_X^p \,\mathop{}\!\mathrm{d}\mu_{\pmb{\alpha}}(x).$$* *Proof.* Writing $$f(x) - \int_{\{-1,1\}^n} f \,\mathop{}\!\mathrm{d}\mu_{\pmb{\alpha}} = P_0^{\pmb{\alpha}}f(x) - P_\infty^{\pmb{\alpha}}f(x) = - \int_0^\infty \mathscr{L}_{\pmb{\alpha}} P_t^{\pmb{\alpha}}f(x)\,\mathop{}\!\mathrm{d}t$$ and using Jensen's inequality and Proposition [Proposition 13](#prop:ident){reference-type="ref" reference="prop:ident"}, we see that for $\theta_1(t),\ldots,\theta_n(t)\in\mathbb{R}$, $$\label{eq:apply-sgp} \begin{split} \Bigg(\int_{\{-1,1\}^n}\Big\| &f(x) - \int_{\{-1,1\}^n} f \,\mathop{}\!\mathrm{d}\mu_{\pmb{\alpha}} \Big\|_X^p \,\mathop{}\!\mathrm{d}\mu_{\pmb{\alpha}}(x) \Bigg)^{1/p}\leq \int_0^\infty \Bigg(\int_{\{-1,1\}^n}\Big\| \mathscr{L}_{\pmb{\alpha}} P_t^{\pmb{\alpha}}f(x) \Big\|_X^p \,\mathop{}\!\mathrm{d}\mu_{\pmb{\alpha}}(x)\Bigg)^{1/p} \,\mathop{}\!\mathrm{d}t \\ & = \int_0^\infty \Bigg(\int_{\{-1,1\}^n}\Bigg\| \mathbb{E}\Big[ \sum_{i=1}^n \eta_t^{\alpha_i}\big(x(i),X_t^{\pmb{\alpha}}(i);\theta_i(t)\big) \partial_i^{\alpha_i} f(X_t^{\pmb{\alpha}}) \ \Big| \ X_0^{\pmb{\alpha}}=x\Big] \Bigg\|_X^p \,\mathop{}\!\mathrm{d}\mu_{\pmb{\alpha}}(x)\Bigg)^{1/p} \,\mathop{}\!\mathrm{d}t \\ & \leq \int_0^\infty \Bigg( \mathbb{E} \Big\| \sum_{i=1}^n \eta_t^{\alpha_i}\big(X_0^{\pmb{\alpha}}(i),X_t^{\pmb{\alpha}}(i);\theta_i(t)\big) \partial_i^{\alpha_i} f(X_t^{\pmb{\alpha}}) \Big\|_X^p \Bigg)^{1/p} \, \mathop{}\!\mathrm{d}t, \end{split}$$ where in the last expectation $X_0^{\pmb{\alpha}}$ is distributed according to $\mu_{\pmb{\alpha}}$. Now, by the reversibility of the chain, this expectation can be written as $$\label{eq:use-rev} \begin{split} \mathbb{E} \Big\| \sum_{i=1}^n \eta_t^{\alpha_i}\big(X_0^{\pmb{\alpha}}(i),X_t^{\pmb{\alpha}}(i);&\theta_i(t)\big) \partial_i^{\alpha_i} f(X_t^{\pmb{\alpha}}) \Big\|_X^p \\ & = \int_{\{-1,1\}^n} \sum_{y\in \{-1,1\}^n} p_t^{\pmb{\alpha}}(x,y) \Big\| \sum_{i=1}^n \eta_t^{\alpha_i}\big(y(i),x(i);\theta_i(t)\big) \partial_i^{\alpha_i} f(x)\Big\|_X^p \, \mathop{}\!\mathrm{d}\mu_{\pmb{\alpha}}(x). \end{split}$$ Fixing $x\in\{-1,1\}^n$, equation [\[eq:center\]](#eq:center){reference-type="eqref" reference="eq:center"} asserts that each $\eta_t^{\alpha_i} (y(i),x(i);\theta_i(t))$ is a centered random variable when $y(i)$ is distributed according to $p_t^{\alpha_i}(x(i),\cdot)$. Therefore, as $p_t^{\pmb{\alpha}}(x,\cdot)$ is a product measure, the Rademacher type condition for sums of centered independent random vectors (see [@LT91 Proposition 9.11]) yields the bound $$\label{eq:use-type} \begin{split} & \int_{\{-1,1\}^n} \sum_{y\in \{-1,1\}^n} p_t^{\pmb{\alpha}}(x,y) \Big\| \sum_{i=1}^n \eta_t^{\alpha_i}\big(y(i),x(i);\theta_i(t)\big) \partial_i^{\alpha_i} f(x)\Big\|_X^p \, \mathop{}\!\mathrm{d}\mu_{\pmb{\alpha}}(x) \\ & \leq \big( 2\mathsf{T}_p(X)\big)^p \int_{\{-1,1\}^n} \sum_{i=1}^n \sum_{y(i)\in \{-1,1\}} p_t^{\alpha_i}\big(x(i),y(i)\big)\ \big|\eta_t^{\alpha_i}\big(y(i),x(i);\theta_i(t)\big)\big|^p \big\|\partial_i^{\alpha_i} f(x)\big\|_X^p \, \mathop{}\!\mathrm{d}\mu_{\pmb{\alpha}}(x) \\ & \leq \big( 2\mathsf{T}_p(X)\big)^p \sum_{i=1}^n \int_{\{-1,1\}^n} \Big(\sum_{y(i)\in \{-1,1\}} p_t^{\alpha_i}\big(x(i),y(i)\big)\ \big|\eta_t^{\alpha_i}\big(y(i),x(i);\theta_i(t)\big)\big|^2\Big)^{p/2} \big\|\partial_i^{\alpha_i} f(x)\big\|_X^p \, \mathop{}\!\mathrm{d}\mu_{\pmb{\alpha}}(x), \end{split}$$ where we also used that $p\leq2$. Now, choosing the $\theta_i(t)$ which minimize the quantity in the left-hand side of [\[eq:second-mom\]](#eq:second-mom){reference-type="eqref" reference="eq:second-mom"} with bias $\alpha_i$, and combining [\[eq:apply-sgp\]](#eq:apply-sgp){reference-type="eqref" reference="eq:apply-sgp"}, [\[eq:use-rev\]](#eq:use-rev){reference-type="eqref" reference="eq:use-rev"} and [\[eq:use-type\]](#eq:use-type){reference-type="eqref" reference="eq:use-type"}, we conclude that $$\begin{split} \Bigg(\int_{\{-1,1\}^n}\Big\| f(x) - & \int_{\{-1,1\}^n} f \,\mathop{}\!\mathrm{d}\mu_{\pmb{\alpha}} \Big\|_X^p \,\mathop{}\!\mathrm{d}\mu_{\pmb{\alpha}}(x) \Bigg)^{1/p} \\ & \leq 2\mathsf{T}_p(X) \int_0^\infty \Bigg( \sum_{i=1}^n \int_{\{-1,1\}^n} \big\| \partial_i^{\alpha_i} f(x)\big\|_X^p \,\mathop{}\!\mathrm{d}\mu_{\pmb{\alpha}}(x) \Bigg)^{1/p} \, \frac{\mathop{}\!\mathrm{d}t}{\sqrt{e^t-1}}, \end{split}$$ which is precisely the desired estimate. ◻ Equipped with the biased Poincaré inequality of Theorem [Theorem 14](#thm:poincare){reference-type="ref" reference="thm:poincare"}, we can conclude the proof. *Proof of Theorem [Theorem 1](#thm:main){reference-type="ref" reference="thm:main"}.* Let $X=(\mathbb{R}^d,\|\cdot\|_X)$ be a $d$-dimensional normed space and suppose that $f:\{-1,1\}^n\to X$ is a function such that $$\label{eq:lip-con} \forall \ x,y\in\{-1,1\}^n, \qquad \rho(x,y) \leq \|f(x)-f(y)\|_X \leq D\rho(x,y)$$ for some $D\geq1$. The conclusion of the theorem follows from [@IVV20] when $d\geq n$ so we shall assume that $d<n$. Let $F:[-1,1]^n\to X$ be the multilinear extension of $f$ given by [\[eq:extension\]](#eq:extension){reference-type="eqref" reference="eq:extension"}. Then, $F$ is clearly continuous as a polynomial and therefore, by Lemma [Lemma 11](#lem:ole2){reference-type="ref" reference="lem:ole2"}, there exists a point $z\in\mathscr{C}_d^n$ such that $F(z)=F(-z)$. As $z$ has at least $n-d$ coordinates equal to 1 in absolute value we shall assume without loss of generality that $|z({d+1})|=\ldots=|z(n)|=1$ and consider the functions $h_+, h_-:\{-1,1\}^d\to X$ which are defined as $$\forall \ x\in\{-1,1\}^d, \qquad h_\pm(x) = f\big( \pm x(1),\ldots, \pm x(d),\pm z(d+1),\ldots,\pm z(n)\big).$$ Consider also the bias vector $\pmb{\alpha}_z = \big( \frac{1+z(1)}{2},\ldots,\frac{1+z(d)}{2}\big)\in(0,1)^d$ and notice that, by the multilinearity of $F$, we have the identity $$\int_{\{-1,1\}^d} h_+(x)\,\mathop{}\!\mathrm{d}\mu_{\pmb{\alpha}_z}(x) = F(z) = F(-z) = \int_{\{-1,1\}^d} h_-(x)\,\mathop{}\!\mathrm{d}\mu_{\pmb{\alpha}_z}(x).$$ Therefore, by the triangle inequality and Theorem [Theorem 14](#thm:poincare){reference-type="ref" reference="thm:poincare"} we get $$\label{eq:main-ineq} \begin{split} \int_{\{-1,1\}^d}\big\| h_+(x) & - h_-(x) \big\|_X^p \,\mathop{}\!\mathrm{d}\mu_{\pmb{\alpha}_z}(x) \\ & \leq 2^{p-1} \int_{\{-1,1\}^d}\big\| h_+(x) - F(z) \big\|_X^p + \big\| h_-(x) - F(-z) \big\|_X^p \,\mathop{}\!\mathrm{d}\mu_{\pmb{\alpha}_z}(x) \\ & \leq 2^{2p-1} \big(\pi \mathsf{T}_p(X)\big)^p \sum_{i=1}^d \int_{\{-1,1\}^d} \big\| \partial_i^{\frac{1+z(i)}{2}} h_+(x)\big\|_X^p+\big\| \partial_i^{\frac{1+z(i)}{2}} h_-(x)\big\|_X^p \,\mathop{}\!\mathrm{d}\mu_{\pmb{\alpha}_z}(x) . \end{split}$$ Now, in view of the lower Lipschitz condition [\[eq:lip-con\]](#eq:lip-con){reference-type="eqref" reference="eq:lip-con"}, we clearly have $$\big\| h_+(x) - h_-(x) \big\|_X = \big\| f\big(x(1),\ldots,x(d),z(d+1),\ldots,z(n)\big) - f\big(-x(1),\ldots,-x(d),-z(d+1),\ldots,-z(n)\big) \big\|_X \geq n$$ for every $x\in\{-1,1\}^d$. On the other hand, for a fixed $i\in\{1,\ldots,d\}$ and $\beta = \frac{1+z(i)}{2}$, we have $$\begin{split} \int_{\{-1,1\}} \big\| \partial_i^\beta & h_+(x)\big\|_X^p\,\mathop{}\!\mathrm{d}\mu_\beta(x(i)) = \beta \|\partial_i^\beta h_+(x(1),\ldots,1,\ldots,x(d))\|_X^p + (1-\beta) \|\partial_i^\beta h_+(x(1),\ldots,-1,\ldots,x(d))\|_X^p \\ & = \big( \beta(1-\beta)^p + (1-\beta) \beta^p \big) \ \big\|h_+(x(1),\ldots, 1,\ldots,x(d)) - h_+(x(1),\ldots, -1,\ldots,x(d))\big\|_X^p \leq \frac{D^p}{2^p}, \end{split}$$ where in the last equality we used that $p\leq 2$ along with the upper Lipschitz condition [\[eq:lip-con\]](#eq:lip-con){reference-type="eqref" reference="eq:lip-con"}. The same bound also holds for $h_-$. Integrating the last two inequalities and combining them with [\[eq:main-ineq\]](#eq:main-ineq){reference-type="eqref" reference="eq:main-ineq"}, we deduce that $$n^p \leq (2\pi \mathsf{T}_p(X))^p d D^p,$$ which completes the proof of the theorem. ◻ **Remark 15**. *The identity of Proposition [Proposition 13](#prop:ident){reference-type="ref" reference="prop:ident"} in the case of the uniform measure $\sigma_n$ (which was obtained in [@IVV20]) is simpler. Let $\xi_1(t),\ldots,\xi_n(t)$ be i.i.d. random variables distributed according to $\mu_{\beta(t)}$, where $\beta(t) = \frac{1+e^{-t}}{2}$. Then, for any point $x\in\{-1,1\}^n$, the corresponding unbiased process $\{X_t(i)\}_{t\geq0}$ with $X_0=x$ has distribution equal to $x(i)\xi_i(t)$ at time $t$. Thus applying formula [\[eq:identi\]](#eq:identi){reference-type="eqref" reference="eq:identi"} with $\alpha_i = \frac{1}{2}$ and $\theta_i (t)= \tfrac{e^{-t}}{2}$, we recover the usual identity $$\label{eq:ivv} \forall \ x\in\{-1,1\}^n,\qquad \mathscr{L}P_t f(x) = - \mathbb{E}\Big[ \sum_{i=1}^n \frac{\xi_i(t)-e^{-t}}{e^t-e^{-t}}\cdot \partial_i f\big(x\xi(t)\big) \Big],$$ where $x\xi(t) = (x(1)\xi_1(t),\ldots,x(n)\xi_n(t))$, as was proven in [@IVV20].* # Proof of Theorem [Theorem 2](#thm:pisier){reference-type="ref" reference="thm:pisier"} {#sec:pisier} Recall that a normed space $(X,\|\cdot\|_X)$ has cotype $q\in[2,\infty)$ with constant $C\in(0,\infty)$ if for every $n\in\mathbb{N}$ and $v_1,\ldots,v_n\in X$, we have $$\int_{\{-1,1\}^n} \Big\| \sum_{i=1}^n x_iv_i\Big\|_X^q \,\mathop{}\!\mathrm{d}\sigma_n(x) \geq \frac{1}{C^q} \sum_{i=1}^n \|v_i\|_X^q.$$ We say that $X$ has finite cotype if it has cotype $q$ for some $q\in[2,\infty)$. The proof of Theorem [Theorem 2](#thm:pisier){reference-type="ref" reference="thm:pisier"} is similar to the arguments of [@IVV20; @CE24] using as input the new identity [\[eq:identi\]](#eq:identi){reference-type="eqref" reference="eq:identi"} for the time derivative of the biased semigroup $\{P_t^{\pmb{\alpha}}\}_{t\geq0}$. In view of that, we shall omit various simple details and we will be less attentive with the values of the implicit constants. We start by proving the (weaker) biased Pisier inequality, in which the Orlicz norm on the left hand side of the conclusions of Theorem [Theorem 2](#thm:pisier){reference-type="ref" reference="thm:pisier"} is replaced by an $L_p$ norm. **Theorem 16**. *For every $\alpha\in(0,1)$, there exists $\mathsf{K}_{\alpha}\in(0,\infty)$ such that the following holds when $p\in[1,\infty)$. For any normed space $(X,\|\cdot\|_X)$ and any $n\in\mathbb{N}$, every function $f:\{-1,1\}^n\to X$ satisfies $$\label{eq:pw1} \bigg\| f - \int_{\{-1,1\}^n} f\,\mathop{}\!\mathrm{d}\mu_\alpha^n\bigg\|_{L_p(\mu_\alpha^n;X)} \leq \mathsf{K}_{\alpha}(\log n+1) \left( \int_{\{-1,1\}^n} \Big\| \sum_{i=1}^n \delta_i \partial_i^\alpha f\Big\|_{L_p(\mu_\alpha^n;X)}^p \,\mathop{}\!\mathrm{d}\sigma_n(\delta)\right)^{1/p}.$$ If additionally $X$ is assumed to be of finite cotype, then there exists $\mathsf{K}_{p,\alpha}(X)$ such that $$\label{eq:pw2} \bigg\| f - \int_{\{-1,1\}^n} f\,\mathop{}\!\mathrm{d}\mu_\alpha^n\bigg\|_{L_p(\mu_\alpha^n;X)} \leq \mathsf{K}_{p,\alpha}(X) \left( \int_{\{-1,1\}^n} \Big\| \sum_{i=1}^n \delta_i \partial_i^\alpha f\Big\|_{L_p(\mu_\alpha^n;X)}^p \,\mathop{}\!\mathrm{d}\sigma_n(\delta)\right)^{1/p}.$$* *Proof.* Let $\pmb{\alpha}=(\alpha,\ldots,\alpha)$ and without loss of generality assume that $\int f \,\mathop{}\!\mathrm{d}\mu_{\pmb{\alpha}}=0$. We start by proving the general inequality [\[eq:pw1\]](#eq:pw1){reference-type="eqref" reference="eq:pw1"}. Notice that, by convexity, $$\big\| \mathscr{L}_{\pmb{\alpha}}f\big\|_{L_p(\mu_{\pmb{\alpha}};X)} \stackrel{\eqref{eq:gen}}{\leq} \sum_{i=1}^n \big\| \partial_i^\alpha f\big\|_{L_p(\mu_{\pmb{\alpha}};X)} \leq 2n \| f \|_{L_p(\mu_{\pmb{\alpha}};X)},$$ where the last inequality follows from the definitions of $\partial_i^\alpha$. Therefore, $$\begin{split} \forall \ t\geq0, \qquad \big\| e^{-t\mathscr{L}_{\pmb{\alpha}}} f\big\|_{L_p(\mu_{\pmb{\alpha}};X)} \leq \sum_{m=0}^\infty \frac{t^m}{m!} \big\| \mathscr{L}_{\pmb{\alpha}}^mf\big\|_{L_p(\mu_{\pmb{\alpha}};X)} \leq\sum_{m=0}^\infty \frac{(2nt)^m}{m!} \|f\|_{L_p(\mu_{\pmb{\alpha}};X)} = e^{2nt}\|f\|_{L_p(\mu_{\pmb{\alpha}};X)}. \end{split}$$ Since $P_t^{\pmb{\alpha}} = e^{t\mathscr{L}_{\pmb{\alpha}}}$ for $t\geq0$, this inequality applied to $P_t^{\pmb{\alpha}}f$ with $t=\tfrac{1}{n}$ implies that $$\|f\|_{L_p(\mu_{\pmb{\alpha}};X)} \leq e^2 \big\|P_{1/n}^{\pmb{\alpha}}f\big\|_{L_p(\mu_{\pmb{\alpha}};X)}.$$ Similarly to [\[eq:apply-sgp\]](#eq:apply-sgp){reference-type="eqref" reference="eq:apply-sgp"} and [\[eq:use-rev\]](#eq:use-rev){reference-type="eqref" reference="eq:use-rev"}, we thus have $$\begin{split} \|f&\|_{L_p(\mu_{\pmb{\alpha}};X)}\leq e^2 \big\|P_{1/n}^{\pmb{\alpha}}f\big\|_{L_p(\mu_{\pmb{\alpha}};X)}. \\ & \leq e^2 \int_{1/n}^\infty \left( \int_{\{-1,1\}^n} \sum_{y\in \{-1,1\}^n} p_t^{\pmb{\alpha}}(x,y) \Big\| \sum_{i=1}^n \eta_t^{\alpha}\big(y(i),x(i);\theta_i(t)\big) \partial_i^{\alpha} f(x)\Big\|_X^p \, \mathop{}\!\mathrm{d}\mu_{\pmb{\alpha}}(x)\right)^{1/p} \,\mathop{}\!\mathrm{d}t \end{split}$$ Fix $t\geq0$ and $x\in\{-1,1\}^n$. Since $p_t^{\pmb{\alpha}}(x,\cdot)$ is a product measure, the centering condition [\[eq:center\]](#eq:center){reference-type="eqref" reference="eq:center"} gives $$\label{use-cent} \begin{split} \sum_{y\in \{-1,1\}^n} p_t^{\pmb{\alpha}}(x,y) & \Big\| \sum_{i=1}^n \eta_t^{\alpha}\big(y(i),x(i);\theta_i(t)\big) \partial_i^{\alpha} f(x)\Big\|_X^p \\ & \leq 2^p \int_{\{-1,1\}^n} \sum_{y\in \{-1,1\}^n} p_t^{\pmb{\alpha}}(x,y) \Big\| \sum_{i=1}^n \delta_i \big| \eta_t^{\alpha}\big(y(i),x(i);\theta_i(t)\big)\big| \partial_i^{\alpha} f(x)\Big\|_X^p \, \mathop{}\!\mathrm{d}\sigma_n(\delta). \end{split}$$ To prove [\[eq:pw1\]](#eq:pw1){reference-type="eqref" reference="eq:pw1"}, set $\theta_i(t) = \frac{e^{-t}}{2}$ and observe that by the contraction principle [@LT91 Theorem 4.4], we can further bound this quantity by $$\begin{split} 2^p \sum_{y\in \{-1,1\}^n} p_t^{\pmb{\alpha}}(x,y) & \int_{\{-1,1\}^n} \Big\| \sum_{i=1}^n \delta_i \big| \eta_t^{\alpha}\big(y(i),x(i);\tfrac{e^{-t}}{2}\big)\big| \partial_i^{\alpha} f(x)\Big\|_X^p \, \mathop{}\!\mathrm{d}\sigma_n(\delta) \\& \leq 2^p \max_{\chi,\psi\in\{-1,1\}} \big| \eta_t^{\alpha}\big(\psi,\chi;\tfrac{e^{-t}}{2}\big)\big|^p \int_{\{-1,1\}^n} \Big\| \sum_{i=1}^n \delta_i \partial_i^{\alpha} f(x)\Big\|_X^p \, \mathop{}\!\mathrm{d}\sigma_n(\delta). \end{split}$$ It is now elementary to check that we have $$\max_{\chi,\psi\in\{-1,1\}}\big|\eta_t^{\alpha}\big(\psi,\chi;\tfrac{e^{-t}}{2}\big)\big| \lesssim_\alpha \begin{cases} \frac{1}{t}, & \mbox{for }t\in(0,1) \\ e^{-t}, & \mbox{for } t\geq1 \end{cases}$$ and thus combining all the above we get $$\|f\|_{L_p(\mu_{\pmb{\alpha}};X)} \lesssim_{\alpha} \left( \int_{1/n}^1 \frac{\mathop{}\!\mathrm{d}t}{t} + \int_1^\infty e^{-t}\,\mathop{}\!\mathrm{d}t \right)\cdot\left( \int_{\{-1,1\}^n} \Big\| \sum_{i=1}^n \delta_i \partial_i f\Big\|_{L_p(\mu_{\pmb{\alpha}};X)}^p \,\mathop{}\!\mathrm{d}\sigma_n(\delta)\right)^{1/p},$$ which readily concludes the proof of [\[eq:pw1\]](#eq:pw1){reference-type="eqref" reference="eq:pw1"}. For the proof of [\[eq:pw2\]](#eq:pw2){reference-type="eqref" reference="eq:pw2"}, we combine [\[eq:apply-sgp\]](#eq:apply-sgp){reference-type="eqref" reference="eq:apply-sgp"} and [\[eq:use-rev\]](#eq:use-rev){reference-type="eqref" reference="eq:use-rev"} with [\[use-cent\]](#use-cent){reference-type="eqref" reference="use-cent"} to get $$\|f\|_{L_p(\mu_{\pmb{\alpha}};X)}\leq 2 \int_{0}^\infty \left( \int_{\{-1,1\}^{2n}}\sum_{y\in \{-1,1\}^n} p_t^{\pmb{\alpha}}(x,y) \Big\| \sum_{i=1}^n \delta_i \big| \eta_t^{\alpha}\big(y(i),x(i);\tfrac{e^{-t}}{2}\big)\big| \partial_i^{\alpha} f(x)\Big\|_X^p \, \mathop{}\!\mathrm{d}\sigma_n(\delta) \mathop{}\!\mathrm{d}\mu_{\pmb{\alpha}}(x)\right)^{1/p}\!\!\!\! \mathop{}\!\mathrm{d}t.$$ Fix $x\in\{-1,1\}^n$ and $t\geq0$. Denoting by $I(x)= \{i:\ x(i)=1\}$, the inner term is bounded above by $$\label{eq:break-to-iid} \begin{split} \int_{\{-1,1\}^n} & \sum_{y\in \{-1,1\}^n} p_t^{\pmb{\alpha}}(x,y) \Big\| \sum_{i=1}^n \delta_i \big| \eta_t^{\alpha}\big(y(i),x(i);\tfrac{e^{-t}}{2}\big)\big| \partial_i^{\alpha} f(x)\Big\|_X^p \, \mathop{}\!\mathrm{d}\sigma_n(\delta) \\ & \leq 2^{p-1} \int_{\{-1,1\}^{I(x)}} \sum_{y\in \{-1,1\}^{I(x)}} p_t^{\pmb{\alpha}}(x,y) \Big\| \sum_{i\in I(x)} \delta_i \big| \eta_t^{\alpha}\big(y(i),1;\tfrac{e^{-t}}{2}\big)\big| \partial_i^{\alpha} f(x)\Big\|_X^p \, \mathop{}\!\mathrm{d}\sigma_{I(x)}(\delta) \\ & + 2^{p-1} \int_{\{-1,1\}^{I(x)^c}} \sum_{y\in \{-1,1\}^{I(x)^c}} p_t^{\pmb{\alpha}}(x,y) \Big\| \sum_{i\notin I(x)} \delta_i \big| \eta_t^{\alpha}\big(y(i),-1;\tfrac{e^{-t}}{2}\big)\big| \partial_i^{\alpha} f(x)\Big\|_X^p\, \mathop{}\!\mathrm{d}\sigma_{I(x)^c}(\delta). \end{split}$$ When $y\in\{-1,1\}^{I(x)}$ is distributed according to $p_t^{\pmb{\alpha}}(x,\cdot)$ and $\delta$ is a uniformly random sign, the random variables $\delta_i|\eta_t^{\alpha}(y(i),1;\tfrac{e^{-t}}{2})|$, $i\in I(x)$, are independent and identically distributed. Therefore, standard comparison principles going back to works of Maurey and Pisier (see, e.g., [@LT91 Proposition 9.14]) show that if $X$ has cotype $q<\infty$ and $r> \max\{p, q\}$, then $$\label{eq:Ix} \begin{split} \int_{\{-1,1\}^{I(x)}} &\sum_{y\in \{-1,1\}^{I(x)}} p_t^{\pmb{\alpha}}(x,y) \Big\| \sum_{i\in I(x)} \delta_i \big| \eta_t^{\alpha}\big(y(i),1;\tfrac{e^{-t}}{2}\big)\big| \partial_i^{\alpha} f(x)\Big\|_X^p \, \mathop{}\!\mathrm{d}\sigma_{I(x)}(\delta) \\ & \lesssim_{r,X} \Big( \sum_{\psi\in\{-1,1\}}p_t^{{\alpha}} \big(1,\psi;\tfrac{e^{-t}}{2}\big) \big| \eta_t^{\alpha}\big(\psi,1;\tfrac{e^{-t}}{2}\big)\big|^r \Big)^{p/r} \int_{\{-1,1\}^{I(x)}} \Big\| \sum_{i\in I(x)} \delta_i \partial_i^{\alpha} f(x)\Big\|_X^p\, \mathop{}\!\mathrm{d}\sigma_{I(x)}(\delta). \end{split}$$ It is again elementary to show that $$\label{rmom} \max_{\chi\in\{-1,1\}} \Big( \sum_{\psi\in\{-1,1\}}p_t^{{\alpha}} \big(\chi,\psi;\tfrac{e^{-t}}{2}\big) \big| \eta_t^{\alpha}\big(\psi,\chi;\tfrac{e^{-t}}{2}\big)\big|^r \Big)^{1/r} \lesssim_{r,\alpha} \begin{cases} \frac{1}{t^{1-\frac1r}}, & \mbox{for }t\in(0,1) \\ e^{-t}, & \mbox{for } t\geq1 \end{cases}.$$ In view of the convergence of the integrals $$\int_0^1 \frac{\mathop{}\!\mathrm{d}t}{t^{1-\frac1r}} + \int_1^\infty e^{-t}\,\mathop{}\!\mathrm{d}t \lesssim_r 1,$$ combining [\[eq:Ix\]](#eq:Ix){reference-type="eqref" reference="eq:Ix"}, the corresponding estimate on $I(x)^c$ and [\[eq:break-to-iid\]](#eq:break-to-iid){reference-type="eqref" reference="eq:break-to-iid"}, we finally get $$\begin{split} \|f\|_{L_p(\mu_{\pmb{\alpha}};X)} \lesssim_{r,\alpha,X} \left(\int_{\{-1,1\}^n} \int_{\{-1,1\}^n} \Big\| \sum_{i\in I(x)} \delta_i\partial_i^{\alpha} f(x)\Big\|_X^p + \Big\| \sum_{i\notin I(x)} \delta_i \partial_i^{\alpha} f(x)\Big\|_X^p \,\mathop{}\!\mathrm{d}\sigma_n(\delta) \, \mathop{}\!\mathrm{d}\mu_{\pmb{\alpha}(x)} \right)^{1/p}. \end{split}$$ Now, if $F,G$ are two independent centered $X$-valued random vectors, then $$\mathbb{E}\|F+G\|_X^p \geq \max\big\{ \mathbb{E}_F \|F+\mathbb{E}_G[G]\|_X^p, \mathbb{E}_G\|\mathbb{E}_F[F]+G\|_X^p\big\} = \max\big\{ \mathbb{E}\|F\|_X^p, \mathbb{E}\|G\|_X^p\big\}.$$ Therefore, $$\begin{split} \Bigg(\int_{\{-1,1\}^n} \int_{\{-1,1\}^n} \Big\| \sum_{i\in I(x)} \delta_i\partial_i^{\alpha} f(x)\Big\|_X^p + \Big\| \sum_{i\notin I(x)} & \delta_i \partial_i^{\alpha} f(x)\Big\|_X^p \,\mathop{}\!\mathrm{d}\sigma_n(\delta) \, \mathop{}\!\mathrm{d}\mu_{\pmb{\alpha}(x)} \Bigg)^{1/p} \\ & \lesssim \left(\int_{\{-1,1\}^n} \Big\| \sum_{i=1}^n \delta_i\partial_i^{\alpha} f(x)\Big\|_{L_p(\mu_{\pmb{\alpha}};X)}^p \,\mathop{}\!\mathrm{d}\sigma_n(\delta) \right)^{1/p} \end{split}$$ and this concludes the proof of the theorem. ◻ We now proceed to discuss the vector-valued $L_p$ logarithmic Sobolev inequality of Theorem [Theorem 2](#thm:pisier){reference-type="ref" reference="thm:pisier"}. Recall that given a function $f:(\Omega,\mu)\to X$, we define the $L_p(\log L)^a$ Orlicz norm of $f$ as $$\|f\|_{L_p(\log L)^a(\mu;X)} = \inf\left\{ \gamma>0: \ \int_\Omega \tfrac{\|f(\omega)\|_X^p}{\gamma^p} \ \log^a\Big(e+\tfrac{\|f(\omega)\|_X^p}{\gamma^p}\Big) \,\mathop{}\!\mathrm{d}\mu(\omega) \leq 1\right\}.$$ In the case of the unbiased cube, Theorem [Theorem 2](#thm:pisier){reference-type="ref" reference="thm:pisier"} was proven recently in [@CE24] and the proof used as a black box the unbiased analogue of Theorem [Theorem 16](#thm:pisier-weak){reference-type="ref" reference="thm:pisier-weak"} which is due to [@Pis86; @IVV20]. As the modifications which are required to derive Theorem [Theorem 2](#thm:pisier){reference-type="ref" reference="thm:pisier"} from Theorem [Theorem 16](#thm:pisier-weak){reference-type="ref" reference="thm:pisier-weak"} in the biased case are almost mechanical, we shall only offer a high-level description of the proof. *Sketch of the proof of Theorem [Theorem 2](#thm:pisier){reference-type="ref" reference="thm:pisier"}.* Given a scalar-valued function $h:\{-1,1\}^n\to\mathbb{R}$, we denote by $\mathsf{M}h:\{-1,1\}^n\to\mathbb{R}_+$ the asymmetric gradient of $h$, given by $$\forall \ x\in\{-1,1\}^n,\qquad \mathsf{M}h(x) = \bigg( \sum_{i=1}^n \partial_ih(x)_+^2\bigg)^{1/2},$$ where $a_+=\max\{a,0\}$ for $a\in\mathbb{R}$. A combination of the triangle inequality with a technical result of Talagrand [@Tal93 Proposition 5.1] on the biased cube implies that if a vector-valued function $f:\{-1,1\}^n\to X$ satisfies $\int f \,\mathop{}\!\mathrm{d}\mu_\alpha^n=0$, then $$\|f\|_{L_p(\log L)^{p/2}(\mu_\alpha^n;X)} \lesssim_{p,\alpha} \big\| \mathsf{M} \|f\|_X \big\|_{L_p(\mu_\alpha^n;X)} + \|f\|_{L_1(\mu_\alpha^n;X)}.$$ The second term can be controlled by the right hand side of [\[eq:p1\]](#eq:p1){reference-type="eqref" reference="eq:p1"} or [\[eq:p2\]](#eq:p2){reference-type="eqref" reference="eq:p2"} using the results of Theorem [Theorem 16](#thm:pisier-weak){reference-type="ref" reference="thm:pisier-weak"}. For the first term, we use the key pointwise inequality of [@CE24], asserting that $$\forall \ x\in\{-1,1\}^n,\qquad \mathsf{M}\|f\|_X(x) \leq \sqrt{2}\Bigg( \int_{\{-1,1\}^n} \Big\| \sum_{i=1}^n \delta_i \partial_i f(x)\Big\|_X^p\, \mathop{}\!\mathrm{d}\sigma_n(\delta) \Bigg)^{1/p}.$$ To further upper bound this by the square function where $\partial_i$ is replaced by $\partial_i^\alpha$, observe that $$\partial_i^\alpha f(x) \in\big\{ 2\alpha \partial_i f(x), 2(1-\alpha) \partial_if(x)\big\}$$ and thus the contraction principle and integration in $x$ yield the desired conclusions. ◻ # Proof of Theorem [Theorem 4](#thm:stable){reference-type="ref" reference="thm:stable"} {#sec:5} In this section we shall prove the equivalence of stable type and metric stable type for norms. The key ingredient in the proof is the following characterization of stable type (see, e.g., [@LT91 Proposition 9.12 (iii)]) which goes back at least to [@MP84]. **Lemma 17**. *A Banach space $(X,\|\cdot\|_X)$ has stable type $p\in[1,2)$ if and only if there exists a constant $\mathsf{ST}_p(X)\in(0,\infty)$ such that for every $n\in\mathbb{N}$ and every vectors $v_1,\ldots,v_n\in X$, we have $$\label{eq:lt} \int_{\{-1,1\}^n} \Big\| \sum_{i=1}^n x_i v_i\Big\|_X^p \,\mathop{}\!\mathrm{d}\sigma_n(x) \leq \mathsf{ST}_p(X)^p \big\|\big(\|v_1\|_X,\ldots,\|v_n\|_X\big)\big\|_{\ell_{p,\infty}^n}^p.$$* *Proof of Theorem [Theorem 4](#thm:stable){reference-type="ref" reference="thm:stable"}.* It is clear from Lemma [Lemma 17](#lem:lt){reference-type="ref" reference="lem:lt"} that any normed space with metric stable type $p$ also has stable type $p$. For the converse implication, we treat the case $p=1$ separately. Assuming that $X$ has stable type 1, it follows from [@Pis74b] that it also has nontrivial Rademacher type and thus (in view of [@Pis73]) finite cotype. Therefore, by the $X$-valued Pisier inequality with a dimension-free constant of [@IVV20], we get $$\begin{split} \int_{\{-1,1\}^n} \big\|f(x)-f(-x)\big\|_X\,\mathop{}\!\mathrm{d}\sigma_n(x) & \leq 2\int_{\{-1,1\}^n} \Big\| f(x) - \int_{\{-1,1\}^n} f\,\mathop{}\!\mathrm{d}\sigma_n\Big\|_X \,\mathop{}\!\mathrm{d}\sigma_n(x) \\ & \lesssim_{X} \int_{\{-1,1\}^{n}}\int_{\{-1,1\}^{n}} \mathbb{E} \Big\|\sum_{i=1}^{n} \delta_i \partial_i f(x)\Big\|_X \, \mathop{}\!\mathrm{d}\sigma_n(\delta) \mathop{}\!\mathrm{d}\sigma_{n}(x). \end{split}$$ Applying Lemma [Lemma 17](#lem:lt){reference-type="ref" reference="lem:lt"} conditionally on $x\in\{-1,1\}^n$, we can further bound this quantity as $$\begin{split} \int_{\{-1,1\}^{n}}\int_{\{-1,1\}^{n}} \mathbb{E} \Big\|\sum_{i=1}^{n} & \delta_i \partial_i f(x)\Big\|_X \, \mathop{}\!\mathrm{d}\sigma_n(\delta) \mathop{}\!\mathrm{d}\sigma_{n}(x) \\ & \leq \mathsf{ST}_1(X) \int_{\{-1,1\}^n} \big\|\big(\|\partial_1f(x)\|_X,\ldots,\|\partial_nf(x)\|_X\big)\big\|_{\ell_{1,\infty}^n} \, \mathop{}\!\mathrm{d}\sigma_n(x) \end{split}$$ and this proves the converse implication since $\mathfrak{d}_if(x) = \|\partial_if(x)\|_X$. While this proof extends to all values of $p$, in the case $p>1$ we present a more cumbersome argument which avoids the $X$-valued Pisier inequality and thus gives better dependence on parameters of $X$. For $t\geq0$, let $\xi_1(t),\ldots,\xi_n(t)$ be i.i.d. random variables distributed according to $\mu_{\beta(t)}$, where $\beta(t)=\tfrac{1+e^{-t}}{2}$, and denote by $\eta_i(t) = \frac{\xi_i(t)-e^{-t}}{e^t-e^{-t}}$. Then, it follows from the semigroup argument leading to [\[eq:apply-sgp\]](#eq:apply-sgp){reference-type="eqref" reference="eq:apply-sgp"} along with identity [\[eq:ivv\]](#eq:ivv){reference-type="eqref" reference="eq:ivv"} of [@IVV20] and Jensen's inequality that $$\begin{split} \Bigg(\int_{\{-1,1\}^n} \big\|f(x)-f(-x)\big\|_X^p\,\mathop{}\!\mathrm{d}\sigma_n(x)\Bigg)^{1/p} & \leq 2\Bigg(\int_{\{-1,1\}^n} \Big\| f(x) - \int_{\{-1,1\}^n} f\,\mathop{}\!\mathrm{d}\sigma_n\Big\|_X^p \,\mathop{}\!\mathrm{d}\sigma_n(x)\Bigg)^{1/p} \\ & \leq 2 \int_0^\infty \Bigg( \int_{\{-1,1\}^n} \mathbb{E} \Big\|\sum_{i=1}^n \eta_i(t) \partial_i f(x)\Big\|_X^p \, \mathop{}\!\mathrm{d}\sigma_n(x)\Bigg)^{1/p} \,\mathop{}\!\mathrm{d}t \end{split}$$ Fixing $x\in\{-1,1\}^n$, the independent random vectors $\eta_1(t)\partial_1f(x),\ldots,\eta_n(t)\partial_nf(x)$ are centered. Thus, by standard symmetrization estimates, we can further bound the last term by $$\begin{split} 2 \int_0^\infty \Bigg( \int_{\{-1,1\}^n} \mathbb{E} \Big\|\sum_{i=1}^n \eta_i(t) \partial_i f(x)\Big\|_X^p& \, \mathop{}\!\mathrm{d}\sigma_n(x)\Bigg)^{1/p} \,\mathop{}\!\mathrm{d}t \\ & \leq 4 \int_0^\infty \Bigg( \int_{\{-1,1\}^n} \mathbb{E} \Big\|\sum_{i=1}^n \delta_i \eta_i(t) \partial_i f(x)\Big\|_X^p \, \mathop{}\!\mathrm{d}\sigma_n(x)\Bigg)^{1/p} \,\mathop{}\!\mathrm{d}t, \end{split}$$ where the expectation on the right hand side is with respect to $(\delta_i,\eta_i(t))$, where the $\delta_i$ are uniformly random signs, independent of the $\eta_i(t)$. Therefore, conditioning first on the values of $\eta_1(t),\ldots,\eta_n(t)$ and using Lemma [Lemma 17](#lem:lt){reference-type="ref" reference="lem:lt"} for the integrand, we get $$\mathbb{E} \Big\|\sum_{i=1}^n \delta_i\eta_i(t) \partial_i f(x)\Big\|_X^p \leq \mathsf{ST}_p(X)^p \mathbb{E} \bigg\|\sum_{i=1}^n \eta_i(t) \big\| \partial_if(x)\big\|_X e_i \bigg\|_{\ell_{p,\infty}^n}^p,$$ where $\{e_1,\ldots,e_n\}$ is the standard basis of $\mathbb{R}^n$. It is well-known that, since $p>1$, $\ell_{p,\infty}^n$ is isomorphic to a normed space up to constants depending only on $p$. Moreover, it has finite cotype with a constant independent of $n$ (for instance because it can be realized as a real interpolation space between $\ell_1^n$ and $\ell_2^n$, see [@BL76 Section 5.3]). Therefore, using again the comparison principle [@LT91 Proposition 9.14], we have $$\begin{split} \mathbb{E} \bigg\|\sum_{i=1}^n \eta_i(t) \big\| \partial_if(x)\big\|_X e_i \bigg\|_{\ell_{p,\infty}^n}^p& \lesssim_{p} \|\eta_1(t)\|_{L_r}^p \mathbb{E} \bigg\|\sum_{i=1}^n \delta_i \big\| \partial_if(x)\big\|_X e_i \bigg\|_{\ell_{p,\infty}^n}^p \\ & = \|\eta_1(t)\|_{L_r}^p \big\|\big(\|\partial_1f(x)\|_X,\ldots,\|\partial_nf(x)\|_X\big)\big\|_{\ell_{p,\infty}^n}^p \end{split}$$ for any $r>\max\{r',p\}$, where $r'$ is the cotype of $\ell_{p,\infty}^n$. Combining all the above along with the fact that $t\mapsto \|\eta_1(t)\|_{L_r}$ is integrable for any $r<\infty$, we conclude the proof. It is worth emphasizing that this proof shows additionally that the metric stable type constant of $X$ is proportional to the parameter $\mathsf{ST}_p(X)$ of [\[eq:lt\]](#eq:lt){reference-type="eqref" reference="eq:lt"} up to constants depending only on $p$, provided that $p>1$. ◻ Finally, we present the simple proof of the distortion bound [\[eq:dist\]](#eq:dist){reference-type="eqref" reference="eq:dist"}. *Proof of Proposition [Proposition 6](#prop:dist){reference-type="ref" reference="prop:dist"}.* Suppose that $(\mathsf{M},d_\mathsf{M})$ has metric stable type $p$ with constant $S$ and suppose that a function $f:\{-1,1\}^n\to\mathsf{M}$ satisfies $$\label{xx} \forall \ x,y\in\{-1,1\}^n,\qquad s \rho_{\bf w}(x,y) \leq d_\mathsf{M}\big(f(x),f(y)\big) \leq sD \rho_{\bf w}(x,y)$$ for some constants $s>0$ and $D\geq1$. Combined with the stable type assumption, this gives $$\begin{split} s^p \|{\bf w}\|_{\ell_1^n}^p \stackrel{\eqref{xx}}{\leq} \int_{\{-1,1\}^n} d_\mathsf{M}\big( f(x)&,f(-x)\big)^p \,\mathop{}\!\mathrm{d}\sigma_n(x) \leq S^p \int_{\{-1,1\}^n} \big\| \big(\mathfrak{d}_1f(x),\ldots,\mathfrak{d}_nf(x)\big)\big\|^p_{\ell_{p,\infty}^n} \,\mathop{}\!\mathrm{d}\sigma_n(x) \\ &\stackrel{\eqref{xx}}{\leq} (sSD)^p \int_{\{-1,1\}^n} \| (w_1,\ldots,w_n)\|^p_{\ell_{p,\infty}^n} \,\mathop{}\!\mathrm{d}\sigma_n(x) = (sSD)^p \|{\bf w}\|_{\ell_{p,\infty}^n}^p. \end{split}$$ Rearranging gives the desired lower bound [\[eq:dist\]](#eq:dist){reference-type="eqref" reference="eq:dist"} for the distortion $D$. ◻ # Discussion and open problems {#sec:disc} The proof of Theorem [Theorem 1](#thm:main){reference-type="ref" reference="thm:main"} in fact implies a Poincaré-type inequality for *restrictions* of functions $f:\{-1,1\}^n\to X$ if $\dim(X)<n$, which in turn yields the refined distortion lower bounds. An inspection of the argument reveals that for every such $f$ there exists a subset $\sigma\subseteq\{1,\ldots,n\}$ with $|\sigma| \leq \dim(X)$, a point $w\in\{-1,1\}^{\sigma^c}$ and a bias vector $\pmb{\alpha} = (\alpha_i)_{i\in\sigma}\in(0,1)^\sigma$ such that $$\label{eq:real-conclusion} \begin{split} \int_{\{-1,1\}^\sigma} \big\| f(x,w)-& f(-x,-w)\big\|_X^p \,\mathop{}\!\mathrm{d}\mu_{\pmb{\alpha}}(x) \\ & \leq 2^{2p-1} \big(\pi \mathsf{T}_p(X)\big)^p \sum_{i\in\sigma} \int_{\{-1,1\}^\sigma} \big\| \partial_i^{\alpha_i} f(x,w)\big\|_X^p + \big\|\partial_i^{\alpha_i}f(-x,-w)\big\|_X^p \,\mathop{}\!\mathrm{d}\mu_{\pmb{\alpha}}(x). \end{split}$$ Such refinements of Poincaré-type inequalities for topological reasons had not been exploited since Oleszkiewicz's original work [@Ole96]. The last decades have seen the development of many metric inequalities on graphs which yield nonembeddability results into normed spaces. We believe that investigating whether the distortion estimates which one obtains this way can be further improved assuming upper bounds for the dimension of the target space is a very worthwhile research program. As examples, we mention the nonembeddability of graphs with large girth into uniformly smooth spaces [@LMN02; @NPSS06], of $\ell_\infty$-grids into spaces of finite cotype [@MN08] and of trees [@Bou86; @LNP09] and diamond graphs [@JS09; @EMN23] into uniformly convex spaces. The results of [@IVV20] in fact imply that any Lipschitz embedding of $\{-1,1\}^n$ into a normed space of Rademacher type $p$ incurs $p$-*average* distortion at least a constant multiple of $\mathsf{T}_p(X)^{-1} n^{1-1/p}$. It would be interesting to understand whether the bound of Theorem [Theorem 1](#thm:main){reference-type="ref" reference="thm:main"} can be extended to average distortion embeddings beyond bi-Lipschitz ones. The Poincaré inequality [\[eq:real-conclusion\]](#eq:real-conclusion){reference-type="eqref" reference="eq:real-conclusion"} implies that any $f:\{-1,1\}^n\to X=(\mathbb{R}^d,\|\cdot\|_X)$ satisfies $$\label{eq:edge-dist} \sup_{(x,y): \ \rho(x,y)=n} \frac{n}{\|f(x)-f(y)\|_X} \sup_{(x,y): \ \rho(x,y)=1} \|f(x)-f(y)\|_X \gtrsim \frac{n}{\mathsf{T}_p(X) d^{1/p}},$$ provided that $d<n$. This is stronger than the lower bound of Theorem [Theorem 1](#thm:main){reference-type="ref" reference="thm:main"} as it implies that $f$ incurs large distortion only on *specific* pairs of points, namely those which are either antipodal or connected by an edge. Moreover though, [\[eq:edge-dist\]](#eq:edge-dist){reference-type="eqref" reference="eq:edge-dist"} is sharp for any value of $d$ and $p$. Indeed, assume without loss of generality that $\tfrac{n}{d}$ is an odd integer and consider a partition $I_1,\ldots,I_d$ of $\{1,\ldots,n\}$ in $d$ parts of size $\tfrac{n}{d}$. Then, the mapping $f:\{-1,1\}^n\to\ell_p^d$, where $p\in[1,2]$, given by $$\forall \ x\in\{-1,1\}^n, \qquad f(x) \stackrel{\mathrm{def}}{=}\Big( \sum_{i\in I_1} x_i, \ldots, \sum_{i\in I_d} x_i\Big)$$ satisfies $$\inf_{(x,y): \ \rho(x,y)=n}\big\|f(x)-f(y)\big\|_{\ell_p^d} = \inf_{x\in\{-1,1\}^n} \bigg( \sum_{k=1}^d \Big| \sum_{i\in I_k} x_i\Big|^p\bigg)^{1/p} = d^{1/p}$$ and $\|f(x)-f(y)\|_{\ell_p^d}=1$ when $\rho(x,y)=1$. The functional inequality [\[eq:real-conclusion\]](#eq:real-conclusion){reference-type="eqref" reference="eq:real-conclusion"} in fact implies that if $\theta\in(0,1)$, then the bi-Lipschitz distortion of the $\theta$-snowflake of $\{-1,1\}^n$ into a finite dimensional normed space $X$ satisfies $$\mathsf{c}_X\big(\{-1,1\}^n, \rho^\theta\big) \gtrsim \frac{n^\theta}{\mathsf{T}_p(X) \min\{n,\dim(X)\}^{1/p}}.$$ As the reasons behind the impossibility of dimension reduction of Theorem [Theorem 1](#thm:main){reference-type="ref" reference="thm:main"} are partly topological, it is natural to ask in what other settings can one deduce similar conclusions. For instance, do $d$-dimensional manifolds of nonpositive or nonnegative curvature (which have Enflo type 2, see [@Oht09]) admit the same distortion bounds as normed spaces of type 2? In this paper we extended the semigroup machinery developed in [@IVV20] to the biased cube, driven by the geometric application obtained in Theorem [Theorem 1](#thm:main){reference-type="ref" reference="thm:main"}. As an aside, this led to biased versions of the vector-valued Poincaré and logarithmic Sobolev inequalities of [@IVV20; @CE24]. The same reasoning also yields biased extensions of the vector-valued versions of Talagrand's influence inequality [@Tal94] which were studied in [@CE23] for the uniform probability measure on the hypercube. Moreover, combining the biased semigroup machinery with the arguments of [@BIM23 Section 2.5], one can derive biased versions of the isoperimetric-type inequalities of Eldan and Gross [@EG22]. As these extensions are mostly mechanical given the material of this paper and in lack of a specific application which may follow from them, we omit them.

arxiv_math

--- abstract: | We show that there exists a constant $c > 0$ such that if $G$ is a planar graph with 5-correspondence assignment $(L,M)$, then $G$ has at least $2^{c\cdot v(G)}$ distinct $(L,M)$-colourings. This confirms a conjecture of Langhede and Thomassen. More broadly, we introduce a general method showing how hyperbolicity theorems for certain families of critical graphs can be used to derive lower bounds on the number of colourings of the associated class of planar graphs. Hence our main result follows from this method plus a technical theorem (that we proved in a previous paper) involving the hyperbolicity of graphs critical for $5$-correspondence colouring. We further demonstrate our method in the case of counting 3-correspondence colourings of planar graphs of girth at least five. Finally, we use these theorems to show analogous results hold in the case of counting 5-correspondence colourings of locally planar graphs, and counting 3-correspondence colourings of locally planar graphs of girth at least five. author: - "Luke Postle[^1]" - Evelyne Smith-Roberge$^\dagger$ bibliography: - bibliog.bib date: - - - title: Exponentially Many Correspondence Colourings of Planar and Locally Planar Graphs --- # Introduction ## Results {#subsec:results} All graphs in this paper are simple and finite. Given a graph $G$ with vertex-set $V(G)$ and edge-set $E(G)$, we denote $|V(G)|$ and $|E(G)|$ by $v(G)$ and $e(G)$, respectively. A *$k$-colouring* of $G$ is a function $\varphi:V(G) \rightarrow \{1, 2, \dots, k\}$ with the property that $\varphi(u) \neq \varphi(v)$ for each $uv \in E(G)$. We say $G$ is *$k$-colourable* if there exists a $k$-colouring of $G$. A *$k$-list assignment* for a graph $G$ is a function $L$ with domain $V(G)$ that assigns to each vertex $v \in V(G)$ a set $L(v)$ (called a *list of colours*) with $|L(v)| \geq k$. We say $G$ is *$L$-colourable* if there exists a colouring $\varphi$ of $G$ with $\varphi(v) \in L(v)$ for each $v \in V(G)$. We say $G$ is *$k$-list-colourable* (or *$k$-choosable*) if there exists an $L'$-colouring for every $k$-list assignment $L'$ of $G$. Many questions in the field of graph colouring involve determining whether or not a graph with specific structure is colourable (for some notion of colouring). Relatedly, we might ask the following: given that a graph *is* colourable, how easy is it to find a colouring? One way to answer this question is to investigate how many distinct colourings of the graph there are. This sort of question has already been studied extensively for list and ordinary colourings: for instance, Birkhoff and Lewis [@birkhoff1946chromatic] showed that if $G$ is a planar graph, then $G$ has at least $60 \cdot 2^{v(G)-3}$ distinct 5-colourings. Similar results were shown for list colouring; to present these results concisely, we introduce the following definition. **Definition 1**. Let $\mathcal{G}$ be a class of graphs. We say $\mathcal{G}$ has *exponentially many $k$-list colourings* if there exists a constant $c > 0$ such that for every graph $G \in \mathcal{G}$ and every $k$-list assignment $L$ of $G$, there exist at least $2^{c\cdot v(G)}$ distinct $L$-colourings of $G$. Thomassen [@thomassen5LC] famously showed in 1994 that planar graphs are 5-list colourable. In 1993, Voigt [@voigt1993list] gave a construction of a planar graph that is not 4-list colourable: thus Thomassen's result is best possible. In 2007, Thomassen [@thomassen2007exponentially] proved that planar graphs have exponentially many 5-list colourings (with $c = \frac{1}{9}$). Recently in 2022, it was observed by Bosek, Grytczuk, Gutowski, Serra, and Zając [@bosek2022graph] that, using the polynomial method (discussed further in Subsection [1.3](#subsec:polmethod){reference-type="ref" reference="subsec:polmethod"}), it is straightforward to improve this bound (to $c = \frac{\log_2 5}{4}$). Analogous results hold for planar graphs of higher girth and with smaller list sizes. Recall that the *girth* of a graph is the length of a shortest cycle in the graph (where if the graph is a forest, we define the girth to be infinite). In 1995, Thomassen proved that planar graphs of girth at least five are 3-list colourable [@thomassen3LC], and later in 2007 [@thomassen2007many], that this class of graphs has exponentially many 3-list colourings (with $c = \frac{1}{10000}$). Again, it was observed by Bosek et al. [@bosek2022graph] that the bound can be further improved (in this case, to $c = \frac{\log_2 3}{6}$). One might wonder whether analogous results hold for other forms of colouring: in particular, whether these results extend to *correspondence colouring.* Correspondence colouring is a natural generalization of list colouring first introduced by Dvořák and the first author in 2015 [@dvovrak2018correspondence]. It is defined as follows. **Definition 2**. Let $G$ be a graph. A *$k$-correspondence assignment for $G$* is a $k$-list assignment $L$ together with a function $M$ that assigns to every edge $e = uv \in E(G)$ a partial matching $M_e$ between $\{u\}\times L(u)$ and $\{v\}\times L(v)$. An $(L,M)$-colouring of $G$ is a function $\varphi$ that assigns to each vertex $v \in V(G)$ a colour $\varphi(v) \in L(v)$ such that for every $e = uv \in E(G)$, the vertices $(u, \varphi(u))$ and $(v, \varphi(v))$ are non-adjacent in $M_e$. We say that $G$ is $(L,M)$-colourable if such a colouring exists, and that $G$ is *$k$-correspondence-colourable* if $G$ is $(L,M)$-colourable for every $k$-correspondence assignment $(L,M)$ for $G$. Since its introduction, there have been over a hundred papers on correspondence colouring; see for example [@bernshteyn2016asymptotic; @bernshteyn2018sharp; @bernshteyn2019differences; @bernshteyn2017dp; @liu2019dp; @zhang2021edge] for a small sample. In the paper in which they introduce correspondence colouring, Dvořák and the first author observed that Thomassen's proofs of the 5-choosability of planar graphs and 3-choosability of planar graphs of girth at least five naturally carry over to the correspondence colouring framework. That is: planar graphs are 5-correspondence colourable, and planar graphs of girth at least five are 3-correspondence colourable. In light of this, it is natural to wonder whether there are exponentially many 5-correspondence colourings of planar graphs. Hence we have the following definition. **Definition 3**. Let $\mathcal{G}$ be a class of graphs. We say $\mathcal{G}$ has *exponentially many $k$-correspondence colourings* if there exists a constant $c > 0$ such that for every graph $G \in \mathcal{G}$ and every $k$-correspondence assignment $(L,M)$ of $G$, there exist at least $2^{c\cdot v(G)}$ distinct $(L,M)$-colourings of $G$. Indeed in 2021, Langhede and Thomassen [@langhede2021exponentially] conjectured as follows. **Conjecture 4** (Langhede and Thomassen, [@langhede2021exponentially]). *Planar graphs have exponentially many 5-correspondence colourings.* In [@langhede2021exponentially], Langhede and Thomassen proved that planar graphs have exponentially many $\mathbb{Z}_5$-colourings; their bound was improved and the result was strengthened to the more general framework of field colourings by Bosek et al. in [@bosek2022graph]. Field colouring generalizes ordinary colouring, but is not a generalization of list colouring; correspondence colouring, however, is a generalization of both list colouring and field colouring. It follows that a proof of Conjecture [Conjecture 4](#thomconj){reference-type="ref" reference="thomconj"} would imply Langhede and Thomassen's result on exponentially many $\mathbb{Z}_5$-colourings as well as Bosek et al.'s result on field colourings (albeit with perhaps a worse value of the constant $c$ in both cases). One of our main results is a proof of Conjecture [Conjecture 4](#thomconj){reference-type="ref" reference="thomconj"}. In particular, we prove the following. thmexpmanycc[\[expmanycc\]]{#expmanycc label="expmanycc"} If $G$ is a planar graph and $(L,M)$ is a 5-correspondence assignment for $G$, then $G$ has at least $2^\frac{v(G)}{67}$ distinct $(L,M)$-colourings. One might wonder whether similar results hold for graphs embedded in other surfaces; in particular, for *locally planar* graphs. **Definition 5**. A *non-contractible cycle* in a surface is a cycle that cannot be continuously deformed to a single point. An embedded graph is *$\rho$-locally planar* if every cycle (in the graph) that is non-contractible (in the surface) has length at least $\rho$. This is closely related to the concept of *edge-width*. The edge-width of an embedded graph is the length of the shortest non-contractible cycle; if a graph is $\rho$-locally planar, it has edge-width at least $\rho$. As the name suggests, the local structure of a locally planar graph is planar. Pleasantly, locally planar graphs sometimes share colouring properties with planar graphs: in 2006, DeVos, Kawarabayashi, and Mohar [@devos2006locally] showed that for every surface $\Sigma$, there exists a constant $\rho= 2^{O(g)}$, where $g$ is the Euler genus of $\Sigma$, such that every $\rho$-locally planar graph that embeds in $\Sigma$ is 5-list-colourable. Thomassen proved a similar result for 5-colourability in 1993 [@thomassen1993five]. In [@postle2018hyperbolic], Postle and Thomas showed that for $k \in \{3,4,5\}$, analogous results for $(8-k)$-list-colouring graphs of girth at least $k$ (with $\rho = \Omega(\log(g))$) are implied by the *hyperbolicity* of certain associated families of graphs. *Hyperbolicity* is defined below. Throughout the rest of the paper, we use $(G, \Sigma)$ to denote a graph $G$ embedded in a surface $\Sigma$. **Definition 6**. Let $\mathcal{F}$ be a family of embedded graphs. We say that $\mathcal{F}$ is *hyperbolic* if there exists a constant $c > 0$ such that if $(G, \Sigma) \in \mathcal{F}$, then for every closed curve $\eta : S^1 \rightarrow \Sigma$ that bounds an open disk $\Delta$ and intersects $G$ only in vertices, if $\Delta$ includes a vertex of $G$, then the number of vertices of $G$ in $\Delta$ is at most $c(|\{x \in S^1 : \eta(x) \in V (G)\}| - 1)$. We say that $c$ is a *Cheeger constant* for $\mathcal{F}$. Note that in the above definition, each vertex $v\in V(G)$ contributes 1 to the quantity $|\{x \in S^1 : \eta(x) \in V (G)\}|$ for each time $\eta$ intersects $v$. Thus for each $k \in \{3,4,5\}$, if $(G,\Sigma)$ is a locally planar embedded graph of girth at least $(8-k)$, then $G$ has exponentially many distinct $L$-colourings for every $k$-list assignment $L$. In the correspondence colouring framework, much less is known. It was shown by Kim, Kostochka, Li, and Zhu in [@kim2020line] and independently by the authors in [@esrlukelocal] that locally planar graphs are 5-correspondence colourable; and the authors also observe in [@esrlukelocal] that locally planar graphs of girth at least five are 3-correspondence colourable. Our results, like those of Postle and Thomas, follow from the hyperbolicity of certain associated graph families. In Section [5](#sec:locallyplanar){reference-type="ref" reference="sec:locallyplanar"}, we prove the following result, showing that not only is it true that locally planar graphs are $5$-correspondence colourable, but in fact they have exponentially many $5$-correspondence colourings. thmthmexpmanyloc[\[thm:expmanylocallyplanar\]]{#thm:expmanylocallyplanar label="thm:expmanylocallyplanar"} For every surface $\Sigma$, there exists a constant $\rho > 0$ with $\rho = O(\log g)$ (where $g$ is the Euler genus of $\Sigma$) such that the following holds: if $G$ is a $\rho$-locally planar graph that embeds in $\Sigma$ and $(L,M)$ is a 5-correspondence assignment for $G$, then $G$ has at least $2^\frac{v(G)}{3484}$ distinct $(L,M)$-colourings. We also give an analogous theorem for 3-correspondence colouring locally planar graphs of girth at least five. Our theorems also follow from the hyperbolicity of certain graph families: thus hyperbolicity theorems can be used to show not only that planar and locally planar graphs are colourable (for some notion of colouring), but also that there exist exponentially many colourings of both planar and locally planar graphs. That exponentially many colourings follows from these hyperbolicity theorems (rather than the related notion of *strong* hyperbolicity, which bounds the number of vertices in annuli rather than disks) was not known. For more interesting implications of hyperbolicity, we refer the reader to [@postle2018hyperbolic]. ## From Hyperbolicity to Exponentially Many Colourings: A General Method {#subsec:hyperbolicity} Another contribution of this paper is a general method to show there exist exponentially many colourings (for some notion of colouring) of graphs on surfaces. Namely our method shows that deficiency versions of the hyperbolicity of the associated family of critical graphs implies exponentially many colourings of planar graphs in the class (and following the work of the first author and Thomas [@postle2018hyperbolic] also of graphs in the class embedded in a fixed surface provided the class is also strongly hyperbolic). The theory of hyperbolic families of graphs was developed by the first author and Thomas in [@postle2018hyperbolic]. Loosely speaking, a family of embedded graphs is *hyperbolic* if every graph in the family satisfies a certain linear isoperimetric inequality (given in the definition above): roughly, that for every disk in the surface, the number of vertices in the disk is linear in the number of vertices in the boundary of the disk. Similarly, a family is *strongly hyperbolic* if the number of vertices in each annulus is a linear function of the number of vertices in its two boundary components. This is defined more formally in [@postle2018hyperbolic]. Per [@postle2018hyperbolic], many results involving colouring graphs on surfaces are implied by the hyperbolicity of the associated class of *critical* graphs (i.e. the minimal non-colourable graphs in the class). Pleasantly, to prove hyperbolicity, it usually suffices to prove a related result for extending a precolouring of the outer cycle of a planar graph (or two cycles, in the case of strong hyperbolicity). Hence many theorems about colouring graphs on surfaces can be reduced to theorems about planar graphs via the theory of hyperbolicity. The implications of such hyperbolicity-related theorems are discussed in more detail in [@lukeevehyperbolicity] and [@postle2018hyperbolic]; they include among others: colouring theorems for locally planar (embedded) graphs, and efficient algorithms for colouring decidability of graphs embedded in fixed surfaces (per the work of Dvořák and Kawarabayashi [@dvovrak2013list]). Thus another main contribution of this paper is to show that such theorems can also be used to prove the existence of exponentially many colourings of planar graphs in the class. Technically, we need a *deficiency* version of a hyperbolicity theorem (one that bounds the number of vertices not only in the number of boundary vertices of the disk but also the number of edges inside the disk; see Section [2](#sec:keyresults){reference-type="ref" reference="sec:keyresults"} for a formal definition). That our results follow from these deficiency hyperbolicity theorems further motivates the study of hyperbolic families. Hence our Theorem [\[expmanycc\]](#expmanycc){reference-type="ref" reference="expmanycc"} will follow from the deficiency hyperbolicity theorem for $5$-correspondence coloring that we proved in a previous paper [@lukeevehyperbolicity]. Our method can also be applied in other colouring settings where the appropriate deficiency hyperbolicity theorem is known: as another demonstration of our method, we show in Section [4](#sec:girth5){reference-type="ref" reference="sec:girth5"} that our technique can also be used to prove there exist exponentially many 3-correspondence colourings of planar graphs of girth at least five. In particular, we show the following. thmexpmanyccgirthfive[\[expmanyccgirthfive\]]{#expmanyccgirthfive label="expmanyccgirthfive"} If $G$ is a planar graph and girth at least five and $(L,M)$ is a 3-correspondence assignment for $G$, then $G$ has at least $2^\frac{v(G)}{282}$ distinct $(L,M)$-colourings. We note that the bound above (which originally appeared in the PhD thesis of the second author [@evethesis]) was recently improved by Dahlberg, Kaul, and Mudrock [@dahlberg2023algebraic] (to $c = \frac{\log_2 3}{6}$) using the polynomial method. We discuss the polynomial method further in the next subsection. Analogously to the 5-correspondence colouring case, we further show Theorem [\[expmanyccgirthfive\]](#expmanyccgirthfive){reference-type="ref" reference="expmanyccgirthfive"} can be used to prove the theorem below. thmthmexpmanylocfive[\[thm:expmanylocallyplanarg5\]]{#thm:expmanylocallyplanarg5 label="thm:expmanylocallyplanarg5"} For every surface $\Sigma$, there exists a constant $\rho > 0$ with $\rho = O(\log g)$ (where $g$ is the Euler genus of $\Sigma$) such that the following holds: if $G$ is a $\rho$-locally planar graph of girth at least five that embeds in $\Sigma$ and $(L,M)$ is a 3-correspondence assignment for $G$, then $G$ has at least $2^\frac{v(G)}{25380}$ distinct $(L,M)$-colourings. ## Comparison to Polynomial Method {#subsec:polmethod} We should remark that the proofs of the hyperbolicity theorems mentioned in the foregoing subsection are often long and technical. This might motivate the skeptical reader to wonder why one should use our method (which relies on the existence of the hyperbolicity theorems) at all, given the existence of (for instance) the polynomial method, which has been used to tackle similar questions involving counting graph colourings and often yields better bounds. Our reasoning is twofold: first, as alluded to above and explained in more detail in [@postle2018hyperbolic], hyperbolicity theorems come with a host of other interesting implications. For that reason, it is of independent interest to prove them; and once they are proved, the remainders of the proofs of Theorems [\[expmanycc\]](#expmanycc){reference-type="ref" reference="expmanycc"} and [\[expmanyccgirthfive\]](#expmanyccgirthfive){reference-type="ref" reference="expmanyccgirthfive"} are quite concise (approximately two pages for each proof). Our method demonstrates yet another interesting implication of these hyperbolicity theorems, and in so doing further motivates their study. Second, our method can be applied to more colouring types and graph class combinations than the polynomial method (for instance, 5-correspondence colouring planar graphs). As mentioned in Subsections [1.1](#subsec:results){reference-type="ref" reference="subsec:results"} and [1.2](#subsec:hyperbolicity){reference-type="ref" reference="subsec:hyperbolicity"}, the polynomial method has been used to attain the best known bounds on $c$ for showing there are exponentially many 5-list colouring planar graphs [@bosek2022graph], exponentially many 3-list colouring planar graphs of girth at least five [@bosek2022graph], and exponentially many 3-correspondence colourings of planar graphs of girth at least five [@dahlberg2023algebraic]. We give a brief summary of the polynomial method. In [@bosek2022graph] and [@dahlberg2023algebraic], the authors use a slightly weaker version of a famous theorem of Alon and Füredi [@alon1993covering] concerning the number of non-vanishing points of a polynomial $P(x_1, x_2, \dots, x_n)$ over a set $B \subseteq \mathbb{F}^n$, where $\mathbb{F}$ is a field. In particular, they use the following. **Theorem 7** ([@bosek2022graph]). *Let $\mathbb{F}$ be an arbitrary field, and let $A_1,A_2,\dots, A_n$ be any nonempty subsets of $\mathbb{F}$ with $S = \sum_{i=1}^n |A_i|$ and $t = \max |A_i|$. Let $B = A_1 \times A_2 \times \dots \times A_n$, and suppose that $P(x_1, \dots, x_n)$ is a polynomial over $\mathbb{F}$ of degree $d$ that does not vanish on all of $B$. Then the number of points in $B$ for which $P$ has a non-zero value is at least $t^{\frac{S-n-d}{t-1}}$ provided $t \geq 2$.* This theorem is the main tool. The general method is to then associate a graph $G$ with vertex set $\{v_1, \dots, v_n\}$ and list assignment $L(v_i) = A_i$ for all $i \in \{1, \dots, n\}$ a polynomial $P(x_1, \dots, x_n) = \prod_{v_iv_j \in E(G) } (x_i-x_j)$. For a point $(a_1, \dots, a_n) \in A_1 \times \dots \times A_n$, we have that $P(a_1, \dots, a_n)$ is non-vanishing if and only if $a_i \neq a_j$ for all $v_iv_j \in E(G)$. Since $a_i \in A_i = L(v_i)$ for all $i \in \{1, \dots, n\}$, the point $(a_1, \dots, a_n)$ corresponds to a valid $L$-colouring of $G$. If $G$ is planar, the value of $d$ (which corresponds to $e(G)$) is appropriately bounded and it then follows from Theorem [Theorem 7](#thm:alonfuredi){reference-type="ref" reference="thm:alonfuredi"} that there are exponentially many $L$-colourings. In [@dahlberg2023algebraic], the authors define a different polynomial to that described above; their polynomial encodes 3-correspondence colouring, rather than the more straightforward 3-list colouring. For a graph $G$ with correspondence assignment $(L,M)$, we may assume without loss of generality that every vertex $v$ has $L(v) = \mathbb{F}_3$, the finite field with three elements. Recall that in the case of correspondence colouring, the potential conflicts between vertices $u,v$ with $uv \in E(G)$ are described by a partial matching $M_{uv}$ between $L(v)$ and $L(u)$. Since $G$ has the fewest possible $(L,M)$-colourings when $M_{uv}$ is a perfect matching for all $uv \in E(G)$, it follows that when lower-bounding the number of $(L,M)$-colourings of $G$ we may assume that $M_{uv}$ describes a permutation $\sigma$ of $\mathbb{F}_3$. The authors observe that for each such permutation $\sigma$, either $z -\sigma(z)$ is the same for all $z \in \mathbb{F}_3$, or $z + \sigma(z)$ is the same for all $z \in \mathbb{F}_3$. This motivates defining the factor of the polynomial $P$ associated with $v_iv_j \in E(G)$ as $(x_i+(-1)^c x_j-a)$, where $c$ and $a$ are chosen so that this factor is zero if and only if the values of $x_i$ and $x_j$ are matched in $M_{v_iv_j}$. However, this strategy does not work in the case of 5-correspondence colouring, as no such observation concerning $\sigma$ (and resulting in linear factors for each of the edges in the graph) holds. Though the polynomial method used by Dahlberg, Kaul, and Mudrock [@dahlberg2023algebraic] and Bosek et al. [@bosek2022graph] gives better explicit bounds on the number of colourings than our method or the more ad-hoc approaches of Thomassen [@thomassen2007exponentially; @thomassen2007many], our method does not have the same algebraic constraints, and thus applies to more diverse notions of colouring. In addition, as we show in Section [5](#sec:locallyplanar){reference-type="ref" reference="sec:locallyplanar"}, we can use our hyperbolicity results to extend these theorems counting planar colourings to counting colourings of locally planar graphs. ## Outline of Paper {#subsec:paperoutline} In Subsection [1.5](#subsec:methodoverview){reference-type="ref" reference="subsec:methodoverview"}, we give a brief overview of our method and the proofs of Theorems [\[expmanycc\]](#expmanycc){reference-type="ref" reference="expmanycc"} and [\[expmanyccgirthfive\]](#expmanyccgirthfive){reference-type="ref" reference="expmanyccgirthfive"}. Section [2](#sec:keyresults){reference-type="ref" reference="sec:keyresults"} contains several key results (some from previous papers) which will be used in the remainder of the paper. In particular, Subsection [2.1](#subsec:deficiencytheorems){reference-type="ref" reference="subsec:deficiencytheorems"} contains theorems related to deficiency and hyperbolicity, and Subsection [2.2](#subsec:corr-delsub){reference-type="ref" reference="subsec:corr-delsub"} introduces *correspondence-deletable subgraphs* and uses the results from Subsection [2.1](#subsec:deficiencytheorems){reference-type="ref" reference="subsec:deficiencytheorems"} to derive further useful tools for the proof of Theorem [\[expmanycc\]](#expmanycc){reference-type="ref" reference="expmanycc"}. Section [3](#sec:girth3){reference-type="ref" reference="sec:girth3"} contains a proof of Theorem [\[expmanycc\]](#expmanycc){reference-type="ref" reference="expmanycc"}. The tools required for the proof of Theorem [\[expmanyccgirthfive\]](#expmanyccgirthfive){reference-type="ref" reference="expmanyccgirthfive"} are found in Subsection [4.1](#subsec:toolsgirth5){reference-type="ref" reference="subsec:toolsgirth5"}; the proof of Theorem [\[expmanyccgirthfive\]](#expmanyccgirthfive){reference-type="ref" reference="expmanyccgirthfive"} is in Subsection [4.2](#subsec:proofsgirth5){reference-type="ref" reference="subsec:proofsgirth5"}. Section [5](#sec:locallyplanar){reference-type="ref" reference="sec:locallyplanar"} contains the proofs of Theorems [\[thm:expmanylocallyplanar\]](#thm:expmanylocallyplanar){reference-type="ref" reference="thm:expmanylocallyplanar"} and [\[thm:expmanylocallyplanarg5\]](#thm:expmanylocallyplanarg5){reference-type="ref" reference="thm:expmanylocallyplanarg5"}. Finally, Section [6](#sec:conclusion){reference-type="ref" reference="sec:conclusion"} contains a discussion on further directions. ## Proof and Method Overview {#subsec:methodoverview} Our main theorem is Theorem [\[expmanycc\]](#expmanycc){reference-type="ref" reference="expmanycc"} (proved in Section [3](#sec:girth3){reference-type="ref" reference="sec:girth3"}). In fact, we prove a more technical version of Theorem [\[expmanycc\]](#expmanycc){reference-type="ref" reference="expmanycc"}Theorem [Theorem 18](#expmanyextensions){reference-type="ref" reference="expmanyextensions"}which involves counting the number of extensions of a precoloured connected subgraph $S$ of a planar graph $G$ to a 5-correspondence colouring of $G$ itself. We show that if the precolouring of $S$ has at least one extension, then it has exponentially many extensions. The precise number of extensions is counted in terms of the deficiency of $G$, defined more formally in Section [2](#sec:keyresults){reference-type="ref" reference="sec:keyresults"}. This stronger, deficiency version of Theorem [\[expmanycc\]](#expmanycc){reference-type="ref" reference="expmanycc"} is more amenable to an inductive proof, as will be clear to the reader in Section [3](#sec:girth3){reference-type="ref" reference="sec:girth3"}. Analogously, in Section [4](#sec:girth5){reference-type="ref" reference="sec:girth5"}, we prove Theorem [\[expmanyccgirthfive\]](#expmanyccgirthfive){reference-type="ref" reference="expmanyccgirthfive"} via Theorem [Theorem 24](#expmanyextensions5){reference-type="ref" reference="expmanyextensions5"}, which concerns extending a precolouring of a connected subgraph $S$ of a planar graph $G$ of girth at least five to a 3-correspondence colouring of $G$. Both the proof of Theorem [Theorem 18](#expmanyextensions){reference-type="ref" reference="expmanyextensions"} and that of Theorem [Theorem 24](#expmanyextensions5){reference-type="ref" reference="expmanyextensions5"} proceed by induction on $v(G) - v(S)$. There are essentially two cases to consider: either there exists a subgraph $H$ with $S \subsetneq H \subsetneq G$ and $v(S)<v(H)<v(G)$ such that every colouring of $H$ extends to a colouring of $G$, or no such subgraph $H$ exists. In the former case, by our choice of $G$, since $v(S)<v(H)<v(G)$ and $S \subsetneq H \subsetneq G$, we have that both that $v(H)-v(S) < v(G) -v(S)$ and that $v(G)-v(H)< v(G)-v(S)$. Theorem [Theorem 18](#expmanyextensions){reference-type="ref" reference="expmanyextensions"} then follows by induction by first counting the extensions of the colouring of $S$ to $H$, and then of each colouring of $H$ to $G$. In the latter case where no such subgraph $H$ exists, we use our deficiency hyperbolicity theorems (Theorems [Theorem 12](#theorem:stronglinear){reference-type="ref" reference="theorem:stronglinear"} and [Observation 19](#girth5:stronglinear){reference-type="ref" reference="girth5:stronglinear"}) to show that $G$ has relatively high deficiency with respect to $S$ (see Theorem [\[d_gepsilonbound\]](#d_gepsilonbound){reference-type="ref" reference="d_gepsilonbound"} in the case of 5-correspondence colouring and Theorem [Lemma 21](#d_gepsilonboundg5){reference-type="ref" reference="d_gepsilonboundg5"} in the case of 3-correspondence colouring). We then pick a vertex $v$ outside $S$ with the most neighbours in $S$, and then consider extending the precolouring of $S$ first to $S+v$ and then to $G$. Since $S + v$ does not have the properties of a graph $H$ as described in the previous paragraph, we use the deficiency theorems to bound the deficiency of the whole graph, at which point the formula follows since there is at least one extension by assumption. Other key tools are Theorems [Theorem 17](#thomtech5cc){reference-type="ref" reference="thomtech5cc"} and [Theorem 23](#thomtech3cc){reference-type="ref" reference="thomtech3cc"} (in the girth three and five case, respectively), which are Thomassen's stronger inductive theorems that imply that planar graphs are 5-correspondence colourable, and that planar graphs of girth at least five are 3-correspondence colourable. These help bound the deficiency in the final steps of the proofs. We note the case where $V(G) = V(S) \cup \{v\}$ is especially important, as it helps determine the coefficient in the exponent of Theorems [\[expmanycc\]](#expmanycc){reference-type="ref" reference="expmanycc"} and [\[expmanyccgirthfive\]](#expmanyccgirthfive){reference-type="ref" reference="expmanyccgirthfive"}. The general method then may be described as follows: the first step is to define the correct notion of deficiency and deletability for the graph class and type of colouring with which we are working. We then prove a deficiency version of a hyperbolicity theorem for the critical graphs in the class under study, and use this to derive a bound on the deficiency of graphs that do not contain deletable subgraphs (see Definition [Definition 14](#def:deletable){reference-type="ref" reference="def:deletable"}). One final key tool is a theorem analogous to Theorem [Theorem 17](#thomtech5cc){reference-type="ref" reference="thomtech5cc"}. # Our Toolbox {#sec:keyresults} In this section, we develop the tools required for the proof of Theorem [\[expmanycc\]](#expmanycc){reference-type="ref" reference="expmanycc"}. In particular, Subsection [2.1](#subsec:deficiencytheorems){reference-type="ref" reference="subsec:deficiencytheorems"} contains our hyperbolicity theorem for 5-correspondence colouring and its stronger, deficiency version from our previous paper [@lukeevehyperbolicity]. This will be our main tool in proving Theorem [\[expmanycc\]](#expmanycc){reference-type="ref" reference="expmanycc"}. Subsection [2.2](#subsec:corr-delsub){reference-type="ref" reference="subsec:corr-delsub"} introduces the notion of *correspondence-deletable subgraphs*, and derives from Theorem [Theorem 12](#theorem:stronglinear){reference-type="ref" reference="theorem:stronglinear"} (our deficiency hyperbolicity theorem for 5-correspondence colourings) useful results concerning said deletable subgraphs. The analogous tools for 3-correspondence colouring used in the proof of Theorem [\[expmanyccgirthfive\]](#expmanyccgirthfive){reference-type="ref" reference="expmanyccgirthfive"} are found in Section [4](#sec:girth5){reference-type="ref" reference="sec:girth5"} (Subsection [4.1](#subsec:toolsgirth5){reference-type="ref" reference="subsec:toolsgirth5"}). ## Deficiency Theorems for Hyperbolicity {#subsec:deficiencytheorems} This subsection introduces our main tool, Theorem [Theorem 12](#theorem:stronglinear){reference-type="ref" reference="theorem:stronglinear"} (from one of our previous papers [@lukeevehyperbolicity]). Theorem [Theorem 12](#theorem:stronglinear){reference-type="ref" reference="theorem:stronglinear"} concerns the hyperbolicity of a specific class of *critical* graphs. We recall the following definition. **Definition 8**. Let $G$ be a planar graph, $S$ a proper subgraph of $G$, and $(L,M)$ a correspondence assignment for $G$. We say $G$ is *$S$-critical with respect to $(L,M)$* if for every proper subgraph $G' \subset G$ such that $S \subseteq G'$, there exists an $(L,M)$-colouring of $S$ that extends to an $(L,M)$-colouring of $G'$, but does not extend to an $(L,M)$-colouring of $G$. If the correspondence assignment is clear from the context, we shorten this and say that $G$ is *$S$-critical.* The main hyperbolicity theorem presented in [@lukeevehyperbolicity] is as follows. **Theorem 9**. *[@lukeevehyperbolicity][\[theorem:mainhypthm\]]{#theorem:mainhypthm label="theorem:mainhypthm"} Let $\varepsilon = \frac{1}{50}$, let $G$ be a 2-connected plane graph, let $C$ be the outer cycle of $G$, and let $(L,M)$ be a 5-correspondence assignment for $G$. If $G$ is $C$-critical with respect to $(L,M)$, then $v(G) \leq \frac{1+\varepsilon}{\varepsilon}\cdot v(C)$.* Theorem [\[theorem:mainhypthm\]](#theorem:mainhypthm){reference-type="ref" reference="theorem:mainhypthm"} is functionally equivalent to proving the hyperbolicity of the family of graphs that are $\emptyset$-critical for some $5$-correspondence assignment and hence can be used to obtain the various applications of hyperbolicity to described in the introduction to $5$-correspondence colouring. Further discussion on this topic can be found in [@lukeevehyperbolicity]. However, Theorem [\[theorem:mainhypthm\]](#theorem:mainhypthm){reference-type="ref" reference="theorem:mainhypthm"} is not quite strong enough to obtain our result lower-bounding the number of 5-correspondence colourings of planar graphs. To prove Theorem [\[expmanycc\]](#expmanycc){reference-type="ref" reference="expmanycc"}, we need a stronger theorem which bounds $v(G)$ not only in terms of $v(C)$, but also in terms of $e(G)-e(C)$. To this end, we define the notion of *deficiency*. In the following definition (and indeed the rest of the paper), for a graph $G$ and subgraph $H \subseteq G$, we will use the notation $v(G|H):= v(G)-v(H)$ and $e(G|H) := e(G)-e(H)$. **Definition 10**. Let $G$ be a graph, and $H$ a subgraph of $G$. For a positive integer $g \geq 3$, we define the *$g$-deficiency of $G$ with respect to $H$* as $\textnormal{def}_g(G|H) := (g-2)e(G|H)-g\cdot v(G|H)$. The following quantity is also useful. **Definition 11**. Let $G$ be a graph, and $H$ a subgraph of $G$. For a positive integer $g \geq 3$ and positive $\varepsilon \in \mathbb{R}$, we define $d_{g,\varepsilon}(G|H) : = (g-2)e(G|H) - (g+\varepsilon)v(G|H)$. Equivalently, $d_{g, \varepsilon} (G|H) := \textnormal{def}_g(G|H)-\varepsilon \cdot v(G|H)$. (Note this nearly matches the definition of $d(\cdot)$ given in [@lukeevehyperbolicity], ignoring the $b(\cdot)$ and $q(\cdot)$ terms.) We will require the following results from our previous paper [@lukeevehyperbolicity]. Theorem [Theorem 12](#theorem:stronglinear){reference-type="ref" reference="theorem:stronglinear"} will be used in the proof of Theorem [\[expmanycc\]](#expmanycc){reference-type="ref" reference="expmanycc"}, and Observation [Observation 19](#girth5:stronglinear){reference-type="ref" reference="girth5:stronglinear"}, in the proof of Theorem [\[expmanyccgirthfive\]](#expmanyccgirthfive){reference-type="ref" reference="expmanyccgirthfive"}. Theorem [Theorem 12](#theorem:stronglinear){reference-type="ref" reference="theorem:stronglinear"} is a weaker version of one of the main theorems in [@lukeevehyperbolicity] (Theorem 3.21), ignoring the boundary and quasiboundary terms (and using $\alpha = \frac{1}{25}$ and $\gamma = \frac{7}{10}$). The deficiency version of our hyperbolicity theorem is as follows. **Theorem 12**. *Let $\varepsilon = \frac{1}{50}$, let $G$ be a 2-connected plane graph, let $C$ be the outer cycle of $G$, and let $(L,M)$ be a 5-correspondence assignment for $G$. If $G$ is $C$-critical with respect to $(L,M)$ and $v(G|C)\ge 2$, then $d_{3, \varepsilon}(G|C)\geq \frac{23}{10}$.* The following easy observation follows directly from our definitions. **Observation 13**. *Let $G$ be a graph; let $H$ be a subgraph of $G$; and let $\varepsilon > 0$. If $d_{3, \varepsilon}(G|H) \geq 0$, then $v(G|H) \leq \varepsilon^{-1}\cdot \textnormal{def}_3(G|H)$.* *Proof.* By definition, $d_{3, \varepsilon}(G|H) = \textnormal{def}_3(G|H)-\varepsilon \cdot v(G|H)$. Since $d_{3,\varepsilon }(G|H) \geq 0$, it follows that $0 \leq \textnormal{def}_3(G|H)-\varepsilon \cdot v(G|H)$. Note that $\varepsilon > 0$; by isolating $v(G|H)$, we obtain the desired result. ◻ Note that the fact that $d_{3, \varepsilon}(G|C) \geq 0$ in Theorem [Theorem 12](#theorem:stronglinear){reference-type="ref" reference="theorem:stronglinear"} implies Theorem [\[theorem:mainhypthm\]](#theorem:mainhypthm){reference-type="ref" reference="theorem:mainhypthm"}: using Euler's formula for graphs embedded in the plane and the definition of deficiency, we have that $\textnormal{def}_3(G|C) \leq v(C)-3$. Combining this and Observation [Observation 13](#deficiencyobs){reference-type="ref" reference="deficiencyobs"}, one obtains Theorem [\[theorem:mainhypthm\]](#theorem:mainhypthm){reference-type="ref" reference="theorem:mainhypthm"}. ## Correspondence-Deletable Subgraphs {#subsec:corr-delsub} This subsection introduces *correspondence-deletable* graphs, defined below. A related notion (*deletable graphs*) was defined for list colouring in [@postlelocalalgs]. We then show how to use our deficiency hyperbolicity theorem (Theorem [Theorem 12](#theorem:stronglinear){reference-type="ref" reference="theorem:stronglinear"}) to obtain useful results concerning correspondence-deletable subgraphs. **Definition 14**. Let $G$ be a graph, and let $H$ be an induced nonempty subgraph of $G$. We say $H$ is *$r$-correspondence-deletable* if for every correspondence assignment $(L,M)$ of $H$ such that $|L(v)| \geq r-(\deg_G(v) - \deg_H(v))$ for each $v \in V(G)$, the graph $H$ has an $(L,M)$-colouring no matter the correspondence assignment $(L,M)$. Note that if a graph is $r$-correspondence-deletable, it follows that it is also $r$-deletable (the list colouring analogue, defined in [@postlelocalalgs]). The converse, however, does not hold. If $H$ is an $r$-correspondence-deletable subgraph of $G$, then every $(L,M)$-colouring of $G\setminus V(H)$ extends to an $(L,M)$-colouring of $H$. In fact, the definition above captures an even stronger notion: for an $r$-correspondence-deletable subgraph $H \subseteq G$, an $(L,M)$-colouring of $G\setminus V(H)$ extends to an $(L,M)$-colouring of $H$ no matter the correspondence assignment $(L,M)$ and no matter the structure of $G \setminus V(H)$. We highlight the relationship between critical graphs and deletable graphs: let $G$ be a graph, and $S$ a subgraph of $G$. If there exists a 5-correspondence assignment $(L,M)$ such that $G$ is $S$-critical with respect to $(L,M)$, then $G$ has no 5-correspondence-deletable subgraph that is vertex-disjoint from $S$. The converse does not hold. However, we can still use our deficiency hyperbolicity theorem to show that $G$ has high deficiency in this case. This result (below) will be instrumental in the proof of Theorem [\[expmanycc\]](#expmanycc){reference-type="ref" reference="expmanycc"}. Theorem [\[d_gepsilonbound\]](#d_gepsilonbound){reference-type="ref" reference="d_gepsilonbound"} will be proved later in this subsection. thmd_gepsilonbound[\[d_gepsilonbound\]]{#d_gepsilonbound label="d_gepsilonbound"} Let $G$ be a plane graph, and let $\varepsilon$ be as in Lemma [Lemma 15](#Hcritical){reference-type="ref" reference="Hcritical"}. If $H$ is a connected subgraph of $G$ such that there does not exist $X \subseteq V(G) \setminus V(H)$ such that $G[X]$ is 5-correspondence-deletable in $G$, then $d_{3,\varepsilon}(G|H) \geq 0$. We remark a list colouring version of Theorem [\[d_gepsilonbound\]](#d_gepsilonbound){reference-type="ref" reference="d_gepsilonbound"} was proved in [@postlelocalalgs] (Lemma 5.22). To prove Theorem [\[d_gepsilonbound\]](#d_gepsilonbound){reference-type="ref" reference="d_gepsilonbound"}, we will need the following two results. Again, list colouring versions of these results are proved in [@postlelocalalgs] (Theorem 5.20 and Proposition 5.21). The proof of Lemma [Lemma 15](#Hcritical){reference-type="ref" reference="Hcritical"} is nearly identical to its list colouring analogue except for the use of Theorem [Theorem 12](#theorem:stronglinear){reference-type="ref" reference="theorem:stronglinear"} (instead of a list colouring version of this theorem). **Lemma 15**. *There exists $\varepsilon > 0$ such that following holds: if $G$ is a plane graph, $H$ is a connected subgraph of $G$, and there exists a 5-correspondence assignment $(L,M)$ for $G$ such that $G$ is $H$-critical with respect to $(L,M)$, then $d_{3,\varepsilon}(G|H) \geq 0$.* *Proof.* In the case where $v(G|H) = 0$, this follows from the definition of $H$-critical (since if $G$ is $H$-critical with $v(G|H) = 0$ then it follows that $e(G|H) \geq 1$). In the case where $v(G|H) =1$, again this follows from the definition of $H$-critical (as the vertex in $V(G) \setminus V(H)$ has at least five neighbours in $H$). The $v(G|H)\geq 2$ case follows from Theorem [Theorem 12](#theorem:stronglinear){reference-type="ref" reference="theorem:stronglinear"}. Note that if $G$ is $H$-critical, we may assume without loss of generality that $G$ is 2-connected: it suffices to delete edges from $H$ until it is a tree; to split each $v \in V(H)$ into $\deg(v)$ copies of itself, where each copy $v'$ is adjacent to copies $u'$, $w'$ of the two vertices $u,w$ that precede and follow it in the boundary walk of the tree obtained from $H$ (with multiplicity), as well as each vertex of $V(G)$ that lies between $u$ and $w$ in the portion of the cyclic ordering of the neighbours of $v$ (in the tree obtained from $H$) containing no other vertices of $H$. After performing this operation, $H$ becomes a facial cycle, and the result follows just as in Lemma 2.5 in [@luke5LC]. ◻ The proof of the following proposition is nearly identical to that of Proposition 5.21 in [@postlelocalalgs]; we include the proof for the purposes of cohesion. **Proposition 16**. *Let $G$ be a graph and $H$ a proper subgraph of $G$ such that $V(H) \neq V(G)$. If $G-V(H)$ is not $r$-correspondence-deletable in $G$, then there exists a subgraph $G_0$ of $G$ containing $H$ and an $r$-correspondence assignment for $G$ such that $G_0$ is $H$-critical.* *Proof (Adapted from Proposition 5.21, [@postlelocalalgs]).* Since $G-V(H)$ is not $r$-correspondence-deletable in $G$, there exists a correspondence assignment $(L_0, M_0)$ such that $|L_0(v)| \geq r-\deg_H(v)$ for each $v \in V(G) \setminus V(H)$ and $G-V(H)$ is not $(L_0, M_0)$-colourable. Define a new correspondence assignment $(L,M)$ of $G$ as follows. For each $v \in V(H)$, define $c_v$ to be a new colour not appearing in any other list. Define $R$ to be a set of $r-1$ distinct colours not appearing in any other list or in $\cup_{v \in V(H)} \{c_v\}$. Set $L(v) = \{c_v\} \cup R$ for each $v \in V(H)$. For each $u \in V(G) \setminus V(H)$, let $L(u) = L_0(u) \cup \{c_v: v \in N(u) \cap V(H)\}$. For each $uv \not \in E(H)$, set $M_{uv} = (M_0)_{uv}$. For each $uv \in E(H)$, set $M_{uv} = \emptyset$. Finally, for each $uv$ with $u \in V(G)\setminus V(H)$ and $v \in V(H)$, set $M_{uv} = \{(u,c_v)(v,c_v)\}$. Now $(L,M)$ is an $r$-correspondence assignment of $G$. Let $\phi$ be the colouring of $H$ given by $\phi(v) = c_v$ for every $v \in V(H)$. Since $G-V(H)$ is not $(L_0,M_0)$-colourable, it follows that $\phi$ does not extend to an $(L,M)$-colouring of $G$. Let $G'$ be an inclusion-wise minimal subgraph of $G$ containing $H$ such that $\phi$ does not extend to an $(L,M)$-colouring of $G'$. By the minimality of $G'$, we have that $\phi$ extends to an $(L,M)$-colouring of every proper subgraph of $G'$ containing $H$. Thus $G'$ is $H$-critical with respect to $(L,M)$, as desired. ◻ We are now equipped to prove Theorem [\[d_gepsilonbound\]](#d_gepsilonbound){reference-type="ref" reference="d_gepsilonbound"}. *Proof of Theorem [\[d_gepsilonbound\]](#d_gepsilonbound){reference-type="ref" reference="d_gepsilonbound"}.* We proceed by induction on $v(G|H)+e(G|H)$. If $V(H) = V(G)$, then $d_{3,\varepsilon}(G|H) \geq 0$ as desired. So we may assume that $V(H)\neq V(G)$. By assumption, $G-V(H)$ is not 5-correspondence-deletable in $G$. By Proposition [Proposition 16](#notdelthencrit){reference-type="ref" reference="notdelthencrit"}, it follows that there exists a subgraph $G_0$ of $G$ containing $H$ and 5-correspondence assignment for $G_0$ such that $G_0$ is $H$-critical. Note that $H$ is a proper subgraph of $G_0$ by definition of $H$-critical. By Lemma [Lemma 15](#Hcritical){reference-type="ref" reference="Hcritical"}, we have that $d_{3,\varepsilon}(G_0|H) \geq 0$. By definition of critical, since $H$ is connected it follows that $G_0$ is connected. Note that $v(G|G_0)+ e(G|G_0) < v(G|H) + e(G|H)$. Hence by induction, $d_{3,\varepsilon}(G|G_0) \geq 0$. By definition of $d_{3, \varepsilon}$, we have that $d_{3,\varepsilon}(G|H) = d_{3,\varepsilon}(G|G_0) + d_{3,\varepsilon}(G_0|H) \geq 0 + 0 = 0$, as desired. ◻ # Counting 5-Correspondence Colourings {#sec:girth3} In this section, we prove Theorem [Theorem 18](#expmanyextensions){reference-type="ref" reference="expmanyextensions"} which we will show afterwards implies Theorem [\[expmanycc\]](#expmanycc){reference-type="ref" reference="expmanycc"}. Theorem [Theorem 18](#expmanyextensions){reference-type="ref" reference="expmanyextensions"} involves counting the number of colouring extensions of a precoloured subgraph $S$ of a graph $G$ to a colouring of $G$ itself. The precise bound on the number of extensions is given in part in terms of the deficiency of $G$ with respect to $S$. The reader may find it helpful to consult Figure [\[fig:expmany\]](#fig:expmany){reference-type="ref" reference="fig:expmany"} while reading for a depiction of the cases considered in the proof. The final tool we will need before proving Theorem [Theorem 18](#expmanyextensions){reference-type="ref" reference="expmanyextensions"} is the following theorem, due to Thomassen. This theorem was originally written in the language of list colouring; however, as pointed out by Dvořák and the first author in [@dvovrak2018correspondence], the proof also carries over to the realm of correspondence colouring. **Theorem 17** (Thomassen, [@thomassen5LC]). *Let $G$ be a plane graph. Let $C$ be the subgraph of $G$ whose edge- and vertex-set are precisely those of the outer face boundary walk of $G$. Let $(L,M)$ be a correspondence assignment for $G$ where $|L(v)| \geq 1$ for a path $S \subseteq C$ with $v(S)\leq 2$; where $|L(v)| \geq 3$ for all $v \in V(C) \setminus V(S)$; and where $|L(v)|\geq 5$ for all $v \in V(G) \setminus V(C)$. Then every $(L,M)$-colouring of $S$ extends to an $(L,M)$-colouring of $G$.* Here is our precolouring extension theorem (that implies Theorem [\[expmanycc\]](#expmanycc){reference-type="ref" reference="expmanycc"}). **Theorem 18**. *Let $\varepsilon$ be as in Theorem [\[d_gepsilonbound\]](#d_gepsilonbound){reference-type="ref" reference="d_gepsilonbound"}. Let $G$ be a plane graph, let $S$ be a connected subgraph of $G$, and let $(L,M)$ be a 5-correspondence assignment for $G$. If $\phi$ is an $(L,M)$-colouring of $S$ that extends to an $(L,M)$-colouring of $G$, then $$\log_2 E(\phi) \geq \frac{v(G|S) - (\varepsilon^{-1}+1)\textnormal{def}_3(G|S)}{67},$$* *where $E(\phi)$ denotes the number of extensions of $\phi$ to $G$.* *Proof.* We proceed by induction on $v(G|S)$. First suppose $v(G|S) = 0$. Then since $\textnormal{def}_3(G|S)$ is positive, the right-hand side of the equation is negative. As $\phi$ extends to an $(L,M)$-colouring of $G$ by assumption, we have that $\log_2 E(\phi)\ge 0$ and the result follows. Next suppose that $v(G|S) = 1$, and let $v \in V(G)\setminus V(S)$. Then $$\frac{v(G|S)-(\varepsilon^{-1}+1)\textnormal{def}_3(G|S)}{67} = \frac{1-(\varepsilon^{-1}+1)(\deg(v)-3)}{67}.$$ Note that when $\deg(v) \geq 4$, the right-hand side is negative since $\varepsilon^{-1} > 0$. Since $\phi$ extends to an $(L,M)$-colouring of $G$ by assumption, it follows that $\log_2 E(\phi) \geq 0$, and so $\log_2 E(\phi) \geq \frac{v(G|S) - (\varepsilon^{-1}+1)\textnormal{def}_3(G|S)}{67}$ holds as desired. We may therefore assume that $\deg(v) \leq 3$. Since $|L(v)| \geq 5$ it follows that $E(\phi) \geq 5-\deg(v)$. Therefore it suffices to show that $\log_2 (5-\deg(v)) \geq \frac{1-(\varepsilon^{-1}+1)(\deg(v)-3)}{67}$, or equivalently, that $67 \geq \frac{1-(\varepsilon^{-1}+1)(\deg(v)-3)}{\log_2 (5-\deg(v))}$. The right-hand side is maximized when $\deg(v) = 0$, in which case it is easy to verify that $67 > \frac{154}{\log_2(5)}$. We may therefore assume that $v(G|S) \geq 2$. Before proceeding with the remainder of the case analysis, we will need the following claim. **Claim 1**. *There does not exist a graph $H \subsetneq G$ with $S \subsetneq H$ and $v(S) < v(H) < v(G)$ such that $G-V(H)$ is a 5-correspondence-deletable subgraph of $G$.* *Proof.* Suppose not. Let $G' := G - V(H)$. Note that $G'$ is an induced subgraph of $G$. Since $v(S) < v(H)< v(G)$, we have that $v(H|S) < v(G|S)$, and so it follows by induction that there are at least $2^\frac{v(H|S)-(\varepsilon^{-1}+1)\textnormal{def}_3(H|S)}{67}$ extensions of $\phi$ to $H$. Since $G'$ is 5-correspondence-deletable, we have by the definition of 5-correspondence-deletable subgraph that each of these extensions of $\phi$ to an $(L,M)$-colouring of $H$ extends further to an $(L,M)$ colouring of $G'$, and thus to $G$. Since $v(S) < v(H)< v(G)$, it follows that $v(G|H) < v(G|S)$, and so by induction for each extension of $\phi$ to an $(L,M)$ colouring $\phi'$ of $H$ there are at least $2^\frac{v(G|H)-(\varepsilon^{-1}+1)\textnormal{def}_3(G|H)}{67}$ extensions of $\phi'$ to $G$. Therefore $$\begin{aligned} \log_2E(\phi) &\geq \frac{v(H|S)-(\varepsilon^{-1}+1)\textnormal{def}_3(H|S)}{67} + \frac{v(G|H)-( \varepsilon^{-1}+1)\textnormal{def}(G|H)}{67} \\ &=\frac{v(H|S)+v(G|H)-(\varepsilon^{-1}+1)(\textnormal{def}_3(H|S)+\textnormal{def}_3(G|H))}{67} \\ &= \frac{v(G|S)-(\varepsilon^{-1}+1)\textnormal{def}_3(G|S)}{67},\end{aligned}$$ as desired. ◻ Among all vertices in $V(G)\setminus V(S)$, choose a vertex $v$ that maximizes $|N(v) \cap V(S)|$. Let $H := S + v$ (see Figure [\[fig:expmany\]](#fig:expmany){reference-type="ref" reference="fig:expmany"}). Since $\phi$ extends to an $(L,M)$-colouring of $G$, there is at least $1 = 2^0$ extension $\phi'$ of $\phi$ to $H$ where $\phi'$ extends further to an $(L,M)$-colouring of $G$. Since $\phi'$ extends to $G$ and $v(G|H) < v(G|S)$, we have by induction that there exist at least $2^\frac{v(G|H)-(\varepsilon^{-1}+1)\textnormal{def}_3(G|H)}{67}$ extensions of $\phi'$ to $G$. Therefore $$\log_2 E(\phi) \geq 0 + \frac{v(G|H)-(\varepsilon^{-1}+1)\textnormal{def}_3(G|H)}{67}.$$ Moreover, since $v(G|H) = v(G|S)-1$ and $\textnormal{def}_3(G|H) = \textnormal{def}_3(G|S)-\deg_H(v)+3$, it follows that $$\log_2 E(\phi) \geq \frac{v(G|S) - 1-(\varepsilon^{-1}+1)(\textnormal{def}_3(G|S)-\deg_H(v)+3)}{67}.$$ If $\deg_H(v) \geq 4$, the desired result immediately holds since $\varepsilon > 0$. Thus we may assume $\deg_H(v) \leq 3$. First suppose $\deg_H(v) \leq 2$. We will show $G':= G-V(H)$ is 5-correspondence-deletable, contradicting Claim [Claim 1](#claimrdeletable){reference-type="ref" reference="claimrdeletable"}. To that end, let $(L',M')$ be a correspondence assignment for $G'$ with $|L'(u)| \geq 5-(\deg_G(u) - \deg_H(u))$ for all $u \in V(G')$. Note that by our choice of $v$, every vertex in $G'$ has $|L'(u)| \geq 2$. We claim $G'$ is $(L',M')$-colourable. To see this, let $T \subseteq V(G')$ be the set of vertices $\{u \in V(G'): |L'(u)| = 2\}$. Note that each vertex in $T$ is adjacent to $v$ in $G$. Let $G'':= G[V(G')\cup \{v\}]$. Let $(L'',M'')$ be a correspondence assignment for $G''$ obtained from $(L',M')$ as follows: define $L''$ by setting $L''(u) = L'(u)$ for all $u \in V(G')\setminus T$; setting $L''(v) = \{c\}$, where $c$ is a new colour not present in $\cup_{u \in V(G')}L(u)$; setting $L''(u) = L'(u) \cup \{c\}$ for all $u \in T$. Finally, define $M''$ by setting $M_{u_1u_2}'' = M'_{u_1u_2}$ for all $u_1u_2 \in E(G')$; and setting $M_{vu}'' = \{(v,c)(u,c)\}$ for all $u \in T$. Note that with the sole exception of $v$, every vertex $u$ in the outer face boundary walk of $G''$ has $|L''(u)| \geq 3$ by definition, and every vertex in $V(G'')\setminus (\{v\} \cup T)$ has $|L''(u)| \geq 5$. By Theorem [Theorem 17](#thomtech5cc){reference-type="ref" reference="thomtech5cc"}, $G''$ has an $(L'',M'')$-colouring $\phi'$. Since no vertex $u \in T$ has $\phi'(u) = c$ by construction, it follows that $\phi'$ is an $(L',M')$-colouring of $G'$. This proves that $G' = G-V(H)$ is 5-correspondence-deletable; and since $v(S) < v(H) < v(G)$, this contradicts Claim [Claim 1](#claimrdeletable){reference-type="ref" reference="claimrdeletable"}. We may therefore assume that $\deg_H(v) = 3$. By Claim [Claim 1](#claimrdeletable){reference-type="ref" reference="claimrdeletable"}, there does not exist $X \subseteq V(G)\setminus V(H)$ such that $G[X]$ is 5-correspondence-deletable in $G$. Thus by Theorem [\[d_gepsilonbound\]](#d_gepsilonbound){reference-type="ref" reference="d_gepsilonbound"}, we have that $d_{3,\varepsilon}(G|H) \geq 0$. Using this and Observation [Observation 13](#deficiencyobs){reference-type="ref" reference="deficiencyobs"}, it follows that $$\label{vghbound} v(G|H) \leq \varepsilon^{-1}\textnormal{def}_3(G|H).$$ Moreover, since $v(G|S) = 1+v(G|H)$ and $\textnormal{def}_3(G|S) = \textnormal{def}_3(G|H) + \textnormal{def}_3(H|S)$, it follows that $$\frac{v(G|S)-(\varepsilon^{-1}+1)\textnormal{def}_3(G|S)}{67} = \frac{1+v(G|H)-(\varepsilon^{-1}+1)(\textnormal{def}_3(G|H)+\textnormal{def}_3(H|S))}{67}.$$ By definition of $\textnormal{def}_3(G|S)$, we have that $\textnormal{def}_3(H|S)= \deg_H(v)-3 = 0$ since $\deg_H(v)=3$. Using this and Observation [Observation 13](#deficiencyobs){reference-type="ref" reference="deficiencyobs"}, we obtain that $$\begin{aligned} \frac{v(G|S)-(\varepsilon^{-1}+1)\textnormal{def}_3(G|S)}{67} & \leq \frac{1+\varepsilon^{-1}\textnormal{def}_3(G|H)-(\varepsilon^{-1}+1)\textnormal{def}_3(G|H)}{67} \\ &= \frac{1-\textnormal{def}_3(G|H)}{67}. \end{aligned}$$ By Theorem [\[d_gepsilonbound\]](#d_gepsilonbound){reference-type="ref" reference="d_gepsilonbound"}, we have that $\textnormal{def}_3(G|H) \geq \varepsilon \cdot v(G|H)$, and since $v(G|S) \geq 2$, it follows that $v(G|H) \geq 1$ and so $\textnormal{def}_3(G|H) \geq \varepsilon$. Since $\textnormal{def}_3(G|H)$ is integral, it follows that $\textnormal{def}_3(G|H) \geq 1$, and thus the right-hand side above is at most 0. Altogether then, we find that $$\frac{v(G|S)-(\varepsilon^{-1}+1)\textnormal{def}_3(G|S)}{67} \leq 0.$$ But since $\phi$ extends to an $(L,M)$-colouring of $G$, we have that $\log_2(E(\phi)) \geq 0$. Thus $$\log_2 E(\phi) \geq \frac{v(G|S)-(\varepsilon^{-1}+1)\textnormal{def}_3(G|S)}{67},$$ as desired. ◻ We end this section by showing how Theorem [Theorem 18](#expmanyextensions){reference-type="ref" reference="expmanyextensions"} implies Theorem [\[expmanycc\]](#expmanycc){reference-type="ref" reference="expmanycc"}. *Proof of Theorem [\[expmanycc\]](#expmanycc){reference-type="ref" reference="expmanycc"}.* Let $S$ be the empty graph, and $\phi$ a trivial colouring of $S$. Let $E(\phi)$ denote the number of extensions of $\phi$ to $G$. Since $G$ is planar, we have that $e(G) \leq 3\cdot v(G)$. By Theorem [Theorem 18](#expmanyextensions){reference-type="ref" reference="expmanyextensions"}, we find that $$\begin{aligned} \log_2{E(\phi)} &\geq \frac{v(G)-(50+1)(e(G)-3\cdot v(G))}{67} \\ &\geq\frac{v(G)}{67},\end{aligned}$$ as desired. ◻ # Counting 3-Correspondence Colourings {#sec:girth5} This section contains a proof of Theorem [\[expmanyccgirthfive\]](#expmanyccgirthfive){reference-type="ref" reference="expmanyccgirthfive"} which states that planar graphs of girth at least five have exponentially many 3-correspondence colourings. As in the girth three case, we prove Theorem [\[expmanyccgirthfive\]](#expmanyccgirthfive){reference-type="ref" reference="expmanyccgirthfive"} via a more technical theorem involving counting the number of colouring extensions of a precoloured subgraph. These proofs are found in Subsection [4.2](#subsec:proofsgirth5){reference-type="ref" reference="subsec:proofsgirth5"}. The tools needed to prove Theorem [\[expmanyccgirthfive\]](#expmanyccgirthfive){reference-type="ref" reference="expmanyccgirthfive"} are found in Subsection [4.1](#subsec:toolsgirth5){reference-type="ref" reference="subsec:toolsgirth5"} and are analogous to those needed to prove Theorem [\[expmanycc\]](#expmanycc){reference-type="ref" reference="expmanycc"}. ## Deficiency and Deletability Theorems for the Girth At Least Five Case {#subsec:toolsgirth5} First, we need a deficiency version of a hyperbolicity theorem, given below. Note this is a slightly weaker version of a similar result from a previous paper of ours: namely, the result below corresponds to Observation 2 in [@lukeevehyperbolicity], ignoring the $q(\cdot)$ term. **Observation 19**. *Let $\varepsilon = \frac{1}{88}$; let $G$ be a planar graph of girth at least five; let $S$ be a connected subgraph of $G$; let $(L,M)$ be a 3-correspondence assignment for $G$. Suppose that $G$ is $S$-critical with respect to $(L,M)$, and that* - *$G$ is not composed of exactly $S$ and one edge not in $S$, and* - *$G$ is not composed of exactly $S$ together with one vertex of degree $3$.* *Then $d_{5, \varepsilon}(G|S) \geq 3$.* Below is a lemma analogous to Lemma [Lemma 15](#Hcritical){reference-type="ref" reference="Hcritical"}. **Lemma 20**. *Let $\varepsilon = \frac{1}{88}$, let $G$ be a plane graph with girth at least five, and let $H$ be a connected subgraph of $G$. If there exists a 3-correspondence assignment $(L,M)$ for $G$ such that $G$ is $H$-critical with respect to $(L,M)$, then $d_{5, \varepsilon}(G|H) \geq 3$.* *Proof.* For $v(G|H) \geq 2$ or $v(G|H) = 1$ and $e(G|H) \neq 3$, this is directly implied by Observation [Observation 19](#girth5:stronglinear){reference-type="ref" reference="girth5:stronglinear"}. Suppose now that $v(G|H) = 0$. Since $G$ is $H$-critical, it follows that $H$ is a proper subgraph of $G$ and so that $e(G|H) \geq 1$. Thus $d_{3,\varepsilon} \geq 3-0 = 3$. If $v(G|H) = 1$ and $e(G|H) = 3$, then $d_{5, \varepsilon}(G|H) = 3\cdot 3-5\cdot 1- \varepsilon \cdot 1 = 4-\varepsilon$. This is at least 3, since $\varepsilon = \frac{1}{88}$. ◻ Using this, we now establish a girth at least five version of Theorem [\[d_gepsilonbound\]](#d_gepsilonbound){reference-type="ref" reference="d_gepsilonbound"}. The proof is nearly identical to that of Theorem [\[d_gepsilonbound\]](#d_gepsilonbound){reference-type="ref" reference="d_gepsilonbound"}, and thus we omit it. **Lemma 21**. *Let $\varepsilon$ be as in Lemma [Lemma 20](#Hcritical5){reference-type="ref" reference="Hcritical5"}. If $G$ is a plane graph with girth at least five and $H$ is a connected subgraph of $G$ such that there does not exist $X \subseteq V(G) \setminus V(H)$ such that $G[X]$ is 3-correspondence-deletable in $G$, then $d_{5, \varepsilon}(G|H) \geq 3$.* Similar to the case of Observation [Observation 13](#deficiencyobs){reference-type="ref" reference="deficiencyobs"}, this trivially implies the following. **Observation 22**. *Let $G$ be a graph of girth at least five, and $H$ a subgraph of $G$. If $d_{5,\varepsilon}(G|H) \geq 0$, then $v(G|H) \leq \varepsilon^{-1}\cdot (\textnormal{def}_5(G|H)-3)$.* Finally, we will need the following theorem, due to Thomassen. As in the girth three case, this theorem was originally written in the language of list colouring; however, as pointed out by Dvořák and the first author [@dvovrak2018correspondence], the proof also carries over to the realm of correspondence colouring. **Theorem 23** (Thomassen, [@thomassen3LCnew]). *Let $G$ be a plane graph of girth at least five. Let $C$ be the subgraph of $G$ whose edge- and vertex-set are precisely those of the outer face boundary walk of $G$. Let $(L,M)$ be a correspondence assignment for $G$ where $|L(v)| \geq 1$ for each vertex $v$ in a path or cycle $S \subseteq C$ with $v(S)\leq 6$; where $|L(v)| = 2$ for each vertex $v$ in an independent set $A$ of vertices in $V(C) \setminus V(S)$; where $|L(v)|\geq 3$ for all $v \in V(G) \setminus (A \cup V(S))$; and where there is no edge between vertices in $A$ and vertices in $S$. Then every $(L,M)$-colouring of $S$ extends to an $(L,M)$-colouring of $G$.* ## Proof of Theorem [\[expmanyccgirthfive\]](#expmanyccgirthfive){reference-type="ref" reference="expmanyccgirthfive"} {#subsec:proofsgirth5} We now prove the following theorem, which is the girth at least five analogue to Theorem [Theorem 18](#expmanyextensions){reference-type="ref" reference="expmanyextensions"}. The reader may find it helpful to consult Figure [\[fig:expmany5\]](#fig:expmany5){reference-type="ref" reference="fig:expmany5"} while reading for a depiction of the cases considered in the proof. **Theorem 24**. *Let $G$ be a plane graph of girth at least five, let $S$ be a connected subgraph of $G$, and let $(L,M)$ be a 3-correspondence assignment for $G$. If $\phi$ is an $(L,M)$-colouring of $S$ that extends to an $(L,M)$-colouring of $G$, then $$\log_2 E(\phi) \ge \frac{v(G|S) - 89\cdot\textnormal{def}_5(G|S)}{282},$$ where $E(\phi)$ denotes the number of extensions of $\phi$ to $G$.* *Proof.* We proceed by induction on $v(G|S)$. First suppose that $v(G|S) = 0$. Then since $\textnormal{def}_5(G|S)$ is positive, the right-hand side of the equation is negative. Since $\phi$ extends to an $(L,M)$-colouring of $G$ by assumption, we have that $\log_2 E(\phi) \geq 0$ and the result follows. Next suppose that $v(G|S) = 1$, and let $v \in V(G)\setminus V(S)$. Then $$\frac{v(G|S)-89\cdot \textnormal{def}_5(G|S)}{282} = \frac{1-89(3\cdot \deg(v)-5)}{282}.$$ Note that when $\deg(v) \geq 2$, the right-hand side is negative. Since $\phi$ extends to an $(L,M)$-colouring of $G$ by assumption, it follows that $\log_2 E(\phi) \geq 0$, and so $$\log_2 E(\phi) \ge \frac{v(G|S) - 89\cdot \textnormal{def}_5(G|S)}{282},$$ as desired. Thus we may assume that $\deg(v) \leq 1$. Since $|L(v)| \geq 3$, we have that $E(\phi) \geq 3-\deg(v)$. Therefore it suffices to show that $\log_2 (3-\deg(v)) \geq \frac{1-89(3\cdot\deg(v)-5)}{282}$; or, equivalently, that $$282 \geq \frac{1-89(3\cdot \deg(v)-5)}{\log_2(3-\deg(v))}.$$ When $\deg(v) = 0$, the right-hand side equals $\frac{446}{\log_2(3)} < 282$. When $\deg(v) = 1$, the right-hand side equals $179$, which again is less than $282$. Thus the inequality above holds. We may therefore assume that $v(G|S) \geq 2$. Before proceeding with the remainder of the case analysis, we will need the following claim. **Claim 2**. *There does not exist a graph $H \subsetneq G$ with $S \subsetneq H$ and $v(S) < v(H) < v(G)$ such that $G-V(H)$ is a 3-correspondence-deletable subgraph of $G$.* *Proof.* Suppose not. Let $G' := G - V(H)$ be a 3-correspondence-deletable subgraph. Note that $G'$ is induced. Since $S \subsetneq H$ and $H \subsetneq G$ and $v(S) < v(H) < v(G)$, it follows that $v(H|S) < v(G|S)$. By induction, there are at least $2^\frac{v(H|S)-89\cdot \textnormal{def}_5(H|S)}{282}$ extensions of $\phi$ to $H$. Since $G'$ is a 3-correspondence-deletable subgraph of $G$, we have by definition that each of these extensions of $\phi$ to an $(L,M)$-colouring of $H$ extends further to an $(L,M)$ colouring of $G'$, and therefore to $G$. Since $H \subsetneq G$ and $S \subsetneq H$ and $v(S) < v(H) < v(G)$, we have that $v(G|H) < v(G|S)$, and so by induction for each extension of $\phi$ to an $(L,M)$ colouring $\phi'$ of $H$ there are at least $2^\frac{v(G|H)-89\cdot \textnormal{def}_5(G|H)}{282}$ extensions of $\phi'$ to $G$. Therefore $$\begin{aligned} \log_2E(\phi) &\geq \frac{v(H|S)-89\cdot \textnormal{def}_5(H|S)}{282} + \frac{v(G|H)-89\cdot \textnormal{def}_5(G|H)}{282} \\ &=\frac{v(H|S)+v(G|H)-89(\textnormal{def}_5(H|S)+\textnormal{def}_5(G|H))}{282} \\ &= \frac{v(G|S)-89\cdot \textnormal{def}_5(G|S)}{282},\end{aligned}$$ as desired. ◻ Among all vertices in $V(G)\setminus V(S)$, choose a vertex $v$ that maximizes $|N(v) \cap V(S)|$. Let $H := S + v$, and let $G' := G- V(H)$. Since $\phi$ extends to an $(L,M)$-colouring of $G$, there is at least $1 = 2^0$ extension $\phi'$ of $\phi$ to $H$ where $\phi'$ extends further to an $(L,M)$-colouring of $G$. Since $\phi'$ extends to $G$ and $v(G|H) < v(G|S)$, by induction we find that there exist at least $2^\frac{v(G|H)-89\cdot \textnormal{def}_5(G|H)}{282}$ extensions of $\phi'$ to $G$. Therefore $$\log_2 E(\phi) \geq 0 + \frac{v(G|H)-89\cdot \textnormal{def}_5(G|H)}{282}.$$ Since $\textnormal{def}_5(G|H) = \textnormal{def}_5(G|S)-3\cdot \deg_H(v)+5$, it follows that $$\begin{aligned} \log_2 E(\phi) &\geq \frac{v(G|S) - 1-89(\textnormal{def}_5(G|S)-3\cdot \deg_H(v)+5)}{282} \\ &= \frac{v(G|S)-1-89\cdot \textnormal{def}_5(G|S)}{282} - \frac{89(5-3\cdot \deg_H(v))}{282}.\end{aligned}$$ If $\deg_H(v) \geq 2$, then $\log_2 E(\phi) \geq \frac{v(G|S)-1-89\cdot \textnormal{def}_5(G|S)}{282}$, as desired. Thus we may assume $\deg_H(v) \leq 1$. Suppose $\deg_H(v) = 0$. We will show $G'$ is 3-correspondence-deletable, contradicting Claim [Claim 2](#claimrdeletable5){reference-type="ref" reference="claimrdeletable5"}. To that end, let $(L',M')$ be a correspondence assignment for $G'$ with $|L'(u)| \geq 3-(\deg_G(u) - \deg_H(u))$ for all $u \in V(G')$. We claim $G'$ is $(L',M')$-colourable. To see this, first note that by our choice of $v$, every vertex in $G'$ has $|L'(u)| \geq 2$. Let $T \subseteq V(G')$ be the set of vertices $\{u \in V(G'): |L'(u)| = 2\}$. Since $\deg_H(v) = 0$, by our choice of $v$ each of the vertices $u \in T$ is adjacent to $v$ in $G$. Since $G$ has girth at least five, it follows that $T$ is an independent set. By Theorem [Theorem 23](#thomtech3cc){reference-type="ref" reference="thomtech3cc"}, $G'$ has an $(L',M')$-colouring $\phi'$. This proves that $G'$ is 3-correspondence-deletable; and since $v(S) < v(H) < v(G)$, this contradicts Claim [Claim 2](#claimrdeletable5){reference-type="ref" reference="claimrdeletable5"}. We may therefore assume that $\deg_H(v) = 1$. First suppose that $v(G|S) = 2$. Let $\{w\} = V(G) \setminus V(H)$. Note that in this case $G'=G- V(H)$ consists only of the vertex $u$. Since $v \in V(G) \setminus V(S)$ was chosen to maximize $|N(v) \cap V(S)|$, it follows that $|N(w) \cap V(S)| \leq 1$. Since $|L(u)| \geq 3$ and $\deg_G(w) \leq 2$, it follows that $G'$ is 3-correspondence-deletable, again contradicting Claim [Claim 2](#claimrdeletable5){reference-type="ref" reference="claimrdeletable5"}. Thus we assume that $v(G|S) \geq 3$. Let $X$ be the set of vertices in $V(G) \setminus V(S)$ with at least one neighbour in $S$. Note that $v \in X$. By our choice of $v$, since $\deg_H(v) = 1$ it follows that every vertex in $X$ has exactly one neighbour in $S$. First suppose that $X$ is an independent set. We will show that $G'$ is a 3-correspondence-deletable subgraph of $G$, again contradicting Claim [Claim 2](#claimrdeletable5){reference-type="ref" reference="claimrdeletable5"}. To that end, let $(L',M')$ be a correspondence assignment for $G'$ with $|L'(u)| \geq 3-(\deg_G(u) - \deg_H(u))$ for all $u \in V(G')$. Note that by our choice of $v$ and the fact that $X$ is an independent set, it follows that every vertex in $G'$ has $|L'(u)| \geq 2$. We claim $G'$ is $(L',M')$-colourable. To see this, let $G'' := G[V(G')\cup \{v\}]$. Note that in $G$, each vertex in $X$ and exactly one vertex in $S$. It follows that the vertices of $X$ are in the outer face boundary walk of $G''$. Let $(L'',M'')$ be a correspondence assignment for $G''$ obtained from $(L',M')$ as follows: define $L''$ by setting $L''(u) = L'(u)$ for all $u \in V(G')\setminus N_{G''}(v)$; setting $L''(v) = \{c\}$, where $c$ is a new colour not present in $\cup_{u \in V(G')}L(u)$; setting $L''(u) = L'(u) \cup \{c\}$ for all $u \in N_{G''}(v)$. Finally, define $M''$ by setting $M_{u_1u_2}'' = M'_{u_1u_2}$ for all $u_1u_2 \in E(G')$; and setting $M_{vu}'' = \{(v,c)(u,c)\}$ for all $u \in N_{G''}(v)$. Note that with the sole exception of $v$, every vertex $u$ in the outer face boundary walk of $G''$ has $|L''(u)| \geq 2$ by definition, and every vertex in $V(G'')\setminus (\{v\} \cup X)$ has $|L''(u)| \geq 3$. Moreover, by assumption, $X$ is an independent set. By Theorem [Theorem 23](#thomtech3cc){reference-type="ref" reference="thomtech3cc"}, $G''$ has an $(L'',M'')$-colouring $\phi'$. Since no vertex $u \in T$ has $\phi'(u) = c$ by construction, it follows that $\phi'$ is an $(L',M')$-colouring of $G'$. This proves that $G' = G-V(H)$ is 3-correspondence-deletable; and since $v(S) < v(H) < v(G)$, this contradicts Claim [Claim 2](#claimrdeletable5){reference-type="ref" reference="claimrdeletable5"}. Thus $X$ is not an independent set, and so there exist vertices $u,w \in X$ such that $uw \in E(G)$. Let $H' := S + u + w$ (see Figure [\[fig:expmany5\]](#fig:expmany5){reference-type="ref" reference="fig:expmany5"}). Since every vertex in $X$ has exactly one neighbour in $S$, it follows that $e(H'|S) = 3$. Thus $\textnormal{def}_5(G|H') = \textnormal{def}_5(G|S)+1$; and by Claim [Claim 2](#claimrdeletable5){reference-type="ref" reference="claimrdeletable5"}, there does not exist $X \subseteq V(G) \setminus V(H')$ such that $G[X]$ is 3-correspondence-deletable in $G$. Thus by Lemma [Lemma 21](#d_gepsilonboundg5){reference-type="ref" reference="d_gepsilonboundg5"}, we have that $d_{5,\varepsilon}(G|H') \geq 3$, and so by Observation [Observation 22](#deficiencyobs5){reference-type="ref" reference="deficiencyobs5"}, $\textnormal{def}_5(G|H') \geq \varepsilon \cdot v(G|H') + 3$. Thus $\textnormal{def}_5(G|S) \geq 2 + \varepsilon \cdot v(G|H')$. It follows that $$\begin{aligned} v(G|S)-89\cdot \textnormal{def}_5(G|S) & \leq v(G|S) -89(2 + \varepsilon \cdot v(G|H)) \\ &= v(G|S)-89(2 + \varepsilon \cdot (v(G|S)-2)) \\ &= v(G|S)-89\varepsilon \cdot v(G|S)-89(2-2\varepsilon).\end{aligned}$$ As $\varepsilon = \frac{1}{88}$, the above is negative. Thus $$0 > \frac{v(G|S)-89\cdot \textnormal{def}_5(G|S)}{282}.$$ Since $\phi$ extends to an $(L,M)$-colouring of $G$, it follows that $\log_2 E(\phi) \geq 0$, and so $\log_2 E(\phi) > \frac{v(G|S)-89\cdot \textnormal{def}_5(G|S)}{282}$, as desired. ◻ As an easy corollary, we obtain Theorem [\[expmanyccgirthfive\]](#expmanyccgirthfive){reference-type="ref" reference="expmanyccgirthfive"}. *Proof of Theorem [\[expmanyccgirthfive\]](#expmanyccgirthfive){reference-type="ref" reference="expmanyccgirthfive"}.* Let $S$ be the empty graph, and $\phi$ a trivial colouring of $S$. Let $E(\phi)$ denote the number of extensions of $\phi$ to $G$. Since $G$ is planar and has girth at least five, $e(G)\leq \frac{5}{3} \cdot v(G)$. By Theorem [Theorem 24](#expmanyextensions5){reference-type="ref" reference="expmanyextensions5"}, $$\begin{aligned} \log_2 E(\phi) & \geq \frac{v(G)-89(3\cdot e(G)-5\cdot v(G))}{282} \\ &\geq \frac{v(G)}{282},\end{aligned}$$ as desired. ◻ # Locally Planar Graphs {#sec:locallyplanar} The main theorems of this section are Theorems [\[thm:expmanylocallyplanar\]](#thm:expmanylocallyplanar){reference-type="ref" reference="thm:expmanylocallyplanar"} and [\[thm:expmanylocallyplanarg5\]](#thm:expmanylocallyplanarg5){reference-type="ref" reference="thm:expmanylocallyplanarg5"}, restated below, which show that for each $k \in \{3,5\}$, locally planar graphs of girth $(8-k)$ have exponentially many $k$-correspondence colourings. We prove these results by first proving technical theorems (Theorem [Theorem 26](#thm:localplanarhypversion){reference-type="ref" reference="thm:localplanarhypversion"} for the 5-correspondence case and [Theorem 27](#thm:localplanarhypversion-g5){reference-type="ref" reference="thm:localplanarhypversion-g5"} for the 3-correspondence case) and combining these technical theorems with a theorem of Postle and Thomas [@postle2018hyperbolic], stated below. Before stating these technical theorems, we require the following definition. **Definition 25**. Let $\varepsilon > 0$, let $G$ be a graph, and let $H$ be an induced subgraph of $G$. We say $H$ is *$\varepsilon$-exponentially-$r$-correspondence-deletable* if for every correspondence assignment $(L,M)$ of $H$ such that $|L(v)| \geq r-(\deg_G(v) - \deg_H(v))$ for each $v \in V(G)$, the graph $H$ has at least $2^{\varepsilon v(H)}$ distinct $(L,M)$-colourings no matter the correspondence assignment $(L,M)$. Here are the technical theorems mentioned above. **Theorem 26**. *Let $\mathcal{F}$ be a family of embedded graphs where for each $(G, \Sigma) \in \mathcal{F}$, the graph $G$ does not contain a subset $X \subseteq V(G)$ such that $G[X]$ is $\frac{1}{3484}$-exponentially-5-correspondence-deletable. Then $\mathcal{F}$ is hyperbolic, and $52$ is a Cheeger constant for $\mathcal{F}$.* **Theorem 27**. *Let $\mathcal{F}$ be a family of embedded graphs of girth at least five where for each $(G, \Sigma) \in \mathcal{F}$, the graph $G$ does not contain a subset $X \subseteq V(G)$ such that $G[X]$ is $\frac{1}{25380}$-exponentially-5-correspondence-deletable. Then $\mathcal{F}$ is hyperbolic, and $270$ is a Cheeger constant for $\mathcal{F}$.* Finally, we will invoke the following theorem due to Postle and Thomas. **Theorem 28** (Postle and Thomas, [@postle2018hyperbolic]). *For every hyperbolic family $\mathcal{F}$ that is closed under curve cutting there exists a constant $k>0$ such that if $(G,\Sigma) \in \mathcal{F}$ and $\Sigma$ has Euler genus $g$, then $G$ contains a non-contractible cycle of length at most $k\log(g+1)$.* Theorem [\[thm:expmanylocallyplanar\]](#thm:expmanylocallyplanar){reference-type="ref" reference="thm:expmanylocallyplanar"} follows easily from Theorem [Theorem 26](#thm:localplanarhypversion){reference-type="ref" reference="thm:localplanarhypversion"}; the proof of Theorem [\[thm:expmanylocallyplanarg5\]](#thm:expmanylocallyplanarg5){reference-type="ref" reference="thm:expmanylocallyplanarg5"} assuming Theorem [Theorem 27](#thm:localplanarhypversion-g5){reference-type="ref" reference="thm:localplanarhypversion-g5"} is identical, and as such we omit it. *Proof of Theorem [\[thm:expmanylocallyplanar\]](#thm:expmanylocallyplanar){reference-type="ref" reference="thm:expmanylocallyplanar"} (assuming Theorem [Theorem 26](#thm:localplanarhypversion){reference-type="ref" reference="thm:localplanarhypversion"})..* Let $\mathcal{F}$ be the family of embedded graphs where for each $(H,\Sigma) \in \mathcal{F}$, the graph $H$ does not contain a subgraph that is $\frac{1}{3484}$-exponentially-5-correspondence-deletable. By Theorem [Theorem 26](#thm:localplanarhypversion){reference-type="ref" reference="thm:localplanarhypversion"}, $\mathcal{F}$ is a hyperbolic family. Fix a surface $\Sigma$ of Euler genus $g$. By Theorem [Theorem 28](#thm:smalledgewidth){reference-type="ref" reference="thm:smalledgewidth"}, there exists a constant $k > 0$ such that every graph in $\mathcal{F}$ embedded in $\Sigma$ has edge-width at most $k \log(g+1)$. Let $(G, \Sigma)$ be an embedded graph with edge-width greater than $k \log (g+1)$, and let $(L,M)$ be a 5-correspondence assignment for $G$. We will prove by induction on $v(G)$ that $G$ has exponentially-many distinct $(L,M)$-colourings. First suppose $v(G) \in \{0,1\}$. Then $G$ trivially has at least $2^{\frac{v(G)}{3484}}$ distinct $(L,M)$-colourings, as desired. Next suppose $v(G) \geq 2$, that $G$ has edge-width greater than $k\log(g+1)$, and that the statement holds for all embedded graphs $(G', \Sigma)$ of edge-width greater than $k \log (g+1)$ with fewer vertices. Since every graph in $\mathcal{F}$ embedded in $\Sigma$ has edge-width at most $k\log(g+1)$, it follows that $G$ contains a subgraph $(H, \Sigma)$ that is $\frac{1}{3484}$-exponentially-5-correspondence-deletable. Note that $G-H$ is a graph embeddable in $\Sigma$ with edge-width greater than $k\log(g+1)$. By induction, $G-H$ has at least $2^\frac{v(G|H)}{3484}$ distinct $(L,M)$-colourings. By definition of $\frac{1}{3484}$-exponentially-5-correspondence-deletable, every $(L,M)$-colouring of $G-H$ extends to at least $2^\frac{v(H)}{3438}$ distinct $(L,M)$-colourings of $H$. Thus $G$ has at least $2^\frac{v(G|H)+v(H)}{3484}$ distinct $(L,M)$-colourings, as desired. ◻ Our final point of order is thus to prove Theorem [Theorem 26](#thm:localplanarhypversion){reference-type="ref" reference="thm:localplanarhypversion"}. We will omit the proof of Theorem [Theorem 27](#thm:localplanarhypversion-g5){reference-type="ref" reference="thm:localplanarhypversion-g5"} as it is nearly identical; key differences will be discussed at the end of this section. Before proceeding, we will require the following easy results which follow from the definition of $r$-correspondence-deletability. **Lemma 29**. *Let $G$ be a graph. If $H$ is an $r$-correspondence-deletable subgraph of $G$, and $G-V(H)$ contains an $r$-correspondence-deletable subgraph $H'$, then $G[V(H) \cup V(H')]$ is an $r$-correspondence-deletable subgraph of $G$.* The following is an immediate corollary of Lemma [Lemma 29](#lemma:maxdel){reference-type="ref" reference="lemma:maxdel"}. **Corollary 30**. *If $H$ is a maximal $r$-correspondence-deletable subgraph of $G$, then $G-V(H)$ does not contain an $r$-correspondence-deletable subgraph.* We now prove Theorem [Theorem 26](#thm:localplanarhypversion){reference-type="ref" reference="thm:localplanarhypversion"}. *Proof of Theorem [Theorem 26](#thm:localplanarhypversion){reference-type="ref" reference="thm:localplanarhypversion"}.* Let $(G, \Sigma) \in \mathcal{F}$, and let $\eta$ be a closed curve in $\Sigma$ that intersects $G$ only in vertices and that bounds an open disk $\Delta$. We may assume that $\eta$ is a simple curve; otherwise, we split vertices intersected by $\eta$ into multiple copies to reduce to this case. Let $D$ be the set of vertices of $G$ intersected by $\eta$. Suppose that $\Delta$ includes a vertex of $G$. We will show the number vertices of $G$ in $\Delta$ is at most $52(|D|-1)$. Let $G_\Delta$ be the subgraph of $G$ embedded in $\Delta$ and its boundary, $\eta$. For each pair of consecutive vertices $u,v$ along $\eta$, add the edge $uv$ if $uv \not \in E(G)$, so that $G_\Delta[D]$ is connected. Let $G_D := G_\Delta[D]$. Let $\varepsilon$ be as in Theorem [\[d_gepsilonbound\]](#d_gepsilonbound){reference-type="ref" reference="d_gepsilonbound"}, and let $\alpha := \frac{\varepsilon}{2\varepsilon + 1}$ (that is, $\varepsilon = \frac{1}{50}$ and hence $\alpha := \frac{1}{52}$). Let $G' \subseteq G_\Delta$ be a maximal connected subgraph with $D \subseteq V(G')$ and $d_{3,\alpha}(G'|G_D) \geq 0$. Note that such a subgraph $G'$ exists, since $d_{3,\alpha}(G_D|G_D) = 0$. Moreover, it follows that $G'$ is an induced subgraph of $G_\Delta$ since adding edges does not decrease $d_{3, \alpha}(G'|G_D)$. Note that by definition of $d_{3,\alpha}$, we have that since $G_D \subseteq G' \subseteq G_\Delta$, $$d_{3,\alpha}(G_\Delta|G_D) = d_{3,\alpha}(G_\Delta|G')+ d_{3,\alpha}(G'|G_D).$$ We aim to show that $d_{3,\alpha}(G_\Delta|G_D) \geq 0$, which we will show later implies that the number of vertices of $G$ in $\Delta$ is at most $52(|D|-1)$. To that end, we prove the following. **Claim 3**. *$d_{3,\alpha}(G_\Delta|G') \geq 0$.* *Proof.* To see this, suppose not. Then since $d_{3,\alpha}(G_\Delta|G') < 0$ and $\alpha < \varepsilon$, it follows that $d_{3,\varepsilon}(G_\Delta|G') < 0$. By the contrapositive of Theorem [\[d_gepsilonbound\]](#d_gepsilonbound){reference-type="ref" reference="d_gepsilonbound"}, there exists a nonempty subset $X \subseteq V(G_\Delta)-V(G')$ such that $G_\Delta[X]$ is 5-correspondence-deletable subgraph of $G_\Delta$. We choose $X \subseteq G_\Delta-V(G')$ to be a maximal such set, and let $H = G[X]$. Note that since $H$ is maximal, by Corollary [Corollary 30](#cor:maxdel){reference-type="ref" reference="cor:maxdel"} there does not exist a 5-correspondence-deletable subgraph of $G_\Delta-V(H)$ disjoint from $G'$. Thus by Theorem [\[d_gepsilonbound\]](#d_gepsilonbound){reference-type="ref" reference="d_gepsilonbound"}, we have that $d_{3, \varepsilon}(G_\Delta-V(H)~|~G') \geq 0$, and so $d_{3, \alpha}(G_\Delta-V(H)~|~G') \geq 0$ since $\alpha < \varepsilon$. Now, since $G_D \subseteq G' \subseteq G_\Delta-V(H)$, by definition of $d_{3,\alpha}$ we have that $d_{3, \alpha}(G_\Delta-V(H)~|~G_D) = d_{3, \alpha}(G_\Delta-V(H)~|~G') + d_{3, \alpha}(G'|G_D)$. As the right hand side is non-negative, so is the left hand side; and so by our choice of $H$, we have that $G_\Delta-V(H) =G'$. Since $(G, \Sigma) \in \mathcal{F}$, we have that (though $H$ is 5-correspondence-deletable) $H$ is not $\frac{1}{3484}$-exponentially-5-correspondence-deletable. Thus there exists a correspondence assignment $(L,M)$ of $H$ where $|L(u)| \geq 5-(\deg_{G_\Delta}(u) - \deg_H(u))$ for each $u \in V(H)$ and such that $H$ has an $(L,M)$-colouring, but does not have at least $2^{\frac{v(H)}{3484}}$ distinct $(L,M)$-colourings. Let $(L',M')$ be the 5-correspondence assignment for $G_{\Delta}$ defined as follows. For each vertex $v \in V(G')$, let $\varphi(v)$ be a new unique colour for $v$. For $v\in V(G')$, we let $L'(v) := \{\phi(v)\}$. For $u\in V(H)$, we let $L'(u) := L(u) \cup \{\varphi(v): v \in V(G') \cap N_{G_\Delta}(u)\}$. For $e=uv\in E(G_{\Delta})$, we let $M'_e = M_e$ if $e\in E(H)$, and $M'_e=\emptyset$ if $e\in E(G')$; finally for $e\in E(G_{\Delta})\setminus (E(H)\cup E(G'))$ where we have that $|\{u,v\}\cap V(H)|=1$, weassuming without loss of generality that $u\in V(H)$ and $v\in V(G')$let $M'_{uv} = \{(u, \varphi(v))(v, \varphi(v))\}$. Recall that $G'$ is connected by definition; and thus by Theorem [Theorem 18](#expmanyextensions){reference-type="ref" reference="expmanyextensions"} applied to $G_\Delta$, $G'$, and $(L',M')$, $$\begin{aligned} \log_2 E(\varphi) &\geq \frac{v(G_\Delta|G')-(\varepsilon^{-1}+1)\textnormal{def}_3(G_\Delta|G ')}{67}\\ &=\frac{v(H)-(\varepsilon^{-1}+1)\textnormal{def}_3(G_\Delta|G')}{67}\end{aligned}$$ Since $d_{3,\alpha}(G_{\Delta}|G') = \textnormal{def}_3(G_{\Delta}|G') - \alpha \cdot v(G_\Delta|G') < 0$, it follows that $\textnormal{def}_3(G_\Delta | G') < \alpha \cdot v(G_\Delta|G')$; equivalently, that $\textnormal{def}_3(G_\Delta|G') < \alpha \cdot v(H)$. Thus, continuing from above, $$\begin{aligned} \log_2 E(\varphi) &> \frac{v(H)-(\varepsilon^{-1}+1)\alpha\cdot v(H)}{67} \\ &= \frac{(1-\alpha(\varepsilon^{-1}+1))v(H)}{67} \\ &= \frac{v(H)}{52 \cdot 67} \textrm{ \hskip 8mm since $\alpha = \frac{1}{52}$ and $\varepsilon = \frac{1}{50}$.}\end{aligned}$$ As every extension of $\varphi$ to $H$ corresponds to a distinct $(L,M)$-colouring of $H$, we have that $H$ has at least $2^{\frac{v(H)}{3484}}$ distinct $(L,M)$-colourings, a contradiction. ◻ Thus $d_{3,\alpha}(G_\Delta|G') \geq 0$, and so $d_{3,\alpha}(G_\Delta|G_D) \geq 0$. We now show this implies the number of vertices in $G_\Delta-D$ is at most $52(|D|-1)$. To see this, let $F(G_\Delta)$ be the set of faces in the embedding of $G_\Delta$, and let $f(G_\Delta) = |F(G_\Delta)|$. Recall that $G_D$ is connected. We claim moreover that $G_\Delta$ is connected; otherwise, by Theorem [\[expmanycc\]](#expmanycc){reference-type="ref" reference="expmanycc"} a component $C$ of $G$ has at least $2^\frac{v(C)}{67}$ distinct $(L,M)$-colourings for every 5-correspondence assignment $(L,M)$. But $C$ is therefore $\frac{1}{3484}$-exponentially-5-correspondence-deletable, contradicting that $G \in \mathcal{F}$. Moreover, we we may assume $G_\Delta$ has at least three vertices: this too follows easily from the fact that $G$ does not contain a $\frac{1}{3484}$-exponentially-5-correspondence-deletable subgraph. Since $G_\Delta$ is connected and $v(G_\Delta) \geq 3$, the boundary walk of every face has length at least three. Let $f_O$ be the outer face of $G_\Delta$. By Euler's formula for graphs embedded in surfaces, $$\begin{aligned} e(G_\Delta)-3v(G_\Delta) &= e(G_\Delta)-3(e(G_\Delta)+2-f(G_\Delta)) \\ &= 3f(G_\Delta)-2e(G_\Delta)-6 \\ &= \sum_{f \in F(G_\Delta)} (3-|f|)-6 \\ &\leq (3-|D|)-6 \textrm{\hskip 4mm since $|f_O| \geq |D|$, and $3-|f| \leq 0$ for all other $f \in F(G_\Delta)$.} \\ &= -|D|-3.\end{aligned}$$ Now, since $\textnormal{def}_3(G_\Delta|G_D) = e(G_\Delta|G_D)-3v(G_\Delta|G_D)$, it follows from above that $\textnormal{def}_3(G_\Delta|G_D) \leq -|D|-3 -e(G_D)+3|D| = 2|D|-e(G_D) -3$. By construction, $e(G_D) \geq |D|-1$, and so $\textnormal{def}_3(G_\Delta|G_D) \leq |D|-2$. Since $d_{3,\alpha}(G_\Delta|G_D) = \textnormal{def}_3(G_\Delta|G_D)-\alpha \cdot v(G_\Delta|G_D)$ and $d_{3, \alpha}(G_\Delta|G_D) \geq 0$, it follows that $(|D|-2)-\alpha \cdot v(G_\Delta|G_D) \geq 0$. Rearranging, we have that $v(G_\Delta|G_D) \leq \alpha^{-1}(|D|-2)$. Thus $\mathcal{F}$ is a hyperbolic family, and $\frac{1}{\alpha} = 52$ is a Cheeger constant for $\mathcal{F}$, as desired. ◻ We conclude this section with a quick word on the differences between the proofs of Theorems [Theorem 26](#thm:localplanarhypversion){reference-type="ref" reference="thm:localplanarhypversion"} and [Theorem 27](#thm:localplanarhypversion-g5){reference-type="ref" reference="thm:localplanarhypversion-g5"}: first, instead of adding edges $uv$ between consecutive vertices along $\eta$ to ensure $G_\Delta[D]$ is connected, we add new vertices $x_1,x_2$ to $D$ and paths $ux_1x_2v$. Call the set $D$ together with all new vertices $D'$. After performing this operation, $G_\Delta[D']$ is connected, and $|D'| \leq 3|D|$. Later, we invoke Theorems [Lemma 21](#d_gepsilonboundg5){reference-type="ref" reference="d_gepsilonboundg5"} and [Theorem 24](#expmanyextensions5){reference-type="ref" reference="expmanyextensions5"} instead of the analogous Theorems [\[d_gepsilonbound\]](#d_gepsilonbound){reference-type="ref" reference="d_gepsilonbound"} and [Theorem 18](#expmanyextensions){reference-type="ref" reference="expmanyextensions"} used in the girth three case. Finally, in the closing arguments where we count the number of vertices in $\Delta$ and invoke Euler's formula, we use the fact that the boundary walk of each face is at least five. This follows from the fact that either $D'$ has at least four vertices and $G$ has girth at least five, or $|D'| = |D| \leq 2$ and so since $G$ has girth at least five and does not contain $\frac{1}{25380}$-exponentially-3-correspondence-deletable subgraph, there are at least three vertices in $G_\Delta - D'$. # Further Directions {#sec:conclusion} It is natural to wonder whether deficiency hyperbolicity theorems exist for other planar graph classes. (As discussed in Subsection [1.2](#subsec:hyperbolicity){reference-type="ref" reference="subsec:hyperbolicity"}, such theorems would have other interesting implications beyond the scope of counting colourings.) For instance, in [@esrlukelocal], we introduce the concept of a *local girth list assignment*. This is defined below, following another necessary definition. The *girth of an edge* and *girth of a vertex* are defined as follows. **Definition 31**. Let $G$ be a graph, and $e\in E(G)$. The *girth of $e$* is denoted by $g(e)$ and is defined as the length of the shortest cycle in $G$ in which $e$ is contained. Similarly, for $v \in V(G)$, the *girth of $v$* is denoted by $g(e)$ and is defined as the length of the shortest cycle in $G$ in which $v$ is contained. If a vertex or edge is not contained in a cycle, its girth is defined as being infinite. **Definition 32**. Let $G$ be a graph with list assignment $L$. We say $L$ is a *local girth list assignment* if every vertex $v \in V(G)$ has $|L(v)| \geq 3$; if every vertex $v \in V(G)$ with $g(v) =4$ has $|L(v)| \geq 4$; and if every vertex $v \in V(G)$ with $g(v)=3$ has $|L(v)| \geq 5$. We say $G$ is *local girth choosable* if $G$ has an $L$-colouring for every local girth list assignment $L$. In [@esrlukelocal] we prove the following theorem. **Theorem 33**. *Every planar graph is local girth choosable.* In light of the existing literature concerning exponentially many list colourings of planar graphs, it is natural to wonder, for each planar graph $G$ with local girth list assignment $L$, whether $G$ has exponentially many distinct $L$-colourings. Using Theorem [Theorem 7](#thm:alonfuredi){reference-type="ref" reference="thm:alonfuredi"}, it is not too difficult to show this. Indeed, we prove this below, following a useful proposition. **Proposition 34**. *If $G$ is a planar graph that contains a cycle, then $$v(G) - \sum_{e \in E(G)} \left(1-\frac{2}{g(e)} \right) \geq 2,$$ where we interpret $\frac{2}{g(e)}$ to be $0$ if $g(e)=\infty$.* *Proof.* We assume that $G$ is a plane graph (that is, we fix a planar embedding of $G$). Note that since $G$ contains at least one cycle, it follows that every face of $G$ contains at least one cycle in its boundary walk. By Euler's formula for graphs embedded in the plane, we have that $v(G) - e(G) + f(G) \geq 2$, where $f(G)$ denotes the number of faces of $G$. Now we assign a charge of +1 to each face and vertex of $G$, and -1 to each edge of $G$. Hence from above, the total sum of all charges is at least $+2$. We discharge according to the following rule: for each face $f$, choose a cycle $C$ in its boundary walk, and send $\frac{1}{e(C)}$ charge to each edge in $C$. Note then that if an edge $e$ is contained in a cycle, it receives at most $+\frac{2}{g(e)}$ charge. If $e$ is not contained in a cycle, then since $g(e)$ is defined as infinite, again $e$ receives at most $\frac{2}{g(e)}$ charge (where $\frac{2}{g(e)} = 0$). Thus after discharging, the sum of the charges is at most $v(G) -e(G) + \sum_{e \in E(G)} \frac{2}{g(e)}$, and so $$\begin{aligned} v(G) - \sum_{e \in E(G)} \left( 1 - \frac{2}{g(e)} \right) \geq 2, \end{aligned}$$ as desired. ◻ **Theorem 35**. *If $G$ is a planar graph and $L$ is a local girth list assignment for $G$, then $G$ has at least $5^\frac{v(G)}{12}$ distinct $L$-colourings.* *Proof.* We may assume without loss of generality that $|L(v)| = 5$ for all $v \in V(G)$ with $g(v) =3$; that $|L(v)| = 4$ for all $v \in V(G)$ with $g(v) = 4$; and that $|L(v)| = 3$ for all $v \in V(G)$ with $g(v) \geq 5$. We follow the polynomial method described in Subsection [1.3](#subsec:polmethod){reference-type="ref" reference="subsec:polmethod"}. First assume $G$ is a forest, in which case $e(G) \leq v(G)-1$. In this case, $|L(v)| \geq 3$ for all $v \in V(G)$, and so since $G$ has an $L$-colouring by Theorem [Theorem 33](#localgirth){reference-type="ref" reference="localgirth"}, it follows by the Alon-Füredi theorem (Theorem [Theorem 7](#thm:alonfuredi){reference-type="ref" reference="thm:alonfuredi"}) that $G$ has at least $3^{\frac{3\cdot v(G) - v(G) -(v(G)-1)}{2}} = 3^\frac{v(G)+1}{2}$, as desired. Thus we may assume $G$ is not a forest, and therefore contains a cycle. Since there exists an $L$-colouring of $G$ by Theorem [Theorem 33](#localgirth){reference-type="ref" reference="localgirth"}, a direct application of the Alon-Füredi theorem (Theorem [Theorem 7](#thm:alonfuredi){reference-type="ref" reference="thm:alonfuredi"}) gives that there are at least $5^\frac{S - v(G) -e(G)}{4}$ distinct $L$-colourings, where $S = \sum_{v \in V(G)} |L(v)|$. For each $i \in \{3,4\}$, let $e_i := |\{e \in E(G): g(e) = i\}|$. Let $e_5 := |\{e \in E(G): g(e) \geq 5\}|$. Analogously, for each $i \in \{3,4\}$, let $v_i := |\{v \in V(G): g(v) = i\}|$, and let $v_5 := |\{v \in V(G): g(v) \geq 5\}|$. Since $L$ is a local girth list assignment, it follows that $S \geq 5v_3 + 4v_4 + 3v_5$, and so from above, we find that there exist at least $5^\frac{4v_3+3v_4+2v_5-e_3-e_4-e_5}{4}$ distinct $L$-colourings of $G$. It remains to bound the numerator in the exponent. To that end, we define two subgraphs of $G$ as follows: let $G_3$ be the graph with $V(G_3):= \{v \in V(G): g(v) = 3\}$ and $E(G_3) := \{e \in E(G): g(e) = 3\}$. Let $G_4$ be the graph with $V(G_4):= \{v \in V(G): g(v) \in \{3,4\}\}$ and $E(G_4) = \{e \in E(G): g(e) \in \{3,4\}\}$. Note that $v(G_3) = v_3$. Hence by Proposition [Proposition 34](#prop:eulergirth){reference-type="ref" reference="prop:eulergirth"} applied to $G_3$, we find that $$\label{eq:1} v_3 \geq \frac{e_3}{3}.$$ Similarly, note that $v(G_4) = v_3 + v_4$. By Proposition [Proposition 34](#prop:eulergirth){reference-type="ref" reference="prop:eulergirth"} applied to $G_4$, we find that $$\label{eq:2} v_3 + v_4 \geq \frac{e_3}{3} + \frac{e_4}{2}.$$ Finally, by Proposition [Proposition 34](#prop:eulergirth){reference-type="ref" reference="prop:eulergirth"} applied to $G$, we find that $$\label{eq:3} v_3 + v_4 + v_5 \geq \frac{e_3}{3} + \frac{e_4}{2} + \frac{3e_5}{5}.$$ Adding Equation [\[eq:1\]](#eq:1){reference-type="ref" reference="eq:1"} with $\frac{1}{3}$ of Equation [\[eq:2\]](#eq:2){reference-type="ref" reference="eq:2"} and $\frac{5}{3}$ of Equation [\[eq:3\]](#eq:3){reference-type="ref" reference="eq:3"}, we obtain $$\begin{aligned} 3v_3 +2v_4 + \frac{5}{3} v_5 \geq e_3 + e_4 + e_5.\end{aligned}$$ Plugging this into our Alon-Füredi bound, we obtain the desired result. ◻ It is also natural to wonder whether an analogous result holds in the correspondence colouring framework. In this framework, it is not clear how one would define the polynomial required to invoke Theorem [Theorem 7](#thm:alonfuredi){reference-type="ref" reference="thm:alonfuredi"}. A more pressing issue is that it is not currently known whether planar graphs are local girth correspondence colourable at all. However, we conjecture this is true. **Conjecture 36**. *If $G$ is a planar graph and $(L,M)$ is a correspondence assignment for $G$ where $L$ is a local girth list assignment, then $G$ is $(L,M)$-colourable.* Our method (using deficiency versions of hyperbolicity theorems) could also be used to show planar graphs have exponentially many local girth correspondence colourings, but proving such a hyperbolicity theorem seems difficult. Indeed, it is not yet known whether there exists a hyperbolicity theorem for local girth list colouring. We conjecture this (and its correspondence colouring analogue) is true. **Conjecture 37**. *There exists a theorem analogous to Theorem [\[theorem:mainhypthm\]](#theorem:mainhypthm){reference-type="ref" reference="theorem:mainhypthm"} for local girth list colouring.* **Conjecture 38**. *There exists a theorem analogous to Theorem [\[theorem:mainhypthm\]](#theorem:mainhypthm){reference-type="ref" reference="theorem:mainhypthm"} for local girth correspondence colouring.* One might also wonder whether there exists a *strong* hyperbolicity theorem for local girth list colouring; however, given the intricacy of the proof of even Theorem [Theorem 33](#localgirth){reference-type="ref" reference="localgirth"}, proving such a theorem seems a daunting task. Returning our attentions to other surfaces: in [@thomassen2006number], Thomassen proved the following for arbitrary embedded graphs. **Theorem 39** (Thomassen, [@thomassen2006number]). *For every surface $\Sigma$, there exists a constant $c > 0$ such that if $(G, \Sigma)$ is an embedded graph and $G$ is 5-colourable, then $G$ has at least $c \cdot 2^n$ distinct 5-colourings.* In [@postle20213] and [@postle2018hyperbolic], the first author and Thomas generalized Thomassen's results to list colouring as follows (in fact, they prove a stronger notion again involving counting the number of extensions of a precoloured subgraph). **Theorem 40** (Postle & Thomas, [@postle2018hyperbolic]). *There exist constants $\epsilon, \alpha > 0$ such that the following holds. Let $G$ be a graph with $n$ vertices embedded in a fixed surface $\Sigma$ of genus $g$, and let $H$ be a proper subgraph of $G$. Suppose that either $L$ is a 5-list-assignment for $G$, or that $G$ has girth at least five and that $L$ is a 3-list-assignment for $G$. If $\phi$ is an $L$-colouring of $H$ that extends to an $L$-colouring of $G$, then $\phi$ extends to at least $2^{\varepsilon(n-\alpha(g+v(H)))}$ distinct $L$-colourings of $G$.* In [@kelly2018exponentially], Kelly and Postle proved an analogous theorem for 4-list colouring embedded graphs of girth at least four. We note that these theorems would also follow directly from the polynomial method described in Subsection [1.3](#subsec:polmethod){reference-type="ref" reference="subsec:polmethod"}. Using the approach of Dahlberg, Kaul, and Mudrock [@dahlberg2023algebraic], analogous results for 3-correspondence colouring embedded graphs of girth at least five also hold. Again, the polynomial method does not seem amenable to proving there are exponentially many 5-correspondence colourings of embedded graphs, though this should follow if one could prove *strong* hyperbolicity theorems for this class of graphs. The first author and Thomas proved such a theorem for 5-list colouring; but as the full proof is over 200 pages, generalizing such a result to the correspondence colouring framework seems a formidable task. #### Acknowledgement. Several results in this paper form part of the doctoral dissertation [@evethesis] of the second author, written under the guidance of the first. [^1]: We acknowledge the support of the Natural Sciences and Engineering Research Council of Canada (NSERC) \[Discovery Grant No. 2019-04304\].\ $^*$Cette recherche a été financée par le Conseil de recherches en sciences naturelles et en génie du Canada (CRSNG)\[Discovery Grant No. 2019-04304\].

arxiv_math

--- abstract: | In this paper, we study a natural discretization of the smooth Gaussian curvature on surfaces, which is defined as the quotient of the angle defect and the area of a geodesic disk at a vertex of a polyhedral surface. It is proved that each decorated piecewise Euclidean metric on surfaces with nonpositive Euler number is discrete conformal to a decorated piecewise Euclidean metric with this discrete curvature constant. We further investigate the prescribing combinatorial curvature problem for a parametrization of this discrete curvature and prove some Kazdan-Warner type results. The main tools are Bobenko-Lutz's discrete conformal theory for decorated piecewise Euclidean metrics on surfaces and variational principles with constraints. address: - School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, P.R.China - School of Mathematics and Statistics, Wuhan University, Wuhan 430072, P.R. China author: - Xu Xu, Chao Zheng title: "**A discrete uniformization theorem for decorated piecewise Euclidean metrics on surfaces, II**" --- [^1] # Introduction The classical Gaussian curvature at a point $p$ in a Riemann surface can be defined as $$R(p)=\lim_{r\rightarrow 0}\frac{12}{\pi r^4}(\pi r^2-A(r)),$$ where $A(r)$ is the area of the geodesic disk of radius $r$ at $p$. If we apply this definition to a vertex $i$ of a piecewise Euclidean surface, this gives a natural discretization of the classical Gaussian curvature (up to a constant) $$\label{Eq: curvature R} R_i=\frac{K_i}{r_i^2},$$ where $K_i=2\pi-\theta_i$ is the angle defect at $i$, $\theta_i$ is the cone angle at $i$ and $r_i$ is the radius of the geodesic disk at $i$. We call $R_i$ as the discrete Gaussian curvature or combinatorial curvature. It is natural to study the discrete uniformization theorem for the discrete Gaussian curvature $R_i$. A good approach to this problem is working in the framework of decorated piecewise Euclidean metrics recently introduced by Bobenko-Lutz [@BL]. Suppose $S$ is a connected closed surface and $V$ is a finite non-empty subset of $S$, the pair $(S, V)$ is called a marked surface. A piecewise Euclidean metric (PE metric) $dist_{S}$ on the marked surface $(S,V)$ is a flat cone metric with the conic singularities contained in $V$. A marked surface with a PE metric is called a PE surface, denoted by $(S,V, dist_{S})$. The points in $V$ are called vertices of the PE surface. A decoration $r$ on a PE surface $(S,V, dist_{S})$ is a choice of circle of radius $r_i$ at each vertex $i\in V$. These circles in the decoration are called vertex-circles. We denote a decorated PE surface by $(S,V, dist_{S}, r)$ and call the pair $(dist_S,r)$ a decorated PE metric on the marked surface $(S,V)$. In this paper, we focus on the case that each pair of vertex-circles is separated. **Theorem 1**. Let $(dist_S,r)$ be a decorated PE metric on a marked surface $(S,V)$ with Euler number $\chi(S)\leq0$. Let $\overline{R}\leq 0$ be a function defined on $V$ satisfying $\overline{R}\not\equiv0$ if $\chi(S)<0$ and $\overline{R}\equiv0$ if $\chi(S)=0$. Then there exists a unique discrete conformal equivalent decorated PE metric $(\widetilde{dist_S},\widetilde{r})$ on $(S,V)$ with the prescribed discrete Gaussian curvature $\overline{R}$ (up to scaling if $\chi(S)=0$). Theorem [Theorem 1](#Thm: existence 1){reference-type="ref" reference="Thm: existence 1"} is a discrete analogue of Kazdan-Warner's results in [@KW1; @KW2]. As a corollary of Theorem [Theorem 1](#Thm: existence 1){reference-type="ref" reference="Thm: existence 1"}, we have the following discrete uniformization theorem for the discrete Gaussian curvature $R_i$ on decorated PE surfaces. **Corollary 2**. For any decorated PE metric $(dist_S,r)$ on a marked surface $(S,V)$ with Euler number $\chi(S)\leq0$, there exists a unique discrete conformal equivalent decorated PE metric $(\widetilde{dist_S},\widetilde{r})$ on $(S,V)$ with constant discrete Gaussian curvature $R$ (up to scaling if $\chi(S)=0$). The combinatorial curvature $R$ in ([\[Eq: curvature R\]](#Eq: curvature R){reference-type="ref" reference="Eq: curvature R"}) was first introduced by Ge-Xu [@GX2] for Thurston's circle packing metrics on triangulated surfaces. After that, there are lots of research activities on the combinatorial curvature $R$ on surfaces. See [@GJ; @GX2; @GX; @IMRN; @GX; @JFA; @Xu1; @XZ; @TAMS; @XZ; @CVPDE] and others for example. Most of these works gave equivalent conditions for the existence of discrete conformal metrics with prescribed combinatorial curvature $R$ via combinatorial curvature flows. In Theorem [Theorem 1](#Thm: existence 1){reference-type="ref" reference="Thm: existence 1"} and Corollary [Corollary 2](#Cor: existence 1){reference-type="ref" reference="Cor: existence 1"}, we give some sufficient conditions for the existence involving only the prescribed combinatorial curvatures and the topology of the surfaces. Following [@GX2], we further introduce the following parameterized combinatorial curvature for the decorated PE metrics on surfaces $$\label{Eq: curvature R_alpha} R_{\alpha, i}=\frac{K_i}{r_i^\alpha},$$ where $\alpha\in \mathbb{R}$ is a constant. If $\alpha=2$, then $R_{2, i}$ is the combinatorial curvature $R_i$ defined by ([\[Eq: curvature R\]](#Eq: curvature R){reference-type="ref" reference="Eq: curvature R"}). We call $R_{\alpha}$ as the combinatorial $\alpha$-curvature. **Theorem 3**. Let $(dist_S,r)$ be a decorated PE metric on a marked surface $(S,V)$. Suppose $\alpha\in \mathbb{R}$ is a constant and $\overline{R}: V\rightarrow\mathbb{R}$ is a given function. There exists a discrete conformal equivalent decorated PE metric $(\widetilde{dist_S},\widetilde{r})$ with combinatorial $\alpha$-curvature $\overline{R}$ if one of the following conditions is satisfied \(1\) : $\chi(S)>0,\ \alpha<0,\ \overline{R}>0$; \(2\) : $\chi(S)<0,\ \alpha\neq0,\ \overline{R}\leq 0,\ \overline{R}\not\equiv 0$; \(3\) : $\chi(S)=0,\ \alpha\neq0,\ \overline{R}\equiv0$; \(4\) : $\alpha=0$, $\overline{R}\in (-\infty, 2\pi)$, $\sum_{i\in V}\overline{R}_{i}=2\pi \chi(S)$. If $\alpha\overline{R}\leq0$, the decorated PE metric $(\widetilde{dist_S},\widetilde{r})$ is unique (up to scaling if $\alpha\overline{R}\equiv 0$). Theorem [Theorem 3](#Thm: existence 2){reference-type="ref" reference="Thm: existence 2"} is a generalization of Theorem [Theorem 1](#Thm: existence 1){reference-type="ref" reference="Thm: existence 1"}. Specially, if $\alpha=2$, then the cases **(2)** and **(3)** in Theorem [Theorem 3](#Thm: existence 2){reference-type="ref" reference="Thm: existence 2"} are reduced to Theorem [Theorem 1](#Thm: existence 1){reference-type="ref" reference="Thm: existence 1"}. By the relationship of the combinatorial $\alpha$-curvature $R_\alpha$ and the angle defect $K$, the cases **(3)** and **(4)** in Theorem [Theorem 3](#Thm: existence 2){reference-type="ref" reference="Thm: existence 2"} are covered by Bobenko-Lutz's work [@BL]. In the following, we just prove the cases **(1)** and **(2)** of Theorem [Theorem 3](#Thm: existence 2){reference-type="ref" reference="Thm: existence 2"}. **Remark 4**. Since Bobenko-Lutz's discrete conformal theory of decorated PE metrics also applies to Luo's vertex scalings and thus generalizes Gu-Luo-Sun-Wu's results in [@Gu1] and Springborn's results in [@Springborn], Theorem [Theorem 3](#Thm: existence 2){reference-type="ref" reference="Thm: existence 2"} also generalizes the authors' results in [@XZ; @CVPDE]. As a corollary of Theorem [Theorem 3](#Thm: existence 2){reference-type="ref" reference="Thm: existence 2"}, we have the following discrete uniformization theorem for the combinatorial $\alpha$-curvature $R_\alpha$. **Corollary 5**. Suppose $(S,V)$ is a marked surface with a decorated PE metric $(dist_S,r)$ and $\alpha\in \mathbb{R}$ is a constant. \(1\) : If $\alpha\chi(S)\leq0$, there exists a unique discrete conformal equivalent decorated PE metric $(\widetilde{dist_S},\widetilde{r})$ with constant combinatorial $\alpha$-curvature $R_\alpha$ (up to scaling if $\alpha\chi(S)=0$). \(2\) : If $\alpha<0$ and $\chi(S)<0$, there exists a discrete conformal equivalent decorated PE metric $(\widetilde{dist_S},\widetilde{r})$ with negative constant combinatorial $\alpha$-curvature $R_\alpha$. The paper is organized as follows. In Section [2](#Sec: DC theory){reference-type="ref" reference="Sec: DC theory"}, we briefly recall Bobenko-Lutz's discrete conformal theory for the decorated PE metrics on surfaces. Then we prove the global rigidity of decorated PE metrics with respect to the combinatorial $\alpha$-curvature on a marked surface. In Section [3](#Sec: existence){reference-type="ref" reference="Sec: existence"}, we first deform the combinatorial $\alpha$-curvature $R_\alpha$ in ([\[Eq: curvature R_alpha\]](#Eq: curvature R_alpha){reference-type="ref" reference="Eq: curvature R_alpha"}) and give Theorem [Theorem 14](#Thm: existence 3){reference-type="ref" reference="Thm: existence 3"}, which is equivalent to Theorem [Theorem 3](#Thm: existence 2){reference-type="ref" reference="Thm: existence 2"}. Then we translate Theorem [Theorem 14](#Thm: existence 3){reference-type="ref" reference="Thm: existence 3"} into an optimization problem with constraints, i.e., Theorem [Theorem 17](#Thm: inequality constraints){reference-type="ref" reference="Thm: inequality constraints"}. Using a classical result from calculus, i.e., Theorem [Theorem 18](#Thm: calculus){reference-type="ref" reference="Thm: calculus"}, we translate Theorem [Theorem 17](#Thm: inequality constraints){reference-type="ref" reference="Thm: inequality constraints"} into Theorem [Theorem 19](#Thm: main 2){reference-type="ref" reference="Thm: main 2"}. In the end, with the help of the asymptotical expression for the energy function $\mathcal{E}$ in Lemma [Lemma 21](#Lem: E decomposition){reference-type="ref" reference="Lem: E decomposition"} obtained by the authors in [@XZ], we prove Theorem [Theorem 19](#Thm: main 2){reference-type="ref" reference="Thm: main 2"}.\ \ **Acknowledgements**\ The first author thanks Professor Feng Luo for his invitation to the workshop "Discrete and Computational Geometry, Shape Analysis, and Applications\" taking place at Rutgers University, New Brunswick from May 19th to May 21st, 2023. The first author also thanks Carl O. R. Lutz for helpful communications during the workshop. # Rigidity of decorated PE metrics {#Sec: DC theory} ## Discrete conformal equivalence and Bobenko-Lutz's discrete conformal theory Let $\mathcal{T}={(V,E,F)}$ be a triangulation of a marked surface $(S, V)$, where $V,E,F$ represent the sets of vertices, edges and faces respectively. A triangulation $\mathcal{T}$ for a PE surface $(S,V, dist_S)$ is a geodesic triangulation if the edges are geodesics in the PE metric $dist_S$. We use one index to denote a vertex (such as $i$), two indices to denote an edge (such as $\{ij\}$) and three indices to denote a face (such as $\{ijk\}$) in the triangulation $\mathcal{T}$. The PE metric $dist_{S}$ on a PE surface with a geodesic triangulation $\mathcal{T}$ defines a length map $l: E\rightarrow \mathbb{R}_{>0}$ such that $l_{ij}, l_{ik}, l_{jk}$ satisfy the triangle inequalities for any triangle $\{ijk\}\in F$. Conversely, given a function $l: E\rightarrow \mathbb{R}_{>0}$ satisfying the triangle inequalities for any face $\{ijk\}\in F$, one can construct a PE metric on a triangulated surface by isometrically gluing Euclidean triangles along edges in pairs. In the following, we use $l: E\rightarrow \mathbb{R}_{>0}$ to denote a PE metric and use $(l,r)$ to denote a decorated PE metric on a triangulated surface $(S,V,\mathcal{T})$. **Definition 6** ([@BL], Proposition 2.2). Let $\mathcal{T}$ be a triangulation of a marked surface $(S,V)$. Two decorated PE metrics $(l,r)$ and $(\widetilde{l},\widetilde{r})$ on $(S,V, \mathcal{T})$ are discrete conformal equivalent if and only if there exists a discrete conformal factor $u\in \mathbb{R}^V$ such that $$\label{Eq: DCE1} \widetilde{r}_i=e^{u_i}r_i,$$ $$\label{Eq: DCE2} \widetilde{l}_{ij}^2 =(e^{2u_i}-e^{u_i+u_j})r^2_i +(e^{2u_j}-e^{u_i+u_j})r^2_j +e^{u_i+u_j}l_{ij}^2$$ for any edge $\{ij\}\in E$. **Remark 7**. For any two circles $C_i$ and $C_j$ in the Euclidean plane, one can define the inversive distance $I_{ij}=\frac{l^2_{ij}-r^2_i-r^2_j}{2r_ir_j}$, where $l_{ij}$ is the distance of the centers of the two circles and $r_i$, $r_j$ are the radii of $C_i, C_j$ respectively. The inversive distance is invariant under Möbius transformations [@Coxeter]. Denote the inversive distance of two vertex-circles in $(l,r)$ and $(\widetilde{l},\widetilde{r})$ as $I$ and $\widetilde{I}$ respectively. If $(l,r)$ and $(\widetilde{l},\widetilde{r})$ are discrete conformal equivalent in the sense of Definition [Definition 6](#Def: DCE){reference-type="ref" reference="Def: DCE"}, it is shown [@BL] that $I=\widetilde{I}$. Since each pair of vertex-circles is required to be separated, it is easy to see that $I>1$. Therefore, the discrete conformal equivalent decorated PE metrics on triangulated surfaces in Definition [Definition 6](#Def: DCE){reference-type="ref" reference="Def: DCE"} can be taken as the separated inversive distance circle packing metrics introduced by Bowers-Stephenson [@BS]. Please refer to [@CLXZ; @Guo; @Luo3; @Xu; @AIM; @Xu; @MRL] for more properties of the inversive distance circle packing metrics on triangulated surfaces. For any decorated triangle $\{ijk\}$, there is a unique circle $C_{ijk}$ simultaneously orthogonal to the three vertex-circles at the vertices $i,j,k$ [@Glickenstein; @preprint]. This circle $C_{ijk}$ is called as the face-circle of the decorated triangle $\{ijk\}$. Denote $\alpha_{ij}^k$ as the interior intersection angle of the face-circle $C_{ijk}$ and the edge $\{ij\}$. The edge $\{ij\}$, shared by two adjacent decorated triangles $\{ijk\}$ and $\{ijl\}$, is called weighted Delaunay if $$\alpha_{ij}^k+\alpha_{ij}^l\leq \pi.$$ The triangulation $\mathcal{T}$ is called weighted Delaunay in the decorated PE metric $(dist_S,r)$ if every edge in the triangulation is weighted Delaunay. Here we take the definition of weighted Delaunay triangulation from Bobenko-Lutz [@BL]. There are other equivalent definitions for the weighted Delaunay triangulation using the signed distance of the center of $C_{ijk}$ to the edges. Please refer to [@CLXZ; @Glickenstein; @DCG; @Glickenstein; @JDG; @Glickenstein; @preprint; @GT] and others. Note that the combinatorial $\alpha$-curvature $R_\alpha$ in ([\[Eq: curvature R_alpha\]](#Eq: curvature R_alpha){reference-type="ref" reference="Eq: curvature R_alpha"}) is independent of the geodesic triangulations of a decorated PE surface $(S,V, dist_{S}, r)$. In general, the existence of decorated PE metrics with prescribed combinatorial $\alpha$-curvatures on triangulated surfaces can not be guaranteed if the triangulation is fixed. In the following, we work with a generalization of the discrete conformal equivalence in Definition [Definition 6](#Def: DCE){reference-type="ref" reference="Def: DCE"}, introduced by Bobenko-Lutz [@BL], which allows the triangulation of the marked surface to be changed under the weighted Delaunay condition. **Definition 8** ([@BL], Definition 4.11). Two decorated PE metrics $(dist_{S},r)$ and $(\widetilde{dist}_{S},\widetilde{r})$ on the marked surface $(S,V)$ are discrete conformal equivalent if there is a sequence of triangulated decorated PE surfaces $(\mathcal{T}^0,l^0,r^0),...,(\mathcal{T}^N,l^N,r^N)$ such that \(1\) : the decorated PE metric of $(\mathcal{T}^0,l^0,r^0)$ is $(dist_{S},r)$ and the decorated PE metric of $(\mathcal{T}^N,l^N,r^N)$ is $(\widetilde{dist}_{S},\widetilde{r})$, \(2\) : each $\mathcal{T}^n$ is a weighted Delaunay triangulation of the decorated PE surface $(\mathcal{T}^n,l^n,r^n)$, \(3\) : if $\mathcal{T}^n=\mathcal{T}^{n+1}$, then there is a discrete conformal factor $u\in \mathbb{R}^V$ such that $(\mathcal{T}^n,l^n,r^n)$ and $(\mathcal{T}^{n+1},l^{n+1},r^{n+1})$ are related by ([\[Eq: DCE1\]](#Eq: DCE1){reference-type="ref" reference="Eq: DCE1"}) and ([\[Eq: DCE2\]](#Eq: DCE2){reference-type="ref" reference="Eq: DCE2"}), \(4\) : if $\mathcal{T}^n\neq\mathcal{T}^{n+1}$, then $\mathcal{T}^n$ and $\mathcal{T}^{n+1}$ are two different weighted Delaunay triangulations of the same decorated PE surface. Definition [Definition 8](#Def: GDCE){reference-type="ref" reference="Def: GDCE"} gives an equivalence relationship for decorated PE metrics on a marked surface. The equivalence class of a decorated PE metric $(dist_S,r)$ on $(S,V)$ is also called as the discrete conformal class of $(dist_S,r)$ and denoted by $\mathcal{D}(dist_S,r)$. **Lemma 9** ([@BL]). The discrete conformal class $\mathcal{D}(dist_S,r)$ of a decorated PE metric $(dist_S,r)$ on the marked surface $(S,V)$ is parameterized by $\mathbb{R}^V=\{u: V\rightarrow \mathbb{R}\}$. For simplicity, for any $(\widetilde{dist}_S,\widetilde{r})\in \mathcal{D}(dist_S,r)$, we denote it by $(dist_S(u),r(u))$ for some $u\in \mathbb{R}^V$. Set $$\mathcal{C}_\mathcal{T}(dist_{S},r) =\{u\in \mathbb{R}^V |\ \mathcal{T}\ \text{is a weighted Delaunay triangulation of}\ (S,V,dist_S(u),r(u))\}.$$ **Lemma 10** ([@BL]). The set $$J=\{\mathcal{T}| \mathcal{C}_{\mathcal{T}}(dist_{S},r)\ \text{has non-empty interior in}\ \mathbb{R}^V\}$$ is a finite set, $\mathbb{R}^V=\cup_{\mathcal{T}_i\in J}\mathcal{C}_{\mathcal{T}_i}(dist_{S},r)$ and each $\mathcal{C}_{\mathcal{T}_i}(dist_{S},r)$ is homeomorphic to a polyhedral cone (with its apex removed) and its interior is homeomorphic to $\mathbb{R}^V$. ## The extended energy function There exist geometric relationships between the decorated triangles and $3$-dimensional generalized hyperbolic polyhedra. Specially, there is a generalized hyperbolic tetrahedra in $\mathbb{H}^3$ with one ideal vertex and three hyper-ideal vertices corresponding to a decorated triangle $\{ijk\}$. Denote $\mathrm{Vol}(ijk)$ as the truncated volume of this generalized hyperbolic tetrahedra. The truncated volume $\mathrm{Vol}(ijk)$ can be characterized by an explicit formula. Please refer to [@BL; @Sp1] for more details. Set $$\begin{aligned} F_{ijk}(u_i,u_j,u_k) =&-2\mathrm{Vol}(ijk)+\theta_{jk}^iu_i+\theta_{ki}^ju_j +\theta_{ij}^ku_k\\ &+(\frac{\pi}{2}-\alpha_{ij}^k)\lambda_{ij} +(\frac{\pi}{2}-\alpha_{ki}^j)\lambda_{ki} +(\frac{\pi}{2}-\alpha_{jk}^i)\lambda_{jk}, \end{aligned}$$ where $\theta_{jk}^i$ is the inner angle of the decorated triangle $\{ijk\}$ at the vertex $i$ and $\lambda_{ij}=\cosh^{-1} I_{ij}$. By the Schläfli formula, we have $$\nabla F_{ijk}=(\theta_{jk}^i,\theta_{ki}^j, \theta_{ij}^k)$$ and $$F_{ijk}((u_i,u_j,u_k)+c(1,1,1))=F_{ijk}(u_i,u_j,u_k)+c\pi$$ for $c\in \mathbb{R}$. On a decorated PE surface $(S,V,l,r)$ with a weighted Delaunay triangulation $\mathcal{T}$, Bobenko-Lutz [@BL] defined the following function $$\label{Eq: F1} \mathcal{H}_{\mathcal{T}}(u) =\sum_{\{ijk\}\in F}F_{ijk}(u_i,u_j,u_k) =-2\sum_{\{ijk\}\in F}\mathrm{Vol}(ijk)+\sum_{i\in V}\theta_iu_i+\sum_{\{ij\}\in E}(\pi-\alpha_{ij})\lambda_{ij},$$ where $\theta_i=\sum_{\{ijk\}\in F}\theta^i_{jk}$ and $\alpha_{ij}=\alpha_{ij}^k+\alpha_{ij}^l$. It should be mentioned that the function $\mathcal{H}_{\mathcal{T}}(u)$ in ([\[Eq: F1\]](#Eq: F1){reference-type="ref" reference="Eq: F1"}) differs from its original definition in [@BL] (Equation 4-9) by some constant. Then $$\label{Eq: property of H ijk} \mathcal{H}_{\mathcal{T}}(u+c\mathbf{1}) =\mathcal{H}_{\mathcal{T}}(u)+c|F|\pi$$ for $c\in \mathbb{R}$. By the definition of $\mathcal{H}_{\mathcal{T}}$, the following energy function $$\mathcal{E}_{\mathcal{T}}(u) =-\mathcal{H}_{\mathcal{T}}(u)+2\pi\sum_{i\in V}u_i$$ is well-defined on $\mathcal{C}_\mathcal{T}(dist_{S},r)$ with $\nabla_{u_i} \mathcal{E}_{\mathcal{T}} =2\pi-\theta_i=K_i$. Moreover, $$\label{Eq: property of E} \mathcal{E}_{\mathcal{T}}(u+c\mathbf{1}) =\mathcal{E}_{\mathcal{T}}(u)+2c\pi\chi(S)$$ for $c\in \mathbb{R}$. **Theorem 11** ([@BL], Proposition 4.13). For a discrete conformal factor $u\in \mathbb{R}^V$, let $\mathcal{T}$ be a weighted Delaunay triangulation of the decorated PE surface $(S,V,dist_S(u),r(u))$. The map $$\label{Eq: extended H} \mathcal{H} :\ \mathbb{R}^V\rightarrow \mathbb{R},\ \quad u\mapsto \mathcal{H}_{\mathcal{T}}(u)$$ is well-defined, concave, and twice continuously differentiable over $\mathbb{R}^V$. Therefore, the function $\mathcal{E}_{\mathcal{T}}(u)$ defined on $\mathcal{C}_\mathcal{T}(dist_{S},r)$ can be extended to be $\mathcal{E}(u)$ defined on $\mathbb{R}^V$ by the following formula $$\label{Eq: extended E} \mathcal{E}(u) =-\mathcal{H}(u)+2\pi\sum_{i\in V}u_i.$$ ## Rigidity of decorated PE metrics {#rigidity-of-decorated-pe-metrics} A basic problem on the combinatorial $\alpha$-curvature is to understand the relationships between the decorated PE metrics and the combinatorial $\alpha$-curvatures. The following theorem shows the global rigidity of decorated PE metrics with respect to the combinatorial $\alpha$-curvature on a marked surface, which corresponds to the rigidity parts of Theorem [Theorem 3](#Thm: existence 2){reference-type="ref" reference="Thm: existence 2"}. **Theorem 12**. Suppose $(S,V)$ is a marked surface with a decorated PE metric $(dist_S,r)$, $\alpha\in \mathbb{R}$ is a constant and $\overline{R}: V\rightarrow\mathbb{R}$ is a given function. \(1\) : If $\alpha\overline{R}\equiv 0$, then there exists at most one discrete conformal factor $u^*\in \mathbb{R}^V$ up to scaling such that the decorated PE metric $(dist_S(u^*),r(u^*))$ in the discrete conformal class $\mathcal{D}(dist_S,r)$ has the combinatorial $\alpha$-curvature $\overline{R}$. \(2\) : If $\alpha\overline{R}\leq 0$ and $\alpha\overline{R}\not\equiv 0$, then there exists at most one discrete conformal factor $u^*\in \mathbb{R}^V$ such that the decorated PE metric $(dist_S(u^*),r(u^*))$ in the discrete conformal class $\mathcal{D}(dist_S,r)$ has the combinatorial $\alpha$-curvature $\overline{R}$. By Theorem [Theorem 11](#Thm: extended H){reference-type="ref" reference="Thm: extended H"}, the following function $$\mathbb{E}(u) =-\mathcal{H}(u)+\int^u_{u_0} \sum_{i\in V}(2\pi-\overline{R}_ir_i^\alpha)du_i$$ is well-defined and twice continuously differentiable over $\mathbb{R}^V$, where $r_i=e^{u_i}r^0_i$ and $r^0$ is the initial data. By direct calculations, we have $$\nabla_{u_i}\mathbb{E} =-\sum_{\{ijk\}\in F}\theta^i_{jk} +2\pi-\overline{R}_ir_i^\alpha =K_i-\overline{R}_ir_i^\alpha.$$ Therefore, for $u^*\in \mathbb{R}^V$, the decorated PE metric in $(dist_S(u^*),r(u^*))$ has the combinatorial $\alpha$-curvature $\overline{R}$ if and only if $\nabla_{u_i}\mathbb{E}(u^*)=0, \forall i\in V$. Moreover, $$%\label{Eq: F2} \mathrm{Hess}_u\ \mathbb{E} =-\mathrm{Hess}_u\ \mathcal{H} -\alpha \left( \begin{array}{ccc} \overline{R}_1r^\alpha_1 & &\\ & \ddots & \\ & & \overline{R}_{|V|}r^\alpha_{|V|} \\ \end{array} \right).$$ The equality ([\[Eq: property of H ijk\]](#Eq: property of H ijk){reference-type="ref" reference="Eq: property of H ijk"}) and Theorem [Theorem 11](#Thm: extended H){reference-type="ref" reference="Thm: extended H"} imply that $\mathrm{Hess}_u\mathcal{H}\leq0$ with kernel $\{c\mathbf{1}^\mathrm{T}\in \mathbb{R}^V|c\in \mathbb{R}\}$. If $\alpha\overline{R}\equiv 0$, then $\mathrm{Hess}_u\ \mathbb{E}$ is positive semi-definite with kernel $\{c\mathbf{1}^\mathrm{T}\in \mathbb{R}^V|c\in \mathbb{R}\}$ and hence $\mathbb{E}$ is convex on $\mathbb{R}^V$ and strictly convex on $\{\sum_{i\in V}u_i=0\}$. If $\alpha\overline{R}\leq 0$ and $\alpha\overline{R}\not\equiv 0$, then $\mathrm{Hess}_u\ \mathbb{E}$ is positive definite and hence $\mathbb{E}$ is strictly convex on $\mathbb{R}^V$. The conclusion follows from the following result from calculus. **Lemma:** If $f:\Omega \rightarrow \mathbb{R}$ is a $C^1$-smooth strictly convex function on an open convex set $\Omega \subset \mathbb{R}^n$, then its gradient $\nabla f:\Omega \rightarrow \mathbb{R}^n$ is injective. **Remark 13**. For a decorated PE surfaces $(S,V,l,r)$ with a fixed triangulation $\mathcal{T}$, the global rigidity of the inversive distance circle packing metrics with respect to the combinatorial $\alpha$-curvature $R_\alpha$ has been proved by Ge-Jiang [@GJ] and Ge-Xu [@GX2]. They extended the function $\mathbb{E}(u)$ by extending the inner angles of a triangle by constants. This approach was introduced by Bobenko-Pinkall-Springborn [@BPS] for Luo's vertex scalings and further developed by Luo [@Luo3] for Bowers-Stephenson's inversive distance circle packings. Here we take another approach introduced by Bobenko-Lutz [@BL] to extend the function $\mathbb{E}(u)$, in which we change the triangulation of the marked surface under the weighted Delaunay condition. This approach was first introduced by Gu-Luo-Sun-Wu [@Gu1] and Gu-Guo-Luo-Sun-Wu [@Gu2] to establish the discrete uniformization theorem for Luo's vertex scalings on surfaces. The first approach can not ensure the triangles being non-degenerate, while the second approach can. # Existence of decorated PE metrics {#Sec: existence} ## Variational principles with constraints In this subsection, to simplify the calculations, we deform the combinatorial $\alpha$-curvature $R_\alpha$ in ([\[Eq: curvature R_alpha\]](#Eq: curvature R_alpha){reference-type="ref" reference="Eq: curvature R_alpha"}) and give Theorem [Theorem 14](#Thm: existence 3){reference-type="ref" reference="Thm: existence 3"} which is equivalent to Theorem [Theorem 3](#Thm: existence 2){reference-type="ref" reference="Thm: existence 2"}. Then we translate Theorem [Theorem 14](#Thm: existence 3){reference-type="ref" reference="Thm: existence 3"} into an optimization problem with inequality constraints by variational principles, which involves the function $\mathcal{E}(u)$ defined in ([\[Eq: extended E\]](#Eq: extended E){reference-type="ref" reference="Eq: extended E"}). For an initial decorated PE metric $(l^0,r^0)$, the combinatorial $\alpha$-curvature is $K^0_i/(r^0_i)^\alpha$. Suppose a decorated PE metric $(l,r)$ is discrete conformal equivalent to $(l^0,r^0)$, then $r_i=e^{u_i}r_i^0$ by ([\[Eq: DCE1\]](#Eq: DCE1){reference-type="ref" reference="Eq: DCE1"}). The combinatorial $\alpha$-curvature of the decorated PE metric $(l,r)$ can be written as $$R_{\alpha,i} =\frac{K_i}{r^\alpha_i} =\frac{K_i}{(r^0_i)^\alpha e^{\alpha u_i}}.$$ For simplicity, set $$\mathcal{R}_{\alpha,i}=R_{\alpha,i}(r^0_i)^\alpha.$$ Then $$\label{Eq: key 4} \mathcal{R}_{\alpha,i}=\frac{K_i}{e^{\alpha u_i}}.$$ We also call $\mathcal{R}_{\alpha}$ as the combinatorial $\alpha$-curvature. Note that $(r^0_i)^\alpha>0$, then the signs of $\mathcal{R}_{\alpha,i}$ and $R_{\alpha,i}$ are the same for any $i\in V$. Denote $\overline{\mathcal{R}}$ as the prescribed combinatorial $\alpha$-curvature corresponding to $\mathcal{R}_{\alpha}$. Then $\overline{\mathcal{R}}_i=\overline{R}_i(r^0_i)^\alpha$ and the signs of $\overline{\mathcal{R}}_i$ and $\overline{R}_i$ are the same. Hence, to prove Theorem [Theorem 3](#Thm: existence 2){reference-type="ref" reference="Thm: existence 2"}, we just need to prove the following theorem. **Theorem 14**. For any decorated PE metric $(dist_S,r)$ on a marked surface $(S,V)$, there is a discrete conformal equivalent decorated PE metric $(\widetilde{dist_S},\widetilde{r})$ with combinatorial $\alpha$-curvature $\overline{\mathcal{R}}$ if one of the following conditions is satisfied \(1\) : $\chi(S)>0,\ \alpha<0,\ \overline{\mathcal{R}}>0$; \(2\) : $\chi(S)<0,\ \alpha\neq0,\ \overline{\mathcal{R}}\leq 0,\ \overline{\mathcal{R}}\not\equiv 0$. Since the angle defect $K$ satisfies the following discrete Gauss-Bonnet formula ([@Chow-Luo], Proposition 3.1) $$\label{Eq: Gauss-Bonnet} \sum_{i\in V}K_i=2\pi\chi(S),$$ then the combinatorial $\alpha$-curvature $\mathcal{R}_\alpha$ in ([\[Eq: key 4\]](#Eq: key 4){reference-type="ref" reference="Eq: key 4"}) satisfies the following discrete Gauss-Bonnet formula $$\sum_{i\in V}\mathcal{R}_ie^{\alpha u_i}=2\pi\chi(S).$$ Therefore, if $\overline{\mathcal{R}}\in \mathbb{R}^V$ is the combinatorial $\alpha$-curvature of some decorated PE metric discrete conformal to $(l,r)$ on $(S,V)$, then $$\sum_{i\in V}\overline{\mathcal{R}}_ie^{\alpha u_i}=2\pi\chi(S).$$ Let $\alpha\in \mathbb{R}$ be a non-zero constant. Set $$\label{A} \mathcal{A}=\{u\in \mathbb{R}^V|0>\sum_{i\in V}\overline{\mathcal{R}}_i e^{\alpha u_i}\geq 2\pi\chi(S),\ \overline{\mathcal{R}}\leq0,\ \overline{\mathcal{R}}\not\equiv0\},$$ $$\label{B} \mathcal{B}=\{u\in \mathbb{R}^V|0<\sum_{i\in V}\overline{\mathcal{R}}_i e^{\alpha u_i}\leq 2\pi\chi(S),\ \overline{\mathcal{R}}>0\},$$ $$\label{C} \mathcal{C}=\{u\in \mathbb{R}^V|\sum_{i\in V} \overline{\mathcal{R}}_i e^{\alpha u_i}\leq 2\pi\chi(S)<0,\ \overline{\mathcal{R}}\leq0,\ \overline{\mathcal{R}}\not\equiv0 \}.$$ **Proposition 15**. The sets $\mathcal{A},\ \mathcal{B}$ and $\mathcal{C}$ are unbounded closed subsets of $\mathbb{R}^V$. We only prove this proposition for the set $\mathcal{A}$ and the proofs for the sets $\mathcal{B}$ and $\mathcal{C}$ are similar. **(I):** To prove the closeness of the set $\mathcal{A}$ in $\mathbb{R}^V$, we just need to show $\mathcal{A}=\overline{\mathcal{A}}$, where $\overline{\mathcal{A}}$ represents the closure of the set $\mathcal{A}$ in $\mathbb{R}^V$. Suppose $\{u_{i,n}\}_{n\in \mathbb{N}}$ is a sequence in $\mathcal{A}$ such that $\lim_{n\rightarrow +\infty}u_{i,n}=\lambda_i\in \mathbb{R}, \forall i\in V$. It is direct to see that $\lim_{n\rightarrow +\infty} \sum_{i\in V}\overline{\mathcal{R}}_i e^{\alpha u_{i,n}} =\sum_{i\in V}\overline{\mathcal{R}}_i e^{\alpha \lambda_i}\geq 2\pi\chi(S).$ Note that the definition of $\mathcal{A}$ in ([\[A\]](#A){reference-type="ref" reference="A"}) shows $\overline{\mathcal{R}}\leq0,\ \overline{\mathcal{R}}\not\equiv0$. This implies that there exists $i_0\in V$ such that $\overline{\mathcal{R}}_{i_0}<0$. Then $$\lim_{n\rightarrow +\infty} \sum_{i\in V}\overline{\mathcal{R}}_i e^{\alpha u_{i,n}} =\sum_{i\in V}\overline{\mathcal{R}}_i e^{\alpha \lambda_{i}}\leq\overline{\mathcal{R}}_{i_0} e^{\alpha \lambda_{i_0}} <0.$$ This implies $\lambda=(\lambda_1,...,\lambda_{|V|})\in \mathcal{A}$ and hence $\mathcal{A}=\overline{\mathcal{A}}$. Therefore, the set $\mathcal{A}$ is a closed subset of $\mathbb{R}^V$. **(II):** If $u\in \mathcal{A}$, for any $c\in \mathbb{R}$, we have $$\sum_{i\in V} \overline{\mathcal{R}}_i e^{\alpha (u_i+c)}=e^{\alpha c}\sum_{i\in V} \overline{\mathcal{R}}_i e^{\alpha u_i}<0.$$ If $\alpha<0$, $u\in \mathcal{A}$, then $$\sum_{i\in V} \overline{\mathcal{R}}_i e^{\alpha (u_i+c)}=e^{\alpha c}\sum_{i\in V} \overline{\mathcal{R}}_i e^{\alpha u_i}\geq 2\pi\chi(S)$$ is equivalent to $$c\geq\frac{1}{\alpha}\log\frac{2\pi\chi(S)} {\sum_{i\in V} \overline{\mathcal{R}}_i e^{\alpha u_i}}.$$ This implies that the ray $\{u+c\mathbf{1}|c\geq\frac{1}{\alpha}\log\frac{2\pi\chi(S)} {\sum_{i\in V} \overline{\mathcal{R}}_i e^{\alpha u_i}},\ \alpha<0 \}$ stays in the set $\mathcal{A}$. Hence, the set $\mathcal{A}$ is unbounded if $\alpha<0$. If $\alpha>0$, for $u\in \mathcal{A}$, we have $$\sum_{i\in V} \overline{\mathcal{R}}_i e^{\alpha (u_i+c)}=e^{\alpha c}\sum_{i\in V} \overline{\mathcal{R}}_i e^{\alpha u_i}\geq 2\pi\chi(S)$$ is equivalent to $$c\leq\frac{1}{\alpha}\log\frac{2\pi\chi(S)} {\sum_{i\in V} \overline{\mathcal{R}}_i e^{\alpha u_i}}.$$ This implies that the ray $\{u+c\mathbf{1}|c\leq\frac{1}{\alpha}\log\frac{2\pi\chi(S)} {\sum_{i\in V} \overline{\mathcal{R}}_i e^{\alpha u_i}}\}$ stays in the set $\mathcal{A}$. Hence, the set $\mathcal{A}$ is unbounded if $\alpha>0$. This completes the proof. According to Proposition [Proposition 15](#Prop: ABC){reference-type="ref" reference="Prop: ABC"}, we have following result. **Lemma 16**. If one of the following three conditions is satisfied \(1\) : $\alpha>0$ and the energy function $\mathcal{E}$ attains a minimum in the set $\mathcal{A}$, \(2\) : $\alpha<0$ and the energy function $\mathcal{E}$ attains a minimum in the set $\mathcal{B}$, \(3\) : $\alpha<0$ and the energy function $\mathcal{E}$ attains a minimum in the set $\mathcal{C}$, then the minimum value point of $\mathcal{E}$ lies in the set $\{u\in \mathbb{R}^V|\sum_{i\in V} \overline{\mathcal{R}}_i e^{\alpha u_i}=2\pi\chi(S)\}$. Suppose $\alpha>0$ and the function $\mathcal{E}$ attains a minimum at $u\in \mathcal{A}$. The definition of $\mathcal{A}$ in ([\[A\]](#A){reference-type="ref" reference="A"}) implies $\chi(S)<0$. Set $$c_0=\frac{1}{\alpha}\log\frac{2\pi\chi(S)} {\sum_{i\in V} \overline{\mathcal{R}}_i e^{\alpha u_i}},$$ then $c_0\geq0$. By the proof of Proposition [Proposition 15](#Prop: ABC){reference-type="ref" reference="Prop: ABC"}, $u+c_0\mathbf{1}\in \mathcal{A}$. Therefore, by the additive property of the function $\mathcal{E}$ in ([\[Eq: property of E\]](#Eq: property of E){reference-type="ref" reference="Eq: property of E"}), we have $$\mathcal{E}(u)\leq \mathcal{E}(u+c_0\mathbf{1}) =\mathcal{E}(u)+2\pi c_0\chi(S),$$ which implies $c_0\leq0$ by $\chi(S)<0$. Hence $c_0=0$ and $\sum_{i\in V} \overline{\mathcal{R}}_i e^{\alpha u_i}=2\pi\chi(S)$. This proves the case $\textbf{(1)}$. The proofs for the cases $\textbf{(2)}$ and $\textbf{(3)}$ are similar, we omit the details here. By Lemma [Lemma 16](#Lem: minimum lies at the boundary){reference-type="ref" reference="Lem: minimum lies at the boundary"}, we translate Theorem [Theorem 14](#Thm: existence 3){reference-type="ref" reference="Thm: existence 3"} into the following theorem, which is a non-convex optimization problem with inequality constraints. **Theorem 17**. Let $(dist_S,r)$ be a decorated PE metric on a marked surface $(S,V)$ with $\chi(S)\neq0$. Suppose $\alpha\in \mathbb{R}$ is a non-zero constant and $\overline{\mathcal{R}}$ is a given function defined on $V$. \(1\) : If $\overline{\mathcal{R}}\leq0$, $\overline{\mathcal{R}}\not\equiv0$, $\alpha>0$ and the energy function $\mathcal{E}$ attains a minimum in $\mathcal{A}$, then there exists a decorated PE metric in the discrete conformal class $\mathcal{D}(dist_S,r)$ with combinatorial $\alpha$-curvature $\overline{\mathcal{R}}$; \(2\) : If $\overline{\mathcal{R}}>0$, $\alpha<0$ and the energy function $\mathcal{E}$ attains a minimum in $\mathcal{B}$, then there exists a decorated PE metric in the discrete conformal class $\mathcal{D}(dist_S,r)$ with combinatorial $\alpha$-curvature $\overline{\mathcal{R}}$; \(3\) : If $\overline{\mathcal{R}}\leq0$, $\overline{\mathcal{R}}\not\equiv0$, $\alpha<0$ and the energy function $\mathcal{E}$ attains a minimum in $\mathcal{C}$, then there exists a decorated PE metric in the discrete conformal class $\mathcal{D}(dist_S,r)$ with combinatorial $\alpha$-curvature $\overline{\mathcal{R}}$. Lemma [Lemma 16](#Lem: minimum lies at the boundary){reference-type="ref" reference="Lem: minimum lies at the boundary"} shows that if $u\in \mathbb{R}^V$ is a minimum of the energy function $\mathcal{E}$ defined on one of these sets, then $\sum_{i\in V} \overline{\mathcal{R}}_i e^{\alpha u_i}= 2\pi\chi(S)$. The conclusion follows from the following claim.\ **Claim :** Up to scaling, the decorated PE metrics with combinatorial $\alpha$-curvature $\overline{\mathcal{R}}$ in the discrete conformal class are in one-to-one correspondence with the critical points of the function $\mathcal{E}$ under the constraint $\sum_{i\in V} \overline{\mathcal{R}}_i e^{\alpha u_i}=2\pi\chi(S)$. We use the method of Lagrange multipliers to prove this claim. Set $$G(u,\mu)=\mathcal{E}(u)-\mu \left(\sum_{i\in V} \overline{\mathcal{R}}_i e^{\alpha u_i}-2\pi\chi(S)\right),$$ where $\mu\in \mathbb{R}$ is a Lagrange multiplier. If $u$ is a critical point of the function $\mathcal{E}$ under the constraint $\sum_{i\in V} \overline{\mathcal{R}}_i e^{\alpha u_i}=2\pi\chi(S)$, then by the fact $\nabla_{u_i} \mathcal{E}=K_i$, we have $$0=\frac{\partial G(u,\mu)}{\partial u_i} =K_i-\mu\alpha\overline{\mathcal{R}}_i e^{\alpha u_i},$$ which implies $$\mathcal{R}_{\alpha, i}=\frac{K_i}{e^{\alpha u_i}} =\mu\alpha\overline{\mathcal{R}}_i.$$ By the discrete Gauss-Bonnet formula ([\[Eq: Gauss-Bonnet\]](#Eq: Gauss-Bonnet){reference-type="ref" reference="Eq: Gauss-Bonnet"}), the Lagrange multiplier $\mu$ satisfies $$\mu=\frac{2\pi \chi(S)}{\alpha\sum_{i\in V} \overline{\mathcal{R}}_i e^{\alpha u_i}}=\frac{1}{\alpha}$$ under the constraint $\sum_{i\in V} \overline{\mathcal{R}}_i e^{\alpha u_i}=2\pi\chi(S)$. This implies the combinatorial $\alpha$-curvature $$\mathcal{R}_{\alpha, i}=\mu\alpha\overline{\mathcal{R}}_i =\frac{2\pi \chi(S)}{\sum_{i\in V} \overline{\mathcal{R}}_i e^{\alpha u_i}}\overline{\mathcal{R}}_i=\overline{\mathcal{R}}_i$$ under the constraint $\sum_{i\in V} \overline{\mathcal{R}}_i e^{\alpha u_i}=2\pi\chi(S)$. ## Reduction to Theorem [Theorem 19](#Thm: main 2){reference-type="ref" reference="Thm: main 2"} {#reduction-to-theorem-thm-main-2} By Theorem [Theorem 17](#Thm: inequality constraints){reference-type="ref" reference="Thm: inequality constraints"}, we just need to prove that the function $\mathcal{E}(u)$ attains the minimum in the sets $\mathcal{A},\ \mathcal{B}$ and $\mathcal{C}$ respectively. Recall the following classical result from calculus. **Theorem 18**. Let $\Omega\subseteq \mathbb{R}^m$ be a closed set and $f: \Omega\rightarrow \mathbb{R}$ be a continuous function. If every unbounded sequence $\{u_n\}_{n\in \mathbb{N}}$ in $\Omega$ has a subsequence $\{x_{n_k}\}_{k\in \mathbb{N}}$ such that $\lim_{k\rightarrow +\infty} f(x_{n_k})=+\infty$, then $f$ attains a minimum in $\Omega$. One can refer to [@Kourimska; @Thesis] (Section 4.1) for a proof of Theorem [Theorem 18](#Thm: calculus){reference-type="ref" reference="Thm: calculus"}. The majority of the conditions in Theorem [Theorem 18](#Thm: calculus){reference-type="ref" reference="Thm: calculus"} are satisfied, including the sets $\mathcal{A},\ \mathcal{B}$ and $\mathcal{C}$ are closed subsets of $\mathbb{R}^V$ by Proposition [Proposition 15](#Prop: ABC){reference-type="ref" reference="Prop: ABC"} and the energy function $\mathcal{E}$ is continuous. To prove Theorem [Theorem 14](#Thm: existence 3){reference-type="ref" reference="Thm: existence 3"}, we just need to prove the following theorem. **Theorem 19**. Suppose $(S,V)$ is a marked surface with a decorated PE metric $(dist_S,r)$, $\alpha\in \mathbb{R}$ is a constant and $\overline{\mathcal{R}}$ is a given function defined on $V$. If one of the following three conditions is satisfied \(1\) : $\alpha>0$ and $\{u_n\}_{n\in \mathbb{N}}$ is an unbounded sequence in $\mathcal{A}$, \(2\) : $\alpha<0$ and $\{u_n\}_{n\in \mathbb{N}}$ is an unbounded sequence in $\mathcal{B}$, \(3\) : $\alpha<0$ and $\{u_n\}_{n\in \mathbb{N}}$ is an unbounded sequence in $\mathcal{C}$, then there exists a subsequence $\{u_{n_k}\}_{k\in \mathbb{N}}$ of $\{u_n\}_{n\in \mathbb{N}}$ such that $\lim_{k\rightarrow +\infty} \mathcal{E}(u_{n_k})=+\infty$. ## Behaviour of sequences of conformal factors Let $\{u_n\}_{n\in \mathbb{N}}$ be an unbounded sequence in $\mathbb{R}^V$. Denote its coordinate sequence at $j\in V$ by $\{u_{j,n}\}_{n\in \mathbb{N}}$. Motivated by [@Kourimska], we call the sequence $\{u_n\}_{n\in \mathbb{N}}$ with the following properties as a "good\" sequence. \(1\) : It lies in one cell $\mathcal{C}_\mathcal{T}(dist_{S},r)$ of $\mathbb{R}^V$; \(2\) : There exists a vertex $i^*\in V$ such that $u_{i^*,n}\leq u_{j,n}$ for all $j\in V$ and $n\in \mathbb{N}$; \(3\) : Each coordinate sequence $\{u_{j,n}\}_{n\in \mathbb{N}}$ either converges, diverges properly to $+\infty$, or diverges properly to $-\infty$; \(4\) : For any $j\in V$, the sequence $\{u_{j,n}-u_{i^*,n}\}_{n\in \mathbb{N}}$ either converges or diverges properly to $+\infty$. By Lemma [Lemma 10](#Lem: finite decomposition){reference-type="ref" reference="Lem: finite decomposition"}, it is obvious that every sequence of discrete conformal factors in $\mathbb{R}^V$ possesses a "good\" subsequence. Hence, the "good\" sequence could be chosen without loss of generality. To prove Theorem [Theorem 19](#Thm: main 2){reference-type="ref" reference="Thm: main 2"}, we further need the following two results obtained by the authors in [@XZ]. **Lemma 20** ([@XZ], Corollary 3.6). For a discrete conformal factor $u\in \mathbb{R}^V$, let $\mathcal{T}$ be a weighted Delaunay triangulation of the decorated PE surface $(S,V,dist_S(u),r(u))$. For any decorated triangle $\{ijk\}\in F$ in $\mathcal{T}$, at least two of the three sequences $\{u_{i,n}-u_{i^*,n}\}_{n\in \mathbb{N}}$, $\{u_{j,n}-u_{i^*,n}\}_{n\in \mathbb{N}}$, $\{u_{k,n}-u_{i^*,n}\}_{n\in \mathbb{N}}$ converge. **Lemma 21** ([@XZ], Lemma 3.12). There exists a convergent sequence $\{D_n\}_{n\in \mathbb{N}}$ such that the function $\mathcal{E}$ satisfies $$\mathcal{E}(u_n)=D_n+2\pi\left(u_{i^*,n}\chi(S)+\sum_{j\in V}(u_{j,n}-u_{i^*,n})\right).$$ **Proof of Theorem [Theorem 19](#Thm: main 2){reference-type="ref" reference="Thm: main 2"}:** Let $\{u_n\}_{n\in \mathbb{N}}$ be an unbounded "good\" sequence. We just need to prove that $\lim_{n\rightarrow +\infty} \mathcal{E}(u_n)=+\infty$. **(1):** Let $\alpha>0$ and $\{u_n\}_{n\in \mathbb{N}}$ be an unbounded sequence in $\mathcal{A}$. The definition of $\mathcal{A}$ in ([\[A\]](#A){reference-type="ref" reference="A"}) implies $\chi(S)<0$, $\overline{\mathcal{R}}\leq0$ and $\overline{\mathcal{R}}\not\equiv0$. Since the sequence $\{u_n\}_{n\in \mathbb{N}}$ lies in $\mathcal{A}$, we have $$\label{Eq: key 1} 0>\sum_{j\in V} \overline{\mathcal{R}}_j e^{\alpha (u_{j,n}-u_{i^*,n})} =e^{-\alpha u_{i^*,n}}\cdot\sum_{j\in V} \overline{\mathcal{R}}_j e^{\alpha u_{j,n}} \geq 2\pi\chi(S) e^{-\alpha u_{i^*,n}}.$$ By the definition of "good\" sequence, the sequence $\left\{\sum_{j\in V}(u_{j,n}-u_{i^*,n})\right\}_{n\in \mathbb{N}}$ converges to a finite positive number or diverges properly to $+\infty$ If $\left\{\sum_{j\in V}(u_{j,n}-u_{i^*,n})\right\}_{n\in \mathbb{N}}$ converges to a finite positive number, then the sequence $\{u_{j,n}-u_{i^*,n}\}_{n\in \mathbb{N}}$ converges for any $j\in V$. This implies $\sum_{j\in V} \overline{\mathcal{R}}_j e^{\alpha (u_{j,n}-u_{i^*,n})}$ converges to a finite negative number by ([\[Eq: key 1\]](#Eq: key 1){reference-type="ref" reference="Eq: key 1"}). Then by $\chi(S)<0$, we have $$-\alpha u_{i^*,n}\geq \ln\frac{\sum_{j\in V} \overline{\mathcal{R}}_j e^{\alpha (u_{j,n}-u_{i^*,n})}}{2\pi\chi(S)}.$$ Hence $\{u_{i^*,n}\}_{n\in \mathbb{N}}$ is bounded from above by $\alpha>0$. This implies $\{u_{i^*,n}\}_{n\in \mathbb{N}}$ converges to a finite number or diverges properly to $-\infty$. If $\{u_{i^*,n}\}_{n\in \mathbb{N}}$ converges to a finite number, then by $\{u_{j,n}-u_{i^*,n}\}_{n\in \mathbb{N}}$ converges for any $j\in V$, we have $\{u_{j,n}\}_{n\in \mathbb{N}}$ is bounded for any $j\in V$. This contradicts the assumption that $\{u_n\}_{n\in \mathbb{N}}$ is unbounded. Therefore, the sequence $\{u_{i^*,n}\}_{n\in \mathbb{N}}$ diverges properly to $-\infty$. Combining this with $\chi(S)<0$ and Lemma [Lemma 21](#Lem: E decomposition){reference-type="ref" reference="Lem: E decomposition"}, we have $\lim_{n\rightarrow +\infty} \mathcal{E}(u_n)=+\infty$. If $\left\{\sum_{j\in V}(u_{j,n}-u_{i^*,n})\right\}_{n\in \mathbb{N}}$ diverges properly to $+\infty$, then there exists at least one vertex $j\in V$ such that the sequence $\{u_{j,n}-u_{i^*,n}\}_{n\in \mathbb{N}}$ diverges properly to $+\infty$. By Lemma [Lemma 20](#Lem: one infty two converge){reference-type="ref" reference="Lem: one infty two converge"}, for any vertex $k\sim j$, the sequence $\{u_{k,n}-u_{i^*,n}\}_{n\in \mathbb{N}}$ converges. Since $\alpha>0$, then $e^{\alpha (u_{j,n}-u_{i^*,n})}$ converges to a finite positive number or diverges properly to $+\infty$ and for at least one vertex $j\in V$ the term $e^{\alpha (u_{j,n}-u_{i^*,n})}$ converges to a finite positive number. Since $\overline{\mathcal{R}}\leq0$ and $\overline{\mathcal{R}}\not\equiv0$, then $\sum_{j\in V} \overline{\mathcal{R}}_j e^{\alpha (u_{j,n}-u_{i^*,n})}$ converges to a finite negative number or diverges properly to $-\infty$. $(i)$ : Suppose $\sum_{j\in V} \overline{\mathcal{R}}_j e^{\alpha (u_{j,n}-u_{i^*,n})}$ converges to a finite negative number. Similar arguments imply $u_{i^*,n}$ is bounded from above, then $u_{i^*,n}\chi(S)$ is bounded from below by $\chi(S)<0$. Combining with the assumption that $\left\{\sum_{j\in V}(u_{j,n}-u_{i^*,n})\right\}_{n\in \mathbb{N}}$ diverges properly to $+\infty$, we have $\lim_{n\rightarrow +\infty} \mathcal{E}(u_n)=+\infty$ by Lemma [Lemma 21](#Lem: E decomposition){reference-type="ref" reference="Lem: E decomposition"}. $(ii)$ : Suppose $\sum_{j\in V}\overline{\mathcal{R}}_j e^{\alpha (u_{j,n}-u_{i^*,n})}$ diverges properly to $-\infty$. Then $2\pi\chi(S) e^{-\alpha u_{i^*,n}}$ diverges properly to $-\infty$ by ([\[Eq: key 1\]](#Eq: key 1){reference-type="ref" reference="Eq: key 1"}). Since $\chi(S)<0$, then $e^{-\alpha u_{i^*,n}}$ diverges properly to $+\infty$. By $\alpha>0$, then $\{u_{i^*,n}\}_{n\in \mathbb{N}}$ diverges properly to $-\infty$. Hence $u_{i^*,n}\chi(S)$ diverges properly to $+\infty$ by $\chi(S)<0$. Then $\lim_{n\rightarrow +\infty} \mathcal{E}(u_n)=+\infty$ by Lemma [Lemma 21](#Lem: E decomposition){reference-type="ref" reference="Lem: E decomposition"}. **(2):** Let $\alpha<0$ and $\{u_n\}_{n\in \mathbb{N}}$ be an unbounded sequence in $\mathcal{B}$. The definition of $\mathcal{B}$ in ([\[B\]](#B){reference-type="ref" reference="B"}) implies $\chi(S)>0$ and $\overline{\mathcal{R}}>0$. Since the sequence $\{u_n\}_{n\in \mathbb{N}}$ lies in $\mathcal{B}$, we have $$\label{Eq: key 2} 0<\sum_{j\in V} \overline{\mathcal{R}}_j e^{\alpha (u_{j,n}-u_{i^*,n})} =e^{-\alpha u_{i^*,n}}\cdot\sum_{j\in V} \overline{\mathcal{R}}_j e^{\alpha u_{j,n}} \leq 2\pi\chi(S) e^{-\alpha u_{i^*,n}}.$$ If $\left\{\sum_{j\in V}(u_{j,n}-u_{i^*,n})\right\}_{n\in \mathbb{N}}$ converges, then the sequence $\{u_{j,n}-u_{i^*,n}\}_{n\in \mathbb{N}}$ converges for any $j\in V$. This implies $\sum_{j\in V} \overline{\mathcal{R}}_j e^{\alpha (u_{j,n}-u_{i^*,n})}$ converges to a finite positive number by ([\[Eq: key 2\]](#Eq: key 2){reference-type="ref" reference="Eq: key 2"}). Since $\chi(S)>0$, then the equation ([\[Eq: key 2\]](#Eq: key 2){reference-type="ref" reference="Eq: key 2"}) implies $$-\alpha u_{i^*,n}\geq \ln \frac{\sum_{j\in V} \overline{\mathcal{R}}_j e^{\alpha (u_{j,n}-u_{i^*,n})}}{2\pi\chi(S)}.$$ By $\alpha<0$, then $\{u_{i^*,n}\}_{n\in \mathbb{N}}$ is bounded from below. This implies $\{u_{i^*,n}\}_{n\in \mathbb{N}}$ converges to a finite number or diverges properly to $+\infty$. Combining this with $\{u_n\}_{n\in \mathbb{N}}$ is unbounded and $\{u_{j,n}-u_{i^*,n}\}_{n\in \mathbb{N}}$ converges for all $j\in V$, we have the sequence $\{u_{i^*,n}\}_{n\in \mathbb{N}}$ diverges properly to $+\infty$. By $\chi(S)>0$ and Lemma [Lemma 21](#Lem: E decomposition){reference-type="ref" reference="Lem: E decomposition"}, we have $\lim_{n\rightarrow +\infty} \mathcal{E}(u_n)=+\infty$. If the sequence $\left\{\sum_{j\in V}(u_{j,n}-u_{i^*,n})\right\}_{n\in \mathbb{N}}$ diverges properly to $+\infty$, then there exists at least one vertex $j\in V$ such that the sequence $\{u_{j,n}-u_{i^*,n}\}_{n\in \mathbb{N}}$ diverges properly to $+\infty$. By Lemma [Lemma 20](#Lem: one infty two converge){reference-type="ref" reference="Lem: one infty two converge"}, for any vertex $k\sim j$, the sequence $(u_{k,n}-u_{i^*,n})_{n\in \mathbb{N}}$ converges. Therefore, $e^{\alpha (u_{j,n}-u_{i^*,n})}$ converges to zero or a finite positive number and for at least one vertex $j\in V$ the term $e^{\alpha (u_{j,n}-u_{i^*,n})}$ converges to a finite positive number. Since $\alpha<0$ and $\overline{\mathcal{R}}>0$, then $\sum_{j\in V} \overline{\mathcal{R}}_j e^{\alpha (u_{j,n}-u_{i^*,n})}$ converges to a finite positive number. This implies $2\pi\chi(S) e^{-\alpha u_{i^*,n}}$ has a positive lower bound by ([\[Eq: key 2\]](#Eq: key 2){reference-type="ref" reference="Eq: key 2"}). By $\alpha<0$ and $\chi(S)>0$, then $\{u_{i^*,n}\}_{n\in \mathbb{N}}$ is bounded from below. Then $u_{i^*,n}\chi(S)$ is bounded from below. Combining this with $\left\{\sum_{j\in V}(u_{j,n}-u_{i^*,n})\right\}_{n\in \mathbb{N}}$ diverges properly to $+\infty$, we have $\lim_{n\rightarrow +\infty} \mathcal{E}(u_n)=+\infty$ by Lemma [Lemma 21](#Lem: E decomposition){reference-type="ref" reference="Lem: E decomposition"}. **(3):** Let $\alpha<0$ and $\{u_n\}_{n\in \mathbb{N}}$ be an unbounded sequence in $\mathcal{C}$. The definition of $\mathcal{C}$ in ($\ref{C}$) implies $\chi(S)<0$ and $\overline{\mathcal{R}}\leq0$ and $\overline{\mathcal{R}}\not\equiv0$. Since the sequence $\{u_n\}_{n\in \mathbb{N}}$ lies in $\mathcal{C}$, we have $$\label{Eq: key 3} \sum_{j\in V} \overline{\mathcal{R}}_j e^{\alpha (u_{j,n}-u_{i^*,n})}=e^{-\alpha u_{i^*,n}}\cdot\sum_{j\in V} \overline{\mathcal{R}}_j e^{\alpha u_{j,n}}\leq 2\pi\chi(S)e^{-\alpha u_{i^*,n}}<0.$$ If $\left\{\sum_{j\in V}(u_{j,n}-u_{i^*,n})\right\}_{n\in \mathbb{N}}$ converges, then the sequence $\{u_{j,n}-u_{i^*,n}\}_{n\in \mathbb{N}}$ converges for all $j\in V$. This implies that $\sum_{j\in V} \overline{\mathcal{R}}_j e^{\alpha (u_{j,n}-u_{i^*,n})}$ converges to a finite negative number by ([\[Eq: key 3\]](#Eq: key 3){reference-type="ref" reference="Eq: key 3"}). Since $\chi(S)<0$, then the equation ([\[Eq: key 3\]](#Eq: key 3){reference-type="ref" reference="Eq: key 3"}) implies $$-\alpha u_{i^*,n}\leq \ln \frac{\sum_{j\in V} \overline{\mathcal{R}}_j e^{\alpha (u_{j,n}-u_{i^*,n})}}{2\pi\chi(S)}.$$ By $\alpha<0$, we have $\{u_{i^*,n}\}_{n\in \mathbb{N}}$ is bounded from above. Combining this with $\{u_n\}_{n\in \mathbb{N}}$ is unbounded and $\{u_{j,n}-u_{i^*,n}\}_{n\in \mathbb{N}}$ converges for all $j\in V$, the sequence $\{u_{i^*,n}\}_{n\in \mathbb{N}}$ diverges properly to $-\infty$. Combining this with $\chi(S)<0$ and Lemma [Lemma 21](#Lem: E decomposition){reference-type="ref" reference="Lem: E decomposition"}, we have $\lim_{n\rightarrow +\infty} \mathcal{E}(u_n)=+\infty$. If $\left\{\sum_{j\in V}(u_{j,n}-u_{i^*,n})\right\}_{n\in \mathbb{N}}$ diverges properly to $+\infty$, then there exists at least one vertex $j\in V$ such that the sequence $\{u_{j,n}-u_{i^*,n}\}_{n\in \mathbb{N}}$ diverges properly to $+\infty$. By Lemma [Lemma 20](#Lem: one infty two converge){reference-type="ref" reference="Lem: one infty two converge"}, for any vertex $k\sim j$, the sequence $\{u_{k,n}-u_{i^*,n}\}_{n\in \mathbb{N}}$ converges. Since $\alpha<0$, then $e^{\alpha (u_{j,n}-u_{i^*,n})}$ converges to zero or a finite positive number and for at least one vertex $j\in V$ the term $e^{\alpha (u_{j,n}-u_{i^*,n})}$ converges to a finite positive number. Note that $\sum_{j\in V} \overline{\mathcal{R}}_j e^{\alpha (u_{j,n}-u_{i^*,n})}<0$ by ([\[Eq: key 3\]](#Eq: key 3){reference-type="ref" reference="Eq: key 3"}). Therefore, $\sum_{j\in V} \overline{\mathcal{R}}_j e^{\alpha (u_{j,n}-u_{i^*,n})}$ converges to zero or a finite negative number. $(i)$ : Suppose $\sum_{j\in V} \overline{\mathcal{R}}_j e^{\alpha (u_{j,n}-u_{i^*,n})}$ converges to zero. Then $2\pi\chi(S) e^{-\alpha u_{i^*,n}}$ converges to zero by ([\[Eq: key 3\]](#Eq: key 3){reference-type="ref" reference="Eq: key 3"}). Since $\alpha<0$ and $\chi(S)<0$, then $\{u_{i^*,n}\}_{n\in \mathbb{N}}$ diverges properly to $-\infty$. Combining this with $\chi(S)<0$ and Lemma [Lemma 21](#Lem: E decomposition){reference-type="ref" reference="Lem: E decomposition"}, we have $\lim_{n\rightarrow +\infty} \mathcal{E}(u_n)=+\infty$. $(ii)$ : Suppose $\sum_{j\in V} \overline{\mathcal{R}}_j e^{\alpha (u_{j,n}-u_{i^*,n})}$ converges to a finite negative number. Then $2\pi\chi(S) e^{-\alpha u_{i^*,n}}$ has a negative lower bound by ([\[Eq: key 3\]](#Eq: key 3){reference-type="ref" reference="Eq: key 3"}). By $\alpha<0$ and $\chi(S)<0$, then $u_{i^*,n}$ is bounded from above. Combining this with $\chi(S)<0$ and $\left\{\sum_{j\in V}(u_{j,n}-u_{i^*,n})\right\}_{n\in \mathbb{N}}$ diverges properly to $+\infty$, we have $\lim_{n\rightarrow +\infty} \mathcal{E}(u_n)=+\infty$ by Lemma [Lemma 21](#Lem: E decomposition){reference-type="ref" reference="Lem: E decomposition"}. 50 A. Bobenko, C. Lutz, *Decorated discrete conformal maps and convex polyhedral cusps*. [arXiv:2305.10988v1\[math.GT\]](https://arxiv.org/abs/2305.10988). A. Bobenko, U. Pinkall, B. Springborn, *Discrete conformal maps and ideal hyperbolic polyhedra*. Geom. Topol. 19 (2015), no. 4, 2155-2215. P. L. Bowers, K. Stephenson, *Uniformizing dessins and Belyĭ maps via circle packing*. Mem. Amer. Math. Soc. 170 (2004), no. 805. Y. Chen, Y. Luo, X. Xu, S. Zhang, *Bowers-Stephenson's conjecture on the convergence of inversive distance circle packings to the Riemann mapping*, [arXiv:2211.07464 \[math.MG\]](https://arxiv.org/abs/2211.07464). B. Chow, F. Luo, *Combinatorial Ricci flows on surfaces*, J. Differential Geom, Volume 63, No. 1 (2003), 97-129. H. S. M. Coxeter. *Inversive distance*. Annali di Matematica, 71(1):73-83, December 1966. H. Ge, W. Jiang, *On the deformation of inversive distance circle packings, III*. J. Funct. Anal. 272 (2017), no. 9, 3596-3609. H. Ge; X. Xu, *A discrete Ricci flow on surfaces with hyperbolic background geometry*, Int. Math. Res. Not. IMRN 2017, no. 11, 3510-3527. H. Ge, X. Xu, *On a combinatorial curvature for surfaces with inversive distance circle packing metrics*. J. Funct. Anal. 275 (2018), no. 3, 523-558. H. Ge, X. Xu, *A combinatorial Yamabe problem on two and three dimensional manifolds*, Calc. Var. Partial Differential Equations 60 (2021), no. 1, 20. D. 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arxiv_math

--- abstract: | In this paper we give an explicit solution of Dzherbashyan-Caputo-fractional Cauchy problems related to equations with derivatives of order $\nu k$, for $k$ non-negative integer and $\nu>0$. The solution is obtained by connecting the differential equation with the roots of the characteristic polynomial and it is expressed in terms of Mittag-Leffler-type functions. Under the some stricter hypothesis the solution can be expressed as a linear combination of Mittag-Leffler functions with common fractional order $\nu$. We establish a probabilistic relationship between the solutions of differential problems with order $\nu/m$ and $\nu$, for natural $m$. Finally, we use the described method to solve fractional differential equations arising in the fractionalization of partial differential equations related to the probability law of planar random motions with finite velocities. author: - | Fabrizio Cinque$^1$ and Enzo Orsingher$^2$\ Department of Statistical Sciences, Sapienza University of Rome, Italy\ $^1$fabrizio.cinque\@uniroma1.it, [ORCID: 0000-0002-9981-149X](https://orcid.org/0000-0002-9981-149X)\ $^2$enzo.orsingher\@uniroma1.it title: Analysis of fractional Cauchy problems with some probabilistic applications --- ***Keywords:*** Dzherbashyan-Caputo derivative, Mittag-Leffler functions, Fourier transforms, Laplace transforms, Random motions. *2020 MSC:* Primary 34A08; Secondary 35R11, 60K99. # Introduction In this paper we consider fractional equations of the form $$\label{problemaIntroduzione} \sum_{k=0}^N \lambda_k\frac{\partial^{\nu k} }{\partial t^{\nu k}}F(t, x) = 0,\ \ t\ge0,\ x\in \mathbb{R},\ \text{ with }\ \nu>0,$$ where the roots of $\sum_{k=0}^N\lambda_k y^k = 0$ are different from $0$, and subject to the general initial conditions $$\label{condizioniInizialiIntroduzione} \frac{\partial^{l} F}{\partial t^{l}}\Big|_{t=0} = f_l(x),\ \ x\in \mathbb{R}^d,\ l=0,\dots,\left\lceil N\nu \right\rceil -1.$$ The fractional derivatives are in the sense of Dzherbashyan-Caputo, that is, for $m\in \mathbb{N}_0$, $$\label{derivataCaputo} \frac{\mathop{}\!\mathrm{d}^\nu}{\mathop{}\!\mathrm{d}t^\nu} f(t) = \begin{cases} \begin{array}{l l} \displaystyle\frac{1}{\Gamma(m-\nu)}\int_0^t (t-s)^{m-\nu-1}\frac{\mathop{}\!\mathrm{d}^m}{\mathop{}\!\mathrm{d}s^m}f(s)\mathop{}\!\mathrm{d}s & \ \text{if}\ m-1<\nu<m\\[9pt] \displaystyle\frac{\mathop{}\!\mathrm{d}^m}{\mathop{}\!\mathrm{d}t^m} f(t) &\ \text{if} \ \nu = m. \end{array}\end{cases}$$ We recall that the Laplace-transform of the fractional derivative of order $\nu>0$ can be expressed as, for suitable $\mu>0$, $$\label{trasformataLaplaceDerivataFrazionariaIntroduzione} \int_0^\infty e^{-\mu t}\frac{\partial^\nu}{\partial t^\nu}f(t) \mathop{}\!\mathrm{d}t= \mu^{\nu} \int_0^\infty e^{-\mu t}f(t) \mathop{}\!\mathrm{d}t - \sum_{l=1}^{\left\lceil \nu \right\rceil} \mu^{\nu-l} \frac{\partial^{l-1}}{\partial t^{l-1}}f\Big|_{t=0}$$ where we assume that $\lim_{t\longrightarrow \infty} e^{-\mu t}\frac{\partial^{l-1}}{\partial t^{l-1}}f(t)=0,\ l\ge1$. Dzherbashyan-Caputo fractional derivatives and the associated Cauchy problems have been intensively studied by many authors in the last decades, see for instance [@CM2008; @KM2004; @P1994] and the more recent papers such as [@K2016; @M2019]. The main interest to such topic is arisen by their applications in several branches of science, such as physics and mechanics, see [@GKMR2014; @P1999]. Fractional derivatives and the study of its related Cauchy problems also appear in the theory of stochastic processes. The main novelty of this work lies in the probabilistic relationship we establish between the solution of fractional Cauchy problems of different order and its application to the study of the fractional version of random motion with finite velocity.\ Our research aim is that of extending the results firstly presented in Orsingher and Beghin [@OB2004], where the authors studied the time-fractional telegraph equation and the probabilistic interpretation of the solution. In particular, they were also able to prove that the probability law of the telegraph process subordinated with a reflecting Brownian motion satisfies the time-fractional differential equation $$%\label{} \frac{\partial^{2\nu}u}{\partial t^{2\nu}} +2\lambda \frac{\partial^\nu u}{\partial t^\nu} = c^2\frac{\partial^2u}{\partial x^2}, \ \ \ \text{with } \nu = \frac{1}{2},$$ subject to the initial condition $u(0,x) =\delta(x)$ and $u_t(0,x)=0, \ x\in \mathbb{R}$. Later, these kinds of relationships were extended in a series of papers, see [@DoOT2014; @OB2009]. In particular, in the paper by Orsingher and Toaldo [@OT2017] the authors studied the time-space-fractional equation $$\label{equazioneOrsingherToaldo} \sum_{j=1}^m \lambda_j \frac{\partial^{\nu_j}u}{\partial t^{\nu_j}} = -c^2(-\Delta)^\beta,\ \ \ 0<\nu_j\le 1,\ \forall\ j,\ \beta \in (0,1],$$ subject to the initial condition $u(0,x) = \delta(x),\ x\in \mathbb{R}^d$. In equation ([\[equazioneOrsingherToaldo\]](#equazioneOrsingherToaldo){reference-type="ref" reference="equazioneOrsingherToaldo"}), $-(-\Delta)^\beta$ denotes the fractional Laplacian (see [@K2017] for further details on this operator). The authors proved the relationship between this kind of equations and the probability law of an isotropic $d$-dimensional stable process, $S^{2\beta}$, subordinated with the inverse of a linear combination of independent stable processes, $L(t) = \inf\{s\ge0\,:\,\sum_{j=1}^m \lambda_j^{1/\nu_j} H_{\nu_j}(s)\ge t\}$, with $\lambda_j>0\ \forall\ j$ and $H_{\nu_j}$ stable processes of order $\nu_j\in(0,1)$.\ The novelty here is that the order of the Dzherbashyan-Caputo fractional derivatives appearing in ([\[problemaIntroduzione\]](#problemaIntroduzione){reference-type="ref" reference="problemaIntroduzione"}) can be arbitrarily large, although of the form $\nu k$. We point out that we state our main results in terms of ordinary fractional differential equations. Then, we are using this result to study partial fractional differential equations by means of the Fourier-transform approach.\ In Section 3, thanks to the use of the Laplace transform method, we show that the solution of the fractional Cauchy problem given by ([\[problemaIntroduzione\]](#problemaIntroduzione){reference-type="ref" reference="problemaIntroduzione"}) and ([\[condizioniInizialiIntroduzione\]](#condizioniInizialiIntroduzione){reference-type="ref" reference="condizioniInizialiIntroduzione"}) can be expressed as a combination of Mittag-Leffler-type functions with order of fractionality equal to $\nu>0$. Then we connect the solutions of problems with different *order of fractionality* by means of a probability expectation such as, with $n\in\mathbb{N}$, $$\label{relazioneValoreAttesoIntroduzione} F_{\nu/n}(t,x) = \mathbb{E}\,F_{\nu}\Biggl(\,\prod_{j=1}^{n-1}G_{j}^{(n)}(t),\,x\Biggr)$$ where $F_{\nu/n}$ and $F_{\nu}$ are respectively the solution of a problem of degree $\nu/n$ and $\nu$ with suitable initial conditions and $G_{j}^{(n)}(t)$ are positive absolutely continuous random variables for each $t\ge0,\ j=1,\dots,n-1$ (see Section [2.2](#sottosezioneVariabiliAleatorieG){reference-type="ref" reference="sottosezioneVariabiliAleatorieG"} for details). The relationship ([\[relazioneValoreAttesoIntroduzione\]](#relazioneValoreAttesoIntroduzione){reference-type="ref" reference="relazioneValoreAttesoIntroduzione"}), where $F_{\nu/n}$ and $F_{\nu}$ are Fourier transforms of probability laws, leads to the equivalence (in terms of finite-dimensional distributions) of two processes, with the second one being time-changed through $\prod_{j=1}^{n-1}G_{j}^{(n)}(t)$.\ The problem we study in this paper was inspired by the fractionalization of the higher order partial differential equations governing the probability distribution of the position of random motion moving with a finite number of velocities. For instance, the fourth-order equation $$\label{equazioneQuartoOrdineIntro} \Bigl(\frac{\partial^2 }{\partial t^2}+2\lambda\frac{\partial}{\partial t} +\lambda^2\Bigr)\biggl(\frac{\partial^2}{\partial t^2} +2\lambda\frac{\partial}{\partial t} -c^2\Bigl(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}\Bigr)\biggr)p +c^4\frac{\partial^4 p}{\partial x^2\partial y^2} = 0,$$ which emerges in the analysis of a planar stochastic dynamics with orthogonal-symmetrically chosen directions (see [@CO2023] for more details). The Fourier transforms of the equation ([\[equazioneQuartoOrdineIntro\]](#equazioneQuartoOrdineIntro){reference-type="ref" reference="equazioneQuartoOrdineIntro"}) has the form $$\label{equazioneQuartoOrdineFourierIntro} \Bigl(\frac{\partial^2 }{\partial t^2}+2\lambda\frac{\partial}{\partial t} +\lambda^2\Bigr)\Bigl(\frac{\partial^2}{\partial t^2} +2\lambda\frac{\partial}{\partial t} +c^2(\alpha^2+\beta^2)\Bigr)F +c^4\alpha^2\beta^2F= 0,$$ and its fractional version, with $\nu>0$, is $$\begin{aligned} \label{equazioneQuartoOrdineFourierFrazionariaIntro} \frac{\partial^{4\nu}F}{\partial t^{4\nu}} + 4\lambda \frac{\partial^{3\nu}F}{\partial t^{3\nu}}+5\lambda^2\frac{\partial^{2\nu}F}{\partial t^{2\nu}}+ 2\lambda^3\frac{\partial^\nu F}{\partial t^\nu} + c^2(\alpha^2+\beta^2)\Bigl(\frac{\partial^2 }{\partial t^2}+2\lambda\frac{\partial}{\partial t} +\lambda^2\Bigr)F+ c^4\alpha^2\beta^2 F= 0.\end{aligned}$$ Equation ([\[equazioneQuartoOrdineFourierFrazionariaIntro\]](#equazioneQuartoOrdineFourierFrazionariaIntro){reference-type="ref" reference="equazioneQuartoOrdineFourierFrazionariaIntro"}) equivalently arises by considering the Fourier transform of the time-fractional version of equation ([\[equazioneQuartoOrdineIntro\]](#equazioneQuartoOrdineIntro){reference-type="ref" reference="equazioneQuartoOrdineIntro"}).\ In the last section of the paper we describe some applications of the theory constructed in Section 3 in the field of random motions with finite velocities. In detail, we study a method to derive the value of the (integer) derivatives of the Fourier transform (also called the characteristic function), in the time origin $t=0$, of the probability law of the position of the moving particle. Thanks to this result we can build the Cauchy problem solved by the characteristic function of a general motion and study its time-fractional counterpart. We provide two examples concerning planar random movements. # Preliminary concepts ## Convolutions of Mittag-Leffler-type functions The generalized Mittag-Leffler (GML) function, also known as three-parameter Mittag-Leffler fucntion, is a generalization of the exponential function. It has been first introduced by Prabhakar [@P1971] and is defined as $$\label{MittagLefflerGeneralizzata} E_{\nu, \delta}^{\gamma} (x) = \sum_{k=0}^\infty \frac{\Gamma(\gamma+k)}{\Gamma(\gamma)\,k!}\frac{x^k}{ \Gamma(\nu k+ \delta)}, \ \ \ \ \ \nu, \gamma,\delta \in \mathbb{C}, Re(\nu), Re(\gamma), Re(\delta) >0, \ x\in \mathbb{R}.$$ By considering $\gamma = 1$, ([\[MittagLefflerGeneralizzata\]](#MittagLefflerGeneralizzata){reference-type="ref" reference="MittagLefflerGeneralizzata"}) reduces to the well-known Mittag-Leffler function, see Pillai [@P1990], Gorenflo *et al.* [@GKMR2014].\ In this paper, as in many others, we are representing the solutions of fractional differential Cauchy problems in terms of Mittag-Leffler-type functions. These applications naturally appear in the fractional calculus, see Mainardi [@M2020]. For our work it is useful to recall the Laplace transform of function ([\[MittagLefflerGeneralizzata\]](#MittagLefflerGeneralizzata){reference-type="ref" reference="MittagLefflerGeneralizzata"}), $$\label{inversaMittagLefferGeneralizzata} \int_0^\infty e^{-\mu x}x^{\delta-1}E_{\nu,\delta}^{\gamma}(\beta x^{\nu}) \mathop{}\!\mathrm{d}x = \frac{\mu^{\nu\gamma-\delta}}{(\mu^\nu-\beta)^\gamma}, \ \ \ \Big|\frac{\mu^\nu}{\beta}\Big|<1.$$ Let $M \in \mathbb{N}$. Below we use the following multivariate analogue of the generalized Mittag-Leffler $$\label{GenMittagLefflerMulti} E_{\nu,\delta}^{\gamma}(x) = \sum_{k_1,\dots,k_M=0}^\infty\,\prod_{j=1}^M\,\frac{\Gamma(\gamma_j+k_j)}{\Gamma(\gamma_j)\,k_j!}\,x_j^{k_j}\,\frac{1}{\Gamma\bigl(\nu\sum_{j=1}^Mk_j+\delta\bigr)},$$ where $\gamma = (\gamma_1,\dots,\gamma_M)\in \mathbb{C}^M,\ \nu,\delta \in \mathbb{C}$, with $Re(\gamma_1),\dots,Re(\gamma_M),Re(\nu)>0$, and $x\in \mathbb{C}^M$. Function ([\[GenMittagLefflerMulti\]](#GenMittagLefflerMulti){reference-type="ref" reference="GenMittagLefflerMulti"}) is a particular case of the multivariate Mittag-Leffler introduced by Saxena *et al.* [@SKRs2011] and used in Cinque [@C2022] to represent the distribution of the sum of independent generalized Mittag-Leffler random variables. **Lemma 1**. *Let $M \in \mathbb{N}$ and $t\ge0$. Also assume that $\gamma_1,\dots,\gamma_M\in \mathbb{C},\ \nu,\delta \in \mathbb{C}\setminus\{0\}$ such that $Re(\gamma_1),\dots,Re(\gamma_M),Re(\nu)>0$and $\eta_1\not=\dots\not = \eta_M\in\mathbb{C}$. Then, $$\Biggl(\mathop{\scalebox{2.3}{\raisebox{-0.2ex}{$\ast$}}}_{j=1}^M x^{\delta_j-1}E_{\nu,\delta_j}^{\gamma_j}(\eta_j x^\nu)\Biggr) (t) = t^{\sum_{j=1}^M \delta_j-1} E_{\nu,\sum_{j=1}^M \delta_j}^{(\gamma_1,\dots,\gamma_M)}\Big(\eta_1t^\nu,\dots,\eta_Mt^\nu\Big)$$ where the convolution is performed with respect to the variable $x\ge0$.* Note that the parameters $\delta_j$ appear just in terms of their summation $\sum_{j=1}^M\delta_j$, meaning that it does not matter how they are distributed among the factors of the convolution. *Proof.* It is sufficient to show that for $n\in \mathbb{N}$ and suitable $\nu,\delta_0,\delta,\gamma_0,\dots,\gamma_n, \eta_0,\dots,\eta_n$, $$\Biggl(x^{\delta_0-1}E_{\nu,\delta_0}^{\gamma_0}(\eta_0x^\nu) \ast E_{\nu,\delta}^{(\gamma_1,\dots,\gamma_n)} (\eta_1x^\nu,\dots,\eta_n x^\nu)\Biggr)(t)= t^{\delta_0+\delta-1}E_{\nu,\delta_0+\delta}^{(\gamma_0,\gamma_1,\dots,\gamma_n)} (\eta_0 t^\nu,\eta_1t^\nu,\dots,\eta_n t^\nu).$$ Indeed, $$\begin{aligned} \Biggl(&x^{\delta_0-1}E_{\nu,\delta_0}^{\gamma_0}(\eta_0x^\nu) \ast E_{\nu,\delta}^{(\gamma_1,\dots,\gamma_n)} (\eta_1x^\nu,\dots,\eta_n x^\nu)\Biggr)(t)\\ &=\sum_{k_0=0}^\infty \frac{\Gamma(\gamma_0+k_0)}{\Gamma(\gamma_0)\,k_0!} \eta_0^{k_0} \sum_{k_1,\dots,k_n = 0}^\infty\, \Biggl(\,\prod_{j=1}^n\,\frac{\Gamma(\gamma_j+k_j)}{\Gamma(\gamma_j)\,k_j!} \eta_j^{k_j}\Biggr) \int_0^t \frac{(t-x)^{\nu k_0+\delta_0-1}x^{\nu\sum_{j=1}^n k_j + \delta -1}}{\Gamma\bigl(\nu k_0+\delta_0\bigr)\Gamma\bigl(\nu\sum_{j=1}^n k_j+\delta\bigr)}\mathop{}\!\mathrm{d}x\\ & =\sum_{k_0=0}^\infty \frac{\Gamma(\gamma_0+k_0)}{\Gamma(\gamma_0)\,k_0!} \eta_0^{k_0} \sum_{k_1,\dots,k_n = 0}^\infty\, \Biggl(\,\prod_{j=1}^n\,\frac{\Gamma(\gamma_j+k_j)}{\Gamma(\gamma_j)\,k_j!} \eta_j^{k_j}\Biggr) \frac{t^{\nu\sum_{j=0}^n k_j+\delta_0+\delta-1}}{\Gamma\bigl(\nu\sum_{j=0}^n k_j+\delta_0+\delta\bigr)}\\ & = \sum_{k_0,\dots,k_n = 0}^\infty\, \Biggl(\,\prod_{j=0}^n\,\frac{\Gamma(\gamma_j+k_j)}{\Gamma(\gamma_j)\,k_j!} \Bigl(\eta_j t^\nu\Bigr)^{k_j}\Biggr) \frac{t^{\delta_0+\delta-1}}{\Gamma\bigl(\nu\sum_{j=0}^n k_j+\delta_0+\delta\bigr)}.\end{aligned}$$ ◻ For the convolution of $M$ two-parameters Mittag-Leffler functions we can derive an expression in terms of a linear combination of $M$ two-parameters Mittag-Leffler functions having all the same parameters. **Proposition 1**. *Let $M \in \mathbb{N}$ and $t\ge0$. Also assume that $\gamma_1,\dots,\gamma_M\in \mathbb{C},\ \nu,\delta \in \mathbb{C}\setminus\{0\}$ such that $Re(\gamma_1),\dots,Re(\gamma_M),Re(\nu)>0$ and $\eta_1\not=\dots\not = \eta_M\in\mathbb{C}$. Then, $$\label{convoluzioneMittagLeffler} \Biggl(\mathop{\scalebox{2.3}{\raisebox{-0.2ex}{$\ast$}}}_{i=1}^M x^{\delta_i-1}E_{\nu,\delta_i}(\eta_i x^\nu)\Biggr)(t) = t^{\sum_{h=1}^M \delta_h-1}\sum_{i=1}^M\frac{\eta_i^{M-1}}{\displaystyle\prod_{\substack{j=1\\j\not=i}}^M(\eta_i-\eta_j)} E_{\nu,\,\sum_{h=1}^M \delta_h}\bigl(\eta_it^\nu\bigr)$$ where the convolution is performed with respect to the non-negative variable $x\ge0$.* *Proof.* First we recall that for $n,M\in\mathbb{N}_0$ and $\eta_1\not=\dots\not=\eta_N\in \mathbb{C}\setminus\{0\}$, $$\label{formulaPerCasoMolteplicitaUnitarie} \sum_{i=1}^M \frac{\eta_i^{n}}{\prod_{\substack{j=1\\j\not=i}}^M(\eta_i-\eta_j)} =0, \ \ \text{with }\ n\le M-2.$$ Then, we also note that the right-hand side of formula ([\[convoluzioneMittagLeffler\]](#convoluzioneMittagLeffler){reference-type="ref" reference="convoluzioneMittagLeffler"}) can be also written as $$\label{convoluzioneMittagLefflerAlternativa} t^{\sum_{h=1}^M \delta_h-1}\sum_{i=1}^M\frac{\eta_i^{M-1}}{\prod_{\substack{j=1\\j\not=i}}^M(\eta_i-\eta_j)} E_{\nu,\,\sum_{h=1}^M \delta_h}\bigl(\eta_it^\nu\bigr) = \sum_{k=0}^\infty \frac{ t^{\nu k + \sum_{h=1}^M \delta_h -1}}{\Gamma\bigl(\nu k + \sum_{h=1}^M \delta_h\bigr)}\sum_{i=1}^M \frac{\eta_i^{k+M-1}}{\prod_{\substack{j=1\\j\not=i}}^M(\eta_i-\eta_j) }.$$ We now proceed by induction. The induction base (M=2) can be found in Orsingher and Beghin [@OB2004]. Now, assume that ([\[convoluzioneMittagLeffler\]](#convoluzioneMittagLeffler){reference-type="ref" reference="convoluzioneMittagLeffler"}) holds for $M-1$. $$\begin{aligned} \Biggl(&\mathop{\scalebox{2.3}{\raisebox{-0.2ex}{$\ast$}}}_{i=1}^M x^{\delta_i-1}E_{\nu,\delta_i}(\eta_i x^\nu)\Biggr)(t) \nonumber\\ & = \sum_{i=1}^{M-1} \frac{\eta_i^{M-2}}{\prod_{\substack{j=1\\j\not=i}}^{M-1}(\eta_i-\eta_j) } \int_0^t x^{\delta_M-1} E_{\nu, \delta_M}\bigl(\eta_Mx^\nu\bigr) (t-x)^{ \sum_{h=1}^{M-1} \delta_h-1}E_{\nu, \sum_{h=1}^{M-1} \delta_h}\Bigl(\eta_i(t-x)^\nu\Bigr)\mathop{}\!\mathrm{d}x\label{usoBaseInduttiva}\\ &= \sum_{i=1}^{M-1} \frac{\eta_i^{M-2}}{\prod_{\substack{j=1\\j\not=i}}^{M-1}(\eta_i-\eta_j) } \sum_{k=0}^\infty \frac{ t^{\nu k + \sum_{h=1}^M \delta_h -1}}{\Gamma\bigl(\nu k + \sum_{h=1}^M \delta_h\bigr)}\Bigl(\frac{\eta_i^{k+1}}{\eta_i-\eta_M}+\frac{\eta_M^{k+1}}{\eta_M-\eta_i}\Bigr)\nonumber\\ & = \sum_{i=1}^{M-1}\, \frac{\eta_i^{M-1}\, t^{\sum_{h=1}^M \delta_h-1}}{\prod_{\substack{j=1\\j\not=i}}^{M}(\eta_i-\eta_j)} \,E_{\nu,\,\sum_{h=1}^M \delta_h}\bigl(\eta_it^\nu\bigr)\nonumber\\ &\ \ \ - t^{\sum_{h=1}^M \delta_h-1}\, E_{\nu,\,\sum_{h=1}^M \delta_h}\bigl(\eta_Mt^\nu\bigr) \,\eta_M \,\sum_{i=1}^{M-1}\, \frac{\eta_i^{M-2}}{\prod_{\substack{j=1\\j\not=i}}^{M}(\eta_i-\eta_j)} \label{ultimoPassaggioConvoluzioneMittagLeffler}\\ & = \sum_{i=1}^{M-1}\, \frac{\eta_i^{M-1}\, t^{\sum_{h=1}^M \delta_h-1}}{\prod_{\substack{j=1\\j\not=i}}^{M}(\eta_i-\eta_j)} \,E_{\nu,\,\sum_{h=1}^M \delta_h}\bigl(\eta_it^\nu\bigr) + t^{\sum_{h=1}^M \delta_h-1}\, E_{\nu,\,\sum_{h=1}^M \delta_h}\bigl(\eta_Mt^\nu\bigr)\frac{\eta_M^{M-1}}{\prod_{\substack{j=1\\j\not=i}}^{M}(\eta_i-\eta_j)} .\nonumber\end{aligned}$$ where in step ([\[usoBaseInduttiva\]](#usoBaseInduttiva){reference-type="ref" reference="usoBaseInduttiva"}) we used the induction base (i.e. with $M=2$) written as in ([\[convoluzioneMittagLefflerAlternativa\]](#convoluzioneMittagLefflerAlternativa){reference-type="ref" reference="convoluzioneMittagLefflerAlternativa"}), and in step ([\[ultimoPassaggioConvoluzioneMittagLeffler\]](#ultimoPassaggioConvoluzioneMittagLeffler){reference-type="ref" reference="ultimoPassaggioConvoluzioneMittagLeffler"}) we suitably used formula ([\[formulaPerCasoMolteplicitaUnitarie\]](#formulaPerCasoMolteplicitaUnitarie){reference-type="ref" reference="formulaPerCasoMolteplicitaUnitarie"}). ◻ ## Generalization of absolute normal distribution {#sottosezioneVariabiliAleatorieG} In [@BO2003] the authors introduced the following absolutely continuous positively distributed random variables. Let $n\in \mathbb{N}$ and $y>0$, $$\label{leggeSingolaG} P\{G_j^{(n)}(t)\in \mathop{}\!\mathrm{d}y\} = \frac{y^{j-1}\mathop{}\!\mathrm{d}y}{n^{\frac{j}{n-1}-1}t^{\frac{j}{n(n-1)}}\Gamma(j/n)}e^{-\frac{y^n}{(n^nt)^{\frac{1}{n-1}}}}, \ \ \ t>0,\ j =1,\ \dots, n-1.$$ Note that in the case of $n=2$ we have only one element and $G^{(2)}_1(t) = |B(2t)|,\ t\ge0,$ with $B$ being a standard Brownian motion. If $G_1^{(n)}, \dots, G_{n-1}^{(n)}$ are independent, then the joint density reads, with $y_1,\dots,y_{n-1}>0$, $$P\Bigg\{\bigcap_{j=1}^{n-1} \big\{G_j^{(n)}(t)\in \mathop{}\!\mathrm{d}y_j\big\}\Bigg\} = \Bigl(\frac{n}{2\pi}\Bigr)^{\frac{n-1}{2}} \frac{1}{\sqrt{t}} \Biggl(\prod_{j=1}^{n-1} y_j^{j-1}\,\mathop{}\!\mathrm{d}y_j\Biggr)\,e^{-( n^nt)^{\frac{-1}{n-1}}\sum_{j=1}^{n-1}y_j^n}.$$ Let $t>0$ and $n\ge 2$. It is easy to derive that the Mellin-transform of distribution ([\[leggeSingolaG\]](#leggeSingolaG){reference-type="ref" reference="leggeSingolaG"}) reads, for $s>0$, $$\int_0^{\infty} y^{s-1} f_{G_j^{(n)} (t)}(y)\mathop{}\!\mathrm{d}y = \Bigl(nt^{1/n}\Bigr)^{\frac{s-1}{n-1}}\frac{\Gamma\bigl(\frac{s+j-1}{n}\bigr)}{\Gamma\bigl(\frac{j}{n}\bigr)}, \ \ \ j=1,\dots, n-1.$$ In the independence case, the Mellin-transform of the density, $f_{G^{(n)}(t)}$, of the product $G^{(n)} (t) = \prod_{j=1}^{n-1} G_{j}^{(n)}(t)$, is, with $s>0$, $$\int_0^{\infty} y^{s-1} f_{G^{(n)} (t)}(y)\mathop{}\!\mathrm{d}y =\prod_{j=1}^{n-1} \Bigl(nt^{1/n}\Bigr)^{\frac{s-1}{n-1}}\frac{\Gamma\bigl(\frac{s+j-1}{n}\bigr)}{\Gamma\bigl(\frac{j}{n}\bigr)} = \frac{t^{\frac{s-1}{n}}}{\Gamma\bigl(\frac{s-1}{n}+1\bigr)}\Gamma(s).$$ where in the last equality we used the following $n$-multiplication formula of Gamma function for $z = 1/n$ and $s/n$, $$\prod_{j=1}^{n-1}\Gamma\Bigl(z+\frac{j-1}{n}\Bigr)= \frac{(2\pi)^{\frac{n-1}{2}}n^{\frac{1}{2}-nz}\,\Gamma(nz)}{\Gamma\Bigl(z+\frac{n-1}{n}\Bigr)}.$$ # Fractional differential Cauchy problem In this section we derive an explicit formula for the solution to the fractional Cauchy problem given by ([\[problemaIntroduzione\]](#problemaIntroduzione){reference-type="ref" reference="problemaIntroduzione"}) and ([\[condizioniInizialiIntroduzione\]](#condizioniInizialiIntroduzione){reference-type="ref" reference="condizioniInizialiIntroduzione"}). Hereafter we are considering functions $f:[0,\infty)\times\mathbb{R}^d\longrightarrow \mathbb{R}$ such that $\lim_{t\longrightarrow \infty} e^{-\mu t}\frac{\partial^{l-1}}{\partial t^{l-1}}f(t)=0\ \forall\ l$. **Theorem 2**. *Let $d,N\in \mathbb{N},\ \nu>0$ and $\lambda_0,\dots,\lambda_N\in \mathbb{R}$. If $$\label{rappresentazionePolinomio} \sum_{k=0}^N \lambda_k x^k = \prod_{j=1}^M(x-\eta_j)^{m_j} \ \ \text{with }\ \eta_1,\dots,\eta_M\in \mathbb{C}\setminus \{0\},$$ then, the solution to the fractional Cauchy problem of parameter $\nu$ $$\label{problemaGenerale} \begin{cases} \displaystyle\sum_{k=0}^N \lambda_k\frac{\partial^{\nu k} }{\partial t^{\nu k}}F(t, x) = 0,\ \ t\ge0,\ x\in \mathbb{R}^d\\[15pt] \displaystyle\frac{\partial^{l} F}{\partial t^{l}}\Big|_{t=0} = f_l(x),\ \ x\in \mathbb{R}^d,\ l=0,\dots,\left\lceil \nu N \right\rceil -1, \end{cases}$$ is the function $F:[0,\infty)\times\mathbb{R}^d\longrightarrow \mathbb{R}$ given by $$\label{tesiGenerale} F(t,x) = \sum_{l=0}^{\left\lceil \nu N \right\rceil-1} f_{l}(x) \sum_{k=k_l}^N \lambda_k\, t^{\nu(N-k)+l} \, E_{\nu, \,\nu(N-k)+l+1}^{(m_1,\dots,m_M)}\Big(\eta_1t^\nu, \dots, \eta_Mt^\nu\Big),$$ with $k_l = \min\{k=1,\dots,N\,:\, \nu k> l\},\ l=0,\dots,\left\lceil \nu N \right\rceil-1$.* Note that $k_0=1$ and $l-1< \nu k \le l$ for all $k_{l-1}\le k< k_l$. Formula ([\[tesiGenerale\]](#tesiGenerale){reference-type="ref" reference="tesiGenerale"}) can be also written inverting the sums into $\sum_{k=1}^N \sum_{l=0}^{\left\lceil \nu k \right\rceil-1}$. Condition ([\[rappresentazionePolinomio\]](#rappresentazionePolinomio){reference-type="ref" reference="rappresentazionePolinomio"}) implies that $\eta_1,\dots,\eta_M$ are the $M$ roots of the $N$-th order polynomial with coefficients $\lambda_0,\dots,\lambda_N$, respectively with algebraic molteplicity $m_1,\dots,m_M\ge1$. In the case $M=N$, all the roots have algebraic molteplicity equal to 1 and the solution can be expressed in terms of a combination of Mittag-Leffler functions (see Theorem [Theorem 4](#teoremaMolteplicitaUnitarie){reference-type="ref" reference="teoremaMolteplicitaUnitarie"}). *Proof.* By means of the $t$-Laplace transform, the differential equation in problem ([\[problemaGenerale\]](#problemaGenerale){reference-type="ref" reference="problemaGenerale"}) turns into, for $\mu\ge0$ (we use the notation $G(\mu,x) =\mathcal{L}(F)(\mu,x)=\int_0^\infty e^{-\mu t}F(t,x)\mathop{}\!\mathrm{d}t$ and keep in mind formula ([\[trasformataLaplaceDerivataFrazionariaIntroduzione\]](#trasformataLaplaceDerivataFrazionariaIntroduzione){reference-type="ref" reference="trasformataLaplaceDerivataFrazionariaIntroduzione"})) $$\begin{aligned} %(\mu, x) 0&= \mathcal{L}\Biggl(\sum_{k=0}^N \lambda_k\frac{\partial^{\nu k} }{\partial t^{\nu k}}F \Biggr) = \sum_{k=0}^N \lambda_k\,\mathcal{L}\Bigl(\frac{\partial^{\nu k} }{\partial t^{\nu k}}F\Bigr) =\lambda_0 G +\sum_{k=1}^{N} \lambda_k \Bigl[\mu^{\nu k}G-\sum_{l=1}^{\left\lceil \nu k \right\rceil}\mu^{\nu k -l} f_{l-1}\Bigr], \nonumber\end{aligned}$$ which gives $$\begin{aligned} \label{trasformataLaplaceGenerale} G(\mu,x) &= \frac{\displaystyle \sum_{k=1}^{N} \lambda_k \sum_{l=1}^{\left\lceil \nu k \right\rceil}\mu^{\nu k -l} f_{l-1}(x)}{\displaystyle\sum_{k=0}^N \lambda_k \mu^{\nu k}} = \frac{\displaystyle\sum_{l=1}^{\left\lceil \nu N \right\rceil} f_{l-1}(x) \sum_{k=k_{l-1}}^N \lambda_k\, \mu^{\nu k -l} }{\displaystyle\prod_{h=1}^M \bigl(\mu^{\nu}-\eta_h\bigr)^{m_h}},\end{aligned}$$ where we used hypothesis ([\[rappresentazionePolinomio\]](#rappresentazionePolinomio){reference-type="ref" reference="rappresentazionePolinomio"}) and $k_{l-1}$ is defined in the statement. We now compute the $\mu$-Laplace inverse of the functions $\mu^{\nu k - l}/ \prod_{h=1}^M \bigl(\mu^{\nu}-\eta_h\bigr)^{m_h}$, for $l=1,\dots,\left\lceil N\nu \right\rceil$ and $k = k_{l-1},\dots,N$, by properly applying formula ([\[inversaMittagLefferGeneralizzata\]](#inversaMittagLefferGeneralizzata){reference-type="ref" reference="inversaMittagLefferGeneralizzata"}). Let us consider $M_k = \min\{n\,:\, \sum_{h=1}^n m_h \ge k\}$, therefore $M_1 = 1$ because $m_1\ge1$ and $M_N = M$. Clearly, $\sum_{h=1}^{M_k} m_h \ge k> \sum_{h=1}^{M_{k}-1}m_h$ and $m_{M_k} \ge k -\sum_{h=1}^{M_{k}-1}m_h$; clearly $\sum_{h=1}^M m_h = N$. We can decompose $\nu k -l$ as follows (it is not the only way): $$\begin{aligned} \nu k - l&=\nu\Bigl(\sum_{h=1}^{M_k-1}m_h + k - \sum_{h=1}^{M_k-1}m_h\Bigr) - l\,\frac{\sum_{h=1}^M m_h}{N} \nonumber\\ &= \nu\sum_{h=1}^{M_k-1}m_h +\nu\Bigl( k - \sum_{h=1}^{M_k-1}m_h\pm \sum_{h=M_k}^{M}m_h\Bigr) - l\,\frac{\sum_{h=1}^M m_h}{N}\nonumber \\ &=\sum_{h=1}^{M_k-1}\Bigl(\nu m_h-l\frac{m_h}{N}\Bigr) + \Biggl[\nu m_{M_k}-\Bigl(\nu m_{M_k} -\nu k + \nu \sum_{h=1}^{M_k-1}m_h + l\frac{m_{M_k}}{N}\Bigr)\Biggr] \nonumber\\ &\ \ \ + \sum_{h=M_k+1}^{M}\Biggl[ \nu m_h -\Bigl(\nu m_h+l\frac{m_h}{N}\Bigr)\Biggr].\label{decomposizioneEsponente}\end{aligned}$$ In view of ([\[decomposizioneEsponente\]](#decomposizioneEsponente){reference-type="ref" reference="decomposizioneEsponente"}) we can write, by denoting with $\mathcal{L}^{-1}$ the inverse $\mu$-Laplace transform operator, for $l=1,\dots,\left\lceil N\nu \right\rceil$ and $k = k_{l-1},\dots,N$ $$\begin{aligned} \mathcal{L}^{-1}& \Biggl(\frac{\mu^{\nu k -l} }{\displaystyle\prod_{h=1}^M \bigl(\mu^{\nu}-\eta_h\bigr)^{m_h}}\Biggr)(t) \nonumber\\ &=\prod_{h=1}^{M_k-1} \mathcal{L}^{-1}\Biggl( \frac{\mu^{\nu m_h-l m_h/N}}{\bigl(\mu^{\nu}-\eta_h\bigr)^{m_h}}\Biggr)(t)\, \mathcal{L}^{-1}\Biggl( \frac{\mu^{\nu m_{M_k}- \bigl(\nu \sum_{h=1}^{M_k}m_h -\nu k + l m_{M_k}/N\bigr)} }{\bigl(\mu^{\nu}-\eta_h\bigr)^{m_{M_k}}}\Biggr)(t) \nonumber\\ & \ \ \ \times \prod_{h=M_k+1}^M \mathcal{L}^{-1}\Biggl( \frac{\mu^{- l m_h/N}}{\bigl(\mu^{\nu}-\eta_h\bigr)^{m_h}} \Biggr)(t)\label{passaggioAntitrasformazioneLaplace}\\ & = \mathop{\scalebox{2.3}{\raisebox{-0.2ex}{$\ast$}}}_{h=1}^{M_k-1} t^{l m_h/N -1} E_{\nu,\,l m_h/N}^{m_h}\bigl(\eta_ht^\nu\bigr)\ast t^{\nu \sum_{h=1}^{M_k}m_h -\nu k +l m_{M_k}/N-1} E_{\nu,\,\nu \sum_{h=1}^{M_k}m_h -\nu k + l m_{M_k}/N}^{m_{M_k}}\bigl(\eta_{M_k}t^\nu\bigr)\nonumber\\ &\ \ \ \ast\mathop{\scalebox{2.3}{\raisebox{-0.2ex}{$\ast$}}}_{h=1}^{M_k-1} t^{\nu m_h - l m_h/N -1} E_{\nu,\,\nu m_h+ lm_h/N}^{m_h}\bigl(\eta_ht^\nu\bigr)\nonumber\\%\label{ultimoPassaggioAntitrasformazioneLaplace}\\ & = t^{\nu (N-k)+l-1}E_{\nu,\,\nu (N-k)+l}^{(m_1,\dots,m_M)}\bigl(\eta_1t^\nu,\dots,\eta_Mt^\nu\bigr),\label{inversioneLaplaceGenerale}\end{aligned}$$ where in step ([\[passaggioAntitrasformazioneLaplace\]](#passaggioAntitrasformazioneLaplace){reference-type="ref" reference="passaggioAntitrasformazioneLaplace"}) we used ([\[inversaMittagLefferGeneralizzata\]](#inversaMittagLefferGeneralizzata){reference-type="ref" reference="inversaMittagLefferGeneralizzata"}) and in the last step we used Lemma [Lemma 1](#lemmaConvMitLefGen){reference-type="ref" reference="lemmaConvMitLefGen"}. Note that in step ([\[passaggioAntitrasformazioneLaplace\]](#passaggioAntitrasformazioneLaplace){reference-type="ref" reference="passaggioAntitrasformazioneLaplace"}) it is necessary to keep the "$\delta$" terms greater than $0$ (see ([\[inversaMittagLefferGeneralizzata\]](#inversaMittagLefferGeneralizzata){reference-type="ref" reference="inversaMittagLefferGeneralizzata"})) and this is the main reason of using the above decomposition of $\nu k -l$.\ By combining ([\[trasformataLaplaceGenerale\]](#trasformataLaplaceGenerale){reference-type="ref" reference="trasformataLaplaceGenerale"}) and ([\[inversioneLaplaceGenerale\]](#inversioneLaplaceGenerale){reference-type="ref" reference="inversioneLaplaceGenerale"}) we readily obtain result ([\[tesiGenerale\]](#tesiGenerale){reference-type="ref" reference="tesiGenerale"}) (after the change of variable $l' = l-1$). ◻ **Remark 1** (Non-homogeneous equation). Under the hypothesis of Theorem [Theorem 2](#teoremaGenerale){reference-type="ref" reference="teoremaGenerale"} we can easily study the Cauchy problem in the case of a non-homogeneous fractional equation. In details, for $g:\mathbb{R}\times\mathbb{R}^d\longrightarrow\mathbb{R}$, such that there exists the $t$-Laplace transform, the solution of $$\begin{cases} \displaystyle\sum_{k=0}^N \lambda_k\frac{\partial^{\nu k} }{\partial t^{\nu k}}F(t, x) = g(t,x),\ \ t\ge0,\ x\in \mathbb{R}^d\\[15pt] \displaystyle\frac{\partial^{l} F}{\partial t^{l}}\Big|_{t=0} = f_l(x),\ \ x\in \mathbb{R}^d,\ l=0,\dots,\left\lceil \nu N \right\rceil -1, \end{cases}$$ reads $$\begin{aligned} &F(t,x) \label{tesiGeneraleNonOmogenea}\\ & = \sum_{l=0}^{\left\lceil N\nu \right\rceil-1} f_{l}(x) \sum_{k=k_l}^N \lambda_k\, t^{\nu(N-k)+l} \, E_{\nu, \,\nu(N-k)+l+1}^{(m_1,\dots,m_M)}\big(\eta t^\nu\big)- \int_0^t g(t-y,x)\,y^{\nu N-1}E_{\nu, \nu N}^{(m_1,\dots,m_M)}\big(\eta y^\nu\big)\mathop{}\!\mathrm{d}y,\nonumber\end{aligned}$$ where $\eta = (\eta_1,\dots,\eta_M)$. The above results easily follows by observing that formula ([\[trasformataLaplaceGenerale\]](#trasformataLaplaceGenerale){reference-type="ref" reference="trasformataLaplaceGenerale"}) becomes $$\label{trasformataLaplaceGeneraleNonOmogenea} G(\mu,x) = \frac{\displaystyle\sum_{l=1}^{\left\lceil N\nu \right\rceil} f_{l-1}(x) \sum_{k=k_{l-1}}^N \lambda_k\, \mu^{\nu k -l} -\mathcal{L}(g)(\mu,x)}{\displaystyle\prod_{h=1}^M \bigl(\mu^{\nu}-\eta_h\bigr)^{m_h}}$$ and we observe that the $\mu$-Laplace inverse of the term concerning the function $g$ is $$\begin{aligned} \mathcal{L}^{-1}\Biggl(\mathcal{L}(g)(\mu,x)\Bigl(\prod_{h=1}^M \bigl(\mu^{\nu}-\eta_h\bigr)^{m_h}\Bigr)^{-1} \Biggr)(t,x) &= \int_0^t g(t-y,x)\,y^{\nu N-1}E_{\nu, \nu N}^{(m_1,\dots,m_M)}\big(\eta y^\nu\big)\mathop{}\!\mathrm{d}y\\ %&=g(x)\,t^{\nu N}E_{\nu, \nu N +1}^{(m_1,\dots,m_M)}\big(\eta t^\nu\big),\end{aligned}$$ where we used that $\mathcal{L}^{-1}\Bigl(\Bigl(\prod_{h=1}^M \bigl(\mu^{\nu}-\eta_h\bigr)^{m_h}\Bigr)^{-1} \Bigr)(t,x) = t^{\nu N-1}E_{\nu, \nu N }^{(m_1,\dots,m_M)}\big(\eta t^\nu\big)$ (obtained by proceeding as shown for ([\[inversioneLaplaceGenerale\]](#inversioneLaplaceGenerale){reference-type="ref" reference="inversioneLaplaceGenerale"})). Note that in the case of $g$ being constant with respect to the variable $t$, the last term of ([\[tesiGeneraleNonOmogenea\]](#tesiGeneraleNonOmogenea){reference-type="ref" reference="tesiGeneraleNonOmogenea"}) reads $-g(x)t^{\nu N}E_{\nu, \nu N +1}^{(m_1,\dots,m_M)}\big(\eta t^\nu\big)$. **Remark 2**. Consider the real sequence $\{\nu_n\}_{n\in \mathbb{N}}$ such that $\nu_n\longrightarrow\nu>0$ and $\left\lceil \nu_n N \right\rceil=\left\lceil \nu N \right\rceil>0\ \forall\ n$. Then, $$\label{risultatoLimite} F_{\nu}(t,x) = \lim_{n\to\infty}F_{\nu_n}(t,x),\ \ t\ge0,\ x\in \mathbb{R}^d,$$ where $F_{\nu},F_{\nu_n}$ are respectively the solutions to the problem of parameter $\nu$ and $\nu_n\ \forall\ n$, with the same initial conditions. This means that we can connect the limit of the solutions (pointwise) to the "limit" of the Cauchy problems (where the initial conditions stay the same because $\left\lceil \nu_n N \right\rceil=\left\lceil \nu N \right\rceil\ \forall\ n$). Result ([\[risultatoLimite\]](#risultatoLimite){reference-type="ref" reference="risultatoLimite"}) comes from the continuity of the function ([\[GenMittagLefflerMulti\]](#GenMittagLefflerMulti){reference-type="ref" reference="GenMittagLefflerMulti"}) with respect to the fractional parameter $\nu>0$. This can be seen as a consequence of the continuity of the Gamma function on the real half-line and a suitable application of the dominated convergence theorem. **Theorem 3**. *Let $d,N,n\in \mathbb{N},\ \nu>0$. Let $\lambda_0,\dots,\lambda_N\in \mathbb{R}$ and $\eta_1,\dots,\eta_M\in \mathbb{C}\setminus \{0\}$ satisfying condition ([\[rappresentazionePolinomio\]](#rappresentazionePolinomio){reference-type="ref" reference="rappresentazionePolinomio"}). Then, the solution $F_{\nu/n}$ of the problem of parameter $\nu/n$ $$\label{problemaOrdineFrazione} \begin{cases} \displaystyle\sum_{k=0}^N \lambda_k\frac{\partial^{\nu k/n} }{\partial t^{\nu k/n}}F(t, x) = 0,\ \ t\ge0,\ x\in \mathbb{R},\\[15pt] \displaystyle\frac{\partial^{l} F}{\partial t^{l}}\Big|_{t=0} = f_l(x),\ \ x\in \mathbb{R}^d,\ l=0,\dots,\left\lceil \frac{N\nu}{n} \right\rceil -1, \end{cases}$$ can be expressed as $$\label{subordinazioneGenerale} F_{\nu/n}(t,x) = \mathbb{E}\,F_{\nu}\Biggl(\,\prod_{j=1}^{n-1}G_{j}^{(n)}(t),\,x\Biggr),$$ where the $G_j^{(n)}(t)$ are the random variables introduced in Section [2.2](#sottosezioneVariabiliAleatorieG){reference-type="ref" reference="sottosezioneVariabiliAleatorieG"} and $F_{\nu}$ is the solution to a problem of parameter $\nu$ with suitable initial condition $$\label{problemaOrdineInteroAssociato} \begin{cases} \displaystyle\sum_{k=0}^N \lambda_k\frac{\partial^{\nu k} }{\partial t^{\nu k}}F(t, x) = 0,\ \ t\ge0,\ x\in \mathbb{R}\\[15pt] \displaystyle\frac{\partial^{l} F}{\partial t^{l}}\Big|_{t=0} = \begin{cases} f_l,\ \ l=hn, \ \text{with } h =0,\dots,\left\lceil N\nu/n \right\rceil -1,\\ 0,\ \ otherwise. \end{cases} \end{cases}$$* Note that the conditions of the problem ([\[problemaOrdineFrazione\]](#problemaOrdineFrazione){reference-type="ref" reference="problemaOrdineFrazione"}) of degree $\nu/n$ appear in the associated problem ([\[problemaOrdineInteroAssociato\]](#problemaOrdineInteroAssociato){reference-type="ref" reference="problemaOrdineInteroAssociato"}) in the derivative whose order is multiple of $n$, while the other initial conditions are assumed equal to $0$. We also point out that all the conditions of the original problem always appear in the related problem since $n\Bigl(\left\lceil \nu N/n \right\rceil-1\Bigr)\le \left\lceil \nu N \right\rceil-1$. *Proof.* We begin by showing a possible way to express the multivariate Mittag-Leffler ([\[GenMittagLefflerMulti\]](#GenMittagLefflerMulti){reference-type="ref" reference="GenMittagLefflerMulti"}) of fractional order $\nu/n$ in terms of that of fractional order $\nu$. Remember that for the gamma function, with $z\in\mathbb{C}$ and $n\in\mathbb{N}$ we can write (thanks to the $n$-multiplication formula of the Gamma function) $$\begin{aligned} \Gamma\Bigl(z+\frac{n-1}{n}\Bigr)^{-1} &= \frac{\prod_{j=1}^{n-1}\Gamma\Bigl(z+\frac{j-1}{n}\Bigr)}{(2\pi)^{\frac{n-1}{2}}n^{\frac{1}{2}-nz}\Gamma(nz)} \nonumber\\ &= \frac{1}{(2\pi)^{\frac{n-1}{2}}n^{\frac{1}{2}-nz}\Gamma(nz)}\prod_{j=1}^{n-1} \int_0^\infty e^{-w_j} w_j^{z+\frac{j-1}{n}-1}\mathop{}\!\mathrm{d}w_j.\label{espressioneFunzioneGamma}\end{aligned}$$ Let $x\in\mathbb{C}^M$ and $L,h>0$, $$\begin{aligned} E_{\frac{\nu}{n},\, \frac{\nu}{n}L+ h}^{(m_1,\dots,m_M)}(x) &= \sum_{k_1,\dots,k_M=0} \Biggl( \prod_{j=1}^M \frac{\Gamma(m_j+k_j)}{\Gamma(m_j)\,k_j!}x_j^{k_j}\Biggr) \Gamma\Biggl(\frac{\nu}{n}\sum_{j=1}^M k_j+\frac{\nu}{n}L+h\Biggr)^{-1} \label{primoPassaggioSubordinazioneGenerale}\\ & = \sum_{k_1,\dots,k_M=0} \Biggl( \prod_{j=1}^M \frac{\Gamma(m_j+k_j)}{\Gamma(m_j)\,k_j!}x_j^{k_j}\Biggr) \frac{\Gamma\Bigl(\nu\sum_{j=1}^M k_j+\nu L+ n h - (n-1)\Bigr)^{-1}}{(2\pi)^{\frac{n-1}{2}}n^{\frac{1}{2}-\bigl(\nu\sum_{h=1}^M k_h+\nu L+n h -(n-1)\bigr)}}\nonumber\\ &\ \ \ \times\prod_{j=1}^{n-1} \int_0^\infty e^{-w_j} w_j^{\frac{1}{n}\bigl(\nu\sum_{h=1}^M k_h+\nu L +n h-n+j\bigr)-1}\mathop{}\!\mathrm{d}w_j \nonumber\\ & = \frac{n^{\nu L+n(h-1)+1/2}}{(2\pi)^{\frac{n-1}{2}}}\int_0^\infty \cdots\int_0^\infty \prod_{j=1}^{n-1} e^{-w_j} w_j^{\frac{1}{n}\bigl(\nu L+n h -n+j\bigr)-1}\mathop{}\!\mathrm{d}w_j \nonumber \\ & \ \ \ \times E_{\nu,\, \nu L+n(h-1)+1}^{(m_1,\dots,m_M)}\Bigl(x\,n^\nu \prod_{j=1}^{n-1} w_j^{\nu/n}\Bigr)\label{rappresentazioneMitLefGen}\end{aligned}$$ where in ([\[primoPassaggioSubordinazioneGenerale\]](#primoPassaggioSubordinazioneGenerale){reference-type="ref" reference="primoPassaggioSubordinazioneGenerale"}) we used ([\[espressioneFunzioneGamma\]](#espressioneFunzioneGamma){reference-type="ref" reference="espressioneFunzioneGamma"}) with $z = \frac{\nu}{n}\sum_{j=1}^M k_j+ h+\frac{\nu}{n}L -\frac{n-1}{n}$. Now we apply ([\[rappresentazioneMitLefGen\]](#rappresentazioneMitLefGen){reference-type="ref" reference="rappresentazioneMitLefGen"}) (with $h=l+1$ and $L= N-k$) to formula ([\[tesiGenerale\]](#tesiGenerale){reference-type="ref" reference="tesiGenerale"}) and derive result ([\[subordinazioneGenerale\]](#subordinazioneGenerale){reference-type="ref" reference="subordinazioneGenerale"}). Let us consider $\eta = (\eta_1,\dots,\eta_M)$ given in the hypotheses, then $$\begin{aligned} F_{\nu/n}(t,x)& = \sum_{l=0}^{\left\lceil \frac{\nu N}{n} \right\rceil-1} f_{l}(x) \sum_{k=k_{l}}^N \lambda_k\, t^{\frac{\nu}{n}(N-k)+l} \, E_{\frac{\nu}{n}, \,\frac{\nu}{n}(N-k)+l+1}^{(m_1,\dots,m_M)}\Big(\eta_1t^{\nu/n}, \dots, \eta_Mt^{\nu/n}\Big)\nonumber\\ & = \sum_{l=0}^{\left\lceil \frac{\nu N}{n} \right\rceil-1} f_{l}(x) \sum_{k=k_{l}}^N \lambda_k\, t^{\frac{\nu}{n}(N-k)+l}\,\frac{n^{\nu (N-k)+nl+1/2}}{(2\pi)^{\frac{n-1}{2}}}\int_0^\infty \cdots\int_0^\infty \Biggl(\prod_{j=1}^{n-1} \mathop{}\!\mathrm{d}w_j\Biggr) \nonumber \\ & \ \ \ \times \Biggl(\prod_{j=1}^{n-1} e^{-w_j} \Biggr) \Biggl(\prod_{j=1}^{n-1} w_j^{\frac{1}{n}\bigl(\nu (N-k)+n l -n+j\bigr)-1}\Biggr) E_{\nu,\, \nu (N-k)+nl+1}^{(m_1,\dots,m_M)}\Biggl(\eta\Bigl(nt^{1/n} \prod_{j=1}^{n-1} w_j^{1/n}\Bigr)^\nu\Biggr)\nonumber\\%\label{passaggioCambioDiVariabileGenerale}\\ & = \Bigl(\frac{n}{2\pi}\Bigr)^{\frac{n-1}{2}} \frac{1}{\sqrt{t}}\int_0^\infty \cdots\int_0^\infty \Biggl(\prod_{j=1}^{n-1} \mathop{}\!\mathrm{d}y_j\Biggr)\Biggl(\prod_{j=1}^{n-1} y_j^{j-1}\Biggr)\Biggl(\prod_{j=1}^{n-1} e^{-\frac{y_j^n}{( n^nt)^{\frac{1}{n-1}}}} \Biggr)\nonumber\\ &\ \ \ \times \sum_{l=0}^{\left\lceil \frac{\nu N}{n} \right\rceil-1} f_{l}(x) \sum_{k=k_{l}}^N \lambda_k \Biggl( \prod_{j=1}^{n-1} y_j\Biggr)^{\nu (N-k)+nl} E_{\nu,\, \nu (N-k)+nl+1}^{(m_1,\dots,m_M)}\Biggl(\eta \prod_{j=1}^{n-1} y_j^\nu\Biggr) \label{rappresentazioneEsplicitaSoluzioneGradoFrazione}\end{aligned}$$ where in the last step we used the change of variables $$nt^{1/n} \prod_{j=1}^{n-1} w_j^{1/n} = \prod_{j=1}^{n-1} y_j \iff w_j =\frac{y_j^n}{\bigl( n^nt\bigr)^{\frac{1}{n-1}}}, \ \forall\ j \implies \prod_{j=1}^{n-1} \mathop{}\!\mathrm{d}w_j = \frac{\prod_{j=1}^{n-1} \mathop{}\!\mathrm{d}y_j\, y_j^{n-1}}{nt}$$ and we performed some simplifications.\ At last, we show that the second line of ([\[rappresentazioneEsplicitaSoluzioneGradoFrazione\]](#rappresentazioneEsplicitaSoluzioneGradoFrazione){reference-type="ref" reference="rappresentazioneEsplicitaSoluzioneGradoFrazione"}) coincides with the time-changed solution $F_{\nu}\Bigl(\,\prod_{j=1}^{n-1}G_{j}^{(n)}(t),\,x\Bigr)$ of the associated problem ([\[problemaOrdineInteroAssociato\]](#problemaOrdineInteroAssociato){reference-type="ref" reference="problemaOrdineInteroAssociato"}). Let us denote with $\tilde{f}_l$ the function appearing in the $l$-th condition of problem ([\[problemaOrdineInteroAssociato\]](#problemaOrdineInteroAssociato){reference-type="ref" reference="problemaOrdineInteroAssociato"}) and with $\tilde{k}_l = \min\{k=1,\dots,N\,:\,\nu k> l\}$ for $l = 0,\dots, \left\lceil \nu N \right\rceil-1$. Then, the solution of the related Cauchy problem reads $$\label{FinteraParte1} F_\nu(s,x) = \sum_{l=0}^{\left\lceil \nu N \right\rceil-1} \tilde{f}_{l}(x) \sum_{k=\tilde{k}_{l}}^N \lambda_k\, s^{\nu(N-k)+l} \, E_{\nu, \,\nu(N-k)+l+1}^{(m_1,\dots,m_M)}\Big(\eta_1t^\nu, \dots, \eta_Ms^\nu\Big),$$ where the functions $\tilde{f}_{l}$ are identically null for $l \not = nh$ with $h = 0,\dots, \left\lceil \nu N/n \right\rceil-1$, therefore we can write (removing the indexes of the null terms and performing the change of variable $l = nh$) $$\label{FinteraParte2} F_\nu(s,x) = \sum_{h=0}^{\left\lceil \frac{\nu N}{n} \right\rceil-1} \tilde{f}_{nh}(x) \sum_{k=\tilde{k}_{nh}+1}^N \lambda_k\, s^{\nu(N-k)+nh} \, E_{\nu, \,\nu(N-k)+nh+1}^{(m_1,\dots,m_M)}\Big(\eta_1s^\nu, \dots, \eta_Ms^\nu\Big).$$ By observing that $\tilde{k}_{nh} = \min\{k=1,\dots,N: \nu k> nh\} = \min\{k=1,\dots,N: \nu k/n> h\} =k_{h} \ \forall \ h$, we obtain the last line of ([\[rappresentazioneEsplicitaSoluzioneGradoFrazione\]](#rappresentazioneEsplicitaSoluzioneGradoFrazione){reference-type="ref" reference="rappresentazioneEsplicitaSoluzioneGradoFrazione"}) by setting $s = \prod_{j=1}^{n-1}y_j$. ◻ **Remark 3** (Brownian subordination). If $n=2$, formula ([\[subordinazioneGenerale\]](#subordinazioneGenerale){reference-type="ref" reference="subordinazioneGenerale"}) becomes $$\label{subordinazioneGeneraleBrowniano} F_{\nu/2}(t,x) = \mathbb{E}\,F_{\nu}\Bigl(\,|B(2t)|,\,x\Bigr),$$ with $B$ standard Brownian motion (see Section [2.2](#sottosezioneVariabiliAleatorieG){reference-type="ref" reference="sottosezioneVariabiliAleatorieG"}).\ Furthermore, by keeping in mind ([\[subordinazioneGeneraleBrowniano\]](#subordinazioneGeneraleBrowniano){reference-type="ref" reference="subordinazioneGeneraleBrowniano"}) and iterating the same argument, we obtain that $$\label{subordinazioneGeneraleBrownianoIterato} F_{\nu/2^{n}}(t,x) = \mathbb{E}\,F_{\nu}\Bigl(\,|B_n(2|B_{n-1}(2|\cdots 2|B_1(2t)| \cdots |\,)\,|\,)\,|,\,x\Bigr),$$ where $B_1,\dots,B_n$ are independent standard Brownian motions and $F_\nu$ solution of the associated problem of the form ([\[problemaOrdineInteroAssociato\]](#problemaOrdineInteroAssociato){reference-type="ref" reference="problemaOrdineInteroAssociato"}) with $2^n$ replacing $n$. ## Algebraic multiplicities equal to 1 In this section we restrict ourselves to the case where the characteristic polynomial in ([\[rappresentazionePolinomio\]](#rappresentazionePolinomio){reference-type="ref" reference="rappresentazionePolinomio"}) has all distinct roots. This hypothesis permits us to present a more elegant result than that of Theorem [Theorem 2](#teoremaGenerale){reference-type="ref" reference="teoremaGenerale"}. **Theorem 4**. *Let $d,N\in \mathbb{N},\ \nu>0$ and $\lambda_0,\dots,\lambda_N\in \mathbb{R}$. If $$\label{rappresentazionePolinomioMolteplicitaUnitarie} \sum_{k=0}^N \lambda_k x^k = \prod_{j=1}^N(x-\eta_j) \ \ \text{with }\ \eta_1,\dots,\eta_N\in \mathbb{C}\setminus \{0\},$$ then, the solution to the fractional Cauchy problem $$\label{problemaDifferenziale} \begin{cases} \displaystyle\sum_{k=0}^N \lambda_k \frac{\partial^{\nu k} }{\partial t^{\nu k}}F(t, x) = 0,\ \ t\ge0,\ x\in \mathbb{R}\\[15pt] \displaystyle\frac{\partial^{l} F}{\partial t^{l}}\Big|_{t=0} = f_l(x),\ \ x\in \mathbb{R}^d,\ l=0,\dots,\left\lceil \nu N \right\rceil -1, \end{cases}$$ is the function $F:[0,\infty)\times\mathbb{R}^d\longrightarrow \mathbb{R}$ given by $$\label{soluzioneProblemaCasoMolteplicitaUnitarie} F(t,x) = \sum_{h=1}^N\sum_{l=0}^{\left\lceil \nu N \right\rceil-1} E_{\nu, l+1}\bigl(\eta_h t^\nu\bigr) f_{l}(x) \,t^{l} \sum_{k=k_{l}}^N \frac{\lambda_k \,\eta_h^{k-1}}{\displaystyle\prod_{\substack{j=1\\j\not=h}}^N(\eta_h-\eta_j)},$$ with $k_l = \min\{k=1,\dots,N\,:\, \nu k> l\},\ l=0,\dots,\left\lceil \nu N \right\rceil-1$.* Note that in result ([\[soluzioneProblemaCasoMolteplicitaUnitarie\]](#soluzioneProblemaCasoMolteplicitaUnitarie){reference-type="ref" reference="soluzioneProblemaCasoMolteplicitaUnitarie"}) the fractional order $\nu$ influences only the fractional order of the Mittag-Leffler function (and the number of initial conditions), so the coefficients of the linear combination are constant (with respect to $\nu$). We point out that the series in ([\[soluzioneProblemaCasoMolteplicitaUnitarie\]](#soluzioneProblemaCasoMolteplicitaUnitarie){reference-type="ref" reference="soluzioneProblemaCasoMolteplicitaUnitarie"}) can be inverted becoming $\sum_{k=1}^N \sum_{l=0}^{\left\lceil \nu k \right\rceil-1}$. *Proof.* First we note that, for $n\in \mathbb{N}_0$ and $l\in\mathbb{C}$, $$\label{rappresentazioneMitLefSemplificata} E_{\nu,\,n\nu+l}(x) = \frac{E_{\nu,l}(x)}{x^n}-\sum_{j=1}^n \frac{x^{-j}}{\Gamma\bigl((n-j)\nu + l\bigr)}.$$ Now, we proceed as in the proof of Theorem [Theorem 2](#teoremaGenerale){reference-type="ref" reference="teoremaGenerale"} and we perform the $t$-Laplace transform of the equation in problem ([\[problemaDifferenziale\]](#problemaDifferenziale){reference-type="ref" reference="problemaDifferenziale"}). In this case, formula ([\[trasformataLaplaceGenerale\]](#trasformataLaplaceGenerale){reference-type="ref" reference="trasformataLaplaceGenerale"}) reads $$\label{trasformataLaplaceGeneraleNonOmogenea} G(\mu,x) = \frac{\displaystyle\sum_{l=1}^{\left\lceil \nu N \right\rceil} f_{l-1}(x) \sum_{k=k_{l-1}}^N \lambda_k\, \mu^{\nu k -l}}{\displaystyle\prod_{h=1}^N \bigl(\mu^{\nu}-\eta_h\bigr)}.$$ We now invert the functions $\mu^{\nu k -l} /\prod_{h=1}^N \bigl(\mu^{\nu}-\eta_h\bigr)$ for $l=1,\dots, \left\lceil \nu N \right\rceil$ and $k = k_{l-1}+1, \dots, N$. We note that $$\nu k - l = \nu - l\frac{N-k+1}{N} + (k-1)\Bigl(\nu- \frac{l}{N}\Bigr)$$ and therefore we write $$\begin{aligned} \mathcal{L}^{-1}& \Biggl(\frac{\mu^{\nu k -l} }{\displaystyle\prod_{h=1}^N \bigl(\mu^{\nu}-\eta_h\bigr)}\Biggr)(t) \nonumber\\ &=\mathcal{L}^{-1}\Biggl( \frac{\mu^{\nu-l(N-k+1)/N} }{\mu^{\nu}-\eta_1}\Biggr)(t)\,\prod_{h=2}^{k} \mathcal{L}^{-1}\Biggl( \frac{\mu^{\nu -l/N}}{\mu^{\nu}-\eta_h}\Biggr)(t)\, \prod_{h=k+1}^N\mathcal{L}^{-1}\Bigl( \frac{1 }{\mu^{\nu}-\eta_h}\Bigr)(t) \nonumber\\ & = t^{l(N-k+1)/N-1}E_{\nu,l\frac{N-k+1}{N}}\bigl(\eta_1 t^\nu\bigr)\ast\mathop{\scalebox{2.3}{\raisebox{-0.2ex}{$\ast$}}}_{h=2}^{k} t^{l/N -1} E_{\nu,\,\frac{l }{N}}\bigl(\eta_ht^\nu\bigr) \ast\mathop{\scalebox{2.3}{\raisebox{-0.2ex}{$\ast$}}}_{h=k+1}^{N} t^{\nu-1}E_{\nu,\,\nu }\bigl(\eta_ht^\nu\bigr)\label{passaggioConvoluzioneMittagLeffler}\\ & = \sum_{h=1}^N \frac{t^{\nu(N-k)+l-1}\eta_h^{N-1}}{\prod_{\substack{j=1\\j\not=h}}^N (\eta_h-\eta_j)} E_{\nu, \nu(N-k) +l}\bigl(\eta_h t^\nu\bigr)\label{passaggioRappresentazioneMitLef}\\ & = \sum_{h=1}^N \frac{t^{l-1}\eta_h^{k-1}}{\prod_{\substack{j=1\\j\not=h}}^N (\eta_h-\eta_j)} E_{\nu, l}\bigl(\eta_h t^\nu\bigr) - \sum_{i=1}^{N-k}\frac{t^{\nu(N-k-i)+l-1}}{\Gamma\bigl(\nu(N-k-i)+l-1\bigr)} \sum_{h=1}^N \frac{\eta_h^{N-1-i}}{\prod_{\substack{j=1\\j\not=h}}^N (\eta_h-\eta_j)}\label{passaggioSemplificazioneEccezionale}\\ & = \sum_{h=1}^N \frac{t^{l-1}\eta_h^{k-1}}{\prod_{\substack{j=1\\j\not=h}}^N (\eta_h-\eta_j)} E_{\nu, l}\bigl(\eta_h t^\nu\bigr), \label{inversioneFattoreCasoMolteplicitaUnitarie}\end{aligned}$$ where in step ([\[passaggioConvoluzioneMittagLeffler\]](#passaggioConvoluzioneMittagLeffler){reference-type="ref" reference="passaggioConvoluzioneMittagLeffler"}) we used Proposition [Proposition 1](#proposizioneConvoluzioneMittagLeffler){reference-type="ref" reference="proposizioneConvoluzioneMittagLeffler"}, in step ([\[passaggioRappresentazioneMitLef\]](#passaggioRappresentazioneMitLef){reference-type="ref" reference="passaggioRappresentazioneMitLef"}) we used ([\[rappresentazioneMitLefSemplificata\]](#rappresentazioneMitLefSemplificata){reference-type="ref" reference="rappresentazioneMitLefSemplificata"}) and changed the order of the sums in the second term, and in step ([\[passaggioSemplificazioneEccezionale\]](#passaggioSemplificazioneEccezionale){reference-type="ref" reference="passaggioSemplificazioneEccezionale"}) we used formula ([\[formulaPerCasoMolteplicitaUnitarie\]](#formulaPerCasoMolteplicitaUnitarie){reference-type="ref" reference="formulaPerCasoMolteplicitaUnitarie"}) (note that $N-i-1\le N- 2$ for each $i = 1,\dots,N-k$). Finally, with formula ([\[inversioneFattoreCasoMolteplicitaUnitarie\]](#inversioneFattoreCasoMolteplicitaUnitarie){reference-type="ref" reference="inversioneFattoreCasoMolteplicitaUnitarie"}) at hand, the inversion of ([\[trasformataLaplaceGeneraleNonOmogenea\]](#trasformataLaplaceGeneraleNonOmogenea){reference-type="ref" reference="trasformataLaplaceGeneraleNonOmogenea"}) yields the claimed result ([\[soluzioneProblemaCasoMolteplicitaUnitarie\]](#soluzioneProblemaCasoMolteplicitaUnitarie){reference-type="ref" reference="soluzioneProblemaCasoMolteplicitaUnitarie"}) (after the change of variable $l'=l-1$). ◻ We observe that in the case where all the initial conditions are equal to null functions, except the first one, result ([\[soluzioneProblemaCasoMolteplicitaUnitarie\]](#soluzioneProblemaCasoMolteplicitaUnitarie){reference-type="ref" reference="soluzioneProblemaCasoMolteplicitaUnitarie"}) simplifies into $$\label{soluzioneProblemaCasoMolteplicitaUnitarieCondizioniNulle} F(t,x) = \sum_{h=1}^N E_{\nu, 1}\bigl(\eta_h t^\nu\bigr) f_{0}(x) \sum_{k=1}^N \frac{\lambda_k \,\eta_h^{k-1}}{\prod_{\substack{j=1\\j\not=h}}^N(\eta_h-\eta_j)}.$$ **Remark 4** (Integer derivatives). From Theorem [Theorem 4](#teoremaMolteplicitaUnitarie){reference-type="ref" reference="teoremaMolteplicitaUnitarie"}, by setting $\nu=1$ in ([\[problemaDifferenziale\]](#problemaDifferenziale){reference-type="ref" reference="problemaDifferenziale"}), we obtain the general solution to the integer order differential Cauchy problem. In particular, under the condition ([\[rappresentazionePolinomioMolteplicitaUnitarie\]](#rappresentazionePolinomioMolteplicitaUnitarie){reference-type="ref" reference="rappresentazionePolinomioMolteplicitaUnitarie"}), we can write $$\label{soluzioneProblemaInteroCasoMolteplicitaUnitarie} F(t,x) = \sum_{h=1}^N e^{\eta_h t}\sum_{l=0}^{N-1}f_{l}(x) t^l\sum_{k=l+1}^N \frac{\lambda_k \,\eta_h^{k-1-l}}{\displaystyle\prod_{\substack{j=1\\j\not=h}}^N(\eta_h-\eta_j)}.$$ Note that in this case $k_l=l+1\ \forall\ l$. Furthermore, for $l\ge1$, we can write $$\label{MittagLefflerEsponenziale} E_{1, l+1}(x) =\frac{1}{x^{l}}\Biggl(e^x- \sum_{i=0}^{l-1}\frac{x^i}{i!}\Biggr).$$ In light of ([\[MittagLefflerEsponenziale\]](#MittagLefflerEsponenziale){reference-type="ref" reference="MittagLefflerEsponenziale"}), formula ([\[soluzioneProblemaCasoMolteplicitaUnitarie\]](#soluzioneProblemaCasoMolteplicitaUnitarie){reference-type="ref" reference="soluzioneProblemaCasoMolteplicitaUnitarie"}) can be written as $$\begin{aligned} F(t,x) &= \sum_{h=1}^N\sum_{l=0}^{N-1} \Biggl(e^{\eta_h t^\nu}- \sum_{i=0}^{l-1}\frac{(\eta_h t)^i}{i!}\Biggr)\frac{f_{l}(x) t^l}{\eta_h^{l}} \sum_{k=k_l}^N \frac{\lambda_k \,\eta_h^{k-1}}{\displaystyle\prod_{\substack{j=1\\j\not=h}}^N(\eta_h-\eta_j)} \nonumber\\ & = \sum_{h=1}^N e^{\eta_h t^\nu}\sum_{l=0}^{N-1} f_{l}(x) t^l \sum_{k=l+1}^N \frac{\lambda_k \,\eta_h^{k-l-1}}{\displaystyle\prod_{\substack{j=1\\j\not=h}}^N(\eta_h-\eta_j)} - \sum_{l=0}^{N-1}f_{l}(x)\sum_{i=0}^{l-1}\frac{t^{i+l}}{i!}\sum_{k=l+1}^N \lambda_k \sum_{h=1}^N\frac{\eta_h^{k-l-1+i}}{\displaystyle\prod_{\substack{j=1\\j\not=h}}^N(\eta_h-\eta_j)}\nonumber\end{aligned}$$ and the last term is equal to $0$ because the last sum is always null thanks to formula ([\[formulaPerCasoMolteplicitaUnitarie\]](#formulaPerCasoMolteplicitaUnitarie){reference-type="ref" reference="formulaPerCasoMolteplicitaUnitarie"}) (in fact, $k-l-1+i\le k-l-1+(l-1)\le k-2\le N-2$ ). Finally, we observe that in the case of null initial conditions, except the first one, formula ([\[soluzioneProblemaInteroCasoMolteplicitaUnitarie\]](#soluzioneProblemaInteroCasoMolteplicitaUnitarie){reference-type="ref" reference="soluzioneProblemaInteroCasoMolteplicitaUnitarie"}) coincides with the solution ([\[soluzioneProblemaCasoMolteplicitaUnitarieCondizioniNulle\]](#soluzioneProblemaCasoMolteplicitaUnitarieCondizioniNulle){reference-type="ref" reference="soluzioneProblemaCasoMolteplicitaUnitarieCondizioniNulle"}) (where $\nu>0$) with the exponential function replacing the Mittag-Leffler function. **Remark 5**. We point out that the result in Theorem [Theorem 3](#teoremaSubordinazione){reference-type="ref" reference="teoremaSubordinazione"} can be directly proved also from formula ([\[soluzioneProblemaCasoMolteplicitaUnitarie\]](#soluzioneProblemaCasoMolteplicitaUnitarie){reference-type="ref" reference="soluzioneProblemaCasoMolteplicitaUnitarie"}). In particular, the case with $\nu/n = 1/n$ follows by suitably applying the following representation of the Mittag-Leffler function, with $h\in \mathbb{N}$, $$\begin{aligned} E_{1/n, h}(x) &= \sqrt{\frac{n}{(2\pi)^{n-1}}} \frac{1}{x^{n(h-1)}}\int_0^\infty \cdots \int_0^\infty \Biggl(\prod_{j=1}^{n-1}e^{-y_j} y_j^{j/n-1} \mathop{}\!\mathrm{d}y_j \Biggr) \Biggl( e^{nx\bigl(\prod_{j=1}^{n-1}y_j\bigr)^{1/n}} \\ &\ \ \ - \sum_{i=0}^{n(h-1)-1} \Bigl(nx\prod_{j=1}^{n-1}y_j^{1/n}\Bigr)^i\frac{1}{i!}\Biggr),\end{aligned}$$ which in the case of $n=2$, after the change of variable $y_1= y^2$, can be written as $$E_{1/2, h}(x) = \frac{2x^{2(1-h)}}{\sqrt{\pi}}\int_0^\infty e^{-y^2}\Biggl( e^{2xy} - \sum_{i=0}^{2h-3}\frac{ (2xy)^i}{i!}\Biggr) \mathop{}\!\mathrm{d}y.$$ The above formulas can be derived as formula (2.9) of [@BO2003]. # Application to random motions with finite velocity Let $\bigl(\Omega, \mathcal{F},\{\mathcal{F}_t\}_{t\ge0}, P\bigr)$ be a filtered probability space and $d \in \mathbb{N}$. In the following we assume that every random object is suitably defined on the above probability space (i.e. if we introduce a stochastic process, this is adapted to the given filtration). Let $N$ be a homogeneous Poisson process with rate $\lambda>0$ and consider the vectors $v_0,\dots, v_M\in\mathbb{R}^d$. Let $V$ be a stochastic process taking values in $\{v_0,\dots,v_M\}\ a.s.$ and such that, for $t\ge0$, $$p_k = P\{V(0)=v_k\},\ \ P\{V(t+\mathop{}\!\mathrm{d}t) = v_k\,|\,V(t) = v_h,\, N(t,t+\mathop{}\!\mathrm{d}t]=1\} = p_{hk},\ \ \ h,k = 0,\dots,M.$$ We say that $V$ is the process describing the velocity of the associated random motion $X$ defined as $$\label{definizioneMoto} X(t) = \int_0^t V(s)\mathop{}\!\mathrm{d}s = \sum_{i=0}^{N(t)-1} \bigl(T_{i+1} -T_i\bigr) V(T_i) + \bigl(t-T_{N(t)}\bigr) V(T_{N(t)}), \ \ t\ge0,$$ where $T_i$ denotes the $i$-th arrival time of $N$ and $V(T_i)$ denotes the random speed after the $i$-th event recorded by $N$, therefore after the potential switch occurring at time $T_i$. The stochastic process $X$ describes the position of a particle moving in a $d$-dimensional (real) space with velocities $v_0, \dots, v_M$ and which can change its current velocity only when the process $N$ records a new event. ## Initial conditions for the characteristic function Denote with $\gamma_{hk}^{(t)}:[0,1]\longrightarrow \mathbb{R}^d$ the segment between $v_h t$ and $v_k t$, that is $\gamma_{hk}^{(t)}(\delta) = v_ht\delta + v_kt(1-\delta),\ \delta\in [0,1]$. Now, it is easy to see that the distribution of the position $X(t)$, conditionally on the occurrence of one Poisson event in $[0,t]$ and the two different velocities taken at time $0$ (say $v_h$) and $t$ (say $v_k$), is uniformly distributed on the segment between the two velocities at time $t$ (that is $\gamma_{hk}^{(t)}$). In formulas, for $h\not=k=0,\dots,M$, $$P\{X(t)\in \mathop{}\!\mathrm{d}x\,|\,V(0) =v_h, V(t) = v_k, N[0,t]=1\} = \frac{\mathop{}\!\mathrm{d}x}{||v_h-v_k||t},\ \ \text{with }x\in \gamma_{hk}^{(t)}.$$ Then, for $t\ge0$ in the neighborhood of $0$ we observe that there can occur at maximum one Poisson event and therefore not more than one change of velocity for the motion $X$. Thus, we can write the Fourier transform of the distribution of $X(t)$, for $\alpha \in \mathbb{R}^d$ (with $<\cdot,\cdot>$ denoting the dot product in $\mathbb{R}^d$), $$\begin{aligned} \mathbb{E} e^{i\,<\alpha, X(t)>} &= \bigl(1-\lambda t\bigr)\sum_{k=0}^M p_k\, e^{i\,<\alpha,v_kt>}+ \lambda t\sum_{k=0}^M p_k \,p_{kk}e^{i\,<\alpha,v_kt>}\nonumber\\ &\ \ \ +\lambda t\sum_{\substack{h,k=0\\h\not=k}}^M p_h\, p_{hk} \int_{\gamma_{hk}^{(t)}}\frac{e^{i\,<\alpha,x>}}{||v_h-v_k||t}\mathop{}\!\mathrm{d}x\nonumber\\ & = \bigl(1-\lambda t\bigr)\sum_{k=0}^M p_k e^{it\,<\alpha,v_k>}+ \lambda t \sum_{h,k=0}^M p_h\, p_{hk}\int_0^1 e^{it\,<\alpha,\,v_h\delta + v_k(1-\delta)>}\mathop{}\!\mathrm{d}\delta.\label{trasformataFourierIntorno0}\end{aligned}$$ By means of ($\ref{trasformataFourierIntorno0})$ we easily derive the values of the derivatives of the Fourier transform of the distribution of the position $X(t)$ in the neighborhood of $0$ and therefore also in $t=0$, which will be used as initial conditions for the Cauchy problem. We point out that function ([\[trasformataFourierIntorno0\]](#trasformataFourierIntorno0){reference-type="ref" reference="trasformataFourierIntorno0"}) is based on the first order approximation of the probability mass of the Poisson process in the neighborhood of $t=0$. However, this approximation is sufficient to provide the characteristic function in the neighborhood of $0$; in fact, the probability law of random motions with finite velocities are derived by requiring only the knowledge at the first order, therefore we do not need a further expansion to obtain the higher order derivatives (in $t=0$).\ In detail we obtain, with $n\in \mathbb{N}_0, \ \alpha\in \mathbb{R}^d$, for $t$ sufficiently close to $0$, $$\begin{aligned} &\frac{\partial^{n} }{\partial t^{n}}\mathbb{E} e^{i\,<\alpha, X(t)>}\nonumber\\ &= \sum_{k=0}^M p_k e^{it<\alpha, v_k>} \Bigl[-n\lambda + (1-\lambda t)i<\alpha,v_k>\Bigr] \bigl(i<\alpha,v_k>\bigr)^{n-1}\nonumber\\ &\ \ \ +\lambda\sum_{h,k=0}^M p_h\, p_{hk}\int_0^1 e^{it\,<\alpha,\,v_h\delta + v_k(1-\delta)>}\Bigl[n+it<\alpha,v_h\delta + v_k(1-\delta)>\Bigr]\bigl(i<\alpha,\,v_h\delta + v_k(1-\delta)>\bigr)^{n-1}\mathop{}\!\mathrm{d}\delta,\end{aligned}$$ which in $t= 0$ simplifies into $$\begin{aligned} \frac{\partial^{n} }{\partial t^{n}}\mathbb{E} e^{i\,<\alpha, X(t)>}\Big|_{t=0}&= \sum_{k=0}^M p_k \Bigl[-n\lambda + i<\alpha,v_k>\Bigr] \bigl(i<\alpha,v_k>\bigr)^{n-1}\nonumber\\ &\ \ \ +n\lambda\sum_{h,k=0}^M p_h\, p_{hk}\int_0^1\bigl(i<\alpha,\,v_h\delta + v_k(1-\delta)>\bigr)^{n-1}\mathop{}\!\mathrm{d}\delta.\label{condizioneInizialeDerivataNesima}\end{aligned}$$ For derivatives of order $0,1,2$ we can write, with $\alpha\in \mathbb{R}^d$, $$\begin{aligned} \mathbb{E} e^{i\,<\alpha, X(0)>} =1 ,\ \ \ &\label{derivataZeroMotoAleatorio}\\ \frac{\partial }{\partial t}\mathbb{E} e^{i\,<\alpha, X(t)>}\Big|_{t=0}\ &= i<\alpha,\sum_{k=0}^Mp_k v_k>,\label{derivataPrimaMotoAleatorio}\\ \frac{\partial^2 }{\partial t^2}\mathbb{E} e^{i\,<\alpha, X(t)>}\Big|_{t=0} &\nonumber\label{derivataSecondaMotoAleatorio}\\ = -2\lambda i&<\alpha,\sum_{k=0}^Mp_k v_k>- \sum_{k=0}^M p_k <\alpha,v_k>^2 +\lambda i<\alpha,\sum_{h,k=0}^Mp_hp_{hk} (v_h + v_k)>.\end{aligned}$$ Formula ([\[derivataZeroMotoAleatorio\]](#derivataZeroMotoAleatorio){reference-type="ref" reference="derivataZeroMotoAleatorio"}) is due to the fact that the particle performing the random motion is always assumed to be in the origin of $\mathbb{R}^d$ at time $t=0$. It is interesting to observe that the first derivative, given in ([\[derivataPrimaMotoAleatorio\]](#derivataPrimaMotoAleatorio){reference-type="ref" reference="derivataPrimaMotoAleatorio"}), is equal to $0$ for all $\alpha\in\mathbb{R}^d$ if and only if $\sum_{k=0}^M p_k v_k=0$. **Example 1** (Orthogonal planar random motion). We consider a random motion $(X,Y)$ governed by a homogeneous Poisson process $N$ with rate $\lambda>0$, moving in the plane with the following orthogonal velocities, $$\label{direzioniMotoOrtogonale} v_k=\Biggl(c\cos\Bigl(\frac{k\pi}{2}\Bigr), c\sin\Bigl(\frac{k\pi}{2}\Bigr)\Biggr),\ \ c>0 \text{ with } k=0,1,2,3,$$ and such that from velocity $v_k$ the particle can uniformly switch either to $v_{k-1}$ or $v_{k+1}$, that is $P\{V(T_{n+1}) = v_{k+1} \,|\,V(T_n ) = v_k\} = P\{V(T_{n+1}) = v_{k-1} \,|\,V(T_n ) = v_k\} =1/2,\ k=0,1,2,3$. Therefore, the particle whose motion is described by $(X,Y)$ lies in the square $S_{ct} = \{(x,y)\in\mathbb{R}^2\,:\,|x|+|y|\le ct\}$ at time $t>0$ and at each Poisson event take a direction orthogonal to the current one (see Figure [\[MPS_1\]](#MPS_1){reference-type="ref" reference="MPS_1"}). We refer to [@CO2023] (and references herein) for further details on planar orthogonal random motions and [@CO2023b] for its three-dimensional version. The probability distribution $p(x,y)\mathop{}\!\mathrm{d}x \mathop{}\!\mathrm{d}y = P\{X(t)\in\mathop{}\!\mathrm{d}x, Y(t)\in\mathop{}\!\mathrm{d}y\},\ t\ge0,\ x,y \in Q_{ct},$ of the position of the motion $(X,Y)$ satisfies the fourth-order differential equation $$\label{equazioneQuartoOrdine} \Bigl(\frac{\partial^2 }{\partial t^2}+2\lambda\frac{\partial}{\partial t} +\lambda^2\Bigr)\biggl(\frac{\partial^2}{\partial t^2} +2\lambda\frac{\partial}{\partial t} -c^2\Bigl(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}\Bigr)\biggr)p +c^4\frac{\partial^4 p}{\partial x^2\partial y^2} = 0,$$ and it is known that the current position $\bigl(X(t),Y(t)\bigr)$ can be represented as a linear combination of two independent telegraph processes, In details, for $t\ge0$, $$\label{decomposizioneMotoOrtogonale} \begin{cases} X(t) = U(t) + V(t),\\ Y(t) = U(t) - V(t), \end{cases}$$ where $U=\{U(t)\}_{t\ge0}$ and $V=\{V(t)\}_{t\ge0}$ are independent one-dimensional telegraph processes moving with velocities $\pm c/2$ and with rate $\lambda/2$ (note that a similar results holds in the case of a non-homogeneous Poisson process as well, see [@CO2023]). The Fourier transforms of the equation ([\[equazioneQuartoOrdine\]](#equazioneQuartoOrdine){reference-type="ref" reference="equazioneQuartoOrdine"}) has the form $$\label{equazioneQuartoOrdineFourier} \Bigl(\frac{\partial^2 }{\partial t^2}+2\lambda\frac{\partial}{\partial t} +\lambda^2\Bigr)\Bigl(\frac{\partial^2}{\partial t^2} +2\lambda\frac{\partial}{\partial t} +c^2(\alpha^2+\beta^2)\Bigr)F +c^4\alpha^2\beta^2F= 0,$$ and by means of formulas ([\[derivataZeroMotoAleatorio\]](#derivataZeroMotoAleatorio){reference-type="ref" reference="derivataZeroMotoAleatorio"}), ([\[derivataPrimaMotoAleatorio\]](#derivataPrimaMotoAleatorio){reference-type="ref" reference="derivataPrimaMotoAleatorio"}), ([\[derivataSecondaMotoAleatorio\]](#derivataSecondaMotoAleatorio){reference-type="ref" reference="derivataSecondaMotoAleatorio"}) and ([\[condizioneInizialeDerivataNesima\]](#condizioneInizialeDerivataNesima){reference-type="ref" reference="condizioneInizialeDerivataNesima"}) the initial conditions are $$\label{condizioniMotoPianoDirezioniOrtogonali} F(0,\alpha,\beta) = 1,\ \ F_t(0,\alpha,\beta) =0,\ \ F_{tt}(0,\alpha,\beta) =-\frac{c^2}{2}\bigl(\alpha^2+\beta^2\bigr), \ \ F_{ttt}(0,\alpha,\beta) =\frac{\lambda c^2}{2}\bigl(\alpha^2+\beta^2\bigr).$$ Now, the fractional version of equation ([\[equazioneQuartoOrdineFourier\]](#equazioneQuartoOrdineFourier){reference-type="ref" reference="equazioneQuartoOrdineFourier"}), written in the form ([\[problemaDifferenziale\]](#problemaDifferenziale){reference-type="ref" reference="problemaDifferenziale"}), with $\nu>0$, is $$\begin{aligned} \frac{\partial^{4\nu}F}{\partial t^{4\nu}} + 4\lambda \frac{\partial^{3\nu}F}{\partial t^{3\nu}}+\Bigl(5\lambda^2+c^2(\alpha^2+\beta^2)\Bigr)\frac{\partial^{2\nu}F}{\partial t^{2\nu}}&+ 2\lambda\Bigl(\lambda^2+c^2(\alpha^2+\beta^2)\Bigr)\frac{\partial^\nu F}{\partial t^\nu}\nonumber\\ & + c^2\Bigl(\lambda^2(\alpha^2+\beta^2)+c^2\alpha^2\beta^2\Bigr) F= 0. \label{equazioneQuartoOrdineFourierFrazionaria}\end{aligned}$$ Let $A = \sqrt{\lambda^2-c^2(\alpha^2-\beta^2)}$ and $B = \sqrt{\lambda^2-c^2(\alpha^2+\beta^2)}$. Note that $c^2(\alpha^2+\beta^2) = \lambda^2-\bigl(A^2+B^2\bigr)/2$. Then, the following equality holds $$\begin{aligned} x^4&+ 4\lambda x^3+\Bigl(5\lambda^2+c^2(\alpha^2+\beta^2)\Bigr)x^2+ 2\lambda\Bigl(\lambda^2+c^2(\alpha^2+\beta^2)\Bigr)x + c^2\Bigl(\lambda^2(\alpha^2+\beta^2)+c^2\alpha^2\beta^2\Bigr) x\\ & = \prod_{k=1}^4 (x-\eta_k),\end{aligned}$$ with $$\label{formulaEta} \eta_1 = -\lambda-\frac{A+B}{2},\ \eta_2 = -\lambda+\frac{A-B}{2},\ \eta_2 = -\lambda-\frac{A-B}{2},\ \eta_4 = -\lambda+\frac{A+B}{2}.$$ With this at hand, by means of Theorem [Theorem 4](#teoremaMolteplicitaUnitarie){reference-type="ref" reference="teoremaMolteplicitaUnitarie"} it is easy to calculate the solution to a fractional Cauchy problem associated with equation ([\[equazioneQuartoOrdineFourierFrazionaria\]](#equazioneQuartoOrdineFourierFrazionaria){reference-type="ref" reference="equazioneQuartoOrdineFourierFrazionaria"}).\ For instance, in the case of initial conditions $F(0,\alpha,\beta) = 1$ and $\frac{\partial^l F}{\partial t^l}\big|_{t=0} = 0$ for all $l$ (whose values depend on $\nu$), the solution reads $$\begin{aligned} F_\nu&(t,\alpha, \beta)\\ & = \Biggl(\lambda^2-\Bigl(\frac{A-B}{2}\Bigr)^2\Biggr)\Bigl(\lambda -\frac{A+B}{2} \Bigr) E_{\nu, 1}(\eta_1 t^\nu) + \Biggl(\lambda^2-\Bigl(\frac{A+B}{2}\Bigr)^2\Biggr)\Bigl(\lambda +\frac{A-B}{2} \Bigr) E_{\nu, 1}(\eta_2 t^\nu) \\ & \ \ \ +\Biggl(\lambda^2-\Bigl(\frac{A+B}{2}\Bigr)^2\Biggr)\Bigl(\lambda -\frac{A-B}{2} \Bigr) E_{\nu, 1}(\eta_3 t^\nu) + \Biggl(\lambda^2-\Bigl(\frac{A-B}{2}\Bigr)^2\Biggr)\Bigl(\lambda +\frac{A+B}{2} \Bigr) E_{\nu, 1}(\eta_4 t^\nu) \end{aligned}$$ with $\eta_i$ given in ([\[formulaEta\]](#formulaEta){reference-type="ref" reference="formulaEta"}).\ In the case of initial conditions given by ([\[condizioniMotoPianoDirezioniOrtogonali\]](#condizioniMotoPianoDirezioniOrtogonali){reference-type="ref" reference="condizioniMotoPianoDirezioniOrtogonali"}) and $3/4<\nu\le1$ (so all the conditions are required), the solution reads $$\begin{aligned} F_\nu(t,\alpha, \beta) &= \frac{1}{4}\Biggl[ \Bigl(1 -\frac{\lambda}{A} \Bigr)\Bigl(1 -\frac{\lambda}{B} \Bigr) E_{\nu, 1}(\eta_1 t^\nu) +\Bigl(1 +\frac{\lambda}{A} \Bigr)\Bigl(1 -\frac{\lambda}{B} \Bigr) E_{\nu, 1}(\eta_2 t^\nu) \\ & \ \ \ +\Bigl(1 - \frac{\lambda}{A} \Bigr)\Bigl(1 + \frac{\lambda}{B} \Bigr) E_{\nu, 1}(\eta_3 t^\nu) +\Bigl(1 +\frac{\lambda}{A} \Bigr)\Bigl(1 +\frac{\lambda}{B} \Bigr) E_{\nu, 1}(\eta_4 t^\nu) \Biggr]. \end{aligned}$$ Note that for $\nu = 1$ this is the Fourier transform of the probability law of the orthogonal planar motion $(X,Y)$. This particular case can be also shown by considering the representation ([\[decomposizioneMotoOrtogonale\]](#decomposizioneMotoOrtogonale){reference-type="ref" reference="decomposizioneMotoOrtogonale"}) in terms of independent one-dimensional telegraph processes and their well-known Fourier transform (see for instance [@OB2004] formula (2.16)). **Example 2** (Planar motion with three directions). Let us consider a planar random motion $(X,Y)$ governed by a homogeneous Poisson process with rate $\lambda>0$ and moving with velocities $$\label{velocitaMotoPianoLO} v_0 = (c,0),v_1 = (-c/2,\sqrt{3}c/2),v_2 = (-c/2,-\sqrt{3}c/2), \ \ \text{ with } c>0.$$ Let us assume that the particle starts moving with a uniformly chosen velocity among the three possible choices in ([\[velocitaMotoPianoLO\]](#velocitaMotoPianoLO){reference-type="ref" reference="velocitaMotoPianoLO"}) and at each Poisson event it uniformly selects the next one (including also the current one). This kind of motion are sometimes called as *complete minimal planar random motion*, see [@CC2023; @LO2004] for further details.\ The support of the position of the stochastic dynamics at time $t\ge0$ is the triangle $T_{ct} =\{(x,y)\in\mathbb{R}^2\,:\,-ct/2\le x\le ct, (x-ct)/\sqrt{3}\le y\le (ct-x)/\sqrt{3} \}$ (see Figure [\[MP3\]](#MP3){reference-type="ref" reference="MP3"}). It is known that the probability distribution $p(x,y)\mathop{}\!\mathrm{d}x \mathop{}\!\mathrm{d}y = P\{X(t)\in\mathop{}\!\mathrm{d}x, Y(t)\in\mathop{}\!\mathrm{d}y\}$ of the position of the motion $(X,Y)$ satisfies the third-order differential equation $$\label{equazioneMotoPianoTreDirezioni} \Bigl(\frac{\partial }{\partial t} + \frac{3\lambda}{2}\Bigr)^3 p -\frac{27\lambda^3}{8}+ \frac{27\lambda^2}{16} \frac{\partial p }{\partial t} -\frac{3}{4}c^2\Delta\Bigl(\frac{\partial }{\partial t} + \frac{3\lambda}{2}\Bigr)p-\frac{3}{4}c^3\frac{\partial^3 p}{\partial x\partial y^2}+\frac{c^3}{4}\frac{\partial^3 p}{\partial x^3}= 0,$$ where $\Delta$ denotes the Laplacian operator. Note that equation ([\[equazioneMotoPianoTreDirezioni\]](#equazioneMotoPianoTreDirezioni){reference-type="ref" reference="equazioneMotoPianoTreDirezioni"}) can be derived by considering formula (1.9) of [@LO2004] and putting $3/2\lambda$ instead of $\lambda$ (this is sufficient by virtue of Remark 3.4 of [@CO2023]).\ The initial conditions of the Cauchy problem related to equation ([\[equazioneMotoPianoTreDirezioni\]](#equazioneMotoPianoTreDirezioni){reference-type="ref" reference="equazioneMotoPianoTreDirezioni"}) follow by suitably applying formulas ([\[derivataZeroMotoAleatorio\]](#derivataZeroMotoAleatorio){reference-type="ref" reference="derivataZeroMotoAleatorio"}), ([\[derivataPrimaMotoAleatorio\]](#derivataPrimaMotoAleatorio){reference-type="ref" reference="derivataPrimaMotoAleatorio"}) and ([\[derivataSecondaMotoAleatorio\]](#derivataSecondaMotoAleatorio){reference-type="ref" reference="derivataSecondaMotoAleatorio"}). In particular, for the Fourier transform of $p$, $F(t,\alpha,\beta) = \int_{\mathbb{R}^2} e^{i (\alpha x+ \beta y)}p(t,x,y)\mathop{}\!\mathrm{d}x\mathop{}\!\mathrm{d}y$, we derive $$\label{condizioniMotoPianoTreDirezioni} F(0,\alpha,\beta) = 1,\ \ F_t(0,\alpha,\beta) =0,\ \ F_{tt}(0,\alpha,\beta) =-\frac{c^2}{2}\bigl(\alpha^2+\beta^2\bigr),$$ obviously the first two conditions imply that $p(0,x,y) = \delta(x,y)$, with $\delta$ denoting the Dirac delta function centered in the origin, and $p_t(0,x,y) = 0\ \forall \ x,y$. We refer to [@LO2004] for further details about this motion, such as the explicit form of $p$ (see also [@CO2023b; @Dc2002]). Now, thanks to Theorem [Theorem 3](#teoremaSubordinazione){reference-type="ref" reference="teoremaSubordinazione"} we can easily give a probabilistic interpretation of the time-fractional version of order $\nu = 1/n, \ m\in\mathbb{N}$ of equation ([\[equazioneMotoPianoTreDirezioni\]](#equazioneMotoPianoTreDirezioni){reference-type="ref" reference="equazioneMotoPianoTreDirezioni"}), subject to the same initial conditions. Note that for $0<\nu\le1/3$ only the first condition is needed, for $1/3<\nu\le1/2$ the first two conditions are required and for $2/3<\nu\le1$ all three conditions are necessary.\ In details, the fractional Cauchy problem after the Fourier transformation is given by $$\label{problemaFourierMotoPianoTreDirezioni} \frac{\partial^3 F_\nu}{\partial t^3} +\frac{9\lambda}{2} \frac{\partial^2F_\nu }{\partial t^2} +\Biggl(\Bigl(\frac{3}{2}\Bigr)^4\lambda^2+\frac{3c^2(\alpha^2+\beta^2)}{4}\Biggr)\frac{\partial F_\nu}{\partial t} + \Bigl(\frac{9\lambda c^2(\alpha^2+\beta^2)}{8}+\frac{3ic^3\alpha\beta^2}{4}-\frac{ic^3\alpha^3}{4}\Bigr)F_\nu = 0$$ subject to the initial conditions in ([\[condizioniMotoPianoTreDirezioni\]](#condizioniMotoPianoTreDirezioni){reference-type="ref" reference="condizioniMotoPianoTreDirezioni"}). By means of Theorem [Theorem 3](#teoremaSubordinazione){reference-type="ref" reference="teoremaSubordinazione"} we have that the Fourier transform $F_{1/n}$ satisfying ([\[problemaFourierMotoPianoTreDirezioni\]](#problemaFourierMotoPianoTreDirezioni){reference-type="ref" reference="problemaFourierMotoPianoTreDirezioni"}) can be expressed as $F_{1/n}(t,x) = \mathbb{E}\,F\Biggl(\,\prod_{j=1}^{n-1}G_{j}^{(n)}(t),\,x\Biggr),$ where $F$ denotes the solution to the Fourier-transformed problem of equation ([\[equazioneMotoPianoTreDirezioni\]](#equazioneMotoPianoTreDirezioni){reference-type="ref" reference="equazioneMotoPianoTreDirezioni"}). Thus, the time-fractional version of ([\[equazioneMotoPianoTreDirezioni\]](#equazioneMotoPianoTreDirezioni){reference-type="ref" reference="equazioneMotoPianoTreDirezioni"}), with $\nu = 1/n$ for natural $n$, describes the probability law of a stochastic process $(X_\nu,Y_\nu)$ that is a time-changed planar motion with velocities ([\[velocitaMotoPianoLO\]](#velocitaMotoPianoLO){reference-type="ref" reference="velocitaMotoPianoLO"}), $$\bigl(X_\nu(t),Y_\nu(t)\bigr) \stackrel{d}{=} \Biggl( X\Bigl(\prod_{j=1}^{n-1}G_{j}^{(n)}(t)\Bigr), Y\Bigl(\prod_{j=1}^{n-1}G_{j}^{(n)}(t)\Bigr) \Biggr),\ \ \ t\ge0.$$ **Declarations**\ \ **Ethical Approval.** This declaration is not applicable.\ **Competing interests.** The authors have no competing interests to declare.\ **Authors' contributions.** Both authors equally contributed in the preparation and the writing of the paper.\ **Funding.** The authors received no funding.\ **Availability of data and materials.** This declaration is not applicable 00 Beghin, L., Orsingher, E. 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--- abstract: | Recently Krylov [@krylov1] established weak existence of solutions to SDEs for integrable drifts in mixed Lebesgue spaces, whose exponents satisfy the condition $1/q+d/p\leqslant 1$, thus going below the celebrated Ladyzhenskaya-Prodi-Serrin condition. We present here a variant of such result, whose proof relies on an alternative technique, based on a *partial* Zvonkin transform; this allows for drifts with growth at infinity and/or in uniformly local Lebesgue spaces.\ \ address: "Lucio Galeati, EPFL, Bâtiment MA, 1015 Lausanne, Switzerland Email: lucio.galeati\\@epfl.ch " author: - Lucio Galeati bibliography: - myBiblio.bib title: A note on weak existence for singular SDEs --- # Introduction Consider a multidimensional SDE on ${\mathbb R}^d$, $d\geqslant 2$, of the form $$\label{eq:intro-sde} \,\mathrm{d}X_t = b_t(X_t) \,\mathrm{d}t + \,\mathrm{d}W_t.$$ where $W$ is a standard Brownian motion. It is by now well established that, even when the drift $b$ is singular, the SDE [\[eq:intro-sde\]](#eq:intro-sde){reference-type="eqref" reference="eq:intro-sde"} may still admits strong, pathwise unique solutions, in a regularization by noise fashion. In particular, a major focus in the literature is devoted to integrable drifts satisfying the Ladyzhenskaya-Prodi-Serrin condition, namely[^1] $$\label{eq:LPS} b\in L^q_t L^p_x, \quad \frac{2}{q}+\frac{d}{p}\leqslant 1. \tag{LPS}$$ The importance of [\[eq:LPS\]](#eq:LPS){reference-type="eqref" reference="eq:LPS"} comes from its connection to advection-diffusion equations, in particular the solvability of $3$D Navier--Stokes equations, as well as the fact that it arises naturally from a scaling argument (see e.g. [@BFGM2019]), hence why it is regarded as a *critical class* of drifts for the solvability of [\[eq:intro-sde\]](#eq:intro-sde){reference-type="eqref" reference="eq:intro-sde"}. The celebrated work of Krylov and Röckner [@krylov2005] came close to [\[eq:LPS\]](#eq:LPS){reference-type="eqref" reference="eq:LPS"}, up to only allowing the strict inequality, but it took several additional years and efforts to understand the critical case, see [@BFGM2019; @krylov2021strong; @RocZha2] and the review [@kinzebulatov2023]. However recently Krylov [@krylov1] pointed out, elaborating on a previous result of Gyöngy and Martínez [@GyoMar2001], that in order to attain weak existence of solutions to [\[eq:intro-sde\]](#eq:intro-sde){reference-type="eqref" reference="eq:intro-sde"} it suffices to consider mixed Lebesgue spaces[^2] with exponents $p,q\in [1,\infty]$ satisfying $$\label{eq:krylov-exponents} \frac{1}{q}+\frac{d}{p} \leqslant 1.$$ He also showed that this condition is optimal, in the sense that for $(p,q)$ satisfying the opposite inequality one can find drifts for which weak existence fails. Finer properties of the Markov process $X$ constructed in this way have then been established in [@krylov2; @krylov3; @krylov4; @krylov5]. This note stems from an attempt to understand condition [\[eq:krylov-exponents\]](#eq:krylov-exponents){reference-type="eqref" reference="eq:krylov-exponents"} from a different perspective, introducing an heuristic which hopefully might be relevant in other settings. In order to explain it, it is useful to momentarily enlarge the class of problems and consider [\[eq:intro-sde\]](#eq:intro-sde){reference-type="eqref" reference="eq:intro-sde"} driven by a fractional Brownian motion $W$ of Hurst parameter $H\in (0,1)$. In this case, running the same scaling argument as in [@BFGM2019], it was predicted in [@GalGer2022 Section 1.1] that drifts $b\in L^q_t C^\alpha_x$ should be critical under the condition $$\label{eq:fbm-coefficients} \alpha = 1-\frac{1}{H q'}, \quad \frac{1}{q'}=1-\frac{1}{q},$$ although a complete rigorous proof of this claim is still missing. The scaling procedure consists in "zooming in" to look at the dynamics at short times; by self-similarity of the driving noise, this is equivalent (in law) to considering the same dynamics on $[0,1]$ but with rescaled drift $b^\lambda(t,x)=\lambda^{1-H} b(\lambda t, \lambda^H x)$. The critical class of drifts is then identified as the one invariant under this transformation, in the sense that $b$ and $b^\lambda$ have (roughly) the same norm; heuristically, the noise and the nonlinearity have "the same strength" and none is overtaking the other at small times. In this sense, the scaling itself doesn't directly predict any wellposedness or illposedness results, rather it informs us on which component is locally driving the dynamics; if this is the drift $b$ (namely we are in the supercritical regime $\alpha<1-1/(Hq')$), then we might expect the dynamics to display similar phenomena as in the absence of noise. This a priori doesn't exclude it from being well-defined, or existence of solutions to hold, which still depends on the drift $b$ in consideration; but it tells us that the noise $W$ shouldn't be too much of help. A different way to look at [\[eq:fbm-coefficients\]](#eq:fbm-coefficients){reference-type="eqref" reference="eq:fbm-coefficients"} is to regard it as an *interpolation class* between two extrema, given respectively by $b\in L^1_t C^1_x$ ($q=1$) and $b\in L^\infty_t C^{1-1/H}_x$ ($q=\infty$)[^3]. Note that the endpoint $L^1_t C^1_x$ is the standard Cauchy-Lipschitz class, for which wellposedness of [\[eq:intro-sde\]](#eq:intro-sde){reference-type="eqref" reference="eq:intro-sde"} holds regardless of the choice of the driving noise $W$; instead the second endpoint $L^\infty_t C^{1-1/H}_x$, with a uniform-in-time regularity condition, is the one dictated by the scaling of the noise. In this sense, if one is just interested in *weak existence* of solutions, rather than their wellposedness, it makes sense to modify the first endpoint with another classical ODE requirement, $b\in L^1_t C^0_x$, under which solutions can be constructed by Peano's theorem (again, this result being valid for any choice of $W$). Interpolating between these two endpoints, one obtains a new class of drifts, for which there is some hope to retain weak existence results. Observe that the range of exponents [\[eq:krylov-exponents\]](#eq:krylov-exponents){reference-type="eqref" reference="eq:krylov-exponents"} can be recovered by the same heuristics, interpolating between $L^1_t L^\infty_x$ for $q=1$ ("almost Peano") and the time-homogeneous LPS class $L^\infty_t L^d_x$. An analogue of [\[eq:krylov-exponents\]](#eq:krylov-exponents){reference-type="eqref" reference="eq:krylov-exponents"} in the fractional Brownian case is currently being obtained in [@ButGal]. The aim of this note is to show that, in the case of Brownian SDEs, this interpolation heuristic can be made rigorous, by employing a *partial Zvonkin transform*. More precisely, given a drift which decomposes as $b=b^1+b^2$, where $b^1$ is a "good drift" for weak existence results, while $b^2$ is a more singular component, we can find a transformation $\Phi$ of the state space (obtained by solving a parabolic PDE) which removes the latter. One then ends up with a new SDE for $Y=\Phi(X)$, driven by a drift $\tilde{b}$ which retains the properties of $b^1$ (e.g. local boundedness and linear growth); this allows to develop a priori estimates, which ultimately lead to existence by a compactness argument. Although Zvonkin transform is by now a well-established tool for solving singular SDEs (see e.g. [@XXZZ2020]), it is usually performed at the level of the whole drift $b$, without isolating its most singular part. In this direction, the only precursors in the literature we are aware of are [@XieZhang] (where $b^1$ instead plays the role of a *coercive* component) and partially [@ZhangYuan2021]. For the sake of simplicity, so far we considered SDEs with additive noise, but our result allows for the presence of a multiplicative diffusion $\sigma$, satisfying the conditions outlined below. In the nest statement, $\tilde{L}^p_x$ denote uniformly local Lebesgue spaces, see the notation section. *Assumption 1*. The drift $b:[0,T]\times {\mathbb R}^d\to {\mathbb R}^d$ is of the form $b=b^1+b^2$, where $$\label{eq:ass-drift} \frac{b^1}{1+|x|}\in L^{1+\varepsilon}_t L^\infty_x, \quad b^2\in L^\infty_t \tilde{L}^{d+\varepsilon}_x \quad \text{for some } \varepsilon\in (0,1).$$ The diffusion matrix $\sigma:[0,T]\times {\mathbb R}^d\to {\mathbb R}^{d\times d}$ is uniformly continuous in space, bounded and nondegenerate. Namely, there exist a constant $K>0$ such that $$\label{eq:ass-diffusion-1} K^{-1} |\xi|^2 \leqslant|\sigma^\ast (t,x) \xi|^2 \leqslant K|\xi|^2 \quad \forall\,\xi\in{\mathbb R}^d,\,(t,x)\in [0,T] \times {\mathbb R}^d.$$ and a modulus of continuity $\omega_\sigma$ such that $$\label{eq:ass-diffusion-2} |\sigma(t,x)-\sigma(t,y)|\leqslant\omega_\sigma(|x-y|) \quad \forall (t,x,y)\in [0,T]\times {\mathbb R}^{2d}.$$ To state our main result, we adopt the following solution concept for SDEs; ${\mathcal P}({\mathbb R}^d)$ denotes the set of probability measures on ${\mathbb R}^d$. *Definition 2*. Let $b:[0,T]\times {\mathbb R}^d\to{\mathbb R}^d$ and $\sigma:[0,T]\times {\mathbb R}^d\to {\mathbb R}^{d\times d}$ be measurable functions, $\mu_0\in {\mathcal P}({\mathbb R}^d)$. A *weak solution* to the SDE $$\label{eq:intro-sde-detailed} \,\mathrm{d}X_t = b_t(X_t) \,\mathrm{d}t + \sigma_t(X_t) \,\mathrm{d}W_t$$ with initial law $\mu_0$ is a tuple $(\Omega,{\mathcal F}, \{{\mathcal F}_t\}_{t\geqslant 0}, {\mathbb P}; X,W)$ given by a filtered probability space, a ${\mathcal F}_t$-adapted process $X$ and a ${\mathcal F}_t$-Brownian motion $W$ such that $$\begin{aligned} \int_0^T |b_s(X_s)| \,\mathrm{d}s + \int_0^T |\sigma_s(X_s)|^2 \,\mathrm{d}s <\infty \quad {\mathbb P}\text{-a.s.},\end{aligned}$$ $X_0$ is distributed as $\mu_0$ and ${\mathbb P}$-a.s. it holds $$\begin{aligned} X_t =X_0 + \int_0^t b_s(X_s) \,\mathrm{d}s + \int_0^t \sigma_s(X_s) \,\mathrm{d}W_s \quad \forall\, t\in [0,T].\end{aligned}$$ Let us set ${\mathcal P}_1({\mathbb R}^d):=\{\nu\in {\mathcal P}({\mathbb R}^d): \int_{{\mathbb R}^d} |x| \nu(\!\,\mathrm{d}x) <\infty \}$. **Theorem 3**. *Let $(b,\sigma)$ satisfy Assumption [Assumption 1](#ass:diffusion){reference-type="ref" reference="ass:diffusion"}; then for any initial distribution $\mu_0\in \mathcal{P}_1({\mathbb R}^d)$, there exists a weak solution $X$ to the SDE [\[eq:intro-sde-detailed\]](#eq:intro-sde-detailed){reference-type="eqref" reference="eq:intro-sde-detailed"}, with initial law $\mu_0$, in the sense of Definition [Definition 2](#defn:weak-solution){reference-type="ref" reference="defn:weak-solution"}.* Here are two relevant consequences of Theorem [Theorem 3](#thm:main-theorem){reference-type="ref" reference="thm:main-theorem"}. **Corollary 4**. *Let $b\in L^q_t \tilde{L}^p_x$ for parameters $(p,q)$ satisfying $$\label{eq:exponents-weaker} \frac{1}{q}+\frac{d}{p}<1$$ Then $b$ admits a decomposition satisfying [\[eq:ass-drift\]](#eq:ass-drift){reference-type="eqref" reference="eq:ass-drift"}, so that Theorem [Theorem 3](#thm:main-theorem){reference-type="ref" reference="thm:main-theorem"} applies.* To state the next corollary, we need to define weak solutions to Fokker-Planck equations. *Definition 5*. Given measurable $b$, $\sigma$, set $a:=\sigma\sigma^\ast$. We say that a flow of measures $t\mapsto \mu_t$ is a weak solution to the Fokker-Planck equation $$\label{eq:FP} \partial_t \mu + \mathord{{\rm div}}(b \mu) = \frac{1}{2} \sum_{i,j} \partial^2_{ij} (a_{ij} \mu)$$ if $t\mapsto \mu_t$ is continuous in the sense of distributions, $b\mu$ and $a_{ij}\mu$ are well defined distributions and for any $\varphi\in C^{\infty}_c((0,T)\times {\mathbb R}^d)$ it holds $$\begin{aligned} \int_{[0,T]} \int_{{\mathbb R}^d} (\partial_t \varphi + b_t\cdot\nabla \varphi + \frac{1}{2}\sum_{i,j} a_{ij} \partial^2_{ij} \varphi)(x) \mu_t(\!\,\mathrm{d}x) \,\mathrm{d}t = 0\end{aligned}$$ **Corollary 6**. *Let $b$, $\sigma$ satisfy Assumption [Assumption 1](#ass:diffusion){reference-type="ref" reference="ass:diffusion"}. Then for any $\mu_0\in {\mathcal P}_1$, there exists a weak solution $\mu$ to the Fokker-Planck equation [\[eq:FP\]](#eq:FP){reference-type="eqref" reference="eq:FP"} in the sense of Definition [Definition 5](#defn:FP){reference-type="ref" reference="defn:FP"}, with the properties that $t\mapsto \mu_t$ is continuous in the weak topology of measures and $\mu\vert_{t=0}=\mu_0$. Moreover $\mu\in L^{\tilde q}_t L^{\tilde p}_{x}$ for any $(\tilde p, \tilde q)$ satisfying $$\label{eq:integrability-density} \frac{1}{\tilde q} + \frac{d}{\tilde p} >d, \quad (\tilde p, \tilde q)\in (1,\infty)^2.$$ In particular, this ensures that $b \mu,\, a \mu \in L^1_t L^1_{loc}$.* Let us give some comments on Theorem [Theorem 3](#thm:main-theorem){reference-type="ref" reference="thm:main-theorem"}. *Remark 7*. Our result presents both advantages and drawbacks compared to the original one from [@krylov1]. On one hand, we can only allow a strict inequality in [\[eq:exponents-weaker\]](#eq:exponents-weaker){reference-type="eqref" reference="eq:exponents-weaker"}, as a consequence of the parameter $\varepsilon>0$ in [\[eq:ass-drift\]](#eq:ass-drift){reference-type="eqref" reference="eq:ass-drift"}; on the other, we can allow for drifts being either unbounded (at most of linear growth) or belonging to localised Lebesgue spaces $\tilde L^p_x$. Finally, contrary to [@krylov1], our condition $b\in L^q_t \tilde L^p_x$ doesn't change depending on whether $q\leqslant p$ or $p\leqslant q$, which makes it slightly more natural in analogy with [\[eq:LPS\]](#eq:LPS){reference-type="eqref" reference="eq:LPS"}. *Remark 8*. Both the result from [@krylov1] and Theorem [Theorem 3](#thm:main-theorem){reference-type="ref" reference="thm:main-theorem"} only establish weak existence of solutions. In fact, counterexamples to uniqueness in law in Besov spaces have been constructed in [@GalGer2022 Section 1.3]; for $d=1$, these include drifts $b\in L^q_t L^p_x$ satisfying $1/q+d/p>1$. In terms of scaling, this answers a problem left open in [@krylov1], although it should be mentioned that therein only $d\geqslant 2$ is considered. *Remark 9*. In light of Remark [Remark 8](#rem:nonuniqueness){reference-type="ref" reference="rem:nonuniqueness"}, it might seem that condition like [\[eq:krylov-exponents\]](#eq:krylov-exponents){reference-type="eqref" reference="eq:krylov-exponents"} is not so interesting; however it might have relevant applications for nonlinear PDEs. To illustrate this, consider the prototypical case of the $3$D Navier--Stokes equations (the same which motivated the interest in [\[eq:LPS\]](#eq:LPS){reference-type="eqref" reference="eq:LPS"}). Leray weak solution satisfy $u\in L^\infty_t L^2_x\cap L^2_t H^1_x$, which by Sobolev embeddings implies $$\label{eq:regularity-navier-stokes} u\in L^q_t L^p_x\quad \text{for} \quad \frac{2}{q}+\frac{3}{p} = \frac{3}{2} \quad \forall\, p\in [2,6]$$ which is considerably far from [\[eq:LPS\]](#eq:LPS){reference-type="eqref" reference="eq:LPS"}. However, by taking $q=2$, $p=6$, condition [\[eq:regularity-navier-stokes\]](#eq:regularity-navier-stokes){reference-type="eqref" reference="eq:regularity-navier-stokes"} interesects with [\[eq:krylov-exponents\]](#eq:krylov-exponents){reference-type="eqref" reference="eq:krylov-exponents"}, allowing to obtain a priori estimates for the associated SDE. It should be also mentioned that, exploiting the divergence free of $u$ and its Sobolev regularity, recently [@ZhaZha2021] and [@zhao2020] were able to construct weak solutions and prove uniqueness of the stochastic Lagrangian flow associated to $u$. In this sense, condition [\[eq:krylov-exponents\]](#eq:krylov-exponents){reference-type="eqref" reference="eq:krylov-exponents"} is just another small piece of the puzzle, hinting that [\[eq:LPS\]](#eq:LPS){reference-type="eqref" reference="eq:LPS"} might not be the end of the story for Navier-Stokes equations. *Remark 10*. We expect our strategy to work in other cases, for instance: i) $b$ of the form $b=b^1+\ldots+b^n$ with $b^i\in L^{q_i}_t \tilde L^{p_i}_x$ with $(q_i,p_i)$ satisfying [\[eq:exponents-weaker\]](#eq:exponents-weaker){reference-type="eqref" reference="eq:exponents-weaker"}; ii) coefficients belonging to mixed normed spaces, i.e. $b\in L^q_t L^{p_1}_{x_1} \ldots L^{p_d}_{x_d}$ with $1/q+\sum_i 1/p_i<1$, in analogy to what was obtained in [@ling2021strong] as a refinement of [@krylov2005]. Something more interesting would be to understand whether one can obtain novel existence and/or uniqueness results by interpolating other classes of drifts. For instance, one could consider $L^1_t L^\infty_x$ and $L^\infty_t C^\gamma_x$ with $\gamma>-1/2$, where for the latter weak existence and uniqueness of solutions was established in [@FlIsRu2017] again by Zvonkin transform. We leave this problem for future investigations. ## Structure of the paper {#structure-of-the-paper .unnumbered} We conclude this introduction by explaining the relevant notations and conventions. In Section [2](#sec:preliminaries){reference-type="ref" reference="sec:preliminaries"} we recall some analytic tools, most notably involving the resolution of parabolic PDEs, invoked throughout the paper. We develop all the relevant a priori estimates for our SDE in Section [3](#sec:apriori){reference-type="ref" reference="sec:apriori"}, by first considering smooth coefficients; then in Section [4](#sec:main-thm){reference-type="ref" reference="sec:main-thm"} we prove our results, by running a compactness argument and passing to the limit. ## Notations and conventions {#notations-and-conventions .unnumbered} We always work on a finite time interval $[0,T]$, although arbitrarily large. We write $a\lesssim b$ to mean that there exists a positive constant $C$ such that $a \leqslant C b$; we use the index $a\lesssim_\lambda b$ to highlight the dependence $C=C(\lambda)$. For any $m\in \mathbb{N}$ and $p\in [1,\infty]$, we denote by $L^p({\mathbb R}^d;{\mathbb R}^m)$ the standard Lebesgue space; when there is no risk of confusion in the parameter $m$, we will simply write $L^p_x$ for short and denote by $\| \cdot\|_{L^p_x}$ the corresponding norm. Similarly, we denote by $L^p_{loc}({\mathbb R}^d;{\mathbb R}^m)=L^p_{loc}$ local Lebesgue spaces, endowed with their natural Frechét topology; finally, we consider uniformly local Lebesgue spaces $\tilde L^p({\mathbb R}^d;{\mathbb R}^m)=\tilde L^p_x$ as defined by $$\begin{aligned} \tilde{L}^p_x:=\Big\{\varphi \in L^p_{loc}({\mathbb R}^d;{\mathbb R}^m): \| f \varphi\|_{\tilde L^p_x}:= \sup_{z\in{\mathbb R}^d} \| \chi^z \varphi \|_{L^p_x}<\infty\Big\};\end{aligned}$$ here $\chi^z:=\chi(\cdot - z)$, where $\chi$ is a smooth nonnegative function such that $\chi(x)= 1$ if $|x|\leqslant 1$ and $\xi(x)=0$ if $|x|\geqslant 2$. For $\alpha\in [0,+\infty)$, $C^\alpha (\mathbb{R}^d;{\mathbb R}^m)=C_x^\alpha$ stands for the usual Hölder continuous function space, made of continuous bounded functions with continuous and bounded derivatives up to order $\lfloor \alpha\rfloor\in\mathbb{N}$ and with globally $\{\alpha\}$-Hölder continuous derivatives of order $\lfloor \alpha\rfloor$. Given a Banach space $E$, we denote by $C([0,T];E)=C_t E$ the set of all continuous functions $\varphi:[0,T]\to E$, endowed with the supremum norm $\| \varphi\|_{C^0_t E}=\sup_{t\in [0,T]} \| \varphi_t\|_E$. Similarly for $\gamma\in (0,1)$ we define $C^\gamma([0,T];E)=C^\gamma_t E$ as the set of $\gamma$-Hölder continuous functions, with associated seminorm and norm $$\llbracket \varphi \rrbracket_{C^\gamma_t E}:=\sup_{s\neq t} \frac{\| \varphi_t-\varphi_s\|_E}{|t-s|^\alpha}, \quad \| \varphi \|_{C^\gamma_t E} := \| \varphi \|_{C^0_t E} + \llbracket \varphi \rrbracket_{C^\gamma_t E}.$$ Given a Frechét space $E$, with topology induced by a countable collection of seminorms $(d_j)_{j\in{\mathbb N}}$, and a parameter $q\in [1,\infty]$, we denote by $L^q(0,T;E)=L^q_t E$ the space of measurable functions $\varphi:[0,T]\to E$ such that $\int_0^T d_j( \varphi_t, 0)^q \,\mathrm{d}t <\infty$ for all $j\in{\mathbb N}$ (with the usual convention for $q=\infty$). Similarly, we say that $\varphi^n\to \varphi$ in $L^q_t E$ if $$\begin{aligned} \lim_{n\to\infty} \int_0^T d_j( \varphi^n_t, \varphi_t)^q \,\mathrm{d}t <\infty \quad \forall\, j\in {\mathbb N}.\end{aligned}$$ The above definitions can be concatenated by choosing different $E$, so that one can define $C^\gamma_t C^0_x$, $L^\infty_t C^1_x$, $L^q_t \tilde L^p_x$ and so on. Whenever $q=p$, we might write for simplicity $L^p_{t,x}$ in place of $L^p_t L^p_x$. When $E={\mathbb R}^d$, for simplicity we will drop it and just write $L^q_t$, $C^\gamma_t$, in place of $L^q_t {\mathbb R}^d$, $C^\gamma_t {\mathbb R}^d$. Whenever we are given a filtered probability space $(\Omega,\mathcal{F},\{{\mathcal F}_t\}_{t\geqslant 0},\mathbb{P})$, we will always assume the filtration $\{{\mathcal F}_t\}_{t\geqslant 0}$ to satisfy the standard assumptions. We denote by ${\mathbb E}$ expectation w.r.t. ${\mathbb P}$; if $X$ is a random variable define on $\Omega$, we denote by ${\mathcal L}(X)={\mathbb P}\circ X^{-1}$ its law under ${\mathbb P}$. # Analytic preliminaries {#sec:preliminaries} As mentioned above, a primary tool in our analysis is the so called Zvonkin transformation, which is related to solving a class of backward parabolic PDEs of the form $$\label{eq:zvonkin-pde} \partial_t u + \frac{1}{2}a:D^2 u + g\cdot\nabla u -\lambda u= -f, \quad u\vert_{t=T}=0.$$ Here we assume we are given $\sigma$ satisfying conditions [\[eq:ass-diffusion-1\]](#eq:ass-diffusion-1){reference-type="eqref" reference="eq:ass-diffusion-1"}-[\[eq:ass-diffusion-2\]](#eq:ass-diffusion-2){reference-type="eqref" reference="eq:ass-diffusion-2"} and we define the associated parameter set $\Theta:=(T,d,K,\omega_\sigma)$; we adopt the notations $a=\sigma \sigma^\ast$, $a:D^2 u = \sum_{i,j} a_{ij} \partial^2_{ij} u$ and $g\cdot \nabla u=\sum_{i}g_i\partial_i u$. If $u$ and $f$ are vector-valued, then [\[eq:zvonkin-pde\]](#eq:zvonkin-pde){reference-type="eqref" reference="eq:zvonkin-pde"} is understood componentwise. **Proposition 11**. *Let $\sigma$ satisfy [\[eq:ass-diffusion-1\]](#eq:ass-diffusion-1){reference-type="eqref" reference="eq:ass-diffusion-1"}-[\[eq:ass-diffusion-2\]](#eq:ass-diffusion-2){reference-type="eqref" reference="eq:ass-diffusion-2"}, $\varepsilon>0$ and $g\in L^\infty_t\tilde L^{d+\varepsilon}_x$. Then there exists $\lambda_0\geqslant 1$, depending on $\Theta$, $\varepsilon$ and $\| g \|_{L^{\infty}_t \tilde L^{d+\varepsilon}_x}$, such that for all $\lambda\geqslant\lambda_0$ and for all $f\in L^\infty_t \tilde L^{d+\varepsilon}_x$ there exists a unique strong solution $u$ to the PDE [\[eq:zvonkin-pde\]](#eq:zvonkin-pde){reference-type="eqref" reference="eq:zvonkin-pde"}. Furthermore there exist $\delta=\delta(\varepsilon)>0$ and $C=C(\Theta,\varepsilon,\| g \|_{L^{\infty}_t \tilde L^{d+\varepsilon}_x})$ such that $$\label{eq:zvonkin-estim} \lambda^\delta \| u\|_{C^0_t C^1_x} + \| u\|_{C^{1/2}_t C^0_x} \leqslant C \| f\|_{L^\infty_t \tilde L^{d+\varepsilon}_x}.$$* *Proof.* Although the result is classical in the case of constant diffusion and classical Lebesgue spaces, we haven't found a direct reference in our setting; we will derive it from [@XXZZ2020 Theorem 3.2], which however makes the proof a bit technical. By time reversal, we can reduce ourselves to the case of a forward parabolic equation with $u\vert_{t=0}=0$. By the hypothesis, we can find $q\in (1,\infty)$ large enough and $\alpha>1$ such that $2\delta:= 2-\alpha-2/q-d/(d+\varepsilon)>0$; applying [@XXZZ2020 Theorem 3.2] for such $\alpha$ and $q$, $p=d+\varepsilon$, $p'=q'=\infty$, we deduce the wellposedness of [\[eq:zvonkin-pde\]](#eq:zvonkin-pde){reference-type="eqref" reference="eq:zvonkin-pde"} as well as the estimate[^4] $$\begin{aligned} \lambda^\delta \| u\|_{\tilde{\mathbb{H}}^{\alpha,\infty}_\infty(T)} + \| \partial_t u\|_{\tilde{\mathbb{L}}^{d+\varepsilon}_q(T)} + \| u\|_{\tilde{\mathbb{H}}^{2,d+\varepsilon}_q(T)} \lesssim \| f\|_{\tilde {\mathbb{L}}^{d+\varepsilon}_q(T)} \lesssim \| f\|_{L^\infty_t \tilde L^{d+\varepsilon}_x}\end{aligned}$$ The estimate for $\| u\|_{C^0_t C^1_x}$ then follows from the embedding $\mathbb{H}^{\alpha,\infty}_\infty(T)\hookrightarrow L^\infty_t C^1_x$. The estimate for $\| u\|_{C^{1/2}_t C^0_x}$ instead follows by interpolation: for $\theta=1/2+1/q$ it holds $$\begin{aligned} \| u\|_{C^{1/2}_t C^0_x} \lesssim \| u\|_{C^{1/2}_t C^{1-2/q-d/(d+\varepsilon)}_x} \lesssim \| \partial_t u\|_{\tilde{\mathbb{L}}^{d+\varepsilon}_q(T)}^{\theta}\, \| u\|_{\tilde{\mathbb{H}}^{2,d+\varepsilon}_q(T)}^{1-\theta} \lesssim \| f\|_{L^\infty_t \tilde{L}^{d+\varepsilon}_x}.\qquad \qedhere\end{aligned}$$ ◻ Let $b=b^1+b^2$ and $\sigma$ as in Assumption [Assumption 1](#ass:diffusion){reference-type="ref" reference="ass:diffusion"}. By virtue of estimate [\[eq:zvonkin-estim\]](#eq:zvonkin-estim){reference-type="eqref" reference="eq:zvonkin-estim"}, we can find $\bar{\lambda}=\bar\lambda(\Theta,\varepsilon,\| b^2\|_{L^\infty_t \tilde{L}^{d+\varepsilon}_x})$ such that the vector-valued solution $u:=u^b$ to the PDE [\[eq:zvonkin-pde\]](#eq:zvonkin-pde){reference-type="eqref" reference="eq:zvonkin-pde"} associated to $f=g=b^2$ and $\bar\lambda$ satisfies $\| u\|_{C^0_t C^1_x} \leqslant 1/2$. Correspondingly, we define the *partial Zvonkin transform* associated to $b$ to be $\Phi_t(x):= x + u^b_t(x)$. **Lemma 12**. *For any $t\in [0,T]$, $\Phi_t$ is a diffeomorphism of ${\mathbb R}^d$ into itself and there exists a constant $C=C(\Theta,\varepsilon,\| b^2\|_{L^\infty_t \tilde{L}^{d+\varepsilon}_x})$ such that for all $x,\,y\in {\mathbb R}^d$ and $s,\,t\in [0,T]$ it holds $$\label{eq:properties-zvonkin} \frac{1}{2} |x-y|\leqslant|\Phi_t(x)-\Phi_t(y)| \leqslant 2 |x-y|, \quad |\Phi_t(x)-\Phi_s(x)|\leqslant C |t-s|^{1/2}.$$ Moreover the same estimate holds with $\Phi_t$ replaced by its inverse $\Phi^{-1}_t$.* *Proof.* The statement for $\Phi_t$ follows by its definition and the available estimates for $u$: it holds $\nabla\Phi_t(x)=I + \nabla u_t(x)$ with $|\nabla u_t(x)|\leqslant 1/2$, yielding the diffeomorphism property and the first estimate in [\[eq:properties-zvonkin\]](#eq:properties-zvonkin){reference-type="eqref" reference="eq:properties-zvonkin"}, while $|\Phi_t(x)-\Phi_s(x)|=|u^b_t(x)-u^b_s(x)|\leqslant|t-s|^{1/2} \| u\|_{C^{1/2}_t C^0_x}$. The bi-Lipschitz property for $\Phi^{-1}_t$ follows similarly; we are left with estimating the Hölder continuity of $t\mapsto \Phi^{-1}_t(x)$. It holds $$\begin{aligned} \sup_x |\Phi^{-1}_t(x)-\Phi^{-1}_s(x)| & = \sup_x | x - \Phi^{-1}_s(\Phi_t(x))| = \sup_x | \Phi^{-1}_s(\Phi_s(x)) - \Phi^{-1}_s(\Phi_t(x))|\\ & \leqslant 2 \sup_x | \Phi_s(x)-\Phi_t(x)| \lesssim |t-s|^{1/2} \| u^b\|_{C^{1/2}_t C^0_x}. \qquad \qedhere\end{aligned}$$ ◻ We conclude this section with a basic result, guaranteeing that any $f$ belonging in mixed Lebesgue spaces can be decomposed as in [\[eq:ass-drift\]](#eq:ass-drift){reference-type="eqref" reference="eq:ass-drift"}. **Lemma 13**. *Let $f\in L^q_t L^p_x$ for some $(q,p)\in [1,\infty]$ satisfying $1/q+d/p<1$. Then there exists $\varepsilon=\varepsilon(p,q)>0$ such that $f$ can be decomposed as $f=f^\leqslant+f^>$, where $$\begin{aligned} f^\leqslant\in L^{1+\varepsilon}_t L^\infty_x, \quad \| f^{\leqslant} \|_{L^{1+\varepsilon}_t L^\infty_x} \leqslant\| f\|_{L^q_t L^p_x}^{\frac{q}{1+\varepsilon}}, \quad f^>\in L^\infty_t L^{d+\varepsilon}_x, \quad \| f^>\|_{L^\infty_t L^{d+\varepsilon}_x} \leqslant 1.\end{aligned}$$ A similar statement holds with $L^p_x$ (resp. $L^{d+\varepsilon}_x$) replaced by $\tilde L^p_x$ (resp. $\tilde L^{d+\varepsilon}_x$).* *Proof.* For notational simplicity, we give the proof in the case $f\in L^q_t L^p_x$, the other case being identical up to keeping track of $\chi^z$ in all the computations. The result is a basic consequence of interpolation theory, but let us give an explicit choice of the decomposition. By the assumption, we can find $\varepsilon>0$ such that $$\label{eq:interpolation-coeff} \frac{1+\varepsilon}{q}+ \frac{d+\varepsilon}{p}=1.$$ For such choice, set $$\begin{aligned} f^\leqslant_t(x):= f_t(x) \mathbbm{1}_{|f_t(x)|\leqslant R_t}, \quad f^>_t(x):= f_t(x) \mathbbm{1}_{|f_t(x)|> R_t}, \quad R_t := \| f_t\|_{L^p_x}^{\frac{p}{p-d-\varepsilon}}.\end{aligned}$$ Then it holds $$\begin{aligned} \| f^>_t\|_{L^{d+\varepsilon}_x} \leqslant\Big( \int_{{\mathbb R}^d} R_t^{d+\varepsilon-p} |f_t(x)|^p \,\mathrm{d}x \Big)^{\frac{1}{d+\varepsilon}} \leqslant R_t^{\frac{d+\varepsilon-p}{d}} \| f_t\|_{L^p_x}^{\frac{p}{d+\varepsilon}}=1 \quad \forall\, t\in [0,T]\end{aligned}$$ while by virtue of [\[eq:interpolation-coeff\]](#eq:interpolation-coeff){reference-type="eqref" reference="eq:interpolation-coeff"} we have $$\begin{aligned} \int_0^T \| f^\leqslant_t\|_{L^\infty_x}^{1+\varepsilon} \,\mathrm{d}t \leqslant\int_0^T R_t^{1+\varepsilon} \,\mathrm{d}t = \int_0^T \| f_t\|_{L^p_x}^q \,\mathrm{d}t < \infty. \qquad \qedhere\end{aligned}$$ ◻ # A priori estimates {#sec:apriori} Throughout this section, we will assume that, in addition to Assumption [Assumption 1](#ass:diffusion){reference-type="ref" reference="ass:diffusion"}, $b$ and $\sigma$ are sufficiently regular; to fix the ideas, we will take $\sigma$ uniformly Lipschitz and $b\in L^1_t C^1_{\mathrm{loc}}$ such that $b/(1+|x|)\in L^1_t L^\infty_x$. In this case, strong existence and pathwise uniqueness of solutions to [\[eq:intro-sde-detailed\]](#eq:intro-sde-detailed){reference-type="eqref" reference="eq:intro-sde-detailed"} is classical; our goal is to devise a priori estimates which only rely on the norms and parameters appearing in Assumption [Assumption 1](#ass:diffusion){reference-type="ref" reference="ass:diffusion"}. Recall that we are also supplied with a random initial condition $X_0$ satisfying ${\mathbb E}[|X_0|]<\infty$ (corresponding to $\mu_0\in {\mathcal P}_1$). We divide our analysis in several steps. *Step 1: Partial Zvonkin transform*. Let $u^b$ be defined as in Section [2](#sec:preliminaries){reference-type="ref" reference="sec:preliminaries"} for suitably chosen $\bar\lambda$ and set $\Phi_t(x):=x + u_t^b(x)$. By Lemma [Lemma 12](#lem:properties-zvonkin){reference-type="ref" reference="lem:properties-zvonkin"}, $\Phi_t$ is a diffeomorphism from ${\mathbb R}^d$ to itself; moreover since $u^b$ solves [\[eq:zvonkin-pde\]](#eq:zvonkin-pde){reference-type="eqref" reference="eq:zvonkin-pde"} for $f=g=b^2$, by construction $\Phi$ solves the PDE $$\begin{aligned} \partial_t \Phi + \frac{1}{2}a:D^2 \Phi + b^2\cdot\nabla \Phi = \bar\lambda u, \quad \Phi_T(x)=x.\end{aligned}$$ Introducing the new variable $Y_t=\Phi_t(X_t)$, we deduce that $Y$ solves $$\begin{aligned} \,\mathrm{d}Y_t & = (\partial_t \Phi + \frac{1}{2}a:D^2\Phi + b\cdot\nabla \Phi)_t(X_t) \,\mathrm{d}t + \sigma_t(X_t) \nabla \Phi_t(X_t) \,\mathrm{d}W_t\\ & = (\bar\lambda u + b^1\cdot\nabla \Phi)_t(X_t) \,\mathrm{d}t + \sigma_t(X_t) \nabla \Phi_t(X_t) \,\mathrm{d}W_t\end{aligned}$$ so that $Y$ solves the SDE $\,\mathrm{d}Y = \tilde b(Y)\,\mathrm{d}t + \tilde\sigma(Y)\,\mathrm{d}W$ with new coefficients $$\begin{aligned} \tilde b := (\bar\lambda u + b^1\cdot\nabla \Phi)\circ \Phi^{-1}, \quad \tilde \sigma := (\sigma_t\nabla \Phi_t)\circ \Phi^{-1}.\end{aligned}$$ It follows from the smallness condition $\|u\|_{C^0_t C^1_x} \leqslant 1/2$ and property [\[eq:properties-zvonkin\]](#eq:properties-zvonkin){reference-type="eqref" reference="eq:properties-zvonkin"} (applied both for $\Phi_t$ and $\Phi_t^{-1}$) that $\tilde b$ is still of linear growth, and in particular $$\label{eq:estim-new-coefficients} \Big\| \frac{\tilde b_t}{1+|x|}\Big\|_{L^\infty_x} \leqslant\bar\lambda + 4\, \Big\| \frac{b_t}{1+|x|}\Big\|_{L^\infty_x} \quad \forall\, t\in [0,T], \quad \| \tilde{\sigma}\|_{L^\infty_{t,x}} \leqslant 2\, \| \sigma\|_{L^\infty_{t,x}}.$$ Let us set $h_t:= \bar\lambda + 4\, \| b_t/(1+|x|)\|_{L^\infty_x}$; by Assumption [\[eq:ass-drift\]](#eq:ass-drift){reference-type="ref" reference="eq:ass-drift"}, it holds $h\in L^{1+\varepsilon}_t$. *Step 2: A priori estimates for $Y$.* Set $Z_t:= \int_0^t \tilde\sigma_s(Y_s) \,\mathrm{d}W_s$, so that $Y$ satisfies $$\begin{aligned} Y_t = Y_0 + \int_0^t \tilde{b}_s(Y_s) \,\mathrm{d}s + Z_t;\end{aligned}$$ since $\tilde{b}_s(x)\leqslant h_s(1+|x|)$, we can apply Grönwall's lemma at a pathwise level to find $$\label{eq:pathwise-gronwall} \| Y(\omega)\|_{C^0_t} \leqslant e^{\| h \|_{L^1_t}} \Big( |Y_0(\omega)| + \sup_{t\in [0,T]} |Z_t(\omega)| \Big) \quad {\mathbb P}\text{-a.s.}$$ Furthermore by the properties of $\tilde{b}$ and Hölder's inequality, it holds $$\begin{aligned} |Y_t-Y_s| & \leqslant(1+\| Y\|_{C^0_t}) \int_s^t h_r \,\mathrm{d}r + |Z_t-Z_s|\\ & \lesssim |t-s|^{\frac{\varepsilon}{1+\varepsilon}} \Big( \| h\|_{L^{1+\varepsilon}_t} + \| h\|_{L^{1+\varepsilon}_t}\, \| Y\|_{C^0_t} + \llbracket Z\rrbracket_{C^{\varepsilon/(1+\varepsilon)}_t} \Big) \quad {\mathbb P}\text{-a.s.};\end{aligned}$$ dividing by $|t-s|^{\varepsilon/(1+\varepsilon)}$, taking supremum and combining this with [\[eq:pathwise-gronwall\]](#eq:pathwise-gronwall){reference-type="eqref" reference="eq:pathwise-gronwall"}, one arrives at $$\label{eq:pathwise-holder-estim} \| Y(\omega) \|_{C^{\varepsilon/(1+\varepsilon)}_t} \lesssim e^{2 \| h \|_{L^{1+\varepsilon}_t} } \Big( |Y_0(\omega)| + \|Z(\omega)\|_{C^{\varepsilon/(1+\varepsilon)}_t} \Big)\quad {\mathbb P}\text{-a.s.}$$ *Step 3: A priori estimates for $X$.* Recall that $X_t=\Phi_t^{-1}(Y_t)$, where by construction $\Phi^{-1}$ satisfies $|\Phi_t^{-1}(x)|\leqslant|x| + 1/2$ and [\[eq:properties-zvonkin\]](#eq:properties-zvonkin){reference-type="eqref" reference="eq:properties-zvonkin"}. It follows that ${\mathbb P}$-a.s. $\sup_t |X_t| \leqslant 1 + \sup_t |Y_t|$ and $$\begin{aligned} |X_t-X_s| \leqslant|\Phi_t^{-1} (Y_t)-\Phi_t^{-1}(Y_s)| + |\Phi_t^{-1} (Y_s)-\Phi_s^{-1}(Y_s)| \lesssim |Y_t-Y_s| + |t-s|^{1/2};\end{aligned}$$ combined with the pathwise bounds [\[eq:pathwise-gronwall\]](#eq:pathwise-gronwall){reference-type="eqref" reference="eq:pathwise-gronwall"}-[\[eq:pathwise-holder-estim\]](#eq:pathwise-holder-estim){reference-type="eqref" reference="eq:pathwise-holder-estim"}, we finally obtain an estimate of the form $$\label{eq:pathwise-estim-X} \| X(\omega)\|_{C^{\varepsilon/(1+\varepsilon)}_t} \lesssim 1 + |X_0(\omega)| + \|Z(\omega)\|_{C^{\varepsilon/(1+\varepsilon)}_t}\quad {\mathbb P}\text{-a.s.}$$ where the hidden constant depends on $\Theta$, $\varepsilon$, $\| b^2\|_{L^\infty_t \tilde{L}^{d+\varepsilon}_x}$, $T$ and $\| h\|_{L^{1+\varepsilon}_t}$. Recall that $Z$ is defined as a stochastic integral, with uniformly bounded $\tilde{\sigma}$; a standard application of Burkholder-Davis-Gundy inequality and Kolmogorov's continuity theorem allows to deduce that $\|Z\|_{C^{\varepsilon/(1+\varepsilon)}_t}$ admits moments of any order, in particular it has finite expectation. In view of the assumptions on $X_0$, we have derived the following result. **Lemma 14**. *There exists a constant $C$ (depending on $\Theta$, $\varepsilon$, $\| b^2\|_{L^\infty_t \tilde{L}^{d+\varepsilon}_x}$, $\| h\|_{L^{1+\varepsilon}_t}$ and $T$) such that $$\label{eq:moments-X} {\mathbb E}\big[ \| X\|_{C^{\varepsilon/(1+\varepsilon)}_t} \big] \leqslant C\big(1 + {\mathbb E}\big[|X_0|\big]\big).$$* *Step 4: A priori estimates on the density.* We have the following: **Lemma 15**. *Let $b$, $\sigma$ be regular coefficients satisfying Assumption [Assumption 1](#ass:diffusion){reference-type="ref" reference="ass:diffusion"}, $X$ the solution to [\[eq:intro-sde-detailed\]](#eq:intro-sde-detailed){reference-type="eqref" reference="eq:intro-sde-detailed"} and set $\mu_t={\mathcal L}(X_t)$. Then for any pair $(\tilde p, \tilde q)$ satisfying [\[eq:integrability-density\]](#eq:integrability-density){reference-type="eqref" reference="eq:integrability-density"} it holds $$\| \mu\|_{L^{\tilde q}_t L^{\tilde p}_x} \lesssim 1+ {\mathbb E}[|X_0|]$$ where the hidden constant depends on $\Theta$, $\varepsilon$, $\tilde p$, $\tilde q$, $\| b^1/(1+|x|)\|_{L^1_t L^\infty_x}$ and $\| b^2\|_{L^\infty_t \tilde{L}^{d+\varepsilon}_x}$.* *Proof.* Let $(\tilde p, \tilde q)$ be fixed and denote by $(\tilde p', \tilde q')$ their conjugate exponents. By the duality relation $(L^{\tilde q}_t L^{\tilde p}_x)^\ast = L^{\tilde q'}_t L^{\tilde p'}_x$, in order to prove the claim it suffices to show that $$\label{eq:density-goal} |\langle f,\mu\rangle| = \bigg| \int_0^T \int_{{\mathbb R}^d} f_s(x) \mu_s(\!\,\mathrm{d}x) \,\mathrm{d}s\bigg| = \bigg| \int_0^T {\mathbb E}[f_s(X_s)] \,\mathrm{d}s\bigg| \lesssim \| f\|_{L^{\tilde q}_t L^{\tilde p}_x} (1 + {\mathbb E}[|X_0|])$$ for all $f\in L^{\tilde q'}_t L^{\tilde p'}_x$; by linearity, we may assume $\| f\|_{L^{\tilde q'}_t L^{\tilde p'}_x}=1$. Observe that $(\tilde q,\tilde p)$ satisfy [\[eq:integrability-density\]](#eq:integrability-density){reference-type="eqref" reference="eq:integrability-density"} if and only if their duals satisfy $1/\tilde q' + d/\tilde p'<1$; we can therefore invoke Lemma [Lemma 13](#lem:decomposition-lemma){reference-type="ref" reference="lem:decomposition-lemma"} to decompose $f=f^{\leqslant} + f^>$ with $\| f^\leqslant\|_{L^{1+\varepsilon}_t L^\infty_x}, \| f^>\|_{L^\infty_t L^{d+\varepsilon}_x} \leqslant 1$. The first term is easy to estimate, since $$\label{eq:density-estim-easy} \bigg| \int_0^T {\mathbb E}[f^{\leqslant}_t(X_t)] \,\mathrm{d}s\bigg| \leqslant\int_0^T \| f^{\leqslant}_t\|_{L^\infty_x} \,\mathrm{d}t \lesssim_T \| f^\leqslant\|_{L^{1+\varepsilon}_t L^\infty_x}.$$ For the second one, fix any value $\lambda$ large enough such that Proposition [Proposition 11](#prop:zvonkin){reference-type="ref" reference="prop:zvonkin"} applies for $g=b^2$ and $f^>$ in place of $f$; let $u$ denote the associated scalar-valued solution to [\[eq:zvonkin-pde\]](#eq:zvonkin-pde){reference-type="eqref" reference="eq:zvonkin-pde"}, which thus satisfies [\[eq:zvonkin-estim\]](#eq:zvonkin-estim){reference-type="eqref" reference="eq:zvonkin-estim"}. Applying Itô's formula on $[0,T]$, we find $$\begin{aligned} u_T(X_T)-u_0(X_0) = \int_0^T (\partial_t u + \frac{1}{2} a:D^2 u + b^2\cdot\nabla u)(X_t)\,\mathrm{d}t + \int_0^T (b^1\cdot\nabla u)(X_t) \,\mathrm{d}t + M_T\end{aligned}$$ for a suitable martingale $M$. Rearranging the terms, applying $u_T\equiv 0$ and taking expectation, we get $$\begin{aligned} \int_0^T {\mathbb E}[ f^>_t(X_t)] \,\mathrm{d}t = {\mathbb E}[u_0(X_0)] + \int_0^T {\mathbb E}[(b^1\cdot\nabla g)(X_t)] \,\mathrm{d}t;\end{aligned}$$ applying assumption [\[eq:ass-drift\]](#eq:ass-drift){reference-type="eqref" reference="eq:ass-drift"} for $b^1$, we then find $$\label{eq:density-estim-hard}\begin{split} \bigg| \int_0^T {\mathbb E}[f^>_t(X_t)] \,\mathrm{d}t \bigg| & \leqslant\| u_0\|_{L^\infty_x} + \| \nabla u\|_{L^\infty_{t,x}} \Big\| \frac{b^1}{1+|x|}\Big\|_{L^1_t L^\infty_x} \int_0^T (1+{\mathbb E}[|X_t|]) \,\mathrm{d}t\\ & \lesssim 1 + {\mathbb E}[ |X_0|] <\infty \end{split}$$ where in the last step we applied Lemma [Lemma 14](#lem:moments-X){reference-type="ref" reference="lem:moments-X"}. Combining [\[eq:density-estim-easy\]](#eq:density-estim-easy){reference-type="eqref" reference="eq:density-estim-easy"} and [\[eq:density-estim-hard\]](#eq:density-estim-hard){reference-type="eqref" reference="eq:density-estim-hard"} yields [\[eq:density-goal\]](#eq:density-goal){reference-type="eqref" reference="eq:density-goal"} and thus the conclusion. ◻ # Proof of the main results {#sec:main-thm} *Proof of Theorem [Theorem 3](#thm:main-theorem){reference-type="ref" reference="thm:main-theorem"}.* The proof is based on classical approximation and compactness arguments. Let $(b,\sigma)$ satisfying Assumption [Assumption 1](#ass:diffusion){reference-type="ref" reference="ass:diffusion"} and $\mu_0\in {\mathcal P}_1$ be given. By mollifying $b^1$, $b^2$ and $\sigma$, we can construct an approximating sequence $(b^{1,n},b^{2,n},\sigma^n)$ satisfying Assumption [Assumption 1](#ass:diffusion){reference-type="ref" reference="ass:diffusion"} uniformly in $n$; more precisely, we require that $$\begin{aligned} \Big\| \frac{b^{1,n}_t}{1+|x|}\Big\|_{L^\infty_x} \leqslant h_t, \quad \sup_n \| b^{2,n}\|_{L^\infty_t \tilde{L}^{d+\varepsilon}_x}<\infty,\end{aligned}$$ where the function $h\in L^{1+\varepsilon}_t$ is independent of $n$, while $\sigma^n$ satisfy conditions [\[eq:ass-diffusion-1\]](#eq:ass-diffusion-1){reference-type="eqref" reference="eq:ass-diffusion-1"}-[\[eq:ass-diffusion-2\]](#eq:ass-diffusion-2){reference-type="eqref" reference="eq:ass-diffusion-2"} for a constant $K$ and a modulus of continuity $\omega_\sigma$ independent of $n$. Furthermore, the sequence can be constructed so that $$\label{eq:properties-approx} \lim_{n\to\infty} \sup_{t,x} |\sigma^n(t,x)-\sigma(t,x)|=0, \quad b^{1,n}\to b^1 \text{ in } L^{1+\varepsilon}_t L^p_{loc}, \quad b^{2,n}\to b^2 \text{ in } L^q_t L^{d+\varepsilon}_{loc}$$ for all $p,q<\infty$. Finally, for fixed $n$ the coefficients $(b^{1,n},b^{2,n},\sigma^n)$ are regular, in the sense that $b^n\in L^1_t C^1_{loc}$ and satisfying linear growth conditions, while $\sigma^n\in L^\infty_t C^1_x$. Consider a filtered probability space $(\Omega,\mathcal{F},\{{\mathcal F}_t\}_{t\geqslant 0},{\mathbb P})$, endowed with some random variables $(\xi,W)$ such that ${\mathcal L}(\xi)=\mu_0$, $\xi$ is ${\mathcal F}_0$-measurable and $W$ is a ${\mathcal F}_t$-Brownian motion. For any $n$, we can construct classically a strong solution to the SDE $$\,\mathrm{d}X^n = b^n_t(X^n_t) \,\mathrm{d}t + \sigma^{n}_t(X^n_t) \,\mathrm{d}W_t, \quad X^n\vert_{t=0}=\xi.$$ Since $(b^n,\sigma^n)$ satisfy Assumption [Assumption 1](#ass:diffusion){reference-type="ref" reference="ass:diffusion"}, all the results from Section [3](#sec:apriori){reference-type="ref" reference="sec:apriori"} apply; in particular, setting $\mu^n_t= {\mathcal L}(X^n_t)$, by Lemmas [Lemma 14](#lem:moments-X){reference-type="ref" reference="lem:moments-X"}-[Lemma 15](#lem:apriori-density){reference-type="ref" reference="lem:apriori-density"} it holds $$\label{eq:uniform-bounds} \sup_n {\mathbb E}\big[ \| X\|_{C^{\varepsilon/(1+\varepsilon)}_t} \big]<\infty, \quad \sup_n \| \mu^n\|_{L^{\tilde q}_t L^{\tilde p}_x} <\infty\ \quad \forall\, (\tilde q, \tilde p) \text{ satisfying \eqref{eq:exponents-weaker}}.$$ Furthermore, by [\[eq:pathwise-estim-X\]](#eq:pathwise-estim-X){reference-type="eqref" reference="eq:pathwise-estim-X"} we have the ${\mathbb P}$-a.s. bounds $$\label{eq:prelim-bound} \| X^n(\omega)\|_{C^0_t} \lesssim 1 + |\xi(\omega)| + \|Z^n(\omega)\|_{C^{\varepsilon/(1+\varepsilon)}_t}$$ with constant independent of $n$ and $Z^n=\int_0^\cdot \tilde\sigma^n_t(Y^n_t) \,\mathrm{d}W_t$. By construction, $\tilde{\sigma}^n$ are uniformly bounded, thus the family of r.v. $\{\|Z^n(\omega)\|_{C^{\varepsilon/(1+\varepsilon)}_t}\}_n$ admits uniformly bounded second moment, making it uniformly integrable. As the same holds for the single r.v. $|\xi|$, we deduce uniform integrability of the r.v.s appearing on the l.h.s. of [\[eq:prelim-bound\]](#eq:prelim-bound){reference-type="eqref" reference="eq:prelim-bound"}, namely $$\label{eq:uniform-integrability} \lim_{R\to\infty} \sup_n {\mathbb E}\big[ \| X^n\|_{C^0_t} \mathbbm{1}_{\| X^n\|_{C^0_t}>R} \big] = 0.$$ The first estimate in [\[eq:uniform-bounds\]](#eq:uniform-bounds){reference-type="eqref" reference="eq:uniform-bounds"}, together with Ascoli-Arzelà's theorem, immediately implies tightness of $\{{\mathcal L}(X^n)\}_n$ in $C^0_t$, thus also tightness of $\{{\mathcal L}(\xi,X^n,W)\}_n$ in ${\mathbb R}^d\times C^0_t\times C^0_t$. By an application of Prokhorov's theorem, we can extract a (not relabelled) subsequence such that $\{{\mathcal L}(\xi,X^n,W)\}_n$ converge in law; by Skorokhod's theorem, we can then construct a new probability space $(\tilde \Omega, \tilde {\mathcal F},\tilde {\mathbb P})$ and a sequence of random variables $(\tilde \xi^n, \tilde X^n, \tilde W^n)$ defined on it such that ${\mathcal L}(\xi, X^n, W)= {\mathcal L}(\xi^n,\tilde X^n,\tilde W^n)$ and $(\tilde \xi^n, \tilde X^n,\tilde W^n)\to (\tilde \xi,\tilde X,\tilde W)$ $\tilde{\mathbb P}$-a.s. in ${\mathbb R}^d\times C^0_t\times C^0_t$. Standard arguments show that $W$ is a Brownian motion w.r.t. the common filtration ${\mathcal G}_t=\sigma(\tilde \xi, \tilde X_r,\tilde W_r: r\leqslant t)$ and that ${\mathcal L}(\xi)=\mu_0$; it remains to show that $(\tilde\xi, \tilde X, \tilde W)$ is the desired weak solution to the SDE [\[eq:intro-sde-detailed\]](#eq:intro-sde-detailed){reference-type="eqref" reference="eq:intro-sde-detailed"}. In order to do so, it suffices to show that we can pass to the limit in each term in the approximations, namely that $$\label{eq:main-proof-goal} \int_0^\cdot b^{i,n}_t(\tilde X^n_t) \,\mathrm{d}t \to \int_0^\cdot b^i_t(\tilde X_t) \,\mathrm{d}t, \quad \int_0^\cdot \sigma^n_t(\tilde X^n_t) \,\mathrm{d}\tilde W^n_t \to \int_0^\cdot \sigma_t(\tilde X_t) \,\mathrm{d}\tilde W_t$$ in probability in $C^0_t$, for $i=1,2$. We first consider the stochastic integrals in [\[eq:main-proof-goal\]](#eq:main-proof-goal){reference-type="eqref" reference="eq:main-proof-goal"}, which are the easiest. By construction $\sigma^n\to \sigma$ uniformly in $(t,x)$ and $\tilde X^n\to \tilde X$ $\tilde {\mathbb P}$-a.s. in $C^0_t$, so that $\sigma^n(\tilde X^n)\to \sigma(\tilde X)$ as well; on the other hand $\tilde W^n\to \tilde W$ in $C^0_t$, and so by applying [@DGHT2011 Lemma 2.1], we conclude that $\int_0^\cdot \sigma^n(\tilde X^n) \,\mathrm{d}\tilde W^n \to \int_0^\cdot \sigma (\tilde X) \,\mathrm{d}\tilde W$ in probability. We claim that, for $i=1,2$, it holds $$\label{eq:convergence-goal} \lim_{n\to\infty} \tilde {\mathbb E}\Big[ \int_0^T |b^{i,n}_t(\tilde X^n_t) - b^i_t(\tilde X_t)| \,\mathrm{d}t \Big] = 0$$ from which [\[eq:main-proof-goal\]](#eq:main-proof-goal){reference-type="eqref" reference="eq:main-proof-goal"} will follow. We only give details for [\[eq:convergence-goal\]](#eq:convergence-goal){reference-type="eqref" reference="eq:convergence-goal"} for $i=1$, the other case being similar. In order to prove [\[eq:convergence-goal\]](#eq:convergence-goal){reference-type="eqref" reference="eq:convergence-goal"}, we divide our analysis in three main steps. We first introduce a cutoff $\psi_R$ in the space variable and show that it gives rise to a negligible error; we then compare the (cutoffs of the) terms associated to $b^{1,n}(\tilde X^n)$ and $b^1(\tilde X^n)$, and finally $b^1(\tilde X^n)$ and $b^1(\tilde X)$. Combining everything with the triangular inequality shows that $$\begin{aligned} \lim_{n\to\infty} \tilde {\mathbb E}\Big[ \int_0^T |b^{1,n}_t(\tilde X^n_t) - b^1_t(\tilde X_t)| \,\mathrm{d}t \Big] \leqslant\delta\end{aligned}$$ for arbitrary $\delta>0$, from which the claim [\[eq:convergence-goal\]](#eq:convergence-goal){reference-type="eqref" reference="eq:convergence-goal"} follows. Recall that $\{X^n\}_n$ satisfy the uniform integrability [\[eq:uniform-integrability\]](#eq:uniform-integrability){reference-type="eqref" reference="eq:uniform-integrability"}, so that the same holds for $\tilde{X}^n$ (as well as $\tilde X$). We can then localize the drifts $b^{1,n}$ by introducing a cutoff function $\psi_R(x):=\psi(|x|/R)$, where $\psi$ is a smooth function satisfying $\psi\equiv 1$ on $[0,1]$ and $\psi\equiv 0$ on $[2,\infty)$. The error committed by this approximation can be made arbitrarily small for sufficiently large $R$, since $$\begin{aligned} \tilde{\mathbb E}\Big[ \int_0^T & |b^{1,n}_t(\tilde X^n_t) - (b^{1,n}_t \psi_R) (\tilde X^n_t)| \,\mathrm{d}t\Big] \leqslant\tilde{\mathbb E}\Big[ \int_0^T |b^{1,n}_t(\tilde X^n_t)| \mathbbm{1}_{|\tilde X^n_t|\geqslant R} \,\mathrm{d}t \Big]\\ & \leqslant\tilde{\mathbb E}\Big[ \int_0^T h_t (1+ |\tilde X^n_t|) \mathbbm{1}_{\|\tilde X^n\|_{C^0_t} \geqslant R} \,\mathrm{d}t\Big] \leqslant\| h\|_{L^1_t} \tilde{\mathbb E}\Big[ (1 + \| \tilde X^n\|_{C^0_t}) \mathbbm{1}_{\|\tilde X^n\|_{C^0_t} \geqslant R} \Big]\end{aligned}$$ where the last term goes to $0$ as $R\to\infty$, uniformly in $n$, by virtue of [\[eq:uniform-integrability\]](#eq:uniform-integrability){reference-type="eqref" reference="eq:uniform-integrability"}. Next, let us fix some $p\in [1,\infty)$ large enough such that $1/(1+\varepsilon)+d/p<1$; by contruction of the approximations, it holds $b^{1,n} \psi_R \to b^1 \psi$ in $L^{1+\varepsilon}_t L^p_x$; on the other hand, by [\[eq:uniform-bounds\]](#eq:uniform-bounds){reference-type="eqref" reference="eq:uniform-bounds"} the measures $\mu^n$ are uniformly bounded in $L^{(1+\varepsilon)/\varepsilon}_t L^{p'}_x$. It follows that $$\begin{aligned} \lim_{n\to\infty} \tilde{\mathbb E}\Big[ \int_0^T |(b^{1,n}_t\psi_R)(\tilde X^n_t) -b^1_t\psi_R)(\tilde X^n_t)| \,\mathrm{d}t \Big] \leqslant\lim_{n\to\infty} \| (b^{1,n}-b^1) \psi_R\|_{L^{1+\varepsilon}_t L^p_x} \| \mu^n\|_{L^{(1+\varepsilon)/\varepsilon}_t L^{p'}_x} = 0\end{aligned}$$ It remains to show that, for any fixed $R$, it holds $$\begin{aligned} \lim_{n\to\infty} {\mathbb E}\Big[ \int_0^T |(b_s \psi_R) (\tilde X^n_s) - (b_s \psi_R)(X_s)| \,\mathrm{d}s \Big] = 0.\end{aligned}$$ The argument is based on approximation arguments similar to the previous ones, so we only sketch it. If $b$ were continuous, the claim would follow from the property that $\tilde X^n\to \tilde X$ in $C^0_t$ and dominated convergence; otherwise, we we can find a continuous $\tilde b$ such that $\| (\tilde b-b)\psi_R\|_{L^{1+\varepsilon}_t L^p_x}\leqslant\delta$ for some finite $p$ large enough s.t. $1/(1+\varepsilon) + d/p<1$ and some $\delta>0$ arbitrarily small. We can then pass to the limit in $\tilde b$ and apply the uniform integrability bounds on $\mu^n$ and $\mu$ to show that the remaining error terms are arbitrarily small. ◻ *Proof of Corollary [Corollary 4](#cor:main-cor-1){reference-type="ref" reference="cor:main-cor-1"}.* It follows immediately from Lemma [Lemma 13](#lem:decomposition-lemma){reference-type="ref" reference="lem:decomposition-lemma"}. ◻ *Proof of Corollary [Corollary 6](#cor:main-cor-2){reference-type="ref" reference="cor:main-cor-2"}.* Consider the approximations $(b^n, \sigma^n,X^n)$ constructed in the proof of Theorem [Theorem 3](#thm:main-theorem){reference-type="ref" reference="thm:main-theorem"}. Clearly $\mu^n_t={\mathcal L}(X^n_t)$ are now solutions to [\[eq:FP\]](#eq:FP){reference-type="eqref" reference="eq:FP"} with $(b,a)$ replaced by $(b^n,a^n)$, where $a^n=\sigma^n (\sigma^n)^\ast$, and $\mu^n$ converge weakly to $\mu_t={\mathcal L}(X_t)$. The continuity of $t\mapsto \mu_t$ in the weak convergence of measures is a direct consequence of the fact that $X$ has continuous paths. The argument employed to show that $X$ is a solution to the SDE also proves that $b^n\mu^n$ (resp. $a^n \mu^n$) converge weakly to $b \mu$ (resp. $a \mu$), which implies by passing to the limit that $\mu$ is a weak solution to [\[eq:FP\]](#eq:FP){reference-type="eqref" reference="eq:FP"}. The $L^{\tilde q}_t L^{\tilde p}_x$-bound for $\mu$ follows from the uniform bound [\[eq:uniform-bounds\]](#eq:uniform-bounds){reference-type="eqref" reference="eq:uniform-bounds"} at the level of the approximations $\mu^n$ and the lower semicontinuity of $L^{\tilde q}_t L^{\tilde p}_x$-norms under weak convergence. ◻ # Acknowledgements {#acknowledgements .unnumbered} This work originates from some stimulating discussions with Oleg Butkovsky, to whom I'm very thankful, who pointed out the existence of the paper [@krylov1] and wondered about the interpretation to give to the condition $1/q+d/p=1$, also in relation to the work [@ButGal]. # Funding information {#funding-information .unnumbered} The author is supported by the SNSF Grant 182565 and by the Swiss State Secretariat for Education, Research and Innovation (SERI) under contract number MB22.00034. [^1]: See the end of the introduction for the definition of $L^q_t L^p_x$ and all other relevant function spaces. [^2]: More precisely, it is required that $b\in L^q_t L^p_x$ if $p\geqslant q$ and $b\in L^p_x L^q_t$ otherwise. [^3]: Besov-Hölder spaces $C^\alpha_x$ are just one option and one might instead consider Lebesgue spaces with the same scaling behaviour. For instance, for $H=1/2$, $C^{-1}_x$ scales like $L^d_x$, which recovers the critical scale $b\in L^\infty_t L^d_x$. In this direction, let us mention [@butkovsky2023stochastic] for weak existence results for SDEs driven by fractional Brownian motion with (autonomous) drift in subcritical Lebesgue scales $L^p_x$. [^4]: We refer to [@XXZZ2020] for the definitions of $\mathbb{H}^{\alpha,p}_q(T)$ and ${\mathbb{L}}^{p}_q(T)$.

arxiv_math

--- abstract: | Let $\mathrm{ex}(G_{n,p}^r,F)$ denote the maximum number of edges in an $F$-free subgraph of the random $r$-uniform hypergraph $G_{n,p}^r$. Building on recent work of Conlon, Lee, and Sidorenko, we prove non-trivial lower bounds on $\mathrm{ex}(G_{n,p}^r,F)$ whenever $F$ is not Sidorenko. This connection between Sidorenko's conjecture and random Turán problems gives new lower bounds on $\mathrm{ex}(G_{n,p}^r,F)$ whenever $F$ is not Sidorenko, and further allows us to bound how "far" from Sidorenko an $r$-graph $F$ is whenever upper bounds for $\mathrm{ex}(G_{n,p}^r,F)$ are known. author: - "Jiaxi Nie[^1]" - "Sam Spiro[^2]" bibliography: - refs.bib title: Sidorenko Hypergraphs and Random Turán Numbers --- # Introduction This paper establishes a connection between two seemingly unrelated problems in extremal combinatorics: Sidorenko's conjecture and random Turán problems. Throughout this paper we consider $r$-uniform hypergraphs, or $r$-graphs for short. For an $r$-graph $H=(V,E)$, we use the notation $v(H):=|V|$ and $e(H):=|E|$. ## Sidorenko's Conjecture Recall that a *homomorphism* from an $r$-graph $F$ to an $r$-graph $H$ is a map $\phi:V(F)\to V(H)$ such that $\phi(e)$ is an edge of $H$ whenever $e$ is an edge of $F$. We let $\hom(F,H)$ denote the number of homomrophisms from $F$ to $H$ and define the *homomorphism density* $$t_F(H)=\frac{\hom(F,H)}{v(H)^{v(F)}},$$ which is equivalently the probability that a random map $\phi:V(F)\to V(H)$ is a homomorphism. An important definition regarding homomorphism densities is the following. **Definition 1**. We say that an $r$-graph $F$ is *Sidorenko* if for every $r$-graph $H$ we have $$t_F(H)\ge t_{K_r^r}(H)^{e(F)},$$ where $K_r^r$ is the $r$-graph consisting of a single edge. This bound is asymptotically best possible by considering random hypergraphs. Observe that if $F$ is not $r$-partite, then $F$ is not Sidorenko (due to $H=K_r^r$, for example). As such, it suffices to only consider $r$-partite $r$-graphs when discussing Sidorenko hypergraphs. In particular, for $r=2$ Sidorenko famously conjectured the following. **Conjecture 1** (Sidorenko's Conjecture [@Sidorenko1991Inequalities; @Sidorenko1993Acorrelation]). *Every bipartite graph is Sidorenko.* A large body of literature is dedicated towards Sidorenko's conjecture [@conlon2010approximate; @conlon2017finite; @conlon2018sidorenko; @conlon2018some; @coregliano2021biregularity; @fox2017local; @hatami2010graph; @kim2016two; @li2011logarithimic; @lovasz2011subgraph; @szegedy2014information], but overall this problem remains very wide open. It is known that Sidorenko's conjecture does not extend to hypergraphs. In particular, Sidorenko [@Sidorenko1993Acorrelation] showed that $r$-uniform loose triangles (see ) are not Sidorenko for $r\ge 3$. Because of this, there seems to have been relatively little interest in studying which $r$-graphs are Sidorenko. This changed very recently with the work of Conlon, Lee, and Sidorenko [@conlon2023extremal] who showed the following connection between non-Sidorenko hypergraphs and Turán numbers. Here we recall that the *Turán number* $\mathrm{ex}(n,F)$ of an $r$-graph $F$ is the maximum number of edges that an $n$-vertex $F$-free $r$-graph can have. **Theorem 2** ([@conlon2023extremal]). *If $F$ is not Sidorenko, then there exists $c=c(F)$ such that $$\mathrm{ex}(n,F)=\Omega\left(n^{r-\frac{v(F)-r}{e(F)-1}+c}\right).$$* We note that this improves upon the trivial lower bound $\mathrm{ex}(n,F)=\Omega(n^{r-\frac{v(F)-r}{e(F)-1}})$ which holds for all $F$ by a standard random deletion argument. ## The Random Turán Problem We now discuss our second problem of interest: the random Turán problem. Define $G_{n,p}^r$ to be the random $r$-graph on $n$ vertices obtained by including each possible edge independently and with probability $p$. When $r=2$ we simply write $G_{n,p}$ instead of $G_{n,p}^2$. We define the *random Turán number* $\mathrm{ex}(G_{n,p}^r,F)$ to be the maximum number of edges in an $F$-free subgraph of $G_{n,p}^r$. Note that when $p=1$ we have $\mathrm{ex}(G_{n,1}^r,F)=\mathrm{ex}(n,F)$, so this can be viewed as a probabilistic analog of the classical Turán number. The asymptotics of $\mathrm{ex}(G_{n,p}^r,F)$ has essentially been determined whenever $F$ is not an $r$-partite $r$-graph due to independent breakthrough work of Schacht [@schacht2016extremal] and of Conlon and Gowers [@conlon2016combinatorial], and as such we will focus only on the case when $F$ is $r$-partite. For $r=2$, McKinely and Spiro [@mckinley2023random] recently conjectured the following, where here we define the $r$-density of an $r$-graph $F$ with $v(F)>r$ by $$m_r(F):=\max_{F'\subseteq F:v(F')>r}\left\{\frac{e(F')-1}{v(F')-r}\right\},$$ and we say that $F$ is *$r$-balanced* if $m_r(F)=\frac{e(F)-1}{v(F)-r}$. **Conjecture 3** ([@mckinley2023random]). *If $F$ is a graph with $\mathrm{ex}(n,F)=\Theta(n^\alpha)$ for some $\alpha\in (1,2]$, then a.a.s. $$\mathrm{ex}(G_{n,p},F)= \begin{cases}\Theta(p^{\alpha-1}n^\alpha) & p\ge n^{\frac{2-\alpha-1/m_2(F)}{\alpha-1}} (\log n)^{O(1)}\\n^{2-1/m_2(F)}(\log n)^{O(1)} & n^{\frac{2-\alpha-1/m_2(F)}{\alpha-1}}(\log n)^{O(1)} \ge p\ge n^{-1/m_2(F)},\\ (1+o(1))p {n\choose 2} & n^{-1/m_2(F)}\gg p\gg n^{-2}.\end{cases}$$* In particular, predicts that for bipartite graphs, $\mathrm{ex}(G_{n,p},F)$ always has a "flat middle range" where $\mathrm{ex}(G_{n,p},F)=n^{2-1/m_2(F)}(\log n)^{O(1)}$ for the entire range $n^{-1/m_2(F)}\le p\le n^{\frac{2-\alpha-1/m_2(F)}{\alpha-1}}(\log n)^{O(1)}$; see for an example. is known to hold (assuming certain conjectures regarding $\mathrm{ex}(n,F)$) for complete bipartite graphs and even cycles [@morris2016number], and for theta graphs [@mckinley2023random], with the lower bound being known to hold for powers of rooted trees [@spiro2022random]. ![The plot of $\mathrm{ex}(G_{n,p},C_4)$ as proven by Füredi [@F]. More generally, predicts that $\mathrm{ex}(G_{n,p},F)$ should have a "flat middle range" for every bipartite graph $F$ starting at $p=n^{-1/m_2(F)}$.](GraphC4.pdf){#fig:enter-label width="40%"} It is known that does not extend to hypergraphs. In particular, Nie, Spiro, and Verstraëte [@nie2021triangle] showed this does not hold for $r$-uniform loose triangles with $r\ge 3$. ## Our Results The reader may notice that the situation for Sidorenko $r$-graphs and for random Turán problems parallel each other quite closely, in particular with regard to loose triangles serving as a counterexample to each problem. The main result of this paper shows that this is not a coincidence: in many cases, $r$-graphs which are not Sidorenko have a stronger lower bound on $\mathrm{ex}(G_{n,p}^r,F)$ than the generalization of would predict. We prove this by extending to give lower bounds for random Turán numbers. In addition to this, we give a quantitative statement which in several cases gives optimal bounds for $\mathrm{ex}(G_{n,p}^r,F)$. To state the quantitative result, we define for an $r$-graph $F$ the quantity $$s(F):=\sup\{s: \exists H,\ t_F(H)=t_{K_r^r}(H)^{s+e(F)}>0\}.$$ Note that $s(F)=0$ if and only if $F$ is Sidorenko, and more generally, $s(F)$ measures how "far" an $r$-graph is from being Sidorenko. With this in mind, we state our main result. **Theorem 4**. *If $F$ is an $r$-graph with $e(F)\ge 2$ and $\frac{v(F)-r}{e(F)-1}<r$, then for any $p=p(n)\ge n^{-\frac{v(F)-r}{e(F)-1}}$, we have a.a.s. $$\mathrm{ex}(G_{n,p}^r,F)\ge n^{r-\frac{v(F)-r}{e(F)-1}-o(1)}(p n^{\frac{v(F)-r}{e(F)-1}})^{\frac{s(F)}{e(F)-1+s(F)}}.$$* We emphasize that this bound is only non-trivial when $s(F)>0$, i.e. when $F$ is not Sidorenko. When $s(F)>0$ and $F$ is $r$-balanced, shows that $\mathrm{ex}(G_{n,p}^r,F)$ does not have a "flat middle range" as predicted in the case of graphs by . As an aside, we note that the bound of is always at least as strong as the (implicit quantitative version of) ; see and the surrounding discussion for more. To get effective bounds in , one must determine how large $s(F)$ is, i.e. how "far" from Sidorenko $F$ is; and we believe this to be a problem of independent of interest. **Problem 5**. *Given an $r$-graph $F$, determine (bounds for) $s(F)$.* allows us to translate between the random Turán problem and the Sidorenko-type problem . Specifically, shows that lower bounds on $s(F)$ give lower bounds for the random Turán problem, and conversely, upper bounds on the random Turán problem give upper bounds on $s(F)$. For example, the non-Sidorenko $r$-graphs established by Conlon, Lee, and Sidorenko [@conlon2023extremal Theorem 3.1] gives the following result for the random Turán problem. **Corollary 6**. *Let $F$ be an $r$-partite $r$-graph of odd girth. Then there exists an $\epsilon>0$ such that if $p\ge n^{-\frac{v(F)-r}{e(F)-1}}$, then a.a.s. $$\mathrm{ex}(G_{n,p}^r,F)\ge n^{r-\frac{v(F)-r}{e(F)-1}-o(1)}(p n^{\frac{v(F)-r}{e(F)-1}})^{\epsilon}.$$* On the other hand, the simplest case of $p=1$ in gives the following general bound on $s(F)$. **Corollary 7**. *If $F$ is an $r$-graph with $\mathrm{ex}(n,F)=O(n^\alpha)$ and $\alpha<r$, then $$s(F)\le \frac{v(F)-\alpha}{r-\alpha}-e(F).$$* While can sometimes provide tight bounds (see ), in general more effective bounds are obtained by considering smaller values of $p$. To state these improved bounds, we make the following definitions which will be used throughout the paper. **Definition 2**. Given a $k$-graph $F$, we define the $r$-expansion $\mathrm{E}^r(F)$ to be the $r$-graph obtained by enlarging each $k$-edge of $F$ with a set of $r-k$ vertices of degree one. When $F$ is the graph cycle $C_\ell$, we write $C_{\ell}^r:=\mathrm{E}^r(C_\ell)$ and refer to this as the $r$-uniform *loose cycle* of length $\ell$. Lastly, we define an $r$-graph $T$ to be a *tight $r$-tree* if its edges can be ordered as $e_1,\dots,~e_t$ so that $$\forall i\ge 2~\exists v\in e_i~and~1\le s\le i-1~such~that~v\not\in\cup_{j=1}^{i-1}e_j~and~e_i-v\subset e_s.$$ The following bounds for $s(F)$ turn out to follow from together with known bounds for random Turán numbers [@mubayi2023random; @nie2023random; @nie2023tur]; see the end of for further details on the derivation. **Corollary 8**. * * - *For $r>k \ge 2$, we have $$s(\mathrm{E}^r(K^k_{k+1}))\le \frac{1}{r-k}.$$* - *For $\ell \ge 1$ and $r\ge 3$, we have $$s(C^r_{2\ell+1})\le\frac{2\ell-1}{r-2}.$$* - *For $r\ge k\ge 2$ and any tight $k$-tree $T$, $s(\mathrm{E}^r(T))=0$. That is, expansions of tight trees are Sidorenko.* - *For $\ell \ge 1$ and $r\ge 3$ we have $s(C_{2\ell}^r)=0$. That is, loose even cycles are Sidorenko.* In addition to these immediate consequences of , we prove two new results related to $s(F)$ and expansions. First, we show that expansions of $F$ which contain $K_{k+1}^k$ are not Sidorenko. **Theorem 9**. *If $F$ is a $k$-graph which contains $K_{k+1}^k$ as a subgraph, then for all $r>k$ we have $$s(\mathrm{E}^r(F))\ge\frac{1}{r-k}.$$ In particular, $\mathrm{E}^r(F)$ is not Sidorenko.* The proof of Theorem [Theorem 9](#thm:containKk){reference-type="ref" reference="thm:containKk"} makes use of a construction of Gower and Janzer [@gowers2021generalizations], which generalized the seminal construction of Ruzsa and Szemerédi[@ruzsa1978]. We note that the quantitative lower bound of combined with gives optimal lower bounds for the random Turán number $\mathrm{ex}(G_{n,p}^r,\mathrm{E}^r(K_{k+1}^k))$; see [@nie2023random; @nie2021triangle]. Moreover, this result together with (a) gives the following. **Corollary 10**. *For $r>k\ge 2$, we have $$s(\mathrm{E}^r(K_{k+1}^k))=\frac{1}{r-k}.$$* We note that the $r=k+1$ case of this corollary gives an example where the general upper bound of is tight. Finally, we establish an upper bound on $s(\mathrm{E}^r(F))$ in terms of $s(F)$. **Theorem 11**. *If $F$ is a $k$-graph with $s(F)<\infty$, then for all $r\ge k$ we have $$s(\mathrm{E}^r(F))\le \frac{v(F)-k}{v(F)-k+(r-k)(s(F)+e(F)-1)}\cdot s(F).$$* For example, one can check that knowing $s(\mathrm{E}^{k+1}(K_{k+1}^k))\le 1$ from together with implies $s(\mathrm{E}^{r}(K_{k+1}^k))\le \frac{1}{r-k}$ for all $r>k$, and similarly one recovers our upper bound for $C_{2\ell+1}^r$ assuming the upper bound for $r=3$. We also obtain the following nice corollary by taking $s(F)=0$. **Corollary 12**. *If $F$ is a Sidorenko $k$-graph, then its expansions $\mathrm{E}^r(F)$ are Sidorenko for all $r\ge k$.* # Proof of and its Corollaries {#sec:main} Our proof of is based off the proof of from [@conlon2023extremal] which relies on the tensor product trick. Given two $r$-graphs $G,H$, we define the *tensor product* $G\otimes H$ to be $r$-graph on $V(G)\times V(H)$ where $((x_1,y_1),\ldots,(x_r,y_r))\in E(G\otimes H)$ if and only if $(x_1,\ldots,x_r)\in E(G)$ and $(y_1,\ldots,y_r)\in E(H)$. For $N$ a positive integer, we define the $N$-fold tensor product $H^{\otimes N}$ inductively by setting $H^{\otimes 1}=H$ and $H^{\otimes N}=H\otimes H^{\otimes(N-1)}$. The key property we need regarding tensor products is the fact that for any $r$-graphs $F,H$ and $N\ge 1$, we have $$t_F(H^{\otimes N})=t_F(H)^N,$$ which is straightforward to verify. By incorporating the tensor product trick from [@conlon2023extremal] together with random homomorphisms, we can show that $r$-graphs $G$ with few copies of $F$ have large $F$-free subgraphs. To this end, given $r$-graphs $G,F$, we define $\mathrm{ex}(G,F)$ to be the maximum number of edges in an $F$-free subgraph of $G$, and we let $\mathcal{N}_F(G)$ denote the number of copies of $F$ in $G$. **Lemma 13**. *If $F$ is an $r$-graph such that there exists an $r$-graph $H$ with $t_{K_r^r}(H)=\alpha$ and $t_{F}(H)=\alpha^{s+e(F)}$, then for all $r$-graphs $G$ and integers $N\ge 1$ we have $$\mathrm{ex}(G,F)\ge \alpha^N e(G)-\alpha^{(s+e(F))N} \mathcal{N}_F(G).$$* *Proof.* Let $\phi:V(G)\to V(H^{\otimes N})$ be chosen uniformly at random, and let $G'\subseteq G$ be the subgraph consisting of all hyperedges $e\in G$ which are mapped bijectively onto an edge of $H^{\otimes N}$. By linearity of expectation, we have $$\mathbb{E}[e(G')]=t_{K_r^r}(H^{\otimes N}) \cdot e(G)=\alpha^N \cdot e(G),$$ and $$\mathbb{E}[\mathcal{N}_F(G')]=t_{F}(H^{\otimes N})\cdot \mathcal{N}_F(G)= \alpha^{(s+e(F))N} \cdot \mathcal{N}_F(G).$$ Thus if we define $G''\subseteq G'$ by deleting an edge from each copy of $F$ in $G'$, then $G''$ is $F$-free and satisfies $$\mathbb{E}[e(G'')]\ge \alpha^N e(G)-\alpha^{(s+e(F))N} \mathcal{N}_F(G),$$ and hence there must exist an $F$-free subgraph of $G$ with at least this many edges, proving the result. ◻ With this we can prove our main result. *Proof of .* Recall that we wish to prove if $F$ is an $r$-graph with $e(F)\ge 2$ and $\frac{v(F)-r}{e(F)-1}<r$, then for any $p=p(n)\ge n^{-\frac{v(F)-r}{e(F)-1}}$, we have a.a.s. $$\mathrm{ex}(G_{n,p}^r,F)\ge n^{r-\frac{v(F)-r}{e(F)-1}-o(1)}(p n^{\frac{v(F)-r}{e(F)-1}})^{\frac{s(F)}{e(F)-1+s(F)}}.$$ If $s(F)=0$ then this result follows by a standard random deletion argument, so from now on we assume $s(F)>0$. Consider any $0<\epsilon\le s(F)$ (which exists by assumption $s(F)>0$). By definition of $s(F)$, there exists a non-empty $r$-graph $H$ with $t_{K_r^r}(H)=\alpha>0$ and $t_{F}(H)=\alpha^{s+e(F)}$ with $0\le s(F)-\epsilon\le s\le s(F)$. By we find $$\mathrm{ex}(G_{n,p}^r,F)\ge \alpha^N e(G_{n,p}^r)-\alpha^{(s+e(F))N} \mathcal{N}_F(G_{n,p}^r),\label{eq:relative}$$ so it suffices to choose an $N$ such that this is sufficiently large a.a.s. Given $p$ and any function $\delta(n)=o(1)$, let $N\ge 1$ be the smallest integer such that $$q:=\delta(n)n^{-\frac{v(F)-r}{e(F)-1+s}}p^{-\frac{e(F)-1}{e(F)-1+s}}\ge \alpha^N.$$ Note that such an integer exists since $0<\alpha<1$. Also note that $\alpha^N\le q\le \alpha^{N-1}$ by the minimality of $N$. Let $A$ denote the event that $e(G_{n,p}^r)\ge \frac{1}{2r!} pn^r$. Because $e(G_{n,p}^r)$ is a binomial random variable and $pn^r\ge n^{r-\frac{v(F)-r}{e(F)-1}}\to \infty$ by hypothesis, the Chernoff bound implies that the event $A$ holds a.a.s. Let $B$ denote the event that $\mathcal{N}_F(G_{n,p}^r)\le \delta(n)^{-1/2} p^{e(F)} n^{v(F)}$. Since $\mathbb{E}[\mathcal{N}_F(G_{n,p}^r)]\le p^{e(F)} n^{v(F)}$, it follows by Markov's inequality and $\delta(n)=o(1)$ that $B$ holds a.a.s. Because $A\cap B$ hold a.a.s., we find that a.a.s. the bound in [\[eq:relative\]](#eq:relative){reference-type="eqref" reference="eq:relative"} is at least $$\begin{aligned} \frac{1}{2r!} \alpha^N pn^r-\delta(n)^{-1/2}\alpha^{(s+e(F))N} p^{v(F)}n^{e(F)}&\ge \frac{1}{2r!} \alpha^N pn^r(1-2r!\delta(n)^{-1/2}q^{s+e(F)-1} p^{e(F)-1}n^{v(F)-r})\\&=\frac{1}{2r!} \alpha^N pn^r(1-2r!\delta(n)^{s+e(F)-3/2}).\end{aligned}$$ Note that $s+e(F)-3/2>0$ since $s\ge 0$ and $e(F)\ge 2$. Thus for $n$ sufficiently large the quantity above is at least $$\begin{aligned} \frac{1}{4r!} \alpha^N pn^r&\ge \frac{\alpha}{4r!} \delta(n) n^{r-\frac{v(F)-r}{e(F)-1}}(p n^{\frac{v(F)-r}{e(F)-1}})^{\frac{s}{e(F)-1+s}}\\ &\ge \frac{\alpha}{4r!} \delta(n) n^{-\frac{\epsilon}{e(F)-1+s(F)}}\cdot n^{r-\frac{v(F)-r}{e(F)-1}}(p n^{\frac{v(F)-r}{e(F)-1}})^{\frac{s(F)}{e(F)-1+s(F)}},\end{aligned}$$ with this last step used $s\ge s(F)-\epsilon$. As $\epsilon>0$ was arbitrary and $\delta(n)$ tends to 0 arbitrarily slowly, we conclude the desired result. ◻ As an aside, the bound of continues to hold in expectation even if $\frac{v(F)-r}{e(F)-1}\ge r$. However, in this case we can not say $G_{n,p}^r$ has any edges a.a.s., and hence no non-trivial lower bound for $\mathrm{ex}(G_{n,p}^r,F)$ can hold a.a.s. Focusing on the $p=1$ case, quickly gives the following. **Corollary 14**. *If $F$ is an $r$-graph such that there exists a non-empty $r$-graph $H$ with $t_F(H)=t_{K_r^r}(H)^{s+e(F)}$, then $$\mathrm{ex}(n,F)=\Omega\left(n^{r-\frac{v(F)-r}{e(F)-1}+\frac{(v(F)-r)s}{(e(F)-1)(s+e(F)-1)}}\right).$$* *Proof.* Take $G=K_n^r$ in , which means $e(G)\ge \frac{1}{2 r!} n^r$ and $\mathcal{N}_F(T)\le n^{v(F)}$, so for $\alpha=t_{K_r^r}(H)$ and any $N\ge 1$ we have $$\mathrm{ex}(n,F)=\mathrm{ex}(G,F)\ge \frac{1}{2r!}\alpha^N n^r\left(1-2r! \alpha^{(s+e(F)-1)N}n^{v(F)-r}\right).$$ We conclude the result by taking $N$ such that $\alpha^N$ is a sufficiently small constant times $n^{-\frac{v(F)-r}{s+e(F)-1}}=n^{(v(F)-r)\left(\frac{-1}{e(F)-1}+\frac{s}{(e(F)-1)(s+e(F)-1)}\right)}$. ◻ As a point of comparison, a more careful analysis of the proof giving yields the following quantitative bound. **Theorem 15** (Quantitative ). *If $F$ is an $r$-graph such that there exists a non-empty $r$-graph $H$ with $t_F(H)=t_{K_r^r}(H)^{s+e(F)}$ and $t_{K_r^r}(H)=v(H)^{-\delta}$, then $$\mathrm{ex}(n,F)=\Omega\left(n^{r-\frac{v(F)-r}{e(F)-1}+\frac{\delta s}{e(F)-1}}\right).$$* It is not difficult to show that $\delta\le \frac{v(F)-r}{s+e(F)-1}$ in (see for a formal proof). If $\delta$ obtains this maximum possible value then matches ; otherwise does strictly better. We now sketch how together with known results for random Turán bounds implies Corollaries [Corollary 7](#cor:General){reference-type="ref" reference="cor:General"} and [Corollary 8](#cor:upperBounds){reference-type="ref" reference="cor:upperBounds"}. *Proof of .* Recall that we wish to show that if $F$ is an $r$-graph with $\mathrm{ex}(n,F)=O(n^\alpha)$, then $$s(F)\le \frac{v(F)-\alpha}{r-\alpha}-e(F).$$ Let $s$ be such that there exists a non-empty $r$-graph $H$ with $t_F(H)=t_{K_r^r}(H)^{s+e(F)}$. By , we have $$\Omega(n^{r-\frac{v(F)-r}{s+e(F)-1}})=\mathrm{ex}(n,F)=O(n^\alpha).$$ This implies $r-\frac{v(F)-r}{s+e(F)-1}\le \alpha$, and rearranging gives $s\le \frac{v(F)-\alpha}{r-\alpha}-e(F)$. As $s(F)$ is the supremum over all such $s$, we conclude the result. ◻ *Proof of .* Throughout we implicitly utilize the fact that every $F$ we consider is $r$-balanced and hence $m_r(F)=\frac{v(F)-r}{e(F)-1}$. We first show (a): for $r>k\ge 2$ that $s(\mathrm{E}^r(K_{k+1}^k))\le \frac{1}{r-k}$. By [@nie2023random Theorem 1.5], we have for $p= n^{-\frac{1}{m_r(\mathrm{E}^r(K_{k+1}^k))}+c}$ with $c=c(k,r)>0$ that a.a.s. $$\mathrm{ex}(G^r_{n,p}, \mathrm{E}^r(K^k_{k+1}))= p^{\frac{1}{(r-k)k+1}}n^{k-1+o(1)},$$ hence by Theorem [Theorem 4](#thm:main){reference-type="ref" reference="thm:main"} $$\frac{s(\mathrm{E}^r(K^k_{k+1}))}{k+s(\mathrm{E}^r(K^k_{k+1}))}\le \frac{1}{(r-k)k+1},$$ which implies the desired upper bound. For (b), by [@nie2023random Theorem 1.3], we have for $p=n^{-\frac{1}{m_r(C_{2\ell+1}^r)}+c}$ with $c=c(\ell,r)>0$ that a.a.s. $$\mathrm{ex}(G^r_{n,p}, C^r_{2\ell+1})\le p^{\frac{2\ell-1}{2\ell(r-1)-1}}n^{2+o(1)},$$ hence by Theorem [Theorem 4](#thm:main){reference-type="ref" reference="thm:main"} $$\frac{s(C^r_{2\ell+1})}{2\ell+s(C^r_{2\ell+1})}\le \frac{2\ell-1}{2\ell(r-1)-1},$$ which implies $$s(C^r_{2\ell+1})\le \frac{2\ell-1}{r-2}.$$ For (c), by [@nie2023random Theorem 1.4], we have $\mathrm{ex}(G_{n,p}^r,\mathrm{E}^r(T))=n^{r-\frac{1}{m_r(\mathrm{E}^r(T))}+o(1)}$ a.a.s. for $p=n^{c-\frac{1}{m_r(\mathrm{E}^r(T))}}$ with $c=c(k,r)>0$, which implies the bound. For (d), it was proven in [@mubayi2023random; @nie2023tur] that $\mathrm{ex}(G_{n,p}^r,C_{2\ell}^r)=n^{r-\frac{1}{m_r(C_{2\ell}^r)}+o(1)}$ a.a.s. for $p=n^{c-\frac{1}{m_r(C_{2\ell}^r)}}$ with $c=c(\ell,r)>0$, from which the bound follows. ◻ # Proof of To prove our lower bounds on $s(\mathrm{E}^r(F))$ when $F$ contains $K_{k+1}^k$, we make use of the following construction of Gowers and Janzer [@gowers2021generalizations], which is a generalization of the seminal construction of Ruzsa ans Szemerédi [@ruzsa1978]. **Theorem 16** ([@gowers2021generalizations; @ruzsa1978]). *For $r>k\ge 2$ and $n\ge 1$, there exists a graph on $n$ vertices with the following two properties:* - *It has $n^ke^{-O(\sqrt{\log n})}$ subgraphs isomorphic to $K_r$;* - *Every $K_k$ is contained in exactly one maximal clique of size at most $r$.* Note that property $(ii)$ is slightly stronger than the original Theorem 1.2 in [@gowers2021generalizations] which states "every $K_k$ is contained in at most one $K_r$\". In fact, property $(ii)$ is inherently implied by the proof of Lemma 3.1 in [@gowers2021generalizations]. Let $G_{n,r,k}$ be the graph guaranteed by Theorem [Theorem 16](#theorem:GJ){reference-type="ref" reference="theorem:GJ"}. Consider an $r$-graph $H_{n,r,k}$ on $V(G_{n,r,k})$ whose edges are the vertex sets of copies of $K_r$ in $G_{n,r,k}$. **Proposition 17**. *For $r>k\ge 2$ and $n\ge1$, $H_{n,r,k}$ has the following properties:* - *$e(H_{n,r,k})\ge n^ke^{(-O(\sqrt{\log n}))}$;* - *Any two edges intersect in at most $k-1$ vertices;* - *$H_{n,r,k}$ does not contain any subgraph isomorphic to $\mathrm{E}^r(K_{k+1}^{k})$.* *Proof.* Property (i) is trivial. Suppose there are two edges intersecting in at least $k$ vertices. This means there exists a copy of $K_k$ contained in two different copies of $K_r$ in $G_{n,r,k}$, contradicting property (ii) of $G_{n,r,k}$. Similarly, if $H_{n,r,k}$ contained a copy of $\mathrm{E}^r(K_{k+1}^{k})$, then in $G_{n,r,k}$, there would be a copy of $K_k$ contained in a $K_{k+1}$ and a $K_r$ not containing the $K_{k+1}$, contradicting property (ii) of $G_{n,r,k}$. ◻ Now we are ready to prove our main result for this section. *Proof of .* Recall that we wish to prove that if $F$ is a $k$-graph which contains $K_{k+1}^k$ as a subgraph, then for all $r>k$ we have $$s(\mathrm{E}^r(F))\ge \frac{1}{r-k}.$$ That is, for any $\epsilon>0$ we want to find an $r$-graph $H$ such that $$t_{\mathrm{E}^r(F)}(H)\le t_{K_r^r}(H)^{e(F)+\frac{1}{r-k}-\epsilon}.$$ Let $H=H_{n,r,k}$ with $n$ to be chosen sufficiently large in terms of $\epsilon$. The crucial observation is the following. **Claim 18**. *If $\phi:V(\mathrm{E}^r(K_{k+1}^k))\to V(H)$ is a homomorphism, then $\phi(e)=\phi(f)$ for all $e,f\in\mathrm{E}^r(K_{k+1}^k)$.* *Proof.* Label the edges of $\mathrm{E}^r(K^k_{k+1})$ as $e_1,\dots,e_{k+1}$ and let $e_i=\{v_1,\dots,v_{k+1},w_{i,1},\dots,w_{i,r-k}\}\setminus\{v_i\}$. Without loss of generality, we suppose for contradiction that $\phi(e_1)\not=\phi(e_2)$. By Proposition [Proposition 17](#proposition:GJ1){reference-type="ref" reference="proposition:GJ1"}(ii) we find $|\phi(e_1)\cap \phi(e_2)|\le k-1$. On the other hand, since $\phi(e_1\cap e_2)\subseteq\phi(e_1)\cap \phi(e_2)$ has size $k-1$, we must have $\phi(e_1\cap e_2)= \phi(e_1)\cap \phi(e_2)$. In particular, $\phi(v_1)\not\in \phi(e_1)$. Note that $v_1\in e_i$ for all $i>1$, and hence $\phi(v_1)\in \phi(e_i)$, for $i>1$. This implies $\phi(e_1)\not=\phi(e_i)$, and hence, $|\phi(e_1)\cap \phi(e_i)|=k-1$, for all $i>1$. By symmetry, we have $|\phi(e_i)\cap \phi(e_j)|=k-1$ for all $i\not=j$. This means $\phi(e_1),\dots,\phi(e_{k+1})$ form a copy of $\mathrm{E}^r(K^{k}_{k+1})$ in $H_{m,r,k}$, which contradicts Proposition [Proposition 17](#proposition:GJ1){reference-type="ref" reference="proposition:GJ1"}(iii). ◻ This allows us to prove the following. **Claim 19**. *If $x\in V(F)$ is contained in a $K_{k+1}^k$, then for any map $\phi:V(F)\setminus\{x\}\to V(H)$, there are at most $O(1)$ homomorphisms $\phi':V(\mathrm{E}^r(F))\to V(H)$ such that the restriction $\phi'|_{V(F)\setminus\{x\}}$ equals $\phi$.* *Proof.* Let $x_1,\ldots,x_k$ be the other vertices of the $K_{k+1}^k$ containing $x$. Because $\mathrm{E}^r(F)$ contains an edge $e$ containing $\{x_1,\ldots,x_k\}$, if the set $X=\{\phi(x_1),\ldots,\phi(x_k)\}$ either has size less than $k$ or is not contained in an edge of $H$, then no homomorphism restricts to $\phi$, so we may assume this is not the case. By (ii), there exists a unique edge $h\in H$ containing $X$, and any homomorphism $\phi'$ which restricts to $\phi$ must map $e$ to $h$. By , we must have $\phi'(x)\in h$. Let $y\in h$ and define $\phi_y:V(F)\to V(H)$ by having $\phi_y(x)=y$ and $\phi_y(z)=\phi(z)$ for all other $z$. By the observation above, any $\phi'$ which restricts to $\phi$ must restrict to $\phi_y$ for one of the at most $O(1)$ choices $y\in h$. We claim that for any $y\in h$ there are at most $O(1)$ homomorphism $\phi'$ which restricts to $\phi_y$, from which the result will follow. Indeed, consider any vertex $z\in V(\mathrm{E}^r(F))\setminus V(F)$, which by definition of the expansion means there is an edge $\{z_1,\ldots,z_k\}\in E(F)$ such that $\{z,z_1,\ldots,z_k\}$ is contained in an edge $e'$ of $\mathrm{E}^r(F)$. If the set $Z=\{\phi_y(z_1),\ldots,\phi_y(z_k)\}$ has size less than $k$ or is not contained in an edge of $H$, then no homomorphism restricts to $\phi_y$, so we may assume this is not the case. By (ii), there exists a unique edge $h'\in H$ containing $Z$, and any homomorphism $\phi'$ which restricts to $\phi_y$ must map $e'$ to $h'$. In conclusion, for any $z\in V(\mathrm{E}^r(F))\setminus V(F)$ there are at most $O(1)$ vertices $z$ can map to in a homomorphism $\phi'$ which restricts to $\phi_y$. Thus there are at most $O(1)$ homomorphisms $\phi'$ which restrict to $\phi_y$, proving the claim. ◻ Observe that the number of maps $\phi:V(F)\setminus\{x\}\to V(H)$ is at most $n^{v(F)-1}$, and hence the claim above implies $$t_{\mathrm{E}^r(F)}(H)\le n^{v(F)-1-v(\mathrm{E}^r(F))}=n^{-1-(r-k)e(F)}.$$ On the other hand, $$t_{K_r^r}(H)^{e(F)+\frac{1}{r-k}-\epsilon}=n^{(k-r-o(1))(e(F)+\frac{1}{r-k}-\epsilon)}=n^{-1-(r-k)e(F)+(r-k)\epsilon-o(1)},$$ and for $n$ sufficiently large in terms of $\epsilon$ this is greater than the bound for $t_{\mathrm{E}^r(F)}(H)$ obtained above, proving the result. ◻ Before going on, we note that one can easily adapt the proof above to give stronger quantitative bounds on $s(F)$ in certain cases. For example, if there exists a subset $V\subseteq V(F)$ such that for every $x\in V$ there exist vertices $x_1,\ldots,x_k\in V(F)\setminus V$ forming a $K_{k+1}^k$ with $x$, then one can prove $$s(\mathrm{E}^r(F))\ge \frac{|V|}{r-k}.$$ # Proof of Here we establish an upper bound on $s(\mathrm{E}^r(F))$ in terms of $F$. For this the following will be useful. **Lemma 20**. *If $F'$ is an $r$-partite $r$-graph and $t_{F'}(H)\le t_{K_r^r}(H)^{s+e(F')}$, then $t_{K_r^r}(H)\ge v(H)^{-\frac{v(F')-r}{s+e(F')-1}}$.* *Proof.* Let $H$ satisfy $t_{F'}(H)\le t_{K_r^r}(H)^{s+e(F')}$ and let $\delta$ be such that $t_{K_r^r}(H)=v(H)^{-\delta}$. This means $H$ has $(r!)^{-1} v(H)^{r-\delta}$ edges, and hence $\hom(F',H)\ge v(H)^{r-\delta}$ (since $F'$ mapping onto a single edge of $H$ is always a homomrphism by assumption of $F'$ being $r$-partite). Hence $$v(H)^{r-\delta-v(F')}\le t_{F'}(H)\le v(H)^{-\delta(s+e(F'))},$$ and rearranging shows $\delta\le \frac{v(F')-r}{s+e(F')-1}$, proving the result. ◻ For our proof, it will be convenient to work with weighted $r$-graphs $W$ where the weight of an $r$-set $\{x_1,\ldots,x_r\}$ is denoted $W(x_1,\ldots,x_r)$. We let $\mathrm{Hom}(F,W)$ denote the set of all maps $\phi:V(F)\to V(W)$ which are injective on $e\in E(F)$. We define the weight $w(\phi)$ of $\phi\in \mathrm{Hom}(F,W)$ to be $\prod_{e\in E(F)} W(\phi(e))$ and we define $\hom(F,W)=\sum_{\phi\in \mathrm{Hom}(F,W)}w(\phi)$. With this we can define the notion of homomorphism densities $t_F(W)$ exactly as before, and it is not difficult to show that if $s(F)=s$ then $t_F(W)\ge t_{K_r^r}(W)^{s+e(F)}$ for all weighted $r$-graphs $W$. *Proof of .* Recall that we wish to show that if $F$ is a $k$-graph with $s(F)<\infty$, then for all $r\ge k$ we have $$s(\mathrm{E}^r(F))\le s:= \frac{v(F)-k}{v(F)-k+(r-k)(s(F)+e(F)-1)}\cdot s(F).$$ Assume for contradiction that there exists an $n$-vertex $r$-graph $H$ such that $t_{\mathrm{E}^r(F)}(H)<t_{K_r^r}(H)^{s+e(F)}$. Since $s(F)<\infty$ by assumption, $F$ must be $k$-partite and hence $\mathrm{E}^r(F)$ must be $r$-partite. Thus by we must have $t_{K_{r}^{r}}(H)\ge n^{-\frac{v(F)+(r-k)e(F)-r}{s+e(F)-1}}$, or equivalently $$n\ge t_{K^r_r}(H)^{-\frac{s+e(F)-1}{v(F)+(r-k)e(F)-r}}.\label{eq:vertBound}$$ We define an auxiliary weighted $k$-graph $W$ on $V(H)$ such that for any $k$-set $X$ we have $W(X)=(r-k)\,!\deg_H(X)$, i.e. $(r-k)!$ times the number of edges of $H$ containing $X$. By definition of $s(F)$ we have $$\begin{aligned} &t_{F}(W)=\frac{\sum_{\phi\in \mathrm{Hom}(F,W)}\prod_{e\in E(F)}(r-k)\,!\deg_H(\phi(e))}{n^{v(F)}}\\ &\ge t_{K^k_k}(W)^{s(F)+e(F)}=\left(\frac{k\,!}{n^k}\sum_{X\in W}(r-k)\,!\deg_H(X)\right)^{s(F)+e(F)}=\left(\frac{r\,!e(H)}{n^k}\right)^{s(F)+e(F)}. \end{aligned}$$ By definition of expansions, every homomorphism $\phi:V(\mathrm{E}^r(F))\to V(H)$ can be formed by first choosing a homomorphism $\phi':V(F)\to V(W)$, and then for each $e'\in F$ with $e\in \mathrm{E}^r(F)$ the edge containing $e'$, one chooses some edge $h\in E(H)$ containing the $k$-set $\phi(e')$ together with a bijection from $e\setminus e'$ to $h\setminus \phi(e)$. Thus we have $$\hom(\mathrm{E}^r(F),H)=\sum_{\phi\in \mathrm{Hom}(F,W)}\prod_{e\in E(F)}(r-k)\,!\deg_H(\phi(e)).$$ Hence, $$\begin{aligned} t_{\mathrm{E}^r(F)}(H)&=\frac{\sum_{\phi\in \mathrm{Hom}(F,W)}\prod_{e\in E(F)}(r-k)\,!\deg_H(\phi(e))}{n^{v(F)+(r-k)e(F)}}\\ &\ge \frac{\left(r\,!e(H) n^{-k}\right)^{s(F)+e(F)}}{n^{(r-k)e(F)}}\\ &=t_{K_r^r}(H)^{s(F)+e(F)}\cdot n^{(r-k)s(F)}\\ &\ge t_{K_r^r}(H)^{s(F)+e(F)}\cdot t_{K_r^r}(H)^{-\frac{(r-k)s(F)(s+e(F)-1)}{v(F)+(r-k)e(F)-r}}, \end{aligned}$$ where this last step used [\[eq:vertBound\]](#eq:vertBound){reference-type="eqref" reference="eq:vertBound"}. One can verify that this final quantity equals $t_{K_r^r}(H)^{s+e(F)}$, a contradiction to our choice of $H$. We conclude the result. ◻ As an aside, it is tempting to generalize the statement of to hold even at $s(F)=\infty$. Indeed, "taking the limit" in suggests that when $s(F)=\infty$ we should have $$s(\mathrm{E}^r(F))\le \frac{v(F)-k}{r-k}.$$ This does hold whenever $\mathrm{E}^r(F)$ satisfies $\mathrm{ex}(n,\mathrm{E}^r(F))=O(n^k)$ by our general upper bound since $$\frac{v(\mathrm{E}^r(F))-k}{r-k}-e(\mathrm{E}^r(F))=\frac{v(F)-k}{r-k}.$$ However, such a result does not hold in general. For example, it certainly fails if $\mathrm{E}^r(F)$ is not $r$-partite, such as when considering $\mathrm{E}^3(K_t)$ with $t>3$. # Concluding Remarks There are many questions left to explore regarding (non-)Sidorenko hypergraphs which we break into three broad categories. **Sidorenko Expansions.** In we showed that expansions of Sidorenko hypergraphs are Sidorenko. This motivates the following conjecture. **Conjecture 21**. *For every bipartite graph $F$, there exists an $r\ge 2$ such that $\mathrm{E}^r(F)$ is Sidorenko.* Note that Sidorenko's conjecture predicts this holds with $r=2$ for all $F$. Moreover, suggests it may be easier to prove for larger values of $r$ (since if it holds for some $r_0$, then it holds for all $r\ge r_0$). As such, can be viewed as a (potentially) weaker version of Sidorenko's conjecture, and it would be particularly interesting if one could verify it for some $r\ge 2$ independent of $F$. Another question asks whether the converse of holds. **Question 22**. *Is it true that $F$ is Sidorenko if and only if all of its expansions $\mathrm{E}^r(F)$ are Sidorenko?* Note that if has an affirmative answer, then would be equivalent to Sidorenko's conjecture. The simplest case that we do not know how to answer is the following. **Question 23**. *If $F$ is a non-bipartite graph, are all of its expansions $\mathrm{E}^r(F)$ not Sidorenko?* We note that [@conlon2023extremal Theorem 3.1] shows this holds if $F$ has odd girth, but beyond this we know nothing. **$k$-linear Hypergraphs**. In we proved that expansions of $k$-graphs containing $K_{k+1}^k$ are not Sidorenko. We conjecture that the following stronger result holds. **Conjecture 24**. *If $F$ is an $r$-graph such that $|e\cap f|<k$ for any distinct $e,f\in F$ and such that $F$ contains an expansion $\mathrm{E}^r(K_{k+1}^k)$ as a subgraph, then $F$ is not Sidorenko.* The case $k=2$ was proven in [@conlon2023extremal], but as they note, their construction does not seem to effectively generalize to higher uniformities. We offer an alternative proof of the $k=2$ case in the appendix of the arXiv version of this paper to serve as another potential source of inspiration towards proving . **Bounds for $s(F)$**. motivates the problem of determining $s(F)$ for non-Sidorenko hypergraphs, especially those for which the random Turán number is unknown. One outstanding case is that of loose odd cycles. **Problem 25**. *Determine $s(C_{2\ell+1}^r)$.* In we showed $s(C_{2\ell+1}^r)\le \frac{2\ell-1}{r-2}$, and by considering $H=K_r^r$ it is possible to prove that $s(C_{2\ell+1}^r)$ is at least roughly $r^{-2\ell-1}$. We believe the upper bound is closer to the truth, but we do not think this is tight. Our best guess (though we would not go so far as to make it a conjecture) is that $$s(C_{2\ell+1}^r)=\frac{\ell}{(r-1)\ell-1}.$$ Indeed, the lower bound $s(C_{2\ell+1}^r)\ge \frac{\ell}{(r-1)\ell-1}$ would follow if there existed $n$-vertex $r$-graphs of girth $2\ell+2$ with $n^{1+1/\ell-o(1)}$ edges, which is the densest such an $r$-graph can be [@collier2018linear]. Such $r$-graphs are only known to exist when $\ell=1$ due to Ruzsa-Szmerédi type constructions, and it is difficult for us to imagine a construction that would give a better lower bound for $s(C_{2\ell+1}^r)$ than this. We also note that the general upper bound $s(C_{2\ell+1}^r)\le \frac{\ell}{(r-1)\ell-1}$ would follow from the $r=3$ result by using . # Appendix: Linear Hypergraphs with Loose Triangles {#appendix-linear-hypergraphs-with-loose-triangles .unnumbered} Here we give an alternative proof to the following result from [@conlon2023extremal]. **Theorem 26** ([@conlon2023extremal]). *If $F$ is linear and contains a loose triangle, then $F$ is not Sidorenko.* As our primary aim is to demonstrate a new approach and not to prove anything novel, we will be somewhat loose with the details. Our proof will rely on the language of graphons; we refer the reader to e.g. [@conlon2023extremal] for a refresher on the relevant definitions. *Proof.* Our proof relies on the following basic claims. **Claim 27**. *Let $W$ be a graphon and $p$ the constant graphon of density $p$. For any $r$-graph $F$, we have $$t_F(p+(1-p)W)=\sum_{F'\subseteq F} p^{e(F)-e(F')}(1-p)^{e(F')} t_{F'}(W).$$* We say that a hypergraph $F$ is a *forest* if one can order its edges $e_1,\ldots,e_m$ such that $e_i$ intersects $\bigcup_{j<i} e_j$ in at most one vertex. **Claim 28**. *There exists a graphon $W$ with $t_F(W)=t_{K_r^r}(W)^k$ whenever $F$ is a forest with $k$ edges and where $t_F(W)<t_{K_r^r}(W)^3$ for any other $F$ which is linear.* *Proof.* Take $W$ to be the graphon corresponding to $K_r^r$, i.e., $W(x_1,\ldots,x_r)=1$ if $|\{x_1,\ldots,x_r\}\cap [(i-1)/r,i/r]|=1$ for all $1\le i\le r$ and $W(x_1,\ldots,x_r)=0$ otherwise. It is not difficult to verify that $t_F(W)=t_{K_r^r}(W)^k$ when $F$ is a forest (this follows because $W$ is a "regular" graphon) and that $t_{C_4^r}(W)<t_{K_r^r}(W)^3$ (because $C_4^r$ contains a non-spanning forest on 3 edges). One can also check that $t_{C_3^r}(W)<t_{K_r^r}(W)^3$, namely because $W$ has the same homomorphism densities as $K_r^r$ and because $$\frac{\hom(C_3^r,K_r^r)}{r^{v(C_3^r)}}=\frac{r(r-1)(r-2)\cdot [(r-2)!]^3}{r^{3r-3}}<\left(\frac{r!}{r^r}\right)^3,$$ since this inequality is equivalent to saying $r(r-1)(r-2)\cdot r^3< (r(r-1))^3$. Now every linear $F$ either contains a forest on at least 4 edges or contains $C_3^r$ or $C_4^r$, from which the result follows. ◻ We now prove the result. Let $F$ be a linear $r$-graph containing a loose triangle. Let $W$ be the graphon from , and possibly by taking $N$-fold tensor powers, we can assume its edge density $q:=t_{K_r^r}(W)$ is sufficiently small and that there exists some $c<1$ sufficiently small with $t_{F'}(W)\le c q^3$ for all $F'\subseteq F$ which are either loose triangles or which contain at least 4 edges. We will show that $F$ is not Sidorenko by considering the graphon $\frac{1}{2}+\frac{1}{2}W$, which by is equivalent to showing $$2^{-e(F)}\sum_k \sum_{F'\subseteq F: e(F')=k} t_{F'}(W)=t_F\left(\frac{1}{2}+\frac{1}{2}W\right)<t_{K_r^r}\left(\frac{1}{2}+\frac{1}{2}W\right)^{e(F)}=2^{-e(F)}\sum_k {e(F)\choose k} q^k.\label{eq:looseTriangle}$$ Because $F$ is linear, every $F'\subseteq F$ with at most 3 edges is either a forest or a loose triangle. By definition of $W$, in the former case we have $t_{F'}(W)=q^k$, and in the latter case we have $t_{F'}(W)\le c q^3$. As $F$ contains at least one loose triangle, we have $$(1-c) q^3+\sum_{k\le 3} \sum_{F'\subseteq F: e(F')=k} t_{F'}(W)\le \sum_{k\le 3} \sum_{F'\subseteq F: e(F')=k} q^k\le \sum_{k\le 3} {e(F)\choose k} q^k.$$ Note that every $F'\subseteq F$ with $e(F')>3$ has $t_{F'}(W)\le c q^3$ by assumption. Thus by taking $c$ sufficiently small we have $\sum_{k>3} \sum_{F'\subseteq F: e(F')=k} t_{F'}(W)<(1-c)q^3$, which combined with the equation above implies [\[eq:looseTriangle\]](#eq:looseTriangle){reference-type="eqref" reference="eq:looseTriangle"} as desired. ◻ We note that a nearly identical proof goes through to show that $F$ is not Sidorenko whenever $F$ has odd girth (which was also proven in [@conlon2023extremal]). # Proofs that Don't Work for {#proofs-that-dont-work-for .unnumbered} Below, we (somewhat informally) demonstrate two natural approaches which do *not* solve . For this lemma, we consider $r$-raphs with loops, i.e. $r$-graphs whose edges are multisets of size $r$. Given an $r$-graph $H$, we define $H^*$ to be the $(r+1)$-graph with loops where $e$ is an edge of $H^*$ if and only if it contains an edge of $H$. For example, if $H=K_2$, then $H^*$ has edges $\{1,1,2\},\{1,2,2\}$; and we note that this is the original example used to show that $C_3^3$ is not Sidorenko [@Sidorenko1993Acorrelation]. We will also say that a hypergraph $H$ *witnesses* that $F$ is not Sidorenko if $t_F(H)<t_{K_r^r}(H)^{e(F)}$. **Lemma 29**. *Let $F,H$ be graphs such that $H$ witnesses that $F$ is not Sidorenko.* - *It is not true in general that $H^+$ witnesses that $F^+$ is not Sidorenko.* - *It is not true in general that $H^*$ witnesses that $F^+$ is not Sidorenko.* *Proof Sketch.* Let $F$ be the disjoint union of a triangle and $\ell$ copies of $C_4$, and take $H=K_2$. Then $H$ witnesses that $F$ is not Sidorenko. However, for $\ell$ sufficiently large we have $$t_{F^+}(H^+)=(6/3^6)(18/3^8)^\ell\ge (2/9)^{3+4\ell}=t_{K_3^3}(H^+).$$ Similarly for the same $F,H$ as above, we have for $\ell$ sufficiently large that $$t_{F^+}(H^*)=(26/2^6) (82/2^8)^\ell\ge (3/4)^{3+4\ell}=t_{K_3^3}(H^*).$$ This proves the result. As an aside, the exact same computations holds if $F$ is taken to be a triangle and $\ell$ $C_4$'s sharing a common vertex, so this lemma continues to hold even if $F$ is assumed to be connected. ◻ [Here was the start of an attempt to write the more general thing, I don't think it's really worth going through the details.]{style="color: red"} # Appendix: Hypergraphs with Odd Girth {#appendix-hypergraphs-with-odd-girth .unnumbered} We say that a hypergraph $F$ has girth $\ell\ge 3$ if it is linear, contains a $C_\ell^r$, and contains no $C_{\ell'}^r$ for any $\ell'<\ell$. Here we give an alternative proof to the following result from [@conlon2023extremal]. **Theorem 30** ([@conlon2023extremal]). *If $F$ is an $r$-graph with odd girth, then $F$ is not Sidorenko.* As our primary aim is to demonstrate a new approach and not to prove anything novel, we will be somewhat loose with the details. Our proof will rely on the language of graphons; we refer the reader to e.g. [@conlon2023extremal] for a refresher on the relevant definitions. *Proof.* Our proof relies on the following basic claims. **Claim 31**. *Let $W$ be a graphon and $p$ the constant graphon of density $p$. For any $r$-graph $F$, we have $$t_F(p+(1-p)W)=\sum_{F'\subseteq F} p^{e(F)-e(F')}(1-p)^{e(F')} t_{F'}(W).$$* We say that a hypergraph $F$ is a *forest* if one can order its edges $e_1,\ldots,e_m$ such that $e_i$ intersects $\bigcup_{j<i} e_j$ in at most one vertex. **Claim 32**. *There exists a graphon $W$ with $t_F(W)=t_{K_r^r}(W)^k$ whenever $F$ is a forest with $k$ edges, and where $t_{F}(W)<t_{K_r^r}(W)^{2\ell+1}$ when $F=C_{2\ell+1}^r$ or $F=C_{2\ell+2}$* *Proof.* Take $W$ to be the graphon corresponding to $K_r^r$, i.e., $W(x_1,\ldots,x_r)=1$ if $|\{x_1,\ldots,x_r\}\cap [(i-1)/r,i/r]|=1$ for all $1\le i\le r$ and $W(x_1,\ldots,x_r)=0$ otherwise. It is not difficult to verify that $t_F(W)=t_{K_r^r}(W)^k$ when $F$ is a forest (this follows because $W$ is a "regular" graphon) and that $t_{C_{2\ell+2}^r}(W)<t_{K_r^r}(W)^{2\ell+1}$ (because $C_{2\ell+2}^r$ contains a non-spanning forest on $2\ell+1$ edges). One can also check that $t_{C_{2\ell+1}^r}(W)<t_{K_r^r}(W)^{2\ell+1}$, namely because $W$ has the same homomorphism densities as $K_r^r$ and because $$\frac{\hom(C_{2\ell+1}^r,K_r^r)}{r^{v(C_3^r)}}=\frac{((r-1)^{2\ell+1}-(r-1))\cdot [(r-2)!]^{2\ell+1}}{r^{(2\ell+1)(r-1)}}<\left(\frac{r!}{r^r}\right)^{2\ell+1},$$ where the equality used that there are $(r-1)^{2\ell+1}-(r-1)$ proper $r$-colorings of $C_{2\ell+1}$ and the inequality is equivalent to saying $(r-1)^{2\ell+1}-(r-1)< (r-1)^{2\ell+1}$. ◻ We now prove the result. Let $F$ be an $r$-graph with girth $2\ell+1$. Let $W$ be the graphon from , and possibly by taking $N$-fold tensor powers, we can assume its edge density $q:=t_{K_r^r}(W)$ is sufficiently small and that there exists some $c<1$ sufficiently small with $t_{F'}(W)\le c q^3$ for all $F'\subseteq F$ which are either loose triangles or which contain at least 4 edges. We will show that $F$ is not Sidorenko by considering the graphon $\frac{1}{2}+\frac{1}{2}W$, which by is equivalent to showing $$2^{-e(F)}\sum_k \sum_{F'\subseteq F: e(F')=k} t_{F'}(W)=t_F\left(\frac{1}{2}+\frac{1}{2}W\right)<t_{K_r^r}\left(\frac{1}{2}+\frac{1}{2}W\right)^{e(F)}=2^{-e(F)}\sum_k {e(F)\choose k} q^k.\label{eq:looseTriangle}$$ Because $F$ is linear, every $F'\subseteq F$ with at most 3 edges is either a forest or a loose triangle. By definition of $W$, in the former case we have $t_{F'}(W)=q^k$, and in the latter case we have $t_{F'}(W)\le c q^3$. As $F$ contains at least one loose triangle, we have $$(1-c) q^3+\sum_{k\le 3} \sum_{F'\subseteq F: e(F')=k} t_{F'}(W)\le \sum_{k\le 3} \sum_{F'\subseteq F: e(F')=k} q^k\le \sum_{k\le 3} {e(F)\choose k} q^k.$$ Note that every $F'\subseteq F$ with $e(F')>3$ has $t_{F'}(W)\le c q^3$ by assumption. Thus by taking $c$ sufficiently small we have $\sum_{k>3} \sum_{F'\subseteq F: e(F')=k} t_{F'}(W)<(1-c)q^3$, which combined with the equation above implies [\[eq:looseTriangle\]](#eq:looseTriangle){reference-type="eqref" reference="eq:looseTriangle"} as desired. ◻ As a final aside, in order to generalize to prove , one would in particular need to find a graphon $W$ which shows $\mathrm{E}^r(K_{k+1}^k)$ is not Sidorenko and which is "$(k-1)$-regular," meaning every set of $k-1$ vertices is contained in the same number of hyperedges (though this alone is not sufficient). [Actually could this be sufficient/does a single edge work again? The idea is that you can look at $p+qW$ with $q=1-p$ with $p$ large. Yes this might just work and have all of this be way simpler since the sum of at least 4 terms is at most $q^4$ which should just be good enough without having to any tensor products, interesting. So yes we might just win if $K_{k+1}^{k+1}$ shows\...right well as noted before we definitely need this to show cones aren't Sidorenko actually to get anywhere.]{style="color: red"} Okay so let's just say we have the $(r+1)$-graph formed by taking $C_3^{r}$ and adding a cone, then the number of homomorphisms here is $(r+1)\cdot \hom(C_3^r(K_r^r))$ so we win if $$\frac{(r+1)r(r-1)(r-2)\cdot [(r-2)!]^3}{(r+1)^{3r-2}}<\left(\frac{(r+1)!}{(r+1)^{r+1}}\right)^3,$$ this is equivalent to saying\...okay this is looking a little worriesome but anyways $$(r+1)r(r-1)(r-2)\cdot (r+1)^2<(r(r-1))^3,$$ which is equivalent to saying $(r+1)^3(r-2)<r^2(r-1)^2$ which does not holdh. [^1]: Shanghai Center for Mathematical Sciences, Fudan University `[email protected]`. [^2]: Dept. of Mathematics, Rutgers University `[email protected]`. This material is based upon work supported by the National Science Foundation Mathematical Sciences Postdoctoral Research Fellowship under Grant No. DMS-2202730.

arxiv_math

--- abstract: | We prove that any hyperbolic group acting properly discontinuously and cocompactly on a $\mathrm{CAT}(0)$ cube complex admits a projective Anosov representation into $\mathrm{SL}(d, \ensuremath{\mathbb{R}})$ for some $d$. More specifically, we show that if $\Gamma$ is a hyperbolic quasiconvex subgroup of a right-angled Coxeter group $C$, then a generic representation of $C$ by reflections restricts to a projective Anosov representation of $\Gamma$. address: - Institut des Hautes Études Scientifiques, 35 route de Chartres, 91440 Bures-sur-Yvette, France - Institut des Hautes Études Scientifiques, 35 route de Chartres, 91440 Bures-sur-Yvette, France - Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA - Department of Mathematics, University of Wisconsin -- Madison, WI 53706, USA author: - Sami Douba - Balthazar Fléchelles - Theodore Weisman - Feng Zhu bibliography: - specialbib.bib title: Cubulated hyperbolic groups admit Anosov representations --- # Introduction The prototypical examples of Gromov-hyperbolic groups are the convex cocompact subgroups of $\mathrm{PO}(n,1) = \mathrm{Isom}(\mathbb{H}^n)$, where $\mathbb{H}^n$ denotes real hyperbolic $n$-space, and more generally, of $\mathrm{Isom}(X)$, where $X$ is a rank-one symmetric space of noncompact type. It is known that not all hyperbolic groups can arise in this fashion: there are hyperbolic groups which do not admit faithful representations into *any* matrix group [@kapovichpolygons; @canary2019new; @tholozan2021linearity; @tholozan2022residually], and even among linear hyperbolic groups there are examples which fail to admit discrete faithful representations in any rank-one Lie group [@tholozan2021linearity; @dt2022anosov]. Apart from the convex cocompact subgroups of linear rank-one Lie groups, a significant source of linear hyperbolic groups is the class of hyperbolic groups that are *virtually special* in the sense of Haglund and Wise [@HW08]. A theorem of Agol [@agolVHC] established that the virtually compact special hyperbolic groups are precisely those hyperbolic groups that are *cubulated*, that is, those that act properly discontinuously and cocompactly on $\mathrm{CAT}(0)$ cube complexes. Many diverse examples of cubulated hyperbolic groups can be found "in nature" and in the literature: see e.g. [@nibloreevescoxeter; @wise2004cubulating; @ollivier2011cubulating; @bergeron2012boundary; @wise2021structure; @giralt]. In fact, there are currently no known examples of noncubulated convex cocompact subgroups of $\mathrm{PO}(n,1)$, and indeed, Wise conjectured that no such subgroups exist [@wise2014cubical Conj. 13.52]. The following question is also due to Wise [@wise2014cubical Prob. 13.53]. **Question 1**. Does every cubulated hyperbolic group embed as a convex cocompact subgroup of $\mathrm{PO}(n,1)$ for some $n \geq 2$? There are positive results towards for some low-dimensional cubulated hyperbolic groups; see e.g. [@kapovichpolygons; @markovic2013criterion; @haissinsky2015hyperbolic]. However, the question remains open even for some of the most standard examples of cubulated hyperbolic groups. For example, it is not known if every hyperbolic right-angled Coxeter group embeds convex cocompactly in $\mathrm{PO}(n,1)$ for some $n$, or even if the latter holds for such Coxeter groups of arbitrarily large virtual cohomological dimension. On the other hand, Danciger--Guéritaud--Kassel [@DGKgeomded] and Lee--Marquis [@leemarquis2019] (see also [@DGKLM]) have shown that every hyperbolic Coxeter group admits an *Anosov* representation. ## Anosov representations Anosov representations were introduced by Labourie [@labourie2006anosov], and their theory subsequently developed by Guichard--Wienhard [@gw2012anosov], Kapovich--Leeb--Porti [@klp2017characterizations; @klp2018domains; @KLP2018], Guéritaud--Guichard--Kassel--Wienhard [@ggkw2017anosov], Bochi--Potrie--Sambarino [@BPS], and many others. They have emerged as a successful generalization of convex cocompact representations in arbitrary rank. An Anosov representation $\rho\colon \Gamma \to \mathop{\mathrm{GL}}(d,\ensuremath{\mathbb{R}})$ gives a quasi-isometric embedding of $\Gamma$ into the target group, and the class of Anosov representations is stable under small perturbations: the set of Anosov representations from $\Gamma$ into $\mathop{\mathrm{GL}}(d,\ensuremath{\mathbb{R}})$ is an open subset of $\mathop{\mathrm{Hom}}(\Gamma,\mathop{\mathrm{GL}}(d,\ensuremath{\mathbb{R}}))$ [@labourie2006anosov]. Anosov representations may also be characterized in terms of the existence of limit maps relating the dynamics of $\Gamma$ acting on its boundary to the dynamics of the image subgroup acting on a flag variety [@klp2017characterizations], and are closely related to convex cocompact actions in real projective space [@dgk2017convex]. The class of groups admitting an Anosov representation is known to be strictly larger than that of groups which admit a convex cocompact embedding in rank one [@dt2022anosov]. However, little is currently understood about the possible isomorphism types of groups admitting Anosov representations. Any such group is necessarily finite-by-linear, and is Gromov-hyperbolic [@klp2017characterizations; @BPS], but as observed by Canary [@canary2021anosov Q. 50.2], there are no other known restrictions: **Question 2**. Does every linear hyperbolic group admit an Anosov representation? Even though is open, there are still surprisingly few constructions available for producing concrete examples of groups admitting Anosov representations, beyond the groups that are already known to admit convex cocompact representations in rank one. Apart from hyperbolic Coxeter groups, Kapovich [@kapovich2007convex] has shown that fundamental groups of certain Gromov--Thurston manifolds admit Anosov representations (note that in high dimensions, groups of the latter form are not commensurable to Coxeter groups [@js2003]). In another direction, some combination theorems for Anosov subgroups have been established in recent years [@DGKexamples; @dkl2019; @dk2022; @dk2023], which can be used to prove that the class of groups admitting Anosov embeddings into $\mathop{\mathrm{SL}}(d, \ensuremath{\mathbb{R}})$ for some $d$ is closed under free products [@dgk2017convex; @dt2022anosov]. In this paper, we prove: **Theorem 3**. *Every cubulated hyperbolic group admits an Anosov representation.* significantly enlarges the family of groups known to admit Anosov representations; see below for some example applications. However, our proof also provides new examples of Anosov representations even for groups already known to admit them, since our main theorem ( below) can provide different Anosov representations for different cubulations of the same group. Before stating the result, we briefly recall the definition of an Anosov representation. For a matrix $g \in \mathop{\mathrm{GL}}(d, \ensuremath{\mathbb{R}})$ and $1 \le k \le d$, we let $\sigma_k(g)$ denote the $k$^th^ largest *singular value* of $g$, i.e. the $k$^th^ largest eigenvalue of the matrix $\sqrt{gg^T}$, counted with multiplicity. **Definition 4**. Let $\Gamma$ be a finitely generated group, equipped with the word metric $|\cdot|$ induced by a finite generating set. A representation $\rho\colon \Gamma \to \mathop{\mathrm{GL}}(d, \ensuremath{\mathbb{R}})$ is *$k$-Anosov* for some $1 \le k < d$ if there are constants $A, B > 0$ so that for all $\gamma \in \Gamma$, $$\log\left(\frac{\sigma_k(\rho(\gamma))}{\sigma_{k+1}(\rho(\gamma))}\right) \ge A|\gamma| - B.$$ A $1$-Anosov representation is also called a *projective* Anosov representation. We will discuss singular values and Anosov representations in further detail in . For now, we remark that bears little resemblance to Labourie's original definiton of an Anosov representation, which was stated in terms of a certain flow on a bundle associated to the representation; the equivalence of the definitions is a theorem due to Kapovich--Leeb--Porti [@klp2017characterizations] and Bochi--Potrie--Sambarino [@BPS]. Our main theorem is as follows: **Theorem 5**. *Let $(C, S)$ be a right-angled Coxeter system, let $\Gamma \hookrightarrow C$ be a quasiconvex embedding of a hyperbolic group $\Gamma$ into $C$, and let $\rho\colon C \to \mathop{\mathrm{SL}}^{\pm}(|S|, \ensuremath{\mathbb{R}})$ be a simplicial representation of $C$ whose Cartan matrix is *fully nondegenerate*. Then the restriction $\rho|_\Gamma$ is $1$-Anosov.* In this paper, a *simplicial representation* of a right-angled Coxeter group $C$ is a deformation of the well-known *geometric representation* $C \to \mathop{\mathrm{SL}}^{\pm}(|S|, \ensuremath{\mathbb{Z}})$ studied by Tits [@bourbaki68]; see for more detail. The geometric representation itself essentially never satisfies the technical condition demanded by , but representations with this property do exist for any right-angled Coxeter group. In fact, these representations form an open dense subset of the space of simplicial representations, and can even be arranged to have image lying in $\mathrm{O}(p, q)$ or $\mathop{\mathrm{SL}}^{\pm}(|S|, \ensuremath{\mathbb{Z}})$ (see ). Thus, our proof of does not reprove the result of Haglund--Wise and Agol that cubulated hyperbolic groups are linear. Rather, we use their characterization of cubulated hyperbolic groups as hyperbolic virtual quasiconvex subgroups of right-angled Coxeter groups, and apply . then follows once we know that admitting an Anosov representation is a commensurability invariant (see [@dt2022anosov Lemma 2.1]). ## Actions of reflection groups on projective space {#sec:refproj} In the special case where the ambient right-angled Coxeter group $C$ in is Gromov-hyperbolic, then work of Danciger--Guéritaud--Kassel--Lee--Marquis [@DGKLM] (see also [@DGKgeomded]) implies that any representation $\rho\colon C \to \mathop{\mathrm{SL}}^{\pm}(|S|, \ensuremath{\mathbb{R}})$ as in the theorem is already $1$-Anosov; in that case it follows easily that the restriction of $\rho$ to any quasiconvex subgroup of $C$ is 1-Anosov also. However, the Haglund--Wise construction typically yields a quasiconvex embedding of a compact special hyperbolic group $\Gamma$ into a *non-hyperbolic* right-angled Coxeter group $C$, and when this occurs we cannot invoke the results in [@DGKgeomded] or [@DGKLM]. Further, we know of no procedure that replaces $C$ with a hyperbolic Coxeter group (although we do not know of any reason such a procedure cannot exist). Our proof of ultimately differs significantly from the approach in [@DGKgeomded] and [@DGKLM], and provides a new proof of the fact that hyperbolic right-angled Coxeter groups admit Anosov representations. Indeed, our methods rely on a completely different characterization of Anosov representations. However, the starting point for both proofs is the same: we consider representations $C \to \mathop{\mathrm{SL}}^{\pm}(|S|, \ensuremath{\mathbb{R}})$ which are *generated by reflections*. The general theory of such representations was developed by Vinberg [@Vinberg1971], generalizing Tits' study of the geometric representation, and centers around the action of $C$ on a *convex domain* in real projective space $\ensuremath{\mathbb{P}}(\ensuremath{\mathbb{R}}^{|S|})$. As an illustrative example, the geometric representation of the free product $\Gamma = \ensuremath{\mathbb{Z}}/2 * \ensuremath{\mathbb{Z}}/2 * \ensuremath{\mathbb{Z}}/2$ realizes $\Gamma$ as a reflection lattice in $\mathrm{O}(2,1)$, acting on the *projective* or *Klein* model of the hyperbolic plane embedded in $\ensuremath{\mathbb{P}}(\ensuremath{\mathbb{R}}^3)$ as a convex ball. Although $\Gamma$ is a hyperbolic group in this case, the geometric representation fails to be Anosov, as the product of any pair of distinct generators is a nontrivial unipotent element in $\mathrm{O}(2,1)$. However, $\Gamma$ also admits *convex cocompact* representations into $\mathrm{O}(2,1)$, where the product of any pair of distinct generators is instead loxodromic. Such representations also preserve another convex domain $\Omega \subset \ensuremath{\mathbb{P}}(\ensuremath{\mathbb{R}}^3)$, called the *Vinberg domain*, which is *not* projectively equivalent to the Klein model for $\ensuremath{\mathbb{H}}^2$. ## Proof idea Our proof of heavily exploits the relationship between the projective geometry of the Vinberg domain $\Omega$ for a right-angled Coxeter system $(C, S)$ acting by reflections, and the combinatorial geometry of the *Davis complex* $\mathrm{D}(C,S)$, a natural $\mathrm{CAT}(0)$ cube complex with a properly discontinuous and cocompact $C$-action. Specifically, we relate *half-spaces* in $\mathrm{D}(C,S)$ to certain convex subsets of projective space, which we call *half-cones*. By examining the nesting properties of half-cones, we are able to prove that if a geodesic $\gamma_n$ in an irreducible right-angled Coxeter group $C$ does not get "stuck" inside of a proper *standard* subgroup in $C$ (i.e. a subgroup generated by a subset of the generating set $S$) then the singular value gaps of the sequence $\rho(\gamma_n)$ grow at a uniform rate. This directly verifies the condition in . Our proof also needs to handle the case where the geodesic $\gamma_n$ spends arbitrarily long amounts of time inside of standard subgroups of $C$, and this is where the majority of the work takes place. The strategy is to induct on the size of the generating set $S$, and assume that the geodesic $\gamma_n$ experiences uniform singular value gap growth on each sub-geodesic lying in a proper standard subgroup of $C$. The challenge is then to "glue together" the singular value gap growth on each of these sub-geodesics. This "gluing" process is somewhat involved, but there are essentially only two different techniques at play. One of them is a *uniform transversality* argument for stable and unstable subspaces of elements in $\mathop{\mathrm{SL}}^{\pm}(|S|, \ensuremath{\mathbb{R}})$, and relies on an understanding of the convex projective geometry of the Vinberg domain for $\rho$. The other technique is to apply the *higher-rank Morse lemma* and *local-to-global principle* of Kapovich--Leeb--Porti [@KLP2018; @KLP2023] (see also [@Riestenberg]), a pair of deep theorems about the geometry of certain quasi-geodesic sequences in higher-rank symmetric spaces. **Remark 6**. An interesting feature of our proof is that it uses the hyperbolicity of the quasiconvex subgroup $\Gamma$ only indirectly, in the form of a condition on the walls in $\mathrm{D}(C,S)$ crossed by an arbitrary geodesic in $\Gamma$ (see ). As a consequence, our proof actually shows that this condition implies hyperbolicity of $\Gamma$, since any group admitting an Anosov representation is necessarily hyperbolic; see . In the special case where $\Gamma = C$, this gives an alternative (albeit inefficient) proof of Gromov's "no empty square" characterization of hyperbolic right-angled Coxeter groups relying on the theory of Anosov representations (this was also accomplished in [@DGKgeomded] and [@leemarquis2019]). ## Examples and applications {#sec:examples} provides evidence that groups admitting Anosov representations are in some sense "abundant." One way to make this precise is Gromov's density model for random finitely presented groups: at density $<\frac{1}{12}$, a random group satisfies the $C'(\frac{1}{6})$ small cancellation condition, and such groups are hyperbolic and cubulated [@wise2004cubulating]; more generally, at density $<\frac{1}{6}$, random groups are hyperbolic and cubulated (see [@gromov1993asymptotic 9.B], [@ollivier2011cubulating]). Hence, by , random finitely presented groups at density $< \frac16$ admit Anosov representations. These results tell us that we cannot control the dimension of the Anosov representations provided by even for torsion-free cubulated hyperbolic groups of bounded cohomological dimension: for fixed $n$, a random group at any positive density has no $n$-dimensional linear representations with finite kernel [@kozmalubotzky], while random groups at density $<\frac{1}{2}$ have cohomological dimension $2$. We mention, however, that not all groups that admit $1$-Anosov representations into $\mathrm{SL}(d,\ensuremath{\mathbb{R}})$---indeed, not all convex cocompact subgroups of rank-one Lie groups---are cubulated. For example, any action of a discrete group with Kazhdan's property (T) on a finite-dimensional $\mathrm{CAT}(0)$ cube complex has a global fixed point [@nibloreeves97], ruling out cubulability for uniform lattices in $\mathrm{Sp}(n,1)$, $n \geq 2$, and $F_{4{(-20)}}$. For subtler reasons, it is also true that uniform lattices in $\mathrm{PU}(n,1)$, $n \geq 2$, fail to be cubulated [@delzantgromov; @py2013coxeter]. ### Strict hyperbolization Recent work of Lafont and Ruffoni [@lafont2022special] has established that the Charney--Davis *strict hyperbolization* process [@charney1995strict] also yields cubulated hyperbolic groups, which allows us to produce examples of Anosov subgroups with various "exotic" properties. For instance, Ontaneda [@ontaneda] has used strict hyperbolization to construct new examples of closed negatively curved Riemannian manifolds in any dimension $n \ge 4$ which are not homeomorphic to any locally symmetric space of rank one. Work of Januszkiewicz--Świa̧tkowski [@js2003] implies that the fundamental groups of these manifolds are not commensurable to any Coxeter group when $n > 61$, meaning that gives the first proof that these groups admit Anosov representations. For another sample application of and strict hyperbolization, recall that if $M$ is a closed negatively curved Riemannian manifold, then the Gromov boundary of $\pi_1(M)$ is a topological sphere. However, Davis--Januszkiewicz [@dj91] showed that the latter may fail if $M$ is merely a closed aspherical manifold with hyperbolic fundamental group. The Davis--Januszkiewicz examples are constructed via strict hyperbolization, so their fundamental groups are cubulated by the work of Lafont--Ruffoni. Combining these results with a theorem of Bestvina [@bestvina96 Thm. 2.8] and Theorem [Theorem 3](#thm:cubulated_anosov){reference-type="ref" reference="thm:cubulated_anosov"} yields the following: **Theorem 7**. *For every $n \geq 4$, there is some $d \in \mathbb{N}$ and an Anosov subgroup of $\mathrm{SL}(d, \mathbb{R})$ whose Gromov boundary is not homeomorphic to an $n$-sphere, but is nevertheless a homology $n$-manifold with the homology of an $n$-sphere.* This theorem may be viewed as a positive answer for each $n \geq 4$ to a variant of a question of Kapovich [@kapovich2008kleinian Q. 9.4], within the broader realm of Anosov groups. It in fact follows from known results that Theorem [Theorem 7](#exotic){reference-type="ref" reference="exotic"} also holds for $n=3$. Indeed, for an example with $n=3$, it suffices to take one of the Anosov representations guaranteed by Danciger--Guéritaud--Kassel [@DGKgeomded] of a right-angled Coxeter group given by a flag no-square triangulation of a nontrivial homology $3$-sphere (see [@davis1983]); for the existence of such triangulations, see [@ps2009flag]. That the latter approach fails as soon as $n \geq 4$ follows from aforementioned work of Januszkiewicz--Świa̧tkowski [@js2003 Sect. 2.2]. **Remark 8**. The situation for groups admitting $1$-Anosov representations into $\mathop{\mathrm{SL}}(d, \ensuremath{\mathbb{R}})$ appears to be strikingly different from the situation for groups admitting *Borel* Anosov representations into $\mathop{\mathrm{SL}}(d, \ensuremath{\mathbb{R}})$ (a representation $\rho\colon \Gamma \to \mathop{\mathrm{SL}}(d, \ensuremath{\mathbb{R}})$ is Borel Anosov if it is $k$-Anosov for every $1 \le k < d$). Indeed, Sambarino conjectured that every group admitting a Borel Anosov representation into $\mathop{\mathrm{SL}}(d, \ensuremath{\mathbb{R}})$ is virtually either a free group or the fundamental group of a closed surface, and this conjecture has been verified for infinitely many $d$ (see [@ct2020], [@tsouvalas2020], [@dey2022]). ## Further questions We have already observed that would no longer hold if we removed the hypothesis regarding fully nondegenerate Cartan matrices (see the end of Section [1.2](#sec:refproj){reference-type="ref" reference="sec:refproj"}). However, it seems plausible that a version of could hold with a considerably relaxed version of this hypothesis, as long as we impose some assumptions on the quasiconvex embedding $\Gamma \hookrightarrow C$. For instance, it might be sufficient to ask for $\Gamma$ to have finite intersection with every standard virtually unipotent subgroup of $C$. For appropriate quasiconvex embeddings, this could allow the conclusion of to hold for an *arbitrary* representation of $C$ by reflections (in particular, for the geometric representation). Even in its current form, still gives us a great deal of freedom to pick the simplicial representation $\rho\colon C \to \mathop{\mathrm{SL}}^{\pm}(d, \ensuremath{\mathbb{R}})$. In particular, for a fixed infinite right-angled Coxeter group $C$, we can pick $\rho$ from a positive-dimensional submanifold $M$ of the representation variety $\mathop{\mathrm{Hom}}(C, \mathop{\mathrm{SL}}^{\pm}(d, \ensuremath{\mathbb{R}}))$. We might want to consider properties of the restriction map $R\colon M \to \mathop{\mathrm{Hom}}(\Gamma, \mathop{\mathrm{SL}}^{\pm}(d, \ensuremath{\mathbb{R}}))$---for instance, it would be interesting to know the dimension of $R(M)$, or whether $R(M)$ contains any representations with Zariski-dense image. ## Acknowledgments The authors would like to thank Gye-Seon Lee, Jason Manning, Lorenzo Ruffoni, and Abdul Zalloum for productive discussion. We are also grateful to the Mathematisches Forschungsinstitut Oberwolfach and the organizers of the Arbeitsgemeinschaft: Higher Rank Teichmüller Theory (2241), where this work was initiated. S.D. was supported by the Huawei Young Talents Program. B.F. was partially supported by the grant NRF-2022R1I1A1A01072169 and received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (ERC starting grant DiGGeS, grant agreement No 715982, and ERC consolidator grant GeometricStructures, grant agreement No 614733). T.W. was partially supported by NSF grant DMS-2202770. F.Z. was partially supported by ISF grant 737/20 and an AMS-Simons Travel Grant. # Cube complexes and right-angled Coxeter groups {#sec:cube_complexes_RACGs} In this section we briefly review some essential background on the topic of nonpositively curved cube complexes, right-angled Coxeter groups, and the Davis complex. We refer to [@HW08], [@Davis] for further detail. Afterwards, we introduce a useful combinatorial framework for working with geodesics in the Davis complex, in the form of *itineraries*. ## CAT(0) cube complexes For our purposes, a *cube complex* $X$ is a finite-dimensional cell complex in which each cell is a cube and attaching maps are combinatorial isomorphisms onto their images. A $d$-cell of $X$ is a *$d$-cube*. A $0$-cube is a *vertex*, a $1$-cube is an *edge*, and a $2$-cube is a *square*. Under the identification of a $d$-cube $c$ of $X$ with $[-1,1]^d$, a *midcube* of $c$ is an intersection of $c$ with a coordinate hyperplane of $\mathbb{R}^d$. By gluing midcubes of adjacent cubes of $X$ whenever they meet, one obtains immersed subspaces of $X$, called *hyperplanes*, each of which also carries a natural cube complex structure. Note that a compact cube complex possesses only finitely many hyperplanes. Two edges of a cube complex $X$ are *elementary parallel* if they appear as opposite edges of a square in $X$. A *wall* of $X$ is a class of the equivalence relation on the edge set of $X$ generated by elementary parallelisms. Two edges of $X$ are *parallel* if they belong to the same wall of $X$ (in other words, if they are dual to the same hyperplane of $X$). Frequently, we will abuse terminology and refer to properties of "walls" when we really mean properties of the corresponding hyperplanes. In particular, when we say that an intersection of walls $W_1 \cap W_2$ is empty or nonempty, we mean to refer to an intersection of hyperplanes. One says a cube complex $X$ is *nonpositively curved* if the link of each vertex of $X$ is a flag simplicial complex; recall that a simplicial complex $L$ is *flag* if each clique $\mathcal{V}$ in $L$ spans a $(|\mathcal{V}|-1)$-simplex. If $X$ is moreover simply connected, then $X$ is said to be $\mathrm{CAT}(0)$. Implicit in this terminology is a theorem of Gromov [@gromov] that the path metric on a cube complex $X$ induced by the Euclidean metric on each of its cubes is $\mathrm{CAT}(0)$ if and only if $X$ is $\mathrm{CAT}(0)$ in the previous combinatorial sense. A key feature of a $\mathrm{CAT}(0)$ cube complex is that each of its hyperplanes is separating. ## Right-angled Coxeter groups Let $\Sigma$ be a finite simplicial graph with vertex set $S$. The *right-angled Coxeter group* $C_\Sigma$ with *nerve* $\Sigma$ is the group given by the presentation with generating set $S$ subject to the relations that two generators $s,t \in S$ commute if and only if $s$ and $t$ are adjacent as vertices in $\Sigma$. The pair $(C_\Sigma, S)$ is a *right-angled Coxeter system*. A conjugate within $C_\Sigma$ of an element of $S$ is a *reflection*. If the complement of $\Sigma$ is connected, we say $(C_\Sigma, S)$ is *irreducible*; this is equivalent to saying that $C_\Sigma$ does not decompose as a nontrivial direct product. If $(C,S)$ is a right-angled Coxeter system and $T \subset S$, we denote by $C(T)$ the subgroup of $C$ generated by $T$. We refer to such subgroups of $C$ as *standard subgroups*. For any $T \subset S$, the pair $(C(T), T)$ is again a right-angled Coxeter system. ### The Davis complex Given a finite simplicial graph $\Sigma$, there is a $\mathrm{CAT}(0)$ cube complex $\mathrm{D}(C_\Sigma, S)$, called the *Davis complex* of $(C_\Sigma, S)$, on which the group $C_\Sigma$ acts by combinatorial automorphisms, that may be constructed as follows. Let $\mathrm{D}'(C_\Sigma, S)$ be the square complex (i.e. $2$-dimensional cube complex) obtained by attaching a square to each labeled $4$-cycle of the form $stst$ in the Cayley graph $\mathrm{Cay}(C_\Sigma,S)$ of $C_\Sigma$ with respect to the generating set $S$, where $s,t \in S$. Then $\mathrm{D}(C_\Sigma, S)$ is the unique nonpositively curved cube complex with $2$-skeleton $\mathrm{D}'(C_\Sigma, S)$. Each wall $W$ in $\mathrm{D}(C_\Sigma, S)$ is fixed by a unique reflection $r$ in $C_\Sigma$, and conversely each reflection $r$ fixes a unique wall. As any reflection $r$ is a conjugate of a unique $s \in S$, the edges comprising any given wall $W$ of $\mathrm{D}(C_\Sigma, S)$ are all labeled with a single generator $s(W) \in S$. We call $s(W)$ the *type* of $W$. For each $s \in S$, we also let $W(s)$ denote the unique wall fixed by the reflection $s$. ### Quasiconvex subgroups Given a right-angled Coxeter system $(C,S)$, a subgroup $\Gamma < C$ is *quasiconvex* (with respect to the standard generating set $S$) if, viewing $\Gamma$ as a subset of the vertices of $\mathrm{D}(C,S)$, there is some $K > 0$ such that every combinatorial geodesic in $\mathrm{D}(C,S)$ with endpoints in $\Gamma$ lies in the combinatorial $K$-neighborhood of $\Gamma$. (For example, if $s_1, t_1, s_2, t_2$ are distinct elements of $S$ such that each element of $\{s_1, t_1\}$ commutes with each element of $\{s_2,t_2\}$, but $s_i$ and $t_i$ do not commute for $i=1,2$, then the cyclic subgroup $\langle s_1t_1s_2t_2 \rangle < C$ is *not* quasiconvex.) A subgroup $\Gamma < C$ is quasiconvex if and only if there is a $\Gamma$-invariant convex subcomplex $\widetilde{Y}$ of $\mathrm{D}(C,S)$ on which $\Gamma$ acts cocompactly; see [@HW08 Cor. 7.8] or [@haglund2008finite Thm. H]. In particular, standard subgroups of $C$ are quasiconvex. This second characterization of quasiconvexity is the one that will be relevant to us. It turns out that the hyperbolic groups that are cubulated are precisely those that virtually embed as quasiconvex subgroups of right-angled Coxeter groups. Indeed, it follows from seminal work of Haglund--Wise [@HW08] and Agol [@agolVHC] that if a hyperbolic group $\Gamma$ acts properly discontinuously and cocompactly on a $\mathrm{CAT}(0)$ cube complex $\widetilde{X}$, then there is a finite-index subgroup $\Gamma' < \Gamma$, a right-angled Coxeter system $(C,S)$, an embedding $\iota\colon \Gamma' \rightarrow C$, and an $\iota$-equivariant embedding of $\widetilde{X}$ as a convex subcomplex of $\mathrm{D}(C,S)$. In more detail, Agol [@agolVHC] showed that there is a finite-index torsion-free subgroup $\Lambda < \Gamma$ such that the cube complex $\widetilde{X} / \Lambda$ is *special* in the sense of Haglund and Wise [@HW08], answering a question of the latter two authors [@HW08 Prob. 11.7]. One can then pass to a deeper finite-index subgroup $\Gamma' < \Lambda$ such that $X:= \widetilde{X} / \Gamma'$ is moreover *$C$-special*; see [@HW08 Prop. 3.10]. The latter condition is designed to ensure the existence of a *local isometry* from the cube complex $X$ to the orbicomplex $\mathrm{D}(C_X, S_X)/C_X$ for some right-angled Coxeter system $(C_X, S_X)$ associated to $X$, inducing an embedding $\iota\colon \Gamma' \rightarrow C_X$ between (orbicomplex) fundamental groups, and lifting to an $\iota$-equivariant embedding of $\widetilde{X}$ as a convex subcomplex of $\mathrm{D}(C_X,S_X)$. (Alternatively, one can find a finite-index subgroup $\Gamma'' < \Lambda$ so that $\widetilde{X} / \Gamma''$ admits a local isometry into the Salvetti complex for a right-angled Artin group, and then apply [@dj2000].) We remark that, even in the case that the action of $\Gamma$ on $\widetilde{X}$ is the action of a hyperbolic right-angled Coxeter group on its Davis complex, the right-angled Coxeter group $C_X$ that one obtains via the process above may *not* be hyperbolic. ## Itineraries Throughout this paper, we will need a good understanding of the combinatorial behavior of geodesics in right-angled Coxeter groups, and especially geodesics lying in quasiconvex hyperbolic subgroups of right-angled Coxeter groups. It is often convenient to work not with the geodesics themselves, but rather some related combinatorial data in the form of an *itinerary*. For the following, we fix a right-angled Coxeter system $(C, S)$. **Definition 9**. Recall that a wall $W$ in the Davis complex $\mathrm{D}(C,S)$ can be viewed as an equivalence class of edges in the Cayley graph $\mathop{\mathrm{Cay}}(C, S)$. An *itinerary* is a sequence of walls $W_1, \ldots, W_n$, such that for some sequence of edges $e_i \in W_i$, the sequence $e_1 \cdots e_n$ is a geodesic edge path in $\mathop{\mathrm{Cay}}(C, S)$. We say that this edge path *follows* the itinerary $\ensuremath{\mathcal{W}}$. A geodesic edge path $e_1 \cdots e_n$ in the Cayley graph $\mathop{\mathrm{Cay}}(C, S)$ always determines a geodesic word in $S$, by reading off the generators $s_i \in S$ labeling each edge $e_i$; conversely, a geodesic word in $S$ gives rise to infinitely many different geodesic edge paths, one for each possible base point of the path in $\mathop{\mathrm{Cay}}(C, S)$. So a geodesic word records strictly less information than a geodesic edge path in $\mathop{\mathrm{Cay}}(C, S)$. On the other hand an itinerary records strictly *more* information than a geodesic word but *less* information than a geodesic edge path: there may be many different edge paths which follow the same itinerary $W_1, \ldots, W_n$, but each of these edge paths determines the same geodesic word, namely the word $$s(W_1) \cdots s(W_n).$$ **Definition 10**. If $\ensuremath{\mathcal{W}}$ is an itinerary $W_1, \ldots, W_n$, we say the geodesic word $s(W_1) \cdots s(W_n)$ is *traversed* by $\ensuremath{\mathcal{W}}$. We also say that $\ensuremath{\mathcal{W}}$ *traverses* the unique element $\gamma \in C$ represented by this geodesic word. If $\alpha \in C$ is the initial vertex of some edge path following $\ensuremath{\mathcal{W}}$, then we say that $\ensuremath{\mathcal{W}}$ *departs from* $\alpha$. Similarly, if $\beta \in C$ is the last vertex of an edge path following $\ensuremath{\mathcal{W}}$, then $\ensuremath{\mathcal{W}}$ *arrives at* $\beta$. If $\alpha, \beta \in C$, and $\ensuremath{\mathcal{W}}$ departs from $\alpha$ and traverses $\alpha^{-1}\beta$, then we say that $\ensuremath{\mathcal{W}}$ *joins* $\alpha$ to $\beta$. Note that this is stronger than saying that $\ensuremath{\mathcal{W}}$ departs from $\alpha$ and arrives at $\beta$. The proposition below follows directly from the fact that walls are defined to be equivalence classes of edges in the Cayley graph of the Coxeter system $(C,S)$: **Proposition 11**. *Let $W_1, \ldots, W_n$ be an itinerary departing from the identity, and let $s_i = s(W_i)$ for all $1 \le i \le n$. Then $W_n = s_1 \cdots s_{n-1} W(s_n)$.* In general a single itinerary can depart from different elements in $C$ (and likewise can arrive at different elements in $C$). However, an itinerary always traverses a unique element. **Definition 12**. If $\ensuremath{\mathcal{U}} = W_1, \ldots, W_n$ is an itinerary, we let $\gamma(\ensuremath{\mathcal{U}}) = \gamma(W_1, \ldots, W_n)$ denote the group element traversed by $\ensuremath{\mathcal{U}}$. If $W_i, W_j$ are walls in $\ensuremath{\mathcal{U}}$, with $W_i$ appearing before $W_j$, we let $\gamma_{\ensuremath{\mathcal{U}}}(W_i, W_j)$ denote the group element traversed by the sub-itinerary of $\ensuremath{\mathcal{U}}$ beginning with $W_i$ and ending with $W_j$. ### Partial order on walls As any wall in $\mathrm{D}(C,S)$ separates $\mathrm{D}(C,S)$ into two convex components, it follows that for any $\alpha, \beta \in C$, the walls appearing in an itinerary $W_1 \ldots , W_n$ joining $\alpha$ to $\beta$ are precisely the walls in $\mathrm{D}(C,S)$ separating $\alpha$ from $\beta$. Motivated by this, we introduce the following notation. **Definition 13**. For $\alpha, \beta \in C$, we let $\mathbf{W}(\alpha, \beta)$ denote the set of walls in $\mathrm{D}(C,S)$ separating $\alpha$ from $\beta$. We write $\mathbf{W}(\alpha)$ for $\mathbf{W}(\mathrm{id}, \alpha)$. The set $\mathbf{W}(\alpha, \beta)$ is endowed with a partial order $<$, defined as follows: if $W_i, W_j \in \mathbf{W}(\alpha, \beta)$, then $W_i < W_j$ if $W_i$ separates $\alpha$ from $W_j$ in $\mathrm{D}(C,S)$ (equivalently, if $W_j$ separates $W_i$ from $\beta$). Recall that two elements $a, b$ in a poset are *incomparable* if neither $a < b$ nor $b < a$ holds. We say that two disjoint subsets $A, B$ of a poset are *completely incomparable* if every element of $A$ is incomparable to every element in $B$. It is immediate that if $\alpha, \beta \in C$ and $W_i, W_j$ are two walls in $\mathbf{W}(\alpha, \beta)$, then $W_i$ and $W_j$ are incomparable with respect to $<$ if and only if $W_i \cap W_j$ is nonempty. When this occurs, the generators $s(W_i)$ and $s(W_j)$ must commute. Every itinerary joining $\alpha$ to $\beta$ determines a total ordering of the set $\mathbf{W}(\alpha, \beta)$ which is compatible with the partial ordering $<$. The proposition below says that all compatible total orderings of $\mathbf{W}(\alpha, \beta)$ arise in precisely this way. **Proposition 14**. *Let $\alpha, \beta \in C$. There is a one-to-one correspondence between the following three sets:* 1. *Itineraries joining $\alpha$ to $\beta$,* 2. *Geodesic words in $S$ representing $\alpha^{-1}\beta$,* 3. *Total orderings of $\mathbf{W}(\alpha, \beta)$ which are compatible with $<$.* *Proof.* The correspondence between the first two sets is immediate, once we recognize that itineraries joining $\alpha$ to $\beta$ are in one-to-one correspondence with geodesic edge paths in the Cayley graph $\mathop{\mathrm{Cay}}(C, S)$ joining $\alpha$ to $\beta$. We have already observed that any itinerary joining $\alpha$ to $\beta$ gives rise to an ordering on $\mathbf{W}(\alpha, \beta)$ compatible with $<$, so we just need to check that any such ordering determines an itinerary. First observe that for any $\gamma \in C$ and any wall $W$ in $\mathrm{D}(C,S)$, if no walls separate $\gamma$ from $W$, then $W$ contains a unique edge incident to $\gamma$. To see this, let $H_\pm$ denote the half-spaces in $\mathrm{D}(C,S)$ bounded by $W$, chosen so that $\gamma \in H_-$. We consider a minimal-length edge path $p$ in $\mathop{\mathrm{Cay}}(C, S)$ joining $\gamma$ to $H_+$. The last edge $e_2$ in $p$ must belong to $W$, since it crosses from $H_-$ to $H_+$. If there is more than one edge in $p$ and $e_1$ is the next-to-last edge, then, by minimality of $p$, the edges $e_1$ and $e_2$ do not lie in a common square of $\mathrm{D}(C,S)$. It follows that the edge path $e_1e_2$ is a geodesic segment for the $\mathrm{CAT}(0)$ metric on $\mathrm{D}(C,S)$ (indeed, it is more generally true that a local isometry between $\mathrm{CAT}(0)$ cube complexes is an isometric embedding with respect to their $\mathrm{CAT}(0)$ metrics; see [@HW08 Lem. 2.11], [@bridsonhaefliger Prop. II.4.14]). The nearest point projection to $e_i$ of the hyperplane $\Pi_i$ dual to $e_i$ is the midpoint of $e_i$, so that the $\Pi_i$ are disjoint. The wall corresponding to $\Pi_1$ thus separates $\gamma$ from $W$, a contradiction. Now, fix an ordering $W_1, \ldots, W_n$ on $\mathbf{W}(\alpha, \beta)$ which is compatible with $<$. The previous claim tells us that $W_1$ contains a unique edge in $\mathop{\mathrm{Cay}}(C, S)$ incident to $\alpha$. If $\alpha'$ is the other endpoint of this edge, then $W_2, \ldots, W_n$ is an ordering on $\mathbf{W}(\alpha', \beta)$, compatible with the partial ordering $<$ on this set. Proceeding iteratively, we then construct an edge path in $\mathop{\mathrm{Cay}}(C, S)$ from $\alpha$ to $\beta$ which crosses exactly the sequence of walls $W_1, \ldots, W_n$, meaning this sequence is an itinerary. ◻ **Definition 15**. We say two itineraries $\ensuremath{\mathcal{U}}, \ensuremath{\mathcal{U}}'$ are *equivalent* if there are elements $\alpha, \beta \in C$ so that both $\ensuremath{\mathcal{U}}$ and $\ensuremath{\mathcal{U}}'$ join $\alpha$ to $\beta$. means that equivalent itineraries $\ensuremath{\mathcal{U}}, \ensuremath{\mathcal{U}}'$ always consist of the same set of walls. And, if $\ensuremath{\mathcal{U}}$ joins $\alpha$ to $\beta$ for some $\alpha, \beta \in C$, then so does any equivalent itinerary $\ensuremath{\mathcal{U}}'$. Thus the third condition of ensures that the definition of "equivalence" actually describes an equivalence relation. ### Efficient itineraries Whenever $W$ and $W'$ are walls in $\mathrm{D}(C,S)$, then there is always some itinerary $\ensuremath{\mathcal{U}}$ whose first wall is $W$ and whose last wall is $W'$. Every wall $W_i$ in $\ensuremath{\mathcal{U}}$ must either separate $W$ from $W'$, or intersect at least one of $W, W'$. tells us that we can always find another itinerary equivalent to $\ensuremath{\mathcal{U}}$ by putting all of the walls in $\ensuremath{\mathcal{U}}$ intersecting $W$ or $W'$ either first or last. That is, if there are any walls in $\ensuremath{\mathcal{U}}$ which intersect either $W$ or $W'$, we can reorder the walls and restrict to a strictly shorter sub-itinerary to get a new (non-equivalent) itinerary whose first wall is $W$ and whose last wall is $W'$. On the other hand, if $\ensuremath{\mathcal{U}}$ is *any* itinerary with initial wall $W$ and final wall $W'$, then $\ensuremath{\mathcal{U}}$ must contain every wall separating $W$ from $W'$. **Definition 16**. We say that an itinerary $\ensuremath{\mathcal{U}} = W_1, \ldots, W_n$ is *efficient* if every wall $W_i$ with $1 < i < n$ is disjoint from both $W_1$ and $W_n$. Given any two distinct walls $W, W'$, the argument above shows that there is always an efficient itinerary with initial wall $W$ and final wall $W'$, and that each efficient itinerary between $W$ and $W'$ must consist of the same set of walls. The itinerary orders these walls in a way which is compatible with the partial ordering: $W_i < W_j$ if $W_i$ separates $W$ from $W_j$. Thus means that any pair of efficient itineraries between $W, W'$ are equivalent, and we can define the following. **Definition 17**. Let $W_1$, $W_2$ be distinct walls in $\mathrm{D}(C,S)$. We let $\gamma(W_1, W_2)$ denote the unique element in $C$ traversed by any efficient itinerary whose first wall is $W_1$ and whose last wall is $W_2$. In general, not every itinerary is equivalent to an efficient itinerary, although this is "almost" true in the special case where $\mathrm{D}(C,S)$ is hyperbolic; see below. # Bounded product projections {#sec:bounded_product_projections} In this section we fix a right-angled Coxeter system $(C, S)$, and use the setup from the previous section to prove some combinatorial results about *hyperbolic subcomplexes* of the Davis complex $\mathrm{D}(C,S)$. Our main aim is to prove , which implies that every geodesic in a hyperbolic subcomplex of $\mathrm{D}(C,S)$ is traversed by an itinerary consisting almost entirely of "regularly-spaced" pairwise disjoint walls. In later sections, we will be able to work with geodesics in $C$ by only considering this set of disjoint walls. **Definition 18**. Suppose that $(C, S)$ is a right-angled Coxeter system. Let $\gamma \in C$, and let $D > 0$. We say a group element $\gamma \in C$ has *$D$-bounded product projections* if every pair of disjoint completely incomparable subsets $A, B \subset \mathbf{W}(\gamma)$ satisfies $\min(|A|, |B|) \le D$. We say that a subgroup $\Gamma \leq C$ has *$D$-bounded product projections* if every $\gamma \in \Gamma$ has $D$-bounded product projections. We just say $\Gamma$ has *bounded product projections* if there exists some $D > 0$ so that $\Gamma$ has $D$-bounded product projections. More intuitively, the elements in $C$ with bounded product projections are precisely those elements $\gamma$ whose geodesic representatives do not "travel diagonally" in a combinatorially embedded Euclidean 2-plane $E \hookrightarrow \mathrm{D}(C,S)$; "diagonally" is in reference to the product structure $E = \ensuremath{\mathbb{R}}\times \ensuremath{\mathbb{R}}$ induced by the cubulation of $E$. A geodesic representing an element with $D$-bounded product projections *may* spend an arbitrary amount of time in a 2-flat, but it must spend all but $D$ of its length traveling parallel to one of the $\ensuremath{\mathbb{R}}$ factors. The following is immediate from [@hagen2012geometry Thm. 4.1.3]. **Lemma 19**. *Let $\Gamma$ be a quasiconvex subgroup of $C$. If $\Gamma$ is hyperbolic, then $\Gamma$ has bounded product projections.* **Remark 20**. It is shown in [@hagen2012geometry] that the converse of also holds, that is, that if $\Gamma$ has bounded product projections, then $\Gamma$ is hyperbolic. In fact, this direction also follows from the proof of Theorem [Theorem 5](#thm:mainthm){reference-type="ref" reference="thm:mainthm"}; our proof shows that if $\Gamma$ has bounded product projections, then certain representations of $C$ restrict to Anosov representations of $\Gamma$, and it follows from [@KLP2018], [@BPS] that any group admitting an Anosov representation is hyperbolic. Whenever $\ensuremath{\mathcal{W}}$ is an itinerary, we let $|\ensuremath{\mathcal{W}}|$ denote the number of walls appearing in $\ensuremath{\mathcal{W}}$. If $\ensuremath{\mathcal{W}} = W_1, \ldots, W_n$ and $\ensuremath{\mathcal{U}} = U_1, \ldots U_m$ are itineraries, we write $\ensuremath{\mathcal{W}}, \ensuremath{\mathcal{U}}$ for the concatenation $$W_1, \ldots, W_n, U_1, \ldots, U_m,$$ as long as this sequence of walls is also an itinerary. **Proposition 21**. *Given $D > 0$, there exists $R > 0$ (depending only on $D$) satisfying the following. Suppose that $\gamma \in C$ has $D$-bounded product projections. Then any itinerary traversing $\gamma$ is equivalent to an itinerary $\ensuremath{\mathcal{U}}$ of the form $$\{W_1\}, \ensuremath{\mathcal{V}}_1, \{W_2\}, \ensuremath{\mathcal{V}}_2, \ldots, \{W_n\}, \ensuremath{\mathcal{V}}_n,$$ such that:* 1. *[\[item:smallsep\]]{#item:smallsep label="item:smallsep"} Every $\ensuremath{\mathcal{V}}_i$ satisfies $|\ensuremath{\mathcal{V}}_i| \le R$,* 2. *[\[item:v_intersect\]]{#item:v_intersect label="item:v_intersect"} Every wall in $\ensuremath{\mathcal{V}}_i$ intersects $W_i$,* 3. *[\[item:disjoint_daviscx\]]{#item:disjoint_daviscx label="item:disjoint_daviscx"} For every $i \ne j$ we have $W_i \cap W_j = \emptyset$,* 4. *[\[item:few_intersections\]]{#item:few_intersections label="item:few_intersections"} Every wall $W_i$ intersects at most $R$ other walls in $\ensuremath{\mathcal{U}}$.* **Remark 22**. Conditions [\[item:v_intersect\]](#item:v_intersect){reference-type="ref" reference="item:v_intersect"} and [\[item:few_intersections\]](#item:few_intersections){reference-type="ref" reference="item:few_intersections"} in the proposition together imply [\[item:smallsep\]](#item:smallsep){reference-type="ref" reference="item:smallsep"}, but we still list [\[item:smallsep\]](#item:smallsep){reference-type="ref" reference="item:smallsep"} above because later we will use [\[item:v_intersect\]](#item:v_intersect){reference-type="ref" reference="item:v_intersect"}, [\[item:disjoint_daviscx\]](#item:disjoint_daviscx){reference-type="ref" reference="item:disjoint_daviscx"} and a stronger form of [\[item:smallsep\]](#item:smallsep){reference-type="ref" reference="item:smallsep"} to prove [\[item:few_intersections\]](#item:few_intersections){reference-type="ref" reference="item:few_intersections"}. tells us in particular that the length of any geodesic in $\mathrm{D}(C,S)$ representing some $\gamma \in C$ with $D$-bounded product projections can be estimated (up to a multiplicative constant) as the maximal length of a *chain* $W_1 < W_2 < \ldots < W_n$ in $\mathbf{W}(\gamma)$. In fact, this weaker statement holds for *arbitrary* $\gamma \in C$; the point of the proposition is that when $\gamma$ has $D$-bounded product projections, every wall in the chain is disjoint from almost every other wall in $\mathbf{W}(\gamma)$. The proof of is purely combinatorial. In fact, it relies only on the poset structure of the set of walls separating a pair of elements in $C$. We first prove a useful lemma: **Lemma 23**. *Suppose that $\gamma = \alpha^{-1}\beta$ is a nontrivial element in $C$ with $D$-bounded product projections. Then there exists a minimal wall $W$ in $\mathbf{W}(\alpha, \beta)$ such that $W \cap W' \ne \emptyset$ for at most $(2D + 1) \cdot 4^D$ walls $W'$ in $\mathbf{W}(\alpha, \beta)$.* In particular, the lemma tells us that any group element $\gamma$ with $D$-bounded product projections is traversed by an itinerary which is "nearly" efficient. *Proof.* Without loss of generality we may assume $\alpha = \mathrm{id}$ and consider the set of walls $\mathbf{W}(\gamma)$. We let $\mathbf{W}_{\min}(\gamma)$ denote the minimal walls in $\mathbf{W}(\gamma)$. Every minimal element in a poset is incomparable with every other minimal element. So, the bounded product projections property implies that the number of walls in $\mathbf{W}_{\min}(\gamma)$ is at most $2D + 1$, since otherwise we could partition $\mathbf{W}_{\min}(\gamma)$ into disjoint completely incomparable subsets, both containing at least $D + 1$ elements. We can think of every wall $W \in \mathbf{W}(\gamma)$ as having one of finitely many "intersection types," determined by the walls in $\mathbf{W}_{\min}(\gamma)$ which $W$ intersects. Precisely, we define an "intersection mapping" $I_{\min}:\mathbf{W}(\gamma) \to 2^{\mathbf{W}_{\min}(\gamma)}$, by: $$I_{\min}(W) = \{V \in \mathbf{W}_{\min}(\gamma) : V \cap W \ne \emptyset\}.$$ We claim the following: **Claim 1**. Let $U_1, U_2$ be subsets of $\mathbf{W}_{\min}(\gamma)$, such that $U_1 \nsubseteq U_2$ and $U_2 \nsubseteq U_1$. Then the sets of walls $A = I_{\min}^{-1}(U_1)$ and $B = I_{\min}^{-1}(U_2)$ are completely incomparable. To prove the claim, observe that if the hypothesis holds, then there is a wall $V_1 \in U_1 \setminus U_2$ and a wall $V_2 \in U_2 \setminus U_1$. Since $V_1$ does not intersect any wall in $B$, and $V_1$ is minimal, we have $V_1 < W_B$ for every $W_B \in B$. Similarly, we have $V_2 < W_A$ for every $W_A \in A$. And, since every wall $W_A \in A$ intersects $V_1$ nontrivially, each $W_A$ is incomparable with $V_1$. So, there is a total ordering of $\mathbf{W}(\gamma)$ (compatible with $<$) where each $W_A \in A$ precedes $V_1$, hence precedes every $W_B \in B$. Arguing symmetrically, there is also a compatible ordering of the walls where every $W_B \in B$ precedes every $W_A \in A$. Thus, $A$ and $B$ are completely incomparable, proving the claim. Next, consider the collection of subsets $T \subset 2^{\mathbf{W}_{\min}(\gamma)}$ given by $$T = \{U \subset \mathbf{W}_{\min}(\gamma) : |I_{\min}^{-1}(U)| > 2D + 1\}.$$ The previous claim, together with the bounded product projections axiom, implies that if $U_1$ and $U_2$ are subsets of $\mathbf{W}_{\min}(\gamma)$, neither of which is a subset of the other, then either $|I_{\min}^{-1}(U_1)| \le D$ or $|I_{\min}^{-1}(U_2)| \le D$---so in particular at most one of $U_1, U_2$ can lie in $T$. This means that there is at most one maximal element in $T$, with respect to the partial ordering on $2^{\mathbf{W}_{\min}(\gamma)}$ given by inclusion. However, the element $\mathbf{W}_{\min}(\gamma) \in 2^{\mathbf{W}_{\min}(\gamma)}$ cannot itself lie in $T$: by definition, every element in $I_{\min}^{-1}(\mathbf{W}_{\min}(\gamma))$ is incomparable with every minimal wall, so every element in $I_{\min}^{-1}(\mathbf{W}_{\min}(\gamma))$ is itself minimal in $\mathbf{W}(\gamma)$ and therefore $|I_{\min}^{-1}(\mathbf{W}_{\min}(\gamma))| \le |\mathbf{W}_{\min}(\gamma)| \le 2D + 1$. We conclude that $T$ is either empty, or it has a unique maximal element which is not all of $\mathbf{W}_{\min}(\gamma)$. In either case, there must be some wall $W_- \in \mathbf{W}_{\min}(\gamma)$ such that $W_- \notin U$ for any $U \in T$. In other words, for any set $U \subset \mathbf{W}_{\min}(\gamma)$ containing $W_-$, we have $|I_{\min}^{-1}(U)| \le 2D + 1$. And by definition, the set of walls in $\mathbf{W}(\gamma)$ intersecting $W_-$ is the union of the sets $I_{\min}^{-1}(U)$ over all subsets $U \subset \mathbf{W}_{\min}(\gamma)$ with $W_- \in U$. Since the number of such sets $U$ is at most $2^{|\mathbf{W}_{\min}(\gamma) - 1|} \le 2^{2D} = 4^D$, we conclude that the number of walls in $\mathbf{W}(\gamma)$ intersecting $W_-$ is at most $(2D + 1) \cdot 4^D$. ◻ *Proof of .* Consider an itinerary traversing $\gamma$. As in the proof of the previous lemma, without loss of generality we may assume that this itinerary joins $\mathrm{id}$ to $\gamma$, and contains exactly the walls in $\mathbf{W}(\gamma)$. Because of , our goal is to find a total ordering on $\mathbf{W}(\gamma)$, compatible with $<$, which gives an itinerary $\ensuremath{\mathcal{U}}$ of the desired form. We will find the desired itinerary $\ensuremath{\mathcal{U}}$ iteratively. We let $R' = (2D + 1) \cdot 4^D$. Using , we choose a minimal wall $W_1 \in \mathbf{W}(\gamma)$ which intersects at most $R'$ other walls in $\mathbf{W}(\gamma)$. Then, we let $\ensuremath{\mathcal{V}}_1$ be an itinerary consisting of the set of walls in $\mathbf{W}(\gamma)$ which intersect $W_1$, arranged in an arbitrary order compatible with $<$. We obtain an itinerary traversing $\gamma$ of the form $$\{W_1\}, \ensuremath{\mathcal{V}}_1, \ensuremath{\mathcal{V}}_1',$$ where $|\ensuremath{\mathcal{V}}_1| \le R'$, and every wall in $\ensuremath{\mathcal{V}}_1'$ is disjoint from $W_1$. Using again, we pick a minimal wall $W_2$ in $\ensuremath{\mathcal{V}}_1'$ which intersects at most $R'$ walls in $\ensuremath{\mathcal{V}}_1'$. We let $\ensuremath{\mathcal{V}}_2$ be the walls in $\ensuremath{\mathcal{V}}_1' -\{W_2\}$ which intersect $W_2$ (again arranged in an arbitrary compatible order), and obtain another equivalent itinerary $$\{W_1\}, \ensuremath{\mathcal{V}}_1, \{W_2\}, \ensuremath{\mathcal{V}}_2, \ensuremath{\mathcal{V}}_2'.$$ We proceeed iteratively in this fashion until we have eventually obtained an itinerary $\ensuremath{\mathcal{U}}$ of the form $$\{W_1\}, \ensuremath{\mathcal{V}}_1, \{W_2\}, \ensuremath{\mathcal{V}}_2, \ldots, \{W_n\}, V_n,$$ such that each $\ensuremath{\mathcal{V}}_k$ satisfies $|\ensuremath{\mathcal{V}}_k| \le R'$, and for any $k, \ell$ with $k < \ell$, the wall $W_k$ is disjoint from both $W_\ell$ and every wall in $\ensuremath{\mathcal{V}}_\ell$. So, as long as we take $R \ge R' = (2D + 1) \cdot 4^D$, the itinerary $\ensuremath{\mathcal{U}}$ satisfies conditions [\[item:smallsep\]](#item:smallsep){reference-type="ref" reference="item:smallsep"}, [\[item:v_intersect\]](#item:v_intersect){reference-type="ref" reference="item:v_intersect"}, and [\[item:disjoint_daviscx\]](#item:disjoint_daviscx){reference-type="ref" reference="item:disjoint_daviscx"} in the statement of the proposition. It remains to show that for some choice of $R$, this itinerary also satisfies condition [\[item:few_intersections\]](#item:few_intersections){reference-type="ref" reference="item:few_intersections"}. We claim that taking $R = R'D + D$ is sufficient. To see this, fix $k$, and consider the set $I(W_k)$ of walls in $\mathbf{W}(\gamma)$ which intersect $W_k$. We wish to show that $|I(W_k)| \le R'D + D$. We know that $W_k$ is disjoint from every $W_i$ for $i \ne k$ and from every wall in $\ensuremath{\mathcal{V}}_\ell$ for every $\ell > k$. So, every wall in $I(W_k)$ is contained in some $\ensuremath{\mathcal{V}}_i$ for $i \le k$. Since each $\ensuremath{\mathcal{V}}_i$ contains at most $R'$ walls, there are at most $R'D$ walls contained in the union $$\bigcup_{j = k - D + 1}^k I(W_k) \cap \ensuremath{\mathcal{V}}_j.$$ So, if $I_{k-D}$ denotes the set of walls $$I_{k-D} = \bigcup_{i=1}^{k - D} I(W_k) \cap \ensuremath{\mathcal{V}}_i,$$ we must have $|I(W_k)| \le |I_{k-D}| + R'D$. For any $i, j$ with $i < j < k$, we have $W_i < W_j < W_k$. This means that if some wall $V$ is incomparable with both $W_i$ and $W_k$, it is also incomparable with $W_j$. Now, if $V \in I_{k - D}$, we know that $V \cap W_i$ is nonempty for some $i \le k - D$, and $V \cap W_k$ is nonempty by assumption, so necessarily $V$ intersects $W_j$ for every $j$ with $i \le j \le k$. In particular, $V$ intersects $W_j$ for every $j$ with $k - D \le j \le k$. That is, every wall in $I_{k-D}$ intersects each of the $D + 1$ walls $W_{k-D}, \ldots, W_k$. But then the fact that $\gamma$ has $D$-bounded product projections implies that $|I_{k-D}| \le D$, hence $|I(W_k)| \le R'D + D$ as required. ◻ **Corollary 24**. *The itinerary $\ensuremath{\mathcal{U}}$ coming from satisfies the following properties:* 1. *[\[item:elts_in_coxbnhd\]]{#item:elts_in_coxbnhd label="item:elts_in_coxbnhd"} For every $i < j$, the group element $\gamma_{\ensuremath{\mathcal{U}}}(W_i, W_j)$ satisfies $$\gamma_{\ensuremath{\mathcal{U}}}(W_i, W_j) = \eta_i \cdot \gamma(W_i, W_j) \cdot \eta_j,$$ where $|\eta_i|, |\eta_j| < R$.* 2. *[\[item:subitineraries_minimal\]]{#item:subitineraries_minimal label="item:subitineraries_minimal"} Suppose $\ensuremath{\mathcal{U}}'$ is equivalent to $\ensuremath{\mathcal{U}}$. Then any sub-itinerary of $\ensuremath{\mathcal{U}}'$ with length greater than $2R$ is equivalent to an itinerary of the form $$\ensuremath{\mathcal{Y}}_i, \ensuremath{\mathcal{W}}_{ij}, \ensuremath{\mathcal{Y}}_j,$$ for some $i \le j$, where $\ensuremath{\mathcal{W}}_{ij}$ is an efficient itinerary between $W_i$ and $W_j$, and $|\ensuremath{\mathcal{Y}}_i|, |\ensuremath{\mathcal{Y}}_j| < R$.* *Proof.* [\[item:elts_in_coxbnhd\]](#item:elts_in_coxbnhd){reference-type="ref" reference="item:elts_in_coxbnhd"} Fix $i < j$, and consider the sub-itinerary $\ensuremath{\mathcal{U}}_{ij} = \{W_i\}, \ensuremath{\mathcal{V}}_i, \ldots \{W_j\}$. We know that at most $R$ walls in this sub-itinerary intersect $W_i$, and at most $R$ walls in this sub-itinerary intersect $W_j$, so there is an equivalent itinerary of the form $\ensuremath{\mathcal{Y}}_i, \{W_i\}, \ensuremath{\mathcal{Y}}, \{W_j\}, \ensuremath{\mathcal{Y}}_j$, where $|\ensuremath{\mathcal{Y}}_i|, |\ensuremath{\mathcal{Y}}_j| < R$, and the walls in $\ensuremath{\mathcal{Y}}_i$ and $\ensuremath{\mathcal{Y}}_j$ are precisely the walls in $\ensuremath{\mathcal{U}}_{ij}$ which respectively intersect $W_i$ and $W_j$. Then the walls in $\ensuremath{\mathcal{Y}}$ are precisely the walls separating $W_i$ from $W_j$, so $$\gamma(\ensuremath{\mathcal{U}}_{ij}) = \gamma(\ensuremath{\mathcal{Y}}_i)\gamma(W_i, W_j)\gamma(\ensuremath{\mathcal{Y}}_j).$$ [\[item:subitineraries_minimal\]](#item:subitineraries_minimal){reference-type="ref" reference="item:subitineraries_minimal"} Let $\ensuremath{\mathcal{U}}'$ be equivalent to $\ensuremath{\mathcal{U}}$, and let $\ensuremath{\mathcal{Y}}$ be a sub-itinerary of $\ensuremath{\mathcal{U}}'$. Let $\mathbf{W}$ be the ordered set of walls $W_1 < \ldots < W_n$. First, suppose that $\ensuremath{\mathcal{Y}}$ contains at least one wall $W_i \in \mathbf{W}$. We can then choose $W_i \le W_j$ to be (respectively) minimal and maximal walls in $\mathbf{W} \cap \ensuremath{\mathcal{Y}}$; then the argument from the previous case shows that $\ensuremath{\mathcal{Y}}$ is equivalent to an itinerary $\ensuremath{\mathcal{Y}}_i, \ensuremath{\mathcal{W}}_{ij}, \ensuremath{\mathcal{Y}}_j$ as required. So, we just need to show that if $|\ensuremath{\mathcal{Y}}| > 2R$ then $\ensuremath{\mathcal{Y}}$ contains at least one wall in $\mathbf{W}$. To see this, we consider the total orderings of the walls in $\ensuremath{\mathcal{U}}$ given by the equivalent itineraries $\ensuremath{\mathcal{U}}$ and $\ensuremath{\mathcal{U}}'$; both of these total orderings must be compatible with the partial order $<$ on all the walls in $\ensuremath{\mathcal{U}}$. For each $W_i \in \mathbf{W}$, we let $n(W_i)$ denote the index of $W_i$ in $\ensuremath{\mathcal{U}}$, and let $n'(W_i)$ denote the index of $W_i$ in $\ensuremath{\mathcal{U}}'$. Now, since each $W_i$ is independent from at most $R$ other walls in $\ensuremath{\mathcal{U}}$ with respect to the partial order $<$, we must have $|n(W_i) - n'(W_i)| \le R$ for every $i$. Then, since every sub-itinerary of $\ensuremath{\mathcal{U}}$ with length at least $R$ contains at least one wall in $\mathbf{W}$, the same is true for every sub-itinerary of $\ensuremath{\mathcal{U}}'$ with length at least $2R$. ◻ # Reflection groups acting on convex projective domains Let $C$ be a right-angled Coxeter group with generating set $S$. In this section, we discuss the theory of representations $\rho\colon C \to \mathop{\mathrm{SL}}^\pm(|S|, \ensuremath{\mathbb{R}})$ which are *generated by linear reflections*, as studied by Vinberg [@Vinberg1971]. Such representations give rise to a "projective model" for the Davis complex of $(C, S)$, in the form of a *convex projective domain* $\Omega$ preserved by $\rho$. We refer to [@Vinberg1971], [@DGKLM] for further background. ## Cartan matrices and simplicial representations {#sec:cartan_simplicial} Let $V$ be a real vector space with dimension $d$. **Definition 25**. A *linear reflection* is an element in $\mathop{\mathrm{SL}}^\pm(V)$ with a $(d-1)$-dimensional eigenspace with eigenvalue 1, and a 1-dimensional eigenspace with eigenvalue $-1$. Equivalently, a linear reflection is any element $r \in \mathop{\mathrm{SL}}^\pm(V)$ which can be written $r = \mathrm{id}- v \otimes\alpha$, where $\alpha \in V^*$, $v \in V$, and $\alpha(v) = 2$. We refer to the $-1$-eigenspace of $r$ (which is spanned by the vector $v$) as the *polar* of $r$. The $1$-eigenspace of $r$, given by $\ker(\alpha)$, is the *reflection hyperplane* of $r$. The vector $v$ and dual vector $\alpha$ are determined up to a choice of scale: we can replace $v$ with $\lambda v$ and $\alpha$ with $\lambda^{-1}\alpha$ for any nonzero $\lambda \in \ensuremath{\mathbb{R}}$ to obtain the same reflection. We say that a representation $\rho\colon C \to \mathop{\mathrm{SL}}^\pm(V)$ is *generated by reflections* if $\rho$ maps each $s \in S$ to some linear reflection in $\mathop{\mathrm{SL}}^\pm(V)$. There is always at least one discrete faithful representation $\rho\colon C \to \mathop{\mathrm{SL}}^{\pm}(|S|, \ensuremath{\mathbb{R}})$ which is generated by reflections, called the *geometric representation*. In fact, whenever $C$ is an infinite right-angled Coxeter group, there is an uncountable (continuous) family of conjugacy classes of such representations. Below, we explain how to construct the representations in this family. **Definition 26**. Let $(C, S)$ be a right-angled Coxeter system, and write $S = \{s_1, \ldots, s_n\}$. We say that an $|S| \times |S|$ real matrix $A$ is a *Cartan matrix* for $(C, S)$ if it satisfies the following three criteria: 1. For every $i$, we have $A_{ii} = 2$.[\[cond:compatibility_diagonal\]]{#cond:compatibility_diagonal label="cond:compatibility_diagonal"} 2. For all $i \ne j$ such that $s_i$ and $s_j$ commute, we have $A_{ij} = 0$. [\[cond:compatibility_commuting\]]{#cond:compatibility_commuting label="cond:compatibility_commuting"} 3. For all $i \ne j$ such that $s_i$ and $s_j$ do *not* commute, we have $A_{ij} < 0$ and $A_{ij}A_{ji} \ge 4$.[\[cond:compatibility_not_commuting\]]{#cond:compatibility_not_commuting label="cond:compatibility_not_commuting"} **Remark 27**. It also makes sense to consider Cartan matrices for an *arbitrary* (i.e. not necessarily right-angled) Coxeter group $C$. The definition is slightly more complicated in this case. We omit it as it is not relevant for the present paper. We can use any Cartan matrix $A$ for a Coxeter group $C$ to define a representation $\rho_A\colon C \to \mathop{\mathrm{SL}}^{\pm}(|S|, \ensuremath{\mathbb{R}})$: we let $n = |S|$, let $\{e_1, \ldots, e_n\}$ be the standard basis for $\ensuremath{\mathbb{R}}^n$, and let $\{e^1, \ldots, e^n\}$ be the corresponding dual basis. For each $i$, we set $\alpha_i = e^i$, and let $v_i$ be the vector $$v_i = \sum_{k=1}^n A_{ki}e_k.$$ Then the group element $\mathrm{id}- v_i \otimes\alpha_i$ is a linear reflection and for every $i,j$ we have $\alpha_i(v_j) = A_{ij}$. One can check directly that the assignment $s_i \mapsto (\mathrm{id}- v_i \otimes\alpha_i)$ determines a representation of the Coxeter group $C$. **Definition 28**. Given a Cartan matrix $A$ for a Coxeter group $C$, we let $\rho_A\colon C \to \mathop{\mathrm{SL}}^{\pm}(n, \ensuremath{\mathbb{R}})$ denote the representation determined by the assignment $$s_i \mapsto (\mathrm{id}- v_i \otimes\alpha_i)$$ described above. We will refer to $\rho_A$ as the *simplicial representation* associated to $A$. The terminology is motivated by the fact that $\rho_A$ induces a discrete and faithful action of $C$ on a convex domain in projective space $\ensuremath{\mathbb{P}}(\ensuremath{\mathbb{R}}^n)$, with fundamental domain a simplex (see below). **Definition 29**. Fix a right-angled Coxeter system $(C, S)$, and let $A$ be the unique Cartan matrix for $(C, S)$ which satisfies $A_{ij} = A_{ji} = -2$ for every $i \ne j$ such that $s_i$ and $s_j$ do not commute. The simplicial representation $\rho_A$ associated to this Cartan matrix is called the *geometric* (or *Tits*) representation of $C$. The geometric representation was first studied by Tits [@bourbaki68], who proved that it is always faithful with discrete image in $\mathop{\mathrm{SL}}^{\pm}(|S|, \ensuremath{\mathbb{R}})$. In [@Vinberg1971], Vinberg proved that the same is true for every simplicial representation $\rho$. **Remark 30**. If the Cartan matrix $A$ is symmetric, then one can define a (possibly degenerate) bilinear form $\langle \cdot, \cdot \rangle$ on $\ensuremath{\mathbb{R}}^{|S|}$ by setting $\langle e_i, e_j \rangle = A_{ij}$. In this case, the simplicial representation $\rho_A$ defined above is isomorphic to the *dual* of the representation $$s_i \mapsto (\mathrm{id}- \langle \cdot, e_i \rangle e_i),$$ which preserves the bilinear form $\langle \cdot , \cdot \rangle$. If the form $\langle \cdot, \cdot \rangle$ is nondegenerate, then it induces an isomorphism from $\ensuremath{\mathbb{R}}^{|S|}$ to its dual, and $\rho_A$ preserves the corresponding (nondegenerate) bilinear form (meaning $\rho_A(C)$ has image lying in some $O(p,q)$ with $p + q = |S|$). ### Nondegeneracy conditions Several times in this paper, we will want to consider the restriction of a simplicial representation $\rho$ of a right-angled Coxeter group $C$ to some standard subgroup $C(T) \leq C$. When $C(T)$ is a proper standard subgroup, the restricted representation cannot be irreducible, as it has an invariant subspace $V_T = \mathop{\mathrm{span}}\{v_s : s \in T\}$. So, for each subset $T \subseteq S$, we let $\rho_T\colon C(T) \to \mathop{\mathrm{SL}}^{\pm}(V_T)$ denote the representation induced by the restriction of $\rho$ to $C(T)$. To facilitate inductive arguments, we would like to have a condition which guarantees that the representation $\rho_T$ is isomorphic to a simplicial representation of $C(T)$, and inherits some nondegeneracy properties of the original representation $\rho$. We know that the representation $\rho_T$ will be isomorphic to a simplicial representation precisely when the set of restrictions $\{\alpha_t|_{V_T} : t \in T\}$ is a basis for $V_T^*$. The Cartan matrix associated to $\rho_T$ is then the principal submatrix of the Cartan matrix for $\rho$ corresponding to the subset $T$. With this in mind, we define the following: **Definition 31**. We will say that a matrix $A$ is *fully nondegenerate* if all of its principal minors are nonzero. The lemma below is completely elementary, but key to several of our later arguments. **Lemma 32**. *Let $\rho\colon C \to \mathop{\mathrm{SL}}_\pm(|S|, \ensuremath{\mathbb{R}})$ be a simplicial representation of a right-angled Coxeter group $C$ with fully nondegenerate Cartan matrix. For any subset $T \subset S$, define $V_T = \mathop{\mathrm{span}}\{v_s : s \in T\}$ and $V_T^\perp = \bigcap_{s \in T} \ker(\alpha_s)$. Then there is a $C(T)$-invariant decomposition $V = V_T \oplus V_T^\perp$, and the representation $\rho_T\colon C(T) \to \mathop{\mathrm{SL}}^{\pm}(V_T)$ is isomorphic to a simplicial representation.* In the terminology of [@DGKLM], says that the simplicial representation $\rho$ associated to a fully nondegenerate Cartan matrix is both *reduced* and *dual-reduced*, and that the same is true for the representations $\rho_T\colon C(T) \to \mathop{\mathrm{SL}}^{\pm}(V_T)$ for every $T \subseteq S$. *Proof.* The nondegeneracy of the Cartan matrix implies that the subsets $\{v_s : s \in T\}$ and $\{\alpha_s : s \in T\}$ are both linearly independent in $\ensuremath{\mathbb{R}}^{|S|}$ and its dual, respectively. This implies that the subspaces $V_T$ and $V_T^\perp$ have complementary dimension. Since the principal minor of the Cartan matrix corresponding to the subset $T$ is nonzero, the set of restrictions $\{\alpha_s|_{V_T} : s \in T\}$ is also linearly independent in the dual $V_T^*$. This implies that $V_T$ and $V_T^\perp$ are transverse, and that $\rho_T$ is isomorphic to a simplicial representation. ◻ **Remark 33**. Given a right-angled Coxeter group $C$, one can always find a fully nondegenerate Cartan matrix $A$ for $C$. In fact, the space of fully nondegenerate matrices is open and dense in the space of Cartan matrices for $C$, since each principal minor of $A$ is a polynomial in the parameters determining $A$ which is not identically zero. If desired, one can also arrange for this fully nondegenerate matrix to be symmetric, or to have integer entries. ## Invariant convex projective domains Let $V$ be a real vector space. Recall that a subset $\tilde{\Omega} \subset V$ is a *convex cone* if it is convex and invariant under multiplication by positive real numbers. A *convex domain* in $\ensuremath{\mathbb{P}}(V)$ is the image of an open convex cone under the projectivization map $V -\{0\} \to \ensuremath{\mathbb{P}}(V)$. A convex domain is *properly convex* if its closure is a convex subset of some affine chart in $\ensuremath{\mathbb{P}}(V)$ (equivalently, if its closure does not contain any projective line). Tits and Vinberg proved that simplicial representations are discrete and faithful by showing that there is a certain $\rho$-invariant convex domain in $\ensuremath{\mathbb{P}}(\ensuremath{\mathbb{R}}^{|S|})$ with nonempty interior, and that the action of $C$ on this domain is faithful and properly discontinuous. The structure of this domain is essential to this paper, so we describe it below. **Definition 34**. Let $A$ be a Cartan matrix for a Coxeter group $C$, which determines vectors $v_s \in \ensuremath{\mathbb{R}}^{|S|}$ and dual vectors $\alpha_s \in (\ensuremath{\mathbb{R}}^{|S|})^*$ for each $s \in S$. - The *fundamental simplicial cone* for the representation $\rho_A$ is the set $$\tilde{\Delta} = \bigcap_{s \in S} \{v \in \ensuremath{\mathbb{R}}^{|S|} : \alpha_s(v) \le 0 \textrm{ for all } s \in S\}.$$ The *fundamental simplex* $\Delta$ is the projectivization of $\tilde{\Delta}$ in $\ensuremath{\mathbb{P}}(\ensuremath{\mathbb{R}}^{|S|})$. - The *Tits cone* is the interior of the set $$\bigcup_{\gamma \in C} \rho_A(\gamma) \tilde{\Delta}.$$ The *Vinberg domain* is the projectivization of the Tits cone in $\ensuremath{\mathbb{P}}(\ensuremath{\mathbb{R}}^{|S|})$. We usually denote the Tits cone by $\tilde{\Omega}_{\mathrm{Vin}}$, and the Vinberg domain by $\Omega_{\mathrm{Vin}}$. **Theorem 35** (See Tits [@bourbaki68], Vinberg [@Vinberg1971]). *Let $\rho$ be a simplicial representation of a Coxeter group $C$. Then:* 1. *The Tits cone $\tilde{\Omega}_{\mathrm{Vin}}$ is a nonempty convex open subset of $\ensuremath{\mathbb{R}}^{|S|}$;* 2. *the action of $C$ on $\Omega_{\mathrm{Vin}}$ is faithful and properly discontinuous;* 3. *the simplex $\Delta \cap \Omega_{\mathrm{Vin}}$ is a fundamental domain for the action;* 4. *the dual graph to the tiling of $\Omega_{\mathrm{Vin}}$ by copies of $\Delta \cap \Omega_{\mathrm{Vin}}$ is equivariantly identified with the Cayley graph of $C$ with generating set $S$.* **Remark 36**. We have only stated for simplicial representations of the Coxeter group $C$. However, Vinberg's result also applies to a broad family of representations of $C$ generated by linear reflections. In particular, a version of the theorem applies to representations $\rho\colon C \to \mathop{\mathrm{SL}}^{\pm}(V)$ where $\dim V$ is not necessarily equal to $|S|$. While the Vinberg domain $\Omega_{\mathrm{Vin}}$ associated to a simplicial representation $\rho$ is always a convex domain, it is not necessarily *properly* convex. However, Vinberg also gave conditions which make it possible to guarantee that this holds in broad circumstances: **Proposition 37** (See [@Vinberg1971 Lemma 15 and Proposition 22]). *Let $A$ be a Cartan matrix for an irreducible Coxeter group $C$, and suppose that $\det(A) \ne 0$. Then the following are equivalent:* 1. *The matrix $A$ is of *negative type*, i.e. $A$ has a negative eigenvalue.* 2. *The Vinberg domain $\Omega_{\mathrm{Vin}}$ for $\rho_A$ is properly convex.* 3. *The Coxeter group $C$ is infinite.* Sometimes, when working with representations of $C$ generated by reflections, it will also be convenient to consider $\rho$-invariant domains $\Omega \subset \ensuremath{\mathbb{P}}(\ensuremath{\mathbb{R}}^{|S|})$ *other* than the Vinberg domain. **Definition 38**. Let $\rho\colon C \to \mathop{\mathrm{SL}}^{\pm}(|S|, \ensuremath{\mathbb{R}})$ be a simplicial representation of an irreducible right-angled Coxeter group $C$. We will call any $\rho$-invariant nonempty convex domain $\Omega \subset \Omega_{\mathrm{Vin}}$ a *reflection domain* for $\rho$. Typically, we lose nothing by requiring reflection domains to lie inside of the Vinberg domain $\Omega_{\mathrm{Vin}}$, instead of merely asking for them to be $\rho$-invariant convex open sets in $\ensuremath{\mathbb{P}}(\ensuremath{\mathbb{R}}^{|S|})$. The reason is the following fact, due to Danciger-Guéritaud-Kassel-Lee-Marquis: **Proposition 39** (See [@DGKLM Proposition 4.1]). *Let $(C,S)$ be an irreducible infinite right-angled Coxeter system with $|S| > 2$, and let $\rho$ be a simplicial representation of $C$ with nonsingular Cartan matrix. Then $\Omega_{\mathrm{Vin}}$ contains every $\rho$-invariant properly convex domain in $\ensuremath{\mathbb{P}}(\ensuremath{\mathbb{R}}^{|S|})$.* ## Dual domains and the dual Vinberg domain {#sec:dual_domains} Let $V$ be a real vector space and let $\tilde{\Omega}$ be an open convex cone in $V$. The cone $\tilde{\Omega}$ determines a *dual cone* $\tilde{\Omega}^* \subset \ensuremath{\mathbb{P}}(V^*)$, defined by $$\tilde{\Omega}^* = \{w \in V^* : w(v) < 0 \quad \forall x \in \overline{\tilde{\Omega}} -\{0\}\}.$$ Then if $\Omega \subset \ensuremath{\mathbb{P}}(V)$ is some convex domain, the dual domain $\Omega^* \subset \ensuremath{\mathbb{P}}(V^*)$ is the projectivization of $\tilde{\Omega}^*$, for some (any) convex cone $\tilde{\Omega}$ projecting to $\Omega$. It follows immediately that $\Omega^*$ is invariant under the dual action of any $g \in \mathop{\mathrm{SL}}^{\pm}(V)$ which preserves $\Omega$. In addition, it is not hard to verify that if $\Omega$ is both properly convex and open, then so is $\Omega^*$. We note further that duality reverses inclusions: if $\Omega_1 \subset \Omega_2$, then $\Omega_2^* \subset \Omega_1^*$. ### Dual simplicial representations We now let $V$ be the vector space $\ensuremath{\mathbb{R}}^{|S|}$, and let $\rho\colon C \to \mathop{\mathrm{SL}}^{\pm}(V)$ be a simplicial representation for a right-angled Coxeter group $C$. If the associated Cartan matrix is nonsingular, then the dual representation $\rho^*:C \to \mathop{\mathrm{SL}}^{\pm}(V^*)$ is isomorphic to a simplicial representation of $C$ whose Cartan matrix is the transpose of the Cartan matrix of $\rho$. Since $\rho$ preserves the Vinberg domain $\Omega_{\mathrm{Vin}}$, the dual representation preserves the dual domain $\Omega_{\mathrm{Vin}}^*$. When $C$ is irreducible, infinite, and not virtually cyclic, implies that $\Omega_{\mathrm{Vin}}^*$ is contained in the Vinberg domain $\mathscr{O}_{\mathrm{Vin}}$ associated to the dual (simplicial) representation $\rho^*$. Explicitly, if we let $\tilde{\mathscr{D}}$ be the cone $$\{w \in V^* : v_s(w) \le 0 \quad \forall s \in S\},$$ and let $\mathscr{D}$ be the projectivization of $\tilde{\mathscr{D}}$, then $\mathscr{O}_{\mathrm{Vin}}$ is the projectivization of the interior of $$\tilde{\mathscr{O}}_{\mathrm{Vin}}:= \bigcup_{\gamma \in C} \rho^*(\gamma) \tilde{\mathscr{D}}.$$ also applies to $\rho^*$, with $\mathscr{D}$ taking the place of $\Delta$ and $\mathscr{O}_{\mathrm{Vin}}$ taking the place of $\Omega_{\mathrm{Vin}}$. We always have: **Proposition 40** (The dual of a reflection domain is a reflection domain). *Let $A$ be a nonsingular Cartan matrix for an infinite irreducible right-angled Coxeter group $C$, and let $\rho_A$ be the associated simplicial representation. Let $\Omega_{\mathrm{Vin}}, \mathscr{O}_{\mathrm{Vin}}$ be the Vinberg domains for $\rho_A, \rho_A^*$, respectively. If $\Omega \subseteq \Omega_{\mathrm{Vin}}$ is a reflection domain for $\rho_A$, then $\Omega^* \subseteq \mathscr{O}_{\mathrm{Vin}}$.* When $C$ is not virtually cyclic this result follows from , and if $C$ is an infinite dihedral group this can be checked directly. # Walls and half-cones in reflection domains In this section we continue to explore some features of the projective geometry of reflection domains. Our main goal is to translate our combinatorial understanding of geodesics in a right-angled Coxeter group into information about the action of the corresponding sequence of group elements in $\mathop{\mathrm{SL}}^{\pm}(n, \ensuremath{\mathbb{R}})$. **Definition 41**. Let $C$ be an irreducible right-angled Coxeter group, and let $\Omega$ be a reflection domain for a simplicial representation $\rho$ of $C$. A *wall* $W$ in $\Omega$ is the fixed-point set in $\Omega$ of a reflection $r$ in $C$. Whenever $W$ is a wall, we let $\overline{W}$ denote the closure of $W$ in $\overline{\Omega}$. Equivalently, $\overline{W}$ is the closure of $W$ in projective space $\ensuremath{\mathbb{P}}(V)$. The *polar* of a wall $W$ is the polar of the reflection $r$ fixing $W$. Every wall in $\Omega$ is the intersection of some projective hyperplane with $\Omega$, so if $\Omega$ is properly convex, then each wall separates $\Omega$ into two convex components. Further, every wall $W$ is a union of codimension-1 faces of tiles in $\Omega$, corresponding exactly to the set of edges in the Cayley graph of $C$ fixed by the reflection that defines $W$. Consequently, we have the following: **Proposition 42**. *Let $\Omega$ be a reflection domain for a simplicial representation of an infinite irreducible right-angled Coxeter group $C$. There is an equivariant one-to-one correspondence between walls in $\Omega$ and walls in the Davis complex $\mathrm{D}(C,S)$, which satisfies the following properties.* 1. *Two walls in $\Omega$ intersect if and only if the corresponding walls in $\mathrm{D}(C,S)$ intersect (equivalently, if and only if the corresponding reflections in $C$ commute).* 2. *Fix a basepoint $x_0$ in the interior of $\Delta \cap \Omega$, and let $\gamma \in C$. The set of walls in $\Omega$ separating $x_0$ from $\rho(\gamma) \cdot x_0$ corresponds precisely to the set of walls in $\mathrm{D}(C,S)$ separating $\mathrm{id}$ from $\gamma$.* tells us that the walls in any reflection domain carry all of the same combinatorial information as the walls in the Davis complex. So, throughout the rest of this section, we will freely apply the combinatorial setup and results regarding walls in Sections [2](#sec:cube_complexes_RACGs){reference-type="ref" reference="sec:cube_complexes_RACGs"} and [3](#sec:bounded_product_projections){reference-type="ref" reference="sec:bounded_product_projections"} to the walls in a reflection domain $\Omega$. In particular, for any group element $\gamma \in C$, we can regard the poset of walls in $\mathrm{D}(C,S)$ separating $\mathrm{id}$ from $\gamma$ as a poset of projective walls in $\Omega$. Moreover, the notion of an *itinerary* traversing an element in $C$ also makes sense in this context. **Definition 43**. If $W_1, \ldots, W_n$ is a sequence of walls in a reflection domain $\Omega$ corresponding to an itinerary of walls in the Davis complex $\mathrm{D}(C,S)$, we will say that $W_1, \ldots, W_n$ is an *itinerary in $\Omega$* or an *$\Omega$-itinerary*. The element $\gamma \in C$ *traversed* by this itinerary is the element traversed by the corresponding itinerary in $\mathrm{D}(C,S)$. ## Disjoint walls in $\Omega_{\mathrm{Vin}}$ and $\overline{\Omega_{\mathrm{Vin}}}$ The projective walls in the Vinberg domain $\Omega_{\mathrm{Vin}}$ actually carry some additional combinatorial information beyond what is encoded in the corresponding walls in the Davis complex $\mathrm{D}(C,S)$: the intersection pattern of the *closures* of the set of walls in $\overline{\Omega_{\mathrm{Vin}}}$ is also informed by the structure of the Coxeter group $C$. Specifically, we have the following: **Lemma 44**. *Let $C$ be a right-angled Coxeter group, and let $\Omega_{\mathrm{Vin}}$ be the Vinberg domain for a simplicial representation of $C$. If $W_1$, $W_2$ are walls in $\Omega_{\mathrm{Vin}}$, then $\overline{W}_1 \cap \overline{W}_2 \ne \emptyset$ if and only if $\gamma(W_1, W_2)$ lies in a proper standard subgroup of $C$.* will follow from and below. For these two results, we assume that $\rho$ is a simplicial representation for a right-angled Coxeter system $(C, S)$. **Lemma 45**. *If $W_1, \ldots, W_n$ is an efficient $\Omega_{\mathrm{Vin}}$-itinerary between $W_1$ and $W_n$, then $\overline{W_1} \cap \overline{W_n} = \bigcap_{i=1}^n \overline{W_i}$.* *Proof.* First, observe that if $W_1 \cap W_n \ne \emptyset$, then $W_1, W_n$ is already an efficient itinerary between $W_1$ and $W_n$. So we can suppose that $W_1 \cap W_n = \emptyset$. It suffices to show that $\overline{W_1} \cap \overline{W_n} \subset \overline{W_i}$ for every $1 < i < n$, so fix a wall $W_i$ strictly between $W_1$ and $W_n$. Since $W_1, \ldots, W_n$ is an efficient itinerary, $W_i$ separates $W_1$ from $W_n$: that is, $\Omega_{\mathrm{Vin}}-W_i$ has exactly two connected components $O_-, O_+$, with $W_1 \subset O_-$ and $W_n \subset O_+$. By convexity, $\overline{\Omega_{\mathrm{Vin}}} -\overline{W_i}$ also has exactly two components $O_-', O_+'$, which satisfy $\overline{O_-} = \overline{O_-'}$ and $\overline{O_+} = \overline{O_+'}$ (as for walls, the closures are taken in projective space). We have $\overline{W_1} \subset \overline{O_-}$ and $\overline{W_n} \subset \overline{O_+}$, and therefore $\overline{W_1} \cap \overline{W_n} \subset \overline{O_-'} \cap \overline{O_+'} = \overline{W_i}$. ◻ **Lemma 46**. *Let $W_1, \ldots, W_n$ be an $\Omega_{\mathrm{Vin}}$-itinerary departing from the identity, and let $s_i = s(W_i)$ for $1 \le i \le n$. Then $\bigcap_{i=1}^n \overline{W(s_i)} = \bigcap_{i=1}^n \overline{W_i}$.* *Proof.* We induct on $n$. The case $n = 1$ is immediate. When $n > 1$, we let $\gamma_{n-1} = s_1 \cdots s_{n-1}$. By we have $W_n = \rho(\gamma_{n-1})W(s_n)$, so inductively we have $$\begin{aligned} \bigcap_{i=1}^n \overline{W_i} = \left(\bigcap_{i=1}^{n-1} \overline{W_i} \right) \cap \overline{W_n} = \left(\bigcap_{i=1}^{n-1} \overline{W(s_i)} \right) \cap \rho(\gamma_{n-1}) \overline{W(s_n)}. \end{aligned}$$ Since $\rho(s_i)$ fixes $\overline{W(s_i)}$ pointwise for each $1 \le i \le n$, both $\rho(\gamma_{n-1})$ and $\rho(\gamma_{n-1}^{-1})$ fix the intersection $\bigcap_{i=1}^{n-1} \overline{W(s_i)}$ pointwise. So, we have $$\begin{aligned} \left(\bigcap_{i=1}^{n-1} \overline{W(s_i)} \right) \cap \rho(\gamma_{n-1}) \overline{W(s_n)} &= \rho(\gamma_{n-1})\left( \left(\rho(\gamma_{n-1}^{-1})\bigcap_{i=1}^{n-1}\overline{W(s_i)}\right) \cap \overline{W(s_n)}\right)\\ &= \rho(\gamma_{n-1}) \bigcap_{i=1}^n \overline{W(s_i)} = \bigcap_{i=1}^n \overline{W(s_i)}. \end{aligned}$$ ◻ *Proof of .* Fix an efficient itinerary $\ensuremath{\mathcal{V}} = V_1, \ldots, V_n$ whose first wall is $W_1$ and whose last wall is $W_2$. For any $h \in C$ we have $\gamma(W_1, W_2) = \gamma(hW_1, hW_2)$. So, after translating $W_1, W_2$ and $\ensuremath{\mathcal{V}}$ by an appropriate element $h$, we can assume that $\ensuremath{\mathcal{V}}$ departs from the identity. Set $s_i = s(V_i)$ for $1 \le i \le n$, so that $\gamma(W_1, W_2) = s_1 \cdots s_n$. From and , we know that $\overline{W_1} \cap \overline{W_2}$ is nonempty if and only if the intersection $\bigcap_{i=1}^n \overline{W(s_i)}$ is nonempty. But since the representation $\rho$ is simplicial, this occurs if and only if the set $S'$ of generators appearing in the geodesic word $s_1 \cdots s_n$ is a proper subset of $S$, i.e. if $\gamma(V_1, V_n) = s_1 \cdots s_n$ lies in a proper standard subgroup of $C$. ◻ ## Half-cones Consider an infinite geodesic sequence $\gamma_n$ in a right-angled Coxeter group $C$. If we fix a reflection domain $\Omega$ for some simplicial representation $\rho$ of $C$, the geodesic sequence corresponds to an (infinite) $\Omega$-itinerary $\ensuremath{\mathcal{W}} = W_1, W_2, \ldots$ Each wall $W_n$ in $\ensuremath{\mathcal{W}}$ cuts $\Omega$ into a pair of connected components, which we call *half-spaces*; we can fix a basepoint $x_0 \in \Omega$, and let $H_n$ be the half-space which does *not* contain $x_0$. By considering the infinite intersection of closed half-spaces $\bigcap_{n=1}^\infty \overline{H_n}$, we can try and find a well-defined "limit point" for the geodesic $\rho(\gamma_n)$ in projective space $\ensuremath{\mathbb{P}}(V)$. However, we will run into some difficulties: while it is often true that pairs of half-spaces are *nested*, (meaning that $H_{n+k} \subset H_n$ for some $n, k > 0$), they will never be *strongly* nested, i.e. they will *never* satisfy $\overline{H_{n+k}} \subset H_n$. This makes it hard to guarantee that the intersections $\bigcap_{i=1}^n \overline{H_n}$ decrease in size at a "uniform rate." We solve this problem by enlarging the half-spaces $H_n$ to subsets of projective space which we call *half-cones*. Half-cones satisfy the same nesting properties as half-spaces (see below). They will often (but not always) additionally satisfy the *strong* nesting property as well, which allows us to employ them in asymptotic arguments later on. We will also be able to give a fairly complete description of when the strong nesting property fails (see ), which will be key to later inductive arguments. For the precise definitions, we fix an irreducible infinite right-angled Coxeter group $C$, let $V$ denote the vector space $\ensuremath{\mathbb{R}}^{|S|}$, and let $\rho\colon C \to \mathop{\mathrm{SL}}^{\pm}(V)$ be a simplicial representation of $C$. We let $\Omega \subset \Omega_{\mathrm{Vin}}$ be a reflection domain for $\rho$, so that $\Omega$ is the projectivization of an invariant convex sub-cone $\tilde{\Omega} \subset \tilde{\Omega}_{\mathrm{Vin}}$ of the Tits cone. We let $\tilde{\Delta}_\Omega$ denote the set $\tilde{\Delta} \cap \tilde{\Omega}$. Then the projectivization $\Delta_\Omega$ of $\tilde{\Delta}_\Omega$ is a fundamental domain for the action of $C$ on $\Omega$. **Definition 47**. Let $W$ be a wall in $\Omega$, so that $W$ is the fixed-point set of a reflection $\mathrm{id}- v \otimes\alpha$ for some $v \in V$, $\alpha \in V^*$ with $\alpha(v) = 2$. Up to replacing $\alpha, v$ with $-\alpha, -v$, we may assume that $\alpha(x) \le 0$ for every $x \in \tilde{\Delta}_\Omega$. - We let $\widetilde{\mathbf{Hs}}_+(W)$ denote the interior of the convex cone $\{x \in \tilde{\Omega} : \alpha(x) \ge 0\}$. The *positive half-space* over $W$, denoted $\mathbf{Hs}_+(W)$, is the projectivization of $\widetilde{\mathbf{Hs}}_+(W)$. Similarly, $\widetilde{\mathbf{Hs}}_-(W)$ is the interior of the cone $\{x \in \tilde{\Omega} : \alpha(x) \le 0\}$, and the *negative half-space* $\mathbf{Hs}_-(W)$ is the projectivization of $\widetilde{\mathbf{Hs}}_-(W)$. - We let $\widetilde{\mathbf{Hc}}_+(W)$ denote the interior of the convex hull (in $V$) of $\ker(\alpha) \cap \tilde{\Omega}$ and $v$. The *positive half-cone* over $W$, denoted $\mathbf{Hc}_+(W)$, is the projectivization of $\widetilde{\mathbf{Hc}}_+(W)$. We define the set $\widetilde{\mathbf{Hc}}_-(W)$ and the *negative half-cone* $\mathbf{Hc}_-(W)$ in the same way, but with $v$ replaced with $-v$. See . For $s \in S$, we let $\mathbf{Hs}_\pm(s)$ and $\mathbf{Hc}_\pm(s)$ denote the positive and negative half-spaces and half-cones over the wall $W(s)$ fixed by $\rho(s)$. Note that the positive half-space $\mathbf{Hs}_+(W)$ over a wall $W$ is precisely the connected component of $\Omega -W$ whose closure does *not* contain the fundamental domain $\Delta_\Omega$. This also gives a way to distinguish positive and negative half-cones (see below). **Lemma 48**. *Let $W_1, \ldots, W_n$ be an $\Omega$-itinerary departing from the identity, traversing a geodesic word $s_1 \cdots s_n$ in $C$. Then $\mathbf{Hs}_\pm(W_n) = \rho(s_1 \cdots s_{n-1})\mathbf{Hs}_\pm(s_n)$ and $\mathbf{Hc}_\pm(W_n) = \rho(s_1 \cdots s_{n-1})\mathbf{Hc}_\pm(s_n)$.* *Proof.* We know from that $W_n$ is the wall $\rho(s_1 \cdots s_{n-1})W(s_n)$. So, if we fix $\alpha \in V^*$ so that $W_n = [\ker \alpha \cap \tilde{\Omega}]$ and $\alpha(\tilde{\Delta}_\Omega) \le 0$, we know that either $\alpha = \rho^*(s_1 \cdots s_{n-1})\alpha_{s_n}$ or $\alpha = -\rho^*(s_1 \cdots s_{n-1})\alpha_{s_n}$. Now, since the walls $W_1, \ldots, W_n$ are precisely the walls separating $\tilde{\Delta}_\Omega$ from $\rho(s_1 \cdots s_n)\tilde{\Delta}_\Omega$, we must have $$\alpha(\rho(s_1 \cdots s_n)\tilde{\Delta}_\Omega) \ge 0,$$ or equivalently $(\rho^*(s_n \cdots s_1)\alpha)(\tilde{\Delta}_\Omega) \ge 0$. So, since $(\rho^*(s_n)\alpha_{s_n})(\tilde{\Delta}_\Omega) = -\alpha_{s_n}(\tilde{\Delta}_\Omega) \ge 0$, we have $\alpha = \rho^*(s_1 \cdots s_{n-1})\alpha_{s_n}$. This proves the desired result for half-spaces. A dual argument then shows that, if $v \in V$ is a lift of the polar of $W$ with $\alpha(v) = 2$, then $\rho(s_1 \cdots s_{n-1})v = v_{s_n}$, which proves the result for half-cones. ◻ **Lemma 49**. *For every wall $W$ in $\Omega$, we have $\mathbf{Hs}_\pm(W) \subset \mathbf{Hc}_\pm(W)$.* *Proof.* By , we just to check that $\mathbf{Hs}_\pm(s) \subset \mathbf{Hc}_\pm(s)$ for every $s \in S$. In fact, since the reflection $\rho(s)$ interchanges $\mathbf{Hs}_+(s)$ with $\mathbf{Hs}_-(s)$, and $\mathbf{Hc}_+(s)$ with $\mathbf{Hc}_-(s)$, we just need to verify that $\mathbf{Hs}_+(s) \subset \mathbf{Hc}_+(s)$. We write $\rho(s) = \mathrm{id}- v_s \otimes\alpha_s$, and consider $x \in \widetilde{\mathbf{Hs}}_+(s) -\ker(\alpha_s)$. Since $\rho(s)$ interchanges $\widetilde{\mathbf{Hs}}_+(s)$ and $\widetilde{\mathbf{Hs}}_-(s)$, we know that $\rho(s)x \in \widetilde{\mathbf{Hs}}_-(s)$. Then since $\tilde{\Omega}$ is a convex cone, the line segment $\ell \subset V$ joining $x$ to $\rho(s)x$ lies in $\tilde{\Omega}$. The segment $\ell$ is $\rho(s)$-invariant, and it must be transverse to $\ker(\alpha_s)$ at a point $x_0 \in \ker(\alpha_s) \cap \tilde{\Omega}$. So, the endpoints of $\ell$ have the form $x_0 \pm t v_s$ for some $t > 0$. But then since $\alpha_s(x) > 0$ by assumption we must have $x = x_0 + tv_s$, hence $x \in \widetilde{\mathbf{Hc}}_+(s)$. ◻ ## Dual walls and dual half-cones **Definition 50**. Given $\Omega$ a reflection domain for a simplicial representation $\rho$, and $W$ a wall in $\Omega$, fixed by a reflection $\rho(\gamma)$ for $\gamma \in C$, we write $W^*$ to denote the wall in the dual reflection domain $\Omega^*$ fixed by the reflection $\rho^*(\gamma)$. We observe the following useful consequence of : **Lemma 51**. *Let $C$ be an irreducible right-angled Coxeter group, and let $\Omega_{\mathrm{Vin}}$ be the Vinberg domain for a simplicial representation $\rho$ of $C$ with nonsingular Cartan matrix. Suppose that $W_1, W_2$ are two walls in $\Omega_{\mathrm{Vin}}$ such that $\overline{W}_1 \cap \overline{W}_2 = \emptyset$. Then $\overline{W_1^*} \cap \overline{W_2^*} = \emptyset$.* *Proof.* Let $\gamma_1$ and $\gamma_2$ be the reflections in $C$ such that $\rho(\gamma_i)$ fixes $W_i$ for $i = 1,2$. By , the walls $W_1$ and $W_2$ have disjoint closures if and only if $\gamma(W_1, W_2)$ does not lie in a proper standard subgroup of $C$. But this condition depends only on $\gamma_1$ and $\gamma_2$, and not on the specific simplicial representation $\rho$. Further, since the Cartan matrix of $\rho$ is nonsingular, the dual representation $\rho^*$ is also a simplicial representation. This means that $W_1$ and $W_2$ have disjoint closures if and only if the walls in $\mathscr{O}_{\mathrm{Vin}}$ fixed by $\rho^*(\gamma_1)$, $\rho^*(\gamma_2)$ have disjoint closures (see ). However, $\mathscr{O}_{\mathrm{Vin}}$ contains $\Omega_{\mathrm{Vin}}^*$ (), which means that these two walls contain $W_1^*$ and $W_2^*$. ◻ If $\tilde{\Omega} \subset \tilde{\Omega}_{\mathrm{Vin}}$ is the invariant cone over a reflection domain for $\rho$, then the dual cone $\tilde{\Omega}^*$ is an invariant sub-cone of $\tilde{\mathscr{O}}_{\mathrm{Vin}}$, projecting to the dual reflection domain $\Omega^*$. So, the previous section tells us that we can define half-spaces $\mathbf{Hs}_\pm(W^*)$ and half-cones $\mathbf{Hc}_\pm(W^*)$ over the wall $W^*$ in $\Omega^*$. The half-cones $\mathbf{Hc}_\pm(W)$ are themselves properly convex domains, which means that we can also define their duals $\mathbf{Hc}_\pm(W)^* \subset \ensuremath{\mathbb{P}}(V^*)$ (see the beginning of ). **Lemma 52**. *For any wall $W$ in a reflection domain $\Omega$, we have $\mathbf{Hc}_+(W)^* = \mathbf{Hc}_-(W^*)$.* *Proof.* We will verify that this holds when $W$ is the projective wall $W_s$ fixed by a reflection $\rho(s)$ for $s \in S$. In this case $W_s^*$ is the projective wall in $\Omega^*$ fixed by $\rho^*(s)$. We can use (and its dual version) to see that the general case follows from this one: let $\ensuremath{\mathcal{W}}$ be an $\Omega$-itinerary $W_1, \ldots, W_n$ departing from the identity, with $W_n = W$, and let $s_1 \cdots s_n$ be the geodesic word traversed by $\ensuremath{\mathcal{W}}$. Then: $$\begin{aligned} \mathbf{Hc}_+(W)^* &= (\rho(s_1 \cdots s_{n-1})\mathbf{Hc}_+(W_{s_n}))^*\\ &= \rho^*(s_1 \cdots s_{n-1})\mathbf{Hc}_+(W_{s_n})^*\\ &= \rho^*(s_1 \cdots s_{n-1})\mathbf{Hc}_-(W_{s_n}^*)\\ &= \mathbf{Hc}_-(W^*). \end{aligned}$$ So, now fix $s \in S$. By definition, we have $$\widetilde{\mathbf{Hc}}_+(W_s) = \{x + tv_s : x \in \ker(\alpha_s) \cap \tilde{\Omega}, t > 0\},$$ which means that $\widetilde{\mathbf{Hc}}_+(W_s)^*$ is the interior of the set $$\{w \in V_S^* : w(x + tv_s) \le 0 \textrm{ for all }x \in \ker(\alpha_s) \cap \tilde{\Omega}, t > 0\}.$$ Now let $w$ be any point in $\widetilde{\mathbf{Hc}}_+(W_s)^*$. Since $v_s$ does not lie in $\ker(\alpha_s)$, we can uniquely write $w = w_1 + w_2$ for $w_1 \in \ensuremath{\mathbb{R}}\alpha_s$ and $w_2 \in \ker(v_s)$ (viewing $v_s$ as an element of $V^{**}$). Since $v_s$ is a nonzero vector in the closure of $\widetilde{\mathbf{Hc}}_+(W_s)$, we know $w(v_s) < 0$. Then because $w(v_s) = w_1(v_s)$, and $\alpha_s(v_s) = 2$, we must have $w_1 = t\alpha_s$ for $t < 0$. And, any $x \in \ker(\alpha_s) \cap \tilde{\Omega}$ lies in the closure of $\widetilde{\mathbf{Hc}}_+(W_s)$ also, so we know $w(x) < 0$, hence $w_2(x) < 0$. By , we know that for any $v \in \tilde{\Omega}$, we can write $v = x + tv_s$ for $x \in \ker(\alpha_s) \cap \tilde{\Omega}$ and $t \in \ensuremath{\mathbb{R}}$. Then $w_2(v) = w_2(x) < 0$, which means that $w_2 \in \ker(v_s) \cap \tilde{\Omega}^*$. This tells us that we can write $\widetilde{\mathbf{Hc}}_+(W_s)^*$ as the interior of $$\{w + t\alpha_s : w \in \ker(v_s) \cap \tilde{\Omega}^*, t < 0\},$$ which is precisely the definition of $\widetilde{\mathbf{Hc}}_-(W_s^*)$. ◻ ## Nested half-cones {#sec:half_cone_nesting} For the rest of the section, we will work exclusively with the Vinberg domain $\Omega_{\mathrm{Vin}}$ for a simplicial representation $\rho$, rather than an arbitrary reflection domain. Our goal is to understand when the half-cones over a pair of walls $W_1, W_2$ in $\Omega_{\mathrm{Vin}}$ are (strongly) nested inside of each other. **Lemma 53** (Half-cones nest). *Let $C$ be an irreducible infinite right-angled Coxeter group, and let $\rho\colon C \to \mathop{\mathrm{SL}}^{\pm}(V)$ be a simplicial representation. Suppose that $W_1$ and $W_2$ are walls in $\Omega_{\mathrm{Vin}}$ such that $W_1 \cap W_2 = \emptyset$ and $\mathbf{Hs}_+(W_2) \subset \mathbf{Hs}_+(W_1)$. Then $\mathbf{Hc}_+(W_2) \subset \mathbf{Hc}_+(W_1)$. Moreover, if $\overline{W_1} \cap \overline{W_2} = \emptyset$, then $\partial \mathbf{Hc}_+(W_1) \cap \overline{\mathbf{Hc}_+(W_2)}$ is contained in $\partial \mathbf{Hc}_+(W_1) \cap \overline{W_2}$.* *Proof.* We let $\gamma_1, \gamma_2$ be the elements of $C$ such that $\rho(\gamma_i)$ fixes $W_i$ for $i = 1,2$. We fix $\alpha_1, \alpha_2 \in V^*$ so that $\widetilde{\mathbf{Hs}}_+(W_1)$ lies in the positive half-space of $V$ defined by $\alpha_1$, and similarly for $\alpha_2, \widetilde{\mathbf{Hs}}_+(W_2)$. Then we let $\tilde{W}_1 = \ker \alpha_1 \cap \tilde{\Omega}_{\mathrm{Vin}}$ and $\tilde{W}_2 = \ker \alpha_2 \cap \tilde{\Omega}_{\mathrm{Vin}}$, and fix vectors $v_1, v_2 \in V$ so that $\widetilde{\mathbf{Hc}}_+(W_i)$ is the interior of the convex hull of $\tilde{W}_i$ and $v_i$ for $i = 1,2$. Since $\mathbf{Hs}_+(W_2) \subset \mathbf{Hs}_+(W_1)$, we know that $\tilde{W}_2 \subset \widetilde{\mathbf{Hc}}_+(W_1)$. We will show that $v_2 \in \overline{\widetilde{\mathbf{Hc}}_+(W_1)}$, and that if $\overline{W_2} \cap \overline{W_1} = \emptyset$, then $v_2 \in \widetilde{\mathbf{Hc}}_+(W_1)$. Then both parts of the lemma will follow from convexity of $\mathbf{Hc}_+(W_1)$. Consider the 2-dimensional subspace spanned $V_{12}$ by $v_1$ and $v_2$. If $V_{12} \cap \overline{\tilde{\Omega}_{\mathrm{Vin}}}$ is trivial, then since $\Omega_{\mathrm{Vin}}$ is properly convex there is some $w \in \tilde{\Omega}_{\mathrm{Vin}}^*$ which vanishes on both $v_1$ and $v_2$. But then $w$ is fixed by both $\rho^*(\gamma_1)$ and $\rho^*(\gamma_2)$, implying that $W_1^* \cap W_2^*$ is nonempty. But this occurs if and only if the reflections fixing $W_1^*$ and $W_2^*$ commute, which we know is not the case because $W_1 \cap W_2 = \emptyset$. We conclude that $V_{12} \cap \overline{\tilde{\Omega}_{\mathrm{Vin}}}$ is nontrivial, and let $x$ be a nonzero point in this intersection. Next, suppose that $V_{12} \cap \tilde{\Omega}_{\mathrm{Vin}}$ is trivial. Repeating the same argument from before, we see there is some nonzero $w \in \overline{\tilde{\Omega}_{\mathrm{Vin}}^*}$ which vanishes on $v_1$ and $v_2$ and therefore $\overline{W_1^*} \cap \overline{W_2^*} \ne \emptyset$. But by this is impossible if $\overline{W_1} \cap \overline{W_2} = \emptyset$. We conclude that if $\overline{W_1} \cap \overline{W_2} = \emptyset$, then we can take our nonzero point $x \in V_{12} \cap \overline{\tilde{\Omega}_{\mathrm{Vin}}}$ to lie in $V_{12} \cap \tilde{\Omega}_{\mathrm{Vin}}$. Now, we can view $v_1$ as a linear functional on $V^*$, whose kernel intersects the interior of the dual domain $\tilde{\Omega}_{\mathrm{Vin}}^*$ in the reflection wall for $\rho^*(\gamma_1)$. Equivalently, $v_1$ does not lie in the closure of the dual domain to $\tilde{\Omega}_{\mathrm{Vin}}^*$ in $V^{**}$. But this dual domain is canonically $\tilde{\Omega}_{\mathrm{Vin}}$, which means that $v_1 \notin \overline{\tilde{\Omega}_{\mathrm{Vin}}}$. Because of this, implies that $x = x_0 + tv_1$ for some $x_0 \in \overline{\tilde{W}_1}$ and $t \in \ensuremath{\mathbb{R}}$, hence $v_2 = x_0 + t'v_1$ for $t' \in \ensuremath{\mathbb{R}}$. Thus, we know that $v_2$ lies in the closure of the set $U = \widetilde{\mathbf{Hc}}_-(W_1) \cup \widetilde{\mathbf{Hc}}_+(W_1) \cup \tilde{W}_1$. Moreover, if $\overline{W_1} \cap \overline{W_2} = \emptyset$, we may take $x_0 \in \tilde{W}_1$, which implies that in fact $v_2 \in U$. Now, since $\widetilde{\mathbf{Hs}}_+(W_1)$ contains $\widetilde{\mathbf{Hs}}_+(W_2)$, we must have $\tilde{W_1} \subset \widetilde{\mathbf{Hs}}_-(W_2)$ and therefore $$\rho(\gamma_2)\tilde{W_1} \subset \widetilde{\mathbf{Hs}}_+(W_2) \subset \widetilde{\mathbf{Hs}}_+(W_1).$$ Letting $y \in \tilde{W_1}$, we see that $\alpha_1(\rho(\gamma_2)y) > 0$, i.e. $\alpha_1(y - \alpha_2(y)v_2) > 0$ and therefore $\alpha_1(v_2) > 0$. Thus, $v_2$ cannot lie in the closure of $\widetilde{\mathbf{Hc}}_-(W_1)$. This proves that $v_2 \in \overline{\widetilde{\mathbf{Hc}}_+(W_1)}$, and that $v_2 \in \widetilde{\mathbf{Hc}}_+(W_1)$ if $\overline{W_1} \cap \overline{W_2} = \emptyset$. ◻ ### Strong nesting {#sec:strong_nesting} Now consider a pair of disjoint walls $W_1, W_2$ in $\Omega_{\mathrm{Vin}}$, such that $\mathbf{Hs}_+(W_2) \subset \mathbf{Hs}_+(W_1)$. We want to know precisely when the half-cone $\mathbf{Hc}_+(W_2)$ is *strongly* nested inside of $\mathbf{Hc}_+(W_1)$, i.e. when the closure of $\mathbf{Hc}_+(W_2)$ is contained in $\mathbf{Hc}_+(W_1)$. It is clear that this strong nesting cannot occur if $\overline{W_1} \cap \overline{W_2}$ is nonempty (meaning, by , that the group element $\gamma(W_1, W_2)$ lies in a proper standard subgroup of $C$). One might hope that this is the only situation in which strong nesting fails, and indeed, if this were true, it would greatly simplify several inductive arguments in the final section of this paper. But it turns out that this is not the case; see for an explicit counterexample. However, in the special case where a geodesic representing the group element $\gamma = \gamma(W_1, W_2)$ lies in a hyperbolic subcomplex of $\mathrm{D}(C,S)$, the lemma below gives us some control over the situations where the half-cones over $W_1, W_2$ fail to strongly nest. **Lemma 54**. *Let $\ensuremath{\mathcal{W}} = W_0, \ldots, W_{n+1}$ be an efficient itinerary in $\Omega_{\mathrm{Vin}}$, and suppose that $\mathbf{Hc}_+(W_{n+1}) \subset \mathbf{Hc}_+(W_0)$ but $\overline{\mathbf{Hc}_+(W_{n+1})} \not\subset \mathbf{Hc}_+(W_0)$. Suppose further that $\gamma(\ensuremath{\mathcal{W}})$ has $D$-bounded product projections. Then, for a constant $R$ depending only on $D$, $\ensuremath{\mathcal{W}}$ is equivalent to an $\Omega_{\mathrm{Vin}}$-itinerary of the form $\ensuremath{\mathcal{U}}_-, \ensuremath{\mathcal{U}}_0, \ensuremath{\mathcal{U}}_+$, such that:* 1. *The itinerary $\ensuremath{\mathcal{U}}_0$ satisfies $|\ensuremath{\mathcal{U}}_0| < 2R$;* 2. *The itineraries $\ensuremath{\mathcal{U}}_-, \ensuremath{\mathcal{U}}_+$ are both efficient;* 3. *Both intersections $\bigcap_{U \in \ensuremath{\mathcal{U}}_+} \overline{U}$ and $\bigcap_{U \in \ensuremath{\mathcal{U}}_-}\overline{U}$ are nonempty.* In light of , the lemma above tells us that a pair of disjoint walls in the Vinberg domain will fail to have strictly nested half-cones only if the group element traversed by an efficient itinerary between them is (roughly) a product of two elements which each lie in a proper standard subgroup. *Proof.* First, if $\overline{W_0} \cap \overline{W_{n+1}}$ is nonempty, then by we can just take $\ensuremath{\mathcal{U}}_- = \ensuremath{\mathcal{W}}$ and $\ensuremath{\mathcal{U}}_0, \ensuremath{\mathcal{U}}_+$ to be empty. So, assume $\overline{W_0} \cap \overline{W_{n+1}} = \emptyset$. From , if $\partial \mathbf{Hc}_+(W_0)$ intersects $\overline{\mathbf{Hc}_+(W_{n+1})}$, then $\partial W_{n+1}$ contains a point $x_{n+1}$ in the boundary of $\mathbf{Hc}_+(W_0)$. Then since $\overline{W_0} \cap \overline{W_{n+1}} = \emptyset$, there is a point $x_0 \in \partial W_0$ and a nontrivial projective segment $\ell \subset \partial \mathbf{Hc}_+(W_0)$ joining $x_0$ to $x_{n+1}$, such that the projective span $L$ of $\ell$ contains the tip $v_0$ of $\mathbf{Hc}_+(W_0)$. By convexity, $\ell$ is contained in $\overline{\Omega_{\mathrm{Vin}}}$. And, since $\ell \subset \partial \mathbf{Hc}_+(W_0)$, implies that $\ell \subset \partial \Omega_{\mathrm{Vin}}$. Further, the reflection $R_0$ fixing $W_0$ acts by a reflection on $L$ fixing $x_0$, and preserves $\partial \Omega_{\mathrm{Vin}}$. So the union $\ell' = \ell \cup R_0\ell$ is a projective segment in $\partial \Omega_{\mathrm{Vin}}$ containing $x_0$ in its interior. Let $v_{n+1}$ be the polar of $W_{n+1}$, and consider the projective subspace $P$ spanned by $L$ and $v_{n+1}$. First, suppose that $P$ does not intersect the interior of $\Omega_{\mathrm{Vin}}$. Then, there is a *supporting hyperplane* $H$ of $\Omega_{\mathrm{Vin}}$ containing $P$, i.e. a hyperplane in $\ensuremath{\mathbb{P}}(V)$ which intersects $\overline{\Omega_{\mathrm{Vin}}}$ but not $\Omega_{\mathrm{Vin}}$. In particular, this supporting hyperplane contains both $v_{n+1}$ and $v_0$, so it is preserved by the reflections fixing $W_0$ and $W_{n+1}$. So, $H$ lies in $\overline{W_0^*} \cap \overline{W_{n+1}^*}$. But, implies that this intersection is empty once $\overline{W_0} \cap \overline{W_{n+1}}$ is empty. We conclude that $P$ intersects the interior of $\Omega_{\mathrm{Vin}}$. In particular $L$ is a proper subspace of $P$, so $P$ is a projective $2$-plane. Since each wall $W_i$ for $1 < i < n$ is disjoint from both $W_0$ and $W_{n+1}$, the intersection $W_i \cap P$ cannot span $P$. So, $W_i \cap P$ is a projective segment $\ell_i$. Each $\ell_i$ must separate $W_0 \cap P$ from $W_{n+1} \cap P$ in $\Omega_{\mathrm{Vin}}\cap P$, so the closure of $\ell_i$ intersects the (closed) segment $\ell$ at a point $x_i$. See . Then, by reordering the walls in $\ensuremath{\mathcal{W}}$, we can obtain an equivalent efficient itinerary where the corresponding ordering of the walls $W_i$ is compatible with the (partial) ordering of the $x_i$ along $\ell$. We partition this itinerary into two pieces $\ensuremath{\mathcal{U}}_-', \ensuremath{\mathcal{U}}_+'$, where $\ensuremath{\mathcal{U}}_-'$ consists of the walls $W_i$ such that $x_i \in \ell -\{x_{n+1}\}$, and $\ensuremath{\mathcal{U}}_+'$ consists of the walls $W_i$ such that $x_i = x_{n+1}$. then says that there is a uniform constant $R$ so that $\ensuremath{\mathcal{U}}_+'$ is equivalent to an itinerary $\ensuremath{\mathcal{U}}_0'', \ensuremath{\mathcal{U}}_+$, where $|\ensuremath{\mathcal{U}}_0''| \le R$ and $\ensuremath{\mathcal{U}}_+$ contains a unique minimal wall. Since $\ensuremath{\mathcal{U}}_+$ also contains the (unique) maximal wall $W_{n+1}$ in $\ensuremath{\mathcal{W}}$, it must be efficient. Similarly, we can find an itinerary $\ensuremath{\mathcal{U}}_-, \ensuremath{\mathcal{U}}_0'$ equivalent to $\ensuremath{\mathcal{U}}'_-$, so that $\ensuremath{\mathcal{U}}_-$ is efficient, and $|\ensuremath{\mathcal{U}}_0'| \le R$. We let $\ensuremath{\mathcal{U}}_0 = \ensuremath{\mathcal{U}}_0', \ensuremath{\mathcal{U}}_0''$, so that $\ensuremath{\mathcal{W}}$ is equivalent to $\ensuremath{\mathcal{U}}_-, \ensuremath{\mathcal{U}}_0, \ensuremath{\mathcal{U}}_+$. Since $x_{n+1} \in \overline{W_i}$ for each $W_i$ in $\ensuremath{\mathcal{U}}_+$, we already know that the intersection $\bigcap_{U \in \ensuremath{\mathcal{U}}_+} \overline{U}$ is nonempty. To see that the intersection $\bigcap_{U \in \ensuremath{\mathcal{U}}_-}\overline{U}$ is also nonempty, let $W_m$ be the (unique) maximal wall in $\ensuremath{\mathcal{U}}_-$, so that $\overline{W_m}$ intersects the interior of the projective line segment $\ell' \subset \partial \Omega_{\mathrm{Vin}}$ at the point $x_m$. Any supporting hyperplane of $\Omega_{\mathrm{Vin}}$ at $x_m$ must contain $\ell'$, which means that any such hyperplane contains the projective line $L$, and in particular contains the point $v_0$. Let $v_m$ be the polar of the wall $W_m$. By , the projective line spanned by $x_m$ and $v_m$ does not intersect $\Omega_{\mathrm{Vin}}$, so it is contained in some supporting hyperplane $H$ of $\Omega_{\mathrm{Vin}}$ at $x_m$. Then $H$ is an element of $\overline{\Omega_{\mathrm{Vin}}^*}$ which contains both $v_m$ and $v_0$, so it is preserved by the reflections fixing $W_0$ and $W_m$. That is, $H \in \overline{W_0^*} \cap \overline{W_m^*}$, so implies that the intersection $\overline{W_0} \cap \overline{W_m}$ is also nonempty. Then we are done after applying . ◻ # Singular values, stable and unstable subspaces, and regularity {#sec:singular_values} The definition of an Anosov representation used in this paper () is rooted in linear algebra---specifically, in the *singular value decomposition* of matrices in $\mathop{\mathrm{GL}}(d, \ensuremath{\mathbb{R}})$. The main purpose of this section is to provide various methods for estimating singular values and singular value gaps, which we will use throughout the rest of the paper. Much of the content of this section can also be interpreted in terms of the geometry and dynamics of semisimple Lie groups or their associated Riemannian symmetric spaces, and some of the cited results rely on this perspective (or other complicated tools) for their proofs. However, we will state all of the estimates we need in elementary terms, and refer to [@BPS Sect. 7] for an overview of the connection between the different viewpoints. ## Singular values {#subsec:singular values} We equip $\ensuremath{\mathbb{R}}^d$ with its standard Euclidean inner product. Given a matrix $g \in \mathop{\mathrm{GL}}(d,\ensuremath{\mathbb{R}})$, recall that the $i$^th^ singular value $\sigma_i(g)$ is the $i$^th^ largest eigenvalue of the linear map $\sqrt{gg^T}$, where $\sqrt{gg^T}$ is the unique positive-definite matrix squaring to the positive-definite matrix $gg^T$. The definition of $\sigma_i(g)$ depends on the choice of positive-definite inner product on $\ensuremath{\mathbb{R}}^d$. We obtain different singular values for $g$ by choosing a different inner product $\langle \cdot, \cdot \rangle$, and using the adjoint of $g$ with respect to $\langle \cdot, \cdot \rangle$ in place of the transpose matrix $g^T$. Geometrically, the $i$^th^ singular value is the length of the $i$^th^ longest axis of the image ellipsoid $$\{gv : v \in \ensuremath{\mathbb{R}}^d, \|v\| = 1\},$$ measured with respect to the chosen inner product on $\ensuremath{\mathbb{R}}^d$. If $\sigma_i(g) > \sigma_{i+1}(g)$, then we define the unstable subspace ${E^+_{i}}(g)$ as the subspace of $\ensuremath{\mathbb{R}}^d$ spanned by the $i$ longest axes of this image ellipsoid. If $\sigma_{d-i}(g) > \sigma_{d-i+1}(g)$, we also have the stable space ${E^-_{i}}(g) = {E^+_{i}}(g^{-1})$, which is the subspace spanned by the $i$ least expanded axes of the unit sphere $\{v \in \ensuremath{\mathbb{R}}^d : \|v\| = 1\}$. All of this information is combined in the singular value decomposition (or $KAK$ decomposition) of $g$: any element $g \in \mathop{\mathrm{GL}}(d, \ensuremath{\mathbb{R}})$ can be written as a product $g = ka\ell$, where $a$ is a diagonal matrix with nonzero entries $\sigma_1, \ldots, \sigma_d$, and $k, \ell \in \mathrm{O}(d)$ are such that $k$ takes the span of the first $i$ standard basis vectors to $E_i^+(g)$, and $\ell$ takes $E_i^-(g)$ to the span of the last $i$ standard basis vectors. We let $\mu_i(g)$ denote $\log\sigma_i(g)$, and collect the logarithms together in the vector $\mu(g) := (\mu_1(g), \mu_2(g), \dots, \mu_d(g))$. We will also write $\mu_{i,i+1}(g)$ as shorthand for $\mu_i(g) - \mu_{i+1}(g)$. ## Additivity estimates for singular value gaps As noted in the introduction, a representation $\rho\colon \Gamma \to \mathop{\mathrm{GL}}(d,\ensuremath{\mathbb{R}})$ of a finitely-generated group $\Gamma$ is $1$-Anosov if and only if there exist constants $A, B > 0$ such that $\mu_{1,2}(\rho(\gamma)) \geq A|\gamma| - B$ for all $\gamma \in \Gamma$. This definition depends on a choice of word metric determined by a choice of finite generating set for the group $\Gamma$, but this does not matter since all such word metrics are quasi-isometrically equivalent. It also depends on the choice of inner product used to define singular values, but this also turns out not to matter. One justification is the following error estimate on logarithms of singular values: **Lemma 55** (Additivity estimate for $\mu$; see [@ggkw2017anosov Fact 2.18]). *For any norm $||\cdot||$ on $\ensuremath{\mathbb{R}}^d$, there is a constant $K > 0$ such that for any $g, h_1, h_2 \in \mathop{\mathrm{SL}}^{\pm}(d, \ensuremath{\mathbb{R}})$, we have $$||\mu(h_1gh_2) - \mu(g)|| \le K(||\mu(h_1)|| + ||\mu(h_2)||).$$ In particular, for any $1 \le i < d$, there is some $K' > 0$ such that $$|\mu_{i,i+1}(h_1gh_2) - \mu_{i,i+1}(g)| \le K'(||\mu(h_1)|| + ||\mu(h_2)||).$$* As any pair of positive-definite inner products on $\ensuremath{\mathbb{R}}^d$ differ by composition with some fixed element $h \in \mathop{\mathrm{GL}}(d, \ensuremath{\mathbb{R}})$, the lemma implies in particular that choosing a different inner product on $\ensuremath{\mathbb{R}}^d$ to define singular values only changes each $\mu_k(g)$ by a uniformly bounded additive amount. The lemma also tells us that if $\Gamma$ is any finitely generated subgroup of $\mathop{\mathrm{SL}}^{\pm}(d, \ensuremath{\mathbb{R}})$, equipped with a word metric $|\cdot|$, then $||\mu(\gamma)||$ is at most $K|\gamma|$ for a uniform constant $K$. Thus, we can also apply the lemma to obtain the following useful result: **Lemma 56** ("Triangle inequality" for $\mu_{i,i+1}$). *Let $\Gamma$ be a finitely generated subgroup of $\mathop{\mathrm{SL}}^{\pm}(d,\ensuremath{\mathbb{R}})$, equipped with a word metric $|\cdot|$. There exists a constant $K > 0$ such that for any $\gamma, \eta_1, \eta_2 \in \Gamma$, we have $$|\mu_{i,i+1}(\eta_1\gamma\eta_2) - \mu_{i,i+1}(\gamma)| \le K(|\eta_1| + |\eta_2|).$$* We will frequently apply this lemma throughout the rest of the paper, as it allows us to estimate singular value gaps for the images of "nearby" elements in a finitely generated group $\Gamma$ under some representation $\rho$. We will also sometimes want to estimate singular value gaps for the product of a pair of *large* elements $\gamma_1, \gamma_2$ in $\Gamma$. This is possible as long as the stable and unstable subspaces of $\rho(\gamma_1), \rho(\gamma_2)$ exist and are *uniformly transverse*. In this situation, we can apply the following: **Lemma 57** (Uniform transversality estimate for $\mu_{1,2}$; see [@BPS Lemma A.7]). *Let $g$, $h$ be two elements of $\mathop{\mathrm{SL}}^{\pm}(d, \ensuremath{\mathbb{R}})$, and suppose that $\mu_{1,2}(g) > 0$ and $\mu_{1,2}(h) > 0$. If $\theta = \angle({E^-_{d-1}}(g), {E^+_{1}}(h))$, then we have $$\mu_{1,2}(gh) \ge \mu_{1,2}(g) + \mu_{1,2}(h) + 2\log(\sin\theta).$$* ## Regularity, uniform gaps, and the local-to-global principle The next estimates in this section are less elementary in nature, but can still be stated in terms of linear algebra. **Definition 58** (Uniform regularity). Let $g_n$ be a (finite or infinite) sequence in $\mathop{\mathrm{GL}}(d, \ensuremath{\mathbb{R}})$. We say that the sequence $g_n$ is *$(A, B)$-regular* for constants $A \ge 0, B \ge 0$ if for all $n, m$, we have $$\label{eq:uniform_regularity} \mu_{1,2}(g_n^{-1}g_m) \ge A\mu_{1,d}(g_n^{-1}g_m) - B.$$ If an infinite sequence $g_n$ is $(A, B)$-regular for some $A, B$, then we say that $g_n$ is *uniformly regular*. Sequences as in are a special case of the "coarsely uniformly regular" sequences defined by Kapovich-Leeb-Porti in [@KLP2018]; in general a different type of "regularity" can be defined for each singular value gap $\mu_{i,i+1}$. For sequences lying in a finitely generated subgroup $\Gamma < \mathop{\mathrm{GL}}(d, \ensuremath{\mathbb{R}})$, we can also strengthen this notion of uniform regularity as follows: **Definition 59**. Let $\Gamma$ be a finitely generated subgroup of $\mathop{\mathrm{GL}}(d,\ensuremath{\mathbb{R}})$, and let $\gamma_n$ be a sequence in $\Gamma$ which is a geodesic with respect to a word metric $|\cdot|$. We say the geodesic $\gamma_n$ *has $(A,B)$-gaps* if $$\mu_{1,2}(\gamma_n^{-1}\gamma_m) \ge A|m - n| - B.$$ If a geodesic $\gamma_n$ has $(A,B)$-gaps for some $A,B$, we also say it has *uniform gaps*. A geodesic $\gamma_n$ with uniform gaps is always uniformly regular: for any $g \in \mathop{\mathrm{GL}}(d, \ensuremath{\mathbb{R}})$, we have $\mu_{1,d}(g) = \log||g|| + \log||g^{-1}||$, where $||\cdot||$ is the operator norm on $\mathop{\mathrm{GL}}(d, \ensuremath{\mathbb{R}})$. This means that there is a constant $K$ (depending only on $\Gamma$ and on the choice of finite generating set determining $|\cdot|$) so that $$\mu_{1,d}(\gamma_n^{-1} \gamma_m) \leq K|m-n|.$$ Geodesics with uniform gaps as in are a special case of the *uniformly regular undistorted* (or *URU*) sequences defined in the work of Kapovich--Leeb--Porti [@klp2017characterizations]; "undistorted" refers to the fact that these sequences map to quasi-geodesics in the Riemannian symmetric space associated to the semisimple Lie group $\mathop{\mathrm{PGL}}(d, \ensuremath{\mathbb{R}})$. We note that, in this language, 1-Anosov representations are precisely those which send sequences in the group which are geodesic with respect to a word metric to uniformly regular and undistorted sequences. A key feature of geodesics with uniform gaps is that they have well-defined "limit points" in both projective space $\ensuremath{\mathbb{P}}(\ensuremath{\mathbb{R}}^d)$ and its dual (indeed, this is one way of defining limit maps for 1-Anosov representations). To state this result, we let $d_{\mathbb{P}}$ denote the metric on projective space obtained by viewing $\ensuremath{\mathbb{P}}(\ensuremath{\mathbb{R}}^d)$ as the quotient of the unit sphere in $\ensuremath{\mathbb{R}}^d$. Then we let $d_{\mathbb{P}}^*$ denote the metric on the Grassmannian $\mathrm{Gr}_{d-1}(\ensuremath{\mathbb{R}}^d)$ obtained by viewing the projectivization of each $(d-1)$-plane in $\mathrm{Gr}_{d-1}(\ensuremath{\mathbb{R}}^d)$ as a subset of $\ensuremath{\mathbb{P}}(\ensuremath{\mathbb{R}}^d)$, and taking Hausdorff distance with respect to $d_{\mathbb{P}}$. **Lemma 60** (Uniform convergence for geodesics with uniform gaps; see [@BPS Proposition 2.4]). *Let $\Gamma$ be a finitely generated subgroup of $\mathop{\mathrm{GL}}(d, \ensuremath{\mathbb{R}})$. If $\gamma_n$ is a geodesic in $\Gamma$ with $(A,B)$-gaps, then the limits $$\lim_{n \to \infty} {E^+_{1}}(\gamma_n), \qquad \lim_{n \to \infty} {E^-_{d-1}}(\gamma_n)$$ exist in $\ensuremath{\mathbb{P}}(\ensuremath{\mathbb{R}}^d)$ and $\mathrm{Gr}_{d-1}(\ensuremath{\mathbb{R}}^d)$, respectively, and are uniform in $A, B$ with respect to the metrics $d_{\mathbb{P}}$ and $d_{\mathbb{P}}^*$.* **Remark 61**. also follows from the *higher-rank Morse lemma* of Kapovich--Leeb--Porti [@KLP2018] (discussed briefly below), but we refer to the result in [@BPS] since it is closer to the form that we actually need. Both of these results are stronger and more technical than what we have stated here---in particular, they imply a version of which does not require the sequence $\gamma_n$ to lie in a finitely generated subgroup of $\mathop{\mathrm{GL}}(d, \ensuremath{\mathbb{R}})$. We refer also to [@ggkw2017anosov Sect. 5] for a related result that holds under the assumption that $\Gamma$ is a word-hyperbolic group. The next estimate in this section follows from several results of Kapovich--Leeb--Porti. The first, proved in [@KLP2018], is a higher-rank version of the Morse lemma. It implies that uniformly regular undistorted sequences in a finitely generated subgroup $\Gamma < \mathop{\mathrm{SL}}^{\pm}(d, \ensuremath{\mathbb{R}})$ are essentially equivalent to uniformly *Morse* quasi-geodesic sequences in the associated Riemannian symmetric space $X$, i.e. quasi-geodesic sequences which stay uniformly close to certain "Morse subsets" in $X$. The next result, proved in [@KLP2014] (or [@KLP2023]), says that these Morse quasi-geodesics satisfy a *local-to-global* principle; a more precise version of this theorem was later proved by Riestenberg [@Riestenberg]. Combining these results gives us a local-to-global principle for geodesics with uniform gaps in a finitely generated group $\Gamma$, which we can state as follows: **Theorem 62** (Local-to-global principle; see [@KLP2018 Theorem 1.3] and [@KLP2023 Theorem 1.1] or [@KLP2014 Theorem 7.18] or [@Riestenberg Theorem 1.1]). *Let $\Gamma$ be a finitely generated subgroup of $\mathop{\mathrm{SL}}^{\pm}(d, \ensuremath{\mathbb{R}})$. Given $A, B > 0$, there exist constants $A', B', \lambda > 0$ satisfying the following. Suppose that $\gamma_n$ is a sequence in $\Gamma$ which is a geodesic with respect to a word metric $|\cdot|$, and that for every $i < j$ with $|i - j| \le \lambda$, the sub-geodesic $\gamma_i, \ldots, \gamma_j$ has $(A, B)$-gaps. Then $\gamma_n$ has $(A',B')$-gaps.* ## Estimating singular value gaps using the Hilbert metric Our last estimate on singular values comes from convex projective geometry. Any properly convex domain $\Omega$ in real projective space $\ensuremath{\mathbb{P}}(\ensuremath{\mathbb{R}}^d)$ can be endowed with a natural metric $d_{\Omega}$ called the *Hilbert metric*. The Hilbert metric on $\Omega$ is always proper, geodesic, and invariant under any projective transformations preserving $\Omega$. We will use the Hilbert metric as a tool to study the behavior of *nested sequences* of properly convex domains in $\ensuremath{\mathbb{P}}(\ensuremath{\mathbb{R}}^d)$. Roughly, we will use the Hilbert metric to measure how much a projective transformation $g$ *fails* to preserve a given properly convex domain $\Omega$, and then use this to estimate the singular value gaps of $g$. We provide this estimate in below, but first we recall some basic definitions. **Definition 63**. Let $\Omega$ be a properly convex domain. For $x, y \in \Omega$, we define the Hilbert distance $d_\Omega(x, y)$ by $$d_{\Omega}(x,y) = \frac12\log([a, b; x, y]),$$ where $a, b$ are the unique points in $\partial \Omega$ such that $a, x, y, b$ lie on a projective line in that order, and $[\cdot, \cdot; \cdot, \cdot]$ is the projective cross-ratio with formula $$= \frac{\norm{a-c}}{\norm{a-d}}\cdot\frac{\norm{b-d}}{\norm{b-c}}.$$ The distances appearing in the cross-ratio formula can be measured using any Euclidean metric on any affine chart containing $a, b, c, d$; the value of the cross-ratio does not depend on the choice of chart or metric. From this, it follows immediately that for any properly convex domain $\Omega \subset \ensuremath{\mathbb{P}}(V)$, any $g \in \mathop{\mathrm{GL}}(V)$, and any $x, y \in \Omega$, we have $$d_{\Omega}(x, y) = d_{g\Omega}(gx, gy).$$ For our purposes, the most important property of the Hilbert metric is the following standard result, which can be verified by a straightforward computation: **Lemma 64** (Expansion of the Hilbert metric on nested domains). *Let $\Omega_1, \Omega_2$ be properly convex domains in $\ensuremath{\mathbb{P}}(\ensuremath{\mathbb{R}}^d)$, and suppose $\Omega_1 \subseteq \Omega_2$. Then for all $x, y \in \Omega_1$, we have $$d_{\Omega_1}(x,y) \ge d_{\Omega_2}(x,y).$$ Further, if the strong inclusion $\overline{\Omega_1} \subset \Omega_2$ also holds, then there is a constant $\lambda > 1$ (depending only on $\Omega_1$ and $\Omega_2$) so that for all $x, y \in \Omega_1$, we have $$d_{\Omega_1}(x,y) \ge \lambda \cdot d_{\Omega_2}(x,y).$$* Our main application of the Hilbert metric in this paper is the following. Here, and elsewhere, if $\Omega$ is a properly convex domain and $X \subseteq \Omega$, then $\mathrm{diam}_{\Omega}(X)$ refers to the diameter of $X$ with respect to the Hilbert metric $d_\Omega$. **Lemma 65** (Hilbert metric estimates singular value gaps). *Fix properly convex domains $\Omega_1, \Omega_2 \subset \ensuremath{\mathbb{P}}(\ensuremath{\mathbb{R}}^d)$. There exists a constant $D > 0$, depending only on $\Omega_1, \Omega_2$, such that for any $g \in \mathop{\mathrm{GL}}(d, \ensuremath{\mathbb{R}})$ with $g\overline{\Omega_1} \subset \Omega_2$, we have $$\label{eq:hilbert_metric_estimates_sv_gaps} \mu_{1,2}(g) \ge -\log( \mathrm{diam}_{\Omega_2}(g\Omega_1)) - D.$$* *Proof.* Let $A_d$ be the standard affine chart $\{[a_1 : \ldots : a_{d-1} : 1] \subset \ensuremath{\mathbb{P}}(\ensuremath{\mathbb{R}}^d) : a_i \in \ensuremath{\mathbb{R}}\}$, equipped with the Euclidean metric induced by the standard inner product on $\ensuremath{\mathbb{R}}^d$. We let $B$ be the unit ball in $A_d$ centered at $[e_d] = [0 : \ldots : 0 : 1]$, and first consider the special case of the lemma where $\Omega_1 = \Omega_2 = B$, and $g$ fixes both $e_d$ and its orthogonal complement $e_d^\perp = \mathop{\mathrm{span}}\{e_1, \ldots, e_{d-1}\}$. That is, we assume that $g$ is block-diagonal, of the form $$\begin{pmatrix} H \\ & \lambda \end{pmatrix}$$ for $H \in \mathop{\mathrm{GL}}(e_d^\perp)$ and $\lambda \in \ensuremath{\mathbb{R}}$. Then $gB$ is an ellipsoid in $A_d$, centered at $[e_d]$, whose semi-major axis has length $\sigma_1(H) / |\lambda|$. Then if $g\overline{B} \subset B$, we have $\sigma_1(H) / |\lambda| < 1$ and thus $\sigma_1(g)/\sigma_2(g) = |\lambda|/\sigma_1(H)$. The diameter of $gB$ with respect to the Hilbert metric $d_B$ is therefore given by $$\begin{aligned} \mathop{\mathrm{diam}}_B(gB) &= \log\left(\frac{1+e^{-\mu_{1,2}(g)}}{1 - e^{-\mu_{1,2}(g)}}\right)\\ &\geq 2\log(1 + e^{-\mu_{1,2}(g)}) \\ &\geq 2e^{-\mu_{1,2}(g)}. \end{aligned}$$ This proves that [\[eq:hilbert_metric_estimates_sv_gaps\]](#eq:hilbert_metric_estimates_sv_gaps){reference-type="eqref" reference="eq:hilbert_metric_estimates_sv_gaps"} holds in this special case, when we take $D = \log 2$. The rest of the proof amounts to reducing the general case to this one. First, we reduce the general situation to the case where $\overline{\Omega_2} \subset \Omega_1$, by choosing some $h_1 \in \mathop{\mathrm{SL}}(d, \ensuremath{\mathbb{R}})$ so that $\overline{\Omega_2} \subset h_1\Omega_1$. Then, if $\Omega_1' = h_1\Omega_1$, we have $$\mathop{\mathrm{diam}}_{\Omega_2}(g\Omega_1) = \mathop{\mathrm{diam}}_{\Omega_2}(gh_1^{-1}\Omega_1'),$$ and by the difference $|\mu_{1,2}(g) - \mu_{1,2}(gh_1^{-1})|$ is bounded by a constant only depending on $h_1$. So, we can replace $\Omega_1$ with $\Omega_1'$ and $g$ with $gh_1^{-1}$. This introduces uniformly bounded additive error to the left-hand side of [\[eq:hilbert_metric_estimates_sv_gaps\]](#eq:hilbert_metric_estimates_sv_gaps){reference-type="eqref" reference="eq:hilbert_metric_estimates_sv_gaps"}, and does not affect the right-hand side. Now, since $g\overline{\Omega_1} \subset \Omega_2 \subset \Omega_1$, the Brouwer fixed-point theorem implies that $g$ fixes some $x \in \Omega_2$. Since duality reverses inclusions, we also know that $g^{-1}\overline{\Omega_2^*} \subset \Omega_1^* \subset \Omega_2^*$, and therefore $g$ also fixes an element of $\Omega_1^*$, which we can view as a hyperplane $U \subset \ensuremath{\mathbb{R}}^d$. Since $\overline{\Omega_2} \subset \Omega_1$, there is a uniform lower bound $\varepsilon> 0$ on the distance between $x$ and $U$, in a fixed metric on $\ensuremath{\mathbb{P}}(\ensuremath{\mathbb{R}}^d)$. So, there is a compact set $K = K(\varepsilon)$ in $\mathop{\mathrm{SL}}(d, \ensuremath{\mathbb{R}})$ such that for some $h \in K$, the spaces $hx, hU$ are orthogonal with respect to the standard inner product on $\ensuremath{\mathbb{R}}^d$. By projective invariance of the Hilbert metric we know that $$\mathop{\mathrm{diam}}_{\Omega_2}(g\Omega_1) = \mathop{\mathrm{diam}}_{h\Omega_2}(hg\Omega_1).$$ So, if we replace $\Omega_1, \Omega_2$ with $h\Omega_1, h\Omega_2$, and replace $g$ with $hgh^{-1}$, the right-hand side of [\[eq:hilbert_metric_estimates_sv_gaps\]](#eq:hilbert_metric_estimates_sv_gaps){reference-type="eqref" reference="eq:hilbert_metric_estimates_sv_gaps"} again stays the same, and (again applying ) the left-hand side only changes by an additive amount, bounded uniformly in terms of $\Omega_1, \Omega_2$. Then, after further conjugating by an orthogonal matrix, we can assume the fixed point of $g$ in $\Omega_2$ is $[e_d]$, and that $\overline{\Omega_1}$ lies in the $g$-invariant affine chart $A_d$. For any $r > 0$, we let $B_r$ denote the open ball of radius $r$ centered at $[e_d]$ in the affine chart $A_d$. Since $\Omega_1, \Omega_2$ are both bounded open convex subsets in $A_d$, we can find radii $r_1, r_2 > 0$ such that $B_{r_1} \subset \Omega_1$, and $\overline{\Omega_2} \subset B_{r_2}$. We let $h_1, h_2$ denote the dilations in $A_d$ by a factor of $r_1, r_2$, respectively, and we let $g' = h_2^{-1}gh_1$. Using , we see that $$\mathop{\mathrm{diam}}_{B_1}(g'B_1) = \mathop{\mathrm{diam}}_{B_{r_2}}(gB_{r_1}) < \mathop{\mathrm{diam}}_{\Omega_2}(g\Omega_1),$$ and one more application of tells us that the difference between $\mu_{1,2}(g)$ and $\mu_{1,2}(g')$ is bounded by a constant depending only on $\Omega_1, \Omega_2$. This reduces the problem to the case that we originally proved. ◻ # Stable and unstable subspaces in half-cones Let $(C, S)$ be an infinite irreducible right-angled Coxeter system with $|S| = d$, and let $\rho\colon C \to \mathop{\mathrm{SL}}^{\pm}(d, \ensuremath{\mathbb{R}})$ be a simplicial representation. As in the previous section, we assume $V = \ensuremath{\mathbb{R}}^d$ is endowed with its standard inner product, which allows us to define $\sigma_k(g)$ and $\mu_k(g)$ for every $g \in \mathop{\mathrm{SL}}^{\pm}(V)$; if $\mu_{k, k+1}(g) \ne 0$ we can also define ${E^+_{k}}(g)$ and ${E^-_{d-k}}(g)$. In this section, we establish some estimates on the location of the subspaces ${E^+_{1}}(\rho(\gamma))$ and ${E^-_{d-1}}(\rho(\gamma))$ for certain elements $\gamma \in C$, in terms of the walls of an $\Omega_{\mathrm{Vin}}$-itinerary $\ensuremath{\mathcal{W}}$ traversing $\gamma$. Ultimately, we will use these estimates to argue that the stable and unstable subspaces of certain group elements $\rho(\gamma_1), \rho(\gamma_2)$ are *uniformly transverse*, which allows us to apply to get an estimate on $\mu_{1,2}(\rho(\gamma_1\gamma_2))$. The idea behind the first estimate is as follows. An unstable subspace ${E^+_{1}}(g)$ should be thought of as an "attracting" subspace for $g$ in $\ensuremath{\mathbb{P}}(V)$. If $\gamma$ is a group element separated from the identity by walls $W_1, \ldots, W_n$, and $x_0$ is a fixed basepoint in a reflection domain $\Omega$, then $\rho(\gamma) \cdot x_0$ is separated from $x_0$ by this same set of walls, and thus an "attracting" subspace for $\rho(\gamma)$ should at least lie in the positive half-space and positive half-cone bounded by the wall $W_1$. In fact, this "attracting" subspace should lie in the positive half-cone bounded by the wall $W_k$, as long as $k$ is much smaller than $n$. To make this precise, we first need a general fact about group actions on convex projective domains: **Lemma 66**. *Let $\Omega$ be a properly convex domain, and let $g_n \in \mathop{\mathrm{SL}}^{\pm}(d, \ensuremath{\mathbb{R}})$ be any sequence of elements such that $g_n\Omega = \Omega$, and $\mu_{1,2}(g_n) \to \infty$. Then for any point $x \in \Omega$, the set of accumulation points of the sequence $g_nx$ is the same as the set of accumulation points of the sequence ${E^+_{1}}(g_n)$.* *Proof.* Fix $x \in \Omega$, and choose an arbitrary subsequence of $g_n$ so that both of the sequences $g_nx$ and ${E^+_{1}}(g_n)$ converge. We will show that these sequences converge to the same point. Let $\mathop{\mathrm{Aut}}(\Omega) < \mathop{\mathrm{SL}}^{\pm}(d, \ensuremath{\mathbb{R}})$ denote the subgroup $\{g \in \mathop{\mathrm{SL}}^{\pm}(d, \ensuremath{\mathbb{R}}) : g \Omega = \Omega\}$. The Hilbert metric on $\Omega$ (see the previous section) is always proper and $\mathop{\mathrm{Aut}}(\Omega)$-invariant, which implies that the action of $\mathop{\mathrm{Aut}}(\Omega)$ on $\Omega$ is proper. Using this, one can check (see e.g. [@IZ21 Proposition 5.6]) that any accumulation point of the sequence of stable subspaces ${E^-_{d-1}}(g_n)$ is a supporting hyperplane of $\Omega$ (recall from the proof of that a supporting hyperplane is a projective hyperplane which intersects $\overline{\Omega}$, but not $\Omega$ itself). Up to subsequence, ${E^-_{d-1}}(g_n)$ converges to a fixed supporting hyperplane $H$. We can then appeal to the singular value decomposition (or $KAK$ decomposition, see ) of $g_n$ to see that if $z$ is any point in $\ensuremath{\mathbb{P}}(V) -H$, the sequence $g_nz$ converges to the limit of ${E^+_{1}}(g_n)$. In particular this holds for $z = x$ and we are done. ◻ Before stating and proving our first estimate, we establish some more notation. As in the previous section, we let $d_{\mathbb{P}}$ denote a Riemannian metric on $\ensuremath{\mathbb{P}}(V)$ induced by the standard inner product on $V$. **Definition 67**. For $x \in \ensuremath{\mathbb{P}}(V)$ and $\varepsilon> 0$, we let $B_\varepsilon(x)$ denote the open $\varepsilon$-ball about $x$, with respect to the metric $d_{\mathbb{P}}$. Similarly, if $Z \subset \ensuremath{\mathbb{P}}(V)$ is a subset, $N_\varepsilon(Z)$ denotes the $\varepsilon$-neighborhood of $Z$ in $\ensuremath{\mathbb{P}}(V)$. For $\varepsilon> 0$, the "$-\varepsilon$-neighborhood" of a set $Z \subset \ensuremath{\mathbb{P}}(V)$ is by convention the complement of the $\varepsilon$-neighborhood of the complement of $Z$, i.e. the set $N_{-\varepsilon}(Z) := \{x \in Z : B_\varepsilon(x) \subset Z\}.$ We now prove the first main estimate we need from this section: **Lemma 68** (Unstable subspaces lie close to half-cones). *For any $\varepsilon> 0$ and any $N > 0$, there exists a constant $M > 0$ satisfying the following. Suppose that $\ensuremath{\mathcal{W}}$ is an $\Omega$-itinerary $W_1, \ldots, W_n$ traversing a group element $\gamma \in C$, and suppose that $\mu_{1,2}(\rho(\gamma)) > M$. Then for any $k \le \min(n, N)$:* 1. *If $\ensuremath{\mathcal{W}}$ departs from the identity, then the unstable subspace ${E^+_{1}}(\rho(\gamma))$ is contained in $N_\varepsilon(\mathbf{Hc}_+(W_k))$.* 2. *If $\ensuremath{\mathcal{W}}$ arrives at the identity, then the stable subspace ${E^-_{d-1}}(\rho(\gamma))$ (viewed as a subset of $\ensuremath{\mathbb{P}}(V)$) is disjoint from $N_{-\varepsilon}(\mathbf{Hc}_-(W_{n-k}))$.* *Proof.* Let $\varepsilon, N$ be given, and fix $k \le N$. We will prove the first statement by contradiction: suppose that there is a sequence of group elements $\gamma_m \in C$ so that as $m \to \infty$, $\mu_{1,2}(\rho(\gamma_m))$ tends to infinity, but for some $\Omega$-itinerary $W_{0, m}, \ldots W_{|\gamma_m|, m}$ joining the identity to $\gamma_m$, the point ${E^+_{1}}(\rho(\gamma_m))$ does *not* lie in the $\varepsilon$-neighborhood of $\mathbf{Hc}_+(W_{k,m})$. Since $W_{k,m}$ is the last wall in an itinerary of length $k$ departing from the identity, after extracting a subsequence we can assume that $W_{k,m} = W_k$ for a wall $W_k$ independent of $m$. Fix a basepoint $x$ in the fundamental domain $\Delta \cap \Omega$. Then by definition, $W_k$ separates $x$ from $\rho(\gamma_m)x$, so $\rho(\gamma_m)x$ lies in the halfspace $\mathbf{Hs}_+(W_k)$, and therefore $\rho(\gamma_m)x$ accumulates in $\overline{\mathbf{Hs}_+(W_k)}$. Then by , ${E^+_{1}}(\rho(\gamma_m))$ also accumulates in $\overline{\mathbf{Hs}_+(W_k)}$. But we know that $\overline{\mathbf{Hs}_+(W_k)} \subset \overline{\mathbf{Hc}_+(W_k)}$ by , so eventually ${E^+_{1}}(\rho(\gamma_m))$ lies in the $\varepsilon$-neighborhood of $\mathbf{Hc}_+(W_k)$, contradiction. The second statement follows from the first via duality. Here are the details. Given $\gamma \in C$, we consider the group element $\rho^*(\gamma^{-1}) \in \mathop{\mathrm{SL}}^{\pm}(V^*)$. If $W_1, \ldots, W_n$ is an $\Omega$-itinerary traversing $\gamma$ and arriving at the identity, then $W_n, \ldots, W_1$ is an $\Omega$-itinerary traversing $\gamma^{-1}$ and departing from the identity. The sequence of dual walls $W_n^*, \ldots, W_1^*$ is an $\Omega^*$-itinerary, also traversing $\gamma^{-1}$ and departing from the identity. Observe that the singular values of $\rho^*(\gamma^{-1})$ are precisely the same as the singular values of $\rho(\gamma)$, so in particular $\mu_{1,2}(\rho^*(\gamma^{-1})) = \mu_{1,2}(\rho(\gamma))$. Then, the first part of the lemma implies that if $\mu_{1,2}(\rho(\gamma))$ is sufficiently large, the unstable subspace ${E^+_{1}}(\rho^*(\gamma^{-1})) \in \ensuremath{\mathbb{P}}(V^*)$ is contained in an arbitrarily small neighborhood of $\mathbf{Hc}_+(W_{n-k}^*) = \mathbf{Hc}_-(W_{n-k})^*$, with respect to the metric $d^*_{\mathbb{P}}$ on $\ensuremath{\mathbb{P}}(V^*)$; recall that this is the metric on $\ensuremath{\mathbb{P}}(V^*)$ obtained by viewing $\ensuremath{\mathbb{P}}(V^*)$ as the space of projective hyperplanes in $\ensuremath{\mathbb{P}}(V)$, and then taking Hausdorff distance with respect to $d_{\mathbb{P}}$. Therefore, if $K \subset \ensuremath{\mathbb{P}}(V)$ is the kernel of ${E^+_{1}}(\rho^*(\gamma^{-1}))$, then as long as $\mu_{1,2}(\rho(\gamma))$ is sufficiently large, there is a projective hyperplane $H$, within Hausdorff distance $\varepsilon$ of $K$, so that every point in the closure of $\mathbf{Hc}_-(W_{n-k})$ is not contained in $H$. The inner product on $V$ induces an isomorphism $V^* \to V$, which identifies $\rho^*(\gamma^{-1}) \in \mathop{\mathrm{SL}}^{\pm}(V^*)$ with the transpose $\rho(\gamma)^T \in \mathop{\mathrm{SL}}^{\pm}(V)$. The kernel $K$ of ${E^+_{1}}(\rho^*(\gamma^{-1}))$ is the orthogonal complement of the subspace ${E^+_{1}}(\rho(\gamma)^T)$, which is in turn the stable subspace ${E^-_{d-1}}(\rho(\gamma))$. We conclude that if $x \in \mathbf{Hc}_-(W_{n-k})$ is contained in ${E^-_{d-1}}(\rho(\gamma))$, then $d_{\mathbb{P}}(x, H) < \varepsilon$, hence $x$ is distance at most $\varepsilon$ from a point in the complement of $\mathbf{Hc}_-(W_{n-k})$. That is, no point in $N_{-\varepsilon}(\mathbf{Hc}_-(W_{n-k}))$ is contained in ${E^-_{d-1}}(\rho(\gamma))$. ◻ ## Bounding unstable subspaces away from walls The estimate given by the previous lemma tells us that unstable subspaces for $\rho(\gamma)$ are located near half-cones over the walls $W_1, \ldots, W_n$ in $\Omega_{\mathrm{Vin}}$ separating a basepoint $x_0$ from $\rho(\gamma)x_0$. We will want to apply this lemma to see that this unstable subspace is uniformly far from some hyperplane in the complement of $\mathbf{Hc}_+(W_1)$. If the sequence of halfcones $\mathbf{Hc}_+(W_k)$ is *strongly nested* (see ), then this poses little problem. However, if the closed walls separating $x_0$ from $\rho(\gamma)x_0$ have a common intersection in the boundary of $\Omega_{\mathrm{Vin}}$, then so do the boundaries of the half-cones over these walls. A priori, the unstable subspace ${E^+_{1}}(\rho(\gamma))$ could lie near this intersection of boundaries, and we would have no way to bound the distance between ${E^+_{1}}(\rho(\gamma))$ and the complement of $\mathbf{Hc}_+(W_1)$. The lemma below tells us that this does not occur. The idea is that, when the closed walls $\overline{W_1}, \ldots, \overline{W_n}$ have a common intersection in $\ensuremath{\partial}\Omega_{\mathrm{Vin}}$, then $\gamma$ lies "close" to a proper standard subgroup $C(T) < C$. The intersection of these walls lies in a "trivial subspace" for the action of this subgroup, which must be far from any attracting subspace for $\gamma$. See . **Lemma 69**. *Let $\Omega$ be a reflection domain for a simplicial representation $\rho$ with fully nondegenerate Cartan matrix. Then there exist $\varepsilon_0, M > 0$ satisfying the following. Suppose that $W_1, \ldots, W_n$ is an $\Omega$-itinerary departing from the identity, satisfying $W_1 \cap W_n = \emptyset$ and $\overline{W_1} \cap \overline{W_n} \ne \emptyset$, and let $\gamma = \gamma(W_1, W_n)$. If $\mu_{1,2}(\rho(\gamma)) > M$, then the distance between ${E^+_{1}}(\rho(\gamma))$ and $\overline{W_1} \cap \overline{W_n}$ is at least $\varepsilon_0$.* *Proof.* We can replace the itinerary $W_1, \ldots, W_n$ with an efficient itinerary from $W_1$ to $W_n$ also departing from the identity, since this does not affect the element $\gamma(W_1, W_n)$. For each $i$, we write $s_i \in S$ for the type $s(W_i)$ of the wall $i$. By the intersection $\overline{W_1} \cap \overline{W_n}$ is equal to the intersection $\bigcap_{i=1}^{n} \overline{W_i}$, and by this intersection is in turn contained in $\bigcap_{i=1}^{n}\ensuremath{\mathbb{P}}(\ker(\alpha_{s_i}))$. Since this intersection is nonempty, the set of elements $s_i$ for $1 \le i \le n$ lie in a proper subset $T \subsetneq S$. Then $\gamma \in C(T)$. Since the Cartan matrix for $\rho$ is fully nondegenerate, implies that $\rho(\gamma)$ acts block-diagonally on $V$ with respect to a decomposition $V_T \oplus V_T^\perp$, where $$V_T = \mathop{\mathrm{span}}\{v_s : s \in T\}, \qquad V_T^\perp = \bigcap_{s \in T} \ker(\alpha_s).$$ Since these subspaces are transverse, we can choose $\varepsilon_0 > 0$ so that the distance between the projective subspaces $\ensuremath{\mathbb{P}}(V_T), \ensuremath{\mathbb{P}}(V_T^\perp) \subset \ensuremath{\mathbb{P}}(V)$ is at least $2\varepsilon_0$. Since there are only finitely many possible choices for $T$, we may choose this $\varepsilon_0$ independently of $\gamma$. Next, we can choose a linear map $h \in \mathop{\mathrm{SL}}(V)$, depending only on $T$, so that the decomposition $hV_T \oplus hV_T^\perp$ is orthogonal. Then the conjugate $g = h\rho(\gamma)h^{-1}$ acts block-diagonally on a pair of orthogonal subspaces, and implies that there is a bound on the difference $|\mu_{1,2}(\rho(\gamma)) - \mu_{1,2}(g)|$, depending only on $h$ (and therefore only on $T$). In particular, this bound is independent of $\gamma$, so if $\mu_{1,2}(\rho(\gamma))$ is sufficiently large, we know that $\mu_{1,2}(g) > 0$. Since $g$ acts by the identity on $hV_T^\perp$, this means that ${E^+_{1}}(g)$ lies in the subspace $hV_T$. Then, Lemma A.4 and A.5 in [@BPS] imply that we have $$\begin{aligned} d_{\mathbb{P}}\left( {E^+_{1}}(\rho(\gamma)), h^{-1} {E^+_{1}}(g) \right) & \leq d_{\mathbb{P}}\left( {E^+_{1}}(\rho(\gamma)), {E^+_{1}}(\rho(\gamma) h^{-1}) \right) + d_{\mathbb{P}}\left( {E^+_{1}}(\rho(\gamma)h^{-1}), h^{-1} {E^+_{1}}(g) \right) \\ & \leq e^{\mu_{1,d}(h) -\mu_{1,2}(\rho(\gamma))} + e^{\mu_{1,d}(h) - \mu_{1,2}(\rho(\gamma) h^{-1})} \\ & \leq e^{-\mu_{1,2}(\rho(\gamma))} \left( e^{\mu_{1,d}(h)} + e^{2\mu_{1,d}(h)} \right) \end{aligned}$$ which is smaller than $\varepsilon_0$ once $\mu_{1,2}(\rho(\gamma))$ is sufficiently large. Since $h^{-1}{E^+_{1}}(g)$ lies in $V_T$, we conclude that the unstable subspace ${E^+_{1}}(\rho(\gamma))$ lies within distance $\varepsilon_0$ of $\ensuremath{\mathbb{P}}(V_T)$, hence it lies at least distance $\varepsilon_0$ from $\ensuremath{\mathbb{P}}(V_T^\perp)$. But we have ensured that the intersection $\overline{W_1} \cap \overline{W_n}$ lies in $\ensuremath{\mathbb{P}}(V_T^\perp)$, so we get the desired result. ◻ # Proof of main theorem {#sec:main_thm} In this section we finally set about proving . We assume that $\Gamma$ is a hyperbolic quasiconvex subgroup of a right-angled Coxeter group $C$ with generating set $S$. We let $d = |S|$, and fix a simplicial representation $\rho\colon C \to \mathop{\mathrm{SL}}^{\pm}(d, \ensuremath{\mathbb{R}})$ with fully nondegenerate Cartan matrix. We want to prove the following uniform inequality: there exist constants $A, B >0$ such that for any $\gamma \in \Gamma$, $$\mu_{1,2}(\rho(\gamma)) \geq A|\gamma| - B.$$ To do so, it will be more convenient to work on a larger subset of $C$ containing $\Gamma$: **Definition 70**. Let $D > 0$. We let $\mathrm{BP}(D) \subset C$ denote the subset of group elements with $D$-bounded product projections (see ). **Remark 71**. If $C$ is not hyperbolic, then $\mathrm{BP}(D)$ is not a subgroup of $C$ for any $D$. says precisely that there exists a uniform constant $D$ such that $\Gamma \subseteq \mathrm{BP}(D)$. We choose to work with the set $\mathrm{BP}(D)$ rather than the subgroup $\Gamma$ because $\mathrm{BP}(D)$ has the following useful property: if $w$ is a geodesic word representing some $\gamma \in \mathrm{BP}(D)$, and $w'$ is a subword of $w$, then $w'$ also represents an element of $\mathrm{BP}(D)$. This means that we can prove statements about group elements in $\mathrm{BP}(D)$ by cutting those elements up into smaller pieces and applying inductive arguments. Recall that for each subset $T \subseteq S$, the $C(T)$-invariant subspace $V_T \subseteq V$ is given by $\mathop{\mathrm{span}}\{v_s : s \in T\}$. We let $\rho_T\colon C(T) \to \mathop{\mathrm{SL}}^{\pm}(V_T)$ denote the representation of $C(T)$ obtained by restricting the $\rho$-action of $C(T)$ to $V_T$. Below, we will show the following. **Proposition 72**. *For any subset $T \subseteq S$ and any $D > 0$, there exist constants $A, B > 0$ such that, if $\gamma \in \mathrm{BP}(D) \cap C(T)$, then $\mu_{1,2}(\rho_T(\gamma)) \ge A |\gamma| - B$.* This gives us the main theorem by taking $T = S$. ## Setting up the inductive statement Since we have assumed that the simplicial representation $\rho$ has fully nondegenerate Cartan matrix, implies that for any subset $T \subseteq S$, the restriction of the representation $\rho$ to the subgroup $C(T)$ decomposes as a product $\rho_{T} \oplus \mathrm{id}$ on $V_{T} \oplus V_T^\perp$, where $\mathrm{id}$ denotes the trivial representation $C(T) \to \mathop{\mathrm{SL}}^{\pm}(V_T^\perp)$. After a bounded change in inner product (which only changes singular values up to bounded multiplicative error), we can assume that the subspaces $V_T, V_T^\perp$ are orthogonal. So, when $\gamma \in C(T)$, the inequality in is satisfied with uniform constants for $\rho_T(\gamma)$ if and only if it is satisfied for $\rho(\gamma)$ (possibly with different constants). Further, the representation $\rho_T$ is also a simplicial representation of the right-angled Coxeter group $C(T)$ with fully nondegenerate Cartan matrix. So, to prove , we assume inductively that there are constants $A, B > 0$ so that for any proper subset $T \subsetneq S$, if $\gamma \in C(T) \cap \mathrm{BP}(D)$, then $$\label{eq:subgroup_inequality} \mu_{1,2}(\rho(\gamma)) \ge A |\gamma| - B.$$ We wish to show that, possibly after changing the constants $A$ and $B$, this inequality also holds for any $\gamma \in \mathrm{BP}(D)$. If $|S| = 1$, the conclusion of is obvious, since in this case $C \cong \ensuremath{\mathbb{Z}}/2$. So we just need to consider the inductive step. Before doing so, we make some simplifying arguments. First, if $C$ is finite, then so is $\mathrm{BP}(D)$, and the desired inequality holds trivially. So, we may assume $C$ is infinite. Next, for any $\gamma \in \mathrm{BP}(D)$, we can find a (possibly trivial) product decomposition of $C$ into $C(S_1) \times C(S_2)$ so that $C(S_1)$ is irreducible and the image of $\gamma$ under the projection $C \to C(S_2)$ has length at most $D$; this holds because every wall in $\mathrm{D}(C,S)$ whose type lies in $S_1$ intersects every wall whose type lies in $S_2$, and vice versa. Then, writing $\gamma = \gamma_1\gamma_2$ for $\gamma_1 \in C(S_1)$, $\gamma_2 \in C(S_2)$, we apply to see that $$\mu_{1,2}(\rho(\gamma)) \ge \mu_{1,2}(\rho(\gamma_1)) - KD$$ for a uniform constant $K$. So, by replacing $\gamma$ with $\gamma_1$, for the purposes of proving we can assume that $\gamma \in \mathrm{BP}(D)$ always lies in a subgroup $C(T)$ of $C$ which is both infinite and irreducible. If this subgroup is proper, then we can directly apply the inductive assumption [\[eq:subgroup_inequality\]](#eq:subgroup_inequality){reference-type="eqref" reference="eq:subgroup_inequality"} to obtain the desired lower bound on $\mu_{1,2}(\rho(\gamma))$. So we may assume that $C$ is itself irreducible. We have therefore reduced the inductive step of our proposition to the following statement: **Proposition 73**. *Suppose that $C$ is infinite and irreducible, and fix $D > 0$. If there exist uniform constants $A_0, B_0 > 0$ such that, for any proper subset $T \subsetneq S$ and any $\eta \in C(T) \cap \mathrm{BP}(D)$, we have $$\label{eq:vertex_sv_gap} \mu_{1,2}(\rho(\eta)) \ge A_0|\eta| - B_0,$$ then there are constants $A, B > 0$ such that for any $\gamma \in \mathrm{BP}(D)$, we have $$\mu_{1,2}(\rho(\gamma)) \ge A|\gamma| - B.$$* The rest of the section is devoted to the proof of this proposition. So, from now on, we assume that the hypotheses of the proposition hold. Since we assume $C$ is infinite and irreducible, we may assume that the Vinberg domain $\Omega_{\mathrm{Vin}}$ is properly convex by . ## Cutting itineraries into pieces Fix an element $\gamma \in \mathrm{BP}(D)$. The first step in proving the estimate in is to cut an itinerary traversing $\gamma$ into several sub-itineraries, as follows. We let $\ensuremath{\mathcal{U}}$ be an itinerary traversing $\gamma$ of the form $$\{W_1\}, \ensuremath{\mathcal{V}}_1, \{W_2\}, \ensuremath{\mathcal{V}}_2, \ldots, \{W_n\}, \ensuremath{\mathcal{V}}_n,$$ which satisfies the conclusions of (and of ). In particular, in the partial order $<$ on walls in $\ensuremath{\mathcal{U}}$, we have $W_1 < \ldots < W_n$. We let $\mathbf{W}$ denote the set $\{W_i : 1 \le i \le n\}$, and for any $W_i, W_j \in \mathbf{W}$ with $W_i < W_j$, we let $\ensuremath{\mathcal{U}}(W_i, W_j)$ denote the sub-itinerary of $\ensuremath{\mathcal{U}}$ starting with $W_i$ and ending with $W_j$. We also assume that the itinerary $\ensuremath{\mathcal{U}}$ departs from the identity; this means that $\mathbf{Hs}_+(W_j) \subset \mathbf{Hs}_+(W_i)$ whenever $i < j$. We then choose a subset $\mathbf{Z} = \{Z_1, \ldots, Z_N\}$ of the walls $W_i$ as follows: we take $Z_1 = W_1$. Then, for each $j > 1$, we take $Z_j$ to be the first wall $W_i > Z_{j-1}$ such that $\overline{\mathbf{Hc}_+(W_i)} \subset \mathbf{Hc}_+(Z_{j-1})$. If there is no such wall, and $Z_{j-1}$ is not already maximal in $\mathbf{W}$, we let $Z_j = Z_N$ be the maximal wall in $\mathbf{W}$. It follows that for every $i < j < N$, we have $$\label{eq:itinerary_strict_halfcones} \overline{\mathbf{Hc}_+(Z_j)} \subset \mathbf{Hc}_+(Z_i).$$ Each pair of walls $Z_i, Z_{i+1}$ now determines a sub-itinerary $\ensuremath{\mathcal{U}}(Z_i, Z_{i+1})$ of $\ensuremath{\mathcal{U}}$. ## Outline of the rest of the proof {#sec:cutting_gluing} We separate our sub-itineraries $\ensuremath{\mathcal{U}}(Z_i, Z_{i+1})$ into "short" and "long" ones, where the distinction depends on some uniform threshold on the length to be specified later. **If our itinerary spends a uniform proportion of its lifetime inside "short" sub-itineraries** (i.e. a proportion above a uniform threshold $\tau$, specified independent of $\gamma$), then the number of "short" sub-itineraries is at least a constant times $|\gamma|$. Using [\[eq:itinerary_strict_halfcones\]](#eq:itinerary_strict_halfcones){reference-type="eqref" reference="eq:itinerary_strict_halfcones"}, we obtain a sequence of strictly nested half-cones. The Hilbert diameters of these half-cones are decreasing at a uniform exponential rate. Then we use to get a corresponding uniform exponential estimate on the singular value gap. **Otherwise, we can assume the geodesic spends a large fraction of its lifetime inside of "long" sub-itineraries.** For this case, we use the results of to observe that each sub-itinerary traverses a group element which (up to uniformly bounded error) either lies in a proper standard subgroup of $C$, or else can be decomposed into into a product of two group elements, each of which lies in a proper standard subgroup of $C$. Using our inductive assumption, we can then show that each of our sub-itineraries traverses a group element whose singular value gap is uniformly exponential in its length. We then need to show that when we "glue together" all of these sub-itineraries, we obtain an itinerary which also traverses a group element with uniformly exponential singular value gap. There are several different "gluing" steps in this process, but each of them essentially relies on just two techniques. First, if two adjacent "long" sub-itineraries are separated by a short sub-itinerary whose initial and final walls are well-separated in $\Omega_{\mathrm{Vin}}$, then we can use this separation to estimate positions of stable and unstable subspaces, and use uniform transversality () to estimate singular value gaps. On the other hand, if we *cannot* separate two adjacent "long" sub-itineraries $\ensuremath{\mathcal{V}}_-, \ensuremath{\mathcal{V}}_+$ by a pair of well-separated walls, then every sub-itinerary which "overlaps" both $\ensuremath{\mathcal{V}}_-$ and $\ensuremath{\mathcal{V}}_+$ must (up to some uniform error) traverse a group element lying in a proper standard subgroup of $C$. Once again, our inductive assumption implies that these overlapping sub-itineraries also have uniformly exponential singular value gaps. By applying the Kapovich-Leeb-Porti local-to-global principle (), we get a uniform gap estimate for the geodesic traversed by the concatenation $\ensuremath{\mathcal{V}}_-, \ensuremath{\mathcal{V}}_+$. The length constant in the local-to-global lemma is ultimately what determines the threshold for a "long" sub-itinerary. By repeatedly using these two gluing arguments, we are eventually able to show that if the proportion of time spent in "long" sub-itineraries is sufficiently close to 1, then we obtain a uniform gap estimate for the entire itinerary. Combining this estimate with the previous case completes the proof. ## Estimating time spent in "short" sub-itineraries Given any itinerary $\ensuremath{\mathcal{V}}$, recall that the length $|\ensuremath{\mathcal{V}}|$ is the word-length of the element $\ensuremath{\mathcal{V}}$ traverses, or equivalently the number of walls appearing in $\ensuremath{\mathcal{V}}$. If $W_i < W_j < W_k$ are three walls in $\mathbf{W}$, then we have $$|\ensuremath{\mathcal{U}}(W_i, W_k)| = |\ensuremath{\mathcal{U}}(W_i, W_j)| + |\ensuremath{\mathcal{U}}(W_j, W_k)| - 1,$$ since the wall $W_j$ is counted twice on the right-hand side. It will therefore be convenient to introduce some additional notation, and let $d_{\mathcal{U}}(W_i, W_j)$ denote $|\ensuremath{\mathcal{U}}(W_i, W_j)| - 1$. With this notation, for any $W_i < W_j < W_k$, we have $$\label{eq:udist_additive} d_{\mathcal{U}}(W_i, W_k) = d_{\mathcal{U}}(W_i, W_j) + d_{\mathcal{U}}(W_j, W_k).$$ Now suppose we have fixed a quantity $L > 0$. For any pair of walls $W_i < W_j$ in $\mathbf{W}$, we define the truncated length $\{d_{\mathcal{U}}(W_i, W_j)\}_{L}$ by $$\{d_{\mathcal{U}}(W_i, W_j)\}_{L} := \begin{cases} d_{\mathcal{U}}(W_i, W_j), &d_{\mathcal{U}}(W_i, W_j) < L\\ 0, &d_{\mathcal{U}}(W_i, W_j) \ge L. \end{cases}$$ Similarly, we define $\{d_{\mathcal{U}}(W_i, W_j)\}^L$ to be $d_{\mathcal{U}}(W_i, W_j)$ if this is at least $L$, and $0$ otherwise. Given the collection of walls $\mathbf{Z} = Z_1 < \ldots < Z_N$, we define the quantity $$r_L(\mathbf{Z}) := \frac{1}{d_{\mathcal{U}}(Z_1, Z_N)}\sum_{i=1}^{N-1} \{d_{\mathcal{U}}(Z_i, Z_{i+1})\}_{L}.$$ Roughly, $r_L(\mathbf{Z})$ is the proportion of time $\ensuremath{\mathcal{U}}$ spends inside of "short" sub-itineraries bounded by elements of $\mathbf{Z}$, where an itinerary is "short" if its length is at most $L$. **Notation 74**. To simplify notation, for the rest of the paper, if $\ensuremath{\mathcal{W}}$ is an itinerary, we abbreviate $\mu_{1,2}(\rho(\gamma(\ensuremath{\mathcal{W}})))$ to $\mu_{1,2}(\ensuremath{\mathcal{W}})$. Further, if $W_1, W_2$ are walls in $\Omega_{\mathrm{Vin}}$, we write $\mu_{1,2}(W_1, W_2)$ for $\mu_{1,2}(\rho(\gamma(W_1, W_2)))$. Similarly, we write ${E^\pm_{i}}(\ensuremath{\mathcal{W}})$ for ${E^\pm_{i}}(\rho(\gamma(\ensuremath{\mathcal{W}})))$, and ${E^\pm_{i}}(W_1, W_2)$ for ${E^\pm_{i}}(\rho(\gamma(W_1, W_2)))$. ## Mostly short sub-itineraries We first want to prove the following: **Proposition 75**. *For any $L > 0$ and $\tau \in (0, 1)$, there are constants $A \ge 0, B > 0$ (depending on $L, \tau$) such that if $r_L(\mathbf{Z}) \ge \tau$, then $\mu_{1,2}(\ensuremath{\mathcal{U}}) \ge A|\ensuremath{\mathcal{U}}| - B$.* *Proof.* We let $L > 0$ and $\tau \in (0, 1)$ be given, and let $m$ denote the number of indices $i$ in $1, \ldots, N$ such that $d_{\mathcal{U}}(Z_i, Z_{i+1}) < L$. Thus, we have $$m \cdot L \ge \sum_{i=1}^{N-1} \{d_{\mathcal{U}}(Z_i, Z_{i+1})\}_{L},$$ and since we assume $r_L(\mathbf{Z}) \ge \tau$, we have $$m \ge \frac{\tau}{L} \cdot d_{\mathcal{U}}(Z_1, Z_N).$$ We now consider the sequence of half-cones $\mathbf{Hc}_+(Z_1), \ldots, \mathbf{Hc}_+(Z_N)$. For every index $n$, let $\gamma_n$ denote the group element traversed by the itinerary $\ensuremath{\mathcal{U}}(Z_1, \ldots, Z_n)$, and let $s_n = s(Z_n)$. Then implies that $$\mathbf{Hc}_+(Z_n) = \rho(\gamma_n) \mathbf{Hc}_-(s_n),$$ where, for any $s \in S$, $\mathbf{Hc}_+(s)$ denotes the positive half-cone over the reflection wall for $\rho(s)$. Let $d_n$ denote the Hilbert metric on the half-cone $\mathbf{Hc}_+(Z_n)$. As the half-cones $\mathbf{Hc}_+(Z_{n-1})$ and $\mathbf{Hc}_+(Z_n)$ are always nested, implies that $d_n \ge d_{n-1}$ for all $n \le N$. When $n < N$, then the half-cones $\mathbf{Hc}_+(Z_{n-1})$ and $\mathbf{Hc}(Z_n)$ are *strongly* nested. Equivalently, $$\label{eq:halfcone_sequence_nest} \rho(\gamma_{n-1}^{-1}\gamma_n)\overline{\mathbf{Hc}_-(s_n)} \subset \mathbf{Hc}_-(s_{n-1}).$$ For any $n$, we have $\gamma_{n-1}^{-1}\gamma_n = s_{n-1}\gamma(Z_{n-1}, Z_n)$. So, if $d_{\mathcal{U}}(Z_{n-1}, Z_n) < L$, there only are finitely many possible choices for the group element $\rho(\gamma_{n-1}^{-1}\gamma_n)$ in the inclusion [\[eq:halfcone_sequence_nest\]](#eq:halfcone_sequence_nest){reference-type="eqref" reference="eq:halfcone_sequence_nest"} above, as well as finitely many possible choices for the half-cones $\mathbf{Hc}_-(s_n)$ and $\mathbf{Hc}_-(s_{n-1})$. We can then apply the sharper form of to see that there is a uniform constant $\lambda > 1$ (depending only on $\rho, L$) so that, if $n < N$, $d_{\mathcal{U}}(Z_{n-1}, Z_n) < L$ and $x, y \in \mathbf{Hc}_+(Z_n)$, then $$\label{eq:hilbert_contract} d_n(x,y) \ge \lambda \cdot d_{n-1}(x,y).$$ Now let $\ell$ be the last index smaller than $N$ such that $d_{\mathcal{U}}(Z_{\ell - 1}, Z_\ell) < L$, and let $D_\ell$ be the diameter of $\mathbf{Hc}_+(Z_\ell)$ with respect to the Hilbert metric on $\mathbf{Hc}_+(Z_{\ell - 1})$, or equivalently the diameter of $\rho(\gamma_{\ell-1}^{-1}\gamma_\ell)\mathbf{Hc}_-(s_\ell)$ with respect to the Hilbert metric on $\mathbf{Hc}_-(s_{\ell-1})$. Since there are only finitely many possible values for these two strongly nested half-cones, we can bound $D_\ell$ beneath a uniform constant $D$, which depends only on $L$. From [\[eq:hilbert_contract\]](#eq:hilbert_contract){reference-type="eqref" reference="eq:hilbert_contract"}, the inequality $d_n \ge d_{n-1}$, and [\[eq:halfcone_sequence_nest\]](#eq:halfcone_sequence_nest){reference-type="eqref" reference="eq:halfcone_sequence_nest"} it follows that the diameter of the half-cone $\mathbf{Hc}_+(Z_N) = \rho(\gamma_N)\mathbf{Hc}_-(s_N)$ with respect to the Hilbert metric on $\mathbf{Hc}_+(Z_1)$ is at most $\lambda^{-(m-1)}D$. As there are only finitely many possible values for $\mathbf{Hc}_+(Z_1)$ and $\mathbf{Hc}_-(s_N)$, we can then apply to see that $$\mu_{1,2}(\rho(\gamma_N)) \ge m \cdot \log(\lambda) - B$$ for some uniform constant $B$. By , the group element $\gamma_N = \gamma_{\ensuremath{\mathcal{U}}}(Z_1, Z_N)$ is within uniformly bounded distance of $\gamma(\ensuremath{\mathcal{U}})$ in the Coxeter group $C$, and $m$ is uniformly linear in $d_{\mathcal{U}}(Z_1, Z_N)$ (which is within uniformly bounded additive error of $|\ensuremath{\mathcal{U}}|$). So we can apply to complete the proof. ◻ ## Mostly long sub-itineraries Our goal for the rest of the section is to prove the following: **Proposition 76**. *There exist constants $A > 0$, $B, L \ge 0$ and $\tau \in (0,1)$ so that if $r_L(\mathbf{Z}) \le \tau$, then $\mu_{1,2}(\ensuremath{\mathcal{U}}) \ge A|\ensuremath{\mathcal{U}}| - B$.* Since the constants $A, B, L, \tau$ above can all be chosen independently of $\ensuremath{\mathcal{U}}$ and $\gamma$, putting together with will imply , which in turn gives us the desired (and thus our main theorem). Throughout the section, we will obtain many intermediate estimates for various sub-itineraries $\ensuremath{\mathcal{U}}'$ of $\ensuremath{\mathcal{U}}$, which all roughly have the form $$\mu_{1,2}(\ensuremath{\mathcal{U}}') \ge A|\ensuremath{\mathcal{U}}'| - B$$ for some constants $A, B$. We will not keep track of the different constants $A,B$ for all of these different estimates. Most of this section involves obtaining and refining various estimates for $\mu_{1,2}(\ensuremath{\mathcal{U}}(W_i, W_j))$ and $\mu_{1,2}(W_i, W_j)$ for certain walls $W_i, W_j \in \mathbf{W}$. Due to .[\[item:elts_in_coxbnhd\]](#item:elts_in_coxbnhd){reference-type="ref" reference="item:elts_in_coxbnhd"} and , we can go back and forth between $\gamma_{\ensuremath{\mathcal{U}}}(W_i, W_j)$ and $\gamma(W_i, W_j)$ when we make these estimates. Precisely, we can say the following: **Lemma 77**. *There are uniform constants $R_1, R_2$ so that for any walls $W_i < W_j$ in $\mathbf{W}$, we have $$|\mu_{1,2}(\ensuremath{\mathcal{U}}(W_i, W_j)) - \mu_{1,2}(W_i, W_j)| < R_1$$ and $$\left| |\gamma(W_i, W_j)| - d_{\mathcal{U}}(W_i, W_j) \right| < R_2.$$* Using and the above, we also obtain the following: **Lemma 78**. *There is a uniform constant $K > 0$ so that for any walls $W_i < W_j < W_k$ in $\mathbf{W}$, we have $$\begin{aligned} |\mu_{1,2}(W_i, W_k) - \mu_{1,2}(W_i, W_j)| &\leq K \cdot d_{\mathcal{U}}(W_j, W_k), \\ |\mu_{1,2}(W_i, W_k) - \mu_{1,2}(W_j, W_k)| &\leq K \cdot d_{\mathcal{U}}(W_i, W_j). \end{aligned}$$* **Definition 79**. Let $W_i, W_\ell$ be walls in $\mathbf{W}$ with $W_i < W_\ell$. For constants $A, B > 0$, we say that the pair of walls $W_i, W_\ell$ *has $(A, B)$-gaps* if for every pair of walls $W_j, W_k$ with $W_i \le W_j < W_k \le W_\ell$, we have $$\label{eq:regular_pair_inequality} \mu_{1,2}(W_j, W_k) \ge A|\gamma(W_j, W_k)| - B.$$ If $\ensuremath{\mathcal{V}}$ is any itinerary equivalent to a sub-itinerary of $\ensuremath{\mathcal{U}}$, we say that $\ensuremath{\mathcal{V}}$ *has $(A, B)$-gaps* if every pair of walls $W_i < W_\ell$ in $\ensuremath{\mathcal{V}} \cap \mathbf{W}$ has $(A, B)$-gaps. **Remark 80**. means that we could equivalently define pairs of walls with $(A,B)$-gaps by replacing $\mu_{1,2}(W_j, W_k)$ with $\mu_{1,2}(\ensuremath{\mathcal{U}}(W_j, W_k))$ and $|\gamma(W_j, W_k)|$ with $|\ensuremath{\mathcal{U}}(W_j, W_k)|$ (or with $d_{\mathcal{U}}(W_j, W_k)$) in [\[eq:regular_pair_inequality\]](#eq:regular_pair_inequality){reference-type="eqref" reference="eq:regular_pair_inequality"} above---if we do, the constants $A, B$ change, but by a controlled amount depending only on the group $C$ and the representation $\rho$. Using and .[\[item:subitineraries_minimal\]](#item:subitineraries_minimal){reference-type="ref" reference="item:subitineraries_minimal"}, one can also see that, if $\ensuremath{\mathcal{V}}$ is a sub-itinerary of some itinerary equivalent to $\ensuremath{\mathcal{U}}$, then $\ensuremath{\mathcal{V}}$ has $(A,B)$-gaps if and only the geodesic following $\ensuremath{\mathcal{V}}$ has uniform gaps in the sense of (with constants depending on $A, B$). ### Finding sub-itineraries with uniform gaps We have used the set of walls $\mathbf{Z}$ to cut the itinerary $\ensuremath{\mathcal{U}}$ into pieces, so our first main task is to show that each piece corresponds to a geodesic segment in $\Gamma$ with uniform gaps. This will follow from: **Proposition 81**. *There are constants $A, B > 0$ so that every pair of walls $W < W'$ in $\mathbf{W}$ which satisfies $\overline{\mathbf{Hc}_+(W')} \not\subset \mathbf{Hc}_+(W)$ has $(A, B)$-gaps.* We need several intermediate lemmas in order to prove . First, we observe that our inductive assumption [\[eq:vertex_sv_gap\]](#eq:vertex_sv_gap){reference-type="eqref" reference="eq:vertex_sv_gap"} from and the quasiconvexity of standard subgroups in $C$ implies the following: **Lemma 82**. *There are uniform constants $A, B$ so that, if $\ensuremath{\mathcal{V}}$ is any itinerary equivalent to a sub-itinerary of $\ensuremath{\mathcal{U}}$, and $\gamma(\ensuremath{\mathcal{V}}) \in C(T)$ for some $T \subsetneq S$, then $\ensuremath{\mathcal{V}}$ has $(A, B)$-gaps.* We next need two different "straightness" lemmas for sub-itineraries of $\ensuremath{\mathcal{U}}$. Both and below essentially say that if two adjacent itineraries $\ensuremath{\mathcal{V}}_-, \ensuremath{\mathcal{V}}_+$ in $\ensuremath{\mathcal{U}}$ have sufficiently large gaps, and there are walls between $\ensuremath{\mathcal{V}}_-$ and $\ensuremath{\mathcal{V}}_+$ which are well-separated in $\Omega_{\mathrm{Vin}}$, then $\mu_{1,2}(\ensuremath{\mathcal{V}})$ is approximately $\mu_{1,2}(\ensuremath{\mathcal{V}}_-) + \mu_{1,2}(\ensuremath{\mathcal{V}}_+)$. The conclusion of both lemmas is nearly the same, but in , we assume slightly weaker separation hypotheses and slightly stronger hypotheses on gaps than in (we refer to for a discussion of the difference between the separation hypotheses in these lemmas). We need both lemmas for the arguments throughout the rest of the section. **Lemma 83** (Additivity for well-separated itineraries, I). *Given $\lambda > 0$, there are constants $M, D > 0$ so that the following holds. Suppose that $W_i < W_j < W_k < W_\ell$ are walls in $\mathbf{W}$ satisfying:* 1. *[\[item:roots_large_I\]]{#item:roots_large_I label="item:roots_large_I"} $\mu_{1,2}(W_i, W_j) > M$ and $\mu_{1,2}(W_k, W_\ell) > M$;* 2. *$\overline{\mathbf{Hc}_+(W_k)} \subset \mathbf{Hc}_+(W_j)$.* 3. *$d_{\mathcal{U}}(W_j, W_k) < \lambda$.* *Then we have $$\mu_{1,2}(W_i, W_\ell) \ge \mu_{1,2}(W_i, W_j) + \mu_{1,2}(W_k, W_\ell) - D.$$* *Proof.* Let $\ensuremath{\mathcal{V}}$ be an efficient itinerary from $W_i$ to $W_\ell$, and write $\ensuremath{\mathcal{V}}$ as a concatenation $\ensuremath{\mathcal{V}}_-, \ensuremath{\mathcal{V}}_+$ so that $\ensuremath{\mathcal{V}}_-$ contains $W_i, W_j$ and $\ensuremath{\mathcal{V}}_+$ contains $W_k, W_\ell$. We change basepoints so that $\ensuremath{\mathcal{V}}_-$ arrives at the identity and $\ensuremath{\mathcal{V}}_+$ departs from the identity. After doing so, the inclusion $\mathbf{Hs}_+(W_k) \subset \mathbf{Hs}_+(W_j)$ becomes $\mathbf{Hs}_+(W_k) \subset \mathbf{Hs}_-(W_j)$, which means we can assume that $\overline{\mathbf{Hc}_+(W_k)} \subset \mathbf{Hc}_-(W_j)$. We write $H_- = \mathbf{Hc}_-(W_j)$ and $H_+ = \mathbf{Hc}_+(W_k)$. We choose $\varepsilon> 0$ so that $N_{2\varepsilon}(H_+) \subset N_{-2\varepsilon}(H_-)$. Since the distance $|\gamma(W_j, W_k)| \le |\ensuremath{\mathcal{U}}(W_j, W_k)|$ is bounded by $\lambda$, this $\varepsilon$ can be chosen uniformly in $\lambda$. Next, we choose $M' > 0$ (depending only on $\varepsilon$ and $\lambda$) as in , so that if $\mu_{1,2}(\ensuremath{\mathcal{V}}_+) > M'$, then ${E^+_{1}}(\rho(\gamma(\ensuremath{\mathcal{V}}_+))) \in N_\varepsilon(H_+)$, and if $\mu_{1,2}(\ensuremath{\mathcal{V}}_-) > M'$, then ${E^-_{d-1}}(\rho(\gamma(\ensuremath{\mathcal{V}}_-)))$ is disjoint from $N_{-\varepsilon}(H_-)$. Finally, since $d_{\mathcal{U}}(W_j, W_k) < \lambda$, we know that $\gamma(\ensuremath{\mathcal{V}}_-) = \gamma(W_i, W_j)\eta_-$ and $\gamma(\ensuremath{\mathcal{V}}_+) = \eta_+\gamma(W_k, W_\ell)$ for $\eta_\pm$ satisfying $|\eta_\pm| < \lambda$. So, by , we can find $M$ so that if $\mu_{1,2}(W_i, W_j) > M$, then $\mu_{1,2}(\ensuremath{\mathcal{V}}_-) > M'$, and similarly for $W_k, W_\ell$ and $\ensuremath{\mathcal{V}}_+$. Now, applying and our hypothesis [\[item:roots_large_I\]](#item:roots_large_I){reference-type="eqref" reference="item:roots_large_I"} above, we see that an $\varepsilon$-neighborhood of ${E^+_{1}}(\ensuremath{\mathcal{V}}_+)$ is disjoint from ${E^-_{d-1}}(\ensuremath{\mathcal{V}}_-)$. That is, this pair of subspaces is uniformly transverse (see ). Thus, the additivity estimate () together with gives us the desired bound. ◻ **Lemma 84** (Additivity for well-separated itineraries, II). *Given $A, \lambda > 0$ and $B \ge 0$, there is a constant $D > 0$ so that the following holds. Suppose $W_i < W_j < W_k < W_\ell$ are walls in $\mathbf{W}$ satisfying:* 1. *The pairs of walls $W_i, W_j$ and $W_k, W_\ell$ both have $(A, B)$-gaps.* 2. *The intersection $\overline{W_j} \cap \overline{W_k}$ is empty.* 3. *We have $d_{\mathcal{U}}(W_j, W_k) < \lambda$.* *Then we have $$\label{eq:disjoint_wall_ineq} \mu_{1,2}(W_i, W_\ell) \ge \mu_{1,2}(W_i, W_j) + \mu_{1,2}(W_k, W_\ell) - D.$$* *Proof.* First, observe that if either $|\gamma(W_i, W_j)|$ or $|\gamma(W_k, W_\ell)|$ is smaller than some constant $N$, then we can use to find some constant $D$ depending only on $N$ so that the inequality [\[eq:disjoint_wall_ineq\]](#eq:disjoint_wall_ineq){reference-type="eqref" reference="eq:disjoint_wall_ineq"} holds. So, throughout the proof, we will be able to assume that both $|\gamma(W_i, W_j)|$ and $|\gamma(W_k, W_\ell)|$ are larger than any given constant that depends only on $A, B, \lambda$. Since we have assumed that the pairs $W_i, W_j$ and $W_k, W_\ell$ have $(A,B)$-gaps, this means we can also assume that $\mu_{1,2}(W_i, W_j)$ and $\mu_{1,2}(W_k, W_\ell)$ are larger than any given constant depending on $A, B, \lambda$. implies that we have a nesting of half-cones $\mathbf{Hc}_+(W_k) \subset \mathbf{Hc}_+(W_j)$. If this nesting is strict, the reasoning from the previous paragraph tells us that we can apply and we will be done. So, we now assume that $\overline{\mathbf{Hc}_+(W_k)}$ is not contained in $\mathbf{Hc}_+(W_j)$. As in the proof of , we let $\ensuremath{\mathcal{V}}$ be an efficient itinerary from $W_i$ to $W_\ell$, and write it as a concatenation $\ensuremath{\mathcal{V}}_-, \ensuremath{\mathcal{V}}_+$ for $\ensuremath{\mathcal{V}}_-$ containing $W_i, W_j$ and $\ensuremath{\mathcal{V}}_+$ containing $W_k, W_\ell$; in fact, we can ensure that the first wall of $\ensuremath{\mathcal{V}}_+$ is $W_k$. We can again assume that $\ensuremath{\mathcal{V}}_-$ arrives at the identity and $\ensuremath{\mathcal{V}}_+$ departs from the identity. As before, set $H_- = \mathbf{Hc}_-(W_j)$ and $H_+ = \mathbf{Hc}_+(W_k)$, so that $H_+$ (but not $\overline{H_+}$) is contained in $H_-$. For each wall $W \in \ensuremath{\mathcal{V}}_+ \cap \mathbf{W}$, we let ${E^+_{1}}(W)$ denote the 1-dimensional unstable subspace of $\rho(\gamma_{\ensuremath{\mathcal{V}}_+}(W))$, where $\gamma_{\ensuremath{\mathcal{V}}_+}(W)$ is the group element traversed by a sub-itinerary of $\ensuremath{\mathcal{V}}_+$ starting with the initial wall of $\ensuremath{\mathcal{V}}_+$ and ending in $W$. Let $\varepsilon_0, M_0$ be the fixed constants from . We know that $\ensuremath{\mathcal{V}}_+ = \ensuremath{\mathcal{U}}(W_k, W_\ell)$ has $(A, B)$-gaps. So, by applying the uniform convergence property for geodesics with uniform gaps in $\mathop{\mathrm{SL}}^{\pm}(d, \ensuremath{\mathbb{R}})$ (), we see that there is a fixed constant $N$ (depending only on $\varepsilon_0$, $A$, $B$) so that if $W$ is any wall in $\mathbf{W}$ separating $W_k$ and $W_\ell$, with $|\gamma(W_k, W)| \ge N$, we have $$\label{eq:uniform_dist_estimate} d_{\mathbb{P}}({E^+_{1}}(W), {E^+_{1}}(\ensuremath{\mathcal{V}}_+)) < \varepsilon_0/2.$$ By increasing $N$ if necessary, we can again apply the fact that $\ensuremath{\mathcal{V}}_+$ has uniform gaps to also ensure that as long as $|\gamma(W_k, W)| \ge N$, then $\mu_{1,2}(W_k, W) > M_0$. Now, we may assume that $N > 2R$, and that $|\gamma(W_k, W_\ell)| \ge N$. Then, [\[item:subitineraries_minimal\]](#item:subitineraries_minimal){reference-type="ref" reference="item:subitineraries_minimal"} implies that there is a uniform $R > 0$ and some wall $W$ in $\mathbf{W}$ with $W_k < W < W_\ell$ and $N < |\gamma(W_k, W)| \le N + R$. We let $H_+'$ denote the half-cone $\mathbf{Hc}_+(W)$; from , we know that $H_+' \subset H_+ \subset H_-$. This tells us that $\ensuremath{\partial}H_+' \cap \ensuremath{\partial}H_- \subset \ensuremath{\partial}H_+ \cap \ensuremath{\partial}H_-$. But, also implies that $\ensuremath{\partial}H_+' \cap \ensuremath{\partial}H_-$ is a subset of $\ensuremath{\partial}W$, and that $\ensuremath{\partial}H_+ \cap \ensuremath{\partial}H_-$ is a subset of $\ensuremath{\partial}W_k$, which means we have $$\label{eq:halfcone_boundary_intersection} \ensuremath{\partial}H_+' \cap \ensuremath{\partial}H_- \subset \overline{W} \cap \overline{W_k}.$$ We now consider two possibilities. Case 1 : $\overline{W_k} \cap \overline{W} = \emptyset$. In this case, the inclusion [\[eq:halfcone_boundary_intersection\]](#eq:halfcone_boundary_intersection){reference-type="eqref" reference="eq:halfcone_boundary_intersection"} means that the inclusion of half-cones $H_+' \subset H_-$ is strict, i.e. that $\overline{H_+'} \subset H_-$. So, we can replace $W_k$ with $W$ and apply (with $\lambda + N + R$ in place of $\lambda$); since the pairs $W_i, W_j$ and $W_k, W_\ell$ have uniform gaps, as long as $|\gamma(W_i, W_j)|$ and $|\gamma(W_k, W_\ell)|$ are sufficiently large, the hypotheses of are satisfied. Case 2 : $\overline{W_k} \cap \overline{W} \ne \emptyset$. In this case, we will show that the subspaces ${E^+_{1}}(\ensuremath{\mathcal{V}}_+)$ and ${E^-_{d-1}}(\ensuremath{\mathcal{V}}_-)$ are uniformly transverse by arguing that ${E^+_{1}}(\ensuremath{\mathcal{V}}_+)$ lies close to $H_+'$ and far from $\overline{W} \cap \overline{W_k}$, and ${E^-_{d-1}}(\ensuremath{\mathcal{V}}_-)$ does not lie too close to $H_-$; see . Since the closures $\overline{H_+'}$ and $\overline{H_-}$ are compact, we can choose some $\varepsilon> 0$ so that the intersection $$N_{2\varepsilon}(H_+') \cap N_{\varepsilon}(\ensuremath{\mathbb{P}}(V) -H_-)$$ is contained in the $\varepsilon_0/4$-neighborhood of $\ensuremath{\partial}H_+' \cap \ensuremath{\partial}H_-$ (and thus, by [\[eq:halfcone_boundary_intersection\]](#eq:halfcone_boundary_intersection){reference-type="eqref" reference="eq:halfcone_boundary_intersection"}, in the $\varepsilon_0/4$-neighborhood of $\overline{W} \cap \overline{W_k}$). Since $|\gamma(W_j, W)| < \lambda + N + R$, we can choose this $\varepsilon$ uniformly in $\lambda, N, R, \varepsilon_0$. We can also ensure that $\varepsilon< \varepsilon_0/4$. We now choose a constant $M$ which is larger than the corresponding $M$ from , taking the constant $\varepsilon$ in the lemma to be our chosen $\varepsilon$, and $N$ to be $\lambda + N + R$. We may assume that both $\mu_{1,2}(\ensuremath{\mathcal{V}}_+)$ and $\mu_{1,2}(\ensuremath{\mathcal{V}}_-)$ are at least $M$. We claim that the distance between ${E^+_{1}}(\ensuremath{\mathcal{V}}_+)$ and ${E^-_{d-1}}(\ensuremath{\mathcal{V}}_-)$ must then be at least $\varepsilon$. To see this, suppose for a contradiction that there is some point $x \in {E^-_{d-1}}(\ensuremath{\mathcal{V}}_-)$ which is contained in the $\varepsilon$-neighborhood of ${E^+_{1}}(\ensuremath{\mathcal{V}}_+)$. implies that ${E^+_{1}}(\ensuremath{\mathcal{V}}_+)$ is contained in $N_\varepsilon(H_+')$ and ${E^-_{d-1}}(\ensuremath{\mathcal{V}}_-)$ is disjoint from $N_{-\varepsilon}(H_-)$. This means that $x$ must lie in the intersection $$N_{2\varepsilon}(H_+') \cap (\ensuremath{\mathbb{P}}(V) -N_{-\varepsilon}(H_-)) = N_{2\varepsilon}(H_+') \cap N_\varepsilon(\ensuremath{\mathbb{P}}(V) -H_-).$$ In turn, this means that $x$ is contained in the $\varepsilon_0/4$-neighborhood of $\overline{W_k} \cap \overline{W}$ and so ${E^+_{1}}(\ensuremath{\mathcal{V}}_+)$ is contained in the $\varepsilon_0/2$-neighborhood of $\overline{W_k} \cap \overline{W}$. Then, using [\[eq:uniform_dist_estimate\]](#eq:uniform_dist_estimate){reference-type="eqref" reference="eq:uniform_dist_estimate"}, we see that ${E^+_{1}}(W)$ is contained in the $\varepsilon_0$-neighborhood of $\overline{W_k} \cap \overline{W}$. However, this contradicts . We finally conclude that the subspaces ${E^+_{1}}(\ensuremath{\mathcal{V}}_+)$ and ${E^-_{d-1}}(\ensuremath{\mathcal{V}}_-)$ are uniformly transverse, which gives us the desired inequality by and .  ◻ The last intermediate lemma we need for the proof of is a direct consequence of the Kapovich--Leeb--Porti local-to-global principle (): **Lemma 85** (Local-to-global for pairs with uniform gaps). *For every $A, B > 0$, there exists $\lambda > 0$ and $A', B' > 0$ satisfying the following. Suppose that $W_i, W_j$ are walls in $\mathbf{W}$ such that for every pair $n, m$ with $d_{\mathcal{U}}(W_n, W_m) < \lambda$ and $W_i \le W_n < W_m \le W_j$, the pair $W_n, W_m$ has $(A,B)$-gaps. Then $W_i, W_j$ has $(A',B')$-gaps.* *Proof of .* Fix walls $W_i < W_j$ in $\mathbf{W}$ so that $W \le W_i < W_j \le W'$. Since $W_i$ and $W_j$ separate $W$ and $W'$, and the assumption $\overline{\mathbf{Hc}_+(W')} \not\subset \mathbf{Hc}_+(W)$ means that the strong inclusion $\overline{\mathbf{Hc}_+(W_j)} \subset \mathbf{Hc}_+(W_i)$ cannot hold. We want to find uniform constants $A, B$ so that $\mu_{1,2}(W_i, W_j) \ge A|\gamma(W_i, W_j)| - B$. If $\overline{W}_n \cap \overline{W}_m \ne \emptyset$, then implies that $\gamma(W_i, W_j) \in C(T)$ for a proper standard subgroup $C(T) < C$, and we are done after applying . So, assume that $\overline{W_i} \cap \overline{W_j} = \emptyset$. We consider an efficient itinerary $\ensuremath{\mathcal{V}}$ between $W_i$ and $W_j$. By , $\ensuremath{\mathcal{V}}$ is equivalent to an itinerary of the form $\ensuremath{\mathcal{W}}_-, \ensuremath{\mathcal{W}}_0, \ensuremath{\mathcal{W}}_+$, where $|\ensuremath{\mathcal{W}}_0| < 2R$ for a uniform constant $R$, the itineraries $\ensuremath{\mathcal{W}}_-$ and $\ensuremath{\mathcal{W}}_+$ are efficient, and the intersections $\bigcap_{W \in \ensuremath{\mathcal{W}}_-} \overline{W}$ and $\bigcap_{W \in \ensuremath{\mathcal{W}}_+} \overline{W}$ are both nonempty. By , both $\gamma(\ensuremath{\mathcal{W}}_-)$ and $\gamma(\ensuremath{\mathcal{W}}_+)$ lie in (possibly different) proper standard subgroups of $C$. We let $\ensuremath{\mathcal{W}}_-'$ be the concatenation $\ensuremath{\mathcal{W}}_-, \ensuremath{\mathcal{W}}_0$, so that $\ensuremath{\mathcal{V}} = \ensuremath{\mathcal{W}}_-', \ensuremath{\mathcal{W}}_+$. By and , we see that $\ensuremath{\mathcal{W}}_-'$ and $\ensuremath{\mathcal{W}}_+$ both have $(A_0, B_0)$-gaps, for uniform constants $A_0, B_0$. We then choose constants $A, B, \lambda$ as in the local-to-global [Lemma 85](#lem:local_to_global_itineraries){reference-type="ref" reference="lem:local_to_global_itineraries"}, depending on $A_0$, $B_0$. Suppose that for every pair of walls $W_n, W_m \in \mathbf{W}$ such that $W_n \in \ensuremath{\mathcal{W}}_-'$, $W_m \in \ensuremath{\mathcal{W}}_+$, and $d_{\mathcal{U}}(W_n, W_m) < \lambda$, we have $\overline{W_n} \cap \overline{W_m} \ne \emptyset$. then implies that each such pair satisfies $\gamma(W_n, W_m) \in C(T)$ for some proper subgroup $C(T) < C$, and then implies that each such pair has $(A_0, B_0)$-gaps. Then implies that the pair $W_i, W_j$ has $(A,B)$-gaps, and we are done. So, we may now suppose that there is some pair of walls $W_n, W_m \in \mathbf{W}$ such that $W_n \in \ensuremath{\mathcal{W}}_-'$, $W_m \in \ensuremath{\mathcal{W}}_+$, and $d_{\mathcal{U}}(W_n, W_m) < \lambda$, but $\overline{W_n} \cap \overline{W_m} = \emptyset$. Since $\ensuremath{\mathcal{W}}_-', \ensuremath{\mathcal{W}}_+$ both have $(A_0, B_0)$-gaps, so do the pairs $W_i, W_n$ and $W_m, W_j$. We can then apply to the walls $W_i < W_n < W_m < W_j$ to complete the proof. ◻ As a consequence of , we obtain: **Proposition 86**. *There are uniform constants $A, B > 0$ so that every pair of consecutive walls $Z_i < Z_{i+1}$ in $\mathbf{Z}$ has $(A,B)$-gaps.* *Proof.* Consider walls $W_n < W_m$ in $\mathbf{W}$ with $Z_i \le W_n < W_m \le Z_{i+1}$. We want to find constants $A, B$ so that $\mu_{1,2}(W_n, W_m) \ge A|\gamma(W_n, W_m)| - B$. We may assume that $W_m > W_{n+1}$: otherwise, by [\[item:v_intersect\]](#item:v_intersect){reference-type="ref" reference="item:v_intersect"} we have $|\gamma(W_n, W_m)| = |\gamma(W_n, W_{n+1})| = 1$, and the desired inequality holds as long as we ensure $B > A$. Now consider the pair of walls $W_n < W_{m - 1}$. We know that $W_{m-1} < Z_{i+1}$, which means that the strong inclusion of half-cones $\overline{\mathbf{Hc}_+(W_{m - 1})} \subset \mathbf{Hc}_+(W_n)$ cannot hold: in this case would imply that also $\overline{\mathbf{Hc}_+(W_{m-1})} \subset \mathbf{Hc}_+(Z_i)$, and $Z_{i+1}$ was chosen to be the minimal wall in $\mathbf{W}$ whose positive half-cone strongly nests into $\mathbf{Hc}_+(Z_i)$. Then implies that $\mu_{1,2}(W_n, W_{m-1}) \ge A|\gamma(W_n, W_{m-1})| - B$ for some uniform constants $A, B$. So, we can apply and the fact that $d_{\mathcal{U}}(W_{m-1}, W_m) < R$ to complete the proof. ◻ ### Combining sub-itineraries with uniform gaps using the local-to-global principle shows that each pair of walls $Z_i, Z_{i+1}$ defines a sub-itinerary $\ensuremath{\mathcal{U}}(Z_i, Z_{i+1})$ which has $(A,B)$-gaps, for uniform $A, B$. For the next step of the proof, we remove walls from the collection $\mathbf{Z}$ to "combine" adjacent sub-itineraries; we do so in a way which ensures that each sub-itinerary bounded by consecutive walls in $\mathbf{Z}$ has $(A',B')$-gaps, for uniform constants $A', B'$ which do not depend on the itinerary $\ensuremath{\mathcal{U}}$. More precisely, we modify $\mathbf{Z}$ via the following procedure. Using and , we choose constants $A, B > 0$ so that every pair of consecutive walls $Z_i, Z_{i+1}$ in $\mathbf{Z}$ has $(A,B)$-gaps, *and* so that every pair of walls $W < W'$ in $\mathbf{W}$ satisfying $\overline{\mathbf{Hc}_+(W')} \not\subset \mathbf{Hc}_+(W)$ also has $(A,B)$-gaps. Then, we fix local-to-global constants $\lambda, A', B'$ from , depending on our chosen $A, B$. Now, suppose we have a sequence of consecutive elements $Z_j, Z_{j+1}, \ldots, Z_{j + n}$ in $\mathbf{Z}$ which satisfies the following property: 1. [\[item:adjacent_nosep\]]{#item:adjacent_nosep label="item:adjacent_nosep"} For each wall $Z_i$ in $\mathbf{Z}$ with $Z_j < Z_i < Z_{j + n}$, and every pair of walls $W, W' \in \mathbf{W}$ such that $W < Z_i < W'$ and $d_{\mathcal{U}}(W, W') < \lambda$, we have $\overline{\mathbf{Hc}_+(W')} \not\subset \mathbf{Hc}_+(W)$. It immediately follows from , and that the pair of walls $Z_j, Z_{j + n}$ has $(A',B')$-gaps. So, for each maximal subsequence $Z_j, \ldots, Z_{j + n}$ in $\mathbf{Z}$ satisfying [\[item:adjacent_nosep\]](#item:adjacent_nosep){reference-type="ref" reference="item:adjacent_nosep"}, we delete all elements except $Z_j$ and $Z_{j + n}$ from $\mathbf{Z}$. After this modification (and after reindexing $\mathbf{Z}$ appropriately), it is still true that each pair of consecutive walls $Z_i, Z_{i+1}$ has $(A, B)$-gaps for uniform $A, B$. Further, since we have only deleted walls from $\mathbf{Z}$ (and we do not delete the walls $Z_1$ or $Z_N$), the quantity $r_L(\mathbf{Z})$ can only decrease for any fixed $L$. So we can still assume that $r_L(\mathbf{Z}) < \tau$ if this was true before we deleted walls from $\mathbf{Z}$. After this modification, however, we gain the following. We now know that our $\mathbf{Z}$ satisfies the following additional property: **Proposition 87** (Long adjacent itineraries are well-separated). *Let $\lambda$ be the local-to-global constant defined above. Suppose that $Z_{i-1}, Z_i, Z_{i+1}$ are three consecutive walls in $\mathbf{Z}$ such that $d_{\mathcal{U}}(Z_{i-1}, Z_i) \ge \lambda$ and $d_{\mathcal{U}}(Z_i, Z_{i+1}) \ge \lambda$. Then there is a pair of walls $W, W' \in \mathbf{W}$ satisfying:* 1. *$Z_{i-1} < W < Z_i < W' < Z_{i+1}$;* 2. *$d_{\mathcal{U}}(W, W') < \lambda$;* 3. *$\overline{\mathbf{Hc}_+(W')} \subset \mathbf{Hc}_+(W)$.* ### Combining well-separated adjacent sub-itineraries For the next step, we again combine sequences of adjacent sub-itineraries by deleting elements from $\mathbf{Z}$. Given $L_0 > 0$, we consider maximal subsequences $Z_j, Z_{j+1}, \ldots, Z_{j + n}$ of $\mathbf{Z}$ which satisfy: 1. [\[item:long_consecutive_itineraries\]]{#item:long_consecutive_itineraries label="item:long_consecutive_itineraries"} for all indices $i$ with $j \le i < j + n$, we have $d_{\mathcal{U}}(Z_i, Z_{i+1}) \ge L_0$. We claim the following: **Lemma 88**. *There are uniform constants $A, L_0 > 0$ so that, if $Z_j, Z_{j + 1}, \ldots, Z_{j + n}$ is a subsequence of $\mathbf{Z}$ satisfying [\[item:long_consecutive_itineraries\]](#item:long_consecutive_itineraries){reference-type="ref" reference="item:long_consecutive_itineraries"}, then $$\mu_{1,2}(Z_j, Z_{j + n}) \ge A \cdot d_{\mathcal{U}}(Z_j, Z_{j + n}).$$* *Proof.* As each pair of consecutive walls $Z_i < Z_{i+1}$ has uniform gaps, we know there are constants $A', B'$ so that $\mu_{1,2}(Z_i, Z_{i+1}) \ge A'd_{\mathcal{U}}(Z_i, Z_{i+1}) - B'$. Let $K$ be the constant from , let $\lambda$ be the local-to-global constant from , and let $M, D$ be the constants from depending on $\lambda$. Then, choose $L_0$ large enough so that $$A'L_0 - 3\lambda K - B' - D > M,$$ and define $A = (A'L_0 - 2\lambda K - B' - D)/L_0$, so that $A L_0 > M + \lambda K$. To prove the proposition, we induct on the length of the sequence $Z_j, \ldots, Z_{j + n}$. In the base case $n = 1$, we have $$\begin{aligned} \mu_{1,2}(Z_j, Z_{j + 1}) &\ge A'd_{\mathcal{U}}(Z_j, Z_{j+1}) - B'\\ &=\left(A + \frac{2\lambda K + B' + D}{L_0}\right)d_{\mathcal{U}}(Z_j, Z_{j+1}) - B'. \end{aligned}$$ Since $d_{\mathcal{U}}(Z_j, Z_{j+1}) \ge L_0$ by assumption, we see that $$\mu_{1,2}(Z_j, Z_{j+1}) \ge A \cdot d_{\mathcal{U}}(Z_j, Z_{j+1}) + 2\lambda K + D > A \cdot d_{\mathcal{U}}(Z_j, Z_{j+1}).$$ We now assume that $n > 1$, so that inductively we have $$\mu_{1,2}(Z_j, Z_{j + n - 1}) \ge A \cdot d_{\mathcal{U}}(Z_j, Z_{j + n - 1}).$$ From , we can find walls $W, W'$ with $W < Z_{j + n - 1} < W'$ satisfying $\overline{\mathbf{Hc}_+(W')} \subset \mathbf{Hc}_+(W)$ and $d_{\mathcal{U}}(W, W') < \lambda$. In particular we know $d_{\mathcal{U}}(W, Z_{j + n - 1}) < \lambda$, so by we know that $$\label{eq:inductive_bound} \mu_{1,2}(Z_j, W) \ge A \cdot d_{\mathcal{U}}(Z_j, Z_{j + n - 1}) - \lambda K.$$ In particular, since $d_{\mathcal{U}}(Z_j, Z_{j + n - 1}) \ge L_0$, it follows that $\mu_{1,2}(Z_j, W) > M$. We can also combine with the argument from the base case and the fact that $d_{\mathcal{U}}(W', Z_{j + n}) < \lambda$ to see that $$\label{eq:length1_bound} \mu_{1,2}(W', Z_{j + n}) \ge A \cdot d_{\mathcal{U}}(Z_{j + n - 1}, Z_{j + n}) + \lambda K + D.$$ In particular, this tells us that $\mu_{1,2}(W', Z_{j + n}) > M + 2\lambda K + D > M$. We now apply to the walls $Z_j < W < W' < Z_{j + n}$ to see that $$\begin{aligned} d_{\mathcal{U}}(Z_{j}, Z_{j + n}) &\ge \mu_{1,2}(Z_{j}, W) + \mu_{1,2}(W', Z_{j + n}) - D. \end{aligned}$$ Putting this inequality together with [\[eq:inductive_bound\]](#eq:inductive_bound){reference-type="eqref" reference="eq:inductive_bound"} and [\[eq:length1_bound\]](#eq:length1_bound){reference-type="eqref" reference="eq:length1_bound"} and using the fact that $d_{\mathcal{U}}(Z_{j}, Z_{j + n}) = d_{\mathcal{U}}(Z_{j}, Z_{j + n - 1}) + d_{\mathcal{U}}(Z_{j + n - 1}, Z_{j + n})$ completes the proof. ◻ We keep the constant $L_0$ from fixed for the rest of the paper. As in the previous step, we now modify the collection $\mathbf{Z}$ by replacing all maximal sequences $Z_j, Z_{j + 1}, \ldots, Z_{j + n}$ satisfying [\[item:long_consecutive_itineraries\]](#item:long_consecutive_itineraries){reference-type="ref" reference="item:long_consecutive_itineraries"} with $Z_j, Z_{j + n}$. Once again, we know that for any $L$, the quantity $r_L(\mathbf{Z})$ can only decrease after this modification. After reindexing, the previous lemma now gives us the following: **Proposition 89**. *For a uniform constant $A_0$, and every $i$ with $1 \le i < N$, if $d_{\mathcal{U}}(Z_i, Z_{i+1}) \ge L_0$, then $\mu_{1,2}(Z_i, Z_{i+1}) > A_0 \cdot d_{\mathcal{U}}(Z_i, Z_{i+1})$.* In addition, $\mathbf{Z}$ now satisfies: **Proposition 90** (No adjacent long sub-itineraries). *Let $Z_{i-1}, Z_i, Z_{i+1}$ be consecutive walls in $\mathbf{Z}$. Then either $d_{\mathcal{U}}(Z_{i-1}, Z_i) < L_0$ or $d_{\mathcal{U}}(Z_i, Z_{i+1}) < L_0$.* ### Choosing the threshold for "long" sub-itineraries Let $N = |\mathbf{Z}|$. For any given $L > 0$, and for each $1 < i < j \le N$, we now define $$t^+_L(i,j) := \sum_{n=i}^j \{d_{\mathcal{U}}(Z_{n-1}, Z_n)\}^{L}.$$ We similarly define $t^-_L(i, j)$ via the truncated length: $$t_L^-(i, j) := \sum_{n=i}^j \{d_{\mathcal{U}}(Z_{n-1}, Z_n)\}_{L}.$$ That is, $t^+_L(i, j)$ is the amount of time the itinerary $\ensuremath{\mathcal{U}}(Z_i, Z_j)$ spends inside of sub-itineraries $\ensuremath{\mathcal{U}}(Z_n, Z_{n+1})$ whose length is at least $L$, and $t^-_L(i, j)$ is the amount of time it spends inside of itineraries with length bounded strictly above by $L$. It follows from [\[eq:udist_additive\]](#eq:udist_additive){reference-type="eqref" reference="eq:udist_additive"} that for any $L$ and any $i,j$ we have $$\label{eq:short_long_sum} t_L^+(i, j) + t_L^-(i, j) = d_{\mathcal{U}}(Z_i, Z_j).$$ We also define the quantity $$r_L(i,j) := t_L^-(i,j) / d_{\mathcal{U}}(Z_i, Z_j).$$ By definition, we have $r_L(\mathbf{Z}) = r_L(1, N)$. **Definition 91**. We now let $A_0, L_0$ be the constants from . We then fix constants $D_0, M_0$ as in , where the constant $\lambda$ in the lemma is chosen to be $L_0$. Then, we define $$L_1 = \max\{M_0/A_0, D_0/A_0, L_0\}$$ and then choose constants $D_1, M_1$ as in again, but this time taking $\lambda = L_1$. **Lemma 92**. *Given $A, B, C > 0$, there exists $L > 0, \tau_1 \in (0,1)$ so that for any $i < j$, if $r_L(i,j) < \tau_1$, then $$A \cdot t_L^+(i,j) - B\cdot t_L^-(i,j) > C.$$* *Proof.* Since $t_L^-(i,j) = d_{\mathcal{U}}(Z_i, Z_j)r_L(i,j)$ and $t_L^+(i,j) = d_{\mathcal{U}}(Z_i, Z_j)(1 - r_L(i,j))$, the left-hand side of the inequality above is given by $$d_{\mathcal{U}}(Z_i, Z_j) \cdot (A (1 - r_L(i,j)) - B r_L(i,j)).$$ If $r_L(i,j) < \tau_1$ for some $\tau_1 \in (0,1)$, the above expression is bounded below by $$d_{\mathcal{U}}(Z_i, Z_j)(A(1 - \tau_1) - B\tau_1).$$ As $\tau_1 \to 0$, the quantity $(A(1 - \tau_1) - B\tau_1)$ approaches $A$. In particular, if $\tau_1 < \frac{A}{2(A + B)}$, then the above expression is at least $d_{\mathcal{U}}(Z_i, Z_j) \cdot \frac{A}{2}$. Finally, the fact that $r_L(i, j) < \tau_1 < 1$ implies that $d_{\mathcal{U}}(Z_i, Z_j) > L$ since there has to be at least one long sub-itinerary, so the lemma follows if we take $L \geq 2C / A$. ◻ **Definition 93**. We define $$B_0 := \max\left\{D_1, D_0, K \right\},$$ where $K$ is the constant from , and then use the previous lemma to fix constants $L, \tau_1$ so that $$\label{eq:uniform_root_ineq} A_0 \cdot t_L^+(i,j) - B_0\cdot t_L^-(i,j) > \max\{M_0, M_1\}$$ whenever $r_L(i,j) < \tau_1$. We also choose $L$ large enough so that $L > L_1 \ge L_0$. The length $L$ will be our threshold for "long" sub-itineraries. ### Combining all remaining sub-itineraries **Proposition 94**. *For any $1 \le i < j \le N$, if $r_L(i,j) < \tau_1$, then $$\mu_{1,2}(Z_i, Z_j) \ge A_0 \cdot t_L^+(i, j) - B_0 \cdot t_L^-(i, j).$$* *Proof.* We will induct on $j - i$. In the base case $j = i + 1$, then since $r_L(i,j) < \tau_1 < 1$ we must have $d_{\mathcal{U}}(Z_i, Z_{i+1}) \ge L$. Since $L \ge L_0$, tells us that $\mu_{1,2}(Z_i, Z_j) \ge A_0d_{\mathcal{U}}(Z_i, Z_j)$ and since $d_{\mathcal{U}}(Z_i, Z_j) = t_L^+(i,j)$ in this case, we are done. So now suppose that $j - i > 1$. First, observe that if $d_{\mathcal{U}}(Z_i, Z_{i + 1}) < L$, then $r_L(i+1, j) < r_L(i, j) < \tau_1$. So, by induction we have $$\mu_{1,2}(Z_{i+1}, Z_j) \ge A_0 \cdot t_L^+(i + 1, j) - B_0 \cdot t_L^-(i + 1, j).$$ In this case, we also know that $t_L^+(i, j) = t_L^+(i + 1, j)$. So, after applying , we have $$\begin{aligned} \mu_{1,2}(Z_i, Z_j) &\ge A_0 \cdot t_L^+(i, j) - B_0 \cdot t_L^-(i + 1, j) - K \cdot d_{\mathcal{U}}(Z_i, Z_{i+1})\\ &\ge A_0 \cdot t_L^+(i, j) - B_0 \cdot t_L^-(i + 1, j) - B_0 \cdot d_{\mathcal{U}}(Z_i, Z_{i+1}). \end{aligned}$$ But we also know that $t_L^-(i,j) = d_{\mathcal{U}}(Z_i, Z_{i+1}) + t_L^-(i + 1, j)$ in this case, so this proves the desired inequality. A similar argument also shows that the inequality holds in the case where $d_{\mathcal{U}}(Z_{j-1}, Z_j) < L$. So, we may now assume that we have $d_{\mathcal{U}}(Z_i, Z_{i+1}) \ge L$ and $d_{\mathcal{U}}(Z_{j-1}, Z_j) \ge L$. Our strategy in this case is to find a suitable pair of walls $Z_{n-1} < Z_n$ to break the itinerary $\ensuremath{\mathcal{U}}(Z_i, Z_j)$ into three pieces: a "short" piece in the middle, and two "mostly long" pieces on either side. Then we will apply induction and one of our additivity lemmas for well-separated walls (). To find a suitable "short" sub-itinerary in the middle of $d_{\mathcal{U}}(Z_i, Z_j)$, we prove the following claim: **Claim 2**. For some $n$ with $i + 1 < n < j$, we have $$\label{eq:pivot_itinerary} r_L(i, n - 1) < \tau_1, \qquad r_L(n, j) < \tau_1,$$ and $d_{\mathcal{U}}(Z_{n-1}, Z_n) < L$. To find such an $n$, first note that since $d_{\mathcal{U}}(Z_{j-1}, Z_j) \ge L > L_0$, implies that $d_{\mathcal{U}}(Z_{j-2}, Z_{j-1}) < L_0 < L$ and thus $j - 2 > i$. Then, if $r_L(i, j-2) < \tau_1$, we can satisfy the claim by taking $n = j-1$, since $r_L(j-1, j) = 0$. So, we can assume that $r_L(i, j - 2) \ge \tau_1$, and let $n$ be the minimal index such that $r_L(i, n) \ge \tau_1$. Now, we have $$\begin{aligned} r_L(i,j) &= \frac{1}{d_{\mathcal{U}}(Z_i, Z_j)} (t_L^-(i, n) + t_L^-(n, j))\\ &= \frac{d_{\mathcal{U}}(Z_i, Z_n)}{d_{\mathcal{U}}(Z_i, Z_j)} r_L(i, n) + \frac{d_{\mathcal{U}}(Z_n, Z_j)}{d_{\mathcal{U}}(Z_i, Z_j)}r_L(n, j). \end{aligned}$$ From [\[eq:udist_additive\]](#eq:udist_additive){reference-type="eqref" reference="eq:udist_additive"}, we also know that $$\frac{d_{\mathcal{U}}(Z_i, Z_n)}{d_{\mathcal{U}}(Z_i, Z_j)} + \frac{d_{\mathcal{U}}(Z_n, Z_j)}{d_{\mathcal{U}}(Z_i, Z_j)} = 1.$$ So, since $r_L(i,n) \ge \tau_1$ we must have $r_L(n, j) < \tau_1$, and by minimality of $n$ we also know that $r_L(i, n - 1) < \tau_1$. But in particular we also know $r_L(i, n-1) < r_L(i, n)$, implying $d_{\mathcal{U}}(Z_{n-1}, Z_n) < L$. This proves the claim, so to finish the inductive step, we need to consider two cases: Case 1: $d_{\mathcal{U}}(Z_{n-1}, Z_n) < L_1$ : In this case, we consider the sub-itineraries $\ensuremath{\mathcal{U}}(Z_i, Z_{n-1})$ and $\ensuremath{\mathcal{U}}(Z_n, Z_j)$. Since $r_L(i, n-1) < \tau_1$ and $r_L(n, j) < \tau_1$, we may assume inductively that $$\begin{aligned} \label{eq:inductive_root_ineq_left} \mu_{1,2}(i, n-1) &\ge A_0t_L^+(i, n-1) - B_0t_L^-(i,n-1),\\ \label{eq:inductive_root_ineq_right} \mu_{1,2}(n, j) &\ge A_0t_L^+(n, j) - B_0t_L^-(n, j). \end{aligned}$$ It then follows directly from and [\[eq:inductive_root_ineq_left\]](#eq:inductive_root_ineq_left){reference-type="eqref" reference="eq:inductive_root_ineq_left"}, [\[eq:inductive_root_ineq_right\]](#eq:inductive_root_ineq_right){reference-type="eqref" reference="eq:inductive_root_ineq_right"} above that $$\begin{aligned} \mu_{1,2}(i, n-1) > M_1, \qquad \mu_{1,2}(n, j) > M_1. \end{aligned}$$ So, since $d_{\mathcal{U}}(Z_{n-1}, Z_n) < L_1$, we can apply to the walls $Z_i, Z_{n-1}, Z_n, Z_j$ to obtain $$\begin{aligned} \mu_{1,2}(Z_i, Z_j) &> \mu_{1,2}(Z_i, Z_{n-1}) + \mu_{1,2}(Z_n, Z_j) - D_1\\ &\ge A_0(t_L^+(i, n-1) + t_L^+(n, j)) - B_0(t_L^-(i, n-1) + t_L^-(n, j)) - D_1\\ &\ge A_0t_L^+(i,j) - B_0(t_L^-(i, n-1) + t_L^-(n, j) + 1), \end{aligned}$$ where the last inequality holds because we have defined $B_0 \geq D_1$. But then since $1 \le d_{\mathcal{U}}(Z_{n-1}, Z_n) < L$, we know that $$t_L^-(i,j) = t_L^-(i, n-1) + t_L^-(n, j) + d_{\mathcal{U}}(Z_{n-1}, Z_n) \ge t_L^-(i,n-1) + t_L^-(n, j) + 1,$$ which proves the desired inequality in this case. Case 2: $d_{\mathcal{U}}(Z_{n-1}, Z_n) \ge L_1$ : In this case, since $L_1 \ge L_0$, implies that the two sub-itineraries $\ensuremath{\mathcal{U}}(Z_{n-2}, Z_{n-1})$, $\ensuremath{\mathcal{U}}(Z_n, Z_{n+1})$ to the left and right of $\ensuremath{\mathcal{U}}(Z_{n-1}, Z_n)$ both have length less than $L_0$. Since we know $r_L(i, n-1) < \tau_1 < 1$ and $r_L(n, j) < \tau_1 < 1$, this also tells us that $i < n - 2$ and $n + 1 < j$, and that $$r_L(i, n-2) < \tau_1, \qquad r_L(n + 1, j) < \tau_1.$$ We then see directly from together with the induction hypothesis that $\mu_{1,2}(Z_i, Z_{n-2}) > M_0$ and $\mu_{1,2}(Z_{n+1}, Z_j) > M_0$. In addition, since $d_{\mathcal{U}}(Z_{n-1}, Z_n) \ge L_1 = \max\{L_0, M_0/A_0, D_0/A_0\}$, implies that $\mu_{1,2}(Z_{n-1}, Z_n) \ge \max\{M_0, D_0\}$. We now apply twice. First, we apply the lemma to the walls $Z_i < Z_{n-2} < Z_{n-1} < Z_n$, which gives the bound $$\label{eq:case2_bound_1} \mu_{1,2}(Z_i, Z_n) \ge \mu_{1,2}(Z_i, Z_{n-2}) + \mu_{1,2}(Z_{n-1}, Z_n) - D_0.$$ Since $\mu_{1,2}(Z_{n-1}, Z_n) \ge D_0$ we see that $\mu_{1,2}(Z_i, Z_n) > M_0$, which means we can then apply to the walls $Z_i < Z_n < Z_{n+1} < Z_j$ to obtain $$\label{eq:case2_bound_2} \mu_{1,2}(Z_i, Z_j) \ge \mu_{1,2}(Z_i, Z_n) + \mu_{1,2}(Z_{n+1}, Z_j) - D_0.$$ Putting [\[eq:case2_bound_1\]](#eq:case2_bound_1){reference-type="eqref" reference="eq:case2_bound_1"} and [\[eq:case2_bound_2\]](#eq:case2_bound_2){reference-type="eqref" reference="eq:case2_bound_2"} together we see that $$\mu_{1,2}(Z_i, Z_j) \ge \mu_{1,2}(Z_i, Z_{n-2}) + \mu_{1,2}(Z_{n-1}, Z_n) + \mu_{1,2}(Z_{n+1}, Z_j) - 2D_0.$$ Since each of $d_{\mathcal{U}}(Z_{n-2}, Z_{n-1})$, $d_{\mathcal{U}}(Z_{n-1}, Z_n)$, and $d_{\mathcal{U}}(Z_n, Z_{n+1})$ is less than $L$, we know that $t_L^+(i, j) = t_L^+(i, n-2) + t_L^+(n+1, j)$. Thus, after applying induction to the terms $\mu_{1,2}(Z_i, Z_{n-2})$ and $\mu_{1,2}(Z_{n+1}, Z_j)$ in the inequality above, and discarding the (nonnegative) $\mu_{1,2}(Z_{n-1}, Z_n)$ term, we obtain $$\begin{aligned} \mu_{1,2}(Z_i, Z_j) &\ge A_0t_L^+(i, j) - B_0(t_L^-(i, n - 2) + t_L^-(n+1, j)) - 2D_0\\ &\ge A_0t_L^+(i, j) - B_0(t_L^-(i, n - 2) + t_L^-(n+1, j) + 2). \end{aligned}$$ For the last line we apply the fact that $B_0 \ge D_0$. Finally, since $t_L^-(n - 2, n - 1) \ge 1$ and $t_L^-(n, n + 1) \ge 1$, we get $$\begin{aligned} t_L^-(i,j) &= t_L^-(i,n-2) + t_L^-(n-2, n - 1) + t_L^-(n - 1, n) + t_L^-(n, n + 1) + t_L^-(n+1, j)\\ &> t_L^-(i, n - 2) + t_L^-(n + 1, j) + 2, \end{aligned}$$ and we obtain the desired inequality in this case as well.  ◻ Finally we obtain the estimate we originally wanted. *Proof of .* We set $$\tau = \min\left\{\tau_1, \frac{A_0}{2(A_0 + B_0)}\right\},$$ where $A_0, B_0$ are the constants from . To simplify notation we write $r = r_L(\mathbf{Z})$. Since all of our modifications to $\mathbf{Z}$ have only decreased $r$, if we had $r < \tau$ before our modifications, this is still true for our current $\mathbf{Z}$. By definition, we have $r\cdot d_{\mathcal{U}}(Z_1, Z_N) = t_L^-(1, N)$, and [\[eq:short_long_sum\]](#eq:short_long_sum){reference-type="eqref" reference="eq:short_long_sum"} implies that $$t_L^+(1, N) = (1 - r)\cdot d_{\mathcal{U}}(Z_1, Z_N).$$ Then, since $r < \tau \le \tau_1$, we can use to obtain: $$\begin{aligned} \mu_{1,2}(Z_1, Z_N) &\ge \left(A_0(1 - r) - B_0 r \right) \cdot d_{\mathcal{U}}(Z_1, Z_N)\\ &\ge \left(A_0(1 - \tau) - B_0 \tau \right) \cdot d_{\mathcal{U}}(Z_1, Z_N)\\ &\ge \frac{A_0}{2} d_{\mathcal{U}}(Z_1, Z_N). \end{aligned}$$ Our construction ensures that $Z_1 = W_1$ and $Z_N$ is always the maximal wall in $\mathbf{W}$. From and the definition of $\ensuremath{\mathcal{U}}$, we know there is a uniform $R > 0$ so that $\gamma(\ensuremath{\mathcal{U}}) = \eta\, \gamma(Z_1, Z_N)\, \eta'$, for some $\eta, \eta' \in C$ with $|\eta|, |\eta'| < R$. Thus the desired estimate follows from . ◻ # Failure of strong nesting for half-cones {#sec:appendix} The purpose of this appendix is to prove the following two claims: **Proposition 95**. *There exists a right-angled Coxeter group $C$, a simplicial representation $\rho$ of $C$, and a pair of walls $W, W'$ in the Vinberg domain $\Omega_{\mathrm{Vin}}$ for $\rho$ which satisfies the following properties:* 1. *$\gamma(W, W')$ does not lie in a proper standard subgroup of $C$ (equivalently, by , $\overline{W} \cap \overline{W'} = \emptyset$);* 2. *$\mathbf{Hs}_+(W') \subset \mathbf{Hs}_+(W)$;* 3. *$\overline{\mathbf{Hc}_+(W')}$ is *not* contained in $\mathbf{Hc}_+(W)$.* **Proposition 96**. *There exists a right-angled Coxeter group $C$ and reflections $R, R'$ in $C$ such that, for any simplicial representation $\rho\colon C \to \mathop{\mathrm{SL}}^{\pm}(|S|, \ensuremath{\mathbb{R}})$ with fully nondegenerate Cartan matrix, and *any* reflection domain $\Omega$ for $\rho$, if $W, W'$ are the walls in $\Omega$ preserved by $R, R'$, then:* 1. *$\gamma(W, W')$ does not lie in a proper standard subgroup of $C$ (so in particular $\overline{W} \cap \overline{W'} = \emptyset$);* 2. *$\mathbf{Hs}_+(W') \subset \mathbf{Hs}_+(W)$;* 3. *$\overline{\mathbf{Hc}_+(W')}$ is *not* contained in $\mathbf{Hc}_+(W)$.* Although the second proposition above implies the first, we will prove these results one at a time, since the construction for is a slightly more complicated variation of the construction for . Note that if were false, then the proofs in of this paper would considerably simplify---in particular, the proof of could be reduced to a direct application of . tells us that we cannot resolve the problem simply by replacing $\Omega_{\mathrm{Vin}}$ in with some other carefully chosen reflection domain. **Remark 97**. If $C$ is a hyperbolic Coxeter group, it follows from [@DGKLM Corollary 1.11] that there is a simplicial representation $\rho\colon C \to \mathop{\mathrm{SL}}^{\pm}(V)$ and a reflection domain $\Omega$ for $\rho$ so that half-cones over any two walls $W, W'$ in $\Omega$ with disjoint closures will strongly nest. It seems likely that this holds even for some examples of simplicial representations of non-hyperbolic Coxeter groups, but we do not pursue this here. ## Proof of Consider the right-angled Coxeter group $C$ with generating set $S = \langle a, b, c, d, e \rangle$, and nerve given in below (recall that there is an edge between two vertices in the nerve precisely when the corresponding generators commute): That is, $C$ is isomorphic to the group $(\ensuremath{\mathbb{Z}}/2 * \ensuremath{\mathbb{Z}}/2)^2 * (\ensuremath{\mathbb{Z}}/2)$, where the $\ensuremath{\mathbb{Z}}/2 * \ensuremath{\mathbb{Z}}/2$ factors are generated by the pairs $a, c$ and $b, d$, and $e$ generates the remaining $\ensuremath{\mathbb{Z}}/2$ free factor. Observe that $C$ is a minimal example of an irreducible non-hyperbolic right-angled Coxeter group: a theorem of Gromov (see [@gromov Section 4.2.C] or [@Davis Chapter 12]) implies that a right-angled Coxeter group fails to be hyperbolic precisely when the "empty square" on the vertices $a, b, c, d$ appears as a full subgraph of the nerve. (The general version of this theorem for Coxeter groups which are not necessarily right-angled is due to Moussong [@Moussong].) Let $V = \ensuremath{\mathbb{R}}^5$. We choose a nonsingular Cartan matrix for $C$ and a simplicial representation determined by this Cartan matrix. We suppress the representation from the notation, and just let $C$ act directly on $\ensuremath{\mathbb{P}}(V)$. For each $s \in \{a, b, c, d, e\}$, we let $H_s \subset \ensuremath{\mathbb{P}}(V)$ denote the (projective) reflection hyperplane for the reflection $s$, and we let $F_s$ denote the closed face of the Tits simplex $\Delta$ fixed by $s$. Further, for any subset $S' \subset S$, we let $H_{S'}$ denote $\bigcap_{s \in S'} H_s$, and $F_{S'}$ denote $\bigcap_{s \in S'}F_s$. Consider the 2-dimensional projective subspace $H_{\{a,c\}} = H_a \cap H_c$. This subspace contains the closed face $F_{\{a,c\}}$ of the fundamental simplex $\Delta$ fixed pointwise by the standard subgroup $C(a,c)$. Since $C(a,c)$ is infinite, $F_{\{a,c\}}$ must be contained in the boundary of $\Omega_{\mathrm{Vin}}$ and hence so are all of its translates under the action of $C$. Now, since the subgroup $C(b, d)$ centralizes $C(a, c)$, it preserves the subspace $H_{\{a,c\}}$, and in fact $b$ and $d$ act on this subspace by projective reflections fixing the lines $H_b \cap H_{\{a,c\}}$ and $H_d \cap H_{\{a,c\}}$. The relative interior of the orbit $C(b,d) \cdot F_{\{a,c\}}$ is an infinite-sided convex polygon $P$ in $H_{\{a,c\}}$, which must be a subset of $\ensuremath{\partial}\Omega_{\mathrm{Vin}}$ (see ). On the other hand, since the generator $e$ does *not* commute with either $a$ or $c$, the intersection $H_{\{a,c\}} \cap e H_{\{a,c\}}$ is the 1-dimensional subspace $H_{\{a,c\}} \cap H_e$. This tells us that the closed polygon $\overline{P}$ is exactly the intersection of $\overline{\Omega_{\mathrm{Vin}}}$ with $H_{\{a,c\}}$, i.e. it is a face of $\Omega_{\mathrm{Vin}}$. Now, consider the geodesic word $w = bdeac$, and let $\ensuremath{\mathcal{W}}$ be an $\Omega_{\mathrm{Vin}}$-itinerary traversing $w$, departing from the identity. Since no pair of consecutive generators in $w$ commutes, every pair of distinct walls in $\ensuremath{\mathcal{W}}$ is disjoint in $\Omega_{\mathrm{Vin}}$. This means that $\ensuremath{\mathcal{W}}$ must be efficient. So, if $W, W'$ are respectively the first and last walls in $\ensuremath{\mathcal{W}}$, we must have $\gamma(W, W') = bdeac$. As $W \cap W' = \emptyset$ and $\ensuremath{\mathcal{W}}$ departs from the identity, we know that $\mathbf{Hs}_+(W)$ contains $\mathbf{Hs}_+(W')$. By we have $W' = bdea \cdot W(c)$, where $W(c)$ is the reflection wall in $\Omega_{\mathrm{Vin}}$ for $c$. From this, it follows that: **Proposition 98**. *The intersection $\overline{W'} \cap \overline{P}$ is given by $bd \cdot F_{\{a,c,e\}}$, where $F_{\{a,c,e\}}$ is the edge of $F_{\{a,c\}}$ fixed by $e$.* *Proof.* Since $\overline{W(c)} = H_c \cap \overline{\Omega_{\mathrm{Vin}}}$, and $\Omega_{\mathrm{Vin}}$ is $C$-invariant, we have $\overline{P} \cap \overline{W'} = \overline{P} \cap bdea \cdot H_c$. Since $P$ is invariant under $b,d$, this intersection is the same as $bd(\overline{P} \cap ea \cdot H_c)$. Then since $e$ does not commute with $c$ or $a$, we have $(ea \cdot H_c) \cap H_{\{a,c\}} = H_{\{a,c,e\}}$ and thus $\overline{P} \cap ea \cdot H_c = F_{\{a,c,e\}}$. ◻ We know that the polar of $b$ lies in the projective subspace $H_{a,c}$ since $b$ commutes with both $a$ and $c$. So, we can use the argument from to see that the boundary of the half-cone $\mathbf{Hc}_+(W)$ contains the connected component of $\overline{P} -\overline{W}$ which does *not* contain $F_{\{a,c\}}$. Thus, the boundary of $\mathbf{Hc}_+(W)$ contains $\overline{W'}\cap H_{a,c}$ and so the half-cones over these walls cannot strongly nest. **Remark 99**. For any given $k \ge 1$, we can also consider the word $w = (bd)^ke(ac)^k$, and an itinerary $\ensuremath{\mathcal{W}}$ traversing $w$ departing from the identity. A nearly identical argument to the above shows that the initial and terminal walls $W, W'$ of $\ensuremath{\mathcal{W}}$ also satisfy the conclusions of . This proves that, in , the group element $\gamma(W, W')$ cannot even be made to lie "close" to a proper standard subgroup: we cannot find a uniform constant $R$ so that $\gamma(W, W') = \eta_1 \gamma \eta_2$ for $\eta_1, \eta_2$ satisfying $|\eta_i| < R$ and $\gamma$ lying in a proper standard subgroup of $C$. ### Modifying the example {#sec:modifying} The argument above also shows that the corresponding half-cones in the Vinberg domain $\mathscr{O}_{\mathrm{Vin}}\subset \ensuremath{\mathbb{P}}((\ensuremath{\mathbb{R}}^5)^*)$ for the dual representation $\rho^*$ (see ) do not strongly nest. By , the corresponding half-cones in the dual $\mathscr{O}_{\mathrm{Vin}}^*$ cannot strongly nest either. By , the reflection domain $\mathscr{O}_{\mathrm{Vin}}^*$ is contained inside of every reflection domain for $\rho$, so we denote it $\Omega_{\min}$. One could still hope to find some reflection domain $\Omega$ lying between $\Omega_{\min}$ and $\Omega_{\mathrm{Vin}}$ where the half-cones over the walls $W \cap \Omega$ and $W' \cap \Omega$ strongly nest. In fact, for the example above, it is possible to find such a domain, by taking $\Omega$ to be a small neighborhood of $\Omega_{\min}$ with respect to the Hilbert metric on $\Omega_{\mathrm{Vin}}$. This strategy works because the segments in $\partial \Omega_{\mathrm{Vin}}$ joining $\partial W$ to $\partial W'$ are not contained in $\partial \Omega_{\min}$; see . But this can fail if the intersection between $\partial \Omega_{\mathrm{Vin}}$ and $\partial \Omega_{\min}$ contains large-dimensional faces, which is what occurs in the next counterexample. ## Proof of {#sec:6dim_counterexample} We consider a right-angled Coxeter group $C$ whose generating set $S$ splits into three disjoint subsets $D = \{d_1, d_2\}$, $T = \{t_1, t_2, t_3\}$, and $E = \{e\}$, with the following relations: - Each $t_i \in T$ commutes with each $d_j \in D$; - The generator $e$ commutes with $t_1$ and $t_3$. There are no other relations among the generators, meaning the system $(C, S)$ has the nerve depicted in below. Observe that the subgroup $C(T)$ is an $(\infty, \infty, \infty)$ triangle group, which commutes with the infinite dihedral subgroup $C(D)$. As for the previous example, we let $V = \ensuremath{\mathbb{R}}^6$, and fix an arbitrary simplicial representation $C \to \mathop{\mathrm{SL}}^{\pm}(6, \ensuremath{\mathbb{R}})$ with fully nondegenerate Cartan matrix. We again omit the representation from the notation and allow $C$ to act directly on $V$ and $\ensuremath{\mathbb{P}}(V)$. Consider the 3-dimensional projective subspace $H_D \subset \ensuremath{\mathbb{P}}(V)$, containing the closed face $F_D$ of $\Delta$. This face is a tetrahedron whose faces span fixed subspaces for the reflections in $T \cup E$. As every point in $F_D$ has infinite stabilizer, it lies in the boundary of the Vinberg domain $\Omega_{\mathrm{Vin}}$. Since the centralizer of the subgroup $C(D)$ is precisely $C(T)$, the interior of the orbit $C(T) \cdot F_D$ is a subset of $H_D$. In fact, one may apply the more general form of given in [@Vinberg1971] to see that $C(T) \cdot F_D$ is an infinite-sided convex polytope $P \subset H_D$, whose closure $\overline{P}$ is a face of the Vinberg domain tiled by copies of $F_D$. See . It is possible to argue as in the proof of to see that there is a pair of walls $W, W'$ in $\Omega_{\mathrm{Vin}}$ with disjoint closures, such that the boundary of $\mathbf{Hc}_+(W)$ contains a component of $P -\overline{W}$ whose closure intersects $\overline{W'}$. We want to see that something similar occurs not just for walls in the Vinberg domain, but in *any* reflection domain for this simplicial representation. For this, we consider the unique minimal reflection domain $\Omega_{\min}$ for the simplicial representation $\rho$, whose existence is guaranteed by the discussion in Section [9.1.1](#sec:modifying){reference-type="ref" reference="sec:modifying"}. We can get a precise description of $\Omega_{\min}$ in our situation using a theorem of Danciger-Guéritaud-Kassel-Lee-Marquis. **Definition 100**. Let $(C, S)$ be an infinite irreducible right-angled Coxeter system, and suppose $C$ acts via a simplicial representation on $V = \ensuremath{\mathbb{R}}^{|S|}$ with nonsingular Cartan matrix. For any subset $S' \subseteq S$, we let $\tilde{\Sigma}_{S'} \subset \overline{\tilde{\Delta}}$ denote the set $$\tilde{\Sigma}_{S'} = \left\{x = \sum_{t \in S'}\lambda_tv_t : \alpha_s(x) \le 0 \textrm{ and } \lambda_t \ge 0 \quad \forall s \in S, t \in S' \right\}.$$ We let $\Sigma_{S'}$ denote the projectivization of $\tilde{\Sigma}_{S'}$ in $\ensuremath{\mathbb{P}}(V)$, and write $\tilde{\Sigma} = \tilde{\Sigma}_S$ and $\Sigma = \Sigma_S$. **Theorem 101** (See [@DGKLM Theorem 5.2]). *Let $(C, S)$ be an infinite irreducible right-angled Coxeter group with $|S| > 2$, acting on $V = \ensuremath{\mathbb{R}}^{|S|}$ by a simplicial representation with nonsingular Cartan matrix. Then the set $\overline{\Omega_{\min}} = \overline{\bigcup_{\gamma \in C} \gamma\Sigma}$ is the closure of the unique reflection domain $\Omega_{\min}\subset \Omega_{\mathrm{Vin}}$ which is contained in every reflection domain for $\rho$.* **Remark 102**. The version of proved in [@DGKLM] holds under considerably weaker hypotheses than what we have stated here. In particular, the result in [@DGKLM] can be applied to non-right-angled Coxeter groups and non-simplicial representations. We now consider the intersection of the sets $\Sigma$ and $\overline{\Omega_{\min}}$ with the projective subspace $H_D$, for the specific right-angled Coxeter group $C$ we described above. Observe that the subgroup $C(T)$ acts on the subspace $V_T = \mathop{\mathrm{span}}\{v_{t_1}, v_{t_2}, v_{t_3}\}$, via a simplicial representation whose Cartan matrix is a (fully nondegenerate) principal submatrix of our original Cartan matrix. Thus $\Sigma_T \subset \Sigma \cap \ensuremath{\mathbb{P}}(V_T)$ is a hexagon, with alternating sides contained in the projective lines $H_{t_i} \cap \ensuremath{\mathbb{P}}(V_T)$ (see ). In particular, since $C(D)$ and $C(T)$ commute, we know that $\ensuremath{\mathbb{P}}(V_T) \subset H_D$, and therefore $\Sigma_T \subset F_D$. In addition, since $t_1$ and $t_3$ commute with all of the generators in $D \cup E$, the subspace $\ensuremath{\mathbb{P}}(V_{t_1, t_3})$ is contained in the 2-dimensional projective subspace $H_{D \cup E} = H_D \cap H_E$, which means that the edge $\Sigma_{t_1, t_3}$ of the hexagon $\Sigma_T$ is contained in $H_{D \cup E}$ (see , left). Since $\Sigma_T \subset F_D$, the relative interior of the orbit $C(T) \cdot \Sigma_T$ is a $C(T)$-invariant convex subset $P_{\min}$ of the polytope $P$, contained in the $C(T)$-invariant subspace $\ensuremath{\mathbb{P}}(V_T)$; it is a copy of the minimal $C(T)$-invariant domain for the simplicial representation of $C(T)$ on $V_T$ (see , right). We now consider the geodesic word $w = t_1t_3t_2ed_1d_2$, and let $\ensuremath{\mathcal{W}}$ be an $\Omega_{\mathrm{Vin}}$-itinerary traversing $w$, departing from the identity. As in the previous example, since no pair of consecutive generators in $w$ commutes, the itinerary $\ensuremath{\mathcal{W}}$ is efficient. Thus, writing $W_{\mathrm{Vin}}$ and $W'_{\mathrm{Vin}}$ for the first and last walls of $\ensuremath{\mathcal{W}}$, $\gamma(W_{\mathrm{Vin}}, W_{\mathrm{Vin}}') = t_1t_3t_2ed_1d_2$. We let $R, R'$ denote the reflections in $C$ fixing $W_{\mathrm{Vin}}, W_{\mathrm{Vin}}'$, and let $\Omega$ be an arbitrary reflection domain for $C$. We let $W, W'$ denote the walls in $\Omega$ fixed by $R, R'$, and let $W_{\min}, W_{\min}'$ denote the walls in $\Omega_{\min}$ fixed by the same pair of reflections. By and , we have $$\Omega_{\min}\subseteq \Omega \subseteq \Omega_{\mathrm{Vin}},$$ and $$W_{\min} \subseteq W \subseteq W_{\mathrm{Vin}}, \qquad W_{\min}' \subseteq W' \subseteq W_{\mathrm{Vin}}'.$$ By , we have $W_{\min}' = t_1t_3t_2ed_1 \cdot (H_{d_2} \cap \Omega_{\min})$. Using this, we show: **Proposition 103**. *The intersection $\overline{W_{\min}'} \cap \overline{P_{\min}}$ contains $t_1t_3t_2 \cdot \Sigma_{t_1, t_3}$.* *Proof.* Since $P_{\min}$ is $C(T)$-invariant and $\Omega_{\min}$ is $C$-invariant, we just need to check that $\overline{P_{\min}} \cap ed_1 \cdot H_{d_2}$ contains $\Sigma_{t_1, t_3}$. Since $e$ does not commute with $d_1$ or $d_2$, we know that $ed_1 \cdot H_{d_2} \cap H_D = H_{D \cup E}$, and as $P_{\min} \subset H_D$ we therefore have $$\overline{P_{\min}} \cap ed_1 \cdot H_{d_2} = \overline{P_{\min}} \cap H_{D \cup E}.$$ We have already seen that the edge $\Sigma_{t_1, t_3}$ of the hexagon $\Sigma_T$ is contained in $H_{D \cup E}$, which gives us the desired intersection since $C(T)$-translates of $\Sigma_T$ tile $P_{\min}$. ◻ Now, since the $\Omega_{\mathrm{Vin}}$-itinerary $\ensuremath{\mathcal{W}}$ departs from the identity, we know that $\mathbf{Hs}_+(W_{\mathrm{Vin}})$ contains $\mathbf{Hs}_+(W'_{\mathrm{Vin}})$ and thus $\mathbf{Hs}_+(W') \subset \mathbf{Hs}_+(W)$. We consider the boundary of the half-cone $\mathbf{Hc}_+(W)$. Note that, since the entire tetrahedron $F_D$ has infinite stabilizer in $C$, any point in $\overline{W} \cap F_D$ must lie in $\ensuremath{\partial}W$. In particular, since $W_{\min} \subset W$, the boundary $\ensuremath{\partial}W$ must contain $\overline{W_{\min}} \cap F_D$, which is a reflection wall in the 2-dimensional domain $P_{\min}$. As $P_{\min}$ is a reflection domain for $C(T)$ in the $C(T)$-invariant subspace $\ensuremath{\mathbb{P}}(V_T)$, tells us that the closure of the half-cone $\mathbf{Hc}_+(\overline{W_{\min}} \cap P_{\min})$ on the domain $P_\text{min}$ contains the component of $P_{\min} -\overline{W_{\min}}$ which does not contain $\Sigma_T$ (see ). In particular, as $W_{\min}$ is a reflection wall for $t_1$, this half-cone contains $t_1t_3t_2 \cdot \Sigma_{t_1, t_3}$. As $\overline{W_{\min}} \cap P_{\min} \subset \ensuremath{\partial}W$, we have $$\mathbf{Hc}_+(\overline{W_{\min}} \cap P_{\min}) \subset \partial \mathbf{Hc}_+(W),$$ meaning that $\partial \mathbf{Hc}_+(W)$ contains $t_1t_3t_2 \cdot \Sigma_{t_1, t_3}$. Then, since $W'_{\min} \subset W'$ we see from that $\overline{W'} \cap \partial \mathbf{Hc}_+(W)$ is nonempty, meaning that the halfcones $\mathbf{Hc}_+(W)$ and $\mathbf{Hc}_+(W')$ cannot strongly nest. **Remark 104**. As in , we can apply a nearly identical argument to the word $(t_1t_3t_2)^ke(d_1d_2)^k$ for any given $k > 0$ to see that the group element $\gamma(W, W')$ can also be made to lie arbitrarily far from any standard subgroup of $C$.

arxiv_math

--- abstract: | Steady states are invaluable in the study of dynamical systems. High-dimensional dynamical systems, due to a separation of time-scales, often evolve towards a lower dimensional manifold $M$. We introduce an approach to locate saddle points (and other fixed points) that utilizes gradient extremals on such *a priori* unknown (Riemannian) manifolds, defined by adaptively sampled point clouds, with local coordinates discovered on-the-fly through manifold learning. The technique, which efficiently biases the dynamical system along a curve (as opposed to exhaustively exploring the state space), requires knowledge of a single minimum and the ability to sample around an arbitrary point. We demonstrate the effectiveness of the technique on the Müller-Brown potential mapped onto an unknown surface (namely, a sphere). Previous work employed a similar algorithmic framework to find saddle points using Newton trajectories and gentlest ascent dynamics; we therefore also offer a brief comparison with these methods. author: - A. Georgiou - H. Vandecasteele - J. M. Bello-Rivas - I. Kevrekidis bibliography: - aipsamp.bib nocite: "[@*]" title: Locating saddle points using gradient extremals on manifolds adaptively revealed as point clouds --- > The energy landscapes for molecular dynamics, protein folding, glassy materials, etc. may exhibit large numbers of minima and transition states. Locating these critical points---especially in high dimensions---is challenging. Current approaches (e.g. nudged elastic band, string method) start with two minima and *a priori* knowledge of collective variables which represent the system well in a lower dimensional setting; other methods attempt to exhaustively explore the state space (metadynamics, adaptive biasing force, iMapD). In this paper, we present a technique to systematically bias dynamical systems from a single minimum to a saddle point (transition state) without *a priori* knowledge of good collective variables. # [\[sec:intro\]]{#sec:intro label="sec:intro"}Introduction Locating saddle points of dynamical systems is of primary importance to a variety of applications, including the location of transition states of chemical systems described at the atomic level [@Bochevarov2013] or the study of transitions between molecular configurations in molecular dynamics. [@leimkuhler2015] Unfortunately, finding these transition states simply via simulation is time-prohibitive: the trajectories spend the majority of their time in a basin of attraction around a stable equilibrium, rarely traveling to a different basin of attraction (potential well). Often, trajectories of the high-dimensional system quickly evolve to, and then move slowly along, a low-dimensional manifold (for example, due to a time-scale separation). The coarse (collective) variables that parameterize the slow manifold are either known *a priori* or can be determined through manifold learning techniques. Many techniques exist to locate saddle points of these systems. The metadynamics [@laio2002] and adaptive biasing force [@darve2008] methods attempt to exhaustively explore the whole space, yet they require *a priori* knowledge of collective variables; iMapD [@chiavazzo2017] alleviates this need by exploiting manifold learning techniques. Another class of methods, namely nudged elastic band or string method variants, [@olender1996; @Jonsson1998; @Weinan2002] trace a curve between two known minima: specifically, they minimize the energy at each point along a path connecting two minima (reactant and product) while attempting to maintain equidistant spacing between the points. These methods require a global set of collective variables. A third class of methods require only a single minimum, utilizing local information to follow a continuous curve to a nearby fixed point (hopefully a saddle). These one-dimensional curves include Newton trajectories, [@quapp1998; @hirsch2004] gentlest ascent dynamics [@Weinan2010] or dimer method [@henkelman1999] trajectories, and gradient extremals, [@basilevsky1982] which we study here. These methods still require a global set of collective variables and were originally formulated for Euclidean space. Gradient extremals have been extended to smooth Riemannian manifolds to locate fixed points. [@Rowe1982; @Filippidis2013] In this work, we forego *a priori* knowledge of good collective variables as well as an explicit formulation of the manifold. Instead, we follow gradient extremals on initially unexplored Riemannian manifolds gradually revealed by adaptively sampled point clouds. We utilize local sets of collective variables that are discovered on-the-fly through manifold learning (here, diffusion maps [@COIFMAN2006]). By sampling along the manifold, discovering a good local chart, resolving the path locally on the chart, and repeating this process at the newly found location, the system is driven from an initial conformation (typically a stable equilibrium) to a saddle point. Our methodology successfully locates new fixed points using a single initial point and without need for *a priori* knowledge of collective variables. The method does not exhaustively explore the state space, but rather follows a single path at a time. In prior work we utilized a similar algorithmic approach, but with the resolved paths being those of Newton trajectories [@Bello-Rivas2023] and gentlest ascent dynamics. [@bellorivas2023gentlest] # [\[sec:GE\]]{#sec:GE label="sec:GE"}Gradient Extremals Gradient extremals are a type of curve that connects fixed points of a dynamical system. We will assume, because of our motivating examples, that the dynamical system is a gradient system, i.e., the vector field $X(x) = -DE(x)$ for some twice-differentiable potential $E$. This assumption can be relaxed to arbitrary vector fields by working with the squared magnitude of the vector field. [@lucia2002] An example of this can be found towards the end of Section [\[sec:grad-ex-euc\]](#sec:grad-ex-euc){reference-type="ref" reference="sec:grad-ex-euc"}. Gradient extremals can be thought of as following the "valleys, ridges, cirques, or cliffs" of the potential $E$. [@hoffman1986] ## [\[sec:grad-ex-euc\]]{#sec:grad-ex-euc label="sec:grad-ex-euc"}Gradient Extremals in Euclidean Space Gradient extremal curves [@basilevsky1982; @hoffman1986; @quapp1989; @lucia2002] are defined as the locus of points that minimize (or maximize) the gradient norm along each level set of the potential $E$ $$\label{eq:lucia_optimization} \underset{x\in \mathbb{R}^n}{\mathop{\mathrm{arg\,min}}} (\max) \{ \| D E(x) \|^2 : E(x)=L\}.$$ The points on the level set highlighted in Figure [\[fig:schematic\]](#fig:schematic){reference-type="ref" reference="fig:schematic"} {#fig:subd width="0.55\\linewidth"} [\[fig:subd\]]{#fig:subd label="fig:subd"} {width="0.35\\linewidth"} that minimize (respectively, maximize) the gradient are the farthest (respectively, closest) from the fixed point. Equivalently, one may formulate gradient extremals as the locus of points where the gradient of the potential is an eigenvector of the Hessian $D^2 E(x)$, $$\label{eq:classical_ge} D^2 E(x) DE(x) = \lambda DE(x),$$ along each level set $E(x)=L$. Here $\lambda$ is the associated eigenvalue. One can verify that both methods are equivalent by applying the method of Lagrange multipliers on [\[eq:lucia_optimization\]](#eq:lucia_optimization){reference-type="eqref" reference="eq:lucia_optimization"}. Gradient extremals can be followed through continuation methods [@continuation2003] where the level set value, $L$, acts as the parameter of interest and [\[eq:lucia_optimization\]](#eq:lucia_optimization){reference-type="eqref" reference="eq:lucia_optimization"} is optimized for each $L$. [@lucia2004] Note that a specific gradient extremal curve may not traverse all fixed points---it might instead connect just a subset of them. [@hirschquapp2004] See Figure [2](#fig:GE_MB){reference-type="ref" reference="fig:GE_MB"} ![Gradient extremals (following the eigenvector corresponding to the smallest eigenvalue of the Hessian of the potential) on the Müller-Brown potential, which is defined in [\[eq:muller-brown\]](#eq:muller-brown){reference-type="eqref" reference="eq:muller-brown"}. Mimima are depicted by yellow circles; saddle points by squares. Note that the top minimum is not directly connected to the bottom two. They are instead connected through a long detour [@ohno2004] that extends beyond the boundaries of the figure.](Figures/Euclidean_Space/GE_MB.png){#fig:GE_MB width="1\\linewidth"} for an example of gradient extremals on the Müller-Brown potential (defined in  [\[eq:muller-brown\]](#eq:muller-brown){reference-type="eqref" reference="eq:muller-brown"}). We also note that a gradient extremal can exhibit turning points (or "meander"), as seen in Figure [5](#fig:yannik){reference-type="ref" reference="fig:yannik"}, ![The gradient extremal following the eigenvector corresponding to the smallest eigenvalue of the Hessian of [\[eq:yannik\]](#eq:yannik){reference-type="eqref" reference="eq:yannik"}. Note how the gradient extremal "meanders," or exhibits turning points.](Figures/Euclidean_Space/Yannik.png "fig:"){#fig:yannik width="0.9\\linewidth"} [\[fig:sube\]]{#fig:sube label="fig:sube"} ![The gradient extremal following the eigenvector corresponding to the smallest eigenvalue of the Hessian of [\[eq:yannik\]](#eq:yannik){reference-type="eqref" reference="eq:yannik"}. Note how the gradient extremal "meanders," or exhibits turning points.](Figures/Euclidean_Space/Yannik_3D_ZoomD.png){#fig:yannik width="0.9\\linewidth"} which depicts a gradient extremal on the potential defined by $$\begin{aligned} \label{eq:yannik} E&(x^1, x^2)= \nonumber \\ &-\sum_{i = 1}^4 A_i \exp\left(a_i (x_i^1-x^1_{0i})^2 + b_i(x^2_i-x^2_{0i})^2 + c_i(x^1_i x^2_i)\right)\end{aligned}$$ with the coefficients listed in Table [1](#tab:Yannik){reference-type="ref" reference="tab:Yannik"}. The superscripts here, and throughout the paper, *refer to the component indices* (Einstein notation) unless otherwise noted. $i$ $A_i$ $a_i$ $b_i$ $c_i$ $x^1_{0i}$ $x^2_{0i}$ ----- -------- ------------------- ------------------- ------------------- ------------ ------------ 1 10 $\frac{-1}{3000}$ $\frac{-1}{1000}$ 0 0 0 2 $-0.4$ $\frac{-1}{300}$ $\frac{-1}{30}$ $\frac{1}{200}$ 15 10 3 0.8 $\frac{-1}{1500}$ $\frac{-1}{1500}$ $\frac{1}{2000}$ 25 100 4 6 $\frac{-1}{3000}$ $\frac{-1}{3000}$ $\frac{3}{20000}$ 40 30 : [\[tab:Yannik\]]{#tab:Yannik label="tab:Yannik"}Coefficients of the potential defined by [\[eq:yannik\]](#eq:yannik){reference-type="eqref" reference="eq:yannik"}. We previously assumed that the dynamical system is of gradient type. However, this assumption can be removed by replacing the potential by the squared magnitude of any given vector field $X$. [@lucia2002] The gradient extremal then becomes the locus of points that solve the constrained optimization problem $$\label{eq:lucia_optimization_general} \underset{x \in \mathbb{R}^n}{\mathop{\mathrm{arg\,min}}}(\max) \ \{ \lVert D (X^TX) \rVert^2: X^TX=L\}.$$ Notice that $E$ in [\[eq:lucia_optimization\]](#eq:lucia_optimization){reference-type="eqref" reference="eq:lucia_optimization"} is replaced by the squared magnitude $X^TX$. We illustrate this technique with the vector field describing a first order, exothermic, irreversible reaction $A\rightarrow B$ taking place in a continuous-stirred tank reactor (CSTR). The governing mass and energy balances are $$\begin{aligned} \label{eq:CSTR} \frac{dx^1}{dt} &= -x^1 + D_a(1-x^1)\exp{\big(x^2\big)} \nonumber \\ \frac{dx^2}{dt} &= -x^2+BD_a(1-x^1)\exp{\big(x^2\big)}-\beta(x^2-x^2_{c})\end{aligned}$$ where $x^1$ is the conversion of species $A$ and $x^2$ is dimensionless temperature as detailed in Ref. . (Please remember that the superscripts in $x^1$ and $x^2$ are indices, not powers, in Einstein notation.) With parameters set to $D_a = 0.085$, $B=22$, $\beta=3$, $x^2_{c}=-0.04$, the dynamical system exhibits two unstable steady states, one stable steady state, and one stable limit cycle. Figure [6](#fig:CSTR){reference-type="ref" reference="fig:CSTR"} ![[\[fig:CSTR\]]{#fig:CSTR label="fig:CSTR"}Gradient extremal (orange curve) on the graph of the squared magnitude, $X^TX$, for the CSTR. The steady states are shown in yellow.](Figures/Euclidean_Space/CSTR_FixedLabel.png){#fig:CSTR width="0.9\\linewidth"} illustrates gradient extremals on the squared magnitude of the vector field [\[eq:CSTR\]](#eq:CSTR){reference-type="eqref" reference="eq:CSTR"}. ## [\[sec:GE_Riemmanian\]]{#sec:GE_Riemmanian label="sec:GE_Riemmanian"}Gradient Extremals on Riemannian Manifolds We now turn our attention to resolving gradient extremals for vector fields defined on a manifold. Recall that the purpose of the proposed algorithm is to connect fixed points (hopefully a local minimum to a kinetically relevant nearby saddle): if one could sample and map a chart (defined as a small portion of the manifold represented with reduced coordinates) that includes both minimum and saddle point, one could directly solve for a gradient extremal by treating the chart as a Euclidean space. The resulting path would not be a true gradient extremal, but would still properly link the two critical points, which was our original intent. However, chances are that multiple local charts will be needed, whether because (a) the curvature of the manifold makes the projection onto a single chart impossible or (b) a single chart does not contain both the initial and final fixed points. In the former case, increased curvature makes an accurate, invertible mapping between manifold and local chart difficult: choosing multiple smaller areas to work with alleviates this issue. In the latter case, unless both connected fixed points lie on the chart, a gradient extremal would terminate at the boundary of the chart without connecting to the next fixed point of interest. To ensure the path transitions properly from one chart to the next, we use differential geometry to resolve the gradient extremal in the Riemannian setting of a $d$-dimensional smooth manifold, $M \subset \mathbb{R}^n$, with Riemannian metric $g$. The Riemannian metric $g$ defines the notions of distances and angles on the manifold by assigning inner products on the tangent spaces of the manifold. Let $E : M \to \mathbb{R}$ be the potential energy function on the manifold. We create a mapping $\varphi: U \subset M \to \mathbb{R}^d$ that defines a system of coordinates of an open subset $U \subset M$ on a local chart in $\mathbb{R}^d$. Let $\psi : \mathbb{R}^d \to U$, $u \mapsto \psi(u)$, be the inverse map of the chart $\varphi$. We define a Hessian matrix in terms of the covariant derivative on $(M, g)$ with respect to the Levi-Civita connection. A connection describes how a vector changes as it moves from one point on a manifold to the next. We use the following notation when working on $(M,g)$, with $Z=\psi^{\star} E: \mathbb{R}^d \to \mathbb{R}$ now representing the pullback energy: $$Z=\psi^{\star} E = E\circ \psi. %\ \ u \in \mathbb{R}^d.$$ Thus, the Euclidean formulation of gradient extremals [\[eq:classical_ge\]](#eq:classical_ge){reference-type="eqref" reference="eq:classical_ge"} now becomes, on a local chart, $$\label{eq:hessian-eigenproblem} \mathop{\mathrm{Hess}}Z \, \mathop{\mathrm{grad}}Z = \lambda \mathop{\mathrm{grad}}Z.$$ First, we define our Riemannian metric, $g$, as $$\label{eq:metric} g = \sum_{i, j = 1}^d g_{ij} \, \text{d}\varphi^i \otimes \text{d} \varphi^j,$$ where $g_{ij} = (D \psi^\top D \psi)_{ij}$ for $i, j = 1, \dotsc, d$. This metric allows us to define an inner product between two tangent vectors $S$ and $T$ defined in local coordinates: $g(S,T) = \sum^d_{i,j=1} g_{ij}S^iT^i$. We can now compute the gradient and Hessian on the manifold. Each component of the gradient becomes $$\mathop{\mathrm{grad}}(Z)^i = \sum^d_{j=1} g^{ij}\frac{\partial Z}{\partial \varphi^j},$$ where $g^{ij}$ refers to element $(i,j)$ of the inverse metric tensor. The Hessian is defined in terms of the covariant derivative: $$\mathop{\mathrm{Hess}}(Z)T = \nabla_{T} \mathop{\mathrm{grad}}({Z})$$ where the covariant derivative, which allows us to specify the parallel transport along tangent vectors of a manifold, is described as $$\label{eq:riemannian-parallel-transport} \nabla_{T} Y = \sum_{k = 1}^d \left\{ \sum_{i = 1}^d \left( \frac{\partial Y^k}{\partial \varphi^i} + \sum_{j = 1}^d \Gamma_{ij}^k Y^j \right) {T^i} \right\} \frac{\partial}{\partial\varphi^k}.$$ The above expression constitutes the covariant derivative of $Y$ in the direction of $T$. The coefficients $\Gamma_{ij}^k$ act as a correction to otherwise straight trajectories on the tangent plane so that they remain on the manifold. These coefficients are unique to a given affine connection. Here we use the Levi-Civita connection, the unique symmetric connection compatible with the metric, for which the coefficients, known as Christoffel symbols, are given by $$\Gamma^\ell_{jk} = \frac{1}{2}\sum_{i = 1}^d g^{\ell i} \left( \frac{\partial g_{ij}}{\partial \varphi^k} + \frac{\partial g_{ik}}{\partial \varphi^j} - \frac{\partial g_{jk}}{\partial \varphi^i} \right).$$ Beginning with an initial point $p_0$, its energy level $L = E(p_0)$, and the smallest (or largest) eigenvalue, $\lambda$, of the Hessian of the pullback energy evaluated at the initial point, we can now use continuation to solve the following system of $d+1$ equations: $$\left\{ \begin{aligned} &Z - L = 0 \\ &\mathop{\mathrm{Hess}}(Z) \mathop{\mathrm{grad}}(Z) - \lambda\ \mathop{\mathrm{grad}}(Z) = 0 \end{aligned} \right. \label{eq:continuation}$$ Note that naively using the Euclidean formulation of gradient extremals as defined in [\[eq:classical_ge\]](#eq:classical_ge){reference-type="eqref" reference="eq:classical_ge"} on the local chart would be erroneous because it fails to take into account the geometry of the manifold. Indeed, we would obtain curves that would not, in general, coincide with the true gradient extremals. This is akin to how a straight line joining New York and Madrid drawn on a (Mercator) map is not actually a "straight-line" on the globe (*i.e.,* a segment of a great circle ---also known as a geodesic). **Example 1**. *We illustrate a gradient extremal resolved on a smooth Riemannian manifold defined by the unit sphere $\mathbb{S}^2 = \{ (x,y,z) \in \mathbb{R}^3 \mid x^2+y^2+z^2=1\}$ with potential energy $E(x,y,z)=xyz$ in Figure [9](#fig:Zorro){reference-type="ref" reference="fig:Zorro"}. We define the mapping $\varphi$ as the stereographic projection from the North pole to the tangent plane at the South pole such that $\varphi(x,y,z) = \left(\frac{x}{1-z}, \frac{y}{1-z}\right) \equiv (u,v)$ and its inverse as $\psi(u, v) = \frac{1}{u^2+v^2+1}\left(2u,2v,u^2+v^2-1\right)$, and the pullback energy as $Z(u,v) = \frac{4uv\left(u^2+v^2-1\right)}{\left(u^2+v^2+1\right)^3}$. All fixed points except the North pole are mapped onto a single chart; in our actual numerical scheme, we relax this assumption to allow for sampling of only a small portion of the manifold at a time. We refer the reader to Ref.  for a full derivation of the gradient and Hessian used in this example. [\[ex:zorro\]]{#ex:zorro label="ex:zorro"}* ![**(a)** Two gradient extremal curves shown on the sphere $\mathbb{S}^2 = \{ (x,y,z) \in \mathbb{R}^3 \mid x^2+y^2+z^2=1\}$ with the potential $E(x,y,z)=xyz$. **(b)** The stereographic projection from the North Pole to the tangent plane at the South pole is shown along with the associated pullback potential energy $\psi^\star E$. Full calculation detailed in Example [\[ex:zorro\]](#ex:zorro){reference-type="ref" reference="ex:zorro"}.](Figures/Zorro_Sphere_Example/3D_Zorro.png "fig:"){#fig:Zorro width="0.6\\linewidth"} [\[fig:subf\]]{#fig:subf label="fig:subf"} ![**(a)** Two gradient extremal curves shown on the sphere $\mathbb{S}^2 = \{ (x,y,z) \in \mathbb{R}^3 \mid x^2+y^2+z^2=1\}$ with the potential $E(x,y,z)=xyz$. **(b)** The stereographic projection from the North Pole to the tangent plane at the South pole is shown along with the associated pullback potential energy $\psi^\star E$. Full calculation detailed in Example [\[ex:zorro\]](#ex:zorro){reference-type="ref" reference="ex:zorro"}.](Figures/Zorro_Sphere_Example/2D_Zorro.png){#fig:Zorro width="0.9\\linewidth"} As we did in Section [\[sec:grad-ex-euc\]](#sec:grad-ex-euc){reference-type="ref" reference="sec:grad-ex-euc"}, we can relax the assumption that $E$ corresponds to the potential energy of a gradient vector field if we work instead with the squared magnitude of a general vector field $X$ on the manifold. If we let $\ell(u)$ be the pullback of the squared magnitude evaluated at point $u$, then the counterpart to [\[eq:lucia_optimization_general\]](#eq:lucia_optimization_general){reference-type="eqref" reference="eq:lucia_optimization_general"} on a Riemannian manifold $(M, g)$ becomes $$\mathop{\mathrm{arg\,min}}_{u \in \ell^{-1}(\{ L \})}(\max) \left\{ g_u(\mathop{\mathrm{grad}}\ell(u), \mathop{\mathrm{grad}}\ell(u)) \right\},$$ where $$\begin{aligned} \ell \colon M &\to \mathbb{R}_{\ge 0} \\ u & \mapsto \ell(u) = g_u(\psi^{\star}X, \psi^{\star}X) = \psi^{\star}(X^TX).\end{aligned}$$ Here, $\mathop{\mathrm{grad}}$ is again the Riemannian gradient and $\ell(u)$ corresponds to the pullback of the term $X^\top X$ in the Euclidean case. As an example of a non-gradient system, let us study the van der Pol oscillator on a Riemannian manifold. First, let $\mathbb{R}^2$ be the Euclidean plane with coordinates $(x^1, x^2)$ and the usual Euclidean metric. The van der Pol oscillator is given by the vector field $X$: $$\begin{aligned} \frac{dx^1}{dt} &= x^2 \nonumber \\ \frac{dx^2}{dt} &= \mu(1-(x^1)^2)x^2 -x^1 \nonumber\end{aligned}$$ for some constant $\mu \in \mathbb{R}$. Next, the Poincaré disk model of the hyperbolic plane is the Riemannian manifold given by $$\mathbb{H}^2 = \left\{ (y^1, y^2) \in \mathbb{R}^2 \, : \, (y^1)^2 + (y^2)^2 < 1 \right\}$$ and the metric $$g = \frac{4 (\mathrm{d}y^1 \otimes \mathrm{d} y^1 + \mathrm{d}y^2 \otimes \mathrm{d} y^2)}{\left( 1 - (y^1)^2 - (y^2)^2 \right)}.$$ We define the van der Pol vector field $X$ on $\mathbb{H}^2$ simply by restricting the Euclidean vector field to the subset $\mathbb{H}^2$. In other words, we retain only the vector field $X$ as defined in the disk $\mathbb{H}^2$ and this yields, in the new hyperbolic metric $g$ above, *a different squared length function* than the Euclidean one. Figure [12](#fig:squared-length-function){reference-type="ref" reference="fig:squared-length-function"} shows the difference between the squared length function of the van der Pol oscillator $X$ in $\mathbb{R}^2$ and its restriction to $\mathbb{H}^2$. {#fig:squared-length-function width="0.85\\linewidth"} [\[fig:suba\]]{#fig:suba label="fig:suba"} {#fig:squared-length-function width="0.9\\linewidth"} # [\[sec:Algorithm\]]{#sec:Algorithm label="sec:Algorithm"}Algorithm We consider a high-dimensional dynamical system $X(x) \in \mathbb{R}^n$ whose trajectories become quickly attracted to a slow $d$-dimensional manifold $M \subset \mathbb{R}^n \ (n \gg d)$. Suppose we have an initial point $p_0 \in M$ near a minimum, obtained by running a single long-enough gradient descent trajectory, such that the fast dynamics have subsided, and movement on the manifold dominates. Next, we draw $N$ samples in a neighborhood $U \subset M$ around $p_0$. Any sampling technique appropriate to the application can be used; in the subsequent example, we randomly perturb $p_0$ in ambient space and run the $N$ trajectories for a short time $\tau$, such that they now effectively lie on the manifold. Another option would be to use umbrella sampling [@Torrie1977; @fiorin2013] based on the reduced local coordinates.[@Bello-Rivas2023] Next, we apply a dimensionality reduction technique such as diffusion maps [@COIFMAN2006] to uncover a set of reduced, latent coordinates that describe the manifold in the neighborhood of the initial point. It is important to emphasize that we use *global* manifold learning techniques on a neighborhood on the manifold: as a result, we obtain a good set of coordinates describing the single, *local* chart. A different set of collective variables may be considered for each subsequent chart, which could be better suited for the associated local neighborhood (which could correspond to a different operating regime, stage, conformation, etc. of the system). We then use a Gaussian process to learn the local mappings between the ambient space and discovered chart $\varphi: U \rightarrow \mathbb{R}^d$ and its inverse $\psi: \mathbb{R}^d \rightarrow U$, as well as a mapping of the reduced coordinates to the potential energy, the pullback energy $Z: \mathbb{R}^d \rightarrow \mathbb{R}$. In the simple examples of this paper, the potential function in ambient space is given. In more complex applications (e.g. effective descriptions of molecular dynamics simulations in terms of collective variables), where this potential function is not explicitly known, some additional local computations (e.g. through umbrella sampling) would be required to estimate its gradient (mean force) at each step. Further, in the case of a high-dimensional ambient space (of higher dimensions than the examples shown here), a Gaussian process may inadequately lift from latent to ambient space. Rather than fitting a function for $\psi$ in these more complex applications, Ref.  presents instead a method using biased sampling by simulating a stochastic differential equation to estimate $\psi(u)$. After learning the mappings between the manifold and the local chart, we can now compute the gradient extremal in the Riemannian setting as explained in Section [\[sec:GE_Riemmanian\]](#sec:GE_Riemmanian){reference-type="ref" reference="sec:GE_Riemmanian"}. Here, we use a pseudo-arclength continuation method [@continuation2003] with adaptive step sizing, stopping when an equilibrium point or chart boundary is reached. Detection of a fixed point can be done by monitoring the norm of the vector field (either using the vector field itself in ambient space, or by inspecting the gradient of the pullback energy on the chart). If using a Gaussian Process to obtain the diffeomorphisms, detection of a chart boundary can be accomplished by monitoring that the norm of the covariance matrix at a given point remains below a chosen threshold. If a boundary is reached, it is necessary to switch charts --- that is, to resample around the point at the boundary of the previous chart and relearn the mappings and pullback energy function. Care has to be taken to continue the curve in the same direction between local charts, potentially oriented in opposing directions; this can be accomplished by mapping the last few ambient points of the curve onto the new local chart and ensuring continuation occurs in the same direction (e.g. the angle between the previous curve mapped onto the new chart and the continuation direction is acute). The full methodology is summarized in Algorithm [\[alg:ge\]](#alg:ge){reference-type="ref" reference="alg:ge"}. initial point $p=p_0$ near minimum, samples per iteration $N$, continuation parameters, threshold $\rho > 0$ used in the convergence criterion. equilibrium $p_\star$ of $X$. Sample $N$ points from a neighborhood $U \subset M$ of $p_{n\text{-}1}$. Use manifold learning to obtain local coordinates $\varphi \colon U \rightarrow \mathbb{R}^d$ Obtain a parameterization $\psi \colon V \rightarrow U$, where $V = \varphi(U) \subseteq \mathbb{R}^d$. Approximate $Z = \psi^\star E$ via Gaussian process regression. $q_n, v_n \leftarrow \textsc{resolve\_extremal\_curve}(q, v)$ $p_n, w_n \leftarrow \psi(q_n), \psi_\star(q_n) v_n$ $p_\star \leftarrow p_n$ Initial point $q_0 \in \mathbb{R}^d$ and velocity $v_0 \in \mathbb{R}^d$. Final point $q$ and velocity $v$ of the extremal curve $\gamma$. Construct numerical continuation curve $\gamma$ for $$\left\{ \begin{aligned} & \mathop{\mathrm{Hess}}Z\mathop{\mathrm{grad}}Z = \lambda \mathop{\mathrm{grad}}Z \\ & Z(\gamma(t)) = L \end{aligned} \right.$$ such that $\gamma(0) = q_0$ and $\dot{\gamma}(0) = v_0$. $L$ is the value of the level set at the current time step. Prolong $\gamma$ until the first instant $t > 0$ such that $\gamma(t) \in V$ but $\gamma(t + s) \not\in V$ for any $s > 0$. $\gamma(t)$, $\dot{\gamma}(t)$ While we believe our algorithm is as parsimonious as possible (i.e. starting from a single minimum, using a set of reduced collective variables discovered on-the-fly, and efficiently exploring in a single direction), it is not without limitations. Here, we detail those nuances and limitations: Exploration of all directions : Given a particular minimum, there exist multiple different directions along which one can initialize a gradient extremal curve. For example, from a minimum on a 2D manifold, one could follow both sides (positive and negative) of both eigenvectors of the Hessian. Algorithm [\[alg:ge\]](#alg:ge){reference-type="ref" reference="alg:ge"} assumes that we always follow a single, pre-chosen direction. A bookkeeping strategy would be required to systematically explore all gradient extremals of a dynamical system. The Global Terrain Algorithm [@lucia2002] and the Navigation Algorithm [@Filippidis2013] utilize a graph structure to conduct such bookkeeping. These techniques could be integrated into Algorithm [\[alg:ge\]](#alg:ge){reference-type="ref" reference="alg:ge"} to provide an orderly exploration of the fixed points of the dynamical system. Disconnected minima : Unfortunately, a single gradient extremal curve is not guaranteed to connect all fixed points of a potential energy surface. Two minima may occasionally lie on separate gradient extremal curves [@hirschquapp2004] or may be connected by a very long detour. [@ohno2004] This is the case in the Müller-Brown potential, where the topmost minimum follows a long detour to the bottom two (see Figure [2](#fig:GE_MB){reference-type="ref" reference="fig:GE_MB"}). An alternative to following the long detour would be to perform a steepest descent trajectory from both sides of the newly located leftmost saddle. Discovery of all saddle points given a single starting minimum is not guaranteed. Continuation parameters : Our implementation hinges on a predictor-corrector scheme for numerical continuation. Care must be taken to set reasonable tolerance values and step size maxima. This is especially true near fixed points, which give rise to numerical difficulties. Continuation sometimes becomes numerically challenging, e.g. in the neighborhood of a new fixed point in the current chart, because the potential energy surface is locally flat there. To locate the fixed point once we approach its neighborhood, Ref.  suggests switching to an equation-solving technique such as quadratic acceleration or conjugate gradients to accurately locate the saddle. These techniques can be employed using the learned mappings in the chart of interest. Sampling and manifold learning : An adequate approximation of the manifold is critical. Poor sampling or failed manifold learning will undermine the success of the algorithm. The more curvature a manifold exhibits, the more challenging it is to create an accurate, invertible mapping between manifold and local chart. Working instead with multiple, smaller charts will alleviate this issue. # Comparison of Potential Curves Connecting Fixed Points The briefly aforementioned paths (gradient extremals, Newton trajectories, gentlest ascent dynamics, and that of the nudged elastic band or string variants) are all examples of paths that connect a minimum to a saddle. Sometimes, they coincide with reaction paths: continuous curves that monotonically ascend towards a saddle and then monotonically decrease to the next minimum. [@bofill2020] (While we use the definition of the reaction path as described in Ref. , we note that there is disagreement in the literature surrounding the formal definition of a reaction path. For example, Ref.  uses the term reaction path as a synonym for a curve in which all points of the reaction path sit in a valley. Further complicating matters, the path described by Ref.  is sometimes referred to as the minimum energy path. We do not use this definition for a reaction path nor a minimum energy path in this work.) Curves that display turning points in $L$, and thus switch between increasing and decreasing in energy (i.e. curves that "meander") are not reaction paths. Here, we briefly compare the paths, which differ in mathematical formulation and how they ascend to the saddle. Our comparison is done in Euclidean space. The path of the nudged elastic band (NEB) [@Jonsson1998] and string variants, [@Weinan2002; @behn2011; @chaffey2012] converge to the steepest descent path from each side of a saddle point. [@sheppard2011] This curve is often---but not always---referred to in literature as the minimum energy path (MEP) as it marks the "path of least resistance" or most probable reaction path between a chemical reactant and product. [@dunitz1975] (An alternative definition for the MEP is a reaction path in which all points sit in a valley. This alternative definition has been well studied by Quapp and Bofill in Refs. . We do not use this definition in this work.) The tangent of this curve is parallel to the gradient except at fixed points. The discovery of the MEP using NEB requires an estimated path initialization and knowledge of two minimum, [@palenik2021; @halgren1977] which prevent it from being used with our algorithmic framework. Nevertheless, we present it here for comparison due to its significance. A Newton trajectory (NT), also known as an isocline or reduced gradient following curve, is a curve in which the normalized vector field at each point along the curve is equal to a chosen, fixed vector. [@quapp1998; @hirsch2004] Unlike the path of the NEB, it carries no physical interpretation. A gradient extremal, as previously described, is a curve comprised of points in which the gradient is always an eigenvector of the Hessian of the potential. A gradient extremal does, however, bear physical significance as it usually follows a valley floor or ridge. Gentlest ascent dynamics forms a new dynamical system that is designed to search for and follow the eigenvector corresponding to the smallest eigenvalue of the Hessian of the potential such that a trajectory ascends towards a saddle point in the least steep manner possible. [@Weinan2010] Note that this trajectory is not guaranteed to converge to a saddle. A GAD curve is different than a gradient extremal, due to the presence of the ascent term unless the gradient extremal is confluent to an MEP. Further comparisons of gradient extremals, GAD, and NTs can be found in Refs. . Figure [16](#fig:Euc_Comparison){reference-type="ref" reference="fig:Euc_Comparison"} ![**(a)** Gradient extremal, the path of NEB (calculated via the String method), GAD (initialized with a velocity equivalent to the smallest eigenvector of the Hessian of the potential), and NT (with a horizontal slope) between the rightmost minimum and nearby saddle on the Müller-Brown potential, which is further defined in [\[eq:muller-brown\]](#eq:muller-brown){reference-type="eqref" reference="eq:muller-brown"}. **(b)** The four curves starting at the bottom minimum of the potential defined by [\[eq:yannik\]](#eq:yannik){reference-type="eqref" reference="eq:yannik"}. Circles mark minima; squares mark saddles. All four curves are unique.](Figures/Euclidean_Space/MB_All.png "fig:"){#fig:Euc_Comparison width="1\\linewidth"} [\[fig:euc_comp_mb\]]{#fig:euc_comp_mb label="fig:euc_comp_mb"} ![**(a)** Gradient extremal, the path of NEB (calculated via the String method), GAD (initialized with a velocity equivalent to the smallest eigenvector of the Hessian of the potential), and NT (with a horizontal slope) between the rightmost minimum and nearby saddle on the Müller-Brown potential, which is further defined in [\[eq:muller-brown\]](#eq:muller-brown){reference-type="eqref" reference="eq:muller-brown"}. **(b)** The four curves starting at the bottom minimum of the potential defined by [\[eq:yannik\]](#eq:yannik){reference-type="eqref" reference="eq:yannik"}. Circles mark minima; squares mark saddles. All four curves are unique.](Figures/Euclidean_Space/Yannik_All.png "fig:"){#fig:Euc_Comparison width="1\\linewidth"} [\[fig:euc_comp_mb1\]]{#fig:euc_comp_mb1 label="fig:euc_comp_mb1"} illustrates the four curves on the Müller-Brown potential and [\[eq:yannik\]](#eq:yannik){reference-type="eqref" reference="eq:yannik"}. We emphasize that the different techniques do produce four distinct curves, and that the presence of "meandering" depends on the potential energy surface.[^1] The gradient extremal curve is the most direct route (aside from that of the string method) to the saddle in the Müller-Brown potential, but is the most indirect on [\[eq:yannik\]](#eq:yannik){reference-type="eqref" reference="eq:yannik"} due to the presence of turning points. We have previously demonstrated the use of Newton trajectories and GAD within the algorithmic framework presented here to locate saddle points on Riemannian manifolds adaptively revealed in Refs.  respectively. The selection of which path is optimal for a given problem hinges on the shape of the potential energy surface, which, in the context of the algorithm, is likely unknown *a priori*. # Numerical Example We demonstrate the algorithm on the 2D Müller-Brown potential surface $$\begin{aligned} \label{eq:muller-brown} U(w^1, w^2)=\sum_{i = 1}^4 A_i \, \exp&\left( a_i (w^1 - w^1_{0i})^2 \right. \nonumber \\ &\left. + b_i (w^1 - w^1_{0i}) (w^2 - w^2_{0i}) \right. \nonumber \\ &\left. + c_i (w^2 - w^2_{0i})^2\right)\end{aligned}$$ mapped onto the unit sphere using the transformation $$V(x,y,z) = (U \circ \kappa)(\arctan{(y/x)}, \arctan{(z/\sqrt{x^2+y^2}})$$ where $\kappa(k^1, k^2)=(1.973521294k^1-1.85, 1.750704373k^2+0.875)$ is an affine mapping. The constants in [\[eq:muller-brown\]](#eq:muller-brown){reference-type="eqref" reference="eq:muller-brown"} are listed in Table [2](#tab:mb){reference-type="ref" reference="tab:mb"}. $i$ $A_i$ $a_i$ $b_i$ $c_i$ $y^1_{0i}$ $y^2_{0i}$ ----- -------- -------- ------- -------- ------------ ------------ 1 $-200$ $-1$ 0 $-10$ 1 0 2 $-100$ $-1$ 0 $-10$ 0 0.5 3 $-170$ $-6.5$ 11 $-6.5$ $-0.5$ $-1.5$ 4 15.0 0.7 0.6 0.7 $-1$ 1 : [\[tab:mb\]]{#tab:mb label="tab:mb"}Coefficients of the planar Müller-Brown potential. Beginning at the rightmost minimum, we sample 500 points in the neighborhood of an initial point as in Figure [20](#fig:Sampling){reference-type="ref" reference="fig:Sampling"}, {#fig:Sampling width="0.4\\linewidth"} {#fig:Sampling width="0.4\\linewidth"} {#fig:Sampling width="0.50\\linewidth"} {#fig:Sampling width="0.47\\linewidth"} learn a local chart using diffusion maps, and learn the mappings between ambient space and charts using Gaussian processes as explained in Section [\[sec:Algorithm\]](#sec:Algorithm){reference-type="ref" reference="sec:Algorithm"}. Figure [23](#fig:MB_step1){reference-type="ref" reference="fig:MB_step1"} {#fig:MB_step1 width="0.9\\linewidth"} [\[fig:sub1\]]{#fig:sub1 label="fig:sub1"} {#fig:MB_step1 width="0.9\\linewidth"} shows the second iteration of a gradient extremal in the learned latent space and its equivalence on the sphere in ambient space; Figure [26](#fig:MB_step5){reference-type="ref" reference="fig:MB_step5"} {#fig:MB_step5 width="0.9\\linewidth"} [\[fig:subc\]]{#fig:subc label="fig:subc"} {#fig:MB_step5 width="0.9\\linewidth"} shows the fifth. We reached the saddle point after iterating through a sequence of 11 successive charts; the full gradient extremal can be seen in Figure [28](#fig:MB_final){reference-type="ref" reference="fig:MB_final"}. {#fig:MB_final width="0.75\\linewidth"} {#fig:MB_final width="0.75\\linewidth"} This example and a generic implementation of our algorithm can be found at <https://github.com/tasiag/gradient-extremals-on-manifolds>. # Conclusion We have presented an algorithmic framework to locate saddle points (and other fixed points) of vector fields on manifolds gradually revealed by point clouds. The methodology assumes that a low-dimensional manifold, which can be uncovered on-the-fly by manifold learning techniques, quickly attracts trajectories of the high-dimensional dynamical system. Gradient extremals, curves that are known to connect fixed points, can then be followed on the manifold to the next point of interest. The technique does not require *a priori* knowledge of the manifold, nor of good collective variables; rather, it requires only an initial starting point close to a minimum and the ability to sample the dynamical system. The technique does not exhaustively search the whole space, but predicts a single new location to sample at, so that the simulator is efficiently and effectively biased towards the next point of interest. We demonstrate the technique on the Müller-Brown potential mapped to a sphere, without prior knowledge of the manifold. The technique is applicable both to (effective) gradient problems, where it helps discover transition states, as well as to general vector fields, where it helps connect isolated solution branches. We also compare how a gradient extremal curve compares to our prior work, which shares algorithmic elements but uses Newton trajectories [@Bello-Rivas2023] and GAD curves [@bellorivas2023gentlest] to locate saddle points. We expect the collection of the three techniques to act as a robust toolkit that can scale to systems of higher complexity. We thank Yannick De Decker (Université Libre de Bruxelles, Belgium) for providing [\[eq:yannik\]](#eq:yannik){reference-type="eqref" reference="eq:yannik"}. The work of A.S.G, J.M.B.-R., and I.G.K. has been partially supported by the US Air Force Office of Scientific Research (AFOSR MURI) and the US Department of Energy. The work of A.S.G. has been partially supported by T32CA153952 from NCI. The work of H.V. has been supported by grant 1179820N from the FWO (Belgium). # Data Availability Statement {#data-availability-statement .unnumbered} The data that support the findings of this study are openly available at the repository gradient-extremals-on-manifolds on Github, [DOI 10.5281/zenodo.8384160](https://zenodo.org/record/8384160). [^1]: See [@bofill2020] for a mathematical description of when NT, GAD, and GE exhibit turning points.

arxiv_math

**Uniform accuracy of implicit-explicit backward differentiation formulas (IMEX-BDF) for linear hyperbolic relaxation systems** Zhiting Ma [^1] Juntao Huang [^2] Wen-An Yong [^3] **Abstract** This work is concerned with the uniform accuracy of implicit-explicit backward differentiation formulas for general linear hyperbolic relaxation systems satisfying the structural stability condition proposed previously by the third author. We prove the uniform stability and accuracy of a class of IMEX-BDF schemes discretized spatially by a Fourier spectral method. The result reveals that the accuracy of the fully discretized schemes is independent of the relaxation time in all regimes. It is verified by numerical experiments on several applications to traffic flows, rarefied gas dynamics and kinetic theory. : # Introduction This paper is concerned with uniformly stable and accurate numerical methods for one-dimensional linear hyperbolic relaxation systems $$\label{eq:PDE-general} U_t + A U_{x} = \frac{1}{\varepsilon}Q U.$$ Here $U=U(x,t)\in\mathbb{R}^n$, $x \in\mathbb{R}$, $t\geq 0$, $A$ and $Q$ are $n\times n$ constant matrices, the subscripts $t$ and $x$ refer to the partial derivatives with respect to $t$ and $x$, and $\varepsilon > 0$ is a small parameter standing for the relaxation time. Such partial differential equations (PDEs) are the linearized version of first-order PDEs with relaxation. The latter models a large number of different irreversible phenomena. Important examples include kinetic theories (moment closure systems [@levermore1996moment; @Di2017nm], discrete-velocity kinetic models [@broadwell1964shock; @platkowski1988discrete]), nonlinear optics [@hanouzet2000approximation], radiation hydrodynamics [@pomraning2005equations; @mihalas2013foundations], traffic flows [@aw2000siam], dissipative relativistic fluid flows [@geroch1990dissipative], chemically reactive flows [@giovangigli2012multicomponent], and invisicid gas dynamics with relaxation [@zeng1999gas]. Due to the small parameter $\varepsilon$, usual numerical schemes are stable only if the time step is of order $O(\varepsilon)$. To overcome this drawback, the so-called implicit-explicit (IMEX) schemes were adapted [@ascher1995siam], where the convection part is treated explicitly and the source term is treated implicitly. The IMEX schemes include the IMEX Runge-Kutta method (IMEX-RK, e.g., [@ascher1997; @kennedy2003additive; @dimarco2013siam; @pareschi2005jsc]) and IMEX multistep method including IMEX backward differentiation formulas (IMEX-BDF, e.g., [@ascher1995siam; @hundsdorfe2007imex; @dimarco2017siam; @albi2020implicit-explicit]). As reported in [@boscarino2007error-analysis; @hu_uniform_2019], many IMEX-RK schemes suffer from accuracy degeneration when $\varepsilon$ goes to zero, while the numerical experiments indicate the uniform accuracy of certain IMEX multistep schemes for a wide range of $\varepsilon$ [@hundsdorfe2007imex; @dimarco2017siam; @albi2020implicit-explicit]. The aim of this work is to clarify the uniform accuracy of the multistep schemes for linear hyperbolic relaxation systems [\[eq:PDE-general\]](#eq:PDE-general){reference-type="eqref" reference="eq:PDE-general"}. For the Jin-Xin model [@jin1995cpam] as a specific relaxation system, the uniform stability and accuracy have been studied in [@hu_uniform_2019] for the IMEX-BDF schemes and in [@hu_uniform_2023] for the IMEX-RK schemes. Thus, our task is to generalize the analysis in [@hu_uniform_2019] for the Jin-Xin model to general hyperbolic relaxation systems satisfying the structural stability condition proposed in [@yong_singular_1999]. As shown in [@yong_singular_1999; @liu2001basic; @Yong2008], the structural stability condition are tacitly respected by many well-developed physical theories. Therefore, our analysis is expected to have a wide range of applications. Under the structural stability condition, we prove the uniform stability and accuracy of the fully discretized IMEX-BDF schemes up to fourth order. The spatial discretization is done by adopting a Fourier spectral method [@hesthaven2007spectral]. The proof invokes a multiplier technique developed in [@dahlquist1978G-stability; @nevanlinna1981multiplier]. Our results hold for any value of the small parameter $\varepsilon$. In other words, the accuracy of the schemes is independent of $\varepsilon$ in all regimes. We also present numerical tests to verify our theoretical results with several specific relaxation systems, including the linearized Aw-Rascle-Zhang traffic model [@aw2000siam; @zhang2002non], the Broadwell model [@broadwell1964shock], and a moment closure system [@grad1949kinetic; @cai2014cpam]. The rest of the paper is organized as follows. In Section [2](#sec:preliminaries){reference-type="ref" reference="sec:preliminaries"}, we introduce the structural stability condition and IMEX-BDF schemes for the relaxation systems [\[eq:PDE-general\]](#eq:PDE-general){reference-type="eqref" reference="eq:PDE-general"}. Section [3](#sec:2-constant){reference-type="ref" reference="sec:2-constant"} is devoted to our main results including uniform-in-$\varepsilon$ stability and accuracy of the IMEX-BDF schemes. Numerical experiments are presented in Section [4](#sec:numerical-tests){reference-type="ref" reference="sec:numerical-tests"} to validate our theoretical findings. # Preliminaries {#sec:preliminaries} In this section, we introduce the structural stability condition and a class of implicit-explict backward differentiation formulas (IMEX-BDF) for system [\[eq:PDE-general\]](#eq:PDE-general){reference-type="eqref" reference="eq:PDE-general"}. ## Structural Stability Condition {#subsec:ssc} The structural stability condition reads as (i) There is an invertible $n\times n$ matrix $P$ and an invertible $r\times r$ $(0<r\le n)$ matrix $\hat{S}$ such that $$\nonumber P Q = \left( \begin{array}{cc} 0 & 0 \\ 0 & \hat{S} \end{array} \right) P.$$ (ii) There exists a symmetric positive-definite (SPD) matrix $A_0$ such that $$\nonumber A_0 A = A^T A_0.$$ (iii) The hyperbolic part and the source term are coupled in the sense: $$\nonumber A_0 Q + Q^T A_0 \le -P^T \left( \begin{array}{cc} 0 & 0 \\ 0 & I_r \end{array} \right) P .$$ Here the superscript $T$ denotes the transpose and $I_r$ is the unit matrix of order $r$. About this set of conditions, we remark as follows. Condition (i) is classical for initial-value problems of systems of ordinary differential equations (ODE, spatially homogeneous systems), while (ii) means the symmetrizable hyperbolicity of the system of first-order partial differential equations (PDE) in [\[eq:PDE-general\]](#eq:PDE-general){reference-type="eqref" reference="eq:PDE-general"}. Condition (iii) characterizes a kind of coupling between the ODE and PDE parts. As shown in [@yong_singular_1999; @liu2001basic; @Yong2008], the structural stability condition has been tacitly respected by many well-developed physical theories. Recently, it is shown in [@Di2017nm; @zhao2017stability; @ma2023nonrelativisti] to be proper for certain moment closure systems. Under the structural stability condition, the existence and stability of the zero relaxation limit of the corresponding initial-value problems have been established in [@yong_singular_1999]. Assuming the structural stability condition, we introduce $\tilde{U} := PU$ and transform system [\[eq:PDE-general\]](#eq:PDE-general){reference-type="eqref" reference="eq:PDE-general"} into its equivalent version $$\nonumber \tilde{U}_t + \tilde{A} \tilde{U}_{x} = \frac{1}{\varepsilon} \left( \begin{array}{cc} 0 & 0 \\ 0 & \hat{S} \end{array} \right) \tilde{U},$$ where $\tilde{A} := P A P^{-1}$. It is easy to see that the above equivalent version satisfies the structural stability condition with $\tilde{P}=I$ and $\tilde{A}_0 = P^{-T}A_0P^{-1}$. Thus, throughout this paper we only consider the transformed version (drop the tilde) $$\label{eq:PDE} U_t + AU_{x} = \frac{1}{\varepsilon}\left( \begin{array}{cc} 0 & 0 \\ 0 & \hat{S} \end{array} \right) U \equiv \frac{1}{\varepsilon} Q U .$$ It was proved in [@yong_singular_1999] (Theorem 2.2) that $P^{-T}A_0P^{-1}$ is a block-diagonal matrix (with the same partition as in (i) and (iii)). Thus, the symmetrizer for [\[eq:PDE\]](#eq:PDE){reference-type="eqref" reference="eq:PDE"} has the following block-diagonal form $$\nonumber A_0 = \left( \begin{array}{cc} A_{01} & 0 \\ 0 & A_{02} \end{array} \right).$$ We further assume that $A_{02}\hat{S}$ is symmetric (negative-definite), which holds true for many physical models [@Yong2008]. ## IMEX-BDF schemes {#subsec:IMEX-BDF} Let $u^{n}=u^n(x)$ denote the numerical solution at time $t_n = T_0 + n\Delta t$, where $T_0$ is the initial time, $n$ is a non-negative integer, and $\Delta t$ is the time step. The $q$-th order IMEX-BDF scheme for system [\[eq:PDE\]](#eq:PDE){reference-type="eqref" reference="eq:PDE"} read as $$\label{eq:BDF-ODE} \sum_{i=0}^q \alpha_i u^{n+i} + \Delta t\sum_{i=0}^{q-1} \gamma_i A u_x^{n+i} = \beta \frac{\Delta t}{\varepsilon}Q u^{n+q}.$$ Here, $\alpha:=(\alpha_0,\dots,\alpha_q)$, $\gamma:=(\gamma_0,\dots,\gamma_{q-1})$ and $\beta > 0$ are constant to be determined by the requirement that [\[eq:BDF-ODE\]](#eq:BDF-ODE){reference-type="eqref" reference="eq:BDF-ODE"} is $q$-th order accurate [@hundsdorfe2007imex]. Examples are - $q=1$: $$u^{n+1} - u^n + \Delta tA u_x^n = \frac{\Delta t}{\varepsilon} Q u^{n+1},$$ - $q=2$: $$\nonumber u^{n+2} - \frac{4}{3}u^{n+1} + \frac{1}{3}u^n + \Delta t\left( \frac{4}{3}A u_x^{n+1} - \frac{2}{3} A u_x^n \right) = \frac{2}{3} \frac{\Delta t}{\varepsilon}Q u^{n+2},$$ - $q=3$: $$\nonumber u^{n+3} - \frac{18}{11} u^{n+2} + \frac{9}{11}u^{n+1} - \frac{2}{11}u^n + \Delta t\left(\frac{18}{11}A u_x^{n+2} - \frac{18}{11}A u_x^{n+1} + \frac{6}{11 } A u_x^n\right) = \frac{6}{11}\frac{\Delta t}{\varepsilon}Q u^{n+3}.$$ To analyze the IMEX-BDF scheme [\[eq:BDF-ODE\]](#eq:BDF-ODE){reference-type="eqref" reference="eq:BDF-ODE"}, we need the following multiplier technique established in [@akrivis2016backward; @hu_uniform_2019]. **Lemma 1**. *Given $\alpha= (\alpha_0,\dots,\alpha_q)$ and $\gamma=(\gamma_0,\dots,\gamma_{q-1})$ in [\[eq:BDF-ODE\]](#eq:BDF-ODE){reference-type="eqref" reference="eq:BDF-ODE"} with $q=1, 2, 3, 4$, there exist a positive-definite quadratic form $$G(u_1, \dots, u_q) = \sum_{i, j=1}^q g_{ij}u_i u_j,$$ a semi-positive-definite quadratic form $$A(u_1, \dots, u_{q-1}) = \sum_{i, j=1}^{q-1} a_{ij} u_i u_j,$$ two linear forms $L_1(u_1, \dots, u_{q-1})$, $L_2(u_1, \dots, u_{q})$ such that $$\nonumber \begin{aligned} &\left(u_q - L_1(u_1, \dots, u_{q-1}\right))\sum_{i=0}^q\alpha_i u_i \\ =& G(u_1, \dots, u_q) - G(u_0, \dots, u_{q-1}) + d_1\left(u_q - L_1(u_1, \dots, u_{q-1}) - d_2\sum_{i=0}^{q-1}\gamma_i u_i\right)^2 \end{aligned}$$ and $$\nonumber \left(u_q - L_1(u_1, \dots, u_{q-1})\right)u_q = A(u_2, \dots, u_{q}) - A(u_1, \dots, u_{q-1}) + L_2^2(u_1, \dots, u_{q}).$$ Here constants $d_1>0$ and all other constants are real.* The proof of this lemma can be found in [@hu_uniform_2019]. Here we list the quadratic forms, the linear forms, and the constants for $q=1, 2$. - $q=1$: $$\nonumber G(u_1) = \frac{1}{2}u_1^2, \quad d_1 = \frac{1}{2}, \quad d_2 = 1, \quad L_2(u_1) = u_1.$$ - $q=2$: $$\nonumber \begin{aligned} &G(u_1, u_2) = \frac{1}{6}u_1^2 - \frac{2}{3}u_1u_2 + \frac{5}{6}u_2^2, \quad A(u_1) = 0, \quad L_1(u_1) = 0,\\[4mm] &L_2(u_1, u_2) = u_2, \quad d_1 = \frac{1}{6}, \quad d_2 = \frac{3}{2}. \end{aligned}$$ For our purpose, we generalize Lemma [Lemma 1](#lem:multiplier-Hu){reference-type="ref" reference="lem:multiplier-Hu"} to the case where $u_j$ are vectors. To do this, we take a symmetric positive-definite (SPD) matrix $H$ and define a weighted inner-product for vectors $u,v\in\mathbb{R}^n$: $$\nonumber (u, v)_{H} := u^T H v$$ and norm $$\nonumber \left\Vert u\right\Vert_{H} := \sqrt{(u,u)_H}.$$ When $H=I_n$, the subscript $H$ will be omitted. The generalized version of Lemma [Lemma 1](#lem:multiplier-Hu){reference-type="ref" reference="lem:multiplier-Hu"} is **Lemma 2**. *Let the coefficients $g_{ij}$, $a_{ij}$ of the quadratic forms $G(u_1, \dots, u_q)$ and $A(u_1, \dots, u_{q-1})$, $L_1(u_1, \dots, u_{q-1}), L_2(u_1, \dots, u_q)$ and $d_1, d_2$ be same as those in Lemma [Lemma 1](#lem:multiplier-Hu){reference-type="ref" reference="lem:multiplier-Hu"}. For $u_j \in \mathbb{R}^n (j=0, \cdots, q)$ with $q=1,2,3,4$, set $$\begin{aligned} G_H(u_1, \dots, u_q) = \sum_{i,j=1}^q g_{ij}(u_i, u_j)_H,\quad A_H(u_1, \dots, u_{q-1}) = \sum_{i,j=1}^{q-1} a_{ij}(u_i, u_j)_H. \end{aligned}$$ Then the following two equalities hold: $$\nonumber \begin{aligned} &\left(u_q - L_1(u_1, \dots, u_{q-1}), \sum_{i=0}^q\alpha_i u_i\right)_H \\ =& G_H(u_1, \dots, u_q) - G_H(u_0, \dots, u_{q-1}) + d_1 \left\Vert u_q - L_1(u_1, \dots, u_{q-1}) - d_2\sum_{i=0}^{q-1}\gamma_i u_i\right\Vert_H^2\\ \end{aligned}$$ and $$\nonumber \left(u_q - L_1(u_1, \dots, u_{q-1}), u_q\right)_H = A_H(u_2, \dots, u_{q}) - A_H(u_1, \dots, u_{q-1}) + \left\Vert L_2(u_1, \dots, u_q)\right\Vert_H^2.$$* *Proof.* It is well-known that for the given SPD matrix $H$, there exists a SPD matrix $M$ such that $H=M^2$. Then, for $u, v\in\mathbb{R}^n$ define $\tilde{u}=Mu$ and $\tilde{v}=Mv$. It holds that $$\nonumber (u, v)_H = (\tilde{u}, \tilde{v}), \quad \left\Vert u\right\Vert_H = \left\Vert\tilde{u}\right\Vert.$$ Thus the right-hand side of the first equality is equal to $$\nonumber \begin{aligned} \textrm{RHS} ={}& \sum_{i,j=1}^q g_{ij}(u_i, u_j)_H - \sum_{i,j=0}^{q-1} g_{ij}(u_i, u_j)_H + d_1 \left\Vert u_q - L_1(u_1, \dots, u_{q-1}) - d_2 \sum_{i=0}^{q-1}\gamma_i u_i\right\Vert_H^2\\ ={}& \sum_{i,j=1}^q g_{ij}(\tilde{u}_i, \tilde{u}_j) - \sum_{i,j=0}^{q-1} g_{ij}(\tilde{u}_i, \tilde{u}_j) + d_1 \left\Vert\tilde{u}_q - L_1(\tilde{u}_1, \dots, \tilde{u}_{q-1}) - d_2 \sum_{i=0}^{q-1}\gamma_i \tilde{u}_i\right\Vert^2 \\ ={}& \left(\tilde{u}_q - L_1(\tilde{u}_1, \dots, \tilde{u}_{q-1}), ~\sum_{i=0}^q\alpha_i \tilde{u}_i\right) \\ ={}& \left(u_q - L_1(u_1, \dots, u_{q-1}), ~\sum_{i=0}^q\alpha_i u_i\right)_H = \textrm{LHS}. \end{aligned}$$ Here the third equality follows from Lemma [Lemma 1](#lem:multiplier-Hu){reference-type="ref" reference="lem:multiplier-Hu"} for each component of the $n$-vectors. Similarly, the second equality can be shown. This completes the proof. ◻ # Uniform accuracy {#sec:2-constant} In this section, we consider system [\[eq:PDE\]](#eq:PDE){reference-type="eqref" reference="eq:PDE"} with periodic boundary conditions. As in [@hu_uniform_2019], we use the Fourier-Galerkin spectral method to the semi-discretized IMEX-BDF scheme [\[eq:BDF-ODE\]](#eq:BDF-ODE){reference-type="eqref" reference="eq:BDF-ODE"} in the spatial direction to obtain $$\label{scheme1} \sum_{i=0}^q\alpha_i U_N^{n+i} + \Delta tA \sum_{i=0}^{q-1}\gamma_i (U_N^{n+i})_x = \frac{\beta\Delta t}{\varepsilon} QU_N^{n+q}.$$ Here $U_N \in P_N := \textrm{span}\{ e^{ikx}| -N\le k\le N \}$ with $N$ being an integer. For $P_N$-functions $U_N$, the following inequality is known [@hesthaven2007spectral]: $$\label{equ:spectral} \left\Vert(U_N)_x\right\Vert^2 \le N^2 \left\Vert U_N\right\Vert^2.$$ Here the notation $\left\Vert\cdot\right\Vert$ denotes the usual $L^2$ norm of the square integrable periodic functions. ## Stability {#subsec:stability} Assume the structural stability condition and the symmetry of the matrix $A_{02}\hat{S}$. In this subsection, we analyze the uniform-in-$\varepsilon$ stability of the fully discretized scheme [\[scheme1\]](#scheme1){reference-type="eqref" reference="scheme1"}. The main idea of our analysis will be illustrated firstly with the first-order scheme. ### First-order scheme For $q=1$, scheme [\[scheme1\]](#scheme1){reference-type="eqref" reference="scheme1"} reads as $$\nonumber U_N^{n+1} - U_N^{n} + \Delta tA (U_N^n)_x = \frac{\Delta t}{\varepsilon} QU_N^{n+1}.$$ Multiplying this scheme with $(U_N^{n+1})^T A_0$ and integrating the resultant equality over $x$ gives $$\label{eq:stablity-imex1} \int(U_N^{n+1})^T A_0(U_N^{n+1} - U_N^{n}) + \Delta t\int (U_N^{n+1})^T A_0 A (U_N^n)_x = \frac{\Delta t}{\varepsilon} \int (U_N^{n+1})^T A_0 QU_N^{n+1} .$$ Since $A_0$ is symmetric, the first term on the LHS of [\[eq:stablity-imex1\]](#eq:stablity-imex1){reference-type="eqref" reference="eq:stablity-imex1"} can be decomposed as $$\nonumber \begin{aligned} & \int(U_N^{n+1})^T A_0(U_N^{n+1} - U_N^{n}) \\ ={}& \frac{1}{2}\int(U_N^{n+1})^T A_0 U_N^{n+1} - \frac{1}{2}\int(U_N^{n})^T A_0 U_N^{n} + \frac{1}{2}\int(U_N^{n+1}-U_N^{n})^T A_0 (U_N^{n+1}-U_N^{n}), \end{aligned}$$ while the second term is $$\nonumber \begin{aligned} & \Delta t\int (U_N^{n+1})^T A_0 A (U_N^n)_x \\ ={}& \Delta t\int (U_N^{n+1}-U_N^n)^T A_0 A (U_N^n)_x + \Delta t\int (U_N^{n})^T A_0 A (U_N^n)_x \\ ={}& \Delta t\int (U_N^{n+1}-U_N^n)^T A_0 A (U_N^n)_x + \frac{1}{2}\Delta t\int ((U_N^{n})^T A_0 A U_N^n)_x \\ ={}& \Delta t\int (U_N^{n+1}-U_N^n)^T A_0 A (U_N^n)_x . \end{aligned}$$ Here we have used the symmetry of $A_0A$ and the periodic boundary conditions. Thanks to the structural stability condition (iii), the RHS in [\[eq:stablity-imex1\]](#eq:stablity-imex1){reference-type="eqref" reference="eq:stablity-imex1"} is non-negative. Thus, it follows from [\[eq:stablity-imex1\]](#eq:stablity-imex1){reference-type="eqref" reference="eq:stablity-imex1"} that $$\nonumber \begin{aligned} \frac{1}{2}\int(U_N^{n+1})^T A_0 U_N^{n+1} - \frac{1}{2}\int(U_N^{n})^T A_0 U_N^{n} & + \frac{1}{2}\int(U_N^{n+1}-U_N^{n})^T A_0 (U_N^{n+1}-U_N^{n}) \\ & + \Delta t\int (U_N^{n+1}-U_N^n)^T A_0 A (U_N^n)_x \le 0. \end{aligned}$$ Define $E^n := \frac{1}{2}\int(U_N^{n})^T A_0 U_N^{n}$ and denote by $2 \kappa$ the smallest eigenvalue of the SPD matrix $A_0$. We deduce from the last inequality and the inequality [\[equ:spectral\]](#equ:spectral){reference-type="eqref" reference="equ:spectral"} that $$\nonumber \begin{aligned} & E^{n+1} - E^n \\ \le{}& -\frac{1}{2}\int(U_N^{n+1}-U_N^{n})^T A_0 (U_N^{n+1}-U_N^{n}) - \Delta t\int (U_N^{n+1}-U_N^n)^T A_0 A (U_N^n)_x \\[4mm] \le{}& -\kappa \left\Vert U_N^{n+1}-U_N^n\right\Vert^2 + \kappa \left\Vert U_N^{n+1}-U_N^n\right\Vert^2 + \frac{C(\Delta t)^2}{\kappa} \left\Vert(U_N^n)_x\right\Vert^2 \\[4mm] \le{}& \frac{C(\Delta t)^2}{\kappa}N^2 \left\Vert U_N^n\right\Vert^2 \leq \frac{C(\Delta t)^2}{\kappa^2}N^2 E^n. \end{aligned}$$ Finally, let $\Delta t\le c_{CFL}/N^2$. Then we have $$\nonumber E^{n} \le (1+C\Delta t) E^{n-1} \le (1+C\Delta t)^n E_0 \le e^{CT}E_0,$$ namely, $$\nonumber \int(U_N^{n})^T A_0 U_N^{n} \le e^{CT}\int(U_N^{0})^T A_0 U_N^{0} .$$ This is the stability of the first-order fully discretized IMEX-BDF scheme. ### Higher-order schemes For other $q$, we have the following similar stability result. **Theorem 1**. *Under the structural stability condition, assume the CFL condition $\Delta t\le c_{CFL}/N^2$ with $c_{CFL}>0$ a constant. Then the IMEX-BDF scheme [\[scheme1\]](#scheme1){reference-type="eqref" reference="scheme1"} with $q=1,2,3,4$ is uniformly stable in the sense that $$\nonumber \left\Vert U_N^n\right\Vert^2\le C\sum_{i=0}^{q-1}\left(\left\Vert U_N^i\right\Vert^2 + \frac{\Delta t}{\varepsilon}\left\Vert W_N^i\right\Vert^2\right)$$ for integer $n$ such that $t_n=T_0+n\Delta t\le T$, where $C$ is a constant independent of $\varepsilon$, $N$ and $\Delta t$, and $U^n_N=\begin{pmatrix} V^n_N\\W^n_N \end{pmatrix}$.* *Proof.* Recall the scheme [\[scheme1\]](#scheme1){reference-type="eqref" reference="scheme1"} $$\nonumber \sum_{i=0}^q\alpha_i U_N^{n+i} + \Delta tA \sum_{i=0}^{q-1}\gamma_i (U_N^{n+i})_x = \frac{\beta\Delta t}{\varepsilon} QU_N^{n+q}.$$ In Lemma [Lemma 2](#lemma:multiplier){reference-type="ref" reference="lemma:multiplier"}, taking $H=A_0$ from the structural stability condition we have $$\nonumber \begin{aligned} & \int \left(U_N^{n+q}-\sum_{i=1}^{q-1}\eta_i U_N^{n+i} \right)^T A_0 \sum_{i=0}^q\alpha_i U_N^{n+i} \\ ={}& \int G_{A_0}(U_N^{n+1}, \dots, U_N^{n+q}) - \int G_{A_0}(U_N^{n}, \dots, U_N^{n+q-1})\\ {}&+ d_1 \left\Vert U_N^{n+q} - \sum_{i=1}^{q-1}\eta_i U_N^{n+i} - d_2 \sum_{i=0}^{q-1}\gamma_i U_N^{n+i}\right\Vert_{A_0}^2. \end{aligned}$$ Thanks to the symmetry of $A_0A$ and the periodic boundary conditions, we deduce from the inequality [\[equ:spectral\]](#equ:spectral){reference-type="eqref" reference="equ:spectral"} that $$\nonumber \begin{aligned} &\left\vert \Delta t\int \left(U_N^{n+q}-\sum_{i=1}^{q-1}\eta_i U_N^{n+i} \right)^T A_0 A \sum_{i=0}^{q-1}\gamma_i (U_N^{n+i})_x \right\vert\\ \leq {}&\left\vert \Delta t\int \left(U_N^{n+q}-\sum_{i=1}^{q-1}\eta_i U_N^{n+i} - d_2 \sum_{i=0}^{q-1}\gamma_i U_N^{n+i} \right)^T A_0 A \sum_{i=0}^{q-1}\gamma_i (U_N^{n+i})_x \right\vert\\ &+\left\vert \Delta t\int d_2\left( \sum_{i=0}^{q-1}\gamma_i U_N^{n+i} \right)^T A_0 A \sum_{i=0}^{q-1}\gamma_i (U_N^{n+i})_x \right\vert\\ \leq {}& \left\vert \Delta t\int \left(U_N^{n+q}-\sum_{i=1}^{q-1}\eta_i U_N^{n+i} - d_2 \sum_{i=0}^{q-1}\gamma_i U_N^{n+i} \right)^T A_0 A \sum_{i=0}^{q-1}\gamma_i (U_N^{n+i})_x \right\vert\\ \le{}& \kappa \left\Vert U_N^{n+q}-\sum_{i=1}^{q-1}\eta_i U_N^{n+i} - d_2 \sum_{i=0}^{q-1}\gamma_i U_N^{n+i}\right\Vert^2 + \frac{C(\Delta t)^2}{\kappa} \left\Vert\sum_{i=0}^{q-1} \gamma_i (U_N^{n+i})_x\right\Vert^2 \\ \le{}& \kappa \left\Vert U_N^{n+q}-\sum_{i=1}^{q-1}\eta_i U_N^{n+i} - d_2 \sum_{i=0}^{q-1}\gamma_i U_N^{n+i}\right\Vert^2 + \frac{C(\Delta t)^2N^2}{\kappa} \sum_{i=0}^{q-1} \left\Vert U_N^{n+i}\right\Vert^2 \\ \end{aligned}$$ with $\kappa>0$. Moreover, the source term can be estimated as $$\nonumber \begin{aligned} & \int \left(U_N^{n+q}-\sum_{i=1}^{q-1}\eta_i U_N^{n+i} \right)^T A_0 \frac{\beta\Delta t}{\varepsilon} QU_N^{n+q} \\ ={}& \frac{\beta\Delta t}{\varepsilon} \int \left(U_N^{n+q}-\sum_{i=1}^{q-1}\eta_i U_N^{n+i} \right)^T A_0 QU_N^{n+q} \\ ={}& \frac{\beta\Delta t}{\varepsilon} \int \left(U_N^{n+q}-\sum_{i=1}^{q-1}\eta_i U_N^{n+i} \right)^T \left( \begin{array}{cc} 0 & 0 \\ 0 & A_{02}\hat{S} \end{array} \right) U_N^{n+q} \\ ={}& -\frac{\beta\Delta t}{\varepsilon} \int \left(W_N^{n+q}-\sum_{i=1}^{q-1}\eta_i W_N^{n+i} \right)^T M W_N^{n+q} \\ ={}& - \frac{\beta\Delta t}{\varepsilon} \left( \int A_{M}(W_N^{n+2}, \dots, W_N^{n+q}) - \int A_{M}(W_N^{n+1}, \dots, W_N^{n+q-1}) + \left\Vert\sum_{i=1}^q c_i W_N^{n+i}\right\Vert^2_M\right) \end{aligned}$$ with $U=\begin{pmatrix} V\\ W \end{pmatrix}$ and $M:= - A_{02}\hat{S}$ a SPD matrix. Combining the last three estimates, we arrive at $$\label{eq:combine-estimate} \begin{aligned} & \int G_{A_0}(U_N^{n+1}, \dots, U_N^{n+q}) - \int G_{A_0}(U_N^{n}, \dots, U_N^{n+q-1})\\ {}&+ d_1 \left\Vert U_N^{n+q} - \sum_{i=1}^{q-1}\eta_i U_N^{n+i} - d_2 \sum_{i=0}^{q-1}\gamma_i U_N^{n+i}\right\Vert_{A_0}^2 \\[4mm] \le{} & \kappa \left\Vert U_N^{n+q}-\sum_{i=1}^{q-1}\eta_i U_N^{n+i} - d_2 \sum_{i=0}^{q-1}\gamma_i U_N^{n+i}\right\Vert^2 + \frac{C(\Delta t)^2 N^2}{\kappa} \sum_{i=0}^{q-1} \left\Vert U_N^{n+i}\right\Vert^2 \\ {}& - \frac{\beta\Delta t}{\varepsilon} \left(\int A_{M}(W_N^{n+2}, \dots, W_N^{n+q}) - \int A_{M}(W_N^{n+1}, \dots, W_N^{n+q-1}) + \left\Vert\sum_{i=1}^q c_i W_N^{n+i}\right\Vert^2_M\right) . \end{aligned}$$ Set $$\nonumber G_{A_0, U}^n = \int G_{A_0}(U_N^{n}, \dots, U_N^{n+q-1}), \quad A_{M, W}^n = \int A_{M}(W_N^{n+1}, \dots, W_N^{n+q-1})$$ and $$\nonumber E^n = G_{A_0, U}^{n} + \frac{\beta\Delta t}{\varepsilon} A_{M, W}^{n}.$$ Note that $$\label{stability-err} C^{-1} \sum_{i=0}^{q-1} \left\Vert U_N^{n+i}\right\Vert^2 \leq G_{A_0, U}^{n}\leq C \sum_{i=0}^{q-1} \left\Vert U_N^{n+i}\right\Vert^2, \quad 0\leq A_{M, W}^{n} \leq C \sum_{i=1}^{q-1} \left\Vert U_N^{n+i}\right\Vert^2.$$ By taking $\kappa=d_1/2$ and $\Delta t\le c_{CFL}/N^2$, it follows from [\[eq:combine-estimate\]](#eq:combine-estimate){reference-type="eqref" reference="eq:combine-estimate"} that $$\nonumber E^{n+1} - E^n \le C\Delta t\sum_{i=0}^{q-1} \left\Vert U_N^{n+i}\right\Vert^2 \le C\Delta tE^n.$$ Therefore, we have $$\nonumber E^{n+1} \le (1+C\Delta t)E^n$$ and furthermore $$\nonumber E^n \le e^{CT}E^0.$$ Hence we have $$\nonumber \left\Vert U_N^n\right\Vert^2\le C\sum_{i=0}^{q-1}\left(\left\Vert U_N^i\right\Vert^2 + \frac{\Delta t}{\varepsilon}\left\Vert W_N^i\right\Vert^2\right)$$ and the proof is complete. ◻ ## Regularity {#subsec:regularity-estimate} To analyze the truncation error of the IMEX-BDF scheme [\[scheme1\]](#scheme1){reference-type="eqref" reference="scheme1"}, we need the following uniform-in-$\varepsilon$ regularity estimate. For this purpose, we multiply the both sides of [\[eq:PDE\]](#eq:PDE){reference-type="eqref" reference="eq:PDE"} with $U^TA_0$ and integrate over $x$ to obtain $$\nonumber \begin{aligned} \int U^TA_0U_t + \int U^TA_0 A U_{x} = \frac{1}{\varepsilon} \int U^TA_0QU \end{aligned}$$ and thereby $$\label{regu-inequ} \begin{aligned} % \int U^TA_0U_t + \int U^TA_0 A U_{x} &= \frac{1}{\varepsilon} \int U^TA_0QU, \\ \frac 1 2\int (U^TA_0U)_t + \frac 1 2\int (U^TA_0A U)_{x} = \frac{1}{\varepsilon} \int W^TA_{02}\hat{S}W \le 0. \end{aligned}$$ Due to the periodic boundary conditions, we have $$\nonumber \int U^T(x,t)A_0U(x,t) dx \le \int U^T(x,0)A_0U(x,0) dx,$$ which implies $$\nonumber \left\Vert U(\cdot,t)\right\Vert\le C \left\Vert U(\cdot,0)\right\Vert, \quad t \geq 0.$$ Here $C$ only depends on the symmetrizer $A_0$. Since [\[eq:PDE\]](#eq:PDE){reference-type="eqref" reference="eq:PDE"} is linear with constant coefficients, the partial derivative $\partial_x^s U$ of order $s$ also satisfies [\[eq:PDE\]](#eq:PDE){reference-type="eqref" reference="eq:PDE"} and therefore $$\label{equ:hs} \left\Vert U(\cdot,t)\right\Vert_{H^s} \le C \left\Vert U(\cdot,0)\right\Vert_{H^s}.$$ Here $\left\Vert U(\cdot,t)\right\Vert_{H^s}$ denotes the standard norm for the Sobolev space $H^s$ of the periodic function $U = U(x,t)$. **Theorem 2**. *For any integer $s\ge0$, the solution to [\[eq:PDE\]](#eq:PDE){reference-type="eqref" reference="eq:PDE"} satisfies* 1. *for all $t\ge 0$, $$\label{eq:regularity-Hs} \left\Vert U(\cdot,t)\right\Vert_{H^s}^2 \le C \left\Vert U(\cdot,0)\right\Vert_{H^s}^2,$$* 2. *for all $t\ge 2\delta_0^{-1} s\varepsilon\log(1/\varepsilon)$, $$\label{eq:regularity-U} \left\Vert\partial_t^{r_1}\partial_x^{r_2}U(\cdot,t)\right\Vert^2 \le C \left\Vert U(\cdot,0)\right\Vert_{H^s}^2, \quad r_1+{r_2}\le s$$ and $$\label{eq:regularity-W} \left\Vert\partial_t^{r_1}\partial_x^{r_2}W(\cdot,t)\right\Vert^2 \le C \varepsilon^2 \left\Vert U(\cdot,0)\right\Vert_{H^s}^2, \quad r_1+{r_2}\le s-1.$$ Here $\delta_0>0$ is a constant determined by the SPD matricies $A_{02}$ and $A_{02}\hat{S}$, $C$ is a generic constant independent of $\varepsilon$, $U=\begin{pmatrix} V\\W \end{pmatrix}$, and $r_1, r_2$ are non-negative integers.* *Proof.* Estimate [\[eq:regularity-Hs\]](#eq:regularity-Hs){reference-type="eqref" reference="eq:regularity-Hs"} is just [\[equ:hs\]](#equ:hs){reference-type="eqref" reference="equ:hs"} and [\[eq:regularity-W\]](#eq:regularity-W){reference-type="eqref" reference="eq:regularity-W"} simply follows from [\[eq:regularity-U\]](#eq:regularity-U){reference-type="eqref" reference="eq:regularity-U"} together with the equation $$\nonumber W = \varepsilon \hat{S}^{-1}(W_t + A_{21}V_x + A_{22}W_x).$$ Next, we prove [\[eq:regularity-U\]](#eq:regularity-U){reference-type="eqref" reference="eq:regularity-U"} by induction on $s$. It is trivial for $s=0$. Assume [\[eq:regularity-U\]](#eq:regularity-U){reference-type="eqref" reference="eq:regularity-U"} for $(s-1)$ and we prove the estimate with $s$. Notice that for any $0\le r\le s-1$, $\partial_t\partial_x^r U$ satisfies the same equation [\[eq:PDE\]](#eq:PDE){reference-type="eqref" reference="eq:PDE"}. As in obtaining [\[regu-inequ\]](#regu-inequ){reference-type="eqref" reference="regu-inequ"}, we have $$\nonumber \begin{aligned} \frac 1 2\int ((\partial_t\partial_x^rU)^TA_0 \partial_t\partial_x^rU)_t ={}& \frac{1}{\varepsilon} \int (\partial_t\partial_x^rW)^TA_{02}\hat{S} \partial_t\partial_x^rW \\ \leq{}& - \frac{\delta_0}{2\varepsilon} \int (\partial_t\partial_x^rW)^T A_{02} \partial_t\partial_x^rW\\ \le{}& -\frac{\delta_0}{2\varepsilon}\int (\partial_t\partial_x^rU)^T A_0 \partial_t\partial_x^rU + \frac{\delta_0}{2\varepsilon}\int (\partial_t\partial_x^rV)^T A_{01} \partial_t\partial_x^rV. \end{aligned}$$ Here $\delta_0>0$ is a constant determined by the SPD matricies $A_{02}$ and $A_{02}\hat{S}$. Denote $$\nonumber E(t) = \int (\partial_t\partial_x^rU)^TA_0 \partial_t\partial_x^rU.$$ The last inequality can be written as $$\nonumber E'(t) \le -\frac{\delta_0}{\varepsilon}E(t) + \frac{C \delta_0}{\varepsilon}\left\Vert\partial_t\partial_x^rV(t)\right\Vert^2.$$ By Gronwall's inequality, we have $$\label{equ:E-estimate} \begin{aligned} E(t) \le{}& e^{-\frac{\delta_0}{\varepsilon}t} E(0) + \frac{C\delta_0}{\varepsilon}\int_0^t e^{\frac{\delta_0}{\varepsilon}(\tau-t)}\left\Vert\partial_t\partial_x^rV(\tau)\right\Vert^2 d\tau. \end{aligned}$$ On the other hand, from the equation for $W$ in [\[eq:PDE\]](#eq:PDE){reference-type="eqref" reference="eq:PDE"} $$\nonumber \begin{aligned} \partial_t\partial_x^r W = -\partial_x^r(A_{21}V_x + A_{22}W_x - \frac{1}{\varepsilon}\hat{S}W) = - A_{21}\partial_x^{r+1}V - A_{22}\partial_x^{r+1}W + \frac{1}{\varepsilon}\hat{S}\partial_x^r W \end{aligned}$$ and estimate [\[eq:regularity-Hs\]](#eq:regularity-Hs){reference-type="eqref" reference="eq:regularity-Hs"}, it follows that $$\nonumber \left\Vert\partial_t\partial_x^r W\right\Vert^2\le C (\frac{1}{\varepsilon^2}+1)\left\Vert U(\cdot,0)\right\Vert_{H^s}^2.$$ Similarly, we have $$\nonumber \left\Vert\partial_t\partial_x^r V\right\Vert^2\le C \left\Vert U(\cdot,0)\right\Vert_{H^s}^2.$$ Thus, it follows from [\[equ:E-estimate\]](#equ:E-estimate){reference-type="eqref" reference="equ:E-estimate"} that $$\nonumber \begin{aligned} E(t) \le{}& Ce^{-\frac{\delta_0}{\varepsilon}t}(\frac{1}{\varepsilon^2}+1)\left\Vert U(\cdot,0)\right\Vert_{H^s}^2 + C(1-e^{-\frac{\delta_0}{\varepsilon}t})\left\Vert U(\cdot,0)\right\Vert_{H^s}^2 \\ \le{}& C (\frac{1}{\varepsilon^2}e^{-\frac{\delta_0}{\varepsilon}t}+1)\left\Vert U(\cdot,0)\right\Vert_{H^s}^2. \end{aligned}$$ Here we have used $$E(0) \leq C \left\Vert\partial_t\partial_x^r V(0)\right\Vert^2 + C \left\Vert\partial_t\partial_x^r W(0)\right\Vert^2 \leq C(\frac{1}{\varepsilon^2}+1) \left\Vert U(\cdot,0)\right\Vert_{H^s}^2.$$ Then for $t_0=2 \delta_0^{-1}\varepsilon\log(1/\varepsilon)$, we have $E(t_0)\le C\left\Vert U(\cdot,0)\right\Vert_{H^s}^2$ and thus $$\nonumber \left\Vert\partial_t\partial_x^r U(t_0)\right\Vert^2\le C\left\Vert U(\cdot,0)\right\Vert_{H^s}^2.$$ Now define $\tilde{U}(t)=\partial_tU(t+t_0)$, then $\tilde{U}$ also satisfies the same equation and $$\nonumber \left\Vert\tilde{U}(0)\right\Vert_{H^{s-1}}^2 \le C\left\Vert U(\cdot,0)\right\Vert_{H^s}^2.$$ By the induction hypothesis $$\nonumber \left\Vert\partial_t^{r_1}\partial_x^{r_2} \tilde{U}(t)\right\Vert^2\le C\left\Vert U(\cdot,0)\right\Vert_{H^s}^2, \quad r_1 + r_2 \le s-1, \quad t \ge 2\delta_0^{-1}(s-1)\varepsilon\log(1/\varepsilon),$$ which implies [\[eq:regularity-U\]](#eq:regularity-U){reference-type="eqref" reference="eq:regularity-U"}. ◻ ## Error estimates {#subsec:uniform-accuracy} In this subsection, we establish our main result on the uniform-in-$\varepsilon$ accuracy of the IMEX-BDF scheme [\[scheme1\]](#scheme1){reference-type="eqref" reference="scheme1"}. As in [@hu_uniform_2019], we consider two types of initial conditions. - **Type 1**: The initial data $U(x, 0)$ satisfies $$\nonumber \left\Vert\partial_t^{q+1}U(\cdot, 0)\right\Vert_{H^1} + \left\Vert\partial_t^{q}U(\cdot, 0)\right\Vert_{H^2} \le C$$ for $q=1,2,3,4$. Such data will be used for the IMEX-BDF scheme starting at $T_0\ge0$. - **Type 2**: The initial data $U(x, 0)$ satisfies $$\nonumber \left\Vert U(\cdot, 0)\right\Vert_{H^{q+2}} \le C$$ for $q=1,2,3,4$. Such data will be used for the IMEX-BDF scheme starting at $T_0\ge 2\delta_0^{-1} (q+2)\varepsilon\log(1/\varepsilon)$. **Lemma 3**. *Let $U^{n}=U(x, t^n)$ is an exact solution to equation [\[eq:PDE\]](#eq:PDE){reference-type="eqref" reference="eq:PDE"} with period initial data $U = U(x, 0)$ above. Then the truncation error of the IMEX-BDF [\[scheme1\]](#scheme1){reference-type="eqref" reference="scheme1"} satisfies $$\nonumber \left\Vert\sum_{i=0}^q\alpha_i U^{n+i} + \Delta tA \sum_{i=0}^{q-1}\gamma_i (U^{n+i})_x - \frac{\beta\Delta t}{\varepsilon} QU^{n+q}\right\Vert\leq C(\Delta t)^{q+1}.$$* *Proof.* Notice that $\partial_t^{r_1}\partial_x^{r_2}U$ satisfies the equation [\[eq:PDE\]](#eq:PDE){reference-type="eqref" reference="eq:PDE"}. The regularity estimate [\[eq:regularity-Hs\]](#eq:regularity-Hs){reference-type="eqref" reference="eq:regularity-Hs"} implies $$\nonumber \left\Vert\partial_t^{q+1}U(t)\right\Vert_{H^1} \le C \left\Vert\partial_t^{q+1}U(0)\right\Vert_{H^1} \le C$$ and $$\nonumber \left\Vert\partial_t^{q}\partial_x U(t)\right\Vert_{H^1} \le \left\Vert\partial_t^{q} U(t)\right\Vert_{H^2} \le C \left\Vert\partial_t^{q} U(0)\right\Vert_{H^2}\le C,$$ for initial data of Type 1. For initial data of Type 2, the regularity estimate [\[eq:regularity-U\]](#eq:regularity-U){reference-type="eqref" reference="eq:regularity-U"} leads to $$\nonumber \left\Vert\partial_t^{r_1}\partial_x^{r_2}U(t)\right\Vert\le C \left\Vert U(t)\right\Vert_{H^{q+2}} \le C, \quad r_1+r_2\le q+2,$$ for any $t\ge 2\delta_0^{-1}(q+2)\varepsilon\log(1/\varepsilon)$. This implies $$\nonumber \left\Vert\partial_t^{q+1}U(t)\right\Vert_{H^1} \le C$$ by taking $r_1=q+1$, $r_2=0, 1$ and $$\nonumber \left\Vert\partial_t^{q}\partial_xU(t)\right\Vert_{H^1} \le C$$ by taking $r_1=q$, $r_2=0, 1, 2$. Moreover, it follows from the Sobolev inequality that $$\nonumber \left\Vert\partial_t^{q+1}U(t)\right\Vert_{L^{\infty}} +\left\Vert\partial_t^{q}\partial_xU(t)\right\Vert_{L^{\infty}} \le C\left( \left\Vert\partial_t^{q+1}U(t)\right\Vert_{H^1} + \left\Vert\partial_t^{q}\partial_xU(t)\right\Vert_{H^1} \right )\le C$$ for the initial data of the two types. On the other hand, from [@hundsdorfe2007imex] we know the following facts related to the IMEX-BDF scheme: $$\nonumber \left\vert\sum_{i=0}^q \alpha_i u^{n+i} - \beta\Delta t\partial_t u^{n+q}\right\vert \le C\Delta t^{q+1}\max_{t\in[T^0,T]}\left\vert\partial_t^{q+1}u^{n+q}\right\vert$$ and $$\nonumber \left\vert\sum_{i=0}^{q-1}\gamma_i\partial_x u^{n+i} - \beta\Delta t\partial_x u^{n+q}\right\vert \le C\Delta t^{q+1}\max_{t\in[T^0,T]}\left\vert\partial_t^{q}\partial_xu^{n+q}\right\vert$$ for any smooth function $u=u(x, t)$, where $u^n:=u(x,t^n)$. Thus, for the spatially periodic function $u=u(x, t)$, we have $$\nonumber \left\Vert\sum_{i=0}^q \alpha_i u^{n+i} - \beta\Delta t\partial_t u^{n+q}\right\Vert \le C\left\Vert\sum_{i=0}^q \alpha_i u^{n+i} - \beta\Delta t\partial_t u^{n+q}\right\Vert_{L^{\infty}} \le C(\Delta t)^{q+1}$$ and $$\nonumber \left\Vert\sum_{i=0}^{q-1}\gamma_i\partial_x u^{n+i} - \beta\Delta t\partial_x u^{n+q}\right\Vert \le C\left\Vert\sum_{i=0}^{q-1}\gamma_i\partial_x u^{n+i} - \beta\Delta t\partial_x u^{n+q}\right\Vert_{L^{\infty}} \le C(\Delta t)^{q+1}.$$ Denote by $R_U^n$ the truncation error of the IMEX-BDF scheme [\[scheme1\]](#scheme1){reference-type="eqref" reference="scheme1"}: $$\nonumber R_U^n = \sum_{i=0}^q\alpha_i U^{n+i} + \Delta tA \sum_{i=0}^{q-1}\gamma_i (U^{n+i})_x - \frac{\beta\Delta t}{\varepsilon} QU^{n+q}.$$ It follows from the last two inequalities that $$\nonumber \begin{aligned} \left\Vert R_U^n\right\Vert& = \left\Vert\sum_{i=0}^q\alpha_i U^{n+i} + \Delta tA \sum_{i=0}^{q-1}\gamma_i (U^{n+i})_x - \frac{\beta\Delta t}{\varepsilon} QU^{n+q}\right\Vert\\[4mm] &\leq \left\Vert\sum_{i=0}^q \alpha_i U^{n+i} - \beta\Delta t\partial_t U^{n+q}\right\Vert + \left\Vert\beta\Delta t\partial_t U^{n+q} + \beta\Delta tA \partial_x U^{n+q} - \frac{\beta\Delta t}{\varepsilon} QU^{n+q} \right\Vert\\[4mm] &~~ + \left\Vert\Delta tA \sum_{i=0}^{q-1}\gamma_i (U^{n+i})_x - \beta\Delta tA \partial_x U^{n+q}\right\Vert\\ &\leq C (\Delta t)^{q+1} + 0 + C (\Delta t)^{q+1}\\ & \le C(\Delta t)^{q+1}. \end{aligned}$$ This completes the proof. ◻ **Theorem 3**. *Under the conditions of Theorem [Theorem 1](#thm:stability-const){reference-type="ref" reference="thm:stability-const"}, the IMEX-BDF scheme [\[scheme1\]](#scheme1){reference-type="eqref" reference="scheme1"} for system [\[eq:PDE\]](#eq:PDE){reference-type="eqref" reference="eq:PDE"} is uniformly $q$-th order accurate, that is $$\nonumber \left\Vert U(\cdot, t_n) - U_N^n\right\Vert^2 \le C\left( (\Delta t)^{2q} + e_{init} \right).$$ Here $U=U(x, t)$ is the exact solution to equation [\[eq:PDE\]](#eq:PDE){reference-type="eqref" reference="eq:PDE"} with initial data above, $C$ is a constant independent of $\varepsilon$, $N$ and $\Delta t$, and $e_{init}$ is related to the initial projection error $$\nonumber e_{init} := \sum_{i=0}^{q-1}\left( \left\Vert V(\cdot, t_i) - V_N^i\right\Vert^2 + (1+\frac{\Delta t}{\varepsilon})\left\Vert W(\cdot, t_i) - W_N^i\right\Vert \right)$$ with $U=\begin{pmatrix} V\\W \end{pmatrix}$ and $U^n_N=\begin{pmatrix} V^n_N\\W^n_N \end{pmatrix}$.* *Proof.* Set $\delta U^n=U(x, t_n) - U_N^n$. It is clear that the error $\delta U^n$ satisfies the scheme [\[scheme1\]](#scheme1){reference-type="eqref" reference="scheme1"} with residue $R_U^n$. Then by repeating the argument of Theorem [Theorem 1](#thm:stability-const){reference-type="ref" reference="thm:stability-const"} and using $$\nonumber \begin{aligned} \int \left( \delta U^{n+q} - \sum_{i=1}^q\eta_i\delta U^{n+i} \right) R_U^n dx \le{}& \kappa\Delta t\left\Vert\delta U^{n+q}\right\Vert^2 + \kappa C\Delta t\sum_{i=1}^{q-1} \left\Vert\delta U^{n+i}\right\Vert^2 + \frac{C}{\kappa}(\Delta t)^{2q+1} \end{aligned}$$ with $\kappa>0$, we obtain $$\label{err-sccuracy} \begin{aligned} E^{n+1} - E^n \le& C \Delta t\sum_{i=0}^{q-1} \left\Vert\delta U^{n+i}\right\Vert^2 - \frac{\beta\Delta t}{\varepsilon} \left\Vert\sum_{i=1}^q c_i \delta W^{n+i}\right\Vert^2_M + \kappa\Delta t\left\Vert\delta U^{n+q}\right\Vert^2 + \frac{C}{\kappa}(\Delta t)^{2q+1}\\ % \leq & C \Delta t G_{A_0, \delta U}^n + \kappa\dt\norm{\delta U^{n+q}}^2 + \frac{C}{\kappa}(\dt)^{2q+1} \end{aligned}$$ where $$\nonumber E^n = G_{A_0, \delta U}^{n} + \frac{\beta\Delta t}{\varepsilon} A_{M, \delta W}^{n}$$ with $$\nonumber G_{A_0, \delta U}^n = \int G_{A_0}(\delta U^{n}, \dots, \delta U^{n+q-1}), \quad A_{M, \delta W}^n = \int A_{M}(\delta W^{n+1}, \dots, \delta W^{n+q-1}).$$ As inequalities [\[stability-err\]](#stability-err){reference-type="eqref" reference="stability-err"}, we have $$\nonumber C^{-1} \sum_{i=0}^{q-1} \left\Vert\delta U^{n+i}\right\Vert^2 \leq G_{A_0, \delta U}^{n}\leq C \sum_{i=0}^{q-1} \left\Vert\delta U^{n+i}\right\Vert^2, \quad 0 \leq A_{M, \delta W}^{n} \leq C \sum_{i=1}^{q-1} \left\Vert\delta U^{n+i}\right\Vert^2.$$ Then inequality [\[err-sccuracy\]](#err-sccuracy){reference-type="eqref" reference="err-sccuracy"} gives $$\nonumber \begin{aligned} E^{n+1} - E^n &\leq C \Delta t G_{A_0, \delta U}^n + C \kappa\Delta tG_{A_0, \delta U}^{n+1} + \frac{C}{\kappa}(\Delta t)^{2q+1}\\ &\leq C \Delta t E^n + C \kappa\Delta tE^{n+1} + \frac{C}{\kappa}(\Delta t)^{2q+1}. \end{aligned}$$ With this, we take $\kappa$ sufficiently small to obtain $$\nonumber E^{n+1} \le (1+C\Delta t)E^n + C(\Delta t)^{2q+1}$$ implying $$\nonumber E^n \le C E^0 + C(\Delta t)^{2q}.$$ Since $$\left\Vert U(\cdot, t_n) - U_N^n\right\Vert^2 = \left\Vert\delta U^n\right\Vert^2 \leq C E^n$$ and $$E^0 \leq C \sum_{i=0}^{q-1} \left( \left\Vert\delta V^{i}\right\Vert^2 + (1+\frac{\Delta t}{\varepsilon})\left\Vert\delta W^{i}\right\Vert^2 \right) = C e_{init},$$ the proof is completed. ◻ We end this section with the following corollary. **Corollary 1**. *Under the conditions of Theorem [Theorem 3](#thm:uniform-accuracy){reference-type="ref" reference="thm:uniform-accuracy"}, if $$\nonumber \left\Vert U(\cdot, T_0)\right\Vert_{H^{2q+1}} + \left\Vert\partial_t U(\cdot, T_0)\right\Vert_{H^{2q}} \le C,$$ the error estimate $$\nonumber \left\Vert U(\cdot, t_n) - U_N^n\right\Vert^2 \le C \left( (\Delta t)^{2q} + \frac{1}{N^{4q}} \right),$$ holds for integer $n$ such that $t_n=T_0+n\Delta t\le T$.* *Proof.* By Theorem [Theorem 3](#thm:uniform-accuracy){reference-type="ref" reference="thm:uniform-accuracy"}, it suffices to prove that $e_{init}\le C/N^{4q}$. To do so, we use the following property of Fourier projection [@hesthaven2007spectral] and Theorem [Theorem 2](#thm:regularity-const){reference-type="ref" reference="thm:regularity-const"} to obtain $$\nonumber \left\Vert U(\cdot, t_i) - U_N^i\right\Vert^2 \le \frac{1}{N^{4q+2}}\left\Vert U(\cdot, t_i)\right\Vert^2_{H^{2q+1}} \le \frac{C}{N^{4q+2}}.$$ Similarly, we have $$\nonumber \left\Vert\partial_x (V(\cdot, t_i) - V_N^i)\right\Vert^2 + \left\Vert\partial_t (W(\cdot, t_i) - W_N^i)\right\Vert^2 \le \frac{C}{N^{4q}}.$$ Then we deduce from the equation for $W$ in [\[eq:PDE\]](#eq:PDE){reference-type="eqref" reference="eq:PDE"} that $$\nonumber \left\Vert W(\cdot, t_i) - W_N^i\right\Vert^2 \le \varepsilon^2 \frac{C}{N^{4q}}.$$ Hence $e_{init}\le C/N^{4q}$ and the conclusion follows. ◻ # Numerical tests {#sec:numerical-tests} In this section, we numerically test the accuracy of the IMEX-BDF schemes applied to several linearized hyperbolic relaxation systems including the Aw-Rascle-Zhang traffic model [@aw2000siam; @zhang2002non], the Broadwell model [@broadwell1964shock], and the Grad's moment system [@grad1949kinetic; @cai2014cpam]. In all the numerical tests, we adopt the Fourier-Galerkin spectral method for spatial discretization with modes $|k|\leq N$ and fix $N=100$ to ensure that the discretization error in space is much smaller than that in time. The reference solution $U_{ref}$ is computed with a much finer time step. ## Aw-Rascle-Zhang traffic model The model [@aw2000siam; @zhang2002non] is $$\nonumber \begin{aligned} &\partial_t \rho + \partial_x (\rho v) = 0,\\ &\partial_t v + \left(v - \rho p'(\rho) \right)\partial_x v = \frac{V(\rho) - v}{\varepsilon}, \end{aligned}$$ with $$\nonumber p(\rho) = c_0 \rho^\gamma, \quad V(\rho) = v_f \left(1 - \frac{\rho}{\rho_m}\right).$$ Here $\rho = \rho(x, t)$ is the traffic density, $v = v(x,t)$ is the traffic speed, and $\varepsilon$ is a relaxation time characterizing the response of the drivers to the traffic situation. The variable $p(\rho)$ is the traffic pressure and the equilibrium velocity-density relationship $V(\rho)$ is given in the Greenshield model [@greenshields1935study]. The linearization of the model around a uniform steady state $(\rho^\star, v^\star)$ is $$\nonumber \begin{aligned} &\partial_t \rho + v^\star \partial_x \rho + \rho^\star \partial_x v = 0,\\ &\partial_t v - \left(\rho^\star p'(\rho^\star) - v^\star \right)\partial_x v = \frac{\rho V'(\rho^\star) - v}{\varepsilon}. \end{aligned}$$ In our numerical test, we take $$\nonumber c_0 =\frac{3}{2}, \quad \gamma = 1, \quad \rho_m = 8, \quad v_f = 4, \quad (\rho^\star, v^\star) = (1, 1).$$ Then the linearized model becomes $$\nonumber \begin{aligned} \partial_t U + A\partial_x U = \frac{1}{\varepsilon} Q U, \end{aligned}$$ with $$\nonumber \begin{aligned} U = (\rho, ~ v)^T, \quad A = \begin{pmatrix} 1 & 1\\ 0 &-\frac{1}{2} \end{pmatrix},\quad Q=\begin{pmatrix} 0 & 0\\ -\frac{1}{2} & -1 \end{pmatrix}. \end{aligned}$$ It is easy to verify that the last system satisfies the structural stability condition with $$\begin{aligned} P = \begin{pmatrix} 1 & 0\\ \frac{1}{2} & 1 \end{pmatrix}, \quad A_0 = \begin{pmatrix} 3 & 2\\ 2 & 4 \end{pmatrix}. \end{aligned}$$ The computational domain is $[0, 1]$ with periodic boundary conditions and the initial data are given by $$\nonumber \rho(x,0) = \sin(2\pi x) + 1.1.$$ For the second-order scheme, we choose the initial data for $v$ as $$\nonumber v(x, 0) = -\frac{1}{2}\rho(x, 0),$$ which is consistent up to $O(1)$. For the third-order scheme, we choose the initial data for $v$ as $$\nonumber v(x, 0) = -\frac{1}{2}\rho(x, 0) - \frac{\varepsilon}{2} \partial_x \rho(x, 0),$$ which is consistent up to $O(\varepsilon)$. For the fourth-order scheme, we choose the initial data for $v$ as $$\nonumber v(x, 0) = -\frac{1}{2}\rho(x, 0) - \frac{\varepsilon}{2} \partial_x \rho(x, 0) - \frac{\varepsilon^2}{4} \partial_{xx} \rho(x, 0),$$ which is consistent up to $O(\varepsilon^2)$. The starting values of the IMEX-BDF scheme at $t = i\Delta t$ with $i =1, \cdots, q-1$, are prepared using the IMEX-RK schemes (ARS(2,2,2) for second- and third-order scheme, ARS(4,4,3) for fourth-order scheme [@ascher1997]) with a much smaller time step $\delta t = \Delta t/500$. We compute the solution to time $T=1$ and estimate the $L^2$ error of the solutions $U_{\Delta t}$ as $\left\Vert U_{\Delta t}-U_{ref}\right\Vert$. Table [1](#tab:ARZ-error){reference-type="ref" reference="tab:ARZ-error"} gives the $L^2$ error and convergence rates with respect to $\Delta t$ of IMEX-BDF schemes of order $q=2, 3, 4$ with $\varepsilon$ ranging from $10^{-7}$ to $1$. We can observe that the numerical results are in perfect agreement with our theoretical analysis for various values of $\varepsilon$. The minor order degeneration in the fourth-order scheme with $\Delta t=1.79\times10^{-4}$ is due to the machine precision limitations. --------------- ------------ -------------- ------- ------------- ------- -------------- ------- $\varepsilon$ $\Delta t$ second order third order fourth order $L^2$-error order $L^2$-error order $L^2$-error order $10^{-7}$ 1.43e-03 4.46e-04 \- 2.25e-06 \- 1.08e-08 \- 7.14e-04 1.11e-04 2.00 2.82e-07 3.00 6.74e-10 4.00 3.57e-04 2.75e-05 2.02 3.52e-08 3.00 4.24e-11 3.99 1.79e-04 6.55e-06 2.07 4.34e-09 3.02 3.14e-12 3.76 $10^{-6}$ 1.43e-03 4.46e-04 \- 2.25e-06 \- 1.08e-08 \- 7.14e-04 1.11e-04 2.00 2.82e-07 3.00 6.74e-10 4.00 3.57e-04 2.75e-05 2.02 3.52e-08 3.00 4.23e-11 4.00 1.79e-04 6.55e-06 2.07 4.34e-09 3.02 3.13e-12 3.76 $10^{-5}$ 1.43e-03 4.46e-04 \- 2.25e-06 \- 1.08e-08 \- 7.14e-04 1.11e-04 2.00 2.82e-07 3.00 6.74e-10 4.00 3.57e-04 2.75e-05 2.02 3.52e-08 3.00 4.24e-11 3.99 1.79e-04 6.56e-06 2.07 4.34e-09 3.02 3.14e-12 3.76 $10^{-4}$ 1.43e-03 4.46e-04 \- 2.25e-06 \- 1.07e-08 \- 7.14e-04 1.12e-04 1.99 2.81e-07 3.00 6.73e-10 4.00 3.57e-04 2.83e-05 1.98 3.52e-08 3.00 4.22e-11 3.99 1.79e-04 7.11e-06 1.99 4.40e-09 3.00 2.97e-12 3.83 $10^{-3}$ 1.43e-03 4.98e-04 \- 2.29e-06 \- 1.05e-08 \- 7.14e-04 1.34e-04 1.89 3.05e-07 2.91 6.76e-10 3.96 3.57e-04 3.52e-05 1.93 4.09e-08 2.90 4.46e-11 3.92 1.79e-04 8.67e-06 2.02 5.31e-09 2.95 3.01e-12 3.89 $10^{-2}$ 1.43e-03 5.81e-04 \- 2.71e-06 \- 7.40e-09 \- 7.14e-04 1.47e-04 1.98 3.48e-07 2.96 4.69e-10 3.98 3.57e-04 3.67e-05 2.00 4.41e-08 2.98 2.89e-11 4.02 1.79e-04 8.77e-06 2.06 5.48e-09 3.01 1.26e-12 4.52 $10^{-1}$ 1.43e-03 4.07e-04 \- 2.26e-06 \- 1.72e-08 \- 7.14e-04 9.97e-05 2.03 2.83e-07 3.00 1.08e-09 4.00 3.57e-04 2.46e-05 2.02 3.53e-08 3.00 6.71e-11 4.00 1.79e-04 5.86e-06 2.07 4.36e-09 3.02 4.15e-12 4.01 $10^{0}$ 1.43e-03 2.77e-03 \- 4.32e-05 \- 1.27e-06 \- 7.14e-04 3.37e-04 3.04 5.41e-06 3.00 7.94e-08 4.00 3.57e-04 8.34e-05 2.02 6.76e-07 3.00 4.97e-09 4.00 1.79e-04 1.97e-05 2.07 8.34e-08 3.02 3.19e-10 3.96 --------------- ------------ -------------- ------- ------------- ------- -------------- ------- : Aw-Rascle-Zhang traffic model: The $L^2$ error of the solutions computed by IMEX-BDF schemes of order $q=2,3,4$. ## Broadwell model The Broadwell model is a simplified discrete velocity model for the Boltzmann equation [@broadwell1964shock]. It describes a two-dimensional (2D) gas as composed of particles of only four velocities with a binary collision law and spatial variation in only one direction. When looking for one-dimensional solutions of the 2D gas, the evolution equations of the model are given by $$\nonumber \begin{aligned} \partial_t f_{+} + \partial_x f_{+} &= -\frac{1}{\varepsilon}(f_{+}f_{-} - f_0^2),\\ \partial_t f_{-} - \partial_x f_{-} &= -\frac{1}{\varepsilon}(f_{+}f_{-} - f_0^2),\\ \partial_t f_{0} &= \frac{1}{\varepsilon}(f_{+}f_{-} - f_0^2). \end{aligned}$$ Here $f_{+}$, $f_{-}$ and $f_0$ denote the particle density function at time $t$, position $x$ with velocity $1$, $-1$ and $0$, respectively, $\varepsilon>0$ is the mean free path. Set $$\nonumber \rho = f_{+} + 2f_0 + f_{-}, \quad m = f_{+} - f_{-}, \quad z = f_{+} + f_{-}.$$ The Broadwell equations can be rewritten as $$\nonumber \begin{aligned} \partial_t \rho + \partial_x m &= 0,\\ \partial_t m + \partial_x z &= 0,\\ \partial_t z + \partial_x m &= \frac{1}{2\varepsilon}( \rho^2 + m^2 - 2\rho z). \end{aligned}$$ A local Maxwellian is the density function that satisfies $z = \frac{1}{2\rho}(\rho^2 + m^2)$. Considering the linearized version at $\rho_\star = 2, m_\star=0, z_\star=1$, we obtain the linearized Broadwell system as follows $$\nonumber \begin{aligned} \partial_t U + A \partial_x U = \frac{1}{\varepsilon}QU, \end{aligned}$$ with $$\nonumber \begin{aligned} U = (\rho, ~m, ~z)^T, \qquad A=\begin{pmatrix} 0 & 1 & 0\\ 0 & 0 & 1\\ 0 & 1 & 0\\ \end{pmatrix}, \qquad Q=\begin{pmatrix} 0 & 0 & 0\\ 0 & 0 & 0\\ 1 & 0 & -2\\ \end{pmatrix}. \end{aligned}$$ It has been shown in [@yong_singular_1999] that the Broadwell model satisfies the structural stability condition. In our numerical test, the computational spatial domain is $[-\pi, \pi]$ with periodic boundary conditions and the initial data of $\rho$ and $m$ are given by $$\nonumber \begin{aligned} \rho(x, 0) = 1 + a_{\rho}\sin(2x), \quad m(x, 0) = \rho(x, 0) \left( \frac{1}{2} + a_{u}\cos(2x) \right)\\ % , \quad z(x, 0) = 0.5\rho(x, 0),\\ % &z(x, 0)=z^{eq}(\rho_0,m_0) = \frac{1}{2}(\rho(x,0)+\rho(x,0)u^2(x, 0))\\ % &a_\rho = 0.3, \quad a_u =0.1, \\ % &z(x, 0) = \frac{1}{2}\rho(x,0)+\frac{\varepsilon}{4}\partial_x m(x, 0),\\ % &z(x, 0) = 0.5\rho(x, 0) + 0.05 \varepsilon \rho(x, 0)\sin(2x) - 0.15\varepsilon u(x,0)\cos(2x). \end{aligned}$$ with $a_\rho = 0.3$ and $a_u =0.1$. For the second-order scheme, we choose the initial data for $z$ as $$\nonumber z(x, 0) = \frac{1}{2}\rho(x, 0),$$ which is consistent up to $O(1)$. For the third-order scheme, we choose the initial data for $z$ as $$\nonumber z(x, 0) = \frac{1}{2}\rho(x,0) - \frac{\varepsilon}{4}\partial_x m(x, 0),$$ which is consistent up to $O(\varepsilon)$. For the fourth-order scheme, we choose the initial data for $z$ as $$\nonumber z(x, 0) = \frac{1}{2}\rho(x,0) - \frac{\varepsilon}{4}\partial_x m(x, 0) - \frac{\varepsilon^2}{16}\partial_{xx} \rho(x, 0),$$ which is consistent up to $O(\varepsilon^2)$. The starting values at $i\Delta t$ with $i =1, \cdots, q-1$ are prepared using ARS(4,4,3) with a much smaller time step $\delta t = \Delta/500$. We compute the solution to time $T=2$ and estimate the error of the solutions $U_{\Delta t}$ as $\left\Vert U_{\Delta t}-U_{ref}\right\Vert$. In Table [2](#tab:Broadwel-error){reference-type="ref" reference="tab:Broadwel-error"}, we present the numerical results of IMEX-BDF schemes of order $q = 2, 3, 4$, and various values of $\Delta t$ and $\varepsilon$. The uniform $q-$th order accuracy is clearly achieved for $q = 2, 3, 4$. This closely aligns with our theoretical analysis. --------------- ------------ -------------- ------- ------------- ------- -------------- ------- $\varepsilon$ $\Delta t$ second order third order fourth order $L^2$-error order $L^2$-error order $L^2$-error order $10^{-7}$ 5.00e-03 4.58e-04 \- 4.07e-06 \- 5.03e-08 \- 2.50e-03 1.14e-05 2.00 5.09e-07 3.00 3.17e-09 3.99 1.25e-03 2.83e-05 2.02 6.36e-08 3.00 1.98e-10 4.00 6.25e-04 6.74e-06 2.07 7.84e-09 3.02 1.24e-11 4.00 $10^{-6}$ 5.00e-03 4.58e-04 \- 4.07e-06 \- 5.03e-08 \- 2.50e-03 1.14e-05 2.00 5.09e-07 3.00 3.17e-09 3.99 1.25e-03 2.83e-05 2.02 6.36e-08 3.00 1.99e-10 4.00 6.25e-04 6.74e-06 2.07 7.84e-09 3.02 1.25e-11 4.00 $10^{-5}$ 5.00e-03 4.59e-04 \- 4.07e-06 \- 5.03e-08 \- 2.50e-03 1.14e-05 2.00 5.09e-07 3.00 3.17e-09 3.99 1.25e-03 2.83e-05 2.02 6.36e-08 3.00 1.98e-10 4.00 6.25e-04 6.75e-06 2.07 7.84e-09 3.02 1.24e-11 4.00 $10^{-4}$ 5.00e-03 4.59e-04 \- 4.07e-06 \- 5.03e-08 \- 2.50e-03 1.15e-05 2.00 5.09e-07 3.00 3.17e-09 3.99 1.25e-03 2.84e-05 2.01 6.36e-08 3.00 1.98e-10 4.00 6.25e-04 6.80e-06 2.06 7.84e-09 3.02 1.25e-11 3.99 $10^{-3}$ 5.00e-03 4.64e-04 \- 4.06e-06 \- 5.02e-08 \- 2.50e-03 1.17e-05 1.98 5.08e-07 3.00 3.16e-09 3.99 1.25e-03 2.97e-05 1.98 6.34e-08 2.99 1.98e-10 4.00 6.25e-04 7.28e-06 2.06 7.83e-09 3.01 1.23e-11 4.00 $10^{-2}$ 5.00e-03 4.98e-04 \- 3.98e-06 \- 4.91e-08 \- 2.50e-03 1.27e-05 1.97 5.01e-07 2.99 3.09e-09 3.99 1.25e-03 3.20e-05 1.99 6.28e-08 2.99 1.94e-10 4.00 6.25e-04 7.68e-06 2.06 7.77e-09 3.01 1.20e-11 4.00 $10^{-1}$ 5.00e-03 5.00e-04 \- 3.69e-06 \- 4.04e-08 \- 2.50e-03 1.25e-05 2.00 4.64e-07 2.99 2.53e-09 3.99 1.25e-03 3.11e-05 2.01 5.81e-08 3.00 1.59e-10 4.00 6.25e-04 7.41e-06 2.07 7.17e-09 3.02 9.80e-12 4.02 $10^{0}$ 5.00e-03 5.13e-04 \- 7.61e-06 \- 1.17e-07 \- 2.50e-03 1.28e-05 2.00 9.51e-07 3.00 7.31e-09 4.00 1.25e-03 3.16e-05 2.02 1.19e-07 3.00 4.57e-10 4.00 6.25e-04 7.52e-06 2.07 1.46e-08 3.02 2.85e-11 4.00 --------------- ------------ -------------- ------- ------------- ------- -------------- ------- : Broadwell system: The $L^2$ error of the solutions computed by IMEX-BDF schemes of order $q=2,3,4$. ## Linearized Grad's moment system The linearized Grad's moment system in 1D [@grad1949kinetic; @cai2014cpam; @zhao2017stability] reads as $$\label{equ:linear-moment-equ} \partial_t U + A\partial_x U = \frac{1}{\varepsilon}QU$$ with $$\nonumber \begin{aligned} U = \begin{pmatrix} \rho \\ w \\ \theta/\sqrt{2} \\ \sqrt{3!}f_3 \\ \vdots \\\sqrt{M!}f_M \end{pmatrix}, A = \begin{pmatrix} 0 & 1 & & & & \\ 1 & 0 & \sqrt{2} & & & \\ & \sqrt{2} & 0 & \sqrt{3} & & \\ & & \sqrt{3} & 0 & \ddots & \\ & & & \ddots & 0 & \sqrt{M} \\ & & & & \sqrt{M} & 0 \end{pmatrix}, Q = -\mbox{diag}(0, 0, 0, \underbrace{1, \cdots, 1}_{M-2}). \end{aligned}$$ In the above equation, $\rho$ is the density, $w$ is the macroscopic velocity, $\theta$ is the temperature and $f_3, \cdots, f_M$ with $M\geq 3$ are high order moments. The moment system is obtained by taking moments on the both sides of the Bhatnagar-Gross-Krook (BGK) model [@Bhatnagar1954511]. It was shown in [@Di2017nm; @zhao2017stability] that the moment system satisfies the structural stability condition. Here we only consider its linearized version. The spatial domain is taken as $x\in [-\pi, \pi]$ with periodic boundary conditions. We solve the linearized Grad's moment system [\[equ:linear-moment-equ\]](#equ:linear-moment-equ){reference-type="eqref" reference="equ:linear-moment-equ"} with $M=5$. The initial data are prepared by $$\nonumber (\rho, ~ w, ~ \theta)(x, 0) = \left( \sin(2x)+1.1, ~ 0, ~ \sqrt{2} \right),\qquad (f_3, f_4, f_5) = (0, ~ 0, ~ 0).$$ The starting values at $i\Delta t, i = 0, \cdots, q-1$, are prepared using an ARS(4,4,3) scheme with a much smaller time step $\delta t = \Delta t/500$. We compute the solution to time $T=1$ and estimate the error of the solution $U_{\Delta t}$ as $\left\Vert U_{\Delta t}-U_{ref}\right\Vert$. Table [3](#tab:LinearMoment-error){reference-type="ref" reference="tab:LinearMoment-error"} show the $L^2$ error of IMEX-BDF schemes of order $q=2, 3, 4$, and various values of $\Delta t$ and $\varepsilon$. Again, we observe the uniform accuracy of the scheme with $\varepsilon$ ranging from $10^{-7}$ to 1. --------------- ------------ -------------- ------- ------------- ------- -------------- ------- $\varepsilon$ $\Delta t$ second order third order fourth order $L^2$-error order $L^2$-error order $L^2$-error order $10^{-7}$ 2.50e-03 1.04e-03 \- 1.34e-05 \- 9.29e-08 \- 1.25e-03 2.62e-04 2.00 1.68e-06 2.99 5.88e-09 3.98 6.25e-04 6.49e-05 2.01 2.10e-07 3.00 3.70e-10 3.99 3.13e-04 1.55e-05 2.07 2.59e-08 3.02 2.34e-11 3.98 $10^{-6}$ 2.50e-03 1.04e-03 \- 1.34e-05 \- 9.29e-08 \- 1.25e-03 2.62e-04 2.00 1.68e-06 2.99 5.88e-09 3.98 6.25e-04 6.49e-05 2.01 2.10e-07 3.00 3.70e-10 3.99 3.13e-04 1.55e-05 2.07 2.59e-08 3.02 2.33e-11 3.99 $10^{-5}$ 2.50e-03 1.04e-03 \- 1.34e-05 \- 9.29e-08 \- 1.25e-03 2.62e-04 2.00 1.68e-06 2.99 5.88e-09 3.98 6.25e-04 6.49e-05 2.01 2.10e-07 3.00 3.70e-10 3.99 3.13e-04 1.55e-05 2.07 2.59e-08 3.02 2.31e-11 4.00 $10^{-4}$ 2.50e-03 1.04e-03 \- 1.34e-05 \- 9.29e-08 \- 1.25e-03 2.62e-04 2.00 1.68e-06 2.99 5.88e-09 3.98 6.25e-04 6.49e-05 2.01 2.10e-07 3.00 3.70e-10 3.99 3.13e-04 1.55e-05 2.07 2.59e-08 3.02 2.31e-11 4.00 $10^{-3}$ 2.50e-03 1.04e-03 \- 1.34e-05 \- 9.31e-08 \- 1.25e-03 2.62e-04 2.00 1.68e-06 2.99 5.92e-09 3.97 6.25e-04 6.49e-05 2.01 2.10e-07 3.00 3.75e-10 3.98 3.13e-04 1.55e-05 2.07 2.59e-08 3.02 2.36e-11 3.99 $10^{-2}$ 2.50e-03 1.05e-03 \- 1.33e-05 \- 9.58e-08 \- 1.25e-03 2.62e-04 2.00 1.67e-06 2.99 6.09e-09 3.98 6.25e-04 6.50e-05 2.01 2.09e-07 3.00 3.84e-10 3.99 3.13e-04 1.55e-05 2.07 2.58e-08 3.02 2.41e-11 4.00 $10^{-1}$ 2.50e-03 1.06e-03 \- 1.31e-05 \- 1.01e-07 \- 1.25e-03 2.66e-04 2.00 1.65e-06 2.99 6.40e-09 3.98 6.25e-04 6.59e-05 2.01 2.06e-07 3.00 4.02e-10 3.99 3.13e-04 1.57e-05 2.07 2.54e-08 3.02 2.52e-11 4.00 $10^{0}$ 2.50e-03 1.07e-03 \- 1.26e-05 \- 1.01e-07 \- 1.25e-03 2.68e-04 2.00 1.58e-06 3.00 6.39e-09 3.99 6.25e-04 6.64e-05 2.01 1.98e-07 3.00 4.01e-10 3.99 3.13e-04 1.58e-05 2.07 2.44e-08 3.02 2.51e-11 4.00 --------------- ------------ -------------- ------- ------------- ------- -------------- ------- : Linearized Grad's moment system: The $L^2$ error of the solutions computed by IMEX-BDF schemes. 10 G. Akrivis and E. Katsoprinakis. 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W.-A. Yong. Singular perturbations of first-order hyperbolic systems with stiff source terms. , 155(1):89--132, 1999. W.-A. Yong. Basic aspects of hyperbolic relaxation systems. , pages 259--305, 2001. W.-A. Yong. An interesting class of partial differential equations. , 49(3):033503, 2008. Y. Zeng. Gas dynamics in thermal nonequilibrium and general hyperbolic systems with relaxation. , 150:225--279, 1999. H. M. Zhang. A non-equilibrium traffic model devoid of gas-like behavior. , 36(3):275--290, 2002. W. Zhao, W.-A. Yong, and L.-S. Luo. Stability analysis of a class of globally hyperbolic moment system. , 15(3):609--633, 2017. [^1]: Yanqi Lake Beijing Institute of Mathematical Sciences and Applications, Beijing 101408, China. E-mail: mazt\@bimsa.cn [^2]: Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX, 79409, USA. E-mail: juntao.huang\@ttu.edu. Research is partially supported by NSF DMS-2309655 and DOE DE-SC0023164. [^3]: Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China; and Yanqi Lake Beijing Institute of Mathematical Sciences and Applications, Beijing 101408, China. E-mail: wayong\@tsinghua.edu.cn. Research is partially supported by National Key Research and Development Program of China (Grant no. 2021YFA0719200) and the National Natural Science Foundation of China (Grant no.12071246).

arxiv_math

--- abstract: | We prove a general local rigidity theorem for pull-backs of homogeneous forms on reductive symmetric spaces under representations of discrete groups. One application of the theorem is that the volume of a closed manifold locally modelled on a reductive homogeneous space $G/H$ is constant under deformation of the $G/H$-structure. The proof elaborates on an argument given by Labourie for closed anti-de Sitter $3$-manifolds. The core of the work is a reinterpretation of old results of Cartan, Chevalley and Borel, showing that the algebra of $G$-invariant forms on $G/H$ is generated by "Chern--Weil forms" and "Chern--Simons forms". address: | ÉNS -- PSL, CNRS\ 45 rue d'Ulm\ 75005 Paris author: - Nicolas Tholozan bibliography: - biblio.bib title: Chern--Simons theory and cohomological invariants of representation varieties --- # Introduction {#introduction .unnumbered} Throughout the paper, we consider $G$ a connected real semisimple Lie group with finite center, $\sigma$ an involutive automorphism of $G$ and $H$ the subgroup of $G$ fixed by $\sigma$. The right quotient $G/H$ is called a *reductive symmetric space*. The involution $\sigma$ is called a *Cartan involution* when $H$ is compact. In that case, $H$ is a maximal compact subgroup of $G$ and $G/H$ is a Riemannian symmetric space of non-compact type sometimes called *the* symmetric space of $G$. By a result of E. Cartan, every $G$-invariant differential form on $G/H$ is closed, hence the cohomology of the complex of $G$-invariant forms on $G/H$ with values in $\mathbb{C}$ is isomorphic to the algebra of $G$-invariant forms, denoted $\Omega^\bullet_\textit{inv}(G/H,\mathbb{C})$. We also fix a smooth connected manifold $M$, with universal cover $\widetilde M$ and fundamental group $\pi_1(M)$. Let $\rho: \pi_1(M) \to G$ be a representation and $\widetilde s: \widetilde M \to G/H$ a smooth $\rho$-equivariant map. This map factors to a smooth section $s$ of the flat $G/H$-bundle $$M\times_\rho (G/H)$$ and the pull-back by $\widetilde s$ of any $\omega\in \Omega^\bullet_\textit{inv}(G/H,\mathbb{C})$ factors to a closed form $s^*\omega$ on $M$ (see Section [1.3](#ss: Flat bundles){reference-type="ref" reference="ss: Flat bundles"}). An application of Stokes' formula shows that the de Rahm cohomology class $[s^*\omega]$ of $s^*\omega$ only depends on the homotopy class of $s$. In particular, since the Riemannian symmetric space of $G$ is contractible, all the sections of the bundle $M\times_\rho (G/H)$ are homotopic. Thus, in that case, $[s^*\omega]$ only depends on the representation $\rho$, and we denote it by $\rho^*\omega$.\ ## Main results The purpose of this paper is to prove that the cohomology classes $[s^* \omega]$ do not vary when the representation $\rho$ moves in the representation variety $\mathrm{Hom}(\pi_1(M), G)$. To be more precise, let $(\rho_t)_{t\in [0,1]}$ be a smooth path in $\mathrm{Hom}(\pi_1(M),G)$. Any smooth section $s_0$ of the flat bundle $M\times_{\rho_0}(G/H)$ extends to a smooth family of sections $s_t$ of $M\times_{\rho_t}(G/H)$, unique up to homotopy (see Section [3.1](#ss: Families of bundles){reference-type="ref" reference="ss: Families of bundles"}). **Theorem 1**. *Let $\omega$ be a $G$-invariant form on $G/H$. For any smooth family $(\rho_t)_{t\in [0,1]}$ of representations of $\pi_1(M)$ into $G$ and any smooth family $(s_t)_{t\in [0,1]}$ of sections of $M\times_{\rho_t} (G/H)$, we have $$[s_t^*\omega] = [s_0^*\omega]$$ for all $t\in [0,1]$.* The general formulation of this theorem is meant to encompass two situations: the one where $K$ is a maximal compact subgroup (in which case the choice of $s$ is irrelevant) and the one where $s$ is a local diffeomorphism and $\omega$ a $G$-invariant volume form. We now detail the consequences of Theorem [Theorem 1](#t: Main Theorem){reference-type="ref" reference="t: Main Theorem"} in these two contexts. ### Pull-back of group cohomology In this section, we take $H = K$ to be a maximal compact subgroup of $G$. The *Van Est isomorphism* $\mathop{\mathrm{VE}}$ identifies the algebra of $G$-invariant forms on $G/K$ to the *continuous cohomology* $\mathrm{H}^\bullet_c(G,\mathbb{C})$ (see Section [3.3](#ss: Continuous group cohomology Proof ){reference-type="ref" reference="ss: Continuous group cohomology Proof "}). Approaching the classifying space of any finitely presented group $\Gamma$ by closed manifolds, we get from Theorem [Theorem 1](#t: Main Theorem){reference-type="ref" reference="t: Main Theorem"} the following corollary: **Corollary 2**. *Let $\Gamma$ be a finitely presented group. Then, for every $\omega \in \mathrm{H}^\bullet_c(G,\mathbb{C})$, the map $$\begin{aligned} \mathrm{Hom}(\Gamma,G) &\to & \mathrm{H}^\bullet(\Gamma, \mathbb{C}) \\ \rho & \mapsto & \rho^*\omega \end{aligned}$$ is locally constant on $\mathrm{Hom}(\Gamma,G)$.* The cohomology classes $\rho^*\omega$ define topological invariants that are constant on connected components of $\mathrm{Hom}(\Gamma, G)$ and can sometimes help distinguish these components. A famous example of this phenomenon is when $\Gamma$ is the fundamental group of a closed oriented surface $S$ of genus $g\geq 2$ and $G= \mathrm{PSL}(2,\mathbb{R})$. For a suitably chosen generator $\omega$ of $\mathrm{H}^\bullet_c(\mathrm{PSL}(2,\mathbb{R}), \mathbb{C}) \simeq \mathbb{C}$, Milnor [@Milnor58] proved that the number $$\int_S \rho^*\omega$$ is an integer contained in the interval $[2-2g, 2g-2]$, called the *Euler class* of the representation $\rho$. Moreover, Goldman [@Goldman88] and Hitchin [@Hitchin87] proved that the Euler class classifies the connected components of $\mathrm{Hom}(\Gamma, \mathrm{PSL}(2,\mathbb{R}))$. ### Volume of locally homogeneous manifolds In this section, $G$ is a semisimple Lie group and $H$ any reductive subgroup. Let $M$ be an orientable manifold of the same dimension as $G/H$. A *$G/H$-structure* on $M$ can be defined as the data of a pair $(\mathrm{dev}, \rho)$ where $\rho:\pi_1(M) \to G$ is a homomorphism called the *holonomy* of the $G/H$-structure and $\mathrm{dev}: \widetilde M \to G/H$ is a local diffeomorphism called the *developing map*. The hypotheses on $G$ and $H$ imply the existence of a $G$-invariant volume forme $vol_{G/H}$ on the homogeneous space $G/H$, whose pull-back by the developping factors to a volume form $\mathrm{dev}^*vol_{G/H}$ on $M$. The *volume* of the $G/H$-structure is by definition the integral of this volume form over $M$ $$\mathbf{Vol}(M, \mathrm{dev}) = \int_M \mathrm{dev}^* vol_{G/H}~.$$ The Ehresmann--Thurston principle (see [@BergeronGelander04]) roughly states that every deformation of the holonomy representation corresponds (in an essentially unique way) to a deformation of the $G/H$-structure of $M$. Theorem [Theorem 1](#t: Main Theorem){reference-type="ref" reference="t: Main Theorem"} implies that the volume is constant along such deformations. **Corollary 3**. *Let $G$ be a semisimple Lie group with finite center and finitely many components and $H$ a reductive subgroup of $G$. Let $M$ be a closed manifold of the same dimension as $G/H$ and $(\mathrm{dev}_t, \rho_t)_{t\in [0,1]}$ a continuous family of $G/H$-structures on $M$. Then $$\mathbf{Vol}(M, \mathrm{dev}) = \mathbf{Vol}(M, \mathrm{dev}_0)$$ for all $t\in [0,1]$.* ## Earlier results Theorem [Theorem 1](#t: Main Theorem){reference-type="ref" reference="t: Main Theorem"} contains as a particular case many topological invariants that have been extensively studied before, such as Toledo invariants of surface group representations (or more generally Kähler groups) into Lie groups of Hermitian type [@BIW10; @Toledo89], volume of representations of hyperbolic lattices [@BBI13; @Francaviglia04; @KimKim13; @BCG07], complex volume of representations of $3$-manifold groups [@GTZ15; @BFG14], or volume of compact quotients of reductive homogeneous spaces. To our knowledge, in all these examples, Theorem [Theorem 1](#t: Main Theorem){reference-type="ref" reference="t: Main Theorem"} is already known, and our intention here is merely to give a (almost) unified presentation of these different results. Let us discuss some of these applications. ### Toledo invariant of surface group representations Let $S$ be a closed surface. The Toledo invariant generalizes the Euler class to representations of $\pi_1(S)$ into a simple Lie group $G$ of Hermitian type. In that case, the cohomology group $\mathrm{H}^2_c(G,\mathbb{C})$ is generated by a class $\omega$ (corresponding to a $G$-invariant Kähler form on the symmetric space). Given a representation $\rho: \pi_1(S)\to G$, the Toledo invariant of $\rho$ is defined by $$\tau(\rho) = \int_S \rho^*\omega~.$$ It is rational, satisfies a Milnor--Wood inequality, and representations with maximal Toledo invariant have strong geometric properties [@Toledo89; @BIW10]. Unlike in the case of $\mathrm{PSL}(2,\mathbb{R})$ however, the Toledo invariant does not always distinguish connected components of $\mathrm{Hom}(\Gamma, G)$ (see [@BGG06]). The local rigidity of the Toledo invariant (and, in fact, its rationality) is a well-known consequences of Chern--Weil theory: one shows that the Toledo invariant is the degree of the pull-back of a complex automorphic line bundle on $G/K$. ### Volume of representations into $\mathrm{Isom}_+(\mathbb{H}^d)$ Another well-studied generalization of the Euler class is when $M$ is a closed $d$-manifold and $G= \mathrm{SO}_\circ(d,1)$ is the group of orientation preserving isometries of the hyperbolic $d$-space $\mathbb{H}^d$. The continuous cohomology of $\mathrm{SO}_\circ(d,1)$ is generated in degree $d$ by a class $\omega$ corresponding to the volume form of the hyperbolic $d$-space. Given $\rho: \pi_1(M) \to\mathrm{SO}_\circ(d,1)$, the number $\int_M \rho^*\omega$ is called the *volume* of the representation $\rho$. For even $d$, the volume of $\rho$ is (up to a universal constant) the Euler class of the pull-back of the tangent bundle $T \mathbb{H}^d$ and is thus an integer. In contrast, for odd $d$, the volume cannot be directly related to a characteristic class, and its values remain mysterious. Besson--Courtois--Gallot proved in [@BCG07] that the volume is locally constant on $\mathrm{Hom}(\pi_1(M), \mathrm{SO}_\circ (d,1))$, using Schläfli's formula for the variation of the volume of a simplex. This proof seems quite specific to hyperbolic geometry. An alternative proof in dimension $3$ follows from the identification of the volume of $\rho$ with the imaginary part of its *Cheeger--Chern--Simons invariant*. While this proof was probably known to experts before Besson--Courtois--Gallot's work, it seems it was only written later in [@GTZ15]. Theorem [Theorem 1](#t: Main Theorem){reference-type="ref" reference="t: Main Theorem"} gives in particular an alternative proof of Besson--Courtois--Gallot's theorem which, in odd dimension, is based on Chern--Simons theory and is strongly related to [@GTZ15]. ### Complex volumes of representations of $3$-manifold groups In [@GTZ15], Garoufalidis, Thurston and Zickert generalize the Cheeger--Chern--Simons invariant to representations of the fundamental group of a closed $3$-manifold $M$ into $\mathrm{SL}(d,\mathbb{C})$ (and, in fact, any simply connected simple complex Lie group). In that case, the group $G$ itself carries a holomorphic bi-invariant $3$-form $\omega$, given on the Lie algebra $\mathfrak{g}$ by $$\omega(u,v,w)= \mathrm{Tr}(u [v,w])~.$$ Let $\rho$ be a representation of $\pi_1(M)$ into $\mathrm{SL}(d,\mathbb{C})$. Since $\mathrm{SL}(d,\mathbb{C})$ is connected and simply connected and $M$ has dimension $3$, the flat bundle $M\times_\rho \mathrm{SL}(d,\mathbb{C})$ always admits a section $s$. Moreover, given two such sections $s_1$ and $s_2$, the difference $\int_M s_1^* \omega - \int_M s_2^*\omega$ is an integral multiple of $4\pi^2$. The *complex volume* of $\rho$ is then defined by $$\mathbf{Vol}_\mathbb{C}(\rho) = - i \int_M s^*\omega\in \mathbb{C}/4\pi^2i\mathbb{Z}~.$$ When $n=3$, the real part of $\mathbf{Vol}_\mathbb{C}(\rho)$ is the volume of $\rho$ defined above. This complex volume can be interpreted as a Chern--Simons class. More precisely, it is the Cheeger--Chern--Simons invariant of the flat connection of monodromy $\rho$ on the trivial complex vector bundle of rank $n$. General arguments of Chern--Simons theory then imply that it is locally constant on $\mathrm{Hom}(\pi_1(M),\mathrm{SL}(n,\mathbb{C}))$. The present work can be viewed as a generalisation of this fact.\ *Remark 1*. Both the Toledo invariant, the volume of representions in $\mathrm{Isom}_+(\mathbb{H}^d)$ and the complex volume of representations of $3$-manifold groups have also been defined and investigated of manifolds $M$ with boundary (see [@BIW10; @BBI13; @KimKim13; @Francaviglia04; @GTZ15; @BFG14]). While some local rigidity theorems have been proven in higher dimension, when $M$ has dimension $2$ or $3$ the Toledo invariant or complex volume are typically not locally constant on character varieties. It leads to a very different story that we do not consider further here. ### Volume anti-de Sitter $3$-manifolds Despite its proximity with the local rigidity results mentioned above, Corollary [Corollary 3](#c: Main Corollary Volume){reference-type="ref" reference="c: Main Corollary Volume"} on the volume of $G/H$-manifolds did not seem to be known before. In the first interesting case where $G/H$ is the *anti-de Sitter space of dimension $3$*, the question whether the volume is invariant under continuous deformations was asked in the influential survey [@QuestionsAdS Question 2.3].\ Recall that the anti-de Sitter $3$-space $\mathrm{AdS}^3$ can be seen as the Lie group $\mathrm{PSL}(2,\mathbb{R})$ equipped with its Killing metric. Its isometry group is (up to finite index) the group $\mathrm{PSL}(2,\mathbb{R})\times \mathrm{PSL}(2,\mathbb{R})$ acting by left and right multiplication. Anti-de Sitter structures on $3$-manifolds have been extensively studied [@KulkarniRaymond85; @Goldman85; @Salein00; @KasselThese; @Tholozan3] and are now well-understood. In particular, Kassel (partly relying on previous works) proved in [@KasselThese] that, up to finite covers, closed anti-de Sitter $3$-manifolds have the form $$(j, \rho)(\pi_1(S))\backslash \mathrm{PSL}(2,\mathbb{R})$$ where $S$ is a closed hyperbolic surface, $j:\pi_1(S)\to \mathrm{PSL}(2,\mathbb{R})$ is the holonomy of the hyperbolic metric on $S$, and $\rho: \pi_1(S)\to \mathrm{PSL}(2,\mathbb{R})$ is another representation such that there exists a $(j,\rho)$-equivariant contracting map from $\mathbb{H}^2$ to $\mathbb{H}^2$. Since this is an open condition on the pair $(j,\rho)$, closed anti-de Sitter $3$-manifolds have a rich deformation theory, which is completely described in my thesis [@TholozanThese Chapter 4] (see also [@Tholozan3]). Using Kassel's description of closed anti-de Sitter $3$-manifolds and with an explicit differential geometric computation, I proved in [@Tholozan5] the following formula for their volume: $$\label{eq: Volume AdS} \mathbf{Vol}\left((j,\rho)(\pi_1(S))\backslash \mathrm{PSL}(2,\mathbb{R})\right) = \frac{\pi^2}{2}(\mathbf{eu}(j) + \mathbf{eu}(\rho))~,$$ where $\mathbf{eu}$ denotes the Euler class of a representation. In particular, the volume is constant along continuous deformations of the anti-de Sitter structure. Upon hearing about this result, Labourie got the intuition that it could be derived from Chern--Simons theory, which lead him to present an alternative proof at an MSRI seminar [@LabourieMSRI]. Labourie's proof was never published, but he explained it to me in details, which it sparked the present work. Let us also mention that another proof of [\[eq: Volume AdS\]](#eq: Volume AdS){reference-type="eqref" reference="eq: Volume AdS"} was given by Alessandrini and Li [@AlessandriniLi15] using Higgs bundles. Finally, I show in [@Tholozan5] that the local rigidity of the volume is not true anymore when one considers finite volume non-compact quotients of $\mathrm{AdS}^3$. ### Volume of compact quotients Generalizing my work on the volume of anti-de Sitter $3$-manifolds, I inverstigated more systematically in [@Tholozan6] the volume of *compact quotients* of reductive symmetric spaces $G/H$, i.e. quotients of $G/H$ by a discrete subgroup $\Gamma$ of $G$ acting freely, properly discontinuously and cocompactly. There, I proved the following formula: $$\label{eq: Volume compact quotients} \mathbf{Vol}(\Gamma \backslash G/H) = \int_{[\Gamma]} \iota^*\omega_{G,H}~,$$ where $\iota: \Gamma \to G$ is the inclusion, $\omega_{G/H}$ is a class in $\mathrm{H}^\bullet_c(G,\mathbb{C})$ depending only on $H$, and $[\Gamma]$ is a certain *fundamental class* in $\mathrm{H}_\bullet(\Gamma, \mathbb{Z})$. In many cases, the class $\omega_{G,H}$ happens to be a characteristic class, and one deduces the rationality of $\mathbf{Vol}(\Gamma \backslash G/H)$ (and, in particular, its local rigidity). Equation [\[eq: Volume compact quotients\]](#eq: Volume compact quotients){reference-type="ref" reference="eq: Volume compact quotients"} also allows to readily apply Corollary [Corollary 2](#c: Main Corollary Cohomology){reference-type="ref" reference="c: Main Corollary Cohomology"} to prove the volume rigidity of compact quotients of $G/H$. Corollary [Corollary 3](#c: Main Corollary Volume){reference-type="ref" reference="c: Main Corollary Volume"} is more general in that it deals with any closed manifold locally modelled on $G/H$, which might not a priori be *complete* (i.e. a quotient of the model). While it is conjectured that every closed manifold locally modelled on a reductive homogeneous space is complete (a variation on Markus' conjecture), this conjecture is far from being solved, and it is not even known in general whether completeness is stable under small deformations. It is thus more satisfying to have a volume rigidity result without any completeness assumption. ## Cohomology of symmetric spaces As we just saw, there are many situations where Theorem [Theorem 1](#t: Main Theorem){reference-type="ref" reference="t: Main Theorem"} was known to follow either from classical arguments in Chern--Weil or Chern--Simons theory. The core of this work will thus be to prove that these arguments in fact account for all the $G$-invariant forms on $G/H$, i.e. that $\Omega^\bullet_\textit{inv}(G/H)$ is generated by *Chern--Weil forms* and *Chern--Simons forms* (which will be properly defined in Section [2](#s: Invariant forms on symmetric spaces){reference-type="ref" reference="s: Invariant forms on symmetric spaces"}). This will be done by reinterpreting old results of Cartan, Chevalley and Borel on the cohomology of compact symmetric spaces in light of Chern--Simons theory.\ Recall that the algebra of $G$-invariant forms on $G/H$ is canonically isomorphic to the cohomology algebra of the *dual compact symmetric space* $G_U/H_U$ (see Section [Proposition 16](#prop: Comparison cohomology compact dual){reference-type="ref" reference="prop: Comparison cohomology compact dual"}). We first establish the following version of Cartan--Borel's structure theorem for this cohomology: **Theorem 4**. *Let $G/H$ be a compact symmetric space. We have $$\mathrm{H}^\bullet(G/H, \mathbb{C}) = \mathrm{H}^\bullet_\textit{even}(G/H, \mathbb{C}) \otimes \mathrm{H}^\bullet_\textit{odd}(G/H,\mathbb{C})~,$$ where* - *the subalgebra $\mathrm{H}^\bullet_\textit{even}(G/H, \mathbb{C})$ is the algebra of characteristic classes of the *tautological principal $H$-bundle* $G \to G/H$;* - *the subalgebra $\mathrm{H}^\bullet_\textit{odd}(G/H, \mathbb{C})$ is the pull-back of $\mathrm{H}^\bullet(G,\mathbb{C})$ under the map $$\label{eq: square map} \begin{array}{cccc} \iota_{G,H}: & G/H & \to & G\\ & gH & \mapsto & g \sigma(g)^{-1}~. \end{array}$$* The map $\iota_{G,H}$ defined in [\[eq: square map\]](#eq: square map){reference-type="eqref" reference="eq: square map"} will play a crucial role throughout the paper. *Remark 2*. While the above theorem is essentially due Cartan and Borel, we do not know if the interpretation of $\mathrm{H}^\bullet_\textit{odd}(G/H,\mathbb{C})$ as a pull-back under the map $\iota_{G,H}$ was explicitly stated before. To understand the odd part of the cohomology of $G/H$ it is thus enough to understand the cohomology of a compact semisimple Lie group $G$. The cohomology of $G$ does not consist of Chern--Weil characteristic classes, but (perhaps unsurprisingly to experts), we will prove that it is generated by *Chern--Simons classes*. More precisely, consider the space $X_G= G$ equipped with the action of $G\times G$ by left and right multiplication, and let $P_G$ be the principal $G$-bundle $$G\times G \to X_G~.$$ The bundle $P_G$ carries two invariant flat connections $\Theta_L$ and $\Theta_R$, corresponding respectively to left and right parallelism on $G$. We will prove the following: **Lemma 5**. *The algebra $\mathrm{H}^\bullet(G,\mathbb{C})$ is generated by the *Chern--Simons classes* associated to the pair of connections $(\Theta_L, \Theta_R)$ on the principal $G$-bundle $P_G$.* ## Assumptions on $G$ It is easy to find counterexamples to Theorem [Theorem 1](#t: Main Theorem){reference-type="ref" reference="t: Main Theorem"} if one removes the assumption that $G$ is semisimple. For instance, let $\mathbb E^2 \overset{\mathrm{def}}{=}\mathrm{O}(2)\ltimes \mathbb{R}^2 / \mathrm{O}(2)$ be the Euclidean plane and $\omega$ its translation invariant area form. Let $M$ be the $2$-torus $\mathbb{R}^2/\mathbb{Z}^2$. Consider the family of $\mathbb E^2$-structures on $M$ given by $$\begin{array}{rrcl} \mathrm{dev}_t : & \widetilde M = \mathbb{R}^2 & \to & \mathbf{E}^2 = \mathbb{R}^2 \\ \ & (x,y) & \mapsto & (tx,ty) \end{array}$$ and $$\begin{array}{rrcl} \rho_t : & \pi_1(M)= \mathbb{Z}^2 & \to & \mathbb{R}^2 \subset \mathrm{Isom}(\mathbf{E}^2) \\ \ & (u,v) & \mapsto & (tu,tv) \end{array} ~.$$ Then one easily sees that $$\int_M \mathrm{dev}_t^* \omega = t^2~,$$ which is thus not constant in $t$. This example can of course be broadly generalized. Let $G/H$ be a homogeneous space such that $G$ admits a non trivial normalizer $N(G)$ in $\mathop{\mathrm{Diff}}(G/H)$. Then $N(G)$ acts on $\Omega_\textit{inv}^\bullet(G/H,\mathbb{C})$. Let $\omega$ be a $G$-invariant form on $G/H$ which is not fixed by $N(G)$. Then conjugating a representation $\rho:\pi_1(M) \to G$ by $N(G)$ will typically change the class $\rho^* \omega$.\ ## Structure of the paper Though many of the results contained here will perhaps be unsurprising to experts, I could not find references that embrace precisely what I need of Chern--Simons theory and cohomology of symmetric spaces. The paper with thus try to be as self-contained as possible. In Section [1](#s: Chern--Weil and Chern--Simons){reference-type="ref" reference="s: Chern--Weil and Chern--Simons"}, after recalling some background, we introduce Chern--Weil and Chern--Simons forms associated to connections on a principal bundle, and their relations with characteristic classes. Section [2](#s: Invariant forms on symmetric spaces){reference-type="ref" reference="s: Invariant forms on symmetric spaces"} is devoted to the description of the algebra $\Omega^\bullet_\textit{inv}(G/H)$. Relying on the work of Cartan and Borel, we prove Theorems [Theorem 4](#t: Cohomology symmetric space){reference-type="ref" reference="t: Cohomology symmetric space"} and Lemma [Lemma 5](#l: Cohomology Lie group Chern-Simons){reference-type="ref" reference="l: Cohomology Lie group Chern-Simons"}, and deduce that $\Omega^\bullet_\textit{inv}(G/H,\mathbb{C})$ is generated by Chern--Weil and Chern--Simons forms. Finally in Section [3](#s:Local rigidity){reference-type="ref" reference="s:Local rigidity"}, we recall the classical rigidity results for Chern--Weil and Chern--Simons classes, leading to the proof of Theorem [Theorem 1](#t: Main Theorem){reference-type="ref" reference="t: Main Theorem"}, and its corollaries. ## Acknoledgements {#acknoledgements .unnumbered} This paper is very much indebted to François Labourie, who explained to me the nuts and bolts of Chern--Simons theory and how it could be used to prove volume rigidity of locally homogeneous manifolds. I thank him more generally for the inspiration that his work has been for my research. # Chern--Weil and Chern--Simons classes {#s: Chern--Weil and Chern--Simons} Recall that $G$ is a real connected semisimple Lie group. Let $\mathfrak{g}$ denote the Lie algebra of $G$. ## Principal bundles A *principal $G$-bundle* over a manifold $M$ is a smooth fiber bundle $$p:P\to M$$ equiped with a smooth right action of $G$ preserving the fibers and acting simply transitively on each fiber. If $V$ is a manifold equipped with a smooth left action of $G$ (for instance, a linear representation of $G$ or a $G$-homogeneous manifold), one can associate to any principal bundle $P$ a fiber bundle $$P\times_G V = P \times V/\langle (p,v)\sim (pg, g^{-1}v), g\in G\rangle$$ called the *associated $V$-bundle*. In particular, the *adjoint bundle* of a principal bundle $P$ is the vector bundle $$\mathrm{Ad}(P) = P\times_G \mathfrak{g}$$ associated to the adjoint representation of $G$. Let $P$ be a principal $G$-bundle over a manifold $M$ and $H$ a Lie subgroup of $G$. The right quotient $P/H$ is canonically isomorphic to the associated bundle $P\times_G (G/H)$. A *reduction of structure group to $H$* of $P$ is a principal $H$-bundle $P'$ equipped with a bundle map $\varphi: P'\to P$ that commutes with the right $H$-action. The image of $\varphi$ contains a unique right $H$-orbit in each fiber of $P$ and thus factors to a section of the associated bundle $P/H$. Conversely, the preimage in $P$ of a section of $P/H$ is a reduction of structure group of $P$ to $H$. A *gauge transformation* of a principal bundle $\pi: P\to M$ is a bundle automorphism (i.e. a diffeomorphism $h:P\to P$ such that $\pi\circ h = \pi$) that commutes with the right action of $G$. The group of gauge transformations is canonically identified with the group of smooth sections of the associated bundle $$\mathrm{Aut}(P) \overset{\mathrm{def}}{=}P\times_G G~,$$ where $G$ acts on itself by conjugation. ## Connections, curvature Every $u\in \mathfrak{g}$ defines a vector field on $P$ by taking the derivative of the $G$-action, that we denote by $X_u$. The vector fields $X_u$ are tangent to the fibers of the bundle, and the map $u\mapsto X_u$ defines a trivialization of the subbundle $T F$ of $TP$ tangent to the fibers. **Definition 3**. A *(principal) connection* on $P$ is a $1$-form $\Theta$ with values in $\mathfrak{g}$, satisfying the following properties: - $\Theta(X_u) =u$ for all $u\in \mathfrak{g}$ - $g^* \Theta = \mathrm{Ad}_g \circ \Theta$ for all $g\in G$. The difference between two principal connections $\Theta$ and $\Theta'$ is a $1$-form with values in $\mathfrak{g}$ satisfying the following properties: - $(\Theta- \Theta')(X_u) = 0$ for all $u\in \mathfrak{h}$, - $g^*(\Theta-\Theta') = \mathrm{Ad}_g\circ (\Theta-\Theta')$ for all $g\in G$. It thus factors to a $1$-form on $M$ with values in $\mathrm{Ad}(P)$. Hence the space $\mathop{\mathrm{Conn}}(P)$ of principal connections on $P$ is an affine space over the space $\Omega^1(M,\mathrm{Ad}(P))$. The kernel of a principal connection is a distribution transverse to $TF$ and defines an Ehresmann connection with holonomy in $G$. The $2$-form $$R_\Theta = \mathrm{d}\Theta + \frac12 [\Theta, \Theta]$$ with values in $\mathfrak{g}$ has the following properties: - $R_\Theta(X_u, \cdot) = 0$ for all $u\in \mathfrak{g}$, - $g^* R_\Theta = \mathrm{Ad}_g \circ R_\Theta$ for all $g\in G$. It follows that this form factors to a $2$-form on $M$ with values in $\mathrm{Ad}(P)$ called the *curvature form* of $\Theta$, and that we still denote $R_\Theta$. Recall that, given a smooth map $f$ from a manifold $N$ to $M$, one can pull-back a principal bundle $P\to M$ by setting $$f^*P = \{(x,p)\in N\times P\mid \pi(p)= f(x)\}~.$$ If $P$ is equipped with a principal connection $\Theta$, then $f^*\Theta$ is a principal connection on $f^*P$ with curvature form $f^*R_\Theta$. ## Flat bundles {#ss: Flat bundles} The curvature form $R_\Theta$ vanishes if and only if the distribution $\ker(\Theta)$ is integrable, in which case the connection is called *flat*. The connection $\Theta$ is flat if and only if $P$ locally admits *parallel sections*, i.e. section $s$ such that $s^*\Theta \equiv 0$. Given $\rho$ a representation of $\pi_1(M)$ into $G$, define $$P_\rho = \widetilde M \times G/ \langle (x,g)\sim (\gamma\cdot x, \rho(\gamma) g), \gamma \in \pi_1(M) \rangle~.$$ Then $P_\rho$ is a principal $G$-bundle and the "trivial" flat connection on $\widetilde M \times G$ factors to a flat connection $\Theta_\rho$ on $P_\rho$. Conversely if $P$ is a principal bundle equipped with a flat connection $\Theta$, local parallel sections globalize over the universal cover, and one deduces that $(P,\Theta)$ is isomorphic to $(P_\rho,\Theta_\rho)$ for a representation $\rho$ called the *holonomy* of the flat connection. Actually, the holonomy is only defined up to conjugation in $G$ (corresponding to the choice of a trivialisation over the universal cover). Given a left action of $G$ on a manifold $V$, we have a canonical isomorphism $$P_\rho \times_G V \simeq M\times_\rho V \overset{\mathrm{def}}{=}\widetilde M \times V/ \langle (x,v)\sim (\gamma\cdot x, \rho(\gamma) g), \gamma \in \pi_1(M) \rangle~.$$ If $s$ is a section of the fiber bundle $M\times_\rho V \to M$, the lift $\widetilde s$ of $s$ to $\widetilde M$ is a map from $\widetilde M$ to $V$ which is *$\rho$-equivariant*, i.e. satisfies $$\widetilde s(\gamma \cdot x) = \rho(\gamma) \cdot \widetilde s(x)$$ for all $\gamma \in \pi_1(M)$. Conversely, any $\rho$-equivariant map $\widetilde s: \widetilde M \to V$ factors to a section $s$ of $M\times_\rho V$. If $\omega$ is a $G$-invariant form on $V$, then the form $\widetilde s^* \omega$ on $\widetilde M$ is $\pi_1(M)$-invariant and thus factors to a form on $M$ denoted $s^*\omega$. ## Characteristic classes and the Chern--Weil homomorphism Let $EG$ be a contractible CW complex equipped with a free and proper right action of the topological group $G$. The quotient space $BG$ is called a *classifying space* for the (topological) group $G$ and $EG$ a *universal principal $G$-bundle*. It is universal in the sense that, given any principal $G$-bundle $P\to M$, there exists a continuous map $f_P:M \to BG$, such that $P$ is isomorphic to the pull-back of the bundle $EG$ by $f_P$. Moreover, $f_P$ is unique up to homotopy. In particular, any two classifying spaces are homotopy equivalent. **Definition 4**. Let $P\to M$ be a principal $G$-bundle. The *algebra of characteristic classes* of $P$ is the image of the homomorphism $$f_P^*: \mathrm{H}^\bullet(BG,\mathbb{C}) \to \mathrm{H}^\bullet(M,\mathbb{C})~.$$ Here, the cohomology can a priori be taken with coefficients in an arbitrary domain. Let $\mathrm{Sym}^\bullet_\textit{inv}(\mathfrak{g}^\vee)$ denote the graded algebra of $\mathbb{C}$-valued polynomials on $\mathfrak{g}$ invariant under the adjoint action, with grading given by twice the degree. We will always identify homogeneosu polynomials of degree $k$ with symmetric $k$-linear forms. Let $P\to M$ be a principal $G$-bundle and $\Theta$ a connection on $P$. For any homogeneous polynomial $f$ in $\mathrm{Sym}^k_\textit{inv}(\mathfrak{g}^\vee)$, the form $$\mathop{\mathrm{CW}}_f(\Theta) \overset{\mathrm{def}}{=}f(R_\Theta)$$ is a well-defined $2k$-form on $M$ with coefficients in $\mathbb{C}$, which we call a *Chern--Weil form*. Note that Chern--Weil forms are natural with respect to pull-backs: if $P\to M$ is a principal bundle equipped with a connection $\Theta$ and $\varphi:N\to M$ is a smooth map, then $$\mathop{\mathrm{CW}}_f(\varphi^*\Theta) = \varphi^* \mathop{\mathrm{CW}}_f(\Theta)~.$$ One can show that $\mathop{\mathrm{CW}}_f(\Theta)$ is closed, and that its de Rham cohomology class $\mathop{\mathrm{cw}}_f(P)$ does not depend on the connection. We thus get a homomorphism of graded algebras $$\label{eq: Morphism de Chern--Weil} \begin{array}{rrcl} \Phi_P : & \mathrm{Sym}^\bullet_\textit{inv}(\mathfrak{g}^\vee) & \to & \mathrm{H}^\bullet(M,\mathbb{C}) \\ \ & f & \mapsto & \mathop{\mathrm{cw}}_f(P) \end{array} ~.$$ **Theorem 5** (Chern--Weil). *There exists a homomorphism of graded algebras $$\Phi_{EG}: \mathrm{Sym}^\bullet_\textit{inv}(\mathfrak{g}^\vee) \to \mathrm{H}^\bullet(BG,\mathbb{C})$$ such that, for any principal bundle $P\to M$, the following diagram commutes: $$\xymatrix{ \mathrm{Sym}^\bullet_\textit{inv}(\mathfrak{g}^\vee) \ar[r]^{\Phi_{EG}} \ar[dr]_{\Phi_P} & \mathrm{H}^\bullet(BG,\mathbb{C}) \ar[d]^{f_P^*} \\ & \mathrm{H}^\bullet(M,\mathbb{C})~. }$$ Moreover, when $G$ is compact, $\Phi_{EG}$ is an isomorphism.* *Remark 6*. The notation $\Phi_{EG}$ is meaningful here: $\Phi_{EG}$ is formally the homomorphism of [\[eq: Morphism de Chern\--Weil\]](#eq: Morphism de Chern--Weil){reference-type="eqref" reference="eq: Morphism de Chern--Weil"} of the universal principal bundle, and it is indeed constructed as an inductive limit of $\Phi_{P_n}$ for principal bundles ${P_n}$ "approaching" $EG$. *Remark 7*. If $G$ is not compact, let $K$ be a maximal compact subgroup of $G$. Then $BK = EG/K$ is a classifying space for $K$ and the fibration $BK\to BG$ is a homotopy equivalence since its fibers $G/K$ are contractible. We thus get that $\mathrm{H}^\bullet(BG,\mathbb{C})\simeq \mathrm{H}^\bullet(BK,\mathbb{C})$. Over a manifold $M$, this translates into the fact that every principal $G$-bundle $P$ admits a reduction of structure group $P'$ to $K$, unique up to homotopy, and the characteristic classes of $P$ are those of $P'$. The integral structure on the cohomology of $BG$ can be transported to $\mathrm{Sym}^\bullet_\textit{inv}(\mathfrak{g}^\vee)$: **Definition 8**. A polynomial $\mathrm{Sym}^\bullet_\textit{inv}(\mathfrak{g}^\vee)$ will be called *integral* (resp. rational) if $\Phi_{EG}(f)$ belongs to the image of $\mathrm{H}^\bullet(BG,\mathbb{Z})$ (resp. $\mathrm{H}^\bullet(BG,\mathbb{Q})$) in $\mathrm{H}^\bullet(BG,\mathbb{C})$. If $f$ is integral, then for every principal bundle $P\to M$ the Chern--Weil class $c_f(P)$ belongs to $\mathrm{H}^\bullet(M,\mathbb{Z})$. More generally, let $c_1,\ldots, c_n$ be a basis of $\mathrm{H}^\bullet(BG,\mathbb{Z})$. Then, for any $f\in \\Sym^\bullet_\textit{inv}(\mathfrak{g}^\vee)$, we can write $$\Phi_{EG}(f) = \sum_{i=1}^n \alpha_i c_i~,\quad \alpha_i\in \mathbb{C}~,$$ and get that, for any principal bundle $P\to M$, the Chern--Weil class $\mathop{\mathrm{cw}}_f(P)$ belongs to the cohomology with coefficients in the submodule $\mathrm{Vect}_\mathbb{Z}(\alpha_1, \ldots, \alpha_n)\subset \mathbb{C}$. This gives strong rational properties to Chern--Weil classes, which live a priori in the de Rham cohomology. ## Chern--Simons forms We recall without proofs the general setting of Chern--Simons theory and refer to the initial paper of Chern--Simons [@ChernSimons74] for details.\ Fix a principal $G$-bundle $P$ over a manifold $M$ and a polynomial $f\in \mathrm{Sym}^k_\textit{inv}(\mathfrak{g}^\vee)$. Given two connections $\Theta_0, \Theta_1$ on $P$, the difference between the Chern--Weil forms $$\mathop{\mathrm{CW}}_f(P,\Theta_1)- \mathop{\mathrm{CW}}_f(P,\Theta_0)$$ is exact. Chern--Simons theory provides a way to construct a primitive to this form, which is well defined up to an *exact* term. Recall that the space $\mathop{\mathrm{Conn}}(P)$ of principal connections on $P$ is an affine space over $\Omega^1(M,\mathrm{Ad}(P))$. Given $\Theta\in \mathop{\mathrm{Conn}}(P)$ and $\Psi \in T_\Theta \mathop{\mathrm{Conn}}(P) = \Omega^1(M,\mathrm{Ad}(P))$, we set $$A_f(\Psi) = k f(\Psi, R_\Theta, \ldots, R_\Theta)~.$$ We see the map $A_f$ as a $1$-form on $\mathop{\mathrm{Conn}}(P)$ with values in $\Omega^{2k-1}(M,\mathbb{C})$. We will denote by $\mathrm D$ the exterior derivative on $\mathop{\mathrm{Conn}}(P)$, to avoid confusion with the exterior derivative on $M$. In particular, $\mathrm D A_f$ is a $2$-form on $\mathop{\mathrm{Conn}}(P)$ with values in $\Omega^{2k-1}(M,\mathbb{C})$, while $$\mathrm{d}A_f: \Psi \mapsto \mathrm{d}(A_f(\Psi))$$ is a $1$-form on $\mathop{\mathrm{Conn}}(P)$ with values in $\Omega^{2k}(M,\mathbb{C})$. With a bit of familiarity with differential calculus on a vector bundle with connection, one can prove the following formulae: $$\label{eq: small d of the Chern--Simons action} \mathrm{d}A_f(\Psi) = k f(\mathrm{d}\Psi + [\Theta, \Psi], R_\Theta, \ldots, R_\Theta) = \frac{\mathrm{d}\ }{\mathrm{d}t}_{\vert t= 0} \mathop{\mathrm{CW}}_f(\Theta + t \Psi)~.$$ $$\label{eq: capital D of the Chern--Simons action} \mathrm D A_f(\Psi_1, \Psi_2) = -k (k-1) \mathrm{d}f(\Psi_1, \Psi_2, \Omega_\Theta, \ldots, \Omega_\Theta)~.$$ The identity [\[eq: small d of the Chern\--Simons action\]](#eq: small d of the Chern--Simons action){reference-type="eqref" reference="eq: small d of the Chern--Simons action"} can be re-written as $$\label{eq: dA = DC} \mathrm{d}A_f= \mathrm D \mathop{\mathrm{CW}}_f~,$$ where $\mathop{\mathrm{CW}}_f$ is seen as a function on $\mathop{\mathrm{Conn}}(P)$ with values in $\Omega^{2k}(M,\mathbb{C})$. The identity [\[eq: capital D of the Chern\--Simons action\]](#eq: capital D of the Chern--Simons action){reference-type="eqref" reference="eq: capital D of the Chern--Simons action"} implies that $\mathrm D A_f$ has coefficients in the space of exact forms on $M$. **Definition 9**. Given a homogeneous $G$-invariant polynomial $f$ on $\mathfrak{g}$, we define the *Chern--Simons form* of a piecewise smooth path $(\Theta_t)_{t\in [0,1]}$ in $\mathop{\mathrm{Conn}}(P)$ by $$\mathop{\mathrm{CS}}_f((\Theta_t)_{t\in [0,1]}) = \int_{t=0}^1 A_f(\dot \Theta_t) \mathrm{d}t~.$$ In particular we define the Chern--Simons form of a pair of connections $(\Theta_0, \Theta_1)\in \mathop{\mathrm{Conn}}(P)^2$ as the Chern--Simons form of the straight path between them: $$\mathop{\mathrm{CS}}_f(\Theta_0, \Theta_1) = \mathop{\mathrm{CS}}_f(((1-t)\Theta_0 + t \Theta_1)_{t\in [0,1]})~.$$ Like Chern--Weil forms, Chern--Simons forms are natural with respect to pull-backs: if $P\to M$ is a principal bundle equiped with a pair of connections $\Theta_0, \Theta_1$ and $\varphi:N\to M$ is a smooth map, then $$\mathop{\mathrm{CS}}_f(\varphi^*\Theta_0, \varphi^*\Theta_1) = \varphi^* \mathop{\mathrm{CS}}_f(\Theta_0,\Theta_1)~.\\ $$ From [\[eq: dA = DC\]](#eq: dA = DC){reference-type="eqref" reference="eq: dA = DC"}, we get that the Chern--Simons form is a primitive of the difference between the Chern--Weil forms: **Proposition 10**. *For any piecewise smooth path $(\Theta_t)_{t\in [0,1]}$ in $\mathop{\mathrm{Conn}}(P)$, we have $$\mathrm{d}\mathop{\mathrm{CS}}_f((\Theta_t)_{t\in [0,1]}) = \mathop{\mathrm{CW}}_f(\Theta_1)- \mathop{\mathrm{CW}}_f(\Theta_0)~.$$* This already shows that the difference between the Chern--Simons forms of two paths with the same endpoints is a closed form. In fact, since $\mathrm D A_f$ takes values into the space of exact forms by [\[eq: capital D of the Chern\--Simons action\]](#eq: capital D of the Chern--Simons action){reference-type="eqref" reference="eq: capital D of the Chern--Simons action"}, the Stokes formula (in the infinite dimensional space $\mathop{\mathrm{Conn}}(P)$) shows that this difference is exact. **Proposition 11**. *For any piecewise smooth path $(\Theta_t)_{t\in [0,1]}$ in $\mathop{\mathrm{Conn}}(P)$, the form $$\mathop{\mathrm{CS}}_f((\Theta_t)_{t\in [0,1]}) - \mathop{\mathrm{CS}}_f(\Theta_0,\Theta_1)$$ is exact.* *In particular, for any $(\Theta_0, \Theta_1, \Theta_2) \in \mathop{\mathrm{Conn}}(P)^3$, the form $$\mathop{\mathrm{CS}}_f(\Theta_0, \Theta_1) + \mathop{\mathrm{CS}}_f(\Theta_1, \Theta_2) - \mathop{\mathrm{CS}}_f(\Theta_0,\Theta_2)$$ is exact.* If, for various reasons, the form $\mathop{\mathrm{CS}}_f(\Theta_0,\Theta_1)$ is closed, then it defines a de Rham cohomology class on $M$ that we denote $\mathop{\mathrm{cs}}_f(\Theta_0,\Theta_1)$. By Proposition [Proposition 11](#prop: Chern-Simons loop exact){reference-type="ref" reference="prop: Chern-Simons loop exact"}, the Chern--Simons form of any smooth path from $\Theta_0$ to $\Theta_1$ gives a representative of this class.\ Let us mention three situations where Chern--Simons forms lead to cohomological invariants. ### Gauge transformations Assume that $\Theta_1= h^* \Theta_0$ for some gauge transformation $h$ of $P$. Then $R_{\Theta_1} = \mathrm{Ad}_h^{-1} \circ R_{\Theta_0}$ and, since $f$ is $G$-invariant, we get that $\mathop{\mathrm{CW}}_f(\Theta_1)= \mathop{\mathrm{CW}}_f(\Theta_0)$. Hence $\mathop{\mathrm{CS}}_f(\Theta_0,h^*\Theta_0)$ is closed. If there exists a smooth path $(h_t)_{t\in [0,1]}$ in the gauge group of $P$ such that $h_0 = \mathrm{Id}_P$ and $h_1 = h$, then one can verify that $$\mathop{\mathrm{cs}}_f(\Theta_0, h^*\Theta_0) = [\mathop{\mathrm{CS}}_f(h_t^*\Theta_0)_{t\in [0,1]}] = 0~.$$ thus Chern--Simons classes can help distinguish connected components in the gauge group. In fact, one can interpretate the Chern--Simons class $\mathop{\mathrm{cs}}_f(\Theta_0, h^*\Theta_0)$ as a Chern--Weil class on $M\times \mathbb{S}^1$. Indeed, define a principal bundle $P_h$ over $M\times \mathbb{S}^1$ by $$P_h = P\times [0,1] / (p,0) \simeq (h (p), 1)~.$$ Let $p_1 : M\times \mathbb{S}^1 \to M$ denote the projection on the first factor and let $[\mathrm{d}t]$ denote the pull-back to $\mathrm{H}^1(M\times \mathbb{S}^1)$ of the generator of $\mathrm{H}^1(\mathbb{S}^1, \mathbb{Z})$. **Proposition 12**. *For every $f\in \mathrm{Sym}^\bullet_\textit{inv}(\mathfrak{g}^\vee)$ and every connection $\Theta$ on $P$, we have $$p_1^* \mathop{\mathrm{cs}}_f(\Theta, h^* \Theta) \wedge [\mathrm{d}t ] = \mathop{\mathrm{cw}}_f(P_h)~.$$* **Corollary 13**. *If $f$ is integral, then $\mathop{\mathrm{cs}}_f(\Theta, h^*\Theta)$ belongs to the image of $\mathrm{H}^\bullet(M,\mathbb{Z})$ in $\mathrm{H}^\bullet(M,\mathbb{C})$.* ### Flat connections When $\Theta_0$ and $\Theta_1$ are flat, $\mathop{\mathrm{CW}}_f(\Theta_0) = \mathop{\mathrm{CW}}_f(\Theta_1) = 0$, hence $\mathop{\mathrm{CS}}_f(\Theta_0,\Theta_1)$ is closed. The following immediate proposition is key in proving local rigidity of Chern--Simons invariants: **Proposition 14**. *Let $(\Theta_t)_{t\in [0,1]}$ be a smooth path of flat connections on $P$. Then $$\mathop{\mathrm{cs}}_f(\Theta_0, \Theta_1) = 0~.$$* *Proof.* By Proposition [Proposition 11](#prop: Chern-Simons loop exact){reference-type="ref" reference="prop: Chern-Simons loop exact"}, we have $$\mathop{\mathrm{cs}}_f(\Theta_0, \Theta_1) = [\mathop{\mathrm{CS}}_f((\Theta_t)_{t\in [0,1]})]~.$$ From the definition, we see that $\mathop{\mathrm{CS}}_f((\Theta_t)_{t\in [0,1]}) = 0$ since $R_{\Theta_t} = 0$ for all $t$. ◻ Assume $f$ is integral. Let $\rho_0$ and $\rho_1$ be two representations of $\pi_1(M)$ into $G$ such that the associated principal bundles $P_{\rho_0}$ and $P_{\rho_1}$ are isomorphic. Then there exist two flat connections $\Theta_0$ and $\Theta_1$ on the same principal bundle with respective holonomies $\rho_1$ and $\rho_2$. Moreover, $\Theta_0$ and $\Theta_1$ are uniquely defined up to a gauge transformation. By Corollary [Corollary 13](#coro: Integral Chern-Simons gauge transformation){reference-type="ref" reference="coro: Integral Chern-Simons gauge transformation"} and Proposition [Proposition 11](#prop: Chern-Simons loop exact){reference-type="ref" reference="prop: Chern-Simons loop exact"}, we thus get a well-defined cohomology class $$\mathop{\mathrm{cs}}_f(\rho_0,\rho_1) \overset{\mathrm{def}}{=}\mathop{\mathrm{cs}}_f(\Theta_0, \Theta_1) \mod \mathbb{Z}~.$$ Thus, while Chern--Weil classes can distinguish connected components in $\mathrm{Hom}(\pi_1(M),G)$ by distinguishing the homeomorphism type of the associated principal bundles, Chern--Simons classes can distinguish between connected components of representations whose associated principal bundles are isomorphic. ### Chern--Simons theory on $3$-manifolds Though this paragraph is not useful to the rest of the paper, we thought that including it would clarify how our presentation of Chern--Simons theory relates to its extensive developments in three-dimensional topology.\ Any simple Lie algebra admits an invariant bilinear symmetric form $$\kappa: (u,v)\mapsto \mathrm{Tr}(\mathrm{ad}_u \mathrm{ad}_v)$$ called the *Killing form*. Moreover there is a constant $a\neq 0$ such that $a\kappa$ is integral in the sense of Definition [Definition 8](#defi: Integral polynomial){reference-type="ref" reference="defi: Integral polynomial"}. Assume moreover that $G$ is connected and simply connected. Because $\pi_2(G) =\{0\}$, every principal $G$-bundle over a $3$-manifold is trivial and thus carries a flat connection with trivial holonomy, that we denote by $D$. Now let $P$ be a principal $G$-bundle over a closed $3$-manifold $M$ with a connection $\Theta$. The form $$\mathop{\mathrm{CS}}_{a\kappa} (D,\Theta)$$ is trivially closed since it is a $3$-form. Moreover, its value modulo $\mathbb{Z}$ does not depend on the choice of the trivialization $D$ since $a\kappa$ is integral. Integrating over $M$, one can thus associate a Chern--Simons invariant $$\mathfrak{cs}(\Theta) \overset{\mathrm{def}}{=}\int_M \mathop{\mathrm{CS}}_{a\kappa} (D,\Theta) \in \mathbb{C}/\mathbb{Z}$$ to any principal $G$-bundle with a connection $\Theta$. One can apply this for instance to the Levi--Civita connection of a Riemannian metric to define the Chern--Simons invariant of a closed Riemannian $3$-manifold. # The algebra of invariant forms on symmetric spaces {#s: Invariant forms on symmetric spaces} Recall that we have fixed a connected semisimple Lie group $G$ and an involutive isomorphism $\sigma$ of $G$, and denoted by $H$ the neutral component of the subgroup fixed by $\sigma$. Up to quotienting $G$, we can assume without loss of generality that $H$ does not contain a non-trivial normal subgroup of $G$, so that $G$ acts faithfully on the symmetric space $G/H$. The involution $\sigma$ induces an involution of $G/H$ which fixes the base point $o \overset{\mathrm{def}}{=}\mathbf{1}_G H$ and acts as $-\mathrm{Id}$ on $T_o G/H$. In this section, we describe the algebra $\Omega^\bullet_\textit{inv}(G/H,\mathbb{C})$ of $G$-invariant forms on $G/H$, relying mostly on the work of Cartan [@Cartan50] and Borel [@Borel53]. Our goal is to prove that this algebra is generated by Chern--Weil forms and Chern--Simons forms associated to $G$-invariant connections on automorphic bundles over $G/H$. ## Invariant forms on symmetric spaces The study of invariant differential forms on symmetric spaces started with Élie Cartan and was carried on by his son Henri. A first elementary but useful result is the following: **Proposition 15** (E. Cartan). *Every $G$-invariant form on $G/H$ is closed.* In other words, the complex of $G$-invariant forms $\Omega^\bullet_\textit{inv}(G/H,\mathbb{C})$ has trivial differential and is thus isomorphic to its cohomology. A series of isomorphisms identifies it canonically with the cohomology of the *dual compact symmetric space*. Define first the complexification $\mathfrak{g}_\mathbb{C}$, $\mathfrak{h}_C$, $G_\mathbb{C}$ and $H_\mathbb{C}$ of $\mathfrak{g}$, $\mathfrak{h}$, $G$ and $H$ respectively in the following way: - $\mathfrak{g}_\mathbb{C}$ is the complex Lie algebra $\mathfrak{g}\otimes_\mathbb{R}\mathbb{C}$, - $\mathfrak{h}_\mathbb{C}$ is the complex Lie subalgebra $\mathfrak{h}\otimes_\mathbb{R}\mathbb{C}$, - $G_\mathbb{C}$ is the neutral component of $\mathrm{Aut}(\mathfrak{g}_\mathbb{C})$, - $H_\mathbb{C}$ is the neutral component of the subgroup preserving $\mathfrak{h}_\mathbb{C}$. Note that, because $\mathfrak{g}_\mathbb{C}$ is a semisimple Lie algebra, it is the Lie algebra of the complex group $G_\mathbb{C}$. Since $\mathfrak{h}_\mathbb{C}$ is the subalgebra fixed by an involution, it is the Lie algebra of its stabilizer.[^1] Recall that a Cartan involution of an algebraic group $G_\mathbb{C}$ is an involutive isomorphism whose set of fixed points is compact. By a result of E. Cartan (see also [@Richardson68]), we can always choose a Cartan involution $\theta$ of $G_\mathbb{C}$ that commutes with $\sigma$. Moreover, it is unique up to conjugation by $H_\mathbb{C}$. Define $G_U$ to be subgroup of $G_\mathbb{C}$ fixed by $\theta$, $H_U = G_U\cap \mathbb{H}_C$, and $\mathfrak{g}_U$ and $\mathfrak{h}_U$ their respective Lie algebras. The groups $G_U$ and $H_U$ are compact real forms of $G_\mathbb{C}$ and $H_\mathbb{C}$. In particular, they are connected (since the complex groups are connected and retract to their maximal compact subgroup). We call $G_U/H_U$ the *dual compact symmetric space*. given a linear representation $V$ of a group $G$ over a field $k$ and $l$ an extension of $k$, denote by $\Lambda^\bullet_l(V^\vee)^G$ the subalgebra of $G$-invariant forms in the algebra of alternate $k$-multilinear forms. **Proposition 16**. *There are canonical isomorphisms $$\begin{array}{ccccc} \Omega^\bullet_\textit{inv}(G/H,\mathbb{C}) &\simeq & \Lambda^\bullet_\mathbb{C}((\mathfrak{g}/\mathfrak{h})^\vee)^H &\simeq &\Lambda^\bullet_\mathbb{C}((\mathfrak{g}_C/\mathfrak{h}_C)^\vee)^{H_\mathbb{C}} \\ &\simeq & \Lambda^\bullet_\mathbb{C}((\mathfrak{g}_U/\mathfrak{h}_U)^\vee)^{H_U} &\simeq & \Omega^\bullet_\textit{inv}(G_U/H_U,\mathbb{C}) \\ & & &\simeq & \mathrm{H}^\bullet(G_U/H_U,\mathbb{C})~. \end{array}$$* *Proof.* The first isomorphism is well-known: every $G$-invariant form on $G/H$ restricts to an alternate form on $\mathfrak{g}/\mathfrak{h}\simeq T_o G/H$ which is invariant under the adjoint action of $H$, and conversely, every such alternate form extends in a unique way to a $G$-invariant form on $G/H$. For the second isomorphism, note first that every $H$-invariant $\mathbb{R}$-multilinear form $\alpha$ on $\mathfrak{g}/\mathfrak{h}$ extends uniquely to an $H$-invariant $\mathbb{C}$-multilinear form $\alpha_\mathbb{C}$ on $\mathfrak{g}_\mathbb{C}/\mathfrak{h}_\mathbb{C}= \mathfrak{g}/\mathfrak{h}\otimes_\mathbb{R}\mathbb{C}$. Since $\alpha_\mathbb{C}$ is $H$-invariant, it satisfies $u\cdot \alpha_\mathbb{C}\overset{\mathrm{def}}{=}\frac{\mathrm{d}\ }{\mathrm{d}t}_{\vert t=0} \exp(-t\mathrm{ad}_u)^* \alpha_\mathbb{C}= 0$ for every $u\in \mathfrak{h}$. But since $\mathfrak{g}_\mathbb{C}$ is a complex Lie algebra and $\alpha_\mathbb{C}$ is $\mathbb{C}$-multilinear, the map $$u\mapsto u \cdot \alpha_\mathbb{C}$$ is $\mathbb{C}$-linear. Hence the relation $u\cdot \alpha_\mathbb{C}$ is satisfied for all $u \in \mathfrak{h}_\mathbb{C}$. Finally, integrating this relation gives $$h^*\alpha_\mathbb{C}= \alpha_\mathbb{C}$$ for all $h\in H_\mathbb{C}$ since $H_\mathbb{C}$ is connected by definition. Hence $\alpha \mapsto \alpha_\mathbb{C}$ gives the second isomorphism. The third and fourth isomorphisms are respectively the second and first one applied to the compact symmetric space $G_U/H_U$, which has the same complexification. Finally, on $G_U/H_U$, every closed form is cohomologous to a unique invariant one obtained by averaging under the action of $G_U$. Hence the natural map $$\Omega^\bullet_\textit{inv}(G_U/H_U,\mathbb{C}) \to \mathrm{H}^\bullet(G_U/H_U,\mathbb{C})$$ is an isomorphism. ◻ The space $G$ equipped with the right action of $H$ is a principal bundle over $G/H$ which we call the *tautological $H$-bundle*. The involution $\sigma$ preserves the Killing form of $G$, and since $\mathfrak{h}$ is the eigenspace of the involution for the eigenvalue $1$, the Killing form is non-degenerate in restriction to $\mathfrak{h}$. **Definition 17**. The *standard principal connection* on the bundle $G\to G/H$ is the left $G$-invariant form $\Theta_{G/H}$ on $G$ with values in $\mathfrak{h}$ whose value at $\mathbf{1}_G$ is the orthogonal projection to $\mathfrak{h}$ (for the Killing form). We denote by $R_{G/H}$ its curvature form. For any $f \in \mathrm{Sym}^\bullet_\textit{inv}(\mathfrak{h}^\vee)$, the Chern--Weil form $$\mathop{\mathrm{CW}}_f(\Theta_{G/H})$$ thus defines a $G$-invariant form on $G/H$. We set $$\Omega^\bullet_{\textit{even}}(G/H,\mathbb{C}) = \{\mathop{\mathrm{CW}}_f(\Theta_{G/H}), f\in \mathrm{Sym}^\bullet_\textit{inv}(\mathfrak{h}^\vee)\}~.$$ Looking at the restriction of the Chern--Weil forms to $\mathfrak{g}/\mathfrak{h}$ and their behaviour under complexification, one proves the following: **Proposition 18**. *The sequence of isomorphisms given in Proposition [Proposition 16](#prop: Comparison cohomology compact dual){reference-type="ref" reference="prop: Comparison cohomology compact dual"} restrict to an isomorphism $$\Omega^\bullet_{\textit{even}}(G/H,\mathbb{C}) \simeq \mathrm{H}^\bullet_{\textit{even}}(G_U/H_U,\mathbb{C})~,$$ where $\mathrm{H}^\bullet_{\textit{even}}(G_U/H_U,\mathbb{C})$ is the algebra of characteristic classes of the tautological principal bundle of $G_U/H_U$.* Cartan and Borel gave the following description of the cohomology of compact symmetric spaces: **Theorem 19** (Cartan, Borel, [@Borel53]). *Let $G_U/H_U$ be a compact symmetric space, with $G_U$ and $H_U$ connected. Then there exists a subalgebra $$\mathrm{H}^\bullet_{\textit{odd}}(G_U/H_U,\mathbb{C})\subset \mathrm{H}^\bullet(G_U/H_U,\mathbb{C})~,$$ generated by forms of odd degree, such that $$\mathrm{H}^\bullet(G_U/H_U,\mathbb{C}) = \mathrm{H}^\bullet_\textit{even}(G_U/U,\mathbb{C}) \otimes \mathrm{H}^\bullet_\textit{odd}(U/U,\mathbb{C})~.$$ Moreover, the pull-back map $$p^*: \mathrm{H}^\bullet_\textit{odd}(U/U,\mathbb{C})\to \mathrm{H}^\bullet (U,\mathbb{C})$$ induced by the projection $p:U\to G_U/H_U$ vanishes on $\mathrm{H}^{>0}_\textit{even}(G_U/H_U,\mathbb{C})$ and is injective on $\mathrm{H}^\bullet_\textit{odd}(G_U/H_U,\mathbb{C})$.* While the subalgebra $\mathrm{H}\bullet_\textit{odd}(G_U/H_U,\mathbb{C})$ is not uniquely determined, Theorem [Theorem 27](#thm: Odd forms G/H){reference-type="ref" reference="thm: Odd forms G/H"} below will give us a canonical way to construct it and to define a corresponding subalgebra $\Omega^\bullet_\textit{odd}(G/H,\mathbb{C}) \subset \Omega^\bullet_\textit{inv}(G/H,\mathbb{C})$. ## Bi-invariant forms on Lie groups A particular class of symmetric spaces will be of interest to us: the semisimple Lie group $G$ itself equipped with the action of $G\times G$ given by $$(g,h)\cdot x = gxh^{-1}~.$$ The corresponding involution $\sigma$ of $G\times G$ is given by $\sigma(g,h) = (h,g)$, the stabilizer of the base point $\mathbf{1}_G$ is the diagonal subgroup $\Delta(G)$, and the central symmetry at $\mathbf{1}_G$ is the map $g\mapsto g^{-1}$. Since it will be convenient to distinguish between $G$ seen as a group and $G$ seen as a symmetric space, we will denote the latter by $X_G$. The dual compact symmetric space of $X_G$ is simply the symmetric space $X_{G_U} = G_U\times G_U /\Delta(G_U)$. In particular, Proposition [Proposition 16](#prop: Comparison cohomology compact dual){reference-type="ref" reference="prop: Comparison cohomology compact dual"} gives an isomorphism $$\Omega^\bullet_\textit{inv}(X_G,\mathbb{C}) \simeq \mathrm{H}^\bullet(G_U,\mathbb{C})~.$$ The projection map $G_U \times G_U \to X_{G_U}$ induces in cohomology a *coproduct* $$\delta: \mathrm{H}^\bullet(G_U,\mathbb{C}) \to \mathrm{H}^\bullet(G_U,\mathbb{C}) \otimes \mathrm{H}^\bullet(G_U,\mathbb{C})~,$$ giving $\mathrm{H}^\bullet(G_U,\mathbb{C})$ the structure of a *Hopf algebra*. Following Borel, we define: **Definition 20**. A class $x \in \mathrm{H}^k(G_U,\mathbb{C})$ is *primitive* if $$\delta(x) = x\otimes \mathbf{1} + (-1)^{k} \mathbf{1}\otimes x~.$$ We denote by $\mathop{\mathrm{Prim}}(G_U)$ the vector space spanned by primitive classes in $\mathrm{H}^\bullet(G_U,\mathbb{C})$, and by $\mathop{\mathrm{Prim}}(X_G)$ the corresponding subspace of $\Omega^\bullet_\textit{inv}(X_G)$. The general structure theorem of Hopf gives the following: **Theorem 21** (Hopf). *The inclusion $\mathop{\mathrm{Prim}}(G_U)\hookrightarrow \mathrm{H}^\bullet(G_U,\mathbb{C})$ induces an isomorphism $$\Lambda^\bullet \mathop{\mathrm{Prim}}(G_U) \simeq \mathrm{H}^\bullet(G_U,\mathbb{C})~,$$ where $\Lambda^\bullet$ denotes the exterior algebra.* Consequently, we also get that $$\Omega^\bullet_\textit{inv}(X_G,\mathbb{C}) = \Lambda^\bullet\mathop{\mathrm{Prim}}(X_G)~.\\ $$ The space of primitive forms is further described by the work of Chevalley. Let $\mu\in \Omega^1(G,\mathfrak{g})$ denote *Maurer--Cartan form* of $G$, i.e. the left-invariant $1$-form which is the identity at $\mathbf{1}_G$. Let $J\subset \Omega^\bullet_\textit{inv}(X_G,\mathbb{C})$ denote the square of the ideal of forms of positive degree, i.e. the ideal generated by forms that can be factored as a product of two forms of lower degree. The following theorem is attributed by Borel to Cartan, Chevalley and Weil. **Theorem 22** (Cartan--Chevalley--Weil). *Let $\tau: \mathrm{Sym}^{>0}_\textit{inv}(\mathfrak{g}^\vee) \to \Omega^\bullet_\textit{inv}(G,\mathbb{C})$ be the linear map sending a symmetric $k$-linear form $f$ to $$f(\alpha, [\alpha,\alpha], \ldots, [\alpha, \alpha])~.$$ Then the kernel of $\tau$ is the ideal $J$ and the image of $\tau$ is the set of primitive forms $\mathop{\mathrm{Prim}}(X_G)$.* *Remark 23*. Chevalley also proved that $\mathrm{Sym}^\bullet_\textit{inv}(\mathfrak{g}^\vee)$ is a polynomial algebra generated by $\mathop{\mathrm{rank}}_\mathbb{C}(G)$ elements. One deduces that $$\dim \mathrm{Sym}^{>0}_\textit{inv}(\mathfrak{g}^\vee)/J = \dim \mathop{\mathrm{Prim}}(X_G) = \mathop{\mathrm{rank}}_\mathbb{C}(G)~.$$ The formal similarity between the Cartan--Chevalley--Weil description of primitive forms and the definition of the Chern--Simons invariants is more than a coincidence. Let $p_1$ and $p_2$ denote the projections of $G\times G$ to the first and second factor and define $$\Theta_L = p_2^* \mu~, \quad \Theta_R= p_1^*\mu~,$$ where $\mu$ is the Maurer--Cartan form. Then $\Theta_R$ and $\Theta_L$ are two flat invariant connections on the tautological bundle $G\times G\to X_G$. The associated adjoint bundle is canonically identified to the tangent bundle to $G$, and the associated connections $\nabla^L$ and $\nabla^R$ respectively make the left-invariant and right-invariant vector fields parallel.[^2] **Theorem 24**. *For every $f \in \mathrm{Sym}^k_\textit{inv}(\mathfrak{g}^\vee)$, we have $$\mathop{\mathrm{CS}}_f(\Theta_L, \Theta_R) = \frac{(-1)^k k! ((k-1)!}{2^{k-1}(2k-1)!} \tau(f)~.$$* Since the image of $\tau$ generates $\Omega^\bullet_\textit{inv}(X_G,\mathbb{C})$, we deduce: **Corollary 25**. *The algebra of invariant forms on $X_G$ is generated by the Chern--Simons forms associated to the pair of connections $(\Theta_L,\Theta_R)$ on the tautological principal bundle.* *Proof of Theorem [Theorem 24](#thm: Primitive forms -> Chern-Simons){reference-type="ref" reference="thm: Primitive forms -> Chern-Simons"}.* Set $\Theta_t= (1-t) \Theta_L + t \Theta_R$ and denote by $R_t = \mathrm{d}\Theta_t + \frac{1}{2}[\Theta_t, \Theta_t]$ its curvature, seen as a $2$-form on $G\times G$ with values in $\mathfrak{g}$. Since $\Theta_L$ and $\Theta_R$ are flat, one computes that $$R_t = \frac{-t(1-t)}{2} \left( [\Theta_R, \Theta_R] + [\Theta_L, \Theta_L]\right) + t (1-t)[\Theta_L,\Theta_R]~.$$ Define $\widehat\mathop{\mathrm{CS}}_f\in \Omega^2(G\times G, \mathfrak{g})$ by $$\label{eq: Computing Chern--Simons} \widehat\mathop{\mathrm{CS}}_f = k \int_{t=0}^1 f(\dot \Theta_t, R_t, \ldots , R_t)~.$$ By construction, $\widehat\mathop{\mathrm{CS}}_f$ is the pull-back to $G\times G$ of the Chern--Weil form $\mathop{\mathrm{CS}}_f(\Theta_L, \Theta_R)$, hence$\mathop{\mathrm{CS}}_f(\Theta_L, \Theta_R)$ is the pull-back of $\widehat\mathop{\mathrm{CS}}_f$ by any section of the bundle. Consider the section $s: g \mapsto (\mathbf{1}_G, g)$. Then $s^*\Theta_R = 0$ and $s^* \Theta_L= \mu$, hence $$s^* \dot \Theta_t = - \mu$$ and $$s^*R_t = \frac{-t(1-t)}{2} [\mu, \mu]~.$$ We can thus compute: $$\begin{aligned} \mathop{\mathrm{CS}}_f(\Theta_L, \Theta_R) &=& k \int_{t=0}^1 f(s^* \dot \Theta_t, s^* R_t, \ldots, R_t)\mathrm{d}t\\ &=& (-1)^k k \left (\int_{t=0}^1 \left(\frac{t(1-t)}{2}\right)^{k-1} \mathrm{d}t\right)\, f(\alpha, [\alpha,\alpha], \ldots, [\alpha, \alpha])\\ &=& \frac{(-1)^k k! ((k-1)!}{2^{k-1}(2k-1)!} \tau(f)~.\end{aligned}$$ ◻ ## The inclusion $\iota_\sigma$ Let us recall the definition of the map $\iota_{G,H}$ given in the introduction: **Definition 26**. The map $\iota_{G,H}: G/H \to X_G$ is defined by $$\iota_{G,H}(gH) = g \sigma(g)^{-1}$$ for every class $gH$ in $G/H$. This does not depend on the representative $g$ since $h\sigma(h)^{-1} = \mathbf{1}_G$ for all $h\in H$. The map $\iota_{G,H}$ is an inclusion of symmetric spaces. It is equivariant with respect to the representation $(\mathrm{Id}, \sigma): G\to G\times G$. In particular, every invariant form on $X_G$ can be pulled-back by $\iota_{G,H}$ to a $G$-invariant form on $G/H$. Moreover, this operation commutes with the comparison isomorphisms, i.e. we have the following diagram: $$\xymatrix{ \Omega^\bullet_\textit{inv}(X_G,\mathbb{C}) \ar[r]^{\iota_{G,H}^*} \ar[d]_\simeq & \Omega^\bullet_\textit{inv}(G/H,\mathbb{C}) \ar[d]^\simeq \\ \mathrm{H}^\bullet(G_U,\mathbb{C}) \ar[r]^{\iota_{G_U,H_U}^*} & \mathrm{H}^\bullet(G_U/H_U,\mathbb{C})~. }$$ We can now state our refined version of Cartan--Borel's theorem for $G$-invariant forms on a (not necessarily compact) symmetric space: **Theorem 27**. *The space $\Omega^\bullet_\textit{inv}(G/H,\mathbb{C})$ is isomorphic to $$\Omega^\bullet_\textit{even}(G/H,\mathbb{C}) \otimes \Omega^\bullet_\textit{odd}(G/H,\mathbb{C})~,$$ where $\Omega^\bullet_\textit{even}(G/H,\mathbb{C})$ is the algebra of Chern--Weil forms of the standard connection on the tautological principal $H$-bundle and $$\Omega^\bullet_\textit{odd}(G/H,\mathbb{C}) = \iota_{G,H}^* \Omega^\bullet_\textit{inv}(G,\mathbb{C})~.$$* Note that this is just a reformulation of Theorem [Theorem 4](#t: Cohomology symmetric space){reference-type="ref" reference="t: Cohomology symmetric space"} from the introduction in the general case where $G/H$ is not assumed compact. The two theorems are equivalent thanks to the comparison isomorphisms.\ To prove the theorem, we will use the following cute and elementary lemma of linear algebra: **Lemma 28**. *Let $E$ be a Euclidean vector space, $F_1$ and $F_2$ any two subspaces, and $\pi_1$ and $\pi_2$ the respective orthogonal projections to $F_1$ and $F_2$. Then $$F_1 = \mathop{\mathrm{im}}{\pi_1}_{\vert F_2} \oplus \ker {\pi_2}_{\vert F_1}~.$$* *Proof.* Let us first compute dimensions: we have $$\begin{aligned} \dim\mathop{\mathrm{im}}{\pi_1}_{\vert F_2} &=& \dim F_2 - \dim \ker F_2 \cap F_1^\perp\\ &=& \dim F_2 - (\dim E - \dim (F_2^\perp + F_1)) + \dim F_1 \cap F_2^\perp\\ &=& \dim F_2 - \dim E + \dim F_2^\perp +\dim F_1 - \dim F_1 \cap F_2^\perp \\ &=& \dim F_2 - \dim \ker {\pi_2}_{\vert F_1} ~.\end{aligned}$$ It is thus enough to prove that $\mathop{\mathrm{im}}{\pi_1}_{\vert F_2} \cap \ker {\pi_2}_{\vert F_1} = \{0\}$. Let $v$ be a vector in $\mathop{\mathrm{im}}{\pi_1}_{\vert F_2} \cap \ker {\pi_2}_{\vert F_1}$, and write $v= \pi_1(w)$, $w\in F_2$. Then $$\begin{aligned} \langle v,v\rangle &=&\langle \pi_1(w), w\rangle \quad \textrm{since $w - \pi_1(w) \in F_1^\perp$}\\ &=& \langle \pi_2 \circ \pi_1(w), w\rangle \quad \textrm{since $w \in F_2$}\\ &=& \langle \pi_2(v), w \rangle = 0~.\end{aligned}$$ Hence $v= 0$ since $E$ is euclidean. ◻ *Proof of Theorem [Theorem 27](#thm: Odd forms G/H){reference-type="ref" reference="thm: Odd forms G/H"}.* Thanks to the comparison isomorphisms, it is enough to prove the theorem for compact groups, for which invariant forms are in bijection with cohomology classes. The fact that the cohomology of $G/H$ is the tensor product of an even part and an odd part is already given by Cartan and Borel, and the only thing we need to prove is that one can choose $$\mathrm{H}^\bullet_\textit{odd}(G/H,\mathbb{C}) = \iota_{G,H}^*\mathrm{H}^\bullet(G,\mathbb{C})$$ in their theorem. Let $I$ denote the ideal in $\mathrm{H}^\bullet(G/H,\mathbb{C})$ generated by $\mathrm{H}^{>0}_\textit{even}(G/H,\mathbb{C})$. Since Theorem [Theorem 19](#thm: Cartan Borel){reference-type="ref" reference="thm: Cartan Borel"} already tells us that $\mathrm{H}^\bullet(G/H,\mathbb{C})$ is a tensor product, what we need to prove is mainly that $\iota_{G,H}^* \mathrm{H}^\bullet(G,\mathbb{C})$ is a complement to $I$. Fix an integer $k$, and let $E$ be the space of $H$-invariant alternate $k$-forms on $\mathfrak{g}$, which we equip with an $H$-invariant scalar product. Consider the following subspaces of $E$: - the subspace $F_1$ of forms that are pulled back from an $H$-invariant form on $\mathfrak{g}/\mathfrak{h}$, - the subspace $F_2$ of $G$-invariant forms. Recall that $F_1$ identifies naturally with $\mathrm{H}^k(G/H,\mathbb{C})$, while $F_2$ identifies naturally with $\mathrm{H}^k(G,\mathbb{C})$. Through this identification, we see the pull-back maps $\pi^*: \mathrm{H}^k(G/H,\mathbb{C}) \to \mathrm{H}^k(G,\mathbb{C})$ and $\iota_{G,H}^*: \mathrm{H}^k(G,\mathbb{C})\to \mathrm{H}^k(G/H,\mathbb{C})$ as maps between $F_1$ and $F_2$. Let $\pi_1$ and $\pi_2$ denote respectively the orthogonal projections to $F_1$ and $F_2$. We claim that we have the following identifications : $$\label{eq: cohomology map from G/H to G} \pi^*= {\pi_2}_{\vert F_1}~,$$ $$\label{eq: cohomology map from G to G/H} \iota_{G,H}^*= 2^k {\pi_1}_{\vert F_2}$$ - *Proof of [\[eq: cohomology map from G/H to G\]](#eq: cohomology map from G/H to G){reference-type="eqref" reference="eq: cohomology map from G/H to G"}:* Let $\omega$ be a $G$-invariant $k$-form on $G/H$ and $\omega'$ its pull-back to $G$, which is a closed left-invariant $k$-form on $G$. It is cohomologous to a unique bi-invariant form $\overline{\omega'}$ on $G$, obtained by averaging $\omega'$ under the right action of $G$ (with respect to the Haar probability measure on $G$). Since $\omega'$ is already left invariant, the restriction of $\overline{\omega'}$ to $T_{\mathbf{1}_G} G = \mathfrak{g}$ is given by $$\overline{\omega'}_{\mathbf{1}_G} = \int_G \mathrm{Ad}_g^* \omega'_{\mathbf{1}_G} \mathrm{d}g~.$$ Since averaging under the adjoint $G$-action is the orthogonal projection to $F_2$, we conclude that $$\pi^*[\omega] = \pi_2([\omega])$$ after the appropriate identifications.\ - *Proof of [\[eq: cohomology map from G to G/H\]](#eq: cohomology map from G to G/H){reference-type="eqref" reference="eq: cohomology map from G to G/H"}*: Denote by $p$ the projection $\mathfrak{g}\to \mathfrak{g}/\mathfrak{h}$ and $j : \mathfrak{g}/\mathfrak{h}\to \mathfrak{g}$ be the section of $p$ with image $\mathfrak{h}^\perp$. Since the derivative of the involution $\sigma$ is the identity on $\mathfrak{h}$ and minus the identity on $\mathfrak{h}^\perp$, we get that $$\mathrm{d}_o \iota_{G,H} = 2 j~,$$ where $o$ denotes the basepoint $\mathbf{1}_G H$. Note that $j \circ p: \mathfrak{g}\to \mathfrak{g}$ is the orthogonal projection to $\mathfrak{h}^\perp$, from which one easily deduces that $p^*j^* : E \to F_1$ is the orthogonal projection $\pi_1$ on $F_1$. Let now $\omega$ be a biinvariant form on $G$ and $\omega'$ its pull-back to $G/H$. Then we have $$p^* (\omega'_o) = 2^k p^*j^*\omega_{\mathbf{1}_G} = 2^k \pi_1(\omega_{\mathbf{1}_G})~.$$ which rewrites $$\iota_{G,H}^*[\omega] = 2^k \pi_1([\omega])$$ after the apropriate identifications. Applying Lemma [Lemma 28](#lem: Pair of projectors){reference-type="ref" reference="lem: Pair of projectors"}, we get that $$\mathrm{H}^\bullet(G/H,\mathbb{C}) = \ker \pi^* \oplus \iota_{G,H}^* \mathrm{H}^\bullet(G,\mathbb{C})~.$$ By Theorem [Theorem 19](#thm: Cartan Borel){reference-type="ref" reference="thm: Cartan Borel"}, the kernel of $\pi^*$ is the ideal $I$ generated by $\mathrm{H}^{>0}_\textit{even}(G/H,\mathbb{C})$. Hence $\mathrm{H}^\bullet(G/H,\mathbb{C})$ is generated by $\mathrm{H}^\bullet_\textit{even}(G/H,\mathbb{C})$ and $\iota_{G,H}^*(G,\mathbb{C})$. The rest is a completely general argument: let $\mathrm{H}^\bullet_\textit{odd}(G/H,\mathbb{C})$ be a subalgebra such that $$\mathrm{H}^\bullet(G/H,\mathbb{C}) = \mathrm{H}^\bullet_\textit{even}(G/H,\mathbb{C}) \otimes \mathrm{H}^\bullet_\textit{odd}(G/H,\mathbb{C})~.$$ quotienting by the ideal $I$ we get that $\mathrm{H}^\bullet_\textit{odd}(G/H,\mathbb{C}) \simeq \iota_{G,H}^* \mathrm{H}^\bullet(G,\mathbb{C})$. The inclusions of $\mathrm{H}^\bullet_\textit{even}(G/H,\mathbb{C})$ and $\iota_{G,H}^*\mathrm{H}^\bullet(G,\mathbb{C})$ induce a morphism $$\mathrm{H}^\bullet_\textit{even}(G/H,\mathbb{C}) \otimes \iota_{G,H}^*\mathrm{H}^\bullet(G,\mathbb{C}) \to \mathrm{H}^\bullet(G/H,\mathbb{C})\simeq \mathrm{H}^\bullet_\textit{even}(G/H,\mathbb{C}) \otimes \mathrm{H}^\bullet_\textit{odd}(G/H,\mathbb{C})$$ which is surjective. Since both algebras have the same dimension over $\mathbb{C}$, it is an isomorphism. ◻ # Local rigidity of cohomological invariants {#s:Local rigidity} This section will conclude the proof of Theorem [Theorem 1](#t: Main Theorem){reference-type="ref" reference="t: Main Theorem"} and its corollaries. ## Smooth families of principal bundles {#ss: Families of bundles} Let us first introduce properly the notion of *smooth family of bundles* and smooth family of sections used informally in the introduction. Here, the term "bundle" is meant to include principal bundles as well as their associated bundles. We call $(E_t)_{t\in [0,1]}$ a *smooth family of bundles* on a manifold $M$ if there is a smooth bundle $E$ over $M\times [0,1]$ such that $E_t$ is the pull-back of $E$ under the map $x\mapsto (x,t)$. If $(E_t)_{t\in [0,1]}$ is a smooth family of principal $G$-bundles on $M$, we call $(s_t)_{t\in [0,1]}$ a *smooth family of sections* of $E_t$ is each $s_t$ is the pull-back under $x\mapsto (x,t)$ of a smooth section $s$ on $E$. The following classical lemma can be attributed to Ehresmann: **Lemma 29**. *Let $M$ be a smooth manifold and $E$ a bundle over $M\times [0,1]$. Then $E$ is isomorphic to $p_1^* E_0$, where $p_1: M\times [0,1] \to M$ is the projection to the first factor.* More informally, any smooth family of vector bundles is trivial. As a consequence we have the following: **Corollary 30**. *Let $(E_t)_{t\in [0,1]}$ be a smooth family of bundles over $M$ and $s$ a smooth section of $E_0$. Then there exists a smooth family of sections $(s_t)_{t\in [0,1]}$ of $(E_t)_{t\in [0,1]}$ such that $s_0=s$. Moreover, if $(s'_t)_{t\in [0,1]}$ is another such family, then $s'_t$ is isotopic to $s_t$ for all $t$.* The main example we will be interested in is smooth families of flat bundles. Recall that, fixing a finite generating set $S$ of $\pi_1(M)$, one can see $\mathrm{Hom}(\pi_1(M),G)$ as an analytic subset of the analytic manifold $G^S$. We call a path $(\rho_t)_{t\in [0,1]}$ in $\mathrm{Hom}(\pi_1(M),G)$ smooth if it $t\mapsto \rho_t$ is a smooth map from $[0,1]$ to $G^S$ (equivalently, if $t\mapsto \rho_t(\gamma)$ is smooth in $G$ for all $t$). The space $\mathrm{Hom}(\pi_1(M),G)$ is locally connected by smooth arcs, in the sense that any two representations $\rho_0$ and $\rho_1$ in the same connected component are the endpoints of a smooth path in $\mathrm{Hom}(\pi_1(M),G)$. Let $V$ be a manifold equipped with a $G$-action. If $(\rho_t)_{t\in [0,1]}$ is a smooth family of representations then the family $$\left(M\times_{\rho_t} V\right)_{t\in [0,1]}$$ is a smooth family of bundles. In particular, $(P_{\rho_t})_{t\in [0,1]}$ is a smooth family of principal bundles. By Lemma [Lemma 29](#lemma: family of principal bundles){reference-type="ref" reference="lemma: family of principal bundles"}, this family is trivial, and we deduce: **Corollary 31**. *Let $(\rho_t)_{t\in [0,1]}$ be a smooth family of representations of $\pi_1(M)$ into $G$. Then there exists a principal $G$-bundle $P$ and a smooth family of flat connections $(\Theta_t)_{t\in [0,1]}$ on $P$ such that $\Theta_t$ has holonomy $\rho_t$.* *Moreover, if $P'$ is another principal $G$-bundle and $(\Theta'_t)_{t\in [0,1]}$ another family of flat connections with monodromies $(\rho_t)_{t\in [0,1]}$, then there exists a smooth family $(\varphi_t)_{t\in [0,1]}$ of bundle isomorphisms from $P$ to $P'$ such that $\Theta'_t = \varphi_t^* \Theta_t$.* In particular, the topology of the bundle $P_\rho$ is constant when $\rho$ varies in a connected component of $\mathrm{Hom}(\pi_1(M),G)$. ## Proof of Theorem [Theorem 1](#t: Main Theorem){reference-type="ref" reference="t: Main Theorem"} {#proof-of-theorem-t-main-theorem} We now have a sufficient understanding of invariant forms on symmetric spaces to prove Theorem [Theorem 1](#t: Main Theorem){reference-type="ref" reference="t: Main Theorem"}. Clearly, it is enough to prove it for $\omega$ in a subset of $\Omega^\bullet_\textit{inv}(G/H,\mathbb{C})$ generating the whole algebra. We will thus prove it separately for $\omega \in \Omega^\bullet_\textit{even}(G/H,\mathbb{C})$ (using Chern--Weil theory) and for $\omega \in \iota_{G,H}^*\mathop{\mathrm{Prim}}(X_G)$ (using Chern--Simons theory). While both arguments are probably well-known it is worth including them here for completeness. ### The even case Let us first reinterpret the definition of $\Omega^\bullet_\textit{even}(G/H,\mathbb{C})$ after pull-back by a section of a flat $G/H$ bundle. Fix $\omega \in\Omega^\bullet_\textit{even}(G/H,\mathbb{C})$ and let $f \in \mathrm{Sym}^\bullet_\textit{inv}(\mathfrak{h}^\vee)$ be such that $$\omega = f(R_{G/H})~,$$ where $R_{G/H}$ is the curvature of the standard connection on the tautological principal $H$-bundle $G\to G/H$. **Proposition 32**. *For any manifold $M$, any representation $\rho: \pi_1(M) \to G$ and any smooth section $s$ of the flat bundle $M\times_\rho (G/H)$, we have $$[s^*\omega] = \mathop{\mathrm{cw}}_f (P_\rho(s))~,$$ where $P_\rho(s)$ is the reduction to $H$ of the flat $G$-bundle $P_\rho$ given by $s$.* *Proof.* Denote by $P_{G/H}$ the tautological principal $H$-bundle over $G/H$ and let $\widetilde s$ be the lift of $s$ to $\widetilde M$. Then $\pi_1(M)$ acts on $\widetilde s^* P_{G/H}$ via $\rho$, and, by construction, the quotient $\pi_1(M) \backslash \widetilde s^* P_{G/H}$ is the principal $H$-bundle $P_\rho(s)$. Now, the standard connection $\Theta_{G/H}$ on $P_{G/H}$ pulls back to a $\pi_1(M)$-invariant connection on $\widetilde s^* P_{G/H}$ which factors to a connection $s^* \Theta_{G/H}$ on $P_\rho(s)$, with curvature $s^* R_{G/H}$. By naturality of the pull-back, we have $$\begin{aligned} \mathop{\mathrm{cw}}_f (P_\rho(s)) &=& [\mathop{\mathrm{CW}}_f(s^* \Theta_{G/H})]\\ &=& [f(s^*R_{G/H})]\\ &=& [s^*\omega]~.\end{aligned}$$ ◻ **Corollary 33**. *For any manifold $M$, any smooth path $(\rho_t)_{t\in [0,1]}$ in $\mathrm{Hom}(\pi_1(M),G)$ and any smooth family of sections $s_t$ of $P_{\rho_t}$, we have $$[s_t^*\omega] = [s_0^*\omega]$$ for all $t$.* *Proof.* The principal $H$-bundles $P_{\rho_t}(s_t)$ form a smooth family of bundles over $M$. By Lemma [Lemma 29](#lemma: family of principal bundles){reference-type="ref" reference="lemma: family of principal bundles"}, they are all isomorphic and, in particular, they have the same characteristic classes. Hence $$[s_t^*\omega] = \mathop{\mathrm{cw}}_f(P_{\rho_t}(s_t)) = \mathop{\mathrm{cw}}_f(P_{\rho_0}(s_0)) = [s_0^* \omega]$$ for all $t\in [0,1]$. ◻ ### The odd case Now, let $\rho_1$, $\rho_2$ be two representations of $\pi_1(M)$ into $G$ and $s$ a section of the flat bundle $M\times_{(\rho_1,\rho_2)} X_G$. Denote by $(P_{\rho_i}, \Theta_{\rho_i})$ the flat principal bundles associated to $\rho_i$, $i=1,2$. Then $s$ induces an isomorphism of principal bundles $$\begin{array}{rrcl} \varphi_s : & P_{\rho_2} & \to & P_{\rho_1} \\ \ & (x,g) & \mapsto & (x, \widetilde s (x) g) \end{array} ~.$$ Choose $\omega\in \mathop{\mathrm{Prim}}(X_G)$ and let $f\in \mathrm{Sym}^k_\textit{inv}(\mathfrak{g}^\vee)$ be such that $\omega = \frac{(-1)^k k! ((k-1)!}{2^{k-1}(2k-1)!} \tau(f)$. **Proposition 34**. *For any manifold $M$, any pair of representations $(\rho_1, \rho_2)\in \mathrm{Hom}(\pi_1(M),G)^2$ and any smooth section $s$ of the flat $X_G$-bundle associated to $(\rho_1,\rho_2)$, we have $$[s^*\omega] = \mathop{\mathrm{cs}}_f(\Theta_{\rho_1}, \varphi_s^*\Theta_{\rho_2})~.$$ In particular, for any representation $\rho \in \mathrm{Hom}(\pi_1(M),G)$ and any smooth section $s$ of the corresponding flat $G/H$-bundle, we have $$[s^* (\iota_{G,H}^* \omega)] = \mathop{\mathrm{cs}}_f(\Theta_\rho, \varphi_{\iota_{G,H}\circ s}^* \Theta_{\sigma \circ \rho})~.$$* *Proof.* Denote by $P_{X_G}$ the tautological principal $G$-bundle over $X_G$ and $\Theta_L$ and $\Theta_R$ the two invariant connections on $P_{X_G}$ given respectively by left and right parallelism. Let $\widetilde s$ be the lift of $s$ to $\widetilde M$. Then $\pi_1(M)$ acts on $\widetilde s^* P_{X_G}$ via $(\rho_1,\rho_2)$ and, by construction, the quotient $\pi_1(M) \backslash \widetilde s^* P_{X_G}$ equipped with the pulled-back connections $s^* \Theta_L$ and $s^*\Theta_R$ is isomorphic to the principal $G$-bundle $P_{\rho_1}$ equipped with the flat connections $\Theta_{\rho_1}$ and $\varphi_s^*\Theta_{\rho_2}$. Now, by Theorem [Theorem 24](#thm: Primitive forms -> Chern-Simons){reference-type="ref" reference="thm: Primitive forms -> Chern-Simons"}, we have $$\omega = \mathop{\mathrm{CS}}_f(\Theta_L, \Theta_R)~,$$ and by naturality under pull-back, we deduce that $$\begin{aligned} &=& [s^*\mathop{\mathrm{CS}}_f(\Theta_L, \Theta_R)]\\ &=& [\mathop{\mathrm{CS}}_f(s^*\Theta_L, s^*\Theta_R)]\\ &=& \mathop{\mathrm{cs}}_f(\Theta_{\rho_1}, \varphi_s^*\Theta_{\rho_2})~.\end{aligned}$$ The second part of the theorem is an immediate consequence of the first part, since $\iota_{G,H} \circ \widetilde s$ is $(\rho, \sigma \circ \rho)$-equivariant. ◻ **Corollary 35**. *For any manifold $M$, any smooth path $(\rho_{1,t},\rho_{2,t})_{t\in [0,1]}$ in $\mathrm{Hom}(\pi_1(M), G\times G)$ and any smooth family of sections $s_t$ of $M \times_{(\rho_{1,t}, \rho_{2,t})} X_G$, we have $$[s_t^*\omega] = [s_0^*\omega]$$ for all $t$.* *In particular, for any smooth path $(\rho_t)_{t\in [0,1]}$ and any smooth family of sections $s_t$ of $M\times_{\rho_t} (G/H)$, we have $$[s_t^* (\iota_{G,H}^*\omega)] = [s_0^* (\iota_{G,H}^*\omega)]$$ for all $t$.* *Proof.* By Lemma [Lemma 29](#lemma: family of principal bundles){reference-type="ref" reference="lemma: family of principal bundles"}, there exists a principal $G$-bundle $P$ over $M$ with two smooth families of connections $(\Theta_{1,t})_{t\in [0,1]}$ and $(\Theta_{2,t})_{t\in [0,1]}$ such that $$(P,\Theta_{1,t},\Theta_{2,t}) \simeq (P_{\rho_{1,t}}, \Theta_{\rho_{1,t}},\varphi_{s_t}^*\Theta_{\rho_{2,t}})$$ for all $t$. By Proposition [Proposition 34](#prop: Pull-back Chern--Simons){reference-type="ref" reference="prop: Pull-back Chern--Simons"}, we have $$\begin{aligned} - [s_0^*\omega]&=& \mathop{\mathrm{cs}}_f(\Theta_{1,t},\Theta_{2,t})-\mathop{\mathrm{cs}}_f(\Theta_{1,0},\Theta_{2,0})\\ &=& \mathop{\mathrm{cs}}_f(\Theta_{2,t}, \Theta_{2,0}) - \mathop{\mathrm{cs}}_f(\Theta_{1,t}, \Theta_{1,0})~.\end{aligned}$$ Since $\Theta_{2,0}$ and $\Theta_{2,t}$ (resp. $\Theta_{1,0}$ and $\Theta_{1,t}$) are joined by a smooth path of flat connections, we conclude that $$[s_t^*\omega] = [s_0^*\omega]$$ by Proposition [Proposition 14](#prop: Vanishing CS flat family){reference-type="ref" reference="prop: Vanishing CS flat family"}. ◻ ### Conclusion of the proof We have proven Theorem [Theorem 1](#t: Main Theorem){reference-type="ref" reference="t: Main Theorem"} for $\omega \in \Omega^\bullet_\textit{even}(G/H,\mathbb{C})$ (Corollary [Corollary 33](#coro: Rigidity Chern Weil){reference-type="ref" reference="coro: Rigidity Chern Weil"}) and for $\omega \in \iota_{G,H}^* \mathop{\mathrm{Prim}}(X_G)$ (Corollary [Corollary 35](#coro: rigidity Chern--Simons){reference-type="ref" reference="coro: rigidity Chern--Simons"}). By Theorem [Theorem 21](#thm: Hopf){reference-type="ref" reference="thm: Hopf"}, $\mathop{\mathrm{Prim}}(X_G)$ generates $\Omega^\bullet_\textit{inv}(X_G)$; hence $\iota_{G,H}^* \mathop{\mathrm{Prim}}(X_G)$ generates $\Omega^\bullet_\textit{odd}(G/H,\mathbb{C})$. Since $\Omega^\bullet_\textit{inv}(G/H,\mathbb{C}) = \Omega^\bullet_\textit{even}(G/H,\mathbb{C})\otimes \Omega^\bullet_\textit{odd}(G/H,\mathbb{C})$ by Theorem [Theorem 27](#thm: Odd forms G/H){reference-type="ref" reference="thm: Odd forms G/H"}, we conclude that Theorem [Theorem 1](#t: Main Theorem){reference-type="ref" reference="t: Main Theorem"} holds for any $G$-invariant form. ## Continuous group cohomology {#ss: Continuous group cohomology Proof } The *continuous cohomology* $\mathrm{H}^\bullet_c(G,\mathbb{C})$ (with constant coefficients) of a Lie group $G$ is the cohomology of the complex $\mathcal C^\bullet_c(G,\mathbb{C})^G$, where $\mathcal C^k_c(G,\mathbb{C})^G$ is the space of $G$-invariant continuous functions on $G^{k+1}$, equipped with the usual differential: $$\mathrm{d}f(g_0, \ldots , g_{k+1}) = \sum_{j=0}^{k+1} (-1)^j f(g_0, \ldots, \widehat g_j, \ldots, g_{k+1})~.$$ When $G$ is connected, The *Van Est isomorphism* identifies $\mathrm{H}^\bullet_c(G,\mathbb{C})$ with the algebra of $G$-invariant forms on the symmetric space $G/K$. To be more precise, recall that if $M$ is a manifold, there is a map $\pi$ from $M$ to a classifying space of $\pi_1(M)$, unique up to homotopy, which induces the identity on the fundamental groups. **Theorem 36** (Van Est). *There is an isomorphism of graded algebras $$\mathop{\mathrm{VE}}: \mathrm{H}^\bullet_c(G,\mathbb{C}) \to \Omega^\bullet_\textit{inv}(G/K,\mathbb{C})$$ such that, for any smooth manifold $M$ and any representation $\rho: \pi_1(M) \to G$, the following diagram commutes: $$\xymatrix{ \mathrm{H}^\bullet_c(G,\mathbb{C}) \ar[r]^{\mathop{\mathrm{VE}}} \ar[d]_{\rho^*} & \Omega^\bullet_\textit{inv}(G/K,\mathbb{C}) \ar[d]^{\rho^*} \\ \mathrm{H}^\bullet(\Gamma,\mathbb{C}) \ar[r]^{\pi^*} & \mathrm{H}^\bullet(M,\mathbb{C})~. }$$* Here the map $\rho^*: \mathrm{H}^\bullet_c(G,\mathbb{C}) \to \mathrm{H}^\bullet(\Gamma,\mathbb{C})$ is the pull-back map on group cohomology while $\rho^*: \Omega^\bullet_\textit{inv}(G/K,\mathbb{C}) \to \mathrm{H}^\bullet(M,\mathbb{C})$ maps $\omega$ to $[s^*\omega]$ for any smooth section of $M\times_\rho (G/K)$. *Proof of Corollary [Corollary 2](#c: Main Corollary Cohomology){reference-type="ref" reference="c: Main Corollary Cohomology"}.* Let $\Gamma$ be a finitely presented group. Assume first that $\Gamma$ is the fundamental group of an aspherical manifold $M$. Then the map $\pi^*: \mathrm{H}^\bullet(\Gamma,\mathbb{C}) \to \mathrm{H}^\bullet(M,\mathbb{C})$ is an isomoprhism. Let $(\rho_t)_{t\in [0,1]}$ be a smooth family of representations of $\Gamma$ into $G$ and let $\alpha$ be a continuous cohomology class in $\mathrm{H}^\bullet_c(G,\mathbb{C})$. Then we have $$\begin{aligned} \pi^* \rho_t^* \alpha &=& \rho_t^* \mathop{\mathrm{VE}}(\alpha) \quad \textrm{by Theorem \ref{thm: Van Est}}\\ &=& \rho_0^* \mathop{\mathrm{VE}}(\alpha) \quad \textrm{by Theorem \ref{t: Main Theorem}}\\ &=& \pi^* \rho_0^* \alpha~,\end{aligned}$$ and we conclude that $\rho_t^* \alpha= \rho_0^* \alpha$ for all $t$ since $\pi^*$ is injective. In general, while $\Gamma$ need not be the fundamental group of an aspherical manifold, one can always find for any $n\in \mathbb{N}$ a manifold $M$ such that $\pi_1(M) = \Gamma$ and $\pi_k(M) = \{0\}$ for all $2\leq k\leq n$. Then $\pi^*: \mathrm{H}^k(\Gamma,\mathbb{C}) \to \mathrm{H}^k(M,\mathbb{C})$ is an isomorphism for all $1\leq k\leq n$. Applying the above arguments thus gives that $\rho_t^* \alpha$ is constant in $t$ for all $\alpha \in \mathrm{H}^{\leq n}(\Gamma,\mathbb{C})$. Since $n$ is arbitrary, the conclusion follows. ◻ ## Volume of locally homogeneous spaces In this section we prove Corollary [Corollary 3](#c: Main Corollary Volume){reference-type="ref" reference="c: Main Corollary Volume"}, namely the volume rigidity of manifolds locally modelled on a reductive homogeneous space $G/H$.\ The Killing form of $\mathfrak{g}$ restricted to $\mathfrak{h}^\perp \simeq \mathfrak{g}/\mathfrak{h}$ extends to a $G$-invariant metric on $G/H$. Moreover, $G/H$ can be oriented since $H$ is assumed to be connected. Hence the metric induces a $G$-invariant volume form $vol_{G/H}$. Let $(\mathrm{dev}_t,\rho_t)_{t\in [0,1]}$ be a smooth family of $G/H$-structures on a closed manifold $M$. By definition, the *volume* of the $G/H$-structure $(\mathrm{dev}_t, \rho_t)$ is the number $$\int_M \mathrm{dev}_t^*vol_{G/H}~.$$ When $G/H$ is symmetric, Corollary [Corollary 3](#c: Main Corollary Volume){reference-type="ref" reference="c: Main Corollary Volume"} is thus a direct application of Theorem [Theorem 1](#t: Main Theorem){reference-type="ref" reference="t: Main Theorem"}, taking $\widetilde s_t = \mathrm{dev}_t$ and $\omega = vol_{G/H}$. We prove the general case where $G$ is semisimple and $H\subset G$ is reductive by lifting the $G/H$-structure to a $X_G$-structure on an $H/\Lambda$-bundle over $M$. *Proof of Corollary [Corollary 3](#c: Main Corollary Volume){reference-type="ref" reference="c: Main Corollary Volume"}.* Fix a uniform lattice $\Lambda$ in $H$ (it exists by Borel--Harish-Chandra's theorem). Let $M$ be a closed manifold and $(\mathrm{dev}_t, \rho_t)$ a smooth family of $G/H$-structures on $M$. Let $\widetilde P\to \widetilde M$ denote the pull-back of the tautological principal $H$-bundle over $G/H$ by $\mathrm{dev}_t$ and and $P\to M$ its quotient under $\pi_1(M)$ (the bundle $P$ does not depend on $t$ by Lemma [Lemma 29](#lemma: family of principal bundles){reference-type="ref" reference="lemma: family of principal bundles"}). By construction, the map $\mathrm{dev}_t$ lifts to a local diffeomorphism $\widehat\mathrm{dev}_t: \widetilde P \to X_G = G$, which satisfies $$\widehat\mathrm{dev}_t(\gamma \cdot p) = \rho_t(\gamma) \widehat\mathrm{dev}_t(p)$$ for $\gamma \in \pi_1(M)$ and $$\widehat\mathrm{dev}_t(p \cdot \lambda) = \widehat\mathrm{dev}_t(p) \lambda$$ for all $\lambda \in \Lambda$. Setting $$\begin{array}{rrcl} \widehat\rho_t : & \pi_1(M)\times \Lambda & \to & G\times G \\ \ & (\gamma,\lambda) & \mapsto & (\rho_t(\gamma), \lambda)~, \end{array}$$ we thus get that $(\widehat\mathrm{dev}_t, \widehat\rho_t)$ defines a $X_G$-structure on the closed manifold $P/ \Lambda$. Applying Theorem [Theorem 1](#t: Main Theorem){reference-type="ref" reference="t: Main Theorem"} to the symmetric space $X_G$, we deduce that $$\int_{P/\Lambda} \widehat\mathrm{dev}_t^*vol_{X_G} = \int_{P/\Lambda} \widehat\mathrm{dev}_0^*vol_{X_G}$$ for all $t$. On the other hand, after normalizing the volume forms of $X_G$, $H$ and $G/H$ compatibly, we have $$\int_{P/\Lambda} \widehat\mathrm{dev}_t^*vol_{X_G} = \mathbf{Vol}(H/\Lambda) \int_M \mathrm{dev}_t^*vol_{G/H}~.$$ We thus conclude that $$\int_M \mathrm{dev}_t^*vol_{G/H} = \int_M\mathrm{dev}_0^* vol_{G/H}$$ for all $t$. ◻ [^1]: While the complexification is perfectly well-defined at the level of Lie algebras, there is something arbitrary in our definition of the complexification of Lie group. For instance, the complexification of $\mathrm{SL}(k,\mathbb{R})$ is the adjoint group $\mathrm{PSL}(k,\mathbb{C})$. We did not settle with a more algebraic definition because we want to consider connected simple Lie groups such as $\mathrm{SO}_\circ (p,1)$, which are not necessarily algebraic. [^2]: Note that both connections are indeed *bi-invariant*, because the right-translate of a left-invariant vector field is another left-invariant vector field.

arxiv_math

--- abstract: | This paper investigates the problem of decentralized resource allocation in the presence of Byzantine attacks. Such attacks occur when an unknown number of malicious agents send random or carefully crafted messages to their neighbors, aiming to prevent the honest agents from reaching the optimal resource allocation strategy. We characterize these malicious behaviors with the classical Byzantine attacks model, and propose a class of Byzantine-resilient decentralized resource allocation algorithms augmented with dual-domain defenses. The honest agents receive messages containing the (possibly malicious) dual variables from their neighbors at each iteration, and filter these messages with robust aggregation rules. Theoretically, we prove that the proposed algorithms can converge to neighborhoods of the optimal resource allocation strategy, given that the robust aggregation rules are properly designed. Numerical experiments are conducted to corroborate the theoretical results. author: - "Runhua Wang, Qing Ling and Zhi Tian [^1] [^2]" title: | Dual-domain Defenses for Byzantine-resilient\ Decentralized Resource Allocation --- Resource allocation, decentralized multi-agent network, Byzantine-resilience # INTRODUCTION {#sec 1} resource allocation has found wide applications in various fields, such as smart grids, transportation systems, wireless sensor networks, etc [@b-Zhong-Fan-2013; @b-Md-Noor-A-Rahim-2022; @b-Ayaz-Ahmad-2015]. Mathematically speaking, it minimizes the average cost of decentralized agents subject to local and global resource constraints, where the optimization variable is the resource allocation strategy. Solving this optimization problem requires collaboration between neighboring agents. Nevertheless, such collaboration is not always reliable since some of the agents could be malicious. The aim of this paper is to develop effective decentralized resource allocation algorithms that are resilient to malicious agents. **Decentralized Resource Allocation Algorithms.** Existing decentralized resource allocation algorithms can be categorized as continuous-time [@b-Shu-Liang-2020; @b-Wenwen-Jia-2022; @b-Yanan-Zhu-2021; @b-Kaihong-Lu-2022] and discrete-time [@b-Hariharan-Lakshmanan-2008; @b-LXiao-2006; @b-Euhanna-Ghadimi-2013; @b-Thinh-T.Doan-2021; @b-Yun-Xu-2017; @b-Jiaqi-Zhang-2020; @b-AngeliaNedic-2018; @b-SulaimanA-Alghunaim-2020]. In this paper, we focus on discrete-time algorithms. The primary challenge in algorithm design is to satisfy the global resource constraint. Weighted gradient methods have been proposed to guarantee global constraint satisfaction with the aid of feasible initialization [@b-Hariharan-Lakshmanan-2008; @b-LXiao-2006; @b-Euhanna-Ghadimi-2013], but they turn out to be sensitive to perturbations. The work of [@b-Hariharan-Lakshmanan-2008] is based on time-varying networks, while [@b-LXiao-2006] considers fixed networks. The work of [@b-Euhanna-Ghadimi-2013] utilizes historical information to accelerate the algorithm. On the other hand, primal-dual algorithms handle the global resource constraint via introducing a dual variable [@b-Thinh-T.Doan-2021; @b-Yun-Xu-2017; @b-Jiaqi-Zhang-2020; @b-AngeliaNedic-2018; @b-SulaimanA-Alghunaim-2020]. The works of [@b-Thinh-T.Doan-2021; @b-Yun-Xu-2017] develop decentralized Lagrangian methods, which precisely solve the primal sub-problems while perform a dual gradient step at each iteration. The work of [@b-Jiaqi-Zhang-2020] employs the push-pull gradient method to solve the dual problem and proposes a dual gradient tracking algorithm for unbalanced networks. For non-smooth resource allocation problems, decentralized proximal primal-dual algorithms are developed in [@b-AngeliaNedic-2018; @b-SulaimanA-Alghunaim-2020]. The decentralized resource allocation algorithms discussed above perform well when all the agents are honest. However, malicious agents, either spontaneously or by manipulation, are always threats to decentralized networks. These agents do not follow the given algorithmic protocol, but send random or crafted messages to their honest neighbors for the sake of misleading the optimization process. To characterize such behaviors, we use the classical Byzantine attacks model and term the malicious agents as Byzantine agents [@b-LeslieLamport-1982; @b-ZhixiongYang-2020]. We briefly review some general Byzantine-resilient decentralized *optimization* algorithms and few Byzantine-resilient *resource allocation* algorithms, as follows. **Byzantine-resilient Algorithms.** Given a general Byzantine-resilient decentralized optimization problem, honest agents cooperate to reach a consensual optimal solution that minimizes their average cost function. This is different to the resource allocation problem, where the honest agents are expected to obtain different optimal solutions (namely, allocated resources). Some works focus on deterministic problems [@b-Lili-Su-2021; @b-Zhixiong-Yang-2019; @b-Shreyas-Sundaram-2019; @b-Lili-Su-2020; @b-Cheng-Fang-2022; @b-Waseem-Abbas-2022; @b-Kananart-Kuwaranancharoen-2020] and some others consider stochastic problems [@b-ZhaoxianWu-2022; @b-LieHe-2022]. Their common feature is to let each honest agent aggregate possibly malicious messages (namely, optimization variables) received from its neighbors in a robust manner. For Byzantine-resilient optimization problems with deterministic cost functions, when the optimization variable is a scalar, [@b-Lili-Su-2021; @b-Zhixiong-Yang-2019] proposes the trimmed mean (TM) robust aggregation rule, with which each honest agent discards the smallest $b$ and the largest $b$ messages received from its neighbors, followed by averaging the remaining messages and its own. Here $b$ is an estimated upper bound of the number of Byzantine neighbors. A similar approach in [@b-Shreyas-Sundaram-2019] lets each honest agent filter $b$ received messages larger and $b$ received messages smaller than its own message, also followed by averaging. For high-dimensional problems, [@b-Lili-Su-2020; @b-Cheng-Fang-2022] extends TM to coordinate-wise TM (CTM), such that each honest agent performs the TM operation at each dimension. The work of [@b-Waseem-Abbas-2022] introduces the notion of centerpoint, which is an extension of the robust median aggregation rule to the high-dimensional scenario. In [@b-Kananart-Kuwaranancharoen-2020], each iteration involves two filtering steps: distance-based and dimension-wise removals. Distance-based removal calculates the Euclidean distances between the received messages and the agent's own message, sorts the distances, and removes $b$ messages with the largest distances. Additionally, messages with extreme values in any dimension are removed. When the cost functions are stochastic, TM and CTM are also applicable. Besides, the work of [@b-ZhaoxianWu-2022] proposes iterative outlier scissor (IOS), in which each honest agent iteratively discards $b$ messages that are the farthest from the average of the remaining received messages. The work of [@b-LieHe-2022] proposes self-centered clipping (SCC), in which each honest agent uses its own optimization variable as the center, clips the received messages, and then runs weighted average. Although the aforementioned Byzantine-resilient decentralized optimization algorithms are proved to be effective, they cannot be directly applied to solve the resource allocation problem. The local optimization variables of the honest agents are coupled with a consensus constraint in the former but with a global resource constraint in the latter. Therefore, in a decentralized resource allocation algorithm, filtering "outliers" from the neighboring optimization variables becomes meaningless. To fill this gap, [@b-Berkay-Turan-2021] proposes a primal-dual Byzantine-resilient resource allocation algorithm from a robust optimization perspective, but the proposed algorithm is only applicable in a distributed network with a central server. A Byzantine-resilient decentralized resource allocation (BREDA) algorithm is developed in [@b-Runhua-Wang-2022]. In addition to the updates of primal and dual variables, each honest agent maintains an auxiliary variable that dynamically tracks the average of all honest agents' primal variables. Then, CTM is applied to aggregate the neighboring auxiliary variables. **Our Contributions.** This paper focuses on the challenging and less-studied Byzantine-resilient decentralized resource allocation problem, and makes the following contributions: **C1)** We propose a class of primal-dual Byzantine-resilient decentralized resource allocation algorithms with dual-domain defenses. The key intuition is that the honest agents should reach a consensual dual variable. Therefore, we can let each honest agent filter the received neighboring dual variables with properly designed robust aggregation rules, including but not limited to CTM, IOS and SCC. **C2)** Compared with BREDA that defends against Byzantine attacks in the primal domain [@b-Runhua-Wang-2022], the proposed algorithms utilize dual-domain defenses, and have the following advantages: (i) maintaining less variables and simpler updates; (ii) allowing more general robust aggregation rules than CTM; (iii) being able to reach dual consensus. **C3)** Theoretically, we prove that if the robust aggregation rules are properly designed, the proposed algorithms converge to neighborhoods of the optimal primal-dual pair, and the honest agents are guaranteed to reach consensus in the dual domain even at presence of Byzantine attacks. With numerical experiments, we verify Byzantine-resilience of the proposed algorithms and its advantages over BREDA. Compared to the short, preliminary version of this paper [@R-Wang-2024-icassp], we have added derivations for the algorithm design, details of the theoretical analysis, and extra numerical experiments. **Paper Organization:** This paper is organized as follows. In Section [2](#sec 2){reference-type="ref" reference="sec 2"}, we formulate the decentralized resource allocation problem under Byzantine attacks. Section [3](#sec 3){reference-type="ref" reference="sec 3"} proposes an attack-free decentralized resource allocation algorithm that operates in the dual domain, and shows its failure under Byzantine attacks. Section [4](#sec 4){reference-type="ref" reference="sec 4"} further proposes a class of Byzantine-resilient decentralized resource allocation algorithms. Section [5](#sec 5){reference-type="ref" reference="sec 5"} establishes convergence of the proposed Byzantine-resilient decentralized resource allocation algorithms. Numerical experiments are given in Section [6](#sec 6){reference-type="ref" reference="sec 6"}. Section [7](#sec 7){reference-type="ref" reference="sec 7"} summarizes this paper and discusses future research directions. **Notation:** Throughout this paper, $(\cdot)^{ \top }$ stands for the transposition of a vector or a matrix, $\|\cdot\|$ stands for the $\ell_{2}$-norm of a vector or a matrix, $\|\cdot\|_{F}$ denotes the Frobenius norm of a matrix, and $\left \langle \cdot , \cdot \right \rangle$ represents the inner product of vectors. We define $\widetilde{\bm{1}}\in \mathbb{R}^{J}$ and $\bm{1}\in \mathbb{R}^{H}$ as all-one column vectors while $I\in \mathbb{R}^{H \times H}$ as an identity matrix, where $J$ is the number of all agents and $H$ is the number of honest agents. # PROBLEM STATEMENT {#sec 2} We consider a decentralized resource allocation problem that involves a network of autonomous agents. The network is modeled as an undirected, connected graph $\widetilde{\mathcal{G}} ( \mathcal{J},\widetilde{\mathcal{E}} )$ with the set of vertices $\mathcal{J}:=\left \{ 1, \cdots,J \right \}$ and the set of edges $\widetilde{\mathcal{E}}$. If $( i,j) \in \widetilde{\mathcal{E}}$, then the two agents $i$ and $j$ are neighbors and can communicate with each other. For agent $i$, define the set of its neighbors as $\mathcal{N}_{i}= \{ j\mid ( i,j ) \in \widetilde{\mathcal{E}} \}$. Each agent $i$ possesses a strongly convex local cost function $f_{i}\left ( \bm{\theta}_{i} \right )$, where $\bm{\theta}_{i} \in \mathbb{R}^{D}$ stands for the amount of local resources and belongs to a compact, convex set $C_{i}$. The average amount of local resources, denoted as $\frac{1}{J}\sum_{i \in \mathcal{J}}\bm{\theta}_{i}$, equals to a constant vector $\bm{s} \in \mathbb{R}^{D}$. When all the agents are honest, the decentralized resource allocation problem is formulated as $$\begin{aligned} \label{eq_primal-problem} \begin{split} \underset{\widetilde{\bm{\Theta}}}{\min} \quad & \widetilde{f}(\widetilde{\bm{\Theta}})=\frac{1}{J}\sum_{i \in \mathcal{J}}f_{i}\left ( \bm{\theta}_{i} \right ),\\ s. t. \quad & \frac{1}{J}\sum_{i \in \mathcal{J}}\bm{\theta}_{i} = \bm{s}, \quad \bm{\theta}_{i} \in C_{i}, ~ \forall i \in \mathcal{J}, \\ \end{split}\end{aligned}$$ where $\widetilde{\bm{\Theta}}=[\bm{\theta}_{1},\cdots, \bm{\theta}_{J}] \in \mathbb{R}^{JD}$ concatenates all the local variables and $\widetilde{C}$ is the Cartesian product of $C_{i}$ for all $i\in \mathcal{J}$. The decentralized resource allocation problem in the form of [\[eq_primal-problem\]](#eq_primal-problem){reference-type="eqref" reference="eq_primal-problem"} arises in, for example, economic dispatch in smart grids [@b-Qiao-Li-2019; @b-Huaqing-Li-2020]. The goal is to obtain an optimal generation strategy that minimizes the total generation cost, while satisfying a global power demand constraint and local generator constraints, through cooperation among a network of generators. We will introduce the economic dispatch problem in detail in Section [6](#sec 6){reference-type="ref" reference="sec 6"}, and focus on the case that some of the generators are malicious. When some of the agents are Byzantine, solving [\[eq_primal-problem\]](#eq_primal-problem){reference-type="eqref" reference="eq_primal-problem"} is an impossible task, because they will not collaborate with the honest agents during the optimization process. Denote the set of Byzantine agents as $\mathcal{B}$ and the set of honest agents as $\mathcal{H}: =\mathcal{J}\setminus \mathcal{B}$. The numbers of Byzantine agents and honest agents are denoted as $B$ and $H$, respectively. Note that the number and identities of Byzantine agents are not known in advance, but we can roughly estimate an upper bound of the number. For notational convenience, we number the honest agents from $1$ to $H$, and the Byzantine agents from $H+1$ to $H+B$. Consider a subgraph $\mathcal{G}(\mathcal{H}, \mathcal{ \mathcal{E} })$ of $\widetilde{\mathcal{G}} ( \mathcal{J},\widetilde{\mathcal{ \mathcal{E} }})$, where $\mathcal{E} = \{(i,j) \in \widetilde{\mathcal{E}}; i, j \in \mathcal{H} \}$ is the set of edges between the honest agents. We assume $\mathcal{G}(\mathcal{H}, \mathcal{ \mathcal{E} })$ to be connected too so that the honest agents can cooperate. The goal of the honest agents is to solve $$\begin{aligned} \label{eq_oracle-problem} \begin{split} \underset{ \bm{\Theta}}{\min} \quad & f\left ( \bm{\Theta} \right ):=\frac{1}{ H }\sum_{i \in \mathcal{H}}f_{i}\left ( \bm{\theta}_{i} \right ),\\ s.t. \quad & \frac{1}{ H }\sum_{i\in \mathcal{H}}\bm{\theta}_{i} = \bm{s}, \quad \bm{\theta}_{i} \in C_{i}, ~ \forall i \in \mathcal{H}, \\ \end{split}\end{aligned}$$ where $\bm{\Theta}=[\bm{\theta}_{1},\cdots, \bm{\theta}_{H}] \in \mathbb{R}^{HD}$ concatenates all the local variables of the honest agents and $C$ is the Cartesian product of $C_{i}$ for all $i\in \mathcal{H}$. However, solving [\[eq_oracle-problem\]](#eq_oracle-problem){reference-type="eqref" reference="eq_oracle-problem"} is still challenging since the honest agents cannot distinguish their Byzantine neighbors, while the latter can send arbitrarily malicious messages during the optimization process. Therefore, in this paper, we focus on developing Byzantine-resilient decentralized resource allocation algorithms to approximately solve [\[eq_oracle-problem\]](#eq_oracle-problem){reference-type="eqref" reference="eq_oracle-problem"}. # ATTACK-FREE DECENTRALIZED RESOURCE ALLOCATION {#sec 3} This section begins with reviewing an attack-free decentralized resource allocation algorithm, which operates in the dual domain, to solve [\[eq_primal-problem\]](#eq_primal-problem){reference-type="eqref" reference="eq_primal-problem"}. ## Algorithm Development The Lagrangian function of ([\[eq_primal-problem\]](#eq_primal-problem){reference-type="ref" reference="eq_primal-problem"}) is $$\begin{aligned} \label{eq_Lagrangian-primal-problem} \begin{split} \widetilde{\mathcal{L}}( \widetilde{\bm{\Theta}} ; \widetilde{\bm{\lambda }}) := \frac{1}{J}\sum_{i \in \mathcal{J}}f_{i}\left ( \bm{\theta}_{i} \right )+ \widetilde{\bm{\lambda }}^{\top} (\frac{1}{J}\sum_{i \in \mathcal{J}}\bm{\theta}_{i}-\bm{s}), \end{split}\end{aligned}$$ where $\widetilde{\bm{\lambda }}\in \mathbb{R}^D$ is the dual variable. Hence, the dual function $\widetilde{d}(\widetilde{\bm{\lambda }}) := \min_{\widetilde{\bm{\Theta}}\in \widetilde{C}} \widetilde{\mathcal{L}}( \widetilde{\bm{\Theta}} ; \widetilde{\bm{\lambda }})$ is given by $$\begin{aligned} \label{eq_dual-function_primal} \widetilde{d}(\widetilde{\bm{\lambda }}):=&\underset{\widetilde{\bm{\Theta}}\in \widetilde{C}}{\min}\{\frac{1}{J}\sum_{i \in \mathcal{J}}f_{i}\left ( \bm{\theta}_{i} \right )+\widetilde{\bm{\lambda }}^{\top}( \frac{1}{J}\sum_{i \in \mathcal{J}}\bm{\theta}_{i}-\bm{s})\} \\ =& \frac{1}{J}\sum_{i \in \mathcal{J}} \underset{\bm{\theta}_{i}\in C_{i}}{\min}\{f_{i}(\bm{\theta}_{i})+\widetilde{\bm{\lambda }}^{\top}\bm{\theta}_{i}\}-\widetilde{\bm{\lambda }}^{\top}\bm{s}\notag \\ =& \frac{1}{J}\sum_{i \in \mathcal{J}}(-\underset{\bm{\theta}_{i}\in C_{i}}{\max}\{-f_{i}(\bm{\theta}_{i})-\widetilde{\bm{\lambda }}^{\top}\bm{\theta}_{i}\})-\widetilde{\bm{\lambda }}^{\top}\bm{s}\notag \\ =& \frac{1}{J} \sum_{i \in \mathcal{J}} -\widetilde{F}_{i}^{*}(-\widetilde{\bm{\lambda }})-\widetilde{\bm{\lambda }}^{\top}\bm{s},\notag\end{aligned}$$ where $\widetilde{F}_{i}^{*}(\widetilde{\bm{\lambda }}):=\max_{\bm{\theta}_{i}\in C_{i}}\{\widetilde{\bm{\lambda }}^{\top}\bm{\theta}_{i}-f_{i}(\bm{\theta}_{i})\}$. With it, we write the dual problem of [\[eq_primal-problem\]](#eq_primal-problem){reference-type="eqref" reference="eq_primal-problem"} as a minimization problem in the form of $$\begin{aligned} \label{eq_dual-problem_primal} \underset{\widetilde{\bm{\lambda }}\in \mathbb{R}^D}{\min} ~ \widetilde{g}(\widetilde{\bm{\lambda }})= -d(\widetilde{\bm{\lambda}})=\sum_{i \in \mathcal{J}}\widetilde{g}_{i}(\widetilde{\bm{\lambda }}),\end{aligned}$$ where $\widetilde{g}_{i}(\widetilde{\bm{\lambda }}):=\frac{1}{J}\widetilde{F}_{i}^{*}(-\widetilde{\bm{\lambda }})+\frac{1}{J}\widetilde{\bm{\lambda }}^{\top}\bm{s}$. Because $f_{i}(\cdot)$ is strongly convex, according to the conjugate correspondence theorem in [@b-Amir-Beck-2017], its conjugate function $\widetilde{F}_{i}^{*}(\cdot)$ is smooth. By Danskin's theorem [@b-Dimitri-P.-Bertsekas-1999], the gradient $\nabla \widetilde{F}^{*}_{i}(\bm{\lambda}_{i})=\arg\max_{\bm{\theta}_{i}\in C_{i}}\{\bm{\lambda }_{i}^{\top}\bm{\theta}_{i}-f_{i}(\bm{\theta}_{i})\}$. Hence, we have $$\begin{aligned} \label{eq_gradient_dual_function_primal} \nabla \widetilde{g}_{i}(\bm{\lambda}_{i})=\frac{1}{J}\bm{s}-\frac{1}{J}\arg\underset{\bm{\theta}_{i}\in C_{i}}{\min}\{\bm{\lambda}_{i}^{\top}\bm{\theta}_{i}+f_{i}(\bm{\theta}_{i})\}.\end{aligned}$$ According to the above discussions, the optimization problem [\[eq_dual-problem_primal\]](#eq_dual-problem_primal){reference-type="eqref" reference="eq_dual-problem_primal"} can be solved through decentralized gradient methods [@b-Thinh-T.Doan-2021; @b-AngeliaNedic-2009; @b-B-Johansson-2008]. To do so, we let each agent holds a local dual variable $\bm{\lambda}_{i} \in \mathbb{R}^D$. The updates of primal and dual variables for all agents $i\in \mathcal{J}$ in the attack-free decentralized resource allocation algorithm at iteration $k+1$ are given by $$\begin{aligned} \bm{\theta}_{i}^{k}&=\arg\underset{\bm{\theta}_{i}\in C_{i}}{\min}\{\bm{\theta}_{i}^{\top}\bm{\lambda}_{i}^{k}+f_{i}(\bm{\theta}_{i})\}, \label{eq_DEGRA-1} \\ \bm{\lambda}_{i}^{k+\frac{1}{2}}&=\bm{\lambda}_{i}^{k}-\gamma^{k}\nabla \widetilde{g}_{i}(\bm{\lambda}_{i}^{k})=\bm{\lambda}_{i}^{k}-\gamma^{k}(\frac{1}{J}\bm{s}-\frac{1}{J}\bm{\theta}_{i}^{k}), \label{eq_DEGRA-2} \\ \bm{\lambda}_{i}^{k+1}&=\sum_{j\in \mathcal{N}_{i}\cup \{i\}}\widetilde{e}_{ij}\bm{\lambda}_{j}^{k+\frac{1}{2}}. \label{eq_DEGRA-3}\end{aligned}$$ Therein, $\gamma^k > 0$ is the step size and $\widetilde{e}_{ij}\geq 0$ is the weight assigned by agent $i$ to agent $j$. Note that $\widetilde{e}_{ij}>0$ if and only if $(i, j)\in \widetilde{\mathcal{E}}$ or $i=j$. We collect these weights in $\widetilde{E}=[\widetilde{e}_{ij}]\in \mathbb{R}^{J\times J}$, which is assumed to be doubly stochastic. Such an attack-free decentralized resource allocation algorithm is summarized in Algorithm [\[alg1\]](#alg1){reference-type="ref" reference="alg1"}. Initialization: All agents $i\in \mathcal{J}$ initialize $\bm{\lambda}_{i}^{0}=\bm{\lambda}^{0}$. Compute $\bm{\theta}_{i}^{k}=\arg\min_{\bm{\theta}_{i}\in C_{i}} \{\bm{\theta}_{i}^{\top}\bm{\lambda}_{i}^{k}+f_{i}(\bm{\theta}_{i})\}$. Compute $\bm{\lambda}_{i}^{k+\frac{1}{2}}=\bm{\lambda}_{i}^{k}-\gamma^{k}(\frac{1}{J}\bm{s}-\frac{1}{J}\bm{\theta}_{i}^{k})$. Broadcast $\bm{\lambda}_{i}^{k+\frac{1}{2}}$ to its neighbors. Receive $\bm{\lambda}_{j}^{k+\frac{1}{2}}$ from its neighbors. Aggregate $\bm{\lambda}_{i}^{k+1}=\sum_{j\in \mathcal{N}_{i}\cup \{i\}}\widetilde{e}_{ij}\bm{\lambda}_{j}^{k+\frac{1}{2}}$. ## Failure of Attack-free Decentralized Resource Allocation Algorithm under Byzantine Attacks When all the agents are honest, the decentralized resource allocation algorithm outlined in [\[eq_DEGRA-1\]](#eq_DEGRA-1){reference-type="eqref" reference="eq_DEGRA-1"}--[\[eq_DEGRA-3\]](#eq_DEGRA-3){reference-type="eqref" reference="eq_DEGRA-3"} can effectively solve [\[eq_primal-problem\]](#eq_primal-problem){reference-type="eqref" reference="eq_primal-problem"}; readers are referred to [@b-Thinh-T.Doan-2021; @b-AngeliaNedic-2009; @b-B-Johansson-2008]. However, it fails in the presence of Byzantine attacks. At iteration $k+1$, each honest agent $i\in \mathcal{H}$ updates $\bm{\lambda}_{i}^{k+1}$ based on $\bm{\lambda}_{i}^{k+\frac{1}{2}}$ from its own and $\bm{\lambda}_{j}^{k+\frac{1}{2}}$ from its neighbors $j \in \mathcal{N}_i$. An honest neighbor $j \in \mathcal{N}_i \cap \mathcal{H}$ faithfully sends the message $\bm{\lambda}_{j}^{k+\frac{1}{2}}$, but a Byzantine neighbor $j \in \mathcal{N}_i \cap \mathcal{B}$ may send an arbitrarily malicious message $*$ instead of the true message $\bm{\lambda}_{j}^{k+\frac{1}{2}}$. We define the message sent by agent $j$ as $$\begin{aligned} \label{broadcasting_message} \check{\bm{\lambda}}_{j}^{k+\frac{1}{2}} =\left\{\begin{matrix}\bm{\lambda}_{j}^{k+\frac{1}{2}}, \quad & j\in \mathcal{H}, \\ *, \quad & j \in \mathcal{B}. \end{matrix}\right.\end{aligned}$$ The malicious messages sent by the Byzantine agents prevent the honest agents from obtaining the optimal dual variable and corresponding resource allocation strategy. We provide a simple example to illustrate their impact. Assume that the local cost function of agent $i$ is $f_{i}(\bm{\theta}_{i})=\bm{\theta}_{i}^{2}$, the local resource constraint set is $C_{i}=[0,100]$, and the average resource is $\bm{s}=50$. The optimal dual variable and resource allocation of agent $i$ are $\bm{\lambda}_{i}^{*}=-100$ and $\bm{\theta}_{i}^{*}=50$, respectively. According to [\[eq_DEGRA-1\]](#eq_DEGRA-1){reference-type="eqref" reference="eq_DEGRA-1"}, the update of $\bm{\theta}_{i}^{k+1}$ is $\bm{\theta}_{i}^{k+1}=\Pi_{[0,100]}(-\frac{\bm{\lambda}_{i}^{k+1}}{2})$, the projection of $-\frac{\bm{\lambda}_{i}^{k+1}}{2}$ onto $[0,100]$. A Byzantine agent $j$ can manipulate $\bm{\lambda}_{i}^{k+1}$ by [\[eq_DEGRA-3\]](#eq_DEGRA-3){reference-type="eqref" reference="eq_DEGRA-3"} to be either $0$ or $-200$ through sending a proper $\check{\bm{\lambda}}_{j}^{k+\frac{1}{2}}$. In consequence, honest agent $i$ will obtain resource allocation of either $\bm{\theta}_{i}^{k+1}=0$ or $\bm{\theta}_{i}^{k+1}=100$, which are faraway from the optimal solution. # BYZANTINE-RESILIENT DECENTRALIZED RESOURCE ALLOCATION {#sec 4} In light of the influence of Byzantine attacks to decentralized resource allocation, we propose a class of Byzantine-resilient decentralized resource allocation algorithms to approximately solve [\[eq_oracle-problem\]](#eq_oracle-problem){reference-type="eqref" reference="eq_oracle-problem"} in this section. ## Algorithm Development As we have shown in Section [3](#sec 3){reference-type="ref" reference="sec 3"}, the decentralized resource allocation algorithm outlined in [\[eq_DEGRA-1\]](#eq_DEGRA-1){reference-type="eqref" reference="eq_DEGRA-1"}--[\[eq_DEGRA-3\]](#eq_DEGRA-3){reference-type="eqref" reference="eq_DEGRA-3"} fails in the presence of Byzantine attacks. This is due to the vulnerability of the weighted average aggregation in [\[eq_DEGRA-3\]](#eq_DEGRA-3){reference-type="eqref" reference="eq_DEGRA-3"} to Byzantine attacks. To address this issue, we replace the weighted average with proper robust aggregation rules, and propose a class of Byzantine-resilient decentralized resource allocation algorithms. The updates of each honest agent $i \in \mathcal{H}$ are given by $$\begin{aligned} \bm{\theta}_{i}^{k}&=\arg\underset{\bm{\theta}_{i}\in C_{i}}{\min}\{\bm{\theta}_{i}^{\top}\bm{\lambda}_{i}^{k}+f_{i}(\bm{\theta}_{i})\},\label{eq_BRE-DEGRA-1}\\ \bm{\lambda}_{i}^{k+\frac{1}{2}}&=\bm{\lambda}_{i}^{k}-\gamma^{k}(\frac{1}{J}\bm{s}-\frac{1}{J}\bm{\theta}_{i}^{k}),\label{eq_BRE-DEGRA-2}\\ \bm{\lambda}_{i}^{k+1}&=AGG_{i}(\bm{\lambda}_{i}^{k+\frac{1}{2}},\{\check{\bm{\lambda}}_{j}^{k+\frac{1}{2}}\}_{j\in \mathcal{N}_{i}}),\label{eq_BRE-DEGRA-3}\end{aligned}$$ where $AGG_{i}(\cdot)$ denotes a certain robust aggregation rule of honest agent $i$. The proposed Byzantine-resilient decentralized resource allocation algorithm is summarized in Algorithm [\[alg2\]](#alg2){reference-type="ref" reference="alg2"}. In this paper, we mainly consider the applications of three well-appreciated robust aggregation rules: CTM, IOS and SCC. Further, we will show that a wide class of robust aggregation rules enable the updates of [\[eq_BRE-DEGRA-1\]](#eq_BRE-DEGRA-1){reference-type="eqref" reference="eq_BRE-DEGRA-1"}--[\[eq_BRE-DEGRA-3\]](#eq_BRE-DEGRA-3){reference-type="eqref" reference="eq_BRE-DEGRA-3"} to converge to neighborhoods of the optimal resource allocation strategy of [\[eq_oracle-problem\]](#eq_oracle-problem){reference-type="eqref" reference="eq_oracle-problem"}. The remaining design is to delineate the conditions for "proper" robust aggregation rules. **Robust Aggregation Rules.** Intuitively, for an honest agent $i$, we expect that the output of $AGG_{i}(\bm{\lambda}_{i}^{k+\frac{1}{2}}, \{\check{\bm{\lambda}}_{j}^{k+\frac{1}{2}} \}_{j\in \mathcal{N}_{i}})$ is close to a proper weighted average of the messages from its honest neighbors and its own local dual variable, denoted as $\bar{\bm{\lambda}}_{i}^{k+\frac{1}{2}}$ $:=\sum_{j\in (\mathcal{N}_{i}\cap\mathcal{H})\cup \{i\}}e_{ij}\bm{\lambda}_{j}^{k+\frac{1}{2}}$ with the weights $\{e_{ij}\}_{j \in \mathcal{H}}$ satisfying $\sum_{j\in (\mathcal{N}_{i}\cap\mathcal{H})\cup \{i\}}e_{ij}=1$. We use the maximal value of $\{\|\bm{\lambda}_{j}^{k+\frac{1}{2}}-\bar{\bm{\lambda}}_{i}^{k+\frac{1}{2}}\| \}_{j\in (\mathcal{N}_{i}\cap \mathcal{H})\cup \{i\}}$ as the metric to quantify the proximity. Therefore, we follow [@b-ZhaoxianWu-2022; @b-Haoxiang-Ye-2023] to characterize a set of robust aggregation rules with a virtual weight matrix and a contraction constant. **Definition 1**. *Consider a set of robust aggregation rules $\{AGG_i\}_{i \in \mathcal{H}}$. If there exist a constant $\rho \geq 0$ and a matrix $E\in \mathbb{R}^{H \times H}$ whose elements satisfy $e_{ij} \in (0,1]$ when $j\in (\mathcal{N}_{i}\cap\mathcal{H})\cup \{i\}$, $e_{ij}=0$ when $j\notin (\mathcal{N}_{i}\cap\mathcal{H})\cup \{i\}$, and $\sum_{j\in (\mathcal{N}_{i}\cap\mathcal{H})\cup \{i\}}e_{ij}=1$ for any $i \in \mathcal{H}$, such that it holds $$\begin{aligned} \label{eq_definition1} & \|AGG_{i}(\bm{\lambda}_{i},\{\check{\bm{\lambda}}_{j}\}_{j\in \mathcal{N}_{i}})-\bar{\bm{\lambda}}_{i}\| \\ \le & \rho \max_{j\in (\mathcal{N}_{i}\cap \mathcal{H})\cup \{i\}}\|\bm{\lambda}_{j}-\bar{\bm{\lambda}}_{i}\| \nonumber\end{aligned}$$ for any $i\in \mathcal{H}$, then $\rho$ is the contraction constant and $E$ is the virtual weight matrix associated with the set of robust aggregation rules $\{AGG_i\}_{i \in \mathcal{H}}$. Here $\bar{\bm{\lambda}}_{i}:=\sum_{j\in (\mathcal{N}_{i}\cap\mathcal{H})\cup \{i\}}e_{ij}\bm{\lambda}_{j}$.* In the next section, we will prove that if a robust aggregation rule satisfies Definition [Definition 1](#d1){reference-type="ref" reference="d1"}, it is "proper" if the associated $\rho$ is small and $E$ is doubly stochastic. **Remark 1**. *The work of [@b-Haoxiang-Ye-2023] has demonstrated that CTM, IOS and SCC all satisfy Definition [Definition 1](#d1){reference-type="ref" reference="d1"}, and specified their corresponding $\rho$ and $E$. Note that the pair of $(\rho,E)$ is not unique. Finding the best pair is beyond the scope of this paper, and we will investigate this issue in our future work.* Initialization: All agents $i$ initialize $\bm{\lambda}_{i}^{0}=\bm{\lambda}^{0}$. Compute $\bm{\theta}_{i}^{k}=\arg\min_{\bm{\theta}_{i}\in C_{i}}\{\bm{\theta}_{i}^{\top}\bm{\lambda}_{i}^{k}+f_{i}(\bm{\theta}_{i})\}$. Compute $\bm{\lambda}_{i}^{k+\frac{1}{2}}=\bm{\lambda}_{i}^{k}-\gamma^{k}(\frac{1}{J}\bm{s}-\frac{1}{J}\bm{\theta}_{i}^{k})$. Broadcast $\bm{\lambda}_{i}^{k+\frac{1}{2}}$ to its neighbors. Receive $\check{\bm{\lambda}}_{j}^{k+\frac{1}{2}}$ from its neighbors. Aggregate $\bm{\lambda}_{i}^{k+1}=AGG_{i}(\bm{\lambda}_{i}^{k+\frac{1}{2}}, \{\check{\bm{\lambda}}_{j}^{k+\frac{1}{2}}\}_{j\in \mathcal{N}_{i}})$. Broadcast $\check{\bm{\lambda}}_{i}^{k+\frac{1}{2}}=*$ to its neighbors. ## Advantages over BREDA Our proposed algorithms have several advantages over BREDA [@b-Runhua-Wang-2022]: simplicity, generality and dual consensus. First, at each iteration of BREDA, each honest agent needs to update a primal variable, a dual variable, and an auxiliary variable that tracks the average of the honest primal variables. By contrast, at each iteration of our proposed algorithms, each honest agent only updates two local variables, one is primal and the other is dual. Second, the robust aggregation rule of BREDA is confined to CTM; using other robust aggregation rules lacks convergence guarantee. However, CTM does not fit for the scenario that an honest agent has a large number of Byzantine neighbors, because the number of discarded messages has to be at least twice. This is unfavorable especially when the underlying network is sparse. Instead, our proposed algorithms allow a wide class of robust aggregation rules that satisfy Definition [Definition 1](#d1){reference-type="ref" reference="d1"}. Third, BREDA guarantees the local auxiliary variables to be nearly consensual, but the local dual variables are not necessarily so. We will validate this fact in the numerical experiments. Since the optimal dual variable stands for the shadow price of the resources [@b-FPKelly-1998], reaching consensus of the local dual variables is important in various applications. Our proposed algorithms have such a guarantee, as shown in the next section. # CONVERGENCE ANALYSIS {#sec 5} This section analyzes convergence of the attack-free and Byzantine-resilient decentralized resource allocation algori- thms, outlined in [\[eq_DEGRA-1\]](#eq_DEGRA-1){reference-type="eqref" reference="eq_DEGRA-1"}--[\[eq_DEGRA-3\]](#eq_DEGRA-3){reference-type="eqref" reference="eq_DEGRA-3"} and [\[eq_BRE-DEGRA-1\]](#eq_BRE-DEGRA-1){reference-type="eqref" reference="eq_BRE-DEGRA-1"}--[\[eq_BRE-DEGRA-3\]](#eq_BRE-DEGRA-3){reference-type="eqref" reference="eq_BRE-DEGRA-3"}, respectively. We begin with several assumptions. **Assumption 1**. *For any $i\in \mathcal{J}$, the local cost function $f_{i}(\cdot)$ is $u_{f}$-strongly convex and $L_{f}$-smooth, and the local constraint set $C_{i}$ is compact and convex.* **Assumption 2**. *There exist $\widetilde{\bm{\Theta}}$ and $\bm{\Theta}$ in the relative interiors of $\widetilde{C}$ and $C$, such that the constraints $\frac{1}{J}\sum_{i \in \mathcal{J}}\bm{\theta}_{i}=\bm{s}$ and $\frac{1}{H}\sum_{i\in \mathcal{H}}\bm{\theta}_{i}=\bm{s}$ satisfy, respectively.* With Assumptions [Assumption 1](#a1){reference-type="ref" reference="a1"} and [Assumption 2](#a2){reference-type="ref" reference="a2"}, the duality gaps of [\[eq_primal-problem\]](#eq_primal-problem){reference-type="eqref" reference="eq_primal-problem"} and [\[eq_oracle-problem\]](#eq_oracle-problem){reference-type="eqref" reference="eq_oracle-problem"} are both 0. In addition, the negative dual functions to minimize are also strongly convex and smooth. **Assumption 3**. *The graphs $\widetilde{\mathcal{G}} ( \mathcal{J},\widetilde{\mathcal{ \mathcal{E} }} )$ and ${\mathcal{G}} ( \mathcal{J},{\mathcal{ \mathcal{E} }} )$ are both undirected and connected. The weight matrices $\widetilde{E}$ and $E$ are doubly stochastic and row stochastic, respectively, and satisfy $$\begin{aligned} \widetilde{\kappa} & := \|\widetilde{E}-\frac{1}{J}\widetilde{\bm{1}}\widetilde{\bm{1}}^{\top}\|^{2} < 1, \label{eq_a6_tttt} \\ \kappa & := \|E-\frac{1}{H}\bm{1}\bm{1}^{\top} E\|^{2} < 1. \label{eq_a6}\end{aligned}$$* We have emphasized that the connectedness of $\widetilde{\mathcal{G}}$ and ${\mathcal{G}}$ is necessary. The requirement [\[eq_a6_tttt\]](#eq_a6_tttt){reference-type="eqref" reference="eq_a6_tttt"} is common in decentralized optimization. It holds when $\widetilde{e}_{ij}>0$ if and only if $(i, j)\in \widetilde{\mathcal{E}}$ or $i=j$. The requirement [\[eq_a6\]](#eq_a6){reference-type="eqref" reference="eq_a6"} on the associated virtual weight matrix $E$ is in the same form of [\[eq_a6_tttt\]](#eq_a6_tttt){reference-type="eqref" reference="eq_a6_tttt"} if $E$ is doubly stochastic, but we allow $E$ to be only row stochastic. ## Convergence of Attack-free Decentralized Resource Allocation Algorithm Denote $(\widetilde{\bm{\Theta}}^{*},\widetilde{\bm{\lambda}}^{*})$ as the optimal primal-dual pair of [\[eq_primal-problem\]](#eq_primal-problem){reference-type="eqref" reference="eq_primal-problem"}, in which $\widetilde{\bm{\Theta}}^{*}\in \mathbb{R}^{JD}$ and $\widetilde{\bm{\lambda}}^{*}\in \mathbb{R}^{D}$. The following theorem shows the convergence of the attack-free decentralized allocation algorithm [\[eq_DEGRA-1\]](#eq_DEGRA-1){reference-type="eqref" reference="eq_DEGRA-1"}--[\[eq_DEGRA-3\]](#eq_DEGRA-3){reference-type="eqref" reference="eq_DEGRA-3"}. **Theorem 1**. *Consider $\widetilde{\bm{\Theta}}^{k+1}$ and $\{\bm{\lambda}_{i}^{k+1}\}_{i\in \mathcal{J}}$ generated by the attack-free decentralized resource allocation algorithm [\[eq_DEGRA-1\]](#eq_DEGRA-1){reference-type="eqref" reference="eq_DEGRA-1"}--[\[eq_DEGRA-3\]](#eq_DEGRA-3){reference-type="eqref" reference="eq_DEGRA-3"} and suppose that no Byzantine agents are present. If Assumptions [Assumption 1](#a1){reference-type="ref" reference="a1"}--[Assumption 3](#a3){reference-type="ref" reference="a3"} hold, then with a proper decreasing step size $\gamma^{k}=O(\frac{1}{k})$, we have\ $a)$ $\lim_{k \to +\infty} \sum_{i\in \mathcal{J}}\|\bm{\lambda}_{i}^{k+1}-\widetilde{\bm{\lambda}}^{*}\|=0$,\ $b)$ $\lim_{k \to +\infty} \|\widetilde{\bm{\Theta}}^{k+1}-\widetilde{\bm{\Theta}}^{*}\|=0$.* Theorem [Theorem 1](#t1){reference-type="ref" reference="t1"} shows that the local primal and dual variables generated by [\[eq_DEGRA-1\]](#eq_DEGRA-1){reference-type="eqref" reference="eq_DEGRA-1"}--[\[eq_DEGRA-3\]](#eq_DEGRA-3){reference-type="eqref" reference="eq_DEGRA-3"} converge to their optima. This matches the classical conclusion for the decentralized gradient method [@b-Thinh-T.Doan-2021; @b-AngeliaNedic-2009; @b-B-Johansson-2008]. Those works assume convex and possibly non-smooth cost functions, while we assume strongly convex and smooth cost functions, with which we have performance guarantee for the ensuing Byzantine-resilient algorithms. The proof of Theorem [Theorem 1](#t1){reference-type="ref" reference="t1"} and the conditions on the step size $\gamma^{k}$ are presented in Appendix [9](#appendix1){reference-type="ref" reference="appendix1"}. ## Convergence of Byzantine-resilient Decentralized Resource Allocation Algorithm Similarly, denote $({\bm{\Theta}}^{*},{\bm{\lambda}}^{*})$ as the optimal primal-dual pair of [\[eq_oracle-problem\]](#eq_oracle-problem){reference-type="eqref" reference="eq_oracle-problem"}, in which ${\bm{\Theta}}^{*}\in \mathbb{R}^{HD}$ and ${\bm{\lambda}}^{*}\in \mathbb{R}^{D}$. The following theorem shows the convergence of the Byzantine-resilient decentralized allocation algorithm [\[eq_BRE-DEGRA-1\]](#eq_BRE-DEGRA-1){reference-type="eqref" reference="eq_BRE-DEGRA-1"}--[\[eq_BRE-DEGRA-3\]](#eq_BRE-DEGRA-3){reference-type="eqref" reference="eq_BRE-DEGRA-3"}. **Theorem 2**. *Consider ${\bm{\Theta}}^{k+1}$ and $\{\bm{\lambda}_{i}^{k+1}\}_{i\in \mathcal{H}}$ generated by the Byzantine-resilient decentralized resource allocation algorithm [\[eq_BRE-DEGRA-1\]](#eq_BRE-DEGRA-1){reference-type="eqref" reference="eq_BRE-DEGRA-1"}--[\[eq_BRE-DEGRA-3\]](#eq_BRE-DEGRA-3){reference-type="eqref" reference="eq_BRE-DEGRA-3"}. Suppose that Byzantine agents are present but the used robust aggregation rule satisfies [\[eq_definition1\]](#eq_definition1){reference-type="eqref" reference="eq_definition1"} in Definition [Definition 1](#d1){reference-type="ref" reference="d1"}. If Assumptions [Assumption 1](#a1){reference-type="ref" reference="a1"}--[Assumption 3](#a3){reference-type="ref" reference="a3"} hold and the contraction constant $\rho$ satisfies $$\rho<\frac{1-\kappa}{8\sqrt{H}},$$ then with a proper decreasing step size $\gamma^{k}=O(\frac{1}{k})$, we have\ $a)$ $\limsup_{k\to +\infty}\sum_{i\in \mathcal{H}}\|\bm{\lambda}_{i}^{k+1}-\bm{\lambda}^{*}\|\le \sqrt{ \frac{192\delta^{2}H^{2}}{\beta^{2}}}\cdot \sqrt{1+\frac{9}{\epsilon^{3}}}\cdot\sqrt{4\rho^{2}H+\chi^{2}},$\ $b)$ $\lim_{k\to +\infty}\sum_{i\in \mathcal{H}}\|\bm{\lambda}_{i}^{k+1}-\bar{\bm{\lambda}}^{k+1}\|= 0,$\ $c)$ $\limsup_{k\to +\infty}\|\bm{\Theta}^{k+1}-\bm{\Theta}^{*}\|\le \frac{1}{u_{f}}\cdot \sqrt{ \frac{192\delta^{2}}{\beta^{2}}}\cdot \sqrt{1+\frac{9}{\epsilon^{3}}}\cdot\sqrt{4\rho^{2}H+\chi^{2}},$\ where $\bar{\bm{\lambda}}^{k+1}:=\frac{1}{H}\sum_{i\in \mathcal{H}}\bm{\lambda}_{i}^{k+1}$, $\beta=\frac{1}{H(u_{f}+L_{f})}$, $\epsilon=\kappa-8\rho\sqrt{H}$, and $\chi^{2}:=\frac{1}{H}\|E^{\top}\bm{1}-\bm{1}\|^{2}$ quantifies the non-doubly stochasticity of $E$.* The proof of Theorem [Theorem 2](#t2){reference-type="ref" reference="t2"} and the conditions on the step size $\gamma^{k}$ are presented in Appendix [8](#appendix2){reference-type="ref" reference="appendix2"}. Theorem [Theorem 2](#t2){reference-type="ref" reference="t2"} demonstrates that if the robust aggregation rule is properly designed such that the associated contraction constant $\rho$ is sufficiently small, then the local primal and dual variables generated by [\[eq_BRE-DEGRA-1\]](#eq_BRE-DEGRA-1){reference-type="eqref" reference="eq_BRE-DEGRA-1"}--[\[eq_BRE-DEGRA-3\]](#eq_BRE-DEGRA-3){reference-type="eqref" reference="eq_BRE-DEGRA-3"} converge to neighborhoods of their optima. Sizes of the neighborhoods are determined by the associated contraction constant $\rho$ and virtual weight matrix $E$ (more precisely, $\chi^2$). Notably, the local dual variables are guaranteed to reach consensus even under Byzantine attacks. Compared to the proof of Theorem [Theorem 1](#t1){reference-type="ref" reference="t1"}, that of Theorem [Theorem 2](#t2){reference-type="ref" reference="t2"} is more challenging. First, under the Byzantine attacks and with the robust aggregation rule, dual-domain consensus is no longer merited. We discover that $\rho$ must be sufficiently small for reaching consensus. Second, due to the imperfectness during the aggregation, each iteration incurs an error determined by $\rho$ and $\chi^2$. We have to handle such an error within the analysis. Note that when $\rho=0$ and $E$ is doubly stochastic, Theorem [Theorem 2](#t2){reference-type="ref" reference="t2"} reduces to Theorem [Theorem 1](#t1){reference-type="ref" reference="t1"}. Our analysis is related to but significantly different from that in [@b-ZhaoxianWu-2022]. The work of [@b-ZhaoxianWu-2022] considers a general Byzantine-resilient decentralized stochastic non-convex *optimization* problem, and analyzes robust aggregation rules that satisfy Definition [Definition 1](#d1){reference-type="ref" reference="d1"} in the primal domain. By contrast, we consider a strongly convex *resource allocation* problem, and analyze in the dual domain. The different assumptions lead to different convergence metrics, and the corresponding technical tools are different, too. # NUMERICAL EXPERIMENTS {#sec 6} In this section, we conduct numerical experiments to show the performance of the proposed Byzantine-resilient decen- tralized resource allocation algorithms. {width="16.7cm"} {width="16.7cm"} {width="16.7cm"} {width="16.7cm"} ## Case 1: Synthetic Problem We first test on a synthetic and scalar case with $D=1$. Consider a randomly generated network consisting of $J=100$ agents, where each agent has $15$ neighbors. The weight $\widetilde{e}_{ij}$ is set to $\frac{1}{16}$ if and only if $(i,j)\in \widetilde{\mathcal{E}}$ or $i=j$. The total amount of resources is $5000$ such that $\bm{s}=50$. The local constraint of each agent $i$ is $\bm{\theta}_{i} \in C_{i} = [0, 100]$. Each agent $i$ has a local cost function $f_{i}\left ( \bm{\theta}_{i} \right )=a_{i}(\bm{\theta}_{i}-b_{i})^{2}$, in which $a_i\sim \mathcal{U} (1,2)$ and $b_i\sim \mathcal{N} (2,0.6^{2})$ with $\mathcal{U}(\cdot, \cdot)$ standing for uniform distribution and $\mathcal{N} (\cdot, \cdot)$ for Gaussian distribution. Such quadratic cost functions is also used in [@b-LXiao-2006; @b-Yun-Xu-2017; @b-Jiaqi-Zhang-2020]. We randomly select $B=6$ Byzantine agents by default, but allow each agent to have at most $4$ Byzantine neighbors. For the proposed algorithms, we test four types of Byzantine attacks: large-value, small-value, large-value Gaussian, and small-value Gaussian. With large-value attacks, a Byzantine agent sets its message as $-0.01$. With small-value attacks, a Byzantine agent sets its message as $-600$. With large-value Gaussian attacks, a Byzantine agent sets its message following a Gaussian distribution with mean $-30$ and variance $5^2$. With small-value Gaussian attacks, a Byzantine agent sets its message following a Gaussian distribution with mean $-300$ and variance $40^2$. We consider three popular robust aggregation rules: CTM, IOS and SCC. The step size is $\gamma^{k}=(k+1)^{-0.1}$, which is faster than the conservative theoretical step size in the order of $O(\frac{1}{k})$. We use the attack-free decentralized resource allocation algorithm [\[eq_DEGRA-1\]](#eq_DEGRA-1){reference-type="eqref" reference="eq_DEGRA-1"}--[\[eq_DEGRA-3\]](#eq_DEGRA-3){reference-type="eqref" reference="eq_DEGRA-3"} as a baseline. Another baseline is BREDA. Note that BREDA defends against Byzantine attacks in the primal domain, whereas our proposed algorithms defend in the dual domain. To enable fair comparisons, for the dual-domain large-value attacks, we generate the corresponding primal-domain attacks such that their effects on the primal variables are almost the same, for our proposed algorithms and BREDA, respectively. Similarly, we also generate the corresponding primal-domain small-value attacks. Thus, with large-value and small-value attacks in BREDA, a Byzantine agent sets its message as $100$ and $0$, respectively. Note that it is difficult to generate the corresponding primal-domain large-value and small-value Gaussian attacks, and we do not compare with BREDA under these attacks. $\rho^{2}$ $\chi^{2}$ $\rho^{2}+\chi^{2}$ ----- ------------ ------------ --------------------- CTM $0.44$ $0.0031$ $0.44$ IOS $0.11$ $0$ $0.11$ SCC $2.75$ $0$ $2.75$ [\[table-1\]]{#table-1 label="table-1"} large-value small-value large-value Gaussian small-value Gaussian -------------- ------------- ------------- ---------------------- ---------------------- BREDA 105.70 121.09 / / proposed+CTM 1.20e-02 1.07e-02 1.20e-02 1.07e-02 proposed+IOS 1.09e-02 1.09e-02 1.09e-02 1.09e-02 proposed+SCC 3.36e-02 3.16e-02 3.36e-02 3.16e-02 [\[table-2\]]{#table-2 label="table-2"} large-value small-value large-value Gaussian small-value Gaussian -------------- ------------- ------------- ---------------------- ---------------------- BREDA 130.48 144.95 / / proposed+CTM 1.95e-02 1.58e-02 1.95e-02 1.58e-02 proposed+IOS 9.08e-02 9.08e-02 9.08e-02 9.08e-02 proposed+SCC 3.65e-02 3.70e-02 3.65e-02 3.70e-02 \ [\[table-3\]]{#table-3 label="table-3"} To observe the sensitivity of CTM, IOS and SCC to their parameters, we consider two scenarios: optimal parameters and non-optimal parameters. For the scenario of optimal parameters, each honest agent sets the parameters $b$ of CTM and IOS as the number of its Byzantine neighbors. In SCC, the clipping threshold $\tau$ is determines according to Theorem 3 in [@b-LieHe-2022]. The results are shown in Figs. [\[fig_1\]](#fig_1){reference-type="ref" reference="fig_1"} and [\[fig_3\]](#fig_3){reference-type="ref" reference="fig_3"}. On the other hand, for the scenario of non-optimal parameters, all honest agents set $b=4$, which corresponds to the upper bound of the number of Byzantine neighbors, in CTM and IOS. In SCC, the clipping threshold is set to $\tau=0.2$. The results are shown in Figs. [\[fig_2\]](#fig_2){reference-type="ref" reference="fig_2"} and [\[fig_4\]](#fig_4){reference-type="ref" reference="fig_4"}. Performance metrics are primal optimality $\|\bm{\Theta}^{k}-\bm{\Theta}^{*}\|$, dual optimality $\sum_{i\in \mathcal{H}}\|\bm{\lambda}_{i}^{k}-\bm{\lambda}^{*}\|$, cost optimality $\|f(\bm{\Theta}^{k})-f(\bm{\Theta}^{*})\|$, constraint violation $\|\frac{1}{\mathcal{H}}\sum_{i\in \mathcal{H}}\bm{\theta}_{i}^{k}-\bm{s}\|$, and dual consensus error $\sum_{i\in \mathcal{H}}\|\bm{\lambda}_{i}^{k}-\bar{\bm{\lambda}}^{k}\|^{2}$. Fig. [\[fig_1\]](#fig_1){reference-type="ref" reference="fig_1"} illustrates that the attack-free decentralized resource allocation algorithm [\[eq_DEGRA-1\]](#eq_DEGRA-1){reference-type="eqref" reference="eq_DEGRA-1"}--[\[eq_DEGRA-3\]](#eq_DEGRA-3){reference-type="eqref" reference="eq_DEGRA-3"} fails under all Byzantine attacks. By contrast, the proposed algorithms and BREDA demonstrate satisfactory Byzantine-resilience. Among the robust aggregation rules used in our proposed algorithms, IOS performs the best and CTM is better than SCC in terms of primal optimality, dual optimality, cost optimality, and constraint violation. To see the reason, recall that Theorem [Theorem 2](#t2){reference-type="ref" reference="t2"} shows the primal optimality and dual optimality are both in the order of $O(\rho^{2}+\chi^{2})$. We calculate the corresponding bounds of $\rho^{2}+\chi^{2}$ in Table [\[table-1\]](#table-1){reference-type="ref" reference="table-1"} according to Lemmas 3--5 in [@b-Haoxiang-Ye-2023]. From the smallest to the largest are IOS, CTM and SCC, which validates our theoretical findings. Fig. [\[fig_1\]](#fig_1){reference-type="ref" reference="fig_1"} also reveals that BREDA is worse than the proposed algorithms with proper robust aggregation rules. To further highlight the advantages of our proposed algorithms, we list the dual consensus errors in Table [\[table-2\]](#table-2){reference-type="ref" reference="table-2"}. No matter the types of Byzantine attacks and robust aggregation rules, the proposed algorithms are all able to achieve nearly perfect dual consensus. By contrast, BREDA cannot guarantee dual consensus. This phenomenon reveals the benefits of the dual-domain defenses. When the parameters are non-optimal, the above conclusions still hold. Fig. [\[fig_2\]](#fig_2){reference-type="ref" reference="fig_2"} and Table [\[table-3\]](#table-3){reference-type="ref" reference="table-3"} demonstrate the advantages of our proposed dual-domain defense algorithms over BREDA in the scenario with non-optimal parameters. In Figs. [\[fig_3\]](#fig_3){reference-type="ref" reference="fig_3"} and [\[fig_4\]](#fig_4){reference-type="ref" reference="fig_4"}, we check the sensitivity of the compared algorithms to the number of Byzantine agents $B$ by setting $B$ as $4$, $5$, $6$, $7$ and $8$. The attack-free decentralized resource allocation algorithm [\[eq_DEGRA-1\]](#eq_DEGRA-1){reference-type="eqref" reference="eq_DEGRA-1"}--[\[eq_DEGRA-3\]](#eq_DEGRA-3){reference-type="eqref" reference="eq_DEGRA-3"} fails for any $B$. By contrast, both BREDA and our proposed algorithms demonstrate satisfactory resilience, and their performance is steady when $B$ varies. {width="16.7cm"} $\rho^{2}$ $\chi^{2}$ $\rho^{2}+\chi^{2}$ ----- ------------ ------------ --------------------- -- -- CTM $0.024$ $0.11$ $0.134$ IOS $0.006$ $0$ $0.006$ SCC $0.965$ $0$ $0.965$ [\[table-4\]]{#table-4 label="table-4"} large-value small-value large-value Gaussian small-value Gaussian -------------- ------------- ------------- ---------------------- ---------------------- BREDA 0.51 0.49 / / proposed+CTM 2.16e-04 3.28e-03 3.48e-03 3.28e-03 proposed+IOS 3.37e-03 3.37e-03 3.37e-03 3.37e-03 proposed+SCC 3.55e-03 3.23e-03 3.54e-03 3.23e-03 [\[table-5\]]{#table-5 label="table-5"} ## Case 2: Economic Dispatch for IEEE 118-Bus Test System We next consider a power dispatch problem for the IEEE 118-bus test system, which contains 54 generators [@b-IEEE-118]. Each generator $i$ has a local power $\bm{\theta}_{i}$ and a corresponding cost function $f_{i}(\bm{\theta}_{i})=\eta_{i}\bm{\theta}_{i}^{2}+\zeta_{i}\bm{\theta}_{i}+\xi_{i}$, where $\eta_{i}\in [0.0024, 0.0697]$, $\zeta_{i}\in [8.3391, 37.6968]$, and $\xi_{i}\in [6.78, 74.33]$. The local constraint of each agent $i$ is $\bm{\theta}_{i}\in [\bm{\theta}_{i}^{\min},\bm{\theta}_{i}^{\max}]$, where $\bm{\theta}_{i}^{\min}\in [5,150]$ and $\bm{\theta}_{i}^{\max}\in [30,420]$. The total amount of resources is set as 6000, such that $\bm{s}=\frac{6000}{54}$ [@b-Thinh-T.Doan-2021]. To test the performance of the proposed algorithms, we randomly select one Byzantine agent out of the 54 generators and apply different types of Byzantine attacks, including large-value, small-value, large-value Gaussian, and small-value Gaussian. For large-value attacks, the Byzantine generator sets its message as $-0.01$, whereas for small-value attacks, the Byzantine generator sets its message as $-100$. For large-value Gaussian attacks, the Byzantine generator sets its message following a Gaussian distribution with mean $-10$ and variance $5^{2}$. For small-value Gaussian attacks, the Byzantine generator sets its message following a Gaussian distribution with mean $-50$ and variance $10^{2}$. We also design the corresponding larger-value and smaller-value attacks for BREDA, where the Byzantine generator sets its message as $420$ and $5$, respectively. The weight matrix $\widetilde{E}$ is constructed according to the Metropolis constant weight rule [@b-Wei-Shi-2015]. The parameters $b$ and $\tau$ are optimal. The step size for the proposed algorithms is determined as $\gamma^{k}=(k+1)^{-0.7}$. Fig. [\[fig_5\]](#fig_5){reference-type="ref" reference="fig_5"} demonstrates the failure of the attack-free decentralized resource allocation algorithm, as well as the resilience of the proposed algorithms and BREDA against various Byzantine attacks. We also calculate the corresponding bounds of $\rho^{2}+\chi^{2}$ of the robust aggregation rules IOS, CTM, and SCC, as presented in Table [\[table-4\]](#table-4){reference-type="ref" reference="table-4"}. Observe that a smaller bound of $\rho^{2}+\chi^{2}$ leads to better performance, which has been predicted by our theoretical findings. According to Fig. [\[fig_5\]](#fig_5){reference-type="ref" reference="fig_5"}, BREDA performs worse than the proposed algorithms with proper robust aggregation rules. We calculate the dual consensus errors of the proposed algorithms with different robust aggregation rules and BREDA, as presented in Table [\[table-5\]](#table-5){reference-type="ref" reference="table-5"}. The proposed algorithms achieve nearly consensual dual variables and BREDA does not. # CONCLUSIONS AND FUTURE WORK {#sec 7} In this paper, we investigate the problem of decentralized resource allocation under Byzantine attacks. We propose a class of Byzantine-resilient algorithms equipped with robust aggregation rules, featured in dual-domain defenses. Given that the robust aggregation rules are properly designed, we prove that the generated primal and dual variables of the honest agents converge to neighborhoods of their optima, while the dual variables are able to reach consensus. The numerical experiments show the resilience of the proposed algorithms to various Byzantine attacks. In the future, we plan to extend our algorithm development and theoretical analysis to stochastic and online decentralized resource allocation problems under Byzantine attacks, which are of particular importance for time-sensitive applications. # Proof of Theorem [Theorem 2](#t2){reference-type="ref" reference="t2"} {#appendix2} ## Part $a$ of Theorem [Theorem 2](#t2){reference-type="ref" reference="t2"} {#part-a-of-theorem-t2} According to the update of $\bm{\lambda}_{i}^{k+1}$ in Algorithm [\[alg2\]](#alg2){reference-type="ref" reference="alg2"}, we have $$\begin{aligned} \label{eq_proof_theorem2-b-1} &\|\bar{\bm{\lambda}}^{k+1}-\bm{\lambda}^{*}\|^{2}\\ =&\|\frac{1}{H}\sum_{i\in \mathcal{H}} AGG_{i}(\bm{\lambda}_{i}^{k+\frac{1}{2}},\{\check{\bm{\lambda}}_{j}^{k+\frac{1}{2}}\}_{j\in \mathcal{N}_{i}})-\bm{\lambda}^{*}\|^{2}\notag\\ =&\|\frac{1}{H}\sum_{i\in \mathcal{H}} AGG_{i}(\bm{\lambda}_{i}^{k+\frac{1}{2}},\{\check{\bm{\lambda}}_{j}^{k+\frac{1}{2}}\}_{j\in \mathcal{N}_{i}})-\bar{\bm{\lambda}}^{k+\frac{1}{2}}+\bar{\bm{\lambda}}^{k+\frac{1}{2}}-\bm{\lambda}^{*}\|^{2}\notag \\ \le& \frac{1}{v_{1}}\|\frac{1}{H}\sum_{i\in \mathcal{H}}AGG_{i}(\bm{\lambda}_{i}^{k+\frac{1}{2}},\{\check{\bm{\lambda}}_{j}^{k+\frac{1}{2}}\}_{j\in \mathcal{N}_{i}})-\bar{\bm{\lambda}}^{k+\frac{1}{2}}\|^{2}\notag\\ &+\frac{1}{1-v_{1}}\|\bar{\bm{\lambda}}^{k+\frac{1}{2}}-\bm{\lambda}^{*}\|^{2} \notag\\ =&\frac{1}{v_{1}}\|\frac{1}{H}\sum_{i\in \mathcal{H}}AGG_{i}(\bm{\lambda}_{i}^{k+\frac{1}{2}},\{\check{\bm{\lambda}}_{j}^{k+\frac{1}{2}}\}_{j\in \mathcal{N}_{i}})-\frac{1}{H}\sum_{i\in\mathcal{H}}\bar{\bm{\lambda}}_{i}^{k+\frac{1}{2}}\notag\\ &+\frac{1}{H}\sum_{i\in\mathcal{H}}\bar{\bm{\lambda}}_{i}^{k+\frac{1}{2}}-\bar{\bm{\lambda}}^{k+\frac{1}{2}}\|^{2}+\frac{1}{1-v_{1}}\|\bar{\bm{\lambda}}^{k+\frac{1}{2}}-\bm{\lambda}^{*}\|^{2}\notag\\ \le&\frac{2}{v_{1}}\|\frac{1}{H}\sum_{i\in \mathcal{H}}AGG_{i}(\bm{\lambda}_{i}^{k+\frac{1}{2}},\check{\bm{\lambda}}_{j}^{k+\frac{1}{2}}(j\in \mathcal{N}_{i}))-\frac{1}{H}\sum_{i\in\mathcal{H}}\bar{\bm{\lambda}}_{i}^{k+\frac{1}{2}}\|^{2}\notag\\ &+\frac{2}{v_{1}}\|\frac{1}{H}\sum_{i\in\mathcal{H}}\bar{\bm{\lambda}}_{i}^{k+\frac{1}{2}}-\bar{\bm{\lambda}}^{k+\frac{1}{2}}\|^{2}+\frac{1}{1-v_{1}}\|\bar{\bm{\lambda}}^{k+\frac{1}{2}}-\bm{\lambda}^{*}\|^{2}\notag\\ \le & \underbrace{\frac{2}{v_{1}H}\sum_{i\in \mathcal{H}}\|AGG_{i}(\bm{\lambda}_{i}^{k+\frac{1}{2}},\{\check{\bm{\lambda}}_{j}^{k+\frac{1}{2}}\}_{j\in \mathcal{N}_{i}})-\bar{\bm{\lambda}}_{i}^{k+\frac{1}{2}}\|^{2}}_{T_{1}}\notag\\ &+\underbrace{\frac{2}{v_{1}}\|\frac{1}{H}\sum_{i\in\mathcal{H}}\bar{\bm{\lambda}}_{i}^{k+\frac{1}{2}}-\bar{\bm{\lambda}}^{k+\frac{1}{2}}\|^{2}}_{T_{2}}+\underbrace{\frac{1}{1-v_{1}}\|\bar{\bm{\lambda}}^{k+\frac{1}{2}}-\bm{\lambda}^{*}\|^{2}}_{T_{3}},\notag\end{aligned}$$ where $v_{1}$ is any positive constant in $(0,1)$. To derive the first inequality, we use $\|\bm{a}+\bm{b}\|^{2}\le \frac{1}{v}\|\bm{a}\|^{2}+\frac{1}{1-v}\|\bm{b}\|^{2}$ for any positive constant $v\in (0,1)$. The last inequality holds because $(a_{1}+\cdots+a_{H})^{2}\le H(a_{1}^{2}+\cdots+a_{H}^{2})$. Next, we analyze $T_{1}$, $T_{2}$ and $T_{3}$ in turn. **Bounding $T_{1}$:** According to [\[eq_definition1\]](#eq_definition1){reference-type="eqref" reference="eq_definition1"} in Definition [Definition 1](#d1){reference-type="ref" reference="d1"}, $T_{1}$ can be bounded by $$\begin{aligned} \label{eq_proof_theorem2-b-T1-1} T_{1}\le& \frac{2}{v_{1}H}\sum_{i\in \mathcal{H}}\rho^{2}\max_{j\in \mathcal{N}_{i}\cap\mathcal{H}\cup \{i\}}\|\bm{\lambda}_{j}^{k+\frac{1}{2}}-\bar{\bm{\lambda}}_{i}^{k+\frac{1}{2}}\|^{2}\\ =&\frac{2\rho^{2}}{v_{1}H}\sum_{i\in \mathcal{H}}\max_{j\in \mathcal{N}_{i}\cap\mathcal{H}\cup \{i\}}\|\bm{\lambda}_{j}^{k+\frac{1}{2}}-\bar{\bm{\lambda}}^{k+\frac{1}{2}}+\bar{\bm{\lambda}}^{k+\frac{1}{2}}-\bar{\bm{\lambda}}_{i}^{k+\frac{1}{2}}\|^{2} \notag \\ \le & \frac{4\rho^{2}}{v_{1}H}\sum_{i\in \mathcal{H}}\max_{j\in \mathcal{N}_{i}\cap\mathcal{H}\cup \{i\}}\|\bm{\lambda}_{j}^{k+\frac{1}{2}}-\bar{\bm{\lambda}}^{k+\frac{1}{2}}\|^{2}\notag\\ &+ \frac{4\rho^{2}}{v_{1}H}\sum_{i\in \mathcal{H}}\|\bar{\bm{\lambda}}_{i}^{k+\frac{1}{2}}-\bar{\bm{\lambda}}^{k+\frac{1}{2}}\|^{2}\notag \\ \le & \frac{4\rho^{2}}{v_{1}H}\sum_{i\in \mathcal{H}}\max_{i\in \mathcal{H}}\|\bm{\lambda}_{i}^{k+\frac{1}{2}}-\bar{\bm{\lambda}}^{k+\frac{1}{2}}\|^{2}\notag\end{aligned}$$ $$\begin{aligned} &+ \frac{4\rho^{2}}{v_{1}H}\sum_{i\in \mathcal{H}}\max_{i\in \mathcal{H}}\|\bm{\lambda}_{i}^{k+\frac{1}{2}}-\bar{\bm{\lambda}}^{k+\frac{1}{2}}\|^{2}\notag\\ =& \frac{8\rho^{2}}{v_{1}}\max_{i\in \mathcal{H}}\|\bm{\lambda}_{i}^{k+\frac{1}{2}}-\bar{\bm{\lambda}}^{k+\frac{1}{2}}\|^{2}.\notag\end{aligned}$$ Define $\Lambda=[\cdots,\bm{\lambda}_{i},\cdots]\in \mathbb{R}^{H\times D}$ that collects $\bm{\lambda}_{i}$ of all honest agents $i\in \mathcal{H}$. Combining the fact $\max_{i\in \mathcal{H}}\|\bm{\lambda}_{i}^{k+\frac{1}{2}}-\bar{\bm{\lambda}}^{k+\frac{1}{2}}\|^{2}\le \|\Lambda^{k+\frac{1}{2}}-\frac{1}{H}\bm{1}\bm{1}^{\top}\Lambda^{k+\frac{1}{2}} \|^{2}_{F}$ and [\[eq_proof_theorem2-b-T1-1\]](#eq_proof_theorem2-b-T1-1){reference-type="eqref" reference="eq_proof_theorem2-b-T1-1"}, we obatin $$\begin{aligned} \label{eq_proof_theorem2-b-T1} T_{1}\le \frac{8\rho^{2}}{v_{1}}\|\Lambda^{k+\frac{1}{2}}-\frac{1}{H}\bm{1}\bm{1}^{\top}\Lambda^{k+\frac{1}{2}} \|^{2}_{F}.\end{aligned}$$ **Bounding $T_{2}$:** By the definition of $\bar{\bm{\lambda}}_{i}=\sum_{j\in \mathcal{N}_{i}\cap\mathcal{H}\cup \{i\}}e_{i,j}\bm{\lambda}_{j}$, we have $$\begin{aligned} \label{eq_proof_theorem2-b-T2-1} T_{2}=& \frac{2}{v_{1}}\|\frac{1}{H}\sum_{i\in\mathcal{H}}\sum_{j\in \mathcal{N}_{i}\cap\mathcal{H}\cup \{i\}}e_{i,j}\bm{\lambda}_{j}^{k+\frac{1}{2}}-\bar{\bm{\lambda}}^{k+\frac{1}{2}}\|^{2}\\ =& \frac{2}{v_{1}}\|\frac{1}{H}\bm{1}^{\top}E\Lambda^{k+\frac{1}{2}}-\frac{1}{H}\bm{1}^{\top}\Lambda^{k+\frac{1}{2}} \|^{2}\notag\\ =& \frac{2}{v_{1}}\|\frac{1}{H}\bm{1}^{\top}(E\Lambda^{k+\frac{1}{2}}-\frac{1}{H}\bm{1}\bm{1}^{\top}\Lambda^{k+\frac{1}{2}}) \|^{2} \notag\\ =& \frac{2}{v_{1}H^{2}} \|\bm{1}^{\top}(E\Lambda^{k+\frac{1}{2}}-\frac{1}{H}\bm{1}\bm{1}^{\top}\Lambda^{k+\frac{1}{2}}) \|^{2} \notag\\ =& \frac{2}{v_{1}H^{2}} \|(\bm{1}^{\top}E-\bm{1}^{\top})(\Lambda^{k+\frac{1}{2}}-\frac{1}{H}\bm{1}\bm{1}^{\top}\Lambda^{k+\frac{1}{2}}) \|^{2} \notag\\ \le&\frac{2}{v_{1}H^{2}} \|E^{\top}\bm{1}-\bm{1}\|^{2}\|\Lambda^{k+\frac{1}{2}}-\frac{1}{H}\bm{1}\bm{1}^{\top}\Lambda^{k+\frac{1}{2}} \|^{2}. \notag\end{aligned}$$ To drive the last equality, we use Definition [Definition 1](#d1){reference-type="ref" reference="d1"} that the virtual weight matrix $E$ is row stochastic. Define $\chi^{2}=\frac{1}{H}\|E^{\top}\bm{1}-\bm{1}\|^{2}$ to quantify how non-column stochastic the virtual weight matrix $E$ is. Applying the fact $\|\cdot\|^{2}\le \|\cdot\|_{F}^{2}$ to the right-hand side of [\[eq_proof_theorem2-b-T2-1\]](#eq_proof_theorem2-b-T2-1){reference-type="eqref" reference="eq_proof_theorem2-b-T2-1"}, we have $$\begin{aligned} \label{eq_proof_theorem2-b-T2} T_{2}\le \frac{2\chi^{2}}{v_{1}H}\|\Lambda^{k+\frac{1}{2}}-\frac{1}{H}\bm{1}\bm{1}^{\top}\Lambda^{k+\frac{1}{2}} \|_{F}^{2}.\end{aligned}$$ **Bounding $T_{3}$:** Averaging both sides of [\[eq_BRE-DEGRA-2\]](#eq_BRE-DEGRA-2){reference-type="eqref" reference="eq_BRE-DEGRA-2"} over $i\in \mathcal{H}$, we have $$\begin{aligned} \label{eq_proof_theorem2-b-T3-1} \bar{\bm{\lambda}}^{k+\frac{1}{2}}=\bar{\bm{\lambda}}^{k}-\frac{\gamma^{k}}{H}\sum_{i\in \mathcal{H}}(\frac{1}{J}\bm{s}-\frac{1}{J}\bm{\theta}_{i}^{k}).\end{aligned}$$ The dual problem of [\[eq_oracle-problem\]](#eq_oracle-problem){reference-type="eqref" reference="eq_oracle-problem"} can be written as a minimization problem in the form of $$\begin{aligned} \label{eq_dual-problem_oracle} \underset{\bm{\lambda }\in \mathbb{R}^D}{\min} ~ g(\bm{\lambda })=\sum_{i\in\mathcal{H}}g_{i}(\bm{\lambda }),\end{aligned}$$ where $g_{i}(\bm{\lambda }):=\frac{1}{H}F_{i}^{*}(-\bm{\lambda })+\frac{1}{H}\bm{\lambda }^{\top}\bm{s}$ and $F_{i}^{*}(\bm{\lambda}):=\max_{\bm{\theta}_{i}\in C_{i}} \{\bm{\lambda}^{\top}\bm{\theta}_{i}-f_{i}(\bm{\theta}_{i})\}$. Based on the definition of $g_{i}(\bm{\lambda }):=\frac{1}{H}F_{i}^{*}(-\bm{\lambda })+\frac{1}{H}\bm{\lambda }^{\top}\bm{s}$ and Danskin's theorem [@b-Dimitri-P.-Bertsekas-1999], we have $$\begin{aligned} \label{eq_gradient_dual_function_oracle} \nabla g_{i}(\bm{\lambda}_{i})=\frac{1}{H}\bm{s}-\frac{1}{H}\arg\underset{\bm{\theta}_{i}\in C_{i}}{\min}\{\bm{\lambda}_{i}^{\top}\bm{\theta}_{i}+f_{i}(\bm{\theta}_{i})\}.\end{aligned}$$ Combining [\[eq_BRE-DEGRA-1\]](#eq_BRE-DEGRA-1){reference-type="eqref" reference="eq_BRE-DEGRA-1"}, [\[eq_proof_theorem2-b-T3-1\]](#eq_proof_theorem2-b-T3-1){reference-type="eqref" reference="eq_proof_theorem2-b-T3-1"} and [\[eq_gradient_dual_function_oracle\]](#eq_gradient_dual_function_oracle){reference-type="eqref" reference="eq_gradient_dual_function_oracle"}, we can obtain $$\begin{aligned} \label{eq_proof_theorem2-b-T3-2} \bar{\bm{\lambda}}^{k+\frac{1}{2}}=\bar{\bm{\lambda}}^{k}-\frac{\gamma^{k}}{J}\sum_{i\in \mathcal{H}}\nabla g_{i}(\bm{\lambda}_{i}^{k}).\end{aligned}$$ Substituting [\[eq_proof_theorem2-b-T3-2\]](#eq_proof_theorem2-b-T3-2){reference-type="eqref" reference="eq_proof_theorem2-b-T3-2"} into $T_{3}$, we have $$\begin{aligned} \label{eq_proof_theorem2-b-T3-3} T_{3}=&\frac{1}{1-v_{1}}\|\bar{\bm{\lambda}}^{k}-\bm{\lambda}^{*}-\frac{\gamma^{k}}{J}\sum_{i\in \mathcal{H}}\nabla g_{i}(\bar{\bm{\lambda}}^{k})\\ &+\frac{\gamma^{k}}{J}\sum_{i\in \mathcal{H}}\nabla g_{i}(\bar{\bm{\lambda}}^{k})-\frac{\gamma^{k}}{J}\sum_{i\in \mathcal{H}}\nabla g_{i}(\bm{\lambda}_{i}^{k})\|^{2}\notag\\ =&\frac{1}{1-v_{1}}\|\bar{\bm{\lambda}}^{k}-\bm{\lambda}^{*}-\frac{\gamma^{k}}{J}\sum_{i\in \mathcal{H}}\nabla g_{i}(\bar{\bm{\lambda}}^{k})\|^{2}\notag\\ &+\frac{(\gamma^{k})^{2}}{1-v_{1}}\|\frac{1}{J}\sum_{i\in \mathcal{H}}(\nabla g_{i}(\bar{\bm{\lambda}}^{k})-\nabla g_{i}(\bm{\lambda}_{i}^{k}))\|^{2}+\frac{2\gamma^{k}}{1-v_{1}}\cdot \notag\\ &\left \langle \bar{\bm{\lambda}}^{k}-\bm{\lambda}^{*}-\frac{\gamma^{k}}{J}\sum_{i\in \mathcal{H}}\nabla g_{i}(\bar{\bm{\lambda}}^{k}), \frac{1}{J}\sum_{i\in \mathcal{H}}(\nabla g_{i}(\bar{\bm{\lambda}}^{k})-\nabla g_{i}(\bm{\lambda}_{i}^{k}))\right \rangle \notag\\ \le& \frac{1}{1-v_{1}}\|\bar{\bm{\lambda}}^{k}-\bm{\lambda}^{*}-\frac{\gamma^{k}}{J}\sum_{i\in \mathcal{H}}\nabla g_{i}(\bar{\bm{\lambda}}^{k})\|^{2}\notag\\ &+\frac{(\gamma^{k})^{2}}{1-v_{1}}\|\frac{1}{J}\sum_{i\in \mathcal{H}}(\nabla g_{i}(\bar{\bm{\lambda}}^{k})-\nabla g_{i}(\bm{\lambda}_{i}^{k}))\|^{2}\notag\\ &+ \frac{v_{2}^{-1}\gamma^{k}}{1-v_{1}}\|\frac{1}{J}\sum_{i\in \mathcal{H}}(\nabla g_{i}(\bar{\bm{\lambda}}^{k})-\nabla g_{i}(\bm{\lambda}_{i}^{k}))\|^{2}\notag\\ &+\frac{v_{2}\gamma^{k}}{1-v_{1}}\|\bar{\bm{\lambda}}^{k}-\bm{\lambda}^{*}-\frac{\gamma^{k}}{J}\sum_{i\in \mathcal{H}}\nabla g_{i}(\bar{\bm{\lambda}}^{k})\|^{2}\notag\\ \le&\frac{1+v_{2}\gamma^{k}}{1-v_{1}}\|\bar{\bm{\lambda}}^{k}-\bm{\lambda}^{*}-\frac{\gamma^{k}}{J}\sum_{i\in \mathcal{H}}\nabla g_{i}(\bar{\bm{\lambda}}^{k})\|^{2}\notag\\ &+\frac{\gamma^{k}(\gamma^{k}+v_{2}^{-1})H}{(1-v_{1})J^{2}}\sum_{i\in \mathcal{H}}\|\nabla g_{i}(\bar{\bm{\lambda}}^{k})-\nabla g_{i}(\bm{\lambda}_{i}^{k})\|^{2},\notag\end{aligned}$$ where $v_{2}>0$ is any positive constant. To drive the first inequality, we use $2\bm{a}^{\top}\bm{b}\le v^{-1}\|\bm{a}\|^{2}+v\|\bm{b}\|^{2}$ for any $v>0$. The last inequality holds because $(a_{1}+\cdots+a_{H})^{2}\le H(a_{1}^{2}+\cdots+a_{H}^{2})$. Next, we analyze the first and second terms at the right-hand side of [\[eq_proof_theorem2-b-T3-3\]](#eq_proof_theorem2-b-T3-3){reference-type="eqref" reference="eq_proof_theorem2-b-T3-3"} in turn. According to the fact that $\sum_{i\in \mathcal{H}}\nabla g_{i}(\bm{\lambda}^{*})=\bm{0}$, we have $$\begin{aligned} \label{eq_proof_theorem2-b-T3-4} &\frac{1+v_{2}\gamma^{k}}{1-v_{1}}\|\bar{\bm{\lambda}}^{k}-\bm{\lambda}^{*}-\frac{\gamma^{k}}{J}\sum_{i\in \mathcal{H}}\nabla g_{i}(\bar{\bm{\lambda}}^{k})\|^{2}\\ =& \frac{1+v_{2}\gamma^{k}}{1-v_{1}}\|\bar{\bm{\lambda}}^{k}-\bm{\lambda}^{*}-\frac{\gamma^{k}}{J}\sum_{i\in \mathcal{H}}\nabla g_{i}(\bar{\bm{\lambda}}^{k})-\frac{\gamma^{k}}{J}\sum_{i\in \mathcal{H}}\nabla g_{i}(\bm{\lambda}^{*})\|^{2}\notag \\ =&\frac{(1+v_{2}\gamma^{k})(\gamma^{k})^{2}H^{2}}{(1-v_{1})J^{2}}\|\frac{1}{H}\sum_{i\in \mathcal{H}}\nabla g_{i}(\bar{\bm{\lambda}}^{k})-\frac{1}{H}\sum_{i\in \mathcal{H}}\nabla g_{i}(\bm{\lambda}^{*})\|^{2}\notag\\ &+\frac{1+v_{2}\gamma^{k}}{1-v_{1}}\|\bar{\bm{\lambda}}^{k}-\bm{\lambda}^{*}\|^{2}- \frac{2\gamma^{k}(1+v_{2}\gamma^{k})H}{(1-v_{1})J}\cdot \notag\\ &\left \langle \bar{\bm{\lambda}}^{k}-\bm{\lambda}^{*},\frac{1}{H}\sum_{i\in \mathcal{H}}\nabla g_{i}(\bar{\bm{\lambda}}^{k})-\frac{1}{H}\sum_{i\in \mathcal{H}}\nabla g_{i}(\bm{\lambda}^{*}) \right \rangle .\notag\end{aligned}$$ For $\left \langle \bar{\bm{\lambda}}^{k}-\bm{\lambda}^{*},\frac{1}{H}\sum_{i\in \mathcal{H}}\nabla g_{i}(\bar{\bm{\lambda}}^{k})-\frac{1}{H}\sum_{i\in \mathcal{H}}\nabla g_{i}(\bm{\lambda}^{*}) \right \rangle$, the last term at the right-hand side of [\[eq_proof_theorem2-b-T3-4\]](#eq_proof_theorem2-b-T3-4){reference-type="eqref" reference="eq_proof_theorem2-b-T3-4"}, we have the following bound. According to Lemma [Lemma 2](#l2){reference-type="ref" reference="l2"}, $g_{i}(\cdot)$ is $\frac{1}{HL_{f}}$-strongly convex and $\frac{1}{Hu_{f}}$-smooth. By Lemma 3 in [@b-Kun-Yuan-2016], since $\frac{1}{H}\sum_{i\in \mathcal{H}}g_{i}(\cdot)$ is $\frac{1}{HL_{f}}$-strongly convex and $\frac{1}{Hu_{f}}$-smooth, we have $$\begin{aligned} \label{eq_proof_theorem2-b-T3-5} &\left \langle \bar{\bm{\lambda}}^{k}-\bm{\lambda}^{*},\frac{1}{H}\sum_{i\in \mathcal{H}}\nabla g_{i}(\bar{\bm{\lambda}}^{k})-\frac{1}{H}\sum_{i\in \mathcal{H}}\nabla g_{i}(\bm{\lambda}^{*}) \right \rangle\\ \ge & \alpha \|\frac{1}{H}\sum_{i\in \mathcal{H}}\nabla g_{i}(\bar{\bm{\lambda}}^{k})-\frac{1}{H}\sum_{i\in \mathcal{H}}\nabla g_{i}(\bm{\lambda}^{*})\|^{2}+\beta\|\bar{\bm{\lambda}}^{k}-\bm{\lambda}^{*}\|^{2},\notag\end{aligned}$$ where $\alpha=\frac{Hu_{f}L_{f}}{u_{f}+L_{f}}$ and $\beta=\frac{1}{H(u_{f}+L_{f})}$. Substituting [\[eq_proof_theorem2-b-T3-5\]](#eq_proof_theorem2-b-T3-5){reference-type="eqref" reference="eq_proof_theorem2-b-T3-5"} into [\[eq_proof_theorem2-b-T3-4\]](#eq_proof_theorem2-b-T3-4){reference-type="eqref" reference="eq_proof_theorem2-b-T3-4"} and rearranging the terms, we have $$\begin{aligned} \label{eq_proof_theorem2-b-T3-6} &\frac{1+v_{2}\gamma^{k}}{1-v_{1}}\|\bar{\bm{\lambda}}^{k}-\bm{\lambda}^{*}-\frac{\gamma^{k}}{J}\sum_{i\in \mathcal{H}}\nabla g_{i}(\bar{\bm{\lambda}}^{k})\|^{2}\\ \le& \frac{(1+v_{2}\gamma^{k})(1-2\gamma^{k}\beta\cdot\frac{H}{J})}{1-v_{1}}\|\bar{\bm{\lambda}}^{k}-\bm{\lambda}^{*}\|^{2}\notag\\ &+\frac{(1+v_{2}\gamma^{k})((\gamma^{k})^{2}\cdot\frac{H}{J}-2\gamma^{k}\alpha)}{1-v_{1}}\cdot\notag\\ &\frac{H}{J}\|\frac{1}{H}\sum_{i\in \mathcal{H}}\nabla g_{i}(\bar{\bm{\lambda}}^{k})-\frac{1}{H}\sum_{i\in \mathcal{H}}\nabla g_{i}(\bm{\lambda}^{*})\|^{2} \notag\\ \le &\frac{(1+v_{2}\gamma^{k})(1-\gamma^{k}\beta\cdot\frac{H}{J})}{1-v_{1}}\|\bar{\bm{\lambda}}^{k}-\bm{\lambda}^{*}\|^{2},\notag\end{aligned}$$ where the last inequality holds with a proper step size $\gamma^{k}$ satisfying $(\gamma^{k})^{2}\cdot\frac{H}{J}-2\gamma^{k}\alpha\le 0$. Since $g_{i}(\cdot)$ is $\frac{1}{Hu_{f}}$-smooth, we obtain $$\begin{aligned} \label{eq_proof_theorem2-b-T3-7} &\frac{\gamma^{k}(\gamma^{k}+v_{2}^{-1})}{(1-v_{1})H}\cdot\frac{H^{2}}{J^{2}}\sum_{i\in \mathcal{H}}\|\nabla g_{i}(\bar{\bm{\lambda}}^{k})-\nabla g_{i}(\bm{\lambda}_{i}^{k})\|^{2}\\ \le& \frac{\gamma^{k}(\gamma^{k}+v_{2}^{-1})}{(1-v_{1})H^{3}u_{f}^{2}}\cdot\frac{H^{2}}{J^{2}}\sum_{i\in \mathcal{H}}\|\bm{\lambda}_{i}^{k}-\bar{\bm{\lambda}}^{k}\|^{2}\notag\\ = &\frac{\gamma^{k}(\gamma^{k}+v_{2}^{-1})}{(1-v_{1})H^{3}u_{f}^{2}}\cdot\frac{H^{2}}{J^{2}}\|\Lambda^{k}-\frac{1}{H}\bm{1}\bm{1}^{\top}\Lambda^{k}\|_{F}^{2}.\notag\end{aligned}$$ Substituting [\[eq_proof_theorem2-b-T3-6\]](#eq_proof_theorem2-b-T3-6){reference-type="eqref" reference="eq_proof_theorem2-b-T3-6"} and [\[eq_proof_theorem2-b-T3-7\]](#eq_proof_theorem2-b-T3-7){reference-type="eqref" reference="eq_proof_theorem2-b-T3-7"} into [\[eq_proof_theorem2-b-T3-3\]](#eq_proof_theorem2-b-T3-3){reference-type="eqref" reference="eq_proof_theorem2-b-T3-3"} and rearranging the terms, we obtain $$\begin{aligned} \label{eq_proof_theorem2-b-T3} T_{3}\le & \frac{(1+v_{2}\gamma^{k})(1-\gamma^{k}\beta\cdot\frac{H}{J})}{1-v_{1}}\|\bar{\bm{\lambda}}^{k}-\bm{\lambda}^{*}\|^{2} \\ &+\frac{\gamma^{k}(\gamma^{k}+v_{2}^{-1})}{(1-v_{1})H^{3}u_{f}^{2}}\cdot\frac{H^{2}}{J^{2}}\|\Lambda^{k}-\frac{1}{H}\bm{1}\bm{1}^{\top}\Lambda^{k}\|_{F}^{2}.\notag\end{aligned}$$ Substituting [\[eq_proof_theorem2-b-T1\]](#eq_proof_theorem2-b-T1){reference-type="eqref" reference="eq_proof_theorem2-b-T1"}, [\[eq_proof_theorem2-b-T2\]](#eq_proof_theorem2-b-T2){reference-type="eqref" reference="eq_proof_theorem2-b-T2"} and [\[eq_proof_theorem2-b-T3\]](#eq_proof_theorem2-b-T3){reference-type="eqref" reference="eq_proof_theorem2-b-T3"} into [\[eq_proof_theorem2-b-1\]](#eq_proof_theorem2-b-1){reference-type="eqref" reference="eq_proof_theorem2-b-1"} and rearranging the terms, we have $$\begin{aligned} \label{eq_proof_theorem2-b-2} &\|\bar{\bm{\lambda}}^{k+1}-\bm{\lambda}^{*}\|^{2}\\ \le & \frac{(1+v_{2}\gamma^{k})(1-\gamma^{k}\beta\cdot\frac{H}{J})}{1-v_{1}}\|\bar{\bm{\lambda}}^{k}-\bm{\lambda}^{*}\|^{2} \notag\\ + & \frac{\gamma^{k}(\gamma^{k}+v_{2}^{-1})}{(1-v_{1})H^{3}u_{f}^{2}}\cdot\frac{H^{2}}{J^{2}}\|\Lambda^{k}-\frac{1}{H}\bm{1}\bm{1}^{\top}\Lambda^{k}\|_{F}^{2}\notag\\ + & \frac{8\rho^{2}}{v_{1}}\|\Lambda^{k+\frac{1}{2}}-\frac{1}{H}\bm{1}\bm{1}^{\top}\Lambda^{k+\frac{1}{2}} \|^{2}_{F}+\frac{2\chi^{2}}{v_{1}H}\|\Lambda^{k+\frac{1}{2}}-\frac{1}{H}\bm{1}\bm{1}^{\top}\Lambda^{k+\frac{1}{2}} \|_{F}^{2}.\notag\end{aligned}$$ Setting $v_{1}=\frac{\gamma^{k}\beta\cdot\frac{H}{J}}{4}\in(0,1)$ and $v_{2}=\frac{\beta\cdot\frac{H}{J}}{2(1-\gamma^{k}\beta\cdot\frac{H}{J})}>0$, from [\[eq_proof_theorem2-b-2\]](#eq_proof_theorem2-b-2){reference-type="eqref" reference="eq_proof_theorem2-b-2"} we have $$\begin{aligned} \label{eq_proof_theorem2-b-3} &\|\bar{\bm{\lambda}}^{k+1}-\bm{\lambda}^{*}\|^{2}\\ \le & (1-\frac{\gamma^{k}\beta\cdot\frac{H}{J}}{4})\|\bar{\bm{\lambda}}^{k}-\bm{\lambda}^{*}\|^{2}+\frac{4\gamma^{k}}{\beta H^{2}u_{f}^{2}J}\|\Lambda^{k}-\frac{1}{H}\bm{1}\bm{1}^{\top}\Lambda^{k}\|_{F}^{2}\notag\\ &+ \frac{8(4\rho^{2}H+\chi^{2})J}{\gamma^{k} \beta H^{2}}\|\Lambda^{k+\frac{1}{2}}-\frac{1}{H}\bm{1}\bm{1}^{\top}\Lambda^{k+\frac{1}{2}} \|_{F}^{2}.\notag\end{aligned}$$ Substituting [\[lemma3-2\]](#lemma3-2){reference-type="eqref" reference="lemma3-2"} in Lemma [Lemma 3](#l3){reference-type="ref" reference="l3"} into [\[eq_proof_theorem2-b-3\]](#eq_proof_theorem2-b-3){reference-type="eqref" reference="eq_proof_theorem2-b-3"} and rearranging the terms, we obtain $$\begin{aligned} \label{eq_proof_theorem2-b-4} &\|\bar{\bm{\lambda}}^{k+1}-\bm{\lambda}^{*}\|^{2}\\ \le & (1-\frac{\gamma^{k}\beta\cdot\frac{H}{J}}{4})\|\bar{\bm{\lambda}}^{k}-\bm{\lambda}^{*}\|^{2}+\frac{4\gamma^{k}}{\beta H^{2}u_{f}^{2}J}\|\Lambda^{k}-\frac{1}{H}\bm{1}\bm{1}^{\top}\Lambda^{k}\|_{F}^{2}\notag\\ &+ \frac{24(4\rho^{2}H+\chi^{2})J}{\gamma^{k} \beta H^{2}}\|\Lambda^{k}-\frac{1}{H}\bm{1}\bm{1}^{\top}\Lambda^{k} \|_{F}^{2}\notag\\ &+ \frac{48\gamma^{k}\delta^{2}(4\rho^{2}H+\chi^{2})H}{\beta J}.\notag\end{aligned}$$ Based on Lemma [Lemma 4](#l4){reference-type="ref" reference="l4"}, we can rewrite [\[eq_proof_theorem2-b-4\]](#eq_proof_theorem2-b-4){reference-type="eqref" reference="eq_proof_theorem2-b-4"} as $$\begin{aligned} \label{eq_proof_theorem2-b-5} &\|\bar{\bm{\lambda}}^{k+1}-\bm{\lambda}^{*}\|^{2}\\ \le & (1-\frac{\gamma^{k}\beta\cdot\frac{H}{J}}{4})\|\bar{\bm{\lambda}}^{k}-\bm{\lambda}^{*}\|^{2}+\frac{72(\gamma^{k})^{3}\delta^{2}H}{\beta\epsilon^{3}u_{f}^{2}J^{3}}\notag\\ &+\frac{432\gamma^{k}(4\rho^{2}H+\chi^{2})\delta^{2}H}{\beta\epsilon^{3} J} + \frac{48\gamma^{k}\delta^{2}(4\rho^{2}H+\chi^{2})H}{\beta J}.\notag\end{aligned}$$ Set a proper decreasing step size $\gamma^{k}=\frac{4}{\beta\cdot\frac{H}{J}(k_{0}+k)}$, where $k_{0}> 1$ is a any positive integer. Thus, $1-\frac{\gamma^{k} \beta\cdot\frac{H}{J}}{4}=1-\frac{1}{k_{0}+k}$ and [\[eq_proof_theorem2-b-5\]](#eq_proof_theorem2-b-5){reference-type="eqref" reference="eq_proof_theorem2-b-5"} can be rewritten as $$\begin{aligned} \label{eq_proof_theorem2-b-6} &\|\bar{\bm{\lambda}}^{k+1}-\bm{\lambda}^{*}\|^{2}\\ \le & (1-\frac{1}{k_{0}+k})\|\bar{\bm{\lambda}}^{k}-\bm{\lambda}^{*}\|^{2}+ \frac{1}{(k_{0}+k)^{3}}\cdot \frac{4608\delta^{2}}{\beta^{4}\epsilon^{3}u_{f}^{2}H^{2}}\notag\\ &+\frac{1}{k_{0}+k}\cdot\frac{192\delta^{2}}{\beta^{2}}\cdot(1+\frac{9}{\epsilon^{3}})\cdot(4\rho^{2}H+\chi^{2}).\notag\end{aligned}$$ Then we rewrite [\[eq_proof_theorem2-b-6\]](#eq_proof_theorem2-b-6){reference-type="eqref" reference="eq_proof_theorem2-b-6"} recursively and obtain $$\begin{aligned} \label{eq_proof_theorem2-b-7} &\|\bar{\bm{\lambda}}^{k+1}-\bm{\lambda}^{*}\|^{2}\\ & \le \underbrace{\prod_{k'=0}^{k}(1-\frac{1}{k_{0}+k-k'})}_{T_{4}}\|\bar{\bm{\lambda}}^{0}-\bm{\lambda}^{*}\|^{2}\notag\\ &+[\prod_{k'=0}^{k-1}(1-\frac{1}{k_{0}+k-k'})\frac{1}{(k_{0}+0)^{3}}+\cdots+\notag\\ &\underbrace{\prod_{k'=0}^{0}\frac{(1-\frac{1}{k_{0}+k-k'})}{(k_{0}+k-1)^{3}}+\frac{1}{(k_{0}+k)^{3}}}_{T_{5}}]\cdot \frac{4608\delta^{2}}{\beta^{4}\epsilon^{3}u_{f}^{2}H^{2}}\notag\\ &+[\prod_{k'=0}^{k-1}(1-\frac{1}{k_{0}+k-k'})\frac{1}{k_{0}+0}+\cdots+\notag\\ &\underbrace{\prod_{k'=0}^{0}\frac{(1-\frac{1}{k_{0}+k-k'})}{k_{0}+k-1}+\frac{1}{k_{0}+k}]}_{T_{6}}\cdot\frac{192\delta^{2}(4\rho^{2}H+\chi^{2})(1+\frac{9}{\epsilon^{3}})}{\beta^{2}}.\notag\end{aligned}$$ Next, we analyze $T_{4}$, $T_{5}$ and $T_{6}$ in turn. For $T_{4}$, we have $$\begin{aligned} \label{eq_proof_theorem2-b-T4} T_{4}=&\prod_{k'=0}^{k}(1-\frac{1}{k_{0}+k-k'})\\ =&\frac{k_{0}+k-1}{k_{0}+k}\cdot\frac{k_{0}+k-2}{k_{0}+k-1}\cdot\cdots\cdot\frac{k_{0}-1}{k_{0}}\notag\\ =&\frac{k_{0}-1}{k_{0}+k}.\notag\end{aligned}$$ For $T_{5}$, we have $$\begin{aligned} \label{eq_proof_theorem2-b-T5} T_{5}=&\frac{k_{0}}{k_{0}+k}\cdot\frac{1}{(k_{0}+0)^{3}}+\frac{k_{0}+1}{k_{0}+k}\cdot\frac{1}{(k_{0}+1)^{3}}+\cdots\\ &+\frac{k_{0}+k-1}{k_{0}+k}\cdot\frac{1}{(k_{0}+k-1)^{3}}+\frac{1}{(k_{0}+k)^{3}}\notag\\ =& \frac{1}{k_{0}+k}[\frac{1}{(k_{0})^{2}}+\frac{1}{(k_{0}+1)^{2}}+\cdots\notag\\ &+\frac{1}{(k_{0}+k-1)^{2}}+\frac{1}{(k_{0}+k)^{2}}]\notag\\ \le& \frac{1}{k_{0}+k}\cdot\frac{1}{k_{0}-1}.\notag\end{aligned}$$ To drive the last inequality, we use $\sum_{k'=k_{0}}^{k}\frac{1}{(k')^{2}}\le \frac{1}{k_{0}-1}$. For $T_{6}$, we have $$\begin{aligned} \label{eq_proof_theorem2-b-T6} T_{6}=&[\frac{k_{0}}{k_{0}+k}\cdot\frac{1}{k_{0}+0}+\frac{k_{0}+1}{k_{0}+k}\cdot\frac{1}{k_{0}+1}+\cdots\\ &+\frac{k_{0}+k-1}{k_{0}+k}\cdot\frac{1}{k_{0}+k-1}+\frac{1}{k_{0}+k}]\notag\\ =& \frac{k+1}{k_{0}+k}.\notag\end{aligned}$$ Substituting [\[eq_proof_theorem2-b-T4\]](#eq_proof_theorem2-b-T4){reference-type="eqref" reference="eq_proof_theorem2-b-T4"}, [\[eq_proof_theorem2-b-T5\]](#eq_proof_theorem2-b-T5){reference-type="eqref" reference="eq_proof_theorem2-b-T5"} and [\[eq_proof_theorem2-b-T6\]](#eq_proof_theorem2-b-T6){reference-type="eqref" reference="eq_proof_theorem2-b-T6"} into [\[eq_proof_theorem2-b-7\]](#eq_proof_theorem2-b-7){reference-type="eqref" reference="eq_proof_theorem2-b-7"}, we have $$\begin{aligned} \label{eq_proof_theorem2-b-8} &\|\bar{\bm{\lambda}}^{k+1}-\bm{\lambda}^{*}\|^{2}\\ \le&\frac{k_{0}-1}{k_{0}+k}\|\bar{\bm{\lambda}}^{0}-\bm{\lambda}^{*}\|^{2}+\frac{1}{(k_{0}+k)(k_{0}-1)}\cdot\frac{4608\delta^{2}}{\beta^{4}\epsilon^{3}u_{f}^{2}H^{2}}\notag\\ &+\frac{k+1}{k_{0}+k}\cdot \frac{192\delta^{2}}{\beta^{2}}\cdot(1+\frac{9}{\epsilon^{3}})\cdot(4\rho^{2}H+\chi^{2}).\notag\end{aligned}$$ Applying the triangle inequality into [\[eq_proof_theorem2-b-8\]](#eq_proof_theorem2-b-8){reference-type="eqref" reference="eq_proof_theorem2-b-8"}, we obtain $$\begin{aligned} \label{eq_proof_theorem2-b-9} &\|\bar{\bm{\lambda}}^{k+1}-\bm{\lambda}^{*}\|\\ \le&\sqrt{\frac{k_{0}-1}{k_{0}+k}}\|\bar{\bm{\lambda}}^{0}-\bm{\lambda}^{*}\|+\sqrt{\frac{1}{(k_{0}+k)(k_{0}-1)}\cdot\frac{4608\delta^{2}}{\beta^{4}\epsilon^{3}u_{f}^{2}H^{2}}}\notag\\ &+\sqrt{\frac{k+1}{k_{0}+k}\cdot \frac{192\delta^{2}}{\beta^{2}}\cdot(1+\frac{9}{\epsilon^{3}})\cdot(4\rho^{2}H+\chi^{2})}.\notag\end{aligned}$$ By Lemma [Lemma 4](#l4){reference-type="ref" reference="l4"} and using the step size $\gamma^{k}=\frac{4}{\beta\cdot\frac{H}{J}(k_{0}+k)}$, we obtain $$\begin{aligned} \label{eq_proof_theorem2-b-10} \sum_{i\in \mathcal{H}}\|\bm{\lambda}_{i}^{k+1}-\bar{\bm{\lambda}}^{k+1}\|^{2}=&\|\Lambda^{k+1}-\frac{1}{|\mathcal{H}|}\bm{1}\bm{1}^{\top}\Lambda^{k+1}\|^{2}_{F}\\ \le &\frac{18(\gamma^{k+1})^{2}\delta^{2}H^{3}}{\epsilon^{3} \cdot J^{2}}\notag\\ =& \frac{288\delta^{2}H}{(k_{0}+k+1)^{2}\epsilon^{3}\beta^{2}}.\notag\end{aligned}$$ By $(a_{1}+\cdots+a_{H})^{2}\le H(a_{1}^{2}+\cdots+a_{H}^{2})$ and [\[eq_proof_theorem2-b-10\]](#eq_proof_theorem2-b-10){reference-type="eqref" reference="eq_proof_theorem2-b-10"}, we have $$\begin{aligned} \label{eq_proof_theorem2-b-11} \sum_{i\in \mathcal{H}}\|\bm{\lambda}_{i}^{k+1}-\bar{\bm{\lambda}}^{k+1}\|=&\sqrt{H}\cdot\sqrt{\sum_{i\in \mathcal{H}}\|\bm{\lambda}_{i}^{k+1}-\bar{\bm{\lambda}}^{k+1}\|^{2}}\\ \le &\sqrt{H}\cdot \sqrt{\frac{288\delta^{2}H}{(k_{0}+k+1)^{2}\epsilon^{3}\beta^{2}}}.\notag\end{aligned}$$ Combining [\[eq_proof_theorem2-b-9\]](#eq_proof_theorem2-b-9){reference-type="eqref" reference="eq_proof_theorem2-b-9"} and [\[eq_proof_theorem2-b-11\]](#eq_proof_theorem2-b-11){reference-type="eqref" reference="eq_proof_theorem2-b-11"} yields $$\begin{aligned} \label{eq_proof_theorem2-b-12} &\sum_{i\in \mathcal{H}}\|\bm{\lambda}_{i}^{k+1}-\bm{\lambda}^{*}\|\\ \le& \sum_{i\in \mathcal{H}}\|\bm{\lambda}_{i}^{k+1}-\bar{\bm{\lambda}}^{k+1}\|+H\cdot\|\bar{\bm{\lambda}}^{k+1}-\bm{\lambda}^{*}\|\notag\\ \le&H\cdot\sqrt{\frac{k_{0}-1}{k_{0}+k}}\|\bar{\bm{\lambda}}^{0}-\bm{\lambda}^{*}\|+\sqrt{\frac{1}{(k_{0}+k)(k_{0}-1)}\cdot\frac{4608\delta^{2}}{\beta^{4}\epsilon^{3}u_{f}^{2}}}\notag\\ &+H\cdot\sqrt{\frac{k+1}{k_{0}+k}\cdot \frac{192\delta^{2}}{\beta^{2}}\cdot(1+\frac{9}{\epsilon^{3}})\cdot(4\rho^{2}H+\chi^{2})}\notag\\ &+\sqrt{H}\cdot \sqrt{\frac{288\delta^{2}H}{(k_{0}+k+1)^{2}\epsilon^{3}\beta^{2}}}.\notag\end{aligned}$$ Taking $k \to +\infty$, we obtain $$\begin{aligned} \label{eq_proof_eq22} &\limsup_{k\to +\infty}\sum_{i\in \mathcal{H}}\|\bm{\lambda}_{i}^{k+1}-\bm{\lambda}^{*}\|\\ \le&\sqrt{\frac{192\delta^{2}H^{2}}{\beta^{2}}\cdot(1+\frac{9}{\epsilon^{3}})\cdot(4\rho^{2}H+\chi^{2})}.\notag\end{aligned}$$ ## Part $b$ of Theorem [Theorem 2](#t2){reference-type="ref" reference="t2"} {#part-b-of-theorem-t2} Based on $\|\Lambda^{k+1}-\frac{1}{H}\bm{1}\bm{1}^{\top}\Lambda^{k+1} \|^{2}_{F}\le \frac{18(\gamma^{k+1})^{2}\delta^{2}H^{3}}{\epsilon^{3} \cdot J^{2}}$ in Lemma [Lemma 4](#l4){reference-type="ref" reference="l4"} and the fact $\|\Lambda^{k+1}-\frac{1}{H}\bm{1}\bm{1}^{\top}\Lambda^{k+1} \|^{2}_{F}=\sum_{i\in \mathcal{H}}\|\bm{\lambda}_{i}^{k+1}-\bar{\bm{\lambda}}^{k+1}\|^{2}$, with a proper decreasing step size $\gamma^{k}=O(\frac{1}{k})$, taking $k \to +\infty$, we obtain $$\begin{aligned} \label{eq-proof-theorem2-c-1} \lim_{k\to +\infty}\sum_{i\in \mathcal{H}}\|\bm{\lambda}_{i}^{k+1}-\bar{\bm{\lambda}}^{k+1}\|= 0.\end{aligned}$$ We conclude by summarizing the conditions on the step size $\gamma^{k}$ in Theorem [Theorem 2](#t2){reference-type="ref" reference="t2"}. It must satisfy $(\gamma^{k})^{2}\cdot\frac{H}{J}-2\gamma^{k}\alpha\le 0$, $\frac{18(\gamma^{k})^{2}}{\epsilon u_{f}^{2}J^{2}}\le \frac{(2-\epsilon)\epsilon^{2}}{3(3-\epsilon)}$, $1\le \frac{(\gamma^{k})^{2}}{(\gamma^{k+1)^{2}}}\le \frac{2}{1+(1-\epsilon^{2})}$, as well as $\gamma^{k}\le \frac{u_{f}J}{2\sqrt{3}}$. The specific step size $\gamma^{k}=\frac{4}{\beta\cdot\frac{H}{J}(k_{0}+k)}$ with $k_{0}\ge \max\{\frac{2}{\alpha\beta},\sqrt{\frac{216(3-\epsilon)}{(2-\epsilon)\epsilon u_{f}^{2}H^{2}\beta^{2}}},\frac{1}{\sqrt{\frac{2}{1+(1-\epsilon^{2})}}-1},\frac{8\sqrt{3}}{u_{f}H \beta}\}$ satisfies these conditions. ## Part $c$ of Theorem [Theorem 2](#t2){reference-type="ref" reference="t2"} {#part-c-of-theorem-t2} The Lagrangian function of [\[eq_oracle-problem\]](#eq_oracle-problem){reference-type="eqref" reference="eq_oracle-problem"} is $$\begin{aligned} \label{eq_Lagrangian-oracle-problem} \begin{split} \mathcal{L}\left ( \bm{\Theta} ; \bm{\lambda} \right ):= \frac{1}{H}\sum_{i\in \mathcal{H}}f_{i}\left ( \bm{\theta}_{i} \right )+ \left \langle \bm{\lambda }, \frac{1}{H}\sum_{i\in \mathcal{H}}\bm{\theta}_{i}-\bm{s} \right \rangle. \end{split}\end{aligned}$$ Since $\bm{\Theta}^{*}$ is the optimal solution of the primal problem [\[eq_oracle-problem\]](#eq_oracle-problem){reference-type="eqref" reference="eq_oracle-problem"}, we have $\frac{1}{H}\sum_{i\in \mathcal{H}}\bm{\theta}_{i}^{*} = \bm{s}$ and $\bm{\theta}^{*}\in C$. According to [\[eq_Lagrangian-oracle-problem\]](#eq_Lagrangian-oracle-problem){reference-type="eqref" reference="eq_Lagrangian-oracle-problem"}, for any dual variable $\bm{\lambda}$ we have $$\begin{aligned} \label{eq_proof_theorem2-a-1} \mathcal{L}\left ( \bm{\Theta}^{*} ; \bm{\lambda} \right )=&\frac{1}{H}\sum_{i\in \mathcal{H}}f_{i}\left ( \bm{\theta}_{i}^{*} \right )+ \left \langle \bm{\lambda }, \frac{1}{H}\sum_{i\in \mathcal{H}}\bm{\theta}_{i}^{*}-\bm{s} \right \rangle\\ =& f(\bm{\Theta}^{*}). \notag\end{aligned}$$ By Assumption [Assumption 2](#a2){reference-type="ref" reference="a2"}, the duality gap is zero. According to [\[eq_dual-problem_oracle\]](#eq_dual-problem_oracle){reference-type="eqref" reference="eq_dual-problem_oracle"}, for any $\bm{\lambda}$ we obtain $g(\bm{\lambda}^{*})=-f(\bm{\Theta}^{*})=-\mathcal{L}\left ( \bm{\Theta}^{*} ; \bm{\lambda} \right )$. Now we introduce a vector $^{\dagger}\bm{\Theta}^{k+1}:=[_{}^{\dagger}\bm{\theta}_{1}^{k+1}; \cdots; _{}^{\dagger}\bm{\theta}_{H}^{k+1}]$, where $_{}^{\dagger}\bm{\theta}_{i}^{k+1}=\arg\min_{\bm{\theta}_{i}\in C_{i}} \{\bm{\theta}_{i}^{\top}\bar{\bm{\lambda}}^{k+1}+f_{i}(\bm{\theta}_{i})\}$ and $\bar{\bm{\lambda}}^{k+1}:=\frac{1}{H}\sum_{i\in \mathcal{H}}\bm{\lambda}_{i}^{k+1}$. Therefore, we have $$\begin{aligned} \label{eq_proof_theorem2-a-2} &g(\bar{\bm{\lambda}}^{k+1})-g(\bm{\lambda}^{*})\\ =&-\inf_{\bm{\Theta}\in C}\mathcal{L}\left ( \bm{\Theta} ; \bar{\bm{\lambda}}^{k+1} \right )+\mathcal{L}\left ( \bm{\Theta}^{*} ; \bar{\bm{\lambda}}^{k+1} \right )\notag\\ =&-\mathcal{L}\left ( ^{\dagger}\bm{\Theta}^{k+1} ; \bar{\bm{\lambda}}^{k+1} \right )+\mathcal{L}\left ( \bm{\Theta}^{*} ; \bar{\bm{\lambda}}^{k+1} \right ).\notag\end{aligned}$$ Assumption [Assumption 1](#a1){reference-type="ref" reference="a1"} shows that the local cost function $f_{i}(\cdot)$ is $u_{f}$-strongly convex. Further using the definition of $\mathcal{L}\left ( \bm{\Theta} ; \bm{\lambda} \right )$ in [\[eq_Lagrangian-oracle-problem\]](#eq_Lagrangian-oracle-problem){reference-type="eqref" reference="eq_Lagrangian-oracle-problem"}, we know that $\mathcal{L}\left ( \bm{\Theta} ; \bm{\lambda} \right )$ is $u_{f}$-strongly convex with respect to $\bm{\Theta}$. Therefore, we have $$\begin{aligned} \label{eq_proof_theorem2-a-3} &\mathcal{L}\left ( \bm{\Theta}^{*} ; \bar{\bm{\lambda}}^{k+1} \right )-\mathcal{L}\left ( ^{\dagger}\bm{\Theta}^{k+1} ; \bar{\bm{\lambda}}^{k+1} \right )\\ \ge &\nabla \mathcal{L}^{\top}\left ( ^{\dagger}\bm{\Theta}^{k+1} ; \bar{\bm{\lambda}}^{k+1} \right )(\bm{\Theta}^{*}-\ ^{\dagger}\bm{\Theta}^{k+1})+\frac{u_{f}}{2}\| ^{\dagger}\bm{\Theta}^{k+1}-\bm{\Theta}^{*}\|^{2}.\notag\end{aligned}$$ Combining [\[eq_proof_theorem2-a-2\]](#eq_proof_theorem2-a-2){reference-type="eqref" reference="eq_proof_theorem2-a-2"} and [\[eq_proof_theorem2-a-3\]](#eq_proof_theorem2-a-3){reference-type="eqref" reference="eq_proof_theorem2-a-3"}, we obtain $$\begin{aligned} \label{eq_proof_theorem2-a-4} &g(\bar{\bm{\lambda}}^{k+1})-g(\bm{\lambda}^{*})\\ \ge& \nabla \mathcal{L}^{\top}\left ( ^{\dagger}\bm{\Theta}^{k+1} ; \bar{\bm{\lambda}}^{k+1} \right )(\bm{\Theta}^{*}-\ ^{\dagger}\bm{\Theta}^{k+1})+\frac{u_{f}}{2}\| ^{\dagger}\bm{\Theta}^{k+1}-\bm{\Theta}^{*}\|^{2}\notag\\ \ge&\frac{u_{f}}{2}\| ^{\dagger}\bm{\Theta}^{k+1}-\bm{\Theta}^{*}\|^{2}.\notag\end{aligned}$$ To drive the last inequality, we use optimality condition of $_{}^{\dagger}\bm{\theta}_{i}^{k+1}=\arg\min_{\bm{\theta}_{i}\in C_{i}} \{\bm{\theta}_{i}^{\top}\bar{\bm{\lambda}}^{k+1}+f_{i}(\bm{\theta}_{i})\}$ [@b-Dimitri-P.-Bertsekas-1999 Proposition 2.1.2]. According to Lemma [Lemma 2](#l2){reference-type="ref" reference="l2"}, $g_{i}(\bm{\lambda})$ is smooth with constant $\frac{1}{Hu_{f}}$. Therefore, function $g(\bm{\lambda })=\sum_{i\in\mathcal{H}}g_{i}(\bm{\lambda })$ is smooth with constant $\frac{1}{u_{f}}$. This fact leads to $$\begin{aligned} \label{eq_proof_theorem2-a-5} &g(\bar{\bm{\lambda}}^{k+1})-g(\bm{\lambda}^{*})\\ \le& \nabla g^{\top}(\bm{\lambda}^{*})(\bar{\bm{\lambda}}^{k+1}-\bm{\lambda}^{*})+\frac{1}{2u_{f}}\|\bar{\bm{\lambda}}^{k+1}-\bm{\lambda}^{*}\|^{2}\notag\\ =& \frac{1}{2u_{f}}\|\bar{\bm{\lambda}}^{k+1}-\bm{\lambda}^{*}\|^{2}.\notag\end{aligned}$$ To drive the last equality, we use the fact that $\nabla g(\bm{\lambda}^{*})=\bm{0}$. Combining [\[eq_proof_theorem2-a-4\]](#eq_proof_theorem2-a-4){reference-type="eqref" reference="eq_proof_theorem2-a-4"} and [\[eq_proof_theorem2-a-5\]](#eq_proof_theorem2-a-5){reference-type="eqref" reference="eq_proof_theorem2-a-5"}, we have $$\begin{aligned} \label{eq_proof_theorem2-a-6} \|^{\dagger}\bm{\Theta}^{k+1}-\bm{\Theta}^{*}\|^{2} \le \frac{1}{(u_{f})^{2}}\|\bar{\bm{\lambda}}^{k+1}-\bm{\lambda}^{*}\|^{2}.\end{aligned}$$ Combining [\[eq_proof_theorem2-b-9\]](#eq_proof_theorem2-b-9){reference-type="eqref" reference="eq_proof_theorem2-b-9"} and [\[eq_proof_theorem2-a-6\]](#eq_proof_theorem2-a-6){reference-type="eqref" reference="eq_proof_theorem2-a-6"}, we obtain $$\begin{aligned} \label{eq_proof_theorem2-a-7} &\| ^{\dagger}\bm{\Theta}^{k+1}-\bm{\Theta}^{*}\| \\ \le&\frac{1}{u_{f}}\cdot[\sqrt{\frac{k_{0}-1}{k_{0}+k}}\|\bar{\bm{\lambda}}^{0}-\bm{\lambda}^{*}\|+\sqrt{\frac{1}{(k_{0}+k)(k_{0}-1)}\cdot\frac{288\delta^{2}}{\beta^{4}\epsilon^{3}u_{f}^{2}H^{2}}}\notag\\ &+\sqrt{\frac{k+1}{k_{0}+k}\cdot \frac{192\delta^{2}}{\beta}\cdot(1+\frac{3}{\epsilon^{3}})\cdot(4\rho^{2}H+\chi^{2})}].\notag\end{aligned}$$ Taking $k\to +\infty$, we obtain $$\begin{aligned} \label{eq_proof_theorem2-a-8} &\limsup_{k\to +\infty}\| ^{\dagger}\bm{\Theta}^{k+1}-\bm{\Theta}^{*}\|\\ \le& \frac{1}{u_{f}}\cdot \sqrt{ \frac{192\delta^{2}}{\beta}\cdot(1+\frac{3}{\epsilon^{3}})\cdot(4\rho^{2}H+\chi^{2})}. \notag\end{aligned}$$ According to Assumption [Assumption 1](#a1){reference-type="ref" reference="a1"}, $f_{i}(\bm{\theta}_{i})$ is $u_{f}$-strongly convex. By the conjugate correspondence theorem in [@b-Amir-Beck-2017], the conjugate function $F_{i}^{*}(\bm{\lambda})=\max_{\bm{\theta}_{i}\in C_{i}}\{\bm{\lambda}^{\top}\bm{\theta}_{i}-f_{i}(\bm{\theta}_{i})\}$ is $\frac{1}{u_{f}}$-smooth. In consequence, the gradient $\nabla F^{*}_{i}(-\bm{\lambda})=\arg\min_{\bm{\theta}_{i}\in C_{i}}\{\bm{\lambda }^{\top}\bm{\theta}_{i}+f_{i}(\bm{\theta}_{i})\}$ is $\frac{1}{u_{f}}$-Lipschitz continuous. According to the definition of Lipschitz continuity, we have $$\begin{aligned} \label{eq_proof_theorem2-a-lipschitz-continuous} \|\nabla F^{*}_{i}(\bm{\lambda}_{i}^{k+1})-\nabla F^{*}_{i}(\bar{\bm{\lambda}}^{k+1})\| \le \frac{1}{u_{f}}\|\bm{\lambda}_{i}^{k+1}-\bar{\bm{\lambda}}^{k+1}\|.\end{aligned}$$ Based on $_{}^{\dagger}\bm{\theta}_{i}^{k+1}=\arg\min_{\bm{\theta}_{i}\in C_{i}} \{\bm{\theta}_{i}^{\top}\bar{\bm{\lambda}}^{k+1}+f_{i}(\bm{\theta}_{i})\}$ and $\bm{\theta}_{i}^{k+1}=\arg\min_{\bm{\theta}_{i}\in C_{i}} \{\bm{\theta}_{i}^{\top}\bm{\lambda}_{i}^{k+1}+f_{i}(\bm{\theta}_{i})\}$, we obtain $\bm{\theta}_{i}^{k+1}=\nabla F^{*}_{i}(\bm{\lambda}_{i}^{k+1})$ and $_{}^{\dagger}\bm{\theta}_{i}^{k+1}=\nabla F^{*}_{i}(\bar{\bm{\lambda}}^{k+1})$. Substituting them into [\[eq_proof_theorem2-a-lipschitz-continuous\]](#eq_proof_theorem2-a-lipschitz-continuous){reference-type="eqref" reference="eq_proof_theorem2-a-lipschitz-continuous"}, we have $$\begin{aligned} \label{eq_proof_theorem2-a-10} \|\bm{\theta}_{i}^{k+1}-\ _{}^{\dagger}\bm{\theta}_{i}^{k+1}\| \le \frac{1}{u_{f}}\|\bm{\lambda}_{i}^{k+1}-\bar{\bm{\lambda}}^{k+1}\|.\end{aligned}$$ Combining [\[eq_proof_theorem2-a-10\]](#eq_proof_theorem2-a-10){reference-type="eqref" reference="eq_proof_theorem2-a-10"}, $^{\dagger}\bm{\Theta}^{k+1}:=[_{}^{\dagger}\bm{\theta}_{1}^{k+1}; \cdots; _{}^{\dagger}\bm{\theta}_{H}^{k+1}]$ and $\bm{\Theta}^{k+1}:=[\bm{\theta}_{1}^{k+1}; \cdots; \bm{\theta}_{H}^{k+1}]$, we obtain $$\begin{aligned} \label{eq_proof_theorem2-a-11} \|\bm{\Theta}^{k+1}-\ ^{\dagger}\bm{\Theta}^{k+1}\| \le \frac{1}{u_{f}}\sum_{i\in \mathcal{H}}\|\bm{\lambda}_{i}^{k+1}-\bar{\bm{\lambda}}^{k+1}\|.\end{aligned}$$ Combining [\[eq_proof_theorem2-b-11\]](#eq_proof_theorem2-b-11){reference-type="eqref" reference="eq_proof_theorem2-b-11"} and [\[eq_proof_theorem2-a-11\]](#eq_proof_theorem2-a-11){reference-type="eqref" reference="eq_proof_theorem2-a-11"}, we have $$\begin{aligned} \label{eq_proof_theorem2-a-12} \|\bm{\Theta}^{k+1}-\ ^{\dagger}\bm{\Theta}^{k+1}\| \le \frac{\sqrt{H}}{u_{f}}\cdot \sqrt{\frac{288\delta^{2}H}{(k_{0}+k+1)^{2}\epsilon^{3}\beta^{2}}}.\end{aligned}$$ Taking $k\to +\infty$, we obtain $$\begin{aligned} \label{eq_proof_theorem2-a-13} \lim_{k\to +\infty}\|\bm{\Theta}^{k+1}-\ ^{\dagger}\bm{\Theta}^{k+1}\| =0.\end{aligned}$$ Combining [\[eq_proof_theorem2-a-8\]](#eq_proof_theorem2-a-8){reference-type="eqref" reference="eq_proof_theorem2-a-8"} and [\[eq_proof_theorem2-a-13\]](#eq_proof_theorem2-a-13){reference-type="eqref" reference="eq_proof_theorem2-a-13"} yields $$\begin{aligned} \label{eq_proof_theorem2-a-9} &\limsup_{k\to +\infty}\|\bm{\Theta}^{k+1}-\bm{\Theta}^{*}\|\\ \le &\limsup_{k\to +\infty}\|\bm{\Theta}^{k+1}-\ ^{\dagger}\bm{\Theta}^{k+1}\|+\limsup_{k\to +\infty}\|^{\dagger}\bm{\Theta}^{k+1}-\bm{\Theta}^{*}\|\notag\\ \le& \frac{1}{u_{f}}\cdot \sqrt{ \frac{192\delta^{2}}{\beta}\cdot(1+\frac{3}{\epsilon^{3}})\cdot(4\rho^{2}H+\chi^{2})}. \notag\end{aligned}$$ ## Supporting Lemmas **Lemma 1**. *Under Assumption [Assumption 1](#a1){reference-type="ref" reference="a1"}, for any $\bm{\lambda} \in \mathbb{R}^D$, the maximum distance between the honest agents' local dual gradients and their average, denoted by $\max_{i\in \mathcal{H}}\|\nabla g_{i}(\bm{\lambda}) - \frac{1}{H} \sum_{i\in \mathcal{H}} \nabla g_{i}(\bm{\lambda}) \|^{2}$, is bounded by some positive constant $\delta^2$.* ***Proof**. Recalling the definition of the local dual gradient $\nabla g_{i}(\bm{\lambda})=-\frac{1}{H}\arg\min_{\bm{\theta}_{i}\in C_{i}} \{\bm{\lambda}^{\top}\bm{\theta}_{i}+f_{i}(\bm{\theta}_{i})\}+\frac{1}{H}\bm{s}$, we have $$\begin{aligned} &\max_{i\in \mathcal{H}}\|\nabla g_{i}(\bm{\lambda}) - \frac{1}{H} \sum_{i\in \mathcal{H}} \nabla g_{i}(\bm{\lambda}) \|^{2}\\ =&\max_{i\in \mathcal{H}}\|-\frac{1}{H}\arg\underset{\bm{\theta}_{i}\in C_{i}}{\min}\{\bm{\lambda}^{\top}\bm{\theta}_{i}+f_{i}(\bm{\theta}_{i})\}+\frac{1}{H}\bm{s}\notag\\ &+\frac{1}{H^{2}}\sum_{i\in \mathcal{H}}\arg\underset{\bm{\theta}_{i}\in C_{i}}{\min}\{\bm{\lambda}^{\top}\bm{\theta}_{i}+f_{i}(\bm{\theta}_{i})\}-\frac{1}{H}\bm{s}\|^{2}\notag\end{aligned}$$ $$\begin{aligned} =&\max_{i\in \mathcal{H}}\|\frac{1}{H}(\frac{1}{H}\sum_{i\in \mathcal{H}}\arg\underset{\bm{\theta}_{i}\in C_{i}}{\min}\{\bm{\lambda}^{\top}\bm{\theta}_{i}+f_{i}(\bm{\theta}_{i})\}\notag\\ &-\arg\underset{\bm{\theta}_{i}\in C_{i}}{\min}\{\bm{\lambda}^{\top}\bm{\theta}_{i}+f_{i}(\bm{\theta}_{i})\})\|^{2}.\notag\end{aligned}$$ According to Assumption [Assumption 1](#a1){reference-type="ref" reference="a1"}, the local constraint sets $C_{i}$ are bounded by hypothesis, and we know that $\max_{i\in \mathcal{H}}\|\nabla g_{i}(\bm{\lambda}) - \frac{1}{H} \sum_{i\in \mathcal{H}} \nabla g_{i}(\bm{\lambda}) \|^{2}$ is also bounded by some positive constant, which we denoted as $\delta^{2}$.* **Lemma 2**. *Under Assumption [Assumption 1](#a1){reference-type="ref" reference="a1"}, for any honest agent $i \in \mathcal{H}$, the local dual function $g_{i}(\bm{\lambda})$ is strongly convex with constant $\frac{1}{HL_{f}}$ and smooth with constant $\frac{1}{Hu_{f}}$.* ***Proof**. According to Assumption [Assumption 1](#a1){reference-type="ref" reference="a1"}, $f_{i}(\cdot)$ is $u_{f}$-strongly convex and $L_{f}$-smooth. By the conjugate correspondence theorem [@b-Amir-Beck-2017], its conjugate function $F_{i}^{*}(\bm{\lambda})=\max_{\bm{\theta}_{i}\in C_{i}} \{\bm{\lambda}^{\top}\bm{\theta}_{i}-f_{i}(\bm{\theta}_{i})\}$ is $\frac{1}{L_{f}}$-strongly convex and $\frac{1}{u_{f}}$-smooth. By the definition $g_{i}(\bm{\lambda})=\frac{1}{H}F_{i}^{*}(-\bm{\lambda })+\frac{1}{H}\bm{\lambda }^{\top}\bm{s}$ $(\forall i \in \mathcal{H})$, we know $g_{i}(\bm{\lambda})$ is $\frac{1}{HL_{f}}$-strongly convex and $\frac{1}{Hu_{f}}$-smooth.* **Lemma 3**. *Define a matrix $\Lambda^{k+\frac{1}{2}}=[\cdots,\bm{\lambda}_{i}^{k+\frac{1}{2}},\cdots]\in \mathbb{R}^{H\times D}$ that collects the dual variables $\bm{\lambda}_{i}^{k+\frac{1}{2}}$ of all honest agents $i\in \mathcal{H}$ generated by Algorithm [\[alg2\]](#alg2){reference-type="ref" reference="alg2"}. Under Assumption [Assumption 1](#a1){reference-type="ref" reference="a1"}, we have $$\begin{aligned} \label{lemma3-1} &\|\Lambda^{k+\frac{1}{2}}-\frac{1}{H}\bm{1}\bm{1}^{\top}\Lambda^{k+\frac{1}{2}} \|^{2}_{F}\\ \le&(\frac{1}{1-v}+\frac{6(\gamma^{k})^{2}}{v\cdot u_{f}^{2}J^{2}})\|\Lambda^{k}-\frac{1}{H}\bm{1}\bm{1}^{\top}\Lambda^{k} \|^{2}_{F}+\frac{3(\gamma^{k})^{2}\delta^{2}H^{3}}{v\cdot J^{2}},\notag\end{aligned}$$ where $v$ is any positive constant in $(0,1)$. If $v=\frac{1}{2}$ and the step size $\gamma^{k}\le \frac{u_{f}J}{2\sqrt{3}}$, this further yields $$\begin{aligned} \label{lemma3-2} &\|\Lambda^{k+\frac{1}{2}}-\frac{1}{H}\bm{1}\bm{1}^{\top}\Lambda^{k+\frac{1}{2}} \|^{2}_{F}\\ \le&3\|\Lambda^{k}-\frac{1}{H}\bm{1}\bm{1}^{\top}\Lambda^{k} \|^{2}_{F}+\frac{6(\gamma^{k})^{2}\delta^{2}H^{3}}{ J^{2}}.\notag\end{aligned}$$* ***Proof**. According to the update of $\bm{\lambda}_{i}^{k+\frac{1}{2}}$ in Algorithm [\[alg2\]](#alg2){reference-type="ref" reference="alg2"} and the fact $\|\Lambda^{k+\frac{1}{2}}-\frac{1}{H}\bm{1}\bm{1}^{\top}\Lambda^{k+\frac{1}{2}} \|^{2}_{F}=\sum_{i\in \mathcal{H}}\|\bm{\lambda}_{i}^{k+\frac{1}{2}}-\bar{\bm{\lambda}}^{k+\frac{1}{2}}\|^{2}$, we have $$\begin{aligned} \label{eq-proof-lemma3-1-1} &\|\Lambda^{k+\frac{1}{2}}-\frac{1}{H}\bm{1}\bm{1}^{\top}\Lambda^{k+\frac{1}{2}} \|^{2}_{F}=\sum_{i\in \mathcal{H}}\|\bm{\lambda}_{i}^{k+\frac{1}{2}}-\bar{\bm{\lambda}}^{k+\frac{1}{2}}\|^{2} \\ =&\sum_{i\in \mathcal{H}}\|\bm{\lambda}_{i}^{k}-\gamma^{k}\cdot \frac{H}{J}\nabla g_{i}(\bm{\lambda}_{i}^{k})-\bar{\bm{\lambda}}^{k}+\gamma^{k}\cdot\frac{1}{J}\sum_{i\in \mathcal{H}}\nabla g_{i}(\bm{\lambda}_{i}^{k})\|^{2}\notag\\ =& \sum_{i\in \mathcal{H}}\|\bm{\lambda}_{i}^{k}-\bar{\bm{\lambda}}^{k}-\gamma^{k}\cdot\frac{H}{J}(\nabla g_{i}(\bm{\lambda}_{i}^{k})-\frac{1}{H}\sum_{i\in \mathcal{H}}\nabla g_{i}(\bm{\lambda}_{i}^{k}))\|^{2}\notag\\ \le&\frac{(\gamma^{k})^{2}}{v}\cdot\frac{H^{2}}{J^{2}}\sum_{i\in \mathcal{H}}\|\nabla g_{i}(\bm{\lambda}_{i}^{k})-\frac{1}{H}\sum_{i\in \mathcal{H}}\nabla g_{i}(\bm{\lambda}_{i}^{k}))\|^{2} \notag\\ &+\frac{1}{1-v}\sum_{i\in \mathcal{H}}\|\bm{\lambda}_{i}^{k}-\bar{\bm{\lambda}}^{k}\|^{2}, \notag\end{aligned}$$ where $v\in (0,1)$ is any positive constant. To drive the last inequality, we use the fact that $\|\bm{a}+\bm{b}\|^{2}\le \frac{1}{v}\|\bm{a}\|^{2}+\frac{1}{1-v}\|\bm{b}\|^{2}$ for any positive constant $v\in (0,1)$.* *For $\frac{(\gamma^{k})^{2}}{v}\cdot\frac{H^{2}}{J^{2}}\sum_{i\in \mathcal{H}}\|\nabla g_{i}(\bm{\lambda}_{i}^{k})-\frac{1}{H}\sum_{i\in \mathcal{H}}\nabla g_{i}(\bm{\lambda}_{i}^{k}))\|^{2}$, the first term at the right-hand side of [\[eq-proof-lemma3-1-1\]](#eq-proof-lemma3-1-1){reference-type="eqref" reference="eq-proof-lemma3-1-1"}, we have $$\begin{aligned} \label{eq-proof-lemma3-1-2} &\frac{(\gamma^{k})^{2}}{v}\cdot\frac{H^{2}}{J^{2}}\sum_{i\in \mathcal{H}}\|\nabla g_{i}(\bm{\lambda}_{i}^{k})-\frac{1}{H}\sum_{i\in \mathcal{H}}\nabla g_{i}(\bm{\lambda}_{i}^{k}))\|^{2}\end{aligned}$$ $$\begin{aligned} =&\frac{(\gamma^{k})^{2}}{v}\cdot\frac{H^{2}}{J^{2}}\sum_{i\in \mathcal{H}}\|\nabla g_{i}(\bm{\lambda}_{i}^{k})-\nabla g_{i}(\bar{\bm{\lambda}}^{k})+\nabla g_{i}(\bar{\bm{\lambda}}^{k})\notag\\ &-\frac{1}{H}\sum_{i\in \mathcal{H}}\nabla g_{i}(\bar{\bm{\lambda}}^{k}))+\frac{1}{H}\sum_{i\in \mathcal{H}}\nabla g_{i}(\bar{\bm{\lambda}}^{k}))-\frac{1}{H}\sum_{i\in \mathcal{H}}\nabla g_{i}(\bm{\lambda}_{i}^{k}))\|^{2} \notag\\ \le&\frac{3(\gamma^{k})^{2}}{v}\cdot\frac{H^{2}}{J^{2}}\sum_{i\in \mathcal{H}}\|\nabla g_{i}(\bm{\lambda}_{i}^{k})-\nabla g_{i}(\bar{\bm{\lambda}}^{k})\|^{2}\notag\\ &+\frac{3(\gamma^{k})^{2}}{v}\cdot\frac{H^{2}}{J^{2}}\sum_{i\in \mathcal{H}}\|\nabla g_{i}(\bar{\bm{\lambda}}^{k})-\frac{1}{H}\sum_{i\in \mathcal{H}}\nabla g_{i}(\bar{\bm{\lambda}}^{k}))\|^{2} \notag\\ &+\frac{3(\gamma^{k})^{2}}{v}\cdot\frac{H^{2}}{J^{2}}\sum_{i\in \mathcal{H}}\|\frac{1}{H}\sum_{i\in \mathcal{H}}\nabla g_{i}(\bar{\bm{\lambda}}^{k}))-\frac{1}{H}\sum_{i\in \mathcal{H}}\nabla g_{i}(\bm{\lambda}_{i}^{k}))\|^{2}.\notag\end{aligned}$$ According to Lemmas [Lemma 1](#l1){reference-type="ref" reference="l1"} and [Lemma 2](#l2){reference-type="ref" reference="l2"}, we further have $$\begin{aligned} \label{eq-proof-lemma3-1-3} &\frac{(\gamma^{k})^{2}}{v}\cdot\frac{H^{2}}{J^{2}}\sum_{i\in \mathcal{H}}\|\nabla g_{i}(\bm{\lambda}_{i}^{k})-\frac{1}{H}\sum_{i\in \mathcal{H}}\nabla g_{i}(\bm{\lambda}_{i}^{k}))\|^{2}\\ \le& \frac{6(\gamma^{k})^{2}}{v\cdot u_{f}^{2}J^{2}}\sum_{i\in\mathcal{H}}\|\bm{\lambda}_{i}^{k}-\bar{\bm{\lambda}}^{k}\|^{2}+\frac{3(\gamma^{k})^{2}\delta^{2}H^{3}}{v\cdot J^{2}}.\notag\end{aligned}$$* *Substituting [\[eq-proof-lemma3-1-3\]](#eq-proof-lemma3-1-3){reference-type="eqref" reference="eq-proof-lemma3-1-3"} into [\[eq-proof-lemma3-1-1\]](#eq-proof-lemma3-1-1){reference-type="eqref" reference="eq-proof-lemma3-1-1"} and then rearranging the terms, we obtain $$\begin{aligned} \label{eq-proof-lemma3-1} &\|\Lambda^{k+\frac{1}{2}}-\frac{1}{H}\bm{1}\bm{1}^{\top}\Lambda^{k+\frac{1}{2}} \|^{2}_{F}=\sum_{i\in \mathcal{H}}\|\bm{\lambda}_{i}^{k+\frac{1}{2}}-\bar{\bm{\lambda}}^{k+\frac{1}{2}}\|^{2}\\ \le& (\frac{1}{1-v}+\frac{6(\gamma^{k})^{2}}{v \cdot u_{f}^{2}J^{2}})\sum_{i\in \mathcal{H}}\|\bm{\lambda}_{i}^{k}-\bar{\bm{\lambda}}^{k}\|^{2}+\frac{3(\gamma^{k})^{2}\delta^{2}H^{3}}{v\cdot J^{2}}\notag\\ =&(\frac{1}{1-v}+\frac{6(\gamma^{k})^{2}}{v \cdot u_{f}^{2}J^{2}})\|\Lambda^{k}-\frac{1}{H}\bm{1}\bm{1}^{\top}\Lambda^{k} \|^{2}_{F}+\frac{3(\gamma^{k})^{2}\delta^{2}H^{3}}{v\cdot J^{2}}.\notag\end{aligned}$$* **Lemma 4**. *Define a matrix $\Lambda^{k+1}=[\cdots,\bm{\lambda}_{i}^{k+1},\cdots]\in \mathbb{R}^{H\times D}$ that collects the dual variables $\bm{\lambda}_{i}^{k+1}$ of all honest agents $i\in \mathcal{H}$ generated by Algorithm [\[alg2\]](#alg2){reference-type="ref" reference="alg2"}. Suppose that the robust aggregation rules $AGG_i$ satisfy [\[eq_definition1\]](#eq_definition1){reference-type="eqref" reference="eq_definition1"} in Definition [Definition 1](#d1){reference-type="ref" reference="d1"}. Under Assumptions [Assumption 1](#a1){reference-type="ref" reference="a1"} and [Assumption 3](#a3){reference-type="ref" reference="a3"}, if the contraction constant $\rho$ satisfies $\rho<\frac{1-\kappa}{8\sqrt{H}}$, we have $$\begin{aligned} \label{lemma4} \|\Lambda^{k+1}-\frac{1}{H}\bm{1}\bm{1}^{\top}\Lambda^{k+1} \|^{2}_{F}\le \frac{18(\gamma^{k+1})^{2}\delta^{2}H^{3}}{\epsilon^{3} \cdot J^{2}},\end{aligned}$$ where $\epsilon:=1-\kappa-8\rho\sqrt{H}$.* ***Proof**. For any positive constant $w\in (0,1)$, we have $$\begin{aligned} \label{eq-proof-lemma4-1} &\|\Lambda^{k+1}-\frac{1}{H}\bm{1}\bm{1}^{\top}\Lambda^{k+1} \|^{2}_{F} \\ =& \|\Lambda^{k+1}-\frac{1}{H}\bm{1}\bm{1}^{\top}\Lambda^{k+1}+E\Lambda^{k+\frac{1}{2}}-E\Lambda^{k+\frac{1}{2}}\notag\\ &+\frac{1}{H}\bm{1}\bm{1}^{\top}E\Lambda^{k+\frac{1}{2}}-\frac{1}{H}\bm{1}\bm{1}^{\top}E\Lambda^{k+\frac{1}{2}}\|^{2}_{F} \notag\\ \le& \underbrace{\frac{1}{1-w}\|E\Lambda^{k+\frac{1}{2}}-\frac{1}{H}\bm{1}\bm{1}^{\top}E\Lambda^{k+\frac{1}{2}}\|^{2}_{F}}_{T_{7}}+\underbrace{\frac{2}{w}\|\Lambda^{k+1}-E\Lambda^{k+\frac{1}{2}}\|^{2}_{F}}_{T_{8}}\notag\\ &+\underbrace{\frac{2}{w}\|\frac{1}{H}\bm{1}\bm{1}^{\top}\Lambda^{k+1}-\frac{1}{H}\bm{1}\bm{1}^{\top}E\Lambda^{k+\frac{1}{2}}\|^{2}_{F}}_{T_{9}}. \notag\end{aligned}$$ Next, we analyze $T_{7}$, $T_{8}$ and $T_{9}$ in turn.* ***Bounding $T_{7}$:** According to Assumption [Assumption 3](#a3){reference-type="ref" reference="a3"}, we have $$\begin{aligned} \label{eq-proof-lemma4-2} T_{7}=&\frac{1}{1-w}\|E\Lambda^{k+\frac{1}{2}}-\frac{1}{H}\bm{1}\bm{1}^{\top}E\Lambda^{k+\frac{1}{2}}\|^{2}_{F}\\ =&\frac{1}{1-w}\|(I-\frac{1}{H}\bm{1}\bm{1}^{\top})E\Lambda^{k+\frac{1}{2}}\|^{2}_{F} \notag\\ =&\frac{1}{1-w}\|(I-\frac{1}{H}\bm{1}\bm{1}^{\top})E(I-\frac{1}{H}\bm{1}\bm{1}^{\top})\Lambda^{k+\frac{1}{2}}\|^{2}_{F}\notag\\ \le&\frac{1}{1-w}\|(I-\frac{1}{H}\bm{1}\bm{1}^{\top})E\|^{2}\|(I-\frac{1}{H}\bm{1}\bm{1}^{\top})\Lambda^{k+\frac{1}{2}}\|^{2}_{F} \notag\\ =&\frac{\kappa}{1-w}\|\Lambda^{k+\frac{1}{2}}-\frac{1}{H}\bm{1}\bm{1}^{\top}\Lambda^{k+\frac{1}{2}} \|^{2}_{F}, \notag\end{aligned}$$ where the last inequality holds because of Assumption [Assumption 3](#a3){reference-type="ref" reference="a3"} and the fact that $\|AB\|_{F}^{2}\le \|A\|^{2}\|B\|_{F}^{2}$.* ***Bounding $T_{8}$:** According to the update of $\bm{\lambda}_{i}^{k+1}$ in Algorithm [\[alg2\]](#alg2){reference-type="ref" reference="alg2"} and [\[eq_definition1\]](#eq_definition1){reference-type="eqref" reference="eq_definition1"} in Definition [Definition 1](#d1){reference-type="ref" reference="d1"}, we have $$\begin{aligned} \label{eq-proof-lemma4-3} T_{8}=&\frac{2}{w}\|\Lambda^{k+1}-E\Lambda^{k+\frac{1}{2}}\|^{2}_{F}\\ =&\frac{2}{w}\sum_{i\in\mathcal{H} }\|\bm{\lambda}_{i}^{k+1}-\bar{\bm{\lambda}}_{i}^{k+\frac{1}{2}}\|^{2} \notag\\ =&\frac{2}{w}\sum_{i\in\mathcal{H} }\|AGG_{i}(\bm{\lambda}_{i}^{k+\frac{1}{2}},\{\check{\bm{\lambda}}_{j}^{k+\frac{1}{2}}\}_{j\in \mathcal{N}_{i}})-\bar{\bm{\lambda}}_{i}^{k+\frac{1}{2}}\|^{2} \notag\\ \le&\frac{2}{w}\sum_{i\in \mathcal{H}}\rho^{2}\max_{j\in \mathcal{N}_{i}\cap\mathcal{H}\cup i}\|\bm{\lambda}_{j}^{k+\frac{1}{2}}-\bar{\bm{\lambda}}_{i}^{k+\frac{1}{2}}\|^{2} \notag\\ =& \frac{2\rho^{2}}{w}\sum_{i\in \mathcal{H}}\max_{j\in \mathcal{N}_{i}\cap\mathcal{H}\cup i}\|\bm{\lambda}_{j}^{k+\frac{1}{2}}-\bar{\bm{\lambda}}^{k+\frac{1}{2}}+\bar{\bm{\lambda}}^{k+\frac{1}{2}}-\bar{\bm{\lambda}}_{i}^{k+\frac{1}{2}}\|^{2} \notag\\ \le& \frac{4\rho^{2}}{w}\sum_{i\in \mathcal{H}}[\max_{j\in \mathcal{N}_{i}\cap\mathcal{H}\cup i}\|\bm{\lambda}_{j}^{k+\frac{1}{2}}-\bar{\bm{\lambda}}^{k+\frac{1}{2}}\|^{2}+\|\bar{\bm{\lambda}}^{k+\frac{1}{2}}-\bar{\bm{\lambda}}_{i}^{k+\frac{1}{2}}\|^{2}]\notag\\ \le&\frac{4\rho^{2}}{w}\sum_{i\in \mathcal{H}}[\max_{i\in \mathcal{H}}\|\bm{\lambda}_{i}^{k+\frac{1}{2}}-\bar{\bm{\lambda}}^{k+\frac{1}{2}}\|^{2}+\max_{i\in \mathcal{H}}\|\bar{\bm{\lambda}}^{k+\frac{1}{2}}-\bar{\bm{\lambda}}_{i}^{k+\frac{1}{2}}\|^{2}]\notag\\ =&\frac{8\rho^{2}H}{w}\max_{i\in \mathcal{H}}\|\bm{\lambda}_{i}^{k+\frac{1}{2}}-\bar{\bm{\lambda}}^{k+\frac{1}{2}}\|^{2}\notag\\ \le&\frac{8\rho^{2}H}{w} \|\Lambda^{k+\frac{1}{2}}-\frac{1}{H}\bm{1}\bm{1}^{\top}\Lambda^{k+\frac{1}{2}} \|^{2}_{F},\notag\end{aligned}$$ where the last inequality holds as $\max_{i\in \mathcal{H}}\|\bm{\lambda}_{i}^{k+\frac{1}{2}}-\bar{\bm{\lambda}}^{k+\frac{1}{2}}\|^{2}\le \|\Lambda^{k+\frac{1}{2}}-\frac{1}{H}\bm{1}\bm{1}^{\top}\Lambda^{k+\frac{1}{2}} \|^{2}_{F}$.* ***Bounding $T_{9}$:** Likewise, according to the update of $\bm{\lambda}_{i}^{k+1}$ in Algorithm [\[alg2\]](#alg2){reference-type="ref" reference="alg2"} and [\[eq_definition1\]](#eq_definition1){reference-type="eqref" reference="eq_definition1"} in Definition [Definition 1](#d1){reference-type="ref" reference="d1"}, we have $$\begin{aligned} \label{eq-proof-lemma4-4} T_{9}=&\frac{2}{w}\|\frac{1}{H}\bm{1}\bm{1}^{\top}\Lambda^{k+1}-\frac{1}{H}\bm{1}\bm{1}^{\top}E\Lambda^{k+\frac{1}{2}}\|^{2}_{F}\\ =&\frac{2}{w}\|\frac{1}{H}\bm{1}\bm{1}^{\top}(\Lambda^{k+1}-E\Lambda^{k+\frac{1}{2}})\|^{2}_{F} \notag\\ \le & \frac{2}{w} \|\frac{1}{H}\bm{1}\bm{1}^{\top}\|^{2}_{F}\|\Lambda^{k+1}-E\Lambda^{k+\frac{1}{2}}\|^{2}_{F}\notag\\ =&\frac{2}{w}\|\Lambda^{k+1}-E\Lambda^{k+\frac{1}{2}}\|^{2}_{F}\notag\\ \le &\frac{8\rho^{2}H}{w}\max_{i\in \mathcal{H}}\|\bm{\lambda}_{i}^{k+\frac{1}{2}}-\bar{\bm{\lambda}}^{k+\frac{1}{2}}\|^{2}\notag\\ \le&\frac{8\rho^{2}H}{w} \|\Lambda^{k+\frac{1}{2}}-\frac{1}{H}\bm{1}\bm{1}^{\top}\Lambda^{k+\frac{1}{2}} \|^{2}_{F}.\notag\end{aligned}$$ To drive the last equality, we use the fact $\|\frac{1}{H}\bm{1}\bm{1}^{\top}\|^{2}_{F}=1$. From the last equality to the last inequality, we use the same technique in deriving [\[eq-proof-lemma4-3\]](#eq-proof-lemma4-3){reference-type="eqref" reference="eq-proof-lemma4-3"}.* *Therefore, substituting [\[eq-proof-lemma4-2\]](#eq-proof-lemma4-2){reference-type="eqref" reference="eq-proof-lemma4-2"}, [\[eq-proof-lemma4-3\]](#eq-proof-lemma4-3){reference-type="eqref" reference="eq-proof-lemma4-3"} and [\[eq-proof-lemma4-4\]](#eq-proof-lemma4-4){reference-type="eqref" reference="eq-proof-lemma4-4"} into [\[eq-proof-lemma4-1\]](#eq-proof-lemma4-1){reference-type="eqref" reference="eq-proof-lemma4-1"} and rearranging the terms, we obtain $$\begin{aligned} \label{eq-proof-lemma4-5} &\|\Lambda^{k+1}-\frac{1}{H}\bm{1}\bm{1}^{\top}\Lambda^{k+1} \|^{2}_{F}\\ \le&(\frac{\kappa}{1-w}+\frac{16\rho^{2}H}{w})\|\Lambda^{k+\frac{1}{2}}-\frac{1}{H}\bm{1}\bm{1}^{\top}\Lambda^{k+\frac{1}{2}} \|^{2}_{F}. \notag\end{aligned}$$ Substituting [\[lemma3-1\]](#lemma3-1){reference-type="eqref" reference="lemma3-1"} in Lemma [Lemma 3](#l3){reference-type="ref" reference="l3"} into [\[eq-proof-lemma4-5\]](#eq-proof-lemma4-5){reference-type="eqref" reference="eq-proof-lemma4-5"} and rearranging the terms, we obtain $$\begin{aligned} \label{eq-proof-lemma4-6} &\|\Lambda^{k+1}-\frac{1}{H}\bm{1}\bm{1}^{\top}\Lambda^{k+1} \|^{2}_{F}\\ \le&(\frac{\kappa}{1-w}+\frac{16\rho^{2}H}{w})(\frac{1}{1-v}+\frac{6(\gamma^{k})^{2}}{v\cdot u_{f}^{2}J^{2}}) \|\Lambda^{k}-\frac{1}{H}\bm{1}\bm{1}^{\top}\Lambda^{k} \|^{2}_{F}\notag\\ &+(\frac{\kappa}{1-w}+\frac{16\rho^{2}H}{w})\cdot\frac{3(\gamma^{k})^{2}\delta^{2}H^{3}}{v\cdot J^{2}}\notag.\end{aligned}$$* *By setting the constant $w=4\rho\sqrt{H}\le 1-\kappa$, $\frac{\kappa}{1-w}\le \kappa+w$ holds. Therefore, we can rewrite [\[eq-proof-lemma4-6\]](#eq-proof-lemma4-6){reference-type="eqref" reference="eq-proof-lemma4-6"} as $$\begin{aligned} \label{eq-proof-lemma4-7} &\|\Lambda^{k+1}-\frac{1}{H}\bm{1}\bm{1}^{\top}\Lambda^{k+1} \|^{2}_{F}\\ \le& (\kappa+8\rho\sqrt{H})(\frac{1}{1-v}+\frac{6(\gamma^{k})^{2}}{v\cdot u_{f}^{2}J^{2} })\|\Lambda^{k}-\frac{1}{H}\bm{1}\bm{1}^{\top}\Lambda^{k} \|^{2}_{F}\notag\\ &+(\kappa+8\rho\sqrt{H})\cdot\frac{3(\gamma^{k})^{2}\delta^{2}H^{3}}{v\cdot J^{2}}\notag\\ =&(1-\epsilon)(\frac{1}{1-v}+\frac{6(\gamma^{k})^{2}}{v\cdot u_{f}^{2}J^{2}})\|\Lambda^{k}-\frac{1}{H}\bm{1}\bm{1}^{\top}\Lambda^{k} \|^{2}_{F}\notag\\ &+(1-\epsilon)\cdot\frac{3(\gamma^{k})^{2}\delta^{2}H^{3}}{v\cdot J^{2}},\notag\end{aligned}$$ where $\epsilon:=1-\kappa-8\rho\sqrt{H}$. The parameter $\rho$ should satisfy $\rho<\frac{1-\kappa}{8\sqrt{H}}$ to guarantee $\epsilon>0$.* *Set $v=\frac{\epsilon}{3}$ and a proper step size $\gamma^{k}$ satisfying $\frac{6(\gamma^{k})^{2}}{v\cdot u_{f}^{2}J^{2}}\le \frac{(2-\epsilon)\epsilon^{2}}{3(3-\epsilon)}=\frac{(\epsilon-v-v\epsilon)v}{1-v}$. Therefore, we have $\frac{1}{1-v}+\frac{6(\gamma^{k})^{2}}{v\cdot u_{f}^{2}J^{2}}\le 1+\epsilon$. In consequence, [\[eq-proof-lemma4-7\]](#eq-proof-lemma4-7){reference-type="eqref" reference="eq-proof-lemma4-7"} can be rewritten as $$\begin{aligned} \label{eq-proof-lemma4-8} &\|\Lambda^{k+1}-\frac{1}{H}\bm{1}\bm{1}^{\top}\Lambda^{k+1} \|^{2}_{F}\\ \le& (1-\epsilon^{2})\|\Lambda^{k}-\frac{1}{H}\bm{1}\bm{1}^{\top}\Lambda^{k} \|^{2}_{F}+\frac{9(\gamma^{k})^{2}\delta^{2}H^{3}}{\epsilon \cdot J^{2}}.\notag\end{aligned}$$ Under the conditions $\rho<\frac{1-\kappa}{8\sqrt{H}}$ and $\epsilon\in(0,1)$, we write [\[eq-proof-lemma4-8\]](#eq-proof-lemma4-8){reference-type="eqref" reference="eq-proof-lemma4-8"} recursively to yield $$\begin{aligned} \label{eq-proof-lemma4-9} &\|\Lambda^{k+1}-\frac{1}{H}\bm{1}\bm{1}^{\top}\Lambda^{k+1} \|^{2}_{F}\\ \le& (1-\epsilon^{2})^{k+1}\|\Lambda^{0}-\frac{1}{H}\bm{1}\bm{1}^{\top}\Lambda^{0} \|^{2}_{F}\notag\\ &+\sum_{k'=0}^{k}(1-\epsilon^{2})^{k-k'}\cdot(\gamma^{k'})^{2}\cdot\frac{9\delta^{2}H^{3}}{\epsilon \cdot J^{2}}.\notag\end{aligned}$$* *With the same initialization $\bm{\lambda}_{i}^{0}$ for all honest agents $i\in \mathcal{H}$, we can rewrite [\[eq-proof-lemma4-9\]](#eq-proof-lemma4-9){reference-type="eqref" reference="eq-proof-lemma4-9"} as $$\begin{aligned} \label{eq-proof-lemma4-10} &\|\Lambda^{k+1}-\frac{1}{H}\bm{1}\bm{1}^{\top}\Lambda^{k+1} \|^{2}_{F}\\ \le& \sum_{k'=0}^{k}(1-\epsilon^{2})^{k-k'}\cdot(\gamma^{k'})^{2}\cdot\frac{9\delta^{2}H^{3}}{\epsilon \cdot J^{2}}.\notag\end{aligned}$$ To bound $\sum_{k'=0}^{k}(1-\epsilon^{2})^{k-k'}\cdot (\gamma^{k'})^{2}$ in [\[eq-proof-lemma4-10\]](#eq-proof-lemma4-10){reference-type="eqref" reference="eq-proof-lemma4-10"}, we define $y^{k}$ as $$\begin{aligned} y^{k}=\sum_{k'=0}^{k-1}(1-\epsilon^{2})^{k-1-k'}\cdot (\gamma^{k'})^{2},\end{aligned}$$ which satisfies the relation $y^{k+1}=(1-\epsilon^{2})y^{k}+(\gamma^{k})^{2}$. Substituting $\psi_{1}=1-\epsilon^{2}\in (0,1)$, $\psi_{2}=1\ge 0$ and $y^{0}=0\le (\gamma^{0})^{2}$ to Lemma [Lemma 5](#l4-appendix1){reference-type="ref" reference="l4-appendix1"}, for integer $k\geq0$ and the step size $\gamma^{k}$ satisfying $1\le \frac{(\gamma^{k})^{2}}{(\gamma^{k+1)^{2}}}\le \frac{2}{1+(1-\epsilon^{2})}$, we have $$\begin{aligned} \label{eq-proof-lemma4-11} y^{k+1}&=\sum_{k'=0}^{k}(1-\epsilon^{2})^{k-k'}\cdot (\gamma^{k'})^{2}\\ &\le \frac{2}{1-(1-\epsilon^{2})}(\gamma^{k+1})^{2}= \frac{2}{\epsilon^{2}}(\gamma^{k+1})^{2}.\notag\end{aligned}$$ With [\[eq-proof-lemma4-11\]](#eq-proof-lemma4-11){reference-type="eqref" reference="eq-proof-lemma4-11"}, we can rewrite [\[eq-proof-lemma4-10\]](#eq-proof-lemma4-10){reference-type="eqref" reference="eq-proof-lemma4-10"} as $$\begin{aligned} \label{eq-proof-lemma3-11} \|\Lambda^{k+1}-\frac{1}{H}\bm{1}\bm{1}^{\top}\Lambda^{k+1} \|^{2}_{F}\le \frac{18(\gamma^{k+1})^{2}\delta^{2}H^{3}}{\epsilon^{3} \cdot J^{2}},\end{aligned}$$ which completes the proof.* **Lemma 5**. *Suppose that for any integer $k\ge 0$, a sequence $\{\gamma^{k}\}$ satisfies $$\begin{aligned} \label{l4-appendix1-1} 1\le \frac{(\gamma^{k})^{2}}{(\gamma^{k+1})^{2}}\le \frac{2}{1+\psi_{1}}\end{aligned}$$ for some $\psi_{1}\in(0,1)$, and another sequence $\{y^{k}\}$ satisfies $$\begin{aligned} \label{l4-appendix1-2} y^{k+1}\le \psi_{1}y^{k}+\psi_{2}(\gamma^{k})^{2} \quad \mbox{and} \quad y^{0}\le \psi_{2}(\gamma^{0})^{2}\end{aligned}$$ for some $\psi_{1}\in(0,1)$ and $\psi_{2}\ge 0$. Then, $y^{k}$ is upper-bounded by $$\begin{aligned} y^{k}\le \frac{2\psi_{2}}{1-\psi_{1}}(\gamma^{k})^{2}. \label{l4-appendix1-3}\end{aligned}$$* ***Proof**. With the conditions $1\le \frac{2}{1+\psi_{1}}$ and $y^{0}\le \psi_{2}(\gamma^{0})^{2}$, we have $y^{0}\le \frac{2\psi_{2}}{1-\psi_{1}}(\gamma^{0})^{2}$. Therefore, when $k=0$, the proposition $y^{k}\le \frac{2\psi_{2}}{1-\psi_{1}}(\gamma^{k})^{2}$ holds.* *Now we prove the conclusion by mathematical induction. Suppose that when $k=k'$, the proposition $y^{k'}\le \frac{2\psi_{2}}{1-\psi_{1}}(\gamma^{k'})^{2}$ holds. We analyze when $k=k'+1$, whether $y^{k'+1}\le \frac{2\psi_{2}}{1-\psi_{1}}(\gamma^{k'+1})^{2}$ holds. Combining $y^{k'+1}\le \psi_{1}y^{k'}+\psi_{2}(\gamma^{k'})^{2}$ and $y^{k'}\le \frac{2\psi_{2}}{1-\psi_{1}}(\gamma^{k'})^{2}$, we obtain $y^{k'+1}\le \frac{(1+\psi_{1})\psi_{2}}{1-\psi_{1}}(\gamma^{k'})^{2}$. Since the step size $\gamma^{k'}$ satisfies $\frac{(\gamma^{k'})^{2}}{(\gamma^{k'+1})^{2}}\le \frac{2}{1+\psi_{1}}$, we have $(\gamma^{k'})^{2}\le \frac{2}{1+\psi_{1}}\cdot(\gamma^{k'+1})^{2}$ and conclude that $y^{k'+1}\le \frac{(1+\psi_{1})\psi_{2}}{1-\psi_{1}}(\gamma^{k'})^{2}\le \frac{2\psi_{2}}{1-\psi_{1}}(\gamma^{k'+1})^{2}$. Hence, when $k=k'+1$, $y^{k'+1}\le \frac{2\psi_{2}}{1-\psi_{1}}(\gamma^{k'+1})^{2}$ holds. This completes the proof.* # Proof of Theorem [Theorem 1](#t1){reference-type="ref" reference="t1"} {#appendix1} ## Part $a$ of Theorem [Theorem 1](#t1){reference-type="ref" reference="t1"} {#part-a-of-theorem-t1} According to the update of $\bm{\lambda}_{i}^{k+1}$ in Algorithm [\[alg1\]](#alg1){reference-type="ref" reference="alg1"}, we have $$\begin{aligned} \label{eq_proof_theorem1-b-1} \|\bar{\bm{\lambda}}^{k+1}-\widetilde{\bm{\lambda}}^{*}\|^{2}=\|\frac{1}{J}\sum_{i\in \mathcal{J}}\sum_{j\in \mathcal{N}_{i}\cup \{i\}}\widetilde{e}_{ij}\bm{\lambda}_{j}^{k+\frac{1}{2}}-\widetilde{\bm{\lambda}}^{*}\|^{2}.\end{aligned}$$ According to Assumption [Assumption 3](#a3){reference-type="ref" reference="a3"}, $\widetilde{E}$ is doubly stochastic. Therefore, we have $\sum_{i\in \mathcal{J}}\sum_{j\in \mathcal{N}_{i}\cup \{i\}}\widetilde{e}_{ij}\bm{\lambda}_{j}^{k+\frac{1}{2}}=\sum_{i\in \mathcal{J}}\bm{\lambda}_{i}^{k+\frac{1}{2}}$. Combining the update of $\bm{\lambda}_{i}^{k+\frac{1}{2}}$ in Algorithm [\[alg1\]](#alg1){reference-type="ref" reference="alg1"} and the fact $\sum_{i\in \mathcal{J}}\sum_{j\in \mathcal{N}_{i}\cup \{i\}}\widetilde{e}_{ij}\bm{\lambda}_{j}^{k+\frac{1}{2}}=\sum_{i\in \mathcal{J}}\bm{\lambda}_{i}^{k+\frac{1}{2}}$, we can rewrite [\[eq_proof_theorem1-b-1\]](#eq_proof_theorem1-b-1){reference-type="eqref" reference="eq_proof_theorem1-b-1"} as $$\begin{aligned} \label{eq_proof_theorem1-b-2} &\|\bar{\bm{\lambda}}^{k+1}-\widetilde{\bm{\lambda}}^{*}\|^{2}\\ =& \|\frac{1}{J}\sum_{i\in \mathcal{J}}[\bm{\lambda}_{i}^{k}-\gamma^{k}\nabla \widetilde{g}_{i}(\bm{\lambda}_{i}^{k})]-\widetilde{\bm{\lambda}}^{*}\|^{2}\notag\\ =&\|\bar{\bm{\lambda}}^{k}-\widetilde{\bm{\lambda}}^{*}-\frac{\gamma^{k}}{J}\sum_{i\in \mathcal{J}}\nabla \widetilde{g}_{i}(\bar{\bm{\lambda}}^{k})\notag\\ &+\frac{\gamma^{k}}{J}\sum_{i\in \mathcal{J}}\nabla \widetilde{g}_{i}(\bar{\bm{\lambda}}^{k})-\frac{\gamma^{k}}{J}\sum_{i\in \mathcal{J}}\nabla \widetilde{g}_{i}(\bm{\lambda}_{i}^{k})\|^{2} \notag\\ =&\|\bar{\bm{\lambda}}^{k}-\widetilde{\bm{\lambda}}^{*}-\frac{\gamma^{k}}{J}\sum_{i\in \mathcal{J}}\nabla \widetilde{g}_{i}(\bar{\bm{\lambda}}^{k})\|^{2}\notag\\ &+\frac{(\gamma^{k})^{2}}{J^{2}}\|\sum_{i\in \mathcal{J}}(\nabla \widetilde{g}_{i}(\bar{\bm{\lambda}}^{k})-\nabla \widetilde{g}_{i}(\bm{\lambda}_{i}^{k}))\|^{2}+2\gamma^{k}\cdot\notag\\ &\left \langle \bar{\bm{\lambda}}^{k}-\widetilde{\bm{\lambda}}^{*}-\frac{\gamma^{k}}{J}\sum_{i\in \mathcal{J}}\nabla \widetilde{g}_{i}(\bar{\bm{\lambda}}^{k}), \frac{1}{J}\sum_{i\in \mathcal{J}}(\nabla \widetilde{g}_{i}(\bar{\bm{\lambda}}^{k})-\nabla \widetilde{g}_{i}(\bm{\lambda}_{i}^{k}))\right \rangle\notag\\ \le&\|\bar{\bm{\lambda}}^{k}-\widetilde{\bm{\lambda}}^{*}-\frac{\gamma^{k}}{J}\sum_{i\in \mathcal{J}}\nabla \widetilde{g}_{i}(\bar{\bm{\lambda}}^{k})\|^{2}\notag\\ &+\frac{(\gamma^{k})^{2}}{J^{2}}\|\sum_{i\in \mathcal{J}}(\nabla \widetilde{g}_{i}(\bar{\bm{\lambda}}^{k})-\nabla \widetilde{g}_{i}(\bm{\lambda}_{i}^{k}))\|^{2}\notag\\ &+ \frac{v^{-1}\gamma^{k}}{J^{2}}\|\sum_{i\in \mathcal{J}}(\nabla \widetilde{g}_{i}(\bar{\bm{\lambda}}^{k})-\nabla \widetilde{g}_{i}(\bm{\lambda}_{i}^{k}))\|^{2}\notag\\ &+v\gamma^{k}\|\bar{\bm{\lambda}}^{k}-\widetilde{\bm{\lambda}}^{*}-\frac{\gamma^{k}}{J}\sum_{i\in \mathcal{J}}\nabla \widetilde{g}_{i}(\bar{\bm{\lambda}}^{k})\|^{2}\notag\\ \le&\underbrace{(1+v\gamma^{k})\|\bar{\bm{\lambda}}^{k}-\widetilde{\bm{\lambda}}^{*}-\frac{\gamma^{k}}{J}\sum_{i\in \mathcal{J}}\nabla \widetilde{g}_{i}(\bar{\bm{\lambda}}^{k})\|^{2}}_{T_{10}}\notag\\ &+\underbrace{\frac{\gamma^{k}(\gamma^{k}+v^{-1})}{J}\sum_{i\in \mathcal{J}}\|\nabla \widetilde{g}_{i}(\bar{\bm{\lambda}}^{k})-\nabla \widetilde{g}_{i}(\bm{\lambda}_{i}^{k})\|^{2}}_{T_{11}},\notag\end{aligned}$$ where $v>0$ is any positive constant. To drive the first inequality, we use $2\bm{a}^{\top}\bm{b}\le v^{-1}\|\bm{a}\|^{2}+v\|\bm{b}\|^{2}$ for any $v>0$. The last inequality holds because $(a_{1}+\cdots+a_{J})^{2}\le J(a_{1}^{2}+\cdots+a_{J}^{2})$. Next, we analyze $T_{10}$ and $T_{11}$ in turn.\ **Bounding $T_{10}$:** According to $\sum_{i\in \mathcal{J}}\nabla \widetilde{g}_{i}(\widetilde{\bm{\lambda}}^{*})=\bm{0}$, we have $$\begin{aligned} \label{eq_proof_theorem1-b-T1-1} & \hspace{-1em} T_{10}= (1+v\gamma^{k})\|\bar{\bm{\lambda}}^{k}-\widetilde{\bm{\lambda}}^{*}-\frac{\gamma^{k}}{J}(\sum_{i\in \mathcal{J}}\nabla \widetilde{g}_{i}(\bar{\bm{\lambda}}^{k})-\sum_{i\in \mathcal{J}}\nabla \widetilde{g}_{i}(\widetilde{\bm{\lambda}}^{*}))\|^{2}\notag\end{aligned}$$ $$\begin{aligned} & \hspace{-1em} = (1+v\gamma^{k})\|\bar{\bm{\lambda}}^{k}-\widetilde{\bm{\lambda}}^{*}\|^{2}\\ & \hspace{-1em} +(1+v\gamma^{k})(\gamma^{k})^{2}\|\frac{1}{J}\sum_{i\in \mathcal{J}}\nabla \widetilde{g}_{i}(\bar{\bm{\lambda}}^{k})-\frac{1}{J}\sum_{i\in \mathcal{J}}\nabla \widetilde{g}_{i}(\widetilde{\bm{\lambda}}^{*})\|^{2}\notag\\ & \hspace{-1em} -2\gamma^{k}(1+v\gamma^{k})\cdot \left \langle \bar{\bm{\lambda}}^{k}-\widetilde{\bm{\lambda}}^{*},\frac{1}{J}\sum_{i\in \mathcal{J}}\nabla \widetilde{g}_{i}(\bar{\bm{\lambda}}^{k})-\frac{1}{J}\sum_{i\in \mathcal{J}}\nabla \widetilde{g}_{i}(\widetilde{\bm{\lambda}}^{*}) \right \rangle. \notag\end{aligned}$$ Now we analyze the last term at the right-hand side of [\[eq_proof_theorem1-b-T1-1\]](#eq_proof_theorem1-b-T1-1){reference-type="eqref" reference="eq_proof_theorem1-b-T1-1"}, $\left \langle \bar{\bm{\lambda}}^{k}-\widetilde{\bm{\lambda}}^{*},\frac{1}{J}\sum_{i\in \mathcal{J}}\nabla \widetilde{g}_{i}(\bar{\bm{\lambda}}^{k})-\frac{1}{J}\sum_{i\in \mathcal{J}}\nabla \widetilde{g}_{i}(\widetilde{\bm{\lambda}}^{*}) \right \rangle$. By Lemma [Lemma 7](#l2-appendix1){reference-type="ref" reference="l2-appendix1"}, $\widetilde{g}_{i}(\cdot)$ is strongly convex with constant $\frac{1}{JL_{f}}$ and smooth with constant $\frac{1}{Ju_{f}}$. According to Lemma 3 in [@b-Kun-Yuan-2016], since $\frac{1}{J}\sum_{i\in \mathcal{J}}\widetilde{g}_{i}(\cdot)$ is $\frac{1}{JL_{f}}$-strongly convex and $\frac{1}{Ju_{f}}$-smooth, we have $$\begin{aligned} \label{eq_proof_theorem1-b-T1-2} &\left \langle \bar{\bm{\lambda}}^{k}-\widetilde{\bm{\lambda}}^{*},\frac{1}{J}\sum_{i\in \mathcal{J}}\nabla \widetilde{g}_{i}(\bar{\bm{\lambda}}^{k})-\frac{1}{J}\sum_{i\in \mathcal{J}}\nabla \widetilde{g}_{i}(\widetilde{\bm{\lambda}}^{*}) \right \rangle \\ &\ge \widetilde{\alpha}\|\frac{1}{J}\sum_{i\in \mathcal{J}}\nabla \widetilde{g}_{i}(\bar{\bm{\lambda}}^{k})-\frac{1}{J}\sum_{i\in \mathcal{J}}\nabla \widetilde{g}_{i}(\widetilde{\bm{\lambda}}^{*})\|^{2}+\widetilde{\beta}\|\bar{\bm{\lambda}}^{k}-\widetilde{\bm{\lambda}}^{*}\|^{2},\notag\end{aligned}$$ where $\widetilde{\alpha}=\frac{J u_{f}L_{f}}{u_{f}+L_{f}}$ and $\widetilde{\beta}=\frac{1}{J(u_{f}+L_{f})}$. Substituting [\[eq_proof_theorem1-b-T1-2\]](#eq_proof_theorem1-b-T1-2){reference-type="eqref" reference="eq_proof_theorem1-b-T1-2"} into [\[eq_proof_theorem1-b-T1-1\]](#eq_proof_theorem1-b-T1-1){reference-type="eqref" reference="eq_proof_theorem1-b-T1-1"} and rearranging the terms, we have $$\begin{aligned} \label{eq_proof_theorem1-b-T1} T_{10}\le& (1+v\gamma^{k})(1-2\gamma^{k}\widetilde{\beta})\|\bar{\bm{\lambda}}^{k}-\widetilde{\bm{\lambda}}^{*}\|^{2}\\ &+(1+v\gamma^{k})((\gamma^{k})^{2}-2\gamma^{k}\widetilde{\alpha})\cdot \notag\\ &\hspace{1em} \|\frac{1}{J}\sum_{i\in \mathcal{J}}\nabla \widetilde{g}_{i}(\bar{\bm{\lambda}}^{k})-\frac{1}{J}\sum_{i\in \mathcal{J}}\nabla \widetilde{g}_{i}(\widetilde{\bm{\lambda}}^{*})\|^{2}\notag\\ \le& (1+v\gamma^{k})(1-2\gamma^{k}\widetilde{\beta})\|\bar{\bm{\lambda}}^{k}-\widetilde{\bm{\lambda}}^{*}\|^{2},\notag\end{aligned}$$ where the last inequality holds with a proper step size $\gamma^{k}$ satisfying $(\gamma^{k})^{2}-2\gamma^{k}\widetilde{\alpha}\le 0$. **Bounding $T_{11}$:** Since $\widetilde{g}_{i}(\cdot)$ is $\frac{1}{Ju_{f}}$-smooth, we obtain $$\begin{aligned} \label{eq_proof_theorem1-b-T2} T_{11}\le \frac{\gamma^{k}(\gamma^{k}+v^{-1})}{J^{3}u_{f}^{2}}\sum_{i\in \mathcal{J}}\|\bm{\lambda}_{i}^{k}-\bar{\bm{\lambda}}^{k}\|^{2}.\end{aligned}$$ Substituting [\[eq_proof_theorem1-b-T1\]](#eq_proof_theorem1-b-T1){reference-type="eqref" reference="eq_proof_theorem1-b-T1"} and [\[eq_proof_theorem1-b-T2\]](#eq_proof_theorem1-b-T2){reference-type="eqref" reference="eq_proof_theorem1-b-T2"} into [\[eq_proof_theorem1-b-2\]](#eq_proof_theorem1-b-2){reference-type="eqref" reference="eq_proof_theorem1-b-2"} and rearranging the terms, we have $$\begin{aligned} \label{eq_proof_theorem1-b-3} &\|\bar{\bm{\lambda}}^{k+1}-\widetilde{\bm{\lambda}}^{*}\|^{2}\\ \le & (1+v\gamma^{k})(1-\gamma^{k}\widetilde{\beta})\|\bar{\bm{\lambda}}^{k}-\widetilde{\bm{\lambda}}^{*}\|^{2}\notag\\ &+\frac{\gamma^{k}(\gamma^{k}+v^{-1})}{J^{3}u_{f}^{2}}\sum_{i\in \mathcal{J}}\|\bm{\lambda}_{i}^{k}-\bar{\bm{\lambda}}^{k}\|^{2}\notag\\ =&(1+v\gamma^{k})(1-\gamma^{k}\widetilde{\beta})\|\bar{\bm{\lambda}}^{k}-\widetilde{\bm{\lambda}}^{*}\|^{2}\notag\\ &+\frac{\gamma^{k}(\gamma^{k}+v^{-1})}{J^{3}u_{f}^{2}}\|\widetilde{\Lambda}^{k}-\frac{1}{J}\widetilde{\bm{1}}\widetilde{\bm{1}}^{\top}\widetilde{\Lambda}^{k}\|_{F}^{2}, \notag\end{aligned}$$ where the matrix $\widetilde{\Lambda}:=[\cdots,\bm{\lambda}_{i},\cdots]\in \mathbb{R}^{J\times D}$ collects $\bm{\lambda}_{i}$ of all agents $i\in \mathcal{J}$. To drive the last equality, we use the fact that $\sum_{i\in \mathcal{J}}\|\bm{\lambda}_{i}^{k}-\bar{\bm{\lambda}}^{k}\|^{2}=\|\widetilde{\Lambda}^{k}-\frac{1}{J}\widetilde{\bm{1}}\widetilde{\bm{1}}^{\top}\widetilde{\Lambda}^{k}\|_{F}^{2}$. Based on Lemma [Lemma 8](#l3-appendix1){reference-type="ref" reference="l3-appendix1"}, we can rewrite [\[eq_proof_theorem1-b-3\]](#eq_proof_theorem1-b-3){reference-type="eqref" reference="eq_proof_theorem1-b-3"} as $$\begin{aligned} \label{eq_proof_theorem1-b-4} &\|\bar{\bm{\lambda}}^{k+1}-\widetilde{\bm{\lambda}}^{*}\|^{2}\\ \le &(1+v\gamma^{k})(1-\gamma^{k}\widetilde{\beta})\|\bar{\bm{\lambda}}^{k}-\widetilde{\bm{\lambda}}^{*}\|^{2}\notag\\ &+\frac{18(\gamma^{k})^{3}\widetilde{\delta}^{2}(\gamma^{k}+v^{-1})}{\sigma^{3}u_{f}^{2}J^{2}}. \notag\end{aligned}$$ Setting $v=\frac{\widetilde{\beta}}{2(1-\gamma^{k}\widetilde{\beta})}$, we can rewrite [\[eq_proof_theorem1-b-4\]](#eq_proof_theorem1-b-4){reference-type="eqref" reference="eq_proof_theorem1-b-4"} as $$\begin{aligned} \label{eq_proof_theorem1-b-5} \hspace{-1em}\|\bar{\bm{\lambda}}^{k+1}-\widetilde{\bm{\lambda}}^{*}\|^{2}\le (1-\frac{\gamma^{k}\widetilde{\beta}}{2})\|\bar{\bm{\lambda}}^{k}-\widetilde{\bm{\lambda}}^{*}\|^{2}+\frac{36(\gamma^{k})^{3}\widetilde{\delta}^{2}}{\sigma^{3}u_{f}^{2}J^{2}\widetilde{\beta}}.\end{aligned}$$ We further set a proper decreasing step size $\gamma^{k}=\frac{2}{\widetilde{\beta}(k_{0}+k)}$, where $k_{0}> 1$ is any positive integer. Thus, $1-\frac{\gamma^{k}\widetilde{\beta}}{2}=1-\frac{1}{k_{0}+k}$, and [\[eq_proof_theorem1-b-5\]](#eq_proof_theorem1-b-5){reference-type="eqref" reference="eq_proof_theorem1-b-5"} can be rewritten as $$\begin{aligned} \label{eq_proof_theorem1-b-6} &\|\bar{\bm{\lambda}}^{k+1}-\widetilde{\bm{\lambda}}^{*}\|^{2}\\ \le&(1-\frac{1}{k_{0}+k})\|\bar{\bm{\lambda}}^{k}-\widetilde{\bm{\lambda}}^{*}\|^{2}+\frac{1}{(k_{0}+k)^{3}}\cdot\frac{288\widetilde{\delta}^{2}}{\widetilde{\beta}^{4}\sigma^{3}u_{f}^{2}J^{2}}.\notag\end{aligned}$$ Then we rewrite [\[eq_proof_theorem1-b-6\]](#eq_proof_theorem1-b-6){reference-type="eqref" reference="eq_proof_theorem1-b-6"} recursively and obtain $$\begin{aligned} \label{eq_proof_theorem1-b-7} &\|\bar{\bm{\lambda}}^{k+1}-\widetilde{\bm{\lambda}}^{*}\|^{2}\\ &\le \underbrace{\prod_{k'=0}^{k}(1-\frac{1}{k_{0}+k-k'})}_{T_{12}}\|\bar{\bm{\lambda}}^{0}-\widetilde{\bm{\lambda}}^{*}\|^{2}\notag\\ &+\big[\prod_{k'=0}^{k-1}(1-\frac{1}{k_{0}+k-k'})\frac{1}{(k_{0}+0)^{3}}+\cdots+\notag\\ &+\underbrace{\prod_{k'=0}^{0}\frac{(1-\frac{1}{k_{0}+k-k'})}{(k_{0}+k-1)^{3}}+\frac{1}{(k_{0}+k)^{3}}}_{T_{13}}\big]\cdot\frac{288\widetilde{\delta}^{2}}{\widetilde{\beta}^{4}\sigma^{3}u_{f}^{2}J^{2}}.\notag\end{aligned}$$ Next we analyze $T_{12}$ and $T_{13}$ in turn. For $T_{12}$, we have $$\begin{aligned} \label{eq_proof_theorem1-b-T3} T_{12}=&\prod_{k'=0}^{k}(1-\frac{1}{k_{0}+k-k'})\\ =&\frac{k_{0}+k-1}{k_{0}+k}\cdot\frac{k_{0}+k-2}{k_{0}+k-1}\cdot\cdots\cdot\frac{k_{0}-1}{k_{0}}\notag\\ =&\frac{k_{0}-1}{k_{0}+k}.\notag\end{aligned}$$ For $T_{13}$, we have $$\begin{aligned} \label{eq_proof_theorem1-b-T4} T_{13}=&\frac{k_{0}}{k_{0}+k}\cdot\frac{1}{(k_{0}+0)^{3}}+\frac{k_{0}+1}{k_{0}+k}\cdot\frac{1}{(k_{0}+1)^{3}}+\cdots\\ &+\frac{k_{0}+k-1}{k_{0}+k}\cdot\frac{1}{(k_{0}+k-1)^{3}}+\frac{1}{(k_{0}+k)^{3}}\notag\\ =& \frac{1}{k_{0}+k}[\frac{1}{(k_{0})^{2}}+\frac{1}{(k_{0}+1)^{2}}+\cdots\notag\\ &+\frac{1}{(k_{0}+k-1)^{2}}+\frac{1}{(k_{0}+k)^{2}}]\notag\\ \le& \frac{1}{k_{0}+k}\cdot\frac{1}{k_{0}-1}.\notag\end{aligned}$$ To drive the last inequality, we use $\sum_{k'=k_{0}}^{k}\frac{1}{{k'}^{2}}\le \frac{1}{k_{0}-1}$. Substituting [\[eq_proof_theorem1-b-T3\]](#eq_proof_theorem1-b-T3){reference-type="eqref" reference="eq_proof_theorem1-b-T3"} and [\[eq_proof_theorem1-b-T4\]](#eq_proof_theorem1-b-T4){reference-type="eqref" reference="eq_proof_theorem1-b-T4"} into [\[eq_proof_theorem1-b-7\]](#eq_proof_theorem1-b-7){reference-type="eqref" reference="eq_proof_theorem1-b-7"}, we have $$\begin{aligned} \label{eq_proof_theorem1-b-8} &\|\bar{\bm{\lambda}}^{k+1}-\widetilde{\bm{\lambda}}^{*}\|^{2}\\ \le &\frac{k_{0}-1}{k_{0}+k}\|\bar{\bm{\lambda}}^{0}-\widetilde{\bm{\lambda}}^{*}\|^{2}+\frac{1}{(k_{0}+k)(k_{0}-1)}\cdot \frac{288\widetilde{\delta}^{2}}{\widetilde{\beta}^{4}\sigma^{3}u_{f}^{2}J^{2}}.\notag\end{aligned}$$ Applying the triangle inequality into [\[eq_proof_theorem1-b-8\]](#eq_proof_theorem1-b-8){reference-type="eqref" reference="eq_proof_theorem1-b-8"}, we obtain $$\begin{aligned} \label{eq_proof_theorem1-b-9} &\|\bar{\bm{\lambda}}^{k+1}-\widetilde{\bm{\lambda}}^{*}\|\\ \le &\sqrt{\frac{k_{0}-1}{k_{0}+k}}\|\bar{\bm{\lambda}}^{0}-\widetilde{\bm{\lambda}}^{*}\|+\sqrt{\frac{1}{(k_{0}+k)(k_{0}-1)}\cdot \frac{288\widetilde{\delta}^{2}}{\widetilde{\beta}^{4}\sigma^{3}u_{f}^{2}J^{2}}}.\notag\end{aligned}$$ According to Lemma [Lemma 8](#l3-appendix1){reference-type="ref" reference="l3-appendix1"} and using the step size $\gamma^{k}=\frac{2}{\widetilde{\beta}(k_{0}+k)}$, we have $$\begin{aligned} \label{eq_proof_theorem1-b-10} \sum_{i\in \mathcal{J}}\|\bm{\lambda}_{i}^{k+1}-\bar{\bm{\lambda}}^{k+1}\|^{2}=&\|\widetilde{\Lambda}^{k+1}-\frac{1}{J}\widetilde{\bm{1}}\widetilde{\bm{1}}^{\top}\widetilde{\Lambda}^{k+1}\|^{2}_{F}\\ \le& \frac{72\widetilde{\delta}^{2}J}{\sigma^{3}\widetilde{\beta}^{2}(k_{0}+k+1)^{2}}.\notag\end{aligned}$$ By $(a_{1}+\cdots+a_{J})^{2}\le J(a_{1}^{2}+\cdots+a_{J}^{2})$ and [\[eq_proof_theorem1-b-10\]](#eq_proof_theorem1-b-10){reference-type="eqref" reference="eq_proof_theorem1-b-10"}, we have $$\begin{aligned} \label{eq_proof_theorem1-b-11} \sum_{i\in \mathcal{J}}\|\bm{\lambda}_{i}^{k+1}-\bar{\bm{\lambda}}^{k+1}\|\le& \sqrt{J}\sqrt{\sum_{i\in \mathcal{J}}\|\bm{\lambda}_{i}^{k+1}-\bar{\bm{\lambda}}^{k+1}\|^{2}}\\ \le& \sqrt{J}\sqrt{\frac{72\widetilde{\delta}^{2}J}{\sigma^{3}\widetilde{\beta}^{2}(k_{0}+k+1)^{2}}}.\notag\end{aligned}$$ Combining [\[eq_proof_theorem1-b-9\]](#eq_proof_theorem1-b-9){reference-type="eqref" reference="eq_proof_theorem1-b-9"} and [\[eq_proof_theorem1-b-11\]](#eq_proof_theorem1-b-11){reference-type="eqref" reference="eq_proof_theorem1-b-11"} yields $$\begin{aligned} \label{eq_proof_theorem1-b-12} &\sum_{i\in \mathcal{J}}\|\bm{\lambda}_{i}^{k+1}-\widetilde{\bm{\lambda}}^{*}\|\\ \le& \sum_{i\in \mathcal{J}}\|\bm{\lambda}_{i}^{k+1}-\bar{\bm{\lambda}}^{k+1}\|+J\cdot \|\bar{\bm{\lambda}}^{k+1}-\widetilde{\bm{\lambda}}^{*}\|\notag\\ \le& J\cdot\sqrt{\frac{k_{0}-1}{k_{0}+k}}\|\bar{\bm{\lambda}}^{0}-\widetilde{\bm{\lambda}}^{*}\|+\sqrt{\frac{1}{(k_{0}+k)(k_{0}-1)} \cdot\frac{288\widetilde{\delta}^{2}}{\widetilde{\beta}^{4}\sigma^{3}u_{f}^{2}}}\notag\\ &+\sqrt{J}\sqrt{\frac{72\widetilde{\delta}^{2}J}{\sigma^{3}\widetilde{\beta}^{2}(k_{0}+k+1)^{2}}}.\notag\end{aligned}$$ Taking $k \to +\infty$, we obtain $$\begin{aligned} \label{eq_proof_theorem1-b-13} \lim_{k\to +\infty}\sum_{i\in \mathcal{J}}\|\bm{\lambda}_{i}^{k+1}-\widetilde{\bm{\lambda}}^{*}\| =0.\end{aligned}$$ ## Part $b$ of Theorem [Theorem 1](#t1){reference-type="ref" reference="t1"} {#part-b-of-theorem-t1} Since $\widetilde{\bm{\Theta}}^{*}$ is the optimal solution of the primal problem [\[eq_primal-problem\]](#eq_primal-problem){reference-type="eqref" reference="eq_primal-problem"}, we have $\frac{1}{J}\sum_{i\in \mathcal{J}}\widetilde{\bm{\theta}}_{i}^{*} = \bm{s}$ and $\widetilde{\bm{\Theta}}^{*}\in \widetilde{C}$. According to [\[eq_Lagrangian-primal-problem\]](#eq_Lagrangian-primal-problem){reference-type="eqref" reference="eq_Lagrangian-primal-problem"}, for any dual variable $\widetilde{\bm{\lambda}}$ we have $$\begin{aligned} \label{eq_proof_theorem1-a-1} \widetilde{\mathcal{L}}\left ( \widetilde{\bm{\Theta}}^{*} ; \widetilde{\bm{\lambda}} \right )=&\frac{1}{J}\sum_{i\in \mathcal{J}}f_{i}\left ( \widetilde{\bm{\theta}}_{i}^{*} \right )+ \left \langle \widetilde{\bm{\lambda} }, \frac{1}{J}\sum_{i\in \mathcal{J}}\widetilde{\bm{\theta}}_{i}^{*}-\bm{s} \right \rangle\\ =& \widetilde{f}(\widetilde{\bm{\Theta}}^{*}).\notag\end{aligned}$$ By Assumption [Assumption 2](#a2){reference-type="ref" reference="a2"}, the duality gap is zero. Therefore, for any $\widetilde{\bm{\lambda}}$ we obtain $\widetilde{g}(\widetilde{\bm{\lambda}}^{*})=-\widetilde{f}(\widetilde{\bm{\Theta}}^{*})=-\widetilde{\mathcal{L}}\left ( \widetilde{\bm{\Theta}}^{*} ; \widetilde{\bm{\lambda}} \right )$. Now we introduce a vector $_{}^{\dagger}\widetilde{\bm{\Theta}}^{k+1}:=[ _{}^{\dagger}\bm{\theta}_{1}^{k+1}; \cdots; _{}^{\dagger}\bm{\theta}_{J}^{k+1}]$, where $_{}^{\dagger}\bm{\theta}_{i}^{k+1}=\arg\min_{\bm{\theta}_{i}\in C_{i}} \{\bm{\theta}_{i}^{\top}\bar{\bm{\lambda}}^{k+1}+f_{i}(\bm{\theta}_{i})\}$ and $\bar{\bm{\lambda}}^{k+1}:=\frac{1}{J}\sum_{i\in \mathcal{J}}\bm{\lambda}_{i}^{k+1}$. Therefore, we have $$\begin{aligned} \label{eq_proof_theorem1-a-2} &\widetilde{g}(\bar{\bm{\lambda}}^{k+1})-\widetilde{g}(\widetilde{\bm{\lambda}}^{*})\\ =&-\inf_{\widetilde{\bm{\Theta}}\in \widetilde{C}}\widetilde{\mathcal{L}}\left ( \widetilde{\bm{\Theta}} ; \bar{\bm{\lambda}}^{k+1} \right )+\widetilde{\mathcal{L}}\left ( \widetilde{\bm{\Theta}}^{*} ; \bar{\bm{\lambda}}^{k+1} \right )\notag\\ =&-\widetilde{\mathcal{L}}\left ( ^{\dagger}\widetilde{\bm{\Theta}}^{k+1} ; \bar{\bm{\lambda}}^{k+1} \right )+\widetilde{\mathcal{L}}\left ( \widetilde{\bm{\Theta}}^{*} ; \bar{\bm{\lambda}}^{k+1} \right ).\notag\end{aligned}$$ Assumption [Assumption 1](#a1){reference-type="ref" reference="a1"} shows that the local cost function $f_{i}(\cdot)$ is $u_{f}$-strongly convex. Further using the definition of $\widetilde{\mathcal{L}}\left ( \widetilde{\bm{\Theta}} ; \widetilde{\bm{\lambda}} \right )$ in [\[eq_Lagrangian-primal-problem\]](#eq_Lagrangian-primal-problem){reference-type="eqref" reference="eq_Lagrangian-primal-problem"}, we know that $\widetilde{\mathcal{L}}\left ( \widetilde{\bm{\Theta}} ; \widetilde{\bm{\lambda}} \right )$ is $u_{f}$-strongly convex with respect to $\widetilde{\bm{\Theta}}$. Therefore, we have $$\begin{aligned} \label{eq_proof_theorem1-a-3} &\widetilde{\mathcal{L}}\left ( \widetilde{\bm{\Theta}}^{*} ; \bar{\bm{\lambda}}^{k+1} \right )-\widetilde{\mathcal{L}}\left ( ^{\dagger}\widetilde{\bm{\Theta}}^{k+1} ; \bar{\bm{\lambda}}^{k+1} \right )\\ \ge& \nabla \widetilde{\mathcal{L}}\left ( ^{\dagger}\widetilde{\bm{\Theta}}^{k+1} ; \bar{\bm{\lambda}}^{k+1} \right )^{\top} (\widetilde{\bm{\Theta}}^{*}-\ ^{\dagger}\widetilde{\bm{\Theta}}^{k+1})+\frac{u_{f}}{2}\|^{\dagger}\widetilde{\bm{\Theta}}^{k+1}-\widetilde{\bm{\Theta}}^{*}\|^{2}.\notag\end{aligned}$$ Combining [\[eq_proof_theorem1-a-2\]](#eq_proof_theorem1-a-2){reference-type="eqref" reference="eq_proof_theorem1-a-2"} and [\[eq_proof_theorem1-a-3\]](#eq_proof_theorem1-a-3){reference-type="eqref" reference="eq_proof_theorem1-a-3"}, we obtain $$\begin{aligned} \label{eq_proof_theorem1-a-4} &\widetilde{g}(\bar{\bm{\lambda}}^{k+1})-\widetilde{g}(\widetilde{\bm{\lambda}}^{*})\\ \ge& \nabla \widetilde{\mathcal{L}}\left ( ^{\dagger}\widetilde{\bm{\Theta}}^{k+1} ; \bar{\bm{\lambda}}^{k+1} \right )^{\top}(\widetilde{\bm{\Theta}}^{*}-\ ^{\dagger}\widetilde{\bm{\Theta}}^{k+1})+\frac{u_{f}}{2}\|^{\dagger}\widetilde{\bm{\Theta}}^{k+1}-\widetilde{\bm{\Theta}}^{*}\|^{2}\notag\\ \ge&\frac{u_{f}}{2}\|^{\dagger}\widetilde{\bm{\Theta}}^{k+1}-\widetilde{\bm{\Theta}}^{*}\|^{2}. \notag\end{aligned}$$ To drive the last inequality, we use the optimality condition of $_{}^{\dagger}\bm{\theta}_{i}^{k+1}=\arg\min_{\bm{\theta}_{i}\in C_{i}} \{\bm{\theta}_{i}^{\top}\bar{\bm{\lambda}}^{k+1}+f_{i}(\bm{\theta}_{i})\}$ [@b-Dimitri-P.-Bertsekas-1999 Proposition 2.1.2]. According to Lemma [Lemma 7](#l2-appendix1){reference-type="ref" reference="l2-appendix1"}, $\widetilde{g}_{i}(\widetilde{\bm{\lambda}})$ is smooth with constant $\frac{1}{Ju_{f}}$. Therefore, $\widetilde{g}(\widetilde{\bm{\lambda }}):=\sum_{i\in\mathcal{J}}\widetilde{g}_{i}(\widetilde{\bm{\lambda }})$ is smooth with constant $\frac{1}{u_{f}}$. This fact leads to $$\begin{aligned} \label{eq_proof_theorem1-a-5} &\widetilde{g}(\bar{\bm{\lambda}}^{k+1})-\widetilde{g}(\widetilde{\bm{\lambda}}^{*})\\ \le& \nabla \widetilde{g}^{\top}(\widetilde{\bm{\lambda}}^{*})(\bar{\bm{\lambda}}^{k+1}-\widetilde{\bm{\lambda}}^{*})+\frac{1}{2u_{f}}\|\bar{\bm{\lambda}}^{k+1}-\widetilde{\bm{\lambda}}^{*}\|^{2}\notag\\ =& \frac{1}{2u_{f}}\|\bar{\bm{\lambda}}^{k+1}-\widetilde{\bm{\lambda}}^{*}\|^{2}.\notag\end{aligned}$$ To drive the last equality, we use the fact that $\nabla \widetilde{g}(\widetilde{\bm{\lambda}}^{*})=\bm{0}$. Combining [\[eq_proof_theorem1-a-4\]](#eq_proof_theorem1-a-4){reference-type="eqref" reference="eq_proof_theorem1-a-4"} and [\[eq_proof_theorem1-a-5\]](#eq_proof_theorem1-a-5){reference-type="eqref" reference="eq_proof_theorem1-a-5"}, we have $$\begin{aligned} \label{eq_proof_theorem1-a-6} \|^{\dagger}\widetilde{\bm{\Theta}}^{k+1}-\widetilde{\bm{\Theta}}^{*}\|^{2} \le \frac{1}{u_{f}^{2}}\|\bar{\bm{\lambda}}^{k+1}-\widetilde{\bm{\lambda}}^{*}\|^{2}.\end{aligned}$$ Combining [\[eq_proof_theorem1-b-9\]](#eq_proof_theorem1-b-9){reference-type="eqref" reference="eq_proof_theorem1-b-9"} and [\[eq_proof_theorem1-a-6\]](#eq_proof_theorem1-a-6){reference-type="eqref" reference="eq_proof_theorem1-a-6"}, we obtain $$\begin{aligned} \label{eq_proof_theorem1-a-7} &\|^{\dagger}\widetilde{\bm{\Theta}}^{k+1}-\widetilde{\bm{\Theta}}^{*}\|\\ \le& \frac{1}{u_{f}^{2}}\cdot \big[\sqrt{\frac{k_{0}-1}{k_{0}+k}}\|\bar{\bm{\lambda}}^{0}-\widetilde{\bm{\lambda}}^{*}\|\notag\\ &+\sqrt{\frac{1}{(k_{0}+k)(k_{0}-1)}\cdot \frac{288\widetilde{\delta}^{2}(1-\sigma )}{\widetilde{\beta}^{4}\sigma^{3}u_{f}^{2}J^{2}}} \big].\notag\end{aligned}$$ Taking $k\to +\infty$, we obtain $$\begin{aligned} \label{eq_proof_theorem1-a-8} \lim_{k\to +\infty}\|^{\dagger}\widetilde{\bm{\Theta}}^{k+1}-\widetilde{\bm{\Theta}}^{*}\|=0.\end{aligned}$$ According to Assumption [Assumption 1](#a1){reference-type="ref" reference="a1"}, $f_{i}(\bm{\theta}_{i})$ is $u_{f}$-strongly convex. By the conjugate correspondence theorem in [@b-Amir-Beck-2017], we know that the conjugate function $\widetilde{F}_{i}^{*}(\widetilde{\bm{\lambda}})=\max_{\bm{\theta}_{i}\in C_{i}}\{\widetilde{\bm{\lambda}}^{\top}\bm{\theta}_{i}-f_{i}(\bm{\theta}_{i})\}$ is $\frac{1}{u_{f}}$-smooth. Therefore, the gradient $\nabla \widetilde{F}^{*}_{i}(-\widetilde{\bm{\lambda}})=\arg\min_{\bm{\theta}_{i}\in C_{i}}\{\widetilde{\bm{\lambda}}^{\top}\bm{\theta}_{i}+f_{i}(\bm{\theta}_{i})\}$ is $\frac{1}{u_{f}}$-Lipschitz continuous. According to the definition of Lipschitz continuity, we have $$\begin{aligned} \label{eq_proof_theorem1-a-lipschitz-continuous} \|\nabla \widetilde{F}^{*}_{i}(\bm{\lambda}_{i}^{k+1})-\nabla \widetilde{F}^{*}_{i}(\bar{\bm{\lambda}}^{k+1})\| \le \frac{1}{u_{f}}\|\bm{\lambda}_{i}^{k+1}-\bar{\bm{\lambda}}^{k+1}\|.\end{aligned}$$ Based on $_{}^{\dagger}\bm{\theta}_{i}^{k+1}=\arg\min_{\bm{\theta}_{i}\in C_{i}} \{\bm{\theta}_{i}^{\top}\bar{\bm{\lambda}}^{k+1}+f_{i}(\bm{\theta}_{i})\}$ and $\bm{\theta}_{i}^{k+1}=\arg\min_{\bm{\theta}_{i}\in C_{i}} \{\bm{\theta}_{i}^{\top}\bm{\lambda}_{i}^{k+1}+f_{i}(\bm{\theta}_{i})\}$, we obtain $\bm{\theta}_{i}^{k+1}=\nabla \widetilde{F}^{*}_{i}(\bm{\lambda}_{i}^{k+1})$ and $_{}^{\dagger}\bm{\theta}_{i}^{k+1}=\nabla \widetilde{F}^{*}_{i}(\bar{\bm{\lambda}}^{k+1})$. Substituting $\bm{\theta}_{i}^{k+1}=\nabla \widetilde{F}^{*}_{i}(\bm{\lambda}_{i}^{k+1})$ and $_{}^{\dagger}\bm{\theta}_{i}^{k+1}=\nabla \widetilde{F}^{*}_{i}(\bar{\bm{\lambda}}^{k+1})$ into [\[eq_proof_theorem1-a-lipschitz-continuous\]](#eq_proof_theorem1-a-lipschitz-continuous){reference-type="eqref" reference="eq_proof_theorem1-a-lipschitz-continuous"}, we have $$\begin{aligned} \label{eq_proof_theorem1-a-10} \|\bm{\theta}_{i}^{k+1}-\ _{}^{\dagger}\bm{\theta}_{i}^{k+1}\| \le \frac{1}{u_{f}}\|\bm{\lambda}_{i}^{k+1}-\bar{\bm{\lambda}}^{k+1}\|.\end{aligned}$$ Combining [\[eq_proof_theorem1-a-10\]](#eq_proof_theorem1-a-10){reference-type="eqref" reference="eq_proof_theorem1-a-10"}, $^{\dagger}\widetilde{\bm{\Theta}}^{k+1}:=[_{}^{\dagger}\bm{\theta}_{1}^{k+1}; \cdots; _{}^{\dagger}\bm{\theta}_{J}^{k+1}]$ and $\widetilde{\bm{\Theta}}^{k+1}:=[\bm{\theta}_{1}^{k+1}; \cdots; \bm{\theta}_{J}^{k+1}]$, we obtain $$\begin{aligned} \label{eq_proof_theorem1-a-11} \|\widetilde{\bm{\Theta}}^{k+1}-\ ^{\dagger}\widetilde{\bm{\Theta}}^{k+1}\| \le \frac{1}{u_{f}}\sum_{i\in \mathcal{J}}\|\bm{\lambda}_{i}^{k+1}-\bar{\bm{\lambda}}^{k+1}\|.\end{aligned}$$ Combining [\[eq_proof_theorem1-b-11\]](#eq_proof_theorem1-b-11){reference-type="eqref" reference="eq_proof_theorem1-b-11"} and [\[eq_proof_theorem1-a-11\]](#eq_proof_theorem1-a-11){reference-type="eqref" reference="eq_proof_theorem1-a-11"}, we have $$\begin{aligned} \label{eq_proof_theorem1-a-12} \|\widetilde{\bm{\Theta}}^{k+1}-\ ^{\dagger}\widetilde{\bm{\Theta}}^{k+1}\| \le \frac{\sqrt{J}}{u_{f}}\cdot\sqrt{\frac{72\widetilde{\delta}^{2}J}{\sigma^{3}\widetilde{\beta}^{2}(k_{0}+k+1)^{2}}}.\end{aligned}$$ Taking $k\to +\infty$, we obtain $$\begin{aligned} \label{eq_proof_theorem1-a-13} \lim_{k\to +\infty}\|\widetilde{\bm{\Theta}}^{k+1}-\ ^{\dagger}\widetilde{\bm{\Theta}}^{k+1}\| =0.\end{aligned}$$ Combining [\[eq_proof_theorem1-a-8\]](#eq_proof_theorem1-a-8){reference-type="eqref" reference="eq_proof_theorem1-a-8"} and [\[eq_proof_theorem1-a-13\]](#eq_proof_theorem1-a-13){reference-type="eqref" reference="eq_proof_theorem1-a-13"} yields $$\begin{aligned} \label{eq_proof_theorem1-a-9} &\lim_{k\to +\infty}\|\widetilde{\bm{\Theta}}^{k+1}-\widetilde{\bm{\Theta}}^{*}\|\\ \leq &\lim_{k\to +\infty}\|\widetilde{\bm{\Theta}}^{k+1}-\ ^{\dagger}\widetilde{\bm{\Theta}}^{k+1}\|+\lim_{k\to +\infty}\|^{\dagger}\widetilde{\bm{\Theta}}^{k+1}-\widetilde{\bm{\Theta}}^{*}\|=0\notag.\end{aligned}$$ We conclude by summarizing the conditions on the step size $\gamma^{k}$ in Theorem [Theorem 1](#t1){reference-type="ref" reference="t1"}. It must satisfy $(\gamma^{k})^{2}-2\gamma^{k}\widetilde{\alpha}\le 0$, $\frac{6(\gamma^{k})^{2}}{ u_{f}^{2}J^{2}}\le\frac{(2-\sigma)\sigma^{2}}{3(3-\sigma)}$ and $1\le \frac{(\gamma^{k})^{2}}{(\gamma^{k+1})^2}\le \frac{2}{1+(1-\sigma^{2})}$. The step size $\gamma^{k}=\frac{2}{\widetilde{\beta}(k_{0}+k)}$ with $k_{0}\ge \max\{\frac{1}{\widetilde{\alpha}\widetilde{\beta}},\sqrt{\frac{72(3-\sigma)}{(2-\sigma)\sigma^{2}u_{f}^{2}J^{2}\widetilde{\beta}^{2}}},\frac{1}{\sqrt{\frac{2}{1+(1-\sigma^{2})}}-1}\}$ satisfies these conditions. ## Supporting Lemmas **Lemma 6**. *Under Assumption [Assumption 1](#a1){reference-type="ref" reference="a1"}, for any $\widetilde{\bm{\lambda}} \in \mathbb{R}^D$, the maximum distance between the local dual gradients and their average, denoted by $\max_{i\in \mathcal{J}}\|\nabla \widetilde{g}_{i}(\widetilde{\bm{\lambda}}) - \frac{1}{J} \sum_{i\in \mathcal{J}} \nabla \widetilde{g}_{i}(\widetilde{\bm{\lambda}}) \|^{2}$, is bounded by some positive constant $\widetilde{\delta}^{2}$.* ***Proof**. Recalling the definition of the local dual gradient $\nabla \widetilde{g}_{i}(\widetilde{\bm{\lambda}})=-\frac{1}{J}\arg\min_{\bm{\theta}_{i}\in C_{i}} \{\widetilde{\bm{\lambda}}^{\top}\bm{\theta}_{i}+f_{i}(\bm{\theta}_{i})\}+\frac{1}{J}\bm{s}$, we have $$\begin{aligned} &\max_{i\in \mathcal{J}}\|\nabla \widetilde{g}_{i}(\widetilde{\bm{\lambda}}) - \frac{1}{J} \sum_{i\in \mathcal{J}} \nabla \widetilde{g}_{i}(\widetilde{\bm{\lambda}}) \|^{2}\\ =&\max_{i\in \mathcal{J}}\|-\frac{1}{J}\arg\underset{\bm{\theta}_{i}\in C_{i}}{\min}\{\widetilde{\bm{\lambda}}^{\top}\bm{\theta}_{i}+f_{i}(\bm{\theta}_{i})\}+\frac{1}{J}\bm{s}\notag\\ &\hspace{2.5em}+\frac{1}{J^{2}}\sum_{i\in \mathcal{J}}\arg\underset{\bm{\theta}_{i}\in C_{i}}{\min}\{\widetilde{\bm{\lambda}}^{\top}\bm{\theta}_{i}+f_{i}(\bm{\theta}_{i})\}-\frac{1}{J}\bm{s}\|^{2}\notag\\ =&\max_{i\in \mathcal{J}}\|\frac{1}{J}(\frac{1}{J}\sum_{i\in \mathcal{J}}\arg\underset{\bm{\theta}_{i}\in C_{i}}{\min}\{\widetilde{\bm{\lambda}}^{\top}\bm{\theta}_{i}+f_{i}(\bm{\theta}_{i})\}\notag\\ &\hspace{2.5em}-\arg\underset{\bm{\theta}_{i}\in C_{i}}{\min}\{\widetilde{\bm{\lambda}}^{\top}\bm{\theta}_{i}+f_{i}(\bm{\theta}_{i})\})\|^{2}.\notag\end{aligned}$$ According to Assumption [Assumption 1](#a1){reference-type="ref" reference="a1"}, the local constraint sets $C_{i}$ are bounded by hypothesis, and we know that $\max_{i\in \mathcal{J}}\|\nabla g_{i}(\widetilde{\bm{\lambda}}) - \frac{1}{J} \sum_{i\in \mathcal{J}} \nabla g_{i}(\widetilde{\bm{\lambda}}) \|^{2}$ is also bounded by some positive constant, which we denote as $\widetilde{\delta}^{2}$.* **Lemma 7**. *Under Assumption [Assumption 1](#a1){reference-type="ref" reference="a1"}, for any agent $i$, the local dual function $\widetilde{g}_{i}(\widetilde{\bm{\lambda}})$ is strongly convex with constant $\frac{1}{JL_{f}}$ and smooth with parameter $\frac{1}{Ju_{f}}$.* ***Proof**. According to Assumption [Assumption 1](#a1){reference-type="ref" reference="a1"}, $f_{i}(\cdot)$ is $u_{f}$-strongly convex and $L_{f}$-smooth. By the conjugate correspondence theorem [@b-Amir-Beck-2017], its conjugate function $\widetilde{F}_{i}^{*}(\widetilde{\bm{\lambda}})=\max_{\bm{\theta}_{i}\in C_{i}}\{\widetilde{\bm{\lambda}}^{\top}\bm{\theta}_{i}-f_{i}(\bm{\theta}_{i})\}$ is $\frac{1}{L_{f}}$-strongly convex and $\frac{1}{u_{f}}$-smooth. By the definition of $\widetilde{g}_{i}(\widetilde{\bm{\lambda}})=\frac{1}{J}\widetilde{F}_{i}^{*}(-\widetilde{\bm{\lambda }})+\frac{1}{J}\widetilde{\bm{\lambda }}^{\top}\bm{s}$, we know $\widetilde{g}_{i}(\widetilde{\bm{\lambda}})$ is $\frac{1}{JL_{f}}$-strongly convex and $\frac{1}{Ju_{f}}$-smooth.* **Lemma 8**. *Define a matrix $\widetilde{\Lambda}^{k+1}:=[\cdots,\bm{\lambda}_{i}^{k+1},\cdots]\in \mathbb{R}^{J\times D}$ that collects the dual variables $\bm{\lambda}_{i}^{k+1}$ of all agents $i\in \mathcal{J}$ generated by Algorithm [\[alg1\]](#alg1){reference-type="ref" reference="alg1"}. Under Assumptions [Assumption 1](#a1){reference-type="ref" reference="a1"} and [Assumption 3](#a3){reference-type="ref" reference="a3"}, we have $$\begin{aligned} \label{lemma3-appendix1} \|\widetilde{\Lambda}^{k+1}-\frac{1}{J}\widetilde{\bm{1}}\widetilde{\bm{1}}^{\top}\widetilde{\Lambda}^{k+1} \|^{2}_{F}\le \frac{18(\gamma^{k+1})^{2}\widetilde{\delta}^{2}J}{\sigma^{3}},\end{aligned}$$ where $\sigma=1- \widetilde{\kappa} \in (0,1)$.* ***Proof**. Define $\nabla \widetilde{g}(\widetilde{\Lambda})=[\cdots,\nabla \widetilde{g}_{i}(\bm{\lambda}_{i}),\cdots]\in \mathbb{R}^{J \times D}$ to collect the dual gradients $\nabla\widetilde{g}_{i}(\bm{\lambda}_{i})$ of all agents $i\in \mathcal{J}$. With these notations, we can rewrite the updates of $\bm{\lambda}_{i}^{k+1}$ and $\bm{\lambda}_{i}^{k+\frac{1}{2}}$ in Algorithm [\[alg1\]](#alg1){reference-type="ref" reference="alg1"} in compact forms of $$\begin{aligned} \label{lemma3-appendix1-proof-1} \widetilde{\Lambda}^{k+\frac{1}{2}}=\widetilde{\Lambda}^{k}-\gamma^{k}\nabla \widetilde{g}(\widetilde{\Lambda}^{k}),\end{aligned}$$ $$\begin{aligned} \label{lemma3-appendix1-proof-2} \widetilde{\Lambda}^{k+1}=\widetilde{E}\widetilde{\Lambda}^{k+\frac{1}{2}}.\end{aligned}$$ Combining [\[lemma3-appendix1-proof-1\]](#lemma3-appendix1-proof-1){reference-type="eqref" reference="lemma3-appendix1-proof-1"} and [\[lemma3-appendix1-proof-2\]](#lemma3-appendix1-proof-2){reference-type="eqref" reference="lemma3-appendix1-proof-2"}, and also using the fact that $\widetilde{E}$ is doubly stochastic by Assumption [Assumption 3](#a3){reference-type="ref" reference="a3"}, we have $$\begin{aligned} \label{lemma3-appendix1-proof-3} &\|\widetilde{\Lambda}^{k+1}-\frac{1}{J}\widetilde{\bm{1}}\widetilde{\bm{1}}^{\top}\widetilde{\Lambda}^{k+1} \|^{2}_{F}\\ =&\|\widetilde{E}(\widetilde{\Lambda}^{k}-\gamma^{k}\nabla \widetilde{g}(\widetilde{\Lambda}^{k}))-\frac{1}{J}\widetilde{\bm{1}}\widetilde{\bm{1}}^{\top}\widetilde{E}(\widetilde{\Lambda}^{k}-\gamma^{k}\nabla \widetilde{g}(\widetilde{\Lambda}^{k}))\|_{F}^{2}\notag\\ =&\|\widetilde{E}\widetilde{\Lambda}^{k}-\frac{1}{J}\widetilde{\bm{1}}\widetilde{\bm{1}}^{\top}\widetilde{\Lambda}^{k}-\widetilde{E}\gamma^{k}\nabla \widetilde{g}(\widetilde{\Lambda}^{k})+\frac{1}{J}\widetilde{\bm{1}}\widetilde{\bm{1}}^{\top}\gamma^{k}\nabla \widetilde{g}(\widetilde{\Lambda}^{k})\|_{F}^{2}\notag\\ \le&\frac{1}{1-v}\|\widetilde{E}\widetilde{\Lambda}^{k}-\frac{1}{J}\widetilde{\bm{1}}\widetilde{\bm{1}}^{\top}\widetilde{\Lambda}^{k}\|_{F}^{2}\notag\\ &+\frac{1}{v}\|\widetilde{E}\gamma^{k}\nabla \widetilde{g}(\widetilde{\Lambda}^{k})-\frac{1}{J}\widetilde{\bm{1}}\widetilde{\bm{1}}^{\top}\gamma^{k}\nabla \widetilde{g}(\widetilde{\Lambda}^{k})\|_{F}^{2} \notag\\ =&\frac{1}{1-v}\|(\widetilde{E}-\frac{1}{J}\widetilde{\bm{1}}\widetilde{\bm{1}}^{\top})(\widetilde{\Lambda}^{k}-\frac{1}{J}\widetilde{\bm{1}}\widetilde{\bm{1}}^{\top}\widetilde{\Lambda}^{k})\|_{F}^{2}\notag\\ &+\frac{(\gamma^{k})^{2}}{v}\|(\widetilde{E}-\frac{1}{J}\widetilde{\bm{1}}\widetilde{\bm{1}}^{\top})(\nabla \widetilde{g}(\widetilde{\Lambda}^{k})-\frac{1}{J}\widetilde{\bm{1}}\widetilde{\bm{1}}^{\top}\nabla \widetilde{g}(\widetilde{\Lambda}^{k}))\|_{F}^{2} \notag\\ \le& \frac{1}{1-v}\|\widetilde{E}-\frac{1}{J}\widetilde{\bm{1}}\widetilde{\bm{1}}^{\top}\|^{2}\|\widetilde{\Lambda}^{k}-\frac{1}{J}\widetilde{\bm{1}}\widetilde{\bm{1}}^{\top}\widetilde{\Lambda}^{k}\|_{F}^{2}\notag\\ &+\frac{(\gamma^{k})^{2}}{v}\|\widetilde{E}-\frac{1}{J}\widetilde{\bm{1}}\widetilde{\bm{1}}^{\top}\|^{2}\|\nabla \widetilde{g}(\widetilde{\Lambda}^{k})-\frac{1}{J}\widetilde{\bm{1}}\widetilde{\bm{1}}^{\top}\nabla \widetilde{g}(\widetilde{\Lambda}^{k})\|_{F}^{2},\notag\end{aligned}$$ where $v\in (0,1)$ is any positive constant. To drive the last inequality, we use the fact that $\|AB\|_{F}^{2}\le \|A\|^{2}\|B\|_{F}^{2}$. By Assumption [Assumption 3](#a3){reference-type="ref" reference="a3"}, $\widetilde{\kappa} := \|\widetilde{E}-\frac{1}{J}\widetilde{\bm{1}}\widetilde{\bm{1}}^{\top}\|^{2} < 1$. Thus, we have $$\begin{aligned} \label{lemma3-appendix1-proof-4} &\|\widetilde{\Lambda}^{k+1}-\frac{1}{J}\widetilde{\bm{1}}\widetilde{\bm{1}}^{\top}\widetilde{\Lambda}^{k+1} \|^{2}_{F}\\ \le& \frac{\widetilde{\kappa}}{1-v}\|\widetilde{\Lambda}^{k}-\frac{1}{J}\widetilde{\bm{1}}\widetilde{\bm{1}}^{\top}\widetilde{\Lambda}^{k}\|_{F}^{2}\notag\\ &+\frac{(\gamma^{k})^{2}\widetilde{\kappa}}{v}\|\nabla \widetilde{g}(\widetilde{\Lambda}^{k})-\frac{1}{J}\widetilde{\bm{1}}\widetilde{\bm{1}}^{\top}\nabla \widetilde{g}(\widetilde{\Lambda}^{k})\|_{F}^{2}.\notag\end{aligned}$$* *We bound the term $\frac{(\gamma^{k})^{2}\widetilde{\kappa}}{v}\|\nabla \widetilde{g}(\widetilde{\Lambda}^{k})-\frac{1}{J}\widetilde{\bm{1}}\widetilde{\bm{1}}^{\top}\nabla \widetilde{g}(\widetilde{\Lambda}^{k})\|_{F}^{2}$ at the right-hand side of [\[lemma3-appendix1-proof-4\]](#lemma3-appendix1-proof-4){reference-type="eqref" reference="lemma3-appendix1-proof-4"} as $$\begin{aligned} \label{lemma3-appendix1-proof-5} &\frac{(\gamma^{k})^{2}\widetilde{\kappa}}{v}\|\nabla \widetilde{g}(\widetilde{\Lambda}^{k})-\frac{1}{J}\widetilde{\bm{1}}\widetilde{\bm{1}}^{\top}\nabla \widetilde{g}(\widetilde{\Lambda}^{k})\|_{F}^{2}\end{aligned}$$ $$\begin{aligned} =&\frac{(\gamma^{k})^{2}\widetilde{\kappa}}{v}\sum_{i\in \mathcal{J}}\|\nabla \widetilde{g}_{i}(\bm{\lambda}_{i}^{k})-\frac{1}{J}\sum_{i\in \mathcal{J}}\nabla \widetilde{g}_{i}(\bm{\lambda}_{i}^{k})\|^{2}\notag\\ =&\frac{(\gamma^{k})^{2}\widetilde{\kappa}}{v}\sum_{i\in \mathcal{J}}\|\nabla \widetilde{g}_{i}(\bm{\lambda}_{i}^{k})-\nabla \widetilde{g}_{i}(\bar{\bm{\lambda}}^{k})+\nabla \widetilde{g}_{i}(\bar{\bm{\lambda}}^{k})\notag\\ &-\frac{1}{J}\sum_{i\in \mathcal{J}}\nabla \widetilde{g}_{i}(\bar{\bm{\lambda}}^{k})+\frac{1}{J}\sum_{i\in \mathcal{J}}\nabla \widetilde{g}_{i}(\bar{\bm{\lambda}}^{k}) -\frac{1}{J}\sum_{i\in \mathcal{J}}\nabla \widetilde{g}_{i}(\bm{\lambda}_{i}^{k})\|^{2} \notag\\ \le&\frac{3(\gamma^{k})^{2}\widetilde{\kappa}}{v}\sum_{i\in \mathcal{J}}\|\nabla \widetilde{g}_{i}(\bm{\lambda}_{i}^{k})-\nabla \widetilde{g}_{i}(\bar{\bm{\lambda}}^{k})\|^{2}\notag\\ &+\frac{3(\gamma^{k})^{2}\widetilde{\kappa}}{v}\sum_{i\in \mathcal{J}}\|\nabla \widetilde{g}_{i}(\bar{\bm{\lambda}}^{k})-\frac{1}{J}\sum_{i\in \mathcal{J}}\nabla \widetilde{g}_{i}(\bar{\bm{\lambda}}^{k})\|^{2}\notag\\ &+\frac{3(\gamma^{k})^{2}\widetilde{\kappa}}{v}\sum_{i\in \mathcal{J}}\|\frac{1}{J}\sum_{i\in \mathcal{J}}\nabla \widetilde{g}_{i}(\bar{\bm{\lambda}}^{k}) -\frac{1}{J}\sum_{i\in \mathcal{J}}\nabla \widetilde{g}_{i}(\bm{\lambda}_{i}^{k})\|^{2}. \notag\end{aligned}$$ According to Lemmas [Lemma 6](#l1-appendix1){reference-type="ref" reference="l1-appendix1"} and [Lemma 7](#l2-appendix1){reference-type="ref" reference="l2-appendix1"}, we have $$\begin{aligned} \label{lemma3-appendix1-proof-6} &\frac{(\gamma^{k})^{2}\widetilde{\kappa}}{v}\|\nabla \widetilde{g}(\widetilde{\Lambda}^{k})-\frac{1}{J}\widetilde{\bm{1}}\widetilde{\bm{1}}^{\top}\nabla \widetilde{g}(\widetilde{\Lambda}^{k})\|_{F}^{2}\\ \le& \frac{6(\gamma^{k})^{2}\widetilde{\kappa}}{v\cdot u_{f}^{2}J^{2}}\sum_{i\in \mathcal{J}}\|\bm{\lambda}_{i}^{k}-\bar{\bm{\lambda}}^{k}\|^{2}+\frac{3(\gamma^{k})^{2}\widetilde{\delta}^{2}\widetilde{\kappa} J}{v}\notag\\ =&\frac{6(\gamma^{k})^{2}\widetilde{\kappa}}{v\cdot u_{f}^{2}J^{2}}\|\widetilde{\Lambda}^{k}-\frac{1}{J}\widetilde{\bm{1}}\widetilde{\bm{1}}^{\top}\widetilde{\Lambda}^{k}\|_{F}^{2}+\frac{3(\gamma^{k})^{2}\widetilde{\delta}^{2}\widetilde{\kappa} J}{v}.\notag\end{aligned}$$* *Substituting [\[lemma3-appendix1-proof-6\]](#lemma3-appendix1-proof-6){reference-type="eqref" reference="lemma3-appendix1-proof-6"} into [\[lemma3-appendix1-proof-4\]](#lemma3-appendix1-proof-4){reference-type="eqref" reference="lemma3-appendix1-proof-4"} and rearranging the terms, we now obtain $$\begin{aligned} \label{lemma3-appendix1-proof-7} &\|\widetilde{\Lambda}^{k+1}-\frac{1}{J}\widetilde{\bm{1}}\widetilde{\bm{1}}^{\top}\widetilde{\Lambda}^{k+1} \|^{2}_{F}\\ \le& (\frac{\widetilde{\kappa}}{1-v}+\frac{6(\gamma^{k})^{2}\widetilde{\kappa}}{v\cdot u_{f}^{2}J^{2}})\|\widetilde{\Lambda}^{k}-\frac{1}{J}\widetilde{\bm{1}}\widetilde{\bm{1}}^{\top}\widetilde{\Lambda}^{k}\|_{F}^{2}+\frac{3(\gamma^{k})^{2}\widetilde{\delta}^{2}\widetilde{\kappa} J}{v}\notag\\ =&(1-\sigma)(\frac{1}{1-v}+\frac{6(\gamma^{k})^{2}}{v\cdot u_{f}^{2}J^{2}})\|\widetilde{\Lambda}^{k}-\frac{1}{J}\widetilde{\bm{1}}\widetilde{\bm{1}}^{\top}\widetilde{\Lambda}^{k}\|_{F}^{2}+\frac{3(\gamma^{k})^{2}\widetilde{\delta}^{2}J}{v},\notag\end{aligned}$$ where $\sigma=1- \widetilde{\kappa} \in (0,1)$. Setting $v=\frac{\sigma}{3}$ and a proper step size $\gamma^{k}$ satisfying $\frac{6(\gamma^{k})^{2}}{ u_{f}^{2}J^{2}}\le\frac{(2-\sigma)\sigma^{2}}{3(3-\sigma)}=\frac{(\sigma-v-v\sigma)v}{1-v}$, we have $\frac{1}{1-v}+\frac{6(\gamma^{k})^{2}}{v\cdot u_{f}^{2}J^{2}}\le 1+\sigma$. With this, we can rewrite [\[lemma3-appendix1-proof-7\]](#lemma3-appendix1-proof-7){reference-type="eqref" reference="lemma3-appendix1-proof-7"} as $$\begin{aligned} \label{lemma3-appendix1-proof-8} &\|\widetilde{\Lambda}^{k+1}-\frac{1}{J}\widetilde{\bm{1}}\widetilde{\bm{1}}^{\top}\widetilde{\Lambda}^{k+1} \|^{2}_{F}\\ \le &(1-\sigma^{2})\|\widetilde{\Lambda}^{k}-\frac{1}{J}\widetilde{\bm{1}}\widetilde{\bm{1}}^{\top}\widetilde{\Lambda}^{k}\|_{F}^{2}+\frac{9(\gamma^{k})^{2}\widetilde{\delta}^{2}J}{\sigma}.\notag\end{aligned}$$ Using [\[lemma3-appendix1-proof-8\]](#lemma3-appendix1-proof-8){reference-type="eqref" reference="lemma3-appendix1-proof-8"} recursively yields $$\begin{aligned} \label{lemma3-appendix1-proof-9} &\|\widetilde{\Lambda}^{k+1}-\frac{1}{J}\widetilde{\bm{1}}\widetilde{\bm{1}}^{\top}\widetilde{\Lambda}^{k+1} \|^{2}_{F}\\ \le &(1-\sigma^{2})^{k+1}\|\widetilde{\Lambda}^{0}-\frac{1}{J}\widetilde{\bm{1}}\widetilde{\bm{1}}^{\top}\widetilde{\Lambda}^{0}\|_{F}^{2}\notag\\ &+\sum_{k'=0}^{k}(1-\sigma^{2})^{k-k'}\cdot(\gamma^{k'})^{2}\cdot\frac{9\widetilde{\delta}^{2}J}{\sigma}.\notag\end{aligned}$$* *With the same initialization $\bm{\lambda}_{i}^{0}$ for all agents $i\in \mathcal{J}$, we can rewrite [\[lemma3-appendix1-proof-9\]](#lemma3-appendix1-proof-9){reference-type="eqref" reference="lemma3-appendix1-proof-9"} as $$\begin{aligned} \label{lemma3-appendix1-proof-10} &\|\widetilde{\Lambda}^{k+1}-\frac{1}{J}\widetilde{\bm{1}}\widetilde{\bm{1}}^{\top}\widetilde{\Lambda}^{k+1} \|^{2}_{F}\\ \le &\sum_{k'=0}^{k}(1-\sigma^{2})^{k-k'}\cdot(\gamma^{k'})^{2}\cdot\frac{9\widetilde{\delta}^{2}J}{\sigma}.\notag\end{aligned}$$ To bound $\sum_{k'=0}^{k}(1-\sigma^{2})^{k-k'}\cdot(\gamma^{k'})^{2}$ in [\[lemma3-appendix1-proof-10\]](#lemma3-appendix1-proof-10){reference-type="eqref" reference="lemma3-appendix1-proof-10"}, we define $\widetilde{y}^{k}$ as $$\begin{aligned} \label{lemma3-appendix1-proof-11} \widetilde{y}^{k}=\sum_{k'=0}^{k-1}(1-\sigma^{2})^{k-1-k'}\cdot (\gamma^{k'})^{2},\end{aligned}$$ which satisfies the relation $\widetilde{y}^{k+1}=(1-\sigma^{2})\widetilde{y}^{k}+(\gamma^{k})^{2}$.* *Substituting $\widetilde{y}^{k+1} = y^{k+1}$, $\psi_{1}=1-\sigma^{2}\in (0,1)$, $\psi_{2}=1\ge 0$ and $\widetilde{y}^{0}=0\le (\gamma^{0})^{2}$ into Lemma [Lemma 5](#l4-appendix1){reference-type="ref" reference="l4-appendix1"}, for any integer $k \ge 0$ and the step size $\gamma^{k}$ satisfying $1\le \frac{(\gamma^{k})^{2}}{(\gamma^{k+1})^2}\le \frac{2}{1+(1-\sigma^{2})}$, we have $$\begin{aligned} \label{lemma3-appendix1-proof-12} \widetilde{y}^{k+1} & =\sum_{k'=0}^{k}(1-\sigma^{2})^{k-k'}\cdot (\gamma^{k'})^{2} \\ & \le \frac{2}{1-(1-\sigma^{2})}(\gamma^{k+1})^{2} = \frac{2}{\sigma^{2}}(\gamma^{k+1})^{2}.\notag\end{aligned}$$ With [\[lemma3-appendix1-proof-12\]](#lemma3-appendix1-proof-12){reference-type="eqref" reference="lemma3-appendix1-proof-12"}, we can rewrite [\[lemma3-appendix1-proof-10\]](#lemma3-appendix1-proof-10){reference-type="eqref" reference="lemma3-appendix1-proof-10"} as $$\begin{aligned} \label{lemma3-appendix1-proof-13} \|\widetilde{\Lambda}^{k+1}-\frac{1}{J}\widetilde{\bm{1}}\widetilde{\bm{1}}^{\top}\widetilde{\Lambda}^{k+1} \|^{2}_{F}\le \frac{18(\gamma^{k+1})^{2}\widetilde{\delta}^{2}J}{\sigma^{3}},\end{aligned}$$ which completes the proof.* 1 R. 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[^1]: Runhua Wang and Qing Ling are with the School of Computer Science and Engineering and the Guangdong Provincial Key Laboratory of Computational Science, Sun Yat-Sen University, Guangzhou, Guangdong 510006, China. Zhi Tian is with the Department of Electrical and Computer Engineering, George Mason University, Fairfax, VA 22030, USA. Corresponding author: Qing Ling (lingqing556\@mail.sysu.edu.cn). [^2]: Qing Ling is supported by National Natural Science Foundation of China grants 61973324, 12126610 and 62373388, Guangdong Basic and Applied Basic Research Foundation grant 2021B1515020094, and Guangdong Provincial Key Laboratory of Computational Science grant 2020B1212060032. A short, preliminary version of this paper has been submitted to ICASSP 2024 [@R-Wang-2024-icassp].

arxiv_math

--- abstract: | Let $(V,E)$ be a finite connected graph. We are concerned about the Chern-Simons Higgs model $$\Delta u=\lambda e^u(e^u-1)+f, \eqno{(0.1)}$$ where $\Delta$ is the graph Laplacian, $\lambda$ is a real number and $f$ is a function on $V$. When $\lambda>0$ and $f=4\pi\sum_{i=1}^N\delta_{p_i}$, $N\in\mathbb{N}$, $p_1,\cdots,p_N\in V$, the equation (0.1) was investigated by Huang, Lin, Yau (Commun. Math. Phys. 377 (2020) 613-621) and Hou, Sun (Calc. Var. 61 (2022) 139) via the upper and lower solutions principle. We now consider an arbitrary real number $\lambda$ and a general function $f$, whose integral mean is denoted by $\overline{f}$, and prove that when $\lambda\overline{f}<0$, the equation $(0.1)$ has a solution; when $\lambda\overline{f}>0$, there exist two critical numbers $\Lambda^\ast>0$ and $\Lambda_\ast<0$ such that if $\lambda\in(\Lambda^\ast,+\infty)\cup(-\infty,\Lambda_\ast)$, then $(0.1)$ has at least two solutions, including one local minimum solution; if $\lambda\in(0,\Lambda^\ast)\cup(\Lambda_\ast,0)$, then $(0.1)$ has no solution; while if $\lambda=\Lambda^\ast$ or $\Lambda_\ast$, then $(0.1)$ has at least one solution. Our method is calculating the topological degree and using the relation between the degree and the critical group of a related functional. Similar method is also applied to the Chern-Simons Higgs system, and a partial result for the multiple solutions of the system is obtained. address: - School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, China - School of Mathematics and Statistics, Guangxi Normal University, Guilin 541004, China - School of Mathematics, Renmin University of China, Beijing 100872, China author: - Jiayu Li - Linlin Sun - Yunyan Yang title: Topological degree for Chern-Simons Higgs models on finite graphs --- Topological degree,Chern-Simons Higgs modle,Finite graph\ 39A12,46E39 # Introduction The Chern-Simons Higgs model, introduced by Hong, Kim, Pac [@HongKP] and Jackiw, Weinberg [@JackiwW], has always attracted the attention of many mathematicians in the fields of geometry and physics, see for examples [@Carffarali; @Chae; @DJLW1; @DJLW2; @LinPY; @Nolasco; @Nolasco-2; @Tarantello; @Wang]. Among many versions, the self-dual Chern-Simons Higgs vortex equation on a flat 2-torus $\Sigma$ can be written as $$\begin{aligned} \label{eeq-1}\Delta u=\frac{4}{k^2}e^u(e^u-1)+4\pi\sum_{i=1}^{k_0}m_i\delta_{p_i},\end{aligned}$$ where $k>0$ is the Chern-Simons constant, $m_i\in\mathbb{N}$, $p_i\in\Sigma$, $i=1,\cdots,k_0$. The solution of the above equation is called a vertex solution, each $p_i$ is called a vertex point, and $m_i$ stands for the multiplicity of $p_i$. From the view of physics, the vortex points are closely related to the local maximum point of the magnetic flux in the Chern-Simons Higgs model. Let $u_0$ be a solution of $$\left\{\begin{array}{lll}\Delta u_0=-\frac{4\pi N}{|\Sigma|}+4\pi\sum_{i=1}^{k_0}m_i\delta_{p_i}\\[1.2ex] \int_\Sigma u_0dv_g=0,\end{array}\right.$$ where $N=\sum_{i=1}^{k_0}m_i$. Set $v=u-u_0$. Then ([\[eeq-1\]](#eeq-1){reference-type="ref" reference="eeq-1"}) can be written in a more favourable form $$\begin{aligned} \label{eeq-2}\Delta v=\lambda he^v(he^v-1)+\frac{4\pi N}{|\Sigma|},\end{aligned}$$ where $\lambda=\frac{4}{k^2}$ and $h=e^{u_0}$ is a positive function on $\Sigma$. A solution $v$ of ([\[eeq-2\]](#eeq-2){reference-type="ref" reference="eeq-2"}) is called of finite energy if $v\in W^{1,2}(\Sigma)$, a usual Sobolev space. Indeed, it is known that the corresponding physical energy of the solution $v$ is finite if $u\in W^{1,2}(\Sigma)$. Thus, solutions of finite energy are physically meaningful in ([\[eeq-2\]](#eeq-2){reference-type="ref" reference="eeq-2"}) and there have been many existence results for $W^{1,2}(\Sigma)$ solutions of ([\[eeq-2\]](#eeq-2){reference-type="ref" reference="eeq-2"}), see [@Carffarali; @DJLW-0; @DJLW1; @Nolasco; @Nolasco-2; @Tarantello; @Wangmeng-2; @Wangmeng] and the references therein. By using the principle of upper and lower solutions, Caffarelli and Yang constructed a maximal solution. In addition to the above references, [@DJLW2; @lanli] also indicated that the equation ([\[eeq-2\]](#eeq-2){reference-type="ref" reference="eeq-2"}) admits a variational structure. Different from the theoretical significance on Riemann surfaces, the analysis on graphs is very important for applications, such as image processing, data mining, network and so on. Among lots of directions, partial differential equations arising in geometry or physics are worth studying on graphs. Various equations, including the heat equation [@Horn; @Huang; @Lin1; @Lin2], the Fokker-Planck and Schrödinger equations [@Chow1; @Chow2], have been studied by many mathematicians. In particular, Grigor'yan, Lin and Yang [@Gri1; @Gri2; @Gri3] studied the existence of solutions for a series of nonlinear elliptic equations on graphs by using the variational methods. In this direction, Zhang, Zhao, Han and Shao [@Han2; @Han; @ZhangZhao] obtained nontrivial solutions to nonlinear Schrödinger equations with potential wells. Similar problems on infinite metric graphs were studied by Akduman-Pankov [@Akduman-Pankov]. The Kazdan-Warner equation was extended by Keller-Schwarz [@Keller-Schwarz] to canonically compactifiable graphs. Semi-linear heat equations on locally finite graphs were studied by Ge, Jiang, Lin and Wu [@Ge-Jiang; @Lin1; @Lin2]. For other related works, we refer the readers to [@Ge-Proc; @SunYH; @Hua; @Hua-2; @Lin-Yang1; @Lin-Yang2; @LiuYang; @LiuY; @ref16; @Zhang-Lin; @Zhang-Lin2; @zhuxb] and the references therein. To describe the Chern-Simons Higgs model in the graph setting, we introduce some notations. Let $(V,E)$ be a connected finite graph, where $V$ is the set of vertices and $E$ is the set of edges. Let $\mu :V\rightarrow (0,+\infty)$ and $\{w_{xy}:xy\in E\}$ be its measure and weights respectively. The weight $w_{xy}$ is always assumed to be positive and symmetric. The Laplacian of a function $u: V\rightarrow \mathbb{R}$ reads as $$\Delta u(x)=\frac{1}{\mu (x)} \sum_{y\sim x} w_{xy}(u(y)-u(x)),$$ where $y\sim x$ means $y$ is adjacent to $x$, i.e. $xy\in E$. The gradient of $u$ is defined as $$\nabla u(x)=\left(\sqrt{\frac{w_{xy_1}}{2\mu(x)}}(u(y_1)-u(x)),\cdots,\sqrt{\frac{w_{xy_{\ell_x}}}{2\mu(x)}}(u(y_{\ell_x})-u(x))\right),$$ where $\{y_1,\cdots,y_{\ell_x}\}$ are all distinct points adjacent to $x$. Clearly, such an $\ell_x$ is unique and $\nabla u(x)\in\mathbb{R}^{\ell_x}$. The integral of $u$ is given by $$\int_{V}ud\mu=\sum_{x\in V}\mu(x)u(x).$$ Now we consider an analog of ([\[eeq-2\]](#eeq-2){reference-type="ref" reference="eeq-2"}) on a connected finite graph, namely $$\begin{aligned} \label{eq-1}\Delta u=\lambda e^u(e^u-1)+f\quad{\rm in}\quad V,\end{aligned}$$ where $\lambda\in \mathbb{R}$, $f:V\rightarrow\mathbb{R}$ is a function. It was proved by Huang, Lin and Yau [@LinYau] that if $\lambda>0$ and $f=4\pi\sum_{i=1}^N\delta_{p_i}$, there exists a critical number $\lambda^\ast>0$ such that ([\[eq-1\]](#eq-1){reference-type="ref" reference="eq-1"}) has a solution when $\lambda>\lambda^\ast$, while ([\[eq-1\]](#eq-1){reference-type="ref" reference="eq-1"}) has no solution when $0<\lambda<\lambda^\ast$. The critical case $\lambda=\lambda^\ast$ was solved by Hou and Sun [@HouSun], who proved that ([\[eq-1\]](#eq-1){reference-type="ref" reference="eq-1"}) has also a solution. Such results are essentially based on the method of upper and lower solutions principle. This together with variational method may lead to existence results for other forms of Chern-Simons Higgs models, see Chao and Hou [@Hou-2]. Recently, a more delicate analysis was employed by Huang, Wang and Yang [@HuangWY] to get existence of solutions of the Chern-Simons Higgs system. Topological degree theory is a powerful tool in studying partial differential equations in the Euclidean space or Riemann surfaces, see for example Li [@Liyanyan]. It was first used by Sun and Wang [@SunW] to solve the Kazdan-Warner equation on finite graphs. Very recently, it was also employed by Liu [@LiuY-3] to deal with the mean field equation. Our aim is to use this powerful tool to study the Chern-Simons Higgs model. The first and most important step is to get a priori estimate for solutions, say **Theorem 1**. *Let $(V,E)$ be a connected finite graph with symmetric weights, i.e. $w_{xy}=w_{yx}$ for all $xy\in E$. Let $\sigma\in[0,1]$, $\lambda$ and $f$ satisfy $$\begin{aligned} \label{cond-0}\Lambda^{-1}\leq |\lambda|\leq \Lambda,\,\,\, \Lambda^{-1}\leq \left|\int_Vfd\mu\right|\leq \Lambda,\,\,\,\|f\|_{L^\infty(V)}\leq \Lambda\end{aligned}$$ for some real number $\Lambda>0$. If $u$ is a solution of $$\begin{aligned} \label{eq-hom}\Delta u=\lambda e^{u}(e^u-\sigma)+f\quad{\rm in}\quad V,\end{aligned}$$ then there exists a constant $C$, depending only on $\Lambda$ and the graph $V$, such that $|u(x)|\leq C$ for all $x\in V$.* When $\sigma=1$, the equation ([\[eq-hom\]](#eq-hom){reference-type="ref" reference="eq-hom"}) is exactly ([\[eq-1\]](#eq-1){reference-type="ref" reference="eq-1"}). In the case $\lambda>0$ and $f=4\pi\sum_{i=1}^N\delta_{p_i}$, where $p_1,\cdots,p_N\in V$ and $N\in\mathbb{N}$, let $\lambda^\ast$ be the critical number in [@LinYau]. Then for any $\lambda_k>\lambda^\ast$ with $\lambda_k\rightarrow\lambda^\ast$ as $k\rightarrow\infty$, there exists a solution $u_{\lambda_k}$ of ([\[eq-1\]](#eq-1){reference-type="ref" reference="eq-1"}) with $\lambda=\lambda_k$, $k=1,2,\cdots$. It follows from Theorem [Theorem 1](#prior-1){reference-type="ref" reference="prior-1"} that $(u_{\lambda_k})$ is uniformly bounded in $V$. Hence up to a subsequence, $(u_{\lambda_k})$ uniformly converges to some $u^\ast$, which is a solution of ([\[eq-1\]](#eq-1){reference-type="ref" reference="eq-1"}) with $\lambda=\lambda^\ast$. This gives another proof of a result of Hou and Sun [@HouSun]. Denote $X=L^\infty(V)$ and define a map $F:X\rightarrow X$ by $$\begin{aligned} \label{map}F(u)=-\Delta u+\lambda e^{u}(e^u-1)+f.\end{aligned}$$ The second step is to calculate the topological degree of $F$ by using its homotopic invariance property. **Theorem 2**. *Let $(V,E)$ be a connected finite graph with symmetric weights, and $F:X\rightarrow X$ be a map defined by ([\[map\]](#map){reference-type="ref" reference="map"}). Suppose that $\lambda\int_V{f}d\mu\not=0$. Then there exists a large number $R_0>0$ such that for all $R\geq R_0$, $$\deg(F,B_R,0)=\left\{ \begin{array}{lll} 1&{\rm if}& \lambda>0,\,\int_V{f}d\mu<0\\[1.2ex] 0&{\rm if}& \lambda\int_V{f}d\mu>0\\[1.2ex] -1&{\rm if}& \lambda<0,\,\int_V{f}d\mu>0, \end{array}\right.$$ where $B_R=\{u\in X:\|u\|_{L^\infty(V)}<R\}$ is a ball in $X$.* As an application of the above topological degree, our existence results for the Chern-Simons Higgs model read as follows: **Theorem 3**. *Let $(V,E)$ be a connected finite graph with symmetric weights. Then we have the following:\ $(\mathsf{a})$ If $\lambda\int_Vfd\mu<0$, then the equation ([\[eq-1\]](#eq-1){reference-type="ref" reference="eq-1"}) has a solution;\ $(\mathsf{b})$ If $\lambda\int_Vfd\mu>0$, then two subcases are distinguished: $(i)$ $\int_Vfd\mu>0$. There exists a real number $\Lambda^\ast>0$ such that when $\lambda>\Lambda^\ast$, ([\[eq-1\]](#eq-1){reference-type="ref" reference="eq-1"}) has at least two different solutions; when $0<\lambda<\Lambda^\ast$, ([\[eq-1\]](#eq-1){reference-type="ref" reference="eq-1"}) has no solution; when $\lambda=\Lambda^\ast$, ([\[eq-1\]](#eq-1){reference-type="ref" reference="eq-1"}) has at least one solution; $(ii)$ $\int_Vfd\mu<0$. There exists a real number $\Lambda_\ast<0$ such that when $\lambda<\Lambda_\ast$, ([\[eq-1\]](#eq-1){reference-type="ref" reference="eq-1"}) has at least two different solutions; when $\Lambda_\ast<\lambda<0$, ([\[eq-1\]](#eq-1){reference-type="ref" reference="eq-1"}) has no solution; when $\lambda=\Lambda_\ast$, ([\[eq-1\]](#eq-1){reference-type="ref" reference="eq-1"}) has at least one solution.* We remark that Case $(\mathsf{b})$ $(i)$ includes $\lambda>0$ and $f=4\pi\sum_{i=1}^N\delta_{p_i}$ as a special case, which was studied in [@Hou-2; @HouSun; @LinYau; @HuangWY]. In the subcase $\lambda>\Lambda^\ast>0$ or $\lambda<\Lambda_\ast<0$, we shall construct a local minimum solution, and then use the topological degree to obtain the existence of another solution. Our arguments are essentially different from those in [@Hou-2; @HuangWY; @LiuYang]. Note that a solution of ([\[eq-1\]](#eq-1){reference-type="ref" reference="eq-1"}) is a critical point of the functional $J_\lambda:X\rightarrow\mathbb{R}$ defined by $$\begin{aligned} \label{funct-1}J_\lambda(u)=\frac{1}{2}\int_V|\nabla u|^2d\mu+\frac{\lambda}{2}\int_V(e^u-1)^2d\mu+\int_Vfud\mu.\end{aligned}$$ Here a local minimum solution of ([\[eq-1\]](#eq-1){reference-type="ref" reference="eq-1"}) means a local minimum critical point of $J_\lambda$.\ Also we consider the Chern-Simons Higgs system $$\begin{aligned} \label{system} \left\{ \begin{array}{lll} \Delta u=\lambda e^v(e^u-1)+f\\[1.2ex] \Delta v=\lambda e^u(e^v-1)+g, \end{array}\right.\end{aligned}$$ where $\lambda$ is a real number, and $f,g$ are functions on $V$. Similar to the single equation, we need also a priori estimate. **Theorem 4**. *Let $(V,E)$ be a connected finite graph with symmetric weights. Suppose that $\sigma\in[0,1]$, $\lambda,\eta$ are two positive real numbers, $f,g$ are two functions verifying that $\int_V{f}d\mu>0$ and $\int_V{g}d\mu>0$. If $(u,v)$ is a solution of the system $$\begin{aligned} \label{system-8} \left\{ \begin{array}{lll} \Delta u=\lambda e^v(e^u-\sigma)+f\\[1.2ex] \Delta v=\eta e^u(e^v-\sigma)+g, \end{array}\right.\end{aligned}$$ then there exists a constant $C$, depending only on $\lambda,\eta,f,g$ and the graph $V$, such that $$\|u\|_{L^\infty(V)}+\|v\|_{L^\infty(V)}\leq C.$$* To compute the topological degree, we define a map $\mathcal{F}:X\times X\rightarrow X\times X$ by $$\begin{aligned} \label{map-syst}\mathcal{F}(u,v)=(-\Delta u+\lambda e^{v}(e^u-1)+f,-\Delta v+\eta e^u(e^v-1)+g).\end{aligned}$$ **Theorem 5**. *Let $(V,E)$ be a connected finite graph with symmetric weights, and $\mathcal{F}$ be a map defined by ([\[map-syst\]](#map-syst){reference-type="ref" reference="map-syst"}). If $\lambda>0$, $\eta>0$, $\int_Vfd\mu>0$ and $\int_Vgd\mu>0$, then there exists a large number $R_0>0$ such that for all $R\geq R_0$, $$\deg(\mathcal{F},B_R,(0,0))=0,$$ where $B_R=\{(u,v)\in X\times X:\|u\|_{L^\infty(V)}+\|v\|_{L^\infty(V)}<R\}$ is a ball in $X\times X$.* Define a functional $\mathcal{J}_\lambda:X\times X\rightarrow\mathbb{R}$ by $$\begin{aligned} \label{functional}\mathcal{J}_\lambda(u,v)=\int_V\nabla u\nabla vd\mu+\lambda\int_V(e^u-1)(e^v-1)d\mu+\int_V(fv+gu)d\mu.\end{aligned}$$ Note that for all $(\phi,\psi)\in X\times X$, $$\begin{aligned} \nonumber \langle\mathcal{J}_\lambda^\prime(u,v),(\phi,\psi)\rangle&=&\left.\frac{d}{dt}\right|_{t=0}\mathcal{J}(u+t\phi,v+t\psi)\\\label{derivative} &=&\int_V\left\{\left(-\Delta v+\lambda e^u(e^v-1)+g\right)\phi+\left(-\Delta u+\lambda e^v(e^u-1)+f\right)\psi\right\}d\mu.\end{aligned}$$ Clearly $(u,v)$ is a critical point of $\mathcal{J}_\lambda$ if and only if it is a solution of the system ([\[system\]](#system){reference-type="ref" reference="system"}). As a consequence of Theorem [Theorem 5](#degree-system){reference-type="ref" reference="degree-system"}, we have the following **Theorem 6**. *Let $(V,E)$ be a connected finite graph with symmetric weights, $\lambda>0$, $\int_Vfd\mu>0$, $\int_Vgd\mu>0$, and $\mathcal{J}_\lambda$ be a functional defined by ([\[functional\]](#functional){reference-type="ref" reference="functional"}). If either $\mathcal{J}_\lambda$ has a non-degenerate critical point, or $\mathcal{J}_\lambda$ has a local minimum critical point, then it must have another critical point.* It should be remarked that Theorem [Theorem 6](#syst-thm){reference-type="ref" reference="syst-thm"} gives another solution of ([\[system\]](#system){reference-type="ref" reference="system"}) under the condition that $\mathcal{J}_\lambda$ has a non-degenerate or a local minimum critical point beforehand. So it is only a partial result for the problem of multiple solutions of the system ([\[system\]](#system){reference-type="ref" reference="system"}).\ The remaining part of this paper is organized as follows: In Section [2](#sec-apri){reference-type="ref" reference="sec-apri"}, we give a priori estimate for solutions of ([\[eq-1\]](#eq-1){reference-type="ref" reference="eq-1"}) (Theorem [Theorem 1](#prior-1){reference-type="ref" reference="prior-1"}); The topological degree of $F:X\rightarrow X$ (Theorem [Theorem 2](#degree){reference-type="ref" reference="degree"}) was calculated in Section [3](#sec-Brouwer){reference-type="ref" reference="sec-Brouwer"}; In Section [4](#sec-existence){reference-type="ref" reference="sec-existence"}, we prove the existence result (Theorem [Theorem 3](#existence){reference-type="ref" reference="existence"}); The priori estimate and existence of solutions of the Chern-Simons Higgs system (Theorems [Theorem 4](#system-apriori){reference-type="ref" reference="system-apriori"}-[Theorem 6](#syst-thm){reference-type="ref" reference="syst-thm"}) are discussed in Section [5](#sec-system){reference-type="ref" reference="sec-system"}. # A priori estimate {#sec-apri} In this section, we shall prove Theorem [Theorem 1](#prior-1){reference-type="ref" reference="prior-1"}. In order to provide readers with a clear understanding of the proof, we demonstrate the entire process from simple cases to complex cases. Precisely the proof will be divided into several lemmas as below.\ The first priori estimate is for fixed $\lambda$ and $f$. **Lemma 7**. *Suppose that $u$ is a solution of ([\[eq-1\]](#eq-1){reference-type="ref" reference="eq-1"}), where $\lambda\not= 0$ and $\int_Vfd\mu\not=0$. Then there exists a constant $C$, depending only on $\lambda$, $f$ and the graph $V$, such that $|u(x)|\leq C$ for all $x\in V$.* If $u$ is a solution of ([\[eq-1\]](#eq-1){reference-type="ref" reference="eq-1"}), then integration by parts gives $$\begin{aligned} \label{eq-2}0=\int_V\Delta ud\mu=\lambda\int_Ve^u(e^u-1)d\mu+\int_Vfd\mu.\end{aligned}$$ Firstly, we show that $u$ has a uniform upper bound. With no loss of generality, we may assume $\max_Vu>0$. For otherwise, $u$ has already upper bound 0. Observing $$\left|\int_{u<0}e^u(e^u-1)d\mu\right|\leq \frac{1}{4}|V|,$$ we derive from ([\[eq-2\]](#eq-2){reference-type="ref" reference="eq-2"}) that $$\int_{u\geq 0}e^u(e^u-1)d\mu\leq a:=\frac{1}{4}|V|+\frac{1}{|\lambda|}\left|\int_Vfd\mu\right|.$$ This together with the fact $$\int_{u\geq 0}e^u(e^u-1)d\mu=\sum_{x\in V,\,u(x)\geq 0}\mu(x)e^{u(x)}(e^{u(x)}-1) \geq\mu_0e^{\max_Vu}(e^{\max_Vu}-1)$$ leads to $$\begin{aligned} \label{upper}\max_V u\leq \log\frac{1+\sqrt{1+4a/\mu_0}}{2},\end{aligned}$$ where $\mu_0=\min_{x\in V}\mu(x)>0$, since $V$ is finite. Secondly, we prove that $u$ has also a uniform lower bound. To see this, in view of ([\[eq-1\]](#eq-1){reference-type="ref" reference="eq-1"}) and ([\[upper\]](#upper){reference-type="ref" reference="upper"}), we calculate for any $x\in V$, $$\begin{aligned} |\Delta u(x)|&\leq&|\lambda|\left|e^{u(x)}(e^{u(x)}-1)\right|+|f(x)|\\ &\leq&|\lambda|(e^{2u(x)}+e^{u(x)})+|f(x)|\\ &\leq&|\lambda|\left(\frac{(1+\sqrt{1+4a/\mu_0})^2}{4}+\frac{1+\sqrt{1+4a/\mu_0}}{2}\right)+\|f\|_{L^\infty(V)}\\ &=:&b.\end{aligned}$$ Hence, there holds $$\begin{aligned} \label{up-2-0}\|\Delta u\|_{L^\infty(V)}\leq b.\end{aligned}$$ We may assume $V=\{x_1,\cdots,x_\ell\}$, $u(x_{1})=\max_Vu$, $u(x_{\ell})=\min_Vu$, and without loss of generality $x_1x_2, x_2x_3,\cdots, x_{\ell-1}x_\ell$ is the shortest path connecting $x_1$ and $x_\ell$. It follows that $$\begin{aligned} \nonumber 0\leq u(x_1)-u(x_\ell)&\leq&\sum_{j=1}^{\ell-1}|u(x_j)-u(x_{j+1})|\\\nonumber &\leq&\frac{\sqrt{\ell-1}}{\sqrt{w_0}}\left(\sum_{j=1}^{\ell-1}w_{x_jx_{j+1}}(u(x_j)-u(x_{j+1}))^2\right)^{1/2}\\\label{ineq-0} &\leq&\frac{\sqrt{\ell-1}}{\sqrt{w_0}}\left(\int_V|\nabla u|^2d\mu\right)^{1/2},\end{aligned}$$ where $w_0=\min_{x\in V,\,y\sim x} w_{xy}>0$. Denoting $\overline{u}=\frac{1}{|V|}\int_Vud\mu$, we obtain by integration by parts $$\begin{aligned} \int_V|\nabla u|^2d\mu&=&-\int_V(u-\overline{u})\Delta ud\mu\\ &\leq&\left(\int_V(u-\overline{u})^2d\mu\right)^{1/2}\left(\int_V(\Delta u)^2d\mu\right)^{1/2}\\ &\leq&\left(\frac{1}{\lambda_1}\int_V|\nabla u|^2d\mu\right)^{1/2}\left(\int_V(\Delta u)^2d\mu\right)^{1/2},\end{aligned}$$ which gives $$\begin{aligned} \label{ineq-1}\int_V|\nabla u|^2d\mu\leq \frac{1}{\lambda_1}\int_V(\Delta u)^2d\mu\leq \frac{1}{\lambda_1} \|\Delta u\|_{L^\infty(V)}^2|V|,\end{aligned}$$ where $\lambda_1=\inf_{\overline{v}=0,\int_Vv^2d\mu=1}\int_V|\nabla v|^2d\mu>0$. Combining ([\[ineq-0\]](#ineq-0){reference-type="ref" reference="ineq-0"}) and ([\[ineq-1\]](#ineq-1){reference-type="ref" reference="ineq-1"}), we conclude $$\begin{aligned} \label{equiv}\max_Vu-\min_V u\leq \sqrt{\frac{(\ell-1)|V|}{w_0\lambda_1}}\|\Delta u\|_{L^\infty(V)}.\end{aligned}$$ We remark that ([\[equiv\]](#equiv){reference-type="ref" reference="equiv"}) holds for arbitrary function $u$, such an inequality was obtained by Sun and Wang [@SunW] by using the equivalence of all norms in a finite dimensional vector space, and here we give an explicit constant instead of $C$. The power of ([\[equiv\]](#equiv){reference-type="ref" reference="equiv"}) is evident. In view of ([\[up-2-0\]](#up-2-0){reference-type="ref" reference="up-2-0"}), we have $$\begin{aligned} \label{scope}\max_Vu-\min_V u\leq c_0:=b\sqrt{\frac{(\ell-1)|V|}{w_0\lambda_1}}.\end{aligned}$$ Coming back to ([\[eq-2\]](#eq-2){reference-type="ref" reference="eq-2"}), we have $$\begin{aligned} \label{eq-3}\int_Ve^u(e^u-1)d\mu=c_1:=-\frac{1}{\lambda}\int_Vfd\mu.\end{aligned}$$ By the assumptions $\lambda\not=0$ and $\int_Vfd\mu\not=0$, we know $c_1\not=0$. Now we *claim* that $$\begin{aligned} \label{A}\max_Vu> -A:=\log\min\left\{1,\frac{|c_1|}{4|V|}\right\}.\end{aligned}$$ For otherwise, $\max_Vu\leq -A$, which together with ([\[eq-3\]](#eq-3){reference-type="ref" reference="eq-3"}) implies $$\begin{aligned} |c_1|&=&\left|\int_Ve^u(e^u-1)d\mu\right|\\ &\leq&\int_V(e^{2u}+e^u)d\mu\\ &\leq&(e^{2\max_Vu}+e^{\max_Vu})|V|\\ &\leq& 2e^{-A}|V|\\ &<&\frac{|c_1|}{2}.\end{aligned}$$ This contradicts $c_1\not=0$, and thus confirms our claim ([\[A\]](#A){reference-type="ref" reference="A"}). Inserting ([\[A\]](#A){reference-type="ref" reference="A"}) into ([\[scope\]](#scope){reference-type="ref" reference="scope"}), we obtain $$-A-c_0\leq \min_Vu\leq\max_V u\leq \log\frac{1+\sqrt{1+4a/\mu_0}}{2},$$ as we desired. $\hfill\Box$\ The second priori estimate is for the changing $\lambda$ and $f$. **Lemma 8**. *Let $u$ be a solution of ([\[eq-1\]](#eq-1){reference-type="ref" reference="eq-1"}). If $\lambda$ and $f$ satisfy ([\[cond-0\]](#cond-0){reference-type="ref" reference="cond-0"}), then there exists a constant $C$, depending only on $\Lambda$ and the graph $V$, such that $|u(x)|\leq C$ for all $x\in V$.* It suffices to modify the argument in the proof of Lemma [Lemma 7](#simp-1){reference-type="ref" reference="simp-1"}. Similar to ([\[upper\]](#upper){reference-type="ref" reference="upper"}), we first have the upper bound estimate $$\begin{aligned} \label{ineq-2}\max_V u\leq \log\frac{1+\sqrt{1+4a/\mu_0}}{2},\end{aligned}$$ where $\mu_0=\min_{x\in V}\mu(x)$ and $a=|V|+\Lambda^2$. Next, instead of ([\[scope\]](#scope){reference-type="ref" reference="scope"}), we have $$\begin{aligned} \label{ineq-3}\max_Vu-\min_V u\leq c_0=b\sqrt{\frac{(\ell-1)|V|}{w_0\lambda_1}},\end{aligned}$$ where $\lambda_1=\inf_{\overline{v}=0,\int_Vv^2d\mu=1}\int_V|\nabla v|^2d\mu$, $\ell$ denotes the number of all points of $V$, $w_0=\min_{x\in V,\,y\sim x}w_{xy}$ and $$b=\Lambda\left(\frac{(1+\sqrt{1+4a/\mu_0})^2}{4}+\frac{1+\sqrt{1+4a/\mu_0}}{2}+1\right).$$ To proceed, we shall show $$\begin{aligned} \label{ineq-4}\max_Vu> -A=\log\min\left\{1,\frac{1}{4|V|\Lambda^2}\right\}.\end{aligned}$$ Suppose not. We have $\max_Vu\leq -A$ and $$\begin{aligned} \frac{1}{\Lambda^2}\leq \left|\frac{1}{\lambda}\int_Vfd\mu\right|&=&\left|\int_Ve^u(e^u-1)d\mu\right|\\ &\leq&\int_V(e^{2u}+e^u)d\mu\\ &\leq& 2e^{-A}|V|\\ &<&\frac{1}{2\Lambda^2},\end{aligned}$$ which is impossible. Thus ([\[ineq-4\]](#ineq-4){reference-type="ref" reference="ineq-4"}) holds. Combining ([\[ineq-2\]](#ineq-2){reference-type="ref" reference="ineq-2"}), ([\[ineq-3\]](#ineq-3){reference-type="ref" reference="ineq-3"}) and ([\[ineq-4\]](#ineq-4){reference-type="ref" reference="ineq-4"}), we get the desired result. $\hfill\Box$\ The third priori estimate is not only for changing $\lambda$ and $f$, but also for the changing parameter $\sigma$. **Lemma 9**. *Let $\sigma\in [0,1]$, $\lambda$ and $f$ satisfy ([\[cond-0\]](#cond-0){reference-type="ref" reference="cond-0"}) for some real number $\Lambda>0$. If $u$ is a solution of ([\[eq-hom\]](#eq-hom){reference-type="ref" reference="eq-hom"}), then there exists a constant $C$, depending only on $\Lambda$ and the graph $V$, such that $|u(x)|\leq C$ for all $x\in V$.* If $u$ is a solution of ([\[eq-hom\]](#eq-hom){reference-type="ref" reference="eq-hom"}), then integration by parts gives $$0=\int_V\Delta ud\mu=\lambda\int_Ve^u(e^u-\sigma)d\mu+\int_Vfd\mu.$$ Similar to ([\[upper\]](#upper){reference-type="ref" reference="upper"}), keeping in mind $\sigma\in[0,1]$, we first have the same upper bound estimate as ([\[ineq-2\]](#ineq-2){reference-type="ref" reference="ineq-2"}), namely $$\max_V u\leq \log\frac{1+\sqrt{1+4a/\mu_0}}{2},$$ where $\mu_0=\min_{x\in V}\mu(x)$ and $a=|V|+\Lambda^2$. Next, we have the same estimates as ([\[ineq-3\]](#ineq-3){reference-type="ref" reference="ineq-3"}) and ([\[ineq-4\]](#ineq-4){reference-type="ref" reference="ineq-4"}), which is independent of the parameter $\sigma\in[0,1]$. In particular $$\max_Vu> -A=\log\min\left\{1,\frac{1}{4|V|\Lambda^2}\right\}.$$ This ends the proof of the lemma, and completes the proof of Theorem [Theorem 1](#prior-1){reference-type="ref" reference="prior-1"}. $\hfill\Box$ # Topological degree {#sec-Brouwer} In this section, we shall prove Theorem [Theorem 2](#degree){reference-type="ref" reference="degree"}. Precisely we shall compute the topological degree of certain maps related to the Chern-Simons Higgs model.\ *Proof of Theorem [Theorem 2](#degree){reference-type="ref" reference="degree"}.* Assume $V=\{x_1,\cdots,x_\ell\}$. let $X=L^\infty(V)$. We may identify $X$ with the Euclidean space $\mathbb{R}^\ell$. Without causing ambiguity, we define a map $F:X\times[0,1]\rightarrow X$ by $$F(u,\sigma)=-\Delta u+\lambda e^u(e^u-\sigma)+f,\quad (u,\sigma)\in X\times [0,1].$$ Obviously, $F$ is a smooth map. For the fixed real number $\lambda$ and the fixed function $f$, since $\lambda\overline{f}\not=0$, there must exist a large number $\Lambda>0$ such that $$\begin{aligned} \label{bound}\Lambda^{-1}\leq |\lambda|\leq \Lambda,\,\,\, \Lambda^{-1}\leq \left|\int_Vfd\mu\right|\leq \Lambda,\,\,\,\|f\|_{L^\infty(V)}\leq \Lambda.\end{aligned}$$ Here and in the sequel, $\overline{f}$ denotes the integral mean of a function $f$. Then it follows from Theorem [Theorem 1](#prior-1){reference-type="ref" reference="prior-1"} that there exists a constant $R_0>0$, depending only on $\Lambda$ and the graph $V$, such that for all $\sigma\in[0,1]$, all solutions of $F(u,\sigma)=0$ satisfy $\|u\|_{L^\infty(V)}< R_0$. Denote a ball centered at $0\in X$ with radius $r$ by $B_r\subset X$, and its boundary by $\partial B_r=\{u\in X:\|u\|_{L^\infty(V)}=r\}$. Thus we conclude $$0\not\in F(\partial B_{R},\sigma),\quad\forall \sigma\in[0,1],\,\,\forall R\geq R_0.$$ By the homotopic invariance of the topological degree, we have $$\begin{aligned} \label{degree-1}\deg(F(\cdot,1),B_{R},0)=\deg(F(\cdot,0),B_{R},0),\quad\forall R\geq R_0.\end{aligned}$$ Given any $\epsilon>0$, we define another smooth map $G_\epsilon:X\times[0,1]\rightarrow X$ by $$G_\epsilon(u,t)=-\Delta u+\lambda e^{2u}+(t+(1-t)\epsilon) f,\quad (u,t)\in X\times [0,1].$$ Notice that $$\left|(t+(1-t)\epsilon)\int_Vfd\mu\right|\geq\min\{1,\epsilon\}\left|\int_Vfd\mu\right|,\quad\forall t\in[0,1].$$ Applying Theorem [Theorem 1](#prior-1){reference-type="ref" reference="prior-1"} again, we find a constant $R_\epsilon>0$, depending only on $\epsilon$, $\Lambda$ and the graph $V$, such that all solutions $u$ of $G_\epsilon(u,t)=0$ satisfy $\|u\|_{L^\infty(V)}< R_\epsilon$ for all $t\in [0,1]$. This implies $$0\not\in G_\epsilon(\partial B_{R_\epsilon},t),\quad\forall t\in[0,1].$$ Hence the homotopic invariance of the topological degree leads to $$\begin{aligned} \label{degree-2}\deg(G_\epsilon(\cdot,1),B_{R_\epsilon},0)=\deg(G_\epsilon(\cdot,0),B_{R_\epsilon},0).\end{aligned}$$ To calculate $\deg(G_\epsilon(\cdot,0),B_{R_\epsilon},0)$, we need to understand the solvability of the equation $$\begin{aligned} \label{eq-4}G_\epsilon(u,0)=-\Delta u+\lambda e^{2u}+\epsilon f=0.\end{aligned}$$ Now we *claim* two properties of solutions of ([\[eq-4\]](#eq-4){reference-type="ref" reference="eq-4"}): $(i)$ If $\lambda\overline{f}<0$, then there exists an $\epsilon_0>0$ such that for any $\epsilon\in (0,\epsilon_0)$, ([\[eq-4\]](#eq-4){reference-type="ref" reference="eq-4"}) has a unique solution $u_\epsilon$, which satisfies $e^{2u_\epsilon}\leq C\epsilon$, where $C$ is a constant depending only on $\Lambda$ and the graph $V$; $(ii)$ If $\lambda\overline f>0$, then ([\[eq-4\]](#eq-4){reference-type="ref" reference="eq-4"}) has no solution for all $\epsilon>0$. To see Claim $(i)$, for any $\epsilon>0$, we let $v_\epsilon$ be the unique solution of the equation $$\left\{\begin{array}{lll} \Delta v=\epsilon f-\epsilon \overline{f}\quad{\rm in}\quad V\\[1.2ex] \overline{v}=0. \end{array}\right.$$ Then the solvability of ([\[eq-4\]](#eq-4){reference-type="ref" reference="eq-4"}) is equivalent to that of the equation $$\begin{aligned} \label{20}\Delta w=\lambda e^{2v_\epsilon}e^{2w}+\epsilon\overline{f}.\end{aligned}$$ Note that the existence of solutions to ([\[20\]](#20){reference-type="ref" reference="20"}), under the assumptions that $\epsilon$ is sufficiently small and $\lambda\overline{f}<0$, follows from ([@Gri1], Theorems 2 and 4). Hence there exists some $\epsilon_1>0$ such that if $0<\epsilon<\epsilon_1$, then the equation ([\[eq-4\]](#eq-4){reference-type="ref" reference="eq-4"}) has a solution $u_\epsilon$. Integrating both sides of ([\[eq-4\]](#eq-4){reference-type="ref" reference="eq-4"}), we have by ([\[bound\]](#bound){reference-type="ref" reference="bound"}), $$\int_Ve^{2u_\epsilon}d\mu=-\frac{\epsilon}{\lambda}\int_Vfd\mu\leq \Lambda^2\epsilon,$$ which leads to $$\begin{aligned} \label{small}e^{2u_\epsilon(x)}\leq \frac{\Lambda^2}{\mu_0}\epsilon,\quad\forall x\in V,\end{aligned}$$ where $\mu_0=\min_{x\in V}\mu(x)$. We also need to prove the uniqueness of the solution. Let $\varphi$ be an arbitrary solution of ([\[eq-4\]](#eq-4){reference-type="ref" reference="eq-4"}), namely it satisfies $$\begin{aligned} \label{var-eq}\Delta\varphi=\lambda e^{2\varphi}+\epsilon f.\end{aligned}$$ The same procedure as above gives $$\begin{aligned} \label{var}\int_Ve^{2\varphi}d\mu\leq \Lambda^2\epsilon,\quad e^{2\varphi(x)}\leq \frac{\Lambda^2}{\mu_0}\epsilon\quad{\rm for\,\,\,all}\quad x\in V.\end{aligned}$$ Subtracting ([\[var-eq\]](#var-eq){reference-type="ref" reference="var-eq"}) from ([\[eq-4\]](#eq-4){reference-type="ref" reference="eq-4"}) and integrating by parts, we have $$0=\int_V\Delta(u_\epsilon-\varphi)d\mu=\lambda\int_V(e^{2u_\epsilon}-e^{2\varphi})d\mu,$$ which leads to $$\min_V(u_\epsilon-\varphi)<0<\max_V(u_\epsilon-\varphi).$$ As a consequence, there holds $$\begin{aligned} \label{abs}|u_\epsilon-\varphi|\leq \max_V(u_\epsilon-\varphi)-\min_V(u_\epsilon-\varphi).\end{aligned}$$ Also we derive from ([\[eq-4\]](#eq-4){reference-type="ref" reference="eq-4"}), ([\[small\]](#small){reference-type="ref" reference="small"}), ([\[var-eq\]](#var-eq){reference-type="ref" reference="var-eq"}), and ([\[var\]](#var){reference-type="ref" reference="var"}), $$\begin{aligned} \nonumber |\Delta(u_\epsilon-\varphi)(x)|&=& \left|\lambda \left(e^{2u_\epsilon(x)}- e^{2\varphi(x)}\right)\right|\\\nonumber &\leq&2\Lambda\left(e^{2u_\epsilon(x)}+ e^{2\varphi(x)}\right)|u_\epsilon(x)-\varphi(x)|\\\label{est} &\leq&\frac{4\Lambda^3}{\mu_0}\epsilon|u_\epsilon(x)-\varphi(x)|.\end{aligned}$$ Combining ([\[equiv\]](#equiv){reference-type="ref" reference="equiv"}), ([\[abs\]](#abs){reference-type="ref" reference="abs"}) and ([\[est\]](#est){reference-type="ref" reference="est"}), we obtain $$\begin{aligned} \label{contr}\max_V(u_\epsilon-\varphi)-\min_V(u_\epsilon-\varphi)\leq \sqrt{\frac{(\ell-1)|V|}{w_0\lambda_1}}\frac{4\Lambda^3}{\mu_0}\epsilon \left(\max_V(u_\epsilon-\varphi)-\min_V(u_\epsilon-\varphi)\right).\end{aligned}$$ Choose $$\epsilon_0=\min\left\{\epsilon_1,\sqrt{\frac{w_0\lambda_1}{(\ell-1)|V|}}\frac{\mu_0}{8\Lambda^3}\right\}.$$ If we take $0<\epsilon<\epsilon_0$, then ([\[contr\]](#contr){reference-type="ref" reference="contr"}) implies $\varphi\equiv u_\epsilon$ on $V$, and thus ([\[eq-4\]](#eq-4){reference-type="ref" reference="eq-4"}) has a unique solution. Hence $(i)$ holds. To see Claim $(ii)$, in the case $\lambda\overline{f}>0$, if ([\[eq-4\]](#eq-4){reference-type="ref" reference="eq-4"}) has a solution $u$, then there holds $$0=\int_V\Delta ud\mu=\lambda\int_Ve^{2u}d\mu+\int_Vfd\mu,$$ which is impossible. This confirms $(ii)$, and our claims hold. Let us continue to prove the theorem. Note that $-\Delta: X\rightarrow X$ is a nonnegative definite symmetric operator, its eigenvalues are written as $$0=\lambda_0<\lambda_1\leq \lambda_2\leq\cdots\leq \lambda_{\ell-1},$$ where $\ell$ is the number of all points in $V$. By Claim $(i)$, in the case $\lambda\overline{f}<0$, we may choose a sufficiently small $\epsilon>0$ such that $G_\epsilon(u,0)=0$ has a unique solution $u_\epsilon$ verifying $$2|\lambda|e^{2u_\epsilon(x)}<\lambda_1.$$ A straightforward calculation shows $$DG_\epsilon(u_\epsilon,0)=-\Delta+2\lambda e^{2u_\epsilon}{\rm I},$$ where we identify the linear operator $-\Delta$ with the $\ell\times\ell$ matrix corresponding to $-\Delta$, and denote the $\ell\times\ell$ diagonal matrix ${\rm diag}[1,1,\cdots,1]$ by ${\rm I}$. Clearly $$\deg(G_\epsilon(\cdot,0),B_{R_\epsilon},0)={\rm sgn\,det}\left(DG_\epsilon(u_\epsilon,0)\right) ={\rm sgn}\left\{2\lambda e^{2u_\epsilon(x)}\Pi_{j=1}^{\ell-1}(\lambda_j+2\lambda e^{2u_\epsilon(x)})\right\}={\rm sgn}\lambda.$$ This together with ([\[degree-1\]](#degree-1){reference-type="ref" reference="degree-1"}) and ([\[degree-2\]](#degree-2){reference-type="ref" reference="degree-2"}) leads to $$\begin{aligned} \deg(F(\cdot,1),B_{R_\epsilon},0)&=&\deg(F(\cdot,0),B_{R_\epsilon},0)\\ &=&\deg(G_\epsilon(\cdot,1),B_{R_\epsilon},0)\\ &=&\deg(G_\epsilon(\cdot,0),B_{R_\epsilon},0)\\ &=&{\rm sgn}\lambda.\end{aligned}$$ By Claim $(ii)$, in the case $\lambda\overline{f}>0$, since $G_\epsilon(u,0)=0$ has no solution, we obtain $$\deg(F(\cdot,1),B_{R_\epsilon},0)=\deg(G_\epsilon(\cdot,0),B_{R_\epsilon},0)=0.$$ Thus the proof of Theorem [Theorem 2](#degree){reference-type="ref" reference="degree"} is completed. $\hfill\Box$ # Existence results {#sec-existence} In this section, we shall prove Theorem [Theorem 3](#existence){reference-type="ref" reference="existence"} by using the topological degree in Theorem [Theorem 2](#degree){reference-type="ref" reference="degree"}.\ *Proof of Theorem [Theorem 3](#existence){reference-type="ref" reference="existence"} $(\mathsf{a})$.* If $\lambda\overline{f}<0$, then by Theorem [Theorem 2](#degree){reference-type="ref" reference="degree"}, we find some large $R_0>1$ such that $$\deg(F,B_{R_0},0)\not=0.$$ Thus the Kronecker's existence theorem implies ([\[eq-1\]](#eq-1){reference-type="ref" reference="eq-1"}) has a solution. $\hfill\Box$\ In the remaining part of this section, we always assume $\lambda\overline{f}>0$. We first prove that ([\[eq-1\]](#eq-1){reference-type="ref" reference="eq-1"}) has a local minimum solution for large $|\lambda|$, say **Lemma 10**. *If $|\lambda|$ is chosen sufficiently large, then the equation $(3)$ has a local minimum solution.* Let us first consider the subcase $\lambda>0$ and $\overline{f}>0$. Set $$\begin{aligned} \label{Llambda}L_\lambda u=-\Delta u+\lambda e^u(e^u-1)+f.\end{aligned}$$ For real numbers $A$ and $\lambda$, there hold $$L_\lambda A=\lambda e^A(e^A-1)+f,\quad L_\lambda \log \frac{1}{2}=-\frac{1}{4}\lambda +f.$$ Clearly, taking sufficiently large $A>1$ and $\lambda>1$, we have $$\begin{aligned} \label{u-l}L_\lambda A>0,\quad L_\lambda \log\frac{1}{2}<0.\end{aligned}$$ Recall the functional $J_\lambda:X=L^\infty(V)\rightarrow\mathbb{R}$ defined by ([\[funct-1\]](#funct-1){reference-type="ref" reference="funct-1"}). Since $X\cong \mathbb{R}^\ell$, $J_\lambda\in C^2(X,\mathbb{R})$, and $\{u\in X:\log\frac{1}{2}\leq u\leq A\}$ is a bounded closed subset of $X$, it is easy to find some $u_\lambda\in X$ satisfying $\log \frac{1}{2}\leq u_\lambda(x)\leq A$ for all $x\in V$ and $$\begin{aligned} \label{J-min}J_\lambda(u_\lambda)=\min_{\log\frac{1}{2}\leq u\leq A}J_\lambda(u).\end{aligned}$$ We *claim* that $$\begin{aligned} \label{strict}\log\frac{1}{2}< u_\lambda(x)< A\quad{\rm for\,\,all}\quad x\in V.\end{aligned}$$ Suppose not. There must hold $u_\lambda(x_0)=\log\frac{1}{2}$ for some $x_0\in V$, or $u_\lambda(x_1)=A$ for some $x_1\in V$. If $u_\lambda(x_0)=\log\frac{1}{2}$, then we take a small $\epsilon>0$ such that $$\log\frac{1}{2}<u_\lambda(x)+t\delta_{x_0}(x)<A,\quad\forall x\in V,\,\forall t\in(0,\epsilon).$$ On one hand, in view of ([\[u-l\]](#u-l){reference-type="ref" reference="u-l"}) and ([\[J-min\]](#J-min){reference-type="ref" reference="J-min"}), we have $$\begin{aligned} \nonumber 0&\leq&\left.\frac{d}{dt}\right|_{t=0}J_\lambda(u_\lambda+t\delta_{x_0})\\\nonumber &=&\int_V\left(-\Delta u_\lambda+\lambda e^{u_\lambda}(e^{u_\lambda}-1)+f\right)\delta_{x_0}d\mu\\\nonumber &=&-\Delta u_\lambda(x_0)+\lambda e^{u_\lambda(x_0)}(e^{u_\lambda(x_0)}-1)+f(x_0)\\\label{less} &<&-\Delta u_\lambda(x_0).\end{aligned}$$ On the other hand, since $u_\lambda(x)\geq u_\lambda(x_0)$ for all $x\in V$, we conclude $\Delta u_\lambda(x_0)\geq 0$, which contradicts ([\[less\]](#less){reference-type="ref" reference="less"}). Hence $u_\lambda(x)>\log\frac{1}{2}$ for all $x\in V$. In the same way, we exclude the possibility of $u_\lambda(x_1)=A$ for some $x_1\in V$. This confirms our claim ([\[strict\]](#strict){reference-type="ref" reference="strict"}). Combining ([\[J-min\]](#J-min){reference-type="ref" reference="J-min"}) and ([\[strict\]](#strict){reference-type="ref" reference="strict"}), we conclude that $u_\lambda$ is a local minimum critical point of $J_\lambda$, in particular, $u_\lambda$ is a solution of ([\[eq-1\]](#eq-1){reference-type="ref" reference="eq-1"}). Now we consider the subcase $\lambda<0$ and $\overline{f}<0$. Let $\varphi$ be the unique solution of $$\left\{\begin{array}{lll} \Delta\varphi=f-\overline{f}\\[1.2ex] \overline{\varphi}=0. \end{array}\right.$$ Using the notation of the operator $L_\lambda$ given by ([\[Llambda\]](#Llambda){reference-type="ref" reference="Llambda"}), we have $$\begin{aligned} \nonumber L_\lambda(\varphi-A)&=&-\Delta\varphi+\lambda e^{\varphi-A}(e^{\varphi-A}-1)+f\\\nonumber &=&\lambda e^{\varphi-A}(e^{\varphi-A}-1)+\overline{f}\\\label{lo-bd} &<&0\end{aligned}$$ and $$\begin{aligned} L_\lambda(\log\frac{1}{2})&=&\lambda e^{\log\frac{1}{2}}(e^{\log\frac{1}{2}}-1)+f\\ &=&-\frac{\lambda}{4}+f\\ &>&0,\end{aligned}$$ provided that $\lambda<4\min_Vf$ and $A>1$ is chosen sufficiently large. Similar to ([\[J-min\]](#J-min){reference-type="ref" reference="J-min"}) and ([\[strict\]](#strict){reference-type="ref" reference="strict"}), there exists some $u_\lambda$ satisfying $\varphi(x)-A< u_\lambda(x)< \log\frac{1}{2}$ for all $x\in V$ and $$J_\lambda(u_\lambda)=\min_{\varphi-A\leq u\leq \log\frac{1}{2}}J_\lambda(u)=\min_{\varphi-A< u< \log\frac{1}{2}}J_\lambda(u).$$ This implies $u_\lambda$ is a local minimum solution of ([\[eq-1\]](#eq-1){reference-type="ref" reference="eq-1"}). $\hfill\Box$\ To proceed, we also need the following: **Lemma 11**. *If $\lambda_1>0$ such that the equation $L_{\lambda_1}u=0$ has a solution $u_{\lambda_1}$, then for any $\lambda>\lambda_1$, we have $$L_{\lambda} \left(u_{\lambda_1}+\log\frac{\lambda_1}{\lambda}\right)<0.$$ Similarly, if $\lambda_2<0$ such that $L_{\lambda_2}u_{\lambda_2}=0$, then for any $\lambda<\lambda_2$, there hods $$L_{\lambda} \left(u_{\lambda_2}+\log\frac{\lambda_2}{\lambda}\right)>0.$$* If $\lambda>\lambda_1>0$, then $$\begin{aligned} L_\lambda \left(u_{\lambda_1}+\log\frac{\lambda_1}{\lambda}\right)&=&-\Delta u_{\lambda_1}+\lambda_1 e^{u_{\lambda_1}} \left(\frac{\lambda_1}{\lambda}e^{u_{\lambda_1}}-1\right)+f \\ &<&-\Delta u_{\lambda_1}+\lambda_1 e^{u_{\lambda_1}} (e^{u_{\lambda_1}}-1)+f\\ &=&0.\end{aligned}$$ If $\lambda<\lambda_2<0$, then $$\begin{aligned} L_\lambda \left(u_{\lambda_2}+\log\frac{\lambda_2}{\lambda}\right)&=&-\Delta u_{\lambda_2}+\lambda_2 e^{u_{\lambda_2}} \left(\frac{\lambda_2}{\lambda}e^{u_{\lambda_2}}-1\right)+f\\ &>&-\Delta u_{\lambda_2}+\lambda_2 e^{u_{\lambda_2}} \left(e^{u_{\lambda_2}}-1\right)+f\\ &=&0,\end{aligned}$$ as we desired. $\hfill\Box$\ As a consequence, we have **Lemma 12**. *Assume $L_{\lambda_1}u_{\lambda_1}=L_{\lambda_2}u_{\lambda_2}=0$ on $V$. If either $\lambda>\lambda_1>0$ or $\lambda<\lambda_2<0$, then the equation (3) has a local minimum solution $u_\lambda$.* Assume $\lambda>\lambda_1>0$. Let $A>1$ be a sufficiently large constant such that $L_\lambda A>0$ and $u_{\lambda_1}+\log\frac{\lambda_1}{\lambda}< A$ on $V$. Then there exists some $u_\lambda$ such that $$J_\lambda(u_\lambda)=\min_{u_{\lambda_1}+\log\frac{\lambda_1}{\lambda}\leq u\leq A}J_\lambda(u).$$ Suppose there is some point $x_0\in V$ satisfying $u_\lambda(x_0)=u_{\lambda_1}(x_0)+\log\frac{\lambda_1}{\lambda}$. Let $\epsilon>0$ be so small that for $t\in(0,\epsilon)$, there holds $$u_{\lambda_1}(x)+\log\frac{\lambda_1}{\lambda}< u_\lambda(x)+t\delta_{x_0}(x)< A\quad{\rm for\,\,all}\quad x\in V.$$ Similarly as we did in the proof of Lemma [Lemma 10](#large){reference-type="ref" reference="large"}, we have by Lemma [Lemma 11](#solution){reference-type="ref" reference="solution"}, $$\begin{aligned} 0&\leq&\left.\frac{d}{dt}\right|_{t=0}J_\lambda(u_\lambda+t\delta_{x_0})\\ &=&-\Delta u_\lambda(x_0)+\lambda e^{u_\lambda(x_0)}(e^{u_\lambda(x_0)}-1)+f(x_0)\\ &=&-\Delta\left(u_\lambda-u_{\lambda_1}\right)(x_0)+L_{\lambda}\left(u_{\lambda_1}+\log{\frac{\lambda_1}{\lambda}}\right)(x_0)\\ &<&-\Delta\left(u_\lambda-u_{\lambda_1}\right)(x_0).\end{aligned}$$ This contradicts the fact that $x_0$ is a minimum point of $u_\lambda-u_{\lambda_1}-\log\frac{\lambda_1}{\lambda}$. Hence $$u_\lambda(x)>u_{\lambda_1}(x)+\log\frac{\lambda_1}{\lambda},\quad\forall x\in V.$$ In the same way we obtain $u(x)<A$ for all $x\in V$. Therefore $u_\lambda$ is a local minimum critical point of $J_\lambda$. Assume $\lambda<\lambda_2<0$. The constant $A>1$ is chosen sufficiently large such that $\varphi-A<u_{\lambda_2}+\log\frac{\lambda_2}{\lambda}$ on $V$, and $\varphi-A$ satisfies ([\[lo-bd\]](#lo-bd){reference-type="ref" reference="lo-bd"}). Clearly there exists some $u_\lambda$ such that $$J_\lambda(u_\lambda)=\min_{\varphi-A\leq u\leq u_{\lambda_2}+\log\frac{\lambda_2}{\lambda}}J_\lambda(u).$$ If there is some point $x_1\in V$ satisfying $u_\lambda(x_1)=u_{\lambda_2}(x_1)+\log\frac{\lambda_2}{\lambda}$, then there is a small $\epsilon>0$ such that for $t\in(0,\epsilon)$, there holds $$\varphi(x)-A< u_\lambda(x)-t\delta_{x_1}(x)< u_{\lambda_2}(x)+\log\frac{\lambda_2}{\lambda}\quad{\rm for\,\,all}\quad x\in V.$$ Thus we have by Lemma [Lemma 11](#solution){reference-type="ref" reference="solution"}, $$\begin{aligned} 0&\leq&\left.\frac{d}{dt}\right|_{t=0}J_\lambda(u_\lambda-t\delta_{x_1})\\ &=&\Delta u_\lambda(x_1)-\lambda e^{u_\lambda(x_1)}(e^{u_\lambda(x_1)}-1)-f(x_0)\\ &=&\Delta\left(u_\lambda-u_{\lambda_2}\right)(x_0)-L_{\lambda}\left(u_{\lambda_2}+\log{\frac{\lambda_2}{\lambda}}\right)(x_0)\\ &<&\Delta\left(u_\lambda-u_{\lambda_2}\right)(x_0).\end{aligned}$$ This contradicts the fact that $x_1$ is a maximum point of $u_\lambda-u_{\lambda_2}-\log\frac{\lambda_2}{\lambda}$. Hence $$u_\lambda(x)<u_{\lambda_2}(x)+\log\frac{\lambda_2}{\lambda},\quad\forall x\in V.$$ In the same way we obtain $u(x)>\varphi(x)-A$ for all $x\in V$. Therefore $u_\lambda$ is a local minimum critical point of $J_\lambda$. Thus we complete the proof of the lemma. $\hfill\Box$\ We conclude from Lemmas [Lemma 10](#large){reference-type="ref" reference="large"} and [Lemma 12](#minimum-sol){reference-type="ref" reference="minimum-sol"} that the following two critical numbers are well defined. $$\begin{aligned} \label{g-1} &\Lambda^\ast=\inf\left\{\lambda>0: \lambda\overline{f}>0, J_\lambda \,\,{\rm has\,\,a\,\,local\,\,minimum\,\,critical\,\,point}\right\}\\[1.2ex] &\Lambda_\ast=\sup\left\{\lambda<0: \lambda\overline{f}>0, J_\lambda \,\,{\rm has\,\,a\,\,local\,\,minimum\,\,critical\,\,point}\right\}.\label{g-2}\end{aligned}$$ **Lemma 13**. *If $\overline{f}>0$, then $\Lambda^\ast\geq 4\overline{f}$; If $\overline{f}<0$, then $\Lambda_\ast\leq 4\overline{f}$.* Suppose $\lambda\not=0$ and $u$ is a solution of $\Delta u=\lambda e^u(e^u-1)+f$. Integration by parts gives $$-\frac{\int_Vfd\mu}{\lambda}=\int_Ve^u(e^u-1)d\mu\geq -\frac{|V|}{4},$$ since $e^u(e^u-1)\geq -\frac{1}{4}$. The conclusion follows from ([\[g-1\]](#g-1){reference-type="ref" reference="g-1"}) and ([\[g-2\]](#g-2){reference-type="ref" reference="g-2"}) immediately. $\hfill\Box$\ We are now ready to complete the proof of the remaining part of the theorem.\ *Proof of Theorem [Theorem 3](#existence){reference-type="ref" reference="existence"} $(\mathsf{b})$.* *We first consider the solvability of the equation ([\[eq-1\]](#eq-1){reference-type="ref" reference="eq-1"}) under the assumption $\lambda\in (0,\Lambda^\ast]\cup[\Lambda_\ast,0)$.* If $\lambda\in (0,\Lambda^\ast)\cup(\Lambda_\ast,0)$, then $(\ref{eq-1})$ has no solution. Indeed, suppose there exists a number $\lambda_1\in (0,\Lambda^\ast)\cup(\Lambda_\ast,0)$ such that ([\[eq-1\]](#eq-1){reference-type="ref" reference="eq-1"}) has a solution at $\lambda=\lambda_1$. With no loss of generality, we assume $\lambda_1\in (\Lambda_\ast,0)$, then by Lemma [Lemma 12](#minimum-sol){reference-type="ref" reference="minimum-sol"}, ([\[eq-1\]](#eq-1){reference-type="ref" reference="eq-1"}) has a local minimum solution at any $\lambda\in [\Lambda_\ast,\lambda_1)$. This contradicts the definition of $\Lambda_\ast$. Hence $(\ref{eq-1})$ has no solution for any $\lambda\in (0,\Lambda^\ast)\cup(\Lambda_\ast,0)$. Note that for any $j\in\mathbb{N}$, there exists a solution $u_j$ of ([\[eq-1\]](#eq-1){reference-type="ref" reference="eq-1"}) with $\lambda=\Lambda_\ast-1/j$. According to Theorem [Theorem 1](#prior-1){reference-type="ref" reference="prior-1"}, $(u_j)$ is uniformly bounded in $V$. Thus up to a subsequence, $(u_j)$ uniformly converges to some function $u^\ast$, a solution of ([\[eq-1\]](#eq-1){reference-type="ref" reference="eq-1"}) with $\lambda=\Lambda_\ast$. In the same way, ([\[eq-1\]](#eq-1){reference-type="ref" reference="eq-1"}) has also a solution at $\lambda=\Lambda^\ast$.\ *We next consider multiple solutions of ([\[eq-1\]](#eq-1){reference-type="ref" reference="eq-1"}) under the assumption $\lambda\in(\Lambda^\ast,+\infty)\cup(-\infty,\Lambda_\ast)$.* If $\lambda \in(\Lambda^\ast,+\infty)\cup(-\infty,\Lambda_\ast)$, by ([\[g-1\]](#g-1){reference-type="ref" reference="g-1"}) and ([\[g-2\]](#g-2){reference-type="ref" reference="g-2"}), we let $u_\lambda$ be a local minimum critical point of $J_\lambda$. With no loss of generality, we may assume $u_\lambda$ is the unique critical point of $J_\lambda$. For otherwise, $J_\lambda$ has already at least two critical points, and the proof terminates. According to ([@Chang93], Chapter 1, page 32), the $q$-th critical group of ${J}_\lambda$ at $u_\lambda$ is defined by $$\begin{aligned} \label{group}\mathsf{C}_q({J}_\lambda,u_\lambda)=\mathsf{H}_q({J}_\lambda^c\cap {U},\{{J}_\lambda^c\setminus\{u_\lambda\}\} \cap {U},\mathsf{G}),\end{aligned}$$ where ${J}_\lambda(u_\lambda)=c$, ${J}_\lambda^c=\{u\in X:{J}_\lambda(u)\leq c\}$, $U$ is a neighborhood of $u_\lambda\in X$, $\mathsf{H}_q$ is the singular homology group with the coefficients groups $\mathsf{G}$, say $\mathbb{Z}$, $\mathbb{R}$. By the excision property of $\mathsf{H}_q$, this definition is not dependent on the choice of $U$. It is easy to calculate $$\begin{aligned} \label{cri}{\mathsf{C}}_q(J_\lambda,u_\lambda)=\delta_{q0}\mathsf{G}.\end{aligned}$$ Note that $J_\lambda$ satisfies the Palais-Smale condition. Indeed, if $J_\lambda(u_j)\rightarrow c\in\mathbb{R}$ and $J^\prime(u_j)\rightarrow 0$ as $j\rightarrow\infty$, then using the method of proving Theorem [Theorem 1](#prior-1){reference-type="ref" reference="prior-1"}, we obtain $(u_j)$ is uniformly bounded. Since $X$ is pre-compact, then up to a subsequence, $(u_j)$ converges uniformly to some $u^\ast$, a critical point $J_\lambda$. Thus the Palais-Smale condition follows. Notice also that $$DJ_\lambda(u)=-\Delta u+\lambda e^{u}(e^u-1)+f=F(u),$$ where $F$ is given as in Theorem [Theorem 2](#degree){reference-type="ref" reference="degree"}. According to ([@Chang93], Chapter 2, Theorem 3.2), in view of ([\[cri\]](#cri){reference-type="ref" reference="cri"}), we have for sufficiently large $R>1$, $$\deg(F,B_{R},0)=\deg(DJ_\lambda,B_R,0)=\sum_{q=0}^\infty (-1)^q{\rm rank}\, \mathsf{C}_q(J_\lambda,u_{\lambda})=1.$$ This contradicts $\deg(F,B_{R},0)=0$ derived from Theorem [Theorem 2](#degree){reference-type="ref" reference="degree"}. Therefore the equation ([\[eq-1\]](#eq-1){reference-type="ref" reference="eq-1"}) has at least two different solutions, and the proof of Theorem [Theorem 3](#existence){reference-type="ref" reference="existence"} $(\mathsf{b})$ is finished. $\hfill\Box$ # Chern-Simons Higgs System {#sec-system} In this section, we shall calculate the topological degree of the map related to the Chern-Simons Higgs system ([\[system\]](#system){reference-type="ref" reference="system"}), and then use the degree to obtain partial results for multiplicity of solutions to the system. In particular, Theorems [Theorem 4](#system-apriori){reference-type="ref" reference="system-apriori"}-[Theorem 6](#syst-thm){reference-type="ref" reference="syst-thm"} will be proved. We first derive a priori estimate for solutions of ([\[system-8\]](#system-8){reference-type="ref" reference="system-8"}), a deformation of ([\[system\]](#system){reference-type="ref" reference="system"}).\ *Proof of Theorem [Theorem 4](#system-apriori){reference-type="ref" reference="system-apriori"}.* Let $\sigma\in[0,1]$, $\lambda>0$, $\eta>0$, $\overline{f}>0$, $\overline{g}>0$, and $(u,v)$ be a solution of the system ([\[system-8\]](#system-8){reference-type="ref" reference="system-8"}). Note that there exist a unique solution $\varphi$ to the equation $$\left\{ \begin{array}{lll} \Delta \varphi=f-\overline{f}\\[1.2ex] \int_V\varphi d\mu=0 \end{array}\right.$$ and a unique solution $\psi$ to the equation $$\left\{ \begin{array}{lll} \Delta \psi=g-\overline{g}\\[1.2ex] \int_V\psi d\mu=0. \end{array}\right.$$ Set $w=u-\varphi$ and $z=v-\psi$. Then we have $$\begin{aligned} \label{system-2} \left\{ \begin{array}{lll} \Delta w=\lambda e^\psi e^z(e^{\varphi}e^w-\sigma)+\overline{f}\\[1.2ex] \Delta z=\eta e^{\varphi}e^w(e^\psi e^z-\sigma)+\overline{g}, \end{array}\right.\end{aligned}$$ We *claim* that $$\begin{aligned} \label{up-2}w(x)<-\min_V\varphi\quad {\rm for\,\, all}\quad x\in V.\end{aligned}$$Suppose not. There necessarily hold $\max_Vw\geq -\min_V\varphi$. Take $x_0\in V$ satisfying $w(x_0)=\max_Vw$. Since $\sigma\in[0,1]$, $\lambda>0$, $\overline{f}>0$ and $\varphi(x_0)+w(x_0)\geq 0$, we have $$0\geq \Delta w(x_0)=\lambda e^{\psi(x_0)} e^{z(x_0)}(e^{\varphi(x_0)}e^{w(x_0)}-\sigma)+\overline{f}\geq\overline{f}> 0,$$ which is impossible. Hence our claim ([\[up-2\]](#up-2){reference-type="ref" reference="up-2"}) follows. Keeping in mind $\eta>0$ and $\overline{g}>0$, in the same way as above, we also have $$\begin{aligned} \label{up-4}z(x)<-\min_V\psi\quad {\rm for\,\, all}\quad x\in V.\end{aligned}$$ Inserting ([\[up-2\]](#up-2){reference-type="ref" reference="up-2"}) and ([\[up-4\]](#up-4){reference-type="ref" reference="up-4"}) into ([\[system-2\]](#system-2){reference-type="ref" reference="system-2"}), we obtain $$\|\Delta w\|_{L^\infty(V)}+\|\Delta z\|_{L^\infty(V)}\leq C$$ for some constant $C$, depending only on $\lambda,\eta,f,g$ and the graph $V$. The most important thing here is that the constant $C$ is not dependent on the parameter $\sigma\in[0,1]$. Coming back to the inequality ([\[equiv\]](#equiv){reference-type="ref" reference="equiv"}), we immediately conclude $$\begin{aligned} \label{diffe-1}\max_V w-\min_V w\leq C\end{aligned}$$ and $$\max_V z-\min_V z\leq C.$$ Observe that integration on both sides of the second equation in ([\[system-2\]](#system-2){reference-type="ref" reference="system-2"}) leads to $$\int_Ve^{\varphi}e^w(e^\psi e^z-\sigma)d\mu=-\frac{\overline{g}}{\eta}|V|.$$ As a consequence, there holds $$\begin{aligned} 0<\frac{\overline{g}}{\eta}\leq e^{\max_Vw}e^{\max_V\varphi}\left(e^{\max_V\psi}+1\right) \leq Ce^{\max_Vw}.\end{aligned}$$ Hence $\max_Vw\geq -C$, and in view of ([\[diffe-1\]](#diffe-1){reference-type="ref" reference="diffe-1"}), $$\begin{aligned} \label{lower}\min_Vw\geq -C.\end{aligned}$$ In the same way, from ([\[diffe-1\]](#diffe-1){reference-type="ref" reference="diffe-1"}) and the first equation of ([\[system-2\]](#system-2){reference-type="ref" reference="system-2"}), we derive $$\begin{aligned} \label{lower-2}\min_Vz\geq -C.\end{aligned}$$ In view of ([\[up-2\]](#up-2){reference-type="ref" reference="up-2"}), ([\[up-4\]](#up-4){reference-type="ref" reference="up-4"}), ([\[lower\]](#lower){reference-type="ref" reference="lower"}) and ([\[lower-2\]](#lower-2){reference-type="ref" reference="lower-2"}), the proof of the theorem is completed. $\hfill\Box$\ Now we calculate the topological degree of the map defined as in ([\[map-syst\]](#map-syst){reference-type="ref" reference="map-syst"}).\ *Proof of Theorem [Theorem 5](#degree-system){reference-type="ref" reference="degree-system"}.* Let $X=L^\infty(V)$. Define a map $\mathcal{F}:X\times X\times[0,1]\rightarrow X\times X$ by $$\mathcal{F}(u,v,\sigma)=(-\Delta u+\lambda e^v(e^u-\sigma)+f,-\Delta v+\eta e^u(e^v-\sigma)+g),\quad\forall (u,v,\sigma)\in X\times X\times [0,1].$$ Obviously $\mathcal{F}\in C^2(X\times X\times[0,1],X\times X)$. On one hand, by Theorem [Theorem 4](#system-apriori){reference-type="ref" reference="system-apriori"}, there exists some $R_0>0$ such that for any $R\geq R_0$, we have $$0\not\in \mathcal{F}(\partial B_R,\sigma),\quad\forall \sigma\in[0,1],$$ and thus the homotopic invariance of the topological degree implies $$\begin{aligned} \label{homotopy}\deg(\mathcal{F}(\cdot,1),B_R,(0,0))=\deg(\mathcal{F}(\cdot,0),B_R,(0,0)).\end{aligned}$$ Here we denote $B_R=\{(u,v)\in X\times X:\|u\|_{L^\infty(V)}+\|v\|_{L^\infty(V)}<R\}$ and $\partial B_R=\{(u,v)\in X\times X:\|u\|_{L^\infty(V)}+\|v\|_{L^\infty(V)}=R\}$, as usual. On the other hand, we calculate $\deg(\mathcal{F}(\cdot,0),B_R,(0,0))$. Since $\lambda>0$ and $\overline{f}>0$, integrating both sides of the first equation of the system $$\begin{aligned} \label{system-0} \left\{ \begin{array}{lll} \Delta u=\lambda e^{u+v}+f\\[1.2ex] \Delta v=\eta e^{u+v}+g, \end{array}\right.\end{aligned}$$ we get a contradiction, provided that ([\[system-0\]](#system-0){reference-type="ref" reference="system-0"}) is solvable. This implies $$\left\{(u,v)\in X\times X: \mathcal{F}(u,v,0)=(0,0)\right\}=\varnothing.$$ As a consequence, there holds $$\begin{aligned} \label{0-deg}\deg(\mathcal{F}(\cdot,0),B_R,(0,0))=0.\end{aligned}$$ Combining ([\[homotopy\]](#homotopy){reference-type="ref" reference="homotopy"}) and ([\[0-deg\]](#0-deg){reference-type="ref" reference="0-deg"}), we get the desired result. $\hfill\Box$\ Let $\mathcal{J}_\lambda:X\times X\rightarrow\mathbb{R}$ be a functional defined as in ([\[functional\]](#functional){reference-type="ref" reference="functional"}). Note that the critical point of $\mathcal{J}_\lambda$ is a solution of the Chern-Simons system ([\[system\]](#system){reference-type="ref" reference="system"}). The following property of $\mathcal{J}_\lambda$ will be not only useful for our subsequent analysis, but also of its own interest. **Lemma 14**. *Under the assumptions $\lambda>0$, $\overline{f}>0$ and $\overline{g}>0$, $\mathcal{J}_\lambda$ satisfies the Palais-Smale condition at any level $c\in\mathbb{R}$.* Let $c\in\mathbb{R}$ and $\{(u_k,v_k)\}$ be a sequence in $X\times X$ such that $\mathcal{J}_\lambda(u_k,v_k)\rightarrow c$ and $$\mathcal{J}_\lambda^\prime(u_k,v_k)\rightarrow(0,0)\quad{\rm in}\quad (X\times X)^\ast\cong \mathbb{R}^{\ell}\times\mathbb{R}^\ell.$$ This together with ([\[derivative\]](#derivative){reference-type="ref" reference="derivative"}) gives $$\begin{aligned} \label{deriv}\left\{\begin{array}{lll} -\Delta u_k+\lambda e^{v_k}(e^{u_k}-1)+f=o_k(1)\\[1.2ex] -\Delta v_k+\lambda e^{u_k}(e^{v_k}-1)+g=o_k(1), \end{array}\right.\end{aligned}$$ where $o_k(1)\rightarrow 0$ uniformly on $V$ as $k\rightarrow\infty$. Comparing ([\[deriv\]](#deriv){reference-type="ref" reference="deriv"}) with the system ([\[system\]](#system){reference-type="ref" reference="system"}), we have by using the same method as in the proof of Theorem [Theorem 4](#system-apriori){reference-type="ref" reference="system-apriori"}, $$\|u_k\|_{L^\infty(V)}+\|v_k\|_{L^\infty(V)}\leq C$$ for some constant $C$, provided that $k\geq k_1$ for some large positive integer $k_1$. Since $V$ is finite, $X$ is pre-compact. Hence, up to a subsequence, $u_k\rightarrow u^\ast$ and $v_k\rightarrow v^\ast$ uniformly in $V$ for some functions $u^\ast$ and $v^\ast$. Obviously $\mathcal{J}_\lambda^\prime(u^\ast,v^\ast)=(0,0)$. Thus $\mathcal{J}_\lambda$ satisfies the $(PS)_c$ condition. $\hfill\Box$\ Finally we prove a partial multiple solutions result for the system ([\[system\]](#system){reference-type="ref" reference="system"}).\ *Proof of Theorem [Theorem 6](#syst-thm){reference-type="ref" reference="syst-thm"}*. We distinguish two hypotheses to proceed. *Case* 1. *$\mathcal{J}_\lambda$ has a non-degenerate critical point $(u_\lambda,v_\lambda)$.* Since $(u_\lambda,v_\lambda)$ is non-degenerate, we have $$\det D^2\mathcal{J}_\lambda(u_\lambda,v_\lambda)\not =0.$$ Suppose $(u_\lambda,v_\lambda)$ is the unique critical point of $\mathcal{J}_\lambda$. Then we conclude for all $R>\|u_\lambda\|_{L^\infty(V)}+\|v_\lambda\|_{L^\infty(V)}$, $$\begin{aligned} \label{non-0}\deg(D\mathcal{J}_\lambda, B_R,(0,0))={\rm sgn}\,\det D^2\mathcal{J}_\lambda(u_\lambda,v_\lambda)\not=0.\end{aligned}$$ Here and in the sequel, as in the proof of Theorem [Theorem 5](#degree-system){reference-type="ref" reference="degree-system"}, $B_R$ is a ball centered at $(0,0)$ with radius $R$. Notice that $D\mathcal{J}_\lambda(u,v)=\mathcal{F}(u,v)$ for all $(u,v)\in X\times X$, where $\mathcal{F}$ is defined as in ([\[map-syst\]](#map-syst){reference-type="ref" reference="map-syst"}). By Theorem [Theorem 5](#degree-system){reference-type="ref" reference="degree-system"}, we have $$\deg(D\mathcal{J}_\lambda,B_R,(0,0))=\deg(\mathcal{F},B_R,(0,0))=0,$$ contradicting ([\[non-0\]](#non-0){reference-type="ref" reference="non-0"}). Hence $\mathcal{J}_\lambda$ must have at least two critical points. *Case $2$. $\mathcal{J}_\lambda$ has a local minimum critical point $(\varphi_\lambda,\psi_\lambda)$.* Similar to ([\[group\]](#group){reference-type="ref" reference="group"}), the $q$-th critical group of $\mathcal{J}_\lambda$ at the critical point $(\varphi_\lambda,\psi_\lambda)$ reads as $$\mathsf{C}_q(\mathcal{J}_\lambda,(\varphi_\lambda,\psi_\lambda))=\mathsf{H}_q(\mathcal{J}_\lambda^c\cap \mathscr{U},\{\mathcal{J}_\lambda^c\setminus\{(\varphi_\lambda,\psi_\lambda)\}\} \cap\mathscr{U},\mathsf{G}),$$ where $\mathcal{J}_\lambda(\varphi_\lambda,\psi_\lambda)=c$, $\mathcal{J}_\lambda^c=\{(u,v)\in X\times X:\mathcal{J}_\lambda(u,v)\leq c\}$, $\mathscr{U}$ is a neighborhood of $(\varphi_\lambda,\psi_\lambda)\in X\times X$, $\mathsf{G}=\mathbb{Z}$ or $\mathbb{R}$ is the coefficient group of $\mathsf{H}_q$. With no loss of generality, we assume $(\varphi_\lambda,\psi_\lambda)$ is the unique critical point of $\mathcal{J}_\lambda$. Since $(\varphi_\lambda,\psi_\lambda)$ is a local minimum critical point, we easily get $$\mathsf{C}_q(\mathcal{J}_\lambda,(\varphi_\lambda,\psi_\lambda))=\delta_{q0}\mathsf{G}.$$ By Lemma [Lemma 14](#PS){reference-type="ref" reference="PS"}, $\mathcal{J}_\lambda$ satisfies the Palais-Smale condition. Then applying ([@Chang93], Chapter 2, Theorem 3.2) and Theorem [Theorem 5](#degree-system){reference-type="ref" reference="degree-system"}, we obtain $$\begin{aligned} 0=\deg{(\mathcal{F},B_R,(0,0))}&=&\deg(D\mathcal{J}_\lambda,B_R,(0,0))\\ &=&\sum_{q=0}^\infty (-1)^q{\rm rank}\, \mathsf{C}_q\left(\mathcal{J}_\lambda,(\varphi_\lambda,\psi_\lambda)\right)\\ &=&1, \end{aligned}$$ provided that $R>\|\varphi_\lambda\|_{L^\infty(V)}+\|\psi_\lambda\|_{L^\infty(V)}$. 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arxiv_math

--- abstract: | Recent constructions have shown that interesting behaviours can be observed in the finiteness properties of Kähler groups and their subgroups. In this work, we push this further and exhibit, for each integer $k$, new hyperbolic groups admiting surjective homomorphisms to ${\mathds Z}$ and to ${\mathds Z}^{2}$, whose kernel is of type $\mathscr{F}_{k}$ but not of type $\mathscr{F}_{k+1}$. By a fibre product construction, we also find examples of nonnormal subgroups of Kähler groups with exotic finiteness properties. address: - Faculty of Mathematics, KIT, Englerstr. 2, 76131 Karlsruhe, Germany - Institut Fourier, Université Grenoble Alpes & CNRS, 38000 Grenoble, France author: - Claudio Llosa Isenrich - Pierre Py bibliography: - References.bib title: Groups with exotic finiteness properties from complex Morse theory --- # Introduction Two of the most basic properties of groups are being finitely generated and being finitely presented. These properties admit a geometric interpretation in terms of classifying spaces, which leads to a higher dimensional generalisation introduced by Wall [@Wal-65]: for a natural number $n$, a group $G$ is called of finiteness type $\mathscr{F}_n$ if it admits a $K(G,1)$ which is a CW-complex with finite $n$-skeleton. Finite generation is then equivalent to $\mathscr{F}_1$ while finite presentability is equivalent to $\mathscr{F}_2$. We say that a group has *exotic* finiteness properties, if it is $\mathscr{F}_n$, but not $\mathscr{F}_{n+1}$ for some integer $n\geq 0$. The existence of such groups is classical for $n=0, 1$ and was proved by Stallings [@Sta-63] for $n=2$ and by Bieri for all $n\geq 3$ [@Bie-76]. Since then, many examples have been constructed, showcasing that exotic finiteness properties can appear under a wide range of additional assumptions on the group. Classical methods used to construct groups with exotic finiteness properties include Bestvina--Brady Morse theory [@BesBra-97] and Brown's criterion [@Bro-87]. For recent use of the latter criterion, see [@SWZ19] and the references there. Starting with the works of Kapovich [@Kap-98] and Dimca, Papadima and Suciu [@DimPapSuc-09-II], it has become increasingly apparent that complex Morse theory provides a powerful method for constructing groups with exotic finiteness properties. The purpose of these works was to prove that fundamental groups of compact Kähler manifolds (*Kähler groups*) can be non-coherent, respectively can have arbitrary exotic finiteness properties. These methods have since been extended to produce a range of examples of Kähler groups with exotic finiteness properties [@Llo-16-II; @Llo-17; @BriLlo-16; @NicPy-21]. These works generalised the construction in [@DimPapSuc-09-II], leading to Kähler groups with exotic finiteness properties which all arise as fundamental groups of generic fibres of holomorphic maps from certain compact Kähler manifolds onto a complex torus. However, the work of Nicolás and Py [@NicPy-21] provides tools for constructing such examples from holomorphic maps with isolated singularities onto arbitrary closed Riemann surfaces (possibly of genus greater than $1$), showing that the potential of these methods stretches beyond the realm of the already known examples. Finally, using again complex Morse theory, we recently produced, for every integer $n\geq 0$, examples of subgroups of hyperbolic groups of type $\mathscr{F}_n$, but not $\mathscr{F}_{n+1}$ [@LloPy-22]. These were the first such examples when $n\ge 4$. This solved an old problem raised by Brady [@Bra-99], where classical methods using Bestvina--Brady Morse theory had so far only been able to provide an answer up to $n=3$ [@Bra-99; @Lod-18; @Kro-21; @LIMP-21]. This showed that indeed the methods from complex Morse theory are sufficiently powerful to construct examples of groups with exotic finiteness properties that are of interest beyond the realm of complex geometry. Let us also mention in this context that recently the combination of Bestvina--Brady Morse theory and real hyperbolic geometry led to the first example of a non-hyperbolic subgroup of a hyperbolic group with a finite classifying space [@IMM-22; @IMM-21]. In this work we provide further examples of groups with exotic finiteness properties built from complex geometry. We emphasise that we produce both Kähler and non-Kähler groups. The first result takes as input a new class of hyperbolic Kähler groups constructed by Stover and Toledo [@StoTol-21-II]. Their groups arise as fundamental groups of certain compact Kähler manifolds which admit a Kähler metric of negative sectional curvature, but are not homotopy equivalent to any locally symmetric manifold. Combining ideas from [@LloPy-22] and [@StoTol-21-II], we shall prove: **Theorem 1**. *For every $n\geq 2$ there is an $n$-dimensional compact Kähler manifold $Y$ which admits a Kähler metric of negative sectional curvature, is not homotopy equivalent to any locally symmetric manifold and which has the following property. There exists a dense open set $O\subset H^{1}(Y,{\mathds R})-\{0\}$ which is invariant by multiplication by nonzero scalars, such that for any homomorphism $\phi: \pi_1(Y)\to \mathbb{Z}$ contained in $O$, the kernel ${\rm{ker}}(\phi)$ is of type $\mathscr{F}_{n-1}$ but not of type ${\rm FP}_n(\mathbb{Q})$.* The definition of the finiteness property ${\rm FP}_n(\mathbb{Q})$ will be recalled in Section [2](#sec:FinProps){reference-type="ref" reference="sec:FinProps"}. This produces many new subgroups of hyperbolic groups of finiteness type $\mathscr{F}_{n-1}$ and not $\mathscr{F}_n$, thus extending the main result from [@LloPy-22]. Using the Cartwright--Steger surface, we can also produce further examples of Kähler groups of type $\mathscr{F}_{2n-1}$ and not $\mathscr{F}_{2n}$ for all $n\geq 2$. See Section [4.3](#sec:ST-CS){reference-type="ref" reference="sec:ST-CS"} for a discussion. Besides producing homomorphisms from certain negatively curved Kähler groups onto $\mathbb{Z}$, whose kernels have exotic finiteness properties, we can also produce similar homomorphisms onto $\mathbb{Z}^{2}$, using results from the theory of *Bieri-Neumann-Strebel-Renz invariants* (in short BNSR-invariants; see Section [2](#sec:FinProps){reference-type="ref" reference="sec:FinProps"} for some background). This is the content of our next main result. **Theorem 2**. *Let $X$ be a closed aspherical Kähler manifold with positive first Betti number and nonzero Euler characteristic. Assume that the Albanese map of $X$ is finite. Let $n={\rm dim}_{{\mathds C}} \, X$. Then there exist surjective morphisms $\pi_{1}(X)\to \mathbb{Z}^{2}$ whose kernel is of type $\mathscr{F}_{n-1}$ but not of type ${\rm FP}_n(\mathbb{Q})$. The set of such homomorphisms is open.* Let us recall that a map $f$ is said to be *finite* if the preimage by $f$ of each point of the target space is a finite set. For the definition of the Albanese torus and Albanese map of a closed Kähler manifold, we refer the reader to [@LloPy-22 §3.1] and [@Voi-book §12.1.3]. As for the topology alluded to in the statement of Theorem [Theorem 2](#thm:zeetwo){reference-type="ref" reference="thm:zeetwo"}, it will be defined in Section [2](#sec:FinProps){reference-type="ref" reference="sec:FinProps"}. As we shall explain in Section [5](#sec:zeetwo){reference-type="ref" reference="sec:zeetwo"}, this theorem applies to certain arithmetic ball quotients as well as to some of the manifolds built by Stover and Toledo in [@StoTol-21-II]. This allows us to deduce the following result. **Corollary 3**. *For every $k\ge 1$ there is a hyperbolic group $G$ and a surjective homomorphism $\phi: G \to \mathbb{Z}^2$ such that ${\rm{ker}}(\phi)$ is of type $\mathscr{F}_{k}$, but not ${\rm FP}_{k+1}(\mathbb{Q})$.* We now proceed to explain a new way of using complex Morse theory to produce groups with exotic finiteness properties as fibre products. An important novelty is that these groups are not constructed as kernels of homomorphisms, making them rather different from most other groups constructed using Morse theory. One main consequence of this approach is the following result. **Theorem 4**. *Let $X_{1}=\Gamma\backslash \mathbb{B}_{\mathbb{C}}^n$ be a compact complex ball quotient with $n\ge 2$, and let $p_1:X_1\to \Sigma$ be a surjective holomorphic map with connected fibres onto a closed hyperbolic Riemann surface. Assume that $p_1$ has a finite non-empty set of critical points. Let $p_2: X_2 \to \Sigma$ be a ramified covering with non-trivial set of singular values that is disjoint from the set of singular values of $p_1$. Assume that $p_{2\ast}: \pi_1(X_2)\to \Sigma$ is surjective.* *Then the group theoretic fibre product $P\leq \pi_1(X_1)\times \pi_1(X_2)$ of the induced surjective homomorphisms $p_{i,\ast}:\pi_1(X_i)\to \pi_1(\Sigma)$, $i=1,~2$, is a non-normal Kähler subgroup of finiteness type $\mathscr{F}_n$ and not ${\rm FP}_{n+1}(\mathbb{Q})$.* We recall that the fibre product $P$ is the subgroup of $\pi_1(X_1)\times \pi_1(X_2)$ defined as follows: $$P:=\{(a,b)\in \pi_1(X_1)\times \pi_1(X_2): p_{1,\ast}(a)=p_{2,\ast}(b)\}.$$ Concrete examples to which the $n=2$ version of Theorem [Theorem 4](#thm:Livne){reference-type="ref" reference="thm:Livne"} can be applied are the so-called *Livné surfaces* [@Liv-81]. We refer to Section [6.1](#sec:Liv-Ex){reference-type="ref" reference="sec:Liv-Ex"} for their definition. For $n\ge 3$ we do not know examples of ball quotients admitting a map $p_1$ as in the theorem. We also observe that our assumption that the critical set of $p_1$ is non-empty is always satisfied, thanks to a theorem due to Koziarz and Mok [@KozMok-2010]. We prefer however to state Theorem [Theorem 4](#thm:Livne){reference-type="ref" reference="thm:Livne"} as above to emphasise the fact that we do need some critical points! Finally, let us mention that our methods from [@LloPy-22] can also be applied to obtain a new proof of the following result of Kochloukova and Vidussi from [@KocVid-22]. **Theorem 5** (Kochloukova--Vidussi [@KocVid-22 Corollary 1.11]). *Let $n\geq 2$. There is an aspherical smooth complex projective variety $X^n$ of dimension $n$ whose fundamental group $\pi_1(X^n)$ is an irreducible polysurface group which contains for every $j\in \left\{1,\dots,n\right\}$ a subgroup of type $\mathscr{F}_{j-1}$, but not of type ${\rm FP}_{j}(\mathbb{Q})$.* The definition of polysurface groups will be recalled in Section [7](#sec:Koc-Vid){reference-type="ref" reference="sec:Koc-Vid"}, where we will also explain our new proof of Theorem [Theorem 5](#thm:KV){reference-type="ref" reference="thm:KV"}. ## Structure {#structure .unnumbered} In Section [2](#sec:FinProps){reference-type="ref" reference="sec:FinProps"}, we give some background on finiteness properties in group theory. In Section [3](#sec:CxTori){reference-type="ref" reference="sec:CxTori"}, we introduce the main construction methods of groups with exotic finiteness properties (Kähler or not) from maps onto complex tori. It can serve as an introductory reference for gaining an overview of the techniques from the works [@DimPapSuc-09-II; @Kap-98; @Llo-16-II; @Llo-17; @BriLlo-16; @NicPy-21; @LloPy-22]. In Section [4](#sec:ST-Examples){reference-type="ref" reference="sec:ST-Examples"}, we illustrate these methods by proving Theorem [Theorem 1](#thm:ST-Intro){reference-type="ref" reference="thm:ST-Intro"}. In Section [5](#sec:zeetwo){reference-type="ref" reference="sec:zeetwo"}, we prove Theorem [Theorem 2](#thm:zeetwo){reference-type="ref" reference="thm:zeetwo"}. In Section [6](#sec:Livne){reference-type="ref" reference="sec:Livne"}, we explain how complex Morse theory can be used to produce non-normal subgroups with exotic finiteness properties, proving Theorem [Theorem 4](#thm:Livne){reference-type="ref" reference="thm:Livne"}. In Section [7](#sec:Koc-Vid){reference-type="ref" reference="sec:Koc-Vid"}, we describe a new proof of Theorem [Theorem 5](#thm:KV){reference-type="ref" reference="thm:KV"}, which was first proved by Kochloukova and Vidussi. Finally, Section [8](#morse-theoretic-remarks){reference-type="ref" reference="morse-theoretic-remarks"} contains a few remarks about the existence of *perfect circle-valued Morse functions* and about alternative proofs of some of our results, relying purely on (real) Morse theory rather than on the theory of BNSR-invariants. ## Acknowledgements {#acknowledgements .unnumbered} We would like to thank Benoît Claudon and Vincent Koziarz who told us about the existence of fibrations with isolated critical points on Livné's surfaces, and Bruno Martelli who pointed out to us the connection of our work to the existence of perfect circle-valued Morse functions. Finally, we thank again Benoît Claudon, for pointing out the reference [@GrKi]. # Finiteness properties of groups {#sec:FinProps} We already introduced the homotopical finiteness property $\mathscr{F}_n$, which requires that a group has a classifying space with finite $n$-skeleton. A second important set of finiteness properties are the homological finiteness properties. For an abelian unital ring $R$ we say that a group $G$ is of finiteness type ${\rm FP}_n(R)$ if there is a projective resolution $$\dots \to P_n\to P_{n-1}\to \dots \to P_0\to R\to 0$$ of the trivial $RG$-module $R$ which is finitely generated up to dimension $n$. Taking the free resolution induced by the cellular complex associated with a classifying space, it is easy to see that $\mathscr{F}_n$ implies property ${\rm FP}_n(R)$ for every $R$. In degree $1$, this is an equivalence: a group $G$ is of type ${\rm FP}_1(R)$ for some ring $R$ if and only if it is of type $\mathscr{F}_1$. However, in higher dimensions Bestvina and Brady [@BesBra-97] have shown that ${\rm FP}_n(R)$ does not imply $\mathscr{F}_n$ and, moreover, for different rings $R_1$ and $R_2$ the properties ${\rm FP}_n(R_1)$ and ${\rm FP}_n(R_2)$ are in general not equivalent. Finally, let us also mention that properties ${\rm FP}_n(\mathbb{Z})$ and $\mathscr{F}_2$ together imply $\mathscr{F}_n$. For a detailed introduction to finiteness properties, we refer the reader to [@Bro-82]. An important source of examples of groups with exotic finiteness properties are kernels of homomorphisms onto free abelian groups. A key reason for this is that the homotopical finiteness properties of such kernels can be studied via Morse theoretical means. These properties are completely encoded by the so-called BNSR-invariants which we now introduce [@BNS-87; @BieRen-88; @Renz-thesis]. There are also homological analogues of these invariants, but we shall not deal with them here. The character sphere of a finitely generated group $G$ is the sphere $$S(G):= \left( {\rm Hom}(G,\mathbb{R})-\left\{0\right\}\right)/\sim,$$ where the equivalence relation $\sim$ is defined as follows. Two nonzero characters $\chi_1,~\chi_2: G\to \mathbb{R}$ are *equivalent* if there is a real number $\lambda>0$ with $\chi_1=\lambda \cdot \chi_2$. For a group $G$ of type $\mathscr{F}_n$, one can define $n$ BNSR-invariants, which are subsets of the sphere $S(G)$. They are denoted by $\Sigma^{i}(G)$ ($1\le i \le n$) and form a decreasing sequence: $$\Sigma^n(G)\subseteq \Sigma^{n-1}(G)\subseteq \dots \subseteq \Sigma^1(G)\subseteq S(G).$$ When $G$ admits a finite classifying space, the invariant $\Sigma^n (G)$ is defined for any natural integer $n$. The definition of these sets is related to the relative connectivity properties of certain "half-spaces\" associated to the characters, in the universal cover of a $K(G,1)$. Their precise definition is slightly technical. Since we will not work with it here, we omit it and refer to [@BNS-87; @BieRen-88; @BS-book]. We simply make two remarks: $\bullet$ The invariant $\Sigma^1 (G)$, for $G$ a finitely generated group, can be defined quite simply as follows. One considers a Cayley graph $\Gamma$ associated to a finite symmetric generating subset of $G$, and a nonzero character $\chi : G \to {\mathds R}$. One declares that $[\chi]\in \Sigma^{1}(G)$ if the subgraph of $\Gamma$ generated by the vertices where $\chi \ge 0$ is connected. See [@BS-book] or [@Py-book Ch. 11] for more details on this definition. $\bullet$ When $G$ is the fundamental group of a closed aspherical manifold $X$, any (nonzero) character $\chi$ is obtained by integration of a closed $1$-form $\alpha$ on $X$. On the universal cover $\widehat{X}$ of $X$, the pull-back of $\alpha$ is exact, and we can fix a primitive $f : \widehat{X}\to {\mathds R}$ for it. The condition $[\chi] \in \Sigma^{k}(G)$ is then a condition on the behavior of the inclusion maps $$\{f\ge C\}\to \{ f \ge D\}$$ (for real numbers $C\ge D$) on homotopy groups in dimension $\le k-1$. See Appendix B in [@BS-book] or Definition 12 in [@LloPy-22]. The main properties that we will require here are summarised by the following results [@BNS-87; @BieRen-88; @BS-book]. **Proposition 6**. *Let $G$ be a group of type $\mathscr{F}_n$. Then the following hold:* 1. *for $0\leq \ell \leq n$ the BNSR-invariant $\Sigma^{\ell}(G)$ is an open subset of $S(G)$;* 2. *for $\chi: G\to \mathbb{Z}$ an integer-valued character, $[ \chi]\in \Sigma^{\ell}(G)\cap -\Sigma^{\ell}(G)$ if and only if ${\rm{ker}}(\chi)$ is of type $\mathscr{F}_{\ell}$.* **Theorem 7**. *Let $k\ge 1$. Let $G$ be a group of type $\mathscr{F}_{n}$ and let $\chi : G \to {\mathds Z}^k$ be a surjective homomorphism. Then the kernel of $\chi$ is of type $\mathscr{F}_{n}$ if and only if for every nonzero homomorphism $u : {\mathds Z}^k \to {\mathds R}$, $[u\circ \chi]\in \Sigma^{n}(G)$.* Let us now elaborate on Theorem [Theorem 7](#thm:finiteness-high-rank){reference-type="ref" reference="thm:finiteness-high-rank"} and introduce the topology on the space of surjective morphisms $G \to {\mathds Z}^k$ that was alluded to in the introduction. Let $\chi$ and $G$ be as in Theorem [Theorem 7](#thm:finiteness-high-rank){reference-type="ref" reference="thm:finiteness-high-rank"}. The classes $$[u\circ \chi],$$ with $u\in {\rm Hom}(\mathbb{Z}^k,\mathbb{R})-\{0\}$, form a $(k-1)$-dimensional subsphere $S(\chi)$ of $S(G)$. Theorem [Theorem 7](#thm:finiteness-high-rank){reference-type="ref" reference="thm:finiteness-high-rank"} then says that the kernel of $\chi$ is of type $\mathscr{F}_{n}$ if and only if the sphere $S(\chi)$ is contained in $\Sigma^{n}(G)$. Note that $S(\chi)$ depends only on the kernel of $\chi$ and it determines that kernel. Hence if $N$ is a normal subgroup of $G$ such that the quotient $G/N$ is isomorphic to ${\mathds Z}^k$, we will write $S(N)$ for the sphere $S(\chi)$ where $\chi : G \to {\mathds Z}^k$ is any surjection obtained by composing the projection $G \to G/N$ with an isomorphism between $G/N$ and ${\mathds Z}^k$. When $k$ is fixed, we then define a topology on the space of subgroups $N\lhd G$ such that $G/N$ is isomorphic to ${\mathds Z}^k$. For two such subgroups $N_1$ and $N_2$, we say that $N_1$ is close to $N_2$ if the sphere $S(N_1)$ is contained in a small enough neighbourhood of $S(N_2)$. The space of surjective morphisms $\chi : G \to {\mathds Z}^k$ is then endowed with the smallest topology making the map $\chi \mapsto {\rm ker}(\chi)$ continuous. Since the BNSR-invariants are open sets, the following proposition is immediate. **Proposition 8**. *Let $G$ be a group of type $\mathscr{F}_{n}$. Let $k\ge 1$ be a natural number. The set of surjective morphisms $\chi : G \to {\mathds Z}^k$ whose kernel is of type $\mathscr{F}_{n}$ is open.* # Construction methods from maps to complex tori {#sec:CxTori} A common denominator of most of the existing constructions from complex geometry for groups with exotic finiteness properties is that they start from a holomorphic map to a complex torus. We will now describe the two main methods of this kind. The first starts from a holomorphic map with isolated singularities to a one-dimensional torus, while the second requires a finite map to a torus of arbitrary dimension. ## Kähler groups from maps with isolated singularities The first construction of Kähler groups with arbitrary exotic finiteness properties is due to Dimca, Papadima and Suciu [@DimPapSuc-09-II]. Their construction starts from an elliptic curve $E$ and $n\geq 3$ ramified double covers $f_i:S_{g_i}\to E$ where $S_{g_{i}}$ is a Riemann surface of genus $g_i\geq 2$. They show that for the map $f=\sum_{i=1}^{n} f_{i} :S_{g_1}\times \dots \times S_{g_n}\to E$ obtained by summation in $E$, the fundamental group $H=\pi_1(f^{-1}(p))$ of a generic fibre of $f$ is of type $\mathscr{F}_{n-1}$ and not ${\rm FP}_{n}({\mathds Q})$ and is canonically isomorphic to ${\rm{ker}}\left(f_{\ast}: \pi_1(S_{g_1})\times \dots \times \pi_1(S_{g_n})\to \pi_1(E)\cong \mathbb{Z}^2\right)$. This construction has since been extended by Llosa Isenrich [@Llo-16-II; @Llo-17], Bridson and Llosa Isenrich [@BriLlo-16] and Nicolás and Py [@NicPy-21], showing its flexibility. The main results from [@DimPapSuc-09-II; @NicPy-21] can be summarised as follows. **Theorem 9** (Dimca--Papadima--Suciu [@DimPapSuc-09-II Theorem C], Nicolás--Py [@NicPy-21 Theorem B]). *[\[thm:NP-DPS\]]{#thm:NP-DPS label="thm:NP-DPS"} Let $M$ be an $n$-dimensional aspherical compact complex manifold with $n\geq 3$, let $S$ be a closed Riemann surface of positive genus and let $f:M\to S$ be a holomorphic map with isolated critical points and connected fibres. Assume that $f$ has at least one critical point. Let $F$ be a smooth generic fibre of $f$. Then $\pi_1(F)$ is of finiteness type $\mathcal{F}_{n-1}$, but not ${\rm FP}_{n}({\mathds Q})$, and is canonically isomorphic to $\mathrm{ker}(f_{\ast}:\pi_1(M)\to \pi_1(S))$.* **Remark 10**. Nicolás and Py prove that $\mathrm{ker}(f_{\ast})$ is not of type ${\rm FP}_{n}({\mathds Q})$ using properties of isolated singularities. If $M$ has a nonzero $n$-th $\ell^2$-Betti number and $S$ has genus $1$, then this also follows from [@LIMP-21 Proposition 14]. The nonvanishing of the middle-dimensional $\ell^2$-Betti number occurs for instance if $M$ is Kähler hyperbolic with nonzero Euler characteristic [@Gro-91; @Pan-96]. However, we emphasise that Theorem [\[thm:NP-DPS\]](#thm:NP-DPS){reference-type="ref" reference="thm:NP-DPS"} applies in a more general context. We observe that in the context of Theorem [Theorem 9](#thm:dpsnp){reference-type="ref" reference="thm:dpsnp"}, the kernel of $f_{\ast}$ is a Kähler group, being isomorphic to the fundamental group of a generic fibre of $f$. This will not be the case in general for the groups constructed in the next section. We will return to this question in Section [5.2](#sec:zeetwo:second){reference-type="ref" reference="sec:zeetwo:second"}. Note that there are generalisations of Theorem [\[thm:NP-DPS\]](#thm:NP-DPS){reference-type="ref" reference="thm:NP-DPS"} which relax the conditions that $f$ has isolated singularities and that the image of $f$ is one-dimensional, see [@BriLlo-16 Theorem 2.2] and [@Llo-17 Section 2]. ## Subgroups of Kähler groups from finite maps {#sec:skgffm} In [@LloPy-22], the authors of this work presented a second construction method of subgroups of Kähler groups with exotic finiteness properties and employed it to show that for every $n\geq 2$ there is a subgroup of a hyperbolic group of type $\mathscr{F}_{n-1}$ but not ${\rm FP}_{n}({\mathds Q})$. This approach is based on ideas of Simpson who studied connectivity properties of sublevel sets of certain harmonic functions $f:\widehat{X}\to \mathbb{R}$ obtained by lifting a harmonic $1$-form to the universal covering of a compact Kähler manifold and taking a primitive. See [@Sim-93], as well as [@LiMaWa-21] for related results. The following theorem summarises the results from [@LloPy-22] that we will require here. **Theorem 11**. *Let $X$ be a closed aspherical Kähler manifold of complex dimension $n\geq 2$. Assume that there exists a finite holomorphic map from $X$ to a complex torus. Then there is a nonzero character $\chi: \pi_1(X)\to \mathbb{Z}$ with kernel of type $\mathscr{F}_{n-1}$. If, moreover, the Euler characteristic of $X$ is non-trivial, then ${\rm{ker}}(\chi)$ is not of type ${\rm FP}_n({\mathds Q})$.* *Proof.* This is an immediate consequence of combining [@LloPy-22 Theorem 6], [@LloPy-22 Proposition 18], and the openness of the BNSR-invariant. ◻ Under the assumptions of Theorem [Theorem 11](#thm:LP-BNSR){reference-type="ref" reference="thm:LP-BNSR"}, there are in fact many characters with kernels having exotic finiteness properties: **Addendum 12**. *Under the assumptions of Theorem [Theorem 11](#thm:LP-BNSR){reference-type="ref" reference="thm:LP-BNSR"}, $\Sigma^{n-1}(\pi_1(X))\cap -\Sigma^{n-1}(\pi_1(X))\subseteq S(\pi_{1}(X))$ is a dense open subset. In particular, the set of characters $\chi:\pi_1(X)\to \mathbb{Z}$ with ${\rm{ker}}(\chi)$ of type $\mathcal{F}_{n-1}$ is dense in $S(\pi_{1}(X))$. If, moreover, the Euler characteristic of $X$ is non-zero, then ${\rm{ker}}(\chi)$ is not of type ${\rm FP}_n(\mathbb{Q})$ for any character $\chi$.* The first two sentences of the addendum are consequences of [@LloPy-22 Theorem 6] and [@LloPy-22 Proposition 18], together with the properties of the BNSR-invariants recalled in Section [2](#sec:FinProps){reference-type="ref" reference="sec:FinProps"}. We now justify the last sentence of the addendum. Under the running assumptions, the manifold $X$ carries a holomorphic $1$-form with finitely many zeros. This follows again from [@LloPy-22 Theorem 6]. Theorem 10 from [@LloPy-22] implies that $X$ satisfies the conclusion of Singer's conjecture, i.e. the $\ell^{2}$-Betti numbers $b^{(2)}_{i}(X)$ vanish except for $i=n$. We thus have $\chi (X)=(-1)^{n}b_{n}^{(2)}(X)$. Proposition 14 from [@LIMP-21] then yields that the kernel of an arbitrary character of $\pi_{1}(X)$ is not of type ${\rm FP}_{n}({\mathds Q})$ if $X$ has nonzero Euler characteristic. **Remark 13**. Let $X$ be a closed Kähler manifold of complex dimension $n$, with finite Albanese map $a_{X} : X\to A(X)$. If $X$ is aspherical, Theorem 10 from [@LloPy-22] implies that $X$ satisfies Singer's conjecture. This can actually be proved without assuming $X$ to be aspherical. Indeed, if $[\omega]$ is a Kähler class on $A(X)$, then $a_{X}^{\ast}[\omega]$ is a Kähler class on $X$ by [@GrKi Prop. 3.6]. Any differential form in the class $[\omega]$ admits, after pull-back to the universal cover of $A(X)$, a primitive of at most linear growth. Hence the same is true for forms in the class $a_{X}^{\ast}[\omega]$, lifted to the universal cover of $X$. The desired conclusion then follows from [@CaXa]. # The Stover--Toledo groups and their subgroups {#sec:ST-Examples} Stover and Toledo recently constructed, in all dimensions $\geq 2$, smooth complex projective varieties admitting a Kähler metric of negative sectional curvature, which are not homotopy equivalent to a locally symmetric manifold, see [@StoTol-21-II]. These are the first such examples in dimension $\geq 4$. Earlier examples had been constructed in dimension $2$ and $3$ by Mostow--Siu, Deraux and Zheng [@MoSi-80; @zheng1; @zheng2; @Deraux-04; @Deraux-05]. Stover and Toledo's examples are obtained by ramified cover of suitable congruence covers of arithmetic ball quotients of the simplest type. We summarise here their work and then apply the construction described in Section [3.2](#sec:skgffm){reference-type="ref" reference="sec:skgffm"} to the fundamental groups of the corresponding negatively curved Kähler manifolds, yielding a proof of Theorem [Theorem 1](#thm:ST-Intro){reference-type="ref" reference="thm:ST-Intro"}. ## Complex ball quotients and the Stover--Toledo construction {#subsec:stto} For $m\geq 1$ we denote by ${\rm PU}(m,1)$ the group of holomorphic isometries of the unit ball $\mathbb{B}^m_{{\mathds C}}$ of ${\mathds C}^m$ equipped with the Bergman metric. We will consider cocompact lattices $\Gamma< {\rm PU}(m,1)$ which are *arithmetic*. We refer the reader to [@Mar-91; @Zim] for the definition of this notion. More specifically, we will be interested in uniform arithmetic lattices of the *simplest type*, whose definition we now recall. Let $F\subset {\mathds R}$ be a totally real number field, let $E\subset {\mathds C}$ be a purely imaginary quadratic extension of $F$ and let $V=E^{n+1}$. Assume that we are given a Hermitian form $H: V\times V \to E$ such that its extension to $V\otimes {\mathds C}$ has signature $(m,1)$. Given an embedding $\sigma: E\to {\mathds C}$ we denote by $H^{\sigma}$ the Hermitian form on ${\mathds C}^{n+1}$ obtained by applying $\sigma$ to the coefficients of the matrix representing $H$ in the canonical basis of $V$. Assume that for every embedding $\sigma: E\to {\mathds C}$ with $\sigma|_F\neq {\rm id}|_F$, the twisted Hermitian form $H^{\sigma}$ has signature $(m+1,0)$. Let $\mathcal{O}_E$ be the ring of integers of $E$ and let $U(H,\mathcal{O}_E)$ be the group of $(m+1)\times (m+1)$-matrices with coefficients in $\mathcal{O}_E$ which preserve $H$. Then $U(H,\mathcal{O}_E)$ is a lattice in the group $U(V\otimes {\mathds C},H)$ of automorphisms of $(V\otimes {\mathds C}, H)$. It is uniform if and only if $F\neq {\mathds Q}$. We call a lattice $\Gamma < {\rm PU}(n,1)$ of the *simplest type* if it is commensurable to a lattice of the form $U(H,\mathcal{O}_E)$. For $m\geq 2$, let $X=\Gamma \backslash \mathds{B}^m_{{\mathds C}}$ be a smooth compact complex hyperbolic $m$-manifold. We call a pair $(X,D)$ a *good pair* if $D=D_1\cup D_2\cup \dots \cup D_k\subset X$ is a non-trivial divisor such that the $D_i$ are pairwise non-intersecting smooth codimension one subvarieties of $X$. We call $D$ *totally geodesic* if the embeddings of the $D_i$ in $X$ are totally geodesic. If $\Gamma$ is of the simplest type, then $X$ admits totally geodesically immersed divisors. Up to passing to a finite cover of $X$, we can find such divisors which are embedded. We will require the following more precise version of this result (see [@StoTol-21-II Section 5]). **Lemma 14**. *Let $\Gamma<{\rm PU}(m,1)$ be a torsion-free congruence arithmetic lattice of the simplest type and let $X=\Gamma \backslash \mathbb{B}^m_{{\mathds C}}$. Then there exists a finite congruence cover $p:X'\to X$ and a divisor $D'\subset X'$ such that $(X',D')$ is a totally geodesic good pair.* Note that conversely, if $m\geq 2$ and $(X,D)$ is a totally geodesic good pair, with $X$ arithmetic, then $\Gamma$ is of the simplest type by [@StoTol-21-II Proposition 3.2]. **Theorem 15** ([@StoTol-21-II Theorem 1.5 and Proposition 5.1]). *Assume that $\Gamma < {\rm PU}(m,1)$ is a cocompact torsion-free congruence arithmetic lattice of the simplest type and let $X=\Gamma\backslash \mathbb{B}^m_{\mathbb{C}}$. Let $D\subset X$ be a divisor such that $(X,D)$ is a totally geodesic good pair. Let $d\geq 2$. Then there is a finite cover $p:X'\to X$ which admits a cyclic $d$-fold ramified cover $Y\to X'$ with ramification locus the totally geodesic divisor $D':=p^{-1}(D)$. The cover $Y$ is a smooth projective variety which admits a metric of negative sectional curvature and is not homotopy equivalent to any locally symmetric manifold.* Let us make a few comments on Theorem [Theorem 15](#thm:StoverToledo){reference-type="ref" reference="thm:StoverToledo"}. The fact that ramified covers of ball quotients along totally geodesic divisors admit negatively curved Kähler metrics was known prior to [@StoTol-21-II] and is due to Zheng [@zheng1], who generalized earlier work by Mostow--Siu [@MoSi-80]. The key contribution made in [@StoTol-21-II] is to show that one can find many arithmetic ball quotients $X$ containing totally geodesic divisors $D$, forming a good pair and such that the integral homology class of $D$ is divisible by some nontrivial integer, thus allowing to build cyclic ramified covers. The new contribution (the divisibility of the homology class $[D]$) relies on deep results on the cohomology of arithmetic groups due to Bergeron, Millson and Moeglin [@BeMiMo-16]. **Remark 16**. Arithmeticity of the lattices under consideration appears in two ways in this work. Firstly, through the results on the cohomology of arithmetic groups used in [@StoTol-21-II], and secondly through properties of the Albanese map of arithmetic ball quotients [@Eys-18; @LloPy-22]. In this second appearance, arithmeticity is used in a much more elementary way. ## New subgroups of hyperbolic groups of type $\mathscr{F}_{m-1}$ and not $\mathscr{F}_m$ We will check that we can apply Theorem [Theorem 11](#thm:LP-BNSR){reference-type="ref" reference="thm:LP-BNSR"} and Addendum [Addendum 12](#add-to-thm:LP-BNSR){reference-type="ref" reference="add-to-thm:LP-BNSR"} to the Stover--Toledo examples. To do so, we first need the following: **Lemma 17**. *Every compact Kähler manifold $Y$ as in Theorem [Theorem 15](#thm:StoverToledo){reference-type="ref" reference="thm:StoverToledo"} has non-zero Euler characteristic.* *Proof.* Recall that the Euler chracteristic is multiplicative under finite etale covers. For ramified covers, there is a similar formula, taking into account the ramification. It reads as follows, the notation being as in Theorem [Theorem 15](#thm:StoverToledo){reference-type="ref" reference="thm:StoverToledo"}: $$\chi (Y)-\chi (D')=d(\chi (X')-\chi (D)).$$ Since $D$ and $D'$ are diffeomorphic, we obtain: $$\label{eq:eulerch} \chi (Y)=d\chi (X')+(1-d)\chi(D).$$ We now use the fact that compact ball quotients of dimension $k$ have nonzero Euler characteristic, of the same sign as $(-1)^k$. Hence, $\chi (X')$ and $\chi (D)$ are both nonzero, of opposite signs. Thus the two terms on the right-hand side of [\[eq:eulerch\]](#eq:eulerch){reference-type="eqref" reference="eq:eulerch"} are nonzero of the same sign and $\chi (Y)\neq 0$. Alternatively, we could have appealed to Gromov's work [@Gro-91] to justify the nonvanishing of $\chi (Y)$, but the above argument is simpler. ◻ We are now ready to state and prove the main result of this section. Theorem [Theorem 1](#thm:ST-Intro){reference-type="ref" reference="thm:ST-Intro"} is a direct consequence. **Theorem 18**. *Let $m\geq 2$, let $\Gamma<{\rm PU}(m,1)$ be a uniform torsion-free congruence arithmetic lattice of the simplest type and let $X=\Gamma \backslash \mathbb{B}^m_{{\mathds C}}$. Let $d\ge 2$. Then there is a finite cover $X'\to X$ such that there exists a finite ramified cover $Y'_d\to X'$ with the following properties:* 1. *$Y'_d$ is a smooth projective variety which admits a metric of negative sectional curvature and is not homotopy equivalent to any locally symmetric manifold,* 2. *the $(m-1)$-th BNSR-invariant $\Sigma^{m-1}(\pi_1(Y'_d))$ is dense in the character sphere $S(\pi_1(Y'_d))$ and every rational character $\xi\in {\rm Hom}(\pi_1(Y'_d),{\mathds Q})-\{0\}$ such that $[\xi]$ is in the dense open set $\Sigma^{m-1}(\pi_1(Y'_d))\cap -\Sigma^{m-1}(\pi_1(Y'_d))$ satisfies that ${\rm{ker}}(\xi)$ is of type $\mathscr{F}_{m-1}$ but not of type ${\rm FP}_m({\mathds Q})$.* *Proof.* By Lemma [Lemma 14](#lem:totgeod){reference-type="ref" reference="lem:totgeod"}, there exists a finite congruence cover $X_1\to X$ and a divisor $D_1\subset X_1$ such that $(X_1,D_1)$ is a totally geodesic good pair. Theorem [Theorem 15](#thm:StoverToledo){reference-type="ref" reference="thm:StoverToledo"} implies that there is a finite congruence cover $p_2: X_2\to X_1$ which admits a $d$-fold cyclic ramified cover $q: Y_d\to X_2$ with ramification locus the totally geodesic divisor $D_2=p_2^{-1}(D_1)$. The manifold $Y_d$ admits a Kähler metric of negative sectional curvature and does not have the homotopy type of a locally symmetric space. By a theorem of Eyssidieux [@Eys-18] (see also [@LloPy-22 Theorem 24]) there is a further finite cover $p_3:X_3\to X_2$ such that the Albanese map $a_{X_3}\colon X_3\to A(X_3)$ is an immersion and thus defines a finite map to a complex torus. The pair $(X_3,D_3=p_3^{-1}(D_2))$ is again a totally geodesic good pair. Moreover, $p_3$ induces a regular cover $Y'_d\to Y_d$ such that there is a $d$-fold ramified cover $Y'_d\to X_3$ with ramification locus $D_3$. In particular, $Y'_d$ is a smooth projective variety admitting a metric of negative sectional curvature. Since finite (possibly ramified) covers are finite maps and compositions of finite maps are finite, the induced holomorphic map $Y'_d\to A(X_3)$ is finite. By Lemma [Lemma 17](#lem:euler){reference-type="ref" reference="lem:euler"}, $Y'_d$ has non-trivial Euler characteristic. Addendum [Addendum 12](#add-to-thm:LP-BNSR){reference-type="ref" reference="add-to-thm:LP-BNSR"} thus completes the proof. ◻ ## New Kähler groups of type $\mathscr{F}_{2n-1}$ and not $\mathscr{F}_{2n}$ {#sec:ST-CS} We sketch here without details a possible construction of groups as in the title of this section. Let $X_{{\rm CS}}$ be the *Cartwright-Steger surface*. It is a quotient of the unit ball of ${\mathds C}^2$ by a uniform congruence arithmetic lattice $\Gamma \leq {\rm PU}(2,1)$ of the simplest type [@Sto-14 p. 89-90] (see also [@StoTol-21-II Remark 3.7]) with $b_1(X)=2$. Consequently its Albanese map $f:X_{{\rm CS}}\to E$ is onto an elliptic curve $E$. It was shown in [@CarKozYeu-17] that this map has isolated singularities. Moreover, $f$ has connected fibres. Let $n\ge 2$ be an integer. Consider the map $$F : X_{{\rm CS}} \times \cdots \times X_{{\rm CS}}\to E$$ from the direct product of $n$ copies of $X_{{\rm CS}}$ to $E$ obtained by summing the map $f$ appplied to each factor. It was proved in [@NicPy-21] that the fundamental group of the generic fibre of the map $F$ is a Kähler group of type $\mathscr{F}_{2n-1}$ but not of type ${\rm FP}_{2n}(\mathbb{Q})$. Applying Stover and Toledo's work, we can consider a congruence cover $p : X' \to X_{{\rm CS}}$ admitting a ramified cover $p' : Y\to X'$ along a totally geodesic divisor. If the ramification locus in $X'$ is chosen in general position, it will project *onto* $E$ via the map $f\circ p$. This implies that the map $$h:=f\circ p\circ p' : Y\to E$$ has isolated critical points. One can thus repeat[^1] the construction from [@NicPy-21] by considering the map $$h+\cdots +h : Y \times \cdots \times Y\to E$$ from the direct product of $n\ge 2$ copies of $Y$. This yields new Kähler groups with exotic finiteness properties, sitting inside a direct product of hyperbolic groups. # Homomorphisms to ${\mathds Z}^{2}$ {#sec:zeetwo} In this section, we prove Theorem [Theorem 2](#thm:zeetwo){reference-type="ref" reference="thm:zeetwo"} (Section [5.1](#sec:zeetwo:first){reference-type="ref" reference="sec:zeetwo:first"}) and then discuss whether the groups with exotic finiteness properties constructed in this article and in [@LloPy-22] are Kähler or not (Section [5.2](#sec:zeetwo:second){reference-type="ref" reference="sec:zeetwo:second"}). ## Hodge theory yields circles in the BNSR-invariants {#sec:zeetwo:first} Let $X$ be a closed Kähler manifold of complex dimension $n$. Let $a\in H^{1}(X,{\mathds C})$ be a cohomology class whose real and imaginary part are independent in $H^{1}(X,{\mathds R})$. Equivalently, we require that $a$ is not a complex multiple of a real class. Associated to $a$, there is a circle $$S^{1}(a)\subset S(\pi_{1}(X))\cong H^{1}(X,{\mathds R})-\{0\}/{\mathds R}_{+}^{\ast}.$$ The circle $S^1(a)$ is made of the projections in $S(\pi_{1}(X))$ of the (de Rham) cohomology classes $${\rm Re}(e^{i\theta}a) \;\;\; (\theta \in {\mathds R}).$$ We now assume that $a$ is the cohomology class of a holomorphic $1$-form $\alpha$ with finitely many zeros. In particular, if $X$ is aspherical, Theorem 6 from [@LloPy-22] can be applied to the cohomology classes $[{\rm Re}(e^{i\theta}\alpha)]$ for each real number $\theta$, since $e^{i\theta}\alpha$ is a holomorphic $1$-form with finitely many zeros. That theorem gives that $[{\rm Re}(e^{i\theta}\alpha)]\in \Sigma^{n-1}(\pi_{1}(X))\cap -\Sigma^{n-1}(\pi_{1}(X))$. In other words, the circle $S^{1}([\alpha])$ is contained in $\Sigma^{n-1}(\pi_{1}(X))\cap -\Sigma^{n-1}(\pi_{1}(X))$. We have proved: **Proposition 19**. *If $X$ is aspherical and if $\alpha$ is a holomorphic $1$-form with finitely many zeros on $X$ we have: $$S^{1}([\alpha])\subset \Sigma^{n-1}(\pi_{1}(X))\cap -\Sigma^{n-1}(\pi_{1}(X)).$$* We are now ready to prove Theorem [Theorem 2](#thm:zeetwo){reference-type="ref" reference="thm:zeetwo"}, combining Proposition [Proposition 19](#prop:cibnsr){reference-type="ref" reference="prop:cibnsr"} with the results recalled in Section [2](#sec:FinProps){reference-type="ref" reference="sec:FinProps"}. *Proof of Theorem [Theorem 2](#thm:zeetwo){reference-type="ref" reference="thm:zeetwo"}.* Let $X$ be as in the statement of Theorem [Theorem 2](#thm:zeetwo){reference-type="ref" reference="thm:zeetwo"}. The set $U\subset H^{0}(X,\Omega^{1}_{X})$ of holomorphic $1$-forms with finitely many zeros is dense (see Propositions 14 and 18 in [@LloPy-22]). Let $\alpha \in U$. By Proposition [Proposition 19](#prop:cibnsr){reference-type="ref" reference="prop:cibnsr"}, the circle $S^{1}([\alpha])$ is contained in $\Sigma^{n-1}(\pi_{1}(X))\cap -\Sigma^{n-1}(\pi_{1}(X))$. Let $a_{1}$ and $a_2$ be rational elements in $H^{1}(X,{\mathds R})$ which are close enough to $[{\rm Re}(\alpha)]$ and $[{\rm Im}(\alpha)]$ respectively. The set $\Sigma^{n-1}(\pi_{1}(X))\cap -\Sigma^{n-1}(\pi_{1}(X))$ being open, we have $$\label{eq:fdetr} [\cos (t)a_1 +\sin (t) a_2] \in \Sigma^{n-1}(\pi_{1}(X))\cap -\Sigma^{n-1}(\pi_{1}(X))$$ for all real numbers $t$, if $a_{1}$ and $a_{2}$ are close enough to $[{\rm Re}(\alpha)]$ and $[{\rm Im}(\alpha)]$. Let $N$ be a large enough integer so that $Na_{1}$ and $Na_{2}$ define morphisms from $\pi_{1}(X)$ to $\mathbb{Z}$. The image $\Lambda$ of the morphism $(Na_1,Na_2) : \pi_{1}(X)\to {\mathds Z}^2$ is isomorphic to ${\mathds Z}^2$. Equation [\[eq:fdetr\]](#eq:fdetr){reference-type="eqref" reference="eq:fdetr"} implies that for any nonzero morphism $u : \Lambda \to {\mathds R}$, we have $[u\circ (Na_1,Na_2)]\in \Sigma^{n-1}(\pi_{1}(X))\cap -\Sigma^{n-1}(\pi_{1}(X))$. Theorem [Theorem 7](#thm:finiteness-high-rank){reference-type="ref" reference="thm:finiteness-high-rank"} then implies that the kernel of the morphism $$(Na_1,Na_2) : \pi_{1}(X)\to \Lambda \cong {\mathds Z}^{2}$$ is of type $\mathscr{F}_{n-1}$. The fact that it is not of type ${\rm FP}_n(\mathbb{Q})$ follows from the last point of Addendum [Addendum 12](#add-to-thm:LP-BNSR){reference-type="ref" reference="add-to-thm:LP-BNSR"}; indeed, the latter property holds for kernels of arbitrary characters. Finally the openness of the set of homomorphisms $\pi_{1}(X)\to {\mathds Z}^{2}$ whose kernel is of type $\mathscr{F}_{n-1}$ follows from Proposition [Proposition 8](#prop:openess){reference-type="ref" reference="prop:openess"}. This concludes the proof of Theorem [Theorem 2](#thm:zeetwo){reference-type="ref" reference="thm:zeetwo"}. ◻ Theorem [Theorem 2](#thm:zeetwo){reference-type="ref" reference="thm:zeetwo"} has the following consequence, which also proves Corollary [Corollary 3](#cor:zeetwo){reference-type="ref" reference="cor:zeetwo"}. **Corollary 20**. *Assume that $Z$ is a $n$-dimensional complex manifold which is either a complex ball quotient by a uniform arithmetic lattice with non-trivial first Betti number, or one of the Stover--Toledo manifolds $Y_d'$ constructed in Theorem [Theorem 18](#thm:ST-main-text){reference-type="ref" reference="thm:ST-main-text"}. Then there is a finite cover $Z'\to Z$ and a surjective homomorphism $\phi:\pi_1(Z')\to \mathbb{Z}^2$ with kernel of type $\mathscr{F}_{n-1}$, but not ${\rm FP}_n(\mathbb{Q})$.* *Proof.* We argue as in the proof of Theorem [Theorem 18](#thm:ST-main-text){reference-type="ref" reference="thm:ST-main-text"}, using Eyssidieux's work [@Eys-18], that in both cases there is a finite cover $Z'\to Z$ with finite Albanese map and non-trivial Euler characteristic. The result is then an immediate consequence of Theorem [Theorem 2](#thm:zeetwo){reference-type="ref" reference="thm:zeetwo"}. ◻ Since the virtual first Betti number of arithmetic lattices $\Gamma\le {\rm PU}(n,1)$ with $b_1(\Gamma)>0$ is infinite [@Ago-06; @Ven-08], it is natural to ask if Corollary [Corollary 20](#cor:zeetwo-main-text){reference-type="ref" reference="cor:zeetwo-main-text"} generalises to abelian quotients of arbitrary rank. **Question 21**. *Let $n\geq 2$ and $k\geq 3$ be integers. Let $Z$ be as in Corollary [Corollary 20](#cor:zeetwo-main-text){reference-type="ref" reference="cor:zeetwo-main-text"} with ${\rm dim}_{\mathbb{C}}(Z)=n$. Is there a finite cover $Z'\to Z$ whose fundamental group admits a surjective homomorphism $\phi:\pi_1(Z')\to \mathbb{Z}^k$ with kernel of type $\mathscr{F}_{n-1}$ and not of type ${\rm FP}_n(\mathbb{Q})$?* A positive answer to this question would also provide a positive answer to the following more general question. **Question 22**. *Let $n\geq 4$ and $k\geq 3$ be integers. Is there a hyperbolic group $G$ together with a surjective homomorphism $\phi: G\to \mathbb{Z}^k$ such that ${\rm{ker}}(\phi)$ is of type $\mathscr{F}_{n-1}$ and not of type ${\rm FP}_n(\mathbb{Q})$?* For $n=2$, this last question can be answered using a classical construction due to Rips, see [@BrHa p. 227] and very recently Kropholler and Llosa Isenrich gave an answer for $n=3$ [@Llo-Kro]. The case $n>3$ remains open. ## Kähler and non-Kähler subgroups {#sec:zeetwo:second} We mentioned in the introduction that complex Morse theory methods allow to construct examples of groups with exotic finiteness properties as subgroups of Kähler groups. These groups are sometimes Kähler but not always. In the next theorem, we justify that some of these groups are not Kähler. We focus on subgroups of lattices in ${\rm PU}(n,1)$ and explain in Remark [Remark 24](#rem:kcnpst){reference-type="ref" reference="rem:kcnpst"} how one could possibly obtain slightly more general results. **Theorem 23**. *Let $\Gamma < {\rm PU}(n,1)$ be a torsion-free cocompact lattice and let $\psi : \Gamma \to {\mathds Z}^{k}$ be a surjective homomorphism.* - *If $k=1$, the kernel of $\psi$ is not a Kähler group.* - *If $k=2$, and if the kernel of $\psi$ is a Kähler group, there exists a holomorphic map $\pi$ with connected fibres from the quotient $\Gamma \backslash\mathbb{B}_{{\mathds C}}^{n}$ onto an elliptic curve $E$, and an isomorphism $\varphi : \pi_{1}(E)\to {\mathds Z}^2$ such that $\psi=\varphi\circ \pi_{\ast}$.* When $n\le 2$, the kernel of a morphism $\psi$ as above cannot be finitely presented. This follows from Corollary 15 in [@LIMP-21]; see also [@fisher]. Hence Theorem [Theorem 23](#thm:nonk){reference-type="ref" reference="thm:nonk"} is relevant only for $n\ge 3$. As a consequence of it, we obtain that in the context of Theorem [Theorem 2](#thm:zeetwo){reference-type="ref" reference="thm:zeetwo"} applied to ball quotients, most of the kernels under consideration are not Kähler. Indeed, let $X$ be a closed ball quotient with finite Albanese map. Let $\psi : \pi_{1}(X)\to {\mathds Z}^2$ be a surjective morphism. The space $H^{1}(X,{\mathds R})$ is endowed with its natural complex structure via the identification $$H^{1,0}(X)\to H^{1}(X,{\mathds R})$$ mapping a holomorphic $1$-form to the class of its real part. If the kernel of $\psi$ is Kähler, then the plane $P\subset H^{1}(X,{\mathds R})$ defined by $\psi$ is a complex line, thanks to Theorem [Theorem 23](#thm:nonk){reference-type="ref" reference="thm:nonk"}. This last condition can be broken by a suitable rational perturbation of $\psi$, yielding many morphisms onto ${\mathds Z}^2$ with non-Kähler kernels. Note that we assumed that the Albanese map of $X$ was finite, hence $b_{1}(X)>2$ and $H^{1}(X,{\mathds Q})$ is large enough to perturb $\psi$. *Proof of Theorem [Theorem 23](#thm:nonk){reference-type="ref" reference="thm:nonk"}.* As observed above, we can assume that $n\ge 2$. We fix $\Gamma$ and $\psi$ as in the statement of the theorem. We write $X=\Gamma \backslash \mathbb{B}_{\mathbb{C}}^{n}$. We assume that there exists a closed Kähler manifold $Y$ whose fundamental group is isomorphic to the kernel of $\psi$. The natural morphism $\varrho : \pi_{1}(Y) \to \pi_{1}(X)$ is induced by a smooth map $h : Y\to X$. By work of Eells and Sampson there exists a harmonic map $h_0 : Y \to X$ homotopic to $h$, see [@EelSam-64] and also [@ABCKT-95 p. 68]. This map is pluriharmonic and the $(1,0)$ part of its differential $$dh_{0}^{1,0} : TY \to TX\otimes {\mathds C}$$ is holomorphic for a suitable holomorphic structure on the bundle $TX\otimes {\mathds C}$, see [@ABCKT-95 Ch. 6] or [@Py-book §9.2.2]. We distinguish two cases. If the complex rank of $dh_{0}^{1,0}$ is equal to $1$, then the harmonic map $h_0$ factors through a Riemann surface: there exists a surjective holomorphic map with connected fibres onto a Riemann surface, denoted by $\pi : Y \to \Sigma$, and a harmonic map $m_{0} : \Sigma \to X$ such that $$h_0=m_{0}\circ \pi.$$ This result is due to Carlson and Toledo [@CarTol-89 §7]. The kernel of the induced morphism $\pi_{\ast}$ is contained in the kernel of $\varrho$. Since $\varrho$ is injective, so is $\pi_{\ast}$. Since $\pi$ has connected fibres, $\pi_{\ast}$ is also surjective, and we obtain that $$\pi_{\ast} : \pi_{1}(Y)\to \pi_{1}(\Sigma)$$ is an isomorphism. However, $\pi_1(Y)$ is not of type ${\rm FP}_n(\mathbb{Q})$ by Addendum [Addendum 12](#add-to-thm:LP-BNSR){reference-type="ref" reference="add-to-thm:LP-BNSR"}, while surface groups are of type $\mathscr{F}_r$ for all $r$, yielding a contradiction. Alternatively, one can also observe that $\Gamma$ fits into an extension $$1 \to \pi_{1}(\Sigma)\to \Gamma \to {\mathds Z}^k \to 1$$ to obtain a contradiction by a theorem due to Bregman and Zhang [@BrZh] (see also the work of Nicolás [@Nic-22] for a different proof of this theorem). This first half of the proof works for arbitrary $k$. We now assume that the complex rank of $dh_{0}^{1,0}$ is greater than $1$. In that case another result due to Carlson and Toledo [@CarTol-89 Cor. 3.7] ensures that the map $h_0$ is holomorphic (after possibly replacing the complex structure on $X$ by its complex conjugate structure). To complete the proof we use arguments similar to those in [@Llo-18-I §3] and [@Llo-17 p.19--20]. The kernel of the pull-back map $$h_{0}^{\ast} : H^{1}(X,{\mathds C})\to H^{1}(Y,{\mathds C})$$ is a sub-Hodge structure of $H^{1}(X,{\mathds C})$. In particular it has even dimension. By the original definition of $\pi_{1}(Y)$, which is isomorphic to the kernel of $\psi$, we obtain a contradiction if $k=1$. This concludes the proof that ${\rm ker}(\psi)$ is not Kähler for $k=1$. When $k=2$, we consider the Albanese tori of $X$ and $Y$, denoted by $A(X)$ and $A(Y)$. We let $a_X : X \to A(X)$ and $a_Y : Y \to A(Y)$ be the corresponding Albanese maps. By the universal property of Albanese maps, there exists a holomorphic map $\theta : A(Y)\to A(X)$ making the following diagram commutative: $$\xymatrix{Y \ar[r]^{h_{0}} \ar[d]^{a_{Y}} & X \ar[d]^{a_{X}} \\ A(Y) \ar[r]^{\theta} & A(X). \\ }$$ We can assume that $\theta$ is a group morphism, after possibly composing $a_X$ by a translation. The codimension of the image of $A(Y)$ in $A(X)$ is equal to the dimension of the kernel of the restriction map $H^{1,0}(X)\to H^{1,0}(Y)$, which is equal to $\frac{k}{2}=1$. Hence $\theta(A(Y))$ is a codimension $1$ subtorus of $A(X)$. We let $E$ be the quotient $A(X)/\theta (A(Y))$ and $\pi$ be the composition of the map $a_X$ with the quotient map $A(X)\to E$. Let $a\in H^{1}(X,{\mathds Z})$. We write $a=a_X^{\ast}b$ with $b\in H^{1}(A(X),{\mathds Z})$. Then $a$ vanishes on $\pi_{1}(Y)$ if and only if $b$ vanishes on the image of $\pi_{1}(A(Y))$ in $\pi_{1}(A(X))$. This is also equivalent to the fact that $b$ comes from a class in $H^{1}(E,{\mathds Z})$. This implies that $\psi=\varphi \circ \pi_{\ast}$ for some morphism $\varphi : \pi_{1}(E)\to {\mathds Z}^2$ which is necessarily an isomorphism. We finally have to justify the fact that $\pi$ has connected fibers. If this is not the case, we can consider the Stein factorization $X\to \Sigma \to E$ of this map. If $\Sigma\neq E$ and $\Sigma$ is hyperbolic, we obtain a contradiction since the kernel of the morphism $\pi_{1}(\Sigma)\to \pi_{1}(E)$ is not finitely generated. If $\Sigma$ has genus one, it cannot be a nontrivial covering space of $E$, otherwise $\pi_{\ast}$ would not be surjective. Hence $\Sigma =E$ and $\pi$ has connected fibers. This concludes the proof of Theorem [Theorem 23](#thm:nonk){reference-type="ref" reference="thm:nonk"}. ◻ **Remark 24**. One could try to prove a statement analogous to Theorem [Theorem 23](#thm:nonk){reference-type="ref" reference="thm:nonk"}, when the lattice $\Gamma < {\rm PU}(n,1)$ is replaced by the fundamental group of one of the negatively curved manifolds built by Stover and Toledo and described in Section [4.1](#subsec:stto){reference-type="ref" reference="subsec:stto"}. This can most probably be done along the following lines. The natural Kähler form on these manifolds (see [@MoSi-80; @zheng1]) is built by pulling back the Kähler form of the ball quotient and adding a suitable $(1,1)$ form supported near the branch locus. This metric is known to have nonpositive Hermitian sectional curvature (see [@ABCKT-95 Ch. 6] or [@Py-book §9.2.2] for this notion). This property is enough to build a harmonic map and to prove that it is pluriharmonic, as in the proof of Theorem [Theorem 23](#thm:nonk){reference-type="ref" reference="thm:nonk"}. Then, one should adapt the results from [@CarTol-89], about harmonic maps to complex hyperbolic manifolds, to the case where the target is one of the manifolds built by Stover--Toledo. # Fibre products over Riemann surfaces {#sec:Livne} In this section we describe a new method that uses complex Morse theory to produce examples of non-normal subgroups with exotic finiteness properties in a direct product of non-positively curved groups. Most previous constructions of subgroups with exotic finiteness properties in non-positively curved groups lead to normal subgroups. Our method might possibly be applied in other situations and could lead to further interesting examples. ## Fibre products with isolated singularities and Livné's surfaces {#sec:Liv-Ex} Assume that $X_1$ and $X_2$ are closed complex manifolds of dimensions $n_i={\rm{dim}}_{{\mathds C}}(X_i)$ with $n_1+n_2\geq 3$ and that $p_i: X_i\to \Sigma$ ($i=1, 2$), is a surjective holomorphic map with isolated critical points onto a closed hyperbolic Riemann surface $\Sigma$. Assume further that $p_1$ and $p_2$ induce surjections on fundamental groups, that the fibres of $p_1$ are connected, and that the sets of critical values of $p_1$ and $p_2$ are both non-empty and have empty intersection. Let $$Z:=\left\{(x_1,x_2)\in X_1\times X_2 \mid p_1(x_1)=p_2(x_2)\right\}\subset X_1\times X_2$$ be the fibre product and observe that $Z=(p_1,p_2)^{-1}(\Delta_{\Sigma})$ for $\Delta_{\Sigma}\hookrightarrow \Sigma \times \Sigma$ the diagonal. Our hypothesis on critical values implies that $Z$ is a smooth connected complex submanifold of $X_1\times X_2$. Denote by $q: Y\to \Sigma\times \Sigma$ the covering space corresponding to the subgroup $\pi_{1}(\Delta_{\Sigma})<\pi_{1}(\Sigma)\times \pi_{1}(\Sigma)$ and let $\widehat{q}:W\to X_1\times X_2$ be the covering space induced by $(p_1,p_2)_{\ast}^{-1}(\pi_1(\Delta_{\Sigma}))$. We fix a hyperbolic metric on $\Sigma$ and equip $Y$ with the metric of non-positive curvature obtained by pulling back the corresponding product metric on $\Sigma \times \Sigma$. Let $\widehat{\Delta}_{\Sigma}$ be the unique *compact* connected component of the preimage of $\Delta_{\Sigma}$ in $Y$. Let $$\begin{array}{cccc} f: & Y & \to & \mathbb{R}_{\geq 0} \\ & y & \mapsto & \left({\rm dist}\left(y,\widehat{\Delta}_{\Sigma}\right)\right)^2 \\ \end{array}$$ be the square of the distance function to $\widehat{\Delta}_{\Sigma}$ in $Y$. Since $\Sigma$ is non-positively curved, for every $y\in Y$ there is a unique shortest geodesic connecting $y$ to a point in $\widehat{\Delta}_{\Sigma}$ (it intersects $\widehat{\Delta}_{\Sigma}$ orthogonally). This implies that the restriction of $f$ to $f^{-1}((0,\infty))= Y \setminus \widehat{\Delta}_{\Sigma}$ is a submersion. One can in fact show that $Y$ is diffeomorphic to the normal bundle of $\widehat{\Delta}_{\Sigma}$, see [@Sik-87 Section 2]. We fix once and for all a map $\pi : W \to Y$ lifting the composition $(p_{1},p_{2})\circ \widehat{q}$, so that we have a commutative diagram: $$\xymatrix{ & W\ar[r]^{\widehat{q}}\ar[ld]_{f\circ \pi}\ar[d]^{\pi}& X_1\times X_2\ar[d]^{(p_1,p_2)}\\ \mathbb{R}_{\geq 0}& Y \ar[l]^{f}\ar[r]_{q} &\Sigma \times \Sigma. }$$ Observe that the maps $f$ and $\pi$ are proper. Thus, the same applies to $f\circ\pi$. Since $p_1$ and $p_2$ induce surjections on fundamental groups, the fibre product of $Y$ and $X_1 \times X_2$ is connected, hence can be identified with $W$. We then define: $$\widehat{Z}:=\pi^{-1}(\widehat{\Delta}_{\Sigma}).$$ The description of $W$ as a fibre product implies that $\widehat{q}$ induces a diffeomorphism between $\widehat{Z}$ and $Z$. In the next two sections, we shall prove the following theorem: **Theorem 25**. *The manifold $W$ and the group $\pi_{1}(Z)$ enjoy the following properties:* 1. *the inclusion $\widehat{Z} \hookrightarrow W$ induces an isomorphism on homotopy groups up to degree $n_{1}+n_{2}-2$, in particular $W$ and $Z$ have isomorphic fundamental groups,* 2. *$H_{n_1+n_2}(W,\mathbb{Q})$ is infinite dimensional,* 3. *$\pi_1(Z)$ is the group theoretic fibre product of the surjective homomorphisms $p_{1\ast}:\pi_1(X_{1})\to \pi_1(\Sigma)$ and $p_{2\ast}:\pi_1(X_{2})\to \pi_1(\Sigma)$,* 4. *if $X_1$ and $X_2$ are aspherical, then $\pi_1(Z)$ is of finiteness type $\mathscr{F}_{n_1+n_2-1}$ but not of type ${\rm FP}_{n_1+n_2}(\mathbb{Q})$.* Theorem [Theorem 4](#thm:Livne){reference-type="ref" reference="thm:Livne"} from the introduction follows immediately from Theorem [Theorem 25](#thm:FibreProduct){reference-type="ref" reference="thm:FibreProduct"}. Let us now turn to the examples. In [@Liv-81], Livné constructed a family of smooth compact complex algebraic surfaces $S_d(N)$ by taking ramified covers of the compactification $E(N)$ of the universal elliptic curve with full level $N$ structure for every integer $N\geq 5$. The holomorphic map $E(N)\to X(N)$ to the compactification of the moduli space of elliptic curves with a full level $N$ structure has isolated singularities. Since the ramified covering is along a divisor whose irreducible components are sections of $E(N)\to X(N)$, the induced holomorphic map $S_d(N)\to X(N)$ also has isolated singularities. Livné proves that for $(N,d)\in \left\{(7,7), (8,4), (9,3), (12,2)\right\}$ the surface $S_d(N)$ is a complex ball quotient. Since $X(N)$ is a closed hyperbolic Riemann surface, this provides examples of 2-dimensional complex ball quotients admitting maps with isolated singularities to closed hyperbolic Riemann surfaces, to which one can apply Theorem [Theorem 4](#thm:Livne){reference-type="ref" reference="thm:Livne"}. The proof of Theorem [Theorem 25](#thm:FibreProduct){reference-type="ref" reference="thm:FibreProduct"} is given in the next two subsections. ## A perturbation of the distance function {#sec:perturbation} In this section, by perturbing the function $f\circ \pi$, we will construct a proper function $$h : W \to {\mathds R}_{\ge 0}$$ which is Morse outside of $\widehat{Z}$ and all of whose critical points outside of $\widehat{Z}$ have index equal to $n_{1}+n_{2}$. This function will be used in Section [6.3](#sec:ComplexMorse){reference-type="ref" reference="sec:ComplexMorse"} to prove Thereom [Theorem 25](#thm:FibreProduct){reference-type="ref" reference="thm:FibreProduct"}. We start by observing that $f\circ \pi|_{W\setminus \widehat{Z}}$ has isolated critical points, coinciding with the zeros of the differential of $\pi$. In the lemma below, we denote by $d_{w}\varphi$ the differential of a smooth function $\varphi$ at a point $w$. **Lemma 26**. *[\[lem:Sing\]]{#lem:Sing label="lem:Sing"} Fix a point $w\in W\setminus \widehat{Z}$. The following are equivalent:* 1. *${\rm d}_{w}(f\circ\pi)=0$* 2. *${\rm d}_w \pi=0$* 3. *${\rm d}_{\widehat{q}(w)}(p_1,p_2)=0$.* *If a function $f' : W \to {\mathds R}_{\ge 0}$ is sufficiently $C^{1}$-close to $f$ near $w$, the conditions above are also equivalent to $d_{w}(f'\circ \pi)=0$.* *Proof.* The equivalence of (2) and (3) is a direct consequence of the fact that $q$ and $\widehat{q}$ are covering maps. By the chain rule, (2) implies (1). We continue the proof by showing that (1) implies (2). Recall that the restriction of $f$ to $Y\setminus \widehat{\Delta}_{\Sigma}$ is a submersion. Let $y\in Y\setminus \widehat{\Delta}_{\Sigma}$ and let $$(s_1,s_2):=q(y)\in \Sigma\times \Sigma.$$ The orthogonal decomposition $$T_{(s_1,s_2)}\Sigma\times\Sigma = T_{s_1}\Sigma \oplus T_{s_2}\Sigma$$ pulls back to a decomposition $T_yY= E_{y,1}\oplus E_{y,2}$, with $E_{y,i}=({\rm d}_{y}q)^{-1}(T_{s_i}\Sigma)$. Let $\mathbb{H}$ be the hyperbolic plane, thought of as the universal cover of $\Sigma$. The orthogonal projection of a point $(a,b)\in \mathbb{H}\times \mathbb{H}$ on the diagonal $\Delta_{\mathbb{H}}$ is the point $(z,z)$ where $z$ is the midpoint on the geodesic going from $a$ to $b$. The gradient flow of the distance to the diagonal $\Delta_{\mathbb{H}}$ is obtained by "flowing\" simultaneously $a$ and $b$ toward $z$. From this one sees easily that the differential of the function $f$ is nonzero when restricted to each of the two subspaces $E_{y, j}$ ($j=1, 2$). Let $w\in W\setminus \widehat{Z}$ with ${\rm d}_w\pi \neq 0$. We write $\widehat{q}(w)=(x_{1},x_{2})$. If $d_w\pi$ is a submersion, then $d_w(f\circ \pi)\neq 0$. Thus, we may assume that ${\rm d}_w \pi$ has ${\mathds C}$-rank $1$. This is the case if exactly one of the $x_i$'s is a critical point of $p_i$, say $x_1$. In that case we have that ${\rm{im}}({\rm d}_w\pi)=E_{\pi(w),2}$ and therefore ${\rm d}_w(f\circ \pi)\neq 0$ by the previous observation applied to $y=\pi (w)$. This completes the proof of the equivalence of (1), (2) and (3). The condition $$d_{\pi (w)}f (E_{\pi (w),j})\neq 0\;\;\;\; (j=1, 2)$$ being $C^1$-open, the assertion about perturbations of $f$ follows immediately. ◻ We now construct the function $h$. Let $${\rm CV}((p_{1},p_{2}))\subset \Sigma \times \Sigma$$ be the product of the set of critical values of $p_1$ with the set of critical values of $p_2$. In other words, ${\rm CV}((p_{1},p_{2}))$ is the image under $(p_{1},p_{2})$ of the set of zeros of the differential of $(p_{1},p_{2})$. This set is finite and does not intersect the diagonal $$\Delta_{\Sigma}\subset \Sigma \times \Sigma,$$ by assumption. Hence the set ${\rm CV}(\pi)=q^{-1}({\rm CV}((p_{1},p_{2})))$ is discrete and does not intersect $\widehat{\Delta}_{\Sigma}$. By compactness it also avoids a closed neighborhood $U$ of $\widehat{\Delta}_{\Sigma}$. We will perturb $f$ near each point of ${\rm CV}(\pi)$ to obtain a function $f' : Y \to {\mathds R}_{\ge 0}$ and will later perturb $f'\circ \pi$ near the zero set of the differential of $\pi$ to obtain the function $h$. We pick a family of disjoint closed balls $$(B(v))_{v\in {\rm CV}(\pi)},$$ which are also disjoint from $U$. We identify $B(v)$ with the closed unit ball $B_{1}$ of ${\mathds C}^{2}$, via some holomorphic coordinates. We denote by $B_r$ the closed ball of radius $r$ and use the symbol $\vert \vert \cdot \vert \vert$ simultaneously for the Euclidean norm on ${\mathds C}^{2}$ and for the dual norm on $({\mathds C}^{2})^{\ast}$ defined by $$\vert \vert \varphi \vert \vert = \underset{\vert \vert v \vert \vert \le 1}{{\rm sup}}\vert \varphi (v)\vert,$$ for $\varphi \in ({\mathds C}^{2})^{\ast}$. We fix once and for all a smooth function $$\chi : {\mathds C}^2\to [0,1]$$ such that $\chi$ is equal to $0$ on a neighborhood of the origin, and is equal to $1$ on a neighborhood of the set $\{\vert \vert z \vert \vert \ge 1\}$. We now pick a constant $D\ge 1$ such that the inequalities: $$\vert \vert d_{z}\chi\vert \vert \le D,$$ $$\vert f(z)- (f(0) + d_{0}f(z))\vert\le D \vert \vert z \vert \vert^{2},$$ $$\vert \vert d_{z}f-d_{0}f\vert \vert \le D \vert \vert z\vert \vert$$ hold for each point $z\in B_1$. For $r\in (0,1)$, we let $$h_{r}(z)=\chi (\frac{z}{r})f(z)+\left(1-\chi (\frac{z}{r})\right)(f(0)+d_{0}f(z)).$$ The function $h_r$ is equal to $f$ near the boundary of $B_r$ and we have on $B_r$: $$\vert f(z)-h_{r}(z)\vert \le D r^2$$ and $$d_{z}h_{r}=d_{0}f+E(z,r)$$ where the norm of the linear form $E(z,r)$ is bounded by $(D+D^{2})r$. In particular, there exists $r_0>0$ such that for $r\le r_{0}$ the differential $d_{z}h_{r}$ cannot vanish for $z\in B_r$. We also assume that $r_0$ is chosen small enough so that $h_{r}>0$ and so that the critical points of $h_{r}\circ \pi$ on $\pi^{-1}(B_r)$ are the zeros of the differential of $\pi$ (for $r\le r_{0}$); this is possible thanks to Lemma [Lemma 26](#lem:crdi){reference-type="ref" reference="lem:crdi"}. Since $h_{r_{0}}$ coincides with $f$ near the boundary of $B_{r_{0}}$ we can modify $f$ by replacing it by $h_{r_{0}}$ in the ball $B_{r_{0}}$. We perform this modification in all the balls $(B(v))_{v\in {\rm CV}(\pi)}$ and obtain a new function $f' : Y \to {\mathds R}_{\ge 0}$. Hence the critical points of the function $$f'\circ \pi : W \to {\mathds R}_{\ge 0}$$ are the zeros of $d\pi$, away from $\widehat{Z}$. Let $w\in W \setminus \widehat{Z}$ be such a critical point. We pick coordinates as before on the ball $B(\pi (w))$. Up to a constant, the map $f'$ is equal to a linear form $\ell$ near $\pi (w)$. We write $\ell ={\rm Re}(A)$ where $A : {\mathds C}^2 \to {\mathds C}$ is a complex linear form. The holomorphic function $A\circ \pi$ has an isolated critical point at $w$. Let $\mu$ be its Milnor number (see Section 7 and Appendix B in [@Mil-68]). We now choose coordinates on a small closed ball $B$ around $w$. Let $B' \Subset B$ be a smaller closed ball. We assume that $A\circ \pi$ has only one critical point in $B$ (i.e. $w$, identified with the origin). There exist complex linear forms $u : {\mathds C}^{n_{1}+n_{2}}\to {\mathds C}$ arbitrarily close to $0$ so that $A\circ \pi+u$ has no critical point near the boundary of $B$ and $\mu$ nondegenerate critical points in the interior of $B$, all contained in $B'$. We modify $f'\circ \pi$ inside $B$ so that it equals ${\rm Re}(A\circ \pi+u)$ inside $B'$, $f'\circ \pi$ near the boundary of $B$, and so that its only critical point are inside $B'$. We can assume that the perturbation still takes positive values and is at distance at most $1$ from $f'\circ \pi$. Finally, we denote by $h$ the function obtained by modifying $f'\circ \pi$ as above in a neighbourhood of each zero of $d\pi$. Since $h$ is the real part of a holomorphic Morse function in a neighbourhood of each of its critical points, they are all non-degenerate of index $n_1 + n_2$. To summarize, we have proved: **Proposition 27**. *There exists a smooth function $$h : W \to {\mathds R}_{\ge 0}$$ such that:* 1. *$h$ coincides with $f\circ \pi$ on a neighborhood of $\widehat{Z}$,* 2. *$h$ is proper,* 3. *the set of critical points of $h$ in $W\setminus \widehat{Z}$ is discrete and each critical point is nondegenerate of index $n_{1}+n_{2}$.* ## Conclusion of the proof {#sec:ComplexMorse} We start this section by two propositions which will easily imply Theorem [Theorem 25](#thm:FibreProduct){reference-type="ref" reference="thm:FibreProduct"}. **Proposition 28**. *The inclusion $\widehat{Z} \hookrightarrow W$ induces an isomorphism on homotopy groups up to degree $n_{1}+n_{2}-2$.* *Proof.* Since $h=f\circ \pi$ close enough to $\widehat{Z}$ and since $f\circ \pi$ has no critical points near $\widehat{Z}$, besides the points of $\widehat{Z}$ themselves, we can pick $\varepsilon >0$ such that the inclusion $\widehat{Z} \hookrightarrow h^{-1}([0,\varepsilon])$ is a homotopy equivalence. This follows from the fact that any tubular neighbourhood of $\widehat{Z}$ contains $h^{-1}([0,\delta])$ for some $\delta >0$ and conversely that for any $\delta >0$, $h^{-1}([0,\delta])$ contains a tubular neighbourhood of $\widehat{Z}$. Since the set of critical points of $h$ is discrete away from $\widehat{Z}$, and since all these critical points have index $n_{1}+n_{2}$, for every regular values $b\ge a$, the space $h^{-1}([0,b])$ is obtained from $h^{-1}([0,a])$, up to homotopy, by attaching finitely many cells of dimension $n_{1}+n_{2}$. In particular the inclusion $h^{-1}([0,\varepsilon])\hookrightarrow W$ induces an isomorphism on homotopy groups up to degree $n_{1}+n_{2}-2$. Combined with the observation from the previous paragraph, this fact implies the conclusion of the proposition. ◻ **Proposition 29**. *The $(n_1+n_2)$-th homology group $H_{n_1+n_2}(W,\mathbb{Q})$ is infinite dimensional.* *Proof.* This is identical to an argument from [@NicPy-21]. We choose an increasing sequence of regular values $(a_{k})_{k\ge 0}$ of $h$ converging to infinity in such a way that $h^{-1}((a_{k},a_{k+1}))$ always contains at least one critical point of $h$. The group $$H_{n_{1}+n_{2}}(W,{\mathds Q})$$ is the direct limit of the groups $H_{n_{1}+n_{2}}(h^{-1}([0,a_{k}]),{\mathds Q})$. We write $W_{k}:=h^{-1}([0,a_{k}])$. Since $W_{k+1}$ is obtained from $W_{k}$ by gluing $(n_{1}+n_{2})$-dimensional cells, up to homotopy, the Betti numbers $$(b_{n_{1}+n_{2}-1}(W_{k}))_{k\ge 0}$$ form a non-increasing sequence. We pick an integer $k_{0}$ such that this sequence is constant for $k\ge k_{0}$. An application of the Mayer-Vietoris sequence then proves that the maps $$H_{n_{1}+n_{2}}(W_{k},{\mathds Q})\to H_{n_{1}+n_{2}}(W_{k+1},{\mathds Q})\;\;\;\; (k\ge k_{0})$$ are injective and that the sequence $(b_{n_{1}+n_{2}}(W_{k}))_{k\ge k_{0}}$ is strictly increasing. This implies that $H_{n_{1}+n_{2}}(W,{\mathds Q})$ is infinite dimensional and concludes the proof. We refer the reader to [@NicPy-21 p. 61--62] for more details. ◻ We can now conclude the proof of Theorem [Theorem 25](#thm:FibreProduct){reference-type="ref" reference="thm:FibreProduct"}. *Proof of Theorem [Theorem 25](#thm:FibreProduct){reference-type="ref" reference="thm:FibreProduct"}.* The first item of the theorem follows from Propositions [Proposition 28](#prop:attaching-n-cells){reference-type="ref" reference="prop:attaching-n-cells"} and from the following remark: since $n_{1}+n_{2}\ge 3$, the inclusion $\widehat{Z} \hookrightarrow W$ induces an isomorphism on fundamental group and since $Z$ and $\widehat{Z}$ are diffeomorphic, $\pi_{1}(Z)$ and $\pi_{1}(W)$ indeed have isomorphic fundamental groups. The second item of the theorem follows from Proposition [Proposition 29](#prop:infinite-homology){reference-type="ref" reference="prop:infinite-homology"}. To prove the third item, we observe that the fundamental group of $W$ is by construction equal to the fiber product: $$\label{eq:fproducts6} \{(a,b)\in \pi_{1}(X_{1})\times \pi_{1}(X_{2}), p_{1,\ast}(a)=p_{2,\ast}(b)\}.$$ Since the projection from $\widehat{Z}$ to $Z$ is an isomorphism, we obtain that the morphism induced by the inclusion of $Z$ in $X_{1}\times X_{2}$ is injective at the level of fundamental groups, with image given by the subgroup [\[eq:fproducts6\]](#eq:fproducts6){reference-type="eqref" reference="eq:fproducts6"}. Finally, if $X_1$ and $X_2$ are aspherical, so is $W$. This means that $W$ is a $K(\pi_1(W),1)$. Proposition [Proposition 29](#prop:infinite-homology){reference-type="ref" reference="prop:infinite-homology"} then implies that $$\pi_{1}(Z)\cong \pi_{1}(W)$$ is not of type ${\rm FP}_{n_{1}+n_{2}}({\mathds Q})$ while Proposition [Proposition 28](#prop:attaching-n-cells){reference-type="ref" reference="prop:attaching-n-cells"} implies that it is of type $\mathscr{F}_{n_{1}+n_{2}-1}$. This completes the proof of Theorem [Theorem 25](#thm:FibreProduct){reference-type="ref" reference="thm:FibreProduct"}. ◻ # Subgroups of $n$-iterated Kodaira fibrations {#sec:Koc-Vid} The goal of this section is to use methods from complex geometry to provide a new proof of a result of Kochloukova and Vidussi on finiteness properties of subgroups of fundamental groups of $n$-iterated Kodaira fibrations. Along the way, our proof shows that certain iterated Kodaira fibrations admit finite Albanese maps. This provides more examples of closed aspherical Kähler manifolds to which the methods of the present article and of [@LloPy-22] can be applied. ## Constructing $n$-iterated Kodaira fibrations We start by recalling the inductive definition of $n$-iterated Kodaira fibrations. **Definition 30**. Let $X$ be a compact complex manifold of dimension $n\geq 1$. If $n=2$, then we call $X$ a *$2$-iterated Kodaira fibration* (or simply a *Kodaira fibration*) if there exists a holomorphic submersion $\pi: X\to Y$ with connected fibres onto a closed hyperbolic Riemann surface, which is not isotrivial. If $n>2$, then we call $X$ an *$n$-iterated Kodaira fibration* if there is a holomorphic submersion $\pi:X\to Y$ with connected fibres onto an $(n-1)$-iterated Kodaira fibration $Y$, which is not isotrivial. We call a group $G$ a *polysurface group* of length $n$ if there is a filtration $$1=G_0\unlhd G_1\unlhd G_2\unlhd \dots \unlhd G_n=G$$ such that $G_i/G_{i-1}$ is the fundamental group of an orientable closed hyperbolic surface for $1\leq i \leq n$. By construction fundamental groups of $n$-iterated Kodaira fibrations are polysurface groups. We call a group $G$ *irreducible* if no finite index subgroup of $G$ decomposes as a direct product of two non-trivial groups. In [@LloPy-21], we inductively constructed $n$-iterated Kodaira fibrations with injective monodromy, starting from the Kodaira--Atiyah examples of ($2$-iterated) Kodaira fibrations, which are known to have injective monodromy. The induction step is given by the following theorem, summarising results from [@LloPy-21 Section 5]. **Theorem 31**. *Let $X$ be an $n$-iterated Kodaira fibration with injective monodromy. Then there are finite coverings $X',~ X''\to X$ such that $X'$ is the base space of an $(n+1)$-iterated Kodaira fibration $Z\to X'$ with injective monodromy and there is a finite map $Z\to X'\times X''$ which defines a ramified covering of its image.* The proof of Theorem [Theorem 31](#thm:ItKodFib){reference-type="ref" reference="thm:ItKodFib"} consists in performing a classical construction due to Kodaira and Atiyah "in family\". It also relies on the fact that a finite cover $X'\to X$ of an $n$-iterated Kodaira fibration $X$ with injective mondromy is again an $n$-iterated Kodaira fibration with injective monodromy, with base a finite cover of the base of $X$ (see the proof of [@LloPy-21 Proposition 39]). In particular, if one starts from the Kodaira--Atiyah fibration, the following is an implicit consequence of the construction in [@LloPy-21 Section 5]. **Theorem 32**. *Fix an integer $n\geq 2$. There exists a sequence of $i$-iterated Kodaira fibrations $X^i$, $2\leq i \leq n$, and a closed hyperbolic Riemann surface $X^1$, together with holomorphic submersions with connected fibres $\pi_{i,i-1}: X^i\to X^{i-1}$, which are not isotrivial and have injective monodromy, with the following properties:* 1. *there is a finite map $f: X^n\to S_1\times \dots \times S_{2^{n-1}}$ to a direct product of $2^{n-1}$ closed hyperbolic Riemann surfaces,* 2. *the group $\pi_1(X^n)$ is irreducible.* *Proof.* By construction [@Ati-69; @Kod-67], the Kodaira-Atiyah fibration $X^2$ is a ramified cover of a direct product $R\times T$ of two closed hyperbolic Riemann surfaces and the projection to either of the factors is a holomorphic submersion with connected fibres and injective monodromy. In particular, the map $X^2\to R\times T$ is finite and we can choose $X^1=R$. Inductively applying Theorem [Theorem 31](#thm:ItKodFib){reference-type="ref" reference="thm:ItKodFib"} and passing to finite covers of the $X^i$ if necessary, we can now construct a sequence of $i$-iterated Kodaira fibrations with the desired properties. The only thing that is not an immediate consequence is the irreducibility statement. For this we will prove by induction on $i$ the following stronger statement: if $G_{1}$ and $G_{2}$ are two commuting subgroups of $\pi_{1}(X^{i})$ such that $G_{1}\cdot G_{2}$ has finite index in $\pi_{1}(X^{i})$, then either $G_1$ or $G_2$ is trivial. When $i=2$ the argument is essentially contained in [@Joh-1994]; we give a complete proof however. We first observe that the previous statement is true in a surface group, implying the case $i=1$, as the reader can check readily. Let now $i\in \{2, \ldots , n\}$ and let $G_1$ and $G_2$ be commuting subgroups of $\pi_{1}(X^{i})$, generating a subgroup of finite index. We consider the fibration $$\pi_{i,i-1} : X^{i}\to X^{i-1}.$$ By induction hypothesis one of the two groups $$\pi_{i,i-1}(G_{1}), \;\; \pi_{i,i-1}(G_{2})$$ is trivial[^2]. We assume that $\pi_{i, i-1}(G_{1})=1$. The group $H:={\rm Ker}(\pi_{i,i-1 \ast})\cap \left(G_{1}\cdot G_{2}\right)$ has finite index in ${\rm Ker}(\pi_{i,i-1 \ast})$, hence is a surface group. The group $G_1$ is normal in $H$ and $G_2$ normalizes both $H$ and $G_{1}$. Since $G_2$ centralizes $G_1$, Lemma 35 from [@LloPy-21] implies that $G_2$ actually centralizes $H$. This in turn implies that $G_{2}$ centralizes ${\rm Ker}(\pi_{i,i-1 \ast})$. Hence $\pi_{i,i-1 \ast}|_{G_2}$ is injective and its image is contained in the kernel of the monodromy representation. This contradicts the injectivity of this representation and concludes the proof of our statement. ◻ ## A complex geometry proof of a Theorem of Kochloukova--Vidussi We now fix an integer $n\ge 2$ and a sequence $X^i$ as in Theorem [Theorem 32](#thm:ItKodFib2){reference-type="ref" reference="thm:ItKodFib2"}. For $i\le j$, we denote by $\pi_{j,i} : X^j\to X^i$ the surjective holomorphic maps with connected fibres obtained by composing the various maps $X^{s}\to X^{s-1}$. For $2\leq i \leq n$, let $F_i$ be a fibre of $\pi_{i,i-1}$. The inclusion $F_i\hookrightarrow X^i$ is $\pi_1$-injective. We deduce that the restrictions $\pi_{j,j-1}: \pi_{j,i}^{-1}(F_i)\to \pi_{j-1,i}^{-1}(F_i)$ define $(j-i+1)$-iterated Kodaira fibrations with injective monodromy for $n\geq j\geq i+1$. We then define $$Y^{i}:=\pi_{n,n-i+1}^{-1}(F_{n-i+1})\subseteq X^{n},$$ and observe that the fundamental group of $Y^i$ injects into that of $X^n$. From this, we deduce the following result. **Proposition 33**. *There exists an $n$-iterated Kodaira fibration $X^n$ with the following properties:* 1. *$X^n$ has injective monodromy and for every integer $i$ with $n\geq i \geq 2$, $X^n$ contains a $i$-iterated Kodaira fibration $Y^i\subseteq X^n$, which is $\pi_{1}$-injected.* 2. *The group $\pi_1(X^n)$ is irreducible.* 3. *There is a finite holomorphic map $X^n\to A$ to a complex torus.* *Proof.* We choose $X^n$ as in Theorem [Theorem 32](#thm:ItKodFib2){reference-type="ref" reference="thm:ItKodFib2"}. The first and second part then follow from the above discussion and Theorem [Theorem 32](#thm:ItKodFib2){reference-type="ref" reference="thm:ItKodFib2"}. Moreover, there is a finite holomorphic map $X^n\to S_1\times \dots \times S_{2^{n-1}}$ to a direct product of $2^{n-1}$ closed hyperbolic Riemann surfaces. Since the Albanese map of a closed hyperbolic Riemann surface is an embedding, choosing $A$ to be the Albanese torus of $S_1\times \dots\times S_{2^{n-1}}$ completes the proof. ◻ As a consequence of Proposition [Proposition 33](#prop:ItKodaira){reference-type="ref" reference="prop:ItKodaira"}, we obtain a new proof of a result of Kochloukova and Vidussi [@KocVid-22 Corollary 1.11]. *Proof of Theorem [Theorem 5](#thm:KV){reference-type="ref" reference="thm:KV"}.* Let $X^n$ be an $n$-iterated Kodaira fibration satisfying the conditions in Proposition [Proposition 33](#prop:ItKodaira){reference-type="ref" reference="prop:ItKodaira"}. The smooth projective variety $X^n$ is obtained by iteratively constructing locally trivial surface bundles starting from a closed Riemann surface. In particular, $X^n$ is aspherical and, by multiplicativity of the Euler characteristic, its Euler characteristic is non-trivial. The same applies for the $i$-iterated Kodaira fibrations $Y^i\subseteq X^n$. Since, moreover, the restriction of the finite holomorphic map $X^n \to A$ to any of the $Y^i$ is finite holomorphic, all of the $Y^i$ satisfy the hypotheses of Theorem [Theorem 11](#thm:LP-BNSR){reference-type="ref" reference="thm:LP-BNSR"}. Thus, for every integer $i\in \left\{2,\dots,n\right\}$, there is a character $\chi_i:\pi_1(Y_i)\to \mathbb{Z}$ with kernel of type $\mathscr{F}_{i-1}$ and not of type ${\rm FP}_i(\mathbb{Q})$. Finally, for $i=1$ the assertion follows, since the fundamental group of the fibre of $\pi_{n,n-1}$ is a hyperbolic surface group. ◻ **Remark 34**. If $\Gamma < {\rm PU}(n,1)$ is an arithmetic lattice of the simplest type such that the Albanese map of the manifold $X=\Gamma\backslash \mathbb{B}^{n}_{{\mathds C}}$ is finite, it is also true (in analogy with Theorem [Theorem 5](#thm:KV){reference-type="ref" reference="thm:KV"}) that for each $i\in \{1, \ldots , n\}$, $\Gamma$ contains a subgroup of type $\mathscr{F}_{i-1}$ which is not of type ${\rm FP}_{i}({\mathds Q})$. This follows from applying the result from the present article (or from [@LloPy-22]) to the fundamental group of suitable complex totally geodesic submanifolds, embedded in $X$ or some finite covering space of $X$. # Further remarks {#morse-theoretic-remarks} As mentioned in the introduction, we explain in this section how to recover some of our results (and results from [@LloPy-22]) using purely real Morse theoretical arguments instead of the theory of BNSR-invariants. This concerns mainly the study of kernels of homomorphisms to ${\mathds Z}$. For homomorphisms to ${\mathds Z}^2$, there are no purely Morse theoretical arguments known to us. We recall that a closed $1$-form on a manifold is said to be Morse if it is locally the differential of a Morse function. We start with the following: **Proposition 35**. *Let $X$ be a closed oriented aspherical $2n$-dimensional manifold. Let $O_{n}\subset H^{1}(X,{\mathds R})-\{0\}$ be the set of nontrivial classes which can be represented by a Morse $1$-form all of whose zeros have index $n$. Then:* 1. *The set $O_n$ is open.* 2. *The projection of $O_n$ in the sphere $S(\pi_{1}(X))$ is contained in $\Sigma^{n-1}(\pi_{1}(X))\cap -\Sigma^{n-1}(\pi_{1}(X))$.* 3. *If $\xi$ is a rational class in $O_n$, the kernel of $\xi$ is of type $\mathscr{F}_{n-1}$.* *Proof.* Let $a$ be a class in $O_n$ represented by a Morse $1$-form $\theta$, all of whose zeros having index $n$. Let $\theta_{1}, \ldots , \theta_{N}$ be closed $1$-forms whose cohomology classes form a basis of $H^{1}(X,{\mathds R})$. There exists a neighbourhood $U$ of the origin in ${\mathds R}^N$ such that if $(x_{1}, \ldots , x_{N})\in U$, then the form $$\theta +\sum_{j=1}^{N}x_{j}\theta_{j}$$ is Morse and all its zeros have index $n$. Hence the classes $$\label{eq:morseforms} \bigg[\theta +\sum_{j=1}^{N}x_{j}\theta_{j}\bigg], \;\;\;\; (x_{1}, \ldots ,x_{N})\in U$$ lie in $O_n$. Since the classes in [\[eq:morseforms\]](#eq:morseforms){reference-type="eqref" reference="eq:morseforms"} form a neighbourhood of $a$, this proves the first claim. The second claim can be shown using arguments from Morse theory; we refer to Section 2.3 in [@LloPy-22] for a proof that is based on ideas from the work of Simpson [@Sim-93]. The proof given there is stated for a closed form which is the real part of a holomorphic $1$-form with finitely many zeros. It applies equally well for a closed (real) Morse $1$-form all of whose zeros have index equal to half the real dimension of the manifold. We finally prove the third claim, using classical arguments which are similar to the ones used in the proof of Proposition [Proposition 28](#prop:attaching-n-cells){reference-type="ref" reference="prop:attaching-n-cells"}. We assume that $n\ge 2$, otherwise there is nothing to prove. If $\xi \in O_n$ is rational, there exists a closed $1$-form $\alpha$ which is Morse and whose critical points are all of index $n$, whose cohomology class is proportional to $\xi$, and such that the integration morphism $$\label{eq:intmo} I_{\alpha} : \pi_{1}(X)\to {\mathds R}$$ defined by $\alpha$ has image equal to ${\mathds Z}$. Let $\widehat{X}\to X$ be the covering space associated to the kernel of $I_{\alpha}$. We fix a primitive $f : \widehat{X}\to {\mathds R}$ of the lift of $\alpha$ to $\widehat{X}$. The function $f$ is a proper Morse function, all of whose critical points have index $n$. Let $c$ be a regular value of $f$ and let $\widehat{X}_{c}:=f^{-1}(c)$. By real Morse theory, for all positive real numbers $t$ such that $c\pm t$ is a regular value of $f$, the space $f^{-1}([c-t, c+t])$ is obtained from the closed manifold $f^{-1}(c)$ up to homotopy equivalence by attaching finitely many $n$-cells. Since $f^{-1}(c)$ can be equipped with a finite CW-complex structure, this implies that the space $\widehat{X}$ is homotopy equivalent to a CW-complex with finite $(n-1)$-skeleton. Since $\widehat{X}$ is a classifying space for ${\rm ker}(\chi)$, we deduce that this group is of type $\mathscr{F}_{n-1}$. This concludes the proof of Proposition [Proposition 35](#prop:morsebnsr){reference-type="ref" reference="prop:morsebnsr"}. ◻ To continue our discussion, we shall need the following classical result. **Proposition 36**. *Let $X$ be a closed complex manifold of complex dimension $n$ and let $\alpha$ be a closed holomorphic $1$-form with finitely many zeros. Then the cohomology class $[{\rm Re}(\alpha)]$ can be represented by a Morse $1$-form all of whose zeros have index $n$.* The proof of this proposition is the same as the second part of the deformation argument in Section [6.2](#sec:perturbation){reference-type="ref" reference="sec:perturbation"}: near each point $p$ which is a zero of $\alpha$, we write $\alpha = dh$ for some holomorphic function $h$ defined on a ball $B_p$ centered at $p$. We can perturb $h$ into a holomorphic function $h' : B_p \to {\mathds C}$ which has finitely many critical points, all nondegenerate, and which has no critical point near the boundary. We then take one more $C^{\infty}$ perturbation $h_p$ which coincides with $h$ near the boundary of the ball and with $h'$ on a smaller ball, in such a way that $h_p=h'$ near each critical point of $h_p$. We consider the $C^{\infty}$ complex-valued form $\beta$ which equals $dh_p$ in $B_p$ and $\alpha$ outside of the union of the balls $B_p$. The forms $\alpha$ and $\beta$ are cohomologous, hence so are their real parts. The only zeros of the real part of $\beta$ are contained in one of the balls $B_p$ (for $p$ a zero of $\alpha$) and are critical points of $h_p$. Since $h_p$ is a holomorphic Morse function near its critical points, the claim follows, observing that the real part of the function $$(z_{1},\ldots , z_{n})\mapsto z_{1}^{2}+\cdots +z_{n}^{2}$$ is given by $(x_{1}^{2}+\cdots +x_{n}^{2})-(y_{1}^{2}+\cdots +y_{n}^{2})$ if $z_j=x_j +iy_j$ with $x_{j}, y_{j}\in {\mathds R}$. We refer the reader to [@NicPy-21 p. 59-60] for more details on the perturbation argument. As announced above, the following proposition can serve as a substitute for the use of BNSR-invariants, when studying kernels of homomorphism to ${\mathds Z}$. Instead of working with the set $\Sigma^{n-1} \cap - \Sigma^{n-1}$, we simply consider the smaller set $O_n$, which is also open, and which is dense under suitable hypotheses. **Proposition 37**. *Let $X$ be a closed Kähler manifold of complex dimension $n$, and let $O_n \subset H^{1}(X,{\mathds R})-\{0\}$ be defined as in Proposition [Proposition 35](#prop:morsebnsr){reference-type="ref" reference="prop:morsebnsr"}. Assume that $X$ has finite Albanese map. Then:* 1. *The set $O_n$ is dense in $H^{1}(X,{\mathds R})-\{0\}$,* 2. *if moreover $X$ is aspherical, any rational class in $O_n$ has kernel of type $\mathscr{F}_{n-1}$ (and not of type ${\rm FP}_{n}(\mathbb{Q})$ if the Euler characteristic of $X$ is nonzero).* *Proof.* Let $a_X : X \to A(X)$ be the Albanese map of $X$. If $\beta$ is a holomorphic $1$-form on $A(X)$ whose restriction to any positive dimensional subtorus does not vanish, then $\alpha =a_X^{\ast}\beta$ has finitely many zeros; moreover the set of forms $\alpha$ obtained in this way is dense in $H^{1,0}(X)$ (see Propositions 14 and 18 in [@LloPy-22]). According to Proposition [Proposition 36](#prop:realmorsety){reference-type="ref" reference="prop:realmorsety"}, we have $[{\rm Re} (\alpha)]\in O_n$ for such an $\alpha$. This proves that $O_n$ is dense. The second point follows from Proposition [Proposition 35](#prop:morsebnsr){reference-type="ref" reference="prop:morsebnsr"} and the last claim in Addendum [Addendum 12](#add-to-thm:LP-BNSR){reference-type="ref" reference="add-to-thm:LP-BNSR"}. ◻ We now turn to the existence of *perfect circle-valued Morse functions* on Kähler manifolds. Let us recall the terminology first. Let $M$ be a smooth closed manifold. We call a map $M \to S^1$ a circle-valued function and say that it is Morse if it coincides locally with a Morse function. A circle-valued Morse function $f : M \to S^1$ allows to study the topology of $M$ by starting with a regular fibre of $f$, thickening it, and attaching handles when passing a critical level set. In particular for such an $f$ we have the formula: $$\chi (M)=\sum_{x} (-1)^{{\rm ind}(x)},$$ where the sum runs over the finitely many critical points of $f$, and ${\rm ind}(x)$ is the index of a critical point $x$. We say that $f$ is perfect if it has $\vert \chi (M)\vert$ critical points. When $M$ is odd-dimensional, or simply of Euler characteristic equal to $0$, this means that $f$ is a fibration. When $M$ is even-dimensional of nonzero Euler characteristic, this happens if and only if the indices of the critical points all have the same parity. Some existence results for perfect circle-valued Morse functions on negatively curved manifolds are available: every closed hyperbolic $3$-manifold has a finite cover fibring over the circle, as follows from the work of Agol [@Ago-13], see also [@Ber-15]; in dimension $5$, Italiano--Martelli--Migliorini built examples of cusped real hyperbolic manifolds fibring over the circle [@IMM-22], and in dimension $4$, Battista and Martelli found finitely many examples (both closed and cusped) of real hyperbolic $4$-manifolds admitting perfect circle-valued Morse functions [@BaMa-22]. We observe here that Propositions [Proposition 36](#prop:realmorsety){reference-type="ref" reference="prop:realmorsety"} and [Proposition 37](#prop:subsproof){reference-type="ref" reference="prop:subsproof"} have the following immediate consequence. **Theorem 38**. *Let $X$ be a closed Kähler manifold of complex dimension $n$, with finite Albanese map. Then any rational ray contained in the dense open set $O_n\subset H^{1}(X,{\mathds R})-\{0\}$ can be represented by a circle-valued Morse function all of whose critical points have index $n$ (in particular such a Morse function is perfect).* Applying the result of Eyssidieux [@Eys-18] already alluded to before (see also [@LloPy-22]), we obtain: **Corollary 39**. *Let $\Gamma < {\rm PU}(n,1)$ be a cocompact torsion-free arithmetic lattice. Assume that $b_{1}(\Gamma)>0$. Then there is a finite index subgroup $\Gamma_0 < \Gamma$ such that for any subgroup of finite index $\Gamma_1 < \Gamma_0$, there is a dense open set $O\subset H^{1}(\Gamma_1,{\mathds R})$ such that any rational class in $O$ can be represented (up to scalar) by a circle-valued Morse function on $\Gamma \backslash \mathbb{B}_{\mathbb{C}}^n$ all of whose critical points have index $n$.* [^1]: One must possibly replace $E$ by a finite cover to ensure that $h$ has connected fibers. [^2]: If $i=2$, this also follows because $\pi_1(X^{1})$ is a surface group.

arxiv_math

--- abstract: | The multivariate arithmetic Tutte polynomial of arithmetic matroids is a generalization of the multivariate Tutte polynomial of matroids. In this note, we give the convolution formulas for the multivariate arithmetic Tutte polynomial of the product of two arithmetic matroids. In particular, the convolution formulas for the multivariate arithmetic Tutte polynomial of an arithmetic matroid are obtained. Applying our results, several known convolution formulas including [@Dupont Theorem 10.9 and Corollary 10.10] and [@Backman Theorems 1 and 4] are proved by a purely combinatorial proof. The proofs presented here are significantly shorter than the previous ones. In addition, we obtain a convolution formula for the characteristic polynomial of an arithmetic matroid. author: - | Tianlong Ma,   Xian'an Jin[^1],  Weiling Yang\ School of Mathematical Sciences\ Xiamen University\ P. R. China\ `Email: [email protected], [email protected], [email protected]` title: Convolution formulas for multivariate arithmetic Tutte polynomials --- matroid; arithmetic matroid; multivariate Tutte polynomial; convolution formula 05C31,05B35 # Introduction Let $M=(X,rk)$ denote a matroid on ground set $X$ with rank function $rk$. For each $e\in X$, let $v_e$ be a variable of $e$, and for $A\subseteq X$, let $\mathbf{v}_A=\{v_e:e\in A\}$. In particular, we write $\mathbf{v}$ for $\mathbf{v}_X$. In [@Sokal], Sokal defined the following multivariate version of the Tutte polynomial of a matroid $M=(X,rk)$ in the variables $q^{-1}$ and $\mathbf{v}$: $$\begin{aligned} \label{equ1.1} Z_{M}(q,\mathbf{v})=\sum_{A\subseteq X}q^{-rk(A)}\prod_{e\in A}v_e. \end{aligned}$$ In particular, if $v_e=v$ for each $e\in X$ in a matroid $M=(X,rk)$, then we write $Z_{M}(q,v)$ for $Z_{M}(q,\mathbf{v})$. If we substitute $q$ and the variable $v_e$ of each $e\in X$ for $(x-1)(y-1)$ and $y-1$ respectively, and multiply by a prefactor $(x-1)^{r(M)}$ into Eq. ([\[equ1.1\]](#equ1.1){reference-type="ref" reference="equ1.1"}), we obtain the standard bivariate Tutte polynomial: $$T_{M}(x,y)=\sum_{A\subseteq X}(x-1)^{rk(X)-rk(A)}(y-1)^{|A|-rk(A)}.$$ A triple $\mathcal{M}=(X,rk,m)$ is called a *multiplicity matroid*, introduced by Moci in [@Moci], if $(X,rk)$ is a matroid and $m$ is a function (called multiplicity) from the family of all subsets of $X$ to the positive integers, that is, $m: 2^{X}\rightarrow \mathbb{Z}_{> 0}$. We say that $m$ is *trivial* if it is identically equal to $1$. A multiplicity matroid $\mathcal{M}=(X,rk,m)$ is called an *arithmetic matroid*, introduced by D'Adderio and Moci in [@D'AdderioMoci], if $m$ satisfies the following axioms: \(1\) : For all $A\subseteq X$ and $e \in X$, if $rk(A\cup \{e\})=rk(A)$, then $m(A\cup \{e\})$ divides $m(A)$; otherwise, $m(A)$ divides $m(A\cup \{e\})$. \(2\) : If $A\subseteq B \subseteq X$ and $B$ is a disjoint union $B=A\cup F\cup T$ such that for all $A\subseteq C\subseteq B$ we have $rk(C)=rk(A)+|C\cap F|$, then $m(A)\cdot m(B)=m(A\cup F)\cdot m(A\cup T).$ \(3\) : If $A\subseteq B \subseteq X$ and $rk(A)=rk(B)$, then $$\sum_{A\subseteq T\subseteq B}(-1)^{|T|-|A|}m(T)\geq 0.$$ \(4\) : If $A\subseteq B \subseteq X$ and $rk^{*}(A)=rk^{*}(B)$, then $$\sum_{A\subseteq T\subseteq B}(-1)^{|T|-|A|}m(X\setminus T)\geq 0,$$ where $rk^{*}(A)=|A|+rk(X\setminus A)-rk(X)$ and $rk^{*}(B)$ is similar. In [@Branden], Brändén and Moci generalized the multivariate Tutte polynomial from matroids to arithmetic matroids. The *multivariate arithmetic Tutte polynomial* of an arithmetic matroid $\mathcal{M}=(X,rk,m)$ is defined by $$\begin{aligned} \label{equ1.2} \mathcal{Z}_{\mathcal{M}}(q,\mathbf{v})=\sum_{A\subseteq X}m(A)q^{-rk(A)}\prod_{e\in A}v_e. \end{aligned}$$ In particular, if $v_e=v$ for each $e\in X$ in a multiplicity matroid $\mathcal{M}=(X,rk,m)$, then we write $\mathcal{Z}_{\mathcal{M}}(q,v)$ for $\mathcal{Z}_{\mathcal{M}}(q,\mathbf{v})$. If we substitute $q$ and the variable $v_e$ of each $e\in X$ for $(x-1)(y-1)$ and $y-1$ respectively, and multiply by a prefactor $(x-1)^{r(\mathcal{M})}$ into Eq. ([\[equ1.2\]](#equ1.2){reference-type="ref" reference="equ1.2"}), we obtain the *arithmetic Tutte polynomial* $$\mathfrak{M}_{\mathcal{M}}(x,y)=\sum_{A\subseteq X}m(A)(x-1)^{rk(X)-rk(A)}(y-1)^{|A|-rk(A)},$$ introduced by D'Adderio and Moci in [@D'AdderioMoci]. If $\mathcal{M}$ is only a multiplicity matroid, $\mathfrak{M}_{\mathcal{M}}(x,y)$ is called the *multiplicity Tutte polynomial* of $\mathcal{M}$, introduced by Moci in [@Moci]. Kook, Reiner and Stanton [@Kook] and Etienne and Las Vergnas[@Etienne] found a well-known convolution formula for the Tutte polynomial $T_{M}(x,y)$ of a matroid $M$: $$T_{M}(x,y)=\sum_{A\subseteq X} T_{M/A}(x,0)T_{M|A}(0,y),$$ where $M/A$ and $M|A$ denote the contraction and restriction of $A$ from $M$, respectively. Kung [@Kung] generalized this formula to subset-corank polynomials which are related to multivariate Tutte polynomial. Motivated by the work of Kung in [@Kung], we first obtain the convolution formulas for multivariate arithmetic Tutte polynomials of the product of two arithmetic matroids in this note. In particular, the convolution formulas for the multivariate arithmetic Tutte polynomial of an arithmetic matroid are given. Secondly, applying our results, several known convolution formulas are proved by a purely combinatorial method. Finally, we obtain a convolution formula for the characteristic polynomial of an arithmetic matroid. # Main results Let $\mathcal{M}=(X,rk,m)$ be a multiplicity matroid. For $T\subseteq X$, the restriction and contraction of $T$ from $\mathcal{M}$ were given in [@D'AdderioMoci]. The multiplicity matroid on $T$ with the rank function and the multiplicity obtained by restricting $rk$ and $m$ to subsets of $T$ respectively, denoted by $\mathcal{M}|T$, is called the *restriction* of $\mathcal{M}$ to $T$. The *contraction* of $T$ from $\mathcal{M}$, denoted by $\mathcal{M}/T$, is the multiplicity matroid $(X\setminus T, rk_{\mathcal{M}/T}, m_{\mathcal{M}/T})$, where $rk_{\mathcal{M}/T}$ and $m_{\mathcal{M}/T}$ are defined by $$rk_{\mathcal{M}/T}(A)=rk(A\cup T)-rk(T)$$ and $$m_{\mathcal{M}/T}(A)=m(A\cup T)$$ for $A\subseteq X\setminus T$. Let $M=(X,rk)$ be a matroid. Recall that $\mathbf{v}_A=\{v_e:e\in A\}$. Let $u_e$ be another variable of $e\in A$ and $\mathbf{u}_A=\{u_e:e\in A\}$ for $A\subseteq X$. We define $(\mathbf{u}\mathbf{v})_{A}=\{u_ev_e:e\in A\}$ for $A\subseteq X$, and write $\mathbf{u}\mathbf{v}$ for $(\mathbf{u}\mathbf{v})_{X}$. For two multiplicity matroids $\mathcal{M}_1=(X,rk,m_1)$ and $\mathcal{M}_2=(X,rk,m_2)$ over a fixed underlying matroid $(X,rk)$, the *product* of $\mathcal{M}_1$ and $\mathcal{M}_2$, denoted by $\mathcal{M}_1\bullet\mathcal{M}_2$, is defined by $(X,rk,m_1m_2)$, where $m_1m_2$ is the product of two multiplicity functions $m_1$ and $m_2$, given by $$m_1m_2(A)=m_1(A)m_2(A)$$ for $A\subseteq X$. Delucchi and Moci [@Delucchi] proved that if both $\mathcal{M}_1$ and $\mathcal{M}_2$ are two arithmetic matroids, then $\mathcal{M}_1\bullet\mathcal{M}_2$ is also an arithmetic matroid. We first have the following convolution formula for the multivariate arithmetic Tutte polynomial of the product of two arithmetic matroids. **Theorem 1**. *Let $\mathcal{M}_1=(X,rk,m_1)$ and $\mathcal{M}_2=(X,rk,m_2)$ be two arithmetic matroids, and let $\mathcal{M}=\mathcal{M}_1\bullet\mathcal{M}_2$. Then $$\begin{aligned} \mathcal{Z}_{\mathcal{M}}(pq,\mathbf{uv})=&\sum_{T\subseteq X}p^{-rk(T)}\prod_{e\in T}(-u_e)\mathcal{Z}_{\mathcal{M}_1|T}(q,-\mathbf{v})\mathcal{Z}_{\mathcal{M}_2/T}(p,\mathbf{u})\\ =&\sum_{T\subseteq X}p^{-rk(T)}\prod_{e\in T}(-u_e)\mathcal{Z}_{\mathcal{M}_2|T}(q,-\mathbf{v})\mathcal{Z}_{\mathcal{M}_1/T}(p,\mathbf{u}). \end{aligned}$$* *Proof.* For two subsets $A$ and $B$ of $X$, we have $$\sum_{T:B\subseteq T\subseteq A}(-1)^{|T|-|B|}= \begin{cases} 1, & \text{if } A=B,\\ 0, & \text{otherwise. } \end{cases}$$ Therefore, we have $$\begin{aligned} \mathcal{Z}_{\mathcal{M}}(pq,\mathbf{uv})&=\sum_{A,B:B\subseteq A\subseteq X}m_1(B)m_2(A)p^{-rk(A)}q^{-rk(B)}\left(\prod_{e\in A}u_e\right)\left(\prod_{e\in B}v_e\right)\sum_{T: B\subseteq T\subseteq A}(-1)^{|T|-|B|}\\ &=\sum_{T: T\subseteq X}(-1)^{|T|}p^{-rk(T)}\left(\prod_{e\in T}u_e\right)\left(\sum_{B: B\subseteq T}(-1)^{|B|}m_{1}(B)q^{-rk(B)}\prod_{e\in B}v_e\right)\cdot\\ &\ \ \ \left(\sum_{A: T\subseteq A\subseteq X}m_2(A)p^{-rk_{M/T}(A\setminus T)}\prod_{e\in A\setminus T}u_e\right)\\ &=\sum_{T: T\subseteq X}p^{-rk(T)}\left(\prod_{e\in T}(-u_e)\right)\left(\sum_{B: B\subseteq T}m_{1}(B)q^{-rk(B)}\prod_{e\in B}(-v_e)\right)\cdot\\ &\ \ \ \left(\sum_{A: T\subseteq A\subseteq X}m_2(A)p^{-rk_{M/T}(A\setminus T)}\prod_{e\in A\setminus T}u_e\right)\\ &=\sum_{T: T\subseteq X}p^{-rk(T)}\left(\prod_{e\in T}(-u_e)\right)\mathcal{Z}_{\mathcal{M}_1|T}(q,-\mathbf{v})\mathcal{Z}_{\mathcal{M}_2/T}(p,\mathbf{u}). \end{aligned}$$ The second equation holds since $\mathcal{M}=\mathcal{M}_1\bullet\mathcal{M}_2=\mathcal{M}_2\bullet\mathcal{M}_1$. ◻ We write $\mathcal{Z}_{\mathcal{M}}(q,uv)$ for $\mathcal{Z}_{\mathcal{M}}(q,\mathbf{uv})$ if $u_e=u$ and $v_e=v$ for each $e\in X$ in an arithmetic matroid $\mathcal{M}=(X,rk,m)$ . We have the following specialization of Theorem [Theorem 1](#mtpc0){reference-type="ref" reference="mtpc0"}. **Theorem 2**. *Let $\mathcal{M}_1=(X,rk,m_1)$ and $\mathcal{M}_2=(X,rk,m_2)$ be two arithmetic matroids, and let $\mathcal{M}=\mathcal{M}_1\bullet\mathcal{M}_2$. Then $$\begin{aligned} \mathcal{Z}_{\mathcal{M}}(pq,uv)=&\sum_{T\subseteq X}p^{-rk(T)}(-u)^{|T|}\mathcal{Z}_{\mathcal{M}_1|T}(q,-v)\mathcal{Z}_{\mathcal{M}_2/T}(p,u)\\ =&\sum_{T\subseteq X}p^{-rk(T)}(-u)^{|T|}\mathcal{Z}_{\mathcal{M}_2|T}(q,-v)\mathcal{Z}_{\mathcal{M}_1/T}(p,u). \end{aligned}$$* For an arithmetic matroid $\mathcal{M}=(X,rk,m)$, we use $M$ to denote the underlying matroid $(X,rk)$ conveniently. Taking one of two multiplicity functions $m_1$ and $m_2$ in Theorems [Theorem 1](#mtpc0){reference-type="ref" reference="mtpc0"} and [Theorem 2](#mtpc1){reference-type="ref" reference="mtpc1"} to be trivial, we obtain: **Corollary 3**. *Let $\mathcal{M}=(X,rk,m)$ be an arithmetic matroid. Then $$\begin{aligned} \mathcal{Z}_{\mathcal{M}}(pq,\mathbf{uv})=&\sum_{T\subseteq X}p^{-rk(T)}\prod_{e\in T}(-u_e)\mathcal{Z}_{\mathcal{M}|T}(q,-\mathbf{v})Z_{M/T}(p,\mathbf{u})\\ =&\sum_{T\subseteq X}p^{-rk(T)}\prod_{e\in T}(-u_e)Z_{M|T}(q,-\mathbf{v})\mathcal{Z}_{\mathcal{M}/T}(p,\mathbf{u}). \end{aligned}$$* **Corollary 4**. *Let $\mathcal{M}=(X,rk,m)$ be an arithmetic matroid. Then $$\begin{aligned} \mathcal{Z}_{\mathcal{M}}(pq,uv)=&\sum_{ T\subseteq X}p^{-rk(T)}(-u)^{|T|}\mathcal{Z}_{\mathcal{M}|T}(q,-v)Z_{M/T}(p,u)\\ =&\sum_{T\subseteq X}p^{-rk(T)}(-u)^{|T|}Z_{\mathcal{M}|T}(q,-v)\mathcal{Z}_{M/T}(p,u). \end{aligned}$$* A generalization of the well-known convolution formula for the Tutte polynomial of a matroid was shown by Kung in [@Kung]; for details, see also [@Ellis-Monaghan Theorem 12.25] or [@Wang0 Theorem 5.3]. Recently, an analogous formula for the arithmetic Tutte polynomial of the product of two arithmetic matroids was obtained by Dupont, Fink and Moci in [@Dupont]. As an application of Theorem [Theorem 2](#mtpc1){reference-type="ref" reference="mtpc1"}, we now give it a purely combinatorial proof. **Corollary 5**. **[@Dupont]* Let $\mathcal{M}_1=(X,rk,m_1)$ and $\mathcal{M}_2=(X,rk,m_2)$ be two arithmetic matroids, and let $\mathcal{M}=\mathcal{M}_1\bullet\mathcal{M}_2$. Then $$\begin{aligned} &\mathfrak{M}_{\mathcal{M}}(1+ab,1+cd)\\ &=\sum_{A\subseteq X}a^{rk(X)-rk(A)}(-d)^{|A|-rk(A)} \mathfrak{M}_{\mathcal{M}_1|A}(1-a,1-c)\mathfrak{M}_{\mathcal{M}_2/A}(1+b,1+d)\\ &=\sum_{A\subseteq X}a^{rk(X)-rk(A)}(-d)^{|A|-rk(A)} \mathfrak{M}_{\mathcal{M}_2|A}(1-a,1-c)\mathfrak{M}_{\mathcal{M}_1/A}(1+b,1+d). \end{aligned}$$* *Proof.* Recall that $$\begin{aligned} \label{equ} \mathfrak{M}_{\mathcal{M}}(x,y)=(x-1)^{rk(X)}\mathcal{Z}_{\mathcal{M}}((x-1)(y-1),y-1). \end{aligned}$$ Then we have $$\mathfrak{M}_{\mathcal{M}}(1+ab,1+cd)=(ab)^{rk(X)}\mathcal{Z}_{\mathcal{M}}(abcd,cd).$$ Taking $p=bd$, $q=ac$, $u=d$ and $v=c$, by Theorem [Theorem 2](#mtpc1){reference-type="ref" reference="mtpc1"}, we have $$\begin{aligned} \mathcal{Z}_{\mathcal{M}}(abcd,cd)=&\sum_{A\subseteq X}(bd)^{-rk(A)}(-d)^{|A|}\mathcal{Z}_{\mathcal{M}_1|A}(ac,-c)\mathcal{Z}_{\mathcal{M}_2/A}(bd,d). \end{aligned}$$ By Eq. ([\[equ\]](#equ){reference-type="ref" reference="equ"}), taking $x=1-a$ and $y=1-c$, we have $$\mathcal{Z}_{\mathcal{M}_1|A}(ac,-c)=(-a)^{-rk(A)}\mathfrak{M}_{\mathcal{M}_1|A}(1-a,1-c),$$ and taking $x=1+b$ and $y=1+d$, we have $$\mathcal{Z}_{\mathcal{M}_2/A}(bd,d)=b^{-rk(\mathcal{M}_2/A)}\mathfrak{M}_{\mathcal{M}_2/A}(1+b,1+d).$$ Therefore $$\begin{aligned} &\mathfrak{M}_{\mathcal{M}}(1+ab,1+cd)\\ &=(ab)^{rk(X)}\sum_{A\subseteq X}(bd)^{-rk(A)}(-d)^{|A|}(-a)^{-rk(A)} b^{-rk(\mathcal{M}_2/A)}\mathfrak{M}_{\mathcal{M}_1|A}(1-a,1-c)\mathfrak{M}_{\mathcal{M}_2/A}(1+b,1+d)\\ &=b^{rk(X)}\sum_{A\subseteq X}b^{-rk(A)-rk(\mathcal{M}_2/A)}a^{rk(X)-rk(A)}(-d)^{|A|-rk(A)} \mathfrak{M}_{\mathcal{M}_1|A}(1-a,1-c)\mathfrak{M}_{\mathcal{M}_2/A}(1+b,1+d). \end{aligned}$$ Note that $rk(X)=rk(A)+rk(\mathcal{M}_2/A)$. Thus the first equation holds. The second equation holds since $\mathcal{M}=\mathcal{M}_1\bullet\mathcal{M}_2=\mathcal{M}_2\bullet\mathcal{M}_1$. ◻ We note that four axioms in the definition of arithmetic matroids are not used in the proofs. Therefore the above results also hold for multiplicity matroids. Recently, Backman and Lenz [@Backman] and Dupont, Fink and Moci [@Dupont] obtained the following formula. We apply Theorem [Theorem 2](#mtpc1){reference-type="ref" reference="mtpc1"} to give it a new and simple proof. **Corollary 6**. **[@Backman; @Dupont]*[\[BackmanCF\]]{#BackmanCF label="BackmanCF"} Let $\mathcal{M}_1=(X,rk,m_1)$ and $\mathcal{M}_2=(X,rk,m_2)$ be two multiplicity matroids, and let $\mathcal{M}=\mathcal{M}_1\bullet\mathcal{M}_2$. Then $$\begin{aligned} \mathfrak{M}_{\mathcal{M}}(x,y)&=\sum_{A\subseteq X} \mathfrak{M}_{\mathcal{M}_1|A}(0,y)\mathfrak{M}_{\mathcal{M}_2/A}(x,0)\\ &=\sum_{A\subseteq X} \mathfrak{M}_{\mathcal{M}_2|A}(0,y)\mathfrak{M}_{\mathcal{M}_1/A}(x,0). \end{aligned}$$* *Proof.* By setting $p=1-x$, $q=1-y$, $u=-1$ and $v=1-y$ and multiplying by $(x-1)^{rk(X)}$ into two equations in Theorem [Theorem 2](#mtpc1){reference-type="ref" reference="mtpc1"}, we have $$\begin{aligned} &(x-1)^{rk(X)}\mathcal{Z}_{\mathcal{M}}((1-x)(1-y),y-1)\\ &=\sum_{ A\subseteq X}(-1)^{-rk(A)}\mathcal{Z}_{\mathcal{M}_1|A}(1-y,y-1)\cdot(x-1)^{rk(X)-rk(A)}\mathcal{Z}_{\mathcal{M}_2/A}(1-x,-1)\\ &=\sum_{A\subseteq X}(-1)^{-rk(A)}\mathcal{Z}_{\mathcal{M}_2|A}(1-y,y-1)\cdot(x-1)^{rk(X)-rk(A)}\mathcal{Z}_{\mathcal{M}_1/A}(1-x,-1). \end{aligned}$$ Note that $rk(X)-rk(A)=rk(M/A)$. Thus, by Eq. ([\[equ\]](#equ){reference-type="ref" reference="equ"}), the result is established. ◻ In [@Backman], Backman and Lenz also obtained the following formula for a multiplicity matroid $\mathcal{M}=(X,rk,m)$: $$\begin{aligned} \mathfrak{M}_{\mathcal{M}}(x,y)&=\sum_{A\subseteq X} \mathfrak{M}_{\mathcal{M}|A}(0,y)T_{M/A}(x,0)\\ &=\sum_{A\subseteq X} T_{M|A}(0,y)\mathfrak{M}_{\mathcal{M}/A}(x,0).\end{aligned}$$ Using the technique similar to the proof of Corollary [\[BackmanCF\]](#BackmanCF){reference-type="ref" reference="BackmanCF"}, Corollary [Corollary 4](#CoroMaT){reference-type="ref" reference="CoroMaT"} gives a purely combinatorial proof of this formula. Certainly, it can be also proved by taking one of two multiplicity functions $m_1$ and $m_2$ in Corollary [\[BackmanCF\]](#BackmanCF){reference-type="ref" reference="BackmanCF"} to be trivial. In [@Wang], Wang, Yeh and Zhou defined the characteristic polynomial $\chi_{\mathcal{M}}(\lambda)$ of an arithmetic matroid $\mathcal{M}=(X,rk,m)$ as follows: $$\chi_{\mathcal{M}}(\lambda)=\sum_{A\subseteq X}(-1)^{|A|}m(A)\lambda^{rk(X)-rk(A)}.$$ Note that if the multiplicity $m$ of an arithmetic matroid $\mathcal{M}=(X,rk,m)$ is trivial, then the characteristic polynomial $\chi_{\mathcal{M}}(\lambda)$ of $\mathcal{M}$ specializes to the classical characteristic polynomial $\chi_{M}(\lambda)$ of $M$. It is easy to see that $$\begin{aligned} \label{chrelation1} \chi_{\mathcal{M}}(\lambda)=\lambda^{rk(X)}\mathcal{Z}_{\mathcal{M}}(\lambda,-1)\end{aligned}$$ and $$\begin{aligned} \label{chrelation2} \chi_{M}(\lambda)=\lambda^{rk(X)}Z_{M}(\lambda,-1).\end{aligned}$$ Applying Corollary [Corollary 4](#CoroMaT){reference-type="ref" reference="CoroMaT"}, we have the following convolution formula for the characteristic polynomial of an arithmetic matroid. **Theorem 7**. *Let $\mathcal{M}=(X,rk,m)$ be an arithmetic matroid. Then $$\begin{aligned} \chi_{\mathcal{M}}(\lambda\xi)&=\sum_{A\subseteq X}\lambda^{rk(X)-rk(A)}\chi_{\mathcal{M}|A}(\lambda)\chi_{M/A}(\xi)\\ &=\sum_{A\subseteq X}\lambda^{rk(X)-rk(A)}\chi_{M|A}(\lambda)\chi_{\mathcal{M}/A}(\xi). \end{aligned}$$* *Proof.* By setting $p=\xi$, $q=\lambda$, $u=-1$ and $v=1$ and multiplying by $(\lambda\xi)^{rk(X)}$ into two equations in Corollary [Corollary 4](#CoroMaT){reference-type="ref" reference="CoroMaT"}, we have $$\begin{aligned} (\lambda\xi)^{rk(X)}\mathcal{Z}_{\mathcal{M}}(\lambda\xi,-1)=&^{}\sum_{ A\subseteq X}\lambda^{rk(X)}\mathcal{Z}_{\mathcal{M}|A}(\lambda,-1)\cdot\xi^{rk(X)-rk(A)}Z_{M/A}(\xi,-1)\\ =&\sum_{A\subseteq X} \lambda^{rk(X)}Z_{\mathcal{M}|A}(\lambda,-1)\cdot\xi^{rk(X)-rk(A)}\mathcal{Z}_{M/A}(\xi,-1). \end{aligned}$$ Note that $rk(X)=rk(A)+rk(M/A)$. Thus the equations hold from Eqs. ([\[chrelation1\]](#chrelation1){reference-type="ref" reference="chrelation1"}) and ([\[chrelation2\]](#chrelation2){reference-type="ref" reference="chrelation2"}). ◻ # Acknowledgements {#acknowledgements .unnumbered} This work is supported by National Natural Science Foundation of China (No. 12171402). 00 S. Backman, M. Lenz, A convolution formula for Tutte polynomials of arithmetic matroids and other combinatorial structures, *Sém. Lothar. Combin.* 78 (2020), Art. B78c, 17 pp. P. Brändén, L. Moci, The multivariate arithmetic Tutte polynomial, *Trans. Amer. Math. Soc.* 366 (10) (2014) 5523--5540. M. D'Adderio, L. Moci, Arithmetic matroids, Tutte polynomial, and toric arrangements, *Adv. Math.* 232 (2013) 335--367. E. Delucchi, L. Moci, Products of arithmetic matroids and quasipolynomial invariants of CW-complexes, *J. Combin. Theory Ser. A* 157 (2018) 28--40. C. Dupont, A. Fink, L. Moci, Universal Tutte characters via combinatorial coalgebras, *Algebraic Combin.* 1 (2018) 603--651. J. Ellis-Monaghan, I. Moffatt (Eds.), Handbook of the Tutte polynomial and related topics, CRC Press, Boca Raton, FL, 2022. G. Etienne, M. Las Vergnas, External and internal elements of a matroid basis, *Discrete Math.* 179 (1--3) (1998) 111--119. W. Kook, V. Reiner, D. Stanton, A convolution formula for the Tutte polynomial, *J. Combin. Theory Ser. B* 76 (2) (1999) 297--300. J.P.S. Kung, Convolution-multiplication identities for Tutte polynomials of graphs and matroids, *J. Combin. Theory Ser. B* 100 (6) (2010) 617--624. L. Moci, A Tutte polynomial for toric arrangements, *Trans. Amer. Math. Soc.* 364 (2012) 1067--1088. A.D. Sokal, The multivariate Tutte polynomial (alias Potts model) for graphs and matroids, In: Surveys in Combinatorics, 327 (2005) 173--226. S. Wang, Möbius conjugation and convolution formulae, *J. Combin. Theory Ser. B* 115 (2015) 117--131. S. Wang, Y. Yeh, F. Zhou, Linear bounds on characteristic polynomials of matroids, *Math. Proc. Camb. Phil. Soc.* 168 (2020) 505--518. [^1]: Corresponding author.

arxiv_math

--- abstract: | Let $A$ be an abelian surface over an algebraically closed field $\overline{k}\xspace$. We describe a rich collection of relations in the kernel of the Albanese map of $A$ arising from hyperelliptic curves. When $\overline{k}\xspace$ admits an embedding $\overline{k}\xspace\hookrightarrow\mathbb{C}\xspace$ and $A$ is isogenous to a product of elliptic curves, we prove that for infinitely many integers $g\geq 2$, there are infinitely many non-isomorphic hyperelliptic curves of genus $g$ mapping birationally into $A$. We use these results to give some progress towards Beilinson's conjecture for zero-cycles, which predicts that for a smooth projective variety $X$ over $\overline{\mathbb{Q}}\xspace$ the kernel of the Albanese map of $X$ is zero. address: - "Department of Mathematics, University of Virginia, 221 Kerchof Hall, 141 Cabell Dr., Charlottesville, VA, 22904, USA. Email: `[email protected]`" - "Department of Mathematics and Statistics, Burnside Hall, Mcgill University, 805 Sherbrooke Street West, Montreal, Quebec H3A 0B9. Email: `[email protected]`" author: - Evangelia Gazaki\* - Jonathan Love\*\* bibliography: - bibfile.bib - bibabelian.bib title: Hyperelliptic curves mapping to abelian varieties and applications to Beilinson's conjecture for zero-cycles --- # Introduction In this article we will be working over an algebraically closed field $\overline{k}\xspace$. For a smooth projective variety $X$ over $\overline{k}\xspace$ we consider the Chow group $\mathop{\mathrm{CH}}_0(X)$ of zero-cycles modulo rational equivalence on $X$. This group has a filtration $$\mathop{\mathrm{CH}}_0(X)\supset F^1(X)\supset F^2(X)\supset 0,$$ where $F^1(X)=\ker\left(\mathop{\mathrm{CH}}_0(X)\xrightarrow{\deg}\mathbb{Z}\xspace\right)$ is the kernel of the degree map, sending the class $[x]$ of a closed point $x\in X$ to $1$, and $F^2(X)=\ker\left(F^1(X)\xrightarrow{\mathop{\mathrm{alb}}_X}\mathop{\mathrm{Alb}}_X(\overline{k}\xspace)\right)$ is the kernel of the Albanese map of $X$. If the field $\overline{k}\xspace$ is transcendental (for example $\overline{k}\xspace=\mathbb{C}\xspace$) and the variety $X$ has positive geometric genus, the Albanese kernel $F^2(X)$ is known to be enormous (see [@Mumford1968; @Bloch1975]) and it cannot be parametrized by an algebraic variety. In contrast, the situation is conjectured to be vastly different over $\overline{\mathbb{Q}}\xspace$. A famous conjecture of Beilinson predicts the following. **Conjecture 1**. *(Beilinson [@Beilinson1984]) Let $X$ be a smooth projective variety over $\overline{\mathbb{Q}}\xspace$. Then the Albanese map is injective. That is, $F^2(X)=0$.* To the author's knowledge there is not a single example of a smooth projective surface with positive geometric genus that is known to satisfy this conjecture. In this article we make substantial progress towards for an abelian surface $A$ by considering hyperelliptic curves mapping to $A$. We say that a point $a\in A$ is *hyperelliptic* if some nonzero multiple of $a$ lies in the image of a morphism $\phi:H\to A$, where $H$ is a hyperelliptic curve over $\overline{k}\xspace$ such that the hyperelliptic involution on $H$ commutes with the negation on $A$ (see ). Our first theorem is the following. **Theorem 1**. *(cf. ) Let $A$ be an abelian surface over an algebraically closed field $\overline{k}\xspace$ with zero element $0\in A$ and let $a,b\in A$. Suppose there exist nonzero integers $m,n$ such that $a$, $b$, and $ma+nb$ are hyperelliptic points. Let $B_{a,b}$ be the divisible hull of the subgroup of $A(\overline{k}\xspace)$ generated by $a,b$ (that is, $x\in B_{a,b}$ if there exists some nonzero $l\in\mathbb{Z}\xspace$ such that $lx\in\langle a,b\rangle$). Then for every $c,d\in B_{a,b}$ the zero-cycle $$z_{c,d}:=[c+d]-[c]-[d]+[0]$$ vanishes in the Albanese kernel $F^2(A)$.* For the purposes of the introduction we chose to state in a simplified form; a much more general statement can be found in . We emphasize that the points $a,b,ma+nb$ may lie in the images of morphisms from three distinct hyperelliptic curves. Thus, can be thought of as an analog of a celebrated result of Beauville and Voisin ([@Beauville/Voisin2004]) who proved that if $X$ is a $K3$ surface over $\overline{k}\xspace$ and $x,y\in X$ are points that lie on some (possibly different) rational curve inside $X$, then $x,y$ are rationally equivalent. The method for proving involves two simple steps: (i) showing that if $a\in A$ is a hyperelliptic point, then $z_{a,a}=0$, and (ii) the zero-cycle $z_{a,b}$ is bilinear on $a,b$. For an abelian surface $A$, the group $F^2(A)$ is generated by zero-cycles of the form $z_{c,d}$ with $c,d\in A$. Thus, yields the following substantial reduction to Beilinson's conjecture for abelian surfaces. **Corollary 1**. *(See for a more general version.) Let $A$ be an abelian surface over $\overline{\mathbb{Q}}\xspace$. Let $\Gamma\leq A(\overline{\mathbb{Q}}\xspace)$ be generated by a sequence $(a_i)_{i\geq 1}$ such that each $a_i$ is a hyperelliptic point, and for each pair $a_i,a_j$ with $i<j$ there is a $\mathbb{Z}\xspace$-linear combination $n_ia_i+n_ja_j$ for some $n_in_j\neq 0$ that is a hyperelliptic point. If the divisible hull of $\Gamma$ is $A(\overline{\mathbb{Q}}\xspace)$, then Beilinson's is true for $A$.* ## Producing large countable collections of hyperelliptic curves The next big task towards Beilinson's conjecture is to find enough hyperelliptic curves $H$ with non-constant morphisms to $A$. In we give a new construction that produces a rich collection of such curves in the case $A$ is isogenous to a product of elliptic curves. **Theorem 2**. *Let $\overline{k}\xspace$ be an algebraically closed field admitting an embedding $\overline{k}\xspace\hookrightarrow\mathbb{C}\xspace$. Let $A$ be an abelian surface over $\overline{k}\xspace$ and suppose $A$ is isogenous to a product of two elliptic curves. There exist infinitely many positive integers $g\geq 2$ with the following property: there exist infinitely many non-isomorphic genus $g$ hyperelliptic curves $H$ mapping birationally into $A$ such that the hyperelliptic involution on $H$ commutes with the negation on $A$.* The basic idea of the proof is to consider the Kummer surface $K$ associated to a product $E\times E'$ of two elliptic curves and give it the structure of an elliptic fibration. Using addition in the Mordell-Weil lattice we produce infinitely many rational curves in $K$. These pull back to hyperelliptic curves in $E\times E'$, with genus lying in a certain explicit interval depending on the canonical height of the corresponding Mordell-Weil element. Within each such interval, we obtain infinitely many distinct hyperelliptic curves by considering the same construction for any isogenous pair of elliptic curves, noting that over an algebraically closed field of characteristic $0$ any isogeny class of elliptic curves has infinitely many isomorphism classes. *Remark 1*. The hyperelliptic curves we construct can be made very explicit. In fact, we can algorithmically write down their Weierstrass equations and the morphisms to $A$. See for examples with genus $g=2$ and $6$. Heuristically, we expect that in most cases the genus achieves the upper bound of the interval we construct, and these upper bounds exhaust all positive integers $g\equiv 2\mod 4$ (see ). We note that when $A=E_1\times E_2$ is a product of two elliptic curves, the Albanese kernel $F^2(A)$ is generated by zero-cycles of the simpler form $$z_{(p,0),(0,q)}:=[(p,q)]-[(p,0)]-[(0,q)]+[(0,0)],$$ with $p\in E_1(\overline{k}\xspace), q\in E_2(\overline{k}\xspace)$. The points $(p,0), (0,q)$ are always hyperelliptic (see ). This gives the following easier analog of . **Corollary 2**. *Let $A=E_1\times E_2$ be a product of elliptic curves over $\overline{\mathbb{Q}}\xspace$. Suppose that $A(\overline{\mathbb{Q}}\xspace)$ is the divisible hull of a subgroup $\Gamma$ generated by a sequence $\{(p_i,q_i)\}_{i\geq 1}$ such that for each $i<j$, there is a hyperelliptic point $(n_ip_i,n_jp_j)$ for some $n_i \cdot n_j\neq 0$. Then Beilinson's Conjecture is true for $A$.* The above Corollary together with the abundance of curves provided by , give some new hope towards Beilinson's conjecture at least for products of elliptic curves. *Remark 2*. We note that for a general abelian surface $A$ over an algebraically closed field $\overline{k}\xspace$ of characteristic zero there are classical constructions that produce countably many genus $2$ curves whose Jacobian is isogenous to $A$ ([@Humbert; @Bost/Mestre]). Thus, even for a simple abelian surface, there is a rich collection of hyperelliptic points (see ). For a product $A=E_1\times E_2$ of two elliptic curves there is also a construction due to Frey and Kani ([@FreyKani]) that under some assumptions on $E_1,E_2$ produces a genus $2$ hyperelliptic curve $H$ having dominant maps $H\to E_i$ for $i=1,2$. To our knowledge, is the first result that produces hyperelliptic curves of arbitrarily high genus mapping to a product $E_1\times E_2$, and even infinitely many non-isomorphic curves of the same genus. ## Relation to Bogomolov's Conjecture The idea of relating Beilinson's conjecture for abelian surfaces to hyperelliptic points was heavily inspired by the analogy with the result of Beauville and Voisin for $K3$'s ([@Beauville/Voisin2004]), and by the general expectation that a projective $K3$ surface over an algebraically closed field $\overline{k}\xspace$ must contain infinitely many rational curves (see [@Bogomolov/Hassett/Tschinkel Conjecture 2]). In fact, a conjecture of Bogomolov from 1981 predicts that every point $x$ in a $K3$ surface over $\overline{\mathbb{Q}}\xspace$ lies in the image of a rational curve (see [@bogomolov_tschinkel p. 2] for a more recent reference). This would imply that every point $a$ in an abelian surface over $\overline{\mathbb{Q}}\xspace$ is hyperelliptic. In particular, Bogomolov's conjecture for the Kummer surface $K$ of $A$ implies Beilinson's conjecture for both $K$ and $A$. Although Bogomolov's conjecture is still a reasonable expectation, there is work that suggests this could be false in general (see [@BaragarMcKinnon; @engel2022maninmumford]). We emphasize that our results require much less than that. Notably, if $A$ is defined over an algebraic number field $k$, takes advantage of the finitely generated structure of the group $A(L)$, where $L/k$ runs in the finite extensions of $k$. ## Results in higher dimensions In we discuss extensions of to abelian varieties of arbitrary dimension. In fact, works verbatim even when $\dim(A)\geq 3$ with a slightly more involved argument (cf. ). Although in dimension $\geq 3$ it might be too optimistic to expect that we have enough hyperelliptic curves, this could be the case in special cases, for example for Jacobians of hyperelliptic curves (see [@bogomolov_tschinkel Theorem 7.1, Remark 7.7] for some motivation towards these expectations). ## Acknowledgement The first author's research was partially supported by the NSF grants DMS-2001605 and DMS-2302196. The second author was supported by a CRM--ISM postdoctoral fellowship. We are truly grateful to Professor Arnaud Beauville who helped us simplify the proof of significantly. We would also like to heartily thank Professors Wayne Raskind, Ari Shnidman and Akshay Venkatesh who showed interest in our work and for useful discussions. # Rational Equivalences on Abelian Surfaces {#abelsurfacesection} For a smooth projective variety $X$ over an algebraically closed field $\overline{k}\xspace$ we consider the Chow group $\mathop{\mathrm{CH}}_0(X)$ of zero-cycles modulo rational equivalence on $X$. **Notation 1**. Following the introduction, we will denote by $F^1(X)$ the kernel of the degree map, $\mathop{\mathrm{CH}}_0(X)\xrightarrow{\deg}\mathbb{Z}\xspace$, and by $F^2(X)$ the kernel of the Albanese map, $F^1(X)\xrightarrow{\mathop{\mathrm{alb}}_X}\mathop{\mathrm{Alb}}_X(\overline{k}\xspace)$. The Albanese variety $\mathop{\mathrm{Alb}}_X$ of $X$ is an abelian variety, which is dual to the Picard variety of $X$. When $X$ is an abelian variety, $\mathop{\mathrm{Alb}}_X=X$, while when $X$ is a $K3$ surface, $\mathop{\mathrm{Alb}}_X=0$. Since $\overline{k}\xspace$ is algebraically closed, Rojtman's theorem (cf. [@Rojtman1980]) gives that the Albanese map induces an isomorphism between the torsion subgroups of $F^1(X)$ and $\mathop{\mathrm{Alb}}_X(\overline{k}\xspace)$. This in particular implies that $F^2(X)$ is torsion-free. This information will be used heavily in the proofs of the main results. Namely, in order to show that a certain zero-cycle $z\in F^2(X)$ vanishes, it is enough to show $z$ is torsion. ## Hyperelliptic points Let $A$ be an abelian surface over $\overline{k}\xspace$. We introduce the notion of a *hyperelliptic point*, which will be used throughout the article. For the purposes of this paper, a *hyperelliptic curve* is a smooth projective curve $H$ over $\overline{k}\xspace$ of genus $g\geq 1$[^1] equipped with a prescribed degree $2$ map $H\to\mathbb{P}^1$. Any such curve comes equipped with a *hyperelliptic involution* $\iota:H\to H$ determined by the property that $H\to\mathbb{P}^1$ is invariant under $\iota$, and the fixed points of $\iota$ in $H(\overline{k}\xspace)$ are *Weierstrass points* of $H$. **Definition 1**. A point $a\in A$ is called hyperelliptic if there exists a hyperelliptic curve $H$ and a morphism $f:H\to A$ with $f\circ \iota=-f$ such that a nonzero multiple of $a$ lies in the image of $f$. **Lemma 1**. *Let $f:H\to A$ be a morphism from a hyperelliptic curve $H/\overline{k}\xspace$ to $A$. Then $f\circ\iota=-f$ if and only if there exists a Weierstrass point of $H$ mapping under $f$ to a $2$-torsion point of $A$.* *Proof.* Fix a base point $q\in H(\overline{k}\xspace)$, so that an embedding $H\to \mathop{\mathrm{Jac}}(H)$ of $H$ in its Jacobian variety is determined by $p\mapsto [p]-[q]$. By the universal property of $\mathop{\mathrm{Jac}}(H)$, $f$ factors through this embedding. Since every morphism of abelian varieties is a homomorphism followed by translation, there exists a homomorphism $\psi:\mathop{\mathrm{Jac}}(H)\to A$ such that $$f(p)=\psi([p]-[q])+f(q).$$ Now for any $p\in H(\overline{k}\xspace)$, the divisor $[\iota p]+[p]-[\iota q]-[q]$ is principal, so $$\begin{aligned} f(\iota p)+f(p)&=\psi([\iota p]+[p]-2[q])+2f(q)\\ &=\psi([\iota q]-[q])+2f(q)\\ &=f(\iota q)+f(q). \end{aligned}$$ Thus the value of $f(\iota p)+f(p)$ is constant[^2] for all $p\in H(\overline{k}\xspace)$, and we have $f\circ\iota=-f$ if and only if this constant is zero. In particular, if $q$ is a Weierstrass point, then $f\circ\iota=-f$ if and only if $f(q)$ is $2$-torsion. ◻ and clearly generalize to higher dimensional abelian varieties and will be used in . The following lemma which is special for abelian surfaces was also obtained in [@bogomolov_tschinkel Lemma 4.1]. For an abelian surface $A$ over $\overline{k}\xspace$ we consider the quotient $A/\langle -1\rangle$ by the negation action. This surface has sixteen singularities corresponding to the sixteen $2$-torsion points of $A$. The Kummer surface $K$ associated to $A$ is the $K3$ surface obtained by resolving the singularities of $A/\langle -1\rangle$. **Lemma 2**. *Let $K$ be the Kummer surface associated to the abelian surface $A$. Then the set of images of non-constant morphisms $\mathbb{P}^1\to K$ is in one-to-one correspondence with the set of images of morphisms $f:H\to A$ from hyperelliptic curves $H$ such that $f\circ\iota=-f$.* *Proof.* If $f:H\to A$ is constant and $f\circ \iota=-f$, then the image of $f$ is a $2$-torsion point. Each such point maps to a singularity of $A/\langle -1\rangle$ and blows up to a rational curve in $K$, so we associate to each $2$-torsion point of $A$ the corresponding blowup in $K$. Now we establish a bijection between images of non-constant maps $f:H\to A$ with $f\circ\iota=-f$ and images of non-constant maps $\mathbb{P}^1\to K$ whose image does not map to a singular point of $A/\langle -1\rangle$. First suppose $f:H\to A$ is non-constant and $f\circ\iota=-f$. The composition $H\to A\to A/\langle -1\rangle$ factors through the double cover $H\to\mathbb{P}^1$, resulting in a non-constant map $\mathbb{P}^1\to A/\langle -1\rangle$. Since the image of this map is not contained in the singular locus, we obtain a rational map $\mathbb{P}^1\dashrightarrow K$, which extends to a morphism $\mathbb{P}^1\to K$ since $\mathbb{P}^1$ is a smooth projective curve and $K$ is projective. In the other direction, if $\mathbb{P}^1\to K$ is a non-constant map with image not contained in the exceptional locus, then $\mathbb{P}^1\to A/\langle -1\rangle$ is non-constant. Taking fiber products we obtain a diagram $$\begin{tikzcd} C\arrow[rr]\arrow[d] & & A\arrow[d] \\ \mathbb{P}^1\arrow[r]& K \arrow[r] & A/\langle -1\rangle \end{tikzcd}$$ for some degree $2$ cover $C\to \mathbb{P}^1$. Let $H$ denote the normalization of $C$. Since $A$ contains no rational curves, $H$ must be irreducible and have genus at least $1$, and the composition $H\to C\to\mathbb{P}^1$ gives $H$ the structure of a hyperelliptic curve. ◻ ## Zero-cycles on Abelian Surfaces Let $A$ be an abelian surface over $\overline{k}\xspace$ with zero element $0\in A$. A straightforward computation gives the following explicit generators for the subgroups $F^1(A), F^2(A)$. $$\begin{aligned} &&F^{1}(A)=\langle[a]-[0]:a\in A\rangle,\\ &&F^{2}(A)=\langle[a+b]-[a]-[b]+[0]:a,\;b\in A\rangle. \end{aligned}$$ **Notation 2**. For $a,b\in A$ we will denote by $z_{a,b}:=[a+b]-[a]-[b]+[0]$. The goal of this subsection is to prove the following more general version of . **Theorem 3**. *Let $A$ be an abelian surface over an algebraically closed field $\overline{k}\xspace$ and let $a,b\in A$. Let $m_i^j\in\mathbb{Z}\xspace$ for $i=1,2$ and $j=1,2,3$ be such that the set of pure symmetric tensors $(m_1^j,m_2^j)\otimes (m_1^j,m_2^j)$ for $j=1,2,3$ spans $\mathop{\mathrm{Sym}}^2\mathbb{Q}\xspace^2$, and suppose the points $m_1^ja+m_2^jb$ for $j=1,2,3$ are each hyperelliptic. Let $B_{a,b}$ be the divisible hull of the subgroup of $A(\overline{k}\xspace)$ generated by $a,b$. Then for every $c,d\in B_{a,b}$ the zero-cycle $$z_{c,d}:=[c+d]-[c]-[d]+[0]$$ vanishes in $F^2(A)$.* *Remark 3*. follows because for any nonzero $m,n\in\mathbb{Z}\xspace$, the points $(1,0)\otimes(1,0)$, $(0,1)\otimes(0,1)$, and $(m,n)\otimes(m,n)$ span $\mathop{\mathrm{Sym}}^2\mathbb{Q}\xspace^2$. The theorem will follow from the following two key lemmas. The first lemma is probably well-known to the experts, and it could have a simpler proof. For completion we include a proof using some results from [@Gazaki2015]. **Lemma 3**. *The element $z_{a,b}$ is bilinear and symmetric on $a,b$.* *Proof.* The symmetry is clear. In [@Gazaki2015] the first author defined for an abelian variety $A$ over an arbitrary field $k$ a filtration $\{F^{r}(A)\}_{r\geq 0}$ extending the filtration $\mathop{\mathrm{CH}}_0(A)\supset F^1(A)\supset F^2(A)$. Among other properties, it follows by [@Gazaki2015 Proposition 3.4] that there exists a well-defined homomorphism $\Psi: A(k)\otimes A(k)\to F^2(A)/F^{3}(A)$, with $\Psi(a\otimes b)=z_{a,b}$. In the case of an abelian surface, it follows by [@Gazaki2015 Corollary 4.5] that the group $F^3(A)$ is torsion, and hence over an algebraically closed field $\overline{k}\xspace$ it vanishes. Thus, we have a linear map $$A(\overline{k}\xspace)\otimes A(\overline{k}\xspace)\to F^2(A), a\otimes b\mapsto z_{a,b},$$ yielding the desired bilinearity. ◻ Since the group $F^2(A)$ is torsion-free, it follows by bilinearity that $z_{p,q}=0$ if $p$ or $q$ is a torsion point. **Corollary 3**. *The Albanese kernel $F^2(A)$ vanishes if and only if $z_{a,a}=0$ for all $a\in A$.* *Proof.* The group $F^2(A)$ is generated by zero-cycles of the form $z_{a,b}$ with $a,b\in A$. Bilinearity and symmetry give $$2z_{a,b}=z_{a,a}+z_{b,b}-z_{a+b,a+b},$$ from where the claim follows. ◻ **Lemma 4**. *Let $a\in A$ be a hyperelliptic point in the sense of . Then $z_{a,a}=0$.* *Proof.* By assumption there exists a morphism $f:H\to A$ from a hyperelliptic curve $H$ such that the hyperelliptic involution on $H$ commutes with the negation on $A$ with $na$ lying in the image of $f$ for some nonzero integer $n$. Since $z_{na,na}=n^2z_{a,a}$, we can reduce to the case $n=1$. For a closed point $q\in H$ denote by $\overline{q}$ the image of $q$ under the hyperelliptic involution. Write $a=\phi(q)$. Let $w$ be a Weierstrass point of $H$. Since the hyperelliptic involution on $H$ commutes with negation on $A$, it follows that $\phi(\overline{q})=-\phi(q)$ and $\phi(w)=p_0$ is a $2$-torsion point of $A$. The proper morphism $\phi$ induces a pushforward homomorphism $$\begin{aligned} &&\phi_\star:\mathop{\mathrm{Pic}}^0(H)\to F^1(A)\\ &&[q]-[w]\mapsto [\phi(q)]-[p_0].\end{aligned}$$ Note that $[q]+[\overline{q}]-2[w]$ is a principal divisor on $H$. This yields, $$0=f_\star([q]+[\overline{q}]-2[w])=[a]+[-a]-2[p_0].$$ But $[a]+[-a]-2[p_0]=-z_{a,-a}+z_{p_0,p_0}$, and $z_{p_0,p_0}=0$ since $p_0$ is a torsion point. Thus, $z_{a,a}=-z_{a,-a}=0$ as desired. ◻ **Corollary 4**. *Suppose that every point $a\in A(\overline{k}\xspace)$ is hyperelliptic. Then $F^2(A)=0$.* *Proof of .* Let $c,d\in B_{a,b}$. By definition there exists $n,m\geq 1$ such that $nc, md$ are $\mathbb{Z}\xspace$-linear combinations of $a,b$. Bilinearity gives $z_{nc,md}=nmz_{c,d}$. Since $nm\neq 0$, it is enough to show $z_{nc,md}=0$, which allows to reduce to the case when $c,d$ lie in the subgroup generated by $a,b$. Let us consider first a special case. Assume that each of the points $a,b,a+b$ is hyperelliptic. It follows by that $z_{a,a}=z_{b,b}=z_{a+b,a+b}=0$, which implies $z_{a,b}=0$. A similar reasoning gives $z_{c,d}=0$ for every $\mathbb{Z}\xspace-$linear combination of $a,b$. The general case is similar. Assume $c=ra+sb$, $d=ta+ub$, for some $r,s,t,u\in\mathbb{Z}\xspace$. Write $$(r,s)\otimes (t,u)=\sum_{i=1}^3 \frac{r_i}{n}(m_1^j,m_2^j)\otimes (m_1^j,m_2^j)$$ for some $r_1,r_2,r_3\in\mathbb{Z}\xspace$ and nonzero $n\in\mathbb{Z}\xspace$. Bilinearity and symmetry allow us to write $$nz_{c,d}=\sum_{i=1}^3 r_iz_{m_1^ja+m_2^jb,m_1^ja+m_2^jb}.$$ By , each term in the above sum is zero, which implies $z_{c,d}$ is torsion, and hence it vanishes as desired. ◻ ## Products of Elliptic Curves {#products-of-elliptic-curves .unnumbered} In the case of a product $A=E_1\times E_2$ of two elliptic curves, we can say something more. In this case we have an equality $$F^2(E_1\times E_2)=\langle [p,q]-[p,0]-[0,q]+[0,0]:p\in E_1, q\in E_2\rangle.$$ **Lemma 5**. *The points $(p,0)$ and $(0,q)$ are hyperelliptic in the sense of .* *Proof.* We can give $E_1$ the structure of a hyperelliptic curve by letting the hyperelliptic involution $\iota:E_1\to E_1$ be given by negation on $E_1$. Then the map $f:E_1\to E_1\times E_2$ determined by $p\mapsto (p,0)$ evidently satisfies $f\circ\iota=-f$. ◻ The above lemma together with yield the following corollary. **Corollary 5**. *Let $A=E_1\times E_2$ be a product of elliptic curves over an algebraically closed field $\overline{k}\xspace$. Let $(p,q)\in A$ be such that $(np,mq)$ is hyperelliptic for some nonzero integers $n,m$. Then the zero-cycle $[(p,q)]-[(p,0)]-[(0,q)]+[(0,0)]$ vanishes.* ## Some Progress towards Beilinson's Conjecture In this section we focus on Beilinson's conjecture, which we recall below. **Conjecture 2**. *(Beilinson, [@Beilinson1984]) Let $X$ be a smooth projective variety over $\overline{\mathbb{Q}}\xspace$. Then the Albanese map is injective. That is, $F^2(X)=0$.* yields the following reduction to . **Corollary 6**. *Let $A$ be an abelian surface over $\overline{\mathbb{Q}}\xspace$. Suppose that the group $A(\overline{\mathbb{Q}}\xspace)$ is the divisible hull of a subgroup $\Gamma$ generated by a sequence $(a_i)_{i\geq 1}$ satisfying the following property. Let $I$ be a collection of integer sequences $\vec{n}=(n_1,n_2,\ldots)$ each with finitely many nonzero terms, such that $$\{\vec{n}\otimes \vec{n}:\vec{n}\in I\}$$ spans $\mathop{\mathrm{Sym}}^2\mathbb{Q}\xspace^\infty$. If $n_1a_1+n_2a_2+\cdots$ is hyperelliptic for all $(n_1,n_2,\ldots)\in I$, then Beilinson's is true for $A$.* *Remark 4*. As a special case, let $e_i$ denote the sequence which is $1$ in the $i$-th component and $0$ otherwise. Then $$I=\{e_i:i\in\mathbb{Z}\xspace_{>0}\}\cup\{n_ie_i+n_je_j:i,j\in\mathbb{Z}\xspace_{>0},\;n_i,n_j\in\mathbb{Z}\xspace_{\neq 0}\}$$ satisfies the conditions of , so holds. *Proof.* It follows by that it is enough to show $z_{c,c}=0$ for every $c\in A$, or equivalently that $z_{Nc,Nc}=0$ for some $N\neq 0$. Let $c\in A(\overline{\mathbb{Q}}\xspace)$. By assumption there exists $N\geq 1$ such that $Nc\in\Gamma$, thus we may reduce to the case $c\in \Gamma$. The claim that $z_{c,c}=0$ for all $c\in\Gamma$ follows exactly like in the proof of . ◻ *Remark 5*. It is a well-known fact that $F^2(X)=0$ for a smooth projective variety $X$ over the algebraic closure $\overline{\mathbb{F}\xspace}_p$ of a finite field (this follows for example from [@Kato/Saito83 Theorem 2]). gives a new proof of this fact for abelian surfaces $A$. Namely, in the course of proving that every point $x$ in the Kummer surface associated to $A$ lies on some rational curve [@bogomolov_tschinkel Theorem 1.1], Bogomolov and Tschinkel demonstrate that every point of $A$ is hyperelliptic; more precisely, they show that there is a single curve $H$ such that every point $a\in A(\overline{\mathbb{F}\xspace}_p)$ is in the image of a morphism $f:H\to A$ with $f\circ\iota=-f$ [@bogomolov_tschinkel Corollary 2.5]. We do not expect an analogue of [@bogomolov_tschinkel Corollary 2.5] to hold over number fields: in the finite field setting, the Frobenius endomorphism of $A$ is used in an essential way to generate sufficiently many morphisms from $H$ to entirely fill out $A(\overline{k}\xspace)$. However, this example illustrates the fact that hyperelliptic points --- even those coming from a single hyperelliptic curve --- can be very plentiful. If we have a much larger collection of hyperelliptic curves mapping into $A/\overline{\mathbb{Q}}\xspace$, as in , it seems plausible that the conditions of may be satisfied, at least for products of elliptic curves. *Remark 6*. Suppose $A=\mathop{\mathrm{Jac}}(C)$ is a Jacobian of a genus $2$ curve. An old construction due to Humbert ([@Humbert]) constructs an explicit curve $H/\overline{k}\xspace$ such that $A/L\simeq \mathop{\mathrm{Jac}}(H)$, where $L$ is a Lagrangian subgroup of $A[2]$. We refer to [@Bost/Mestre] for a more recent reference. Humbert's construction uses Gauss' arithmetic-geometric mean to achieve "doubling of the period matrix\". Since every genus $2$ curve is hyperelliptic, one can iterate the above construction to produce countably many hyperelliptic curves $H_n$ (pairwise non-isomorphic, because the period matrices are distinct) such that there is an isogeny $A\xrightarrow{\phi_n}\mathop{\mathrm{Jac}}(H_n)$ with kernel of order equal to a power of $4$. Since every abelian surface over an algebraically closed field is isogenous to a Jacobian, the above construction shows that we can always find countably many non-isomorphic genus $2$ hyperelliptic curves mapping to a given abelian surface $A$. # Producing curves in abelian surfaces {#curvessection} We continue to assume all varieties are defined over an algebraically closed field $\overline{k}\xspace$. From now on we assume additionally that $\overline{k}\xspace$ admits an embedding $\overline{k}\xspace\hookrightarrow\mathbb{C}\xspace$. The goal of this section is to show that when an abelian surface $A$ is isogenous to a product of elliptic curves, there is an abundance of hyperelliptic curves $f:H\to A$ with negation on $A$ acting as the hyperelliptic involution on $H$. Given a morphism $f:V\to W$ of schemes, let $f(V)$ denote the scheme-theoretic image of $f$; we say $f$ is *birational onto its image*, if there exists a dense open $U\subseteq V$ such that $f(U)$ is open in $f(V)$ and $f|_U:U\to f(U)$ is an isomorphism. The following definition will be used so that we can avoid considering multiple curves $H$ that have the same image in $A$. **Definition 2**. Let $H/\overline{k}\xspace$ be a smooth hyperelliptic curve and $A$ an abelian variety. A map $f:H\to A$ is *compatible* if the hyperelliptic involution on $H$ commutes with negation on $A$, and $f$ maps $H$ birationally onto its image. If $C\subseteq A$ is the image of any non-constant map $f:H\to A$ from a hyperelliptic curve $H$ with $f\circ\iota=-f$, then the normalization map $\widehat{C}\to C$ is a compatible map, and $f$ factors through the normalization (cf. ). In particular, every hyperelliptic point of $A$ () is in the image of a compatible map. The main goal of this section is to prove the following more general version of . **Theorem 4**. *Let $A/\overline{k}\xspace$ be an abelian variety, and suppose there is a homomorphism with finite kernel from a product of two elliptic curves into $A$. For infinitely many positive integers $g\geq 2$, there exist infinitely many non-isomorphic genus $g$ hyperelliptic curves with compatible maps to $A$.* The proof is given in after establishing a series of lemmas. The most important step is the following construction. **Proposition 1**. *Let $E_0,E_0'$ be elliptic curves over $\overline{k}\xspace$, and $n$ a positive integer. There exist infinitely many isomorphism classes of pairs of elliptic curves $E,E'/\overline{k}\xspace$ satisfying the following:* 1. *there is an isogeny $\psi:E\times E'\to E_0\times E_0'$;* 2. *there is a compatible map $f:H\to E\times E'$ from a genus $g$ hyperelliptic curve $H$ satisfying $\frac16n-10\leq g\leq 4n-2$;* 3. *the degree of $\psi$ is relatively prime to $g-1$;* 4. *the degrees of the projection maps $H\to E$ and $H\to E'$ are equal to $2n$.* The proof of this result is the focus of and , but we summarize the argument here. Over an algebraically closed field of characteristic zero, isogeny classes of elliptic curves contain infinitely many isomorphism classes, so there are infinitely many choices of $E,E'/\overline{k}\xspace$ with an isogeny $E\times E'\to E_0\times E_0'$; this remains true even if we restrict to isogenies with degree coprime to $g-1$ for all integers $\frac16n-10\leq g\leq 4n-2$. We can conclude (a), and once we establish (b), (c) will also follow. Given such a pair $E,E'$, let $\pi_{-1}$ be the quotient of $E\times E'$ by $-1$. The $2$-torsion of $E\times E'$ map to singularities under $\pi_{-1}$, and the Kummer surface $K$ of $E\times E'$ is obtained by blowing up these points. Let $D$ be the divisor of $K$ consisting of these $16$ lines. We give $K$ the structure of an elliptic fibration $K\to\mathbb{P}^1$, in such a way that nine of the sixteen rational curves in $D$ are sections of this fibration. Using addition in the Mordell-Weil lattice we can produce many more sections, and produce an explicit formula for their intersection multiplicities with $D$ (). By pulling back such a rational curve $\phi:\mathbb{P}^1\to K$ along $\pi_{-1}$, we obtain a curve $C\to E\times E'$, and the normalization $H$ of $C$ is a hyperelliptic curve mapping compatibly to $E\times E'$. To summarize, we obtain the following diagram, where vertical maps are degree $2$ and horizontal maps are birational onto their images: $$\label{eq:Htikz} \begin{tikzcd} H\ar{r} & C\ar{rr}\ar{d} & & E\times E'\arrow["\pi_{-1}"]{d} \\ & \mathbb{P}^1\ar["\phi"]{r} & K\ar{r} & (E\times E')/\langle -1\rangle. \end{tikzcd}$$ Away from the locus of points mapping to $D$, every point of $\mathbb{P}^1$ pulls back to two smooth points of $C$, so the number of Weierstrass points of $H$ is bounded above by the intersection multiplicity $\mathop{\mathrm{im}}\phi\cdot D$. On the other hand, if $\mathop{\mathrm{im}}\phi$ intersects $D$ transversely at a point $P\in K(\overline{k}\xspace)$, then the preimage of $P$ in $C$ is smooth, and so corresponds to a Weierstrass point of $H$; thus the number of Weierstrass points of $H$ is bounded below by the number of transverse intersections between $\mathop{\mathrm{im}}\phi$ and $D$. We can compute a lower bound on the number of transverse intersections (), and can therefore compute upper and lower bounds on the genus $g$ of $H$. This allows us to establish (b). To prove (d), we use the fact that the degree of a projection $H\to E$ equals the intersection multiplicity in $E\times E'$ of the image of $H$ with a fiber over a point of $E$. This can be computed using intersection theory on $K$. Now assuming , we discuss how to conclude . First fix elliptic curves $E_0,E_0'$ with a homomorphism of finite kernel to $A$; let $m$ denote the degree of this map. We show in that by allowing the curves $E,E'$ in to vary, we obtain infinitely many non-isomorphic hyperelliptic curves $H$, and that the composition of the compatible map $H\to E\times E'$ with the étale map $E\times E'\to E_0\times E_0'$ is compatible. This establishes the in the case $A=E_0\times E_0'$. In general, composing a compatible map $H\to E_0\times E_0'$ with the homomorphism $E_0\times E_0'\to A$ may no longer be compatible. However, the resulting map $H\to A$ will factor into an étale map $H\to H'$ of degree at most $m$ followed by a compatible map $H'\to A$. Since there are infinitely many isomorphism classes of $H$, and any curve $H'$ has finitely many étale covers of bounded degree, we can conclude that there are infinitely many non-isomorphic curves $H'$ with compatible maps to $A$ and genus in a certain finite interval. Since the lower bound for this interval goes to infinity with $n$, the theorem follows. ## Kummer surface as elliptic fibration {#Kummerfibration} We follow the setup described in Sections 3 to 5 of [@shioda07]; see also [@kuwatashioda] for a summary. Since $\overline{k}\xspace$ has characteristic $0$, every elliptic curve over $\overline{k}\xspace$ is isomorphic to a curve in *Legendre form*, $$E_\lambda:y^2=f_\lambda(x):=x(x-1)(x-\lambda)$$ for some $\lambda\in\overline{k}\xspace-\{0,1\}$. Let $E_a,E_b$ be elliptic curves over $\overline{k}\xspace$ for some $a,b\in\overline{k}\xspace-\{0,1\}$. For $i=0,1,2,3$, let $P_i\in E_a[2]$ and $Q_j\in E_b[2]$ be $$\begin{aligned} P_0&=O_a, & P_1&=(0,0), & P_2&=(1,0), & P_3&=(a,0),\\ Q_0&=O_b, & Q_1&=(0,0), & Q_2&=(1,0), & Q_3&=(b,0).\end{aligned}$$ We describe a certain collection of rational curves in the Kummer surface $K$ of $E_a\times E_b$. - For $0\leq i\leq 3$, $P_i\times E_b$ maps to a rational curve in $(E_a\times E_b)/\langle -1\rangle$; its pullback in $K$ is denoted $F_i$. - For $0\leq j\leq 3$, $E_a\times Q_j$ maps to a rational curve in $(E_a\times E_b)/\langle -1\rangle$; its pullback in $K$ is denoted $G_j$. - For $0\leq i,j\leq 3$, $(P_i,Q_j)$ maps to a singularity in $(E_a\times E_b)/\langle -1\rangle$; its blowup in $K$ is denoted $H_{ij}$. If we remove the curves $$\begin{aligned} \label{eq:inf_fiber} F_1,F_2,F_3,G_0,H_{00},H_{10},H_{20},H_{30}\end{aligned}$$ (corresponding to points $(x_1,y_1,x_2,y_2)\in E_a\times E_b$ with $y_1=0$ or $y_2=\infty$) we obtain an affine chart of $K$ defined by the equation $$\label{affinepatch} f_a(x_1)r^2=f_b(x_2),$$ with the rational map $E_a\times E_b\dashrightarrow K$ given by $$(x_1,y_1,x_2,y_2)\mapsto \left(x_1,x_2,\frac{y_2}{y_1}\right).$$ We have a map $\tau:K\to\mathbb{P}^1$ defined by $(x_1,x_2,r)\mapsto r$; this determines a fibration of $K$ called *Inose's pencil*. The fiber at $r=\infty$ is $$3G_0+2(H_{10}+H_{20}+H_{30})+F_1+F_2+F_3,$$ of Kodaira type $IV^*$. The fiber at $r=0$ is $$3F_0+2(H_{01}+H_{02}+H_{03})+G_1+G_2+G_3,$$ also of Kodaira type $IV^*$. There may be other singular fibers (depending on $a$ and $b$), but if $E_a$ and $E_b$ have distinct $j$-invariants, then the fibers at $r=0,\infty$ are the only reducible fibers [@shioda07 Proposition 5.1]. For $1\leq i,j\leq 3$, the curve $H_{ij}$ is a section of the fibration. The curve $H_{00}$ is a multisection, intersecting every fiber with multiplicity $3$. If we fix $H_{11}$ as the zero section, $K$ obtains the structure of an elliptic fibration. We use $\oplus$ to denote Mordell-Weil addition, to distinguish it from the sum of curves as divisors. The remaining eight sections satisfy the relations [@Scholten Section 6] $$\begin{aligned} \label{eq:Hij_relations} \begin{split} H_{13}&=H_{22}\oplus H_{32}\\ H_{12}&=H_{23}\oplus H_{33}\\ H_{21}&=H_{32}\oplus H_{33}\\ H_{31}&=H_{22}\oplus H_{23} \end{split}\end{aligned}$$ while $H_{22},H_{32},H_{23},H_{33}$ are independent for general $a,b$. Let $\cdot$ denote the intersection multiplicity of divisors on $K$, and define the divisor $$D:=\sum_{0\leq i,j\leq 3} H_{ij}.$$ **Lemma 6**. *Suppose $E_a,E_b$ are not isomorphic to each other, and neither has $j$-invariant $0$. Fix $p,q,r,s\in\mathbb{Z}\xspace$, and define the section $$\mathcal{P}:=pH_{22}\oplus qH_{33}\oplus rH_{23}\oplus sH_{32}.$$ Then $$\mathcal{P}\cdot D=8\big(p(p-1)+q(q-1)+r(r-1)+s(s-1)+pq+rs\big)-2.$$* *Proof.* We follow [@Scholten Section 6] which carries out the computation in the special case $p=q=1$, $r=s=0$. First we determine the intersection of $\mathcal{P}$ with the fibers at $r=0,\infty$. As a section of the fibration, $\mathcal{P}$ intersects these fibers each with multiplicity $1$, so it cannot intersect any components that occur with multiplicity greater than $1$. Hence $\mathcal{P}$ has zero intersection with each of $F_0,G_0,H_{01},H_{02},H_{03},H_{10},H_{20},H_{30}$. So $\mathcal{P}$ intersects exactly one of $F_1,F_2,F_3$, and exactly one of $G_1,G_2,G_3$. Now for each $0\leq i\leq 3$, the divisors $D_i:=2F_i+\sum_j H_{ij}$ are fibers of the map $K\to \mathbb{P}^1$ given by $(x_1,x_2,r)\mapsto x_1$ [@shioda07 Equation (3.3)], and hence they have equal intersection product with $\mathcal{P}$. Therefore $3\mathcal{P}\cdot D_0=\mathcal{P}\cdot (D_1+D_2+D_3)$, which simplifies to $$3\mathcal{P}\cdot H_{00}=2\mathcal{P}\cdot (F_1+F_2+F_3)+\sum_{1\leq i,j\leq 3} \mathcal{P}\cdot H_{ij}.$$ Since $\mathcal{P}\cdot (F_1+F_2+F_3)=1$ we obtain $$\mathcal{P}\cdot D=\frac 23+\frac43\sum_{1\leq i,j\leq 3} \mathcal{P}\cdot H_{ij}.$$ This expresses $\mathcal{P}\cdot D$ entirely in terms of intersection products of sections of Inose's pencil, so we can now apply a formula relating intersection multiplicity of sections to the Néron-Tate pairing [@shioda90 Theorem 8.6]. We obtain $$\begin{aligned} \sum_{1\leq i,j\leq 3} \mathcal{P}\cdot H_{ij}&=9\chi+9(\mathcal{P}\cdot H_{11})\\ &\quad+\sum_{1\leq i,j\leq 3} \left((H_{ij}\cdot H_{11})-\langle \mathcal{P}, H_{ij}\rangle-\text{contr}_0(\mathcal{P},H_{ij})-\text{contr}_\infty(\mathcal{P},H_{ij})\right), \end{aligned}$$ where $\chi$ is the arithmetic genus of $K$, and $\text{contr}_v$ is a function on pairs of sections that depends only on which components of the fiber at $v$ they each intersect. Since $K$ is a K3 surface we have $\chi=2$ and $H_{11}\cdot H_{11}=-\chi=-2$, as well as $H_{ij}\cdot H_{11}=0$ for all other $i,j$, so this simplifies to $$\begin{aligned} \label{eq:intersection_sum} \sum_{1\leq i,j\leq 3} \mathcal{P}\cdot H_{ij}=&16+9(\mathcal{P}\cdot H_{11})-\sum_{1\leq i,j\leq 3} \left(\langle \mathcal{P}, H_{ij}\rangle+\text{contr}_0(\mathcal{P},H_{ij})+\text{contr}_\infty(\mathcal{P},H_{ij})\right). \end{aligned}$$ To compute the $\text{contr}_\infty$ terms, we note that Mordell-Weil addition induces a $\mathbb{Z}\xspace/3\mathbb{Z}\xspace$ structure on the multiplicity $1$ components $F_1,F_2,F_3$ of the fiber at $\infty$ (with $F_1$ being the identity). If $\mathcal{P}$ intersects $F_k$ then $$\begin{aligned} \text{contr}_\infty(\mathcal{P},H_{ij})&=\left\{\begin{array}{ll} 0,& \text{if }i=1\text{ or }k=1,\\ 2/3, & \text{if }i,k\neq 1\text{ and }i\neq k,\\ 4/3, & \text{if }i,k\neq 1\text{ and }i=k. \end{array}\right. \end{aligned}$$ [@shioda90 Equation (8.16)]. We can therefore conclude that $$\sum_{1\leq i,j\leq 3}\text{contr}_\infty(\mathcal{P},H_{ij})=\left\{\begin{array}{ll} 0,& \text{if }\mathcal{P}\text{ intersects }F_1,\\ 6, & \text{otherwise.} \end{array}\right.$$ Similarly, $$\sum_{1\leq i,j\leq 3}\text{contr}_0(\mathcal{P},H_{ij})=\left\{\begin{array}{ll} 0,& \text{if }\mathcal{P}\text{ intersects }G_1,\\ 6, & \text{otherwise.} \end{array}\right.$$ To compute $\mathcal{P}\cdot H_{11}$, we again use the relation between intersection and Néron-Tate pairings: $$\label{PH11} \mathcal{P}\cdot H_{11}=\frac12\langle\mathcal{P},\mathcal{P}\rangle-2+\frac12\text{contr}_0(\mathcal{P},\mathcal{P})+\frac12\text{contr}_\infty(\mathcal{P},\mathcal{P}),$$ with $\text{contr}_\infty(\mathcal{P},\mathcal{P})=0$ if $\mathcal{P}$ intersects $F_1$ and $\frac43$ otherwise, and likewise $\text{contr}_0(\mathcal{P},\mathcal{P})=0$ if $\mathcal{P}$ intersects $G_1$ and $\frac43$ otherwise. Plugging this into and noting the wonderful coincidence $9\cdot\frac12\cdot\frac43=6$, the $\text{contr}_v$ terms cancel, so the equation simplifies to $$\begin{aligned} \sum_{1\leq i,j\leq 3} \mathcal{P}\cdot H_{ij}=&\frac92\langle\mathcal{P},\mathcal{P}\rangle-2-\sum_{1\leq i,j\leq 3} \langle \mathcal{P}, H_{ij}\rangle. \end{aligned}$$ Noting that $$\bigoplus_{1\leq i,j\leq 3} H_{ij}=3H_{22}\oplus 3H_{23}\oplus 3H_{32}\oplus 3H_{33}$$ by , and using the canonical height computations from [@Scholten Section 6], we have $$\begin{aligned} \mathcal{P}\cdot D&=\frac23 +\frac43\left(-2+\frac92\langle\mathcal{P},\mathcal{P}\rangle-\langle \mathcal{P},3H_{22}\oplus 3H_{23}\oplus 3H_{32}\oplus 3H_{33}\rangle \right)\\ &=-2+\frac43\begin{pmatrix} p & q & r & s \end{pmatrix}\begin{pmatrix} 4/3 & 2/3 & 0 & 0\\ 2/3 & 4/3 & 0 & 0\\ 0 & 0 & 4/3 & 2/3\\ 0 & 0 & 2/3 & 4/3 \end{pmatrix}\left(\frac92 \begin{pmatrix} p \\ q \\ r \\ s \end{pmatrix}-\begin{pmatrix} 3 \\ 3 \\ 3 \\ 3 \end{pmatrix}\right).\qedhere \end{aligned}$$ ◻ **Corollary 7**. *Suppose $E_a,E_b$ are not isomorphic to each other, and neither has $j$-invariant $0$. For any positive integer $n$, there exists a section $\mathcal{P}$ of Inose's pencil $K\to\mathbb{P}^1$ satisfying $\mathcal{P}\cdot D=8n-2$.* *Proof.* It suffices to show that for any positive integer $n$, there exists $p,q,r,s\in\mathbb{Z}\xspace$ satisfying $$\label{pqrsn} p(p-1)+q(q-1)+r(r-1)+s(s-1)+pq+rs=n.$$ To this end, for $n\in\mathbb{Z}\xspace$ let $b(n)$ denote the number of quadruples $(p,q,r,s)\in\mathbb{Z}\xspace^4$ satisfying ; we must show $b(n)>0$ for all $n>0$. Let $A_2\subseteq\mathbb{R}^2$ denote the hexagonal lattice spanned by $(1,0)$ and $(\frac12,\frac{\sqrt{3}}2)$. The point $u=(\frac12, \frac{1}{2\sqrt{3}})$ is a "deep hole" of the lattice (a point maximally distant from every point of the lattice), and for all $p,q\in\mathbb{Z}\xspace$ we have $$\|p(1,0)+q(\tfrac12,\tfrac{\sqrt{3}}2)-(\tfrac12, \tfrac{1}{2\sqrt{3}})\|^2=p(p-1)+q(q-1)+pq+\frac13.$$ Thus we can define a shifted theta function $$\begin{aligned} \theta_{A_2-u}(t)=\sum_{p,q\in\mathbb{Z}\xspace}t^{p(p-1)+q(q-1)+pq+\frac13}, \end{aligned}$$ so that $$\begin{aligned} \theta_{A_2-u}(t)^2&=\sum_{p,q,r,s\in\mathbb{Z}\xspace}t^{p(p-1)+q(q-1)+r(r-1)+s(s-1)+pq+rs+\frac23}\\ &=t^{\frac23}\sum_{n\in\mathbb{Z}\xspace}b(n)t^{n}. \end{aligned}$$ On the other hand, the coefficient of $t^{n+1/3}$ in $\theta_{A_2-u}(t)$ is given by $3(d_{1,3}(3n+1)-d_{2,3}(3n+1))$, where $d_{i,3}(n)$ denotes the number of divisors of $n$ that are $i$ modulo $3$ (by the equations involving $c(q)$ on pages 696 and 697 of [@borweinborwein]). Thus, by comparing equation (1.4), equation (1.1), equation (1.2), and Theorem 2.1 of [@wang2016], we can conclude that $b(n)=3\sigma_1(3n+2)$ for all $n\geq 0$, where $\sigma_1(n)$ denotes the sum of positive integer divisors of $n$. We conclude $b(n)>0$. ◻ ## Computing the genus of $H$ {#genusH} Let $\phi:\mathbb{P}^1\to K$ be birational onto its image, and recall from that we can produce a curve $C\to E_a\times E_b$ by pullback: $$\begin{tikzcd} C\ar{rr}\ar{d} & & E_a\times E_b\arrow["\pi_{-1}"]{d} \\ \mathbb{P}^1\ar["\phi"]{r} & K\ar{r} & (E_a\times E_b)/\langle -1\rangle. \end{tikzcd}$$ Our goal is to compute the genus of the normalization $H\to C$, and we do this by counting Weierstrass points. Ramification points of the map $C\to\mathbb{P}^1$ do not necessarily correspond to Weierstrass points of the normalization; in particular, a singular ramification point may have two distinct preimages that are sent to each other by the hyperelliptic involution. However, any ramification point in the smooth locus of $C$ will lift to a Weierstrass point on $H$. **Lemma 7**. *Let $\phi$ and $C$ be as above, and $x\in \mathbb{P}^1(\overline{k}\xspace)$. If $\mathop{\mathrm{im}}\phi$ intersects $D$ transversely at $\phi(x)$, then $x$ has a unique smooth preimage in $C$.* *Proof.* Since $\phi(x)\in D$, $x$ maps to a singularity of $(E_a\times E_b)/\langle -1\rangle$, which is the image of a unique $2$-torsion point of $E_a\times E_b$. So the pullback of $x$ in $C$ is a single point $y$. Now consider a tangent vector to $\mathbb{P}^1$ at $x$. Since $\phi$ is a section of a fibration, it acts injectively on tangent spaces, so $\mathop{\mathrm{im}}\phi$ has a well-defined tangent in $K$ at $\phi(x)$. Since $\mathop{\mathrm{im}}\phi$ intersects $D$ transversely, its image in $(E_a\times E_b)/\langle -1\rangle$ also has a well-defined tangent vector. Since pullback along $\pi_{-1}$ corresponds to adjoining a square root to the function field, there is a well-defined uniformizer at $y\in C(\overline{k}\xspace)$ corresponding to the square root of the uniformizer at $x$; this proves $y$ is smooth. ◻ As mentioned in the outline in the beginning of , for a non-torsion section $\mathcal{P}$ we want to give a lower bound on the number of transverse intersections between $\mathcal{P}$ and $D$. Notice that this number is bounded below by the number of transverse intersections between $\mathcal{P}$ and the identity section $H_{11}$. **Lemma 8**. *Let $\mathcal{P}$ be a non-torsion section of the elliptic fibration $K\to \mathbb{P}^1$ defined in . The number of transverse intersections between $\mathcal{P}$ and the identity section $H_{11}$ is greater than or equal to $\frac12\langle \mathcal{P},\mathcal{P}\rangle-18$.* *Proof.* For the proof of this lemma we use results from [@ulmerurzua]. Let $m_t(\mathcal{P},H_{11})$ denote the local intersection multiplicity between $\mathcal{P}$ and $H_{11}$ at $t$, so that $$\mathcal{P}\cdot H_{11}=\sum_{t\in \overline{k}\xspace}m_t(\mathcal{P},H_{11}).$$ Let $I(\mathcal{P},t)$ denote the intersection multiplicity of $\mathcal{P}$ with the *Betti foliation* of $K$, as defined in [@ulmerurzua Section 4.1]. The main property we need to know about the Betti foliation is that $H_{11}$ is a leaf of this foliation, and so $I(\mathcal{P},t)\geq m_t(\mathcal{P},H_{11})$. Consequently, we have $$\begin{aligned} \mathcal{P}\cdot H_{11}-|\{t\in\overline{k}\xspace:m_t(\mathcal{P},H_{11})=1\}|&=\sum_{\substack{t\in \overline{k}\xspace\\ m_t(\mathcal{P},H_{11})\geq 2}}m_t(\mathcal{P},H_{11})\\ &\leq 2\sum_{m_t(\mathcal{P},H_{11})\geq 2}(m_t(\mathcal{P},H_{11})-1)\\ &\leq 2\sum_{I(\mathcal{P},t)\geq 2}(I(\mathcal{P},t)-1)\\ &\leq 2\delta+2\sum_{t\in\overline{k}\xspace}(I(\mathcal{P},t)-1), \end{aligned}$$ where $\delta$ is the number of singular fibers of $K\to\mathbb{P}^1$; these are the only fibers for which it is possible to have $I(\mathcal{P},t)=0$. By [@ulmerurzua Theorem 7.1], we have $$\sum_{t\in\overline{k}\xspace}(I(\mathcal{P},t)-1)\leq -2$$ (note that $d\geq 0$ because $\omega^{\otimes 12}$ has a section, see [@ulmerurzua Page 1]). By we have $$\begin{aligned} \mathcal{P}\cdot H_{11}&=\frac12\langle \mathcal{P},\mathcal{P}\rangle -2+\frac12\text{contr}_0(\mathcal{P},\mathcal{P})+\frac12\text{contr}_0(\mathcal{P},\mathcal{P})\\ &\geq \frac12\langle \mathcal{P},\mathcal{P}\rangle-2 \end{aligned}$$ since the $\text{contr}_v$ terms are non-negative. Finally, we have $\delta\leq 10$ by [@shioda07 Proposition 5.1], so combining the above results we obtain $$\begin{aligned} |\{t\in\overline{k}\xspace:m_t(\mathcal{P},H_{11})=1\}|&\geq \mathcal{P}\cdot H_{11}-2\delta-2\sum_{t\in\overline{k}\xspace}(I(\mathcal{P},t)-1)\\ &\geq \frac12\langle \mathcal{P},\mathcal{P}\rangle-2-20+4 \end{aligned}$$ as desired. ◻ We now recall the statement of . **Proposition 2**. *Let $E_0,E_0'/\overline{k}\xspace$ be elliptic curves and $n$ a positive integer. There exist infinitely many isomorphism classes of pairs of elliptic curves $E,E'/\overline{k}\xspace$ satisfying the following:* 1. *there is an isogeny $\psi:E\times E'\to E_0\times E_0'$;* 2. *there is a compatible map $f:H\to E\times E'$ from a genus $g$ hyperelliptic curve $H$ with $\frac16n-10\leq g\leq 4n-2$;* 3. *the degree of $\psi$ is relatively prime to $g-1$;* 4. *the degrees of the projection maps $H\to E$ and $H\to E'$ are equal to $2n$.* *Proof.* We first show that there are infinitely many distinct isomorphism classes of curves $E$ with an isogeny to $E_0$ of degree coprime to $N:=\text{lcm}(g-1:\frac16n-10\leq g\leq 4n-2)$, following an argument from [@petelecture]. If $\mathop{\mathrm{End}}E_0\simeq \mathbb{Z}\xspace$, let $\mathbf{P}$ denote the set of all prime numbers coprime to $N$; if instead $\mathop{\mathrm{End}}E_0$ is isomorphic to an order in an imaginary quadratic field $K/\overline{\mathbb{Q}}\xspace$, let $\mathbf{P}$ denote the set of all prime numbers coprime to $N$ that are inert in $K/\overline{\mathbb{Q}}\xspace$. In either case $\mathbf{P}$ is infinite, and given distinct $\ell,\ell'\in \mathbf{P}$, there is no element of norm $\ell\ell'$ in $\mathop{\mathrm{End}}E$. Now for each $\ell\in \mathcal{P}$, let $h_\ell:E_0\to E^{(\ell)}$ be a degree $\ell$ isogeny. If $\ell\neq \ell'$ but there exists an isomorphism $\gamma:E^{(\ell)}\to E^{(\ell')}$, then $h_{\ell'}^\vee\circ \gamma\circ h_\ell$ would be a degree $\ell\ell'$ endomorphism of $E_0$, a contradiction; hence the curves $E^{(\ell)}$ for $\ell\in\mathbf{P}$ are pairwise non-isomorphic, and are isogenous to $E_0$ by an isogeny of degree relatively prime to $g-1$ for all $\frac16n-10\leq g\leq 4n-2$. A similar argument holds for $E_0'$. Now there are infinitely many isomorphism classes of curves $E$ with an isogeny $E\to E_0$ of degree coprime to $N$ and nonzero $j$-invariant, and for each such $E$, infinitely many isomorphism classes of curves $E'$ with an isogeny $E'\to E_0'$ of degree coprime to $N$, $j$-invariant nonzero, and $E'\not\simeq E$. For each such pair $(E,E')$, shows that there exists a section $\mathcal{P}$ in the Kummer surface of $E\times E'$ satisfying $\mathcal{P}\cdot D=8n-2$. In particular, there are at most $8n-2$ points of $\mathbb{P}^1$ that pull back to Weierstrass points on $H$. On the other hand, the number of transverse intersections between $\mathcal{P}$ and $D$ is at least the number of transverse intersections between $\mathcal{P}$ and $H_{11}$. Since $\mathcal{P}$ is non-torsion (for instance, because the calculation in the proof of shows that $\langle \mathcal{P},\mathcal{P}\rangle\neq 0$), we can apply to show that the number of transverse intersections between $\mathcal{P}$ and $H_{11}$ is at least $$\begin{aligned} \label{eq:12PP} \frac12\langle\mathcal{P},\mathcal{P}\rangle-18&=\frac23(p^2+pq+q^2+r^2+rs+s^2)-18\\ &\geq\frac13(p(p-1)+q(q-1)+r(r-1)+s(s-1)+pq+rs)-18. \end{aligned}$$ The last inequality follows from the fact that $p^2+pq+q^2\geq -p-q$ for all $p,q\in\mathbb{Z}\xspace$ (which can be checked by noting that $p^2+pq+q^2+p+q$ takes on a minimum value of $-\frac13$ for $p,q\in\mathbb{R}$) and a similar inequality for $r,s$. In conclusion, the number of transverse intersections between $\mathcal{P}$ and $D$ is greater than or equal to $\frac13n-18$. By , each transverse intersection between $\mathcal{P}$ and $D$ lifts to a smooth ramification point in $C$, and therefore corresponds to a Weierstrass point of $H$. Thus the number of Weierstrass points of $H$ is at least $\frac13n-18$ and at most $8n-2$. The number of Weierstrass points also equals $2g+2$, where $g$ is the genus of $H$. Thus the genus of $H$ satisfies $$\frac16n-10\leq g\leq 4n-2$$ as desired. Finally, as in the proof of , we observe that $D_i=2F_i+\sum_j H_{ij}$ is a fiber of the map $K\to\mathbb{P}^1$ given by $(x_1,x_2,r)\mapsto x_1$, and $$\begin{aligned} \mathcal{P}\cdot D_1&=\frac13\mathcal{P}\cdot (D_1+D_2+D_3)\\ &=\frac23+\frac13\sum_{1\leq i,j\leq 3} \mathcal{P}\cdot H_{ij}\\ &=\frac23+\frac13\left(\frac92\langle \mathcal{P},\mathcal{P}\rangle-2-\sum_{1\leq i,j\leq 3}\langle\mathcal{P},H_{ij}\rangle\right)\\ &=\frac23+\frac13\left(\frac92\langle \mathcal{P},\mathcal{P}\rangle-2-\sum_{1\leq i,j\leq 3}\langle\mathcal{P},H_{ij}\rangle\right)\\ &=2n. \end{aligned}$$ Pulling back to $E_a\times E_b$, we find that the intersection multiplicity of $C$ with a fiber of the map to $\mathbb{P}^1$ given by taking the $x_1$ coordinate is equal to $4n$. Thus the intersection multiplicity with a fiber of the projection map to $E_1$ is equal to $2n$, and this computes the degree of the projection to $E_1$. Likewise, the degree of the projection to $E_2$ is $2n$. ◻ *Remark 7*. While the proof above shows that we can take $\frac16n-10\leq g\leq 4n-2$, it is possible to obtain much tighter bounds on $g$ (for instance by bounding tangencies with the sections $H_{ij}$ for other $1\leq i,j\leq 3$). In fact, we predict that remains true if we replace the inequalities in (b) with the equality $g=4n-2$ (and hence that holds for the set of all positive integers $g\equiv 2\pmod 4$). Tangencies between $\mathcal{P}$ and $D$ should be very unusual (they are an "unexpected intersection;" see Remark 1.2 and Theorem 1.7 of [@ulmerurzua-transversality]) and so heuristically, for almost all choices of $E_a$ and $E_b$, *all* intersections of $\mathcal{P}$ with $D$ should be transverse. This heuristic is borne out by evidence: we computed the curves $H$ for many choices of curves $E_a,E_b$ and coefficients $p,q,r,s$, and in every case the genus of $H$ was $4n-2$. *Remark 8*. The assumption $\overline{k}\xspace\hookrightarrow\mathbb{C}\xspace$ is only used in : the computations in [@ulmerurzua-transversality] use analytic methods that only hold for varieties defined over the complex numbers. If a similar bound holds more generally (as we expect, cf. ), then holds over any algebraically closed field of characteristic $0$. If we further require the elliptic curves in and to be ordinary, then holds over any algebraically closed field of characteristic not equal to $2$ or $3$. Using we are left with a compatible map $H\to E\times E'$, an isogeny $E\times E'\to E_0\times E_0'$, and a homomorphism $E_0\times E_0'\to A$ with finite kernel. In order to study the composition of these maps we need some more general algebraic geometry tools. ## Maps birational onto their image and étale maps {#sec:birat_etale} The results of this section hold over any field $k$. All morphisms are separated and finite type, and all schemes are separated and finite type over $k$. A variety is a reduced and irreducible scheme, and a curve is a $1$-dimensional variety. Note that arbitrary fiber products of schemes are well-defined under these restrictions because separated and finite type morphisms are stable under base change and composition. **Lemma 9**. *Let $f:V\to W$ and $g:W\to Z$ be maps of reduced schemes, where $V$ and $W$ have the same dimension. If $g\circ f$ and $g$ are birational onto their images, then $f$ is birational onto its image.* *Proof.* Let $U\subseteq V$ be a dense open that maps isomorphically onto the open $(g\circ f)(U)\subseteq (g\circ f)(V)$, and $U'\subseteq W$ be a dense open that maps isomorphically onto the open $g(U')\subseteq g(W)$. Since $$\dim W\geq \dim f(U)\geq \dim (g\circ f)(U)=\dim U=\dim V=\dim W,$$ we can conclude that $\overline{f(U)}=\overline{f(V)}$ is a union of irreducible components of $W$. Therefore $U'\cap f(U)$ is a dense open in $f(U)$, so $g(U')\cap (g\circ f)(U)$ is dense in $(g\circ f)(U)$. Using the fact that $(g\circ f)|_U$ is an isomorphism, we can conclude that $f^{-1}(U')\cap U$ is dense in $U$, and therefore also dense in $V$. Considering the maps $$f^{-1}(U')\cap U\xrightarrow{f} U'\cap f(U)\xrightarrow{g} g(U')\cap (g\circ f)(U),$$ the composition is an isomorphism and the second map is an isomorphism, so we can conclude the first map is an isomorphism as well. ◻ The following lemma says that in an appropriate sense, étale maps commute with maps that are birational onto their images. **Lemma 10**. *Let $C$ be a smooth projective curve over $k$, and $V,W$ smooth varieties over $k$. Suppose $f:C\to V$ maps $C$ birationally onto its image, and $\phi:V\to W$ is étale. Then there is a diagram $$\begin{tikzcd} C\arrow[r,"f"]\arrow[d,"\psi"] & V\arrow[d,"\phi"]\\ D\arrow[r,"h"] & W. \end{tikzcd}$$ where $D$ is a smooth projective curve over $k$, $\psi$ is étale, and $h$ maps $D$ birationally onto its image. If additionally $\phi$ is a Galois cover, then $\deg\psi\mid\deg\phi$.* *Proof.* Let $D'\to W$ denote the scheme-theoretic image of $\phi\circ f$, and $D\to D'$ the normalization. The map $h:D\to D'\to W$ is a normalization map followed by a closed immersion, and is therefore birational onto its image in $W$. Since $C$ is smooth, the map $C\to D'$ factors through $D$. By taking fiber products we obtain a commutative diagram $$\begin{tikzcd} C\arrow[dr]\arrow[drr]\arrow[drrr,"f"]\arrow[ddr,"\psi"] & & & \\ & D_\phi \arrow[r]\arrow[d,"et"] & D'_\phi\arrow[r,hookrightarrow]\arrow[d,"et"] & V\arrow[d,"et"]\\ & D\arrow[r] & D'\arrow[r, hookrightarrow] & W \end{tikzcd}$$ where $D_\phi:=D\times_W V$ and $D_\phi':=D'\times_W V$, the hooked arrows are closed immersions, and the arrows labelled "et" are étale. As étale covers of reduced schemes, $D'_\phi$ and $D_\phi$ are reduced, and since normalization commutes with smooth base change, $D_\phi\to D'_\phi$ is a birational morphism. Since the composition $C\to D_\phi\to V$ is birational onto its image, and $D_\phi\to V$ is birational onto its image, we can conclude that $C\to D_\phi$ is birational onto its image by . Since $D_\phi$ is smooth, and a birational morphism between smooth projective irreducible curves is an isomorphism, $C\to D_\phi$ maps $C$ isomorphically onto a connected component of $D_\phi$. We can therefore conclude that $\psi:C\to D$ is étale. Finally, assume $\phi$ is a Galois cover, with Galois group $G$ (so that $|G|=\deg\phi$). The action of $G$ on $V$ induces an action on $D_\phi$ which restricts to a transitive action on the connected components of $D_\phi$. By the orbit-stabilizer theorem, the stabilizer of the image of $C$ in $D_\phi$ has order dividing $|G|$, and this stabilizer is the Galois group of the étale cover $\psi:C\to D$. ◻ **Corollary 8**. *Let $H$ be a hyperelliptic curve over $k$, $\phi:A\to B$ an isogeny of abelian varieties over $k$, and $f:H\to A$ a compatible map. Let $H'$ be the normalization of the image of $\phi\circ f:H\to B$. Then $H'$ is a hyperelliptic curve, $H'\to B$ is compatible, and there is an étale map $H\to H'$ with degree dividing $\gcd(\deg\phi,g(H)-1)$.* *In particular, if $\gcd(\deg\phi,g(H)-1)=1$, then $H\simeq H'$ and $\phi\circ f$ is a compatible map from $H$ to $B$.* *Proof.* By , $\phi\circ f$ factors as $h\circ\psi$, where $\psi:H\to H'$ is étale, and $h:H'\to B$ maps $H'$ birationally onto its image. Since negation on $A$ preserves $f(H)$ and $\phi$ is an isogeny, negation on $B$ preserves $(\phi\circ f)(H)$; this action induces an involution $\iota'$ on $H'$ satisfying $h\circ\iota'=-h$. Since $$h\circ\psi\circ \iota=\phi\circ f\circ \iota=-\phi\circ f=-h\circ \psi=h\circ \iota'\circ\psi,$$ and $h$ is birational onto its image, we can conclude that $\psi\circ \iota=\iota'\circ \psi$. Thus $\psi$ induces a surjective map $H/\iota\to H'/\iota'$. Since $H/\iota\simeq\mathbb{P}^1$, we must have $H'/\iota'\simeq\mathbb{P}^1$ as well, so that $H'$ is hyperelliptic. Since $h$ is birational onto its image and $h\circ\iota'=-h$, $h$ is a compatible map. By Riemann-Hurwitz applied to the étale cover $\psi$, we have $$2g(H)-2=(\deg\psi)(2g(H')-2).$$ In particular, this implies $\deg\psi$ divides $g(H)-1$. We also have $\deg\psi\mid \deg\phi$, and therefore $\deg\psi\mid\gcd(\deg\phi,g(H)-1)$. If $\gcd(\deg\phi,g(H)-1)=1$ then $\deg\psi=1$, so $\psi$ is an isomorphism and $\phi\circ f=g\circ\psi$ maps $H$ birationally onto its image into $B$. ◻ *Remark 9*. A special case of is when $g(H)=2$, in which case we don't need to make any assumptions on $\deg\phi$: curves of genus $2$ do not occur as nontrivial étale covers, even if the curves in question are allowed to be singular. By contrast, genus $1$ curves can occur as étale covers of other genus $1$ curves via isogenies of arbitrary degree, and a curve of genus $g\geq 3$ can be a degree $\displaystyle\frac{g-1}{g'-1}$ cover of a genus $g'$ curve whenever $g'\geq 2$ and $(g'-1)\mid (g-1)$. ## Proof of {#infcurves_proof} Let $A/\overline{k}\xspace$ be an abelian variety, and $E_0,E_0'/\overline{k}\xspace$ elliptic curves with a homomorphism $E_0\times E_0'\to A$ having finite kernel. Replacing $A$ with the image of this homomorphism if needed we may assume that the map $\tau:E_0\times E_0'\to A$ is an isogeny. Now fix a positive integer $n$. By , we can exhibit an infinite set $\mathcal{S}$ consisting of quintuples $(E,E',H,\psi,f)$ as described in the proposition: that is, we have $$H\xrightarrow{f}E\times E'\xrightarrow{\psi}E_0\times E_0',$$ and no two quintuples have the same isomorphism class for both $E$ and $E'$. We have degree $2n$ maps $H\to E$ and $H\to E'$; since a given curve $H$ has only finitely many elliptic subcovers of bounded degree [@tamme1972 Satz 1'], each isomorphism class $H$ can occur in only finitely many of these quintuples. Therefore there must be an infinite set of quintuples with pairwise non-isomorphic $H$. For any such quintuple we have $\deg f$ coprime to $g-1$, so the composition $$\psi\circ f:H\to E_0\times E_0'$$ is a compatible map by . Applying again, we can conclude that the composition $$\tau\circ \psi\circ f:H\to A$$ can be written as an étale map $H\to H'$ of degree dividing $m:=\deg\tau$ followed by a compatible map $H'\to A$. By Riemann-Hurwitz the genus of $H'$ is at most $4n-2$ and at least $\frac1m(\frac16n-10)$. Since there are infinitely many curves $H$, and each curve $H'$ has only finitely many étale covers of degree dividing $m$, there must be infinitely many isomorphism classes of curves $H'$. In conclusion, we have identified an infinite collection of isomorphism classes of hyperelliptic curves $H'$ with genus in the interval $[\frac1m(\frac16n-10),4n-2]$ that have compatible maps to $A$. Thus there exists an integer $g$ in this interval for which infinitely many non-isomorphic curves of genus $g$ have compatible maps to $A$. Since $m$ is fixed but $n$ is arbitrary, follows. ## Explicit Examples {#example:explicit} Pick Legendre curves $E_a,E_b$ as in , with $$E_a:y_1^2=x_1(x_1-1)(x_1-a),\qquad E_b:y_2^2=x_2(x_2-1)(x_2-a).$$ If we have a section $\mathbb{P}^1\to K$ given by $r\mapsto (h_1(r),h_2(r),r)$ for some rational functions $h_1,h_2\in\overline{k}\xspace(r)$, then the curve $$C:y^2=h_1(x)(h_1(x)-1)(h_1(x)-a)$$ fits into a commutative diagram $$\begin{tikzcd} C\arrow["(x{,}y)\mapsto (h_1(x){,}y{,}h_2(x){,}xy)"]{rrrr}\arrow["(x{,}y)\mapsto x"]{d} & & & & E_a\times E_b\arrow["(x_1{,}y_1{,}x_2{,}y_2)\mapsto \left(x_1{,}y_1{,}\frac{y_2}{y_1}\right)"]{d}\\ \mathbb{P}^1\arrow["r\mapsto (h_1(r){,}h_2(r){,}r)"]{rrrr} & & & & K, \end{tikzcd}$$ and the normalization of $C$ will be a hyperelliptic curve with a compatible map to $E_a\times E_b$. The sections $\mathbb{P}^1\to K$ we consider will be of the form $$pH_{22}+qH_{33}+rH_{23}+sH_{32}$$ as in . ### Genus $2$ construction {#sec:explicitgen2} ($p=q=0$ and $r=s=1$.) Using Mordell-Weil addition we obtain $$\mathcal{P}=(1,b,r)+(a,1,r)=\left(\frac{(a-b)(b-1)^2}{(a-1)^3r^2-(b-1)^3},\;\frac{(a-b)(a-1)^2r^2}{(a-1)^3r^2-(b-1)^3},r\right).$$ By the discussion above, we obtain the curve $$\begin{aligned} C:y^2&=\left((a-1)^3r^2-(b-1)^3\right)\left((a-b)(b-1)^2-(a-1)^3r^2+(b-1)^3\right)\\ &\qquad\left((a-b)(b-1)^2-a(a-1)^3r^2+a(b-1)^3)\right)\end{aligned}$$ mapping to $E_a\times E_b$. A change of variables takes this to the curve $$H:y^2=\left(x^2-\frac{b-1}{a-1}\right)(x^2-1)\left(x^2-\frac{b}{a}\right),$$ which defines a smooth genus $2$ curve for all $a,b\in\overline{k}\xspace-\{0,1\}$ with $a\neq b$. This is the curve computed by Scholten in [@Scholten Section 7], up to a change of variables. ### Genus $6$ {#sec:explicitgen6} ($p=2$ and $q=r=s=0$.) In this case we have $$\mathcal{P}=2(1,1,r)=(h_1(r),h_2(r),r)$$ with $$h_1(r)=\frac{a(a-1)^3 r^4+(a-1)^2(b-1)(ab-a+b)r^2+(a-1)b(b-1)^3}{(a-1)^3r^4-(ab-a-b-2)(a-1)^2(b-1)^2r^2+(b-1)^3}$$ and $h_2(r)$ some other rational function. As described above, the curve $C:y^2=h_1(x)(h_1(x)-1)(h_1(x)-a)$ maps to $E_a\times E_b$; after a change of variables, we obtain $$\begin{aligned} y^2&=\left(x^2-\frac{b(1 - b)}{a}\right)\left(x^2+(b-1)^2\right)\left(x^2+\frac{b-1}{a-1}\right)\\ &\qquad\left((a - 1) x^4 - (a b - a - b - 2) (a - 1) (b - 1)^2 x^2 + (b - 1)^3\right)\\ &\qquad\left(x^2+\frac{(b-1)^2(ab-b-1)}{a-1}\right)\left(x^2+\frac{(b-1)^2}{(a-1)(ab-a-1)}\right).\end{aligned}$$ For almost all values of $a,b$, the roots of this polynomial are distinct, so we have a genus $6$ hyperelliptic curve mapping birationally into $E_a\times E_b$. # Higher dimensions {#abvars} In this section we explore to what extent the results of can be extended to higher dimensional abelian varieties. We saw that for an abelian surface $A$ over $\overline{\mathbb{Q}}\xspace$ Beilinson's conjecture is equivalent to showing $z_{a,a}=0$ for all $a\in A$ (see ). Our goal is to obtain a result of similar flavor for an abelian variety $A/\overline{\mathbb{Q}}\xspace$ of dimension $d$. For that we need to recall certain facts about motivic filtrations of the group $\mathop{\mathrm{CH}}_0(A)$. ## The Pontryagin Filtration Let $A$ be an abelian variety of dimension $d$ over an algebraically closed field $\overline{k}\xspace$. The group law of $A$ makes $\mathop{\mathrm{CH}}_0(A)$ into a group ring by defining for closed points $a,b\in A$ the Pontryagin product $$[a]\odot[b]:=[a+b].$$ We denote by $G^1(A)$ the augmentation ideal of $\mathop{\mathrm{CH}}_0(A)$ and for $r\geq 1$, $G^r(A):=(G_1(A))^r$ its $r$-th power. We also set $G^0(A)=\mathop{\mathrm{CH}}_0(A)$. A straightforward computation shows that $G^1(A)$ is precisely the subgroup $F^1(A)$ of zero-cycles of degree $0$ and $G^2(A)$ coincides with the kernel $F^2(A)$ of the Albanese map. The filtration $\{G^r(A)\}_{r\geq 0}$ is known as the *Pontryagin filtration* of $\mathop{\mathrm{CH}}_0(A)$ and it has been previously studied by Beauville and Bloch ([@Beauville1983; @Beauville1986; @Bloch1976]). These groups have the following explicit generators. $$\begin{aligned} &&G^{1}(A)=\langle[a]-[0]:a\in A\rangle=\ker(\deg),\\ &&G^{2}(A)=\langle[a+b]-[a]-[b]+[0]:a,\;b\in A\rangle,\\ &&G^{3}(A)=\langle[a+b+c]-[a+b]-[b+c]-[a+c]+[a]+[b]+[c]-[0]:a,\;b,;c\in A\rangle,\\ &&...\\ &&G^{r}(A)=\left\langle\sum_{j=0}^{r}(-1)^{r-j}\sum_{1\leq\nu_{1}<\dots<\nu_{j}\leq r}[a_{\nu_{1}}+\dots+a_{\nu_{j}}]:a_{1},\dots,a_{r}\in A\right\rangle. \end{aligned}$$ **Notation 3**. We will denote by $z_{a_1,\ldots,a_r}$ the generator of $G^r(A)$ corresponding to points $a_1,\ldots,a_r\in A$. Notice that $z_{a_1,\ldots,a_r}=([a_1]-[0])\odot\cdots\odot([a_r]-[0]).$ Beauville and Bloch both showed a vanishing $G^{d+1}(A)=0$. Moreover, it follows by the main theorem of [@Beauville1986] and its proof that we have a decomposition rationally $$\mathop{\mathrm{CH}}_0(A)\otimes\mathbb{Q}\xspace\simeq \bigoplus_{r=0}^d\frac{G^r(A)\otimes\mathbb{Q}\xspace}{G^{r+1}(A)\otimes\mathbb{Q}\xspace}.$$ ## The Gazaki filtration {#gazakifil} The first author described in [@Gazaki2015] the graded quotients $\displaystyle\frac{G^r(A)\otimes\mathbb{Q}\xspace}{G^{r+1}(A)\otimes\mathbb{Q}\xspace}$ as symmetric products of the abelian variety $A$. Namely, she defined a second integral filtration $\{F^r(A)\}_{r\geq 0}$ of $\mathop{\mathrm{CH}}_0(A)$ with the following properties. 1. $G^r(A)\subseteq F^r(A)$ for every $r\geq 0$ (see [@Gazaki2015 Proposition 3.3]). 2. $\displaystyle G^r(A)\otimes\mathbb{Z}\xspace\left[\frac{1}{(r-1)!}\right]=F^r(A)\otimes\mathbb{Z}\xspace\left[\frac{1}{(r-1)!}\right]$ for every $r\geq 1$ (see [@Gazaki2015 Proposition 4.1]). In particular, $F^1(A)=G^1(A)$ and $F^2(A)=G^2(A)$ are the usual subgroups. 3. For every $r\geq 0$ there is a homomorphism $$\Psi_r: \overbrace{A(\overline{k}\xspace)\otimes\cdots\otimes A(\overline{k}\xspace)}^r\to F^r(A)/F^{r+1}(A), \;\;a_1\otimes\cdots\otimes a_r\mapsto z_{a_1,\ldots,a_r}$$ (see [@Gazaki2015 Theorem 1.3]). The map $\Psi_r$ factors through a certain symmetric Somekawa $K$-group $S_r(\overline{k}\xspace;A)$ which we review below. **Definition 3**. Let $A$ be an abelian variety over $\overline{k}\xspace$. For $r\geq 1$ the Somekawa $K$-group $K_r(\overline{k}\xspace;A):=K(\overline{k}\xspace;\overbrace{A,\ldots,A}^r)$ attached to $r$ copies of $A$ is the quotient $$K_r(\overline{k}\xspace;A)=\frac{\overbrace{A(\overline{k}\xspace)\otimes\cdots\otimes A(\overline{k}\xspace)}^r}{(\textbf{WR})},$$ where $(\textbf{WR})$ is the subgroup generated by the following type of elements. Let $C$ be a smooth projective curve over $\overline{k}\xspace$ that admits regular maps $g_i:C\to A$ for $i=1,\ldots,r$. Let $f\in \overline{k}\xspace(C)^\times$. Then we require $$\label{eq:WR} \sum_{x\in C}\mathop{\mathrm{ord}}_x(f)g_1(x)\otimes\cdots\otimes g_r(x)\in(\textbf{WR}).$$ The symmetric $K$-group $S_r(\overline{k}\xspace;A)$ is the quotient of $K_r(\overline{k}\xspace;A)$ by the action of the symmetric group in $r$ variables. The relation $(\textbf{WR})$ is known as *Weil Reciprocity*. We will denote the generator of $S_r(\overline{k}\xspace;A)$ corresponding to the tensor $a_1\otimes\cdots\otimes a_r$ as a symbol $\{a_1,\ldots,a_r\}$. The following proposition is the analog of for higher dimensions and it is an easy consequence of the above analysis. **Proposition 3**. *Let $A$ be an abelian variety over an algebraically closed field $\overline{k}\xspace$. If the symmetric $K$-group $S_2(\overline{k}\xspace;A)$ is torsion, then $F^2(A)=0$. In particular, if $\{a,a\}$ is torsion for every $a\in A$, then $F^2(A)=0$.* *Proof.* First we claim that if $S_2(\overline{k}\xspace;A)$ is torsion, then $S_r(\overline{k}\xspace;A)$ is torsion for all $r\geq 2$. This follows by a product formula for Somekawa $K$-groups proved by the authors in [@GazakiLove2022 Proposition 3.1]. This proposition gives a homomorphism $$\rho:K_2(\overline{k}\xspace;A)\otimes \overbrace{A(\overline{k}\xspace)\otimes\cdots\otimes A(\overline{k}\xspace)}^{r-2}\rightarrow K_r(\overline{k}\xspace;A)$$ given by concatenation of symbols, $\rho(\{a_1,a_2\}\otimes a_3\otimes\cdots\otimes a_n)=\{a_1,\ldots, a_n\}$. We see that $\rho$ is surjective by construction. The homomorphism $\rho$ clearly induces a surjective homomorphism $$S_2(\overline{k}\xspace;A)\otimes A(\overline{k}\xspace)\otimes\cdots\otimes A(\overline{k}\xspace)\rightarrow S_r(\overline{k}\xspace;A),$$ and hence if $S_2(\overline{k}\xspace;A)$ is torsion, then so is $S_r(\overline{k}\xspace;A)$ for all $r\geq 3$. From now on suppose that $S_r(\overline{k}\xspace;A)$ is torsion for all $r\geq 2$. It follows by [\[gazakifil\]](#gazakifil){reference-type="eqref" reference="gazakifil"} (c) that there is a homomorphism $\Psi_d: S_d(\overline{k}\xspace;A)\to F^d(A)/F^{d+1}(A)$ sending the symbol $\{a_1,\ldots,a_r\}$ to $z_{a_1,\ldots,a_r}$. Moreover, it follows by (b) that $\displaystyle F^{d+1}(A)\otimes\mathbb{Z}\xspace\left[\frac{1}{d!}\right]=0$, since $G^{d+1}(A)=0$. Since $F^{d+1}(A)$ is a subgroup of the Albanese kernel, which is torsion-free, it follows that $F^{d+1}(A)=0$, and hence we have a well-defined homomorphism $\Psi_d:S_d(\overline{k}\xspace;A)\to F^d(A).$ Since $S_d(\overline{k}\xspace;A)$ is torsion and $F^d(A)$ is torsion-free, it follows that $\Psi_d=0$. Notice that by construction the map $\Psi_d$ surjects onto $G^d(A)\subseteq F^d(A)$. Thus, we conclude that $G^d(A)=0$, and by torsion-freeness also $F^d(A)=0$. We now proceed by reverse induction. Having proved that $G^r(A)=F^r(A)=0$ for some $r\geq 3$, we get that $G^{r-1}(A)=F^{r-1}(A)=0$ by considering the homomorphism $$\Psi_{r-1}: S_{r-1}(A)\rightarrow F^{r-1}(A)$$ which surjects onto $G^{r-1}(A)$. The last claim follows by symmetry. The group $S_2(\overline{k}\xspace;A)$ is torsion if an only if $\{a,b\}$ is torsion for all $a,b\in A$. But $2\{a,b\}=\{a,a\}+\{b,b\}-\{a+b,a+b\}$. ◻ The following is the analog of and the proof is along the same lines. **Proposition 4**. *Let $A$ be an abelian variety of dimension $d$ over $\overline{k}\xspace$ and let $a\in A$ be a hyperelliptic point (in the sense of ). Then $\{a,a\}\in S_2(\overline{k}\xspace;A)$ is torsion.* *Proof.* By assumption there exists some nonzero multiple of $a$ that lies in the image of a morphism $\phi:H\to A$ from a hyperelliptic curve such that the involution on $H$ commutes with the negation on $A$. As usual, using bilinearity we may assume that $a=\phi(q)$ is itself in the image of $\phi$. Denote by $\overline{q}$ the image of $q$ under the involution of $H$. Then $[q]+[\overline{q}]-2[w]$ is a principal divisor on $H$, where $w$ is a Weierstrass point of $H$. Let $f\in \overline{k}\xspace(C)^\times$ be the function whose divisor is $[q]+[\overline{q}]-2[w]$. We apply the Weil Reciprocity relation $(\textbf{WR})$ of $S_2(\overline{k}\xspace;A)$ for the following choices: $g_1=g_2=\phi$ and $f\in\overline{k}\xspace(C)^\times$ as above. Given that $\phi(q)=-\phi(\overline{q})$, it follows $$\{a,a\}+\{-a,-a\}=0\Rightarrow 2\{a,a\}=0.$$ ◻ Because of , generalizes verbatim to higher dimensions. **Corollary 9**. *Let $A$ be an abelian variety of dimension $d\geq 2$ over $\overline{\mathbb{Q}}\xspace$. Suppose that the group $A(\overline{\mathbb{Q}}\xspace)$ is the divisible hull of a subgroup $\Gamma$ generated by a sequence $(a_i)_{i\geq 1}$ satisfying the following property. Let $I$ be a collection of integer sequences $\vec{n}=(n_1,n_2,\ldots)$ each with finitely many nonzero terms, such that $$\{\vec{n}\otimes \vec{n}:\vec{n}\in I\}$$ spans $\mathop{\mathrm{Sym}}^2\mathbb{Q}\xspace^\infty$. If $n_1a_1+n_2a_2+\cdots$ is hyperelliptic for all $(n_1,n_2,\ldots)\in I$, then Beilinson's is true for $A$.* *Remark 10*. The reason we chose to state most of our results for abelian surfaces is because it seems unlikely that for a general abelian variety $A/\overline{\mathbb{Q}}\xspace$ of $\dim(A)\geq 3$ relations arising from hyperelliptic curves will be enough to prove Beilinson's conjecture. For, in $\dim\geq 3$ the quotient $A/\langle -1\rangle$ by the negation action is in general not Calabi-Yau, and hence it does not contain rational curves (see [@bogomolov_tschinkel Remark 7.7]), which is our main method to construct hyperelliptic curves mapping to $A$. However, this could still be the case for special classes of abelian varieties. For example, when $J$ is the Jacobian of a hyperelliptic curve of genus $g\geq 2$ Bogomolov and Tschinkel proved the analog of Bogomolov's conjecture for $J/\overline{\mathbb{F}\xspace}_p$ ([@bogomolov_tschinkel Theorem 7.1])). [^1]: We allow hyperelliptic curves to be genus $1$ in order to simplify the proofs of results such as and . In these cases it is important to remember that the hyperelliptic involution and the set of Weierstrass points depend on the choice of map $H\to\mathbb{P}^1$. [^2]: This also follows immediately from Weil reciprocity ( with $r=1$).

arxiv_math

--- abstract: | Given a $d$-dimensional vector space $V \subset \mathbb{C}[u]$ of polynomials, its Wronskian is the polynomial $(u + z_1) \cdots (u + z_n)$ whose zeros $-z_i$ are the points of $\mathbb{C}$ such that $V$ contains a nonzero polynomial with a zero of order at least $d$ at $-z_i$. Equivalently, $V$ is a solution to the Schubert problem defined by osculating planes to the moment curve at $z_1, \dots, z_n$. The *inverse Wronski problem* involves finding all $V$ with a given Wronskian $(u + z_1) \cdots (u + z_n)$. We solve this problem by providing explicit formulas for the Grassmann--Plücker coordinates of the general solution $V$, as commuting operators in the group algebra $\mathbb{C}[\mathfrak{S}_n]$ of the symmetric group. The Plücker coordinates of individual solutions over $\mathbb{C}$ are obtained by restricting to an eigenspace and replacing each operator by its eigenvalue. This generalizes work of Mukhin, Tarasov, and Varchenko (2013) and of Purbhoo (2022), which give formulas in $\mathbb{C}[\mathfrak{S}_n]$ for the differential equation satisfied by $V$. Moreover, if $z_1, \dots, z_n$ are real and nonnegative, then our operators are positive semidefinite, implying that the Plücker coordinates of $V$ are all real and nonnegative. This verifies several outstanding conjectures in real Schubert calculus, including the positivity conjectures of Mukhin and Tarasov (2017) and of Karp (2021), the disconjugacy conjecture of Eremenko (2015), and the divisor form of the secant conjecture of Sottile (2003). The proofs involve the representation theory of $\mathfrak{S}_n$, symmetric functions, and $\tau$-functions of the KP hierarchy. author: - Steven N. Karp and Kevin Purbhoo bibliography: - ref.bib title: Universal Plücker coordinates for the Wronski map and positivity in real Schubert calculus --- # Introduction For a system of real polynomial equations with finitely many solutions, we normally expect that some --- but not all --- of the solutions are real, while the remaining solutions come in complex-conjugate pairs. The precise number of real solutions usually depends in a complicated way on the coefficients of the equations. However, in some rare cases, it is possible to obtain a better understanding of the real solutions. A remarkable example occurs in the Schubert calculus of the Grassmannian $\mathrm{Gr}(d,m)$, for Schubert problems defined by flags osculating a rational normal curve. In 1993, Boris and Michael Shapiro conjectured that all such Schubert problems with real parameters have only real solutions. The corresponding systems of equations arise in various guises throughout mathematics, from algebraic curves [@eisenbud_harris83; @kharlamov_sottile03] to differential equations [@mukhin_varchenko04] to pole-placement problems [@rosenthal_sottile98; @eremenko_gabrielov02a]. The conjecture was eventually proved by Mukhin, Tarasov, and Varchenko [@mukhin_tarasov_varchenko09a], using a reformulation in terms of Wronski maps, and machinery from quantum integrable systems and representation theory. While the details of the Mukhin--Tarasov--Varchenko proof are rather intricate, the basic idea is relatively straightforward. They consider a family of commuting linear operators arising from the Gaudin model, and show that they satisfy algebraic equations defining a Schubert problem. Hence, by considering the spectra of these operators, they are able to infer some basic properties of the solutions to the Schubert problem. In this paper we extend these results, making the connection between the commuting operators and the corresponding solutions more explicit and concrete. Consequently, we obtain stronger results in real algebraic geometry, including several generalizations of the Shapiro--Shapiro conjecture. Namely, we resolve the divisor form of the secant conjecture of Sottile (2003), the disconjugacy conjecture of Eremenko [@eremenko15], and the positivity conjectures of Mukhin--Tarasov (2017) and Karp [@karp]. ## The Wronski map and the Bethe algebra {#sec:wronskiintro} Let $\mathrm{Gr}(d,m)$ denote the Grassmannian of all $d$-dimensional linear subspaces of $\mathbb{C}^m$. It is often more convenient to work with the $m$-dimensional vector space $\mathbb{C}_{m-1}[u]$, of univariate polynomials of degree at most $m-1$, rather than $\mathbb{C}^m$. We explicitly identify $\mathbb{C}^m$ with $\mathbb{C}_{m-1}[u]$, via the isomorphism $$\label{eq:isomorphism} (a_1, \dots, a_m) \leftrightarrow \sum_{j=1}^m a_j \frac{u^{j-1}}{(j-1)!} \,.$$ In particular, we also view $\mathrm{Gr}(d,m)$ as the space of all $d$-dimensional subspaces of $\mathbb{C}_{m-1}[u]$. Now fix a nonnegative integer $n$, and let $\nu$ be a partition of $n$ with at most $d$ parts; that is, $\nu = (\nu_1, \dots, \nu_d)$ is a tuple of nonnegative integers such that $\nu_1 \geq \dots \geq \nu_d \geq 0$, and $|\nu| := \nu_1 + \dots + \nu_d = n$. The ***Schubert cell*** $\mathcal{X}^{\nu}$ is the space of all $d$-dimensional linear subspaces of $\mathbb{C}[u]$ that have a basis $(f_1, \dots, f_d)$, with $\deg(f_i) = \nu_i+d-i$. As a scheme, $\mathcal{X}^{\nu}$ is isomorphic to $n$-dimensional affine space. We take $m \geq d+\nu_1$, so that $\mathcal{X}^{\nu}\subseteq \mathrm{Gr}(d,m)$. Let $\mathcal{P}_n\subseteq \mathbb{C}[u]$ denote the $n$-dimensional affine space of monic polynomials of degree $n$. Given $V \in \mathcal{X}^{\nu}$, choose any basis $(f_1, \dots, f_d)$ for $V$. We define $\mathrm{Wr}(V)$ to be the unique monic polynomial which is a scalar multiple of the Wronskian $\mathrm{Wr}(f_1, \dots, f_d)$. It is not hard to see that $\mathrm{Wr}(V) \in \mathcal{P}_n$ is a polynomial of degree $n$, and is independent of the choice of basis. Thus we obtain a map $\mathrm{Wr}: \mathcal{X}^{\nu}\to \mathcal{P}_n$, called the ***Wronski map*** on $\mathcal{X}^{\nu}$. Abstractly, this is a finite morphism from $n$-dimensional affine space to itself. Suppose $g(u) = (u+z_1)\dotsm (u+z_n) \in \mathcal{P}_n$, where $z_1, \dots, z_n$ are complex numbers. The ***inverse Wronski problem*** is to compute the fibre $\mathrm{Wr}^{-1}(g) \subseteq \mathcal{X}^{\nu}$. In their study of the Gaudin model for $\mathfrak{gl}_n$, Mukhin, Tarasov, and Varchenko [@mukhin_varchenko04; @mukhin_varchenko05; @mukhin_tarasov_varchenko06; @mukhin_tarasov_varchenko09a; @mukhin_tarasov_varchenko09b] discovered a connection between the inverse Wronski problem, and the problem of diagonalizing the Gaudin Hamiltonians [@gaudin76]. We will focus on the version of this story from [@mukhin_tarasov_varchenko13], in which the Gaudin Hamiltonians generate the *Bethe algebra (of Gaudin type)* $\mathcal{B}_n(z_1, \dots, z_n)\subseteq \mathbb{C}[\mathfrak{S}_{n}]$, which is a commutative subalgebra of the group algebra of the symmetric group. Let $M^{\nu}$ be the Specht module (i.e. irreducible $\mathfrak{S}_{n}$-representation) associated to the partition $\nu$. Then $\mathcal{B}_n(z_1, \dots, z_n)$ acts on $M^{\nu}$, and the image of this action defines a commutative subalgebra $\mathcal{B}_\nu(z_1, \dots, z_n)\subseteq \mathrm{End}(M^{\nu})$. The following result is stated more precisely as [Theorem 44](#thm:precise){reference-type="ref" reference="thm:precise"}: **Theorem 1** (Mukhin, Tarasov, and Varchenko [@mukhin_tarasov_varchenko13]). *The eigenspaces $E \subseteq M^{\nu}$ of the algebra $\mathcal{B}_\nu(z_1, \dots, z_n)$ are in one-to-one correspondence with the points $V_E \in \mathrm{Wr}^{-1}(g)$. The eigenvalues of the generators of $\mathcal{B}_\nu(z_1, \dots, z_n)$ are coordinates for $V_E$ in some coordinate system.* (There are also scheme-theoretic analogues of [Theorem 1](#thm:vague){reference-type="ref" reference="thm:vague"}, which we discuss in [1.2.1](#sec:schemeintro){reference-type="ref" reference="sec:schemeintro"}.) Unfortunately, [Theorem 1](#thm:vague){reference-type="ref" reference="thm:vague"}/[Theorem 44](#thm:precise){reference-type="ref" reference="thm:precise"} is poorly suited to studying certain properties of the Wronski map. This is because the generators of $\mathcal{B}_n(z_1, \dots, z_n)$ correspond to a somewhat unusual coordinate system for $\mathcal{X}^{\nu}$. Namely, given $V \in \mathcal{X}^{\nu}$, there is a unique ***fundamental differential operator*** $$D_V = \partial_u^d + \psi_1(u) \partial_u^{d-1} + \dots + \psi_d(u)$$ with coefficients $\psi_j(u) \in \mathbb{C}(u)$, such that $V$ is the space of solutions to the differential equation $D_V f(u) = 0$. The coefficients of $D_V$ can be regarded as a coordinate system on $\mathcal{X}^{\nu}$. In the precise formulation (see [Theorem 44](#thm:precise){reference-type="ref" reference="thm:precise"}), the point $V_E\in\mathrm{Wr}^{-1}(g)$ is computed in these coordinates. In order to express $V_E$ in standard coordinates, we need to solve a differential equation, resulting in highly non-linear formulas. The main result of this paper is [Theorem 3](#thm:main){reference-type="ref" reference="thm:main"} below, which is a new version of [Theorem 1](#thm:vague){reference-type="ref" reference="thm:vague"}. Rather than using the fundamental differential operator coordinates, it computes $V_E\in\mathrm{Wr}^{-1}(g)$ in the ***Plücker coordinates***, which are the $d\times d$ minors of a $d\times m$ matrix whose rows form a basis for $V_E$. We introduce (by explicit formulas) a new set of generators $\beta^\lambda$ for $\mathcal{B}_n(z_1, \dots, z_n)$, which are indexed by partitions $\lambda$. For any eigenspace $E \subseteq M^{\nu}$, the corresponding eigenvalues of the $\beta^\lambda$'s are the Plücker coordinates of $V_E$. There are three major advantages of this formulation. First, we obtain a more direct description of $V_E$ which does not require solving a differential equation; the implicit part of our construction lies entirely in understanding the representation theory of $\mathfrak{S}_{n}$. Second, many natural objects of interest are given by *linear* functions of the Plücker coordinates. For example, we readily obtain explicit bases for $V_E$; the Wronskian and the fundamental differential operator coordinates are given as linear functions of the Plücker coordinates; and Schubert varieties and Schubert intersections are defined by linear equations in the Plücker coordinates. Third, basic properties of the operators $\beta^\lambda$ imply positivity results about the Plücker coordinates of $V_E$. This enables us to resolve several conjectures in real algebraic geometry, as we explain in [1.3](#sec:conjectures){reference-type="ref" reference="sec:conjectures"}. ## Universal Plücker coordinates {#sec:universalintro} We now state our main theorem. For every partition $\lambda$, define $$\label{eq:betadef} \beta^\lambda(t) := \sum_{\substack{X \subseteq [n], \\ |X| = |\lambda|}} \, \sum_{\sigma \in \mathfrak{S}_{X}} \chi^\lambda(\sigma) \sigma \prod_{i \in [n] \setminus X} (z_i + t) \,.$$ Here $[n] = \{1, \dots, n\}$, $\mathfrak{S}_{X} \subseteq \mathfrak{S}_{n}$ is the group of permutations of $X$, and $\chi^\lambda : \mathfrak{S}_{X} \to \mathbb{C}$ is the character of the Specht module $M^{\lambda}$. Note that $\beta^\lambda(t)$ is nonzero if and only if $|\lambda| \leq n$. Set $\beta^\lambda := \beta^\lambda(0)$. **Example 2**. If $\lambda = (1,1)$, then $\chi^\lambda$ is the sign character on $\mathfrak{S}_{2}$. When $n=3$, we get $$\beta^{11} = (\mathbbm{1}_{\mathfrak{S}_{3}} - \sigma_{1,2})z_3 + (\mathbbm{1}_{\mathfrak{S}_{3}} - \sigma_{1,3})z_2 + (\mathbbm{1}_{\mathfrak{S}_{3}} - \sigma_{2,3})z_1\,,$$ where $\mathbbm{1}_{\mathfrak{S}_{3}}$ denotes the identity element of $\mathfrak{S}_{3}$, and $\sigma_{i,j} := (i \;\; j)$ is the transposition swapping $i$ and $j$. **Theorem 3**. *Let $z_1, \dots, z_n \in \mathbb{C}$, and set $g(u) := (u+z_1) \dotsm (u+z_n) \in \mathbb{C}[u]$. The operators $\beta^\lambda(t) \in \mathbb{C}[\mathfrak{S}_{n}]$ satisfy the following algebraic identities:* (i) *[\[main_commutativity\]]{#main_commutativity label="main_commutativity"} Commutativity relations: $$\label{eq:commutativity} \beta^\lambda(s) \beta^\mu(t) = \beta^\mu(t) \beta^\lambda(s) \qquad \text{for all partitions $\lambda$ and $\mu$}\,.$$* (ii) *[\[main_translation\]]{#main_translation label="main_translation"} Translation identity: $$\label{eq:translationidentity} \beta^\mu(s+t) = \sum_{\lambda \supseteq \mu} \frac{\mathsf{f}^{\lambda/\mu}}{|\lambda/\mu|!} t^{|\lambda/\mu|} \beta^\lambda(s) \qquad \text{for all partitions } \mu\,,$$ where $\mathsf{f}^{\lambda/\mu}$ denotes the number of standard Young tableaux of shape $\lambda/\mu$.* (iii) *[\[main_pluckers\]]{#main_pluckers label="main_pluckers"} The quadratic Plücker relations [[([\[eq:prelspartitions\]](#eq:prelspartitions){reference-type="ref" reference="eq:prelspartitions"})]{.upright}](#eq:prelspartitions).* *Furthermore:* (iv) *[\[main_generates\]]{#main_generates label="main_generates"} For every partition $\lambda$ and $t \in \mathbb{C}$, we have $\beta^\lambda(t) \in \mathcal{B}_n(z_1, \dots, z_n)$. The set $\{\beta^\lambda \mid |\lambda| \leq n\}$ generates $\mathcal{B}_n(z_1, \dots, z_n)$ as an algebra.* (v) *[\[main_eigenspace\]]{#main_eigenspace label="main_eigenspace"} If $E \subseteq M^{\nu}$ is any eigenspace of $\mathcal{B}_\nu(z_1, \dots, z_n)$, then the corresponding eigenvalues of the operators $\beta^\lambda$ are the Plücker coordinates of a point $V_E \in \mathcal{X}^{\nu}\subseteq \mathrm{Gr}(d,m)$ such that $\mathrm{Wr}(V_E) = g$. Every point of $\mathrm{Wr}^{-1}(g)$ corresponds in this way to some eigenspace $E \subseteq M^{\nu}$ of $\mathcal{B}_\nu(z_1, \dots, z_n)$.* (vi) *[\[main_multiplicity\]]{#main_multiplicity label="main_multiplicity"} The multiplicity of $V_E$ as a point of $\mathrm{Wr}^{-1}(g)$ is equal to $\dim \widehat{E}$, where $\widehat{E} \subseteq M^{\nu}$ is the generalized eigenspace of $\mathcal{B}_\nu(z_1, \dots, z_n)$ containing $E$.* We note that while the translation identity in part [\[main_translation\]](#main_translation){reference-type="ref" reference="main_translation"} is linear, parts [\[main_commutativity\]](#main_commutativity){reference-type="ref" reference="main_commutativity"} and [\[main_pluckers\]](#main_pluckers){reference-type="ref" reference="main_pluckers"} both involve quadratic expressions in $\mathcal{B}_n(z_1, \dots, z_n)$, making them intractable to prove directly. In both of these cases we proceed by reducing the problem to --- and then proving --- an easier identity, using a diverse set of algebraic tools. For part [\[main_commutativity\]](#main_commutativity){reference-type="ref" reference="main_commutativity"}, we use properties of $\mathcal{B}_n(z_1, \dots, z_n)$ and combinatorial ideas which appeared in [@purbhoo]. For part [\[main_pluckers\]](#main_pluckers){reference-type="ref" reference="main_pluckers"}, we employ the translation identity, properties of the exterior algebra, new combinatorial identities of symmetric functions, and the theory of $\tau$-functions of the KP hierarchy. Once identities [\[main_commutativity\]](#main_commutativity){reference-type="ref" reference="main_commutativity"}--[\[main_pluckers\]](#main_pluckers){reference-type="ref" reference="main_pluckers"} are established, parts [\[main_generates\]](#main_generates){reference-type="ref" reference="main_generates"}--[\[main_multiplicity\]](#main_multiplicity){reference-type="ref" reference="main_multiplicity"} are relatively straightforward consequences. While the proof of [Theorem 3](#thm:main){reference-type="ref" reference="thm:main"} uses some of the same mathematical constructions as the proof of [Theorem 1](#thm:vague){reference-type="ref" reference="thm:vague"}/[Theorem 44](#thm:precise){reference-type="ref" reference="thm:precise"} in [@mukhin_tarasov_varchenko13], it does not use the result itself. In fact, our proof of [Theorem 3](#thm:main){reference-type="ref" reference="thm:main"} also provides a new proof of [Theorem 44](#thm:precise){reference-type="ref" reference="thm:precise"} (see [Remark 68](#rmk:newproof){reference-type="ref" reference="rmk:newproof"}). The two arguments are interconnected, which plays a key role in establishing the Plücker relations. **Example 4**. We illustrate [Theorem 3](#thm:main){reference-type="ref" reference="thm:main"} in the case $n=2$, for the Grassmannian $\mathrm{Gr}(2,4)$. Writing $\mathfrak{S}_{2} = \{\mathbbm{1}_{\mathfrak{S}_{2}}, \sigma_{1,2}\}$, we have $$\beta^0 = \mathbbm{1}_{\mathfrak{S}_{2}} \, z_1z_2\,, \qquad \beta^1 = \mathbbm{1}_{\mathfrak{S}_{2}} (z_1 + z_2)\,, \qquad \beta^2 = \mathbbm{1}_{\mathfrak{S}_{2}} + \sigma_{1,2}\,, \qquad \beta^{11} = \mathbbm{1}_{\mathfrak{S}_{2}} - \sigma_{1,2}\,,$$ and $\beta^\lambda = 0$ for all other partitions $\lambda$. Note that the $\beta^\lambda$'s satisfy the equation $$- \beta^0\beta^{22} + \beta^1\beta^{21} - \beta^{11}\beta^2 = 0\,,$$ which is the first non-trivial Plücker relation [[([\[eq:firstplucker\]](#eq:firstplucker){reference-type="ref" reference="eq:firstplucker"})]{.upright}](#eq:firstplucker). There are two Specht modules for $\mathfrak{S}_{2}$, namely $M^{2}$ and $M^{11}$, which are both $1$-dimensional. In $M^{2}$, both $\mathbbm{1}_{\mathfrak{S}_{2}}$ and $\sigma_{1,2}$ act with eigenvalue $1$, and so $$\label{eq:examplepluckers} \beta^0 \rightsquigarrow z_1z_2\,, \qquad \beta^1 \rightsquigarrow z_1 + z_2\,, \qquad \beta^2 \rightsquigarrow 2\,, \qquad \beta^{1,1} \rightsquigarrow 0\,.$$ These are the Plücker coordinates (under the identification [[([\[eq:identification\]](#eq:identification){reference-type="ref" reference="eq:identification"})]{.upright}](#eq:identification)) of the element $$V = \left\langle 1,\, z_1z_2u + \frac{z_1 + z_2}{2}u^2 + \frac{1}{3}u^3 \right\rangle \in \mathcal{X}^{2} \subseteq \mathrm{Gr}(2,4)\,;$$ see [Example 15](#ex:pluckers){reference-type="ref" reference="ex:pluckers"} for further explanation. On the other hand, in $M^{11}$, the element $\mathbbm{1}_{\mathfrak{S}_{2}}$ acts with eigenvalue $1$ and $\sigma_{1,2}$ acts with eigenvalue $-1$, giving the solution $$V = \left\langle \frac{z_1 + z_2}{2} + u,\, -z_1z_2 + u^2 \right\rangle \in \mathcal{X}^{1,1} \subseteq \mathrm{Gr}(2,4)\,.$$ We can check that both elements of $\mathrm{Gr}(2,4)$ have Wronskian $g(u) = (u+z_1)(u+z_2)$. **Example 5**. We illustrate parts [\[main_commutativity\]](#main_commutativity){reference-type="ref" reference="main_commutativity"} and [\[main_pluckers\]](#main_pluckers){reference-type="ref" reference="main_pluckers"} of [Theorem 3](#thm:main){reference-type="ref" reference="thm:main"} in the case $n=4$. Consider the $2$-dimensional representation $M^{\nu}$ of $\mathfrak{S}_{4}$, $\nu = (2,2)$. Following the conventions used by `Sage` [@sagemath], the simple transpositions $\sigma_{1,2}$ and $\sigma_{3,4}$ both act as $(\begin{smallmatrix}1 & 0 \\ 1 & -1\end{smallmatrix})$, and $\sigma_{2,3}$ acts as $(\begin{smallmatrix}0 & -1 \\ -1 & 0\end{smallmatrix})$. Let $\beta^\lambda_\nu \in \mathrm{End}(M^{\nu})$ denote the operator $\beta^\lambda$ acting on $M^{\nu}$, which we regard as a $2\times 2$ matrix. Then $$\begin{gathered} \beta^0_\nu = z_1z_2z_3z_4\,\scalebox{0.85}{$\begin{pmatrix}1 & 0 \\ 0 & 1\end{pmatrix}$}\,, \qquad \beta^1_\nu = (z_1z_2z_3 + z_1z_2z_4 + z_1z_3z_4 + z_2z_3z_4)\,\scalebox{0.85}{$\begin{pmatrix}1 & 0 \\ 0 & 1\end{pmatrix}$}\,, \\[2pt] \beta^2_\nu = \scalebox{0.85}{$\begin{pmatrix} 2z_1z_2 + z_1z_4 + z_2z_3 + 2z_3z_4 & z_1z_3 - z_1z_4 - z_2z_3 + z_2z_4 \\ z_1z_2 - z_1z_4 - z_2z_3 + z_3z_4 & 2z_1z_3 + z_1z_4 + z_2z_3 + 2z_2z_4 \end{pmatrix}$}\,, \\[2pt] \beta^{11}_\nu = \scalebox{0.85}{$\begin{pmatrix} 2z_1z_3 + z_1z_4 + z_2z_3 + 2z_2z_4 & -z_1z_3 + z_1z_4 + z_2z_3 - z_2z_4 \\ -z_1z_2 + z_1z_4 + z_2z_3 - z_3z_4 & 2z_1z_2 + z_1z_4 + z_2z_3 + 2z_3z_4 \end{pmatrix}$}\,, \\[2pt] \beta^{21}_\nu = 3(z_1 + z_2 + z_3 + z_4)\,\scalebox{0.85}{$\begin{pmatrix}1 & 0 \\ 0 & 1\end{pmatrix}$}\,, \qquad \beta^{22}_\nu = 12\,\scalebox{0.85}{$\begin{pmatrix}1 & 0 \\ 0 & 1\end{pmatrix}$}\,,\end{gathered}$$ and $\beta^\lambda_\nu = 0$ for all other partitions $\lambda$ (see [Proposition 12](#prop:betapsd){reference-type="ref" reference="prop:betapsd"}[\[betapsd4\]](#betapsd4){reference-type="ref" reference="betapsd4"}). We can see that the $\beta^\lambda_\nu$'s pairwise commute and satisfy the Plücker relation [[([\[eq:firstplucker\]](#eq:firstplucker){reference-type="ref" reference="eq:firstplucker"})]{.upright}](#eq:firstplucker): $- \beta^0_\nu\beta^{22}_\nu + \beta^1_\nu\beta^{21}_\nu - \beta^{11}_\nu\beta^2_\nu = 0\,.$ We now briefly discuss several additional results related to [Theorem 3](#thm:main){reference-type="ref" reference="thm:main"}, before going into detail about applications to real algebraic geometry in [1.3](#sec:conjectures){reference-type="ref" reference="sec:conjectures"}. ### Scheme-theoretic results {#sec:schemeintro} The eigenspaces of $\mathcal{B}_\nu(z_1, \dots, z_n)$ can also be regarded as the points of the spectrum of the algebra, $\mathop{\mathrm{Spec}}\mathcal{B}_\nu(z_1, \dots, z_n)$. [\[thm:vague,thm:main\]](#thm:vague,thm:main){reference-type="ref" reference="thm:vague,thm:main"} both set-theoretically identify $\mathop{\mathrm{Spec}}\mathcal{B}_\nu(z_1, \dots, z_n)$ with the fibre of the Wronski map $\mathrm{Wr}^{-1}(g) \subseteq \mathcal{X}^{\nu}$. We discuss a more precise scheme-theoretic version of this correspondence in [5.1](#sec:scheme){reference-type="ref" reference="sec:scheme"}. Namely, in the case where $z_1, \dots, z_n$ are distinct, Mukhin, Tarasov, and Varchenko [@mukhin_tarasov_varchenko13] prove that $\mathop{\mathrm{Spec}}\mathcal{B}_\nu(z_1, \dots, z_n)$ and $\mathrm{Wr}^{-1}(g)$ are in fact isomorphic as schemes; equivalently, $\mathcal{B}_\nu(z_1, \dots, z_n)$ is isomorphic to the coordinate ring of $\mathrm{Wr}^{-1}(g)$. We reformulate this result as [Theorem 69](#thm:distinct){reference-type="ref" reference="thm:distinct"}, expressing the isomorphism in terms of Plücker coordinates and the operators $\beta^\lambda$. If $z_1, \dots, z_n$ are not distinct, then $\mathop{\mathrm{Spec}}\mathcal{B}_\nu(z_1, \dots, z_n)$ and $\mathrm{Wr}^{-1}(g)$ are not necessarily isomorphic as schemes. Instead, $\mathop{\mathrm{Spec}}\mathcal{B}_\nu(z_1, \dots, z_n)$ is scheme-theoretically identified with a particular union of Schubert intersections, involving flags osculating a rational normal curve; the latter is set-theoretically (but not always scheme-theoretically) the same as $\mathrm{Wr}^{-1}(g)$. We formulate this scheme-theoretic isomorphism precisely as [Theorem 71](#thm:nondistinct){reference-type="ref" reference="thm:nondistinct"}, and prove it using results from [@mukhin_tarasov_varchenko09b]. Together, [\[thm:distinct,thm:nondistinct\]](#thm:distinct,thm:nondistinct){reference-type="ref" reference="thm:distinct,thm:nondistinct"} give the precise scheme-theoretic formulation of [Theorem 3](#thm:main){reference-type="ref" reference="thm:main"}[\[main_eigenspace\]](#main_eigenspace){reference-type="ref" reference="main_eigenspace"}. ### Bases for $V$ in a fibre of the Wronski map {#sec:basesintro} Using [Theorem 3](#thm:main){reference-type="ref" reference="thm:main"}, we obtain two explicit bases for any element $V \in \mathrm{Wr}^{-1}(g)$, in terms of our operators $\beta^\lambda(t)$ acting on the associated eigenspace $E$. The first basis (see [Theorem 75](#thm:betabasis){reference-type="ref" reference="thm:betabasis"}) depends on the Schubert cell $\mathcal{X}^{\nu}$ containing $V$, corresponding to a matrix representative in reduced row-echelon form. The second basis (see [Theorem 79](#thm:generalbasis){reference-type="ref" reference="thm:generalbasis"}) is independent of the Schubert cell, and only involves the operators $\beta^k(t)$ associated to single-row partitions (i.e. where the corresponding character $\chi^k$ is trivial). ### Geometric transformations {#sec:transformationsintro} The Grassmannians $\mathrm{Gr}(d,m)$ and $\mathrm{Gr}(m-d,m)$ are dual to each other. In [5.3](#sec:transformations){reference-type="ref" reference="sec:transformations"}, we use [Theorem 3](#thm:main){reference-type="ref" reference="thm:main"} to show that this duality corresponds to an automorphism of the algebra $\mathcal{B}_n(z_1, \dots, z_n)$ (see [Proposition 86](#prop:dualityautomorphism){reference-type="ref" reference="prop:dualityautomorphism"}), which recovers a result from [@purbhoo]. We also show that when the partition $\nu$ is a rectangle, the Bethe algebra $\mathcal{B}_\nu(z_1, \dots, z_n)\subseteq \mathrm{End}(M^{\nu})$ is invariant under the action of $\mathrm{PGL}_2$ on the parameters $z_1, \dots, z_n$ (see [Corollary 88](#cor:invariance){reference-type="ref" reference="cor:invariance"}). This also follows from [Theorem 3](#thm:main){reference-type="ref" reference="thm:main"}, and corresponds to the fact that there is a natural $\mathrm{PGL}_2$-action on $\mathrm{Gr}(d,m)$. This action characterizes the Wronski map [@gillespie_levinson_purbhoo23 Section 3.2] and is also related to the combinatorics of promotion and evacuation on standard Young tableaux [@purbhoo13]. ### A $\tau$-function of the KP hierarchy The *KP hierarchy* is a system of differential equations which arose out of the study of solitary waves. As we recall in [2.3.2](#sec:kpbackground){reference-type="ref" reference="sec:kpbackground"}, its solutions are encoded by *$\tau$-functions*. These are symmetric functions satisfying the Hirota equation [[([\[eq:KP\]](#eq:KP){reference-type="ref" reference="eq:KP"})]{.upright}](#eq:KP), or equivalently, functions whose coefficients in the Schur basis satisfy the Plücker relations [[([\[eq:prelspartitions\]](#eq:prelspartitions){reference-type="ref" reference="eq:prelspartitions"})]{.upright}](#eq:prelspartitions). We can then equivalently rephrase [Theorem 3](#thm:main){reference-type="ref" reference="thm:main"}[\[main_pluckers\]](#main_pluckers){reference-type="ref" reference="main_pluckers"} as the statement that the symmetric function $$\label{eq:tauintro} \sum_{X \subseteq [n]} \, \sum_{\sigma \in \mathfrak{S}_{X}} \sigma \otimes \mathsf{p}_{\mu_1}\dotsm \,\mathsf{p}_{\mu_s} \cdot\prod_{i \in [n] \setminus X}z_i$$ with coefficients in $\mathbb{C}[\mathfrak{S}_{n}]$ is a $\tau$-function of the KP hierarchy, where $\mu_1, \dots, \mu_s$ denote the lengths of the cycles of $\sigma \in \mathfrak{S}_{X}$, and $\mathsf{p}_k$ is the $k$th power sum symmetric function. (See [\[thm:taufunction,thm:tauinfinity\]](#thm:taufunction,thm:tauinfinity){reference-type="ref" reference="thm:taufunction,thm:tauinfinity"} for precise statements.) As we mentioned above, our proof of [Theorem 3](#thm:main){reference-type="ref" reference="thm:main"}[\[main_pluckers\]](#main_pluckers){reference-type="ref" reference="main_pluckers"} in fact uses the theory of symmetric functions and $\tau$-functions. ## Conjectures in real algebraic geometry {#sec:conjectures} We continue to assume that $\nu$ is a partition of $n$ with at most $d$ parts, and $m \geq d+\nu_1$, so that the Schubert cell $\mathcal{X}^{\nu}$ is contained in $\mathrm{Gr}(d,m)$. The Schubert variety $\smash{\overline{\mathcal{X}}}^{\nu}\subseteq \mathrm{Gr}(d,m)$ is the closure of $\mathcal{X}^{\nu}$. We write ${\mathchoice% {\scalebox{1.35}{\raisebox{-0.02em}{$\sqsubset\!\!\sqsupset$}}} {\scalebox{1.35}{\raisebox{-0.02em}{$\sqsubset\!\!\sqsupset$}}} {\sqsubset\!\!\sqsupset} {\sqsubset\!\!\sqsupset} }$ for the rectangular partition $(m-d)^d = (m-d, \dots, m-d)$. In this case, $\smash{\overline{\mathcal{X}}}^{{\mathchoice% {\scalebox{1.35}{\raisebox{-0.02em}{$\sqsubset\!\!\sqsupset$}}} {\scalebox{1.35}{\raisebox{-0.02em}{$\sqsubset\!\!\sqsupset$}}} {\sqsubset\!\!\sqsupset} {\sqsubset\!\!\sqsupset} }}= \mathrm{Gr}(d,m)$. We will be mainly concerned with the following Schubert problem. Given $W_1, \dots, W_n$ in $\mathrm{Gr}(m-d,m)$, determine all $d$-planes $V$ such that $$\label{eq:schubertproblem} V \in \smash{\overline{\mathcal{X}}}^{\nu}\qquad \text{and} \qquad V \cap W_i \neq \{0\}\; \text{ for all $i =1, \dots, n$} \,.$$ When $W_1, \dots, W_n$ are sufficiently general, the number of distinct solutions $V$ to the Schubert problem [[([\[eq:schubertproblem\]](#eq:schubertproblem){reference-type="ref" reference="eq:schubertproblem"})]{.upright}](#eq:schubertproblem) is exactly $\mathsf{f}^{\nu} = \dim M^{\nu}$. We will be concerned with solving [[([\[eq:schubertproblem\]](#eq:schubertproblem){reference-type="ref" reference="eq:schubertproblem"})]{.upright}](#eq:schubertproblem) over the real numbers when $W_1, \dots, W_n$ are real, and especially with instances for which *all* the solutions are real. The interest in algebraic problems with only real solutions dates back at least to Fulton [@fulton84 Section 7.2], who wrote, "The question of how many solutions of real equations can be real is still very much open, particularly for enumerative problems." Note that the property of having only real solutions is extremely rare; for example, for a 'random' Schubert problem on $\mathrm{Gr}(d,m)$ defined over $\mathbb{R}$, the number of real solutions is roughly the square root of the number of complex solutions [@burgisser_lerario20]. We refer to [@sottile11] for a detailed survey of real enumerative geometry. ### The Shapiro--Shapiro conjecture {#sec:shapiroshapiro} The ***moment curve*** $\gamma : \mathbb{C}\to \mathbb{C}_{m-1}[u]$ is the parametric curve $$\label{eq:moment} \gamma(t) := \frac{(u+t)^{m-1}}{(m-1)!} \,.$$ The closure of the image of $\gamma$ in $\mathbb{P}^{m-1}$ is a rational normal curve. A $d$-plane $V \in \mathrm{Gr}(d,m)$ ***osculates*** $\gamma$ at $w \in \mathbb{C}$ if $(\gamma(w), \gamma'(w), \gamma''(w), \dots, \gamma^{(d-1)}(w))$ is a basis for $V$. Osculating planes to the moment curve are related to the Wronski map by the following fact (see [Proposition 22](#prop:schubertwronskian2){reference-type="ref" reference="prop:schubertwronskian2"} for a more detailed formulation): **Proposition 6**. *Suppose $W \in \mathrm{Gr}(m-d,m)$ and $V \in \mathrm{Gr}(d,m)$. If $W$ osculates $\gamma$ at $w$, then $V \cap W \neq \{0\}$ if and only if $-w$ is a zero of $\mathrm{Wr}(V)$.* The ***Shapiro--Shapiro conjecture*** can be stated as follows: **Theorem 7** (Mukhin, Tarasov, and Varchenko [@mukhin_tarasov_varchenko09a]). *Let $z_1, \dots, z_n$ be distinct real numbers. For $i =1, \dots, n$, let $W_i \in \mathrm{Gr}(m-d,m)$ be the osculating $(m-d)$-plane to $\gamma$ at $z_i$. Then there are exactly $\mathsf{f}^{\nu}$ distinct solutions to the Schubert problem [[([\[eq:schubertproblem\]](#eq:schubertproblem){reference-type="ref" reference="eq:schubertproblem"})]{.upright}](#eq:schubertproblem), and all solutions are real.* [Theorem 7](#thm:ssc){reference-type="ref" reference="thm:ssc"} was conjectured by Boris and Michael Shapiro in 1993, and extensively tested and popularized by Sottile [@sottile00]. It was proved in the cases $d \le 2$ and $m-d \le 2$ by Eremenko and Gabrielov [@eremenko_gabrielov02], and in general by Mukhin, Tarasov, and Varchenko [@mukhin_tarasov_varchenko09a]. Their proof was later restructured and simplified in [@purbhoo]. A very different proof, based on geometric and topological arguments, is given in [@levinson_purbhoo21]. A notable feature of [Theorem 7](#thm:ssc){reference-type="ref" reference="thm:ssc"} is that the Schubert problem has $\mathsf{f}^{\nu}$ distinct solutions, despite the fact that $W_1, \dots, W_n$ are explicitly specified, and hence not assumed to be general. In fact, this follows from the claim that all solutions are real (see [@sottile11 Theorem 3.2]). A more general form of [Theorem 7](#thm:ssc){reference-type="ref" reference="thm:ssc"} describes how the story changes in the limit as $z_1, \dots, z_n$ become non-distinct, as we discuss in [\[sec:dimensions,sec:generalssc\]](#sec:dimensions,sec:generalssc){reference-type="ref" reference="sec:dimensions,sec:generalssc"}. For the situation described in [Theorem 7](#thm:ssc){reference-type="ref" reference="thm:ssc"}, the Schubert problem [[([\[eq:schubertproblem\]](#eq:schubertproblem){reference-type="ref" reference="eq:schubertproblem"})]{.upright}](#eq:schubertproblem) is equivalent (by [Proposition 6](#prop:schubertwronskian){reference-type="ref" reference="prop:schubertwronskian"}/[Proposition 22](#prop:schubertwronskian2){reference-type="ref" reference="prop:schubertwronskian2"}) to $V\in\mathcal{X}^{\nu}$ and $\mathrm{Wr}(V) = g$, where $g(u) = (u+z_1) \dotsm (u+z_n)$. Thus, [Theorem 7](#thm:ssc){reference-type="ref" reference="thm:ssc"} can be rephrased as follows: if $V \subseteq \mathbb{C}[u]$ is a finite-dimensional vector space of polynomials, and $\mathrm{Wr}(V)$ has only real roots, then $V$ is real. Mukhin, Tarasov and Varchenko deduce [Theorem 7](#thm:ssc){reference-type="ref" reference="thm:ssc"} from [Theorem 1](#thm:vague){reference-type="ref" reference="thm:vague"} using this reformulation. The key point is that when $z_1, \dots, z_n \in \mathbb{R}$, the Gaudin Hamiltonians are self-adjoint operators with respect to a Hermitian inner product. Hence their eigenvalues, which determine the points of $\mathrm{Wr}^{-1}(g)$, are all real. Using [Theorem 3](#thm:main){reference-type="ref" reference="thm:main"}, we obtain a number of generalizations of [Theorem 7](#thm:ssc){reference-type="ref" reference="thm:ssc"}: ### The divisor form of the secant conjecture {#sec:secantconjecture} Let $I \subseteq \mathbb{R}$ be an interval. An $(m-d)$-plane $W \in \mathrm{Gr}(m-d,m)$ is a ***secant*** to $\gamma$ along $I$ if there exist distinct points $w_1, \dots, w_{m-d} \in I$ such that $(\gamma(w_1), \dots, \gamma(w_{m-d}))$ is a basis for $W$. More generally, $W$ is a ***generalized secant*** to $\gamma$ along $I$ if there exist distinct points $w_1, \dots, w_k \in I$ and positive integers $m_1, \dots, m_k$, such that $m_1+ \dots + m_k = m-d$ and $$\big(\gamma(w_1), \gamma'(w_1), \dots, \gamma^{(m_1-1)}(w_1), \ \dots\ , \gamma(w_k), \gamma'(w_k), \dots, \gamma^{(m_k-1)}(w_k)\big)$$ is a basis for $W$. Working projectively, these definitions naturally extend to cyclic intervals of $\mathbb{RP}^1 = \mathbb{R}\cup \{\infty\}$, where the interval is allowed to wrap around infinity. When one of the points $w_i\in I$ is $\infty$, $\gamma^{(j)}(w_i)$ is replaced by $u^j$. Around 2003, Frank Sottile formulated the ***secant conjecture***, which asserts in particular that [Theorem 7](#thm:ssc){reference-type="ref" reference="thm:ssc"} remains true when $W_1, \dots, W_n$ are generalized secants to $\gamma$ along disjoint intervals of $\mathbb{R}$. This statement is what we call the ***divisor form*** of the secant conjecture, since it arises from intersecting Schubert varieties of codimension one, i.e., *Schubert divisors*; the general form of the secant conjecture involves intersecting Schubert varieties of arbitrary codimension, as we discuss in [5.7.2](#sec:generalsecant){reference-type="ref" reference="sec:generalsecant"}. Note that this case of the secant conjecture is a generalization of the Shapiro--Shapiro conjecture, since an osculating plane to $\gamma$ is a special case of a generalized secant. The secant conjecture appeared in [@ruffo_sivan_soprunova_sottile06] (cf. [@sottile11 Section 13.4]), and it was extensively tested experimentally in a project led by Sottile [@garcia-puente_hein_hillar_martin_del_campo_ruffo_sottile_teitler12], as described in [@hillar_garcia-puente_martin_del_campo_ruffo_teitler_johnson_sottile10]. It has also been proved in special cases: Eremenko, Gabrielov, Shapiro, and Vainshtein [@eremenko_gabrielov_shapiro_vainshtein06 Section 3] established the case $m-d\le 2$; and Mukhin, Tarasov, and Varchenko [@mukhin_tarasov_varchenko09c] (cf. [@garcia-puente_hein_hillar_martin_del_campo_ruffo_sottile_teitler12 Section 3.1]) verified the case of the divisor form when there exists $r > 0$ such that every $W_i$ is a (non-generalized) secant where $w_1, \dots, w_{m-d} \in I_i$ are an arithmetic progression of step size $r$. We show that the divisor form of the secant conjecture is true in general: **Theorem 8** (Secant conjecture, divisor form). *Let $I_1, \dots, I_n \subseteq \mathbb{R}$ be pairwise disjoint real intervals. For $i =1, \dots, n$, let $W_i \in \mathrm{Gr}(m-d,m)$ be a generalized secant to $\gamma$ along $I_i$. Then there are exactly $\mathsf{f}^{\nu}$ distinct solutions to the Schubert problem [[([\[eq:schubertproblem\]](#eq:schubertproblem){reference-type="ref" reference="eq:schubertproblem"})]{.upright}](#eq:schubertproblem), and all solutions are real.* This verifies the secant conjecture in the first non-trivial case of interest for a Schubert problem on an arbitrary Grassmannian. As we discuss in [5.7.2](#sec:generalsecant){reference-type="ref" reference="sec:generalsecant"}, we do not yet know how to address the general form of the secant conjecture with our methods. We mention that when $\nu = {\mathchoice% {\scalebox{1.35}{\raisebox{-0.02em}{$\sqsubset\!\!\sqsupset$}}} {\scalebox{1.35}{\raisebox{-0.02em}{$\sqsubset\!\!\sqsupset$}}} {\sqsubset\!\!\sqsupset} {\sqsubset\!\!\sqsupset} }$ (and $n = d(m-d)$), [Theorem 8](#thm:secant){reference-type="ref" reference="thm:secant"} remains true if $I_1, \dots, I_n$ are cyclic intervals of $\mathbb{RP}^1$, i.e., one of the intervals is allowed to wrap around infinity. The secant conjecture is sometimes phrased in this way. This follows from [Theorem 8](#thm:secant){reference-type="ref" reference="thm:secant"} as stated, using the $\mathrm{PGL}_2$-action on $\mathrm{Gr}(d,m)$. ### The disconjugacy conjecture Suppose that $V$ is a $d$-dimensional vector space of real analytic functions, defined on an interval $I\subseteq\mathbb{R}$. Disconjugacy is concerned with the question of how many zeros a function in $V$ can have. By linear algebra, there always exists a nonzero function $f \in V$ such that $f$ has at least $d-1$ zeros on $I$. We say that $V$ is ***disconjugate*** on $I$ if every nonzero function in $V$ has at most $d-1$ zeros on $I$ (counted with multiplicities). Disconjugacy has long been studied because it is related to explicit solutions for linear differential equations; see [@coppel71], as well as [@karp Section 4.1] and the references therein. It is not always straightforward to decide if $V$ is disconjugate on $I$. However, a necessary condition is that $\mathrm{Wr}(V)$ has no zeros on $I$. This is because $\mathrm{Wr}(V)$ has a zero at $w$ if and only if there exists a nonzero $f \in V$ such that $f$ has a zero at $w$ of multiplicity at least $d$. In general, the converse is false; for example, $V = \langle \cos u, \sin u \rangle$ is not disconjugate on $I = \mathbb{R}$, and $\mathrm{Wr}(V) = 1$. Eremenko [@eremenko15; @eremenko19] conjectured that the converse statement is actually correct under very special circumstances. This is known as the ***disconjugacy conjecture***, which we state now as a theorem: **Theorem 9** (Disconjugacy conjecture). *Let $V \subseteq \mathbb{R}[u]$ be a finite-dimensional vector space of polynomials such that $\mathrm{Wr}(V)$ has only real zeros. Then $V$ is disconjugate on every interval which avoids the zeros of $\mathrm{Wr}(V)$.* The disconjugacy conjecture has been verified in the case that $\dim(V) \le 2$ [@eremenko_gabrielov_shapiro_vainshtein06] (cf. [@eremenko15 p. 341]). Eremenko furthermore showed that the disconjugacy conjecture (along with the Shapiro--Shapiro conjecture) implies [Theorem 8](#thm:secant){reference-type="ref" reference="thm:secant"}; in fact, his motivation was to generalize the argument used to prove the $m-d \le 2$ case of the secant conjecture [@eremenko_gabrielov_shapiro_vainshtein06 Section 3]. The main idea is encapsulated in [Lemma 14](#lem:topological){reference-type="ref" reference="lem:topological"}, and explained further in [1.3.5](#sec:conjectureproofs){reference-type="ref" reference="sec:conjectureproofs"}. ### Positivity conjectures A $d$-plane $V \in \mathrm{Gr}(d,m)$ is called ***totally nonnegative*** if all of its Plücker coordinates are real and nonnegative (up to rescaling). Similarly, $V$ is called ***totally positive in $\mathcal{X}^{\nu}$*** if $V\in\mathcal{X}^{\nu}$ and all of its Plücker coordinates which are not trivially zero on $\mathcal{X}^{\nu}$ are positive, i.e., $$\label{eq:positivenupluckers} \Delta^\lambda > 0 \;\text{ for all } \lambda\subseteq\nu \qquad \text{and} \qquad \Delta^\lambda = 0 \;\text{ for all } \lambda\not\subseteq\nu\,.$$ For example, each element $V\in\mathrm{Gr}(2,4)$ from [Example 4](#ex:main){reference-type="ref" reference="ex:main"} is totally nonnegative if and only if $z_1, z_2 \ge 0$, and is totally positive in its Schubert cell if and only if $z_1, z_2 > 0$. The totally nonnegative part of $\mathrm{Gr}(d,m)$ is a totally nonnegative partial flag variety in the sense of Lusztig [@lusztig94; @lusztig98] (see [@bloch_karp23 Section 1] for further discussion), and was studied combinatorially by Postnikov [@postnikov06]. Total positivity in Schubert cells was considered by Berenstein and Zelevinsky [@berenstein_zelevinsky97]. These and similar totally positive spaces have been extensively studied in the past few decades, with connections to representation theory [@lusztig94], combinatorics [@postnikov06], cluster algebras [@fomin_williams_zelevinsky], soliton solutions to the KP equation [@kodama_williams14], scattering amplitudes [@arkani-hamed_bourjaily_cachazo_goncharov_postnikov_trnka16], Schubert calculus [@knutson14], topology [@galashin_karp_lam22], and many other topics. Total positivity also provided one of the original motivations for the Shapiro--Shapiro conjecture, since the moment curve $\gamma$ is an example of a *totally positive* (or *convex*) *curve*; see [5.7.3](#sec:totalreality){reference-type="ref" reference="sec:totalreality"} and cf. [@sottile00 Section 4]. Mukhin--Tarasov and Karp conjectured that the reality statements discussed in [\[sec:shapiroshapiro,sec:secantconjecture\]](#sec:shapiroshapiro,sec:secantconjecture){reference-type="ref" reference="sec:shapiroshapiro,sec:secantconjecture"} have totally positive analogues. We verify these conjectures in slightly greater generality: **Theorem 10** (Positive Shapiro--Shapiro conjecture). *Let $z_1, \dots, z_n$ and $W_1, \dots, W_n$ be as in [Theorem 7](#thm:ssc){reference-type="ref" reference="thm:ssc"}.* (i) *[\[positive1\]]{#positive1 label="positive1"} If $z_1, \dots, z_n \in [0,\infty)$, then all solutions to the Schubert problem [[([\[eq:schubertproblem\]](#eq:schubertproblem){reference-type="ref" reference="eq:schubertproblem"})]{.upright}](#eq:schubertproblem) are real and totally nonnegative.* (ii) *[\[positive2\]]{#positive2 label="positive2"} If $z_1, \dots, z_n \in (0,\infty)$, then all solutions to the Schubert problem [[([\[eq:schubertproblem\]](#eq:schubertproblem){reference-type="ref" reference="eq:schubertproblem"})]{.upright}](#eq:schubertproblem) are real and totally positive in $\mathcal{X}^{\nu}$.* **Theorem 11** (Positive secant conjecture, divisor form). *Let $I_1, \dots, I_n$ and $W_1, \dots, W_n$ be as in [Theorem 8](#thm:secant){reference-type="ref" reference="thm:secant"}.* (i) *[\[positivesecant1\]]{#positivesecant1 label="positivesecant1"} If $I_1, \dots, I_n \subseteq [0,\infty)$, then there are exactly $\mathsf{f}^{\nu}$ distinct solutions to the Schubert problem [[([\[eq:schubertproblem\]](#eq:schubertproblem){reference-type="ref" reference="eq:schubertproblem"})]{.upright}](#eq:schubertproblem), and all solutions are real and totally nonnegative.* (ii) *[\[positivesecant2\]]{#positivesecant2 label="positivesecant2"} If $I_1, \dots, I_n \subseteq (0,\infty)$, then there are exactly $\mathsf{f}^{\nu}$ distinct solutions to the Schubert problem [[([\[eq:schubertproblem\]](#eq:schubertproblem){reference-type="ref" reference="eq:schubertproblem"})]{.upright}](#eq:schubertproblem), and all solutions are real and totally positive in $\mathcal{X}^{\nu}$.* In the special case $\nu = {\mathchoice% {\scalebox{1.35}{\raisebox{-0.02em}{$\sqsubset\!\!\sqsupset$}}} {\scalebox{1.35}{\raisebox{-0.02em}{$\sqsubset\!\!\sqsupset$}}} {\sqsubset\!\!\sqsupset} {\sqsubset\!\!\sqsupset} }$, [Theorem 10](#thm:positive){reference-type="ref" reference="thm:positive"}[\[positive1\]](#positive1){reference-type="ref" reference="positive1"} was conjectured by Evgeny Mukhin and Vitaly Tarasov in 2017, and [\[thm:positive,thm:positivesecant\]](#thm:positive,thm:positivesecant){reference-type="ref" reference="thm:positive,thm:positivesecant"} were conjectured independently in [@karp]. It was shown in [@karp] that the four statements in [\[thm:positive,thm:positivesecant\]](#thm:positive,thm:positivesecant){reference-type="ref" reference="thm:positive,thm:positivesecant"} in the case $\nu={\mathchoice% {\scalebox{1.35}{\raisebox{-0.02em}{$\sqsubset\!\!\sqsupset$}}} {\scalebox{1.35}{\raisebox{-0.02em}{$\sqsubset\!\!\sqsupset$}}} {\sqsubset\!\!\sqsupset} {\sqsubset\!\!\sqsupset} }$ are all pairwise equivalent, and that they are moreover equivalent to the disconjugacy conjecture ([Theorem 9](#thm:disconj){reference-type="ref" reference="thm:disconj"}). We briefly recall from [@karp] why [Theorem 10](#thm:positive){reference-type="ref" reference="thm:positive"}[\[positive1\]](#positive1){reference-type="ref" reference="positive1"} implies the disconjugacy conjecture (the converse is much more subtle, but we do not need it here). Let $V\in\mathrm{Gr}(d,m)$ be such that $\mathrm{Wr}(V)$ has only real zeros, and let $I \subseteq \mathbb{R}$ be an interval which avoids the zeros of $\mathrm{Wr}(V)$, which we may assume is closed. We apply the $\mathrm{PGL}_2$-action so that $I\subseteq (0,\infty)$ and the zeros of $\mathrm{Wr}(V)$ are all negative. Then by [Theorem 10](#thm:positive){reference-type="ref" reference="thm:positive"}[\[positive1\]](#positive1){reference-type="ref" reference="positive1"}, $V$ is totally nonnegative. Equivalently, by a classical result of Gantmakher and Krein [@gantmaher_krein50 Theorem V.3], the sequence of coefficients of every $f\in V$ changes sign at most $d-1$ times. By Descartes's rule of signs, $f$ has at most $d-1$ zeros on $(0,\infty)$, as required. As we have mentioned, the disconjugacy conjecture in turn implies the divisor form of the secant conjecture ([Theorem 8](#thm:secant){reference-type="ref" reference="thm:secant"}). Therefore to prove all of these statements, it suffices to establish [\[thm:positive,thm:positivesecant\]](#thm:positive,thm:positivesecant){reference-type="ref" reference="thm:positive,thm:positivesecant"}. We now explain how to do so. ### Proof of conjectures {#sec:conjectureproofs} We give the proofs of [\[thm:positive,thm:positivesecant\]](#thm:positive,thm:positivesecant){reference-type="ref" reference="thm:positive,thm:positivesecant"}. We begin with the former, which is a direct corollary of our main result ([Theorem 3](#thm:main){reference-type="ref" reference="thm:main"}). We need the following properties of the operators $\beta^\lambda$, which we will prove in [2.3.1](#sec:repthy){reference-type="ref" reference="sec:repthy"}. They are straightforward consequences of the definitions and some well-known results in representation theory. Recall that $M^{\nu}$ can be equipped with a Hermitian inner product, such that every $\sigma \in \mathfrak{S}_{n}$ acts as a unitary operator. **Proposition 12**. *Let $\beta^\lambda_\nu \in \mathrm{End}(M^{\nu})$ denote the operator $\beta^\lambda$ acting on $M^{\nu}$.* (i) *[\[betapsd1\]]{#betapsd1 label="betapsd1"} If $z_1, \dots, z_n \in \mathbb{R}$, then $\beta^\lambda_\nu$ is a self-adjoint operator.* (ii) *[\[betapsd2\]]{#betapsd2 label="betapsd2"} If $z_1, \dots, z_n \in [0,\infty)$, then $\beta^\lambda_\nu$ is positive semidefinite for all $\lambda, \nu$.* (iii) *[\[betapsd3\]]{#betapsd3 label="betapsd3"} If $z_1, \dots, z_n \in (0,\infty)$ and $\lambda \subseteq \nu$, then $\beta^\lambda_\nu$ is positive definite.* (iv) *[\[betapsd4\]]{#betapsd4 label="betapsd4"} If $\lambda \not \subseteq \nu$, then $\beta^\lambda_\nu = 0$.* **Remark 13**. We point out that [Proposition 12](#prop:betapsd){reference-type="ref" reference="prop:betapsd"}[\[betapsd1\]](#betapsd1){reference-type="ref" reference="betapsd1"} is consistent with [Example 5](#ex:22){reference-type="ref" reference="ex:22"}, despite the fact that $\beta^2_\nu$ and $\beta^{11}_\nu$ are not Hermitian matrices (for $z_1, \dots, z_4 \in \mathbb{R}$). This is because in [Example 5](#ex:22){reference-type="ref" reference="ex:22"}, we are not working with an orthonormal basis of $M^{\nu}$. We can change to such a basis in which $\sigma_{1,2}$ and $\sigma_{3,4}$ act as the unitary matrix $(\begin{smallmatrix} 1 & 0 \\ 0 & -1\end{smallmatrix})$, and $\sigma_{2,3}$ acts as $\frac{1}{2}(\begin{smallmatrix}-1 & \sqrt{3} \\ \sqrt{3} & 1\end{smallmatrix})$; then every $\beta^\lambda_\nu$ is Hermitian. *Proof of [Theorem 10](#thm:positive){reference-type="ref" reference="thm:positive"}.* Let $V$ be a solution to the Schubert problem [[([\[eq:schubertproblem\]](#eq:schubertproblem){reference-type="ref" reference="eq:schubertproblem"})]{.upright}](#eq:schubertproblem). Equivalently, by [Proposition 6](#prop:schubertwronskian){reference-type="ref" reference="prop:schubertwronskian"}, we have $V \in \mathcal{X}^{\nu}$ and $\mathrm{Wr}(V) = g$. By [Theorem 3](#thm:main){reference-type="ref" reference="thm:main"}[\[main_eigenspace\]](#main_eigenspace){reference-type="ref" reference="main_eigenspace"}, we can write $V = V_E$ for some eigenspace $E \subseteq M^{\nu}$ of $\mathcal{B}_\nu(z_1, \dots, z_n)$. This means that the Plücker coordinates $[\Delta^\lambda : \lambda \subseteq {\mathchoice% {\scalebox{1.35}{\raisebox{-0.02em}{$\sqsubset\!\!\sqsupset$}}} {\scalebox{1.35}{\raisebox{-0.02em}{$\sqsubset\!\!\sqsupset$}}} {\sqsubset\!\!\sqsupset} {\sqsubset\!\!\sqsupset} }]$ of $V$ are the eigenvalues of the operators $\beta^\lambda_\nu$ on $E$. If $z_1, \dots, z_n \in [0,\infty)$, then [Proposition 12](#prop:betapsd){reference-type="ref" reference="prop:betapsd"}[\[betapsd2\]](#betapsd2){reference-type="ref" reference="betapsd2"} implies that the eigenvalues of $\beta^\lambda_\nu$ are real and nonnegative, so $V$ is totally nonnegative. This proves part [\[positive1\]](#positive1){reference-type="ref" reference="positive1"}. Similarly, if $z_1, \dots, z_n\in (0,\infty)$, then parts [\[betapsd3\]](#betapsd3){reference-type="ref" reference="betapsd3"} and [\[betapsd4\]](#betapsd4){reference-type="ref" reference="betapsd4"} of [Proposition 12](#prop:betapsd){reference-type="ref" reference="prop:betapsd"} imply that [[([\[eq:positivenupluckers\]](#eq:positivenupluckers){reference-type="ref" reference="eq:positivenupluckers"})]{.upright}](#eq:positivenupluckers) holds, so $V$ is totally positive in $\mathcal{X}^{\nu}$. This proves part [\[positive2\]](#positive2){reference-type="ref" reference="positive2"}. ◻ [Theorem 11](#thm:positivesecant){reference-type="ref" reference="thm:positivesecant"} now follows from topological arguments used in [@eremenko15; @karp], which we apply in the following form: **Lemma 14**. *Let $I_1, \dots, I_n$ and $W_1, \dots, W_n$ be as in [Theorem 8](#thm:secant){reference-type="ref" reference="thm:secant"}, and suppose that the disconjugacy conjecture is true. Then there are exactly $\mathsf{f}^{\nu}$ distinct solutions to the Schubert problem [[([\[eq:schubertproblem\]](#eq:schubertproblem){reference-type="ref" reference="eq:schubertproblem"})]{.upright}](#eq:schubertproblem), and all solutions are real. Moreover, for each solution $V$, we can write $\mathrm{Wr}(V) = (u+z_1) \dotsm (u+z_n)$ for some real numbers $z_1\in I_1, \dots, z_n\in I_n$.* *Proof.* In the case that $\nu = {\mathchoice% {\scalebox{1.35}{\raisebox{-0.02em}{$\sqsubset\!\!\sqsupset$}}} {\scalebox{1.35}{\raisebox{-0.02em}{$\sqsubset\!\!\sqsupset$}}} {\sqsubset\!\!\sqsupset} {\sqsubset\!\!\sqsupset} }$, this is precisely [@karp Lemma 4.15]. In fact, the proof of [@karp Lemma 4.15] applies to an arbitrary $\nu \subseteq {\mathchoice% {\scalebox{1.35}{\raisebox{-0.02em}{$\sqsubset\!\!\sqsupset$}}} {\scalebox{1.35}{\raisebox{-0.02em}{$\sqsubset\!\!\sqsupset$}}} {\sqsubset\!\!\sqsupset} {\sqsubset\!\!\sqsupset} }$, using [Theorem 7](#thm:ssc){reference-type="ref" reference="thm:ssc"}. ◻ *Proof of [Theorem 11](#thm:positivesecant){reference-type="ref" reference="thm:positivesecant"}.* We have proved [Theorem 10](#thm:positive){reference-type="ref" reference="thm:positive"}, so the disconjugacy conjecture is true. Hence we can apply [Lemma 14](#lem:topological){reference-type="ref" reference="lem:topological"}, which along with [Theorem 10](#thm:positive){reference-type="ref" reference="thm:positive"} yields the result. ◻ ## Outline The remainder of this paper is organized as follows. In [2](#sec:background){reference-type="ref" reference="sec:background"}, we recall background required for the proof of [Theorem 3](#thm:main){reference-type="ref" reference="thm:main"}. Our discussion spans several topics, including: Plücker coordinates and the Plücker relations; the Wronski map, and its relationship to Schubert varieties, the $\mathrm{PGL}_2$-action on $\mathrm{Gr}(d,m)$, and differential operators; applications of symmetric function theory, including representation theory of symmetric groups, the proof of [Proposition 12](#prop:betapsd){reference-type="ref" reference="prop:betapsd"}, $\tau$-functions of the KP hierarchy, and several symmetric function identities and their proofs; and Bethe subalgebras of $\mathbb{C}[\mathfrak{S}_{n}]$, including the definition of $\mathcal{B}_n(z_1, \dots, z_n)$ and the precise statement of [Theorem 1](#thm:vague){reference-type="ref" reference="thm:vague"} ([Theorem 44](#thm:precise){reference-type="ref" reference="thm:precise"}). [\[sec:commutativitytranslation,sec:pr\]](#sec:commutativitytranslation,sec:pr){reference-type="ref" reference="sec:commutativitytranslation,sec:pr"} are devoted to the proof of [Theorem 3](#thm:main){reference-type="ref" reference="thm:main"}, which we structure as follows. In [3](#sec:commutativitytranslation){reference-type="ref" reference="sec:commutativitytranslation"} we establish some basic properties of the operators $\beta^\lambda(t)$. We prove part [\[main_translation\]](#main_translation){reference-type="ref" reference="main_translation"} (the translation identity), followed by part [\[main_commutativity\]](#main_commutativity){reference-type="ref" reference="main_commutativity"} (the commutativity relations). Combining these parts and some of the arguments involved in their proofs, we also establish [Lemma 46](#lem:comparealgebras){reference-type="ref" reference="lem:comparealgebras"}, which is related to --- but slightly weaker than --- part [\[main_generates\]](#main_generates){reference-type="ref" reference="main_generates"}. We use all of these basic properties, in [4](#sec:pr){reference-type="ref" reference="sec:pr"}, to prove the remaining parts of [Theorem 3](#thm:main){reference-type="ref" reference="thm:main"}. [\[sec:part1proof,sec:part2\]](#sec:part1proof,sec:part2){reference-type="ref" reference="sec:part1proof,sec:part2"} contain the proof of part [\[main_pluckers\]](#main_pluckers){reference-type="ref" reference="main_pluckers"} (the Plücker relations), which is the most technical part of the proof of [Theorem 3](#thm:main){reference-type="ref" reference="thm:main"}. We then deduce parts [\[main_eigenspace\]](#main_eigenspace){reference-type="ref" reference="main_eigenspace"} and [\[main_multiplicity\]](#main_multiplicity){reference-type="ref" reference="main_multiplicity"}, which we use to establish part [\[main_generates\]](#main_generates){reference-type="ref" reference="main_generates"}, in [4.3](#sec:finalsteps){reference-type="ref" reference="sec:finalsteps"}. Finally, in [5](#sec:discussion){reference-type="ref" reference="sec:discussion"}, we discuss several results related to [Theorem 3](#thm:main){reference-type="ref" reference="thm:main"} and its consequences in real algebraic geometry, as well as a variety of open problems. We give the precise scheme-theoretic version of [Theorem 3](#thm:main){reference-type="ref" reference="thm:main"}[\[main_eigenspace\]](#main_eigenspace){reference-type="ref" reference="main_eigenspace"}, as discussed in [1.2.1](#sec:schemeintro){reference-type="ref" reference="sec:schemeintro"}; in particular, this yields a general formula for the dimension of the Bethe algebras $\mathcal{B}_n(z_1, \dots, z_n)$ and $\mathcal{B}_\nu(z_1, \dots, z_n)$. As discussed in [1.2.2](#sec:basesintro){reference-type="ref" reference="sec:basesintro"}, we use [Theorem 3](#thm:main){reference-type="ref" reference="thm:main"} to exhibit two different bases of the solutions $V_E \in \mathrm{Wr}^{-1}(g)$ to the inverse Wronski problem. We explain how Grassmann duality and the $\mathrm{PGL}_2$-action on $\mathrm{Gr}(d,m)$ are reflected in the structure of $\mathcal{B}_n(z_1, \dots, z_n)$ and $\mathcal{B}_\nu(z_1, \dots, z_n)$. We discuss the combinatorial meaning of the commutativity relations [[([\[eq:commutativity\]](#eq:commutativity){reference-type="ref" reference="eq:commutativity"})]{.upright}](#eq:commutativity), and an extension of the $\tau$-function of the KP hierarchy [[([\[eq:tauintro\]](#eq:tauintro){reference-type="ref" reference="eq:tauintro"})]{.upright}](#eq:tauintro) to the infinite symmetric group $\mathfrak{S}_{\infty}$. Finally, we discuss open problems and longstanding conjectures relating to [\[thm:ssc,thm:secant,thm:positive\]](#thm:ssc,thm:secant,thm:positive){reference-type="ref" reference="thm:ssc,thm:secant,thm:positive"}, including the general form of the secant conjecture and the total reality conjecture for convex curves. #### Acknowledgements. {#acknowledgements. .unnumbered} We thank Frank Sottile for helpful discussions. Calculations for this project were carried out using `Sage` [@sagemath], including verification of [Theorem 3](#thm:main){reference-type="ref" reference="thm:main"}[\[main_pluckers\]](#main_pluckers){reference-type="ref" reference="main_pluckers"} up to $n=7$; we thank Mike Zabrocki for assistance with some parts of the code. # Background {#sec:background} We recall some background on Plücker coordinates [@fulton97 Chapter 9], Schubert varieties [@fulton97 Chapter 9], Wronskians [@karp; @purbhoo10; @sottile11], symmetric functions [@stanley99 Chapter 7], representation theory [@sagan01; @serre98], and Bethe algebras [@mukhin_tarasov_varchenko13; @purbhoo]. See the listed references for further details. ## Plücker coordinates {#sec:plucker} For a $d$-plane $V \in \mathrm{Gr}(d,m)$, we can represent $V$ as the row space of a $d \times m$ matrix $A$, which is unique up to left multiplication by $\mathrm{GL}_d$. Recall that $[m] = \{1, 2, \dots, m\}$, and define $\binom{[m]}{d}$ to be the set of $d$-element subsets of $[m]$. We also write such subsets as tuples $(i_1, \dots, i_d)$, with $1 \leq i_1 < \dots <i_d \leq m$. For each $I \in {[m] \choose d}$, let $\Delta_I$ denote the $d \times d$ minor of $A$ with column set $I$. The projective coordinates $\big[\Delta_I : I \in {[m] \choose d}\big]$ are (up to a scalar multiple) independent of the choice of matrix $A$, and are called the ***Plücker coordinates*** of $V$. **Example 15**. Let $V := \left\langle 1,\, z_1z_2u + \frac{z_1 + z_2}{2}u^2 + \frac{1}{3}u^3 \right\rangle \in \mathrm{Gr}(2,4)$, as in [Example 4](#ex:main){reference-type="ref" reference="ex:main"}. Recalling the isomorphism [[([\[eq:isomorphism\]](#eq:isomorphism){reference-type="ref" reference="eq:isomorphism"})]{.upright}](#eq:isomorphism), we can represent $V$ by the $2\times 4$ matrix $$A := \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & z_1z_2 & z_1+z_2 & 2 \end{pmatrix}\,.$$ The Plücker coordinates of $V$ are the $2\times 2$ minors of $A$: $$\Delta_{(1,2)} = z_1z_2, \qquad \Delta_{(1,3)} = z_1+z_2, \qquad \Delta_{(1,4)} = 2, \qquad \Delta_{(2,3)} = \Delta_{(2,4)} = \Delta_{(3,4)} = 0\,,$$ in agreement with [Example 4](#ex:main){reference-type="ref" reference="ex:main"}. (In general, we can construct a matrix $A$ from the Plücker coordinates of $V$ using [Proposition 18](#prop:pluckerbasis){reference-type="ref" reference="prop:pluckerbasis"}.) ### Plücker relations The Plücker coordinates define an embedding $\delta : \mathrm{Gr}(d,m) \hookrightarrow \mathbb{P}^{{m \choose d} -1}$ of the Grassmannian into projective space. To describe the image of $\delta$, it is useful to extend the indexing set for $\Delta_I$ from ${[m] \choose d}$ to $[m]^d$ by the alternating property. That is, if $i_1 < \dots < i_d$, put $$\Delta_{(i_{\sigma(1)}, \dots, i_{\sigma(d)})} := \mathrm{sgn}(\sigma) \Delta_{(i_1, \dots, i_d)} \qquad \text{for all } \sigma \in \mathfrak{S}_{d};$$ if $j_1, \dots, j_d$ are not distinct, put $\Delta_{(j_1, \dots, j_d)} := 0$. Thus for every $I \in [m]^d$, $\Delta_I$ is either zero, or plus or minus some Plücker coordinate. If $I = (i_1, \dots, i_{d+1}) \in [m]^{d+1}$, $J = (j_1, \dots, j_{d-1}) \in [m]^{d-1}$, and $k \in [m]$, write $$\label{eq:deletesubscript} \Delta_{I-k} := \begin{cases} (-1)^{d+1-s}\Delta_{(i_1, \dots, \widehat{i_s}, \dots i_{d+1})}, &\quad \text{if $k = i_s$ for a unique $s \in [d+1]$;} \\ 0, &\quad \text{otherwise} \end{cases}$$ and $$\label{eq:addsubscript} \Delta_{J+k} := \Delta_{(j_1, \dots, j_{d-1},k)} \,.$$ The ***Plücker relations*** for $\mathrm{Gr}(d,m)$ [@fulton97 Section 9.1] are the equations $$\label{eq:prels} \sum_{k = 1}^m \Delta_{I-k} \Delta_{J+k} = 0 \qquad \text{for $I \in [m]^{d+1}$ and $J \in [m]^{d-1}$}\,.$$ The equations [[([\[eq:prels\]](#eq:prels){reference-type="ref" reference="eq:prels"})]{.upright}](#eq:prels) define the image of $\mathrm{Gr}(d,m)$ in $\mathbb{P}^{{m \choose d} -1}$ under the embedding $\delta$, as a scheme. In particular, for every $V \in \mathrm{Gr}(d,m)$, the Plücker coordinates of $V$ satisfy the equations [[([\[eq:prels\]](#eq:prels){reference-type="ref" reference="eq:prels"})]{.upright}](#eq:prels). **Example 16**. Taking $I = (1,2,3)$ and $J = (4)$, we obtain the unique non-trivial Plücker relation for $\mathrm{Gr}(2,4)$: $$\label{eq:gr24plucker} - \Delta_{(1,2)}\Delta_{(3,4)} + \Delta_{(1,3)}\Delta_{(2,4)} - \Delta_{(2,3)}\Delta_{(1,4)} = 0 \,.$$ Other choices for $I,J$ give the same equation (up to sign), or the trivial equation $0=0$. The preceding facts can be reformulated in terms of the exterior algebra of $\mathbb{C}^m$. Let $(e_1, \dots, e_m)$ denote the standard basis for $\mathbb{C}^m$, and for $I = (i_1, \dots, i_d) \in [m]^d$, write $e_I := e_{i_1} \wedge \dots \wedge e_{i_d} \in \mathsf{\Lambda}^{d} \mathbb{C}^m$. If $(v_1, \dots, v_d)$ is a basis for $V\in\mathrm{Gr}(d,m)$, then we have $$v_1 \wedge \dots \wedge v_d = \sum_{I \in {[m] \choose d}} \Delta_I e_I,$$ where the $\Delta_I$'s are the Plücker coordinates of $V$. **Proposition 17** ([@fulton97 Section 9.1]). *Let $\omega := \sum_{I \in {[m] \choose d}} \Delta_I e_I \in \mathsf{\Lambda}^{d} \mathbb{C}^m$, where $\Delta_I\in\mathbb{C}$ for $I\in\binom{[m]}{d}$. Then the following are equivalent:* (a) *[\[exteriorplucker1\]]{#exteriorplucker1 label="exteriorplucker1"} the coefficients $\Delta_I$ of $\omega$ satisfy the Plücker relations [[([\[eq:prels\]](#eq:prels){reference-type="ref" reference="eq:prels"})]{.upright}](#eq:prels);* (b) *[\[exteriorplucker2\]]{#exteriorplucker2 label="exteriorplucker2"} $\omega = v_1 \wedge \dots \wedge v_d$ for some $v_1, \dots, v_d \in \mathbb{C}^m$.* *Furthermore, if [\[exteriorplucker2\]](#exteriorplucker2){reference-type="ref" reference="exteriorplucker2"} holds and $\omega \neq 0$, then $\big[\Delta_I : I \in {[m] \choose d}\big]$ are the Plücker coordinates of $V = \langle v_1, \dots, v_d\rangle$.* ### A basis from the Plücker coordinates Given a basis for $V \in \mathrm{Gr}(d,m)$, by definition we obtain the Plücker coordinates as minors of the matrix of coefficients in the standard basis $(e_1, \dots, e_m)$. Conversely, if we know the Plücker coordinates of $V$, there is a straightforward way to obtain a basis. Namely, for $I= (i_1, \dots, i_d) \in [m]^d$ and $j, k \in [m]$, we define $$\Delta_{(I-j)+k} := \begin{cases} (-1)^{d-s}\Delta_{(i_1, \dots, \widehat{i_s}, \dots i_d, k)}, &\quad \text{if $j = i_s$ for a unique $s \in [d]$;} \\ 0, &\quad \text{otherwise.} \end{cases}$$ Then we have: **Proposition 18**. *Suppose that $V \in \mathrm{Gr}(d,m)$ has Plücker coordinates $\big[\Delta_I : I \in {[m] \choose d}\big]$, and take $J \in {[m] \choose d}$ such that $\Delta_J \neq 0$. Then $\big( \sum_{k=1}^m \Delta_{(J-j)+k} e_k : j \in J \big)$ is a basis for $V$.* *Proof.* Take a $d\times m$ matrix $A$ whose row span is $V$, and let $g$ be the $d\times d$ submatrix of $A$ with column set $J$. Since $\Delta_J\neq 0$, we have $g\in\mathrm{GL}_d$, so $g^{-1}A$ also represents $V$. We can verify that the rows (up to rescaling) of $g^{-1}A$ form the desired basis. ◻ ### Indexing by partitions {#sec:partitions} For our purposes, it is also useful to index Plücker coordinates by partitions. A ***partition*** $\lambda = (\lambda_1, \dots, \lambda_s)$ is a weakly decreasing sequence of positive integers $\lambda_1 \geq \dots \geq \lambda_s > 0$. The ***length*** and ***size*** of $\lambda$ are $\ell(\lambda)=s$ and $|\lambda| = \lambda_1 + \dots + \lambda_s$, respectively, and the notation $\lambda \vdash k$ means $|\lambda| = k$. When convenient, we use exponential notation, e.g., $4^321^4 = (4,4,4,2,1,1,1,1)$. We adopt the convention that $\lambda_j = 0$ for all $j > \ell(\lambda)$, and partitions may be written with any number of trailing zeros, e.g., $(3,3,1)$ and $(3,3,1,0,0,0,0)$ are considered to be the same partition. The ***diagram*** of a partition $\lambda$ is the array of $|\lambda|$ left-justified boxes with $\lambda_i$ boxes in row $i$ for all $i \ge 1$; see [\[fig:partition\]](#fig:partition){reference-type="ref" reference="fig:partition"}. If $\lambda$ and $\mu$ are partitions, we write $\mu \subseteq \lambda$ if the diagram of $\mu$ is contained in the diagram of $\lambda$, i.e., $\mu_i \leq \lambda_i$ for all $i \ge 1$. We introduce a variable $\Delta^\lambda$ for every partition $\lambda$ (using a superscript to distinguish $\Delta^\lambda$ from $\Delta_I$). If $I = (i_1, \dots, i_d) \in {[m] \choose d}$, we identify $$\label{eq:identification} \Delta_I \equiv \Delta^\lambda, \qquad \text{where } \lambda = (i_d-d, \dots, i_2-2, i_1-1) \,;$$ see [\[fig:partition\]](#fig:partition){reference-type="ref" reference="fig:partition"}. We emphasize that this identification depends on the choice of $d$ (but not of $m$). Thus, a partition $\lambda$ indexes a Plücker coordinate of $\mathrm{Gr}(d,m)$ if and only if $\lambda \subseteq {\mathchoice% {\scalebox{1.35}{\raisebox{-0.02em}{$\sqsubset\!\!\sqsupset$}}} {\scalebox{1.35}{\raisebox{-0.02em}{$\sqsubset\!\!\sqsupset$}}} {\sqsubset\!\!\sqsupset} {\sqsubset\!\!\sqsupset} }$, where ${\mathchoice% {\scalebox{1.35}{\raisebox{-0.02em}{$\sqsubset\!\!\sqsupset$}}} {\scalebox{1.35}{\raisebox{-0.02em}{$\sqsubset\!\!\sqsupset$}}} {\sqsubset\!\!\sqsupset} {\sqsubset\!\!\sqsupset} }= (m-d)^d$. For example, indexing by partitions, the Plücker relation [[([\[eq:gr24plucker\]](#eq:gr24plucker){reference-type="ref" reference="eq:gr24plucker"})]{.upright}](#eq:gr24plucker) can be rewritten as $$\label{eq:firstplucker} - \Delta^0 \Delta^{22} + \Delta^{1} \Delta^{21} - \Delta^{11}\Delta^{2} = 0 \,.$$ When indexed by partitions, the Plücker relations [[([\[eq:prels\]](#eq:prels){reference-type="ref" reference="eq:prels"})]{.upright}](#eq:prels) are stable. This means that if $d' \geq d$ and $m'-d' \geq m-d$, then the Plücker relations for $\mathrm{Gr}(d,m)$ are a subset of the Plücker relations for $\mathrm{Gr}(d',m')$. Furthermore, if we set $\Delta^\lambda$ to $0$ for all $\lambda \nsubseteq {\mathchoice% {\scalebox{1.35}{\raisebox{-0.02em}{$\sqsubset\!\!\sqsupset$}}} {\scalebox{1.35}{\raisebox{-0.02em}{$\sqsubset\!\!\sqsupset$}}} {\sqsubset\!\!\sqsupset} {\sqsubset\!\!\sqsupset} }= (m-d)^d$, then every Plücker relation for $\mathrm{Gr}(d',m')$ becomes a (possibly trivial) Plücker relation for $\mathrm{Gr}(d,m)$. Hence taking the union of all non-trivial Plücker relations for all Grassmannians gives the complete list of ***all Plücker relations***, which are valid for all $\mathrm{Gr}(d,m)$'s. Explicitly, for a partition $\lambda$ and $i \geq 1$, let $c = 0$ if $i > \lambda_1$, and otherwise let $c$ be the unique positive integer such that $\lambda_c \geq i > \lambda_{c+1}$. Let $\lambda^{(i)}$ and $\lambda^{(-i)}$ denote the partitions $$\begin{aligned} \lambda^{(i)} &:= (\lambda_1-1, \dots, \lambda_c-1,\, i-1,\, \lambda_{c+1}, \lambda_{c+2}, \dots ) \,, \\ \lambda^{(-i)} &:= (\lambda_1+1, \dots, \lambda_{i-1}+1, \, \lambda_{i+1}, \lambda_{i+2}, \dots) \,;\end{aligned}$$ see [\[fig:operations\]](#fig:operations){reference-type="ref" reference="fig:operations"}. With this notation, the complete list of Plücker relations can be written as follows [@carrell_goulden10 Theorem 4.1]: $\lambda = \,\ydiagram{3,2}$ $\lambda^{(3)} = \,\ydiagram{2,2,2}$ $\lambda^{(-2)} = \,\ydiagram{4}$ $$\label{eq:prelspartitions} \sum_{\substack{i,j \geq 1, \\ |\lambda^{(-i)}| + |\mu^{(j)}| = |\lambda| + |\mu|+ 1}} (-1)^{|\mu|-|\mu^{(j)}|+i+j} \Delta^{\lambda^{(-i)}} \Delta^{\mu^{(j)}} = 0 \qquad{\text{for all partitions $\lambda$ and $\mu$}} \,.$$ (The condition $|\lambda^{(i)}| + |\mu^{(-j)}| = |\lambda| + |\mu|+ 1$ implies that the sum is finite.) For example, taking $\lambda = 0$ and $\mu = 3$ yields the Plücker relation [[([\[eq:firstplucker\]](#eq:firstplucker){reference-type="ref" reference="eq:firstplucker"})]{.upright}](#eq:firstplucker). [Theorem 3](#thm:main){reference-type="ref" reference="thm:main"}[\[main_pluckers\]](#main_pluckers){reference-type="ref" reference="main_pluckers"} asserts that for all $t \in \mathbb{C}$, the operators $\beta^\lambda(t)$ satisfy all of the equations [[([\[eq:prelspartitions\]](#eq:prelspartitions){reference-type="ref" reference="eq:prelspartitions"})]{.upright}](#eq:prelspartitions). Since $\beta^\lambda(t) = 0$ for $|\lambda| > n$, this is equivalent to asserting that they satisfy the Plücker relations [[([\[eq:prels\]](#eq:prels){reference-type="ref" reference="eq:prels"})]{.upright}](#eq:prels) for $\mathrm{Gr}(n,2n)$. Furthermore, it suffices to prove these for $t=0$, since $\beta^\lambda \mapsto \beta^\lambda(t)$ under the change of parameters $(z_1, \dots, z_n) \mapsto (z_1+t, \dots, z_n+t)$. ### Single-column and single-row Plücker relations {#sec:scprels} The Plücker relations [[([\[eq:prels\]](#eq:prels){reference-type="ref" reference="eq:prels"})]{.upright}](#eq:prels) corresponding to $I = (1,2,\dots, d+1)$ for some Grassmannian $\mathrm{Gr}(d,m)$ will play a special role. We refer to these as the ***single-column Plücker relations***, since when they are rewritten in terms of partitions, the first of the two indexing partitions (i.e. $\lambda^{(-i)}$) has at most one column. Equivalently, these are the Plücker relations [[([\[eq:prelspartitions\]](#eq:prelspartitions){reference-type="ref" reference="eq:prelspartitions"})]{.upright}](#eq:prelspartitions) corresponding to $\lambda = 0$. For example, [[([\[eq:firstplucker\]](#eq:firstplucker){reference-type="ref" reference="eq:firstplucker"})]{.upright}](#eq:firstplucker) is a single-column relation. One of the main steps in the proof of [Theorem 3](#thm:main){reference-type="ref" reference="thm:main"}[\[main_pluckers\]](#main_pluckers){reference-type="ref" reference="main_pluckers"} will be to show explicitly that the operators $\beta^\lambda$ satisfy all of the single-column Plücker relations. Similarly, the ***single-row Plücker relations*** are the relations [[([\[eq:prels\]](#eq:prels){reference-type="ref" reference="eq:prels"})]{.upright}](#eq:prels) corresponding to $J = (1,2, \dots, d-1)$ for some Grassmannian $\mathrm{Gr}(d,m)$, or equivalently, the relations [[([\[eq:prelspartitions\]](#eq:prelspartitions){reference-type="ref" reference="eq:prelspartitions"})]{.upright}](#eq:prelspartitions) corresponding to $\mu = 0$. The single-row Plücker relations will play a role in our discussion of bases for the spaces $V_E$ in [5.2](#sec:bases){reference-type="ref" reference="sec:bases"}. ## Schubert varieties and the Wronski map For partitions $\mu \subseteq \lambda$, put $|\lambda/\mu| := |\lambda| - |\mu|$, and let $\lambda/\mu$ be the array of $|\lambda/\mu|$ boxes formed by the set difference of the diagrams of $\lambda$ and $\mu$. We define $\mathsf{f}^{\lambda/\mu}$ as the number of ***standard Young tableaux*** of shape $\lambda/\mu$, that is, the number of ways to fill the boxes of $\lambda/\mu$ with the numbers $1, \dots, |\lambda/\mu|$ (each used exactly once) such that numbers increase along both rows (left to right) and columns (top to bottom). In particular, $\mathsf{f}^{\lambda} := \mathsf{f}^{\lambda/0}$ is the number of standard Young tableaux of shape $\lambda$. These numbers play a prominent role in describing the geometry of the Wronski map, arising as both degrees of Wronski maps, and as coefficients in explicit formulas in terms of Plücker coordinates. For any $d \ge \ell(\lambda)$, we have the following formula for $\mathsf{f}^{\lambda}$ [@sagan01 Exercise 3.20]: $$\label{eq:hookformula} \frac{\mathsf{f}^{\lambda}}{|\lambda|!} = \frac{\prod_{1 \le i < j \le d}(\lambda_i - i - \lambda_j + j)}{\prod_{i=1}^d(\lambda_i - i + d)!}\,.$$ We also have a determinantal formula for $\mathsf{f}^{\lambda/\mu}$ [@stanley99 Corollary 7.16.3]: $$\label{eq:numsyt} \frac{\mathsf{f}^{\lambda/\mu}}{|\lambda/\mu|!} = \det \left( \frac{1}{(\lambda_i -i -\mu_j+j)!} \right)_{1 \leq i,j \leq d} \,,$$ where $|\lambda/\mu| := |\lambda|-|\mu|$, and by convention, $\frac{1}{k!} := 0$ if $k$ is a negative integer. ### Schubert varieties {#sec:schubert} A ***complete flag*** in $\mathbb{C}_{m-1}[u]$ is a tuple $F_\bullet : F_0 \subsetneq \dots \subsetneq F_m$ of nested subspaces, where $\dim F_j = j$ for all $j$. For each partition $\lambda \subseteq {\mathchoice% {\scalebox{1.35}{\raisebox{-0.02em}{$\sqsubset\!\!\sqsupset$}}} {\scalebox{1.35}{\raisebox{-0.02em}{$\sqsubset\!\!\sqsupset$}}} {\sqsubset\!\!\sqsupset} {\sqsubset\!\!\sqsupset} }$, we have a ***Schubert variety*** of $\mathrm{Gr}(d,m)$ relative to $F_\bullet$: $$X_\lambda F_\bullet := \{V \in \mathrm{Gr}(d,m) \mid \dim(V \cap F_{m-d+i-\lambda_i}) \geq i) \text{ for all $i \in [d]$}\} \,.$$ A ***Schubert problem*** on $\mathrm{Gr}(d,m)$ is to find the points (or just the number of points) in an intersection of the form $$\label{eq:generalschubertintersection} \smash{\overline{\mathcal{X}}}^{\nu} \cap X_{\mu_1} F^{(1)}_\bullet \cap \dots \cap X_{\mu_s} F^{(s)}_\bullet \,,$$ where $F^{(1)}_\bullet, \dots, F^{(s)}_\bullet$ are flags, and $\nu, \mu_1, \dots, \mu_s$ are partitions such that $|\nu| = |\mu_1| + \cdots + |\mu_s|$. When the intersection [[([\[eq:generalschubertintersection\]](#eq:generalschubertintersection){reference-type="ref" reference="eq:generalschubertintersection"})]{.upright}](#eq:generalschubertintersection) is transverse, the number of points in the intersection is the coefficient of the Schur function $\mathsf{s}_\nu$ in the product $\mathsf{s}_{\mu_1} \dotsm \mathsf{s}_{\mu_s}$ [@fulton97 Section 9.4]. The Schubert problem [[([\[eq:schubertproblem\]](#eq:schubertproblem){reference-type="ref" reference="eq:schubertproblem"})]{.upright}](#eq:schubertproblem) corresponds to the intersection $$\label{eq:schubertintersection} \smash{\overline{\mathcal{X}}}^{\nu} \cap X_1 F^{(1)}_\bullet \cap \dots \cap X_1 F^{(n)}_\bullet,$$ where $n = |\nu|$ and the flags $F^{(1)}_\bullet, \dots, F^{(n)}_\bullet$ are such that $F^{(i)}_{m-d} = W_i$. When the intersection [[([\[eq:schubertintersection\]](#eq:schubertintersection){reference-type="ref" reference="eq:schubertintersection"})]{.upright}](#eq:schubertintersection) is transverse, it contains $\mathsf{f}^{\nu}$ points. For $w \in \mathbb{C}$, let $$F_j(w) := \langle \gamma(w), \gamma'(w), \dots, \gamma^{(j-1)}(w) \rangle = \langle (u+w)^{m-1}, (u+w)^{m-2}, \dots, (u+w)^{m-j} \rangle$$ be the osculating $j$-plane to the moment curve [[([\[eq:moment\]](#eq:moment){reference-type="ref" reference="eq:moment"})]{.upright}](#eq:moment) at $w$. Then $F_\bullet(w) : F_0(w) \subsetneq \dots \subsetneq F_m(w)$ is a complete flag, called the ***osculating flag*** to $\gamma$ at $w$. For these flags, we use the shorthand notation $$X_\lambda(w) := X_\lambda F_\bullet(w) \,.$$ We also put $F_\bullet(\infty) := \lim_{w \to \infty} F_\bullet(w)$, which is just the ***standard flag***, with $F_j(\infty) = \langle 1, u, \dots, u^{j-1} \rangle$. The Schubert variety $\smash{\overline{\mathcal{X}}}^{\nu}$ is in fact $$\smash{\overline{\mathcal{X}}}^{\nu}= X_{\nu^\vee} (\infty),$$ where $\nu^\vee := (m-d-\nu_d, \dots, m-d-\nu_1)$ is the complement of $\nu$ inside ${\mathchoice% {\scalebox{1.35}{\raisebox{-0.02em}{$\sqsubset\!\!\sqsupset$}}} {\scalebox{1.35}{\raisebox{-0.02em}{$\sqsubset\!\!\sqsupset$}}} {\sqsubset\!\!\sqsupset} {\sqsubset\!\!\sqsupset} }$. Our conventions are such that $\mathop{\mathrm{codim}}X_\lambda F_\bullet = |\lambda|$ for any flag $F_\bullet$, whereas $\dim \smash{\overline{\mathcal{X}}}^{\nu}= \dim \mathcal{X}^{\nu}= |\nu|$. In terms of Plücker coordinates, $\mathcal{X}^{\nu}$ and $\smash{\overline{\mathcal{X}}}^{\nu}$ have straightforward descriptions: **Proposition 19** ([@fulton97 Section 9.4]). *The Schubert variety $\smash{\overline{\mathcal{X}}}^{\nu}$ is the closed subscheme of $\mathrm{Gr}(d,m)$ defined by $\Delta^\lambda = 0$ for all $\lambda \not \subseteq \nu$. The Schubert cell $\mathcal{X}^{\nu}$ is the open subscheme of $\smash{\overline{\mathcal{X}}}^{\nu}$ defined by $\Delta^\nu \neq 0$.* Henceforth, when we refer to the Plücker coordinates of a $d$-plane $V \in \smash{\overline{\mathcal{X}}}^{\nu}$, we will frequently consider only the Plücker coordinates indexed by partitions $\lambda \subseteq \nu$, and write these as $[\Delta^\lambda : \lambda \subseteq \nu]$. By [Proposition 19](#thm:schubertplucker){reference-type="ref" reference="thm:schubertplucker"} all other Plücker coordinates are zero. If $V \in \mathcal{X}^{\nu}$, define the ***normalized Plücker coordinates*** of $V$ to be the unique scaling $(\Delta^\lambda : \lambda \subseteq \nu)$ of the Plücker coordinates such that $\Delta^\nu = \frac{|\nu|!}{\mathsf{f}^{\nu}}$. **Remark 20**. More generally, every Schubert variety $X_\lambda F_\bullet$ is defined as a scheme by a system of linear equations in the Plücker coordinates (we omit the proof). For example, $X_\mu(0)$ is defined by the equations $\Delta^\lambda = 0$ for all $\lambda \not \supseteq \mu$. ### The Wronski map on $\mathrm{Gr}(d,m)$ Recall that if $V$ is a finite-dimensional vector space of polynomials, $\mathrm{Wr}(V)$ is defined to the monic polynomial which is a scalar multiple of $$\mathrm{Wr}(f_1, \dots, f_d) := \begin{vmatrix} f_1 & f_1' & f_1'' & \dots & f_1^{(d-1)} \\ f_2 & f_2' & f_2'' & \dots & f_2^{(d-1)} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\\ f_d & f_d' & f_d'' & \dots & f_d^{(d-1)} \\ \end{vmatrix} \,,$$ where $(f_1, \dots, f_d)$ is any basis for $V$. If $V \in \mathrm{Gr}(d,m)$, then $\mathrm{Wr}(V)$ is a polynomial of degree at most $d(m-d)$, and we obtain a projective morphism $\mathrm{Wr}: \mathrm{Gr}(d,m) \to \mathbb{P}^{d(m-d)}$ called the ***Wronski map*** on $\mathrm{Gr}(d,m)$. Here $\mathbb{P}^{d(m-d)}$ is identified with the projectivization of $\mathbb{C}_{d(m-d)}[u]$ via [[([\[eq:isomorphism\]](#eq:isomorphism){reference-type="ref" reference="eq:isomorphism"})]{.upright}](#eq:isomorphism). Eisenbud and Harris [@eisenbud_harris83 Theorem 2.3] showed that this is a finite morphism of degree $\mathsf{f}^{{\mathchoice% {\scalebox{1.35}{\raisebox{-0.02em}{$\sqsubset\!\!\sqsupset$}}} {\scalebox{1.35}{\raisebox{-0.02em}{$\sqsubset\!\!\sqsupset$}}} {\sqsubset\!\!\sqsupset} {\sqsubset\!\!\sqsupset} }}$, the number of standard Young tableaux of shape ${\mathchoice% {\scalebox{1.35}{\raisebox{-0.02em}{$\sqsubset\!\!\sqsupset$}}} {\scalebox{1.35}{\raisebox{-0.02em}{$\sqsubset\!\!\sqsupset$}}} {\sqsubset\!\!\sqsupset} {\sqsubset\!\!\sqsupset} }$. Restricting the Wronski map to the Schubert cell $\mathcal{X}^{\nu}$, we obtain a finite morphism of affine schemes $\mathrm{Wr}: \mathcal{X}^{\nu}\to \mathcal{P}_n$, where $\mathcal{P}_n$ is the space of monic polynomials of degree $n = |\nu|$. In this case, both the domain and codomain are isomorphic to $n$-dimensional affine space, and the restricted Wronski map has degree $\mathsf{f}^{\nu}$. We have an explicit formula for $\mathrm{Wr}(V)$ in terms of Plücker coordinates: **Proposition 21**. *Let $V \in \mathcal{X}^{\nu}$, where $\nu \vdash n$. If $(\Delta^\lambda : \lambda \subseteq \nu)$ are the normalized Plücker coordinates of $V$, then $$\label{eq:pluckerwronskian} \mathrm{Wr}(V) = \sum_{\lambda \subseteq \nu} \frac{\mathsf{f}^{\lambda}}{|\lambda|!} \Delta^\lambda u^{|\lambda|} \,.$$* *Proof.* By [@purbhoo10 Proposition 2.3], we have $$\mathrm{Wr}(V) = \sum_{\lambda \subseteq \nu}\frac{\prod_{1 \le i < j \le d}(\lambda_i - i - \lambda_j + j)}{\prod_{i=1}^d(\lambda_i - i + d)!} \Delta^\lambda u^{|\lambda|} \,.$$ (The denominator $\prod_{i=1}^d(\lambda_i - i + d)!$ above accounts for the fact that [@purbhoo10] uses a different convention for the isomorphism $\mathbb{C}^m \xrightarrow{\simeq} \mathbb{C}_{m-1}[u]$; namely, the denominator $(j-1)!$ in [[([\[eq:isomorphism\]](#eq:isomorphism){reference-type="ref" reference="eq:isomorphism"})]{.upright}](#eq:isomorphism) is omitted.) We obtain [[([\[eq:pluckerwronskian\]](#eq:pluckerwronskian){reference-type="ref" reference="eq:pluckerwronskian"})]{.upright}](#eq:pluckerwronskian) by applying [[([\[eq:hookformula\]](#eq:hookformula){reference-type="ref" reference="eq:hookformula"})]{.upright}](#eq:hookformula). ◻ The Wronski map is closely related to Schubert varieties for osculating flags of $\gamma$: **Proposition 22**. *Let $V \in \mathrm{Gr}(d,m)$.* (i) *[\[schubertwronskian2_divides\]]{#schubertwronskian2_divides label="schubertwronskian2_divides"} For $w \in \mathbb{C}$, $(u+w)^k$ divides $\mathrm{Wr}(V)$ if and only if $V \in X_\lambda(w)$ for some $\lambda \vdash k$, $\lambda \subseteq {\mathchoice% {\scalebox{1.35}{\raisebox{-0.02em}{$\sqsubset\!\!\sqsupset$}}} {\scalebox{1.35}{\raisebox{-0.02em}{$\sqsubset\!\!\sqsupset$}}} {\sqsubset\!\!\sqsupset} {\sqsubset\!\!\sqsupset} }$.* (ii) *[\[schubertwronskian2_infinity\]]{#schubertwronskian2_infinity label="schubertwronskian2_infinity"} We have $\deg \mathrm{Wr}(V) = n$ if and only if $V \in \mathcal{X}^{\nu}$ for some $\nu \vdash n$, $\nu \subseteq {\mathchoice% {\scalebox{1.35}{\raisebox{-0.02em}{$\sqsubset\!\!\sqsupset$}}} {\scalebox{1.35}{\raisebox{-0.02em}{$\sqsubset\!\!\sqsupset$}}} {\sqsubset\!\!\sqsupset} {\sqsubset\!\!\sqsupset} }$.* *Proof.* By translation, it suffices to prove part [\[schubertwronskian2_divides\]](#schubertwronskian2_divides){reference-type="ref" reference="schubertwronskian2_divides"} when $w = 0$. This case follows from [[([\[eq:pluckerwronskian\]](#eq:pluckerwronskian){reference-type="ref" reference="eq:pluckerwronskian"})]{.upright}](#eq:pluckerwronskian), using [Remark 20](#rmk:0schubertconditions){reference-type="ref" reference="rmk:0schubertconditions"}. Part [\[schubertwronskian2_infinity\]](#schubertwronskian2_infinity){reference-type="ref" reference="schubertwronskian2_infinity"} follows directly from [[([\[eq:pluckerwronskian\]](#eq:pluckerwronskian){reference-type="ref" reference="eq:pluckerwronskian"})]{.upright}](#eq:pluckerwronskian). ◻ The $k=1$ case of part [\[schubertwronskian2_divides\]](#schubertwronskian2_divides){reference-type="ref" reference="schubertwronskian2_divides"} of [Proposition 22](#prop:schubertwronskian2){reference-type="ref" reference="prop:schubertwronskian2"} is equivalent to [Proposition 6](#prop:schubertwronskian){reference-type="ref" reference="prop:schubertwronskian"}. Part [\[schubertwronskian2_infinity\]](#schubertwronskian2_infinity){reference-type="ref" reference="schubertwronskian2_infinity"} should be regarded as the $w=\infty$ case of part [\[schubertwronskian2_divides\]](#schubertwronskian2_divides){reference-type="ref" reference="schubertwronskian2_divides"}. As a consequence of [Proposition 22](#prop:schubertwronskian2){reference-type="ref" reference="prop:schubertwronskian2"}, every fibre of the Wronski map can be described as a union of Schubert intersections. Namely, suppose $g(u) = (u+z_1)^{\kappa_1} \dotsm (u+z_s)^{\kappa_s}$, where $z_1, \dots, z_s$ are distinct complex numbers and $\kappa_1+ \dots + \kappa_s = n$. Then for the Wronski map $\mathrm{Wr}: \mathcal{X}^{\nu}\to \mathcal{P}_n$, we have $$\label{eq:fibre} \mathrm{Wr}^{-1}(g) = \bigcup_{\mu_1 \vdash \kappa_1, \dots, \mu_s \vdash \kappa_s} \smash{\overline{\mathcal{X}}}^{\nu}\cap X_{\mu_1}(z_1) \cap \dots \cap X_{\mu_s}(z_s) \,.$$ In general, this equality only holds set-theoretically. It also holds scheme-theoretically if and only if $\nu$ equals $n$ or $1^n$, or $\kappa_1, \dots, \kappa_s \leq 2$ (i.e. $g(u)$ has no roots of multiplicity greater than $2$); see [Corollary 74](#cor:schemeequality){reference-type="ref" reference="cor:schemeequality"}. As we will discuss in [5.1.3](#sec:dimensions){reference-type="ref" reference="sec:dimensions"}, this accounts for the discrepancy between the scheme structures of $\mathrm{Wr}^{-1}(g)$ and $\mathop{\mathrm{Spec}}\mathcal{B}_\nu(z_1, \dots, z_n)$. ### $\mathrm{PGL}_2(\mathbb{C})$-action and translation action For all $k \geq 0$, the group $\mathrm{GL}_2(\mathbb{C})$ acts on $\mathbb{C}_k[u]$ by Möbius transformations. If $\phi = \big(\begin{smallmatrix} \phi_{11} & \phi_{12} \\ \phi_{21} & \phi_{22} \end{smallmatrix}\big) \in \mathrm{GL}_2(\mathbb{C})$ and $f(u) \in \mathbb{C}_k[u]$, the action is given by $$\phi f(u) := (\phi_{21} u + \phi_{11})^k f\Big(\frac{\phi_{22} u + \phi_{12}}{\phi_{21} u + \phi_{11}}\Big) \,.$$ This induces a $\mathrm{PGL}_2(\mathbb{C})$-action on $\mathrm{Gr}(d,m)$ and $\mathbb{P}^{d(m-d)}$. The Wronski map on $\mathrm{Gr}(d,m)$ is $\mathrm{PGL}_2(\mathbb{C})$-equivariant with respect to these actions. The additive group $(\mathbb{C},+)$ is isomorphic the unipotent subgroup consisting of matrices of the form $(\begin{smallmatrix} 1 & t \\ 0 & 1 \end{smallmatrix})$, and therefore also acts on $\mathbb{C}_k[u]$ and $\mathrm{Gr}(d,m)$. We call this the ***translation action***, since $(\begin{smallmatrix} 1 & t \\ 0 & 1 \end{smallmatrix}) f(u) = f(u+t)$. For $V \in \mathrm{Gr}(d,m)$, we write $$V(t) := (\begin{smallmatrix} 1 & t \\ 0 & 1 \end{smallmatrix}) V \,.$$ Note that $(\begin{smallmatrix} 1 & t \\ 0 & 1 \end{smallmatrix}) X_\lambda(w) = X_\lambda(w+t)$. Furthermore, if $\mathrm{Wr}(V) = g(u)$, then $\mathrm{Wr}(V(t)) = g(u+t)$, and so the translation action of gives an isomorphism between the fibres of the Wronski map over $g(u)$ and $g(u+t)$. The translation action also preserves the Schubert cell $\mathcal{X}^{\nu}$. Explicitly, in terms of Plücker coordinates, we have: **Proposition 23**. *Let $V \in \mathcal{X}^{\nu}$, where $\nu \vdash n$. For $t \in \mathbb{C}$, let $(\Delta^\lambda(t) : \lambda \subseteq \nu)$ be the normalized Plücker coordinates of $V(t)$, and let $\Delta^\lambda(t) = 0$ for $\lambda \not \subseteq \nu$. Then for all $s, t \in \mathbb{C}$ and all partitions $\mu$, we have $$\label{eq:geometrictranslation} \Delta^\mu(s+t) = \sum_{\lambda \supseteq \mu} \frac{\mathsf{f}^{\lambda/\mu}}{|\lambda/\mu|!} \Delta^\lambda(s) t^{|\lambda/\mu|} \,.$$* *Proof.* Let $\phi_t\in\mathrm{End}(\mathbb{C}_{m-1}[u])$ act by translation by $t$. We can represent $\phi_t$ as an $m\times m$ matrix, whose $(i,j)$-entry we calculate via [[([\[eq:isomorphism\]](#eq:isomorphism){reference-type="ref" reference="eq:isomorphism"})]{.upright}](#eq:isomorphism): $$(\phi_t)_{i,j} = \Big[\frac{u^{i-1}}{(i-1)!}\Big]\,\frac{(u+t)^{j-1}}{(j-1)!}\, = \frac{1}{(j-i)!}\,,$$ where the operator $[\frac{u^{i-1}}{(i-1)!}]$ extracts the coefficient of $\frac{u^{i-1}}{(i-1)!}$. If $A$ is a $d\times m$ matrix which represents $V(s)$, then $V(s+t)$ is represented by $A\phi_t^T$. We obtain [[([\[eq:geometrictranslation\]](#eq:geometrictranslation){reference-type="ref" reference="eq:geometrictranslation"})]{.upright}](#eq:geometrictranslation) by applying the Cauchy--Binet identity and [[([\[eq:numsyt\]](#eq:numsyt){reference-type="ref" reference="eq:numsyt"})]{.upright}](#eq:numsyt). ◻ Note that [[([\[eq:geometrictranslation\]](#eq:geometrictranslation){reference-type="ref" reference="eq:geometrictranslation"})]{.upright}](#eq:geometrictranslation) has the same form as the translation identity [[([\[eq:translationidentity\]](#eq:translationidentity){reference-type="ref" reference="eq:translationidentity"})]{.upright}](#eq:translationidentity). This explains the significance of the translation identity: it asserts that the operators $\beta^\lambda(t)$ behave exactly like Plücker coordinates under the translation action. We also note the similarity between equations [[([\[eq:pluckerwronskian\]](#eq:pluckerwronskian){reference-type="ref" reference="eq:pluckerwronskian"})]{.upright}](#eq:pluckerwronskian) and [[([\[eq:geometrictranslation\]](#eq:geometrictranslation){reference-type="ref" reference="eq:geometrictranslation"})]{.upright}](#eq:geometrictranslation). In the notation of [Proposition 23](#prop:geometrictranslation){reference-type="ref" reference="prop:geometrictranslation"}, equation [[([\[eq:pluckerwronskian\]](#eq:pluckerwronskian){reference-type="ref" reference="eq:pluckerwronskian"})]{.upright}](#eq:pluckerwronskian) says that $\mathrm{Wr}(V) = \Delta^0(u)$. ### Inclusions of Grassmannians Let $\nu \vdash n$ be a partition such that $\nu \subseteq {\mathchoice% {\scalebox{1.35}{\raisebox{-0.02em}{$\sqsubset\!\!\sqsupset$}}} {\scalebox{1.35}{\raisebox{-0.02em}{$\sqsubset\!\!\sqsupset$}}} {\sqsubset\!\!\sqsupset} {\sqsubset\!\!\sqsupset} }= (m-d)^d$. Then $\mathcal{X}^{\nu}$ can be regarded as Schubert cell of $\mathrm{Gr}(d,m)$, or of any other Grassmannian $\mathrm{Gr}(d',m')$ with $d' \geq d$ and $m'-d' \geq m-d$. We now argue that it does not matter which Grassmannian we choose to work inside, as far as the geometry of the Schubert variety, Plücker coordinates, total positivity, the Wronski map, and the translation action are concerned. In the following proposition, we write $\smash{\overline{\mathcal{X}}}^{\nu}_{d',m'}$ to specifically mean the Schubert variety $\smash{\overline{\mathcal{X}}}^{\nu}\subseteq \mathrm{Gr}(d',m')$, and $\mathrm{Wr}_{d',m'}$ to mean the Wronski map on $\mathrm{Gr}(d',m')$. **Proposition 24**. *Let $\imath_1 : \mathrm{Gr}(d,m) \hookrightarrow \mathrm{Gr}(d, m+1)$ and $\imath_2 : \mathrm{Gr}(d,m) \hookrightarrow \mathrm{Gr}(d+1, m+1)$ denote the inclusions of Grassmannians defined by $$\imath_1(V) := V \qquad\text{and}\qquad \imath_2(V) := \{f(u) \in \mathbb{C}[u] \mid f'(u) \in V\} \,.$$* (i) *[\[changegr1\]]{#changegr1 label="changegr1"} The maps $\imath_1$ and $\imath_2$ restrict to isomorphisms of Schubert varieties $$\imath_1 : \smash{\overline{\mathcal{X}}}^{\nu}_{d,m} \xrightarrow{\simeq} \smash{\overline{\mathcal{X}}}^{\nu}_{d,m+1} \qquad\text{and}\qquad \imath_2 : \smash{\overline{\mathcal{X}}}^{\nu}_{d,m} \xrightarrow{\simeq} \smash{\overline{\mathcal{X}}}^{\nu}_{d+1,m+1} \,.$$* (ii) *[\[changegr2\]]{#changegr2 label="changegr2"} Both isomorphisms in [\[changegr1\]](#changegr1){reference-type="ref" reference="changegr1"} are, in terms of Plücker coordinates, defined by the identity map: $\Delta^\lambda \mapsto \Delta^\lambda$, $\lambda \subseteq \nu$. That is, $V$, $\imath_1(V)$, and $\imath_2(V)$ all have the same Plücker coordinates.* (iii) *[\[changegr3\]]{#changegr3 label="changegr3"} If any one of the three elements $V$, $\imath_1(V)$, and $\imath_2(V)$ is totally nonnegative (respectively, totally positive in its Schubert cell), then so are the other two.* (iv) *[\[changegr4\]]{#changegr4 label="changegr4"} $\mathrm{Wr}_{d,m+1} \circ \imath_1 = \mathrm{Wr}_{d,m}$ and $\mathrm{Wr}_{d+1,m+1} \circ \imath_2 = \mathrm{Wr}_{d,m}$.* (v) *[\[changegr5\]]{#changegr5 label="changegr5"} Both $\imath_1$ and $\imath_2$ are equivariant with respect to the translation action.* *Proof.* We can verify parts [\[changegr4\]](#changegr4){reference-type="ref" reference="changegr4"} and [\[changegr5\]](#changegr5){reference-type="ref" reference="changegr5"} directly. Parts [\[changegr1\]](#changegr1){reference-type="ref" reference="changegr1"}--[\[changegr3\]](#changegr3){reference-type="ref" reference="changegr3"} follow from the fact that if $V\in\mathrm{Gr}(d,m)$ is represented by the $d\times m$ matrix $A$, then $\imath_1(V)$ and $\imath_2(V)$ are represented by the matrices $$\begin{pmatrix} & & & 0 \\ & A & & \vdots \\ & & & 0 \end{pmatrix} \qquad\text{and}\qquad \begin{pmatrix} 1 & 0 & \cdots & 0 \\ 0 & & & \\ \vdots & & A & \\ 0 & & & \end{pmatrix},$$ respectively. ◻ Note that the maps $\imath_1$ and $\imath_2$ are not $\mathrm{PGL}_2(\mathbb{C})$-equivariant, and therefore if we are concerned with the full $\mathrm{PGL}_2(\mathbb{C})$-action, the choice of Grassmannian is still relevant. However, for the most part, we will only be concerned with the translation action. We can then regard $\mathcal{X}^{\nu}$ as a subvariety of any sufficiently large Grassmannian $\mathrm{Gr}(d,m)$, and the geometry we consider will independent of this choice. When $\nu \vdash n$, it will be particularly convenient to work inside $\mathrm{Gr}(n,2n)$. ### Fundamental differential operators We now recall some background from [@purbhoo]. Consider linear differential operators of the form $$D = \psi_0(u) \partial_u^d + \psi_1(u)\partial_u^{d-1} + \dots + \psi_{d-1}(u)\partial_u+ \psi_d(u) \,,$$ where $\psi_0(u), \dots, \psi_d(u) \in \mathbb{C}(u)$ and $\partial_u$ is the differentiation operator. The operator $D$ acts on functions $f(u)$ as $$Df(u) := \psi_0(u) f^{(d)}(u) + \psi_1(u) f^{(d-1)}(u) + \dots + \psi_{d-1}(u) f'(u) + \psi_d(u) f(u) \,.$$ If $\psi_0(u) \neq 0$, we say that $D$ has ***order*** $d$, and $D$ is ***monic*** if $\psi_0(u) = 1$. We will only be concerned with solutions to $Df = 0$ where $f$ is a polynomial. Define $$\mathop{\mathrm{pker}}D := \{f(u) \in \mathbb{C}[u] \mid Df(u) = 0\} \,.$$ A basic fact about a linear differential operator is that its kernel has dimension less than or equal to its order (see, e.g., [@ince27 Section 3.3.2]), so in particular, we have: **Proposition 25**. *If $D$ is a nonzero linear differential operator of order $d$ with coefficients in $\mathbb{C}(u)$, then $$\dim \mathop{\mathrm{pker}}D \leq d \,.$$* Now suppose that $V \in \mathcal{X}^{\nu}\subseteq \mathrm{Gr}(d,m)$, and let $(f_1, \dots, f_d)$ be any basis for $V$. Consider $$D_Vf := \frac{\mathrm{Wr}(f_1, \dots, f_d, f)}{\mathrm{Wr}(f_1, \dots, f_d)} \,.$$ Expanding the determinant for $\mathrm{Wr}(f_1, \dots, f_d, f)$ along the bottom row, we see that $D_V$ is a monic linear differential operator of order $d$ with coefficients in $\mathbb{C}(u)$. Specifically, we can write $D_V$ in the form $$D_V = \frac{1}{\mathrm{Wr}(V)} \sum_{k=0}^d \sum_{\ell =0}^{n-k} (-1)^k \psi_{k,\ell} u^{n-k-\ell} \partial_u^{d-k} \,,$$ where $\psi_{k,\ell} \in \mathbb{C}$, and $n = |\nu|$. $D_V$ is called the ***fundamental differential operator*** of $V$. **Proposition 26**. *$D_V$ is the unique monic linear operator of order $d = \dim V$ with the property that $V = \mathop{\mathrm{pker}}D_V$.* *Proof.* Let $(f_1, \dots, f_d)$ be a basis for $V$. We have $$D_V f = 0 \qquad\iff\qquad \mathrm{Wr}(f_1, \dots, f_d, f) = 0 \qquad\iff\qquad f\in V,$$ so $V = \mathop{\mathrm{pker}}D_V$. If there were another monic operator $D_V' \neq D_V$ of order $d$ such that $V = \mathop{\mathrm{pker}}D_V'$, then we would have $V \subseteq \mathop{\mathrm{pker}}(D_V - D_V')$, contradicting [Proposition 25](#prop:Dnullity){reference-type="ref" reference="prop:Dnullity"}. ◻ Thus, the numbers ${\mathchoice {\mbox{\boldmath{$\psi$}}} {\mbox{\boldmath{$\psi$}}} {\mbox{\scriptsize\boldmath{$\psi$}}} {\mbox{\tiny\boldmath{$\psi$}}} }:= (\psi_{k,\ell} : 0 \le k \leq d,\ 0 \le \ell \leq n-k)$ uniquely determine (and are uniquely determined by) $V$. We may regard ${\mathchoice {\mbox{\boldmath{$\psi$}}} {\mbox{\boldmath{$\psi$}}} {\mbox{\scriptsize\boldmath{$\psi$}}} {\mbox{\tiny\boldmath{$\psi$}}} }$ as a system of affine coordinates on the Schubert cell $\mathcal{X}^{\nu}$. We refer to these as ***FDO coordinates***, since they come from the fundamental differential operator. Unlike the Plücker coordinates, it is difficult to state the precise relations among FDO coordinates, though in principle it is possible. The main fact we will need is that we can express the normalized Plücker coordinates of $V$ as polynomial functions of ${\mathchoice {\mbox{\boldmath{$\psi$}}} {\mbox{\boldmath{$\psi$}}} {\mbox{\scriptsize\boldmath{$\psi$}}} {\mbox{\tiny\boldmath{$\psi$}}} }$: **Lemma 27**. *Let $\nu \vdash n$. For $V \in \mathcal{X}^{\nu}$, let $(\Delta^\lambda_V : \lambda \subseteq \nu)$ denote the normalized Plücker coordinates of $V$, and let ${\mathchoice {\mbox{\boldmath{$\psi$}}} {\mbox{\boldmath{$\psi$}}} {\mbox{\scriptsize\boldmath{$\psi$}}} {\mbox{\tiny\boldmath{$\psi$}}} }_V$ denote the FDO coordinates of $V$. Then there exist multivariate polynomials $p^\lambda_\nu$ with rational coefficients such that for all $V \in \mathcal{X}^{\nu}$, we have $$\Delta^\lambda_V = p^\lambda_\nu({\mathchoice {\mbox{\boldmath{$\psi$}}} {\mbox{\boldmath{$\psi$}}} {\mbox{\scriptsize\boldmath{$\psi$}}} {\mbox{\tiny\boldmath{$\psi$}}} }_V) \qquad \text{for all $\lambda \subseteq \nu$.}$$* *Proof.* By [@purbhoo Lemma 7.5], there exist formulas for FDO coordinates as polynomial functions of coefficients of a basis for $V$; this change of coordinates is manifestly triangular and therefore polynomially invertible. Hence there exist polynomials which give coefficients of a basis for $V$ in terms of the FDO coordinates. The normalized Plücker coordinates are minors of the coefficient matrix for any basis (up to a global scalar), and hence are also given by polynomials in the FDO coordinates. ◻ **Remark 28**. The polynomials $p^\lambda_\nu$ have high degree and (as described above) depend on $\nu$. By contrast, the formula for changing from Plücker coordinates to FDO coordinates is linear and independent of $\nu$. Specifically, let $(\Delta^\lambda(t) : \lambda \subseteq \nu)$ be the normalized Plücker coordinates of $V(t)$, which are given by [[([\[eq:geometrictranslation\]](#eq:geometrictranslation){reference-type="ref" reference="eq:geometrictranslation"})]{.upright}](#eq:geometrictranslation) with $s=0$. Then the FDO coordinates of $V$ are given by the formula $$\psi_{k,\ell} = [u^{n-k-\ell}]\Delta^{1^k}(u) \,.$$ This can be proved directly from the definition; it also follows from [Lemma 66](#lem:FDOcoords){reference-type="ref" reference="lem:FDOcoords"}. ## Symmetric functions Let $\Lambda = \Lambda_\mathbb{C}(\mathbf{x})$ denote the algebra of symmetric functions over $\mathbb{C}$, in infinitely many variables $x_1, x_2, \dots$. We assume the reader is familiar with symmetric functions; we refer to [@stanley99 Chapter 7] for background. There are several standard bases for $\Lambda$, indexed by partitions: the monomial basis $(\mathsf{m}_\lambda)$, the elementary basis $(\mathsf{e}_\lambda)$, the homogeneous basis $(\mathsf{h}_\lambda)$, the power sum basis $(\mathsf{p}_\lambda)$, and the Schur basis $(\mathsf{s}_\lambda)$. It is sometimes useful to regard $\Lambda$ as being a free polynomial algebra in the power sum generators, $\Lambda = \mathbb{C}[\mathsf{p}_1, \mathsf{p}_2, \mathsf{p}_3, \dots]$. We will also sometimes consider the formal completion $\widehat \Lambda$ of $\Lambda$, in which infinite linear combinations of the basis elements of $\Lambda$ are allowed; equivalently, $\widehat \Lambda = \mathbb{C}[[\mathsf{p}_1, \mathsf{p}_2, \mathsf{p}_3, \dots]]$. The Hall inner product $\langle \cdot, \cdot \rangle$ on $\Lambda$ is the unique Hermitian inner product such that the Schur functions form an orthonormal basis: $\langle \mathsf{s}_\lambda, \mathsf{s}_\mu \rangle = \delta_{\lambda,\mu}$. We also have $\langle \mathsf{m}_\lambda, \mathsf{h}_\mu \rangle = \delta_{\lambda,\mu}$, and $\langle \mathsf{p}_\lambda, \mathsf{p}_\mu \rangle = 0$ for $\lambda \neq \mu$. The inner product $\langle \mathsf{p}_\lambda, \mathsf{p}_\lambda \rangle$ is a positive integer, denoted $\mathsf{z}_\lambda$. We take the convention that $\langle \cdot, \cdot \rangle$ is linear in the first component and antilinear in the second component (unlike in [@stanley99 Section 7.9], where $\langle\cdot,\cdot\rangle$ is bilinear). However, in practice, the symmetric functions appearing in the second component will always have rational coefficients. ### Representation theory of $\mathfrak{S}_{n}$ {#sec:repthy} We recall some background from [@sagan01]. For a permutation $\sigma \in \mathfrak{S}_{n}$, let $\mathsf{cyc}(\sigma)$ denote the cycle type of $\sigma$, which is a partition of $n$. For $\lambda \vdash n$, write $\mathcal{C}_\lambda = \{\sigma \in S_n \mid \mathsf{cyc}(\sigma) = \lambda\}$ for the $\mathfrak{S}_{n}$-conjugacy class consisting of all permutations of cycle type $\lambda$. We have $\mathsf{z}_\lambda = \langle \mathsf{p}_\lambda, \mathsf{p}_\lambda \rangle = \frac{n!}{|\mathcal{C}_\lambda|}$. If $X \subseteq [n]$ and $\sigma \in \mathfrak{S}_{X} \subseteq \mathfrak{S}_{n}$, we write $\mathsf{cyc}(\sigma_X)$ to mean the cycle type of $\sigma$ regarded as a permutation of the set $X$. Hence, $\mathsf{cyc}(\sigma_X) \vdash |X|$ and we do not count fixed points in $[n] \setminus X$ as part of the cycle type. For $\lambda \vdash n$, we have the Specht module $M^{\lambda}$, which is the irreducible representation of $\mathfrak{S}_{n}$ associated to $\lambda$. The character of $M^{\lambda}$ is denoted $\chi^\lambda : \mathfrak{S}_{n}\to \mathbb{C}$, and $\dim M^{\lambda} = \chi^\lambda(\mathbbm{1}_{\mathfrak{S}_{n}}) = \mathsf{f}^{\lambda}$. The Frobenius character formula states that $\chi^\lambda(\sigma) = \langle \mathsf{s}_\lambda, \mathsf{p}_{\mathsf{cyc}(\sigma)} \rangle$. We write $\chi^\lambda_\mu := \langle \mathsf{s}_\lambda, \mathsf{p}_\mu \rangle = \chi^\lambda(\sigma)$ for any $\sigma \in \mathcal{C}_\mu$, so that we have the change of basis formulas $$\label{eq:spchangeofbasis} \mathsf{s}_\lambda = \sum_{\mu \vdash |\lambda|} \frac{\chi^\lambda_\mu}{\mathsf{z}_\mu}\mathsf{p}_\mu \qquad \text{and} \qquad \mathsf{p}_\mu = \sum_{\lambda \vdash |\mu|} \chi^\lambda_\mu \mathsf{s}_\lambda \,.$$ Also, since conjugacy classes of $\mathfrak{S}_{n}$ are closed under inversion, $\chi^\lambda(\sigma) = \chi^\lambda(\sigma^{-1})$ for all $\sigma\in\mathfrak{S}_{n}$; equivalently, $\chi^\lambda$ is real-valued. More generally, if $X \subseteq [n]$ and $\lambda \vdash |X|$, we define $\chi^\lambda : \mathfrak{S}_{X} \to \mathbb{C}$, by identifying $\mathfrak{S}_{X}$ with $\mathfrak{S}_{|X|}$. We denote the Specht module of $\mathfrak{S}_{X}$ corresponding to $\lambda$ as $M^{\lambda}_X$, which is identified with $M^{\lambda}$ under the isomorphism $\mathfrak{S}_{X} \simeq \mathfrak{S}_{|X|}$. Let $$\label{eq:alphadef} \alpha^\lambda_X := \sum_{\sigma \in \mathfrak{S}_{X}} \chi^\lambda(\sigma) \sigma \in \mathbb{C}[\mathfrak{S}_{X}] \,.$$ Properties of the $\alpha^\lambda_X$'s will allow us to prove [Proposition 12](#prop:betapsd){reference-type="ref" reference="prop:betapsd"}, using the fact that $$\label{eq:alphatobeta} \beta^\lambda = \sum_{\substack{X \subseteq [n], \\ |X| = |\lambda|}} \alpha^\lambda_X \cdot \prod_{i \in [n]\setminus X} z_i \,.$$ Recall that every finite-dimensional representation $N$ of a finite group $G$ admits a Hermitian inner product such that every $g\in G$ acts as a unitary operator [@serre98 Section 1.3]. The following is a result for general $G$ applied to the case $G = \mathfrak{S}_{X}$ (using the fact that $\chi^\lambda$ is real-valued): **Proposition 29** ([@serre98 Theorem 8]). *Let $N$ be a finite-dimensional representation of $\mathfrak{S}_{X}$ equipped with a unitary inner product. Then $\frac{\mathsf{f}^{\lambda}}{|\lambda|!}\alpha^\lambda_X$ acts on $N$ as orthogonal projection onto its $\lambda$-isotypic component (i.e. the sum of all submodules isomorphic to $M^{\lambda}_X$).* **Corollary 30**. *For all partitions $\nu \vdash n$ and $\lambda \vdash |X|$, the element $\frac{\mathsf{f}^{\lambda}}{|\lambda|!}\alpha^\lambda_X$ acts on $M^{\nu}$ as a real orthogonal projection. In particular, $\alpha^\lambda_X$ is positive semidefinite. Also, it is nonzero if and only if $\lambda \subseteq \nu$.* *Proof.* Since $\alpha^\lambda_X \in \mathbb{C}[\mathfrak{S}_{X}]$, we may regard $M^{\nu}$ as a representation of $\mathfrak{S}_{X}$ by restriction. According to the branching rule [@sagan01 Theorem 2.8.3], this restriction decomposes into irreducibles as $\bigoplus_{\mu\vdash |X|,\mu\subseteq\nu} (M^{\mu}_X)^{\oplus\mathsf{f}^{\nu/\mu}}$. The result follows from [Proposition 29](#prop:projection){reference-type="ref" reference="prop:projection"}. ◻ *Proof of [Proposition 12](#prop:betapsd){reference-type="ref" reference="prop:betapsd"}.* Let $\alpha^\lambda_{X,\nu}$ denote the operator $\alpha^\lambda_X$ on $M^{\nu}$. By [Corollary 30](#cor:alphapsd){reference-type="ref" reference="cor:alphapsd"}, $\alpha^\lambda_{X,\nu}$ is self-adjoint and positive semidefinite. By [[([\[eq:alphatobeta\]](#eq:alphatobeta){reference-type="ref" reference="eq:alphatobeta"})]{.upright}](#eq:alphatobeta), if $z_1, \dots, z_n \in \mathbb{R}$, then $\beta^\lambda_\nu$ is a real linear combination of self-adjoint operators, and hence is itself self-adjoint. This proves part [\[betapsd1\]](#betapsd1){reference-type="ref" reference="betapsd1"}. Similarly, if $z_1, \dots z_n \geq 0$, then $\beta^\lambda_\nu$ is a nonnegative linear combination of positive-semidefinite operators, and hence is itself positive semidefinite. This proves part [\[betapsd2\]](#betapsd2){reference-type="ref" reference="betapsd2"}. If $\lambda \not \subseteq \nu$, then by [Corollary 30](#cor:alphapsd){reference-type="ref" reference="cor:alphapsd"} we have $\alpha^\lambda_{X,\nu} = 0$. Hence $\beta^\lambda_\nu = 0$, which proves part [\[betapsd4\]](#betapsd4){reference-type="ref" reference="betapsd4"}. Finally, for part [\[betapsd3\]](#betapsd3){reference-type="ref" reference="betapsd3"}, suppose that $z_1, \dots, z_n > 0$ and $\lambda \subseteq\nu$. In this case, by [Corollary 30](#cor:alphapsd){reference-type="ref" reference="cor:alphapsd"}, the operators $\alpha^\lambda_{X,\nu}$ are nonzero. Since $\beta^\lambda_\nu$ is positive semidefinite, it is positive definite if and only if $\ker \beta^\lambda_\nu = \{0\}$. Furthermore, since $\beta^\lambda_\nu$ is a positive linear combination of positive-semidefinite operators, we have $$\ker \beta^\lambda_\nu = \bigcap_{\substack{X \subseteq [n], \\ |X| = |\lambda|}} \ker \alpha^\lambda_{X,\nu} \,.$$ Since the operators $\alpha^\lambda_{X,\nu}$ are nonzero, we have $\beta^\lambda_\nu \neq 0$. Note that the right-hand side above is independent of $z_1, \dots, z_n$. Therefore, it is sufficient to compute $\ker \beta^\lambda_\nu$ for $z_1 = \dots = z_n = 1$. In this case $\beta^\lambda$ belongs to the centre of $\mathbb{C}[\mathfrak{S}_{n}]$, and so by Schur's lemma [@serre98 Proposition 4], $\beta^\lambda_\nu$ is a multiple of the identity operator on $M^{\nu}$. Since $\beta^\lambda_\nu \neq 0$, we deduce that $\ker \beta^\lambda_\nu = \{0\}$, as required. ◻ ### The KP hierarchy {#sec:kpbackground} The *KP equation* $$\frac{\partial}{\partial x}\left(-4\frac{\partial u}{\partial t} + 6u\frac{\partial u}{\partial x} + \frac{\partial^3 u}{\partial x^3}\right) + 3\frac{\partial^2 u}{\partial y^2} = 0$$ was introduced by Kadomtsev and Petviashvili [@kadomtsev_petviashvili70] as a ($2$+$1$)-dimensional generalization of the KdV equation, to model solitary waves under the effect of weak transverse perturbations (such as shallow water waves). It has since been widely studied; we refer to [@kodama17] for further background. The KP equation is the first equation in a system known as the *KP hierarchy*. Sato [@sato81] discovered that solutions of the KP hierarchy correspond to points in an infinite-dimensional Grassmannian. We recall an algebraic version of this correspondence, following Carrell and Goulden [@carrell_goulden10] (cf. [@miwa_jimbo_date00]). We identify a symmetric function $f \in \Lambda$ with the multiplication operator $f : \Lambda \to \Lambda$, $g \mapsto fg$. The linear operator $f^\perp : \Lambda \to \Lambda$ is the adjoint of $f$ under $\langle \cdot, \cdot \rangle$, and is called a *skewing operator*. Viewing $\Lambda = \mathbb{C}[\mathsf{p}_1, \mathsf{p}_2, \mathsf{p}_3, \dots]$ as a polynomial algebra in the power sum generators, $f^\perp$ is a differential operator with constant coefficients; for example, $\mathsf{p}_k^\perp = k\frac{\partial}{\partial p_k}$. Equations involving skewing operators are partial differential equations. We consider the multiplication operators $\mbox{\small\sf{H}}(t)$, $\mbox{\small\sf{E}}(t)$ and their adjoints $\mbox{\small\sf{H}}^\perp(t)$, $\mbox{\small\sf{E}}^\perp(t)$, defined as follows: $$\begin{aligned} \mbox{\small\sf{H}}(t) &:= \sum_{k \geq 0} \mathsf{h}_k t^k\,, & \mbox{\small\sf{E}}(t) &:= \sum_{k \geq 0} \mathsf{e}_k t^k\,, \\[4pt] \mbox{\small\sf{H}}^\perp(t) &:= \sum_{k \geq 0} \mathsf{h}_k^\perp t^k\,, & \mbox{\small\sf{E}}^\perp(t) &:= \sum_{k \geq 0} \mathsf{e}_k^\perp t^k \,,\end{aligned}$$ where $t$ is a formal indeterminate. Note that $$\label{eq:E} \mbox{\small\sf{E}}(t) = \sum_S t^{|S|} \cdot \prod_{i\in S} x_i,$$ where the sum is taken over all finite subsets $S$ of the positive integers. The operator $\mbox{\small\sf{H}}^\perp(t)$ has a particularly nice alternate description: **Proposition 31** (cf. [@bergeron_haiman13 (14)]). *For every symmetric function $f = f(x_1, x_2, \dots ) \in \Lambda$, we have $$\mbox{\small\sf{H}}^\perp(t)f = f(t, x_1, x_2, \dots) \,.$$ In particular, $\mbox{\small\sf{H}}^\perp(t)$ is a homomorphism.* The ***Bernstein operator*** $\mbox{\small\sf{B}}(t)$ and its adjoint $\mbox{\small\sf{B}}^\perp(t)$ are the following operators: $$\mbox{\small\sf{B}}(t) := \mbox{\small\sf{H}}(t)\mbox{\small\sf{E}}^\perp(-t^{-1}) \qquad \text{and} \qquad \mbox{\small\sf{B}}^\perp(t) := \mbox{\small\sf{E}}(-t^{-1})\mbox{\small\sf{H}}^\perp(t) \,.$$ Let $\tau \in \widehat \Lambda$ be a series in the formal completion of the algebra of symmetric functions. Then $\mbox{\small\sf{B}}(t) \tau$ and $\mbox{\small\sf{B}}^\perp(t^{-1}) \tau$ are series with coefficients in the field ${\mathbb{C}((t^{-1}))}$ of formal Laurent series in $t^{-1}$, that is, $\mbox{\small\sf{B}}(t) \tau,\,\mbox{\small\sf{B}}^\perp(t^{-1}) \tau \in \widehat \Lambda_{{\mathbb{C}((t^{-1}))}}$. We say that $\tau$ is a ***$\tau$-function of the KP hierarchy*** if the following *Hirota bilinear equation* holds: $$\label{eq:KP} [t^{-1}]\, \big( \mbox{\small\sf{B}}(t) \tau \big) \otimes_{\mathbb{C}((t^{-1}))}\big( \mbox{\small\sf{B}}^\perp(t^{-1}) \tau \big) = 0 \,,$$ where $[t^{-1}]$ extracts the coefficient of $t^{-1}$. We can interpret the elementary tensor $f\otimes g$ as the product $f(x_1, x_2, \dots)g(y_1, y_2, \dots)$ of $f$ and $g$ in two different sets of variables. **Theorem 32** (Sato [@sato81 Section 4]; cf. [@miwa_jimbo_date00 Chapter 10]). *For $\tau \in \widehat \Lambda$, the following are equivalent:* (a) *the coefficients of $\tau$ in the Schur basis, $\Delta^\lambda = \langle \tau, \mathsf{s}_\lambda \rangle$, satisfy the Plücker relations [[([\[eq:prelspartitions\]](#eq:prelspartitions){reference-type="ref" reference="eq:prelspartitions"})]{.upright}](#eq:prelspartitions);* (b) *$\tau$ is a $\tau$-function of the KP hierarchy.* Hence, [[([\[eq:KP\]](#eq:KP){reference-type="ref" reference="eq:KP"})]{.upright}](#eq:KP) is a convenient way to encode all of the Plücker relations into a single generating function equation. An alternate proof of [Theorem 32](#thm:KPplucker){reference-type="ref" reference="thm:KPplucker"} was given by Carrell and Goulden [@carrell_goulden10 Theorem 5.2], based an explicit formula for $\mbox{\small\sf{B}}(t)\mathsf{s}_\lambda$. We recall the relevant details of this argument: *Proof of [Theorem 32](#thm:KPplucker){reference-type="ref" reference="thm:KPplucker"}.* By [@carrell_goulden10 Corollaries 3.5 and 3.6], the left-hand side of [[([\[eq:KP\]](#eq:KP){reference-type="ref" reference="eq:KP"})]{.upright}](#eq:KP) is equal to $\sum_{\lambda, \mu} b_{\lambda,\mu} \mathsf{s}_\lambda \otimes \mathsf{s}_\mu$, where $$\label{eq:KPlhs} b_{\lambda,\mu} = \sum_{\substack{i,j \geq 1, \\ |\lambda^{(-i)}| + |\mu^{(j)}| = |\lambda| + |\mu|+ 1}} (-1)^{|\mu|-|\mu^{(j)}|+i+j} \Delta^{\lambda^{(-i)}} \Delta^{\mu^{(j)}} \,.$$ Thus [[([\[eq:KP\]](#eq:KP){reference-type="ref" reference="eq:KP"})]{.upright}](#eq:KP) holds if and only if $b_{\lambda,\mu} = 0$ for all $\lambda$ and $\mu$, i.e., if and only if [[([\[eq:prelspartitions\]](#eq:prelspartitions){reference-type="ref" reference="eq:prelspartitions"})]{.upright}](#eq:prelspartitions) holds. ◻ We will need a similar equation which encodes only the single-column Plücker relations. These turn out to correspond to the constant term in the first tensor factor of [[([\[eq:KP\]](#eq:KP){reference-type="ref" reference="eq:KP"})]{.upright}](#eq:KP): **Lemma 33**. *For $\tau \in \widehat \Lambda$, the following are equivalent:* (a) *the coefficients of $\tau$ in the Schur basis, $\Delta^\lambda = \langle \tau, \mathsf{s}_\lambda \rangle$, satisfy all the single-column Plücker relations;* (b) *$\tau$ satisfies the equation $$\label{eq:scKP} [t^{-1}]\, \langle \mbox{\small\sf{B}}(t) \tau , 1 \rangle \cdot \mbox{\small\sf{B}}^\perp(t^{-1}) \tau = 0 \,.$$* *Proof.* In the notation of [[([\[eq:KPlhs\]](#eq:KPlhs){reference-type="ref" reference="eq:KPlhs"})]{.upright}](#eq:KPlhs), the single-column Plücker relations are the equations $b_{0,\mu} =0$. Since $\sum_{\lambda, \mu} b_{\lambda,\mu} \mathsf{s}_\lambda \otimes \mathsf{s}_\mu$ is the left-hand side of [[([\[eq:KP\]](#eq:KP){reference-type="ref" reference="eq:KP"})]{.upright}](#eq:KP), the left-hand side of [[([\[eq:scKP\]](#eq:scKP){reference-type="ref" reference="eq:scKP"})]{.upright}](#eq:scKP) is $\sum_{\lambda, \mu} b_{\lambda, \mu} \langle \mathsf{s}_\lambda, 1 \rangle \cdot \mathsf{s}_\mu = \sum_{\mu} b_{0,\mu} \mathsf{s}_\mu$. The result follows. ◻ We note that since $\mbox{\small\sf{H}}^\perp(t)1 = 1$, we have $$\langle \mbox{\small\sf{B}}(t) \tau , 1 \rangle = \langle \tau , \mbox{\small\sf{B}}^\perp(t) 1 \rangle = \langle \tau , \mbox{\small\sf{E}}(-t^{-1}) \rangle = \sum_{k \geq 0} (-t)^{-k} \langle \tau, \mathsf{e}_k \rangle \,,$$ and thus [[([\[eq:scKP\]](#eq:scKP){reference-type="ref" reference="eq:scKP"})]{.upright}](#eq:scKP) can be rewritten as $$\label{eq:scKP2} [t^{-1}]\mbox{\small\sf{B}}^\perp(t^{-1})\sum_{k \geq 0} (-t)^{-k} \langle \tau, \mathsf{e}_k \rangle \tau = 0 \,.$$ **Remark 34**. Let $V\in\mathcal{X}^{\nu}\subseteq \mathrm{Gr}(d,m)$ have the normalized Plücker coordinates $(\Delta^\lambda : \lambda \subseteq \nu)$. By [Theorem 32](#thm:KPplucker){reference-type="ref" reference="thm:KPplucker"}, we have the associated $\tau$-function of the KP hierarchy $$\tau := \sum_{\lambda \subseteq \nu}\Delta^\lambda \mathsf{s}_\lambda\in\Lambda.$$ Then the Wronskian of $V$ is the ***exponential specialization*** of $\tau$. Namely, define the unital $\mathbb{C}$-algebra homomorphism $\mathop{\mathrm{ex}}: \Lambda \to \mathbb{C}[u]$ as follows [@stanley99 Section 7.8]: $$\mathop{\mathrm{ex}}(\mathsf{p}_1) := u \qquad \text{and} \qquad \mathop{\mathrm{ex}}(\mathsf{p}_k) := 0 \;\text{ for all } k\ge 2\,.$$ Then $\mathop{\mathrm{ex}}(\mathsf{s}_\lambda) = \frac{\mathsf{f}^{\lambda}}{|\lambda|!}u^{|\lambda|}$ (cf. [@stanley99 p. 344]), so $\mathop{\mathrm{ex}}(\tau) = \mathrm{Wr}(V)$ by [[([\[eq:pluckerwronskian\]](#eq:pluckerwronskian){reference-type="ref" reference="eq:pluckerwronskian"})]{.upright}](#eq:pluckerwronskian). ### Scaled monomial symmetric functions The ***scaled monomial symmetric functions*** (cf. [@merca15]) are defined to be $\widetilde{\mathsf{m}}_\lambda := \langle \mathsf{p}_\lambda, \mathsf{h}_\lambda \rangle \mathsf{m}_\lambda$, i.e., $\widetilde{\mathsf{m}}_\lambda$ is the $\mathsf{m}_\lambda$-term in the monomial expansion of $\mathsf{p}_\lambda$. For example, $\widetilde{\mathsf{m}}_{1^{n}} = n! \, \mathsf{m}_{1^n}$. Equivalently, if $\ell(\lambda) = k$, then $$\label{eq:sm} \widetilde{\mathsf{m}}_\lambda = \sum_{\substack{a_1, \dots, a_k \geq 1\\ \text{(distinct)}}} x_{a_1}^{\lambda_1} \dotsm x_{a_k}^{\lambda_k} \,,$$ where the sum is taken over all $k$-tuples of pairwise distinct positive integers. We will sometimes index the scaled monomial symmetric functions using compositions instead of partitions. A ***composition*** $\kappa = (\kappa_1, \dots, \kappa_k)$ is an arbitrary tuple of positive integers. As with partitions, we write $|\kappa| = \kappa_1 + \dots + \kappa_k$, $\ell(\kappa) = k$, and $\kappa_j = 0$ for $j > k$. We write $\kappa \leq \lambda$ to mean $\ell(\kappa) = \ell(\lambda)$ and $\kappa_i \leq \lambda_i$ for all $i$. Let $\mathsf{sort}(\kappa)$ denote the unique partition of the form $(\kappa_{\sigma(1)}, \dots, \kappa_{\sigma(k)})$ for some $\sigma \in \mathfrak{S}_{k}$. We put $\widetilde{\mathsf{m}}_\kappa := \widetilde{\mathsf{m}}_{\mathsf{sort}(\kappa)}$ for all compositions $\kappa$. Also, if $I \subseteq [k]$, we let $\kappa \setminus \kappa_I$ denote the composition obtained from $\kappa$ by deleting the parts $\kappa_i$ for $i\in I$. For some purposes, the scaled monomial basis is more natural than the monomial basis. For example, we can express $\mbox{\small\sf{H}}^\perp(t)\widetilde{\mathsf{m}}_\kappa$ as follows: **Proposition 35**. *For any composition $\kappa$ of length $k$, we have $$\mbox{\small\sf{H}}^\perp(t) \widetilde{\mathsf{m}}_\kappa = \widetilde{\mathsf{m}}_\kappa + \sum_{i=1}^k t^{\kappa_i}\widetilde{\mathsf{m}}_{\kappa\setminus\kappa_i}\,.$$* *Proof.* This follows from [Proposition 31](#prop:Hperp){reference-type="ref" reference="prop:Hperp"} by a direct calculation. ◻ We will also need the power sum expansion of $\widetilde{\mathsf{m}}_\kappa$: **Proposition 36** (Doubilet [@doubilet72 Theorem 2(i)]; cf. [@merca15 Theorem 2]). *Let $\kappa$ be a composition of length $k$. For $\sigma \in \mathfrak{S}_{k}$, let $\kappa[\sigma]$ denote the partition obtained by amalgamating the parts of $\kappa$ that are in the same cycle of $\sigma$. That is, for each cycle $(i_1 \;\; i_2 \; \cdots \; i_s)$ of $\sigma$, we produce a part of $\kappa[\sigma]$ of size $\kappa_{i_1} + \dots + \kappa_{i_s}$. Then $$\widetilde{\mathsf{m}}_\kappa = \sum_{\sigma \in \mathfrak{S}_{k}} \mathrm{sgn}(\sigma) \mathsf{p}_{\kappa[\sigma]} \,.$$* ### Two symmetric-function identities We record two identities which will help us understand the kernel of the operator $[t^{-1}]\mbox{\small\sf{B}}^\perp(t^{-1}): \Lambda_{\mathbb{C}((t^{-1}))}\to \Lambda$. The first identity describes the commutation relation between $\mbox{\small\sf{B}}^\perp(t^{-1})$ and the multiplication operator $(1-t^j\mathsf{p}_j)$: **Lemma 37**. *For $j \geq 0$, we have $$\big[(1-t^j\mathsf{p}_j), \mbox{\small\sf{B}}^\perp(t^{-1})\big] = \mbox{\small\sf{B}}^\perp(t^{-1}) \,.$$ Equivalently, for any symmetric function $f \in \Lambda$, $$\mbox{\small\sf{B}}^\perp(t^{-1}) (1-t^j \mathsf{p}_j) f = - t^j\mathsf{p}_j \cdot \mbox{\small\sf{B}}^\perp(t^{-1}) f \,.$$* *Proof.* By definition, $\mbox{\small\sf{B}}^\perp(t^{-1}) = \mbox{\small\sf{E}}(-t)\mbox{\small\sf{H}}^\perp(t^{-1})$. Now $\mbox{\small\sf{E}}(-t)$ is a multiplication operator, and $\mbox{\small\sf{H}}^\perp(t^{-1})$ is a homomorphism, so we have the following general identity: for all $f,g \in \Lambda$, $$\mbox{\small\sf{B}}^\perp(t^{-1}) fg = \big(\mbox{\small\sf{B}}^\perp(t^{-1}) f\big) \cdot \big(\mbox{\small\sf{H}}^\perp(t^{-1}) g\big) \,.$$ Taking $g = 1-t^j\mathsf{p}_j$ in the identity above yields the result, using [Proposition 31](#prop:Hperp){reference-type="ref" reference="prop:Hperp"}. ◻ Hence for $\phi(t) \in \widehat \Lambda_{{\mathbb{C}((t^{-1}))}}$, we have $\phi(t) (1-t^j \mathsf{p}_j) \in \ker [t^{-1}]\mbox{\small\sf{B}}^\perp(t^{-1})$ if and only if $t^j \phi(t) \in \ker [t^{-1}]\mbox{\small\sf{B}}^\perp(t^{-1})$. The second identity gives an explicit family of symmetric functions which are in the kernel of $[t^{-1}]\mbox{\small\sf{B}}^\perp(t^{-1})$: **Lemma 38**. *For any composition $\lambda$, consider the symmetric function $$\phi_\lambda(t) := \sum_{\kappa \leq \lambda} t^{|\kappa| - |\lambda|- \ell(\lambda)} \widetilde{\mathsf{m}}_\kappa \,,$$ where the sum is taken over all compositions $\kappa \leq \lambda$. Then $[t^{-1}]\mbox{\small\sf{B}}^\perp(t^{-1})\phi_\lambda(t) = 0$.* **Example 39**. If $\lambda = (3,2)$, the compositions $\kappa \leq \lambda$ are $(3,2)$, $(3,1)$, $(2,2)$, $(2,1)$, $(1,2)$, and $(1,1)$. In this case, [Lemma 38](#lem:Bm-identity){reference-type="ref" reference="lem:Bm-identity"} asserts that $$[t^{-1}]\mbox{\small\sf{B}}^\perp(t^{-1}) \big( t^{-2} \widetilde{\mathsf{m}}_{32} + t^{-3} \widetilde{\mathsf{m}}_{31} + t^{-3} \widetilde{\mathsf{m}}_{22} + 2t^{-4} \widetilde{\mathsf{m}}_{21} + t^{-5} \widetilde{\mathsf{m}}_{11} \big) = 0 \,.$$ *Proof of [Lemma 38](#lem:Bm-identity){reference-type="ref" reference="lem:Bm-identity"}.* Let $\mathcal{T}_\lambda$ denote the set of triples $(\kappa, \pi, \mathbf{a})$ such that: - $\kappa \leq \lambda$ is a composition; - $\pi = \pi_1 \pi_2 \cdots \pi_{\ell(\pi)}$ is a $\{0,1\}$-string of length $\ell(\pi) \geq \ell(\lambda)$, where $\pi_i = 1$ for all $\ell(\lambda) < i \leq \ell(\pi)$; and - $\mathbf{a}= (a_1, \dots, a_{\ell(\pi)})$ is a tuple of distinct positive integers with $a_{\ell(\lambda)+1} < \dots < a_{\ell(\pi)}$. Note that $1 \leq \kappa_i + \pi_i \leq \lambda_i+1$ for all $1 \le i \le \ell(\pi)$. Write $|\pi| := \pi_1 + \dots + \pi_{\ell(\pi)}$. From [[([\[eq:sm\]](#eq:sm){reference-type="ref" reference="eq:sm"})]{.upright}](#eq:sm) and [[([\[eq:E\]](#eq:E){reference-type="ref" reference="eq:E"})]{.upright}](#eq:E), we obtain $$\label{eq:Ephi} \mbox{\small\sf{E}}(-t) \phi_\lambda(t) = \sum_{(\kappa, \pi, \mathbf{a}) \in \mathcal{T}_\lambda} (-1)^{|\pi|}t^{|\kappa|+|\pi|-|\lambda|-\ell(\lambda)} x_{a_1}^{\kappa_1+\pi_1} \dotsm x_{a_{\ell(\pi)}}^{\smash{\kappa_{\ell(\pi)}+\pi_{\ell(\pi)}}} \,.$$ Here the set $S$ in [[([\[eq:E\]](#eq:E){reference-type="ref" reference="eq:E"})]{.upright}](#eq:E) is encoded as $S = \{a_i \mid \pi_i = 1\}$. Let $\mathcal{T}'_\lambda \subseteq \mathcal{T}_\lambda$ be the subset of triples $(\kappa, \pi, \mathbf{a})$ such that $\kappa_i +\pi_i\in \{1, \lambda_i+1\}$ for all $1 \le i \le \ell(\pi)$. If $(\kappa, \pi, \mathbf{a}) \in \mathcal{T}_\lambda \setminus \mathcal{T}'_\lambda$, let $j$ be the smallest integer such that $\kappa_j +\pi_j \notin \{1, \lambda_j+1\}$, and let $(\kappa', \pi', \mathbf{a}) \in \mathcal{T}_\lambda \setminus \mathcal{T}'_\lambda$ be obtained by changing $\pi_j$ to $1-\pi_j$ and $\kappa_j$ to $\kappa_j+2\pi_j-1$. This defines a sign-reversing involution on the terms in the sum [[([\[eq:Ephi\]](#eq:Ephi){reference-type="ref" reference="eq:Ephi"})]{.upright}](#eq:Ephi) corresponding to $\mathcal{T}_\lambda \setminus \mathcal{T}'_\lambda$. Hence we can replace $\mathcal{T}_\lambda$ by $\mathcal{T}'_\lambda$ in [[([\[eq:Ephi\]](#eq:Ephi){reference-type="ref" reference="eq:Ephi"})]{.upright}](#eq:Ephi). For any composition $\mu$, define $$\label{eq:Flambda} F_\mu(t; \mathbf{x}) := \sum_{p \geq \ell(\mu)} (-1)^{p-\ell(\mu)}t^{p-|\mu|-\ell(\mu)} p(p-1) \dotsm (p-\ell(\mu)+1) \,\mathsf{e}_p(\mathbf{x}) \,.$$ Reorganizing [[([\[eq:Ephi\]](#eq:Ephi){reference-type="ref" reference="eq:Ephi"})]{.upright}](#eq:Ephi) into square-free and non-square-free parts, we have $$\label{eq:Ephi2} \mbox{\small\sf{E}}(-t)\phi_\lambda(t) = \sum_{\substack{ J = (j_1, \dots, j_s), \\ \mathbf{a}= (a_1, \dots, a_s)} } (-1)^s x_{a_1}^{\lambda_{j_1}+1} \dotsm x_{a_s}^{\lambda_{j_s}+1} \cdot F_{\lambda \setminus \lambda_J}(t;\mathbf{x}\setminus \mathbf{x}_\mathbf{a}) \,,$$ where the sum is taken over all subsets $J \subseteq [\ell(\lambda)]$ (encoding $\{j \in [\ell(\lambda)] \mid \kappa_j + \pi_j = \lambda_j + 1\}$ for $(\kappa, \pi, \mathbf{a}) \in \mathcal{T}'_\lambda$) and all $|J|$-tuples $\mathbf{a}$ of distinct positive integers. The notation $\mathbf{x}\setminus \mathbf{x}_\mathbf{a}$ means we omit the variables $x_{a_1}, \dots, x_{a_s}$. Using [Proposition 35](#prop:hperpsm){reference-type="ref" reference="prop:hperpsm"}, we have $\mbox{\small\sf{H}}^\perp(t^{-1}) \phi_\lambda(t) = \phi_\lambda(t) + \sum_{i=1}^{\ell(\lambda)} \lambda_i t^{-\lambda_i-1} \phi_{\lambda \setminus \lambda_i}(t) \,.$ Combining this with [[([\[eq:Ephi2\]](#eq:Ephi2){reference-type="ref" reference="eq:Ephi2"})]{.upright}](#eq:Ephi2) gives $$\mbox{\small\sf{B}}^\perp(t^{-1}) \phi_\lambda(t) = \sum_{\substack{ J = (j_1, \dots, j_s), \\ \mathbf{a}= (a_1, \dots, a_s)} } (-1)^s x_{a_1}^{\lambda_{j_1}+1} \dotsm x_{a_s}^{\lambda_{j_s}+1} \cdot G_{\lambda \setminus \lambda_J}(t;\mathbf{x}\setminus \mathbf{x}_\mathbf{a}) \,,$$ where $G_\mu(t; \mathbf{x}) := F_\mu(t;\mathbf{x}) + \sum_{i=1}^{\ell(\mu)} \mu_i t^{-\mu_i-1} F_{\mu \setminus \mu_i}(t;\mathbf{x}).$ Finally, using [[([\[eq:Flambda\]](#eq:Flambda){reference-type="ref" reference="eq:Flambda"})]{.upright}](#eq:Flambda) it is straightforward to check that $[t^{-1}] G_\mu(t; \mathbf{x}) = 0$ for all $\mu$, and the result follows. ◻ ## The Bethe subalgebra of $\mathbb{C}[\mathfrak{S}_{n}]$ {#sec:bethe} Fix a positive integer $n$, and let $z_1, \dots, z_n \in \mathbb{C}$ be complex numbers. As before, let $g(u) = (u+z_1) \dotsm (u+z_n)$. For a subset $X \subseteq [n]$, let $z_X := \prod_{i \in X} z_i$. We recall the operators $\beta^\lambda(t)$ defined in [[([\[eq:betadef\]](#eq:betadef){reference-type="ref" reference="eq:betadef"})]{.upright}](#eq:betadef) and $\alpha^\lambda_X$ defined in [[([\[eq:alphadef\]](#eq:alphadef){reference-type="ref" reference="eq:alphadef"})]{.upright}](#eq:alphadef). By [[([\[eq:alphatobeta\]](#eq:alphatobeta){reference-type="ref" reference="eq:alphatobeta"})]{.upright}](#eq:alphatobeta), we have $$\beta^\lambda(t) = \sum_{\ell=0}^{n-|\lambda|} \beta^\lambda_\ell t^{n-|\lambda|-\ell}, \qquad\text{where }\;\beta^\lambda_{\ell} := \sum_{\substack{X \cap Y = \emptyset, \\ |X| = |\lambda|, |Y| = \ell}} \alpha^\lambda_X z_Y \in \mathbb{C}[\mathfrak{S}_{n}] \;\text{ for } \ell\ge 0 \,.$$ In particular, $\beta^\lambda = \beta^\lambda(0) = \beta^\lambda_{n-|\lambda|}$. ### Generators While most of the operators $\beta^\lambda(t)$ are new, the operators $\beta^{1^{\smash{k}}}(t)$ coincide with those introduced by Mukhin, Tarasov, and Varchenko in [@mukhin_tarasov_varchenko13]. In the remainder of this section we will only be concerned with the partitions $\lambda = 1^k$, $k \leq n$. In this case, for $|X| = k$ the Specht module $M^{1^k}_X$ is the one-dimensional sign representation of $\mathfrak{S}_{X}$, so $$\alpha^{1^{\smash{k}}}_X = \sum_{\sigma \in \mathfrak{S}_{X}} \mathrm{sgn}(\sigma) \sigma \,.$$ Mukhin, Tarasov, and Varchenko define the ***Bethe subalgebra*** (of *Gaudin type*) of $\mathbb{C}[\mathfrak{S}_{n}]$ to be the subalgebra $\mathcal{B}_n(z_1, \dots, z_n)\subseteq \mathbb{C}[\mathfrak{S}_{n}]$ generated by the elements $\beta^{1^{\smash{k}}}_\ell$ for $k, \ell \geq 0$. Equivalently, $\mathcal{B}_n(z_1, \dots, z_n)$ is generated by the elements $\beta^{1^{\smash{k}}}(t)$ for $k \geq 0$, $t \in \mathbb{C}$. **Remark 40**. In the notation of [@mukhin_tarasov_varchenko13], what we call $\beta^{1^{\smash{k}}}(t)$ and $\beta^{1^{\smash{k}}}_\ell$ can be expressed as, respectively, $(-1)^{n-k}\mathit{\Phi}^{[n]}_k(-t)$ and $(-1)^\ell\mathit{\Phi}^{[n]}_{k,\ell}$. Our notation more closely follows [@purbhoo], where our $\beta^{1^{\smash{k}}}(t)$ and $\beta^{1^{\smash{k}}}_\ell$ are denoted $\beta^-_{k,n-k}(t)$ and $\beta^-_{k,\ell}$, respectively. **Remark 41**. When $z_1, \dots, z_n$ are distinct, $\mathcal{B}_n(z_1, \dots, z_n)$ is generated by the *Gaudin elements* $$\sum_{\substack{1 \le j \le n, \\ j \neq k}}\,\frac{\sigma_{j,k}}{z_k - z_j} \in \mathbb{C}[\mathfrak{S}_{n}] \qquad (k = 1, \dots, n)$$ studied in [@gaudin76 Section 2]; see [@mukhin_tarasov_varchenko13 Theorem 4.4] and cf. [@kirillov16a Section 3.3]. The Gaudin elements may be regarded as deformed Jucys--Murphy elements, so that $\mathcal{B}_n(z_1, \dots, z_n)$ is a deformation of the *Gelfand--Tsetlin subalgebra* of $\mathbb{C}[\mathfrak{S}_{n}]$. **Theorem 42** (Mukhin, Tarasov, and Varchenko [@mukhin_tarasov_varchenko13 Sections 2 and 4]). *The algebra $\mathcal{B}_n(z_1, \dots, z_n)$ has the following properties.* (i) *[\[bethebasics1\]]{#bethebasics1 label="bethebasics1"} It is a commutative algebra.* (ii) *[\[bethebasics2\]]{#bethebasics2 label="bethebasics2"} If $z_1, \dots, z_n \in \mathbb{C}$ are generic, it is semisimple and a maximal commutative subalgebra of $\mathbb{C}[\mathfrak{S}_{n}]$, and has dimension $\sum_{\nu \vdash n} \mathsf{f}^{\nu}$.* (iii) *[\[bethebasics3\]]{#bethebasics3 label="bethebasics3"} It contains the centre of $\mathbb{C}[\mathfrak{S}_{n}]$.* (iv) *[\[bethebasics4\]]{#bethebasics4 label="bethebasics4"} It is translation invariant: $\mathcal{B}_n(z_1, \dots, z_n)= \mathcal{B}_n(z_1+t, \dots, z_n+t)$ for all $t \in \mathbb{C}$.* **Remark 43**. Many of the properties of $\mathcal{B}_n(z_1, \dots, z_n)$ depend on the choice of parameters $z_1, \dots, z_n$ (cf. [@mukhin_tarasov_varchenko13]). For example, $\mathcal{B}_n(z_1, \dots, z_n)$ is not always semisimple, and its dimension depends discontinuously on $z_1, \dots, z_n$. When $z_1, \dots, z_n$ are distinct, $\dim \mathcal{B}_n(z_1, \dots, z_n)= \sum_{\nu \vdash n} \mathsf{f}^{\nu}$, even if $\mathcal{B}_n(z_1, \dots, z_n)$ is not semisimple. However, $\dim \mathcal{B}_n(z_1, \dots, z_n)$ can also be strictly smaller. For example, if $z_1 = \dots = z_n$, then $\mathcal{B}_n(z_1, \dots, z_n)$ is equal the centre of $\mathbb{C}[\mathfrak{S}_{n}]$, which has dimension equal to the number of partitions of $n$. We give a general formula for $\dim \mathcal{B}_n(z_1, \dots, z_n)$ in [[([\[eq:dimensionformula\]](#eq:dimensionformula){reference-type="ref" reference="eq:dimensionformula"})]{.upright}](#eq:dimensionformula). If $u$ is a formal indeterminate, then $\beta^{1^{\smash{k}}}(u) \in \mathbb{C}[\mathfrak{S}_{n}] \otimes \mathbb{C}[u]$ is an element of the group algebra of $\mathfrak{S}_{n}$ with coefficients in $\mathbb{C}[u]$, or equivalently, a polynomial in $u$ with coefficients in $\mathbb{C}[\mathfrak{S}_{n}]$. We combine the elements $\beta^{1^{\smash{k}}}(u)$ to produce a linear differential operator, $\mathcal{D}_n : \mathbb{C}[\mathfrak{S}_{n}] \otimes \mathbb{C}(u) \to \mathbb{C}[\mathfrak{S}_{n}] \otimes \mathbb{C}(u)$, defined as $$\label{eq:diffop} \mathcal{D}_n := \frac{1}{g(u)} \sum_{k=0}^n (-1)^k \beta^{1^{\smash{k}}}(u) \partial_u^{n-k} \,.$$ ### Eigenspaces For any partition $\nu \vdash n$, $\mathcal{B}_n(z_1, \dots, z_n)$ acts on the Specht module $M^{\nu}$. The subalgebra $\mathcal{B}_\nu(z_1, \dots, z_n)\subseteq \mathrm{End}(M^{\nu})$ is the algebra defined by the image of this action. We let $\beta^{1^{\smash{k}}}_{\ell,\nu},\, \beta^{1^{\smash{k}}}_\nu(t) \in \mathcal{B}_\nu(z_1, \dots, z_n)$ denote the image of the generators $\beta^{1^{\smash{k}}}_\ell,\,\beta^{1^{\smash{k}}}(t)$. Since $\mathcal{B}_n(z_1, \dots, z_n)$ is commutative, so is $\mathcal{B}_\nu(z_1, \dots, z_n)$. Because $\mathcal{B}_n(z_1, \dots, z_n)$ contains the centre of $\mathbb{C}[\mathfrak{S}_{n}]$, which includes projections onto each $M^{\nu}$ (by [Proposition 29](#prop:projection){reference-type="ref" reference="prop:projection"}), we have a direct product decomposition $$\label{eq:directproductbethe} \mathcal{B}_n(z_1, \dots, z_n)\simeq \prod_{\nu \vdash n} \mathcal{B}_\nu(z_1, \dots, z_n) \,.$$ An ***eigenspace*** of $\mathcal{B}_\nu(z_1, \dots, z_n)$ is a maximal subspace $E \subseteq M^{\nu}$ such that every operator $\xi \in \mathcal{B}_\nu(z_1, \dots, z_n)$ acts as a scalar $\xi_E$ on $E$; $\xi_E$ is the ***eigenvalue*** of $\xi$ on $E$. The eigenspaces of $\mathcal{B}_\nu(z_1, \dots, z_n)$ correspond naturally to the points of $\mathop{\mathrm{Spec}}\mathcal{B}_\nu(z_1, \dots, z_n)$. The ***generalized eigenspace*** containing $E$ is the maximal $\mathcal{B}_\nu(z_1, \dots, z_n)$-submodule $\widehat{E} \subseteq M^{\nu}$ on which $\xi-\xi_E$ acts nilpotently for all $\xi \in \mathcal{B}_\nu(z_1, \dots, z_n)$. For any such eigenspace $E$, we write $\beta^{1^{\smash{k}}}_{\ell,E}$ and $\beta^{1^{\smash{k}}}_E(t)$ for the eigenvalues of $\beta^{1^{\smash{k}}}_{\ell,\nu}$ and $\beta^{1^{\smash{k}}}_\nu(t)$, respectively. Restricting the differential operator $\mathcal{D}_n$ to $E$, we obtain a scalar-valued differential operator $\mathcal{D}_E : \mathbb{C}(u) \to \mathbb{C}(u)$, given by $$\mathcal{D}_E = \frac{1}{g(u)} \sum_{k=0}^n (-1)^k \beta^{1^{\smash{k}}}_E(u) \partial_u^{n-k} \,.$$ Recall that for a finite-dimensional vector space $V \subseteq \mathbb{C}[u]$, $D_V$ denotes the fundamental differential operator of $V$. We now state [Theorem 1](#thm:vague){reference-type="ref" reference="thm:vague"} more precisely (for simplicity, we work inside the Grassmannian $\mathrm{Gr}(n, 2n)$): **Theorem 44** (Mukhin, Tarasov, and Varchenko [@mukhin_tarasov_varchenko13 Theorem 4.3(iv)]). *Let $\nu \vdash n$, and consider the Schubert cell $\mathcal{X}^\nu \subseteq \mathrm{Gr}(n,2n)$. For every eigenspace $E \subseteq M^{\nu}$ of $\mathcal{B}_\nu(z_1, \dots, z_n)$, there exists a unique point $V_E \in \mathrm{Wr}^{-1}(g)$ such that $\mathcal{D}_E = D_{V_E}$. This defines a bijective correspondence between the eigenspaces of $\mathcal{B}_\nu(z_1, \dots, z_n)$ and the points of $\mathrm{Wr}^{-1}(g)$.* **Remark 45**. One advantage of working in $\mathrm{Gr}(n,2n)$ is that $\mathcal{D}_E$ and $D_V$ are both differential operators of order $n$. If $d \neq n$, and we instead regard $\mathcal{X}^\nu$ as a Schubert cell in $\mathrm{Gr}(d,m)$, then differential operators $\mathcal{D}_E$ and $D_{V_E}$ are no longer of the same order, and therefore cannot be equal. Nevertheless, [Theorem 44](#thm:precise){reference-type="ref" reference="thm:precise"} is true more generally for $\mathrm{Gr}(d,m)$, namely, $\mathcal{D}_E\partial_u^d = D_{V_E} \partial_u^n$. This formulation, however, is awkward to work with. ### Regarding the parameters $z_1, \dots, z_n$ In the definition of $\mathcal{B}_n(z_1, \dots, z_n)$, the parameters $z_1,\dots, z_n$ are arbitrary fixed complex numbers; specifically, $-z_1, \dots, -z_n$ are the roots the polynomial $g(u) \in \mathbb{C}[u]$. This is the perspective assumed in the statement of [Theorem 3](#thm:main){reference-type="ref" reference="thm:main"}. However, at various points in the proof of [Theorem 3](#thm:main){reference-type="ref" reference="thm:main"}, we will take two other perspectives on the parameters $z_1, \dots, z_n$, viewing them as generic complex numbers (i.e. $(z_1, \dots, z_n)$ is a general point of $\mathbb{C}^n$), or as formal indeterminates. Any polynomial identity that we can prove under any of these three perspectives must also be true under the other two. This applies only to polynomial identities, and it is not true for abstract properties of Bethe algebras (e.g. see [Remark 43](#rmk:parameterdependence){reference-type="ref" reference="rmk:parameterdependence"}), so some care is required when shifting perspectives. Nevertheless, we use this to our advantage. For example, if $z_1, \dots, z_n$ are assumed to be generic, then $\mathcal{B}_n(z_1, \dots, z_n)$ is a maximal commutative subalgebra of $\mathbb{C}[\mathfrak{S}_{n}]$ by [Theorem 42](#thm:bethebasics){reference-type="ref" reference="thm:bethebasics"}[\[bethebasics2\]](#bethebasics2){reference-type="ref" reference="bethebasics2"}. Or, if $z_1, \dots, z_n$ are formal indeterminates, then we can substitute $z_i \mapsto z_i+t$ into any valid equation. These additional possibilities facilitate certain arguments that would be invalid if $z_1, \dots, z_n \in \mathbb{C}$ are always assumed to be arbitrary and fixed. # Commutativity relations and the translation identity {#sec:commutativitytranslation} The purpose of this section is to prove parts [\[main_commutativity\]](#main_commutativity){reference-type="ref" reference="main_commutativity"} and [\[main_translation\]](#main_translation){reference-type="ref" reference="main_translation"} of [Theorem 3](#thm:main){reference-type="ref" reference="thm:main"}, i.e., commutativity relations and the translation identity for the operators $\beta^\lambda(t)$. We begin by formulating a weak form of [Theorem 3](#thm:main){reference-type="ref" reference="thm:main"}[\[main_generates\]](#main_generates){reference-type="ref" reference="main_generates"} in [3.1](#sec:weakform){reference-type="ref" reference="sec:weakform"}; we then prove it alongside the translation identity ([3.2](#sec:translation){reference-type="ref" reference="sec:translation"}) and the commutativity relations ([3.3](#sec:commutativity){reference-type="ref" reference="sec:commutativity"}). ## A weak form of [Theorem 3](#thm:main){reference-type="ref" reference="thm:main"}[\[main_generates\]](#main_generates){reference-type="ref" reference="main_generates"} {#sec:weakform} Let $\overline{\mathcal{B}}_n(z_1, \dots, z_n)$ denote the subalgebra of $\mathbb{C}[\mathfrak{S}_{n}]$ generated by the elements $\beta^\lambda$. Part [\[main_generates\]](#main_generates){reference-type="ref" reference="main_generates"} of [Theorem 3](#thm:main){reference-type="ref" reference="thm:main"} asserts that $\overline{\mathcal{B}}_n(z_1, \dots, z_n)= \mathcal{B}_n(z_1, \dots, z_n)$; however, we will not deduce this until the end of the proof of [Theorem 3](#thm:main){reference-type="ref" reference="thm:main"}. In the meantime, we will spend the remainder of this section proving the following slightly weaker result, which will be enough to get us through until then: **Lemma 46**. *The algebra $\overline{\mathcal{B}}_n(z_1, \dots, z_n)$ has the following properties.* (i) *[\[comparealgebras1\]]{#comparealgebras1 label="comparealgebras1"} It is a commutative algebra.* (ii) *[\[comparealgebras2\]]{#comparealgebras2 label="comparealgebras2"} If $z_1, \dots, z_n \in \mathbb{C}$ are generic, it equals $\mathcal{B}_n(z_1, \dots, z_n)$.* (iii) *[\[comparealgebras3\]]{#comparealgebras3 label="comparealgebras3"} It contains $\mathcal{B}_n(z_1, \dots, z_n)$.* (iv) *[\[comparealgebras4\]]{#comparealgebras4 label="comparealgebras4"} It is translation invariant: $\overline{\mathcal{B}}_n(z_1, \dots, z_n)= \overline{\mathcal{B}}_n(z_1+t, \dots, z_n+t)$ for all $t \in \mathbb{C}$.* Note that for any fixed $t \in \mathbb{C}$, $\overline{\mathcal{B}}_n(z_1+t, \dots, z_n+t)$ is generated by the elements $\beta^\lambda(t)$. Thus parts [\[comparealgebras1\]](#comparealgebras1){reference-type="ref" reference="comparealgebras1"} and [\[comparealgebras4\]](#comparealgebras4){reference-type="ref" reference="comparealgebras4"} of [Lemma 46](#lem:comparealgebras){reference-type="ref" reference="lem:comparealgebras"} together imply [Theorem 3](#thm:main){reference-type="ref" reference="thm:main"}[\[main_commutativity\]](#main_commutativity){reference-type="ref" reference="main_commutativity"}. In particular, [Lemma 46](#lem:comparealgebras){reference-type="ref" reference="lem:comparealgebras"} tells us that $\overline{\mathcal{B}}_n(z_1, \dots, z_n)$ has all of the same basic properties as $\mathcal{B}_n(z_1, \dots, z_n)$, described in [Theorem 42](#thm:bethebasics){reference-type="ref" reference="thm:bethebasics"}. Therefore, once the lemma is proved, we will begin to use the same notation as we used for $\mathcal{B}_n(z_1, \dots, z_n)$. We write $\overline{\mathcal{B}}_\nu(z_1, \dots, z_n)\subseteq \mathrm{End}(M^{\nu})$ to denote the image of the action $\overline{\mathcal{B}}_n(z_1, \dots, z_n)$ on $M^{\nu}$. As a consequence of [Lemma 46](#lem:comparealgebras){reference-type="ref" reference="lem:comparealgebras"}[\[comparealgebras3\]](#comparealgebras3){reference-type="ref" reference="comparealgebras3"}, we have the direct product decomposition $$\label{eq:directproductaltbethe} \overline{\mathcal{B}}_n(z_1, \dots, z_n)\simeq \prod_{\nu \vdash n} \overline{\mathcal{B}}_\nu(z_1, \dots, z_n) \,.$$ We write $\beta^\lambda_{\ell,\nu},\, \beta^\lambda_\nu(t) \in \overline{\mathcal{B}}_\nu(z_1, \dots, z_n)$ for the operators $\beta^\lambda_\ell,\, \beta^\lambda(t)$ acting on $M^{\nu}$. For each eigenspace $E \subseteq M^{\nu}$ of $\overline{\mathcal{B}}_\nu(z_1, \dots, z_n)$, we write $\beta^\lambda_{\ell,E}$ and $\beta^\lambda_E(t)$ for the corresponding eigenvalues of $\beta^\lambda_{\ell,\nu}$ and $\beta^\lambda_\nu(t)$, respectively. ## Translation identity {#sec:translation} We first prove the translation identity, [Theorem 3](#thm:main){reference-type="ref" reference="thm:main"}[\[main_translation\]](#main_translation){reference-type="ref" reference="main_translation"}, which is required for all parts of [Lemma 46](#lem:comparealgebras){reference-type="ref" reference="lem:comparealgebras"}. Recall the elements $\alpha^\lambda_X$ defined in [[([\[eq:alphadef\]](#eq:alphadef){reference-type="ref" reference="eq:alphadef"})]{.upright}](#eq:alphadef). **Lemma 47**. *Let $\mu\vdash k$, and let $N$ denote the representation of $\mathfrak{S}_{n}$ induced by the Specht module $M^{\mu}$ of $\mathfrak{S}_{k}$ (under the containment $\mathfrak{S}_{k}\subseteq\mathfrak{S}_{n}$). Let $\chi^N:\mathfrak{S}_{n}\to\mathbb{C}$ denote the character of $N$. Then $$\label{eq:induced} \sum_{\sigma\in\mathfrak{S}_{n}}\chi^N(\sigma)\sigma = (n-k)!\sum_{X\in\binom{[n]}{k}}\alpha^\mu_X = \sum_{\substack{\lambda\vdash n, \\ \lambda\supseteq \mu}}\mathsf{f}^{\lambda/\mu}\alpha^\lambda_{[n]}\,.$$* *Proof.* By the branching rule [@sagan01 Theorem 2.8.3], we have $$\chi^N = \sum_{\substack{\lambda\vdash n, \\ \lambda\supseteq \mu}}\mathsf{f}^{\lambda/\mu}\chi^\lambda\,,$$ so the first and third expressions in [[([\[eq:induced\]](#eq:induced){reference-type="ref" reference="eq:induced"})]{.upright}](#eq:induced) are equal. On the other hand, by the definition of the induced representation [@serre98 Theorem 12], we have $$\chi^N(\sigma) = \sum_{\substack{\pi\in C, \\ \pi^{-1}\sigma\pi\in\mathfrak{S}_{k}}}\chi^\mu(\pi^{-1}\sigma\pi) \quad \text{ for all } \sigma\in\mathfrak{S}_{n}\,,$$ where $C\subseteq\mathfrak{S}_{n}$ is any set of coset representatives of the quotient $\mathfrak{S}_{n}/\mathfrak{S}_{k}$. Note that for any $\pi\in\mathfrak{S}_{n}$, we have $\pi\mathfrak{S}_{k}\pi^{-1} = \mathfrak{S}_{\pi([k])}$. Conversely, for any $X\in\binom{[n]}{k}$, there exist precisely $(n-k)!$ elements $\pi\in C$ such that $\pi([k]) = X$. Hence $$\chi^N(\sigma) = (n-k)!\sum_{\substack{X\in\binom{[n]}{k},\\ \sigma\in\mathfrak{S}_{X}}}\chi^\mu(\sigma)\,,$$ where inside the sum, $\chi^\mu$ is a character of $\mathfrak{S}_{X}$. Therefore the first and second expressions in [[([\[eq:induced\]](#eq:induced){reference-type="ref" reference="eq:induced"})]{.upright}](#eq:induced) are equal. ◻ *Proof of [Theorem 3](#thm:main){reference-type="ref" reference="thm:main"}[\[main_translation\]](#main_translation){reference-type="ref" reference="main_translation"}.* We regard $z_1, \dots, z_n$ as formal indeterminates. By translating $(z_1, \dots, z_n) \mapsto (z_1+s, \dots, z_n+s)$, it suffices to prove the identity [[([\[eq:translationidentity\]](#eq:translationidentity){reference-type="ref" reference="eq:translationidentity"})]{.upright}](#eq:translationidentity) when $s = 0$, i.e., $$\label{eq:translationzero} \beta^\mu(t) = \sum_{\lambda \supseteq \mu} \frac{\mathsf{f}^{\lambda/\mu}}{|\lambda/\mu|!} t^{|\lambda/\mu|} \beta^\lambda \,.$$ Let $k := |\mu|$, and note that [[([\[eq:translationzero\]](#eq:translationzero){reference-type="ref" reference="eq:translationzero"})]{.upright}](#eq:translationzero) is symmetric and square-free in $z_1, \dots, z_n$, and homogeneous of total degree $n-k$ in $z_1, \dots, z_n, t$. Therefore it suffices to prove that the coefficients of $z_{\ell+1}\cdots z_nt^{\ell-k}$ on both sides are equal for all $k \le \ell \le n$. This is the second equality of [[([\[eq:induced\]](#eq:induced){reference-type="ref" reference="eq:induced"})]{.upright}](#eq:induced) (after replacing $n$ by $\ell$ and rescaling by $\frac{1}{(\ell-k)!}$). ◻ We now prove parts [\[comparealgebras4\]](#comparealgebras4){reference-type="ref" reference="comparealgebras4"} and [\[comparealgebras3\]](#comparealgebras3){reference-type="ref" reference="comparealgebras3"} of [Lemma 46](#lem:comparealgebras){reference-type="ref" reference="lem:comparealgebras"}. By definition, the algebra $\overline{\mathcal{B}}_n(z_1+t, \dots, z_n+t)$ is generated by the elements $\beta^\lambda(t)$. From [[([\[eq:translationzero\]](#eq:translationzero){reference-type="ref" reference="eq:translationzero"})]{.upright}](#eq:translationzero), we see that $\beta^\mu(t) \in \overline{\mathcal{B}}_n(z_1, \dots, z_n)$ for all $\mu$. Conversely, from [[([\[eq:translationidentity\]](#eq:translationidentity){reference-type="ref" reference="eq:translationidentity"})]{.upright}](#eq:translationidentity) we get $$\beta^\mu = \beta^\mu(t-t) = \sum_{\lambda \supseteq \mu} \frac{\mathsf{f}^{\lambda/\mu}}{|\lambda/\mu|!} (-t)^{|\lambda/\mu|} \beta^\lambda(t) \,,$$ implying that $\beta^\mu \in \overline{\mathcal{B}}_n(z_1+t, \dots, z_n+t)$. This proves part [\[comparealgebras4\]](#comparealgebras4){reference-type="ref" reference="comparealgebras4"}. In the special case where $\lambda = 1^k$, we have that $\beta^{1^{\smash{k}}}(t) \in \overline{\mathcal{B}}_n(z_1, \dots, z_n)$. Since these elements generate of $\mathcal{B}_n(z_1, \dots, z_n)$, this proves part [\[comparealgebras3\]](#comparealgebras3){reference-type="ref" reference="comparealgebras3"}. ## Commutativity {#sec:commutativity} For every partition $\mu$ and $\ell \geq 0$, we define elements $\varepsilon^\mu_\ell \in \overline{\mathcal{B}}_n(z_1, \dots, z_n)$ by $$\varepsilon^\mu_\ell := \sum_{\substack{X \cap Y = \emptyset, \\ |X| = |\mu|, |Y| = \ell}}\; \sum_{\substack{\sigma \in \mathfrak{S}_{X}, \\ \mathsf{cyc}(\sigma) = \mu}} \sigma \, z_Y \,.$$ We write $\varepsilon^\mu(t) := \sum_{\ell = 0}^n \varepsilon^\mu_\ell t^{n-|\mu|-\ell}$, and set $\varepsilon^\mu := \varepsilon^\mu(0) = \varepsilon^\mu_{n-|\mu|}$. These are related to the elements $\beta^\lambda(t)$ by the following change of basis formulas: $$\label{eq:changeofbasis} \beta^\lambda(t) = \sum_{\mu \vdash |\lambda|} \chi^\lambda_\mu \varepsilon^\mu(t) \qquad \text{and} \qquad \varepsilon^\mu(t) = \sum_{\lambda \vdash |\mu|} \frac{\chi^\lambda_\mu}{\mathsf{z}_\mu} \beta^\lambda(t) \,.$$ The first formula follows by definition, whence the second formula follows from [[([\[eq:spchangeofbasis\]](#eq:spchangeofbasis){reference-type="ref" reference="eq:spchangeofbasis"})]{.upright}](#eq:spchangeofbasis). In particular, the elements $\varepsilon^\mu$ are an alternate set of generators for $\overline{\mathcal{B}}_n(z_1, \dots, z_n)$. The $\varepsilon^\mu$'s are combinatorially simpler than the $\beta^\lambda$'s, in that they have fewer terms and do not involve characters. However, even working with this new set of generators, it is difficult to prove the commutativity relations $\varepsilon^\lambda \varepsilon^\mu = \varepsilon^\mu \varepsilon^\lambda$ directly (see [5.4](#sec:commcomb){reference-type="ref" reference="sec:commcomb"} for further discussion). We reduce the problem to a different identity, which follows from combinatorial arguments in [@purbhoo]. ### Reduction {#sec:creduction} **Lemma 48**. *The following are equivalent:* (a) *[\[commutativityreduction_a\]]{#commutativityreduction_a label="commutativityreduction_a"} for all $z_1, \dots, z_n\in\mathbb{C}$, the algebra $\overline{\mathcal{B}}_n(z_1, \dots, z_n)$ is commutative;* (b) *[\[commutativityreduction_b\]]{#commutativityreduction_b label="commutativityreduction_b"} for all $z_1, \dots, z_n\in\mathbb{C}$, the algebra $\overline{\mathcal{B}}_n(z_1, \dots, z_n)$ commutes with $\mathcal{B}_n(z_1, \dots, z_n)$;* (c) *[\[commutativityreduction_c\]]{#commutativityreduction_c label="commutativityreduction_c"} for all $k, \ell \geq 0$ and all partitions $\mu$, we have $$\label{eq:reducedcommutativity} \varepsilon^\mu_\ell \beta^{1^{\smash{k}}} = \beta^{1^{\smash{k}}} \varepsilon^\mu_\ell \,.$$* *Proof.* [\[commutativityreduction_a\]](#commutativityreduction_a){reference-type="ref" reference="commutativityreduction_a"} $\Rightarrow$ [\[commutativityreduction_b\]](#commutativityreduction_b){reference-type="ref" reference="commutativityreduction_b"}: This follows immediately from [Lemma 46](#lem:comparealgebras){reference-type="ref" reference="lem:comparealgebras"}[\[comparealgebras3\]](#comparealgebras3){reference-type="ref" reference="comparealgebras3"}. [\[commutativityreduction_b\]](#commutativityreduction_b){reference-type="ref" reference="commutativityreduction_b"} $\Rightarrow$ [\[commutativityreduction_a\]](#commutativityreduction_a){reference-type="ref" reference="commutativityreduction_a"}: Suppose that [\[commutativityreduction_b\]](#commutativityreduction_b){reference-type="ref" reference="commutativityreduction_b"} holds. Since commutativity is preserved under limits, it suffices to prove [\[commutativityreduction_a\]](#commutativityreduction_a){reference-type="ref" reference="commutativityreduction_a"} when $z_1, \dots, z_n \in \mathbb{C}$ are generic. Then $\mathcal{B}_n(z_1, \dots, z_n)$ is a maximal commutative subalgebra of $\mathbb{C}[\mathfrak{S}_{n}]$ by [Theorem 42](#thm:bethebasics){reference-type="ref" reference="thm:bethebasics"}[\[bethebasics2\]](#bethebasics2){reference-type="ref" reference="bethebasics2"}. By definition, this means that any element of $\mathbb{C}[\mathfrak{S}_{n}]$ that commutes with $\mathcal{B}_n(z_1, \dots, z_n)$ must belong to $\mathcal{B}_n(z_1, \dots, z_n)$. Hence $\overline{\mathcal{B}}_n(z_1, \dots, z_n)\subseteq \mathcal{B}_n(z_1, \dots, z_n)$, which implies [\[commutativityreduction_a\]](#commutativityreduction_a){reference-type="ref" reference="commutativityreduction_a"}. [\[commutativityreduction_b\]](#commutativityreduction_b){reference-type="ref" reference="commutativityreduction_b"} $\Leftrightarrow$ [\[commutativityreduction_c\]](#commutativityreduction_c){reference-type="ref" reference="commutativityreduction_c"}: We will prove this treating $z_1, \dots, z_n$ as formal indeterminates. Statement [\[commutativityreduction_b\]](#commutativityreduction_b){reference-type="ref" reference="commutativityreduction_b"} means that every generator of $\overline{\mathcal{B}}_n(z_1, \dots, z_n)$ commutes with every generator of $\mathcal{B}_n(z_1, \dots, z_n)$, i.e., $$\varepsilon^\mu \beta^{1^{\smash{k}}}(t) = \beta^{1^{\smash{k}}}(t) \varepsilon^\mu \qquad \text{for all } k, t, \mu\,.$$ Since $z_1, \dots, z_n$ are formal indeterminates, this is equivalent to the statement obtained if we translate $(z_1, \dots, z_n) \mapsto (z_1-t, \dots, z_n-t)$: $$\varepsilon^\mu(-t) \beta^{1^{\smash{k}}} = \beta^{1^{\smash{k}}} \varepsilon^\mu(-t) \qquad \text{for all } k, t, \mu\,.$$ Comparing coefficients of $t$ on both sides of the equation above, this is equivalent to [\[commutativityreduction_c\]](#commutativityreduction_c){reference-type="ref" reference="commutativityreduction_c"}. ◻ ### Bijective proof of the relations [[([\[eq:reducedcommutativity\]](#eq:reducedcommutativity){reference-type="ref" reference="eq:reducedcommutativity"})]{.upright}](#eq:reducedcommutativity) {#sec:cbijection} We now explain how [[([\[eq:reducedcommutativity\]](#eq:reducedcommutativity){reference-type="ref" reference="eq:reducedcommutativity"})]{.upright}](#eq:reducedcommutativity) follows from [@purbhoo]; this shows that $\overline{\mathcal{B}}_n(z_1, \dots, z_n)$ is commutative. A ***supported permutation*** of $[n]$ is formally a pair $(\sigma, X)$, which we write as $\sigma_X$, where $\sigma \in \mathfrak{S}_{n}$, and $X \subseteq [n]$ is a set such that $\sigma$ belongs to the subgroup $\mathfrak{S}_{X} \subseteq \mathfrak{S}_{n}$ (equivalently, $[n] \setminus X$ is a subset of the fixed points of $\sigma$). The set $X$ is called the ***support*** of $\sigma_X$, and $\mathsf{cyc}(\sigma_X)$ denotes the partition of $|X|$ which is the cycle type of $\sigma$ restricted to $X$. Let $\mathrm{SP}_n$ denote the set of all supported permutations of $[n]$. With this notation, we can rewrite the generators of $\overline{\mathcal{B}}_n(z_1, \dots, z_n)$ as follows: $$\label{eq:betadef3} \beta^\lambda = \sum_{\substack{\sigma_X \in \mathrm{SP}_n, \\ |X| = |\lambda|}} \chi^\lambda(\sigma) \sigma \,z_{[n] \setminus X}\,, \qquad \varepsilon^\mu = \sum_{\substack{\sigma_X \in \mathrm{SP}_n, \\ \mathsf{cyc}(\sigma_X) = \mu}} \sigma \, z_{[n] \setminus X} \,.$$ We similarly expand both sides of [[([\[eq:reducedcommutativity\]](#eq:reducedcommutativity){reference-type="ref" reference="eq:reducedcommutativity"})]{.upright}](#eq:reducedcommutativity). Let $E^\mu_\ell$ denote the set of pairs $(\sigma_X,Y)$ such that $\sigma_X\in\mathrm{SP}_n$ with $\mathsf{cyc}(\sigma_X) = \mu$, and $Y\subseteq [n]\setminus X$ with $|Y| = \ell$. Let $B_k$ denote the set of pairs $(\sigma'_{X'},Y')$ such that $\sigma'_{X'}\in\mathrm{SP}_n$ with $|X'| = k$, and $Y' = [n] \setminus X'$. Then $$\varepsilon^\mu_\ell = \sum_{(\sigma_X, Y) \in E^\mu_\ell} \sigma \, z_Y \qquad \text{and} \qquad \beta^{1^{\smash{k}}} = \sum_{(\sigma'_{X'}, Y') \in B_k} \mathrm{sgn}(\sigma') \sigma' z_{Y'} \,,$$ and [[([\[eq:reducedcommutativity\]](#eq:reducedcommutativity){reference-type="ref" reference="eq:reducedcommutativity"})]{.upright}](#eq:reducedcommutativity) is the statement $$\label{eq:reducedcommutativity2} \sum_{(\sigma_X, Y; \sigma'_{X'}, Y') \in E^\mu_\ell \times B_k} \mathrm{sgn}(\sigma') \sigma \sigma' z_Y z_{Y'} = \sum_{(\bar\sigma'_{\bar X'}, \bar Y'; \bar\sigma_{\bar X}, \bar Y) \in B_k \times E^\mu_\ell} \mathrm{sgn}(\bar \sigma') \bar \sigma' \bar \sigma \, z_{\bar Y} z_{\bar Y'} \,.$$ To prove [[([\[eq:reducedcommutativity\]](#eq:reducedcommutativity){reference-type="ref" reference="eq:reducedcommutativity"})]{.upright}](#eq:reducedcommutativity), we want to match up each term on the left-hand side of [[([\[eq:reducedcommutativity2\]](#eq:reducedcommutativity2){reference-type="ref" reference="eq:reducedcommutativity2"})]{.upright}](#eq:reducedcommutativity2) with an equal term on the right-hand side. That is, we seek a bijection $$\begin{aligned} E^\mu_\ell \times B_k &\to B_k \times E^\mu_\ell\,, \\ (\sigma_X, Y; \sigma'_{X'}, Y') &\mapsto (\bar\sigma'_{\bar X'}, \bar Y'; \bar\sigma_{\bar X}, \bar Y)\end{aligned}$$ such that $\mathrm{sgn}(\sigma') = \mathrm{sgn}(\bar \sigma')$, $\sigma \sigma' = \bar \sigma' \bar\sigma$, and $z_Yz_{Y'} = z_{\bar Y} z_{\bar Y'}$. A bijection with precisely these properties is established in [@purbhoo Proposition 5.3]. Hence [[([\[eq:reducedcommutativity\]](#eq:reducedcommutativity){reference-type="ref" reference="eq:reducedcommutativity"})]{.upright}](#eq:reducedcommutativity) holds. 0◻ ### Proof of [Lemma 46](#lem:comparealgebras){reference-type="ref" reference="lem:comparealgebras"} {#proof-of-lemcomparealgebras} We now complete the proof of [Lemma 46](#lem:comparealgebras){reference-type="ref" reference="lem:comparealgebras"}. We have already established parts [\[comparealgebras3\]](#comparealgebras3){reference-type="ref" reference="comparealgebras3"} and [\[comparealgebras4\]](#comparealgebras4){reference-type="ref" reference="comparealgebras4"} in [3.2](#sec:translation){reference-type="ref" reference="sec:translation"}, and [\[comparealgebras1\]](#comparealgebras1){reference-type="ref" reference="comparealgebras1"} is proved in [3.3.2](#sec:cbijection){reference-type="ref" reference="sec:cbijection"}. It remains to prove part [\[comparealgebras2\]](#comparealgebras2){reference-type="ref" reference="comparealgebras2"}. The containment $\mathcal{B}_n(z_1, \dots, z_n)\subseteq \overline{\mathcal{B}}_n(z_1, \dots, z_n)$ follows from part [\[comparealgebras3\]](#comparealgebras3){reference-type="ref" reference="comparealgebras3"}, and the reverse containment follows from the proof of the implication [\[commutativityreduction_b\]](#commutativityreduction_b){reference-type="ref" reference="commutativityreduction_b"} $\Rightarrow$ [\[commutativityreduction_a\]](#commutativityreduction_a){reference-type="ref" reference="commutativityreduction_a"} in [3.3.1](#sec:creduction){reference-type="ref" reference="sec:creduction"}. 0◻ **Remark 49**. It is interesting to note that although commutativity is deduced by taking limits from the generic case, we cannot similarly deduce that $\overline{\mathcal{B}}_n(z_1, \dots, z_n)= \mathcal{B}_n(z_1, \dots, z_n)$, even though we know this to be true generically. This is because equality of algebras is not necessarily preserved under taking limits of generators. The most we can deduce from limiting arguments is that $\overline{\mathcal{B}}_n(z_1, \dots, z_n)$ and $\mathcal{B}_n(z_1, \dots, z_n)$ have the same generalized eigenspaces on any representation of $\mathfrak{S}_{n}$. In order to show that $\overline{\mathcal{B}}_n(z_1, \dots, z_n)= \mathcal{B}_n(z_1, \dots, z_n)$, we will argue in [4.3.2](#sec:altbetheequalsbethe){reference-type="ref" reference="sec:altbetheequalsbethe"} that there exist polynomial formulas expressing the generators of $\overline{\mathcal{B}}_n(z_1, \dots, z_n)$ in terms of the generators of $\mathcal{B}_n(z_1, \dots, z_n)$. As these formulas are preserved under taking limits, this will imply the result. # Plücker relations {#sec:pr} In this section we prove the remaining parts (i.e. [\[main_pluckers\]](#main_pluckers){reference-type="ref" reference="main_pluckers"}--[\[main_multiplicity\]](#main_multiplicity){reference-type="ref" reference="main_multiplicity"}) of [Theorem 3](#thm:main){reference-type="ref" reference="thm:main"}. The bulk of argument involves showing that the operators $\beta^\lambda(t)$ satisfy the Plücker relations ([Theorem 3](#thm:main){reference-type="ref" reference="thm:main"}[\[main_pluckers\]](#main_pluckers){reference-type="ref" reference="main_pluckers"}). We divide this into two parts, which we state now as lemmas: **Lemma 50**. *The operators $\beta^\lambda$ satisfy all single-column Plücker relations.* **Lemma 51**. *[Lemma 50](#lem:part1){reference-type="ref" reference="lem:part1"} and the translation identity [[([\[eq:translationidentity\]](#eq:translationidentity){reference-type="ref" reference="eq:translationidentity"})]{.upright}](#eq:translationidentity) together imply that the operators $\beta^\lambda(t)$ satisfy the Plücker relations ([Theorem 3](#thm:main){reference-type="ref" reference="thm:main"}[\[main_pluckers\]](#main_pluckers){reference-type="ref" reference="main_pluckers"}).* The two parts are very different in character. The proof of [Lemma 50](#lem:part1){reference-type="ref" reference="lem:part1"} uses $\tau$-functions of the KP hierarchy, and involves a combinatorial analysis of factorizations of permutations, along with several symmetric-function identities. To prove [Lemma 51](#lem:part2){reference-type="ref" reference="lem:part2"}, we use the exterior algebra, and some of the abstract properties of $\overline{\mathcal{B}}_n(z_1, \dots, z_n)$ from [Theorem 42](#thm:bethebasics){reference-type="ref" reference="thm:bethebasics"} and [Lemma 46](#lem:comparealgebras){reference-type="ref" reference="lem:comparealgebras"}. We prove [\[lem:part1,lem:part2\]](#lem:part1,lem:part2){reference-type="ref" reference="lem:part1,lem:part2"} in [\[sec:part1proof,sec:part2\]](#sec:part1proof,sec:part2){reference-type="ref" reference="sec:part1proof,sec:part2"}, respectively. We conclude in [4.3](#sec:finalsteps){reference-type="ref" reference="sec:finalsteps"} by finishing the proof of [Theorem 3](#thm:main){reference-type="ref" reference="thm:main"}, by showing that parts [\[main_generates\]](#main_generates){reference-type="ref" reference="main_generates"}--[\[main_multiplicity\]](#main_multiplicity){reference-type="ref" reference="main_multiplicity"} hold. ## Proof of [Lemma 50](#lem:part1){reference-type="ref" reference="lem:part1"}: single-column Plücker relations {#sec:part1proof} We turn to the proof of [Lemma 50](#lem:part1){reference-type="ref" reference="lem:part1"}, using $\tau$-functions of the KP hierarchy. ### A $\tau$-function of the KP hierarchy Define $\tau_n \in \mathbb{C}[\mathfrak{S}_{n}] \otimes \Lambda$ to be the following symmetric function with coefficients in the commutative algebra $\overline{\mathcal{B}}_n(z_1, \dots, z_n)\subseteq \mathbb{C}[\mathfrak{S}_{n}]$: $$\label{eq:taudef} \tau_n := \sum_\lambda \beta^\lambda \otimes \mathsf{s}_\lambda \,,$$ where the sum is taken over all partitions $\lambda$. Since $\beta^\lambda = 0$ for $|\lambda| > n$, this is a finite sum over all $|\lambda| \le n$. We mention that because the coefficients of $\tau_n$ in the Schur basis are the operators $\beta^\lambda$, we can use [Theorem 32](#thm:KPplucker){reference-type="ref" reference="thm:KPplucker"} to rephrase [Theorem 3](#thm:main){reference-type="ref" reference="thm:main"}[\[main_pluckers\]](#main_pluckers){reference-type="ref" reference="main_pluckers"} as follows: **Theorem 52**. *For all $n \geq 0$ and all $z_1, \dots, z_n \in \mathbb{C}$, the symmetric function $\tau_n$ in [[([\[eq:taudef\]](#eq:taudef){reference-type="ref" reference="eq:taudef"})]{.upright}](#eq:taudef) is a $\tau$-function of the KP hierarchy, with coefficients in (a commutative subalgebra of) $\mathbb{C}[\mathfrak{S}_{n}]$.* Our approach will instead be to use $\tau_n$ to prove the single-column Plücker relations. We need the following formula for $\tau_n$ in the power sum basis: **Proposition 53**. *We have $$\label{eq:tau} \tau_n = \sum_\mu \varepsilon^\mu \otimes \mathsf{p}_\mu = \sum_{\sigma_X \in \mathrm{SP}_n} \sigma \, z_{[n] \setminus X} \otimes \mathsf{p}_{\mathsf{cyc}(\sigma_X)} \,.$$* *Proof.* The first equality follows from [[([\[eq:spchangeofbasis\]](#eq:spchangeofbasis){reference-type="ref" reference="eq:spchangeofbasis"})]{.upright}](#eq:spchangeofbasis) and [[([\[eq:changeofbasis\]](#eq:changeofbasis){reference-type="ref" reference="eq:changeofbasis"})]{.upright}](#eq:changeofbasis). Then [[([\[eq:betadef3\]](#eq:betadef3){reference-type="ref" reference="eq:betadef3"})]{.upright}](#eq:betadef3) gives the second equality. ◻ By [Lemma 33](#lem:singlecolumneqs){reference-type="ref" reference="lem:singlecolumneqs"}, the operators $\beta^\lambda$ satisfy the single-column Plücker relations if and only if equation [[([\[eq:scKP2\]](#eq:scKP2){reference-type="ref" reference="eq:scKP2"})]{.upright}](#eq:scKP2) is satisfied. Since $\langle \tau_n, \mathsf{e}_k \rangle = \langle \tau_n, \mathsf{s}_{1^k} \rangle = \beta^{1^{\smash{k}}}$, in our case [[([\[eq:scKP2\]](#eq:scKP2){reference-type="ref" reference="eq:scKP2"})]{.upright}](#eq:scKP2) is the statement that $$[t^{-1}]\mbox{\small\sf{B}}^\perp(t^{-1}) \sum_{k=0}^n (-t)^{-k} \beta^{1^{\smash{k}}} \cdot \tau_n = 0 \,.$$ Using [[([\[eq:betadef3\]](#eq:betadef3){reference-type="ref" reference="eq:betadef3"})]{.upright}](#eq:betadef3) and [[([\[eq:tau\]](#eq:tau){reference-type="ref" reference="eq:tau"})]{.upright}](#eq:tau), the sum $\sum_{k=0}^n (-t)^{-k} \beta^{1^{\smash{k}}} \cdot \tau_n$ expands into $$\label{eq:scoperand} \sum_{\sigma_X \in \mathrm{SP}_n} \, \sum_{\pi_Y \in \mathrm{SP}_n} \mathrm{sgn}(\sigma) \cdot \sigma \pi \, z_{[n] \setminus X} z_{[n] \setminus Y} \otimes (-t)^{-|X|} \mathsf{p}_{\mathsf{cyc}(\pi_Y)} \,.$$ Therefore, to prove [Lemma 50](#lem:part1){reference-type="ref" reference="lem:part1"}, we must show that [[([\[eq:scoperand\]](#eq:scoperand){reference-type="ref" reference="eq:scoperand"})]{.upright}](#eq:scoperand) is in the kernel of the operator $[t^{-1}]\mbox{\small\sf{B}}^\perp(t^{-1})$. ### $Z$-factorizations {#sec:zfac} We will now treat $z_1, \dots, z_n$ as formal indeterminates. Since the operator $[t^{-1}]\mbox{\small\sf{B}}^\perp(t^{-1})$ is linear, and treats the first tensor factor of [[([\[eq:scoperand\]](#eq:scoperand){reference-type="ref" reference="eq:scoperand"})]{.upright}](#eq:scoperand) as a scalar, we can prove [Lemma 50](#lem:part1){reference-type="ref" reference="lem:part1"} coefficient-by-coefficient. That is, it is necessary and sufficient to show that for all $\theta \in \mathfrak{S}_{n}$ and all $j_1, \dots, j_n \in \{0,1,2\}$, the $\theta z_1^{j_1} \dotsm z_n^{j_n}$-coefficient of [[([\[eq:scoperand\]](#eq:scoperand){reference-type="ref" reference="eq:scoperand"})]{.upright}](#eq:scoperand) is in $\ker [t^{-1}]\mbox{\small\sf{B}}^\perp(t^{-1})$. Furthermore, it suffices to prove this for coefficients which are square-free in the $z_i$'s. This is because each non-square-free coefficient is equal to a square-free coefficient for a smaller value of $n$. Specifically, the sum of all terms which contain a factor of $z_n^2$ is $$z_n^2 \cdot \sum_{\sigma_X \in \mathrm{SP}_{n-1}} \, \sum_{\pi_Y \in \mathrm{SP}_{n-1}} \mathrm{sgn}(\sigma) \cdot \sigma \pi \, z_{[n-1] \setminus X} z_{[n-1] \setminus Y} \otimes (-t)^{-|X|} \mathsf{p}_{\mathsf{cyc}(\pi_Y)} \,,$$ which (apart from the factor of $z_n^2$) is just [[([\[eq:scoperand\]](#eq:scoperand){reference-type="ref" reference="eq:scoperand"})]{.upright}](#eq:scoperand) with $n$ replaced by $n-1$. Therefore, we fix $\theta \in \mathfrak{S}_{n}$ and $Z \subseteq [n]$, and let $C_{\theta,Z}$ denote the coefficient of $\theta z_{[n] \setminus Z}$ in [[([\[eq:scoperand\]](#eq:scoperand){reference-type="ref" reference="eq:scoperand"})]{.upright}](#eq:scoperand). Our goal is now to show that $C_{\theta,Z} \in \ker [t^{-1}]\mbox{\small\sf{B}}^\perp(t^{-1})$. We begin by obtaining a useful formula for $C_{\theta,Z}$. Since [[([\[eq:scoperand\]](#eq:scoperand){reference-type="ref" reference="eq:scoperand"})]{.upright}](#eq:scoperand) is a quadratic expression in $\mathbb{C}[\mathfrak{S}_{n}]$, we can write $C_{\theta,Z}$ as a sum over certain factorizations of $\theta$. Namely, define a ***$Z$-factorization*** of $\theta$ to be a pair of supported permutations $(\sigma_X, \pi_Y)$ such that $X\cup Y = [n]$, $X \cap Y = Z$, and $\sigma \pi = \theta$. We see that $$C_{\theta,Z} = \sum_{(\sigma_X, \pi_Y)} \mathrm{sgn}(\sigma) (-t)^{-|X|} \mathsf{p}_{\mathsf{cyc}(\pi_Y)} \,,$$ where the sum is over all $Z$-factorizations of $\theta$. It will be more convenient to write $C_{\theta,Z}$ purely in terms of $\pi$ and $Y$. We say that $\pi_Y$ is a ***right $Z$-factor*** of $\theta$, if there exists $\sigma_X$ such that $(\sigma_X, \pi_Y)$ is a $Z$-factorization of $\theta$. Note that if $\sigma_X$ exists, then it is unique, as we must have $\sigma = \theta \pi^{-1}$ and $X = [n] \setminus (Y \setminus Z)$. (However, for some $\pi_Y$, this construction may not produce a valid supported permutation $\sigma_X$.) Using $\mathrm{sgn}(\theta) = \mathrm{sgn}(\sigma)\mathrm{sgn}(\pi)$ and $|Z| = |X|+|Y|-n$, we obtain $$C_{\theta,Z} = \mathrm{sgn}(\theta) \sum_{\pi_Y} \mathrm{sgn}(\pi) (-t)^{|Y|-|Z|-n} \mathsf{p}_{\mathsf{cyc}(\pi_Y)} \,,$$ where the sum is taken over all right $Z$-factors $\pi_Y$ of $\theta$. Finally, for a given $Y \subseteq [n]$, let $\mathrm{RF}(Y)$ denote the set of all permutations $\pi \in \mathfrak{S}_{Y}$ such that $\pi_Y$ is a right $Z$-factor of $\theta$. With this notation, we write $$\label{eq:thetaZcoeff} C_{\theta,Z} = \mathrm{sgn}(\theta) \sum_{Y \subseteq [n]} (-t)^{|Y|-|Z|-n} \sum_{\pi \in \mathrm{RF}(Y)} \mathrm{sgn}(\pi) \mathsf{p}_{\mathsf{cyc}(\pi_Y)} \,.$$ ### $Z$-strips Next, we evaluate the inner sum of [[([\[eq:thetaZcoeff\]](#eq:thetaZcoeff){reference-type="ref" reference="eq:thetaZcoeff"})]{.upright}](#eq:thetaZcoeff): $$\sum_{\pi \in \mathrm{RF}(Y)} \mathrm{sgn}(\pi) \mathsf{p}_{\mathsf{cyc}(\pi_Y)} \,.$$ To accomplish this, we need a precise understanding of which supported permutations are right $Z$-factors of $\theta$. Define a ***$Z$-strip*** of $\theta$ to be a sequence of the form $$\big(a, \theta(a), \theta^2(a), \dots, \theta^{r}(a)\big)\,,$$ where $a \in Z$, $r\ge 0$, $\theta^{r+1}(a) \in Z$, and $\theta(a), \dots, \theta^{r}(a) \notin Z$. A ***$Z$-substrip*** of $\theta$ is a nonempty left-substring of a $Z$-strip, i.e., $$\big( a, \theta(a), \theta^2(a), \dots, \theta^{r'}(a) \big) \qquad \text{ for some }\, 0 \le r' \le r\,.$$ Abusing notation, we will sometimes think of these as sets: this is reasonable because the elements of any $Z$-substrip are distinct, and for any set $S \subseteq [n]$, there is at most one ordering of the elements of $S$ which forms a $Z$-substrip. Let $S_1, \dots, S_k$ be the $Z$-strips of $\theta$. Note that the $S_i$'s are pairwise disjoint and $k = |Z|$, as each element of $Z$ defines a unique $Z$-strip. Put $S := S_1 \cup \dots \cup S_k$. Let $\lambda_i := |S_i|$, and write $\lambda =(\lambda_1, \dots, \lambda_k)$. Thus $\lambda$ is a composition of size $|\lambda| = |S|$. Let $T := [n] \setminus S$ be the set of points which are not in any $Z$-strip. Then $T \cap Z = \emptyset$ and $\theta(T) = T$. Write $\theta|_T \in \mathfrak{S}_{T}$ to denote the permutation $\theta$ restricted to $T$. **Example 54**. Let $\theta := (1 \;\; 2 \;\; 3 \;\; 4 \;\; 5)(6 \;\; 7 \;\; 8)(9 \;\; 10)(11)$ and $Z := \{3,5,7\}$. The $Z$-strips of $\theta$ are $$34,\ 512, \text{ and } 786\,.$$ We have $T = \{9,10,11\}$ and $\theta|_T = (9 \;\; 10)(11)$. We say that a subset $Y \subseteq [n]$ is ***valid*** for $(\theta, Z)$ if the following conditions hold: - $Z \subseteq Y$; - $S_i \cap Y$ is a $Z$-substrip of $\theta$ for all $i \in [k]$; and - $\theta(T \cap Y) = T \cap Y$. **Lemma 55**. *Let $\pi \in \mathfrak{S}_{Y}$.* (i) *[\[RF1\]]{#RF1 label="RF1"} If $Y$ is not valid, then $\mathrm{RF}(Y)$ is empty.* (ii) *[\[RF2\]]{#RF2 label="RF2"} If $Y$ is valid, then $\pi \in \mathrm{RF}(Y)$ if and only if the following conditions hold:* (a) *[\[RF_a\]]{#RF_a label="RF_a"} $S_1 \cap Y, \dots, S_k \cap Y$ are the $Z$-strips of $\pi$;* (b) *[\[RF_b\]]{#RF_b label="RF_b"} $\theta|_{T \cap Y} = \pi|_{T \cap Y}$.* *Proof.* First suppose that $\pi \in \mathrm{RF}(Y)$. We will show that $Y$ is valid (thereby proving part [\[RF1\]](#RF1){reference-type="ref" reference="RF1"}) and that [\[RF_a\]](#RF_a){reference-type="ref" reference="RF_a"} and [\[RF_b\]](#RF_b){reference-type="ref" reference="RF_b"} hold (thereby proving the forward direction of part [\[RF2\]](#RF2){reference-type="ref" reference="RF2"}). We regard $\pi$ as a permutation of $[n]$ which fixes $[n]\setminus Y$ pointwise. Since $\pi \in \mathrm{RF}(Y)$, we have $Z \subseteq Y$ and that $\theta\pi^{-1}$ fixes $Y\setminus Z$ pointwise. That is, for all $a \in [n]$, we have $\pi^{-1}(a) = a$ if $a \notin Y$, and $\pi^{-1}(a) = \theta^{-1}(a)$ if $a \in Y\setminus Z$ (and there is no condition if $a \in Z$). Let $(a_1 \; \cdots \; a_r)$ be a cycle of $\theta|_T$, and set $a_0 := a_r$. Then for all $i\in [r]$, we have that $\pi^{-1}(a_i)$ equals either $a_i$ (if $a_i \notin Y$) or $a_{i-1}$ (if $a_i \in Y$). If we have $a_i \in Y$ for some $i$, then $\pi^{-1}(a_i) = a_{i-1}$, so $\pi^{-1}(a_{i-1}) = a_{i-2}$ since $\pi$ is a bijection, and by induction $\pi$ coincides with $\theta$ on the cycle. Therefore $\theta(T\cap Y) = T\cap Y$ and [\[RF_b\]](#RF_b){reference-type="ref" reference="RF_b"} holds. Now let $(a_0, a_1, \dots, a_r)$ be a $Z$-substrip of $\theta$ with $a_r \in Y$. Since $a_1, \dots, a_r \notin Z$, we have $\pi^{-1}(a_r) = a_{r-1}$, and a similar induction shows that $\pi^{-1}(a_i) = a_{i-1}$ (and hence $a_i\in Y$) for all $i\in [r]$. Thus $Y$ is valid and [\[RF_a\]](#RF_a){reference-type="ref" reference="RF_a"} holds. The backward direction of part [\[RF2\]](#RF2){reference-type="ref" reference="RF2"} follows using similar arguments. ◻ Informally, [Lemma 55](#lem:RF){reference-type="ref" reference="lem:RF"}[\[RF2\]](#RF2){reference-type="ref" reference="RF2"} says that for any given valid $Y$, we can form all permutations $\pi \in \mathrm{RF}(Y)$ by assembling the substrips $S_1 \cap Y, \dots, S_k \cap Y$ into cycles in all possible ways. In addition, $\pi$ must include all cycles of $\theta|_{T \cap Y}$. We illustrate this with an example: **Example 56**. For $(\theta, Z)$ from [Example 54](#ex:strips1){reference-type="ref" reference="ex:strips1"}, consider the valid $Y \hspace*{-1.5pt}:=\hspace*{-1.5pt} \{1,\hspace*{-0.1pt} 3,5,6,7,8,9,10\}$. The substrips $S_i \cap Y$ are $$3,\ 51, \text{ and } 786\,,$$ and $\theta|_{T \cap Y} = (9 \;\; 10)$. There are $3!=6$ permutations in $\mathrm{RF}(Y)$, corresponding to the different ways to assemble the three substrips above into cycles: $$\begin{aligned} &(3)(5 \;\; 1)(7 \;\; 8 \;\; 6)(9 \;\; 10)\,, \\ &(3 \;\; 5 \;\; 1)(7 \;\; 8 \;\; 6)(9 \;\; 10)\,, \\ &(3 \;\; 7 \;\; 8 \;\; 6)(5 \;\; 1)(9 \;\; 10)\,, \end{aligned} \qquad \qquad \begin{aligned} &(3)(5 \;\; 1 \;\; 7 \;\; 8 \;\; 6)(9 \;\; 10)\,, \\ &(3 \;\; 5 \;\; 1 \;\; 7 \;\; 8 \;\; 6)(9 \;\; 10)\,, \\ &(3 \;\; 7 \;\; 8 \;\; 6 \;\; 5 \;\; 1)(9 \;\; 10)\,. \end{aligned}$$ Note that $(9 \;\; 10)$ is a cycle of each of these permutations. **Lemma 57**. *Fix a valid $Y$. Let $\kappa_i := |S_i \cap Y|$, and write $\kappa = (\kappa_1, \dots, \kappa_k)$. Let $\mu := \mathsf{cyc}(\theta|_{T\cap Y})$. Then $$\label{eq:innersum} \sum_{\pi \in \mathrm{RF}(Y)} \mathrm{sgn}(\pi) \mathsf{p}_{\mathsf{cyc}(\pi_Y)} = (-1)^{|\kappa|-k + |\mu| - \ell(\mu)} \widetilde{\mathsf{m}}_\kappa \mathsf{p}_\mu \,.$$* **Example 58**. For $(\theta, Z)$ and $Y$ from [\[ex:strips1,ex:strips2\]](#ex:strips1,ex:strips2){reference-type="ref" reference="ex:strips1,ex:strips2"}, equation [[([\[eq:innersum\]](#eq:innersum){reference-type="ref" reference="eq:innersum"})]{.upright}](#eq:innersum) is $$(-\mathsf{p}_{321} + \mathsf{p}_{33} + \mathsf{p}_{42} + \mathsf{p}_{51} -2\mathsf{p}_{6}) (-\mathsf{p}_2) = \widetilde{\mathsf{m}}_{321}\mathsf{p}_2 \,.$$ The left-hand side is written with $(-1)^{|\mu|-\ell(\mu)}\mathsf{p}_\mu = -\mathsf{p}_2$ factored out; this factor corresponds to the cycle $(9 \;\; 10)$, which is common to all $\pi \in \mathrm{RF}(Y)$. *Proof of [Lemma 57](#lem:innersum){reference-type="ref" reference="lem:innersum"}.* By [Lemma 55](#lem:RF){reference-type="ref" reference="lem:RF"}[\[RF2\]](#RF2){reference-type="ref" reference="RF2"}[\[RF_b\]](#RF_b){reference-type="ref" reference="RF_b"}, we can factor out $\mathrm{sgn}(\theta|_{T\cap Y})\mathsf{p}_{\mathsf{cyc}(\theta|_{T\cap Y})} = (-1)^{|\mu| - \ell(\mu)}\mathsf{p}_\mu$ from the left-hand side of [[([\[eq:innersum\]](#eq:innersum){reference-type="ref" reference="eq:innersum"})]{.upright}](#eq:innersum). Cancelling this factor from both sides reduces the proof of [[([\[eq:innersum\]](#eq:innersum){reference-type="ref" reference="eq:innersum"})]{.upright}](#eq:innersum) to the case when $T\cap Y = \emptyset$. In this case, by [Lemma 55](#lem:RF){reference-type="ref" reference="lem:RF"}[\[RF2\]](#RF2){reference-type="ref" reference="RF2"}[\[RF_a\]](#RF_a){reference-type="ref" reference="RF_a"}, any element $\pi\in\mathrm{RF}(Y)$ is given by arranging the substrips $S_i\cap Y$ into cycles; this corresponds to a permutation $\sigma \in \mathfrak{S}_{k}$, where we regard $S_i\cap Y$ as the element $i\in [k]$. Since $\mathrm{sgn}(\pi) = (-1)^{|\kappa| - k}\mathrm{sgn}(\sigma)$ and $\mathsf{cyc}(\pi_Y) = \kappa[\sigma]$, the equation follows from [Proposition 36](#prop:scaledmonomials){reference-type="ref" reference="prop:scaledmonomials"}. ◻ ### Simplifying $C_{\theta,Z}$ We now use [[([\[eq:innersum\]](#eq:innersum){reference-type="ref" reference="eq:innersum"})]{.upright}](#eq:innersum) to further simplify our formula [[([\[eq:thetaZcoeff\]](#eq:thetaZcoeff){reference-type="ref" reference="eq:thetaZcoeff"})]{.upright}](#eq:thetaZcoeff) for $C_{\theta,Z}$, and complete the proof of [Lemma 50](#lem:part1){reference-type="ref" reference="lem:part1"}. To specify a valid $Y$, it is enough to choose a substrip of each $S_i$ and a subset of the cycles of $\theta|_T$. A substrip of $S_i$ is uniquely determined by its length $\kappa_i$, where $1 \leq \kappa_i \leq \lambda_i$. Thus we have a one-to-one correspondence between valid subsets $Y$ and pairs $(\kappa, H)$, where $\kappa$ is a composition such that $\kappa \leq \lambda$ and $H$ is a subset of the cycles of $\theta|_T$. Abusing notation, we denote the latter condition as $H \subseteq \theta|_T$. Under this correspondence $|Y| = |\kappa| + |\mu_H|$, where $\mu_H$ is the partition listing the lengths of the cycles in $H$. Thus from [[([\[eq:thetaZcoeff\]](#eq:thetaZcoeff){reference-type="ref" reference="eq:thetaZcoeff"})]{.upright}](#eq:thetaZcoeff) and [[([\[eq:innersum\]](#eq:innersum){reference-type="ref" reference="eq:innersum"})]{.upright}](#eq:innersum), we have $$\begin{aligned} C_{\theta,Z} &= \mathrm{sgn}(\theta) \sum_{Y \subseteq [n]} (-t)^{|Y|-k-n} \sum_{\pi \in \mathrm{RF}(Y)} \mathrm{sgn}(\pi) \mathsf{p}_{\mathsf{cyc}(\pi_Y)} \\[4pt] &= \mathrm{sgn}(\theta) \sum_{\kappa \leq \lambda}\, \sum_{H \subseteq \theta|_T} (-t)^{|\kappa|+|\mu_H|-k-n} \cdot (-1)^{|\kappa|-k+|\mu_H|-|H|} \widetilde{\mathsf{m}}_\kappa \,\mathsf{p}_{\mu_H} \\[4pt] &= \mathrm{sgn}(\theta) (-1)^{n} \sum_{\kappa \leq \lambda} t^{|\kappa|-k-n} \widetilde{\mathsf{m}}_\kappa \cdot \sum_{H \subseteq \theta|_T} (-1)^{|H|} t^{|\mu_H|} \mathsf{p}_{\mu_H} \\ &= \mathrm{sgn}(\theta) (-1)^n \sum_{\kappa \leq \lambda} t^{|\kappa|-k-n} \widetilde{\mathsf{m}}_\kappa \cdot \prod_{i=1}^s (1- t^{\eta_i}\mathsf{p}_{\eta_i}) \,,\end{aligned}$$ where $\eta = (\eta_1, \dots, \eta_s) := \mathsf{cyc}(\theta|_T)$. We conclude the proof of [Lemma 50](#lem:part1){reference-type="ref" reference="lem:part1"} as follows. Recall that our goal is to show that $C_{\theta,Z} \in \ker [t^{-1}]\mbox{\small\sf{B}}^\perp(t^{-1})$. By [Lemma 37](#lem:Bp-identity){reference-type="ref" reference="lem:Bp-identity"}, this is true if and only if $$t^{|\eta|} \cdot \sum_{\kappa \leq \lambda} t^{|\kappa|-k-n} \widetilde{\mathsf{m}}_\kappa \in \ker [t^{-1}]\mbox{\small\sf{B}}^\perp(t^{-1}) \,.$$ Since $|\eta| = n - |\lambda|$, this is precisely the statement of [Lemma 38](#lem:Bm-identity){reference-type="ref" reference="lem:Bm-identity"}. 0◻ ## Proof of [Lemma 51](#lem:part2){reference-type="ref" reference="lem:part2"}: general Plücker relations {#sec:part2} We now show that [Lemma 50](#lem:part1){reference-type="ref" reference="lem:part1"}, when combined with the translation identity [[([\[eq:translationidentity\]](#eq:translationidentity){reference-type="ref" reference="eq:translationidentity"})]{.upright}](#eq:translationidentity), implies that the operators $\beta^\lambda(t)$ satisfy the Plücker relations. We extend our convention [[([\[eq:identification\]](#eq:identification){reference-type="ref" reference="eq:identification"})]{.upright}](#eq:identification) for indexing Plücker coordinates to the operators $\beta^\lambda(t)$. For $I = (i_1, \dots, i_n) \in [2n]^n$, define $\beta_I(t)$ as follows: if $i_{\sigma(1)} < \dots < i_{\sigma(n)}$ for some $\sigma \in \mathfrak{S}_{n}$, put $\beta_I(t) := \mathrm{sgn}(\sigma) \beta^\lambda(t)$, where $\lambda = (i_{\sigma(n)}-n, \dots, i_{\sigma(1)}-1)$; if $i_1, \dots, i_n$ are not distinct, put $\beta_I(t) := 0$. We write $\beta_I := \beta_I(0)$. We will work with the differential operator $\mathcal{D}_n$ from [[([\[eq:diffop\]](#eq:diffop){reference-type="ref" reference="eq:diffop"})]{.upright}](#eq:diffop); however, for our purposes, it will be more convenient to omit the leading factor of $\frac{1}{g(u)}$. Hence, we define $$\label{eq:polyDdef} \overline{\mathcal{D}}_n(t) := \sum_{k=0}^n (-1)^k \beta^{1^{\smash{k}}}(u+t) \partial_u^{n-k} \qquad \text{and} \qquad \overline{\mathcal{D}}_n := \overline{\mathcal{D}}_n(0) = g(u)\mathcal{D}_n \,.$$ We will often restrict the domain of these linear operators from $\mathbb{C}[u]$ to $\mathbb{C}_{2n-1}[u]$. ### Reformulation of the translation identity For every nonnegative integer $i$ and $t \in \mathbb{C}$, define the polynomials $$\label{eq:ei} e_i(u) := \frac{u^{i-1}}{(i-1)!} \qquad \text{and} \qquad e_{i,t}(u) := (\begin{smallmatrix} 1 & t \\ 0 & 1 \end{smallmatrix})e_i(u) = e_i(u+t) = \frac{(u+t)^{i-1}}{(i-1)!} \,.$$ Thus, $(e_{1,t}, \dots, e_{2n,t})$ is a basis for $\mathbb{C}_{2n-1}[u]$, and $(e_1, \dots, e_{2n})$ corresponds to the standard basis for $\mathbb{C}^{2n}$ under the isomorphism [[([\[eq:isomorphism\]](#eq:isomorphism){reference-type="ref" reference="eq:isomorphism"})]{.upright}](#eq:isomorphism). For $I = (i_1, \dots, i_n) \in [2n]^n$, write $$e_{I,t} := e_{i_1,t} \wedge \dots \wedge e_{i_n,t} \in \mathsf{\Lambda}^{n} \mathbb{C}_{2n-1}[u] \,,$$ and put $e_I := e_{I,0}$. Following our convention [[([\[eq:identification\]](#eq:identification){reference-type="ref" reference="eq:identification"})]{.upright}](#eq:identification), we write $e^\lambda_t := e_{(\lambda_n+1, \lambda_{n-1}+2, \dots, \lambda_1+n),t}$ for any partition $\lambda \subseteq {\mathchoice% {\scalebox{1.35}{\raisebox{-0.02em}{$\sqsubset\!\!\sqsupset$}}} {\scalebox{1.35}{\raisebox{-0.02em}{$\sqsubset\!\!\sqsupset$}}} {\sqsubset\!\!\sqsupset} {\sqsubset\!\!\sqsupset} }= n^n$. By [Proposition 17](#prop:exteriorplucker){reference-type="ref" reference="prop:exteriorplucker"} and [[([\[eq:geometrictranslation\]](#eq:geometrictranslation){reference-type="ref" reference="eq:geometrictranslation"})]{.upright}](#eq:geometrictranslation), we have $$\label{eq:basistranslation} e^\lambda_t = \sum_{\mu \subseteq \lambda} \frac{\mathsf{f}^{\lambda/\mu}}{|\lambda/\mu|!} t^{|\lambda/\mu|}e^\mu \qquad \text{for every } \lambda\subseteq{\mathchoice% {\scalebox{1.35}{\raisebox{-0.02em}{$\sqsubset\!\!\sqsupset$}}} {\scalebox{1.35}{\raisebox{-0.02em}{$\sqsubset\!\!\sqsupset$}}} {\sqsubset\!\!\sqsupset} {\sqsubset\!\!\sqsupset} }\,.$$ For $t\in\mathbb{C}$, consider the element $\Omega(t) \in \mathsf{\Lambda}^{n} \mathbb{C}_{2n-1}[u] \otimes \overline{\mathcal{B}}_n(z_1, \dots, z_n)$ defined by $$\label{eq:omega} \Omega(t) := \sum_{\lambda \subseteq {\mathchoice% {\scalebox{1.35}{\raisebox{-0.02em}{$\sqsubset\!\!\sqsupset$}}} {\scalebox{1.35}{\raisebox{-0.02em}{$\sqsubset\!\!\sqsupset$}}} {\sqsubset\!\!\sqsupset} {\sqsubset\!\!\sqsupset} }} e^\lambda \otimes \beta^\lambda(t) = \sum_{I \in {[2n] \choose n}} e_I \otimes \beta_I(t) \,,$$ and put $\Omega := \Omega(0)$. The following is a reformulation of the translation identity [[([\[eq:translationidentity\]](#eq:translationidentity){reference-type="ref" reference="eq:translationidentity"})]{.upright}](#eq:translationidentity): **Lemma 59**. *For all $s, t \in \mathbb{C}$, we have $$\Omega(s+t) = \sum_{\lambda\subseteq{\mathchoice% {\scalebox{1.35}{\raisebox{-0.02em}{$\sqsubset\!\!\sqsupset$}}} {\scalebox{1.35}{\raisebox{-0.02em}{$\sqsubset\!\!\sqsupset$}}} {\sqsubset\!\!\sqsupset} {\sqsubset\!\!\sqsupset} }} e^\lambda_t \otimes \beta^\lambda(s) \,.$$* *Proof.* Upon expanding $e^\lambda_t$ as in [[([\[eq:basistranslation\]](#eq:basistranslation){reference-type="ref" reference="eq:basistranslation"})]{.upright}](#eq:basistranslation), this becomes equivalent to [[([\[eq:translationidentity\]](#eq:translationidentity){reference-type="ref" reference="eq:translationidentity"})]{.upright}](#eq:translationidentity). ◻ ### Derivations on the exterior algebra We recall how to extend any linear map $L : V \to W$ to a derivation $L_\#$ on the exterior algebra $\mathsf{\Lambda}^{\bullet}V$: **Lemma 60**. *Let $V$ and $W$ be vector spaces, and let $L : V \to W$ be a linear map.* (i) *[\[exterior1\]]{#exterior1 label="exterior1"} There exists a unique linear map $$L_\# : \mathsf{\Lambda}^{\bullet}V \to \mathsf{\Lambda}^{\bullet}V \otimes W \,,$$ such that for all $k \geq 0$ and $v_1, \dots, v_k \in V$, we have $$\label{eq:Lsharp} L_\#(v_1 \wedge \dots \wedge v_k) = \sum_{i=1}^k (-1)^{k-i} (v_1 \wedge \cdots \wedge \widehat{v}_i \wedge \cdots \wedge v_k) \otimes L(v_i) \,.$$* (ii) *[\[exterior2\]]{#exterior2 label="exterior2"} $L_\#$ is a derivation: for all $\omega \in \mathsf{\Lambda}^{k} V$ and $\omega' \in \mathsf{\Lambda}^{l} V$, we have $$L_\#(\omega \wedge \omega') = \omega \wedge L_\# \omega' + (-1)^{kl} \omega' \wedge L_\# \omega \,.$$* (iii) *[\[exterior3\]]{#exterior3 label="exterior3"} We have $\ker(L_\#) = \mathsf{\Lambda}^{\bullet}\ker(L)$.* *Proof.* The map $V^k \to \mathsf{\Lambda}^{k-1}V \otimes W$, defined by mapping $(v_1, \dots, v_k)$ to the right-hand side of [[([\[eq:Lsharp\]](#eq:Lsharp){reference-type="ref" reference="eq:Lsharp"})]{.upright}](#eq:Lsharp) is $k$-linear and alternating, and therefore extends to a unique linear map on $\mathsf{\Lambda}^{k} V$. This proves part [\[exterior1\]](#exterior1){reference-type="ref" reference="exterior1"}, and part [\[exterior2\]](#exterior2){reference-type="ref" reference="exterior2"} follows directly. For part [\[exterior3\]](#exterior3){reference-type="ref" reference="exterior3"}, it is clear that $\mathsf{\Lambda}^{\bullet}\ker(L) \subseteq \ker(L_\#)$. For the reverse inclusion, after replacing $W$ by $L(V)$, we may assume that $L$ is surjective. First note that if $L$ is invertible, then $L_\# : \mathsf{\Lambda}^{k} V \to \mathsf{\Lambda}^{k-1} V \otimes W$ is the zero map if $k=0$ and injective if $k \geq 1$. The former is true by definition, and the latter is true because $$v_1 \wedge \dots \wedge v_{k-1} \otimes w \mapsto \frac{1}{k} v_1 \wedge \dots \wedge v_{k-1} \wedge L^{-1}(w)$$ defines the left-inverse map. Thus, when $L$ is invertible, we have $\ker(L_\#) = \mathsf{\Lambda}^{0}V$. In general, there exists a direct sum decomposition $V = V' \oplus V''$, with $V' = \ker(L)$. Thus $L$ restricted to $V'$ is the zero map, and $L$ restricted to $V''$ is invertible. Given $\omega \in \mathsf{\Lambda}^{\bullet}V$, we can write $\omega = \sum_{i=1}^s \omega'_i \wedge \omega''_i$, where $\omega'_i \in \mathsf{\Lambda}^{\bullet}V'$ and $\omega''_i \in \mathsf{\Lambda}^{\bullet}V''$ for all $i\in [s]$, and $(\omega'_1, \dots, \omega'_s)$ is linearly independent. By part [\[exterior2\]](#exterior2){reference-type="ref" reference="exterior2"}, we have $L_\# \omega = \sum_{i=1}^s \omega'_i \wedge L_\# \omega''_i$. Finally, if $L_\# \omega = 0$, then $L_\# \omega''_i = 0$ for all $i \in [s]$. Since $L$ restricted to $V''$ is invertible, this implies that $\omega''_i \in \mathsf{\Lambda}^{0} V''$ for all $i \in [s]$. Hence $\omega \in \mathsf{\Lambda}^{\bullet}V'$, which proves $\ker(L_\#) \subseteq \mathsf{\Lambda}^{\bullet}\ker(L)$. ◻ **Corollary 61**. *Let $L : V \to W$ be as in [Lemma 60](#lem:exterior){reference-type="ref" reference="lem:exterior"}, and let $\omega \in \mathsf{\Lambda}^{n} V$. Suppose that $$L_\# \omega = 0 \qquad\text{and}\qquad \dim \ker(L) \leq n \,.$$ Then $\omega = v_1 \wedge \dots \wedge v_n$ for some $v_1, \dots, v_n \in \ker(L)$. Furthermore, if $\omega \neq 0$, then $\dim \ker(L) = n$ and $(v_1, \dots, v_n)$ is a basis for $\ker(L)$.* *Proof.* The result is trivial if $\omega = 0$, so assume $\omega \neq 0$. By [Lemma 60](#lem:exterior){reference-type="ref" reference="lem:exterior"}, we have $\omega \in \mathsf{\Lambda}^{n} \ker(L)$. If $\dim \ker(L) < n$, then $\mathsf{\Lambda}^{n} \ker(L)$ is zero-dimensional, and cannot contain a nonzero vector. Therefore we must have $\dim \ker(L) =n$, whence $\dim \mathsf{\Lambda}^{n} \ker(L) = 1$, and every nonzero vector is of the form $v_1 \wedge \dots \wedge v_n$, where $(v_1, \dots, v_n)$ is a basis for $\ker(L)$. ◻ ### Reduction to single-column Plücker relations {#sec:screduction} If $R$ is a unital commutative $\mathbb{C}$-algebra, we extend the definition of $L_\#$ to $R$-linear maps $L : V \otimes R \to W \otimes R$. In this case, we obtain an $R$-linear derivation $$\label{eq:derivation} L_\# : \mathsf{\Lambda}^{\bullet}V \otimes R \to \mathsf{\Lambda}^{\bullet}V \otimes W \otimes R$$ characterized by [[([\[eq:Lsharp\]](#eq:Lsharp){reference-type="ref" reference="eq:Lsharp"})]{.upright}](#eq:Lsharp). In particular, taking $R = \overline{\mathcal{B}}_n(z_1, \dots, z_n)$, we consider the derivation $$(\overline{\mathcal{D}}_n)_\# : \mathsf{\Lambda}^{\bullet}\mathbb{C}_{2n-1}[u] \otimes \overline{\mathcal{B}}_n(z_1, \dots, z_n)\to \mathsf{\Lambda}^{\bullet}\mathbb{C}_{2n-1}[u] \otimes \mathbb{C}[u] \otimes \overline{\mathcal{B}}_n(z_1, \dots, z_n) \,$$ associated to $\overline{\mathcal{D}}_n : \mathbb{C}_{2n-1}[u] \otimes \overline{\mathcal{B}}_n(z_1, \dots, z_n)\to \mathbb{C}[u] \otimes \overline{\mathcal{B}}_n(z_1, \dots, z_n)$ from [[([\[eq:polyDdef\]](#eq:polyDdef){reference-type="ref" reference="eq:polyDdef"})]{.upright}](#eq:polyDdef). (We have switched the order of the tensor factors from [[([\[eq:diffop\]](#eq:diffop){reference-type="ref" reference="eq:diffop"})]{.upright}](#eq:diffop), to be consistent with [[([\[eq:derivation\]](#eq:derivation){reference-type="ref" reference="eq:derivation"})]{.upright}](#eq:derivation).) For $t \in \mathbb{C}$ and $\mathbb{C}$-vector spaces $V_1$ and $V_2$, let $\mathrm{ev}_t : V_1 \otimes \mathbb{C}[u] \otimes V_2 \to V_1 \otimes V_2$ denote the evaluation map $f \mapsto f(t)$ on the tensor factor of $\mathbb{C}[u]$. **Lemma 62**. *Let $z_1, \dots, z_n$ be formal indeterminates. Then the following are equivalent:* (a) *[\[translationmagic_a\]]{#translationmagic_a label="translationmagic_a"} $(\overline{\mathcal{D}}_n)_\# \Omega = 0$;* (b) *[\[translationmagic_b\]]{#translationmagic_b label="translationmagic_b"} $\mathrm{ev}_0 \big((\overline{\mathcal{D}}_n)_\# \Omega\big) = 0$;* (c) *[\[translationmagic_c\]]{#translationmagic_c label="translationmagic_c"} the operators $\beta^\lambda$ satisfy the single-column Plücker relations for $\mathrm{Gr}(n,2n)$.* *Proof.* [\[translationmagic_a\]](#translationmagic_a){reference-type="ref" reference="translationmagic_a"} $\Leftrightarrow$ [\[translationmagic_b\]](#translationmagic_b){reference-type="ref" reference="translationmagic_b"}: Clearly [\[translationmagic_a\]](#translationmagic_a){reference-type="ref" reference="translationmagic_a"} implies [\[translationmagic_b\]](#translationmagic_b){reference-type="ref" reference="translationmagic_b"}. Conversely, suppose that [\[translationmagic_b\]](#translationmagic_b){reference-type="ref" reference="translationmagic_b"} is true. Then [\[translationmagic_b\]](#translationmagic_b){reference-type="ref" reference="translationmagic_b"} remains true if we perform the translation $(z_1, \dots, z_n) \mapsto (z_1+t, \dots, z_n+t)$ for $t \in \mathbb{C}$: $$\mathrm{ev}_0 \big((\overline{\mathcal{D}}_n(t))_\# \Omega(t)\big) = 0 \,.$$ Now perform the change of variables $u \mapsto u-t$ on the equation above. Under this change of variables, $\mathrm{ev}_0 \mapsto \mathrm{ev}_t$, $\beta^{1^{\smash{k}}}(u+t) \mapsto \beta^{1^{\smash{k}}}(u)$, $\partial_u\mapsto \partial_u$, and $\Omega(t) \mapsto \Omega$ (by [Lemma 59](#lem:exteriortranslation){reference-type="ref" reference="lem:exteriortranslation"}). Thus we obtain $$\mathrm{ev}_t \big((\overline{\mathcal{D}}_n)_\# \Omega \big) = 0 \,.$$ Since this is true for all $t \in \mathbb{C}$, we deduce that [\[translationmagic_a\]](#translationmagic_a){reference-type="ref" reference="translationmagic_a"} holds. [\[translationmagic_b\]](#translationmagic_b){reference-type="ref" reference="translationmagic_b"} $\Leftrightarrow$ [\[translationmagic_c\]](#translationmagic_c){reference-type="ref" reference="translationmagic_c"}: By direct calculation, we have $$\begin{aligned} \mathrm{ev}_0 \big((\overline{\mathcal{D}}_n)_\# \Omega \big) &= \sum_{I \in {[2n] \choose n}} \mathrm{ev}_0 \big((\overline{\mathcal{D}}_n)_\# e_I\big) \cdot \beta_I \\ &= \sum_{J \in {[2n] \choose n-1}} e_J \otimes \sum_{k=1}^{2n} \mathrm{ev}_0 \big(\overline{\mathcal{D}}_n(e_k) \big) \cdot \beta_{J+k} \\ &= \sum_{J \in {[2n] \choose n-1}} e_J \otimes \sum_{k=1}^{n+1} (-1)^{n+1-k} \beta^{1^{\smash{n+1-k}}} \cdot \beta_{J+k} \,,\end{aligned}$$ where $\beta_{J+k}$ is defined analogously to [[([\[eq:addsubscript\]](#eq:addsubscript){reference-type="ref" reference="eq:addsubscript"})]{.upright}](#eq:addsubscript). Therefore, $\mathrm{ev}_0 \big((\overline{\mathcal{D}}_n)_\# \Omega \big) = 0$ if and only if $$\sum_{k=1}^{n+1} (-1)^{n+1-k} \beta^{1^{\smash{n+1-k}}} \beta_{J+k} = 0 \, \qquad \text{for all $J \in \textstyle{[2n] \choose n-1}$} \,.$$ These are precisely the single-column Plücker relations for $\mathrm{Gr}(n,2n)$. ◻ We now complete the proof of [Lemma 51](#lem:part2){reference-type="ref" reference="lem:part2"}. We want to show that the operators $\beta^\lambda(t)$ satisfy the Plücker relations. Recall from [2.1.3](#sec:partitions){reference-type="ref" reference="sec:partitions"} that we only need to consider the Plücker relations for $\mathrm{Gr}(n,2n)$. Furthermore, by translation and continuity, it is enough to prove the result when $t=0$ and $(z_1, \dots, z_n) \in \mathbb{C}^n$ is generic. In particular, by [Lemma 46](#lem:comparealgebras){reference-type="ref" reference="lem:comparealgebras"}[\[comparealgebras2\]](#comparealgebras2){reference-type="ref" reference="comparealgebras2"} and [Theorem 42](#thm:bethebasics){reference-type="ref" reference="thm:bethebasics"}[\[bethebasics2\]](#bethebasics2){reference-type="ref" reference="bethebasics2"}, we may assume that $\overline{\mathcal{B}}_n(z_1, \dots, z_n)$ is semisimple. Hence it suffices to prove that for all partitions $\nu$ and eigenspaces $E \subseteq M^{\nu}$ of $\overline{\mathcal{B}}_\nu(z_1, \dots, z_n)$, the eigenvalues $\beta^\lambda_E$ satisfy the Plücker relations for $\mathrm{Gr}(n,2n)$. By [Lemma 50](#lem:part1){reference-type="ref" reference="lem:part1"}, the operators $\beta^\lambda$ satisfy all single-column Plücker relations. By [Lemma 62](#lem:translationmagic){reference-type="ref" reference="lem:translationmagic"}, we deduce that $(\overline{\mathcal{D}}_n)_\# \Omega = 0$. Thus we have $(\overline{\mathcal{D}}_E)_\# \Omega_E = 0$, where $$\label{eq:deomegae} \overline{\mathcal{D}}_E := \sum_{k=0}^{n} (-1)^k \beta^{1^{\smash{k}}}_E \partial_u^{n-k} \qquad\text{and}\qquad \Omega_E := \sum_{\lambda \subseteq {\mathchoice% {\scalebox{1.35}{\raisebox{-0.02em}{$\sqsubset\!\!\sqsupset$}}} {\scalebox{1.35}{\raisebox{-0.02em}{$\sqsubset\!\!\sqsupset$}}} {\sqsubset\!\!\sqsupset} {\sqsubset\!\!\sqsupset} }} \beta^\lambda_E e^\lambda \,.$$ Since $\overline{\mathcal{D}}_E : \mathbb{C}_{2n-1}[u] \to \mathbb{C}[u]$ is a linear differential operator of order $n$, by [Proposition 25](#prop:Dnullity){reference-type="ref" reference="prop:Dnullity"} we have $\dim \ker (\overline{\mathcal{D}}_E) \leq n$. Therefore by [Corollary 61](#cor:exterior){reference-type="ref" reference="cor:exterior"}, we have $\Omega_E = v_1 \wedge \dots \wedge v_n$ for some $v_1, \dots, v_n \in \mathbb{C}_{2n-1}[u]$. It follows from [Proposition 17](#prop:exteriorplucker){reference-type="ref" reference="prop:exteriorplucker"} that the coefficients $\beta^\lambda_E$ satisfy the Plücker relations for $\mathrm{Gr}(n,2n)$. 0◻ ## Final steps in the proof of [Theorem 3](#thm:main){reference-type="ref" reference="thm:main"} {#sec:finalsteps} We have so far proved parts [\[main_commutativity\]](#main_commutativity){reference-type="ref" reference="main_commutativity"}--[\[main_pluckers\]](#main_pluckers){reference-type="ref" reference="main_pluckers"} of [Theorem 3](#thm:main){reference-type="ref" reference="thm:main"}. We now complete the proof by proving parts [\[main_generates\]](#main_generates){reference-type="ref" reference="main_generates"}--[\[main_multiplicity\]](#main_multiplicity){reference-type="ref" reference="main_multiplicity"}. Most of part [\[main_generates\]](#main_generates){reference-type="ref" reference="main_generates"} is established by [Lemma 46](#lem:comparealgebras){reference-type="ref" reference="lem:comparealgebras"}: all that remains is to show that $\mathcal{B}_n(z_1, \dots, z_n)= \overline{\mathcal{B}}_n(z_1, \dots, z_n)$ in the non-generic case. We first prove parts [\[main_eigenspace\]](#main_eigenspace){reference-type="ref" reference="main_eigenspace"} and [\[main_multiplicity\]](#main_multiplicity){reference-type="ref" reference="main_multiplicity"}, and then address the last case of part [\[main_generates\]](#main_generates){reference-type="ref" reference="main_generates"}. In the process, we also obtain a new proof of [Theorem 44](#thm:precise){reference-type="ref" reference="thm:precise"}. ### Fibres of the Wronski map and eigenspaces We now prove parts [\[main_eigenspace\]](#main_eigenspace){reference-type="ref" reference="main_eigenspace"} and [\[main_multiplicity\]](#main_multiplicity){reference-type="ref" reference="main_multiplicity"} of [Theorem 3](#thm:main){reference-type="ref" reference="thm:main"}, but using $\overline{\mathcal{B}}_n(z_1, \dots, z_n)$ in place of $\mathcal{B}_n(z_1, \dots, z_n)$. Once we have established part [\[main_generates\]](#main_generates){reference-type="ref" reference="main_generates"}, this will give the results as stated. By [Proposition 24](#prop:changegr){reference-type="ref" reference="prop:changegr"}, it is enough to prove these results in $\mathrm{Gr}(n,2n)$. We have already shown that if $E \subseteq M^{\nu}$ is an eigenspace of $\overline{\mathcal{B}}_\nu(z_1, \dots, z_n)$, the complex numbers $\beta^\lambda_E$ satisfy the Plücker relations for $\mathrm{Gr}(n,2n)$. In order to deduce that they are the Plücker coordinates of some point $V_E \in \mathrm{Gr}(n,2n)$, we need to furthermore check these numbers are not all zero. This is implied by the next lemma: **Lemma 63**. *Let $E \subseteq M^{\nu}$ be an eigenspace of $\overline{\mathcal{B}}_\nu(z_1, \dots, z_n)$. Then $\beta^\nu_E = \frac{n!}{\mathsf{f}^{\nu}}$, and $\beta^\lambda_E = 0$ for all $\lambda \not \subseteq \nu$.* *Proof.* Since $|\nu| = n$, we have $\beta^\nu = \alpha^\nu_{[n]}$. By [Proposition 29](#prop:projection){reference-type="ref" reference="prop:projection"}, $\frac{\mathsf{f}^{\nu}}{n!} \beta^\nu$ acts on $M^{\nu}$ as the identity operator. In particular, for every eigenspace $E$, we have $\beta^\nu_E = \frac{n!}{\mathsf{f}^{\nu}}$. If $\lambda \not \subseteq \nu$, then $\beta^\lambda_\nu = 0$ by [Proposition 12](#prop:betapsd){reference-type="ref" reference="prop:betapsd"}[\[betapsd4\]](#betapsd4){reference-type="ref" reference="betapsd4"}, and hence $\beta^\lambda_E = 0$. ◻ In particular, since $\beta^\nu_E \neq 0$, there exists a point $V_E \in \mathrm{Gr}(n,2n)$ with Plücker coordinates $[\beta^\lambda_E: \lambda \subseteq {\mathchoice% {\scalebox{1.35}{\raisebox{-0.02em}{$\sqsubset\!\!\sqsupset$}}} {\scalebox{1.35}{\raisebox{-0.02em}{$\sqsubset\!\!\sqsupset$}}} {\sqsubset\!\!\sqsupset} {\sqsubset\!\!\sqsupset} }]$. Moreover, $V_E$ is contained in the Schubert cell $\mathcal{X}^{\nu}$: **Corollary 64**. *Let $E \subseteq M^{\nu}$ be an eigenspace of $\overline{\mathcal{B}}_\nu(z_1, \dots, z_n)$. Then $V_E \in \mathcal{X}^{\nu}$, and $(\beta^\lambda_E : \lambda \subseteq \nu)$ are the normalized Plücker coordinates of $V_E$.* *Proof.* This follows immediately from [Lemma 63](#lem:identifyscell){reference-type="ref" reference="lem:identifyscell"} and [Proposition 19](#thm:schubertplucker){reference-type="ref" reference="thm:schubertplucker"}. ◻ Next we show that $V_E \in \mathrm{Wr}^{-1}(g)$ for $g(u) = (u+z_1)\dotsm (u+z_n)$: **Lemma 65**. *Let $E \subseteq M^{\nu}$ be an eigenspace of $\overline{\mathcal{B}}_n(z_1, \dots, z_n)$. Then $\mathrm{Wr}(V_E) = g$.* *Proof.* By [Corollary 64](#cor:identifyscell){reference-type="ref" reference="cor:identifyscell"} and [[([\[eq:pluckerwronskian\]](#eq:pluckerwronskian){reference-type="ref" reference="eq:pluckerwronskian"})]{.upright}](#eq:pluckerwronskian), we have $$\mathrm{Wr}(V_E) = \sum_{\lambda} \frac{\mathsf{f}^{\lambda}}{|\lambda|!} \beta^\lambda_E u^{|\lambda|} \,.$$ (We can remove the condition $\lambda \subseteq \nu$ in the summation, since $\beta^\lambda_E = 0$ for all $\lambda \not \subseteq \nu$.) By the translation identity [[([\[eq:translationidentity\]](#eq:translationidentity){reference-type="ref" reference="eq:translationidentity"})]{.upright}](#eq:translationidentity), the right-hand side above equals $\beta^0_E(u)$. By definition $\beta^0(u) = g(u) \cdot \mathbbm{1}_{\mathfrak{S}_{n}}$, and so $\beta^0_E(u) = g(u)$, as required. ◻ We have shown that $V_E \in \mathcal{X}^{\nu}$ and $\mathrm{Wr}(V_E) = g$, which are the first two claims of part [\[main_eigenspace\]](#main_eigenspace){reference-type="ref" reference="main_eigenspace"}. We now argue that the correspondence $E \leftrightarrow V_E$ is bijective. By continuity of the fibres of the Wronski map and the eigenvalues of the operators $\beta^\lambda_\nu$, it suffices to prove this when $z_1, \dots, z_n \in \mathbb{C}$ are generic. In this case, [Lemma 46](#lem:comparealgebras){reference-type="ref" reference="lem:comparealgebras"}[\[comparealgebras2\]](#comparealgebras2){reference-type="ref" reference="comparealgebras2"} and [Theorem 42](#thm:bethebasics){reference-type="ref" reference="thm:bethebasics"}[\[bethebasics2\]](#bethebasics2){reference-type="ref" reference="bethebasics2"} imply that $\overline{\mathcal{B}}_\nu(z_1, \dots, z_n)= \mathcal{B}_\nu(z_1, \dots, z_n)$ is semisimple and is a maximal commutative subalgebra of $\mathrm{End}(M^{\nu})$, and hence has $\mathsf{f}^{\nu }= \dim M^{\nu}$ distinct eigenspaces. First we show that the map $E \mapsto V_E$ is injective. Let $E$ and $E'$ be eigenspaces such that $[\beta^\lambda_E : \lambda \subseteq \nu] = [\beta^\lambda_{E'} : \lambda \subseteq \nu]$ as projective coordinates. By [Corollary 64](#cor:identifyscell){reference-type="ref" reference="cor:identifyscell"} these coordinates are normalized, so $\beta^\lambda_E = \beta^\lambda_{E'}$ for all partitions $\lambda$. Since the elements $\beta^\lambda_\nu$ generate $\overline{\mathcal{B}}_\nu(z_1, \dots, z_n)$, this implies that $\xi_E = \xi_{E'}$ for all $\xi \in \overline{\mathcal{B}}_\nu(z_1, \dots, z_n)$. Hence $E = E'$, as required. Now recall that the degree of the Wronski map $\mathrm{Wr}: \mathcal{X}^{\nu}\to \mathcal{P}_n$ is $\mathsf{f}^{\nu}$, so the map $E \mapsto V_E$ is also surjective. This completes the proof of [Theorem 3](#thm:main){reference-type="ref" reference="thm:main"}[\[main_eigenspace\]](#main_eigenspace){reference-type="ref" reference="main_eigenspace"}. Furthermore, since $\mathrm{Wr}: \mathcal{X}^{\nu}\to \mathcal{P}_n$ is a finite morphism, this argument also establishes part [\[main_multiplicity\]](#main_multiplicity){reference-type="ref" reference="main_multiplicity"}, since both multiplicities of points in $\mathrm{Wr}^{-1}(g)$ and dimensions of generalized eigenspaces behave additively under taking limits. 0◻ ### $\overline{\mathcal{B}}_n(z_1, \dots, z_n)= \mathcal{B}_n(z_1, \dots, z_n)$ {#sec:altbetheequalsbethe} We now complete the proof of [Theorem 3](#thm:main){reference-type="ref" reference="thm:main"}[\[main_generates\]](#main_generates){reference-type="ref" reference="main_generates"}, by showing that $\overline{\mathcal{B}}_n(z_1, \dots, z_n)= \mathcal{B}_n(z_1, \dots, z_n)$ for all $z_1, \dots, z_n \in \mathbb{C}$. We already have $\mathcal{B}_n(z_1, \dots, z_n)\subseteq \overline{\mathcal{B}}_n(z_1, \dots, z_n)$ from [Lemma 46](#lem:comparealgebras){reference-type="ref" reference="lem:comparealgebras"}[\[comparealgebras3\]](#comparealgebras3){reference-type="ref" reference="comparealgebras3"}; we now establish the reverse inclusion. **Lemma 66**. *If $E \subseteq M^{\nu}$ is an eigenspace of $\overline{\mathcal{B}}_\nu(z_1, \dots, z_n)$, then the fundamental differential operator of $V_E$ is $$D_{V_E} = \mathcal{D}_E = \frac{1}{\mathrm{Wr}(V_E)} \sum_{k=0}^n \, \sum_{\ell=0}^{n-k} (-1)^k\beta^{1^{\smash{k}}}_{\ell,E} u^{n-k-\ell} \partial_u^{n-k} \,.$$* *Proof.* Recall from [[([\[eq:deomegae\]](#eq:deomegae){reference-type="ref" reference="eq:deomegae"})]{.upright}](#eq:deomegae) that $(\overline{\mathcal{D}}_E)_\# \Omega_E = 0$ and $\dim \ker (\overline{\mathcal{D}}_E) \leq n$, where $\overline{\mathcal{D}}_E$ is regarded as a linear map $\overline{\mathcal{D}}_E : \mathbb{C}_{2n-1}[u] \to \mathbb{C}[u]$. By [Lemma 63](#lem:identifyscell){reference-type="ref" reference="lem:identifyscell"} we have $\beta^\nu_E \neq 0$, so $\Omega_E \neq 0$. By [Corollary 61](#cor:exterior){reference-type="ref" reference="cor:exterior"} we furthermore deduce that $\dim \ker \overline{\mathcal{D}}_E = n$ and $\Omega_E = v_1 \wedge \dots \wedge v_n$, where $(v_1, \dots, v_n)$ is a basis for $\ker \overline{\mathcal{D}}_E$. By [Proposition 17](#prop:exteriorplucker){reference-type="ref" reference="prop:exteriorplucker"}, we have $V_E = \langle v_1, \dots, v_n\rangle = \ker \overline{\mathcal{D}}_E$. Since $\mathcal{D}_E = \frac{1}{\mathrm{Wr}(V_E)} \overline{\mathcal{D}}_E$ we have $\overline{\mathcal{D}}_E f = 0$ if and only if $\mathcal{D}_E f = 0$. Thus $V_E \subseteq \mathop{\mathrm{pker}}\mathcal{D}_E$, and [Proposition 25](#prop:Dnullity){reference-type="ref" reference="prop:Dnullity"} then gives $V_E = \mathop{\mathrm{pker}}\mathcal{D}_E$. Finally, since $\mathcal{D}_E$ is monic, uniqueness of the fundamental differential operator ([Proposition 26](#prop:FDOunique){reference-type="ref" reference="prop:FDOunique"}) implies that $D_{V_E} = \mathcal{D}_E$. ◻ **Lemma 67**. *Let ${\mathchoice {\mbox{\boldmath{$\psi$}}} {\mbox{\boldmath{$\psi$}}} {\mbox{\scriptsize\boldmath{$\psi$}}} {\mbox{\tiny\boldmath{$\psi$}}} }_\nu := (\beta^{1^{\smash{k}}}_{\ell,\nu} : 0 \le k \le n, 0 \le \ell \le n-k)$, and let $p^\lambda_\nu$ be the polynomials from [Lemma 27](#lem:FDOplucker){reference-type="ref" reference="lem:FDOplucker"}. Treating $z_1, \dots, z_n$ as formal indeterminates, we have $$\beta^\lambda_\nu = p^\lambda_\nu({\mathchoice {\mbox{\boldmath{$\psi$}}} {\mbox{\boldmath{$\psi$}}} {\mbox{\scriptsize\boldmath{$\psi$}}} {\mbox{\tiny\boldmath{$\psi$}}} }_\nu) \,.$$* *Proof.* It suffices to prove the result when $(z_1, \dots, z_n) \in \mathbb{C}^n$ is generic. In particular, by [Lemma 46](#lem:comparealgebras){reference-type="ref" reference="lem:comparealgebras"}[\[comparealgebras2\]](#comparealgebras2){reference-type="ref" reference="comparealgebras2"} and [Theorem 42](#thm:bethebasics){reference-type="ref" reference="thm:bethebasics"}[\[bethebasics2\]](#bethebasics2){reference-type="ref" reference="bethebasics2"}, we may assume that $\overline{\mathcal{B}}_\nu(z_1, \dots, z_n)$ is semisimple. For an eigenspace $E \subseteq M^{\nu}$ of $\overline{\mathcal{B}}_\nu(z_1, \dots, z_n)$, let ${\mathchoice {\mbox{\boldmath{$\psi$}}} {\mbox{\boldmath{$\psi$}}} {\mbox{\scriptsize\boldmath{$\psi$}}} {\mbox{\tiny\boldmath{$\psi$}}} }_E := (\beta^{1^{\smash{k}}}_{\ell,E} : 0 \le k \le n, 0 \le \ell \le n-k)$. [Lemma 66](#lem:FDOcoords){reference-type="ref" reference="lem:FDOcoords"} asserts that ${\mathchoice {\mbox{\boldmath{$\psi$}}} {\mbox{\boldmath{$\psi$}}} {\mbox{\scriptsize\boldmath{$\psi$}}} {\mbox{\tiny\boldmath{$\psi$}}} }_E$ are the FDO coordinates of $V_E$. Since $(\beta^\lambda_E : \lambda \subseteq \nu)$ are the normalized Plücker coordinates of $V_E$, we have $\beta^\lambda_E = p^\lambda_\nu({\mathchoice {\mbox{\boldmath{$\psi$}}} {\mbox{\boldmath{$\psi$}}} {\mbox{\scriptsize\boldmath{$\psi$}}} {\mbox{\tiny\boldmath{$\psi$}}} }_E)$, by the definition of $p^\lambda_\nu$. Since $\overline{\mathcal{B}}_\nu(z_1, \dots, z_n)$ is semisimple, the result follows. ◻ [Lemma 67](#lem:FDOplucker2){reference-type="ref" reference="lem:FDOplucker2"} shows that every generator of $\overline{\mathcal{B}}_\nu(z_1, \dots, z_n)$ is given by a polynomial in the generators of $\mathcal{B}_\nu(z_1, \dots, z_n)$. This proves that $\overline{\mathcal{B}}_\nu(z_1, \dots, z_n)\subseteq \mathcal{B}_\nu(z_1, \dots, z_n)$ for all $z_1, \dots, z_n \in \mathbb{C}$. From the direct product decompositions [[([\[eq:directproductbethe\]](#eq:directproductbethe){reference-type="ref" reference="eq:directproductbethe"})]{.upright}](#eq:directproductbethe) and [[([\[eq:directproductaltbethe\]](#eq:directproductaltbethe){reference-type="ref" reference="eq:directproductaltbethe"})]{.upright}](#eq:directproductaltbethe), we deduce that $\overline{\mathcal{B}}_n(z_1, \dots, z_n)\subseteq \mathcal{B}_n(z_1, \dots, z_n)$, as required. 0◻ **Remark 68**. Combining [Lemma 66](#lem:FDOcoords){reference-type="ref" reference="lem:FDOcoords"} with parts [\[main_generates\]](#main_generates){reference-type="ref" reference="main_generates"} and [\[main_eigenspace\]](#main_eigenspace){reference-type="ref" reference="main_eigenspace"} of [Theorem 3](#thm:main){reference-type="ref" reference="thm:main"}, we immediately obtain a new proof of [Theorem 44](#thm:precise){reference-type="ref" reference="thm:precise"}. # Discussion and open problems {#sec:discussion} We conclude the paper by discussing several related results and open problems. ## Scheme-theoretic statements {#sec:scheme} We now give the more precise scheme-theoretic version of [Theorem 3](#thm:main){reference-type="ref" reference="thm:main"}[\[main_eigenspace\]](#main_eigenspace){reference-type="ref" reference="main_eigenspace"}. We consider the case when $g(u) = (u+z_1) \dotsm (u+z_n)$ has distinct roots in [5.1.1](#sec:distinct){reference-type="ref" reference="sec:distinct"}, and the general case in [5.1.2](#sec:nondistinct){reference-type="ref" reference="sec:nondistinct"}. The ***Plücker relations for $\smash{\overline{\mathcal{X}}}^{\nu}$*** are the Plücker relations, where we substitute $\Delta^\lambda = 0$ for every partition $\lambda \not\subseteq \nu$. Let $S_\nu := \mathbb{C}[\Delta^\lambda : \lambda \subseteq \nu]/\mathcal{I}_\nu$, where $\mathcal{I}_\nu$ is the ideal generated by the Plücker relations for $\smash{\overline{\mathcal{X}}}^{\nu}$. By [Proposition 19](#thm:schubertplucker){reference-type="ref" reference="thm:schubertplucker"}, the Schubert variety $\smash{\overline{\mathcal{X}}}^{\nu}$ is identified with $\mathop{\mathrm{Proj}}S_\nu$. We identify the Schubert cell $\mathcal{X}^{\nu}$ with $\mathop{\mathrm{Spec}}S_\nu^\circ$, where $S_\nu^\circ := S_\nu / \langle \Delta^\nu - \frac{n!}{\mathsf{f}^{\nu}}\rangle$. Under this identification the ring elements $(\Delta^\lambda : \lambda \subseteq \nu)$ are the normalized Plücker coordinates on $\mathcal{X}^{\nu}$. By [Theorem 3](#thm:main){reference-type="ref" reference="thm:main"}[\[main_pluckers\]](#main_pluckers){reference-type="ref" reference="main_pluckers"} and [Lemma 63](#lem:identifyscell){reference-type="ref" reference="lem:identifyscell"}, the elements $\beta^\lambda_\nu$ satisfy the relations in the ideal $\mathcal{I}_\nu + \langle \Delta^\nu - \frac{n!}{\mathsf{f}^{\nu}}\rangle$. Thus we have a well-defined surjective $\mathbb{C}$-algebra homomorphism $$\Phi_\nu : S_\nu^\circ \to \mathcal{B}_\nu(z_1, \dots, z_n), \quad \Delta^\lambda \mapsto \beta^\lambda_\nu\,,$$ which induces a closed embedding $$\Phi^*_\nu : \mathop{\mathrm{Spec}}\mathcal{B}_\nu(z_1, \dots, z_n)\to \mathcal{X}^{\nu} \,.$$ [Theorem 3](#thm:main){reference-type="ref" reference="thm:main"}[\[main_eigenspace\]](#main_eigenspace){reference-type="ref" reference="main_eigenspace"} says that as sets, the image of $\Phi^*_\nu$ is $\mathrm{Wr}^{-1}(g)$, the fibre of the Wronski map $\mathrm{Wr}: \mathcal{X}^{\nu}\to \mathcal{P}_n$ over $g$. That is, $\mathop{\mathrm{Spec}}S_\nu^\circ/\ker(\Phi_\nu) \subseteq \mathcal{X}^{\nu}$ and $\mathrm{Wr}^{-1}(g) \subseteq \mathcal{X}^{\nu}$ have the same points. ### Distinct roots {#sec:distinct} When $g$ has distinct roots, the preceding statement is also true scheme-theoretically: **Theorem 69**. *If $z_1, \dots, z_n \in \mathbb{C}$ are distinct, then the scheme-theoretic image of the closed embedding $\Phi^*_\nu$ is the fibre $\mathrm{Wr}^{-1}(g)$. That is, $S_\nu^\circ /\ker(\Phi_\nu)$ is the coordinate ring of $\mathrm{Wr}^{-1}(g)$.* *Proof.* Let ${\mathchoice {\mbox{\boldmath{$\psi$}}} {\mbox{\boldmath{$\psi$}}} {\mbox{\scriptsize\boldmath{$\psi$}}} {\mbox{\tiny\boldmath{$\psi$}}} }:= (\psi_{k,\ell} : 0 \le k \leq d,\ 0 \le \ell \leq n-k)$ denote the FDO coordinates on the Schubert cell $\mathcal{X}^{\nu}$. Mukhin, Tarasov, and Varchenko [@mukhin_tarasov_varchenko13 Theorem 4.3] prove that the $\mathbb{C}$-algebra homomorphism $S_\nu^\circ \to \mathcal{B}_\nu(z_1, \dots, z_n), \psi_{k,\ell} \mapsto \beta^{1^{\smash{k}}}_\ell$ induces a scheme-theoretic isomorphism $\mathop{\mathrm{Spec}}\mathcal{B}_\nu(z_1, \dots, z_n)\to \mathrm{Wr}^{-1}(g)$. By [\[lem:FDOplucker,lem:FDOplucker2\]](#lem:FDOplucker,lem:FDOplucker2){reference-type="ref" reference="lem:FDOplucker,lem:FDOplucker2"}, this $\mathbb{C}$-algebra homomorphism is $\Phi_\nu$. ◻ ### Non-distinct roots {#sec:nondistinct} We now consider the case when $g$ has repeated roots. Write $g(u) = (u+z_1)^{\kappa_1} \dotsm (u+z_s)^{\kappa_s}$, where $z_1, \dots, z_s \in \mathbb{C}$ are distinct and $\kappa := (\kappa_1, \dots, \kappa_s)$ is a composition of $n$. Consider the Bethe algebra $$\mathcal{B}_{n}(\mathbf{z}_\kappa) := \mathcal{B}_{n}( \underbrace{z_1, \dots, z_1}_{\kappa_1} \,,\, \underbrace{z_2, \dots, z_2}_{\kappa_2} \,,\,\dots\,,\, \underbrace{z_s, \dots, z_s}_{\kappa_s}) \,.$$ Let $\mathfrak{S}_{\kappa} := \mathfrak{S}_{\kappa_1} \times \dots \times \mathfrak{S}_{\kappa_s} \subseteq \mathfrak{S}_{n}$ be the Young subgroup associated to $\kappa$. We write ${\mathchoice {\mbox{\boldmath{$\mu$}}} {\mbox{\boldmath{$\mu$}}} {\mbox{\scriptsize\boldmath{$\mu$}}} {\mbox{\tiny\boldmath{$\mu$}}} }\mathrel{\,\Vdash}\kappa$ to mean that ${\mathchoice {\mbox{\boldmath{$\mu$}}} {\mbox{\boldmath{$\mu$}}} {\mbox{\scriptsize\boldmath{$\mu$}}} {\mbox{\tiny\boldmath{$\mu$}}} }= (\mu_1, \dots, \mu_s)$ is an $s$-tuple of partitions such that $\mu_i \vdash \kappa_i$ for $i=1, \dots, s$. The irreducible representations of $\mathfrak{S}_{\kappa}$ are of the form $$\mathbf{M}^{{\mathchoice {\mbox{\boldmath{$\mu$}}} {\mbox{\boldmath{$\mu$}}} {\mbox{\scriptsize\boldmath{$\mu$}}} {\mbox{\tiny\boldmath{$\mu$}}} }}:= M^{\mu_1} \otimes \dots \otimes M^{\mu_s} \qquad \text{ for} \; {\mathchoice {\mbox{\boldmath{$\mu$}}} {\mbox{\boldmath{$\mu$}}} {\mbox{\scriptsize\boldmath{$\mu$}}} {\mbox{\tiny\boldmath{$\mu$}}} }\mathrel{\,\Vdash}\kappa \,.$$ By Schur's lemma, the Specht module $M^{\nu}$ decomposes as a representation of $\mathfrak{S}_{\kappa}$ as $$\label{eq:spechtdecomposition} M^{\nu}\simeq \bigoplus_{{\mathchoice {\mbox{\boldmath{$\mu$}}} {\mbox{\boldmath{$\mu$}}} {\mbox{\scriptsize\boldmath{$\mu$}}} {\mbox{\tiny\boldmath{$\mu$}}} }\mathrel{\,\Vdash}\kappa} \mathrm{Hom}_{\mathfrak{S}_{\kappa}}(\mathbf{M}^{{\mathchoice {\mbox{\boldmath{$\mu$}}} {\mbox{\boldmath{$\mu$}}} {\mbox{\scriptsize\boldmath{$\mu$}}} {\mbox{\tiny\boldmath{$\mu$}}} }}, M^{\nu}) \otimes \mathbf{M}^{{\mathchoice {\mbox{\boldmath{$\mu$}}} {\mbox{\boldmath{$\mu$}}} {\mbox{\scriptsize\boldmath{$\mu$}}} {\mbox{\tiny\boldmath{$\mu$}}} }} \,.$$ It is not hard to check that $\mathfrak{S}_{\kappa}$ commutes with $\mathcal{B}_{n}(\mathbf{z}_\kappa)$. Thus the action of $\mathcal{B}_{n}(\mathbf{z}_\kappa)$ on $M^{\nu}$ respects the decomposition [[([\[eq:spechtdecomposition\]](#eq:spechtdecomposition){reference-type="ref" reference="eq:spechtdecomposition"})]{.upright}](#eq:spechtdecomposition), preserving each summand $\mathrm{Hom}_{\mathfrak{S}_{\kappa}}(\mathbf{M}^{{\mathchoice {\mbox{\boldmath{$\mu$}}} {\mbox{\boldmath{$\mu$}}} {\mbox{\scriptsize\boldmath{$\mu$}}} {\mbox{\tiny\boldmath{$\mu$}}} }}, M^{\nu}) \otimes \mathbf{M}^{{\mathchoice {\mbox{\boldmath{$\mu$}}} {\mbox{\boldmath{$\mu$}}} {\mbox{\scriptsize\boldmath{$\mu$}}} {\mbox{\tiny\boldmath{$\mu$}}} }}$, and acting trivially on the second tensor factor. That is, $\mathrm{Hom}_{\mathfrak{S}_{\kappa}}(\mathbf{M}^{{\mathchoice {\mbox{\boldmath{$\mu$}}} {\mbox{\boldmath{$\mu$}}} {\mbox{\scriptsize\boldmath{$\mu$}}} {\mbox{\tiny\boldmath{$\mu$}}} }}, M^{\nu})$ is a module for $\mathcal{B}_{n}(\mathbf{z}_\kappa)$. Let $\mathcal{B}_{\nu,{\mathchoice {\mbox{\boldmath{$\mu$}}} {\mbox{\boldmath{$\mu$}}} {\mbox{\scriptsize\boldmath{$\mu$}}} {\mbox{\tiny\boldmath{$\mu$}}} }}(\mathbf{z}_\kappa)$ denote the subalgebra of $\mathrm{End}\big(\mathrm{Hom}_{\mathfrak{S}_{\kappa}}(\mathbf{M}^{{\mathchoice {\mbox{\boldmath{$\mu$}}} {\mbox{\boldmath{$\mu$}}} {\mbox{\scriptsize\boldmath{$\mu$}}} {\mbox{\tiny\boldmath{$\mu$}}} }}, M^{\nu})\big)$ generated by the action, and let $\beta^\lambda_{\nu,{\mathchoice {\mbox{\boldmath{$\mu$}}} {\mbox{\boldmath{$\mu$}}} {\mbox{\scriptsize\boldmath{$\mu$}}} {\mbox{\tiny\boldmath{$\mu$}}} }}, \beta^{\lambda}_{\ell, \nu, {\mathchoice {\mbox{\boldmath{$\mu$}}} {\mbox{\boldmath{$\mu$}}} {\mbox{\scriptsize\boldmath{$\mu$}}} {\mbox{\tiny\boldmath{$\mu$}}} }} \in \mathcal{B}_{\nu,{\mathchoice {\mbox{\boldmath{$\mu$}}} {\mbox{\boldmath{$\mu$}}} {\mbox{\scriptsize\boldmath{$\mu$}}} {\mbox{\tiny\boldmath{$\mu$}}} }}(\mathbf{z}_\kappa)$ denote the images of $\beta^\lambda, \beta^\lambda_\ell \in \mathcal{B}_{n}(\mathbf{z}_\kappa)$. **Proposition 70**. *Up to a scalar multiple, the operator $$\beta^{\mu_1}_\nu(-z_1) \dotsm \beta^{\mu_s}_\nu(-z_s) \in \mathrm{End}(M^{\nu})$$ is the orthogonal projection onto $\mathrm{Hom}_{\mathfrak{S}_{\kappa}}(\mathbf{M}^{{\mathchoice {\mbox{\boldmath{$\mu$}}} {\mbox{\boldmath{$\mu$}}} {\mbox{\scriptsize\boldmath{$\mu$}}} {\mbox{\tiny\boldmath{$\mu$}}} }}, M^{\nu}) \otimes \mathbf{M}^{{\mathchoice {\mbox{\boldmath{$\mu$}}} {\mbox{\boldmath{$\mu$}}} {\mbox{\scriptsize\boldmath{$\mu$}}} {\mbox{\tiny\boldmath{$\mu$}}} }}$. In particular, each such orthogonal projection is an element of $\mathcal{B}_{\nu}(\mathbf{z}_\kappa)$, and we have the direct product decomposition $$\mathcal{B}_{\nu }(\mathbf{z}_\kappa)= \prod_{{\mathchoice {\mbox{\boldmath{$\mu$}}} {\mbox{\boldmath{$\mu$}}} {\mbox{\scriptsize\boldmath{$\mu$}}} {\mbox{\tiny\boldmath{$\mu$}}} }\mathrel{\,\Vdash}\kappa} \mathcal{B}_{\nu,{\mathchoice {\mbox{\boldmath{$\mu$}}} {\mbox{\boldmath{$\mu$}}} {\mbox{\scriptsize\boldmath{$\mu$}}} {\mbox{\tiny\boldmath{$\mu$}}} }}(\mathbf{z}_\kappa) \,.$$* *Proof.* By [[([\[eq:alphatobeta\]](#eq:alphatobeta){reference-type="ref" reference="eq:alphatobeta"})]{.upright}](#eq:alphatobeta), we can write $\beta^{\mu_i}(-z_i)$ as $c_i\frac{\mathsf{f}^{\mu_i}}{|\mu_i|!}\alpha^{\mu_i}_{X_i}$ for some nonzero scalar $c_i$, where $X_i := \{\kappa_1 {+} \cdots {+} \kappa_{i-1} {+} 1, \dots, \kappa_1 {+} \cdots {+} \kappa_i\}$. Hence, by [Proposition 29](#prop:projection){reference-type="ref" reference="prop:projection"}, $\frac{1}{c_i}\beta^{\mu_i}_\nu(-z_i)$ acts as the scalar $\delta_{\lambda_i,\mu_i}$ on the ${\mathchoice {\mbox{\boldmath{$\lambda$}}} {\mbox{\boldmath{$\lambda$}}} {\mbox{\scriptsize\boldmath{$\lambda$}}} {\mbox{\tiny\boldmath{$\lambda$}}} }$-isotypic component of $M^{\nu}$. Therefore $\frac{1}{c_1 \cdots c_s}\beta^{\mu_1}_\nu(-z_1) \dotsm \beta^{\mu_s}_\nu(-z_s)$ acts on $\mathrm{Hom}_{\mathfrak{S}_{\kappa}}(\mathbf{M}^{{\mathchoice {\mbox{\boldmath{$\lambda$}}} {\mbox{\boldmath{$\lambda$}}} {\mbox{\scriptsize\boldmath{$\lambda$}}} {\mbox{\tiny\boldmath{$\lambda$}}} }}, M^{\nu}) \otimes \mathbf{M}^{{\mathchoice {\mbox{\boldmath{$\lambda$}}} {\mbox{\boldmath{$\lambda$}}} {\mbox{\scriptsize\boldmath{$\lambda$}}} {\mbox{\tiny\boldmath{$\lambda$}}} }}$ as the scalar $\delta_{{\mathchoice {\mbox{\boldmath{$\lambda$}}} {\mbox{\boldmath{$\lambda$}}} {\mbox{\scriptsize\boldmath{$\lambda$}}} {\mbox{\tiny\boldmath{$\lambda$}}} },{\mathchoice {\mbox{\boldmath{$\mu$}}} {\mbox{\boldmath{$\mu$}}} {\mbox{\scriptsize\boldmath{$\mu$}}} {\mbox{\tiny\boldmath{$\mu$}}} }}$, as required. ◻ Since $\mathcal{B}_{\nu,{\mathchoice {\mbox{\boldmath{$\mu$}}} {\mbox{\boldmath{$\mu$}}} {\mbox{\scriptsize\boldmath{$\mu$}}} {\mbox{\tiny\boldmath{$\mu$}}} }}(\mathbf{z}_\kappa)$ is a quotient of $\mathcal{B}_{\nu}(\mathbf{z}_\kappa)$, we obtain a surjective $\mathbb{C}$-algebra homomorphism $$\Phi_{\nu,{\mathchoice {\mbox{\boldmath{$\mu$}}} {\mbox{\boldmath{$\mu$}}} {\mbox{\scriptsize\boldmath{$\mu$}}} {\mbox{\tiny\boldmath{$\mu$}}} }} : S_\nu^\circ \to \mathcal{B}_{\nu,{\mathchoice {\mbox{\boldmath{$\mu$}}} {\mbox{\boldmath{$\mu$}}} {\mbox{\scriptsize\boldmath{$\mu$}}} {\mbox{\tiny\boldmath{$\mu$}}} }}(\mathbf{z}_\kappa), \quad \Delta^\lambda \mapsto \beta^\lambda_{\nu,{\mathchoice {\mbox{\boldmath{$\mu$}}} {\mbox{\boldmath{$\mu$}}} {\mbox{\scriptsize\boldmath{$\mu$}}} {\mbox{\tiny\boldmath{$\mu$}}} }}\,,$$ which induces a closed embedding $$\Phi^*_{\nu,{\mathchoice {\mbox{\boldmath{$\mu$}}} {\mbox{\boldmath{$\mu$}}} {\mbox{\scriptsize\boldmath{$\mu$}}} {\mbox{\tiny\boldmath{$\mu$}}} }} : \mathop{\mathrm{Spec}}\mathcal{B}_{\nu,{\mathchoice {\mbox{\boldmath{$\mu$}}} {\mbox{\boldmath{$\mu$}}} {\mbox{\scriptsize\boldmath{$\mu$}}} {\mbox{\tiny\boldmath{$\mu$}}} }}(\mathbf{z}_\kappa) \to \mathcal{X}^{\nu} \,.$$ **Theorem 71**. *The scheme-theoretic image of the closed embedding $\Phi^*_{\nu,\mu}$ is the Schubert intersection $$\label{eq:generalssc} \smash{\overline{\mathcal{X}}}^{\nu}\cap X_{\mu_1}(z_1) \cap \dots \cap X_{\mu_s}(z_s) \,.$$ The scheme-theoretic image of $\Phi^*_\nu$ is the union of Schubert intersections $$\label{eq:specbethe} \bigcup_{{\mathchoice {\mbox{\boldmath{$\mu$}}} {\mbox{\boldmath{$\mu$}}} {\mbox{\scriptsize\boldmath{$\mu$}}} {\mbox{\tiny\boldmath{$\mu$}}} }\mathrel{\,\Vdash}\kappa} \smash{\overline{\mathcal{X}}}^{\nu}\cap X_{\mu_1}(z_1) \cap \dots \cap X_{\mu_s}(z_s) \,.$$* *Proof.* The second statement follows from the first, by [Proposition 70](#prop:schubertprojections){reference-type="ref" reference="prop:schubertprojections"}. For the first statement, we recall some additional background from [@mukhin_tarasov_varchenko09b; @mukhin_tarasov_varchenko13]. To every partition $\lambda$ with $\ell(\lambda) \leq n$, we associate a $\mathfrak{gl}_n$-module $V^\lambda$, which is an irreducible polynomial representation of the Lie algebra $\mathfrak{gl}_n= \mathfrak{gl}_n(\mathbb{C})$. This representation has (up to scalar multiple) a unique singular vector, of weight $\lambda \in \mathbb{Z}^n$. Now, consider the algebra $\mathfrak{gl}_n[t] := \mathfrak{gl}_n \otimes \mathbb{C}[t]$. Given any $\mathfrak{gl}_n$-module $V$ and $w \in \mathbb{C}$, we can extend the $\mathfrak{gl}_n$-action to a $\mathfrak{gl}_n[t]$-action, by letting $t$ act as multiplication by $w$. The resulting $\mathfrak{gl}_n[t]$-module is called an *evaluation module* of $\mathfrak{gl}_n[t]$, denoted by $V(w)$. The $\mathfrak{gl}_n$-Bethe algebra $\widehat{\mathcal{B}}_n\subseteq U(\mathfrak{gl}_n[t])$ is a commutative subalgebra of the universal enveloping algebra of $\mathfrak{gl}_n[t]$, which commutes with the subalgebra $U(\mathfrak{gl}_n) \subseteq U(\mathfrak{gl}_n[t])$. The algebra $\widehat{\mathcal{B}}_n$ is generated by the coefficients of the *universal differential operator*. (We will not need the precise formula for this operator here; see [@mukhin_tarasov_varchenko09b Section 2.7] for complete details.) We can obtain $\widehat{\mathcal{B}}_n$-modules by restricting any $\mathfrak{gl}_n[t]$-module to $\widehat{\mathcal{B}}_n$. Since $\widehat{\mathcal{B}}_n$ commutes with $U(\mathfrak{gl}_n)$, the $\mathfrak{gl}_n$-weight spaces, the spaces of $\mathfrak{gl}_n$-singular vectors, and spaces of singular vectors of any weight are $\widehat{\mathcal{B}}_n$-submodules. For a $\mathfrak{gl}_n[t]$-module $W$, we write $\widehat{\mathcal{B}}_n(W) \subseteq \mathrm{End}(W)$ for the algebra defined by the action of $\widehat{\mathcal{B}}_n$ on $W$. Similarly, we write $\widehat{\mathcal{B}}_n^\mathrm{sing}(W)$ and $\widehat{\mathcal{B}}^\mathrm{sing}_{n,\lambda}(W)$ for the algebras defined by the action on the singular vectors in $W$, and the singular vectors of weight $\lambda$ in $W$, respectively. Note that if $\widehat{\mathcal{B}}_n$ acts trivially on some vector space $M$, then $\widehat{\mathcal{B}}_n(W) \simeq \widehat{\mathcal{B}}_n(W \otimes M)$. Since $\widehat{\mathcal{B}}_n$ and $U(\mathfrak{gl}_n)$ commute, this implies that $\widehat{\mathcal{B}}_n^\mathrm{sing}(W) \simeq \widehat{\mathcal{B}}_n(W)$ as $\mathbb{C}$-algebras. We are mainly concerned with $\mathfrak{gl}_n[t]$-modules which are tensor products of evaluation modules. In particular, for $z_1, \dots, z_s \in \mathbb{C}$ and ${\mathchoice {\mbox{\boldmath{$\mu$}}} {\mbox{\boldmath{$\mu$}}} {\mbox{\scriptsize\boldmath{$\mu$}}} {\mbox{\tiny\boldmath{$\mu$}}} }\mathrel{\,\Vdash}\kappa$, consider the $\mathfrak{gl}_n[t]$-module $$\mathbf{V}^{\mathchoice {\mbox{\boldmath{$\mu$}}} {\mbox{\boldmath{$\mu$}}} {\mbox{\scriptsize\boldmath{$\mu$}}} {\mbox{\tiny\boldmath{$\mu$}}} }(z_1, \dots, z_s):= V^{\mu_1}(z_1) \otimes \dots \otimes V^{\mu_s}(z_s) \,.$$ In the algebras $\widehat{\mathcal{B}}_n\big(\mathbf{V}^{\mathchoice {\mbox{\boldmath{$\mu$}}} {\mbox{\boldmath{$\mu$}}} {\mbox{\scriptsize\boldmath{$\mu$}}} {\mbox{\tiny\boldmath{$\mu$}}} }(z_1, \dots, z_s)\big) \simeq \widehat{\mathcal{B}}_n^\mathrm{sing}\big(\mathbf{V}^{\mathchoice {\mbox{\boldmath{$\mu$}}} {\mbox{\boldmath{$\mu$}}} {\mbox{\scriptsize\boldmath{$\mu$}}} {\mbox{\tiny\boldmath{$\mu$}}} }(z_1, \dots, z_s)\big)$ and $\widehat{\mathcal{B}}^\mathrm{sing}_{n,\nu}\big(\mathbf{V}^{\mathchoice {\mbox{\boldmath{$\mu$}}} {\mbox{\boldmath{$\mu$}}} {\mbox{\scriptsize\boldmath{$\mu$}}} {\mbox{\tiny\boldmath{$\mu$}}} }(z_1, \dots, z_s)\big)$, the universal differential operator takes the form $$\frac{1}{g(u)}\sum_{k=0}^n (-1)^k \smash{\widehat{\beta}}_{k,\ell} u^{n-k-\ell} \partial_u^{n-k} \,,$$ where $\smash{\widehat{\beta}}_{k,\ell}$ are generators of the algebra, and $g(u) = (u+z_1)^{\kappa_1} \dotsm (u+z_s)^{\kappa_s}$. Mukhin, Tarasov, and Varchenko prove that $\mathop{\mathrm{Spec}}\widehat{\mathcal{B}}^\mathrm{sing}_{n,\nu}\big(\mathbf{V}^{\mathchoice {\mbox{\boldmath{$\mu$}}} {\mbox{\boldmath{$\mu$}}} {\mbox{\scriptsize\boldmath{$\mu$}}} {\mbox{\tiny\boldmath{$\mu$}}} }(z_1, \dots, z_s)\big)$ is scheme-theoretically identified with the Schubert intersection [[([\[eq:generalssc\]](#eq:generalssc){reference-type="ref" reference="eq:generalssc"})]{.upright}](#eq:generalssc), under the ring homomorphism $S^\nu \to \widehat{\mathcal{B}}^\mathrm{sing}_{n,\nu}\big(\mathbf{V}^{\mathchoice {\mbox{\boldmath{$\mu$}}} {\mbox{\boldmath{$\mu$}}} {\mbox{\scriptsize\boldmath{$\mu$}}} {\mbox{\tiny\boldmath{$\mu$}}} }(z_1, \dots, z_s)\big)$, $\psi_{k,\ell} \mapsto \smash{\widehat{\beta}}_{k,\ell}$ [@mukhin_tarasov_varchenko09b Theorem 5.13]. Therefore, our task is to show that $\widehat{\mathcal{B}}^\mathrm{sing}_{n,\nu}(\mathbf{V}^{\mathchoice {\mbox{\boldmath{$\mu$}}} {\mbox{\boldmath{$\mu$}}} {\mbox{\scriptsize\boldmath{$\mu$}}} {\mbox{\tiny\boldmath{$\mu$}}} }(z_1, \dots, z_s)) \simeq \mathcal{B}_{\nu,{\mathchoice {\mbox{\boldmath{$\mu$}}} {\mbox{\boldmath{$\mu$}}} {\mbox{\scriptsize\boldmath{$\mu$}}} {\mbox{\tiny\boldmath{$\mu$}}} }}(\mathbf{z}_\kappa)$ under an isomorphism sending $\smash{\widehat{\beta}}_{k,\ell} \mapsto \beta^{1^{\smash{k}}}_{\ell,\nu,{\mathchoice {\mbox{\boldmath{$\mu$}}} {\mbox{\boldmath{$\mu$}}} {\mbox{\scriptsize\boldmath{$\mu$}}} {\mbox{\tiny\boldmath{$\mu$}}} }}$. The argument in the proof of [Theorem 69](#thm:distinct){reference-type="ref" reference="thm:distinct"} then shows that $\mathop{\mathrm{Spec}}\mathcal{B}_{\nu,{\mathchoice {\mbox{\boldmath{$\mu$}}} {\mbox{\boldmath{$\mu$}}} {\mbox{\scriptsize\boldmath{$\mu$}}} {\mbox{\tiny\boldmath{$\mu$}}} }}(\mathbf{z}_\kappa)$ is scheme-theoretically identified with the intersection [[([\[eq:generalssc\]](#eq:generalssc){reference-type="ref" reference="eq:generalssc"})]{.upright}](#eq:generalssc) under $\Phi^*_{\nu,{\mathchoice {\mbox{\boldmath{$\mu$}}} {\mbox{\boldmath{$\mu$}}} {\mbox{\scriptsize\boldmath{$\mu$}}} {\mbox{\tiny\boldmath{$\mu$}}} }}$. For $z_1, \dots, z_n \in \mathbb{C}$, consider the $\mathfrak{gl}_n[t]$-module $$\label{eq:glnt-module} W(z_1, \dots, z_n) := \mathbf{V}^{(1,\dots, 1)}(z_1, \dots, z_n)= \mathbb{C}^n(z_1) \otimes \dots \otimes \mathbb{C}^n(z_n) \,.$$ Note that $\mathbb{C}[\mathfrak{S}_{n}]$ also acts on $W(z_1, \dots, z_n)$ (which, as a $\mathfrak{gl}_n$-module, is just $(\mathbb{C}^n)^{\otimes n}$) by permuting the tensor factors. Since $\widehat{\mathcal{B}}_n$ commutes with $U(\mathfrak{gl}_n)$, by Schur--Weyl duality, the action of any element of $\widehat{\mathcal{B}}_n$ on $W(z_1, \dots, z_n)$ is equivalently given by some element of $\mathbb{C}[\mathfrak{S}_{n}]$. Thus Schur--Weyl duality identifies $\widehat{\mathcal{B}}_n\big(W(z_1, \dots, z_n)\big)$ with some subalgebra of $\mathbb{C}[\mathfrak{S}_{n}]$. Specifically, $\widehat{\mathcal{B}}_n\big(W(z_1, \dots, z_n)\big)$ is identified with $\mathcal{B}_n(z_1, \dots, z_n)\subseteq \mathbb{C}[\mathfrak{S}_{n}]$, and the elements $\smash{\widehat{\beta}}_{k,\ell} \in \widehat{\mathcal{B}}_n\big(W(z_1, \dots, z_n)\big)$ are identified with $\beta^{1^{\smash{k}}}_\ell \in \mathcal{B}_n(z_1, \dots, z_n)$ [@mukhin_tarasov_varchenko13 Theorem 3.2]. In particular, for the $\mathfrak{gl}_n[t]$-module $W(\mathbf{z}_\kappa)$, we have $\mathcal{B}_{n}(\mathbf{z}_\kappa) \simeq \widehat{\mathcal{B}}_n\big(W(\mathbf{z}_\kappa)\big)$. In this case, we can rewrite the right-hand side of [[([\[eq:glnt-module\]](#eq:glnt-module){reference-type="ref" reference="eq:glnt-module"})]{.upright}](#eq:glnt-module) as $$\label{eq:tensordecomp} (\mathbb{C}^n)^{\otimes \kappa_1}(z_1) \otimes \dots \otimes (\mathbb{C}^n)^{\otimes \kappa_s}(z_k) \simeq \bigoplus_{{\mathchoice {\mbox{\boldmath{$\mu$}}} {\mbox{\boldmath{$\mu$}}} {\mbox{\scriptsize\boldmath{$\mu$}}} {\mbox{\tiny\boldmath{$\mu$}}} }\mathrel{\,\Vdash}\kappa}\, \mathbf{V}^{\mathchoice {\mbox{\boldmath{$\mu$}}} {\mbox{\boldmath{$\mu$}}} {\mbox{\scriptsize\boldmath{$\mu$}}} {\mbox{\tiny\boldmath{$\mu$}}} }(z_1, \dots, z_s)\otimes \mathbf{M}^{{\mathchoice {\mbox{\boldmath{$\mu$}}} {\mbox{\boldmath{$\mu$}}} {\mbox{\scriptsize\boldmath{$\mu$}}} {\mbox{\tiny\boldmath{$\mu$}}} }} \,,$$ where $\mathfrak{gl}_n[t]$ acts trivially on $\mathbf{M}^{{\mathchoice {\mbox{\boldmath{$\mu$}}} {\mbox{\boldmath{$\mu$}}} {\mbox{\scriptsize\boldmath{$\mu$}}} {\mbox{\tiny\boldmath{$\mu$}}} }}$ and $\mathfrak{S}_{\kappa}$ acts trivially on $\mathbf{V}^{\mathchoice {\mbox{\boldmath{$\mu$}}} {\mbox{\boldmath{$\mu$}}} {\mbox{\scriptsize\boldmath{$\mu$}}} {\mbox{\tiny\boldmath{$\mu$}}} }(z_1, \dots, z_s)$. Therefore $$\label{eq:bethesingdecomp} \widehat{\mathcal{B}}_n^\mathrm{sing}\big(W(\mathbf{z}_\kappa)\big) \simeq \prod_{{\mathchoice {\mbox{\boldmath{$\mu$}}} {\mbox{\boldmath{$\mu$}}} {\mbox{\scriptsize\boldmath{$\mu$}}} {\mbox{\tiny\boldmath{$\mu$}}} }\mathrel{\,\Vdash}\kappa} \widehat{\mathcal{B}}_n^\mathrm{sing}\big(\mathbf{V}^{\mathchoice {\mbox{\boldmath{$\mu$}}} {\mbox{\boldmath{$\mu$}}} {\mbox{\scriptsize\boldmath{$\mu$}}} {\mbox{\tiny\boldmath{$\mu$}}} }(z_1, \dots, z_s)\big) \simeq \prod_{\substack{\nu \vdash n, \\ {\mathchoice {\mbox{\boldmath{$\mu$}}} {\mbox{\boldmath{$\mu$}}} {\mbox{\scriptsize\boldmath{$\mu$}}} {\mbox{\tiny\boldmath{$\mu$}}} }\mathrel{\,\Vdash}\kappa}} \widehat{\mathcal{B}}^\mathrm{sing}_{n,\nu}\big(\mathbf{V}^{\mathchoice {\mbox{\boldmath{$\mu$}}} {\mbox{\boldmath{$\mu$}}} {\mbox{\scriptsize\boldmath{$\mu$}}} {\mbox{\tiny\boldmath{$\mu$}}} }(z_1, \dots, z_s)\big) \,.$$ The first isomorphism in [[([\[eq:bethesingdecomp\]](#eq:bethesingdecomp){reference-type="ref" reference="eq:bethesingdecomp"})]{.upright}](#eq:bethesingdecomp) is obtained by projecting onto the ${\mathchoice {\mbox{\boldmath{$\mu$}}} {\mbox{\boldmath{$\mu$}}} {\mbox{\scriptsize\boldmath{$\mu$}}} {\mbox{\tiny\boldmath{$\mu$}}} }$-isotypic component of the $\mathfrak{S}_{\kappa}$-action in the decomposition [[([\[eq:tensordecomp\]](#eq:tensordecomp){reference-type="ref" reference="eq:tensordecomp"})]{.upright}](#eq:tensordecomp), for each ${\mathchoice {\mbox{\boldmath{$\mu$}}} {\mbox{\boldmath{$\mu$}}} {\mbox{\scriptsize\boldmath{$\mu$}}} {\mbox{\tiny\boldmath{$\mu$}}} }\mathrel{\,\Vdash}\kappa$. The second isomorphism in [[([\[eq:bethesingdecomp\]](#eq:bethesingdecomp){reference-type="ref" reference="eq:bethesingdecomp"})]{.upright}](#eq:bethesingdecomp) is obtained by further projecting onto the singular vectors of weight $\nu$, for each $\nu \vdash n$; by Schur--Weyl duality this is the same as projecting onto the $M^{\nu}$-isotypic component of the $\mathfrak{S}_{n}$-action on $(\mathbb{C}^n)^{\otimes n}$. (These projections are contained in $\widehat{\mathcal{B}}_n\big(W(\mathbf{z}_\kappa)\big)$ by [Proposition 70](#prop:schubertprojections){reference-type="ref" reference="prop:schubertprojections"}.) But this is the same way that we obtain the decomposition $\mathcal{B}_{n}(\mathbf{z}_\kappa) \simeq \prod_{\nu, {\mathchoice {\mbox{\boldmath{$\mu$}}} {\mbox{\boldmath{$\mu$}}} {\mbox{\scriptsize\boldmath{$\mu$}}} {\mbox{\tiny\boldmath{$\mu$}}} }} \mathcal{B}_{\nu, {\mathchoice {\mbox{\boldmath{$\mu$}}} {\mbox{\boldmath{$\mu$}}} {\mbox{\scriptsize\boldmath{$\mu$}}} {\mbox{\tiny\boldmath{$\mu$}}} }}(\mathbf{z}_\kappa)$. Thus, the identification $\widehat{\mathcal{B}}_n^\mathrm{sing}\big(W(\mathbf{z}_\kappa)\big) \simeq \widehat{\mathcal{B}}_n\big(W(\mathbf{z}_\kappa)\big) \simeq \mathcal{B}_{n}(\mathbf{z}_\kappa)$ also identifies components $\widehat{\mathcal{B}}^\mathrm{sing}_{n,\nu}\big(\mathbf{V}^{\mathchoice {\mbox{\boldmath{$\mu$}}} {\mbox{\boldmath{$\mu$}}} {\mbox{\scriptsize\boldmath{$\mu$}}} {\mbox{\tiny\boldmath{$\mu$}}} }(z_1, \dots, z_s)\big) \simeq \mathcal{B}_{\nu,{\mathchoice {\mbox{\boldmath{$\mu$}}} {\mbox{\boldmath{$\mu$}}} {\mbox{\scriptsize\boldmath{$\mu$}}} {\mbox{\tiny\boldmath{$\mu$}}} }}(\mathbf{z}_\kappa)$. Since the latter identification is obtained by projections of the former, the generators $\smash{\widehat{\beta}}_{k,\ell} \in \widehat{\mathcal{B}}^\mathrm{sing}_{n,\nu}\big(\mathbf{V}^{\mathchoice {\mbox{\boldmath{$\mu$}}} {\mbox{\boldmath{$\mu$}}} {\mbox{\scriptsize\boldmath{$\mu$}}} {\mbox{\tiny\boldmath{$\mu$}}} }(z_1, \dots, z_s)\big)$ are identified with generators $\beta^{1^{\smash{k}}}_{\ell,\nu,{\mathchoice {\mbox{\boldmath{$\mu$}}} {\mbox{\boldmath{$\mu$}}} {\mbox{\scriptsize\boldmath{$\mu$}}} {\mbox{\tiny\boldmath{$\mu$}}} }} \in \mathcal{B}_{\nu,{\mathchoice {\mbox{\boldmath{$\mu$}}} {\mbox{\boldmath{$\mu$}}} {\mbox{\scriptsize\boldmath{$\mu$}}} {\mbox{\tiny\boldmath{$\mu$}}} }}(\mathbf{z}_\kappa)$, as required. ◻ We take this opportunity to state a natural problem: **Problem 72**. *Find explicit formulas for universal Plücker coordinates in the $\mathfrak{gl}_n$-Bethe algebra, which coincide with the operators $\beta^\lambda(s)$ on the $\mathfrak{gl}_n[t]$-modules $\mathbf{V}^{\mathchoice {\mbox{\boldmath{$\mu$}}} {\mbox{\boldmath{$\mu$}}} {\mbox{\scriptsize\boldmath{$\mu$}}} {\mbox{\tiny\boldmath{$\mu$}}} }(z_1, \dots, z_s)$.* ### Dimensions of Bethe algebras {#sec:dimensions} We can now calculate the dimensions of the Bethe algebras $\mathcal{B}_{\nu}(\mathbf{z}_\kappa)$ and $\mathcal{B}_{n}(\mathbf{z}_\kappa)$: **Theorem 73**. *For $k \geq 0$, let $\mbox{\small\sf{S}}_k := \sum_{\lambda \vdash k} \mathsf{s}_\lambda$. Then $$\label{eq:dimensionformula} \dim \mathcal{B}_{\nu}(\mathbf{z}_\kappa) = \langle \mathsf{s}_\nu \,,\, \mbox{\small\sf{S}}_{\kappa_1} \dotsm \mbox{\small\sf{S}}_{\kappa_s} \rangle \qquad \text{and} \qquad \dim \mathcal{B}_{n}(\mathbf{z}_\kappa) = \langle \mbox{\small\sf{S}}_n \,,\, \mbox{\small\sf{S}}_{\kappa_1} \dotsm \mbox{\small\sf{S}}_{\kappa_s} \rangle \,.$$* *Proof.* By [Theorem 71](#thm:nondistinct){reference-type="ref" reference="thm:nondistinct"}, $\dim \mathcal{B}_{\nu, {\mathchoice {\mbox{\boldmath{$\mu$}}} {\mbox{\boldmath{$\mu$}}} {\mbox{\scriptsize\boldmath{$\mu$}}} {\mbox{\tiny\boldmath{$\mu$}}} }}(\mathbf{z}_\kappa)$ is the length of the Schubert intersection [[([\[eq:generalssc\]](#eq:generalssc){reference-type="ref" reference="eq:generalssc"})]{.upright}](#eq:generalssc), as a scheme. As in [2.2.1](#sec:schubert){reference-type="ref" reference="sec:schubert"}, this length is $\langle \mathsf{s}_\nu, \mathsf{s}_{\mu_1} \dotsm \mathsf{s}_{\mu_s}\rangle$ if the intersection is transverse. Since the Schubert varieties are smooth at the points of intersection, this remains true whenever the intersection is proper [@fulton98 Proposition 8.2]. Hence, $$\dim \mathcal{B}_{\nu, {\mathchoice {\mbox{\boldmath{$\mu$}}} {\mbox{\boldmath{$\mu$}}} {\mbox{\scriptsize\boldmath{$\mu$}}} {\mbox{\tiny\boldmath{$\mu$}}} }}(\mathbf{z}_\kappa) = \langle \mathsf{s}_\nu\,, \mathsf{s}_{\mu_1} \dotsm \mathsf{s}_{\mu_s}\rangle \,.$$ Summing over all ${\mathchoice {\mbox{\boldmath{$\mu$}}} {\mbox{\boldmath{$\mu$}}} {\mbox{\scriptsize\boldmath{$\mu$}}} {\mbox{\tiny\boldmath{$\mu$}}} }\mathrel{\,\Vdash}\kappa$ (and for the second formula, also over $\nu \vdash n$) we obtain the formulas [[([\[eq:dimensionformula\]](#eq:dimensionformula){reference-type="ref" reference="eq:dimensionformula"})]{.upright}](#eq:dimensionformula). ◻ We now explain why $\mathop{\mathrm{Spec}}\mathcal{B}_{\nu}(\mathbf{z}_\kappa)$ is (in some cases) scheme-theoretically different from the fibre $\mathrm{Wr}^{-1}(g)$, for $g(u) = (u+z_1)^{\kappa_1} \dotsm (u+z_s)^{\kappa_s}$. Note that [[([\[eq:specbethe\]](#eq:specbethe){reference-type="ref" reference="eq:specbethe"})]{.upright}](#eq:specbethe) is the right-hand side of [[([\[eq:fibre\]](#eq:fibre){reference-type="ref" reference="eq:fibre"})]{.upright}](#eq:fibre), which equals $\mathrm{Wr}^{-1}(g)$ set-theoretically, and so $\mathop{\mathrm{Spec}}\mathcal{B}_{\nu}(\mathbf{z}_\kappa)$ is always set-theoretically identified with $\mathrm{Wr}^{-1}(g)$. The Wronski map is a finite morphism of degree $\mathsf{f}^{\nu}$, so the fibre $\mathrm{Wr}^{-1}(h)$ is a finite scheme of length $\mathsf{f}^{\nu}$ for all $h \in \mathcal{P}_n$. On the other hand, by definition, $\mathop{\mathrm{Spec}}\mathcal{B}_{\nu}(\mathbf{z}_\kappa)$ is a finite scheme of length $\dim \mathcal{B}_{\nu}(\mathbf{z}_\kappa)$, which (depending on $\kappa$) may be strictly less than $\mathsf{f}^{\nu}$. Since [[([\[eq:specbethe\]](#eq:specbethe){reference-type="ref" reference="eq:specbethe"})]{.upright}](#eq:specbethe) is always a subscheme of $\mathrm{Wr}^{-1}(g)$, the two scheme structures coincide if and only if $\dim \mathcal{B}_{\nu}(\mathbf{z}_\kappa) = \mathsf{f}^{\nu}$. The discrepancy between the two scheme structures is accounted for by the following. Let $E \subseteq M^{\nu}$ be an eigenspace of $\mathcal{B}_{\nu}(\mathbf{z}_\kappa)$. By [Theorem 3](#thm:main){reference-type="ref" reference="thm:main"}[\[main_multiplicity\]](#main_multiplicity){reference-type="ref" reference="main_multiplicity"}, the multiplicity of $V_E$, viewed as a point of the fibre $\mathrm{Wr}^{-1}(g)$, is $\dim \widehat E$. Now, by [[([\[eq:fibre\]](#eq:fibre){reference-type="ref" reference="eq:fibre"})]{.upright}](#eq:fibre) the corresponding point $V_E$ belongs to the intersection [[([\[eq:generalssc\]](#eq:generalssc){reference-type="ref" reference="eq:generalssc"})]{.upright}](#eq:generalssc) for some ${\mathchoice {\mbox{\boldmath{$\mu$}}} {\mbox{\boldmath{$\mu$}}} {\mbox{\scriptsize\boldmath{$\mu$}}} {\mbox{\tiny\boldmath{$\mu$}}} }\mathrel{\,\Vdash}\kappa$. Under the decomposition [[([\[eq:spechtdecomposition\]](#eq:spechtdecomposition){reference-type="ref" reference="eq:spechtdecomposition"})]{.upright}](#eq:spechtdecomposition), we must have $E, \widehat E \subseteq \mathrm{Hom}_{\mathfrak{S}_{\kappa}}(\mathbf{M}^{{\mathchoice {\mbox{\boldmath{$\mu$}}} {\mbox{\boldmath{$\mu$}}} {\mbox{\scriptsize\boldmath{$\mu$}}} {\mbox{\tiny\boldmath{$\mu$}}} }}, M^{\nu}) \otimes \mathbf{M}^{{\mathchoice {\mbox{\boldmath{$\mu$}}} {\mbox{\boldmath{$\mu$}}} {\mbox{\scriptsize\boldmath{$\mu$}}} {\mbox{\tiny\boldmath{$\mu$}}} }}$. Since $\mathcal{B}_{\nu}(\mathbf{z}_\kappa)$ acts trivially on the second tensor factor, $E = E_0 \otimes \mathbf{M}^{{\mathchoice {\mbox{\boldmath{$\mu$}}} {\mbox{\boldmath{$\mu$}}} {\mbox{\scriptsize\boldmath{$\mu$}}} {\mbox{\tiny\boldmath{$\mu$}}} }}$ and $\widehat E = \widehat E_0 \otimes \mathbf{M}^{{\mathchoice {\mbox{\boldmath{$\mu$}}} {\mbox{\boldmath{$\mu$}}} {\mbox{\scriptsize\boldmath{$\mu$}}} {\mbox{\tiny\boldmath{$\mu$}}} }}$, for some subspaces $E_0, \widehat E_0 \subseteq \mathrm{Hom}_{\mathfrak{S}_{\kappa}}(\mathbf{M}^{{\mathchoice {\mbox{\boldmath{$\mu$}}} {\mbox{\boldmath{$\mu$}}} {\mbox{\scriptsize\boldmath{$\mu$}}} {\mbox{\tiny\boldmath{$\mu$}}} }}, M^{\nu})$. In fact, $E_0$ is an eigenspace of the algebra $\mathcal{B}_{\nu, {\mathchoice {\mbox{\boldmath{$\mu$}}} {\mbox{\boldmath{$\mu$}}} {\mbox{\scriptsize\boldmath{$\mu$}}} {\mbox{\tiny\boldmath{$\mu$}}} }}(\mathbf{z}_\kappa)$, and $\widehat E_0$ is the corresponding generalized eigenspace. By [Theorem 71](#thm:nondistinct){reference-type="ref" reference="thm:nondistinct"}, the multiplicity of $V_E = V_{E_0}$, viewed as a point of the Schubert intersection [[([\[eq:generalssc\]](#eq:generalssc){reference-type="ref" reference="eq:generalssc"})]{.upright}](#eq:generalssc), is $\dim \widehat E_0$. Hence the two notions of multiplicity differ by a factor of $\dim \mathbf{M}^{{\mathchoice {\mbox{\boldmath{$\mu$}}} {\mbox{\boldmath{$\mu$}}} {\mbox{\scriptsize\boldmath{$\mu$}}} {\mbox{\tiny\boldmath{$\mu$}}} }}$. Informally, in the fibre $\mathrm{Wr}^{-1}(g)$, there is some non-trivial geometry associated with the multiplicity spaces $\mathbf{M}^{\mu}$ of the decomposition [[([\[eq:spechtdecomposition\]](#eq:spechtdecomposition){reference-type="ref" reference="eq:spechtdecomposition"})]{.upright}](#eq:spechtdecomposition), but since $\mathcal{B}_{\nu}(\mathbf{z}_\kappa)$ acts trivially on these spaces, this is not reflected in the geometry of $\mathop{\mathrm{Spec}}\mathcal{B}_{\nu}(\mathbf{z}_\kappa)$. We note the following corollary, which was also effectively observed in [@mukhin_tarasov_varchenko13]: **Corollary 74** (cf. [@mukhin_tarasov_varchenko13 Remark p. 776]). *The following are equivalent:* (a) *[\[schemeequality1\]]{#schemeequality1 label="schemeequality1"} the equality [[([\[eq:fibre\]](#eq:fibre){reference-type="ref" reference="eq:fibre"})]{.upright}](#eq:fibre) holds scheme-theoretically;* (b) *[\[schemeequality2\]]{#schemeequality2 label="schemeequality2"} $\dim \mathcal{B}_{\nu}(\mathbf{z}_\kappa) = \mathsf{f}^{\nu}$;* (c) *[\[schemeequality3\]]{#schemeequality3 label="schemeequality3"} $\nu$ equals $n$ or $1^n$, or $\kappa_i \le 2$ for all $1 \le i \le s$.* *Proof.* The equivalence of [\[schemeequality1\]](#schemeequality1){reference-type="ref" reference="schemeequality1"} and [\[schemeequality2\]](#schemeequality2){reference-type="ref" reference="schemeequality2"} is discussed above. The equivalence of [\[schemeequality2\]](#schemeequality2){reference-type="ref" reference="schemeequality2"} and [\[schemeequality3\]](#schemeequality3){reference-type="ref" reference="schemeequality3"} follows since both conditions are equivalent to $\mathrm{Hom}_{\mathfrak{S}_{\kappa}}(\mathbf{M}^{{\mathchoice {\mbox{\boldmath{$\mu$}}} {\mbox{\boldmath{$\mu$}}} {\mbox{\scriptsize\boldmath{$\mu$}}} {\mbox{\tiny\boldmath{$\mu$}}} }}, M^{\nu}) = 0$ for all ${\mathchoice {\mbox{\boldmath{$\mu$}}} {\mbox{\boldmath{$\mu$}}} {\mbox{\scriptsize\boldmath{$\mu$}}} {\mbox{\tiny\boldmath{$\mu$}}} }\mathrel{\,\Vdash}\kappa$ such that $\dim \mathbf{M}^{{\mathchoice {\mbox{\boldmath{$\mu$}}} {\mbox{\boldmath{$\mu$}}} {\mbox{\scriptsize\boldmath{$\mu$}}} {\mbox{\tiny\boldmath{$\mu$}}} }} > 1$. Alternatively, we can apply [[([\[eq:dimensionformula\]](#eq:dimensionformula){reference-type="ref" reference="eq:dimensionformula"})]{.upright}](#eq:dimensionformula), by expanding $\mbox{\small\sf{S}}_{\kappa_1} \dotsm \mbox{\small\sf{S}}_{\kappa_s}$ in the Schur basis using the Littlewood--Richardson rule [@stanley99 Section A1.3]. ◻ ## Bases for $V_E$ {#sec:bases} [Theorem 3](#thm:main){reference-type="ref" reference="thm:main"} gives us the points $V_E \in \mathrm{Wr}^{-1}(g)$ in terms of their Plücker coordinates. We now describe two ways to obtain a basis for $V_E$ (see [\[thm:betabasis,thm:generalbasis\]](#thm:betabasis,thm:generalbasis){reference-type="ref" reference="thm:betabasis,thm:generalbasis"}), and make several related remarks. ### Plücker-coordinate basis {#sec:basisplucker} We can obtain a basis for $V_E \in \mathcal{X}^{\nu}\subseteq \mathrm{Gr}(d,m)$ using [Proposition 18](#prop:pluckerbasis){reference-type="ref" reference="prop:pluckerbasis"}, corresponding to a matrix representative in reduced row-echelon form. Following our convention [[([\[eq:identification\]](#eq:identification){reference-type="ref" reference="eq:identification"})]{.upright}](#eq:identification), we index the Plücker coordinates $\beta^\lambda_E$ using $\binom{[m]}{d}$ (extended to $[m]^d$ by the alternating property). So in this notation, $\big(\beta_{I,E}: I \in {m \choose [d]}\big)$ are the normalized Plücker coordinates of $V_E$, by [Corollary 64](#cor:identifyscell){reference-type="ref" reference="cor:identifyscell"}. **Theorem 75**. *Let $E \subseteq M^{\nu}$ be an eigenspace of $\mathcal{B}_\nu(z_1, \dots, z_n)$, and let $J = (j_1, \dots, j_d) := (\nu_d+1, \dots, \nu_1+d) \in\binom{[m]}{d}$ correspond to $\nu$ as in [[([\[eq:identification\]](#eq:identification){reference-type="ref" reference="eq:identification"})]{.upright}](#eq:identification). For $i \in [d]$, define $$\label{eq:betabasis} f_i(u) := \frac{\mathsf{f}^{\nu}}{n!} \sum_{k=1}^{j_i} \beta_{(J-j_i)+k, E} \frac{u^{k-1}}{(k-1)!} \,.$$ Then $(f_1, \dots, f_d)$ is the unique basis for $V_E$ such that for all $i\in [d]$ and $k \in J\setminus \{j_i\}$, we have $$f_i(u) = \frac{u^{j_i-1}}{(j_i-1)!} +\, \text{lower-degree terms} \qquad \text{and} \qquad [u^{k-1}] f_i(u) = 0 \,.$$* *Proof.* Since $\big(\beta_{I,E}: I \in {m \choose [d]}\big)$ are the normalized Plücker coordinates of $V_E \in \mathcal{X}^{\nu}$, the fact that $(f_1, \dots, f_d)$ is a basis for $V_E$ follows from [Proposition 18](#prop:pluckerbasis){reference-type="ref" reference="prop:pluckerbasis"}. By [Proposition 19](#thm:schubertplucker){reference-type="ref" reference="thm:schubertplucker"} we have $\beta_{(J-j_i)+k,E} =0$ for all $k > j_i$, so we can reduce the upper index of summation from $m$ to $j_i$ in the definition of $f_i$. By definition of $\beta_{(J-j_i)+k}$, we have $[u^{k-1}] f_i(u) = 0$ if $k \in J\setminus \{j_i\}$. Since the Plücker coordinates are normalized, the constant factor of $\frac{\mathsf{f}^{\nu}}{n!}$ ensures that $\big[\frac{u^{j_i-1}}{(j_i-1)!}\big]f_i(u) = 1$. ◻ **Example 76**. For $V \in \mathcal{X}^{2} \subseteq \mathrm{Gr}(2,4)$ as in [Example 15](#ex:pluckers){reference-type="ref" reference="ex:pluckers"}, let us find the basis $(f_1, f_2)$ from [Theorem 75](#thm:betabasis){reference-type="ref" reference="thm:betabasis"}. We have $J = (1, 4)$, and the Plücker coordinates $\beta_{I,E}$ (where $V = V_E$) are given by [[([\[eq:examplepluckers\]](#eq:examplepluckers){reference-type="ref" reference="eq:examplepluckers"})]{.upright}](#eq:examplepluckers). We calculate that $f_1(u) = \frac{1}{2}\beta_{(J - 1) + 1, E} = 1$ and $$f_2(u) = \scalebox{0.94}{$\displaystyle\frac{1}{2}\Big(\beta_{(J - 4) + 1, E} + \beta_{(J - 4) + 2, E}u + \beta_{(J - 4) + 3, E}\frac{u^2}{2} + \beta_{(J - 4) + 4, E}\frac{u^3}{6}\Big) = \frac{1}{6}u^3 + \frac{z_1+z_2}{4}u^2 + \frac{z_1z_2}{2}u$}\,.$$ ### A Markov basis exhibiting disconjugacy {#sec:basismarkov} Let $V \subseteq \mathbb{R}[u]$ be finite-dimensional vector space of real polynomials, and let $I\subseteq\mathbb{R}$ be an interval. We say that an ordered basis $(f_1, \dots, f_d)$ for $V$ is a ***Markov basis on $I$*** if $$\mathrm{Wr}(f_1, \dots, f_i)\, \text{ is nonzero on $I$ for $i = 1, \dots, d$}\,.$$ If $V$ has a Markov basis on $I$ then $V$ is disconjugate on $I$, and the converse holds if $I$ is open or compact. (This follows from work of Markov [@markoff04 Section 1], Hartman [@hartman69 Proposition 3.1], and Zielke [@zielke79 Theorem 23.3]; see [@karp Section 4.1] for further discussion and background.) Recall that the disconjugacy conjecture ([Theorem 9](#thm:disconj){reference-type="ref" reference="thm:disconj"}) asserts that if $\mathrm{Wr}(V)$ has only real zeros, then $V$ is disconjugate on every interval $I\subseteq\mathbb{R}$ which avoids the zeros of $\mathrm{Wr}(V)$. Without loss of generality, we may assume that $I$ open. Then using the $\mathrm{PGL}_2(\mathbb{R})$-action, it suffices to consider the case $I = (0,\infty)$. The general theory cited above implies that $V$ has a Markov basis on $I$, but it does not explicitly provide us with one. We observe that [Theorem 75](#thm:betabasis){reference-type="ref" reference="thm:betabasis"} provides just such a basis: **Proposition 77**. *Let $V \in \mathcal{X}^{\nu}\subseteq \mathrm{Gr}(d,m)$ such that all the zeros of $\mathrm{Wr}(V)$ are real and nonpositive. Then the basis $(f_1, \dots, f_d)$ for $V$ defined in [[([\[eq:betabasis\]](#eq:betabasis){reference-type="ref" reference="eq:betabasis"})]{.upright}](#eq:betabasis) is a Markov basis on $(0,\infty)$.* *Proof.* We adopt the notation of [Theorem 75](#thm:betabasis){reference-type="ref" reference="thm:betabasis"}. For $i\in [d]$, let $V_i \in \mathrm{Gr}(i,j_i)$ be the span of $(f_1, \dots, f_i)$. Then by construction, we have $$\Delta_I(V_i) = \Delta_{I \cup \{j_{i+1}, \dots, j_d\}}(V) \qquad \text{for all } I\in\textstyle\binom{[j_i]}{i}\,.$$ Since $V$ is totally nonnegative (by [Theorem 10](#thm:positive){reference-type="ref" reference="thm:positive"}[\[positive1\]](#positive1){reference-type="ref" reference="positive1"}), so too is $V_i$. Therefore $\mathrm{Wr}(V_i)$ has nonnegative coefficients by [[([\[eq:pluckerwronskian\]](#eq:pluckerwronskian){reference-type="ref" reference="eq:pluckerwronskian"})]{.upright}](#eq:pluckerwronskian), and hence it is nonzero on $(0,\infty)$. ◻ The proof of [Proposition 77](#prop:disconjbasis){reference-type="ref" reference="prop:disconjbasis"} appears to establish a stronger property of the basis $(f_1, \dots, f_d)$ than claimed: not only is $\mathrm{Wr}(f_1, \dots, f_i)$ nonzero on $(0,\infty)$, it has nonnegative coefficients. In fact, the existence of such a basis is guaranteed by the results of [@karp]. However, we do not know whether we can find a basis with the even stronger property that $\mathrm{Wr}(f_1, \dots, f_i)$ has only real roots: **Problem 78**. *Let $V \subseteq \mathbb{R}[u]$ be a finite-dimensional vector space of polynomials such that all the zeros of $\mathrm{Wr}(V)$ are real and contained in the interval $I \subseteq \mathbb{R}$. Does there exist a basis $(f_1, \dots, f_d)$ for $V$ such that for $1 \le i \le d$, all the zeros of $\mathrm{Wr}(f_1, \dots, f_i)$ are real and contained in $I$?* ### A basis independent of the Schubert cell {#sec:basisindependent} We now give a second basis for $V_E \in \mathrm{Gr}(d,m)$, which does not depend on the ambient Schubert cell $\mathcal{X}^{\nu}$. For $t \in \mathbb{C}$, consider the polynomial $$h_t(u) := \sum_{k=d}^{m} \beta^{k-d}(-t) \otimes e_{k,t}(u) = \sum_{k=d}^{m} \beta^{k-d}(-t) \otimes \frac{(u+t)^{k-1}}{(k-1)!}$$ in $\mathcal{B}_n(z_1, \dots, z_n)\otimes \mathbb{C}_{m-1}[u]$, for $e_{i,t}(u)$ as in [[([\[eq:ei\]](#eq:ei){reference-type="ref" reference="eq:ei"})]{.upright}](#eq:ei). The coefficients $\beta^{k}(-t)$ are indexed by single-row partitions. Similarly, for an eigenspace $E \subseteq M^{\nu}$ of $\mathcal{B}_\nu(z_1, \dots, z_n)$, let $$\label{eq:generalbasisdef} h_{t,E}(u) := \sum_{k=d}^{m} \beta_E^{k-d}(-t) \frac{(u+t)^{k-1}}{(k-1)!} \in \mathbb{C}_{m-1}[u] \,.$$ Viewing $h_t(u)$ as a function of $t$, let $h^{(j)}_{t}(u) := \frac{\partial^j}{\partial t^j} h_{t}(u)$ and $h^{(j)}_{t,E}(u) := \frac{\partial^j}{\partial t^j} h_{t,E}(u)$ denote the $j$th partial derivatives with respect to $t$. **Theorem 79**. *Let $V_E \in \mathrm{Gr}(d,m)$ correspond to an eigenspace $E \subseteq M^{\nu}$ of $\mathcal{B}_\nu(z_1, \dots, z_n)$. Then $V_E$ is spanned by the polynomials $h_{t,E}(u)$ for $t \in \mathbb{C}$. Furthermore, $$\label{eq:generalbasis} \big(h_{t,E}(u), h^{(1)}_{t,E}(u), \dots, h^{(d-1)}_{t,E}(u)\big)$$ is a basis for $V_E$ for every $t \in \mathbb{C}\setminus\{z_1, \dots, z_n\}$.* We consider an example of [Theorem 79](#thm:generalbasis){reference-type="ref" reference="thm:generalbasis"} before giving its proof. **Example 80**. We illustrate [Theorem 79](#thm:generalbasis){reference-type="ref" reference="thm:generalbasis"} in the case $n=2$, for $\mathrm{Gr}(2,4)$. Calculating as in [Example 4](#ex:main){reference-type="ref" reference="ex:main"}, we find $$h_t(u) = (z_1 - t)(z_2 - t) \otimes (u+t) + (z_1 + z_2 - 2t) \otimes \frac{(u+t)^2}{2} + (1 + \sigma_{1,2}) \otimes \frac{(u + t)^3}{6}\,,$$ where we have identified $\mathbbm{1}_{\mathfrak{S}_{2}}$ with $1$ for convenience. The two associated eigenspaces are $E = M^2$ and $E = M^{11}$, whose corresponding elements $V_E \in \mathrm{Gr}(2,4)$ were discussed in [Example 4](#ex:main){reference-type="ref" reference="ex:main"}. [Theorem 79](#thm:generalbasis){reference-type="ref" reference="thm:generalbasis"} asserts that $(h_{t,E}(u), h^{(1)}_{t,E}(u))$ is a basis for $V_E$ for any $t\in\mathbb{C}\setminus\{z_1, z_2\}$. For $E = M^2$, the element $\sigma_{1,2} \in \mathfrak{S}_{2}$ acts as $1$, and so we obtain the following basis for $V_E$: $$\begin{aligned} h_{t,E}(u) &= (z_1 - t)(z_2 - t)(u+t) + \frac{z_1 + z_2 - 2t}{2}(u+t)^2 + \frac{1}{3}(u + t)^3\,, \\ h^{(1)}_{t,E}(u) &= \frac{\partial}{\partial t}h_{t,E}(u) = (z_1 - t)(z_2 - t)\,.\end{aligned}$$ Similarly, for $E = M^{11}$ we obtain the following basis for $V_E$: $$\begin{aligned} h_{t,E}(u) &= (z_1 - t)(z_2 - t)(u+t) + \frac{z_1 + z_2 - 2t}{2}(u+t)^2 \,, \\ h^{(1)}_{t,E}(u) &= \frac{\partial}{\partial t}h_{t,E}(u) = (z_1 - t)(z_2 - t) - (u+t)^2\,.\end{aligned}$$ We now prove [Theorem 79](#thm:generalbasis){reference-type="ref" reference="thm:generalbasis"}. The polynomials $h^{(j)}_t(u)$ satisfy a translation property: sending $(z_1, \dots, z_n) \mapsto (z_1-t, \dots, z_n-t)$ and $u\mapsto u+t$ together takes $h^{(j)}_s(u)$ to $h^{(j)}_{s+t}(u)$. Hence it will suffice to work with $h^{(j)}_t(u)$ when $t=0$. Also note that $$\label{eq:inlinearspan} h^{(j)}_t(u) \in \langle h_s(u) \mid s \in \mathbb{C}\rangle \qquad \text{for all $j\ge 0$ and $t\in\mathbb{C}$}\,.$$ **Lemma 81**. *Let $z_1, \dots, z_n \in\mathbb{C}\setminus\{0\}$. Then for $0 \le j \le d-1$, the term in $h^{(j)}_0(u)$ of minimum $u$-degree is $\beta^0 \otimes \frac{u^{d-j-1}}{(d-j-1)!} = (z_1z_2 \cdots z_n)\mathbbm{1}_{\mathfrak{S}_{n}}\otimes \frac{u^{d-j-1}}{(d-j-1)!}$.* *Proof.* This follows by a direct calculation. ◻ For the element $\Omega$ defined in [[([\[eq:omega\]](#eq:omega){reference-type="ref" reference="eq:omega"})]{.upright}](#eq:omega), we have the following result: **Proposition 82**. *Suppose that $(d,m) = (n, 2n)$. Then $$\Omega \wedge h^{(j)}_t(u) = 0 \qquad \text{ for all $j\ge 0$ and $t \in \mathbb{C}$}\,.$$* *Proof.* It suffices to establish the result when $z_1, \dots, z_n$ are formal indeterminates. First we prove that $\Omega \wedge h_0(u) = 0$. We calculate that $$\label{eq:singlerow} \Omega \wedge h_0(u) = \sum_{J \in {[2n] \choose n}}\,\sum_{k=n}^{2n} e_J \wedge e_k \otimes \beta_J\beta^{k-n} = \sum_{I \in \binom{[2n]}{n+1}}e_I \otimes \sum_{k=n}^{2n} \beta_{I-k}\beta^{k-n}\,,$$ where $\beta_{I-k}$ is defined analogously to [[([\[eq:deletesubscript\]](#eq:deletesubscript){reference-type="ref" reference="eq:deletesubscript"})]{.upright}](#eq:deletesubscript). This equals $0$ if and only if the operators $\beta^\lambda$ satisfy all the single-row Plücker relations. These hold by [Theorem 3](#thm:main){reference-type="ref" reference="thm:main"}[\[main_pluckers\]](#main_pluckers){reference-type="ref" reference="main_pluckers"}. Now for $t\in\mathbb{C}$, apply the change of variables $(z_1, \dots, z_n) \mapsto (z_1-t, \dots, z_n-t)$ and $u\mapsto u+t$. This takes $h_0(u)$ to $h_t(u)$, and $\Omega$ remains unchanged by [Lemma 59](#lem:exteriortranslation){reference-type="ref" reference="lem:exteriortranslation"}. Hence the equation $\Omega \wedge h_0(u) = 0$ implies $\Omega \wedge h_t(u) = 0$. By [[([\[eq:inlinearspan\]](#eq:inlinearspan){reference-type="ref" reference="eq:inlinearspan"})]{.upright}](#eq:inlinearspan), we deduce that $\Omega \wedge h^{(j)}_t(u) = 0$ for all $j\ge 0$. ◻ **Corollary 83**. *Let $V_E \in \mathrm{Gr}(d,m)$ correspond to an eigenspace $E \subseteq M^{\nu}$ of $\mathcal{B}_\nu(z_1, \dots, z_n)$. Then $h^{(j)}_{t,E}(u) \in V_E$ for all $j\ge 0$ and $t\in\mathbb{C}$.* *Proof.* In the case that $(d,m) = (n,2n)$, this follows from [Theorem 3](#thm:main){reference-type="ref" reference="thm:main"}[\[main_eigenspace\]](#main_eigenspace){reference-type="ref" reference="main_eigenspace"} and [Proposition 82](#prop:omegawedge){reference-type="ref" reference="prop:omegawedge"}. Now we explain why the statement does not depend on the choice of $(d,m)$ such that $\nu \subseteq {\mathchoice% {\scalebox{1.35}{\raisebox{-0.02em}{$\sqsubset\!\!\sqsupset$}}} {\scalebox{1.35}{\raisebox{-0.02em}{$\sqsubset\!\!\sqsupset$}}} {\sqsubset\!\!\sqsupset} {\sqsubset\!\!\sqsupset} }= (m-d)^d$, whence the general result follows. Since $\nu_1 \le m-d$, by [Proposition 12](#prop:betapsd){reference-type="ref" reference="prop:betapsd"}[\[betapsd4\]](#betapsd4){reference-type="ref" reference="betapsd4"} we have $\beta^k_E = 0$ for all $k > m-d$. Then we see from [[([\[eq:generalbasisdef\]](#eq:generalbasisdef){reference-type="ref" reference="eq:generalbasisdef"})]{.upright}](#eq:generalbasisdef) that $h^{(j)}_{t,E}(u)$ does not depend on $m$. Similarly, sending $d \mapsto d-1$ takes $h^{(j)}_{t,E}(u) \mapsto \partial_uh^{(j)}_{t,E}(u)$, and sending $d\mapsto d+1$ takes $h^{(j)}_{t,E}(u) \mapsto \int_{-t}^uh^{(j)}_{t,E}(s)ds$. Therefore by [Proposition 24](#prop:changegr){reference-type="ref" reference="prop:changegr"}, the statement does not depend on $d$. ◻ *Proof of [Theorem 79](#thm:generalbasis){reference-type="ref" reference="thm:generalbasis"}.* First we show that [[([\[eq:generalbasis\]](#eq:generalbasis){reference-type="ref" reference="eq:generalbasis"})]{.upright}](#eq:generalbasis) is a basis for $V_E$ for every $t \in \mathbb{C}\setminus\{z_1, \dots, z_n\}$. By translation, it suffices to consider the case when $t=0$ (and $z_1, \dots, z_n \neq 0$). By [Lemma 81](#lem:leadingterm){reference-type="ref" reference="lem:leadingterm"}, for all $0 \le j \le d-1$ we have $$\label{eq:leadingterm} h^{(j)}_{0,E}(u) = z_1z_2 \cdots z_n \frac{u^{d-j-1}}{(d-j-1)!} +\, \text{higher-degree terms}\,.$$ Since $z_1z_2 \cdots z_n \neq 0$, we deduce that [[([\[eq:generalbasis\]](#eq:generalbasis){reference-type="ref" reference="eq:generalbasis"})]{.upright}](#eq:generalbasis) is linearly independent. By [Corollary 83](#cor:contains){reference-type="ref" reference="cor:contains"}, the set [[([\[eq:generalbasis\]](#eq:generalbasis){reference-type="ref" reference="eq:generalbasis"})]{.upright}](#eq:generalbasis) is contained in $V_E$, so it is a basis for $V_E$. Now let $W \subseteq \mathbb{C}_{m-1}[u]$ be the subspace spanned by the polynomials $h_{t,E}(u)$ for $t\in\mathbb{C}$. It remains to show that $W = V_E$. By [Corollary 83](#cor:contains){reference-type="ref" reference="cor:contains"}, we have $W \subseteq V_E$. Conversely, by [[([\[eq:inlinearspan\]](#eq:inlinearspan){reference-type="ref" reference="eq:inlinearspan"})]{.upright}](#eq:inlinearspan), we have $h^{(j)}_{t,E}(u) \in W$ for all $j\ge 0$ and $t\in\mathbb{C}$. Since [[([\[eq:generalbasis\]](#eq:generalbasis){reference-type="ref" reference="eq:generalbasis"})]{.upright}](#eq:generalbasis) spans $V_E$ for any $t \in \mathbb{C}\setminus\{z_1, \dots, z_n\}$, we get $V_E \subseteq W$. ◻ **Remark 84**. In [Theorem 79](#thm:generalbasis){reference-type="ref" reference="thm:generalbasis"}, the assumption $t\neq z_1, \dots, z_n$ is necessary in order for [[([\[eq:generalbasis\]](#eq:generalbasis){reference-type="ref" reference="eq:generalbasis"})]{.upright}](#eq:generalbasis) to be a basis when $d\ge 2$. Indeed, [Theorem 79](#thm:generalbasis){reference-type="ref" reference="thm:generalbasis"} implies that $$\mathrm{Wr}\big(h_{t,E}(u), h^{(1)}_{t,E}(u), \dots, h^{(d-1)}_{t,E}(u)\big) = c(t) \mathrm{Wr}(V_E) \qquad \text{for some } c(t)\in\mathbb{C}\,.$$ Evaluating at $u = -t$ and using [[([\[eq:leadingterm\]](#eq:leadingterm){reference-type="ref" reference="eq:leadingterm"})]{.upright}](#eq:leadingterm), we find $c(t) = (-1)^{\binom{d}{2}}(z_1-t)^{d-1} \cdots (z_n-t)^{d-1}$. If $t\in\{z_1, \dots, z_n\}$ then the Wronskian of [[([\[eq:generalbasis\]](#eq:generalbasis){reference-type="ref" reference="eq:generalbasis"})]{.upright}](#eq:generalbasis) is zero, and so [[([\[eq:generalbasis\]](#eq:generalbasis){reference-type="ref" reference="eq:generalbasis"})]{.upright}](#eq:generalbasis) is linearly dependent. ### Dual proof of [Theorem 3](#thm:main){reference-type="ref" reference="thm:main"}[\[main_pluckers\]](#main_pluckers){reference-type="ref" reference="main_pluckers"} {#dual-proof-of-thmmainmain_pluckers} An alternative approach to proving [Theorem 3](#thm:main){reference-type="ref" reference="thm:main"}[\[main_pluckers\]](#main_pluckers){reference-type="ref" reference="main_pluckers"} takes [Theorem 79](#thm:generalbasis){reference-type="ref" reference="thm:generalbasis"} as the definition of $V_E$. The argument is essentially dual to the proof in [4](#sec:pr){reference-type="ref" reference="sec:pr"}, in the sense of [5.3.1](#sec:duality){reference-type="ref" reference="sec:duality"}. We give a brief sketch of the main ideas. We work in $\mathrm{Gr}(n,2n)$. Whereas the proof in [4](#sec:pr){reference-type="ref" reference="sec:pr"} is based on the identity $(\overline{\mathcal{D}}_n)_\# \Omega = 0$, we instead proceed by showing that $\Omega \wedge h_t(u) = 0$ for all $t \in \mathbb{C}$. As in [[([\[eq:singlerow\]](#eq:singlerow){reference-type="ref" reference="eq:singlerow"})]{.upright}](#eq:singlerow), this is equivalent to the operators $\beta^\lambda$ satisfying all single-row Plücker relations. These hold by an argument very similar to our proof of the single-column Plücker relations in [4.1](#sec:part1proof){reference-type="ref" reference="sec:part1proof"}. We deduce that $\Omega_E \wedge h_{t, E}(u) = 0$ for every eigenspace $E$. We now define $V_E \subseteq \mathbb{C}_{2n-1}[u]$ to be the span of the polynomials $h_{t,E}(u)$ for $t \in \mathbb{C}$. As in the proof of [Theorem 79](#thm:generalbasis){reference-type="ref" reference="thm:generalbasis"}, we can argue that $\dim V_E \geq n$. Then standard properties of the exterior algebra imply that $\dim V_E = n$ and $\Omega_E = v_1 \wedge \dots \wedge v_n$, where $(v_1, \dots, v_n)$ is a basis for $V_E$. Hence by [Proposition 17](#prop:exteriorplucker){reference-type="ref" reference="prop:exteriorplucker"}, the coefficients of $\Omega_E$ satisfy the Plücker relations. The remainder of the proof is identical to the one given in [4.2.3](#sec:screduction){reference-type="ref" reference="sec:screduction"}. However, using this alternate definition of $V_E$ creates some challenges for the proof of [Theorem 3](#thm:main){reference-type="ref" reference="thm:main"}[\[main_generates\]](#main_generates){reference-type="ref" reference="main_generates"}, because it does not establish a direct connection with the defining generators $\beta^{1^{\smash{k}}}(t)$ of $\mathcal{B}_n(z_1, \dots, z_n)$. (Of course, [Theorem 79](#thm:generalbasis){reference-type="ref" reference="thm:generalbasis"} ensures that the two definitions of $V_E$ give the same point of $\mathrm{Gr}(n,2n)$.) This can be overcome using the results of [@purbhoo Section 7]. Alternatively, we can show that $h_t(u)$ is in the kernel of the operator $\overline{\mathcal{D}}_n$ from [[([\[eq:polyDdef\]](#eq:polyDdef){reference-type="ref" reference="eq:polyDdef"})]{.upright}](#eq:polyDdef), as follows. By translation, it suffices to prove this when $t = 0$: $$\sum_{0 \le k,\ell \le n} (-1)^k \beta^{1^{\smash{k}}}(u) \beta^{\ell} e_{k+\ell}(u) = 0\,.$$ This turns out to be precisely the equation obtained by taking the coefficient of $e_{[n+1]}$ in $\Omega \wedge h_t(u) = 0$, and then setting $t=u$ and translating by $u$. ## Bethe algebras under geometric transformations {#sec:transformations} In this section, we discuss two examples of natural geometric transformations, and how they manifest in the Bethe algebras $\mathcal{B}_n(z_1, \dots, z_n)$ and $\mathcal{B}_\nu(z_1, \dots, z_n)$. ### Grassmann duality {#sec:duality} The Grassmannians $\mathrm{Gr}(d,m)$ and $\mathrm{Gr}(m-d,m)$ are dual to each other. We explain how to set up this duality to be compatible with the Wronski map and translation; cf. [@karp Section 2.4] (which uses less natural conventions) and [@purbhoo Section 7]. Define the non-degenerate bilinear pairing $(\cdot,\cdot)$ on $\mathbb{C}^m$ by $$(a,b) := \sum_{j=1}^m (-1)^{j-1}a_jb_{m+1-j}\,.$$ Given $V\in\mathrm{Gr}(d,m)$, its ***dual*** is the subspace $$V^* := \{a\in\mathbb{C}^m : (a,b) = 0 \text{ for all } b\in V\} \in \mathrm{Gr}(m-d,m)\,.$$ Then $V\mapsto V^*$ defines an isomorphism $\mathrm{Gr}(d,m) \to \mathrm{Gr}(m-d,m)$. Also, given a partition $\lambda$, we let $\lambda^*$ denote its ***conjugate***, whose diagram is the transpose of that of $\lambda$. **Proposition 85**. *Let $V\in\mathrm{Gr}(d,m)$.* (i) *[\[duality_pluckers\]]{#duality_pluckers label="duality_pluckers"} Taking duals preserves Plücker coordinates: $\Delta^\lambda(V) = \Delta^{\lambda^*}(V^*)$ for all partitions $\lambda$.* (ii) *[\[duality_wronskian\]]{#duality_wronskian label="duality_wronskian"} Taking duals preserves Wronskians: $\mathrm{Wr}(V) = \mathrm{Wr}(V^*)$.* (iii) *[\[duality_translation\]]{#duality_translation label="duality_translation"} Taking duals commutes with translation: $V(t)^* = V^*(t)$ for all $t\in\mathbb{C}$.* (iv) *[\[duality_schubert\]]{#duality_schubert label="duality_schubert"} Taking duals acts on Schubert varieties: $(X_\lambda(w))^* = X_{\lambda^*}(w)$ and $(\smash{\overline{\mathcal{X}}}^{\nu})^* = \smash{\overline{\mathcal{X}}}^{\nu^*}$.* *Proof.* Part [\[duality_pluckers\]](#duality_pluckers){reference-type="ref" reference="duality_pluckers"} follows from [@karp17 Lemma 1.11(ii)]. Then parts [\[duality_wronskian\]](#duality_wronskian){reference-type="ref" reference="duality_wronskian"} and [\[duality_translation\]](#duality_translation){reference-type="ref" reference="duality_translation"} follow from [[([\[eq:pluckerwronskian\]](#eq:pluckerwronskian){reference-type="ref" reference="eq:pluckerwronskian"})]{.upright}](#eq:pluckerwronskian) and [[([\[eq:geometrictranslation\]](#eq:geometrictranslation){reference-type="ref" reference="eq:geometrictranslation"})]{.upright}](#eq:geometrictranslation), respectively, using the fact that $\mathsf{f}^{\lambda/\mu} = \mathsf{f}^{\lambda^*/\mu^*}$ for all $\mu \subseteq \lambda$. For part [\[duality_schubert\]](#duality_schubert){reference-type="ref" reference="duality_schubert"}, by translation, it suffices to prove the first equality when $w=0$; this case follows from [Remark 20](#rmk:0schubertconditions){reference-type="ref" reference="rmk:0schubertconditions"} and [Proposition 85](#prop:duality){reference-type="ref" reference="prop:duality"}[\[duality_pluckers\]](#duality_pluckers){reference-type="ref" reference="duality_pluckers"}. The second equality then follows by taking $w\to\infty$ with $\lambda = \nu^\vee$. ◻ This notion of duality is also compatible with the Bethe algebra. Namely, we have an involutive $\mathbb{C}$-algebra automorphism $\star : \mathbb{C}[\mathfrak{S}_{n}] \to \mathbb{C}[\mathfrak{S}_{n}]$ given by $$\star\sigma = \mathrm{sgn}(\sigma)\sigma \qquad \text{for all } \sigma\in\mathfrak{S}_{n}\,.$$ As shown in [@purbhoo Section 7], the involution $\star$ restricts to an automorphism of the Bethe algebra $\mathcal{B}_n(z_1, \dots, z_n)$. Indeed, we have the following result: **Proposition 86**. *The map $\star$ restricted to $\mathcal{B}_n(z_1, \dots, z_n)$ is an involutive algebra automorphism, sending $\beta^\lambda(t) \mapsto \beta^{\lambda^*}(t)$ for all partitions $\lambda$.* *Proof.* We only need to check that $\star\beta^\lambda(t) = \beta^{\lambda^*}(t)$ for every partition $\lambda$. This follows from the fact that $\mathrm{sgn}(\sigma)\chi^\lambda(\sigma) = \chi^{\lambda^*}(\sigma)$ for all $\sigma\in\mathfrak{S}_{n}$. ◻ ### $\mathrm{PGL}_2(\mathbb{C})$-invariance {#sec:invariance} We have previously seen, in [Theorem 42](#thm:bethebasics){reference-type="ref" reference="thm:bethebasics"}[\[bethebasics4\]](#bethebasics4){reference-type="ref" reference="bethebasics4"} and [Lemma 46](#lem:comparealgebras){reference-type="ref" reference="lem:comparealgebras"}[\[comparealgebras4\]](#comparealgebras4){reference-type="ref" reference="comparealgebras4"}, that the Bethe algebra $\mathcal{B}_n(z_1, \dots, z_n)$ is translation invariant, i.e., $\mathcal{B}_n(z_1, \dots, z_n)= \mathcal{B}_n(z_1+t, \dots, z_n+t)$ for all $t \in \mathbb{C}$. Also, since $\beta^\lambda$ is homogeneous of degree $n-|\lambda|$ in the parameters $z_1, \dots, z_n$, it is clear that $\mathcal{B}_n(z_1, \dots, z_n)$ is scaling invariant, i.e., $\mathcal{B}_n(z_1, \dots, z_n)= \mathcal{B}_{n}(s z_1, \dots, s z_n)$ for all $s \neq 0$. These invariance identities respect the direct product decomposition [[([\[eq:directproductbethe\]](#eq:directproductbethe){reference-type="ref" reference="eq:directproductbethe"})]{.upright}](#eq:directproductbethe), i.e., we also have $\mathcal{B}_\nu(z_1, \dots, z_n)= \mathcal{B}_\nu(z_1+t, \dots, z_n+t)$ and $\mathcal{B}_\nu(z_1, \dots, z_n)= \mathcal{B}_{\nu}(s z_1, \dots, s z_n)$. The two types of invariance above correspond to the translation and scaling actions on the Schubert variety $\smash{\overline{\mathcal{X}}}^{\nu}$. If $E \subseteq M^{\nu}$ is an eigenspace of $\mathcal{B}_\nu(z_1, \dots, z_n)$, then $(\beta^\lambda_E(t) : \lambda \subseteq \nu)$ are the normalized Plücker coordinates of $V_E(t) = (\begin{smallmatrix} 1 & t \\ 0 & 1 \end{smallmatrix}) V_E$, and $(s^{n-|\lambda|}\beta^\lambda_E : \lambda \subseteq \nu)$ are the normalized Plücker coordinates of $\{f(s^{-1}u) \mid f(u) \in V_E\} = (\begin{smallmatrix} s & 0 \\ 0 & 1 \end{smallmatrix}) V_E$. In the case where $\nu = {\mathchoice% {\scalebox{1.35}{\raisebox{-0.02em}{$\sqsubset\!\!\sqsupset$}}} {\scalebox{1.35}{\raisebox{-0.02em}{$\sqsubset\!\!\sqsupset$}}} {\sqsubset\!\!\sqsupset} {\sqsubset\!\!\sqsupset} }= (m-d)^d$ is a rectangle, we have the larger group $\mathrm{PGL}_2(\mathbb{C})$ acting on $\smash{\overline{\mathcal{X}}}^{{\mathchoice% {\scalebox{1.35}{\raisebox{-0.02em}{$\sqsubset\!\!\sqsupset$}}} {\scalebox{1.35}{\raisebox{-0.02em}{$\sqsubset\!\!\sqsupset$}}} {\sqsubset\!\!\sqsupset} {\sqsubset\!\!\sqsupset} }}= \mathrm{Gr}(d,m)$. This suggests that when $\nu$ is a rectangle, there may be a more general invariance statement for $\mathcal{B}_\nu(z_1, \dots, z_n)$ corresponding to the $\mathrm{PGL}_2(\mathbb{C})$-action. We now show that this is the case. We first consider the element $(\begin{smallmatrix} 0 & 1 \\ 1 & 0 \end{smallmatrix})\in \mathrm{PGL}_2(\mathbb{C})$, which acts on $\mathbb{P}^1$ by inversion: $w \mapsto w^{-1}$. Let $\beta^{\lambda, \mathrm{inv}}$ denote the element $\beta^\lambda$ with parameters $z_1, \dots, z_n \neq 0$ replaced by their inverses $z_1^{-1}, \dots, z_n^{-1}$. Hence the elements $\beta^{\lambda, \mathrm{inv}}$ are generators of $\mathcal{B}_{n}(z_1^{-1}, \dots, z_n^{-1})$. **Theorem 87**. *Suppose $\nu = {\mathchoice% {\scalebox{1.35}{\raisebox{-0.02em}{$\sqsubset\!\!\sqsupset$}}} {\scalebox{1.35}{\raisebox{-0.02em}{$\sqsubset\!\!\sqsupset$}}} {\sqsubset\!\!\sqsupset} {\sqsubset\!\!\sqsupset} }$ is a rectangle, and $z_1, \dots, z_n \in \mathbb{C}\setminus \{0\}$.* (i) *[\[inversioninvariance1\]]{#inversioninvariance1 label="inversioninvariance1"} For every partition $\lambda \subseteq \nu$, we have $$\frac{\mathsf{f}^{\lambda}}{|\lambda|!} \cdot \beta^{\lambda, \mathrm{inv}}_\nu = z_1^{-1} \dotsm z_n^{-1} \cdot \frac{\mathsf{f}^{\lambda^\vee}}{|\lambda^\vee|!} \cdot \beta^{\lambda^\vee}_\nu \,,$$ where $\lambda^\vee$ denotes the complement of $\lambda$ in the rectangle $\nu$.* (ii) *[\[inversioninvariance2\]]{#inversioninvariance2 label="inversioninvariance2"} If $E \subseteq M^{\nu}$ is an eigenspace of $\mathcal{B}_\nu(z_1, \dots, z_n)$, then $(\beta^{\lambda, \mathrm{inv}}_E : \lambda \subseteq \nu)$ are the normalized Plücker coordinates of $(\begin{smallmatrix} 0 & 1 \\ 1 & 0 \end{smallmatrix})V_E$.* (iii) *[\[inversioninvariance3\]]{#inversioninvariance3 label="inversioninvariance3"} $\mathcal{B}_\nu(z_1, \dots, z_n)= \mathcal{B}_{\nu}(z_1^{-1}, \dots, z_n^{-1})$ .* *Proof.* Let $X \subseteq [n]$ and $Y := [n] \setminus X$. As in the proof of [Proposition 12](#prop:betapsd){reference-type="ref" reference="prop:betapsd"}, let $\alpha^\lambda_{X,\nu}$ denote the operator $\alpha^\lambda_X$ acting on $M^{\nu}$. We claim that as representations of $\mathfrak{S}_{X} \times \mathfrak{S}_{Y}$, we have the decomposition $$\label{eq:rectangledecomposition} M^{\nu}\simeq \bigoplus_{\lambda \vdash |X|} M^{\lambda}_X \otimes M^{\lambda^\vee}_Y \,.$$ Indeed, by Frobenius reciprocity, [[([\[eq:rectangledecomposition\]](#eq:rectangledecomposition){reference-type="ref" reference="eq:rectangledecomposition"})]{.upright}](#eq:rectangledecomposition) is equivalent to the fact that the *Littlewood--Richardson coefficient* $c^{\nu}_{\lambda,\mu}$ (cf. [@stanley99 Section 7.15]) equals $1$ if $\mu = \lambda^\vee$, and $0$ otherwise. This follows from, e.g., [@fulton97 (9.11)]. Therefore, by [Proposition 29](#prop:projection){reference-type="ref" reference="prop:projection"}, $\frac{\mathsf{f}^{\lambda}}{|\lambda|!} \alpha^\lambda_{X,\nu}$ and $\frac{\mathsf{f}^{\lambda^\vee}}{|\lambda^\vee|!} \alpha^{\lambda^\vee}_{Y,\nu}$ are both equal to the orthogonal projection onto the summand $M^{\lambda}_X \otimes M^{\lambda^\vee}_Y$. In particular, we have $$\frac{\mathsf{f}^{\lambda}}{|\lambda|!} \cdot \alpha^\lambda_{X,\nu} = \frac{\mathsf{f}^{\lambda^\vee}}{|\lambda^\vee|!} \cdot \alpha^{\lambda^\vee}_{[n] \setminus X,\nu} \qquad\text{for all $X \subseteq [n]$}\,.$$ Part [\[inversioninvariance1\]](#inversioninvariance1){reference-type="ref" reference="inversioninvariance1"} now follows directly from [[([\[eq:alphatobeta\]](#eq:alphatobeta){reference-type="ref" reference="eq:alphatobeta"})]{.upright}](#eq:alphatobeta). If $V \in \mathrm{Gr}(d,m)$ has Plücker coordinates $[\Delta^\lambda : \lambda \subseteq {\mathchoice% {\scalebox{1.35}{\raisebox{-0.02em}{$\sqsubset\!\!\sqsupset$}}} {\scalebox{1.35}{\raisebox{-0.02em}{$\sqsubset\!\!\sqsupset$}}} {\sqsubset\!\!\sqsupset} {\sqsubset\!\!\sqsupset} }]$, then a direct calculation using [[([\[eq:hookformula\]](#eq:hookformula){reference-type="ref" reference="eq:hookformula"})]{.upright}](#eq:hookformula) shows that $\big[\frac{|\lambda|!}{\mathsf{f}^{\lambda}}\cdot \frac{\mathsf{f}^{\lambda^\vee}}{|\lambda^\vee|!} \cdot \Delta^{\lambda^\vee} : \lambda \subseteq {\mathchoice% {\scalebox{1.35}{\raisebox{-0.02em}{$\sqsubset\!\!\sqsupset$}}} {\scalebox{1.35}{\raisebox{-0.02em}{$\sqsubset\!\!\sqsupset$}}} {\sqsubset\!\!\sqsupset} {\sqsubset\!\!\sqsupset} }\big]$ are the Plücker coordinates of $(\begin{smallmatrix} 0 & 1 \\ 1 & 0 \end{smallmatrix})V \in \mathrm{Gr}(d,m)$. Part [\[inversioninvariance2\]](#inversioninvariance2){reference-type="ref" reference="inversioninvariance2"} therefore follows from part [\[inversioninvariance1\]](#inversioninvariance1){reference-type="ref" reference="inversioninvariance1"} and [Corollary 64](#cor:identifyscell){reference-type="ref" reference="cor:identifyscell"}. Finally, part [\[inversioninvariance3\]](#inversioninvariance3){reference-type="ref" reference="inversioninvariance3"} follows from part [\[inversioninvariance1\]](#inversioninvariance1){reference-type="ref" reference="inversioninvariance1"} and [Theorem 3](#thm:main){reference-type="ref" reference="thm:main"}[\[main_generates\]](#main_generates){reference-type="ref" reference="main_generates"}. ◻ As a corollary, we obtain the following more general $\mathrm{PGL}_2(\mathbb{C})$-invariance result: **Corollary 88**. *Suppose $\nu = {\mathchoice% {\scalebox{1.35}{\raisebox{-0.02em}{$\sqsubset\!\!\sqsupset$}}} {\scalebox{1.35}{\raisebox{-0.02em}{$\sqsubset\!\!\sqsupset$}}} {\sqsubset\!\!\sqsupset} {\sqsubset\!\!\sqsupset} }$ is a rectangle, $z_1, \dots, z_n \in \mathbb{C}$, and $\phi \in \mathrm{PGL}_2(\mathbb{C})$ is such that $\phi(z_i) \neq \infty$ for $i=1, \dots, n$. Then $$\mathcal{B}_\nu(z_1, \dots, z_n)= \mathcal{B}_{\nu}(\phi(z_1), \dots, \phi(z_n)) \,.$$* *Proof.* This follows from the preceding invariance statements, and the fact that the matrices $(\begin{smallmatrix} 1 & t \\ 0 & 1 \end{smallmatrix})$, $(\begin{smallmatrix} s & 0 \\ 0 & 1 \end{smallmatrix})$, and $(\begin{smallmatrix} 0 & 1 \\ 1 & 0 \end{smallmatrix})$ generate $\mathrm{PGL}_2(\mathbb{C})$. ◻ ## Combinatorics of commutativity {#sec:commcomb} For $\theta \in \mathfrak{S}_{n}$ and $Z \subseteq [n]$, recall from [4.1.2](#sec:zfac){reference-type="ref" reference="sec:zfac"} that a $Z$-factorization of $\theta$ is a pair of supported permutations $(\sigma_X, \pi_Y)$ such that $X\cup Y = [n]$, $X \cap Y = Z$, and $\sigma \pi = \theta$. Hence a $Z$-factorization of $\theta$ is a factorization of $\theta$, together with some additional data relating to fixed points of the factors. The commutativity relation $\varepsilon^\lambda \varepsilon^\mu = \varepsilon^\mu \varepsilon^\lambda$ can be reformulated as a non-trivial combinatorial identity: **Theorem 89**. *For $\theta \in \mathfrak{S}_{n}$, $Z \subseteq [n]$, and partitions $\lambda, \mu$, let $\mathrm{Fac}_{\theta,Z}(\lambda,\mu)$ denote the set of $Z$-factorizations $(\sigma_X, \pi_Y)$ of $\theta$ such that $\mathsf{cyc}(\sigma_X) = \lambda$ and $\mathsf{cyc}(\pi_Y) = \mu$. Then $$\label{eq:Zfcyc} \#\mathrm{Fac}_{\theta,Z}(\lambda, \mu) = \#\mathrm{Fac}_{\theta,Z}(\mu,\lambda) \,.$$* *Proof.* Compare coefficients of $\theta z_{[n]\setminus Z}$ on both sides of $\varepsilon^\lambda \varepsilon^\mu = \varepsilon^\mu \varepsilon^\lambda$. ◻ A direct proof of [[([\[eq:Zfcyc\]](#eq:Zfcyc){reference-type="ref" reference="eq:Zfcyc"})]{.upright}](#eq:Zfcyc) would imply the commutativity relations [[([\[eq:commutativity\]](#eq:commutativity){reference-type="ref" reference="eq:commutativity"})]{.upright}](#eq:commutativity), without going through the reduction in [3.3.1](#sec:creduction){reference-type="ref" reference="sec:creduction"}. We pose this as an open problem: **Problem 90**. *Give a bijective proof of [[([\[eq:Zfcyc\]](#eq:Zfcyc){reference-type="ref" reference="eq:Zfcyc"})]{.upright}](#eq:Zfcyc).* The identity [[([\[eq:reducedcommutativity\]](#eq:reducedcommutativity){reference-type="ref" reference="eq:reducedcommutativity"})]{.upright}](#eq:reducedcommutativity) is much easier to prove bijectively than [[([\[eq:Zfcyc\]](#eq:Zfcyc){reference-type="ref" reference="eq:Zfcyc"})]{.upright}](#eq:Zfcyc), because we only have to keep track of the cycle type of one of the two factors. ## A $\tau$-function with coefficients in $\mathbb{C}[\mathfrak{S}_{\infty}]$ {#sec:tauinfinity} Since $[n-1] \subseteq [n]$, we have natural inclusions $$\mathfrak{S}_{1} \hookrightarrow \mathfrak{S}_{2} \hookrightarrow \mathfrak{S}_{3} \hookrightarrow \cdots \,.$$ The group $\mathfrak{S}_{\infty}$ is defined to be the direct limit of this sequence of inclusions. Equivalently, $\mathfrak{S}_{\infty}$ is the subgroup of permutations of $\{1,2,3, \dots\}$ that fix all but finitely many positive integers. We say that $\sigma_X$ is a ***supported permutation of $\mathfrak{S}_{\infty}$*** if $X$ is a *finite* subset of $\{1,2,3, \dots\}$, and $\sigma \in \mathfrak{S}_{X}$. Hence $\mathrm{SP}_\infty := \bigcup_{n=0}^\infty \mathrm{SP}_n$ is the set of all supported permutations of $\mathfrak{S}_{\infty}$. Equation [[([\[eq:KP\]](#eq:KP){reference-type="ref" reference="eq:KP"})]{.upright}](#eq:KP) is unaffected by rescaling $\tau$ by a constant. If $z_1, \dots, z_n \in \mathbb{C}\setminus \{0\}$, we can rescale the $\tau$-function $\tau_n$ from [[([\[eq:tau\]](#eq:tau){reference-type="ref" reference="eq:tau"})]{.upright}](#eq:tau) by $z_1^{-1} \dotsm z_n^{-1}$: $$z_1^{-1} \dotsm z_n^{-1} \tau_n = \sum_{\sigma_X \in \mathrm{SP}_n} \sigma \, z_{X}^{-1} \otimes \mathsf{p}_{\mathsf{cyc}(\sigma_X)}.$$ This is still a $\tau$-function of the KP hierarchy, which extends formally to $n=\infty$: **Theorem 91**. *Let $z_1, z_2, \dots$ be formal indeterminates. The series $$\tau_\infty := \sum_{\sigma_X \in \mathrm{SP}_\infty} \sigma \, z_{X}^{-1} \otimes \mathsf{p}_{\mathsf{cyc}(\sigma_X)}$$ is a $\tau$-function of the KP hierarchy, with coefficients in (a commutative subalgebra of) $\mathbb{C}[\mathfrak{S}_{\infty}][[z_1^{-1}, z_2^{-1}, \dots]]$.* *Proof.* As $z_1, z_2, \dots$ are formal, the result is true if and only if it is true whenever all but finitely many $z_i^{-1}$ are set to zero. Hence the result is equivalent to [Theorem 52](#thm:taufunction){reference-type="ref" reference="thm:taufunction"}. ◻ Recall from [Theorem 32](#thm:KPplucker){reference-type="ref" reference="thm:KPplucker"} that if $\tau\in\Lambda$ is a $\tau$-function of the KP hierarchy, then it defines a point in $\mathrm{Gr}(d,m)$ whenever $d$ and $m-d$ are sufficiently large. This may no longer hold when $\tau \in \widehat\Lambda$; rather, $\tau$ defines a point in the infinite-dimensional *Sato Grassmannian*. The Wronskian of a point in the Sato Grassmannian does not have a determinantal definition. However, we can define it as a formal series as in [[([\[eq:pluckerwronskian\]](#eq:pluckerwronskian){reference-type="ref" reference="eq:pluckerwronskian"})]{.upright}](#eq:pluckerwronskian), or equivalently, as the exponential specialization $\mathop{\mathrm{ex}}(\tau)$ as in [Remark 34](#rmk:taufunction){reference-type="ref" reference="rmk:taufunction"}. It may be interesting to study $\tau_\infty$ and $\mathop{\mathrm{ex}}(\tau_\infty)$ as analytic objects, and connect them to the inverse Wronski problem for spaces of analytic functions (rather than just polynomials): **Problem 92**. *Find an analytic version of [Theorem 91](#thm:tauinfinity){reference-type="ref" reference="thm:tauinfinity"} when $(z_1, z_2, \dots)$ is an infinite sequence of nonzero complex numbers satisfying an appropriate convergence condition.* ## Higher-degree positivity for Plücker coordinates {#sec:asw} In the case of $\mathrm{Gr}(1,m)$, [Theorem 10](#thm:positive){reference-type="ref" reference="thm:positive"}[\[positive1\]](#positive1){reference-type="ref" reference="positive1"} can be reformulated as the following (rather obvious) statement: if $f\in\mathbb{C}[u]$ is a polynomial of degree at most $m-1$ whose zeros are all are real and nonpositive, then (up to rescaling) $f$ has nonnegative real coefficients. This is a much weaker statement than the following result, which gives a complete characterization of polynomials with nonpositive real zeros: **Theorem 93** (Aissen, Schoenberg, and Whitney [@aissen_schoenberg_whitney52]). *Let $f(u) = \sum_{i=0}^d a_iu^i \in\mathbb{C}[u]$ be a polynomial with at least one nonnegative real coefficient. Then all the zeros of $f$ are real and nonpositive if and only if the infinite Toeplitz matrix $$\label{eq:toeplitz} \begin{pmatrix} a_0 & 0 & 0 & \cdots \\ a_1 & a_0 & 0 & \cdots \\ a_2 & a_1 & a_0 & \cdots \\ \vdots & \vdots & \vdots & \ddots \end{pmatrix}$$ is totally nonnegative (i.e. all its finite minors are real and nonnegative).* Note that nonnegativity of the $1\times 1$ minors of [[([\[eq:toeplitz\]](#eq:toeplitz){reference-type="ref" reference="eq:toeplitz"})]{.upright}](#eq:toeplitz) is equivalent to the sequence of coefficients $(a_0, \dots, a_d)$ being nonnegative, and nonnegativity of the $2\times 2$ minors implies that the sequence is log-concave. We pose the problem of simultaneously generalizing [Theorem 10](#thm:positive){reference-type="ref" reference="thm:positive"}[\[positive1\]](#positive1){reference-type="ref" reference="positive1"} (from the case of linear inequalities to higher-order inequalities) and [Theorem 93](#thm:asw){reference-type="ref" reference="thm:asw"} (from $\mathrm{Gr}(1,m)$ to $\mathrm{Gr}(d,m)$): **Problem 94**. *Give a characterization of the subset of elements $V\in\mathrm{Gr}(d,m)$ such that all zeros of $\mathrm{Wr}(V)$ are real and nonpositive, in terms of polynomial inequalities in the Plücker coordinates.* ## Conjectures in real Schubert calculus {#sec:openconjectures} We briefly discuss the more general forms of the Shapiro--Shapiro conjecture and the secant conjecture, as well as the total reality conjecture for convex curves. ### General form of the Shapiro--Shapiro conjecture {#sec:generalssc} The more general form of the Shapiro--Shapiro conjecture is the following statement: **Theorem 95** (Mukhin, Tarasov, and Varchenko [@mukhin_tarasov_varchenko09b Corollary 6.3]). *Let $\nu \vdash n$ and ${\mathchoice {\mbox{\boldmath{$\mu$}}} {\mbox{\boldmath{$\mu$}}} {\mbox{\scriptsize\boldmath{$\mu$}}} {\mbox{\tiny\boldmath{$\mu$}}} }\mathrel{\,\Vdash}\kappa$, where $\kappa = (\kappa_1, \dots, \kappa_s)$ is a composition of $n$. Also let $z_1, \dots, z_s$ be distinct real numbers. Then the Schubert intersection [[([\[eq:generalssc\]](#eq:generalssc){reference-type="ref" reference="eq:generalssc"})]{.upright}](#eq:generalssc) is real and scheme-theoretically reduced.* The proof below is a reformulation of the argument in [@mukhin_tarasov_varchenko09b]: *Proof.* By [Proposition 12](#prop:betapsd){reference-type="ref" reference="prop:betapsd"}[\[betapsd1\]](#betapsd1){reference-type="ref" reference="betapsd1"}, the operators $\beta^\lambda_\nu \in \mathcal{B}_{\nu}(\mathbf{z}_\kappa)$ are self-adjoint, and hence the eigenvalues of $\beta^\lambda_{\nu,{\mathchoice {\mbox{\boldmath{$\mu$}}} {\mbox{\boldmath{$\mu$}}} {\mbox{\scriptsize\boldmath{$\mu$}}} {\mbox{\tiny\boldmath{$\mu$}}} }}$ are real. This implies that the points of the intersection [[([\[eq:generalssc\]](#eq:generalssc){reference-type="ref" reference="eq:generalssc"})]{.upright}](#eq:generalssc) are real. By [Theorem 71](#thm:nondistinct){reference-type="ref" reference="thm:nondistinct"}, the intersection is scheme-theoretically reduced if and only if $\mathcal{B}_{\nu,{\mathchoice {\mbox{\boldmath{$\mu$}}} {\mbox{\boldmath{$\mu$}}} {\mbox{\scriptsize\boldmath{$\mu$}}} {\mbox{\tiny\boldmath{$\mu$}}} }}(\mathbf{z}_\kappa)$ is semisimple. This again follows from the fact that the generators are self-adjoint. ◻ The reality statement can also be deduced from [Theorem 7](#thm:ssc){reference-type="ref" reference="thm:ssc"}, since the Schubert intersection [[([\[eq:generalssc\]](#eq:generalssc){reference-type="ref" reference="eq:generalssc"})]{.upright}](#eq:generalssc) is contained in the fibre of the Wronskian [[([\[eq:fibre\]](#eq:fibre){reference-type="ref" reference="eq:fibre"})]{.upright}](#eq:fibre), which is a limit of fibres $\mathrm{Wr}^{-1}(g)$ where $g$ has distinct real roots. The reducedness, however, is more subtle, and does not readily follow by limiting arguments. We also obtain the following generalization of [Theorem 10](#thm:positive){reference-type="ref" reference="thm:positive"}: **Corollary 96** (Positive Shapiro--Shapiro conjecture). *Let $z_1, \dots, z_s \in \mathbb{C}$ be distinct.* (i) *[\[generalpositivenu1\]]{#generalpositivenu1 label="generalpositivenu1"} If $z_1, \dots, z_s \in [0,\infty)$, then all points of the Schubert intersection [[([\[eq:generalssc\]](#eq:generalssc){reference-type="ref" reference="eq:generalssc"})]{.upright}](#eq:generalssc) are totally nonnegative.* (ii) *[\[generalpositivenu2\]]{#generalpositivenu2 label="generalpositivenu2"} If $z_1, \dots, z_s \in (0,\infty)$, then all points of the Schubert intersection [[([\[eq:generalssc\]](#eq:generalssc){reference-type="ref" reference="eq:generalssc"})]{.upright}](#eq:generalssc) are totally positive in $\mathcal{X}^{\nu}$.* *Proof.* This follows from [[([\[eq:fibre\]](#eq:fibre){reference-type="ref" reference="eq:fibre"})]{.upright}](#eq:fibre) and the proof of [Theorem 10](#thm:positive){reference-type="ref" reference="thm:positive"} in [1.3.5](#sec:conjectureproofs){reference-type="ref" reference="sec:conjectureproofs"}. ◻ ### General form of the secant conjecture {#sec:generalsecant} Similarly, there is the general form of the secant conjecture, as formulated by Sottile around 2003 (see [1.3.2](#sec:secantconjecture){reference-type="ref" reference="sec:secantconjecture"} for a discussion of the history). For an interval $I \subseteq \mathbb{R}$, we say that a complete flag $F_\bullet : F_0 \subsetneq \dots \subsetneq F_m$ in $\mathbb{C}^m$ is a ***generalized secant flag*** to the moment curve $\gamma$ along $I$ if each subspace $F_i$ is a generalized secant to $\gamma$ along $I$. **Conjecture 97** (Secant conjecture, general form). *Let $\nu \vdash n$ and ${\mathchoice {\mbox{\boldmath{$\mu$}}} {\mbox{\boldmath{$\mu$}}} {\mbox{\scriptsize\boldmath{$\mu$}}} {\mbox{\tiny\boldmath{$\mu$}}} }\mathrel{\,\Vdash}\kappa$, where $\kappa = (\kappa_1, \dots, \kappa_s)$ is a composition of $n$. Let $I_1, \dots, I_s \subseteq \mathbb{R}$ be pairwise disjoint real intervals. If $F^{(1)}_\bullet, \dots , F^{(s)}_\bullet$ are generalized secant flags to $\gamma$ along $I_1, \dots , I_s$, respectively, then the Schubert intersection [[([\[eq:generalschubertintersection\]](#eq:generalschubertintersection){reference-type="ref" reference="eq:generalschubertintersection"})]{.upright}](#eq:generalschubertintersection) is real and scheme-theoretically reduced.* [Theorem 8](#thm:secant){reference-type="ref" reference="thm:secant"} addresses the divisor case of [Conjecture 97](#conj:generalsecant){reference-type="ref" reference="conj:generalsecant"}, i.e., where $\mu_i = 1$ for all $i$ in [[([\[eq:generalschubertintersection\]](#eq:generalschubertintersection){reference-type="ref" reference="eq:generalschubertintersection"})]{.upright}](#eq:generalschubertintersection). (We mention that the proof can be generalized in a straightforward way to handle the case where for each $i$, we have $\mu_i = 1$ or $F^{(i)}_\bullet$ is an osculating flag.) Unfortunately, in general, neither the reality nor the reducedness statements of [Conjecture 97](#conj:generalsecant){reference-type="ref" reference="conj:generalsecant"} follow readily from [Theorem 8](#thm:secant){reference-type="ref" reference="thm:secant"}. In particular, the Schubert intersections described in [Conjecture 97](#conj:generalsecant){reference-type="ref" reference="conj:generalsecant"} cannot be realized as limits of the Schubert intersections in [Theorem 8](#thm:secant){reference-type="ref" reference="thm:secant"}, since the intervals $I_i$ are not allowed to overlap. Also, [Conjecture 97](#conj:generalsecant){reference-type="ref" reference="conj:generalsecant"} does not appear to follow easily from Theorems [Theorem 9](#thm:disconj){reference-type="ref" reference="thm:disconj"} or [Theorem 10](#thm:positive){reference-type="ref" reference="thm:positive"}. For now, therefore, the general form of the secant conjecture remains open. ### Total reality conjecture for convex curves {#sec:totalreality} Recall that the Shapiro--Shapiro conjecture ([Theorem 7](#thm:ssc){reference-type="ref" reference="thm:ssc"}) asserts that the solutions to the Schubert problem [[([\[eq:schubertproblem\]](#eq:schubertproblem){reference-type="ref" reference="eq:schubertproblem"})]{.upright}](#eq:schubertproblem) are all real when each $W_i\in\mathrm{Gr}(m-d,m)$ is an osculating plane to the moment curve $\gamma$ at a real point. There is a generalization of this conjecture due to Boris Shapiro in the 1990's (cf. [@sedykh_shapiro05 Section 1]), called the ***total reality conjecture for convex curves***. Namely, a real continuous curve $\gamma : \mathbb{R}\to \mathbb{R}_{m-1}[u]$ is called ***convex*** if any choice of $m$ distinct points along $\gamma$ are linearly independent. By continuity, this is equivalent to the condition that for all $r\ge m$ and $w_1 < \cdots < w_r$ in $\mathbb{R}$, $$\begin{pmatrix} \vline & & \vline \\ \gamma(w_1) & \cdots & \gamma(w_r) \\ \vline & & \vline \end{pmatrix} \;\text{ represents a totally positive element of } \mathrm{Gr}(m,r)\,,$$ i.e., its $m\times m$ minors all have the same (nonzero) sign. The fact that the moment curve is convex follows from Vandermonde's formula. **Conjecture 98** (Total reality conjecture for convex curves). *Let $\gamma : \mathbb{R}\to \mathbb{R}_{m-1}[u]$ be a convex curve, and let $z_1, \dots, z_{d(m-d)}$ be distinct real numbers. For $i = 1, \dots, d(m-d)$, let $W_i\in\mathrm{Gr}(m-d,d)$ be the osculating $(m-d)$-plane to $\gamma$ at $z_i$. Then there are exactly $\mathsf{f}^{{\mathchoice% {\scalebox{1.35}{\raisebox{-0.02em}{$\sqsubset\!\!\sqsupset$}}} {\scalebox{1.35}{\raisebox{-0.02em}{$\sqsubset\!\!\sqsupset$}}} {\sqsubset\!\!\sqsupset} {\sqsubset\!\!\sqsupset} }}$ distinct solutions to the Schubert problem [[([\[eq:schubertproblem\]](#eq:schubertproblem){reference-type="ref" reference="eq:schubertproblem"})]{.upright}](#eq:schubertproblem) (with $\nu = {\mathchoice% {\scalebox{1.35}{\raisebox{-0.02em}{$\sqsubset\!\!\sqsupset$}}} {\scalebox{1.35}{\raisebox{-0.02em}{$\sqsubset\!\!\sqsupset$}}} {\sqsubset\!\!\sqsupset} {\sqsubset\!\!\sqsupset} }$), and all solutions are real.* In analogy with the Shapiro--Shapiro conjecture, there are several natural extensions of [Conjecture 98](#conj:totalreality){reference-type="ref" reference="conj:totalreality"}, such as to an arbitrary Schubert cell $\mathcal{X}^{\nu}$, to the case that $W_1, \dots, W_n$ are are generalized secants to $\gamma$ along disjoint real intervals, and to Schubert conditions of arbitrary codimension. [Conjecture 98](#conj:totalreality){reference-type="ref" reference="conj:totalreality"} was long thought to be false, due to a counterexample which was only recently found to be erroneous [@sedykh_shapiro05; @shapiro_shapiro22]. It is true for $\mathrm{Gr}(2,4)$, which follows from work of Arkani-Hamed, Lam, and Spradlin on scattering amplitudes [@arkani-hamed_lam_spradlin21 Section 4]; cf. [@shapiro_shapiro22 Theorem 2]. It is not immediately clear how to apply our techniques to address [Conjecture 98](#conj:totalreality){reference-type="ref" reference="conj:totalreality"}, because the connection between the Schubert problem [[([\[eq:schubertproblem\]](#eq:schubertproblem){reference-type="ref" reference="eq:schubertproblem"})]{.upright}](#eq:schubertproblem) and the Wronski map is particular to the case that $\gamma$ is the moment curve. It would be interesting to explain the role played by total positivity. Namely, in the total reality conjecture, positivity is used to define the Schubert conditions; and in [\[thm:positive,thm:positivesecant\]](#thm:positive,thm:positivesecant){reference-type="ref" reference="thm:positive,thm:positivesecant"}, positivity is a property of the solutions of the Schubert problem. The fact that positivity appears in both places appears for now to be a coincidence.

arxiv_math

--- abstract: | We propose a new method to compare survival data based on Higher Criticism (HC) of P-values obtained from many exact hypergeometric tests. The method can accommodate censorship and is sensitive to moderate differences in some unknown and relatively few time intervals, attaining much better power against such differences than the log-rank test and other tests that are popular under non-proportional hazard alternatives. We demonstrate the usefulness of the HC-based test in detecting rare differences compared to existing tests using simulated data and using actual gene expression data. Additionally, we analyze the asymptotic power of our method under a piece-wise homogeneous exponential decay model with rare and weak departures, describing two groups experiencing failure rates that are usually identical over time except in a few unknown instances in which the second group's failure rate is higher. Under an asymptotic calibration of the model's parameters, the HC-based test's power experiences a phase transition across the plane involving the rarity and intensity parameters that mirrors the phase transition in a two-sample rare and weak normal means setting. In particular, the phase transition curve of our test indicates a larger region in which it is fully powered than the corresponding region of the log-rank test. bibliography: - survival.bib title: Detecting rare and weak deviations of non-proportional hazard in survival analysis --- \[To do: - We should review the concept of non-homogeneous hazards and tools, and explain that these tools are ineffective when the locations of the differences are unknown to us. Perhaps implement the test from [@yang2010improved] and [@zhang2021cauchycp]. In particular, [@yang2010improved] describes an adaptive test, reviews an interesting model from [@yang2005semiparametric], and provides a nice literature summary. - Perhaps add a figure sketch of the type of hazard difference we envision. Compare with the scenarios of [@yang2010improved]. - Change significance level to $0.01$. - Add a section about randomizing the P-values (for global testing, not for HCT). Randomizing the test leads to (1) fuzzy decisions or (2) to rejection when the number of rejections is significantly larger than the test's size. Another option (3) is to record the random P-value of HC and then average the result. This approach uses the exact distribution of HC under the null. Lastly, we can (4) randomize both null and alternative and then use a two-sample test -- this approach is less powerful because we should have the exact distribution of the null. \] \[ - Explain that we target situations of multiple ties; the method still applies to individual events but the asymptotic analysis does not hold. - This situation occurs when we pool time instances to search for effects localized in time (excess risk over a certain interval. eg. excessive risk for the first 30 days, first 5 years, etc., but we do not know in advance where excessive risk arises). - Example datasets: marriage survival which only gives upper and lower intervals for divorce data, as occurred [@lillard2003aml]. - Additional datasets are available at [@drysdale2022]: `[’hdfail’, ’UnempDur’, ’Aids2’, ’Z243’, ’support2’, ’vlbw’, ’prostate’, ’acath’, ’actg’, ’Rossi’, ’Unemployment’, ’grace’]` - Also explain the meaning of the Poisson number of events in this case: uniformly scattered across the interval. \] # Introduction {#sec:intro} ## Survival data with rare hazard deviations Suppose that we have survival measurements from two groups, say, the Control Group $x$ and the Treatment Group $y$. We want to determine whether the treatment has a significant effect on survival in the sense that the global difference between groups' failure rates goes beyond expected fluctuations. In general, this is a topic studied for many decades with plenty of scientific and industrial applications [@kiefer1988economic; @armitage2008statistical; @kalbfleisch2011statistical]. One notable tool to compare survival data is the log-rank (LR) test introduced by Mantel in [@mantel1966evaluation], which can accommodate arbitrary right-censorship in the data and is asymptotically equivalent to the likelihood ratio test under the Cox proportional hazard risk model [@peto1972asymptotically; @kalbfleisch2011statistical]. Nevertheless, as we explain below, existing tools such as the LR test are typically ineffective in situations where differences in hazard between the groups are rare (sparse). Namely, in situations where the signal separating the two groups corresponds to a few time intervals experiencing excessive or reduced failure rates while those intervals are unknown to us before collecting the data. The goal of this paper is to develop a tool that can reliably detect such temporarily rare and weak hazard differences in survival data, that can rigorously handle right-censored data [@prentice1978linear], and that is very direct to apply. Figure [\[fig:data_intro\]](#fig:data_intro){reference-type="ref" reference="fig:data_intro"} illustrates an example of survival data with the kind of rare excessive hazard signal we target that is detected by our new method but not by the log-rank test. As a by-product, our tool can indicate those time intervals suspected of experiencing increased or decreased hazard, indicated in Figure [\[fig:data_intro\]](#fig:data_intro){reference-type="ref" reference="fig:data_intro"} by the gray bars. Several examples in which our method can be particularly useful compared to existing methods are listed below. - Discovering age-specific effects. Suppose that some genetic property may cause disease in certain ages of an organism, but we do not know in advance at what ages the effect will occur. Therefore, to decide whether the effect is significant, we look for differences in the rate of developing the disease across all ages. For example, this situation seems relevant to the study of life span quantitative trait loci in Drosophila melanogaster [@10.1534/genetics.104.038331]. - Comparing the rate of decay of radioactive materials. The decay rate of some radioactive materials appears to fluctuate in time due to several possible causes such as "space weather" [@milian2020fluctuations]. In order to identify specific causes, we may compare the survival curve of the material in the exposed environment to that of the same material in a controlled environment. Due to the potential burstiness of space weather, if any effect exists it may manifest through rare and relatively weak fluctuations in the decay rate of the exposed material. Our test is designed to detect such fluctuations. - Identifying possibly opposing temporarily localized hazard trends. As we explain below, our method is also naturally suited to detect non-proportional hazard situations in which hazard differences between the two groups may be positive in some time intervals but negative in others (and potentially zero in most). For example, the study [@johansson2015family] suggests that the effect of pregnancy on breast cancer risk is positive in the short term and negative in the long term. Since the opposing trends may cancel each other, it is challenging to detect a global hazard difference using methods based on averaging like the LR test and its generalizations. Our method is particularly useful compared to other methods in these situations when the direction and the location of the differences are unknown in advance, so in the example above the concepts \"short\" and \"long\" can be objectively determined from the data. The situation described here appears to occur also in the studies [@tsodikov2002semi; @sasaki2005temporal; @yang2005semiparametric; @dekker2008survival; @daniels2017examining; @gregson2019nonproportional] and in effects of the type "what doesn't kill you makes you stronger" [@stenton2022effects; @mathew2011autophagy; @Dongjuan2017]. ## Existing methods and rare effects The most popular methods for discovering differences or excessive risk in survival data are arguably based on averaging some observed quantities across all event times as in the LR test mentioned earlier and its non-homogenous weighted versions [@mantel1966evaluation; @peto1972asymptotically; @gill1980censoring; @harrington1982class; @pepe1991weighted; @galili2021stability]. Therefore, it may be the case that intervals of excessive or reduced risk occur in the data but these are so rare that they go undetected by these averaging-based approaches -- even if the effect's direction is coherence in all non-null intervals which is the typical situation that we address here. More specifically, weighted versions of the log-rank test can be useful against non-proportional hazard alternatives only when the hazard pattern is known in advance; see the additional discussion and references in [@yang2010improved] and [@chauvel2014tests]. Selecting the weights from the data may improve the power against non-proportional hazard alternatives under certain hazard difference functions models [@yang2010improved; @chauvel2014tests], but such adaptive selection necessarily results in some loss of power when this function cannot be estimated reliably as in our case of very sparse and weak differences; see a related discussion in the context of the Gaussian sequence model in [@jin2003detecting]. If an excessive or reduced hazard occurred in a certain interval, it may still be individually *weak* so that the global effect is generally undetectable in a Bonferroni analysis involving the significance of individual time intervals [@jin2016rare]. For the same reasons, methods based on the maximum of several standardized tests are also ineffective in general (again, unless the hazard pattern is prescribed) [@breslow1984two; @self1991adaptive; @fleming1987supremum; @fleming2013counting]. In contrast, we propose in this work to combine signals from individual time intervals using Higher Criticism, which is reputed to be effective in detecting rare and individually weak effects [@donoho2015special]. ## Setting We have two series of positive integers $\{n_x(t)\}_{t=0}^T$ and $\{n_y(t)\}_{t=0}^T$ of equal length, describing the number of subjects at risk at times $t=1,\ldots,T$ in groups $x$ and $y$, respectively. We are also given the sequences $\{o_x(t)\}_{t=1}^T$, and $\{o_y(t)\}_{t=1}^T$, describing the number of events occurring in each group over time, so that $$o_x(t) \leq n_x(t-1) - n_x(t), \qquad o_y(t) \leq n_y(t-1) - n_y(t),\qquad t=1,\ldots,T,$$ with equality only if none of the subjects within the corresponding group were censored between time $t-1$ and $t$. We assume throughout that censorship times are uninformative as in standard log-rank analysis. Denote by $c_{\star}(t)$ the number of censored subjects up to time $t$ in group $\star \in \{x,y\}$. The survival proportion (aka estimated survival probability) at time $t$ is $$\hat{s}_{\star}(t) := \frac{n_{\star}(t)}{n_{\star}(0)- c_{\star}(t)},\qquad t=1,\ldots,T.$$ The Kaplan-Meir survival curve associated with the group ${\star}$ is the graph $$\begin{aligned} \label{eq:Kaplan_Meir} \left\{ \left(t, \hat{s}_{\star}(t)\right) \right\}_{t=1,\ldots,T}, \qquad{\star} \in \{x,y\}.\end{aligned}$$ This curve describes the proportion of at-risk subjects at time $t$ in Group ${\star}$ with censored subjects removed. Figure [\[fig:data_intro\]](#fig:data_intro){reference-type="ref" reference="fig:data_intro"} shows an example of synthetic survival data of two groups and their corresponding Kaplan-Meier curves. ## Method description {#sec:method} We now describe our statistical test for comparing the survival of Group $x$ and Group $y$. Our test uses the Higher Criticism of P-values obtained from many exact hypergeometric tests as per the explanation below. ### Hypergeometric P-values and Survival Analysis The hypergeometric distribution $\mathrm{HyG}(M, N, n)$ has the probability mass function $$\begin{aligned} \label{eq:hyg_pmf} \Pr\left(\mathrm{HyG}(M, N, n)=k \right) = \frac{\binom{N}{k}\binom{M-N}{n-k}}{\binom{M}{n}},\end{aligned}$$ describing the probability of observing $k$ type-$A$ items in a random sample of $n$ items without replacements from a population of size $M$, initially containing $N$ type-$A$ items. Given an observed value $m \in \mathbb N$, the one-sided P-value of the exact hypergeometric test is $$p_{\mathrm{HyG}}(m; M, N, n) := \Pr\left( \mathrm{HyG}(M, N, n) \geq m \right) = \sum_{k=m}^{n} \frac{\binom{N}{k}\binom{M-N}{n-k}}{\binom{M}{n}}.$$ Back to survival analysis. For every $t=1,\ldots,T$, we evaluate: $$\begin{aligned} \label{eq:pvals_def} & p_t := p_{\mathrm{HyG}}( m_t; M_t, N_t, n_t),\end{aligned}$$ with $$m_t = o_y(t), \qquad M_t = n_x(t-1) + n_y(t-1),\quad N_t = n_y(t-1),\quad n_t = o_x(t) + o_y(t).$$ In words, $p_t$ is a P-value under the model proposing that the number of failure events $o_y(t)$ observed in Group $y$ at time $t$ is obtained by sampling without replacements $o_x(t) + o_y(t)$ subjects from a pool of $n_x(t-1) + n_y(t-1)$ subjects, out of which $n_y(t-1)$ subjects are 'at-risk' in Group $y$. The hypergeometric P-value $p_t$ is small if $o_y(t)$ is much larger than the expected number of such events under this model. An example of synthetic survival data and its corresponding hypergeometric P-values are given in Figure [\[fig:data_intro\]](#fig:data_intro){reference-type="ref" reference="fig:data_intro"}. ### Higher Criticism In this work, we combine the P-values $p_1,\ldots,p_T$ using the HC statistic [@donoho2004higher; @donoho2008higher]. Specifically, set $$\begin{aligned} \mathrm{HC}_{i;T}(p_1,\ldots,p_T) := \sqrt{T} \frac{i/T - p_{(i)}}{\sqrt{p_{(i)}(1-p_{(i)})}},\quad i=1,\ldots,T,\end{aligned}$$ where $p_{(i)}\leq \ldots \leq p_{(T)}$ are the ordered P-values observed in the data. The Higher Criticism statistic of Hyper Geometric P-values (HCHG) is defined as $$\begin{aligned} \mathrm{HCHG}_T:= \mathrm{HC}\left(p_1,\ldots,p_T;\gamma_0\right) := \max_{1\leq i \leq T\gamma_0} \mathrm{HC}_{i;T}(p_1,\ldots,p_T) \label{eq:HC_def}.\end{aligned}$$ Here $\gamma_0 \in (0,1]$ is a tunable parameter that does not change the large sample properties of $\mathrm{HCHG}_T$ under either hypothesis [@donoho2004higher]; typically, we take $\gamma_0 = 0.3$. Our test rejects the null hypothesis of equal population survival rates for large values of $\mathrm{HCHG}_T$. ### Critical test values {#sec:critical_values} We are interested in characterizing critical values for testing using $\mathrm{HCHG}_T$ at a prescribed significance level $\alpha$. We propose to obtain these values by simulating samples of $\mathrm{HCHG}_T$ under a proposed null model $H_0$. Namely, given a large simulated sample, we consider the empirical $1-\alpha$ quantile as an estimate of the true quantile $$q_0^{1-\alpha}(\mathrm{HCHG}_T) := \inf \{q \,:\, \Pr\left({\mathrm{HCHG}_T\leq q|H_0} \right) \geq 1-\alpha \}.$$ This estimate of $q_0^{1-\alpha}(\mathrm{HCHG}_T)$ serves as the critical value above which we reject the null at level $\alpha$. Our experience shows that a test based on the simulated $q_0^{1-\alpha}(\mathrm{HCHG}_T)$ has much better power than a test that is based on simulating the $1-\alpha$ quantile of $\mathrm{HC}$ of P-values that are uniformly distributed over $(0,1)$. Indeed, because the data is discrete, the distribution under the null of the hypergeometric P-values is in many cases significantly stochastically larger than uniform hence the null values of $\mathrm{HCHG}_T$ can be significantly smaller than those obtained when the P-values follow a uniform distribution. Consequently, the $1-\alpha$ quantile of a sample of HC of uniform P-values is overly conservative for an $\alpha$-level test. The max Brownian bridge distribution to which HC asymptotes also leads to overly conservative critical values due to the same reason and also due to the slow convergence of HC to its asymptotic distribution from below [@donoho2004higher; @gontscharuk2015intermediates; @moscovich2016exact]. We note that a standard decision-theory solution to improve power while controlling the test level when working with discrete data is to randomize the hypergeometric tests so that the P-values are uniform under any null model for the data [@cox1979theoretical p. 101], [@habiger2011randomised; @chen2020false]. However, in the context of large-scale inference, randomizing individual tests introduces additional issues concerning decision-making and interpreting instances of departure [@kulinskaya2009fuzzy]. Therefore, we propose here not to randomize the hypergeometric tests. Instead, in analyzing real data, we simulate the null distribution of $\mathrm{HCHG}_T$ by randomly assigning group membership to subjects, i.e., ignoring the actual group membership associated with biological traits. We discuss this further in Section [\[sec:simulations\]](#sec:simulations){reference-type="ref" reference="sec:simulations"}. ## Effect direction In most applications we are interested in testing whether the risk in Group $y$ (say) is larger than that of Group $x$, hence we use one-sided P-values in [\[eq:pvals_def\]](#eq:pvals_def){reference-type="eqref" reference="eq:pvals_def"}. Replacing the one-sided P-values with two-sided ones may be justified when we are interested in detecting a global effect that can go either way, coherently or incoherently, over time. Even with one-sided P-values, a significant value of $\mathrm{HCHG}_T$ might remain significant after replacing the roles of $x$ and $y$. If this situation occurs, then our data may experience a global effect that changes over time, e.g., excessive hazard in the short term and reduced hazard in the long term. Effects of this type were studied in [@dekker2008survival; @johansson2015family; @gregson2019nonproportional; @daniels2017examining; @stenton2022effects; @mathew2011autophagy; @Dongjuan2017]. Additionally, we are often interested in a *strictly one-sided effect* in which one group experiences an excessive hazard compared to the other group. We declare that "Group $y$ experiences an increased failure rate compared to Group $x$\" only if HCHG rejects when testing against increased mortality in Group $y$ but does not reject when testing against increased mortality in Group $x$. If each HCHG test is of significance level $\alpha$, the combined test clearly rejects at significance level $\alpha$ or smaller. We summarize the general one-sided procedure in Algorithm [\[alg:1\]](#alg:1){reference-type="ref" reference="alg:1"} and its strictly one-sided variant in Algorithm [\[alg:2\]](#alg:2){reference-type="ref" reference="alg:2"} $M_t \leftarrow n_x(t) + n_y(t)$ $n_t \leftarrow o_x(t) + o_y(t)$ $p_t \leftarrow p_{\mathrm{HyG}}\left( o_y(t); M_t, n_y(t), n_t\right)$ $\mathrm{HCHG}_T\leftarrow \mathrm{HC}\left(p_1,\ldots,p_T \right)$ p $\mathcal S_{(x,y)} \leftarrow \{n_x(t), n_y(t), o_x(t), o_y(t) \}_{t=0}^T$ $\mathcal S_{(y,x)} \leftarrow \{n_y(t), n_x(t), o_y(t), o_x(t) \}_{t=0}^T$ ## Identifying instances of departure [\[sec:hct\]]{#sec:hct label="sec:hct"} Our testing procedure targets scenarios in which differences in the risk of one of the groups may be driven by a small number of time intervals. It follows from previous studies of similar multiple-hypothesis testing situations that a set of individual tests thought to provide the best evidence against the null hypothesis is provided by the so-called higher criticism threshold [@donoho2008higher; @donoho2009feature], defined as the index $t^\star$ of the P-value maximizing the higher criticism functional $\mathrm{HC}_{t;T}$. The aforementioned set is defined as $$\begin{aligned} \label{eq:Delta_def} \Delta^\star := \{t \, : p_{t} \leq p_{(t^*)}\},\qquad \mathrm{HC}_{t^*:T} = \mathrm{HCHG}_T\end{aligned}$$ Figure [\[fig:data_intro\]](#fig:data_intro){reference-type="ref" reference="fig:data_intro"} illustrates membership in $\Delta^\star$ using an example survival data set. In our situation of comparing survival curves, $\Delta^*$ contains the smallest $t^\star \geq 1$ hypergeometric P-values; these P-values derive the global difference between the curves. The significance of this identification depends on the context. For instance, time intervals underlying the global change are likely to be instances of truly excessive risk and therefore potentially mark important points for intervention. To summarize, $\mathrm{HCHG}_T$ offers a single procedure for survival analysis that combines global testing with the identification of time intervals suspected of excessive risk selection. The multiple testing approach to survival analysis we promote in this manuscript perhaps suggests additional analysis using other feature selection procedures, e.g., controlling the false discovery rate (FDR) using the Benjamini-Hochberg (BH) procedure [@benjamini1995controlling]. Nevertheless, our theoretical analysis in Section [2](#sec:analysis){reference-type="ref" reference="sec:analysis"} shows that global testing based on BH is asymptotically powerless in some regimes when our method is still asymptotically powerful. Consequently, in these regimes, BH would report zero discoveries (at any false discovery rate parameter) while HCHG would indicate that the two survival functions of the groups are significantly different at power tending to $1$ as the Type I error tends to $0$. We refer to [@donoho2008higher; @donoho2009feature] for additional discussion about the difference between the two methods in a more general context. ## Asymptotic power analysis and phase transition We analyze a test based on HCHG and compare it to other tests using a theoretical framework involving survival data experiencing rare temporal hazard differences. The objective of this analysis is to validate that our method enjoys certain information-theoretic optimality properties under a model that isolates the type of signal we are trying to capture. Such comparisons are common in mathematical statistics and often lead to useful data analysis tools, i.e., they can detect non-null effects even under some violations of the model [@donoho2004higher; @mukherjee2015hypothesis; @arias2015sparse; @jin2016rare; @pilliat2023optimal]. In our framework, the differences in the random mechanism generating the data of both groups are rare and relatively weak. We assume no censorship in our analysis for simplicity, although it is straightforward to modify our framework to simple censorship models in which the censorship distribution is uninformative and conditionally independent of mortality events. Under our framework, the number of at-risk subjects in Group $x$ at time $t$, denoted as $N_x(t)$, follows a homogeneous random exponential decay in which the reduction in at-risk subjects at time $t$ follows a Poisson distribution whose rate is $N_x(t) \bar{\lambda}$. Here $\bar{\lambda}$ is some global baseline rate that indicates the average time between failure events. The number of at-risk subjects in Group $y$ largely follows the same behavior, except in roughly $\epsilon \cdot T$ of apriori unspecified instances; here $\epsilon = T^{-\beta}$ where $\beta \in (0,1)$ is a parameter controlling the *rarity* of the departures. In those instances, the rate of the Poisson distribution is $N_y(t) \bar{\lambda}'$ where $\bar{\lambda}'$ is obtained by perturbing $\bar{\lambda}$ upwards by an amount controlled by the *intensity* parameter $r \geq 0$. Rarity models of this type are quite common in mathematical statistics and provide a theoretical framework for comparing statistical procedures. The usefulness of our method is not limited to survival data obeying this model as we demonstrate in Section [3.2](#sec:sim_gene_expression){reference-type="ref" reference="sec:sim_gene_expression"} using actual survival data associated with gene expression. We calibrate the model's parameters to $T$ such that perturbations appear on the moderate deviation scale, under which $\mathrm{HCHG}_T$ experiences a phase transition in the following sense. There exists a curve $\left\{\left(\beta, \rho(\beta)\right)\right\}_{\beta \in (0,1)}$ that divides the $(r,\beta)$-plane into two regions. For values $r > \rho(\beta)$, our HC-based test is asymptotically powerful in the sense that there exists a sequence of thresholds for a test based on $\mathrm{HCHG}_T$ under which the sum of Type I and Type II errors goes to zero. For values of $r < \rho(\beta)$ the sum of Type I and Type II errors goes to one under any sequence of thresholds for a test based on $\mathrm{HCHG}_T$. The phase transition curve defined by $\rho(\beta)$ turns out to be equal to the phase transition curve of HC in the two-sample normal and Poisson means models under rare and weak perturbations described in [@DonohoKipnis2020]. We also discuss the asymptotic properties of additional testing procedures like a test based on Bonferroni's correction and the false discovery rate procedure. For comparison, our analysis implies that the log-rank test is asymptotically powerless in the entire range of severe rarity $\beta \in (1/2,1)$, whereas HCHG is asymptotically powerful in this range whenever $r > \rho(\beta)$; this situation is illustrated in Figure [\[fig:theoretical_PT\]](#fig:theoretical_PT){reference-type="ref" reference="fig:theoretical_PT"}. In Section [\[sec:simulations\]](#sec:simulations){reference-type="ref" reference="sec:simulations"}, we exemplify our theoretic derivations numerically by illustrating the Monte-Carlo simulated power of these tests over a grid of configurations of $\beta$ and $r$. We note that although the resulting asymptotic power of HCHG and other tests mirrors previously studied rare and weak signal detection settings [@jin2016rare; @kipnis2021logchisquared], our piece-wise exponential decay model is arguably more complicated than any of those earlier-studied cases due to the inherent dependent of time-instances and the decrease in the effect's size over time. Specifically, our analysis relies on a framework for studying rare and weak signal detection models in which individual effects are moderately large recently developed in [@kipnis2021logchisquared] and [@DonohoKipnis2020b]. The major novelty of our analysis is the successful calibration of the model's parameters in a way that the hypergeometric P-values corresponding to non-null hazard departure experience log-chisquared distributions with moderately large non-centrality parameters. ## Structure of the Paper The rest of this paper is organized as follows. In Section [2](#sec:analysis){reference-type="ref" reference="sec:analysis"} we analyze our method in a theoretical framework and compare it to the log-rank test and other multiple testing methods. In Section [\[sec:simulations\]](#sec:simulations){reference-type="ref" reference="sec:simulations"}, we report on empirical results supporting our theoretical findings. Additional discussion is provided in Section [\[sec:discussion\]](#sec:discussion){reference-type="ref" reference="sec:discussion"}. All proofs are provided in the Supplementary Material [@survival2023supp]. # Power Analysis under Piece-wise Exponential Decay with Rare and Weak Departures {#sec:analysis} We introduce a theoretical framework for analyzing the performance of a test based on HCHG and for comparing it to other tests. Let $x_0$ and $y_0$ be two deterministic constants describing the initial groups' sizes. For $t=0,\ldots, T$, denote by $n_{\star}(t)$ the number of at-risk subjects in Group $\star \in \{x,y\}$. Suppose that there are no censored events and that the sequences $\{O_x(t),O_y(t),N_x(t), N_y(t)\}$ obey [\[eq:sampling_model\]]{#eq:sampling_model label="eq:sampling_model"} $$\begin{aligned} N_x(0) = x_0\qquad \text{and} \qquad N_y(0) = y_0,\end{aligned}$$ and for $t=1,\ldots,T$, $$\begin{aligned} \begin{cases} O_x(t) \sim \mathrm{Pois}( N_x(t-1) \bar{\lambda}_x(t) ) \\ N_x(t) = N_x(t-1) - O_x(t) \end{cases} \quad \text{and}\quad \begin{cases} O_y(t) \sim \mathrm{Pois}( N_y(t-1) \bar{\lambda}_y(t)) \\ N_y(t) = N_y(t-1) - O_y(t). \end{cases} \label{eq:data_model}\end{aligned}$$ The model [\[eq:sampling_model\]](#eq:sampling_model){reference-type="eqref" reference="eq:sampling_model"} describes a piece-wise exponential decay in time of the number of at-risk subjects in either group. We consider testing the null hypothesis of a homogeneous exponential decay in both groups [\[eq:hyp\]]{#eq:hyp label="eq:hyp"} $$\begin{aligned} H_0 \, :\, \bar{\lambda}_x(t) = \bar{\lambda}_y(t) = \bar{\lambda}, \qquad \forall t\in \{1,\ldots,T\}, \end{aligned}$$ against a situation in which Group $y$ experiences some instances of excessive risk: $$\begin{aligned} \label{eq:hyp_perturbation} H_1 \,:\, \bar{\lambda}_x(t) =\bar{\lambda} \quad \text{and} \quad \bar{\lambda}_y(t) = \sqrt{\bar{\lambda}} + \begin{cases} \sqrt{\delta(t)} & t \in I \\ 0 & t \notin I. \end{cases}\end{aligned}$$ Here $\delta(t) \geq 0$ controls the excessive risk in the set of non-null instances $I \subset \{1,\ldots,T\}$. To reflect the fact that we do not know apriori which instances are perturbed, we assume that the membership in $I$ is also random: each $t$ is included in $I$ with probability $\epsilon$ independently of the other $t$'s. We can write [\[eq:sampling_model\]](#eq:sampling_model){reference-type="eqref" reference="eq:sampling_model"} and [\[eq:hyp\]](#eq:hyp){reference-type="eqref" reference="eq:hyp"} under the randomness in $I$ as follows. [\[eq:model_full\]]{#eq:model_full label="eq:model_full"} $$\begin{aligned} N_x(0) = x_0 \quad \text{and} \quad N_y(0) = y_0.\end{aligned}$$ $$\begin{aligned} H_0\,:\, \begin{cases} N_x(t-1) - N_x(t) \sim \mathrm{Pois}\left( N_x(t) \bar{\lambda} \right) \\ N_y(t-1) - N_y(t) \sim \mathrm{Pois}\left(N_y(t) \bar{\lambda} \right) \end{cases} \forall t=1,\ldots,T. \end{aligned}$$ $$\begin{aligned} H_1\,& :\, \begin{cases} N_x(t-1) - N_x(t) \sim & \mathrm{Pois}\left( N_x(t-1) \bar{\lambda} \right) \\ N_y(t-1) - N_y(t) \sim & (1-\epsilon) \cdot \mathrm{Pois}\left(N_y(t-1) \bar{\lambda} \right) \\ & \quad + \epsilon \cdot \mathrm{Pois}\left(N_y(t-1) \bar{\lambda}'(t) \right) \end{cases} \forall t=1,\ldots,T, \end{aligned}$$ where $$\begin{aligned} \bar{\lambda}'(t) := \left(\sqrt{\bar{\lambda}} + \sqrt{\delta(t)} \right)^2 \label{eq:lambda_prime_def}\end{aligned}$$ In words, our model says that the number of events in each group at time $t$ is a random variable that follows a Poisson distribution whose expected value is proportional to the group's size at time $t-1$. Under the null hypothesis, there exists a global \"base\" failure rate $\bar{\lambda}$. This global rate governs both groups, $x$ and $y$. Under the alternative, there are roughly $T \cdot \epsilon$ apriori unspecified instances in which the rate of events in Group $y$ is larger than $\bar{\lambda}$ by an amount of $\delta(t)$ in a square root perturbation which is natural in analyzing count data [@simpson1987minimum]. Additional remarks are in order. - The Poisson distribution implies that the time between two failure events in Group $x$ in the interval $(t, t+1]$ follows an exponential distribution with mean $\bar{\lambda} N_x(t)$; we will assume below that this number is relatively large hence the total number of events across any interval is relatively large (e.g., it is rarely 0). - The time between two failure events in Group $y$ in the interval $(t, t+1]$ follows an exponential distribution with mean $\bar{\lambda} N_y(t)$ with probability $1-\epsilon$ and mean $\bar{\lambda}'(t)N_y(t)$ with probability $\epsilon$. We will assume below that $\bar{\lambda}'(t)/\bar{\lambda}$ asymptotes to $1$ as $T\to \infty$, so that the terminal numbers of at-risk subjects $N_x(T)$ and $N_y(T)$ cannot distinguish between $H_0$ and $H_1$. Piece-wise exponential decay models as in [\[eq:sampling_model\]](#eq:sampling_model){reference-type="eqref" reference="eq:sampling_model"} are common in survival analysis [@feigl1965estimation; @friedman1982piecewise], [@rodriguez2007lecture Ch. 7]; the departures model [\[eq:hyp\]](#eq:hyp){reference-type="eqref" reference="eq:hyp"} appears to be new in this context and is analogous to previously-studied high-dimensional heterogeneous detection models involving rare and weak effects [@ingster1996some; @donoho2004higher; @hall2008properties; @delaigle2009higher; @cai2014optimal; @donoho2015special; @mukherjee2015hypothesis; @kipnis2021logchisquared]. As we explain next, a natural calibration of the model's parameters provides a framework for comparing the asymptotic performance of statistical procedures for testing [\[eq:hyp\]](#eq:hyp){reference-type="eqref" reference="eq:hyp"}. We anticipate that a test based on $\mathrm{HCHG}_T$ would experience similar properties under some extensions of [\[eq:model_full\]](#eq:model_full){reference-type="eqref" reference="eq:model_full"} that allow for time-varying risk rate in the control group, i.e. dropping the homogeneous assumption $\bar{\lambda}_x(t) \equiv \bar{\lambda}$. Extensions to more complex scenarios are often possible based on intuition from previously studied rare and weak multiple testing settings. For example, the analysis in [@hall2010innovated] might be useful for handling certain kinds of conditional dependencies in the Poisson samples across time. We leave these extensions as future work. ## Calibration We consider an asymptotic setting in which $T\to \infty$, while $\epsilon$, $\delta$, $x_0$, $y_0$ and $\bar{\lambda}$ are calibrated to $T$. Our calibration is chosen in such a way that the hypergeometric P-values of [\[eq:pvals_def\]](#eq:pvals_def){reference-type="eqref" reference="eq:pvals_def"} correspond to a rare moderate departures model [@kipnis2021logchisquared]. We introduce additional parameters $r$ and $\beta$ to describe this calibration, as per Table [1](#tbl:calibration){reference-type="ref" reference="tbl:calibration"}. parameter description calibrating parameter reference ----------------- ------------------------------------------------------------------- ----------------------- ------------------------------------------------------------------------------------------------------------ $\epsilon$ proportion of non-null effects $\beta \in (0,1)$ [\[eq:calibration_eps\]](#eq:calibration_eps){reference-type="eqref" reference="eq:calibration_eps"} $\delta$ effect size (Hellinger shift in the risk of non-null occurrences) $r \geq 0$ [\[eq:calibration_delta\]](#eq:calibration_delta){reference-type="eqref" reference="eq:calibration_delta"} $\bar{\lambda}$ baseline risk [\[eq:calibration_rates\]](#eq:calibration_rates){reference-type="eqref" reference="eq:calibration_rates"} : Parameters of the theoretical framework of piece-wise homogeneous exponential decay under rare and weak departures. We assume that the initial group sizes $x_0$ and $y_0$ obey [\[eq:calibration\]]{#eq:calibration label="eq:calibration"} $$\begin{aligned} \label{eq:calibrtion_initial} x_0 \to \infty \quad \text{and} \quad y_0 \to \infty, \quad\text{as} \quad T \to \infty,\end{aligned}$$ while they are asymptotically equivalent in the sense that $$\begin{aligned} \label{eq:calibrtion_kappa} \frac{x_0}{x_y} \to 1,\quad \text{as}\quad T \to \infty .\end{aligned}$$ We calibrate the rarity parameter $\epsilon$ to $T$ according to $$\begin{aligned} \label{eq:calibration_eps} \epsilon := \epsilon(T) = T^{-\beta},\quad \beta \in (0,1).\end{aligned}$$ We calibrate the effect size parameter $\delta$ to $T$ according to $$\begin{aligned} \label{eq:calibration_delta} \delta : = \delta(t;T) = \frac{r}{2}\frac{ \log(T)}{n(t)},\qquad n(t) := \frac{x_0 + y_0}{2} e^{-\bar{\lambda} t}.\end{aligned}$$ Note that, as $T\to \infty$ we have that $$2\frac{N_x(t) N_y(t)}{N_x(t) + N_y(t)} \frac{1}{n(t)} \to 1$$ in probability uniformly in $t\leq T$ hence $n(t)$ is approximately the overall number of at-risk subjects at time $t$ in either group. We further assume that $$\begin{aligned} \label{eq:calibration_rates} %\min_{t \leq T} x_0 \lambda_x(t) \to \infty, \quad \text{while}\quad \max_{t \leq T} \lambda_x(t) T/\log(T) \to 0, \qquad \bar{\lambda} x_0 / \log(T) \to \infty \quad \text{while}\quad \bar{\lambda} T \leq M,\end{aligned}$$ for some finite $M$ that is independent of $T$. Hence, the decay rate $\bar{\lambda}$ is asymptotically not too small and not too large. Note that [\[eq:calibration_rates\]](#eq:calibration_rates){reference-type="eqref" reference="eq:calibration_rates"} implies $x_0/(T \log(T)) \to \infty$ and the expected number of failure events in Group $x$ obeys $$\ensuremath{\mathbb{E}\left[ O_x(t)\right]} = \ensuremath{\mathbb{E}\left[ \bar{\lambda} N_x(t-1)\right]} = \bar{\lambda} x_0(1-\bar{\lambda})^t \approx \bar{\lambda}x_0 e^{-\bar{\lambda}t} \geq \bar{\lambda}x_0 e^{-M} \to \infty.$$ Therefore, [\[eq:calibration_rates\]](#eq:calibration_rates){reference-type="eqref" reference="eq:calibration_rates"} implies that $\ensuremath{\mathbb{E}\left[ O_x(t)\right]} \to \infty$ and likewise $\ensuremath{\mathbb{E}\left[ O_y(t)\right]} \to \infty$ at rates faster than $\log(T)$. Furthermore, the expected proportion relative to the initial size of at-risk subjects in each group at time $t$ converges in probability to $e^{-\bar{\lambda} t}$ under [\[eq:calibration\]](#eq:calibration){reference-type="eqref" reference="eq:calibration"} (see the relevant Lemma in the Supplementary Material) Consequently, the temporally local proportion $N_y(t)/(N_x(t) + N_y(t))$ remains roughly constant in $t$ at around $1/2$, and it is generally impossible to tell the difference between the failure rates by considering only the terminal number of at-risk patients. ## Asymptotic power and phase transition A statistic $U(T)$ based on the data $\{N_x(t), N_y(t), O_x(t), O_y(t)\}_{t=0}^T$ under the setting [\[eq:hyp\]](#eq:hyp){reference-type="eqref" reference="eq:hyp"} is said to be asymptotically *powerful if there exists a sequence of thresholds $\{h(T)\}_{T=1,2,\ldots,}$ such that $$\begin{aligned} \Pr\left(U(T) \geq h(T) | H_0 \right) + \Pr\left(U_n < h(T) | H_1 \right) \to 0.\end{aligned}$$ Conversely, $U(T)$ is said to be asymptotically *powerless if $$\begin{aligned} \Pr\left(U(T) \geq h(T) | H_0 \right) + \Pr\left(U_n < h(T) | H_1 \right) \to 1,\end{aligned}$$ for any sequence of threshold $\{h(T)\}_{T=1,2,\ldots,}$. In words, the asymptotic powerfulness of $U(T)$ says that there exists a sequence of tests for [\[eq:hyp\]](#eq:hyp){reference-type="eqref" reference="eq:hyp"} based on $U(T)$ whose powers tend to $1$ while their sizes tend to $0$. Asymptotic powerlessness says that any test based on $U(T)$ asymptotically has trivial power. See reference [@donoho2004higher; @arias2005near; @donoho2015special] for additional discussions of these definitions.** Under the setting [\[eq:model_full\]](#eq:model_full){reference-type="eqref" reference="eq:model_full"} and the calibration [\[eq:calibration\]](#eq:calibration){reference-type="eqref" reference="eq:calibration"}, $\mathrm{HCHG}_T$ and other statistics experience phase transition phenomena in the following sense. There exists a curve $\rho(\beta)$ such that $\mathrm{HCHG}_T$ is asymptotically powerful if $r > \rho(\beta)$ and asymptotically powerless if $r < \rho(\beta)$. ## Asymptotic power of HCHG The following two theorems imply that the phase transition curve of $\mathrm{HCHG}_T$ is given by $$\begin{aligned} \label{eq:rho} \rho(\beta) &:= \begin{cases} %%0 & 0 < \beta \leq \frac{1}{2}, \\ 2(\beta - 1/2) & \frac{1}{2} < \beta < \frac{3}{4}, \\ 2\left(1-\sqrt{1 -\beta }\right)^2 & \frac{3}{4} \leq \beta < 1, \end{cases}\end{aligned}$$ for $\beta \in (1/2,1)$. For $\beta \leq 1/2$, $\mathrm{HCHG}_T$ and all other test statistics we consider are asymptotically powerful for any $r>0$; we focus our discussion on the case $\beta \in (1/2,1)$. The proofs of these theorems are given in the Supplementary Material. **Theorem 1**. *Consider testing $H_0$ versus $H_1$ as in [\[eq:hyp\]](#eq:hyp){reference-type="eqref" reference="eq:hyp"} when $x_0$, $y_0$, $\bar{\lambda}$, $\epsilon$, and $\delta$ are calibrated to $T$ as in [\[eq:calibration\]](#eq:calibration){reference-type="eqref" reference="eq:calibration"}. If $r > \rho(\beta)$, $\mathrm{HCHG}_T$ of [\[eq:HC_def\]](#eq:HC_def){reference-type="eqref" reference="eq:HC_def"} is asymptotically powerful.* **Theorem 2**. *Consider testing $H_0$ versus $H_1$ as in [\[eq:hyp\]](#eq:hyp){reference-type="eqref" reference="eq:hyp"} when $x_0$, $y_0$, $\bar{\lambda}$, $\epsilon$, and $\delta$ are calibrated to $T$ as in [\[eq:calibration\]](#eq:calibration){reference-type="eqref" reference="eq:calibration"}. Let $\tilde{p}_1,\ldots,\tilde{p}_T$ be P-values obtained from randomizing the hypergeometric tests [\[eq:pvals_def\]](#eq:pvals_def){reference-type="eqref" reference="eq:pvals_def"} such that the distribution of $\tilde{p}_t$ is uniform over $(0,1)$ under $H_0$ and has a density $f_t$ under $H_1$ [@cox1979theoretical p. 101]. Furthermore, assume that there exists $C < \infty$ independent of $T$ such that $\|f_t\|_{\infty} < \infty$ for all $t=1,\ldots,T$. If $r < \rho(\beta)$, $\mathrm{HCHG}_T$ of $\tilde{p}_1,\ldots,\tilde{p}_T$ is asymptotically powerless.* ## Asymptotic power of the log-rank test The Log-Rank test is defined as follows. Set $n(t):= n_x(t) + n_y(t)$, $o(t):=o_x(t) + o_y(t)$, and $$\begin{aligned} e_t & := \frac{n_y(t-1)}{n(t-1)} o(t),\\ v_t & :=\frac{n_y(t-1)n_x(t-1)}{n(t-1)-1} \frac{o(t)}{n(t-1)} \left(1 - \frac{o(t)}{n(t-1)} \right). \end{aligned}$$ The log-rank test statistic is $$\begin{aligned} \label{eq:logrank} \mathrm{LR}_T:= \frac{\sum_{t=1}^T o_y(t) - \sum_{t=1}^T e_t}{\sqrt{\sum_{t=1}^T v_t}}\end{aligned}$$ and we reject for large values of $\mathrm{LR}_T$ [@mantel1966evaluation]. **Theorem 3**. *Consider testing $H_0$ versus $H_1$ as in [\[eq:hyp\]](#eq:hyp){reference-type="eqref" reference="eq:hyp"} when $x_0$, $y_0$, $\bar{\lambda}$, $\epsilon$, and $\delta$ are calibrated to $T$ as in [\[eq:calibration\]](#eq:calibration){reference-type="eqref" reference="eq:calibration"}. The log-rank statistic $\mathrm{LR}_T$ of [\[eq:logrank\]](#eq:logrank){reference-type="eqref" reference="eq:logrank"} is asymptotically powerless if $\beta > 1/2$.* We note that the asymptotic behavior of the log-rank statistic is analogous to that of the Fisher combination statistics $$F_T := 2\sum_{t=1}^T \log(1/p_t)$$ in general rare moderate departures models [@kipnis2021logchisquared]. ## Asymptotic power of other multiple testing procedures The hypergeometric P-values of [\[eq:pvals_def\]](#eq:pvals_def){reference-type="eqref" reference="eq:pvals_def"} under the hypothesis testing setting [\[eq:model_full\]](#eq:model_full){reference-type="eqref" reference="eq:model_full"} and the calibration [\[eq:calibration\]](#eq:calibration){reference-type="eqref" reference="eq:calibration"} correspond to a RMD model with log-chisquared noncentrality parameter $\rho = 1/2$ in the sense of [@kipnis2021logchisquared]. Specifically, define $$\alpha(q,\rho):= (\sqrt{q}-\sqrt{\rho})^2.$$ In the Supplementary Material, we show that the P-values of [\[eq:pvals_def\]](#eq:pvals_def){reference-type="eqref" reference="eq:pvals_def"} obey $$\begin{aligned} \label{eq:pvalue_tail} \lim_{T \to 0} \max_{t=1,\ldots T} \left| \frac{-2\log(\Pr\left( -2\log(p_t) \geq 2q\log(T)| t \in I \right))}{\log(T)} - \alpha(q,r/2) \right| = 0,\end{aligned}$$ for $q > r/2 > 0$. The limit in [\[eq:pvalue_tail\]](#eq:pvalue_tail){reference-type="eqref" reference="eq:pvalue_tail"} says that for time instances $t$ in which the risk in Group y is indeed higher (indicated by $t\in I$), the tail of $-2\log(p_t)$ on the moderate deviations scale behaves as the tail of a non-central chisquared random variable over one degree of freedom. Namely $$-2\log(p_t) \overset{D}{\approx} \left(\sqrt{r \log(T)} + Z \right)^2,\qquad Z\sim \mathcal{N}(0,1)$$ with the approximation in the sense of [\[eq:pvalue_tail\]](#eq:pvalue_tail){reference-type="eqref" reference="eq:pvalue_tail"}. Asymptotic characterizations of several multiple testing procedures involving P-values obeying the RMD formulations are available in [@kipnis2021logchisquared]. These include the Bonferroni type test that is based on $p_{(1)}$ and a test that is based on Benjamini-Hochberg's false discovery rate (FDR) functional $$\begin{aligned} \mathrm{FDR}^*(p_1,\ldots,p_T) := \min_{1\leq t \leq T} \frac{p_{(t)}}{t}, \end{aligned}$$ both of which have the phase transition curve $$\begin{aligned} \rho_{\mathsf{Bonf}}(\beta) := 2\left(1 - \sqrt{1 -\beta }\right)^2,\qquad 1/2 < \beta < 1.\end{aligned}$$ Namely, whenever $3/4 < \beta < 1$, a test based either on the minimal P-value or on $\mathrm{FDR}^*$ are asymptotically powerful in the same region in which $\mathrm{HCHG}_T$ is asymptotically powerful. On the other hand, when $1/2< \beta < 3/4$, there exists a region in which $\mathrm{HCHG}_T$ is asymptotically powerful but these other two tests are asymptotically powerless. This situation is analogous to other two-sample multiple testing settings under rare and moderately large departures [@DonohoKipnis2020; @kipnis2021logchisquared]. We summarize the regions in which these tests are asymptotically powerful in Table [2](#tbl:power){reference-type="ref" reference="tbl:power"}. description test statistic region of powerfulness ---------------------- ------------------------------ -------------------------------------------------------- Higher Criticism $\mathrm{HCHG}_T$ $\beta \in (1/2,1)$, $r > \rho(\beta)$ Bonferroni $1 / p_{(1)}$ $\beta \in (1/2,1)$, $r > \rho_{\mathsf{Bonf}}(\beta)$ False discovery rate $\max_{t} \frac{t}{p_{(t)}}$ $\beta \in (1/2,1)$, $r > \rho_{\mathsf{Bonf}}(\beta)$ Log-rank $\mathrm{LR}_T$ $\beta < 1/2$, $r>0$ Fisher's method $2\sum_{t=1}^T \log(1/p_t)$ $\beta< 1/2$, $r>0$ : Summary of the regions of powerfulness of several test statistics under the two-sample homogeneous Poisson decay model with rare and weak departures. Higher criticism has the largest region of powerfulness among the tests in the table. # Empirical Results [\[sec:simulations\]]{#sec:simulations label="sec:simulations"} ## Simulated Data and Empirical Phase Transition We conduct a sequence of Monte-Carlo experiments with $x_0 = y_0 = T \log(T)$ and the baseline rate $\bar{\lambda} = 2/T$ in a manner similar to [@DonohoKipnis2020]. We consider points $(\beta,r)$ in a grid $I_r \times I_\beta$ covering the range $I_r \subset [0,2.1]$, $I_\beta \subset (0.45,1)$. For each test statistic $U$, we first find the $1-\alpha$ empirical quantile of $U$ under the null hypothesis $r=0$ using $N_0 = {100,000}$ Monte-Carlo experiments. We denote this quantile as $\hat{q}^{1-\alpha}(U)$. Next, for each configuration $(\beta,r)$, we conduct $N=1,000$ Monte-Carlo experiments and define the (Monte-Carlo simulated) *power* of $U$ as the fraction of instances in which $U$ exceeds $\hat{q}^{1-\alpha}(U)$. We denote this power by $\hat{B}(U,\alpha,\beta,r)$. We declare that $\hat{B}(U,\alpha,\beta,r)$ is *substantial* if we can reject the hypothesis $$\begin{aligned} H_{0,\alpha}~~:~~ N \cdot \hat{B}(T,\alpha,\beta,r) \sim \mathrm{Binomial}(N,\alpha)\end{aligned}$$ at level $\alpha_1$. Namely, $\hat{B}(T,\alpha,\beta,r)$ is substantial if $$\Pr \left( \mathrm{Binomial}(N,\alpha) \geq N \cdot \hat{B}(T,\alpha,\beta,r) \right) \leq \alpha_1.$$ Next, we fix $\beta \in I_\beta$ and focus on the strip $\{(\beta,r)\}_{r \in I_r}$. We construct the binary-valued vector indicating those values of $r$ for which $\hat{B}(T,\alpha,\beta,r)$ is substantial. To this vector, we fit the logistic response model $$\Pr\left( \hat{B}(T,\alpha,\beta,r) \text{ substantial} \right) = \frac{1}{1+e^{\theta_1(\beta) r + \theta_0(\beta)}}.$$ The phase transition point of the strip $\{(\beta,r),\, r \in I_r\}$ is defined as the point $r=\hat{\rho}(\beta)$ at which $$\frac{1}{1+e^{\theta_1(\beta) r + \theta_0(\beta)}} = \frac{1}{2}.$$ The empirical phase transition curve is defined as $\{ \hat{\rho}(\beta)\}_{\beta \in I_{\beta}}$. The top panels of Figure [3](#fig:sim){reference-type="ref" reference="fig:sim"} illustrate the Monte-Carlo simulated power and the empirical phase transition curves of $\mathrm{HCHG}_T$ and $\mathrm{LR}_T$ along with their theoretical counterparts. The results illustrated in these figures support our theoretical finding in Theorems [Theorem 1](#thm:HC_powerful){reference-type="ref" reference="thm:HC_powerful"} and [Theorem 2](#thm:HC_powerless){reference-type="ref" reference="thm:HC_powerless"}, establishing $\rho(\beta)$ of [\[eq:rho\]](#eq:rho){reference-type="eqref" reference="eq:rho"} as the boundary between the region where HCHG has asymptotically maximal power and the region where it has asymptotically no power. We also show the Monte-Carlo simulated power and empirical phase transition of the log-rank test; the region of powerfulness is smaller than that of $\mathrm{HCHG}_T$, in agreement with our theoretical result in Theorem [Theorem 3](#thm:LR_powerless){reference-type="ref" reference="thm:LR_powerless"}. The empirical phase transitions of some weighted versions of the log-rank test we experimented with are similar but typically inferior to that of the log-rank. These statistics include: Tarone-Ware [@tarone1977distribution], Gehan-Wilcoxon [@gehan1965generalized], and Fleming-Harrington with $(p,q)=(1,0)$ and $(p,q)=(1,1)$ [@harrington1982class]. The bottom panel in Figure [3](#fig:sim){reference-type="ref" reference="fig:sim"} indicates configurations on the grid $I_\beta \times I_r$ in which the empirical power of a test based on $\mathrm{HCHG}_T$ at the level $0.05$ is significantly better or worse than a test based on different statistics. These illustrations support the theoretical results that imply that $HCHG$ is asymptotically more powerful than existing methods for temporarily rare and relatively weak non-proportional hazard deviations. ![Top: Empirical power of Higher Criticism (Left) and log-rank (Right). The curves $\rho(\beta)$ and $\hat{\rho}(\beta)$ are the theoretical and Monte-Carlo simulated phase transitions of $\mathrm{HCHG}_T$, respectively. The line $\beta=0.5$ and the curve $\hat{\rho}_{\mathrm{LR}_T}(\beta)$ are the theoretical and Monte-Carlo simulated phase transition of the log-rank statistics of [\[eq:logrank\]](#eq:logrank){reference-type="eqref" reference="eq:logrank"}, respectively. Bottom: Configurations with significant empirical power differences between a test based on HCHG and tests based on other statistics. We used $N=1,000$ experiments in each $(r,\beta)$ configuration. Each experiment simulates a sample from the piece-wise homogeneous exponential decay model with rare and weak departures over $T=1,000$ time intervals. ](./Figs/sig_power_diff_HC_Log-rank.png "fig:"){#fig:sim} ![Top: Empirical power of Higher Criticism (Left) and log-rank (Right). The curves $\rho(\beta)$ and $\hat{\rho}(\beta)$ are the theoretical and Monte-Carlo simulated phase transitions of $\mathrm{HCHG}_T$, respectively. The line $\beta=0.5$ and the curve $\hat{\rho}_{\mathrm{LR}_T}(\beta)$ are the theoretical and Monte-Carlo simulated phase transition of the log-rank statistics of [\[eq:logrank\]](#eq:logrank){reference-type="eqref" reference="eq:logrank"}, respectively. Bottom: Configurations with significant empirical power differences between a test based on HCHG and tests based on other statistics. We used $N=1,000$ experiments in each $(r,\beta)$ configuration. Each experiment simulates a sample from the piece-wise homogeneous exponential decay model with rare and weak departures over $T=1,000$ time intervals. ](./Figs/sig_power_diff_HC_Tarone-Ware.png "fig:"){#fig:sim} ![Top: Empirical power of Higher Criticism (Left) and log-rank (Right). The curves $\rho(\beta)$ and $\hat{\rho}(\beta)$ are the theoretical and Monte-Carlo simulated phase transitions of $\mathrm{HCHG}_T$, respectively. The line $\beta=0.5$ and the curve $\hat{\rho}_{\mathrm{LR}_T}(\beta)$ are the theoretical and Monte-Carlo simulated phase transition of the log-rank statistics of [\[eq:logrank\]](#eq:logrank){reference-type="eqref" reference="eq:logrank"}, respectively. Bottom: Configurations with significant empirical power differences between a test based on HCHG and tests based on other statistics. We used $N=1,000$ experiments in each $(r,\beta)$ configuration. Each experiment simulates a sample from the piece-wise homogeneous exponential decay model with rare and weak departures over $T=1,000$ time intervals. ](./Figs/sig_power_diff_HC_minP.png "fig:"){#fig:sim} ## Demonstration for gene expression data {#sec:sim_gene_expression} ### Setup The SCANB dataset [@saal2015sweden] records mortality events over time of 3,069  breast cancer patients. It also includes the expression level of 9,259  in each patient. We removed $557$ genes whose response contains repeated values. For each gene $g$ of the remaining ${8,702}$, we divide the patients into two groups: Group $x$ consists of patients whose expression for $g$ is at or below the median value for $g$, and Group $y$ contains all patients whose expression is larger than this median. This process creates a partition of the patients into two groups of roughly equal sizes which we denote by $x_0(g)$ and $y_0(g)$, 8,702  partitions overall. In each partition, we consolidated all events into $T={82}$ intervals of equal length of approximately 28 days each. ### Simulating null distribution Over $N={50,000}$ iterations, we randomly assign half of the patients to Group $x$ and the other half to Group $y$. This assignment leaves the original correspondence between censorship and event times but removes group associations that, in the actual data, are driven by biology (gene expression). We evaluated the $0.95$ empirical quantile of the size-$N$ sample of values of $\mathrm{HCHG}_T$ of [\[eq:HC_def\]](#eq:HC_def){reference-type="eqref" reference="eq:HC_def"}, denoted $\hat{q}_{0}^{0.95}(\mathrm{HCHG}_T)$. The full histogram of $\mathrm{HCHG}_T$ values is provided in Figure [\[fig:histogram\]](#fig:histogram){reference-type="ref" reference="fig:histogram"}. The difference between the simulated null values of $\mathrm{HCHG}_T$ under random group assignments and uniformly sampled P-values, e.g., as reported in [@donoho2004higher], follows from the super uniformity of the hypergeometric P-values. Consequently, the Z-scores in the HC calculation [\[eq:HC_def\]](#eq:HC_def){reference-type="eqref" reference="eq:HC_def"} are biased downwards hence their maximum is also much smaller and may even be negative. ### Testing We applied our testing procedure in Algorithms [\[alg:1\]](#alg:1){reference-type="ref" reference="alg:1"} and [\[alg:2\]](#alg:2){reference-type="ref" reference="alg:2"} to each gene $g$ to check for increased failures in either group. Namely, we report the existence of any effect associated with $g$ if $\mathrm{HCHG}_T(g)$ exceeds $\hat{q}^{0.95}_0(\mathrm{HCHG}_T)$ or if $\mathrm{Rev}[\mathrm{HCHG}_T(g)]$ exceeds $\hat{q}^{0.95}_0(\mathrm{HCHG}_T)$, where $\mathrm{Rev}[\mathrm{HCHG}_T(g)]$ is obtained from Algorithm [\[alg:1\]](#alg:1){reference-type="ref" reference="alg:1"} after switching the roles of the $x$-series and the $y$-series. We report the existence of a strictly one-sided effect associated with $g$ if $\mathrm{HCHG}_T(g)$ exceeds $\hat{q}^{0.95}_0(\mathrm{HCHG}_T)$ while $\mathrm{Rev}[\mathrm{HCHG}_T(g)]$ does not exceed $\hat{q}^{0.95}_0(\mathrm{HCHG}_T)$, or vice versa. We also used the log-rank test based on $\hat{q}^{0.95}_0(\mathrm{LR}_T)$, the simulated $0.95$ quantile of the log-rank statistic $\mathrm{LR}_T$ of [\[eq:logrank\]](#eq:logrank){reference-type="eqref" reference="eq:logrank"} under $H_0$, as well as several weighted versions of the log-rank test proposed in the literature to discover non-proportional hazard departures [@yang2010improved]. ![Number of genes with expression levels significantly ($\alpha=0.05$) associated with survival according to the $\mathrm{HCHG}_T$ (red) and other tests out of $8,702$ genes. Results from the SCANB data [@saal2015sweden], using median class partitions. In all cases, we report on an effect on either side or both sides simultaneously (Algorithm [\[alg:1\]](#alg:1){reference-type="ref" reference="alg:1"}). Testing for a strictly one-sided effect (Algorithm [\[alg:2\]](#alg:2){reference-type="ref" reference="alg:2"}) leads to a similar diagram with up to 8 discoveries removed from some groups. The tests Tarone-Ware, Gehan-Wilcoxon, and Feliming-Harrington correspond to different weights in the family of weighted log-rank test [@pepe1991weighted]. ](Figs/Venn/venn_HCHG_LR_either.png "fig:"){#fig:venn_2sided} ![Number of genes with expression levels significantly ($\alpha=0.05$) associated with survival according to the $\mathrm{HCHG}_T$ (red) and other tests out of $8,702$ genes. Results from the SCANB data [@saal2015sweden], using median class partitions. In all cases, we report on an effect on either side or both sides simultaneously (Algorithm [\[alg:1\]](#alg:1){reference-type="ref" reference="alg:1"}). Testing for a strictly one-sided effect (Algorithm [\[alg:2\]](#alg:2){reference-type="ref" reference="alg:2"}) leads to a similar diagram with up to 8 discoveries removed from some groups. The tests Tarone-Ware, Gehan-Wilcoxon, and Feliming-Harrington correspond to different weights in the family of weighted log-rank test [@pepe1991weighted]. ](Figs/Venn/venn_HCHG_others_either_side1.png "fig:"){#fig:venn_2sided} ![Number of genes with expression levels significantly ($\alpha=0.05$) associated with survival according to the $\mathrm{HCHG}_T$ (red) and other tests out of $8,702$ genes. Results from the SCANB data [@saal2015sweden], using median class partitions. In all cases, we report on an effect on either side or both sides simultaneously (Algorithm [\[alg:1\]](#alg:1){reference-type="ref" reference="alg:1"}). Testing for a strictly one-sided effect (Algorithm [\[alg:2\]](#alg:2){reference-type="ref" reference="alg:2"}) leads to a similar diagram with up to 8 discoveries removed from some groups. The tests Tarone-Ware, Gehan-Wilcoxon, and Feliming-Harrington correspond to different weights in the family of weighted log-rank test [@pepe1991weighted]. ](Figs/Venn/venn_HCHG_others_either_side2.png "fig:"){#fig:venn_2sided} ### Results We report on the number of genes found significant at the level $0.05$ by each testing method in Figure [6](#fig:venn_2sided){reference-type="ref" reference="fig:venn_2sided"}. For example, in testing for a strictly one-sided effect, $\mathrm{HCHG}_T$ identified $397$ significant genes that the log-rank test did not report as significant. We list some of these genes in Table [\[tbl:results_SCANB\]](#tbl:results_SCANB){reference-type="ref" reference="tbl:results_SCANB"}, where we also report on the empirical P-values with respect to each test statistic associated with these genes. The Kaplan-Meier survival curves corresponding to two example genes are illustrated in Figure [\[fig:real_data\]](#fig:real_data){reference-type="ref" reference="fig:real_data"}. In these figures, we also indicate event times driving change between the groups according to the HC thresholding procedure of [\[eq:Delta_def\]](#eq:Delta_def){reference-type="eqref" reference="eq:Delta_def"}. # Discussion [\[sec:discussion\]]{#sec:discussion label="sec:discussion"} ## Temporal and time-varying effects In previous sections, we discussed the sensitivity of HCHG to detect a global effect 'hiding' in only a few time intervals. Here we emphasize two additional properties of HCHG that are advantageous in analyzing signals of the temporarily rare type compared to other methods. First, as described in Section [\[sec:hct\]](#sec:hct){reference-type="ref" reference="sec:hct"}, $\mathrm{HCHG}_T$ offers a mechanism to identify time intervals thought to constitute evidence for a global excessive or reduced risk. These instances may have important interpretations in some applications. For example, time intervals in which intervention or extra monitoring in medical treatment may be beneficial for on of the groups. Additionally, $\mathrm{HCHG}_T$ can distinguish between effects varying over time that may have opposite trends, e.g., a short-term effect that is different than a long-term one. The studies reported in [@dekker2008survival; @johansson2015family; @gregson2019nonproportional; @daniels2017examining] describe different situations of opposite trends that a procedure based on HCHG can potentially identify. ## Pooling across time intervals When failure events are independent, merging counts over bins of several consecutive time intervals generally reduces the power of a test based on $\mathrm{HCHG}_T$ when the departures are very rare. The intuition here is that the number of non-null instances in one bin is so small that their combined effect diminishes as we increase the bin's size and average the response over its members. This phenomenon is well-understood through previous studies involving rare and weak signal detection models in other settings [@arias2011global]. When failure events are not independent, pooling across time intervals may improve the detection using $\mathrm{HCHG}_T$. For example, suppose the presence of an effect causes an increased risk in group $y$ in some period encompassing several consecutive time intervals. In this case, merging counts across bins of time intervals is particularly useful if the bin size roughly matches the effect duration. When the effect duration is unknown, multi-scale approaches for signal detection might be useful [@arias2005near; @hall2010innovated; @pilliat2023optimal]. In the extreme case when the effects are small and scattered across many instances while the bin size is large, tests based on averaging like the log-rank test or Fisher's combination of the hypergeometric P-values would be preferred. ## Low risk and rare failures The HCHG procedure may be significantly sub-optimal in detecting differences when the risk in both groups is very low so that failure events are very rare, e.g., they almost never occur more than once or twice in an interval. Indeed, this case of very few failure events is very different than the calibration [\[eq:calibration\]](#eq:calibration){reference-type="eqref" reference="eq:calibration"} in which the number of failure events in each time instance goes to infinity. Specifically, the case of few failure events is analogous to the low-counts case of [@DonohoKipnis2020] which does not correspond to departures on the moderate scale. To the best of our knowledge, testing procedures that are sensitive to rare departures when the risk is low is yet an unexplored topic. # Proofs [\[sec:proofs\]]{#sec:proofs label="sec:proofs"} This part contains the proofs of Theorems [Theorem 1](#thm:HC_powerful){reference-type="ref" reference="thm:HC_powerful"},[Theorem 2](#thm:HC_powerless){reference-type="ref" reference="thm:HC_powerless"} and [Theorem 3](#thm:LR_powerless){reference-type="ref" reference="thm:LR_powerless"}. These proofs are based on previous results on the ability and impossibility to detect rare and relatively weak mixtures specified in terms of the P-values from [@kipnis2021logchisquared] and [@DonohoKipnis2020b]. In the section below, we first state and prove a series of technical lemmas that are needed to establish the connection between the hypergeometric P-values in our setting [\[eq:model_full\]](#eq:model_full){reference-type="eqref" reference="eq:model_full"} and the general RMD setting. The proof of Theorems [Theorem 1](#thm:HC_powerful){reference-type="ref" reference="thm:HC_powerful"},[Theorem 2](#thm:HC_powerless){reference-type="ref" reference="thm:HC_powerless"} and [Theorem 3](#thm:LR_powerless){reference-type="ref" reference="thm:LR_powerless"} are provided in subsequent sections. ## Asymptotic Notation Some of the technical lemmas concern arrays of real numbers and random variables (RVs) indexed by $T$ and $t \leq T$ and their asymptotic properties as $T$ goes to infinity. In such cases, we often use the notation $o(1)$ to indicate some deterministic sequence converging to zero uniformly in $t$ and the notation $o_p(1)$ to indicate some sequence of random variables that converge to zero in probability uniformly in $t$. ## Technical Lemmas **Lemma 4**. *For non-negative integers $x,y,n_x,n_y$ define $$\pi^+(x,y;n_x, n_y) = \Pr\left( \mathrm{HyG}(n_x + n_y, n_y ,x+y) \geq y + 1 \right),$$ where $$\Pr\left( \mathrm{HyG}(M, N, n) \geq m \right) = \sum_{k=m}^{n} \frac{\binom{N}{k}\binom{M-N}{n-k}}{\binom{M}{n}}.$$ Let $\{\lambda(T)\}$ and $\{a(T)\}$ be positive sequences indexed by $T$ such that $\lambda(T) / \log(T) \to \infty$, $a(T) \lambda(T) \geq \log^2(T)$, and $a(T)/\log(T) \to 0$. For $q>0$, set $\tilde{y}(T,q,x) = \left( \sqrt{x} + \sqrt{q \log(T) - a(T) } \right)^2$. Let sequences $\{n_x(T)\}$ and $\{n_y(T)\}$ obey $x/n_x(T) \to 0$ and $n_x(T)/n_y(T) \to 1$. There exists $T_0(q)$ such that $$\pi(x,\tilde{y}(T,q,x);n_x(T), n_y(T)) \geq T^{-q},$$ for all $T \geq T_0(q)$ and $x \geq \lambda(T) - \sqrt{a(T) \lambda(T)}$.* ### Proof of Lemma [Lemma 4](#lem:pi_upper_bound){reference-type="ref" reference="lem:pi_upper_bound"} {#proof-of-lemma-lempi_upper_bound} We have $$\begin{aligned} & \pi^+(x,y;n_x,n_y) = \sum_{k=y+1}^{x+y} \frac{\binom{n_y}{k}\binom{n_x}{x+y-k}}{\binom{n_x + n_y}{x+y}} \geq \frac{\binom{n_y}{y+1}\binom{n_x}{x}}{\binom{n_x + n_y}{x+y}}\\ & \qquad = \frac{ \frac{1}{2\pi \sqrt{x (y+1)}}\left( \frac{n_y}{y+1}e \right)^{y+1} \exp\left\{-\frac{(y+1)^2}{2n_y}(1+o(1)) \right\} \left( \frac{n_x}{x}e \right)^x \exp\left\{-\frac{x^2}{2n_x}(1+o(1)) \right\}} { \frac{1}{\sqrt{2\pi (x+y)}}\left( \frac{n_x + n_y}{x+y}e \right)^{x+y} \exp\left\{-\frac{(x+y)^2}{2(n_x+n_y)}(1+o(1)) \right\} };\end{aligned}$$ the last step by Stirling's approximation when $x/n_x \to 0$ and $y/n_y \to 0$. As $n_x,n_y\to \infty$ in such a way that $n_x/n_y \to 1$, we obtain $$\begin{aligned} \pi(x,y;n_x,n_y) & \geq \frac{1+o(1)}{\sqrt{2\pi}} 2^{-(x+y)} \sqrt{\frac{x+y}{x(y+1)}}\left(1 + \frac{y+1}{x} \right)^x \left( 1+ \frac{x-1}{y+1}\right)^{y+1}.\end{aligned}$$ Set $x^*(T) := \lambda(T) - \sqrt{a(T) \lambda(T)}$ and $\tilde{y}^*(T) := \tilde{y}(T, x^*(T), q)$. Since $x \leq \tilde{y}(T,x,q)$, we get $$\begin{aligned} & \inf_{x \geq \lambda(T) - \sqrt{a(T) \lambda(T)}} T^q \pi(x, \tilde{y}(T,x,q);n_x,n_y) \\ & = T^{q} \pi(x^*(T),\tilde{y}^*(T);n_x(T),n_y(T)) \\ & \qquad \qquad \geq T^q e^{o(1)} 2^{-(x^*(T)+\tilde{y}^*(T)+1)} \frac{(1+o(1))}{\sqrt{2 \pi}} \\ \qquad \qquad \qquad & \times \sqrt{\frac{x^*(T)+\tilde{y}^*(T)}{x^*(T)\tilde{y}^*(T)}} \left(1+\frac{\tilde{y}^*(T)}{x^*(T)} \right)^{x^*(T)} \left(1+\frac{x^*(T)}{\tilde{y}^*(T)} \right)^{\tilde{y}^*(T)}.\end{aligned}$$ Under the assumptions on $\{a(T)\}$ and $\{\lambda(T)\}$ the last expression goes to infinity as $T$ goes to infinity. The claim in the lemma follows. 0◻ **Lemma 5**. *For non-negative integers $x,y,n_x,n_y$ define $$\pi(x,y;n_x, n_y) = \Pr\left( \mathrm{HyG}(n_x + n_y, n_y ,x +y) \geq y \right),$$ and for $q,s \in \mathbb R$, define $\alpha(q,s) := \left( \sqrt{q} - \sqrt{s} \right)^2$. Consider the arrays $n_x = n_x(t,T)$ and $n_y = n_y(t,T)$ indexed by $T$ and $t \leq T$ and suppose that $\inf_{t\leq T} n_x \to \infty$ and $n_x / n_y = 1 + o(1)$. Let $\bar{\lambda}:=\bar{\lambda}(T)$ be a sequence obeying $\inf_{t \leq T} n_x\bar{\lambda}/\log(T) \to \infty$. Set $\bar{\lambda}'(t,T) :=(\sqrt{\bar{\lambda}} + \sqrt{\delta})^2$, where $\delta$ satisfies $$\delta := \delta(t,T) := \frac{r \log(T)}{ 4\frac{n_x n_y}{n_x + n_y} }(1+o(1)).$$ For any $q > r/2 \geq 0$, the Poisson RVs $X \sim \mathrm{Pois}(\bar{\lambda}n_x)$ and $Y \sim \mathrm{Pois}(\bar{\lambda}n_y)$ obey $$\begin{aligned} \label{eq:lem:pvalue_tail_basic} \frac{-\log(\Pr\left( \pi(X,Y;n_x, n_y) < T^{-q}) \right) }{\log(T)} - \alpha(q,r/2) = o(1).\end{aligned}$$* ### Proof of Lemma [Lemma 5](#lem:pvalue_tail_basic){reference-type="ref" reference="lem:pvalue_tail_basic"} {#proof-of-lemma-lempvalue_tail_basic} For a fixed $x$, $n_x$, $n_y$, and $a>0$, denote by $y^*(x,a; n_x, n_y)$ the minimal $y$ satisfying $$\pi(x,y; n_x, n_y) \leq e^{-a}.$$ We use the Chernoff inequality for $H \sim \mathrm{HyG}(M,N,n)$ $$\begin{aligned} \label{eq:cheroff} \Pr\left(\sqrt{n} \left(\frac{H}{n} - \frac{N}{M} \right) \geq t \right) \leq e^{-2 t^2}\end{aligned}$$ in order to bound $y^*(x,a;n_x, n_y)$ (e.g., [\[eq:cheroff\]](#eq:cheroff){reference-type="eqref" reference="eq:cheroff"} follows from [@serfling1974probability Corollary 1.1]). It follows from [\[eq:cheroff\]](#eq:cheroff){reference-type="eqref" reference="eq:cheroff"} that if $$\frac{2}{x+y} \left(y - (x+y)\frac{n_y}{n_x+n_y}\right)^2 \geq a,$$ then $\pi(x,y;n_x, n_y) \leq e^{-a}$. Solving for $y>x\geq 0$, we obtain $$y^*(x, a; n_x, n_y) \geq \frac{\sqrt{8(1-\tilde{\kappa}) a x + a^2 } + 4 \tilde{\kappa}(1-\tilde{\kappa}) x + a }{4 (1-\tilde{\kappa})^2},$$ where $\tilde{\kappa}:=n_y/(n_x + n_y)$. Setting $a=q \log(T)$ and conditioning on $N_x=n_x$ and $N_y=n_y$, we have $$\begin{aligned} & \Pr\left(\pi(X,Y;n_x, n_y) < T^{-q} \right) \geq \Pr\left(Y \geq y^*(x, a;n_x, n_y) \right) \nonumber \\ \quad & = \Pr\left( Y \geq \frac{ 4 \tilde{\kappa}(1-\tilde{\kappa}) X + q \log(T) + \sqrt{ 8 (1-\tilde{\kappa}) X q\log(T) + q^2 \log^2(T)}} {4 (1-\tilde{\kappa})^2} \right) \nonumber \\ \quad & = \Pr\left( Y \geq \frac{ 4 \tilde{\kappa}(1-\tilde{\kappa}) X \left(1 + \frac{q \log(T)}{X}\right) + \sqrt{ 8 (1-\tilde{\kappa}) X q\log(T) \left(1 + \frac{q \log(T)}{8 (1-\tilde{\kappa}) X}\right) }} {4 (1-\tilde{\kappa})^2} \right). \label{eq:last_disp}\end{aligned}$$ By Markov's inequality and for any $b \in \mathbb R$, $$\begin{aligned} \Pr\left(X \leq b \right) & = \Pr\left(\left(X-\bar{\lambda} n_x\right)^2 \leq \left(b - \bar{\lambda} n_x\right)^2 \right) \\ & \leq \frac{\bar{\lambda} n_x}{\left(b - \bar{\lambda}n_x\right)^2} = \frac{1}{ n_x \bar{\lambda}} \frac{1}{\left(1 - \bar{\lambda}n_x/b\right)^2}. \end{aligned}$$ Since $n_x \bar{\lambda} / \log(T) \to \infty$ (convergence in probability), there exists a sequence $b(T)$ such that $\log(T)/b(T) \to 0$ and $\Pr\left(X \geq b(T) \right)\to 1$. Consequently, $X/\log(T) \overset{p}{\to} 0$ uniformly in $t\leq T$. Since $Y$ is independent of $X$, we may replace elements involving $q \log(T)/X$ on the right-hand side of [\[eq:last_disp\]](#eq:last_disp){reference-type="eqref" reference="eq:last_disp"} with the notation $o_p(1)$ that indicates a sequence of RVs tending to zero in probability uniformly in $t\leq T$. In addition, $\bar{\kappa}= 1/2 + o(1)$. We obtain $$\begin{aligned} & \Pr\left(\pi(X,Y;n_x, n_y) < T^{-q} \right) \nonumber \\ & \quad \geq \Pr\left( Y \geq X \left(1 + o_p(1)\right) + \sqrt{ 4 X q\log(T) \left(1 + o_p(1)\right) } \right) \nonumber \\ & \quad = \Pr\left( \frac{Y}{2} \geq \frac{X}{2} \left(1 + o_p(1)\right) + 2 \sqrt{ \frac{ \frac{X}{2} q\log(T) \left(1 + o_p(1)\right) } {2}} \right) \nonumber \\ & \quad = \Pr\left( \frac{Y}{2} \geq \left[ \sqrt{ \frac{X}{2}\left(1 + o_p(1)\right)} + \sqrt{ \frac{q\log(T) } {2 }} + o_p(1) \right]^2 \right) \nonumber \\ & \quad = \Pr\left( \sqrt{2 Y } - \sqrt{2 X\left(1 + o_p(1)\right)} + o_p(1) \geq \sqrt{2 q \log(T)} \right). \label{eq:pvalue_tail_basic:1}\end{aligned}$$ By the normal approximation to the Poisson and the delta method, the RVs $\sqrt{2X}$ and $\sqrt{2Y}$ are variance stabilized and satisfy $$\begin{aligned} \sqrt{2 X(1+o_p(1))} + o_p(1) & \overset{D}{=} \mathcal{N}(\sqrt{2 \bar{\lambda}n_x + o(1) }, 1/2) \overset{D}{=} :\sqrt{2 \bar{\lambda} n_x + o(1) } + Z_x /\sqrt{2} , \\ \sqrt{2 Y} + o_p(1) & \overset{D}{=} \mathcal{N}(\sqrt{2 \bar{\lambda}'(t;T)n_y }, 1/2 ) \overset{D}{=}: \sqrt{2 \bar{\lambda}'(t;T)n_y } + Z_y / \sqrt{2}, \end{aligned}$$ where $Z_x,Z_y \overset{\mathsf{iid}}{\sim}\mathcal{N}(0,1)$. With $Z\sim \mathcal{N}(0,1)$, we obtain $$\begin{aligned} \sqrt{2 Y} - \sqrt{2 X (1 + o_p(1))} & \overset{D}{=} o_p(1) + Z + 2 \left( \sqrt{ n_y \bar{\lambda}'(t;T)/ 2 } - \sqrt{ n_x \bar{\lambda} / 2 } \right) \\ & = o_p(1) + Z + 2 \sqrt{\frac{n_x n_y}{n_x + n_y} } \left( \sqrt{\bar{\lambda}} + \sqrt{\delta} - \sqrt{\bar{\lambda}} \right) \\ & = o_p(1) + Z + \sqrt{2 \frac{n_x n_y}{n_x + n_y} \cdot 2 \delta } = o_p(1) + Z + \sqrt{r \log(T)}.\end{aligned}$$ From here, a moderate deviation estimate of $\sqrt{2 Y} - \sqrt{2 X}$ as in [\[eq:pvalue_tail_basic:1\]](#eq:pvalue_tail_basic:1){reference-type="eqref" reference="eq:pvalue_tail_basic:1"} implies [@RubinSethuraman1965; @zeitouni1998large] $$\begin{aligned} \frac{\log \Pr\left(\pi(X,Y;n_x, n_y) < T^{-q} \right)}{\log(T)} +o(1) & \geq \lim_{T \to \infty} \frac{-\left( \sqrt{2q \log(T)} - \sqrt{r \log(T)} \right)^2}{2 \log(T)} \nonumber \\ & = -\left(\sqrt{q} - \sqrt{r / 2 } \right)^2 = -\alpha(q,r/2) \label{eq:asymp_inequality_1}\end{aligned}$$ whenever $q> r / 2$. For the reverse bound, we use Lemma [Lemma 4](#lem:pi_upper_bound){reference-type="ref" reference="lem:pi_upper_bound"} with $\lambda(T) = \bar{\lambda}n_x$ and the sequence $a(T) = \log^2(T)/ (\bar{\lambda}n_x) + \sqrt{\log(T)}$, which satisfies $a(T) \to \infty$ as well as the conditions in Lemma [Lemma 4](#lem:pi_upper_bound){reference-type="ref" reference="lem:pi_upper_bound"}. We obtain $$2(y^*(x, T^{-q})-1) \leq \left( \sqrt{2 x} + \sqrt{2 q \log(T)(1+o(1))} \right)^2,$$ for all $x$ such that $x \geq n_x \bar{\lambda} - \sqrt{a(T) n_x \bar{\lambda}}$. Note that $\pi^+(x,y;n_x,n_y) \leq \pi(x,y;n_x,n_y)$ whenever $y \geq x$. Therefore, conditioned on the event $A_{t,T} := \{X \geq n_x \bar{\lambda} - \sqrt{a(T) n_x \bar{\lambda}} \}$, we have $$\begin{aligned} \Pr\left(\pi(X,Y; n_x, n_y) < T^{-q} \right) & \leq \Pr\left(\pi^+(X,Y; n_x, n_y) < T^{-q} \right) \\ & \leq \left( \sqrt{2 Y} - \sqrt{2X} + \sqrt{2q \log(T)(1+o_p(1))} \right). \end{aligned}$$ From here, the same arguments following [\[eq:pvalue_tail_basic:1\]](#eq:pvalue_tail_basic:1){reference-type="eqref" reference="eq:pvalue_tail_basic:1"} above imply $$\frac{\log \Pr\left(\pi(X,Y;n_x, n_y)< T^{-q} \mid A_{t,T} \right) }{\log(T)} + o(1) \leq -\alpha(q,r/2).$$ Since $a(T) \to \infty$, the normal approximation to the Poisson RV $X$ gives $$\Pr\left( \frac{X - n_x \bar{\lambda} }{\sqrt{n_X \bar{\lambda}}} \geq -\sqrt{a(T)} | N_x=n_x \right) \to 1.$$ It follows that $\Pr\left(A_{t,T} \right) \to 1$, and thus we have the unconditioned asymptotic inequality $$\begin{aligned} \label{eq:asump_inequality_2} \frac{\log \Pr\left(\pi(X,Y;n_x, n_y)< T^{-q} \right) }{\log(T)} + o(1) \leq -\alpha(q,r/2). \end{aligned}$$ Equations [\[eq:asymp_inequality_1\]](#eq:asymp_inequality_1){reference-type="eqref" reference="eq:asymp_inequality_1"} and [\[eq:asump_inequality_2\]](#eq:asump_inequality_2){reference-type="eqref" reference="eq:asump_inequality_2"} imply [\[eq:lem:pvalue_tail_basic\]](#eq:lem:pvalue_tail_basic){reference-type="eqref" reference="eq:lem:pvalue_tail_basic"}. 0◻ **Lemma 6**. *Let $x_0$, $y_0$, $\delta$, $\bar{\lambda}$ and $\bar{\lambda}'$ be calibrated to $T$ as in [\[eq:calibrtion_initial\]](#eq:calibrtion_initial){reference-type="eqref" reference="eq:calibrtion_initial"}, [\[eq:calibrtion_kappa\]](#eq:calibrtion_kappa){reference-type="eqref" reference="eq:calibrtion_kappa"}, [\[eq:calibration_delta\]](#eq:calibration_delta){reference-type="eqref" reference="eq:calibration_delta"}, and [\[eq:calibration_rates\]](#eq:calibration_rates){reference-type="eqref" reference="eq:calibration_rates"}. Let $\{N_x(t)\}_{t=1}^T$ and $\{N_y(t)\}_{t=1}^T$ as in [\[eq:sampling_model\]](#eq:sampling_model){reference-type="eqref" reference="eq:sampling_model"}. As $T \to \infty$, $$\begin{aligned} \label{eq:lem:kappa} \max_{t\leq T} T \left| \frac{N_x(t)}{x_0} e^{\bar{\lambda} t} - 1 \right| \to 0 \quad \text{and} \quad \max_{t\leq T} T^{\beta} \left|\frac{N_y(t)}{y_0} e^{ \bar{\lambda} t} - 1\right| \to 0 \end{aligned}$$ in probability. In particular, $$\begin{aligned} \label{eq:lem:ratio_concentration} 2\frac{N_x(t) N_y(t)}{N_x(t) + N_y(t)} = x_0 e^{-\bar{\lambda} t}(1 + o(1)) = \frac{x_0+y_0}{2} e^{-\bar{\lambda} t}(1 + o_p(1)), \end{aligned}$$ where $o_p(1) \to 0$ uniformly in $t$ as $T \to \infty$.* ### Proof of Lemma [Lemma 6](#lem:const_ratio){reference-type="ref" reference="lem:const_ratio"} {#proof-of-lemma-lemconst_ratio} We only need to show the second convergence in [\[eq:lem:kappa\]](#eq:lem:kappa){reference-type="eqref" reference="eq:lem:kappa"} as the first one is obtained in the limit $\beta \to 1$. By Markov's inequality and since the maximization is over a finite set, for $\eta>0$ we have $$\begin{aligned} \Pr\left(\max_{t\leq T} T^\beta \left| \frac{N_y(t)}{y_0e^{-\bar{\lambda}t }} - 1\right| > \eta \right) & \leq \frac{T^\beta}{\eta^2} \ensuremath{\mathbb{E}_{H_1}\left[ \max_{t \leq T} \left( \frac{N_y(t)}{y_0e^{-\bar{\lambda}t }} - 1\right)^2\right]} \nonumber \\ & = \frac{T^\beta}{\eta^2} \max_{t \leq T} \ensuremath{\mathbb{E}_{H_1}\left[ \left( \frac{N_y(t)}{y_0e^{-\bar{\lambda}t }} - 1\right)^2\right]}. \label{eq:lem:const_ratio_to_show}\end{aligned}$$ For $\Upsilon \sim \mathrm{Pois}(n a)$ for some $n>0$ and $a>0$, we have that $$\ensuremath{\mathbb{E}\left[ \left(\frac{n-\Upsilon}{b} \right)^2\right]} = \frac{n a}{b^2} + \left( \frac{n(1-a)}{b} - 1 \right)^2.$$ Since $N_y(t)$ given $N_y(t-1)$ is distributed as $(1-\epsilon)\mathrm{Pois}(N_y(t-1)\bar{\lambda}) + \epsilon \mathrm{Pois}(N_y(t-1)\bar{\lambda}')$, we get $$\begin{aligned} & \ensuremath{\mathbb{E}\left[ \left(\frac{N_y(t)}{y_0(1-\bar{\lambda})^t} \right)^2 \mid N_y(t-1)\right]} = \frac{\bar{\lambda} N_y(t-1) }{y_0^2(1-\bar{\lambda})^{2t}} + \left( \frac{(1-\bar{\lambda}) N_y(t-1)}{y_0(1-\bar{\lambda})^t} - 1 \right)^2 \\ & \quad + \epsilon \left[ \frac{N_y(t-1)(\bar{\lambda}'-\bar{\lambda}) }{y_0^2(1-\bar{\lambda})^{2t}} - \left( \frac{N_y(t-1)(1-\bar{\lambda}') }{y_0(1-\bar{\lambda})^t} - 1 \right)^2 + \left( \frac{N_y(t-1)(1-\bar{\lambda}) }{y_0(1-\bar{\lambda})^t} - 1 \right)^2\right] \\ & = \frac{\bar{\lambda} N_y(t-1) }{y_0^2(1-\bar{\lambda})^{2t}} + \left( \frac{ N_y(t-1)}{y_0(1-\bar{\lambda})^{t-1}} - 1 \right)^2 \\ & \quad + \epsilon \cdot \frac{N_y(t-1)(\bar{\lambda}'-\bar{\lambda})}{y_0^2(1-\bar{\lambda})^{2t}} \left( 1 - N_y(t-1)(2 - \bar{\lambda} - \bar{\lambda}' ) - 2y_0(1-\bar{\lambda})^t\right).\end{aligned}$$ Next, note that [\[eq:calibration_delta\]](#eq:calibration_delta){reference-type="eqref" reference="eq:calibration_delta"} implies, $$\begin{aligned} \frac{\delta(t-1)}{\bar{\lambda}} & = \frac{\frac{r}{2} \log(T) }{2\bar{\lambda} \frac{x_0}{2} e^{-{\bar{\lambda}t}}} \leq \frac{r e^{M}}{2 } \frac{\log(T) }{\bar{\lambda} x_0} \to 0, \end{aligned}$$ hence $$\begin{aligned} \max_{t \leq T} \frac{\bar{\lambda}'(t) - \bar{\lambda}}{\bar{\lambda}} & = \max_{t \leq T} \left\{2 \sqrt{\frac{\delta(t-1)}{\bar{\lambda}} } + \frac{\delta(t-1)}{\bar{\lambda}} \right\} = o(1). \label{eq:lem_const_ratio:1}\end{aligned}$$ We use [\[eq:lem_const_ratio:1\]](#eq:lem_const_ratio:1){reference-type="eqref" reference="eq:lem_const_ratio:1"} and that $\bar{\lambda}\leq 1$, $(1-\bar{\lambda})^t \geq e^{-\bar{\lambda}n} \geq e^{-M}$, and $N_y(t-1)\leq y_0$, to conclude $$\begin{aligned} & \ensuremath{\mathbb{E}\left[ \left(\frac{N_y(t)}{y_0(1-\bar{\lambda})^t} -1\right)^2 \mid N_y(t-1)\right]} \leq o(1/y_0) + \left( \frac{N_y(t-1)}{y_0(1-\bar{\lambda})^{t-1}} - 1 \right)^2 \\ & \qquad + \epsilon \cdot \frac{\bar{\lambda}'-\bar{\lambda}}{\bar{\lambda}} \frac{ \bar{\lambda}}{ y_0 e^{-M}} \left|2N_y(t-1) + 2y_0 + 1 \right| \\ & = \left( \frac{N_y(t-1)}{y_0(1-\bar{\lambda})^{t-1}} - 1 \right)^2 + o(1/y_0) + \epsilon \cdot o(1). \end{aligned}$$ By induction on $t =1,2,\ldots$, $$\begin{aligned} \ensuremath{\mathbb{E}\left[ \left(\frac{N_y(t)}{y_0(1-\bar{\lambda})^t} - 1\right)^2\right]} & = \ensuremath{\mathbb{E}\left[ \ensuremath{\mathbb{E}\left[ \left(\frac{N_y(t)}{y_0(1-\bar{\lambda})^t} - 1\right)^2 \mid N_y(t-1)\right]}\right]} \\ & \leq o(1/y_0) + \epsilon \cdot o(1). \end{aligned}$$ Since $\epsilon = T^{-\beta}$ and $y_0/T \to \infty$ by [\[eq:calibration_rates\]](#eq:calibration_rates){reference-type="eqref" reference="eq:calibration_rates"}, we obtain $$\begin{aligned} T^{\beta}\ensuremath{\mathbb{E}\left[ \left(\frac{N_y(t)}{y_0(1-\bar{\lambda})^t} - 1\right)^2\right]} = o(1). \label{eq:lem:const_ratio:2}\end{aligned}$$ Finally, for $x>0$ as $t \to \infty$, $$1 \leq \frac{e^{-xt}}{(1-x)^{t}} = e^{t(x^2+o(x^3))},$$ hence under [\[eq:calibration\]](#eq:calibration){reference-type="eqref" reference="eq:calibration"}, we have $$1 \leq \frac{e^{-\bar{\lambda}t}}{(1-\bar{\lambda})^{t}} = e^{t(\bar{\lambda}^2+o(\bar{\lambda}^3))} \leq e^{M^2/T + M^2/T o(\bar{\lambda})} = 1 + o(1/T).$$ Therefore, [\[eq:lem:const_ratio:2\]](#eq:lem:const_ratio:2){reference-type="eqref" reference="eq:lem:const_ratio:2"} implies $$\begin{aligned} T^{\beta}\ensuremath{\mathbb{E}\left[ \left(\frac{N_y(t)}{y_0e^{-t \bar{\lambda}}} - 1\right)^2\right]} = o(1),\end{aligned}$$ and thus the right-hand side of [\[eq:lem:const_ratio_to_show\]](#eq:lem:const_ratio_to_show){reference-type="eqref" reference="eq:lem:const_ratio_to_show"} goes to zero. **Lemma 7**. *Let $\bar{\lambda}$ and $\delta$ be calibrated to $T$ as in [\[eq:calibration_eps\]](#eq:calibration_eps){reference-type="eqref" reference="eq:calibration_eps"}-[\[eq:calibration_delta\]](#eq:calibration_delta){reference-type="eqref" reference="eq:calibration_delta"} and $N_x(t)$ and $N_y(t)$ obey [\[eq:model_full\]](#eq:model_full){reference-type="eqref" reference="eq:model_full"}. Suppose that $$P_t = p_\mathrm{HyG}( \Upsilon'(t); N_x(t) + N_y(t), N_y(t), \Upsilon(t) + \Upsilon'(t) ),$$ where, given $N_x(t)$ and $N_y(t)$, $\Upsilon'(t) \sim \mathrm{Pois}(\bar{\lambda}'(t)N_y(t))$ and $\Upsilon(t) \sim \mathrm{Pois}(\bar{\lambda}N_x(t))$. For $q > r/2 > 0$, we have $$\begin{aligned} \lim_{T \to 0} \max_{t=1,\ldots T} \left| \frac{-\log(\Pr\left( P_t \leq n^{-q} \right))}{\log(T)} - \alpha(q,r/2) \right| = 0.\end{aligned}$$* ### Proof of Lemma [Lemma 7](#lem:pvalue_tail){reference-type="ref" reference="lem:pvalue_tail"} {#proof-of-lemma-lempvalue_tail} Define the sequence of events $$\begin{aligned} \label{eq:AT_def} A_T & = \left\{ \inf_{t\leq T} N_x(t) \geq x_0(T) e^{-(M+1)} \right\} \cap \left\{ \inf_{t\leq T} N_y(t) \geq y_0(T) e^{-(M+1)} \right\} \\ & \qquad \cap \left\{ \sup_{t\leq T} \left| \frac{N_x(t)}{N_y(t)} - \frac{x_0(T)}{y_0(T)} \right| < \frac{x_0(T)}{y_0(T)} \frac{2T^{-\beta}}{1-T^{-\beta}} \right\} \nonumber\end{aligned}$$ (this sequence is independent of $t$). Note that Lemma [Lemma 6](#lem:const_ratio){reference-type="ref" reference="lem:const_ratio"} implies $$\delta = \frac{(r/2) \log(T)}{ \frac{x_0+y_0}{2} e^{-\bar{\lambda}t}} = \frac{r \log(T)}{ 4\frac{N_x(t)N_y(t)}{N_x(t)+N_y(t)} }(1+o_p(1)).$$ Conditioning on $A_T$ and using [\[eq:calibrtion_initial\]](#eq:calibrtion_initial){reference-type="eqref" reference="eq:calibrtion_initial"}, Lemma [Lemma 5](#lem:pvalue_tail_basic){reference-type="ref" reference="lem:pvalue_tail_basic"} implies $$\begin{aligned} \label{eq:lem:pvalue_tail} \lim_{T \to 0} \max_{t=1,\ldots T} \left| \frac{-\log(\Pr\left( P_t \leq n^{-q} \mid A_T \right))}{\log(T)} - \alpha(q,r/2) \right| = 0.\end{aligned}$$ for $q > r/2 > 0$. The claim in the lemma now follows by arguing that $\Pr\left(A_T \right) \to 1$. Indeed, by Lemma [Lemma 6](#lem:const_ratio){reference-type="ref" reference="lem:const_ratio"}, $$\begin{aligned} N_x(t) & = x_0e^{-\bar{\lambda}t}\left( 1 + o_p(1) \right) \geq x_0 e^{-M} \left( 1 + o_p(1) \right)\\ N_y(t) & = y_0 e^{-\bar{\lambda}t} \left( 1 + o_p(1) \right) \geq y_0 e^{-M} \left( 1 + o_p(1) \right) \end{aligned}$$ hence it follows from [\[eq:calibration_rates\]](#eq:calibration_rates){reference-type="eqref" reference="eq:calibration_rates"} that $\inf_{t \leq T} \bar{\lambda}N_x(t)/\log(T) \to \infty$ and $\inf_{t \leq T} \bar{\lambda}N_y(t)/\log(T) \to \infty$ in probability. In addition, Lemma [Lemma 6](#lem:const_ratio){reference-type="ref" reference="lem:const_ratio"} implies that for all $T$ large enough and $t \leq T$ $$\frac{1-T^{-1}}{1+T^{-\beta}} \leq \frac{N_x(t)}{N_y(t)} \frac{y_0(T)}{x_0(T)} \leq \frac{1+T^{-1}}{1-T^{-\beta}}.$$ Consequently, $$\frac{-2T^{-\beta}}{1-T^{-\beta}} \leq -\frac{T^{-\beta}+T^{-1}}{1 + T^{-\beta}} \leq \frac{N_x(T)}{N_y(T)}\frac{y_0(T)}{x_0(T)} - 1 \leq \frac{T^{-\beta}+T^{-1}}{1 - T^{-\beta}} \leq \frac{-2T^{-\beta}}{1-T^{-\beta}}$$ and $$\left|\frac{N_x(t)}{N_y(t)}\frac{y_0(T)}{x_0(T)} - 1 \right| \leq \frac{-2T^{-\beta}}{1-T^{-\beta}},\qquad \forall t \leq T.$$ 0◻ ## [\[sec:proof:LR_powerless\]]{#sec:proof:LR_powerless label="sec:proof:LR_powerless"} Proof of Theorem [Theorem 3](#thm:LR_powerless){reference-type="ref" reference="thm:LR_powerless"} {#secprooflr_powerless-proof-of-theorem-thmlr_powerless} Denote $$\bar{\kappa}_t := \frac{N_y(t)}{N_x(t) + N_y(t)}.$$ Under either hypothesis, $$\begin{aligned} \sum_{t=1}^T O_y(t) - \sum_{t=1}^T E_t & = \sum_{t=1}^T \left( (1-\bar{\kappa}_{t-1})O_y(t) - \bar{\kappa}_{t-1} O_x(t) \right)\end{aligned}$$ and $$\begin{aligned} \sum_{t=1}^T V_t & = \sum_{t=1}^T \bar{\kappa}_{t-1}(1-\bar{\kappa}_{t-1})(1+o_p(1)) \left(O_x(t)+O_y(t)\right)\left( 1 - \frac{O_x(t)+O_y(t)}{N_y(t)}\bar{\kappa}_{t-1} \right),\end{aligned}$$ hence $$\begin{aligned} \mathrm{LR}_T= \frac{\sum_{t=1}^T (1-\bar{\kappa}_{t-1})O_y(t)-\bar{\kappa}_{t-1}O_x(t) }{\sqrt{\sum_{t=1}^T \bar{\kappa}_{t-1}(1-\bar{\kappa}_{t-1})(1+o_p(1)) \left( O_x(t) + O_y(t) \right) \left( 1 - \bar{\kappa}_{t-1} \frac{O_x(t) + O_y(t)}{N_y(t-1)} \right) }} \label{eq:proof:LR}\end{aligned}$$ Under $H_1$, it follows from Lemma [Lemma 6](#lem:const_ratio){reference-type="ref" reference="lem:const_ratio"} that $$\begin{aligned} \begin{split} \label{eq:LR_proof:1} N_x(t) & = x_0 e^{-\bar{\lambda}t} \left( 1 + o_p(1)/T \right) \\ N_y(t) & = y_0e^{-\bar{\lambda}t}\left( 1 + o_p(1)/ T^{\beta} \right), \end{split}\end{aligned}$$ hence by [\[eq:calibration_rates\]](#eq:calibration_rates){reference-type="eqref" reference="eq:calibration_rates"}, $$\bar{\kappa}_t := \frac{1}{2}\left(1 + o_p(1)/T^{\beta}\right).$$ Consequently, $$\begin{aligned} \sum_{t=1}^T E_t & = \sum_{t=1}^T\bar{\kappa}_{t-1} O_t = \frac{1+o_p(1)/T^\beta}{2}\sum_{t=1}^T O_t \\ & = \frac{(1+o_p(1)/T^\beta)}{2}(N_y(0) - N_y(T) +N_x(0)-N_x(T)) \\ & = \frac{(1+o_p(1)/T^\beta)}{2} \left[y_0 \left((1-e^{-\bar{\lambda}T}(1+o_p(1)/T^\beta)\right) \right. \\ & \qquad \qquad \left. + x_0\left(1-e^{-\bar{\lambda}T}(1+o_p(1)/T))\right) \right] \\ & = \frac{y_0 + x_0}{2}(1-e^{-\bar{\lambda}T})(1+o_p(1)/T^{\beta}).\end{aligned}$$ Furthermore, because $$\frac{N_x(t-1)}{N_x(t-1)+N_y(t-1)-1}\left( 1 - \frac{O_x(t)+O_y(t)}{N_x(t-1)+N_y(t-1)} \right) = \frac{1 + o_p(1) T^{-\beta}}{2}(1+o_p(1)),$$ we have $$\begin{aligned} \sum_{t=1}^T V_t & = \frac{1+ o_p(1)}{2}\sum_{t=1}^T E_t \\ & = \frac{1+ o_p(1)}{2} \frac{y_0 + x_0}{2}(1-e^{-\bar{\lambda}T})(1+o_p(1)/T^{\beta}) \\ & = \frac{y_0 + x_0}{4}(1-e^{-\bar{\lambda}T})(1+o_p(1))\end{aligned}$$ hence $$\sqrt{\sum_{t=1}^T V_t} = \sqrt{(1+o_p(1))\frac{y_0+x_0}{2}(1-e^{-\bar{\lambda}T}) }.$$ We conclude that $$\ensuremath{\mathbb{E}\left[ \mathrm{LR}_T|H_1\right]} = \frac{\sum_{t=1}^T \ensuremath{\mathbb{E}\left[ O_y(t)|H_1\right]} - \ensuremath{\mathbb{E}\left[ E_t |H_1\right]}}{\frac{y_0+x_0}{2}(1-e^{-\bar{\lambda}T}) }(1+o(1)) = o(T^{-\beta})$$ and $$\mathrm{Var}\left[ \mathrm{LR}_T|H_1\right] = \frac{\mathrm{Var}[\sum_{t=1^T} O_y(t) - E_t |H_1]}{\frac{y_0+x_0}{2}(1-e^{-\bar{\lambda}T}) }(1+o(1)) \geq T \log(T)(1+o(T^{-\beta}) + o(1).$$ The situation under $H_0$ is obtained from the expressions above with $\beta = 1$. We conclude that $$\begin{aligned} \label{eq:LR:proof2} \frac{\mathrm{Var}[\mathrm{LR}_T|H_1]}{\mathrm{Var}[\mathrm{LR}_T|H_0]} \to 1, \end{aligned}$$ and $$\begin{aligned} \label{eq:LR:proof1} \frac{\ensuremath{\mathbb{E}\left[ \mathrm{LR}_T|H_1\right]} - \ensuremath{\mathbb{E}\left[ \mathrm{LR}_T|H_0\right]}}{\sqrt{\mathrm{Var}[\mathrm{LR}_T|H_0]}} = o(T^{-\beta+1/2}) \to 0,\end{aligned}$$ for $\beta > 1/2$. Since $\mathrm{LR}_T$ is asymptotically normal, [\[eq:LR_proof:1\]](#eq:LR_proof:1){reference-type="eqref" reference="eq:LR_proof:1"} and [\[eq:LR:proof2\]](#eq:LR:proof2){reference-type="eqref" reference="eq:LR:proof2"} implies that $\mathrm{LR}_T$ is asymptotically powerless (c.f. [@arias2017distribution]). ## Asymptotically Rare Moderate Departures Setting [\[sec:RMD\]]{#sec:RMD label="sec:RMD"} Consider the RVs $$\begin{aligned} P_t & = p_{\mathrm{HyG}}\left( O_y(t); N_x(t) + N_y(t), N_y(t), O_x(t) + O_y(t) \right), \qquad t=1,\ldots,T,\end{aligned}$$ where $p_{\mathrm{HyG}}$ is defined in [\[eq:hyg_pmf\]](#eq:hyg_pmf){reference-type="eqref" reference="eq:hyg_pmf"}. Given $N_x(t-1)=n_x(t)$ and $N_y(t-1)=n_y(t)$, $P_t$ is a RV whose distribution is independent of $P_1,\ldots,P_{t-1}$ and obeys $$\begin{aligned} P_t \overset{D}{=} p_\mathrm{HyG}( \Upsilon_y(t); n_x(t) + n_y(t), n_y(t), \Upsilon_x + \Upsilon_y(t) ), \end{aligned}$$ where $$\begin{aligned} \Upsilon_y(t) \sim \mathrm{Pois}(n_y(t) \bar{\lambda}_y(t) ), \qquad \Upsilon_x(t) \sim \mathrm{Pois}(n_x(t) \bar{\lambda}(t)), \end{aligned}$$ Therefore, considering a sequence of hypothesis testing problems indexed by $T$ and the probability law of $P_t$ given $\{N_x(t), N_y(t)\}_{s \leq t}$, we get the following hypothesis testing problem. $$\begin{aligned} \label{eq:hyp_proof} \begin{split} H_0 \,:&\, P_t \overset{\mathsf{iid}}{\sim}\mathcal{U}_t^{(T)}\\ H_1 \,:&\, P_T \overset{\mathsf{iid}}{\sim}(1-\epsilon)\mathcal{U}_t^{(T)} + \epsilon \mathcal{Q}_t^{(T)}. \end{split}\end{aligned}$$ Here $\mathcal{U}_t^{(n)}$ is the distribution of the $t$-th P-value under the null in [\[eq:hyp\]](#eq:hyp){reference-type="eqref" reference="eq:hyp"}, and $\mathcal{Q}_t^{(n)}$ is the distribution of $$p_\mathrm{HyG}( \Upsilon'(t); n_x(t) + n_y(t), n_y(t), \Upsilon(t) + \Upsilon'(t) ),$$ where $\Upsilon'(t) \sim \mathrm{Pois}(\bar{\lambda}'(t)n_y(t)$ and $\Upsilon(t) \sim \mathrm{Pois}(\bar{\lambda}n_x(t))$. The HC test will turn out to be asymptotically powerful for [\[eq:hyp_proof\]](#eq:hyp_proof){reference-type="eqref" reference="eq:hyp_proof"} whenever $r > \rho(\beta)$, hence it is also asymptotically powerful for [\[eq:hyp\]](#eq:hyp){reference-type="eqref" reference="eq:hyp"} in this regime. Let $U_t \sim \mathcal{U}_t^{(T)}$, $Q_t \sim \mathcal{Q}_t^{(T)}$. Since $P_t$ is a P-value under [\[eq:hyp\]](#eq:hyp){reference-type="eqref" reference="eq:hyp"}, the distribution of $P_t$ is super uniform [@lehmann2006testing] hence $$\begin{aligned} \label{eq:hyp_proof_U} \lim_{T\to \infty} \max_{t=1,\ldots,T} \frac{-\log\Pr\left(-2 \log(U_t) \geq 2 q \log(T) \right)}{\log(T)} \leq q.\end{aligned}$$ In addition, it follows from Lemma [Lemma 7](#lem:pvalue_tail){reference-type="ref" reference="lem:pvalue_tail"} that $$\begin{aligned} \label{eq:hyp_proof_Q} \lim_{T\to \infty} \max_{t=1,\ldots,T}\left|\frac{-\log\Pr\left(-2 \log(Q_t) \geq 2 q \log(T) \right)}{\log(T)} - \alpha(q,r/2) \right| = 0,\end{aligned}$$ with $\alpha(q,s)=(\sqrt{r}-\sqrt{s})^2$. Equation [\[eq:hyp_proof_Q\]](#eq:hyp_proof_Q){reference-type="eqref" reference="eq:hyp_proof_Q"} says that on the moderate deviation asymptotic scale, $\{-2\log(Q_t)\}$ behaves as the non-central chisquared distribution over one degree of freedom $$\chi^2(r/2,1) \overset{D}{=} (Z + \sqrt{r \log(T)})^2,\quad Z \sim \mathcal{N}(0,1).$$ Hypothesis testing problems involving rare mixtures of P-values or asymptotic P-values of the form [\[eq:hyp_proof\]](#eq:hyp_proof){reference-type="eqref" reference="eq:hyp_proof"} with mixture components obeying [\[eq:hyp_proof_U\]](#eq:hyp_proof_U){reference-type="eqref" reference="eq:hyp_proof_U"} and [\[eq:hyp_proof_Q\]](#eq:hyp_proof_Q){reference-type="eqref" reference="eq:hyp_proof_Q"} were studied in [@kipnis2021logchisquared]. Theorem [Theorem 1](#thm:HC_powerful){reference-type="ref" reference="thm:HC_powerful"} follows from [@kipnis2021logchisquared Thm. 1.3], and the asymptotic power of tests based on $\mathrm{FDR}^*(p_1,\ldots,p_T)$, $p_{(1)}$, and $F_T$ reported in Table [2](#tbl:power){reference-type="ref" reference="tbl:power"} follows from Theorems 1.5-1.7 of [@kipnis2021logchisquared]. ## Proof of Theorem [Theorem 2](#thm:HC_powerless){reference-type="ref" reference="thm:HC_powerless"} {#proof-of-theorem-thmhc_powerless} A randomized version of the Hypergeometric test $$\begin{aligned} \pi(x, y; n_x, n_y) = \Pr\left(\mathrm{HyG}(n_x + n_y, n_x, x+y) \geq y \right) \end{aligned}$$ is given as $$\begin{aligned} \label{eq:pval_randomized} \tilde{\pi}(x, y; n_x, n_y) & = \Pr\left(\mathrm{HyG}(n_x + n_y, n_y, x+y) \geq y \right) \\ & \quad - U \cdot \Pr\left(\mathrm{HyG}(n_x + n_y, n_x, x+y) = y \right)\nonumber\end{aligned}$$ where $U$ is uniformly distributed over $(0,1)$. For $y \geq x$, we have that $$\begin{aligned} \tilde{\pi}(x;n_x, n_y) & \leq \Pr\left(\mathrm{HyG}(n_x+n_y, n_y, x+y) > y \right) \\ & = \pi^+(x, y; n_x, n_y),\end{aligned}$$ hence the second part of the proof of Lemma [Lemma 5](#lem:pvalue_tail_basic){reference-type="ref" reference="lem:pvalue_tail_basic"} applies when we replace $\pi$ by $\tilde{\pi}$, leading to (the counterpart of [\[eq:asump_inequality_2\]](#eq:asump_inequality_2){reference-type="eqref" reference="eq:asump_inequality_2"}) $$\begin{aligned} \frac{\log \Pr\left(\tilde{\pi}(X,Y;n_x, n_y)< T^{-q} \right) }{\log(T)} + o(1) \leq -\alpha(q,r/2), \qquad q \geq r/2,\end{aligned}$$ or $$\begin{aligned} \Pr\left(\tilde{\pi}(X,Y;n_x, n_y)< T^{-q} \right) \leq T^{-\alpha(q, r/2)(1+o(1))},\quad q \geq r/2. \end{aligned}$$ We now follow similar steps as in the proof of Lemma [Lemma 7](#lem:pvalue_tail){reference-type="ref" reference="lem:pvalue_tail"} and condition on the set $A_T$ of [\[eq:AT_def\]](#eq:AT_def){reference-type="eqref" reference="eq:AT_def"} to deduce that $\tilde{P}_t := \tilde{\pi}(\Upsilon'(t); N_x(t)+N_y(t), N_y(t), \Upsilon(t) + \Upsilon'(t))$ satisfies $$\begin{aligned} \Pr\left(\tilde{P}_t < T^{-q} \right) \leq T^{-\alpha(q, r/2)(1+o(1))},\qquad q \geq r/2. \label{eq:pval_proof_random}\end{aligned}$$ Denote the distribution of $\tilde{P}_t$ by $\tilde{\mathcal{Q}}_t^{(T)}$. We concluded that the randomized version of [\[eq:pvals_def\]](#eq:pvals_def){reference-type="eqref" reference="eq:pvals_def"} according to [\[eq:pval_randomized\]](#eq:pval_randomized){reference-type="eqref" reference="eq:pval_randomized"} are of the form [\[eq:hyp_proof\]](#eq:hyp_proof){reference-type="eqref" reference="eq:hyp_proof"} with $\mathcal{U}_t^{(T)} = \mathrm{Unif}(0,1)$ and $\mathcal{Q}_t^{(T)} = \tilde{Q}_t^{(T)}$, where $P_t \sim \tilde{Q}_t^{(T)}$ satisfies [\[eq:pval_proof_random\]](#eq:pval_proof_random){reference-type="eqref" reference="eq:pval_proof_random"}. This situation falls under the setting of [@DonohoKipnis2020b], hence Theorem [Theorem 2](#thm:HC_powerless){reference-type="ref" reference="thm:HC_powerless"} follows from [@DonohoKipnis2020b Thm. 2.1], provided here in the notation of our setting. **Theorem 8**. *[@DonohoKipnis2020b Thm. 2.1] Given fixed $\beta$ and $r$, consider testing $H_0$ against $H_1$ as in [\[eq:hyp_proof\]](#eq:hyp_proof){reference-type="eqref" reference="eq:hyp_proof"} where $\mathcal{U}_t^{(T)} = \mathrm{Unif}(0,1)$ and $\mathcal{Q}_t^{(T)}$ obeys [\[eq:pval_proof_random\]](#eq:pval_proof_random){reference-type="eqref" reference="eq:pval_proof_random"} and has a continuous density $f_t$, and that, for some $C< \infty$, $\|f_t\|_{\infty} = C$ for all $t=1,\ldots,T$. If $r < \rho(\beta)$, then $\mathrm{HCHG}_T$ of [\[eq:HC_def\]](#eq:HC_def){reference-type="eqref" reference="eq:HC_def"} is asymptotically powerless.*

arxiv_math

--- abstract: | The main aim of this paper is to define a weakest topology $\sigma$ on a linear topological space $(E, \tau)$ such that each $\delta$-continuous functional on $(E, \tau)$ is $\delta$-continuous functional on $(E, \sigma)$ and to find out the relation between the set of these $\delta$-continuous functionals on $(E, \tau)$ and the set of all $\delta$-continuous functionals on $(E, \sigma)$. Also we find out the closure of a subset $A$ of $E$ on that weakest topology by the $\delta$-closure of $A$ with respect to the given topology. --- **A Few Properties of $\delta$-Continuity and $\delta$-Closure on** **Delta Weak Topological Spaces** **Sanjay Roy** Department of Mathematics, Uluberia College Uluberia, Howrah- 711315, West Bengal, India e-mail: sanjaypuremath\@gmail.com **Keywords:** *Delta open set, Delta closed set, Delta Continuous mapping, Delta topological dual, Delta weak topology, Delta bounded set, Delta bounded linear mapping, Delta closure.*\ **2010 Mathematics Subject Classification:** 46A03, 58K15\ # **Introduction** In functional analysis, weak topology on a linear topological space plays an important role to characterize the set of all continuous functionals. As we are all familiar with weak topology, we recall only a few notions of weak topology which will be needed for the discussion of this paper. A linear topology on a linear space $E$ over $K (=\mathbb{R}$ or $\mathbb{C})$ is a topology that makes each operation of $E$ continuous. A locally convex space is a linear topological space that produces a local base containing convex neighbourhoods of null vector $\theta$. To define a weak topology on a linear topological space, we use a well known theorem: Let $E$ be a linear space over $K (=\mathbb{R}$ or $\mathbb{C})$ and $\{P_\alpha:\,\alpha\in \Lambda\}$ be the collection of some seminorms on $E$. Let $\mathcal{V}=\lbrace B_{P_\alpha}(\theta, r_\alpha):\, r_\alpha>0, \, \alpha\in \Lambda\rbrace$, where $B_{P_\alpha}(\theta, r_\alpha)=\{x\in E: P_\alpha(x)<r_\alpha\}$ $\forall\, \alpha\in \Lambda$ and $\forall \,r_\alpha> 0$. Then the collection $\mathcal{U}$ of all finite intersection of the members of $\mathcal{V}$ forms an absolutely convex local base for the weakest locally convex topology on $E$ such that each $p_\alpha$ is continuous for $\alpha\in \Lambda$. If $\Gamma$ is the collection of some linear functional on a linear space $E$ over $K\,(=\mathbb{R}$ or $\mathbb{C})$ then $P_f(x)=|f(x)|$ is a seminorm for all $f\in \Gamma$. So by these collection of seminorms, we can always construct a weakest locally convex topology on $E$ such that each $P_f$ is continuous. This topology is called the $\Gamma$-topology on $E$ and is denoted by $\sigma (E, \Gamma)$. Now in particular, if $E$ is a linear topological space and $\Gamma=E^{*},$ the collection of all continuous linear functionals on $E$, the $\Gamma$-topology is called the weak topology $\sigma(E, E^{*})$ on $E$. In a linear space $E$, it can be shown that $(E, \sigma(E, \Gamma))^{*}=\Gamma$ if $\Gamma$ is a subspace of $E^{\prime}$, the linear space of all linear functionals. So, in particular $(E, \sigma(E, E^{*}))^{*}=E^{*}$. Now our target is to define another type of $\Gamma$-topology namely, delta weak topology. To define this type of $\Gamma$-topology, we first introduce a few notions which are already established. In 1979, T. Noiri [@Noiri] introduced the concept of $\delta$-cluster point. Then he defined $\delta$-closure of a set by the collection of all $\delta$-cluster points. Thereafter he introduced the definition of $\delta$-closed set with the help of $\delta$-closure and defined a $\delta$-open set as a complement of $\delta$-closed set. There he had given the notion of $\delta$-continuous function and established its few properties. According to Noiri, a function $f:X\rightarrow Y$ is said to be $\delta$-continuous if for each $x\in X$ and each open neighbourhood $V$ of $f(x)$, there exists an open neighbourhood $U$ of $x$ such that $f(int[cl(U)])\subseteq int[cl(V)]$. In 1993, S. Raychaudhuri and M. N. Mukherjee [@Raychaudhuri] generalized the concepts of $\delta$-continuity and $\delta$-open set and called these general definitions $\delta$-almost continuity and $\delta$-preopen set respectively. In 1993, N. Palaniappan and K. C. Rao [@Palaniappan] introduced a concepts of regular generalized closed set and regular generalized open set. Thereafter many works have been done on another concepts of $\delta$-semiopen set [@Park; @Caldas; @1], $\delta$-set [@Saleh], Semi$^{*}\delta$-open set [@Missier; @2], regular$^{*}$ open set [@Missier; @1; @Missier; @3]. In 2005, M. Caldas [@Caldas; @2] has investigated to find out some applications of $\delta$-preopen sets. In 2021, R. M. Latif [@Latif] has defined a $\delta$-continuous function in an another way. In this paper we have taken the definition of $\delta$-continuous function as stated by R.M. Latif and by these collection of $\delta$-continuous functions, we have introduced the concept of delta weak topology and tried to find out the behaviour of the collection of all $\delta$-continuous functionals and of the $\delta$-closure of a set on a delta weak topological space. # Preliminaries Here we have given a few basic definitions and theorems which will be needed in the sequel. Throughout this paper we have taken the field $K$ as either the field of real numbers $\mathbb{R}$ or the field of complex numbers $\mathbb{C}$. **Definition 1**. *[@Noiri] An open set $U$ of a topological space $(X,\tau)$ is said to be a Regular Open Set if $U=$ $int[cl(U)]$.* **Definition 2**. *[@Noiri] A subset $V$ of a topological space $(X,\tau)$ is said to be regular closed if $V=$ $cl[int(V)]$.* **Theorem 3**. *[@Roy] $V$ is a regular closed set iff $X\setminus V$ is a regular open set.* **Result 4**. *[@Roy] Let $(X, \tau)$ be a topological space and $U\in \tau$. Then $int[cl(U)]$ is regular open.* **Note 5**. *[@Roy] A subset $U$ of $(X,\tau)$ is called $\delta$-open set if for each $x\in U$, $\exists$ a regular open set $P$ such that $x\in P$ $\subseteq U$.* **Theorem 6**. *[@Roy] $V$ is $\delta$-closed iff $V$ is the intersection of all regular closed set containing $V$.* **Theorem 7**. *[@Roy] The intersection of finitely many regular open sets in any topological space is regular open.* **Theorem 8**. *[@Roy] The arbitrary union of $\delta$-open sets in any topological spaces is $\delta$-open.* **Theorem 9**. *[@Roy] The intersection of finitely many $\delta$-open sets in any topological spaces is $\delta$-open.* **Theorem 10**. *[@Roy] The intersection of finite collection of $\delta$-closed sets is $\delta$-closed.* **Definition 11**. *[@Latif] A mapping $f:(X,\tau)\rightarrow (Y,\sigma)$ is said to be $\delta$-continuous if the pre-image of every open subset of $Y$ is a $\delta$-open subset of $X$.* **Definition 12**. *Let $M$ be an absorbing subset of a linear space $E$ over $K$. Let $C_M(x)=\{\lambda>0: \frac{x}{\lambda}\in M\}$. Clearly $C_M(x)\neq \phi$. The mapping $p_M: E\rightarrow \mathbb{R}^+\cup\{0\}$ defined by $p_M(x)=\inf C_M(x)$ for all $x\in E$ is called the Minkowski functional of $M$.* **Definition 13**. *A real valued function $p$ defined on a linear space $E$ over $K$ is called a seminorm if the following conditions are satisfied :\ $(i)$ $p(x+y)\leq p(x)+p(y)$ for all $x, y\in E$,\ $(ii)$ $p(\alpha x)=|\alpha| p(x)$ for all $\alpha\in K$ and for all $x\in E$.* **Theorem 14**. *Let $M$ be an absorbing subset of a linear space $E$ over $K$ and $p_M$ be the Minkowski functional of $M$. Then\ $(i)$ $p_M(t x)=t \,p_M(x)$ for all $x\in E$ and for all $t> 0$,\ $(ii)$ $M$ is convex $\Rightarrow$ $p_M(x+y)\leq p_M(x)+p_M(y)$ for all $x, y\in E$,\ $(iii)$ $M$ is absolutely convex $\Rightarrow$ $M$ is a seminorm.* **Theorem 15**. *In a locally convex space $E$ over $K$, the absolutely convex closed neighbourhoods of $\theta$ forms a local base at $\theta$.* **Theorem 16**. *Let $E$ be a linear topological space and $A\subseteq E$. Then\ $(i)$ $A$ is balanced and $\theta\in int A\Rightarrow int A$ is balanced.\ $(ii)$ $A$ is convex $\Rightarrow int A$ is convex.* **Theorem 17**. *Let $E$ be a complex vector space. If $f$ is a complex linear functional, its real part is a real linear functional. Also, every real linear functional $f_0: E\rightarrow \mathbb{R}$ is the real part of a unique complex linear functional $f: E\rightarrow \mathbb{C}$ where $f(x)=f_0(x)-if_0(ix)$ for all $x\in E$.* **Theorem 18**. *$($ Hahn-Banach theorem $)$\ Suppose $(i)$ $M$ is a subspace of a real linear space $E$,\ $(ii)$ $p:E\rightarrow \mathbb{R}$ is a function satisfying $p(x+y)\leq p(x)+p(y)$ and $p(tx)=t p(x)$ for all $x, y\in E$ and $t\geq 0$,\ $(iii)$ $f_0$ is a linear functional on $M$ such that $f_0(x)\leq p(x)$ for all $x\in M$.\ Then there exists a real linear functional $f$ on $E$ such that $f(x)=f_0(x)$ for all $x\in M$ and $-p(-x)\leq f(x)\leq p(x)$ for all $x\in E$.* # Delta Weak Topology **Definition 19**. *Let $(E, \tau)$ be a linear topological space over the field $K= (\mathbb{R}$ or $\mathbb{C})$. The collection of all linear $\delta$-continuous functionals is called $\delta$- topological dual of $E$ and it is denoted by $\delta$-$E^{*}$. As every $\delta$-continuous mapping $f$ is continuous, $\delta$-$E^{*}\subseteq E^{*}$.* **Definition 20**. *Let $(E, \tau)$ be a linear topological space over the field $K= (\mathbb{R}$ or $\mathbb{C})$. Let $\delta$-$E^{*}$ be the $\delta$- topological dual of $E$. For each element $f\in \delta$-$E^{*}$ we define a seminorm $p_f(x)=|f(x)|$ where $x\in E$. Then the collection of all such seminorms $\{p_f:\, f\in \delta$-$E^{*}\}$ induces a topology on $E$. This topology is called the delta weak topology $(\delta$- weak topology $)$ and it is denoted by $\sigma (E, \delta$-$E^{*})$.* **Remark 21**. *From the above Definition [Definition 20](#d2){reference-type="ref" reference="d2"}, it is clear that $\sigma (E, \delta$-$E^{*})$ is the smallest locally convex topology on $E$ defined by the seminorms $\{p_f:\, f\in \delta$-$E^{*}\}$ such that each $p_f$ is continuous on $(E, \sigma (E, \delta$-$E^{*}))$ where $p_f(x)=|f(x)|$ for all $x\in E$. A local base at $\theta$, the null vector of $E$ on $(E, \sigma (E, \delta$-$E^{*}))$ consists the sets of the form $v(f_1, f_2,\cdots, f_n; r_1, r_2,\cdots,r_n)=\cap_{i=1}^{n}\{x\in E: \,|f_i(x)|<r_i\}$ where $n\in \mathbb{N}$, $f_i\in \delta$-$E^{*}$ and $r_i> 0$ for $i=1, 2, \cdots, n$.* **Theorem 22**. *If $(E, \tau)$ is a linear topological space then $\sigma (E, \delta$-$E^{*})\subseteq\sigma (E, E^{*})$, where $\sigma (E, E^{*})$ is the weak topology on $E$.* *Proof.* Since $f\in\delta$-$E^{*}$ implies that $f\in E^*$, the subbasic open neighbourhood $B_f^\delta(\theta, r)=\{x\in E: |f(x)|< r\}$ of $\theta$ in $\sigma (E, \delta$-$E^{*})$ is a subbasic open neighbourhood of $\theta$ in $\sigma (E, E^{*})$. So, $\sigma (E, \delta$-$E^{*})\subseteq\sigma (E, E^{*})$. ◻ **Note 23**. *If $(E, \tau)$ is a linear topological space then we know that $\sigma (E, E^{*})\subseteq \tau$. So from the above Theorem [Theorem 22](#th1){reference-type="ref" reference="th1"}, it follows that $\sigma (E, \delta$-$E^{*})\subseteq\sigma (E, E^{*})\subseteq \tau$.* **Note 24**. *Let $(E, \tau)$ be a linear topological space. Then $(E, \sigma (E, \delta$-$E^{*}))^*=\delta$-$E^{*}$ as $\delta$-$E^{*}$ is a linear subspace of $E^{'}$, the linear space of all linear functionals, that is, the collection of all continuous linear functionals of $(E, \sigma (E, \delta$-$E^{*}))$ is the set $\delta$-$E^{*}$.* **Theorem 25**. *Let $(E, \tau)$ be a linear topological space over $K$. Then $\sigma (E, \delta$-$E^{*})$ is the weakest topology on $E$ such that each member of $\delta$-$E^{*}$ is $\delta$-continuous.* *Proof.* Let $f\in \delta$-$E^{*}$. We first show that $f$ is $\delta$-continuous on $(E, \sigma (E, \delta$-$E^{*}))$. As $(E, \sigma (E, \delta$-$E^{*}))$ is a Locally convex space, it is enough to show that for any open neighbourhood $U$ of $0$ in $K$, there exists $V\in \sigma (E, \delta$-$E^{*})$ such that $int_\sigma cl_\sigma V=V$ and $\theta\in V\subseteq f^{-1}(U)$, where $int_\sigma$ and $cl_\sigma$ are denoted respectively by interior and closure operators of $(E, \sigma (E, \delta$-$E^{*}))$. Since $f\in \delta$-$E^{*}$, there exists $W\in \tau$ such that $int_\tau cl_\tau W=W$ and $\theta\in W\subseteq f^{-1}(U)$ where $int_\tau$ and $cl_\tau$ are denoted respectively by interior and closure operators of $(E, \tau)$. Then there exists a balanced neighbourhood $B$ of $\theta$ on $(E, \tau)$ such that $\theta\in B\subseteq W\subseteq f^{-1}(U)$. Since $f$ is linear on a linear topological space, $f$ is an open mapping on $(E, \tau)$. So $f(B)$ is a balanced neighbourhood of $0$ in $U$. So there exists $r>0$ such that $B\subseteq f^{-1}(-r, r)=\{x\in E: |f(x)|<r\}\subseteq W\subseteq f^{-1}(U)$ and $(-r, r)\subseteq U$. Let $V=\{x\in E: |f(x)|<r\}$. Then $V$ is a subbasic open neighbourhood of $\theta$ on $(E, \sigma (E, \delta$-$E^{*}))$ and $\theta\in V\subseteq f^{-1}(U)$. Now obviously $V\subseteq int_\sigma cl_\sigma V$.\ Now since $p_f$ is continuous on $(E, \sigma (E, \delta$-$E^{*}))$, $\{x\in E: |f(x)|=p_f(x)\leq r\}$ is closed in $\sigma (E, \delta$-$E^{*})$. So, $cl_\sigma V\subseteq \{x\in E: |f(x)|\leq r\}$. Then $int_\sigma cl_\sigma V\subseteq int_\sigma\{x\in E: |f(x)|\leq r\}\subseteq int_\tau \{x\in E: |f(x)|\leq r\}= \{x\in E: |f(x)|< r\}=V$ as $f$ is continuous on $(E,\tau)$. Thus $V$ is a regular open set on $(E, \sigma (E, \delta$-$E^{*}))$. So each member of $\delta$-$E^{*}$ is $\delta$-continuous on $(E, \sigma (E, \delta$-$E^{*}))$.\ Now let $\gamma$ be any topology on $E$ such that each member of $\delta$-$E^{*}$ is $\delta$-continuous on $(E, \gamma)$. Let $\{x\in E: |f(x)|< r\}$ be a subbasic $\sigma (E, \delta$-$E^{*})$-open neighbourhood of $\theta$, where $f\in \delta$-$E^{*}$. Since $f\in \delta$-$E^{*}$, $f$ is $\delta$-continuous on $(E, \gamma)$, that is, continuous on $(E, \gamma)$. Thus $\{x\in E: |f(x)|< r\}\in \gamma$. This completes the proof. ◻ **Theorem 26**. *Let $(E, \tau)$ be a linear topological space over $K$. Then $\delta$-$(E, \sigma (E, \delta$-$E^{*}))^{*}= \delta$-$E^{*}$.* *Proof.* From the above Theorem [Theorem 25](#th2){reference-type="ref" reference="th2"}, we can say that $\delta$-$E^{*}\subseteq\delta$-$(E, \sigma (E, \delta$-$E^{*}))^{*}$. Let $f\in \delta$-$(E, \sigma (E, \delta$-$E^{*}))^{*}$ and $U$ be any open neighbourhood of $f(\theta)$. Then there exists $V\in \sigma (E, \delta$-$E^{*})$ such that $\theta\in V\subseteq f^{-1}(U)$ and $int_\sigma cl_\sigma V=V$. Since $V\in \sigma (E, \delta$-$E^{*})$, there exist $f_1, f_2, \cdots, f_n \in \delta$-$E^{*}$ such that $V=\cap_{i=1}^n\{x\in E: |f_i(x)|<r_i\}$, where $r_i> 0$ for $i=1, 2, \cdots, n$. Since each $f_i\in \delta$-$E^{*}$, $\{x\in E: |f_i(x)|<r_i\}= f_i^{-1}(B(0, r_i))$ is a $\delta$-open on $(E, \tau)$ for $i=1, 2, \cdots, n$. So there exists regular open set $V_i$ on $(E, \tau)$ such that $\theta\subseteq V_i\subseteq \{x\in E: |f_i(x)|<r_i\}$ for $i=1, 2, \cdots, n$. So $\theta\in \cap_{i=1}^n V_i\subseteq V\subseteq f^{-1}(U)$.\ Obviously, $\cap_{i=1}^n V_i\subseteq int_\tau cl_\tau[ \cap_{i=1}^n V_i]$. Now $int_\tau cl_\tau [\cap_{i=1}^n V_i]\subseteq int_\tau [ \cap_{i=1}^ncl_\tau V_i]\subseteq \cap_{i=1}^n[ int_\tau cl_\tau V_i]= \cap_{i=1}^n V_i$ as each $V_i$ is regular on $(E, \tau)$. So for any open neighbourhood $U$ of $f(\theta)$, there exists a regular open set $\cap_{i=1}^n V_i$ on $(E,\tau)$ such that $\theta\in \cap_{i=1}^n V_i\subseteq f^{-1}(U)$. Again since $E$ is a linear topological space and $f$ is linear, $f\in \delta$-$E^{*}$. Thus $\delta$-$(E, \sigma (E, \delta$-$E^{*}))^{*}\subseteq \delta$-$E^{*}$. Hence $\delta$-$(E, \sigma (E, \delta$-$E^{*}))^{*}= \delta$-$E^{*}$. ◻ **Theorem 27**. *Let $(E, \tau)$ be a linear topological space. Then $\delta$-$(E, \sigma (E, \delta$-$E^{*}))^{*}= \delta$-$(E, \sigma(E, E^{*}))^{*}$.* *Proof.* From Theorem [Note 23](#n2){reference-type="ref" reference="n2"}, we have $\sigma (E, \delta$-$E^{*})\subseteq\sigma (E, E^{*})\subseteq\tau$. Let $f\in \delta$-$(E, \sigma (E, E^{*}))^{*}$ and $U$ be any open neighbourhood of $f(\theta)$. Then there exists a regular open set $V$ in $\sigma (E, E^{*})$ such that $\theta\in V\subseteq f^{-1}(U)$. Since $V\in\sigma (E, E^{*})$, there exist continuous functionals $f_1, f_2, \cdots, f_n$ on $(E, \tau)$ such that $V=\cap_{i=1}^n\{x\in E: |f_i(x)|<r_i\}$ where $r_i>0$ for $i=1, 2, \cdots, n$. Obviously $V\in\tau$. So, $V\subseteq int_\tau cl_\tau V$. Again $int_\tau cl_\tau V=int_\tau cl_\tau [\cap_{i=1}^n\{x\in E: |f_i(x)|<r_i\}]\subseteq int_\tau [\cap_{i=1}^n cl_\tau\{x\in E: |f_i(x)|<r_i\}]\subseteq \cap_{i=1}^n int_\tau [cl_\tau\{x\in E: |f_i(x)|<r_i\}]=\cap_{i=1}^n \{x\in E: |f_i(x)|<r_i\}=V$ as each $f_i$ is continuous on $(E, \tau)$. So, there exists a regular open set $V$ on $(E, \tau)$ such that $\theta\in V\subseteq f^{-1}(U)$. Again since $E$ is a linear topological space and $f$ is linear, $f\in \delta$-$E^{*}$. Again by Theorem [Theorem 26](#th3){reference-type="ref" reference="th3"} we have $\delta$-$(E, \sigma (E, \delta$-$E^{*}))^{*}= \delta$-$E^{*}$. So, $f\in \delta$-$(E, \sigma (E, \delta$-$E^{*}))^{*}$. Thus $\delta$-$(E, \sigma (E, E^{*}))^{*}\subseteq \delta$-$(E, \sigma (E, \delta$-$E^{*}))^{*}$.\ Let $f\in\delta$-$(E, \sigma (E, \delta$-$E^{*}))^{*}$ and $U$ be any open neighbourhood of $f(\theta)$. Then there exists a regular open set $V$ in $\sigma (E, \delta$-$E^{*})$ such that $\theta\in V\subseteq f^{-1}(U)$. Since $V\in\sigma (E, \delta$-$E^{*})$, $V\in\sigma(E, E^{*})$ as $\sigma (E, \delta$-$E^{*})\subseteq \sigma(E, E^{*})$ by Theorem [Theorem 22](#th1){reference-type="ref" reference="th1"}. So, there exist continuous functionals $f_1, f_2, \cdots, f_n$ on $(E, \tau)$ such that $V=\cap_{i=1}^n\{x\in E: |f_i(x)|<r_i\}$ where $r_i>0$ for $i=1, 2, \cdots, n$. Since $(E, \sigma (E, E^{*}))^{*}= E^{*}$, each $f_i\in (E, \sigma (E, E^{*}))^{*}$. So, $\{x\in E: |f_i(x)|<r_i\}$ is regular open set on $(E, \sigma (E, E^{*}))$, that is, $int^\sigma cl^\sigma\{x\in E: |f_i(x)|<r_i\}=\{x\in E: |f_i(x)|<r_i\}$ for all $i=1, 2, \cdots, n$ where $int^\sigma$ and $cl^\sigma$ are denoted respectively by interior and closure operators on $(E, \sigma (E, E^{*}))$. Since $V\in\sigma (E, E^{*})$, $V\subseteq int^\sigma cl^\sigma V$. Now $int^\sigma cl^\sigma V= int^\sigma cl^\sigma [\cap_{i=1}^n\{x\in E: |f_i(x)|<r_i\}]\subseteq int^\sigma [\cap_{i=1}^ncl^\sigma\{x\in E: |f_i(x)|<r_i\}]\subseteq \cap_{i=1}^n int^\sigma cl^\sigma\{x\in E: |f_i(x)|<r_i\}=\cap_{i=1}^n \{x\in E: |f_i(x)|<r_i\}=V$. So for any open neighbourhood $U$ of $f(\theta)$, there exists a regular open set $V$ on $(E,\sigma (E, E^{*}))$ such that $\theta\in V\subseteq f^{-1}(U)$. Again since $(E,\sigma (E, E^{*}))$ is a locally convex space and $f$ is linear, $f\in \delta$-$(E, \sigma(E, E^{*}))^{*}$. Thus $\delta$-$(E, \sigma (E, \delta$-$E^{*}))^{*}\subseteq \delta$-$(E, \sigma(E, E^{*}))^{*}$. Hence $\delta$-$(E, \sigma (E, \delta$-$E^{*}))^{*}= \delta$-$(E, \sigma(E, E^{*}))^{*}$. ◻ **Remark 28**. *Let $(E, \tau)$ be a linear topological space. Then\ $\delta$-$(E, \sigma (E, \delta$-$E^{*}))^{*}= \delta$-$(E, \sigma(E, E^{*}))^{*}=\delta$-$E^{*}$* **Theorem 29**. *Let $(E,\tau)$ and $(F, \tau_1)$ be two linear topological spaces over the field $K$ and $T: E\rightarrow F$ be linear. If $T: (E, \tau)\rightarrow (F, \tau_1)$ is $\delta$-continuous then $T: (E, \sigma(E, \delta$-$E^{*}))\rightarrow (F, \sigma(F, F^{*}))$ is $\delta$-continuous.* *Proof.* Let $V$ be any basic open neighbourhood of $\theta$ on $(F, \sigma(F, F^{*}))$. Then $V=\cap_{i=1}^n\{y\in F: |g_i(y)|<r_i\}$ where $r_i>0$ and $g_i\in F^{*}$ for $i=1, 2, \cdots, n$. Now $T^{-1}(V)=T^{-1}[\cap_{i=1}^n\{y\in F: |g_i(y)|<r_i\}]=\cap_{i=1}^n T^{-1}\{y\in F: |g_i(y)|<r_i\}=\cap_{i=1}^n \{x\in E: |g_i(T(x))|<r_i\}$. To show that $T^{-1}(V)\in \sigma(E, \delta$-$E^{*})$, it is enough to show that $g_i\circ T\in \delta$-$E^{*}$ for $i\in\{1, 2, \cdots, n\}$. It is easy to see that each $g_i\circ T$ is a linear functional on $E$. Let $i\in\{1, 2, \cdots, n\}$ and $U$ be any open set in $K$. Now $(g_i\circ T)^{-1}(U)= T^{-1}(g_i^{-1}(U))$. Since $g_i\in F^{*}$, $g_i^{-1}(U)$ is an open set on $(F, \tau_1)$. So, $T^{-1}(g_i^{-1}(U))$ is a delta open set on $(E, \tau)$. Thus $g_i\circ T\in \delta$-$E^{*}$ for $i\in\{1, 2, \cdots, n\}$. This completes the proof. ◻ **Corollary 30**. *Let $(E,\tau)$ and $(F, \tau_1)$ be two linear topological spaces over the field $K$ and $T: E\rightarrow F$ be linear. If $T: (E, \tau)\rightarrow (F, \tau_1)$ is $\delta$-continuous then $T: (E, \sigma(E, \delta$-$E^{*}))\rightarrow (F, \sigma(F, \delta$-$F^{*}))$ is $\delta$-continuous.* *Proof.* Follows from the Theorem [Theorem 29](#th4){reference-type="ref" reference="th4"}. ◻ **Definition 31**. *A Subset $B$ of a linear topological space $(E, \tau)$ is said to be $\delta$- bounded if for every $\delta$-open neighbourhood $U$ of $\theta$ there exists $\lambda_0>0$ such that $\lambda B\subseteq U$ for all $|\lambda|\leq \lambda_0$.* **Definition 32**. *Let $(E,\tau)$ and $(F, \tau_1)$ be two linear topological spaces over the field $K$. A linear mapping $T: (E, \tau)\rightarrow (F, \tau_1)$ is said to be $\delta$- bounded if $T$ maps every bounded set of $E$ to a $\delta$-bounded set of $F$.* **Theorem 33**. *Let $(E,\tau)$ be a linear topological space over the field $K$. A set $B\subseteq E$ is $\sigma(E, \delta$-$E^{*})$ $\delta$-bounded iff for each $f\in \delta$-$E^{*}$, $\sup_{x\in B}|f(x)|<\infty$.* *Proof.* Let $B\subseteq E$ be a $\sigma(E, \delta$-$E^{*})$ $\delta$-bounded and $f\in \delta$-$E^{*}$. Then by the Remark [Remark 28](#r1){reference-type="ref" reference="r1"}, $f\in\delta$-$(E, \sigma (E, \delta$-$E^{*}))^{*}$. So, $V(f,\, 1)=\{x\in E: |f(x)|<1\}$ is a $\delta$-open neighbourhood of $\theta$ on $(E, \sigma(E, \delta$-$E^{*}))$. Since $B\subseteq E$ is $\sigma(E, \delta$-$E^{*})$ $\delta$-bounded, there exists $\lambda_0>0$ such that $\lambda_0 B\subseteq V(f,\, 1)$. So, $|f(\lambda_0\, x)|<1$ for all $x\in B$, that is, $|f( x)|<\frac{1}{\lambda_0}$ for all $x\in B$, that is, $\sup_{x\in B}|f(x)|\leq \frac{1}{\lambda_0}$.\ Conversely, suppose the given condition is satisfied. Let $V$ be a basic $\delta$-open neighbourhood of $\theta$ in $\sigma(E, \delta$-$E^{*})$. Then there exist $f_i\in \delta$-$E^{*}$ for $i=1, 2, \cdots, n$ such that $V=\cap_{i=1}^n\{x\in E: |f_i(x)|<r_i\}$ where $r_i>0$ for $i=1, 2, \cdots, n$. Let $\sup_{x\in B}|f_i(x)|<k_i$ for $i=1, 2, \cdots, n$. Let $\lambda_0>0$ be chosen so that $k_i\lambda_0<r_i$ for $i=1, 2, \cdots, n$. Then for $|\lambda|\leq\lambda_0$ and $x\in B$, $|f_i(\lambda x)|=|\lambda||f_i(x)|\leq\lambda_0k_i<r_i$ for all $i=1, 2, \cdots, n$. So, $\lambda x\in V$ for all $|\lambda|\leq\lambda_0$ and $x\in B$. So, $B$ is $\sigma(E, \delta$-$E^{*})$ $\delta$-bounded set. ◻ **Theorem 34**. *Let $A$ and $B$ are non-void convex subsets of a linear topological space $E$ over $K$ and $A\cap B=\phi$. If $A$ is $\delta$-open then there exists $f\in\delta$-$E^{*}$ and $r\in \mathbb{R}$ such that $Re (f(x))<r\leq Re f(y)$ for all $x\in A$ and for all $y\in B$.* *Proof.* It is suffices to prove this theorem for real scalars. For, if the scalar field be $\mathbb{C}$ and $f_0$ be a $\delta$-continuous real linear functional that gives the required inequalities then $f(x)=f_0(x)-if_0(ix)$ is the required unique $\delta$-continuous complex linear functional of the theorem with $Re f=f_0$. Thus we may take $K=\mathbb{R}$.\ Let $a_0\in A$ and $b_0\in B$ and $x_0=b_0-a_0$. Then $x_0\neq\theta$. Let $C=x_0+(A-B)$. Obviously, $\theta\in C$. Also $C$ is $\delta$-open and convex. As $C$ is open and $\theta\in C$, the Minkowski functional $p$ of $C$ can be defined. Then $p(x+y)\leq p(x)+p(y)$ and $p(tx)=t.p(x)$ for any $x, y\in E$ and $t\geq 0$. Since $A\cap B=\phi$, $\theta\notin A- B$ and hence $x_0\notin C$. Consequently, $p(x_0)\geq 1$.\ Let $M=\{tx_0: t\in \mathbb{R}\}$ be the one dimensional subspace spanned by $x_0$. Let $f_0:M\rightarrow K$ be a linear functional defined by $f_0(tx_0)=t$.\ If $t\geq 0$, $f_0(t x_0)=t\leq t p(x_0)=p(t x_0)$.\ If $t< 0$, $f_0(t x_0)=t< 0\leq p(t x_0).$ Thus $f_0(y)\leq p(y)$ for all $y\in M$. Then by Hahn Banach theorem there exists a linear functional $f$ on $E$ such that $f(x)=f_0(x)$ for $x\in M$ and $-p(-x)\leq f(x)\leq p(x)$ for all $x\in E$. Now since $C\subseteq \{x\in E: p(x)\leq 1\}$, $p(x)\leq 1$ for all $x\in C$ and hence $f(x)\leq 1$ for all $x\in C$. Again for $x\in -C$, $-x\in C$ and hence $f(-x)\leq 1$, that is $f(x)\geq -1$. Thus $|f(x)|\leq 1$ for all $x\in C\cap (-C)$. Now since $C$ is $\delta$-open, $-C$ is $\delta$-open and hence $C\cap (-C)$ is $\delta$-open neighbourhood of $\theta$. So $f$ is bounded in a $\delta$-open neighbourhood of $\theta$.\ We now show that $f$ is $\delta$-continuous. Let $B(0, \epsilon)=\{\lambda\in K: |\lambda|<\epsilon\}$ be any neighbourhood of $0$ in $K$. Let $a\in f^{-1}(B(0, \epsilon)).$ Then $|f(a)|< \epsilon$. So there exists $r> 0$ such that $|f(a)|+ r< \epsilon$. Let $W=\frac{r}{2} V$ where $V=C\cap (-C).$ Then $W$ is also $\delta$-open neighbourhood of $\theta$ and for $\frac{r}{2}x\in W$, $|f(\frac{r}{2} x)|=\frac{r}{2} |f(x)|\leq \frac{r}{2}< r$. So, $a\in a+W\subseteq f^{-1}(B(0, \epsilon))$. Thus $f^{-1}(B(0, \epsilon))$ is a $\delta$-open neighbourhood of $a$ and hence $f$ is $\delta$-continuous.\ Now let $a\in A$ and $b\in B$. Then $f(a)-f(b)+1=f(a)-f(b)+f(x_0)= f(a-b+x_0)\leq p(a-b+x_0)\leq 1$ as $a-b+x_0\in C$. So, $f(a)\leq f(b)$ for any $a\in A$ and $b\in B$. Also Since $f$ is linear, $f(A)$ and $f(B)$ are convex subsets of $\mathbb{R}$, that is, intervals in $\mathbb{R}$ with $f(A)$ to the left of $f(B)$. Again since any linear functional is open, $f(A)$ is open. So there exists $r>0$ such that $f(x)<r\leq f(y)$ for $x\in A$ and $y\in B$. ◻ **Definition 35**. *Let $(E, \tau)$ be a topological space and $A\subseteq E$. The delta closure of $A$ is denoted by $\delta$-$cl(A)$ and is defined by\ $\delta$-$cl(A)=\{x\in E:\,$ every delta neighbourhood of $x$ intersects $A\}$.* **Note 36**. *Let $(E, \tau)$ be a topological space and $A\subseteq E$. Then $cl(A)\subseteq \delta$-$cl(A)$.* **Theorem 37**. *Let $(E, \tau)$ be a topological space and $A\subseteq E$. Then $\delta$-$cl(A)$ is $\delta$-closed.* *Proof.* Let $x\notin \delta$-$cl(A)$. Then there exists a $\delta$-open neighbourhood $U$ of $x$ such that $U\cap A=\phi$. Now if possible let $y\in U\cap \delta$-$cl(A)$ then $y\in U$ and $y\in \delta$-$cl(A)$ which implies that $U\cap A\neq\phi$, a contradiction. Thus $U\cap \delta$-$cl(A)=\phi$. Then $x\in U\subseteq E\setminus\delta$-$cl(A)$. Hence $E\setminus\delta$-$cl(A)$ is $\delta$-open, that is, $\delta$-$cl(A)$ is $\delta$-closed set. ◻ **Theorem 38**. *Let $A$ and $B$ are non-void convex subsets of a locally convex space $E$ over $K$ and $A\cap B=\phi$. Then there exists $f\in\delta$-$E^{*}$ and real numbers $r_1$ and $r_2$ with $r_1< r_2$ such that $Re f(x)<r_1<r_2< Re f(y)$ for all $x\in A$ and for all $y\in B$ if and only if $\theta\notin \delta$-$cl(A- B)$.* *Proof.* It is suffices to prove this theorem for real scalars. For, if the scalar field be $\mathbb{C}$ and $f_0$ be a $\delta$-continuous real linear functional that gives the required inequalities then $f(x)=f_0(x)-if_0(ix)$ is the required unique $\delta$-continuous complex linear functional of the theorem with $Re f=f_0$. Thus we may take $K=\mathbb{R}$.\ Suppose that there exists $f\in\delta$-$E^{*}$ and real numbers $r_1$ and $r_2$ with $r_1< r_2$ such that $f(x)<r_1<r_2< f(y)$ for all $x\in A$ and for all $y\in B$. If possible let $\theta\in \delta$-$cl(A- B)$. Then every delta neighbourhood of $\theta$ intersects $A - B$. Since $f\in\delta$-$E^{*}$, $\{x\in E: |f(x)|<r_2-r_1\}$ is a delta neighbourhood of $\theta$. So, $\{x\in E: |f(x)|<r_2-r_1\}\cap (A-B)\neq \phi$. Therefore there exist $a\in A$ and $b\in B$ such that $|f(a-b)|<r_2-r_1$. Thus $|f(b)- f(a)|<r_2-r_1$ which contradicts our assumption.\ Conversely, Let $\theta\notin \delta$-$cl(A- B)$. Then $(\delta$-$cl(A- B))^c$ is an $\delta$-open neighbourhood of $\theta$. Since $E$ is locally convex, by Theorem [Theorem 15](#th5){reference-type="ref" reference="th5"} there exists an absolutely convex closed neighbourhood $U$ of $\theta$ such that $U\subseteq (\delta$-$cl(A- B))^c$. Then $int [cl \,U]\subseteq (\delta$-$cl(A- B))^c$ as $cl\, U=U$, that is, $int [cl\, U]\cap \delta$-$cl(A- B)=\phi$. Then $int [cl\, U]\cap (A- B)=\phi$. Let $V= int [cl \,U]$. Then $V$ is regular open, that is, $V$ is $\delta$-open set and $V\cap (A-B)=\phi$ and also $V$ is convex. Then by Theorem [Theorem 34](#th6){reference-type="ref" reference="th6"}, there exists $f\in\delta$-$E^{*}$ and $r\in \mathbb{R}$ such that $f(v)<r\leq f(b)-f(a)$ for every $v\in V$, $a\in A$ and $b\in B$. Since $f$ is open and $V$ is an open convex neighbourhood of $\theta$, $f(V)$ is an open interval containing $0$ in $\mathbb{R}$. Let $\sup_{v\in V}f(v)=r_0$. Then $0<r_0\leq r$. Now $f(a)+r_0\leq f(a)+r\leq f(b)$ for all $a\in A$ and $b\in B$ implies that, $\sup_{a\in A}f(a)+r_0\leq f(b)$ for all $b\in B$. Then $\sup_{a\in A}f(a)+r_0\leq \inf_{b\in B}f(b)$. So for any $x\in A$ and $y\in B$ we have\ $f(x)\leq \sup_{a\in A}f(a)< \sup_{a\in A}f(a)+\frac{1}{2}r_0=r_1< \sup_{a\in A}f(a)+\frac{2}{3}r_0=r_2< \sup_{a\in A}f(a)+r_0\leq\inf_{b\in B}f(b)\leq f(y)$, that is, $f(x)<r_1<r_2< f(y)$ for every $x\in A$ and $y\in B$. ◻ **Theorem 39**. *Let $\tau_1$ and $\tau_2$ be two locally convex topological spaces on a linear space $E$ over $K$ such that they have the same $\delta$-continuous linear functionals, that is, $\delta$-$(E, \tau_1)^{*}=\delta$-$(E, \tau_2)^{*}$. Then an absolutely convex set $A\subseteq E$ is $\tau_1$ $\delta$-closed = $\tau_2$ $\delta$-closed.* *Proof.* Suppose a convex set $A\subseteq E$ is $\tau_1$ $\delta$-closed and $a\notin A$. Then $A-a$ is $\tau_1$ $\delta$-closed and $\theta\notin A-a$, that is, $\theta\notin \delta$-$cl_{\tau_1}(A-a)$. Then by Theorem [Theorem 38](#th7){reference-type="ref" reference="th7"}, there exists $\tau_1$ $\delta$-continuous functional $f$ and real numbers $r_1$, $r_2$ with $r_1<r_2$ such that $Re f(x)<r_1<r_2< Re f(a)$ for all $x\in A$. So, $\sup_{x\in A}Re f(x)\leq r_1<r_2< Re f(a)$. Since $A$ is absolutely convex and $f$ is linear, $f(A)$ an is absolutely convex subset of $K$. So, $f(A)$ is a circular disc with centre at $0$. Thus $|f(x)|\leq r_1<r_2< Re f(a)\leq |f(a)|$ for all $x\in A$, that is, $\sup_{x\in A}|f(x)|\leq r_1<r_2< |f(a)|$. So there exists $\epsilon>0$ such that $\sup_{x\in A}|f(x)|+ \epsilon < |f(a)|$. So, $|f(x)|+ \epsilon < |f(a)|$ for all $x\in A$, or, $|f(x)-f(a)|\geq |f(a)|-|f(x)|>\epsilon$ for all $x\in A$. Let $U=\{z\in E:\, |f(z)|<\epsilon\}$. By hypothesis $f$ is a $\tau_2$ $\delta$-continuous functional and hence $U$ is a $\tau_2$ $\delta$-open neighbourhood of $\theta$. So $a+U$ is a $\tau_2$ $\delta$-open neighbourhood of $a$. We now show that $(a+U)\cap A=\phi$. If $b\in A$ with $a+u=b$ for some $u\in U$. Then $|f(b)-f(a)|>\epsilon$, that is, $|f(b-a)|>\epsilon$, that is, $|f(u)|>\epsilon$ for some $u\in U$ which is a contradiction. Thus $a+U$ is a $\tau_2$ $\delta$-open neighbourhood of $a$ disjoint from $A$. Hence $A$ is $\tau_2$ $\delta$-closed.\ Interchanging $1$ and $2$, we can similarly show that $A$ is $\tau_2$ $\delta$-closed implies $A$ is $\tau_1$ $\delta$-closed. ◻ **Theorem 40**. *Let $(E,\tau)$ be a locally convex space over the field $K$. If $B$ is a convex subset of $E$, then $\sigma(E, \delta$-$E^{*})$-closure of $B=$delta $\tau$-closure of $B$, that is, $cl_{\sigma(E, \delta-E^{*})}(B)=\delta$-$cl_\tau(B)$.* *Proof.* Let $a\notin \delta$-$cl_\tau(B)$. Since $B$ and $\{a\}$ are disjoint convex sets in $E$ and $\theta\notin \delta$-$cl_\tau(B)-a=\delta$-$cl_\tau( \delta$-$cl_\tau(B)-a)$, by Theorem [Theorem 38](#th7){reference-type="ref" reference="th7"}, there exist $f\in\delta$-$E^{*}$ and real numbers $r_1$ and $r_2$ with $r_1< r_2$ such that $Re f(x)<r_1<r_2< Re f(a)$ for all $x\in \delta$-$cl_\tau(B)$. Since $f\in\delta$-$E^{*}$, by Remark [Remark 28](#r1){reference-type="ref" reference="r1"}, $f\in \delta$-$(E, \sigma (E, \delta$-$E^{*}))^{*}$. Then $Re f\in \delta$-$(E, \sigma (E, \delta$-$E^{*}))^{*}$. So $U=\{x\in E:\,|Re f(x)|<r_2-r_1\}$ is a delta $\sigma (E, \delta$-$E^{*})$ neighbourhood of $\theta$. Then $a+U$ is a delta $\sigma (E, \delta$-$E^{*})$ neighbourhood of $a$. We now show that $(a+U)\cap B=\phi$. If possible let $b=a+u$ for some $b\in B$ and $u\in U$. Then $|Re f(b-a)|<r_2-r_1$, that is, $|Re f(b)-Re f(a)|<r_2-r_1$ for some $b\in B\subseteq \delta$-$cl_\tau(B)$ which is a contradiction. Thus $a\notin cl_{\sigma(E, \delta-E^{*})}(B)$. So, $cl_{\sigma(E, \delta-E^{*})}(B)\subseteq \delta$-$cl_\tau(B)$.\ Again let $a\in\delta$-$cl_\tau(B)$ and $U$ be any $\sigma(E, \delta$-$E^{*})$ open neighbourhood of $\theta$. Then there exist $f_i\in \delta$-$E^{*}$ and $r_i>0$ for $i=1, 2, \cdots, n$ such that $U=\cap_{i=1}^n\{x\in E: f_i(x)<r_i\}$. Since $f_i\in \delta$-$E^{*}, \,\{x\in E: f_i(x)<r_i\}$ is delta open set on $(E, \tau)$ for $i=1, 2, \cdots, n$. So, $U$ is delta open set on $(E, \tau)$ containing $\theta$ and hence $(a+U)\cap B\neq\phi$. Therefore $a\in cl_{\sigma(E, \delta-E^{*})}(B)$. Thus $\delta$-$cl_\tau(B)\subseteq cl_{\sigma(E, \delta-E^{*})}(B)$. Hence $cl_{\sigma(E, \delta-E^{*})}(B)=\delta$-$cl_\tau(B)$. ◻ **Corollary 41**. *If $(E, \tau)$ be a locally convex space then\ $(i)$ a convex subset $B$ of $E$ is delta $\tau$-closed if and only if it is $\sigma(E, \delta$-$E^{*})$-closed.* *Proof.* Follows from the Theorem [Theorem 40](#th8){reference-type="ref" reference="th8"}. ◻ # Conclusion {#conclusion .unnumbered} To define delta weak topology, we have used the definition of $\delta$-continuity as stated by R. M. Latif [@Latif]. Then find out the characteristic of the set of $\delta$-continuities and $\delta$-closure of a set on delta weak topological spaces. So, one can try to define another type of $\Gamma$-topology with the help of T. Noiri's [@Noiri] definition of $\delta$-continuity and to find out the behaviour of various properties on these spaces. # Acknowledgement {#acknowledgement .unnumbered} I acknowledge my wife Mrs. Krishna Roy for her valuable suggestions in linguistic and grammatical parts of this paper. 0 *T. Noiri, On $\delta$-continuous functions*, J. Korean Math. Soc., 16(2), 1979, 161-166. *S. Raychaudhuri and M.N. Mukherjee, On $\delta$-almost continuity and $\delta$-preopen sets*, Bulletin of the Institute of Mathematics, Academia Sinica 21(4), 1993, 357-366. *N. Palaniappan and K. Chandrasekara Rao, Regular generalized closed sets*, Kyungpook Math., 33, 1993, 211-219. *J.H. Park, B. Y. Lee and M.J. Son, On $\delta$-semiopen sets in topological space*, J. Indian Acad. Math., 19(1), 1997, 59-67. *M. Saleh, Some applications of $\delta$-sets to H-closed spaces*, Questions and Answer in General Topology 17, 1999, 203-211. *M. Caldas, D.N. Georgian, S. Jafari and T. Noiri, More on $\delta$-semiopen sets*, Note di Mathematica, 22(2) 2003, 113-126. *M. Caldas, T. Fukutake, S. Jafari And T. Noiri, Some Applications Of $\delta$-Preopen Sets In Topological Spaces*, Bulletin Of The Institute Of Mathematics Academia Sinica 33(3), 2005, 261-276. *S. Pious Missier, M. Annalakshmi, Between Regular Open Sets And Open Sets*, International Journal of Mathematical Archive, 7(5), 2016, 128-133. *S. Pious Missier, C. Reena, On Semi$^{*}\delta$- Open Sets in Topological Spaces*, IOSR Journal of Mathematics (IOSR-JM), 12(5), 2016, 01-06. *S.Pious Missier, M.Annalakshmi and G. Mahadevan, On Regular$^{*}$-- Open Sets*, Global Journal of Pure and Applied Mathematics, 13(9), 2017, 5717-5726. *R. Mohammad Latif, Delta -- Open Sets And Delta -- Continuous Functions*, International Journal Of Pure Mathematics, 8 2021, 1-23. *S. Roy, S. Mondal, S. S. Roy and B. Mandal, A Study on Delta Compact Spaces*, Communicated.

arxiv_math

--- abstract: | We study the notion of degeneration for affine schemes associated to systems of algebraic differential equations with coefficients in the fraction field of a multivariate formal power series ring. In order to do this, we use an integral structure of this field that arises as the unit ball associated to the tropical valuation, first introduced in the context of tropical differential algebra. This unit ball turns out to be a particular type of integral domain, known as Bézout domain. By applying to these systems a translation map along a vector of weights that emulates the one used in classical tropical algebraic geometry, the resulting translated systems will have coefficients in this unit ball. When the resulting quotient module over the unit ball is torsion-free, then it gives rise to integral models of the original system in which every prime ideal of the unit ball defines an initial degeneration, and they can be found as a base-change to the residue field of the prime ideal. In particular, the closed fibres of our integral models can be rightfully called initial degenerations, since we show that the maximal ideals of this unit ball naturally correspond to monomial orders. We use this correspondence to define initial forms of differential polynomials and initial ideals of differential ideals, and we show that they share many features of their classical analogues. address: - Instituto de Matemáticas Unidad Oaxaca, Universidad Nacional Autónoma de México, Oaxaca, Mexico. - Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany. - Centro de Investigación en Matemáticas, A.C. (CIMAT), Jalisco S/N, Col. Valenciana CP. 36023 Guanajuato, Gto, México. - Bernoulli Institute, University of Groningen, The Netherlands author: - Lara Bossinger - Sebastian Falkensteiner - Cristhian Garay-López - Marc Paul Noordman bibliography: - bib.bib title: Tropical initial degeneration for systems of algebraic differential equations --- # Introduction In algebraic geometry, the study of algebraic geometric objects has long shifted from studying a single object to studying families of objects with certain characteristics in common. More specifically, degeneration techniques are a basic tool used broadly in algebraic geometry and are the foundation for subjects such as *the theory of integral models* for varieties, or *deformation theory*. The aim of the present article is to introduce these techniques to study the spaces of formal power series solutions of systems of algebraic differential equations. In the algebraic case, when studying solutions to polynomial equations, Gröbner theory provides the necessary tools to construct an *initial degeneration* of the polynomial system: these are flat families over the affine line $\pi:\mathcal X\to \mathbb A^1$ whose fibre $\mathcal X_t=\pi^{-1}(t)$ over (almost) any non zero value $t\in \mathbb A^1$ is (isomorphic to) the set of solutions of the original system, and whose fibre over the origin $\mathcal X_0=\pi^{-1}(0)$ is the set of solutions to a simplified polynomial system (for example, the polynomial equations are truncated to binomial or monomial equations). Gröbner theory is closely related to tropical geometry as the fundamental theorem of tropical (algebraic) geometry demonstrates [@maclagan2015introduction]. More precisely, every initial degeneration with irreducible special fibre corresponds to a point in the tropical variety (see, e.g. [@Bos_initial]). The generalization of the fundamental theorem to polynomial differential systems was initiated in [@AGT16] (the ordinary case) and taken further to the partial case in [@FGH20; @mereta_fund]. Our construction originates from these results, more precisely from the study of tropical solutions to tropical systems of algebraic differential equations, and our findings may be phrased in terms of non-Archimedean valuations or seminorms, as we explain below. We would like to mention that this paper is not the first exploring possible generalizations of algebraic methods to the differential case. In particular, Gröbner theoretical methods have been employed in a similar setting in the works [@ArocaRond; @Aroca-Ilardi; @FT20]. ## A new example of Bézout non-Archimedean norm We consider the fraction field $K(\!(\mathbf{t})\!)$ of the ring of formal power series in $m\geq1$ variables $\mathbf{t}=(t_1,\ldots,t_m)$ and with coefficients in a field of characteristic zero $K$. We study the algebraic and geometric properties of a recently introduced non-Archimedean norm $\operatorname{trop}$ defined on $K(\!(\mathbf{t})\!)$ having values in the (idempotent) fraction semifield of the semiring of *vertex polynomials* $V\mathbb{B}(\mathbf{t})$ ([@CGL; @FGH20], see §[2.3](#Sect_valuationsFormalPowerSeriesRings){reference-type="ref" reference="Sect_valuationsFormalPowerSeriesRings"} precisely [\[eq: def tropical seminorm\]](#eq: def tropical seminorm){reference-type="eqref" reference="eq: def tropical seminorm"}. This is the tropical seminorm, and it is not a classical Krull valuation, but rather a new example of the concept of Bézout $\ell$-valuation[^1], which was introduced in [@RY]. Even if the target semifield $V\mathbb{B}(\mathbf{t})$ is not totally ordered, this setting behaves in a similar way to the Krull valuations; in particular, the subring $K(\!(\mathbf{t})\!)^\circ\subset K(\!(\mathbf{t})\!)$ of elements having $\operatorname{trop}$ norm bounded by $1\in V\mathbb{B}(\mathbf{t})$ turns out to be a Bézout domain. In this context, this subring is the analogue of the valuation ring of a Krull valuation, and we call it the *unit ball* of the seminorm. This yields one of the first concrete applications of Bézout $\ell$-valuations through tropical geometry. We choose to present this concept in the language of non-Archimedean seminorms using idempotent semiring theory, and we recall connections to Bézout $\ell$-valuations. ## Application Consider the differential ring $(K[\![\mathbf{t}]\!],D)$, where $D=\{\tfrac{\partial}{\partial t_i}\::\:i=1,\ldots,m\}$ is the set of usual derivations. The tropical seminorm $\operatorname{trop}$ appeared first in the tropical aspect of systems of differential algebraic equations with coefficients in $K[\![\mathbf{t}]\!]$, see [@FGH20]. The differential field $(K(\!(\mathbf{t})\!),D)$ is a differential extension of $(K[\![\mathbf{t}]\!],D)$, and when $m>1$, the Bézout domain $K(\!(\mathbf{t})\!)^\circ\subset K(\!(\mathbf{t})\!)$ is a proper extension of $K[\![\mathbf{t}]\!]$. We apply these previous results to set up the theory of tropical initial degeneration of systems of algebraic differential equations with coefficients in $K(\!(\mathbf{t})\!)$. ## Initial degenerations {#SS:ID} Given an algebraic variety $X$ over a field $K$ and an integral domain $R$ with fraction field $\operatorname{Frac}(R)=K$, a *model* of $X$ over the base $B=\operatorname{Spec}(R)$ is a flat morphism $\pi:\mathcal X \to B$ such that $X\cong \mathcal X\times_B\operatorname{Spec}(K)$. Denote by $F_{m,n}$, respectively $R_{m,n}$, the ring of polynomials in the variables $\{x_{i,J}\::\:i=1,\ldots,n,\:J\in\mathbb{N}^m\}$ with coefficients in $K(\!(\mathbf{t})\!)$, respectively in the unit ball $K(\!(\mathbf{t})\!)^\circ$. For a given weight vector $w=(w_1,\ldots,w_n)\in \mathbb B[\![\mathbf{t}]\!]^n$ (here $\mathbb B$ denotes the Boolean semifield) and a differential polynomial $P\in F_{m,n}$, we define its $w$-translation $P_w\in R_{m,n}$ in [\[eq:init form\]](#eq:init form){reference-type="eqref" reference="eq:init form"}. Broadly speaking we are generalizing the ordinary case which appeared in [@FT20; @HuGao2020]. The $w$-translated ideal of a given ideal in $F_{m,n}$, is the ideal in $R_{m,n}$ generated by the $w$-translations of all its elements (Definition [Definition 29](#def:init ideal){reference-type="ref" reference="def:init ideal"}). An initial degeneration is specified by choosing a prime ideal $\mathfrak{p}\subset K(\!(\mathbf{t})\!)^\circ$. **Theorem 1**. *(Proposition [Proposition 30](#Proposition_Model){reference-type="ref" reference="Proposition_Model"} and Theorem [Theorem 32](#thm_degeneration){reference-type="ref" reference="thm_degeneration"}) Let $w\in \mathbb B[\![\mathbf{t}]\!]^n$ and $G \subset F_{m,n}$ be an ideal such that the quotient $R_{m,n}/G_w$ of $R_{m,n}$ by the $w$-translated ideal $G_w$ is a torsion-free $K(\!(\mathbf{t})\!)^\circ$-module. Then $\mathcal X(w):=\operatorname{Spec}(R_{m,n}/G_w)$ is a model for $\mathcal X(w)_{\eta}:=\operatorname{Spec}(F_{m,n}/G)$ over $K(\!(\mathbf{t})\!)^\circ$. Moreover, given any maximal ideal $\mathfrak m\in \operatorname{maxSpec}( K(\!(\mathbf{t})\!)^\circ)$ the corresponding closed fibre satisfies $$\mathcal X(w)_{\mathfrak{m}}\cong \operatorname{Spec}\left(K\{x_{i,J}\}/\overline{G}_w\!\right)$$ where $\overline{G}_w$ is the induced ideal in $K\{x_{i,J}\}$ (see the Proof of Lemma [Lemma 25](#lem:fiber_general){reference-type="ref" reference="lem:fiber_general"} for the precise definition of $\overline{G}_w$).* A concept closely related to the fibre of a model is the one of initial ideals. We use the $w$-translation map together with a maximal ideal $\mathfrak{m}\subset K(\!(\mathbf{t})\!)^{\circ}$ to define the initial form $\text{in}_{(w,\mathfrak{m})}(P)$ of $P\in F_{m,n}$ with respect to the pair $(w,\mathfrak{m})$, and the initial ideal $\text{in}_{(w,\mathfrak{m})}(G)$ with respect to the pair $(w,\mathfrak{m})$ of an ideal $G\subset F_{m,n}$. We show that for a polynomial $P\in F_{m,n}$, our expressions [\[eq:explicit_translation\]](#eq:explicit_translation){reference-type="eqref" reference="eq:explicit_translation"} for its $w$-translation $P_w\in R_{m,n}$, and [\[eq:explicit_initial_deg\]](#eq:explicit_initial_deg){reference-type="eqref" reference="eq:explicit_initial_deg"} for its initial form $\text{in}_{(w,\mathfrak{m})}(P)$ are natural generalizations of the same constructions for the case of standard tropical algebra, c.f. [@gubler2013guide §5]. The second half of the above theorem follows from a careful analysis of the set of maximal ideals in $K(\!(\mathbf{t})\!)^\circ$ denoted by $\operatorname{maxSpec}( K(\!(\mathbf{t})\!)^\circ)$. We find the following result which is proven in a constructive manner in both directions. **Theorem 2**. *([Theorem 41](#thm_characterization){reference-type="ref" reference="thm_characterization"}) There exists an explicit one-to-one correspondence between maximal ideals of $K(\!(\mathbf{t})\!)^{\circ}$ and monomial orders on $\mathbb{N}^m$.* This characterization is important since it allows us to give explicit formulas for computing the initial form $\text{in}_{(w,\mathfrak{m})}(P)$ of $P\in F_{m,n}$ with respect to the pair $(w,\mathfrak{m})$, see for instance [\[eq:concrete_reduction\]](#eq:concrete_reduction){reference-type="eqref" reference="eq:concrete_reduction"}, [\[eq:explicit:fpwcoeff\]](#eq:explicit:fpwcoeff){reference-type="eqref" reference="eq:explicit:fpwcoeff"}. ## Outline In [2](#Sect_Preliminaries){reference-type="ref" reference="Sect_Preliminaries"} we introduce non-Archimedean seminorms, idempotent semiring theory and Bézout $\ell$-valuations. We give results on these concepts which will be necessary for defining initial degeneration. Of uttermost importance hereby is the correspondence theorem between ideals in the domain and $k$-ideals in the image of a Bézout valuation (see [Theorem 12](#ideals_are_the_same){reference-type="ref" reference="ideals_are_the_same"}) which is new up to our knowledge. In Subsection [2.3](#Sect_valuationsFormalPowerSeriesRings){reference-type="ref" reference="Sect_valuationsFormalPowerSeriesRings"} we show that $\operatorname{trop}$ is indeed a Bézout seminorm on $K(\!(\mathbf{t})\!)$ ([Proposition 19](#Proposition_trop_is_Bezout){reference-type="ref" reference="Proposition_trop_is_Bezout"}) such that the previous result can be applied. Moreover, in [Proposition 22](#Proposition_nonNoetherian){reference-type="ref" reference="Proposition_nonNoetherian"} we show that the ring $K(\!(\mathbf{t})\!)^{\circ}$ defined by $\operatorname{trop}$ is in the multivariate case not Noetherian. In [3](#Sect_models_deg){reference-type="ref" reference="Sect_models_deg"}, the notions from tropical differential algebra are recalled and results from [2](#Sect_Preliminaries){reference-type="ref" reference="Sect_Preliminaries"} are used to show that models are found in this setting ([Proposition 30](#Proposition_Model){reference-type="ref" reference="Proposition_Model"}) such that initial degenerations can be properly defined, see [Theorem 32](#thm_degeneration){reference-type="ref" reference="thm_degeneration"}. We give the definitions of initial form and initial ideal at the pair $(w,\mathfrak{m})$ in Definition [Definition 36](#def:new_initials){reference-type="ref" reference="def:new_initials"}. The maximal ideals of $K(\!(\mathbf{t})\!)^{\circ}$ are studied detailed in the subsequent [4](#section_maximalIdeals){reference-type="ref" reference="section_maximalIdeals"}. In Proposition [Proposition 50](#prop:multiplicative){reference-type="ref" reference="prop:multiplicative"}, we show that taking the initial form at the pair $(w,\mathfrak{m})$ of a polynomial is a multiplicative map, as it is the case in classical tropical algebraic geometry. # Preliminaries {#Sect_Preliminaries} ## Non-Archimedean seminorms Let $R$ be a (commutative) ring and let $S$ be an idempotent (commutative) semiring. On $S$ we define the order $a \le b$ if and only if $a+b=b$. Order considerations on idempotent semirings will be made with respect to this order if no further remark is made. **Definition 1**. A *(non-Archimedean) seminorm* is a map $v:R\xrightarrow{}S$ from a ring $R$ to an idempotent semiring $S$ that satisfies 1. (unit) $v(0)=0$ and $v(1)=1$; 2. (sign) $v(-1)=1$; 3. (submultiplicativity) $v(ab)\leq v(a)v(b)$; and 4. (subaditivity) $v(a+b)\leq v(a)+v(b)$; The seminorm $v$ is a *norm* if every $a \ne 0$ fulfills $v(a) \ne 0$, and it is called multiplicative if $v(ab)= v(a)v(b)$ holds for every $a,b \in R$. A multiplicative norm is called a *valuation*. Let $v:R\xrightarrow{}S$ be a seminorm. The set $R^{\circ}:=\{a \in R : v(a) \leq 1\}$ is called the *unit ball of $v$*, and is a subring of $R$. By a little abuse of notation we also refer to the set $S^\circ:=\{x\in S:x\le 1\}$ as the unit ball of $S$; it is a subsemiring of $S$. **Definition 2**. A seminorm $v:R\xrightarrow{}S$ is called *integral* if $v(a) \leq 1$ for all $a \in R$. A seminorm $v:R\xrightarrow{}S$ is integral if and only if $R^\circ=R$. By restricting the domain of definition of $v$ to $R^{\circ}$, an integral seminorm $v^{\circ}:R^\circ\to S^\circ$ is induced. **Definition 3**. A subset $I$ of a semiring $S$ is an ideal if $0$, $a+b$ and $ac$ are elements of $I$ whenever $a,b\in I$ and $c\in S$. An ideal $I$ of $S$ is 1. a $k$-*ideal* or a *subtractive ideal*, if whenever $a+b\in I$, $a \in I$ and $b \in S$, then $b \in I$. 2. *prime*, if its complement $S\setminus I$ is a multiplicative subset of $S$. **Lemma 4**. *Let $S$ be an idempotent semiring. An ideal $I$ of $S$ is *downward closed* if whenever $b\in I$ and $a\le b$, we have $a\in I$. Subtractive ideals are equivalent to downward closed ideals.* *Proof.* Assume $I\subset S$ is subtractive and take $b\in I$ and $a\in S$ with $a\le b$. Then $a+b=b\in I$ which implies $a\in I$ as $I$ is subtractive. To the contrary, if $I$ is downward closed, consider $b\in I, a\in S$ and $a+b\in I$. Since $a+(a+b) = a+b$, it follows that $a \le a+b$, and because $I$ is downward closed, $a \in I$. ◻ Let $R$ be a ring. Let $\operatorname{Id}(R)$, respectively $\operatorname{fgId}(R)$, denote the set of ideals of $R$, respectively finitely generated ideals of $R$. Note that $\operatorname{Id}(R)$ and $\operatorname{fgId}(R)$ are semirings with respect to the sum and product of ideals. If $S$ is a semiring, we denote by $\operatorname{Id}_k(S)$ the set of $k$-ideals of $S$. By [@BG Corollary 3.8] the map $u_R:R\to \operatorname{fgId}(R)$ sending an element $a\in R$ to its principal ideal $(a)$ is a surjective valuation that induces an isomorphism of semirings $\operatorname{Id}(R)\cong \operatorname{Id}_k(\operatorname{fgId}(R))$. We use this correspondence for describing the maximal ideals of the unit balls $R^{\circ}$ and $V\mathbb{B}(\mathbf{t})^\circ$ of the tropical valuation $\operatorname{trop}:K(\!(\mathbf{t})\!)\rightarrow V\mathbb{B}(\mathbf{t})$ in terms of monomial orders in Section [4](#section_maximalIdeals){reference-type="ref" reference="section_maximalIdeals"}. ## Bézout $\ell$-valuations as seminorms If $S$ is a semiring, we denote by $U(S)\subset (S,\times,1)$ its group of multiplicatively invertible elements. We continue with the following concepts. **Definition 5**. An (integral) domain $R$ is called a *Bézout domain* if every finitely generated ideal is principal. Let $R$ be a Bézout domain. The greatest common divisor (gcd) and least common multiple (lcm) of two elements $a,b\in R$ always exist, and they satisfy $$ab=u\,\text{gcd}(a,b)\,\text{lcm}(a,b)$$ for some $u\in U(R)$. We write $\Gamma(R):=R/U(R)$ and denote by $\pi:R\to \Gamma(R)$ the quotient projection, sending $a\in R$ to its class $\pi(a)=[a]$. Note that $\Gamma(R)$ endowed with the product $[a][b]=[ab]$ and addition $[a]+[b]=[\text{gcd}(a,b)]$ is an idempotent semiring, and $\pi:R\to \Gamma(R)$ becomes an integral valuation. **Remark 6**. If $R$ is a Bézout domain, then there is an isomorphism of semirings $\operatorname{fgId}(R)\cong \Gamma(R)$, thus the norm $u_R:R\to \operatorname{fgId}(R)$ coincides with the quotient projection $\pi:R\to \Gamma(R)$. If $R$ is a Bézout domain, then the semiring $\Gamma(R)$ is multiplicatively cancellative and the fraction semifield $\operatorname{Frac}(\Gamma(R))$ exists. The valuation $\pi:R\to \Gamma(R)$ can be extended to a valuation $\operatorname{Frac}(\pi):\operatorname{Frac}(R)\to \operatorname{Frac}(\Gamma(R))$ with the usual definition: $\operatorname{Frac}(\pi)(\tfrac{a}{b})=\tfrac{\pi(a)}{\pi(b)}$. **Definition 7**. Let $R$ be a Bézout domain. Its divisibility semiring is the semiring $\Gamma(R)$, and its divisibility semifield is the semifield $\operatorname{Frac}(\Gamma(R))$. Throughout the remaining part of section, $R$ denotes a field and $S$ a semifield. In this case, $\operatorname{Frac}(R) = R$, $\Gamma(R)=\{ [0],[1] \}$, and $\pi:R\to \Gamma(R)$ is the trivial norm. **Definition 8**. Let $R$ be a field and let $S$ be a semifield. We say that a multiplicative seminorm $v:R\to S$ is *Bézout* if for every $a,b\in R$, there exists $x,y\in R^\circ$ such that $v(xa+yb)=v(a)+v(b)$. Since $R$ is a field, it follows that a Bézout seminorm is automatically a norm, hence, it is a valuation after Definition [Definition 1](#def_gen_nav){reference-type="ref" reference="def_gen_nav"}. Let $v:R\to S$ be Bézout valuation with induced integral norm $v^{\circ}:R^{\circ}\to S^{\circ}$. We start with the following observation about principal ideals of $R^{\circ}$. **Lemma 9**. *Given $a\in R^\circ$, we have $$(a) = \{b \in R^\circ : v(b) \leq v(a)\}.$$* *Proof.* The statement is clear if $a = 0$, so assume that $a \neq 0$. Then a straightforward computation shows that $$\begin{aligned} v(b) \leq v(a) \Longleftrightarrow v\left(\frac{b}{a}\right)=\frac{v(b)}{v(a)}\leq1 \Longleftrightarrow \frac{b}{a} \in R^\circ \Longleftrightarrow b \in (a).\quad\qedhere\end{aligned}$$ ◻ This already has the consequence that ideals are determined by seminorms, in the following sense. **Corollary 10**. *Let $I$ be an ideal of $R^\circ$ and $a \in I$. Then for any $b \in R^\circ$ with $v(b) \leq v(a)$, we have $b \in I$.* *Proof.* If $v(b) \leq v(a)$ then by [Lemma 9](#ideals_are_downwards_closed_gen){reference-type="ref" reference="ideals_are_downwards_closed_gen"}, we have $b \in (a) \subset I$. ◻ Therefore, an ideal $I \subset R^\circ$ is uniquely determined by the subset $v(I) \subseteq S^\circ$ and we have $$\label{eq:I and v(I)} I = \{a \in R : v(a) \in v(I)\}.$$ It follows from [Lemma 9](#ideals_are_downwards_closed_gen){reference-type="ref" reference="ideals_are_downwards_closed_gen"} that $v(I)$ is a downward closed subset of $S^\circ$. In particular, $v(I)$ is closed under multiplication with elements in $S^\circ$: if $b \in v(I)$ and $a \in S^\circ$, then $ab \leq b$ (since $a \leq 1$) and so $ab \in v(I)$. **Lemma 11**. *Let $v: R \to S$ be a Bézout valuation such that $v^{\circ}$ is surjective. Let $I$ be an ideal of $R^\circ$ and let $x, y \in v(I)$. Then $x + y \in v(I)$.* *Proof.* As $v^\circ$ is surjective there exist $f,g\in R^\circ$ such that $v(f)=x$ and $v(g)=y$. Moreover, by [\[eq:I and v(I)\]](#eq:I and v(I)){reference-type="eqref" reference="eq:I and v(I)"} we have that $f,g\in I$. As $v$ is Bézout there exist $\alpha,\beta\in R^\circ$ such that $$v(\alpha f+\beta g)=v(f)+v(g)=x+y.$$ Notice that $\alpha f+\beta g\in I$ as $I$ is an ideal and then again by [\[eq:I and v(I)\]](#eq:I and v(I)){reference-type="eqref" reference="eq:I and v(I)"} we conclude $x+y\in v(I)$. ◻ [Lemma 11](#closed_under_addition_gen){reference-type="ref" reference="closed_under_addition_gen"} implies that $v(I)$ is closed under summation, hence it is subtractive by Lemma [Lemma 4](#lem:substractive=downward closed){reference-type="ref" reference="lem:substractive=downward closed"}. Thus, if $I\subset R^{\circ}$ is an ideal, then $v(I)\subset S^\circ$ is a $k$-ideal. **Theorem 12** (Correspondence theorem). *Let $v : R \to S$ be a Bézout valuation such that $v^\circ:R^\circ\to S^\circ$ is surjective. There is a one-to-one correspondence between ideals $I\subset R^{\circ}$ and $k$-ideals $J \subset S^\circ$ given by $I=v^{-1}(J)$ and $J=v(I)$. This correspondence preserves prime ideals.* *Proof.* The only thing left to verify is that $v^{-1}(J)$ is an ideal of $R^{\circ}$. Let $f,g \in v^{-1}(J)=\{ g \in R^{\circ} : v(g) \in J\}$ and $r \in R^{\circ}$. Then $f+g \in v^{-1}(J)$ and $rf \in v^{-1}(J)$, since $v(f+g) \le v(f)+v(g) \in J$ and $v(rf) \le v(r)v(f) \in J$ and $J$ is downwards closed. For the second statement, let $I$ and $J$ be two ideals such that $I=v^{-1}(J)$ and $v(I)=J$. Since $f\notin I$ if and only if $v(f)\notin J$ and $v: R^{\circ} \to S^{\circ}$ is surjective, the statement follows. ◻ **Remark 13**. As we pointed out before, the correspondence Theorem [Theorem 12](#ideals_are_the_same){reference-type="ref" reference="ideals_are_the_same"} is related to [@BG Corollary 3.8], the difference is that in our case, the direct image $v(I)\subset S^\circ$ of an ideal $I\subset R^\circ$ under the surjective valuation $v^\circ$ is already a $k$-ideal of $S^\circ$, and there is no need to take its $k$-closure. Although we will not need a generalization of the correspondence theorem to radical and primary ideals, we will give a proof here for the sake of completeness. **Proposition 14**. *The correspondence induced by a Bézout valuation $v$ preserves radical and primary ideals.* *Proof.* First, let us show the preservation of radical ideals. Any radical ideal can be written as intersection of minimal prime ideals, see e.g. [@Eisenbud_commutative Corollary 2.12]. Since $v$ preserves primes, it remains to show that minimal primes are preserved. For this purpose, let $I$ be a minimal prime ideal in $R^{\circ}$, i.e., for every prime ideal $J$ with $J \subseteq I$ it holds that $J=I$, and let $J' \subseteq v(I)$ be a prime ideal. Then there is a prime ideal $J$ such that $v(J)=J'$. Thus, $v(J) \subseteq v(I)$ and therefore $J \subseteq I$. By the minimality, $J=I$. The same argument can be used for $v^{-1}$ and minimal prime $k$-ideals in $S^{\circ}$ and the statement follows. Now, let us show the preservation of primary ideals. Let $J\subset S^{\circ}$ be a primary $k$-ideal and let $I=v^{-1}(J)$. If $fg\in I$, then $v(fg)=v(f)v(g)\in J$. For $v(f)\notin J$, we have $f \notin I$. Then $v(g)^n=v(g^n)\in J$ and $g^n\in I$ for some $n \in \mathbb{N}$. Conversely, let $I\subset R^{\circ}$ be a primary ideal and let $J=v(I)$. Let $xy\in J$ and $f \in v^{-1}(x), g \in v^{-1}(y)$ be arbitrary. Since $fg \in v^{-1}(xy) \subset I$, then $f \in I$ or $g^n \in I$ for some natural number $n$. Thus, $v(f) \in J$ or $v(g^n)=v(g)^n \in J$. ◻ The next result shows that for a surjective Bézout valuation $v:R\to S$, the unit ball $R^\circ$ is Bézout and $v$ is characterized by the norm $\pi:R^\circ\to \Gamma(R^\circ).$ **Lemma 15**. *Let $v : R \to S$ be a surjective Bézout valuation. Then $v$ is isomorphic to the norm $\operatorname{Frac}(\pi):\operatorname{Frac}(R^\circ)\to \operatorname{Frac}(\Gamma(R^\circ))$.* *Proof.* First we show that $S=\operatorname{Frac}(S^\circ)$. Given some $x \in S$ we want to write it as $x = a/b$ with $a, b \leq 1$. For this we just set $a = x/(1+x)$ and $b = 1/(1+x)$. Since $x \leq 1 + x$ and $1 \leq 1 + x$ we have $a, b \leq 1$ and also clearly $x = a/b$. So $S^\circ\subseteq S \subseteq \operatorname{Frac}(S^\circ)$, and the inclusion $\operatorname{Frac}(S^\circ) \subseteq S$ follows from the fact that $S$ is a semifield. Assume that $v$ is surjective. Then for $x\in S^\circ$ there exists $a\in R$ such that $v(a)=x$, but $a\in R^{\circ}$ by definition. Thus, $v^\circ$ is surjective. Since $\operatorname{Frac}(R^\circ)=R$ [@RY Proposition 1], there are $a,b\in R^\circ$ such that $x=\tfrac{a}{b}$. So $v(x)=\tfrac{v(a)}{v(b)}=\tfrac{v^\circ(a)}{v^\circ(b)}$, and $v:R\to S$ is the extension $\operatorname{Frac}(v^\circ)$ of $v^{\circ}:R^{\circ}\to S^{\circ}$. If $v:R\to S$ is a Bézout valuation, then $R^\circ$ is a Bézout domain by [@RY Proposition 1]. Then every finitely generated ideal of $R^{\circ}$ is principal, and the isomorphism $\operatorname{Id}(R^\circ)\cong \operatorname{Id}_k(S^\circ)$ of Theorem [Theorem 12](#ideals_are_the_same){reference-type="ref" reference="ideals_are_the_same"} restricts to a isomorphism $\Gamma(R^\circ)\cong\operatorname{fgId}(R^\circ)\cong \operatorname{fgId}_k(S^\circ)$ sending $(a)$ to $v^{\circ}((a))$. Note that $v^{\circ}((a))=(v^{\circ}(a))_k$: the $k$-closure of the principal ideal generated by $v^{\circ}(a)$. We now make the identification $S^\circ\cong \operatorname{fgId}_k(S^\circ)$ by sending $s$ to $(s)_k$. This is clearly surjective, and it is injective since $(s)_k=(t)_k$ implies $s\leq t$ and $t\leq s$, so $t=s$. Thus $\phi:\Gamma(R^\circ) \to S^\circ$ given by $\phi((a))=v^{\circ}(a)$ is an isomorphism satisfying $v^\circ=\phi\circ\pi$, and $v=\operatorname{Frac}(v^\circ)=\operatorname{Frac}(\phi)\circ \operatorname{Frac}(\pi)$. ◻ **Corollary 16**. *Let $v : R \to S$ be a Bézout valuation. Then $v$ is surjective if and only if $v^\circ$ is surjective.* *Proof.* Let $v^\circ$ be surjective. Since $S=\operatorname{Frac}(S^\circ)$ (see proof of [Lemma 15](#cor:structure){reference-type="ref" reference="cor:structure"}), for any $y\in S$ there are $y_1,y_2\in S^\circ$ such that $y=\tfrac{y_1}{y_2}$, and since $v^\circ$ is surjective, there exists $a_1,a_2\in R$ such that $v(\tfrac{a_1}{a_2})=\tfrac{v^\circ(a_1)}{v^\circ(a_2)}=\tfrac{y_1}{y_2}=y$. Thus, $v$ is surjective. The reverse direction is proven in [Lemma 15](#cor:structure){reference-type="ref" reference="cor:structure"}. ◻ **Corollary 17**. *Let $v : R \to S$ be a surjective Bézout valuation. Then $U(R^\circ)=\{x\in R\::\: v(x)=1\}$.* *Proof.* If $x\in R$ satisfies $v(x)=1$, then $x\in R^\circ$. Since $v$ is surjective, we have that $\operatorname{Frac}(R^\circ)=R$, so there exist $y,z\in R^\circ$ such that $x=\tfrac{y}{z}\neq0$, then $v(y)=v(z)$, and $x^{-1}=\tfrac{z}{y}\in R^\circ$. Conversely, if $x\in U(R^\circ)$, then $v(x)$ is invertible in $S^\circ$ (since $v$ is multiplicative), but $U(S^\circ)=\{1\}$ since it is simple (and thus sharp). ◻ In the next section we apply these results to a concrete example of a Bézout valuation. ## Valuations in the context of formal power series rings {#Sect_valuationsFormalPowerSeriesRings} Let $\mathbb B=\{0,1\}$ denote the Boolean idempotent semifield. Fix an integer $m\ge 1$ and a tuple of variables $\mathbf{t}=(t_1,\dots,t_m)$. We denote by $\mathbb{B}[\![\mathbf{t}]\!]$ the semiring of formal Boolean power series, and by $V\mathbb{B}[\mathbf{t}]$ the idempotent semiring of vertex polynomials as in [@CGL]. **Remark 18**. Recall that $\mathbb{B}[\![\mathbf{t}]\!]$ is isomorphic to $\mathcal{P}(\mathbb{N}^m)$ and $V\mathbb{B}[\mathbf{t}]$ is isomorphic to $\mathbb{T}_m$ from [@FGH20] with the isomorphism given by taking supports. More precisely, the elements of $V\mathbb{B}[\mathbf{t}]$ are subsets of $\mathbb N^m$ that are equal to the vertices of the Newton polyhedra they generate. The addition $\oplus$ on $V\mathbb{B}[\mathbf{t}]$ is given by taking the set union of two vertex polynomials and then projecting onto the vertices of the outcome. Similarly, the product $\odot$ on $V\mathbb{B}[\mathbf{t}]$ is given by taking the Minkowski sum of the vertex polynomials and then projecting onto the vertices of the outcome. Since $V\mathbb{B}[\mathbf{t}]$ is integral (i.e. multiplicatively cancellative, that is whenever $a\odot b=a\odot c$ then either $a=0$ or $b=c$), we can construct the fraction semifield $V\mathbb{B}(\mathbf{t}):=\text{Frac}(V\mathbb{B}[\mathbf{t}])$ as follows: the elements of $V\mathbb{B}(\mathbf{t})$ are of the form $\frac{a}{b}$ with $a,b\in V\mathbb{B}[\mathbf{t}]$ and $b\neq0$, the map $V\mathbb{B}[\mathbf{t}]\xrightarrow[]{} V\mathbb{B}(\mathbf{t})$ sending $a$ to $\frac{a}{1}$ is an embedding, and $\frac{a}{b}=\frac{c}{d}$ if and only if $a \odot d=b \odot c$. The sum and product of fractions are defined as usual: $\frac{a}{b}\odot\frac{c}{d}=\frac{a\odot c}{b\odot d}$ and $\frac{a}{b} \oplus \frac{c}{d}=\frac{a \odot d\oplus b\odot c}{b\odot d}$. Note that on $V\mathbb{B}(\mathbf{t})$ we have the order $\frac{a}{b}\leq\frac{c}{d}$ if and only if $a\odot d\leq b\odot c$. Let $K[\![\mathbf{t}]\!]$ be the ring of formal power series in the variables $\mathbf{t}=(t_1,\ldots,t_m)$ with coefficients in the field $K$. We define the tropical seminorm $$\label{eq: def tropical seminorm} \operatorname{trop}:K[\![\mathbf{t}]\!]\rightarrow V\mathbb{B}[\mathbf{t}]$$ as the composition of the maps $\operatorname{Supp}: K[\![\mathbf{t}]\!]\rightarrow \mathbb{B}[\![\mathbf{t}]\!]$ given by taking the support set of a power series, composed with $V:\mathbb{B}[\![\mathbf{t}]\!]\rightarrow V\mathbb{B}[\mathbf{t}]$ given by projecting onto the vertex set of the Newton polyhedron generated by an element of $\mathbb{B}[\![\mathbf{t}]\!]$. The tropical seminorm is a surjective valuation, so we will call it the tropical valuation. Recall that the map $\operatorname{Frac}(\operatorname{trop}):K(\!(\mathbf{t})\!)\rightarrow V\mathbb{B}(\mathbf{t})$ is defined by $\operatorname{Frac}(\operatorname{trop})(\frac{\varphi}{\psi}):=\frac{\operatorname{trop}(\varphi)}{\operatorname{trop}(\psi)}$. This map is also a surjective valuation by [@CGL Corollary 7.3], thus we will call it also the tropical valuation and will be denoted by $\operatorname{trop}$. Moreover, we have: **Proposition 19**. *The tropical valuation $\operatorname{trop}: K(\!(\mathbf{t})\!)\to V\mathbb{B}(\mathbf{t})$ is a surjective $K$-algebra Bézout valuation.* *Proof.* Let $\varphi=\sum_Ia_I\mathbf{t}^I$ and $\psi=\sum_Ib_I\mathbf{t}^I$ be nonzero elements in $K[\![\mathbf{t}]\!]$. If $\operatorname{trop}(\varphi+\psi)\neq\operatorname{trop}(\varphi)\oplus\operatorname{trop}(\psi)$, there exists $I\in \operatorname{trop}(\varphi)\oplus\operatorname{trop}(\psi)$ such that $a_I+b_I=0$. Since $\operatorname{trop}(\varphi)\oplus\operatorname{trop}(\psi)$ is a polynomial, we can choose $M\in\mathbb{N}$ such that $a_I+Mb_I\neq0$ for all $I\in \operatorname{trop}(\varphi)\oplus\operatorname{trop}(\psi)$. It follows that $\operatorname{trop}(\varphi+M\mathbf{t}^0\psi)=\operatorname{trop}(\varphi)\oplus\operatorname{trop}(\psi)$ and $\operatorname{trop}(M\mathbf{t}^0)=1$. Now consider $\frac{\varphi_1}{\varphi_2}$ and $\frac{\psi_1}{\psi_2}$ elements in $K(\!(\mathbf{t})\!)^*$. We apply the above argument to $\varphi=\varphi_1\psi_2$ and $\psi=\varphi_2\psi_1$ to find $$\operatorname{trop}\left(\frac{\varphi_1}{\varphi_2}+M\mathbf{t}^0\frac{\psi_1}{\psi_2}\right) =\frac{\operatorname{trop}(\varphi+M\mathbf{t}^0\psi)}{\operatorname{trop}(\varphi_2\psi_2)} =\operatorname{trop}\left(\frac{\varphi_1}{\varphi_2}\right)\oplus\operatorname{trop}\left(\frac{\psi_1}{\psi_2}\right),$$ which finishes the proof. ◻ The following result gives a concrete characterization of the tropical valuation $\operatorname{trop}:K(\!(\mathbf{t})\!)\to V\mathbb{B}(\mathbf{t})$. **Corollary 20**. *Let $\operatorname{trop}^\circ:K(\!(\mathbf{t})\!)^\circ\to V\mathbb{B}(\mathbf{t})^\circ$ be the induced integral valuation. Then $V\mathbb{B}(\mathbf{t})^\circ\cong \{\operatorname{trop}(x)\leq 1\}/\{\operatorname{trop}(x)=1\}$, and $\operatorname{trop}^\circ$ is isomorphic to the resulting quotient projection.* *Proof.* Since $\operatorname{trop}: K(\!(\mathbf{t})\!)\to V\mathbb{B}(\mathbf{t})$ is a surjective Bézout seminorm, by [Proposition 19](#Proposition_trop_is_Bezout){reference-type="ref" reference="Proposition_trop_is_Bezout"} and [Lemma 15](#cor:structure){reference-type="ref" reference="cor:structure"}, it follows that $$V\mathbb{B}(\mathbf{t})^\circ\cong\Gamma(K(\!(\mathbf{t})\!)^\circ)=K(\!(\mathbf{t})\!)^\circ/U(K(\!(\mathbf{t})\!)^\circ),$$ and $U(K(\!(\mathbf{t})\!)^\circ)=\{x\in K(\!(\mathbf{t})\!)\::\:\operatorname{trop}(x)=1\}$ by Corollary [Corollary 17](#cor:unit_ball){reference-type="ref" reference="cor:unit_ball"}. ◻ The correspondence Theorem [Theorem 12](#ideals_are_the_same){reference-type="ref" reference="ideals_are_the_same"} says that we have a semiring isomorphism $$\operatorname{Id}(K(\!(\mathbf{t})\!)^\circ)\cong \operatorname{Id}_k(V\mathbb{B}(\mathbf{t})^\circ)\cong \operatorname{Id}_k(K(\!(\mathbf{t})\!)^\circ/\{\operatorname{trop}(x)=1\}).$$ We use this correspondence to characterize the maximal $k$-ideals of the semiring $V\mathbb{B}(\mathbf{t})^\circ$ in Corollary [Corollary 49](#cor:max:ideals:unitball){reference-type="ref" reference="cor:max:ideals:unitball"}. **Example 21**. Suppose that $m=1$. Since for $f \in K[\![t]\!]$ it holds that $\operatorname{trop}(f)=\min(\operatorname{Supp}(f))$, we have that $$\begin{aligned} K(\!(t)\!)^\circ &= \{ f/g \in K(\!(t)\!) : \min(\min(\operatorname{Supp}(f)),\min(\operatorname{Supp}(g))) = \min(\operatorname{Supp}(g)) \} \\ &= \{ f/g \in K(\!(t)\!) : \min(\operatorname{Supp}(f)) \ge \min(\operatorname{Supp}(g)) \} =K[\![t]\!],\end{aligned}$$ which is Noetherian. **Proposition 22**. *For $m>1$, the ring $K(\!(\mathbf{t})\!)^\circ$ is not Noetherian.* *Proof.* We demonstrate the statement in the case of $\mathbf{t}=(t,u)$, the general case follows. Define, for each $n \geq 1$, the element $$\omega_n := %\frac{\{(2n+1,0), (0,2n+1)\}}{\{(2n+1,0), (n,n), (0,2n+1)\}} \frac{t^{2n+1}+u^{2n+1}}{t^{2n+1}+t^nu^n+u^{2n+1}} \in V\mathbb{B}(t,u).$$ Then for each $n$ we have that $\omega_n \in V\mathbb{B}(t,u)^\circ$, and also $\omega_n < \omega_{n+1}$. Define $$I_n = \{q \in K(\!(\mathbf{t})\!)^\circ : \operatorname{trop}(q) \leq \omega_n\}.$$ It follows from the correspondence Theorem [Theorem 12](#ideals_are_the_same){reference-type="ref" reference="ideals_are_the_same"} that $I_n$ defines an ideal in $K(\!(\mathbf{t})\!)^{\circ}$, which can also be proven directly. Then it follows that $I_n \subsetneq I_{n+1}$ and the $I_n$ form a strictly increasing sequence of ideals in $K(\!(\mathbf{t})\!)^\circ$. ◻ # Models over the unit ball and initial degenerations {#Sect_models_deg} For a fixed integer $m\geq1$, we consider the Bézout-valued field $(K(\!(\mathbf{t})\!),\operatorname{trop})$ together with its unit ball $K(\!(\mathbf{t})\!)^\circ\subset K(\!(\mathbf{t})\!)$ from the previous section. The purpose of this section is to create models over $K(\!(\mathbf{t})\!)^\circ$ of certain schemes $X$ defined over $K(\!(\mathbf{t})\!)$, as well as initial degenerations of certain $K(\!(\mathbf{t})\!)$-algebras which appear in the theory of differential algebra. In this first section, we follow [@gubler2013guide §4]. ## Models over the unit ball Let $R$ be an integral domain such that $\operatorname{Frac}(R)=K$. An $R$-*model* of a scheme $X$ over the field $K$ is a flat scheme $\mathcal X$ over $R$ with generic fiber $\mathcal{X}_\eta=\mathcal{X}_K=X$. See [@gubler2013guide Definition 4.1]. Equivalently, if $B=\operatorname{Spec}(R)$, then a model of $X$ over $B$ is a flat morphism[^2] $\pi:\mathcal X \to B$ such that $X= \mathcal X\times_B\operatorname{Spec}(K)$. Then $X$ is called the *generic fibre* of $\mathcal X$. Sometimes we also ask the map $\pi:\mathcal X \to B$ to be proper. **Lemma 23**. *A module over $K(\!(\mathbf{t})\!)^\circ$ is flat if and only if it is torsion-free.* *Proof.* Any flat module is torsion-free. The converse follows from the fact that torsion-free modules over Prüfer domains, and hence Bézout domains, are flat (see e.g. [@bazzoni2006prufer Theorem 3.3]). ◻ For another fixed integer $n\geq 1$, we consider the set of variables $\{x_{i,J}\::\:1\leq i\leq n,\:J\in\mathbb{N}^m\}$, which we abbreviate simply by $\{x_{i,J}\}$. Let us denote by $R_{m,n}$ the ring of polynomials in the variables $\{x_{i,J}\}$ with coefficients in the unit ball $K(\!(\mathbf{t})\!)^\circ$. It is customary to express the elements of $R_{m,n}$ as finite sums $P=\sum_Ma_ME_M$ where $a_M\in K(\!(\mathbf{t})\!)^\circ$ and $E_M$ are finite products of the variables $\{x_{i,J}\}$. We consider flat schemes over $K(\!(\mathbf{t})\!)^\circ$ of the form $\mathcal{X}=Spec(A)$ for $A=R_{m,n}/J$, thus its generic fiber $X=\mathcal{X}_\eta$ is of the form $X=Spec(A_{K(\!(\mathbf{t})\!)})$ for $A_{K(\!(\mathbf{t})\!)}:=A\otimes_{K(\!(\mathbf{t})\!)^{\circ}}K(\!(\mathbf{t})\!)$. By flatness we have $A\subset A_{K(\!(\mathbf{t})\!)}$. A closed subscheme $Y$ of $X$ is given by an ideal $I_Y\subset A_{K(\!(\mathbf{t})\!)}$. The closure $\overline{Y}$ of $Y$ in $\mathcal{X}$ (in the Zariski topology) is defined as the closed subscheme of $\mathcal{X}$ given by the ideal $I_Y\cap A$. The following result and its proof are straightforward generalizations of [@gubler2013guide Proposition 4.4]. **Proposition 24**. *The closure $\overline{Y}$ of $Y$ in $\mathcal{X}$ is the unique closed subscheme of $\mathcal{X}$ with generic fiber $Y$ which is flat over $K(\!(\mathbf{t})\!)^\circ$* In particular, we can apply Proposition [Proposition 24](#prop:creation_of_models){reference-type="ref" reference="prop:creation_of_models"} to $A=R_{m,n}$, so that $A_{K(\!(\mathbf{t})\!)}=F_{m,n}$ is the ring of polynomials in the variables $\{x_{i,J}\}$ and coefficients in $K(\!(\mathbf{t})\!)$. Thus, given an ideal $I_X\subset F_{m,n}$, its contraction $I_X\cap R_{m,n}$ gives already a model $\mathcal{X}:=\operatorname{Spec}(R_{m,n}/I_X\cap R_{m,n})$ over $K(\!(\mathbf{t})\!)^\circ$ for $X=\operatorname{Spec}(F_{m,n}/I_X)$. This shows that we can construct models effectively by taking the closure of ideals $I\subset F_{m,n}$. We will introduce the motivation from differential algebra behind these particular choices in [3.2](#section:TDA){reference-type="ref" reference="section:TDA"}. We denote by $\kappa(\mathfrak{p})$ the residue field of a point $\mathfrak{p}\in \operatorname{Spec}(K(\!(\mathbf{t})\!) ^\circ)$. If $\mathcal{X}=\operatorname{Spec}(R_{m,n}/J)$ is a flat scheme over $K(\!(\mathbf{t})\!) ^\circ$, the fibre $\mathcal{X}_{\mathfrak{p}}$ of $\mathcal{X}$ over $\mathfrak{p}$ is the spectrum of the ring $(R_{m,n}/J)\otimes_{K(\!({\bf t})\!)^\circ}\kappa(\mathfrak{p})$. If the prime ideal under consideration is a maximal ideal $\mathfrak{m}\subset K(\!(\mathbf{t})\!)^{\circ}$, then $\kappa(\mathfrak{m})=K(\!({\bf t})\!)^\circ/\mathfrak{m}$. Let $\mathfrak{m}\subset K(\!(\mathbf{t})\!)^{\circ}$ be a maximal ideal and consider the quotient projection $\pi:K(\!({\bf t})\!)^\circ\xrightarrow[]{}K(\!({\bf t})\!)^\circ/\mathfrak{m}$. By Corollary [Corollary 46](#quotient_by_maximal_ideal_is_K){reference-type="ref" reference="quotient_by_maximal_ideal_is_K"} below, there exists an isomorphism $\psi_{\mathfrak m}:K(\!({\bf t})\!)^\circ/\mathfrak{m}\to K$, so there is an induced morphism $\psi_{\mathfrak m}\circ\pi:K(\!({\bf t})\!)^\circ\xrightarrow[]{}K$ (which with the help of Theorem [Theorem 41](#thm_characterization){reference-type="ref" reference="thm_characterization"} can be described concretely as in [\[eq:concrete_reduction\]](#eq:concrete_reduction){reference-type="eqref" reference="eq:concrete_reduction"}). This in turn induces a morphism $$\label{eq:reduction} \psi_{\mathfrak m}\circ\pi:R_{m,n}\xrightarrow[]{}K[x_{i,J}]$$ by applying $\psi_{\mathfrak m}\circ\pi$ coefficient-wise: if $P=\sum_Ma_ME_M$ is an element in $R_{m,n}$, then $\psi_{\mathfrak m}\circ\pi(P)=\sum_M\psi_{\mathfrak m}\circ\pi(a_M)E_M$. **Lemma 25**. *Let $J\subset R_{m,n}$ be an ideal and let $\mathfrak{m}\subset K(\!(\mathbf{t})\!)^{\circ}$ be a maximal ideal. Then $(R_{m,n}/J)\otimes_{K(\!({\bf t})\!)^\circ} (K(\!({\bf t})\!)^\circ/\mathfrak m) \cong K[x_{i,J}]/\overline{J}_{\mathfrak{m}}$, where $\overline{J}_{\mathfrak{m}}\subset K[x_{i,J}]$ denotes the extended ideal under the morphism [\[eq:reduction\]](#eq:reduction){reference-type="eqref" reference="eq:reduction"}.* *Proof.* By little abuse of notation we denote by $\mathfrak m$ also the ideal it generates in $R_{m,n}$ and also the induced map $\psi_{\mathfrak m}:R_{m,n}/\mathfrak{m}\to K[x_{i,J}]$. Then $$\begin{aligned} (R_{m,n}/J)\otimes_{K(\!({\bf t})\!)^\circ} (K(\!({\bf t})\!)^\circ/\mathfrak m) &\cong& (R_{m,n}/J)/(R_{m,n}/J \cdot \mathfrak m) \\ &\cong& (R_{m,n}/\mathfrak m)/(J \cdot \mathfrak m) \\ &\cong& K[x_{i,J}]/\overline{J}_{\mathfrak{m}} \end{aligned}$$ where $\overline{J}_{\mathfrak{m}}$ is the ideal generated by $\psi_{\mathfrak m}(J\cdot \mathfrak m)\subset K[x_{i,J}]$. ◻ In particular, if $\mathcal{X}=\operatorname{Spec}(R_{m,n}/J)$ is a flat scheme over $K(\!(\mathbf{t})\!) ^\circ$ and $\mathfrak{m}\subset K(\!(\mathbf{t})\!)^{\circ}$ is a maximal ideal, then the closed fiber $\mathcal{X}_{\mathfrak{m}}$ of the model $\mathcal{X}$ is simply $\operatorname{Spec}(K[x_{i,J}]/\overline{J}_{\mathfrak{m}})$, where $\overline{J}_{\mathfrak{m}}$ is the ideal in $K[x_{i,J}]$ generated by $\{\psi_{\mathfrak m}\circ\pi(P)\::\:P\in J\}$. ## Tropical differential algebra {#section:TDA} Let $K$ be a field of characteristic zero and let $n,m\geq1$ be integers. Recall that $F_{m,n}$ denotes the ring of polynomials in the variables $\{x_{i,J}\}$ and coefficients in $K(\!(\mathbf{t})\!)$. In order to define our notion of (initial) degeneration in this setting, we need some concepts from tropical differential algebra, for which we follow [@CGL §7.1]. We denote by $D=\{\tfrac{\partial}{\partial t_i}\::\:i=1,\ldots,n\}$ the usual partial derivations defined on $K[\![\mathbf{t}]\!]$. If $E$ is a monomial in the variables $\{x_{i,J}\}$ and $\varphi=(\varphi_1,\ldots,\varphi_n)\in K[\![\mathbf{t}]\!]^n$, we define the differential evaluation $E(\varphi)$ by replacing the variable $x_{i,J}$ for $J=(j_1,\ldots,j_m)$ with $\tfrac{\partial^{j_1+\cdots +j_m}\varphi_i}{\partial t_1^{j_1} \cdots \partial t_m^{j_m}}$. If $P=\sum_M\frac{a_M}{b_M}E_M$ is an element in $F_{m,n}$, where $a_M,b_M \in K[\![\mathbf{t}]\!]$ and $E_M$ are differential monomials, we define $\text{ev}_P(\varphi)=\sum_M\frac{a_M}{b_M}E_M(\varphi)$. Denote by $e_K:\mathbb{B}[\![\mathbf{t}]\!]\to K[\![\mathbf{t}]\!]$ the section of the map $\operatorname{Supp}: K[\![\mathbf{t}]\!]\rightarrow \mathbb{B}[\![\mathbf{t}]\!]$ that sends $A\subset\mathbb{N}^m$ to the formal power series $e_K(A)=\sum_{I\in A}\mathbf{t}^I$. If $E$ is a monomial in the variables $\{x_{i,J}\}$ and $w=(w_1,\ldots,w_n)\in\mathbb{B}[\![\mathbf{t}]\!]^n$, then we have $$\label{eq:ev_mon} V(E(w))=\operatorname{trop}(E(e_K(w))),$$ where $e_K(w)=(e_K(w_1),\ldots,e_K(w_n))\in K[\![\mathbf{t}]\!]^n$. The next step is to use these tools from differential algebra together with the tropical seminorm $\operatorname{trop}: K(\!(\mathbf{t})\!)\to V\mathbb{B}(\mathbf{t})$ to define a seminorm $\operatorname{trop}_w:F_{m,n}\longrightarrow V\mathbb{B}(\mathbf{t})$ that depends on a fixed weight vector $w=(w_1,\ldots,w_n)\in\mathbb{B}[\![\mathbf{t}]\!]^n$. **Definition 26**. Given a fixed weight vector $w\in\mathbb{B}[\![\mathbf{t}]\!]^n$, we define the map $\operatorname{trop}_w:F_{m,n}\longrightarrow V\mathbb{B}(\mathbf{t})$ by sending $P=\sum_M\frac{a_M}{b_M}E_M$ to $$\label{eq:tropw} \operatorname{trop}_w(P):=\bigoplus_M\operatorname{trop}\biggl(\frac{a_M}{b_M}E_M(e_K(w))\biggr)=\bigoplus_M\biggl(\frac{\operatorname{trop}(a_M)}{\operatorname{trop}(b_M)}\odot V(E_M(w))\biggr).$$ The equality between the two expressions appearing in [\[eq:tropw\]](#eq:tropw){reference-type="eqref" reference="eq:tropw"} follows from [\[eq:ev_mon\]](#eq:ev_mon){reference-type="eqref" reference="eq:ev_mon"}. It was shown in [@CGL] that $\operatorname{trop}_w$ is a $K$-algebra seminorm. Given a weight vector $w=(w_1,\ldots,w_n)\in \mathbb{B}[\![\mathbf{t}]\!]^n$ we will now use the seminorm [\[eq:tropw\]](#eq:tropw){reference-type="eqref" reference="eq:tropw"} to construct a $w$-translation map $$\label{eq:wtr} \operatorname{tr}_w:F_{m,n}\xrightarrow[]{}R_{m,n}.$$ The following construction is inspired by the works [@FT20; @HuGao2020] in the ordinary case. We will also see in [\[eq:explicit_translation\]](#eq:explicit_translation){reference-type="eqref" reference="eq:explicit_translation"} that the explicit expression for the map [\[eq:wtr\]](#eq:wtr){reference-type="eqref" reference="eq:wtr"} does not fall very far to the usual expression of the translation to the origin of a torus, which is operative in classical tropical geometry, see [@gubler2013guide §5]. Let us go back to the differential evaluation $E(e_K(w))$ which appears in the definition of [\[eq:tropw\]](#eq:tropw){reference-type="eqref" reference="eq:tropw"}. Recall that for $\varphi\in K[\![\mathbf{t}]\!]\setminus\{0\}$, we denote by $\overline{\varphi}\in K[\![\mathbf{t}]\!]$ the restriction of $\varphi$ to the vertices of its Newton polygon. Then $\varphi=\overline{\varphi}+\widetilde{\varphi}$, where the *initial form* of $\varphi$ is $\overline{\varphi}$, and it is a polynomial. For every variable $x_{i,J}$ appearing in $E$, we write $\Theta(J)(e_K(w_i))=\frac{\partial^{j_1+\cdots +j_m}e_K(w_i)}{\partial t_1^{j_1} \cdots \partial t_m^{j_m}}$. Thus the differential evaluation $E(e_K(w))$ equals the usual algebraic evaluation of $E$ at the vector $(\Theta(J)(e_K(w_i))\::\:i,J)$, and we have a decomposition $$\label{eq:initial_data_dm} E(e_K(w))=E(\overline{\Theta(J)(e_K(w_i)})\::\:i,J)+R,$$ where again the *initial form* of $E(e_K(w))$ is stored in the first term of the right hand side of [\[eq:initial_data_dm\]](#eq:initial_data_dm){reference-type="eqref" reference="eq:initial_data_dm"}. If $P\in F_{m,n}$ satisfies $\operatorname{trop}_w(P)=\frac{a}{b}\neq0$ with $A=\operatorname{trop}(a)$ and $B=\operatorname{trop}(b)$, we set $$T(\operatorname{trop}_w(P))^{-1}:= {\frac{B}{A}} \in K(\!(\mathbf{t})\!)$$ where $A,B$ are considered in $K(\!(\mathbf{t})\!)$ via the natural embedding. Since $A$ and $B$ are uniquely determined by $a$ and $b$, respectively, $T(\operatorname{trop}_w(P))^{-1}$ exists. Moreover, $T(\operatorname{trop}_w(P))^{-1}$ is well-defined by taking into account the multiplicativity of $\operatorname{trop}$. We now specify the polynomial $\operatorname{tr}_w(P)=P_w$ by substituting every instance of $x_{i,J}$ in $P$ by $\overline{\Theta(J)(e_K(w_i))}x_{i,J}$ and then multiplying the result by $T(\operatorname{trop}_w(P))^{-1}$: $$\label{eq:init form} P_w:=\begin{cases} T(\operatorname{trop}_w(P))^{-1}P(\overline{\Theta(J)(e_K(w_i))}x_{i,J}),&\text{ if }\operatorname{trop}_w(P)\neq0\\ 0,&\text{ if }\operatorname{trop}_w(P)=0. \\ \end{cases}$$ **Proposition 27**. *If $P=\sum_Ma_ME_M\in F_{m,n}$, then $P_w$ is an element in $R_{m,n}$. Also, if $\operatorname{trop}_w(P)\neq0$, then $$\label{eq:explicit_translation} P_w=\sum_M[T(\operatorname{trop}_w(P))^{-1}a_ME_M(\overline{\Theta(J)(e_K(w_i)}))]E_M,$$ where $E_M(\overline{\Theta(J)(e_K(w_i)}))$ denotes the usual algebraic evaluation of the monomial $E_M$ at the vector $(\overline{\Theta(J)(e_K(w_i)})\:i,J)$.* *Proof.* It was shown in [@CGL Lemma 7.14] that [\[eq:init form\]](#eq:init form){reference-type="eqref" reference="eq:init form"} is an element in $R_{m,n}$ when instead of using $(\overline{\Theta(J)(e_K(w_i))}\::\:i,J)$ one uses $(T(w_i,J)\::\:i,J)$, where $T(w_i,J)=e_K(\operatorname{Vert}((w_i-J)_{\geq0}))$. These two expressions are linked as follows $$T(w_i,J)=e_K(\operatorname{Supp}(\overline{\Theta(J)(e_K(w_i)}))),\quad \text{for all }i,J.$$ Thus the first part follows since $T(w_i,J)$ and $\overline{\Theta(J)(e_K(w_i))}$ have the same support for all $i,J$ and non-negative integer coefficients. The second part follows from [\[eq:initial_data_dm\]](#eq:initial_data_dm){reference-type="eqref" reference="eq:initial_data_dm"}, since replacing every instance of $x_{i,J}$ in $P$ by $\overline{\Theta(J)(e_K(w_i))}x_{i,J}$ sends the monomial $E_M$ of $P$ to $E_M(\overline{\Theta(J)(e_K(w_i)}))E_M$. ◻ **Remark 28**. Note that the following polynomial $$\widetilde{P_w}=\sum_M[T(\operatorname{trop}_w(P))^{-1}a_ME_M(e_K(w))]E_M,$$ is a modified version of [\[eq:explicit_translation\]](#eq:explicit_translation){reference-type="eqref" reference="eq:explicit_translation"}, and it represents the exact copy in this setting of the translation to the origin of a torus appearing in [@gubler2013guide §5]. However, our definition is simpler since it extracts only the initial data (which is a polynomial) of the evaluations $E_M(e_K(w))$ (which are series), as we showed above. **Definition 29**. Let $w=(w_1,\ldots,w_n)\in\mathbb{B}[\![\mathbf{t}]\!]^n$. For a given ideal $G\subset F_{m,n}$, we define its *$w$-translation* by $G_w$ as the ideal in $R_{m,n}$ generated by $\{P_w\::\:P\in G\}$. **Proposition 30**. *Let $w=(w_1,\ldots,w_n)\in\mathbb{B}[\![\mathbf{t}]\!]^n$ and $G\subset F_{m,n}$ be an ideal. If $R_{m,n}/G_w$ is torsion-free, then $$\mathcal{X}(w):=\operatorname{Spec}(R_{m,n}/G_w)$$ is a model over $K(\!(\mathbf{t})\!) ^\circ$ for the generic fibre $\mathcal{X}(w)_\eta$.* *Proof.* This is a consequence of Lemma [Lemma 23](#lem:flat){reference-type="ref" reference="lem:flat"}. ◻ Let $w\in\mathbb{B}[\![\mathbf{t}]\!]^n$ and $G\subset F_{m,n}$ be such that $R_{m,n}/G_w$ is torsion-free. The fibre $\mathcal{X}(w)_{\mathfrak{p}}$ of $\mathcal{X}(w)$ over $\mathfrak{p}\in \operatorname{Spec}(K(\!(\mathbf{t})\!) ^\circ)$ is the spectrum of the ring $(R_{m,n}/G_w)\otimes_{K(\!({\bf t})\!)^\circ}\kappa(\mathfrak{p})$. **Definition 31**. Let $w=(w_1,\ldots,w_n)\in\mathbb{B}[\![\mathbf{t}]\!]^n$ and $G\subset F_{m,n}$ be such that $R_{m,n}/G_w$ is torsion-free. Further let $\mathcal{X}(w)=\operatorname{Spec}(R_{m,n}/G_w)$ with generic fibre $X=\mathcal{X}(w)_\eta$. Given a point $\mathfrak{p}\in \operatorname{Spec}(K(\!(\mathbf{t})\!) ^\circ)\setminus\{\eta\}$ the fibre $\mathcal{X}(w)_{\mathfrak{p}}$ of $\mathcal{X}(w)$ over $\mathfrak{p}\in \operatorname{Spec}(K(\!(\mathbf{t})\!) ^\circ)$ is called the *initial degeneration* of $X$ at the pair $(w,\mathfrak{p})$. **Theorem 32**. *Let $w=(w_1,\ldots,w_n)\in\mathbb{B}[\![\mathbf{t}]\!]^n$ and $G\subset F_{m,n}$ be an ideal such that $R_{m,n}/G_w$ is torsion-free. Then for any maximal ideal $\mathfrak{m}\subset K(\!(\mathbf{t})\!)^{\circ}$ we have that $$\mathcal X(w)_{\mathfrak{m}}\cong \operatorname{Spec}\left(K[x_{i,J}]/\overline{G}_w\!\right)$$ where $\overline{G}_w\subset K[x_{i,J}]$ denotes the extended ideal under the morphism [\[eq:reduction\]](#eq:reduction){reference-type="eqref" reference="eq:reduction"}.* *Proof.* We have that $\mathcal X(w)_{\mathfrak m}$ is the spectrum of $(R_{m,n}/G_w)\otimes_{K(\!({\bf t})\!)^\circ} (K(\!({\bf t})\!)^\circ/\mathfrak m)$, which is isomorphic to $K[x_{i,J}]/(\overline{G_w})_{\mathfrak{m}}$ by Lemma [Lemma 25](#lem:fiber_general){reference-type="ref" reference="lem:fiber_general"}. ◻ **Remark 33**. In the case of $m=1$ (i.e. the case of discrete valuation rings) since $K[\![t]\!]= K(\!(t)\!)^{\circ}\subset K(\!(t)\!)$ it follows that $\operatorname{Spec}(K(\!(t)\!)^{\circ})=\{(0),\mathfrak{m}=(t)\}$. So there is only one initial degeneration $\mathcal{X}(w)_\mathfrak{m}$ for every $w=(w_1,\ldots,w_n)\in\mathbb{B}[\![t]\!]^n$. As the residue field of $K(\!(t)\!)^{\circ}$ is $\kappa(\mathfrak{m})=K$ we have that $\mathcal{X}(w)_\mathfrak{m}$ is a scheme over $K$. We will study the maximal spectrum of $K(\!(\mathbf{t})\!)^{\circ}$ for the general case of $m>1$ in §[4](#section_maximalIdeals){reference-type="ref" reference="section_maximalIdeals"}. ## Initial ideals along maximal ideals Let $G\subset F_{m,n}$ be any ideal and $w\in\mathbb{B}[\![\mathbf{t}]\!]^n$, so that $G_w$ is an ideal of $R_{m,n}$. If $\mathfrak{m}\subset K(\!(\mathbf{t})\!)^{\circ}$ is a maximal ideal, in this section we will focus on the extended ideal $(\overline{G_w})_{\mathfrak{m}}\subset K[x_{i,J}]$ under the morphism [\[eq:reduction\]](#eq:reduction){reference-type="eqref" reference="eq:reduction"}, no matter if $R_{m,n}/G_w$ is flat or not. We will show that these extended ideals share many properties with the construction of initial ideals in classical tropical geometry. We start with the following result **Proposition 34**. *Let $G\subset F_{m,n}$ be an ideal, $w=(w_1,\ldots,w_n)\in\mathbb{B}[\![\mathbf{t}]\!]^n$ and $\mathfrak{m}\subset K(\!(\mathbf{t})\!)^{\circ}$ a maximal ideal. Then the extended ideal $(\overline{G_w})_{\mathfrak{m}}\subset K[x_{i,J}]$ under the morphism [\[eq:reduction\]](#eq:reduction){reference-type="eqref" reference="eq:reduction"} is the ideal in $K[x_{i,J}]$ generated by $\{\psi_{\mathfrak m}\circ\pi(P_w)\::\:P\in G\}$.* *Proof.* If $P=\sum_Ma_ME_M\in F_{m,n}$ and $\operatorname{trop}_w(P)\neq0$, we have from [\[eq:explicit_translation\]](#eq:explicit_translation){reference-type="eqref" reference="eq:explicit_translation"} that $P_w=\sum_M[T(\operatorname{trop}_w(P))^{-1}a_ME_M(\overline{\Theta(J)(e_K(w_i)}))]E_M$, thus $$\label{eq:explicit_initial_deg} \psi_{\mathfrak m}\circ\pi(P_w)=\sum_M\psi_{\mathfrak m}\circ\pi[T(\operatorname{trop}_w(P))^{-1}a_ME_M(\overline{\Theta(J)(e_K(w_i)}))]E_M.$$ Now, it is clear that the ideal in $K[x_{i,J}]$ generated by $\{\psi_{\mathfrak m}\circ\pi(P_w)\::\:P\in G\}$ is contained in $(\overline{G_w})_{\mathfrak{m}}$. Conversely, an element of $G_w$ is a finite sum $R=\sum_P Q_PP_w$ with $P\in G$ and $Q_P\in R_{m,n}$, thus every generator $(\overline{G_w})_{\mathfrak{m}}$ is of the form $$\psi_{\mathfrak m}\circ\pi(R)=\psi_{\mathfrak m}\circ\pi\bigl(\sum_P Q_PP_w\bigr)=\sum_P \psi_{\mathfrak m}\circ\pi(Q_P)\psi_{\mathfrak m}\circ\pi(P_w),$$ since the map $\psi_{\mathfrak m}\circ\pi$ from [\[eq:reduction\]](#eq:reduction){reference-type="eqref" reference="eq:reduction"} is a homomorphism of rings. ◻ **Remark 35**. Note that the following polynomial $$\psi_{\mathfrak m}\circ\pi( \widetilde{P_w})=\sum_M\psi_{\mathfrak m}\circ\pi[T(\operatorname{trop}_w(P))^{-1}a_ME_M(e_K(w))]E_M,$$ is a modified version of [\[eq:explicit_initial_deg\]](#eq:explicit_initial_deg){reference-type="eqref" reference="eq:explicit_initial_deg"}, and it would be the exact copy in this setting of the initial form appearing in [@gubler2013guide Remark 5.7], which is the usual definition of initial form which is operative in tropical geometry. Once again, our definition is simpler for the same reasons described in Remark [Remark 28](#rem:two_versions){reference-type="ref" reference="rem:two_versions"}. **Definition 36**. For a pair $(w,\mathfrak{m})$ of a weight $w=(w_1,\ldots,w_n)\in\mathbb{B}[\![\mathbf{t}]\!]^n$ and a maximal ideal $\mathfrak{m}\subset K(\!(\mathbf{t})\!)^{\circ}$, we denote [\[eq:explicit_initial_deg\]](#eq:explicit_initial_deg){reference-type="eqref" reference="eq:explicit_initial_deg"} by $\text{in}_{(w,\mathfrak{m})}(P):=\psi_{\mathfrak m}\circ\pi(P_w)$, and call it the *initial form of $P\in F_{m,n}$ at $(w,\mathfrak{m})$*. Similarly, if $G\subset F_{m,n}$ is an ideal, we denote by $\text{in}_{(w,\mathfrak{m})}(G)$ the ideal in $K[x_{i,J}]$ generated by $\{\text{in}_{(w,\mathfrak{m})}(P)\::\:P\in G\}$, and call it the *initial ideal of $G$ at $(w,\mathfrak{m})$*. We now return to the differential setting. The set $D=\{\tfrac{\partial}{\partial t_i}\::\:i=1,\ldots,n\}$ of partial derivations defined on $K[\![\mathbf{t}]\!]$ can be extended to $K(\!(\mathbf{t})\!)$ in the usual way, and also to $F_{m,n}$. If we denote them also by $D$, the pair $(F_{m,n},D)$ is a differential ring, and an ideal $G\subset F_{m,n}$ is differential if it is closed under the action of $D$. We are particularly interested in the case in which $G\subset F_{m,n}$ is a differential ideal and $m>1$. **Example 37**. Let $P=x_{(1,1)}-tx_{(0,0)} \in K(\!(t,u)\!)\{x\}=F_{2,1}$ and $w = \mathbb{N}^2 \setminus \{(1,1)\}\in\mathbb{B}[\![t,u]\!]$. Then $\operatorname{trop}_w(P)=\{(1,0),(0,1)\}$ and $$P_w = x_{(1,1)} - \frac{t}{t+u}x_{(0,0)}.$$ Consider a maximal ideal $\mathfrak{m}\subset K(\!(t,u)\!)^{\circ}$. We see from Example [Example 47](#exp_continuation){reference-type="ref" reference="exp_continuation"} that $$P_w\!\mod \mathfrak{m}=\left\{\begin{matrix} x_{(1,1)}, & \frac{t}{t+u}\in\mathfrak m\\ x_{(1,1)}- x_{(0,0)}, & \frac{t}{t+u}\notin \mathfrak m\end{matrix}\right.$$ The derivatives of $P$ are of the form $$P_{(j_1,j_2)}= x_{(1+j_1,1+j_2)} - tx_{(j_1,j_2)} - j_1x_{(j_1-1,j_2)} .$$ We obtain for every $(j_1,j_2) \ne (0,0)$ that $\operatorname{trop}_w(P_{(j_1,j_2)})=\{(0,0)\}$ and $$(P_{(j_1,j_2)})_w = \begin{cases} x_{(1+j_1,1+j_2)} - t(t+u)x_{(j_1,j_2)} - j_1x_{(j_1-1,j_2)}, & (j_1,j_2)=(1,1) \\ x_{(1+j_1,1+j_2)} - tx_{(j_1,j_2)} - j_1(t+u)x_{(j_1-1,j_2)}, & (j_1,j_2)=(2,1) \\ x_{(1+j_1,1+j_2)} - tx_{(j_1,j_2)} - j_1x_{(j_1-1,j_2)}, & \text{otherwise.} \end{cases}$$ Since the differential ideal generated by $P$ in $F_{2,1}$, denoted as $[P]$, is prime, the algebraic ideal generated by the $(P_{(j_1,j_2)})_w$ already gives us $[P]_w$. Moreover, there is no element in $[P]$ that is in $K(\!(t,u)\!)[x_{(0,0)},x_{(1,0)},x_{(0,1)}]$ and since all differential monomials of $(P_{(j_1,j_2)})_w$ are the same as that of $P_{(j_1,j_2)}$, there is also no such element in $[P]_w$. All differential monomials involving higher derivatives can be reduced to one of order one or zero such that we obtain $$R_{2,1}/[P]_w \cong K(\!(t,u)\!)^{\circ}[x_{(0,0)},x_{(1,0)},x_{(0,1)}].$$ A characterization of prime ideals of such a polynomial ring is given e.g. in [@ferrero1997prime]. # Maximal ideals of the unit ball {#section_maximalIdeals} In §[3](#Sect_models_deg){reference-type="ref" reference="Sect_models_deg"} we saw that for any point $\mathfrak{p}\in \operatorname{Spec}(K(\!(\mathbf{t})\!) ^\circ)\setminus\{\eta\}$, we can construct a degeneration of a generic fibre. Among the most important prime ideals are the maximal ideals, and in this section we characterize the maximal ideals of the Bézout domain $K(\!(\mathbf{t})\!)^\circ$. We fix an integer $m\geq1$. First we recall the definition of monomial order. **Definition 38**. Let $<$ be a total order on the monoid $(\mathbb{N}^m,+,0)$. Then $<$ is a *monomial order* if - $0<a$ for all $a\in \mathbb{N}^m\setminus \{0\}$, - $a<b$ implies $a+c<b+c$ for all $c\in \mathbb{N}^m$. **Lemma 39**. *Every monomial order on $\mathbb{N}^m$ satisfies that for any non-empty subset $S \subset \mathbb{N}^m$ there is a unique minimum $\min_<(S)$ which moreover is a vertex of the convex hull of $S$. In particular, a monomial order is a well-order.* *Proof.* Consider $a\in \mathbb{N}^m$ and observe that, by Definition [Definition 38](#def:mono order){reference-type="ref" reference="def:mono order"}(b), $a=\min_<\{a+\mathbb{N}^m\}.$ For a subset $\emptyset\neq S\subset \mathbb{N}^m$, this observation generalizes as follows. Let $P(S)$ be the convex hull of $S+\mathbb{N}^m$ in $\mathbb R^m$ and let $U(S)$ be the union of all bounded faces of $P(S)$. Then $U(S)\cap \mathbb{N}^m$ contains the vertex set of $S$ (which agrees with the set of vertices of $P(S)$) and moreover $$\min{}_<\{U(S)\cap \mathbb{N}^m\} = \min{}_<\{S\}.$$ We need to show that this minimum is achieved at a unique vertex of $S$. As $U(S)\cap \mathbb{N}^m$ is finite, the monomial order $<$ can be represented by a weight vector $w\in \mathbb R^m$, i.e. $\min_<\{U(S)\cap \mathbb{N}^m\}=\min_w\{U(S)\cap \mathbb{N}^m\}$ for some $w\in \mathbb R^m$. Let $m_1<\dots<m_k$ be the elements of $U(S)\cap \mathbb{N}^m$. Then any $w\in \mathbb R^m$ satisfying $w\cdot m_1<w\cdot m_2<\dots<w\cdot m_k$ represents $<$; moreover, we may assume $w\in \mathbb{N}^m_{> 0}$ (see e.g. Lemma 3.1.1 in [@HerzogHibi]). Suppose $m=\min_<\{U(S)\cap \mathbb{N}^m\}$ is not a vertex. Then there exist $0<\lambda_i<1$ with $\sum_{i=1}^k \lambda_i = 1$ such that $m=\sum_{i=1}^k \lambda_i m_i$. In particular, $$w\cdot m= \sum_{i=1}^k \lambda_i\, w\cdot m_i.$$ So either $\lambda_{1}=\cdots=\lambda_{k}$ and $w\cdot m= w\cdot m_1 = \cdots = w \cdot m_k$, but this contradicts $w\cdot m_1<\dots<w\cdot m_k$; or $w\cdot m_i<w\cdot m<w\cdot m_j$ for some $1 \le i,j \le k$ which contradicts the assumption that $m=\min_w\{U(S)\cap \mathbb{N}^m\}$.  ◻ **Remark 40**. Lemma [Lemma 39](#lem:mono is vertex){reference-type="ref" reference="lem:mono is vertex"} is closely related to [@ArocaRond Lemma 3.7], yet slightly different: Aroca and Rond are dealing with total orders $<$ in $\mathbb{R}^m$ compatible with its group structure, and which can be seen as monomial orders on $\mathbb R^m$. In loc. cit. it is shown that for any such order $<$ and any rational polyhedral cone $\sigma\subset \mathbb R^m$ which is non-negative, i.e., $0 \le a$ for all $a \in \sigma$, the set $\sigma\cap \mathbb Z^m$ is well ordered. The aim of this section is to establish the following bijection between the set of monomial orders on the monoid $(\mathbb{N}^m,+,0)$ and the set of maximal ideals of $K(\!(\mathbf{t})\!)^\circ$. **Theorem 41**. *There is an identification between maximal ideals $\mathfrak{m}$ of $K(\!(\mathbf{t})\!)^\circ$ and monomial orders on $\mathbb{N}^m$ defined, for $I,J \in \mathbb{N}^m$, as $$\label{eq_characterization} I < J \quad \Longleftrightarrow \quad \frac{\mathbf{t}^J}{\mathbf{t}^I + \mathbf{t}^J} \in \mathfrak{m} \quad \Longleftrightarrow \quad \frac{\mathbf{t}^I}{\mathbf{t}^I + \mathbf{t}^J} \notin \mathfrak{m}.$$* We will use the remaining of this section to prove this result; the respective identifications are given in [Proposition 42](#prop_maxideal){reference-type="ref" reference="prop_maxideal"} and [Proposition 48](#Proposition_MaxIdeal_gives_Order){reference-type="ref" reference="Proposition_MaxIdeal_gives_Order"}. Given $f\in K[\![\mathbf{t}]\!]$ and $I\in\mathbb{N}^m$, we may denote by $f_I$ the coefficient of $\mathbf{t}^I$ in $f$. **Proposition 42**. *Given a monomial order $<$ on $\mathbb N^m$, the set $$\mathfrak{m}_{<} := \{f/g \in K(\!(\mathbf{t})\!)^\circ : f_{\min\!{}_{<} ( \operatorname{Supp}(g) )}=0\}$$ is a maximal ideal of $K(\!(\mathbf{t})\!)^\circ$.* *Proof.* We claim that $\mathfrak{m}_{<}$ is the kernel of the map $$\begin{aligned} c_< : K(\!(\mathbf{t})\!)^\circ &\to K\\ \frac{f}{g} &\longmapsto \frac{f_{\min\!{}_{<} ( \operatorname{Supp}(g) )}}{g_{\min\!{}_{<} ( \operatorname{Supp}(g) )}}. \end{aligned}$$ Let us first show that the map $c_<$ is well-defined. Let $f/g=h/\ell$ with $f,g,h,\ell \in K[\![\mathbf{t}]\!]$, we need to show that $$f_{\min\!{}_{<} ( \operatorname{Supp}(g) )}\ell_{\min\!{}_{<} ( \operatorname{Supp}(\ell))}=h_{\min\!{}_{<} ( \operatorname{Supp}(\ell) )}g_{\min\!{}_{<} ( \operatorname{Supp}(g) )}.$$ If $f_{\min\!{}_{<}(\operatorname{Supp}(g))}=0$ or $h_{\min\!{}_{<}(\operatorname{Supp}(\ell))}=0$, the equality follows easily. Let us suppose that $f_{\min\!{}_{<}(\operatorname{Supp}(g))}\neq0$ and $h_{\min\!{}_{<}(\operatorname{Supp}(\ell))}\neq0$. Note that $\min\!{}_{<}(\operatorname{Supp}(g))\leq \min\!{}_{<}(\operatorname{Supp}(f))$ and $\min\!{}_{<}(\operatorname{Supp}(g))< \min\!{}_{<}(\operatorname{Supp}(f))$ if and only if $f_{\min\!{}_{<}(\operatorname{Supp}(g))}=0$. Thus, $\min\!{}_{<}(\operatorname{Supp}(f))=\min\!{}_{<}(\operatorname{Supp}(g))$ and $\min\!{}_{<}(\operatorname{Supp}(h))=\min\!{}_{<}(\operatorname{Supp}(\ell))$ for the same reason. Since $f\ell=gh$, we have $$X:=\operatorname{trop}(f\ell)=\operatorname{trop}(f)\operatorname{trop}(\ell)=\operatorname{trop}(hg)=\operatorname{trop}(h)\operatorname{trop}(g),$$ and $I\in X$ if and only if there exist unique $J,K,L,M$ in the respective vertex sets such that $I=J+K=L+M$. In particular, for every $I\in X$ we have $$(f\ell)_I=f_J\ell_K=g_Lh_M=(gh)_I.$$ We set $$I_< := \min\!{}_{<}(\operatorname{Supp}(f))+\min\!{}_{<}(\operatorname{Supp}(\ell))=\min\!{}_{<}(\operatorname{Supp}(g))+\min\!{}_{<}(\operatorname{Supp}(h)).$$ We need show that $I_<\in X$. Suppose that $I_<=A+B$ for some $A\in \operatorname{trop}(f)$ and $B\in\operatorname{trop}(\ell)$, then $\min\!{}_{<}(\operatorname{Supp}({f}))\leq A$ and $\min\!{}_{<}(\operatorname{Supp}(\ell))\leq B$. Thus, $$I_<=\min\!{}_{<}(\operatorname{Supp}({f}\ell))=\min\!{}_{<}(\operatorname{Supp}({f}))+\min\!{}_{<}(\operatorname{Supp}(\ell))\leq A+B=I_<,$$ which holds only for the uniquely determined $A=\min\!{}_{<}(\operatorname{Supp}({f}))$ and $B=\min\!{}_{<}(\operatorname{Supp}(\ell))$. Thus, $I_<\in X$. The additivity and multiplicativity of $c_<$ can be shown in a similar way. Thus, $c_<$ defines a surjective homomorphism such that, by the first isomorphism theorem, $$K(\!(\mathbf{t})\!)^{\circ}/\mathfrak{m}_{<} = K(\!(\mathbf{t})\!)^{\circ}/\text{ker}(c_<) \cong K,$$ which is the case exactly if $\mathfrak{m}_{<}$ is a maximal ideal. ◻ Note that given a monomial order $<$ on $\mathbb N^m$, expressions of the form $\frac{\mathbf{t}^J}{\mathbf{t}^I + \mathbf{t}^J}$ are in $\mathfrak{m}_<$ if and only if $I<J$. Later we will use exactly those fractions to define a monomial order on $\mathbb N^m$ from a given maximal ideal $\mathfrak{m}\subset K(\!(\mathbf{t})\!)^\circ$. Before we do that, we first establish that $K(\!(\mathbf{t})\!)^\circ / \mathfrak{m} \cong K$ for every maximal ideal $\mathfrak{m}$ of $K(\!(\mathbf{t})\!)^\circ$. **Definition 43**. Let $a\leq b$ in $V\mathbb{B}[\mathbf{t}]$, with supports $A$,$B$ respectively. We say that $a$ is *irrelevant* for $b$ if $A\cap B=\emptyset$, and we write $a\ll b$. Otherwise we say that $a$ is relevant for $b$. If $\frac{a}{b}\leq \frac{c}{d}$ in $V\mathbb{B}(\mathbf{t})$, then $\frac{a}{b}\ll \frac{c}{d}$ if and only if $a\odot d\ll c\odot b$ in $V\mathbb{B}[\mathbf{t}]$. **Lemma 44**. *Let $q \in K(\!(\mathbf{t})\!)^\circ$ be such that $\operatorname{trop}(q) \ll 1$. Then $q$ is contained in every maximal ideal of $K(\!(\mathbf{t})\!)^\circ$.* *Proof.* Let $\mathfrak{m}$ be a maximal ideal of $K(\!(\mathbf{t})\!)^\circ$. Suppose it does not contain $q$. Then $(1) = \mathfrak{m} + (q)$ by maximality of $\mathfrak{m}$. So there are $r \in \mathfrak{m}$ and $s \in K(\!(\mathbf{t})\!)^\circ$ such that $1 = r + qs$. Since $\operatorname{trop}(q) \ll 1$ and $\operatorname{trop}(s) \leq 1$ we have $\operatorname{trop}(qs) \ll 1$. Now write $r = f/g$ and $qs = h/g$ for $f,g,h \in K[\![\mathbf{t}]\!]$ (we have already equalized the denominators here). Then $1 = r+qs = (f + h)/g$. But $\operatorname{trop}(h/g) \ll 1$ means that the support of $h$ has no vertices of $g$. So $f$ has in its support all the vertices of $g$, meaning that $\operatorname{trop}(f) = \operatorname{trop}(g)$, so that $r = f/g$ is a unit in $K(\!(\mathbf{t})\!)^\circ$. But $r \in \mathfrak{m}$, so $\mathfrak{m} = (1)$, which is not a maximal ideal of $K(\!(\mathbf{t})\!)^\circ$. ◻ **Lemma 45**. *Given $q \in K(\!(\mathbf{t})\!)^\circ$, there exists an integer $n \geq 1$ and elements $\alpha_1, \ldots, \alpha_n \in K$ such that $\operatorname{trop}\left(\prod_{k = 1}^n ( q - \alpha_k) \right) \ll 1.$* *Proof.* Let $q = f/g$ with $f,g \in K[\![\mathbf{t}]\!]$. If $f\ll g$, then $(q-0)=q\ll1$. Otherwise, if $\{I_1\ldots, I_n\}$ is the vertex set of $g$, then $\operatorname{trop}(f)\cap \{I_1\ldots, I_n\}\neq\emptyset$. For each $k = 1, \ldots, n$, we set $\alpha_k := \frac{f_{I_k}}{g_{I_k}}$. Notice that the denominator $g_{I_k}$ is never zero, but the numerator, and thus $\alpha_k$, might be zero. Let $r = \prod_{k = 1}^n (q - \alpha_k)$. We claim that $\operatorname{trop}(r) \ll 1$. First, we compute $$r = \prod_{k = 1}^n (q - \alpha_k) = \frac{\prod_{k=1}^n [f - \alpha_k g]}{g^n}.$$ We need to show that $\operatorname{trop}(\prod_{k=1}^n [f - \alpha_k g]) \ll \operatorname{trop}(g^n)$. Note that the vertices of $g^n$ are of the form $n I_k$ for $k = 1,\ldots, n$. Fix a value for $k$; we prove that $nI_k$ is not a vertex of $(f - \alpha_1 g)\cdots (f - \alpha_n g)$: If $I_k\notin \operatorname{trop}(f)\cap \{I_1\ldots, I_n\}$, then $\alpha_k=0$ and $(f-\alpha_kg)_{I_k}=f_{I_k}=0$. If $I_k\in \operatorname{trop}(f)\cap \{I_1\ldots, I_n\}$, then $(f-\alpha_kg)_{I_k}=0$. Pick any weight vector $w \in \mathbb R^m$ such that $I_k$ is the $w$-minimal vertex of $\operatorname{Supp}(g)$. Since $\operatorname{trop}(f) \leq \operatorname{trop}(g)$ we have that $\operatorname{trop}(f - \alpha_i g) \leq \operatorname{trop}(g)$ for each $i$, and so either $I_k$ appears in $\operatorname{trop}(f - \alpha_i g)$ in which case it is the $w$-minimal vertex of it, or $I_k$ does not appear, in which case the $w$-minimal vertex of $f - \alpha_i g$ is strictly bigger than $I_k$. The latter happens at least once, namely for $i = k$, because by construction, the coefficient $(f - \alpha_k g)_{I_k}$ is zero. The $w$-minimal vertex of the product $\prod_i (f_i - \alpha_i g_i)$ is therefore strictly bigger (with respect to the $w$-ordering) than $nI_k$. Since $nI_k$ is the $w$-minimal vertex of $\operatorname{Supp}(g^n)$, it follows that $nI_k$ is not a vertex of the product. ◻ **Corollary 46**. *Let $\mathfrak{m}$ be a maximal ideal of $K(\!(\mathbf{t})\!)^\circ$. Then the composition of the inclusion $K \hookrightarrow K(\!(\mathbf{t})\!)^\circ$ with the quotient map $K(\!(\mathbf{t})\!)^\circ \to K(\!(\mathbf{t})\!)^\circ / \mathfrak{m}$ is an isomorphism. Consequently, $K(\!(\mathbf{t})\!)^\circ / \mathfrak{m} \cong K$.* *Proof.* Let $q \in K(\!(\mathbf{t})\!)^\circ$ and pick $\alpha_1,\ldots, \alpha_n \in K$ as in [Lemma 45](#product_of_differences){reference-type="ref" reference="product_of_differences"}. Then [Lemma 44](#irrelevant_contained_in_maximal_ideal){reference-type="ref" reference="irrelevant_contained_in_maximal_ideal"} implies that $\prod_{i} (q - \alpha_i) \in \mathfrak{m}$. Since $\mathfrak{m}$ is prime, at least one of the factors is in $\mathfrak{m}$. Thus for every $q \in K(\!(\mathbf{t})\!)^\circ$ there is an $\alpha \in K$ such that $q - \alpha \in \mathfrak{m}$. This is equivalent to the statement that the composition $K \to K(\!(\mathbf{t})\!)^\circ \to K(\!(\mathbf{t})\!)^\circ / \mathfrak{m}$ is surjective and an isomorphism as injectivity follows from the fact that $K\cap \mathfrak m=\{0\}$. ◻ Now let us show that a maximal ideal $\mathfrak{m}$ of $K(\!(\mathbf{t})\!)^\circ$ induces a monomial order on $\mathbb{N}^m$. For any $I, J \in \mathbb{N}^m$ we define $$I \preceq_{\mathfrak{m}} J \quad \Longleftrightarrow \quad \frac{\mathbf{t}^I}{\mathbf{t}^I + \mathbf{t}^J} \notin \mathfrak{m}.$$ **Example 47**. We continue with Example [Example 37](#exp){reference-type="ref" reference="exp"}. Consider a maximal ideal $\mathfrak{m}\subset K(\!(t)\!)^{\circ}$. Then either $u\prec_{\mathfrak{m}}t$ or $t\prec_{\mathfrak{m}}u$. In the first case, we have $\frac{t}{t+u}\in \mathfrak m$, so $P_w\mod \mathfrak{m}=x_{(1,1)}$. In the second case, $\frac{u}{t+u}\in \mathfrak m$ and so $$\frac{t}{t+u}-1=\frac{u}{t+u}\in \mathfrak m.$$ Hence, in this case $P_w\mod \mathfrak m=x_{(1,1)}-x_{(0,0)}$. **Proposition 48**. *The relation $\preceq_{\mathfrak{m}}$ on $\mathbb{N}^m$ is a monomial order.* *Proof.* We note that the relations $I \preceq_{\mathfrak{m}} I$ and $0 \preceq_{\mathfrak{m}} J$ follow from the observations that $1/2$ and $1/(1 + \mathbf{t}^J)$ are units in $K(\!(\mathbf{t})\!)^\circ$ and therefore not in $\mathfrak{m}$. Let $I, J \in \mathbb{N}^m$ with $I \preceq_{\mathfrak{m}} J$ and let $K\in \mathbb{N}^m$. Since $\frac{\mathbf{t}^I}{\mathbf{t}^I + \mathbf{t}^J} \notin \mathfrak{m}$, we obtain that $$\frac{\mathbf{t}^K}{\mathbf{t}^K} \cdot \frac{\mathbf{t}^I}{\mathbf{t}^I + \mathbf{t}^J} = \frac{\mathbf{t}^{I+K}}{\mathbf{t}^{I+K} + \mathbf{t}^{J+K}} \notin \mathfrak{m}$$ and thus, $I+K \preceq_{\mathfrak{m}} J+K$. Now let $I, J \in \mathbb{N}^m$ be arbitrary. Note that $$\frac{\mathbf{t}^I}{\mathbf{t}^I + \mathbf{t}^J} + \frac{\mathbf{t}^J}{\mathbf{t}^I + \mathbf{t}^J} = 1 \notin \mathfrak{m}$$ so at least one of the two terms is not in $\mathfrak{m}$. So we have $I \preceq_{\mathfrak{m}} J$ or $J \preceq_{\mathfrak{m}} I$. Suppose that both terms are not in $\mathfrak{m}$. If $I \neq J$ then we have $\operatorname{trop}( \mathbf{t}^J/(\mathbf{t}^I + \mathbf{t}^J) ) = \operatorname{trop}( \mathbf{t}^J / (\mathbf{t}^I - \mathbf{t}^J) )$, so by [Corollary 10](#trop_determines_elements_of_ideal){reference-type="ref" reference="trop_determines_elements_of_ideal"} we find that $\mathbf{t}^J / (\mathbf{t}^I - \mathbf{t}^J)$ is also not in $\mathfrak{m}$. Now consider the product $$q = \frac{\mathbf{t}^I}{\mathbf{t}^I + \mathbf{t}^J} \cdot \frac{\mathbf{t}^J}{\mathbf{t}^I - \mathbf{t}^J} = \frac{\mathbf{t}^{I + J}}{\mathbf{t}^{2I} - \mathbf{t}^{2J}}.$$ This is a product of two elements of $K(\!(\mathbf{t})\!)^\circ \setminus \mathfrak{m}$, and therefore not in $\mathfrak{m}$. But we have $\operatorname{trop}(q) \ll 1$ since $I + J$ is not a vertex of $\{2I, 2J\}$. This contradicts [Lemma 44](#irrelevant_contained_in_maximal_ideal){reference-type="ref" reference="irrelevant_contained_in_maximal_ideal"}. So we conclude that $I = J$, which yields the anti-symmetry. Now it remains to show that $\preceq_{\mathfrak{m}}$ is transitive. Let $I,J,L \in \mathbb{N}^m$ all distinct with $I \preceq_{\mathfrak{m}} J$ and $J \preceq_{\mathfrak{m}} L$. We have $$\frac{\mathbf{t}^{J}}{\mathbf{t}^I + \mathbf{t}^J + \mathbf{t}^L} = \frac{\mathbf{t}^J}{\mathbf{t}^I + \mathbf{t}^J}\frac{\mathbf{t}^I + \mathbf{t}^J}{\mathbf{t}^I + \mathbf{t}^J + \mathbf{t}^L} \in \mathfrak{m}$$ since the first factor is in $\mathfrak{m}$ and the second in $K(\!(\mathbf{t})\!)^\circ$. Similarly, we have $$\frac{\mathbf{t}^{L}}{\mathbf{t}^I + \mathbf{t}^J + \mathbf{t}^L} = \frac{\mathbf{t}^L}{\mathbf{t}^J + \mathbf{t}^L}\frac{\mathbf{t}^J + \mathbf{t}^L}{\mathbf{t}^I + \mathbf{t}^J + \mathbf{t}^L} \in \mathfrak{m}$$ Since $1 = (\mathbf{t}^I + \mathbf{t}^J + \mathbf{t}^L)/(\mathbf{t}^I + \mathbf{t}^J + \mathbf{t}^L)$ is not in $\mathfrak{m}$, we must have that $\mathbf{t}^I / (\mathbf{t}^I + \mathbf{t}^J + \mathbf{t}^L)$ is not in $\mathfrak{m}$. From the same factorization $$\frac{\mathbf{t}^{I}}{\mathbf{t}^I + \mathbf{t}^J + \mathbf{t}^L} = \frac{\mathbf{t}^I}{\mathbf{t}^I + \mathbf{t}^L}\frac{\mathbf{t}^I + \mathbf{t}^L}{\mathbf{t}^I + \mathbf{t}^J + \mathbf{t}^L}$$ it then follows that $\mathbf{t}^I / (\mathbf{t}^I + \mathbf{t}^L)$ is not in $\mathfrak{m}$, so that $I \preceq_{\mathfrak{m}} L$. ◻ For the case $m=1$, we have that $K(\!(t)\!)^\circ=K[\![t]\!]$ and $\mathfrak m=(t)$, and we recover the fact that the unique maximal ideal encodes the unique monomial order on $\mathbb{N}$. **Corollary 49**. *Every maximal $k$-ideal of $V\mathbb{B}(\mathbf{t})^\circ$ is of the form $$\mathfrak{m}_<^\dag=\{\tfrac{a}{b}\in V\mathbb{B}(\mathbf{t})^\circ\::\:a_{\min_<(b)}=0\}$$ for some monomial order $<$ on $\mathbb{N}^m$.* *Proof.* By Theorem [Theorem 41](#thm_characterization){reference-type="ref" reference="thm_characterization"} and Proposition [Proposition 42](#prop_maxideal){reference-type="ref" reference="prop_maxideal"}, every maximal ideal of $K(\!(\mathbf{t})\!)^\circ$ is of the form $\mathfrak{m}_<=\{f/g \in K(\!(\mathbf{t})\!)^\circ : f_{\min\!{}_{<} ( \operatorname{Supp}(g) )}=0\}$ for some monomial order $<$ on $\mathbb{N}^m$. By Theorem [Theorem 12](#ideals_are_the_same){reference-type="ref" reference="ideals_are_the_same"} and Proposition [Proposition 19](#Proposition_trop_is_Bezout){reference-type="ref" reference="Proposition_trop_is_Bezout"}, every maximal ideal of $V\mathbb{B}(\mathbf{t})^\circ$ is the image under trop of a maximal ideal of $K(\!(\mathbf{t})\!)^\circ$, and we have $\operatorname{trop}(\mathfrak{m}_<)=\mathfrak{m}_<^\dag$. ◻ An important consequence is that by construction taking initial forms commutes with multiplication. We derive this fact from Theorem [Theorem 41](#thm_characterization){reference-type="ref" reference="thm_characterization"} as follows. **Proposition 50**. *Consider a pair $(w,\mathfrak{m})$ of a weight $w=(w_1,\ldots,w_n)\in\mathbb{B}[\![\mathbf{t}]\!]^n$ and a maximal ideal $\mathfrak{m}\subset K(\!(\mathbf{t})\!)^{\circ}$. If $P,Q\in F_{m,n}$, then $$\text{in}_{(w,\mathfrak{m})}(PQ)=\text{in}_{(w,\mathfrak{m})}(P)\text{in}_{(w,\mathfrak{m})}(Q).$$* *Proof.* Let $P=\sum_Ma_ME_M$ and $Q=\sum_Nb_NE_N$, with $\operatorname{trop}_w(P)=\tfrac{a}{b}$ and $\operatorname{trop}_w(Q)=\tfrac{c}{d}$, so that $T(\operatorname{trop}_w(P))^{-1}=\tfrac{B}{A}$ and $T(\operatorname{trop}_w(Q))^{-1}=\tfrac{D}{C}$. Then by [\[eq:explicit_translation\]](#eq:explicit_translation){reference-type="eqref" reference="eq:explicit_translation"}, we have $$\begin{aligned} P_wQ_w&=[\sum_M[\tfrac{B}{A}a_ME_M(\overline{\Theta(J)(e_K(w_i)}))]E_M][\sum_N[\tfrac{D}{C}b_NE_N(\overline{\Theta(J)(e_K(w_i)}))]E_N]= \\&=\sum_O[\sum_{M+N=O}\tfrac{BD}{AC}a_Mb_NE_M(\overline{\Theta(J)(e_K(w_i)}))E_N(\overline{\Theta(J)(e_K(w_i)}))]E_ME_N. \end{aligned}$$ Now let $PQ=\sum_O(\sum_{M+N=O}a_Mb_N)E_ME_N$ with $\operatorname{trop}_w(PQ)=\tfrac{e}{f}$, so that $T(\operatorname{trop}_w(PQ))^{-1}=\tfrac{F}{E}$. Then $$\begin{aligned} (PQ)_w=\sum_O[\sum_{M+N=O}\tfrac{F}{E}a_Mb_NE_M(\overline{\Theta(J)(e_K(w_i)}))E_N(\overline{\Theta(J)(e_K(w_i)}))]E_ME_N. \end{aligned}$$ Let $<_{\mathfrak m}$ be the monomial order obtained by applying Theorem [Theorem 41](#thm_characterization){reference-type="ref" reference="thm_characterization"}, then by Proposition [Proposition 42](#prop_maxideal){reference-type="ref" reference="prop_maxideal"}, the reduction map $\psi_{\mathfrak m}\circ\pi:K(\!(\mathbf{t})\!)^{\circ}\xrightarrow[]{}K$ appearing in [\[eq:reduction\]](#eq:reduction){reference-type="eqref" reference="eq:reduction"} can be described concretely by $$\label{eq:concrete_reduction} \psi_{\mathfrak m}\circ\pi(\tfrac{f}{g})=\frac{f_{\min\!{}_{<_{\mathfrak m}} ( \operatorname{Supp}(g) )}}{g_{\min\!{}_{<_{\mathfrak m}} ( \operatorname{Supp}(g) )}}.$$ Clearly $P_wQ_w$ and $(PQ)_w$ only differ by the constants $\tfrac{BD}{AC}$ and $\tfrac{F}{E}$. Now we have $\frac{a}{b}\odot\frac{c}{d}=\frac{e}{f}$, since $\operatorname{trop}_w$ is multiplicative. In particular we have $AC=E+R$ and $BD=F+S$ where $R$ (respectively $S$) does not have any vertex of $E$ (respectively of $F$). Thus $(BD)_{\min\!{}_{<_{\mathfrak m}} ( \operatorname{Supp}(AC) )}=F_{\min\!{}_{<_{\mathfrak m}} ( \operatorname{Supp}(E) )}$ and $(AC)_{\min\!{}_{<_{\mathfrak m}} ( \operatorname{Supp}(AC) )}=E_{\min\!{}_{<_{\mathfrak m}} ( \operatorname{Supp}(E) )}$, which yields $\psi_{\mathfrak m}\circ\pi(\tfrac{BD}{AC})=\psi_{\mathfrak m}\circ\pi(\tfrac{F}{E})$. We now apply [\[eq:explicit_initial_deg\]](#eq:explicit_initial_deg){reference-type="eqref" reference="eq:explicit_initial_deg"}, which finishes the proof. ◻ One last remark concerning the concept of initial defined on [@FGH20+] for polynomials with coefficients in $K[\![\mathbf{t}]\!]$ is convenient. Given a weight $w=(w_1,\ldots,w_n)\in\mathbb{B}[\![\mathbf{t}]\!]^n$ and $P=\sum_Ma_ME_M\in K[\![\mathbf{t}]\!][x_{i,J}]$, its initial was defined as $$\operatorname{in}_w(P)= \sum_{\substack{\operatorname{trop}_w(a_{M}E_M)\cap \operatorname{trop}_w(P)\neq \emptyset}} \overline{a_{M}}E_M.$$ Given a maximal ideal $\mathfrak{m}\subset K(\!(\mathbf{t})\!)^{\circ}$ corresponding to the monomial order $<_{\mathfrak m}$, by [\[eq:concrete_reduction\]](#eq:concrete_reduction){reference-type="eqref" reference="eq:concrete_reduction"} we have $$\label{eq:explicit:fpwcoeff} in_{(w,\mathfrak{m})}(P)=\sum_{I+J=\min\!{}_{<_{\mathfrak m}} (\operatorname{trop}_w(P))\in \operatorname{trop}_w(a_ME_M)}[c(w)_J(a_M)_I]E_M$$ where $c(w)_J\in\mathbb{N}$ is a constant that depends only on the weight vector $w$. Certainly, every $\text{in}_{(w,\mathfrak{m})}(P)$ can be recovered from $\operatorname{in}_w(P)$. It remains future work to verify whether non-maximal prime ideals of $K(\!(\mathbf{t})\!)^\circ$ can also be encoded as some type of orders of $\mathbb{N}^m$. # Acknowledgments {#acknowledgments .unnumbered} This paper was finished during a research visit from S.F. and C.G. at the Unidad Oaxaca IM-UNAM in the context of a CIMPA School in Oaxaca June 2023. We are thankful for the financial support of the Apoyo Especial Alfonso Nápoles Gándara (IM-UNAM) and the great hospitality during this time. We would like to acknowledge the support and input of Mercedes Haiech in the beginning of this project. L.B. is partially supported by PAPIIT project IA100122 dgapa UNAM. S.F. is partially supported by the grant PID2020-113192GB-I00 (Mathematical Visualization: Foundations, Algorithms and Applications) from the Spanish MICINN and by the OeAD project FR 09/2022. [^1]: Recall that an $\ell$-valuation is a seminorm $v:K\xrightarrow[]{}S$ defined on a field $K$ and taking values in an idempotent semifield; it differs from Krull valuations in the sense that $S$ does not need to be totally-ordered. [^2]: Here *flat* is in the sense of Definition II-28 and *proper* as defined on page 95 in [@EH_GeomSch].

arxiv_math

--- abstract: | We consider the nonlinear optimization problem with least $\ell_1$-norm measure of constraint violations and introduce the concepts of the D-stationary point, the DL-stationary point and the DZ-stationary point with the help of exact penalty function. If the stationary point is feasible, they correspond to the Fritz-John stationary point, the KKT stationary point and the singular stationary point, respectively. In order to show the usefulness of the new stationary points, we propose a new exact penalty sequential quadratic programming (SQP) method with inner and outer iterations and analyze its global and local convergence. The proposed method admits convergence to a D-stationary point and rapid infeasibility detection without driving the penalty parameter to zero, which demonstrates the commentary given in \[SIAM J. Optim., 20 (2010), 2281--2299\] and can be thought to be a supplement of the theory of nonlinear optimization on rapid detection of infeasibility. Some illustrative examples and preliminary numerical results demonstrate that the proposed method is robust and efficient in solving infeasible nonlinear problems and a degenerate problem without LICQ in the literature. author: - "Xin-Wei Liu[^1]" - "Yu-Hong Dai[^2]" title: Exact penalty method for D-stationary point of nonlinear optimization --- Nonlinear programming, least constraint violations, infeasible detection, exact penalty method, global and local convergence 49M37, 65K05, 90C26, 90C30, 90C55 Exact penalty method for D-stationary point of nonlinear optimization # Introduction {#intro} Consider the general nonlinear optimization $$\begin{aligned} \min~ f(x) \quad\hbox{s.t.}~ h(x)=0,~g(x)\ge 0, \label{prob1-1}\end{aligned}$$ where $x\in\Re^n$ is the unknown, $f: \Re^n\to\Re$, $h: \Re^n\to\Re^{m_{\cal E}}$, $g: \Re^n\to\Re^{m_{\cal I}}$ are twice continuously differentiable real-valued functions defined on $\Re^n$, $m_{\cal E}$ and $m_{\cal I}$ are two nonnegative integers with $\max\,\{m_{\cal E}, m_{\cal I}\}>0$. When $m_{\cal E}=0$ and $m_{\cal I}>0$, the problem is an inequality constrained optimization. If $m_{\cal E}>0$ and $m_{\cal I}=0$, the problem is an optimization problem with only equality constraints. For the general problem ([\[prob1-1\]](#prob1-1){reference-type="ref" reference="prob1-1"}), a point $x^*$ is called a **Fritz-John point** if $x^*$ is feasible and there exist a scalar $\rho^*\ge 0$ and multiplier vectors $(\mu^*,\lambda^*)\in\Re^{m_{\cal E}}\times\Re_+^{m_{\cal I}}$ such that $$\begin{aligned} &\,\,&\,\,\rho^*\nabla f(x^*)-\nabla h(x^*)\mu^*-\nabla g(x^*)\lambda^*=0, \label{wes1b}\\ &\,\,&\,\,\mu_i^*h_i(x^*)=0,~i=1,\ldots,m_{\cal E}, \\ &\,\,&\,\,\lambda_i^*g_i(x^*)=0,~i=1,\ldots,m_{\cal I}.\label{wes3b} \end{aligned}$$ Furthermore, if $\rho^*>0$, $x^*$ is called a **KKT point**; if $\rho^*=0$, $x^*$ is called a **singular stationary point**. A minimizer of problem ([\[prob1-1\]](#prob1-1){reference-type="ref" reference="prob1-1"}) is always a Fritz-John point, and is a KKT point under either the linear independence constraint qualification (LICQ) or the Mangasarian-Fromowitz constraint qualification (MFCQ). Now let us consider the following simple one-dimensional example $$\begin{aligned} \min~x\quad\hbox{s.t.}~x^2-1\ge 0,~x-2\ge 0. \label{s-example}\end{aligned}$$ Surprisingly, when starting from a standard initial point $x_0=-4$, many methods including some state-of-the-art solvers can only converge to an infeasible point $x^*=-1$, as observed in [@ByrMaN; @LiuDHS; @LiuSun01; @WacBie04]. How can we understand this point? Since the point is even not feasible to problem ([\[s-example\]](#s-example){reference-type="ref" reference="s-example"}), it is not a Fritz-John point at all. This means that, to have a comprehensive understandings on general nonlinear optimization problems, we have to face the fundamental feasibility/infeasibility issue. On one hand, the linearization of feasible nonlinear constraints may lead to inconsistent subproblems during the solution procedure (for example, see [@ByrCuN10; @Char1978; @ColCon80; @Fletch87; @GouOrT15; @HanMan79; @Piet1969]). On the other hand, the original nonlinear optimization problem may be infeasible in itself (for example, see [@BurCuW14; @DLS17; @LDHmm; @NOW]). Remarkably, to deal with the infeasibility issue, Burke and Han [@BurHan89] proposed the concept of **infeasible stationary point**, which minimizes the constraint violation under some kind of measure. As Byrd, Curtis and Nocedal [@ByrCuN10] pointed out, there are various types of infeasible stationary points, some of which are strict isolated local minimizers of the infeasibility measure and others may belong to a set of minimizers of the measure or may simply be stationary points of the measure. Furthermore, Byrd, Curtis and Nocedal [@ByrCuN10] presented a set of conditions to guarantee the superlinear convergence of their SQP algorithm to an infeasible stationary point. Burke, Curtis and Wang [@BurCuW14] considered the general program with equality and inequality constraints, and proved that their SQP algorithm has strong global convergence and rapid convergence to a KKT point, and has superlinear/quadratic convergence to an infeasible stationary point. Recently, Dai, Liu and Sun [@DLS17] proposed a primal-dual interior-point method, which can be superlinearly or quadratically convergent to a KKT point if the original problem is feasible, and can be superlinearly or quadratically convergent to an infeasible stationary point when the problem is infeasible. As an infeasible stationary point is just to minimize the constraint violation under some kind of measure, it has nothing to do with the objective function. Motivated by the trajectory optimization problem in aerospace engineering, Dai and Zhang [@DZ] proposed the basic model of optimization with least constraint violation. If the least $\ell_2$-norm measure of constraint violation is used, they provided the optimality conditions for $M$-stationary point and $L$-stationary point and proved that the classical penalty method and the smoothing Fischer-Burmeister function method still work when the problem is infeasible. Marvelously, for convex optimization problem with least constraint violation, Dai and Zhang [@DZa] demonstrated that the dual of the problem has an unbounded solution set, the optimality condition can be characterized by the augmented Lagrangian, and the augmented Lagrangian method is able to find an approximate solution when the least violated shift is in the domain of the subdifferential of the optimal value function. More related work can be found in [@CD; @ChiGil]. Following the above line, we consider the nonlinear optimization with least $\ell_1$-norm measure of constraint violations. With the help of exact penalty function (see Section 2 for details), we propose the following concept of $D$-stationary points for problem ([\[prob1-1\]](#prob1-1){reference-type="ref" reference="prob1-1"}), which can be regarded as the generalizations of the well-known Fritz-John points, KKT points and singular stationary points of feasible optimization to infeasible optimization. A point $x^*$, which may be either feasible or infeasible to problem ([\[prob1-1\]](#prob1-1){reference-type="ref" reference="prob1-1"}), is a called **D-stationary point** of problem ([\[prob1-1\]](#prob1-1){reference-type="ref" reference="prob1-1"}) if it is a stationary point for minimizing the $\ell_1$-norm of constraint violations and there exist a scalar $\rho^*\ge 0$ and multiplier vectors $(\mu^*,\lambda^*)\in\Re^{m_{\cal E}}\times\Re_+^{m_{\cal I}}$ such that $$\begin{aligned} &\,\,&\,\,\rho^*\nabla f(x^*)-\nabla h(x^*)\mu^*-\nabla g(x^*)\lambda^*=0, \label{wes1a}\\ &\,\,&\,\,\mu_i^*h_i(x^*)+|h_i(x^*)|=0,~i=1,\ldots,m_{\cal E}, \\ &\,\,&\,\,\lambda_i^*g_i(x^*)+\max\{0,-g_i(x^*)\}=0,~i=1,\ldots,m_{\cal I}.\label{wes3a} \end{aligned}$$ Furthermore, if $\rho^*>0$, $x^*$ is called a **DL-stationary point**. If $\rho^*=0$, $x^*$ is called a **DZ-stationary point**. The definitions of the D-stationary point, the DL-stationary point and the DZ-stationary point can be applied no matter whether problem ([\[prob1-1\]](#prob1-1){reference-type="ref" reference="prob1-1"}) is feasible or infeasible. More exactly, we can see that a feasible D-stationary point is a Fritz-John point, a feasible DL-stationary point is a KKT point, and a feasible DZ-stationary point is a singular stationary point. In addition, an infeasible DZ-stationary point is an infeasible stationary point whose characteristics have been given in Burke and Han [@BurHan89] and Byrd, Curtis and Nocedal [@ByrCuN10]. For the simple one-dimensional problem ([\[s-example\]](#s-example){reference-type="ref" reference="s-example"}), we can check that $x^*=-1$ is an infeasible D-stationary point (or an infeasible DL-stationary point) with $\rho^*=0.1$ and $\lambda^*=(0.45, 1.0)$. Significantly, equations ([\[wes1a\]](#wes1a){reference-type="ref" reference="wes1a"})--([\[wes3a\]](#wes3a){reference-type="ref" reference="wes3a"}) correspond to the Fritz-John conditions of the feasible optimization with relaxation of least constraint violations (see ([\[relax5\]](#relax5){reference-type="ref" reference="relax5"})). In order to show the usefulness of our new stationary points, we shall propose a new exact penalty sequential quadratic programming (SQP) method with inner and outer iterations and analyze its global and local convergence. The proposed method generates inner iterations by an SQP method for the exact penalty subproblem with fixed penalty parameter and the outer iterations are responsible for update of the parameter. As a result, the method admits convergence to a D-stationary point and rapid infeasibility detection even though the penalty parameter is not small enough, which is distinguished from the existing methods with rapid detection of infeasibility such as [@BurCuW14; @ByrCuN10; @DLS17; @LiuDHS]. The latter requires the penalty parameter tend to zero (or infinity as the sequence of parameters is nondecreasing). Some illustrative examples and preliminary numerical results demonstrate that the proposed method is robust and efficient in solving infeasible nonlinear problems and a degenerate problem without LICQ in the literature. The paper is organized as follows. In section 2, the D-stationary point, DL-stationary point and DZ-stationary are introduced based on optimization with least $\ell_1$-norm constraint violation. Some simple illustrative examples are also given to show their relations to the exact penalty optimization. In section 3, we reformulate the original optimization problem into a new feasible problem with a parameter which is an smooth equivalent of the $\ell_1$ exact penalty optimization. Some basic results on the feasible reformulation and the optimization with least $\ell_1$-norm constraint violation are proved as well. We present our exact penalty SQP method in section 4. The global and local convergence are analyzed in sections 5 and 6, respectively. Some numerical results are reported in section 7. We conclude our paper in the last section. Throughout the paper, we use standard notations from the literature. All vectors are column vectors. For $u, v\in\Re^s$, $\max\{u,v\}$ and $|u|$ are two $s$-dimensional vectors with components $\max\{u_i,v_i\}$ and $|u_i|$ for $i=1, \ldots, s$, respectively. For any vectors $x$ and $y$, $w=(x,y)$ means $w=[x^T\quad y^T]^T$. A letter with subscript $k$ is related to the $k$th iteration, the subscript $i$ indicates the $i$th component of a vector, and the subscript $kj$ is the $j$th iteration in solving the $k$th subproblem. The equation $\theta_k={O}(t_k)$ means that there exists a scalar $M$ independent of $k$ such that $|\theta_k|\le M|t_k|$ for all $k$ large enough, and equation $\theta_k={o}(t_k)$ means that $|\theta_k|\le\epsilon_k|t_k|$ for all $k$ large enough with $\lim_{k\to 0}\epsilon_k=0$. If it is not specified, letter $I$ is the identity matrix, ${\cal E}=\{1,2,\ldots,m_{\cal E}\}$ and ${\cal I}=\{1,2,\ldots,m_{\cal I}\}$ are two index sets, $\|\cdot\|$ be the Euclidean norm. Some unspecified notations may be identified from the context. # D-stationary points for problem ([\[prob1-1\]](#prob1-1){reference-type="ref" reference="prob1-1"}) and some illustrative examples The stationary properties play a key role in recognizing optimal solutions and developing efficient algorithms for nonlinear optimization ([\[prob1-1\]](#prob1-1){reference-type="ref" reference="prob1-1"}). For feasible optimization, the Fritz-John conditions and the KKT conditions are those of our well-known and extensively used stationary properties, where the Fritz-John conditions see ([\[wes1b\]](#wes1b){reference-type="ref" reference="wes1b"})--([\[wes3b\]](#wes3b){reference-type="ref" reference="wes3b"}). The others are the conditions on singular stationary points. We restate the conditions on the KKT point and the singular stationary point so that the readers can have a better understanding on our introduction of new stationary points. [\[def0\]]{#def0 label="def0"} Consider nonlinear optimization ([\[prob1-1\]](#prob1-1){reference-type="ref" reference="prob1-1"}). Suppose that functions $f$, $g$ and $h$ are continuously differentiable on $\Re^n$, $x^*\in\Re^n$ is given.\ (1) The point $x^*$ is a KKT point if it is feasible to the problem and there exist $\mu^*\in\Re^{m_{\cal E}}$ and $\lambda^*\in\Re_+^{m_{\cal I}}$ such that $$\begin{aligned} &\,\,&\,\,\nabla f(x^*)-\nabla h(x^*)\mu^*-\nabla g(x^*)\lambda^*=0, \nonumber\\ &\,\,&\,\,\lambda_i^*g_i(x^*)=0,~i=1,\ldots,m_{\cal I}. \nonumber\end{aligned}$$ (2) The point $x^*$ is a singular stationary point if it is feasible to the problem and there exist $\mu^*\in\Re^{m_{\cal E}}$ and $\lambda^*\in\Re_+^{m_{\cal I}}$ such that $(\mu^*,\lambda^*)\ne 0$, and $$\begin{aligned} &\,\,&\,\,-\nabla h(x^*)\mu^*-\nabla g(x^*)\lambda^*=0, \nonumber\\ &\,\,&\,\,\lambda_i^*g_i(x^*)=0,~i=1,\ldots,m_{\cal I}. \nonumber\end{aligned}$$ Based on the motivation to deal with infeasible optimization, we give the following definition on the DL-stationary point and the DZ-stationary point, which can be thought as a generalization of the conditions on the KKT point and the singular stationary point. In addition, some illustrative examples are given to show the significance of the stationary points for optimization with least $\ell_1$-norm constraint violations. [\[def1\]]{#def1 label="def1"} Consider the nonlinear optimization problem ([\[prob1-1\]](#prob1-1){reference-type="ref" reference="prob1-1"}). Suppose that functions $f$, $g$ and $h$ are continuously differentiable on $\Re^n$ and that $x^*\in\Re^n$ is a given point, which may be either feasible or infeasible to the problem.\ (1) The point $x^*$ is referred to as a *DL-stationary point* of nonlinear optimization ([\[prob1-1\]](#prob1-1){reference-type="ref" reference="prob1-1"}) if it is a stationary point of minimizing $\|h(x)\|_1+\|\max\{0, -g(x)\}\|_1$, and there exist multiplers $(\mu^*,\lambda^*)\in\Re^{m_{\cal E}}\times\Re_+^{m_{\cal I}}$ such that $$\begin{aligned} &\,\,&\,\,\nabla f(x^*)-\nabla h(x^*)\mu^*-\nabla g(x^*)\lambda^*=0, \label{wes1}\\ &\,\,&\,\,\mu_i^*h_i(x^*)=-|h_i(x^*)|,~i=1,\ldots,m_{\cal E}, \\ &\,\,&\,\,\lambda_i^*g_i(x^*)=-\max\{0,-g_i(x^*)\},~i=1,\ldots,m_{\cal I}.\label{wes3} \end{aligned}$$ (2) The point $x^*$ is a *DZ-stationary point* if it is a stationary point for minimizing $\|h(x)\|_1+\|\max\{0, -g(x)\}\|_1$, and there exist $\mu^*\in\Re^{m_{\cal E}}$ and $\lambda^*\in\Re_+^{m_{\cal I}}$ such that $$\begin{aligned} &\,\,&\,\,-\nabla h(x^*)\mu^*-\nabla g(x^*)\lambda^*=0, \label{wes1c}\\ &\,\,&\,\,\mu_i^*h_i(x^*)=-|h_i(x^*)|,~i=1,\ldots,m_{\cal E}, \\ %\mu_i^*\in [-1,1],~ &\,\,&\,\,\lambda_i^*g_i(x^*)=-\max\{0,-g_i(x^*)\},~i=1,\ldots,m_{\cal I}. \label{wes3c}\end{aligned}$$ Similar to that both the KKT point and the singular stationary point are the Fritz-John point, both the DL-stationary point and the DZ-stationary point belong to the set of D-stationary points. In essence, if $x^*$ is a stationary point for minimizing $\|h(x)\|_1+\|\max\{0, -g(x)\}\|_1$, then the conditions ([\[wes1c\]](#wes1c){reference-type="ref" reference="wes1c"})--([\[wes3c\]](#wes3c){reference-type="ref" reference="wes3c"}) hold naturally with some $\mu^*\in\Re^{m_{\cal E}}$ and $\lambda^*\in\Re_+^{m_{\cal I}}$. Thus, a DZ-stationary point is a singular stationary point if the point is feasible and it is an infeasible stationary point if the point is infeasible. What is more interesting and important, when the optimization problem has inherent violations $(v_1^h,\ldots,v_{m_{\cal E}}^h)$ for equality constraints and $(v_1^g,\ldots,v_{m_{\cal I}}^g)$ for inequality constraints, the conditions ([\[wes1\]](#wes1){reference-type="ref" reference="wes1"})--([\[wes3\]](#wes3){reference-type="ref" reference="wes3"}) with all $|h_i(x^*)|$ and $\max\{0,-g_i(x^*)\}$ being replaced, respectively, by $v_i^h$ and $v_i^g$ correspond to the KKT conditions of the feasible optimization, a relaxation of the original problem, $$\begin{aligned} \min~f(x)\quad \hbox{s.t.}~h_i(x)=v_i^h,~\forall i\in{\cal E};~g_i(x)\ge v_i^g,~\forall i\in{\cal I}. \label{relax5}\end{aligned}$$ Thus we can see that D-stationary point, DL-stationary point and DZ-stationary point are generalization of Fritz-John point, KKT point and singular stationary point, respectively, from the feasible case to the general case. In order to show the usefulness of our new stationary points, based on robustness of the exact penalty optimization, we investigate the stationary properties of an exact penalty optimization. To this aim, consider the $\ell_1$ exact penalty approach on the original nonlinear optimization ([\[prob1-1\]](#prob1-1){reference-type="ref" reference="prob1-1"}), $$\begin{aligned} \min_{x\in\Re^n}~P_{\rho}(x)=:\rho f(x)+c(x), \label{prob1L1}\end{aligned}$$ where $c: \Re^n\to\Re$ and $c(x)=\|h(x)\|_1+\|\max\{0, -g(x)\}\|_1$ is the $\ell_1$-norm measure of constraint violations, $\rho>0$ is a penalty parameter. Apparently, $c(x)$ is nonsmooth even though both $h(x)$ and $g(x)$ are twice differentiable. Moreover, $c(x)\ge 0$ for all $x\in\Re^n$. It is well-known that $P_{\rho}(x)$ is an exact penalty function in the sense that, if the original problem ([\[prob1-1\]](#prob1-1){reference-type="ref" reference="prob1-1"}) is feasible, its local minima can be the minima of problem ([\[prob1L1\]](#prob1L1){reference-type="ref" reference="prob1L1"}) under second-order sufficient conditions for all penalty parameters smaller than certain threshold. We further reformulate the non-smooth $\ell_1$ exact penalty optimization ([\[prob1L1\]](#prob1L1){reference-type="ref" reference="prob1L1"}) as a smooth feasible problem with only inequality constraints, $$\begin{aligned} \min_{x,y,z}&\,\,&\,\,\rho f(x)+\sum_{i=1}^{m_{\cal E}} y_i+\sum_{i=1}^{m_{\cal I}} z_i, \label{prob2-1}\\ \hbox{s.t.}&\,\,&\,\,y-h(x)\ge 0,~ y+h(x)\ge 0, \label{prob2-2}\label{prob2-3}\\ &\,\,&\,\,z+g(x)\ge 0,~ z\ge 0, \label{prob2-4}\label{prob2-5}\end{aligned}$$ where $x\in\Re^n$, $y\in\Re^{m_{\cal E}}$, $z\in\Re^{m_{\cal I}}$, $\rho>0$ is the penalty parameter. Since the constraints ([\[prob2-3\]](#prob2-3){reference-type="ref" reference="prob2-3"}) and ([\[prob2-5\]](#prob2-5){reference-type="ref" reference="prob2-5"}) guarantee $y\ge 0$ and $z\ge 0$, one has $\|y\|_1=\sum_{i=1}^{m_{\cal E}}y_i$ and $\|z\|_1=\sum_{i=1}^{m_{\cal I}}z_i$. The same reformulation appears in (2.5) of [@GouOrT15] and is an extension of (2.2) in [@ByrCuN10]. It is noted that [@BurCuW14; @NocWri99] use a different reformulation with both equality and inequality constraints for $\ell_1$ exact penalty problem ([\[prob1L1\]](#prob1L1){reference-type="ref" reference="prob1L1"}), where they need double not single additional nonnegative variable as ours for every equality constraint. For convenience of statement, we denote $$\begin{aligned} &\,\,&\,\,F_{\rho}(x,y,z)=\rho f(x)+\sum_{i=1}^{m_{\cal E}} y_i+\sum_{i=1}^{m_{\cal I}} z_i, \nonumber\\ &\,\,&\,\,G(x,y,z)=(y-h(x),y+h(x),z+g(x),z). \nonumber\end{aligned}$$ Let us look at three simple examples, by which we may have an intuitive understanding on our new stationary points and their relations to the original optimization, the exact penalty optimization and the optimization with least $\ell_1$ constraint violations. **Example 1**. *Consider the nonlinear constrained optimization problem $$\begin{aligned} \min_x~f(x):=x^2+4x \quad\hbox{s.t.}~ h(x):=x-1=0. \nonumber\end{aligned}$$* Obviously, the problem is feasible and has a solution $x^*=1$. The reformulation ([\[prob2-1\]](#prob2-1){reference-type="ref" reference="prob2-1"})--([\[prob2-5\]](#prob2-5){reference-type="ref" reference="prob2-5"}) has the form $$\begin{aligned} \min_{x,y}&\,\,&\,\,F_{\rho}(x,y):=\rho x^2+4\rho x+y \nonumber\\ \hbox{s.t.}&\,\,&\,\,G(x,y):=\left[\begin{array}{l} y-x+1 \\ y+x-1 \end{array}\right]\ge 0. \nonumber\label{probe2-2}\end{aligned}$$ If $\rho>\frac{1}{6}$, one has $(x^*,y^*)=(\frac{1}{2\rho}-2,3-\frac{1}{2\rho})$, $(u^*,v^*)=(0,1)$; otherwise, $(x^*,y^*)=(1,0)$, $(u^*,v^*)=(\frac{1}{2}-3\rho,\frac{1}{2}+3\rho)$. Moreover, $$\begin{aligned} F_{\rho}(x^*,y^*)=\left\{\begin{array}{ll} -\frac{1}{4\rho}-4\rho+3, & \rho>\frac{1}{6}; \\[2mm] 5\rho, & \rho\le\frac{1}{6}. \end{array}\right. \nonumber\end{aligned}$$ Finally, $x^*=1$ is a KKT point of the problem. The associated Lagrange multiplier is $\mu^*=6$. The threshold is $\rho^*=\frac{1}{6}$. **Example 1**. *Consider an infeasible nonlinear constrained optimization problem $$\begin{aligned} \min_x~f(x):=x^2+4x\quad\hbox{s.t.}~ h(x):=\left[\begin{array}{l} x-1\\ x+1 \end{array}\right]=0. \nonumber\end{aligned}$$* Corresponding to the reformulation ([\[prob2-1\]](#prob2-1){reference-type="ref" reference="prob2-1"})--([\[prob2-5\]](#prob2-5){reference-type="ref" reference="prob2-5"}), we have the problem $$\begin{aligned} \min_{x,y}&\,\,&\,\,F_{\rho}(x,y):=\rho x^2+4\rho x+(y_1+y_2) \nonumber\\ \hbox{s.t.}&\,\,&\,\,G(x,y):=\left[\begin{array}{l} y_1-x+1 \\ y_1+x-1 \\ y_2-x-1 \\ y_2+x+1 \end{array}\right]\ge 0. \nonumber\end{aligned}$$ If $\rho>1$, the KKT point of the reformulation problem is $(x^*,y_1^*,y_2^*)=(\frac{1}{\rho}-2,3-\frac{1}{\rho},1-\frac{1}{\rho})$, and $(u_1^*,v_1^*,u_2^*,v_2^*)=(0,1,0,1)$ is the associated Lagrange multiplier vector. If $\rho\le1$, the KKT point and the associated Lagrange multiplier are $(x^*,y_1^*,y_2^*)=(-1, 2, 0)$ and $(u_1^*,v_1^*,u_2^*,v_2^*)=(0,1,1-\rho,\rho)$, respectively. Thus, $$\begin{aligned} F_{\rho}(x^*,y^*)=\left\{\begin{array}{ll} -\frac{1}{\rho}-4\rho+4, & \rho>1; \\[2mm] 2-3\rho, & \rho\le1. \end{array}\right. \nonumber\end{aligned}$$ Note that $[-1,1]$ is the set of solutions for minimizing $|x-1|+|x+1|$. By Definition [\[def1\]](#def1){reference-type="ref" reference="def1"}, both $x^*=-1$ and $x^*=1$ are the DL-stationary points, where our solution $x^*=-1$ is the unique global minimizer of optimization with least $\ell_1$-norm constraint violations $$\begin{aligned} \min~ x^2+4x\quad\hbox{s.t.}~ x~\hbox{minimizes}~|x-1|+|x+1|. \nonumber\end{aligned}$$ **Example 1**. *Consider an inequality constrained optimization with inconsistent constraints (see also Dai and Zhang [@DZ]) $$\begin{aligned} \min~ x_1^2+x_2^2 \quad\hbox{s.t.}~ -x_1-x_2+1\ge 0, ~ x_1+x_2-2\ge 0. \nonumber\end{aligned}$$* The exact penalty approach solves the problem by solving its smooth and feasible reformulation $$\begin{aligned} \min&\,\,&\,\,\rho x_1^2+\rho x_2^2+z_1+z_2 \nonumber\\ \hbox{s.t.}&\,\,&\,\,z_1-x_1-x_2+1\ge 0, \nonumber\\ &\,\,&\,\,z_2+x_1+x_2-2\ge 0, \nonumber\\ &\,\,&\,\,z_1\ge 0,~z_2\ge 0. \nonumber\end{aligned}$$ Although the original problem is infeasible, the preceding problem is well posed. By solving the KKT conditions, we have $x_1^*=x_2^*=0.5$, $z_1^*=0$, $z_2^*=1$, $s_1^*=1-\rho$, $s_2^*=1$, $t_1^*=\rho$, $t_2^*=0$ provided $\rho\le 1$. This means that, for $0<\rho\le 1$, we find a point with $f(x^*)=0.5$ at which the $\ell_1$-norm and the $\ell_2$-norm of constraint violations are $1$. In contrast, the method in [@DZ] got the solution $x_1^*=x_2^*=0.75$, $f(x^*)=\frac{9}{8}$, which has the same $\ell_1$-norm constraint violations with ours and has less $\ell_2$-norm constraint violations $\frac{\sqrt{2}}{2}$ and larger objective value $\frac{9}{8}$. In other words, our method finds a point with much better objective value than the one proposed in [@DZ] under the same $\ell_1$-norm constraint violations. In addition, if $\rho>1$, one has $x^*=(\frac{1}{2\rho},\frac{1}{2\rho})$, $z^*=(0,2-\frac{1}{\rho})$, $s^*=(0,1)$, $t^*=(1,0)$. Finally, one has $$\begin{aligned} F_{\rho}(x^*,z^*)=\left\{\begin{array}{ll} -\frac{1}{2\rho}+2, & \rho>1; \\[2mm] 0.5\rho+1, & \rho\le 1. \end{array}\right. \nonumber\end{aligned}$$ It is easy to examine that all points satisfying one of the two constraints are the ones minimizing the $\ell_1$-norm constraint violations, among which our solution $(0.5,0.5)$ is a DL-stationary point and is the unique global minimizer of optimization with least $\ell_1$-norm constraint violations. # The exact penalty optimization and the optimization with least constraint violations In this section, we investigate the relations among the original nonlinear optimization, the exact penalty optimization and its smooth reformulation, and the optimization with least constraint violations theoretically. Moreover, we provide a certificate on the solution of optimization with least $\ell_1$-norm constraint violations. The certificate is important in addressing convergence to the D-stationary points and the rapid infeasibility detection as the penalty parameter is not small. Corresponding to the problem ([\[prob1-1\]](#prob1-1){reference-type="ref" reference="prob1-1"}), the optimization with least $\ell_1$-norm measure of constraint violations is a bi-level optimization problem $$\begin{aligned} \min~ f(x)\quad\hbox{s.t.}~ x\in\hbox{argmin}_{x\in\Re^n}~c(x):=\|h(x)\|_1+\|\max\{0, -g(x)\}\|_1. \label{prob1b1-2}\label{prob1b1-1}\end{aligned}$$ If problem ([\[prob1-1\]](#prob1-1){reference-type="ref" reference="prob1-1"}) is feasible, then it is equivalent to the problem ([\[prob1b1-1\]](#prob1b1-1){reference-type="ref" reference="prob1b1-1"}) in sense that both of them have the same minima. However, like problem ([\[prob1L1\]](#prob1L1){reference-type="ref" reference="prob1L1"}) but not the original problem, the formulation ([\[prob1b1-1\]](#prob1b1-1){reference-type="ref" reference="prob1b1-1"}) admits infeasible constraints. Since it is subject to the set of solutions of an unconstrained non-smooth problem, the formulation ([\[prob1b1-1\]](#prob1b1-1){reference-type="ref" reference="prob1b1-1"}) is difficult and complicated in how to describe its optimality conditions and design efficient algorithms. These difficulties may be alleviated when the original problem ([\[prob1-1\]](#prob1-1){reference-type="ref" reference="prob1-1"}) is convex, see [@CD; @ChiGil; @DZ; @DZa] and references therein. For general nonconvex nonlinear optimization, we investigate the stationary points of problem ([\[prob1b1-1\]](#prob1b1-1){reference-type="ref" reference="prob1b1-1"}) with the help of exact penalty optimization ([\[prob1L1\]](#prob1L1){reference-type="ref" reference="prob1L1"}), which results in our new stationary points of nonlinear optimization. The following results show the relations between exact penalty optimization ([\[prob1L1\]](#prob1L1){reference-type="ref" reference="prob1L1"}) and the original problem, where the second part of results can see Theorem 4.4 of [@HanMan79] and Theorem 17.3 of [@NocWri99]. If $x^*$ is a local minimizer of problem ([\[prob1L1\]](#prob1L1){reference-type="ref" reference="prob1L1"}) and $c(x^*)=0$, $\rho>0$ is any penalty parameter, then $x^*$ is a local minimizer of the original nonlinear optimization ([\[prob1-1\]](#prob1-1){reference-type="ref" reference="prob1-1"}). Conversely, if $x^*$ is a strict local minimizer of the original problem ([\[prob1-1\]](#prob1-1){reference-type="ref" reference="prob1-1"}) at which the KKT conditions are satisfied with Lagrange multipliers $\mu^*\in\Re^{m_{\cal E}}$ and $\lambda^*\in \Re_+^{m_{\cal I}}$, then $x^*$ is a local minimizer of problem ([\[prob1L1\]](#prob1L1){reference-type="ref" reference="prob1L1"}) for all $\rho<\rho^*$, where $\rho^*=\frac{1}{\max\{\|\mu^*\|_{\infty},\|\lambda\|^*_{\infty}\}}$. If, in addition, the second-order sufficient conditions of problem ([\[prob1-1\]](#prob1-1){reference-type="ref" reference="prob1-1"}) hold at the triple $(x^*,\mu^*,\lambda^*)$ and $\rho<\rho^*$, then $x^*$ is a strict local minimizer of problem ([\[prob1L1\]](#prob1L1){reference-type="ref" reference="prob1L1"}). *Proof.* If the point $x^*$ is a local minimizer of problem ([\[prob1L1\]](#prob1L1){reference-type="ref" reference="prob1L1"}), we have for some $\delta>0$ and all $x\in{\cal N}(x^*,\delta)$, $$\begin{aligned} \rho f(x^*)+c(x^*)\le\rho f(x)+c(x), \nonumber\end{aligned}$$ where $\delta$ is a scalar and ${\cal N}(x^*,\delta)$ is a $\delta$-neighborhood of $x^*$. If $c(x^*)=0$, the preceding inequality is reduced to $$\begin{aligned} \rho f(x^*)\le\rho f(x)+c(x),\quad\forall x\in{\cal N}(x^*,\delta). \nonumber\end{aligned}$$ Hence, $f(x^*)\le f(x)$ for those $x\in{\cal N}(x^*,\delta)\cap\{x|c(x)=0\}$. That is, $x^*$ is a local minimizer of nonlinear optimization ([\[prob1-1\]](#prob1-1){reference-type="ref" reference="prob1-1"}). The proof for the remaining results can see that of Theorem 4.4 of [@HanMan79]. ◻ Let us revisit problem ([\[prob2-1\]](#prob2-1){reference-type="ref" reference="prob2-1"})--([\[prob2-5\]](#prob2-5){reference-type="ref" reference="prob2-5"}), the smooth equivalent of problem ([\[prob1L1\]](#prob1L1){reference-type="ref" reference="prob1L1"}). Apparently, for every $x\in\Re^n$, $(x,y,z)$ with $y=|h(x)|$ and $z=\max\{0,-g(x)\}$ is a feasible point of problem ([\[prob2-1\]](#prob2-1){reference-type="ref" reference="prob2-1"})--([\[prob2-5\]](#prob2-5){reference-type="ref" reference="prob2-5"}). Thus, problem ([\[prob2-1\]](#prob2-1){reference-type="ref" reference="prob2-1"})--([\[prob2-5\]](#prob2-5){reference-type="ref" reference="prob2-5"}) is always feasible no matter whether problem ([\[prob1-1\]](#prob1-1){reference-type="ref" reference="prob1-1"}) is feasible or not. Similar to the reformulation (1.3) of [@GouOrT15], we have the regular properties on problem ([\[prob2-1\]](#prob2-1){reference-type="ref" reference="prob2-1"})--([\[prob2-5\]](#prob2-5){reference-type="ref" reference="prob2-5"}) as follows. [\[lemma1\]]{#lemma1 label="lemma1"} Problem ([\[prob2-1\]](#prob2-1){reference-type="ref" reference="prob2-1"})--([\[prob2-5\]](#prob2-5){reference-type="ref" reference="prob2-5"}) is always feasible. Moreover, the following results are true.\ (1) MFCQ holds at any feasible point $(x,y,z)$ of problem ([\[prob2-1\]](#prob2-1){reference-type="ref" reference="prob2-1"})--([\[prob2-5\]](#prob2-5){reference-type="ref" reference="prob2-5"});\ (2) If a point $\hat x$ is feasible to the problem ([\[prob1-1\]](#prob1-1){reference-type="ref" reference="prob1-1"}) and LICQ holds at $\hat x$, then $(\hat x,\hat y,\hat z)$ with $\hat y=0$ and $\hat z=0$ is a feasible point of problem ([\[prob2-1\]](#prob2-1){reference-type="ref" reference="prob2-1"})--([\[prob2-5\]](#prob2-5){reference-type="ref" reference="prob2-5"}) and LICQ holds at $(\hat x,\hat y,\hat z)$;\ (3) Let $(\hat x,\hat y,\hat z)$ be a feasible point of problem ([\[prob2-1\]](#prob2-1){reference-type="ref" reference="prob2-1"})--([\[prob2-5\]](#prob2-5){reference-type="ref" reference="prob2-5"}), $\hat{\cal I}_h=\{i| h_i(\hat x)=0\}$, $\hat{\cal I}_g=\{i| g_i(\hat x)=0\}$. If $\nabla h_i(\hat x) (i\in\hat{\cal I}_h)$, $\nabla g_i(\hat x) (i\in\hat{\cal I}_g)$ are linearly independent, then for problem ([\[prob2-1\]](#prob2-1){reference-type="ref" reference="prob2-1"})--([\[prob2-5\]](#prob2-5){reference-type="ref" reference="prob2-5"}), LICQ holds at $(\hat x,\hat y,\hat z)$. *Proof.* For every $x\in\Re^n$, let $y=|h(x)|$, $z=\max\{0, -g(x)\}$. Then all constraints in problem ([\[prob2-1\]](#prob2-1){reference-type="ref" reference="prob2-1"})--([\[prob2-5\]](#prob2-5){reference-type="ref" reference="prob2-5"}) are satisfied with $(x,y,z)$. Thus, problem ([\[prob2-1\]](#prob2-1){reference-type="ref" reference="prob2-1"})--([\[prob2-5\]](#prob2-5){reference-type="ref" reference="prob2-5"}) is feasible for all $x\in\Re^n$ and suitably selected $y\in\Re^{m_{\cal E}}$ and $z\in\Re^{m_{\cal I}}$. \(1\) Note that $$\begin{aligned} \nabla G(x,y,z)=\left[\begin{array}{cccc} -\nabla h(x) & \nabla h(x) & \nabla g(x) & 0 \\ I_{m_{\cal E}} & I_{m_{\cal E}} & 0 & 0 \\ 0 & 0 & I_{m_{\cal I}} & I_{m_{\cal I}}\end{array}\right]. \nonumber\end{aligned}$$ For any $\Delta w=(\Delta x,\Delta y,\Delta z)$, $$\begin{aligned} \nabla G(x,y,z)^T\Delta w>0~\hbox{if and only if}~\left\{\begin{array}{l} \Delta y-\nabla h(x)^T\Delta x>0, \\ \Delta y+\nabla h(x)^T\Delta x>0, \\ \Delta z+\nabla g(x)^T\Delta x>0, \\ \Delta z>0. \end{array}\right.\nonumber\end{aligned}$$ Thus, for any $\Delta x$, there are $\Delta y>|\nabla h(x)^T\Delta x|$ and $\Delta z>\max\{0,-\nabla g(x)^T\Delta x\}$ such that $\nabla G(x,y,z)^T\Delta w>0$, which implies that MFCQ holds immediately. \(2\) and (3) can be verified by examining the columns of $\nabla G(x,y,z)$. ◻ Problem ([\[prob2-1\]](#prob2-1){reference-type="ref" reference="prob2-1"})--([\[prob2-5\]](#prob2-5){reference-type="ref" reference="prob2-5"}) is robust in the sense that it is feasible even though the constraints in the original problem ([\[prob1-1\]](#prob1-1){reference-type="ref" reference="prob1-1"}) are inconsistent, and its local minima are always the KKT points since MFCQ naturally holds at any feasible points. Furthermore, the associated Lagrange multipliers of any KKT points of the smooth problem ([\[prob2-1\]](#prob2-1){reference-type="ref" reference="prob2-1"})--([\[prob2-5\]](#prob2-5){reference-type="ref" reference="prob2-5"}) are bounded. [\[pro230116a\]]{#pro230116a label="pro230116a"} If the exact penalty funtion $P_{\rho}:\Re^n\to\Re$ is lower bounded and $x^*$ is a local minimizer, then there exist Lagrange multipliers $u^*\in\Re_+^{m_{\cal E}}$, $v^*\in\Re_+^{m_{\cal E}}$, $s^*\in\Re_+^{m_{\cal I}}$, $t^*\in\Re_+^{m_{\cal I}}$ such that the KKT conditions of problem ([\[prob2-1\]](#prob2-1){reference-type="ref" reference="prob2-1"})--([\[prob2-5\]](#prob2-5){reference-type="ref" reference="prob2-5"}) hold at $(x^*,y^*,z^*)$ with $y^*=|h(x^*)|$ and $z^*=\max\{0,-g(x^*)\}$. Moreover, $u^*\le e_{m_{\cal E}}$, $v^*\le e_{m_{\cal E}}$, $s^*\le e_{m_{\cal I}}$, $t^*\le e_{m_{\cal I}}$, where $e_{m_{\cal E}}\in\Re^{m_{\cal E}}$ and $e_{m_{\cal I}}\in\Re^{m_{\cal I}}$ are all-one vectors. *Proof.* Due to the equivalence between problems ([\[prob1L1\]](#prob1L1){reference-type="ref" reference="prob1L1"}) and ([\[prob2-1\]](#prob2-1){reference-type="ref" reference="prob2-1"})--([\[prob2-5\]](#prob2-5){reference-type="ref" reference="prob2-5"}), $x^*$ is a minimizer of problem ([\[prob1L1\]](#prob1L1){reference-type="ref" reference="prob1L1"}) if and only if $(x^*,y^*,z^*)$ is a minimizer of problem ([\[prob2-1\]](#prob2-1){reference-type="ref" reference="prob2-1"})--([\[prob2-5\]](#prob2-5){reference-type="ref" reference="prob2-5"}). Since MFCQ holds at $(x^*,y^*,z^*)$, it follows from the first-order necessary optimality conditions of general constrained optimization (see, for example, [@NocWri99; @SunYua06]) that any minimizer $(x^*,y^*,z^*)$ of problem ([\[prob2-1\]](#prob2-1){reference-type="ref" reference="prob2-1"})--([\[prob2-5\]](#prob2-5){reference-type="ref" reference="prob2-5"}) satisfies the KKT conditions $$\begin{aligned} &\,\,&\,\,\rho\nabla f(x^*)+\nabla h(x^*)(u^*-v^*)-\nabla g(x^*)s^*=0, \label{kkt2.9-1}\\ &\,\,&\,\,1-u_i^*-v_i^*=0,\ i=1,\ldots,m_{\cal E}, \label{kkt2.9-2}\\ &\,\,&\,\,1-s_i^*-t_i^*=0,\ i=1,\ldots,m_{\cal I}, \label{kkt2.9-3}\\ %\dd\dd u^*\ge 0,\ v^*\ge 0,\ s^*\ge 0,\ t^*\ge 0, \label{kkt2.9-4}\\ &\,\,&\,\,(u^*)^T(y^*-h(x^*))=0,~ (v^*)^T(y^*+h(x^*))=0, \\ &\,\,&\,\,(s^*)^T(z^*+g(x^*))=0,~ (t^*)^Tz^*=0, \label{kkt2.9-8}\end{aligned}$$ where $u^*\ge 0$, $v^*\ge 0$, $s^*\ge 0$, $t^*\ge 0$ are associated Lagrange multipliers. The boundedness of $u^*$, $v^*$, $s^*$ and $t^*$ derive from equations ([\[kkt2.9-2\]](#kkt2.9-2){reference-type="ref" reference="kkt2.9-2"}), ([\[kkt2.9-3\]](#kkt2.9-3){reference-type="ref" reference="kkt2.9-3"}) and the nonnegativity immediately. ◻ Similar results have been presented in [@BurCuW14; @ByrCuN10; @GouOrT15]. For example, Gould et al. in [@GouOrT15] think that the reformulation is surprisingly regular. Let $L(x,\mu,\lambda)=f(x)-\mu^Th(x)-\lambda^Tg(x)$ be the Lagrangian of problem ([\[prob1-1\]](#prob1-1){reference-type="ref" reference="prob1-1"}), where $\mu$ and $\lambda$ are Lagrange multipliers associated with the equality and inequality constraints, respectively. In what follows, we describe the relations of minimizers between problem ([\[prob1-1\]](#prob1-1){reference-type="ref" reference="prob1-1"}) and its robust counterpart ([\[prob2-1\]](#prob2-1){reference-type="ref" reference="prob2-1"})--([\[prob2-5\]](#prob2-5){reference-type="ref" reference="prob2-5"}). [\[lemma2\]]{#lemma2 label="lemma2"} Let $x^*$ be a local minimizer of problem ([\[prob1-1\]](#prob1-1){reference-type="ref" reference="prob1-1"}) at which LICQ and the second-order sufficient conditions (SOSCs) are satisfied with Lagrange multipliers $\mu^*\in\Re^{m_{\cal E}}$ and $\lambda^*\in\Re_{+}^{m_{\cal I}}$. Then there exists a threshold value $\rho^*>0$ such that, for all $0<\rho\le\rho^*$, $(x^*,y^*,z^*)$ together with $y^*=0$ and $z^*=0$ is a strict local minimizer of problem ([\[prob2-1\]](#prob2-1){reference-type="ref" reference="prob2-1"})--([\[prob2-5\]](#prob2-5){reference-type="ref" reference="prob2-5"}), and there exist Lagrange multipliers $u^*\in\Re_+^{m_{\cal E}}$, $v^*\in\Re_+^{m_{\cal E}}$, $s^*\in\Re_+^{m_{\cal I}}$, $t^*\in\Re_+^{m_{\cal I}}$ such that for problem ([\[prob2-1\]](#prob2-1){reference-type="ref" reference="prob2-1"})--([\[prob2-5\]](#prob2-5){reference-type="ref" reference="prob2-5"}), LICQ, the KKT conditions and SOSCs are satisfied at $(x^*,y^*,z^*)$. *Proof.* The result follows from the equivalence of problem ([\[prob2-1\]](#prob2-1){reference-type="ref" reference="prob2-1"})--([\[prob2-5\]](#prob2-5){reference-type="ref" reference="prob2-5"}) to the exact penalty optimization and Theorem 17.3 of [@NocWri99]. In particular, $(x^*,0,0)$ is feasible to the problem ([\[prob2-1\]](#prob2-1){reference-type="ref" reference="prob2-1"})--([\[prob2-5\]](#prob2-5){reference-type="ref" reference="prob2-5"}), and the LICQ at $(x^*,0,0)$ is implied by LICQ at $x^*$ for problem ([\[prob1-1\]](#prob1-1){reference-type="ref" reference="prob1-1"}). The penalty parameter $\rho^*=\frac{1}{\max\{\|\mu^*\|_{\infty},\|\lambda^*\|_{\infty}\}}$, and $u_i^*=\frac{1}{2}{(1-\rho\mu_i^*)},~v_i^*=\frac{1}{2}{(1+\rho\mu_i^*)}~\hbox{for}~ i=1,\ldots,m_{\cal E},$ $s_i^*={\rho}{\lambda_i^*},~t_i^*=1-{\rho\lambda_i^*}~\hbox{for}~i=1,\ldots,m_{\cal I}.$ ◻ The preceding proposition shows that, when the original nonlinear constrained optimization ([\[prob1-1\]](#prob1-1){reference-type="ref" reference="prob1-1"}) is feasible, the reformulation problem ([\[prob2-1\]](#prob2-1){reference-type="ref" reference="prob2-1"})--([\[prob2-5\]](#prob2-5){reference-type="ref" reference="prob2-5"}) can be equivalent to it when the parameter $\rho \le\frac{1}{\max\{\|\mu^*\|_{\infty},\|\lambda^*\|_{\infty}\}}$. However, this threshold is unknown before solving the original problem, and certain adaptive update procedure on the penalty parameter should be introduced to promote convergence of the methods based on solving the reformulation problem. The following result is important since it provides a certificate on that a solution of the reformulation ([\[prob2-1\]](#prob2-1){reference-type="ref" reference="prob2-1"})--([\[prob2-5\]](#prob2-5){reference-type="ref" reference="prob2-5"}) can be a solution of optimization with least $\ell_1$-norm constraint violations ([\[prob1b1-1\]](#prob1b1-1){reference-type="ref" reference="prob1b1-1"}). If the solution is not a solution of an equality constrained optimization problem based the active set of constraints at the solution of problem ([\[prob2-1\]](#prob2-1){reference-type="ref" reference="prob2-1"})--([\[prob2-5\]](#prob2-5){reference-type="ref" reference="prob2-5"}), it is not a solution of optimization with least $\ell_1$-norm constraint violations. Thus, it establishes a foundation for designing methods for nonlinear optimization based on optimization with least $\ell_1$-norm constraint violations. [\[theorem2\]]{#theorem2 label="theorem2"} For any fixed $\rho>0$, suppose that $(x^*,y^*,z^*)$ with $y^*=|h(x^*)|$ and $z^*=\max\{0, -g(x^*)\}$ is a KKT point of problem ([\[prob2-1\]](#prob2-1){reference-type="ref" reference="prob2-1"})--([\[prob2-5\]](#prob2-5){reference-type="ref" reference="prob2-5"}), $(u^*,v^*,s^*,t^*)$ with $u^*\in\Re_+^{m_{\cal E}}$, $v^*\in\Re_+^{m_{\cal E}}$, $s^*\in\Re_+^{m_{\cal I}}$, $t^*\in\Re_+^{m_{\cal I}}$ are the associated Lagrange multipliers. Let ${\cal I}_h^*=\{i| h_i(x^*)=0\}$, ${\cal I}_g^*=\{i| g_i(x^*)=0\}$. If $x^*$ is a local solution of problem $$\begin{aligned} \min_{x\in\Re^n}~\|h(x)\|_1+\|\max\{0, -g(x)\}\|_1, \label{L1p}\end{aligned}$$ then it is a KKT point of problem $$\begin{aligned} \min_x&\,\,&\,\,f(x) \label{kkt2.13a-1}\\ \hbox{s.t.}&\,\,&\,\,h_i(x)=0,~i\in{\cal I}_h^*, \label{kkt2.13a-2}\\ &\,\,&\,\,g_i(x)=0,~i\in{\cal I}_g^*. \label{kkt2.13a-3}\end{aligned}$$ Furthermore, if gradients $\nabla h_i(x^*) (i\in{\cal I}_h^*)$, $\nabla g_i(x^*) (i\in{\cal I}_g^*)$ are linearly independent, and there holds $d^T\nabla_{xx}^2\hat L(x^*,\hat\mu^*,\hat\lambda^*)d>0$ for all $d\ne 0$ satisfying $\nabla h_i(x^*)^Td=0$ for $i\in{\cal I}_h^*$ and $\nabla g_i(x^*)^Td=0$ for $i\in{\cal I}_g^*$, where $\hat\mu_i^* (i\in{\cal I}_h^*)$ and $\hat\lambda_i^* (i\in{\cal I}_g^*)$ are the Lagrange multipliers of problem ([\[kkt2.13a-1\]](#kkt2.13a-1){reference-type="ref" reference="kkt2.13a-1"})--([\[kkt2.13a-3\]](#kkt2.13a-3){reference-type="ref" reference="kkt2.13a-3"}) and $\hat L(x^*,\hat\mu^*,\hat\lambda^*)=f(x^*)-\sum_{i\in{\cal I}_h^*}\hat\mu_i^* h_i(x^*)-\sum_{i\in{\cal I}_g^*}\hat\lambda_i^* g_i(x^*)$, then $x^*$ is a strict local minimizer of optimization with least $\ell_1$-norm constraint violations ([\[prob1b1-1\]](#prob1b1-1){reference-type="ref" reference="prob1b1-1"}) (i.e., $x^*$ is a local solution for minimizing $\ell_1$-norm constraint violations and is a strict local minimizer of $f(x)$ in certain neighborhood of $x^*$). *Proof.* Note that $(x^*,y^*,z^*,u^*,v^*,s^*,t^*)$ satisfies the KKT conditions ([\[kkt2.9-1\]](#kkt2.9-1){reference-type="ref" reference="kkt2.9-1"})--([\[kkt2.9-8\]](#kkt2.9-8){reference-type="ref" reference="kkt2.9-8"}) of problem ([\[prob2-1\]](#prob2-1){reference-type="ref" reference="prob2-1"})--([\[prob2-5\]](#prob2-5){reference-type="ref" reference="prob2-5"}). Then $u_i^*=1,~v_i^*=0$ for $i\in\{i|h_i(x^*)>0\}$, $u_i^*=0,~v_i^*=1$ for $i\in\{i|h_i(x^*)<0\}$, $s_i^*=0,~t_i^*=1$ for $i\in\{i|g_i(x^*)>0\}$, $s_i^*=1,~t_i^*=0$ for $i\in\{i|g_i(x^*)<0\}$. The fact that $x^*$ is a solution of problem ([\[L1p\]](#L1p){reference-type="ref" reference="L1p"}) implies that there exist $\xi_i^*\in [-1,1], i\in{\cal I}_h^*$ and $\eta_i^*\in [-1,0], i\in{\cal I}_g^*$ such that $$\begin{aligned} &\,\,&\,\,\sum_{i\in\{i|h_i(x^*)>0\}}\nabla h_i(x^*)-\sum_{i\in\{i|h_i(x^*)<0\}}\nabla h_i(x^*)+\sum_{i\in{\cal I}_h^*}\xi_i^*\nabla h_i(x^*) \nonumber\\ &\,\,&\,\,\quad\quad-\sum_{i\in\{i|g_i(x^*)<0\}}\nabla g_i(x^*)+\sum_{i\in{\cal I}_g^*}\eta_i^*\nabla g_i(x^*)=0. \label{210912a}\end{aligned}$$ Subtracting ([\[210912a\]](#210912a){reference-type="ref" reference="210912a"}) from the left-hand-side of ([\[kkt2.9-1\]](#kkt2.9-1){reference-type="ref" reference="kkt2.9-1"}), we derive $$\begin{aligned} \rho\nabla f(x^*)-\sum_{i\in{\cal I}_h^*}\tilde\mu_i^*\nabla h_i(x^*)-\sum_{i\in{\cal I}_g^*}\tilde\lambda_i^*\nabla g_i(x^*)=0, \nonumber\end{aligned}$$ where $\tilde\mu_i^*=\xi_i^*-u_i^*+v_i^*$, $\tilde\lambda_i^*=\eta_i^*+s_i^*$. Thus, $x^*$ is a KKT point of problem ([\[kkt2.13a-1\]](#kkt2.13a-1){reference-type="ref" reference="kkt2.13a-1"})--([\[kkt2.13a-3\]](#kkt2.13a-3){reference-type="ref" reference="kkt2.13a-3"}), $\hat\mu_i^*=\frac{\tilde\mu_i^*}{\rho}, i\in{\cal I}_h^*$ and $\hat\lambda_i^*=\frac{\tilde\lambda_i^*}{\rho}, i\in{\cal I}_g^*$ are the associated Lagrange multipliers. It is well known that conditions of the proposition will guarantee $x^*$ to be a strict local minimizer of problem ([\[kkt2.13a-1\]](#kkt2.13a-1){reference-type="ref" reference="kkt2.13a-1"})--([\[kkt2.13a-3\]](#kkt2.13a-3){reference-type="ref" reference="kkt2.13a-3"}) (see [@NocWri99; @SunYua06]). Since by continuity all points around $x^*$ is a subset of the feasible set of problem ([\[kkt2.13a-1\]](#kkt2.13a-1){reference-type="ref" reference="kkt2.13a-1"})--([\[kkt2.13a-3\]](#kkt2.13a-3){reference-type="ref" reference="kkt2.13a-3"}), the desired result follows immediately. ◻ # An exact penalty SQP algorithm The SQP approach is very effective in solving general nonlinear optimization. In this section, we present an exact penalty SQP method for the original problem ([\[prob1-1\]](#prob1-1){reference-type="ref" reference="prob1-1"}) based on solving the subproblem ([\[prob2-1\]](#prob2-1){reference-type="ref" reference="prob2-1"})--([\[prob2-5\]](#prob2-5){reference-type="ref" reference="prob2-5"}). Our method can be taken as a modified line search S$\ell_1$QP which not only is applicable to the feasible nonlinear constrained optimization like the generic S$\ell_1$QP (see [@Fletch87]), but also is applicable to the infeasible problems. It is also a modification of the method [@BurCuW14] where we use a simple rule for penalty update and inner-outer iterative framework to replace the sophisticated rule and single iterative framework. Moreover, our QP subproblems are different from those of [@BurCuW14] in coping with equality constraints. The proposed method shares similarity to the methods in [@BurCuW14; @ByrCuN10] which are based on solving QP subproblems and can be rapidly detecting the infeasibility of nonlinear constrained optimization. It is also similar to that of [@ByrCuN10] which may be necessary for solving several QP subproblems before updating the penalty parameter, although the aims and results are not the same where we use every direction generated by QP subproblems to obtain a new inner iteration point. Our gains from these modifications are the global and local convergence of the exact penalty approach to the stationary points of nonlinear optimization with least constraint violations. For problem ([\[prob2-1\]](#prob2-1){reference-type="ref" reference="prob2-1"})--([\[prob2-5\]](#prob2-5){reference-type="ref" reference="prob2-5"}), general SQP solves the QP subproblem $$\begin{aligned} \min_{p\in\Re^{n}\times\Re^{m_{\cal E}}\times\Re^{m_{\cal I}}}&\,\,&\,\,\nabla F_{\rho}(x,y,z)^Tp+\frac{1}{2}p^TH p \label{210706a}\\ \hbox{s.t.}&\,\,&\,\,G(x,y,z)+\nabla G(x,y,z)^Tp\ge 0, \label{210706b}\end{aligned}$$ where $H$ can be the exact Lagrangian Hessian $$\left[\begin{array}{cc} H_{\rho}(x,u,v,s) & 0_{n\times ({m}_{\cal E}+m_{\cal I})} \\ 0_{({m}_{\cal E}+m_{\cal I})\times n} & 0_{({m}_{\cal E}+m_{\cal I})\times ({m}_{\cal E}+m_{\cal I})} \end{array}\right]$$ with $H_{\rho}(x,u,v,s)=\rho\nabla^2 f(x)+\sum_{i=1}^{m_{\cal E}}(u_i-v_i)\nabla^2 h_i(x)-\sum_{i=1}^{m_{\cal I}}s_i\nabla^2 g_i(x)\in\Re^{n\times n}$ or its appropriate approximation, $(x,y,z)$ is the current iterate, $(u,v,s,t)$ is the estimate of the associated Lagrange multiplier vector. In particular, let $p=(d,y^+-y,z^+-z)$, $B_{\rho}$ be a positive definite approximation to $H_{\rho}(x,u,v,s)$, then subproblem ([\[210706a\]](#210706a){reference-type="ref" reference="210706a"})--([\[210706b\]](#210706b){reference-type="ref" reference="210706b"}) is the following feasible QP $$\begin{aligned} \min_{d,y^+,z^+}&\,\,&\,\,e_{m_{\cal E}}^Ty^++e_{m_{\cal I}}^Tz^++\rho \nabla f(x)^Td+\frac{1}{2}d^TB_{\rho}d \label{qp1a1}\\ \hbox{s.t.}&\,\,&\,\,y^+-h(x)-\nabla h(x)^Td\ge 0, \label{qp1a2}\\ &\,\,&\,\,y^++h(x)+\nabla h(x)^Td\ge 0, \label{qp1a3}\\ &\,\,&\,\,z^++g(x)+\nabla g(x)^Td\ge 0, \label{qp1a4}\\ &\,\,&\,\,z^+\ge 0, \label{qp1a5}\end{aligned}$$ where $e_{m_{\cal E}}$ and $e_{m_{\cal I}}$ are all-one vectors. Let $$c'(x;d)=\|h(x)+\nabla h(x)^Td\|_1+\|\max\{0,-(g(x)+\nabla g(x)^Td)\}\|_1.$$ The preceding problem is an equivalent of non-smooth $\ell_1$QP subproblem of [@Fletch87], $$\begin{aligned} \min_d&\,\,&\,\,\rho\nabla f(x)^Td+\frac{1}{2}d^TB_{\rho}d+c'(x;d). \nonumber\end{aligned}$$ If $(d_{\ell},y_{\ell}^+,z_{\ell}^+)$ be a solution of subproblem ([\[qp1a1\]](#qp1a1){reference-type="ref" reference="qp1a1"})--([\[qp1a5\]](#qp1a5){reference-type="ref" reference="qp1a5"}) at $(x_{\ell},y_{\ell},z_{\ell})$ with parameter $\rho_k$ and $B_{\rho}=B_{\ell}$, then $y_{\ell}^+=|h(x_{\ell})+\nabla h(x_{\ell})^Td_{\ell}|,~ z_{\ell}^+=\max\{0,-(g(x_{\ell})+\nabla g(x_{\ell})^Td_{\ell})\}$, and $$\begin{aligned} \rho_k\nabla f(x_{\ell})^Td_{\ell}+\frac{1}{2}d_{\ell}^TB_{\ell}d_{\ell}+(\|y_{\ell}^+\|_1\|+\|z_{\ell}^+\|_1)-(\|y_{\ell}\|_1\|+\|z_{\ell}\|_1)\le 0. \nonumber\end{aligned}$$ Now we are ready to present our exact penalty SQP algorithm with inner and outer iterations. Step 1 $x_0\in\Re^{n}$, $B_0\in{\cal S}_{++}^{n}$, $\rho_0>0$, $\sigma\in (0,1)$, $\tau\in (0,1)$, and the tolerance $\epsilon$. Let $y_0=|h(x_0)|$, $z_0=\max\{0,-g(x_0)\}$. Set $k:=0$. 1 For given $\rho_k$, solve the smooth feasible optimization ([\[prob2-1\]](#prob2-1){reference-type="ref" reference="prob2-1"})--([\[prob2-5\]](#prob2-5){reference-type="ref" reference="prob2-5"}) by SQP method approximately. **Step 1.0** Set $\ell:=0$, $(x_{\ell},y_{\ell},z_{\ell})=(x_k,y_k,z_k)$, $B_{\ell}=B_k$. **Step 1.1** Solve the QP subproblem ([\[qp1a1\]](#qp1a1){reference-type="ref" reference="qp1a1"})--([\[qp1a5\]](#qp1a5){reference-type="ref" reference="qp1a5"}) to derive $(d_{\ell},y_{\ell}^+,z_{\ell}^+)$ and Lagrange multipliers $(u_{\ell+1},v_{\ell+1},s_{\ell+1},t_{\ell+1})$, where $(x,y,z)=(x_{\ell},y_{\ell},z_{\ell})$, $B=B_{\ell}$, $\rho=\rho_k$. **Step 1.2** Evaluate $r_{\ell}:=(\|y_{\ell}^+\|_1+\|z_{\ell}^+\|_1)-(\|y_{\ell}\|_1+\|z_{\ell}\|_1)$. If $r_{\ell}\ge-\epsilon$, set $(x_{k+1},y_{k+1},z_{k+1})=(x_{\ell},y_{\ell},z_{\ell})$, $\mu_{k+1}=(v_{\ell+1}-u_{\ell+1})$, $\lambda_{k+1}=s_{\ell+1}$, $B_{k+1}=B_{\ell}$, stop the inner SQP algorithm and go to Step 2. **Step 1.3** Compute $P_{\rho_k}^{'}(x_{\ell};d_{\ell}):=\rho_{k}\nabla f(x_{\ell})^Td_{\ell}+r_{\ell}$. Select a nonnegative integer $\imath$ as small as possible such that $$\begin{aligned} P_{\rho_{k}}(x_{\ell}+\tau^{\imath}d_{\ell})-P_{\rho_{k}}(x_{\ell})\le\sigma\tau^{\imath}P_{\rho_{k}}^{'}(x_{\ell};d_{\ell}) \label{stepsize}\end{aligned}$$ and let $\alpha_{\ell}=\tau^{\imath}$. **Step 1.4** Let $x_{\ell+1}=x_{\ell}+\alpha_{\ell}d_{\ell},~y_{\ell+1}=|h(x_{\ell+1})|,~z_{\ell+1}=\max\{0,-g(x_{\ell+1})\}$. Update $B_{\ell}$ to $B_{\ell+1}$, set $\ell:=\ell+1$ and go to Step 1.1. 2 (**Update the parameter**). Let $d_{k+1}$ be a solution of the regularized linear $\ell_1$ minimization $$\begin{aligned} \min_d~\frac{1}{2}d^TB_{k+1}d+c'(x_{k+1};d), \label{LS}\end{aligned}$$ $y_{k+1}^+=|h(x_{k+1})+\nabla h(x_{k+1})^Td_{k+1}|$, $z_{k+1}^+=\max\{0,-(g(x_{k+1})+\nabla g(x_{k+1})^Td_{k+1})\}$, and $r_{k+1}=(\|y_{k+1}^+\|_1+\|z_{k+1}^+\|_1)-(\|y_{k+1}\|_1+\|z_{k+1}\|_1)$. If $\|d_{k+1}\|_{\infty}>\epsilon$, select a nonnegative integer $\imath$ as small as possible such that $x_{k+1}^0=x_{k+1}+\alpha_{k+1}d_{k+1}$ with $\alpha_{k+1}=\tau^{\imath}$ satisfies $$\begin{aligned} c(x_{k+1}^0)-c(x_{k+1})\le\sigma\tau^{\imath}r_{k+1}, \label{stepsize1}\end{aligned}$$ and update $\rho_k$ to $\rho_{k+1}= \min\{0.01\rho_k,\frac{c(x_{k+1})-c(x_{k+1}^0)}{f(x_{k+1}^0)-f(x_{k+1})}\}$ provided $\rho_kf(x_{k+1}^0)+c(x_{k+1}^0)>\rho_kf(x_{k+1})+c(x_{k+1})$, otherwise $\rho_{k+1}= \min\{0.1\rho_k,\rho_k^{1.5}\}$. Update $B_{k+1}$, set $(x_{k+1},y_{k+1},z_{k+1})=(x_{k+1}^0,y_{k+1}^0,z_{k+1}^0)$. Else if $\|d_{\ell}\|_{\infty}>\epsilon$, set $\rho_{k+1}=\min\{0.01\rho_k,\rho_k^{1.5}\}$. If $\max\{\|d_{k+1}\|_{\infty},\|d_{\ell}\|_{\infty}\}\le\epsilon$, stop the algorithm. Otherwise, set $k:=k+1$ and go to Step 1. Our algorithm consists of inner iterations (Step 1) and Outer iterations (Step 2), where the former approximately solves the smooth equivalent of the exact penalty problem ([\[prob1L1\]](#prob1L1){reference-type="ref" reference="prob1L1"}), and the latter updates the penalty parameter and provides suitable initial points for inner iterations based on feedback of the former. The whole algorithm will terminate at Step 3 with $\|d_{\ell}\|_{\infty}$ and $\|d_k\|_{\infty}$ sufficiently small. The inner iterations terminate when $r_{\ell}\ge-\epsilon$. This condition will be satisfied as $\|d_{\ell}\|$ is small enough, while it admits the termination of inner algorithm when $\|d_{\ell}\|$ is not very small but the constraint violations could not be reduced apparently. Our algorithm has similarity to the existing SQP methods with rapid detection of infeasibility [@BurCuW14; @ByrCuN10] in that it is proposed based an exact penalty function and uses line search methods along directions obtained by solving QP subproblems. However, they are distinct in the way for updating the penalty parameter, which brings about the distinct global and local convergence results. # Global convergence Our algorithm consists of the inner loop and the outer loop, in which the inner loop is an SQP algorithm for solving a smooth feasible optimization ([\[prob2-1\]](#prob2-1){reference-type="ref" reference="prob2-1"})--([\[prob2-5\]](#prob2-5){reference-type="ref" reference="prob2-5"}) with a fixed penalty parameter and the outer algorithm updates the penalty parameter based on the feedback of inner algorithm. In particular, since MFCQ always holds, every local minimizer of optimization ([\[prob2-1\]](#prob2-1){reference-type="ref" reference="prob2-1"})--([\[prob2-5\]](#prob2-5){reference-type="ref" reference="prob2-5"}) is a KKT point and the Lagrange multipliers are bounded by the parameter. ## Global convergence of the inner SQP In this subsection, we analyze convergence of the inner SQP algorithm. It is assumed that the inner SQP algorithm does not terminate in finite number of iterations. Our analysis will show that $d_{\ell}\to 0$ as $\ell\to\infty$ under this supposition, which further implies $r_{\ell}\to 0$ and the finite termination of inner loop with $r_{\ell}\ge-\epsilon$. Based on the supposition, the infinite sequences $\{x_{\ell}\}$, $\{y_{\ell}\}$, $\{z_{\ell}\}$, $\{B_{\ell}\}$, $\{d_{\ell}\}$ are generated. The fixed penalty parameter is $\rho_k$. Before doing our global analysis, we need the following blanket assumptions. **Assumption 1**. *(1) The iterative sequence $\{x_{\ell}\}$ is in a open bounded set of $\Re^n$ (which may depend on $\rho_k$). (2) The sequence of approximate matrices $\{B_{\ell}\}$ is uniformly bounded and symmetric positive definite for all $\ell$.* The conditions (1) and (2) in Assumption [Assumption 1](#ass1){reference-type="ref" reference="ass1"} are quite general conditions extensively used in convergence analysis for nonlinear optimization. Assumption [Assumption 1](#ass1){reference-type="ref" reference="ass1"} (2) is a moderate and easy satisfied condition since, as an example, the approximate Hessian $B_{\ell}$ can be simply chosen to be a multiple of the identity matrix. The first two results show that every QP subproblem for the exact penalty optimization is well-behaved, and the line search procedure is well-defined. Under Assumption [Assumption 1](#ass1){reference-type="ref" reference="ass1"}, for any given $\rho_k$, the solution $(d_{\ell},y_{\ell}^+,z_{\ell}^+)$ of the QP subproblem is unique and the associated vectors of Lagrange multipliers $(u_{\ell+1},v_{\ell+1},s_{\ell+1},t_{\ell+1})$ are bounded in $[0,1]$. Furthermore, if all gradients $\nabla h_i(x_{\ell}), ~i\in\{i|h_i(x_{\ell})=0\}$ and $\nabla g_i(x_{\ell}), ~i\in\{i| g_i(x_{\ell})=0\}$ are linearly independent, then the vector of multipliers $(u_{\ell+1},v_{\ell+1},s_{\ell+1},t_{\ell+1})$ is also unqiue. *Proof.* For simplicity of statement, let $$\begin{aligned} M_{\ell}(d)=&\,\,&\,\,\rho_k\nabla f(x_{\ell})^Td+\frac{1}{2}d^TB_{\ell}d+(\|h(x_{\ell})+\nabla h(x_{\ell})^Td\|_1 \nonumber\\ &\,\,&\,\,+\|\max\{0,-(g(x_{\ell})+\nabla g(x_{\ell})^Td)\}\|_1). \nonumber\end{aligned}$$ We first prove that $M_{\ell}(d)$ is a strongly convex function. By some calculation, for any $d_1\in\Re^n$, $d_2\in\Re^n$, and any $\alpha\in[0,1]$, one has $$\begin{aligned} M_{\ell}(\alpha d_1+(1-\alpha)d_2)\le\alpha M_{\ell}(d_1)+(1-\alpha)M_{\ell}(d_2)-\frac{1}{2}\alpha(1-\alpha)(d_1-d_2)^TB_{\ell}(d_1-d_2) \nonumber\end{aligned}$$ since $\|h(x_{\ell})+\nabla h(x_{\ell})^T(\alpha d_1+(1-\alpha)d_2)\|_1 \le\alpha\|h(x_{\ell})+\nabla h(x_{\ell})^Td_1\|_1+(1-\alpha)\|h(x_{\ell})+\nabla h(x_{\ell})^Td_2\|_1$ and $$\begin{aligned} &\,\,&\,\,\|\max\{0,-(g(x_{\ell})+\nabla g(x_{\ell})^T(\alpha d_1+(1-\alpha)d_2))\}\|_1 \nonumber\\ &\,\,&\,\,\le\alpha\|\max\{0,-(g(x_{\ell})+\nabla g(x_{\ell})^Td_1)\}\|_1+(1-\alpha)\|\max\{0,-(g(x_{\ell})+\nabla g(x_{\ell})^Td_2)\}\|_1.\nonumber\end{aligned}$$ Then the strong convexity of $M_{\ell}(d)$ follows from the uniform symmetric positive definiteness of $B_{\ell}$ (see Assumption [Assumption 1](#ass1){reference-type="ref" reference="ass1"} (2)). The fact that $M_{\ell}(d)$ is strongly convex implies that $d_{\ell}$ is unique, which further implies that $$\begin{aligned} y^+_{\ell}=|h(x_{\ell})+\nabla h(x_{\ell})^Td_{\ell}|\quad\hbox{and}\quad z^+_{\ell}=|\max\{0,-(g(x_{\ell})+\nabla g(x_{\ell})^Td_{\ell})\}| \nonumber\end{aligned}$$ are unique. In addition, it follows from the KKT conditions of QP subproblems that $$\begin{aligned} u_{\ell+1,i}+v_{\ell+1,i}=1,~\forall i=1,\ldots,m_{\cal E};~s_{\ell+1,i}+t_{\ell+1,i}=1,~\forall i=1,\ldots,m_{\cal I}, \nonumber\end{aligned}$$ and $(u_{\ell+1},v_{\ell+1},s_{\ell+1},t_{\ell+1})\ge 0$. Thus, $\|(u_{\ell+1},v_{\ell+1},s_{\ell+1},t_{\ell+1})\|_{\infty}\le1$. If the vectors of Lagrange multipliers $(u_{\ell+1},v_{\ell+1},s_{\ell+1},t_{\ell+1})$ are not unique, and an other vectors $(u_{\ell+1}^{'},v_{\ell+1}^{'},s_{\ell+1}^{'},t_{\ell+1}^{'})\ne(u_{\ell+1},v_{\ell+1},s_{\ell+1},t_{\ell+1})$ also satisfy the KKT conditions of QP subproblems, then $$\begin{aligned} \nabla h(x_{\ell})(u_{\ell+1}^{'}-v_{\ell+1}^{'}-u_{\ell+1}+v_{\ell+1})-\nabla g(x_{\ell})(s_{\ell+1}^{'}-s_{\ell+1})=0. \nonumber \end{aligned}$$ Noting that $u_{\ell+1,i}^{'}-v_{\ell+1,i}^{'}-u_{\ell+1,i}+v_{\ell+1,i}=0$ for $i\notin\{i|h_i(x_{\ell})=0\}$ and $s_{\ell+1,i}^{'}-s_{\ell+1,i}=0$ for $i\notin\{i| g_i(x_{\ell})=0\}$, the preceding equation contradicts that the gradients $\nabla h_i(x_{\ell}), ~i\in\{i|h_i(x_{\ell})=0\}$ and $\nabla g_i(x_{\ell}), ~i\in\{i| g_i(x_{\ell})=0\}$ are linearly independent, which completes our proof. ◻ [\[alp\]]{#alp label="alp"} Under Assumption [Assumption 1](#ass1){reference-type="ref" reference="ass1"}, there exists a scalar $\beta_k\in (0,1]$ independent of $\ell$ such that for $\alpha_{\ell}\in (0,\beta_k]$, ([\[stepsize\]](#stepsize){reference-type="ref" reference="stepsize"}) always holds. Thus, $\{\alpha_{\ell}\}$ is bounded away from zero. *Proof.* We firstly prove that $\{d_{\ell}\}$ is bounded by contradiction. Due to $$\begin{aligned} \rho_k\nabla f(x_{\ell})+B_{\ell}d_{\ell}+\nabla h(x_{\ell})(u_{\ell+1}-v_{\ell+1})-\nabla g(x_{\ell})s_{\ell+1}=0, \nonumber\end{aligned}$$ and the boundedness of $B_{\ell}$, $\nabla f(x_{\ell})$, $\nabla h(x_{\ell})$, and $(u_{\ell+1},v_{\ell+1},s_{\ell+1},t_{\ell+1})$, if $\{d_{\ell}\}$ is unbounded, dividing $\|d_{\ell}\|$ and taking the limit $\ell\to\infty$ on both sides of the preceding equation will result in $\lim_{\ell\to\infty}B_{\ell}\frac{d_{\ell}}{\|d_{\ell}\|}=0$, which contradicts the uniform positive definiteness of $B_{\ell}$. For every $\ell\ge 0$, one has $P^{'}_{\rho_k}(x_{\ell};d_{\ell})\le-\frac{1}{2}d_{\ell}^TB_{\ell}d_{\ell}\le-\frac{1}{2}\lambda_{\min}(B_{\ell})\|d_{\ell}\|^2$ and $$\begin{aligned} &\,\,&\,\,P_{\rho_k}(x_{\ell}+\alpha d_{\ell})-P_{\rho_k}(x_{\ell})-\sigma\alpha P^{'}_{\rho_k}(x_{\ell};d_{\ell}) \nonumber\\ &\,\,&\,\,=\rho_k f(x_{\ell}+\alpha d_{\ell})-\rho_k f(x_{\ell})-\sigma\alpha\rho_k\nabla f(x_{\ell})^Td_{\ell} \nonumber\\ &\,\,&\,\,\quad+(\|h(x_{\ell}+\alpha d_{\ell})\|_1-\|h(x_{\ell})\|_1-\sigma\alpha(\|h(x_{\ell})+\nabla h(x_{\ell})^Td_{\ell}\|_1-\|h(x_{\ell})\|_1)) \nonumber\\ &\,\,&\,\,\quad+(\|\max\{0,-g(x_{\ell}+\alpha d_{\ell})\}\|_1-\|\max\{0,-g(x_{\ell})\}\|_1 \nonumber\\ &\,\,&\,\,\quad\quad\quad-\sigma\alpha(\|\max\{0,-(g(x_{\ell})+\nabla g(x_{\ell})^Td_{\ell})\}\|_1-\|\max\{0,-g(x_{\ell})\}\|_1)) \nonumber\\ &\,\,&\,\,\le(1-\sigma)\alpha P^{'}_{\rho_k}(x_{\ell};d_{\ell})+O(\alpha^2\|d_{\ell}\|^2). \label{230405a}\end{aligned}$$ Then, due to the boundedness of $\|d_{\ell}\|$ and $\sigma\in (0,1)$, there exists a scalar $\beta_k\in (0,1]$ independent of inner iterations such that the left side of ([\[230405a\]](#230405a){reference-type="ref" reference="230405a"}) is non-positive for all $\alpha\in (0,\beta_k]$. Consequently, the result follows immediately. ◻ The next result indicates the finite termination of inner iterations. Under Assumption [Assumption 1](#ass1){reference-type="ref" reference="ass1"}, $\{P_{\rho_k}(x_{\ell})\}$ is lower bounded. Then one has $\lim_{\ell\to\infty} \|d_{\ell}\|=0$. That is, for any given $\epsilon>0$, $r_{\ell}\ge-\epsilon$ will be satisfied in a finite number of iterations. *Proof.* Note that the sequence $\{P_{\rho_k}(x_{\ell})\}$ is monotonically non-increasing. Thus, one has $\lim_{\ell\to\infty}P_{\rho_k}(x_{\ell})=P_k^*$ with $P_k^*$ being finite or $-\infty$. Since $f$ is lower bounded, $P_{\rho_k}(x_{\ell})$ is lower bounded, which rules the case $P_k^*=-\infty$ out. By taking the limit $\ell\to\infty$ on both hand-sides of ([\[stepsize\]](#stepsize){reference-type="ref" reference="stepsize"}) and noting the result of Lemma [\[alp\]](#alp){reference-type="ref" reference="alp"} and the inequality $P^{'}_{\rho_k}(x_{\ell};d_{\ell})\le-\frac{1}{2}\lambda_{\min}(B_{\ell})\|d_{\ell}\|^2$, one has the limit $\lim_{\ell\to\infty} \|d_{\ell}\|=0$ which further implies $\lim_{\ell\to\infty} |r_{\ell}|=0$. Our proof is completed. ◻ ## Global convergence of the whole algorithm In this subsection, we prove that for small $\rho_k$, $x_k$ can be either an approximate KKT point or an approximate DL-stationary point of the original problem. Otherwise, $\rho_k\to 0$ and there exists a cluster point of $\{x_k\}$ which is either a singular stationary point or a DZ-stationary point of problem ([\[prob1-1\]](#prob1-1){reference-type="ref" reference="prob1-1"}). Firstly, by our definitions, it is easy to know that if $\|y_k\|_1=0$ and $\|z_k\|_1=0$, then $x_k$ is a KKT point of the original optimization problem ([\[prob1-1\]](#prob1-1){reference-type="ref" reference="prob1-1"}). When $\|y_k\|_1+\|z_k\|_1\ne 0$, $x_k$ may be a DL-stationary point of the original optimization problem ([\[prob1-1\]](#prob1-1){reference-type="ref" reference="prob1-1"}) when $d_k=0$. [\[s4.2.2\]]{#s4.2.2 label="s4.2.2"} If $x_k$ is a KKT point of the smooth feasible problem ([\[prob2-1\]](#prob2-1){reference-type="ref" reference="prob2-1"})--([\[prob2-5\]](#prob2-5){reference-type="ref" reference="prob2-5"}) and $y_k=z_k=0$, then $x_k$ is a KKT point of the nonlinear optimization problem ([\[prob1-1\]](#prob1-1){reference-type="ref" reference="prob1-1"}). *Proof.* Due to $y_k=z_k=0$, $x_k$ is feasible to the problem ([\[prob1-1\]](#prob1-1){reference-type="ref" reference="prob1-1"}). Then the KKT conditions ([\[kkt2.9-1\]](#kkt2.9-1){reference-type="ref" reference="kkt2.9-1"})--([\[kkt2.9-8\]](#kkt2.9-8){reference-type="ref" reference="kkt2.9-8"}) of problem ([\[prob2-1\]](#prob2-1){reference-type="ref" reference="prob2-1"})--([\[prob2-5\]](#prob2-5){reference-type="ref" reference="prob2-5"}) are reduced to those for problem ([\[prob1-1\]](#prob1-1){reference-type="ref" reference="prob1-1"}). ◻ If for some $k\ge 0$, the inner iterations $\{x_{\ell}\}$ of the SQP algorithm converge to $x_{k+1}$ which is infeasible to the problem ([\[prob1-1\]](#prob1-1){reference-type="ref" reference="prob1-1"}), and $d_{k+1}=0$ is a solution of the regularized linear $\ell_1$ minimization subproblem ([\[LS\]](#LS){reference-type="ref" reference="LS"}), then $x_{k+1}$ is a DL-stationary point of the original optimization problem ([\[prob1-1\]](#prob1-1){reference-type="ref" reference="prob1-1"}). *Proof.* The fact $d_{k+1}=0$ shows $x_{k+1}$ is a stationary point of the $\ell_1$ minimization problem of constraint violations. Thus by Definition [\[def1\]](#def1){reference-type="ref" reference="def1"}, $x_{k+1}$ is a DL-stationary point of the original optimization problem ([\[prob1-1\]](#prob1-1){reference-type="ref" reference="prob1-1"}). ◻ We are left to show the results on $\rho_k$ to be sufficiently small, which will be given in the next lemma. Let $\{x_k\}$ be an infinite sequence generated by the outer algorithm. Then $k\to\infty$ and $\rho_k\to 0$. Suppose that ${\cal K}$ is an infinite set of indices and $\{x_k| k\in{\cal K}\}$ is any convergent subsequence of $\{x_k\}$, $\|y_k\|_1+\|z_k\|_1\to 0$ as $k\in{\cal K}$ and $k\to\infty$, then the limit point of $\{x_k| k\in{\cal K}\}$ is a singular stationary point of problem ([\[prob1-1\]](#prob1-1){reference-type="ref" reference="prob1-1"}). Otherwise, $\|y_k\|_1+\|z_k\|_1\not\to 0$ as $k\in{\cal K}$ and $k\to\infty$ and every cluster point of $\{x_k| k\in{\cal K}\}$ is a DZ-stationary point of problem ([\[prob1-1\]](#prob1-1){reference-type="ref" reference="prob1-1"}). *Proof.* For $k\in{\cal K}$, the limit $\|y_k\|_1+\|z_k\|_1\to 0$ as $k\to\infty$ implies that the limit point of $\{x_k| k\in{\cal K}\}$ is a feasible point of problem ([\[prob1-1\]](#prob1-1){reference-type="ref" reference="prob1-1"}). Combining with the KKT conditions of problem ([\[prob2-1\]](#prob2-1){reference-type="ref" reference="prob2-1"})--([\[prob2-5\]](#prob2-5){reference-type="ref" reference="prob2-5"}) and $\rho_k\to 0$ results in that the stationary conditions given in Definition [\[def0\]](#def0){reference-type="ref" reference="def0"} (2) hold at the limit point. Thus, the limit point of $\{x_k| k\in{\cal K}\}$ is a singular stationary point of problem ([\[prob1-1\]](#prob1-1){reference-type="ref" reference="prob1-1"}). Similarly, if $\|y_k\|_1+\|z_k\|_1\not\to 0$ as $k\in{\cal K}$ and $k\to\infty$, then any cluster point of $\{x_k| k\in{\cal K}\}$ is infeasible to the problem ([\[prob1-1\]](#prob1-1){reference-type="ref" reference="prob1-1"}), which brings about DZ-stationary points of problem ([\[prob1-1\]](#prob1-1){reference-type="ref" reference="prob1-1"}) since both the KKT conditions of problem ([\[prob2-1\]](#prob2-1){reference-type="ref" reference="prob2-1"})--([\[prob2-5\]](#prob2-5){reference-type="ref" reference="prob2-5"}) and the limit $\rho_k\to 0$ imply the conditions given in Definition [\[def1\]](#def1){reference-type="ref" reference="def1"} (2). ◻ In summary, we have the main global convergence results on Algorithm [\[alg1\]](#alg1){reference-type="ref" reference="alg1"} as follows. Suppose that $\{(x_k,y_k,z_k)\}$ is a sequence generated by the outer algorithm. If $(x_k,y_k,z_k)$ is the terminating point with $\rho_k$ far from zero, then $x_k$ is an approximate KKT point of problem ([\[prob1-1\]](#prob1-1){reference-type="ref" reference="prob1-1"}) provided both $\|y_k\|$ and $\|z_k\|$ small enough, otherwise is an approximate DL-stationary point of problem ([\[prob1-1\]](#prob1-1){reference-type="ref" reference="prob1-1"}). If $(x_k,y_k,z_k)$ is the terminating point with $\rho_k$ close to zero enough, then $x_k$ is an approximate singular stationary point of problem ([\[prob1-1\]](#prob1-1){reference-type="ref" reference="prob1-1"}) provided both $\|y_k\|$ and $\|z_k\|$ small enough, otherwise is an approximate DZ-stationary point of problem ([\[prob1-1\]](#prob1-1){reference-type="ref" reference="prob1-1"}). *Proof.* The results follow from the preceding three lemmas of this subsection immediately. ◻ # Local convergence We have proved in the preceding section that under very mild conditions, our algorithm may either terminate finitely at certain outer iteration with $\rho_k$ far from zero or generate a sequence of outer iterations with $\rho_k\to 0$. When the outer iteration terminates with $\rho_k$ far from zero, the inner iterations may converge to either a KKT point or a DL-stationary point; otherwise, the outer iterations may converge to either a singular stationary point or a DZ-stationary point. In Section 6.1, it is proved that under suitable assumptions, our algorithm can be quadratically convergent to a KKT point when the original problem is feasible, and can be quadratically convergent to a DL-stationary point when it is infeasible (that is, *our algorithm may rapidly detect infeasibility of the problem without requiring $\rho_k$ close to zero*). In Section 6.2, our algorithm can be rapidly convergent to a DZ-stationary point as $\rho_k\to 0$, which is similar to the results of [@BurCuW14; @ByrCuN10]. Like all local convergence analysis, we need to suppose that all functions $f(x)$, $g(x)$ and $h(x)$ are twice differentiable, and their second derivatives are Lipschitz continuous around the limit $x^*$. ## Rapid convergence to either a KKT point or a DL-stationary point This subsection focuses on rapid convergence of the inner algorithm for fixed $\rho_k>0$. It is well known that SQP can be locally quadratically/superlinearly convergent when the sequence of Hessians of subproblems is suitably approximated to the exact Lagrangian Hessian, LICQ and the strict complementarity conditions hold at the solution. Thus, we will examine these conditions on problem ([\[prob2-1\]](#prob2-1){reference-type="ref" reference="prob2-1"})--([\[prob2-5\]](#prob2-5){reference-type="ref" reference="prob2-5"}). **Assumption 1**. *Let $\{(x_{\ell},y_{\ell},z_{\ell})\}$ be the iterative sequence generated by the inner algorithm, and $\{(u_{\ell},v_{\ell},s_{\ell},t_{\ell})\}$ is the associated sequence of Lagrangian multipliers. There hold $(x_{\ell},y_{\ell},z_{\ell})\to (x^*,y^*,z^*)$ and $(u_{\ell},v_{\ell},s_{\ell},t_{\ell})\to(u^*,v^*,s^*,t^*)$ as $\ell\to\infty$, where $(x^*,y^*,z^*)$ is a KKT point of problem ([\[prob2-1\]](#prob2-1){reference-type="ref" reference="prob2-1"})--([\[prob2-5\]](#prob2-5){reference-type="ref" reference="prob2-5"}) in which $x^*$ is either a KKT point (correspondingly, $y^*+z^*=0$) or a DL-stationary point ($y^*+z^*>0$) of the original problem ([\[prob1-1\]](#prob1-1){reference-type="ref" reference="prob1-1"}), $(u^*,v^*,s^*,t^*)$ is the associated Lagrange multiplier vector.* Under Assumption [Assumption 1](#ass2a){reference-type="ref" reference="ass2a"}, all components of the Lagrange multiplier vectors $u^*$, $v^*$, $s^*$, $t^*$ are in the interval $[0,1]$. Let ${\cal I}_h^{\ell}=\{i\in{\cal E}| h_i(x_{\ell})=0\}$, ${\cal I}_g^{\ell}=\{i\in{\cal I}| g_i(x_{\ell})=0\}$, ${\cal I}_h^{*}=\{i\in{\cal E}| h_i(x^*)=0\}$, ${\cal I}_g^{*}=\{i\in{\cal I}| g_i(x^*)=0\}$, and $\mu_{\ell}=\frac{v_{\ell}-u_{\ell}}{\rho_k}$, $\lambda_{\ell}=\frac{s_{\ell}}{\rho_k}$, $\mu^*=\frac{v^*-u^*}{\rho_k}$, $\lambda^*=\frac{s^*}{\rho_k}$. Then $\mu_{\ell}\to\mu^*$ and $\lambda_{\ell}\to\lambda^*$ as $\ell\to\infty$. **Assumption 1**. * \ (1) The gradients $\nabla h_i(x^*), i\in{\cal I}_h^*$ and $\nabla g_i(x^*), i\in{\cal I}_g^*$ are linearly independent;\ (2) The multipliers $u_i^*\in (0,1)$ and $v_i^*\in (0,1)$, $\forall~i\in{\cal I}_h^*$, and $s_i^*\in (0,1)$, $\forall~i\in{\cal I}_g^*$;\ (3) The Lagrangian Hessian satisfies $$d^T\nabla^2_{xx}L(x^*,\mu^*,\lambda^*)d\ge\gamma\|d\|^2,$$ $\forall d\in\{d| \nabla h_i(x^*)^Td=0,~i\in{\cal I}_h^*; \nabla g_i(x^*)^Td=0,~i\in{\cal I}_g^*\}$, where $L(x,\mu,\lambda)=f(x)-\mu^Th(x)-\lambda^Tg(x)$, and $\gamma>0$ is a constant.* Assumption [Assumption 1](#ass2c1){reference-type="ref" reference="ass2c1"} (1) holds if and only if LICQ holds with problem ([\[prob2-1\]](#prob2-1){reference-type="ref" reference="prob2-1"})--([\[prob2-5\]](#prob2-5){reference-type="ref" reference="prob2-5"}). Since $u_i^*+v_i^*=1$ for all $i\in{\cal E}$, Assumption [Assumption 1](#ass2c1){reference-type="ref" reference="ass2c1"} (2) implies the strict complementarity conditions for problem ([\[prob2-1\]](#prob2-1){reference-type="ref" reference="prob2-1"})--([\[prob2-5\]](#prob2-5){reference-type="ref" reference="prob2-5"}). Correspondingly, $|\rho_k\mu^*_i|<1,~i\in{\cal I}_h^*$ and $\rho_k\lambda^*_i\in(0,1),~i\in{\cal I}_g^*$ for the original problem. The following results show that under Assumptions [Assumption 1](#ass2a){reference-type="ref" reference="ass2a"} and [Assumption 1](#ass2c1){reference-type="ref" reference="ass2c1"}, $d_{\ell}$ can be a superlinearly or quadratically convergent step. [\[thms1\]]{#thms1 label="thms1"} Suppose that Assumptions [Assumption 1](#ass2a){reference-type="ref" reference="ass2a"} and [Assumption 1](#ass2c1){reference-type="ref" reference="ass2c1"} hold. \(1\) If $\|(B_{\ell}-\nabla^2_{xx} L(x^*,\mu^*,\lambda^*))d_{\ell}\|=o(\|d_{\ell}\|)$, then $$\begin{aligned} \lim_{\ell\to\infty}\frac{\|x_{\ell}+d_{\ell}-x^*\|}{\|x_{\ell}-x^*\|}=0. \label{local1}\end{aligned}$$ \(2\) If $B_{\ell}=\nabla^2_{xx}L(x_{\ell},\mu_{\ell},\lambda_{\ell})$, then $\|x_{\ell}+d_{\ell}-x^*\|=O(\|x_{\ell}-x^*\|^2).$ *Proof.* Let $A_*=[\nabla h_{{\cal I}_h^*}(x^*)~\nabla g_{{\cal I}_g^*}(x^*)]$, $A_{\ell}=[\nabla h_{{\cal I}_h^{\ell}}(x_{\ell})~\nabla g_{{\cal I}_g^{\ell}}(x_{\ell})]$, $P_*=I-A_*(A_*^TA_*)^{-1}A_*^T$ and $P_{\ell}=I-A_{\ell}(A_{\ell}^TA_{\ell})^{-1}A_{\ell}^T$, where $I$ is the $n\times n$ identity matrix. Consider the system $$\begin{aligned} &\,\,&\,\, \left[\begin{array}{c} P_*\nabla_{xx}^2L(x^*,\mu^*,\lambda^*) \\ A_*^T\end{array}\right]d=0. \label{080915c}\end{aligned}$$ Let $d^*\in\Re^n$ be any one of its solutions. If $d^*\ne 0$, then $$\begin{aligned} (d^*)^TP_*\nabla_{xx}^2L(x^*,\mu^*,\lambda^*)d^*=0, \quad A_*^Td^*=0, \nonumber\end{aligned}$$ so we have $(d^*)^T\nabla_{xx}^2L(x^*,\mu^*,\lambda^*)d^*=0,$ which contradicts Assumption [Assumption 1](#ass2c1){reference-type="ref" reference="ass2c1"} (3). This contradiction shows that the coefficient matrix of the system ([\[080915c\]](#080915c){reference-type="ref" reference="080915c"}) has full column rank. Therefore, by Assumptions [Assumption 1](#ass2a){reference-type="ref" reference="ass2a"} and [Assumption 1](#ass2c1){reference-type="ref" reference="ass2c1"}, for all sufficiently large ${\ell}$, the matrix $$\begin{aligned} \left[\begin{array}{c} P_{\ell}\nabla_{xx}^2L(x^*,\mu^*,\lambda^*) \\ A_{\ell}^T\end{array}\right] \nonumber\end{aligned}$$ is of full column rank. Since $\nabla f(x_{\ell})+B_{\ell}d_{\ell}-\nabla h_{{\cal I}_h^{\ell}}(x_{\ell})\mu_{{\cal I}_h^{\ell}}-\nabla g_{{\cal I}_g^{\ell}}(x_{\ell})\lambda_{{\cal I}_g^{\ell}}=0,$ there holds $$\begin{aligned} P_{\ell}B_{\ell}d_{\ell} &\,\,&\,\,=-P_{\ell}(\nabla f(x_{\ell})-\nabla h_{{\cal I}_h^{\ell}}(x_{\ell})\mu_{{\cal I}_h^{\ell}}-\nabla g_{{\cal I}_g^{\ell}}(x_{\ell})\lambda_{{\cal I}_g^{\ell}}) \nonumber\\ &\,\,&\,\,=-P_{\ell}(\nabla f(x_{\ell})-\nabla h_{{\cal I}_h^{\ell}}(x_{\ell})\mu_{{\cal I}_h^*}-\nabla g_{{\cal I}_g^{\ell}}(x_{\ell})\lambda_{{\cal I}_g^*}) \nonumber\\ &\,\,&\,\, =-P_{\ell}\nabla_{xx}^2L(x^*,\mu^*,\lambda^*)(x_{\ell}-x^*)+{O}(\|x_{\ell}-x^*\|^2). \nonumber \end{aligned}$$ Thus, $$\begin{aligned} &\,\,&\,\,P_{\ell}(B_{\ell}-\nabla_{xx}^2L(x^*,\mu^*,\lambda^*))d_{\ell} \nonumber\\ &\,\,&\,\,=-P_{\ell}\nabla_{xx}^2L(x^*,\mu^*,\lambda^*) (x_{\ell}+d_{\ell}-x^*)+{O}(\|x_{\ell}-x^*\|^2). \label{080915a}\end{aligned}$$ Thanks to $h_{{\cal I}_h^{\ell}}(x^*)=0$ and $h_{{\cal I}_h^{\ell}}(x_{\ell})=h_{{\cal I}_h^{\ell}}(x_{\ell})-h_{{\cal I}_h^{\ell}}(x^*) =\nabla h_{{\cal I}_h^{\ell}}(x^*)^T(x_{\ell}-x^*)+{O}(\|x_{\ell}-x^*\|^2)$, and $g_{{\cal I}_g^{\ell}}(x^*)=0$ and $g_{{\cal I}_g^{\ell}}(x_{\ell})=g_{{\cal I}_g^{\ell}}(x_{\ell})-g_{{\cal I}_g^{\ell}}(x^*) =\nabla g_{{\cal I}_g^{\ell}}(x^*)^T(x_{\ell}-x^*)+{O}(\|x_{\ell}-x^*\|^2)$, there holds $$\begin{aligned} A_{{\ell}}^T(x_{\ell}+d_{\ell}-x^*)=\left[\begin{array}{c} h_{{\cal I}_h^{\ell}}(x_{\ell}) \\ g_{{\cal I}_g^{\ell}}(x^*)\end{array}\right]+A_{{\ell}}^Td_{\ell}+{O}(\|x_{\ell}-x^*\|^2). \label{080915b}\end{aligned}$$ Putting ([\[080915a\]](#080915a){reference-type="ref" reference="080915a"}) and ([\[080915b\]](#080915b){reference-type="ref" reference="080915b"}) together in a matrix, we can obtain $$\begin{aligned} &\,\,&\,\,\left[\begin{array}{c} P_{\ell}\nabla_{xx}^2L(x^*,\mu^*,\lambda^*) \\ A_{{\ell}}^T\end{array}\right](x_{\ell}+d_{\ell}-x^*)\nonumber\\ &\,\,&\,\,=\left[\begin{array}{c} -P_{\ell}(B_{\ell}-\nabla_{xx}^2L(x^*,\mu^*,\lambda^*))d_{\ell} \\ h_{{\cal I}_h^{\ell}}(x_{\ell})+\nabla h_{{\cal I}_h^{\ell}}(x_{\ell})^Td_{\ell} \\ g_{{\cal I}_g^{\ell}}(x_{\ell})+\nabla g_{{\cal I}_g^{\ell}}(x_{\ell})^Td_{\ell} \end{array}\right]+{O}(\|x_{\ell}-x^*\|^2), \label{080915d}\end{aligned}$$ where the coefficient matrix has previously been proved to be of full column rank for all sufficiently large ${\ell}$. \(1\) If $\|(B_{\ell}-\nabla^2_{xx} L(x^*,\mu^*,\lambda^*))d_{\ell}\|=o(\|d_{\ell}\|)$, then $$\begin{aligned} \left\|\left[\begin{array}{c} -P_{\ell}(B_{\ell}-\nabla_{xx}^2L(x^*,\mu^*,\lambda^*))d_{\ell} \\ h_{{\cal I}_h^{\ell}}(x_{\ell})+\nabla h_{{\cal I}_h^{\ell}}(x_{\ell})^Td_{\ell} \\ g_{{\cal I}_g^{\ell}}(x_{\ell})+\nabla g_{{\cal I}_g^{\ell}}(x_{\ell})^Td_{\ell} \end{array}\right]\right\|={o}(\|d_{\ell}\|). \nonumber\end{aligned}$$ Hence, by ([\[080915d\]](#080915d){reference-type="ref" reference="080915d"}) and since $x_{\ell}\to x^*$ as ${\ell}\to\infty$, there holds $$\begin{aligned} \lim_{\ell\to\infty}\frac{\|x_{\ell}+d_{\ell}-x^*\|}{\|x_{\ell}-x^*\|}=\lim_{{\ell}\to\infty}\frac{{o}(\|d_{\ell}\|)}{\|x_{\ell}-x^*\|}, \nonumber \end{aligned}$$ which shows $$\begin{aligned} \lim_{{\ell}\to\infty}\frac{\|d_{\ell}\|}{\|x_{\ell}-x^*\|}=1 \label{081104a}\end{aligned}$$ and thus ([\[local1\]](#local1){reference-type="ref" reference="local1"}) follows immediately. \(2\) If $B_{\ell}=\nabla^2_{xx}L(x_{\ell},\mu_{\ell},\lambda_{\ell})$, then $$\begin{aligned} \left\|\left[\begin{array}{c} -P_{\ell}(B_{\ell}-\nabla_{xx}^2L(x^*,\mu^*,\lambda^*))d_{\ell} \\ h_{{\cal I}_h^{\ell}}(x_{\ell})+\nabla h_{{\cal I}_h^{\ell}}(x_{\ell})^Td_{\ell} \\ g_{{\cal I}_g^{\ell}}(x_{\ell})+\nabla g_{{\cal I}_g^{\ell}}(x_{\ell})^Td_{\ell} \end{array}\right]\right\|=O(\|x_{\ell}-x^*\|\|d_{\ell}\|), \nonumber\end{aligned}$$ which implies that ([\[081104a\]](#081104a){reference-type="ref" reference="081104a"}) holds. Hence, it follows from ([\[080915d\]](#080915d){reference-type="ref" reference="080915d"}) that $$\begin{aligned} \lim_{\ell\to\infty}\frac{\|x_{\ell}+d_{\ell}-x^*\|}{\|x_{\ell}-x^*\|^2}=\lim_{{\ell}\to\infty}\frac{{O}(\|d_{\ell}\|)}{\|x_{\ell}-x^*\|}+O(1)=O(1), \nonumber\end{aligned}$$ which completes our proof. ◻ ## Rapid convergence to a DZ-stationary point We have known that convergence to a DZ-stationary point can happen only when $k\to\infty$ and $\rho_k\to 0$. For convenience of statement, we denote $w^*=(x^*,\mu^*,\lambda^*)$, $\hat w^*=(x^*,u^*,v^*,s^*,t^*)$, and $w_k=(x_k,\mu_k,\lambda_k)$, $\hat w_k=(x_k,u_k,v_k,s_k,t_k)$ for all $k\ge 0$. When applying to optimization with only inequality constraints, the approach for local analysis in this subsection is almost the same as that of [@ByrCuN10]. Thus, it can be thought of as an extension of the analysis of [@ByrCuN10] to general constrained optimization. The following assumptions are similar to those of [@ByrCuN10]. **Assumption 1**. * \ (1) $w_k\rightarrow w^*$ and $\rho_k\rightarrow 0$ as $k\rightarrow \infty$;\ (2) The gradients $\nabla h_i(x^*), i\in{\cal I}_h^*$ and $\nabla g_i(x^*), i\in{\cal I}_g^*$ are linearly independent;\ (3) The multipliers $u_i^*\in (0,1)$ and $v_i^*\in (0,1)$, $\forall~i\in{\cal I}_h^*$, and $s_i^*\in (0,1)$, $\forall~i\in{\cal I}_g^*$;\ (4) $d^TB^*d>0$ for all $d\ne 0$ such that $\nabla h_i(x^*)^Td=0$ $\forall i\in{\cal I}_h^*$ and $\nabla g_i(x^*)^Td=0$ $\forall i\in{\cal I}_g^*$, where $B^*=\sum_{i=1}^{m_{\cal E}}\mu_i^{*}\nabla^2h_i(x^*)-\sum_{i=1}^{m_{\cal I}}\lambda_i^{*}\nabla^2g_i(x^*)$, $\mu_i^*=\hbox{sign}(h_i(x^*))$ as $h_i(x^*)\ne 0$ and $\mu_i^*=u_i^*-v_i^*$ for $i\in{\cal I}_h^*$, $\lambda_i^*=1$ as $g_i(x^*)<0$, $\lambda_i^*=0$ as $g_i(x^*)>0$ and $\lambda_i^*=s_i^*$ for $i\in{\cal I}_g^*$.* Assumption [Assumption 1](#ass2b){reference-type="ref" reference="ass2b"} (1) and (3) imply that for problems ([\[prob2-1\]](#prob2-1){reference-type="ref" reference="prob2-1"})--([\[prob2-5\]](#prob2-5){reference-type="ref" reference="prob2-5"}), LICQ holds at $(x_k,y_k,z_k)$ for all sufficiently large $k$. Thus, $(u_k,v_k,s_k,t_k)$ and $(\mu_k,\lambda_k)$ are unique for all sufficiently large $k$. Let ${\cal I}_{h+}^*=\{i| h_i(x^*)>0,~i=1,\ldots,m_{\cal E}\}$, ${\cal I}_{h-}^*=\{i| h_i(x^*)<0,~i=1,\ldots,m_{\cal E}\}$, ${\cal I}_{h}^*=\{i| h_i(x^*)=0,~i=1,\ldots,m_{\cal E}\}$, ${\cal I}_{g-}^*=\{i| g_i(x^*)<0,~i=1,\ldots,m_{\cal I}\}$, ${\cal I}_{g}^*=\{i| g_i(x^*)=0,~i=1,\ldots,m_{\cal I}\}$. Consider the system $$\begin{aligned} &\,\,&\,\,\rho_k \nabla f(x)+\sum_{i\in{\cal I}_{h+}^*}\nabla h_i(x)-\sum_{i\in{\cal I}_{h-}^*}\nabla h_i(x)+\sum_{i\in{\cal I}_{h}^*}\mu_i\nabla h_i(x) \nonumber\\ &\,\,&\,\,\quad\quad\quad\quad-\sum_{i\in{\cal I}_{g-}^*}\nabla g_i(x)-\sum_{i\in{\cal I}_{g}^*}\lambda_i\nabla g_i(x)=0, \nonumber\\ &\,\,&\,\,h_i(x)=0,~i\in{\cal I}_h^*, \nonumber\\ &\,\,&\,\,g_i(x)=0,~i\in{\cal I}_g^*. \nonumber\end{aligned}$$ Let $x_k^{\rho_k}$ be one of its solutions for given value of penalty parameter $\rho_k$. Correspondingly, $w_k^{\rho_k}=(x_k^{\rho_k},\mu_k^{\rho_k},\lambda_k^{\rho_k})$, $\hat w_k^{\rho_k}=(x_k^{\rho_k},u_k^{\rho_k},v_k^{\rho_k},s_k^{\rho_k},t_k^{\rho_k})$, and like those for $x^*$, we have the index sets ${\cal I}_{h+}^{\rho_k}$, ${\cal I}_{h-}^{\rho_k}$, ${\cal I}_h^{\rho_k}$, ${\cal I}_{g+}^{\rho_k}$, ${\cal I}_{g-}^{\rho_k}$, ${\cal I}_g^{\rho_k}$. The following results can be proved in a similar way to Lemmas 3.3 and 3.5 of [@ByrCuN10]. [\[230413a\]]{#230413a label="230413a"} Under Assumption [Assumption 1](#ass2b){reference-type="ref" reference="ass2b"}, there exists a $\rho^*>0$ such that the preceding system of equations has a unique solution $w_k^{\rho_k}$ for all $\rho_k\le\rho^*$, and there holds $$\|w_k^{\rho_k}-w^*\|\le {\rho_k}M_1<\epsilon, \label{20140415e1}$$ where $\epsilon>0$ is small enough and $M_1$ is a positive constant independent of $\rho_k$. Moreover, ${\cal I}_{h+}^{\rho_k}={\cal I}_{h+}^*$, ${\cal I}_{h-}^{\rho_k}={\cal I}_{h-}^*$, ${\cal I}_h^{\rho_k}={\cal I}_h^*$, ${\cal I}_{g+}^{\rho_k}={\cal I}_{g+}^*$, ${\cal I}_{g-}^{\rho_k}={\cal I}_{g-}^*$, ${\cal I}_g^{\rho_k}={\cal I}_g^*$, and $$\begin{aligned} u_i^{\rho_k}\in (0,1)~\hbox{and}~v_i^{\rho_k}\in (0,1)~\forall~i\in{\cal I}_h^{\rho_k},\quad s_i^{\rho_k}\in (0,1)~\forall~i\in{\cal I}_g^{\rho_k}. \nonumber\end{aligned}$$ [\[230413b\]]{#230413b label="230413b"} Under Assumption [Assumption 1](#ass2b){reference-type="ref" reference="ass2b"}, there exist sufficiently small scalars $\rho^*>0$ and $\epsilon>0$ such that for all $\rho_k\le\rho^*$ and $\|w_k-w^*\|<\epsilon$, $$\|w_{k+1}-w_k^{\rho_k}\|\le M_2\|w_k-w_k^{\rho_k}\|, \label{20140415ea}$$ where we take $x_{k+1}=x_k+d_k$ in $w_{k+1}$, $M_2>0$ is a constant independent of $\rho_k$. In accordance with the preceding two lemmas, one can have the next main result of this subsection. Under Assumption [Assumption 1](#ass2b){reference-type="ref" reference="ass2b"}, $$\|w_{k+1}-w^*\|=O({\rho_k})+O(\|w_{k}-w^*\|^2).$$ Therefore, if ${\rho_k}=O(\|w_k-w^*\|)^2$, then the convergence is quadratic; otherwise, if instead ${\rho_k}=o(\|w_k-w^*\|)$, the convergence is superlinear. *Proof.* The result is straightforward from Lemmas [\[230413a\]](#230413a){reference-type="ref" reference="230413a"} and [\[230413b\]](#230413b){reference-type="ref" reference="230413b"}. ◻ # Numerical experiments Our targets in this section are to show that our method is usable and to demonstrate that our theoretical results are achievable. In this sense, we will not attempt to compare our method with any recognized software, but use our method to solve some small benchmark test examples in the literature for nonlinear programs, for example, [@ByrCuN10; @DLS17; @HocSch81; @LDHmm; @LiuDHS; @LiuSun01]. We have solved five examples in our numerical experiments, where the first three examples are infeasible and are solved by the methods in Byrd, Curtis and Nocedal [@ByrCuN10] and our paper [@DLS17] for observing the rapid detection of infeasibility. The fourth and the fifth are feasible examples, where the former is challenging since many effective solvers based on SQP and IPM can only find an infeasible point which has been known in this paper to be a DL-stationary point when starting from some infeasible points, and the latter is one for which the minimizer is a singular stationary point and the linear independence constraint qualification (LICQ) does not hold. It is noted that, for problem (TP3), our algorithm identifies that the infeasible stationary point is a DL-stationary point, which brings about the rapid detection of infeasibility without driving the penalty parameter $\rho_k$ to zero. This realizes the commentary made by Byrd, Curtis and Nocedal at section 3.2 of their pioneering work [@ByrCuN10] and demonstrates the validity of our convergence theory on D-stationary points in this paper. In addition, we also show that convergence to the degenerate solution of nonlinear optimization with high accuracy is applicable for problem (TP5). In our implementation, we use the standard starting points for $x_0$ in all test problems. The initial estimates for all Lagrange multipliers (that is, the components of $\mu_0$ and $\lambda_0$) are set to be one. The initial parameters are selected as $\rho_0=1.0$, $\sigma=0.01$, $\tau=0.5$, $\epsilon=10^{-8}$ ($10^{-6}$ in [@ByrCuN10]). The initial approximate of the Hessian $B_0$ of Lagrangian $L(x,\lambda)=\rho f(x)-\mu^Th(x)-\lambda^Tg(x)$ is simply chosen to be the identity matrix, and the step-size is computed by the Armijo line-search procedure. According to Definitions [\[def0\]](#def0){reference-type="ref" reference="def0"} and [\[def1\]](#def1){reference-type="ref" reference="def1"}, in order to ascertain what are the terminating points when implementing our algorithm, we calculate the measures $$\begin{aligned} &\,\,&\,\,E_k^{\hbox{dual}}=\|\rho_{k-1}\nabla f(x_k)-\nabla h(x_k)\mu_k-\nabla g(x_k)\lambda_k\|_{\infty}, \nonumber\\ &\,\,&\,\,E_k^{\hbox{compl}}=\max\{\|\mu_k\circ h(x_k)+|h(x_k)|\|_{\infty}, \|\lambda_k\circ g(x_k)+\max\{0,-g(x_k)\}\|_{\infty}\}, \nonumber\\ &\,\,&\,\,E_k^{\hbox{feas}}=\max\{\|h(x_k)\|_{\infty},\|\max\{0,-g(x_k)\}\|_{\infty}\}, \nonumber\end{aligned}$$ where $\circ$ is the Hadamard product of two vectors. In addition, we will report the values of objective functions and the number of solved QP subproblems for inner iterations, the number of evaluations of functions and gradients in inner iterations. The first test problem is referred as *unique* in [@ByrCuN10]: $$\begin{aligned} ({\rm TP1}) \quad\min~ x_1+x_2 \quad\hbox{s.t.}~ x_2-x_1^2-1\ge 0, ~ 0.3(1-e^{x_2})\ge 0. \nonumber\end{aligned}$$ The standard initial point is $x_0=(3,2)$, an approximate DZ-stationary point close to $x^*=(0,1)$, which is also an approximate strict minimizer of the $\ell_1$-norm constraint violations, was found in [@ByrCuN10]. Our algorithm terminates at $k=6$ with an approximate solution to $x^*$. The output of our algorithm is given in Table [1](#tab1){reference-type="ref" reference="tab1"}, where "iter-sb\" represents the number of inner iterations for solving the $k$-th subproblem with fixed parameter $\rho_k$. Algorithm [\[alg1\]](#alg1){reference-type="ref" reference="alg1"} takes the full step at almost all iterates. It is easy to observe from Table [1](#tab1){reference-type="ref" reference="tab1"} that both $E_{k}^{\hbox{dual}}$ and $E_k^{\hbox{compl}}$ are small enough, which show the rapid convergence to the stationary point. $k$ $f_k$ $E_{k}^{\hbox{dual}}$ $E_{k}^{\hbox{compl}}$ $E_{k}^{\hbox{feas}}$ iter-sb $\rho_k$ $\hbox{numf}_k$ $\hbox{numg}_k$ ----- --------- ----------------------- ------------------------ ----------------------- --------- ------------ ----------------- ----------------- 0 5 7 0 8 \- 1.0 1 1 1 -0.8032 1.0816 5.0886e-09 1.8456 4 0.01 4 4 2 0.9941 2.1018e-04 3.0986e-08 0.5155 5 1.0000e-04 6 5 3 0.9999 6.1454e-08 2.8401e-11 0.5155 2 1.0000e-06 2 2 4 1.0000 1.9815e-05 1.0122e-09 0.5155 1 1.0000e-09 1 1 5 1.0000 3.3072e-09 5.6710e-11 0.5155 1 1.0000e-09 1 1 : Output for test problem (TP1): $13$ inner iterations needed. The second problem is the *isolated* problem of [@ByrCuN10]: $$\begin{aligned} \min&\,\,&\,\,x\sb{1}+x_2 \nonumber\\ ({\rm TP2}) \quad\quad \hbox{s.t.}&\,\,&\,\,-x_{1}^{2}+x\sb{2}-1\ge 0, \nonumber\\ &\,\,&\,\,-x_{1}^{2}-x\sb{2}-1\ge 0, \nonumber\\ &\,\,&\,\,x_{1}-x\sb{2}^2-1\ge 0, \nonumber\\ &\,\,&\,\,-x_{1}-x\sb{2}^2-1\ge 0. \nonumber\end{aligned}$$ Starting from the same initial point $x_0=(3,2)$ as problem (TP1), the algorithm in [@ByrCuN10] found an approximate DZ-stationary point close to $x^*=(0,0)$, a strict minimizer of the infeasibility measure in $\ell_1$ and $\ell_2$ norms. Our algorithm terminates at an approximate point to it. The output of our algorithm for problem (TP2) is reported in Table [2](#tab2){reference-type="ref" reference="tab2"}, which shows the rapid convergence to the stationary point. $k$ $f_k$ $E_{k}^{\hbox{dual}}$ $E_{k}^{\hbox{compl}}$ $E_{k}^{\hbox{feas}}$ iter-sb $\rho_k$ $\hbox{numf}_k$ $\hbox{numg}_k$ ----- ------------ ----------------------- ------------------------ ----------------------- --------- ------------ ----------------- ----------------- 0 5 13 0 12 \- 1.0 1 1 1 -0.4947 0.0141 5.0286e-12 1.3089 8 0.01 11 8 2 -0.0050 2.9454e-08 2.6117e-10 1.0025 4 1.0000e-04 4 4 3 1.1633e-05 1.4189e-04 2.3552e-12 1.0000 1 1.0000e-06 1 1 4 2.5621e-11 1.0276e-06 2.3759e-14 1.0000 1 1.0000e-09 1 1 5 2.5621e-11 2.8630e-08 0 1.0000 1 1.0000e-09 0 0 : Output for test problem (TP2): $15$ inner iterations needed. The third test problem is the *nactive* problem in [@ByrCuN10]: $$\begin{aligned} \hbox{min}&\,\,&\,\,x\sb{1} \nonumber\\ ({\rm TP3}) \quad\quad \hbox{s.t.} &\,\,&\,\,\frac{1}{2}(-x\sb{1}-x\sb{2}^2-1)\ge 0, \nonumber\\ &\,\,&\,\,x\sb{1}-x_2^2\ge 0, \nonumber\\ &\,\,&\,\,-x\sb{1}+x_2^2\ge 0. \nonumber\end{aligned}$$ This problem is still infeasible. The given initial point is $x_0=(-20,10)$. The point $x^*=(0,0)$ was an infeasible stationary point with $\|\max\{0,-g(x^*)\}\|=0.5$ and is also a DL-stationary point. Algorithm [\[alg1\]](#alg1){reference-type="ref" reference="alg1"} terminates at an approximate point $x_k$ with $k=2$ and $\rho_k=0.01$, and takes full steps at all inner iterates. The output is shown in Table [3](#tab3){reference-type="ref" reference="tab3"} where we need to solve $12$ QP subproblems. $k$ $f_k$ $E_{k}^{\hbox{dual}}$ $E_{k}^{\hbox{compl}}$ $E_{k}^{\hbox{feas}}$ iter-sb $\rho_k$ $\hbox{numf}_k$ $\hbox{numg}_k$ ----- ------------- ----------------------- ------------------------ ----------------------- --------- ---------- ----------------- ----------------- 0 -20 10 120 120 \- 1.0 1 1 1 -21.7061 7.9913e-11 1.8034e-11 21.7061 5 0.01 5 5 2 -4.2698e-13 1.4457e-10 1.0461e-13 0.5000 7 0.01 7 7 : Output for test problem (TP3): $12$ inner iterations needed. We also use Algorithm [\[alg1\]](#alg1){reference-type="ref" reference="alg1"} to solve the challenging example given in ([\[s-example\]](#s-example){reference-type="ref" reference="s-example"}): $$\begin{aligned} ({\rm TP4})\quad\hbox{min}~x \quad\hbox{s.t.}~x\sp{2}-1\ge 0,~x-2\ge 0. \nonumber\end{aligned}$$ This problem has a unique global minimizer $x^*=2$, at which both LICQ and MFCQ hold, and the second-order sufficient optimality conditions are satisfied. However, like most of the methods for nonlinear optimization, including various SQP and interior-point methods, our method terminates at $x^*=-1.0000$, a DL-stationary point when starting from an infeasible point $x_0=-4$. Our numerical tests show that the point $x^*$ cannot be improved by changing the initial penalty parameter. Some new techniques should be incorporated for this difficulty. $k$ $f_k$ $E_{k}^{\hbox{dual}}$ $E_{k}^{\hbox{compl}}$ $E_{k}^{\hbox{feas}}$ iter-sb $\rho_k$ $\hbox{numf}_k$ $\hbox{numg}_k$ ----- --------- ----------------------- ------------------------ ----------------------- --------- ---------- ----------------- ----------------- 0 -4 8 15 6 \- 1.0 1 1 1 -4 4.1933e-10 7.8351e-10 6 1 0.1000 0 0 2 -1.0000 8.8707e-14 5.2231e-10 3.0000 6 0.1000 6 6 : Output for test problem (TP4): $7$ inner iterations needed. In the final experiment, we solve a standard test problem taken from [@HocSch81 Problem 13]: $$\begin{aligned} ({\rm TP5}) \quad\hbox{min}~ (x\sb{1}-2)\sp{2}+x\sb{2}\sp{2} \quad\hbox{s.t.}~ (1-x\sb{1})\sp{3}-x\sb{2}\geq 0, ~ x\sb{1}\geq 0, ~ x\sb{2}\geq 0. \nonumber\end{aligned}$$ This problem is obviously feasible, and has the optimal solution $x^*=(1,0)$ which is not a KKT point but a singular stationary point, at which the gradients of active constraints are linearly dependent. It is a challenging problem since the convergence of many algorithms depends on the LICQ. There is not much detail on the solution of this problem in the literature. The standard initial point $x_0=(-2,-2)$ is an infeasible point. For this problem, we select $\rho_0=10^3$. Algorithm [\[alg1\]](#alg1){reference-type="ref" reference="alg1"} terminates at $x_{k}\approx(1.0000004967,-0.0000000000)$ with $k=26$, an approximate solution close to the minimizer $x^*$ very sufficiently. The parameter $\rho_k\to 0$ very quickly. The output of the algorithm is given in Table [5](#tab5){reference-type="ref" reference="tab5"}. $k$ $f_k$ $E_{k}^{\hbox{dual}}$ $E_{k}^{\hbox{compl}}$ $E_{k}^{\hbox{feas}}$ iter-sb $\rho_k$ $\hbox{numf}_k$ $\hbox{numg}_k$ ----- --------- ----------------------- ------------------------ ----------------------- --------- ------------ ----------------- ----------------- 0 20 18 29 2 \- 1.0e+3 1 1 1 20 7973 29 2 1 100 0 0 2 10.0000 588.0000 9.0000 1.0000 1 10 1 1 3 9.7936 47.6284 8.4966 1.0322 1 1 1 1 4 3.3733 2.1970 1.1540 0.9907 3 0.1000 3 3 5 0.6151 0.0172 0.0197 0.0098 11 1.0000e-03 14 11 6 0.9497 1.0767e-07 1.6575e-05 8.2905e-06 8 1.0000e-05 8 8 7 0.9930 2.0267e-06 3.7378e-08 2.3394e-08 6 3.1623e-08 6 6 8 0.9970 3.2532e-06 4.6188e-09 3.0816e-09 2 5.6234e-12 2 2 9 0.9982 1.1838e-06 1.5704e-09 1.0471e-09 1 1.3335e-17 1 1 10 0.9989 4.6321e-07 3.2247e-10 2.1499e-10 1 4.8697e-26 1 1 11 0.9993 1.7528e-07 7.9826e-11 5.3217e-11 1 1.0000e-30 1 1 12 0.9996 6.7188e-08 1.8493e-11 1.2329e-11 1 1.0000e-30 1 1 13 0.9997 2.5629e-08 4.3963e-12 2.9308e-12 1 1.0000e-30 1 1 14 0.9998 9.7941e-09 1.0350e-12 6.9001e-13 1 1.0000e-30 1 1 15 0.9999 3.7404e-09 2.4458e-13 1.6305e-13 1 1.0000e-30 1 1 16 0.9999 1.4289e-09 1.2431e-13 3.8479e-14 1 1.0000e-30 1 1 17 1.0000 5.4590e-10 8.7394e-14 9.0863e-15 1 1.0000e-30 1 1 18 1.0000 2.0863e-10 6.1632e-14 2.1455e-15 1 1.0000e-30 1 1 19 1.0000 7.9797e-11 4.5121e-14 5.0686e-16 1 1.0000e-30 1 1 20 1.0000 3.0582e-11 3.5889e-14 1.1985e-16 1 1.0000e-30 1 1 21 1.0000 1.1778e-11 3.2477e-14 2.8405e-17 1 1.0000e-30 1 1 22 1.0000 4.5916e-12 3.4053e-14 6.7710e-18 1 1.0000e-30 1 1 23 1.0000 1.8443e-12 4.0137e-14 1.6396e-18 1 1.0000e-30 1 1 24 1.0000 7.9742e-13 4.9823e-14 4.1665e-19 1 1.0000e-30 1 1 25 1.0000 4.1254e-13 6.0513e-14 1.2596e-19 1 1.0000e-30 1 1 26 1.0000 3.0213e-13 6.7940e-14 6.4726e-20 1 1.0000e-30 1 1 : Output for test problem (TP5): $51$ inner iterations needed. In summary, the preceding numerical results not only demonstrate global convergence to the stationary points with least constraint violations on Algorithm [\[alg1\]](#alg1){reference-type="ref" reference="alg1"} for infeasible and degenerate nonlinear programs, but also illustrate that our algorithm is capable of rapidly detecting infeasibility of nonlinear programs without driving the penalty parameter $\rho_k$ to zero for some infeasible optimization. # Conclusion We introduce the stationary points of nonlinear optimization with least constraint violations including the D-stationary point, the DL-stationary point and the DZ-stationary point for dealing with possible infeasible optimization problems. These stationary points can be thought of as a generalization of the classic Fritz-John point, KKT point and singular stationary point for feasible optimization to infeasible optimization, where the DL-stationary point has a dependence on the objective like the KKT point, and the DZ-stationary point corresponds to the singular stationary point. To examine the usefulness of our new stationary points, based on robustness of the exact penalty optimization, we present an exact penalty SQP method with inner and outer iterations for nonlinear optimization, and analyze its global and local convergence. It is shown that when the solution of an infeasible optimization is a DL-stationary point, the rapid infeasibility detection can happen without driving the penalty parameter to zero. This demonstrates the commentary of Byrd, Curtis and Nocedal [@ByrCuN10] and is a supplement of the theory of nonlinear optimization on rapid detection of infeasibility. 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Schittkowski, *Test Examples for Nonlinear Programming Codes, Lecture Notes in Eco. and Math. Systems 187, Springer-Verlag, Berlin, New York, 1981. X.-W. Liu, Y.-H. Dai and Y.-K. Huang, *A primal-dual interior-point relaxation method with global and rapidly local convergence for nonlinear programs, Math. Meth. Oper. Res., 96 (2022), 351--382. X.-W. Liu, Y.-H. Dai, Y.-K. Huang and J. Sun, *A novel augmented Lagrangian method of multipliers for optimization with general inequality constraints, Math. Comput., 92 (2023), 1301--1330. X.-W. Liu and J. Sun, *A robust primal-dual interior point algorithm for nonlinear programs, SIAM J. Optim., 14 (2004), 1163--1186. X.-W. Liu and Y.-X. Yuan, *A robust algorithm for optimization with general equality and inequality constraints, SIAM J. Sci. Comput., 22 (2000), 517--534. X.-W. Liu and Y.-X. Yuan, *A null-space primal-dual interior-point algorithm for nonlinear optimization with nice convergence properties, Math. Program., 125 (2010), 163--193. J. Nocedal and S. Wright, Numerical Optimization, Second Edition, Springer Science+Business Media, LLC, 2006. J. Nocedal, F. Öztoprak and R.A. Waltz, *An interior point method for nonlinear programming with infeasibility detection capabilities, Optim. Methods Softw., 29 (2014), 837--854. T. Pietrzykowski, *An exact potential method for constrained maxima, SIAM J. Numer. Anal., 6 (1969), 299--304. W.Y. Sun and Y.-X. Yuan, Optimization Theory and Methods: Nonlinear Programming, Springer, New York, 2006. A. Wächter and L.T. Biegler, *Line search filter methods for nonlinear programming: Motivation and global convergence, SIAM J. Optim., 16 (2005), 1--31. Y. Yuan, *On the convergence of a new trust region algorithm, Numer. Math., 70 (1995), 515--539.************************* [^1]: Institute of Mathematics, Hebei University of Technology, Tianjin 300401, China (). The research is supported by the NSFC grants (nos. 12071108 and 11671116). [^2]: Corresponding author. LSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China; School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China ([http://lsec.cc.ac.cn/\\string\~dyh/](http://lsec.cc.ac.cn/\string~dyh/)). This author is supported by the NSFC grants (nos. 12021001, 11991021, 11991020 and 11971372) and the Strategic Priority Research Program of Chinese Academy of Sciences (no. XDA27000000).

arxiv_math

--- abstract: | In this paper, we provide a volume growth estimate for complete manifold with gradient Ricci almost soliton structures. With this, we can get some rigidity results by selecting special test functions $p(x)$ in volume growth inequality and then get the Volume comparision theorem. By the volume comparision theorem, we can get some useful tools in the classification of Ricci almost solitons. Finally, we obtain a precise estimate of the growth of potential functions of complete noncompact shrinking almost soliton, and then get the volume growth result and classification of it. address: School of Mathematical Sciences , East China Normal University, Shanghai 200241, China author: - Li Wen-Qi title: New Volume Comparision Results and Classification of Gradient Ricci Almost Solitons --- # Introduction Let $(M^n,g)$ be a complete Riemannian manifold, if there is a smooth vector feild $X$ exists on $M^n$ that the Ricci curvature tensor of $M^n$ is $$Ric+\frac{1}{2}L_{X}g=\rho g,$$ for some constants $\rho$, then we called $M^n$ a Ricci soliton. The symbol $L_X$ stands for the Lie derivative in the direction of $X$. In the special case where $X=\nabla f$ for some $f\in C^{\infty}$, the soliton equation becomes $$Ric+Hessf=\rho g.$$ At this time we call $M^n$ a gradient Ricci soliton. The Ricci solitons were first introduced by R.Hamiltons in [@Ha3] as the self-similar solutions of Ricci flow [@Ha2], which play an improtant rule in the proof Poincáre conjecture by Perelmann [@Pel1; @Pel2]. In recent years, there have been many results on the classification of gradient solitons [@CCZ; @N; @NW; @Pw1; @Pw2; @Z]. In [@PRR], Pigola.S, Rigoli.M and Rimoldi.M extended the definition of gradient soliton by considering the constant $\rho$ as a smooth function then called it gradient Ricci almost solitons. So the soliton equation became $$Ric+Hessf=\rho(x) g.$$ We call $(M^n,g,f,\rho)$ is a gradient Ricci almost solitons with potential function $f$ and soliton function $\rho$. Gradient almost solitons can be seen as a mixed type of Ricci solitons and Yamabe solitons(in some cases we call it Ricci-Yamabe solitons) and can be considered as a special genelized of Einstein eqution if $\rho(x)$ is the scaler curvature. The soliton of Ricci--Bourguigon flow [@CCD] is an important example of Ricci almost solitons which is different from Ricci soliton or Yamabe soliton. For compact Ricci almost solitons, A.Barros and E.Ribeiro Jr have already got some classification results by using maximal principle to the scaler curvature in [@AE; @ABR]. In our paper, we will get some classification results of complete noncompact Ricci almost solitons by proving a volume comparision theorem for manifold with $$Ric+\nabla^2f\ge \rho g .$$ The first result we are ready to mention is that the volume growth of the geodesic ball of a complete gradinate Ricci almost soliton can be controlled by leading in a test function $p(x)\in C^2(M)$. **Theorem 1**. *Let $(M^n,g,f,\rho)$ be a complete manifold whose Ricci curvature satisfying $f\in C^2(M^n),\rho\in C(M^n)$, $Ric+Hessf\ge\rho(x)g$. Assume $o\in M$ a fixed point. Let $p(x)\in C^2(M)$ satisfies $p(x)>0$ and $\dot{p}(r)\to 2r+o(r)$ when $r\to 0$. Then for any $r>0$, there exists postive constants $A$ and $d_0=min\{1,inj(M^n)\}$ that the volume of geodesic ball $B_{o}(r)$ satisfying $$Vol(B_o(r))\le A\int_{\mathbb{S}^n}\int_{0}^{r}\exp \{\{\int_{d_0}^{k}\frac{1}{p(s)}(\int_{0}^{s}p(t)(\ddot{f}(t)-\rho(t))+\frac{(n-1)\dot{p}^2(t)}{4p(t)}dt)ds\}\}dkd\Theta$$* **Remark 2**. *The function $\dot{p}(x)=<\partial_r,\nabla p>(x)$, $\partial_r$ is the gradinate of a normal geodesic.* We lead in the test function in order to find the optimal volume growth result. It is interesting that when the equality satisfying, the test function $p(x)$ is a solution of a Ricati equation $$\frac{4(\rho(x)-\ddot{f}(x)}{n-1}=\frac{2(\ddot{p}(x)p(x)-\dot{p}^{2}(x))}{p^2(x)}+\frac{\dot{p}^2(x)}{p^2(x)}.$$ And the metric of geodesic ball is the same as geodesic ball in a simply connect manifold with metric $g=dr^2+p(r,\theta)ds^{2}_{n-1}$. Then we get the volume comparision theorem **Theorem 3**. *$(M^n,g,f,\rho)$ is a complete manifold whose Ricci curvature with injective radius $d_0$ satisfying $f\in C^2(M^n)$, $\rho\in C(M^n)$, $Ric+Hessf\ge\rho(x)g$. Assume $o\in M$ a fixed point. If the equation $$\frac{4(\rho(x)-\ddot{f}(x)}{n-1}=\frac{2(\ddot{p}(x)p(x)-\dot{p}^{2}(x))}{p^2(x)}+\frac{\dot{p}^2(x)}{p^2(x)}$$ has a solution $p(x)$ satisfying $\dot{p}(r)\to 2r+o(r)$ when $r\to 0$. Then for $r\in[0,d_0]$, we have $$Vol(B_o(r))\le\int_{0}^{r}\int_{\mathbb{S}^n}p^{(n-1)/2}(k,\theta)d\theta dk.$$ And if the equality is satisfied, $B_o(r)$ is isometric to the geodesic ball in simply connect manifold with metric $g=dr^2+p(r,\theta)ds^{2}_{n-1}$.* Considering $\rho=C$, we can get a similar result as Wei and Wylie in [@WW]. And with this volume comparision theorem, we can get some useful rigidity results of complete noncompact almost solitons. And for compact almost solitons, we prove a Bonnet-Myers type theorem with maximal diameter rigidity. Finally, we extend the result of [@CZ] to shrinking Ricci almost solitons with special soliton functions. According to the results of [@CCZ], if the soliton function is bounded and hypeharmonic, then the scaler curvature of it is positive, and we can get an estimation of potential functions and finally the volume growth result as [@CZ] **Theorem 4**. *$(M^n,g,f,\rho)$ is a complete noncompact almost shrinking solitons, the soliton function $\rho$ satisfying $K_1\ge\rho\ge K_2>0$, $\Delta\rho\le0$, $\rho\to C$ when $r\to\infty$, $\rho\le C+\frac{C'}{r^2}$ holds for all $r$ and positive constants $C$, $C'$. The potential function $f$ satisfies $\int_{0}^{\infty}\dot{\rho}f\ge K>-\infty$. Then the volume growth of shrinking almost solitons satisfies $Vol(B_o(r))\le C''r^n(C'>0)$. When the equality holds, $(M^n,g,f,\rho)$ is a scale of Gaussian.* In the process of proving this theorem, we also get a volume growth theorem for almost soliton with bounded $\rho$ and scaler curvature. # Volume Growth of Gradient Ricci almost Solitons In this section, we will prove the main theorem of volume growth of the geodesic ball and provide an equivalent condition for the equality holding in theorem[Theorem 1](#thm1.1){reference-type="ref" reference="thm1.1"}. ## Proof of Theorem1.1   Let $o$ be a point chosen from $M^n$, $(r,\theta)$ is a polar coordinate start from $o$ and $r$ is the distance function. Intinally, considering $x=(r,\theta)$ not in cut locus $\mathcal{C}(o)$, $\omega(r,\theta)=\partial_{r}\log{J}=\frac{\partial_{r}J}{J}$ is the mean curvature of the boundary of the geodesic ball $B_o(r)$. Under the polar coordinate, let $p(x)\in C^2(M^n)$, if $\gamma(t)$ is a minimal geodesic with normal speed,we use $f(t)$ to reperesent $f(\gamma(t))$. As the same, we use $\int_{s_0}^{s_1} f(t)dt$(or sometimes we use $\int_{s_0}^{s_1} f(x)dt$) to represent the path integral of $f$ on $\gamma(t)$and we may leave out the variable $t$ due to typesetting. *Proof.* By the variation formula of the volume of geodesic ball in [@Li], we have $$\dot{\omega}+\frac{\omega^2}{n-1}+Ric(\partial_r,\partial_r)\le 0\label{ineq3.1}.$$ Assume $p(x)\in C^2(B_o(r))$ satisfies $p(x)>0$, $p(o)=0$ and $\dot{p}(r)\to 2r+o(r)$ when $r\to 0$. Multiplying equation([\[ineq3.1\]](#ineq3.1){reference-type="ref" reference="ineq3.1"}) by $p(x)$ and integrating it from $\varepsilon$ to r yields $$\int_{\varepsilon}^{r}p(s)\dot{\omega}ds+\int_{\varepsilon}^{r}\frac{p(s)\omega^2}{n-1}ds\le -\int_{\varepsilon}^{r}p(s)Ric(\partial_r,\partial_r)ds,$$ let $\varepsilon\to 0$, we have $$\label{ineq3.2} p(r)\omega(r)\le \int_{0}^{r}\frac{(n-1)\dot{p^2}(s)}{4p(s)}-p(s)Ric(\partial_r,\partial_r)ds,$$ which leads $$p(r)\partial_r\log \frac{J(r)}{exp(F(r))}\le \int_{0}^{r}-p(s)Ric(\partial_r,\partial_r)ds.$$ $$\partial_rF(r)=\frac{n-1}{4p(r)}(\int_{0}^{r}\frac{\dot{p}^2}{p}dt)$$ We know that when $r\to 0$, $\frac{n-1}{4p(r)}(\int_{0}^{r}\frac{\dot{p}^2}{p}dt)\to \frac{n-1}{r}$, so $F(r)\to -\infty$. Assume there exist a constant satisfies $C'\le inj(M^n)$ and $F(C')=0$ holds for any $\theta$. Then we obtain $$\partial_r\log\frac{ J}{\exp(\int_{C'}^{r}\frac{1}{p}((\frac{n-1}{4})\int_{0}^{s}\frac{\dot{p}^2}{p}dt)ds)}\le \frac{1}{p}\int_{0}^{r}-pRic(\partial_r,\partial_r)ds,$$ integrating it from $0$ to $r$ yields $$\log \frac{ J}{J(C',\theta)\exp(\int_{C'}^{r}\frac{1}{p}((\frac{n-1}{4})\int_{0}^{s}\frac{\dot{p}^2}{p}dt)ds)}\le \int_{C'}^{r}\frac{1}{p}\int_{0}^{s}-pRic(\partial_r,\partial_r)dtds,$$ so we get an uper bound of $J$ by function $p(x)$, $f(x)$, $\rho(x)$ $$\begin{footnotesize} J(r,\theta)\le J(C',\theta)\exp \{\{\int_{C'}^{r}\frac{1}{p(s)}(\int_{0}^{s}p(t)(\ddot{f}(t)-\rho(t))+\frac{(n-1)\dot{p}^2(t)}{4p(t)}dt)ds\}\} \end{footnotesize}$$ It is easy to check that when $r\to 0$, $$\exp \{\{\int_{C'}^{r}\frac{1}{p(s)}(\int_{0}^{s}p(t)(\ddot{f}(t)-\rho(t))+\frac{(n-1)\dot{p}^2(t)}{4p(t)}dt)ds\}\}\to 0,$$ which means $J\to 0$ when $r\to 0$. So $Vol(B_o(r))\to 0(r\to 0)$. Then we have $$\label{eq3.7} \begin{aligned} Vol(B_o(r))&\le \int_{\mathbb{S}^n}\int_{0}^{r}J(C',\theta)\exp \{\{\int_{C'}^{k}\frac{1}{p(s)}(\int_{0}^{s}p(t)(\ddot{f}(t)-\rho(t))+\frac{(n-1)\dot{p}^2(t)}{4p(t)}dt)ds\}\}dkd\Theta \\&=A\int_{\mathbb{S}^n}\int_{0}^{r}\exp \{\{\int_{d_0}^{k}\frac{1}{p(s)}(\int_{0}^{s}p(t)(\ddot{f}(t)-\rho(t))+\frac{(n-1)\dot{p}^2(t)}{4p(t)}dt)ds\}\}dkd\Theta. \end{aligned}$$ $A$ is a postive constant. ◻ By Theorem[Theorem 1](#thm1.1){reference-type="ref" reference="thm1.1"}, we can get many different volume growth results by choosing different $p(x)$, so it is interesting to investigate for what $p(x)$ and $C'$, the optimal upper bound of [\[eq3.7\]](#eq3.7){reference-type="ref" reference="eq3.7"} can be obtained and what will happen if the equality is satisfied. In [@CRZ] Cheng X, E.Ribeiro Jr and Detang Zhou proved that if $\rho=1/2$(in this case $M$ is called gradient shrinking solitons), let $p(x)=r^2$, we can get a sharp upper bound, and if the upper bound is satisfied, the soliton is isometric to the Euclidean space(Gaussian). So if we find such $p(x)$, we may get the rigidity result. ## The equality in theorem[Theorem 1](#thm1.1){reference-type="ref" reference="thm1.1"} {#the-equality-in-theoremthm1.1} **Corollary 5**. *The equality in Thm[Theorem 1](#thm1.1){reference-type="ref" reference="thm1.1"} holds for distance $r$ and a positive constant $C'$ if and only if $(M^n,g,f,\rho)$ satisfies the system $$\begin{aligned} \omega(r,\theta)&=\frac{(n-1)\dot{p}(x)}{2p(x)}\\ \frac{4(\rho(x)-\ddot{f}(x))}{n-1} &=\frac{2(\ddot{p}(x)p(x)-\dot{p}^{2}(x))}{p^2(x)}+\frac{\dot{p}^2(x)}{p^2(x)} \end{aligned}$$ that have a solution on $M^n/o$, and the solution $p(x)$ satisfies $\dot{p}(r)\to 2r+o(r)$, $\varepsilon>0$ when $r\to 0$.* *Proof.* If the equality of [\[ineq3.2\]](#ineq3.2){reference-type="ref" reference="ineq3.2"} holds, we can get $$\int_{0}^{r}p(s)(\omega-\frac{(n-1)\dot{p}(s)}{2p(s)})^2ds=0,$$ , which leads the first part of this corollary.\ By the equality of volume form, we can get $$\int_{0}^{r}p(s)(\dot{\omega}+\frac{\omega^2}{n-1}+Ric(\partial_r,\partial_r))ds=0,$$ and we know that $p(x)\ge 0$, so the equality in [\[ineq3.1\]](#ineq3.1){reference-type="ref" reference="ineq3.1"} holds. Through the first part of the corollary, we can easily get the second part. And by the proof of Theorem[Theorem 1](#thm1.1){reference-type="ref" reference="thm1.1"}, we know how $p(x)$ is defined, then we get the last part of this corollary. ◻ It is easy to check that if the equality in Thm[Theorem 1](#thm1.1){reference-type="ref" reference="thm1.1"} holds, $$J(C_0,\theta)\exp \{\{\int_{C_0}^{C_1}\frac{1}{p(s)}(\int_{0}^{s}p(t)(\ddot{f}(t)-\rho(t))+\frac{(n-1)\dot{p}^2(t)}{4p(t)}dt)ds\}\}=J(C_1,\theta)$$ holds for any $C_0,C_1>0$. So the choice of $C'$ won't affect the final result. And we know that $M^n$ is a complete manifold, $r$ is the distance function, so $J(r,\theta)\to r^{(n-1)}$ when $r\to 0$. Then $J(r,\theta)=p^{(n-1)}(x)$ refers to any $x$ in segment domain. At this time the volume of geodesic ball satisfies $$Vol(B_o(r))=\int_{0}^{r}\int_{\mathbb{S}^n}p^{(n-1)/2}(k,\theta)d\theta dk$$ # Rigidity result In this section, we will continue to study the condition when the equality satisfying in theorem[Theorem 1](#thm1.1){reference-type="ref" reference="thm1.1"}. we will prove a similar volume comparision theorem as manifold with Ricci curvature bound and then get Bonnet-Myers type theorem with maximal diameter rigidity. We can also get some rigidity results about noncompact almost solitons which we can use in last section. ## Volume comparison theorem on almost solitons with rigidity *Proof.* If the equality in[Theorem 1](#thm1.1){reference-type="ref" reference="thm1.1"} is satisfied, by the second variation formula, we have $$\dot{\omega}+\frac{\omega^2}{n-1}+Ric(\partial_r,\partial_r)=0.$$ With the result of bochner formula and representation of [\[ineq3.1\]](#ineq3.1){reference-type="ref" reference="ineq3.1"} in [@Pe], we have $$|\mathop{\mathrm{Hess}}r|^2=\frac{(\Delta r)^2}{n-1},$$ which means $$\mathop{\mathrm{Hess}}r=\frac{\Delta r}{n-1} g_{\theta}.$$ And we have $$\Delta r=\omega(x).$$ So the metric satisfies $$L_{\partial_r}g=\frac{(n-1)\dot{p}(x)}{2p(x)}g_{\theta}.$$ And assume $\bar{\Delta}$ as the Laplacian on manifold with metric $g=dr^2+p(r,\theta)ds^2_{n-1}$, $\bar{r}$ is the distance function, it is easy to check that $$\bar{\Delta}\bar{r}=\frac{(n-1)\dot{p}(x)}{2p(x)}=\Delta r$$ So by Laplacian comparision theorem, when $r$ smaller than the injective radius of $M^n$, $B_o(T)$ is isometric to the geodesic ball in simply connect manifold with metric $g=dr^2+p(r,\theta)ds^{2}_{n-1}$. ◻ Then we get a Volume comparison theorem **Theorem 6**. *$(M^n,g,f,\rho)$ is a complete manifold with injective radius $d_0$ whose Ricci curvature satisfying $Ric+Hessf\ge\rho(x)g$, $f\in C^2(M^n)$, $\rho\in C(M^n)$. Assume $o\in M$ a fixed point. If the equation $$\frac{4(\rho(x)-\ddot{f}(x))}{n-1}=\frac{2(\ddot{p}(x)p(x)-\dot{p}^{2}(x))}{p^2(x)}+\frac{\dot{p}^2(x)}{p^2(x)}$$ has a solution $p(x)$ satisfying $p(o)=0$, $\dot{p}(r)\to 2r+o(r)$, when $r\to 0$. Then for $r\in[0,d_0]$, we have $$Vol(B_o(r))\le\int_{0}^{r}\int_{\mathbb{S}^n}p^{(n-1)/2}(k,\theta)d\theta dk.$$ And if the equality is satisfied, $B_o(r)$ is isometric to the geodesic ball in simply connect manifold with metric $g=dr^2+p(r,\theta)ds^{2}_{n-1}$.* When $M^n$ is an almost soliton, $f,\rho\in C^{\infty}(M^n)$. So by the existence theorem of Ricati equation, we can always find a $p(x)$ satisfies equation in Corollary[Corollary 5](#Cor2.1){reference-type="ref" reference="Cor2.1"}. And when $r\to 0$, there exsits a constant $C_0$ that $C_0+\epsilon\le\ddot{f}(t)-\rho(t)\le C_0+\epsilon$ which means $r^2+O(r^{2+\epsilon})\le p(r)\le r^2+O'(r^{2+\epsilon})$. Then we know that $p(x)\to r^2+o(r^{2+\varepsilon}), \varepsilon>0$ when $r\to 0$. So if we want to get some global rigidity results, the only thing we need to do is finding some ways to remove the restriction of injective radius. Same as Cartan-Hadamard manifold, if $r\in[0,T], \dot{p}\ge 0$, $$L_{\partial_r}g(\partial_\theta,\partial_\theta)>0,$$ which means $o$ has no conjugate point on $B_o(T)$. Then we can get **Theorem 7**. *$(M^n,g,f,\rho)$ is a complete almost soliton satisfying the equality in Thm[Theorem 6](#thm3.1){reference-type="ref" reference="thm3.1"} holds for all $r\ge0$ and $\dot{p}\ge 0$. Then the universal coverring of $M^n$ is diffmorphism to $R^n$. Hence the homotopy group $\pi_i(M^n)$ vanish for $i>1$.* So if $M^n$ is a complete noncompact with $\dot{p}\ge 0$ holds for all $r\ge0$, consider it's universial Riemiann covering $\bar{M}^n$. It is easy to know that $\bar{M}^n$ is a simply connect complete noncompact manifold without conjugate point. Which means the injective radius of $\bar{M}^n$ is infty. So if the equality in Thm[Theorem 6](#thm3.1){reference-type="ref" reference="thm3.1"} holds, $\bar{M}^n$ is a simply connect manifold with metric $g=dr^2+p(r,\theta)ds^{2}_{n-1}$. If $\ddot{f}(x)-\rho(x)\le0$ applies to all $r\ge0$, by the result of Ricati comparision, we can obviously know that for all $x$, $\dot{p}(x)\ge 0$ holds. It means that if the equality in [Theorem 6](#thm3.1){reference-type="ref" reference="thm3.1"} satisfies for all $r\ge0$, then $L_{\partial_r}g(\partial_\theta,\partial_\theta)>0$. So we have **Theorem 8**. *$(M^n,g,f,\rho)$ is a complete noncompact almost soliton satisfying $\rho(x)-\ddot{f}(x)\le0$ for all $r\ge0$ with the equality that holds in Thm[Theorem 6](#thm3.1){reference-type="ref" reference="thm3.1"} . Then the universial covering of $(M^n,g,f,\rho)$ is a simply connect manifold with metric $g=dr^2+p(r,\theta)ds^{2}_{n-1}$.* If $M^n$ is an almost soliton with special volume growth, We can also get the rigidity result like **Corollary 9**. *If $(M^n,g,f,\rho)$ is a complete noncompact almost soliton which has the same volume growth as the simply connect manifold with constant curvature $K\le0$. Then the almost soliton is isometric to it.* *Proof.* It is easy to check that if we choose $p(x)=sn_{K}(r)$, then the equality in Thm[Theorem 6](#thm3.1){reference-type="ref" reference="thm3.1"} always holds. And at this time $\rho(x)-\ddot{f}(x)=(n-1)K\le0$. So by Thm[Corollary 9](#thm3.2){reference-type="ref" reference="thm3.2"}, we know that $M^n$ is isometric to the simply connect manifold with constant curvature $K\le0$. ◻ ## Max Diameter Rigidity   As the manifold with Ricci curvature bound, we can also prove the Bonnet -Myers type theorem with max diameter rigidity if the potential function and the soliton function have some special relationship. **Theorem 10**. *If $(M^n,g,f,\rho)$ is a complete almost soliton satisfying $\rho(x)-\ddot{f}(x)\ge (n-1)K>0$ holds for all $r\ge0$, then $diam(M^n,g)\le \frac{\pi}{\sqrt{K}}$. Furthermore, $(M^n,g)$ has finite fundemental group.* *Proof.* Let $c:[0,l]\to M^n$ be a unit speed geodesic start from $o$ . Along $c$, considering the variational field $$V_i(t)=sin(\frac{\pi}{l}t)E_i(t), E_i=\frac{\partial\theta_i}{p(c(t))}.$$ By the second variation formula, we know that $$\begin{aligned} \sum_{i=2}^{n}\frac{d^2E}{ds^2}&=\sum_{i=2}^{n}\int_{0}^{l}|\dot{V}_{i}^{2}(t)|-R(V_i,\dot{c},\dot{c},V_i)dt\\&=(n-1)(\frac{\pi}{l})^2\int_{0}^{l}cos^2(\frac{\pi}{l}t)dt-\int_{0}^{l}sin^2(\frac{\pi}{l}t)Ric(\partial_r,\partial_r)dt\\&\le(n-1)(\frac{\pi}{l})^2\int_{0}^{l}cos^2(\frac{\pi}{l}t)dt-(n-1)K\int_{0}^{l}cos^2(\frac{\pi}{l}t)dt \end{aligned}$$ So when $l>\frac{\pi}{\sqrt{K}}$, $c$ can't be a minimal geodesic which means $diam(M^n,g)\le \frac{\pi}{\sqrt{K}}$. And we know that the universal Riemiann covering of $M^n$ is still an almost soliton with $\rho(x)-\ddot{f}(x)\ge (n-1)K>0$ which means it is compact. Then the fundemental group of $M^n$ is finity. ◻ And if we consider bounded $f$ and $\rho=(n-1)H>0$ like [@WW], we can get the similar result. **Corollary 11**. *$(M^n,g,f,\rho)$ is a complete almost soliton satisfying $|f|\le K$ and $\rho=(n-1)H>0$, then $diam(M^n,g)\le \pi\sqrt{\frac{4k+n-1}{(n-1)H}}$.* *Proof.* Same as last theorem, let $c:[0,l]\to M^n$ be a unit speed geodesic start from $o$. Considering the variational field $$V_i(t)=sin(\frac{\pi}{l}t)E_i(t), E_i=\frac{\partial\theta_i}{p(c(t))}.$$ We have $$\begin{aligned} \sum_{i=2}^{n}\frac{d^2E}{ds^2}&=\sum_{i=2}^{n}\int_{0}^{l}|\dot{V}_{i}^{2}(t)|-R(V_i,\dot{c},\dot{c},V_i)dt\\ &=(n-1)(\frac{\pi}{l})^2\int_{0}^{l}cos^2(\frac{\pi}{l}t)dt-\int_{0}^{l}sin^2(\frac{\pi}{l}t)Ric(\partial_r,\partial_r)dt\\ &\le(n-1)(\frac{\pi}{l})^2\int_{0}^{l}cos^2(\frac{\pi}{l}t)dt-((n-1)K-\frac{4k\pi^2}{l^2})\int_{0}^{l}cos^2(\frac{\pi}{l}t)dt \end{aligned}$$ Then we know that $diam(M^n,g)\le \pi\sqrt{\frac{4k+n-1}{(n-1)H}}$. ◻ With the Volume comparision theorem we have already got, we can get the maximal diameter rigidity result. **Theorem 12**. *If $(M^n,g,f,\rho)$ is a complete almost soliton satisfying $\rho(x)-\ddot{f}(x)\ge (n-1)K>0$ holds for all $r\ge0$ and $diam(M^n,g)=\frac{\pi}{\sqrt{K}}$. Then $M^n$ is isometric to the sphere with constant curvature $K$.* *Proof.* Assume $o$, $q\in M^n$, $d(o,q)=\frac{\pi}{\sqrt{K}}=d$. Then for any $r<d$, $B(o,r)\cap B(q,d-r)=\emptyset$. Let $V(o,r)=V(B(o,r)), V(q,r)=V(B(q,r))$, $V_k(r)$ is the volume of geodesic ball in sphere with constant curvature $K$ and radiu $r$. By Volume comparision theorem we can get the inequlity $$\begin{aligned} V(M)&\ge V(o,r)+V(q,d-r)\\ &=\frac{V(o,r)}{V_k(r)}V_k(r)+\frac{V(o,d-r)}{V_k(d-r)}V_k(d-r)\\ &\ge \frac{V(o,d)}{V_k(d)}V_k(r)+\frac{V(o,d)}{V_k(d)}V_k(d-r)\\ &=V(M) \end{aligned}.$$ Which means $V(o,r)=V_K(r)$ holds for any $r\in[0,d]$. So we know that $B(o,r)$ is isometric to $B_K(r))$ holds for any $r\in[0,d]$. Then we can easily get the result that $M^n$ is isometric to the sphere with constant curvature $K$. ◻ # The estimate of potential function and volume growth of shrinking almost solitons In [@CZ], the author bulid an estimate of potential function on shrinking Ricci solitons by using the property that the scaler curvature of the shrinking Ricci solitons is nonnagetive. And according to [@PRR], we know that if the potential function $\rho(x)$ is bounded and $\Delta \rho\le 0$, the scaler curvature is nonnagetive.As a result, we can arrive at a same estimate. With this esimation, we find a volume growth result of shrinking almost solitons and with volume comparision theorem we have already got, we can obtain the rigidity result and then a classification result of complete noncompact shrinking almost solitons . First, the scaler curvature of gradinate Ricci almost solitons obeys the same type equation as the gradinate Ricci solitons. **Proposition 13**. *The scaler curvature of Ricci almost soliton satisfies $$\partial_iR=2R_{ij}\partial_jf+2(n-1)\rho_i.$$* *Proof.* by the properties of almost solitons, we have $$\partial_iR_{jk}=-\nabla_i\nabla_j(\nabla_k f)+\rho_ig_{jk}$$ $$\partial_jR_{ik}=-\nabla_j\nabla_i(\nabla_k f)+\rho_jg_{ik},$$ subtracting these two equations we have $$\partial_iR_{jk}-\partial_jR_{ik}=R_{ijlk}\partial_lf+\rho_ig_{jk}-\rho_jg_{ik}.$$ Taking the trace on j and k we get $$\partial_iR-\partial_jR_{ij}=R_{il}\partial_lf+(n-1)\rho_i,$$ then we have $$\label{eq5.3} \partial_iR=2R_{ij}\partial_jf+2(n-1)\rho_i.$$ ◻ **Remark 14**. *In the process of proving this theory, we used Einstein summation conciliation.* Just let $R_{ij}=\rho(x)g_{ij}-f_{ij}$ in ([\[eq5.3\]](#eq5.3){reference-type="ref" reference="eq5.3"}). And integrating it on a minimal geodesic so that we can get **Corollary 15**. *$$\label{eq5.4} \left\{ \begin{array}{lr} &R+|\nabla f|^2-F=c\\ &2\rho f_i+2(n-1)\rho_i=F_i \end{array} \right.$$ ($c=R(o)+|\nabla f|^2(o)-F(o)$ for some $o\in M$)* by simply caculating, we have $$2\rho \dot{f}+2(n-1)\dot{\rho}=\dot{F},$$ then $$\label{dotfest} |\nabla f|^2\le \int_{0}^{r}(2\rho \dot{f}+2(n-1)\dot{\rho})ds+R(p)+|\nabla f|^2(p).$$ Considering the case where $f(x)>0$, if $f$ and $\rho$ satisfying $\int_{0}^{r}\dot{\rho}fds\ge K>-\infty$ holds for sufficiently $r$, then let $0<\rho\le K_1$ there will exist a constant $c'>0$ that $$\begin{aligned} |\nabla f|^2\le 2K_1(f+c'),\end{aligned}$$ we know that $|\nabla f|^2=|\nabla(f+c')|^2$ so $$\label{eq5.5} |\nabla(f+c')|^2\le 2K_1(f+c'),$$ which means $$|\nabla\sqrt{(f+c')}|^2\le \frac{K_1}{2},$$ so we get the uper bound of the potential function $$f(x)\le (\sqrt{\frac{K_1}{2}}r+C')^2,$$ which proves that **Theorem 16**. *$(M^n,g,f,\rho)$ is an almost shrinking solitons that satisfies $\rho(x)\le K_1$, $\Delta \rho\le 0$ and $\int_{0}^{r}\dot{\rho}fds\ge K>-\infty$ holds for sufficiently $r$, then the potential function satisfying $f(x)\le (K_1r+C')^2$ holds for a constant $C'$.* And as the same way in[@CZ], we can get the lower bound of the potential function by using its upper bound and the lower bound of soliton functions $\rho(x)$ . **Theorem 17**. *$(M^n,g,f,\rho)$ is an almost shrinking solitons satisfying $K_2\le\rho(x)\le K_1$, $\Delta \rho\le 0$ and $\int_{0}^{r}\dot{\rho}fds\ge K>-\infty$ for sufficiently $r$, then the potential function $f(x)$ satisfying $$\frac{(K_2r-C^{''})^2}{2K_1}\le f(x)\le (\sqrt{\frac{K_1}{2}}r+C')^2$$ holds for sufficiently large $r$.* *Proof.* Considering a minimizing normal geodesic $\gamma(t)$, $0\le t\le t_0$ holds for some arbitrary large $t_0>0$, $x_0=\gamma(0)$. $X(t)=\dot{\gamma}(t)$ is denoted as the unite vector along $\gamma(t)$. Then, by the second variation of arc length, we have $$\int_{0}^{s_0}\phi^2Ric(X,X)dt\le \int_{0}^{s_0}|\dot{\phi}(t)|^2dt$$ holds for any nonnegative function $\phi(x)$ defined on the interval $[0,t_0]$. As Hamilton's way in [@Ha1], we choose $\phi(t)$ by $$\phi(t)=\left\{ \begin{array}{lcr} t, &t\in[0,1], \\1, &t\in[1,t_0-1] \\t_0-t, &t\in[t_0-1,t_0] \end{array} \right.$$ then $$\begin{aligned} \int_{0}^{s_0}Ric(X,X)dt&=\int_{0}^{s_0}\phi^2Ric(X,X)dt+\int_{0}^{s_0}(1-\phi^2)Ric(X,X)dt \\&\le \int_{0}^{s_0}|\dot{\phi}(t)|^2dt+\int_{0}^{s_0}(1-\phi^2)Ric(X,X)dt \\&\le2(n-1)+\underset{B_{x_0}(1)}{max}|Ric|+\underset{B_{\gamma(s_0)}(1)}{max}|Ric|.\end{aligned}$$ On the other hand we have $$\nabla_X\dot{f}=\nabla_X\nabla_X f=\rho(x)-Ric(X,X).$$ So we have $$\begin{aligned} \dot{f}(\gamma(s_0))-\dot{f}(x_0)&=\int_{0}^{s_0}\rho(\gamma(t))dt-\int_{0}^{s_0}Ric(X,X)dt \\&\ge K_2s_0-2(n-1)+\underset{B_{x_0}(1)}{max}|Ric|+\underset{B_{\gamma(s_0)}(1)}{max}|Ric|.\end{aligned}$$ So if $M^n$ have bounded Ricci curvature $|Ric|\le C$, we can get an under bound of $f(x)$ that $$f(x)\ge (\frac{K_2}{2}r-C')^2$$ holds for sufficiently large $r$. To remove the restrictions on Ricci curvature bound, we have to modify the above argument. Considering different value of $\dot{f}(x)$ from $\gamma(s_0-1)$ to $\gamma(1)$, we have $$\begin{aligned} \dot{f}(\gamma(s_0-1))-\dot{f}(\gamma(1))&=\int_{1}^{s_0-1}\nabla_X\dot{f}dt \\&\ge K_2(s_0-2)-\int_{1}^{s_0-1}\phi^2Ric(X,X)dt \\&\ge K_2(s_0-2)-2(n-1)-\underset{B_{x_0}(1)}{max}|Ric|+\int_{s_0-1}^{s_0}\phi^2Ric(X,X)dt.\end{aligned}$$ And we know that $Ric(X,X)=\rho(x)-\nabla_X\dot{f}(x)$, so we have $$\begin{aligned} \int_{s_0-1}^{s_0}\phi^2Ric(X,X)dt&=\int_{s_0-1}^{s_0}\phi^2\rho(\gamma(t))dt-\int_{s_0-1}^{s_0}\phi^2\nabla_X\dot{f}(\gamma(t))dt\\&\ge \frac{K_2}{3}+\dot{f}(\gamma(s_0-1))-2\int_{s_0-1}^{s_0}\phi\dot{f}(\gamma(t))dt.\end{aligned}$$ Therefore $$2\int_{s_0-1}^{s_0}\phi\dot{f}(\gamma(t))dt\ge K_2s_0-c^{''}(c^{''}\ge 0),$$ which leads $$\underset{t\in [s_0-1,s_0]}{max}|\dot{f}(\gamma(t))|\ge K_2s_0-c^{''}.$$ And by [\[eq5.5\]](#eq5.5){reference-type="ref" reference="eq5.5"} we know that $$|\dot{f}|\le\sqrt{2K_1f}+K',$$ and $$|f(\gamma(s))-f(\gamma(s_0))|\le K_1(s_0-s)\le K_1.$$ So we have $$\sqrt{2K_1f(\gamma(s_0))}\ge K_2s_0-C^{''}(C^{''}> 0),$$ which leads $$f(\gamma(s))\ge\frac{(K_2s-C^{''})^2}{2K_1}$$ satisfying for sufficiently large $s$ ◻ Now we can prove theorem[Theorem 4](#thm1.3){reference-type="ref" reference="thm1.3"} *Proof.* If $(M^n,g,f,\rho)$ is a complete noncompact almost shrinking soliton, by theorem[Theorem 1](#thm1.1){reference-type="ref" reference="thm1.1"}, we know that the volume growth of geodesic ball satisfies $$Vol(B_o(r))\le A\int_{\mathbb{S}^n}\int_{0}^{r}\exp \{\{\int_{d_0}^{k}\frac{1}{p(s)}(\int_{0}^{s}p(t)(\ddot{f}(t)-\rho(t))+\frac{(n-1)\dot{p}^2(t)}{4p(t)}dt)ds\}\}dkd\Theta .$$ Considering $p(x)=r^2$, we know that $$\exp \{\{\int_{d_0}^{k}\frac{1}{p(s)}(\int_{0}^{s}\frac{(n-1)\dot{p}^2(t)}{4p(t)}dt)ds\}\}=r^{n-1}$$ So the only thing we need to prove is $$\exp \{\{\int_{d_0}^{k}\frac{1}{p(s)}(\int_{0}^{s}p(t)(\ddot{f}(t)-\rho(t))dt)ds\}\}\le C.$$[\[exp\]]{#exp label="exp"} Using integration by parts, we can turn it into another form $$\label{eq4.12} \begin{aligned} \int_{d_0}^{k}\frac{1}{s^2}(\int_{0}^{s}t^2(\ddot{f}(t)-\rho(t))dt)ds&=C+\int_{d_0}^{k}\frac{1}{s}(\ddot{f}(s)-\rho(s))ds-\frac{1}{k}(\int_{d_0}^{k}t^2(\ddot{f}(t)-\rho(t))dt)\\&=C+\frac{1}{k}(k(\int_{d_0}^{k}\frac{1}{s}(\ddot{f}(s)-\rho(s))ds-(\int_{d_0}^{k}t^2(\ddot{f}(t)-\rho(t))dt))\\&=C+\frac{1}{k}(\int_{d_0}^{k}\int_{0}^{m}n(\ddot{f}(n)-\rho(n))dndm)\\&=C+\frac{1}{k}(\int_{d_0}^{k}m\dot{f}-f-\int_{0}^{m}n\rho(n)dndm)\\&\le C+\frac{1}{k}(\int_{d_0}^{k}\frac{m^2}{4}+\dot{f}^2-f-\int_{0}^{m}n\rho(n)dndm). \end{aligned}$$ And we have already known that $\int_{0}^{k}\dot{\rho}fds\ge K$, then by [\[dotfest\]](#dotfest){reference-type="ref" reference="dotfest"} we can get $$\begin{aligned} |\dot{f}|^2&\le \int_{0}^{r}(2\rho \dot{f}+2(n-1)\dot{\rho})ds+C'\\ &\le 2\rho f+2(n-1)\rho+C'. \end{aligned}$$ Which leads $$\frac{1}{k}(\int_{d_0}^{k}\frac{m^2}{4}+\dot{f}^2-f-\int_{0}^{m}n\rho(n)dndm)\le C''+\frac{1}{k}(\int_{d_0}^{k}\frac{m^2}{4}+(2\rho-1)f+2(n-1)\rho-\int_{0}^{m}n\rho(n)dndm)$$ So when $\rho\le \frac{1}{2}+\frac{C}{r^2}$, $\rho\to \dfrac{1}{2}$, we have the estimate $$C''+\frac{1}{k}(\int_{d_0}^{k}\frac{m^2}{4}+(2\rho-1)f+2(n-1)\rho-\int_{0}^{m}n\rho(n)dndm)\le C_1\frac{1}{r}\int_{0}^{r}\frac{Cf}{s^2}ds.$$ Then by the estimate of potential function $f$, we obtain the estimate [\[exp\]](#exp){reference-type="ref" reference="exp"}. If $\rho\to C\neq\frac{1}{2}$, we can get the same result because the scale of a shrinking almost soliton is also a shrinking almost soliton. So the choice of constant won't affect final result. When the equality holds, by rigidity result we have already got, $M^n$ is the scale of Euclidean space. And by [\[eq4.12\]](#eq4.12){reference-type="ref" reference="eq4.12"}, we know that $\dot{f}=\frac{r}{2}$, $f(p)=0$, which means $f=\frac{r^2}{4}$. Then $(M^n,g,f,\rho)$ is Gaussian. ◻ The reason why we give so much restrictions to soliton functions $\rho$ is to control the volume growth by Eucliden volume growth. If we only need a volume growth result, the restrictions for $\rho$ will be greatly relaxed. **Theorem 18**. *$(M^n,g,f,\rho)$ is a complete noncompact almost soliton with bounded $\rho$ and bounded scaler curvature. The potential function $f$ satisfies $\int_{0}^{\infty}\dot{\rho}f\ge K>-\infty$. Then the volume growth of shrinking almost solitons satisfying $$Vol(B_o(r))\le \int_{M^n}exp(Ar^2+Br+C)drd\Theta$$ holds for positive constants $A, B, C$.* *Proof.* Assume $|\rho|\le K$, same as the proof of last theorem, we can also get inequality $$|\nabla f|^2\le 2K_1(f+c'),$$ which means $$f(x)\le (\sqrt{\frac{K_1}{2}}r+C')^2.$$ Same as the proof of last theorem, assume $p(x)=r^2$, we can obtain that $$Vol(B_o(r))\le \int_{M^n}exp(C_1f+C_2r^2+C_3r+C_4)dx\le \int_{M^n}exp(Ar^2+Br+C)drd\Theta$$ At this time the inequality won't be sharped because $p(x)=r^2$ may not be the best test function. ◻ It is very interesting to study the optimal volume growth and the rigidity problem for complete noncompact almost solitons, because it's a way to classify noncompact almost solitons by volume comparision theorem. So at last we will leave an very interesting question: for which $\rho$, we can control the volume growth by polynomial volume growth? 99 ## some A.Barros, E.Ribeiro Jr, *Some characterizations for compact almost Ricci solitons*\[J\]. Proc. Amer. Math. Soc. 140 (2012) 1033-1040. A.Barros,R.Batista, E.Ribeiro Jr, *Compact almost Ricci solitons with constant scalar curvature are gradient.*\[J\]. Monatshefte f¨ur Mathematik, 174, 29--39 (2014). G. Catino, L. Cremaschi, Z. Djadli, C. Mantegazza, and L. Mazzieri, The Ricci-Bourguignon flow, Pacific J. Math. 287 (2017), no. 2, 337--370. Cao Huai-Dong, Detang Zhou *On complete gradient shrinking Ricci solitons* Journal of Differential Geometry 85.2 (2010): 175-186. Cao Huai-Dong, Chen Binglong, Zhu Xiping, *Recent Developments on Hamilton's Ricci flow*, Surveys in Differential Geometry XII, 2008. Cheng X, E.Ribeiro Jr, Detang Zhou *Volume growth estimates for Ricci solitons and quasi-Einstein manifolds*\[J\]. The Journal of Geometric Analysis, 2022, 32(2): 62. R.S.Hamilton, *The formation of singularities in the Ricci flow*, Surveys in Differential Geometry (Cambridge, MA, 1993), 2, 7-136, International Press, Combridge, MA,1995. R.S.Hamilton, *Three-manifolds with positive Ricci curvature*. J. Differ. Geom.17(2), 255--306 (1982) R.S.Hamilton, *The Ricci flow on surfaces*, Contemporary Mathematics 71 (1988) 237-261. Peter Li, *Geometric analysis,* Cambridge Studies in Advanced Mathematics, vol. 134, Cambridge University Press, Cambridge, 2012. A.Naber, Noncompact shrinking four solitons with nonnegative curvature, J. Reine.Angew. Math. 645 (2010), 125-153. Ni Lei and N.Wallach, *on a classification of gradient shrinking solitons*, Math. Res.Lett. 15:5 (2008), 941-955 G.Perelmann, *The entropy formula for the Ricci flow and its geometric applications*,arXiv:math.DG/0211159. G.Perelmann, *Ricci flow with surgery on three manifolds* arXiv:math.DG/0303109. P.Petersen, *Riemannian geometry\[M\]*. New York: Springer, 2006. P.Petersen, W.Wylie, *On the classification of gradient Ricci solitons*, Geom. Topol. 14 (2010), 2277--2300. P.Petersen, W.Wylie, *Rigidity of gradient Ricci solitons*, Pacific J. Math. 241 (2009), 329--345. S.Pigola, M.Rigoli, M.Rimoldi, Alberto.G.Setti *Ricci almost solitons*\[J\]. Annali della Scuola Normale Superiore di Pisa-Classe di Scienze, 2011, 10(4): 757-799. G.Wei, W.Wylie, *Comparison geometry for the Bakry-Emery Ricci tensor*, Journal of Differential Geometry 83 (2009), 377--405. Z.H.Zhang, *On the completeness of gradient Ricci solitons*, Proc. Amer. Math. Soc. 137(2009), 2755--2759.

arxiv_math

--- abstract: | It is often of interest to know which systems will approach a periodic trajectory when given a periodic input. Results are available for certain classes of systems, such as contracting systems, showing that they always entrain to periodic inputs. In contrast to this, we demonstrate that there exist systems which are globally exponentially stable yet do not entrain to a periodic input. This could be seen as surprising, as it is known that globally exponentially stable systems are in fact contracting with respect to some Riemannian metric. author: - "Alon Duvall$^{1}$ and Eduardo Sontag$^{2}$ [^1] [^2] [^3]" bibliography: - mybib.bib title: " **Global exponential stability (and contraction of an unforced system) does not imply entrainment to periodic inputs** " --- # INTRODUCTION Entrainment to a periodic input can be roughly described (a precise definition is given below) as the property that a time-independent system will converge to a periodic trajectory with the same period as a given forcing periodic input. When this occurs is often of interest in physical systems, see, for instance, [@article]. Of course, this property is true for stable linear systems. More generally, it is known that if a system is contractive with respect to a logarithmic norm, then it must entrain to periodic inputs, and a similar result holds for systems that are contractive with respect to arbitrary Riemannian structures [@7039986] [@FB-CTDS]. Furthermore, it is also known that if a system is globally exponentially stable to an equilibrium (GES), then the system is contractive with respect to a suitable Riemannian metric [@slotine1998; @Andrieu2017]. Stated in this vague fashion, it would seem that any GES system must entrain to periodic inputs. We show by means of a counterexample that this implication is false, exhibiting a GES system and a periodic input to which the system does not entrain. We identify the gap in the above reasoning: GES implies contractivity with respect to an arbitrary Riemannian metric, but contractivity in the absence of an input is not equivalent to contractivity when an input is present. On the other hand, for constant metrics, defined by logarithmic norms, contractivity of the unforced system implies contractivity of the system with inputs. We go on to show that the key property needed is uniform contractivity with respect to any constant input. # Background and notation First, we will rigorously define entrainment: **Definition 1**. *We say a function $v: [0,\infty) \rightarrow \mathbb{R}^p$ is periodic with period $T>0$ if $v(t) = v(t+T)$ for all $t\geq0$.* We will apply the above definition both to inputs ($p=m$ below) and states ($p=n$). **Definition 2**. *Consider a system $\dot{x} = f(x,u)$, with $f: \mathbb{R}^n\times\mathbb{R}^m\to \mathbb{R}^n$. We say that this system *entrains to periodic inputs* if the following property holds: given a function (an "input" or "control") $u: [0,\infty) \to \mathbb{R}^m$ which is periodic with period $T$, all solutions of $\dot{x}(t) = f(x(t),u(t))$ converge to a unique limit cycle with period $T$.* We assume that the system dynamics $f$ satisfies conditions for existence and uniqueness of solutions, and forcing functions are measurable essentially bounded, see e.g. [@mct]. We next define globally exponentially stable systems: **Definition 3**. *Consider a system $\dot{x} = f(x,t)$, with $f: \mathbb{R}^n\times\mathbb{R}\to \mathbb{R}^n$ and $f(0,t)=0$ for all $t$. We say that this system is *globally exponentially stable*, or just *GES*, if there exists a number $\lambda > 0$ such that, for any trajectory $x(t)$ and any time $t_0$, $$\|x(t)\| \leq e^{-\lambda (t-t_0)} \|x(t_0)\| \; \mbox{for all}\, t\geq t_0.$$* Here $\|.\|$ is the usual Euclidean norm. We use Euclidean norm only for simplicity, since our purpose is to construct counterexamples, but more general norms could be used as well, or simply distances from $x(t)$ to $0$ in an arbitrary metric space. In our examples, the vector field $f$ will be independent of $t$, in which case we only need to consider $t_0=0$. # Losing entrainment and contraction ## Example of a GES system which does not entrain Consider the following two-dimensional system with two-dimensional input ($n=m=2$): $$\begin{aligned} \dot{x} &= &-x + \frac{x}{2} \sin(x^2 + y^2) -y \;+\; u_1(t)\\ \dot{y} &=& -y + \frac{y}{2} \sin(x^2 + y^2) + x \;+\; u_2(t)\,.\end{aligned}$$ Let $r^*$ be any positive value of $r$ at which the function $$f(r) = - r + \frac{r}{2}\sin(r^2)$$ attains a strict local maximum. In other words, $r^*$ must satisfy $$\frac{\sin\left(r^2\right)}{2}+r^2\,\cos\left(r^2\right)=1$$ and $$3r\cos\left(r^2\right)-2r^2\,\sin\left(r^2\right) < 0\,.$$ The smallest local maximum is found numerically to be approximately $$r^* = 2.79098840365914 \,.$$ Consider the periodic control (of period $2\pi$) $$u(t) = \begin{pmatrix} u_1(t) \\ u_2(t) \end{pmatrix} = -\begin{pmatrix} (- r^* + \frac{r^*}{2}\sin((r^*)^2)) \cos(t) \\ (- r^* + \frac{r^*}{2}\sin((r^*)^2)) \sin(t) \end{pmatrix}\,.$$ The resulting system, once that this input is plugged in, is as follows: $$\begin{aligned} \dot{x} &=& -x + \frac{x}{2} \sin(x^2 + y^2) -y \nonumber \\ && - (- r^* + \frac{r^*}{2}\sin((r^*)^2)) \cos(t) \label{equ:system equations 1} \\ \dot{y} &=& -y + \frac{y}{2} \sin(x^2 + y^2) + x \nonumber \\ && - (- r^* + \frac{r^*}{2}\sin((r^*)^2)) \sin(t) \label{equ:system equations 2}\end{aligned}$$ We show next that the curve $\gamma(t) = (r^*\cos(t),r^*\sin(t))$ is a periodic trajectory (of period $2 \pi$) of our system with the given input. Indeed, plugging in the values $(r^*\cos(t),r^*\sin(t))$ we get: $$\begin{aligned} \dot{x} \!\! &=& \!\!\!\! -r^*\cos(t) + \frac{r^*\cos(t)}{2} \sin((r^*\cos(t))^2 + (r^*\sin(t))^2)\\ && \quad -r^*\sin(t) - (- r^* + \frac{r^*}{2}\sin((r^*)^2)) \cos(t) \\ &=& -r^* \sin(t)\\ \dot{y} \!\! &=& \!\!\!\! -r^*\sin(t) + \frac{r^*\sin(t)}{2} \sin((r^*\cos(t))^2 + (r^*\sin(t))^2) \\ && \quad + r^*\cos(t) - (- r^* + \frac{r^*}{2}\sin((r^*)^2)) \sin(t)\\ &=& r^* \cos(t)\,.\end{aligned}$$ Note that the right hand side of the above two equations is just the tangent vector to $\gamma(t)$, and thus we can conclude that $\gamma(t)$ is in fact a trajectory. Its image is the circle of radius $r^*$ centered at the origin. Next, let us consider an arbitrary trajectory that starts at a point of the form $(x,y)$ such that $x^2 + y^2 <(r^*)^2$. We will use polar coordinates ($r(t),\theta(t))$ to represent this trajectory, so that $x(t) = r(t) \cos(\theta(t))$ and $y(t) = r(t) \sin(\theta(t))$. Thus $r(t)=\sqrt{x(t)^2+y(t)^2}$ along this trajectory and the initial condition $(x(0),y(0))$ has the form in polar coordinates $(r(0),\theta(0))$, with $r^2(0) = x(0)^2 + y(0)^2$. We will show that $r(t)$ is decreasing whenever $r(t) < r^*$ is sufficiently close to $r^*$. We have that (note we are now suppressing that $x,y$ and $r$ are all functions of $t$): $$\begin{aligned} \frac{\dot{r^2}}{2} &= x \dot{x} + y\dot{y} \\ &= -(x^2 + y^2) + \frac{x^2 + y^2}{2} \sin(x^2 + y^2) \\ &\phantom{=} - (x \cos(t) + y \sin(t))(- r^* + \frac{r^*}{2}\sin((r^*)^2)) \\ &= -r^2 + \frac{r^2}{2} \sin(r^2) \\ &\phantom{=} - (r \cos(t - \theta)) (- r^* + \frac{r^*}{2}\sin((r^*)^2)).\end{aligned}$$ If $r$ is close enough to $r^*$ such that $0 > f(r) - f(r^*)$, then $$\begin{aligned} &-(r^2) + \frac{r^2}{2} \sin(r^2) - (r \cos(t - \theta)) (- r^* + \frac{r^*}{2}\sin((r^*)^2)) \\ &\leq -(r^2) + \frac{r^2}{2} \sin(r^2) - (r) (- r^* + \frac{r^*}{2}\sin((r^*)^2)).\end{aligned}$$ $$= r(f(r) - f(r^*))< 0 \,.$$ Thus we see that any points in the interior of the circle $r = r^*$ and close enough to the trajectory $\gamma(t)$ actually move away from from $\gamma(t)$, and thus we do not approach $\gamma(t)$. Figure [1](#fig:non_entraining){reference-type="ref" reference="fig:non_entraining"} illustrates this. ![We plot two trajectories for our system defined by equations [\[equ:system equations 1\]](#equ:system equations 1){reference-type="ref" reference="equ:system equations 1"} and [\[equ:system equations 2\]](#equ:system equations 2){reference-type="ref" reference="equ:system equations 2"}: One with initial conditions at $(r^*,0)$ and one with initial conditions at $(r^*-0.1,0)$. We see the trajectory starting at $(r^*-0.1,0)$ does not approach, and in fact diverges from, the periodic orbit corresponding to the circle of radius $r^*$.](not_converging_image_2.png){#fig:non_entraining width="50%"} This system is equivalent to the following system in polar coordinates: $$\begin{aligned} \dot{r} &=& f(r) \;=\; - r + \frac{r}{2}\sin(r^2)\\ \dot{\theta} &=& 1\,.\end{aligned}$$ Indeed, the motivation for this system was that we simply want to "force" an unstable periodic orbit, such that points slightly closer to the origin move away from this periodic orbit, showing that there is no global convergence to a unique periodic orbit. This form of the system also makes it easy to see that it is in fact GES, since we have that $f(r) \leq -\frac{r}{2}$ for all nonnegative $r$, which implies that, for all solutions, $$\|x(t)\| \leq e^{-\frac{1}{2} t } \|x(0)\| \;\; \mbox{for all} \; t\geq 0\,.$$ ## Example of losing contraction upon translation A definition of contraction with respect to a state-dependent (and time-independent) Riemannian metric on $\mathbb{R}^n$ is given in [@slotine1998]. We prefer to use the form given in [@parillo_slotine_2008], Definition 1: Given the $n$-dimensional autonomous system $$\dot{x} \;=\; f(x) \,,$$ a contraction metric is an $n \times n$ symmetric matrix $M(x)$ that is uniformly positive definite (that is, $v^T M(x) v \geq a$ for all $x$ and $v$ in $R^n$, for some $a>0$) such that $$\frac{\partial f}{\partial x}^T M(x) + M(x) \frac{\partial f}{\partial x} + \dot M (x)$$ is uniformly negative definite, where the notation $\dot M (x)$ is shorthand for the matrix whose $(i,j)$th entry is $$\dot{M}(x)_{ij} = \frac{\partial M_{ij}}{\partial x}^T f(x) \,.$$ Theorem 1 in [@parillo_slotine_2008] says that if a contraction matrix exists and there is some equilibrium, then all trajectories converge to it. Moreover, under the stronger assumption that $$\frac{\partial f}{\partial x}^T M(x) + M(x) \frac{\partial f}{\partial x} + \dot M (x) \leq \;-\; \alpha M(x)$$ for all $x$, for some $\alpha>0$ (where $A\leq B$ means that $B-A$ is positive definite) then there is automatically an equilibrium. The second result follows from the fact that $d(x(t),y(t)) \leq e^{- (\alpha/2)t} d(x(0),y(0))$ along trajectories, where $d$ is the geodesic distance associated to the metric $M(x)$. Let us consider now as an example the scalar system $\dot{x} = f(x) = - x + \frac{x}{2}\sin(x^2)$ where $n=1$. One needs a function $m:\mathbb{R}\rightarrow \mathbb{R}$ such that $m(x)\geq a > 0$ for all $x$ and $$m'(x)f(x) + 2f'(x)m(x) \;\leq\; - \beta m(x)$$ for all $x$, where $\beta> 0$. In our case, we pick $m(x) = 1/(\sin(x^2)/2 - 1)^2$, which is larger than 4/9 for all $x$, and with $f(x) = (x \sin(x^2))/2 - x$ we have that $$\begin{aligned} m'(x)f(x) + 2f'(x)m(x) &=& \frac{4}{\sin(x^2) - 2}\\ &\leq& \frac{-1/3}{(\sin(x^2)/2 - 1)^2} \,.\end{aligned}$$ Thus we see that for $\beta = 1/3$ our inequality is satisfied. Indeed, this follows since $$\frac{4(\sin(x^2)/2 - 1)^2}{\sin(x^2) - 2} \leq \frac{1}{\sin(x^2) - 2} \leq -\frac{1}{3} \,.$$ We next show that contractivity breaks down for this metric if we add a constant input, i.e., we will consider the system $\dot{x} = f(x) + c$. In this case we have that $$\begin{aligned} m'(x) (f(x)+c) + 2f'(x) m(x) &=& \frac{4}{\sin(x^2) - 2} \\ &&\quad + c\frac{16 x \cos(x^2)}{(2 -\sin(x^2))^3} \\ &\geq& c\frac{16 x \cos(x^2)}{(2 -\sin(x^2))^3} - 4 \\ &\geq& c\frac{16 x \cos(x^2)}{27} - 4\,.\end{aligned}$$ If we pick $c = \frac{27}{16}$ and $x = 4\sqrt{2 \pi }$ we see that our final expression is clearly positive, and thus our system does not contract everywhere with respect to this metric (despite the fact that when $c=0$, our system *is* contractive with respect to this metric). # Constant and non-constant inputs, and entrainment In this section, we will first remark that uniform contractivity (and hence entrainment) under *constant* (additive) inputs that take values in a given set $B$ implies uniform contractivity (and hence entrainment) under *arbitrary* inputs taking values in $B$. We will then prove, however, that if $B=\mathbb{R}^n$ then the only metrics for which this property can hold are constant, but if $B$ is bounded, there do exist non-constant metrics that provide contractivity. ## Connecting constant and non-constant inputs **Theorem 1**. *Consider the system with additive inputs $\dot{x} = f(x) + u$, where the inputs take values $u(t)\in U \subset \mathbb{R}^n$. Suppose that this system is uniformly contractive with respect to a metric $M(x)$, uniformly over all constant inputs $u(t)\equiv c\in U$. Then the system is uniformly contractive with respect to a metric $M(x)$, uniformly on all inputs $u$ with values in $U$.* .  The assumption means that there is a constant $\beta>0$ such that, for all $x$, and all $g(x) = f(x)+c$, $c\in U$, the following inequality holds: $$\frac{\partial g}{\partial x}^T M + M \frac{\partial g}{\partial x} + \dot{M} \leq -\beta M$$ (where $M = M(x)$). Here $$\dot{M} = \dot{M}_1 + \dot{M}^{[c]}_2\,,$$ where we define $$\{\dot{M}_1\}_{ij} := \frac{\partial M_{ij}}{\partial x} f(x),\, \{\dot{M}^{[c]}_2\}_{ij} := \frac{\partial M_{ij}}{\partial x} c \,.$$ Thus we have an estimate, uniform on $x$ and $c$, $$\frac{\partial f}{\partial x}^T M + M \frac{\partial f}{\partial x} + \dot{M}_1 + \dot{M}^{[c]}_2 \leq -\beta M.$$ Now given an arbitrary input $u$ with values in $U$, we'd like to have $$\frac{\partial f}{\partial x}^T M + M \frac{\partial f}{\partial x} + \dot{M}_1 + \dot{M}^{[u(t)]}_2 \leq -\beta M$$ for all $t$, where $\{\dot{M}^{[u(t)]}_2\}_{ij} := \frac{\partial M_{ij}}{\partial x} u(t)$. This is true because $u(t)\in U$. $\rule{1ex}{1.4ex}$ As a corollary we have, under the same assumptions, entrainment to any periodic input $u(t)$ satisfying $u(t) \in U$ for all $t \geq 0$. This follows from a result from [@slotine1998] which states that if a system $\dot{x} = f(x,u(t))$ is contracting then we have entrainment. In Section [5.2](#sec:metric){reference-type="ref" reference="sec:metric"} we will provide a version of Theorem [Theorem 1](#thm:additive){reference-type="ref" reference="thm:additive"} which applies to non-additive inputs, and is formulated in an abstract metric space setup. ## Metrics must be constant for contractivity under arbitrary additive constant inputs We next show that when $U=\mathbb{R}^n$, uniform contraction metrics under aditive inputs must be constant. **Theorem 2**. *Suppose we have a system $\dot{x} = f(x)$ which is contractive with respect to some metric $M(x)$, where $M(x)$ is not constant. Then there exists a constant input $c \in \mathbb{R}^n$ such that $\dot{x} = f(x) +c$ is no longer contractive with respect to $M(x)$.* .  Consider the system $\dot{x} = f(x) + c$, where $c$ is a constant vector. Since we are free to choose $c$ as we please, we will occasionally change $c$ in the course of the proof. For this system to be contractive we must have that $$\frac{\partial f}{\partial x}^T M + M \frac{\partial f}{\partial x} + \dot{M} \leq 0.$$ let us write $\dot{M} = \dot{M}_1 + \dot{M}_2$ where $\{\dot{M}_1\}_{ij} = \frac{\partial M_{ij}}{\partial x} f(x)$ and $\{\dot{M}_2\}_{ij} = \frac{\partial M_{ij}}{\partial x} c$. Now we have that $$\frac{\partial f}{\partial x}^T M + M \frac{\partial f}{\partial x} + \dot{M} = \frac{\partial f}{\partial x}^T M + M \frac{\partial f}{\partial x} + \dot{M}_1 + \dot{M}_2.$$ Since $M(x)$ is not constant we can always find a vector $x$ and a vector $c$ such that $\dot{M_2}(x)$ is nonzero, as well as a vector $z$ such that $\alpha = z^t\dot{M}_2(x) z \neq 0$. If $\alpha < 0$, then replace $c$ with $-c$ so that we will instead get that $\alpha > 0$. Now assume $\alpha > 0$ and set $$\beta = z^t (\frac{\partial f}{\partial x}^T M + M \frac{\partial f}{\partial x} + \dot{M}_1) z$$ Pick $N$ such that $N\alpha > |\beta|$. Upon replacing $c$ with $Nc$, we have that $$z^t (\frac{\partial f}{\partial x}^T M + M \frac{\partial f}{\partial x} + \dot{M}_1 + \dot{M}_2) z = \beta + N \alpha > 0.$$ Thus we see that given that $M(x)$ is nonconstant, we can always find a constant input $c$ for which our system is no longer contractive everywhere with respect to $M(x)$. $\rule{1ex}{1.4ex}$ ## Bounded additive controls have non-constant contractions One might wonder if we can do better than the previous theorem. Along these lines, one might ask: Does there exist a *bounded* set $B$ such that if we are uniformly contractive for inputs taking values in $B$ with respect to a certain metric $M$, then our metric *must* be constant? This is not true, as we will now show by means of a counterexample. **Proposition 1**. *Given an arbitrary bounded set $B \subseteq \mathbb{R}$ there exists a one dimensional system $\dot{x} = f(x) + u(t)$ such that if $u(t) \in B$ for all $t \geq 0$, then there exists a non-constant metric for which this system is contractive.* .  Using the notation from Theorem [Theorem 2](#thm: all c implies constant M){reference-type="ref" reference="thm: all c implies constant M"} we want to establish the inequality $$\frac{\partial f}{\partial x}^T M + M \frac{\partial f}{\partial x} + \dot{M}_1 + \dot{M}_2 \leq -\beta M$$ for some particular choices of (non-constant) $M$ and $f$. Consider the system where $f(x) = -x + c$. For this our equation looks like $$(c - x) \frac{\partial M}{\partial x} \leq( 2-\beta) M \,.$$ Suppose $\beta = 1$ and $M(x) = 1 + \epsilon(x)$. Then we have that our inequality is equivalent to $$(c - x) \epsilon'(x) \leq 1 + \epsilon(x)\,.$$ Consider the quantity $\epsilon_m(x) = e^{-x^2/m}$. Substituting $\epsilon_m(x)$ for $\epsilon(x)$ in our inequality we have that $$(c - x) \epsilon_m'(x) = \frac{-2x(c-x)}{m} e^{-x^2/m}.$$ Note that for large enough $m$, this expression goes uniformly to 0 (because $B$ is bounded), and so for large enough $m$ we will have that $(c - x) \epsilon_m'(x) \leq 1 + \epsilon_m(x)$. Thus we have that $M(x) = 1 + \epsilon_m(x)$ is a valid contraction metric for large enough $m$. $\rule{1ex}{1.4ex}$ # Generalizations and comments We discuss now several directions in which our results can be generalized. ## A more general sufficient condition forcing a metric to be constant We will use the following notation: 1. $\lambda_{\max}(A)$ is the maximum eigenvalue of an arbitrary matrix $A$. 2. $\|v\|_2$ is simply the Euclidean norm of an arbitrary vector $v$. 3. $\|A\|_2 = \sqrt{\lambda_{\max} (A^T A)}$ for an arbitrary matrix $A$. 4. $\mu_2(A) = \lambda_{\max}\left(\frac{A + A^t}{2} \right)$ for an arbitrary matrix $A$. The last three operations are a vector norm, its induced matrix norm, and its induced logarithmic norm, respectively. Logarithmic norms are routinely used in contraction theory, and an early exposition is in the control textbook [@DesoerVidyasagar]. The following elementary facts are well-known, but we include proofs to make the exposition self-contained. **Lemma 1**. *If $A$ and $B$ are symmetric and $A \leq B$, then $\mu_2(A) \leq \mu_2(B)$.* .  First note $\mu_2(A)$ is simply the maximum value of its eigenvalues, for a symmetric matrix $A$. Suppose $A$'s maximum eigenvalue is larger than $B$'s maximum eigenvalue. Then we have that if $x$ is the corresponding eigenvector for $A$ that $$\begin{aligned} \frac{x^t (B- A) x}{x^t x} &=& \frac{x^t B x}{x^t x} - \lambda_{\max}(A)\\ &\leq& \lambda_{\max}(B) - \lambda_{\max}(A)< 0.\end{aligned}$$ This a contradiction, and so we are done. $\rule{1ex}{1.4ex}$ **Lemma 2**. *We have that $\mu_2(AB) \leq \|A \|_2 \|B \|_2$.* .  Note that for a symmetric matrix $S$ we have that $\lambda_{\max}(S) \leq |\lambda_{\max}(S)| \leq \|S\|_2$. We have that $$\begin{aligned} \mu_2(AB) &=& \lambda_{\max} \left(\frac{AB + (AB)^t}{2}\right) \leq \left\| \frac{AB + (AB)^t}{2} \right\|_2 \\ &\leq& \|A \|_2 \|B \|_2 \end{aligned}$$ $\rule{1ex}{1.4ex}$ Recall that the *convex closure* of a set of point $P$ is indicated by $\mbox{conv}(P)$ and is the set of all points $p$ such that we can find a finite set of points $O \subset P$ such that $p = \sum_{i} \lambda_i o_i$, where $\lambda_i \geq 0, \sum_i \lambda_i = 1$ and $o_i \in O$. **Theorem 3**. *Suppose we have:* 1. *An $n$ dimensional system $\dot{x} = f(x,u)$.* 2. *$k$ infinite sequences of constant inputs $\{u_{i,j}\}$ where $1 \leq i$ and $1 \leq j \leq k$.* 3. *For each fixed $x$ there exists a large enough $i_0$ so that $\mbox{conv}(\{\frac{f(x,u_{i,j})}{\|f(x,u_{i,j})\|_2}\}_{1 \leq j \leq k})$ always contains a certain open sphere $O$ centered at 0 for $i \geq i_0$.* 4. *For any fixed $j$ and for all $x$ we have that. $$\lim_{i \rightarrow \infty} \frac{\left\| \frac{\partial f(x,u_{i,j})}{\partial x}\right\|_2 }{\|f(x,u_{i,j})\|_2} = 0.$$* *Then if our system is contractive with respect to a Riemannian metric for each of our constant inputs, the metric must be constant.* .  First we fix an arbitrary $x \in \mathbb{R}^n$. We will carry out all our computations at this set $x$. We have our necessary equation for contractivity: $$\frac{\partial f}{\partial x}^T M + M \frac{\partial f}{\partial x} + \dot{M} \leq 0.$$ Now we can rearrange to get $$\dot{M} \leq -\frac{\partial f}{\partial x}^TM - M \frac{\partial f}{\partial x} .$$ Now upon taking the logarithmic norm $\mu_2$ of both sides we have that $$\begin{aligned} \mu( \dot{M} )_2 &\leq & \mu_2 \left(-\frac{\partial f}{\partial x}^TM - M \frac{\partial f}{\partial x} \right) \\ &\leq & \mu_2 \left(-\frac{\partial f}{\partial x}^TM \right) + \mu_2\left(- M \frac{\partial f}{\partial x}\right) \\ &\leq & 2\left\|-\frac{\partial f}{\partial x}^T \right\|_2 \|M\|_2 \\ &=& 2\left\|\frac{\partial f}{\partial x}^T \right\|_2 \|M\|_2. %\end{aligned}$$ Dividing by $\|f(x,u_{i,j})\|_2$ we get $$\mu_2\left(\frac{ \dot{M} }{\|f(x,u_{i,j})) \|_2}\right) \leq \frac{2 \|M\|_2 \| \frac{\partial f}{\partial x}^T \|_2 }{\|f(x,u_{i,j})\|_2} \xrightarrow[i \rightarrow \infty]{} 0.$$ Thus we see that we have $\lim_{i \rightarrow \infty} \mu_2\left(\frac{ \dot{M} }{\|f(x,u_{i,j})\|_2} \right) \leq 0$. Now the $lk$'th entry of $\frac{ \dot{M} }{\|f(x,u_{i,j})\|_2}$ is $\frac{\partial M_{lk}}{\partial x} \frac{f(x,u_{i,j})}{\|f(x,u_{i,j})\|_2}$. Here $\frac{\partial M_{lk}}{\partial x}$ is the gradient of $M_{lk}$ and is a row vector with $n$ entries. Suppose we have a matrix $G$ such that its $lk$'th entry is $\{G\}_{lk} = \frac{\partial M_{lk}}{\partial x} v$ where $v \in \mathbb{R}^n$. Let $\alpha$ be positive real number such that $v \in \alpha O$. Now we can write $$v = \sum_j \alpha_{i,j} \frac{f(x,u_{i,j})}{\|f(x,u_{i,j} \|},$$ where we have that $0 \leq \alpha_{i,j} \leq \alpha$ for large enough $i$ and all $j$ (this follows from the fact that $v \in \alpha O \in \alpha* \mbox{conv}(\{\frac{f(x,u_{i,j})}{\|f(x,u_{i,j})\|_2}\}_{1 \leq j \leq k})$). Thus we have that $$\{G\}_{lk} = \frac{\partial M_{lk}}{\partial x} v = \sum_j \alpha_{i,j} \frac{\partial M_{lk}}{\partial x} \frac{f(x,u_{i,j})}{\|f(x,u_{i,j})\|_2}.$$ Now let $M^{i,j}$ be the matrix with $\{M^{i,j}\}_{lk} = \frac{\partial M_{lk}}{\partial x} \frac{f(x,u_{i,j})}{\|f(x,u_{i,j})\|_2}$. We have that $$\mu_2(G) \leq \alpha \sum_j \mu_2(M^{i,j}) \xrightarrow[i \rightarrow \infty]{} 0.$$ Thus we have that $\mu_2(G) \leq 0$ and by replacing $v$ with $-v$ we also have $\mu_2(-G) \leq 0$ which only happens when $G = 0$. Thus we have that $\frac{\partial M_{lk}}{\partial x} v = 0$ for all $v$, and so we must have $\frac{\partial M_{lk}}{\partial x} = 0$. Since this is true at all $x$ we must have that $M(x)$ is the constant matrix. ### Example Consider the system $$\begin{bmatrix} \dot{x} \\ \dot{y} \\ \dot{z} \end{bmatrix} = \begin{bmatrix} -1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \\ \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} + \begin{bmatrix} 1 & 0 & 0 & -1 \\ 0 & 1 & 0 & -1 \\ 0 & 0 & 1 & -1 \end{bmatrix} \begin{bmatrix} u_1 \\ u_2 \\ u_3 \\ u_4 \end{bmatrix}.$$ Suppose we have that for all $k$ that $u_k \geq 0$, and at most one of $u_1,u_2,u_3$ or $u_4$ are nonzero. Take as four sequences of controls $$\{u_{k,1}, u_{k,2}, u_{k,3}, u_{k,4} \} = \left\{\begin{bmatrix} k \\ 0 \\ 0 \\ 0 \end{bmatrix},\begin{bmatrix} 0 \\ k \\ 0 \\ 0 \end{bmatrix},\begin{bmatrix} 0 \\ 0 \\ k \\ 0 \end{bmatrix},\begin{bmatrix} 0 \\ 0 \\ 0 \\ k \end{bmatrix} \right\}.$$ Where $k$ is any positive integer. This sequence of controls satisfies the conditions of our theorem. Indeed, we have that $\frac{f(x,u_{k,j})}{\|f(x,u_{k,j})\|_2}$ is simply a constant vector for each $j$, and thus the convex closure of all 4 of these constant vectors always contains an open ball. We also have that $$\lim_{i \rightarrow \infty} \frac{\left\| \frac{\partial f(x,u_{k,j})}{\partial x}\right\|_2 }{\|f(x,u_{k,j})\|_2} = 0$$ since the numerator is always a constant, while the denominator always goes to infinity. Thus while we are contractive with regard to a constant metric (such as the usual Euclidean metric), we are not contractive with respect to a nonconstant metric. ### Example Consider the system $$\dot{x} = f(x) + c.$$ Here $c \in \mathbb{R}^n$ is our constant control, which we allow to take on any value. We see that we have $$\lim_{\|c\|_2 \rightarrow \infty} \frac{\left\| \frac{\partial f(x,c)}{\partial x}\right\|_2 }{\|f(x,c)\|_2} = \lim_{\|c\|_2 \rightarrow \infty} \frac{\left\| \frac{\partial f(x)}{\partial x}\right\|_2 }{\|f(x)+c\|_2} = 0$$ since the numerator does not depend on $c$, and the denominator goes to infinity as $\|c\|_2$ tends to infinity. We can also always find $n+1$ sequences of constant inputs $c_{i,j}$ such that $\lim_{i \rightarrow \infty} \|c_{i,j}\|_2 = \infty$ and $\mbox{conv}(\{\frac{f(x) + c_{i,j}}{\|f(x) + c_{i,j}\|_2} \}_{1 \leq j \leq n+1 })$ always contains a fixed open ball (e.g., take $c_{i,j} = i v_j$, where $\mbox{conv}(\{v_j\}_{1 \leq j \leq n+1})$ contains an open ball). Thus the conditions of our theorem are satisfied, and thus we have another proof of Theorem $\ref{thm: all c implies constant M}$. $\rule{1ex}{1.4ex}$ ## Contraction on a metric space {#sec:metric} Suppose we are given a metric space $\mathbb{M}$ with metric $d$. Refer to the space of mappings of $\mathbb{M}$ to itself as $C(M)$. **Definition 4**. *The *topology induced by its supremum distance* on $C(M)$ is the topology induced by the metric $\mathcal{D}$ on $C(M)$ defined by $$\mathcal{D}(f,g) = \sup\{d(f(x),g(x)| x \in M\}.$$* **Definition 5**. *Given two functions $u,v: \mathbb{R} \rightarrow \mathbb{R}^n$ their *concatenation* $uv$ at $t_0$ is the function defined as$$uv = \begin{cases} u & \text{when } t < t_0 \\ v & \text{when } t \geq t_0 \end{cases}$$* **Theorem 4**. *Suppose we have:* 1. *A metric space $\mathbb{M}$ with metric $d$.* 2. *The space of mappings of $\mathbb{M}$ to itself $C(\mathbb{M})$ (give this space the topology induced by its supremum distance).* 3. *The space of measurable, bounded, and locally integrable functions from $\mathbb{R} \rightarrow \mathbb{R}^n$ (call this space $L_M$, give this space its supremum norm, where the norm is Euclidean).* 4. *A mapping $\phi:L_M \times \mathbb{R} \times \mathbb{R} \rightarrow C(\mathbb{M})$ satisfying that* 1. *It is continuous in $L_M$ (given two arbitrary and fixed entries for the other arguments)* 2. *$\phi(g,t_2,t_3) \circ \phi(f,t_1,t_2) = \phi(gf,t_1,t_3)$ where $gf$ is the concatenation of the two functions at $t_2$.* 3. *$\phi(f(t),t_1,t_2) = \phi(f(t+T), t_1 -T, t_2 - T )$ for all $T,t,t_1,t_2 \in \mathbb{R}$.* 4. *Whenever we have a compact set $U \subseteq L_M$ of constant mappings then there exists $\lambda < 0$ such that $d(\phi(f,0,t)(x),\phi(f,0,t)(y)) \leq e^{\lambda t}d(x,y)$ for all $x,y \in \mathbb{M}$, all $t\geq 0$, and all $f \in U$.* *Let $L_{ct}$ be the space of mappings produced by concatenating finitely many times functions from $U$, and let $\overline{L}_{ct}$ be the closure of this set. If we have $f \in \overline{L}_{ct}$ and $t_1 > t_2$ then $\phi(f,t_1,t_2)$ is a contraction.* .  First we will argue that all piecewise constant functions also give us contractions. This is clear from the requirement that $\phi(g,t_2,t_3) \circ \phi(f,t_1,t_2) = \phi(gf,t_1,t_3)$. Indeed, letting $\phi_g = \phi(g,t_2,t_3)$ and $\phi_f = \phi(f,t_1,t_2)$ we have if $d(\phi_f(x),\phi_f(y)) \leq \lambda_1 d(x,y)$ and $d(\phi_g(x),\phi_g(y)) \leq \lambda_2 d(x,y)$ then we have $$d(\phi_g (\phi_f(x)),\phi_g(\phi_f(y))) \leq \lambda_2 \lambda_1 d(x,y).$$ Thus the composition of $\phi_f$ and $\phi_g$ still gives us a contraction and so $\phi(gf,t_1,t_3)$ must be a contraction. It follows that if we concatenation finitely many functions into a piecewise constant function $F$, we will still have that $\phi(F,t_1,t_2)$ is a contraction. Since any piecewise constant function (taking on finitely many different values) can be produced in this manner, we can conclude that for all $F \in L_{ct}$ that $\phi(F,t_1,t_2)$ is in fact a contraction. Now suppose we consider a compact set $B \subseteq U$ of constant functions, taking their image from a bounded codomain in $\mathbb{R}^n$. For each $f \in B$ we have that $d(\phi(f,t_1,t_2)(x),\phi(f,t_1,t_2)(x)) \leq e^{\lambda (t_2-t_1)} d(x,y)$ where $\lambda$ is the contraction constant we know will work for all our constant functions by condition 4(d). Now suppose we have a piecewise constant function $f$ on the interval $[t_1,t_2]$ and set $\phi(f(x),t_1, t_2) = \phi_f$. Suppose our function takes on value $c_i$ on interval $i$ of length $\alpha_i(t_2-t_1)$, where $\sum_i \alpha_i = 1$ and the $c_i$ are all contained in some compact set $B$. Composing all these pieces as in property 4(b) of our conditions, we have that $$\begin{aligned} d(\phi_f(x),\phi_f(y)) &\leq& (\prod_i e^{\lambda \alpha_i(t_2 -t_1)}) d(x,y)\\ &=& e^{\lambda (t_2 - t_1)} d(x,y). \end{aligned}$$ Thus not only are our piecewise constant mappings contractions, but there is a contraction constant $\lambda$ that they all satisfy (assuming we are taking our constant functions from a bounded set). Now for a general function $f$ that can be approximated (in supremum norm) by piecewise constant functions, note that due to $f$ being bounded its values are contained in a compact set. Thus it can be approximated by piecewise constant functions taking on values from a compact set. Thus all these approximating functions have a universal contraction coefficient $\lambda < 0$, and by continuity of $\phi$ we have that $\phi(f,t_1,t_2)$ must also be a contraction. Indeed, if we have a sequence of piecewise constant $f_i$ converging to $f$ so that $\phi_{f_i}$ converges to $\phi_f$, then we have that for a given pair of points $x,y \in \mathbb{M}$ and for some $\epsilon > 0$ that $$\begin{aligned} d(\phi_f(x),\phi_f(y)) &\leq& d(\phi_{f}(y),\phi_{f_i}(y)) + d(\phi_{f_i}(x),\phi_{f_i}(y)) \\ && + \; d(\phi_{f_i}(x),\phi_{f}(x)) \\ &\leq& e^{-\lambda(t_1 - t_2)} d(x,y) + \epsilon. \end{aligned}$$ Thus since we can pick $\epsilon$ to be arbitrarily small (it becomes small as $i \rightarrow \infty$) we have that $$d(\phi_f(x),\phi_f(y)) \leq e^{-\lambda(t_1 - t_2)} d(x,y).$$ $\rule{1ex}{1.4ex}$ This theorem, while abstract, applies concretely to ordinary differential equations. If we have a system $\dot{x} = f(x) + u(t)$ where $u(t)$ is in a compact set $B$, $f(x)$ is smooth, and for $u(t)$ constant our system is contractive, then by Theorem 55 in [@mct] and Theorem 2 in [@slotine1998] we have that $\phi$ satisfies the required conditions and so we can use our theorem. Thus if $\dot{x} = f(x) + c$ is a contraction for $c$ in some compact set $B$ and if $u(t) \in B$ for all $t \geq 0$ then our system will still be a contraction. Thus if $u(t)$ is periodic we will have entrainment. # Conclusion and future directions In this conference paper we studied the connections between global exponential stability, contractions with respect to constant and nonconstant metrics, and entrainment. In the full version of this paper, we will describe additional sufficient conditions for which a globally exponentially stable system would in fact entrain to periodic inputs. [^1]: This work was partially supported by grants AFOSR FA9550-21-1-0289 and NSF/DMS-2052455 [^2]: $^{1}$Northeastern University `[email protected]` [^3]: $^{2}$Northeastern University `[email protected], [email protected]`

arxiv_math

--- abstract: | We show that for an Erdős-Rényi graph on $N$ vertices with expected degree $d$ satisfying $\log^{-1/9}N\leq d\leq \log^{1/40}N$, the largest eigenvalues can be precisely determined by small neighborhoods around vertices of close to maximal degree. Moreover, under the added condition that $d\geq\log^{-1/15}N$, the corresponding eigenvectors are localized, in that the mass of the eigenvector decays exponentially away from the high degree vertex. This dependence on local neighborhoods implies that the edge eigenvalues converge to a Poisson point process. These theorems extend a result of Alt, Ducatez, and Knowles, who showed the same behavior for $d$ satisfying $(\log\log N)^4\ll d\leq (1-o_{N}(1))\frac{1}{\log 4-1}\log N$. To achieve high accuracy in the constant degree regime, instead of attempting to guess an approximate eigenvector of a local neighborhood, we analyze the true eigenvector of a local neighborhood, and show it must be localized and depend on local geometry. author: - "Ella Hiesmayr[^1]" - "Theo McKenzie[^2]" bibliography: - ref.bib title: The Spectral Edge of Constant Degree Erdős-Rényi Graphs --- # Introduction {#sec:intro} Finding the spectrum and eigenvectors of the adjacency matrix of graphs is an ubiquitous problem in combinatorics and spectral theory, with important applications to computer science and mathematical physics, see [@chung1997spectral; @kottos1997quantum; @alon1998spectral] for general overviews on their applicability. More specifically, researchers have studied the "typical" behavior of eigenvalues and eigenvectors through random models. The most well-studied of these is the Erdős-Rényi model $\mathcal{G}(N,\frac dN)$, where each of the possible $\binom N2$ edges is included independently with probability $\frac dN$. Therefore the expected degree of a vertex is $\frac{N-1}{N}d\approx d$. See the monograph of Guionnet for an overview of known results and the state of the field for this model [@guionnet2021bernoulli]. In this paper, we focus on the specific behavior of the eigenvalues near the edge of the spectrum. This spectral edge has received much attention, as it has its own specific applications. For example, the edge governs the mixing rate of Markov chains, and graph partitioning (as shown in [@hoory2006expander]), and also achieves localization in the Anderson model ([@anderson1958absence], further discussed later).[^3] The behavior of the extreme eigenvalues and eigenvectors in the Erdős-Rényi model is known to go through various phase transitions. When $d\gg N^{1/3}$ these edge eigenvalues have Tracy-Widom fluctuations,[^4] similar to the fluctuations of the eigenvalues of a GOE matrix [@erdHos2012spectral; @erdHos2013spectral; @lee2018local]. When $N^\epsilon \leq d\ll N^{1/3}$, for some fixed $\epsilon>0$, the top eigenvalues lose GOE behavior and edge eigenvalues become Gaussian distributed [@huang2020transition; @he2021fluctuations]. For sparser Erdős-Rényi graphs, Krivelevich and Sudakov showed using a graph decomposition that eigenvalues are governed by the statistics surrounding the highest degree vertices [@krivelevich2003largest]. Therefore, if $\mu_k$ is the expected number of vertices of degree $k$, we define $$\label{eq:udef} \mathfrak{u}:=\arg\min_{k\in \mathbb{Z}} \left \{\max \left \{ \mu_k,\mu_k^{-1} \right \} \right \}.$$ $\mathfrak{u}$ is roughly the largest degree we expect to occur in the graph. Krivelevich and Sudakov showed the largest eigenvalue of an Erdős-Rényi graph is $(1+o_N(1))\max\{d,\sqrt{\mathfrak{u}}\}$. The balls and bins paradigm implies that for $\log^{-1}N\ll d\ll\log N$, $\mathfrak{u}=\Theta( \frac{\log N }{\log \log N})$. Therefore, this shows that there is a phase transition in the largest eigenvalue at $d=\sqrt{\frac{\log N}{\log\log N}}$. In fact, we begin to see the local affect of high degree vertices at $d\asymp \log N$. This is well known to be the threshold for connectivity (as was shown in the original work of Erdős and Rényi [@erdHos1960evolution]), but is also the threshold for large fluctuations in the degree sequence, as opposed to greater concentration seen for larger $d$. Specifically, Benaych-Georges, Bordenave, and Knowles showed that when $d\gg \log N$, edge eigenvalues converge to the edge of the support of the asymptotic eigenvalue distribution, but when $d\ll \log N$, roughly, edge eigenvalues are "governed" by the largest degree vertices of the adjacency matrix [@benaych2019largest; @benaych2020spectral], with the specific threshold later given by Alt, Ducatez, and Knowles [@alt2021extremal]. Alt, Ducatez, and Knowles further studied this problem, and managed to obtain impressively detailed results. Through the works of [@alt2021delocalization; @alt2022completely; @alt2023poisson; @alt2023localized], the authors show a transition between the occurrence of delocalized eigenvectors in the bulk of the spectrum, and localized eigenvectors near the edge for $d\lesssim\log N$ (the specific bounds on $d$ vary paper to paper, but all results are for this sparse regime). We focus specifically on [@alt2023poisson]. In this result, Alt, Ducatez, and Knowles show that the largest eigenvalues of the graph are determined by two combinatorial statistics around the high degree vertices. As was shown previously through [@krivelevich2003largest] and [@benaych2019largest], the primary term is the degree of the high degree vertex, which we denote here by $\alpha_x$. To gain the necessary levels of accuracy, they also track the secondary term $\beta_x$, which is the number of vertices of distance exactly 2 from a high degree vertex $x$ in the graph.[^5] Note that this notation slightly differs from the one in [@alt2023poisson], where $\alpha_x$ and $\beta_x$ are normalized by $d$. Reinterpreting their result, they show the following. **Theorem 1** ([@alt2023poisson]). *For $\zeta> 4$ and sufficiently small constant $\xi>0$, assume that $(\log\log N)^\zeta\leq d\leq (\frac{1}{\log 4-1}-(\log N)^{-\xi})\log N$. For $K:=d^{1/2-2/\zeta-16\xi}$ there are some $\delta, \epsilon>0$ such that the first $K$ eigenvalues are of the form $$\label{eq:adk} \frac{\alpha_x}{\sqrt{\alpha_x-\frac{\beta_x}{2\alpha_x}(\frac{\alpha_x}{d}+\frac{\beta_x}{\alpha_x d})+\frac{\beta_x}{2\alpha_x}\sqrt{(\frac{\alpha_x}{d}+\frac{\beta_x}{d\alpha_x})^2-4\frac{\alpha_x}{d}}}}+O(d^{-\epsilon})$$ for $K$ vertices of degree at least $\mathfrak{u}-\frac{d^{-\delta}}{\log \mathfrak{u}}$.* This is done by, given $\alpha_x$ and $\beta_x$, making an educated guess for the structure of the eigenvector. To use this approximate eigenvector it is crucial that the statistics of the local neighborhoods of high degree vertices are concentrated, in particular that the degrees of the vertices in the neighborhood is reasonably close to $d$, which is approximately their expected value. According to [\[eq:adk\]](#eq:adk){reference-type="eqref" reference="eq:adk"}, to be able to properly estimate eigenvalues using this strategy it is crucial that $d \gg 1$. ## Main results Finding similar behavior when $d$ is constant has been an open question asked by Guionnet [@guionnet2021bernoulli]. In this paper, we use new techniques to show that in fact, the same behavior holds in the constant $d$ regime, and continues until the average degree is subconstant. Namely, we prove the following. **Theorem 2**. *Consider a $G\sim \mathcal{G}(N,\frac dN)$ graph with $\log^{-1/9}N\leq d\leq \log^{1/40}N$. With high probability, for each of the $e^{\log ^{1/8}N}$ largest eigenvalues $\lambda$ of the adjacency matrix, there is some vertex $x$ such that $$\lambda=\sqrt{\alpha_x+\frac{\beta_x}{\alpha_x}+\frac{d^2+d}{\alpha_x}}+O((d^{3/2}+1)\mathfrak{u}^{-11/6}).$$ Moreover, for $k\leq e^{\log^{1/8}N}$, the vertex $x$ corresponding to the $k$th largest eigenvalue is the $k$th vertex in the lexicographic ordering $(\alpha_x, \beta_x)$.* As we will show, these high degree vertices are spaced throughout the graph, and have almost independent local statistics. Therefore, similar to [@alt2023poisson], the distribution of the highest eigenvalues is described by a Poisson point process with density given by the probability of existence of an $(\alpha,\beta)$ pair, where $\alpha$ is close to maximal among all vertices. To this end, define the discrete intensity measure $\rho:\mathbb{R}\rightarrow \mathbb{R}$, $$\rho(\frac{s}{\mathfrak{u}}):= N\sum_{\ell=0}^{2\log^{1/8}N}\left(\frac{e^{-d}d^{\mathfrak{u}-\ell}}{(\mathfrak{u}-\ell)!}\frac{e^{-d(\mathfrak{u}-\ell)}(d(\mathfrak{u}-\ell))^{(s-\mathfrak{u}+\ell)(\mathfrak{u}-\ell)}}{ ((s-\mathfrak{u}+\ell)(\mathfrak{u}-\ell))!} \mathbf{1}_{\langle s(u-\ell)\rangle=0}\mathbf{1}_{(s-u+\ell)(u-\ell)\leq d(\mathfrak{u}-\ell)+\mathfrak{u}^{7/8}}\right).$$ where $\mathbf{1}_{\langle x\rangle =0}$ is the indicator that $x$ is a whole number. $\rho$ induces a Poisson point process $\Psi$. The meaning of $\rho$ is that it is the intensity measure of $\alpha+\frac{\beta}{\alpha}$ if $\alpha\sim Pois(d)$ and $\beta\sim Pois(d\alpha)$, restricted to $\alpha\in[\mathfrak{u}-2\log^{1/8}N,\mathfrak{u}],\beta\in[0,d\alpha+\mathfrak{u}^{7/8}]$. As we will see, Theorem [Theorem 2](#thm:maineigenvalue){reference-type="ref" reference="thm:maineigenvalue"} implies $\Psi$ approximates the density of $\lambda_x^2$ at the edge of the spectrum. Formally, we we will consider proximity in Lévy-Prokhorov distance, which is a metrization of the weak topology. Namely we define, for two Borel measures $\nu_1,\nu_2$ on $\mathbb{R}$, $$D(\nu_1,\nu_2)=\inf\{\epsilon>0|\forall A\in \mathfrak B,\nu_1(A)\leq \nu_2(A_\epsilon)+\epsilon\textnormal{ and }\nu_2(A)\leq \nu_1(A_\epsilon)+\epsilon\}$$ where $\mathfrak B$ is the set of Borel measurable sets in $\mathbb{R}$ and $A_\epsilon$ is the neighborhood of radius $\epsilon$ around $A$. Moreover, for $K>0$, define $\kappa(K)$ as $$\label{eq:kappadef} \kappa(K)=\inf\{s\in\mathbb{R},\rho([s,\infty))\leq K\}.$$ **Theorem 3**. *Set $K=e^{\log^{1/8}N}$, and recall the definition of $\kappa(K)$ from [\[eq:kappadef\]](#eq:kappadef){reference-type="eqref" reference="eq:kappadef"}. Consider the density function $$\Phi:=\sum_{\lambda\in spec(A)}\delta_{\mathfrak{u}(\lambda^2-\frac{d^2+d}{\mathfrak{u}})}.$$ Then for $d$ as defined in Theorem [Theorem 2](#thm:maineigenvalue){reference-type="ref" reference="thm:maineigenvalue"}, $$\lim_{N\rightarrow\infty}D(\Phi\mathbf{1}_{[\kappa(K),\infty)},\Psi\mathbf{1}_{[\kappa(K),\infty)})\rightarrow0.$$* We multiplied both point processes by $\mathfrak{u}$ to emphasize that the approximation of $\lambda^2$ in Theorem [Theorem 2](#thm:maineigenvalue){reference-type="ref" reference="thm:maineigenvalue"} is $o(1/\mathfrak{u})$. As shown in Theorem [Theorem 2](#thm:maineigenvalue){reference-type="ref" reference="thm:maineigenvalue"}, the largest eigenvalues are approximately the square root of a rational function on local statistics, therefore it is simpler to consider the point process of $\lambda^2$, as it leads to a nicer expression. Much of our analysis will be on the squared eigenvalue $\lambda^2$. Theorem [Theorem 3](#thm:process){reference-type="ref" reference="thm:process"} implies the fluctuations of the top eigenvalue. Similar to if there is increasing degree as in [@alt2023poisson], fluctuations are determined at two scales. If the expected number of vertices of degree $\mathfrak{u}$ (that is $\mu_\mathfrak{u}$) is constant, then the maximum degree has nonzero variance and will dominate the fluctuation of the maximum eigenvalue. Therefore, in this case the top eigenvalue will fluctuate at a large scale, as its first order fluctuation is a shifted Bernoulli, based on whether there is a vertex of degree $\mathfrak{u}$ or not. If the expected number of vertices of degree $\mathfrak{u}$ is subconstant or superconstant, then the largest degree of the graph becomes deterministic, and the fluctuations are determined by $\beta$. As $\beta$ is then distributed as a Poisson, the fluctuations become much smaller and become those of the maximum of Poissons. The fact that these eigenvalues are almost completely determined by local neighborhoods is intrinsically linked to the fact that eigenvectors are Anderson localized (see [@anderson1958absence]). That is, they decay exponentially around a fixed vertex. We show the following, which implies eigenvectors are close to as localized as possible. **Theorem 4**. *Define $B_r(x)$ to be the ball of radius $r$ around $x$ in $G$. For $c\leq 1/15$, consider $\log^{-c}N\leq d\leq \log^{1/40}N$, and $K_2=\log^{o_N(1)} N$. Moreover, we fix $r'\geq 1$, and if $c>0$, we add the requirement that $r'\leq 1/(3c)$. With high probability, the eigenvectors $\mathbf{v}$ corresponding to the $K_2$ largest eigenvalues are *exponentially localized* in the sense that for each $\mathbf{v}$ there is some vertex $x$ such that $$\mathbf{v}|_{x}=\frac1{\sqrt 2}+O((1+d^{-1/2})\mathfrak{u}^{-1/3})$$ and for $1\leq i\leq r'$, $$\|\mathbf{v}|_{S_i(x)}\| =\left(\frac{d}{\alpha}\right)^{(i-1)/2}\frac1{\sqrt 2}(1+O((1+d^{-1/2}+d^{-i+1})\mathfrak{u}^{-1/3}))$$ and $$\|\mathbf{v}|_{[N]\backslash B_i(x)}\|=\left(\frac{d}{\alpha}\right)^{i/2}\frac{1}{\sqrt 2}(1+O((1+d^{-1/2}+d^{-r'+1})\mathfrak{u}^{-1/3})).$$* Our desire to keep our result as general as possible has resulted in this long expression for our error. There are multiple error terms and for $d\asymp 1$, the ratio of $d,i,$ and $\mathfrak{u}$ governs which type of error will dominate. We consider a significantly smaller number of eigenvalues in this theorem than in Theorem [Theorem 2](#thm:maineigenvalue){reference-type="ref" reference="thm:maineigenvalue"} as in order to show eigenvector localization, we must quantify the gaps of the eigenvalues induced by Theorem [Theorem 2](#thm:maineigenvalue){reference-type="ref" reference="thm:maineigenvalue"} and Theorem [Theorem 3](#thm:process){reference-type="ref" reference="thm:process"}, whereas the previous theorems do not require such gaps. We do this only for the region given above, as our focus is the spectral edge, but most likely further analysis could extend this to further eigenvalues. ## Idea of Proof We follow the general framework of the proof of [@alt2023poisson]. Our overall goal is to show that the largest eigenvalues are determined by the local geometry of the highest degree vertices. Once we show this, the Poisson point process and eigenvector structure follow from the randomness of the graph. We show this determination by classifying vertices by degree, and associating an eigenvector with each high degree vertex. The main issue with generalizing the classification in [@alt2023poisson] to constant degree is that in this regime, the fluctuations of statistics of the balls surrounding individual entries become too large to guess the eigenvector based on the inputs $\alpha_x,\beta_x$ only. Such a formula already is quite technical, so we avoid directly coming up with a more involved equation that gives a more accurate guess. Instead of making a guess, *we use the true top eigenvector of the ball of radius $r$ around the high degree vertex*. The advantage of such a method is that the only error in the eigenvector equation comes from truncating at level $r$. Therefore, if we can show the eigenvector is localized away from level $r$, then the error from this truncation can be drastically smaller. The disadvantage is that, considering we are not making a fixed guess, we initially have no information about the eigenvector or eigenvalue, even whether it is localized or not. The neighborhoods of the highest degree vertices are typically tree-like, and the central vertex has much higher degree than all others. Therefore, by analyzing the eigenvector equation at each vertex, we create a system of linear equations for the eigenvector and eigenvalue. Because the neighborhood is a tree, this simplifies into a recursive equation for the eigenvalue. Moreover, because the central vertex has much higher degree, we can show this equation can be truncated up to small error with a rational function of local statistics, giving near exact dependence. In summary, as opposed to guessing a specific approximate eigenvector, we show that with high probability whatever eigenvector *does* occur can give the approximation in Theorem [Theorem 2](#thm:maineigenvalue){reference-type="ref" reference="thm:maineigenvalue"}. When we fully write out the equation for the eigenvalue, the first two terms are $\alpha_x$ and $\frac{\beta_x}{\alpha_x}$, which are used in [@alt2023poisson] to completely determine the eigenvector. We show, explicitly giving the next few terms, that the equation for the eigenvalue beyond $\alpha,\beta$ concentrates. Specifically, although the fluctuation of statistics increase as degree decreases, the dependence on the fluctuating statistics decreases at a quicker rate. In fact, in our regime, the eigenvalue decays quickly enough that it implies the lexicographic ordering of Theorem [Theorem 2](#thm:maineigenvalue){reference-type="ref" reference="thm:maineigenvalue"}. Given the dependence on statistics of the eigenvalue and eigenvectors of local neighborhoods, we translate this into statements about the overall graph. By standard perturbation theoretic arguments, this reduces to showing these local statistics are well separated for vertices corresponding to the edge of the spectrum. The requisite statistics are binomially distributed, which are approximately Poisson. Therefore it becomes useful to give precise tail bounds on the Poisson distribution. Tao recently gave such a tight, two-sided bound on his blog [@tao2022improved], which is sufficient to show that $(\alpha_x,\beta_x)$ that are close to lexicographically maximizing are well separated. Once we have proper control over these high degree vertices, we need to control the contribution of the rest of the spectrum. To do this, for vertices of still somewhat large degree, we proceed as per [@alt2023poisson] and take a localized test vector supported on a pruned graph, where edges are pruned in such a way that high degree vertices are separated and neighborhoods are tree-like. For all other vertices, we can use the decomposition of Krivelevich and Sudakov, who show that after deleting a subgraph of disjoint stars surrounding high degree vertices, the spectral radius is far from $\sqrt{\mathfrak{u}}$ [@krivelevich2003largest]. This makes it much simpler to bound the spectral radius of the operator away from the largest eigenvectors of the highest degree vertices, for which we also use that the second . ## Related Work There are many results concerning the bulk of the spectrum of random matrices. Focusing specifically on sparse Erdős-Rényi graphs, Khorunzhy, Shcherbina, and Vengerovsky, then Zakharevich, analyzed the moments of the limiting distribution of Wigner matrices of general models that include constant degree Erdős-Rényi graphs in order to study the limiting measure of the spectral distribution [@khorunzhy2004eigenvalue; @zakharevich2006generalization]. Benaych-Georges, Guionnet, and Male give a central limit theorem for linear statistics of a model that includes constant degree Erdős-Renyi graphs [@benaych2014central]. Along with these bulk results, Bhaswar Bhattacharya and Ganguly, then Bhaswar Bhattacharya, Sohom Bhattacharya, and Ganguly, give a large deviation principle for the edge eigenvalues of very sparse Erdős-Rényi graphs [@bhattacharya2020upper; @bhattacharya2021spectral]. These random matrices have been studied as a model for quantum physics, specifically Hamiltonians of disordered systems. We see similar eigenvector localization in the edge of the spectrum in the Anderson model (see [@anderson1958absence]), where vertices on an integer lattice are given random potential, and we study the spectrum of the resulting Schrödinger operator. Eigenvectors near the edge of the spectrum are known to be localized for various models (e.g. [@goldsheid1977random; @frohlich1983absence; @aizenman1993localization; @ding2020localization]) whereas there has been less progress on the structure of eigenvectors in the bulk. Lévy matrices, a model of Wigner matrices where entries are sampled from distributions with heavy tails, have also proved to be a useful model for studying eigenvector localization. Focusing specifically on results concerning the edge of the spectrum, there is a transition from delocalized eigenvectors in the bulk to localized eigenvectors at the edge [@bordenave2013localization; @bordenave2017delocalization]. Moreover, similar to the sparse Erdős-Rényi model, eigenvalues near the edge of the spectrum in sparse Lévy models are known to converge to a Poisson point process [@soshnikov2004poisson; @auffinger2009poisson]. ## Negative eigenvalues The discussion above concerns the largest eigenvalues of the adjacency matrix, however, by the exact same analysis we can consider the most negative eigenvalues. Under the high probability assumption that the neighborhood of every high degree vertex is a tree, by the bipartite nature of a tree, every positive eigenvalue of a neighborhood of a tree has a corresponding negative eigenvalue that is of the same magnitude and has the same localization properties. Therefore, Theorems [Theorem 2](#thm:maineigenvalue){reference-type="ref" reference="thm:maineigenvalue"}, [Theorem 3](#thm:process){reference-type="ref" reference="thm:process"}, and [Theorem 4](#thm:maineigenvector){reference-type="ref" reference="thm:maineigenvector"} all apply to the most negative eigenvalues as well. ## Extension of results We believe that by increasing the analysis from our given set of local statistics to higher moments, our methods can be used to give even more accurate formulae for the largest eigenvalues based on the degree sequence of the highest degree vertices. Such an argument could show separation of the largest $\log^aN$ eigenvalues for any fixed $a\geq 0$, giving a more specific (and more complicated) Poisson point process and showing eigenvector localization for all these $\log^aN$ eigenvectors. However, for simplicity of the argument, in this work we only consider $K=\log^{o_N(1)}N$. Using this same argument, we may be able to improve the necessary lower bound on $d$. Some estimates and concentration results required us to lower bound $d$ by $\log^{-c} N$ for some $c < 1$, and error bounds simplify given our concrete assumption on $d$, but there are also important structural consideration for smaller $d$. If $d=\log^{-c}N$ for $c>0$, then the connected components surrounding high degree vertices can be of small radius, which means that neighborhoods of high degree vertices could be identical, leading to the eigenvalues of those neighborhoods being identical, and thus the eigenvectors would no longer be localized around one high-degree vertex. This implies that we cannot remove the dependence of $d$ on $r$ in Theorem [Theorem 4](#thm:maineigenvector){reference-type="ref" reference="thm:maineigenvector"}. On the other hand, including more terms in our expression for $\lambda_x$ could improve the lower bound on $d$ for Theorem [Theorem 2](#thm:maineigenvalue){reference-type="ref" reference="thm:maineigenvalue"} and Theorem [Theorem 3](#thm:process){reference-type="ref" reference="thm:process"}. However, in our opinion we will need new methods to achieve the threshold of $d=e^{-(\log\log N)^2}$ appearing in [@krivelevich2003largest] and [@bhattacharya2021spectral]. Such a threshold is natural as for $d\leq e^{-(\log\log N)^2}$, all connected components are of size $(1+o(1))\mathfrak{u}$, making localization and independence trivial. Similarly, by using slightly tighter bounds on the probabilities of some tail events, we expect we can improve the upper bound on $d$ in Theorem [Theorem 2](#thm:maineigenvalue){reference-type="ref" reference="thm:maineigenvalue"}. However, there is a natural barrier at $\log^{1/2}N$, both in the exponential rate of decay of eigenvectors, and the application of the method in [@krivelevich2003largest]. We are not motivated to optimize our technique for the upper bound considering results are already known for larger $d$ from [@alt2023poisson]. ## Overview of the paper In Section [2](#sec:prelim){reference-type="ref" reference="sec:prelim"}, we overview the notation and give some preliminary theorems we will use throughout. In Section [3](#sec:regimes){reference-type="ref" reference="sec:regimes"}, we define the different vertex degree regimes, where vertices are classified by degree. Our goal will be to be to give an explicit unitary transform $U$ such that $UAU^{*}$ has a block decomposition that is close to diagonal. Each diagonal block corresponding to high degree vertices will correspond to a space spanned by vectors localized around these vertices. This is similar to the decomposition from[@alt2023poisson], and it also bears resemblance to the more directly combinatorial decompositions of other sparse matrix results [@krivelevich2003largest; @bhattacharya2021spectral]. The majority of the rest of the proof is dedicated to analyzing each of these regimes. In Section [4](#sec:fine){reference-type="ref" reference="sec:fine"}, we analyze the eigenvector and eigenvalue of the largest eigenvalue of the ball of radius $r$ surrounding the highest degree vertices, and we show that it is localized and well approximated by a formula involving only $\alpha, \beta,$ and $d$. In Section [5](#sec:rough){reference-type="ref" reference="sec:rough"}, we use a test vector to show that vectors corresponding to vertices of high, but not too high degree, do not have large eigenvalues. In Section [6](#sec:finaldecomp){reference-type="ref" reference="sec:finaldecomp"}, we analyze the bulk using the decomposition from [@krivelevich2003largest], and we show the given block decomposition is such that the highest degree vertices dominate. This gives Theorem [Theorem 2](#thm:maineigenvalue){reference-type="ref" reference="thm:maineigenvalue"}. In Section [7](#sec:anticoncentration){reference-type="ref" reference="sec:anticoncentration"}, we use this formula to prove the Poisson process, and in Section [8](#sec:eigenvector){reference-type="ref" reference="sec:eigenvector"} we show this implies Theorem [Theorem 4](#thm:maineigenvector){reference-type="ref" reference="thm:maineigenvector"}. ## Acknowledgements {#acknowledgements .unnumbered} We thank Shirshendu Ganguly for introducing us to this question and for many helpful discussions. # Preliminaries {#sec:prelim} In this section we introduce the notation we will use, define the exact parameters we will use, as well as relevant results about Erdős-Rényi graphs for the regime we are interested in. We then introduce distributional results we will use, in particular about the Binomial and Poisson distributions, and comparisons between them. Finally we state some spectral properties of graphs that we will use repeatedly. ## Notation We consider a graph $G=(V,E)$ sampled from the Erdős-Rényi distribution $\mathcal{G}(N,\frac dN)$ with adjacency matrix $A_G$. When $G$ is clear from the context, we write this as $A$. We consider our vertex set to be $[N]$. Each of the possible $\binom N2$ edges is included independently with probability $\frac dN$. In this paper, we work in the following regime. **Definition 5** (Choice of parameters). *We fix $c>0$ and we take the average degree $d(N)$ as any function such that for Theorem [Theorem 2](#thm:maineigenvalue){reference-type="ref" reference="thm:maineigenvalue"} and Theorem [Theorem 3](#thm:process){reference-type="ref" reference="thm:process"}, $\log^{-1/9}N\leq d\leq \log^{1/40} N$, and for Theorem [Theorem 4](#thm:maineigenvector){reference-type="ref" reference="thm:maineigenvector"}, $\log^{-c}N\leq d\leq \log^{1/40} N$ for $c\leq 1/15$.* *Our analysis will be based on considering balls of radius $r$ around the highest degree vertices. Most results are true for sufficiently large $r$, but in fact, it is enough to take $r = 5$ in order to prove Theorems [Theorem 2](#thm:maineigenvalue){reference-type="ref" reference="thm:maineigenvalue"} and [Theorem 3](#thm:process){reference-type="ref" reference="thm:process"}. For Theorem [Theorem 4](#thm:maineigenvector){reference-type="ref" reference="thm:maineigenvector"}, we need a slightly larger radius. So for the rest of this paper it is sufficient to take $r:=\max \left \{ 5, 2r' \right \}$, where $r'$ is the parameter from Theorem [Theorem 4](#thm:maineigenvector){reference-type="ref" reference="thm:maineigenvector"}.* For $i\geq 0$, we denote by $B_i(x)$ the ball of radius $i$ around $x$, rooted at $x$. Moreover, we define $S_i(x)$ to be the set of vertices $y$ such that the shortest path from $x$ to $y$ is of length $i$. In other words $S_i(x)$ are all vertices that are not in $B_{i-1}(x),$ and that are connected to $x$ by a path of length $i$. We also call $S_i(x)$ the sphere of radius $i$ around $x$. Given a root vertex $x$, we define a partial ordering on vertices by writing $u \leq v$ if there is a shortest path from $x$ to $v$ that goes through $u$. We also write for $y\in [N]$, $$\label{eq:Ndef} N_y:=\left|\{u\in [N]:u\geq y, u\sim y\}\right|$$ as the number of children of $y$ in the rooted graph. Similarly, for a rooted or unrooted graph, for a vertex $y\in [N]$ we define $\Gamma_y=\{z\in[N]:z\sim y\}$ to be the neighborhood of $y$. The following parameters will be used to approximate the largest eigenvalues. **Definition 6**. *Our parameters are defined as follows.* 1. *$\alpha_x:=|\Gamma_x|$, the degree of the vertex $x$* 2. *$\beta_x:=\sum_{y\sim x}N_y$, the number of vertices in $S_2(x)$,* 3. *$\beta^{(1,1)}_x=\sum_{y_2\in S_2(x)}N_{y_2}$, the number of vertices in $S_3(x)$, and* 4. *$\beta^{(2)}_x$=$\sum_{y\sim x}N_y^2$.* As we will see, the eigenvalue is mostly determined by these four statistics. In our regime, the last two statistics (as well as all others) are well concentrated enough that we can write an accurate enough formula for the eigenvalue based on only $\alpha,\beta$. The combinatorial aspects of an Erdős-Rényi graph are governed by binomial distributions. Therefore, by $\textnormal{Bin}(k;N,p)$ we denote the probability that a binomial random variable with $N$ trials and success probability $p$ is equal to $k$. Recall the definition of $\mathfrak{u}$ from [\[eq:udef\]](#eq:udef){reference-type="eqref" reference="eq:udef"}. As $d$ is small, the maximum degree is almost deterministic. **Lemma 7** ([@bollobas01randomgraphs], Theorem 3.7). *Define $\mu_k=N\textnormal{Bin}(k;N-1,\frac dN)$ to be the expected number of vertices of degree $k$ in the graph. Define $\mathfrak{u}$ to be the positive integer value of $k$ that minimizes $\max\{\mu_k,\mu_k^{-1} \}$. Then if $d=o(\log N)$, with high probability the maximum degree is in $\{\mathfrak{u}-1,\mathfrak{u}\}$.* In order to calculate $\mathfrak{u}$, note that by the Stirling approximation, having $\mu_k\approx 1$ implies $$(1+o_N(1))\log N-\mathfrak{u}\log \mathfrak{u}+ \mathfrak{u}-d+\mathfrak{u}\log d-\frac12\log(2\pi \mathfrak{u})=0.$$ Therefore, in our regime of $d$, $$\label{eq:uapprox} \mathfrak{u}=(1+o_N(1))\frac{\log N}{\log\log N-\log d}.$$ and $\mathfrak{u}= \Theta \left (\frac{\log N}{\log\log N} \right )$. Throughout, $\mathbf{1}_X, \mathbf{1}(X)$ denote the indicator on the event $X$ occurring. For a vector $v \in \mathbb{R}^2$ we denote by $\| v \|$ its Euclidean norm. For a matrix $A$ in $\mathbb{R}^{m \times n}$, we denote by $\| A \|$ the operator norm, i.e. $\| A \| = \sup_{v \in \mathbb{R}^n} \frac{ \| A v \|_2}{\| v\|_2}.$ ## Distributional notation and comparisons We will estimate our binomial distributions with Poisson distributions, as the large $N$ limit of a binomial is a Poisson when $p$ is small. The following approximations are standard, with the proofs given in the appendix in Section [9](#sec:estimates){reference-type="ref" reference="sec:estimates"}. **Lemma 8**. *If $X\sim Binom(n,p)$ and $Y\sim Pois(np)$, and if $k,np\leq \sqrt n$, then $$\mathbb{P}(X=k)= \left (1+O \left (\frac{k^2+(np)^2+1}{n} \right ) \right )\mathbb{P}(Y=k).$$* This implies that the tails are also the same up to a small error. **Corollary 9**. *If $X\sim Binom(n,p)$ and $Y\sim Pois(np)$, and if $k\leq \sqrt n$, $np\leq n^{1/2-c}$ for some fixed constant $c$, then $$\mathbb{P}(X\geq k)= \left (1+O \left (\frac{k^2+(np)^2+1}{n} \right ) \right ) \mathbb{P}(Y\geq k)+O\left(\left(e p\sqrt{n}\right)^{\sqrt n}\right).$$* Having established these comparisons, we use very tight bounds on the Poisson tail. Tao gives such a tight bound on his blog [@tao2022improved], where he notes that forms of this bound are given previously [@glynn1987upper; @talagrand1995concentration]. In the post, Tao gives the proof of the upper bound and leaves the proof of the lower bound to the reader. We prove both sides in the appendix in Section [9](#sec:estimates){reference-type="ref" reference="sec:estimates"}. **Lemma 10**. *For $X \sim \text{Pois}(\lambda)$ and $\delta\geq \frac1{\sqrt{\lambda}}$, for sufficiently large $\lambda$ $$\mathbb{P}\big ( X \geq \lambda ( 1 + \delta ) \big ) \leq \frac{ e^{ -\lambda h ( \delta ) } }{ \sqrt{\lambda\min\{\delta, \delta^2\} } },$$ where $h(\delta) = (\delta+1) \log ( \delta + 1 ) - \delta$.* *Moreover, if $\lambda(1+\delta)$ is an integer, then there is a universal constant $c_{\scaleto{\ref{lem:sharp_poisson_tail}}{3pt}}$ such that for sufficiently large $\lambda$ $$\mathbb{P}\big ( X \geq \lambda ( 1 + \delta ) \big ) \geq c_{\scaleto{\ref{lem:sharp_poisson_tail}}{3pt}}\frac{ e^{ -\lambda h ( \delta ) } }{ \sqrt{\lambda\min\{\delta, \delta^2\} } }.$$* The integrality assumption is necessary as for very large $\delta$, the difference in probability between $\mathbb{P}( X \geq \lambda (1+\delta) )$ and $\mathbb{P}(X\geq\lambda(1+\delta)+1)$ is large enough that for very small $c>0$, this two sided bound could not possibly hold for $\lceil\lambda(1+\delta)\rceil$ and $\lfloor\lambda(1+\delta)\rfloor+c$ simultaneously. For its use in our paper, the integrality assumption is irrelevant, as we will only need the lower bound in the small $\delta$ regime. **Corollary 11**. *For $X\sim Pois(\lambda)$, if $\lambda\delta^3=o_\lambda(1)$ and $\delta\geq \frac1{\sqrt{\lambda}}$, then $$\label{eq:tightpoisson} (1 - o_\lambda(1)) c_{\scaleto{\ref{lem:sharp_poisson_tail}}{3pt}} \frac{ e^{ -\frac{\lambda\delta^2}{2}} }{\delta \sqrt{ \lambda } } \leq \mathbb{P}\big ( X \geq \lambda ( 1 + \delta ) \big ) \leq (1+o_\lambda(1))\frac{ e^{ -\frac{\lambda\delta^2}{2}} }{ \delta\sqrt{ \lambda } }.$$* *Proof.* For the upper bound, $$\mathbb{P}\big ( X \geq \lambda ( 1 + \delta ) \big ) \leq \frac{ e^{ -\lambda h ( \delta ) } }{ \sqrt{\lambda\min\{\delta, {\delta}^2\} } }\leq (1+o_\lambda(1))\frac{ e^{ -\frac{\lambda\delta^2}{2}} }{ \delta\sqrt{ \lambda } }$$ by the Taylor expansion of $h(\delta)$. For the lower bound, define $\delta'=\delta+\frac1\lambda$. Then $\lambda(1+\delta)+1=\lambda(1+\delta')$. We have $$\mathbb{P}\big ( X \geq \lambda ( 1 + \delta ) \big ) = \mathbb{P}\big ( X \geq \lceil\lambda ( 1 + \delta )\rceil \big ) \geq c_{\scaleto{\ref{lem:sharp_poisson_tail}}{3pt}}\frac{ e^{ -\lambda h ( \delta' ) } }{ \sqrt{\lambda\min\{\delta', {\delta'}^2\} } }\geq (1-o_\lambda(1))c_{\scaleto{\ref{lem:sharp_poisson_tail}}{3pt}}\frac{ e^{ -\frac{\lambda{\delta}^2}{2}} }{ \delta\sqrt{ \lambda } }.$$ ◻ Because we will need these bounds repeatedly we also state the following estimate for the tails of binomials, which follows from Corollary [Corollary 9](#cor:bintopoistail){reference-type="ref" reference="cor:bintopoistail"} and Lemma [Lemma 10](#lem:sharp_poisson_tail){reference-type="ref" reference="lem:sharp_poisson_tail"}. **Corollary 12**. *If $X \sim \textnormal{Binom}(n,p)$, $np = o( \sqrt{n})$ and $\tau = o(\sqrt{n})$ then $$\mathbb{P}\left ( X \geq \tau \right ) \leq (1+o(1)) e^{- \tau \log \tau + \tau \log( np ) - \tau + np}.$$* We will use a weaker, simpler version of this bound for low probability events. **Lemma 13** ([@janson2011random] Theorem 2.1). *If $X\sim Binom(N,p)$, and $\lambda=Np$, then for $t\geq 0$ $$\mathbb{P}(X-\lambda\geq t)\leq \exp\left(-\frac{t^2}{2\lambda+2t/3}\right),~~ \mathbb{P}(X-\lambda\leq -t)\leq \exp\left(-\frac{t^2}{2\lambda}\right).$$[\[lem:weakertail\]]{#lem:weakertail label="lem:weakertail"}* Because we are dealing with sparser matrices than [@alt2023poisson] and need more precise estimates, we need to bound the sum of *squares* of the degrees of neighbors of high degree vertices. Therefore we require estimates for the sum of distributions with heavy Weibull tails. Such a bound follows from [\[lem:weakertail\]](#lem:weakertail){reference-type="ref" reference="lem:weakertail"} and the tail results in [@bakhshizadeh2023sharp]. Justification for this generalization is given in the appendix in Section [9](#sec:estimates){reference-type="ref" reference="sec:estimates"}. **Lemma 14**. *For any $n>0$ and $d=o(n^{1/3})$, consider $n$ independent i.i.d. samples $X_1,\ldots X_n\sim \textnormal{Binom}(N,\frac dN)$. There is some constant $c_{\scaleto{\ref{lem:weibullbound}}{3pt}}>0$ such that for any $t >n^{2/3}$, $$\mathbb{P}\left ( \left |\sum_{i=1}^n X_i^2-\mathbb{E}\left [ \sum_{i=1}^n X_i^2 \right ]\right |>t \right )\leq 2n\exp\left(-\frac{c_{\scaleto{\ref{lem:weibullbound}}{3pt}}}{d^3+1}\sqrt{t}\right).$$ Moreover, if $t>2(d^2+1)n^{2/3}$, $$\mathbb{P}\left ( \left |\sum_{i=1}^n X_i^2-\mathbb{E}\left [ \sum_{i=1}^n X_i^2 \right ]\right |>t \right )\leq 2n\exp\left(-c_{\scaleto{\ref{lem:weibullbound}}{3pt}}\sqrt{t}\right).$$* ## Spectral properties of graphs We use the following spectral bound for trees. **Lemma 15**. *[@kesten1959symmetric] [\[lem:forest-bound\]]{#lem:forest-bound label="lem:forest-bound"} If $T$ is a forest with maximum degree bounded by $\Delta$, then $\lambda_{\max}(A_T)\le2\sqrt{\Delta-1}$.* We use the following result to quantify the proximity of true eigenvalues and eigenvectors to approximate ones. **Lemma 16** (See [@alt2023poisson] Lemma E.1). *Consider a self-adjoint matrix $M$ and $\Delta, \epsilon>0$ satisfying $5\epsilon\leq \Delta$. For $\lambda\in \mathbb{R}$, assume $M$ has a unique eigenvalue $\mu$ in the interval $[\lambda-\Delta,\lambda+\Delta]$ with eigenvector $\mathbf{w}$. If there is a normalized vector $\mathbf{v}$ such that $\|(M-\lambda)\mathbf{v}\|\leq \epsilon$, then $$\mu-\lambda=\langle \mathbf{v},(M-\lambda)\mathbf{v}\rangle+O\left (\frac{\epsilon^2}{\Delta}\right ),\|\mathbf{w}-\mathbf{v}\|=O\left (\frac{\epsilon}{\Delta}\right)$$* In particular this lemma will be used in the following form: **Lemma 17**. *For $i\geq 2$, let $A$ be the adjacency matrix of the ball of radius $i$ around a vertex $x$ of degree $\alpha$, such that $B_i(x)$ is a tree, $\frac{|S_2(x)|}{\alpha} \leq s(n) \ll \alpha$ and the degree of each vertex in $B_i(x)\setminus{\{x\}}$ is at most $t(n) \leq \frac{\alpha}{5}$.* *Then the maximum eigenvalue $\mu$ of $A$ satisfies $$\mu = \sqrt{\alpha} + O \left ( \frac{s(n) }{ \sqrt{\alpha} } \right ).$$* *Proof.* We take as our test vector $\mathbf{w}$ the eigenvector corresponding to the star graph consisting of the central vertex $x$ and its neighbors. Thus $\mathbf{w}|_x = \frac1{\sqrt 2}$ and for $y \sim x$, $\mathbf{w}|_y = \frac1{\sqrt{2 \alpha}}$. Since $B_i(x)$ is a tree, each $z \in S_2(x)$ has exactly one neighbor in $S_1(x)$, so $(A\mathbf{w})|_z=\frac{1}{\sqrt{2\alpha}}$. Moreover the number of non-zero entries in $A \mathbf{w}- \sqrt{\alpha} \mathbf{w}$ is $|S_2(x)| \leq \alpha s(n)$. The above implies that $$\| A \mathbf{w}- \sqrt{\alpha} \mathbf{w}\| \leq \sqrt { \alpha s(n) \frac{1}{2 \alpha }} = \sqrt{ \frac{s(n)}{2} },$$ which corresponds to $\varepsilon$ in Lemma [Lemma 16](#lem:estimator){reference-type="ref" reference="lem:estimator"}. To utilize Lemma [Lemma 16](#lem:estimator){reference-type="ref" reference="lem:estimator"} we require $\Delta$ such that $A$ has a unique eigenvalue in $[\lambda - \Delta, \lambda + \Delta].$ For this we use eigenvalue interlacing: after deleting the row and column of $A$ corresponding to $x$, the matrix is the adjacency matrix of a forest with degree at most $t(n)$. By the spectral radius of a tree from Lemma [\[lem:forest-bound\]](#lem:forest-bound){reference-type="ref" reference="lem:forest-bound"}, the maximum eigenvalue of this submatrix is at most $2 \sqrt{ t(n) }.$ Thus $A$ has at most one eigenvalue in the interval $\left [ 2.1 \sqrt{ t(n) }, 2 \sqrt{\alpha} - 2.1 \sqrt{ t(n) } \right ]$, namely, if any, its maximum eigenvalue. Thus we can take $\Delta = \sqrt{\alpha} - 2.1 \sqrt{ t(n) } \geq \left ( 1 - 2.1/\sqrt{5} \right ) \sqrt{\alpha}$ in Lemma [Lemma 16](#lem:estimator){reference-type="ref" reference="lem:estimator"}. The estimates on the errors now simply follow from plugging in our values for $\varepsilon$ and $\Delta$, since $\varepsilon= \sqrt{\frac{s(n)}{2}} \ll \Delta$ by assumption. ◻ # Regimes {#sec:regimes} In a first step we will analyze the spectral contribution of high degree vertices, which we separate into three regimes. After defining these regimes we analyze their sizes. Finally we state a theorem about the approximate diagonalization of the adjacency matrix $A$, that we will use to prove some of our main theorems. **Definition 18**. *We define the following sets of high degree vertices in our graph.* *We define the sets for $m\geq 0$ $$\mathcal{X}_m := \left \{x \in [N]: \alpha_x \geq \mathfrak{u}- m \right \}.$$ This then gives the regimes* 1. *The *fine regime*: $\mathcal{W} := \mathcal{X}_{\mathfrak{u}^{1/4}}$.* 2. *The *intermediate regime*: $\mathcal{V} :=\mathcal{X}_{\mathfrak{u}^{2/3}}$.* 3. *The *rough regime*: $\mathcal{U} :=\mathcal{X}_{\mathfrak{u}/2}$.* The precise thresholds are not significant and rather of technical nature, our analysis would continue to work if we replaced these thresholds with $\mathfrak{u}-\mathfrak{u}^{c_1}, \mathfrak{u}-\mathfrak{u}^{c_2}, c_3 \mathfrak{u}$ for some constants satisfying $0<c_1<1/2, 1/2<c_2<1$ and $0<c_3<1$. For large $N$, we have $\mathcal{W}\subset \mathcal{V}\subset \mathcal{U}$. First we bound the sizes of these sets. The upper bounds will let us perform union bounds, whereas the lower bound on $|\mathcal{X}_m|$ tells us that all of our highest degree vertices have almost the same degree. **Lemma 19**. *For $m\geq 0$, with probability $1-O((\frac{\mathfrak{u}}{d})^{-(m+1/2)})$ $$\begin{aligned} \label{eq:sizeupper} |\mathcal{X}_m| \leq \frac32\left(\frac{\mathfrak{u}}{d}\right)^{m+1/2}.\end{aligned}$$* *Moreover, for $1\leq m \leq \mathfrak{u}^{c}$ with $c<1/2$, with probability $1-O \left (\left(\frac{\mathfrak{u}}{d}\right)^{-(m-1/2)}\right )$, $$\begin{aligned} \label{eq:sizeslower} \ |\mathcal{X}_m|\geq \frac12\left(\frac{\mathfrak{u}}{d}\right)^{m-1/2}. \end{aligned}$$* *Proof.* By Lemma 3.11 in [@bollobas01randomgraphs], we know that $\textnormal{Var}(|\mathcal{X}_m|) = O(\mathbb{E}[|\mathcal{X}_m|] )$. By a Chebyshev inequality, we have that with probability $1-O(\frac{1}{\mathbb{E}[|\mathcal{X}_m|]})$,$$\label{eq:bollobasvar} \frac{1}{2}\mathbb{E}[|\mathcal{X}_m|]\leq |\mathcal{X}_m|\leq \frac32\mathbb{E}[|\mathcal{X}_m|].$$ Therefore, it is sufficient to show that $\mathbb{E}[|\mathcal{X}_m|]$ and $\mathbb{E}[|\mathcal{U}|]$ satisfy the above bounds, and that each is $\omega_N(1)$. Recall that $\mu_k := N \mathbb{P}( \text{ a vertex is of degree $k$ }) = N \textnormal{Bin}(k;N-1,p)$. Thus for any $k \in \mathbb{N}$, $$\label{eq:expratio} \frac{\mu_{k+1}}{\mu_k} = \frac{\textnormal{Bin}(k+1;N-1,p)}{\textnormal{Bin}(k;N-1,p)} = \frac{(N-k-1)p}{(k+1)(1-p)}.$$ By the definition of $\mathfrak{u}$, and the fact that $\mu_k$ monotonically decreases in $k$, $$\begin{aligned} \mu_{\mathfrak{u}}^{-1} \leq \mu_{\mathfrak{u}-1}&=& \frac{N\mathfrak{u}(1-\frac dN)}{d(N-\mathfrak{u})}\mu_{\mathfrak{u}},\\ \mu_{\mathfrak{u}} \leq \mu_{\mathfrak{u}+1}^{-1}&=& \frac{N(\mathfrak{u}+1)(1-\frac dN)}{d(N-\mathfrak{u}-1)}\mu_{\mathfrak{u}}^{-1}\end{aligned}$$ Therefore $$(1-o_N(1))\sqrt{\frac{d}{\mathfrak{u}}}\leq \mu_\mathfrak{u}\leq (1+o_N(1))\sqrt{\frac{\mathfrak{u}}{d}}.$$ We have by [\[eq:expratio\]](#eq:expratio){reference-type="eqref" reference="eq:expratio"}, for $m \leq \mathfrak{u},$ $$\begin{aligned} \mu_{\mathfrak{u}-m } & = \mu_\mathfrak{u}\left ( \frac{N-d}{d} \right )^{m } \prod_{ i = 1}^{m } \frac{\mathfrak{u}-i+1}{N-\mathfrak{u}+i-1} \\ &= (1+o_N(1))\mu_\mathfrak{u}\frac{\mathfrak{u}^m}{d^m}\prod_{i=1}^m(1-\frac{i-1}{\mathfrak{u}}). \end{aligned}$$ An upper bound on this is $(1+o_N(1))(\frac{\mathfrak{u}}{d})^{m+1/2}$. Summing over all $0\leq n\leq m$ gives [\[eq:sizeupper\]](#eq:sizeupper){reference-type="eqref" reference="eq:sizeupper"}. Assuming that $m\leq \mathfrak{u}^c$ for $c<1/2$, $$(1+o_N(1))\mu_\mathfrak{u}\frac{\mathfrak{u}^m}{d^m}\prod_{i=1}^m(1-\frac{i-1}{\mathfrak{u}})\geq (1+o_N(1))\mu_\mathfrak{u}\frac{\mathfrak{u}^m}{d^m} e^{-m^2/\mathfrak{u}}\geq (1+o_N(1))\mu_\mathfrak{u}\frac{\mathfrak{u}^m}{d^m},$$ giving [\[eq:sizeslower\]](#eq:sizeslower){reference-type="eqref" reference="eq:sizeslower"}. ◻ **Corollary 20**. *For our regimes this implies that with high probability $$\begin{aligned} | \mathcal{W}| \ll e^{u^c} \text{ for any } c > \frac{1}{4} \\ | \mathcal{V}| \ll e^{u^c} \text{ for any } c > \frac{2}{3} \\ | \mathcal{U}| \ll N^c \text{ for any } c > \frac{1}{2} \end{aligned}$$* **Remark 21**. *Note that in the proof we derive bounds on the expected values of the sizes of these sets, which immediately imply bounds on the probability that a given vertex falls into one of the sets, since for any set $\mathcal{T}$, $\mathbb{E}[|\mathcal{T}|] = N \mathbb{P}( \text{ vertex } 1 \in \mathcal{T}).$* Given this, we can introduce a structure theorem, subdividing $A$ according to the different regimes. We will decompose our matrix using a unitary transform $U$ made clear later. Here $D_\mathcal{W}, D_{\mathcal{V}\backslash \mathcal{W}},D_{\mathcal{U}\backslash \mathcal{V}}$ are diagonal operators associated with the balls surrounding vertices in $\mathcal{U}$. A summary of the results concerning this decomposition is as follows. **Theorem 22**. *With high probability, there is a unitary transformation $U:\mathbb{R}^{N}\rightarrow\mathbb{R}^N$ such that we can write $$\label{eq:blockdecomp} A=U\left(\begin{array}{cccc} D_{\mathcal{W}}&0&0&E_{\mathcal{W}}^*\\ 0& D_{\mathcal{V}\backslash \mathcal{W}}&0&E_{\mathcal{V}\backslash \mathcal{W}}^*\\ 0&0& D_{\mathcal{U}\backslash \mathcal{V}}+\mathcal{E}_{\mathcal{U}\backslash \mathcal{V}}&E_{\mathcal{U}\backslash \mathcal{V}}^*\\ E_{\mathcal{W}}& E_{\mathcal{V}\backslash \mathcal{W}}&E_{\mathcal{U}\backslash \mathcal{V}}&\mathcal{X} \end{array}\right)U^{*}$$ where [\[eq:blockdecomp\]](#eq:blockdecomp){reference-type="eqref" reference="eq:blockdecomp"} satisfies the following* 1. *The *fine regime* operator $D_{\mathcal{W}}$ is diagonal, of dimension $2|\mathcal{W}|$, and has at least $e^{\log^{1/8}N}$ eigenvalues of value at least $\sqrt{\mathfrak{u}}-O(\mathfrak{u}^{-3/8})$.* 2. *The *intermediate regime* operator $D_{\mathcal{V}\backslash \mathcal{W}}$ is diagonal, of dimension $2|\mathcal{V}\backslash \mathcal{W}|$, and $\|D_{\mathcal{V}\backslash \mathcal{W}}\|\leq \sqrt{\mathfrak{u}}-\Omega(\mathfrak{u}^{-1/4})$.* 3. *The *rough regime* operator $D_{\mathcal{U}\backslash \mathcal{V}}$ is diagonal and of dimension $2|\mathcal{U}\backslash \mathcal{V}|$ and satisfies $\|D_{\mathcal{U}\backslash \mathcal{V}}\|\leq \sqrt{\mathfrak{u}}-\mathfrak{u}^{1/6-o_N(1)}$.* 4. *The *bulk* operator $\mathcal{X}$ satisfies $\|\mathcal{X}\|=(\frac{1}{\sqrt{2}}+o_N(1))\sqrt{\mathfrak{u}}$.* 5. *The error terms satisfy $\|E_\mathcal{W}\|,\|E_{\mathcal{V}\backslash \mathcal{W}}\|=O( (d^{r}+1)\mathfrak{u}^{-r/2+1})$, $\|\mathcal{E}_{\mathcal{U}\backslash \mathcal{V}}\|+\|E_{\mathcal{U}\backslash \mathcal{V}}\|=O((\log\log N)^2)$.* These results, along with results concerning the structure of the eigenvectors associated with the operator surrounding vertices of $\mathcal{W}$, imply that the edge eigenvectors and eigenvalues come from the operator associated with $D_\mathcal{W}$. This in turn will give the theorems from Section [1](#sec:intro){reference-type="ref" reference="sec:intro"}. Much of the remainder of the paper is dedicated to showing this decomposition. # Fine and Intermediate Regime {#sec:fine} In this section we start by proving structural results about the balls around vertices in $\mathcal{V}$, which we then use to derive a recursion for the largest eigenvalue and eigenvector of the balls around the vertices in that regime. We also derive a first approximation of the largest eigenvalue for all vertices in $\mathcal{V}$. In part [4.2](#sec:fine_eigenvector){reference-type="ref" reference="sec:fine_eigenvector"} we then use these ingredients to prove the exponential decay of this eigenvector for all vertices in $\mathcal{V}$. In the last part [4.3](#sec:fine_eigenvalue){reference-type="ref" reference="sec:fine_eigenvalue"} we then derive a more precise expression for the eigenvalue of vertices in $\mathcal{W}$, based on concentration results from part [4.1](#sec:fine_general){reference-type="ref" reference="sec:fine_general"}. ## General Structure {#sec:fine_general} In this section, we create a test vector and test eigenvalue for the neighborhoods of vertices in $\mathcal{V}$. In order to do this, we first define an event under which the balls around vertices in $\mathcal{V}$ have a nice structure. While the eigenvalue and eigenvector will be defined using $B_r(x)$, the following structural results are for slightly larger balls so that we can also bound the error coming from truncating the balls and guarantee that there is no intersection with balls of radius 3 around vertices from $\mathcal{U}.$ **Definition 23**. *Define $\Omega_{\scaleto{\ref{dfn:omegadef}}{3pt}}$ to be the event that the following are true.* 1. *For all $x \neq y\in \mathcal{V}$, $B_{r+3}(x)\cap B_{r+3}(y)=\emptyset$.* 2. *For all $x\in \mathcal{V}$, $B_{r+3}(x)$ is a tree.* 3. *For $1\leq i\leq r$ and every vertex $x\in \mathcal{V}$, $$\left||S_{i}(x)|-d^{i-1}\alpha_x\right|= O (d^{i - 3/2} + 1) \mathfrak{u}^{\frac{7}{8}}$$* *Moreover, for every vertex $x\in \mathcal{W}$, $$\left||S_{i}(x)|-d^{i-1}\alpha_x\right|\leq O (d^{i - 3/2} + 1) \mathfrak{u}^{\frac23}.$$* 4. *For $x\in \mathcal{V}$, every $y\in B_{r+3}(x) \setminus \{ x \},$ satisfies $N_y\leq \mathfrak{u}^{3/4}$.* *Moreover, for $x\in \mathcal{W}$, every $y\in B_{r+3}(x)\setminus \{ x \}$ satisfies $N_y\leq \mathfrak{u}^{1/3}$.* 5. *For every $x\in \mathcal{V}$, $$\left|\sum_{y\in S_1(x)} N_y^2 -(d^2+d)\alpha_x\right|\leq O \left ( \mathfrak{u}^{3/2} \right ).$$ Moreover, for every $x\in \mathcal{W}$, $$\left|\sum_{y\in S_1(x)} N_y^2-(d^2+d)\alpha_x\right|\leq O \left ( \mathfrak{u}^{2/3} \right ).$$* Note that statement 3 in [Definition 23](#dfn:omegadef){reference-type="ref" reference="dfn:omegadef"} implies that $|S_2(x) | \leq 2 d \alpha_x$ for our regime of $d$. If $d$ was smaller than $\log^{-c} N,$ for some $c > \frac{1}{4},$ this would no longer be true. Because of the sparsity of the graph and concentration of independent binomials, we can show that the event $\Omega_{\scaleto{\ref{dfn:omegadef}}{3pt}}$ almost always occurs. Considering statements similar to most of these bounds have appeared previously, in, e.g. [@alt2023poisson], we defer the proof of the following lemma to the appendix in Section [10](#sec:structure){reference-type="ref" reference="sec:structure"}. **Lemma 24**. *The event $\Omega_{\scaleto{\ref{dfn:omegadef}}{3pt}}$ occurs with high probability.* We also know from Lemma [Lemma 7](#lem:max_degree){reference-type="ref" reference="lem:max_degree"}, that with high probability the maximum degree is bounded by $\mathfrak{u}$. Therefore, we will consistently use these structural properties. **Assumption 25**. *For the rest of Section [4](#sec:fine){reference-type="ref" reference="sec:fine"}, we assume that $\Omega_{\scaleto{\ref{dfn:omegadef}}{3pt}}$ occurs. Moreover we assume the high probability event that the maximum degree in $G$ is in $\{ \mathfrak{u}-1, \mathfrak{u}\}$.* Under this event we know enough about the structure of the neighborhoods around vertices in the intermediate regime, to analyze its top eigenvalue and eigenvector, which we use as the test eigenvector and eigenvalue. **Definition 26**. *For $x\in \mathcal{V}$, we define $\lambda_x$ to be the top eigenvalue of $A_{B_r(x)}$. Moreover, define $\mathbf{w}_{+}(x)$ to be the eigenvector corresponding to $\lambda_x$ and $\mathbf{w}_{-}(x)$ to be the eigenvector of the most negative eigenvalue of $A_{B_r(x)}$. We use the notation $\mathbf{w}_{\pm}(x)$, when a statement is true for both $\mathbf{w}_{+}(x)$ and $\mathbf{w}_{-}(x)$. Depending on the context, we will also use $\mathbf{w}_{\pm}(x)$ to denote the above eigenvector padded with $0$'s to make it a vector in $\mathbb{R}^{N}$.* Under $\Omega_{\scaleto{\ref{dfn:omegadef}}{3pt}}$, $B_r(x)$ is a tree, therefore it is bipartite. Therefore the most negative eigenvalue is $-\lambda_x$, and if $y\in B_r(x)$, then $$\mathbf{w}_{-}(x)|_y=(-1)^{d(x,y)}\mathbf{w}_{+}(x)|_y.$$ Moreover, as $B_r(x)$ is a tree, its eigenvectors and eigenvalues satisfy a nice recursion. Consider an eigenvector $\mathbf{w}$ of $A_{B_r(x)}$ with eigenvalue $\lambda$. We consider the eigenvector equation at $x$: $$\lambda \mathbf{w}|_x=\sum_{y\sim x} \mathbf{w}|_{y}$$ which, if $\mathbf{w}|_x\neq 0$, can be rewritten as $$\label{eq:rootvertex} \lambda=\sum_{y\sim x} \frac{\mathbf{w}|_{y}}{ \mathbf{w}|_x}.$$ More generally, we have for $v\sim u, v\geq u$, i.e. for $v$ a child of $u$ in the tree rooted at $x$, if $\mathbf{w}|_v\neq 0$, $$\label{eq:eigenvector} \lambda \mathbf{w}|_v = \mathbf{w}|_u + \sum_{y_1\sim v, y_1\geq v} \mathbf{w}|_{y_1} \Rightarrow \mathbf{w}|_v=\frac1{\lambda-\sum_{y_1\sim v, y_1\geq v}\frac{\mathbf{w}|_{y_1}}{\mathbf{w}|_v}}\mathbf{w}|_u.$$ Plugging in [\[eq:eigenvector\]](#eq:eigenvector){reference-type="eqref" reference="eq:eigenvector"} into [\[eq:rootvertex\]](#eq:rootvertex){reference-type="eqref" reference="eq:rootvertex"} for every $y_1\sim x$ gives $$\lambda=\sum_{y_1\sim x} \frac{\frac1{\lambda-\sum_{y_2\sim y_1, y_2\geq y}\frac{\mathbf{w}|_{y_2}}{\mathbf{w}|_{y_1}}}\mathbf{w}|_x}{ \mathbf{w}|_x}=\sum_{y_1\sim x} \frac1{\lambda-\sum_{y_2\sim y_1, y_2\geq y_1}\frac{\mathbf{w}|_{y_2}}{\mathbf{w}|_{y_1}}}.$$ Repeating this process for all vertices gives that $$\label{eq:eigdef} \lambda=\sum_{y_1\sim x}\frac{1}{\lambda-\sum_{y_2\sim y_1, y_2\geq y_1} \frac{1}{\lambda-\sum_{y_3\sim y_2, y_3\geq y_2} \frac{1}{\lambda-\sum_{y_4\sim y_3, y_4\geq y_3} \cdots }}},$$ where the right hand side is a continued fraction of at most $r$ levels. Since $A_{B_r(x)}$ is a connected graph, its adjacency matrix is irreducible and this implies by the Perron-Frobenius theorem, that the top eigenvector $\mathbf{w}|_{+}(x)$ of $B_{r}(x)$ is the only positive eigenvector, implying in particular that [\[eq:eigdef\]](#eq:eigdef){reference-type="eqref" reference="eq:eigdef"} does not contain any 0 denominators. This means that we can use [\[eq:eigdef\]](#eq:eigdef){reference-type="eqref" reference="eq:eigdef"} for our definition of $\lambda_x$. To further examine this, we require an initial two sided bound, based on $B_r(x)$ being close to a star graph. This will be enough to bound the contribution of balls around vertices in $\mathcal{V}\setminus \mathcal{W}$, and for vertices in $\mathcal{W}$, we will eventually bootstrap it into a tighter bound in Lemma [\[lem:lambdaexpression\]](#lem:lambdaexpression){reference-type="ref" reference="lem:lambdaexpression"}. **Lemma 27**. *For any vertex $x\in \mathcal{V}$,* *$$\alpha_x\leq \lambda^2_x \leq \alpha_x +O(d).$$* *Proof.* As the spectral radius of a star is the square root of the degree of the central vertex, $\lambda^2_x\geq \alpha_x$. For the upper bound, we apply Lemma [Lemma 17](#lem:basic_eigenvalue_approximation){reference-type="ref" reference="lem:basic_eigenvalue_approximation"}. By the definition of $\Omega_{\scaleto{\ref{dfn:omegadef}}{3pt}}$, (3) we can take $s(n) = 2 d$, and by the definition of $\Omega_{\scaleto{\ref{dfn:omegadef}}{3pt}}$, (4) we can take $t(n) = \mathfrak{u}^{3/4}$. Lemma [Lemma 17](#lem:basic_eigenvalue_approximation){reference-type="ref" reference="lem:basic_eigenvalue_approximation"} thus implies that $\lambda_x = \sqrt{\alpha_x} + O \left (\frac{d}{\sqrt{\alpha_x}} \right )$ implying that $\lambda_x^2 = \alpha_x + O(d).$ ◻ ## Eigenvector structure {#sec:fine_eigenvector} We now prove that the entries of the eigenvector decay exponentially with the distance from the central vertex. For easier readability, we will suppress $x$ in the notation for the rest of the section, so we write $\alpha:=\alpha_x$, $\lambda:=\lambda_x$, $\mathbf{w}_{\pm}=\mathbf{w}_{\pm}(x),$ and $S_1 = S_i(x).$ Moreover we define $N_{y^*}=\max_{y\in B_{r}(x)}N_y$. **Lemma 28**. *If $x\in \mathcal{V}$, then for $u,v\in B_r(x)$ such that $u \sim v$ and $u \leq v$, $$\mathbf{w}_+|_u= \left (1+O\left (\mathfrak{u}^{-1/4}\right ) \right )\lambda\mathbf{w}_+|_v$$ and if $x\in \mathcal{W}$, then for $u\in B_r(x)$, $$\mathbf{w}_+|_u=\left (1+O \left (\mathfrak{u}^{-2/3} \right ) \right )\lambda\mathbf{w}_+|_v.$$* *Proof.* We will show that for a vertex $u$ such that $u\sim v$, $u\leq v$, it holds that $$\label{eq:rougheigequation} \mathbf{w}_+|_u= \left (1+O \left ( \frac{N_{y^*}}{\lambda^2} \right ) \right ) \lambda\mathbf{w}_+|_v.$$ Together with Lemma [Lemma 27](#lem:eigenvalueapproximation){reference-type="ref" reference="lem:eigenvalueapproximation"} and Lemma [Definition 23](#dfn:omegadef){reference-type="ref" reference="dfn:omegadef"}, this implies the result. By the eigenvector equation [\[eq:eigenvector\]](#eq:eigenvector){reference-type="eqref" reference="eq:eigenvector"}, we must have $\mathbf{w}_+|_u\leq \lambda\mathbf{w}_+|_v$. For a lower bound on $\mathbf{w}_+|_u$, we proceed by induction on the distance from $x$, starting from the leaves in $B_r(x)$ (note that these are not necessarily leaves in $G$). Any leaf $v$ only has one neighbor $u$, making this base case trivial, as there is only one neighbor and $\mathbf{w}_+|_u=\lambda \mathbf{w}_+|_v.$ Now, assume [\[eq:rougheigequation\]](#eq:rougheigequation){reference-type="eqref" reference="eq:rougheigequation"} is true for all $z\geq u$. Then, applying [\[eq:eigenvector\]](#eq:eigenvector){reference-type="eqref" reference="eq:eigenvector"} to $v$, we get $$\begin{aligned} \nonumber\mathbf{w}_+|_u&=&\left({\lambda-\sum_{y\sim v, y\geq v}\frac{\mathbf{w}_+|_y}{\mathbf{w}_+|_v}}\right) \mathbf{w}_+|_v \\ \nonumber &\geq& \left(\lambda-\frac{N_{y^*}}{\lambda-O \left (\frac{N_{y^*}}{\lambda} \right )}\right)\mathbf{w}_+|_v\\ &\geq&\left(1-O \left (\frac{N_{y^*}}{\lambda^2} \right ) \right)\lambda\mathbf{w}_+|_v,\label{eq:eigenlocal}\end{aligned}$$ where we used the inductive hypothesis to get the first inequality and then used that by Lemma [Lemma 24](#lem:fine_balls_disjoint){reference-type="ref" reference="lem:fine_balls_disjoint"} the rooted trees around $\lambda$ satisfy $N_{y*}\ll \lambda^2$. ◻ Such a tight bound implies exponential decay on various levels. We can now bound the error from approximating these eigenvectors using the truncation. **Proposition 29**. *We define $$\Lambda:=\sum_{x\in \mathcal{V}} \bigg(\lambda_x\mathbf{w}_{+}(x)\mathbf{w}_{+}(x)^*-\lambda_x\mathbf{w}_{-}(x)\mathbf{w}_{-}(x)^*\bigg).$$ Then $$\max_{x\in \mathcal{V}, \sigma\in\{\pm1\}}\|(A-\Lambda)\mathbf{w}_{\sigma}(x)\|=O( (d^{r/2}+1)\mathfrak{u}^{-(r-1)/2}).$$* *Proof.* For any $x\in \mathcal{V}, \sigma\in \{\pm1\}$, $\mathbf{w}_\sigma(x)$ satisfies $(A-\Lambda)\mathbf{w}_{\sigma}(x)=(A-A_{B_r(x)})\mathbf{w}_{\sigma}(x)$. The only nonzero entries of this vector are supported on $S_{r+1}(x)$. This holds because the only rows of $(A-A_{B_r(x)})$ that have non-zero entries corresponding to $B_r(x)$ are vertices in $S_{r+1}(x)$, since $B_{r+1}(x)$ are disjoint trees by $\Omega_{\scaleto{\ref{dfn:omegadef}}{3pt}}$. By Lemma [Lemma 27](#lem:eigenvalueapproximation){reference-type="ref" reference="lem:eigenvalueapproximation"} and Lemma [Lemma 28](#lem:vertchange){reference-type="ref" reference="lem:vertchange"}, each entry in $\mathbf{w}_\sigma(x)$ corresponding to vertices in $S_{r}(x)$ has value at most $(1+o_N(1))\mathfrak{u}^{-r/2}$. Moreover, under $\Omega_{\scaleto{\ref{dfn:omegadef}}{3pt}}$, the number of vertices in the $r+1$ level is $d^{r}\alpha_x+O(d^{r - 1/2} + 1) \mathfrak{u}^{\frac{7}{8}} \leq O( (d^r + 1) \mathfrak{u})$. Therefore, $\|(A-\Lambda)\mathbf{w}_{\sigma(x)}\| = O( (d^{r/2}+1) \mathfrak{u}^{-r/2+1/2})$. ◻ This error term gives us sufficient information about $\mathcal{V}\backslash \mathcal{W}$, and we can now focus only on the fine regime $\mathcal{W}$. The following proposition gives bounds on the total mass the eigenvector assigns to each sphere. **Proposition 30**. *For all $x\in \mathcal{W}$, the eigenvector $\mathbf{w}_+$ satisfies $$\label{eq:centerbound} \mathbf{w}_+|_x=\frac1{\sqrt 2}+O \left ((1+d^{-1/2})\mathfrak{u}^{-1/3} \right ).$$ and for $1\leq i \leq r$, $$\label{eq:annulusbound} \|\mathbf{w}_+|_{S_i}\| =\left(\frac d\alpha\right)^{(i-1)/2}\frac1{\sqrt 2} \left (1+O \left ((1+d^{-1/2}+d^{-i+1})\mathfrak{u}^{-1/3} \right ) \right )$$ and $$\label{eq:complementnorm} \|\mathbf{w}_+|_{[N]\backslash B_{i}}\|= \sqrt{\frac{1}{1-\frac d\alpha}}\left(\frac{d}{\alpha}\right)^{i/2}\frac{1}{\sqrt 2}(1+O((1+d^{-1/2}+d^{-r+1})\mathfrak{u}^{-1/3})).$$* *Proof.* In this proof we repeatedly use the approximation of $\lambda$ from Lemma [Lemma 27](#lem:eigenvalueapproximation){reference-type="ref" reference="lem:eigenvalueapproximation"} in order to replace $\lambda$ by $\alpha^\frac{1}{2}$ or vice-versa up to some small multiplicative error. First note that Lemma [Lemma 28](#lem:vertchange){reference-type="ref" reference="lem:vertchange"} implies that for each $v\in S_{i}$, $$\mathbf{w}_+|_{v}=(1+O(\mathfrak{u}^{-2/3}))\lambda^{-i}(\mathbf{w}_+|_{x}).$$ Therefore, as we know by Lemma [Lemma 24](#lem:fine_balls_disjoint){reference-type="ref" reference="lem:fine_balls_disjoint"} that $|S_i|=d^{i-1}\alpha+O((d^{i-3/2}+1)\mathfrak{u}^{2/3})$, $$\begin{aligned} \|\mathbf{w}_+|_{S_i}\|&=& d^{(i-1)/2}\alpha^{1/2}(1+O((d^{-1/2}+d^{-i+1})\mathfrak{u}^{-1/3})\lambda^{-i}\mathbf{w}_+|_{x} \label{eq:firstexpo}.\end{aligned}$$ Note that for this approximation to hold and for the error term to go to 0, we require $d\geq \mathfrak{u}^{-\frac{1}{3r}}$ The complicated error term is necessary as different terms could maximize for different regimes of $d$. Specifically, this means that for the normalized vector $\mathbf{w}_+$, $$\begin{aligned} (\mathbf{w}_+|_x)^2+\|\mathbf{w}_+|_{S_1}\|^2&\geq& 1-O\left(\frac d{\mathfrak{u}}(1+(d^{-1/2}+d^{-1})\mathfrak{u}^{-1/3})\right)\\ &\geq& 1-O\left(\frac d\mathfrak{u}\right).\end{aligned}$$ Using this together with [\[eq:firstexpo\]](#eq:firstexpo){reference-type="eqref" reference="eq:firstexpo"} for $\mathbf{w}_+|_{S_1}$ and solving for $\mathbf{w}_+|_x$ gives $$\begin{aligned} (\mathbf{w}_+|_x)^2=\frac12+O((1+d^{-1/2})\mathfrak{u}^{-1/3}+d\mathfrak{u}^{-1})\\ \mathbf{w}_+|_x=\frac1{\sqrt 2}+O((1+d^{-1/2})\mathfrak{u}^{-1/3}).\end{aligned}$$ Combining [\[eq:firstexpo\]](#eq:firstexpo){reference-type="eqref" reference="eq:firstexpo"} and [\[eq:centerbound\]](#eq:centerbound){reference-type="eqref" reference="eq:centerbound"}, $$\begin{aligned} \|\mathbf{w}_+|_{S_i}\|&=& d^{(i-1)/2}\alpha^{1/2}(1+O((d^{-1/2}+d^{-i+1})\mathfrak{u}^{-1/3}))\lambda^{-i}(\frac1{\sqrt 2}+O((1+d^{-1/2})\mathfrak{u}^{-1/3}))\\ &=&d^{(i-1)/2}\alpha^{-(i-1)/2}\frac1{\sqrt 2}(1+O((1+d^{-1/2}+d^{-i+1})\mathfrak{u}^{-1/3})).\end{aligned}$$ Similarly, for [\[eq:complementnorm\]](#eq:complementnorm){reference-type="eqref" reference="eq:complementnorm"}, we have by [\[eq:firstexpo\]](#eq:firstexpo){reference-type="eqref" reference="eq:firstexpo"}, $$\begin{aligned} \|\mathbf{w}_+|_{[N]\backslash B_{i}}\|^2&=&\sum_{j = i+1}^r \|\mathbf{w}_+|_{S_j}\|^2\\ &=&\frac{1}{1-\frac{d}{\alpha}}d^{i}\alpha^{-i}\frac1{2}(1+O((1+d^{-1/2}+d^{-r+1})\mathfrak{u}^{-1/3})).\end{aligned}$$ ◻ Note that Proposition [Proposition 30](#prop:expdecay){reference-type="ref" reference="prop:expdecay"} implies that almost all mass of the vector $\mathbf{w}_+$ is on $x$ and $S_1.$ ## Eigenvalue structure {#sec:fine_eigenvalue} Along with the eigenvector, we further analyze the eigenvalue. To do this, we expand [\[eq:eigdef\]](#eq:eigdef){reference-type="eqref" reference="eq:eigdef"} as an infinite sum which we get by repeatedly using the expansion $\frac{1}{\lambda-q}=\sum_{k=0}^\infty \frac{q^k}{\lambda^{k+1}}$. We will bound the eigenvalue $\lambda$ through the moments of the degree sequence surrounding $x\in \mathcal{W}$. We recall the definitions of $\beta^{(2)}$ and $\beta^{(1,1)}$ from Definition [Definition 6](#dfn:betadef){reference-type="ref" reference="dfn:betadef"}. Note that $\beta^{(1,1)} = |S_3|$ and $\beta^{(2)} = \sum_{y \sim x} N_y^2.$ **Lemma 31**. *With high probability, for any $x\in \mathcal{W}$, $$\label{eq:lambdasquaredfinal} \lambda^2=\alpha+\lambda^{-2}\beta+\lambda^{-4}(\beta^{(2)}+\beta^{(1,1)})+O((d^2+d)\mathfrak{u}^{-5/3}).$$* *Proof.* We can rewrite [\[eq:eigdef\]](#eq:eigdef){reference-type="eqref" reference="eq:eigdef"} as $$\lambda^2=\sum_{y_1\sim x}\frac{1}{1-\frac{1}{\lambda^2}\sum_{y_2\sim y_1, y_2\geq y_1} \frac{1}{1-\frac{1}{\lambda^2}\sum_{y_3\sim y_2, y_3\geq y_2} \frac{1}{1-\frac{1}{\lambda^2}\sum_{y_4\sim y_3, y_4\geq y_3} \cdots }}}.$$ and expand this as $$\lambda^2=\sum_{y_1\sim x}\sum_{k_1=0}^{\infty} \left(\frac{1}{\lambda^2}\sum_{y_2\sim y_1, y_2\geq y_1} \frac{1}{1-\frac{1}{\lambda^2}\sum_{y_3\sim y_2, y_3\geq y_2} \frac{1}{1-\frac{1}{\lambda^2}\sum_{y_4\sim y_3, y_4\geq y_3} \cdots }}\right)^{k_1}.$$ and once again as $$\label{eq:lambdaexpanded} \lambda^2=\sum_{y_1\sim x}\sum_{k_1=0}^{\infty} \left(\frac{1}{\lambda^2}\sum_{y_2\sim y_1, y_2\geq y_1} \sum_{k_2=0}^\infty \left(\frac{1}{\lambda^2}\sum_{y_3\sim y_2, y_3\geq y_2} \frac{1}{1-\frac{1}{\lambda^2}\sum_{y_4\sim y_3, y_4\geq y_3} \cdots }\right)^{k_2}\right)^{k_1}.$$ Of course we could repeat this process, but this is sufficient accuracy for our purposes. We do the same analysis as before, starting at the innermost level, corresponding to the leaves, and then inducting our way back up the tree. For any vertex $v$, we can write a recursive equation by defining $$F(v):=\frac{1}{1-\frac1{\lambda^2}\sum_{y\sim v, y\geq v} F(y)}$$ which gives that $$\lambda^2=\sum_{y_1\sim x}\sum_{k_1=0}^{\infty} \left(\frac{1}{\lambda^2}\sum_{y_2\sim y_1, y_2\geq y_1} \sum_{k_2=0}^\infty \left(\frac{1}{\lambda^2}\sum_{y_3\sim y_2, y_3\geq y_2} F(y_3)\right)^{k_2}\right)^{k_1}.$$ We now estimate $F_{v}$ for $v\in B_{r}(x)$. For any leaf $v$, as there are no $y\geq v$, $F(v)=1$. For the rest, we use induction. Recall that $N_y^*$ is the maximum degree in $B_r(x) \setminus \{ x \},$ and that $N_{y^*} \ll \lambda^2$ under the event $\Omega_{\scaleto{\ref{dfn:omegadef}}{3pt}}$. Assume that for all $y\sim v, y \geq v$, $F(y)=1+O \left (\frac{N_{y^*}}{\lambda^2} \right )$. Then $$F(v)=\frac{1}{1-\frac1{\lambda^2}\sum_{y\sim v, y\geq v}F(y)} \leq \frac{1}{1-\frac{N_{y^*}}{\lambda^2} \left (1+O \left (\frac{N_{y^*}}{\lambda^2} \right ) \right )} =1+O \left (\frac{N_{y^*}}{\lambda^2} \right ) .$$ Therefore [\[eq:lambdaexpanded\]](#eq:lambdaexpanded){reference-type="eqref" reference="eq:lambdaexpanded"} becomes $$\lambda^2=\sum_{y_1\sim x}\sum_{k_1=0}^{\infty} \left(\frac{1}{\lambda^2}\sum_{y_2\sim y_1, y_2\geq y_1} \sum_{k_2=0}^\infty \left(\frac{1}{\lambda^2}\sum_{y_3\sim y_2, y_3\geq y_2} 1+O \left (\frac{N_{y^*}}{\lambda^2} \right ) \right)^{k_2}\right)^{k_1}.$$ We want to show this expansion relies only on the terms in Definition [Definition 6](#dfn:betadef){reference-type="ref" reference="dfn:betadef"}. Therefore we rewrite the first few terms as $$\begin{aligned} \alpha&=&\sum_{y_1\sim x}1\\ \lambda^{-2}\beta&=&\sum_{y_1\sim x}\left(\frac{1}{\lambda^2}\sum_{y_2\sim y_1, y_2\geq y_1} \left(\frac{1}{\lambda^2}\sum_{y_3\sim y_2, y_3\geq y_2} 1+O \left (\frac{N_{y^*}}{\lambda^2} \right )\right)^{0}\right)^1\\ \lambda^{-4}\beta^{(1,1)}&=&\sum_{y_1\sim x}\left(\frac{1}{\lambda^2}\sum_{y_2\sim y_1, y_2\geq y_1}^\infty \left(\frac{1}{\lambda^2}\sum_{y_3\sim y_2, y_3\geq y_2} 1 \right)^{1}\right)^1.\end{aligned}$$ The contribution of $\beta^{(2)}$ is more complicated, but as $\beta^{(2)}=\sum_{y_1\sim x} N_y^2$, we can write $$\lambda^{-4}\beta^{(2)}=\sum_{y_1\sim x}\left(\frac{1}{\lambda^2}\sum_{y_2\sim y_1, y_2\geq y_1} \left(\frac{1}{\lambda^2}\sum_{y_3\sim y_2, y_3\geq y_2} 1+O \left (\frac{N_{y^*}}{\lambda^2} \right )\right)^{0}\right)^2$$ meaning we have $$\begin{gathered} \sum_{y_1\sim x}\left(\frac{1}{\lambda^2}\sum_{y_2\sim y_1, y_2\geq y_1} \sum_{k_2=0}^\infty \left(\frac{1}{\lambda^2}\sum_{y_3\sim y_2, y_3\geq y_2} 1+O \left (\frac{N_{y^*}}{\lambda^2} \right )\right)^{k_2}\right)^{2}-\lambda^{-4}\beta^{(2)} \\ =\sum_{y_1\sim x}\left( \frac{1}{\lambda^2} \sum_{y_2\sim y_1, y_2\geq y_1} \sum_{k_2=1}^\infty \left(\frac{1}{\lambda^2}\sum_{y_3\sim y_2, y_3\geq y_2} 1+O \left (\frac{N_{y^*}}{\lambda^2} \right ) \right)^{k_2}\right)^{2} \\ +2\lambda^{-4}\sum_{y_1\sim x}N_y\sum_{y_2\sim y_1, y_2\geq y_1} \sum_{k_2=1}^\infty \left(\frac{1}{\lambda^2}\sum_{y_3\sim y_2, y_3\geq y_2} 1+O \left (\frac{N_{y^*}}{\lambda^2} \right ) \right)^{k_2}.\end{gathered}$$ Therefore subtracting these terms from [\[eq:lambdaexpanded\]](#eq:lambdaexpanded){reference-type="eqref" reference="eq:lambdaexpanded"} gives $$\begin{aligned} \nonumber\lambda^2-\alpha-\lambda^{-2}\beta-\lambda^{-4}\beta^{(2)}-\lambda^{-4}\beta^{(1,1)}= & & \sum_{y_1\sim x}\sum_{k_1=3}^{\infty} \left(\frac{1}{\lambda^2}\sum_{y_2\sim y_1, y_2\geq y_1} \sum_{k_2=0}^\infty \left(\frac{1}{\lambda^2}\sum_{y_3\sim y_2, y_3\geq y_2} 1+O \left (\frac{N_{y^*}}{\lambda^2} \right ) \right)^{k_2}\right)^{k_1}\\ \nonumber&+&\sum_{y_1\sim x}\frac{1}{\lambda^2}\sum_{y_2\sim y_1, y_2\geq y_1} \sum_{k_2=2}^\infty \left(\frac{1}{\lambda^2}\sum_{y_3\sim y_2, y_3\geq y_2} 1+O \left (\frac{N_{y^*}}{\lambda^2} \right ) \right)^{k_2}\\ \nonumber&+&\sum_{y_1\sim x}\frac{1}{\lambda^2}\sum_{y_2\sim y_1, y_2\geq y_1} \frac{1}{\lambda^2} \sum_{y_3\sim y_2, y_3\geq y_2} O \left (\frac{N_{y^*}}{\lambda^2} \right ) \\ \nonumber&+&\sum_{y_1\sim x}\left(\frac{1}{\lambda^2}\sum_{y_2\sim y_1, y_2\geq y_1} \sum_{k_2=1}^\infty \left(\frac{1}{\lambda^2}\sum_{y_3\sim y_2, y_3\geq y_2} 1+O \left (\frac{N_{y^*}}{\lambda^2} \right ) \right)^{k_2}\right)^{2}\\ \nonumber &+& 2\lambda^{-4}\sum_{y_1\sim x}N_y\sum_{y_2\sim y_1, y_2\geq y_1} \sum_{k_2=1}^\infty \left(\frac{1}{\lambda^2}\sum_{y_3\sim y_2, y_3\geq y_2} 1+O \left (\frac{N_{y^*}}{\lambda^2} \right ) \right)^{k_2}.\end{aligned}$$ Each of our error terms now has coefficient $\lambda^{-6}$ or smaller. Therefore, by the same exponential decay and the fact that $\lambda^{-6} \leq \alpha^{-3}$ and $N_{y^*} \ll \lambda^2,$ each of the five sums can be written as $$\label{eq:lambdaerror} O(\alpha^{-3} N_{y^*}(\beta^{(1,1)}+\beta^{(2)})).$$ By the assumptions of $\Omega_{\scaleto{\ref{dfn:omegadef}}{3pt}}$, namely the bound on the maximal degree and the approximations of $\beta^{(1,1)}$ and $\beta^{(2)},$ as well as our bounds for $d$, for $x\in \mathcal{W}$ $$\begin{aligned} O(\alpha^{-3} N_{y^*}(\beta^{(1,1)}+\beta^{(2)}))&=&O(\mathfrak{u}^{-3}\mathfrak{u}^{1/3}(\beta^{( 2)}+\beta^{(1,1)}))\\ &=&O(\mathfrak{u}^{-8/3}((d^2+d)\mathfrak{u}+\mathfrak{u}^{2/3}+d^2\mathfrak{u}+(d^{3/2}+1)\mathfrak{u}^{2/3}))\\ &=&O((d^2+d)\mathfrak{u}^{-5/3}).\end{aligned}$$ ◻ We now solve for $\lambda$ in this equation. Note that while $\lambda^2 = \alpha (1+o(1))$ by Lemma [Lemma 27](#lem:eigenvalueapproximation){reference-type="ref" reference="lem:eigenvalueapproximation"}, this by itself does not give a precise enough approximation. Instead we bootstrap this initial approximation using the structural properties from $\Omega_{\scaleto{\ref{dfn:omegadef}}{3pt}}.$ **Lemma 32**. *For $x\in \mathcal{W}$, [\[lem:lambdaexpression\]]{#lem:lambdaexpression label="lem:lambdaexpression"} $$\label{eq:lambdasquaredfinal2} \lambda^2=\alpha+\frac{\beta}{\alpha}+\frac{\beta^{(1,1)}+\beta^{(2)}}{\alpha^2}-\frac{(\beta)^2}{\alpha^3}+O((d^2+d)\mathfrak{u}^{-5/3}).$$* *Proof.* We rewrite [\[eq:lambdasquaredfinal\]](#eq:lambdasquaredfinal){reference-type="eqref" reference="eq:lambdasquaredfinal"} as $$\lambda^2=\alpha+\frac{1}{\alpha+(\lambda^2-\alpha)}\beta + \frac{1}{(\alpha+(\lambda^2-\alpha))^2}\beta^{(1,1)}+\frac{1}{(\alpha+(\lambda^2-\alpha))^2}\beta^{(2)}+O \left ( \left (d^2+d \right )\mathfrak{u}^{-5/3} \right ).$$ Moreover, by Lemma [Lemma 27](#lem:eigenvalueapproximation){reference-type="ref" reference="lem:eigenvalueapproximation"}, $\lambda^2-\alpha=O(d)$, so we can approximate $$\frac{1}{\alpha+(\lambda^2-\alpha)}=\frac1{\alpha}\frac{1}{1+\frac{(\lambda^2-\alpha)}{\alpha}}=\frac1\alpha \left (1 + O \left ( \frac{d}{\mathfrak{u}} \right ) \right ).$$ Therefore $$\lambda^2 = \alpha+\frac{1}{\alpha}\beta \left (1+O \left (\frac d{\mathfrak{u}} \right ) \right ) +\frac{1}{\alpha^2} \left (\beta^{(1,1)}+\beta^{(2)} \right ) \left (1+O \left (\frac d{\mathfrak{u}} \right ) \right) +O \left ( \left ( d^2+d \right )\mathfrak{u}^{-5/3} \right ) = \alpha+\frac{\beta}\alpha +O \left ( \frac{ d^2 + d }{ \mathfrak{u}} \right ).$$ This implies that $\frac{1}{\lambda^2} = \frac{1}{\alpha + \frac{\beta}{\alpha} + \left ( \lambda^2 - \alpha + \frac{\beta}{\alpha} \right ) } = \frac{1}{\alpha - \frac{\beta}{\alpha} } \left ( 1 + O \left ( \frac{d^2 + d}{\mathfrak{u}^2} \right ) \right ).$ Plugging this more precise approximation for $\lambda^2$ once again into [\[eq:lambdasquaredfinal\]](#eq:lambdasquaredfinal){reference-type="eqref" reference="eq:lambdasquaredfinal"} we get $$\begin{aligned} \lambda^2 &=&\alpha + \frac{1}{\alpha+\frac{\beta}{\alpha}}\beta \left ( 1 + O \left ( \frac{d^2 + d}{\mathfrak{u}^2} \right ) \right ) + \frac{1}{(\alpha+\frac{\beta}{\alpha})^2}(\beta^{(1,1)}+\beta^{(2)}) \left ( 1 + O \left ( \frac{d^2 + d}{\mathfrak{u}^2} \right ) \right ) + O((d^2+d)\mathfrak{u}^{-5/3})\\ &=& \alpha+\frac{1}{\alpha+\frac{\beta}{\alpha}}\beta+\frac{1}{\left ( \alpha+\frac{\beta}{\alpha} \right )^2}\left ( \beta^{(1,1)}+\beta^{(2)} \right ) + O\left ( \left (d^2+d \right )\mathfrak{u}^{-5/3} \right ).\end{aligned}$$ Due to our bounds on $d$ from [Definition 5](#dfn:rdfn){reference-type="ref" reference="dfn:rdfn"}, we can expand $\frac1{\alpha+\frac{\beta}{\alpha}}=\frac1\alpha-\frac{\beta}{\alpha^3}+O \left ( \frac{d^2}{\mathfrak{u}^3} \right )$, which gives us $$\begin{aligned} \lambda^2&=&\alpha+\frac{\beta}\alpha+\frac{\beta^{(1,1)}+\beta^{(2)}}{\alpha^2}-\frac{(\beta)^2}{\alpha^3} + O \left ( \left ( d^2+d \right )\mathfrak{u}^{-5/3} \right )\end{aligned}$$ as desired. ◻ Given this much tighter approximation of $\lambda_x$ for $x \in \mathcal{W}$, we can now show that the order of the eigenvalues corresponding to $B_r(x)$ for $x \in \mathcal{W}$ is the same as the lexicographic order of $\alpha_x$ and $\beta_x$. **Lemma 33**. *For two vertices $u,v\in \mathcal{W}$, if $\alpha_u \geq \alpha_v,$ $\lambda^2_u-\lambda^2_v \geq \alpha_u-\alpha_v + O((d^{1/2}+1)\mathfrak{u}^{-1/3})$. Moreover, if $\alpha_u = \alpha_v$, and $\beta_u \geq \beta_v,$ then $\lambda_u^2-\lambda_v^2 \geq \frac{\beta_u-\beta_v}\mathfrak{u}+O((1+d^{3/2})\mathfrak{u}^{-4/3})$. Therefore, for $x\in \mathcal{W}$, $\lambda_x^2$ are ordered according to the lexicographic ordering of $(\alpha_x,\beta_x)$.* *Proof.* By [\[eq:lambdasquaredfinal2\]](#eq:lambdasquaredfinal2){reference-type="eqref" reference="eq:lambdasquaredfinal2"}, $$\lambda^2_u-\lambda^2_v \geq \alpha_u-\alpha_v - \left| \frac{\beta_u}{\alpha_u}-\frac{\beta_v}{\alpha_v} \right | + \left| \frac{\beta^{(1,1)}_u}{\alpha^2_u}-\frac{\beta^{(1,1)}_v}{\alpha^2_v} \right | + \left | \frac{\beta^{(2)}_u}{\alpha^2_u}-\frac{\beta^{(2)}_v}{\alpha^2_v} \right | + \left | \frac{(\beta_u)^2}{\alpha_u^3}-\frac{(\beta_v)^2}{\alpha^3_v} \right| +O \left ( \left ( d^2+d \right ) \mathfrak{u}^{-5/3} \right ).$$ Using the definition of $\Omega_{\scaleto{\ref{dfn:omegadef}}{3pt}}$ (3-5), namely the concentration of $\beta, \beta^{(1,1)}$ and $\beta^{(2)}$, this implies $$\lambda^2_u-\lambda^2_v \geq \alpha_u-\alpha_v + O \left ( \left ( d^{1/2} + 1 \right ) \mathfrak{u}^{-1/3} \right ).$$ Now, assume that $\alpha_u = \alpha_v$. Then by [\[eq:lambdasquaredfinal2\]](#eq:lambdasquaredfinal2){reference-type="eqref" reference="eq:lambdasquaredfinal2"}, $$\begin{aligned} \lambda^2_u-\lambda^2_v &\geq& \frac{\beta_u-\beta_v} {\alpha_u} - \left|\frac{\beta^{(1,1)}_u}{\alpha^2_u}-\frac{\beta^{(1,1)}_v}{\alpha^2_v} \right | + \left| \frac{\beta^{(2)}_u}{\alpha^2_u}-\frac{\beta^{(2)}_v}{\alpha^2_v} \right | + \left| \frac{(\beta_u)^2}{\alpha_u^3}-\frac{(\beta_v)^2}{\alpha^3_v} \right| +O \left ( \left ( d^2+d \right ) \mathfrak{u}^{-5/3} \right ).\end{aligned}$$ Once again, by the definition of $\Omega_{\scaleto{\ref{dfn:omegadef}}{3pt}}$ (3-5), this implies $$\begin{aligned} \lambda_u^2 - \lambda_v^2 & \geq & \frac{\beta_u-\beta_v}\mathfrak{u}+O \left ( \left (d^{3/2} + 1 \right ) \mathfrak{u}^{-4/3} \right ).\end{aligned}$$ To see that this induces a lexicographic ordering, if $\alpha_u\neq \alpha_v$, then $\alpha_u - \alpha_v \geq 1 \gg (d^{1/2}+1)\mathfrak{u}^{-1/3}$. Similarly, if $\alpha_u = \alpha_v$, but $\beta_u \neq \beta_v$, then $\frac{\beta_u-\beta_v}{\mathfrak{u}} \geq \frac1\mathfrak{u}\gg(1+d^{3/2})\mathfrak{u}^{-4/3}$, by our assumptions on $d$ from Definition [Definition 5](#dfn:rdfn){reference-type="ref" reference="dfn:rdfn"}. ◻ # Rough Regime {#sec:rough} In this section we construct approximate eigenvectors corresponding to small balls around vertices in $\mathcal{U}\setminus \mathcal{V}$, and we derive a less precise approximation for the eigenvalues corresponding to those balls. The approach used in this section is very similar to the approach in Section 6.4 of [@alt2023poisson]. Note a result like Lemma [Lemma 27](#lem:eigenvalueapproximation){reference-type="ref" reference="lem:eigenvalueapproximation"}, which we will eventually use to show that eigenvalues from vertices in $\mathcal{V}\setminus \mathcal{W}$ cannot compete with the largest ones from vertices in $\mathcal{W}$, cannot directly be derived for all vertices in $\mathcal{U}.$ The main obstructions are that the growth of the spheres and the maximum degree in the balls cannot be bounded as tightly as for vertices in $\mathcal{V}.$ More precisely our goal for this section is to show the following. **Lemma 34**. *For $x\in \mathcal{U}$, we can create a set of unit vectors $\mathbf{w}_\sigma(x)$ with $\sigma\in \{\pm1\}$ such that* 1. *For $u,v \in \mathcal{U}, u \neq v$, and $\sigma_1, \sigma_2 \in \{\pm1\}$ we have $supp(\mathbf{w}_{\sigma_1}(u))\cap supp(\mathbf{w}_{\sigma_2}(v))=\emptyset$.* 2. *We have $\| A \mathbf{w}_\sigma(x) - \sigma \lambda_x \mathbf{w}_\sigma(x) \| = O ( \log\log N )$, where $\lambda_x = \sqrt{\alpha_x+\frac{\beta_x}{\alpha_x}}$.* Towards this end, we will first prove two weaker structural lemmas around vertices in $\mathcal{U}$. Firstly we have weaker bounds on the neighborhood growth and the fluctuations of the degrees of the neighbors. The following lemma has a similar proof as Lemma [Lemma 24](#lem:fine_balls_disjoint){reference-type="ref" reference="lem:fine_balls_disjoint"}, and we defer the proof to Section [10](#sec:structure){reference-type="ref" reference="sec:structure"} of the appendix. **Lemma 35**. *We have that with high probability for any vertex $x \in \mathcal{U}$ simultaneously and any $i \leq 3$, the sphere $S_i(x)$ at distance $i$ from $x$ satisfies* *$$\left |S_i(x) \right | = O \left ((d + \log \log N)^{i-1}\mathfrak{u}\right ) .$$ Moreover, $$\sum_{ y \in S_1(x) } \left ( N_y - d \right )^2\leq O \left ( \left ( \log N \right )^2 \right ).$$* Next, we show the balls around vertices in $\mathcal{U}$ are close to disjoint trees: with high probability the neighborhoods around vertices in the rough regime are almost trees and contain few disjoint paths that contain other vertices from the rough regime. This result basically corresponds to Lemma 5.5 and Lemma 7.3 in [@alt2021extremal], albeit for a different regime of $d$. The proof is also very similar and is deferred to Section [10](#sec:structure){reference-type="ref" reference="sec:structure"} of the appendix. **Lemma 36**. *Let $\mathcal{U}_\eta = \left \{ x \in [N]: \alpha_x \geq \eta \mathfrak{u}\right \},$ and $s$ be some positive integer, then with high probability for some constants $C_1$ and $C_2$ that only depend on $\eta$, simultaneously for all $x \in \mathcal{U}_\eta$,* 1. *$|E \left ( B_s(x) \right )| < \left | V \left ( B_s(x) \right ) \right | - 1 + C_1$ and* 2. *$B_s(x)$ contains less than $C_2$ edge disjoint paths in $B_s(x)$ containing other vertices from $\mathcal{U}_\eta$.* Note that $\mathcal{U}= \mathcal{U}_\frac{1}{2}$ and that it is enough to take constants $C_1 = 2$ and $C_2 > \frac{2}{\eta}$. We now construct a "pruned" graph in which the neighborhoods of vertices in $\mathcal{U}$ are disjoint trees. The construction works in the same manner as in Lemma 7.2 of [@alt2021extremal] and we use it to prove a statement similarly to Proposition 6.19 in [@alt2023poisson]. Once more the proof can be found in the appendix. **Lemma 37**. *Recall that we denote by $G$ the random graph sampled from $\mathcal{G}(N,\frac dN)$. With high probability, there is a subgraph $\hat G\subset G$ such that for all vertices $x \in \mathcal{U},$* 1. *Balls of radius 3 around $x$ in $\hat{G}$, which we denote by $\hat{B}_3(x)$, are disjoint;* 2. *The subgraphs induced by $\hat{B}_3(x)$ are trees;* 3. *The maximum degree of $G - \hat G$ is bounded;* 4. *For $i\leq 3$, the spheres $\hat{S}_i(x)$ in the pruned graph $\hat{G}$ satisfy $$\left |\hat{S}_{i}(x) \right | = O \left ( (d + \log \log N)^{i-1}\mathfrak{u}\right ) ;$$* 5. *We have that $$\sum_{y \in \hat{S}_1(x)} \left ( \hat N_y(x)- \frac{\hat{\beta}}{\hat \alpha} \right )^2\leq O \left ( (\log N)^2 \right ).$$* *Proof of Lemma [Lemma 34](#lem:roughconstruction){reference-type="ref" reference="lem:roughconstruction"}.* Define $\hat\alpha_x,\hat\beta_x$ to be the parameters of the pruned graph $\hat G$. We use the same test vector as [@alt2023poisson], restated with our parameters this is the unit vector $$\label{eq:rough_eigenvector} \mathbf{w}_{\sigma}(x):= \frac{1}{\sqrt{2}}\left(\frac{\sqrt{\hat \alpha_x}}{\sqrt{\hat\alpha_x+\frac{\hat\beta_x}{\hat\alpha_x}}} \mathbf{1}_{x} +\sigma\frac{1}{\sqrt{\hat\alpha_x}}\mathbf{1}_{\hat S_1(x)} +\frac{1}{\sqrt{\hat\alpha_x(\hat\alpha_x +\frac{\hat\beta_x}{\hat\alpha_x})}}\mathbf{1}_{\hat S_2(x)}\right).$$ The first statement of the Lemma now follows by Lemma [Lemma 37](#lem:prunedgraph){reference-type="ref" reference="lem:prunedgraph"}. We now fix $x$ and $\sigma$ and drop them from our notation for better readability. To prove the second statement we define $\hat\lambda = \sqrt{\hat\alpha+\frac{\hat\beta}{\hat\alpha}}$ and $\hat{A} = A_{\hat{G}}$ and use a triangle inequality to bound $$\| A \mathbf{w}- \sigma \lambda \mathbf{w}\| \leq \left \|A \mathbf{w}- \hat{A} \mathbf{w}\right \| + \left \|\hat{A} \mathbf{w}- \sigma \hat{\lambda} \mathbf{w}\right \| + \left \|\hat{\lambda} \mathbf{w}- \lambda \mathbf{w}\right \|.$$ The first term on the right hand side is at most constant, as by Lemma [Lemma 37](#lem:prunedgraph){reference-type="ref" reference="lem:prunedgraph"} the maximum degree of $G - \hat G$ is bounded by a constant and since the maximum row sum is an upper bound for the maximum eigenvalue of a positive symmetric matrix. Similarly the last term is bounded since by Lemma [Lemma 37](#lem:prunedgraph){reference-type="ref" reference="lem:prunedgraph"}, $\hat{\alpha}$ differs from $\alpha$ by at most a constant, and $\beta$ and $\hat{beta}$ can both be bounded by $(d + \log \log N) \mathfrak{u}.$ This implies that $$|\lambda - \hat{\lambda}| = \sqrt{\alpha +\frac {\beta }{\alpha }}-\sqrt{\hat \alpha +\frac {\hat\beta }{\hat\alpha }}=\sqrt{\alpha }\sqrt{1+\frac {\beta }{\alpha ^2}}-\sqrt{\alpha }\sqrt{1+\frac{\hat \alpha -\alpha }{\alpha }+\frac {\hat\beta }{\alpha \hat\alpha }}\ll 1.$$ The second term can be computed as $$\begin{aligned} & \sqrt{2} \left ( \hat A \mathbf{w}-\sigma \hat \lambda \mathbf{w}\right ) \\ & = \left( \sigma \sqrt{\hat{\alpha}}- \sigma \sqrt{\hat{\alpha}}\right)\mathbf{1}_{x} + \sum_{y\in \hat{S}_1} \left(\frac{\sqrt{\hat{\alpha}}}{\hat\lambda}+\frac{\hat{N}_y}{\sqrt{\hat{\alpha}}\hat{\lambda}}-\frac{\hat{\lambda}}{\sqrt{\hat{\alpha}}}\right)\mathbf{1}_y +\sum_{y\in \hat S_2} \left(\frac{\sigma}{\sqrt{\hat{\alpha}}} - \frac{\sigma}{\sqrt{\hat\alpha}}\right) \mathbf{1}_y + \sum_{y\in \hat S_3} \frac{1}{\sqrt{\hat{\alpha}} \hat \lambda}\mathbf{1}_y.\end{aligned}$$ Therefore, using Lemma [Lemma 37](#lem:prunedgraph){reference-type="ref" reference="lem:prunedgraph"} and the lower bound on $\alpha_x$ for $x \in \mathcal{U}$, we get $$\begin{aligned} 2 \left \|\hat A \mathbf{w}-\sigma \hat \lambda \mathbf{w}\right \|^2 &\leq& \frac{1}{\hat{\alpha} \left (\hat\alpha+\frac{\hat\beta}{\hat\alpha} \right )}\sum_{y\in \hat S_1}\left(\hat{N}_y-\frac{\hat{\beta}}{\hat\alpha}\right)^2+|\hat{S}_3|\frac{1}{\hat{\alpha}\left ( \hat{\alpha}+\frac{\hat\beta}{\hat\alpha}\right )}\\ &\leq& O \left ( (\log\log N)^2 +\frac{(d+\log\log N)^2}{\mathfrak{u}} \right ).\end{aligned}$$ Putting these three bounds together and using our expression [\[eq:uapprox\]](#eq:uapprox){reference-type="eqref" reference="eq:uapprox"} for $\mathfrak{u}$ and the bounds in [Definition 5](#dfn:rdfn){reference-type="ref" reference="dfn:rdfn"} for $d$, we get that, $$\|A \mathbf{w}-\sigma \lambda \mathbf{w}\| = O(\log\log N).$$ ◻ # Block Decomposition {#sec:finaldecomp} This section is devoted to proving Theorem [Theorem 22](#thm:structure){reference-type="ref" reference="thm:structure"}, for which we use results from Sections [4](#sec:fine){reference-type="ref" reference="sec:fine"} and [5](#sec:rough){reference-type="ref" reference="sec:rough"}. For this we first bound the contribution of the remainder of the matrix, i.e. from vectors that are orthogonal to the largest eigenvectors of small balls around the high-degree vertices. We end this section by using the approximate diagonalizetion to prove Theorem [Theorem 2](#thm:maineigenvalue){reference-type="ref" reference="thm:maineigenvalue"}. ## Bulk vectors We now prove that there is no contribution from any other vector. To do this, we use the decomposition of Krivelevich and Sudakov [@krivelevich2003largest]. This lets us reduce to only considering stars around high degree vertices. Here, we state a structure theorem that combines elements of the proof of Theorem 1.1 and Lemma 2.2 in [@krivelevich2003largest]. To do this, recall that $\Gamma_x$ are all vertices adjacent to $x$ and consider the sets of vertices $$\begin{aligned} \mathcal{Y}_1&:=&\left\{x\in[N]:\alpha_x\geq \mathfrak{u}^{3/4}\right\}\\ \mathcal{Y}_2&:=&\left\{x\in[N]:\Gamma_x\cap \mathcal{Y}_1\neq \emptyset\right\}\end{aligned}$$ **Proposition 38** ([@krivelevich2003largest]). *For $G\sim \mathcal{G}(N,\frac{d}{N})$ graph, if $d=o(\log^{1/2} N)$, then with high probability, there is a subgraph $\mathcal{H}\subset G$ such that* 1. *$\mathcal{H}$ is contained in the bipartite subgraph induced by $\mathcal{Y}_1$ and $\mathcal{Y}_2$,* 2. *$\mathcal{H}$ is a union of stars on disjoint vertices,* 3. *$\|A_{G\backslash \mathcal{H}}\|=O(d+\mathfrak{u}^{7/16})$.* Note that $\mathcal{H}$ is the graph $G_6-H$ in [@krivelevich2003largest]. This is strong enough to show that no other vector interferes in the largest eigenvalues. Define $U_{\mathcal{U}}$ to be the space spanned by $\mathbf{w}_{\pm}(x)$ as defined in Definition [Definition 26](#dfn:wdef){reference-type="ref" reference="dfn:wdef"} for $x\in \mathcal{V}$, and $\mathbf{w}_{\pm}(x)$ as defined in Lemma [Lemma 34](#lem:roughconstruction){reference-type="ref" reference="lem:roughconstruction"} for $x\in \mathcal{U}\backslash \mathcal{V}$. **Lemma 39**. *For any vector $\mathbf{v}\in \mathbb{R}^N$ that satisfies $\|v |\ = 1$ and $\mathbf{v}\perp U_\mathcal{U}$, $$\langle \mathbf{v}, A \mathbf{v}\rangle \leq (1+o_N(1))\frac{1}{\sqrt{2}}\sqrt{\mathfrak{u}}.$$* *Proof.* By Proposition [Proposition 38](#prop:ks){reference-type="ref" reference="prop:ks"}, we have that $$\langle \mathbf{v},A\mathbf{v}\rangle \leq \max_{x\in \mathcal{Y}_1} \langle \mathbf{v},A_{\tilde B_1(x)}\mathbf{v}\rangle+O(d+\mathfrak{u}^{7/16})$$ where $\tilde B_{1}(x)$ is the ball of radius $1$ around $x$ in $\mathcal H$ from Proposition [Proposition 38](#prop:ks){reference-type="ref" reference="prop:ks"} and $A_{\tilde B_{1}(x)}$ is the adjacency matrix of the graph on the vertices $[N]$ induced by the ball $\tilde B_{1}(x).$ Therefore we split into cases based on the degree of $x$. For $x\in \mathcal{V}$, we know that $\mathbf{v}\perp \frac1{\sqrt{2}}(\mathbf{w}_+(x)+\mathbf{w}_-(x))$. Therefore $$\langle \mathbf{v}, \mathbf{1}_x\rangle =\langle \mathbf{v},\frac1{\sqrt{2}}(\mathbf{w}_+(x)+\mathbf{w}_-(x))\rangle +\langle \mathbf{v},\mathbf{1}_x-\frac1{\sqrt{2}}(\mathbf{w}_+(x)+\mathbf{w}_-(x))\rangle\leq \|\mathbf{1}_x-\frac1{\sqrt{2}}(\mathbf{w}_+(x)+\mathbf{w}_-(x))\|.$$ By Proposition [Proposition 30](#prop:expdecay){reference-type="ref" reference="prop:expdecay"}, $$\|\mathbf{1}_x-\frac1{\sqrt{2}}(\mathbf{w}_+(x)+\mathbf{w}_-(x))\|=O(\mathfrak{u}^{-1/3}).$$ We then have $$\langle \mathbf{v},A_{\tilde B_{1}(x)} \mathbf{v}\rangle\leq 2|\langle \mathbf{v}, \mathbf{1}_x\rangle|\sum_{y\sim x}|\langle \mathbf{v}, \mathbf{1}_y\rangle|=O(\mathfrak{u}^{-1/3}\cdot \sqrt{\mathfrak{u}}).$$ Similarly, for $x\in \mathcal{U}\backslash \mathcal{V}$, we have $\langle \mathbf{v}, \mathbf{1}_x\rangle\leq \|\mathbf{1}_x-\frac1{\sqrt{2}}(\mathbf{w}_+(x)+\mathbf{w}_-(x))\|$. By the definition of the eigenvector in [\[eq:rough_eigenvector\]](#eq:rough_eigenvector){reference-type="eqref" reference="eq:rough_eigenvector"}, and using properties from Lemma [Lemma 37](#lem:prunedgraph){reference-type="ref" reference="lem:prunedgraph"}, we have that $$\left \|\mathbf{1}_x-\frac1{\sqrt{2}}(\mathbf{w}_+(x)+\mathbf{w}_-(x)) \right \| = O \left ( \sqrt{ \left [ \frac{\sqrt{\hat{\alpha}+\frac{\hat{\beta}}{\hat{\alpha}}}-\sqrt{\hat{\alpha}} }{\sqrt{\hat{\alpha}+\frac{\hat{\beta}}{\hat{\alpha}}}} \right ]^2 + \hat{\beta} \frac{1}{\hat{\alpha} \left ( \hat{\alpha} + \frac{ \hat{\beta}}{\hat{\alpha}} \right )} } \right ) = O \left ( \sqrt{ \frac{ \beta^2}{\alpha^3} + \frac{\beta}{\alpha^{2}} } \right) = O \left ( \frac{ d+ \log \log N}{\mathfrak{u}} \right ).$$ Therefore, by the same argument as before $$\langle \mathbf{v},A_{\tilde B_1(x)}\mathbf{v}\rangle= O \left ( \frac{ d + \log \log N}{\sqrt{\mathfrak{u}}} \right ) =o_N \left ( 1 \right ).$$ For any vertex $x\in \mathcal{Y}_1\backslash \mathcal{U}$, the maximum degree is $\mathfrak{u}/2$ and the spectral norm is given by the spectral radius of a star graph, namely for any vector $\mathbf{v}$ such that $\| \mathbf{v}\| = 1,$ $$\langle \mathbf{v},A_{\tilde B_1(x)}\mathbf{v}\rangle\leq \sqrt{\frac{\mathfrak{u}}{2}}.$$ Combining these cases gives the result. ◻ ## Structure Theorem We now have all the ingredients to prove the structure theorem. *Proof of Theorem [Theorem 22](#thm:structure){reference-type="ref" reference="thm:structure"}.* We can now fully define the block decomposition from [\[eq:blockdecomp\]](#eq:blockdecomp){reference-type="eqref" reference="eq:blockdecomp"}, $$A=U\left(\begin{array}{cccc} D_{\mathcal{W}}&0&0&E_{\mathcal{W}}^*\\ 0& D_{\mathcal{V}\backslash \mathcal{W}}&0&E_{\mathcal{V}\backslash \mathcal{W}}^*\\ 0&0& D_{\mathcal{U}\backslash \mathcal{V}}+\mathcal{E}_{\mathcal{U}\backslash \mathcal{V}}&E_{\mathcal{U}\backslash \mathcal{V}}^*\\ E_{\mathcal{W}}& E_{\mathcal{V}\backslash \mathcal{W}}&E_{\mathcal{U}\backslash \mathcal{V}}&\mathcal{X} \end{array}\right)U^{*}$$ We first define the unitary matrix $U$. We set the first $2|\mathcal{W}|$ columns of $U$ to vectors $\mathbf{w}_{\pm}(x)$ for $x\in \mathcal{W}$, and denote this part of the matrix by $U_\mathcal{W}$, then we set the next $2|\mathcal{V}\backslash \mathcal{W}|$ columns to $\mathbf{w}_{\pm}(x)$ for $x \in \mathcal{V}\backslash \mathcal{W}$ and denote this part of the matrix by $U_{\mathcal{V}\setminus \mathcal{W}}$, for $\mathbf{w}_{\pm}(x)$ as defined in Definition [Definition 26](#dfn:wdef){reference-type="ref" reference="dfn:wdef"}. The next $2 | \mathcal{U}\setminus \mathcal{V}|$ columns are the vectors $\mathbf{w}_\pm(x)$ for $x \in \mathcal{U}\setminus \mathcal{V}$ as defined in Lemma [Lemma 34](#lem:roughconstruction){reference-type="ref" reference="lem:roughconstruction"}, and we denote this part of the matrix by $U_{\mathcal{U}\setminus \mathcal{V}}$. We denote these three parts of the matrix together by $U_{\mathcal{U}}$. We then complete $U$ arbitrarily with a basis of the rest of $\mathbb{R}^N$, namely $U_{\mathcal{U}^\perp}\subset \mathbb{R}^N$. It is implied by the definition of $U$ that the diagonal matrices $D_\mathcal{W}$ and $D_{\mathcal{V}\setminus \mathcal{W}}$ have entries $\sigma \lambda_x$ on the diagonal, i.e. the eigenvalue of the truncated balls corresponding to each $w_\sigma(x)$. The diagonal operator $D_{\mathcal{U}\backslash \mathcal{V}}$, is defined to have entries $\sigma \lambda_x$, from Lemma [Lemma 34](#lem:roughconstruction){reference-type="ref" reference="lem:roughconstruction"}. $0$'s exist in the requisite places as we can assume by $\Omega_{\scaleto{\ref{dfn:omegadef}}{3pt}}$ that balls of vertices in $\mathcal{V}$ are disjoint, and for each $x\in \mathcal{V}$, the maximum degree of a vertex in $B_{r+3}(x)\backslash x$ is $\mathfrak{u}^{3/4}$, implying that there are no intersections with balls of radius 3 around vertices in $\mathcal{U}\setminus \mathcal{V}$. By Lemma [Lemma 19](#lem:sizes){reference-type="ref" reference="lem:sizes"}, with high probability, the $e^{\log^{1/8}N}$ vertices of largest degree have degree at least $\mathfrak{u}-2\log^{1/8}N$. Therefore the eigenvalues corresponding to these vertices have value at least $\sqrt{\mathfrak{u}-2\log^{1/8}N}=\sqrt{\mathfrak{u}}-O(\log^{-3/8}N)$ by Lemma [Lemma 27](#lem:eigenvalueapproximation){reference-type="ref" reference="lem:eigenvalueapproximation"}. To get a bound on $D_{\mathcal{V}\setminus \mathcal{W}},$ we use the upper bound from Lemma [Lemma 27](#lem:eigenvalueapproximation){reference-type="ref" reference="lem:eigenvalueapproximation"}. This gives that for any vertex $x \in \mathcal{V},$ and for the range of $d$ defined in [Definition 5](#dfn:rdfn){reference-type="ref" reference="dfn:rdfn"}, $$\lambda_x \leq \sqrt{\mathfrak{u}- \mathfrak{u}^\frac{1}{4} + O(d)} = \sqrt{\mathfrak{u}} - \frac{ \mathfrak{u}^{-\frac{1}{2}}}{2} + O \left ( \frac{d}{\sqrt{\mathfrak{u}}} \right ) \leq \sqrt{\mathfrak{u}} - \Theta \left ( \mathfrak{u}^{-\frac{1}{2}} \right ).$$ By Lemma [Proposition 29](#lem:fineerror){reference-type="ref" reference="lem:fineerror"}, for any $x\in \mathcal{V}$, $\sigma\in \{\pm 1\}$, $\|(A-\Lambda)\mathbf{w}_{\sigma}(x)\|=O((d^{r/2}+1)\mathfrak{u}^{-(r-1)/2})$. We will now show that this implies a bound on $\| E_W \|:$ Using that $U^*_{\mathcal{W}}$ is a surjective projection of $\mathbb{R}^N$ onto $\mathbb{R}^{2 | \mathcal{W}|}$ and $U_{\mathcal{U}^\perp}$ is an injective embedding of $\mathbb{R}^{N - 2|\mathcal{U}|}$ onto $\mathbb{R}^N,$ we can transform $E_W$, which maps $\mathbb{R}^{2|\mathcal{W}|}$ to $\mathbb{R}^{N - 2 |\mathcal{U}|},$ into an operator from $\mathbb{R}^N$ to $\mathbb{R}^N$, with the same spectral properties. Therefore, using additionally that outside of the choice of $\sigma\in \{\pm1\}$, the supports of $\mathbf{w}_\sigma(x)$ are independent, $$\begin{gathered} \|E_{\mathcal{W}}\|= \|U_{\mathcal{U}^\perp}E_{\mathcal{W}}U_\mathcal{W}^*\| =\max_{\mathbf{v}\in span(U_\mathcal{W}), \| \mathbf{v}\| = 1}\|U_{\mathcal{U}^\perp}E_\mathcal{W}U_\mathcal{W}^*\mathbf{v}\| \leq \max_{x\in \mathcal{W},\sigma\in \{\pm 1\}} 2 \|U_{\mathcal{U}^\perp}E_\mathcal{W}U_\mathcal{W}^*\mathbf{w}_\sigma (x)\|\\ = \max_{x\in \mathcal{W},\sigma\in \{\pm 1\}} 2 \|(A-U_{\mathcal{W}}D_\mathcal{W}U_\mathcal{W}^*)\mathbf{w}_\sigma (x)\| = \max_{x\in \mathcal{W},\sigma\in \{\pm 1\}}2\|(A-\Lambda)\mathbf{w}_{\sigma}(x)\| = O( (d^{r/2}+1)\mathfrak{u}^{-(r-1)/2}).\end{gathered}$$ Here we use the definition of $\Lambda$ from Lemma [Proposition 29](#lem:fineerror){reference-type="ref" reference="lem:fineerror"}. The same is true for $\|E_{\mathcal{V}\backslash \mathcal{W}}\|$. Instead of bounding the operator norms of $E_{\mathcal{U}\setminus \mathcal{V}}$ and $\mathcal{E}_{\mathcal{U}\setminus \mathcal{V}}$ individually, we bound the operator norm of their concatenation, which will be an upper bound for both. Similarly to before we can write $$\left \| \begin{bmatrix} \mathcal{E}_{\mathcal{U}\setminus \mathcal{V}} \\ E_{\mathcal{U}\setminus \mathcal{V}} \end{bmatrix} \right \| = \left \| \begin{bmatrix} U_{\mathcal{U}\setminus \mathcal{V}} U_{\mathcal{U}^\perp} \end{bmatrix} \begin{bmatrix} \mathcal{E}_{\mathcal{U}\setminus \mathcal{V}} \\ E_{\mathcal{U}\setminus \mathcal{V}} \end{bmatrix} U_{\mathcal{U}\setminus \mathcal{V}}^* \right\|.$$ By subsequently proceeding in the same way as for the error coming from the fine regime, Lemma [Lemma 34](#lem:roughconstruction){reference-type="ref" reference="lem:roughconstruction"}, 2., gives the desired bound. Finally for the bulk, $\|\mathcal{X}\|\leq (\frac1{\sqrt2}+o_N(1))\sqrt\mathfrak{u}$ by Lemma [Lemma 39](#lem:bulk){reference-type="ref" reference="lem:bulk"}. ◻ This immediately gives the following. **Corollary 40**. *$$\left\|\left(\begin{array}{ccc} D_{\mathcal{V}\backslash \mathcal{W}}&0&E_{\mathcal{V}\backslash \mathcal{W}}^*\\ 0& D_{\mathcal{U}\backslash \mathcal{V}}+\mathcal{E}_{\mathcal{U}\backslash \mathcal{V}}&E_{\mathcal{U}\backslash \mathcal{V}}^*\\ E_{\mathcal{V}\backslash \mathcal{W}}&E_{\mathcal{U}\backslash \mathcal{V}}&\mathcal{X} \end{array}\right)\right\|\leq \sqrt{\mathfrak{u}}-\Theta(\mathfrak{u}^{-1/4}).$$* *Proof.* This norm is at most $$\begin{gathered} \max \left \{\|D_{\mathcal{V}\backslash \mathcal{W}}\|,\left\|\left(\begin{array}{cc} D_{\mathcal{U}\backslash \mathcal{V}}+\mathcal{E}_{\mathcal{U}\backslash \mathcal{V}}&E_{\mathcal{U}\backslash \mathcal{V}}^*\\ E_{\mathcal{U}\backslash \mathcal{V}}&\mathcal{X} \end{array}\right)\right\| \right \}+\|E_{\mathcal{V}\backslash \mathcal{W}}\|\\ \leq \max \Big \{\|D_{\mathcal{V}\backslash \mathcal{W}}\|,\max \big \{\|D_{\mathcal{U}\backslash \mathcal{V}} + \mathcal{E}_{\mathcal{U}\setminus \mathcal{V}} \|,\|\mathcal{X}\| \big \}+\|E_{\mathcal{U}\backslash \mathcal{V}} \Big \}+\|E_{\mathcal{V}\backslash \mathcal{W}}\|.\end{gathered}$$ The bound then follows from Theorem [Theorem 22](#thm:structure){reference-type="ref" reference="thm:structure"} and our bounds on $d$ from [Definition 5](#dfn:rdfn){reference-type="ref" reference="dfn:rdfn"}. ◻ With this, we can show that the top eigenvalues correspond to $\mathcal{W}$. **Proposition 41**. *For every $k\leq e^{\log^{1/8}N}$, the $k$th largest eigenvalue $\lambda$ of $A$ corresponds to the $k$th largest lexicographic maximizer $x\in \mathcal{W}$ of $(\alpha_x,\beta_x)$ in that $\lambda=\lambda_x+O( (d^r+1)\mathfrak{u}^{-r+1})$.* *Proof.* We start with the matrix $$U\left(\begin{array}{cccc} D_{\mathcal{W}}&0&0&0\\ 0& D_{\mathcal{V}\backslash \mathcal{W}}&0&E_{\mathcal{V}\backslash \mathcal{W}}^*\\ 0&0& D_{\mathcal{U}\backslash \mathcal{V}}+\mathcal{E}_{\mathcal{U}\backslash \mathcal{V}}&E_{\mathcal{U}\backslash \mathcal{V}}^*\\ 0& E_{\mathcal{V}\backslash \mathcal{W}}&E_{\mathcal{U}\backslash \mathcal{V}}&\mathcal{X} \end{array}\right)U^{*}$$ We then make the transformation by performing the summation $$U\left(\begin{array}{cccc} D_{\mathcal{W}}&0&0&0\\ 0& D_{\mathcal{V}\backslash \mathcal{W}}&0&E_{\mathcal{V}\backslash \mathcal{W}}^*\\ 0&0& D_{\mathcal{U}\backslash \mathcal{V}}+\mathcal{E}_{\mathcal{U}\backslash \mathcal{V}}&E_{\mathcal{U}\backslash \mathcal{V}}^*\\ 0& E_{\mathcal{V}\backslash \mathcal{W}}&E_{\mathcal{U}\backslash \mathcal{V}}&\mathcal{X} \end{array}\right)U^{*} +U \left(\begin{array}{cccc} 0&0&0&E_{\mathcal{W}}^*\\ 0&0&0&0\\ 0&0&0&0\\ E_{\mathcal{W}}&0&0&0. \end{array}\right)U^{*}$$ By perturbation theory, e.g. [@baumgartel1985analytic 7.1.1], [@bamieh2020tutorial Equation 23], each eigenvalue has changed by at most $O(\|E_\mathcal{W}\|^2)=O( (d^r+1)\mathfrak{u}^{-r+1})$. Moreover, as $r\geq 5$, by Theorem [Theorem 22](#thm:structure){reference-type="ref" reference="thm:structure"} and Corollary [Corollary 40](#claim:errorbound){reference-type="ref" reference="claim:errorbound"}, after the perturbation, nothing outside of $D_{\mathcal{W}}$ can correspond to one of the $e^{\log^{1/8}N}$ largest eigenvalues. Moreover, by Lemma [Lemma 33](#lem:difference){reference-type="ref" reference="lem:difference"}, the ordering of eigenvalues must match the ordering in $D_W$, inducing the lexicographic ordering. ◻ *Proof of Theorem [Theorem 2](#thm:maineigenvalue){reference-type="ref" reference="thm:maineigenvalue"}.* Consider the vertex $x$ corresponding to one of the $e^{\log^{1/8}N}$ largest eigenvalues. By Lemma [\[lem:lambdaexpression\]](#lem:lambdaexpression){reference-type="ref" reference="lem:lambdaexpression"} and the concentration results from the definition of $\Omega_{\scaleto{\ref{dfn:omegadef}}{3pt}}$, (3-5), we have that $$\lambda_x^2=\alpha_x+\frac{\beta_x}{\alpha_x}+\frac{d^2}{\alpha_x}+\frac{d^2+d}{\alpha_x}-\frac{d^2}{\alpha_x}+O(\frac{d^{3/2}+1}{\mathfrak{u}^{4/3}}).$$ Therefore by Proposition [Proposition 41](#prop:eigstructure){reference-type="ref" reference="prop:eigstructure"}, the true eigenvalue $\lambda$ satisfies $$\begin{aligned} \lambda&=&\sqrt{\alpha_x+\frac{\beta_x}{\alpha_x}+\frac{d^2+d}{\alpha_x}+O(\frac{d^{3/2}+1}{\mathfrak{u}^{4/3}})}+O( (d^{r}+1)\mathfrak{u}^{-{(r-1)}}).\\ &=&\sqrt{\alpha_x+\frac{\beta_x}{\alpha_x}+\frac{d^2+d}{\alpha_x}}+O((d^{3/2}+1)\mathfrak{u}^{-11/6}+ (d^{r}+1)\mathfrak{u}^{-{(r-1)}})\\ &=&\sqrt{\alpha_x+\frac{\beta_x}{\alpha_x}+\frac{d^2+d}{\alpha_x}}+O((d^{3/2}+1)\mathfrak{u}^{-11/6}).\end{aligned}$$ as we have assumed $r\geq5$. The lexicographic ordering follows immediately from Proposition [Proposition 41](#prop:eigstructure){reference-type="ref" reference="prop:eigstructure"}. ◻ # Anticoncentration {#sec:anticoncentration} In this section we prove that the distribution for $\lambda^2$ is anticoncentrated at the edge of the spectrum. To start, we show that the joint distribution of $\{\alpha_x\}_{x\in \mathcal{W}}$ and $\{\beta_x\}_{x\in \mathcal{W}}$ is approximately that of independent Poissons. This result is similar to [@alt2023poisson] Lemma 7.1, but proven in a somewhat different way. We then proceed similarly to [@alt2023poisson] to derive that this implies Theorem [Theorem 3](#thm:process){reference-type="ref" reference="thm:process"}, which says that that the maximal pairs $(\alpha_x, \beta_x)$ are close to the maximal values of a Poisson process. This result is then used to show Lemma [Lemma 45](#lem:eigenvaluespacing){reference-type="ref" reference="lem:eigenvaluespacing"}, which gives a lower bound on the distance between the size of the 2-spheres around vertices with maximal or almost maximal degree. In the final lemma of this section we show that this implies spacing of the largest eigenvalues. **Lemma 42**. *For $d$ according to [Definition 5](#dfn:rdfn){reference-type="ref" reference="dfn:rdfn"}, let $G$ be a graph generated from the Erdős-Rényi graph distribution $\mathcal{G}\left ( N, \frac{d}{N} \right )$. Moreover, for $k \leq e^{log^{2/3} N}$, consider vertices $z_1,\ldots z_k\in[N]$, along with $\frac12\mathfrak{u}\leq v_1,\ldots, v_k\leq 2\mathfrak{u}$ and $w_1,\ldots w_k$ such that $1\leq w_i\leq dv_i+\mathfrak{u}^{7/8}$ for $1\leq i\leq k$. Then define i.i.d. $X_1,X_2,\ldots X_k\sim Pois(d)$ and independent $Y_{v_1}\sim Pois(dv_1),Y_{v_2}\sim Pois(dv_2),\ldots, Y_{v_k} \sim Pois(dv_k)$. If $A$ is the event that there are no intersections between the balls of radius 1 around the vertices $z_1, \dots, z_k$, and no edges from $S_1(z_i)$ to $S_1(z_j)$ for any $i,j$ (including i=j), then $$\label{eq:decorr} \mathbb{P}\left(\bigcap_{i=1}^k \left \{ \alpha_{z_i} = v_i, \beta_{z_i} =w_i\right \}\cap A\right) = \left(1+N^{-1+o_N(1)} \right)\prod_{i=1}^k\mathbb{P}(X_i=v_i) \mathbb{P}(Y_{v_i}=w_i)$$* *Proof.* Let $Z = \{ z_1, \dots, z_k\}$, with $A_Z$ being the adjacency matrix of $Z$. Recall that $\Gamma_{z_i}$ denotes the neighbors of a vertex $z_i$. We analyse the event that $\cap_i \{ \alpha_{z_i} = v_i \}$ and that there are no edges between any $z_i$, as well as no intersection between the neighborhoods of the $z_i$, sequentially. That $A_Z = 0$ happens with probability $(1-d/N)^{\binom{k}{2}}.$ Then we first need to choose exactly $v_1$ vertices among $[N] \setminus Z$ connected to $z_1$. Subsequently we need to choose exactly $v_2$ vertices among $[N] \setminus ( Z \cup \Gamma_{z_1} )$ and make sure that there are no edges between $z_2$ and $\Gamma_{z_1}$, and so on. Note that this way the edges we consider at each step are independent of the previously considered events and moreover the number of edges between $z_i$ and $[N] \setminus ( Z \cup \cup_{j=1}^{i-1} \Gamma_j)$ is binomially distributed with parameters $N - k - \sum_{j=1}^{i-1} v_i$ and $d/N.$ This gives $$\begin{aligned} & \mathbb{P}\left ( \cap_{i=1}^k \{ \alpha_{z_i} = v_i \} \cap A_Z = 0 \cap \{ \cap_{i=1}^k \Gamma_{z_i} = \emptyset \} \right ) \\ & = \left [ \prod_{i = 1}^k \mathbb{P}\left ( \textnormal{Binom}\left ( N - k - \sum_{j=1}^{i-1} v_j, \frac{d}{N} \right ) = v_i \right ) \left ( 1 - \frac{d}{N} \right )^{\sum_{j=1}^{i-1} v_i} \right ] \left ( 1 - \frac{d}{N} \right )^{\binom{k}{2}}. \end{aligned}$$ We now use Lemma [Lemma 8](#lem:binomtopois){reference-type="ref" reference="lem:binomtopois"} to approximate the binomial probabilities, the bound on the error terms follows from our assumptions on $v_i$, $k$ and the bounds on $d$ from [Definition 5](#dfn:rdfn){reference-type="ref" reference="dfn:rdfn"}. $$\begin{aligned} \mathbb{P}\left ( \textnormal{Binom}\left ( N - k - \sum_{j=1}^{i-1} v_j, \frac{d}{N} \right ) = v_i \right ) & = \left ( 1 + \tilde{O} \left ( N^{-1} \right ) \right ) \mathbb{P}\left ( \text{Pois}\left ( d - \frac{ d \left (k + \sum_{j=1}^{i-1} v_j \right )}{N} \right ) = v_i \right ) \\ & = \left( 1 + N^{-1+o_N(1)} \right ) \mathbb{P}\left (\text{Pois}(d) = v_i \right ). \end{aligned}$$ Moreover $\left ( 1 - \frac{d}{N} \right )^{\sum_{j=1}^{i-1} v_i}$ as well as $\left ( 1 - \frac{d}{N} \right )^{\binom{k}{2}}$ can also be written as $1 + N^{-1+o_N(1)}$. There are $N^{o_N(1)}$ such error terms, which all together implies that $$\begin{aligned} \label{eq:alpha_approx} \mathbb{P}\left ( \cap_{i=1}^k \{ \alpha_{z_i} = v_i \} \cap \{ A_Z = 0 \} \ \cap \{ \cap_{i=1}^k \Gamma_{z_i} = \emptyset \} \right ) & = \left( 1 + N^{-1+o_N(1)} \right ) \mathbb{P}\left (\text{Pois}(d) = v_i \right ). \end{aligned}$$ We now condition on the event $\cap_{i=1}^k \{ \alpha_{z_i} = v_i \} \cap \{ A_Z = 0 \} \cap \{ \cap_{i=1}^k \Gamma_{z_i} = \emptyset \}$ and similarly analyse $\{\beta_{z_i} = w_i \}$. Note that the event $A$ does not require that the $S_2(z_i)$ are all disjoint which makes the analysis slightly simpler. The number of edges from $S_1(z_i)$ to $[N] \setminus ( Z \cup \Gamma_Z )$ is Binomial with parameters $v_i ( N - k - \sum_{j=1}^k v_j)$ and $\frac{d}{N}$, and those random variables are independent since we condition on $A_Z = 0$ and $\cap_{i=1}^k \Gamma_{z_i} = \emptyset$. Finally, the probability that there are no edges within and across any $\Gamma_{z_i}$ is equal to $(1-d/N)^{\sum_{i \neq j} v_iv_j + \sum_i \binom{v_i}{2}}$. Thus, defining $A_\Gamma$ to be the adjacency matrix of $\Gamma_Z = \cup_{i=1}^k \Gamma_{z_i},$ we get $$\begin{aligned} & \mathbb{P}\left ( \cap_{i=1}^k \{ \beta_{z_i} = w_i \} \cap A_\Gamma = 0 \big | \cap_{i=1}^k \{ \alpha_{z_i} = v_i \} \cap \{ A_Z = 0 \} \cap \left \{ \cap_{i=1}^k \Gamma_{z_i} = \emptyset \right \} \right ) \\ & = \left [ \prod_{i = 1}^{k} \mathbb{P}\left ( \textnormal{Binom}\left ( v_i \left ( N - k - \sum_{j = 1}^k v_j \right ), \frac{d}{N} \right ) = w_i \right ) \right ] \left ( 1 - \frac{d}{N} \right )^{ \sum_{i,j=1, i \neq j}^k v_iv_j + \sum_{i=1}^k \binom{v_i}{2} }\end{aligned}$$ The last term can as before be written as $1 + N^{-1+o_N(1)}$. When $v_i=0$, we can immediately replace the Binomial random variables by Poisson random variables with parameter 0, since they are both constant 0. For $v_i>0$, we once more use the Poisson approximation from Lemma [Lemma 8](#lem:binomtopois){reference-type="ref" reference="lem:binomtopois"}, which together with our bounds on $v_i$, $w_i$, $k$ and $d$, gives $$\begin{aligned} \mathbb{P}\left ( \textnormal{Binom}\left ( v_i \left ( N - k - \sum_{j = 1}^k v_j \right ), \frac{d}{N} \right ) = w_i \right ) & = \left ( 1 + \tilde{O}( N^{-1} )\right ) \mathbb{P}\left ( \text{Pois}\left ( v_i d \left ( 1 - \frac{ k + \sum_{j=1}^k v_j }{N} \right ) \right ) = w_i \right ) \\ & = \left ( 1 + N^{-1+o_N(1)}\right ) \mathbb{P}\left ( \text{Pois}\left ( v_i d ) = w_i \right ) \right )\end{aligned}$$ Combining this with [\[eq:alpha_approx\]](#eq:alpha_approx){reference-type="eqref" reference="eq:alpha_approx"}, implies the result. ◻ We need to extend our analysis from a small number of vertices to the entire set. Therefore, we once again emulate the argument of [@alt2023poisson]. Define the parameter $(Z_x)_{x\in[N]}$ to be an exchangeable family of random variables in a measurable space $\mathcal{Z}$. We then define for $F\subset \mathcal Z^k$ and the point process $\Phi$, $$q_{\Phi}(F):=N(N-1)\cdots (N-k+1)\mathbb{P}((Z_1,\ldots Z_k)\in F).$$ **Lemma 43** ([@alt2023poisson] Lemma 7.8). *For $n,m\in \mathbb{N}$ and disjoint, measurable $I_1,\ldots I_n$ $$\begin{gathered} \mathbb{P}\left ( \Phi(I_1)=k_1,\ldots,\Phi(I_n)=k_n \right ) =\frac1{k_1!\cdots k_{n}!}\sum_{\substack{ \ell_1,\ldots,\ell_n \\ \sum \ell_i \leq m}} \frac{(-1)^{\sum_{i}\ell_i}}{\ell_1!\cdots \ell_n!}q_\Phi \left ( I_1^{k_1+\ell_1}\times \cdots \times I_n^{k_n+\ell_n} \right )\\ +O \left (\frac1{k_1!\cdots k_{n}!}\sum_{\substack{ \ell_1,\ldots,\ell_n \\ \sum \ell_i = m+1}} \frac{(-1)^{\sum_{i}\ell_i}}{\ell_1!\cdots \ell_n!}q_\Phi \left (I_1^{k_1+\ell_1}\times \cdots \times I_n^{k_n+\ell_n} \right ) \right ).\end{gathered}$$* We can then use this to show that edge eigenvalues form a Poisson process. *Proof of Theorem [Theorem 3](#thm:process){reference-type="ref" reference="thm:process"}.* By Lemma [Lemma 19](#lem:sizes){reference-type="ref" reference="lem:sizes"}, Theorem [Theorem 2](#thm:maineigenvalue){reference-type="ref" reference="thm:maineigenvalue"}, and Lemma [Lemma 33](#lem:difference){reference-type="ref" reference="lem:difference"}, with high probability, only vertices of degree between $\mathfrak{u}-2\log^{1/8}N$ and $\mathfrak{u}$ contribute to the top $e^{\log^{1/8}N}$ eigenvalues. Similarly, for every relevant vertex $v$, $\beta\leq d\alpha_v+\mathfrak{u}^{7/8}$ for every relevant vertex $v$ with high probability by Lemma [Lemma 24](#lem:fine_balls_disjoint){reference-type="ref" reference="lem:fine_balls_disjoint"}. In this region, $\lambda^2=\alpha_v+\beta_v/\alpha_v+{(d^2+d)}/{\alpha_v}+O((d^{3/2}+1)\mathfrak{u}^{-4/3})$. Moreover, with high probability, neighborhoods of size $r$ around such vertices $v$ are disjoint and treelike by Lemma [Lemma 24](#lem:fine_balls_disjoint){reference-type="ref" reference="lem:fine_balls_disjoint"}. Therefore, define $\mathcal{A}(x,y)$ to be the event that for the ordered pair $(x,y)\in \mathbb{N}^2$, $\mathfrak{u}-2\log^{1/8}N\leq x\leq \mathfrak{u}$ and $0\leq y\leq dx+\mathfrak{u}^{7/8}$, and let $\mathcal{B}$ be the event that the neighborhoods around all such vertices $v \in [N]$ are disjoint tree-like. If we define $\Lambda(\alpha,\beta)=\alpha+\beta/\alpha$, then with high probability $\Phi$ is contained in the intensity measure of $(\Lambda(\alpha_v,\beta_v)+\epsilon(d,N))\mathbf{1}_{\mathcal{A}(\alpha_v,\beta_v) \cap \mathcal{B}}$, where $\epsilon=O(\frac{d^{3/2}+1}{\mathfrak{u}^{4/3}})$. Moreover, $\Psi$ is contained in the intensity measure of $\Lambda(X,Y_{X})\mathbf{1}_{\mathcal{A}(X,Y_X)}$, where $X\sim Pois(d)$ and $Y_X\sim Pois(dX)$. It is then sufficient to show that for $X\sim Pois(d)$, $$\label{eq:tvar} d_{TV}\left((\alpha_v,\beta_v)\mathbf{1}_{\mathcal{A}(\alpha_v,\beta_v) \cap \mathcal{B}},(X,Y_X)\mathbf{1}_{\mathcal{A}(X,Y_X)}\right)=o_N(1).$$ Define the two dimensional point process ${\Phi'}$ given by $(\alpha_v,\beta_v)\mathbf{1}_{\mathcal{A}(\alpha_v,\beta_v) \cap \mathcal{B}}$, and the Poisson process $\Psi'$ as induced by $(X,Y_X)\mathbf{1}_{\mathcal{A}(X,Y_X)}$. We consider all potential ordered pairs $(x,y)$ such that $\mathfrak{u}-2\log^{1/8}N\leq x\leq \mathfrak{u}$ and $0\leq y\leq dx+\mathfrak{u}^{7/8}$. Therefore, if $n_x:=2\log^{1/8}N+1, n_y=d\mathfrak{u}+\mathfrak{u}^{7/8}$+1, there are at most $n:=n_xn_y$ possibilities. Define $|\Phi'((x,y))|$ to be the number of points the point process $\Phi'$ has at $(x,y)$. We consider an event $E$, which for some set $K_E$ of vectors in $\mathbb{N}^{n}$, is defined as follows. We write $\mathcal{E}(X)$ here to mean that the event $X$ occurs. $$E:=\bigsqcup_{\mathbf{k}\in K_E} \mathcal{E}\left ( \left |\Phi'((x_{1},y_1)) \right |=k_1,\ldots, \left |\Phi'((x_{n_x},y_{n_y})) \right | = k_n \right ).$$ By Lemma [Lemma 43](#lem:smalltoall){reference-type="ref" reference="lem:smalltoall"}, $$\begin{gathered} \Pr \left ( |{\Phi'}((x_1,y_1))|=k_1,\ldots, |\Phi'((x_{n_x},y_{n_y}))|=k_n \right ) =\frac1{k_1!\cdots k_{n}!}\sum_{\substack{\ell_1,\ldots,\ell_{n}\\ \sum l_i \leq e^{\log^{2/3}N}/2-1}} \frac{(-1)^{\sum_{i}\ell_i}}{\ell_1!\cdots \ell_{n}!} q_{\Phi'} \left ( I_1^{k_1+\ell_1}\times \cdots \times I_{n}^{k_n + \ell_{n}} \right )\\ +O\left(\frac{1}{k_1!\cdots k_{n}!}\sum_{\substack{\ell_1,\ldots,\ell_{n}\\ \sum \ell_i =e^{\log^{2/3}N}/2}} q_{\Phi'} \left ( I_1^{k_1+\ell_1}\times \cdots \times I_{n}^{k_n+\ell_{n}} \right ) \right )\end{gathered}$$ where $I_i$ is the lattice point $(x_i,y_i)$. We use this threshold for $\sum k_i,$ as by Lemma [Lemma 19](#lem:sizes){reference-type="ref" reference="lem:sizes"} and Markov's inequality, with probability $1-e^{-\Omega(\log^{2/3}N)}$, there are at most $e^{\log^{2/3}N}/2-1$ vertices with degree larger than $\mathfrak{u}- 2 \log^{1/8}N$, which implies that we only need to consider vectors such that $k:=\sum_{i=1}^{n} k_i\leq e^{\log^{2/3}N}/2-1$. By Lemma [Lemma 42](#lem:totalvariation){reference-type="ref" reference="lem:totalvariation"}, $$\begin{gathered} \sum_{\mathbf{k}\in K_{E}}\frac1{k_1!\cdots k_{n}!}\sum_{\substack{\ell_1,\ldots,\ell_{n}\\ \sum l_i \leq e^{\log^{2/3}N}/2-1}} \frac{(-1)^{\sum_{i}\ell_i}}{\ell_1!\cdots \ell_{n}!}q_{\Phi'}(I_1^{k_1+\ell_1}\times \cdots \times I_{n}^{k_n + \ell_{n}})\\=(1+N^{-1+o_N(1)}) \sum_{\mathbf{k}\in K_{E}}\frac1{k_1!\cdots k_{n}!}\sum_{\substack{\ell_1,\ldots,\ell_{n}\\ \sum l_i \leq e^{\log^{2/3}N}/2-1}} N^{k+\ell} \frac{(-1)^{\sum_{i}\ell_i}}{\ell_1!\cdots \ell_{n}!} \prod_{i=1}^n\left(\mathbb{P}(Pois(d)=x_i) \mathbb{P}(Pois(dx_i)=y_i)\right)^{k_i + \ell_i}.\end{gathered}$$ Also, $$\begin{gathered} \frac{1}{k_1!\cdots k_{n} !}\sum_{\substack{\ell_1,\ldots,\ell_{n}\\ \sum \ell_i =e^{\log^{2/3}N}/2}} q_{\Phi'}(I_1^{k_1+\ell_1}\times \cdots \times I_{n}^{k_n + \ell_{n}})\\ =O\left(\frac{1}{k_1!\cdots k_{n}!}\sum_{\substack{\ell_1,\ldots,\ell_{n}\\ \sum \ell_i =e^{\log^{2/3}N}/2}} \frac{1}{(\ell/n)!^n}N^{k+\ell}\mathbb{P}\left ( \text{Pois}(d)=\mathfrak{u}-2\log^{1/8}N \right ) ^{k+\ell} \right).\end{gathered}$$ where $\ell=\sum_{i} \ell_i=e^{\log^{2/3}N}/2$. By the definition of the Poisson, $$\frac{1}{k_1!\cdots k_{n}!}\sum_{\substack{\ell_1,\ldots,\ell_{n}\\ \sum \ell_i =e^{\log^{2/3}N}/2}} \frac{1}{((\ell/n)!)^n}N^{k+\ell}\mathbb{P}\left (Pois(d)=\mathfrak{u}-2\log^{1/8}N \right )^{k+\ell} =e^{-\Omega(e^{\log^{2/3}N})}.$$ By using Lemma [Lemma 43](#lem:smalltoall){reference-type="ref" reference="lem:smalltoall"} once again, $$\begin{gathered} \frac1{k_1!\cdots k_{n}!}\sum_{\sum \ell_1\ldots,\ell_{n}\leq e^{\log^{2/3}N}/2-1}N^{k+\ell} \frac{(-1)^{\sum_{i}\ell_i}}{\ell_1!\cdots \ell_{n}!}\prod_{i=1}^n\left(\mathbb{P}(Pois(d)=x_i) \mathbb{P}(Pois(dx_i)=y_i)\right)^{k_i+\ell_i}\\ =(1+N^{-1+o_N(1)})\Pr(\Psi'(I_1)=k_1+\ell_1,\ldots, \Psi'(I_n)=k_n+\ell_n,\Psi'(I_{n})=k_{n})+e^{-\Omega(e^{\log^{2/3}N})}.\end{gathered}$$ We now wish to pass from this error to total variation distance. The total number of possibilities of $k$ for $\sum_{i=1}^n k_i\leq e^{\log^{2/3}N}/2$ is given by the balls and bins paradigm as $\sum_{k=0}^{e^{\log^\frac{2}{3} N}/2} \binom{n + k - 1}{k - 1} \leq e^{\log^{3}N}$. Therefore, the error for any event is at most $$\label{eq:2dPoissonProcess} d_{TV}\left((\alpha_v,\beta_v)\mathbf{1}_{\mathcal{A}(\alpha_v,\beta_v) \cap \mathcal{B}},(X,Y_X)\mathbf{1}_{\mathcal{A}(X,Y_X) } \right)=e^{\log^3 N}e^{-\Omega(e^{\log^{2/3}N})}+N^{-1+o_N(1)}=o_N(1).$$ ◻ Now we can simply work with independent Poissons, for which the distribution of the maximum is easier to analyze. We start by determining an interval into which the maximizers of the $Y_i$, which approximate the $\beta_x$, fall. **Lemma 44**. *Consider any function $\zeta(N)=\omega_N(1)$ and $1\leq K=\zeta^{o_N(1)}$. For fixed $m>0$, $\mathfrak{a}=\Theta(\mathfrak{u})$, and i.i.d. $Y_1, \dots, Y_\zeta\sim Pois(d\mathfrak{a})$, with probability $1-O_N(\frac{1}{\sqrt{\log \zeta}})$, the $K$ largest values $Y_{(1)},\ldots,Y_{(K)}$ are such that for every $1\leq i\leq K$, $$Y_{(i)}\in \left [ d\mathfrak{a}+ \sqrt{ 2 d\mathfrak{a}\log \zeta} - \sqrt{d\mathfrak{a}}\frac{\log K+\frac12\log2-\log c_{\scaleto{\ref{lem:sharp_poisson_tail}}{3pt}}+\frac32\log \log \zeta }{\sqrt{2\log \zeta}}, d\mathfrak{a}+ \sqrt{2 d\mathfrak{a}\log \zeta} \right ]$$ where $c_{\scaleto{\ref{lem:sharp_poisson_tail}}{3pt}}$ is the constant from Lemma [Lemma 10](#lem:sharp_poisson_tail){reference-type="ref" reference="lem:sharp_poisson_tail"}.* *Proof.* If $Y_{(K)}$ is less than some value $T$, then there are at least $\zeta-K+1$ $Y_i$'s less than $T$. Therefore, $$\mathbb{P}(Y_{(K)} \geq T) \geq 1 - \binom{\zeta}{K-1}( 1 - \mathbb{P}(Y_1 \geq T))^{\zeta-K+1} \geq 1 - \binom{\zeta}{K-1}e^{-(\zeta-K+1) \mathbb{P}(Y_1 \geq T)}\geq 1-e^{(K-1)\log \zeta-(\zeta-K+1) \mathbb{P}(Y_1 \geq T)}.$$ To bound $\mathbb{P}(Y_1 \geq T)$ for $T = d\mathfrak{a}+\sqrt{ 2 d \mathfrak{a}\log \zeta} - \frac{\sqrt{ d \mathfrak{a}}(\log K+\frac12\log2-\log c_{\scaleto{\ref{lem:sharp_poisson_tail}}{3pt}}+\frac32\log \log \zeta )}{\sqrt{2\log \zeta}}$, we use the tail bound from Corollary [Corollary 11](#eq:poistailex){reference-type="ref" reference="eq:poistailex"}, with $$\delta := \frac{1}{\sqrt{d\mathfrak{a}}} \left ( \sqrt{ 2 \log \zeta} - \frac{\log K+\frac12\log2-\log c_{\scaleto{\ref{lem:sharp_poisson_tail}}{3pt}}+\frac32\log \log \zeta }{\sqrt{2\log \zeta}}\right ).$$ As $$d\mathfrak{a}\delta^2/2 = \log \zeta- \log K+\frac12\log2+\log c_{\scaleto{\ref{lem:sharp_poisson_tail}}{3pt}}-\frac32\log \log \zeta + \frac{(\log K+\frac12\log2-\log c_{\scaleto{\ref{lem:sharp_poisson_tail}}{3pt}}+\frac32\log \log \zeta)^2 }{4\log \zeta},$$ Corollary [Corollary 11](#eq:poistailex){reference-type="ref" reference="eq:poistailex"} gives that $$\begin{aligned} &\mathbb{P}\left ( Y_1 \geq d\mathfrak{a}+ \sqrt{d\mathfrak{a}} \left ( \sqrt{ 2 \log \zeta} - \frac{\log K+\frac12\log2-\log c_{\scaleto{\ref{lem:sharp_poisson_tail}}{3pt}}+\frac32\log \log \zeta }{\sqrt{2\log \zeta}}\right )\right)\\ &\geq \frac{c_{\scaleto{\ref{lem:sharp_poisson_tail}}{3pt}}}{\zeta} \frac{e^{ \log K+\frac12\log2-\log c_{\scaleto{\ref{lem:sharp_poisson_tail}}{3pt}}+\frac32\log \log \zeta - \frac{(\log K+\frac12\log2-\log c_{\scaleto{\ref{lem:sharp_poisson_tail}}{3pt}}+\frac32\log \log \zeta)^2 }{4\log \zeta} }}{ \sqrt{ 2 \log \zeta} - \frac{\log K+\frac12\log2-\log c_{\scaleto{\ref{lem:sharp_poisson_tail}}{3pt}}+\frac32\log \log \zeta }{\sqrt{2\log \zeta}} }\\ &=(1+o_N(1))\frac{K\log \zeta}{\zeta}. \end{aligned}$$ Thus $$1- e^{(K-1)\log \zeta - (\zeta-K+1) \mathbb{P}( Y_1 \geq T)} = 1-\zeta^{-1+o_N(1)}.$$ To prove the upper bound we proceed similarly, using that by a union bound $$\label{eq:maxupper} \mathbb{P}(Y_{(1)} \geq T ) \leq \zeta\mathbb{P}( Y_1 \geq T).$$ Using once more the tail bound from Corollary [Corollary 11](#eq:poistailex){reference-type="ref" reference="eq:poistailex"} with $\delta = \sqrt{\frac{2 \log \zeta}{d\mathfrak{a}}}$, we obtain $$\begin{aligned} \mathbb{P}(Y_1 \geq T) \leq \frac{e^{-\log \zeta}}{\sqrt{2 \log \zeta}}, \end{aligned}$$ which means that [\[eq:maxupper\]](#eq:maxupper){reference-type="eqref" reference="eq:maxupper"} can be upper bounded by $\frac{1}{\sqrt{2\log \zeta}}.$ ◻ The following lemma will imply spacing between eigenvalues. **Lemma 45**. *Fix $\mathfrak{a}\in \{\mathfrak{u}-1,\mathfrak{u}\}$. For any $K=\log^{o(1)}N$, with high probability the maximum $K+1$ values of $\beta^{(1)}_x$ of vertices with degree $\mathfrak{a}$ are separated by at least $\frac{(d\mathfrak{u})^{1/2}}{\log(\frac{\mathfrak{u}}{d})^3(K+1)^{3\log\log \log N}}$.* *Proof.* Denote by $\zeta$ the number of vertices of degree $\mathfrak{a}$. We will split into two cases, based on whether $\zeta$ is small relative to $K$. As we will see, if $\zeta$ is small, then we can bound the probability using the anticoncentration of the Poisson. If $\zeta$ is larger, we can shift our focus to the regime of Lemma [Lemma 44](#lem:largest-k-sphere){reference-type="ref" reference="lem:largest-k-sphere"}. It is sufficient to split our cases according to $(K+1)^{\log\log\log(N)}$. Consider two vertices $u,v$ such that $\alpha_u=\alpha_v=\mathfrak{a}$. If $\zeta \leq (K+1)^{\log\log\log(N)}$, then [\[eq:2dPoissonProcess\]](#eq:2dPoissonProcess){reference-type="eqref" reference="eq:2dPoissonProcess"} in the proof of Theorem [Theorem 3](#thm:process){reference-type="ref" reference="thm:process"} implies that the distribution of the $\beta_x$'s approaches the distribution of Poissons, so the probability that $|\beta^{(1)}_u-\beta^{(1)}_v|\leq \eta$ is at most $\frac{2\eta}{\sqrt{d\mathfrak{a}}}+\tilde O(N^{-1/2})$, considering the mode of a Poisson is at its mean, with probability at most $\frac{1}{\sqrt{d\mathfrak{a}}}$. Therefore the probability that any pair is within distance $\eta$ is at most $$\binom{\zeta}2 \frac{2\eta}{\sqrt{d\mathfrak{a}}}\leq \frac{\eta(K+1)^{2\log\log \log N}}{\sqrt{d\mathfrak{a}}}$$ This converges to 0 for $\eta=\frac{(d\mathfrak{u})^{1/2}}{(K+1)^{3\log\log \log N}}$. Otherwise, if $\zeta \geq (K+1)^{\log\log\log(N)}$, referring once more to [\[eq:2dPoissonProcess\]](#eq:2dPoissonProcess){reference-type="eqref" reference="eq:2dPoissonProcess"} in the proof of Theorem [Theorem 3](#thm:process){reference-type="ref" reference="thm:process"} and Lemma [Lemma 44](#lem:largest-k-sphere){reference-type="ref" reference="lem:largest-k-sphere"}, with high probability the $K$ maximizers $x$ of $\beta^{(1)}_x$ satisfy $$\label{eq:window} \beta^{(1)}_x \in \left [ d\mathfrak{a}+\sqrt{ 2 d\mathfrak{a}\log \zeta} - \sqrt{d\mathfrak{a}}\frac{\log K+\frac12\log2-\log c_{\scaleto{\ref{lem:sharp_poisson_tail}}{3pt}}+\frac32\log \log \zeta }{\sqrt{2\log \zeta}},d\mathfrak{a}+\sqrt{2(\log\zeta) d\mathfrak{a}} \right ].$$ To show the improvement in density, we consider the probability that $\beta^{(1)}_x=d\mathfrak{a}+t$, for $|t| = (1+o(_N1)) \sqrt{2(\log \zeta)d\mathfrak{a}}.$ We then have by the Stirling approximation, $$\frac{e^{-d\mathfrak{a}}(d\mathfrak{a})^{d\mathfrak{a}+t}}{(d\mathfrak{a}+t)!}= (1+o_N(1))\frac{e^{t}}{(1+\frac{t}{d\mathfrak{a}})^{d\mathfrak{a}+t}\sqrt{2\pi( d\mathfrak{a}+t)}}.$$ To approximate this, we have $$\begin{aligned} \left ( 1+\frac{t}{d\mathfrak{a}} \right )^{d\mathfrak{a}+t} &=& e^{\log \left ( 1 + \frac{t}{d \mathfrak{a}} \right ) (d \mathfrak{a}+ t )} = e^{\left ( \frac{t}{d \mathfrak{a}} - \frac{t^2}{2(d\mathfrak{a})^2} + O \left ( \frac{t^3}{(d\mathfrak{a})^3} \right ) \right ) (d \mathfrak{a}+ t)} = e^{t + \frac{t^2}{2d \mathfrak{a}} + O \left ( \frac{ t^3}{(da)^2} \right )}\end{aligned}$$ In our window, $t\geq \sqrt{2(\log\zeta)d\mathfrak{a}}-\sqrt{d\mathfrak{a}}\frac{\log K+\frac12\log2-\log c_{\scaleto{\ref{lem:sharp_poisson_tail}}{3pt}}+\frac32\log \log \zeta }{\sqrt{2\log \zeta}}$, and we have $$\frac{e^{-d\mathfrak{a}}(d\mathfrak{a})^{d\mathfrak{a}+t}}{(d\mathfrak{a}+t)!}= \frac{1}{e^{(1+o_N(1))t^2/(2d\mathfrak{a}))}\sqrt{2\pi( d\mathfrak{a}+t)}}\leq \frac{e^{\log K+\frac12\log2-\log c_{\scaleto{\ref{lem:sharp_poisson_tail}}{3pt}}+\frac32\log \log \zeta}}{\zeta^{1-O(\sqrt{\frac{\log \zeta}{d\mathfrak{a}}})} \sqrt{2\pi( d\mathfrak{a}+ t)}}.$$ Here we must have $\sqrt{\frac{\log\zeta}{d\mathfrak{u}}}\rightarrow 0$, therefore, we use the assumption that $d\gg \frac{(\log\log N)^2}{\log N}$. The probability that there are at least two vertices in a window of length $2\eta$ around some $d \mathfrak{a}+ t$ with $t = (1+o_N(1)) \sqrt{2 da \log \zeta}$ is therefore $$\label{eq:windowprob} \binom{\zeta}{2}\left (\frac{2\eta e^{\log K+\frac12\log2-\log c_{\scaleto{\ref{lem:sharp_poisson_tail}}{3pt}}+\frac32\log \log \zeta}}{ \zeta^{1-O(\sqrt{\frac{\log \zeta}{d\mathfrak{a}}})}\sqrt{2\pi(d\mathfrak{a}+t)}}\right )^2\leq4c_{\scaleto{\ref{lem:sharp_poisson_tail}}{3pt}}^{-1} K^2 \eta^2 \log(\frac{\mathfrak{u}}{d})^3(d\mathfrak{u})^{-1}$$ for sufficiently large $N$, considering that with high probability $\zeta\leq (\frac{\mathfrak{u}}{d})^{3/2}$ by Lemma [Lemma 19](#lem:sizes){reference-type="ref" reference="lem:sizes"}. To translate this into distance between $\beta^{(1)}_u$ and $\beta^{(1)}_v$, we cover the large interval corresponding to [\[eq:window\]](#eq:window){reference-type="eqref" reference="eq:window"} with small intervals of length $2\eta$, and centers spaced at distance $\eta$. To cover this large interval, we need at most $\eta^{-1}\sqrt{d\mathfrak{u}}$ small intervals. Therefore, union bounding the probability [\[eq:windowprob\]](#eq:windowprob){reference-type="eqref" reference="eq:windowprob"} gives that $$\begin{aligned} &\mathbb{P}\bigg(\exists u,v\in [N]:\alpha_u=\alpha_v=\mathfrak{a},|\beta^{(1)}_u-\beta^{(1)}_v|\leq \eta\bigg)\mathbf{1}\left(\zeta\geq (K+1)^{\log\log\log N}\right)\\ &\leq4c_{\scaleto{\ref{lem:sharp_poisson_tail}}{3pt}}^{-1} K^2 \eta^2 \log(\frac{\mathfrak{u}}{d})^3(d\mathfrak{u})^{-1} \eta^{-1}\sqrt{d\mathfrak{u}}\\ &\leq5c_{\scaleto{\ref{lem:sharp_poisson_tail}}{3pt}}^{-1} K^2 \eta \log(\frac{\mathfrak{u}}{d})^3(d\mathfrak{u})^{-1/2}\end{aligned}$$ This probability converges to 0 for $\eta=\frac{(d\mathfrak{u})^{1/2}}{\log(\frac{\mathfrak{u}}{d})^3(K+1)^{3\log\log \log N}}$. ◻ **Lemma 46**. *With high probability, for $u\neq v\in \mathcal{W}$ corresponding to the largest $K+1$ $\lambda$'s, we have $|\lambda_u-\lambda_v|\geq \frac{d^{1/2}}{3\mathfrak{u}\log(\frac{\mathfrak{u}}{d})^3(K+1)^{3\log\log \log N}}$.* *Proof.* By Lemma [Lemma 33](#lem:difference){reference-type="ref" reference="lem:difference"}, this is immediately true if $\alpha_u \neq \alpha_v$. Therefore assume $\alpha_u=\alpha_v$. By Lemma [Lemma 45](#lem:eigenvaluespacing){reference-type="ref" reference="lem:eigenvaluespacing"}, with high probability the $K+1$ maximizers of $\beta^{(1)}_u$ are spaced at distance $\frac{(d\mathfrak{u})^{1/2}}{\log(\frac{\mathfrak{u}}{d})^3(K+1)^{3\log\log \log N}}$. Therefore, by Lemma [Lemma 33](#lem:difference){reference-type="ref" reference="lem:difference"}, for $u\neq v$ and sufficiently large $N$, and as $d\gg \log^{-5/3} N$, $$\begin{aligned} \nonumber|\lambda_u-\lambda_v|&=&\frac{|\lambda^2_u-\lambda^2_v|}{|\lambda_u+\lambda_v|}\\ \nonumber&\geq&\left(\frac{\frac{(d\mathfrak{u})^{1/2}}{\log(\frac{\mathfrak{u}}{d})^3(K+1)^{3\log\log \log N}}}{\mathfrak{u}} +O((1+d^{3/2})\mathfrak{u}^{-4/3})\right)\frac{1}{2\sqrt{\mathfrak{u}}+O(\frac d{\sqrt{\mathfrak{u}}})}\\ \label{eq:eigenvaluespacing} &\geq&\frac{d^{1/2}}{3\mathfrak{u}\log(\frac{\mathfrak{u}}{d})^3(K+1)^{3\log\log \log N}} \end{aligned}$$ by the lower bound on $d$. ◻ # Eigenvector Structure {#sec:eigenvector} **Proposition 47**. *For $k\leq K=\log^{o_N(1)}N$, define $x$ to be the vertex corresponding to the $k$th largest eigenvalue of $A$, as per Theorem [Theorem 2](#thm:maineigenvalue){reference-type="ref" reference="thm:maineigenvalue"}. The eigenvector $\mathbf{v}$ of $\lambda$ satisfies* *$$\|\mathbf{v}-\mathbf{w}_+(x)\|=O \left ( \mathfrak{u}^{-r/2+2} \right ).$$* *Proof.* By Theorem [Theorem 2](#thm:maineigenvalue){reference-type="ref" reference="thm:maineigenvalue"}, there is a correspondence between the top $K+1$ eigenvalues and eigenvectors of the matrix $A$ and the top $K+1$ eigenvalues $\lambda_x$ of the truncated balls around vertices $x \in \mathcal{W}$ together with their eigenvectors. Moreover, by Lemma [Lemma 46](#lem:lambdaubound){reference-type="ref" reference="lem:lambdaubound"}, the difference between each pair of these $K+1$ eigenvalues is at least $\frac{d^{1/2}}{4\mathfrak{u}\log(\frac\mathfrak{u}d)^3(K+1)^{3\log\log\log N}}$. Standard perturbation theory (see [@greenbaum2020first] Theorem 2 and the remarks following it) gives that, if we fix the index $1\leq i\leq K$, $$\begin{aligned} \|\mathbf{v}-\mathbf{w}_{+}(x)\|&\leq& \|E_W\|\cdot \Big (\min_{j\neq i}|\lambda-\lambda_j| \Big )^{-1}\\ &\leq& O \left ( \frac{\mathfrak{u}\log(\frac{\mathfrak{u}}{d})^3(K+1)^{3\log\log \log N}}{d^{1/2}}(d^{r}+1)\mathfrak{u}^{-r/2+1/2} \right )\\ &=&O \left (\mathfrak{u}^{-r/2+2} \right ).\end{aligned}$$ by our assumptions on $d$ from Definition [Definition 5](#dfn:rdfn){reference-type="ref" reference="dfn:rdfn"} and $K$, and the bound on $\| E_W \|$ from Theorem [Theorem 22](#thm:structure){reference-type="ref" reference="thm:structure"}. ◻ *Proof of Theorem [Theorem 4](#thm:maineigenvector){reference-type="ref" reference="thm:maineigenvector"}.* By Proposition [Proposition 30](#prop:expdecay){reference-type="ref" reference="prop:expdecay"}, Proposition [Proposition 47](#prop:vecstructure){reference-type="ref" reference="prop:vecstructure"}, and the triangle inequality, $$\begin{aligned} \mathbf{v}|_{x}&=&\frac1{\sqrt 2}+O \left ( \left (1+d^{-1/2} \right )\mathfrak{u}^{-1/3}+\mathfrak{u}^{-r/2+2} \right )\\ &=&\frac1{\sqrt 2}+O \left ( \left (1+d^{-1/2} \right )\mathfrak{u}^{-1/3} \right ) .\end{aligned}$$ as $r\geq 5$. Moreover, as $r\geq 2r'$, for $1 \leq i \leq r',$ $$\begin{aligned} \left \|\mathbf{v}|_{S_i(x)}\right \| &= &\left(\frac{d}{\alpha}\right)^{(i-1)/2}\frac1{\sqrt 2} \left (1+O \left ( \left (1+d^{-1/2}+d^{-(i-1)} \right )\mathfrak{u}^{-1/3} \right) \right )+O \left (\mathfrak{u}^{-r/2+2} \right )\\ &=&\left(\frac{d}{\alpha}\right)^{(i-1)/2}\frac1{\sqrt 2}\left (1+O \left ( \left (1+d^{-1/2}+d^{-i+1} \right )\mathfrak{u}^{-1/3} \right) \right)\end{aligned}$$ as desired. Similarly, $$\begin{aligned} \left \|\mathbf{w}_+|_{[N]\backslash B_{i}(x)} \right \| = \left(\frac{d}{\alpha}\right)^{i/2} \left (1+O \left ( \left ( 1+d^{-1/2}+d^{-r+1} \right )\mathfrak{u}^{-1/3} \right) \right ). \end{aligned}$$ ◻ # Estimates {#sec:estimates} ## Binomial estimates Although we will mostly approximate the degrees by Poisson random variables it is sometimes more convenient to work with the precise distribution. To this end we state here a classical tail bound we will use. We start by reproducing a classic tail bound for Binomial random variables. **Lemma 48** (Lemma 4.7.2 in [@ash65]). *Let $X \sim \textnormal{Binom}( n, p )$, and define $I_p(q) := q \log \frac{q}{p} + (1-q) \log \frac{1-q}{1-p}$. Then for $k > np$ $$\mathbb{P}( X \geq k ) \leq e^{-n I_p \left ( \frac{k}{n} \right )},$$ where $I_p(q) = q \log \frac{q}{p} + (1-q) \log \frac{1-q}{1-p}$.* By considering $n-X$, the bound above implies that similarly for $k < np$, $\mathbb{P}( X \leq k ) \leq e^{-n I_p \left ( \frac{k}{n} \right )}.$ As we will be interested in vertices of large degree in our graph, and the degrees follow a binomial distribution, we will repeatedly use tail bounds such as the following. **Lemma 49**. *Let $m = n + o(n)$ and $p = \frac{d}{N}$, and define $X \sim \textnormal{Bin}(m,p)$ then, for $\tau = o(n),$ it holds for some constant $c$, that for $n$ large enough, $$\mathbb{P}\left ( X \geq \tau \right ) \leq e^{-\tau \log (\tau) + c \tau }.$$* *Proof.* By Lemma [Lemma 48](#lem:ash-binomial){reference-type="ref" reference="lem:ash-binomial"}, $$\mathbb{P}\left ( X \geq \tau \right ) \leq e^{- m I_p(\frac{\tau}{m})}.$$ Using that $\log(x) \geq \frac{x}{1+x}$ for $x > -1$, and that $\frac{d-\tau}{n-\tau} > - \frac{1}{2}$ for $n$ large enough, we get that $$\begin{aligned} I_p \left (\frac{\tau}{m} \right ) & = \frac{\tau}{m} \log \left ( \frac{\tau}{d } \frac{n}{m} \right ) + \left ( 1 - \frac{\tau}{m} \right ) \log \left ( \frac{ n - \tau }{ n - d} \frac{m-\tau}{n-\tau} \frac{n}{m} \right ) \\ & \geq \frac{\tau}{m} \log (\tau) - \frac{\tau}{m} \log (d) - 2 \left ( 1 - \frac{\tau}{m} \right ) \frac{d-\tau}{n-\tau} + \left ( 1 - \frac{\tau}{m} \right ) \log \left ( \frac{m-\tau}{n-\tau} \right ) + \log \left ( \frac{n}{m} \right )\end{aligned}$$ Now note that, after multiplication by $m$, all but the first term are $O\left ( \tau \right )$, which implies the bound. ◻ ## Poisson Approximation *Proof of Lemma [Lemma 8](#lem:binomtopois){reference-type="ref" reference="lem:binomtopois"}.* We simplify using Sterling's approximation and the fact that $\frac{e^c}{(1+\frac cn)^n}=1+O(\frac{c^2}{n})$ for $|c| < n$: $$\begin{aligned} \mathbb{P}(X=k)&=&\binom{n}{k} p^k (1-p)^{n-k}\\ &=& \left (1+O \left (\frac1n \right ) \right )\frac{1}{k!}\frac{n^n}{(n-k)^{n-k}e^{k}}\sqrt{\frac{1}{1-\frac{k}{n}}}p^k(1-p)^{n-k}\\ &=& \left (1+O \left (\frac{k^2+(np)^2+1}n \right ) \right )\frac{e^{-np}(np)^k}{k!}\\ &=&\left (1+O \left (\frac{k^2+(np)^2+1}n \right ) \right )\mathbb{P}(Y=k)\end{aligned}$$ ◻ *Proof of Lemma [Corollary 9](#cor:bintopoistail){reference-type="ref" reference="cor:bintopoistail"}.* We have $$\begin{aligned} \mathbb{P}(X\geq k)&=& \left (1+O \left (\frac{k^2+(np)^2+1}{n} \right ) \right )\mathbb{P}(Y\geq k)+O\left(\mathbb{P}\left (X\geq \sqrt n \right )+\mathbb{P}\left (Y\geq \sqrt n \right )\right)\end{aligned}$$ and the latter term satisfies $$\begin{aligned} O\left(\mathbb{P}\left (X\geq \sqrt n \right )+\mathbb{P}\left (Y\geq \sqrt n \right )\right) = O\left(\binom{n}{\sqrt{n}}p^{\sqrt n}+\frac{(np)^{\sqrt n}}{\sqrt n!}\right) = O\left(\left( ep \sqrt{n} \right)^{\sqrt n}\right).\end{aligned}$$ ◻ *Proof of Lemma [Lemma 10](#lem:sharp_poisson_tail){reference-type="ref" reference="lem:sharp_poisson_tail"}.* $$\begin{aligned} \mathbb{P}\big ( X \geq \lambda ( 1 + \delta ) \big ) & = \sum_{ k=\lambda ( 1 + \delta ) }^{\infty} \mathbb{P}( X = k ) \\ &=e^{ -\lambda } \sum_{ k=\lambda ( 1 + \delta ) }^{\infty} \frac{ \lambda^k }{ k! }. \end{aligned}$$ Note that for $k$ in our range, $\lambda/k\leq 1/(1+\delta)$. Therefore, we can upper bound this with a geometric series. $$\begin{aligned} e^{ -\lambda } \sum_{ k=0 }^{\infty} \frac{ \lambda^k }{ k! }&\leq &e^{ -\lambda }\frac{\lambda^{\lambda ( 1 + \delta ) }}{(\lambda ( 1 + \delta ) )!} \sum_{ k=0 }^{\infty} \frac{1 }{ (1+\delta)^k }\\ &\leq&e^{ -\lambda }\frac{\lambda^{\lambda ( 1 + \delta ) }}{(\lambda ( 1 + \delta ) )!} (\frac{1+\delta}{\delta}).\end{aligned}$$ Using a Stirling approximation, we obtain $$\begin{aligned} e^{ -\lambda }\frac{\lambda^{\lambda ( 1 + \delta ) }}{(\lambda ( 1 + \delta ) )!}(\frac{1+\delta}{\delta}) &=& (1+o_\lambda(1))e^{-\lambda}\frac{e^{\lambda(1+\delta)}}{\sqrt{2\pi \lambda(1+\delta)}}\frac{\lambda^{\lambda ( 1 + \delta )}}{(\lambda ( 1 + \delta ) )^{\lambda ( 1 + \delta )}}(\frac{1+\delta}{\delta})\\ &=& (1+o_\lambda(1)) \frac{\exp(-\lambda (1+\delta)\log(1+\delta)+\lambda(1+\delta)-\lambda)}{\sqrt{2\pi \lambda(1+\delta)}\frac{\delta}{1+\delta}}\\ &\leq& \frac{\exp(-\lambda (1+\delta)\log(1+\delta)+\lambda \delta )}{\sqrt{\lambda}\min\{\sqrt{\delta},\delta\}}\end{aligned}$$ for sufficiently large $\lambda$. **Lower bound:** We once again have that $$\begin{aligned} \mathbb{P}\big ( X \geq \lambda ( 1 + \delta ) \big ) & = \sum_{ k=\lambda ( 1 + \delta ) }^{\infty} \mathbb{P}( X = k ) \\ &=e^{ -\lambda } \sum_{ k=\lambda ( 1 + \delta ) }^{\infty} \frac{ \lambda^k }{ k! }\\ &=\frac{(1+o_\lambda(1))e^{ -\lambda }}{\sqrt{2\pi}}\sum_{k\geq \lambda(1+\delta)}\frac{\lambda^ke^k}{k^{k+1/2}} \end{aligned}$$ by a Stirling approximation. We write $$c:=\lambda(1+\delta),~~~~~~f(x):=-(x+1/2)\log x+x\log \lambda+x.$$ Therefore, we have $$f'(x)=-\log x+\log \lambda +\frac{1}{2x},~~~~~f''(x)=-\frac{1}{x}+\frac{1}{2x^2}$$ We then approximate our probability by an integral, which serves as a lower bound as our function is decreasing. We then perform a Laplace method type bound. $$\begin{aligned} \sum_{k= \lambda(1+\delta)}^{{c+c^{1/3}}}\frac{\lambda^ke^k}{k^{k+1/2}}&\geq&\int_{c}^{c+c^{1/3}} \frac{\lambda^{x}e^{x}}{(x)^{{x}+1/2}}dx\\ &=&\int_c^{c+c^{1/3}} \exp\left(f(x)\right)dx\\ &=&e^{f(c)}\int_c^{c+c^{1/3}} \exp\left(f(x)-f(c) \right)dx\\ &=&e^{f(c)}\int_c^{c+c^{1/3}} \exp\left(f'(c)(x-c) +O_\lambda(f''(c)(x-c)^2 )\right)dx\end{aligned}$$ where the last statement follows from Taylor expanding and the formula of $f''(x)$. By the choice of our window, $f''(c)(x-c)^2=O(\lambda^{-1/3})$. Therefore, this is $$\begin{aligned} e^{f(c)}\int_c^{c+c^{1/3}} \exp\left(f'(c)(x-c) +O_\lambda(f''(c)(x-c)^2 )\right)dx&=&(1+o_\lambda(1))e^{f(c)-cf'(c)}\int_{c}^{c+c^{1/3}} \exp(f'(c)x) dx\\ &=&-(1+o_\lambda(1))\frac{e^{f(c)-cf'(c)+cf'(c)}}{f'(c)}(1-e^{c^{1/3}f'(c)})\\ &=&-(1+o_\lambda(1))\frac{e^{f(c)}}{f'(c)}.\end{aligned}$$ By our choice of $c$, and under the assumption that $\delta>\frac{1}{\sqrt{\lambda}}$, we have $$-\frac{1}{f'(c)}=(1+o_\lambda(1))\frac{1}{\log (1+\delta)}.$$ Putting this together with the fact that $\mathbb{P}\big ( X \geq \lambda ( 1 + \delta ) \big )\geq \mathbb{P}\big ( X = \lambda ( 1 + \delta ) \big )$, we have that $$\begin{aligned} \mathbb{P}\big ( X \geq \lambda ( 1 + \delta ) \big )&\geq&(1+o_\lambda(1))e^{-\lambda}\frac{\lambda^{\lambda(1+\delta)}e^{\lambda(1+\delta)}}{(\lambda(1+\delta))^{\lambda(1+\delta)+1/2}}\cdot \max\left\{\frac{1}{\log (1+\delta)},1\right\}\\ &=&(1+o_\lambda(1))c_{\scaleto{\ref{lem:sharp_poisson_tail}}{3pt}}\frac{\exp(-\lambda h(\delta))}{\sqrt{\lambda}\min\{\sqrt{\delta},\delta\}}\end{aligned}$$ for some constant $c_{\scaleto{\ref{lem:sharp_poisson_tail}}{3pt}}$. ◻ *Proof of [Lemma 14](#lem:weibullbound){reference-type="ref" reference="lem:weibullbound"}.* We will only work with the upper tail as the lower tail is similar. The bound we get from Lemma [\[lem:weakertail\]](#lem:weakertail){reference-type="ref" reference="lem:weakertail"} is $$\Pr \left (X>t+\mathbb{E}(X) \right ) \leq \exp\left(-\frac{t^2}{2d+\frac23t}\right).$$ Therefore, $$\Pr \left ( X^2\geq t^2+2t\mathbb{E}[X]+\mathbb{E}[X]^2 \right )\leq \exp\left(-\frac{t^2}{2d+\frac23t}\right).$$ which we rewrite as $$\Pr \left ( X^2-(d^2+d)\geq t^2+(2t-1)\mathbb{E}[X] \right )\leq \exp\left(-\frac{t^2}{2d+\frac23t}\right).$$ By using the substitution $t=\sqrt{y+d^2+d}-d$, we can rewrite this as $$\Pr \left ( X^2-(d^2+d)\geq y \right )\leq \exp \left (-\frac{y+2d^2+d-2d\sqrt{y+d^2+d}}{\frac23\sqrt{y+d^2+d}+\frac43d} \right ).$$ As mentioned, this will follow from [@bakhshizadeh2023sharp], Theorem 1. First, we wish to show that, using the notation in the paper, $v(L,\beta)\rightarrow \textnormal{Var}(X^2)$ for large $L$. We first show that the strategy of proof of Lemma 4 extends to our scenario. The probability that $X^2-(d^2+d)\geq y$ is at most $e^{-\frac{\sqrt{y}}{12}}$ for $y\geq 4(d^2+d)$. Therefore, we choose our $Y$ to be $$Y=(X^2-\mathbb{E}(X^2))^2\exp( X/12)\mathbf{1}_{X^2>E(X^2)}.$$ $Y$ is integrable as the moment generating function $\mathbb{E}(e^{cX})$ is finite for all $c$. Therefore, for sufficiently large $mt$, $v(L,\beta)\approx \textnormal{Var}(X^2)$. This makes the formulation of Theorem 1 in [@bakhshizadeh2023sharp] much simpler, and $t_{\max}$ is such that $$t_{\max}=\textnormal{Var}(X^2)\frac{mt_{\max}+2d^2+d-2d\sqrt{mt_{\max}+d^2+d}}{\frac23 mt_{\max}\sqrt{mt_{\max}+d^2+d}+\frac43d}.$$ Therefore $t_{\max}=(1+o_m(1))(\frac32\textnormal{Var}(X^2))^{2/3}m^{-1/3}$. Using this on the four terms in Theorem 1 from [@bakhshizadeh2023sharp], as well as the bound that $I(t)\geq \sqrt{t}/12$ if $t\geq 4(d^2+d)$, gives the result. ◻ # Structure surrounding vertices {#sec:structure} *Proof of Lemma [Lemma 24](#lem:fine_balls_disjoint){reference-type="ref" reference="lem:fine_balls_disjoint"}, 1.* We essentially follow the proof of Lemma 7.3 in [@alt2021extremal]. If the balls of radius $r$ around vertices of $\mathcal{V}$ are not disjoint there must be two vertices $x$ and $y$ in $\mathcal{V}$, connected by a simple path of length at least 1 (if the two vertices are connected by an edge) and at most $2r$. Let us denote the vertices on this path by $z_1, \dots, z_s$, where $s$ can range from 0 to $2r-1$. If we define $z_0 = x$ and $z_{s+1} = y$, then we must have an edge from $z_i$ to $z_{i+1}$ for $i = 0, \dots, s$. Eventually we will take a union bound over all $x, y$ and paths between them, but let us first compute the probability of a single such event. For convenience of notation let $\tau =\mathfrak{u}- \mathfrak{u}^{2/3}$, the degree lower bound for the intermediate regime. For a given $s$ as well as distinct vertices $x,y, z_1, \dots, z_s$ in $[N]$, $$\begin{aligned} & \mathbb{P}\left (x, y \in \mathcal{V}, (z_i, z_{i+1}) \in E(G) \text{ for } i \in \{0,\dots, s\} \right ) \\ & \leq \mathbb{P}\left ( \left |\Gamma_x\backslash \{y,z_1\} \right |\geq \tau - 2, \left |\Gamma_y\backslash \{x,z_s\} \right | \geq \tau - 2, (z_i, z_{i+1}) \in E(G) \text{ for } i \in \{0, \dots, s\} \right ) \\ & = \mathbb{P}\left ( \left |\Gamma_x\backslash \{y,z_1\} \right |\geq \tau - 2 \right ) \mathbb{P}\left ( \left |\Gamma_y\backslash \{x,z_s\} \right |\geq \tau-2 \right ) \left (\frac dN \right )^{s+1} \end{aligned}$$ as now all events are independent and the path contains $s+1$ edges. As $|\Gamma_x\backslash \{y,z_1\}|$ and $|\Gamma_y\backslash \{x,z_s\}|$ are distributed as $\textnormal{Bin}(N-2, d/n)$, we can use Lemma [Corollary 12](#lem:binom_heavy_tail){reference-type="ref" reference="lem:binom_heavy_tail"} to get $$\begin{aligned} \mathbb{P}\left ( |\Gamma_x\backslash \{y,z_1\}| \geq \tau-2 \right ) \leq (1+o_N(1)) e^{-\tau\log\tau+\tau\log d + \tau - d }\leq e^{O(\mathfrak{u}^{2/3})-\log N}\end{aligned}$$ by Lemma [Lemma 19](#lem:sizes){reference-type="ref" reference="lem:sizes"}. Thus by using a union bound, we can bound the probability that two balls of radius $r$ around two vertices in $\mathcal{V}$ intersect, by considering that we can choose $x,y$ in $\binom{N}{2}$ ways and then we need to choose the path between $x$ and $y$, i.e. for some $s$ between $0$ and $2r$, we need $s$ ordered vertices, for which there are $(N-2)_s$ ways. Combining this with the above bounds gives that the probability of two vertices in $\mathcal{V}$ being connected by such a path is bounded by $$\begin{aligned} \binom{N}{2} \sum_{s=0}^{2r} (N-2)_s \left(\frac dN\right)^{s+1} e^{-(2+o_N(1))\log N} & \leq N^2 \sum_{s=0}^{2r} N^s \left ( \frac{d}{N} \right )^{s+1}e^{-(1+o_N(1))2\log N} \\ & \leq f(d,r) e^{-(1+o_N(1))\log N}\end{aligned}$$ where we bounded $\sum_{s=0}^{2r} d^{s+1}$ by $f(d,r) := d\frac{d^{2r+1}-1}{d-1}$ when $d \neq 1$ and $f(d,r) = 2r+1$ when $d = 1$. Thus for any constant $r$ the event holds with high probability. ◻ In order to prove that the balls around vertices in the fine regime are with high probability trees, we start by bounding the probability that the ball around a fixed vertex contains $m$ excess edges, this result and its proof are almost identical to Lemma 5.5 in [@alt2021extremal] but we chose to include them for sake of completeness. **Lemma 50**. *For a vertex $x \in [N]$, any integer $C_1 \geq 1$ and any constant $s$ it holds that $$\mathbb{P}\left ( E(B_s(x)) \geq V( B_s(x) -1 + C_1 | S_1(x) \right ) \leq C d^{3 C_1} \left ( \frac{ |S_1|}{N} \right )^{C_1},$$ where $C$ is a constant that depends on the constants $s$ and $C_1$.* *Proof.* We use the argument from Lemma 5.5 in [@alt2021extremal], but obtain a slightly different bound that is better suited for our regime of $d$. Let $T$ be a spanning tree of $B_s(x)$. If $B_s(x)$ contains at least $C_1$ excess edges, there are $C_1$ edges in $B_s(x)$ not contained in $T$, denote those by $E_E$. Let $V_E$ denote the vertices incident to those edges and $E_P$ the edges on the unique paths in $T$ from $x$ to the vertices of $V_E$. Finally let $V_P$ denote the vertices incident to edges in $E_P$. (See Figure 4 in [@alt2021extremal] for an illustration.) We define $H$ to be the graph with vertices $V_E \cup V_P$ and edges $E_E \cup E_P$. Let $S^F_r(x)$ denote that sphere of radius $r$ around $x$ in the graph $F$. Then the graph $H$ is a graph on the vertices $[N]$ that satisfies the following properties: 1. $x \in F$, 2. $S_1^F(x) \subseteq S_1^G(x)$, 3. $|S_1^F(x)| \geq 1$ 4. $E(F) = V(F) - 1 + C_1$ and 5. $V(F) \leq 2 C_1 r + 1$. The last property holds since the edges $E_E$ incident to at most $2C_1$ distinct vertices and the paths in $T$ from $x$ to those vertices are of length at most $r$, which implies that $V_P$ contains at most $2 C_1 r$ additional distinct vertices besides $x$. Thus we can bound the probability that $E( B_s(x) ) \geq V( B_s(x) ) - 1 + C_1$ by the probability that $B_s(x)$ contains a subgraph $F$ satisfying properties 1.-5. above. For a given $x \in \mathcal{V}$, we start by conditioning on $S_1(x)$. Then let $\mathbb{F}(x)$ denote the subgraphs satisfying properties 1.-5. Recalling that $G$ denotes our Erdős-Rényi graph $\mathcal{G}\left ( N, \frac{d}{N} \right )$, we can bound $$\begin{aligned} \mathbb{P}\left ( E \left ( B_s(x) \right ) \geq V \left ( B_s(x) \right ) - 1 + C_1 | S_1(x) \right ) & \leq \mathbb{P}\left ( \cup_{F \in \mathbb{F}(x)} F \subseteq G \big | S_1(x) \right ) \\ & \leq \sum_{ F \in \mathbb{F}(x) } \mathbb{P}\left ( F \subseteq G \big | S_1(x) \right ) . \end{aligned}$$ Now note that conditioned on $S_1$ we can construct any graph $F \in \mathbb{F}(x)$ by first choosing $1 \leq s \leq C_1$ vertices from $S_1(x)$, then choosing $0 \leq t \leq 2 C_1 r - s$ (we lose the $+1$ since $x$ is always part of $F$) vertices from $[N] \setminus \mathcal{B}_1(x)$, and then building a tree with these $s + t + 1$ vertices such that the first $s$ are neighbors of $x$ and the remaining $t$ vertices connect to that graph (but not to $x$), and then adding $C_1$ additional edges. We can bound the number of such graphs by the number of labeled trees on $s+t+1$ vertices (for which Cayley's formula gives that there are $(s+t+1)^{s+t-1}$) times the number of ways of choosing $C_1$ edges, which can be bounded by $(s+t+1)^{2C_1}.$ The probability that such a graph is contained in $G$ is then equal to $\left ( \frac{d}{N} \right )^{ t + C_1}$ since the number of edges in $F$ without those between $x$ and vertices in $S_1(x)$ is equal to $t + C_1.$ So continuing from above, we get $$\begin{aligned} & \leq \sum_{s = 1}^{C_1} \sum_{t = 0}^{ 2 C_1 r - s} \binom{|S_1|}{s} \binom{ N - |S_1(x)| - 1}{t} (s+t+1)^{s+t-1 + 2C_1} \left ( \frac{d}{N} \right )^{ t + C_1} \\ & \leq \sum_{s = 1}^{C_1} \sum_{t = 0}^{ 2 C_1 r - s} \frac{ |S_1|^s}{s!} \frac{N^t}{t!} (s+t+1)^{s+t + 2C_1 -1} \left ( \frac{d}{N} \right )^{ t + C_1} \\ & \leq \frac{1}{N^{C_1}} \left ( d( 2C_1 r + 1 )^2 \right )^{C_1} \sum_{s = 1}^{C_1} \frac{ |S_1|^s}{s!} (2 C_1 r + 1 )^{s} \sum_{t = 0}^{ 2 C_1 r - s} \frac{1}{t!} (d(2 C_1 r + 1 ))^{t} \\ & \leq \frac{1}{N^{C_1}} \left ( d(2 C_1 r + 1 )^2 \right )^{C_1} C_1 \left ( |S_1| (2C_1 r + 1) \right )^{C_1} 2 C_1 r \left ((d(2 C_1 r + 1 ))^2 \right )^{2C_1r} \\ & \leq \frac{1}{N^{C_1}} 2 r C_1^{ 2 } d^{3 C_1} (2C_1 r + 1)^{5 C_1 } |S_1| ^{C_1} \\ & \leq C d^{3 C_1} \left ( \frac{ |S_1|}{N} \right )^{C_1}, \end{aligned}$$ where $C$ is a constant that depends on the constants $r$ and $C_1$. ◻ *Proof of Lemma [Lemma 24](#lem:fine_balls_disjoint){reference-type="ref" reference="lem:fine_balls_disjoint"}, 2.* Note that by a union bound over all vertices, we get $$\begin{aligned} & \mathbb{P}\left ( \exists x \in \mathcal{V}: \mathcal{B}_r(x) \text{ is not a tree} \right ) \\ \leq & \sum_{x \in [N]} \mathbb{P}\left ( \mathcal{B}_r(x) \text{ is not a tree}, x \in \mathcal{V}, \mathfrak{u}- \mathfrak{u}^{\frac{2}{3}} \leq \alpha_x < 2 \frac{\log N}{\log \log N} \right ) + \mathbb{P}\left ( \alpha_x \geq 2 \frac{\log N}{\log \log N} \right ) \\ \leq & \sum_{x \in [N]} \mathbb{E}\left [ \mathbb{P}\left ( \mathcal{B}_r(x) \text{ is not a tree} | S_1(x) \right ) \mathbf{1}\left (\mathfrak{u}- \mathfrak{u}^{\frac{2}{3}} \leq \alpha_x < 2 \frac{\log N}{\log \log N} \right ) \right ] + \mathbb{P}\left ( d_x \geq 2 \frac{\log N}{\log \log N} \right ) \\ \leq & \sum_{x \in [N]} \mathbb{E}\left [ C d^{3 C_1} \frac{ |S_1| }{ N } \mathbf{1}\left (\mathfrak{u}- \mathfrak{u}^{\frac{2}{3}} \leq \alpha_x < 2 \frac{\log N}{\log \log N} \right ) \right ] + \mathbb{P}\left ( d_x \geq 2 \frac{\log N}{\log \log N} \right ) \\ \leq & N C d^{3 C_1} \frac{2 \log N}{N} \frac{e^{d+ \mathfrak{u}^{\frac{2}{3} \log \mathfrak{u}}}}{N} + N^{-\frac{3}{2}},\end{aligned}$$ where we apply the bound from Lemma [Lemma 50](#lem:excess_edges){reference-type="ref" reference="lem:excess_edges"} with $C_1 = 1$, as well as the bound on $|\mathcal{V}|$ from [Lemma 19](#lem:sizes){reference-type="ref" reference="lem:sizes"} and then used Lemma [Corollary 12](#lem:binom_heavy_tail){reference-type="ref" reference="lem:binom_heavy_tail"} for the second term. ◻ *Proof of Lemma [Lemma 24](#lem:fine_balls_disjoint){reference-type="ref" reference="lem:fine_balls_disjoint"}, 3.* We show this for $\mathcal{V}$. The proof for $\mathcal{W}$ is idendical. For a vertex $x$, and constants $C_i$ that will be set later, let us define the events $$\mathcal{G}_i(x) : = \left \{ \left||S_{i}(x)|-d^{i-1} \alpha_x \right|\leq C_i \left ( d^{i - \frac{3}{2}} + 1 \right ) \mathfrak{u}^{\frac{7}{8}} \right \}$$ and $\mathcal{F}_i(x) := \bigcap_{j = 1}^i \mathcal{G}_i(x).$ We will write $\mathcal{G}_i$ and $\mathcal{F}_i$ whenever it is clear from context which vertex they refer to. First note that under $\mathcal{F}_i(x)$, $|B_i(x)| \leq \sqrt{N}.$ Now fix a vertex $x$. $\mathcal{G}_1$ holds trivially by the definition of $\alpha_x$. For $i \geq 2$ we now first show that conditional on $S_1$ the probability that $S_i$ is large given that $S_{i-1}$ is small is small. More precisely we show that $$\mathbb{P}\left ( \mathcal{G}_i^c \cap \mathcal{F}_{i-1} | S_1 \right ) \leq 2 \exp \left \{ - \mathfrak{u}^\frac{3}{4} \right \}$$ To show the above equation first observe that conditioned on $B_{i-1}$, $S_i$ consists of all the neighbors of vertices in $S_{i-1}$ that are not in $B_{i-1}$. Thus, conditionally on $B_{i-1}$, $|S_i|$ is distributed as $\textnormal{Binom}(|S_{i-1}|(N-|B_{i-1}|),d/N)$. Note that this implies that $$\mathbb{E}[ |S_{i}| | B_{i-1} ] = d |S_{i-1}| - d \frac{|S_{i-1}||B_{i-1}|}{N}.$$ Thus under the event $\mathcal{F}_i$, by Lemma [\[lem:weakertail\]](#lem:weakertail){reference-type="ref" reference="lem:weakertail"}, because $|B_i| \leq \sqrt{N},$ $$\begin{aligned} & \mathbb{P}\left ( \left | |S_i| - \mathbb{E}\left [ |S_i| \big | B_{i-1} \right ] \right | \geq \sqrt{d |S_{i-1}|} \mathfrak{u}^\frac{3}{8} + \mathfrak{u}^\frac{7}{8} \Big | B_{i-1} \right ) \\ & \leq 2 \exp \left \{ - \frac { \left ( \sqrt{d |S_{i-1}|} \mathfrak{u}^\frac{3}{8} + \mathfrak{u}^\frac{7}{8} \right )^2} {2 \left (d |S_{i-1}| - d \frac{|S_{i-1}||B_{i-1}|}{N} \right ) + \frac{2}{3} \left ( \sqrt{d |S_{i-1}|} \mathfrak{u}^\frac{3}{8} + \mathfrak{u}^\frac{7}{8} \right ) } \right \} \\ & \leq 2 \exp \left \{ - \frac { \left ( \sqrt{d |S_{i-1}|} \mathfrak{u}^\frac{3}{8} + \mathfrak{u}^\frac{7}{8} \right )^2} {2 \left (d |S_{i-1}| - d \right ) + \frac{2}{3} \left ( \sqrt{d |S_{i-1}|} \mathfrak{u}^\frac{3}{8} + \mathfrak{u}^\frac{7}{8} \right ) } \right \} \\ & \leq 2 \exp \left \{ - C \mathfrak{u}^\frac{3}{4} \right \}\end{aligned}$$ for some constant $C$ that does not depend on $i$. Now we need to transform the above inequality in the one that we are actually trying to prove. For this we need to estimate some quantities: Let us define $\delta_{i-1} = |S_{i-1} | - d^{i-2} \alpha_x$, note that under $\mathcal{F}_{i-1}$ $$d \delta_{i-1} \leq 2 C_{i-1} (d^{ i - \frac{3}{2} } + 1) u^\frac{7}{8}$$ and generally $d \leq \left ( d^{ i - \frac{3}{2}} + 1 \right ) \mathfrak{u}^\frac{7}{8}$. For easier readability set $\varepsilon_i = C_i \left ( d^{ i - \frac{3}{2}} + 1 \right ) \mathfrak{u}^\frac{7}{8}$. Then $$\begin{aligned} \left | | S_i | - d^{i-1} \alpha_x \right | \geq \varepsilon_i & \Rightarrow \big| |S_i| - d|S_{i-1}| \big | \geq \varepsilon_i - d \delta_{i-1} \\ & \Rightarrow \big| |S_i| - \mathbb{E}[ |S_i| | B_{i-1} ] \big | \geq \varepsilon_i - d \delta_{i-1} - d \\ & \Rightarrow \big| |S_i| - \mathbb{E}[ |S_i| | B_{i-1} ] \big | \geq ( C_i - 2 C_{i-1} - 1) \left ( d^{ i - \frac{3}{2}} + 1 \right ) \mathfrak{u}^\frac{7}{8}\end{aligned}$$ When $\mathcal{F}_{i-1}$ holds and $\alpha_x \leq 2 \mathfrak{u}$, $$\begin{aligned} \label{eq:S_ierrorbound} \sqrt{d |S_{i-1}|} u ^\frac{3}{8} + \mathfrak{u}^\frac{7}{8} & \leq \sqrt{d \left ( d^{i-2} \alpha_x + C_{i-1} \left ( d^{i- \frac{5}{2}} + 1 \right ) \mathfrak{u}^\frac{7}{8} \right ) }u^\frac{3}{8} + \mathfrak{u}^\frac{7}{8} \\ & \leq \left (\sqrt{ 2 (C_{i-1} + 1) } + 1 \right ) ( d^{i-\frac{3}{2}} + 1) u^\frac{7}{8}.\end{aligned}$$ If we set $C_i$ such that $C_i - 2 C_{i-1} -1 \geq \left (\sqrt{ 2 (C_{i-1} + 1) } + 1 \right )$, then whenever $cF_{i-1}$ holds and $\alpha_x \leq 2 \mathfrak{u}$, $$\mathbb{P}\left ( \left | | S_i | - d^{i-1} \alpha_x \right | \geq C_i ( d^{i - \frac{3}{2}} + 1 ) u^\frac{7}{8} \big |B_{i-1} \right ) \leq \mathbb{P}\left ( \big| |S_i| - \mathbb{E}[ |S_i| | B_{i-1} ] \big | \geq \sqrt{d |S_{i-1}|} u ^\frac{3}{8} + \mathfrak{u}^\frac{7}{8} \big | B_{i-1} \right ).$$ Finally we put all of this together in a union bound $$\begin{aligned} & \mathbb{P}\Big ( \exists x \in \mathcal{V}: \cup_{i = 1}^{r+3} \mathcal{G}_i^c(x) \Big ) \\ & \leq N \mathbb{E}\left [ \mathbb{P}\left ( \cup_{i = 1}^{r+3} \mathcal{G}_i^c(x) \big | S_1(x) \right ) \mathbf{1}(x \in \mathcal{V}) \right ] \\ & \leq N \mathbb{E}\left [ \sum_{i = 1}^{r+3} \mathbb{P}\left ( \mathcal{G}^c_i(x) \cap \mathcal{F}_{i-1}(x) \big | S_1(x) \right ) \mathbf{1} \left (\mathfrak{u}- \mathfrak{u}^\frac{2}{3} \leq \alpha_x \leq 2 \mathfrak{u}\right ) \right ] + N\mathbb{P}\left ( \alpha_x > 2 \mathfrak{u}\right ) \\ & \leq N \mathbb{E}\left [ \sum_{i = 1}^{r+3} \mathbb{E}\left [ \mathbb{P}\left ( \big | |S_i| - d^{i-1} \alpha_x \big | \geq \varepsilon_i \bigg | B_{i-1} \right ) \mathbf{1} \left ( \mathcal{F}_{i-1} \right ) \Big | S_1 \right ] \mathbf{1} \left (\mathfrak{u}- \mathfrak{u}^\frac{2}{3} \leq \alpha_x \leq 2 \mathfrak{u}\right ) \right ] + N\mathbb{P}\left ( \alpha_x > 2 \mathfrak{u}\right ) \\ & \leq N \mathbb{E}\left [ \sum_{i = 2}^{r+3} 2 \exp \left \{ - C \mathfrak{u}^\frac{3}{4} \right \} \mathbf{1} \left ( \mathfrak{u}- \mathfrak{u}^\frac{2}{3} \leq \alpha_x \leq 2 \mathfrak{u}\right ) \right ] + N\mathbb{P}\left ( \alpha_x > 2 \mathfrak{u}\right ) \\ & \leq N (r+3) e^{-C \mathfrak{u}^\frac{3}{4}}\mathbb{P}\left ( \mathfrak{u}- \mathfrak{u}^\frac{2}{3} \leq \alpha_x \right ) + N \mathbb{P}\left ( \alpha_x > 2 \mathfrak{u}\right ) \\ &\leq N (r+3) e^{-C \mathfrak{u}^\frac{3}{4}} \frac{1}{N} \frac32e^{d+ \mathfrak{u}^{\frac{2}{3}} \log \mathfrak{u}}\sqrt{\frac{\mathfrak{u}}{d}} + N \frac{1}{N^\frac{3}{2}}\end{aligned}$$ where we used that $\mathfrak{u}= \Theta \left ( \frac{\log N}{\log \log N} \right )$ by [\[eq:uapprox\]](#eq:uapprox){reference-type="eqref" reference="eq:uapprox"} and then applied Lemma [Corollary 12](#lem:binom_heavy_tail){reference-type="ref" reference="lem:binom_heavy_tail"} for the second term and the bound on $|\mathcal{V}|$ from Lemma [Lemma 19](#lem:sizes){reference-type="ref" reference="lem:sizes"}for the first term. ◻ *Proof of Lemma [Lemma 24](#lem:fine_balls_disjoint){reference-type="ref" reference="lem:fine_balls_disjoint"}, 4.* We will prove this for $\mathcal{V}$, the proof with $\mathcal{W}$ is identical. First note that by Lemma [\[lem:weakertail\]](#lem:weakertail){reference-type="ref" reference="lem:weakertail"}, for $X \sim \textnormal{Binom}(N, d/N)$, $\mathbb{P}\left ( X \geq \mathfrak{u}^{\frac{3}{4}} \right ) \leq e^{-\Omega(\mathfrak{u}^{\frac{3}{4}})},$ since $d \leq (\log N)^\frac{1}{5}$ by [Definition 5](#dfn:rdfn){reference-type="ref" reference="dfn:rdfn"}. The basic idea now is that by Lemma [Lemma 24](#lem:fine_balls_disjoint){reference-type="ref" reference="lem:fine_balls_disjoint"} 3, there are $O((r+3)d^{r+2}\mathfrak{u})$ vertices in $B_{r+3}(x)$ and by Lemma [Lemma 19](#lem:sizes){reference-type="ref" reference="lem:sizes"} there are $e^{O(\mathfrak{u}^\frac{2}{3}\log \mathfrak{u})}$ vertices in $\mathcal{V}$, so union bounding over all those vertices implies the result. Let us now make this precise. We show this level by level. For all $y \in S_i(x)$, conditioned on $B_i$, the $N_y$ are independent and distributed as $\textnormal{Binom}(N - |B_{i}|, d/N),$ which is stochastically dominated by $\textnormal{Binom}(N, d/N)$. Thus the probability that any $N_y \geq u^\frac{3}{4}$ is bounded by $e^{- \Omega(u^\frac{3}{4})}$. Putting everything together and using the notation $\mathcal{F}_i$ as defined in the previous proof, we first get $$\begin{aligned} & \mathbb{P}\left ( \exists x \in \mathcal{V}: \exists y \in B_{r+3}(x): N_y > \mathfrak{u}^\frac{3}{4} \right ) & \leq \mathbb{P}\left ( \exists x \in \mathcal{V}: \exists y \in B_{r+3}(x): N_y > \mathfrak{u}^\frac{3}{4},\alpha_x \leq 2 \mathfrak{u}\right ) + \mathbb{P}\left ( \exists x: \alpha_x > 2 \mathfrak{u}\right )\end{aligned}$$ and we know by [Lemma 24](#lem:fine_balls_disjoint){reference-type="ref" reference="lem:fine_balls_disjoint"}, 3. that the latter event happens with low probability. The first term on the other hand we can bound by $$\begin{aligned} & \sum_{x \in [N]} \mathbb{P}\left (x \in \mathcal{V}, \cup_{i = 1}^{r+3} \left \{ \exists y \in S_i(x): N_y > \mathfrak{u}^\frac{3}{4} \right \}, \alpha_x \leq 2 \mathfrak{u}\right ) \\ & \leq \sum_{x \in [N]} \sum_{i = 1}^{r+3} \mathbb{P}\left ( \exists y \in S_i(x): N_y > \mathfrak{u}^\frac{3}{4} , \mathcal{F}_i(x) , x \in \mathcal{V}, \alpha_x \leq 2 \mathfrak{u}\right ) + \sum_{x \in [N]} \sum_{i = 1}^{r+3} \mathbb{P}\left ( \mathcal{F}_i^c(x), x \in \mathcal{V}, \alpha_x \leq 2 \mathfrak{u}\right ) \\\end{aligned}$$ Now the latter term is small by the previous proof (note that we used a union bound there as well.) For the former term we proceed as follows: $$\begin{aligned} & \leq \sum_{x \in [N]} \sum_{i = 1}^{r+3} \mathbb{E}\left [ \mathbb{E}\left [ \mathbf{1} \left ( \exists y \in S_i(x): N_y > \mathfrak{u}^\frac{3}{4} \right ) \big | B_{i} \right ] \mathbf{1}_{ \mathcal{F}_i } \mathbf{1}_{x \in \mathcal{V}} \mathbf{1}_{\alpha_x \leq 2 \mathfrak{u}} \right ] \\ & \leq \sum_{x \in [N]} \sum_{i = 1}^{r+3} e^{-\Omega(u^\frac{3}{4})} \mathbb{E}\left [ |S_i| \mathbf{1}_{\mathcal{F}_i(x)} \mathbf{1}_{x \in \mathcal{V}} \mathbf{1}_{\alpha_x \leq 2 \mathfrak{u}} \right ] \\ & \leq \sum_{x \in [N]} e^{-\Omega(u^\frac{3}{4})} \sum_{i=1}^{r+3} \mathbb{E}\left [ \left ( d^{i-1} \alpha + O( d^{i-\frac{3}{2}} \mathfrak{u}^\frac{7}{8} + \mathfrak{u}^\frac{7}{8} ) \right ) \mathbf{1}_{\alpha_x \leq 2 \mathfrak{u}} \mathbf{1}_{x \in \mathcal{V}} \right ]\\ & \leq e^{-\Omega(u^\frac{3}{4})} O \left ( (r+3) (1+d^{r+2}) \mathfrak{u}\right ) \frac32e^{d+ \mathfrak{u}^{\frac{2}{3}} \log \mathfrak{u}}\sqrt{\frac{\mathfrak{u}}{d}},\end{aligned}$$ which is small as $N \to \infty.$ ◻ *Proof of Lemma [Lemma 24](#lem:fine_balls_disjoint){reference-type="ref" reference="lem:fine_balls_disjoint"}, 5.* We prove this for $\mathcal{W}$, the proof for $\mathcal{V}$ is equivalent. Here we use Lemma [Lemma 14](#lem:weibullbound){reference-type="ref" reference="lem:weibullbound"} with $t=2\mathfrak{u}^{2/3}$. The probability bound we obtain is $$\exp(-\Omega(\mathfrak{u}^{-1/3}/(d^3+1))).$$ Therefore, for our range of $d$, it is possible to union bound over all vertices in $|\mathcal{W}|$, as this gives us $$\exp(\mathfrak{u}^{1/4})\exp(-\Omega(\mathfrak{u}^{-1/3}/(d^3+1)))=\exp(-\Omega(\mathfrak{u}^{-1/3}/(d^3+1)))$$ by the bound in Lemma [Lemma 19](#lem:sizes){reference-type="ref" reference="lem:sizes"}. ◻ *Proof of Lemma [Lemma 35](#lem:size_balls){reference-type="ref" reference="lem:size_balls"}.* We show that with high probability, for all vertices in $\mathcal{U}$, $$|S_i| \leq 4^{i-1} (d + \log \log N - \log d )^{i-1} \mathfrak{u}$$ which implies the statement by our bounds on $d$ from Definition [Definition 5](#dfn:rdfn){reference-type="ref" reference="dfn:rdfn"}. The strategy is similar as in the proof of Lemma [Lemma 24](#lem:fine_balls_disjoint){reference-type="ref" reference="lem:fine_balls_disjoint"}, 3: First note that by Lemma [\[lem:weakertail\]](#lem:weakertail){reference-type="ref" reference="lem:weakertail"}, under the event $\mathcal{F}_{i-1}$, $$\begin{aligned} \mathbb{E}\left [|S_{i-1}| \big | B_{i-1} \right ] = d |S_{i-1} | - | B_{i-1} | |S_{i-1} | \frac{ d }{ N } \leq d |S_{i-1} | - d\end{aligned}$$ using this bound and the fact that $d \leq d + \log \log N - \log d$, we get that $$\begin{aligned} & \mathbb{P}\left ( \big | |S_i| - \mathbb{E}\left [ |S_i | B_{i-1} \right ] \big | \geq \left ( (d + \log \log N - \log d) \sqrt{2 |S_{i-1}| u} + (d + \log \log N - \log d) \mathfrak{u}\right ) \big | B_{i-1} \right ) \\ & \leq 2 \exp \left \{ - \frac { \left ( (d + \log \log N - \log d) \sqrt{2 |S_{i-1}| u} + (d + \log \log N - \log d) \mathfrak{u}\right )^2} {2 \mathbb{E}\left [ |S_i| | B_{i-1} \right ] + \frac{2}{3} \left ( (d + \log \log N - \log d) \sqrt{|S_{i-1}| u} + (d + \log \log N - \log d) \mathfrak{u}\right ) } \right \} \\ & \leq 2 e^{ - (d+ \log \log N - \log d) \mathfrak{u}} \\ & \leq 2 N^{-1+o(1)}\end{aligned}$$ by considering which term in the denominator is smaller and then using the approximation from [\[eq:uapprox\]](#eq:uapprox){reference-type="eqref" reference="eq:uapprox"} for $\mathfrak{u}$. Now note that under the event $\mathcal{F}_i$, $$\begin{aligned} \mathbb{E}\left [|S_{i-1}| \big | B_{i-1} \right ] \leq d 4^{i-2}( d + \log \log N - \log d )^{i-2} \mathfrak{u}\leq 4^{i-2} ( d + \log \log N - \log d )^{i-1} \mathfrak{u}\end{aligned}$$ such that $$\begin{aligned} & |S_i| \geq 4^{i-1} ( d + \log \log N - \log d)^{i-1} \mathfrak{u}\\ \Rightarrow & |S_i| - \mathbb{E}\left [ |S_i| \big | B_{i-1} \right ] \geq 3 \cdot 4^{i-2} ( d + \log \log N - \log d)^{i-1} \mathfrak{u}\\ \Rightarrow & |S_i| - \mathbb{E}\left [ |S_i| \big | B_{i-1} \right ] \geq ( d + \log \log N - d ) \sqrt{ 2 |S_{i-1}| \mathfrak{u}} + (d + \log \log N - \log d ) \mathfrak{u}.\end{aligned}$$ This implies that $$\begin{aligned} \mathbb{P}\left ( |S_i| \geq 4^{i-2} ( d + \log \log N - \log d )^{i-1} \mathfrak{u}| B_i \right ) \leq 2 N^{-1 + o(1)}.\end{aligned}$$ We now proceed as in the end of the proof of Lemma [Lemma 24](#lem:fine_balls_disjoint){reference-type="ref" reference="lem:fine_balls_disjoint"}, 3, by using the bound on $\mathcal{U}$ from Lemma [Lemma 19](#lem:sizes){reference-type="ref" reference="lem:sizes"}. For the second statement of the Lemma, note that it is sufficient to bound $\sum_{y\sim x}N_y^2$. We once more use Lemma [Lemma 14](#lem:weibullbound){reference-type="ref" reference="lem:weibullbound"} as in the proof of Lemma [Lemma 24](#lem:fine_balls_disjoint){reference-type="ref" reference="lem:fine_balls_disjoint"}, 5, setting $t=c_{\scaleto{\ref{lem:weibullbound}}{3pt}}^{-2}\log^2N$. ◻ *Proof of Lemma [Lemma 36](#lem:rough_tree_disjoint){reference-type="ref" reference="lem:rough_tree_disjoint"}.* We use the arguments from the proofs of Lemma 5.5 and Lemma 7.3 in [@alt2021extremal], but choose to include them here for sake of completeness. For any $x \in [N]$ let $\mathcal{E}_x$ denote the event that there are at least $C_1$ excess edges in $B_s(x)$. Then by a union bound $$\begin{aligned} \mathbb{P}\left ( \exists x \in \mathcal{U}: \mathcal{E}_x \right ) & \leq \sum_{ x \in [N]} \mathbb{P}\left ( \mathcal{E}_x, x \in \mathcal{U}, \alpha_x < 2 \mathfrak{u}\right ) + \mathbb{P}( \alpha_x \geq 2 \mathfrak{u}), \end{aligned}$$ where the second summand can be bounded by $N^{-\frac{3}{2}}$ according to Lemma [Corollary 12](#lem:binom_heavy_tail){reference-type="ref" reference="lem:binom_heavy_tail"}. In order to bound the first term, we want to condition on $S_1(x)$, and then apply Lemma [Lemma 50](#lem:excess_edges){reference-type="ref" reference="lem:excess_edges"} for this we write $$\begin{aligned} \mathbb{P}\left ( \mathcal{E}_x, x \in \mathcal{U}, \alpha_x < 2 \mathfrak{u}\right ) & = \mathbb{E}\left [ \mathbb{P}( \mathcal{E}_x |S_1(x) ) \mathbf{1}(\{ x \in \mathcal{U}, \alpha_x < 2 \mathfrak{u}\}) \right ] \\ & \leq \mathbb{E}\left [ C d^{3 C_1} \left ( \frac{ |S_1|}{N} \right )^{C_1} \mathbf{1} \left ( \eta \mathfrak{u}\leq \alpha_x \leq 2 \mathfrak{u}\right ) \right ] \\ & \leq C \left ( 2d^3 \right )^{C_1} \left ( \frac{ \log N}{N} \right )^{C_1} \frac{1}{N^\frac{\eta}{2}} \end{aligned}$$ Thus by taking $C_1 \geq 2$ and then doing a union bound over all $x$ we get the desired result. For the second statement of the Lemma, we proceed as in the proof of Lemma 7.3 in [@alt2021extremal] and write $\mathcal{I}_x$, the event that there are at least $C_2$ disjoint paths in $B_s(x)$ ending at vertices in $\mathcal{U}_\eta$, as a union over the specific paths: $$\begin{aligned} \mathcal{I}_x = \bigcup_{\mathbf{y}, \mathbf{z}} \Gamma^{(C_2)}_{ \mathbf{y}, \mathbf{z} }, \end{aligned}$$ where the union is taken over all vectors $\mathbf{y} = (y_1, \dots, y_{C_2} )$ with distinct entries in $[N] \setminus \{ x \}$ and the $C_2$-tuples $\mathbf{z}$ of disjoint vectors $(z^{(1)}, \dots, z^{(C_2)})$ of length $r_j \in \{ 0, \dots, s \}$ for $j \in [C_2]$, and $$\Gamma_{ \mathbf{y}, \mathbf{z} } = \left \{ y_j \in \mathcal{U}_\eta, \{x,z_1^{(j)} \}, \{ z_i^{(j)}, z_{i+1}^{(j)} \}, \{ z_{r_j}^{(j)}, y^{j} \} \in E(G) \forall i \in [r_j - 1], j \in [k] \right \}.$$ For some fixed $\mathbf{y}$ and $\mathbf{z}$, and thus fixed set of $(r_1, \dots, r_{C_2})$, since all paths are disjoint, when we denote by $\mathcal{N}_x$ the neighborhood of a vertex $x$, and use the independence of the edges, we get that $$\begin{aligned} \mathbb{P}\left ( \Gamma_{\mathbf{y}, \mathbf{z}} \right ) \leq & \mathbb{P}\left ( \big |\mathcal{N}_x \cap ([N] \setminus \mathbf{y}) \big | \geq \eta \mathfrak{u}- C_2 \right ) \\ & \prod_{j=1}^{C_2} \mathbb{P}\left ( \big | \mathcal{N}_y \cap ([N] \setminus \{ x \} \cup \mathbf{y}) \} \big | \geq \eta \mathfrak{u}- C_2 - 1 \right ) \\ & \left ( \frac{d}{N} \right )^{\sum_{ j = 1}^{C_2} r_j + 1}. \end{aligned}$$ We now apply Lemma [Lemma 10](#lem:sharp_poisson_tail){reference-type="ref" reference="lem:sharp_poisson_tail"} and Corollary [Corollary 9](#cor:bintopoistail){reference-type="ref" reference="cor:bintopoistail"} to bound the remaining probabilities: since $C_2$ is constant all these probabilities will be bounded by $$e^{-\eta u \log u + c \eta u} = e^{- (1+o(1)) \eta \log N}.$$ This implies that the above probability is bounded by $$\frac{d^{\sum_{ j = 1}^{C_2} r_j + 1}}{ N^{\left (\sum_{j=1}^{C_2} r_j + 1 \right ) + \eta C_2 (1+o(1))} }.$$ To complete the union bound we need to count the number of terms, i.e. possible paths, for each sequence of $r_j$s. To do this we note that given that $x$ is fixed, there are $\binom{N - 1}{C_2}$ ways of picking $\mathbf{y}$ and for the $z^{(j)}_i$ on each path there are $\binom{N - k - \sum_{i = 1}^{j-1} r_i}{r_j}$ ways of picking them. Thus $$\begin{aligned} & \mathbb{P}\left ( \mathcal{I}_x \right ) \\ & \leq \binom{N - 1}{C_2} \sum_{r_1 = 0}^s \cdots \sum_{r_{C_2} = 0}^s \binom{N-C_2 - 1 }{r_1} \cdots \binom{N - C_2 - \sum_{i = 1}^{C_2-1 r_i}}{r_{C_2}} \frac{d^{\sum_{ j = 1}^{C_2} r_j + 1}}{ N^{\left (\sum_{j=1}^{C_2} r_j + 1 \right ) + \eta C_2 (1+o(1))} } \\ & \leq C \frac{ d^{C_2 (s + 1)}}{N^{\eta C_2 (1 + o(1) )} } \end{aligned}$$ where $C$ is a constant that depends on the constants $s$ and $C_2$. By taking $C_2 > \frac{2}{\eta},$ and then taking a union bound over all $x$, this implies that all $B_s(x)$ only contain a constant number of disjoint paths ending at other vertices from $\mathcal{U}$ with high probability. ◻ *Proof of Lemma [Lemma 37](#lem:prunedgraph){reference-type="ref" reference="lem:prunedgraph"}.* To construct the pruned graph $\hat{G}$ we delete edges in the same manner as in Lemma 7.2 in [@alt2021extremal]: For every vertex $x \in \mathcal{U}$, and its neighbor $y$, consider the set of vertices $T_y$ that are connected to $y$ by a path of length at most $3$, without traversing the edge $(x,y)$. If $x$ is in this set, or the graph induced by $T_y$ on $G$ is not a tree, then we prune the edge $(x,y)$. Denote the set of edges that are pruned in this way $P_x$. According to Lemma [Lemma 36](#lem:rough_tree_disjoint){reference-type="ref" reference="lem:rough_tree_disjoint"}, with high probability, each vertex $x \in \mathcal{U}$ has less than $C_1$ "excess" edges that create cycles in $B_3(x)$. Thus by the above procedure we prune at most $C_1 - 1$ edges that are adjacent to $x$. In the second step, we work with the graph on $[N]$ with edges $E(G) \setminus P_x$, in which $B_3(x)$ is a tree. In that graph we consider for each neighbor $y$ of $x$, the vertices $V_y$ in $B_3(x)$ that are connected to $y$ by a path that does not use the edge $(x,y)$. If any of the vertices in $V_y$ is in $\mathcal{U}$, we prune the edge $(x,y)$ and add it to $P_x$. By Lemma [Lemma 36](#lem:rough_tree_disjoint){reference-type="ref" reference="lem:rough_tree_disjoint"} we prune at most $C_2 - 1$ edges adjacent to $x$ by doing this procedure. We then apply these steps, by choosing an arbitrary order of vertices in $\mathcal{U}$, then pruning edges surrounding these vertices sequentially. Let $H$ be the graph on $[N]$ that only consists of the edges $\cup_{ x \in cU} P_x$ that we pruned. We then define our pruned graph $\hat{G}$ to be the graph $G$ with edges $E(G) \setminus \cup_{ x \in cU} P_x$. By construction $\hat{G}$ satisfies 1. and 2. Note that only vertices $x \in \mathcal{U}$ and vertices $y \in \cup_{x \in \mathcal{U}} S_1(x)$ are not isolated in $H$. It is clear that at each step of this procedure we prune at most $C_2 + C_1 - 2$ edges adjacent to some $x$. Moreover, note that that any subsequent step cannot affect the degree of $x$ in $H$: otherwise, if we have already pruned for $x \in \mathcal{U}$, if in a subsequent pruning for $x' \in \mathcal{U}$ we were to delete an edge adjacent to $x$, this would mean that $(x',x)$ is an edge in $G$, in which case we would already have pruned it when doing the pruning for $x$. Now let $y \in \cup_{x \in \mathcal{U}} S_1(x) \setminus \mathcal{U}$, i.e. let $y$ be a vertex that is not in $\mathcal{U}$ and is a neighbor of some vertex $x \in \mathcal{U}$. By Lemma [Lemma 36](#lem:rough_tree_disjoint){reference-type="ref" reference="lem:rough_tree_disjoint"}, $y$ can be adjacent to at most $C_2 - 1$ additional vertices from $\mathcal{U}$, since otherwise $B_2(x)$ contains more than $C_2 - 1$ vertices from $\mathcal{U}$. Thus we prune at most $C_2 - 1$ edges adjacent to $y$. Hence the maximal degree of the graph $H$ is $C_1 + C_2 - 2,$ implying 3. Recall the assumption that the maximum degree is at most $\mathfrak{u}$. Thus for each edge $(x,y)$ that we prune, $\beta_x$ is reduced by at most $\mathfrak{u}$. Additionally for each vertex $y \in S_1(x)$, we delete at most $C_2 - 1$ edges by doing the pruning procedure for other $x' \in \mathcal{U}$. This implies that $0 \leq \beta_x - \hat{\beta}_x \leq \alpha_x (C_2 - 1) + (C_1 + C_2 - 2) \mathfrak{u}= O( \mathfrak{u}),$ which implies 4. Lemma [Lemma 35](#lem:size_balls){reference-type="ref" reference="lem:size_balls"} gives a bound on the growth of the spheres in the original graph and since $\alpha_x$ and $\hat{\alpha}_x$ are of the same order, 5. follows immediately. For the last statement we rewrite $$\sum_{y \in \hat{S}_1(x)} \left ( \hat { N}_y - \frac{ \hat{\beta}}{\hat{\alpha}} \right )^2 \leq 3 \left [ \sum_{y \in \hat{S}_1(x)} \left ( \hat { N}_y - N_y \right )^2 + \sum_{y \in \hat{S}_1(x)} \left ( N_y - d \right )^2 + \sum_{y \in \hat{S}_1(x)} \left ( d - \frac{ \hat{\beta}}{\hat{\alpha}} \right )^2 \right ].$$ The first term can be bounded by $O(\hat{\alpha})$ since $G - \hat G$ has a bounded degree by Lemma [Lemma 37](#lem:prunedgraph){reference-type="ref" reference="lem:prunedgraph"}, 3, for the second term we use Lemma [Lemma 35](#lem:size_balls){reference-type="ref" reference="lem:size_balls"} and the last term can be bounded by $\hat{\alpha} \left ( d - \frac{\hat{\beta}}{\hat{\alpha}} \right )^2$ and then be bounded using Lemma [Lemma 37](#lem:prunedgraph){reference-type="ref" reference="lem:prunedgraph"}, 3. ◻ [^1]: `[email protected]`. University of California, Berkeley. Supported by the Citadel Securities Berkeley Statistics PhD Fellowship. [^2]: `[email protected]`. Stanford University. Supported by NSF GRFP Grant DGE-1752814 and NSF Grant DMS-2212881. [^3]: Unless otherwise specified, theorems stated in the introduction are of phenomena that occur with high probability. [^4]: Here and throughout, when writing $x\ll y$, we mean that $x=o(y)$, $x\lesssim y$ means $x=O(y)$, and $x\asymp y$ means $x=\Theta(y)$. [^5]: We have translated their parameters into the unnormalized versions we use in our proof.

arxiv_math

--- abstract: | Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$. A subset $I$ of $V(G)$ is an independent vertex subset if no two vertices in $I$ are adjacent in $G$. We study the number, $\sigma_1(G)$, of all subsets of $v(G)$ that contain exactly one pair of adjacent vertices. We call those subsets 1-nearly independent vertex subsets. Recursive formulas of $\sigma_1$ are provided, as well as some cases of explicit formulas. We prove a tight lower (resp. upper) bound on $\sigma_1$ for graphs of order $n$. We deduce as a corollary that the star $K_{1,n-1}$ (the tree with degree sequence $(n-1,1,\dots,1)$) is the $n$-vertex tree with smallest $\sigma_1$, while it is well known that $K_{1,n-1}$ is the $n$-vertex tree with largest number of independent subsets. author: - | Eric Ould Dadah Andriantiana\ Department of Mathematics (Pure and Applied)\ Rhodes University\ Makhanda, 6140 South Africa\ `Email: [email protected] Zekhaya B. Shozi Department of Mathematical Sciences Sol Plaatje University Kimberley, 8301 South Africa Email: [email protected] ` bibliography: - references.bib title: The number of $1$-nearly independent vertex subsets --- **Keywords:** $1$-nearly independent vertex subset; Minimal connected graphs; Maximal graphs. \ # Introduction A simple and undirected graph $G$ is an ordered pair of sets $G=(V(G),E(G))$, where every element of the set of $E(G)$ of edges is a $2$-element subset of the set of vertices $V(G)$. $|V(G)|$ is the order of $G$, and $|E(G)|$ its size. For graph theory notation and terminology, we generally follow [@henning2013total]. We use the standard notation $[k] = \{1,\ldots,k\}$. An *independent (vertex) subset* of a graph $G$, with vertex set $V(G)$ and edge set $E(G)$, is any subset of $V(G)$ that does not contain adjacent vertices. The number of independent subsets of a graph is a popular graph invariant with rich literature. See the survey in [@wagner2010maxima], where it is called the *Merrifield-Simmons index*. See also the book [@li2012shi] for an even more extensive survey. The book itself is on the energy of graphs but given the strong connection between the number of independent vertex subsets and the energy of graphs, the book also contains a wealth of results on the number of independent vertex subsets. Merrifield and Simmons used [@Merrifield198055] the number independent subsets of molecular graphs and as a measure of molecular complexity, bond strength and boiling point of the associated molecules. This boosted the interest of both chemists and mathematicians to study the invariant. Obtained results go beyond just the class of molecular graphs. In [@Andriantiana2013724] the structure of the tree with a given degree sequence $D$ that has the largest number of independent subsets is fully characterised. The result implies as corollaries characterisations of trees with largest number of independent subsets in various other classes like trees with fixed order, or with fixed order and given maximum degree. Various ways of generalisation of the notion of independent subsets have been attempted. For example [@jagota2001generalization] generalised the concept of maximal-independent set, by considering the $k$-insulated set $S$ of a graph $G$ defined as a subset of its vertices such that each vertex in $S$ is adjacent to at most $k$ other vertices in $S$ and each vertex not in $S$ is adjacent to at least $k+1$ vertices in $S$. See also [@drmota1991generalized], which studies subsets which do not contain pair of vertices with distance shorter than a specified integer $k$. In this paper, we propose a new other generalisation. $H$ is a *subgraph* of $G$ if $V(H)\subseteq V(G)$ and $E(H)\subseteq E(G)$. If furthermore $$E(H)=E(G)\cap \{\{u,v\}:u,v\in V(G)\},$$ then we say that $H$ is an *induced subgraph* of $G$. This means that for a given set of vertices $V(H)$ it has all possible edges of $G$ that it can have. Note that to any independent subset $I$ of $G$ corresponds an induced subgraph $(I,\emptyset)$ of $G$. We propose one way to generalise the notion of independent subsets. We call any induced subgraph of size $k$ a $k$-nearly independent subset of $G$. Let $\sigma_k(G)$ denotes the number of $k$-nearly independent subsets of $G$. $\sigma_0(G)$ is the number of independent susbets of $G$. In this paper, we study $\sigma_1$. At least for the classes of graphs we investigated, the behaviour of $\sigma_1$ has little in common with that of $\sigma_0$. Among all graphs of order $n$, while the edgeless graph $\overline{K_n}$ has the largest $\sigma_0$, it has the smallest $\sigma_1$ as $\sigma_1(\overline{K_n})=0$. The complete graph $K_n$, that has all possible edges, which has the smallest $\sigma_0$ does not reach an extremal value for $\sigma_1$ if $n\geq 6$: not the minimum nor the maximum. The rest of the paper is structured as follows. We start with a preliminary sections that contains technical formulas, some of which will be used in this paper, while others are included because we expect that they will be useful in further studies of $\sigma_1$. We also investigate the effect of an edge removal in that section and provide explicit expression of $\sigma_1$ for selected types of graphs. The main results are in Section [3](#Sec:Main){reference-type="ref" reference="Sec:Main"}. There, we provide a full characterisation of the family of graphs that reach the minimum $\sigma_1$ among all graphs of order $n$ size $m$. Several corollaries follows from this. For example, if $m=n-1$, the star $K_{1,n-1}$ (the connected $n$-vertex graph with $n-1$ vertices of degree $1$) is a member of the family. Hence it has the minimum $\sigma_1$ among all trees, while it is known to have the maximum $\sigma_0$. The characterisation of the graph of order $n$ and maximum $\sigma_1$ that we characterise at the end of the section is a rather unusual graph. For $n\geq 8$ the maximum $\sigma_1$ is attained by two graph both with maximum degree $1$, one has size $3$ and the other has size $4$. # Preliminary Let $G$ be a graph with vertex set $V(G)$, edge set $E(G)$, order $n = |V(G)|$ and size $m = |E(G)|$. We denote the degree of a vertex $v$ in $G$ by $\deg_Gv$. The maximum degrees in $G$ is denoted by $\Delta(G).$ The complement of $G$ is denoted by $\overline{G}$ and defined as $$\overline{G}=(V(G),\{uv: u,v\in V(G), u\neq v \text{ and }uv\notin E(G)\}).$$ For a subset $S$ of vertices of a graph $G$, we denote by $G - S$ the graph obtained from $G$ by deleting the vertices in $S$ and all edges incident to them. If $S = \{v\}$, then we simply write $G - v$ rather than $G - \{v\}$. For any graphs $G$ and $H$, we define the *join* $$G+H=(V(G)\cup V(H), E(G)\cup E(H)\cup\{uv: u\in V(G)\text{ and } v\in V(H) \})$$ obtained by adding all possible edges between $G$ and $H$. The subgraph of $G$ induced by the set $S$ is denoted by $\langle S \rangle_G$. We denote the path graph, cycle graph, wheel graph and complete graph on $n$ vertices by $P_n$, $C_n$, $W_n$ and $K_n$, respectively. For positive integers $r$ and $s$, we denote by $K_{r,s}$ the bipartite graph with partite sets $X$ and $Y$ such that $|X|=r$ and $|Y|=s$. A bipartite graph $K_{1,n-1}$ is also called a *star*. Let $k\ge 2$ be an integer. A *broom graph*, denoted $B_n^k$, is the tree of order $n$ obtained from the path, $P_k$, of order $k$ by adding $n-k$ new vertices and then joining them to exactly one end-vertex of $P_k$. A *lollipop graph*, denoted $L_n^k$, is the graph of order $n$ obtained from a path $P_k$ of order $k$, by adding a complete graph, $K_{n-k}$, of order $n-k$ and then joining every vertex of $K_{n-k}$ to exactly one end-vertex of $P_k$. A *tadpole*, denoted $T_n^k$, is the graph of order $n$ obtained from a path, $P_k$, of order $k$ by adding another path, $P_{n-k}$, of order $n-k$ and then joining the end-vertices of $P_{n-k}$ to exactly one end-vertex of $P_k$. ## Recursive formula Suppose that $G$ and $H$ are disjoint graphs and $K=G\cup H$. Then, every $1$-nearly independent subsets $I$ of $K$ splits uniquely into two parts: the part $I_G$ that is in $G$ and the part $I_H$ that is in $H$. If the edge in $I$ is from $G$ then $I_G$ is a $1$-nearly independent subset and $I_H$ an independent subset. If the edge in $I$ is from $H$, then $I_H$ is a $1$-nearly independent subset while $I_G$ is an independent subset. Hence we have $$\sigma_1(G\cup H) =\sigma_1(G)\sigma_0(H)+\sigma_0(G)\sigma_1(H).$$ For any vertex $v$ of a graph $G$ $$\begin{aligned} \label{Eq:Rec} \sigma_1(G) =\sigma_1(G-v)+\sigma_1(G-N[v])+\sum_{u\in N(v)}\sigma_0(G-(N[u]\cup N[v]))>\sigma_1(G-v),\end{aligned}$$ where $\sigma_1(G-v)$ counts the number of $1$-nearly independent subsets that do not contain $v$, $\sigma_1(G-N[v])$ counts those that contain $v$ as a single vertex, and $\sum_{\in N(v)}\sigma_0(G-(N[u]\cup N[v]))$ counts those that contain $v$ is as an edge. ## Effect of an edge or a vertex removal on $\sigma_1$ From Equation [\[Eq:Rec\]](#Eq:Rec){reference-type="ref" reference="Eq:Rec"}, we see that removing a vertex $v$ from a graph $G$ decreases its $\sigma_1$, or keeps it unchanged if $\sigma_1(G-N[v])=0$ and $N(v)=\emptyset$. This condition means that $v$ is an isolated vertex and $G-N[v]=G-v$ has no edge. The graph $G$ is edge-less in this case. Removing an edge might leave the value of $\sigma_1$ unchanged, decreasing or increasing. Below are some examples of situations when some of these possibilities arises. Note that for any edge $e$ of $K_3$ we have $$\sigma_1(K_3)=3>\sigma_1(K_3-e)=\sigma_1(P_3)=2,$$ while for any edge $e'$ of $P_3$ we have $$\sigma_1(P_3)=2=\sigma_1(P_3-e')=2\sigma_1(K_2),$$ and if $e''$ is the middle edge of $P_4$, then $$\sigma_1(P_4)=5<\sigma_1(P_4-e'')=\sigma_1(2K_2)=6.$$ ## Explicit formulas for $\sigma_1$ of some graphs Using known formulas for $\sigma_0$ and some of the recursive formulas in the previous subsection, we present in this section explicit formulas for $\sigma_1$ of a complete graph $K_n$, an unicyclic graph $U_n$, a path $P_n$, a cycle $C_n$, and a wheel graph $W_n$ in terms of $n$. More formulas of the $\sigma_1$ of a broom graph $B_n^k$, a lollipop graph $L_n^k$, and a tadpole $T_n^k$ in terms of $n$, $k$ and $\sigma_1$ of paths are also presented. It is convenient to set $\sigma_1(P_t)=0$ whenever $t\leq 0$ and $\sigma_0(P_t)=1$ whenever $t\leq 0$. Let $\alpha = \frac{1+\sqrt{5}}{2}$ and $\beta = \frac{1-\sqrt{5}}{2}$. Then, we have $\alpha + \beta =1$, $\alpha - \beta = \sqrt{5}$, $\alpha\cdot \beta = -1$. The following formulas are well known, see for example [@andriantiana2010number] and [@wagner2010maxima]. **Theorem 1** (cf. [@andriantiana2010number]). *For $n \in \mathbb{N}$, we have $$\begin{aligned} \label{eq:sigma-of-a-path-by-eric} \sigma_0(P_n) = \frac{1}{\sqrt{5}} \left( \alpha^{n+2} - \beta^{n+2} \right). \end{aligned}$$* **Theorem 2** (cf. [@wagner2010maxima]). *If $G_1, G_2, \ldots, G_r$ are the connected components of a graph $G$, then $$\begin{aligned} \label{eq:sigma-union-is-product-of-sigmas} \sigma_0(G) = \sigma_0\left(\bigcup\limits_{i=1}^r G_i \right) = \displaystyle \prod\limits_{i=1}^r \sigma_0(G_i). \end{aligned}$$* ### $\sigma_1$ of $K_n$, $P_n$, $U_n$, $C_n$, and $W_n$ Every edge of $K_n$ can only be contained in exactly one $1$-nearly independent subsets. Hence, we have $$\begin{aligned} \sigma_1(K_n) = |E(K_n)| = \binom{n}{2} = \frac{n(n-1)}{2}.\end{aligned}$$ Let $P_n = (\{v_1, \dots,v_n\},\{e_i=v_iv_{i+1}: 1\leq i\leq n-1\})$. Since the number of $1$-nearly independent subsets containing $e_i$ in $P_n$ is exactly $\sigma_0(P_{i-2} \cup P_{n-i-2})$, we have $$\begin{aligned} \label{eq:sum-equation-of-sigma-1-of-Pn} \sigma_1(P_n) &= \displaystyle \sum\limits_{i=1}^{n-1} \sigma_0(P_{i-2} \cup P_{n-i-2}) \nonumber\\ &=\displaystyle \sum\limits_{i=1}^{n-1} \sigma_0(P_{i-2}) \sigma_0(P_{n-i-2}) & \text{ by Theorem \ref{thm:sigma-union-is-product-of-sigmas}}\nonumber\\ &= \frac{1}{5} \displaystyle \sum\limits_{i=1}^{n-1} (\alpha^i - \beta^i)(\alpha^{n-i} - \beta^{n-i}) & \text{ by Theorem \ref{thm:sigma-of-a-path-by-eric}} \nonumber\\ &= \frac{1}{5} \displaystyle \sum\limits_{i=1}^{n-1} \left[ \alpha^n - \beta^n \left(\frac{\alpha}{\beta}\right)^i - \alpha^n \left(\frac{\beta}{\alpha} \right)^i + \beta^n \right] \nonumber \\ &=\frac{1}{5}(n-1)\left(\alpha^n + \beta^n \right) -\frac{1}{5} \left[ \beta^n \displaystyle \sum\limits_{i=1}^{n-1} \left(\frac{\alpha}{\beta}\right)^i + \alpha^n \displaystyle \sum\limits_{i=1}^{n-1} \left(\frac{\beta}{\alpha} \right)^i \right].\end{aligned}$$ The sum $$\begin{aligned} \displaystyle \sum\limits_{i=1}^{n-1} \left(\frac{\alpha}{\beta}\right)^i = \frac{ \frac{\alpha}{\beta} \left[ \left( \frac{\alpha}{\beta} \right)^{n-1} -1 \right] }{ \left( \frac{\alpha}{\beta} -1\right)},\end{aligned}$$ implies that $$\begin{aligned} \label{eq:beta-n-sum-geometric-series} \beta^n \displaystyle \sum\limits_{i=1}^{n-1} \left(\frac{\alpha}{\beta}\right)^i = \alpha \beta \left( \frac{ \alpha^{n-1} - \beta^{n-1} }{\alpha -\beta} \right) = - \frac{1}{\sqrt{5}}\left( \alpha^{n-1} - \beta^{n-1}\right).\end{aligned}$$ Similarly, $$\begin{aligned} \displaystyle \sum\limits_{i=1}^{n-1} \left(\frac{\beta}{\alpha}\right)^i = \frac{ \frac{\beta}{\alpha} \left[ \left( \frac{\beta}{\alpha} \right)^{n-1} -1 \right] }{ \left( \frac{\beta}{\alpha} -1\right)},\end{aligned}$$ leads to $$\begin{aligned} \label{eq:alpha-n-sum-geometric-series} \alpha^n \displaystyle \sum\limits_{i=1}^{n-1} \left(\frac{\beta}{\alpha}\right)^i = \alpha \beta \left( \frac{ \alpha^{n-1} - \beta^{n-1} }{\alpha -\beta} \right) = - \frac{1}{\sqrt{5}}\left( \alpha^{n-1} - \beta^{n-1}\right).\end{aligned}$$ Thus, substituting Equations [\[eq:beta-n-sum-geometric-series\]](#eq:beta-n-sum-geometric-series){reference-type="eqref" reference="eq:beta-n-sum-geometric-series"} and [\[eq:alpha-n-sum-geometric-series\]](#eq:alpha-n-sum-geometric-series){reference-type="eqref" reference="eq:alpha-n-sum-geometric-series"} into Equation [\[eq:sum-equation-of-sigma-1-of-Pn\]](#eq:sum-equation-of-sigma-1-of-Pn){reference-type="eqref" reference="eq:sum-equation-of-sigma-1-of-Pn"} yields $$\begin{aligned} \sigma_1(P_n) &= \frac{1}{5}(n-1)\left(\alpha^n + \beta^n \right) -\frac{1}{5} \left[- \frac{1}{\sqrt{5}}\left( \alpha^{n-1} - \beta^{n-1}\right) - \frac{1}{\sqrt{5}}\left( \alpha^{n-1} - \beta^{n-1}\right) \right]\\ &= \frac{1}{5}\left[(n-1)\left(\alpha^n + \beta^n \right) + \frac{2}{\sqrt{5}} \left( \alpha^{n-1} -\beta^{n-1} \right) \right].\end{aligned}$$ Let $U_n$ be the graph obtained from the tree $K_{1,n-1}$ by adding one more edge to make it a unicyclic graph. Then, clearly, $\Delta(U_n) = n-1$. Let $v$ be the vertex of maximum degree in $U_n$. Then, we have $$\begin{aligned} \sigma_1(U_n)&=\sigma_1(U_n-v) + \sigma_1(U_n - N_G[v]) + \displaystyle \sum\limits_{u\in N_G(v)}\sigma_0(U_n-(N_G[u]\cup N_G[v]))\\ &=\sigma_1(P_2\cup \overline{K_{n-3}}) + \sigma_1(\emptyset) + (n-1)\sigma_0(\emptyset)\\ &=\sigma_1(P_2)\sigma_0(\overline{K_{n-3}}) + \sigma_1(\overline{K_{n-3}})\sigma_0(P_2)+\sigma_1(\emptyset) + (n-1)\sigma_0(\emptyset)\\ &=n-1+2^{n-3}.\end{aligned}$$ Let $C_n=(\{v_1,\dots,v_n\},\{e_i=v_iv_{i+1}:1\leq i\leq n-1\}\cup \{e_n=v_nv_1\})$. Note that if we remove consecutive vertices of $C_n$, we obtain a path of order less than $n$. Every edge of $C_n$ is contained in exactly $\sigma_0(P_{n-4})$ $1$-nearly independent subsets. Thus, $$\begin{aligned} \sigma_1(C_n) &= \displaystyle \sum\limits_{i=1}^n \sigma_0(P_{n-4})=n\sigma_0(P_{n-4}) = \frac{n}{\sqrt{5}}\left( \alpha^{n-2} - \beta^{n-2} \right).\end{aligned}$$ A *wheel graph* $W_n$, of order $n$ is given by $W_n = C_{n-1} + K_1$. Note that in $W_n$, a vertex $v$ is central if and only if $\deg_{W_n}v = n-1$. Let $v$ be a central vertex of $W_n$. Then $$\begin{aligned} \sigma_1(W_n) &= \sigma_1(W_n-v) + \sigma_1(W_n - N_G[v]) + \displaystyle \sum\limits_{u\in N_G(v)}\sigma_0(W_n-(N_G[u]\cup N_G[v]))\\ &=\sigma_1(C_{n-1}) + \sigma_1(\emptyset) + (n-1)\sigma_0(\emptyset)\\ &=\frac{1}{\sqrt{5}}(n-1)\left(\alpha^{n-3} -\beta^{n-3} \right) + (n-1)\\ &=(n-1)\left[1 + \frac{1}{\sqrt{5}} \left( \alpha^{n-3} -\beta^{n-3} \right) \right].\end{aligned}$$ ### $\sigma_1$ of $B_n^k$, $L_n^k$, and $T_n^k$ Let $v$ be a vertex of $B_n^k$ of maximum degree. Then $\deg_{B_n^k} v = n-k+1$, and so $$\begin{aligned} \sigma_1(B_n^k) = &\sigma_1(B_n^k-v) + \sigma_1(B_n^k - N_G[v]) + \displaystyle \sum\limits_{u\in N_G(v)}\sigma_0(B_n^k-(N_G[u]\cup N_G[v]))\\ =&\sigma_1(P_{k-1} \cup \overline{K_{n-k}}) + \sigma_1(P_{k-2}) + \sigma_0(P_{k-3}) + (n-k)\sigma_0(P_{k-2})\\ =&\sigma_1(P_{k-1})\sigma_0(\overline{K_{n-k}}) + \sigma_1(\overline{K_{n-k}})\sigma_0(P_{k-1}) + \sigma_1(P_{k-2}) + \sigma_0(P_{k-3}) \\ &+ (n-k)\sigma_0(P_{k-2})\\ =&\sigma_1(P_{k-1})\cdot 2^{n-k} + \sigma_1(P_{k-2}) + \sigma_0(P_{k-3}) + (n-k)\sigma_0(P_{k-2}).\end{aligned}$$ Let $v$ be a vertex of $L_n^k$ of maximum degree. Then $\deg_{L_n^k} v = n-k+1$, and so $$\begin{aligned} \sigma_1(L_n^k) = &\sigma_1(L_n^k-v) + \sigma_1(L_n^k - N_G[v]) + \displaystyle \sum\limits_{u\in N_G(v)}\sigma_0(L_n^k-(N_G[u]\cup N_G[v]))\\ =&\sigma_1(P_{k-1}\cup K_{n-k}) + \sigma_1(P_{k-2}) + \sigma_0(P_{k-3}) + (n-k)\sigma_0(P_{k-2})\\ =&\sigma_1(P_{k-1})\sigma_0(K_{n-k}) + \sigma_1(K_{n-k})\sigma_0(P_{k-1}) + \sigma_1(P_{k-2}) + \sigma_0(P_{k-3}) \\ &+ (n-k)\sigma_0(P_{k-2})\\ =&(n-k+1)\sigma_1(P_{k-1}) + \binom{n-k}{2} \sigma_0(P_{k-1}) + \sigma_1(P_{k-2}) + \sigma_0(P_{k-3})\\ &+(n-k)\sigma_0(P_{k-2}).\end{aligned}$$ Let $v$ be a vertex of $T_n^k$ of maximum degree. Then $\deg_{T_n^k} v = 3$, and so $$\begin{aligned} \sigma_1(T_n^k) = &\sigma_1(T_n^k-v) + \sigma_1(T_n^k - N_G[v]) + \displaystyle \sum\limits_{u\in N_G(v)}\sigma_0(T_n^k-(N_G[u]\cup N_G[v]))\\ =&\sigma_1(P_{k-1}\cup P_{n-k}) + \sigma_1(P_{k-2}\cup P_{n-k-2}) + \sigma_0(P_{k-3}\cup P_{n-k-2})\\ &+ 2\sigma_0(P_{k-2}\cup P_{n-k-3})\\ =&\sigma_1(P_{k-1})\sigma_0(P_{n-k}) + \sigma_1(P_{n-k})\sigma_0(P_{k-1}) + \sigma_1(P_{k-2})\sigma_0(P_{n-k-2}) \\ &+ \sigma_1(P_{n-k-2})\sigma_0(P_{k-2}) + \sigma_0(P_{k-3})\sigma_0(P_{n-k-2}) + 2\sigma_0(P_{k-2})\sigma_0(P_{n-k-3}).\end{aligned}$$ # Main result {#Sec:Main} In this section we present some bounds on the number of $1$-nearly vertex independent subsets of a graph $G$. In particular, we present a lower bound on the number of $1$-nearly vertex independent subsets of a connected graph $G$ with $m$ edges. Thereafter, we present an upper bound on the number of $1$-nearly vertex independent subsets of a graph $G$ that is not necessarily connected. The families of graphs that achieve equality on each of these bounds are characterised. ## Minimal connected graphs In this subsection we characterise the family of connected graphs with $m$ edges and minimum number of $1$-nearly independent vertex subsets. This will be used to determine the connected graph, and then the tree, with given order and minimum number of $1$-nearly independent vertex subsets. **Definition 1**. *Let $G = (V(G), E(G))$ be a graph with vertex set $V(G)$ and edge set $E(G)$. If $e=uv \in E(G)$, then $e$ is a good edge if $N_G[u]\cup N_G[v] = V(G)$. The graph $G$ is a good graph if for every edge $e \in E(G)$, $e$ is a good edge. Let $$\mathcal{H} = \{ G \mid G \text{ is a good graph} \}.$$* It follows from the definition that a good graph has to be connected. **Theorem 3**. *If $G$ is a connected graph of size $m$, then $$\begin{aligned} \label{if-G-is-a-connected-graph-of-order-n-size-m-then-sigma1G-atleast-m} \sigma_1(G) \ge m, \end{aligned}$$ with equality if and only if $G \in \mathcal{H}$.* *Proof.* Inequality ([\[if-G-is-a-connected-graph-of-order-n-size-m-then-sigma1G-atleast-m\]](#if-G-is-a-connected-graph-of-order-n-size-m-then-sigma1G-atleast-m){reference-type="ref" reference="if-G-is-a-connected-graph-of-order-n-size-m-then-sigma1G-atleast-m"}) follows from the fact that the two ends of any edge of a graph $G$ form a $1$-nearly independent vertex subset of $G$. If $G$ is an $(n,m)$-graph that is not good, then there is an edge $uv$ of $G$ that is not a good edge. That is, $V(G)\setminus(N_G[u]\cup N_G[v])\neq \emptyset$. In addition to the $1$-nearly independent vertex subsets made of the two ends of the edge $uv$, the subset $\{u,v,z\}$ is also a $1$-nearly independent vertex subset, for any $z\in V(G)\setminus (N_G[u]\cup N_G[v]).$ Hence, we have $\sigma_1(G)>m$. ◻ Definition [Definition 1](#Def:Good){reference-type="ref" reference="Def:Good"} already gives a full characterisation of the family $\mathcal{H}$. However, using that description, it is still hard to imagine how the structure of an element of $\mathcal{H}$ with a large number of vertices should look like. The rest of this section is a further investigation of $\mathcal{H}$ aiming to understand the structures of its elements. **Lemma 1**. *The family $\mathcal{H}$ is closed under the join operation.* *Proof.* Let $G_1 = (V(G_1), E(G_1))$ and $G_2 = (V(G_2), E(G_2))$ be two graphs in $\mathcal{H}$. Then $G_1$ and $G_2$ are good graphs. We want to show that $G_1+G_2\in \mathcal{H}$. Since $G_1$ and $G_2$ are good graphs, we have $N_{G_1}[s] \cup N_{G_1}[t] = V(G_1)$ for all $st\in E(G_1)$, and $N_{G_2}[u] \cup N_{G_2}[v] = V(G_2)$ for all $uv\in E(G_2)$. Let $wx \in E(G_1+G_2)$. Then either $wx \in E(G_1)$ or $wx \in E(G_2)$ or $w \in V(G_1)$ and $x\in V(G_2)$. If $wx \in E(G_1)$, then, since $G_1$ is a good graph, $wx$ is a good edge in $G_1$. By the definition of the join operation, $N_{G_1+G_2}[w] = N_{G_1}[w] \cup V(G_2)$ and $N_{G_1+G_2}[x] = N_{G_1}[x] \cup V(G_2)$. Thus, $$N_{G_1+G_2}[w] \cup N_{G_1+G_2}[x] = N_{G_1}[w] \cup N_{G_1}[x] \cup V(G_2) = V(G_1)\cup V(G_2) = V(G_1+G_2).$$ By the same reasoning, if $wx \in E(G_2)$, then, since $G_2$ is a good graph, $wx$ is a good edge in $G_2$, and by the definition of the join operation, $wx$ is also a good edge in $G_1+G_2$. Hence, we assume that $w \in V(G_1)$ and $x \in V(G_2)$. By the definition of the join operation, $N_{G_1+G_2}[w] = N_{G_1}[w] \cup V(G_2)$ and $N_{G_1+G_2}[x] = N_{G_2}[x] \cup V(G_1)$. Since $N_{G_1}[w] \subseteq V(G_1)$ and $N_{G_2}[x] \subseteq V(G_2)$, we have $$N_{G_1+G_2}[w] \cup N_{G_1+G_2}[x] = V(G_1)\cup V(G_2)=V(G_1+G_2),$$ implying that $wx$ is also a good edge in $G_1+G_2$. Since $wx$ was arbitrarily chosen, the graph $G_1+G_2$ is a good graph, and thus $G_1 + G_2 \in \mathcal{H}$. ◻ **Lemma 2**. *If $G \in \mathcal{H}$, then for any integer $\ell \ge 1$, $G + \overline{K_\ell} \in \mathcal{H}$, where $\overline{K_\ell}$ is an edgeless graph of order $\ell$.* *Proof.* Let $G \in \mathcal{H}$. Then $G$ is a good graph, and so $N_G[u] \cup N_G[v] = V(G)$ for all $uv \in E(G)$. Let $\overline{K_\ell}$ be the empty graph of order $\ell$ with vertex set $V(\overline{K_\ell}) = \{w_1, w_2, \ldots, w_\ell\}$. By the join operation, $N_{G+\overline{K_\ell}} [w_i] = V(G)\cup \{w_i\}$ for all $i \in [\ell]$. Let $vx \in E(G+\overline{K_\ell})$. Since $\overline{K_\ell}$ is an edgeless graph, $vx \notin E(\overline{K_\ell})$. If $vx \in E(G)$, then, since $G$ is a good graph, $vx$ is a good edge in $G$. By the definition of the join operation, $N_{G+\overline{K_\ell}}[v] = N_{G}[v] \cup V(\overline{K_\ell})$ and $N_{G+\overline{K_\ell}}[x] = N_{G}[x] \cup V(\overline{K_\ell})$. Thus, $$N_{G+\overline{K_\ell}}[v] \cup N_{G+\overline{K_\ell}}[x] = N_{G}[v] \cup N_{G}[x] \cup V(\overline{K_\ell}) = V(G) \cup V(\overline{K_\ell}) = V(G + \overline{K_\ell}),$$ implying that $vx$ is also a good edge in $G+\overline{K_\ell}$. Hence, we assume that $v\in V(G)$ and $x\in V(\overline{K_\ell})$. Thus, $x=w_i$ for some $i\in [\ell]$. By the definition of the join operation, $N_{G+\overline{K_\ell}} [v] = N_G[v] \cup V(\overline{K_\ell})$. Since $N_G[v] \subseteq V(G)$, we have $$N_{G+\overline{K_\ell}} [v] \cup N_{G+\overline{K_\ell}} [x] = N_{G+\overline{K_\ell}} [v] \cup N_{G+\overline{K_\ell}} [w_i] = V(G) \cup V(\overline{K_\ell})= V(G+\overline{K_\ell}),$$ implying that $vx = vw_i$ is a good edge in $G+\overline{K_\ell}$. Since $wx$ was arbitrarily chosen, the graph $G+\overline{K_\ell}$ is a good graph, and thus $G+\overline{K_\ell} \in \mathcal{H}$. ◻ Let $\mathcal{H}_1=\{K_1\}\cup \{K_{r,s}\mid r,s\in \mathbb{N}\}$. For any integer $k\geq 2$, we define $$\mathcal{H}_k=\{K+H\mid K,H\in \mathcal{H}_{k-1}\}\cup \{G+\overline{K_\ell}\mid G\in \mathcal{H}_{k-1} \text{ and }\ell\in\mathbb{N} \}.$$ **Theorem 4**. *$$\mathcal{H}=\bigcup_{k\in\mathbb{N}}\mathcal{H}_k.$$* *Proof.* It is easy to check that $\mathcal{H}_1\subseteq \mathcal{H}$. Also, by Lemma [Lemma 1](#Lem:ClosedH){reference-type="ref" reference="Lem:ClosedH"} and Lemma [Lemma 2](#Lem:ExtendH){reference-type="ref" reference="Lem:ExtendH"}, it follows that $\bigcup_{k\in\mathbb{N}}\mathcal{H}_k\subseteq \mathcal{H}$. Thus, it remains for us to prove the reverse inclusion. Suppose that $G$ is a good graph. Then $G$ is connected. If $G$ has only one vertex, then $G \in \mathcal{H}_1 \subseteq \mathcal{H}$. Hence, we may assume that $G$ has more than one vertex. Thus $G$ has at least one edge. We now consider the subgraph $H$ of $G$ with largest number of vertices, and such that $H=K+L$ for some induced subgraphs $K$ and $L$ of $G$. Suppose that $H$ is not $G$. Since $G$ is connected, there is a vertex $u$ in $G-H$ that is adjacent to a vertex $v$ in $H$. Without loss of generality, we can assume that $v\in V(K)$. Then, the edge $uv$ has to be a good edge; that is, $V(H) = V(K)\cup V(L)\subseteq N_G[u]\cup N_G[v]=V(G)$. If there is a vertex $w\in V(K)$ that is not adjacent to $u$, then every vertex in $L$ has to be adjacent to $u$ (because every edge $ws$ for any $s\in V(L)$ has to be a good edge). In this case, we could add $u$ to $K$ and obtain a bigger subgraph of $G$, namely $H'=\langle V(K)\cup\{u\}\rangle_G+L$. However, this contradicts our choice of $H$. Hence, we may assume that all the vertices in $K$ are adjacent to $u$. In this case, we can again add $u$ to $L$ and have a bigger subgraph of $G$, namely $H''=K+\langle V(L)\cup\{u\}\rangle_G$. Once again, this is a contradiction to our choice of $H$. Hence, we must have $V(H)=V(G)$. It is only left to prove that each of $K$ and $L$ is either an edgeless graph or a good graph. It is sufficient to prove that if $K$ (or $L$) has an edge then it is a good graph. Suppose that $xy$ is an edge in $K$. Then it has to be a good edge in $G$; that is, $N_G[x]\cup N_G[y]=V(G)$. Thus, $N_K[x]\cup N_K[y]= (N_G[x]\cup N_G[y])\cap V(K)=V(G)\cap V(K)=V(K)$, implying that $xy$ also a good edge in $K$. ◻ Since the star $K_{1,n-1}$ is the element of $\mathcal{H}$ with fewest edges among all elements of $\mathcal{H}$ with $n$ vertices, and it is a tree, we obtain the following corollaries: **Corollary 1**. *If $G$ is a connected graph of order $n$, then $$\begin{aligned} %\label{sigm-1-atleast-n-minus-1} \sigma_1(G) \ge n-1, \end{aligned}$$ with equality if and only if $G \cong K_{1, n-1}$.* **Corollary 2**. *Among all trees $T$ of order $n$, we have $$\begin{aligned} %\label{sigm-1-atleast-n-minus-1} \sigma_1(T) \ge n-1, \end{aligned}$$ with equality if and only if $G \cong K_{1, n-1}$.* Note that $K_{1,n-1}$ has the largest value if one counts the number of independent vertex subsets instead. ## Maximal graphs In this subsection we characterise the family of graphs with $n$ vertices and maximum number of $1$-nearly independent vertex subsets. The edgeless graph $\overline{K_n}$, that has the maximum $\sigma_0$ among all graphs of order $n$, has the minimum $\sigma_1$ as $\sigma_1(\overline{K_n})=0$. We observe that for $n\geq 7$, $\sigma_1(U_n) =2^{n-3}+n-1> n(n-1)/2=\sigma_1(K_n)$. So, $K_n$ is neither be a minimal graph nor a maximal graph in this case. **Theorem 5**. *If $G$ is a graph of order $n \ge 6$, then $$\begin{aligned} \sigma_1(G) \le \frac{27}{64}\cdot 2^n, \end{aligned}$$ with equality if and only if $G \cong 3K_2 \cup (n-6)K_1$ or $G \cong 4K_2 \cup (n-8)K_1$.* *Proof.* Note that for $n\geq 8$, we have $$\sigma_1(3K_2\cup (n-6)K_1)=3\cdot 3^22^{n-6}=\frac{27}{64}\cdot 2^n=4\cdot 3^32^{n-8}=\sigma_1(4K_2\cup (n-8)K_1).$$ We use induction on the order $n$ of a graph $G$. If $n= 6$, then, by Table [3](#sigma-1-in-graphs-of-order-6){reference-type="ref" reference="sigma-1-in-graphs-of-order-6"}, $\sigma_1(G) \le 27 = \frac{27}{64}\cdot 2^n$, where $3K_2$ is the only one that reaches $27$. this proves the base case. Now we may assume that the result holds for all graphs $G'$ of order $n<k$, where $k \ge 7$. Let $G$ be a graph of order $n=k$ and maximum $\sigma_1$. We proceed with the following series of claims. **Claim 1**. *If $G$ contains an isolated vertex, then $\sigma_1(G) \le \frac{27}{64}\cdot 2^n$, with equality only if $G\in \{3K_2\cup (n-6)K_1,4K_2\cup (n-8)K_1\}$.* *Proof.* Suppose that $G$ contains an isolated vertex $v$. Then $$\begin{aligned} \sigma_1(G) &= 2\sigma_1(G-v)\le 2 \left(\frac{27}{64}\cdot 2^{n-1}\right)= \frac{27}{64}\cdot 2^n.\end{aligned}$$ The equality only holds if $G-v\in\{3K_2\cup ((n-1)-6)K_1),\sigma_1(4K_2\cup ((n-1)-8)K_1)\}$, in which case $G\in\{3K_2\cup (n-6)K_1),\sigma_1(4K_2\cup (n-8)K_1)\}$. This completes the proof of Claim [Claim 1](#if-G-contains-an-isolated-vertex-then-sigma-1-atmost-max){reference-type="ref" reference="if-G-contains-an-isolated-vertex-then-sigma-1-atmost-max"}. ◻ Thus, by Claim [Claim 1](#if-G-contains-an-isolated-vertex-then-sigma-1-atmost-max){reference-type="ref" reference="if-G-contains-an-isolated-vertex-then-sigma-1-atmost-max"}, we may assume that $G$ does not contain an isolated vertex. **Claim 2**. *If $\Delta(G) = 1$, then $\sigma_1(G) \le \frac{27}{64}\cdot 2^n$, where equality only holds if $n\in\{6,8\}$.* *Proof.* Suppose $\Delta(G) = 1$. Then, since $G$ does not contain any isolated vertex, $G \cong \left(\frac{n}{2} \right)K_2$, and we have $\sigma_1(G) = \left(\frac{n}{2}\right) (3)^{\frac{n}{2} -1} = \left(\frac{n}{6}\right) (\sqrt{3})^n$. Consider the function $$\begin{aligned} f(n) = \frac{1}{2^n} \left( \sigma_1(G) - \frac{27}{64}\cdot 2^n \right) = \left(\frac{n}{6}\right) \left(\frac{\sqrt{3}}{2}\right)^n - \frac{27}{64}.\end{aligned}$$ Then $$\begin{aligned} f'(n) = \frac{1}{6}\left(\frac{\sqrt{3}}{2}\right)^n + \left(\frac{n}{6}\right) \left(\frac{\sqrt{3}}{2}\right)^n \ln\left(\frac{\sqrt{3}}{2}\right) = \frac{1}{6} \left( 1 + n\ln\left(\frac{\sqrt{3}}{2}\right) \right) \left(\frac{\sqrt{3}}{2}\right)^n.\end{aligned}$$ Since $\ln\left(\frac{\sqrt{3}}{2}\right) < -0.14$, we have $n\ln\left(\frac{\sqrt{3}}{2}\right) < -1$ for all $n\ge 10$, and so $f'(n) < 0$ for all $n\ge 10$. Therefore, for all $n\ge 10$, we have $f(n) < f(10) < -0.1 \le 0$. We deduce, therefore, that for $n\ge 10$, if $G$ is a graph of order $n$ with no isolated vertices, and with $\Delta(G) =1$, then $\sigma_1(G) < \frac{27}{64}\cdot 2^n$. ◻ By Claim [Claim 1](#if-G-contains-an-isolated-vertex-then-sigma-1-atmost-max){reference-type="ref" reference="if-G-contains-an-isolated-vertex-then-sigma-1-atmost-max"} and Claim [Claim 2](#if-max-degree-one-then-sigma-1-is-atmost-max){reference-type="ref" reference="if-max-degree-one-then-sigma-1-is-atmost-max"}, we may assume that $G$ does not contain an isolated vertex and $\Delta(G) \ge 2$. Let $v$ be a vertex of $G$ with maximum degree $q\ge 2$. **Claim 3**. *If $q \ge 4$, then $\sigma_1(G) < \frac{27}{64}\cdot 2^n$.* *Proof.* Suppose $q\ge 4$, and let $v$ be a vertex of $G$ with maximum degree $q$. Thus, $$\begin{aligned} \sigma_1(G) &= \sigma_1(G-v) + \sigma_1(G - N_G[v]) + \displaystyle \sum\limits_{u\in N_G(v)}\sigma_0(G-(N_G[u]\cup N_G[v]))\\ &\le \frac{27}{64}\cdot 2^{n-1} + \frac{27}{64}\cdot 2^{n-1-q} + q\cdot 2^{n-1-q}\\ &=\frac{27}{64}\cdot 2^n \left( \frac{1}{2} + \frac{1}{2^{q+1}} \left( 1 + \frac{64}{27}q \right) \right).\end{aligned}$$ Now it remains for us to show that for all $q \in \mathbb{Z}$ such that $q \ge 4$, $\frac{1}{2^{q+1}} \left( 1 + \frac{64}{27}q \right) < \frac{1}{2}$. We use induction on $q\ge 4$. If $q=4$, then $\frac{1}{2^{4+1}} \left( 1+ \frac{64}{27}(4) \right) = 0.327 < \frac{1}{2}$, thereby proving the base case. Assume that the result holds for $q = k \ge 4$. That is, assume that $\frac{1}{2^{k+1}} \left( 1 + \frac{64}{27}k \right) < \frac{1}{2}$. Thus, if $q=k+1$, we have $$\begin{aligned} \frac{1}{2^{k+1+1}} \left( 1 + \frac{64}{27}(k+1) \right) &= \frac{1}{2} \cdot \frac{1}{2^{k+1}} \left( 1 + \frac{64}{27}k + \frac{64}{27} \right)\\ &= \frac{1}{2} \cdot \frac{1}{2^{k+1}} \left( 1 + \frac{64}{27}k\right) + \frac{64}{54} \cdot \frac{1}{2^{k+1}}.\\ &\le \frac{1}{2}\cdot \frac{1}{2} + \frac{1}{27} & \text{since } k \ge 4\\ &=\frac{31}{108}< \frac{1}{2}.\end{aligned}$$ Thus, by mathematical induction, $\frac{1}{2^{q+1}} \left( 1 + \frac{64}{27}q \right) < \frac{1}{2}$ for all $q \in \mathbb{Z}$ such that $q \ge 4$. Since $\frac{1}{2^{q+1}} \left( 1 + \frac{64}{27}q \right) < \frac{1}{2}$ for all $q \in \mathbb{Z}$ such that $q \ge 4$, we deduce, therefore, that if $\Delta(G) = q \ge 4$, then $\sigma_1(G) < \frac{27}{64}\cdot 2^n$. ◻ By Claim [Claim 3](#if-q-at-least-4-then-sigma-1-at-most-max){reference-type="ref" reference="if-q-at-least-4-then-sigma-1-at-most-max"} (as well as Claim [Claim 1](#if-G-contains-an-isolated-vertex-then-sigma-1-atmost-max){reference-type="ref" reference="if-G-contains-an-isolated-vertex-then-sigma-1-atmost-max"} and Claim [Claim 2](#if-max-degree-one-then-sigma-1-is-atmost-max){reference-type="ref" reference="if-max-degree-one-then-sigma-1-is-atmost-max"}), we may assume that $2\le q \le 3$. **Claim 4**. *If $G$ is a connected graph with $\Delta(G)=q$, where $2\le q \le 3$, then $\sigma_1(G) < \frac{27}{64}\cdot 2^n$.* *Proof.* Let $G$ be a connected graph of order $n\ge 7$ with $\Delta(G)=q$, where $2\le q\le 3$. If $q=2$, then $G \cong P_n$ or $G \cong C_n$. Let $v$ be a central vertex in $G$. Then $\deg_Gv =2$, and we have $$\begin{aligned} \sigma_1(G) &= \sigma_1(G-v) + \sigma_1(G - N_G[v]) + \displaystyle \sum\limits_{u\in N_G(v)}\sigma_0(G-(N_G[u]\cup N_G[v]))\\ &\le \frac{27}{64}\cdot 2^{n-1} + \frac{27}{64}\cdot 2^{n-3} + 2(2^{n-4})\\ &\le \frac{27}{64}\cdot 2^{n}\left( \frac{1}{2} + \frac{1}{8} + \frac{64}{27}\cdot \frac{1}{8} \right) <\frac{27}{64}\cdot 2^n.\end{aligned}$$ Hence, we may assume that $q=3$. Let $v$ be a vertex of degree $3$ in $G$. Since $n\ge 7$, there must exist at least one neighbour of $v$ having degree at least $2$. Thus, we have $$\begin{aligned} \sigma_1(G) &= \sigma_1(G-v) + \sigma_1(G - N_G[v]) + \displaystyle \sum\limits_{u\in N_G(v)}\sigma_0(G-(N_G[u]\cup N_G[v]))\\ &\le \frac{27}{64}\cdot 2^{n-1} + \frac{27}{64}\cdot 2^{n-4} + 2(2^{n-4}) + 2^{n-5}\\ &\le \frac{27}{64}\cdot 2^{n}\left( \frac{1}{2} + \frac{1}{16} + \frac{64}{27}\cdot \frac{1}{8} + \frac{64}{27}\cdot \frac{1}{32} \right) <\frac{27}{64}\cdot 2^n.\end{aligned}$$ This completes the proof of Claim [Claim 4](#if-q-2-or-q-3-and-G-connected-then-sigma1-max){reference-type="ref" reference="if-q-2-or-q-3-and-G-connected-then-sigma1-max"}. ◻ Hence, by Claim [Claim 4](#if-q-2-or-q-3-and-G-connected-then-sigma1-max){reference-type="ref" reference="if-q-2-or-q-3-and-G-connected-then-sigma1-max"}, we may assume that $2 \le q \le 3$ and $G$ has at least two components. **Claim 5**. *If $G'$ is a component of $G$, then neither $G'$ nor $G-G'$ contains $3K_2$, otherwise $\sigma_1(G)$ does not reach the maximum value.* *Proof.* Suppose that $G'$ is a component of $G$ with $t$ vertices, and one of $G'$ and $G-G'$, say $G'$, contains $3K_2$. Remove edges from $G'$ to make it a $3K_3\cup (|V(G)|-6)K_1$. By the inductive hypothesis (and the the fact that removing any edge increases $\sigma_0$), we have $\sigma_1(G')\leq \sigma_1(3K_3\cup (t-6)K_1)$ and $\sigma_0(G')<\sigma_0(3K_3\cup (t-6)K_1)$. Thus, $$\begin{aligned} \sigma_1(G) &=\sigma_1(G')\sigma_0(G-G')+\sigma_0(G')\sigma_1(G-G')\\ &<\sigma_1(3K_3\cup (t-6)K_1)\sigma_0(G-G')+\sigma_0(3K_3\cup (t-6)K_1)\sigma_1(G-G')=\sigma_1(G''), \end{aligned}$$ where $G''$ is obtained from $G$ by replacing $G'$ by $3K_3\cup (t-6)K_1$. ◻ Therefore, Claim [Claim 5](#if-G'-is-a-component-then-it-must-be-3K_2-free){reference-type="ref" reference="if-G'-is-a-component-then-it-must-be-3K_2-free"} (combined with the fact that $G$ cannot contain an isolated vertex) implies that $G$ has at most three connected components, otherwise three of the component would contain $3K_2$). Suppose that $G$ has three components, namely $G_1$, $G_2$ and $G_3$. Then, in each of the components, any pair of edges are adjacent (otherwise two of the components would have a $3K_2$). This implies that, for $i\in [3]$, $G_i$ is a star or a complete graph $K_3$. Let $G_1$ be the largest component of $G$. Then $G_1$ has at least $3$ vertices, since the order of $G$ is at least $7$. If $G_1$ is a star, let $v$ be a vertex of degree $1$ in $G_1$ attached at a vertex $u$ of degree $q$, where $2\le q\le 3$. Then $G-G_1$ contains $2K_2$ and thus $$\begin{aligned} \sigma_1(G) &= \sigma_1(G-v) + \sigma_1(G - N_G[v]) + \displaystyle \sum\limits_{u\in N_G(v)}\sigma_0(G-(N_G[u]\cup N_G[v]))\\ &=\sigma_1(G-v)+\sigma_1(G-v-u)+\sigma_0(G-v-N[u])\\ &\le \frac{27}{64}\cdot 2^{n-1} + \frac{27}{64}\cdot 2^{n-2} + 3^2\cdot 2^{n-1-q-4}\\ &=\frac{27}{64}\cdot 2^n \left(\frac{1}{2} + \frac{1}{4} + \frac{64}{27}\cdot 9 \cdot \frac{1}{2^{5+q}} \right)<\frac{27}{64}\cdot 2^n, & \text{ since } q \ge 2.\end{aligned}$$ So, $G$ would not have maximum $\sigma_1$ in this case. If $G_1$ is a complete graph $K_3$ with vertices $v,u$ and $z$, then we have $$\begin{aligned} \sigma_1(G) &= \sigma_1(G-v) + \sigma_1(G - N_G[v]) + \displaystyle \sum\limits_{u\in N_G(v)}\sigma_0(G-(N_G[u]\cup N_G[v]))\\ &=\sigma_1(G-v)+\sigma_1(G-v-u-z)+2\sigma_0(G-v-u-z)\\ &\le \frac{27}{64}\cdot 2^{n-1} + \frac{27}{64}\cdot 2^{n-3} + 2\cdot 3^2 \cdot 2^{n-1-2-4}\\ &=\frac{27}{64}\cdot 2^n \left( \frac{1}{2} + \frac{1}{8} + \frac{1}{3}\right)<\frac{27}{64}\cdot 2^n.\end{aligned}$$ Again, $G$ does not have the maximum $\sigma_1$. Suppose that $G$ has two components, namely $G_1$ and $G_2$. Then, neither of them is a $K_1$ nor contains $3K_2$. Suppose that $G_1$ has a vertex with degree $\Delta(G)$. Suppose that $\Delta(G) = q =3$. Since $G-G_1$ contains $K_2$, we have $$\begin{aligned} \sigma_1(G) \sigma_1(G) &= \sigma_1(G-v) + \sigma_1(G - N_G[v]) + \displaystyle \sum\limits_{u\in N_G(v)}\sigma_0(G-(N_G[u]\cup N_G[v]))\\ &\le \frac{27}{64}\cdot 2^{n-1} + \frac{27}{64}\cdot 2^{n-1-3} + 3\cdot 3 \cdot 2^{n-1-3-2}\\ &= \frac{27}{64}\cdot 2^n \left( \frac{1}{2} + \frac{1}{16} + \frac{1}{3} \right)<\frac{27}{64}\cdot 2^n.\end{aligned}$$ Again, $G$ does not have the maximum $\sigma_1$. Hence, we may assume that $\Delta(G) = q = 2$. Then, each component $G_i$, of $G$ (for $i\in [2]$) is a path or a cycle. Since $G_i$ cannot contain $3K_2$, we deduce, therefore, that the order of $G_i$ is at most $5$, and hence the order of $G = G_1\cup G_2$ is at most $10$. Suppose one component, say $G_1$, is a path, and let $v$ be an end-vertex of $G_1$. Then, $G-G_1$ contains $K_2$, and we have $$\begin{aligned} \sigma_1(G) &= \sigma_1(G-v) + \sigma_1(G - N_G[v]) + \displaystyle \sum\limits_{u\in N_G(v)}\sigma_0(G-(N_G[u]\cup N_G[v]))\\ &\le \frac{27}{64}\cdot 2^{n-1} + \frac{27}{64}\cdot 2^{n-2} + 3\cdot 2^{n-1-1-2}\\ &=\frac{27}{64}\cdot 2^n \left( \frac{1}{2} + \frac{1}{4} + \frac{2}{9} \right)<\frac{27}{64}\cdot 2^n.\end{aligned}$$ Hence, we may assume that every component $G_i$, for $i \in [2]$, is a cycle. We, therefore, compute $\sigma_1(G)$ exhaustively as follows: - For $n=7$ we have $\sigma_1(C_3\cup C_4)=37<54=\frac{27}{64}\cdot 2^7.$ - For $n=8$ we have $\sigma_1(C_4\cup C_4)=56 < \sigma_1(C_3\cup C_5)=85<108=\frac{27}{64}\cdot 2^8.$ - For $n=9$ we have $\sigma_1(C_4\cup C_5)=130 < \sigma_1(C_3\cup C_6) = 165 < 216 = \frac{27}{64}\cdot 2^9.$ - For $n=10$ we have $\sigma_1(C_4\cup C_6) = 250 < \sigma_1(C_5\cup C_5)=300 =260<432=\frac{27}{64}\cdot 2^{10}.$ This completes the proof of Theorem [Theorem 5](#if-G-is-a-graph-of-order-n-6-then-sigma-1-at-most-max){reference-type="ref" reference="if-G-is-a-graph-of-order-n-6-then-sigma-1-at-most-max"}. ◻ # Appendix ## $\sigma_1$ in graphs of small order In this appendix, we exhaustively compute $\sigma_1(G)$, where $G$ is any graph of order $n$, where $1\le n\le 6$. If $1\le n \le 4$, then $\sigma_1(G)$ is computed in Table [1](#sigma-1-in-graphs-of-small-order){reference-type="ref" reference="sigma-1-in-graphs-of-small-order"}. $|V(G)|$ $G$ $\mathbf{\sigma_1(G)}$ ---------- ----- ------------------------ $1$ $0$ $2$ $0$ $1$ $3$ $0$ $2$ $2$ $3$ $4$ $0$ $3$ $4$ $4$ $4$ $5$ $5$ $5$ $6$ $6$ $6$ : $\sigma_1$ in graphs of order $n$, where $1\le n \le 4$. If $n = 5$, then $\sigma_1(G)$ is computed in Table [2](#sigma-1-in-graphs-of-order-5){reference-type="ref" reference="sigma-1-in-graphs-of-order-5"}. **A graph $G$** $\mathbf{\sigma_1(G)}$ **A graph $G$** $\mathbf{\sigma_1(G)}$ **A graph $G$** $\mathbf{\sigma_1(G)}$ ----------------- ------------------------ ----------------- ------------------------ ----------------- ------------------------ $0$ $8$ $10$ $4$ $9$ $10$ $6$ $9$ $10$ $6$ $9$ $10$ $7$ $9$ $11$ $8$ $9$ $11$ $8$ $9$ $12$ $8$ $9$ $12$ $8$ $10$ $12$ $8$ $10$ $12$ $8$ $10$ $8$ $10$ : $\sigma_1$ in graphs of order $n=5$. If $n = 6$, then $\sigma_1(G)$ is computed in Table [3](#sigma-1-in-graphs-of-order-6){reference-type="ref" reference="sigma-1-in-graphs-of-order-6"}. **A graph $G$** $\mathbf{\sigma_1(G)}$ **A graph $G$** $\mathbf{\sigma_1(G)}$ **A graph $G$** $\mathbf{\sigma_1(G)}$ ----------------- ------------------------ ----------------- ------------------------ ----------------- ------------------------ $0$ $5$ $8$ $8$ $9$ $9$ $11$ $11$ $11$ $11$ $12$ $12$ $12$ $12$ $12$ $12$ $12$ $13$ $13$ $13$ $13$ $13$ $13$ $13$ $13$ $13$ $13$ $13$ $13$ $14$ $14$ $14$ $14$ $14$ $14$ $14$ $14$ $14$ $14$ $14$ $14$ $14$ **A graph $G$** $\mathbf{\sigma_1(G)}$ **A graph $G$** $\mathbf{\sigma_1(G)}$ **A graph $G$** $\mathbf{\sigma_1(G)}$ ----------------- ------------------------ ----------------- ------------------------ ----------------- ------------------------ $14$ $14$ $14$ $14$ $15$ $15$ $15$ $15$ $15$ $15$ $15$ $15$ $15$ $15$ $15$ $15$ $15$ $15$ $15$ $15$ $15$ $15$ $15$ $15$ $16$ $16$ $16$ $16$ $16$ $16$ $16$ $16$ $16$ $16$ $16$ $16$ $16$ $16$ $16$ $16$ $16$ $16$ **A graph $G$** $\mathbf{\sigma_1(G)}$ **A graph $G$** $\mathbf{\sigma_1(G)}$ **A graph $G$** $\mathbf{\sigma_1(G)}$ ----------------- ------------------------ ----------------- ------------------------ ----------------- ------------------------ $16$ $16$ $16$ $16$ $17$ $17$ $17$ $17$ $17$ $17$ $17$ $17$ $17$ $17$ $17$ $17$ $17$ $17$ $17$ $18$ $18$ $18$ $18$ $18$ $18$ $18$ $18$ $18$ $18$ $18$ $18$ $18$ $18$ $18$ $18$ $18$ $18$ $18$ $19$ $19$ $19$ $19$ **A graph $G$** $\mathbf{\sigma_1(G)}$ **A graph $G$** $\mathbf{\sigma_1(G)}$ **A graph $G$** $\mathbf{\sigma_1(G)}$ ----------------- ------------------------ ----------------- ------------------------ ----------------- ------------------------ $19$ $20$ $20$ $20$ $20$ $20$ $20$ $20$ $20$ $20$ $20$ $20$ $20$ $20$ $20$ $20$ $21$ $21$ $22$ $22$ $22$ $23$ $23$ $23$ $24$ $24$ $24$ $24$ $26$ $27$ : $\sigma_1$ in graphs of order $n=6$.

arxiv_math

--- abstract: | We present here a new splitting method to solve Lyapunov equations of the type $AP + PA^T=-BB^T$ in a Kronecker product form. Although that resulting matrix is of order $n^2$, each iteration of the method demands only two operations with the matrix $A$: a multiplication of the form $(A-\sigma I) \hat{B}$ and a inversion of the form $(A-\sigma I)^{-1}\hat{B}$. We see that for some choice of a parameter the iteration matrix is such that all their eigenvalues are in absolute value less than 1, which means that it should converge without depending on the starting vector. Nevertheless we present a theorem that enables us how to get a good starting vector for the method. author: - Licio Hernanes Bezerra and Felipe Wisniewski (\*) date: | Departamento de Matemática,\ Universidade Federal de S. Catarina, Florianópolis, SC\ Brazil 88040-900 licio.bezerra\@ufsc.br\ (\*) Universidade Estadual do Paraná, União da Vitória, PR\ Brazil 84600-185 felipewisniewski\@yahoo.com.br title: An iterative method to solve Lyapunov equations --- # Introduction Consider the following Lyapunov equation $$\label{lyap} AP+PA^T=-BB^T,$$ where $A \in \mathbb{R}^{n \times n}$ and $B \in \mathbb{R}^{n \times p}$. Let $\lambda(A)$ denote the spectrum of $A$. If all the eigenvalues of $A$ have negative real part, we call $A$ a stable matrix. In this case, there is an unique solution $P$ of equation [\[lyap\]](#lyap){reference-type="eqref" reference="lyap"} and $P$ is a symmetric semi-positive defined matrix. The equation [\[lyap\]](#lyap){reference-type="eqref" reference="lyap"} appears in several engineering problems as can be seen in [@Sim2016]. In Mathematics, for instance, it appears in the calculation of Gram matrices $P$ e $Q$ associated to a dynamical system given by the following equations: $$\Sigma\equiv\left\{\begin{array}{l} \dot x(t)=Ax(t) + Bu(t), \\ y(t)=Cx(t)+Du(t), \quad t \geq t_0, x(t_0)=x_0, \end{array}\right.$$ where $A \in \mathbb{R}^{n\times n},$ $B \in \mathbb{R}^{n\times m}$, $C \in \mathbb{R}^{p\times n},$ $D \in \mathbb{R}^{p\times m},$ $x(t) \in \mathbb{R}^n$, and $u(t) \in \mathbb{R}^m$, which are well analysed in [@Ant]. In [@Sim2016] the author presents an overview of the current state methods of resolution of Lyapunov equation. For large-scale problems we can point out methods based on Krilov subspaces, which consist of projecting the problem onto a significantly smaller Krylov space and then solving the reduced order matrix equation. In [@Sim2007] the author proposes to project onto a block-wise Krylov subspace of the type $$\label{Krylov_racional_introducao} \mathbf{K}_l(A,B)=span\{B, A^{-1}B, AB, A^{-2}B, A^{2}B, A^{-3}B,..., A^{l-2}B, A^{-(l-1)}B\}.$$ Another method of resolution of equation [\[lyap\]](#lyap){reference-type="eqref" reference="lyap"} that is widely used is the ADI (Alternating Direction Implicit) method. It appeared in 1968 [@Smith] and in 1988 a technique that consists in using a control parameter was added, which motivated the name of the method [@Wachspress]. Since then some variations of the method have arisen, for instance, [@Penzl], [@FRM]. In [@Sim2011] it has been proved that under certain conditions ADI is equivalent to a Krylov method with the rational basis given in [\[Krylov_racional_introducao\]](#Krylov_racional_introducao){reference-type="eqref" reference="Krylov_racional_introducao"}. Here we introduce a new stationary iterative methods for solving the Lyapunov equation formulated as a vector form by means of the Kronecker product. In [@Sim2016], it is said that approaches based on the Kronecker formulations were abandoned since that linear system is of order $n^2$, which would be very expensive computationally speaking. However, each iteration of our method demands only two algebraic operations: $(A-\sigma I)^{-1}\hat B$ and $A$ $(A+\sigma I)\tilde B$, where $\hat B$ and $\tilde B$ are matrices of the same order than B, as discussed in the next section. Like Jacobi, Gauss-Seidel and other splitting methods, our method has limitations as regards convergence. Anyway, it is an available simple method to solve Lyapunov equations. # The method We want to calculate a low rank matrix $P$ which is an approximate solution of ([\[lyap\]](#lyap){reference-type="ref" reference="lyap"}). Note that the equation ([\[lyap\]](#lyap){reference-type="ref" reference="lyap"}) can be written as the following standard linear equation $\tilde{A}{p}={b}$, where $\tilde{A}=(I \otimes A +A \otimes I),$ ${p}=\text{vec}(P)$ and ${b}=\text{vec}(-BB^T)$. The symbol $\otimes$ denotes here the Kronecker product of matrices and $\text(vec)(X)$ is the usual representation of a matrix $X$ as a column vector. Let $\sigma$ be a positive real number. Note that the equation ([\[lyap\]](#lyap){reference-type="ref" reference="lyap"}) is also equivalent to $\tilde{A}_{\sigma}{p}={b}$, where $$\tilde{A}_{\sigma}=\left[I\otimes (A-\sigma I)+(A+\sigma I) \otimes I\right].$$ From this we can define the splitting as follows: $$\tilde{A}_{\sigma}=M_{\sigma}-N_{\sigma},$$ where $M_{\sigma}=I\otimes (A-\sigma I)$ and $N_{\sigma}=-(A+\sigma I) \otimes I$. Note that if $A$ is a stable matrix and $\sigma$ is positive, then $(A-\sigma I)$ is inversible, and so is the matriz $M_{\sigma}$. Moreover, $$M_{\sigma}^{-1}=I \otimes (A-\sigma I)^{-1}.$$ From a starting vector ${p}_0 \in \mathbb{R}^{n^2}$, we define for $k=0,1,2,...$ the following iterative method: $$\label{splittin_lyap_geral} {p}_{k+1}=M_{\sigma}^{-1}N_{\sigma}{p}_{k}+M_{\sigma}^{-1}{b},$$ where $$M_{\sigma}^{-1}N_{\sigma}=-(A+\sigma I)\otimes (A-\sigma I)^{-1}.$$ A very well-known theorem about this class of iteration [@GOL] tells us that, for all starting vector $p_0$, the method converges if and only if the spectrum of $M_{\sigma}^{-1}N_{\sigma}$ is less than 1. In our case, **Theorem 1**. *The sequence defined by [\[splittin_lyap_geral\]](#splittin_lyap_geral){reference-type="eqref" reference="splittin_lyap_geral"} converges to the solution of equation [\[lyap\]](#lyap){reference-type="eqref" reference="lyap"} if and only if $$\label{razao_alpha} \left\vert \frac{\lambda_i+\sigma}{\lambda_j-\sigma} \right\vert < 1, \quad \forall \lambda_i,\lambda_j \in \lambda(A).$$* Now, by analysing the iterations of the method, we can state the following result. **Proposition 1**. *Let $P_0= 0$. Then $$\begin{aligned} \label{iteradas} P_{k+1}&=&\sum_{i=1}^{k+1}(-1)^{i+1}(A-\sigma I)^{-i}BB^T\left((A+\sigma I)^T\right)^{(i-1)} \nonumber \\ &=& P_k + (-1)^{k+2}(A-\sigma I)^{-k-1}BB^T\left((A+\sigma I)^T\right)^{k}\end{aligned}$$* *Proof.* $p_1 = M_{\sigma}^{-1}b = \left( I \otimes (A-\sigma I)^{-1}\right) b$, which is equivalent to $$P_1 = -(A-\sigma I)^{-1} BB^T.$$ Suppose that for $k\ge 1$ $$P_k = P_{k-1} + (-1)^{k+1}(A-\sigma I)^{-k}BB^T\left((A+\sigma I)^T\right)^{k-1}.$$ Since for all $k\ge 0$ ${p}_{k+1}=-(A+\sigma I)\otimes (A-\sigma I)^{-1}{p}_{k}+ p_1$, that is, $$P_{k+1} = - (A-\sigma I)^{-1} P_k (A+\sigma I)^T + P_1,$$ we have that $$P_{k+1} = - (A-\sigma I)^{-1} P_{k-1} (A+\sigma I)^T +$$ $$+ (-1)^{k+2} (A-\sigma I)^{-1} (A-\sigma I)^{-k}BB^T\left((A+\sigma I)^T\right)^{k-1} (A+\sigma I)^T + P_1=$$ $$= - (A-\sigma I)^{-1} P_{k-1} (A+\sigma I)^T + (-1)^{k+2} (A-\sigma I)^{-k-1}BB^T\left((A+\sigma I)^T\right)^{k} + P_1.$$ Hence, $$P_{k+1} = P_k + (-1)^{k+2} (A-\sigma I)^{-k-1}BB^T\left((A+\sigma I)^T\right)^{k},$$ ◻ **Remark 2**. *Note that for each step it suffices to compute $(A-\sigma I)^{-1}[(A-\sigma I)^{-k}B]$ and $(A+\sigma I)[(A+\sigma I)^{k-1}] B$, where $[(A-\sigma I)^{-k}B]$ and $[(A+\sigma I)^{k-1}B]$ have already been calculated. Therefore, although the method has been developed to solve a system of order $n^2$, each iteration demands only two algebraic operations with a matrix of order $n$.* Simple calculation yields the following result. **Proposition 2**. *Let $\lambda(A) = \{\lambda_1,\lambda_2,...,\lambda_n\}$, that is, the spectrum of $A$. Suppose $A$ is stable, that is, $Re \, (\lambda)<0$ for all $\lambda \in \lambda(A)$. Then the sequence defined by [\[splittin_lyap_geral\]](#splittin_lyap_geral){reference-type="eqref" reference="splittin_lyap_geral"} converges to the solution of equation [\[lyap\]](#lyap){reference-type="eqref" reference="lyap"} if and only if for all $1\le i, j\le n$ $$\sigma > \frac{|\lambda_j|^2 - |\lambda_i|^2}{2 \, (Re\, \lambda_j + Re\, \lambda_i)}.$$* **Remark 3**. *Since the matrix $A$ is real, without loss of generality we can suppose in the above inequality that $Im \, \lambda_i \ge 0$ and $Im \, \lambda_j \ge 0$. Therefore, $$\frac{|\lambda_j|^2 - |\lambda_i|^2}{2 \, (Re\, \lambda_j + Re\, \lambda_i)} = \frac{|\lambda_j| + |\lambda_i|}{2 \, (Re\, \lambda_j + Re\, \lambda_i)} (|\lambda_j| - |\lambda_i|) \le$$ $$\le \frac{|Re \, \lambda_j + Re\, \lambda_i| + (Im \, \lambda_j + Im \, \lambda_i)} {2 \, |Re\, \lambda_j + Re\, \lambda_i|} \left| |\lambda_j| - |\lambda_i|\right|. \label{sigma}$$ From another point of view, $$\frac{|\lambda_j|^2 - |\lambda_i|^2}{2 \, (Re\, \lambda_j + Re\, \lambda_i)} = \frac{(Re \, \lambda_j)^2 - (Re \, \lambda_i)^2 + (Im \, \lambda_j)^2 - (Im \, \lambda_i)^2}{2 \, (Re\, \lambda_j + Re\, \lambda_i)} =$$ $$=\frac{Re \, \lambda_j - Re \, \lambda_i}{2} + \frac{Im \, \lambda_j + Im \, \lambda_i}{2 \, (Re\, \lambda_j + Re\, \lambda_i)} . (Im \, \lambda_j - Im \, \lambda_i).$$* *In practical studies of stability, we just consider stable matrices that have their spectra contained in the left half-plane limited by lines of the type $y = \pm k x$. For instance, it is usual to take $k\le 20$ in Brazilian electrical power systems stability analysis as safety margin (damping ratio of 5%) [@GMP]. In this case, from [\[sigma\]](#sigma){reference-type="ref" reference="sigma"}, it suffices that $$\sigma \ge \min \left( \frac{21}{2} r(A),\max_{\lambda \in \lambda(A)} \frac{|Re \, \lambda|}{2} + 10 \max_{\lambda \in \lambda(A)} |Im \, \lambda|\right).$$ where $r(A)$ is the spectral radius of $A$, in order to guarantee convergence of the splitting method described above. These parameters can be calculated, for instance, by methods that use Möbius transforms [@lic]. MATLAB function $eigs$ also computes some subsets of eigenvalues, e.g., $eigs(A,k,'largestreal')$ returns the k largest real part eigenvalues [@leh].* # Concluding remarks We introduced here an iteration method that starts with $p_0=0$. The general formulae of the $kth$ iteration from another starting value is similar to the formulae ([\[iteradas\]](#iteradas){reference-type="ref" reference="iteradas"}) obtained with $p_0=0$. We could say that the formulae ([\[lyap\]](#lyap){reference-type="ref" reference="lyap"}) truely resulted from $p_1 = - M^{-1} b$. Moreover, if $rank (B) = p$, each iteration realizes a rank-$p$ update. We have computed approximate solutions of different Lyapunov equations by using our method, typically applying to large sparse matrices that occur in the modelling of power system stability problems. For starting vectors, we have followed the theorem below to calculate a good one. **Theorem 4**. *Suppose that the columns of the $n\times p$ matrix $B$ in equation [\[lyap\]](#lyap){reference-type="eqref" reference="lyap"} are spanned by only $k$ eigenvectors of $A$, that is, $B = V_k R_k$, where $V_k$ is a $n\times k$ matrix formed with those $k$ eigenvectors as columns, and $R_k$ is a $k\times p$ matrix. Then the solution $P$ can be exactly computed.* *Proof.* Let $P=V_kX_kV_k^H$. Let $D_k$ the diagonal matrix such that $AV_k = V_kD_k$. Then $AP + PA^T=-BB^T$ can be written as $$V_kD_kX_kV_k^H + V_kX_k \overline{D_k} V_k^H = - V_k R_k R_k^H V_k^H.$$ Hence, $$D_kX_k + X_k \overline{D_k} = - R_k R_k^H.$$ Therefore, $X_k = - C_k \circ R_k R_k^H$, where $C_k$ is the $k\times k$ Cauchy matrix $$C_k = \left( \begin{array}{cccc} \frac{1}{\lambda_1 + \overline{\lambda_1}} & \frac{1}{\lambda_1 + \lambda_1} & \cdots & \frac{1}{\lambda_1 + \overline{\lambda_k}} \\ \frac{1}{\overline{\lambda_1} + \overline{\lambda_1}} & \frac{1}{\overline{\lambda_1}+\lambda_1} & \cdots & \frac{1}{\overline{\lambda_1} + \overline{\lambda_k}} \\ \vdots & \vdots 7 \vdots & \vdots \\ \frac{1}{\lambda_k + \overline{\lambda_1}} & \frac{1}{\lambda_k + \lambda_1} & \cdots & \frac{1}{\lambda_k + \overline{\lambda_k}} \end{array} \right),$$ and $\circ$ denotes the Hadamard product of matrices. ◻ From Theorem [\[start\]](#start){reference-type="eqref" reference="start"}, if we first calculate an invariant subspace of $A$ close to the subspace spanned by the columns of $B$, it generally results a good starting vector for our method, that is a vector that yields a fast convergence of the method after a few iterations. There are tests and comparisons between our methods and ADI, performances of the method with acceleration etc in [@the], where the method was first introduced from what we have known up to now. 10 A. C. Antoulas, *Approximation of Large-Scale Dynamical Systems*, SIAM, Philadelphia, PA, 2005. L. H. Bezerra and C. Tomei, *Spectral Transformation Algorithms for Computing Unstable Modes of Large ScalePower Systems*, Comput. Appl. Math. **18: 1** (1999), 1--14. F. Freitas, J. Rommes, and N. Martins, *Gramian-based reduction method applied to large sparse power system descriptor models*, IEEE Pow. Ener. Soc. Ge. **23: 3** (2009), 1258-1270. G. H. Golub and C. F. Van Loan, *Matrix Computations*, Johns Hopkins University Press, Baltimore, MD, 3rd Ed., 1996. S. Gomes Jr., N. Martins, and C. Portela, *Computing Small-Signal Stability Boundaries for Large-Scale Power Systems*, IEEE Trans. Power Syst. **18: 2** (2003), 747--752. R. B. Lehoucq, D. C. Sorenson, and C. Yang. *ARPACK Users' Guide. Philadelphia*, SIAM, Philadelphia,PA, 1998. T. Penzl, *A Cyclic Low-Rank Smith Method for Large Sparse Lyapunov Equations*, SIAM J. Sci. Comput. **21: 4** (1999) 1401--1418. V. Simoncini, *A new iterative method for solving large-scale Lyapunov matrix equations* , SIAM J. Sci. Comput. **29: 3** (2007), 1268--1288. V. Druski, L. Knizhnerman, and V. Simoncini, *Analysis of the Rational Krylov Subspace and ADI Methods for Solving the Lyapunov Equation*, SIAM J. Numer. Anal. **49: 5** (2011) 1875--1898. V. Simoncini, *Computational Methods for Linear Matrix Equations* SIAM Rev. **58: 3** (2016), 377--441. R. A. Smith, *Matrix Equation \$XA + BX = C\$*, SIAM J. Appl. Math. *16: 1* (1968), 198--201. F. Wisniewski, *Métodos de Resolução da Equação de Lyapunov e Aplicações em Redução de Modelo*, Dr. thesis, Universidade Federal de Santa Catarina, Florianópolis, Brasil, 2019. E. L. Wachspress, *Iterative solution of the Lyapunov matrix equation* Appl. Math. Lett. **1: 1** (1988), 87--90.

arxiv_math

--- abstract: | This paper introduces a novel higher order Adams inequality that incorporates an exact growth condition for a class of weighted Sobolev spaces. Our rigorous proof confirms the validity of this inequality and provides insights into the optimal nature of the critical constant and the exponent within the denominator. Furthermore, we apply this inequality to study a class of ordinary differential equations (ODEs), where we successfully derive both a concept of the weak solution and a comprehensive regularity theory. address: - Dep. Mathematics, Federal University of Paraı́ba -900, João Pessoa-PB, Brazil - Dep. Mathematics, University of Connecticut , Storrs-CT, United States of America - Dep. Mathematics, Federal University of Paraı́ba -900, João Pessoa-PB, Brazil author: - João Marcos do Ó - Guozhen Lu - Raonı́ Ponciano title: SHARP HIGHER ORDER ADAMS' INEQUALITY WITH EXACT GROWTH CONDITION ON WEIGHTED SOBOLEV SPACES --- # Introduction Our main goal is to establish an Adams-type inequality for a class of weighted Sobolev spaces, including the exact growth condition and sharp constants. Before we present our main results, let us introduce some related classical results. For the case of the first derivative, J. Moser [@MR0301504] refined a result by N. Trudinger [@MR0216286] (see also [@MR0140822] and [@MR0192184]) proving the following theorem. **Theorem 1**. *Let $\Omega\subset\mathbb R^N$ with $N\geq2$ and $u\in W^{1,N}(\Omega)$ such that $\|\nabla u\|_{L^{N}(\Omega)}\leq1$. Then there exists $C(N)>0$ such that $$\int_\Omega\exp(\alpha|u|^{\frac{N}{N-1}})\mathrm dx\leq C(N)|\Omega|$$ for any $\alpha\leq\alpha_N:=N\omega_{N-1}^{N-1}$, where $\omega_{N-1}$ is the area of the surface of the unit ball in $\mathbb R^N$. The constant $\alpha_N$ is sharp because if $\alpha>\alpha_N$, then the integral on the left can be arbitrarily large by an appropriate choice of $u$.* Several generalizations of Theorem [Theorem 1](#theomoser){reference-type="ref" reference="theomoser"} have been developed; see, for instance, [@MR1163431; @MR3053467; @MR1865413; @MR0960950] and references therein. When $\Omega=\mathbb R^N$, many results have been aimed to obtain a similar estimate as in Theorem [Theorem 1](#theomoser){reference-type="ref" reference="theomoser"}. For such cases, we need to replace the function $\exp$ by its series representative without the first terms $$\exp_N(t):=\mathrm e^t-\sum_{j=0}^{N-2}\dfrac{t^j}{j!},\quad N\in\mathbb N,$$ since we only have the embedding $W^{1,N}(\mathbb R^N)\hookrightarrow L^q(\mathbb R^N)$ for $N\leq q<\infty$. Consequently, there exists $u\in W^{1,N}(\mathbb R^N)$ such that $\int_{\mathbb R^N}\exp(\alpha|u|^{\frac{N}{N-1}})\mathrm dx=\infty$ for any $\alpha>0$. We also cite the work due to S. Adachi and K. Tanaka [@MR1646323] (see also [@MR1163431; @MR1704875]) for a class of scale invariant inequality in whole $\mathbb R^N$ involving the Dirichlet norm $\|\nabla u\|_{L^N(\mathbb R^N)},$ **Theorem 2**. *Let $0<\alpha<\alpha_N$. There exists a constant $C(\alpha,N)>0$ such that $$\sup_{\underset{\|\nabla u\|_{L^N(\mathbb R^N)}\leq1}{u\in W^{1,N}(\mathbb R^N)}}\int_{\mathbb R^N}\exp_N(\alpha|u|^{\frac{N}{N-1}})\mathrm dx\leq C(\alpha,N)\|u\|_{L^N(\mathbb R^N)}^N.$$ Moreover, the constant $\alpha_N$ is sharp in the sense that if $\alpha\geq\alpha_N$, the supremum is infinite.* In [@MR2400264], Y. Li and B. Ruf obtained an inequality involving the full Sobolev norm $\|u\|_{W^{1,N}(\mathbb R^N)}:=\|\nabla u\|_{L^N(\mathbb R^N)}+\|u\|_{L^N(\mathbb R^N)}$: **Theorem 3**. *There exists a constant $C(N)>0$ such that for any domain $\Omega\subset\mathbb R^N$, $$\sup_{\underset{\|u\|_{W^{1,N}(\Omega)}\leq1}{u\in W^{1,N}(\Omega)}}\int_\Omega\exp_N(\alpha_N|u|^{\frac{N}{N-1}})\mathrm dx\leq C(N).$$ Moreover, the constant $\alpha_N$ is sharp in the sense that if $\alpha_N$ is replaced by any $\alpha>\alpha_N$, the supremum is infinite.* We mention that a symmetrization-free argument without using the Pólya-Szegö inequality was found for both critical and subcritical Moser-Trudinger inequalities by N. Lam et al. in [@MR3053467], [@MR2980499] and [@MR3130507]. More recently, in [@MR3269875; @MR3729597; @MR4117991] the authors were able to prove that results of the type of Theorem [Theorem 2](#theoat){reference-type="ref" reference="theoat"}, [Theorem 3](#theolr){reference-type="ref" reference="theolr"} and the inequaliies proved in [@MR1163431; @MR1704875] are indeed equivalent. Remember that the Trudinger-Moser problem arose from the search for the optimal Orlicz space $L_\Phi$ such that $W^{1,N}_0(\Omega)\hookrightarrow L_\Phi(\Omega)$, since $W^{1,N}_0(\Omega)\hookrightarrow L^q(\Omega)$ for all $q\geq1$ does not provide the best Orlicz space. Also, note that Theorem [Theorem 2](#theoat){reference-type="ref" reference="theoat"} does not achieve the best constant $\alpha_N$, and Theorem [Theorem 3](#theolr){reference-type="ref" reference="theolr"} achieves it only when assuming the full norm $\|u\|_{W^{1,N}(\mathbb R^N)}\leq1$. Therefore, a natural question arises: *Is it possible to achieve the best constant $\alpha_N$ when\ we only require the restriction $\|\nabla u\|_{L^N(\mathbb R^N)}\leq1$?* S. Ibrahim, N. Masmoudi, and K. Nakanishi [@MR3336837] gave a positive answer to this question for the two dimensional case. Afterward, N. Masmoudi and F. Sani [@MR3355498] generalized the result in $\mathbb{R}^N$ for all dimensions $N$ and H. Tang and the second author [@MR3472818] established independently the same result in hyperbolic space $\mathbb{H}^N$ for all dimension $N$. We only state it in the Euclidean space $\mathbb{R}^N$ here. **Theorem 4**. *There exists a constant $C(N)>0$ such that for any $u\in W^{1,N}(\mathbb R^N)$ with $\|\nabla u\|_{L^N(\mathbb R^N)}\leq1$, $$\int_{\mathbb R^N}\dfrac{\exp_N(\alpha_N |u|^{\frac{N}{N-1}})}{(1+|u|)^{\frac{N}{N-1}}}\mathrm dx\leq C(N)\|u\|_{L^N(\mathbb R^N)}^N.$$ Moreover, this inequality fails if the power $\frac{N}{N-1}$ in the denominator is replaced with any $p<\frac{N}{N-1}$.* In the paper [@MR3943303], N. Lam et al established the singular version of Trudinger-Moser inequality with the exact growth under mixed norm in the Soblev type spaces $u\in D^{N,q}(\mathbb{R}^N)$, where $D^{N,q} (\mathbb{R}^{N}),~q\geq1$ is the completion of $C_{0}^{\infty}\left(\mathbb{R}^{N}\right)$ under the norm $\left\Vert \nabla u\right\Vert _{N}+\left\Vert u\right\Vert _{q}$. We note that when $q=N$, then $D^{N,q}\left(\mathbb{R}^{N}\right) =W^{1,N}\left(\mathbb{R}^{N}\right)$. Then the following singular Trudinger-Moser inequality with exact growth was established by N. Lam, L. Zhang, and the second author in [@MR3943303 Theorem 15]. **Theorem 5**. *Let $N\geq2$, $0\leq \beta<N$, and $p\geq q\geq1$. Then there exists a constant $C=C\left( N,p,q,\beta\right) >0$ such that for all $u\in D^{N,q}\left(\mathbb{R}^{N}\right)$ with $\left\Vert \nabla u\right\Vert _{N}\leq1,$ there holds $${\displaystyle\int\limits_{\mathbb{R}^{N}}}\frac{\Phi_{N,q,\beta}\left( \alpha_{N}\left( 1-\frac{\beta}{N}\right)u^{\frac{N}{N-1}}\right) }{\left( 1+\left\vert u\right\vert ^{\frac{p}{N-1}\left( 1-\frac{\beta}{N}\right) }\right) \left\vert x\right\vert ^{\beta}}dx\leq C\left\Vert u\right\Vert _{q}^{q\left( 1-\frac{\beta}{N}\right) },$$ *where $\Phi_{N,q,\beta}$ is the Taylor series of the exponential without the first terms given by the Equation (F) in [@MR3943303]. Moreover, the inequality does not hold when* $p<q.$* In the paper [@MR3587065], affine Trudinger-Moser inequalities with the exact growth were proved by N. Lam and the second author (see Theorem 1.3 in [@MR3587065]). N. Masmoudi and F. Sani [@MR3225631] also derived the following second order Adams' inequality with the exact growth condition in $\mathbb R^4$: **Theorem 6**. *There exists a constant $C>0$ such that for any $u\in W^{2,2}(\mathbb R^4)$ with $\|\Delta u\|_{L^2(\mathbb R^4)}\leq1$, $$\int_{\mathbb R^4}\dfrac{e^{32\pi^2u^2}-1}{(1+|u|)^2}\mathrm dx\leq C\|u\|^2_{L^2(\mathbb R^2)}.$$ Moreover, this fails if the power 2 in the denominator is replaced with any $p<2$.* H. Tang, M. Zhu, and the second author [@MR3405815] established the second order Adams' inequality with the exact growth condition in $\mathbb{R}^N$ for all dimensions $N\geq3$ by demonstrating the following results: **Theorem 7**. *There exists a constant $C(N)>0$ such that for all $u\in W^{2,\frac{N}2}(\mathbb R^N)$ ($N\geq3$) satisfying $\|\Delta u\|_{L^{\frac{N}2}(\mathbb R^N)}\leq1$, $$\int_{\mathbb R^N}\dfrac{\exp_N(\beta_N|u|^{\frac{N}{N-2}})}{(1+|u|)^{\frac{N}{N-2}}}\leq C(N)\|u\|_{L^{\frac{N}{2}}(\mathbb R^N)}^{\frac{N}2},$$ where $\beta_N=\frac{N}{\omega_{N-1}}\left(\frac{\pi^{\frac{N}2}4}{\Gamma(\frac{N}2-1)}\right)^{\frac{N}{N-2}}$.* **Theorem 8**. *If the power $\frac{N}{N-2}$ in the denominator is replaced by any $p<\frac{N}{N-2}$, there exists a sequence of functions $(u_n)$ such that $\|\Delta u_n\|_{L^{\frac{N}2}(\mathbb R^N)}\leq1$, but $$\dfrac{1}{\|u_n\|^{\frac{N}2}_{L^{\frac{N}2}(\mathbb R^N)}}\int_{\mathbb R^N}\dfrac{\exp_N(\beta_N|u_n|^{\frac{N}{N-2}})}{(1+|u_n|)^p}\mathrm dx\to\infty.$$ Moreover, if $\beta>\beta_N$, there exists a sequence of functions $(u_n)$ such that $\|\Delta u_n\|_{L^{\frac{N}2}(\mathbb R^N)}\leq1$, but $$\dfrac{1}{\|u_n\|^{\frac{N}2}_{L^{\frac{N}2}(\mathbb R^N)}}\int_{\mathbb R^N}\dfrac{\exp_N(\beta|u_n|^{\frac{N}{N-2}})}{(1+|u_n|)^p}\mathrm dx\to\infty,$$ for any $p\geq0$.* A singular version of Adams' inequality with the exact growth was also proved in $W^{2, 2}(\mathbb{R}^4)$ by N. Lam and G. Lu in [@MR3587065] (see Theorem 1.2 there). **Theorem 9**. *Let $0\leq\beta<4$ and $0<\alpha\leq32\pi^{2}\left( 1-\frac{\beta}{4}\right)$. Then there exists a constant $C=C\left( \alpha,\beta\right) >0$ such that $${\displaystyle\int\limits_{\mathbb{R}^{4}}}\frac{e^{\alpha u^{2}}-1}{\left( 1+\left\vert u\right\vert ^{2-\beta/2}\right) \left\vert x\right\vert ^{\beta}}dx\leq C\left\Vert u\right\Vert_{2}^{2-\frac{\beta}{2}}~\text{for all }u\in W^{2,2}\left( \mathbb{R}^{4}\right) :\left\Vert \Delta u\right\Vert _{2}\leq1.$$ *Moreover, the power* $2-\beta/2$ *in the denominator cannot be replaced with any* $q<2-\beta/2$*.** To conclude, we note that N. Masmoudi and F. Sani [@MR3848068] obtained the exact growth inequality with higher order derivatives by considering the $k$-generalized gradient $$\nabla^ku=\left\{\begin{array}{ll} (-\Delta)^{\frac{k}{2}}u,&\mbox{if }m\mbox{ is even}, \\ \nabla(-\Delta)^{\frac{k-1}2}u,&\mbox{if }m\mbox{ is odd}, \end{array}\right.$$ and the critical value $\beta_{N,k}$ defined by $$\beta_{N,k}=\dfrac{N}{\omega_{N-1}}\left\{\begin{array}{ll} \left[\dfrac{\pi^{\frac{N}2}2^k\Gamma\left(\frac{k}2\right)}{\Gamma\left(\frac{N-k}2\right)}\right]^{\frac{N}{N-k}}&\mbox{if }m\mbox{ is even}, \\ \left[\dfrac{\pi^{\frac{N}2}2^k\Gamma\left(\frac{k+1}2\right)}{\Gamma\left(\frac{N-k+1}2\right)}\right]^{\frac{N}{N-k}}&\mbox{if }m\mbox{ is odd}, \end{array}\right.$$ $\omega_{N-1}$ is the surface measure of the unit $N$-ball. Let $\lceil x\rceil:=\min\{n\in\mathbb Z\colon n\geq x\}$ denote the celling function. They precisely proved the following theorem. **Theorem 10**. *Let $k$ be a positive integer with $2<k<N$. There exists a constant $C_{N,k}>0$ such that $$\int_{\mathbb R^N}\dfrac{\exp_{\lceil\frac{N}{k}-2\rceil}\left(\beta_{N,k}|u|^{\frac{N}{N-k}}\right)}{(1+|u|)^{\frac{N}{N-k}}}\mathrm dx\leq C_{N,k}\|u\|^{\frac{N}k}_{L ^{\frac{N}k}(\mathbb R^N)},$$ for all $u\in W^{k,\frac{N}{k}}(\mathbb R^N)$ with $\|\nabla^ku\|_{L^{\frac{N}k}(\mathbb R^N)}\leq1$. Moreover, the above inequality fails if the power $\frac{N}{N-k}$ in the denominator is replaced with any $p<\frac{N}{N-k}$.* We also mention in passing that Trudinger-Moser type inequality with exact growth with Riesz type potential have also been studied in [@Qin] and [@MQ], and the Trudinger-Moser and Adams trace inequalities with exact growth on half spaces were studied in [@CLYZ]. ## Main Results In this work, we prove an inequality involving exact growth for the weighted Sobolev space $X^{k,p}_\infty$, which will be defined in the following paragraph. In recent times, the weighted Sobolev space $X^{k,p}_R$ has been extensively studied due to its applicability in radial elliptic problems for the operator $Lu=-r^{-\gamma}(r^\alpha|u'|^\beta u')'$ which includes the $p$-Laplacian and the $k$-Hessian as a particular case. For more details, see [@MR1422009; @MR3670473; @MR1929156; @MR1069756; @MR1982932], and references therein. Specifically, by choosing the parameters properly, we have $\mathrm{(i)}$ $L$ is the Laplacian for $\alpha=\gamma=N-1$ and $\beta=0$;\ $\mathrm{(ii)}$ $L$ is the $p$-Laplacian for $\alpha=\gamma=N-1$ and $\beta=p-2$;\ $\mathrm{(iii)}$ $L$ is the $k$-Hessian for $\alpha=N-k$, $\gamma=N-1$ and $\beta=k-1$. Therefore, the importance of this space arises from the fact that $X^{1,p}_{0,R}$ is a suitable space to work on problems like: $$\left\{\begin{array}{l} Lu=f(r,u)\mbox{ in }(0,R), \\ u'(0)=u(R)=0,\\ u>0\mbox{ in }(0,R). \end{array}\right.$$ Before we state our results, for easy reference, we introduce some notations and the functions spaces used throughout this work. For each nonnegative integer $\ell$ and $0<R\leq\infty$, let $AC_{\mathrm{loc}}^\ell(0,R)$ be the set of all functions $u\colon(0,R)\to \mathbb R$ such that $u^{(\ell)}\in AC_{\mathrm{loc}}(0,R)$, where $u^{(\ell)}=\mathrm d^\ell u/\mathrm dr^\ell$. For $p\geq1$ and $\alpha$ real numbers, we denote by $L^p_\alpha=L^p_\alpha(0,R)$ the weighted Lebesgue space of measurable functions $u\colon(0,R)\to\mathbb R$ such that $$\|u\|_{L^p_\alpha}=\left(\int_0^R|u|^pr^\alpha\mathrm dr\right)^{1/p}<\infty,$$ which is a Banach space under the standard norm $\|u\|_{L_\alpha^p}$. For any positive integer $k$ and $(\alpha_0,\ldots,\alpha_k)\in \mathbb R^{k+1}$, with $\alpha_j>-1$ for $j=0,1,\ldots,k$, in [@MR4112674], the authors considered the weighted Sobolev spaces for higher order derivatives $X^{k,p}_{0,R}=X^{k,p}_{0,R}(\alpha_0,\ldots,\alpha_k)$ given by all functions $u\in AC^{k-1}_{\mathrm{loc}}(0,R)$ such that $$\lim_{r\to R}u^{(j)}(r)=0,\quad j=0,\ldots,k-1\mbox{ and }u^{(j)}\in L^p_{\alpha_j},\quad j=0,\ldots,k.$$ Recently, in [@arXiv:2302.02262] was considered the weighted Sobolev spaces for higher order derivatives without boundary conditions denoted by $$X_R^{k,p}\!=\!X_R^{k,p}(\alpha_0,\ldots,\alpha_k)\!=\!\{u\colon(0,R)\to\mathbb R : u^{(j)}\in L^p_{\alpha_j},\ j=0,1,\ldots,k\},$$ for any positive integer $k$ and $(\alpha_0,\ldots,\alpha_k)\in\mathbb R^{k+1}$. Using [@arXiv:2302.02262 Proposition 2.2], one can obtain $u\in AC_{\mathrm{loc}}^{k-1}(0,R)$ for all $u\in X^{k,p}_R$. The spaces $X_R^{k,p}$ and $X^{k,p}_{0,R}$ are complete under the norm $$\|u\|_{X_R^{k,p}}=\left(\sum_{j=0}^k\|u^{(j)}\|^p_{L^p_{\alpha_j}}\right)^{1/p}.$$ The study of weighted Sobolev spaces has attracted significant attention due to their essential role in understanding various partial differential equations that involve radial functions. For more details about the theory of weighted Sobolev spaces and embeddings related to such spaces, please refer to [@MR2838041; @MR3209335; @MR3575914; @MR3957979; @arXiv:2108.04977; @arXiv:2203.14181; @MR1929156] and references therein. Considering the elliptic operator $L_{\theta,\gamma}u=-r^{-\theta}(r^{\gamma}u')'$ (which corresponds to the Laplacian when $\theta=\gamma=N-1$ and $u$ is radial), we can define the $k$-generalized operator in a similar manner as defined by D. R. Adams [@MR0960950]: $$\nabla_L^ku=\left\{\begin{array}{ll} L_{\theta,\gamma}^{\frac{k}2}u,&\mbox{if }k\mbox{ is even}, \\ (L_{\theta,\gamma}^{\frac{k-1}2}u)',&\mbox{if }k\mbox{ is odd}. \end{array}\right.$$ Let $0<R<\infty$. We define the weighted Sobolev space with Navier boundary condition, denoted by $X^{k,p}_{\mathcal N,L,R}$, as follows: $$X^{k,p}_{\mathcal N,L,R}=\left\{u\in X_R^{k,p}(\alpha_0,\ldots,\alpha_k)\colon L_{\theta,\gamma}^j u(R)=0\quad \forall j=0,\ldots,\left\lfloor\frac{k-1}2\right\rfloor\right\},$$ where $\lfloor x\rfloor$ denotes the floor function defined as the largest integer less than or equal to $x\in\mathbb R$. When $R=\infty$, $X^{k,p}_{\mathcal N,L,\infty}$ denotes the space $X^{k,p}_\infty$. In our first theorem, we establish the equivalence between the norms $\|\nabla^k_L\cdot\|_{L^p_{\nu}}$ and $\|\cdot\|_{X^{k,p}_R}$ on $X^{k,p}_{\mathcal N,L,R}$ under the following condition: $$\label{hipthetagamma} \left\{\begin{array}{ll} \gamma-1+\lfloor\frac{k-2}{2}\rfloor(\gamma-\theta-2)-\dfrac{\alpha_k-kp+1}{p}>0,&\mbox{if }\theta+2\geq\gamma, \\ \gamma-1-\dfrac{\alpha_k-kp+1}{p}>0,&\mbox{if }\theta+2<\gamma. \end{array}\right.$$ **Theorem 1**. *Assume $0<R\leq\infty$, $k\geq2$, $\alpha_k-(k-1)p+1\geq0$, [\[hipthetagamma\]](#hipthetagamma){reference-type="eqref" reference="hipthetagamma"}, and $\alpha_i\geq\alpha_k-(k-i)p$ for all $i=0,\ldots,k$. The norm $\|\nabla^k_L\cdot\|_{L^p_{\nu}}$ is equivalent to the norm $\|\cdot\|_{X^{k,p}_R}$ on $X^{k,p}_{\mathcal N,L,R}(\alpha_0,\ldots,\alpha_k)$, where $\nu=\alpha_k+\lfloor\frac{k}{2}\rfloor(\theta-\gamma)p$.* The objective of this paper is to provide a comprehensive study to obtain a necessary and sufficient condition on $\beta$ and $q$ for the following inequality to hold: $$\int_0^\infty\dfrac{\exp_p(\beta|u|^{\frac{p}{p-1}})}{(1+|u|)^{q}}r^\eta\mathrm dr\leq C\|u\|^p_{L^p_\eta}$$ for $u\in X^{k,p}_\infty$ satisfying $\|\nabla^k_Lu\|_{L^p_\nu}\leq1$, where $\alpha_k=kp-1$, $\nu=\alpha_k+\lfloor\frac{k}2\rfloor(\theta-\gamma)p$, and $$\exp_p(t):=\sum_{j=0}^\infty\dfrac{t^{p-1+j}}{\Gamma(p+j)}.$$ For more details about the choice of $\exp_p$ instead of $\exp_{\lceil p\rceil}$, see [@arXiv:2306.00194 Remark 1.2]. Firstly, we provide Trudinger-Moser and Adams inequalities with the exact growth condition on weighted Sobolev spaces, precisely for the case of first order derivatives and for the case of second order derivatives. **Theorem 2**. *Let $p>1$, and consider $X^{1,p}_\infty(\alpha_0,\alpha_1)$ with $\alpha_1=p-1$ and $\alpha_0\geq-1$. If $\eta>-1$, then there exists a constant $C=C(p,\eta)>0$ such that for all $u\in X^{1,p}_\infty$ satisfying $\|u'\|_{L^p_{\alpha_1}}\leq 1$, $$\int_0^\infty\dfrac{\exp_p(\beta_{0,1}|u|^{\frac{p}{p-1}})}{(1+|u|)^{\frac{p}{p-1}}}r^{\eta}\mathrm dr\leq C\|u\|^{p}_{L^{p}_{\eta}},$$ where $\beta_{0,1}=\eta+1$.* **Theorem 3**. *Let $p>1$, and consider $X^{2,p}_\infty(\alpha_0,\alpha_1,\alpha_2)$ with $\alpha_2=2p-1$, $\alpha_1\geq p-1$, and $\alpha_0\geq\alpha_1-p$. If $\eta>-1$, $\gamma=(\alpha_2+(p-1)\eta)/p$, and $\theta\in\mathbb R$, then there exists a constant $C=C(p,\eta)>0$ such that for all $u\in X^{2,p}_\infty$ satisfying $\|L_{\theta,\gamma}u\|_{L^p_{\alpha_2+(\theta-\gamma)p}}\leq 1$, $$\int_0^\infty\dfrac{\exp_p(\beta_{0,2}|u|^{\frac{p}{p-1}})}{(1+|u|)^{\frac{p}{p-1}}}r^{\eta}\mathrm dr\leq C\|u\|^{p}_{L^{p}_{\eta}},$$ where $\beta_{0,2}=(\eta+1)\left(\gamma-1\right)^{\frac{p}{p-1}}$.* Using Theorem [Theorem 3](#theo1){reference-type="ref" reference="theo1"} we are able to establish an Adams inequality with the exact growth condition for higher order derivatives, as present in Theorem [Theorem 4](#theoexactk){reference-type="ref" reference="theoexactk"} below. Furthermore, in Theorem [Theorem 5](#theosuperc){reference-type="ref" reference="theosuperc"}, we demonstrate the sharpness of the constant $\beta_{0,k}$ and the power $\frac{p}{p-1}$ in the denominator. **Theorem 4**. *Consider $X^{k,p}_\infty(\alpha_0,\ldots,\alpha_k)$ with $\alpha_k-kp+1=0$. Assume $\alpha_i\geq\alpha_k-(k-i)p$ for all $i=1,\ldots,k$. Let $p>1$, $\eta>-1$, $\theta+2>\gamma$, $\theta>\lfloor\frac{k}{2}\rfloor(\theta+2-\gamma)-1$, and $\gamma=(2p-1+(p-1)\eta)/p$. There exists a constant $C=C(k,p,\eta)>0$ such that for all $u\in X^{k,p}_\infty$ satisfying $\|\nabla_L^ku\|_{L^p_{\nu}}\leq1$ (where $\nu=\alpha_k+\lfloor\frac{k}{2}\rfloor(\theta-\gamma)p$), $$\int_0^\infty\dfrac{\exp_p(\beta_{0,k}|u|^{\frac{p}{p-1}})}{(1+|u|)^{\frac{p}{p-1}}}r^\eta\mathrm dr\leq C\|u\|^p_{L^p_\eta},$$ where $$\beta_{0,k}=\left\{\begin{array}{ll} (\eta+1)\left[(\gamma-1)(\theta+2-\gamma)^{k-2}\dfrac{\Gamma\left(\frac{k}2\right)\Gamma\left(\frac{\gamma-1}{\theta+2-\gamma}\right)}{\Gamma\left(\frac{\gamma-1}{\theta+2-\gamma}-\frac{k-2}{2}\right)}\right]^{\frac{p}{p-1}}&\mbox{if }k\mbox{ is even}, \\ (\eta+1)\left[(\gamma-1)(\theta+2-\gamma)^{k-2}\dfrac{\Gamma\left(\frac{k+1}2\right)\Gamma\left(\frac{\gamma-1}{\theta+2-\gamma}\right)}{\Gamma\left(\frac{\gamma-1}{\theta+2-\gamma}-\frac{k-3}{2}\right)}\right]^{\frac{p}{p-1}}&\mbox{if }k\mbox{ is odd}. \end{array}\right.$$* We note that Theorem [Theorem 4](#theoexactk){reference-type="ref" reference="theoexactk"} is a similar result to Theorem [Theorem 10](#theoexacthigher){reference-type="ref" reference="theoexacthigher"} but, instead of involving the classical Sobolev space $W^{k,\frac{N}k}(\mathbb R^N)$, it involves the weighted Sobolev space $X^{k,p}_\infty$. We develop two corollaries that adapt the results of Theorem [Theorem 2](#theoat){reference-type="ref" reference="theoat"} and Theorem [Theorem 3](#theolr){reference-type="ref" reference="theolr"} to the context of weighted Sobolev spaces. Note that Corollary [Corollary 1](#cor1){reference-type="ref" reference="cor1"} generalizes [@MR3670473 Theorem 1.1]. **Corollary 1**. *Under the assumptions of Theorem [Theorem 4](#theoexactk){reference-type="ref" reference="theoexactk"}, for any $\beta<\beta_{0,k}$, there exists $C_\beta=C(\beta,k,p,\eta)>0$ such that $$\int_0^\infty\exp_p(\beta|u|^{\frac{p}{p-1}})r^\eta\mathrm dr\leq C_\beta\|u\|^p_{L^p_\eta}.$$* **Corollary 2**. *Under the assumptions of Theorem [Theorem 4](#theoexactk){reference-type="ref" reference="theoexactk"} and $p\geq2$, for any $\tau>0$, there exists a constant $C_\tau>0$ such that $$\sup_{\underset{\|\nabla^k_Lu\|^p_{L^p_\nu}+\tau\|u\|^p_{L^p_\eta}\leq1}{u\in X^{k,p}_\infty}}\int_0^\infty\exp_p(\beta_{0,k}|u|^{\frac{p}{p-1}})r^\eta\mathrm dr\leq C_\tau.$$* **Theorem 5**. *Consider $X^{k,p}_\infty(\alpha_0,\ldots,\alpha_k)$ with $\alpha_k-kp+1=0$ and $p>1$. Then there exists a sequence $(u_n)$ such that $\|\nabla^k_Lu_n\|_{L^p_{\alpha_k+\lfloor\frac{k}{2}\rfloor(\theta-\gamma)p}}\leq 1$ and $$\label{eqsuc1} \dfrac{1}{\|u_n\|^p_{L^p_{\eta}}}\int_0^\infty\dfrac{\exp_p\left(\beta_{0,k}|u_n|^{\frac{p}{p-1}}\right)}{(1+|u_n|)^q}r^\eta\mathrm dr\overset{n\to\infty}\longrightarrow \infty,\quad\forall q<\frac{p}{p-1}.$$ Moreover, for all $\beta>\beta_{0,k}$ and $q\geq0$, we have $$\label{eqsuc2} \int_0^\infty\dfrac{\exp_p\left(\beta|u_n|^{\frac{p}{p-1}}\right)}{(1+|u_n|)^q}r^\eta\mathrm dr\overset{n\to\infty}\longrightarrow \infty.$$* ## Application By using Theorem [Theorem 3](#theo1){reference-type="ref" reference="theo1"} in combination with the minimax argument, we obtain the existence and regularity of solutions for a class of the fourth order problem. Precisely, **Theorem 6**. *Assume $\eta>1$ and $f\colon[0,\infty)\times\mathbb R\to\mathbb R$ such that $f(r,\cdot)$ is an odd function with $f(r,t)\geq0$ for all $t\geq0$. Let $\theta=(\eta+3)/2$ and $F(r,t):=\int_0^sf(r,t)\mathrm dt$ satisfying [\[h1\]](#h1){reference-type="eqref" reference="h1"} and [\[h2\]](#h2){reference-type="eqref" reference="h2"}. Then there exists $u_0\in C^4(0,\infty)\cap C^3([0,\infty))$ a nontrivial classical solution of $$\label{eqprobleml} \left\{\begin{array}{l} \Delta^2_{\theta}u=\lambda^{-1}f(r,u)r^{\eta-\theta}\mbox{ in }(0,\infty),\\ u'(0)=(\Delta_\theta u)'(0)=0, \end{array}\right.$$ with $\lambda=\int_0^\infty f(r,u_0)u_0r^\eta\mathrm dr$ and $\Delta_\theta=L_{\theta,\theta}$. Moreover, $\Delta_\theta u_0\in C^2(0,\infty)\cap C^1([0,\infty))$, $u_0''(0)=-\Delta_\theta u_0(0)/(\theta+1)$, and $u_0'''(0)=\lim_{r\to\infty}u_0(r)=\lim_{r\to\infty}\Delta_\theta u_0(r)=0$.* As an immediate consequence, when $\theta=\eta=3$, we obtain the following corollary **Corollary 3**. *Consider the dimensional $N=4$. Assume $f\colon\mathbb R^4\times\mathbb R\to\mathbb R$ radial on $x$ such that $f(x,\cdot)$ is an odd function with $f(x,t)\geq0$ for all $t\geq0$. Let $F(x,t):=\int_0^sf(x,t)\mathrm dt$ satisfying [\[h1\]](#h1){reference-type="eqref" reference="h1"} and [\[h2\]](#h2){reference-type="eqref" reference="h2"}. Then there exists $u_0\in C^4(\mathbb R^4)$ a nontrivial classical solution of $$\Delta^2u=\lambda^{-1}f(x,u)\mbox{ in }\mathbb R^4,$$ with $\lambda=\int_{\mathbb R^N} f(x,u_0)u_0\mathrm dx/|\mathbb S^{3}|$. Moreover, $\lim_{r\to\infty}u_0(r)=\lim_{r\to\infty}\Delta u_0(r)=0$.* ## Organization of the Paper In Section [2](#pre){reference-type="ref" reference="pre"}, we introduce some preliminaries regarding the weighted Sobolev space $X^{k,p}_R$. We will establish in Section [3](#secequi){reference-type="ref" reference="secequi"} the equivalence between the norms $\|\nabla^k_L\cdot\|_{L^p_\nu}$ and $\|\cdot\|_{X^{k,p}_R}$ on $X^{k,p}_{\mathcal N,L,R}$. Moving on to Section [4](#symmandlemma){reference-type="ref" reference="symmandlemma"}, we work on the half $\mu_{\eta,\nu}$-symmetrization, which generalizes the one considered by E. Abreu and L. Fernandez [@MR4097244]. Moreover, in this same section, we establish a crucial lemma (Lemma [Lemma 11](#lemmakey){reference-type="ref" reference="lemmakey"}) that is decisive to the proof of our Theorem [Theorem 3](#theo1){reference-type="ref" reference="theo1"}. To present the main result for the first and second order cases, we delve into Section [5](#mainresults){reference-type="ref" reference="mainresults"}, wherein we prove Theorems [Theorem 2](#theok1){reference-type="ref" reference="theok1"} and [Theorem 3](#theo1){reference-type="ref" reference="theo1"}. In Section [6](#sectionk){reference-type="ref" reference="sectionk"}, we present the conclusive result of the higher order Adams' inequality with exact growth on weighted Sobolev spaces (Theorem [Theorem 4](#theoexactk){reference-type="ref" reference="theoexactk"}) and its corollaries. We obtain, in Section [7](#sectionsuperc){reference-type="ref" reference="sectionsuperc"}, the sharpness of the constants $\beta_{0,k}$ and the exponent $p/(p-1)$ in Theorem [Theorem 5](#theosuperc){reference-type="ref" reference="theosuperc"}. Finally, we provide the proof for the application given by Theorem [Theorem 6](#theoapp){reference-type="ref" reference="theoapp"} in Section [8](#secapp){reference-type="ref" reference="secapp"}. # Preliminaries {#pre} We begin by recalling some definitions following the notation of [@MR1069756]. We say that a function $u\in AC_{\mathrm{loc}}(0,R)$ belongs to class $AC_L(0,R)$ if $\lim_{r\to0}u(r)=0$. Analogously, $u\in AC_{\mathrm{loc}}(0,R)$ belongs to $AC_R(0,R)$ if $\lim_{r\to R}u(r)=0$. In [@MR1069756 Example 6.8], the following Hardy-type inequality was proved: **Proposition 1**. *Given $p,q\in[1,\infty)$ and $\theta,\alpha\in\mathbb R$ the inequality $$\left(\int_0^R|u|^qr^\theta\mathrm dr\right)^{\frac{1}{q}}\leq C\left(\int_0^R|u'|^pr^\alpha\mathrm dr\right)^{\frac{1}{p}}$$ holds for some constant $C=C(p,q,\theta,\alpha,R)>0$ under the following conditions:* *$\mathrm{(i)}$ for $u\in AC_L(0,R)$ if and only if one of the following two conditions are fulfilled:* 1. *$1\leq p\leq q<\infty$, $q\geq\frac{(\theta+1)p}{\alpha-p+1}$, and $\alpha-p+1<0$.* 2. *$1\leq q<p<\infty$, $q>\frac{(\theta+1)p}{\alpha-p+1}$, and $\alpha-p+1<0$.* *$\mathrm{(ii)}$ for $u\in AC_R(0,R)$ if and only if one of the following two conditions is fulfilled:\ * 1. *If $1\leq p\leq q<\infty$, then $$1\leq p\leq q\leq\dfrac{(\theta+1)p}{\alpha-p+1}\mbox{ and }\alpha-p+1>0,$$ or $$\theta>-1\mbox{ and }\alpha-p+1\leq0.$$* 2. *If $1\leq q<p$, then $$1\leq q<p<\infty,\mbox{ with }q<\dfrac{(\theta+1)p}{\alpha-p+1}\mbox{ and }\alpha-p+1>0,$$ or $$\theta>-1\mbox{ and }\alpha-p+1\leq0.$$* We also enunciate the case when $R=\infty$ given by [@MR1069756 Example 6.7]: **Proposition 2**. *Given $1\leq p\leq q<\infty$ and $\theta,\alpha\in\mathbb R$ the inequality $$\left(\int_0^\infty|u|^qr^\theta\mathrm dr\right)^{\frac{1}{q}}\leq C\left(\int_0^\infty|u'|^pr^\alpha\mathrm dr\right)^{\frac{1}{p}}$$ holds for some constant $C=C(p,q,\theta,\alpha)>0$ under the following conditions:* *$\mathrm{(i)}$ for $u\in AC_L(0,R)$ if and only if $\alpha<p-1$ and $\theta=\alpha\frac{q}p-\frac{q(p-1)}p-1$;\ $\mathrm{(ii)}$ for $u\in AC_R(0,R)$ if and only if $\alpha>p-1$ and $\theta=\alpha\frac{q}p-\frac{q(p-1)}p-1$.* Now we recall some previous embeddings and radial lemmas obtained for $X^{k,p}_R$ with $R<\infty$ in [@arXiv:2302.02262] and for $X^{k,p}_\infty$ in [@arXiv:2306.00194]. Moreover, we also present a density result in $X^{k,p}_\infty$. **Theorem 7** (Theorem 1.1 in [@arXiv:2302.02262]). *Let $p\geq1$, $0<R<\infty$, and $\theta\geq\alpha_k-kp$.* *$\mathrm{(a)}$ (Sobolev case) If $\alpha_k-kp+1>0$, then the continuous embedding holds $$X^{k,p}_R(\alpha_0,\ldots,\alpha_k)\hookrightarrow L^q_\theta(0,R)\quad \text{for all}\quad 1\leq q\leq p^*:=\dfrac{(\theta+1)p}{\alpha_k-kp+1}.$$ Moreover, the embedding is compact if $q<p^*.$\ $\mathrm{(b)}$ (Sobolev Limit case) If $\alpha_k-kp+1=0$, then the compact embedding holds $$X^{k,p}_R(\alpha_0,\ldots,\alpha_k)\hookrightarrow L^q_\theta(0,R)\quad \text{for all} \quad q\in[1,\infty).$$\ $\mathrm{(c)}$ (Morrey case) If $\alpha_k-kp+1<0, \; p>1, \; \alpha_k\geq0$ then the continuous embedding holds $$X^{k,p}_R(\alpha_0,\ldots,\alpha_k)\hookrightarrow C^{k-\lfloor\frac{\alpha_k+1}{p}\rfloor-1,\gamma}([0,R]),$$ where $\gamma=\min\left\{1+\left\lfloor\frac{\alpha_k+1}{p}\right\rfloor-\frac{\alpha_k+1}{p},1-\frac1p\right\}$ if $\frac{\alpha_k+1}{p}\notin\mathbb Z$ and $\gamma \in (0,1)$ if $\frac{\alpha_k+1}{p}\in\mathbb Z$.* **Lemma 1** (Propositions 2.3, 2.4 and 2.5 in [@arXiv:2302.02262]). *Let $X^{k,p}_R(\alpha_0,\ldots,\alpha_k)$ be the weighted Sobolev space with $0<R<\infty$ and $1\leq p<\infty$. For all $u\in X^{k,p}_R$ and $r\in(0,R]$ we have $$\label{RL1} |u(r)|\leq C\dfrac1{r^{\frac{\alpha_k-kp+1}{p}}}\|u\|_{X_R^{k,p}},\mbox{ if }\alpha_k-kp+1>0;$$ and $$\label{RL2} |u(r)|\leq C|\log (r/R)|^{\frac{p-1}p}\|u\|_{X^{k,p}_R},\mbox{ if }\alpha_k-kp+1=0\mbox{ and }p>1,$$ where $C=C(\alpha_0,\ldots,\alpha_{k},p,k,R)>0$ is a constant. Moreover, if $\alpha_k-kp+1=0$ and $p=1$, then $$\label{RL3} X^{k,p}_R\hookrightarrow C([0,R]).$$* **Theorem 8** (Theorem 1.1 in [@arXiv:2306.00194]). *Let the weighted Sobolev space $X^{k,p}_\infty(\alpha_0,\ldots,\alpha_k)$ such that $p\geq1$ and $\alpha_{j}\geq\alpha_k-(k-j)p$ for all $j=0,\ldots,k-1$. Consider $\alpha_k-kp\leq\theta\leq\alpha_0$.* *$\mathrm{(a)}$ (Sobolev case) If $\alpha_k-kp+1>0$, then the following continuous embedding holds: $$X^{k,p}_\infty\hookrightarrow L^q_\theta,\quad p\leq q\leq p^*,$$ where $$p^*:=\dfrac{(\theta+1)p}{\alpha_k-kp+1}.$$ Moreover, it is a compact embedding if one of the following two conditions is fulfilled:* 1. *$\theta=\alpha_0$ and $p<q<p^*$;* 2. *$\theta<\alpha_0$ and $p\leq q<p^*$.* *$\mathrm{(b)}$ (Sobolev Limit case) If $\alpha_k-kp+1=0$, then the following continuous embedding holds: $$X^{k,p}_\infty\hookrightarrow L^q_\theta,\quad p\leq q<\infty.$$ Moreover, it is compact embedding if one of the following two conditions is fulfilled:* 1. *$\theta=\alpha_0$, $q>p>1$ and $\alpha_0>-1$;* 2. *$\theta<\alpha_0$ and $q\geq p\geq1$.* **Lemma 2** (Lemma 2.1 in [@arXiv:2306.00194]). *The space $\Upsilon:=\{u|_{[0,\infty)} : u \in C_0^\infty(\mathbb{R})\}$ is dense in the weighted Sobolev space $X^{k,p}_\infty$ under the assumptions $p\geq1$ and $\alpha_{j-1}\geq\alpha_j-p$ for all $j=1,\ldots,k$.* # Equivalence between $\|\nabla_L^k\cdot\|_{L^p_\nu}$ and $\|\cdot\|_{X^{k,p}_R}$ on $X^{k,p}_{\mathcal N,L,R}$ {#secequi} Now we focus on proving Theorem [Theorem 1](#propequivnormkgrad){reference-type="ref" reference="propequivnormkgrad"}, which guarantees the equivalence between the norms $\|\nabla^k_L\cdot\|_{L^p_\nu}$ and $\|\cdot\|_{X^{k,p}_R}$ on $X^{k,p}_{\mathcal N,L,R}$ with $\nu=\alpha_k+\lfloor\frac{k}2\rfloor(\theta-\gamma)p$. Before we present the proof, we need the following four lemmas. The last one (Lemma [Lemma 6](#qww){reference-type="ref" reference="qww"}) is a regularity result for the weighted Sobolev space, which, under some circumstances, implies $L_{\theta,\gamma}u\in X^{k,p}_R\Rightarrow u\in X^{k+2,p}_R$. **Lemma 3**. *Let $0<R\leq\infty$, $j=0,\ldots,k$, and $X^{k,p}_R(\alpha_0,\ldots,\alpha_k)$ be the weighted Sobolev space such that $\alpha_i-(i-1)p+1\geq0$ for all $i=j,\ldots,k$. In the case of $R=\infty$ we also assume $\alpha_{\ell-i}\geq\alpha_\ell-ip$ for all $\ell=j,\ldots,k$ and $i=0,\ldots,\ell-1$. For each $u\in X^{k,p}_R$, we have $\nabla^j_L u\in X^{k-j,p}_R(\alpha_j+\lfloor\frac{j}{2}\rfloor(\theta-\gamma)p,\ldots,\alpha_k+\lfloor\frac{j}{2}\rfloor(\theta-\gamma)p)$ with $$\|\nabla_L^ju\|_{X^{k-j,p}_R}\leq C\|u\|_{X^{k,p}_R},$$ where $C>0$ does not depend on $u$.* *Proof.* We can assume $j\geq2$; otherwise, the lemma is trivial. By induction on $j$ and $\ell$, we can prove that $$\label{doubleinduction} \left(\nabla^j_L u\right)^{(\ell)}(r)=\sum_{i=0}^{j+\ell-1}C_{ij\ell}u^{(j+\ell-i)}r^{\lfloor\frac{j}{2}\rfloor(\gamma-\theta)-i},\quad\forall \ell=0,\ldots,k-j,$$ for some $C_{ij\ell}=C_{ij\ell}(\theta,\gamma)\in\mathbb R$. It is sufficient to check that $$\label{eqbabubabu} \|(\nabla^j_Lu)^{(\ell)}\|_{L^p_{\nu}}\leq C\|u\|_{X^{k,p}_R},$$ where $\nu=\alpha_{j+\ell}+\lfloor\frac{j}{2}\rfloor(\theta-\gamma)p$. Since [\[doubleinduction\]](#doubleinduction){reference-type="eqref" reference="doubleinduction"} and $\alpha_{j+\ell}-(j+\ell-1)p+1\geq0$ hold, we apply Theorem [Theorem 7](#theo32){reference-type="ref" reference="theo32"} (on $u^{(j+\ell-i)}\in X^{i,p}_R(\alpha_{j+\ell-i},\ldots,\alpha_{j+\ell})$) for $0<R<\infty$ and Theorem [Theorem 8](#theoimersaoinfinito){reference-type="ref" reference="theoimersaoinfinito"} for $R=\infty$ to obtain $$\|(\nabla^j_L u)^{(\ell)}\|_{L^p_{\nu}}^p\leq C\sum_{i=0}^{j+\ell-1}\int_0^R\left|u^{(j+\ell-i)}\right|^pr^{\alpha_{j+\ell}-ip}\mathrm dr\leq C\sum_{i=0}^{j+\ell-1}\|u^{(j+\ell-i)}\|^p_{X^{i,p}_R}\leq C\|u\|_{X^{j+\ell,p}_R}^p.$$ This concludes [\[eqbabubabu\]](#eqbabubabu){reference-type="eqref" reference="eqbabubabu"} and the proof of the lemma. ◻ **Lemma 4**. *$X^{k,p}_{\mathcal N,L,R}(\alpha_0,\ldots,\alpha_k)$ is a Banach space with the norm $\|\cdot\|_{X^{k,p}_R}$.* *Proof.* We only need to consider the case when $0<R<\infty$. Let $(u_n)$ be a Cauchy sequence in $X^{k,p}_{\mathcal N,L,R}$. Then $u_n\to u$ in $X^{k,p}_R$ for some $u\in X^{k,p}_R$. By [\[doubleinduction\]](#doubleinduction){reference-type="eqref" reference="doubleinduction"}, $$|L_{\theta,\gamma}^ju(R)|=|\nabla^{2j}_Lu_n(R)-\nabla^{2j}_{L}u(R)|\leq \sum_{i=0}^{2j-1}C_{ij}|u_n^{(2j-i)}(R)-u^{(2j-i)}(R)|R^{j(\gamma-\theta)-i},$$ for all $0\leq j\leq\lfloor\frac{k-1}{2}\rfloor$ and $n\in\mathbb N$. Using [@arXiv:2302.02262 Lemma 2.1] we conclude $$|L_{\theta,\gamma}^ju(R)|\leq \sum_{i=0}^{2j-1}C_{ij}\widetilde C_{ij}\|u_n^{(2j-i)}-u^{(2j-i)}\|_{X^{1,p}_R}R^{j(\gamma-\theta)-i}\overset{n\to\infty}\longrightarrow 0.$$ Therefore, we proved that $u\in X^{k,p}_{\mathcal N,L,R}$ which concludes the completeness of $X^{k,p}_{\mathcal N,L,R}$. ◻ **Lemma 5**. *Let $\alpha_2\in\mathbb R$ and $v\in L^p_{\alpha_2}(0,R)$ with $0<R\leq\infty$ and $1\leq p<\infty$. Define $$\label{expressionu} u(r)=\int_r^Rt^{-\gamma}\int_0^tv(s)s^\theta\mathrm ds\mathrm dt\quad\mathrm{a.e.}\ r\in(0,R).$$ If $\theta>(\alpha_2-p+1)/p$, then $u\in X^{2,p}_R(\alpha_2+(\gamma-\theta-2)p,\alpha_2+(\gamma-\theta-1)p,\alpha_2+(\gamma-\theta)p)$ with $\|u\|_{X^{2,p}_R}\leq C\|v\|_{L^p_{\alpha_2}}$, where $C=C(\theta,\gamma,\alpha_2,p,R)>0$.* *Proof.* The case $R<\infty$ is a direct consequence of [@arXiv:2302.02262 Lemma 4.1] with $\widetilde\alpha_2=\alpha_2+(\gamma-\theta)p$. Note that $$\theta>\frac{\alpha_2-p+1}p\Leftrightarrow\gamma>\frac{\widetilde\alpha_2-p+1}p.$$ For the case $R=\infty$, the same argument as [@arXiv:2302.02262 Lemma 4.1] follows using Proposition [Proposition 2](#prop21JMBOinfty){reference-type="ref" reference="prop21JMBOinfty"} instead of Proposition [Proposition 1](#prop21JMBO){reference-type="ref" reference="prop21JMBO"}. ◻ **Lemma 6**. *Let $v\in X^{k,p}_R(\alpha_k-kp,\ldots,\alpha_k)$ and $$u(r):=\int_r^Rt^{-\gamma}\int_0^tv(s)s^{\theta}\mathrm ds\mathrm dt,\quad r\in(0,R),$$ where $\theta>(\alpha_k-kp-p+1)/p$. Then $$u\in X^{k+2,p}_R(\alpha_k+(\gamma-\theta-k-2)p,\alpha_k+(\gamma-\theta-k-1)p,\ldots,\alpha_k+(\gamma-\theta)p).$$* *Proof.* We claim that $$\label{qw1} v(r)r^{\theta-\gamma-k+i}\in X^{i,p}_R(\alpha_k+(\gamma-\theta-i)p,\ldots,\alpha_k+(\gamma-\theta)p),\quad\forall i=0,\ldots,k.$$ Indeed, the case $i=0$ in [\[qw1\]](#qw1){reference-type="eqref" reference="qw1"} follows by $v\in L^p_{\alpha_k-kp}$. Suppose [\[qw1\]](#qw1){reference-type="eqref" reference="qw1"} holds for any $j=0,\ldots,i$. Our task is to prove that $$v(r)r^{\theta-\gamma-k+i+1}\in X^{i+1,p}_R(\alpha_k+(\gamma-\theta-i-1)p,\ldots,\alpha_k+(\gamma-\theta)p).$$ Since $v(r)r^{\theta-\gamma-k+i+1}\in L^p_{\alpha_k+(\gamma-\theta-i-1)p}$, we only need to check that $$v'(r)r^{\theta-\gamma-k+i+1}+(\theta-\gamma-k+i+1)v(r)r^{\theta-\gamma-k+i}=\left(v(r)r^{\theta-\gamma-k+i+1}\right)'\in X^{i,p}_R.$$ Using induction hypothesis on $v'\in X^{k-1,p}_R$ and $v\in X^{k,p}_R$ we conclude [\[qw1\]](#qw1){reference-type="eqref" reference="qw1"}. Let us verify that $$\label{qw2} r^{-(\gamma+i)}\int_0^rv(s)s^{\theta}\mathrm ds\in L^p_{\alpha_k+(\gamma-\theta+i-k-1)p},\quad\forall i\in\mathbb Z.$$ Denote $w(r)=\int_0^rv(s)s^{\theta}\mathrm ds$. By $\alpha_k-(\theta+k+1)p+1<0$ with Propositions [Proposition 1](#prop21JMBO){reference-type="ref" reference="prop21JMBO"} and [Proposition 2](#prop21JMBOinfty){reference-type="ref" reference="prop21JMBOinfty"}, $$\begin{aligned} \int_0^R\left|r^{-(\gamma+i)}\int_0^rv(s)s^{\theta}\mathrm ds\right|^pr^{\alpha_k+(\gamma-\theta+i-k-1)p}\mathrm dr&=\int_0^R|w(r)|^pr^{\alpha_k-(\theta+k+1)p}\mathrm dr\\ &\leq C\int_0^R|w'(r)|^pr^{\alpha_k-(\theta+k)p}\mathrm dr\\ &=C\|v\|^p_{L^p_{\alpha_k-kp}},\quad\forall i\in\mathbb Z.\end{aligned}$$ We already have $u\in X^{2,p}_R(\alpha_k+(\gamma-\theta-k-2)p,\alpha_k+(\gamma-\theta-k-1)p,\alpha_k+(\gamma-\theta-k)p)$, by Lemma [Lemma 5](#lemmafjs){reference-type="ref" reference="lemmafjs"}. Then it is enough to prove that $$u''(r)=\gamma r^{-(\gamma+1)}\int_0^rv(s)s^{\theta}\mathrm ds-v(r)r^{\theta-\gamma}\in X^{k,p}_R(\alpha_k+(\gamma-\theta-k)p,\ldots,\alpha_k+(\gamma-\theta)p).$$ In view of [\[qw1\]](#qw1){reference-type="eqref" reference="qw1"}, the proof is completed by showing that $$\label{qw3} r^{-(\gamma+1)}\int_0^rv(s)s^{\theta}\mathrm ds\in X^{k,p}_R(\alpha_k+(\gamma-\theta-k)p,\ldots,\alpha_k+(\gamma-\theta)p).$$ Indeed, for each $i\in\mathbb N\cup\{0\}$, set $$w_i(r):=r^{-(\gamma+i+1)}\int_0^rv(s)s^{\theta}\mathrm ds.$$ [\[qw2\]](#qw2){reference-type="eqref" reference="qw2"} guarantees $w_i\in L^p_{\alpha_k+(\gamma-\theta+i-k)p}$. Note that, using [\[qw1\]](#qw1){reference-type="eqref" reference="qw1"}, $$\begin{aligned} w_0\in X^{k,p}_R&\Leftrightarrow-(\gamma+1)r^{-(\gamma+2)}\int_0^rv(s)s^{\theta}\mathrm ds+v(r)r^{\theta-\gamma-1}=w_0'\in X^{k-1,p}_R\\ &\Leftrightarrow w_1\in X^{k-1,p}_R\\ &\Leftrightarrow w_1'\in X^{k-2,p}_R\\ &\ \ \vdots\\ &\Leftrightarrow w_k\in L^p_{\alpha_k+(\gamma-\theta)p}.\end{aligned}$$ This concludes [\[qw3\]](#qw3){reference-type="eqref" reference="qw3"} and therefore the lemma. ◻ *Proof of Theorem [Theorem 1](#propequivnormkgrad){reference-type="ref" reference="propequivnormkgrad"}.* Firstly, note that we can rewrite [\[hipthetagamma\]](#hipthetagamma){reference-type="eqref" reference="hipthetagamma"} as follows: $$\left\{\begin{array}{ll} \theta+1+\lfloor\frac{k}2\rfloor(\gamma-\theta-2)-\dfrac{\alpha_k-kp+1}{p}>0,&\mbox{if }\theta+2\geq\gamma, \\ \theta+1+(\gamma-\theta-2)-\dfrac{\alpha_k-kp+1}{p}>0,&\mbox{if }\theta+2<\gamma. \end{array}\right.$$ Therefore, we have $$\label{hipgammatheta} \theta>\dfrac{\alpha_k-kp+i(\theta+2-\gamma)p-p+1}p,\quad\forall i=1,\ldots,\left\lfloor\frac{k}2\right\rfloor.$$ As did in the proof of [@arXiv:2302.02262 Proposition 4.1], our task is to prove that the mapping $$\begin{aligned} \phi\colon X^{k,p}_{\mathcal N,L,R}&\longrightarrow &L^p_{\alpha_k+\lfloor\frac{k}{2}\rfloor(\theta-\gamma)p}\\ u&\longmapsto&\nabla^k_Lu\end{aligned}$$ is a linear isomorphism. Note that $\phi$ is linear and, by Lemma [Lemma 3](#lemmajaosn){reference-type="ref" reference="lemmajaosn"}, continuous. By the Open Mapping Theorem and Lemma [Lemma 4](#corxnrbanach){reference-type="ref" reference="corxnrbanach"}, it suffices to prove that $\phi$ is bijective. Let $u\in X^{k,p}_{\mathcal N,L,R}$ with $\phi(u)=\nabla^k_{L}u=0$. We claim that $$\label{eqclaimhsu} r^{\gamma}\nabla^i_{L}u\overset{r\to0}\longrightarrow0\quad\forall i\in\{1,\ldots,k-1\}\mbox{ odd number}.$$ Indeed, by [\[doubleinduction\]](#doubleinduction){reference-type="eqref" reference="doubleinduction"}, we have $$\label{eq34} \left|r^{\gamma}\nabla^i_{L}u\right|\leq\sum_{\ell=0}^{i-1}C_{i\ell}|u^{(i-\ell)}|r^{\gamma+\frac{i-1}{2}(\gamma-\theta)-\ell}.$$ Suppose first that $\alpha_k-(k-1)p+1>0$. Since $u^{(i-\ell)}\in X^{k-i+\ell,p}_R$, the radial lemma [\[RL1\]](#RL1){reference-type="eqref" reference="RL1"} implies $|u^{(i-\ell)}(r)|\leq C\|u\|_{X^{k,p}_R}r^{-\frac{\alpha_k-(k-i+\ell)p+1}{p}}$. Then $$\left|r^{\gamma}\nabla^i_{L}u\right|\leq C\|u\|_{X^{k,p}_R}\sum_{\ell=0}^{i-1}r^{\gamma+\frac{i-1}{2}(\gamma-\theta)-\ell-\frac{\alpha_k-(k-i+\ell)p+1}p}\leq C\|u\|_{X^{k,p}_R}r^{\gamma-1+\frac{(i-1)(\gamma-\theta-2)}{2}-\frac{\alpha_k-kp+1}p}.$$ Thus, [\[eqclaimhsu\]](#eqclaimhsu){reference-type="eqref" reference="eqclaimhsu"} follows from [\[hipthetagamma\]](#hipthetagamma){reference-type="eqref" reference="hipthetagamma"}. For the case $\alpha_k-(k-1)p+1=0$, we only need to consider the term with $\ell=i-1$ in [\[eq34\]](#eq34){reference-type="eqref" reference="eq34"} because $u^{(i-\ell)}\in X^{k-i+\ell,p}_R$ fits in the Sobolev case when $\ell<i-1$ and it was solved in the other case. Using radial lemmas [\[RL2\]](#RL2){reference-type="eqref" reference="RL2"} and [\[RL3\]](#RL3){reference-type="eqref" reference="RL3"} instead of [\[RL1\]](#RL1){reference-type="eqref" reference="RL1"}, we obtain $$\left|u^{(i-\ell)}r^{\gamma+\frac{i-1}2(\gamma-\theta)-i+1}\right|\leq C\|u\|_{X^{k,p}_R} r^{\gamma+\frac{(i-1)(\gamma-\theta-2)}{2}}|\log (r/R)|^{\frac{p-1}p}.$$ By [\[hipthetagamma\]](#hipthetagamma){reference-type="eqref" reference="hipthetagamma"} and $\alpha_k-(k-1)p+1=0$, we have $$\left\{\begin{array}{ll} \gamma+\frac{(k-2)(\gamma-\theta-2)}2>0,&\mbox{if }\theta+2\geq\gamma, \\ \gamma>0,&\mbox{if }\theta+2<\gamma. \end{array}\right.$$ Therefore, we conclude [\[eqclaimhsu\]](#eqclaimhsu){reference-type="eqref" reference="eqclaimhsu"}. Let $j$ be the integer such that $k=2j$ or $k=2j+1$. From $\nabla^k_{L} u=0$ and $u\in X^{k,p}_{\mathcal N,L,R}$, we have $\nabla^{2j}_{L}u=L_{\theta,\gamma}^ju=0$. The conditions $u\in X^{k,p}_{\mathcal N,L,R}$ and [\[eqclaimhsu\]](#eqclaimhsu){reference-type="eqref" reference="eqclaimhsu"} guarantee that we can apply the following result $j$-times on $L_{\theta,\gamma}^ju=0$ to obtain $u=0$: $$L_{\theta,\gamma}v=0,\ r^{\gamma}v'\overset{r\to0}\longrightarrow0\mbox{ and }v(R)=0\Rightarrow v=0.$$ This concludes that $\phi$ is injective. Now let us prove that $\phi$ is surjective. Initially, suppose $k=2j$. Given $v\in L^p_{\alpha_k+\lfloor\frac{k}2\rfloor(\theta-\gamma)p}$, Lemma [Lemma 5](#lemmafjs){reference-type="ref" reference="lemmafjs"} implies that there exists $$u_1\!\in\! X^{2,p}_R\!\left(\alpha_k+(j-1)(\theta-\gamma)p-2p,\alpha_k+(j-1)(\theta-\gamma)p-p,\alpha_k+(j-1)(\theta-\gamma)p\right)\cap X^{1,p}_{0,R}$$ such that $L_{\theta,\gamma}u_1=v$. Again using Lemma [Lemma 5](#lemmafjs){reference-type="ref" reference="lemmafjs"} we obtain $$u_2\in X^{2,p}_R(\alpha_k+(j-2)(\theta-\gamma)p-4p,\alpha_k+(j-2)(\theta-\gamma)p-3p,\alpha_k+(j-2)(\theta-\gamma)p-2p)\cap X^{1,p}_{0,R}$$ with $L_{\theta,\gamma}u_2=u_1$. By Lemma [Lemma 6](#qww){reference-type="ref" reference="qww"} and $u_1\in X^{1,p}_{0,R}$, we have $$u_2\in X^{4,p}_{\mathcal N,L,R}(\alpha_k+(j-2)(\theta-\gamma)p-4p,\ldots,\alpha_k+(j-2)(\theta-\gamma)p)$$ and $L^2_{\theta,\gamma}u_2=v$. Proceeding with this argument, we obtain $u_j\in X^{2j,p}_{\mathcal N,L,R}(\alpha_k-2jp,\ldots,\alpha_k)$ such that $\phi(u_j)=L^j_{\theta,\gamma}u_j=v$, because the hypothesis on $\theta$ (see equation [\[hipgammatheta\]](#hipgammatheta){reference-type="eqref" reference="hipgammatheta"}) guarantees $$\theta>\dfrac{\alpha_k-2jp+i(\theta+2-\gamma)p-p+1}{p},\quad\forall i=1,\ldots,j.$$ Now suppose $k=2j+1$. Given $v\in L^p_{\alpha_k+\lfloor{\frac{k}{2}\rfloor(\theta-\gamma)p}}$, set $$\widetilde u(r)=-\int_r^Rv(s)\mathrm ds.$$ By Proposition [Proposition 1](#prop21JMBO){reference-type="ref" reference="prop21JMBO"}, $\widetilde u\in X^{1,p}_{0,R}(\alpha_k+\lfloor{\frac{k}{2}\rfloor(\theta-\gamma)p}-p,\alpha_k+\lfloor{\frac{k}{2}\rfloor(\theta-\gamma)p})$, and $\|\widetilde u\|_{L^p_{\alpha_k+\lfloor{\frac{k}{2}\rfloor(\theta-\gamma)p}-p}}\leq C\|v\|_{L^p_{\alpha_k+\lfloor{\frac{k}{2}\rfloor(\theta-\gamma)p}}}$. As in the proof for $k=2j$ with [\[hipgammatheta\]](#hipgammatheta){reference-type="eqref" reference="hipgammatheta"}, we obtain $u\in X^{2j+1,p}_{\mathcal N,L,R}(\alpha_k-kp,\ldots,\alpha_k)$ such that $L^j_{\theta,\gamma}u=\widetilde u$. Therefore, $\phi(u)=\nabla^k_{L}u=v$, which concludes that $\phi$ is surjective. ◻ # Weighted Symmetrization and Crucial Lemma {#symmandlemma} ## Half-Weighted Symmetrization {#symmandlemma1} In this subsection, we extend the previous work in [@MR4097244] on a half-weighted Schwarz symmetrization using a single measure $\mu_\eta$. However, we now consider a more general scenario by incorporating two different measures: $\mu_\eta$ and $\mu_\nu$. The measure $\mu_\eta$ is defined as $\mathrm d\mu_\eta(r)=r^\eta\mathrm dr$ with $\eta>-1$. Additionally, if $M\subset\mathbb R$ is a measurable set with finite $\mu_\eta-$measure, we let $\nu>-1$ and $M^*=[0,R]$ with $R\in(0,\infty]$ such that $$\mu_\nu((0,R))=\mu_\eta(M).$$ Furthermore, if $u\colon M\to\mathbb R$ is a measurable function satisfying $$\mu_{\eta,u}(t):=\mu_\eta(\{x\in M\colon|u(x)|>t\})<\infty\mbox{ for all }t>0,$$ we denote the half-weighted Schwarz symmetrization of $u$ as $u^*_{\eta,\nu}\colon[0,R]\to\mathbb R$, or simply the half $\mu_{\eta,\nu}-$symmetrization of $u$, given by $$u^*_{\eta,\nu}(r)=\inf\{t\in\mathbb R\colon\mu_\eta(\{x\in M\colon|u(x)|>t\})<\mu_\nu(0,r)\},\quad\forall r\in(0,R),$$ with $u^*_{\eta,\nu}(0):=\lim_{r\to0}u^*_{\eta,\nu}(r)=\mathrm{ess\ sup}(u)$ and $u^*_{\eta,\nu}(R):=\lim_{r\to R}u^*_{\eta,\nu}(r)=0$. For simplicity, we denote $\{u>t\}$ as the set $\{x\in M\colon u(x)>t\}$. Building upon the classical results for Schwarz symmetrization, we establish the following two propositions. **Proposition 3**. *The function $u^*_{\eta,\nu}$ is nonincreasing and left-continuous.* *Proof.* The monotonicity of $u^*_{\eta,\nu}$ follows directly from $$\{t\in\mathbb R\colon\mu_\eta(\{|u|>t\})<\mu_\nu(0,r_1)\}\subset \{t\in\mathbb R\colon\mu_\eta(\{|u|>t\})<\mu_\nu(0,r_2)\},\quad\forall r_1<r_2.$$ Let $r\in[0,R]$. By definition of $u^*_{\eta,\nu}$, given $\varepsilon>0$, there exists a $t$ such that $u^*_{\eta,\nu}(r)\leq t<u^*_{\eta,\nu}(r)+\varepsilon$ and $\mu_\eta(\{|u|>t\})<\mu_\nu(0,r)$. Take $\delta>0$ such that $\mu_\eta(\{|u|>t\})<\mu_\nu(0,r-\delta)$. Then, for all $s\in(r-\delta,r]$, we have $\mu_\eta(\{|u|>t\})<\mu_\nu(0,s)$ and so $u^*_{\eta,\nu}(r)\leq u^*_{\eta,\nu}(s)\leq t<u^*_{\eta,\nu}(r)+\varepsilon$. This proves that $u^*_{\eta,\nu}$ is left-continuous. ◻ **Proposition 4**. *The functions $u\colon M\to\mathbb R$ and $u^*_{\eta,\nu}\colon[0,R]\to\mathbb R$, where $\mu_\eta(M)=\mu_\nu((0,R))$, are equimeasurable with respect to $\mu_\eta$ and $\mu_{\nu}$ respectively, i.e., $$\label{eqhaks} \mu_\eta(\{x\in M\colon |u(x)|>t\})=\mu_\nu(\{x\in [0,R]\colon u^*_{\eta,\nu}(x)>t\}),\quad\forall t\in\mathbb R.$$* *Proof.* Let $t\in\mathbb R$. We can suppose $0\leq t\leq u^*_{\eta,\nu}(0)$. By the definition of $u^*_{\eta,\nu}$, we obtain $$\{r\colon u^*_{\eta,\nu}(r)>t\}\subset\{r\colon\mu_\eta(\{|u|>t\})\geq\mu_\nu(0,r)\}.$$ Set $R_0:=\sup\{r\colon u^*_{\eta,\nu}(r)>t\}$. Since $u^*_{\eta,\nu}$ is nonincreasing and left-continuous, we have $[0,R_0)\subset\{u^*_{\eta,\nu}>t\}\subset[0,R_0]$. Then $$\mu_\nu(\{u^*_{\eta,\nu}>t\})=\mu_\nu(0,R_0)\leq \mu_\eta(\{|u|>t\}).$$ Suppose, by contradiction, that $\mu_\nu(\{u^*_{\eta,\nu}>t\})<\mu_\eta(\{|u|>t\})$. Since $[0,R_0)\subset\{u^*_{\eta,\nu}>t\}$, we have $\mu_\nu(0,R_0)<\mu_\eta(\{|u|>t\})$. Let $\delta>0$ such that $$\mu_{\nu}(0,R_0+\delta)=\mu_\eta(\{|u|>t\}).$$ Thus by $\{u^*_{\eta,\nu}>t\}\subset[0,R_0]$ we obtain $u^*_{\eta,\nu}(R_0+\delta)=t$. Since $u^*_{\eta,\nu}$ is nonincreasing, $u^*_{\eta,\nu}(r)=t$ for all $r\in(R_0,R_0+\delta]$. Making $r\to R_0$ with $r>R_0$ in $$\mu_\eta(\{|u|>t\})\leq\mu_\nu(0,r),$$ we obtain $\mu_\eta(\{|u|>t\})\leq\mu_\nu(\{u^*_{\eta,\nu}>t\})$, which is a contradiction. Therefore the equality [\[eqhaks\]](#eqhaks){reference-type="eqref" reference="eqhaks"} holds. ◻ **Corollary 4**. *Let $u\colon M\to\mathbb R$ be measurable and $\Psi\colon \mathbb R\to[0,\infty)$ be nonnegative and measurable. Then $$\int_M\Psi(|u(r)|)r^\eta\mathrm dr=\int_0^{R}\Psi(u^*_{\eta,\nu}(r))r^\nu\mathrm dr,$$ where $u^*_{\eta,\nu}$ is the half $\mu_{\eta,\nu}$-symmetrization of $u$ and $R\in(0,\infty]$ is such that $\mu_\eta(M)=\mu_\nu((0,R))$.* *Proof.* Proposition [Proposition 4](#prop21){reference-type="ref" reference="prop21"} proved the result for $\Psi$ equal to the characteristic function of a set such as $(t,\infty)$. Then the corollary holds if $\Psi$ is the characteristic function of a Borel set and hence if $\Psi$ is a simple function nonnegative. By the Monotone Convergence Theorem, we conclude for all $\Psi\colon \mathbb R\to[0,\infty)$ nonnegative and measurable. ◻ Let $u^{**}_{\eta,\nu}\colon[0,\infty)\to\mathbb [0,\infty)$ be the maximal function of the rearrangement of $u^*_{\eta,\nu}$, defined as $$\label{plb} u^{**}_{\eta,\nu}(t):=\dfrac{\nu+1}{t^{\nu+1}}\int_0^tu^*_{\eta,\nu}(s)s^\nu\mathrm ds.$$ It follows that $u^{**}_{\eta,\nu}$ is also nonincreasing and $u^*_{\eta,\nu}\leq u^{**}_{\eta,\nu}$. Based on [@MR0928802 Lemma 3.9], we can estimate the norm $\|u^{**}_{\eta,\nu}\|_{L^p_\alpha}$ as stated in the following lemma. **Lemma 7**. *Let $u^*_{\eta,\nu}$ be the half $\mu_{\eta,\nu}$-symmetrization of $u$. If $1<p<\infty$ and $\alpha<\nu p+p-1$, then $$\int_0^\infty|u^{**}_{\eta,\nu}|^pt^\alpha\mathrm dt\leq (\nu+1)^p\left(\dfrac{p}{p-1+\nu p-\alpha}\right)^p\int_0^\infty|u^*_{\eta,\nu}|^pt^\alpha\mathrm dr.$$* *Proof.* Let $a=(\alpha+1-\nu p)(p-1)/p^2$ to simplify notation. By writing $u^*_{\eta,\nu}(s)=s^{-a}s^au^*_{\eta,\nu}(s)$ and applying Hölder's inequality, we obtain (note that $1-ap/(p-1)>0$ because $\alpha<\nu p+p-1$) $$\begin{aligned} u^{**}_{\eta,\nu}(t)&\leq\dfrac{\nu+1}{t^{\nu+1}}\int_0^tu^*_{\eta,\nu}(s)s^\nu\mathrm ds\leq \dfrac{\nu+1}{t^{\nu+1}}\left(\int_0^ts^{-\frac{ap}{p-1}}\mathrm ds\right)^{\frac{p-1}p}\left(\int_0^t|u^*_{\eta,\nu}|^ps^{\nu p+ap}\mathrm ds\right)^{\frac1p}\\ &=(\nu+1)\left(\dfrac{p}{p-1+\nu p-\alpha}\right)^{\frac{p-1}p}t^{-\frac{1}p-\nu-a}\left(\int_0^t|u^*_{\eta,\nu}|^ps^{\nu p+ap}\mathrm ds\right)^{\frac1p}. \end{aligned}$$ Thus, $$|u^{**}_{\eta,\nu}(t)|^p\leq(\nu+1)^p\left(\dfrac{p}{p-1+\nu p-\alpha}\right)^{p-1}t^{-1-\nu p-ap}\int_0^t|u^*_{\eta,\nu}|^ps^{\nu p+ap}\mathrm ds.$$ Using Fubini's Theorem, we get (note that $\alpha<\nu p+p-1$ implies $\alpha<\nu p+ap$) $$\begin{aligned} \int_0^\infty|u^{**}_{\eta,\nu}|^pt^{\alpha}\mathrm dt&\leq (\nu+1)^p\left(\dfrac{p}{p-1+\nu p-\alpha}\right)^{p-1}\\ &\qquad\int_0^\infty \int_s^\infty t^{\alpha-1-\nu p-ap}|u^*_{\eta,\nu}(s)|^ps^{\nu p+ap}\mathrm dt\mathrm ds\\ &=(\nu+1)^p\left(\dfrac{p}{p-1+\nu p-\alpha}\right)^{p-1}\dfrac{1}{\nu p+ap-\alpha}\int_0^\infty|u^*_{\eta,\nu}(s)|^ps^{\alpha}\mathrm ds\\ &=(\nu+1)^p\left(\dfrac{p}{p-1+\nu p-\alpha}\right)^{p}\int_0^\infty|u^*_{\eta,\nu}(s)|^ps^{\alpha}\mathrm ds. \end{aligned}$$ ◻ Next, we present another technical lemma. **Lemma 8**. *Let $p>1$, $\beta>-\alpha_1/(p-1)$, and $\gamma>\beta-1+(\alpha_1-\beta)/p$. If $u\in X^{1,p}_{R}(\alpha_0,\alpha_1)$, then $$1\leq\dfrac{(-\mu_{\beta,u}'(t))^{p-1}}{[(\gamma+1)\mu_{\gamma,u}(t)]^{\frac{p\beta+\alpha_1-\beta}{\gamma+1}}}\left(-\dfrac{\mathrm d}{\mathrm dt}\int_{\{|u|>t\}}|u'|^pr^{\alpha_1}\mathrm dr\right).$$* *Proof.* Set $\theta=\beta+(\alpha_1-\beta)/p$. For fixed $t,h>0$, applying Hölder's inequality, we get $$\dfrac1h\int_{\{t<|u|\leq t+h\}}|u'|r^\theta\mathrm dr\leq\left(\dfrac1h\int_{\{t<|u|\leq t+h\}}|u'|^pr^{\alpha_1}\right)^{\frac1p}\left(\dfrac{\mu_{\beta,u}(t)-\mu_{\beta,u}(t+h)}{h}\right)^{\frac{p-1}p}.$$ By letting $h\downarrow0^+$, we obtain $$\label{eqra1} -\dfrac{\mathrm d}{\mathrm dt}\int_{\{t<|u|\}}|u'|r^\theta\mathrm dr\leq\left(-\dfrac{\mathrm d}{\mathrm dt}\int_{\{t<|u|\}}|u'|^pr^{\alpha_1}\mathrm dr\right)^{\frac1p}\left(-\mu_{\beta,u}'(t)\right)^{\frac{p-1}p}.$$ On the other hand, from the coarea formula and the isoperimetric inequality, $$\int_{\{t<|u|\}}|u'|r^{\theta}\mathrm dr=\int_t^\infty\int_{u^{-1}(s)}r^\theta dH_0(r)\mathrm ds+\int_{-\infty}^{-t}\int_{u^{-1}(s)}r^\theta \mathrm dH_0(r)\mathrm ds.$$ Let $\Omega_t=\{t<|u|\}$, then $\partial\Omega_t=u^{-1}(t)\cup u^{-1}(-t)$. Thus $$\label{eqra2} -\dfrac{\mathrm d}{\mathrm dt}\int_{\{t<|u|\}}|u'|r^\theta\mathrm dr=\int_{\partial\Omega_t}r^\theta\mathrm dH_0(r).$$ Since $\gamma+1>\theta$, we can apply the Isoperimetric Inequality on $\mathbb R$ with weight (see Theorem 6.1 in [@MR3619238]) to obtain $$\label{eqra3} \int_{\partial\Omega}|r|^\theta\mathrm dH_0(r)\geq \left((\gamma+1)\int_\Omega|r|^\gamma\mathrm dr\right)^{\frac{\theta}{\gamma+1}}$$ for any $\Omega\subset\mathbb R$. By using [\[eqra1\]](#eqra1){reference-type="eqref" reference="eqra1"}, [\[eqra2\]](#eqra2){reference-type="eqref" reference="eqra2"}, and [\[eqra3\]](#eqra3){reference-type="eqref" reference="eqra3"}, we conclude the lemma. ◻ Let $f\in L^p_\sigma$ with $p>1$. We consider the following problem with $0<R<\infty$: $$\label{eq7} \left\{\begin{array}{l} L_{\eta,\gamma} u=f\mbox{ in }(0,R), \\ u'(0)=u(R)=0. \end{array}\right.$$ We say that $u\in X^{1,2}_{0,R}(\alpha_0,\gamma)$ is a weak solution of [\[eq7\]](#eq7){reference-type="eqref" reference="eq7"} if $$\int_0^Ru'\varphi'r^{\gamma}\mathrm dr=\int_0^Rf\varphi r^{\eta}\mathrm dr,\quad\forall \varphi\in A,$$ where $A=\{\varphi\in L^{\frac{p}{p-1}}_{\frac{p\eta-\sigma}{p-1}}\colon \varphi'\in L^2_{\gamma}\}$. **Lemma 9**. *Let $u\in X^{1,2}_{0,R}(\alpha_0,\gamma)$ be a weak solution to [\[eq7\]](#eq7){reference-type="eqref" reference="eq7"}. If $\eta>(\sigma-p+1)/p$, then $$-\dfrac{\mathrm d}{\mathrm{dt}}\int_{\{|u|>t\}}|u'|^2r^{{\gamma}}\mathrm dr\leq \int_0^{[(\nu+1)\mu_{\eta,u}(t)]^{\frac1{\nu+1}}}f^*_{\eta,\nu}(r)r^{\nu}\mathrm dr\quad\mbox{a.e. }t>0,$$ where $f^*_{\eta,\nu}$ is the half $\mu_{\eta,\nu}$-symmetrization of $f$.* *Proof.* For fixed $t,h>0$, we define $$\varphi(r):=\left\{\begin{array}{ll} 0&\mbox{if }|u|\leq t, \\ (|u|-t)\ \mathrm{sign}(u)&\mbox{if }t\leq|u|\leq t+h,\\ h\ \mathrm{sign}(u)&\mbox{if }t+h<|u|. \end{array}\right.$$ Since $\eta>(\sigma-p+1)/p$, we have $\varphi\in A$. By considering that $u$ is a weak solution, we obtain $$\int_0^Ru'\varphi'r^{{\gamma}}\mathrm dr=\int_0^Rf\varphi r^\eta\mathrm dr.$$ Thus, $$\begin{aligned} \underset{\{t<|u|\leq t+h\}}{\int}&|u'|^2r^{{\gamma}}\mathrm dr\nonumber\\ &=\underset{\{t<|u|\leq t+h\}}{\int}f(|u|-t)\ \mathrm{sign}(u)r^{\eta}\mathrm dr+\underset{\{|u|>t+h\}}{\int}fh\ \mathrm{sign}(u)r^{\eta}\mathrm dr\nonumber\\ &\leq\underset{\{t<|u|\leq t+h\}}{\int}|f||u|r^{\eta}\mathrm dr-t\underset{\{t<|u|\leq t+h\}}{\int}|f|r^{\eta}\mathrm dr+h\underset{\{|u|>t+h\}}{\int}|f|r^{\eta}\mathrm dr.\label{eq8}\end{aligned}$$ Dividing [\[eq8\]](#eq8){reference-type="eqref" reference="eq8"} by $h$ and letting $h\downarrow0^+$, we obtain $$-\dfrac{\mathrm d}{\mathrm{d}t}\underset{\{|u|>t\}}{\int}|u'|^2r^{\gamma}\mathrm dr\leq-\dfrac{\mathrm d}{\mathrm dt}\underset{\{|u|>t\}}{\int}|f|(|u|-t)r^\eta\mathrm dr.$$ By Leibniz integral rule and Corollary [Corollary 4](#cor21){reference-type="ref" reference="cor21"}, $$-\dfrac{\mathrm d}{\mathrm{d}t}\underset{\{|u|>t\}}{\int}|u'|^2r^{\gamma}\mathrm dr\leq\underset{\{|u|>t\}}{\int}|f|r^{\eta}\mathrm dr=\int_0^{[(\nu+1)\mu_{\eta,u}(t)]^{\frac1{\nu+1}}}f^*_{\eta,\nu}(r)r^{\nu}\mathrm dr.$$ ◻ The next proposition is an adaptation of [@MR3225631 Proposition 3.4] to the half $\mu_{\eta,\nu}$-symmetrization. **Proposition 5**. *Let $u\in X^{1,2}_{0,R}(\alpha_0,\gamma)$ be a weak solution of [\[eq7\]](#eq7){reference-type="eqref" reference="eq7"} with $f\in L^p_\sigma$. If $\eta>(\sigma-p+1)/p$ and $1<\gamma<\eta+2$, then $$u^*_{\eta,\nu}(r_1)-u^*_{\eta,\nu}(r_2)\leq\dfrac{(\nu+1)^{\frac{\gamma-1}{\eta+1}}}{(\eta+1)^{\frac{\eta+\gamma}{\eta+1}}}\int_{r_1}^{r_2}f^{**}_{\eta,\nu}(\xi)\xi^{\nu-\frac{\gamma-1}{\eta+1}(\nu+1)}\mathrm d\xi,\quad\forall 0<r_1\leq r_2\leq R,$$ where $u^*_{\eta,\nu}$ is the half $\mu_{\eta,\nu}$-symmetrization and $f^{**}_{\eta,\nu}$ is given by [\[plb\]](#plb){reference-type="eqref" reference="plb"}.* *Proof.* From $\eta>(\nu-p+1)/p$ and $1<\gamma<\eta+2$ we can use Lemmas [Lemma 8](#lemma22){reference-type="ref" reference="lemma22"} and [Lemma 9](#lemma23){reference-type="ref" reference="lemma23"} to obtain $$1\leq\dfrac{-\mu_{\eta,u}'(t)}{[(\eta+1)\mu_{\eta,u}(t)]^{\frac{\eta+\gamma}{\eta+1}}}\int_0^{[(\nu+1)\mu_{\eta,u}(t)]^{\frac{1}{\nu+1}}}f^*_{\eta,\nu}(r)r^{\nu}\mathrm dr.$$ Considering $0<s'<s$ and integrating on $t$ from $s'$ to $s$, we have $$s-s'\leq\int_s^{s'}\left[(\eta+1)\mu_{\eta,u}(t)\right]^{-\frac{\eta+\gamma}{\eta+1}}\int_0^{[(\nu+1)\mu_{\eta,u}(t)]^{\frac{1}{\nu+1}}}f^*_{\eta,\nu}(r)r^{\nu}\mathrm dr\mu'_{\eta,u}(t)\mathrm dt.$$ Making the variable changing $\xi=[(\nu+1)\mu_{\eta,u}(t)]^{\frac{1}{\nu+1}}$ and using $\int_0^\xi f^*_{\eta,\nu}(r)r^{\nu}\mathrm dr=\xi^{\nu+1}f^{**}_{\eta,\nu}(\xi)/(\nu+1)$, we get $$\label{eq10} s-s'\leq\dfrac{(\nu+1)^{\frac{\gamma-1}{\eta+1}}}{(\eta+1)^{\frac{\eta+\gamma}{\eta+1}}}\int_{[(\nu+1)\mu_{\eta,u}(s)]^{\frac{1}{\nu+1}}}^{[(\nu+1)\mu_{\eta,u}(s')]^{\frac{1}{\nu+1}}}f^{**}_{\eta,\nu}(\xi)\xi^{\nu-\frac{\gamma-1}{\eta+1}(\nu+1)}\mathrm d\xi.$$ Let $0<r_1<r_2\leq R$. Without loss of generality, we may assume $u^*_{\eta,\nu}(r_2)<u^*_{\eta,\nu}(r_1)$. Then there exists $\overline\delta>0$ such that $$u^*_{\eta,\nu}(r_2)+\delta<u^*_{\eta,\nu}(r_1),\quad\forall\delta\in[0,\overline\delta].$$ Fixing $\delta\in(0,\overline \delta]$, we apply equation [\[eq10\]](#eq10){reference-type="eqref" reference="eq10"} for $$s=u^*_{\eta,\nu}(r_1)-\dfrac\delta2\mbox{ and }s'=u^*_{\eta,\nu}(r_2)+\dfrac{\delta}2$$ to obtain $$u^*_{\eta,\nu}(r_1)-u^*_{\eta,\nu}(r_2)-\delta\leq\dfrac{(\nu+1)^{\frac{\gamma-1}{\eta+1}}}{(\eta+1)^{\frac{\eta+\gamma}{\eta+1}}}\int_{[(\nu+1)\mu_{\eta,u}(s)]^{\frac{1}{\nu+1}}}^{[(\nu+1)\mu_{\eta,u}(s')]^{\frac{1}{\nu+1}}}f^{**}_{\eta,\nu}(\xi)\xi^{\nu-\frac{\gamma-1}{\eta+1}(\nu+1)}\mathrm d\xi.$$ By $u^*_{\eta,\nu}(r_1)>s$ and $u^*_{\eta,\nu}(r_2)<s'$, we get $\mu_{\eta,u}(s)\geq r_1^{\nu+1}/(\nu+1)$ and $\mu_{\eta,u}(s')< r_2^{\nu+1}/(\nu+1)$. Then $$u^*_{\eta,\nu}(r_1)-u^*_{\eta,\nu}(r_2)-\delta\leq\dfrac{(\nu+1)^{\frac{\gamma-1}{\eta+1}}}{(\eta+1)^{\frac{\eta+\gamma}{\eta+1}}}\int_{r_1}^{r_2}f^{**}_{\eta,\nu}(\xi)\xi^{\nu-\frac{\gamma-1}{\eta+1}(\nu+1)}\mathrm d\xi.$$ Letting $\delta\to0$, we conclude the result. ◻ ## Crucial Lemma In this subsection, we establish a crucial lemma (Lemma [Lemma 11](#lemmakey){reference-type="ref" reference="lemmakey"}) that enables us to derive Adams' inequality with exact growth for the second derivative case. However, before presenting the lemma, we need to recall a technical result (Lemma [Lemma 10](#lemmakey2){reference-type="ref" reference="lemmakey2"}): **Lemma 10** (Lemma 3.4 in [@MR3336837]). *For any given sequence $a=(a_k)_{k\geq0}$, let $p>1$, $\|a\|_1=\sum_{k=0}^\infty|a_k|$, $\|a\|_p=(\sum_{k=0}^\infty|a_k|^p)^{1/p}$, $\|a\|_{(e)}=(\sum_{k=0}^{\infty}|a_k|^pe^k)^{1/p}$, and $\mu(h)=\inf\{\|a\|_{(e)}\colon\|a\|_1=h,\|a\|_p\leq 1\}$. Then, for $h>1$, there exist positive constants $C_1$ and $C_2$ such that $$C_1\mu(h)\leq\dfrac{e^{\frac{h^{\frac{p}{p-1}}}{p}}}{h^{\frac{1}{p-1}}}\leq C_2\mu(h).$$* Now, we can state the main result in this subsection: **Lemma 11**. *Assume $\alpha_2=2p-1>1$, $\alpha_0\geq-1$, $\eta>-1$, $\theta\in\mathbb R$, and $\gamma=(\alpha_2+(p-1)\eta)/p$. Let $u\in X^{2,p}_\infty(\alpha_0,\alpha_1,\alpha_2)$ with compact support in $[0,\infty)$. If $u^*_\eta(R)\geq1$ and $g:=r^{\theta-\eta}L_{\theta,\gamma} u$ satisfies $g\in L^p_{\alpha_2+(\eta-\gamma)p}$ and $$\int_{ R}^\infty|g^{**}_\eta|^pr^\eta\mathrm dr\leq \left(\dfrac{p}{p-1}\right)^p,$$ then $$\dfrac{\exp(\beta_{0,2}|u^*_\eta(R)|^{\frac{p}{p-1}})}{|u^*_\eta(R)|^{\frac{p}{p-1}}}R^{\eta+1}\leq C\int_R^\infty |u^*_\eta(r)|^pr^\eta\mathrm dr,$$ where $C=C(p,\eta)>0$, $g^{**}_\eta$ is given by [\[plb\]](#plb){reference-type="eqref" reference="plb"} with $\nu=\eta$, and $u^*_\eta$ represents the half $\mu_{\eta,\eta}$-symmetrization.* *Proof.* Let $\overline R>0$ such that $\mathrm{supp}(u)\subset[0,\overline R]$. By Theorem [Theorem 7](#theo32){reference-type="ref" reference="theo32"}, we obtain that $u\in X^{1,2}_{0,\overline R}(\alpha_0,\gamma)$ and is a weak solution of $$\left\{\begin{array}{l} L_{\eta,\gamma} u=g\mbox{ in }(0, \overline R), \\ u'(0)=u( \overline R)=0. \end{array}\right.$$ Using Proposition [Proposition 5](#prop22){reference-type="ref" reference="prop22"} on this problem with $\gamma=(\alpha_2+(p-1)\eta)/p$, we have $$\label{eql1} u^*_\eta(r_1)-u^*_\eta(r_2)\leq\dfrac{1}{\eta+1}\int_{ r_1}^{ r_2}g^{**}_\eta(r)r^{\frac{\eta-p+1}{p}}\mathrm dr,\quad\forall 0<r_1\leq r_2\leq \overline R.$$ Let $h_k=c_0u^*_\eta(Re^{\frac{k}{\eta+1}})$ for each $k\in\mathbb N\cup\{0\}$, where $c_0=(\eta+1)^{\frac{2p-1}p}(p-1)/p$. Define $a_k=h_k-h_{k+1}$ and $a=(a_k)$. Then $a_k\geq0$ and $$\|a\|_1=\sum_{k=0}^\infty|a_k|=h_0=c_0u^*_\eta(R).$$ By [\[eql1\]](#eql1){reference-type="eqref" reference="eql1"} and Hölder's inequality, $$\begin{aligned} a_k&=c_0|u^*_\eta(Re^{\frac{k}{\eta+1}})-u^*_\eta(Re^{\frac{k+1}{\eta+1}})|\\ &\leq \dfrac{c_0}{\eta+1}\left(\int_{ Re^{\frac{k}{\eta+1}}}^{ Re^{\frac{k+1}{\eta+1}}}|g^{**}_\eta(r)|^pr^\eta\mathrm dr\right)^{\frac{1}{p}}(\eta+1)^{\frac{1-p}p}\\ &\leq\dfrac{p-1}p\left(\int_{Re^{\frac{k}{\eta+1}}}^{Re^{\frac{k+1}{\eta+1}}}|g^{**}_\eta(r)|^pr^\eta\mathrm dr\right)^{\frac{1}{p}}.\end{aligned}$$ Then $$\|a\|_p=\left(\sum_{k=0}^\infty|a_k|^p\right)^{\frac1p}\leq\left[\left(\dfrac{p-1}p\right)^{p} \int_R^\infty|g^{**}_\eta(r)|^pr^\eta\mathrm dr\right]^{\frac1p}\leq 1.$$ On the other hand, $$\begin{aligned} R^{-\eta-1}\int_R^\infty|u^*_\eta(r)|^pr^\eta\mathrm dr&\geq R^{-\eta-1}\sum_{k=0}^\infty\int_{Re^{\frac{k}{\eta+1}}}^{Re^{\frac{k+1}{\eta+1}}}|u^*_\eta(Re^{\frac{k+1}{\eta+1}})|^pr^\eta\mathrm dr\\ &=\dfrac{1-e^{-1}}{\eta+1}\sum_{k=0}^\infty|u^*_\eta(Re^{\frac{k+1}{\eta+1}})|^pe^{k+1}\\ &\geq C\sum_{k=0}^\infty |h_{k+1}|^pe^{k+1}\geq C\sum_{k=1}^\infty |a_{k}|^pe^k.\end{aligned}$$ Thus, $$\label{eqpt10} \|a\|_{(e)}^p=a_0^p+\sum_{k=1}^\infty |a_k|^pe^k\leq h_0^p+CR^{-\eta-1}\int_R^\infty|u^*_\eta(r)|^pr^\eta\mathrm dr.$$ Let $R<r<Re^b$, where $b=[(\eta+1)(p-1)/2p]^{\frac{p}{p-1}}$. Note that, by [\[eql1\]](#eql1){reference-type="eqref" reference="eql1"}, $$\begin{aligned} h_0-c_0u^*_\eta(r)&\leq\dfrac{c_0}{\eta+1}\int_{ R}^{ r}g^{**}_\eta(s)s^{\frac{\eta-p+1}{p}}\mathrm ds\leq\dfrac{c_0}{\eta+1}\left(\int_{ R}^{ r}|g^{**}_\eta(s)|^ps^\eta\mathrm ds\right)^{\frac1p}b^{\frac{p-1}p}\\ &\leq \dfrac{b^{\frac{p-1}p}c_0}{\eta+1}\dfrac{p}{p-1}\leq\dfrac{h_0}{2}.\end{aligned}$$ Then $h_0\leq Cu^*_\eta(r)$ for all $R<r<Re^b$. Using $$h_0^p\leq C\dfrac{\int_R^{Re^b}h_0^pr^\eta\mathrm dr}{R^{\eta+1}}\leq C\dfrac{\int_R^{\infty}|u^*_\eta(r)|^pr^\eta\mathrm dr}{R^{\eta+1}}$$ along with [\[eqpt10\]](#eqpt10){reference-type="eqref" reference="eqpt10"}, we have $$\|a\|_{(e)}^p\leq C\dfrac{\int_R^{\infty}|u^*_\eta(r)|^pr^\eta\mathrm dr}{R^{\eta+1}}.$$ From Lemma [Lemma 10](#lemmakey2){reference-type="ref" reference="lemmakey2"} and $c_0^{\frac{p}{p-1}}=\beta_{0,2}$, we conclude $$C\dfrac{\int_R^{\infty}|u^*_\eta(r)|^pr^\eta\mathrm dr}{R^{\eta+1}}\geq\left[\frac{\exp\left(\frac{h_0^{\frac{p}{p-1}}}{p}\right)}{h_0^{\frac{1}{p-1}}}\right]^p=\dfrac{\exp\left(\beta_{0,2}|u^*_\eta(R)|^{\frac{p}{p-1}}\right)}{c_0^{\frac{p}{p-1}}|u^*_\eta(R)|^{\frac{p}{p-1}}}.$$ Therefore, $$\dfrac{\exp(\beta_{0,2}|u^*_\eta(R)|^{\frac{p}{p-1}})}{|u^*_\eta(R)|^{\frac{p}{p-1}}}R^{\eta+1}\leq C\int_R^\infty |u^*_\eta(r)|^pr^\eta\mathrm dr,$$ which completes the proof of Lemma [Lemma 11](#lemmakey){reference-type="ref" reference="lemmakey"}. ◻ # Proof of the Exact Growth for First and Second Order {#mainresults} As mentioned in [@MR3405815; @MR3848068; @MR3225631; @MR3355498], establishing the exact growth inequality requires a crucial lemma like Lemma [Lemma 11](#lemmakey){reference-type="ref" reference="lemmakey"}. While the proof of both cases shares similarities, we provide a comprehensive proof specifically for the second derivative case due to its additional intricacies. Once we have proven the second order result (Theorem [Theorem 3](#theo1){reference-type="ref" reference="theo1"}), we will address the approach for handling the first derivative case. *Proof of Theorem [Theorem 3](#theo1){reference-type="ref" reference="theo1"}.* From Lemma [Lemma 2](#lemmacompsuppdense){reference-type="ref" reference="lemmacompsuppdense"} and $\|L_{\theta,\gamma} u\|_{L^p_{\alpha_2+(\theta-\gamma)p}}\leq C\|u\|_{X^{2,p}_\infty}$ (using $\alpha_1\geq p-1$ and Lemma [Lemma 3](#lemmajaosn){reference-type="ref" reference="lemmajaosn"}) we can assume that $\mathrm{supp}(u)\subset[0,\infty)$ is compact and $u\in C^\infty(0,\infty)$. From Corollary [Corollary 4](#cor21){reference-type="ref" reference="cor21"}, it is enough to show that $$\int_0^\infty\dfrac{\exp_p(\beta_{0,2}|u^*_{\eta}|^{\frac{p}{p-1}})}{(1+|u^*_{\eta}|)^{\frac{p}{p-1}}}r^\eta\mathrm dr\leq C\|u^*_{\eta}\|^{p}_{L^{p }_{\eta}},$$ where $u^*_{\eta}$ is the half $\mu_{\eta,\eta}$-symmetrization of $u$. We split the integral into two parts $$\int_0^\infty\dfrac{\exp_p(\beta_{0,2}|u^*_{\eta}|^{\frac{p}{p-1}})}{(1+|u^*_{\eta}|)^{\frac{p}{p-1}}}r^\eta\mathrm dr=\int_0^{R_0}\dfrac{\exp_p(\beta_{0,2}|u^*_{\eta}|^{\frac{p}{p-1}})}{(1+|u^*_{\eta}|)^{\frac{p}{p-1}}}r^\eta\mathrm dr+\int_{R_0}^\infty\dfrac{\exp_p(\beta_{0,2}|u^*_{\eta}|^{\frac{p}{p-1}})}{(1+|u^*_{\eta}|)^{\frac{p}{p-1}}}r^\eta\mathrm dr,$$ where $R_0=\inf\{r\geq0\colon u^*_{\eta}(r)\leq1\}\in[0,\infty)$. Note that $u^*_{\eta}(r)\leq1$ for all $r\in(R_0,\infty)$ and then $\exp_p(\beta_{0,2}|u^*_{\eta}|^{\frac{p}{p-1}})\leq C|u^*_{\eta}|^p$ in $r\in(R_0,\infty)$. Thus, $$\int_{R_0}^\infty\dfrac{\exp_p(\beta_{0,2}|u^*_{\eta}|^{\frac{p}{p-1}})}{(1+|u^*_{\eta}|)^{\frac{p}{p-1}}}r^\eta\mathrm dr\leq C\|u^*_{\eta}\|^p_{L^p_\eta}.$$ Therefore, we just need to consider the integral on $(0,R_0)$. Let $R_1>0$ such that $\mathrm{supp}(u)\subset[0,R_1]$. We define $f:=L_{\theta,\gamma} u\in L^p_{\alpha_2+(\theta-\gamma)p}$ and $g:=r^{\theta-\eta}f\in L^p_{\alpha_2+(\eta-\gamma)p}$. Using Theorem [Theorem 7](#theo32){reference-type="ref" reference="theo32"}, we have $u\in X^{1,2}_{0,R_1}(\alpha_0,\gamma)$ and it is a weak solution of the following equation: $$\label{eqpt1} \left\{\begin{array}{l} L_{\eta,\gamma} u=g\mbox{ in }(0,R_1), \\ u'(0)=u(R_1)=0. \end{array}\right.$$ Using $\gamma=(\alpha_2+(p-1)\eta)/p$ and Proposition [Proposition 5](#prop22){reference-type="ref" reference="prop22"} on the problem [\[eqpt1\]](#eqpt1){reference-type="eqref" reference="eqpt1"}, we obtain $$\label{eq14} u^*_\eta(r_1)-u^*_\eta(r_2)\leq\dfrac{1}{\eta+1}\int_{r_1}^{r_2}g^{**}_\eta(r)r^{\frac{\eta-p+1}{p}}\mathrm dr,\quad\forall 0<r_1\leq r_2\leq R_1,$$ where $g^{**}_\eta$ is given by [\[plb\]](#plb){reference-type="eqref" reference="plb"} with $\nu=\eta$. Defining $\alpha:=\int_0^\infty |g^{**}_\eta|^pr^{\eta}\mathrm dr$, Lemma [Lemma 7](#lemma21){reference-type="ref" reference="lemma21"} and Corollary [Corollary 4](#cor21){reference-type="ref" reference="cor21"} imply $$\label{eqalpha} \alpha\leq \left(\dfrac{p}{p-1}\right)^p\int_0^\infty|g^*_\eta|^pr^{\eta}\mathrm dr=\left(\dfrac{p}{p-1}\right)^{p}\int_0^\infty|f|^pr^{p\theta-(p-1)\eta}\mathrm dr\leq\left(\dfrac{p}{p-1}\right)^{p},$$ where we used that $\gamma=(\alpha_2+(p-1)\eta)/p$ and $\|f\|_{L^p_{\alpha_2+(\theta-\gamma)p}}\leq1$. Let $0<\varepsilon_0<1$ be fixed, and define $R_2$ such that $$\label{eqpt3} \int_0^{ R_2}|g^{**}_\eta|^pr^{\eta}\mathrm dr=\alpha\varepsilon_0\mbox{ and }\int_{ R_2}^\infty|g^{**}_\eta|^pr^{\eta}\mathrm dr=\alpha(1-\varepsilon_0).$$ From [\[eq14\]](#eq14){reference-type="eqref" reference="eq14"}, we obtain $$\label{eqpt2} u^*_\eta(r_1)-u^*_\eta(r_2)\leq\dfrac1{\eta+1}\left(\int_{ r_1}^{ r_2}|g^{**}_\eta(r)|^pr^{\eta}\mathrm dr\right)^{\frac1p}\left(\log\dfrac{r_2}{r_1}\right)^{\frac{p-1}p},$$ for all $0<r_1\leq r_2\leq R_1$. Using [\[eqpt3\]](#eqpt3){reference-type="eqref" reference="eqpt3"} and $\gamma=(\alpha_2+\eta(p-1))/p$, [\[eqpt2\]](#eqpt2){reference-type="eqref" reference="eqpt2"} becomes $$\label{eqpt4} u^*_\eta(r_1)-u^*_\eta(r_2)\leq\dfrac{\varepsilon_0^{\frac1p}}{\gamma-1}\left(\log \dfrac{r_2}{r_1}\right)^{\frac{p-1}p}\quad\forall 0<r_1\leq r_2\leq R_2.$$ $$\label{eqpt5} u^*_\eta(r_1)-u^*_\eta(r_2)\leq\dfrac{(1-\varepsilon_0)^{\frac1p}}{\gamma-1}\left(\log \dfrac{r_2}{r_1}\right)^{\frac{p-1}p}\quad\forall R_2\leq r_1\leq r_2.$$ We split the proof into the cases $R_2\geq R_0$ and $R_2<R_0$. Firstly we consider the case $R_2\geq R_0$. By [\[eqpt4\]](#eqpt4){reference-type="eqref" reference="eqpt4"}, for all $r\in(0,R_0]$ we have $$u^*_\eta(r)\leq 1+\dfrac{\varepsilon_0^{\frac1p}}{\gamma-1}\left(\log \dfrac{R_0}{r}\right)^{\frac{p-1}p}.$$ Given $\varepsilon>0$, it is known that $$\label{eqineqepsilon} (1+x)^{\frac{p}{p-1}}\leq (1+\varepsilon)x^{\frac{p}{p-1}}+C_\varepsilon,\quad\forall x>0,$$ where $C_\varepsilon=(1-(1+\varepsilon)^{1-p})^{\frac{1}{1-p}}$. Thus, $$\label{eqpt6} |u^*_\eta(r)|^{\frac{p}{p-1}}\leq (1+\varepsilon)\dfrac{\varepsilon_0^{\frac{1}{p-1}}}{(\gamma-1)^{\frac{p}{p-1}}}\log\dfrac{R_0}r+C_\varepsilon.$$ Taking $\varepsilon>0$ small with $(1+\varepsilon)\varepsilon_0^{\frac1{p-1}}<1$ and using $\beta_{0,2}=(\eta+1)(\gamma-1)^{\frac{p}{p-1}}$ and [\[eqalpha\]](#eqalpha){reference-type="eqref" reference="eqalpha"}, we obtain $$\beta_{0,2}(1+\varepsilon)\dfrac{\varepsilon_0^{\frac{1}{p-1}}}{(\gamma-1)^{\frac{p}{p-1}}}<\eta+1.$$ Then, by [\[eqpt6\]](#eqpt6){reference-type="eqref" reference="eqpt6"} and $\exp_p(t)\leq e^t$ (see [@arXiv:2306.00194 Lemma 3.1]), $$\begin{aligned} \int_0^{R_0}\dfrac{\exp_p(\beta_{0,2}|u^*_\eta|^{\frac{p}{p-1}})}{(1+|u^*_\eta|)^{\frac{p}{p-1}}}r^\eta\mathrm dr&\leq e^{C_\varepsilon}\int_0^{R_0}\exp\left(\beta_{0,2}(1+\varepsilon)\dfrac{\varepsilon_0^{\frac{1}{p-1}}}{(\gamma-1)^{\frac{p}{p-1}}}\log\dfrac{R_0}r\right)r^\eta\mathrm dr\\ &\leq e^{C_\varepsilon}R_0^{\beta_{0,2}(1+\varepsilon)\frac{\varepsilon_0^{\frac{1}{p-1}}}{(\gamma-1)^{\frac{p}{p-1}}}}\int_0^{R_0}r^{\eta-\beta_{0,2}(1+\varepsilon)\frac{\varepsilon_0^{\frac{1}{p-1}}}{(\gamma-1)^{\frac{p}{p-1}}}}\mathrm dr\\ &=CR_0^{\eta+1}\leq C\|u^*_\eta\|_{L^p_\eta}^p.\end{aligned}$$ This concludes the case $R_2\geq R_0$. For the case $R_2<R_0$, we observe that: $$\int_0^{R_0}\dfrac{\exp_p(\beta_{0,2}|u^*_\eta|^{\frac{p}{p-1}})}{(1+|u^*_\eta|)^{\frac{p}{p-1}}}r^\eta\mathrm dr=\int_{R_2}^{R_0}\dfrac{\exp_p(\beta_{0,2}|u^*_\eta|^{\frac{p}{p-1}})}{(1+|u^*_\eta|)^{\frac{p}{p-1}}}r^\eta\mathrm dr+\int_0^{R_2}\dfrac{\exp_p(\beta_{0,2}|u^*_\eta|^{\frac{p}{p-1}})}{(1+|u^*_\eta|)^{\frac{p}{p-1}}}r^\eta\mathrm dr.$$ To estimate the integral over $(R_2,R_0)$, we use [\[eqpt5\]](#eqpt5){reference-type="eqref" reference="eqpt5"} to obtain $$u^*_\eta(r)\leq 1+\dfrac{(1-\varepsilon_0)^{\frac1p}}{\gamma-1}\left(\log\dfrac{R_0}r\right)^{\frac{p-1}p},\quad\forall r\in(R_2,R_0).$$ By setting $\varepsilon_1>0$ small such that $(1+\varepsilon_1)(1-\varepsilon_0)^{\frac{1}{p}}<1$, [\[eqineqepsilon\]](#eqineqepsilon){reference-type="eqref" reference="eqineqepsilon"} implies: $$|u^*_\eta(r)|^{\frac{p}{p-1}}\leq(1+\varepsilon_1)\dfrac{(1-\varepsilon_0)^{\frac{1}{p-1}}}{(\gamma-1)^{\frac{p}{p-1}}}\log\dfrac{R_0}r+C_{\varepsilon_1}.$$ Hence, using the expression $\beta_{0,2}=(\eta+1)(\gamma-1)^{\frac{p}{p-1}}$ and $\exp_p(t)\leq e^t$ (refer to [@arXiv:2306.00194 Lemma 3.1]), we can derive the following inequality: $$\begin{aligned} \int_{R_2}^{R_0}\dfrac{\exp_p(\beta_{0,2}|u^*_\eta|^{\frac{p}{p-1}})}{(1+|u^*_\eta|)^{\frac{p}{p-1}}}r^\eta\mathrm dr&\leq e^{C_{\varepsilon_1}}\int_{R_2}^{R_0}\exp\left(\beta_{0,2}(1+\varepsilon_1)\dfrac{(1-\varepsilon_0)^{\frac{1}{p-1}}}{(\gamma-1)^{\frac{p}{p-1}}}\log\dfrac{R_0}r\right)r^\eta\mathrm dr\\ &\leq e^{C_{\varepsilon_1}}R_0^{\beta_{0,2}(1+\varepsilon_1)\frac{(1-\varepsilon_0)^{\frac{1}{p-1}}}{(\gamma-1)^{\frac{p}{p-1}}}}\int_0^{R_0}r^{\eta-\beta_{0,2}(1+\varepsilon_1)\frac{(1-\varepsilon_0)^{\frac{1}{p-1}}}{(\gamma-1)^{\frac{p}{p-1}}}}\mathrm dr\\ &=CR_0^{\eta+1}\leq C\|u^*_\eta\|_{L^p_\eta}^p.\end{aligned}$$ Consequently, the integral over the interval $(0,R_2)$ remains to be considered. Notably, according to [\[eqineqepsilon\]](#eqineqepsilon){reference-type="eqref" reference="eqineqepsilon"}, we have $$|u^*_\eta(r)|^{\frac{p}{p-1}}\leq(1+\varepsilon_2)|u^*_\eta(r)-u^*_\eta(R_2)|^{\frac{p}{p-1}}+C_{\varepsilon_2}|u^*_\eta(R_2)|^{\frac{p}{p-1}},$$ for all $\varepsilon_2>0$ and $0<r<R_2$. Hence, we obtain $$\begin{aligned} \int_{0}^{R_2}\dfrac{\exp_p(\beta_{0,2}|u^*_\eta|^{\frac{p}{p-1}})}{(1+|u^*_\eta|)^{\frac{p}{p-1}}}r^\eta\mathrm dr&\leq \dfrac1{|u^*_\eta(R_2)|^{\frac{p}{p-1}}}\int_0^{R_2}\exp(\beta_{0,2}|u^*_\eta|^{\frac{p}{p-1}})r^\eta\mathrm dr\\ &\leq\dfrac{\exp(C_{\varepsilon_2}\beta_{0,2}|u^*_\eta(R_2)|^{\frac{p}{p-1}})}{|u^*_\eta(R_2)|^{\frac{p}{p-1}}}\\ &\quad\times\int_0^{R_2}\exp(\beta_{0,2}(1+\varepsilon_2)|u^*_\eta(r)-u^*_\eta(R_2)|^{\frac{p}{p-1}})r^\eta\mathrm dr.\end{aligned}$$ Using the expression [\[eq14\]](#eq14){reference-type="eqref" reference="eq14"}, we have $$\begin{aligned} \int_{0}^{R_2}&\dfrac{\exp_p(\beta_{0,2}|u^*_\eta|^{\frac{p}{p-1}})}{(1+|u^*_\eta|)^{\frac{p}{p-1}}}r^\eta\mathrm dr\leq\dfrac{\exp(C_{\varepsilon_2}\beta_{0,2}|u^*_\eta(R_2)|^{\frac{p}{p-1}})}{|u^*_\eta(R_2)|^{\frac{p}{p-1}}}\nonumber\\ &\quad\times\int_0^{R_2}\exp\left[\dfrac{\beta_{0,2}^{\frac{p-1}{p}}(1+\varepsilon_2)^{\frac{p-1}p}}{\eta+1}\int_{ r}^{ R_2}g^{**}_\eta(s)s^{\frac{\eta-p+1}p}\mathrm ds\right]^{\frac{p}{p-1}}r^\eta\mathrm dr\nonumber\\ &=R_2^{\eta+1}\dfrac{\exp(C_{\varepsilon_2}\beta_{0,2}|u^*_\eta(R_2)|^{\frac{p}{p-1}})}{(\eta+1)|u^*_\eta(R_2)|^{\frac{p}{p-1}}}\nonumber\\ &\quad\times\int_0^\infty\exp\left[\dfrac{\beta_{0,2}^{\frac{p-1}p}(1+\varepsilon_2)^{\frac{p-1}p}}{\eta+1}\int_{ R_2e^{-\frac{t}{\eta+1}}}^{ R_2}g^{**}_\eta(s)s^{\frac{\eta-p+1}p}\mathrm ds\right]^{\frac{p}{p-1}}e^{-t}\mathrm dt\label{eqpt7},\end{aligned}$$ where the change of variable $r=R_2e^{-\frac{t}{\eta+1}}$ has been applied. Let $\phi\colon\mathbb R\to \mathbb R$ be defined as $\phi(t)=0$ for $t\in(-\infty,0]$, and for $t\in(0,\infty)$, we define $$\phi(t)=\dfrac{\beta_{0,2}^{\frac{p-1}p}(1+\varepsilon_2)^{\frac{p-1}p}R_2^{\frac{\eta+1}{p}}}{(\eta+1)^2}g^{**}_\eta(R_2e^{-\frac{t}{\eta+1}})e^{-\frac{t}{p}}.$$ By choosing $\varepsilon_2=\varepsilon_0^{\frac1{1-p}}-1$, we can apply [\[eqalpha\]](#eqalpha){reference-type="eqref" reference="eqalpha"}, [\[eqpt3\]](#eqpt3){reference-type="eqref" reference="eqpt3"}, and $\gamma=(\alpha_2+(p-1)\eta)/p$ to obtain $$\begin{aligned} \int_{-\infty}^\infty|\phi(t)|^p\mathrm dt&=\dfrac{\beta_{0,2}^{p-1}(1+\varepsilon_2)^{p-1}R_2^{\eta+1}}{(\eta+1)^{2p}}\int_0^{\infty}|g^{**}_\eta( R_2e^{-\frac{t}{\eta+1}})|^pe^{-t}\mathrm dt\nonumber\\ &=\dfrac{\beta_{0,2}^{p-1}(1+\varepsilon_2)^{p-1}}{(\eta+1)^{2p-1}}\int_0^{ R_2}|g^{**}_\eta(r)|^pr^{\eta}\mathrm dr\nonumber\\ &\leq (1+\varepsilon_2)^{p-1}\varepsilon_0\dfrac{\beta_{0,2}^{p-1}}{(\eta+1)^{2p-1}}\left(\dfrac{p}{p-1}\right)^p=1.\label{eqpt8}\end{aligned}$$ Consider the following lemma due to D. R. Adams [@MR0960950 Lemma 1]. **Lemma 12**. *Let $a(s,t)$ be a nonnegative measurable function on $(-\infty,\infty)\times[0,\infty)$ such that (a.e.) $$a(s,t)\leq1,\mbox{ when }0<s<t,$$ $$\sup_{t>0}\left(\int_{-\infty}^0+\int_t^\infty a(s,t)^{\frac{p}{p-1}}\mathrm ds\right)^{\frac{p-1}p}=b<\infty.$$ Then there is a constant $c_0=c_0(p,b)$ such that if $\phi\geq0$, $$\int_{-\infty}^{\infty}\phi(s)^p\mathrm ds\leq1,$$ then $$\int_0^\infty e^{-F(t)}\mathrm dt\leq c_0,$$ where $$F(t)=t-\left(\int_{-\infty}^\infty a(s,t)\phi(s)\mathrm ds\right)^{\frac{p}{p-1}}.$$* From [\[eqpt8\]](#eqpt8){reference-type="eqref" reference="eqpt8"}, we can apply the above lemma to the function $a(s,t)=\chi_{(0,t)}(s)$ ($\chi$ denotes the characteristic function) to obtain $$\int_0^\infty\exp\left(\int_0^t\phi(s)\mathrm ds\right)^{\frac{p}{p-1}}e^{-t}\mathrm dt\leq c_0.$$ Using the change of variable $r= R_2e^{-\frac{s}{\eta+1}}$, we have $$\begin{aligned} c_0&\geq \int_0^\infty\exp\left(\int_0^t\phi(s)\mathrm ds\right)^{\frac{p}{p-1}}e^{-t}\mathrm dt\\ &=\int_0^\infty\exp\left(\dfrac{\beta_{0,2}^{\frac{p-1}p}(1+\varepsilon_2)^{\frac{p-1}p}( R_2)^{\frac{\eta+1}{p}}}{(\eta+1)^2}\int_{0}^{t}g^{**}_\eta( R_2e^{-\frac{s}{\eta+1}})e^{-\frac{s}{p}}\mathrm ds\right)^{\frac{p}{p-1}}e^{-t}\mathrm dt\\ &=\int_0^\infty\exp\left(\dfrac{\beta_{0,2}^{\frac{p-1}p}(1+\varepsilon_2)^{\frac{p-1}p}}{\eta+1}\int_{ R_2 e^{-\frac{t}{\eta+1}}}^{ R_2}g^{**}_\eta(r)r^{\frac{\eta-p+1}p}\mathrm dr\right)^{\frac{p}{p-1}}e^{-t}\mathrm dt.\end{aligned}$$ Here, $\varepsilon_2=\varepsilon_0^{\frac{1}{1-p}}-1$ implies $C_{\varepsilon_2}=(1-\varepsilon_0)^{\frac{1}{1-p}}$. Now, using [\[eqpt7\]](#eqpt7){reference-type="eqref" reference="eqpt7"}, we have $$\begin{aligned} \int_{0}^{R_2}\dfrac{\exp_p(\beta_{0,2}|u^*_\eta|^{\frac{p}{p-1}})}{(1+|u^*_\eta|)^{\frac{p}{p-1}}}r^\eta\mathrm dr&\leq c_0R_2^{\eta+1}\dfrac{\exp((1-\varepsilon_0)^{\frac{1}{1-p}}\beta_{0,2}|u^*_\eta(R_2)|^{\frac{p}{p-1}})}{(\eta+1)|u^*_\eta(R_2)|^{\frac{p}{p-1}}}\\ &=CR_2^{\eta+1}\dfrac{\exp((1-\varepsilon_0)^{\frac{1}{1-p}}\beta_{0,2}|u^*_\eta(R_2)|^{\frac{p}{p-1}})}{|u^*_\eta(R_2)|^{\frac{p}{p-1}}}.\end{aligned}$$ Since $\int_{R_2}^\infty|g^{**}_\eta|^pr^{\eta}\mathrm dr\leq(\frac{p}{p-1})^p(1-\varepsilon_0)$, we can apply Lemma [Lemma 11](#lemmakey){reference-type="ref" reference="lemmakey"} to $u(1-\varepsilon_0)^{-\frac1p}$ and obtain $$R_2^{\eta+1}\dfrac{\exp((1-\varepsilon_0)^{\frac{1}{1-p}}\beta_{0,2}|u^*_\eta(R_2)|^{\frac{p}{p-1}})}{|u^*_\eta(R_2)|^{\frac{p}{p-1}}}\leq C(1-\varepsilon_0)^{\frac{2-p}{p-1}}\int_{R_2}^\infty |u^*_\eta(r)|^pr^\eta\mathrm dr.$$ Therefore, we have $$\int_0^{R_2}\dfrac{\exp_p(\beta_{0,2}|u^*_\eta|^{\frac{p}{p-1}})}{(1+|u^*_\eta|)^{\frac{p}{p-1}}}r^\eta\mathrm dr\leq C\|u^*_\eta\|_{L^p_\eta}^p.$$ ◻ The proof of Theorem [Theorem 2](#theok1){reference-type="ref" reference="theok1"} follows a similar approach to the proof of Theorem [Theorem 3](#theo1){reference-type="ref" reference="theo1"}, but with a slight modification. Instead of relying on Proposition [Proposition 5](#prop22){reference-type="ref" reference="prop22"}, we derive a similar result as [\[eqpt2\]](#eqpt2){reference-type="eqref" reference="eqpt2"} through the following expression: $$\label{eqk1} u^*_\eta(r_1)-u^*_\eta(r_2)=-\int_{r_1}^{r_2}(u^*_\eta)'(r)\mathrm dr\leq\left(\int_{r_1}^{r_2}\left|(u^*_\eta)'\right|^pr^{p-1}\mathrm dr\right)^{\frac{1}{p}}\left(\log\dfrac{r_2}{r_1}\right)^{\frac{p-1}{p}},$$ for all $0<r_1\leq r_2$. Applying the Pólya-Szegö inequality for weighted Sobolev spaces, as given by [@MR4097244 Theorem 3.2], to $u\in X^{1,p}_\infty(\eta,\alpha_1)$, we obtain $$\int_0^ \infty|(u_\eta^*)'|^pr^{p-1}\mathrm dr\leq\int_0^\infty|u'|^pr^{p-1}\mathrm dr\leq1.$$ Furthermore, by employing the same argument as in the proof of Lemma [Lemma 11](#lemmakey){reference-type="ref" reference="lemmakey"}, but using [\[eqk1\]](#eqk1){reference-type="eqref" reference="eqk1"} instead of [\[eqpt2\]](#eqpt2){reference-type="eqref" reference="eqpt2"}, we can establish the following lemma: **Lemma 13**. *Assume $\alpha_1=p-1>1$, $\alpha_0\geq-1$, and $\eta>-1$. Let $u\in X^{1,p}_\infty(\alpha_0,\alpha_1)$ with compact support in $[0,\infty)$. If $u^*_\eta(R)\geq1$ and $$\int_{ R}^\infty|u^{*}_\eta|^pr^{p-1}\mathrm dr\leq 1,$$ then $$\dfrac{\exp(\beta_{0,1}|u^*_\eta(R)|^{\frac{p}{p-1}})}{|u^*_\eta(R)|^{\frac{p}{p-1}}}R^{\eta+1}\leq C\int_R^\infty |u^*_\eta(r)|^pr^\eta\mathrm dr,$$ where $C=C(p,\eta)>0$ and $u^*_\eta$ is the half $\mu_{\eta,\eta}$-symmetrization.* # Higher Order Derivative Case {#sectionk} In this section, we demonstrate Adams' inequality with exact growth for higher order derivatives (Theorem [Theorem 4](#theoexactk){reference-type="ref" reference="theoexactk"}) by utilizing the second order derivative case (Theorem [Theorem 3](#theo1){reference-type="ref" reference="theo1"}). Prior to proving the main theorem, we first establish three lemmas. The first lemma (Lemma [Lemma 14](#lemmahardycons){reference-type="ref" reference="lemmahardycons"}) provides a bound on the norm $\|u\|_{L^p_{\alpha-p}}$ in terms of the norm $\|u'\|_{L^p_\alpha}$, while the second lemma (Lemma [Lemma 15](#lemmahardyL){reference-type="ref" reference="lemmahardyL"}) estimates in terms of $\|L_{\theta,\gamma}u\|_{L^p_{\nu}}$. Finally, the third lemma (Lemma [Lemma 16](#lemmaLjnorm){reference-type="ref" reference="lemmaLjnorm"}) employs the estimate from the second lemma to establish a bound on $\|L_{\theta,\gamma}u\|_{L^p_\alpha}$ based on $\|L^j_{\theta,\gamma}u\|_{L^p_\nu}$. Towards the end of the section, we prove the Corollaries [Corollary 1](#cor1){reference-type="ref" reference="cor1"} and [Corollary 2](#cor2){reference-type="ref" reference="cor2"}. By employing these lemmas, we prove the main theorem, which establishes Adams' inequality with exact growth for higher order derivatives. We now proceed to present the three lemmas, followed by the proof of the main theorem. **Lemma 14**. *Let $u\in AC_{\mathrm{loc}}(0,R)$ with $\lim_{r\to R}u(r)=0$. If $\alpha-p+1>0$, then $$\left(\int_0^R|u|^pr^{\alpha-p}\mathrm dr\right)^{\frac1p}\leq\dfrac{p}{\alpha-p+1}\left(\int_0^R|u'|^pr^\alpha\mathrm dr\right)^{\frac1p}.$$* *Proof.* According to [@MR1069756 Theorem 6.2], it suffices to verify that $$\dfrac{p}{\alpha-p+1}=\dfrac{p}{(p-1)^{\frac{p-1}{p}}}\sup_{0<r<R}\|r^{\frac{\alpha-p}p}\|_{L^p(0,r)}\|r^{-\frac{\alpha}p}\|_{L^{\frac{p}{p-1}}(r,R)}.$$ The proof follows because $$\|r^{\frac{\alpha-p}p}\|_{L^p(0,r)}\|r^{-\frac{\alpha}p}\|_{L^{\frac{p}{p-1}}(r,R)}=\dfrac{(p-1)^{\frac{p-1}p}}{\alpha-p+1}\left(1-R^{-\frac{\alpha-p+1}{p-1}}r^{(\alpha-p+1)\frac{p}{p-1}}\right)^{\frac{p}{p-1}}.$$ ◻ **Lemma 15**. *Let $p>1$, $R\in(0,\infty)$, $\alpha>-1$, and the elliptic operator $L_{\theta,\gamma}u=-r^{-\theta}(r^\gamma u')'$ with $\gamma>1$ and $\alpha+1<p(\gamma-1)$. For any $u\in AC^1_{\mathrm{loc}}(0,R)$ such that $\lim_{r\to R}u(r)=\lim_{r\to0}r^\gamma u'(r)=0$, we have $$\left(\int_0^R|u|^pr^\alpha\mathrm dr\right)^{\frac1p}\leq C_{\gamma,\alpha,p}\left(\int_0^R|L_{\theta,\gamma}u|^pr^{p(\theta+2-\gamma)+\alpha}\mathrm dr\right)^{\frac1p},$$ where $$C_{\gamma,\alpha,p}=\dfrac{p^2}{(\alpha+1)[p(\gamma-1)-\alpha-1]}.$$* *Proof.* Let $w(t)=u(Rt^{-\frac{1}{\gamma-1}})$. Then $$\label{eqita3} \int_0^R|u|^pr^\alpha\mathrm dr=\dfrac{R^{\alpha+1}}{\gamma-1}\int_1^\infty|w(t)|^pt^{-\frac{\alpha+\gamma}{\gamma-1}}\mathrm dt$$ and $$\label{eqita4} \int_0^R|L_{\theta,\gamma}u|^pr^{p(\theta+2-\gamma)+\alpha}\mathrm dr=(\gamma-1)^{2p-1}R^{\alpha+1}\int_1^\infty|w''(t)|^pt^{\frac{2p\gamma-\gamma-2p-\alpha}{\gamma-1}}\mathrm dt.$$ Since $\lim_{r\to R}u(r)=\lim_{r\to0}r^\gamma u'(r)=0$, we have $$w(t)=\int_1^t\int_z^\infty-w''(s)\mathrm ds\mathrm dz.$$ Set $$a=\frac{p-1}{p^2}\left(2p-\frac{\alpha+1}{\gamma-1}\right).$$ From the conditions $\alpha>-1$, $\gamma>1$, $p>1$, and $\alpha+1<p(\gamma-1)$, we obtain that $$\label{eqita} 1<\dfrac{ap}{p-1}<2$$ and $$\label{eqita2} 2p-1-\dfrac{\alpha+\gamma}{\gamma-1}<ap<2p-\dfrac{\alpha+\gamma}{\gamma-1}.$$ Thus, by [\[eqita\]](#eqita){reference-type="eqref" reference="eqita"}, $$\begin{aligned} |w(t)|^p&=\left(\int_1^t\int_z^\infty-w''(s)\frac{s^a}{s^a}\mathrm ds\mathrm dz\right)^p\\ &\leq\left[\left(\int_1^t\int_z^\infty|w''(s)|^ps^{ap}\mathrm ds\mathrm dz\right)^{\frac1p}\left(\int_1^t\int_z^\infty s^{-\frac{ap}{p-1}}\mathrm ds\mathrm dz\right)^{\frac{p-1}p}\right]^p\\ &=\int_1^t\int_z^\infty|w''(s)|^ps^{ap}\mathrm ds\mathrm dz\left(\frac{p(\gamma-1)}{p(\gamma-1)-\alpha-1}\int_1^tz^{1-\frac{ap}{p-1}}\mathrm dz\right)^{p-1}\\ &\leq\left(\dfrac{p^2(\gamma-1)^2}{(\alpha+1)[p(\gamma-1)-\alpha-1]}\right)^{p-1}\int_1^t\int_z^\infty|w''(s)|^ps^{ap}\mathrm ds\mathrm dzt^{2p-2-ap},\end{aligned}$$ for all $t>0$. Using [\[eqita2\]](#eqita2){reference-type="eqref" reference="eqita2"}, we have $$\begin{aligned} \int_1^\infty&|w(t)|^pt^{-\frac{\alpha+\gamma}{\gamma-1}}\mathrm dt\\ &\leq\left(\dfrac{p^2(\gamma-1)^2}{(\alpha+1)[p(\gamma-1)-\alpha-1]}\right)^{p-1}\int_1^\infty\int_1^t\int_z^\infty|w''(s)|^ps^{ap}t^{2p-2-\frac{\alpha+\gamma}{\gamma-1}-ap}\mathrm ds\mathrm dz\mathrm dt\\ &=\left(\dfrac{p^2(\gamma-1)^2}{(\alpha+1)[p(\gamma-1)-\alpha-1]}\right)^{p-1}\int_1^\infty|w''(s)|^ps^{ap}\int_1^s\int_z^\infty t^{2p-2-\frac{\alpha+\gamma}{\gamma-1}-ap}\mathrm dt\mathrm dz\mathrm ds\\ &\leq\left(\dfrac{p^2(\gamma-1)^2}{(\alpha+1)[p(\gamma-1)-\alpha-1]}\right)^{p}\int_1^\infty|w''(s)|^ps^{2p-\frac{\alpha+\gamma}{\gamma-1}}\mathrm ds.\end{aligned}$$ Therefore, using [\[eqita3\]](#eqita3){reference-type="eqref" reference="eqita3"} and [\[eqita4\]](#eqita4){reference-type="eqref" reference="eqita4"}, we obtain $$\int_0^R|u|^pr^\alpha\mathrm dr\leq\left(\dfrac{p^2}{(\alpha+1)[p(\gamma-1)-\alpha-1]}\right)^{p} \int_0^R|L_{\theta,\gamma}u|^pr^{p(\theta+2-\gamma)+\alpha}\mathrm dr.$$ ◻ The following lemma was also obtained in [@arXiv:2302.02262 Lemma 4.8] for the particular case $\gamma=\theta$. **Lemma 16**. *Let $p,\gamma>1,\alpha,\theta\in\mathbb R$, and $j\geq2$ be an integer such that $$\label{eqLjnorm} \left\{\begin{array}{ll} -1<\alpha<p(\gamma-1)+(2-j)(\theta+2-\gamma)p-1,&\mbox{if }\theta+2\geq\gamma, \\ (2-j)(\theta+2-\gamma)p-1<\alpha<p(\gamma-1)-1,&\mbox{if }\theta+2<\gamma. \end{array}\right.$$ Suppose $u\in AC_{\mathrm{loc}}^{2j-1}(0,R)$ with $\lim_{r\to R}L_{\theta,\gamma}^iu(r)=\lim_{r\to0}r^\gamma(L_{\theta,\gamma}^iu)'(r)=0$ for all $i=1,\ldots,j-1$. Then $$\|L_{\theta,\gamma}u\|_{L^p_{\alpha}}\leq\left(\prod_{i=1}^{j-1}C_i\right)\|L^j_{\theta,\gamma}u\|_{L^p_{\alpha+(j-1)(\theta+2-\gamma)p}},$$ where, for each $i=1,\ldots,j-1$, $$C_i=\dfrac{p^2}{\left[\alpha+(i-1)(\theta+2-\gamma)p+1\right]\left[p(\gamma-1)-\alpha-1-(i-1)(\theta+2-\gamma)p\right]}.$$* *Proof.* Note that [\[eqLjnorm\]](#eqLjnorm){reference-type="eqref" reference="eqLjnorm"} is equivalent to $$0<\alpha+(i-1)(\theta+2-\gamma)p+1<p(\gamma-1),\quad\forall i=1,\ldots,j-1.$$ Therefore, we can apply Lemma [Lemma 15](#lemmahardyL){reference-type="ref" reference="lemmahardyL"} $j-1$ times to conclude the result. ◻ *Proof of Theorem [Theorem 4](#theoexactk){reference-type="ref" reference="theoexactk"}.* From Lemma [Lemma 2](#lemmacompsuppdense){reference-type="ref" reference="lemmacompsuppdense"} and Lemma [Lemma 3](#lemmajaosn){reference-type="ref" reference="lemmajaosn"} (using $\alpha_i\geq\alpha_k-(k-i)p$ for all $i=1,\ldots,k$), we can assume $\mathrm{supp}(u)\subset[0,\infty)$ is compact and $u\in C^\infty(0,\infty)$. Let $j$ be such that $k=2j$ or $k=2j+1$ ($j=\lfloor\frac{k}2\rfloor$). Since $\theta+2>\gamma$ and $\theta>j(\theta+2-\gamma)-1$, we can apply Lemma [Lemma 16](#lemmaLjnorm){reference-type="ref" reference="lemmaLjnorm"} to obtain $$\label{eqbessa} \|L_{\theta,\gamma} u\|_{L^p_{2p-1+(\theta-\gamma)p}}\leq\left(\prod_{i=1}^{j-1}C_i\right)\|L_{\theta,\gamma}^{j}u\|_{L^p_{2jp-1+j(\theta-\gamma)p}},$$ where $$C_i=\dfrac{1}{i(\theta+2-\gamma)[\gamma-1-i(\theta+2-\gamma)]},\quad\forall i=1,\ldots,j-1.$$ Now we slit the proof into two cases: $k=2j$ and $k=2j+1$. [Case $k=2j$:]{.ul} We define $v:=\prod_{i=1}^{j-1}C_i^{-1}u$. From [\[eqbessa\]](#eqbessa){reference-type="eqref" reference="eqbessa"} and $\|\nabla^k_L u\|_{L^p_\nu}\leq1$, we have $\|L_{\theta,\gamma}v\|_{L^p_{2p-1}}\leq1$. Applying Theorem [Theorem 3](#theo1){reference-type="ref" reference="theo1"} to $v$, we obtain $$\int_0^\infty\dfrac{\exp_p(\beta_{0,2}|v|^{\frac{p}{p-1}})}{(1+|v|)^{\frac{p}{p-1}}}r^\eta\mathrm dr\leq C\|v\|^p_{L^p_{\eta}}.$$ Thus, $$\int_0^\infty\dfrac{\exp_p(\beta_{0,2}\prod_{i=1}^{j-1}C_i^{-\frac{p}{p-1}}|u|^{\frac{p}{p-1}})}{(1+|u|)^{\frac{p}{p-1}}}r^\eta\mathrm dr\leq C\|u\|^p_{L^p_{\eta}}.$$ We are left with the task of proving $$\label{eqkg1} (\eta+1)\left[(\gamma-1)(\theta+2-\gamma)^{k-2}\dfrac{\Gamma\left(\frac{k}2\right)\Gamma\left(\frac{\gamma-1}{\theta+2-\gamma}\right)}{\Gamma\left(\frac{\gamma-1}{\theta+2-\gamma}-\frac{k-2}{2}\right)}\right]^{\frac{p}{p-1}}=\beta_{0,2}\prod_{i=1}^{j-1}C_i^{-\frac{p}{p-1}}.$$ Note that $$\begin{aligned} \prod_{i=1}^{j-1}C_i^{-1}&=(j-1)!(\theta+2-\gamma)^{j-1}\prod_{i=1}^{j-1}[\gamma-1-i(\theta+2-\gamma)]\nonumber\\ &=\Gamma(j)(\theta+2-\gamma)^{2j-2}\prod_{i=1}^{j-1}\left(\frac{\gamma-1}{\theta+2-\gamma}-i\right)\nonumber\\ &=(\theta+2-\gamma)^{2j-2}\dfrac{\Gamma(j)\Gamma\left(\frac{\gamma-1}{\theta+2-\gamma}\right)}{\Gamma\left(\frac{\gamma-1}{\theta+2-\gamma}-j+1\right)}.\label{eqkg2}\end{aligned}$$ Using the expression of $\beta_{0,2}$ together with $j=\frac{k}{2}$ and [\[eqkg2\]](#eqkg2){reference-type="eqref" reference="eqkg2"}, we conclude [\[eqkg1\]](#eqkg1){reference-type="eqref" reference="eqkg1"} as desired. [Case $k=2j+1$:]{.ul} We define $v:=j(\theta+2-\gamma)\prod_{i=1}^{j-1}C_i^{-1}u$. From [\[eqbessa\]](#eqbessa){reference-type="eqref" reference="eqbessa"}, Lemma [Lemma 14](#lemmahardycons){reference-type="ref" reference="lemmahardycons"}, and $\|\nabla^k_L u\|_{L^p_\nu}\leq1$, we have $$\|L_{\theta,\gamma}v\|_{L^p_{2p-1}}\leq j(\theta+2-\gamma)\|L^j_{\theta,\gamma}u\|_{L^p_{2jp-1+j(\theta-\gamma)p}}\leq \|\left(L^j_{\theta,\gamma}u\right)'\|_{L^p_{\alpha_k+\lfloor\frac{k}{2}\rfloor(\theta-\gamma)p}}\leq1.$$ Applying Theorem [Theorem 3](#theo1){reference-type="ref" reference="theo1"} to $v$, we obtain $$\int_0^\infty\dfrac{\exp_p(\beta_{0,2}|v|^{\frac{p}{p-1}})}{(1+|v|)^{\frac{p}{p-1}}}r^\eta\mathrm dr\leq C\|v\|^p_{L^p_{\eta}}.$$ Thus, $$\int_0^\infty\dfrac{\exp_p(\beta_{0,2}j^{\frac{p}{p-1}}(\theta+2-\gamma)^{\frac{p}{p-1}}\prod_{i=1}^{j-1}C_i^{-\frac{p}{p-1}}|u|^{\frac{p}{p-1}})}{(1+|u|)^{\frac{p}{p-1}}}r^\eta\mathrm dr\leq C\|u\|^p_{L^p_{\eta}}.$$ Using the expression of $\beta_{0,2}$, we are left with the task of proving $$\label{eqkg3} (\theta+2-\gamma)^{k-2}\dfrac{\Gamma\left(\frac{k+1}2\right)\Gamma\left(\frac{\gamma-1}{\theta+2-\gamma}\right)}{\Gamma\left(\frac{\gamma-1}{\theta+2-\gamma}-\frac{k-3}{2}\right)}=j(\theta+2-\gamma)\prod_{i=1}^{j-1}C_i^{-1}.$$ By [\[eqkg2\]](#eqkg2){reference-type="eqref" reference="eqkg2"} and $j=\frac{k-1}2$, we obtain $$\begin{aligned} j(\theta+2-\gamma)\prod_{i=1}^{j-1}C_i^{-1}&=(\theta+2-\gamma)^{2j-1}\dfrac{\Gamma(j+1)\Gamma\left(\frac{\gamma-1}{\theta+2-\gamma}\right)}{\Gamma\left(\frac{\gamma-1}{\theta+2-\gamma}-j+1\right)}\\ &=(\theta+2-\gamma)^{k-2}\dfrac{\Gamma\left(\frac{k+1}2\right)\Gamma\left(\frac{\gamma-1}{\theta+2-\gamma}\right)}{\Gamma\left(\frac{\gamma-1}{\theta+2-\gamma}-\frac{k-3}{2}\right)},\end{aligned}$$ which concludes [\[eqkg3\]](#eqkg3){reference-type="eqref" reference="eqkg3"} and the proof of Theorem [Theorem 4](#theoexactk){reference-type="ref" reference="theoexactk"}. ◻ *Proof of Corollary [Corollary 1](#cor1){reference-type="ref" reference="cor1"}.* The proof directly follows from the inequality: $$\label{erqgnal} \exp(\beta t^{\frac{p}{p-1}})\leq C_\beta\dfrac{\exp_p(\beta_{0,k}t^{\frac{p}{p-1}})}{(1+t)^{\frac{p}{p-1}}}\quad\forall \beta\in[0,\beta_{0,k}), t\geq0.$$ To prove the inequality [\[erqgnal\]](#erqgnal){reference-type="eqref" reference="erqgnal"}, we define $\phi(t)=(1+t)^{\frac{p}{p-1}}\exp(\beta t^{\frac{p}{p-1}})/\exp_p(\beta_{0,k}t^{\frac{p}{p-1}})$ for $t>0$ and use the fact that $\phi(t)\overset{t\to0}\longrightarrow\beta/\beta_{0,k}$ and $\phi(t)\overset{t\to\infty}{\longrightarrow}0$ (see [@arXiv:2306.00194 Lemma 3.1]). ◻ *Proof of Corollary [Corollary 2](#cor2){reference-type="ref" reference="cor2"}.* Let $u\in X^{k,p}_\infty\backslash\{0\}$ be such that $\|\nabla^k_Lu\|^p_{L^p_\nu}+\tau\|u\|^p_{L^p_\eta}\leq1$. Choose $\theta\in(0,1)$ with $\|u\|^p_{L^p_\eta}=\theta$ and $\|\nabla^k_Lu\|^p_{L^p_\nu}\leq1-\tau\theta$. In particular, $\theta\leq \frac{1}{\tau}$. [Case $\theta\geq\frac{p-1}{\tau p}$:]{.ul} Define $v=p^{\frac1p}u$. Then $\|\nabla^k_Lv\|^p_{L^p_\nu}=p\|\nabla^k_Lu\|^p_{L^p_\nu}\leq p(1-\tau \theta)\leq1$. By Corollary [Corollary 1](#cor1){reference-type="ref" reference="cor1"}, we have $$\int_0^\infty\exp_p\left(p^{\frac1{p-1}}\beta|u|^{\frac{p}{p-1}}\right)r^\eta\mathrm dr\leq C_\beta,\quad\forall \beta\in(0,\beta_{0,k}).$$ Since $0<\beta_{0,k}p^{-\frac{1}{p-1}}<\beta_{0,k}$, we can choose $\beta=\beta_{0,k}p^{-\frac{1}{p-1}}$ in the above inequality to conclude this case. [Case $\theta<\frac{p-1}{\tau p}$:]{.ul} Let $A:=\{r>0\colon|u(r)|\geq1\}$. Since $|u(r)|<1$ for $r\in(0,\infty)\backslash A$ and $\exp_p(t)\leq Ct^{p-1}$ for all $t\in[0,\beta_{0,k}]$, we have $$\int_{(0,\infty)\backslash A}\exp_p\left(\beta_{0,k}|u|^{\frac{p}{p-1}}\right)r^\eta\mathrm dr\leq C\beta_{0,k}^{p-1}\int_{(0,\infty)\backslash A}|u|^pr^\eta\mathrm dr\leq C\beta_{0,k}^{p-1}.$$ We now are left with the task to prove that $$\label{eqc2} \int_A\exp_p\left(\beta_{0,k}|u|^{\frac{p}{p-1}}\right)r^\eta\mathrm dr\leq C.$$ Note that $\left[\exp_p(t)\right]^q\leq\exp_p(qt)$ for all $t\geq0$ and $q\geq1$. This follows from the fact that $t\mapsto\left[\exp_p(t)\right]^q/\exp_p(qt)$ is a nonincreasing function that converges to 1 as $t$ goes to infinity (see [@arXiv:2306.00194 Lemma 3.1]). Using Hölder's inequality with $q=\frac{p-1}{p-1-\tau\theta}>1$, we obtain $$\begin{aligned} \int_A\exp_p\left(\beta_{0,k}|u|^{\frac{p}{p-1}}\right)r^\eta\mathrm dr&\leq\!\left(\!\int_A\dfrac{\exp_p\left(q\beta_{0,k}|u|^{\frac{p}{p-1}}\right)}{(1+|u|)^{\frac{p}{p-1}}}r^\eta\mathrm dr\!\right)^{\frac1q}\!\left(\!\int_A\!(1+|u|)^{\frac{p}{(p-1)(q-1)}}r^\eta\mathrm dr\!\right)^{\frac{q-1}q}\nonumber\\ &\leq2^{\frac{p}{(p-1)q}}\left(q\int_A\dfrac{\exp_p\left(\beta_{0,k}|v|^{\frac{p}{p-1}}\right)}{(1+|v|)^{\frac{p}{p-1}}}r^\eta\mathrm dr\right)^{\frac1q}\left\||u|^{\frac{p}{p-1}}\right\|^{\frac{1}q}_{L^{\frac{1}{q-1}}_\eta}\label{eqc4},\end{aligned}$$ where $v=q^{\frac{p-1}p}u$. From Corollary [Corollary 1](#cor1){reference-type="ref" reference="cor1"}, we have $$\label{eqc3} \left\||u|^{\frac{p}{p-1}}\right\|_{L^s_\eta}\leq Cs\|u\|^{\frac{p}s}_{L^p_\eta},\quad\forall u\in X^{k,p}_\infty\mbox{ with }\|\nabla^k_Lu\|_{L^p_\nu}\leq1,$$ for all $s$ in the form of $p-1+n$ with $n\in\mathbb N\cup\{0\}$. Using interpolation of weighted Lebesgue spaces, we can extend this inequality for all $s\geq p-1$. Then, by [\[eqc3\]](#eqc3){reference-type="eqref" reference="eqc3"}, $$\label{eqc5} \left\||u|^{\frac{p}{p-1}}\right\|^{\frac1q}_{L^{\frac{1}{q-1}}_\eta}\leq C\left(\dfrac{1}{q-1}\right)^{\frac1q}\|u\|_{L^p_\eta}^{\frac{p(q-1)}q}.$$ From the definition of $q$, $\theta\leq\frac{1}\tau$, $p\geq2$, and $(1-x)^s\leq1-sx$ for all $s,x\in[0,1]$, we have $$\|\nabla^k_Lv\|^p_{L^p_\eta}\leq q^{p-1}(1-\tau\theta)=\left[\dfrac{(1-\tau\theta)^{\frac{1}{p-1}}}{1-\frac{\tau\theta}{p-1}}\right]^{p-1}\leq1.$$ Thus we can apply Theorem [Theorem 4](#theoexactk){reference-type="ref" reference="theoexactk"} to $v\in X^{k,p}_\infty$ to obtain $$\int_0^\infty\dfrac{\exp_p\left(q\beta_{0,k}|v|^{\frac{p}{p-1}}\right)}{(1+|v|)^{\frac{p}{p-1}}}r^\eta\mathrm dr\leq Cq^{p-1}\|u\|^p_{L^p_\eta}.$$ Using this along with [\[eqc4\]](#eqc4){reference-type="eqref" reference="eqc4"} and [\[eqc5\]](#eqc5){reference-type="eqref" reference="eqc5"}, we obtain $$\begin{aligned} \int_A\exp_p\left(\beta_{0,k}|u|^{\frac{p}{p-1}}\right)r^\eta\mathrm dr&\leq Cq^{\frac{p}q}\|u\|^{\frac{p}{q}}_{L^p_\eta}\left(\dfrac{1}{q-1}\right)^{\frac1q}\|u\|_{L^p_\eta}^{\frac{p(q-1)}q}=Cq^{\frac{p-1}q}\left(\frac{q}{q-1}\right)^{\frac1q}\theta^p\\ &=C\tau^{\frac{\tau\theta}{p-1}-1}(p-1)^{\frac{p-1-\tau\theta}{p-1}}\dfrac{\theta^{p-1+\frac{\tau\theta}{p-1}}}{\left(1-\frac{\tau\theta}{p-1}\right)^{p-1-\tau\theta}}.\end{aligned}$$ This concludes [\[eqc2\]](#eqc2){reference-type="eqref" reference="eqc2"} and the proof because the right term is bounded on $0<\theta<\frac{p-1}{\tau p}$. ◻ # Proof of the Supercritical Case {#sectionsuperc} In this final section, we focus on the behavior of the supremum in two scenarios: when $\beta>\beta_{0,k}$ and when $\beta=\beta_{0,k}$ while the power $q$ in the denominator satisfies $q<\frac{p}{p-1}$. More specifically, we provide the proof of Theorem [Theorem 5](#theosuperc){reference-type="ref" reference="theosuperc"}, establishing the sharpness of the constant $\beta_{0,k}$ and the exponent $\frac{p}{p-1}$. *Proof of Theorem [Theorem 5](#theosuperc){reference-type="ref" reference="theosuperc"}.* The construction of the sequence follows a similar argument to that used by D. R. Adams in [@MR0960950]. Let $\phi\in C^\infty[0,1]$ be such that $$\left\{\begin{array}{ll} \phi(0)=\phi'(0)=\cdots=\phi^{(k+1)}(0)=0,\ \phi'\geq0,\\ \phi(1)=\phi'(1)=1,\ \phi''(1)=\cdots\phi^{(k-1)}(1)=0. \end{array}\right.$$ Consider $0<\varepsilon<1/2$ and define $$H(t)=\left\{\begin{array}{llll} \varepsilon\phi\left(\dfrac{t}{\varepsilon}\right),&\mbox{if }0<t\leq\varepsilon \\ t,&\mbox{if }\varepsilon<t\leq1-\varepsilon,\\ 1-\varepsilon\phi\left(\dfrac{1-t}{\varepsilon}\right),&\mbox{if }1-\varepsilon<t\leq1,\\ 1,&\mbox{if }t>1. \end{array}\right.$$ Let $n\in\mathbb N$, $R\in(0,\infty)$ fixed, and define the sequence $(\psi_{n,\varepsilon})_n$ as follows $$\psi_{n,\varepsilon}(r)=H\left(\dfrac{\log\frac{R}{r}}{\log n}\right),\quad 0<r<R.$$ It is not hard to see that $\psi_{n,\varepsilon}\in X^{k,p}_{0,R}\subset X^{k,p}_{\infty}$. By induction on $m\in\mathbb N$ we have $$\label{eqsc0} L^m_{\theta,\gamma}\psi_{n,\varepsilon}(r)=r^{m(\gamma-2-\theta)}\sum_{i=1}^{2m}\dfrac{c_{im}}{(\log n)^i}H^{(i)}\left(\dfrac{\log\frac{R}{r}}{\log n}\right),$$ where $$\left\{\begin{array}{l} c_{11}=\gamma-1,\ c_{21}=-1;\\ c_{1m+1}=-m(\gamma-2-\theta)(\gamma-1+m(\gamma-2-\theta))c_{1m}; \\ c_{2m+1}=-m(\gamma-2-\theta)(\gamma-1+m(\gamma-2-\theta))c_{2m}+(\gamma-1+2m(\gamma-2-\theta))c_{1m};\\ c_{2m+1m+1}=(\gamma-1+2m(\gamma-2-\theta))c_{2mm}-c_{2m-1m}\\ c_{2m+2m+1}=-c_{2mm}. \end{array}\right.$$ and, for each $i=3,\ldots,2m$, $$c_{im+1}=-m(\gamma-2-\theta)(\gamma-1+m(\gamma-2-\theta))c_{im}+(\gamma-1+2m(\gamma-2-\theta))c_{i-1m}-c_{i-2m}.$$ From $c_{11}=\gamma-1$ and $c_{1m+1}=-m(\gamma-2-\theta)(\gamma-1+m(\gamma-2-\theta))c_{1m}$ we obtain $$\begin{aligned} c_{1m}&=(\gamma-1)\prod_{i=1}^{m-1}\left[-i(\gamma-2-\theta)(\gamma-1+i(\gamma-2-\theta))\right]\nonumber\\ &=(\gamma-1)(\theta+2-\gamma)^{m-1}(m-1)!\prod_{i=1}^{m-1}(\theta+2-\gamma)\left(\dfrac{\gamma-1}{\theta+2-\gamma}-i\right)\nonumber\\ &=(\gamma-1)(\theta+2-\gamma)^{2m-2}\Gamma(m)\dfrac{\Gamma\left(\frac{\gamma-1}{\theta+2-\gamma}\right)}{\Gamma\left(\frac{\gamma-1}{\theta+2-\gamma}-m+1\right)},\quad\forall m\in\mathbb N.\label{eqsc1}\end{aligned}$$ We make the following claim: $$\label{eq44} \|\psi_{n,\varepsilon}\|_{L^p_{\eta}}^p=O\left((\log n)^{-p}\right).$$ To prove this, we begin by introducing a change of variables $t=\log\frac{R}{r}/\log n$, which gives us: $$\begin{aligned} \|\psi_{n,\varepsilon}&\|_{L^p_{\eta}}^p=\int_0^R\left|H\left(\dfrac{\log\frac{R}{r}}{\log n}\right)\right|^pr^{\eta}\mathrm dr\nonumber\\ &=R^{\eta+1}\log n\Bigg[\int_0^\varepsilon\left|\varepsilon\phi\left(\dfrac{t}\varepsilon\right)\right|^pn^{-(\eta+1)t}\mathrm dt+\int_\varepsilon^{1-\varepsilon}t^pn^{-(\eta+1)t}\mathrm dt\nonumber\\ &\quad+\int_{1-\varepsilon}^1\left|1-\varepsilon\phi\left(\dfrac{1-t}{\varepsilon}\right)\right|^pn^{-(\eta+1)t}\mathrm dt+\int_1^\infty n^{-(\eta+1)t}\mathrm dt\Bigg]\nonumber\\ &\leq R^{\eta+1}\log n\left[\int_0^\varepsilon\left|\varepsilon\phi\left(\dfrac{t}\varepsilon\right)\right|^pn^{-(\eta+1)t}\mathrm dt+\int_\varepsilon^\infty n^{-(\eta+1)t}\mathrm dt\right]\nonumber\\ &=R^{\eta+1}\log n\int_0^\varepsilon\left|\varepsilon\phi\left(\dfrac{t}\varepsilon\right)\right|^pn^{-(\eta+1)t}\mathrm dt+O\left((\log n)^{-p}\right).\label{eq451}\end{aligned}$$ Let $\nu\in (\frac{1}{k+1},1)$ fixed. Since $\phi\left((\log n)^{-\nu}\right)\leq (\log n)^{-\nu (k+1)}$ for large $n$, we can estimate the integral (for large $n$) as follows $$\begin{aligned} \int_0^\varepsilon\left|\varepsilon\phi\left(\dfrac{t}\varepsilon\right)\right|^pn^{-(\eta+1)t}\mathrm dt&\leq\varepsilon^p\int_0^{\varepsilon(\log n)^{-\nu}}\left|\phi\left((\log n)^{-\nu}\right)\right|^pn^{-(\eta+1)t}\mathrm dt\nonumber\\ &\qquad+\varepsilon^p\int_{\varepsilon(\log n)^{-\nu}}^\varepsilon n^{-(\eta+1)t}\mathrm dt\nonumber\\ &\leq \varepsilon^p(\log n)^{-\nu p(k+1)}\dfrac1{(\eta+1)\log n}\nonumber\\ &\qquad+\varepsilon^p\dfrac{1}{(\eta+1)e^{\varepsilon(\eta+1)(\log n)^{1-\nu}}\log n}\nonumber\\ &=O\left((\log n)^{-p-1}\right).\label{eq46}\end{aligned}$$ Using [\[eq451\]](#eq451){reference-type="eqref" reference="eq451"} along with [\[eq46\]](#eq46){reference-type="eqref" reference="eq46"}, we conclude [\[eq44\]](#eq44){reference-type="eqref" reference="eq44"}. [Case $k=2j$:]{.ul} By equation [\[eqsc0\]](#eqsc0){reference-type="eqref" reference="eqsc0"}, we have $$L^j_{\theta,\gamma}\psi_{n,\varepsilon}(r)=r^{j(\gamma-2-\theta)}\frac{c_{1j}}{\log n}H'\left(\dfrac{\log\frac{R}{r}}{\log n}\right)+r^{j(\gamma-2-\theta)}O((\log n)^{-2}).$$ Then, $$\begin{aligned} \|\nabla_L^k\psi_{n,\varepsilon}\|^p_{L^p_{\alpha_k+j(\theta-\gamma)p}}&=\left|\dfrac{c_{1j}}{\log n}\right|^p\int_{\frac{R}n}^R\left|H'\left(\dfrac{\log\frac{R}{r}}{\log n}\right)+O\left((\log n)^{-1}\right)\right|^pr^{\alpha_k+j(\theta-\gamma)p+j(\gamma-2-\theta)p}\mathrm dr\nonumber\\ &\leq\left|\dfrac{c_{1j}}{\log n}\right|^p\Bigg[\int_{\frac{R}{n}}^{Rn^{\varepsilon-1}}\left|\|\phi'\|_\infty+O\left((\log n)^{-1}\right)\right|^pr^{-1}\mathrm dr\nonumber\\ &\quad+\int_{Rn^{\varepsilon-1}}^{Rn^{-\varepsilon}}r^{-1}\mathrm dr+\int_{Rn^{-\varepsilon}}^R\left|\|\phi'\|_{\infty}+O\left((\log n)^{-1}\right)\right|^pr^{-1}\Bigg]\nonumber\\ &\leq|c_{1j}|^p(\log n)^{1-p}\left[1+2\varepsilon\left|\|\phi'\|_\infty+O\left((\log n)^{-1}\right)\right|^p\right]\nonumber\\ &=|c_{1j}|^p(\log n)^{1-p}A_{\varepsilon,n},\label{eqsc2}\end{aligned}$$ where $A_{\varepsilon,n}:=1+2\varepsilon\left|\|\phi'\|_\infty+O\left((\log n)^{-1}\right)\right|^p$. Set $$u_{n,\varepsilon}(r)=\dfrac{\psi_{n,\varepsilon}(r)}{|c_{1j}|(\log n)^{\frac{1-p}{p}}A_{\varepsilon,n}^{\frac{1}{p}}}.$$ Using [\[eqsc2\]](#eqsc2){reference-type="eqref" reference="eqsc2"} and [\[eq44\]](#eq44){reference-type="eqref" reference="eq44"}, we have $\|\nabla^k_Lu_{n,\varepsilon}\|_{L^p_{\alpha_k+j(\theta-\gamma)p}}\leq1$ and $$\label{gja} \|u_{n,\varepsilon}\|^p_{L^p_\eta}=O((\log n)^{-1}).$$ Given $q\geq0$ and $\beta\geq\beta_{0,k}$, equation [\[eqsc2\]](#eqsc2){reference-type="eqref" reference="eqsc2"} guarantees $$\begin{aligned} \int_0^\infty&\dfrac{\exp_p(\beta|u_{n,\varepsilon}|^{\frac{p}{p-1}})}{(1+|u_{n,\varepsilon}|)^{q}}r^\eta\mathrm dr\geq\int_0^{\frac{R}{n}}\dfrac{\exp_p(\beta|u_{n,\varepsilon}|^{\frac{p}{p-1}})}{(1+|u_{n,\varepsilon}|)^{q}}r^\eta\mathrm dr\nonumber\\ &\geq C\exp\left(\beta|c_{1j}|^{-\frac{p}{p-1}}A_{\varepsilon,n}^{\frac{1}{1-p}}\log n\right)\left(\log n\right)^{-q\frac{p-1}p}\frac{R^{\eta+1}}{n^{\eta+1}}\nonumber\\ &=C\exp\left[(\eta+1)\log n\left(\frac{\beta|c_{1j}|^{-\frac{p}{p-1}}A_{\varepsilon,n}^{\frac{1}{1-p}}}{\eta+1}-1\right)\right]\left(\log n\right)^{-q\frac{p-1}p}.\label{eq444}\end{aligned}$$ From [\[eq444\]](#eq444){reference-type="eqref" reference="eq444"}, we can conclude [\[eqsuc2\]](#eqsuc2){reference-type="eqref" reference="eqsuc2"} because if assuming $\beta>(\eta+1)|c_{1j}|^{\frac{p}{p-1}}=\beta_{0,k}$, we can choose a small $\varepsilon>0$ such that $\beta|c_{1j}|^{-\frac{p}{p-1}}A_{\varepsilon,n}^{\frac{1}{1-p}}>\eta+1$. Now, let's assume $\beta=\beta_{0,k}$ and $q<p/(p-1)$ to prove [\[eqsuc1\]](#eqsuc1){reference-type="eqref" reference="eqsuc1"}. By using [\[gja\]](#gja){reference-type="eqref" reference="gja"} and [\[eq444\]](#eq444){reference-type="eqref" reference="eq444"}, we have $$\begin{aligned} \dfrac{1}{\|u_{n,\varepsilon}\|_{L^p_{\eta}}^p}\int_0^\infty\dfrac{\exp_p(\beta_{0,k}|u_{n,\varepsilon}|^{\frac{p}{p-1}})}{(1+|u_{n,\varepsilon}|)^{q}}r^\eta\mathrm dr&\geq C(\log n)^{1-q\frac{p-1}p}\overset{n\to\infty}\longrightarrow\infty.\end{aligned}$$ [Case $k=2j+1$:]{.ul} This case is analogous to the previous case; however, by referring to equation [\[eqsc0\]](#eqsc0){reference-type="eqref" reference="eqsc0"}, we obtain $$(L^j_{\theta,\gamma}\psi_{n,\varepsilon})'(r)=r^{j(\gamma-2-\theta)-1}\frac{j(\gamma-2-\theta)c_{1j}}{\log n}H'\left(\dfrac{\log\frac{R}{r}}{\log n}\right)+r^{j(\gamma-2-\theta)-1}O((\log n)^{-2}),$$ where we are denoting $j(\gamma-2-\theta)c_{1j}=1$ when $j=0$ to include the first derivative case in this theorem. Then, $$\begin{aligned} \|\nabla_L^k\psi_{n,\varepsilon}\|^p_{L^p_{\alpha_k+j(\theta-\gamma)p}}&=\left|\dfrac{j(\gamma-2-\theta)c_{1j}}{\log n}\right|^p\int_{\frac{R}n}^R\left|H'\left(\dfrac{\log\frac{R}{r}}{\log n}\right)+O\left((\log n)^{-1}\right)\right|^pr^{\alpha_k-2jp-p}\mathrm dr\nonumber\\ &\leq\left|\dfrac{j(\gamma-2-\theta)c_{1j}}{\log n}\right|^p\Bigg[\int_{\frac{R}{n}}^{Rn^{\varepsilon-1}}\left|\|\phi'\|_\infty+O\left((\log n)^{-1}\right)\right|^pr^{-1}\mathrm dr\nonumber\\ &\quad+\int_{Rn^{\varepsilon-1}}^{Rn^{-\varepsilon}}r^{-1}\mathrm dr+\int_{Rn^{-\varepsilon}}^R\left|\|\phi'\|_{\infty}+O\left((\log n)^{-1}\right)\right|^pr^{-1}\Bigg]\nonumber\\ &\leq|j(\gamma-2-\theta)c_{1j}|^p(\log n)^{1-p}\left[1+2\varepsilon\left|\|\phi'\|_\infty+O\left((\log n)^{-1}\right)\right|^p\right]\nonumber\\ &=|j(\gamma-2-\theta)c_{1j}|^p(\log n)^{1-p}A_{\varepsilon,n},\label{eqsc22}\end{aligned}$$ where $A_{\varepsilon,n}:=1+2\varepsilon\left|\|\phi'\|_\infty+O\left((\log n)^{-1}\right)\right|^p$. Set $$u_{n,\varepsilon}(r)=\dfrac{\psi_{n,\varepsilon}(r)}{|j(\gamma-2-\theta)c_{1j}|(\log n)^{\frac{1-p}{p}}A_{\varepsilon,n}^{\frac{1}{p}}}.$$ Using [\[eqsc22\]](#eqsc22){reference-type="eqref" reference="eqsc22"} and [\[eq44\]](#eq44){reference-type="eqref" reference="eq44"}, we have $\|\nabla^k_Lu_{n,\varepsilon}\|_{L^p_{\alpha_k+j(\theta-\gamma)p}}\leq1$ and $$\label{gja2} \|u_{n,\varepsilon}\|^p_{L^p_\eta}=O((\log n)^{-1}).$$ Given $q\geq0$ and $\beta\geq\beta_{0,k}$, [\[eqsc22\]](#eqsc22){reference-type="eqref" reference="eqsc22"} guarantees $$\begin{aligned} \int_0^\infty&\dfrac{\exp_p(\beta|u_{n,\varepsilon}|^{\frac{p}{p-1}})}{(1+|u_{n,\varepsilon}|)^{q}}r^\eta\mathrm dr\geq\int_0^{\frac{R}{n}}\dfrac{\exp_p(\beta|u_{n,\varepsilon}|^{\frac{p}{p-1}})}{(1+|u_{n,\varepsilon}|)^{q}}r^\eta\mathrm dr\nonumber\\ &\geq C\exp\left(\beta|j(\gamma-2-\theta)c_{1j}|^{-\frac{p}{p-1}}A_{\varepsilon,n}^{\frac{1}{1-p}}\log n\right)\left(\log n\right)^{-q\frac{p-1}p}\frac{R^{\eta+1}}{n^{\eta+1}}\nonumber\\ &=C\exp\left[(\eta+1)\log n\left(\frac{\beta|j(\gamma-2-\theta)c_{1j}|^{-\frac{p}{p-1}}A_{\varepsilon,n}^{\frac{1}{1-p}}}{\eta+1}-1\right)\right]\left(\log n\right)^{-q\frac{p-1}p}.\label{eq4442}\end{aligned}$$ From [\[eq4442\]](#eq4442){reference-type="eqref" reference="eq4442"}, we conclude [\[eqsuc2\]](#eqsuc2){reference-type="eqref" reference="eqsuc2"} because assuming $\beta>(\eta+1)|j(\gamma-2-\theta)c_{1j}|^{\frac{p}{p-1}}=\beta_{0,k}$, we can take $\varepsilon>0$ small with $\beta|j(\gamma-2-\theta)c_{1j}|^{-\frac{p}{p-1}}A_{\varepsilon,n}^{\frac{1}{1-p}}>\eta+1$. To establish [\[eqsuc1\]](#eqsuc1){reference-type="eqref" reference="eqsuc1"}, we will assume $\beta=\beta_{0,k}$ and $q<p/(p-1)$. Utilizing [\[gja2\]](#gja2){reference-type="eqref" reference="gja2"} and [\[eq4442\]](#eq4442){reference-type="eqref" reference="eq4442"}, we obtain $$\begin{aligned} \dfrac{1}{\|u_{n,\varepsilon}\|_{L^p_{\eta}}^p}\int_0^\infty\dfrac{\exp_p(\beta_{0,k}|u_{n,\varepsilon}|^{\frac{p}{p-1}})}{(1+|u_{n,\varepsilon}|)^{q}}r^\eta\mathrm dr&\geq C(\log n)^{1-q\frac{p-1}p}\overset{n\to\infty}\longrightarrow\infty,\end{aligned}$$ which completes the proof of [\[eqsuc1\]](#eqsuc1){reference-type="eqref" reference="eqsuc1"} and therefore the proof of Theorem [Theorem 5](#theosuperc){reference-type="ref" reference="theosuperc"}. ◻ # Application on an ODE {#secapp} In this section, we work on the following problem $$\label{eqproblem} \left\{\begin{array}{l} \Delta^2_{\theta}u=f(r,u)r^{\eta-\theta}\mbox{ in }(0,\infty),\\ u'(0)=(\Delta_\theta u)'(0)=0, \end{array}\right.$$ where $\eta>-1$, $\theta=\gamma=(\eta+3)/2$, $\Delta_\theta:=L_{\theta,\theta}$, and $f\colon[0,\infty)\times\mathbb R\to\mathbb R$ continuous satisfying the growth condition $$\label{hipf} |f(r,t)|\leq C\left(e^{\beta t^2}-1\right),\quad\mbox{for some }\beta>0.$$ Specifically, we prove the existence of a weak solution for the problem [\[eqprobleml\]](#eqprobleml){reference-type="eqref" reference="eqprobleml"} and develop the regularity theory for the more general problem [\[eqproblem\]](#eqproblem){reference-type="eqref" reference="eqproblem"}. Throughout this section, we consider the weighted Sobolev space $X^{2,2}_\infty(\alpha_0,\alpha_1,\alpha_2)$, where $\alpha_0\geq-1$, $\alpha_1\geq1$, and $\alpha_2=3$. Based on the following identity: $$\int_0^\infty \Delta_{\theta}uvr^\theta\mathrm dr=\int_0^\infty u'v'r^{\theta}\mathrm dr=\int_0^\infty u\Delta_{\theta}vr^\theta\mathrm dr,$$ for all $v\in X^{2,2}_\infty$ with compact support on $[0,\infty)$, we define $u\in X^{2,2}_\infty(\alpha_0,\alpha_1,\alpha_2)$ to be a weak solution of [\[eqproblem\]](#eqproblem){reference-type="eqref" reference="eqproblem"} if it satisfies $$\int_0^\infty \Delta_{\theta}u\Delta_{\theta}vr^{\theta}\mathrm dr=\int_0^\infty f(r,u)vr^\eta\mathrm dr,\quad\forall v\in X^{2,2}_\infty.$$ By $\theta=\gamma=(\eta+3)/2$ and Lemma [Lemma 3](#lemmajaosn){reference-type="ref" reference="lemmajaosn"}, we can demonstrate that the left term is well-defined. On the other hand, the right term is well-defined by applying the inequalities $[\exp_2(t)]^p\leq\exp_2(pt)$, along with [\[hipf\]](#hipf){reference-type="eqref" reference="hipf"} and the following lemma: **Lemma 17**. *Consider $X^{k,p}_\infty(\alpha_0,\ldots,\alpha_k)$ with $p>1$ and $\alpha_i\geq\alpha_k-(k-i)p$ for all $i=0,\ldots,k-1$. Let $\theta\in[-1,\alpha_0]$. If $\alpha_k-kp+1=0$, then, for each $u\in X^{k,p}_\infty$, $$\int_0^\infty\exp_p\left(\mu|u|^{\frac{p}{p-1}}\right)r^\theta\mathrm dr<\infty,\quad\forall \mu\geq0.$$* *Proof.* Using Fubini's theorem, we obtain that $$\begin{aligned} \int_0^\infty\exp_p\left(\mu|u|^{\frac{p}{p-1}}\right)r^\theta\mathrm dr&=\sum_{j=0}^\infty\dfrac{\mu^{p-1+j}}{\Gamma(p-1+j)}\int_0^\infty|u|^{\frac{p}{p-1}(p-1+j)}r^\theta\mathrm dr\nonumber\\ &\leq\sum_{j=0}^\infty\dfrac{\left(\mu C_j^{\frac{p}{p-1}}\|u\|_{X^{k,p}_\infty}^{\frac{p}{p-1}}\right)^{p-1+j}}{\Gamma(p-1+j)}\label{eqasfkjna},\end{aligned}$$ where $C_j$ is the constant given by Theorem [Theorem 8](#theoimersaoinfinito){reference-type="ref" reference="theoimersaoinfinito"} such that $$\left(\int_0^\infty|u|^{\frac{p}{p-1}(p-1+j)}r^\theta\mathrm dr\right)^{\frac{p-1}{p(p-1+j)}}\leq C_j\|u\|_{X^{k,p}_\infty},\quad\forall j\in\mathbb N\cup\{0\}.$$ Finally, we conclude our lemma by applying the d'Alembert's ratio test on [\[eqasfkjna\]](#eqasfkjna){reference-type="eqref" reference="eqasfkjna"} together with the expression of $C_j$. ◻ Our approach involves using a minimization argument to solve this problem. Therefore, we need to find a minimizer under a constraint for the associate functional $\Phi\colon X^{2,2}_\infty\to\mathbb R$, which is given by $$\Phi(u)=\dfrac12\int_0^\infty\left|\Delta_{\theta}u\right|^2r^{\theta}\mathrm dr-\int_0^\infty F(r,u)r^\eta\mathrm dr,$$ where $F(r,s)=\int_0^sf(r,t)\mathrm dt$. However, before proving the existence of a weak solution, we need the subsequent compactness theorem. **Theorem 9**. *Suppose the assumptions of Theorem [Theorem 4](#theoexactk){reference-type="ref" reference="theoexactk"}, $\alpha_0>-1$, and $F\colon[0,\infty)\times[0,\infty)\to[0,\infty)$ is a function such that $F(X)$ is bounded for $X\subset[0,\infty)\times[0,\infty)$ bounded. Furthermore, assume that $$\label{h1} \lim_{t\to0}F(r,t)t^{-p}=0,\mbox{ uniformly on }r,$$ and $$\label{h2} \lim_{t\to\infty}F(r,t)t^{\frac{p}{p-1}}\exp\left(-\beta_{0,k}t^{\frac{p}{p-1}}\right)=0,\mbox{ uniformly on }r,$$ where $\beta_{0,k}$ is given in Theorem [Theorem 4](#theoexactk){reference-type="ref" reference="theoexactk"}. For any sequence $(u_n)$ weakly converging to some $u_0$ in $X^{k,p}_\infty$ satisfying $\|\nabla^k_Lu_n\|_{L^p_{\alpha_k+\lfloor\frac{k}2\rfloor(\theta-\gamma)p}}\leq1$, we have $$\int_0^\infty F(r,|u_n|)r^\eta\mathrm dr\overset{n\to\infty}\longrightarrow\int_0^\infty F(r,|u|)r^\eta\mathrm dr.$$* *Proof.* From a radial lemma [@arXiv:2306.00194 Lemma 2.2] and $\alpha_0>-1$, we obtain $u_n(r)\to0$ as $r\to\infty$ uniformly on $n$. This, together with [\[h1\]](#h1){reference-type="eqref" reference="h1"}, imply that for any $\varepsilon>0$, there exists $R>0$ such that $$\label{lu33} \int_R^\infty F(r,|u_n|)r^\eta\mathrm dr\leq \varepsilon\int_0^\infty|u_n|^pr^\eta\mathrm dr\leq C\varepsilon\mbox{ and }\int_R^\infty F(r,|u_0|)r^\eta\mathrm dr\leq\varepsilon.$$ On the other hand, for any $\varepsilon>0$, using [\[h2\]](#h2){reference-type="eqref" reference="h2"} and [@arXiv:2306.00194 Lemma 3.1], we obtain $A>0$ such that $$F(r,t)\leq \varepsilon\dfrac{\exp_p\left(\beta_{0,k}t^{\frac{p}{p-1}}\right)}{(1+t)^{\frac{p}{p-1}}},\quad\forall t>A,r\in[0,\infty).$$ Then, by Theorem [Theorem 4](#theoexactk){reference-type="ref" reference="theoexactk"}, $$\label{lu34} \int_{|u_n|>A}F(r,|u_n|)r^\eta\mathrm dr\leq C\varepsilon\int_0^\infty|u_n|^pr^\eta\mathrm dr\leq C\varepsilon\mbox{ and }\int_{|u_n|>A}F(r,|u_0|)r^\eta\mathrm dr\leq C\varepsilon.$$ Combining [\[lu33\]](#lu33){reference-type="eqref" reference="lu33"} with [\[lu34\]](#lu34){reference-type="eqref" reference="lu34"}, we get $$\left|\int_0^\infty F(r,|u_n|)r^\eta\mathrm dr-\int_0^\infty F(r,|u_0|)r^\eta\mathrm dr\right|\leq C\varepsilon+\left|\int_{|u_n|\leq A,r\leq R}\left[F(r,|u_n|)-F(r,|u_0|)\right]r^\eta\mathrm dr\right|,$$ for all $n\in\mathbb N$. Finally, applying the Lebesgue Dominated Convergence Theorem, we conclude $$\int_0^\infty F(r,|u_n|)r^\eta\mathrm dr\overset{n\to\infty}\longrightarrow\int_0^\infty F(r,|u_0|)r^\eta\mathrm dr$$ ◻ We have obtained the weak solution for [\[eqprobleml\]](#eqprobleml){reference-type="eqref" reference="eqprobleml"}, which serves as a specific instance of the case [\[eqproblem\]](#eqproblem){reference-type="eqref" reference="eqproblem"}. This outcome is presented in Proposition [Proposition 6](#prop81){reference-type="ref" reference="prop81"}. Additionally, we establish the regularity theory for the general problem [\[eqproblem\]](#eqproblem){reference-type="eqref" reference="eqproblem"}, achieving regularity throughout the interval $(0,\infty)$ as demonstrated by Proposition [Proposition 7](#propcs){reference-type="ref" reference="propcs"}, and completing the regularity at the origin as described by Proposition [Proposition 8](#propclsol1){reference-type="ref" reference="propclsol1"}. **Proposition 6**. *Assume that $t\mapsto f(r,t)$ is an odd function with $f(r,t)\geq0$, and $F$ satisfies the conditions of Theorem [Theorem 9](#theocomp){reference-type="ref" reference="theocomp"}. Set $$m_\infty:=\sup_{\underset{\|\Delta_\theta u\|_{L^2_{\theta}}=1}{u\in X^{2,2}_\infty}}\int_0^\infty F(r,u)r^\eta\mathrm dr.$$ Then, $m_\infty$ is attained by a function $u_0\in X^{2,2}_\infty$ with $\|\Delta_{\theta}u_0\|_{L^2_{\theta}}=1$. Moreover, $u_0$ is a weak solution of $$\left\{\begin{array}{l} \Delta^2_{\theta}u=\lambda^{-1}f(r,u)r^{\eta-\theta}\mbox{ in }(0,\infty),\\ u(0)=u'(0)=0, \end{array}\right.$$ where $\lambda=\int_0^Rf(r,u_0)u_0r^\eta\mathrm dr$.* *Proof.* Let $(u_n)$ be a maximizing sequence for $m_\infty$ with $\|\Delta_\theta u_n\|_{L^2_{\theta}}=1$. From Theorem [Theorem 1](#propequivnormkgrad){reference-type="ref" reference="propequivnormkgrad"}, we have that $(u_n)$ is bounded on $X^{2,2}_\infty$. Using Theorem [Theorem 9](#theocomp){reference-type="ref" reference="theocomp"}, we obtain $u_0\in X^{2,2}_\infty$ with $\|\Delta_\theta u_0\|_{L^2_{\theta}}\leq1$ such that $$\label{eq816} m_\infty=\int_0^\infty F(r,|u_0|)r^\eta\mathrm dr.$$ Since $f(r,t)=-f(r,-t)\geq0$ for all $t\geq0$, we obtain $F(r,u)\geq0$ and $F(r,\tau u_0)\geq F(r,u_0)$ for all $\tau\geq1$. We can assume that $F(r,u)$ is not always zero. Then, $u_0\in X_{\infty}^{2,2}\backslash\{0\}$ because $u_0$ is a maximum of $m_\infty$. We can assume $\|\Delta_\theta u_0\|_{L^2_{\theta}}=1$. Otherwise, defining $\widetilde u_0=u_0/\|\Delta_\theta u_0\|_{L^2_{\theta}}$, we have $\|\Delta_\theta \widetilde u_0\|_{L^2_{\theta}}=1$ and $$\int_0^\infty F(r,\widetilde u_0)r^\eta\mathrm dr\geq\int_0^\infty F(r,u_0)r^\eta\mathrm dr=\sup_{\underset{\|\Delta_\theta u\|_{L^2_{\theta}}=1}{u\in X^{2,2}_\infty}}\int_0^\infty F(r,u)r^\eta\mathrm dr.$$ Therefore, we obtain $u_0\in X^{2,2}_{\infty}$ that attains the supremum with $\|\Delta_\theta u_0\|_{L^2_{\theta}}=1$. Moreover, applying Lagrange Multipliers in [\[eq816\]](#eq816){reference-type="eqref" reference="eq816"}, we conclude $$\int_0^\infty \Delta_\theta u_0\Delta_\theta vr^{\theta}\mathrm dr=\lambda^{-1}\int_0^Rf(r,u_0)vr^\eta\mathrm dr\quad\forall v\in X_{\infty}^{2,2},$$ where $\lambda=\int_0^Rf(r,u_0)u_0r^\eta\mathrm dr$. ◻ **Proposition 7**. *Suppose $u_0$ is a weak solution of [\[eqproblem\]](#eqproblem){reference-type="eqref" reference="eqproblem"}. Then $u_0\in C^{4}(0,\infty)$, $\Delta_\theta u_0\in C^{2}(0,\infty)$, and $\Delta^2_{\theta}u_0=f(r,u_0)r^{\eta-\theta}$ $\forall r>0$. Moreover, $u_0(r)\overset{r\to\infty}\longrightarrow0$ if $\alpha_0\geq\alpha_1-2\geq-1$, and $\Delta_\theta u_0(r)\overset{r\to\infty}\longrightarrow0$ if $\alpha_0\in(-1,2\theta-3)$ and $\alpha_1\in[1,2\theta-1)$.* *Proof.* We claim that $\Delta_\theta u_0$ has a weak derivative given by $$\label{wdlu} (\Delta_\theta u_0)'(r)=-r^{-\theta}\int_0^rf(s,u_0(s))s^\eta\mathrm ds.$$ Note that the right term is well-defined by Lemma [Lemma 17](#proptobeproved){reference-type="ref" reference="proptobeproved"} and Theorem [Theorem 3](#theo1){reference-type="ref" reference="theo1"}. Using [\[wdlu\]](#wdlu){reference-type="eqref" reference="wdlu"} and that the right term of [\[wdlu\]](#wdlu){reference-type="eqref" reference="wdlu"} is $C^1(0,\infty)$, we conclude our proposition except by $u_0(r)\overset{r\to\infty}\longrightarrow0$ and $\Delta_\theta u_0(r)\overset{r\to\infty}\longrightarrow0$. Firstly, let us prove [\[wdlu\]](#wdlu){reference-type="eqref" reference="wdlu"}. Let $\varphi\in C^\infty_0(0,\infty)$. Writing $v(r)=-\int_r^\infty\varphi s^{-\theta}\mathrm ds$, we have $\varphi=r^\theta v'$ and $v\in X_\infty^{2,2}$. Using that $u_0$ is a weak solution and Fubini's Theorem, we get $$\begin{aligned} \int_0^\infty \Delta_\theta u_0\varphi'\mathrm dr&=-\int_0^\infty \Delta_\theta u_0\Delta_\theta vr^\theta\mathrm dr\\ &=\int_0^\infty\int_r^\infty f(r,u_0(r))\varphi(s)s^{-\theta}r^{\eta}\mathrm ds\mathrm dr\\ &=\int_0^\infty s^{-\theta}\int_0^sf(r,u_0(r))r^\eta\mathrm dr\varphi(s)\mathrm ds.\end{aligned}$$ This concludes [\[wdlu\]](#wdlu){reference-type="eqref" reference="wdlu"}. The fact that $u_0(r)\overset{r\to\infty}\longrightarrow0$ follows by applying [@arXiv:2306.00194 Lemma 2.2] with $\alpha_0\geq\alpha_1-2\geq-1$. Finally, let us check that $\Delta_\theta u_0(r)\overset{r\to\infty}\longrightarrow 0$. Fix $v\colon[0,\infty)\to\mathbb R$ smooth such that $v\equiv0$ in $[0,1]$ and $v(r)=r^{1-\theta}/(\theta-1)$ in $[2,\infty)$. Note that $v\in X^{2,2}_\infty$ by $-1<\alpha_0<2\theta-3$ and $-1\leq\alpha_1<2\theta-1$. Using that $u_0$ is a weak solution of [\[eqproblem\]](#eqproblem){reference-type="eqref" reference="eqproblem"}, we have $$\begin{aligned} \int_0^\infty f(r,u_0)vr^\eta\mathrm dr&=\int_0^\infty \Delta_\theta u_0\Delta_\theta vr^\theta\mathrm dr=-\int_0^\infty \Delta_\theta u_0(r^\theta v')'\mathrm dr\\ &=\lim_{R\to\infty}\Delta_\theta u_0(R)+\int_0^\infty r^\theta(\Delta_\theta u_0)'v'\mathrm dr\\ &=\lim_{R\to\infty}\Delta_\theta u_0(R)-\int_0^\infty\left(r^\theta(\Delta_\theta u_0)'\right)'v\mathrm dr\\ &=\lim_{R\to\infty}\Delta_\theta u_0(R)+\int_0^\infty \Delta_\theta ^2u_0vr^\theta\mathrm dr.\end{aligned}$$ Since $\Delta_\theta^2u_0=r^{\eta-\theta}f(r,u_0)$, we conclude that $\lim_{R\to\infty}\Delta_\theta u_0(R)=0$ and the proof of the proposition. ◻ **Proposition 8**. *Suppose $u_0$ is a weak solution of [\[eqproblem\]](#eqproblem){reference-type="eqref" reference="eqproblem"} with $\eta>1$. Then $u_0\in C^4(0,\infty)\cap C^3([0,\infty))$ is a classical solution of [\[eqproblem\]](#eqproblem){reference-type="eqref" reference="eqproblem"}. Moreover, $\Delta_\theta u_0\in C^2(0,\infty)\cap C^1([0,\infty))$, and the following initial conditions hold: $u'_0(0)=(\Delta_\theta u_0)'(0)=0$. Additionally, we find that $u''_0(0)=-\Delta_\theta u_0(0)/(\alpha+1)$ and $u'''_0(0)=0$.* *Proof.* We claim that $$\label{cl1} \Delta_\theta^2u_0=r^{\eta-\theta}f(r,u_0)\in L^p_{3p-1-\theta p},\quad\forall p>1.$$ Indeed, through the application of $[\exp_2(t)]^p\leq\exp_2(pt)$, we deduce $$\int_0^\infty|r^{\eta-\theta}f(r,u_0)|^pr^{3p-1-\theta p}\mathrm dr\leq C\int_0^\infty \exp_2\left(\beta p u_0^2\right)r^{-1}\mathrm dr.$$ Hence, Lemma [Lemma 17](#proptobeproved){reference-type="ref" reference="proptobeproved"} guarantees the validity of [\[cl1\]](#cl1){reference-type="eqref" reference="cl1"}. Using Lemma [Lemma 5](#lemmafjs){reference-type="ref" reference="lemmafjs"} in [\[cl1\]](#cl1){reference-type="eqref" reference="cl1"}, we obtain $\Delta_{\theta}u_0\in X^{2,p}_\infty(p-1-\theta p,2p-1-\theta p,3p-1-\theta p)$. Similarly, from Lemma [Lemma 5](#lemmafjs){reference-type="ref" reference="lemmafjs"}, we deduce that $u_0\in X^{2,p}_\infty(-p-1-\theta p,-1-\theta p,p-1-\theta p)$. As a result, the Morrey case of Theorem [Theorem 7](#theo32){reference-type="ref" reference="theo32"} guarantees $u_0\in C^1([0,\infty))$. Proposition [Proposition 7](#propcs){reference-type="ref" reference="propcs"} implies that $u_0\in C^4(0,\infty)\cap C^1([0,\infty))$ with $\Delta_\theta u_0\in C^2(0,\infty)$ and $\Delta_\theta^2u_0=r^{\eta-\theta}f(r,u_0)$ for all $r\in(0,\infty)$. We proceed to verify the initial condition $u_0'(0)=(\Delta_\theta u_0)'(0)=0$. Applying L'Hopital's rule in [\[wdlu\]](#wdlu){reference-type="eqref" reference="wdlu"} with $\eta>1$, we have $$\label{ganojdfna} \lim_{r\to0}(\Delta_\theta u_0)'(r)=-\lim_{r\to0}\dfrac{f(r,u_0(r))r^\eta}{\theta r^{\theta-1}}=0.$$ Using $u_0'(r)=-r^{-\theta}\int_0^r\Delta_\theta u_0(s)s^\theta\mathrm ds$ and L'Hopital's rule twice, we obtain $$\lim_{r\to0}u'_0(r)=-\theta^{-1}\lim_{r\to0}\Delta_\theta u_0(r)r=\theta^{-1}\lim_{r\to0}(\Delta_\theta u_0)'(r)r^2=0.$$ By [\[wdlu\]](#wdlu){reference-type="eqref" reference="wdlu"} and [\[ganojdfna\]](#ganojdfna){reference-type="eqref" reference="ganojdfna"}, we conclude that $\Delta_\theta u_0\in C^1([0,\infty))$. Now we are left to show that $u_0\in C^3([0,\infty))$ with $u_0''(0)=-\Delta_\theta u_0(0)/(\theta+1)$ and $u_0'''(0)=0$. We notice that $$u_0''(r)=-\theta\dfrac{u_0'(r)}r-\Delta_\theta u_0(r)=\theta r^{-\theta-1}\int_0^r\Delta_\theta u_0(s)s^\theta\mathrm ds-\Delta_\theta u_0(r)$$ and hence $\lim_{r\to0}u''_0(r)=-\Delta_\theta u_0(0)/(\theta+1)$. Furthermore, $$\begin{aligned} u_0'''(r)&=\theta r^{-1}\Delta_\theta u_0(r)-\theta(\theta+1)r^{-\theta-2}\int_0^r\Delta_\theta u_0(s)s^\theta\mathrm ds-(\Delta_\theta u_0)'(r)\\ &=\dfrac{\theta(\theta+1)}{r^{\theta+2}}\int_0^r\left(\Delta_\theta u_0(r)-\Delta_\theta u_0(s)\right)s^\theta\mathrm ds-(\Delta_\theta u_0)'(r).\end{aligned}$$ Given $\varepsilon>0$, $(\Delta_\theta u_0)'(0)=0$ implies that there exists $\delta>0$ such that $|(\Delta_\theta u_0)'(r)|\leq\varepsilon$ and $|\Delta_\theta u_0(r)-\Delta_\theta u_0(s)|\leq\varepsilon(r-s)$ for all $0<s\leq r<\delta$. 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arxiv_math

--- abstract: | In this manuscript, motivated and inspired by results of Best proximity point of generalized $F$-proximal non-self contractions, we introduce the concept of generalized $\theta-\phi-$proximal contraction and prove new best proximity results for these contractions in the setting of a metric space. Our results generalize and extend many recent results appearing in the literature. An example is being given to demonstrate the usefulness of our results. address: - $^{1}$Department of Mathematics, Faculty of Sciences Dhar El Mahraz, University Sidi Mohamed Ben Abdellah, B. P. 1796 Fes Atlas, Morocco - $^{2}$Department of Mathematics, Faculty of Sciences Ben M'Sik, Hassan II University, B. P. 7955 Casablanca, Morocco author: - Mohamed Rossafi$^{1*}$ and Abdelkarim Kari$^{2}$ title: Best proximity point of generalized $\theta-\phi-$proximal non-self contractions --- # Introduction It is well known that the Banach contraction theorem is the first outstanding result in the field of the fixed point theory that ensure the existence of unique fixed point in complete metric spaces. Due to its importance, various mathematics steadied many interesting extensions and generalizations [@KARI; @KARO; @RK; @ZH]. One of the famous generalizations of the Banach contraction principle [@S] for existence of fixed point for self-mapping on metric space is the theorem by Zheng et al. [@ZH] and the contraction introduced by Jleli and Samet in [@JK]. Best proximity point theorem analyses the condition under which the optimisation problem, namely $\inf_{x \in A } d(x,Tx)$, has a solution. The point $x$ is called the best proximity of $T: A\rightarrow B$, if $d(x, Tx)=d(A, B)$, where $\lbrace d(A, B)=\inf d(x,y): x \in A, y \in B \rbrace$. Note that the best proximity point reduces to a fixed point if $T$ is a self-mapping. Various best proximity point results were established on such spaces [@HA; @HAY; @RK]. Sankar Raj [@RJ] and Zhang et al. [@ZHN] defined the notion of $P-$property and weak $P-$property respectively. Beg et al. [@BG] defined the concept of generalized $F$-proximal non-self contractions and obtained some best proximity point theorems for self-mappings. In this paper, inspired by the idea of generalized $F$-proximal non-self contractions, introduced by Beg et al. [@BG] in metric spaces, we prove a new existence of best proximity point for generalized $\theta-\phi-$proximal contraction defined on a closed subset of a complete metric space. Our theorems extend, generalize and improve many existing results. # Preliminaries Let $(A, B)$ be a pair of non empty subsets of a metric space $(X,d)$. We adopt the following notations: $d(A,B)=\lbrace\inf d\left(a,b\right):a\in A, b\in B\rbrace$; $A_0=\lbrace$ $a\in A$ there exists $b\in A$ such that $d\left(a,b\right)= d\left(A,B\right)\rbrace$; $B_0=\lbrace$ $b\in B$ there exists $a\in A$ such that $d\left(a,b\right)= d\left(A,B\right)\rbrace$. **Definition 1**. [@KI] Let $T: A\rightarrow B$ be a mapping. An element $x^{\ast}$ is said to be a best proximity point of $T$ if $$d\left(x^{\ast },Tx^{\ast }\right)=d\left(A,B\right).$$ **Definition 2**. [@RJ] Let $(A,B)$ be a pair of non empty subsets of a metric space $(X,d)$ such that $A_0$ is non empty. Then the pair $(A,B)$ is to have $P$-property if and only if $$\left\lbrace \begin{aligned} d\left(x_1,y_1\right)&=d\left(A,B\right)\\ d\left(x_2,y_2\right)&=d\left(A,B\right) \end{aligned} \right.\Rightarrow d(x_1,x_2)=d(y_1,y_2)$$ where $x_1, x_2 \in A_{0}$ and $y_1, y_2\in B_{0}$. **Definition 3**. [@BS] A set $B$ is called approximately compact with respect to $A$ if every sequence $\lbrace x_{n} \rbrace$ of $B$ with $d(y, x_{n}) \rightarrow d(y,B)$ for some $y \in A$ has a convergent subsequence. **Definition 4**. [@JK] Let $\Theta$ be the family of all functions $\theta$ : $\left] 0,+\infty \right[$ $\rightarrow \left] 1 ,+\infty \right[$ such that - $\theta$ is strictly increasing; - For each sequence $x_n\in \left] 0,+\infty \right[$; $$\lim_{n\rightarrow 0}x_{n}=0,\,\,\,\text{ if and only if}\,\,\,\lim_{n\rightarrow \infty }\theta\left( x_{n}\right) =1;$$ - $\theta$ is continuous. **Definition 5**. [@ZH] Let $\Phi$ be the family of all functions $\phi$: $\left[ 1,+\infty \right[$ $\rightarrow \left[ 1,+\infty \right[$, such that - $\phi$ is increasing; - For each $t\in \left] 1,+\infty \right[$, $lim_{n\rightarrow \infty }\phi^{n}( t) =1$; - $\phi$ is continuous. **Lemma 6**. *[@ZH] [\[LZ\]]{#LZ label="LZ"} If $\phi$ $\in \Phi$ Then $\phi( 1)$=1, and $\phi( t) < t$.* **Definition 7**. [@ZH]. Let $(X,d)$ be a metric space and $T:X\rightarrow X$ be a mapping. $T$ is said to be a $\theta-\phi-$contraction if there exist $\theta \in \Theta$ and $\phi \in \Phi$ such that for any $x,y\in X,$ $$d\left( Tx,Ty\right) >0\Rightarrow \theta \left[ d\left( Tx,Ty\right) \right] \leq \phi \left [\theta \left(d\left( x,y\right) \right )\right],$$ # Main result In this section, inspired by the notion of $F$-proximal contraction of the first kind and second kind, we introduce new generalized $\theta-\phi$-proximal first kind and second kind on complete metric space. **Definition 8**. The mapping $T: A\rightarrow B$ is said to be a generalized $\theta-\phi$-proximal contraction of first kind if there exist $\theta \in \Theta$, $\phi \in\Phi$ and $a,b,c,h\geq 0$ with $a+b++2ch$, $c\neq 1$ such that $$\left\lbrace \begin{aligned} d\left(u_1,Tv_1\right)&=d\left(A,B\right)\\ d\left(u_2,Tv_2\right)&=d\left(A,B\right) \end{aligned} \right.\Rightarrow \theta(d(u_1,u_2))\\ \leq \phi\left[ \theta\left[ a d\left(v_1,v_2\right)+bd\left(u_1,v_1\right)+cd\left(u_2,v_2\right)+h\left( d\left(v_1,u_2\right)+d\left(v_2,u_1\right)\right)\right] \right]$$ for all $u_1, u_2,v_1, v_2\in A$ and $u_1\neq v_1$. **Definition 9**. The mapping $T:A\rightarrow B$ is said to be a generalized $\theta-\phi$-proximal contraction of second kind if there exist $\theta \in \Theta$, $\phi \in\Phi$ and $a,b,c,h\geq 0$ with $a+b++2ch$, $c\neq 1$ such that $$\left\lbrace \begin{aligned} d\left(u_1,Tv_1\right)&=d\left(A,B\right)\\ d\left(u_2,Tv_2\right)&=d\left(A,B\right) \end{aligned} \right.\Rightarrow \theta(d(Tu_1,Tu_2))\\ $$ $$\leq \phi\left[ \theta\left[ a d\left(Tv_1,Tv_2\right)+bd\left(Tu_1,Tv_1\right)+cd\left(Tu_2,Tv_2\right)+h\left( d\left(Tv_1,Tu_2\right)+d\left(Tv_2,Tu_1\right)\right)\right] \right]$$ for all $u_1, u_2,v_1, v_2\in A$ and $Tu_1\neq Tv_1$. **Theorem 10**. *Let $(X,d)$ be a complete metric space and $(A,B)$ be a pair of non-void closed subsets of $(X,d)$. If $B$ is approximately compact with respect to $A$ and $T: A\rightarrow B$ satisfy the following conditions :* - *$T\left( A_0\right) \in B_0$ and the pair $\left( A,B\right)$ satisfies the weak $P$-property;* - *$T$ is a generalized $\theta-\phi$-proximal contraction of first kind.* *Then there exists a unique $u\in A$ such that $d(u, Tu)=d(A,B)$. In addition, for any fixed element $u_{0 }\in A_{0}$, sequence $\lbrace u_n\rbrace$ defined by $$d( u_{n+1 },T u_{n })= d(A,B),$$ converges to the proximity point.* *Proof.* Choose an element $u_{0}\in A_{0}$. As, $T\left( A_0\right) \in B_0$, therefore there is an element $u_{1} \in A_{0}$ satisfying $$d(u_1,T u_0) = d(A,B).$$ Since $T(A_0) \in B_0$, there exists $u_2\in A_0$ such that $$d(u_2,Tu_1) =d(A,B).$$ Again, since $T(A_0) \in B_0$, there exists $u_3\in A_0$ such that $$d(u_3,Tu_2) =d(A,B).$$ Continuing this process, by induction, we construct a sequence $x_n\in A_0$ such that $$\label{q} d\left(u_{n+1},Tu_{n}\right)= d(A,B), \forall n\in \mathbb{N}.$$ Since $(A,B)$ satisfies the $P$ property, we conclude that $$\label{cc} d(u_n,u_{n+1})= d(Tu_{n},Tu_{n+1}), \forall n\in \mathbb{N}.$$ If $u_{n_{0}}= u_{n_{0}+1}$ for some $n_{0}\in\mathbb{N}$, from ([\[q\]](#q){reference-type="ref" reference="q"}) one obtains $$d\left(u_{n_{0}},Tu_{n_{0}}\right)=d\left(u_{n_{0}+1},Tu_{n_{0}}\right)= d(A,B)$$ that is, $u_{n_{0}}\in BPP$. Thus, we suppose that $d(u_n,x_{n+1}) >0$ for all $n\in\mathbb{N}.$\ We shall prove that the sequence $u_n$ is a Cauchy sequence. Let us first prove that $$\lim_{n\rightarrow \infty }d\left( u_{n},u_{n+1}\right) =0.$$ As $T$ is generalized $(\theta,\phi)$-proximal contraction of the first kind, we have that $$\begin{aligned} \theta \left(d\left(u_{n} ,u_{n+1}\right)\right)&\leq \phi\left[ \theta\left[ a d\left(u_{n-1} ,u_{n} \right)+bd\left(u_{n-1} ,u_{n} \right)+cd\left(u_{n} ,u_{n+1} \right)+h\left( d\left(u_{n-1} ,u_{n+1} \right)+d\left(u_{n} ,u_{n} \right)\right)\right] \right]\\ &=\phi\left[ \theta\left[ a d\left(u_{n-1} ,u_{n} \right)+bd\left(u_{n-1} ,u_{n} \right)+cd\left(u_{n} ,u_{n+1} \right)+h\left( d\left(u_{n-1} ,u_{n+1} \right)\right)\right] \right]\\ & \leq \phi\left[ \theta\left[ a d\left(u_{n-1} ,u_{n} \right)+bd\left(x_{n-1} ,x_{n} \right)+cd\left(u_{n} ,u_{n+1} \right)+h\left( d\left(u_{n-1} ,u_{n} \right)+d\left(u_{n} ,u_{n+1} \right)\right)\right] \right]\\ &= \phi\left[ \theta\left[ (a+b+h) d\left(u_{n-1} ,u_{n} \right)+(c+h)d\left(u_{n} ,u_{n+1} \right)\right] \right]\end{aligned}$$ Since $\theta$ is strictly increasing and by Lemma $\ref{LZ}$, we deduce $$d\left(x_{n} ,x_{n+1}\right)<(a+b+h) d\left(x_{n-1} ,x_{n} \right)+(c+h)d\left(x_{n} ,x_{n+1} \right).$$ Thus $$d\left(u_{n} ,u_{n+1}\right)<\frac{a+b+h}{1-c-h}( d\left(u_{n-1} ,x_{n} \right)).$$ If $b+b+c+2h=1$, we have $0< 1-c-h$ and so $$d\left(u_{n} ,u_{n+1}\right)\leq \frac{a+b+h}{1-c-h}( d\left(u_{n-1} ,u_{n} \right))=d\left(u_{n-1} ,u_{n} \right), \forall n \in\mathbb{N};$$ Consequently, $$\theta\left( d\left(u_{n} ,u_{n+1}\right)\right) \leq \phi\left[ \theta \left( d\left(u_{n-1} ,u_{n} \right)\right)\right]$$ If $b+b+c+2h<1$, we have $0< 1-c-h$ and so $$d\left(u_{n} ,u_{n+1}\right)<d\left(u_{n-1} ,u_{n} \right), \forall n \in\mathbb{N};$$ Consequently, $$\theta\left( d\left(u_{n} ,u_{n+1}\right)\right) \leq \phi\left[ \theta \left( d\left(u_{n-1} ,u_{n} \right)\right)\right]$$ It implies $$\begin{aligned} \theta \left(d\left(u_{n} ,u_{n+1}\right)\right) &\leq \phi\left[ \theta \left(d(x_{n-1},u_{n} \right)\right] \\ &\leq \phi^{2}\left[ \theta\left(d(u_{n-2},u_{n-1} \right)\right] \\ &\leq...\leq \phi ^{n}\left[ \theta \left(d(u_{0},u_{1} \right)\right] .\end{aligned}$$ Taking the limit as $n\rightarrow \infty$, we have $$1\leq \theta(d\left( u_{n},u_{n+1}\right))\leq \lim_{n\rightarrow \infty }\phi ^{n}\left[ \theta(d\left( u_{0},u_{1}\right))\right] =1.$$ Since $\theta\in\Theta$, we obtain $$\lim_{n\rightarrow \infty }d\left( u_{n},u_{n+1}\right) =0.$$ Next, we shall prove that $\left\lbrace u_{n}\right\rbrace _{n\in \mathbb{N}}$ is a Cauchy sequence, i.e, $\lim_{n\rightarrow \infty }d\left( u_{n,}u_{m}\right) =0,$ for all $n\in \mathbb{N}$. Suppose to the contrary that exists $\varepsilon$ $>0$ and sequences $n_{\left( k\right) }$ and $m_{\left( k\right)}$ of natural numbers such that $$\label{5A} m_{\left( k\right) }> n_{\left( k\right) }>k,\ \ d\left( x_{m_{\left( k\right) }},x_{n_{\left( k\right) }}\right) \geq \varepsilon , \ \ D\left( u_{m_{\left( k\right) -1}},u_{n_{\left( k\right) }}\right) <\varepsilon .$$ Using the triangular inequality, we find that, $$\begin{aligned} \label{6A} \varepsilon \leq d\left( u_{m_{\left( k\right) }},u_{n_{\left( k\right)}}\right) &\leq d\left( u_{m_{\left( k\right) }},u_{n\left( k\right)-1}\right) +d\left( x_{n\left( k\right)-1},x_{n_{\left( k\right)}}\right)\\ & <\varepsilon+d\left( u_{n\left( k\right)-1},u_{n_{\left( k\right)}}\right).\end{aligned}$$ Then, by $\ref{5A}$ and $\ref{6A}$, it follows that $$\label{7} \lim_{k\rightarrow \infty }d\left( u_{m_{\left( k\right) }},u_{n_{\left( k\right)}}\right) =\varepsilon .$$ Using the triangular inequality, we find that, $$\begin{aligned} \label{66A} \varepsilon \leq d\left( u_{m_{\left( k\right) }},u_{n_{\left( k\right)}}\right) &\leq d\left( u_{m_{\left( k\right) }},u_{n\left( k\right)+1}\right) +d\left( x_{n\left( k\right)+1},u_{n_{\left( k\right)}}\right)\end{aligned}$$ and $$\begin{aligned} \label{666A} \varepsilon \leq d\left( u_{m_{\left( k\right) }},u_{n_{\left( k\right)+1}}\right) &\leq d\left( u_{m_{\left( k\right) }},u_{n\left( k\right)}\right) +d\left( u_{n\left( k\right)},u_{n_{\left( k\right)+1}}\right)\end{aligned}$$ Then, by $(\ref{66A})$ and $(\ref{666A})$, it follows that $$\label{77} \lim_{k\rightarrow \infty }d\left(u_{ m_{\left( k\right)} },u_{n_{\left( k\right)+1}}\right) =\varepsilon .$$ Similarly method, we conclude that $$\label{777} \lim_{k\rightarrow \infty }d\left( u_{m_{\left( k\right) +1}},u_{n_{\left( k\right)}}\right) =\varepsilon .$$ Using again the triangular inequality, $$\label{88} d\left( u_{m_{\left( k\right) +1}},u_{n_{\left( k\right) +1}}\right) \leq d\left( x_{m_{\left( k\right) +1}},x_{m_{\left( k\right)} }\right) +d\left( u_{m\left( k\right) },u_{n_{\left( k\right) }}\right) +d\left( u_{n_{\left(k\right)}},u_{n_{\left( k\right) +1}}\right).$$ On the other hand, using triangular inequality, we have $$\label{99} d\left( u_{m_{\left( k\right) }},u_{n_{\left( k\right) }}\right) \leq d\left( u_{m_{\left( k\right) }},u_{m_{\left( k\right) +1}}\right) +d\left(u_{m_{\left( k\right) +1}},u_{n_{\left( k\right) +1}}\right) +d\left(u_{n_{\left( k\right) +1}},u_{n_{\left( k\right) }}\right).$$ Letting $k\rightarrow \infty$ in inequality $(\ref{88} )$ and $(\ref{99})$, we obtain $$\label{10} \lim_{k\rightarrow \infty }d\left( u_{m_{\left( k\right) +1}},u_{n_{\left(k\right) +1}}\right) =\varepsilon .$$ Substituting $u_1= x_{m_{\left( k\right)+1}},u_2= x_{n_{\left( k\right)+1}},v_1= u_{m_{\left( k\right)}}$ and $v_1= u_{n_{\left( k\right)}}$ in assumption of the theorem, we get $$\label{xx} \theta \left( {d\left( u_{m_{\left( k\right) +1}},u_{n_{\left( k\right)+1}}\right) }\right)\leq \phi\left\lbrace \theta\left\lbrace \begin{aligned} &ad\left( u_{m_{\left( k\right)}},u_{n_{\left( k\right)}}\right)\\ &+bd\left( u_{m_{\left( k\right)}1},u_{n_{\left( k\right)}}\right)\\ &+cd\left( u_{n_{\left( k\right)}+1},u_{n_{\left( k\right)}}\right)\\ &+h(d\left( u_{m_{\left( k\right)}},u_{n_{\left( k\right)}+1}\right)+d\left( u_{n_{\left( k\right)}},u_{m_{\left( k\right)}+1}\right)) \end{aligned} \right\rbrace \right\rbrace$$ Letting Letting $k\rightarrow \infty$ in ([\[xx\]](#xx){reference-type="ref" reference="xx"}), and using $\left( \theta _{1}\right)$, $\left( \theta _{3}\right),$ $\left( \phi _{3}\right)$ and Lemma ([\[LZ\]](#LZ){reference-type="ref" reference="LZ"}) we obtain $$\theta \left(\varepsilon\right) \leq \phi\left[ \theta\left( a\varepsilon +b\varepsilon+c\varepsilon+2h\varepsilon\right)\right].$$ We derive $$\varepsilon <\varepsilon.$$ Which is a contradiction. Thus $\lim_{n,m\rightarrow \infty }d\left( u_{n},u_{m}\right)= 0$, which shows that $\lbrace x_{n} \rbrace$ is a Cauchy sequence. Then there exists $z\in A$ such that $$\lim_{n\rightarrow \infty }d\left( u_{n},u\right)=0 .$$ Also, $$\begin{aligned} d\left( u,B\right)&\leq d\left( u,Tu_{n}\right)\\ &\leq d\left( u,x_{n+1}\right)+d\left( u_{n+1},Tu_{n}\right)\\ &=d\left( u,u_{n+1}\right)+d\left( A,B\right)\\ &\leq d\left( u,u_{n+1}\right)+d\left( u,B\right).\end{aligned}$$ Therefore, $d\left( u,Tu_{n}\right)\rightarrow d\left( u,B\right).$ In spite of the fact that $B$ is approximately compact with respect to $A$ , the sequence $\lbrace Tu_{n}\rbrace$ has a subsequence $\lbrace Tu_{n_{k}}\rbrace$ converging to some element $v \in B.$ So it turns out that $$\label{dd} d(u,v)= \lim_{n\rightarrow \infty }d\left( u_{n_{k}+1},Tu_{n_{k}}\right)=d(A,B).$$ Thus $u$ must be an element of $A_{0}$. Again, since $T(A_0) \in B_0$, there exists $t\in A_0$ such that $$d(t,Tu) =d(A,B)$$ for some element $t$ in $A$. Using the weak $p$-property and ([\[dd\]](#dd){reference-type="ref" reference="dd"}) we have $$d(u_{n_{k}+1}, t) = d(Pu_{n_{k}}, Pu), \forall n_k \in\mathbb{N}.$$ If for some $n_0$, $d(t, u_{n_{0}+1}) = 0$, consequently $d(P u_{n_{0}}, Tu) = 0$. So $P u_{n_{0}} =Tu$, hence $d(A,B) = d(u, Tu)$. Thus the conclusion is immediate. So let for any $n \geq 0$, $d(t, u_{n+1}) > 0$. Since $T$ is a generalized $(\theta,\phi)$-proximal contraction of the first kind, it follows from this that $$\label{gg} \theta(d(t,u_{n+1})) \leq \phi\left[ \theta\left[ a d\left(u,u_{n}\right)+bd\left(t,u\right)+cd\left(u_{n},u_{n+1}\right)+h\left( d\left(u,u_{n+1}\right)+d\left(u_{n},t\right)\right)\right] \right]$$ Since $\theta$ and $\phi$ are two continuous functions, by letting $n\rightarrow \infty$ in inequality $(\ref{gg})$, we obtain $$\begin{aligned} \theta(d(t,u))&\leq \phi\left[ \theta\left[ (b+h)\left( d\left(u,t\right)\right)\right] \right]\\ &\leq \phi\left[ \theta\left[ \left( d\left(u,t\right)\right)\right] \right]\\ &<\theta(d(t,u)). \end{aligned}$$ It is a contradiction. Therefore, $u=t$, that $$d(u, Tu) = d(t, Tu) = d(A,B).$$ Uniqueness: Suppose that there is another best proximity point $z$ of the mapping $T$ such that $$d(z, Tz)=d(A,B).$$ Since $T$ is a generalized $(\theta,\phi)$-proximal contraction of the first kind, it follows from this that $$\begin{aligned} \theta(d(z,u))&\leq \phi\left[ \theta\left[ a d\left(z,u\right)+bd\left(z,z\right)+cd\left(u,u\right)+h\left( d\left(z,u\right)+d\left(z,u\right)\right)\right] \right]\\ &=\phi\left[ \theta\left[ (a+2h) d\left(z,u\right)\right] \right], \end{aligned}$$ which is a contradiction. Thus, $z$ and $u$ must be identical. Hence, $T$ has a unique best proximity point. ◻ Next, we state and prove the best proximity point theorem for non-self generalized $(\theta,\phi)$-proximal contraction of the second kind. **Theorem 11**. *Let $(X,d)$ be a complete metric space and $(A,B)$ be a pair of non-void closed subsets of $(X,d)$. If $A$ is approximately compact with respect to $B$ and $T: A\rightarrow B$ satisfy the following conditions :* - *$T\left( A_0\right) \in B_0$ and the pair $\left( A,B\right)$ satisfies the weak $P$-property;* - *$T$ is continuous generalized $(\theta,\phi)$-proximal contraction of second kind.* *Then there exists a unique $u\in A$ such that $d(u, Tu)=d(A,B)$ and $u_n \rightarrow u$, where $u_0$ is any fixed point in $A_0$ and $d( u_{n+1 },T u_{n })= d(A,B)$ for $n \geq 0$. Further, if $z$ is another best proximity point of $T$, then $Tu = Tz$.* *Proof.* Similar to Theorem [Theorem 10](#T1){reference-type="ref" reference="T1"}, we can find a sequence $\lbrace u_n\rbrace$ in $A_0$ such that $$\label{SS} d( u_{n+1 },T u_{n })= d(A,B).$$ for all non-negative integral values of $n$. From the $p$-property and ([\[SS\]](#SS){reference-type="ref" reference="SS"}) we get $$d(u_{n }, u_{n+1 }) = d(Tu_{n-1 }, Tu_{n }), \forall n \in\mathbb{N}.$$ If for some $n_{0}$, $d(u_{n _{0+1 }}, u_{n _{0+2 }})= 0$, consequently $d(Tu_{n _{0 }}, Tu_{n _{0 }+1})= 0$. So $Tu_{n _{0 }}= Tu_{n _{0 }+1}$, hence $d(A,B)=d(Tu_{n _{0 }}, T_{n _{0 }+1}).$ Thus the conclusion is immediate. So let for any $n \geq 0$, $d(Tu_{n }, Tu_{n +1}) > 0$. We shall prove that the sequence $u_n$ is a Cauchy sequence. Let us first prove that $$\lim_{n\rightarrow \infty }d\left( u_{n},u_{n+1}\right) =0.$$ As $T$ is generalized $(\theta,\phi)$-proximal contraction of the second kind, we have that $$\begin{aligned} \theta \left(d\left(Tu_{n} ,Tu_{n+1}\right)\right)&\leq \phi\left[ \theta\left[ a d\left(Tu_{n-1} ,Tu_{n} \right)+bd\left(Tu_{n-1} ,Tu_{n} \right)+cd\left(Tu_{n} ,Tu_{n+1} \right)+h\left( d\left(Tu_{n-1} ,Tu_{n+1} \right)+d\left(Tu_{n} ,Tu_{n} \right)\right)\right] \right]\\ &=\phi\left[ \theta\left[ a d\left(Tu_{n-1} ,Tu_{n} \right)+bd\left(Tu_{n-1} ,Tu_{n} \right)+cd\left(Tu_{n} ,Tu_{n+1} \right)+h\left( d\left(Tu_{n-1} ,Tu_{n+1} \right)\right)\right] \right]\\ & \leq \phi\left[ \theta\left[ a d\left(Tu_{n-1} ,Tu_{n} \right)+bd\left(Tu_{n-1} ,Tu_{n} \right)+cd\left(Tu_{n} ,Tu_{n+1} \right)+h\left( d\left(Tu_{n-1} ,Tu_{n} \right)+d\left(Tu_{n} ,Tu_{n+1} \right)\right)\right] \right]\\ &= \phi\left[ \theta\left[ (a+b+h) d\left(Tu_{n-1} ,Tu_{n} \right)+(c+h)d\left(Tu_{n} ,Tu_{n+1} \right)\right] \right]\end{aligned}$$ Since $\theta$ is strictly increasing and by Lemma $\ref{LZ}$, we deduce $$d\left(Tu_{n} ,Tu_{n+1}\right)<(a+b+h) d\left(Tu_{n-1} ,Tu_{n} \right)+(c+h)d\left(Tu_{n} ,Tu_{n+1} \right).$$ Thus $$d\left(Tu_{n} ,Tu_{n+1}\right)<\frac{a+b+h}{1-c-h}( d\left(Tu_{n-1} ,Tu_{n} \right)).$$ If $b+b+c+2h=1$, we have $0< 1-c-h$ and so $$d\left(Tu_{n} ,Tu_{n+1}\right)\leq \frac{a+b+h}{1-c-h}( d\left(Tu_{n-1} ,Tu_{n} \right))=d\left(Tu_{n-1} ,Tu_{n} \right), \forall n \in\mathbb{N};$$ Consequently, $$\theta\left( d\left(Tu_{n} ,Tu_{n+1}\right)\right) \leq \phi\left[ \theta \left( d\left(Tu_{n-1} ,Tu_{n} \right)\right)\right]$$ If $b+b+c+2h<1$, we have $0< 1-c-h$ and so $$d\left(Tu_{n} ,Tu_{n+1}\right)<d\left(Tu_{n-1} ,Tu_{n} \right), \forall n \in\mathbb{N};$$ Consequently, $$\theta\left( d\left(Tu_{n} ,Tu_{n+1}\right)\right) \leq \phi\left[ \theta \left( d\left(Tu_{n-1} ,Tu_{n} \right)\right)\right]$$ It implies $$\begin{aligned} \theta \left(d\left(Tu_{n} ,Tu_{n+1}\right)\right) &\leq \phi\left[ \theta \left(d(Tu_{n-1},Tu_{n} \right)\right] \\ &\leq \phi^{2}\left[ \theta\left(d(Tu_{n-2},Tu_{n-1} \right)\right] \\ &\leq...\leq \phi ^{n}\left[ \theta \left(d(Tu_{0},Tu_{1} \right)\right] .\end{aligned}$$ Taking the limit as $n\rightarrow \infty$, we have $$1\leq \theta(d\left( Tu_{n},Tu_{n+1}\right))\leq \lim_{n\rightarrow \infty }\phi ^{n}\left[ \theta(d\left( Tu_{0},Tu_{1}\right))\right] =1.$$ Since $\theta\in\Theta$, we obtain $$\lim_{n\rightarrow \infty }d\left( Tu_{n},Tu_{n+1}\right) =0.$$ Next, we shall prove that $\left\lbrace Tu_{n}\right\rbrace _{n\in \mathbb{N}}$ is a Cauchy sequence, i.e, $\lim_{n\rightarrow \infty }d\left( Tu_{n,}Tu_{m}\right) =0,$ for all $n\in \mathbb{N}$. Suppose to the contrary that exists $\varepsilon$ $>0$ and sequences $Tn_{\left( k\right) }$ and $Tm_{\left( k\right)}$ of natural numbers such that $$\label{5AA} Tm_{\left( k\right) }> Tn_{\left( k\right) }>k,\ \ d\left( Tu_{m_{\left( k\right) }},Tu_{n_{\left( k\right) }}\right) \geq \varepsilon , \ \ d\left( Tu_{m_{\left( k\right) -1}},Tu_{n_{\left( k\right) }}\right) <\varepsilon .$$ Using the triangular inequality, we find that, $$\begin{aligned} \label{6A} \varepsilon \leq d\left( Tu_{m_{\left( k\right) }},Tu_{n_{\left( k\right)}}\right) &\leq d\left( Tu_{m_{\left( k\right) }},Tx_{n\left( k\right)-1}\right) +d\left( Tu_{n\left( k\right)-1},Tu_{n_{\left( k\right)}}\right)\\ & <\varepsilon+d\left( Tu_{n\left( k\right)-1},Tu_{n_{\left( k\right)}}\right).\end{aligned}$$ Then, by $\ref{5A}$ and $\ref{6A}$, it follows that $$\label{7} \lim_{k\rightarrow \infty }d\left( Tu_{m_{\left( k\right) }},Tu_{n_{\left( k\right)}}\right) =\varepsilon .$$ Using the triangular inequality, we find that, $$\begin{aligned} \label{66A} \varepsilon \leq d\left( Tu_{m_{\left( k\right) }},Tu_{n_{\left( k\right)}}\right) &\leq d\left( Tu_{m_{\left( k\right) }},Tu_{n\left( k\right)+1}\right) +d\left(T u_{n\left( k\right)+1},Tu_{n_{\left( k\right)}}\right)\end{aligned}$$ and $$\begin{aligned} \label{666AA} \varepsilon \leq d\left( Tu_{m_{\left( k\right) }},Tu_{n_{\left( k\right)+1}}\right) &\leq d\left( Tu_{m_{\left( k\right) }},Tu_{n\left( k\right)}\right) +d\left(T u_{n\left( k\right)},Tu_{n_{\left( k\right)+1}}\right)\end{aligned}$$ Then, by $(\ref{66A})$ and $(\ref{666A})$, it follows that $$\label{77} \lim_{k\rightarrow \infty }d\left(Tu_{ m_{\left( k\right) }},Tu_{n_{\left( k\right)+1}}\right) =\varepsilon .$$ Similarly method, we conclude that $$\label{7777} \lim_{k\rightarrow \infty }d\left( Tu_{m_{\left( k\right) +1}},Tu_{n_{\left( k\right)}}\right) =\varepsilon .$$ Using again the triangular inequality, $$\label{888} d\left( Tu_{m_{\left( k\right) +1}},Tu_{n_{\left( k\right) +1}}\right) \leq d\left( u_{m_{\left( k\right) +1}},Tu_{m_{\left( k\right)} }\right) +d\left( Tu_{m\left( k\right) },Tu_{n_{\left( k\right) }}\right) +d\left( Tu_{n_{\left(k\right)}},Tu_{n_{\left( k\right) +1}}\right).$$ On the other hand, using triangular inequality, we have $$\label{999} d\left( Tu_{m_{\left( k\right) }},Tu_{n_{\left( k\right) }}\right) \leq d\left( Tu_{m_{\left( k\right) }},Tu_{m_{\left( k\right) +1}}\right) +d\left(Tu_{m_{\left( k\right) +1}},Tu_{n_{\left( k\right) +1}}\right) +d\left(Tu_{n_{\left( k\right) +1}},Tu_{n_{\left( k\right) }}\right).$$ Letting $k\rightarrow \infty$ in inequality $(\ref{888} )$ and $(\ref{999})$, we obtain $$\label{100} \lim_{k\rightarrow \infty }d\left( Tu_{m_{\left( k\right) +1}},Tu_{n_{\left(k\right) +1}}\right) =\varepsilon .$$ Substituting $u_1= Tu_{m_{\left( k\right)+1}},u_2= Tu_{n_{\left( k\right)+1}},v_1= Tu_{m_{\left( k\right)}}$ and $v_1= Tu_{n_{\left( k\right)}}$ in assumption of the theorem, we get $$\label{xxX} \theta \left( {d\left( Tu_{m_{\left( k\right) +1}},Tu_{n_{\left( k\right)+1}}\right) }\right)\leq \phi\left\lbrace \theta\left\lbrace \begin{aligned} &ad\left( Tu_{m_{\left( k\right)}},Tu_{n_{\left( k\right)}}\right)\\ &+bd\left( Tu_{m_{\left( k\right)}1},Tu_{n_{\left( k\right)}}\right)\\ &+cd\left(T u_{n_{\left( k\right)}+1},Tu_{n_{\left( k\right)}}\right)\\ &+h(d\left( Tu_{m_{\left( k\right)}},Tu_{n_{\left( k\right)}+1}\right)+d\left( Tu_{n_{\left( k\right)}},Tu_{m_{\left( k\right)}+1}\right)) \end{aligned} \right\rbrace \right\rbrace$$ Letting Letting $k\rightarrow \infty$ in ([\[xxX\]](#xxX){reference-type="ref" reference="xxX"}), and using $\left( \theta _{1}\right)$, $\left( \theta _{3}\right),$ $\left( \phi _{3}\right)$ and Lemma ([\[LZ\]](#LZ){reference-type="ref" reference="LZ"}) we obtain $$\theta \left(\varepsilon\right) \leq \phi\left[ \theta\left( a\varepsilon +b\varepsilon+c\varepsilon+2h\varepsilon\right)\right].$$ We derive $$\varepsilon <\varepsilon.$$ Which is a contradiction. Thus $\lim_{n,m\rightarrow \infty }d\left( Tu_{n},Tu_{m}\right)= 0$, which shows that $\lbrace Tu_{n} \rbrace$ is a Cauchy sequence. Then there exists $v\in B$ such that $$\lim_{n\rightarrow \infty }d\left( Tu_{n},v\right)=0 .$$ Also, $$\begin{aligned} d\left( v,A\right)&\leq d\left( v,Tu_{n}\right)\\ &\leq d\left( v,u_{n+1}\right)+d\left( u_{n+1},Tu_{n}\right)\\ &=d\left( v,u_{n+1}\right)+d\left( A,B\right)\\ &\leq d\left( v,u_{n+1}\right)+d\left( v,A\right).\end{aligned}$$ Therefore, $d\left( v,Tu_{n}\right)\rightarrow d\left( v,A\right).$ Since $A$ is approximately compact with respect to $B$ , the sequence $\lbrace u_{n}\rbrace$ has a subsequence $\lbrace u_{n_{k}}\rbrace$ converging to some element $u \in A.$ So it turns out that $$\label{dd} d(u,v)= \lim_{n\rightarrow \infty }d\left( u_{n_{k}+1},Tu_{n_{k}}\right)=d(A,B).$$ Because $T$ is a continuous mapping, $$d(u, Tu) = \lim_{n\rightarrow \infty } d(u_{n+1}, Tu_{n}) = d(A,B).$$ Uniqueness: Suppose that there is another best proximity point $z$ of the mapping $T$ such that $$d(z, Tz)=d(A,B).$$ Since $T$ is a generalized $(\theta,\phi)$-proximal contraction of the first second, it follows from this that $$\begin{aligned} \theta(d(Tz,Tu))&\leq \phi\left[ \theta\left[ a d\left(Tz,Tu\right)+bd\left(Tz,Tz\right)+cd\left(Tu,Tu\right)+h\left( d\left(Tz,Tu\right)+d\left(Tz,Tu\right)\right)\right] \right]\\ &=\phi\left[ \theta\left[ (a+2h) d\left(Tz,Tu\right)\right] \right], \end{aligned}$$ which is a contradiction. Thus, $z$ and $u$ must be identical. Hence, $T$ has a unique best proximity point. **Theorem 12**. *Let $(X,d)$ be a complete metric space and $(A,B)$ be a pair of non-void closed subsets of $(X,d)$. Let $T: A\rightarrow B$ satisfy the following conditions :* - *$T\left( A_0\right) \in B_0$ and the pair $\left( A,B\right)$ satisfies the weak $P$-property;* - *$T$ is a generalized $(\theta,\phi)$-proximal contraction of the first kind as well as a generalized $(\theta,\phi)$-proximal contraction of the second kind.* *Then there exists a unique $u\in A$ such that $d(u, Tu)=d(A,B)$ and $u_n \rightarrow u$, where $u_0$ is any fixed point in $A_0$ and $d( u_{n+1 },T u_{n })= d(A,B)$ for $n \geq 0$.* *Proof.* Similar to Theorem [Theorem 10](#T1){reference-type="ref" reference="T1"}, we find a sequence $\lbrace u_{n } \rbrace$ in $A_0$ such that $$d( u_{n+1 }, T u_{n }) = d(A,B)$$ for all non-negative integral values of $n$. Similar to Theorem [Theorem 10](#T1){reference-type="ref" reference="T1"}, we can show that sequence $\lbrace u_{n } \rbrace$ is a Cauchy sequence. Thus converges to some element $u$ in $A$. As in Theorem [Theorem 11](#T2){reference-type="ref" reference="T2"}, it can be shown that the sequence $\lbrace Tu_{n } \rbrace$ is a Cauchy sequence and converges to some element $v$ in $B$. Therefore, $$\label{BB} d(u, v) = \lim_{n\rightarrow \infty }d( u_{n+1 }, T u_{n }) = d(A,B).$$ Eventually, u becomes an element of $A_0$. In light of the fact that $T(A_0)\in B_0$, $$d(t, Tu) = d(A,B)$$ for some element $t$ in $A$. From the $p$-property framework and ([\[BB\]](#BB){reference-type="ref" reference="BB"},) we get $$d( u_{n+1 }, t) = d(T u_{n }, Tu), \forall n \in \mathbb{N}.$$ If for some $n_0$, $d(t, u_{n_{ 0}+1}) = 0$, consequently $d(Tu_{n_{ 0}}, Tu) = 0$ . So $Tu_{n_{ 0}} =Tu$, hence $d(A,B) = d(u, Tu)$. Thus the conclusion is immediate. So let for any $n \geq 0$, $d(t, u_{n+1 }) > 0$. Since $T$ is a generalized $(\theta,\phi)$-proximal contraction of the first kind, it can be seen that $$\begin{aligned} \label{gggg} \theta(d(t, u_{n+1}))&\leq \phi\left[ \theta \left( ad(u, u_{n }) + bd(t, u) + cd( u_{n }, u_{n+1 }) + h[d(u, u_{n+1 }) + d( u_{n }, t)\right)\right] .\end{aligned}$$ Since $\theta$ and $\phi$ are two continuous functions, by letting $n\rightarrow \infty$ in inequality $(\ref{gggg})$, we obtain, $d(u, Tu) = d(t, Tu) = d(A,B).$ Also, as in the theorem [Theorem 10](#T1){reference-type="ref" reference="T1"}, the uniqueness of the best proximity point of mapping $T$ follows. ◻  ◻ **Example 13**. Let $X=\lbrace \lambda_{n }: n\in\mathbb{N} \rbrace$ with the metric $d(x, y) = \vert x-y \vert$ for all $x, y \in X$, where the sequence $G_{n }$, defined by $$\begin{aligned} &\lambda_{1 } = 1\\ &\lambda_{2 } = 1 + 2\\ &\lambda_{3 } = 1 + 2 + 3\\ &...\\ &\lambda_{n } = 1 + 2 + 3 + . . . + n.\end{aligned}$$ We know, $(X, d)$ is a complete metric space. Let $A = G_{3n } : n\in\mathbb{N}$ and $B = G_{3n-1 } : n\in\mathbb{N}$. It is easy to see that $d(A,B) = 3$, $A_0 = A$ and $B_0 =B$. Define a mappings $T : A \rightarrow B$, by $T(\lambda_{3n }) = \lambda_{3n-1 }$ for all $n \geq 1$. It is clear that $A$ is approximately compact with respect to $B$, $(A,B)$ satisfies the $p$-property, $T$ is continuous and $T(A_0) \subseteq B_0$ . We will show that $T$ is an $(\theta,\phi)$-proximal contraction with $\theta\in\Theta$ and $\phi\in\Phi$ that is $\theta(t)=e^{t}$ and $\phi(t)=t^{\frac{1}{2}}$. Observe that, With out of generality, we may assume that $n < m$, and since $$\begin{aligned} &\lambda_{3n-1 } = 1 + 2 + 3 + . . . + 3n - 1,\\ &\lambda_{3m-1 } = 1 + 2 + 3 + . . . + 3m - 1,\\ &\lambda_{3n } = 1 + 2 + 3 + . . . + 3n - 1+3n,\\ &\lambda_{3m } = 1 + 2 + 3 + . . . + 3m - 1+3m.\end{aligned}$$ It follow that, $$\begin{aligned} d(T(\lambda_{3n }), T(\lambda_{3m}))& = \vert \lambda_{3n-1 }- \lambda_{3m-1}\vert\\ &= 3n + (3n + 1) + . . . + (3m - 1),\end{aligned}$$ $$\begin{aligned} d(\lambda_{3n }, \lambda_{2m})& = \vert \lambda_{2n }- \lambda_{2m}\vert\\ &= 3n + (3n + 1) + . . . + (3m ),\end{aligned}$$ and $$\begin{aligned} d(T(\lambda_{2n }), T(\lambda_{3m}))-d(\lambda_{3n }, \lambda_{3m})& = \vert \lambda_{3n-1 }- \lambda_{3m-1}\vert -\vert \lambda_{3n }- \lambda_{3m}\vert\\ &= 3n -3m.\end{aligned}$$ So that, $$\begin{aligned} e^{d(T(\lambda_{3n }), T(\lambda_{3m}))-d(\lambda_{3n }, \lambda_{3m}) })&= \frac{e^{ d(T(\lambda_{3n }), T(\lambda_{3m})) }}{e^{ d(\lambda_{3n }, \lambda_{3m}) }}\\ &=e^{ 3n -3m})\\ &=e^{ -3(m -n)})\\ &\leq e^{ -3}= \frac{1}{e^{ 3}}.\end{aligned}$$ So that, $$\begin{aligned} e^{ d(T(\lambda_{3n }), T(\lambda_{3m})) }+1&=\theta(d(T(\lambda_{3n }), T(\lambda_{3m})))\\ & \leq e^{ d(\lambda_{3n }, \lambda_{3m}) }\frac{1}{e^{ 3}}+1\\ &\leq \frac{e^{ d(\lambda_{3n }, \lambda_{3m}) }+2}{2}\\ &=\phi\left[ \theta(d(\lambda_{3n }, \lambda_{3m}))\right] .\end{aligned}$$ Consequently, $T$ is an generalized $(\theta,\phi)$-proximal contraction of the second kind with $a=1$, $b = c = h = 0$. Thus, all the conditions of Theorem [Theorem 11](#T2){reference-type="ref" reference="T2"} are satisfied. Hence, $T$ has a unique best proximity point and there exist $\lambda_ { 3}\in A$ such that $$d(\lambda_ { 3}, T\lambda_ { 3}) = d(\lambda_ { 3}, \lambda_ { 2}) = 3 = d(A,B)$$ 99 H. Aydi, H. Lakzian, Z. Mitrović, S. Radenović, Best Proximity Points of MT-Cyclic Contractions with Property UC, Numerical Functional Analysis and Optimization; Volume 41, 2020. S. Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales. Fundam. Math. 3(1), 133--181 (1922). S. S. Basha, P. Veeramani, Best proximity pair theorems for multifunctions with open fibres. J. Approx. Theory 103, 119--129 (2000). I. Beg, G. Mani, A. J. Gnanaprakasam, Best proximity point of generalized F-proximal non-self contractions Journal of Fixed Point Theory and Applications 23 (4), 1-11 A. Eldred, W. Kirk, P. Veeramani, Proximal normal structure and relatively nonexpansive mappings, Stud. Math. 171(3), 283-293 (2005). M. Jleli, B. Samet, A new generalization of the Banach contraction principle, J. Inequal. Appl., 2014 (2014), 8 pages. A. Kari, M. Rossafi, E. Marhrani, M. Aamri, Fixed point theorem for Nonlinear $F-$contraction via $w-$distance, Adv. Math. Phys, 2020(2020), Article ID 6617517. A. Kari, M. Rossafi, E. Marhrani, M. Aamri, $\theta-\phi-$contraction on $\left(\alpha,\eta \right)-$complete rectangular $b-$metric spaces, Int. J. Math. Mathematical Sciences, (2020), Article ID 5689458. V. Parvaneh, M. Reza Haddadi and H. Aydi, On Best Proximity Point Results for Some Type of Mappings. Hindawi Journal of Function Spaces Volume 2020, Article ID 6298138, 6 pages. V. S. Raj, A best proximity point theorem for weakly contractive non-self-mappings. Nonlinear Anal. 2011, 74, 4804--4808. M. Rossafi, A. Kari, Some fixed point theorems for $F$-expansive mapping in generalized metric spaces, Open J. Math. Anal. 2021, 5(2), 17-30. M. Rossafi, A. Kari, Best Proximity Point Theorems for $\alpha$-Proximal $\theta,\phi$-non-self mappings, Asian Journal of Mathematics and Applications, 2021. J. Zhang, Y. Su, Q. Cheng, A note on 'A best proximity point theorem for Geraghty-contractions'. Fixed Point Theory Appl. 2013, 99. D. Zheng , Z. Cai , P. Wang, New fixed point theorems for ($\theta-\phi$)-contraction in complete metric. spaces. Journal of Non linear Sciences and Applications. 2017;10(5):2662, and Information Processing 6 (2008), 433-446.

arxiv_math

--- abstract: | Using Khinchin's inequality, Ger$\check{\mbox{s}}$gorin's theorem and the atomic decomposition of Bergman spaces, we estimate the norm and essential norm of Stević-Sharma type operators from weighted Bergman spaces $A_\omega^p$ to $A_\mu^q$ and the sum of weighted differentiation composition operators with different symbols from weighted Bergman spaces $A_\omega^p$ to $H^\infty$. The estimates of those between Bergman spaces remove all the restrictions of a result in \[Appl. Math. Comput., **217**(2011), 8115--8125\]. As a by-product, we also get an interpolation theorem for Bergman spaces induced by doubling weights. [^1] [^2] : Stević-Sharma type operator; differentiation composition operator; Bergman space; doubling weight. address: - | Juntao Du\ Department of mathematics, Guangdong University of Petrochemical Technology, Maoming, Guangdong, 525000, P. R. China. - | Songxiao Li\ Department of mathematics, Shantou University, Shantou, Guangdong, 515063, P. R. China. - | Zuoling Liu\ Department of Mathematics, Shantou University, Shantou, Guangdong, 515063, P. R. China author: - Juntao Du, Songxiao Li$\dagger$ and Zuoling Liu title: Stević-Sharma type operators between Bergman spaces induced by doubling weights --- # Introduction Let $H(\mathbb{D})$ denote the space of all analytic functions in the open unit disc $\mathbb{D}=\{z\in \mathbb{C}:|z|<1\}$. For a nonnegative function $\omega\in L^1([0,1])$, the extension to $\mathbb{D}$, defined by $\omega(z)=\omega(|z|)$ for all $z\in\mathbb{D}$, is called a radial weight. The set of doubling weights, denoted by $\hat{\mathcal{D}}$, consists of all radial weights $\omega$ such that (see [@Pj2015]) $$\hat{\omega}(r)\leq C \hat{\omega}\left(\frac{r+1}{2}\right)$$ for all $0\leq r <1$ and constant $C=C(\omega)\geq1$. Here $\hat{\omega}(z)=\int_{|z|}^1\omega(t)dt$. Moreover, if $\omega\in\hat{\mathcal{D}}$ and satisfies $$\begin{aligned} \label{0405-1} \hat{\omega}(r)\geq C\hat{\omega}\left(1-\frac{1-r}{K}\right)\end{aligned}$$ for all $0\leq r <1$ and some constants $K=K(\omega)>1$ and $C=C(\omega)>1$, we refer to $\omega$ as a two-sides doubling weight and denote it by $\omega\in\mathcal{D}$. For any $\lambda\in\mathbb{D}$, the Carleson square at $\lambda\in\mathbb{D}$ is defined by $$S(\lambda)=\left\{re^{\mathrm{i\theta}}:|\lambda|\leq r<1, |\mbox{Arg } \lambda-\theta|<\frac{1-|\lambda|}{2}\right\}.$$ For a radial weight $\omega$, let $\omega(S(\lambda))=\int_{S(\lambda)}\omega(z)dA(z)$. Obviously, $\omega(S(\lambda))\approx (1-|\lambda|)\hat{\omega}(\lambda)$. See [@Pj2015; @PjRj2021adv; @PjRjSk2021jga] and references therein for more properties of doubling weight. For $0<p<\infty$ and a given $\omega\in\hat{\mathcal{D}}$, the Bergman space $A_\omega^p$ with doubling weight consists of all functions $f\in H(\mathbb{D})$ such that $$\|f\|_{A_\omega^p}^p=\int_\mathbb{D} |f(z)|^p\omega(z)dA(z)<\infty,$$ where $dA$ is the normalized area measure on $\mathbb{D}$. As usual, we denote $A_\alpha^p$ as the standard weighted Bergman space induced by the radial weight $\omega(z)=(\alpha+1)(1 - |z|^2)^\alpha$ with $-1<\alpha<\infty$. Throughout this paper, we assume that $\hat{\omega}(z) >0$ for all $z\in\mathbb{D}$. Otherwise $A_\omega^p=H(\mathbb{D})$. Let $H^\infty$ denote the bounded analytic function space, i.e., $$H^\infty=\left\{f\in H(\mathbb{D}):\|f\|_{H^\infty}=\sup_{z\in\mathbb{D}}|f(z)|<\infty\right\}.$$ Let $S(\mathbb{D})$ be the set of all analytic self-maps of $\mathbb{D}$. For $n\in\mathbb N\cup\{0\},\varphi\in S(\mathbb{D})$, and $u\in H(\mathbb{D})$, the generalized weighted composition operator $uD_\varphi^{(n)}$ is defined by $$uD_\varphi^{(n)} f=u \left(f^{(n)}\circ \varphi\right), \quad f\in H(\mathbb{D}).$$ The operator $uD_\varphi^{(n)}$ was introduced by Zhu in [@ZXL2007]. The generalized weighted composition operator is also called a weighted differentiation composition operator (see [@st1; @st2; @st3]). When $n=0$, $uD_\varphi^{(n)}$ is the weighted composition operator $uC_\varphi$. In particular, when $n=0$ and $u\equiv 1$, $uD_\varphi^{(n)}$ is the composition operator $C_\varphi$. By using the pull-back measure, the first two authors of this paper and Shi [@DjLsSy2020ms] estimated the norm and essential norm of weighted composition operators between Bergman spaces induced by doubling weights. At the same time, Liu [@Lb2021bams] independently characterized the boundedness and compactness of weighted differentiation composition operator $uD_\varphi^{(n)}:A_\omega^p\to L_\upsilon^q$ when $0<p,q<\infty$, $\omega\in\mathcal{D}$ and $\upsilon$ is a positive Borel measure on $\mathbb{D}$. For more discussion on composition operators and weighted composition operators, we refer to [@CM1995; @EK2023; @GMR2023; @SsSaBa2011amc; @SU2014; @Zhu1] and the references therein. When $u\equiv 1$, $uD_\varphi^{(n)}$ is the differentiation composition operator $D_\varphi^{(n)}$. When $u\equiv 1$ and $\varphi(z)=z$, $uD_\varphi^{(n)}$ is the $n$-th differentiation operator $D^{(n)}$. So, the generalized weighted composition operator attracted a lot of attentions since it covers a lot of classical operators. See [@st1; @st2; @st3; @ZXL2007; @zxl2; @zxl5; @ZXL2019] for further information and results on generalized weighted composition operators on analytic function spaces. In 2011, Stević, Sharma and Bhat [@SsSaBa2011amc] introduced a operator $T_{u_0,u_1,\varphi}$ as follows. $$T_{u_0,u_1,\varphi}=u_0D_\varphi^{(0)}+u_1 D_\varphi^{(1)}.$$ This operator and its extension $\sum_{k=0}^n u_k D_{\varphi_k}^{(n)}$ are called Stević-Sharma type operators by some authors, where $\{u_k\}_{k=0}^n \subset H(\mathbb{D})$ and $\{\varphi_k\}_{k=0}^n\subset S(\mathbb{D})$. In [@SsSaBa2011amc], Stević, Sharma and Bhat characterized the boundedness of $T_{u_0,u_1,\varphi}:A_\alpha^p\to A_\alpha^p$ with the assumption $$\begin{aligned} \label{0914-1} u_0\in H^\infty\,\,\,\mbox{or}\,\,\,\sup_{z\in\mathbb{D}}\frac{|u_1(z)|}{1-|\varphi(z)|^2}<\infty. \end{aligned}$$ Two natural questions are raised. 1. Whether the condition ([\[0914-1\]](#0914-1){reference-type="ref" reference="0914-1"}) can be removed? 2. What about the operator $u_0 D_{\varphi_0}^{(0)}+u_1D_{\varphi_1}^{(1)}$ when $\varphi_0\neq \varphi_1$? See, for example, [@FG2022; @G2022f; @WWG2020; @WWG2020B; @YyLy2015caot; @ZfLy2018caot] for some investigations about these operators. By using Khinchin's inequality, Ger$\check{\mbox{s}}$gorin's theorem, and the atomic decomposition of weighted Bergman spaces, we give a positive answer to question **Q1**. Moreover, we extend it to a more general case and completely estimate the norm and essential norm of the operator $$T_{n,\varphi,\vec{u}} =\sum_{k=0}^n u_k D_\varphi^{(k)}$$ from $A_\omega^p$ to $A_\mu^q$ with $\omega,\mu\in\mathcal{D}$ and $1\leq p,q<\infty$, where $\vec{u}=(u_0, u_1,\cdot\cdot\cdot, u_n)$ with $\{u_k\}_{k=0}^n\subset H(\mathbb{D})$ and $\varphi\in S(\mathbb{D})$. The first result of this paper is stated as follows. **Theorem 1**. *Suppose $1\leq p,q<\infty,\omega,\mu\in\mathcal{D}$, $\varphi\in S(\mathbb{D})$ and $\{u_k\}_{k=0}^n\subset H(\mathbb{D})$. Then, $$\|T_{n,\varphi,\vec{u}}\|_{A_\omega^p\to A_\mu^q}\approx \sum_{k=0}^n\|u_k D_\varphi^{(k)}\|_{A_\omega^p\to A_\mu^q}.$$ Moreover, if $T_{n,\varphi,\vec{u}}:A_\omega^p\to A_\mu^q$ is bounded, then $$\|T_{n,\varphi,\vec{u}}\|_{e,A_\omega^p\to A_\mu^q}\approx \sum_{k=0}^n\|u_kD_\varphi^{(k)}\|_{e,A_\omega^p\to A_\mu^q}.$$* By Remark [Remark 6](#0916-2){reference-type="ref" reference="0916-2"} and Theorem [Theorem 1](#thB){reference-type="ref" reference="thB"} in Section 2, Theorem 1 completely characterizes the norm and essential norm of $T_{n,\varphi,\vec{u}}:A_\omega^p\to A_\mu^q$ Recall that the essential norm of a bounded operator $T:X\to Y$ is defined by $$\|T\|_{e,X\to Y}=\inf\Big\{\|T-K\|_{X\to Y};K:X\to Y \mbox{ is compact}\Big\}.$$ Here $X$ and $Y$ are Banach spaces. Obviously, $T$ is compact if and only if $\|T\|_{e,X\to Y}=0$. For the question **Q2**, Acharyya and Ferguson [@AsFt2019caot] characterized the compactness of the operator $\mathcal{T}_{n,\vec{\varphi},\vec{u}}:A_\alpha^p\to H^\infty$, where $$\mathcal{T}_{n,\vec{\varphi},\vec{u}}=\sum_{k=0}^n u_k D_{\varphi_k}^{(k)} .$$ Here $\vec{u}=(u_0, u_1,\cdot\cdot\cdot, u_n)$ with $\{u_k\}_{k=0}^n\subset H(\mathbb{D})$ and $\vec{\varphi}=(\varphi_0, \varphi_1,\cdot\cdot\cdot, \varphi_n)$ with $\{\varphi_k\}_{k=0}^n\subset S(\mathbb{D})$. In this paper, we extend [@AsFt2019caot Theorem 2] to the case of Bergman spaces $A_\omega^p$ with $\omega\in\hat{\mathcal{D}}$. **Theorem 2**. *Suppose $n\in\mathbb N\cup\{0\}$, $1\leq p<\infty$, $\omega\in\hat{\mathcal{D}}$, $\{u_k\}_{k=0}^n\subset H(\mathbb{D})$ and $\{\varphi_k\}_{k=0}^n\subset S(\mathbb{D})$. Then, $$\|\mathcal{T}_{n,\vec{\varphi},\vec{u}}\|_{A_\omega^p\to H^\infty}\approx \sum\limits_{k=0}^n \sup\limits_{z\in\mathbb{D}} \frac{|u_k(z)|}{(1-|\varphi_k(z)|^2)^k\omega(S(\varphi_k(z)))^\frac{1}{p}}.$$ Moreover, if $\mathcal{T}_{n,\vec{\varphi},\vec{u}}:{A_\omega^p\to H^\infty}$ is bounded, then $$\|\mathcal{T}_{n,\vec{\varphi},\vec{u}}\|_{e,A_\omega^p\to H^\infty}\approx \sum\limits_{k=0}^n \limsup\limits_{|\varphi_k(z)|\to 1} \frac{|u_k(z)|}{(1-|\varphi_k(z)|^2)^k\omega(S(\varphi_k(z)))^\frac{1}{p}}.$$* By Theorem [Theorem 2](#th2){reference-type="ref" reference="th2"}, it is easy to check that $$\|u_kD_{\varphi_k}^{(k)}\|_{A_\omega^p\to H^\infty} \approx \sup_{z\in\mathbb{D}} \frac{|u_k(z)|}{(1-|\varphi_k(z)|^2)^k\omega(S(\varphi_k(z)))^\frac{1}{p}}$$ and $$\|u_kD_{\varphi_k}^{(k)}\|_{e,A_\omega^p\to H^\infty} \approx \limsup_{|\varphi_k(z)|\to 1} \frac{|u_k(z)|}{(1-|\varphi_k(z)|^2)^k\omega(S(\varphi_k(z)))^\frac{1}{p}}.$$ Therefore, Theorem [Theorem 2](#th2){reference-type="ref" reference="th2"} can be stated similarly as Theorem [Theorem 1](#th1){reference-type="ref" reference="th1"}. **Theorem [Theorem 2](#th2){reference-type="ref" reference="th2"}$^{\prime}$ 1**. *Suppose $n\in\mathbb N\cup\{0\}$, $1\leq p<\infty$, $\omega\in\hat{\mathcal{D}}$, $\{u_k\}_{k=0}^n\subset H(\mathbb{D})$ and $\{\varphi_k\}_{k=0}^n\subset S(\mathbb{D})$. Then, $$\|\mathcal{T}_{n,\vec{\varphi},\vec{u}}\|_{A_\omega^p\to H^\infty}\approx \sum\limits_{k=0}^n \|u_kD_{\varphi_k}^{(k)}\|_{A_\omega^p\to H^\infty}.$$ Moreover, if $\mathcal{T}_{n,\vec{\varphi},\vec{u}}:{A_\omega^p\to H^\infty}$ is bounded, then $$\|\mathcal{T}_{n,\vec{\varphi},\vec{u}}\|_{e,A_\omega^p\to H^\infty}\approx \sum\limits_{k=0}^n \|u_kD_{\varphi_k}^{(k)}\|_{e,A_\omega^p\to H^\infty}.$$* The sufficiency of Theorems [Theorem 1](#th1){reference-type="ref" reference="th1"} and [Theorem 2](#th2){reference-type="ref" reference="th2"} are easy to verify. To establish the necessity part, we need the following interpolation theorem, which has its own interesting. Here and henceforth, $\delta_{ij}$ is the Dirac function, that is, $\delta_{ij}=1$ when $i=j$ and $\delta_{ij}=0$ when $i\neq j$. **Theorem 3**. *Let $1\leq p<\infty$, $n\in\mathbb N\cup\{0\}$, $\omega\in \hat{\mathcal{D}}$. Then there is a positive constant $C$ such that for all $\Lambda=\{\lambda_j\}_{j=0}^n\subset\mathbb{D}$ and $J\in\{0,1,\cdots,n\}$, there exists $f_{\Lambda,J}\in A_\omega^p$ satisfying $\|f_{\Lambda,J}\|_{A_\omega^p}\leq C$ and $$\begin{aligned} \label{IC} f_{\Lambda,J}^{(j)}(\lambda_j)=\frac{\delta_{jJ}}{(1-|\lambda_j|^2)^j\omega(S(\lambda_j))^\frac{1}{p}},\,\,j=0,1,\cdots,n.\end{aligned}$$ Moreover, if $J$ is fixed, the functions $\{f_{\Lambda,J}\}$ converge to 0 uniformly on compact subsets of $\mathbb{D}$ as $|\lambda_J|\to 1$.* The rest of this paper is organized as follows. In Section 2, we will gather the necessary preliminaries. Sections 3, 4, and 5 are dedicated to the proofs of Theorem [Theorem 1](#th1){reference-type="ref" reference="th1"}, Theorem [Theorem 3](#th3){reference-type="ref" reference="th3"}, and Theorem [Theorem 2](#th2){reference-type="ref" reference="th2"}, respectively. Throughout this paper, the letter $C$ will represent constants, which may vary from one occurrence to another. For two positive functions $f$ and $g$, we use the notation $f \lesssim g$ to denote that there exists a positive constant $C$, independent of the arguments, such that $f \leq Cg$. Similarly, $f \approx g$ indicates that $f \lesssim g$ and $g \lesssim f$.\ # preliminaries In this section, we state some lemmas which will be used in the proof of main results of this paper. For brief, for any given $\alpha>0$, let $$\omega_{[\alpha]}(z)=(1-|z|^2)^\alpha\omega(z),\,\,\,\,z\in\mathbb{D}.$$ **Lemma 4**. *Suppose $\alpha>0, \omega\in\mathcal{D}$. Then $\omega_{[\alpha]}\in\mathcal{D}$ and $\widehat{\omega_{[\alpha]}}\approx \hat{\omega}_{[\alpha]}$.* *Proof.* For brief, let $\eta=\omega_{[\alpha]}$. Let $C$ and $K$ be those in ([\[0405-1\]](#0405-1){reference-type="ref" reference="0405-1"}). For all $t\in[0,1)$, we have $\hat{\eta}(t)\lesssim (1-t)^{\alpha}\hat{\omega}(t)$ and $$\hat{\eta}(t)\geq \int_t^{1-\frac{1-t}{K}}(1-s^2)^{\alpha}\omega(s)ds \approx (1-t)^{\alpha}\left(\hat{\omega}(t)-\hat{\omega}\bigg(1-\frac{1-t}{K}\bigg) \right)\gtrsim (1-t)^{\alpha}\hat{\omega}(t).$$ Then, we have $\hat{\eta}(t)\approx (1-t)^{\alpha}\hat{\omega}(t)$ and then $$\hat{\eta}(t)\lesssim (1-t)^\alpha \hat{\omega}\bigg(\frac{1+t}{2}\bigg)\approx \hat{\eta}\bigg(\frac{1+t}{2}\bigg).$$ Thus, $\eta\in\hat{\mathcal{D}}$. Since $\frac{\hat{\eta}(t)}{(1-t)^\alpha}$ is essentially decreasing, by Lemma B in [@PjRe2022afm], $\eta\in\mathcal{D}$. The proof is complete. ◻ In [@PjRj2021adv], the authors characterized the Littlewood-Paley formula on Bergman spaces induced radial weights. For the benefit of readers, we state it as follows. **Theorem 1**. *Let $\omega$ be a radial weight, $0<p<\infty$ and $k\in\mathbb N$. Then, for all $f\in H(\mathbb{D})$, $$\int_\mathbb{D}|f(z)|^p\omega(z)dA(z)\approx \sum_{j=0}^{k-1}|f^{(j)}(0)|^p+\int_\mathbb{D}|f^{(k)}(z)|^p(1-|z|^2)^{kp}\omega(z)dA(z)$$ if and only if $\omega\in\mathcal{D}$.* **Lemma 5**. *Assume $1\leq p<\infty,n\in\mathbb N, \omega\in\mathcal{D}$ and $Y$ is a Banach space. Let $T: A_{\omega_{[np]}}^p\to Y$ be a bounded linear operator. Then the following statements hold. $$\|TD^{(n)}\|_{A_\omega^p\to Y}\approx \|T\|_{A_{\omega_{[np]}}^p\to Y},\,\,\, \|TD^{(n)}\|_{e,A_\omega^p\to Y} \approx \|T\|_{e,A_{\omega_{[np]}}^p\to Y}.$$* *Proof.* Let $\eta=\omega_{[np]}$. Since $$\begin{aligned} \label{0429-1} \|TD^{(n)}\|_{A_\omega^p\to Y}=\sup_{f\not\equiv 0}\frac{\|TD^{(n)}f\|_{Y}}{\|f\|_{A_\omega^p}} =\sup_{f^{(n)}\not\equiv 0}\frac{\|TD^{(n)}f\|_{Y}}{\|f\|_{A_\omega^p}},\end{aligned}$$ by Theorem [Theorem 1](#thA){reference-type="ref" reference="thA"} we have $$\|TD^{(n)}\|_{A_\omega^p\to Y}\geq \sup_{f\not\equiv 0 \atop{f(0)=\cdots=f^{(n-1)}(0)=0}}\frac{\|TD^{(n)}f\|_{Y}}{\|f\|_{A_\omega^p}} \approx \sup_{f^{(n)}\not\equiv 0}\frac{\|T f^{(n)}\|_{Y}}{\|f^{(n)}\|_{A_{\eta}^p}}=\|T\|_{A_{\eta}^p\to Y},$$ and $$\|TD^{(n)}\|_{A_\omega^p\to Y}\lesssim \sup_{f^{(n)}\not\equiv 0}\frac{\|T f^{(n)}\|_{Y}}{\|f^{(n)}\|_{A_{\eta}^p}}=\|T\|_{A_{\eta}^p\to Y}.$$ Therefore, $$\begin{aligned} \label{0406-1} \|TD^{(n)}\|_{A_\omega^p\to Y}\approx \|T\|_{A_\eta^p\to Y}.\end{aligned}$$ Suppose $K:A_\omega^p\to Y$ is compact. Let $(If)(z)=\int_0^z f(\xi)d\xi$. By Theorem [Theorem 1](#thA){reference-type="ref" reference="thA"}, $I^n:A_\eta^p\to A_\omega^p$ is bounded. So, $KI^n:A_\eta^p\to Y$ is compact. By ([\[0406-1\]](#0406-1){reference-type="ref" reference="0406-1"}), $$\|T\|_{e,A_\eta^p\to Y}\leq \|T-KI^n\|_{A_\eta^p\to Y}\approx \|TD^{(n)}-KI^nD^{(n)}\|_{A_\omega^p\to Y}.$$ By ([\[0429-1\]](#0429-1){reference-type="ref" reference="0429-1"}) and Theorem [Theorem 1](#thA){reference-type="ref" reference="thA"}, $$\begin{aligned} \|TD^{(n)}-KI^nD^{(n)}\|_{A_\omega^p\to Y}&=\sup_{f^{(n)}\not\equiv 0}\frac{\|(TD^{(n)}-KI^nD^{(n)})f \|_{Y}}{\|f\|_{A_\omega^p}} \\ &\lesssim \sup_{f^{(n)}\not\equiv 0 \atop{f(0)=\cdots=f^{(n-1)}(0)=0}}\frac{\|(TD^{(n)}-KI^nD^{(n)})f\|_{Y}}{\|f\|_{A_\omega^p}} \\ &=\sup_{f^{(n)}\not\equiv 0 \atop{f(0)=\cdots=f^{(n-1)}(0)=0}}\frac{\|(TD^{(n)}-K)f\|_{Y}}{\|f\|_{A_\omega^p}} \\ &\leq \|TD^{(n)}-K\|_{A_\omega^p\to Y}.\end{aligned}$$ Thus, $\|T\|_{e,A_\eta^p\to Y} \lesssim\|TD^{(n)}\|_{e,A_\omega^p\to Y}$. Conversely, suppose $K':A_\eta^p\to Y$ is compact. By Theorem [Theorem 1](#thA){reference-type="ref" reference="thA"}, $D^{(n)}:A_\omega^p\to A_\eta^p$ is bounded. So, $K'D^{(n)}:A_\omega^p\to Y$ is compact. By ([\[0406-1\]](#0406-1){reference-type="ref" reference="0406-1"}), $$\|TD^{(n)}\|_{e,A_\omega^p\to Y}\leq \|TD^{(n)}-K'D^{(n)}\|_{A_\omega^p\to Y} \approx \|T-K'\|_{A_\eta^p\to Y}.$$ Therefore, $\|TD^{(n)}\|_{e,A_\omega^p\to Y} \lesssim \|T\|_{e,A_\eta^p\to Y}$. The proof is complete. ◻ **Remark 6**. *If $1\leq p,q<\infty, \omega,\eta\in\mathcal{D}$, $k\in\mathbb N\cup\{0\}$, for any $u\in H(\mathbb{D})$ and $\varphi\in S(\mathbb{D})$, by Lemma [Lemma 5](#0405-3){reference-type="ref" reference="0405-3"}, we have $$\begin{aligned} \|uD_{\varphi}^{(k)}\|_{A_\omega^p\to A_\eta^q}\approx \|uC_\varphi\|_{A_{\omega_{[kp]}}^p\to A_\eta^q},\,\,\,\, \|uD_{\varphi}^{(k)}\|_{e,A_\omega^p\to A_\eta^q}\approx \|uC_\varphi\|_{e,A_{\omega_{[kp]}}^p\to A_\eta^q}.\end{aligned}$$* The norm and essential norm of $uC_\varphi:A_\omega^p\to A_\mu^q$ were investigated in [@DjLsSy2020ms]. To state them, we need some more notations. When $\omega\in\hat{\mathcal{D}}$ and $0<p<\infty$, if $\gamma>0$ is large enough, let $$\begin{aligned} f_{\lambda,\gamma,\omega,p}(z)=\left(\frac{1-|\lambda|^2}{1-\overline{\lambda}z}\right)^{\gamma}\frac{1}{\omega(S(\lambda))^\frac{1}{p}},\, \,\lambda,z\in\mathbb{D}.\end{aligned}$$ According to [@Pj2015 Lemma 3.1], $\|f_{\lambda,\gamma,\omega,p}\|_{A_\omega^p}\approx 1$. For brevity, we denote $f_{\lambda,\gamma,\omega,p}$ by $f_{\lambda,\gamma}$. When $u\in H(\mathbb{D}),\varphi\in S(\mathbb{D}), 0<q<\infty, \mu\in\hat{\mathcal{D}}$, for any measurable set $E\subset\mathbb{D}$, let $$\nu_{u,\varphi,q,\mu}(E)=\int_{\varphi^{-1}(E)} |u(z)|^q\mu(z)dA(z).$$ Then, $\|uC_\varphi f\|_{A_\mu^q}= \|f\|_{L_{\nu_{u,\varphi,q,\mu}}^q}$. The maximum function of $\nu_{u,\varphi,q,\mu}$ is defined by $$M_\omega(\nu_{u,\varphi,q,\mu})(z)=\sup_{z\in S(a)}\frac{\nu_{u,\varphi,q,\mu}(S(a))}{\omega(S(a))}, \,\,\,a,z\in\mathbb{D}.$$ **Theorem 1**. *Assume $\omega,\mu\in\hat{\mathcal{D}}$, $u\in H(\mathbb{D})$ and $\varphi\in S(\mathbb{D})$.* (i) *When $0<p\leq q<\infty$, the following estimates hold: $$\|uC_\varphi\|_{A_\omega^p\to A_\mu^q}^q \approx \sup_{\lambda\in\mathbb{D}} \int_\mathbb{D}|f_{\lambda,\gamma}(\varphi(z))|^q |u(z)|^q \mu(z) dA(z),$$ and $$\|uC_\varphi\|_{e,A_\omega^p\to A_\mu^q}^q \approx \limsup_{|\lambda|\to 1} \int_\mathbb{D}|f_{\lambda,\gamma}(\varphi(z))|^q |u(z)|^q \mu(z) dA(z).$$* (ii) *When $0<q<p<\infty$, the following statements are equivalent:* 1. *$uC_\varphi:{A_\omega^p\to A_\mu^q}$ is bounded;* 2. *$uC_\varphi:{A_\omega^p\to A_\mu^q}$ is compact;* 3. *$\|M_\omega(\nu_{u,\varphi,q,\mu})\|_{L_\omega^\frac{p}{p-q}}<\infty.$* *Moreover, $$\|uC_\varphi\|^q _{A_\omega^p\to A_\mu^q}\approx \|M_\omega(\nu_{u,\varphi,q,\mu})\|_{L_\omega^\frac{p}{p-q}}.$$* For a positive number $\gamma$ and $j\in\mathbb N$, let $(\gamma)_j=\gamma(\gamma+1)\cdots(\gamma+j-1)$ and $(\gamma)_0=1$. The following lemma is a refinement of a statement in the proof of Theorem 3 in [@AsFt2019caot]. **Lemma 7**. *Suppose $n\in\mathbb N\cup\{0\}$ and $M\geq 1$. There exists a strictly increasing sequence $\{\gamma_k\}_{k=0}^n$ such that $\gamma_0$ is large enough and $$\begin{aligned} \label{0406-3} M+M^2\sum_{k\neq j,{0\leq k\leq n}} \gamma_k^{\frac{1}{2}-k}(\gamma_k)_j<\gamma_j^{\frac{1}{2}-j}(\gamma_j)_j, \,\,\,\,j=0,1,\cdots,n.\end{aligned}$$* *Proof.* Suppose $\gamma_k>n$ for all $k=0,1,\cdots,n$. Since $\gamma_j^\frac{1}{2}\leq \gamma_j^{\frac{1}{2}-j}(\gamma_j)_j$ and $$\begin{aligned} M+M^2\sum_{k\neq j,{0\leq k\leq n}} \gamma_k^{\frac{1}{2}-k}(\gamma_k)_j &=M+M^2\sum_{k=0}^{j-1} \gamma_k^{\frac{1}{2}-k}(\gamma_k)_j +M^2\sum_{k=j+1}^n \gamma_k^{\frac{1}{2}-k}(\gamma_k)_j \\ &\leq M+M^2\sum_{k=0}^{j-1} \gamma_k^{\frac{1}{2}-k}(\gamma_k)_j +M^2\sum_{k=j+1}^n \gamma_k^{\frac{1}{2}-k}(2\gamma_k)^j \\ &\leq M+M^2\sum_{k=0}^{j-1} \gamma_k^{\frac{1}{2}-k}(\gamma_k)_j +n2^nM^2,\end{aligned}$$ it is enough to choose $\{\gamma_k\}$ such that $$\begin{aligned} M+M^2\sum_{k=0}^{j-1} \gamma_k^{\frac{1}{2}-k}(\gamma_k)_j +n2^n M^2 < \gamma_j^{\frac{1}{2}},\,j=0,1,\cdots,n.\end{aligned}$$ When $j=0$, let $\gamma_0>(M+n2^nM^2)^2$. Suppose $\{\gamma_k\}_{k=0}^{j-1}$ is chosen. Then we can choose $$\gamma_{j}>\left(M+M^2\sum_{k=0}^{j-1} \gamma_k^{\frac{1}{2}-k}(\gamma_k)_j +n2^nM^2 \right)^2.$$ By mathematic induction, we get the desired result. The proof is complete. ◻ **Lemma 8**. *[@AsFt2019caot Lemma 3.2] Let $\{\lambda_j\}_{j=0}^n\subset\mathbb{D}$ and $\{z_j\}_{j=0}^n\subset\mathbb{D}$. There exists a constant $C$ depending only on $n$ such that there exists a polynomial $p$ with $\|p\|_{H^\infty}<C$ and $p^{(j)}(\lambda_j)=z_j$ for all $j=0,1,2,\cdots,n$.* As usual, let $\beta(\cdot,\cdot)$ be the Bergman metric, i.e., for all $\xi,\eta\in\mathbb{D}$, $$\beta(\xi,\eta)=\frac{1}{2}\log\frac{1+|\varphi_\xi(\eta)|}{1-|\varphi_\xi(\eta)|},\,\,\mbox{ where }\,\,\varphi_\xi(\eta)=\frac{\xi-\eta}{1-\overline{\xi}\eta}.$$ **Lemma 9**. *[@AsFt2019caot Lemma 3.5] Let $c>1, \varepsilon>0, J,M\in\mathbb N$ and $N\in\mathbb N\cup\{0\}$ be given. Then there is a constant $C$ such that for all $\{z_j\}_{j=1}^J\subset\mathbb{D}$ and $\{w_m\}_{m=1}^M\subset \mathbb{D}$ satisfying $$\beta(z_j,z_1)<\varepsilon, \,\,\,\,\beta(w_m,z_1)>c\varepsilon ,\,\,(1\leq j\leq J,\,\,1\leq m \leq M),$$ there exists a function $f\in H(\mathbb{D})$ satisfying $$\|f\|_{H^\infty}\leq C,\,\,\, f^{(n)}(z_j)=\delta_{0n},\,\,\,\,f^{(n)}(w_m)=0$$ for all $0\leq n\leq N, 1\leq j\leq J, 1\leq m\leq M$.* The following lemma can be obtained by a standard argument, see Lemma 2.10 in [@Tm1996] for example, and we omit its proof here. **Lemma 10**. *Suppose $0<p,q<\infty$, $\omega,\mu\in\hat{\mathcal{D}}$. Let $Y$ be $A_\mu^q$ or $H^\infty$. If $T:A_\omega^p\to Y$ is bounded, then $T$ is compact if and only if $\|T{f_n}\|_Y\to 0$ as $n\to \infty$ whenever $\{f_n\}$ is bounded in $A_\omega^p$ and uniformly converges to 0 on any compact subset of $\mathbb{D}$ as $n\to \infty$.* # proof of Theorem [Theorem 1](#th1){reference-type="ref" reference="th1"} {#proof-of-theorem-th1} *Proof of Therem [Theorem 1](#th1){reference-type="ref" reference="th1"}.* It is obvious that $$\begin{aligned} \|T_{n,\varphi,\vec{u}}\|_{A_\omega^p\to A_\mu^q}\lesssim \sum_{k=0}^n\|u_k D_\varphi^{(k)}\|_{A_\omega^p\to A_\mu^q} \,\,\mbox{ when }\,\, 1\leq p, q<\infty\end{aligned}$$ and $$\begin{aligned} \|T_{n,\varphi,\vec{u}}\|_{e,A_\omega^p\to A_\mu^q}\lesssim \sum_{k=0}^n\|u_k D_\varphi^{(k)}\|_{e,A_\omega^p\to A_\mu^q} \,\,\mbox{ when }\,\, 1\leq p\leq q<\infty.\end{aligned}$$ Next, we only need to prove the inverse of the above inequalities. We first claim that $$\begin{aligned} \label{0405-4} \|u_0D_\varphi^{(0)}\|_{A_\omega^p\to A_\mu^q}\lesssim \|T_{n,\varphi,\vec{u}}\|_{A_\omega^p\to A_\mu^q}\,\,\mbox{ when }\,\, 1\leq p, q<\infty\end{aligned}$$ and $$\begin{aligned} \label{0912-2}\ \|u_0D_\varphi^{(0)}\|_{e,A_\omega^p\to A_\mu^q}\lesssim \|T_{n,\varphi,\vec{u}}\|_{e, A_\omega^p\to A_\mu^q} \,\,\mbox{ when }\,\, 1\leq p\leq q<\infty.\end{aligned}$$ Take these for granted for a moment. Let $\widetilde{T}_{n-1,\varphi,\vec{u}}=\sum_{j=0}^{n-1} u_{j+1}D_\varphi^{(j)}.$ By Lemmas [Lemma 4](#0405-2){reference-type="ref" reference="0405-2"} and [Lemma 5](#0405-3){reference-type="ref" reference="0405-3"}, we have $\omega_{[p]}\in \mathcal{D}$ and $$\|\widetilde{T}_{n-1,\varphi,\vec{u}}\|_{A_{\omega_{[p]}}^p\to A_\mu^q} %\left\|\sum_{j=0}^{n-1} u_{j+1}D_\vp^{(j)}\right\|_{A_{\om_{[p]}}^p\to A_\mu^q} \approx \Big\|\sum_{j=1}^n u_jD_\varphi^{(j)}\Big\|_{A_\omega^p\to A_\mu^q}=\|T_{n,\varphi,\vec{u}}-u_0D_{\varphi}^{(0)}\|_{A_\omega^p\to A_\mu^q}.$$ Then, ([\[0405-4\]](#0405-4){reference-type="ref" reference="0405-4"}) and Triangle Inequality deduce $$\|\widetilde{T}_{n-1,\varphi,\vec{u}} \|_{A_{\omega_{[p]}}^p\to A_\mu^q}\lesssim \|T_{n,\varphi,\vec{u}}\|_{A_\omega^p\to A_\mu^q}.$$ Since $\omega_{[p]}\in \mathcal{D}$, using Lemma [Lemma 5](#0405-3){reference-type="ref" reference="0405-3"} and ([\[0405-4\]](#0405-4){reference-type="ref" reference="0405-4"}) again, we obtain $$\begin{aligned} \|u_1 D_\varphi^{(1)}\|_{A_\omega^p\to A_\mu^q} \approx\|u_1 D^{(0)}_\varphi\|_{A_{\omega_{[p]}}^p\to A_\mu^q} &\lesssim& \left\|\widetilde{T}_{n-1,\varphi,\vec{u}}\right\|_{A_{\omega_{[p]}}^p\to A_\mu^q} \lesssim \|T_{n,\varphi,\vec{u}}\|_{A_\omega^p\to A_\mu^q}.\end{aligned}$$ Then, mathematical induction deduces $$\|u_j D_\varphi^{(j)}\|_{A_\omega^p\to A_\mu^q} \lesssim \|T_{n,\varphi,\vec{u}}\|_{A_\omega^p\to A_\mu^q},\,\,\,\, j=2,3,\cdots,n,$$ and therefore $$\sum_{j=0}^n \|u_j D_\varphi^{(j)}\|_{A_\omega^p\to A_\mu^q} \lesssim \|T_{n,\varphi,\vec{u}}\|_{A_\omega^p\to A_\mu^q}.$$ Similarly, we have $$\sum_{j=0}^n \|u_j D_\varphi^{(j)}\|_{e,A_\omega^p\to A_\mu^q} \lesssim \|T_{n,\varphi,\vec{u}}\|_{e, A_\omega^p\to A_\mu^q}\,\,\mbox{ when }\,\, 1\leq p\leq q<\infty.$$ Here, we omit the proof of the essential norm of $T_{n,\varphi,\vec{u}}:A_\omega^p\to A_\mu^q$ when $q<p$. The above proof of this theorem, Theorem [Theorem 1](#thB){reference-type="ref" reference="thB"}, Lemmas [Lemma 4](#0405-2){reference-type="ref" reference="0405-2"} and [Lemma 5](#0405-3){reference-type="ref" reference="0405-3"} ensure that $T_{n,\varphi,\vec{u}}$ and $u_kD_{\varphi}^{(k)}$ are all compact when $T_{n,\varphi,\vec{u}}$ is bounded. It remains to prove ([\[0405-4\]](#0405-4){reference-type="ref" reference="0405-4"}) and ([\[0912-2\]](#0912-2){reference-type="ref" reference="0912-2"}). To do this, let $T_{n,\varphi,\vec{u}}:A_\omega^p\to A_\mu^q$ be bounded and $\{\gamma_k\}_{k=0}^n$ be those in Lemma [Lemma 7](#0331-1){reference-type="ref" reference="0331-1"} for $M=1$ and large enough. For any $\lambda\in\mathbb{D}$ and $k=0,1,\cdots,n$, let $$f_{\lambda,\gamma_k}(z)=\left(\frac{1-|\lambda|^2}{1-\overline{\lambda}z}\right)^{\gamma_k}\frac{1}{\omega(S(\lambda))^\frac{1}{p}}, ~~~z\in H(\mathbb{D}).$$ Then we have $$\begin{aligned} \label{0326-1} \|T_{n,\varphi,\vec{u}} f_{\lambda,\gamma_k}\|_{A_\mu^q}^q = \int_\mathbb{D}\left|\sum_{j=0}^n \frac{(\gamma_k)_j(\overline{\lambda})^j u_j(z)}{(1-\overline{\lambda}\varphi(z))^j}\right|^q \left|\frac{1-|\lambda|^2}{1-\overline{\lambda}\varphi(z)}\right|^ {q\gamma_k} \frac{1}{\omega(S(\lambda))^{\frac{q}{p}}}\mu(z) dA(z).\end{aligned}$$ Since $\{\gamma_k\}_{k=0}^n$ is increasing, when $k<n$, we obtain $$\begin{aligned} \label{0330-1} \left|\frac{1-|\lambda|^2}{1-\overline{\lambda}\varphi(z)}\right|^ {q\gamma_n} \leq 2^{q(\gamma_n-\gamma_k)} \left|\frac{1-|\lambda|^2}{1-\overline{\lambda}\varphi(z)}\right|^ {q\gamma_k}.\end{aligned}$$ Thus, for all $k=0,1,2,\cdots,n$, $$\begin{aligned} \label{0331-2} \|T_{n,\varphi,\vec{u}} f_{\lambda,\gamma_k}\|_{A_\mu^q}^q \gtrsim \int_\mathbb{D}\left|\sum_{j=0}^n \frac{(\gamma_k)_j(\overline{\lambda})^j u_j(z)}{(1-\overline{\lambda}\varphi(z))^j}\right|^q \left|\frac{1-|\lambda|^2}{1-\overline{\lambda}\varphi(z)}\right|^ {q\gamma_n} \frac{1}{\omega(S(\lambda))^{\frac{q}{p}}}\mu(z) dA(z).\end{aligned}$$ Let $\Delta_{j,k}=\gamma_k^{\frac{1}{2}-k}(\gamma_k)_j, j,k=0,1,\cdots,n$ and $A=(\Delta_{j,k})$. By Ger$\check{\mbox{s}}$gorin's theorem, see [@HrJc1985 Theorem 6.1.1] for example, $|\det(A)|>1$. So, there exists a sequence $\{c_k\}_{k=0}^n$ such that $$\begin{aligned} \label{0331-3} A \left( \begin{array}{c} c_0 \\ c_1\\ \vdots \\ c_n\\ \end{array} \right) = \left( \begin{array}{c} 1 \\ 0\\ \vdots \\ 0 \end{array} \right).\end{aligned}$$ **Case (a). $1\leq p\leq q<\infty.$** Using ([\[0331-3\]](#0331-3){reference-type="ref" reference="0331-3"}), it is easy to check that $$\begin{aligned} \|u_0D_\varphi^{(0)} f_{\lambda,\gamma_n}\|_{A_\mu^q}^q =&\int_\mathbb{D}\left|\left(\sum_{k=0}^n c_k \gamma_k^{\frac{1}{2}-k}(\gamma_k)_0\right) u_0(z)f_{\lambda,\gamma_n}(\varphi(z)) \right|^q \mu(z) dA(z) \nonumber\\ =&\int_\mathbb{D}\left|\sum_{j=0}^n \left( \sum_{k=0}^n c_k \gamma_k^{\frac{1}{2}-k}(\gamma_k)_j\right) \left(\frac{(\overline{\lambda})^j u_j(z)}{(1-\overline{\lambda}\varphi(z))^j} f_{\lambda,\gamma_n}(\varphi(z))\right)\right|^q \mu(z) dA(z) \nonumber\\ \lesssim& \sum_{k=0}^n |c_k|^q \gamma_k^{\frac{q}{2}-kq} \int_\mathbb{D}\left| \sum_{j=0}^n \left( \frac{(\gamma_k)_j(\overline{\lambda})^j u_j(z)}{(1-\overline{\lambda}\varphi(z))^j} f_{\lambda,\gamma_n}(\varphi(z))\right)\right|^q \mu(z) dA(z). \label{0912-1}%\\\end{aligned}$$ Then, ([\[0331-2\]](#0331-2){reference-type="ref" reference="0331-2"}) implies $$\|u_0D_\varphi^{(0)}f_{\lambda,\gamma_n}\|_{A_\mu^q}^q \lesssim \sum_{k=0}^n |c_k|^q \gamma_k^{\frac{q}{2}-kq}\|T_{n,\varphi,\vec{u}} f_{\lambda,\gamma_k}\|_{A_\mu^q}^q \lesssim \left(\sum_{k=0}^n |c_k|^q \gamma_k^{\frac{q}{2}-kq}\right)\|T_{n,\varphi,\vec{u}} \|_{A_\omega^p\to A_\mu^q}^q.$$ By Theorem [Theorem 1](#thB){reference-type="ref" reference="thB"}, we see that $u_0D_\varphi^{(0)}:A_\omega^p\to A_\mu^q$ is bounded and $$\|u_0D_\varphi^{(0)}\|_{A_\omega^p\to A_\mu^q}\lesssim \|T_{n,\varphi,\vec{u}}\|_{A_\omega^p\to A_\mu^q}.$$ If $K:A_\omega^p\to A_\mu^q$ is bounded, similarly to the proof of ([\[0912-1\]](#0912-1){reference-type="ref" reference="0912-1"}), we get $$\begin{aligned} &\|(u_0D_\varphi^{(0)}-K)f_{\lambda,\gamma_n}\|_{A_\mu^q}^q \nonumber\\ =&\int_\mathbb{D}\left|\left(\sum_{k=0}^n c_k \gamma_k^{\frac{1}{2}-k}(\gamma_k)_0\right) \Bigg( u_0(z)f_{\lambda,\gamma_n}(\varphi(z)) -(Kf_{\lambda,\gamma_n})(z)\Bigg)\right|^q \mu(z) dA(z) \nonumber\\ =&\int_\mathbb{D}\left|\sum_{j=0}^n \left( \sum_{k=0}^n c_k \gamma_k^{\frac{1}{2}-k}(\gamma_k)_j\right) \left(\frac{(\overline{\lambda})^j u_j(z)}{(1-\overline{\lambda}\varphi(z))^j} f_{\lambda,\gamma_n}(\varphi(z))-(Kf_{\lambda,\gamma_n})(z)\right)\right|^q \mu(z) dA(z) \nonumber\\ \lesssim& \sum_{k=0}^n |c_k|^q \gamma_k^{\frac{q}{2}-kq} \int_\mathbb{D}\left| \sum_{j=0}^n \left( \frac{(\gamma_k)_j(\overline{\lambda})^j u_j(z)}{(1-\overline{\lambda}\varphi(z))^j} f_{\lambda,\gamma_n}(\varphi(z))-(\gamma_k)_j(Kf_{\lambda,\gamma_n})(z)\right)\right|^q \mu(z) dA(z). \label{0429-2}%\\\end{aligned}$$ Since $$\begin{aligned} &\left| \sum_{j=0}^n \left(\frac{(\gamma_k)_j(\overline{\lambda})^j u_j(z)}{(1-\overline{\lambda}\varphi(z))^j} f_{\lambda,\gamma_n}(\varphi(z))-(r_k)_j(Kf_{\lambda,\gamma_n})(z) \right) \right|^q \\ \lesssim &\left| \sum_{j=0}^n \frac{(\gamma_k)_j(\overline{\lambda})^j u_j(z)}{(1-\overline{\lambda}\varphi(z))^j} f_{\lambda,\gamma_k}(\varphi(z))-(Kf_{\lambda,\gamma_k})(z) \right|^q \left|\frac{1-|\lambda|^2}{1-\overline{\lambda}\varphi(z)}\right|^ {q(\gamma_n-\gamma_k)}\\ &+ \left|\frac{1-|\lambda|^2}{1-\overline{\lambda}\varphi(z)}\right|^ {q(\gamma_n-\gamma_k)} |(Kf_{\lambda,\gamma_k})(z)|^q + \sum_{j=0}^n|(\gamma_k)_j|^q|(Kf_{\lambda,\gamma_n})(z)|^q ,\end{aligned}$$ by ([\[0429-2\]](#0429-2){reference-type="ref" reference="0429-2"}) we have $$\|(u_0D_\varphi^{(0)}-K)f_{\lambda,\gamma_n}\|_{A_\mu^q}^q \lesssim \sum_{k=0}^n |c_k|^q \gamma_k^{\frac{q}{2}-kq} \left(\|(T_{n,\varphi,\vec{u}}-K) f_{\lambda,\gamma_k}\|_{A_\mu^q}^q +\|K f_{\lambda,\gamma_k}\|_{A_\mu^q}^q+\|K f_{\lambda,\gamma_n}\|_{A_\mu^q}^q\right).$$ Then, Triangle Inequality deduces $$\|u_0D_\varphi^{(0)}f_{\lambda,\gamma_n}\|_{A_\mu^q}^q \lesssim \sum_{k=0}^n |c_k|^q \gamma_k^{\frac{q}{2}-kq} \left(\|(T_{n,\varphi,\vec{u}}-K) f_{\lambda,\gamma_k}\|_{A_\mu^q}^q +\|K f_{\lambda,\gamma_k}\|_{A_\mu^q}^q+\|K f_{\lambda,\gamma_n}\|_{A_\mu^q}^q\right).$$ By Lemma [Lemma 10](#0406-2){reference-type="ref" reference="0406-2"}, we obtain $$\begin{aligned} \limsup_{|\lambda|\to 1}\|u_0D_\varphi^{(0)}f_{\lambda,\gamma_n}\|_{A_\mu^q}^q \lesssim \sum_{k=0}^n \limsup_{|\lambda|\to 1} \|(T_{n,\varphi,\vec{u}}-K) f_{\lambda,\gamma_k}\|_{A_\mu^q}^q \lesssim \|T_{n,\varphi,\vec{u}}-K\|_{A_\omega^p\to A_\mu^q}^q.\end{aligned}$$ By Theorem [Theorem 1](#thB){reference-type="ref" reference="thB"} and the arbitrary of $K$, we have $$\|u_0D_\varphi^{(0)}\|_{e, A_\omega^p\to A_\mu^q} \lesssim \|T_{n,\varphi,\vec{u}}\|_{e,A_\omega^p\to A_\mu^q}.$$ **Case (b): $1\leq q<p<\infty.$** Let $\{\lambda_i\}_{i=1}^\infty$ be those $\{\xi_{j,l}^k\}$ in [@PjRjSk2021jga Thoerem 2](also see [@ZxXlFhLj2014amsc Theorem 3.2]) and $r_k(t)$ be Rademacher functions. We have the following statements: 1. For any $a=\{a_i\}_{i=1}^\infty\in l^p$, $\|g_{a,\gamma_k,t}\|_{A_\omega^p}\lesssim \|\{a_i\}_{i=1}^\infty\|_{l^p}$, where $$g_{a,\gamma_k,t}(z)=\sum_{i=1}^\infty a_i r_i(t)f_{\lambda_i,\gamma_k}(z),\,\,k=0,1,\cdots,n.$$ 2. For any $g\in A_\omega^{\frac{p}{2}}$, there exists $\{b_i\}_{i=1}^{\infty}\in l^\frac{p}{2}$ such that $$\begin{aligned} \label{0912-3} g(z)=\sum_{i=1}^\infty b_i\left(\frac{1-|\lambda_i|^2}{1-\overline{\lambda_i}z}\right)^{2\gamma_n}\frac{1}{\omega(S(\lambda_i))^{\frac{2}{p}}}, \,\,\,\,\|g\|_{A_\omega^\frac{p}{2}}\approx \|\{b_i\}\|_{l^\frac{p}{2}}.\end{aligned}$$ By Fubini's theorem, Khinchin's inequality, ([\[0326-1\]](#0326-1){reference-type="ref" reference="0326-1"}) and ([\[0330-1\]](#0330-1){reference-type="ref" reference="0330-1"}), we have $$\begin{aligned} &\int_0^1 \|T_{n,\varphi,\vec{u}} g_{a,\gamma_k,t}\|_{A_\mu^q}^qdt\\ =&\int_\mathbb{D}\int_0^1 \left|\sum_{i=1}^\infty a_i r_i(t) (T_{n,\varphi,\vec{u}} f_{\lambda_i,\gamma_k})(z) \right|^q dt \mu(z) dA(z)\\ \approx& \int_\mathbb{D}\left( \sum_{i=1}^\infty |a_i|^2|(T_{n,\varphi,\vec{u}} f_{\lambda_i,\gamma_k})(z)|^2\right)^\frac{q}{2} \mu(z) dA(z)\\ =& \int_\mathbb{D}\left( \sum_{i=1}^\infty \left| a_i \left(\sum_{j=0}^n \frac{(\gamma_k)_j(\overline{\lambda_i})^j u_j(z)}{(1-\overline{\lambda_i}\varphi(z))^j} \right) \left(\frac{1-|\lambda_i|^2}{1-\overline{\lambda_i}\varphi(z)}\right)^ {\gamma_k} \frac{1}{\omega(S(\lambda_i))^{\frac{1}{p}}} \right|^2 \right)^\frac{q}{2} \mu(z) dA(z)\end{aligned}$$ $$\begin{aligned} \gtrsim& \int_\mathbb{D}\left( \sum_{i=1}^\infty \left| a_i \left(\sum_{j=0}^n \frac{(\gamma_k)_j(\overline{\lambda_i})^j u_j(z)}{(1-\overline{\lambda_i}\varphi(z))^j} \right) \left(\frac{1-|\lambda_i|^2}{1-\overline{\lambda_i}\varphi(z)}\right)^ {\gamma_n} \frac{1}{\omega(S(\lambda_i))^{\frac{1}{p}}} \right|^2 \right)^\frac{q}{2} \mu(z) dA(z) \\ =& \int_\mathbb{D}\left( \sum_{i=1}^\infty |A_{k,i}(z)|^2\right)^\frac{q}{2}\mu(z) dA(z).\end{aligned}$$ Here $$A_{k,i}(z)=a_i \left(\sum_{j=0}^n \frac{(\gamma_k)_j(\overline{\lambda_i})^j u_j(z)}{(1-\overline{\lambda_i}\varphi(z))^j} \right) \left(\frac{1-|\lambda_i|^2}{1-\overline{\lambda_i}\varphi(z)}\right)^ {\gamma_n} \frac{1}{\omega(S(\lambda_i))^{\frac{1}{p}}}.$$ Thus, by the statement (bi), $$\begin{aligned} \int_\mathbb{D}\left( \sum_{i=1}^\infty |A_{k,i}(z)|^2\right)^\frac{q}{2}\mu(z) dA(z) &\lesssim \int_0^1 \|T_{n,\varphi,\vec{u}} g_{a,\gamma_k,t}\|_{A_\mu^q}^qdt\\ &\lesssim \|T_{n,\varphi,\vec{u}}\|_{A_\omega^p\to A_\mu^q}^q\int_0^1 \| g_{a,\gamma_k,t}\|_{A_\omega^p}^qdt\\ & \lesssim \|T_{n,\varphi,\vec{u}}\|_{A_\omega^p\to A_\mu^q}^q \|\{a_i\}\|_{l^p}^q.\end{aligned}$$ Recalling that $\{c_k\}_{k=0}^n$ was decided by ([\[0331-3\]](#0331-3){reference-type="ref" reference="0331-3"}), we have $$\begin{aligned} &\int_\mathbb{D}\left( \sum_{i=1}^\infty \left|a_i \right|^2 \left|(u_0D_\varphi^{(0)} f_{\lambda_i,\gamma_n})(z)\right|^2 \right)^\frac{q}{2} \mu(z) dA(z)\nonumber\\ =& \int_\mathbb{D}\left( \sum_{i=1}^\infty \left|\sum_{k=0}^n c_k \gamma_k^{\frac{1}{2}-k}A_{k,i}(z)\right|^2\right)^\frac{q}{2}\mu(z) dA(z) \nonumber\\ \lesssim& \int_\mathbb{D}\left( \sum_{i=1}^\infty \sum_{k=0}^n |A_{k,i}(z)|^2\right)^\frac{q}{2}\mu(z) dA(z) \nonumber\\ \lesssim& \sum_{k=0}^n\int_\mathbb{D}\left( \sum_{i=1}^\infty |A_{k,i}(z)|^2\right)^\frac{q}{2}\mu(z) dA(z) \nonumber \\ \lesssim& \|T_{n,\varphi,\vec{u}}\|_{A_\omega^p\to A_\mu^q}^q \|\{a_i\}\|_{l^p}^q. \label{0330-2}%\|a\|_{T_2^p(\{\lambda_k\},\om)}^q.\end{aligned}$$ For any $g\in A_\omega^{\frac{p}{2}}$, by the statement (bii), there exists $\{b_i\}_{i=1}^\infty\in l^\frac{p}{2}$ such that ([\[0912-3\]](#0912-3){reference-type="ref" reference="0912-3"}) holds. Let $a_i=b_i^\frac{1}{2}$. So, $\|\{b_i\}\|_{l^\frac{p}{2}}=\|\{a_i\}\|_{l^p}^2$. By ([\[0330-2\]](#0330-2){reference-type="ref" reference="0330-2"}), we get $$\begin{aligned} \|u_0^2 D_\varphi^{(0)} g\|_{A_\mu^\frac{q}{2}}^\frac{q}{2} \leq& \int_\mathbb{D}\left( \sum_{i=1}^\infty \left|a_i\right|^2 \left|(u_0 D_\varphi^{(0)} f_{\lambda_i,\gamma_n})(z)\right|^2 \right)^\frac{q}{2} \mu(z) dA(z)\\ \lesssim& \|T_{n,\varphi,\vec{u}}\|_{A_\omega^p\to A_\mu^q}^q \|\{a_i\}\|_{l^p}^q =\|T_{n,\varphi,\vec{u}}\|_{A_\omega^p\to A_\mu^q}^q \|\{b_i\}\|_{l^\frac{p}{2}}^\frac{q}{2}\\ \approx &\|T_{n,\varphi,\vec{u}}\|_{A_\omega^p\to A_\mu^q}^q \|g\|_{A_\omega^\frac{p}{2}}^\frac{q}{2}.\end{aligned}$$ That is to say, $u_0^2 D_\varphi^{(0)}:A_\omega^\frac{p}{2}\to A_\mu^\frac{q}{2}$ is bounded. By Theorem [Theorem 1](#thB){reference-type="ref" reference="thB"}, $u_0D_\varphi^{(0)}:A_\omega^p\to A_\mu^q$ is bounded and $$\|u_0 D_\varphi^{(0)}\|_{A_\omega^p\to A_\mu^q}\lesssim \|T_{n,\varphi,\vec{u}}\|_{A_\omega^p\to A_\mu^q}.$$ The proof is complete. ◻ # Proof of Theorem [Theorem 3](#th3){reference-type="ref" reference="th3"} {#proof-of-theorem-th3} *Proof of Theorem [Theorem 3](#th3){reference-type="ref" reference="th3"}.* For brief, let $\upsilon(a)=\omega(S(a))^\frac{1}{p}$. By Chapter 1 in [@PjRj2014book], there exist $\alpha,\beta>0$ such that $\frac{\upsilon(t)}{(1-t)^\alpha}$ and $\frac{\upsilon(t)}{(1-t)^\beta}$ are essentially increasing and essentially decreasing on $[0,1)$, respectively. By (4.8) in [@Zhu1], there is a constant $M_0>4$, whenever $\beta(z,w)<\frac{1}{2}$, $$\begin{aligned} \frac{2}{\sqrt{M_0}}<\frac{\upsilon(z)}{\upsilon(w)}<\frac{\sqrt{M_0}}{2}.\end{aligned}$$ By Lemma [Lemma 7](#0331-1){reference-type="ref" reference="0331-1"}, we can choose $\{\gamma_j\}_{j=0}^n$ large enough such that ([\[0406-3\]](#0406-3){reference-type="ref" reference="0406-3"}) holds for $M=M_0$. By Proposition 4.5 in [@Zhu1], there exist $0<\varepsilon<\frac{1}{4}<R<1$ such that whenever $|z|>R$, $\xi,\eta\in D(z,\varepsilon)$ and $0\leq k,j\leq n$, $$\begin{aligned} \label{0516-1} \frac{1}{\sqrt{M_0}}<|\xi|^n, |\eta|^n<1,\end{aligned}$$ $$\begin{aligned} \frac{1}{2}< \frac{(1-|\xi|^2)^{\gamma_j}(1-|\eta|^2)^k}{|1-\overline{\xi}\eta|^{\gamma_j+k}}<2,\end{aligned}$$ and then, $$\begin{aligned} \label{0218-1} \frac{1}{\sqrt{M_0}}< \frac{\upsilon(\eta)}{\upsilon(\xi)} \frac{(1-|\xi|^2)^{\gamma_j}(1-|\eta|^2)^k}{|1-\overline{\xi}\eta|^{\gamma_j+k}}<\sqrt{M_0}.\end{aligned}$$ **Case (a):** $|\lambda_J|>R$ and $\sup\limits_{0\leq j\leq n}\beta(\lambda_j,\lambda_J)<\varepsilon$. Let $f_{\lambda,\gamma}(z)=\left(\frac{1-|\lambda|^2}{1-\overline{\lambda}z}\right)^\gamma\frac{1}{\upsilon(\lambda)}$ and $$f_{\Lambda,J}(z)=\sum_{k=0}^n b_k \gamma_k^{\frac{1}{2}-k}f_{\lambda_k,\gamma_k}(z).$$ Then, the equations ([\[IC\]](#IC){reference-type="ref" reference="IC"}) can be written as $Ab=\delta_J$, in which $$a_{jk}=\gamma_k^{\frac{1}{2}-k}(\gamma_k)_j\overline{\lambda_k}^j \frac{\upsilon(\lambda_j)}{\upsilon(\lambda_k)} \frac{(1-|\lambda_k|^2)^{\gamma_k}(1-|\lambda_j|^2)^j}{(1-\overline{\lambda_k}\lambda_j)^{\gamma_k+j}},$$ and $$b=(b_0,b_1,\cdots,b_n)^T,\,\,\,\,\,\, \delta_J=(\delta_{0J},\,\, \delta_{1J}, \cdots, \,\, \delta_{nJ})^T, \,\,\,\,\, A=(a_{jk}).$$ By ([\[0406-3\]](#0406-3){reference-type="ref" reference="0406-3"}), ([\[0516-1\]](#0516-1){reference-type="ref" reference="0516-1"}) and ([\[0218-1\]](#0218-1){reference-type="ref" reference="0218-1"}), for any $0\leq j\leq n$, $$\begin{aligned} 1+\sum_{k\neq j,0\leq k\leq n}|a_{jk}| < 1+\sqrt{M_0}\sum_{k\neq j,0\leq k\leq n} \gamma_k^{\frac{1}{2}-k} (\gamma_k)_j < \frac{1}{M_0}\gamma_j^{\frac{1}{2}-j}(\gamma_j)_j <|a_{jj}|.\end{aligned}$$ By Ger$\check{\mbox{s}}$gorin's theorem, $|\det(A)|>1$. Meanwhile, since all elements in $A$ are bounded independent of $\{\lambda_j\}_{j=0}^n$, the elements of adjoint matrix $A^*$ of $A$ are also bounded. So, there exists a constant $M_1$ independent of $\{\lambda_j\}_{j=0}^n$ such that $b=A^{-1}\delta_J$ and $\sum_{k=0}^n |b_k|<M_1$. So, $\|f_{\Lambda,J}\|_{A_\omega^p}\lesssim M_1$. For a fixed $J\in\{0,1,\cdots,n\}$ and any given $\delta\in (0,1)$ and $0\leq j\leq n$, set $$E_{j,\delta}=\left\{\lambda_j:\Lambda=\{\lambda_k\}_{k=0}^n\subset \mathbb{D}, |\lambda_J|>\delta, \sup_{0\leq i\leq n}\beta(\lambda_i,\lambda_J)<\varepsilon\right\}.$$ As $\delta$ approaches $1$, $E_{j,\delta}$ approaches the boundary of $\mathbb{D}$. Therefore, the functions $\{f_{\lambda_j,\gamma_j}\}_{\lambda_j\in E_{j,\delta}}$ converge to $0$ uniformly on any compact subset of $\mathbb{D}$. By the arbitrary of $0\leq j\leq n$ and $\sum_{k=0}^n |b_k|<M_1$, the functions $\{f_{\Lambda,J}\}$ converge to $0$ uniformly on any compact subset of $\mathbb{D}$ as $|\lambda_J|$ approaches $1$. **Case (b):** $|\lambda_J|>R$ and $\sup\limits_{0\leq k\leq n}\beta(\lambda_k,\lambda_J)\geq \varepsilon$. Let $\varepsilon^{\prime}=\frac{\varepsilon}{n+1}$. By Pigeonhole Principle, there exists $L\in\{0,1,\cdots,n\}$ such that $$\Big\{\lambda_j:L\varepsilon^{\prime}\leq \beta(\lambda_j,\lambda_J)<(L+1)\varepsilon^{\prime}, j=0,1,2,\cdots,n\Big\}$$ is empty. Set $\Lambda_1=\{z_k\}_{k=0}^n$, where $$z_k=\left\{ \begin{array}{cc} \lambda_k, & \mbox{ if }\,\, \beta(\lambda_k,\lambda_J)<L\varepsilon^{\prime}, \\ \lambda_J, & \mbox{ if }\,\, \beta(\lambda_k,\lambda_J)\geq (L+1)\varepsilon^{\prime}. \end{array} \right.$$ By the proof above, we have a function $f_{\Lambda_1,J}$ such $\|f_{\Lambda_1,J}\|_{A_\omega^p}\lesssim M_1$ and $$\begin{aligned} \label{0517-1} f_{\Lambda_1,J}^{(j)}(z_j)=\frac{\delta_{jJ}}{(1-|z_j|^2)^{j}\upsilon(z_j)},\,\,0\leq j\leq n.\end{aligned}$$ Let $w_1,w_2,\cdots,w_{n^{\prime}}$ be the elements of $\{\lambda_k\}_{k=0}^n \backslash \{z_k\}_{k=0}^n$. By Lemma [Lemma 9](#lm3.5){reference-type="ref" reference="lm3.5"}, there exist a constant $M_2$, independent of $\{\lambda_j\}_{j=0}^n$, $J$, and $h\in H(\mathbb{D})$ such that for all $0\leq k,j\leq n$, $0\leq i\leq n^{\prime}$, we have $$\begin{aligned} \label{0218-3} \|h\|_{H^\infty}<M_2,\,\,\,\, h^{(k)}(w_i)=0, h^{(k)}(z_j)=\delta_{0k}.\end{aligned}$$ Letting $f_{\Lambda,J}=f_{\Lambda_1,J}h$, for all $j=0,1,\cdots,n$, we have $$f_{\Lambda,J}^{(j)}(z)=\sum_{k=0}^j C_j^k f_{\Lambda_1,J}^{(k)}(z)h^{(j-k)}(z).$$ When $\lambda_j=z_j$, by ([\[0517-1\]](#0517-1){reference-type="ref" reference="0517-1"}) and ([\[0218-3\]](#0218-3){reference-type="ref" reference="0218-3"}), we have $$f_{\Lambda,J}^{(j)}(\lambda_j)=f_{\Lambda,J}^{(j)}(z_j)=f_{\Lambda_1,J}^{(j)}(z_j) =\frac{\delta_{jJ}}{(1-|z_j|^2)^{j}\upsilon(z_j)} =\frac{\delta_{jJ}}{(1-|\lambda_j|^2)^{j}\upsilon(\lambda_j)};$$ otherwise, $\lambda_j$ could not be $\lambda_J$ and there exists $w_i$ such that $\lambda_j=w_i$. By ([\[0218-3\]](#0218-3){reference-type="ref" reference="0218-3"}), $$f_{\Lambda,J}^{(j)}(\lambda_j)=f_{\Lambda,J}^{(j)}(w_i)=0 =\frac{\delta_{jJ}}{(1-|\lambda_j|^2)^{j}\upsilon(\lambda_j)}.$$ So, $f_{\Lambda,J}=f_{\Lambda_1,J}h$ is the desired and $\|f_{\Lambda,J}\|_{A_\omega^p}\lesssim M_1 M_2$. Moreover, by the proof of case (a) and $\|h\|_{H^\infty}<M_2$, the functions $\{f_{\Lambda,J}\}$ converge to $0$ uniformly on any compact subset of $\mathbb{D}$ when $|\lambda_J|$ approaches $1$. **Case (c):** $|\lambda_J|\leq R$. By Lemma [Lemma 8](#lm3.2){reference-type="ref" reference="lm3.2"}, there is a constant $M_3$, for all $\{\lambda_j\}_{j=0}^n$ and $0\leq J\leq n$, there is a function $p\in H^\infty$ such that $\|p\|_{H^\infty}<M_3$ and $p^{(j)}(\lambda_j)=\frac{1}{2}\delta_{jJ}$. Then $f(z)=\frac{2p(z)}{(1-|\lambda_J|^2)^J\upsilon(\lambda_J)}$ is the desired. By the above proof, we see that the functions $\{f_{\Lambda,J}\}$ converge to $0$ uniformly on compact subsets of $\mathbb{D}$ as $|\lambda_J|\to 1$. The proof is complete. ◻ # Proof of Theorem [Theorem 2](#th2){reference-type="ref" reference="th2"} {#proof-of-theorem-th2} *Proof of Theorem [Theorem 2](#th2){reference-type="ref" reference="th2"}.* First we consider the norm of $\mathcal{T}_{n,\vec{\varphi},\vec{u}}:A_\omega^p\to H^\infty$. By the assumption and Lemma 3 in [@ZxDj2019mia], we see that for any $f\in A_\omega^p$ and $k=0,1,\cdots,n+1$, $$\begin{aligned} \label{0407-3} |f^{(k)}(z)|\lesssim \frac{\|f\|_{A_\omega^p}}{(1-|z|^2)^k\omega(S(z))^\frac{1}{p}}, ~~z\in\mathbb{D}.\end{aligned}$$ After a calculation, by ([\[0407-3\]](#0407-3){reference-type="ref" reference="0407-3"}) we get $$\begin{aligned} \|u_kD_{\varphi_k}^{(k)}\|_{A_\omega^p\to H^\infty} \lesssim \sup_{z\in\mathbb{D}} \frac{|u_k(z)|}{(1-|\varphi_k(z)|^2)^k\omega(S(\varphi_k(z)))^\frac{1}{p}}.\end{aligned}$$ Therefore, $$\|\mathcal{T}_{n,\vec{\varphi},\vec{u}}\|_{A_\omega^p\to H^\infty}\lesssim \sum_{k=0}^n \sup_{z\in\mathbb{D}} \frac{|u_k(z)|}{(1-|\varphi_k(z)|^2)^k\omega(S(\varphi_k(z)))^\frac{1}{p}}.$$ Conversely, suppose $\mathcal{T}_{n,\vec{\varphi},\vec{u}}:A_\omega^p\to H^\infty$ is bounded. By Theorem [Theorem 3](#th3){reference-type="ref" reference="th3"}, there exists $M'$ such that, for any $\lambda\in\mathbb{D}$ and $J\in\{0,1,\cdots,n\}$, there is a function $f_{\Lambda,J}\in A_\omega^p$ satisfying $\|f_{\Lambda,J}\|_{A_\omega^p}\leq M'$ and $$\begin{aligned} f_{\Lambda,J}^{(j)}(\varphi_j(\lambda))=\frac{\delta_{jJ}}{(1-|\varphi_j(\lambda)|^2)^j\omega(S(\varphi_j(\lambda)))^\frac{1}{p}},\,\,j=0,1,\cdots,n.\end{aligned}$$ Here, $\Lambda=\{\varphi_k(\lambda)\}_{k=0}^n$. Then we have $$\begin{aligned} \Big|\frac{u_J(\lambda)}{(1-|\varphi_J(\lambda)|^2)^J\omega(S(\varphi_J(\lambda)))^{\frac{1}{p}}} \Big| =|(\mathcal{T}_{n,\vec{\varphi},\vec{u}} )f_{\Lambda,J}(\lambda)| \leq \|\mathcal{T}_{n,\vec{\varphi},\vec{u}} f_{\Lambda,J}\|_{H^\infty} \lesssim \|\mathcal{T}_{n,\vec{\varphi},\vec{u}}\|_{A_\omega^p\to H^\infty}.\end{aligned}$$ Therefore, $$\sum_{k=0}^n \sup_{z\in\mathbb{D}} \frac{|u_k(z)|}{(1-|\varphi_k(z)|^2)^k\omega(S(\varphi_k(z)))^\frac{1}{p}} \lesssim \|\mathcal{T}_{n,\vec{\varphi},\vec{u}}\|_{A_\omega^p\to H^\infty},$$ as desired. Next, we estimate the essential norm of $\mathcal{T}_{n,\vec{\varphi},\vec{u}}:A_\omega^p\to H^\infty$. Suppose $\mathcal{T}_{n,\vec{\varphi},\vec{u}}:A_\omega^p\to H^\infty$ is bounded. By the above proof, we see that $$\sup_{z\in\mathbb{D}}|u_k(z)|<\infty, \,\,k=0,1,2,\cdots,n.$$ For any given $r\in[0,1)$, let $(K_r f)(z)=f(rz)$. By Lemma [Lemma 10](#0406-2){reference-type="ref" reference="0406-2"}, it is compact on $A_\omega^p$. So, $u_kD_{\varphi_k}^{(k)}K_r:A_\omega^p\to H^\infty$ is also compact. Let $0<\delta<1$. For $f\in A_\omega^p$, we have $$\begin{aligned} \|u_kD_{\varphi_k}^{(k)} f-u_kD_{\varphi_k}^{(k)}K_r f\|_{H^\infty} &\leq \left(\sup_{|\varphi_k(z)|\leq \delta} +\sup_{\delta<|\varphi_k(z)|<1 }\right)|u_k(z)D_{\varphi_k}^{(k)}f(z)-r^ku_k(z)D_{r\varphi_k}^{(k)}f(z)|\\ &\leq \sup_{|\varphi_k(z)|\leq \delta} |u_k(z)D_{\varphi_k}^{(k)}f(z)-r^ku_k(z)D_{r\varphi_k}^{(k)}f(z)|\\ & ~~~+ \sup_{\delta<|\varphi_k(z)|<1 } |u_k(z)D_{\varphi_k}^{(k)}f(z)-r^ku_k(z)D_{r\varphi_k}^{(k)}f(z)|\\ &:= I+II.\end{aligned}$$ By ([\[0407-3\]](#0407-3){reference-type="ref" reference="0407-3"}), there exists a constant $M$ independent of $f,r,\delta,u_k,\varphi_k$ such that $$\begin{aligned} I\leq& \sup_{|\varphi_k(z)|\leq \delta} (1-r^k)|u_k(z)f^{(k)}(r\varphi_k(z))| +\sup_{|\varphi_k(z)|\leq \delta}|u_k(z)|\left|\int_{r\varphi_k(z)}^{\varphi_k(z)} f^{(k+1)}(\xi)d\xi\right| \\ \leq& (1-r^k+1-r)\sup_{|\varphi_k(z)|\leq \delta} \frac{M|u_k(z)|\|f\|_{A_\omega^p}}{(1-|\varphi_k(z)|^2)^{k+1}\omega(S(\varphi_k(z)))^\frac{1}{p}}\end{aligned}$$ and $$\begin{aligned} II\leq& \sup_{\delta<|\varphi_k(z)|<1 }|u_k(z)f^{(k)}(\varphi_k(z))| + \sup_{\delta<|\varphi_k(z)|<1 }|u_k(z)f^{(k)}(r\varphi_k(z))| \\ \leq& \sup_{\delta<|\varphi_k(z)|<1} \frac{M|u_k(z)|\|f\|_{A_\omega^p}}{(1-|\varphi_k(z)|^2)^k\omega(S(\varphi_k(z)))^\frac{1}{p}}.\end{aligned}$$ So, for any given $\varepsilon>0$, we can choose $r\in(0,1)$ such that $$\begin{aligned} \|u_kD_{\varphi_k}^{(k)} f-u_kD_{\varphi_k}^{(k)}K_r f\|_{H^\infty} \leq \varepsilon \|f\|_{A_\omega^p} +\sup_{\delta<|\varphi_k(z)|<1} \frac{M|u_k(z)|\|f\|_{A_\omega^p}}{(1-|\varphi_k(z)|^2)^k\omega(S(\varphi_k(z)))^\frac{1}{p}}.\end{aligned}$$ Letting $\varepsilon\to 0$ and $\delta\to 1$, we have $$\|u_kD_{\varphi_k}^{(k)}\|_{e,A_\omega^p\to H^\infty} \lesssim \limsup_{|\varphi_k(z)|\to 1} \frac{|u_k(z)|}{(1-|\varphi_k(z)|^2)^k\omega(S(\varphi_k(z)))^\frac{1}{p}}.$$ Therefore, $$\|\mathcal{T}_{n,\vec{\varphi},\vec{u}}\|_{e, A_\omega^p\to H^\infty}\lesssim \sum_{k=0}^\infty \limsup_{|\varphi_k(z)|\to 1} \frac{|u_k(z)|}{(1-|\varphi_k(z)|^2)^k\omega(S(\varphi_k(z)))^\frac{1}{p}}.$$ Finally, we prove that $$\begin{aligned} \sum_{j=0}^n\limsup_{|\varphi_j(\lambda)|\to 1} \frac{|u_j(\lambda)|}{(1-|\varphi_j(\lambda)|^2)^j\omega(S(\varphi_j(\lambda)))^{\frac{1}{p}}} \lesssim \|\mathcal{T}_{n,\vec{\varphi},\vec{u}}\|_{e,A_\omega^p\to H^\infty}.\end{aligned}$$ Suppose $K:A_\omega^p\to H^\infty$ is compact. By Theorem [Theorem 3](#th3){reference-type="ref" reference="th3"}, there exists $M'$ such that, for any $\lambda\in\mathbb{D}$ and $J\in\{0,1,\cdots,n\}$, there is a function $f_{\Lambda,J}\in A_\omega^p$ satisfying $\|f_{\Lambda,J}\|_{A_\omega^p}\leq M'$ and $$\begin{aligned} f_{\Lambda,J}^{(j)}(\varphi_j(\lambda))=\frac{\delta_{jJ}}{(1-|\varphi_j(\lambda)|^2)^j\omega(S(\varphi_j(\lambda)))^\frac{1}{p}},\,\,j=0,1,\cdots,n.\end{aligned}$$ Here, $\Lambda=\{\varphi_k(\lambda)\}_{k=0}^n$. Then we have $$\begin{aligned} \Big|\frac{u_J(\lambda)}{(1-|\varphi_J(\lambda)|^2)^J\omega(S(\varphi_J(\lambda)))^{\frac{1}{p}}} -(Kf_{\Lambda,J})(\lambda)\Big| &=|(\mathcal{T}_{n,\vec{\varphi},\vec{u}}-K)f_{\Lambda,J}(\lambda)|\\ &\leq \|\mathcal{T}_{n,\vec{\varphi},\vec{u}} f_{\Lambda,J}-Kf_{\Lambda,J}\|_{H^\infty}\\ &\lesssim \|\mathcal{T}_{n,\vec{\varphi},\vec{u}}-K\|_{A_\omega^p\to H^\infty}.\end{aligned}$$ By Lemma [Lemma 10](#0406-2){reference-type="ref" reference="0406-2"} and Theorem [Theorem 3](#th3){reference-type="ref" reference="th3"}, $\|Kf_{\Lambda,J}\|_{H^\infty}\to 0$ as $|\varphi_J(\lambda)|\to 1$. Thus, $$\begin{aligned} \limsup_{|\varphi_J(\lambda)|\to 1}\frac{|u_J(\lambda)|}{(1-|\varphi_J(\lambda)|^2)^J\omega(S(\varphi_J(\lambda)))^{\frac{1}{p}}} \lesssim \|\mathcal{T}_{n,\vec{\varphi},\vec{u}}-K\|_{A_\omega^p\to H^\infty}.\end{aligned}$$ Since $K$ and $J$ are arbitrary, we have $$\begin{aligned} \sum_{j=0}^n\limsup_{|\varphi_j(\lambda)|\to 1} \frac{|u_j(\lambda)|}{(1-|\varphi_j(\lambda)|^2)^j\omega(S(\varphi_j(\lambda)))^{\frac{1}{p}}} \lesssim \|\mathcal{T}_{n,\vec{\varphi},\vec{u}}\|_{e,A_\omega^p\to H^\infty}.\end{aligned}$$ The proof is complete. ◻ aa S. Acharyya and T. Ferguson, Sums of weighted differentiation composition operators, *Complex Anal. Oper. Theory,* **13** (2019), 1465--1479. C. Cowen and B. MacCluer, *Composition Operators on Spaces of Analytic Functions*, CRC Press, Boca Raton, 1995. J. Du, S. Li and Y. Shi, Weighted composition operators on weighted Bergman spaces induced by doubling weights, *Math. Scand.*, **126** (2020), 519--539. K. Esmaeili and K. 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Du, Weighted composition operators from weighted Bergman spaces to Stević-type spaces, *Math. Inequal. Appl.*, **22** (2019), 361--376. [^1]: $\dagger$ Corresponding author. [^2]: The work was supported by NNSF of China (Nos. 12371131 and 12271328), Guangdong Basic and Applied Basic Research Foundation (No. 2022A1515012117) and Projects of Talents Recruitment of GDUPT(No. 2022rcyj2008), Project of Science and Technology of Maoming (No. 2023417) and STU Scientific Research Initiation Grant (No. NTF23004).

arxiv_math

--- abstract: | Un pré-feuilletage $\mathscr{F}=\ell\boxtimes\mathcal{F}$ de co-degré $1$ et de degré $d$ sur $\mathbb{P}^{2}_{\mathbb{C}}$ est la donnée d'une droite $\ell$ de $\mathbb{P}^{2}_{\mathbb{C}}$ et d'un feuilletage holomorphe $\mathcal{F}$ sur $\mathbb{P}^{2}_{\mathbb{C}}$ de degré $d-1.$ Nous étudions les pré-feuilletages de co-degré $1$ sur $\mathbb{P}^{2}_{\mathbb{C}}$ ayant une transformée de Legendre (tissu dual) plate. Après avoir établi des résultats généraux sur la platitude du $d$-tissu dual d'un pré-feuilletage homogène de co-degré $1$ et de degré $d$, nous en décrivons quelques exemples explicites et nous montrons qu'à automorphisme de $\mathbb{P}^{2}_{\mathbb{C}}$ près, il y a deux familles et six exemples de pré-feuilletages homogènes de co-degré $1$ et de degré $3$ sur $\mathbb{P}^{2}_{\mathbb{C}}$ ayant cette propriété. Ceci nous permet de prouver un analogue pour les pré-feuilletages de co-degré $1$ et de degré $3$ d'un résultat obtenu en commun avec D. Marı́n sur les feuilletages de degré $3$ à singularités non-dégénérées et de transformée de Legendre plate. Nous montrons par ailleurs que le tissu dual d'un pré-feuilletage convexe réduit de co-degré $1$ sur $\mathbb{P}^{2}_{\mathbb{C}}$ est plat, ce qui constitue un analogue d'un résultat sur les feuilletages de $\mathbb{P}^{2}_{\mathbb{C}}$ dû à D. Marı́n et J. V. Pereira. address: Faculté de Mathématiques, USTHB, BP $32$, El-Alia, $16111$ Bab-Ezzouar, Alger, Algérie author: - "Samir [Bedrouni]{.smallcaps}" title: Pré-feuilletages de co-degré $1$ sur $\mathbb{P}^{2}_{\mathbb{C}}$ ayant une transformée de Legendre plate --- A pre-foliation $\mathscr{F}=\ell\boxtimes\mathcal{F}$ of co-degree $1$ and degree $d$ on $\mathbb{P}^{2}_{\mathbb{C}}$ is the data of a line $\ell$ of $\mathbb{P}^{2}_{\mathbb{C}}$ and a holomorphic foliation $\mathcal{F}$ on $\mathbb{P }^{2}_{\mathbb{C}}$ of degree $d-1.$ We study pre-foliations of co-degree $1$ on $\mathbb{P}^{2}_{\mathbb{ C}}$ with a flat Legendre transform (dual web). After having established some general results on the flatness of the dual $d$-web of a homogeneous pre-foliation of co-degree $1$ and degree $d$, we describe some explicit examples and we show that up to automorphism of $\mathbb{P}^{2}_{\mathbb{C}}$ there are two families and six examples of homogeneous pre-foliations of co-degree $1$ and degree $3$ on $\mathbb {P}^{2}_{\mathbb{C}}$ with a flat dual web. This allows us to prove an analogue for pre-foliations of co-degree $1$ and degree $3$ of a result, obtained in collaboration with D. Marı́n, on foliations of degree $3$ with non-degenerate singularities and a flat Legendre transform. We also show that the dual web of a reduced convex pre-foliation of co-degree $1$ on $\mathbb{P}^{2}_{\mathbb{C}}$ is flat. This is an analogue of a result on foliations of $\mathbb{P}^{2}_{\mathbb{C}}$ due to D. Marı́n and J. V. Pereira. # Introduction {#introduction .unnumbered} Cet article se rattache à une série de travaux en commun avec D. [Marı́n]{.smallcaps} [@BM18Bull; @BM20Bull; @BM21Four; @BM22arxiv] sur les feuilletages holomorphes du plan projectif complexe. Pour les définitions et les notations utilisées (tissu, discriminant $\Delta(\mathcal{W})$, feuilletage homogène, diviseur d'inflexion $\mathrm{I}_{\mathcal{F}}$, singularité radiale, etc.) nous renvoyons à [@BM18Bull Sections 1 et 2]. **Définition 1**. Soient $0\leq k\leq d$ des entiers. Un *pré-feuilletage holomorphe $\mathscr{F}$ sur $\mathbb{P}^{2}_{\mathbb{C}}$ de co-degré $k$ et de degré $d$*, ou simplement *de type $(k,d)$*, est la donnée d'une courbe projective complexe réduite $\mathcal{C}\subset\mathbb{P}^{2}_{\mathbb{C}}$ de degré $k$ et d'un feuilletage holomorphe $\mathcal{F}$ sur $\mathbb{P}^{2}_{\mathbb{C}}$ de degré $d-k.$ On note $\mathscr{F}=\mathcal{C}\boxtimes\mathcal{F}.$ Nous dirons que $\mathcal{C}$ (resp. $\mathcal{F}$) est la *courbe associée* (resp. le *feuilletage associé*) à $\mathscr{F}.$ Un tel objet est alors donné en coordonnées homogènes $[x:y:z]\in\mathbb{P}^{2}_{\mathbb{C}}$ par une $1$-forme du type $\Omega=F(x,y,z)\Omega_0,$ où $\mathbb{C}[x,y,z]_{k}\ni F(x,y,z)=0$ est une équation homogène de la courbe $\mathcal{C}$ et $\Omega_0$ est une $1$-forme homogène de degré $d-k+1$ définissant le feuilletage $\mathcal{F}$, *i.e.* $$\Omega_0=a(x,y,z)\mathrm{d}x+b(x,y,z)\mathrm{d}y+c(x,y,z)\mathrm{d}z,$$ où $a,$ $b$ et $c$ sont des polynômes homogènes de degré $d-k+1$ sans facteur commun satisfaisant la condition d'[Euler]{.smallcaps} $i_{\mathrm{R}}\Omega_0=0$, où $\mathrm{R}=x\frac{\partial{}}{\partial{x}}+y\frac{\partial{}}{\partial{y}}+z\frac{\partial{}}{\partial{z}}$ désigne le champ radial et $i_{\mathrm{R}}$ le produit intérieur par $\mathrm{R}.$ Les pré-feuilletages de type $(0,d)$ sont précisément les feuilletages de degré $d$ sur $\mathbb{P}^{2}_{\mathbb{C}}.$ En vertu de [@MP13], à tout pré-feuilletage $\mathscr{F}=\mathcal{C}\boxtimes\mathcal{F}$ de degré $d\geq1$ et de co-degré $k<d$ sur $\mathbb{P}^{2}_{\mathbb{C}}$ on peut associer un $d$-tissu de degré $1$ sur le plan projectif dual $\mathbb{\check{P}}^{2}_{\mathbb{C}},$ appelé *transformée de [Legendre]{.smallcaps}* (ou tissu dual) de $\mathscr{F}$ et noté $\mathrm{Leg}\mathscr{F}$[\[not:Leg-pref\]]{#not:Leg-pref label="not:Leg-pref"}; si $\mathscr{F}$ est donné dans une carte affine $(x,y)$ de $\mathbb{P}^{2}_{\mathbb{C}}$ par une $1$-forme $\omega=f(x,y)\left(A(x,y)\mathrm{d}x+B(x,y)\mathrm{d}y\right)$ alors, dans la carte affine $(p,q)$ de $\mathbb{\check{P}}^{2}_{\mathbb{C}}$ correspondant à la droite $\{y=px-q\}\subset\mathbb{P}^{2}_{\mathbb{C}},$ $\mathrm{Leg}\mathscr{F}$ est décrit par l'équation différentielle implicite $$F(p,q,x):=f(x,px-q)\left(A(x,px-q)+pB(x,px-q)\right)=0, \qquad \text{avec} \qquad x=\frac{\mathrm{d}q}{\mathrm{d}p}.$$ Lorsque $k\geq1$, $\mathrm{Leg}\mathscr{F}$ se décompose en $\mathrm{Leg}\mathscr{F}=\mathrm{Leg}\mathcal{C}\boxtimes\mathrm{Leg}\mathcal{F}$ où $\mathrm{Leg}\mathcal{C}$ est le $k$-tissu algébrique de $\mathbb{\check{P}}^{2}_{\mathbb{C}}$ défini par l'équation $f(x,px-q)=0$ et $\mathrm{Leg}\mathcal{F}$ est le $(d-k)$-tissu irréductible de degré $1$ de $\mathbb{\check{P}}^{2}_{\mathbb{C}}$ donné par $A(x,px-q)+pB(x,px-q)=0.$ Inversement, tout $d$-tissu décomposable de degré $1$ sur $\mathbb{\check{P}}^{2}_{\mathbb{C}}$ est nécessairement la transformée de [Legendre]{.smallcaps} d'un certain pré-feuilletage sur $\mathbb{P}^{2}_{\mathbb{C}}$ de type $(k,d)$, avec $1\leq k<d.$ Ainsi comprendre la géométrie des $d$-tissus décomposables de degré $1$ sur $\mathbb{\check{P}}^{2}_{\mathbb{C}}$ revient à comprendre la géométrie des pré-feuilletages sur $\mathbb{P}^{2}_{\mathbb{C}}$ de type $(k,d)$, avec $1\leq k<d.$ Nous nous intéressons ici au problème de la platitude des tissus duaux des pré-feuilletages de co-degré $1$, *i.e.* dont la courbe associée est une droite. Dans [@BM18Bull Sections 3, 4 et 5] les auteurs ont étudié la platitude des tissus duaux des feuilletages homogènes de $\mathbb{P}^{2}_{\mathbb{C}}$, puis ils ont montré qu'il est possible de ramener l'étude de la platitude des tissus duaux de certains feuilletages inhomogènes au cadre homogène, *voir* [@BM18Bull Section 6]. Il nous a semblé naturel d'adapter cette démarche au cas des pré-feuilletages de co-degré $1.$ **Définition 2**. Un pré-feuilletage sur $\mathbb{P}^{2}_{\mathbb{C}}$ est dit *homogène* s'il existe une carte affine $(x,y)$ de $\mathbb{P}^{2}_{\mathbb{C}}$ dans laquelle il est invariant sous l'action du groupe des homothéties complexes $(x,y)\longmapsto \lambda(x,y)$, $\lambda\in \mathbb{C}^{*}.$ Un pré-feuilletage homogène $\mathscr{H}$ de type $(1,d)$ sur $\mathbb{P}^{2}_{\mathbb{C}}$ est alors de la forme $\mathscr{H}=\ell\boxtimes\mathcal{H}$, où $\mathcal{H}$ est un feuilletage homogène de degré $d-1$ sur $\mathbb{P}^{2}_{\mathbb{C}}$ et où $\ell$ est une droite passant par l'origine $O$ ou $\ell=L_\infty.$ Le Théorème 3.1 de [@BM18Bull] affirme que le tissu $\mathrm{Leg}\mathcal{H}$ est plat si et seulement si sa courbure est holomorphe sur la partie transverse de son discriminant $\Delta(\mathrm{Leg}\mathcal{H}).$[\[not:Delta-W\]]{#not:Delta-W label="not:Delta-W"} Nous démontrons au paragraphe §[3](#sec:etude-platitude-tissu-dual-pre-feuilletage-homogene){reference-type="ref" reference="sec:etude-platitude-tissu-dual-pre-feuilletage-homogene"} un résultat similaire (Théorème [Théorème 9](#thm:holomorphie-courbure-G(I^tr)-D-ell){reference-type="ref" reference="thm:holomorphie-courbure-G(I^tr)-D-ell"}) pour le tissu $\mathrm{Leg}\mathscr{H}.$ Lorsque $\ell$ passe par l'origine, nous établissons des critères effectifs de l'holomorphie de la courbure de $\mathrm{Leg}\mathscr{H}$ sur certaines composantes irréductibles du discriminant $\Delta(\mathrm{Leg}\mathscr{H})$ (Théorèmes [Théorème 15](#thm:holomorphie-courbure-homogene-D-neq-D-ell){reference-type="ref" reference="thm:holomorphie-courbure-homogene-D-neq-D-ell"} et [Théorème 20](#thm:holomorphie-courbure-homogene-D-ell){reference-type="ref" reference="thm:holomorphie-courbure-homogene-D-ell"}). En fait, les Théorèmes [Théorème 9](#thm:holomorphie-courbure-G(I^tr)-D-ell){reference-type="ref" reference="thm:holomorphie-courbure-G(I^tr)-D-ell"}, [Théorème 15](#thm:holomorphie-courbure-homogene-D-neq-D-ell){reference-type="ref" reference="thm:holomorphie-courbure-homogene-D-neq-D-ell"} et [Théorème 20](#thm:holomorphie-courbure-homogene-D-ell){reference-type="ref" reference="thm:holomorphie-courbure-homogene-D-ell"} fournissent une caractérisation complète de la platitude de $\mathrm{Leg}\mathscr{H}.$ Lorsque $\ell=L_\infty$ nous montrons (Théorème [Théorème 3](#thm:K-Leg-H-egale-K-Leg-L-infini-H){reference-type="ref" reference="thm:K-Leg-H-egale-K-Leg-L-infini-H"}) que les tissus $\mathrm{Leg}\mathcal{H}$ et $\mathrm{Leg}\mathscr{H}$ ont la même courbure; en particulier la platitude de $\mathrm{Leg}\mathscr{H}$ est équivalente à celle de $\mathrm{Leg}\mathcal{H}.$ Plus particulièrement, en degré $d=3$ le tissu $\mathrm{Leg}\mathscr{H}$ est plat (Corollaire [Corollaire 4](#cor:Leg-L-infini-H2-plat){reference-type="ref" reference="cor:Leg-L-infini-H2-plat"}). Rappelons (*voir* [@MP13]) qu'un feuilletage holomorphe sur $\mathbb{P}^{2}_{\mathbb{C}}$ est dit *convexe* si ses feuilles qui ne sont pas des droites n'ont pas de points d'inflexion. Notons (*voir* [@Per01]) que si $\mathcal{F}$ est un feuilletage de degré $d\geq1$ sur $\mathbb{P}^{2}_{\mathbb{C}},$ alors $\mathcal{F}$ ne peut avoir plus de $3d$ droites invariantes (distinctes). De plus, si cette borne est atteinte, alors $\mathcal{F}$ est nécessairement convexe; dans ce cas $\mathcal{F}$ est dit *convexe réduit*. Nous étendons naturellement les notions de convexité et de convexité réduite des feuilletages aux pré-feuilletages en posant: **Définition 3**. Soit $\mathscr{F}=\mathcal{C}\boxtimes\mathcal{F}$ un pré-feuilletage sur $\mathbb{P}^{2}_{\mathbb{C}}.$ Nous dirons que $\mathscr{F}$ est *convexe* (resp. *convexe réduit*) si le feuilletage $\mathcal{F}$ est convexe (resp. convexe réduit) et si de plus la courbe $\mathcal{C}$ est invariante par $\mathcal{F}.$ De cette définition et du Théorème [Théorème 9](#thm:holomorphie-courbure-G(I^tr)-D-ell){reference-type="ref" reference="thm:holomorphie-courbure-G(I^tr)-D-ell"} nous déduirons le corollaire suivant, qui est un analogue du Corollaire 3.4 de [@BM18Bull]. **Corollaire 4** (Corollaire [Corollaire 13](#cor:platitude-pre-feuilletage-homogene-convexe){reference-type="ref" reference="cor:platitude-pre-feuilletage-homogene-convexe"}). **Le tissu dual d'un pré-feuilletage homogène convexe de co-degré $1$ sur $\mathbb{P}^{2}_{\mathbb{C}}$ est plat.** Au §[4](#sec:application-homogene-deg-type-2){reference-type="ref" reference="sec:application-homogene-deg-type-2"} nous donnons une application des résultats du §[3](#sec:etude-platitude-tissu-dual-pre-feuilletage-homogene){reference-type="ref" reference="sec:etude-platitude-tissu-dual-pre-feuilletage-homogene"} aux pré-feuilletages homogènes $\mathscr{H}=\ell\boxtimes\mathcal{H}$ de co-degré $1$ tels que le degré de type de $\mathcal{H}$ soit égal à $2$, *i.e.* $\deg\mathcal{T}_{\mathcal{H}}=2$ (*voir* [@BM18Bull Définition 2.3] pour les définitions du type $\mathcal{T}_{\mathcal{H}}$[\[not:Type-H\]]{#not:Type-H label="not:Type-H"} et du degré de type $\deg\mathcal{T}_{\mathcal{H}}$). Plus précisément, nous décrivons, à automorphisme de $\mathbb{P}^{2}_{\mathbb{C}}$ près, tous les pré-feuilletages homogènes $\mathscr{H}=\ell\boxtimes\mathcal{H}$ de co-degré $1$ et de degré $d\geq3$ tels que $\deg\mathcal{T}_{\mathcal{H}}=2$ et le $d$-tissu $\mathrm{Leg}\mathscr{H}$ soit plat (Proposition [Proposition 33](#pro:class-pre-homogenes-plats-co-degre-1-degre-d-geq-4-degre-Type-2){reference-type="ref" reference="pro:class-pre-homogenes-plats-co-degre-1-degre-d-geq-4-degre-Type-2"}). Nous obtenons en particulier, pour $d=3$, la classification à automorphisme près des pré-feuilletages homogènes de type $(1,3)$ sur $\mathbb{P}^{2}_{\mathbb{C}}$ dont le $3$-tissu dual est plat: *à automorphisme de $\mathbb{P}^{2}_{\mathbb{C}}$ près, il y a deux familles et six exemples de pré-feuilletages homogènes de co-degré $1$ et de degré $3$ sur $\mathbb{P}^{2}_{\mathbb{C}}$ ayant une transformée de [Legendre]{.smallcaps} plate*, *voir* Corollaire [Corollaire 34](#cor:class-pre-homogenes-plats-co-degre-1-degre-3){reference-type="ref" reference="cor:class-pre-homogenes-plats-co-degre-1-degre-3"}. En $2013$ [Marı́n]{.smallcaps} et [Pereira]{.smallcaps} [@MP13 Théorème 4.2] ont montré que le tissu dual d'un feuilletage convexe réduit sur $\mathbb{P}^{2}_{\mathbb{C}}$ est plat. Nous prouvons au §[5](#sec:pre-pre-feuill-convexe-reduit){reference-type="ref" reference="sec:pre-pre-feuill-convexe-reduit"} le résultat analogue suivant pour les pré-feuilletages de co-degré $1.$ **Théorème 5**. **Soit $\mathscr{F}=\ell\boxtimes\mathcal{F}$ un pré-feuilletage convexe réduit de co-degré $1$ et de degré $d\geq3$ sur $\mathbb{P}^{2}_{\mathbb{C}}.$ Alors le $d$-tissu $\mathrm{Leg}\mathscr{F}$ est plat.** Se pose alors le problème suivant: **Problème 1**. * *Soient $\mathcal{F}$ un feuilletage convexe réduit de degré supérieur ou égal à $2$ sur $\mathbb{P}^{2}_{\mathbb{C}}$ et $\ell$ une droite de $\mathbb{P}^{2}_{\mathbb{C}}$ non invariante par $\mathcal{F}.$ Déterminer la position relative de la droite $\ell$ par rapport aux droites invariantes de $\mathcal{F}$ pour que le tissu dual du pré-feuilletage $\ell\boxtimes\mathcal{F}$ soit plat.* * À notre connaissance les seuls feuilletages convexes réduits connus dans la littérature sont ceux qui sont présentés dans [@MP13 Table 1.1]: le feuilletage de [Fermat]{.smallcaps} $\mathcal{F}_{0}^{d-1}$ de degré $d-1$, le feuilletage de [Hesse]{.smallcaps} $\mathcal{F}_{\hspace{-0.4mm}\raisebox{-0.2mm}{\tiny{$H$}}}^{4}$ de degré $4$, le feuilletage modulaire de [Hilbert]{.smallcaps} $\mathcal{F}_{\hspace{-0.4mm}\raisebox{-0.2mm}{\tiny{$H$}}}^{5}$ de degré $5$ et le feuilletage de [Hesse]{.smallcaps} $\mathcal{F}_{\hspace{-0.4mm}\raisebox{-0.2mm}{\tiny{$H$}}}^{7}$ de degré $7$ définis respectivement en carte affine par les $1$-formes $${\mspace{2mu}\overline{\mspace{-1.4mu}\omega\mspace{-1.4mu}}\mspace{2mu}}_{0}^{d-1}=x\mathrm{d}y-y\mathrm{d}x+y^{d-1}\mathrm{d}x-x^{d-1}\mathrm{d}y,$$ $$\omega_{\scalebox{0.64}{\ensuremath H}}^{4}=y(2x^{3}-y^{3}-1)\mathrm{d}x+x(2y^{3}-x^{3}-1)\mathrm{d}y,$$ $$\omega_{\scalebox{0.64}{\ensuremath H}}^{5}=(y^2-1)(y^2-(\sqrt{5}-2)^2)(y+\sqrt{5}x)\mathrm{d}x-(x^2-1)(x^2-(\sqrt{5}-2)^2)(x+\sqrt{5}y)\mathrm{d}y,$$ $$\omega_{\scalebox{0.64}{\ensuremath H}}^{7}=(y^3-1)(y^3+7x^3+1)y\mathrm{d}x-(x^3-1)(x^3+7y^3+1)x\mathrm{d}y. \vspace{1mm}$$ Les deux propositions suivantes, démontrées au §[5](#sec:pre-pre-feuill-convexe-reduit){reference-type="ref" reference="sec:pre-pre-feuill-convexe-reduit"}, donnent une réponse au problème ci-dessus dans le cas du feuilletage de [Fermat]{.smallcaps} $\mathcal{F}_{0}^{d-1}$ et du feuilletage de [Hesse]{.smallcaps} $\mathcal{F}_{\hspace{-0.4mm}\raisebox{-0.2mm}{\tiny{$H$}}}^{4}.$ **Proposition 6**. **Soient $d\geq3$ un entier et $\ell$ une droite de $\mathbb{P}^{2}_{\mathbb{C}}.$ Supposons que $\ell$ ne soit pas invariante par le feuilletage de [Fermat]{.smallcaps} $\mathcal{F}_{0}^{d-1}$ et que le $d$-tissu $\mathrm{Leg}(\ell\boxtimes\mathcal{F}_{0}^{d-1})$ soit plat. Alors $d\in\{3,4\}$ et la droite $\ell$ joint deux (resp. trois) singularités (nécessairement non radiales) de $\mathcal{F}_{0}^{d-1}$ si $d=3$ (resp. si $d=4$).* * Positions relatives de la droite $\ell$ (en bleu) par rapport aux droites invariantes (en rouge) des feuilletages de [Fermat]{.smallcaps} $\mathcal{F}_{0}^{2}$ et $\mathcal{F}_{0}^{3}.$ Le feuilletage $\mathcal{F}_{0}^{2}$ ($d=3$) possède $4$ singularités radiales (points en rouge) et $3$ singularités non radiales (points en bleu) à invariant de [Baum]{.smallcaps}-[Bott]{.smallcaps} $0.$ Le feuilletage $\mathcal{F}_{0}^{3}$ ($d=4$) admet $7$ singularités radiales, $4$ d'ordre un (points en orange) et $3$ d'ordre deux (points en rouge) et $6$ singularités non radiales à invariant de [Baum]{.smallcaps}-[Bott]{.smallcaps} $-\frac{1}{2}.$ **Proposition 7**. **Soit $\ell$ une droite de $\mathbb{P}^{2}_{\mathbb{C}}$ non invariante par le feuilletage de [Hesse]{.smallcaps} $\mathcal{F}_{\hspace{-0.4mm}\raisebox{-0.2mm}{\tiny{$H$}}}^{4}.$ Supposons que le $5$-tissu $\mathrm{Leg}(\ell\boxtimes\mathcal{F}_{\hspace{-0.4mm}\raisebox{-0.2mm}{\tiny{$H$}}}^{4})$ soit plat. Alors la droite $\ell$ passe par quatre singularités (forcément non radiales) de $\mathcal{F}_{\hspace{-0.4mm}\raisebox{-0.2mm}{\tiny{$H$}}}^{4}.$* * L'idée des démonstrations des Propositions [Proposition 6](#proalph:ell-non-invariante-Fermat){reference-type="ref" reference="proalph:ell-non-invariante-Fermat"} et [Proposition 7](#proalph:ell-non-invariante-Hesse){reference-type="ref" reference="proalph:ell-non-invariante-Hesse"} consistera à se ramener au cas homogène, en montrant que les adhérences des $\mathrm{Aut}(\mathbb{P}^{2}_{\mathbb{C}})$-orbites des pré-feuilletages $\ell\boxtimes\mathcal{F}_{0}^{d-1}$ et $\ell\boxtimes\mathcal{F}_{\hspace{-0.4mm}\raisebox{-0.2mm}{\tiny{$H$}}}^{4}$ contiennent des pré-feuilletages homogènes. Le Théorème 6.1 de [@BM18Bull] dit que tout feuilletage de degré $3$ sur $\mathbb{P}^{2}_{\mathbb{C}}$ à singularités non-dégénérées et de transformée de [Legendre]{.smallcaps} plate est linéairement conjugué au feuilletage de [Fermat]{.smallcaps} $\mathcal{F}_{0}^{3}.$ Nous montrons au §[6](#sec:pre-feuilletages-codegre-1-degre-3){reference-type="ref" reference="sec:pre-feuilletages-codegre-1-degre-3"} le résultat similaire suivant pour les pré-feuilletages de co-degré $1$ et de degré $3.$ **Théorème 8**. **Soit $\mathscr{F}=\ell\boxtimes\mathcal{F}$ un pré-feuilletage de co-degré $1$ et de degré $3$ sur $\mathbb{P}^{2}_{\mathbb{C}}.$ Supposons que le feuilletage $\mathcal{F}$ soit à singularités non-dégénérées et que le $3$-tissu $\mathrm{Leg}\mathscr{F}$ soit plat. Alors $\mathcal{F}$ est linéairement conjugué au feuilletage de [Fermat]{.smallcaps} $\mathcal{F}_{0}^{2},$ et la droite $\ell$ est ou bien invariante par $\mathcal{F}$ ou bien joint deux singularités non radiales de $\mathcal{F}.$* * La démonstration de ce théorème utilisera de façon essentielle la classification des pré-feuilletages homogènes de type $(1,3)$ sur $\mathbb{P}^{2}_{\mathbb{C}}$ dont le tissu dual est plat (Corollaire [Corollaire 34](#cor:class-pre-homogenes-plats-co-degre-1-degre-3){reference-type="ref" reference="cor:class-pre-homogenes-plats-co-degre-1-degre-3"}). # Rappels sur la forme fondamentale et la courbure d'un tissu {#subsec:courbure-platitude} Dans ce paragraphe, on rappelle brièvement les définitions de la forme fondamentale et de la courbure d'un $d$-tissu $\mathcal{W}.$ On suppose d'abord que $\mathcal{W}$ est un germe de $d$-tissu de $(\mathbb{C}^{2},0)$ complètement décomposable, $\mathcal{W}=\mathcal{F}_{1}\boxtimes\cdots\boxtimes\mathcal{F}_{d}.$ Pour $i=1,\ldots,d,$ soit $\omega_{i}$ une $1$-forme à singularité isolée en $0$ définissant le feuilletage $\mathcal{F}_{i}.$ D'après [@PP08], pour tout triplet $(r,s,t)$ avec $1\leq r<s<t\leq d,$ on définit $\eta_{rst}=\eta(\mathcal{F}_{r}\boxtimes\mathcal{F}_{s}\boxtimes\mathcal{F}_{t})$ comme l'unique $1$-forme méromorphe satisfaisant les égalités suivantes: $$\label{equa:eta-rst} {\left\{\begin{array}[c]{lll} \mathrm{d}(\delta_{st}\,\omega_{r}) &=& \eta_{rst}\wedge\delta_{st}\,\omega_{r}\\ \mathrm{d}(\delta_{tr}\,\omega_{s}) &=& \eta_{rst}\wedge\delta_{tr}\,\omega_{s}\\ \mathrm{d}(\delta_{rs}\,\omega_{t}) &=& \eta_{rst}\wedge\delta_{rs}\,\omega_{t} \end{array} \right.}$$ où $\delta_{ij}$ désigne la fonction définie par $\omega_{i}\wedge\omega_{j}=\delta_{ij}\,\mathrm{d}x\wedge\mathrm{d}y.$ On appelle *forme fondamentale* du tissu $\mathcal{W}=\mathcal{F}_{1}\boxtimes\cdots\boxtimes\mathcal{F}_{d}$ la $1$-forme[\[not:eta-W\]]{#not:eta-W label="not:eta-W"} $$\label{equa:eta} \hspace{7mm}\eta(\mathcal{W})=\eta(\mathcal{F}_{1}\boxtimes\cdots\boxtimes\mathcal{F}_{d})=\sum_{1\le r<s<t\le d}\eta_{rst}.$$ On vérifie sans peine que $\eta(\mathcal{W})$ est une $1$-forme méromorphe à pôles le long du discriminant $\Delta(\mathcal{W})$ de $\mathcal{W},$ et qu'elle est bien définie à l'addition près d'une $1$-forme fermée logarithmique $\dfrac{\mathrm{d}f}{f}$ avec $f\in\mathcal{O}^*(\mathbb{C}^{2},0)$ (*cf.* [@Rip05; @BM18Bull]). Si maintenant $\mathcal{W}$ est un $d$-tissu sur une surface complexe $M$ (non nécessairement complètement décomposable), alors on peut le transformer en un $d$-tissu complètement décomposable au moyen d'un revêtement galoisien ramifié. L'invariance de la forme fondamentale de ce nouveau tissu par l'action du groupe de [Galois]{.smallcaps} permet de la redescendre en une $1$-forme $\eta(\mathcal{W})$ méromorphe globale sur $M,$ à pôles le long du discriminant de $\mathcal{W}$ (*voir* [@MP13]). La *courbure* du tissu $\mathcal{W}$[\[not:K-W\]]{#not:K-W label="not:K-W"} est par définition la $2$-forme $$\begin{aligned} &K(\mathcal{W})=\mathrm{d}\,\eta(\mathcal{W}).\end{aligned}$$ C'est une $2$-forme méromorphe à pôles le long du discriminant $\Delta(\mathcal{W}),$ canoniquement associée à $\mathcal{W}$; plus précisément, pour toute application holomorphe dominante $\varphi,$ on a $K(\varphi^{*}\mathcal{W})=\varphi^{*}K(\mathcal{W}).$ Un $d$-tissu $\mathcal{W}$ est dit *plat* si sa courbure $K(\mathcal{W})$ est identiquement nulle. Notons enfin qu'un $d$-tissu $\mathcal{W}$ sur $\mathbb{P}^{2}_{\mathbb{C}}$ est plat si et seulement si sa courbure est holomorphe le long des points génériques des composantes irréductibles de $\Delta(\mathcal{W}).$ Ceci résulte de l'holomorphie de $K(\mathcal{W})$ sur $\mathbb{P}^{2}_{\mathbb{C}}\setminus\Delta(\mathcal{W})$ et du fait qu'il n'existe pas de $2$-forme holomorphe sur $\mathbb{P}^{2}_{\mathbb{C}}$ autre que la $2$-forme nulle. # Discriminant du tissu dual d'un pré-feuilletage de co-degré $1$ Rappelons que si $\mathcal{F}$ est un feuilletage sur $\mathbb{P}^{2}_{\mathbb{C}}$, l'application de [Gauss]{.smallcaps} est l'application rationnelle $\mathcal{G}_{\mathcal{F}}\hspace{1mm}\colon\mathbb{P}^{2}_{\mathbb{C}}\dashrightarrow \mathbb{\check{P}}^{2}_{\mathbb{C}}$ définie en tout point régulier $m$ de $\mathcal{F}$ par $\mathcal{G}_{\mathcal{F}}(m)=\mathrm{T}^{\mathbb{P}}_{m}\mathcal{F},$[\[not:Gauss-F\]]{#not:Gauss-F label="not:Gauss-F"} où $\mathrm{T}^{\mathbb{P}}_{m}\mathcal{F}$ désigne la droite tangente à la feuille de $\mathcal{F}$ passant par $m.$ Si $\mathcal{C}\subset\mathbb{P}^{2}_{\mathbb{C}}$ est une courbe passant par certains points singuliers de $\mathcal{F}$, on définit $\mathcal{G}_{\mathcal{F}}(\mathcal{C})$ comme étant l'adhérence de $\mathcal{G}_{\mathcal{F}}(\mathcal{C}\setminus\mathrm{Sing}\mathcal{F}).$[\[not:Sing-F\]]{#not:Sing-F label="not:Sing-F"} **Lemme 1**. **Soit $\mathscr{F}=\ell\boxtimes\mathcal{F}$ un pré-feuilletage de co-degré $1$ sur $\mathbb{P}^{2}_{\mathbb{C}}.$** ****1.*** Si la droite $\ell$ est invariante par $\mathcal{F}$, alors $$\begin{aligned} \Delta(\mathrm{Leg}\mathscr{F})=\Delta(\mathrm{Leg}\mathcal{F})\cup\check{\Sigma}_{\mathcal{F}}^{\ell},\end{aligned}$$ où $\check{\Sigma}_{\mathcal{F}}^{\ell}$ désigne l'ensemble des droites duales des points de $\Sigma_{\mathcal{F}}^{\ell}:=\mathrm{Sing}\mathcal{F}\cap\ell.$* ****2.*** Si la droite $\ell$ n'est pas invariante par $\mathcal{F}$, alors $$\begin{aligned} \Delta(\mathrm{Leg}\mathscr{F})=\Delta(\mathrm{Leg}\mathcal{F})\cup\mathcal{G}_{\mathcal{F}}(\ell).\end{aligned}$$* **Démonstration*.* Nous avons $$\begin{aligned} \Delta(\mathrm{Leg}\mathscr{F})=\Delta(\mathrm{Leg}\mathcal{F})\cup\mathrm{Tang}(\mathrm{Leg}\ell,\mathrm{Leg}\mathcal{F}).\end{aligned}$$ Lorsque $\ell$ n'est pas invariante par $\mathcal{F},$ nous obtenons par un argument de [@Bel14 page 33] que $$\mathrm{Tang}(\mathrm{Leg}\ell,\mathrm{Leg}\mathcal{F})=\mathcal{G}_{\mathcal{F}}(\ell).$$ Supposons que $\ell$ soit invariante par $\mathcal{F}$ et montrons que $\mathrm{Tang}(\mathrm{Leg}\ell,\mathrm{Leg}\mathcal{F})=\check{\Sigma}_{\mathcal{F}}^{\ell}.$ Soit $s\in\Sigma_{\mathcal{F}}^{\ell}$; le fait que $s\in\ell$ (resp. $s\in\mathrm{Sing}\mathcal{F}$) implique que la droite $\check{s}$ duale de $s$ est invariante par $\mathrm{Leg}\ell$ (resp. par $\mathrm{Leg}\mathcal{F}$). Ainsi $\check{s}\subset\mathrm{Tang}(\mathrm{Leg}\ell,\mathrm{Leg}\mathcal{F})$, d'où $\check{\Sigma}_{\mathcal{F}}^{\ell}\subset\mathrm{Tang}(\mathrm{Leg}\ell,\mathrm{Leg}\mathcal{F}).$ Réciproquement, soit $\mathcal{C}$ une composante irréductible de $\mathrm{Tang}(\mathrm{Leg}\ell,\mathrm{Leg}\mathcal{F}).$ Montrons que $\mathcal{C}$ est invariante par $\mathrm{Leg}\mathcal{F}$; supposons par l'absurde que $\mathcal{C}$ soit transverse à $\mathrm{Leg}\mathcal{F}.$ Soit $m$ un point générique de $\mathcal{C}.$ Désignons par $\check{\ell}\in\mathbb{\check{P}}^{2}_{\mathbb{C}}$ le point dual de $\ell$; alors la droite $(\check{\ell}m)$ n'est pas invariante par $\mathrm{Leg}\mathcal{F}$ et est tangente à $\mathrm{Leg}\mathcal{F}$ en $m.$ Comme $\ell$ est $\mathcal{F}$-invariante, le point $\check{\ell}$ est singulier pour $\mathrm{Leg}\mathcal{F}$; c'est donc aussi un point de tangence entre $\mathrm{Leg}\mathcal{F}$ et $(\check{\ell}m).$ Le nombre de points de tangence entre $\mathrm{Leg}\mathcal{F}$ et $(\check{\ell}m)$ est alors $\geq2$, ce qui contredit l'égalité $\deg(\mathrm{Leg}\mathcal{F})=1,$ d'où l'invariance de $\mathcal{C}$ par $\mathrm{Leg}\mathcal{F}.$ Alors $\mathcal{C}$ est aussi invariante par $\mathrm{Leg}\ell$ et est donc une droite passant par $\check{\ell}.$ Il existe donc $s\in\mathrm{Sing}\mathcal{F}$ tel que $\mathcal{C}=\check{s}$; comme $\check{\ell}\in\mathcal{C}$, $s\in \ell$ et donc $s\in\Sigma_{\mathcal{F}}^{\ell}.$ Par conséquent $\mathcal{C}\subset\check{\Sigma}_{\mathcal{F}}^{\ell}.$ ◻ Nous allons maintenant appliquer le lemme précédent au cas d'un pré-feuilletage homogène $\mathscr{H}=\ell\boxtimes\mathcal{H}$ de co-degré $1$ sur $\mathbb{P}^{2}_{\mathbb{C}}.$ Si $\deg\mathscr{H}=d$, le feuilletage homogène $\mathcal{H}$ est donné, pour un bon choix de coordonnées affines $(x,y),$ par une $1$-forme homogène $$\hspace{1mm}\omega=A(x,y)\mathrm{d}x+B(x,y)\mathrm{d}y,\quad \text{où}\hspace{1.5mm}A,B\in\mathbb{C}[x,y]_{d-1}\hspace{1.5mm}\text{et}\hspace{1.5mm}\mathrm{pgcd}(A,B)=1.$$ Si $\ell=L_\infty$ alors $\ell$ est invariante par $\mathcal{H}$ et le Lemme [Lemme 1](#lem:Delta-Leg-ell-F){reference-type="ref" reference="lem:Delta-Leg-ell-F"} assure que $$\Delta(\mathrm{Leg}\mathscr{H})=\Delta(\mathrm{Leg}\mathcal{H})\cup\check{\Sigma}_{\mathcal{H}}^{\infty},$$ où $\check{\Sigma}_{\mathcal{H}}^{\infty}$ désigne l'ensemble des droites duales des points de $\Sigma_{\mathcal{H}}^{\infty}:=\mathrm{Sing}\mathcal{H}\cap L_\infty.$ Supposons que $\ell$ passe par l'origine. Si $\ell$ n'est pas invariante par $\mathcal{H}$, alors (Lemme [Lemme 1](#lem:Delta-Leg-ell-F){reference-type="ref" reference="lem:Delta-Leg-ell-F"}) $$\Delta(\mathrm{Leg}\mathscr{H})=\Delta(\mathrm{Leg}\mathcal{H})\cup\mathcal{G}_{\mathcal{H}}(\ell).$$ Si $\ell$ est invariante par $\mathcal{H}$, alors le point $s:=L_\infty\cap\ell$ est singulier pour $\mathcal{H}$ et, d'après [@BM18Bull Proposition 2.2], nous avons $\Sigma_{\mathcal{H}}^{\ell}=\{O,s\}$; en notant $\check{O}$ et $\check{s}$ les droites duales des points $O$ et $s$ respectivement, le Lemme [Lemme 1](#lem:Delta-Leg-ell-F){reference-type="ref" reference="lem:Delta-Leg-ell-F"} implique donc que $$\Delta(\mathrm{Leg}\mathscr{H})=\Delta(\mathrm{Leg}\mathcal{H})\cup\check{O}\cup\check{s}=\Delta(\mathrm{Leg}\mathcal{H})\cup\check{s},$$ car $\check{O}\subset\Delta(\mathrm{Leg}\mathcal{H}).$ En fait, d'après [@BM18Bull Lemme 3.2], le discriminant de $\mathrm{Leg}\mathcal{H}$ se décompose en $$\Delta(\mathrm{Leg}\mathcal{H})=\mathcal{G}_{\mathcal{H}}(\mathrm{I}_{\mathcal{H}}^{\hspace{0.2mm}\mathrm{tr}})\cup\check{\Sigma}_{\mathcal{H}}^{\mathrm{rad}}\cup\check{O},$$ où $\mathrm{I}_{\mathcal{H}}^{\hspace{0.2mm}\mathrm{tr}}$[\[not:Inflex-Transverse-F\]]{#not:Inflex-Transverse-F label="not:Inflex-Transverse-F"} désigne le diviseur d'inflexion transverse de $\mathcal{H}$ et $\check{\Sigma}_{\mathcal{H}}^{\mathrm{rad}}$ l'ensemble des droites duales des singularités radiales de $\mathcal{H}$ (*voir* [@BM18Bull §1.3] pour les définitions précises de ces notions). Rappelons cependant qu'au feuilletage homogène $\mathcal{H}$ on peut aussi associer l'application rationnelle ${\mspace{2mu}\underline{\mspace{-2mu}\mathcal{G}\mspace{-2mu}}\mspace{2mu}}_{\mathcal{H}}\hspace{1mm}\colon\mathbb{P}^{1}_{\mathbb{C}}\rightarrow \mathbb{P}^{1}_{\mathbb{C}}$ définie par $${\mspace{2mu}\underline{\mspace{-2mu}\mathcal{G}\mspace{-2mu}}\mspace{2mu}}_{\mathcal{H}}([y:x])=[-A(x,y):B(x,y)],$$ et que celle-ci permet de déterminer complètement le diviseur $\mathrm{I}_{\mathcal{H}}^{\hspace{0.2mm}\mathrm{tr}}$ et l'ensemble $\Sigma_{\mathcal{H}}^{\mathrm{rad}}$ (*voir* [@BM18Bull Section 2]): - $\Sigma_{\mathcal{H}}^{\mathrm{rad}}$ est formé des $[b:a:0]\in L_{\infty}$ tels que $[a:b]\in\mathbb{P}^{1}_{\mathbb{C}}$ soit un point critique fixe de ${\mspace{2mu}\underline{\mspace{-2mu}\mathcal{G}\mspace{-2mu}}\mspace{2mu}}_{\mathcal{H}}$; - $\mathrm{I}_{\mathcal{H}}^{\hspace{0.2mm}\mathrm{tr}}=\prod\limits_{i}T_{i}^{n_i}$, où $T_{i}=(b_i\hspace{0.2mm}y-a_i\hspace{0.2mm}x=0)$ et $[a_i:b_i]\in\mathbb{P}^{1}_{\mathbb{C}}$ est un point critique non fixe de ${\mspace{2mu}\underline{\mspace{-2mu}\mathcal{G}\mspace{-2mu}}\mspace{2mu}}_{\mathcal{H}}$ de multiplicité $n_i.$ Des considérations précédentes nous tirons l'énoncé suivant. **Lemme 2**. **Soit $\mathscr{H}=\ell\boxtimes\mathcal{H}$ un pré-feuilletage homogène de co-degré $1$ sur $\mathbb{P}^{2}_{\mathbb{C}}.$** ****1.*** Si $\ell=L_\infty$ alors $$\begin{aligned} \Delta(\mathrm{Leg}\mathscr{H})=\Delta(\mathrm{Leg}\mathcal{H})\cup\check{\Sigma}_{\mathcal{H}}^{\infty}=\mathcal{G}_{\mathcal{H}}(\mathrm{I}_{\mathcal{H}}^{\hspace{0.2mm}\mathrm{tr}})\cup\check{\Sigma}_{\mathcal{H}}^{\infty}\cup\check{O}.\end{aligned}$$* ****2.*** Si la droite $\ell$ passe par l'origine, alors $$\Delta(\mathrm{Leg}\mathscr{H})=\Delta(\mathrm{Leg}\mathcal{H})\cup D_\ell=\mathcal{G}_{\mathcal{H}}(\mathrm{I}_{\mathcal{H}}^{\hspace{0.2mm}\mathrm{tr}})\cup\check{\Sigma}_{\mathcal{H}}^{\mathrm{rad}}\cup\check{O}\cup D_\ell,$$ où la composante $D_\ell$ est définie comme suit. Si $\ell$ est invariante par $\mathcal{H}$, alors $D_\ell:=\check{s}$ est la droite duale du point $s=L_\infty\cap\ell\in\mathrm{Sing}\mathcal{H}.$ Si $\ell$ n'est pas invariante par $\mathcal{H},$ alors $D_\ell:=\mathcal{G}_{\mathcal{H}}(\ell).$* # Étude de la platitude du tissu dual d'un pré-feuilletage homogène de co-degré $1$ {#sec:etude-platitude-tissu-dual-pre-feuilletage-homogene} Notre premier résultat montre que, pour un feuilletage homogène $\mathcal{H}$ sur $\mathbb{P}^{2}_{\mathbb{C}},$ les tissus $\mathrm{Leg}\mathcal{H}$ et $\mathrm{Leg}(L_\infty\boxtimes\mathcal{H})$ ont la même courbure, de sorte que l'on a équivalence entre la platitude de $\mathrm{Leg}\mathcal{H}$ et celle de $\mathrm{Leg}(L_\infty\boxtimes\mathcal{H}).$ **Théorème 3**. * *Soient $d\geq3$ un entier et $\mathcal{H}$ un feuilletage homogène de degré $d-1$ sur $\mathbb{P}^{2}_{\mathbb{C}}.$ Alors $$K(\mathrm{Leg}(L_\infty\boxtimes\mathcal{H}))=K(\mathrm{Leg}\mathcal{H}).$$ En particulier, le $d$-tissu $\mathrm{Leg}(L_\infty\boxtimes\mathcal{H})$ est plat si et seulement si le $(d-1)$-tissu $\mathrm{Leg}\mathcal{H}$ est plat.* * **Corollaire 4**. **Soit $\mathcal{H}$ un feuilletage homogène de degré $2$ sur $\mathbb{P}^{2}_{\mathbb{C}}.$ Alors le $3$-tissu $\mathrm{Leg}(L_\infty\boxtimes\mathcal{H})$ est plat.** Pour démontrer le Théorème [Théorème 3](#thm:K-Leg-H-egale-K-Leg-L-infini-H){reference-type="ref" reference="thm:K-Leg-H-egale-K-Leg-L-infini-H"}, nous aurons besoin de la définition et du théorème suivants. **Définition 5** ([@MPP06]). Soit $\mathcal{W}=\mathcal{F}_1\boxtimes\cdots\boxtimes\mathcal{F}_d$ un $d$-tissu régulier de $(\mathbb{C}^2,0).$ Une *symétrie transverse* de $\mathcal{W}$ est la donnée d'un germe de champ de vecteurs $\mathrm{X}$ transverse aux feuilletages $\mathcal{F}_i$ ($i=1,\ldots,d$) et dont le flot local $\exp(t\mathrm{X})$ préserve les $\mathcal{F}_i.$ **Théorème 6**. **Soient $d\geq3$ un entier et $\mathcal{W}_{d-1}$ un $(d-1)$-tissu régulier de $(\mathbb{C}^2,0)$ admettant une symétrie transverse $\mathrm{X}$; notons $\mathcal{F}_\mathrm{X}$ le feuilletage décrit par $\mathrm{X}.$ Alors $$K(\mathcal{F}_\mathrm{X}\boxtimes\mathcal{W}_{d-1})=K(\mathcal{W}_{d-1}).$$ En particulier, le $d$-tissu $\mathcal{F}_\mathrm{X}\boxtimes\mathcal{W}_{d-1}$ est plat si et seulement si le $(d-1)$-tissu $\mathcal{W}_{d-1}$ est plat.* * Avant de prouver ce théorème, rappelons brièvement la définition du rang $\mathrm{rg}(\mathcal{W})$ d'un $d$-tissu régulier $\mathcal{W}=\mathcal{F}_1\boxtimes\cdots\boxtimes\mathcal{F}_d$ de $(\mathbb{C}^2,0).$ Pour $1\leq i\leq d,$ soit $\omega_i$ une $1$-forme définissant le feuilletage $\mathcal{F}_i$; on définit le $\mathbb{C}$-espace vectoriel $\mathcal{A}(\mathcal{W})$ des *relations abéliennes* de $\mathcal{W}$ par $$\begin{aligned} \mathcal{A}(\mathcal{W}):=\Big\{(\eta_1,\ldots,\eta_d)\in(\Omega^1(\mathbb{C}^2,0))^d \hspace{1mm}\Big|\hspace{1mm} \forall i=1,\ldots,d,\hspace{1mm} \mathrm{d}\eta_i=0,\hspace{1mm} \eta_i\wedge\omega_i=0 \hspace{1mm}\text{ et }\hspace{1mm} \sum_{i=1}^d\eta_i=0\Big\}.\end{aligned}$$ Alors $\mathrm{rg}(\mathcal{W}):=\dim_{\mathbb{C}}\mathcal{A}(\mathcal{W})$;[\[not:rg-W\]]{#not:rg-W label="not:rg-W"} on dispose de la majoration optimale suivante (*cf.* [@PP15 Chapitre 2]): $$\mathrm{rg}(\mathcal{W})\leq\pi_d:=\frac{(d-1)(d-2)}{2}.$$ Rappelons aussi que tout $d$-tissu de rang maximal (*i.e.* de rang $\pi_d$) est nécessairement plat par le critère de [Mihăileanu]{.smallcaps} (*cf.* [@PP15 Théorème 6.3.4]). Le Théorème [Théorème 6](#thm:K-FX-W-egale-K-W){reference-type="ref" reference="thm:K-FX-W-egale-K-W"} est en fait un analogue pour les tissus plats d'un résultat sur les tissus de rang maximal, dû à [Marı́n]{.smallcaps}-[Pereira]{.smallcaps}-[Pirio]{.smallcaps}, à savoir: **Théorème 7** ([@MPP06], Théorème 1). **Avec les notations du Théorème [Théorème 6](#thm:K-FX-W-egale-K-W){reference-type="ref" reference="thm:K-FX-W-egale-K-W"}, on a $$\mathrm{rg}(\mathcal{F}_\mathrm{X}\boxtimes\mathcal{W}_{d-1})=\mathrm{rg}(\mathcal{W}_{d-1})+(d-2).$$ En particulier, $\mathcal{F}_\mathrm{X}\boxtimes\mathcal{W}_{d-1}$ est de rang maximal si et seulement si $\mathcal{W}_{d-1}$ est de rang maximal.* * La preuve du Théorème [Théorème 6](#thm:K-FX-W-egale-K-W){reference-type="ref" reference="thm:K-FX-W-egale-K-W"} consiste essentiellement en l'application de ce résultat pour $d=3.$ **Démonstration du Théorème [Théorème 6](#thm:K-FX-W-egale-K-W){reference-type="ref" reference="thm:K-FX-W-egale-K-W"}*.* En écrivant $\mathcal{W}_{d-1}=\mathcal{F}_1\boxtimes\cdots\boxtimes\mathcal{F}_{d-1}$, on a $$\begin{aligned} K(\mathcal{F}_\mathrm{X}\boxtimes\mathcal{W}_{d-1})=K(\mathcal{W}_{d-1})+\sum\limits_{1\leq i<j\leq d-1}K(\mathcal{W}_{3}^{i,j}),\end{aligned}$$ où $\mathcal{W}_{3}^{i,j}:=\mathcal{F}_\mathrm{X}\boxtimes\mathcal{F}_i\boxtimes\mathcal{F}_j.$ Par ailleurs, comme $\mathrm{X}$ est une symétrie transverse du $2$-tissu $\mathcal{F}_i\boxtimes\mathcal{F}_j$ et comme tout $2$-tissu est de rang maximal, égal à $0$, le Théorème 1 de [@MPP06] (*cf.* Théorème [Théorème 7](#thm:Marin-Pereira-Pirio){reference-type="ref" reference="thm:Marin-Pereira-Pirio"} ci-dessus) implique que le $3$-tissu $\mathcal{W}_{3}^{i,j}$ est de rang maximal, égal à $1,$ de sorte que $K(\mathcal{W}_{3}^{i,j})\equiv0$, d'où l'égalité annoncée. ◻ **Démonstration du Théorème [Théorème 3](#thm:K-Leg-H-egale-K-Leg-L-infini-H){reference-type="ref" reference="thm:K-Leg-H-egale-K-Leg-L-infini-H"}*.* En vertu de [@BM18Bull Section 2], on peut décomposer localement le $d$-tissu $\mathrm{Leg}(L_\infty\boxtimes\mathcal{H})$ sous la forme $\mathrm{Leg}(L_\infty\boxtimes\mathcal{H})=\mathrm{Leg}(L_\infty)\boxtimes\mathcal{W}_{d-1},$ où $\mathcal{W}_{d-1}=\mathcal{F}_1\boxtimes\cdots\boxtimes\mathcal{F}_{d-1}$ et, pour tout $i\in\{1,\ldots,d-1\},$ $\mathcal{F}_i$ est donné par $\check{\omega}_i:=\lambda_{i}(p)\mathrm{d}q-q\mathrm{d}p$, avec $\lambda_{i}(p)=p-p_{i}(p)$ et $\{p_{i}(p)\}={\mspace{2mu}\underline{\mspace{-2mu}\mathcal{G}\mspace{-2mu}}\mspace{2mu}}_{\mathcal{H}}^{-1}(p).$ Or, le champ $\mathrm{X}:=q\frac{\partial }{\partial q}$ définit le feuilletage radial $\mathrm{Leg}(L_\infty)$ et est une symétrie transverse du tissu $\mathcal{W}_{d-1}.$ Donc, $K(\mathrm{Leg}(L_\infty\boxtimes\mathcal{H}))=K(\mathrm{Leg}\mathcal{H})$ par le Théorème [Théorème 6](#thm:K-FX-W-egale-K-W){reference-type="ref" reference="thm:K-FX-W-egale-K-W"}. ◻ *Remarque 8*. On peut aussi raisonner directement, sans faire appel à des résultats sur les tissus de rang maximal. En effet, en posant, pour tous $i,j\in\{1,\ldots,d-1\}$ avec $i\neq j,$ $\mathcal{W}_{3}^{i,j}:=\mathrm{Leg}(L_\infty)\boxtimes\mathcal{F}_i\boxtimes\mathcal{F}_j$, on a $$\begin{aligned} K(\mathrm{Leg}(L_\infty\boxtimes\mathcal{H}))=K(\mathrm{Leg}\mathcal{H})+\sum\limits_{1\leq i<j\leq d-1}K(\mathcal{W}_{3}^{i,j}).\end{aligned}$$ Le feuilletage $\mathrm{Leg}(L_\infty)$ étant décrit par $\check{\omega}_0:=\mathrm{d}p,$ un calcul direct utilisant la formule ([\[equa:eta-rst\]](#equa:eta-rst){reference-type="ref" reference="equa:eta-rst"}) montre que $$\begin{aligned} \eta(\mathcal{W}_{3}^{i,j})=\frac{\mathrm{d}\Big((\lambda_{i}\lambda_{j})(p)\Big)}{(\lambda_{i}\lambda_{j})(p)}+\frac{\mathrm{d}q}{q},\end{aligned}$$ de sorte que $K(\mathcal{W}_{3}^{i,j})=\mathrm{d}\eta(\mathcal{W}_{3}^{i,j})\equiv0,$ d'où $K(\mathrm{Leg}(L_\infty\boxtimes\mathcal{H}))=K(\mathrm{Leg}\mathcal{H}).$ Le théorème suivant donne une caractérisation importante de la platitude du tissu dual d'un pré-feuilletage homogène de co-degré $1.$ **Théorème 9**. **Soit $\mathscr{H}=\ell\boxtimes\mathcal{H}$ un pré-feuilletage homogène de type $(1,d)$ sur $\mathbb{P}^{2}_{\mathbb{C}}$ avec $d\geq3.$ Si la droite $\ell$ est invariante (resp. non invariante) par $\mathcal{H}$ alors le $d$-tissu $\mathrm{Leg}\mathscr{H}$ est plat si et seulement si sa courbure $K(\mathrm{Leg}\mathscr{H})$ est holomorphe sur $\mathcal{G}_{\mathcal{H}}(\mathrm{I}_{\mathcal{H}}^{\hspace{0.2mm}\mathrm{tr}})$ (resp. sur $\mathcal{G}_{\mathcal{H}}(\mathrm{I}_{\mathcal{H}}^{\hspace{0.2mm}\mathrm{tr}})\cup D_{\ell}=\mathcal{G}_{\mathcal{H}}(\mathrm{I}_{\mathcal{H}}^{\hspace{0.2mm}\mathrm{tr}}\cup\ell)$).* * Pour prouver ce théorème, nous aurons besoin du lemme suivant, qui est une reformulation du Lemme 3.1 de [@BFM14] en termes de pré-feuilletages homogènes. **Lemme 10** ([@BFM14], Lemme 3.1). * *Soit $\mathscr{H}$ un pré-feuilletage homogène sur $\mathbb{P}^{2}_{\mathbb{C}}.$ Si la courbure de $\mathrm{Leg}\mathscr{H}$ est holomorphe sur $\mathbb{\check{P}}^{2}_{\mathbb{C}}\hspace{-0.3mm}\setminus\hspace{-0.3mm} \check{O},$ alors $\mathrm{Leg}\mathscr{H}$ est plat.* * Nous aurons aussi besoin de la proposition suivante, qui a son intérêt propre. **Proposition 11**. **Soit $\mathcal{W}_{\nu}$ un germe de $\nu$-tissu de $(\mathbb{C}^{2},0),\,\nu\geq2.$ Supposons que $\Delta(\mathcal{W}_{\nu})$ possède une composante irréductible $C$ totalement invariante par $\mathcal{W}_{\nu}$ et de multiplicité minimale $\nu-1.$ Soit $\mathcal{F}$ un germe de feuilletage de $(\mathbb{C}^{2},0)$ laissant $C$ invariante et soit $\mathcal{W}_{d-\nu-1}$ un germe de $(d-\nu-1)$-tissu régulier de $(\mathbb{C}^{2},0)$ transverse à $C.$ Alors la courbure du $d$-tissu $\mathcal{W}=\mathcal{F}\boxtimes\mathcal{W}_{\nu}\boxtimes\mathcal{W}_{d-\nu-1}$ est holomorphe le long de $C.$* * **Démonstration*.* Comme dans le début de la preuve de [@MP13 Proposition 2.6], on choisit un système de coordonnées locales $(U,(x,y))$ tel que $C\cap U=\{y=0\},$ $\mathrm{T}\mathcal{F}|_{U}=\{\mathrm{d}y+yh(x,y)\mathrm{d}x=0\},$ $$\begin{aligned} \mathrm{T}\mathcal{W}_{\nu}|_{U}=\left\{\mathrm{d}y^{\nu}+y\Big(a_{\nu-1}(x,y)\mathrm{d}y^{\nu-1}\mathrm{d}x+\cdots+a_{0}(x,y)\mathrm{d}x^{\nu}\Big)=0\right\} &&{\fontsize{11}{11pt}\text{et}}&& \mathrm{T}\mathcal{W}_{d-\nu-1}|_{U}=\left\{\prod\limits_{l=1}^{d-\nu-1}(\mathrm{d}x+g_{l}(x,y)\mathrm{d}y)=0\right\}.\end{aligned}$$ Puis, en passant au revêtement ramifié $\pi\hspace{1mm}\colon(x,y)\mapsto(x,y^{\nu})$, on obtient que $\pi^{*}\mathcal{F}=\mathcal{F}_0,$  $\pi^{*}\mathcal{W}_{\nu}=\boxtimes_{k=1}^{\nu}\mathcal{F}_k$ et $\pi^{*}\mathcal{W}_{d-\nu-1}=\boxtimes_{l=1}^{d-\nu-1}\mathcal{F}_{\nu+l},$ où $$\begin{aligned} &\mathcal{F}_0\hspace{0.1mm}:\hspace{0.1mm}\mathrm{d}y+\tfrac{1}{\nu}yh(x,y^{\nu})\mathrm{d}x=0,&& \mathcal{F}_k\hspace{0.1mm}:\hspace{0.1mm}\mathrm{d}x+y^{\nu-2}f(x,\zeta^{k}y)\zeta^{-k}\mathrm{d}y=0,&& \mathcal{F}_{\nu+l}\hspace{0.1mm}:\hspace{0.1mm}\mathrm{d}x+\nu y^{\nu-1}g_{l}(x,y^{\nu})\mathrm{d}y=0,\end{aligned}$$ avec $\zeta=\exp(\tfrac{2\mathrm{i}\pi}{\nu}).$ On a donc $$\begin{aligned} K(\pi^{*}\mathcal{W})=K\big(\pi^{*}(\mathcal{W}_{\nu}\boxtimes\mathcal{W}_{d-\nu-1})\big)+\sum\limits_{1\leq i<j\leq d-1}K(\mathcal{F}_0\boxtimes\mathcal{F}_i\boxtimes\mathcal{F}_j).\end{aligned}$$ Or, d'une part, [@MP13 Proposition 2.6] assure que $K(\mathcal{W}_{\nu}\boxtimes\mathcal{W}_{d-\nu-1})$ est holomorphe le long de $\{y=0\}$, de sorte qu'il en est de même de $K\big(\pi^{*}(\mathcal{W}_{\nu}\boxtimes\mathcal{W}_{d-\nu-1})\big)=\pi^*\big(K(\mathcal{W}_{\nu}\boxtimes\mathcal{W}_{d-\nu-1})\big).$ D'autre part, comme $\{y=0\}$ est invariante par $\mathcal{F}_0$ et que $\{y=0\}\not\subset\mathrm{Tang}(\mathcal{F}_0,\mathcal{F}_i\boxtimes\mathcal{F}_j)$, $K(\mathcal{F}_0\boxtimes\mathcal{F}_i\boxtimes\mathcal{F}_j)$ est holomorphe sur $\{y=0\}$ par application de [@MP13 Théorème 1], *voir* aussi [@BFM14 Théorème 1.1] ou [@Bed17 Corollaire 1.30]. Il en résulte que $\pi^*K(\mathcal{W})=K(\pi^{*}\mathcal{W})$ est holomorphe sur $\{y=0\}.$ Par conséquent $K(\mathcal{W})$ est holomorphe le long de $C$. ◻ *Remarque 12*. On obtient de la même façon un résultat analogue à la Proposition [Proposition 11](#pro:holomorphie-courbure-F-W-nu-W-d-nu-1){reference-type="ref" reference="pro:holomorphie-courbure-F-W-nu-W-d-nu-1"} si on remplace le feuilletage $\mathcal{F}$ par un $2$-tissu $\mathcal{W}_2=\mathcal{F}_1\boxtimes\mathcal{F}_2$ laissant la composante $C\subset\Delta(\mathcal{W}_{\nu})$ totalement invariante. **Démonstration du Théorème [Théorème 9](#thm:holomorphie-courbure-G(I^tr)-D-ell){reference-type="ref" reference="thm:holomorphie-courbure-G(I^tr)-D-ell"}*.* ***i.*** Supposons d'abord que $\ell=L_\infty.$ Alors le Théorème [Théorème 3](#thm:K-Leg-H-egale-K-Leg-L-infini-H){reference-type="ref" reference="thm:K-Leg-H-egale-K-Leg-L-infini-H"} assure que $K(\mathrm{Leg}\mathscr{H})=K(\mathrm{Leg}\mathcal{H}).$ Or, on sait, d'après [@BM18Bull Théorème 3.1], que la platitude du tissu $\mathrm{Leg}\mathcal{H}$ est caractérisée par l'holomorphie de sa courbure $K(\mathrm{Leg}\mathcal{H})$ sur $\mathcal{G}_{\mathcal{H}}(\mathrm{I}_{\mathcal{H}}^{\hspace{0.2mm}\mathrm{tr}}).$ Il en est donc de même du tissu $\mathrm{Leg}\mathscr{H},$ *i.e.* $\mathrm{Leg}\mathscr{H}$ est plat si et seulement si $K(\mathrm{Leg}\mathscr{H})$ est holomorphe le long de $\mathcal{G}_{\mathcal{H}}(\mathrm{I}_{\mathcal{H}}^{\hspace{0.2mm}\mathrm{tr}}).$ ***ii.*** Supposons maintenant que $\ell$ passe par l'origine. Fixons $s\in\Sigma_{\mathcal{H}}^{\infty}$ et décrivons le $d$-tissu $\mathrm{Leg}\mathscr{H}$ au voisinage d'un point générique $m$ de la droite $\check{s}$ duale de $s.$ Notons $\nu-1\geq0$ l'ordre de radialité de $s$; en vertu de [@MP13 Proposition 3.3], au voisinage de $m$, $\mathrm{Leg}\mathscr{H}$ peut se décomposer en $$\begin{aligned} \label{equa:Leg-preh-Leg-ell-W-nu-W-d-nu-1} \mathrm{Leg}\mathscr{H}=\mathrm{Leg}\ell\boxtimes\mathcal{W}_{\nu}\boxtimes\mathcal{W}_{d-\nu-1},\end{aligned}$$ où $\mathcal{W}_{\nu}$ est un $\nu$-tissu irréductible laissant $\check{s}$ invariante et dont la multiplicité du discriminant $\Delta(\mathcal{W}_{\nu})$ le long de $\check{s}$ est minimale, égale à $\nu-1,$ et où $\mathcal{W}_{d-\nu-1}$ est un $(d-\nu-1)$-tissu transverse à $\check{s}.$ Plus explicitement, à conjugaison linéaire près, on peut écrire $\ell=(y=\alpha\,x)$, $s=[1:\rho:0]$, $\check{s}=\{p=\rho\}$, $m=(\rho,q)$, ${\mspace{2mu}\underline{\mspace{-2mu}\mathcal{G}\mspace{-2mu}}\mspace{2mu}}_{\mathcal H}^{-1}(\rho)=\{\rho,r_1,\ldots,r_{d-\nu-1}\},$ de sorte que (*voir* [@BM18Bull Section 2]) $$\begin{aligned} & \mathrm{Leg}\ell:(p-\alpha)\mathrm{d}q-q\mathrm{d}p=0, && \mathcal{W}_{\nu}\Big|_{\check{s}}:\mathrm{d}p=0, && \mathcal{W}_{d-\nu-1}\Big|_{\check{s}}:\prod_{i=1}^{d-\nu-1}\Big((\rho-r_i)\mathrm{d}q-q\mathrm{d}p\Big)=0.\end{aligned}$$ Nous en tirons, en particulier, les deux propriétés suivantes - si $\check{s}\not\subset\mathcal{G}_{\mathcal{H}}(\mathrm{I}_{\mathcal{H}}^{\hspace{0.2mm}\mathrm{tr}})$, le tissu $\mathcal{W}_{d-\nu-1}$ est régulier au voisinage de $m$, car on a alors $r_i\neq r_j$ si $i\neq j$; - si $\check{s}\neq D_\ell=\{p={\mspace{2mu}\underline{\mspace{-2mu}\mathcal{G}\mspace{-2mu}}\mspace{2mu}}_{\mathcal H}(\alpha)\},$ alors $\mathrm{Leg}\ell$ est transverse à $\check{s}$ et $\check{s}\not\subset\mathrm{Tang}(\mathrm{Leg}\ell,\mathcal{W}_{d-\nu-1}).$ Si $s\in\Sigma_{\mathcal{H}}^{\mathrm{rad}}$ est tel que $\check{s}\not\subset\mathcal{G}_{\mathcal{H}}(\mathrm{I}_{\mathcal{H}}^{\hspace{0.2mm}\mathrm{tr}})\cup D_\ell$, alors les propriétés ($\mathfrak{a}$) et ($\mathfrak{b}$) assurent que le $(d-\nu)$-tissu $\mathcal{W}_{d-\nu}:=\mathrm{Leg}\ell\boxtimes\mathcal{W}_{d-\nu-1}$ est transverse à $\check{s}$ et est régulier au voisinage de $m.$ Donc la courbure de $\mathrm{Leg}\mathscr{H}=\mathcal{W}_\nu\boxtimes\mathcal{W}_{d-\nu}$ est holomorphe au voisinage de $m$ par application de [@MP13 Proposition 2.6]. Il en résulte que $K(\mathrm{Leg}\mathscr{H})$ est holomorphe sur $\check{\Sigma}_{\mathcal{H}}^{\mathrm{rad}}\setminus(\mathcal{G}_{\mathcal{H}}(\mathrm{I}_{\mathcal{H}}^{\hspace{0.2mm}\mathrm{tr}})\cup D_\ell).$ Ainsi, d'après la seconde assertion du Lemme [Lemme 2](#lem:Delta-Leg-H){reference-type="ref" reference="lem:Delta-Leg-H"} et le Lemme [Lemme 10](#lem:holomo-O){reference-type="ref" reference="lem:holomo-O"}, $\mathrm{Leg}\mathscr{H}$ est plat si et seulement si $K(\mathrm{Leg}\mathscr{H})$ est holomorphe le long de $\mathcal{G}_{\mathcal{H}}(\mathrm{I}_{\mathcal{H}}^{\hspace{0.2mm}\mathrm{tr}})\cup D_\ell.$ Montrons que dans le cas particulier où $\ell$ est invariante par $\mathcal{H}$, la platitude de $\mathrm{Leg}\mathscr{H}$ est équivalente à l'holomorphie de $K(\mathrm{Leg}\mathscr{H})$ sur $\mathcal{G}_{\mathcal{H}}(\mathrm{I}_{\mathcal{H}}^{\hspace{0.2mm}\mathrm{tr}}).$ D'après ce qui précède, il suffit de prouver que si $D_\ell$ n'est pas contenue dans $\mathcal{G}_{\mathcal{H}}(\mathrm{I}_{\mathcal{H}}^{\hspace{0.2mm}\mathrm{tr}})$, alors $K(\mathrm{Leg}\mathscr{H})$ est holomorphe sur $D_\ell.$ L'invariance de $\ell$ par $\mathcal{H}$ implique l'existence de $s\in\Sigma_{\mathcal{H}}^{\infty}$ tel que $\ell=(Os)$; alors $D_\ell=\check{s}$ est invariante par le feuilletage radial $\mathrm{Leg}\ell.$ De plus, la condition $D_\ell\not\subset\mathcal{G}_{\mathcal{H}}(\mathrm{I}_{\mathcal{H}}^{\hspace{0.2mm}\mathrm{tr}})$ entraîne que $\mathcal{W}_{d-\nu-1}$ est régulier au voisinage de tout point générique $m$ de $D_\ell$ (Propriété ($\mathfrak{a}$)). En appliquant le Théorème 1 de [@MP13] si $\nu=1$ et la Proposition [Proposition 11](#pro:holomorphie-courbure-F-W-nu-W-d-nu-1){reference-type="ref" reference="pro:holomorphie-courbure-F-W-nu-W-d-nu-1"} si $\nu\geq2,$ nous en déduisons que $K(\mathrm{Leg}\mathscr{H})$ est holomorphe le long de $D_\ell.$ ◻ Du Théorème [Théorème 9](#thm:holomorphie-courbure-G(I^tr)-D-ell){reference-type="ref" reference="thm:holomorphie-courbure-G(I^tr)-D-ell"} nous déduisons les deux corollaires suivants. **Corollaire 13**. **Soit $\mathscr{H}$ un pré-feuilletage homogène convexe de co-degré $1$ et de degré $d\geq3$ sur $\mathbb{P}^{2}_{\mathbb{C}}.$ Alors le $d$-tissu $\mathrm{Leg}\mathscr{H}$ est plat.** **Corollaire 14**. **Soit $\mathscr{H}=\ell\boxtimes\mathcal{H}$ un pré-feuilletage homogène de co-degré $1$ et de degré $d\geq3$ sur $\mathbb{P}^{2}_{\mathbb{C}}.$ Supposons que le feuilletage homogène $\mathcal{H}$ soit convexe et que la droite $\ell$ ne soit pas invariante par $\mathcal{H}.$ Alors le $d$-tissu $\mathrm{Leg}\mathscr{H}$ est plat si et seulement si sa courbure $K(\mathrm{Leg}\mathscr{H})$ est holomorphe sur $D_{\ell}=\mathcal{G}_{\mathcal{H}}(\ell).$* * Le théorème suivant est un critère effectif de l'holomorphie de la courbure du tissu dual d'un pré-feuilletage homogène $\mathscr{H}=\ell\boxtimes\mathcal{H}$ (avec $O\in\ell$) le long d'une composante irréductible de $\Delta(\mathrm{Leg}\mathcal{H})\setminus(D_{\ell}\cup\check{O}).$ **Théorème 15**. **Soit $\mathscr{H}=\ell\boxtimes\mathcal{H}$ un pré-feuilletage homogène de co-degré $1$ et de degré $d\geq3$ sur $\mathbb{P}^{2}_{\mathbb{C}}$, défini par la $1$-forme $$\begin{aligned} &\hspace{1.5cm}\omega=(\alpha\,x+\beta\,y)\left(A(x,y)\mathrm{d}x+B(x,y)\mathrm{d}y\right),\quad A,B\in\mathbb{C}[x,y]_{d-1},\hspace{2mm}\mathrm{pgcd}(A,B)=1.\end{aligned}$$ Soit $(p,q)$ la carte affine de $\mathbb{\check{P}}^{2}_{\mathbb{C}}$ associée à la droite $\{y=px-q\}\subset{\mathbb{P}^{2}_{\mathbb{C}}}$ et soit $D=\{p=p_0\}$ une composante irréductible de $\Delta(\mathrm{Leg}\mathcal{H})\setminus(D_{\ell}\cup\check{O}).$ Écrivons ${\mspace{2mu}\underline{\mspace{-2mu}\mathcal{G}\mspace{-2mu}}\mspace{2mu}}_{\mathcal H}^{-1}([p_0:1])=\{[a_1:b_1],\ldots,[a_n:b_n]\}$ et notons $\nu_i$ l'indice de ramification de ${\mspace{2mu}\underline{\mspace{-2mu}\mathcal{G}\mspace{-2mu}}\mspace{2mu}}_{\mathcal H}$ au point $[a_i:b_i]\in\mathbb{P}_{\mathbb{C}}^{1}.$ Pour $i\in\{1,\ldots,n\}$ définissons les polynômes $P_i\in\mathbb{C}[x,y]_{d-\nu_i-1}$ et $Q_i\in\mathbb{C}[x,y]_{2d-\nu_i-3}$ par** *$$\begin{aligned} \label{equa:Pi-Qi} P_i(x,y;a_i,b_i):=\frac{\left| \begin{array}{cc} A(x,y) & A(b_i,a_i) \\ B(x,y) & B(b_i,a_i) \end{array} \right|}{(b_iy-a_i\hspace{0.2mm}x)^{\nu_i}} \quad{\fontsize{11}{11pt}\text{et}}\quad Q_i(x,y;a_i,b_i):=(\nu_i-2)\left(\dfrac{\partial{B}}{\partial{x}}-\dfrac{\partial{A}}{\partial{y}}\right)P_i(x,y;a_i,b_i)+2(\nu_i+1) \left|\begin{array}{cc} \dfrac{\partial{P_i}}{\partial{x}} & A(x,y) \vspace{2mm} \\ \dfrac{\partial{P_i}}{\partial{y}} & B(x,y) \end{array} \right|.\end{aligned}$$* *Alors la courbure de $\mathrm{Leg}\mathscr{H}$ est holomorphe sur $D$ si et seulement si $$\begin{aligned} \sum_{i=1}^{n}\left(1-\frac{1}{\nu_{i}}\right)\Big(p_0\hspace{0.2mm}b_i-a_i\Big) \left(\frac{Q_i(b_i,a_i;a_i,b_i)}{B(b_i,a_i)P_i(b_i,a_i;a_i,b_i)}+\frac{3\nu_i(\alpha+p_0\,\beta)}{\alpha\,b_i+\beta\,a_i}\right)=0.\end{aligned}$$* **Démonstration*.* Soit $\delta\in\mathbb{C}$ tel que $\beta+\alpha\delta\neq0$ et $(b_i-a_i\delta)\neq0$ pour tout $i=1,\ldots,n$. Quitte à conjuguer $\omega$ par la transformation linéaire $(x+\delta\,y,y)$, nous pouvons supposer qu'aucune des droites $\ell=(\alpha x+\beta y=0)$ et $L_i=(b_i\hspace{0.2mm}y-a_i\hspace{0.2mm}x=0)$ n'est verticale, *i.e.* que $\beta\neq0$ et $b_i\neq0$ pour tout $i=1,\ldots,n.$ Posons alors $\rho:=-\frac{\alpha}{\beta}$ et $r_i:=\frac{a_i}{b_i}$; nous avons ${\mspace{2mu}\underline{\mspace{-2mu}\mathcal{G}\mspace{-2mu}}\mspace{2mu}}_{\mathcal H}^{-1}(p_0)=\{r_1,\ldots,r_n\}$ avec ${\mspace{2mu}\underline{\mspace{-2mu}\mathcal{G}\mspace{-2mu}}\mspace{2mu}}_{\mathcal H}(z)=-\dfrac{A(1,z)}{B(1,z)}.$ D'après [@BM22arxiv Lemme 3.5], il existe donc une constante $c\in\mathbb{C}^*$ telle que $$-A(1,z)=p_0B(1,z)-c\prod_{i=1}^{n}(z-r_i)^{\nu_i}.$$ Par ailleurs, le $d$-tissu $\mathrm{Leg}\mathscr{H}$ est donné dans la carte affine $(p,q)$ par l'équation différentielle $$\begin{aligned} \label{equa:LegH} \Big((p-\rho)x-q\Big)\Big(A(x,px-q)+pB(x,px-q)\Big)=0, \qquad \text{avec} \qquad x=\frac{\mathrm{d}q}{\mathrm{d}p};\end{aligned}$$ comme $A,B\in\mathbb{C}[x,y]_{d-1},$ cette équation se réécrit alors $$\begin{aligned} 0 &=x^{d-1}\Big((p-\rho)x-q\Big)\Big(A(1,p-\tfrac{q}{x})+pB(1,p-\tfrac{q}{x})\Big)\\ &=x^{d}\Big(p-\frac{q}{x}-\rho\Big)\Big((p-p_0)B(1,p-\tfrac{q}{x})+c\prod_{i=1}^{n}(p-\tfrac{q}{x}-r_i)^{\nu_i}\Big), \qquad \text{avec} \qquad x=\frac{\mathrm{d}q}{\mathrm{d}p}.\end{aligned}$$ Posons $\check{x}:=q$,  $\check{y}:=p-p_0$  et  $\check{p}:=\dfrac{\mathrm{d}\check{y}}{\mathrm{d}\check{x}}=\dfrac{1}{x}$; dans ces nouvelles coordonnées $D=\{\check{y}=0\}$ et $\mathrm{Leg}\mathscr{H}$ est décrit par l'équation différentielle $$\begin{aligned} F(\check{x},\check{y},\check{p}):= \Big(\check{y}+p_0-\check{p}\check{x}-\rho\Big) \Big(\check{y}B(1,\check{y}+p_0-\check{p}\check{x})+c\prod_{i=1}^{n}(\check{y}+p_0-\check{p}\check{x}-r_i)^{\nu_i}\Big)=0.\end{aligned}$$ Nous avons $F(\check{x},0,\check{p})=c(-\check{x})^d\big(\check{p}-\varphi_{0}(\check{x})\big)\prod_{i=1}^{n}\big(\check{p}-\varphi_{i}(\check{x})\big)^{\nu_i},$ où $\varphi_{0}(\check{x})=\dfrac{p_0-\rho}{\check{x}}$  et  $\varphi_{i}(\check{x})=\dfrac{p_0-r_i}{\check{x}}$; l'hypothèse que $D\neq D_{\ell}=\{p={\mspace{2mu}\underline{\mspace{-2mu}\mathcal{G}\mspace{-2mu}}\mspace{2mu}}_{\mathcal H}(\rho)\}$ se traduit par le fait que, pour tout $i\in\{1,\ldots,n\},$ $r_i\neq\rho$ et donc $\varphi_{i}\not\equiv\varphi_{0}.$ Notons que si $\nu_i\geq2,$ alors $\partial_{\check{y}}F\big(\check{x},0,\varphi_{i}(\check{x})\big)=(r_i-\rho)B(1,r_i)\neq0$; comme $\partial_{\check{p}}F\big(\check{x},0,\varphi_{0}(\check{x})\big)\not\equiv0$ et $\partial_{\check{p}}F\big(\check{x},0,\varphi_{i}(\check{x})\big)\not\equiv0$ si $\nu_i=1,$ il en résulte que la surface[\[not:S-W\]]{#not:S-W label="not:S-W"} $$\begin{aligned} S_{\mathrm{Leg}\mathscr{H}}:=\left\{(\check{x},\check{y},\check{p})\in\mathbb{P}\mathrm{T}^{*}\mathbb{\check{P}}^{2}_{\mathbb{C}}\hspace{1mm}\vert\hspace{1mm}F(\check{x},\check{y},\check{p})=0\right\}\end{aligned}$$ est lisse le long de $D=\{\check{y}=0\}.$ Ainsi, d'après [@BM22arxiv Théorème 2.1], la courbure de $\mathrm{Leg}\mathscr{H}$ est holomorphe sur $D=\{\check{y}=0\}$ si et seulement si $\sum_{i=1}^{n}(\nu_{i}-1)\varphi_{i}(\check{x})\psi_{i}(\check{x})\equiv0$  et  $\sum_{i=1}^{n}(\nu_{i}-1)\frac{\mathrm{d}}{\mathrm{d}\check{x}}\psi_{i}(\check{x})\equiv0,$ où, pour tout $i\in\{1,\ldots,n\}$ tel que $\nu_i\geq2,$ $$\begin{aligned} \psi_{i}(\check{x})=\frac{1}{\nu_{i}} \left[ (\nu_{i}-2)\left(d-\varphi_{i}(\check{x})\dfrac{\partial_{\check{p}}\partial_{\check{y}}F\big(\check{x},0,\varphi_{i}(\check{x})\big)}{\partial_{\check{y}}F\big(\check{x},0,\varphi_{i}(\check{x})\big)}\right) -2(\nu_{i}+1)\left(\frac{\varphi_{0}(\check{x})}{\varphi_{i}(\check{x})-\varphi_{0}(\check{x})}+\hspace{-3.5mm}\sum\limits_{\hspace{3.5mm}j=1,j\neq i}^{n}\frac{\nu_{j}\varphi_{j}(\check{x})}{\varphi_{i}(\check{x})-\varphi_{j}(\check{x})}\right) \right].\end{aligned}$$ Or, si $\nu_i\geq3$ alors $\partial_{\check{p}}\partial_{\check{y}}F\big(\check{x},0,\varphi_{i}(\check{x})\big)=-\check{x}\Big(B(1,r_i)+(r_i-\rho)\partial_{y}B(1,r_i)\Big).$ Nous en déduisons que $$\begin{aligned} \psi_{i}(\check{x})=\psi_i:=\frac{1}{\nu_{i}} \left[ (\nu_{i}-2)\left(d+\frac{\Big(p_0-r_i\Big)\Big(B(1,r_i)+(r_i-\rho)\partial_{y}B(1,r_i)\Big)}{(r_i-\rho)B(1,r_i)}\right) +2(\nu_{i}+1)\left(\frac{p_0-\rho}{r_i-\rho}+\hspace{-3.5mm}\sum\limits_{\hspace{3.5mm}j=1,j\neq i}^{n}\frac{\nu_{j}(p_0-r_j)}{r_i-r_j}\right) \right].\end{aligned}$$ Donc $K(\mathrm{Leg}\mathscr{H})$ est holomorphe sur $D=\{\check{y}=0\}$ si et seulement si $\sum_{i=1}^{n}(\nu_{i}-1)\varphi_{i}(\check{x})\psi_{i}\equiv0.$ D'autre part, en raisonnant comme dans la démonstration de [@BM22arxiv Théorème 3.1], nous obtenons que $$\begin{aligned} \hspace{-3.5mm}\sum\limits_{\hspace{3.5mm}j=1,j\neq i}^{n}\frac{\nu_{j}(p_0-r_j)}{r_i-r_j} =\frac{\left|\begin{array}{cc} \partial_{x}P_i(1,r_i;r_i,1) & A(1,r_i) \vspace{2mm} \\ \partial_{y}P_i(1,r_i;r_i,1) & B(1,r_i) \end{array} \right|}{B(1,r_i)P_i(1,r_i;r_i,1)}\end{aligned}$$ et que, pour tout $i\in\{1,\ldots,n\}$ tel que $\nu_i\geq2,$ $$\begin{aligned} (d-1)B(1,r_i)+(p_0-r_i)\partial_{y}B(1,r_i)=\partial_{x}B(1,r_i)-\partial_{y}A(1,r_i),\end{aligned}$$ de sorte que $$\begin{aligned} \psi_i&=\frac{1}{\nu_{i}} \left[ (\nu_{i}-2)\left(\frac{p_0-\rho}{r_i-\rho}+\frac{\partial_{x}B(1,r_i)-\partial_{y}A(1,r_i)}{B(1,r_i)}\right) +2(\nu_{i}+1)\left( \frac{p_0-\rho}{r_i-\rho}+ \frac{\left|\begin{array}{cc} \partial_{x}P_i(1,r_i;r_i,1) & A(1,r_i) \vspace{2mm} \\ \partial_{y}P_i(1,r_i;r_i,1) & B(1,r_i) \end{array} \right|}{B(1,r_i)P_i(1,r_i;r_i,1)} \right) \right] \\ &= \dfrac{Q_{i}(1,r_i;r_i,1)}{\nu_{i}B(1,r_i)P_{i}(1,r_i;r_i,1)}+\frac{3(p_0-\rho)}{r_i-\rho}.\end{aligned}$$ Par suite, $K(\mathrm{Leg}\mathscr{H})$ est holomorphe le long de $D=\{\check{y}=0\}$ si et seulement si $$\begin{aligned} \frac{1}{\check{x}}\sum_{i=1}^{n}\left(1-\frac{1}{\nu_{i}}\right)\Big(p_0-r_i\Big) \left(\frac{Q_i(1,r_i;r_i,1)}{B(1,r_i)P_i(1,r_i;r_i,1)}+\frac{3\nu_i(p_0-\rho)}{r_i-\rho}\right)=0,\end{aligned}$$ d'où l'énoncé. ◻ *Remarques 16*. - \(i\) On retrouve le fait (*cf.* étape ***ii.*** de la démonstration du Théorème [Théorème 9](#thm:holomorphie-courbure-G(I^tr)-D-ell){reference-type="ref" reference="thm:holomorphie-courbure-G(I^tr)-D-ell"}) que la courbure de $\mathrm{Leg}\mathscr{H}$ est toujours holomorphe le long de $\check{\Sigma}_{\mathcal{H}}^{\mathrm{rad}}\setminus(\mathcal{G}_{\mathcal{H}}(\mathrm{I}_{\mathcal{H}}^{\hspace{0.2mm}\mathrm{tr}})\cup D_{\ell}).$ En effet, si $D$ est contenue dans $\check{\Sigma}_{\mathcal{H}}^{\mathrm{rad}}\setminus(\mathcal{G}_{\mathcal{H}}(\mathrm{I}_{\mathcal{H}}^{\hspace{0.2mm}\mathrm{tr}})\cup D_{\ell}),$ alors l'application ${\mspace{2mu}\underline{\mspace{-2mu}\mathcal{G}\mspace{-2mu}}\mspace{2mu}}_{\mathcal H}$ n'a aucun point critique non fixe dans la fibre ${\mspace{2mu}\underline{\mspace{-2mu}\mathcal{G}\mspace{-2mu}}\mspace{2mu}}_{\mathcal H}^{-1}([p_0:1])$, de sorte que l'on a $p_0\hspace{0.2mm}b_i-a_i=0$ si $\nu_i\geq2,$ ce qui implique (Théorème [Théorème 15](#thm:holomorphie-courbure-homogene-D-neq-D-ell){reference-type="ref" reference="thm:holomorphie-courbure-homogene-D-neq-D-ell"}) que $K(\mathrm{Leg}\mathscr{H})$ est holomorphe sur $D.$ - On sait, d'après [@BM22arxiv Théorème 3.1], que la courbure de $\mathrm{Leg}\mathcal{H}$ est holomorphe sur $D$ si et seulement si $$\begin{aligned} \sum_{i=1}^{n}\left(1-\frac{1}{\nu_{i}}\right)\frac{(p_0\hspace{0.2mm}b_i-a_i)Q_i(b_i,a_i;a_i,b_i)}{B(b_i,a_i)P_i(b_i,a_i;a_i,b_i)}=0.\end{aligned}$$ De ce résultat et du Théorème [Théorème 15](#thm:holomorphie-courbure-homogene-D-neq-D-ell){reference-type="ref" reference="thm:holomorphie-courbure-homogene-D-neq-D-ell"}, nous déduisons les propriétés suivantes: - Si la courbure de $\mathrm{Leg}\mathcal{H}$ est holomorphe sur $D$, alors la courbure de $\mathrm{Leg}\mathscr{H}$ est holomorphe sur $D$ si et seulement si $$\begin{aligned} (\alpha+p_0\,\beta)\sum_{i=1}^{n}\frac{(\nu_i-1)(p_0\hspace{0.2mm}b_i-a_i)}{\alpha\,b_i+\beta\,a_i}=0.\end{aligned}$$ - En particulier, lorsque $d=3$ la fibre ${\mspace{2mu}\underline{\mspace{-2mu}\mathcal{G}\mspace{-2mu}}\mspace{2mu}}_{\mathcal H}^{-1}([p_0:1])$ est réduite à un point, soit $[a:b],$ et l'holomorphie de la courbure de $\mathrm{Leg}\mathscr{H}$ sur $D$ est équivalente à $(\alpha+p_0\,\beta)(p_0\hspace{0.2mm}b-a)=0$, *i.e.* à $\alpha+p_0\,\beta=0$ ou $[a:b]=[p_0:1]$, et donc à $(1,p_0)\in\ell$ ou $[p_0:1]$ est fixe par ${\mspace{2mu}\underline{\mspace{-2mu}\mathcal{G}\mspace{-2mu}}\mspace{2mu}}_{\mathcal H}.$ - Si $(1,p_0)\in\ell$ alors on a équivalence entre l'holomorphie sur $D$ de $K(\mathrm{Leg}\mathscr{H})$ et celle de $K(\mathrm{Leg}\mathcal{H}).$ - Supposons que $\nu_{i}=\nu\geq2$ pour tout $i\in\{1,\ldots,n\}.$ Alors la courbure de $\mathrm{Leg}\mathscr{H}$ est holomorphe sur $D$ si et seulement si $$\begin{aligned} \sum_{i=1}^{n}\left(p_0\hspace{0.2mm}b_i-a_i\right) \left(\frac{\big(\nu-2\big)\big(\partial_{x}B(b_i,a_i)-\partial_{y}A(b_i,a_i)\big)}{B(b_i,a_i)}+\frac{3\nu(\alpha+p_0\,\beta)}{\alpha\,b_i+\beta\,a_i}\right)=0.\end{aligned}$$ En effet, il suffit, dans la démonstration précédente, de poser $\delta_{i,j}=\frac{(p_0-r_i)(p_0-r_j)}{(r_i-r_j)}$ et de remarquer que $$\begin{aligned} \sum_{i=1}^{n}\left((\nu-1)\varphi_{i}(\check{x})\hspace{-3.5mm}\sum\limits_{\hspace{3.5mm}j=1,j\neq i}^{n}\frac{\nu(p_0-r_j)}{r_i-r_j}\right) =\frac{\nu(\nu-1)}{\check{x}}\sum_{i=1}^{n}\hspace{-3.5mm}\sum\limits_{\hspace{3.5mm}j=1,j\neq i}^{n}\delta_{i,j} =\frac{\nu(\nu-1)}{\check{x}}\sum_{1\leq i<j\leq n}(\delta_{i,j}+\delta_{j,i})\equiv0.\end{aligned}$$ En particulier, si la fibre ${\mspace{2mu}\underline{\mspace{-2mu}\mathcal{G}\mspace{-2mu}}\mspace{2mu}}_{\mathcal H}^{-1}([p_0:1])$ contient un seul point critique non fixe de ${\mspace{2mu}\underline{\mspace{-2mu}\mathcal{G}\mspace{-2mu}}\mspace{2mu}}_{\mathcal H}$, soit $[a:b]$, alors - ou bien ${\mspace{2mu}\underline{\mspace{-2mu}\mathcal{G}\mspace{-2mu}}\mspace{2mu}}_{\mathcal H}^{-1}([p_0:1])=\{[a:b]\}$, auquel cas $\nu=d-1$; - ou bien $\#{\mspace{2mu}\underline{\mspace{-2mu}\mathcal{G}\mspace{-2mu}}\mspace{2mu}}_{\mathcal H}^{-1}([p_0:1])=2$, auquel cas $d$ est nécessairement impair, $d=2k+1,$ et $\nu=k.$ Dans les deux cas, la courbure de $\mathrm{Leg}\mathscr{H}$ est holomorphe sur $D$ si et seulement si $$(\nu-2)(\alpha\,b+\beta\,a)\Big(\partial_{x}B(b,a)-\partial_{y}A(b,a)\Big)+3\nu(\alpha+p_0\,\beta)B(b,a)=0.$$ *Exemple 17*. Considérons le pré-feuilletage homogène $\mathscr{H}=\ell\boxtimes\mathcal{H}$ de co-degré $1$ et de degré impair $2k+1\geq5$ sur $\mathbb{P}^{2}_{\mathbb{C}}$ défini par la $1$-forme $$\begin{aligned} \hspace{1cm}&\omega=\left(x-\tau\,y\right)\left(y^k(y-x)^k\mathrm{d}x+(y-\lambda\,x)^k(y-\mu\,x)^k\mathrm{d}y\right),&& \text{où }\lambda,\mu\in\mathbb{C}\setminus\{0,1\}\hspace{2mm}\text{et}\hspace{2mm}\tau\in\mathbb{C}\setminus\{1\}.\end{aligned}$$ On sait, d'après [@BM22arxiv Exemple 3.4], que $D:=\{p=0\}\subset\Delta(\mathrm{Leg}\mathcal{H})$ et que la fibre ${\mspace{2mu}\underline{\mspace{-2mu}\mathcal{G}\mspace{-2mu}}\mspace{2mu}}_{\mathcal H}^{-1}([0:1])$ est formée des deux points $[0:1]$ et $[1:1]$: le point $[0:1]$ (resp. $[1:1]$) est critique fixe (resp. non fixe) de ${\mspace{2mu}\underline{\mspace{-2mu}\mathcal{G}\mspace{-2mu}}\mspace{2mu}}_{\mathcal{H}}$ de multiplicité $k-1.$ De plus, comme $\tau\neq1$, nous avons $[1:\tau]\not\in{\mspace{2mu}\underline{\mspace{-2mu}\mathcal{G}\mspace{-2mu}}\mspace{2mu}}_{\mathcal H}^{-1}([0:1])$, de sorte que $D\neq D_\ell=\{[p:1]={\mspace{2mu}\underline{\mspace{-2mu}\mathcal{G}\mspace{-2mu}}\mspace{2mu}}_{\mathcal H}([1:\tau]\}.$ En vertu de la Remarque [Remarques 16](#rems:holomorphie-courbure-homogene-D-neq-D-ell){reference-type="ref" reference="rems:holomorphie-courbure-homogene-D-neq-D-ell"} (iii), nous en déduisons que la courbure de $\mathrm{Leg}\mathscr{H}$ est holomorphe le long de $D$ si et seulement si $$\begin{aligned} 0=(k-2)(1-\tau)\Big(\partial_{x}B(1,1)-\partial_{y}A(1,1)\Big)+3k B(1,1) =k(1-\lambda)^k(1-\mu)^k\left(\frac{(k-2)(\tau-1)(\lambda+\mu-2\lambda\mu)}{(\lambda-1)(\mu-1)}+3\right),\end{aligned}$$ si et seulement si le quadruplet $(k,\lambda,\mu,\tau)$ vérifie l'équation $(k-2)(\tau-1)(\lambda+\mu-2\lambda\mu)+3(\lambda-1)(\mu-1)=0.$ Notons que, d'après [@BM22arxiv Exemple 3.4], l'holomorphie de la courbure de $\mathrm{Leg}\mathcal{H}$ le long de $D$ est caractérisée par l'équation $(k-2)(\lambda+\mu-2\lambda\mu)=0$. Il en résulte, en particulier, que si la courbure de $\mathrm{Leg}\mathcal{H}$ est holomorphe sur $D$, alors la courbure de $\mathrm{Leg}\mathscr{H}$ ne l'est pas. **Corollaire 18**. **Soit $\mathscr{H}=\ell\boxtimes\mathcal{H}$ un pré-feuilletage homogène de co-degré $1$ et de degré $d\geq3$ sur $\mathbb{P}^{2}_{\mathbb{C}}$, défini par la $1$-forme $$\begin{aligned} &\hspace{1.5cm}\omega=(\alpha\,x+\beta\,y)\left(A(x,y)\mathrm{d}x+B(x,y)\mathrm{d}y\right),\quad A,B\in\mathbb{C}[x,y]_{d-1},\hspace{2mm}\mathrm{pgcd}(A,B)=1.\end{aligned}$$ Supposons que $\mathcal{H}$ possède une droite d'inflexion $T=(ax+by=0)$ transverse d'ordre $\nu-1$. Supposons de plus que $[-a:b]\in\mathbb{P}^{1}_{\mathbb{C}}$ soit le seul point critique non fixe de ${\mspace{2mu}\underline{\mspace{-2mu}\mathcal{G}\mspace{-2mu}}\mspace{2mu}}_{\mathcal H}$ dans sa fibre ${\mspace{2mu}\underline{\mspace{-2mu}\mathcal{G}\mspace{-2mu}}\mspace{2mu}}_{\mathcal H}^{-1}({\mspace{2mu}\underline{\mspace{-2mu}\mathcal{G}\mspace{-2mu}}\mspace{2mu}}_{\mathcal H}([-a:b]))$ et que $[-\alpha:\beta]\not\in{\mspace{2mu}\underline{\mspace{-2mu}\mathcal{G}\mspace{-2mu}}\mspace{2mu}}_{\mathcal H}^{-1}({\mspace{2mu}\underline{\mspace{-2mu}\mathcal{G}\mspace{-2mu}}\mspace{2mu}}_{\mathcal H}([-a:b])).$ Alors la courbure de $\mathrm{Leg}\mathscr{H}$ est holomorphe sur $T'=\mathcal{G}_{\mathcal{H}}(T)$ si et seulement si $$(\alpha\,b-\beta\,a)Q(b,-a;a,b)+3\nu\Big(\alpha\,B(b,-a)-\beta\,A(b,-a)\Big)P(b,-a;a,b)=0,$$ où** *$$\begin{aligned} Q(x,y;a,b):=(\nu-2)\left(\dfrac{\partial{B}}{\partial{x}}-\dfrac{\partial{A}}{\partial{y}}\right)P(x,y;a,b)+2(\nu+1) \left|\begin{array}{cc} \dfrac{\partial{P}}{\partial{x}} & A(x,y) \vspace{2mm} \\ \dfrac{\partial{P}}{\partial{y}} & B(x,y) \end{array} \right| \quad\text{et}\quad P(x,y;a,b):=\frac{\left| \begin{array}{cc} A(x,y) & A(b,-a) \\ B(x,y) & B(b,-a) \end{array} \right|}{(ax+by)^{\nu}}.\end{aligned}$$* **Démonstration*.* À conjugaison linéaire près, on peut supposer que $T'\neq L_\infty$; alors $T'$ a pour équation $p=p_0$, où $p_0=-\frac{A(b,-a)}{B(b,-a)}$. Par le Théorème [Théorème 15](#thm:holomorphie-courbure-homogene-D-neq-D-ell){reference-type="ref" reference="thm:holomorphie-courbure-homogene-D-neq-D-ell"}, la courbure de $\mathrm{Leg}\mathscr{H}$ est holomorphe sur $T'$ si et seulement si $$\begin{aligned} \left(1-\tfrac{1}{\nu}\right)\left(p_0\hspace{0.2mm}b+a\right) \left(\frac{Q(b,-a;a,b)}{B(b,-a)P(b,-a;a,b)}+\frac{3\nu(\alpha+p_0\,\beta)}{\alpha\,b-\beta\,a}\right)=0.\end{aligned}$$ Or, l'hypothèse que le point $[-a:b]$ est non fixe par ${\mspace{2mu}\underline{\mspace{-2mu}\mathcal{G}\mspace{-2mu}}\mspace{2mu}}_{\mathcal H}$ se traduit par $p_0\hspace{0.2mm}b+a\neq0.$ Il en résulte que $K(\mathrm{Leg}\mathscr{H})$ est holomorphe sur $T'$ si et seulement si $$\begin{aligned} \frac{Q(b,-a;a,b)}{P(b,-a;a,b)}+\frac{3\nu\big(\alpha\,B(b,-a)-\beta\,A(b,-a)\big)}{\alpha\,b-\beta\,a}=0,\end{aligned}$$ d'où le résultat. ◻ En particulier, on a le: **Corollaire 19**. **Soit $\mathscr{H}=\ell\boxtimes\mathcal{H}$ un pré-feuilletage homogène de co-degré $1$ et de degré $d\geq3$ sur $\mathbb{P}^{2}_{\mathbb{C}}$, défini par la $1$-forme $$\begin{aligned} &\hspace{1.5cm}\omega=(\alpha\,x+\beta\,y)\left(A(x,y)\mathrm{d}x+B(x,y)\mathrm{d}y\right),\quad A,B\in\mathbb{C}[x,y]_{d-1},\hspace{2mm}\mathrm{pgcd}(A,B)=1.\end{aligned}$$ Supposons que $\mathcal{H}$ admette une droite d'inflexion $T=(ax+by=0)$ transverse d'ordre maximal $d-2$ et que $T\neq\ell.$ Alors la courbure de $\mathrm{Leg}\mathscr{H}$ est holomorphe le long de $T'=\mathcal{G}_{\mathcal{H}}(T)$ si et seulement si $$(d-3)(\alpha\,b-\beta\,a)\Big(\partial_{x}B(b,-a)-\partial_{y}A(b,-a)\Big)+3(d-1)\Big(\alpha\,B(b,-a)-\beta\,A(b,-a)\Big)=0.$$* * Le théorème suivant est un critère effectif de l'holomorphie de la courbure du tissu dual d'un pré-feuilletage homogène $\mathscr{H}=\ell\boxtimes\mathcal{H}$ (avec $O\in\ell$) le long de la composante $D_{\ell}\subset\Delta(\mathrm{Leg}\mathscr{H}).$ **Théorème 20**. **Soit $\mathscr{H}=\ell\boxtimes\mathcal{H}$ un pré-feuilletage homogène de co-degré $1$ et de degré $d\geq3$ sur $\mathbb{P}^{2}_{\mathbb{C}}$, défini par la $1$-forme $$\begin{aligned} &\hspace{1.5cm}\omega=(\alpha\,x+\beta\,y)\left(A(x,y)\mathrm{d}x+B(x,y)\mathrm{d}y\right),\quad A,B\in\mathbb{C}[x,y]_{d-1},\hspace{2mm}\mathrm{pgcd}(A,B)=1.\end{aligned}$$ Écrivons ${\mspace{2mu}\underline{\mspace{-2mu}\mathcal{G}\mspace{-2mu}}\mspace{2mu}}_{\mathcal H}^{-1}({\mspace{2mu}\underline{\mspace{-2mu}\mathcal{G}\mspace{-2mu}}\mspace{2mu}}_{\mathcal H}([-\alpha:\beta]))=\{[-\alpha:\beta],[a_1:b_1],\ldots,[a_n:b_n]\}$ et notons $\nu_i$ (resp. $\nu_0$) l'indice de ramification de ${\mspace{2mu}\underline{\mspace{-2mu}\mathcal{G}\mspace{-2mu}}\mspace{2mu}}_{\mathcal H}$ au point $[a_i:b_i]$ (resp. $[-\alpha:\beta]$). Définissons les polynômes $P_0\in\mathbb{C}[x,y]_{d-\nu_0-1}$ et $Q_0\in\mathbb{C}[x,y]_{2d-\nu_0-3}$ par** *$$\begin{aligned} P_0(x,y;\alpha,\beta):=\frac{\left| \begin{array}{cc} A(x,y) & A(\beta,-\alpha) \\ B(x,y) & B(\beta,-\alpha) \end{array} \right|}{(\alpha\,x+\beta\,y)^{\nu_0}} \quad{\fontsize{11}{11pt}\text{et}}\quad Q_0(x,y;\alpha,\beta):=(\nu_0-1)\left(\dfrac{\partial{B}}{\partial{x}}-\dfrac{\partial{A}}{\partial{y}}\right)P_0(x,y;\alpha,\beta)+(2\nu_0+1) \left|\begin{array}{cc} \dfrac{\partial{P_0}}{\partial{x}} & A(x,y) \vspace{2mm} \\ \dfrac{\partial{P_0}}{\partial{y}} & B(x,y) \end{array} \right|.\end{aligned}$$* *Supposons que ${\mspace{2mu}\underline{\mspace{-2mu}\mathcal{G}\mspace{-2mu}}\mspace{2mu}}_{\mathcal H}([-\alpha:\beta])\neq\infty$ et soit $p_0\in\mathbb{C}$ tel que $[p_0:1]={\mspace{2mu}\underline{\mspace{-2mu}\mathcal{G}\mspace{-2mu}}\mspace{2mu}}_{\mathcal H}([-\alpha:\beta]).$ Alors la courbure de $\mathrm{Leg}\mathscr{H}$ est holomorphe sur $D_\ell$ si et seulement si* *$$\begin{aligned} \left(1+\frac{1}{\nu_{0}}\right)\frac{(\alpha+p_0\,\beta)Q_0(\beta,-\alpha;\alpha,\beta)}{B(\beta,-\alpha)P_0(\beta,-\alpha;\alpha,\beta)}+ \sum_{i=1}^{n}\left(1-\frac{1}{\nu_{i}}\right)\Big(p_0\hspace{0.2mm}b_i-a_i\Big) \left(\frac{Q_i(b_i,a_i;a_i,b_i)}{B(b_i,a_i)P_i(b_i,a_i;a_i,b_i)}+\frac{3\nu_i(\alpha+p_0\,\beta)}{\alpha\,b_i+\beta\,a_i}\right)=0,\end{aligned}$$* *où les $P_i$ et les $Q_i$ ($i=1,\ldots,n$) sont les polynômes donnés par ([\[equa:Pi-Qi\]](#equa:Pi-Qi){reference-type="ref" reference="equa:Pi-Qi"}).* Notons que le $d$-tissu $\mathrm{Leg}\mathscr{H}=\mathrm{Leg}\ell\boxtimes\mathrm{Leg}\mathcal{H}$ n'est pas lisse le long de la composante $D_\ell\subset\mathrm{Tang}(\mathrm{Leg}\ell,\mathrm{Leg}\mathcal{H})$ et on ne peut donc pas appliquer le Théorème 2.1 de [@BM22arxiv] à $\mathrm{Leg}\mathscr{H}$ comme dans la démonstration du Théorème [Théorème 15](#thm:holomorphie-courbure-homogene-D-neq-D-ell){reference-type="ref" reference="thm:holomorphie-courbure-homogene-D-neq-D-ell"}. Pour démontrer le Théorème [Théorème 20](#thm:holomorphie-courbure-homogene-D-ell){reference-type="ref" reference="thm:holomorphie-courbure-homogene-D-ell"}, nous allons d'abord établir, pour un feuilletage $\mathcal{F}$ et un tissu $\mathcal{W}$ lisse le long d'une composante irréductible $D$ de $\mathrm{Tang}(\mathcal{F},\mathcal{W}),$ un critère effectif de l'holomorphie de la courbure de $\mathcal{F}\boxtimes\mathcal{W}$ le long de $D.$ **Théorème 21**. **Soit $\mathcal{W}$ un $(d-1)$-tissu holomorphe sur une surface complexe $M.$ Soit $\mathcal{F}$ un feuilletage holomorphe sur $M.$ Supposons que $\mathcal{W}$ soit lisse le long d'une composante irréductible $D$ de $\mathrm{Tang}(\mathcal{F},\mathcal{W}).$ Alors la forme fondamentale $\eta(\mathcal{F}\boxtimes\mathcal{W})$ est à pôles simples le long de $D.$ Plus précisément, choisissons un système de coordonnées locales $(x,y)$ sur $M$ dans lequel $D=\{y=0\}$ et soit $F(x,y,p)=0,$ $p=\frac{\mathrm{d}y}{\mathrm{d}x},$ une équation différentielle implicite décrivant $\mathcal{W}.$ Écrivons $F(x,0,p)=a_{0}(x)\prod\limits_{\alpha=1}^{n}(p-\varphi_{\alpha}(x))^{\nu_{\alpha}},$ avec $\varphi_{\alpha}\not\equiv\varphi_{\beta}$ si $\alpha\neq\beta,$ et supposons que $\mathcal{F}$ soit donné par une $1$-forme $\omega$ du type $\omega=\mathrm{d}y-\left(\varphi_{1}(x)+yf(x,y)\right)\mathrm{d}x.$ Définissons $Q(x,p)$ par $F(x,0,p)=(p-\varphi_{1}(x))^{\nu_{1}}Q(x,p)$ et posons** *$$\begin{aligned} & h(x)=\frac{1}{\nu_{1}} \left[ (\nu_{1}-1)\left(d-1-\varphi_{1}(x)\dfrac{\partial_{p}\partial_{y}F(x,0,\varphi_{1}(x))+ 2\delta_{\nu_1,2}f(x,0)Q(x,\varphi_{1}(x))}{\partial_{y}F(x,0,\varphi_{1}(x))}\right) -(2\nu_{1}+1)\sum_{\alpha=2}^{n}\frac{\nu_{\alpha}\varphi_{\alpha}(x)}{\varphi_{1}(x)-\varphi_{\alpha}(x)} \right]\end{aligned}$$* *(). Soit $\psi_{\alpha}$ une fonction de la coordonnée $x$ définie, pour tout $\alpha\in\{1,\ldots,n\}$ tel que $\nu_{\alpha}\geq2,$ par $$\begin{aligned} \psi_{\alpha}(x)=\frac{1}{\nu_{\alpha}} \left[ (\nu_{\alpha}-2) \left( d-1-\varphi_{\alpha}(x)\dfrac{\partial_{p}\partial_{y}F\big(x,0,\varphi_{\alpha}(x)\big)}{\partial_{y}F\big(x,0,\varphi_{\alpha}(x)\big)} \right) -2(\nu_{\alpha}+1)\hspace{-3.5mm}\sum\limits_{\hspace{3.5mm}\beta=1,\beta\neq\alpha}^{n}\frac{\nu_{\beta}\varphi_{\beta}(x)}{\varphi_{\alpha}(x)-\varphi_{\beta}(x)} \right].\end{aligned}$$ Alors la $1$-forme $\eta(\mathcal{F}\boxtimes\mathcal{W})-\frac{\theta}{6y}$ est holomorphe le long de $D=\{y=0\}$, où* *$$\begin{aligned} \theta=(\nu_1+1)\left[h(x)\Big(\mathrm{d}y-\varphi_1(x)\mathrm{d}x\Big)+(\nu_1-1)\mathrm{d}y\right]+ \sum_{\alpha=2}^{n}(\nu_\alpha-1)\left[\left(\psi_\alpha(x)+\frac{3\varphi_1(x)}{\varphi_1(x)-\varphi_\alpha(x)}\right) \big(\mathrm{d}y-\varphi_\alpha(x)\mathrm{d}x\big)+(\nu_\alpha-2)\mathrm{d}y\right].\end{aligned}$$* *En particulier, la courbure $K(\mathcal{F}\boxtimes\mathcal{W})$ est holomorphe le long de $D$ si et seulement si $$\begin{aligned} &\hspace{-4cm} (\nu_1+1)\varphi_1(x)h(x)+ \sum_{\alpha=2}^{n}(\nu_{\alpha}-1)\varphi_{\alpha}(x)\left(\psi_{\alpha}(x)+\frac{3\varphi_{1}(x)}{\varphi_{1}(x)-\varphi_{\alpha}(x)}\right)\equiv0 \\ &\hspace{-4.95cm}\text{et} \\ &\hspace{-4cm} \frac{\mathrm{d}}{\mathrm{d}x}\left((\nu_1+1)h(x)+ \sum_{\alpha=2}^{n}(\nu_{\alpha}-1)\left(\psi_{\alpha}(x)+\frac{3\varphi_{1}(x)}{\varphi_{1}(x)-\varphi_{\alpha}(x)}\right)\right)\equiv0.\end{aligned}$$* **Démonstration*.* Au voisinage de tout point générique $m$ de $D$ le tissu $\mathcal{W}$ se décompose en $\mathcal{W}=\boxtimes_{\alpha=1}^{n}\mathcal{W}_{\alpha},$ où $\mathcal{W}_{\alpha}=\boxtimes_{i=1}^{\nu_{\alpha}}\mathcal{F}_{i}^{\alpha}$ et $\mathcal{F}_{i}^{\alpha}|_{y=0}:\mathrm{d}y-\varphi_{\alpha}(x)\mathrm{d}x=0.$ Alors $\eta(\mathcal{F}\boxtimes\mathcal{W})=\eta(\mathcal{W})+\eta_1+\eta_2+\eta_3+\eta_4,$ où $$\begin{aligned} & \eta_1 =\sum_{1\le i<j\le\nu_{1}}\eta(\mathcal{F}\boxtimes\mathcal{F}_{i}^{1}\boxtimes\mathcal{F}_{j}^{1}), && \eta_2=\sum_{\alpha=2}^{n} \sum_{\raisebox{2mm}{${\underset{1\le j\le\nu_{\alpha}}{\underset{1\le i\le\nu_{1}}{}}}$}}\eta(\mathcal{F}\boxtimes\mathcal{F}_{i}^{1}\boxtimes\mathcal{F}_{j}^{\alpha}), \\ & \eta_3=\sum_{\alpha=2}^{n}\sum_{1\le i<j\le\nu_{\alpha}}\eta(\mathcal{F}\boxtimes\mathcal{F}_{i}^{\alpha}\boxtimes\mathcal{F}_{j}^{\alpha}), && \eta_4=\sum_{2\le\alpha<\beta\le n} \sum_{\raisebox{2mm}{${\underset{1\le j\le\nu_{\beta}}{\underset{1\le i\le\nu_{\alpha}}{}}}$}}\eta(\mathcal{F}\boxtimes\mathcal{F}_{i}^{\alpha}\boxtimes\mathcal{F}_{j}^{\beta}).\end{aligned}$$ D'après [@BM22arxiv Théorème 2.1], la partie polaire de la série de [Laurent]{.smallcaps} de $\eta(\mathcal{W})$ en $y=0$ est donnée par $\frac{\theta_0}{y}$, où $$\begin{aligned} \theta_0=\frac{1}{6}\sum\limits_{\alpha=1}^{n}\big(\nu_{\alpha}-1\big)\Big[\psi_{\alpha}(x)\big(\mathrm{d}y-\varphi_{\alpha}(x)\mathrm{d}x\big) +\big(\nu_{\alpha}-2\big)\mathrm{d}y\Big].\end{aligned}$$ Quant aux $1$-formes $\eta_i$, notons d'abord que, comme dans la preuve de [@BM22arxiv Théorème 2.1], la pente $p_{\hspace{-0.2mm}j}$ ($j=1,\ldots,\nu_{\alpha}$) de $\mathrm{T}_{(x,y)}\mathcal{F}_{j}^{\alpha}$ s'écrit sous la forme $$\begin{aligned} &\hspace{1cm}p_{\hspace{-0.2mm}j}=\lambda_{\alpha,j}(x,y):=\varphi_{\alpha}(x)+\sum_{k\geq1}f_{\alpha,k}(x)\zeta_{\alpha}^{jk}y^{\frac{k}{\nu_{\alpha}}}, \quad\text{où}\hspace{1mm} f_{\alpha,k}\in\mathbb{C}\{x\},\end{aligned}$$ avec $f_{\alpha,1}\not\equiv0$ et $\mathrm{\zeta_{\alpha}}=\exp(\tfrac{2\mathrm{i}\pi}{\nu_{\alpha}}).$ De plus, pour $\alpha=1$, si $\nu_1\geq2,$ alors $$\begin{aligned} \label{equa:preuve-thm-nu1} &\hspace{-3.6cm}(f_{1,1}(x))^{\nu_1}=-\frac{\partial_{y}F\big(x,0,\varphi_{1}(x)\big)}{Q(x,\varphi_{1}(x))}\end{aligned}$$ et, pour tout $\alpha\in\{1,\ldots,n\}$ tel que $\nu_\alpha\geq2$, on a $$\begin{aligned} \label{equa:preuve-thm-nu-alpha} &\hspace{1.4cm}\frac{f_{\alpha,2}(x)}{(f_{\alpha,1}(x))^2} =\frac{1}{\nu_{\alpha}}\left[\dfrac{\partial_{p}\partial_{y}F\big(x,0,\varphi_{\alpha}(x)\big)}{\partial_{y}F\big(x,0,\varphi_{\alpha}(x)\big)} -\hspace{-3.5mm}\sum\limits_{\hspace{3.5mm}\beta=1,\beta\neq\alpha}^{n}\frac{\nu_{\beta}}{\varphi_{\alpha}(x)-\varphi_{\beta}(x)}\right].\end{aligned}$$ Posons $\lambda_0(x,y)=\varphi_{1}(x)+yf(x,y)$; d'après [@BM22arxiv Lemme 2.8], on a $\eta(\mathcal{F}\boxtimes\mathcal{F}_{i}^{1}\boxtimes\mathcal{F}_{j}^{1})=a_{i,j}(x,y)\mathrm{d}x+b_{i,j}(x,y)\mathrm{d}y,$ où $$\begin{aligned} &\hspace{-3.3cm}a_{i,j}= -\frac{\left(\partial_y(\lambda_{1,i}\lambda_{1,j})-\partial_x\lambda_{0}\right)\lambda_{0}}{(\lambda_{1,i}-\lambda_{0})(\lambda_{1,j}-\lambda_{0})} -\frac{\left(\partial_y(\lambda_{1,i}\lambda_{0})-\partial_x\lambda_{1,j}\right)\lambda_{1,j}}{(\lambda_{1,i}-\lambda_{1,j})(\lambda_{0}-\lambda_{1,j})} -\frac{\left(\partial_y(\lambda_{1,j}\lambda_{0})-\partial_x\lambda_{1,i}\right)\lambda_{1,i}}{(\lambda_{1,j}-\lambda_{1,i})(\lambda_{0}-\lambda_{1,i})} \\ &\hspace{-4.3cm}{\fontsize{11}{11pt}\text{et}} \\ &\hspace{-3.3cm}b_{i,j}= \frac{\partial_y(\lambda_{1,i}\lambda_{1,j})-\partial_x\lambda_{0}}{(\lambda_{1,i}-\lambda_{0})(\lambda_{1,j}-\lambda_{0})} +\frac{\partial_y(\lambda_{1,i}\lambda_{0})-\partial_x\lambda_{1,j}}{(\lambda_{1,i}-\lambda_{1,j})(\lambda_{0}-\lambda_{1,j})} +\frac{\partial_y(\lambda_{1,j}\lambda_{0})-\partial_x\lambda_{1,i}}{(\lambda_{1,j}-\lambda_{1,i})(\lambda_{0}-\lambda_{1,i})}.\end{aligned}$$ En écrivant $f(x,y)=\sum_{k\geq0}f_{0,k}(x)y^k$ et en posant $w_{1}=y^{\frac{1}{\nu_{1}}}$, un calcul élémentaire conduit aux égalités suivantes: $$\begin{aligned} &\hspace{-1.6cm}\partial_y(\lambda_{1,i}\lambda_{1,j})-\partial_x\lambda_{0}=\frac{1}{\nu_1y} \left[ (\zeta_{1}^{i}+\zeta_{1}^{j})\varphi_{1}f_{1,1}w_1+ 2\left(\zeta_{1}^{i+j}f_{1,1}^{2}+(\zeta_{1}^{2i}+\zeta_{1}^{2j})\varphi_{1}f_{1,2}-\delta_{\nu_1,2}\varphi_{1}^{'}\right)w_{1}^{2}+\cdots \right],\\ &\hspace{-1.6cm}\partial_y(\lambda_{1,i}\lambda_{0})-\partial_x\lambda_{1,j}=\frac{1}{\nu_1y} \left[ \zeta_{1}^{i}\varphi_1f_{1,1}w_1+2\left(\zeta_{1}^{2i}\varphi_1f_{1,2}+(\varphi_1f_{0,0}-\varphi_{1}^{'})\delta_{\nu_1,2}\right)w_{1}^{2}+\cdots \right],\\ &\hspace{-1.6cm}(\lambda_{1,i}-\lambda_{0})(\lambda_{1,j}-\lambda_{0})=\zeta_{1}^{i+j}f_{1,1}^{2}w_{1}^{2}+(\zeta_{1}^{2i+j}+\zeta_{1}^{i+2j})f_{1,1}f_{1,2}w_{1}^{3}+\cdots,\\ &\hspace{-1.6cm}(\lambda_{1,i}-\lambda_{1,j})(\lambda_{0}-\lambda_{1,j})=(\zeta_{1}^{2j}-\zeta_{1}^{i+j})f_{1,1}^{2}w_{1}^{2}+ f_{1,1}\left(\big(2\zeta_{1}^{3j}-\zeta_{1}^{2i+j}-\zeta_{1}^{i+2j}\big)f_{1,2}-2\delta_{\nu_1,2}f_{0,0}\right)w_{1}^{3}+\cdots.\end{aligned}$$ Ces égalités permettent de vérifier que $a_{i,j}$ et $b_{i,j}$ s'écrivent $$\begin{aligned} &\hspace{-0.6cm}a_{i,j}=\frac{\varphi_{1}\left(\varphi_{1}f_{1,2}-f_{1,1}^{2}-\delta_{\nu_1,2}\varphi_{1}f_{0,0}\right)+w_1A_{i,j}}{\nu_1yf_{1,1}^{2}}, && b_{i,j}=\frac{2f_{1,1}^{2}-\varphi_{1}f_{1,2}+\delta_{\nu_1,2}\varphi_{1}f_{0,0}+w_1B_{i,j}}{\nu_1yf_{1,1}^{2}},\end{aligned}$$ où $A_{i,j},B_{i,j}\in\mathbb{C}\{x,w_{1}\}.$ Comme $\eta_1$ est une $1$-forme uniforme et méromorphe, nous en déduisons que la partie polaire de la série de [Laurent]{.smallcaps} de $\eta_1$ en $y=0$ est donnée par $\frac{\theta_1}{y}$, où $$\begin{aligned} \theta_1&=\binom{{\nu_{1}}}{{2}} \left(\frac{\varphi_{1}(x)\Big(\varphi_{1}(x)f_{1,2}(x)-f_{1,1}(x)^{2}-\delta_{\nu_1,2}\varphi_{1}(x)f_{0,0}(x)\Big)}{\nu_1f_{1,1}(x)^{2}}\mathrm{d}x+ \frac{2f_{1,1}(x)^{2}-\varphi_{1}(x)f_{1,2}(x)+\delta_{\nu_1,2}\varphi_{1}(x)f_{0,0}(x)}{\nu_1f_{1,1}(x)^{2}}\mathrm{d}y \right) \\ &=\frac{1}{2}(\nu_{1}-1) \left[ \varphi_{1}(x) \left(\frac{\delta_{\nu_1,2}f_{0,0}(x)}{f_{1,1}(x)^{2}}-\frac{f_{1,2}(x)}{f_{1,1}(x)^{2}}\right)\big(\mathrm{d}y-\varphi_{1}(x)\mathrm{d}x\big)+2\mathrm{d}y-\varphi_{1}(x)\mathrm{d}x \right].\end{aligned}$$ En vertu de ([\[equa:preuve-thm-nu1\]](#equa:preuve-thm-nu1){reference-type="ref" reference="equa:preuve-thm-nu1"}), ([\[equa:preuve-thm-nu-alpha\]](#equa:preuve-thm-nu-alpha){reference-type="ref" reference="equa:preuve-thm-nu-alpha"}) et de l'égalité $f_{0,0}(x)=f(x,0)$, la $1$-forme $\theta_1$ se réécrit $$\begin{aligned} \theta_1=\frac{1}{2}\left(1-\frac{1}{\nu_1}\right) \left(d-1-\varphi_{1}(x)\dfrac{\partial_{p}\partial_{y}F(x,0,\varphi_{1}(x))+2\delta_{\nu_1,2}f(x,0)Q(x,\varphi_{1}(x))}{\partial_{y}F(x,0,\varphi_{1}(x))} +\sum_{\alpha=2}^{n}\frac{\nu_{\alpha}\varphi_{\alpha}(x)}{\varphi_{1}(x)-\varphi_{\alpha}(x)} \right)\big(\mathrm{d}y-\varphi_{1}(x)\mathrm{d}x\big)+\frac{1}{2}(\nu_1-1)\mathrm{d}y.\end{aligned}$$ Passons maintenant à $\eta_2.$ Posons $w_{\alpha,1}=y^{\frac{1}{\nu_{1}\nu_{\alpha}}}$; toujours d'après [@BM22arxiv Lemme 2.8], on a $\eta(\mathcal{F}\boxtimes\mathcal{F}_{i}^{1}\boxtimes\mathcal{F}_{j}^{\alpha})=a_{i,j}^{\alpha}(x,y)\mathrm{d}x+b_{i,j}^{\alpha}(x,y)\mathrm{d}y,$ où $$\begin{aligned} \hspace{-1.6cm}a_{i,j}^{\alpha}&= -\frac{\left(\partial_y(\lambda_{1,i}\lambda_{\alpha,j})-\partial_x\lambda_{0}\right)\lambda_{0}}{(\lambda_{1,i}-\lambda_{0})(\lambda_{\alpha,j}-\lambda_{0})} -\frac{\left(\partial_y(\lambda_{1,i}\lambda_{0})-\partial_x\lambda_{\alpha,j}\right)\lambda_{\alpha,j}}{(\lambda_{1,i}-\lambda_{\alpha,j})(\lambda_{0}-\lambda_{\alpha,j})} -\frac{\left(\partial_y(\lambda_{\alpha,j}\lambda_{0})-\partial_x\lambda_{1,i}\right)\lambda_{1,i}}{(\lambda_{\alpha,j}-\lambda_{1,i})(\lambda_{0}-\lambda_{1,i})}\\ \hspace{-1.6cm}&=\frac{1}{\nu_{1}y} \left(\frac{\varphi_{1}\varphi_{\alpha}}{\varphi_{1}-\varphi_{\alpha}}+w_{\alpha,1}A_{i,j}^{\alpha}\right)\end{aligned}$$ et $$\begin{aligned} \hspace{-2.2cm}b_{i,j}^{\alpha}&= \frac{\partial_y(\lambda_{1,i}\lambda_{\alpha,j})-\partial_x\lambda_{0}}{(\lambda_{1,i}-\lambda_{0})(\lambda_{\alpha,j}-\lambda_{0})}+ \frac{\partial_y(\lambda_{1,i}\lambda_{0})-\partial_x\lambda_{\alpha,j}}{(\lambda_{1,i}-\lambda_{\alpha,j})(\lambda_{0}-\lambda_{\alpha,j})}+ \frac{\partial_y(\lambda_{\alpha,j}\lambda_{0})-\partial_x\lambda_{1,i}}{(\lambda_{\alpha,j}-\lambda_{1,i})(\lambda_{0}-\lambda_{1,i})}\\ \hspace{-2.2cm}&=-\frac{1}{\nu_{1}y} \left(\frac{\varphi_{\alpha}}{\varphi_{1}-\varphi_{\alpha}}+w_{\alpha,1}B_{i,j}^{\alpha}\right),\end{aligned}$$ où $A_{i,j}^{\alpha},B_{i,j}^{\alpha}\in\mathbb{C}\{x,w_{\alpha,1}\}.$ La $1$-forme $\eta_2$ étant uniforme et méromorphe, il en résulte que la partie polaire de la série de [Laurent]{.smallcaps} de $\eta_2$ en $y=0$ est donnée par $\frac{\theta_2}{y}$, où $$\begin{aligned} \theta_2&= \sum_{\alpha=2}^{n}\nu_{1}\nu_{\alpha} \left(\frac{\varphi_{1}(x)\varphi_{\alpha}(x)}{\nu_{1}\left(\varphi_{1}(x)-\varphi_{\alpha}(x)\right)}\mathrm{d}x-\frac{\varphi_{\alpha}(x)}{\nu_{1}\left(\varphi_{1}(x)-\varphi_{\alpha}(x)\right)}\mathrm{d}y\right)\\ &=-(\mathrm{d}y-\varphi_{1}(x)\mathrm{d}x)\sum_{\alpha=2}^{n}\frac{\nu_{\alpha}\varphi_{\alpha}(x)}{\varphi_{1}(x)-\varphi_{\alpha}(x)}.\end{aligned}$$ De façon analogue, en posant $w_{\alpha}=y^{\frac{1}{\nu_{\alpha}}}$ et en utilisant [@BM22arxiv Lemme 2.8], on obtient que $$\begin{aligned} \eta(\mathcal{F}\boxtimes\mathcal{F}_{i}^{\alpha}\boxtimes\mathcal{F}_{j}^{\alpha})=\frac{1}{\nu_{\alpha}y} \left[ \left(-\frac{\varphi_{1}(x)\varphi_{\alpha}(x)}{\varphi_{1}(x)-\varphi_{\alpha}(x)}+w_{\alpha}\,\tilde{A}_{i,j}^{\alpha}(x,w_{\alpha})\right)\mathrm{d}x+ \left(\frac{\varphi_{1}(x)}{\varphi_{1}(x)-\varphi_{\alpha}(x)}+w_{\alpha}\,\tilde{B}_{i,j}^{\alpha}(x,w_{\alpha})\right)\mathrm{d}y \right],\end{aligned}$$ où $\tilde{A}_{i,j}^{\alpha},\tilde{B}_{i,j}^{\alpha}\in\mathbb{C}\{x,w_{\alpha}\}$, de sorte que la partie polaire de la série de [Laurent]{.smallcaps} de $\eta_3$ en $y=0$ est donnée par $\frac{\theta_3}{y}$, où $$\begin{aligned} \theta_3&= \sum_{\alpha=2}^{n}\binom{{\nu_{\alpha}}}{{2}}\left( -\frac{\varphi_{1}(x)\varphi_{\alpha}(x)}{\nu_{\alpha}\left(\varphi_{1}(x)-\varphi_{\alpha}(x)\right)}\mathrm{d}x+ \frac{\varphi_{1}(x)}{\nu_{\alpha}\left(\varphi_{1}(x)-\varphi_{\alpha}(x)\right)}\mathrm{d}y \right)\\ &=\frac{1}{2}\sum_{\alpha=2}^{n}\frac{(\nu_{\alpha}-1)\varphi_1(x)\left(\mathrm{d}y-\varphi_{\alpha}(x)\mathrm{d}x\right)}{\varphi_1(x)-\varphi_{\alpha}(x)}.\end{aligned}$$ Enfin, comme $(\varphi_{1}-\varphi_{\alpha})(\varphi_{\alpha}-\varphi_{\beta})(\varphi_{\beta}-\varphi_{1})\not\equiv0$ pour tous $\beta>\alpha\geq2$, [@BM22arxiv Lemme 2.8] entraîne que la $1$-forme $\eta(\mathcal{F}\boxtimes\mathcal{F}_{i}^{\alpha}\boxtimes\mathcal{F}_{j}^{\beta})$ n'a pas de pôles le long de $y=0$; il en est donc de même de la $1$-forme $\eta_4.$ Par suite, la partie polaire de la série de [Laurent]{.smallcaps} de $\eta(\mathcal{F}\boxtimes\mathcal{W})$ en $y=0$ est donnée par $\frac{\widehat{\theta}}{y}$, où $$\begin{aligned} \widehat{\theta}&=\theta_0+\theta_1+\theta_2+\theta_3\\ &=\frac{1}{6}\Big((\nu_1+1)h(x)-(\nu_1-1)\psi_{1}(x)\Big)\big(\mathrm{d}y-\varphi_{1}(x)\mathrm{d}x\big)+\frac{1}{2}(\nu_1-1)\mathrm{d}y+ \frac{1}{6}\sum\limits_{\alpha=1}^{n}\big(\nu_{\alpha}-1\big)\Big[\psi_{\alpha}(x)\big(\mathrm{d}y-\varphi_{\alpha}(x)\mathrm{d}x\big) +\big(\nu_{\alpha}-2\big)\mathrm{d}y\Big]\\ &\hspace{4.5mm}+\frac{1}{2}\sum_{\alpha=2}^{n}\frac{(\nu_{\alpha}-1)\varphi_1(x)\left(\mathrm{d}y-\varphi_{\alpha}(x)\mathrm{d}x\right)}{\varphi_1(x)-\varphi_{\alpha}(x)}\\ &=\frac{1}{6}(\nu_1+1)\left[h(x)\Big(\mathrm{d}y-\varphi_1(x)\mathrm{d}x\Big)+(\nu_1-1)\mathrm{d}y\right]+ \frac{1}{6}\sum_{\alpha=2}^{n}(\nu_\alpha-1)\left[\left(\psi_\alpha(x)+\frac{3\varphi_1(x)}{\varphi_1(x)-\varphi_\alpha(x)}\right) \big(\mathrm{d}y-\varphi_\alpha(x)\mathrm{d}x\big)+(\nu_\alpha-2)\mathrm{d}y\right]\\ &=\frac{\theta}{6},\end{aligned}$$ d'où l'énoncé. ◻ **Démonstration du Théorème [Théorème 20](#thm:holomorphie-courbure-homogene-D-ell){reference-type="ref" reference="thm:holomorphie-courbure-homogene-D-ell"}*.* Comme dans la démonstration du Théorème [Théorème 15](#thm:holomorphie-courbure-homogene-D-neq-D-ell){reference-type="ref" reference="thm:holomorphie-courbure-homogene-D-neq-D-ell"}, à conjugaison linéaire près, nous pouvons supposer que $\beta\neq0$ et $b_i\neq0$ pour tout $i\in\{1,\ldots,n\}.$ Puis, en posant $r_0:=-\frac{\alpha}{\beta}$ et $r_i:=\frac{a_i}{b_i}$ pour $i\in\{1,\ldots,n\},$ [@BM22arxiv Lemme 3.5] implique l'existence d'une constante $c\in\mathbb{C}^*$ telle que $$-A(1,z)=p_0B(1,z)-c\prod_{i=0}^{n}(z-r_i)^{\nu_i}.$$ Comme $A,B\in\mathbb{C}[x,y]_{d-1},$ l'équation ([\[equa:LegH\]](#equa:LegH){reference-type="ref" reference="equa:LegH"}) décrivant $\mathrm{Leg}\mathscr{H}$ dans la carte affine $(p,q)$ devient alors $$\begin{aligned} x^{d}\Big(p-\tfrac{q}{x}-r_0\Big)\Big((p-p_0)B(1,p-\tfrac{q}{x})+c\prod_{i=0}^{n}(p-\tfrac{q}{x}-r_i)^{\nu_i}\Big)=0, \qquad \text{avec} \qquad x=\frac{\mathrm{d}q}{\mathrm{d}p}.\end{aligned}$$ Posons $\check{x}:=q$,  $\check{y}:=p-p_0$  et  $\check{p}:=\dfrac{\mathrm{d}\check{y}}{\mathrm{d}\check{x}}=\dfrac{1}{x}$; dans ce nouveau système de coordonnées $D_\ell=\{\check{y}=0\}$ et $\mathrm{Leg}\mathscr{H}=\mathrm{Leg}\ell\boxtimes\mathrm{Leg}\mathcal{H}$ est donné par l'équation différentielle $(\check{y}+p_0-\check{p}\check{x}-r_0)F(\check{x},\check{y},\check{p})=0$, où $$\begin{aligned} F(\check{x},\check{y},\check{p})=\check{y}B(1,\check{y}+p_0-\check{p}\check{x})+c\prod_{i=0}^{n}(\check{y}+p_0-\check{p}\check{x}-r_i)^{\nu_i}.\end{aligned}$$ Nous avons $F(\check{x},0,\check{p})=c(-\check{x})^{d-1}\prod_{i=0}^{n}\big(\check{p}-\varphi_{i}(\check{x})\big)^{\nu_i}$, où $\varphi_{i}(\check{x})=\frac{p_0-r_i}{\check{x}}.$ De plus le feuilletage radial $\mathrm{Leg}\ell$ est décrit par $\check{\omega}_0=\mathrm{d}\check{y}-\big(\varphi_{0}(\check{x})+\frac{\check{y}}{\check{x}}\big)\mathrm{d}\check{x}$; en particulier nous avons $D_\ell\subset\mathrm{Tang}(\mathrm{Leg}\ell,\mathrm{Leg}\mathcal{H}).$ Notons que si $\nu_i\geq2$, alors $\partial_{\check{y}}F\big(\check{x},0,\varphi_{i}(\check{x})\big)=B(1,r_i)\neq0$; comme $\partial_{\check{p}}F\big(\check{x},0,\varphi_{i}(\check{x})\big)\not\equiv0$ si $\nu_i=1$, nous en déduisons que la surface $$\begin{aligned} S_{\mathrm{Leg}\mathcal{H}}:=\left\{(\check{x},\check{y},\check{p})\in\mathbb{P}\mathrm{T}^{*}\mathbb{\check{P}}^{2}_{\mathbb{C}}\hspace{1mm}\vert\hspace{1mm}F(\check{x},\check{y},\check{p})=0\right\}\end{aligned}$$ est lisse le long de $D_\ell=\{\check{y}=0\}.$ Donc, d'après le Théorème [Théorème 21](#thm-critere-holomorphie-FW){reference-type="ref" reference="thm-critere-holomorphie-FW"}, la courbure de $\mathrm{Leg}\mathscr{H}$ est holomorphe sur $D_\ell=\{\check{y}=0\}$ si et seulement si $$\begin{aligned} &\hspace{-5cm} (\nu_0+1)\varphi_0(\check{x})h(\check{x})+ \sum_{i=1}^{n}(\nu_{i}-1)\varphi_{i}(\check{x})\left(\psi_{i}(\check{x})+\frac{3\varphi_{0}(\check{x})}{\varphi_{0}(\check{x})-\varphi_{i}(\check{x})}\right)\equiv0 \\ &\hspace{-6.14cm}{\fontsize{11}{11pt}\text{et}} \\ &\hspace{-5cm} \frac{\mathrm{d}}{\mathrm{d}\check{x}}\left((\nu_0+1)h(\check{x})+ \sum_{i=1}^{n}(\nu_{i}-1)\left(\psi_{i}(\check{x})+\frac{3\varphi_{0}(\check{x})}{\varphi_{0}(\check{x})-\varphi_{i}(\check{x})}\right)\right)\equiv0,\end{aligned}$$ où $$\begin{aligned} &\hspace{0.6cm} h(\check{x})=\frac{1}{\nu_{0}} \left[ (\nu_{0}-1)\left(d-1-\varphi_{0}(\check{x})\dfrac{\partial_{\check{p}}\partial_{\check{y}}F\big(\check{x},0,\varphi_{0}(\check{x})\big) -2c\hspace{0.2mm}\delta_{\nu_0,2}(-\check{x})^{d-2}\prod_{j=1}^{n}\big(\varphi_{0}(\check{x})-\varphi_{j}(\check{x})\big)^{\nu_j}}{\partial_{\check{y}}F\big(\check{x},0,\varphi_{0}(\check{x})\big)}\right) -(2\nu_{0}+1)\sum\limits_{j=1}^{n}\frac{\nu_{j}\varphi_{j}(\check{x})}{\varphi_{0}(\check{x})-\varphi_{j}(\check{x})} \right]\end{aligned}$$ et, pour tout $i\in\{1,\ldots,n\}$ tel que $\nu_i\geq2,$ $$\begin{aligned} &\hspace{-3.8cm} \psi_{i}(\check{x})=\frac{1}{\nu_{i}} \left[ (\nu_{i}-2)\left(d-1-\varphi_{i}(\check{x})\dfrac{\partial_{\check{p}}\partial_{\check{y}}F\big(\check{x},0,\varphi_{i}(\check{x})\big)}{\partial_{\check{y}}F\big(\check{x},0,\varphi_{i}(\check{x})\big)}\right) -2(\nu_{i}+1)\hspace{-3.5mm}\sum\limits_{\hspace{3.5mm}j=0,j\neq i}^{n}\frac{\nu_{j}\varphi_{j}(\check{x})}{\varphi_{i}(\check{x})-\varphi_{j}(\check{x})} \right].\end{aligned}$$ Or, si $\nu_i\geq2$ alors $\partial_{\check{p}}\partial_{\check{y}}F\big(\check{x},0,\varphi_{i}(\check{x})\big)=-\check{x}\Big(\partial_{y}B(1,r_i)+2c\hspace{0.2mm}\delta_{\nu_i,2}\prod\limits_{j=0,j\neq i}^{n}\big(r_i-r_j\big)^{\nu_{j}}\Big).$ Il en résulte que $$\begin{aligned} &\hspace{-2.6cm} h(\check{x})=h_0:=\frac{1}{\nu_{0}} \left[ (\nu_{0}-1)\left(d-1+\frac{(p_0-r_0)\partial_{y}B(1,r_0)}{B(1,r_0)}\right) +(2\nu_{0}+1)\sum\limits_{j=1}^{n}\frac{\nu_{j}(p_0-r_j)}{r_0-r_j} \right] \\ &\hspace{-3.85cm}{\fontsize{11}{11pt}\text{et}} \\ &\hspace{-2.6cm} \psi_{i}(\check{x})=\psi_i:=\frac{1}{\nu_{i}} \left[ (\nu_{i}-2)\left(d-1+\frac{(p_0-r_i)\partial_{y}B(1,r_i)}{B(1,r_i)}\right) +2(\nu_{i}+1)\hspace{-3.5mm}\sum\limits_{\hspace{3.5mm}j=0,j\neq i}^{n}\frac{\nu_{j}(p_0-r_j)}{r_i-r_j} \right].\end{aligned}$$ Ainsi $K(\mathrm{Leg}\mathscr{H})$ est holomorphe le long de $D_\ell=\{\check{y}=0\}$ si et seulement si $$\begin{aligned} (\nu_0+1)(p_0-r_0)h_0+\sum_{i=1}^{n}(\nu_i-1)(p_0-r_i)\left(\psi_i+\tfrac{3(p_0-r_0)}{r_i-r_0}\right)=0.\end{aligned}$$ Par ailleurs, nous avons (*cf.* preuve de [@BM22arxiv Théorème 3.1]) $$\begin{aligned} & \sum\limits_{j=1}^{n}\frac{\nu_{j}(p_0-r_j)}{r_0-r_j} =\frac{\left|\begin{array}{cc} \partial_{x}P_0(1,r_0;-r_0,1) & A(1,r_0) \vspace{2mm} \\ \partial_{y}P_0(1,r_0;-r_0,1) & B(1,r_0) \end{array}\right|}{B(1,r_0)P_0(1,r_0;-r_0,1)}, && \hspace{-3.5mm}\sum\limits_{\hspace{3.5mm}j=0,j\neq i}^{n}\frac{\nu_{j}(p_0-r_j)}{r_i-r_j} =\frac{\left|\begin{array}{cc} \partial_{x}P_i(1,r_i;r_i,1) & A(1,r_i) \vspace{2mm} \\ \partial_{y}P_i(1,r_i;r_i,1) & B(1,r_i) \end{array}\right|}{B(1,r_i)P_i(1,r_i;r_i,1)}\quad \big({\fontsize{10}{10pt}\text{pour }} i=1,\ldots,n\big)\end{aligned}$$ et, pour tout $i\in\{0,\ldots,n\}$ tel que $\nu_i\geq2,$ $$\begin{aligned} (d-1)B(1,r_i)+(p_0-r_i)\partial_{y}B(1,r_i)=\partial_{x}B(1,r_i)-\partial_{y}A(1,r_i).\end{aligned}$$ Nous en déduisons, par définition des polynômes $Q_i,$ que $$\begin{aligned} h_0=\frac{Q_{0}(1,r_0;-r_0,1)}{\nu_{0}B(1,r_0)P_{0}(1,r_0;-r_0,1)} \qquad\hspace{2mm}\text{et}\qquad\hspace{2mm} \psi_i=\frac{Q_{i}(1,r_i;r_i,1)}{\nu_{i}B(1,r_i)P_{i}(1,r_i;r_i,1)}.\end{aligned}$$ Par conséquent, $K(\mathrm{Leg}\mathscr{H})$ est holomorphe sur $D_\ell=\{\check{y}=0\}$ si et seulement si $$\begin{aligned} \left(1+\frac{1}{\nu_0}\right)\frac{(p_0-r_0)Q_{0}(1,r_0;-r_0,1)}{B(1,r_0)P_{0}(1,r_0;-r_0,1)} +\sum_{i=1}^{n}\left(1-\frac{1}{\nu_i}\right)\left(p_0-r_i\right) \left(\frac{Q_{i}(1,r_i;r_i,1)}{B(1,r_i)P_{i}(1,r_i;r_i,1)}+\frac{3\nu_{i}(p_0-r_0)}{r_i-r_0}\right)=0,\end{aligned}$$ ce qui termine la démonstration. ◻ **Corollaire 22**. * *Soit $\mathscr{H}=\ell\boxtimes\mathcal{H}$ un pré-feuilletage homogène de co-degré $1$ et de degré $d\geq3$ sur $\mathbb{P}^{2}_{\mathbb{C}}$, défini par la $1$-forme $$\begin{aligned} &\hspace{1.5cm}\omega=(\alpha\,x+\beta\,y)\left(A(x,y)\mathrm{d}x+B(x,y)\mathrm{d}y\right),\quad A,B\in\mathbb{C}[x,y]_{d-1},\hspace{2mm}\mathrm{pgcd}(A,B)=1.\end{aligned}$$ Supposons que la droite $\ell$ ne soit pas invariante par $\mathcal{H}$ et que la fibre ${\mspace{2mu}\underline{\mspace{-2mu}\mathcal{G}\mspace{-2mu}}\mspace{2mu}}_{\mathcal{H}}^{-1}\big({\mspace{2mu}\underline{\mspace{-2mu}\mathcal{G}\mspace{-2mu}}\mspace{2mu}}_{\mathcal{H}}([-\alpha:\beta])\big)$ ne contienne aucun point critique non fixe de ${\mspace{2mu}\underline{\mspace{-2mu}\mathcal{G}\mspace{-2mu}}\mspace{2mu}}_{\mathcal H}.$ Alors la courbure de $\mathrm{Leg}\mathscr{H}$ est holomorphe sur $D_{\ell}=\mathcal{G}_{\mathcal{H}}(\ell)$ si et seulement si $Q(\beta,-\alpha;\alpha,\beta)=0,$ où $$\begin{aligned} \label{equa:polynome-Q} & Q(x,y;\alpha,\beta):=\left| \begin{array}{cc} \dfrac{\partial{P}}{\partial{x}} & A(\beta,-\alpha) \vspace{2mm} \\ \dfrac{\partial{P}}{\partial{y}} & B(\beta,-\alpha) \end{array}\right| &&\text{et}&& P(x,y;\alpha,\beta):=\frac{\left| \begin{array}{cc} A(x,y) & A(\beta,-\alpha) \\ B(x,y) & B(\beta,-\alpha) \end{array} \right|}{\alpha\,x+\beta\,y}.\end{aligned}$$* * *Remarque 23*. En particulier, en degré $d=3$, la courbure de $\mathrm{Leg}\mathscr{H}$ est holomorphe le long de $D_{\ell}$ si et seulement si la droite d'équation $A(\beta,-\alpha)x+B(\beta,-\alpha)y=0$ est invariante par $\mathcal{H}$, ou, de façon équivalente, si et seulement si ${\mspace{2mu}\underline{\mspace{-2mu}\mathcal{G}\mspace{-2mu}}\mspace{2mu}}_{\mathcal{H}}\big({\mspace{2mu}\underline{\mspace{-2mu}\mathcal{G}\mspace{-2mu}}\mspace{2mu}}_{\mathcal{H}}([-\alpha:\beta])\big)={\mspace{2mu}\underline{\mspace{-2mu}\mathcal{G}\mspace{-2mu}}\mspace{2mu}}_{\mathcal{H}}([-\alpha:\beta]).$ En effet, en posant $a=A(\beta,-\alpha)$, $b=B(\beta,-\alpha)$ et $P(x,y;\alpha,\beta)=f(\alpha,\beta)x+g(\alpha,\beta)y$ on obtient $$\begin{aligned} Q(\beta,-\alpha\hspace{0.2mm};\alpha,\beta)=f(\alpha,\beta)b-g(\alpha,\beta)a=P(b,-a\hspace{0.2mm};\alpha,\beta) =-\frac{bA(b,-a)-aB(b,-a)}{\beta\,a-\alpha\,b} =-\frac{\mathrm{C}_{\hspace{-0.3mm}\mathcal{H}}\left(b,-a\right)}{\mathrm{C}_{\hspace{-0.3mm}\mathcal{H}}(\beta,-\alpha)},\end{aligned}$$ où $\mathrm{C}_{\hspace{-0.3mm}\mathcal{H}}=x\hspace{0.2mm}A+yB$[\[not:C-H\]]{#not:C-H label="not:C-H"} désigne le cône tangent de $\mathcal{H}$ en l'origine $O,$ *voir* [@BM18Bull Section 2]. En combinant les Corollaires [Corollaire 14](#cor:platitude-homogene-convexe-ell-non-invariante-non-effectif){reference-type="ref" reference="cor:platitude-homogene-convexe-ell-non-invariante-non-effectif"} et [Corollaire 22](#cor:holomorphie-courbure-homogene-D-ell-fibre-sans-point-critique-non-fixe){reference-type="ref" reference="cor:holomorphie-courbure-homogene-D-ell-fibre-sans-point-critique-non-fixe"}, nous obtenons le: **Corollaire 24**. * *Soit $\mathscr{H}=\ell\boxtimes\mathcal{H}$ un pré-feuilletage homogène de co-degré $1$ et de degré $d\geq3$ sur $\mathbb{P}^{2}_{\mathbb{C}}$, défini par la $1$-forme $$\begin{aligned} &\hspace{1.5cm}\omega=(\alpha\,x+\beta\,y)\left(A(x,y)\mathrm{d}x+B(x,y)\mathrm{d}y\right),\quad A,B\in\mathbb{C}[x,y]_{d-1},\hspace{2mm}\mathrm{pgcd}(A,B)=1.\end{aligned}$$ Supposons que le feuilletage homogène $\mathcal{H}$ soit convexe et que la droite $\ell$ ne soit pas invariante par $\mathcal{H}.$ Alors le $d$-tissu $\mathrm{Leg}\mathscr{H}$ est plat si et seulement si $Q(\beta,-\alpha;\alpha,\beta)=0,$ où $Q$ est le polynôme donné par ([\[equa:polynome-Q\]](#equa:polynome-Q){reference-type="ref" reference="equa:polynome-Q"}).* * Voici deux exemples qui nous seront utiles au paragraphe §[5](#sec:pre-pre-feuill-convexe-reduit){reference-type="ref" reference="sec:pre-pre-feuill-convexe-reduit"}. *Exemple 25*. Considérons le feuilletage homogène $\mathcal{H}_{0}^{d-1}$ défini en carte affine $z=1$ par la $1$-forme[\[not:omega0\]]{#not:omega0 label="not:omega0"} $$\omega_{\hspace{0.2mm}0}^{\hspace{0.2mm}d-1}=(d-2)y^{d-1}\mathrm{d}x+x\left(x^{d-2}-(d-1)y^{d-2}\right)\mathrm{d}y.$$ On sait d'après [@BM18Bull Exemple 6.5] que $\mathcal{H}_{0}^{d-1}$ est convexe, de type $1\cdot\mathrm{R}_{d-2}+(d-2)\cdot\mathrm{R}_{1}$ et à diviseur d'inflexion[\[not:Inflex-F\]]{#not:Inflex-F label="not:Inflex-F"}[\[not:Inflex-Invariant-F\]]{#not:Inflex-Invariant-F label="not:Inflex-Invariant-F"} $$\mathrm{I}_{\mathcal{H}_{0}^{d-1}} =\mathrm{I}_{\mathcal{H}_{0}^{d-1}}^{\hspace{0.2mm}\mathrm{inv}} =-(d-1)(d-2)x\hspace{0.2mm}z\hspace{0.2mm}y^{d-1}\big(y^{d-2}-x^{d-2}\big)^2.$$ Si $\ell$ est l'une des droites invariantes de $\mathcal{H}_{0}^{d-1}$, *i.e.* si $\ell\in\{xyz(y-\zeta^{k} x)=0,\,k=0,\ldots,d-3\},$ où $\zeta=\mathrm{exp}\left(\frac{2\mathrm{i}\pi}{d-2}\right),$ alors le $d$-tissu $\mathrm{Leg}(\ell\boxtimes\mathcal{H}_{0}^{d-1})$ est plat par le Corollaire [Corollaire 13](#cor:platitude-pre-feuilletage-homogene-convexe){reference-type="ref" reference="cor:platitude-pre-feuilletage-homogene-convexe"}. Si $\ell=(y-\rho x=0)$ n'est pas invariante par $\mathcal{H}_{0}^{d-1}$, alors le $d$-tissu $\mathrm{Leg}(\ell\boxtimes\mathcal{H}_{0}^{d-1})$ est plat si et seulement si $\rho^{d-2}=\frac{1}{2(d-2)}$, *i.e.* si et seulement si $\ell\in\left\{y-\rho_{0}\zeta^{k}\,x=0,\,k=0,\ldots,d-3\right\},$ où $\rho_{0}=\sqrt[d-2]{\frac{1}{2\,d-4}}.$ En effet, avec les notations du Corollaire [Corollaire 22](#cor:holomorphie-courbure-homogene-D-ell-fibre-sans-point-critique-non-fixe){reference-type="ref" reference="cor:holomorphie-courbure-homogene-D-ell-fibre-sans-point-critique-non-fixe"}, nous avons $$\begin{aligned} & Q(x,y;-\rho,1)=\left(1-(d-1)\rho^{d-2}\right)\frac{\partial{P}}{\partial{x}}-(d-2)\rho^{d-1}\frac{\partial{P}}{\partial{y}} \\ &\hspace{-1.17cm}{\fontsize{11}{11pt}\text{et}} \\ & P(x,y;-\rho,1)=-(d-2)\left((d-1)(\rho y)^{d-2}-\frac{y^{d-1}-(\rho x)^{d-1}}{y-\rho x}\right) =-(d-2)\left((d-1)(\rho y)^{d-2}-\sum_{i=0}^{d-2}\rho^ix^iy^{d-2-i}\right),\end{aligned}$$ de sorte qu'en vertu du Corollaire [Corollaire 24](#cor:platitude-homogene-convexe-ell-non-invariante-effectif){reference-type="ref" reference="cor:platitude-homogene-convexe-ell-non-invariante-effectif"}, la platitude de $\mathrm{Leg}(\ell\boxtimes\mathcal{H}_{0}^{d-1})$ est caractérisée par $$\begin{aligned} 0=Q(1,\rho;-\rho,1)=\frac{1}{2}(d-1)(d-2)^2\rho^{d-2}\left(\rho^{d-2}-1\right)\left((2\,d-4)\rho^{d-2}-1\right) \Longleftrightarrow\rho^{d-2}=\frac{1}{2(d-2)}.\end{aligned}$$ Dans tous les cas, pour toute droite $\ell\subset\mathbb{P}^{2}_{\mathbb{C}}$ telle que $O\in\ell$ ou $\ell=L_\infty,$ le $d$-tissu $\mathrm{Leg}(\ell\boxtimes\mathcal{H}_{0}^{d-1})$ est plat si et seulement si, à conjugaison linéaire près, $\ell=L_\infty$ ou $\ell\in\left\{xy(y-x)(y-\rho_{0}\,x)=0\right\}.$ En effet, en posant $\varphi(x,y)=(x,\zeta^{k}y)$, nous avons $$\begin{aligned} \varphi^*\left(\big(y-\zeta^{k}x\big)\omega_{0}^{d-1}\right)=\zeta^{2k}\big(y-x\big)\omega_{0}^{d-1} &&\hspace{2mm}\text{et}\hspace{2mm}&& \varphi^*\left(\big(y-\rho_{0}\zeta^{k}\,x\big)\omega_{0}^{d-1}\right)=\zeta^{2k}\big(y-\rho_{0}\,x\big)\omega_{0}^{d-1}.\end{aligned}$$ *Exemple 26*. Pour $d\geq4$, soit $\mathcal{H}_{4}^{d-1}$ le feuilletage homogène défini en carte affine $z=1$ par la $1$-forme[\[not:omega4\]]{#not:omega4 label="not:omega4"} $$\begin{aligned} \hspace{7mm} \omega_{\hspace{0.2mm}4}^{\hspace{0.2mm}d-1}=y(\sigma_d\,x^{d-2}-y^{d-2})\mathrm{d}x+x(\sigma_d\,y^{d-2}-x^{d-2})\mathrm{d}y, \quad\quad\text{où}\hspace{1mm}\sigma_d=1+\tfrac{2}{d-3}.\end{aligned}$$ Ce feuilletage est convexe de type $(d-2)\cdot\mathrm{R}_2$; en effet un calcul élémentaire montre que $$\mathrm{I}_{\mathcal{H}_{4}^{d-1}} =\mathrm{I}_{\mathcal{H}_{4}^{d-1}}^{\hspace{0.2mm}\mathrm{inv}} =\sigma_d(\sigma_d-1)xyz\big(x^{d-2}+y^{d-2}\big)^3.$$ Soit $\ell$ une droite de $\mathbb{P}^{2}_{\mathbb{C}}$ telle que $O\in\ell$ ou $\ell=L_\infty.$ Si $\ell$ est invariante par $\mathcal{H}_{4}^{d-1},$ le Corollaire [Corollaire 13](#cor:platitude-pre-feuilletage-homogene-convexe){reference-type="ref" reference="cor:platitude-pre-feuilletage-homogene-convexe"} assure que $\mathrm{Leg}(\ell\boxtimes\mathcal{H}_{4}^{d-1})$ est plat, et on a $\ell\in\{xyz(y-\xi^{2k+1}\,x)=0,\,k=0,\ldots,d-3\},$ où $\xi=\mathrm{exp}\left(\frac{\mathrm{i}\pi}{d-2}\right).$ Si $\ell$ n'est pas invariante par $\mathcal{H}_{4}^{d-1}$, alors $\ell=\{y-\rho\,x=0\}$ avec $\rho(\rho^{d-2}+1)\neq0$; en appliquant le Corollaire [Corollaire 24](#cor:platitude-homogene-convexe-ell-non-invariante-effectif){reference-type="ref" reference="cor:platitude-homogene-convexe-ell-non-invariante-effectif"}, on obtient que le $d$-tissu $\mathrm{Leg}(\ell\boxtimes\mathcal{H}_{4}^{d-1})$ est plat si et seulement si $$0=Q(1,\rho;-\rho,1)=-\sigma_{d}(d-2)(\rho^{d-2}+1)^2(\rho^{d-2}-1),$$ donc si et seulement si $\rho^{d-2}=1,$ ce qui équivaut à $\ell\in\{y-\xi^{2k}x=0,\,k=0,\ldots,d-3\}.$ Notons que, dans tous les cas, le $d$-tissu $\mathrm{Leg}(\ell\boxtimes\mathcal{H}_{4}^{d-1})$ est plat si et seulement si, à conjugaison linéaire près, $\ell=L_\infty$ ou $\ell\in\left\{x(y-x)(y-\xi\,x)=0\right\}.$ En effet, en posant $\varphi(x,y)=(y,x)$ et $\psi(x,y)=(x,\xi^{2k}y),$ on a $$\begin{aligned} \varphi^*(y\omega_{4}^{d-1})=x\omega_{4}^{d-1},&&\hspace{3mm} \psi^*\left(\big(y-\xi^{2k}x\big)\omega_{4}^{d-1}\right)=\xi^{4k}\big(y-x\big)\omega_{4}^{d-1},&&\hspace{3mm} \psi^*\left(\big(y-\xi^{2k+1}\,x\big)\omega_{4}^{d-1}\right)=\xi^{4k}\big(y-\xi\,x\big)\omega_{4}^{d-1}.\end{aligned}$$ **Corollaire 27**. **Soient $d\geq3$ un entier et $\mathcal{H}$ un feuilletage homogène de degré $d-1$ sur $\mathbb{P}^{2}_{\mathbb{C}}$ défini par la $1$-forme $$\begin{aligned} &\hspace{1.5cm}\omega=A(x,y)\mathrm{d}x+B(x,y)\mathrm{d}y,\quad A,B\in\mathbb{C}[x,y]_{d-1},\hspace{2mm}\mathrm{pgcd}(A,B)=1.\end{aligned}$$ Supposons que $\mathcal{H}$ admette une droite d'inflexion $\ell=(\alpha\,x+\beta\,y=0)$ transverse d'ordre $\nu-1$. Supposons en outre que $[-\alpha:\beta]\in\mathbb{P}^{1}_{\mathbb{C}}$ soit le seul point critique non fixe de ${\mspace{2mu}\underline{\mspace{-2mu}\mathcal{G}\mspace{-2mu}}\mspace{2mu}}_{\mathcal H}$ dans sa fibre ${\mspace{2mu}\underline{\mspace{-2mu}\mathcal{G}\mspace{-2mu}}\mspace{2mu}}_{\mathcal H}^{-1}({\mspace{2mu}\underline{\mspace{-2mu}\mathcal{G}\mspace{-2mu}}\mspace{2mu}}_{\mathcal H}([-\alpha:\beta])).$ Posons $\mathscr{H}:=\ell\boxtimes\mathcal{H}.$ Alors la courbure de $\mathrm{Leg}\mathscr{H}$ est holomorphe le long de $D_\ell$ si et seulement si $Q(\beta,-\alpha;\alpha,\beta)=0,$ où** *$$\begin{aligned} Q(x,y;\alpha,\beta):=(\nu-1)\left(\dfrac{\partial{B}}{\partial{x}}-\dfrac{\partial{A}}{\partial{y}}\right)P(x,y;\alpha,\beta)+(2\nu+1) \left|\begin{array}{cc} \dfrac{\partial{P}}{\partial{x}} & A(x,y) \vspace{2mm} \\ \dfrac{\partial{P}}{\partial{y}} & B(x,y) \end{array} \right| \quad{\fontsize{11}{11pt}\text{et}}\quad P(x,y;\alpha,\beta):=\frac{\left| \begin{array}{cc} A(x,y) & A(\beta,-\alpha) \\ B(x,y) & B(\beta,-\alpha) \end{array} \right|}{(\alpha\,x+\beta\,y)^{\nu}}.\end{aligned}$$* **Corollaire 28**. **Soient $d\geq3$ un entier et $\mathcal{H}$ un feuilletage homogène de degré $d-1$ sur $\mathbb{P}^{2}_{\mathbb{C}}$ défini par la $1$-forme $$\begin{aligned} &\hspace{1.5cm}\omega=A(x,y)\mathrm{d}x+B(x,y)\mathrm{d}y,\quad A,B\in\mathbb{C}[x,y]_{d-1},\hspace{2mm}\mathrm{pgcd}(A,B)=1.\end{aligned}$$ Supposons que $\mathcal{H}$ possède une droite d'inflexion $\ell=(\alpha\,x+\beta\,y=0)$ transverse d'ordre maximal $d-2$. Posons $\mathscr{H}:=\ell\boxtimes\mathcal{H}.$ Alors la courbure de $\mathrm{Leg}\mathscr{H}$ est holomorphe le long de $D_\ell$ si et seulement si la $2$-forme $\mathrm{d}\omega$ s'annule sur la droite $\ell.$* * *Remarque 29*. Lorsque $d\geq4$ la condition .3ex$\scriptscriptstyle\langle\!\langle$ $\mathrm{d}\omega$ s'annule sur la droite $\ell$.3ex $\!\scriptscriptstyle\,\rangle\!\rangle$  exprime aussi l'holomorphie de la courbure de $\mathrm{Leg}\mathcal{H}$ le long de $D_\ell$, en vertu de [@BM18Bull Théorème 3.8]. Le Corollaire [Corollaire 28](#cor:holomorphie-ell-droite-inflexion-maximal){reference-type="ref" reference="cor:holomorphie-ell-droite-inflexion-maximal"} établit ainsi l'équivalence entre l'holomorphie sur $D_\ell$ de $K(\mathrm{Leg}\mathscr{H})$ et celle de $K(\mathrm{Leg}\mathcal{H}).$ # Platitude et pré-feuilletages homogènes $\ell\boxtimes\mathcal{H}$ de co-degré $1$ tels que $\deg\mathcal{T}_{\mathcal{H}}=2$ {#sec:application-homogene-deg-type-2} Nous nous proposons dans ce paragraphe de classifier, à automorphisme de $\mathbb{P}^{2}_{\mathbb{C}}$ près, tous les pré-feuilletages homogènes $\mathscr{H}=\ell\boxtimes\mathcal{H}$ de co-degré $1$ et de degré $d\geq3$ sur $\mathbb{P}^{2}_{\mathbb{C}}$ tels que $\deg\mathcal{T}_{\mathcal{H}}=2$ et le $d$-tissu $\mathrm{Leg}\mathscr{H}$ soit plat. L'égalité $\deg\mathcal{T}_{\mathcal{H}}=2$ est réalisée si et seulement si le type $\mathcal{T}_{\mathcal{H}}$ de $\mathcal{H}$ est de l'une des trois formes suivantes: $2\cdot\mathrm{R}_{d-2}$, $2\cdot\mathrm{T}_{d-2}$, $1\cdot\mathrm{R}_{d-2}+1\cdot\mathrm{T}_{d-2}.$ D'après [@BM18Bull Proposition 4.1], tout feuilletage homogène de type $2\cdot\mathrm{R}_{d-2}$ est conjugué au feuilletage convexe $\mathcal{H}_{1}^{d-1}$ défini par la $1$-forme[\[not:omega1\]]{#not:omega1 label="not:omega1"} $$\omega_{1}^{d-1}=y^{d-1}\mathrm{d}x-x^{d-1}\mathrm{d}y.$$ Les feuilletages homogènes de type $2\cdot\mathrm{T}_{d-2}$, resp. $1\cdot\mathrm{R}_{d-2}+1\cdot\mathrm{T}_{d-2},$ sont donnés, à conjugaison linéaire près, par[\[not:omega2\]]{#not:omega2 label="not:omega2"}[\[not:omega3\]]{#not:omega3 label="not:omega3"} $$\begin{aligned} &\omega_{2}^{d-1}(\lambda,\mu)=(x^{d-1}+\lambda\,y^{d-1})\mathrm{d}x+(\mu\,x^{d-1}-y^{d-1})\mathrm{d}y, \quad\text{où}\hspace{1mm} \lambda,\mu\in\mathbb{C}, \text{avec}\hspace{1mm}\lambda\mu\neq-1, \\ \text{resp.}\hspace{1mm}&\omega_{3}^{d-1}(\lambda)=(x^{d-1}+\lambda y^{d-1})\mathrm{d}x+x^{d-1}\mathrm{d}y, \quad\text{où}\hspace{1mm}\lambda\in\mathbb{C}^*,\end{aligned}$$ *cf.* preuve de [@BM18Bull Proposition 4.1]. Nous noterons $\mathcal{H}_{2}^{d-1}(\lambda,\mu)$, resp. $\mathcal{H}_{3}^{d-1}(\lambda)$ le feuilletage défini par $\omega_{2}^{d-1}(\lambda,\mu)$, resp. par $\omega_{3}^{d-1}(\lambda).$ Dans les trois lemmes qui suivent, $\ell$ désigne une droite de $\mathbb{P}^{2}_{\mathbb{C}}$ telle que $O\in\ell$ ou $\ell=L_\infty.$ **Lemme 30**. **Le $d$-tissu $\mathrm{Leg}(\ell\boxtimes\mathcal{H}_{1}^{d-1})$ est plat si et seulement si, à conjugaison linéaire près, $\ell=L_\infty$ ou $\ell\in\{x(y-x)(y-\xi\,x)=0\},$ où $\xi=\mathrm{exp}\left(\frac{\mathrm{i}\pi}{d-2}\right)$.** **Démonstration*.* Notons tout d'abord que le feuilletage $\mathcal{H}_{1}^{d-1}$ est à diviseur d'inflexion $$\mathrm{I}_{\mathcal{H}_{1}^{d-1}}=\mathrm{I}_{\mathcal{H}_{1}^{d-1}}^{\hspace{0.2mm}\mathrm{inv}}=(d-1)z\,x^{d-1}y^{d-1}\big(y^{d-2}-x^{d-2}\big).$$ ***i.*** Si $\ell$ est invariante par $\mathcal{H}_{1}^{d-1}$, alors $\ell\in\{xyz(y-\xi^{2k} x)=0,\,k=0,\ldots,d-3\}$ et le $d$-tissu $\mathrm{Leg}(\ell\boxtimes\mathcal{H}_{1}^{d-1})$ est plat (Corollaire [Corollaire 13](#cor:platitude-pre-feuilletage-homogene-convexe){reference-type="ref" reference="cor:platitude-pre-feuilletage-homogene-convexe"}). ***ii.*** Supposons que $\ell$ ne soit pas invariante par $\mathcal{H}_{1}^{d-1}$; alors $\ell=(y-\rho x=0)$ avec $\rho(\rho^{d-2}-1)\neq0.$ D'après le Corollaire [Corollaire 24](#cor:platitude-homogene-convexe-ell-non-invariante-effectif){reference-type="ref" reference="cor:platitude-homogene-convexe-ell-non-invariante-effectif"}, le $d$-tissu $\mathrm{Leg}(\ell\boxtimes\mathcal{H}_{1}^{d-1})$ est plat si et seulement si $Q(1,\rho;-\rho,1)=0,$ où $$\begin{aligned} Q(x,y;-\rho,1)=-\frac{\partial{P}}{\partial{x}}-\rho^{d-1}\frac{\partial{P}}{\partial{y}} \qquad\text{et}\qquad P(x,y;-\rho,1)=-\frac{y^{d-1}-(\rho x)^{d-1}}{y-\rho x}=-\sum_{i=0}^{d-2}\rho^i x^i y^{d-2-i}.\end{aligned}$$ Ainsi $Q(1,\rho;-\rho,1)=\frac{1}{2}(d-1)(d-2)\rho^{d-2}(\rho^{d-2}+1)$, et la platitude de $\mathrm{Leg}(\ell\boxtimes\mathcal{H}_{1}^{d-1})$ est équivalente à $\rho^{d-2}=-1$ et donc à $\ell\in\{y-\xi^{2k+1} x=0,\,k=0,\ldots,d-3\}.$ Dans les deux cas considérés, $\mathrm{Leg}(\ell\boxtimes\mathcal{H}_{1}^{d-1})$ est plat si et seulement si, à conjugaison près, $\ell=L_\infty:=(z=0)$ ou $\ell\in\{x(y-x)(y-\xi\,x)=0\}.$ En effet, en posant $\varphi_1(x,y)=(y,x)$ et $\varphi_2(x,y)=(x,\xi^{2k}y)$, nous avons $$\begin{aligned} \varphi_1^*(y\omega_{1}^{d-1})=-x\omega_{1}^{d-1},&&\hspace{1mm} \varphi_2^*\left((y-\xi^{2k} x)\omega_{1}^{d-1}\right)=\xi^{4k}(y-x)\omega_{1}^{d-1},&&\hspace{1mm} \varphi_2^*\left((y-\xi^{2k+1} x)\omega_{1}^{d-1}\right)=\xi^{4k}(y-\xi\,x)\omega_{1}^{d-1}.\end{aligned}$$  ◻ **Lemme 31**. **Le $d$-tissu $\mathrm{Leg}\big(\ell\boxtimes\mathcal{H}_{2}^{d-1}(\lambda,\mu)\big)$ est plat si et seulement si, à conjugaison linéaire près, un des cas suivants a lieu:** - *$\ell=L_\infty$ et $d=3$;* - *$\ell=L_\infty,$ $d\geq4$ et $\lambda=\mu=0$;* - *$\ell=(x=0)$ et $\lambda=\mu=0$;* - *$\ell=(y-x=0),$ $d\geq4$ et $(\lambda,\mu)=\left(\frac{3}{d},-\frac{3}{d}\right)$;* - *$\ell=(y-\xi'\,x=0),$ $d\geq4$ et $(\lambda,\mu)=\left(\frac{3\xi'}{d},-\frac{3}{d\xi'}\right)$, où $\xi'=\mathrm{exp}\left(\frac{\mathrm{i}\pi}{d}\right)$.* **Démonstration*.* Nous avons $\omega_{2}^{d-1}(\lambda,\mu)=A(x,y)\mathrm{d}x+B(x,y)\mathrm{d}y,$ où $A(x,y)=x^{d-1}+\lambda\,y^{d-1}$ et $B(x,y)=\mu\,x^{d-1}-y^{d-1};$ un calcul immédiat montre que $$\begin{aligned} \mathrm{I}_{\mathcal{H}_{2}^{d-1}(\lambda,\mu)}^{\hspace{0.2mm}\mathrm{inv}}=z(x^d+\mu\,x^{d-1}y+\lambda\,xy^{d-1}-y^d) &&\text{et}&& \mathrm{I}_{\mathcal{H}_{2}^{d-1}(\lambda,\mu)}^{\hspace{0.2mm}\mathrm{tr}}=x^{d-2}y^{d-2}.\end{aligned}$$ ***1.*** Si $\ell=L_\infty$ et $d=3,$ alors le tissu $\mathrm{Leg}(\ell\boxtimes\mathcal{H}_{2}^{2}(\lambda,\mu))$ est plat par le Corollaire [Corollaire 4](#cor:Leg-L-infini-H2-plat){reference-type="ref" reference="cor:Leg-L-infini-H2-plat"}. ***2.*** Supposons que $\ell=L_\infty$ et $d\geq4.$ Alors, d'après [@BM18Bull Théorèmes 3.1 et 3.8], le tissu $\mathrm{Leg}(\mathcal{H}_{2}^{d-1}(\lambda,\mu))$ est plat si et seulement si $\mathrm{d}\big(\omega_{2}^{d-1}(\lambda,\mu)\big)$ s'annule sur les deux droites $xy=0.$ Or, $$\begin{aligned} \mathrm{d}\big(\omega_{2}^{d-1}(\lambda,\mu)\big)\Big|_{x=0}=-(d-1)\lambda\,y^{d-2}\mathrm{d}x\wedge\mathrm{d}y \qquad\text{et}\qquad \mathrm{d}\big(\omega_{2}^{d-1}(\lambda,\mu)\big)\Big|_{y=0}=(d-1)\mu\,x^{d-2}\mathrm{d}x\wedge\mathrm{d}y.\end{aligned}$$ Donc $\mathrm{Leg}(\mathcal{H}_{2}^{d-1}(\lambda,\mu))$ est plat si et seulement si $\lambda=\mu=0$; il en est donc de même de $\mathrm{Leg}(\ell\boxtimes\mathcal{H}_{2}^{d-1}(\lambda,\mu))$ (Théorème [Théorème 3](#thm:K-Leg-H-egale-K-Leg-L-infini-H){reference-type="ref" reference="thm:K-Leg-H-egale-K-Leg-L-infini-H"}). ***3.*** Considérons le cas où $\ell\in\{xy=0\}.$ Quitte à permuter les coordonnées $x$ et $y$, nous pouvons supposer que $\ell=(x=0).$ D'après le Théorème [Théorème 9](#thm:holomorphie-courbure-G(I^tr)-D-ell){reference-type="ref" reference="thm:holomorphie-courbure-G(I^tr)-D-ell"}, le $d$-tissu $\mathrm{Leg}(\ell\boxtimes\mathcal{H}_{2}^{d-1}(\lambda,\mu))$ est plat si et seulement si sa courbure est holomorphe sur $\mathcal{G}_{\mathcal{H}_{2}^{d-1}(\lambda,\mu)}(\{xy=0\}).$ Or, d'une part, $K(\mathrm{Leg}(\ell\boxtimes\mathcal{H}_{2}^{d-1}(\lambda,\mu)))$ est holomorphe sur $D_\ell=\mathcal{G}_{\mathcal{H}_{2}^{d-1}(\lambda,\mu)}(\ell)$ si et seulement si $\mathrm{d}\big(\omega_{2}^{d-1}(\lambda,\mu)\big)$ s'annule sur $\ell=(x=0)$ (Corollaire [Corollaire 28](#cor:holomorphie-ell-droite-inflexion-maximal){reference-type="ref" reference="cor:holomorphie-ell-droite-inflexion-maximal"}), *i.e.* si et seulement si $\lambda=0.$ D'autre part, d'après le Corollaire [Corollaire 19](#cor:holomorphie-courbure-droite-inflex-maximal-d-2-T-neq-ell){reference-type="ref" reference="cor:holomorphie-courbure-droite-inflex-maximal-d-2-T-neq-ell"}, $K(\mathrm{Leg}(\ell\boxtimes\mathcal{H}_{2}^{d-1}(\lambda,\mu)))$ est holomorphe sur $\mathcal{G}_{\mathcal{H}_{2}^{d-1}(\lambda,\mu)}(\{y=0\})$ si et seulement si $$0=(d-3)\big(\partial_{x}B(1,0)-\partial_{y}A(1,0)\big)+3(d-1)B(1,0)=d(d-1)\mu\Longleftrightarrow\mu=0.$$ Il en résulte que $\mathrm{Leg}(\ell\boxtimes\mathcal{H}_{2}^{d-1}(\lambda,\mu))$ est plat si et seulement si $\lambda=\mu=0.$ ***4.*** Examinons le cas où $\ell=(y-\rho\,x=0)$ avec $\rho\neq0.$ D'après le Corollaire [Corollaire 19](#cor:holomorphie-courbure-droite-inflex-maximal-d-2-T-neq-ell){reference-type="ref" reference="cor:holomorphie-courbure-droite-inflex-maximal-d-2-T-neq-ell"}, $K(\mathrm{Leg}(\ell\boxtimes\mathcal{H}_{2}^{d-1}(\lambda,\mu)))$ est holomorphe sur $\mathcal{G}_{\mathcal{H}_{2}^{d-1}(\lambda,\mu)}(\{xy=0\})$ si et seulement si $$\begin{aligned} \left\{ \begin{array}[l]{l} 0=-(d-3)\Big(\partial_{x}B(0,-1)-\partial_{y}A(0,-1)\Big)-3(d-1)\Big(A(0,-1)+\rho B(0,-1)\Big)=(-1)^{d}(d-1)(d\lambda-3\rho) \\ \\ 0=-\rho(d-3)\Big(\partial_{x}B(1,0)-\partial_{y}A(1,0)\Big)-3(d-1)\Big(A(1,0)+\rho B(1,0)\Big)=-(d-1)(d\rho\mu+3), \end{array} \right.\end{aligned}$$ si et seulement si $\lambda=\lambda_0:=\frac{3\rho}{d}$,  $\mu=\mu_0:=-\frac{3}{d\rho}$  et  $d\neq3$, car $\lambda\mu\neq-1.$ Maintenant, nous allons distinguer deux cas suivant que $\ell$ soit invariante ou non par $\mathcal{H}_{2}^{d-1}(\lambda_0,\mu_0).$ ***4.1.*** Supposons que $\ell$ soit invariante par $\mathcal{H}_{2}^{d-1}(\lambda_0,\mu_0).$ Alors le tissu dual de $\ell\boxtimes\mathcal{H}_{2}^{d-1}(\lambda_0,\mu_0)$ est plat par le Théorème [Théorème 9](#thm:holomorphie-courbure-G(I^tr)-D-ell){reference-type="ref" reference="thm:holomorphie-courbure-G(I^tr)-D-ell"}. Comme $\mathrm{I}_{\mathcal{H}_{2}^{d-1}(\lambda_0,\mu_0)}^{\hspace{0.2mm}\mathrm{inv}}\Big|_{y=\rho x}=\left(\frac{3}{d}-1\right)\left(\rho^d-1\right)z\,x^d,$ l'invariance de $\ell$ par $\mathcal{H}_{2}^{d-1}(\lambda_0,\mu_0)$ est équivalente à $\rho^d=1$; par conséquent $(\rho,\lambda_0,\mu_0)\in\left\{\Big(\xi'^{2k},\frac{3\xi'^{2k}}{d},-\frac{3}{d\xi'^{2k}}\Big),k=0,\ldots,d-1\right\}.$ À conjugaison près, $(\rho,\lambda_0,\mu_0)=(1,\frac{3}{d},-\frac{3}{d})$; en effet, en posant $\varphi(x,y)=(x,\xi'^{2k}y)$ nous avons $$\begin{aligned} \varphi^*\left(\big(y-\xi'^{2k}x\big)\omega_{2}^{d-1}\left(\tfrac{3\xi'^{2k}}{d},-\tfrac{3}{d\xi'^{2k}}\right)\right) =\xi'^{2k}\left(y-x\right)\omega_{2}^{d-1}\left(\tfrac{3}{d},-\tfrac{3}{d}\right).\end{aligned}$$ ***4.2.*** Supposons que $\ell$ ne soit pas invariante par $\mathcal{H}_{2}^{d-1}(\lambda_0,\mu_0).$ Alors, en vertu du Théorème [Théorème 9](#thm:holomorphie-courbure-G(I^tr)-D-ell){reference-type="ref" reference="thm:holomorphie-courbure-G(I^tr)-D-ell"} et du Corollaire [Corollaire 22](#cor:holomorphie-courbure-homogene-D-ell-fibre-sans-point-critique-non-fixe){reference-type="ref" reference="cor:holomorphie-courbure-homogene-D-ell-fibre-sans-point-critique-non-fixe"}, la platitude de $\mathrm{Leg}(\ell\boxtimes\mathcal{H}_{2}^{d-1}(\lambda_0,\mu_0))$ se traduit par $Q(1,\rho;-\rho,1)=0,$ où $$\begin{aligned} &\hspace{-4cm}Q(x,y;-\rho,1)=\left(\mu_0-\rho^{d-1}\right)\frac{\partial{P}}{\partial{x}}-\left(\lambda_0\rho^{d-1}+1\right)\frac{\partial{P}}{\partial{y}} \\ &\hspace{-5.17cm}{\fontsize{11}{11pt}\text{et}} \\ &\hspace{-4cm}P(x,y;-\rho,1)=\frac{\left(\lambda_0\mu_0+1\right)\left(y^{d-1}-(\rho x)^{d-1}\right)}{y-\rho x} =\left(\lambda_0\mu_0+1\right)\sum_{i=0}^{d-2}\rho^i x^i y^{d-2-i}.\end{aligned}$$ D'où $Q(1,\rho;-\rho,1)=\frac{1}{2}\left(\frac{3}{d}-1\right)\left(\frac{3}{d}+1\right)^2(d-1)(d-2)\rho^{d-3}(\rho^d+1)$ , et par suite $\mathrm{Leg}(\ell\boxtimes\mathcal{H}_{2}^{d-1}(\lambda_0,\mu_0))$ est plat si et seulement si $\rho^d=-1$, donc si et seulement si $(\rho,\lambda_0,\mu_0)\in\left\{\Big(\xi'^{2k+1},\frac{3\xi'^{2k+1}}{d},-\frac{3}{d\xi'^{2k+1}}\Big),k=0,\ldots,d-1\right\}$, autrement dit si et seulement si, à conjugaison près, $(\rho,\lambda_0,\mu_0)=\Big(\xi',\frac{3\xi'}{d},-\frac{3}{d\xi'}\Big),$ car $$\begin{aligned} \varphi^*\left(\big(y-\xi'^{2k+1}x\big)\omega_{2}^{d-1}\left(\tfrac{3\xi'^{2k+1}}{d},-\tfrac{3}{d\xi'^{2k+1}}\right)\right) =\xi'^{2k}\left(y-\xi'x\right)\omega_{2}^{d-1}\left(\tfrac{3\xi'}{d},-\tfrac{3}{d\xi'}\right).\end{aligned}$$ ◻ **Lemme 32**. **Le $d$-tissu $\mathrm{Leg}\big(\ell\boxtimes\mathcal{H}_{3}^{d-1}(\lambda)\big)$ est plat si et seulement si l'un des cas suivants se produit:** - *$\ell=L_\infty$ et $d=3$;* - *$\ell=(dy+3x=0),$ $d\geq4$ et $\lambda=\dfrac{(-1)^d(d-3)d^{d-2}}{3^{d-1}}$;* - *$\ell=(dy+3x=0)$ et $\lambda=\dfrac{(-1)^d(d+3)d^{d-2}}{3^{d-1}}.$* **Démonstration*.* Nous avons $\omega_{3}^{d-1}(\lambda)=A(x,y)\mathrm{d}x+B(x,y)\mathrm{d}y$, où $A(x,y)=x^{d-1}+\lambda y^{d-1}$ et $B(x,y)=x^{d-1}$; un calcul immédiat conduit à $$\begin{aligned} \mathrm{I}_{\mathcal{H}_{3}^{d-1}(\lambda)}^{\hspace{0.2mm}\mathrm{inv}}=z\,x^{d-1}\big(x^{d-1}+x^{d-2}y+\lambda\,y^{d-1}\big) &&\text{et}&& \mathrm{I}_{\mathcal{H}_{3}^{d-1}(\lambda)}^{\hspace{0.2mm}\mathrm{tr}}=y^{d-2}.\end{aligned}$$ ***1.*** Supposons que $\ell=L_\infty.$ Si $d=3$, alors le tissu $\mathrm{Leg}(\ell\boxtimes\mathcal{H}_{3}^{2}(\lambda))$ est plat, en vertu du Corollaire [Corollaire 4](#cor:Leg-L-infini-H2-plat){reference-type="ref" reference="cor:Leg-L-infini-H2-plat"}. Pour $d\geq4,$ les tissus $\mathrm{Leg}(\mathcal{H}_{3}^{d-1}(\lambda))$ et $\mathrm{Leg}(\ell\boxtimes\mathcal{H}_{3}^{d-1}(\lambda))$ ont la même courbure (Théorème [Théorème 3](#thm:K-Leg-H-egale-K-Leg-L-infini-H){reference-type="ref" reference="thm:K-Leg-H-egale-K-Leg-L-infini-H"}) et ne peuvent pas être plats. En effet, nous avons $$\mathrm{d}\big(\omega_{3}^{d-1}(\lambda)\big)\Big|_{y=0}=(d-1)x^{d-2}\mathrm{d}x\wedge\mathrm{d}y\not\equiv0\hspace{0.5mm};$$ ceci implique, d'après [@BM18Bull Théorème 3.8], que $K(\mathrm{Leg}(\mathcal{H}_{3}^{d-1}(\lambda)))$ ne peut pas être holomorphe sur $\mathcal{G}_{\mathcal{H}_{3}^{d-1}(\lambda)}(\{y=0\}).$ ***2.*** Si $\ell=(y=0),$ alors le fait que $\mathrm{d}\big(\omega_{3}^{d-1}(\lambda)\big)$ ne s'annule pas sur $\ell$ entraîne, d'après le Corollaire [Corollaire 28](#cor:holomorphie-ell-droite-inflexion-maximal){reference-type="ref" reference="cor:holomorphie-ell-droite-inflexion-maximal"}, que $K(\mathrm{Leg}(\ell\boxtimes\mathcal{H}_{3}^{d-1}(\lambda)))$ ne peut pas être holomorphe sur $\mathcal{G}_{\mathcal{H}_{3}^{d-1}(\lambda)}(\ell),$ de sorte que $\mathrm{Leg}(\ell\boxtimes\mathcal{H}_{3}^{d-1}(\lambda))$ ne peut pas être plat. ***3.*** Supposons que $\ell=(x-\rho\,y=0),$ où $\rho\in\mathbb{C}.$ D'après le Corollaire [Corollaire 19](#cor:holomorphie-courbure-droite-inflex-maximal-d-2-T-neq-ell){reference-type="ref" reference="cor:holomorphie-courbure-droite-inflex-maximal-d-2-T-neq-ell"}, $K(\mathrm{Leg}(\ell\boxtimes\mathcal{H}_{3}^{d-1}(\lambda)))$ est holomorphe sur $\mathcal{G}_{\mathcal{H}_{3}^{d-1}(\lambda)}(\{y=0\})$ si et seulement si $$\begin{aligned} 0=(d-3)\Big(\partial_{x}B(1,0)-\partial_{y}A(1,0)\Big)+3(d-1)\Big(B(1,0)+\rho A(1,0)\Big)=(d-1)(3\rho+d),\end{aligned}$$ donc si et seulement si $\rho=-\frac{d}{3}$, ce qui équivaut à $\ell=\ell_0$ où $\ell_0=(dy+3x=0).$ Deux cas sont alors à distinguer: ***3.1.*** Cas où $\ell_0$ est invariante par $\mathcal{H}_{3}^{d-1}(\lambda).$ Le Théorème [Théorème 9](#thm:holomorphie-courbure-G(I^tr)-D-ell){reference-type="ref" reference="thm:holomorphie-courbure-G(I^tr)-D-ell"} assure alors que le $d$-tissu $\mathrm{Leg}(\ell_0\boxtimes\mathcal{H}_{3}^{d-1}(\lambda))$ est plat; comme $$\begin{aligned} \mathrm{I}_{\mathcal{H}_{3}^{d-1}(\lambda)}^{\hspace{0.2mm}\mathrm{inv}}\Big|_{x=-\frac{d}{3} y}=-d^{d-1}z\left(\frac{y}{3}\right)^{2d-2}\left((-1)^{d}3^{d-1}\lambda-(d-3)d^{d-2}\right),\end{aligned}$$ l'invariance de $\ell_0$ par $\mathcal{H}_{3}^{d-1}(\lambda)$ est caractérisée par $\lambda=\dfrac{(-1)^d(d-3)d^{d-2}}{3^{d-1}}$ et $d\neq3$, car $\lambda\neq0.$ ***3.2.*** Cas où $\ell_0$ n'est pas invariante par $\mathcal{H}_{3}^{d-1}(\lambda).$ Alors, d'après le Théorème [Théorème 9](#thm:holomorphie-courbure-G(I^tr)-D-ell){reference-type="ref" reference="thm:holomorphie-courbure-G(I^tr)-D-ell"} et le Corollaire [Corollaire 22](#cor:holomorphie-courbure-homogene-D-ell-fibre-sans-point-critique-non-fixe){reference-type="ref" reference="cor:holomorphie-courbure-homogene-D-ell-fibre-sans-point-critique-non-fixe"}, le $d$-tissu $\mathrm{Leg}(\ell_0\boxtimes\mathcal{H}_{3}^{d-1}(\lambda))$ est plat si et seulement si $Q(d,-3;3,d)=0,$ où $$\begin{aligned} &\hspace{-5.1cm}Q(x,y;3,d)=d^{d-1}\frac{\partial{P}}{\partial{x}}-\left(d^{d-1}+(-3)^{d-1}\lambda\right)\frac{\partial{P}}{\partial{y}} \\ &\hspace{-6.35cm}{\fontsize{11}{11pt}\text{et}} \\ &\hspace{-5.1cm}P(x,y;3,d)=\frac{\lambda\left((dy)^{d-1}-(-3x)^{d-1}\right)}{dy+3x}=\lambda\sum_{i=0}^{d-2}(-3x)^i(dy)^{d-2-i}.\end{aligned}$$ Ainsi $Q(d,-3;3,d)=-\frac{1}{6}\lambda(d-1)(d-2)(3d)^{d-2}\Big(3^{d-1}\lambda-(-1)^{d}(d+3)d^{d-2}\Big)$ et la platitude de $\mathrm{Leg}(\ell_0\boxtimes\mathcal{H}_{3}^{d-1}(\lambda))$ se traduit par $\lambda=\dfrac{(-1)^d(d+3)d^{d-2}}{3^{d-1}}.$ ◻ Les Lemmes [Lemme 30](#lem:platitude-Leg-ell-H1){reference-type="ref" reference="lem:platitude-Leg-ell-H1"}, [Lemme 31](#lem:platitude-Leg-ell-H2){reference-type="ref" reference="lem:platitude-Leg-ell-H2"} et [Lemme 32](#lem:platitude-Leg-ell-H3){reference-type="ref" reference="lem:platitude-Leg-ell-H3"} impliquent la proposition suivante. **Proposition 33**. **Soit $\mathscr{H}=\ell\boxtimes\mathcal{H}$ un pré-feuilletage homogène de co-degré $1$ et de degré $d\geq3$ sur $\mathbb{P}^{2}_{\mathbb{C}}.$ Supposons que $\deg\mathcal{T}_{\mathcal{H}}=2,$ autrement dit que l'application ${\mspace{2mu}\underline{\mspace{-2mu}\mathcal{G}\mspace{-2mu}}\mspace{2mu}}_{\mathcal H}$ ait exactement deux points critiques. Alors, pour $d\geq4$ le tissu $\mathrm{Leg}\mathscr{H}$ est plat si et seulement si $\mathscr{H}$ est linéairement conjugué à l'un des dix pré-feuilletages suivants** - *$\mathscr{H}_{1}^{d}=L_\infty\boxtimes\mathcal{H}_{1}^{d-1}$;* - *$\mathscr{H}_{2}^{d}=\{x=0\}\boxtimes\mathcal{H}_{1}^{d-1}$;* - *$\mathscr{H}_{3}^{d}=\{y-x=0\}\boxtimes\mathcal{H}_{1}^{d-1}$;* - *$\mathscr{H}_{4}^{d}=\{y-\xi\,x=0\}\boxtimes\mathcal{H}_{1}^{d-1}$,  où $\xi=\mathrm{exp}\left(\frac{\mathrm{i}\pi}{d-2}\right)$;* - *$\mathscr{H}_{5}^{d}=\{x=0\}\boxtimes\mathcal{H}_{2}^{d-1}(0,0)$;* - *$\mathscr{H}_{6}^{d}=\{dy+3x=0\}\boxtimes\mathcal{H}_{3}^{d-1}(\lambda_0)$,  où $\lambda_0=\frac{(-1)^d(d+3)d^{d-2}}{3^{d-1}}$;* - *$\mathscr{H}_{7}^{d}=\{dy+3x=0\}\boxtimes\mathcal{H}_{3}^{d-1}(\lambda_1)$,  où $\lambda_1=\frac{(-1)^d(d-3)d^{d-2}}{3^{d-1}}$;* - *$\mathscr{H}_{8}^{d}=L_\infty\boxtimes\mathcal{H}_{2}^{d-1}(0,0)$;* - *$\mathscr{H}_{9}^{d}=\{y-x=0\}\boxtimes\mathcal{H}_{2}^{d-1}\big(\tfrac{3}{d},-\tfrac{3}{d}\big)$;* - *$\mathscr{H}_{10}^{d}=\{y-\xi'\,x=0\}\boxtimes\mathcal{H}_{2}^{d-1}\big(\tfrac{3\xi'}{d},-\tfrac{3}{d\xi'}\big)$,  où $\xi'=\mathrm{exp}\left(\frac{\mathrm{i}\pi}{d}\right)$.* *Pour $d=3$ le tissu $\mathrm{Leg}\mathscr{H}$ est plat si et seulement si, à conjugaison linéaire près, ou bien $\mathscr{H}$ est l'un des six pré-feuilletages $\mathscr{H}_{1}^{3},\mathscr{H}_{2}^{3},\ldots,\mathscr{H}_{6}^{3},$ ou bien $\mathscr{H}$ est de l'un des deux types suivants* - *$\mathscr{H}_{7}^{3}(\lambda)=L_\infty\boxtimes\mathcal{H}_{3}^{2}(\lambda)$,  où $\lambda\in\mathbb{C}^*$;* - *$\mathscr{H}_{8}^{3}(\lambda,\mu)=L_\infty\boxtimes\mathcal{H}_{2}^{2}(\lambda,\mu)$,  où $\lambda,\mu\in\mathbb{C}$ avec $\lambda\mu\neq-1.$* En combinant la Proposition [Proposition 33](#pro:class-pre-homogenes-plats-co-degre-1-degre-d-geq-4-degre-Type-2){reference-type="ref" reference="pro:class-pre-homogenes-plats-co-degre-1-degre-d-geq-4-degre-Type-2"} avec le fait que tout feuilletage homogène de degré $2$ sur $\mathbb{P}^{2}_{\mathbb{C}}$ est de degré de type $2$, nous obtenons la classification à automorphisme de $\mathbb{P}^{2}_{\mathbb{C}}$ près des pré-feuilletages homogènes de type $(1,3)$ sur $\mathbb{P}^{2}_{\mathbb{C}}$ dont le tissu dual est plat. **Corollaire 34**. **À automorphisme de $\mathbb{P}^{2}_{\mathbb{C}}$ près, il y a six exemples et deux familles de pré-feuilletages homogènes de co-degré $1$ et de degré $3$ sur $\mathbb{P}^{2}_{\mathbb{C}}$ ayant une transformée de [Legendre]{.smallcaps} plate, à savoir:** - *$\mathscr{H}_{1}^{3}=L_\infty\boxtimes\mathcal{H}_{1}^{2}$;* - *$\mathscr{H}_{2}^{3}=\{x=0\}\boxtimes\mathcal{H}_{1}^{2}$;* - *$\mathscr{H}_{3}^{3}=\{y-x=0\}\boxtimes\mathcal{H}_{1}^{2}$;* - *$\mathscr{H}_{4}^{3}=\{y+x=0\}\boxtimes\mathcal{H}_{1}^{2}$;* - *$\mathscr{H}_{5}^{3}=\{x=0\}\boxtimes\mathcal{H}_{2}^{2}(0,0)$;* - *$\mathscr{H}_{6}^{3}=\{y+x=0\}\boxtimes\mathcal{H}_{3}^{2}(-2)$;* - *$\mathscr{H}_{7}^{3}(\lambda)=L_\infty\boxtimes\mathcal{H}_{3}^{2}(\lambda)$,  où $\lambda\in\mathbb{C}^*$;* - *$\mathscr{H}_{8}^{3}(\lambda,\mu)=L_\infty\boxtimes\mathcal{H}_{2}^{2}(\lambda,\mu)$,  où $\lambda,\mu\in\mathbb{C}$ avec $\lambda\mu\neq-1.$* Nous aurons besoin au paragraphe §[6](#sec:pre-feuilletages-codegre-1-degre-3){reference-type="ref" reference="sec:pre-feuilletages-codegre-1-degre-3"}, pour $\mathcal{H}\in\{\mathcal{H}_{1}^{2},\mathcal{H}_{2}^{2}(0,0),\mathcal{H}_{3}^{2}(-2)\},$ des valeurs des indices de [Camacho-Sad]{.smallcaps} $\mathrm{CS}(\mathcal{H},L_{\infty},s)$,[\[not:indice-CS\]]{#not:indice-CS label="not:indice-CS"} $s\in\mathrm{Sing}\mathcal{H}\cap L_{\infty}$. Pour cela, nous avons calculé, pour chacun de ces trois feuilletages, le polynôme suivant (dit *polynôme de [Camacho-Sad]{.smallcaps} du feuilletage homogène* $\mathcal{H}$) $$\begin{aligned} \mathrm{CS}_{\mathcal{H}}(\lambda)=\prod\limits_{s\in\mathrm{Sing}\mathcal{H}\cap L_{\infty}}(\lambda-\mathrm{CS}(\mathcal{H},L_{\infty},s)).\end{aligned}$$ Le tableau suivant récapitule les types et les polynômes de [Camacho-Sad]{.smallcaps} des feuilletages $\mathcal{H}_{1}^{2}$, $\mathcal{H}_{2}^{2}(0,0)$ et $\mathcal{H}_{3}^{2}(-2)$. $\mathcal{H}$ $\mathcal{T}_{\mathcal{H}}$ $\mathrm{CS}_{\mathcal{H}}(\lambda)$ ---------------------------- ----------------------------------------- --------------------------------------------------------- $\mathcal{H}_{1}^{2}$ $2\cdot\mathrm{R}_1$ $(\lambda-1)^{2}(\lambda+1)$ $\mathcal{H}_{2}^{2}(0,0)$ $2\cdot\mathrm{T}_1$ $(\lambda-\frac{1}{3})^{3}$ $\mathcal{H}_{3}^{2}(-2)$ $1\cdot\mathrm{R}_1+1\cdot\mathrm{T}_1$ $(\lambda-1)(\lambda-\frac{1}{3})(\lambda+\frac{1}{3})$ : Types et polynômes de [Camacho-Sad]{.smallcaps} des feuilletages $\mathcal{H}_{1}^{2}$, $\mathcal{H}_{2}^{2}(0,0)$ et $\mathcal{H}_{3}^{2}(-2)$ # Pré-feuilletages de co-degré $1$ dont le feuilletage associé est convexe réduit {#sec:pre-pre-feuill-convexe-reduit} Nous allons maintenant donner les démonstrations du Théorème [Théorème 5](#thmalph:ell-invariante-convexe-reduit-plat){reference-type="ref" reference="thmalph:ell-invariante-convexe-reduit-plat"} et des Propositions [Proposition 6](#proalph:ell-non-invariante-Fermat){reference-type="ref" reference="proalph:ell-non-invariante-Fermat"} et [Proposition 7](#proalph:ell-non-invariante-Hesse){reference-type="ref" reference="proalph:ell-non-invariante-Hesse"} annoncés dans l'Introduction. **Démonstration du Théorème [Théorème 5](#thmalph:ell-invariante-convexe-reduit-plat){reference-type="ref" reference="thmalph:ell-invariante-convexe-reduit-plat"}*.* Comme $\mathcal{F}$ est par hypothèse convexe réduit, toutes ses singularités sont non-dégénérées ([@BM18Bull Lemme 6.8]). D'après [@BFM14 Lemme 2.2], le discriminant de $\mathrm{Leg}\mathcal{F}$ est alors formé des droites duales des singularités radiales de $\mathcal{F}.$ La première assertion du Lemme [Lemme 1](#lem:Delta-Leg-ell-F){reference-type="ref" reference="lem:Delta-Leg-ell-F"} implique donc que $$\begin{aligned} \Delta(\mathrm{Leg}\mathscr{F})=\check{\Sigma}_{\mathcal{F}}^{\mathrm{rad}}\cup\check{\Sigma}_{\mathcal{F}}^{\ell}.\end{aligned}$$ Pour montrer que la courbure de $\mathrm{Leg}\mathscr{F}$ est identiquement nulle, il suffit donc de montrer qu'elle est holomorphe le long de la droite duale de tout point de $\Sigma_{\mathcal{F}}^{\mathrm{rad}}\cup\Sigma_{\mathcal{F}}^{\ell}.$ Soit donc $s$ un point quelconque de $\Sigma_{\mathcal{F}}^{\mathrm{rad}}\cup\Sigma_{\mathcal{F}}^{\ell}.$ Notons $\nu=\tau(\mathcal{F},s)$[\[not:tau-F-s\]]{#not:tau-F-s label="not:tau-F-s"} l'ordre de tangence de $\mathcal{F}$ avec une droite générique passant par $s$; alors $\nu-1$ désigne l'ordre de radialité de $s,$ et $s\in\Sigma_{\mathcal{F}}^{\mathrm{rad}}$ si et seulement si $\nu\geq2$, *voir* [@BM18Bull §1.3]. Localement près de la droite $\check{s}$ duale de $s$, la Proposition 3.3 de [@MP13] permet de décomposer le tissu $\mathrm{Leg}\mathcal{F}$ sous la forme $\mathrm{Leg}\mathcal{F}=\mathcal{W}_{\nu}\boxtimes\mathcal{W}_{d-\nu-1},$ où $\mathcal{W}_{\nu}$ est un $\nu$-tissu irréductible laissant $\check{s}$ invariante et dont la multiplicité du discriminant $\Delta(\mathcal{W}_{\nu})$ le long de $\check{s}$ est minimale, égale à $\nu-1,$ et où $\mathcal{W}_{d-\nu-1}$ est un $(d-\nu-1)$-tissu transverse à $\check{s}.$ De plus, la convexité de $\mathrm{Leg}\mathcal{F}$ implique, par un argument de la démonstration de [@MP13 Théorème 4.2], que le tissu $\mathcal{W}_{d-\nu-1}$ est régulier près de $\check{s},$ *i.e.* que par un point générique de $\check{s}$ passent $(d-\nu-1)$ tangentes distinctes à $\mathcal{W}_{d-\nu-1}.$ Ainsi, près de la droite $\check{s},$ nous avons la décomposition $$\begin{aligned} \mathrm{Leg}\mathscr{F}=\mathrm{Leg}\ell\boxtimes\mathcal{W}_{\nu}\boxtimes\mathcal{W}_{d-\nu-1}.\end{aligned}$$ Distinguons maintenant deux cas: ***1.*** Cas où $s\in\ell.$ Alors $\check{s}$ est invariante par $\mathrm{Leg}\ell.$ En appliquant le Théorème 1 de [@MP13] si $\nu=1$ et la Proposition [Proposition 11](#pro:holomorphie-courbure-F-W-nu-W-d-nu-1){reference-type="ref" reference="pro:holomorphie-courbure-F-W-nu-W-d-nu-1"} si $\nu\geq2,$ il en résulte que $K(\mathrm{Leg}\mathscr{F})$ est holomorphe le long de $\check{s}.$ ***2.*** Cas où $s\not\in\ell$; alors $s\in\Sigma_{\mathcal{F}}^{\mathrm{rad}}\setminus\Sigma_{\mathcal{F}}^{\ell}.$ Dans ce cas le feuilletage radial $\mathrm{Leg}\ell$ est transverse à $\check{s}.$ D'après ce qui précède, le $(d-\nu)$-tissu $\mathcal{W}_{d-\nu}:=\mathrm{Leg}\ell\boxtimes\mathcal{W}_{d-\nu-1}$ est donc aussi transverse à $\check{s}$ et nous avons $\mathrm{Leg}\mathscr{F}=\mathcal{W}_{\nu}\boxtimes\mathcal{W}_{d-\nu}.$ Par ailleurs, comme $\ell$ est $\mathcal{F}$-invariante, $\mathrm{Tang}(\mathrm{Leg}\ell,\mathrm{Leg}\mathcal{F})=\check{\Sigma}_{\mathcal{F}}^{\ell}$ (*cf.* démonstration du Lemme [Lemme 1](#lem:Delta-Leg-ell-F){reference-type="ref" reference="lem:Delta-Leg-ell-F"}); en particulier, $\mathrm{Tang}(\mathrm{Leg}\ell,\mathcal{W}_{d-\nu-1})\subset\check{\Sigma}_{\mathcal{F}}^{\ell}$ et donc $\check{s}\not\subset\mathrm{Tang}(\mathrm{Leg}\ell,\mathcal{W}_{d-\nu-1}).$ Il s'en suit que le tissu $\mathcal{W}_{d-\nu}$ est régulier près de $\check{s},$ car $\mathcal{W}_{d-\nu-1}$ l'est. Par conséquent la courbure de $\mathrm{Leg}\mathscr{F}$ est holomorphe le long de $\check{s}$ par application de [@MP13 Proposition 2.6]. ◻ **Démonstration de la Proposition [Proposition 6](#proalph:ell-non-invariante-Fermat){reference-type="ref" reference="proalph:ell-non-invariante-Fermat"}*.* Le feuilletage de [Fermat]{.smallcaps} $\mathcal{F}_{0}^{d-1}$ est décrit en coordonnées homogènes par la $1$-forme $${\mspace{2mu}\overline{\mspace{-1.4mu}\Omega\mspace{-1.4mu}}\mspace{2mu}}_{0}^{d-1}=x^{d-1}(y\mathrm{d}z-z\mathrm{d}y)+y^{d-1}(z\mathrm{d}x-x\mathrm{d}z)+z^{d-1}(x\mathrm{d}y-y\mathrm{d}x).$$ Ses $3(d-1)$ droites invariantes sont: $$\begin{aligned} &\hspace{2.4cm} xyz(y-\zeta^k x)(y-\zeta^k z)(x-\zeta^k z)=0, \quad\text{où}\hspace{1mm}k\in\{0,\ldots,d-3\}\hspace{2mm}\text{et}\hspace{2mm}\zeta=\exp(\tfrac{2\mathrm{i}\pi}{d-2}).\end{aligned}$$ Les coordonnées $x,y$ et $z$ jouant un rôle symétrique et $\ell$ étant non invariante par $\mathcal{F}_{0}^{d-1}$, nous pouvons supposer que $\ell=\{\alpha\,x+\beta\,y-z=0\}$ avec $\beta\neq0.$ Alors $\overline{\mathcal{O}(\ell\boxtimes\mathcal{F}_{0}^{d-1})}$[\[not:orbite-pref\]]{#not:orbite-pref label="not:orbite-pref"} contient les pré-feuilletages homogènes suivants: $$\begin{aligned} \mathscr{H}_1=\{y-\alpha\,x=0\}\boxtimes\mathcal{H}_{1}^{d-1},&& \mathscr{H}_2=\{y-\beta\,x=0\}\boxtimes\mathcal{H}_{1}^{d-1},&& \mathscr{H}_3=\big\{x-(\alpha+\beta)y=0\big\}\boxtimes\mathcal{H}_{0}^{d-1}.\end{aligned}$$ En effet, $\ell\boxtimes\mathcal{F}_{0}^{d-1}$ est donné dans la carte affine $z=1$ par $\omega=(\alpha\,x+\beta\,y-1){\mspace{2mu}\overline{\mspace{-1.4mu}\omega\mspace{-1.4mu}}\mspace{2mu}}_{0}^{d-1}$; en posant $\varphi_1=\Big(\frac{x}{y},\frac{\varepsilon}{y}\Big),$ $\varphi_2=\Big(\frac{\varepsilon}{y},\frac{x}{y}\Big)$ et $\varphi_3=\Big(\frac{y+\varepsilon}{x},\frac{y}{x}\Big),$ nous obtenons que $$\begin{aligned} \lim_{\varepsilon\to 0}\varepsilon^{-1}y^{d+2}\varphi_1^*\omega=(y-\alpha\,x)\omega_{1}^{d-1}, &&\hspace{3mm} \lim_{\varepsilon\to 0}\varepsilon^{-1}y^{d+2}\varphi_2^*\omega=(\beta\,x-y)\omega_{1}^{d-1}, &&\hspace{3mm} \lim_{\varepsilon\to 0}\varepsilon^{-1}x^{d+2}\varphi_3^*\omega=\Big((\alpha+\beta)y-x\Big)\omega_{0}^{d-1}.\end{aligned}$$ L'hypothèse que $\mathrm{Leg}(\ell\boxtimes\mathcal{F}_{0}^{d-1})$ est plat entraîne qu'il en est de même des tissus $\mathrm{Leg}\mathscr{H}_1,$ $\mathrm{Leg}\mathscr{H}_2$ et $\mathrm{Leg}\mathscr{H}_3.$ Montrons que la platitude de $\mathrm{Leg}\mathscr{H}_1$ et $\mathrm{Leg}\mathscr{H}_2$ implique qu'à conjugaison linéaire près $$\begin{aligned} (\alpha,\beta)\in E:=\Big\{(0,\xi),(1,1),(1,\xi),(\xi,\xi)\Big\}, \quad\text{où}\hspace{1mm}\xi=\mathrm{exp}\left(\tfrac{\mathrm{i}\pi}{d-2}\right).\end{aligned}$$ Tout d'abord, le $d$-tissu $\mathrm{Leg}\mathscr{H}_1,$ resp. $\mathrm{Leg}\mathscr{H}_2$, est plat si et seulement si (*cf.* démonstration du Lemme [Lemme 30](#lem:platitude-Leg-ell-H1){reference-type="ref" reference="lem:platitude-Leg-ell-H1"}) $$\begin{aligned} \alpha(\alpha^{d-2}-1)(\alpha^{d-2}+1)=0, \qquad\qquad\hspace{2mm} \text{resp.}\hspace{1mm}(\beta^{d-2}-1)(\beta^{d-2}+1)=0,\end{aligned}$$ *i.e.* si et seulement si $\alpha\in\{0,\zeta^{k},\xi\zeta^{k},\,k=0,\ldots,d-3\},$ resp. $\beta\in\{\zeta^{k},\xi\zeta^{k},\,k=0,\ldots,d-3\}.$ Si $\alpha=0$ alors $\beta\neq\zeta^k$, car sinon $\ell$ serait invariante par $\mathcal{F}_{0}^{d-1}.$ Nous en déduisons que $$\begin{aligned} (\alpha,\beta)\in \Big\{ (0,\xi\zeta^{k}),(\zeta^k,\zeta^{k'}),(\zeta^k,\xi\zeta^{k'}),(\xi\zeta^{k},\zeta^{k'}),(\xi\zeta^{k},\xi\zeta^{k'}),\hspace{1.5mm}k,k'=0,\ldots,d-3 \Big\}.\end{aligned}$$ Si, pour $k,k'\in\{0,\ldots,d-3\},$ $$\begin{aligned} (\alpha,\beta)=(0,\xi\zeta^{k}), &&\text{resp.}\hspace{1mm} (\alpha,\beta)\in\Big\{(\zeta^k,\zeta^{k'}),(\zeta^k,\xi\zeta^{k'}),(\xi\zeta^{k},\xi\zeta^{k'})\Big\}, && &&\text{resp.}\hspace{1mm} (\alpha,\beta)=(\xi\zeta^{k},\zeta^{k'}),\end{aligned}$$ alors, en conjuguant $\omega$ par $\Big(x,\frac{y}{\zeta^{k}}\Big)$, resp. $\Big(\frac{x}{\zeta^{k}},\frac{y}{\zeta^{k'}}\Big),$ resp. $\Big(\frac{y}{\zeta^{k}},\frac{x}{\zeta^{k'}}\Big),$ nous nous ramenons à $(\alpha,\beta)=(0,\xi)$, resp. $(\alpha,\beta)\in\{(1,1),(1,\xi),(\xi,\xi)\},$ resp. $(\alpha,\beta)=(1,\xi).$ Ainsi, à conjugaison près, $(\alpha,\beta)$ appartient bien à $E.$ Par ailleurs, d'après l'Exemple [Exemple 25](#eg:H0){reference-type="ref" reference="eg:H0"}, la platitude de $\mathrm{Leg}\mathscr{H}_3$ est équivalente à $$0=(\alpha+\beta)\Big((\alpha+\beta)^{d-2}-1\Big)\Big((\alpha+\beta)^{d-2}-2(d-2)\Big)=:f_d(\alpha,\beta).$$ Comme $$\begin{aligned} & f_d(0,\xi)=2\xi(2d-3)\neq0,&&\quad\quad f_d(1,1)=4(2^{d-2}-1)(2^{d-3}-d+2)=0\Longleftrightarrow d\in\{3,4\},\\ & f_d(\xi,\xi)=2\xi(2^{d-2}+1)(2^{d-2}+2d-4)\neq0,&&\quad\quad f_d(1,\xi)=(\xi+1)\Big((\xi+1)^{d-2}-1\Big)\Big((\xi+1)^{d-2}-2(d-2)\Big)=0\Longleftrightarrow d=3,\end{aligned}$$ il en résulte que $d\in\{3,4\}$ et, à conjugaison près, $$\begin{aligned} &(\alpha,\beta)\in\{(1,1),(1,-1)\}\hspace{1mm}\text{si}\hspace{1mm} d=3 &&\text{et}&& (\alpha,\beta)=(1,1)\hspace{1mm}\text{si}\hspace{1mm} d=4,\end{aligned}$$ *i.e.*, en posant $\ell_1=\{x+y-z=0\}$  et  $\ell_2=\{x-y-z=0\}$, $\ell\in\{\ell_1,\ell_2\}$ si $d=3$   et   $\ell=\ell_1$ si $d=4.$ Pour terminer, il suffit de remarquer que $\ell_1=(s_1s_2s_4)$ et $\ell_2=(s_1s_3)$, où $s_1=[1:0:1]$, $s_2=[0:1:1]$, $s_3=[1:1:0]$ et $s_4=[-1:1:0]$: les points $s_1,s_2$ et $s_3$ sont singuliers pour $\mathcal{F}_{0}^{d-1}$, et $s_4\in\mathrm{Sing}\mathcal{F}_{0}^{d-1}$ si et seulement si $d$ est pair; en particulier le point $s_4$ est singulier pour $\mathcal{F}_{0}^{3}$ mais pas pour $\mathcal{F}_{0}^{2}.$ ◻ **Démonstration de la Proposition [Proposition 7](#proalph:ell-non-invariante-Hesse){reference-type="ref" reference="proalph:ell-non-invariante-Hesse"}*.* Le feuilletage $\mathcal{F}_{\hspace{-0.4mm}\raisebox{-0.2mm}{\tiny{$H$}}}^{4}$ est donné en coordonnées homogènes par la $1$-forme $$\Omega_{\scalebox{0.64}{\ensuremath H}}^{4}=yz(2\,x^3-y^3-z^3)\mathrm{d}x+xz(2y^3-x^3-z^3)\mathrm{d}y+xy(2z^3-x^3-y^3)\mathrm{d}z.$$ Il possède les $12$ droites invariantes suivantes: $$\begin{aligned} xyz(x+y+z)(\zeta\,x+y+z)(x+\zeta\,y+z)(x+y+\zeta\,z)(\zeta^2x+y+z)(x+\zeta^2y+z)(x+y+\zeta^2z)(\zeta^2x+\zeta\,y+z)(\zeta\,x+\zeta^2y+z)=0,\end{aligned}$$ où $\zeta=\exp(\tfrac{2\mathrm{i}\pi}{3}).$ Comme précédemment, on peut supposer que $\ell=\{\alpha\,x+\beta\,y-z=0\}$ avec $\beta\neq0.$ Alors l'adhérence de la $\mathrm{Aut}(\mathbb{P}^{2}_{\mathbb{C}})$-orbite de $\ell\boxtimes\mathcal{F}_{\hspace{-0.4mm}\raisebox{-0.2mm}{\tiny{$H$}}}^{4}$ contient les trois pré-feuilletages homogènes suivants: $$\begin{aligned} \mathscr{H}_1=\{y-\alpha\,x=0\}\boxtimes\mathcal{H}_{4}^{4},&& \mathscr{H}_2=\{y-\beta\,x=0\}\boxtimes\mathcal{H}_{4}^{4},&& \mathscr{H}_3=\big\{a\,x+by=0\big\}\boxtimes\mathcal{H}_{4}^{4},\end{aligned}$$ où $a=\alpha+\beta-1$ et $b=\alpha+\zeta^2\beta-\zeta.$ En effet, dans la carte affine $z=1$, le pré-feuilletage $\ell\boxtimes\mathcal{F}_{\hspace{-0.4mm}\raisebox{-0.2mm}{\tiny{$H$}}}^{4}$ est décrit par $\omega=(\alpha\,x+\beta\,y-1)\omega_{\scalebox{0.64}{\ensuremath H}}^{4}$; en posant $\psi_1=\Big(\frac{x}{y},\frac{\varepsilon}{y}\Big),$ $\psi_2=\Big(\frac{\varepsilon}{y},\frac{x}{y}\Big)$ et $\psi_3=\Big(\frac{x+y}{x+\zeta\,y+\varepsilon},\frac{x+\zeta^2y}{x+\zeta\,y+\varepsilon}\Big),$ un calcul élémentaire montre que $$\begin{aligned} \lim_{\varepsilon\to 0}\varepsilon^{-1}y^{7}\psi_1^*\omega=(\alpha\,x-y)\omega_{4}^{4}, &&\hspace{3mm} \lim_{\varepsilon\to 0}\varepsilon^{-1}y^{7}\psi_2^*\omega=(\beta\,x-y)\omega_{4}^{4}, &&\hspace{3mm} \lim_{\varepsilon\to 0}\varepsilon^{-1}(x+\zeta\,y+\varepsilon)^7\psi_3^*\omega=-9\zeta(a\,x+by)\omega_{4}^{4}.\end{aligned}$$ Le $5$-tissu $\mathrm{Leg}(\ell\boxtimes\mathcal{F}_{\hspace{-0.4mm}\raisebox{-0.2mm}{\tiny{$H$}}}^{4})$ étant plat, il en est de même des $5$-tissus $\mathrm{Leg}\mathscr{H}_i,\,i=1,2,3.$ Or, d'après l'Exemple [Exemple 26](#eg:H4){reference-type="ref" reference="eg:H4"}, pour toute droite $\ell_0$ passant par l'origine, $\mathrm{Leg}(\ell_0\boxtimes\mathcal{H}_{4}^{4})$ est plat si et seulement si $\ell_0=\{x=0\}$ ou $\ell_0=\{y-\rho\,x=0\}$ avec $\rho(\rho^3-1)(\rho^3+1)=0,$ *i.e.* $\rho\in E:=\{0,\zeta^k,-\zeta^k,\,k=0,1,2\}.$ Donc, la platitude de $\mathrm{Leg}\mathscr{H}_1$ (resp. $\mathrm{Leg}\mathscr{H}_2$) est équivalente à $\alpha\in E$ (resp. $\beta\in E\setminus\{0\}$ car $\beta\neq0$). Remarquons que $(\alpha,\beta)\neq(-\zeta^k,-\zeta^{k'})$ sinon $\ell$ serait invariante par $\mathcal{F}_{\hspace{-0.4mm}\raisebox{-0.2mm}{\tiny{$H$}}}^{4}$. Par suite $$\begin{aligned} (\alpha,\beta)\in\Big\{(0,\zeta^{k}),(0,-\zeta^{k}),(\zeta^k,\zeta^{k'}),(\zeta^k,-\zeta^{k'}),(-\zeta^k,\zeta^{k'}),\hspace{1.5mm}k,k'=0,1,2\Big\}.\end{aligned}$$ Si, pour $k,k'\in\{0,1,2\},$ $$\begin{aligned} (\alpha,\beta)\in\Big\{(0,\zeta^k),(0,-\zeta^k)\Big\}, &&\text{resp.}\hspace{1mm} (\alpha,\beta)\in\Big\{(\zeta^k,\zeta^{k'}),(\zeta^k,-\zeta^{k'})\Big\}, && &&\text{resp.}\hspace{1mm} (\alpha,\beta)=(-\zeta^k,\zeta^{k'}),\end{aligned}$$ alors, en conjuguant $\omega$ par $\Big(x,\frac{y}{\zeta^{k}}\Big)$, resp. $\Big(\frac{x}{\zeta^{k}},\frac{y}{\zeta^{k'}}\Big),$ resp. $\Big(\frac{y}{\zeta^{k}},\frac{x}{\zeta^{k'}}\Big),$ on se ramène à $(\alpha,\beta)\in\{(0,1),(0,-1)\}$, resp. $(\alpha,\beta)\in\{(1,1),(1,-1)\},$ resp. $(\alpha,\beta)=(1,-1).$ Il en résulte qu'à conjugaison linéaire près $$(\alpha,\beta)\in F:=\big\{(0,1),(0,-1),(1,1),(1,-1)\big\}.$$ Or, pour $(\alpha,\beta)\in F,$ $\mathrm{Leg}\mathscr{H}_3$ est plat si et seulement si $(\alpha,\beta)=(0,1),$ puisqu'en posant $\tau(\alpha,\beta)=-\frac{a}{b},$ on a $$\begin{aligned} \tau(0,1)=0\in E,&& \tau(0,-1)=2\not\in E,&& \tau(1,1)=\tfrac{\zeta^2}{2}\not\in E,&& \tau(1,-1)=\tfrac{1}{2}\not\in E.\end{aligned}$$ Donc, à conjugaison près, $(\alpha,\beta)=(0,1)$, *i.e.* $\ell=\{y-z=0\}$; alors $\ell$ passe par quatre points singuliers de $\mathcal{F}_{\hspace{-0.4mm}\raisebox{-0.2mm}{\tiny{$H$}}}^{4},$ à savoir les points $s_1=[1:0:0],$ $s_2=[1:1:1]$, $s_3=[\zeta:1:1]$ et $s_4=[\zeta^2:1:1].$ ◻ # Pré-feuilletages de type $(1,3)$ dont le feuilletage associé est à singularités non-dégénérées {#sec:pre-feuilletages-codegre-1-degre-3} Dans ce paragraphe, nous allons démontrer le Théorème [Théorème 8](#thmalph:Fermat2){reference-type="ref" reference="thmalph:Fermat2"} annoncé dans l'Introduction. Pour ce faire, nous avons besoin de deux résultats intermédiaires, dont le premier est valable en degré quelconque. Rappelons d'abord qu'au §[5](#sec:pre-pre-feuill-convexe-reduit){reference-type="ref" reference="sec:pre-pre-feuill-convexe-reduit"} nous avons démontré les Propositions [Proposition 6](#proalph:ell-non-invariante-Fermat){reference-type="ref" reference="proalph:ell-non-invariante-Fermat"} et [Proposition 7](#proalph:ell-non-invariante-Hesse){reference-type="ref" reference="proalph:ell-non-invariante-Hesse"} en se ramenant au cas homogène; en fait cet argument repose implicitement sur la proposition suivante. **Proposition 35**. **Soit $\mathscr{F}=\ell\boxtimes\mathcal{F}$ un pré-feuilletage de co-degré $1$ et de degré $d\geq2$ sur $\mathbb{P}^{2}_{\mathbb{C}}.$ Supposons que le feuilletage $\mathcal{F}$ possède une droite invariante $D$ et que toutes ses singularités sur $D$ soient non-dégénérées. Il existe un pré-feuilletage homogène $\mathscr{H}=\ell_0\boxtimes\mathcal{H}$ de co-degré $1$ et de degré $d$ sur $\mathbb{P}^{2}_{\mathbb{C}}$ tel que:** - *$\mathscr{H}\in\overline{\mathcal{O}(\mathscr{F})}$ et $\mathcal{H}\in\overline{\mathcal{O}(\mathcal{F})}$;* - *si $\ell=D$ (resp. $\ell\neq D$), alors $\ell_0=L_\infty$ (resp. $\ell_0\neq L_\infty)$;* - *$D$ est invariante par $\mathcal{H}$;* - *$\mathrm{Sing}\mathcal{H}\cap D=\mathrm{Sing}\mathcal{F}\cap D$;* - *$\forall\hspace{0.5mm}s\in\mathrm{Sing}\mathcal{H}\cap D,\hspace{1mm} \mu(\mathcal{H},s)=1\label{not:nombre-milnor} \hspace{2mm}\text{et}\hspace{2mm} \mathrm{CS}(\mathcal{H},D,s)=\mathrm{CS}(\mathcal{F},D,s)$.* *Si de plus $\mathrm{Leg}\mathscr{F}$ (resp. $\mathrm{Leg}\mathcal{F}$) est plat, alors $\mathrm{Leg}\mathscr{H}$ (resp. $\mathrm{Leg}\mathcal{H}$) l'est aussi.* Cette proposition est un analogue pour les pré-feuilletages de co-degré $1$ de la Proposition 6.4 de [@BM18Bull] sur les feuilletages de $\mathbb{P}^{2}_{\mathbb{C}}.$ **Démonstration*.* Choisissons des coordonnées homogènes $[x:y:z]\in\mathbb{P}^{2}_{\mathbb{C}}$ telles que $D=L_\infty=(z=0)$; puisque $D$ est $\mathcal{F}$-invariante, $\mathcal{F}$ est défini dans la carte affine $z=1$ par une $1$-forme $\omega$ du type $$\omega=\sum_{i=0}^{d-1}(A_i(x,y)\mathrm{d}x+B_i(x,y)\mathrm{d}y),$$ où les $A_i,\,B_i$ sont des polynômes homogènes de degré $i$. D'après [@BM18Bull Proposition 6.4], comme toutes les singularités de $\mathcal{F}$ sur $D$ sont non-dégénérées, la $1$-forme $\omega_{d-1}=A_{d-1}(x,y)\mathrm{d}x+B_{d-1}(x,y)\mathrm{d}y$ définit bien un feuilletage homogène $\mathcal{H}$ de degré $d-1$ sur $\mathbb{P}^{2}_{\mathbb{C}}$ appartenant à $\overline{\mathcal{O}(\mathcal{F})}$ et vérifiant les propriétés (iii), (iv) et (v) annoncées. Maintenant, écrivons $\ell=\{\alpha\,x+\beta y+\gamma\,z=0\}$; en coordonnées homogènes, $\mathscr{F},$ resp. $\mathcal{H},$ est donné par $$\begin{aligned} &\Omega_{d+1}=(\alpha\,x+\beta y+\gamma\,z)\sum_{i=0}^{d-1}z^{d-i-1}\Big(A_i(x,y)(z\mathrm{d}x-x\mathrm{d}z)+B_i(x,y)(z\mathrm{d}y-y\mathrm{d}z)\Big),&&\\ \text{resp}.\hspace{1.5mm} &\Omega_{d}=A_{d-1}(x,y)(z\mathrm{d}x-x\mathrm{d}z)+B_{d-1}(x,y)(z\mathrm{d}y-y\mathrm{d}z).\end{aligned}$$ En posant $\varphi=\varphi_{\varepsilon}=\left[\frac{x}{\varepsilon}:\frac{y}{\varepsilon}:z\right],$ nous constatons que si $(\alpha,\beta)=(0,0),$ resp. $(\alpha,\beta)\neq(0,0),$ alors $$\begin{aligned} &\hspace{0.5cm}\lim_{\varepsilon\to 0}\varepsilon^{d}\gamma^{-1}\varphi^*\Omega_{d+1}=z\hspace{0.3mm}\Omega_{d}, &&\text{resp.}\hspace{1mm} \lim_{\varepsilon\to 0}\varepsilon^{d+1}\varphi^*\Omega_{d+1}=(\alpha\,x+\beta y)\Omega_{d}.\end{aligned}$$ Il en résulte que l'adhérence de la $\mathrm{Aut}(\mathbb{P}^{2}_{\mathbb{C}})$-orbite de $\mathscr{F}$ contient le pré-feuilletage homogène $\mathscr{H}=\ell_0\boxtimes\mathcal{H},$ où $\ell_0=L_\infty$ si $\ell=D$ et $\ell_0=\{\alpha\,x+\beta y=0\}\neq L_\infty$ si $\ell\neq D.$ ◻ Le lemme technique suivant est un analogue pour les pré-feuilletages de type $(1,3)$ du Lemme 6.7 de [@BM18Bull] sur les feuilletages de degré $3.$ Il joue un rôle clé dans la démonstration du Théorème [Théorème 8](#thmalph:Fermat2){reference-type="ref" reference="thmalph:Fermat2"}. **Lemme 36**. **Soit $\mathscr{F}=\ell\boxtimes\mathcal{F}$ un pré-feuilletage de co-degré $1$ et de degré $3$ sur $\mathbb{P}^{2}_{\mathbb{C}}.$ Supposons que le $3$-tissu $\mathrm{Leg}\mathscr{F}$ soit plat et que le feuilletage $\mathcal{F}$ possède une singularité $m$ non-dégénérée vérifiant $\mathrm{BB}(\mathcal{F},m)\neq4.$[\[not:invariant-BB\]]{#not:invariant-BB label="not:invariant-BB"} Alors par le point $m$ passent exactement deux droites invariantes par $\mathcal{F}.$** **Démonstration*.* Les hypothèses $\mu(\mathcal{F},m)=1$ et $\mathrm{BB}(\mathcal{F},m)\neq4$ assurent l'existence d'une carte affine $(x,y)$ de $\mathbb{P}^{2}_{\mathbb{C}}$ dans laquelle $m=(0,0)$ et $\mathcal{F}$ est défini par une $1$-forme $\omega_0$ du type $\omega_0=\omega_{0,1}+\omega_{0,2}+\omega_{0,3},$ où $$\begin{aligned} \omega_{0,1}=\lambda\hspace{0.1mm}y\mathrm{d}x+\mu\hspace{0.1mm}x\mathrm{d}y,&& \omega_{0,2}=\left(\sum_{i=0}^{2}a_{i}\hspace{0.1mm}x^{2-i}y^{i}\right)\mathrm{d}x+\left(\sum_{i=0}^{2}b_{i}\hspace{0.1mm}x^{2-i}y^{i}\right)\mathrm{d}y,&& \omega_{0,3}=\left(\sum_{i=0}^{2}c_{i}\hspace{0.1mm}x^{2-i}y^{i}\right)(x\mathrm{d}y-y\mathrm{d}x),\end{aligned}$$ avec $\lambda\mu(\lambda+\mu)\neq0.$ Les seules droites passant par $m$ et qui puissent être invariantes par $\mathcal{F}$ sont $(x=0)$ et $(y=0).$ En effet, notons $\mathrm{R}=x\frac{\partial{}}{\partial{x}}+y\frac{\partial{}}{\partial{y}}$ le champ radial centré en $m$; si $L=(ux+vy=0)$ est $\mathcal{F}$-invariante, alors $ux+vy$ doit diviser le cône tangent $\mathrm{C}_{\omega_{0,1}}:=\omega_{0,1}(\mathrm{R})=(\lambda+\mu)xy,$ d'où $u=0$ ou $v=0.$ Nous allons montrer qu'effectivement $(x=0)$ et $(y=0)$ sont invariantes par $\mathcal{F},$ ce qui établira le lemme. Il s'agit de prouver que $a_0=b_2=0,$ vu que l'invariance par $\mathcal{F}$ de $(x=0)$, resp. $(y=0),$ est équivalente à la nullité de $b_2$, resp. $a_0.$ Si $\ell=\{\alpha\,x+\beta\,y+\gamma=0\}$ alors, dans la carte affine $(p,q)$ de $\mathbb{\check{P}}^{2}_{\mathbb{C}}$ correspondant à la droite $\{y=px-q\}\subset{\mathbb{P}^{2}_{\mathbb{C}}},$ le $3$-tissu $\mathrm{Leg}\mathscr{F}$ est décrit par la $3$-forme symétrique $$\begin{aligned} &\hspace{0.5cm}\check{\omega}=\big((\gamma-\beta\,q)\mathrm{d}p+(\alpha+\beta\,p)\mathrm{d}q\big)\check{\omega}_{0}, \\ &\hspace{0cm}{\fontsize{11}{11pt}\text{où}} \\ &\hspace{0.5cm}\check{\omega}_{0}=\mu\,p\mathrm{d}p\mathrm{d}q +(a_0+b_0p+c_0q)\mathrm{d}q^2 +\Big(\lambda\,\mathrm{d}p+(a_1+b_1p+c_1q)\mathrm{d}q\Big)(p\mathrm{d}q-q\mathrm{d}p) +\big(a_2+b_2p+c_2q\big)\big(p\mathrm{d}q-q\mathrm{d}p\big)^2.\end{aligned}$$ Supposons par l'absurde que $a_0\neq0.$ Considérons la famille d'automorphismes $\varphi=\varphi_{\varepsilon}=(a_0\hspace{0.1mm}\varepsilon\hspace{0.1mm}p,\hspace{0.1mm}a_0\hspace{0.1mm}\varepsilon^{2}\hspace{0.1mm}q).$ Nous constatons que si $\gamma\neq0$,  resp. $\gamma=0$ et $\alpha\neq0,$  resp. $\gamma=\alpha=0,$ alors $$\begin{aligned} &\hspace{0.5cm}\lim_{\varepsilon\to 0}\varepsilon^{-5}\gamma^{-1}a_{0}^{-4}\varphi^*\check{\omega}=\theta_1\eta, &&\text{resp.}\hspace{1mm} \lim_{\varepsilon\to 0}\varepsilon^{-6}\alpha^{-1}a_{0}^{-4}\varphi^*\check{\omega}=\theta_2\eta, &&\text{resp.}\hspace{1mm} \lim_{\varepsilon\to 0}\varepsilon^{-7}\beta^{-1}a_{0}^{-5}\varphi^*\check{\omega}=\theta_3\eta,\end{aligned}$$ où $$\begin{aligned} &\hspace{-0.14cm}\theta_1=\mathrm{d}p, && \theta_2=\mathrm{d}q, && \theta_3=p\mathrm{d}q-q\mathrm{d}p, && \eta=-\lambda\,q\mathrm{d}p^2+(\lambda+\mu)p\mathrm{d}p\mathrm{d}q+\mathrm{d}q^2.\end{aligned}$$ Pour $i=1,2,3,$ posons $\mathcal{W}_{3}^{(i)}=\mathcal{F}_i\boxtimes\mathcal{W}_2,$ où $\mathcal{W}_2$ (resp. $\mathcal{F}_i$) désigne le $2$-tissu (resp. le feuilletage) défini par $\eta$ (resp. par $\theta_i$). Il en résulte que si $\gamma\neq0$,  resp. $\gamma=0$ et $\alpha\neq0,$  resp. $\gamma=\alpha=0,$ alors l'adhérence de la $\mathrm{Aut}(\mathbb{\check{P}}^{2}_{\mathbb{C}})$-orbite de $\mathrm{Leg}\mathscr{F}$ contient le $3$-tissu $\mathcal{W}_{3}^{(1)},$ resp. $\mathcal{W}_{3}^{(2)},$ resp. $\mathcal{W}_{3}^{(3)}.$ Or, comme $\mathrm{Leg}\mathscr{F}$ est plat par hypothèse, tout $3$-tissu appartenant à $\overline{\mathcal{O}(\mathrm{Leg}\mathscr{F})}$ est aussi plat. Pour aboutir à une contradiction, il suffit donc de montrer que pour tout $i=1,2,3,$ $\mathcal{W}_{3}^{(i)}$ n'est pas plat. Comme $\Delta(\eta)=f(p,q):=4\lambda\,q+(\lambda+\mu)^2p^2$, il suffit encore de montrer que pour tout $i=1,2,3,$ la courbure de $\mathcal{W}_{3}^{(i)}$ n'est pas holomorphe le long de la composante $\mathcal{C}=\{f(p,q)=0\}\subset\Delta(\mathcal{W}_2),$ qui est une parabole, car $\lambda(\lambda+\mu)\neq0.$ Tout d'abord, notons que $\mathcal{C}$ n'est pas invariante par $\mathcal{W}_2$, puisqu'en posant $\eta_0=(\lambda+\mu)p\mathrm{d}p+2\mathrm{d}q,$ nous avons $$\begin{aligned} &\hspace{-0.4cm}\eta\big|_{\mathcal{C}}=\left(\frac{\eta_0}{2}\right)^2 \qquad\qquad\text{et}\qquad\qquad \eta_0\wedge\mathrm{d}f=-4\mu(\lambda+\mu)p\mathrm{d}p\wedge\mathrm{d}q\not\equiv0.\end{aligned}$$ Considérons le cas où $i\in\{1,2\}.$ Comme $$\begin{aligned} &\hspace{1.27cm}\eta_0\wedge\theta_1\Big|_{\mathcal{C}}=-2\mathrm{d}p\wedge\mathrm{d}q\not\equiv0 \qquad\qquad\text{et}\qquad\qquad \eta_0\wedge\theta_2\Big|_{\mathcal{C}}=(\lambda+\mu)p\mathrm{d}p\wedge\mathrm{d}q\not\equiv0,\end{aligned}$$ nous avons $\mathcal{C}\not\subset\mathrm{Tang}(\mathcal{W}_2,\mathcal{F}_i).$ Donc, d'après [@MP13 Théorème 1] (*cf.* [@BFM14 Théorème 1.1]), la courbure $K(\mathcal{W}_{3}^{(i)})$ est holomorphe sur $\mathcal{C}$ si et seulement si $\mathcal{C}$ est invariante par $\mathcal{F}_i,$ ce qui est impossible, car chaque $\mathcal{F}_i$ est un pinceau de droites et ne peut donc pas admettre une parabole comme courbe invariante. Examinons maintenant le cas où $i=3.$ Dans ce cas $\mathcal{C}\subset\mathrm{Tang}(\mathcal{W}_2,\mathcal{F}_3)$ si et seulement si $\lambda=\mu,$ car $$\begin{aligned} \eta_0\wedge\theta_3\Big|_{\mathcal{C}}=\frac{1}{2\lambda}(\lambda-\mu)(\lambda+\mu)p^2\mathrm{d}p\wedge\mathrm{d}q\equiv0\Longleftrightarrow\lambda=\mu.\end{aligned}$$ Si $\lambda\neq\mu$, alors on peut comme précédemment appliquer le Théorème 1 de [@MP13] et affirmer que $K(\mathcal{W}_{3}^{(3)})$ ne peut pas être holomorphe sur $\mathcal{C}.$ Supposons donc $\lambda=\mu$ et prouvons que $K(\mathcal{W}_{3}^{(3)})\not\equiv0.$ L'image réciproque de $\mathcal{W}_{3}^{(3)}$ par l'application rationnelle $\psi(p,q)=\left(p,\mu(q^2-p^2)\right)$ s'écrit $\psi^*\mathcal{W}_{3}^{(3)}=\mathcal{F}_{3}^{(1)}\boxtimes\mathcal{F}_{3}^{(2)}\boxtimes\mathcal{F}_{3}^{(3)}$, où $$\begin{aligned} & \mathcal{F}_{3}^{(1)}\hspace{0.1mm}:\hspace{0.1mm}(p^2+q^2)\mathrm{d}p-2pq\mathrm{d}q=0,&& \mathcal{F}_{3}^{(2)}\hspace{0.1mm}:\hspace{0.1mm}(p+q)\mathrm{d}p-2q\mathrm{d}q=0,&& \mathcal{F}_{3}^{(3)}\hspace{0.1mm}:\hspace{0.1mm}(p-q)\mathrm{d}p-2q\mathrm{d}q=0.\end{aligned}$$ En utilisant la formule ([\[equa:eta-rst\]](#equa:eta-rst){reference-type="ref" reference="equa:eta-rst"}), un calcul direct conduit à $$\begin{aligned} &\hspace{-1cm} \eta(\psi^{*}\mathcal{W}_{3}^{(3)})=-\frac{p\mathrm{d}p}{q^2}+\frac{4\mathrm{d}q}{q}+\frac{\mathrm{d}(p^2-q^2)}{p^2-q^2},\end{aligned}$$ de sorte que $$K(\psi^{*}\mathcal{W}_{3}^{(3)})=\mathrm{d}\eta(\psi^{*}\mathcal{W}_{3}^{(3)})=-\frac{2p}{q^3}\mathrm{d}p\wedge\mathrm{d}q\not\equiv0,$$ d'où $\psi^{*}K(\mathcal{W}_{3}^{(3)})=K(\psi^{*}\mathcal{W}_{3}^{(3)})\not\equiv0$ et donc $K(\mathcal{W}_{3}^{(3)})\not\equiv0.$ Nous avons ainsi démontré l'égalité $a_0=0$, d'où l'invariance de la droite $(y=0)$ par $\mathcal{F}.$ En échangeant les rôles des coordonnées $x$ et $y$, le même argument montre que $b_2=0$, *i.e.* que la droite $(x=0)$ est aussi invariante par $\mathcal{F}.$ ◻ Avant de commencer la preuve du Théorème [Théorème 8](#thmalph:Fermat2){reference-type="ref" reference="thmalph:Fermat2"}, rappelons (*cf.* [@Bru15]) que si $\mathcal{F}$ est un feuilletage de degré $d$ sur $\mathbb{P}^{2}_{\mathbb{C}}$ alors $$\begin{aligned} \label{equa:Darboux-BB} &\sum_{s\in\mathrm{Sing}\mathcal{F}}\mu(\mathcal{F},s)=d^2+d+1 &&\text{et}&& \sum_{s\in\mathrm{Sing}\mathcal{F}}\mathrm{BB}(\mathcal{F},s)=(d+2)^2.\end{aligned}$$ **Démonstration du Théorème [Théorème 8](#thmalph:Fermat2){reference-type="ref" reference="thmalph:Fermat2"}*.* Écrivons $\mathrm{Sing}\mathcal{F}=\Sigma^{1}\cup\Sigma^{2}$, où $$\begin{aligned} \Sigma^{1}=\{s\in\mathrm{Sing}\mathcal{F}\hspace{1mm}\colon \mathrm{BB}(\mathcal{F},s)=4\} &&\text{et}&& \Sigma^{2}=\mathrm{Sing}\mathcal{F}\setminus\Sigma^{1}\hspace{0.5mm};\end{aligned}$$ notons $\kappa_i=\#\hspace{0.5mm}\Sigma^{i},\,i=1,2.$ Par hypothèse, nous avons $\deg\mathcal{F}=2$ et, pour tout $s\in\mathrm{Sing}\mathcal{F},$ $\mu(\mathcal{F},s)=1.$ Les formules ([\[equa:Darboux-BB\]](#equa:Darboux-BB){reference-type="ref" reference="equa:Darboux-BB"}) donnent alors $$\begin{aligned} \label{equa:Dar-BB-2} &\#\hspace{0.5mm}\mathrm{Sing}\mathcal{F}=\kappa_1+\kappa_2=7 &&\text{et} &&4\kappa_1+\sum_{s\in\Sigma^{2}}\mathrm{BB}(\mathcal{F},s)=16\hspace{0.5mm};\end{aligned}$$ nous en déduisons que $\Sigma^{2}$ est non vide. Soit $m$ un point de $\Sigma^{2}$; d'après le Lemme [Lemme 36](#lem:2-droites-invariantes){reference-type="ref" reference="lem:2-droites-invariantes"} par le point $m$ passent exactement deux droites $D_{m}^{(1)}$ et $D_{m}^{(2)}$ invariantes par $\mathcal{F}$. Alors, pour $i=1,2$, la Proposition [Proposition 35](#pro:pref-degenere-preh){reference-type="ref" reference="pro:pref-degenere-preh"} assure l'existence d'un pré-feuilletage homogène $\mathscr{H}^{(i)}_{m}=\ell_{m}^{(i)}\boxtimes\mathcal{H}^{(i)}_{m}$ de type $(1,3)$ sur $\mathbb{P}^{2}_{\mathbb{C}}$ appartenant à $\overline{\mathcal{O}(\mathscr{F})}$ et tel que la droite $D_{m}^{(i)}$ soit $\mathcal{H}^{(i)}_{m}$-invariante. Comme $\mathrm{Leg}\mathscr{F}$ est par hypothèse plat, il en est de même de $\mathrm{Leg}\mathscr{H}^{(1)}_{m}$ et $\mathrm{Leg}\mathscr{H}^{(2)}_{m}.$ Donc, $\mathscr{H}^{(i)}_{m}$ ($i=1,2$) est linéairement conjugué à l'un des huit modèles du Corollaire [Corollaire 34](#cor:class-pre-homogenes-plats-co-degre-1-degre-3){reference-type="ref" reference="cor:class-pre-homogenes-plats-co-degre-1-degre-3"}. Pour $i=1,2,$ la Proposition [Proposition 35](#pro:pref-degenere-preh){reference-type="ref" reference="pro:pref-degenere-preh"} assure aussi que - si $\ell\neq D_{m}^{(i)}$, alors $\ell_{m}^{(i)}\neq L_\infty$; - $\mathrm{Sing}\mathcal{F}\cap D_{m}^{(i)}=\mathrm{Sing}\mathcal{H}^{(i)}_{m}\cap D_{m}^{(i)}$; - $\forall\hspace{0.5mm}s\in\mathrm{Sing}\mathcal{H}^{(i)}_{m}\cap D_{m}^{(i)},\quad \mu(\mathcal{H}^{(i)}_{m},s)=1\hspace{2mm}\text{et}\quad \mathrm{CS}(\mathcal{H}^{(i)}_{m}, D_{m}^{(i)},s)=\mathrm{CS}(\mathcal{F}, D_{m}^{(i)},s)$. Puisque $\mathrm{CS}(\mathcal{F}, D_{m}^{(1)},m)\mathrm{CS}(\mathcal{F}, D_{m}^{(2)},m)=1,$ nous avons $$\begin{aligned} \label{equa:produit-CS-1} \mathrm{CS}(\mathcal{H}^{(1)}_{m}, D_{m}^{(1)},m)\mathrm{CS}(\mathcal{H}^{(2)}_{m}, D_{m}^{(2)},m)=1.\end{aligned}$$ Supposons dans un premier temps que $\ell\neq D_{m}^{(i)}$ pour $i=1,2$; c'est évidemment le cas si $\ell$ n'est pas invariante par $\mathcal{F}.$ Alors, d'après ($\mathfrak{a}$), nous avons $\ell_{m}^{(i)}\neq L_\infty$ pour $i=1,2.$ Donc, chacun des $\mathscr{H}^{(i)}_{m}$ est conjugué à l'un des cinq pré-feuilletages $\mathscr{H}_{j}^{3},j=2,\ldots,6,$ de sorte que chacun des $\mathcal{H}^{(i)}_{m}$ est conjugué à l'un des trois feuilletages $\mathcal{H}_{1}^{2}$, $\mathcal{H}_{2}^{2}(0,0),$ $\mathcal{H}_{3}^{2}(-2)$ (Corollaire [Corollaire 34](#cor:class-pre-homogenes-plats-co-degre-1-degre-3){reference-type="ref" reference="cor:class-pre-homogenes-plats-co-degre-1-degre-3"}). En consultant la Table [1](#tab:CS(lambda)){reference-type="ref" reference="tab:CS(lambda)"} et en utilisant l'égalité ([\[equa:produit-CS-1\]](#equa:produit-CS-1){reference-type="ref" reference="equa:produit-CS-1"}) ainsi que les relations ($\mathfrak{b}$) et ($\mathfrak{c}$), nous constatons que $$\begin{aligned} \label{equa:CS-moins-1-Sigma-i-cap-D-m-i} \mathrm{CS}(\mathcal{H}^{(1)}_{m}, D_{m}^{(1)},m)=\mathrm{CS}(\mathcal{H}^{(2)}_{m}, D_{m}^{(2)},m)=-1,&&\hspace{2mm} \#\hspace{0.5mm}(\Sigma^{1}\cap D_{m}^{(1)})=\#\hspace{0.5mm}(\Sigma^{1}\cap D_{m}^{(2)})=2,&&\hspace{2mm} \Sigma^{2}\cap D_{m}^{(1)}=\Sigma^{2}\cap D_{m}^{(2)}=\{m\}.\end{aligned}$$ Supposons maintenant que la droite $\ell$ soit égale à l'une des droites $D_{m}^{(i)}$; soit par exemple $\ell=D_{m}^{(2)}.$ Montrons que les égalités ([\[equa:CS-moins-1-Sigma-i-cap-D-m-i\]](#equa:CS-moins-1-Sigma-i-cap-D-m-i){reference-type="ref" reference="equa:CS-moins-1-Sigma-i-cap-D-m-i"}) sont encore vérifiées. Comme $\ell\neq D_{m}^{(1)}$, $\mathcal{H}^{(1)}_{m}$ est conjugué à l'un des feuilletages $\mathcal{H}_{1}^{2}$, $\mathcal{H}_{2}^{2}(0,0),$ $\mathcal{H}_{3}^{2}(-2).$ De plus $\Sigma^2\cap D_{m}^{(1)}=\{m\}$; en effet, si $\Sigma^2\cap D_{m}^{(1)}$ contenait un autre point $m'\neq m,$ on aurait $\ell\neq D_{m'}^{(i)}$ pour $i=1,2,$ de sorte que $\{m'\}=\Sigma^{2}\cap D_{m'}^{(i)}=\Sigma^{2}\cap D_{m}^{(1)}\supset\{m,m'\}$, ce qui est impossible. Il en résulte, d'après la Table [1](#tab:CS(lambda)){reference-type="ref" reference="tab:CS(lambda)"}, que $$\begin{aligned} &\mathrm{CS}(\mathcal{H}^{(1)}_{m}, D_{m}^{(1)},m)=\mathrm{CS}(\mathcal{H}^{(2)}_{m},\ell,m)=-1 \qquad\qquad\text{et}\qquad\qquad \#\hspace{0.5mm}(\Sigma^{1}\cap D_{m}^{(1)})=2,\end{aligned}$$ d'où $$\begin{aligned} &\hspace{-6.26cm}\mathrm{CS}(\mathcal{F}, D_{m}^{(1)},m)=\mathrm{CS}(\mathcal{F},\ell,m)=-1.\end{aligned}$$ Comme ces égalités sont valables pour tout choix de $m\in\Sigma^2\cap\ell$ et puisque toute droite de $\mathbb{P}^{2}_{\mathbb{C}}$ ne peut contenir plus de $\deg\mathcal{F}+1=3$ points singuliers de $\mathcal{F},$ la formule de [Camacho]{.smallcaps}-[Sad]{.smallcaps} (*voir* [@CS82]) $\sum_{s\in\mathrm{Sing}\mathcal{F}\cap \ell}\mathrm{CS}(\mathcal{F},\ell,s)=1$ implique que $$\begin{aligned} \#\hspace{0.5mm}(\Sigma^{1}\cap\ell)=2 \qquad\qquad\text{et}\qquad\qquad \Sigma^{2}\cap\ell=\{m\}.\end{aligned}$$ Les égalités ([\[equa:CS-moins-1-Sigma-i-cap-D-m-i\]](#equa:CS-moins-1-Sigma-i-cap-D-m-i){reference-type="ref" reference="equa:CS-moins-1-Sigma-i-cap-D-m-i"}) sont ainsi établies dans tous les cas. Il s'en suit en particulier que $\mathrm{BB}(\mathcal{F},m)=0.$ Le point $m\in\Sigma^{2}$ étant arbitraire, $\Sigma^{2}$ est formé des $s\in\mathrm{Sing}\mathcal{F}$ tels que $\mathrm{BB}(\mathcal{F},s)=0$. Le système ([\[equa:Dar-BB-2\]](#equa:Dar-BB-2){reference-type="ref" reference="equa:Dar-BB-2"}) se réécrit alors $\kappa_1+\kappa_2=7$  et  $4\kappa_1=16$ dont l'unique solution est $(\kappa_1,\kappa_2)=(4,3)$, c'est-à-dire que $\mathrm{Sing}\mathcal{F}=\Sigma^{1}\cup\Sigma^{2},\hspace{2mm}\#\hspace{0.5mm}\Sigma^{1}=4$ et $\#\hspace{0.5mm}\Sigma^{2}=3.$ Comme $\Sigma^2\cap (D_{m}^{(1)}\cup D_{m}^{(2)})=\{m\}$, $\mathcal{F}$ possède $3\cdot2=6$ droites invariantes, ce qui signifie que $\mathcal{F}$ est convexe réduit. Il découle alors de la classification des feuilletages convexes de degré deux (*cf.* [@FP15 Proposition 7.4] ou [@BM20Bull Théorème A]) que $\mathcal{F}$ est linéairement conjugué au feuilletage de [Fermat]{.smallcaps} $\mathcal{F}_{0}^{2}.$ Nous concluons en remarquant que si la droite $\ell$ n'est pas invariante par $\mathcal{F}$, la platitude de $\mathrm{Leg}\mathscr{F}$ et la Proposition [Proposition 6](#proalph:ell-non-invariante-Fermat){reference-type="ref" reference="proalph:ell-non-invariante-Fermat"} entraînent que $\ell$ doit joindre deux singularités non radiales de $\mathcal{F}.$ ◻ # Index des notations {#index-des-notations .unnumbered} +:-------------------------------------------------------+:----------------------------------------------------------------------------------------------------------------------------------+ | ::: {.small} | ::: {.small} | | $\mathrm{Leg}\mathscr{F}$ | transformée de [Legendre]{.smallcaps} du pré-feuilletage $\mathscr{F}$ | | ::: | ::: | +--------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------------------------+ | ::: {.small} | ::: {.small} | | $\Delta(\mathcal{W})$ | discriminant du tissu $\mathcal{W}$ | | ::: | ::: | +--------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------------------------+ | ::: {.small} | ::: {.small} | | $\mathcal{T}_{\mathcal{H}}$ | type du feuilletage homogène $\mathcal{H}$ | | ::: | ::: | +--------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------------------------+ | ::: {.small} | ::: {.small} | | $\eta(\mathcal{W})$ | forme fondamentale du tissu $\mathcal{W}$ | | ::: | ::: | +--------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------------------------+ | ::: {.small} | ::: {.small} | | $K(\mathcal{W})$ | courbure du tissu $\mathcal{W}$ | | ::: | ::: | +--------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------------------------+ | ::: {.small} | ::: {.small} | | $\mathcal{G}_{\mathcal{F}}$ | application de [Gauss]{.smallcaps} associée au feuilletage $\mathcal{F}$ | | ::: | ::: | +--------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------------------------+ | ::: {.small} | ::: {.small} | | $\mathrm{Sing}\mathcal{F}$ | lieu singulier du feuilletage $\mathcal{F}$ | | ::: | ::: | +--------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------------------------+ | ::: {.small} | ::: {.small} | | $\mathrm{I}_{\mathcal{F}}^{\hspace{0.2mm}\mathrm{tr}}$ | partie transverse du diviseur d'inflexion $\mathrm{I}_{\mathcal{F}}$ | | ::: | ::: | +--------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------------------------+ | ::: {.small} | ::: {.small} | | $\mathrm{rg}(\mathcal{W})$ | rang du tissu $\mathcal{W}$ | | ::: | ::: | +--------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------------------------+ | ::: {.small} | ::: {.small} | | $S_\mathcal{W}$ | surface caractéristique du tissu $\mathcal{W}$ | | ::: | ::: | +--------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------------------------+ | ::: {.small} | ::: {.small} | | $\mathrm{C}_{\hspace{-0.3mm}\mathcal{H}}$ | cône tangent en l'origine du feuilletage homogène $\mathcal{H}$ | | ::: | ::: | +--------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------------------------+ | ::: {.small} | ::: {.small} | | $\mathrm{I}_{\mathcal{F}}$ | diviseur d'inflexion du feuilletage $\mathcal{F}$ | | ::: | ::: | +--------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------------------------+ | ::: {.small} | ::: {.small} | | $\mathrm{I}_{\mathcal{F}}^{\mathrm{inv}}$ | partie invariante du diviseur d'inflexion $\mathrm{I}_{\mathcal{F}}$ | | ::: | ::: | +--------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------------------------+ | ::: {.small} | ::: {.small} | | $\omega_{\hspace{0.2mm}0}^{\hspace{0.2mm}d-1}$ | $(d-2)y^{d-1}\mathrm{d}x+x\left(x^{d-2}-(d-1)y^{d-2}\right)\mathrm{d}y$ | | ::: | ::: | +--------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------------------------+ | ::: {.small} | ::: {.small} | | $\omega_{\hspace{0.2mm}4}^{\hspace{0.2mm}d-1}$ | $y(\sigma_d\,x^{d-2}-y^{d-2})\mathrm{d}x+x(\sigma_d\,y^{d-2}-x^{d-2})\mathrm{d}y$ | | ::: | ::: | +--------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------------------------+ | ::: {.small} | ::: {.small} | | $\omega_{1}^{d-1}$ | $y^{d-1}\mathrm{d}x-x^{d-1}\mathrm{d}y$ | | ::: | ::: | +--------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------------------------+ | ::: {.small} | ::: {.small} | | $\omega_{2}^{d-1}(\lambda,\mu)$ | $(x^{d-1}+\lambda\,y^{d-1})\mathrm{d}x+(\mu\,x^{d-1}-y^{d-1})\mathrm{d}y$ | | ::: | ::: | +--------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------------------------+ | ::: {.small} | ::: {.small} | | $\omega_{3}^{d-1}(\lambda)$ | $(x^{d-1}+\lambda y^{d-1})\mathrm{d}x+x^{d-1}\mathrm{d}y$ | | ::: | ::: | +--------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------------------------+ | ::: {.small} | ::: {.small} | | $\mathrm{CS}(\mathcal{F},\mathcal{C},s)$ | indice de [Camacho]{.smallcaps}-[Sad]{.smallcaps} du feuilletage $\mathcal{F}$ au point $s$ par rapport à la courbe $\mathcal{C}$ | | ::: | ::: | +--------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------------------------+ | ::: {.small} | ::: {.small} | | $\tau(\mathcal{F},s)$ | ordre de tangence du feuilletage $\mathcal{F}$ avec une droite générique passant par le point $s$ | | ::: | ::: | +--------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------------------------+ | ::: {.small} | ::: {.small} | | $\mathcal{O}(\mathscr{F})$ | orbite du pré-feuilletage $\mathscr{F}$ sous l'action de $\mathrm{Aut}(\mathbb{P}^{2}_{\mathbb{C}})$ | | ::: | ::: | +--------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------------------------+ | ::: {.small} | ::: {.small} | | $\mu(\mathcal{F},s)$ | nombre de [Milnor]{.smallcaps} du feuilletage $\mathcal{F}$ au point singulier $s$ | | ::: | ::: | +--------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------------------------+ | ::: {.small} | ::: {.small} | | $\mathrm{BB}(\mathcal{F},s)$ | invariant de [Baum]{.smallcaps}-[Bott]{.smallcaps} du feuilletage $\mathcal{F}$ au point singulier $s$ | | ::: | ::: | +--------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------------------------+ 12 A. Beltrán. . PhD thesis, Pontificia Universidad Católica del Perú, 2014. Available on <https://tesis.pucp.edu.pe/repositorio/handle/20.500.12404/5658>. A. Beltrán, M. Falla Luza, and D. Marı́n. Flat 3-webs of degree one on the projective plane. , 23(4):779--796, 2014. S. Bedrouni. . PhD thesis, University of Sciences and Technology Houari Boumediene, 2017. Available on <https://arxiv.org/abs/1712.03895>. S. Bedrouni and D. Marı́n. Tissus plats et feuilletages homogènes sur le plan projectif complexe. , 146(3):479--516, 2018. S. Bedrouni and D. Marı́n. Une nouvelle démonstration de la classification des feuilletages convexes de degré deux sur $\mathbb{P}^{2}_{\mathbb{C}}$. , 148(4):613--622, 2020. S. Bedrouni and D. Marı́n. Classification of foliations of degree three on $\mathbb{P}^{2}_{\mathbb{C}}$ with a flat Legendre transform. , 71(4):1757--1790, 2021. S. Bedrouni and D. Marı́n. Critère d'holomorphie de la courbure des tissus plans lisses et applications aux tissus duaux des feuilletages homogènes sur $\mathbb{P}^{2}_{\mathbb{C}}$, , 2022. M. Brunella. , volume 1 of *IMPA Monographs*. Springer, Cham, 2015. F. Cano, D. Cerveau, and J. Déserti. . Echelles. Belin, 2013. C. Camacho and P. Sad. Invariant varieties through singularities of holomorphic vector fields. , 115(3):579--595, 1982. C. Favre and J. V. Pereira. Webs invariant by rational maps on surfaces. , 64(3):403--431, 2015. D. Marı́n, J. V. Pereira, and L. Pirio. On planar webs with infinitesimal automorphisms. Inspired by S. S. Chern, *Nankai Tracts Math.*, 11:351--364, 2006. D. Marı́n and J. V. Pereira. Rigid flat webs on the projective plane. 17(1):163--191, 2013. J. V. Pereira. Vector fields, invariant varieties and linear systems. , 51(5):1385--1405, 2001. J. V. Pereira and L. Pirio. Classification of exceptional CDQL webs on compact complex surfaces. , 12:2169--2282, 2010. J. V. Pereira and L. Pirio. , volume 2 of *IMPA Monographs*. Springer, Cham, 2015. O. Ripoll. . Thèse de Doctorat de l'Université Bordeaux 1, 2005. Disponible sur <http://tel.archives-ouvertes.fr/tel-00011928>.

arxiv_math

--- abstract: | In this paper, we investigate metric Jordan algebras, and follow the lines of the paper (J. Milnor: Curvatures of left invariant metrics on Lie groups. Adv. Math. (1976)). Firstly, we define the Jordan-Levi-Civita connection, then we show that every metric Jordan algebra admits a unique Jordan-Levi-Civita connection. Secondly, using the Jordan-Levi-Civita connection, we introduce three natural curvature tensors on metric Jordan algebras, and obtain the corresponding curvature formulas. Thirdly, based on these curvature formulas, we prove that every formally real Jordan algebra admits both a metric of non-positive Jordan curvature, and a Jordan-Einstein metric of negative Jordan scalar curvature. Besides, for nilpotent Jordan algebras, we prove that they admit no Jordan-Einstein metrics. address: - School of Mathematics, Southeast University, Nanjing 210096, P.R. China - School of Mathematics and Statistics, Ningbo University, Ningbo, Zhejiang Province, 315211, People's Republic of China - School of Mathematics and Statistics, Guangdong University of Technology, Guangzhou 510520, P.R. China author: - Hui Zhang - Zaili Yan - Zhiqi Chen$^{*}$ title: Curvatures of metric Jordan algebras --- [^1] # Introduction Jordan algebra was first introduced by P. Jordan in an attempt to generalize the formalism of quantum mechanics ([@J1933; @JVP1934]). Specifically, an algebra $\mathscr{A}$ is said to be a *Jordan algebra* if, for all $X,Y$ in $\mathscr{A}$, $$\begin{aligned} X\circ Y &= Y\circ X,\\ X\circ((X\circ X)\circ Y) &= (X\circ X)\circ (X\circ Y).\end{aligned}$$ Jordan algebras have been proved to be very versatile in both mathematics and physics. For instance, the link between Jordan algebras, symmetric spaces and harmonic analysis [@Bertram2000; @Chu2008; @Chu2012; @Chu2021; @Chu2022; @FK1994; @Koecher1999], the connection between Jordan algebras and quantum theories [@Baez2022; @FFMP2014], and the role Jordan algebras in information geometry [@CJS2023]. We also refer to [@Ior2011] for the applications of Jordan algebras in other fields. In this paper, we focus on *metric Jordan algebras*, i.e., Jordan algebras with an inner product. We mention that metric algebras together with their variants are of great interests in geometry and physics, which have been explored by many authors, see [@BG2018; @BGZ2019; @DFMR2009; @Drin1983; @KO2006; @MP1985] and the references therein. The main goal of this paper is to investigate metric Jordan algebras, and follow the lines of the paper [@M76], where metric Lie algebras in homogeneous Riemannian geometry have been studied. Assume that $(G,\langle\cdot,\cdot\rangle)$ is a connected Lie group with a left-invariant Riemannian metric. It is well known that we may identify $(G,\langle\cdot,\cdot\rangle)$ with the metric Lie algebra $({\mathfrak g},\langle\cdot,\cdot\rangle)$ when studying the curvatures involved problems. In particular, the Ricci operator of $({\mathfrak g},\langle\cdot,\cdot\rangle)$ can be written (see [@besse87book; @Lauret2010]) $$\begin{aligned} \label{Ricciformula} \textnormal{Ric}=\textnormal{M}-\frac{1}{2}B-S(\operatorname{ad}H).\end{aligned}$$ The formula ([\[Ricciformula\]](#Ricciformula){reference-type="ref" reference="Ricciformula"}) together with its generalization is of great importance in finding distinguished metrics, e.g., Einstein metrics and soliton metrics ([@besse87book; @Heber; @Lauret2011]). Besides, it also establishes a closed connection between geometry and algebra. In fact, Lauret found in [@Lauret03; @Lauret2003] that the map [M]{.nodecor} coincides up to a scalar with the *moment map* for the variety of Lie algebras. By further studying the moment map in Lie algebras, he proved that every Einstein solvmanifold is standard ([@Lauret2010]), and later, provided a characterization of solvsolitons ([@Lauret2011]). Moreover, based on Lauret's pioneer work and applications of real geometric invariant theory, B$\ddot{\textnormal{o}}$hm and Lafuente solved the longstanding Alekseevskii conjecture ([@BL2023]). As mentioned by Lauret, the moment map can be naturally extended in other classes of algebras, such as associative algebras, Jordan algebras and etc ([@Lauret2020]), which have recently been considered in some works and we refer to [@GKM; @ZY; @ZCL] for more details in this direction. Naturally, we may ask **Question 1**. *Is there an analogous 'Ricci curvature' on other classes of metric algebras?* On the other hand, we note that many authors have been working towards resolving the challenging problem: Generalizing of Lie's 'third theorem' to the setting of algebras beyond Lie algebras, see [@BGST2023; @BF2017; @GLY; @KW2001] and the references therein. From this point of view, the above question comes naturally again. In this paper, we study Question [Question 1](#Ques){reference-type="ref" reference="Ques"} together with its generalization in the frame of metric Jordan algebras. See TABLE [1](#LieJor){reference-type="ref" reference="LieJor"} below for a comparison of the well known results in metric Lie algebras with the ones obtained in this paper for metric Jordan algebras. [\[LieJor\]]{#LieJor label="LieJor"} **Metric Lie algebras** **V.S.** **Metric Jordan algebras** -------------------------------------------------------------------------------------------------------------------------------------------------- ---------- ------------------------------------------------------------------------------------------------------------------------------------------------- Every metric Lie algebra admits a unique Levi-Civita connection Every metric Jordan algebra admits a unique Jordan-Levi-Civita connection Riemann curvature tensor, Ricci curvature tensor, scalar curvature Jordan curvature tensor, Jordan Ricci curvature tensor, Jordan scalar curvature For an ad-invariant inner product on ${\mathfrak g}$, the Levi-Civita connection is: $\nabla_XY=\frac{1}{2}[X,Y]$, $\forall X,Y\in{\mathfrak g}$ For an associative inner product on $\mathscr{A}$, the Jordan-Levi-Civita connection is: $\nabla_XY=\frac{1}{2}XY$, $\forall X,Y\in\mathscr{A}$ Every compact Lie algebra admits a metric of non-negative Riemann curvature Every formally real Jordan algebra admits a metric of non-positive Jordan curvature Every compact simple Lie algebra admits a Einstein metric of positive scalar curvature, which is given by the Killing form Every simple formally real Jordan algebra admits a Jordan-Einstein metric of negative Jordan scalar curvature A nonzero nilpotent Lie algebra admits no Einstein metrics A nonzero nilpotent Jordan algebra admits no Jordan-Einstein metrics : Lie structures v.s. Jordan structures In detail, the paper is organised as follows: In Sect. [2](#Basic){reference-type="ref" reference="Basic"}, we recall some basic concepts and results of Ricci curvature of metric Lie algebras, the moment map in algebras, and the Jordan algebras, respectively. In Sect. [3](#Connection){reference-type="ref" reference="Connection"}, we first introduce the concept of Jordan-Levi-Civita connection (Def. [Definition 10](#JLCC){reference-type="ref" reference="JLCC"}), then we show that every metric Jordan algebra admits a unique Jordan-Levi-Civita connection (Thm. [Theorem 12](#unique){reference-type="ref" reference="unique"}). Using the Jordan-Levi-Civita connection, we introduce three natural concepts on metric Jordan algebras, i.e., Jordan curvature tensor (Def. [Definition 13](#JCT){reference-type="ref" reference="JCT"}), Jordan Ricci curvature tensor (Def. [Definition 18](#JRT){reference-type="ref" reference="JRT"}), and Jordan scalar curvature (Def. [Definition 25](#JSC){reference-type="ref" reference="JSC"}). We obtain the specific formulas for Jordan Ricci curvature tensor (Thm. [Theorem 22](#RF){reference-type="ref" reference="RF"}) and Jordan scalar curvature ([\[scf\]](#scf){reference-type="ref" reference="scf"}), and the compatibility with the setting of metric Lie algebras provides a solution Question [Question 1](#Ques){reference-type="ref" reference="Ques"}. In Sect. [4](#Curvature){reference-type="ref" reference="Curvature"}, we explore Jordan curvature tensor of *formally real Jordan algebras* (also called *Euclidean Jordan algebras* in the literature), which can be regarded as the counterparts of compact real forms in complex semisimple Lie algebras (see Appendix [7](#Appendix){reference-type="ref" reference="Appendix"}). We first formulate an inequality on formally real Jordan algebras (Lemma [Lemma 27](#inequality){reference-type="ref" reference="inequality"}), which is of independent interest. Then we show that every formally real Jordan algebra admits a metric of non-positive Jordan curvature (Thm. [Theorem 29](#nonpositive){reference-type="ref" reference="nonpositive"}). In Sect. [5](#Ricci){reference-type="ref" reference="Ricci"}, we investigate Jordan-Einstein metrics on Jordan algebras. We show that simple formally real Jordan algebras of dimension at least two admit a Jordan-Einstein metric of negative Jordan scalar curvature (Thm. [Theorem 30](#JEM){reference-type="ref" reference="JEM"} and Cor. [Corollary 35](#Corofss){reference-type="ref" reference="Corofss"}). For nonzero nilpotent Jordan algebras, we prove that they admit no Jordan-Einstein metrics (Thm. [Theorem 37](#nilpotent){reference-type="ref" reference="nilpotent"}). Besides, we construct a non-nilpotent Jordan algebra of the semidirect form which admits a flat Jordan-Einstein metric (Ex. [Example 38](#semidirect){reference-type="ref" reference="semidirect"}), and show that there is a two-dimensional complex semisimple Jordan algebra, containing precisely two real forms $\psi_1$, $\psi_2$ which satisfy that $\psi_1$ admits a flat Jordan-Einstein metric, and $\psi_2$ admits a Jordan-Einstein metric of positive Jordan scalar curvature (Ex. [Example 36](#Ric0+){reference-type="ref" reference="Ric0+"}). In Sect. [6](#Summary){reference-type="ref" reference="Summary"}, we make a few comments on metric Jordan algebras, and collect some natural questions for further study. Besides, for the use of this paper we also summarize some useful results of formally real Jordan algebras in Appendix [7](#Appendix){reference-type="ref" reference="Appendix"}. # Preliminaries {#Basic} In this section, we recall some basic concepts and results of Ricci curvature of metric Lie algebras, the moment map in algebras, and the Jordan algebras, respectively. The ambient field is always assumed to be the real number field $\mathbb{R}$ unless otherwise stated. ## Curvatures of metric Lie algebras Let $G$ be a connected Lie group with the Lie algebra ${\mathfrak g}$ consisting of left-invariant vector fields, and $\langle\cdot,\cdot\rangle$ be a left-invariant Riemannian metric on $G$. It is well-known that we may identify $(G,\langle\cdot,\cdot\rangle)$ with the metric Lie algebra $({\mathfrak g},\langle\cdot,\cdot\rangle)$ (see [@besse87book; @Heber]). Let $\nabla$ be the Levi-Civita connection associated with $({\mathfrak g},\langle\cdot,\cdot\rangle)$ and $X, Y, Z, U, V \in {\mathfrak g}$. Then the torsion-free property and the metric-preserving property of $\nabla$ are respectively given by $$\begin{aligned} &=\nabla_XY-\nabla_YX, \label{tor-f}\\ \langle \nabla_XY, Z&\rangle+\langle Y,\nabla_XZ\rangle=0.\label{metric-skew}\end{aligned}$$ By Koszul's formula, one knows that the Levi-Civita connection $\nabla$ of $({\mathfrak g},\langle\cdot,\cdot\rangle)$ is uniquely determined by the equations ([\[tor-f\]](#tor-f){reference-type="ref" reference="tor-f"}) and ([\[metric-skew\]](#metric-skew){reference-type="ref" reference="metric-skew"}), that is, $$\begin{aligned} \langle \nabla_XY,&Z\rangle=\frac{1}{2}(\langle [X,Y],Z\rangle)-\langle [Y,Z],X\rangle)+\langle [Z,X],Y\rangle).\end{aligned}$$ Associated with the Levi-Civita connection $\nabla$, the Riemann curvature tensor of $({\mathfrak g},\langle\cdot,\cdot\rangle)$ is defined by $$\begin{aligned} R(X,Y)Z=-\nabla_X\nabla_YZ+\nabla_Y\nabla_XZ+\nabla_{[X,Y]}Z,\end{aligned}$$ or equivalently, $R(X,Y)=\nabla_{[X,Y]}-[\nabla_X,\nabla_Y]$. We remark that a frequently encountered definition of Riemann curvature tensor in the literature differs from the above by a sign! The $(0,2)$-type Ricci tensor [Ric]{.nodecor} of $({\mathfrak g},\langle\cdot,\cdot\rangle)$ is a trace or contraction of $R$, i.e., $$\begin{aligned} \textnormal{Ric}=\operatorname{Tr} {(X\rightarrow R(U,X)V)}.\end{aligned}$$ By the fundamental symmetry properties of $R$, one knows that [Ric]{.nodecor} is a symmetric tensor. Furthermore, by taking the trace of [Ric]{.nodecor}, we obtain the scalar curvature [sc]{.nodecor} of $({\mathfrak g},\langle\cdot,\cdot\rangle)$, i.e., $$\begin{aligned} \textnormal{sc}=\operatorname{Tr}_{\langle\cdot,\cdot\rangle}\textnormal{Ric}.\end{aligned}$$ In the frame of metric Lie algebra $({\mathfrak g},\langle\cdot,\cdot\rangle)$, there is a *mean curvature vector* $H\in{\mathfrak g}$, which is defined by $$\begin{aligned} \langle H, X\rangle=\operatorname{Tr}\operatorname{ad}X, ~\forall X\in{\mathfrak g}.\end{aligned}$$ Note that the Lie algebra ${\mathfrak g}$ is *unimodular* if and only if $H=0$. **Lemma 2**. *Let $\{E_i\}$ be an arbitrary orthonormal basis of $({\mathfrak g},\langle\cdot,\cdot\rangle).$ Then $$\begin{aligned} \textnormal{Ric}(X,Y)=-&\frac{1}{2}\sum_{i,j}\langle[X,E_i],E_j\rangle\langle[Y,E_i],E_j\rangle +\frac{1}{4}\sum_{i,j}\langle[E_i,E_j],X\rangle\langle[E_i,E_j],Y\rangle\\ &-\frac{1}{2}B(X,Y)-\frac{1}{2}(\langle [H,X], Y\rangle+\langle X, [H,Y]\rangle), ~~\forall X,Y\in{\mathfrak g},\end{aligned}$$ where $B$ is the Killing form of ${\mathfrak g},$ and $H$ is the mean curvature vector of $({\mathfrak g},\langle\cdot,\cdot\rangle)$.* The Ricci operator Ric (by abuse of notation) of $({\mathfrak g},\langle\cdot,\cdot\rangle)$ is defined by $\langle\textnormal{Ric}(\cdot),(\cdot)\rangle=\textnormal{Ric}(\cdot,\cdot).$ It follows that the Ricci operator is given by (see [@besse87book; @Lauret2010]) $$\begin{aligned} \label{Riccif} \textnormal{Ric}=\textnormal{M}-\frac{1}{2}B-S(\operatorname{ad}H),\end{aligned}$$ where $B$ denotes the symmetric map defined by the Killing form relative to $\langle\cdot,\cdot\rangle,$ $S(\operatorname{ad}H)=\frac{1}{2}(\operatorname{ad}H+(\operatorname{ad}H)^t)$ and [M]{.nodecor} is the symmetric operator defined by $$\begin{aligned} \label{Mformula} \langle\textnormal{M}X,Y\rangle=-\frac{1}{2}\sum_{i,j}\langle[X,E_i],E_j\rangle\langle[Y,E_i],E_j\rangle +\frac{1}{4}\sum_{i,j}\langle[E_i,E_j],X\rangle\langle[E_i,E_j] ,Y\rangle,~~\forall X,Y\in{\mathfrak g}.\end{aligned}$$ We note that for a nilpotent metric Lie algebra $(\mathfrak{n},\langle\cdot,\cdot\rangle)$, the Ricci operator of $(\mathfrak{n},\langle\cdot,\cdot\rangle)$ coincides with $\textnormal{M}.$ ## The moment map in algebras In this subsection, we briefly recall the moment map for the variety of algebras, and we refer to [@Lauret2011; @ZY] for more details. Let $\mathbb{R}^n$ be the usual $n$-dimensional real vector space, $V_n=\otimes^2(\mathbb{R}^n)^* \otimes \mathbb{R}^n$ be the space of all bilinear maps, and $$\mathfrak{M}_n=\{\langle\cdot, \cdot\rangle:\langle\cdot, \cdot\rangle \text { is an inner product on } \mathbb{R}^n\}$$ be the moduli space of all inner products on $\mathbb{R}^n$, respectively. Consider the natural action of $\textnormal{GL}(n)=\textnormal{GL}(\mathbb{R}^n)$ on $V_n$ which corresponds to 'change of basis' $$\begin{aligned} \label{G-action} g.\mu(X, Y)=g \mu(g^{-1} X, g^{-1} Y),\end{aligned}$$ where $g \in \textnormal{GL}(n), \mu \in V_n, X, Y \in \mathbb{R}^n$. Then one immediately sees that the orbit $\textnormal{GL}(n).\mu$ is precisely the isomorphism class of $\mu$. Differentiating ([\[G-action\]](#G-action){reference-type="ref" reference="G-action"}), we obtain the natural action $\mathfrak{gl}(n)$ on $V_n$, i.e., $$\begin{aligned} \label{g-action} (\pi(A) \mu)(X, Y)=A \mu(X, Y)-\mu(\pi(A) X, Y)-\mu(X, \pi(A) Y), ~~ A \in \mathfrak{gl}(n) .\end{aligned}$$ Then it follows that $\pi(A) \mu=0$ if and only if $A \in \operatorname{Der}(\mu)$, the derivation algebra of $\mu$. On the other hand, one knows that the linear group $\textnormal{GL}(n)$ also naturally acts on $\mathfrak{M}_n$, that is $$\begin{aligned} g.\langle\cdot, \cdot\rangle=\langle g^{-1}(\cdot), g^{-1}(\cdot)\rangle,~~ g \in \textnormal{GL}(n),\end{aligned}$$ and this action is obviously transitive. By the notation above, it is not hard to verify that the map $$\begin{aligned} g^{-1}: (g.\mu, \langle\cdot, \cdot\rangle)\rightarrow (\mu, g^{-1}.\langle\cdot, \cdot\rangle),\end{aligned}$$ is an *isometry* between metric algebras, that is, preserving the brackets and the inner products simultaneously. This is precisely the idea: *varying brackets instead of metrics* for the study of metric algebras, which was first introduced by Lauret in [@lauret2001], and since, has motivated much of the recent study of homogeneous Riemannian geometry ([@BL2023; @lauret2001; @Lauret2010; @Lauret2011]). In the sequel, we fix an inner product $\langle\cdot, \cdot\rangle$ on $\mathbb{R}^n$ as a background metric. Then it makes each $\mu \in V_n$ a metric algebra. The fixed inner product $\langle\cdot, \cdot\rangle$ on $\mathbb{R}^n$ also naturally induces a $\textnormal{O}(n)$-invariant inner product on $V_n$ as follows $$\begin{aligned} \label{metric} \langle\mu,\lambda\rangle=\sum_{i,j,k}\langle\mu(X_i,X_j),X_{k}\rangle\langle\lambda(X_i,X_j),X_{k}\rangle,\quad~\mu,\lambda\in V_n,\end{aligned}$$ where $\left\{E_1, E_2, \cdots, E_n\right\}$ is an arbitrary orthonormal basis of $(\mathbb{R}^n,\langle\cdot, \cdot\rangle)$. Moreover, there is an $\textnormal{Ad}(\textnormal{O}(n))$-invariant inner product on $\mathfrak{gl}(n)$, i.e., $$\begin{aligned} \label{gl-metric} (A, B)=\operatorname{Tr} A B^t, ~~A, B \in \mathfrak{gl}(n)\end{aligned}$$ where $\cdot^t$ denotes the transpose relative to $(\mathbb{R}^n,\langle\cdot, \cdot\rangle)$. Decompose $\mathfrak{gl}(n)=\mathfrak{so}(n)+\operatorname{sym}(n)$ into the direct sum of skew matrices and symmetric matrices, respectively, where $$\begin{aligned} \mathfrak{so}(n)=\{A \in \mathfrak{g l}(n): A^t=-A\}, \quad \operatorname{sym}(n)=\{A \in \mathfrak{gl}(n): A^t=A\}.\end{aligned}$$ Then the function $m: V_n \backslash\{0\} \rightarrow \operatorname{sym}(n)$ defined by $$\begin{aligned} \label{MomentDef} (m(\mu), A)=\frac{\langle\pi(A) \mu, \mu\rangle}{\|\mu\|^2}, ~~~~ 0 \neq \mu \in V_n, A \in \operatorname{sym}(n),\end{aligned}$$ is called the *moment map* for the representation $V_n$ of $\mathfrak{gl}(n)$. We note that in the complex case, the function $m$ is precisely the moment map from symplectic geometry, corresponding to the Hamiltonian action of $\textnormal{U}(n)$ on the symplectic manifold $\mathbb{P} V_n$ (see [@MF94]). For each $\mu \in V_n$, we associate it with a map $\textnormal{M}_\mu \in \operatorname{sym}(n)$ as follows $$\begin{aligned} \label{M} \textnormal{M}_\mu=\sum_i L_{E_i}^\mu(L_{E_i}^\mu)^t-\sum_i(L_{E_i}^\mu)^t L_{E_i}^\mu-\sum_i(R_{E_i}^\mu)^t R_{X_i}^\mu,\end{aligned}$$ where $\left\{E_1, E_2, \cdots, E_n\right\}$ is an arbitrary orthonormal basis of $(\mathbb{R}^n,\langle\cdot, \cdot\rangle)$, $L_X^\mu$ and $R_X^\mu: \mathbb{R}^n \rightarrow \mathbb{R}^n$ are, respectively, given by $L_X^\mu(Y)=\mu(X, Y)$ and $R_X^\mu(Y)=\mu(Y, X), \forall Y \in \mathbb{R}^n$. The following result characterizes the relation between $m(\mu)$ and $\textnormal{M}_\mu.$ **Lemma 3** ([@Lauret2011; @ZY]). *Let the notations be as above. Then $m(\mu)=\frac{\textnormal{M}_\mu}{\|\mu\|^2}$ for any $0 \neq \mu \in V_n$, and moreover, $$\begin{aligned} \langle\textnormal{M}_\mu X, Y\rangle= & \sum_{i, j}\langle\mu(E_i, E_j), X\rangle\langle\mu(E_i, E_j), Y\rangle-\sum_{i, j}\langle\mu(E_i, X), E_j\rangle\langle\mu(E_i, Y), E_j\rangle \notag\\ & -\sum_{i, j}\langle\mu(X, E_i), E_j\rangle\langle\mu(Y, E_i), E_j\rangle,~~ \forall X, Y \in \mathbb{R}^n, \label{Mf}\end{aligned}$$ where $\left\{E_1, E_2, \cdots, E_n\right\}$ is an arbitrary orthonormal basis of $(\mathbb{R}^n,\langle\cdot, \cdot\rangle)$.* **Remark 4**. By Lemma [Lemma 3](#M-formu){reference-type="ref" reference="M-formu"}, we know that if $(\mu,\langle\cdot,\cdot\rangle)$ is a metric Lie algebra, then the map $\textnormal{M}_\mu$ in ([\[Mf\]](#Mf){reference-type="ref" reference="Mf"}) coincides up to a scalar with the map $\textnormal{M}$ in ([\[Riccif\]](#Riccif){reference-type="ref" reference="Riccif"}) for $(\mu,\langle\cdot, \cdot\rangle)$. As we shall see later, this also holds for metric Jordan algebras. ## Jordan algebras Jordan algebras were introduced in an attempt to generalize the formalism of quantum mechanics. Specifically, an algebra $\mathscr{A}$ is said to be a Jordan algebra if, for all $X, Y$ in $\mathscr{A}$ : $$\begin{aligned} XY&=YX, \label{commu}\\ X(X^2 Y)&=X^2(XY). \label{weakass}\end{aligned}$$ Using the notation $L_X(Y)=XY$ for all $X, Y \in \mathscr{A}$, and $[S, T]=S T-T S$ for any two endomorphisms of $\mathscr{A}$, the property ([\[weakass\]](#weakass){reference-type="ref" reference="weakass"}) can be written $[L_X, L_{X^2}]=0$ for all $X \in \mathscr{A}$. **Remark 5**. If $(V, \diamond)$ is an associative algebra, then one can naturally define on $V$ a Lie algebra structure and a Jordan algebra structure, respectively, as follows $$\begin{aligned} :=X\diamond Y-Y\diamond X, ~~\forall X, Y \in V,\quad XY:=\frac{1}{2}(X\diamond Y+Y\diamond X), ~~\forall X, Y \in V.\end{aligned}$$ In general a Jordan algebra is not associative. A Jordan algebra $\mathscr{A}$ is called *simple* if $\mathscr{A}^2 \neq 0$ and $\mathscr{A}$ has no nontrivial ideals, and it is called *semisimple* if it is a direct product of simple Jordan algebras. **Definition 6**. Let $\mathscr{A}$ be a Jordan algebra. The Jordan algebra $\mathscr{A}$ is called *solvable* if $\mathscr{A}^{(m)}=0$ for some $m \in \mathbb{N}$, where $\mathscr{A}^{(0)}=\mathscr{A}, \mathscr{A}^{(k+1)}=\mathscr{A}^{(k)} \mathscr{A}^{(k)}, k \geq 0$. The Jordan algebra $\mathscr{A}$ is called *nilpotent* if $\mathscr{A}^m=0$ for some $m \in \mathbb{N}$, where $\mathscr{A}^0=\mathscr{A}, \mathscr{A}^{k+1}=\mathscr{A} \mathscr{A}^k, k \geq 0$. Unlike in the setting of Lie algebras, the concepts of solvability and nilpotency in Jordan algebras turn out to be equivalent (see [@Albert1946]). **Lemma 7** ([@FK1994]). *Let $\mathscr{A}$ be a Jordan algebra, then the following symmetric bilinear form $$\begin{aligned} \label{tauf} \tau(X, Y):=\operatorname{Tr} L_{XY}, \forall X, Y \in \mathscr{A},\end{aligned}$$ is associative, that is, $\tau(X Y, Z)=\tau(X, Y Z)$ for all $X, Y, Z \in \mathscr{A}$.* For a given Jordan algebra, the maximal nilpotent ideal is necessarily unique, which is called the *radical* and denoted by $\mathscr{N}$. It is proved by A. Albert that the radical $\mathscr{N}$ of a Jordan algebra actually coincides with the kernel of the symmetric bilinear form $\tau$ in ([\[tauf\]](#tauf){reference-type="ref" reference="tauf"}). **Theorem 8** (Wedderburn Principal Theorem). *Any real Jordan algebra $\mathscr{A}$ can be written as a vector space direct sum $\mathscr{A}=\mathscr{S}+\mathscr{N}$, where $\mathscr{N}$ is the radical of $\mathscr{A}$, and $\mathscr{S}$ is a maximal semisimple subalgebra of $\mathscr{A}$ isomorphic to $\mathscr{A}/\mathscr{N}$.* It follows Theorem [Theorem 8](#WPT){reference-type="ref" reference="WPT"} that $\mathscr{A}$ is semisimple if and only if the bilinear form $\tau$ is non-degenerate. From this and Lemma [Lemma 7](#tau){reference-type="ref" reference="tau"}, one can easily prove that every semisimple Jordan algebra necessarily carries an identity element. Now, let $\mathscr{A}$ be a Jordan algebra with an identity element $E$, and $\mathbb{R}[Y]$ denote the polynomials in the variable $Y \in \mathscr{A}$ with coefficients in $\mathbb{R}$. Since $\mathbb{R}[Y]$ is the subalgebra of $\mathscr{A}$ generated by $Y$ and the identity element, the number $\operatorname{dim} \mathbb{R}[Y]$ is clearly bounded. We define the *rank* of $\mathscr{A}$ as follows $$\begin{aligned} \label{rank} r=\max~\{\operatorname{dim} \mathbb{R}[Y]: Y \in \mathscr{A}\} .\end{aligned}$$ An element of $X \in \mathscr{A}$ is called *regular*, if $\operatorname{dim} \mathbb{R}[X]$ is maximal among $\operatorname{dim} \mathbb{R}[Y]$ for all $Y \in \mathscr{A}$. The *reduced trace* of a regular element $X \in \mathscr{A}$ is defined by $$\begin{aligned} \label{tr} \operatorname{tr}(X)=\operatorname{Tr} {L_X|_{\mathbb{R}[X]}}.\end{aligned}$$ Moreover, since the set of regular elements is dense in $\mathscr{A},$ the function ([\[tr\]](#tr){reference-type="ref" reference="tr"}) can be uniquely extended to a linear map $\operatorname{tr}:\mathscr{A} \rightarrow \mathbb{R},$ and in particular, we have $\operatorname{tr}(E)=r$ (see [@FK1994]). **Lemma 9** ([@FK1994]). *Let $\mathscr{A}$ be a Jordan algebra with an identity element. Then the symmetric bilinear form $\operatorname{tr}(X Y)$ is associative, that is, $\operatorname{tr}((XY)Z)=\operatorname{tr}(X(Y Z))$ for all $X, Y, Z \in \mathscr{A}$.* For a Jordan algebra $\mathscr{A}$, the Killing form on $\mathscr{A}$ is defined as follows $$\begin{aligned} \label{Killing} B(X, Y):=\operatorname{Tr} L_X L_Y,\end{aligned}$$ for all $X, Y \in \mathscr{A}$. We note that the Killing form of a Jordan algebra is symmetric, but in general not associative. # Connections on metric Jordan algebras {#Connection} In this section, we first introduce the concept of Jordan-Levi-Civita connection in an analogy of Levi-Civita connection on metric Lie algebras, then we show that every metric Jordan algebra admits a unique Jordan-Levi-Civita connection. By the Jordan-Levi-Civita connection, we introduce three natural concepts: Jordan curvature tensor, Jordan Ricci curvature tensor, and Jordan scalar curvature. Moreover, we obtain specific formulas for Jordan Ricci curvature tensor and Jordan scalar curvature. ## Jordan-Levi-Civita Connection **Definition 10** (Jordan-Levi-Civita Connection). Let $(\mathscr{A},\langle\cdot, \cdot\rangle)$ a metric Jordan algebra. Define a bilinear map as follows $$\begin{aligned} \nabla: \mathscr{A} \times \mathscr{A} \rightarrow \mathscr{A},\end{aligned}$$ and write $\nabla_X Y:=\nabla(X, Y)$ for any $X, Y \in \mathscr{A}$. If $\nabla$ satisfies the following conditions $$\begin{aligned} \nabla_XY&+\nabla_YX=XY, \label{c}\\ \langle\nabla_X Y, Z&\rangle-\langle Y, \nabla_X Z\rangle=0,\label{metric-sym}\end{aligned}$$ for all $X, Y, Z \in \mathscr{A}$, then we call $\nabla$ a *Jordan-Levi-Civita connection* on $(\mathscr{A},\langle\cdot, \cdot\rangle)$. **Remark 11**. For a metric Lie algebra $(\mathfrak{g},\langle\cdot, \cdot\rangle)$, the property ([\[metric-skew\]](#metric-skew){reference-type="ref" reference="metric-skew"}) is equivalent to $\nabla_X$ being skew-symmetric for all $X \in \mathfrak{g}$, while for a metric Jordan algebra $(\mathscr{A},\langle\cdot, \cdot\rangle)$, the equation ([\[metric-sym\]](#metric-sym){reference-type="ref" reference="metric-sym"}) is equivalent to $\nabla_X$ being symmetric for all $X \in \mathscr{A}$. **Theorem 12**. *Every metric Jordan algebra $(\mathscr{A},\langle\cdot, \cdot\rangle)$ admits a unique Jordan-Levi-Civita connection $\nabla$, which is given by $$\begin{aligned} \langle\nabla_X Y, Z\rangle=\frac{1}{2}(\langle X Y, Z\rangle-\langle Y Z, X\rangle+\langle Z X, Y\rangle),\end{aligned}$$ for all $X, Y, Z \in \mathscr{A}$.* *Proof.* Assume that $\nabla$ a Jordan-Levi-Civita connection on $(\mathscr{A},\langle\cdot, \cdot\rangle)$, then $$\begin{aligned} \langle\nabla_X Y, Z\rangle-\langle Y, \nabla_X Z\rangle=0, \label{1} \\ \langle\nabla_Y Z, X\rangle-\langle Z, \nabla_Y X\rangle=0, \label{2} \\ \langle\nabla_Z X, Y\rangle-\langle X, \nabla_Z Y\rangle=0, \label{3} \end{aligned}$$ for any $X, Y, Z \in \mathscr{A}$. Adding ([\[1\]](#1){reference-type="ref" reference="1"}) and ([\[2\]](#2){reference-type="ref" reference="2"}) and subtracting ([\[3\]](#3){reference-type="ref" reference="3"}), we have $$\begin{aligned} \langle\nabla_X Y-\nabla_Y X, Z\rangle-\langle\nabla_Z X+\nabla_X Z, Y\rangle+\langle\nabla_Y Z+\nabla_Z Y, X\rangle=0.\end{aligned}$$ Using the property ([\[c\]](#c){reference-type="ref" reference="c"}), we have $$\begin{aligned} \label{Koz} \langle\nabla_X Y, Z\rangle=\frac{1}{2}(\langle X Y, Z\rangle-\langle Y Z, X\rangle+\langle Z X, Y\rangle).\end{aligned}$$ So the Jordan-Levi-Civita connection $\nabla$ is uniquely determined by ([\[c\]](#c){reference-type="ref" reference="c"}) and ([\[metric-sym\]](#metric-sym){reference-type="ref" reference="metric-sym"}). To prove the existence, we define $\nabla$ by ([\[Koz\]](#Koz){reference-type="ref" reference="Koz"}). It is easy to verify that $\nabla$ is well-defined and that it satisfies the desired conditions ([\[c\]](#c){reference-type="ref" reference="c"}) and ([\[metric-sym\]](#metric-sym){reference-type="ref" reference="metric-sym"}). ◻ ## The Jordan Curvature Tensor **Definition 13**. Let $(\mathscr{A},\langle\cdot, \cdot\rangle)$ a metric Jordan algebra and $\nabla$ be the Jordan-Levi-Civita connection. The Jordan curvature tensor of $(\mathscr{A},\langle\cdot, \cdot\rangle)$ is defined by $$\begin{aligned} R(X, Y) Z=-\nabla_X \nabla_Y Z-\nabla_Y \nabla_X Z+\nabla_{X Y} Z,\end{aligned}$$ for all $X, Y, Z \in \mathscr{A}$, or equivalently, $R(X, Y)=\nabla_{X Y}-(\nabla_X \nabla_Y+\nabla_Y \nabla_X)$. Clearly, $R$ is symmetric in the first two positions. Moreover if $R=0$, then $\nabla$ naturally induces a Jordan algebra representation of $\mathscr{A}$. **Remark 14**. Consider the two-dimensional Jordan algebra $\mathscr{A}: e_1 e_1=e_1, e_1 e_2=e_2$. Endow $\mathscr{A}$ with the metric $\langle\cdot,\cdot\rangle$ so that $\{e_1, e_2\}$ is an orthonormal basis. Then by a straightforward calculation, we have $\nabla_{e_1} e_1=e_1, \nabla_{e_1} e_2=e_2$ and $\nabla_{e_2} e_1=0=\nabla_{e_2} e_2$. It follows that the only nontrivial term $R(e_i, e_j)e_k$ is $R(e_1, e_1) e_2=-e_2$. So $R(e_1, e_1)e_2+R(e_1, e_2)e_1 +R(e_2, e_1)e_1=-e_2$. This shows that the sum $R(X, Y)Z+$ $R(Y, Z)X+R(Z, X)Y$ in general does not vanish. In the following, we write $R(X,Y,Z,W)=\langle R(X,Y)Z,W\rangle$ for any $X, Y, Z, W\in \mathscr{A}$ **Lemma 15**. *Let $(\mathscr{A},\langle\cdot, \cdot\rangle)$ a metric Jordan algebra. Then* 1. *$R(X,Y,Z,W)=R(Y,X,Z,W);$* 2. *$R(X,Y,Z,W)=R(X,Y,W,Z).$* *Proof.* Obviously, (a) holds. For (b), it follows from that $\nabla_{X}$ is symmetric for all $X\in\mathscr{A}.$ ◻ We note that in general $R(X, Y, Z, W) \neq R(Z, W, X, Y)$, see Remark [Remark 14](#2dim){reference-type="ref" reference="2dim"}. **Definition 16**. The Jordan curvature of $(\mathscr{A},\langle\cdot,\cdot\rangle)$ is defined as follows $$\begin{aligned} \textnormal{Jc}(X, Y)=\frac{\langle R(X, Y) X, Y\rangle}{\langle X, X\rangle\langle Y, Y\rangle-\langle X, Y\rangle^2}.\end{aligned}$$ for any linearly independent $X, Y \in \mathscr{A}$. If moreover, $\textnormal{Jc}$ is a constant, $(\mathscr{A},\langle\cdot, \cdot\rangle)$ is called of constant Jordan curvature. **Example 17**. Consider the Jordan algebra $\mathscr{A}: e_1 e_1=e_1, e_2 e_2=e_2, \cdots, e_n e_n=e_n$. It is easy to verify that $\mathscr{A}$ admits a metric of constant zero Jordan curvature. ## The Jordan Ricci Curvature tensor For a metric Jordan algebra $(\mathscr{A},\langle\cdot, \cdot\rangle)$, one may define the Jordan Ricci curvature tensor as follows $$\begin{aligned} \label{ric} \operatorname{ric}(U, V)=\operatorname{Tr}{(X \mapsto R(U, X) V)},\end{aligned}$$ for all $U, V \in \mathscr{A}$. We point out that the tensor $\operatorname{ric}$ is in general not symmetric. It is thus natural to introduce the following (symmetric) Jordan Ricci curvature tensor **Definition 18**. The Jordan Ricci curvature tensor of $(\mathscr{A},\langle\cdot, \cdot\rangle)$ is defined as follows $$\begin{aligned} \label{SJTR} \operatorname{Ric}(U, V)=\frac{1}{2}(\operatorname{ric}(U, V)+\operatorname{ric}(V, U)), \forall U, V \in \mathscr{A}.\end{aligned}$$ If moreover, $\operatorname{Ric}$ is a constant multiple of $\langle\cdot\rangle$, then we call $\langle\cdot,\cdot,\rangle$ a Jordan-Einstein metric on $\mathscr{A}$. As we shall see later, [ric]{.nodecor} and [Ric]{.nodecor} are closely related to each other by the following so called *mean curvature vector*. **Definition 19**. For a metric Jordan algebra $(\mathscr{A},\langle\cdot, \cdot\rangle)$, we define $H \in \mathscr{A}$ by $$\begin{aligned} H=\sum_i E_i^2,\end{aligned}$$ where $\{E_i\}$ is an orthonormal basis of $(\mathscr{A},\langle\cdot, \cdot\rangle)$. The vector $H$ is obviously independent of the choice of the orthonormal basis, called the mean curvature vector of $(\mathscr{A},\langle\cdot, \cdot\rangle)$. For any $U \in \mathscr{A}$, we define two linear maps $\nabla_U$ and $\nabla U$ on $(\mathscr{A},\langle\cdot, \cdot\rangle)$, respectively, as follows $$\begin{aligned} \nabla_U(X):=\nabla_UX,\quad \nabla U(X):=\nabla_XU, \end{aligned}$$ where $X\in\mathscr{A}$. **Lemma 20**. *The Jordan Ricci tensor of $(\mathscr{A},\langle\cdot, \cdot\rangle)$ satisfies $$\begin{aligned} \textnormal{Ric}(U,U)=\operatorname{Tr}{(\nabla U)^2}-\frac{1}{2}\operatorname{Tr}{\nabla U^2},\end{aligned}$$ for any $U\in\mathscr{A}$.* *Proof.* Assume that $\{E_i\}$ is an orthonormal basis of $(\mathscr{A},\langle\cdot, \cdot\rangle)$, then we have $$\begin{aligned} \textnormal{Ric}(U,U)=\operatorname{Tr}{(X \mapsto R(U, X)U)}=\sum_i \langle R(U, E_i)U, E_i\rangle=\sum_i \langle R(E_i,U)U, E_i\rangle.\end{aligned}$$ It follows that $$\begin{aligned} \operatorname{Ric}(U, U) & =\sum_i\langle-\nabla_{E_i} \nabla_UU-\nabla_U \nabla_{E_i} U+\nabla_{{E_i}U}U, E_i\rangle \\ & =\sum_i\langle-\nabla_U \nabla_{E_i} U+\nabla_{{E_i}U}U, E_i\rangle-\sum_i\langle\nabla_{E_i} \nabla_UU, E_i\rangle \\ & =: \textnormal{I}-\textnormal{II}.\end{aligned}$$ Note that $E_i U=\nabla_{E_i} U+\nabla_U E_i$, then $$\begin{aligned} \mathrm{I} & =\sum_i\langle-\nabla_U \nabla_{E_i} U, E_i\rangle+\sum_i\langle\nabla_{\nabla_U E_i} U, E_i\rangle+\sum_i\langle\nabla_{\nabla_{E_i} U} U, E_i\rangle \\ & =-\operatorname{Tr} \nabla_U \circ \nabla U+\operatorname{Tr} \nabla U \circ \nabla_U+\operatorname{Tr} \nabla U \circ \nabla U \\ & =\operatorname{Tr}{(\nabla U)^2}.\end{aligned}$$ Since $\nabla_U U+\nabla_U U=U^2$, then $$\textnormal{II}=\sum_i(\nabla_{E_i} \nabla_U U, E_i\rangle=\frac{1}{2} \sum_i(\nabla_{E_i} \nabla U^2, E_i\rangle=\frac{1}{2} \operatorname{Tr} \nabla U^2.$$ This completes the proof. ◻ **Remark 21**. It follows from a similar calculation that $$\begin{aligned} \operatorname{ric}(U, V)=\operatorname{Tr} \nabla U\circ\nabla V-\operatorname{Tr} \nabla \nabla_UV,\end{aligned}$$ for any $U, V \in \mathscr{A}$. **Theorem 22**. *Let $\left\{E_i\right\}$ be an orthonormal basis of $(\mathscr{A},\langle\cdot,\cdot\rangle).$ Then $$\begin{aligned} \operatorname{Ric}(X, Y)=-&\frac{1}{2} \sum_{i, j}\langle XE_i, E_j\rangle\langle YE_i, E_j\rangle+\frac{1}{4}\sum_{i, j}\langle E_iE_j, X\rangle\langle E_iE_j, Y\rangle \\ & +\frac{1}{2}B(X, Y)-\frac{1}{4}\langle H, XY\rangle,~~\forall X, Y \in \mathscr{A},\end{aligned}$$ where $B$ is the Killing form of $\mathscr{A}, H$ is the mean curvature vector of $(\mathscr{A},\langle\cdot,\cdot\rangle)$.* *Proof.* Let $\{E_i\}$ be an orthonormal basis of $(\mathscr{A},\langle\cdot, \cdot\rangle)$. Since by Theorem [Theorem 12](#unique){reference-type="ref" reference="unique"} $$\begin{aligned} \label{k} \langle\nabla_{E_i} U, E_j\rangle=\frac{1}{2}(\langle UE_i, E_j\rangle-\langle UE_j, E_i\rangle+\langle E_iE_j, U\rangle),\end{aligned}$$ for all $i, j$, and $U \in \mathscr{A}$, then $$\begin{aligned} \operatorname{Tr}{(\nabla U)^2} & =\sum_j\langle\nabla_{\nabla_{E_j} U}U, E_j\rangle \\ & =\sum_{i, j}\langle\nabla_{E_i}U, E_j\rangle\langle\nabla_{E_j}U, E_i\rangle \\ & =\frac{1}{4} \sum_{i, j}\langle E_iE_j, U\rangle^2-\underbrace{\frac{1}{4} \sum_{i, j}(\langle UE_i, E_j\rangle -\langle UE_j, E_i\rangle)^2}_{\textnormal{I}}.\end{aligned}$$ It follows that $$\begin{aligned} \textnormal{I} & =\frac{1}{4} \sum_{i, j}\langle UE_i, E_j\rangle^2-\frac{1}{2} \sum_{i, j}\langle UE_i, E_j\rangle\langle UE_j, E_i\rangle+\frac{1}{4} \sum_{i, j}\langle UE_j, E_i\rangle^2 \\ & =\frac{1}{2} \sum_{i, j}\langle UE_i, E_j\rangle^2-\frac{1}{2} \sum_i\langle U(UE_i), E_i\rangle \\ & =\frac{1}{2} \sum_{i, j}\langle UE_i, E_j\rangle^2-\frac{1}{2} \operatorname{Tr} L_UL_U .\end{aligned}$$ So $$\begin{aligned} \operatorname{Tr}{(\nabla U)^2}=\frac{1}{4} \sum_{i, j}\langle E_iE_j, U\rangle^2-\frac{1}{2} \sum_{i, j}\langle UE_i, E_j\rangle^2+\frac{1}{2} B(U, U).\end{aligned}$$ Note that $$\begin{aligned} \operatorname{Tr} {\nabla U^2}=\sum_i\langle\nabla_{E_i} U^2, E_i\rangle=\sum_i\langle U^2, \nabla_{E_i}E_i\rangle=\sum_i\langle U^2, \frac{1}{2} E_i^2\rangle=\frac{1}{2}\langle U^2, H\rangle.\end{aligned}$$ Then by Lemma [Lemma 20](#JRTrF){reference-type="ref" reference="JRTrF"}, we have $$\begin{aligned} \operatorname{Ric}(U, U)=\operatorname{Tr}{(\nabla U)^2}-\frac{1}{2} \operatorname{Tr} {\nabla U^2}=-\frac{1}{2} \sum_{i, j}\langle UE_i, E_j\rangle^2+\frac{1}{4}\sum_{i, j}\langle E_iE_j, U\rangle^2+\frac{1}{2}B(U, U)-\frac{1}{4}\langle H, U^2\rangle.\end{aligned}$$ This completes the proof. ◻ **Remark 23**. Similarly, by Remark [Remark 21](#ricTrf){reference-type="ref" reference="ricTrf"}, one obtains the following result $$\begin{aligned} \operatorname{ric}(X, Y)=- & \frac{1}{2} \sum_{i, j}\langle XE_i, E_j\rangle\langle YE_i, E_j\rangle+\frac{1}{4} \sum_{i, j}\langle E_iE_j, X\rangle\langle E_iE_j, Y\rangle \\ & +\frac{1}{2} B(X, Y)-\frac{1}{2}\langle H, \nabla_X Y\rangle, ~~\forall X, Y \in \mathscr{A}.\end{aligned}$$ Moreover, $\textnormal{ric}=\textnormal{Ric}$ if and only if the operator $L_H$ is self-adjoint. In particular, $\textnormal{ric}=\textnormal{Ric}$ when $H=0$ or a constant multiple of the identity element. **Remark 24**. Suppose that $(\mathscr{A},\langle\cdot,\cdot\rangle)$ is of constant Jordan curvature, then $(\mathscr{A},\langle\cdot,\cdot\rangle)$ is necessarily Jordan-Einstein. Indeed, by Definition [Definition 16](#JC){reference-type="ref" reference="JC"} we know that there exists a constant $c$ such that $$\begin{aligned} \langle R(X, Y) X, Y\rangle=c(\langle X, X\rangle\langle Y, Y\rangle-\langle X, Y\rangle^2), \end{aligned}$$ for all $X, Y \in \mathscr{A}$. So $$\begin{aligned} \operatorname{Ric}(U, U)=\sum_i\langle R(U, E_i)U, E_i\rangle=\sum_i c(\langle U, U\rangle\langle E_i, E_i\rangle-\langle U, E_i\rangle^2)=c(\operatorname{dim}\mathscr{A}-1)\langle U, U\rangle, \forall U \in \mathscr{A}.\end{aligned}$$ That is, $\langle\cdot, \cdot\rangle$ is a Jordan-Einstein metric. The converse is not true, see Remark [Remark 14](#2dim){reference-type="ref" reference="2dim"}. ## Jordan scalar curvature By Definition [Definition 18](#JRT){reference-type="ref" reference="JRT"}, we introduce the Jordan scalar curvature for metric Jordan algebras, which is a trace of $\textnormal{Ric}$, i.e., **Definition 25**. For a metric Jordan algebra $(\mathscr{A},\langle\cdot,\cdot\rangle)$, the Jordan scalar curvature is $\textnormal{sc}=\textnormal{Tr}_{\langle\cdot,\cdot\rangle}{\textnormal{Ric}}.$ By Theorem [Theorem 22](#RF){reference-type="ref" reference="RF"}, it is easy to see that the Jordan scalar curvature is given by $$\begin{aligned} \label{scf} \textnormal{sc}=-\frac{1}{4} \sum_{i, j, k}\langle E_iE_j, E_k\rangle^2+\frac{1}{2}\sum_{i}B(E_i,E_i)-\frac{1}{4}\langle H,H\rangle.\end{aligned}$$ where $\{E_i\}$ is an orthonormal basis of $(\mathscr{A},\langle\cdot,\cdot\rangle)$. It follows that the Jordan scalar curvature [sc]{.nodecor} of a nilpotent metric Jordan algebra satisfies $\textnormal{sc}\leq 0$, and the equality holds if and only if $\mathscr{A}$ is trivial, i.e., $XY=0$ for any $X, Y \in \mathscr{A}.$ # The Jordan curvature of metric Jordan algebras {#Curvature} In this section, we study the Jordan curvature of formally real Jordan algebras, which are counterparts of compact real forms in complex semisimple Lie algebras (see Appendix [7](#Appendix){reference-type="ref" reference="Appendix"} for a comparison). Note that by Proposition [Proposition 40](#FRJAC){reference-type="ref" reference="FRJAC"}, a Jordan algebra $\mathscr{A}$ with an identity element is formally real, if and only if $\mathscr{A}$ admits an associative inner product. **Lemma 26**. *Let $(\mathscr{A},\langle\cdot, \cdot\rangle)$ be a metric Jordan algebra. Assume that $\langle\cdot, \cdot\rangle$, is associative, then the Jordan-Levi-Civita connection is given by $$\begin{aligned} \nabla_XY=\frac{1}{2}XY,\end{aligned}$$ for all $X, Y \in \mathscr{A}$, and moreover, $$\begin{aligned} R(X,Y)Z+R(Y, Z)X+R(Z,X)Y=0,\end{aligned}$$ for all $X, Y, Z \in \mathscr{A}$.* *Proof.* Since $\langle\cdot, \cdot\rangle$ is associative, then $$\begin{aligned} \langle X Y, Z\rangle=\langle X, Y Z\rangle, ~~\forall X, Y, Z \in \mathscr{A}.\end{aligned}$$ By Theorem [Theorem 12](#unique){reference-type="ref" reference="unique"}, we have $$\begin{aligned} \label{12xy} \nabla_X Y=\frac{1}{2} XY, ~~\forall X,Y \in \mathscr{A}.\end{aligned}$$ This proves the first statements. Moreover, it follows from ([\[12xy\]](#12xy){reference-type="ref" reference="12xy"}) that $$\begin{aligned} R(X,Y)Z&=\nabla_{XY}Z-\nabla_X\nabla_YZ-\nabla_Y\nabla_XZ \\ &=\frac{1}{2}(XY)Z-\frac{1}{2}\nabla_XYZ-\frac{1}{2} \nabla_YXZ \\ &=\frac{1}{2}(XY)Z-\frac{1}{4} X(YZ)-\frac{1}{4}Y(XZ) \\ &=\frac{1}{2}(XY)Z-\frac{1}{4}(YZ)X-\frac{1}{4}(ZX)Y.\end{aligned}$$ Similarly, $$\begin{aligned} R(Y,Z)X&=\frac{1}{2}(YZ)X-\frac{1}{4}(ZX)Y-\frac{1}{4}(XY)Z,\\ R(Z,X)Y&=\frac{1}{2}(ZX)Y-\frac{1}{4}(XY)Z-\frac{1}{4}(YZ)X.\end{aligned}$$ So $$\begin{aligned} R(X,Y)Z+R(Y, Z)X+R(Z,X)Y=0,\end{aligned}$$ for all $X, Y, Z \in \mathscr{A}$. This completes the proof. ◻ **Lemma 27**. *Let $\mathscr{A}$ be a formally real Jordan algebra, and $\langle\cdot, \cdot\rangle$ be an associative inner product on it. Then we have the following inequality $$\begin{aligned} \label{XYXY} \langle XY, XY\rangle \leq\langle X^2, Y^2\rangle, ~~\forall X, Y \in \mathscr{A}.\end{aligned}$$* *Proof.* By Theorem [Theorem 41](#FRJSS){reference-type="ref" reference="FRJSS"} and Proposition [Proposition 42](#simpleuni){reference-type="ref" reference="simpleuni"}, it suffices to prove the lemma in the case that $\mathscr{A}$ is simple. Now, let $\mathscr{A}$ be a simple formally real Jordan algebra, and $r$ be the rank of $\mathscr{A}$. By the associativity of the inner product and the commutativity of $\mathscr{A}$ $$\begin{aligned} \langle XY, XY\rangle=\langle X(XY), Y\rangle=\langle L_X^2(Y), Y\rangle,\end{aligned}$$ and $$\begin{aligned} \langle X^2, Y^2\rangle=\langle X^2Y, Y\rangle=\langle L_{X^2}(Y), Y\rangle.\end{aligned}$$ Fix $X$, then both sides are quadratic forms in $Y$. Consider the spectral decomposition of $X$: $$\begin{aligned} X=\sum_{i=1}^r \lambda_iH_i,\end{aligned}$$ where $\{H_1, \cdots, H_r\}$ is a Jordan frame and $\lambda_1, \cdots, \lambda_r \in \mathbb{R}$. According to the Peirce decomposition of $\mathscr{A}$ with respect to $\left\{H_1, \cdots, H_r\right\}$ (see Theorem [Theorem 44](#Peirce){reference-type="ref" reference="Peirce"}), we have the following orthogonal direct sum $$\begin{aligned} \mathscr{A}=\bigoplus_{i=1}^r\mathbb{R}H_i\bigoplus_{i<j} \mathscr{A}_{ij}.\end{aligned}$$ Let $$\begin{aligned} Y=\sum_{i=1}^r \mu_iH_i+\sum_{i<j} Y_{ij},\end{aligned}$$ where $\mu_i \in \mathbb{R}$ and $Y_{ij} \in \mathscr{A}_{i j}$. Then $$\begin{aligned} L_X(Y)=\sum_{i=1}^r\lambda_i\mu_i H_i+\sum_{i<j}\frac{\lambda_i+\lambda_j}{2} Y_{ij}.\end{aligned}$$ It follows that $$\begin{aligned} \langle L_X^2(Y), Y\rangle& =\sum_{i=1}^r \lambda_i^2\mu_i^2+\sum_{i<j}(\frac{\lambda_i+\lambda_j}{2})^2\|Y_{ij}\|^2, \\ \langle L_{X^2}(Y), Y\rangle&=\sum_{i=1}^r \lambda_i^2\mu_i^2+\sum_{i<j}\frac{\lambda_i^2+\lambda_j^2}{2}\|Y_{ij}\|^2 .\end{aligned}$$ The inequality follows since $$\begin{aligned} (\frac{\lambda_i+\lambda_j}{2})^2 \leq \frac{\lambda_i^2+\lambda_j^2}{2}.\end{aligned}$$ This completes the proof. ◻ **Remark 28**. By the proof of Lemma [Lemma 27](#inequality){reference-type="ref" reference="inequality"}, we know that if $\langle XY, XY\rangle=\langle X^2, Y^2\rangle$ for all $X, Y \in \mathscr{A}$, then the formally real Jordan algebra $\mathscr{A}$ is necessarily a direct sum of the one-dimensional simple Jordan algebra (i.e., $e_{1}e_1=e_1$, $r=1$). **Theorem 29**. *Let $\mathscr{A}$ be a formally real Jordan algebra, and $\langle\cdot, \cdot\rangle$ be an associative inner product on it. Then the Jordan curvature of $(\mathscr{A},\langle\cdot, \cdot\rangle)$ is non-positive. Moreover, the Jordan curvature vanishes identically if and only $\mathscr{A}$ is a direct sum of the one-dimensional simple Jordan algebra.* *Proof.* By Lemma [\[12xy\]](#12xy){reference-type="ref" reference="12xy"}, we know that $$\begin{aligned} R(X,Y)X=\frac{1}{2}(XY)X-\frac{1}{4}(YX)X-\frac{1}{4}(X^2)Y=\frac{1}{4}(XY)X-\frac{1}{4}(X^2)Y,\end{aligned}$$ for any $X, Y \in \mathscr{A}$. Since the inner product $\langle\cdot, \cdot\rangle$ is associative, we have $$\begin{aligned} \langle R(X, Y)X, Y\rangle=\frac{1}{4}\langle(XY)X-(X^2)Y, Y\rangle=\frac{1}{4}\langle XY, XY\rangle-\frac{1}{4}\langle X^2, Y^2\rangle.\end{aligned}$$ The theorem is completed by Lemma [Lemma 27](#inequality){reference-type="ref" reference="inequality"} and Remark [Remark 28](#XY=){reference-type="ref" reference="XY="}. ◻ # The Jordan Ricci curvature of metric Jordan algebras {#Ricci} In this section, we explore Jordan-Einstein metrics on Jordan algebras. We show that every simple formally real Jordan algebra of dimension at least two admits a Jordan-Einstein metric of negative Jordan scalar curvature. For nonzero nilpotent Jordan algebras, we prove that they admit no Jordan-Einstein metrics. Besides, we construct some examples of Jordan-Einstein metrics. ## Jordan-Einstein metrics on formally real Jordan algebras Let $\mathscr{A}$ be a simple formally real Jordan algebra of rank $r$ and $E$ be the identity element. Then with respect to a Jordan frame $\{H_1, H_2, \cdots, H_r\},$ $\mathscr{A}$ has the following Peirce decomposition (see Theorem [Theorem 44](#Peirce){reference-type="ref" reference="Peirce"}) $$\begin{aligned} \mathscr{A}=\bigoplus_{i\leq j}\mathscr{A}_{ij}=\bigoplus_{i=1}^r\mathbb{R}H_i\bigoplus_{i<j} \mathscr{A}_{ij}.\end{aligned}$$ For $r \geq 3$, it is known that the number $d=\operatorname{dim} \mathscr{A}_{ij}$ does not depend on $i,j$. So $$\begin{aligned} \label{dim} n=\dim\mathscr{A}=\dim\bigoplus_{i\leq j}\mathscr{A}_{ij}=\sum_{i=1}^r \dim \mathscr{A}_{ii}+\sum_{1\leq i<j\leq r}\dim\mathscr{A}_{ij}=r+\frac{r(r-1)}{2}d.\end{aligned}$$ For $r=2$, we set $d=n-2$. It follows from the trace formula in [@FK1994] that $$\begin{aligned} \label{KilTr} B(X,Y)=\operatorname{Tr} L_XL_Y=\alpha\operatorname{tr}(XY)+\frac{d}{4}\operatorname{tr}(X)\cdot\operatorname{tr}(Y), ~~\forall X, Y \in \mathscr{A},\end{aligned}$$ where $\alpha=1+\frac{(r-2)}{4}d$. Now, we state the main result of this subsection. **Theorem 30**. *Every simple formally real Jordan algebra admits a Jordan-Einstein metric.* In order to prove Theorem [Theorem 30](#JEM){reference-type="ref" reference="JEM"}, we need some preparation work: For a simple formally real Jordan algebra, by Lemma [Lemma 9](#trass){reference-type="ref" reference="trass"} and Proposition [Proposition 40](#FRJAC){reference-type="ref" reference="FRJAC"} we know that $\operatorname{tr}(XY)$ a positive definite symmetric bilinear form on $\mathscr{A}$ which is associative. Using ([\[ratio\]](#ratio){reference-type="ref" reference="ratio"}), we have the following lemma. **Lemma 31**. *Let $(\mathscr{A},\langle\cdot,\cdot\rangle_0)$ be a simple formally real algebra of fixed dimension $n$ and rank $r$, where $$\begin{aligned} \langle X, Y\rangle_0:=\operatorname{tr}(XY)=\frac{r}{n}\tau(X, Y)=\frac{r}{n} \operatorname{Tr}L_{XY}, ~~\forall X, Y \in \mathscr{A}.\end{aligned}$$ Then the mean curvature vector of $(\mathscr{A},\langle\cdot, \cdot\rangle_0)$ is $$\begin{aligned} H_0=\frac{n}{r}E=\left(1+\frac{(r-1)}{2}d\right)E.\end{aligned}$$ Moreover, the Jordan Ricci tensor $\textnormal{Ric}_0=\mathrm{ric}_0$ of $(\mathscr{A},\langle\cdot,\cdot\rangle_0)$ is given by $$\begin{aligned} \textnormal{Ric}_0(X, Y)=-\frac{rd}{16}\langle X, Y\rangle_0+\frac{d}{16}\textnormal{tr}(X)\cdot\textnormal{tr}(Y),\end{aligned}$$ for all $X, Y \in \mathscr{A}$.* *Proof.* Let $\{E_i\}$ be an orthonormal basis of $(\mathscr{A},\langle\cdot,\cdot\rangle_0)$, then by Definition [Definition 19](#MCV){reference-type="ref" reference="MCV"}, the mean curvature vector of $(\mathscr{A},\langle\cdot,\cdot\rangle_0)$ is $$\begin{aligned} H_0=\sum_i E_i^2\end{aligned}$$ Since $\langle\cdot,\cdot\rangle_0$ is associative, we have $$\begin{aligned} \langle X, H_0\rangle_0=\sum_i\langle X, E_i^^2\rangle_0=\sum_i\langle XE_i, E_i\rangle_0=\operatorname{Tr} L_X=\frac{n}{r}\langle X, E\rangle_0,~~\forall X \in \mathscr{A}.\end{aligned}$$ In particular, $H_0=\frac{n}{r}E$. By ([\[dim\]](#dim){reference-type="ref" reference="dim"}), $n=r+\frac{r(r-1)}{2}d$, so we have $H_0=\left(1+\frac{(r-1)}{2}d\right)E$. This proves the first statement. For the second statement. By Remark [Remark 23](#ric=Ric){reference-type="ref" reference="ric=Ric"}, we have $\textnormal{Ric}_0= \textnormal{ric}_0$. It follows from Theorem [Theorem 22](#RF){reference-type="ref" reference="RF"} that the Jordan Ricci tensor is given by $$\begin{aligned} \textnormal{Ric}_0(X, Y)=\langle\textnormal{M}_0 X, Y\rangle_0+\frac{1}{2} B(X, Y)-\frac{1}{4}\langle H_0, XY\rangle_0, ~~\forall X, Y \in \mathscr{A},\end{aligned}$$ where $$\begin{aligned} \langle\textnormal{M}_0 X, Y\rangle_0& =-\frac{1}{2} \sum_{i,j}\langle XE_i, E_j\rangle_0\langle YE_i, E_j\rangle_0+\frac{1}{4} \sum_{i, j}\langle E_iE_j, X\rangle_0\langle E_iE_j, Y\rangle_0 \\ &=-\frac{1}{2} \operatorname{Tr} {L_XL_Y}+\frac{1}{4}\operatorname{Tr} {L_XL_Y} \\ &=-\frac{1}{4} B(X,Y).\end{aligned}$$ Since $H_0=\frac{n}{r}E=\left(1+\frac{(r-1)}{2}d\right)E$, then $$\begin{aligned} \langle H_0, X Y\rangle_0=\frac{n}{r}\langle E, XY\rangle_0=\frac{n}{r}\langle X, Y\rangle_0=\left(1+\frac{(r-1)}{2} d\right)\langle X, Y\rangle_0.\end{aligned}$$ So $$\begin{aligned} \textnormal{Ric}_0(X, Y)=\frac{1}{4} B(X, Y)-\frac{1}{4}\left(1+\frac{(r-1)}{2} d\right)\langle X, Y\rangle_0.\end{aligned}$$ On the other hand, by ([\[KilTr\]](#KilTr){reference-type="ref" reference="KilTr"}) we have $$\begin{aligned} B(X, Y)=\operatorname{Tr} {L_XL_Y}=\left(1+\frac{(r-2)}{4}d\right) \operatorname{tr}(XY)+\frac{d}{4} \operatorname{tr}(X)\cdot\operatorname{tr}(Y).\end{aligned}$$ It follows that $$\begin{aligned} \textnormal{Ric}_0(X, Y)=-\frac{rd}{16}\langle X, Y\rangle_0+\frac{d}{16} \operatorname{tr}(X)\cdot\operatorname{tr}(Y).\end{aligned}$$ This completes the proof of the Lemma. ◻ **Remark 32**. It is not hard to see that $\textnormal{Ric}_0 \leq 0$ and $\textnormal{Ric}_0(E, X)=\textnormal{Ric}(X, E)=0$ for any $X \in \mathscr{A}$. In the sequel, we show that it is possible to scale the inner product $\langle\cdot,\cdot\rangle_0$ in the direction $E$ so that it is a Jordan-Einstein metric on $\mathscr{A}$. **Step one:** Consider the following basis of $\left(\mathscr{A},(\cdot, \cdot\rangle_0\right)$ $$\begin{aligned} \label{E-basis} E_1=H_1,E_2=H_2,\cdots, E_r=H_r, \underbrace{E_{11}^{1},\cdots, E_{11}^{d}}_{\mathscr{A}_{11}},\cdots,\underbrace{E_{1r}^{1},\cdots, E_{1r}^{d}}_{\mathscr{A}_{1r}},\cdots, \underbrace{E_{(r-1)r}^{1},\cdots, E_{(r-1)r}^{d}}_{\mathscr{A}_{(r-1)r}}.\end{aligned}$$ where $\{H_1, H_2, \cdots, H_r\}$ is a Jordan frame of $\mathscr{A}$, and $\{E_{ij}^1, \cdots, E_{ij}^d\}$ is an orthonormal basis of $(\mathscr{A}_{ij}, \langle\cdot, \cdot\rangle_0)$ for $1\leq i<j\leq r$. By Theorem [Theorem 44](#Peirce){reference-type="ref" reference="Peirce"}, we know that ([\[E-basis\]](#E-basis){reference-type="ref" reference="E-basis"}) is an orthonormal basis of $(\mathscr{A},\langle\cdot, \cdot\rangle_0)$. With respect to this basis, it is easy to see the Jordan Ricci tensor is represented by $$\begin{aligned} \textnormal{Ric}_0=-\frac{rd}{16}I+\frac{d}{16}\left(\begin{array}{cccc|c} 1&1&\cdots & 1 & \textbf{0} \\ 1&1&\cdots & 1 & \textbf{0} \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 1 & 1 & \cdots & 1 & \textbf{0} \\ \hline \textbf{0}&\textbf{0}&\cdots&\textbf{0}&\textbf{0} \end{array}\right),\end{aligned}$$ where the square matrix $(\textbf{1})$ is of order $r.$ **Step two:** Let $\{F_1, F_2, \cdots, F_r\}$ be an orthonormal basis of $\textnormal{span}\{E_1, E_2, \cdots, E_r\}$, such that $F_1$ is a positive constant multiple of the identity element $E$. It follows that $$\begin{aligned} \label{F-basis} F_1,F_2,\cdots, F_r, \underbrace{E_{11}^{1},\cdots, E_{11}^{d}}_{\mathscr{A}_{11}},\cdots,\underbrace{E_{1r}^{1},\cdots, E_{1r}^{d}}_{\mathscr{A}_{1r}},\cdots, \underbrace{E_{(r-1)r}^{1},\cdots, E_{(r-1)r}^{d}}_{\mathscr{A}_{(r-1)r}},\end{aligned}$$ is an orthonormal basis of $(\mathscr{A},\langle\cdot, \cdot\rangle_0)$. Since $\langle E, E\rangle_0=\textnormal{tr}(E)=r$, we conclude that $F_1=\frac{1}{\sqrt{r}}E$. Moreover, $F_1, F_2, \cdots, F_r$ being orthogonal imlies $$\begin{aligned} \operatorname{tr}(F_2)=\cdots=\operatorname{tr}(F_r)=0.\end{aligned}$$ With respect to the basis ([\[F-basis\]](#F-basis){reference-type="ref" reference="F-basis"}), it is not hard to verify that $$\begin{aligned} \textnormal{M}_0=-\frac{1}{4}B=-\frac{1}{4}\left(1+\frac{(r-2)}{4} d\right) I-\frac{d}{16}\left(\begin{array}{cccc|c} r & 0 & \cdots & 0 & \textbf{0} \\ 0 & 0 & \cdots & 0 & \textbf{0} \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & \cdots & 0 & \textbf{0} \\ \hline \textbf{0} & \textbf{0} & \cdots & \textbf{0} & \textbf{0} \end{array}\right),\end{aligned}$$ $$\begin{aligned} \frac{1}{2} B=\frac{1}{2}\left(1+\frac{(r-2)}{4} d\right) I+\frac{d}{8}\left(\begin{array}{cccc|c} r & 0 & \cdots & 0 & \textbf{0} \\ 0 & 0 & \cdots & 0 & \textbf{0} \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & \cdots & 0 & \textbf{0} \\ \hline \textbf{0} & \textbf{0} & \cdots & \textbf{0} & \textbf{0} \end{array}\right),\end{aligned}$$ $$\begin{aligned} -\frac{1}{4}\langle H_0, XY\rangle_0=-\frac{n}{4r}\langle X, Y\rangle_0=-\frac{1}{4}\left(1+\frac{(r-1)}{2}d\right)\langle X, Y\rangle_0.\end{aligned}$$ In particular, the Jordan Ricci tensor is represented by $$\begin{aligned} \textnormal{Ric}_0=-\frac{r d}{16} I+\frac{d}{16} \left(\begin{array}{cccc|c} r & 0 & \cdots & 0 & \textbf{0} \\ 0 & 0 & \cdots & 0 & \textbf{0}\\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & \cdots & 0 & \textbf{0} \\ \hline \textbf{0} & \textbf{0} & \cdots & \textbf{0} & \textbf{0} \end{array}\right).\end{aligned}$$ **Step three:** With respect to the basis ([\[F-basis\]](#F-basis){reference-type="ref" reference="F-basis"}), we define a one-parameter subgroup $g_t \in \textnormal{GL}(n)$ $$\begin{aligned} g=\left(\begin{array}{cccc} e^{-t} & & & \\ & 1 & & \\ & & \ddots & \\ & & & 1 \end{array}\right), ~~t \in \mathbb{R},\end{aligned}$$ and a family of metrics $$\begin{aligned} \label{t-metric} \langle\cdot,\cdot\rangle_t=\langle g_t^{-1}(\cdot), g_t^{-1}(\cdot)\rangle_0, ~~t \in \mathbb{R}.\end{aligned}$$ Then it is easy to see that $$\begin{aligned} \label{G-basis} G_1=e^{-t}F_1,G_2=F_2,\cdots, G_r=F_r, \underbrace{E_{11}^{1},\cdots, E_{11}^{d}}_{\mathscr{A}_{11}},\cdots,\underbrace{E_{1r}^{1},\cdots, E_{1r}^{d}}_{\mathscr{A}_{1r}},\cdots, \underbrace{E_{(r-1)r}^{1},\cdots, E_{(r-1)r}^{d}}_{\mathscr{A}_{(r-1)r}},\end{aligned}$$ is an orthonormal basis of $(\mathscr{A},\langle\cdot, \cdot\rangle_t)$ for any $t \in \mathbb{R}$. **Lemma 33**. *Let the notations be as above. For the metric Jordan algebra $(\mathscr{A},\langle\cdot, \cdot\rangle_t)$, the mean curvature vector is given by $$\begin{aligned} H_t=\frac{e^{-2 t}+n-1}{r}E=\frac{e^{-2t}+n-1}{\sqrt{r}}F_1,\end{aligned}$$ Moreover, with respect to the orthonormal basis [([\[G-basis\]](#G-basis){reference-type="ref" reference="G-basis"})]{.nodecor}, we have $$\begin{aligned} \textnormal{M}_t=\frac{1}{4}\left(\begin{array}{cccc} -\frac{2n-1}{r}e^{-2t}+\frac{n-1}{r} e^{2t} & & & \\ & -\frac{2}{r}e^{2t}+c & & \\ & & \ddots& \\ & & &-\frac{2}{r}e^{2t}+c \end{array}\right),\end{aligned}$$ where $c=\frac{2}{r}-1-\frac{(r-2)}{4}d$. $$\begin{aligned} \frac{1}{2}B=\left(\begin{array}{cccc} \frac{1}{2}\left(1+\frac{(r-1)}{2} d\right) e^{-2t} & & & \\ & \frac{1}{2}\left(1+\frac{(r-2)}{4}d\right) & & \\ & & \ddots & \\ & & & \frac{1}{2}\left(1+\frac{(r-2)}{4}d\right) \end{array}\right),\end{aligned}$$ and $$\begin{aligned} -\frac{1}{4}(\langle H_t,\cdot\rangle)= \left(\begin{array}{cccc} -\frac{e^{-2t}+n-1}{4r} & & & \\ & -\frac{e^{-2t}+n-1}{4r} e^{2t} & & \\ & & \ddots & \\ & & & -\frac{e^{-2t}+n-1}{4r} e^{2t} \end{array}\right).\end{aligned}$$* *Proof.* Proof. Since ([\[G-basis\]](#G-basis){reference-type="ref" reference="G-basis"}) is an orthonormal basis of $(\mathscr{A},\langle\cdot, \cdot\rangle_t)$, then $$\begin{aligned} H_t& =\sum_{i=1}^r G_i^2+\sum_{i<j}\sum_{k=1}^d E_{ij}^k \\ &=G_1^2+\sum_{j=2}^rG_j^2+\sum_{i<j}\sum_{k=1}^d E_{ij}^k \\ &=e^{-2t}F_1^2+\sum_{j=2}^r F_i^2+\sum_{i<j}\sum_{k=1}^d E_{ij}^k \\ &=e^{-2t}F_1^2+(H_0-F_1^2) \\ &=e^{-2t}\frac{1}{r}E+\left(\frac{n}{r} E-\frac{1}{r}E\right) \\ &=\frac{e^{-2t}+n-1}{r}E \\ &=\frac{e^{-2t}+n-1}{\sqrt{r}}F_1.\end{aligned}$$ It follows that $$\begin{aligned} \langle H_t, XY\rangle_t=\langle\frac{e^{-2t}+n-1}{\sqrt{r}}F_1, XY\rangle_t=e^{2t}\langle\frac{e^{-2 t}+n-1}{\sqrt{r}} F_1, X Y\rangle_0=\frac{e^{-2t}+n-1}{r} e^{2t}\langle E, XY\rangle_0=\frac{e^{-2t}+n-1}{r}e^{2t}\langle X, Y\rangle_0,\end{aligned}$$ for any $X, Y \in \mathscr{A}$. In particular, $$\begin{aligned} \langle H_t, G_1G_1\rangle_t& =\frac{e^{-2t}+n-1}{r}e^{2t}\langle G_1, G_1\rangle_0=\frac{e^{-2 t}+n-1}{r}. \\ \langle H_t, G_jG_j\rangle_t& =\frac{e^{-2t}+n-1}{r}e^{2t}\langle F_j, F_j\rangle_0=\frac{e^{-2t}+n-1}{r}e^{2t}, ~j \geq 2. \\ \langle H_t, G_iG_j\rangle_t& =0, ~i \neq j. \\ \langle H_t, E_{ij}^p E_{kl}^q\rangle_t&=\delta_{pq}\delta_{ik}\delta_{jl}\frac{e^{-2t}+n-1}{r}e^{2t}.\end{aligned}$$ Moreover $$\begin{aligned} B(G_1, G_1)&=B(e^{-t}F_1, e^{-t}F_1)=\left(1+\frac{(r-1)}{2}d\right)e^{-2t}. \\ B(G_j, G_j)&=B(F_j, F_j)=1+\frac{(r-2)}{4}d, ~j \geq 2. \\ B(G_i, G_j)&=B(F_i, F_j)=0, ~i \neq j. \\ B(E_{ij}^p, E_{kl}^q) & =\delta_{pq}\delta_{ik}\delta_{jl}\left(1+\frac{(r-2)}{4}d\right).\end{aligned}$$ For the calculation of $\textnormal{M}_t$, it follows from a similar discussion as in [@GKM]. Note that $\textnormal{M}_t$ differs from the moment map in $\textnormal{M}$ in ([\[Mf\]](#Mf){reference-type="ref" reference="Mf"}) by a constant multiple of $\frac{1}{4}$. This completes the proof of the Lemma. ◻ Now, we are in a position to prove Theorem [Theorem 30](#JEM){reference-type="ref" reference="JEM"}. ***The proof of Theorem [Theorem 30](#JEM){reference-type="ref" reference="JEM"}**.* By Lemma [Lemma 33](#Rict){reference-type="ref" reference="Rict"}, we know that the Jordan Ricci tensor $\operatorname{Ric}_t$ of $(\mathscr{A},\langle\cdot,\rangle_t)$ is given by $$\begin{aligned} \textnormal{Ric}_t(X, Y)=\langle\textnormal{M}_tX, Y\rangle_t+\frac{1}{2}B(X, Y)-\frac{1}{4}\langle H_t, XY\rangle_t, ~~\forall X, Y \in \mathscr{A}.\end{aligned}$$ To prove the theorem, it suffices to show that there exists some $t \in \mathbb{R}$ such that the following two functions $$\begin{aligned} \label{LSH} \frac{1}{4}\left(-\frac{2n-1}{r} e^{-2t}+\frac{n-1}{r} e^{2t}\right)+\frac{1}{2}\left(1+\frac{(r-1)}{2}d\right)e^{-2t}-\frac{e^{-2t}+n-1}{4r},\end{aligned}$$ and $$\begin{aligned} \label{RSH} \frac{1}{4}\left(-\frac{2}{r}e^{2t}+\frac{2}{r}-1-\frac{(r-2)}{4}d\right)+\frac{1}{2}\left(1+\frac{(r-2)}{4}d\right)-\frac{e^{-2 t}+n-1}{4r} e^{2t}.\end{aligned}$$ have the same value. The term of ([\[LSH\]](#LSH){reference-type="ref" reference="LSH"}) equals to $$\begin{aligned} \left(-\frac{n}{2r}+\frac{1}{2}+\frac{(r-1)}{4}d\right)e^{-2 t}+\frac{(n-1)}{4r}e^{2t}-\frac{(n-1)}{4r}=\frac{(n-1)}{4r}e^{2 t}-\frac{(n-1)}{4r},\end{aligned}$$ and the term of ([\[RSH\]](#RSH){reference-type="ref" reference="RSH"}) equals to $$\begin{aligned} -\frac{(n+1)}{4r}e^{2t}+\frac{1}{4r}+\frac{1}{4}+\frac{(r-2)}{16}d.\end{aligned}$$ The solution of $$\begin{aligned} \frac{(n-1)}{4r}e^{2t}-\frac{(n-1)}{4r}=-\frac{(n+1)}{4r} e^{2 t}+\frac{1}{4r}+\frac{1}{4}+\frac{(r-2)}{16}d\end{aligned}$$ is $$\begin{aligned} t &=\frac{1}{2} \ln \left(\frac{1}{2}+\frac{r}{2n}\left(1+\frac{(r-2)}{4} d\right)\right) \\ & =\frac{1}{2} \ln \left(1-\frac{r d}{8+4(r-1) d}\right) \\ & <0,\end{aligned}$$ for $r\geq 2$ by Remark [Remark 45](#classification){reference-type="ref" reference="classification"}. This completes the theorem. ◻ **Remark 34**. Let $\mathscr{A}$ be a simple formally real Jordan algebra of rank $\geq 2$. Then the Jordan-Einstein metric constructed in the proof of Theorem [Theorem 30](#JEM){reference-type="ref" reference="JEM"} has negative scalar curvature. **Corollary 35**. *Formally real Jordan algebras containing no one-dimensional simple ideal admit a Jordan-Einstein metric of negative scalar curvature.* **Example 36**. In the two dimension case, there are precisely two real semisimple Jordan algebras, i.e., $$\begin{aligned} & \psi_1: e_1 e_1=e_1, ~e_2 e_2=e_2. \\ & \psi_2: e_1 e_1=e_1, ~e_2 e_2=-e_1, ~e_1 e_2=e_2. \end{aligned}$$ The semisimple Jordan algebras $\psi_1$ and $\psi_2$ have the same complexification. Moreover, $\psi_1$ is formally real, and $\psi_2$ is simple. Endow $\psi_1$ and $\psi_2$ with the metric $\langle\cdot, \cdot\rangle$ so that $\left\{e_1, e_2\right\}$ is an orthonormal basis. Then by Theorem [Theorem 22](#RF){reference-type="ref" reference="RF"} and a straightforward calculation, we have $$\begin{aligned} \textnormal{Ric}_{\psi_1}=0, \quad \textnormal{Ric}_{\psi_2}=\frac{1}{4}\langle\cdot, \cdot\rangle.\end{aligned}$$ That is, $\left(\psi_1,\langle\cdot, \cdot\rangle\right)$ and $\left(\psi_2,\langle\cdot, \cdot\rangle\right)$ are both Jordan-Einstein. ## Jordan-Einstein metrics on formally real Jordan algebras Let $(\mathscr{N},\langle\cdot, \cdot\rangle)$ be a nilpotent metric Jordan algebra. Since the Killing form of $\mathscr{N}$ necessarily vanishes, then by Theorem [Theorem 22](#RF){reference-type="ref" reference="RF"} we have $$\begin{aligned} \textnormal{Ric}(X, Y)=\langle\textnormal{M}X, Y\rangle-\frac{1}{4}\langle H, XY\rangle, ~~\forall X, Y \in \mathscr{N},\end{aligned}$$ where $H=\sum_{i=1}^n E_i^2$ is the mean curvature vector, $\{E_i\}$ is an orthonormal basis of $(\mathscr{N},\langle\cdot, \cdot\rangle)$ and $$\begin{aligned} \langle\textnormal{M}X, Y\rangle=-\frac{1}{2}\sum_{i,j}\langle XE_i, E_j\rangle\langle YE_i, E_j\rangle+\frac{1}{4} \sum_{i,j}\langle E_iE_j, X\rangle\langle E_iE_j, Y\rangle, ~~\forall X, Y \in \mathscr{N} .\end{aligned}$$ It is clear that we have $$\begin{aligned} \operatorname{Tr} \textnormal{M}=-\frac{1}{4}\sum_{i,j,k}\langle E_iE_j, E_k\rangle^2 \leq 0,\end{aligned}$$ and the equality holds if and only $\mathscr{N}$ is the trivial algebra. Moreover, if $0\ne Z$ lies in the annihilator of $\mathscr{N}$, then $$\begin{aligned} \langle\textnormal{M}Z, Z\rangle=\frac{1}{4}\sum_{i,j}\langle E_iE_j, Z\rangle\langle E_i E_j, Z\rangle \geq 0.\end{aligned}$$ **Theorem 37**. *A nonzero nilpotent Jordan algebra admits no Jordan-Einstein metrics.* *Proof.* Suppose to the contrary that $(\mathscr{N},\langle\cdot, \cdot\rangle)$ is a nonzero nilpotent metric Jordan algebra with $$\begin{aligned} \textnormal{Ric}=c\langle\cdot, \cdot\rangle,\end{aligned}$$ for some constant $c \in \mathbb{R}$. Since the Killing form $B$ of $\mathscr{N}$ vanishes, then $$\begin{aligned} c\cdot\operatorname{dim}\mathscr{N}=\operatorname{Tr}\textnormal{Ric}=\operatorname{Tr} \textnormal{M}+\frac{1}{2} \operatorname{Tr}B -\frac{1}{4}\langle H, H\rangle=\operatorname{Tr} \textnormal{M}-\frac{1}{4}\langle H, H\rangle<0.\end{aligned}$$ Consequently, $c<0$. On the the hand, let $0 \neq Z$ be an element lying in the annihilator of $\mathscr{N}$. Then $$\begin{aligned} c\langle Z, Z\rangle=\textnormal{Ric}(Z, Z)=\langle\textnormal{M} Z, Z\rangle-\frac{1}{4}\langle H, Z^2\rangle=\langle\textnormal{M}Z, Z\rangle \geq 0.\end{aligned}$$ It follows that $c\geq0$, which is contradiction. So a nonzero nilpotent Jordan algebra admits no Jordan-Einstein metrics. This proves the theorem. ◻ **Example 38**. Consider the $n$-dimensional Jordan algebra $\mathscr{A}$ $$\begin{aligned} e_1 e_1=e_1,~ e_1 e_2=e_2, ~\cdots, ~e_1 e_n=e_n.\end{aligned}$$ It is neither nilpotent nor semisimple. Endow $\mathscr{A}$ with a metric $\langle\cdot, \cdot\rangle$ so that $\{e_1, e_2, \cdots, e_n\}$ is an orthonormal basis. Then by a straightforward calculation, we have $$\begin{aligned} \textnormal{Ric}=0.\end{aligned}$$ That is, $\langle\cdot, \cdot\rangle$ is a flat Jordan-Einstein metric on $\mathscr{A}$. # Summary and comments {#Summary} For a metric Jordan algebra, we have introduced the following natural concepts: Jordan-Levi-Civita connection, Jordan curvature tensor, Jordan Ricci curvature tensor, and Jordan scalar curvature. See TABLE [1](#LieJor){reference-type="ref" reference="LieJor"} for summarized results and the comparison with the metric Lie algebra case. We mention that in a metric Lie algebra $({\mathfrak g},\langle\cdot, \cdot\rangle)$, there is an important notion called *algebraic Ricci solitons*, i.e., the Ricci operator satisfies $$\textnormal{Ric}=cI+D$$ for some $c\in \mathbb{R}$ and $D \in \operatorname{Der}(\mathfrak{g})$. The notion arises as limits under Ricci flow can be naturally seen as a generalization of Einstein metric ([@Jablonski2014; @Jablonski2015H; @lauret2001]). It is well-known that nilpotent Lie algebras admit no Einstein metrics, however, many of them admit *a nilsoliton metric* (i.e., algebraic Ricci soliton in the nilpotent case). This is also true for nilpotent Jordan algebras (see Theorem [Theorem 37](#nilpotent){reference-type="ref" reference="nilpotent"} and [@GKM]). For nilpotent Lie algebras, nilsolitons can be characterized by a solvable Einstein extension, and we don't know whether it holds for nilpotent metric Jordan algebras or not. Besides, the question: *classify Jordan-Einstein metrics on semisimple Jordan algebras up to isometry and scaling*, is also very interesting. # Formally real Jordan algebras {#Appendix} In this section, we recall some fundamental results of formally real Jordan algebras, and details of the proof can be found in [@FK1994]. **Lemma 39** ([@FK1994]). *Let $\mathscr{A}$ be an arbitrary Jordan algebra. Then the following identities hold:* 1. *$[L_X, L_{Y^2}]+2[L_Y, L_{XY}]=0$,* 2. *$[L_X, L_{YZ}]+[L_Y, L_{ZX}]+[L_Z, L_{XY}]=0$,* 3. *$L_{X^2Y}-L_{X^2}L_Y=2(L_{XY}-L_XL_Y)L_X$,* *for any $X, Y, Z \in \mathscr{A}$.* A Jordan algebra $\mathscr{A}$ over $\mathbb{R}$ with an identity element $E$ is called a *formally real Jordan algebra* if $X^2+Y^2=0 \Rightarrow X=0, Y=0$. We note that formally real Jordan algebras also called Euclidean Jordan algebras in the literature. The following proposition characterizes formally real Jordan algebras. **Proposition 40** ([@FK1994]). *Let $\mathscr{A}$ be a Jordan algebra over $\mathbb{R}$ with an identity element. The following statements are equivalent* 1. *$\mathscr{A}$ is a formally real Jordan algebra.* 2. *There exists a positive definite symmetric bilinear form on $\mathscr{A}$ which is associative.* 3. *The symmetric bilinear form $\operatorname{tr}(XY)$ is positive definite.* 4. *The symmetric bilinear form $\tau(X, Y)=\operatorname{Tr} L_{XY}$ is positive definite.* The proof of (ii) $\Rightarrow$ (i) in Proposition [Proposition 40](#FRJAC){reference-type="ref" reference="FRJAC"} follows from the fact that if $\langle\cdot,\cdot\rangle$ is an associative inner product on $\mathscr{A}$ and $X^2+Y^2=0$, then $\langle X^2+Y^2, E\rangle=\langle X, X\rangle+\langle Y, Y\rangle=0$, so $X=Y=0$. **Theorem 41** ([@FK1994]). *Every formally real Jordan algebra is semisimple, which decomposes, in a unique way, a direct sum of simple ideals. Besides, every complex semisimple Jordan algebra is the complexification of some formally real Jordan algebra.* Furthermore, **Proposition 42** ([@FK1994]). In a simple formally real Jordan algebra, the associative symmetric bilinear form is unique up to a scalar. In particular, every associative symmetric bilinear form is a scalar multiple of $\operatorname{tr}(XY).$ **Remark 43**. Combining the results above, one may reasonably consider formally real Jordan algebras as the counterparts of compact real forms in complex semisimple Lie algebras. In the sequel, we recall the Jordan frame and the corresponding Peirce decomposition. Assume that $\mathscr{A}$ is a simple formally real Jordan algebra of fixed dimension $n$ and rank $r \geq 2$. By Lemma [Lemma 7](#tau){reference-type="ref" reference="tau"}, Lemma [Lemma 9](#trass){reference-type="ref" reference="trass"} and Proposition [Proposition 42](#simpleuni){reference-type="ref" reference="simpleuni"}, we have $$\begin{aligned} \label{ratio} \operatorname{tr}(XY)=\frac{r}{n} \tau(X, Y)=\frac{r}{n} \operatorname{Tr}L_{XY}, ~~\forall X, Y \in \mathscr{A}.\end{aligned}$$ A non-zero element $C$ of $\mathscr{A}$ is called a *primitive idempotent*, if $C^2=C$ and $C$ cannot be written as the sum of two non-zero idempotents. It is known that there exists a *Jordan frame*, i.e., a set $\{H_1, H_2, \cdots, H_r\}$ of primitive idempotents of $\mathscr{A}$ such that $H_{1}+H_2+\cdots+H_r$ is the identity element $E$, and $H_iH_j=0$ for all $i \neq j$. Jordan frames in $\mathscr{A}$ are unique up to automorphisms. Moreover, for any $X \in \mathscr{A}$, we can always find a Jordan frame $\{H_1, H_2, \cdots, H_r\}$ such that $$\begin{aligned} X=\lambda_1H_1+\cdots+\lambda_rH_r,\end{aligned}$$ where the numbers $\lambda_1, \cdots, \lambda_r \in \mathbb{R}$ are uniquely determined by $X$. In this case, we have $\operatorname{tr}(X)=\lambda_1+\cdots+\lambda_r$ (compare ([\[tr\]](#tr){reference-type="ref" reference="tr"})). In particular, $\operatorname{tr}(H_i)=1$ for all $1 \leq i \leq r$, and $\operatorname{tr}(E)=r$. Now, let us fix a Jordan frame $\{H_1, H_2, \cdots, H_r\}$ of $\mathscr{A}$. By Proposition [Lemma 39](#Fundamenta){reference-type="ref" reference="Fundamenta"} (i), we have $[L_{H_i}, L_{H_j}]=0$ for all $1 \leq i, j \leq r$. Using Proposition [Lemma 39](#Fundamenta){reference-type="ref" reference="Fundamenta"} (iii) for $X=Y=H_i$, we obtain $$\begin{aligned} 2L_{H_i}^3-3 L_{H_i}^2+L_{H_i}=0.\end{aligned}$$ Therefore, an eigenvalue $\lambda$ of $L_{H_i}$ is a solution of $$\begin{aligned} 2\lambda^3-3 \lambda^2+\lambda=0,\end{aligned}$$ whose roots are $0, \frac{1}{2}$ and $1$. Noting that the operators $L_{H_i}$ are diagonalizable and commute with each other, they admit simultaneously diagonalization. Consider the following subspaces of $\mathscr{A}$ $$\begin{aligned} \mathscr {A}_{ii}&=\mathscr{A}(H_i,1)=\mathbb{R}H_i, \\ \mathscr {A}_{ij}&=\mathscr{A}(H_i, \frac{1}{2}) \cap \mathscr{A}(H_j, \frac{1}{2}), ~~i < j,\end{aligned}$$ where $\mathscr{A}(H_i,1)$ and $\mathscr{A}(H_i, \frac{1}{2})$ denote the eigenspaces of $L_{H_i}$ corresponding to eigenvalues $1$ and $\frac{1}{2}$, respectively. Then we have the following famous Peirce decomposition of $\mathscr{A}$. **Theorem 44** ([@FK1994]). *Let $\mathscr{A}$ be a simple formally real Jordan algebra of rank $r$. Then $\mathscr{A}$ decomposes in the following orthogonal direct sum (with respect to an associative inner product, thus for all) $$\begin{aligned} \mathscr{A}=\bigoplus_{i\leq j}\mathscr{A}_{ij}=\bigoplus_{i=1}^r\mathbb{R}H_i\bigoplus_{i<j} \mathscr{A}_{ij}.\end{aligned}$$ Moreover, $$\begin{aligned} \mathscr{A}_{ij}\mathscr{A}_{ij} &\subset \mathscr{A}_{ii}+\mathscr{A}_{jj}, \\ \mathscr{A}_{ij}\mathscr{A}_{jk} &\subset \mathscr{A}_{ik}, \\ \mathscr{A}_{ij}\mathscr{A}_{kl}&=\{0\}, \text { if }\{i, j\} \cap\{k, I\}=\varnothing,\end{aligned}$$ where $1 \leq i, j, k, l \leq r$.* Note that $H_1+H_2+\cdots+H_r=E$, then we have $\operatorname{Tr} L_X=0=\operatorname{tr}(X)$ for any $X \in \oplus_{i<j}\mathscr{A}_{ij}$. **Remark 45**. Every simple formally real Jordan algebra is isomorphic to one of the following list, where the number $d:=\operatorname{dim} \mathscr{A}_{ij}$ does not depend on $i,j.$ [\[classificationlist\]]{#classificationlist label="classificationlist"} **$\mathscr{A}$** **Objects** **Multiplication rule** **Rank** **$d$** ------------------------------------- ------------------------------------------------------------------------------------------------------------------- ---------------------------------- ---------- --------- $\operatorname{Sym}(n,\mathbb{R})$ $n\times n$ self-adjoint real matrices $A \circ B=\frac{1}{2}(A B+B A)$ $n$ $1$ $\operatorname{Herm}(n,\mathbb{C})$ $n\times n$ self-adjoint complex matrices $A \circ B=\frac{1}{2}(A B+B A)$ $n$ $2$ $\operatorname{Herm}(n,\mathbb{H})$ $n\times n$ self-adjoint quatemionic matrices $A \circ B=\frac{1}{2}(A B+B A)$ $n$ $4$ $\operatorname{Herm}(3,\mathbb{O})$ $3\times 3$ self-adjoint octonionic matrices $A \circ B=\frac{1}{2}(A B+B A)$ $3$ $8$ The spin factors $\mathbb{R} \times \mathbb{R}^{n-1}$ and $f$ is a positive symmetric bilinear form on $\mathbb{R}^{n-1},~n\geq 3$ $(s, x)(t, y) $2$ $n-2$ =(st+f(u, v), tx+s y)$ : Simple formally real Jordan algebra # Acknowledgement {#acknowledgement .unnumbered} We would like to thank Professor Jacques Faraut for providing us an elegant proof of Lemma [Lemma 27](#inequality){reference-type="ref" reference="inequality"}. This paper is partially supported by NSFC (Grant Nos. 11701300, 11931009 and 12131012), the Fundamental Research Funds for the Central Universities (Grant No. 4007012303), and Guangdong Basic and Applied Basic Research Foundation (Grant No. 2023A1515010001) 99 A. Albert: On Jordan algebras of linear transformations. Trans. Amer. Math. Soc. **59** (1946), 524--555. C. Bai, L. Guo, Y. Sheng, and R. Tang: Post-groups, (Lie-)Butcher groups and the Yang-Baxter equation. Math. Ann. (2023). J. Baez: Getting to the bottom of Noether's theorem. In The Philosophy and Physics of Noether's Theorems: A Centenary Volume, pages 66--99. Cambridge University Press, (2022). O. Baues and W. Globke: Rigidity of compact pseudo-Riemannian homogeneous spaces for solvable Lie groups. Int. Math. Res. Not. IMRN 2018, no. 10, 3199--3223. O. Baues, W. Globke and A. Zeghib: Isometry Lie algebras of indefinite homogeneous spaces of finite volume. Proc. Lond. Math. Soc. (3) **119** (2019), no. 4, 1115--1148. W. 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arxiv_math

--- abstract: | In this work, we investigate internal congruences modulo arbitrary powers of $3$ for two functions arising from Ramanujan's classical theta functions $\varphi(q)$ and $\psi(q)$. By letting $$\begin{aligned} \sum_{n\ge 0} ph_3(n) q^n:=\dfrac{\varphi(-q^3)}{\varphi(-q)}\qquad\text{and}\qquad \sum_{n\ge 0} ps_3(n) q^n:=\dfrac{\psi(q^3)}{\psi(q)},\end{aligned}$$ we prove that for any $m\ge 1$ and $n\ge 0$, $$\begin{aligned} ph_3\big(3^{2m-1}n\big)\equiv ph_3\big(3^{2m+1}n\big)\pmod{3^{m+2}},\end{aligned}$$ and $$\begin{aligned} ps_3{\left(3^{2m-1}n+\frac{3^{2m}-1}{4}\right)}\equiv ps_3{\left(3^{2m+1}n+\frac{3^{2m+2}-1}{4}\right)}\pmod{3^{m+2}},\end{aligned}$$ thereby substantially generalizing the previous results of Bharadwaj et al. and Gireesh et al., respectively. address: - Department of Mathematics and Statistics, Dalhousie University, Halifax, NS, B3H 4R2, Canada - School of Mathematical Sciences, Chongqing Normal University, Chongqing 401331, P.R. China author: - Shane Chern - Dazhao Tang title: Ramanujan's theta functions and internal congruences modulo arbitrary powers of $3$ --- # Introduction The *classical theta function* $f(a,b)$ was introduced by Ramanujan in his *Notebooks* [@Ram1957 p. 197, Entry 18]: $$\begin{aligned} f(a,b):=\sum_{k=-\infty}^\infty a^{k(k+1)/2}b^{k(k-1)/2},\qquad|ab|<1,\end{aligned}$$ and it also takes the product form by the *Jacobi triple product identity* [@Coo2017 p. 6, Eq. (0.15)]: $$\begin{aligned} f(a,b)=(-a,ab)_\infty (-b;ab)_\infty(ab;ab)_\infty,\end{aligned}$$ wherein the conventional *$q$-Pochhammer symbol* is adopted: $$\begin{aligned} (A;q)_\infty := \prod_{k\ge 0} (1-Aq^k).\end{aligned}$$ In particular, we are interested in the following two specializations: $$\begin{aligned} {4} \varphi(q) &\,:=\,\,&& \;f(q,q) &&\,=\,\,&& \sum_{k=-\infty}^\infty q^{k^2},\\ \psi(q) &\,:=\,\,&& f(q,q^3) &&\,=\,\,&& \,\,\sum_{k\ge 0} q^{k(k+1)/2}.\end{aligned}$$ Alternatively, $$\begin{aligned} \varphi(-q)=\frac{(q;q)_\infty^2}{(q^2;q^2)_\infty}\qquad\textrm{and}\qquad \psi(q)=\frac{(q^2;q^2)_\infty^2}{(q;q)_\infty},\end{aligned}$$ where we choose $\varphi(-q)$ rather than $\varphi(q)$ in the former to emphasize the similarity of the two product expressions. A particularly notable topic in the world of $q$-series revolves around the arithmetic properties of the coefficients $c(n)$ generated by $$\begin{aligned} \sum_{n\ge 0} c(n)q^n := \prod_\delta (q^\delta;q^\delta)_\infty^{r_\delta}.\end{aligned}$$ Namely, we look for congruences of the form $$\begin{aligned} c(An+B) \equiv C \pmod{M},\end{aligned}$$ holding for any $n\ge 0$, in which the modulus $M$ and the parameters $A$, $B$ and $C$ are fixed, with $C$ usually being $0$. The study of this problem was initiated by the celebrated congruences modulo $5$, $7$ and $11$ due to Ramanujan [@Ram1919; @Ram1921] for the *partition function* $p(n)$, which counts the number of integer partitions of a natural number $n$ and has the generating function [@And1998]: $$\begin{aligned} \sum_{n\ge 0} p(n)q^n = \frac{1}{(q;q)_\infty}.\end{aligned}$$ Such congruences were later extended to moduli of an arbitrary power of $5$, $7$ and $11$ as conjectured by Ramanujan [@Ram1988]: For $\ell\in\{5,7,11\}$ and $\alpha\ge 1$, $$\begin{aligned} p\big(\ell^\alpha n+\delta_{\alpha,\ell}\big)\equiv\begin{cases} 0 \pmod{\ell^\alpha} &\quad\ell=5,11,\\ 0 \pmod{7^{\lceil\frac{\alpha+1}{2}\rceil}} &\quad\ell=7,\end{cases}\end{aligned}$$ with $0\le \delta_{\alpha,\ell} \le \ell^\alpha-1$ being such that $$\begin{aligned} 24\delta_{\alpha,\ell}\equiv1\pmod{\ell^\alpha}.\end{aligned}$$ Here Watson [@Wat1938] proved the cases of powers of $5$ and $7$, while Atkin [@Atk1967] confirmed the case of powers of $11$. In the meantime, we are also interested in *internal* congruences of the form $$\begin{aligned} c(An+B) \equiv c(A'n+B')\pmod{M},\end{aligned}$$ and we expect that the above two quantities are *not* congruent to a fixed number modulo $M$ for all $n$, so as to make this relation more nontrivial. In many cases, the sequence $A'n+B'$ is rendered as a subsequence of $An+B$, and such an internal congruence usually allows us to derive an infinite family of congruences under the fixed modulus $M$. However, internal congruences modulo an arbitrary power of a number are not widely recognized. In this work, the first object of our interest is $$\begin{aligned} \label{eq:ph-gf} \sum_{n\ge 0}ph_3(n)q^n :=\dfrac{\varphi(-q^3)}{\varphi(-q)}.\end{aligned}$$ From a partition-theoretic perspective, $ph_3(n)$ counts the number of $3$-regular overpartitions of $n$ as introduced by Lovejoy [@Lov2003], and this function is usually written as $\overline{A}_3(n)$ in the literature. In 2018, Bharadwaj, Hemanthkumar and Naika [@BHN2018] established the following internal congruences: $$\begin{aligned} ph_3(27n) &\equiv ph_3(3n)\pmod{27},\\ ph_3(243n) &\equiv ph_3(27n)\pmod{81}.\end{aligned}$$ Herein, we offer a substantial generalization by extending the modulus to an arbitrary power of $3$: **Theorem 1**. *For any $m\ge 1$ and $n\ge 0$, $$\begin{aligned} \label{eq:ph-cong} ph_3\big(3^{2m+1}n\big)\equiv ph_3\big(3^{2m-1}n\big) \pmod{3^{m+2}}.\end{aligned}$$* Similar to [\[eq:ph-gf\]](#eq:ph-gf){reference-type="eqref" reference="eq:ph-gf"}, we shall also consider $$\begin{aligned} \label{eq:ps-gf} \sum_{n\ge 0} ps_3(n) q^n:= \frac{\psi(q^3)}{\psi(q)}.\end{aligned}$$ The function $ps_3(n)$ is closely tied with the $\operatorname{pod}_3(n)$ function, which counts the number of partitions of $n$ into non-multiples of $3$ in which the odd parts are distinct, through the relation $ps_3(n)=(-1)^n\operatorname{pod}_3(n)$. According to the work of Gireesh, Hirschhorn and Naika [@GHN2017], the following internal congruences are true: $$\begin{aligned} ph_3(27n+20) &\equiv ph_3(3n+2)\pmod{27},\\ ph_3(243n+182) &\equiv ph_3(27n+20)\pmod{81}.\end{aligned}$$ In the same vein, the above two congruences will be extended in this work: **Theorem 2**. *For any $m\ge 1$ and $n\ge 0$, $$\begin{aligned} \label{eq:ps-cong} ps_3{\left(3^{2m+1}n+\frac{3^{2m+2}-1}{4}\right)} \equiv ps_3{\left(3^{2m-1}n+\frac{3^{2m}-1}{4}\right)}\pmod{3^{m+2}}.\end{aligned}$$* It is remarkable (and indeed unexpected!) that once we have proved [\[eq:ph-cong\]](#eq:ph-cong){reference-type="eqref" reference="eq:ph-cong"} using the strategy in Sect. [4](#sec:ph-cong-proof){reference-type="ref" reference="sec:ph-cong-proof"}, the congruence [\[eq:ps-cong\]](#eq:ps-cong){reference-type="eqref" reference="eq:ps-cong"} follows automatically, as shown in Sect. [5](#sec:ps-cong-proof){reference-type="ref" reference="sec:ps-cong-proof"}. However, an interesting fact is that, in regard to the opposite direction, there likely exists an insuperable obstacle, and we will present a concrete discussion in Remark [Remark 16](#rmk:ps-to-ph){reference-type="ref" reference="rmk:ps-to-ph"}. The remainder of this paper is organized as follows. First, in Sect. [2](#sec:phi-notation){reference-type="ref" reference="sec:phi-notation"} we introduce required auxiliary series associated with $\varphi(-q)$ and establish their corresponding modular equations. Then Sect. [3](#sec:3-adic){reference-type="ref" reference="sec:3-adic"} is devoted to an initial $3$-adic analysis. With such preliminary knowledge, we are about to prove the internal congruences for $ph_3(n)$ in Sect. [4](#sec:ph-cong-proof){reference-type="ref" reference="sec:ph-cong-proof"}. Finally, in Sect. [5](#sec:ps-cong-proof){reference-type="ref" reference="sec:ps-cong-proof"} we show how the internal congruences for $ps_3(n)$ will automatically hold by constructing a different set of auxiliary functions involving $\psi(q)$, which, surprisingly, possess the same modular equations. # Auxiliary functions and modular equations {#sec:phi-notation} Let us write $$\begin{aligned} F(q):=\dfrac{\varphi(-q^3)}{\varphi(-q)}.\end{aligned}$$ In the meantime, we define the following two auxiliary functions: $$\begin{aligned} \label{eq:gamma-def} \gamma=\gamma(q):=\frac{F(q)}{F(q^9)},\end{aligned}$$ and $$\begin{aligned} \label{eq:xi-def} \xi=\xi(q):=\dfrac{\varphi(-q^9)}{\varphi(-q)}.\end{aligned}$$ Finally, let $U$ be the *unitizing operator of degree three*, given by $$\begin{aligned} U{\left(\sum_n a_n q^n\right)}:= \sum_n a_{3n} q^n.\end{aligned}$$ In this section, our main objective is the following: **Theorem 3**. *For every $i\ge 0$, both $U\big(\xi^i\big)$ and $U\big(\gamma \xi^i\big)$ can be expressed as a polynomial in $\mathbb{Z}[\xi]$. In particular, if we write $$\begin{aligned} \label{eq:X-coeff} U\big(\xi^i\big) = \sum_{j} X_{i}(j)\xi^j,\end{aligned}$$ then $X_i(j)=0$ whenever $0\le j< \left\lceil \frac{i}{3}\right\rceil$, where the ceiling function $\lceil x\rceil$ denotes the least integer greater than or equal to $x$.* We first focus on $U\big(\xi^i\big)$, and we shall start with the following initial cases. **Lemma 4**. *We have $$\begin{aligned} U\big(\xi\big) &=\xi-3\xi^2+3\xi^3,\label{eq:xi1}\\ U\big(\xi^2\big) &=-2\xi+9\xi^2-24\xi^3+45\xi^4 -54\xi^5+27\xi^6,\label{eq:xi2}\\ U\big(\xi^3\big) &= \xi - 12 \xi^2 + 66 \xi^3 - 216 \xi^4 + 486 \xi^5 - 810 \xi^6\notag\\ &\quad + 972 \xi^7 - 729 \xi^8 + 243 \xi^9.\label{eq:xi3}\end{aligned}$$* *Proof.* These relations can be shown by a direct cusp analysis as we note that $\xi$ is a modular function on the classical modular curve $X_0(18)$, which has genus $0$. Alternatively, we may perform a more automated proof by first utilizing Smoot's [Mathematica]{.sans-serif} implementation `RaduRK` [@Smo2021] of the Radu--Kolberg algorithm [@Rad2015] to express each of $U\big(\xi\big)$, $U\big(\xi^2\big)$ and $U\big(\xi^3\big)$ in terms of a Hauptmodul on $X_0(18)$, and then applying Garvan's [Maple]{.sans-serif} package `ETA` [@Gar1999], which allows us to certify the equality of two linear combinations of eta-products. See the proof of [@CS2023 Theorem 2.3] for a detailed instance. ◻ Now we move on to general $U\big(\xi^i\big)$. For brevity, we write for $i\ge 0$, $$\begin{aligned} \mathcal{X}_i := U\big(\xi^i\big).\end{aligned}$$ **Lemma 5**. *For any $i\ge 3$, $$\begin{aligned} \label{eq:cX-rec} \mathcal{X}_i=\big(\xi-3\xi^2+3\xi^3\big)\big(3\mathcal{X}_{i-1}-3\mathcal{X}_{i-2}+\mathcal{X}_{i-3}\big).\end{aligned}$$* *Proof.* Let $\omega:=e^{\frac{2\pi i}{3}}$ be a primitive cubic root of unity and denote for $k\in\{0,1,2\}$: $$\begin{aligned} \xi_k:=\xi(\omega^k q).\end{aligned}$$ We start by noting that $$\begin{aligned} \sigma_1 &:= \xi_0 + \xi_1 + \xi_2\\ &\ = 3U\big(\xi\big),\\ \sigma_2 &:= \xi_0\xi_1 + \xi_1\xi_2 + \xi_2\xi_0\\ &\ =\tfrac{1}{2}\big[\big(\xi_0 + \xi_1 + \xi_2\big)^2-\big(\xi_0^2 + \xi_1^2 + \xi_2^2\big)\big]\\ &\ =\tfrac{1}{2}\big[9U\big(\xi\big)^2-3U\big(\xi^2\big)\big],\\ \sigma_3 &:= \xi_0\xi_1\xi_2\\ &\ = \tfrac{1}{6}\big[\big(\xi_0 + \xi_1 + \xi_2\big)^3-3\big(\xi_0 + \xi_1 + \xi_2\big)\big(\xi_0^2 + \xi_1^2 + \xi_2^2\big)+2\big(\xi_0^3 + \xi_1^3 + \xi_2^3\big)\big]\\ &\ = \tfrac{1}{6}\big[27U\big(\xi\big)^3-27U\big(\xi\big)U\big(\xi^2\big)+6U\big(\xi^3\big)\big].\end{aligned}$$ These are instances of Newton's identities for elementary symmetric functions [@Mea1992]. In light of Lemma [Lemma 4](#le:xi-1-3){reference-type="ref" reference="le:xi-1-3"}, it is clear that all of $\sigma_1$, $\sigma_2$ and $\sigma_3$ are in $\mathbb{Z}[\xi]$, and in particular, $$\begin{aligned} \sigma_1 &=3\xi-9\xi^2+9\xi^3,\\ \sigma_2 &=3\xi-9\xi^2+9\xi^3,\\ \sigma_3 &=\xi-3\xi^2+3\xi^3.\end{aligned}$$ Next, we observe that $X=\xi_0$, $\xi_1$ and $\xi_2$ are the three roots of $$\begin{aligned} \big(X-\xi_0\big)\big(X-\xi_1\big)\big(X-\xi_2\big) =X^3-\sigma_1X^2+\sigma_2X-\sigma_3.\end{aligned}$$ So for $k\in\{0,1,2\}$, $$\begin{aligned} \xi_k^3-\sigma_1\xi_k^2+\sigma_2\xi_k-\sigma_3=0,\end{aligned}$$ which further implies that for $i\ge 3$, $$\begin{aligned} \xi_k^i=\sigma_1\xi_k^{i-1}-\sigma_2\xi_k^{i-2}+\sigma_3 \xi_k^{i-3}.\end{aligned}$$ Finally, we note that for $j\ge 0$, $$\begin{aligned} \mathcal{X}_j=U\big(\xi^j\big)=3\big(\xi(q)^j+\xi(\omega q)^j+\xi(\omega^2 q)^j\big) =3\big(\xi_0^j+\xi_1^j+\xi_2^j\big).\end{aligned}$$ Hence, summing both sides of the previous relation over $k\in\{0,1,2\}$ yields the required result. ◻ Next, we establish a relation between $U\big(\gamma\xi^i\big)$ and $U\big(\xi^{i+1}\big)$: **Lemma 6**. *For any $i\ge 0$, $$\begin{aligned} \label{eq:U-gamma-cX} U\big(\gamma \xi^i\big) = \xi^{-1} \mathcal{X}_{i+1}.\end{aligned}$$* *Proof.* In light of [\[eq:gamma-def\]](#eq:gamma-def){reference-type="eqref" reference="eq:gamma-def"} and [\[eq:xi-def\]](#eq:xi-def){reference-type="eqref" reference="eq:xi-def"}, $$\begin{aligned} \gamma=\dfrac{F(q)}{F(q^9)}=\dfrac{\varphi(-q^3)}{\varphi(-q)} \dfrac{\varphi(-q^9)}{\varphi(-q^{27})} =\dfrac{\varphi(-q^3)}{\varphi(-q^{27})} \cdot\dfrac{\varphi(-q^9)}{\varphi(-q)}=\dfrac{\xi(q)}{\xi(q^3)}.\end{aligned}$$ Thus, $$\begin{aligned} U\big(\gamma\xi^i\big)=U{\left(\dfrac{\xi(q)}{\xi(q^3)}\cdot\xi(q)^i\right)} =\dfrac{1}{\xi(q)}\cdot U\big(\xi(q)^{i+1}\big), \end{aligned}$$ as required. ◻ Finally, we conclude our proof of Theorem [Theorem 3](#th:xi-poly){reference-type="ref" reference="th:xi-poly"}. *Proof of Theorem [Theorem 3](#th:xi-poly){reference-type="ref" reference="th:xi-poly"}.* It is plain that $\mathcal{X}_0=1\in \mathbb{Z}[\xi]$. When $i\ge 1$, every $\mathcal{X}_i$ can be expressed as a polynomial in $\mathbb{Z}[\xi]$ according to the recurrence [\[eq:cX-rec\]](#eq:cX-rec){reference-type="eqref" reference="eq:cX-rec"} with the initial cases in Lemma [Lemma 4](#le:xi-1-3){reference-type="ref" reference="le:xi-1-3"} kept in mind. In addition, by an inductive argument, it is also clear from the recurrence [\[eq:cX-rec\]](#eq:cX-rec){reference-type="eqref" reference="eq:cX-rec"} that the minimal $\xi$-power in $\mathcal{X}_i$ is at least $\xi^{\lceil \frac{i}{3}\rceil}$, which, particularly, means that whenever $i\ge 1$, the constant term in the $\xi$-polynomial representation of $\mathcal{X}_i$ vanishes. This fact, together with [\[eq:U-gamma-cX\]](#eq:U-gamma-cX){reference-type="eqref" reference="eq:U-gamma-cX"}, further certifies that $U\big(\gamma \xi^{i-1}\big)\in \mathbb{Z}[\xi]$. ◻ # $3$-Adic analysis {#sec:3-adic} Throughout, let $\nu(n)$ denote the *$3$-adic evaluation* of $n$, which is defined as the *largest* nonnegative integer $\alpha$ such that $3^\alpha\mid n$. Also, as a convention, we assume that $\nu(0)=\infty$. Recall from Theorem [Theorem 3](#th:xi-poly){reference-type="ref" reference="th:xi-poly"} that, as a polynomial in $\mathbb{Z}[\xi]$, the $\xi$-powers in $\mathcal{X}_i$ of degree lower than $$\begin{aligned} d_i := \left\lceil \frac{i}{3}\right\rceil\end{aligned}$$ all vanish. Now we perform the following $3$-adic analysis for the remaining terms: **Lemma 7**. *Let the coefficients $X_i$ be as in [\[eq:X-coeff\]](#eq:X-coeff){reference-type="eqref" reference="eq:X-coeff"}. Then for any $i\ge 1$, $$\begin{aligned} \label{eq:3-adic-0} \nu\big(X_i(d_i)\big)=0,\end{aligned}$$ and further for any $j\ge 1$, $$\begin{aligned} \label{eq:3-adic-j} \nu\big(X_i(d_i+j)\big)\ge\left\lfloor\frac{j+1}{2}\right\rfloor.\end{aligned}$$* *Proof.* In light of Lemma [Lemma 4](#le:xi-1-3){reference-type="ref" reference="le:xi-1-3"}, our statement holds for $i=1$, $2$ and $3$. Now let us assume that the statement is true for $i=3I+1$, $3I+2$ and $3I+3$ with a certain $I\ge 0$. We first certify the claim for $i=3I+4$. To see this, we start with [\[eq:cX-rec\]](#eq:cX-rec){reference-type="eqref" reference="eq:cX-rec"}: $$\begin{aligned} \mathcal{X}_{3I+4}=\big(\xi-3\xi^2+3\xi^3\big) \big(3\mathcal{X}_{3I+3}-3\mathcal{X}_{3I+2}+\mathcal{X}_{3I+1}\big).\end{aligned}$$ Noting that $$\begin{aligned} d_{3I+4} = I+2,\end{aligned}$$ and that $$\begin{aligned} d_{3I+1}=d_{3I+2}=d_{3I+3}=I+1,\end{aligned}$$ we have $$\begin{aligned} X_{3I+4}(d_{3I+4})=3X_{3I+3}(d_{3I+3})-3X_{3I+2}(d_{3I+2}) +X_{3I+1}(d_{3I+1}),\end{aligned}$$ which is not a multiple of $3$ since $X_{3I+1}(d_{3I+1})$ is not divisible by $3$ as assumed. Thus, [\[eq:3-adic-0\]](#eq:3-adic-0){reference-type="eqref" reference="eq:3-adic-0"} holds for $i=3I+4$. Similarly, $$\begin{aligned} X_{3I+4}(d_{3I+4}+1) &=3X_{3I+3}(d_{3I+3}+1)-3X_{3I+2}(d_{3I+2}+1) +X_{3I+1}(d_{3I+1}+1)\\ &\quad-9X_{3I+3}(d_{3I+3})+9X_{3I+2}(d_{3I+2})-3X_{3I+1}(d_{3I+1}).\end{aligned}$$ Since it is already assumed that $\nu\big(X_{3I+1}(d_{3I+1}+1)\big)\ge \lfloor\frac{1+1}{2}\rfloor=1$, we conclude that [\[eq:3-adic-j\]](#eq:3-adic-j){reference-type="eqref" reference="eq:3-adic-j"} holds for $j=1$. Finally, for $j\ge 2$, $$\begin{aligned} &X_{3I+4}(d_{3I+4}+j)\\ &=3X_{3I+3}(d_{3I+3}+j) - 3X_{3I+2}(d_{3I+2}+j) + X_{3I+1}(d_{3I+1}+j)\\ &\quad - 9X_{3I+3}(d_{3I+3}+j-1) + 9X_{3I+2}(d_{3I+2}+j-1) - 3X_{3I+1}(d_{3I+1}+j-1)\\ &\quad + 9X_{3I+3}(d_{3I+3}+j-2) - 9X_{3I+2}(d_{3I+2}+j-2) + 3X_{3I+1}(d_{3I+1}+j-2).\end{aligned}$$ Recalling the assumption that $\nu\big(X_i(d_i+j)\big)\ge\left\lfloor\frac{j+1}{2}\right\rfloor$ is valid for every $j\ge 0$ when $i=3I+1$, $3I+2$ and $3I+3$, we are able to confirm [\[eq:3-adic-j\]](#eq:3-adic-j){reference-type="eqref" reference="eq:3-adic-j"} for $j\ge 2$ since $$\begin{aligned} \nu\big(X_{3I+4}(d_{3I+4}+j)\big) &\ge\min\Bigg\{\left\lfloor\dfrac{j+1}{2}\right\rfloor+1, \left\lfloor\dfrac{j+1}{2}\right\rfloor+1, \left\lfloor\dfrac{j+1}{2}\right\rfloor+0,\\ &\qquad\qquad\left\lfloor\dfrac{j+0}{2}\right\rfloor+2, \left\lfloor\dfrac{j+0}{2}\right\rfloor+2, \left\lfloor\dfrac{j+0}{2}\right\rfloor+1,\\ &\qquad\qquad\left\lfloor\dfrac{j-1}{2}\right\rfloor+2, \left\lfloor\dfrac{j-1}{2}\right\rfloor+2, \left\lfloor\dfrac{j-1}{2}\right\rfloor+1\Bigg\}\\ &=\left\lfloor\dfrac{j+1}{2}\right\rfloor.\end{aligned}$$ Moreover, a similar analysis may be carried out for $i=3I+5$ and $3I+6$, and thus our desired claim holds by induction. ◻ **Remark 8**. By recourse to [\[eq:3-adic-0\]](#eq:3-adic-0){reference-type="eqref" reference="eq:3-adic-0"}, it is immediate that for $i\ge 1$, the coefficient $X_i(d_i)$ never vanishes; otherwise, it is divisible by $3$, thereby contradicting [\[eq:3-adic-0\]](#eq:3-adic-0){reference-type="eqref" reference="eq:3-adic-0"}. Recalling the second part of Theorem [Theorem 3](#th:xi-poly){reference-type="ref" reference="th:xi-poly"}, we conclude that the minimal $\xi$-power in the polynomial representation of $\mathcal{X}_i$ is **exactly** of degree $d_i=\left\lceil \frac{i}{3}\right\rceil$. # Internal congruences for $ph_3(n)$ {#sec:ph-cong-proof} We are about to study the following family of series for $M\ge 1$: $$\begin{aligned} \Phi_M(q):=\begin{cases} \displaystyle\dfrac{1}{F(q^3)}\sum_{n\ge 0}ph_3\big(3^{2m-1}n\big)q^n, &\quad\textrm{if $M=2m-1$},\\ \displaystyle\dfrac{1}{F(q)}\sum_{n\ge 0}ph_3\big(3^{2m}n\big)q^n, &\quad\textrm{if $M=2m$}. \end{cases}\end{aligned}$$ It is clear that for any $m\ge 1$, $$\begin{aligned} \Phi_{2m} &=U\big(\Phi_{2m-1}\big),\label{eq:Phi-even}\\ \Phi_{2m+1} &=U\big(\gamma\Phi_{2m}\big).\label{eq:Phi-odd}\end{aligned}$$ Let us first consider a general setting, wherein the notation is as in Sect. [2](#sec:phi-notation){reference-type="ref" reference="sec:phi-notation"}. **Lemma 9**. *For any $\Lambda\in \mathbb{Z}[\xi]$ such that the minimal $\xi$-power in $\Lambda$ is no lower that $\xi^L$, then the following statements are true:* 1. *$U\big(\Lambda\big)$ is in $\mathbb{Z}[\xi]$, and the minimal $\xi$-power in this polynomial representation is of degree at least $\left\lceil\frac{L}{3}\right\rceil$;* 2. *$U\big(\gamma\Lambda\big)$ is in $\mathbb{Z}[\xi]$, and the minimal $\xi$-power in this polynomial representation is of degree at least $\left\lceil\frac{L-2}{3}\right\rceil$.* *Proof.* Assume that $$\begin{aligned} \Lambda=\sum_{\ell\ge L}a_\ell\xi^\ell.\end{aligned}$$ We have $$\begin{aligned} \label{eq:U-Lambda} U\big(\Lambda\big)=\sum_{\ell\ge L}a_\ell U\big(\xi^\ell\big) =\sum_{\ell\ge L}a_\ell\mathcal{X}_\ell.\end{aligned}$$ Since the ceiling function is non-decreasing, the minimal $\xi$-power in $U\big(\Lambda\big)$ is no lower than that in $\mathcal{X}_L$, which is $\xi^{\lceil\frac{L}{3}\rceil}$ in light of Remark [Remark 8](#rmk:cX-min-power){reference-type="ref" reference="rmk:cX-min-power"}. Next, $$\begin{aligned} \label{eq:U-gamma*Lambda} U\big(\gamma\Lambda\big)=\sum_{\ell\ge L}a_\ell U\big(\gamma\xi^\ell\big) =\sum_{\ell\ge L}a_\ell\xi^{-1}\mathcal{X}_{\ell+1},\end{aligned}$$ where we have utilized [\[eq:U-gamma-cX\]](#eq:U-gamma-cX){reference-type="eqref" reference="eq:U-gamma-cX"}. Now the minimal $\xi$-power in $U\big(\gamma\Lambda\big)$ is no lower than that in $\xi^{-1}\mathcal{X}_{L+1}$, while we note that $\left\lceil\frac{L+1}{3}\right\rceil-1 =\left\lceil\frac{L-2}{3}\right\rceil$. ◻ Now we observe that $$\begin{aligned} \Phi_1=\dfrac{1}{F(q^3)}\sum_{n\ge 0}ph_3\big(3n+2\big)q^n =U\big(\gamma\big).\end{aligned}$$ In light of [\[eq:xi1\]](#eq:xi1){reference-type="eqref" reference="eq:xi1"} and [\[eq:U-gamma-cX\]](#eq:U-gamma-cX){reference-type="eqref" reference="eq:U-gamma-cX"}, we have $$\begin{aligned} \label{eq:Phi1} \Phi_1=1-3\xi+3\xi^2.\end{aligned}$$ By virtue of [\[eq:Phi-even\]](#eq:Phi-even){reference-type="eqref" reference="eq:Phi-even"}, we apply [\[eq:U-Lambda\]](#eq:U-Lambda){reference-type="eqref" reference="eq:U-Lambda"} and get $$\begin{aligned} \label{eq:Phi2} \Phi_2=1-9\xi+36\xi^2-81\xi^3+135\xi^4-162\xi^5+81\xi^6.\end{aligned}$$ Furthermore, utilizing [\[eq:Phi-odd\]](#eq:Phi-odd){reference-type="eqref" reference="eq:Phi-odd"} and [\[eq:U-gamma\*Lambda\]](#eq:U-gamma*Lambda){reference-type="eqref" reference="eq:U-gamma*Lambda"} yields $$\begin{aligned} \label{eq:Phi3} \Phi_3 &=55-2163\xi+34509\xi^2-330318\xi^3+2227338\xi^4-11501919\xi^5\notag\\ &\quad+47744397\xi^6-164234952\xi^7+477601434\xi^8-1189266543\xi^9\notag\\ &\quad+2554873083\xi^{10}-4751141589\xi^{11} +7644778785\xi^{12}-10594276335\xi^{13}\notag\\ &\quad+12526595811\xi^{14}-12440502369\xi^{15} +10115979435\xi^{16}-6457008150\xi^{17}\notag\\ &\quad+3013270470\xi^{18}-903981141\xi^{19}+129140163\xi^{20}.\end{aligned}$$ In general, by iterating the above process, the following claim is clear. **Theorem 10**. *For any $M\ge 1$, we have $\Phi_M \in \mathbb{Z}[\xi]$.* Now we recall that our objective is to prove the congruence [\[eq:ph-cong\]](#eq:ph-cong){reference-type="eqref" reference="eq:ph-cong"}. Hence, we shall work on a new family of series for $m\ge 1$: $$\begin{aligned} \widehat{\Phi}_m:=\dfrac{1}{F(q^3)}\sum_{n\ge 0} {\left(ph_3\big(3^{2m+1}n\big)-ph_3\big(3^{2m-1}n\big)\right)}q^n.\end{aligned}$$ In other words, $$\begin{aligned} \label{eq:Phi-hat-exp-2} \widehat{\Phi}_m=\Phi_{2m+1}-\Phi_{2m-1}.\end{aligned}$$ It is clear that the congruence [\[eq:ph-cong\]](#eq:ph-cong){reference-type="eqref" reference="eq:ph-cong"} in Theorem [Theorem 1](#th:ph-cong){reference-type="ref" reference="th:ph-cong"} is a straightforward consequence of the following result: **Theorem 11**. *For any $m\ge 1$, we have $\widehat{\Phi}_m\in \mathbb{Z}[\xi]$. Furthermore, if we write $$\begin{aligned} \label{eq:Psi-hat-coeff} \widehat{\Phi}_m=\sum_{k}C_m(k)\xi^k,\end{aligned}$$ then for any $k\ge 0$, $$\begin{aligned} \label{eq:nu-C-bound} \nu\big(C_m(k)\big)\ge m+2+\left\lfloor\dfrac{k}{2}\right\rfloor.\end{aligned}$$* *Proof.* We perform our proof by induction on $m$. Clearly, the $m=1$ case can be confirmed by invoking [\[eq:Phi1\]](#eq:Phi1){reference-type="eqref" reference="eq:Phi1"} and [\[eq:Phi3\]](#eq:Phi3){reference-type="eqref" reference="eq:Phi3"}. Now assuming the validity of the statement for a certain $m$, we shall prove [\[eq:nu-C-bound\]](#eq:nu-C-bound){reference-type="eqref" reference="eq:nu-C-bound"} for $m+1$. Our starting point is the following observation: $$\begin{aligned} \widehat{\Phi}_{m+1}=U\big(\gamma U\big(\widehat{\Phi}_m\big)\big).\end{aligned}$$ This can be shown by sequentially acting the operators $U\big(\bullet\big)$ and $U\big(\gamma\cdot \bullet\big)$ to both sides of [\[eq:Phi-hat-exp-2\]](#eq:Phi-hat-exp-2){reference-type="eqref" reference="eq:Phi-hat-exp-2"}, while for the right-hand side, we also require [\[eq:Phi-even\]](#eq:Phi-even){reference-type="eqref" reference="eq:Phi-even"} and [\[eq:Phi-odd\]](#eq:Phi-odd){reference-type="eqref" reference="eq:Phi-odd"}. Now, by [\[eq:X-coeff\]](#eq:X-coeff){reference-type="eqref" reference="eq:X-coeff"}, $$\begin{aligned} U\big(\widehat{\Phi}_m\big) &=C_m(0)+\sum_{k\ge 1}C_m(k)\mathcal{X}_k\\ &=C_m(0)+\sum_{k\ge 1}C_m(k)\sum_{j\ge 0}X_k(d_k+j)\xi^{d_k+j}.\end{aligned}$$ Let us write $$\begin{aligned} U\big(\widehat{\Phi}_m\big)=\sum_{\ell}\widetilde{C}_m(\ell)\xi^\ell.\end{aligned}$$ We shall show that for any $\ell\ge 0$, $$\begin{aligned} \label{eq:nu-C'-bound} \nu\big(\widetilde{C}_m(\ell)\big) \ge m+2+\left\lfloor\frac{\ell}{2}\right\rfloor.\end{aligned}$$ To see this, we note that $C_m(0)$ contributes to $\widetilde{C}_m(0)\xi^0$, while according to our assumption, $$\begin{aligned} \nu\big(C_m(0)\big)\ge m+2+\left\lfloor\frac{0}{2}\right\rfloor.\end{aligned}$$ Also, for the each term $C_m(k)X_k(d_k+j)\xi^{d_k+j}$ contributing to $\widetilde{C}_m(d_k+j)\xi^{d_k+j}$, we note that $k\ge d_k=\left\lceil \frac{k}{3}\right\rceil$ whenever $k\ge 1$, and conclude that $$\begin{aligned} \nu\big(C_m(k)X_k(d_k+j)\big) &=\nu\big(C_m(k)\big)+\nu\big(X_k(d_k+j)\big)\\ &\ge\left(m+2+\left\lfloor\frac{k}{2}\right\rfloor\right) +\left\lfloor\frac{j+1}{2}\right\rfloor\\ &\ge m+2+\left\lfloor\frac{d_k}{2}\right\rfloor +\left\lfloor\frac{j+1}{2}\right\rfloor\\ &\ge m+2+\left\lfloor\frac{d_k+j}{2}\right\rfloor,\end{aligned}$$ where $\nu\big(C_m(k)\big)$ is bounded by the inductive hypothesis and $\nu\big(X_k(d_k+j)\big)$ is bounded by Lemma [Lemma 7](#le:3-adic-X){reference-type="ref" reference="le:3-adic-X"}. Therefore, [\[eq:nu-C\'-bound\]](#eq:nu-C'-bound){reference-type="eqref" reference="eq:nu-C'-bound"} is established. Next, we have $$\begin{aligned} \widehat{\Phi}_{m+1} &=U\big(\gamma U\big(\widehat{\Phi}_m\big)\big)\\ &=\sum_{\ell\ge 0}\widetilde{C}_m(\ell)U\big(\gamma \xi^\ell\big)\\ &=\sum_{\ell\ge 0}\widetilde{C}_m(\ell)\xi^{-1}\mathcal{X}_{\ell+1}\\ &=\sum_{\ell\ge 0}\widetilde{C}_m(\ell)\sum_{j\ge 0} X_{\ell+1}(d_{\ell+1}+j)\xi^{d_{\ell+1}+j-1}.\end{aligned}$$ Here we have applied [\[eq:U-gamma-cX\]](#eq:U-gamma-cX){reference-type="eqref" reference="eq:U-gamma-cX"} for the third equality. Taking out the summands with $\ell=0$ and $1$ gives $$\begin{aligned} \label{eq:Phi-hat-expand-m+1} \widehat{\Phi}_{m+1} &=\widetilde{C}_m(0)\big(1-3\xi+3\xi^2\big)\notag\\ &\quad+\widetilde{C}_m(1) \big({-2}+9\xi-24\xi^2+45\xi^3-54\xi^4+27\xi^5\big)\notag\\ &\quad+\sum_{\ell\ge 2}\widetilde{C}_m(\ell) \sum_{j\ge 0}X_{\ell+1}(d_{\ell+1}+j)\xi^{d_{\ell+1}+j-1},\end{aligned}$$ where $U\big(\gamma\big)=\xi^{-1}U\big(\xi\big)$ and $U\big(\gamma\xi\big)=\xi^{-1}U\big(\xi^2\big)$ can be deduced by [\[eq:xi1\]](#eq:xi1){reference-type="eqref" reference="eq:xi1"} and [\[eq:xi2\]](#eq:xi2){reference-type="eqref" reference="eq:xi2"}, respectively. Recall that $$\begin{aligned} \widehat{\Phi}_{m+1}=\sum_{k}C_{m+1}(k)\xi^k.\end{aligned}$$ To show [\[eq:nu-C-bound\]](#eq:nu-C-bound){reference-type="eqref" reference="eq:nu-C-bound"} for the case of $m+1$, we need to prove that $$\begin{aligned} \label{eq:nu-C-bound-m+1} \nu\big(C_{m+1}(k)\big)\ge(m+1)+2+\left\lfloor\dfrac{k}{2}\right\rfloor =m+3+\left\lfloor\frac{k}{2}\right\rfloor.\end{aligned}$$ For the right-hand side of [\[eq:Phi-hat-expand-m+1\]](#eq:Phi-hat-expand-m+1){reference-type="eqref" reference="eq:Phi-hat-expand-m+1"}, the easier cases are those with $\ell\ge 2$. Note that in such scenarios, $\ell\ge d_{\ell+1}+1=\left\lceil\frac{\ell+1}{3}\right\rceil+1$. Thus, $$\begin{aligned} \nu\big(\widetilde{C}_m(\ell)X_{\ell+1}(d_{\ell+1}+j)\big) &=\nu\big(\widetilde{C}_m(\ell)\big)+\nu\big(X_{\ell+1}(d_{\ell+1}+j)\big)\\ &\ge \left(m+2+\left\lfloor\frac{\ell}{2}\right\rfloor\right) +\left\lfloor\frac{j+1}{2}\right\rfloor\\ &\ge m+2+\left\lfloor\frac{d_{\ell+1}+1}{2}\right\rfloor +\left\lfloor\frac{j+1}{2}\right\rfloor\\ &\ge m+2+\left\lfloor\frac{d_{\ell+1}+j+1}{2}\right\rfloor\\ &=m+3+\left\lfloor\frac{d_{\ell+1}+j-1}{2}\right\rfloor.\end{aligned}$$ Meanwhile, a routine computation reveals that the contributions from $$\begin{aligned} \widetilde{C}_m(0)\cdot (-3\xi)\end{aligned}$$ and $$\begin{aligned} \widetilde{C}_m(1)\cdot(9\xi),\qquad \widetilde{C}_m(1)\cdot(45\xi^3), \qquad \widetilde{C}_m(1)\cdot(-54\xi^4),\qquad \widetilde{C}_m(1)\cdot(27\xi^5)\end{aligned}$$ all match with [\[eq:nu-C-bound-m+1\]](#eq:nu-C-bound-m+1){reference-type="eqref" reference="eq:nu-C-bound-m+1"}. Hence, it will suffice to verify that $$\begin{aligned} \nu\big(\widetilde{C}_m(0)-2\widetilde{C}_m(1)\big) &\ge m+3+\left\lfloor\dfrac{0}{2}\right\rfloor,\label{eq:nu-C'-extra-0}\\ \nu\big(3\widetilde{C}_m(0)-24\widetilde{C}_m(1)\big) &\ge m+3+\left\lfloor\dfrac{2}{2}\right\rfloor.\label{eq:nu-C'-extra-2}\end{aligned}$$ We begin by observing from [\[eq:nu-C\'-bound\]](#eq:nu-C'-bound){reference-type="eqref" reference="eq:nu-C'-bound"} that $$\begin{aligned} \nu\big(3\widetilde{C}_m(0)-24\widetilde{C}_m(1)\big)\ge1+(m+2)=m+3.\end{aligned}$$ Thus, modulo $3^{m+3}$, $$\begin{aligned} \widehat{\Phi}_{m+1}(q)\equiv \widetilde{C}_m(0)-2\widetilde{C}_m(1)\pmod{3^{m+3}}.\end{aligned}$$ In particular, $$\begin{aligned} \label{eq:Phi-hat-0-cong-1} \widehat{\Phi}_{m+1}(0)\equiv \widetilde{C}_m(0)-2\widetilde{C}_m(1)\pmod{3^{m+3}}.\end{aligned}$$ Meanwhile, we note that the constant term in $$\begin{aligned} \widehat{\Phi}_{m+1}(q)=\dfrac{1}{F(q^3)}\sum_{n\ge 0} \left(ph_3\big(3^{2m+3}n\big)-ph_3\big(3^{2m+1}n\big)\right)q^n\end{aligned}$$ vanishes since $$\begin{aligned} ph_3\big(3^{2m+3}0\big)-ph_3\big(3^{2m+1}0\big)=ph_3(0)-ph_3(0)=0.\end{aligned}$$ Thus, $$\begin{aligned} \label{eq:Phi-hat-0-cong-2} \widehat{\Phi}_{m+1}(0) = 0.\end{aligned}$$ Combining [\[eq:Phi-hat-0-cong-1\]](#eq:Phi-hat-0-cong-1){reference-type="eqref" reference="eq:Phi-hat-0-cong-1"} and [\[eq:Phi-hat-0-cong-2\]](#eq:Phi-hat-0-cong-2){reference-type="eqref" reference="eq:Phi-hat-0-cong-2"} gives $$\begin{aligned} \label{eq:Phi-hat-0-cong-1&2} \widetilde{C}_m(0)-2\widetilde{C}_m(1) \equiv 0 \pmod{3^{m+3}}.\end{aligned}$$ This yields [\[eq:nu-C\'-extra-0\]](#eq:nu-C'-extra-0){reference-type="eqref" reference="eq:nu-C'-extra-0"}. In the meantime, recalling that $\widetilde{C}_m(1)\equiv0\pmod{3^{m+2}}$ by [\[eq:nu-C\'-bound\]](#eq:nu-C'-bound){reference-type="eqref" reference="eq:nu-C'-bound"}, we have $-6\widetilde{C}_m(1)\equiv 0 \pmod{3^{m+3}}$, so that $$\begin{aligned} \widetilde{C}_m(0)-8\widetilde{C}_m(1)\equiv0\pmod{3^{m+3}},\end{aligned}$$ or equivalently, $$\begin{aligned} 3\widetilde{C}_m(0)-24\widetilde{C}_m(1) \equiv 0 \pmod{3^{m+4}}.\end{aligned}$$ Now [\[eq:nu-C\'-extra-2\]](#eq:nu-C'-extra-2){reference-type="eqref" reference="eq:nu-C'-extra-2"} is also confirmed. Putting the above arguments together, we arrive at [\[eq:nu-C-bound-m+1\]](#eq:nu-C-bound-m+1){reference-type="eqref" reference="eq:nu-C-bound-m+1"}, and thus close the requested inductive step. ◻ # Internal congruences for $ps_3(n)$ {#sec:ps-cong-proof} Herein, we define $$\begin{aligned} G(q):=\dfrac{\psi(q^3)}{\psi(q)}.\end{aligned}$$ and introduce another set of auxiliary functions: $$\begin{aligned} \label{eq:delta-def} \delta=\delta(q):= q^{-2}\frac{G(q)}{G(q^9)},\end{aligned}$$ and $$\begin{aligned} \label{eq:zeta-def} \zeta=\zeta(q):= q\frac{\psi(q^9)}{\psi(q)}.\end{aligned}$$ Furthermore, for $i\ge 0$, define $$\begin{aligned} \mathcal{Z}_i:= U\big(\zeta^i\big).\end{aligned}$$ We have the following result that exhibits the resemblance between $U\big(\zeta^i\big)$ and $U\big(\xi^i\big)$, as well as between $U\big(\delta\zeta^i\big)$ and $U\big(\gamma\xi^i\big)$: **Theorem 12**. *For every $i\ge 0$, $$\begin{aligned} \label{eq:Z-coeff} \mathcal{Z}_i = \sum_{j} X_{i}(j)\zeta^j \in \mathbb{Z}[\zeta],\end{aligned}$$ where the coefficients $X_i(j)$ are **identical** to those given in [\[eq:X-coeff\]](#eq:X-coeff){reference-type="eqref" reference="eq:X-coeff"} for $\mathcal{X}_i$. In addition, $$\begin{aligned} \label{eq:U-delta-cZ} U\big(\delta \zeta^i\big) = \zeta^{-1} \mathcal{Z}_{i+1}.\end{aligned}$$* *Proof.* Analogous to Lemma [Lemma 4](#le:xi-1-3){reference-type="ref" reference="le:xi-1-3"}, we have the following initial evaluations for $\mathcal{Z}_i$ by either a cusp analysis or an automated computer-aided verification: $$\begin{aligned} \mathcal{Z}_1 &= \zeta - 3 \zeta^2 + 3 \zeta^3,\label{eq:zeta1}\\ \mathcal{Z}_2 &= -2 \zeta + 9 \zeta^2 - 24 \zeta^3 + 45 \zeta^4 - 54 \zeta^5 + 27 \zeta^6,\label{eq:zeta2}\\ \mathcal{Z}_3 &= \zeta - 12 \zeta^2 + 66 \zeta^3 - 216 \zeta^4 + 486 \zeta^5 - 810 \zeta^6\notag\\ &\quad + 972 \zeta^7 - 729 \zeta^8 + 243 \zeta^9.\label{eq:zeta3}\end{aligned}$$ Particularly, these polynomials on the right are identical to those in [\[eq:xi1\]](#eq:xi1){reference-type="eqref" reference="eq:xi1"}, [\[eq:xi2\]](#eq:xi2){reference-type="eqref" reference="eq:xi2"} and [\[eq:xi3\]](#eq:xi3){reference-type="eqref" reference="eq:xi3"} after replacing $\zeta$ with $\xi$. We may continue to copy the argument built on Newton's identities for Lemma [Lemma 5](#le:cX-rec){reference-type="ref" reference="le:cX-rec"}, and then [\[eq:Z-coeff\]](#eq:Z-coeff){reference-type="eqref" reference="eq:Z-coeff"} becomes plain. Finally, note that $$\begin{aligned} \delta=q^{-2}\dfrac{G(q)}{G(q^9)}=q^{-2}\dfrac{\psi(q^3)}{\psi(q)} \dfrac{\psi(q^9)}{\psi(q^{27})}=q^{-3}\dfrac{\psi(q^3)}{\psi(q^{27})} \cdot q\dfrac{\psi(q^9)}{\psi(q)}=\dfrac{\zeta(q)}{\zeta(q^3)},\end{aligned}$$ which immediately indicates [\[eq:U-delta-cZ\]](#eq:U-delta-cZ){reference-type="eqref" reference="eq:U-delta-cZ"}. ◻ Define, for $M\ge 1$: $$\begin{aligned} \Psi_M(q):=\begin{cases} \displaystyle\dfrac{1}{G(q^3)}\sum_{n\ge 0} ps_3{\left(3^{2m-1}n+\frac{3^{2m}-1}{4}\right)}q^n, &\quad\textrm{if $M=2m-1$},\\ \displaystyle\dfrac{1}{G(q)}\sum_{n\ge 0} ps_3{\left(3^{2m}n+\frac{3^{2m}-1}{4}\right)}q^n, &\quad\textrm{if $M=2m$}. \end{cases}\end{aligned}$$ Then we also have that, for any $m\ge 1$, $$\begin{aligned} \Psi_{2m} &=U\big(\Psi_{2m-1} \big),\label{eq:Psi-even}\\ \Psi_{2m+1} &=U\big(\delta\Psi_{2m} \big).\label{eq:Psi-odd}\end{aligned}$$ In a parallel way, the following result is true: **Theorem 13**. *For any $M\ge 1$, if we write $$\begin{aligned} \Phi_M = \sum_{k} c_m(k) \xi^k \in \mathbb{Z}[\xi]\end{aligned}$$ according to Theorem [Theorem 10](#th:Phi-poly){reference-type="ref" reference="th:Phi-poly"}, then $$\begin{aligned} \Psi_M = \sum_{k} c_m(k) \zeta^k \in \mathbb{Z}[\zeta]\end{aligned}$$ with the **same** coefficients in the two polynomial representations.* We continue to define, for $m\ge 1$, $$\begin{aligned} \widehat{\Psi}_m:=\dfrac{1}{G(q^3)}\sum_{n\ge 0} \left(ps_3\big(3^{2m+1}n+\tfrac{3^{2m+2}-1}{4}\big) -ps_3\big(3^{2m-1}n+\tfrac{3^{2m}-1}{4}\big)\right)q^n.\end{aligned}$$ That is, $$\begin{aligned} \label{eq:Psi-hat-exp-2} \widehat{\Psi}_m = \Psi_{2m+1}-\Psi_{2m-1}.\end{aligned}$$ The following analogy is also clear. **Theorem 14**. *For any $m\ge 1$, $$\begin{aligned} \widehat{\Psi}_m = \sum_{k} C_m(k) \zeta^k \in \mathbb{Z}[\zeta],\end{aligned}$$ where the coefficients $C_m(k)$ are **identical** to those given in [\[eq:Psi-hat-coeff\]](#eq:Psi-hat-coeff){reference-type="eqref" reference="eq:Psi-hat-coeff"} for $\widehat{\Phi}_m$.* Finally, invoking the $3$-adic evaluations for the coefficients $C_m(k)$ in [\[eq:nu-C-bound\]](#eq:nu-C-bound){reference-type="eqref" reference="eq:nu-C-bound"}, Theorem [Theorem 2](#th:ps-cong){reference-type="ref" reference="th:ps-cong"} is immediately established. **Remark 16**. Consulting the resemblance between the strategy for the proofs of Theorems [Theorem 1](#th:ph-cong){reference-type="ref" reference="th:ph-cong"} and [Theorem 2](#th:ps-cong){reference-type="ref" reference="th:ps-cong"}, it is natural to ask if we could proceed in the opposite direction by first presenting a direct proof of the internal congruences for $ps_3(n)$. Clearly, the arguments in Sect. [4](#sec:ph-cong-proof){reference-type="ref" reference="sec:ph-cong-proof"} can be transplanted readily by working instead on $\mathbb{Z}[\zeta]$. However, one issue occurs at the very end. Namely, one crucial property for $\widehat{\Phi}_m$ we have utilized is [\[eq:Phi-hat-0-cong-2\]](#eq:Phi-hat-0-cong-2){reference-type="eqref" reference="eq:Phi-hat-0-cong-2"}: $$\begin{aligned} \widehat{\Phi}_{m}(0)=0.\end{aligned}$$ However, for $\widehat{\Psi}_m$, we only have $$\begin{aligned} \widehat{\Psi}_{m}(0)=ps_3{\left(\frac{3^{2m+2}-1}{4}\right)} -ps_3{\left(\frac{3^{2m}-1}{4}\right)},\end{aligned}$$ the value of which is indefinite. This means that we cannot have an immediate conclusion of a congruence parallel to [\[eq:Phi-hat-0-cong-1&2\]](#eq:Phi-hat-0-cong-1&2){reference-type="eqref" reference="eq:Phi-hat-0-cong-1&2"}. Herein, we leave a question to the interested reader. **Problem 15**. *Find a **direct** proof of the following congruence: $$\begin{aligned} ps_3{\left(\frac{3^{2m}-1}{4}\right)}\equiv ps_3{\left(\frac{3^{2m+2}-1}{4}\right)}\pmod{3^{m+2}},\end{aligned}$$ whenever $m\geq1$.* # Declarations {#declarations .unnumbered} ## Competing interests {#competing-interests .unnumbered} The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. # Acknowledgements {#acknowledgements .unnumbered} Dazhao Tang was partially supported by the National Natural Science Foundation of China (No. 12201093), the Natural Science Foundation Project of Chongqing CSTB (No. CSTB2022NSCQ--MSX0387), the Science and Technology Research Program of Chongqing Municipal Education Commission (No. KJQN202200509) and the Doctoral start-up research Foundation (No. 21XLB038) of Chongqing Normal University. 99 G. E. Andrews, *The Theory of Partitions*, Cambridge University Press, Cambridge, 1998. A. O. L. Atkin, Proof of a conjecture of Ramanujan, *Glasgow Math. J.* **8** (1967), 14--32. H. S. S. Bharadwaj, B. Hemanthkumar, and M. S. M. Naika, On $3$- and $9$-regular overpartitions modulo powers of $3$, *Colloq. Math.* **154** (2018), no. 1, 121--130. S. Chern and J. A. Sellers, An infinite family of internal congruences modulo powers of $2$ for partitions into odd parts with designated summands, submitted. Available at arXiv:2308.04348. S. Cooper, *Ramanujan's Theta Functions*, Springer, Cham, 2017. F. Garvan, A $q$-product tutorial for a $q$-series MAPLE package, *Sém. Lothar. Combin.* **42** (1999), Art. B42d, 27 pp. D. S. Gireesh, M. D. Hirschhorn, and M. S. M. Naika, On $3$-regular partitions with odd parts distinct, *Ramanujan J.* **44** (2017), no. 1, 227--236. J. Lovejoy, Gordon's theorem for overpartitions, *J. Combin. Theory Ser. A* **103** (2003), no. 2, 393--401. D. G. Mead, Newton's identities, *Amer. Math. Monthly* **99** (1992), no. 8, 749--751. C.-S. Radu, An algorithmic approach to Ramanujan--Kolberg identities, *J. Symbolic Comput.* **68** (2015), part 1, 225--253. S. Ramanujan, Some properties of $p(n)$, the number of partitions of $n$, *Proc. Cambridge Philos. Soc.* **19** (1919), 207--210. S. Ramanujan, Congruence properties of partitions, *Math. Z.* **9** (1921), no. 1-2, 147--153. S. Ramanujan, *Notebooks. Vol. 2*, Tata Institute of Fundamental Research, Bombay, 1957. S. Ramanujan, *The Lost Notebook and Other Unpublished Papers*, Narosa Publishing House, New Delhi, 1988. N. A. Smoot, On the computation of identities relating partition numbers in arithmetic progressions with eta quotients: an implementation of Radu's algorithm, *J. Symbolic Comput.* **104** (2021), 276--311. G. N. Watson, Ramanujans Vermutung über Zerfällungszahlen, *J. Reine Angew. Math.* **179** (1938), 97--128.

arxiv_math

--- abstract: | In this paper, we completely classify $3$-dimensional complete self-expanders with constant norm $S$ of the second fundamental form and constant $f_{3}$ in Euclidean space $\mathbb R^{4}$, where $h_{ij}$ are components of the second fundamental form, $S=\sum_{i,j}h^{2}_{ij}$ and $f_{3}=\sum_{i,j,k}h_{ij}h_{jk}h_{ki}$. address: - | Zhi Li\ College of Mathematics and Information Science, Henan Normal University, , Xinxiang, Henan, China. - | Guoxin Wei\ School of Mathematical Sciences, South China Normal University, , Guangzhou, China. author: - Zhi Li and Guoxin Wei title: A rigidity theorem of self-expander --- # introduction Let $X: M^{n}\to \mathbb{R}^{n+p}$ be an $n$-dimensional submanifold in the $(n+p)$-dimensional Euclidean space $\mathbb{R}^{n+p}$. If a family of hypersurfaces for $t \in [0, T)$, satisfies $$\dfrac{\partial X(t)}{\partial t}=\vec{H(t)}, \ \ X(0)=X,$$ it is called mean curvature flow (MCF), where $\vec H(t)$ denotes mean curvature vector. An $n$-dimensional smooth immersed submanifold $X: M^{n}\to \mathbb{R}^{n+p}$ is called self-expanders of MCF if its mean curvature vector $\vec H$ satisfies the equation $$\label{1.1-1} \vec H= X^{\perp},$$ where $X^{\perp}$ and $\vec H$ denote the normal component of the position vector $X$ and mean curvature vector, respectively. $M ^{n}$ is a self-expander if and only if $\sqrt{2t} M ^{n}$ is a MCF, $t\in(0,\infty)$. Self-expanders have a very important role in the study of MCF. They describe the asymptotic longtime behavior for MCF and the local structure of MCF after the singularities in the very short time. In the case of codimension one and under certain assumptions on the initial hypersurface at infinity, Ecker and Huisken [@EH] showed that the solutions of mean curvature flow of entire graphs in euclidean space exist for all times $t>0$ and converges to a self-expander. In [@Sta], Stavrou proved the same result under weaker hypotheses that the initial hypersurface has a unique tangent cone at infinity. Self-expanders also appear in the mean curvature evolution of cones. Ilmanen [@Ilm] studied the existence of E-minimizing self-expanding hypersurfaces which converge to prescribed closed cones at infinity in Euclidean space. Ding [@D] studied self-expanders and their relationship to minimal cones in Euclidean space. Bernstein and Wang ([@BW1] and [@BW2]) obtained various results on asymptotically conical self-expanders. In [@CZ], Cheng and Zhou studied some properties of complete properly immersed self-expanders and proved some results related to the spectrum of the drifted Laplacian for self-expanders in higher codimension. Recently, Ancari and Cheng [@AC] mainly studied self-expander hypersurfaces in Euclidean space whose mean curvatures have some linear growth controls. Ishimura [@Ish] and Halldorsson [@H] have completed the classification problem of self-expander curves in $\mathbb{R}^{2}$. Ancari and Cheng [@AC] prove that the surfaces $\Gamma\times \mathbb{R}$ with the product metric are the only complete self-expander surfaces immersed in $\mathbb{R}^{3}$ with constant scalar curvature, where $\Gamma$ is a complete self-expander curve immersed in $\mathbb{R}^{2}$. There are many other works about self-expanders (see [@AIC], [@FM], [@Smo], [@XY] and references therein). Recently, by studying the infimum of $H^{2}$, Li and Wei [@LW2] gave a classification for complete self-expander surfaces immersed in $\mathbb{R}^{3}$ with constant squared norm of the second fundamental form by making use of the generalized maximum principle. For the higher dimension $n$, it is not easy to classify self-expander in Euclidean space with constant squared norm of the second fundamental form. In this paper, for $n=3$, by studying the infimum of $H$, we prove **Theorem 1**. *Let $X: M^{3}\to \mathbb{R}^{4}$ be a $3$-dimensional complete self-expanders in $\mathbb R^{4}$. If the squared norm $S$ of the second fundamental form and $f_{3}$ are constant, then $X: M^{3}\to \mathbb{R}^{4}$ is a hyperplane $\mathbb {R}^{3}$ through the origin, where $h_{ij}$ are components of the second fundamental form, $S=\sum_{i,j}h^{2}_{ij}$ and $f_{3}=\sum_{i,j,k}h_{ij}h_{jk}h_{ki}$.* # Preliminaries Let $X: M^{n} \rightarrow\mathbb{R}^{n+1}$ be an $n$-dimensional hypersurface of $(n+1)$-dimensional Euclidean space $\mathbb{R}^{n+1}$. We choose a local orthonormal frame field $\{e_{A}\}_{A=1}^{n+1}$ in $\mathbb{R}^{n+1}$ with dual coframe field $\{\omega_{A}\}_{A=1}^{n+1}$, such that, restricted to $M^{n}$, $e_{1},\cdots, e_{n}$ are tangent to $M^{n}$. From now on, we use the following conventions on the ranges of indices, $$1\leq i,j,k,l\leq n.$$ $\sum_{i}$ means taking summation from $1$ to $n$ for $i$. Then we have $$dX=\sum_i\limits \omega_{i} e_{i}, \ \ de_{i}=\sum_{j}\limits \omega_{ij}e_{j}+\omega_{i n+1}e_{n+1},$$ $$de_{n+1}=\omega_{n+1 i}e_{i}, \ \ \omega_{n+1i}=-\omega_{in+1},$$ where $\omega_{ij}$ is the Levi-Civita connection of the hypersurface. By restricting these forms to $M$, we get $$\label{2.1-1} \omega_{n+1}=0.$$ Taking exterior derivatives of [\[2.1-1\]](#2.1-1){reference-type="eqref" reference="2.1-1"}, we obtain $$\label{2.1-2} \omega_{in+1}=\sum_{j} h_{ij}\omega_{j},\quad h_{ij}=h_{ji}.$$ $$h=\sum_{i,j}h_{ij}\omega_i\otimes\omega_j, \ \ H= \sum_i\limits h_{ii}$$ are called second fundamental form and mean curvature of $X: M\rightarrow\mathbb{R}^{n+1}$, respectively. Let $S=\sum_{i,j}\limits (h_{ij})^2$ be the squared norm of the second fundamental form of $X: M\rightarrow\mathbb{R}^{n+1}$. The induced structure equations of $M$ are given by $$d\omega_{i}=\sum_j \omega_{ij}\wedge\omega_j, \quad \omega_{ij}=-\omega_{ji},$$ $$d\omega_{ij}=\sum_k \omega_{ik}\wedge\omega_{kj}-\frac12\sum_{k,l} R_{ijkl} \omega_{k}\wedge\omega_{l},$$ where $R_{ijkl}$ denote components of the curvature tensor of the hypersurface. Hence, the Gauss equations are given by $$\label{2.1-3} R_{ijkl}=h_{ik}h_{jl}-h_{il}h_{jk}.$$ Defining the covariant derivative of $h_{ij}$ by $$\label{2.1-4} \sum_{k}h_{ijk}\omega_k=dh_{ij}+\sum_kh_{ik}\omega_{kj} +\sum_k h_{kj}\omega_{ki}.$$ we obtain the Codazzi equations $$\label{2.1-5} h_{ijk}=h_{ikj}.$$ By taking exterior differentiation of [\[2.1-4\]](#2.1-4){reference-type="eqref" reference="2.1-4"}, and defining $$\label{2.1-6} \sum_lh_{ijkl}\omega_l=dh_{ijk}+\sum_lh_{ljk}\omega_{li} +\sum_lh_{ilk}\omega_{lj}+\sum_l h_{ijl}\omega_{lk},$$ we have the following Ricci identities $$\label{2.1-7} h_{ijkl}-h_{ijlk}=\sum_m h_{mj}R_{mikl}+\sum_m h_{im}R_{mjkl}.$$ Defining $$\label{2.1-8} \begin{aligned} \sum_mh_{ijklm}\omega_m&=dh_{ijkl}+\sum_mh_{mjkl}\omega_{mi} +\sum_mh_{imkl}\omega_{mj}+\sum_mh_{ijml}\omega_{mk}\\ &\ \ +\sum_mh_{ijkm}\omega_{ml} \end{aligned}$$ and taking exterior differentiation of [\[2.1-6\]](#2.1-6){reference-type="eqref" reference="2.1-6"}, we get $$\label{2.1-9} \begin{aligned} h_{ijklq}-h_{ijkql}&=\sum_{m} h_{mjk}R_{milq} + \sum_{m}h_{imk}R_{mjlq}+ \sum_{m}h_{ijm}R_{mklq}. \end{aligned}$$ The $\mathcal{L}$-operator is defined by $$\mathcal{L}f=\Delta f+\langle X,\nabla f\rangle,$$ where $\Delta$ and $\nabla$ denote the Laplacian and the gradient operator, respectively. We define the function $f_{3}$ as follows: $$f_{3}=\sum_{i,j,k}h_{ij}h_{jk}h_{ki},$$ then we get $$\label{2.1-10} \aligned &\nabla_{l}f_{3}=3\sum_{i,j,k}h_{ijl}h_{jk}h_{ki}, \\ &\nabla_{p}\nabla_{l}f_{3}=3\sum_{i,j,k}h_{ijlp}h_{jk}h_{ki}+6\sum_{i,j,k}h_{ijl}h_{jkp}h_{ki}, \ \ \ l,p=1, 2, \cdots, n. \endaligned$$ We next suppose that $X: M\rightarrow\mathbb{R}^{n+1}$ is a self-expanders, that is, $H=\langle X, e_{n+1}\rangle$. By a simple calculation, we have the following basic formulas. $$\label{2.1-11} \aligned \nabla_{i}H =&-\sum_{k}h_{ik}\langle X, e_{k}\rangle, \\ \nabla_{j}\nabla_{i}H =&-\sum_{k}h_{ijk}\langle X, e_{k}\rangle-h_{ij}-H\sum_{k}h_{ik}h_{kj}. \endaligned$$ Using the above formulas and the Ricci identities, we can get the following Lemma: **Lemma 1**. *Let $X:M^n\rightarrow \mathbb{R}^{n+1}$ be an $n$-dimensional complete self-expanders in $\mathbb R^{n+1}$. We have $$\label{2.1-12} \mathcal{L}H=-H(S+1),$$ $$\label{2.1-13} \frac{1}{2}\mathcal{L}S =\sum_{i,j,k}h_{ijk}^{2}-S(S+1),$$ $$\label{2.1-14} \frac{1}{3}\mathcal{L}f_{3} =2\sum_{i,j,k}h_{ijp}h_{jkp}h_{ki}-f_{3}(S+1),$$* **Lemma 2**. *Let $X:M^{n}\rightarrow \mathbb{R}^{n+1}$ be an $n$-dimensional complete self-expander in $\mathbb R^{n+1}$. If $S$ is constant, we have $$\label{2.1-15} \aligned &\sum_{i,j,k,l}(h_{ijkl})^{2}=(S+2)\sum_{i,j,k}(h_{ijk})^{2}\\ &-6\sum_{i,j,k,l,p}h_{ijk}h_{il}h_{jp}h_{klp} +3\sum_{i,j,k,l,p}h_{ijk}h_{ijl}h_{kp}h_{lp}. \endaligned$$* *Proof.* By making use of the Ricci identities [\[2.1-7\]](#2.1-7){reference-type="eqref" reference="2.1-7"}, [\[2.1-9\]](#2.1-9){reference-type="eqref" reference="2.1-9"} and a direct calculation, we can obtain [\[2.1-15\]](#2.1-15){reference-type="eqref" reference="2.1-15"}. ◻ In order to prove our results, we need the following generalized maximum principle due to Omori [@O] and Yau [@Y]. **Lemma 3**. *Let $(M^{n},g)$ be a complete Riemannian manifold with Ricci curvature bounded from below. For a $C^{2}$-function $f$ bounded from above, there exists a sequence of points $\{p_{t}\}\in M^{n}$, such that $$\lim_{t\rightarrow\infty} f(p_{t})=\sup f,\quad \lim_{t\rightarrow\infty} |\nabla f|(p_{t})=0,\quad \limsup_{t\rightarrow\infty}\Delta f(p_{t})\leq 0.$$* # Proof of Theorem 1.1 As we all know, if $S\equiv0$, $X:M^{3}\rightarrow \mathbb{R}^{4}$ is $\mathbb{R}^{3}$. Next, we consider that $S>0$. We will prove the following proposition. **Proposition 1**. *For a $3$-dimensional complete self-expander $X:M^{3}\rightarrow \mathbb{R}^{4}$ with non-zero constant squared norm $S$ of the second fundamental form, if $f_{3}$ is constant, we have that $\sup H=\frac{3f_{3}}{2S}$.* *Proof.* We choose $e_{1}$ and $e_{2}$, at each point $p\in M^{3}$, such that $$h_{ij}=\lambda_i\delta_{ij}.$$ From the definitions of $S$ and $H$, we obtain $$H^{2}\leq 3S.$$ Since $S$ is constant, from the Gauss equations, we know that the Ricci curvature of $X:M^{3} \to \mathbb{R}^{4}$ is bounded from below. By applying the generalized maximum principle to the function $H$. Thus, there exists a sequence $\{p_{t}\}$ in $M^{3}$ such that $$\label{3.1-1} \lim_{t\rightarrow\infty} H(p_{t})=\sup H,\quad \lim_{t\rightarrow\infty} |\nabla H|(p_{t})=0,\quad \limsup_{t\rightarrow\infty} \Delta H(p_{t})\leq 0.$$ Since $S$ is constant, by [\[2.1-13\]](#2.1-13){reference-type="eqref" reference="2.1-13"} and [\[2.1-15\]](#2.1-15){reference-type="eqref" reference="2.1-15"}, we know that $\{h_{ij}(p_{t})\}$, $\{h_{ijk}(p_{t})\}$ and $\{h_{ijkl}(p_{t})\}$ are bounded sequences for $i, j, k, l = 1,2,3$. Hence, we can assume that they convergence if necessary, taking a subsequence. $$\begin{aligned} &\lim_{t\rightarrow\infty}H(p_{t})=\bar H, \ \ \lim_{t\rightarrow\infty}h_{ij}(p_{t})=\bar h_{ij}=\bar \lambda_i\delta_{ij}, \\ &\lim_{t\rightarrow\infty}h_{ijk}(p_{t})=\bar h_{ijk}, \ \ \lim_{t\rightarrow\infty}h_{ijkl}(p_{t})=\bar h_{ijkl}, \ \ i, j, k,l=1, 2, 3. \end{aligned}$$ By taking exterior derivative of $H$, from [\[3.1-1\]](#3.1-1){reference-type="eqref" reference="3.1-1"} and by taking limits, we know $$\label{3.1-2} \bar H_{,i}=\bar h_{11i}+\bar h_{22i}+\bar h_{33i}=0, \ \ i=1, 2, 3.$$ According to the definition of the self-expander, [\[3.1-1\]](#3.1-1){reference-type="eqref" reference="3.1-1"} yields $$H_{,i}=-\sum_{k}h_{ik}\langle X, e_{k}\rangle, \ \ \ \ i=1, 2, 3,$$ $$H_{,ij}=-\sum_{k}h_{ijk}\langle X, e_{k}\rangle-h_{ij}-H\sum_{k}h_{ik}h_{kj}, \ \ i,j=1, 2, 3.$$ Thus, we get $$\label{3.1-3} \bar H_{,i}=\bar h_{11i}+\bar h_{22i}+\bar h_{33i}=-\bar \lambda_{i}\lim_{t\rightarrow\infty} \langle X, e_{i} \rangle(p_{t}),\ \ \ \ i=1, 2, 3$$ and $$\label{3.1-4} \begin{cases} \begin{aligned} &\bar H_{,ii}=-\sum_{k}\bar h_{iik}\lim_{t\rightarrow\infty} \langle X, e_{k} \rangle(p_{t})-\bar \lambda_{i}-\bar H\bar \lambda^{2}_{i},\ \ i=1,2,3, \\ &\bar H_{,ij}=-\sum_{k}\bar h_{ijk}\lim_{t\rightarrow\infty} \langle X, e_{k} \rangle(p_{t}),\ \ i\neq j, \ \ i,j=1,2,3. \end{aligned} \end{cases}$$ Since $S$ is constant, we obtain $$\sum_{i,j}h_{ij}h_{ijk}=0, \ \ \ k=1, 2, 3,$$ $$\sum_{i,j}h_{ij}h_{ijkl}+\sum_{i,j}h_{ijk}h_{ijl}=0, \ \ \ k,l=1, 2, 3.$$ Under the processing by taking limits, we have $$\label{3.1-5} \bar\lambda_{1}\bar h_{11k}+\bar\lambda_{2}\bar h_{22k}+\bar\lambda_{3}\bar h_{33k}=0, \ \ \ k=1, 2, 3$$ and $$\label{3.1-6} \begin{cases} \begin{aligned} &\sum_{i}\bar \lambda_{i}\bar h_{iikk}=-\sum_{i,j}\bar h^{2}_{ijk}, \ \ \ k=1, 2, 3, \\ &\sum_{i}\bar \lambda_{i}\bar h_{iikl}=-\sum_{i,j}\bar h_{ijk}\bar h_{ijl}, \ \ k\neq l, \ \ \ k,l=1, 2, 3. \end{aligned} \end{cases}$$ It follows from Ricci identities [\[2.1-7\]](#2.1-7){reference-type="eqref" reference="2.1-7"} that $$\bar h_{ijkl}-\bar h_{ijlk}=\bar\lambda_{i}\bar\lambda_{j}\bar\lambda_{k}\delta_{il}\delta_{jk} -\bar\lambda_{i}\bar\lambda_{j}\bar\lambda_{l}\delta_{ik}\delta_{jl} +\bar\lambda_{i}\bar\lambda_{j}\bar\lambda_{k}\delta_{ik}\delta_{jl} -\bar\lambda_{i}\bar\lambda_{j}\bar\lambda_{l}\delta_{il}\delta_{jk}.$$ That is, $$\label{3.1-7} \begin{cases} \begin{aligned} & \bar h_{1122}-\bar h_{2211}=\bar \lambda_{1}\bar \lambda_{2}(\bar \lambda_{1}-\bar \lambda_{2}),\ \ \bar h_{1133}-\bar h_{3311}=\bar \lambda_{1}\bar \lambda_{3}(\bar \lambda_{1}-\bar \lambda_{3}),\\ &\bar h_{2233}-\bar h_{3322}=\bar \lambda_{2}\bar \lambda_{3}(\bar \lambda_{2}-\bar \lambda_{3}), \ \ \bar h_{iikl}-\bar h_{iilk}=0, \ \ \ i,k,l=1, 2, 3. \end{aligned} \end{cases}$$ Since $f_{3}$ is constant, by [\[2.1-10\]](#2.1-10){reference-type="eqref" reference="2.1-10"}, we know that $$\label{3.1-8} \bar \lambda^{2}_{1}\bar h_{11k}+\bar \lambda^{2}_{2}\bar h_{22k}+\bar \lambda^{2}_{3}\bar h_{33k}=0, \ \ k=1, 2, 3$$ and $$\label{3.1-9} \begin{cases} \begin{aligned} &\sum_{i}\bar \lambda^{2}_{i}\bar h_{iikk}=-2\sum_{i,j}\bar \lambda_{i}\bar h^{2}_{ijk}, \ \ \ k=1, 2, 3, \\ &\sum_{i}\bar \lambda^{2}_{i}\bar h_{iikl}=-2\sum_{i,j}\bar \lambda_{i}\bar h_{ijk}\bar h_{ijl}, \ \ k\neq l, \ \ \ k,l=1, 2, 3. \end{aligned} \end{cases}$$ From the above equations [\[3.1-2\]](#3.1-2){reference-type="eqref" reference="3.1-2"} to [\[3.1-9\]](#3.1-9){reference-type="eqref" reference="3.1-9"}, we can prove the following claim that $$\sup H=\bar H=\frac{3f_{3}}{2S}.$$ In fact, if $\bar \lambda_1=\bar \lambda_2 =\bar \lambda_3$, from [\[3.1-6\]](#3.1-6){reference-type="eqref" reference="3.1-6"} and [\[3.1-9\]](#3.1-9){reference-type="eqref" reference="3.1-9"}, we have $$\begin{cases} \begin{aligned} \bar \lambda_{1}\sum_{i}\bar h_{ii11} =&-(\bar h^{2}_{111}+\bar h^{2}_{221}+\bar h^{2}_{331})-2(\bar h^{2}_{112}+\bar h^{2}_{113}+\bar h^{2}_{123}),\\ \bar \lambda_{1}\sum_{i}\bar h_{ii22} =&-(\bar h^{2}_{112}+\bar h^{2}_{222}+\bar h^{2}_{332})-2(\bar h^{2}_{221}+\bar h^{2}_{223}+\bar h^{2}_{123}),\\ \bar \lambda_{1}\sum_{i}\bar h_{ii33} =&-(\bar h^{2}_{113}+\bar h^{2}_{223}+\bar h^{2}_{333})-2(\bar h^{2}_{331}+\bar h^{2}_{332}+\bar h^{2}_{123}) \end{aligned} \end{cases}$$ and $$\begin{cases} \begin{aligned} &\bar \lambda^{2}_{1}\sum_{i}\bar h_{ii11} =-2\bar \lambda_{1}(\bar h^{2}_{111}+\bar h^{2}_{221}+\bar h^{2}_{331})-4\bar \lambda_{1}(\bar h^{2}_{112}+\bar h^{2}_{113}+\bar h^{2}_{123}),\\ &\bar \lambda^{2}_{1}\sum_{i}\bar h_{ii22} =-2\bar \lambda_{1}(\bar h^{2}_{112}+\bar h^{2}_{222}+\bar h^{2}_{332})-4\bar \lambda_{1}(\bar h^{2}_{221}+\bar h^{2}_{223}+\bar h^{2}_{123}),\\ &\bar \lambda^{2}_{1}\sum_{i}\bar h_{ii33} =-2\bar \lambda_{1}(\bar h^{2}_{113}+\bar h^{2}_{223}+\bar h^{2}_{333})-4\bar \lambda_{1}(\bar h^{2}_{331}+\bar h^{2}_{332}+\bar h^{2}_{123}). \end{aligned} \end{cases}$$ Thus, we infer $$\bar h_{ijk}=0, \ \ i,j,k=1, 2, 3.$$ Then by [\[2.1-13\]](#2.1-13){reference-type="eqref" reference="2.1-13"}, we know that $S=0$. It is a contradiction. If two of $\bar \lambda_1$, $\bar \lambda_2$ and $\bar \lambda_3$ are equal, without loss of generality, we assume that $\bar \lambda_{1}\neq \bar \lambda_{2}=\bar \lambda_{3}$. Then we get from [\[3.1-2\]](#3.1-2){reference-type="eqref" reference="3.1-2"} and [\[3.1-5\]](#3.1-5){reference-type="eqref" reference="3.1-5"} that $$\label{3.1-10} \bar h_{11k}=0, \ \ \bar h_{22k}+\bar h_{33k}=0,\ \ \ k=1, 2, 3.$$ If $\bar \lambda_{1}=0$, then $\bar \lambda_{2}=\bar \lambda_{3} \neq 0$ since $S\neq 0$. By making use of equations [\[3.1-6\]](#3.1-6){reference-type="eqref" reference="3.1-6"}, [\[3.1-9\]](#3.1-9){reference-type="eqref" reference="3.1-9"} and [\[3.1-10\]](#3.1-10){reference-type="eqref" reference="3.1-10"}, we know that $$\begin{cases} \begin{aligned} &\bar \lambda_{2}(\bar h_{2211}+\bar h_{3311})=-2(\bar h^{2}_{221}+\bar h^{2}_{123}),\\ &\bar \lambda_{2}(\bar h_{2222}+\bar h_{3322})=-2(\bar h^{2}_{222}+\bar h^{2}_{223})-2(\bar h^{2}_{221}+\bar h^{2}_{123}) \end{aligned} \end{cases}$$ and $$\begin{cases} \begin{aligned} \bar \lambda^{2}_{2}(\bar h_{2211}+\bar h_{3311})=&-4\bar\lambda_{2}(\bar h^{2}_{221}+\bar h^{2}_{123}),\\ \bar \lambda^{2}_{2}(\bar h_{2222}+\bar h_{3322})=&-4\bar\lambda_{2}(\bar h^{2}_{222}+\bar h^{2}_{223})-2\bar\lambda_{2}(\bar h^{2}_{221}+\bar h^{2}_{123}). \end{aligned} \end{cases}$$ Hence, we get $$\bar h_{221}=\bar h_{123}=0, \ \ \bar h_{222}=\bar h_{223}=0.$$ Namely, we obtain $$\bar h_{ijk}=0, \ \ i,j,k=1, 2, 3.$$ Then by [\[2.1-13\]](#2.1-13){reference-type="eqref" reference="2.1-13"}, we know that $S=0$. It is a contradiction. If $\bar \lambda_{1}\neq0, \ \ \bar \lambda_{2}=\bar \lambda_{3}=0$, combining it with [\[3.1-6\]](#3.1-6){reference-type="eqref" reference="3.1-6"}, [\[3.1-9\]](#3.1-9){reference-type="eqref" reference="3.1-9"} and [\[3.1-10\]](#3.1-10){reference-type="eqref" reference="3.1-10"}, we have $$\begin{cases} \begin{aligned} &\bar \lambda_{1}\bar h_{1111} =-2(\bar h^{2}_{221}+\bar h^{2}_{123}),\\ &\bar \lambda_{1}\bar h_{1122} =-2(\bar h^{2}_{222}+\bar h^{2}_{223})-2(\bar h^{2}_{221}+\bar h^{2}_{123}) \end{aligned} \end{cases}$$ and $$\begin{cases} \begin{aligned} &\bar \lambda^{2}_{1}\bar h_{1111}=0,\\ &\bar \lambda^{2}_{1}\bar h_{1122}=-2\bar \lambda_{1}(\bar h^{2}_{221}+\bar h^{2}_{123}). \end{aligned} \end{cases}$$ We infer $$\bar h_{221}=\bar h_{123}=0, \ \ \bar h_{222}=\bar h_{223}=0.$$ Thus, we conclude $$\bar h_{ijk}=0, \ \ i,j,k=1, 2, 3.$$ Then by [\[2.1-13\]](#2.1-13){reference-type="eqref" reference="2.1-13"}, we know that $S=0$. It is a contradiction. If $\bar \lambda_{1}\neq 0, \ \ \bar \lambda_{2}=\bar \lambda_{3} \neq 0$, combining [\[3.1-2\]](#3.1-2){reference-type="eqref" reference="3.1-2"} with [\[3.1-3\]](#3.1-3){reference-type="eqref" reference="3.1-3"}, we obtain $$\label{3.1-11} \lim_{t\rightarrow\infty} \langle X, e_{k} \rangle(p_{t})=0,\ \ \ k=1, 2, 3.$$ It follows from [\[3.1-4\]](#3.1-4){reference-type="eqref" reference="3.1-4"}, [\[3.1-6\]](#3.1-6){reference-type="eqref" reference="3.1-6"}, [\[3.1-9\]](#3.1-9){reference-type="eqref" reference="3.1-9"}, [\[3.1-10\]](#3.1-10){reference-type="eqref" reference="3.1-10"} and [\[3.1-11\]](#3.1-11){reference-type="eqref" reference="3.1-11"} that $$\label{3.1-12} \begin{cases} \begin{aligned} &\bar h_{1111}+\bar h_{2211}+\bar h_{3311}=-\bar \lambda_{1}-\bar H\bar \lambda^{2}_{1},\\ &\bar h_{1122}+\bar h_{2222}+\bar h_{3322}=-\bar \lambda_{2}-\bar H\bar \lambda^{2}_{2},\\ &\bar h_{1112}+\bar h_{2212}+\bar h_{3312}= 0,\\ &\bar h_{1113}+\bar h_{2213}+\bar h_{3313}= 0, \end{aligned} \end{cases}$$ $$\label{3.1-13} \begin{cases} \begin{aligned} \bar \lambda_{1}\bar h_{1111}+\bar \lambda_{2}(\bar h_{2211}+\bar h_{3311})&=-2(\bar h^{2}_{221}+\bar h^{2}_{123}),\\ \bar \lambda_{1}\bar h_{1122}+\bar \lambda_{2}(\bar h_{2222}+\bar h_{3322})&=-2(\bar h^{2}_{221}+\bar h^{2}_{123})-2(\bar h^{2}_{222}+\bar h^{2}_{223}) ,\\ \bar \lambda_{1}\bar h_{1112}+\bar \lambda_{2}(\bar h_{2212}+\bar h_{3312})&=-2(\bar h_{221}\bar h_{222}+\bar h_{223}\bar h_{123}),\\ \bar \lambda_{1}\bar h_{1113}+\bar \lambda_{2}(\bar h_{2213}+\bar h_{3313})&=-2(\bar h_{221}\bar h_{223}-\bar h_{222}\bar h_{123}) \end{aligned} \end{cases}$$ and $$\label{3.1-14} \begin{cases} \begin{aligned} \bar \lambda^{2}_{1}\bar h_{1111}+\bar \lambda^{2}_{2}(\bar h_{2211}+\bar h_{3311})=&-4\bar\lambda_{2}(\bar h^{2}_{221}+\bar h^{2}_{123}),\\ \bar \lambda^{2}_{1}\bar h_{1122}+\bar \lambda^{2}_{2}(\bar h_{2222}+\bar h_{3322})=&-2(\bar\lambda_{1}+\bar\lambda_{2})(\bar h^{2}_{221}+\bar h^{2}_{123})\\ &-4\bar\lambda_{2}(\bar h^{2}_{222}+\bar h^{2}_{223}),\\ \bar \lambda^{2}_{1}\bar h_{1112}+\bar \lambda^{2}_{2}(\bar h_{2212}+\bar h_{3312})=&-4\bar\lambda_{2}(\bar h_{221}\bar h_{222}+\bar h_{223}\bar h_{123}),\\ \bar \lambda^{2}_{1}\bar h_{1113}+\bar \lambda^{2}_{2}(\bar h_{2213}+\bar h_{3313})=&-4\bar\lambda_{2}(\bar h_{221}\bar h_{223}-\bar h_{222}\bar h_{123}). \end{aligned} \end{cases}$$ Besieds, [\[3.1-12\]](#3.1-12){reference-type="eqref" reference="3.1-12"} yields $$\label{3.1-15} \bar h_{2212}+\bar h_{3312}= -\bar h_{1112},\ \ \bar h_{2213}+\bar h_{3313}= -\bar h_{1113}.$$ Inserting [\[3.1-15\]](#3.1-15){reference-type="eqref" reference="3.1-15"} into [\[3.1-13\]](#3.1-13){reference-type="eqref" reference="3.1-13"} and [\[3.1-14\]](#3.1-14){reference-type="eqref" reference="3.1-14"}, we derive to $$\label{3.1-16} \bar h_{221}\bar h_{222}+\bar h_{223}\bar h_{123}=0, \ \ \bar h_{221}\bar h_{223}-\bar h_{222}\bar h_{123}=0.$$ It follows from [\[3.1-13\]](#3.1-13){reference-type="eqref" reference="3.1-13"} and [\[3.1-14\]](#3.1-14){reference-type="eqref" reference="3.1-14"} that $$\label{3.1-17} \begin{cases} \begin{aligned} &\bar \lambda_{1}(\bar \lambda_{2}-\bar \lambda_{1})\bar h_{1111}= 2\bar\lambda_{2}(\bar h^{2}_{221}+\bar h^{2}_{123}),\\ &\bar \lambda_{1}(\bar \lambda_{2}-\bar \lambda_{1})\bar h_{1122}= 2\bar\lambda_{2}(\bar h^{2}_{222}+\bar h^{2}_{223})+2\bar\lambda_{1}(\bar h^{2}_{221}+\bar h^{2}_{123}). \end{aligned} \end{cases}$$ The following situations can be obtained from [\[3.1-16\]](#3.1-16){reference-type="eqref" reference="3.1-16"}: $\bar h^{2}_{221}+\bar h^{2}_{123} =0$, or $\bar h^{2}_{222}+\bar h^{2}_{223}=0$. If both $\bar h^{2}_{221}+\bar h^{2}_{123}=0$ and $\bar h^{2}_{222}+\bar h^{2}_{223} =0$, we have $$\bar h_{ijk}=0, \ \ i,j,k=1, 2, 3.$$ Then by [\[2.1-13\]](#2.1-13){reference-type="eqref" reference="2.1-13"}, we know that $S=0$. It is a contradiction. If $\bar h^{2}_{221}+\bar h^{2}_{123}=0$ and $\bar h^{2}_{222}+\bar h^{2}_{223} \neq 0$, by making use of [\[3.1-17\]](#3.1-17){reference-type="eqref" reference="3.1-17"}, we have $$\bar \lambda_{1}(\bar \lambda_{2}-\bar \lambda_{1})\bar h_{1111}= 0,\ \ \bar \lambda_{1}(\bar \lambda_{2}-\bar \lambda_{1})\bar h_{1122}= 2\bar\lambda_{2}(\bar h^{2}_{222}+\bar h^{2}_{223}).$$ Thus, $$\label{3.1-18} \bar h_{1111}= 0, \ \ \bar h_{1122}=\frac{2\bar\lambda_{2}}{\bar \lambda_{1}(\bar \lambda_{2}-\bar \lambda_{1})}(\bar h^{2}_{222}+\bar h^{2}_{223}).$$ Noting that $\bar h_{1111}=0$ and by the first equation of [\[3.1-13\]](#3.1-13){reference-type="eqref" reference="3.1-13"}, we know $$\bar h_{2211}+\bar h_{3311}=0, \ \ \bar H_{,11}=0.$$ Hence, by the first equation of [\[3.1-12\]](#3.1-12){reference-type="eqref" reference="3.1-12"}, we have $$\label{3.1-19} \bar \lambda_{1}\bar H+1=0.$$ Combining the second equation of [\[3.1-12\]](#3.1-12){reference-type="eqref" reference="3.1-12"}, the second equation of [\[3.1-13\]](#3.1-13){reference-type="eqref" reference="3.1-13"} with [\[3.1-18\]](#3.1-18){reference-type="eqref" reference="3.1-18"}, we know $$\begin{aligned} -2(\bar h^{2}_{222}+\bar h^{2}_{223})&=\bar \lambda_{1}\bar h_{1122}+\bar \lambda_{2}\Big(-\bar \lambda_{2}-\bar H\bar \lambda^{2}_{2}-\bar h_{1122}\Big) \\ &=\bar \lambda_{2}\Big(-\bar \lambda_{2}-\bar H\bar \lambda^{2}_{2}\Big)+(\bar \lambda_{1}-\bar \lambda_{2}) \cdot \frac{2\bar\lambda_{2}}{\bar \lambda_{1}(\bar \lambda_{2}-\bar \lambda_{1})}(\bar h^{2}_{222}+\bar h^{2}_{223}) \\ &=\bar \lambda_{2}\Big(-\bar \lambda_{2}-\bar H\bar \lambda^{2}_{2}\Big)-\frac{2\bar\lambda_{2}}{\bar \lambda_{1}}(\bar h^{2}_{222}+\bar h^{2}_{223}). \end{aligned}$$ Then by[\[3.1-19\]](#3.1-19){reference-type="eqref" reference="3.1-19"}, we obtain $$\label{3.1-20} \bar h^{2}_{222}+\bar h^{2}_{223}=\frac{\bar \lambda^{2}_{2}}{2}.$$ According to [\[2.1-13\]](#2.1-13){reference-type="eqref" reference="2.1-13"} and [\[3.1-20\]](#3.1-20){reference-type="eqref" reference="3.1-20"}, we have that $$\sum_{i,j,k}h_{ijk}^{2}=S(S+1), \ \ \sum_{i,j,k}h_{ijk}^{2}=4(\bar h^{2}_{222}+\bar h^{2}_{223})=2\bar \lambda^{2}_{2}.$$ Thus, $$S(S+1)=2\bar \lambda^{2}_{2}<\bar \lambda^{2}_{1}+2\bar \lambda^{2}_{2}=S.$$ It is impossible. If $\bar h^{2}_{221}+\bar h^{2}_{123} \neq 0$ and $\bar h^{2}_{222}+\bar h^{2}_{223}=0$, according to [\[3.1-17\]](#3.1-17){reference-type="eqref" reference="3.1-17"} , we have $$\label{3.1-21} \begin{aligned} &\bar \lambda_{1}(\bar \lambda_{2}-\bar \lambda_{1})\bar h_{1111}= 2\bar\lambda_{2}(\bar h^{2}_{221}+\bar h^{2}_{123}),\\ &\bar \lambda_{1}(\bar \lambda_{2}-\bar \lambda_{1})\bar h_{1122}= 2\bar\lambda_{1}(\bar h^{2}_{221}+\bar h^{2}_{123}). \end{aligned}$$ Thus, $$\label{3.1-22} \bar h_{1111}= \frac{2\bar\lambda_{2}}{\bar \lambda_{1}(\bar \lambda_{2}-\bar \lambda_{1})}(\bar h^{2}_{221}+\bar h^{2}_{123}), \ \ \bar h_{1122}=\frac{2}{\bar \lambda_{2}-\bar \lambda_{1}}(\bar h^{2}_{221}+\bar h^{2}_{123}).$$ Combining the second equation of [\[3.1-12\]](#3.1-12){reference-type="eqref" reference="3.1-12"}, the second equation of [\[3.1-13\]](#3.1-13){reference-type="eqref" reference="3.1-13"} with [\[3.1-22\]](#3.1-22){reference-type="eqref" reference="3.1-22"}, we know $$\begin{aligned} -2(\bar h^{2}_{221}+\bar h^{2}_{123})&=\bar \lambda_{1}\bar h_{1122}+\bar \lambda_{2}\big(-\bar \lambda_{2}-\bar H\bar \lambda^{2}_{2}-\bar h_{1122}\big) \\ &=\bar \lambda_{2}\big(-\bar \lambda_{2}-\bar H\bar \lambda^{2}_{2}\big)+(\bar \lambda_{1}-\bar \lambda_{2})\bar h_{1122} \\ &=-\bar \lambda^{2}_{2}\big(1+\bar H\bar \lambda_{2}\big)-2(\bar h^{2}_{221}+\bar h^{2}_{123}). \end{aligned}$$ Hence, $$\label{3.1-23} \bar \lambda_{2}\bar H+1=0.$$ By making use of the first equation of [\[3.1-12\]](#3.1-12){reference-type="eqref" reference="3.1-12"}, the first equation of [\[3.1-13\]](#3.1-13){reference-type="eqref" reference="3.1-13"}, [\[3.1-22\]](#3.1-22){reference-type="eqref" reference="3.1-22"} and [\[3.1-23\]](#3.1-23){reference-type="eqref" reference="3.1-23"}, we know $$\begin{aligned} -2(\bar h^{2}_{221}+\bar h^{2}_{123})&=\bar \lambda_{1}\bar h_{1111}+\bar \lambda_{2}\big(-\bar \lambda_{1}-\bar H\bar \lambda^{2}_{1}-\bar h_{1111}\big) \\ &=\bar \lambda_{2}\big(-\bar \lambda_{1}-\bar H\bar \lambda^{2}_{1}\big)+(\bar \lambda_{1}-\bar \lambda_{2})\bar h_{1111} \\ &=-\bar \lambda_{1}\bar \lambda_{2}\big(1-\frac{\bar \lambda_{1}}{\bar \lambda_{2}}\big)-\frac{2\bar \lambda_{2}}{\bar \lambda_{1}}(\bar h^{2}_{221}+\bar h^{2}_{123}). \end{aligned}$$ Thus, $$\bar h^{2}_{221}+\bar h^{2}_{123}=-\frac{\bar \lambda^{2}_{1}}{2}.$$ It is a contradiction. If $\bar \lambda_1$, $\bar \lambda_{2}$ and $\bar \lambda_3$ are distinct, by use of [\[3.1-2\]](#3.1-2){reference-type="eqref" reference="3.1-2"}, [\[3.1-5\]](#3.1-5){reference-type="eqref" reference="3.1-5"} and [\[3.1-8\]](#3.1-8){reference-type="eqref" reference="3.1-8"}, we have that $$\bar h_{11k}=\bar h_{22k}=\bar h_{33k}=0, \ \ k=1, 2, 3,$$ and $$\sum_{i,j,k}\bar h_{ijk}^{2}=6\bar h^{2}_{123}, \ \ 2\sum_{i,j,k,l}\bar h_{ijl}\bar h_{jkl}\bar h_{ki} =4\bar H\bar h^{2}_{123}.$$ Then it follows from [\[2.1-13\]](#2.1-13){reference-type="eqref" reference="2.1-13"} and [\[2.1-14\]](#2.1-14){reference-type="eqref" reference="2.1-14"} in the Lemma [Lemma 1](#lemma 2.1){reference-type="ref" reference="lemma 2.1"} that $$6\bar h^{2}_{123}=S(S+1), \ \ 4\bar H\bar h^{2}_{123}=(S+1)f_{3},$$ namely, $$(2\bar H S-3f_{3})(S+1)=0.$$ Hence, $$\bar H =\frac{3f_{3}}{2S}.$$ ◻ By applying the generalized maximum principle due to Omori and Yau to the function $-H$, we can obtain the following **Proposition 2**. *For a $3$-dimensional complete self-expander $X:M^{3}\rightarrow \mathbb{R}^{4}$ with non-zero constant squared norm $S$ of the second fundamental form, if $f_{3}$ is constant, we have that $\inf H=\frac{3f_{3}}{2S}$.* *Proof of Theorem [Theorem 1](#theorem 1.1){reference-type="ref" reference="theorem 1.1"}*. If $S=0$, we know that $X: M^{3}\to \mathbb{R}^{4}$ is a hyperplane $\mathbb {R}^{3}$ through the origin. If $S\neq0$, from the Proposition [Proposition 1](#proposition 3.1){reference-type="ref" reference="proposition 3.1"} and the Proposition [Proposition 2](#proposition 3.2){reference-type="ref" reference="proposition 3.2"}, we have that $\sup H=\inf H=\frac{3f_{3}}{2S}$. That is, the mean curvature $H$ and the principal curvatures are constant. Then by a classification theorem due to Lawson [@Law], $X: M^{3}\to \mathbb{R}^{4}$ is $\mathbb{S}^{k}(r)\times \mathbb{R}^{3-k}, \ \ k=1,2,3$. However, for the constant mean curvature $H$, we infer that $H=0$ from [\[2.1-12\]](#2.1-12){reference-type="eqref" reference="2.1-12"}. It is a contradiction. So the main theorem of the present paper is proved. $\square$ The first author was partially supported by the China Postdoctoral Science Foundation Grant No.2022M711074. The second author was partly supported by grant No.12171164 of NSFC, GDUPS (2018), Guangdong Natural Science Foundation Grant No.2023A1515010510. There are no conflicts of interest with third parties. Data sharing not applicable to this article as no datasets were generated or analysed during the current study. 99 S. Ancari and X. Cheng, Volume properties and rigidity on self-expanders of mean curvature flow, Geom. Dedicata, **216**,(2022), no. 2, Paper No. 24, 25 pp. S. Angenent, T. Ilmanen and D. L. Chopp, A computed example of nonuniqueness of mean curvature flow in $\mathbb{R}^{3}$, Comm. Partial Differential Equations, **20** (1995), no. 11-12, 1937-1958. J. Bernstein and L. Wang, The space of asymptotically conical self-expanders of mean curvature flow, Math. Ann., **380**(2021), no. 1-2, 175-230. J. Bernstein and L. Wang, Smooth compactness for spaces of asymptotically conical self-expanders of mean curvature flow, Int. Math. Res. Not., **2021**(2021), no. 12, 9016-9044. X. Cheng and D. Zhou, Spectral properties and rigidity for self-expanding solutions of the mean curvature flows, Math. Ann., **371**(2018), no. 1-2, 371-389. K. Ecker and G.Huisken, Mean curvature evolution of entire graphs, Ann. Math., **130**(1989), no.3, 453-471. Q. Ding, Minimal cones and self-expanding solutions for mean curvature flows, Math. Ann., **376**(2020), no. 1-2, 359-405. F. T. -H. Fong and P. McGrath, Rotational symmetry of asymptotically conical mean curvature flow self-expanders, Comm. Anal. Geom., **27**(2019), no. 3, 599-618. H. Halldorsson, Self-similar solutions to the curve shortening flow, Trans. Amer. Math. Soc., **364**(2012), no. 10, 5285-5309. G. Huisken, Asymptotic behavior for singularities of the mean curvature flow, J. Differential Geom., **31**(1990), 285-299. T. Ilmanen, Lectures on Mean Curvature Flow and Related Equations (Trieste Notes), (1995). N. Ishimura, Curvature evolution of plane curves with prescribed opening angle, Bull Austral. Math. Soc., **52** 1995, 287-296. H. B. Lawson, Local rigidity theorems for minimal hypersurfaces, Ann. Math., **89**(1969), 187-197. Z. Li and G. Wei, A classification of complete $2$-dimensional self-expanders in the Euclidean space $\mathbb R^{3}$, preprint. H. Omori, Isometric immersion of Riemannian manifolds, J. Math. Soc. Japan, **19**(1967), 205-214. K. Smoczyk, Self-expanders of the mean curvature flow, Vietnam J. Math., **49**(2021), 433-445. N. Stavrou, Selfsimilar solutions to the mean curvature flow, J. Reine Angew. Math., **499**(1998), 189-198. J. M. Xie and J. T. Yu, Convexity of $2$-convex translating and expanding solitons to the mean curvature flow in $\mathbb{R}^{n+1}$, J. Geom. Anal., **33**(2023), no. 8, Paper No. 252, 19 pp. S. T. Yau, Harmonic functions on complete Riemannian manifolds, Comm. Pure and Appl. Math., **28**(1975), 201-228.

arxiv_math

--- abstract: | We prove that certain families of compact Coxeter polyhedra in 4- and 5-dimensional hyperbolic space constructed by Makarov give rise to infinitely many commensurability classes of reflection groups in these dimensions. address: - Department of Computer and Mathematical Sciences, University of Toronto Scarborough, 1265 Military Trail, Toronto, ON M1C 1A4, Canada - Institut des Hautes Études Scientifiques, Université Paris-Saclay, 35 route de Chartres, 91440 Bures-sur-Yvette, France - Institut de Mathématiques de Marseille, UMR 7373, CNRS, Aix-Marseille Université author: - Nikolay Bogachev - Sami Douba - Jean Raimbault bibliography: - bib.bib title: Infinitely many commensurability classes of compact Coxeter polyhedra in $\mathbb{H}^4$ and $\mathbb{H}^5$ --- # Introduction Among the most attractive and concrete examples of discrete groups of isometries of hyperbolic space $\mathbb{H}^d$ are the hyperbolic reflection groups, i.e., those generated by reflections. That being said, the existence of *cocompact* such groups---in other words, the existence of compact hyperbolic Coxeter polyhedra---is ultimately a low-dimensional phenomenon. Indeed, Vinberg [@vinberg81nonexistence] showed that there are no such polyhedra in dimensions $d \geq 30$, and the highest dimension in which such a polyhedron has been exhibited is $8$; see Bugaenko [@bugaenko84; @bugaenko92]. We may nevertheless ask whether these polyhedra abound in the dimensions where they do exist. It is clear, for example, that a compact right-angled polyhedron in $\mathbb{H}^d$ gives rise to infinitely many such polyhedra by successively doubling along walls. While compact right-angled polyhedra cease to exist in $\mathbb{H}^d$ as soon as $d > 4$ as follows from the Nikulin inequality [@nikulin81] (see [@potyagailovinberg]), this doubling method can be applied to a more general family of polyhedra than those that are right-angled, an observation that was exploited by Allcock [@All06] to construct infinitely many compact Coxeter polyhedra in $\mathbb{H}^d$ for each $d \leq 6$. However, this method falls short of addressing the following question (see [@164099]). **Question 1**. *Given $d <30$, are there infinitely many *commensurability classes* of compact Coxeter polyhedra in $\mathbb{H}^d$?* Here, we say two compact Coxeter polyhedra in $\mathbb{H}^d$ are *commensurable* if the corresponding reflection groups $\Gamma_1, \Gamma_2 < \mathrm{Isom}(\mathbb{H}^d) \cong \mathbf{PO}(d,1)$ are commensurable (in the wide sense), that is, if there is some $g \in \mathbf{PO}(d,1)$ such that $\Gamma_1 \cap g \Gamma_2 g^{-1}$ is a lattice in $\mathbf{PO}(d,1)$. We will routinely conflate hyperbolic Coxeter polyhedra with their associated reflection groups. The dimensions $d$ for which Question [Question 1](#question){reference-type="ref" reference="question"} is known to have a positive answer include those in which one can find a sequence of compact hyperbolic Coxeter polyhedra with arbitrarily small dihedral angles, since the (adjoint) trace fields of such polyhedra (which are commensurability invariants) have unbounded degree. Such sequences are known to exist for $d=2$ by the Gauss--Bonnet theorem, and for $d=3$ by Andreev's theorem [@andreev1970; @rhd07], but their existence in higher dimensions is open. Note that there can even be uncountably many commensurability classes of compact hyperbolic Coxeter *polygons* of a fixed combinatorial type, whereas the combinatorics of a compact Coxeter polyhedron in $\mathbb{H}^d$ determine its geometry for $d \geq 3$ by Mostow rigidity. A natural way (which may be viewed as a generalization of the doubling method) to try to build new polyhedra is to glue copies of smaller polyhedra along isometric facets, forming so-called *garlands*. For $d=4,5$, Makarov [@Mak68] discovered compact Coxeter polyhedra $P_i^d \subset \mathbb{H}^d$, $i=1,2$, whose garlands yield infinitely many compact Coxeter polyhedra in $\mathbb{H}^d$. Felikson and Tumarkin [@feliksontumarkin2014] have suggested that Makarov's garlands in dimensions $4$ and $5$ constitute infinite families even up to commensurability. We prove that this is indeed the case, and hence settle Question [Question 1](#question){reference-type="ref" reference="question"} in the affirmative for $d=4,5$. **Theorem 2**. *There are infinitely many commensurability classes of compact Coxeter polyhedra in $\mathbb{H}^d$ for $d=4,5$.* Indeed, by applying techniques of the last author [@croissmax] to Makarov's garlands in dimension $4$, we obtain the following quantitative result (compare with [@All06 Thm. 1.2] and [@croissmax Cor. 1.2]). **Theorem 3**. *There is some $c>1$ such that the number of pairwise incommensurable compact Coxeter polyhedra in $\mathbb{H}^4$ of volume $\leq V$ is at least $c^V$.* In view of Theorem [Theorem 2](#Main){reference-type="ref" reference="Main"}, the only dimension $d$ in which infinitely many compact hyperbolic Coxeter polyhedra have been exhibited but for which Question [Question 1](#question){reference-type="ref" reference="question"} remains open is $d=6$. We remark that, while in any infinite family of pairwise incommensurable compact hyperbolic Coxeter polyhedra only finitely many members are arithmetic [@ABSW; @Nikulin; @Fisher_Hurtado; @coxeter_finiteness], not a single nonarithmetic compact Coxeter polyhedron in $\mathbb{H}^6$ is known, as pointed out by Vinberg [@vinberg2014]. When we set out to prove Theorem [Theorem 2](#Main){reference-type="ref" reference="Main"}, our initial hope was that the polyhedra $P_1^d$ and $P_2^d$ were arithmetic and incommensurable, so that we could directly apply arguments of the last author [@croissmax], which in turn build on techniques used by Gromov and Piatetski-Shapiro [@GPS] to verify nonarithmeticity of their "hybrid\" lattices. While it turns out that *none* of the $P_i^d$ are arithmetic, they are all nevertheless *quasi-arithmetic* (see Section [2](#QAlattices){reference-type="ref" reference="QAlattices"} for the definition, as well as [@BK22] for a broader discussion on quasi-arithmetic reflection groups). Arguments as in [@croissmax] then go through once we have verified that $P_1^d$ and $P_2^d$ are not only incommensurable, but that their ambient groups are distinct; for $d=4$, the latter, as well as proper quasi-arithmeticity of the $P_i^4$, were in fact already established by Dotti [@dotti Example 3.16]. (Note that that to say $P_1^d$ and $P_2^d$ have distinct ambient groups would have been the same as saying that $P_1^d$ and $P_2^d$ are incommensurable had they been arithmetic.) By distinguishing the ambient groups of the $P_i^d$, we can also conclude the following (compare with Thomson [@Thomson Thm. 1.6], cf. also [@BGV23 Thm. 1.1]). **Theorem 4**. *A Makarov garland in $\mathbb{H}^d$, $d=4,5$, containing both a $P_1^d$ piece and a $P_2^d$ piece is *not* quasi-arithmetic, so that we may take all the polyhedra in Theorem [Theorem 2](#Main){reference-type="ref" reference="Main"} (and Theorem [Theorem 3](#quantitative){reference-type="ref" reference="quantitative"}) to be non-quasi-arithmetic.* We remark that it is not known for any $3 \leq d < 30$ whether there are infinitely many commensurability classes of quasi-arithmetic Coxeter polyhedra in $\mathbb{H}^d$. On the other hand, and in contrast to the arithmetic setting, Dotti and Kolpakov [@dottikolpakov2022] exhibited infinitely many commensurability classes of quasi-arithmetic compact hyperbolic Coxeter *polygons*, and indeed, a family of such polygons the degrees of whose trace fields are unbounded. It follows from work of Mila [@Mila] that there are only finitely many possibilities for the trace fields of Makarov's garlands, so that we do not address the following variant of Question [Question 1](#question){reference-type="ref" reference="question"}. **Question 5**. *Given $4 \leq d <30$, are there infinitely many options for the trace fields of compact Coxeter polyhedra in $\mathbb{H}^d$?* As observed in [@dottikolpakov2022], it follows from work of Vinberg [@vinberg84absence] that there are only finitely many options for the trace fields of compact quasi-arithmetic Coxeter polyhedra (if any) in $\mathbb{H}^d$ for $d \geq 14$. ## Structure of the paper {#structure-of-the-paper .unnumbered} In Section [2](#QAlattices){reference-type="ref" reference="QAlattices"} we review trace fields and ambient groups for discrete groups of isometries and prove some facts about quasi-arithmetic lattices. We describe Makarov's polyhedra in Section [3](#desc_polytopes){reference-type="ref" reference="desc_polytopes"} and compute their trace fields and ambient groups there. In Section [4](#proofs){reference-type="ref" reference="proofs"} we deduce Theorems [Theorem 2](#Main){reference-type="ref" reference="Main"}, [Theorem 3](#quantitative){reference-type="ref" reference="quantitative"}, and [Theorem 4](#non-quasi-arit){reference-type="ref" reference="non-quasi-arit"}. ## Acknowledgments {#acknowledgments .unnumbered} We thank Pavel Tumarkin for pointing out to us Dotti's example [@dotti Example 3.16]. We are also grateful to Daniel Allcock and Pierre Py for pointing out misprints in a previous draft of this article. S. D. was supported by the Huawei Young Talents Program. J. R. was supported by grant AGDE - ANR-20-CE40-0010-01. # Quasi-arithmetic lattices {#QAlattices} ## Definitions Let $G$ be a semisimple Lie group and $k \subset \mathbb R$ a totally real number field. A $k$-group $\mathbf{G}$ is said to be *admissible* for $G$ if $\mathbf{G}(\mathbb R)$ is isogenous to $G$ and $\mathbf{G}(k \otimes_\sigma \mathbb R)$ is compact for any non-identity embedding $\sigma :\: k \to \mathbb R$. A lattice $\Gamma$ in $G$ is said to be *quasi-arithmetic* if there exist $k$ and $\mathbf{G}$ with $\mathbf{G}$ admissible for $G$ and $\Gamma \cap \mathbf{G}(k)$ a lattice in $G$. If one can find such $k$ and $\mathbf{G}$ such that moreover $\Gamma \cap \mathbf{G}(\mathcal{O}_k)$ is a lattice in $G$, i.e., $\Gamma$ is commensurable to $\mathbf{G}(\mathcal{O}_k)$, where $\mathcal{O}_k$ is the ring of integers of $k$, then $\Gamma$ is *arithmetic*. If $\Gamma$ is quasi-arithmetic but not arithmetic, one says $\Gamma$ is *properly quasi-arithmetic*. Another characterization of quasi-arithmeticity is as follows. Let $\Gamma$ be a lattice in $G$, and $k(\Gamma)$ the subfield of $\mathbb R$ generated by the traces $\operatorname{tr}(\operatorname{Ad}g)$ for $g \in \Gamma$. There is (up to $k(\Gamma)$-isogeny) a unique $k(\Gamma)$-group $\mathbf{G}_\Gamma$ such that $\mathbf{G}_\Gamma(\mathbb{R})$ is isogenous to $G$ and $\Gamma$ is virtually contained in $\mathbf{G}_\Gamma(k(\Gamma))$ via this isogeny. The field $k(\Gamma)$ and the group $\mathbf{G}_\Gamma$ are the *adjoint trace field* and the *ambient group* of $\Gamma$, respectively. (See [@Vinberg_trace] for the existence and unicity of these invariants, the so-called *Vinberg invariants* of $\Gamma$.) Then $\Gamma$ is quasi-arithmetic if and only if $k(\Gamma)$ is a totally real number field and $\mathbf{G}_\Gamma$ is admissible for $G$. If $\Gamma$ is quasi-arithmetic, then $\Gamma$ is arithmetic if and only if $\operatorname{tr}(\operatorname{Ad}g)$ is an algebraic integer for each $g \in \Gamma$. We remark that it follows from Margulis's arithmeticity theorem for irreducible higher-rank lattices [@margulis91discrete Ch. IX], arithmeticity of quaternionic and octonionic hyperbolic lattices [@corlette; @gromovschoen], and integrality of traces of complex hyperbolic lattices [@MR3826464; @baldiullmo; @bfms] that proper quasi-arithmeticity is ultimately a real hyperbolic phenomenon; more precisely, if $G$ admits an irreducible properly quasi-arithmetic lattice, then $G$ is isogenous to $\mathbf{PO}(d,1)$ for some $d \geq 2$. Moreover, as observed by Thomson [@Thomson], it follows from work of Belolipetsky--Thomson [@belolipetskythomson], or a result of Bergeron--Haglund--Wise [@bergeronhaglundwise], that a construction of Agol [@agol2006systoles] yields properly quasi-arithmetic lattices in $\mathbf{PO}(d,1)$ for each $d \geq 2$, so that the latter Lie groups are precisely those $G$ admitting irreducible properly quasi-arithmetic lattices. ## Algebraic rigidity of quasi-arithmetic lattices The following result is similar to [@Thomson Proposition 3.2]. **Proposition 6**. *Let $G$ be a semisimple Lie group and suppose for some subfield $k \subset \mathbb{R}$ one has $k$-groups ${\bf G}_1, {\bf G}_2$ and Lie group isomorphisms $\rho_i :{\bf G}_i(\mathbb{R}) \rightarrow G$, $i= 1,2$, such that $\rho_1({\bf G}_1(k)) \cap \rho_2({\bf G}_2(k))$ is Zariski-dense in $G$. Then the ${\bf G}_i$ are $k$-isogenous.* *Proof.* Let $\Lambda = \rho_1({\bf G}_1(k)) \cap \rho_2({\bf G}_2(k))$ and $\mathrm{Ad}: G \rightarrow \mathrm{GL}(\mathfrak{g})$ be the adjoint representation of $G$, where $\mathfrak{g}$ denotes the Lie algebra of $G$. The adjoint trace field of $\Lambda$ is contained in $k$ since $\rho_1^{-1}(\Lambda) \subset {\bf G}_1(k)$. By [@Vinberg_trace Thm. 1], it follows that there is a basis for $\mathfrak{g}$ with respect to which $\mathrm{Ad}(\Lambda)$ has entries in $k$. Identify $\mathrm{GL}(\mathfrak{g})$ with $\mathrm{GL}_n(\mathbb{R})$ via this basis, where $n= \mathrm{dim}(\mathfrak{g})$. Since $\Lambda$ is assumed to be Zariski-dense in $G$, we have that $\mathrm{Ad}(G) \subset {\bf G}(\mathbb{R})$, where ${\bf G}$ is the $k$-closure of $\mathrm{Ad}(\Lambda)$. Now each $\mathrm{Ad}\circ \rho_i : {\bf G}_i \rightarrow {\bf G}$ is an $\mathbb{R}$-isogeny mapping the Zariski-dense subgroup $\rho_i^{-1}(\Lambda)$ of ${\bf G}_i(k)$ into ${\bf G}(k)$. It follows that $\mathrm{Ad}\circ \rho_i$ is moreover a $k$-isogeny for $i=1,2$; see, for instance, [@Zimmer_book Prop. 3.1.10]. ◻ This implies the following generalisation of [@GPS 1.6] to quasi-arithmetic lattices. **Corollary 1**. *If $\Gamma_1$ and $\Gamma_2$ are two quasi-arithmetic lattices in $G$ and $\Gamma_1 \cap \Gamma_2$ is Zariski-dense in $G$ then the ambient groups of $\Gamma_1$ and $\Gamma_2$ coincide.* *Proof.* To conclude Corollary [Corollary 1](#qa_hybrid){reference-type="ref" reference="qa_hybrid"} from Proposition [Proposition 6](#similartothomson){reference-type="ref" reference="similartothomson"}, we need only prove that the adjoint trace fields of $\Gamma_1$ and $\Gamma_2$ are equal. To that end, let $\Lambda = \Gamma_1 \cap \Gamma_2$ and $\ell$ be the adjoint trace field of $\Lambda$; we prove that the adjoint trace field $k$ of $\Gamma_1$ is equal to $\ell$. We have $\ell \subset k$. If $\ell$ is a proper subfield of $k$ then by the Galois correspondence (applied to some Galois number field containing $k$) there is an embedding $\sigma \not= \mathrm{Id}$ of $k$ into $\mathbb R$ such that $\sigma|_\ell = \mathrm{Id}$. It follows that $\operatorname{Ad}(\Lambda) = \sigma(\operatorname{Ad}(\Lambda)) \subset \sigma(\operatorname{Ad}(\mathbf{G}_1(k))$, and the latter is contained in the compact group $\operatorname{Ad}(\mathbf{G}_1(k \otimes_\sigma \mathbb R))$; here $\mathbf{G}_1$ denotes the ambient group of $\Gamma_1$. We conclude that $\mathrm{Ad}(\Lambda)$ is precompact in $\mathrm{Ad}(G)$, but this contradicts Zariski-density of $\Lambda$ in $G$. ◻ # Makarov's polyhedra {#desc_polytopes} ## Vinberg invariants for hyperbolic reflection groups Let $P$ be a Coxeter polyhedron in $\mathbb{H}^d$ and let $H_1, \ldots, H_N$ be the hyperplanes in $\mathbb{H}^d$ supporting the facets, i.e., codimension-1 faces, of $P$. In the hyperboloid model for $\mathbb{H}^d$ each $H_i$ is associated with a subspace of signature $(d-1, 1)$ in the Lorentz space $\mathbb R^{d, 1}$; the orthogonal to this subspace is a positive line, so we can choose a unit vector $e_i$ orthogonal to this hyperplane.[^1] The Gram matrix $G(P)$ for $P$ is then the Gram matrix of the vectors $e_i$, that is, $$G(P) = \left( (e_i, e_j)_{\mathbb R^{d, 1}} \right)_{1 \le i, j \le N}.$$ From the latter, two invariants are defined: - The field $k(P)$ is that generated by all cyclic products of the matrix $G(P)$ (a cyclic product of a matrix $(a_{ij})_{1\le i, j \le N}$ is a product $a_{i_1i_2}a_{i_2i_3}\cdots a_{i_ki_1}$). In general $k(P)$ is a (possibly proper) subfield of the field generated by the entries of $G(P)$. - The quadratic form $Q_P$ is the non-singular summand of the quadratic form on $\mathbb R^N$ associated with the matrix $G(P)$. If $P$ is a Coxeter polyhedron in $\mathbb{H}^d$ we denote by $\Gamma_P$ the associated reflection group. The following statement follows from work of Vinberg, see [@Vin67 Lemma 7] and [@Vinberg_trace Section 4, Theorem 5]. **Theorem 7** (Vinberg). *Let $P$ be a Coxeter polyhedron in $\mathbb{H}^d$ and suppose $\Gamma_P$ is Zariski-dense in $\mathbf{PO}(d,1)$. Then the adjoint trace field and ambient group of $\Gamma_P$ are $k(P)$ and $\mathbf{PO}(Q_P)$, respectively.* The following criterion is useful for determining when two orthogonal groups are isomorphic (see [@GPS 2.6]), and we will use it to check the conditions in Proposition [Proposition 9](#commclass_garlands){reference-type="ref" reference="commclass_garlands"}. **Lemma 8**. *Let $k$ be a field of characteristic $0$, $m \ge 2$, and $Q, Q'$ be two non-degenerate quadratic forms on $k^m$. Then $\mathbf{PO}(Q)$ is $k$-isogenous to $\mathbf{PO}(Q')$ if and only if $Q, Q'$ are similar over $k$, that is, if and only if there is some $\lambda \in k^\times$ such that the forms $Q$ and $\lambda Q'$ are isometric over $k$.* ## In four dimensions {#4d_Makarov} ### Description We follow Vinberg [@Vinberg_polytopes p.62]. Consider the abstract Coxeter simplex given by the following diagram: Computing the Gram matrix, we see that this abstract simplex can be realized as a hyperbolic simplex $S_1^4 \subset \mathbb{H}^4$. Three of the vertices have spherical link and are hence within $\mathbb{H}^4$. The remaining two, which are the right- and leftmost in the diagram, have links corresponding to the diagrams which represent compact simplices $T_1, T_2$ in $\mathbb{H}^3$, respectively; it follows that these vertices are hyperideal and that there are two hyperplanes $H_1, H_2$ truncating $S_1^4$ into a compact polyhedron $\overline{S_1^4}$, two facets of which are orthogonal to all facets meeting them and are isometric respectively to $T_1$ and $T_2$. We let $P_1^4$ be the double of $\overline{S_1^4}$ along the $T_1$ facet. The polyhedron $P_2^4$ is constructed in a similar manner: consider the Coxeter diagram which likewise represents a simplex $S_2^4$ in $\mathbb{H}^4$. As before, three of the vertices have spherical link, and the remaining two have links with diagrams representing compact simplices $T_3, T_2$ in $\mathbb{H}^3$, respectively. Using the same cutting/doubling (this time along $T_3$), we obtain $P_2^4$. To summarize, we have described compact polyhedra $P_1^4, P_2^4$ in $\mathbb{H}^4$ such that each has two nonadjacent facets isometric to the hyperbolic 3-simplex $T_2$ and orthogonal to all adjacent facets. ### Arithmetic invariants Computing Gram matrices, we see that $S_1^4$ and $P_1^4$ (and $\overline{S_1^4}$) all share the same Vinberg field $k = \mathbb Q(\sqrt 5)$. Since $\Gamma_{S_1^4}$ is Zariski-dense in $\mathbf{PO}(4,1)$, the ambient group of $\Gamma_{S_1^4}$ is the same as that of $\Gamma_{P_1^4}$. The Gram matrix of $S_1^4$ is given by $$\left(\begin{array}{rrrrr} 1 & -\sqrt 2/2 & 0 & 0 & 0 \\ -\sqrt 2/2 & 1 & -\frac{1}{2} & 0 & 0 \\ 0 & -\frac{1}{2} & 1 & -\frac{1}{2} a & 0 \\ 0 & 0 & -\frac{1}{2} a & 1 & -\frac{1}{2} \\ 0 & 0 & 0 & -\frac{1}{2} & 1 \end{array}\right),$$ where $a = 2\cos(\pi/5) = \tfrac{\sqrt 5 + 1}2$. By multiplying the first basis vector by $\sqrt 2$, we obtain the equivalent matrix $$Q_1^4 = \left(\begin{array}{rrrrr} 2 & -1 & 0 & 0 & 0 \\ -1 & 1 & -\frac{1}{2} & 0 & 0 \\ 0 & -\frac{1}{2} & 1 & -\frac{1}{2} a & 0 \\ 0 & 0 & -\frac{1}{2} a & 1 & -\frac{1}{2} \\ 0 & 0 & 0 & -\frac{1}{2} & 1 \end{array}\right),$$ so the ambient group of $\Gamma_{S_1^4}$ is isomorphic to $\mathbf{PO}(Q_1^4)$. This group is admissible for $\mathbf{PO}(4, 1)$ and so $\Gamma_{P_1^4}$ is quasi-arithmetic (in fact, we have checked that $\Gamma_{P_1^4}$ is properly quasi-arithmetic; this was indeed already observed by Dotti [@dotti Example 3.16]). Similarly, the Vinberg fields of $S_2^4$ and $P_2^4$ are both equal to $k$, and the ambient group of $\Gamma_{P_2^4}$ is the same as that of $\Gamma_{S_2^4}$. The Gram matrix of $S_2^4$ is $$Q_2^4 = \left(\begin{array}{rrrrr} 1 & -\frac{1}{2} a & 0 & 0 & 0 \\ -\frac{1}{2} a & 1 & -\frac{1}{2} & 0 & 0 \\ 0 & -\frac{1}{2} & 1 & -\frac{1}{2} a & 0 \\ 0 & 0 & -\frac{1}{2} a & 1 & -\frac{1}{2} \\ 0 & 0 & 0 & -\frac{1}{2} & 1 \end{array}\right),$$ so the ambient group for $\Gamma_{S_2^4}$ is $\mathbf{PO}(Q_2^4)$. It follows that $\Gamma_{P_2^4}$ is quasi-arithmetic (and again, one can check, properly quasi-arithmetic). Now we prove that the ambient groups $\mathbf{PO}(Q_1^4)$ and $\mathbf{PO}(Q_2^4)$ are not $k$-isogenous using Lemma [Lemma 8](#isogeny){reference-type="ref" reference="isogeny"}. This was already established by Dotti [@dotti Example 3.16], but we include the argument here for the convenience of the reader. A straightforward computation in SAGE [@sagemath] shows that the Hasse invariants of $Q_1^4$ and $Q_2^4$ differ at the prime $\mathfrak p_5 = (1-2a)$, i.e., the ideal generated by a square root of 5 in $k$. Recall that the Hasse invariant of a quadratic form $Q$ over a field $K$ is defined to be $$\label{defn_hasse} h_K(Q) = \prod_{i < j} (a_i, a_j)_K$$ where $Q$ is isometric to a diagonal form with coefficients $a_i$, and $(x, y)_K$ is the Hilbert symbol over $K$ (which equals $1$ if the corresponding quaternion algebra is $K$-split and $-1$ otherwise, see e.g. [@McR] for the definition). Assume that $Q_1^4$ is isometric over $k$ to $\lambda Q_2^4$ for some $\lambda\in k^\times$. Since $\tfrac{\det(Q_1^4)}{\det(Q_2^4)} = \frac{6-2a}5$ and the factorisation of $(6-2a)$ into prime ideals is $(2) \cdot \mathfrak p_5$, we see that $\lambda = 2u\mathfrak p_5$ modulo squares, where $u$ is a unit in the integer ring $\mathbb Z[a]$. Let $k_5$ be the localisation of $k$ at $\mathfrak p_5$. By Theorem 2.6.6(3) in [@McR], we have that $(\lambda, \lambda)_{k_5} = 1$ if and only if $-1$ is a square modulo $\mathfrak p_5$. This is indeed the case since $\mathbb Z[a]/\mathfrak p_5 = \mathbb F_5$. So the Hasse invariant of $\lambda Q_2^4$ over $k_5$ is equal to that of $Q_2^4$ and hence distinct from that of $Q_1^4$, so that the forms $Q_1^4$ and $Q_2^4$ cannot be isometric over $k_5$, a fortiori not over $k$. ## In five dimensions {#5d_Makarov} ### Description Each of the Coxeter diagrams gives a simplex in $\mathbb{H}^5$ with a single hyperideal vertex with link corresponding to the diagram which represents a compact simplex in $\mathbb{H}^4$. Truncating these vertices yields compact Coxeter polyhedra $P_1^5, P \subset \mathbb{H}^5$, respectively, each possessing a facet orthogonal to all adjacent facets and which is isometric to the polyhedron with the latter Coxeter diagram. Moreover, there is a nonadjacent facet $F$ of $P$ forming even angles (of $\frac{\pi}{2}$ or $\frac{\pi}{4}$) with all remaining facets. Let $P_2^5$ be the double of $P$ along $F$. ### Arithmetic invariants As in the $4$-dimensional case, the Vinberg field of $S_i^5$ (which coincides with the Vinberg field of $P_i^5$) is $k = \mathbb{Q}(\sqrt{5})$ for $i=1,2$. The Gram matrix of $S_1^5$ is $$Q_1^5 = \begin{pmatrix} 1 & -\frac{1}{2}a & 0 & 0 & 0 & 0 \\ -\frac{1}{2}a & 1 & -\frac{1}{2} & 0 & 0 & 0 \\ 0 & -\frac{1}{2} & 1 & -\frac{1}{2} & 0 & 0 \\ 0 & 0 & -\frac{1}{2} & 1 & -\frac{1}{2} & 0 \\ 0 & 0 & 0 & -\frac{1}{2} & 1 & -\frac{1}{2} \\ 0 & 0 & 0 & 0 & -\frac{1}{2} & 1 \end{pmatrix},$$ so the ambient group for $\Gamma_{S_1^5}$ is $\mathbf{PO}(Q_1^5)$. Since this group is admissible for $\mathbf{PO}(5,1)$, we have that $\Gamma_{P_1^5}$ is quasi-arithmetic (one can again check that $\Gamma_{P_1^5}$ is in fact properly quasi-arithmetic). The Gram matrix of $S_2^5$ is $$\begin{pmatrix} 1 & -\frac{1}{2}a & 0 & 0 & 0 & 0 \\ -\frac{1}{2}a & 1 & -\frac{1}{2} & 0 & 0 & 0 \\ 0 & -\frac{1}{2} & 1 & -\frac{1}{2} & 0 & 0 \\ 0 & 0 & -\frac{1}{2} & 1 & -\frac{1}{2} & 0 \\ 0 & 0 & 0 & -\frac{1}{2} & 1 & -\sqrt{2}/2 \\ 0 & 0 & 0 & 0 & -\sqrt{2}/2 & 1 \end{pmatrix}.$$ By multiplying the last basis vector by $\sqrt{2}$ we obtain the equivalent matrix $$Q_2^5= \begin{pmatrix} 1 & -\frac{1}{2}a & 0 & 0 & 0 & 0 \\ -\frac{1}{2}a & 1 & -\frac{1}{2} & 0 & 0 & 0 \\ 0 & -\frac{1}{2} & 1 & -\frac{1}{2} & 0 & 0 \\ 0 & 0 & -\frac{1}{2} & 1 & -\frac{1}{2} & 0 \\ 0 & 0 & 0 & -\frac{1}{2} & 1 & -1 \\ 0 & 0 & 0 & 0 & -1 & 2 \end{pmatrix},$$ so the ambient group of $\Gamma_{S_2}^5$ is $\mathbf{PO}(Q_2^5)$. This group is admissible for $\mathbf{PO}(5,1)$, and so $\Gamma_{P_2^5}$ is also quasi-arithmetic (and again, one can check, properly quasi-arithmetic). By Lemma [Lemma 8](#isogeny){reference-type="ref" reference="isogeny"}, the $k$-groups $\mathbf{PO}(Q_1^5)$ and $\mathbf{PO}(Q_2^5)$ are $k$-isogenous only if $\frac{\det(Q_1^5)}{\det(Q_2^5)} = \frac{8-2a}{4}$ is a square in $k$. The latter is true if and only if $8-2a$ is a square in $k$, hence in its integer ring $\mathbb{Z}[a]$. We have $8-2a = 2(4-a)$, and the norm of $(4-a)$ is 11. So the norm of $(8-2a)$ is $44$, which is not a square in $\mathbb Z$, so that $8-2a$ is not a square in $\mathbb Z[a]$. # Coxeter polyhedra and commensurability {#proofs} ## Garlands Let $P_1, P_2$ be two compact Coxeter polyhedra in $\mathbb{H}^d$ satisfying the following hypothesis: **Assumption 1**. *Each of the $P_i$ contains two nonadjacent facets isometric to the same Coxeter polyhedron $R$ in $\mathbb{H}^{d-1}$ and orthogonal to all adjacent facets.* Given any sequence $\alpha \in \{1, 2\}^n$ we can form a compact polyhedron $P_\alpha$ by gluing copies of $P_1, P_2$ according to $\alpha$ (what Vinberg [@Vinberg_polytopes] calls a *garland*). Formally, we choose an indexing of the two $R$-facets of $P_i$ which we denote by $\partial^\pm P_i$ respectively, and then we obtain $P_\alpha$ by identifying each $\partial^+ P_{\alpha_i}$ with $\partial^- P_{\alpha_{i+1}}$ for $1\le i \le n-1$. As a consequence of Assumption [Assumption 1](#2side_gluing){reference-type="ref" reference="2side_gluing"}, the resulting polyhedron remains a Coxeter polyhedron. We can also perform a more restricted construction under a weaker hypothesis. **Assumption 2**. *The polyhedron $P_1$ contains a facet isometric to a Coxeter polyhedron $R$ in $\mathbb{H}^{d-1}$ and orthogonal to all adjacent facets, while $P_2$ contains two nonadjacent facets isometric to $R$ and orthogonal to all adjacent facets.* Under this assumption we can form for each $n \ge 1$ a Coxeter polyhedron $L_n$ by gluing $n$ copies of $P_2$ and then capping one end with a copy of $P_1$. ## Infinitely many commensurability classes **Proposition 9**. *Suppose $P_1, P_2$ are two compact quasi-arithmetic Coxeter polyhedra in $\mathbb{H}^d$, $d \geq 3$, such that the $\Gamma_{P_i}$ have distinct ambient groups. Suppose that $P_1, P_2$ satisfy Assumption [Assumption 2](#1side_gluing){reference-type="ref" reference="1side_gluing"}. Then the $L_n$, $n \geq 1$, are pairwise incommensurable.* *Proof.* Suppose there is a closed hyperbolic orbifold $M$ with covers $\pi : M \to L_n$, $\pi' : M \to L_{n'}$. Let $L_n^1$ (resp., $L_{n'}^1$) be the copy of $P_1$ in $L_n$ (resp., $L_{n'}$), thought of as a compact hyperbolic orbifold with connected totally geodesic boundary. By Proposition [Corollary 1](#qa_hybrid){reference-type="ref" reference="qa_hybrid"} and Lemma 3.2 in [@croissmax][^2], we have $\pi^{-1}(L_n^1) = \pi'^{-1}(L_{n'}^1)$. Since $L_n^1$ and $L_{n'}^1$ share the same volume, namely, that of $P_1$, it follows that $\pi$ and $\pi'$ share the same degree. But then $L_n$ and $L_{n'}$ share the same volume, and so $n=n'$. ◻ ## Quantitative estimate for the number of commensurability classes If $\alpha \in \{1, 2\}^n$ we denote by $\bar\alpha$ its mirror sequence (that is, $\bar\alpha_i = \alpha_{n+1-i}$). The following proposition is similar to [@croissmax Proposition 2.1], and we will deduce it from this result. **Proposition 10**. *If $P_1, P_2 \subset \mathbb{H}^d$, $d \geq 3$, satisfying Assumption [Assumption 1](#2side_gluing){reference-type="ref" reference="2side_gluing"} are quasi-arithmetic with distinct ambient groups, then for any $n \ge 1$ and any sequences $\alpha , \beta \in \{1, 2\}^n$ such that $\Gamma_{P_\alpha}$ is commensurable with $\Gamma_{P_\beta}$, we have that $\beta$ or $\bar\beta$ is a subsequence of $(\alpha, \bar{\alpha})$.* An immediate consequence is the following more precise version of Proposition [Proposition 9](#commclass_garlands){reference-type="ref" reference="commclass_garlands"}. **Corollary 2**. *There are at least $\tfrac{2^n}{2n}$ distinct commensurability classes among those of the $P_\alpha$ for $\alpha \in \{1, 2\}^n$.* *Proof of Proposition [Proposition 10](#commclass_garlands2){reference-type="ref" reference="commclass_garlands2"}.* Let $n \ge 1$ and $\alpha \in \{1, 2\}^n$. Let $\sigma_1, \sigma_n$ be the reflections in the facets $\partial^- P_{\alpha_1}$ and $\partial^+ P_{\alpha_n}$. Since these facets are nonadjacent and orthogonal to all adjacent facets, there is a surjective morphism $\varphi$ from $\Gamma_{P_\alpha}$ to the infinite dihedral group $\langle \sigma_1, \sigma_n\rangle$. Let $\varepsilon$ be the morphism $\langle \sigma_1, \sigma_n\rangle \to \mathbb Z/2\mathbb Z$ sending both $\sigma_1$ and $\sigma_n$ to the generator. We define $M_\alpha$ be the double orbifold cover of $P_\alpha$ corresponding to the morphism $\varepsilon\circ \varphi$. Topologically, the orbifold $M_\alpha$ is obtained by first viewing $P_\alpha$ as an orbifold with totally geodesic boundary $\partial^- P_{\alpha_1}\cup \partial^+ P_{\alpha_n}$ (so that the orbifold fundamental group of $P_\alpha$ is generated by the reflections in all facets of $P_\alpha$ apart from $\partial^- P_{\alpha_1}$ and $\partial^+ P_{\alpha_n}$), and then doubling along $\partial^- P_{\alpha_1}\cup \partial^+ P_{\alpha_n}$. This is the setting[^3] of [@croissmax], so if some $P_\beta$ is commensurable with $P_\alpha$ (so that $M_\alpha, M_\beta$ are commensurable to each other as well) we can apply Proposition 2.1 in loc. cit. to conclude that the sequences $(\alpha,\bar \alpha)$ and $(\beta, \bar\beta)$ are cyclic permutations of one another (these sequences are both palindromic so we need not apply a symmetry). In particular $\beta$ or $\bar\beta$ is a subsequence of $(\alpha, \bar\alpha)$. ◻ *Remark 11*. Under the conditions of Proposition [Proposition 10](#commclass_garlands2){reference-type="ref" reference="commclass_garlands2"}, we may in fact conclude that $\beta$ on the nose is a subsequence of $(\alpha, \bar{\alpha})$, so that it is even true that the $P_\alpha$ represent at least $\frac{2^n}{n}$ distinct commensurability classes as $\alpha$ varies within $\{1,2\}^n$. Indeed, if $(\alpha, \bar\alpha)$ and $(\beta, \bar\beta)$ are cyclic permutations of one another and $\beta$ is neither $\alpha$ nor $\bar \alpha$, then $(\alpha, \bar{\alpha})$ is invariant under some nontrivial dihedral group of permutations, and in particular, under some (possibly trivial) cyclic permutation shifting $\beta$ into $(\alpha, \bar\alpha)$. ## Proof of the main results In the notation of Section [3.2](#4d_Makarov){reference-type="ref" reference="4d_Makarov"}, the polyhedra $P_1 = P_1^4$ and $P_2 = P_2^4$ each have two nonadjacent facets isometric to $R= T_2$ and orthogonal to all adjacent facets, and hence satisfy Assumption [Assumption 1](#2side_gluing){reference-type="ref" reference="2side_gluing"}. Since $\Gamma_{P_1^4}$ and $\Gamma_{P_2^4}$ are both quasi-arithmetic with distinct ambient groups, the conditions of Proposition [Proposition 10](#commclass_garlands2){reference-type="ref" reference="commclass_garlands2"} are satisfied, and Theorem [Theorem 3](#quantitative){reference-type="ref" reference="quantitative"} follows from Corollary [Corollary 2](#counting){reference-type="ref" reference="counting"}. In the notation of Section [3.3](#5d_Makarov){reference-type="ref" reference="5d_Makarov"}, we can take $P_1 = P_1^5$ and $P_2 = P_2^5$. Then $P_1, P_2$ satisfy Assumption [Assumption 2](#1side_gluing){reference-type="ref" reference="1side_gluing"}, and the other assumptions of Proposition [Proposition 9](#commclass_garlands){reference-type="ref" reference="commclass_garlands"}. This proves the case $d=5$ of Theorem [Theorem 2](#Main){reference-type="ref" reference="Main"}. ## Proof of Theorem [Theorem 4](#non-quasi-arit){reference-type="ref" reference="non-quasi-arit"} {#proof-of-theorem-non-quasi-arit} The proof is similar to that of Corollary [Corollary 1](#qa_hybrid){reference-type="ref" reference="qa_hybrid"}. Let $i=4,5$ and let $P$ be a garland containing copies of both $P_1^i$ and $P_2^i$. Assume that $\Gamma_P$ is quasi-arithmetic; then there exists a totally real $k$-group $\mathbf{G}$ which is admissible for $\mathbf{PO}(d, 1)$ and such that $\Gamma_P \subset \mathbf{G}(k)$. Since $\Gamma_P$ contains Zariski-dense subgroups of both $\Gamma_{P_j^i}$, the field $k$ contains $\mathbb Q(\sqrt 5)$ and there are $k$-isogenies $\mathbf{G}_j \to \mathbf{G}$ for $j=1, 2$. Since $\mathbf{G}$ is admissible, we have that $k=\mathbb Q(\sqrt 5)$, and it follows that $\mathbf{G}, \mathbf{G}_1, \mathbf{G}_2$ are all $k$-isogenous, which is not the case. [^1]: The sign of $e_i$ is not important for us. [^2]: The latter is stated only for manifolds in this reference, but the proof works as written for orbifolds. [^3]: The results in this reference are stated for manifolds but apply equally well to orbifolds.

arxiv_math

--- abstract: | As we are aware, various types of methods have been proposed to approximate the Caputo fractional derivative numerically. A common challenge of the methods is the non-local property of the Caputo fractional derivative which leads to the slow and memory consuming methods. Diffusive representation of fractional derivative is an efficient tool to overcome the mentioned challenge. This paper presents two new diffusive representations to approximate the Caputo fractional derivative of order $0<\alpha<1$. Error analysis of the newly presented methods together with some numerical examples are provided at the end. author: - "Hassan Khosravian-Arab [^1]" - "Mehdi Dehghan [^2]" title: The sine and cosine diffusive representations for the Caputo fractional derivative --- ***Keyword:*** Caputo fractional derivative; Non-locality property; Diffusive representation; Infinite state representation; Memory free formulation; Error analysis. ***Mathematics Subject Classifications (2000):*** 26A33;  65D30;  65D25;  65D32. # Introduction Nowadays, there is an international awareness on the importance of fractional calculus as well as their broad applications in various areas such as: mathematics, statistics, physics, chemistry, electronic, engineering, biology and etc [@Ji2017; @Lei2017; @Castillo2018; @Du2020; @MR2218073; @MR4030088]. This means that many real world problems have been modeled by the following fractional differential equation (FDEs): $$\label{Eq_1} {}^{C}D_{a^+}^{\alpha}y(t)=F(t,y(t)),\ y(a)=y_a,\ 0<\alpha\leq1,$$ where ${}^{C}D_{a^+}^{\alpha}$ used for the Caputo fractional derivative of order $\alpha$ with starting point $a$ [@MR2680847; @MR2218073]: $$\label{Eq_2} {}^{C}D_{a^+}^{\alpha}y(t)=\frac{1}{\Gamma(1-\alpha)}\int_{a}^{t}(t-\tau)^{-\alpha}y'(\tau)\,d\tau,\ 0<\alpha<1,$$ where $\Gamma(.)$ is the Euler's Gamma function. The first and most important step to solve FDEs [\[Eq_1\]](#Eq_1){reference-type="eqref" reference="Eq_1"} numerically, is to approximate the Caputo fractional derivative(s) ${}^{C}D_{a^+}^{\alpha}y(t)$. Unfortunately, due to the non-locality property of the Caputo fractional derivative, there is a significant computational challenge to approximate this operator numerically [@Diethelm2021; @Diethelm2022]. The review of existing literature on the numerical solutions of FDEs [\[Eq_1\]](#Eq_1){reference-type="eqref" reference="Eq_1"} shows that, all the methods are based on the following two ideas [@Li2015]: - **Direct Methods**: In these methods the Caputo fractional derivative can be approximated directly to obtain the numerical schemes. - **Indirect Methods**: In these methods, the original problem [\[Eq_1\]](#Eq_1){reference-type="eqref" reference="Eq_1"} is transformed into the fractional integral equation and then using a suitable method to discretize the fractional integral, the numerical schemes can be obtained. The Direct Methods (DM) can be also divided into two main categories: - Nodal Methods. - Modal Methods. For the readers convenience, we summarized the existing DM to approximate Caputo fractional derivative [\[Eq_2\]](#Eq_2){reference-type="eqref" reference="Eq_2"} in Fig. [\[DM\]](#DM){reference-type="ref" reference="DM"}. The main drawback of these methods is that to handle the non-locality of the fractional differential operators, they require a relatively large amount of time and/or computer memory [@Diethelm2022; @Diethelm2021]. Specially, when we approximate the Caputo fractional derivative of order $\alpha$ at points $\{x_j\}_{j=0}^n$ based on the nodal methods, the most common difficulty which we face with is that the computational complexity of these methods is proportional to $\mathcal{O}(n^2)$ (for the classical convolution types), $\mathcal{O}(n \log^2 n)$ (for some modification types) or $\mathcal{O}(n \log n)$ (by the use of the fast Fourier transform) and then for large values of $n$, the computational complexity of these methods increases very fast. To overcome this drawback, a new representation of the Caputo fractional derivative (which is so-called as diffusive representation (DR), infinite state representation (ISR) or memory free formulation (MFF)) was introduced by Yuan and Agrawal in [@Yuan2002; @Agrawal2009]. In fact, they have shown that the Caputo fractional derivative can be reformulated as: $$\label{YARep_1} {}^{C}D_{0^+}^{\alpha}y(t)=\int_{0}^{+\infty}\phi(\omega,t)\,d\omega,$$ where $\phi(\omega,t)$ for $\omega\in(0,+\infty)$ called the observed system's infinite states at time $t$ and also satisfies the inhomogeneous first order differential equation in the following form: $$\label{YARep_Diff} \frac{\partial}{\partial t}\phi(\omega,t)=h_1(\omega)\phi(\omega,t)+h_2(\omega)y'(t),\ \phi(\omega,0)=0,$$ with certain functions $h_1; h_2: (0,+\infty) \to \mathbb{R}$. Some new improvements and modifications of the diffusive representation have been introduced in [@Liu2018; @MR3936246; @Trinks2002; @Schmidt2006; @Diethelm2022; @Hinze2019; @MR4392021; @MR3474912; @Birk2010; @Yuan2002; @Audounet; @Matignon2009; @Agrawal2009]. One of the most important features of these methods is to reduce the computational complexity of fractional differential solvers effectively to just $\mathcal{O}(n)$ (See [@Diethelm2021; @Ford2001; @Garrappa2018; @AnewDiff]). [\[DM\]]{#DM label="DM"} The outline of this paper is as follows. In Section [2](#Sec_2){reference-type="ref" reference="Sec_2"} two new diffusive representations (DR) for the Caputo fractional derivative, numerical parts of the new methods and error analysis of them are presented. In Section [3](#Sec_3){reference-type="ref" reference="Sec_3"} some numerical results together with an improvement of the new method are given. Finally, in Section [4](#Sec_4){reference-type="ref" reference="Sec_4"} concluding remarks and some future works are proposed. # The sine and cosine diffusive representations {#Sec_2} In this section, we introduce two new diffusive representations (DRs) which we will call the sine and cosine diffusive representations (SDR and CDR) for Caputo fractional derivative. To reach our aim, we commence with the preliminary definitions and theorems. **Definition 1**. *[@MR2218073; @MR2680847] Let $\alpha>0$. The Caputo fractional derivative is defined as: $$\label{Cap} {}^{C}D_{0^+}^{\alpha}y(t)=\frac{1}{\Gamma(\lceil \alpha \rceil-\alpha)}\int_{0}^{t}(t-\tau)^{\lceil \alpha \rceil-\alpha-1}y^{(\lceil \alpha \rceil)}(\tau)\,d\tau,$$ where $\lceil .\rceil$ stands for the ceiling function, that rounds up to the next integer not less than its argument.* *For $0<\alpha<1$ we have: $$\label{Cap1} {}^{C}D_{0^+}^{\alpha}y(t)=\frac{1}{\Gamma(1-\alpha)}\int_{0}^{t}(t-\tau)^{-\alpha}y'(\tau)\,d\tau.$$* For more information and properties of the Caputo fractional derivative see [@MR2218073; @MR2680847]. Before going to introduce two new diffusive representations of Caputo fractional derivative of order $0<\alpha<1$, we recall Yuan and Agrawal approach [@Yuan2002]. **Theorem 1**. *(Yuan and Agrawal approach (YA)) Let $0<\alpha<1$. We have the following diffusive representation of the Caputo fractional derivative: $$\label{Eq_0} {}^{C}D_{0^+}^{\alpha}y(t)=\int_{0}^{\infty}z^{2\alpha-1} \omega^{YA}(z,t)\,dz,$$ where $$\omega^{YA}(z,t)=\frac{2\sin(\pi\alpha)}{\pi}\left(\int_{0}^{t}e^{-(t-\tau)z^2}y'(\tau)\,d\tau\right).$$ It is easy to verify that $\omega^{YA}(z,t)$ satisfies the following differential equation: $$\label{ODE1} \frac{\partial \omega^{YA}}{\partial t}+z^2\omega^{YA}=\frac{2\sin({\pi\alpha})}{\pi}y'(t),\ \omega^{YA}(z,0)=0.$$* *Proof.* See [@Yuan2002; @Diethelm2008] for the proof of the theorem. ◻ **Lemma 1**. *[@91138334] Let $0<\alpha<1$ and $b\in\Bbb{R}^+$. Then we have: $$\Gamma(\alpha)=\frac{b^\alpha}{\cos\left(\frac{\pi }{2}\alpha\right)}\int_{0}^{+\infty}t^{\alpha-1}\cos(bt)\,dt,$$ and $$\Gamma(\alpha)=\frac{b^\alpha}{\sin\left(\frac{\pi }{2}\alpha\right)}\int_{0}^{+\infty}t^{\alpha-1}\sin(bt)\,dt.$$* *Proof.* See [@91138334] for the proof of this lemma. ◻ Now and in this position, we will going to define two new DRs to approximate the Caputo fractional derivative. **Theorem 2**. *(The cosine diffusive representation (CDR)). For $0<\alpha<1$, one can see $$\label{CDR} {}^{C}D_{0^+}^{\alpha}y(t)=\frac{2\sin(\tfrac{\pi\alpha}{2})}{\pi}\int_{0}^{\infty}z^{\alpha-1}\left(\int_{0}^{t}\cos\left((t-\tau)z\right){y'(\tau)}\,d\tau\right)\,dz=\int_{0}^{\infty}z^{\alpha-1}\omega^C(z,t)\,dz,$$ where $$\label{Kernel_1} \omega^C(z,t)=\frac{2\sin(\tfrac{\pi\alpha}{2})}{\pi}\left(\int_{0}^{t}\cos\left((t-\tau)z\right){y'(\tau)}\,d\tau\right).$$* **Proof.* We first note that: $$\Gamma(\alpha)\Gamma(1-\alpha)=\frac{\pi}{\sin(\pi \alpha)},\ 0<\alpha<1.$$ Then we have: $$\begin{aligned} \label{Eq_5_3} {}^{C}D_{0^+}^{\alpha}y(t)&=&\frac{1}{\Gamma(1-\alpha)}\int_{0}^{t}(t-\tau)^{-\alpha}y'(\tau)\,d\tau\nonumber\\ &=&\frac{\Gamma(\alpha)}{\Gamma(1-\alpha)\Gamma(\alpha)}\int_{0}^{t}(t-\tau)^{-\alpha}y'(\tau)\,d\tau\nonumber\\ &=&\frac{\sin(\pi\alpha)\Gamma(\alpha)}{\pi}\int_{0}^{t}(t-\tau)^{-\alpha}y'(\tau)\,d\tau\nonumber\\ &=&\frac{\sin(\pi\alpha)x^{\alpha}}{\pi\cos\left(\tfrac{\pi \alpha}{2}\right)}\int_{0}^{t}\left(\int_{0}^{\infty}\theta^{\alpha-1}\cos(x\theta)\,d\theta\right)(t-\tau)^{-\alpha}y'(\tau)\,d\tau\nonumber\\ &\stackrel{\theta=(t-\tau)z}{=}&\frac{2x^{\alpha}\sin\left(\tfrac{\pi \alpha}{2}\right)}{\pi}\int_{0}^{\infty}z^{\alpha-1}\left(\int_{0}^{t}\cos\left(x(t-\tau)z\right){y'(\tau)}\,d\tau\right)\,dz\nonumber. \end{aligned}$$ If we take $x=1$, theorem is proved. ◻* **Theorem 3**. *(The sine diffusive representation (SDR)). For $0<\alpha<1$, we have $$\label{SDR} {}^{C}D_{0^+}^{\alpha}y(t)=\frac{2\cos(\tfrac{\pi\alpha}{2})}{\pi}\int_{0}^{\infty}z^{\alpha-1}\left(\int_{0}^{t}\sin\left((t-\tau)z\right){y'(\tau)}\,d\tau\right)\,dz=\int_{0}^{\infty}z^{\alpha}\omega^S(z,t)\,dz,$$ where $$\label{Kernel_2} \omega^S(z,t)=\frac{2\cos(\tfrac{\pi\alpha}{2})}{z\pi} \left(\int_{0}^{t}\sin\left((t-\tau)z\right){y'(\tau)}\,d\tau\right).$$* **Proof.* The proof is similar to the proof of Theorem [Theorem 2](#Cos){reference-type="ref" reference="Cos"}. ◻* An important property of CDR and SDR is given in the next theorems. **Theorem 4**. *Let $0<\alpha<1$.* - *For a given function $y$ for which its second derivative exists on $[0,T]$, $\omega^C(z,t)$ (for fixed $z>0$) satisfies the following second-order differential equation: $$\label{ODE2} \begin{cases} \displaystyle\frac{\partial^2 \omega^C}{\partial t^2}+z^2\omega^C=\frac{2\sin(\tfrac{\pi\alpha}{2})}{\pi} y''(t),\\ \displaystyle\omega^C(z,0)=0,\ \frac{\partial}{\partial t}\omega^C(z,0)=\frac{2\sin(\tfrac{\pi\alpha}{2})}{\pi} y'(0). \end{cases}$$* - *For a given function $y$ for which its first derivative exists on $[0,T]$, $\omega^S(z,t)$ (for fixed $z>0$) satisfies the following second-order differential equation: $$\label{ODE3} \begin{cases} \displaystyle\frac{\partial^2 \omega^S}{\partial t^2}+z^2\omega^S=\frac{2\cos(\tfrac{\pi\alpha}{2})}{\pi}\ y'(t),\\ \displaystyle \omega^S(z,0)=\frac{\partial}{\partial t}\omega^S(z,0)=0. \end{cases}$$* **Proof.* The proofs are straightforward. ◻* In the following some important remarks concerning the mentioned second-order differential equations [\[ODE2\]](#ODE2){reference-type="eqref" reference="ODE2"} and [\[ODE3\]](#ODE3){reference-type="eqref" reference="ODE3"} are given. **Remark 1**. *The following remarks should be noted here:* - *In contrast to the other types of the DRs which coupled with a first-order differential equation, the SDR and CDR are coupled with a second-order differential equations.* - *As we know, the second-order differential equations [\[ODE2\]](#ODE2){reference-type="eqref" reference="ODE2"} and [\[ODE3\]](#ODE3){reference-type="eqref" reference="ODE3"} could be converted to a system of first-order differential equations. Thus, the SDR and CDR can be also considered as the classical DRs.* - *The second derivative of the given function $y(t)$, which appears in Eq. [\[ODE2\]](#ODE2){reference-type="eqref" reference="ODE2"} seems as a bad point of CDR, but, in fact, as we will see later, in application we don't need to evaluate $y''(t)$.* Due to the fact that the classical DRs of Caputo fractional derivative are usually coupled with a first-order differential equation, it is worthy to convert [\[ODE2\]](#ODE2){reference-type="eqref" reference="ODE2"} and [\[ODE3\]](#ODE3){reference-type="eqref" reference="ODE3"} to the system of first-order differential equations. **Theorem 5**. *Let $0<\alpha<1$.* - *For a given function $y$ for which its second derivative exists on $[0,T]$, and $\omega^C(z,t)=x_1(z,t)$ (for fixed $z>0$), then $x_1(z,t)$ satisfies in the following system of first-order differential equations: $$\label{SysODE2} \begin{cases} \displaystyle\frac{\partial x_1}{\partial t}=x_2(z,t),\\ \displaystyle\frac{\partial x_2}{\partial t}=-z^2x_1(z,t)+\frac{2\sin(\tfrac{\pi\alpha}{2})}{\pi} y''(t),\\ \displaystyle x_1(z,0)=0,\ x_2(z,0)=\frac{2\sin(\tfrac{\pi\alpha}{2})}{\pi} y'(0). \end{cases}$$* - *For a given function $y$ for which its first derivative exists on $[0,T]$, and assume $\omega^S(z,t)=x_1(z,t)$ (for fixed $z>0$), where $x_1(z,t)$ satisfies in the following system of first-order differential equations: $$\label{SysODE3} \begin{cases} \displaystyle\frac{\partial x_1}{\partial t}=x_2(z,t),\\ \displaystyle\frac{\partial x_2}{\partial t}=-z^2x_1(z,t)+\frac{2\cos(\tfrac{\pi\alpha}{2})}{\pi}\ y'(t),\\ \displaystyle x_1(z,0)=x_2(z,0)=0. \end{cases}$$* **Proof.* The proofs are straightforward. ◻* The numerical parts of the newly presented methods CDR and SDR are given in the next subsection. ## The numerical parts of CDR and SDR {#Sec_2.1} An important question which arises here is that: How can one approximate the Caputo fractional derivative of order $0<\alpha<1$ based on CDR and SDR? In fact, the procedure to construct the desired approximation for the Caputo fractional derivative of order $0<\alpha<1$ of the given function $y(t)$ has two steps: 1. In the first step, we need to choose a suitable quadrature formula to approximate the following semi-infinite integrals numerically: $$\label{IntCos_1} \int_{0}^{+\infty}z^{\alpha-1}\omega^C(z,t)\,dz\approx\sum_{k=1}^{N}c_k\ \omega^C(z_k^c,t),$$ and $$\label{IntSin_1} \int_{0}^{+\infty}z^{\alpha}\omega^S(z,t)\,dz\approx\sum_{k=1}^{N}s_k\ \omega^S(z_k^s,t),$$ where $z_k^c$, $z_k^s$ and $c_k$, $s_k$ are the nods and weights of the quadrature formulae, respectively. 2. In the second step, we need to approximate $\omega^C(z_k^s,t)$ and $\omega^S(z_k^s,t)$ numerically. To do so, we solve the obtained systems of first-order [\[SysODE2\]](#SysODE2){reference-type="eqref" reference="SysODE2"} and [\[SysODE3\]](#SysODE3){reference-type="eqref" reference="SysODE3"} for $z=z_k^c$ and $z=z_k^s$, respectively. In practice, various numerical algorithms with step size $h$ like as Runge-Kutta scheme or a linear multistep method can be used to solve these systems of differential equations. We denote $\omega^C_h(z_k^s,t_i)$ and $\omega^S_h(z_k^s,t_i)$ as the numerical approximations of $\omega^C(z_k^s,t)$ and $\omega^S(z_k^s,t)$, respectively, obtained from systems [\[SysODE2\]](#SysODE2){reference-type="eqref" reference="SysODE2"} and [\[SysODE3\]](#SysODE3){reference-type="eqref" reference="SysODE3"}, where $t\in[0,T]$ and $h$ is the step size with $$h=\frac{T}{n-1},\ n\in\Bbb{N},$$ and denote by $t_k=(k-1)h$, so $t_1=0$ and $t_n=T$. Now, substituting $\omega^C_h(z_k^s,t_i)$ and $\omega^S_h(z_k^s,t_i)$ into [\[IntCos_1\]](#IntCos_1){reference-type="eqref" reference="IntCos_1"} and [\[IntSin_1\]](#IntSin_1){reference-type="eqref" reference="IntSin_1"}, respectively, the Caputo fractional derivatives of order $0<\alpha<1$ of the given function $y(t)$ at $t_i,\ i=1,2,\cdots,n$ are obtained as: $$\label{IntCos_1Frac} {}^{C}D_{0^+}^{\alpha}y(t)\Big|_{t=t_i}=\int_{0}^{+\infty}z^{\alpha-1}\omega^C(z,t_i)\,dz\approx\sum_{k=1}^{N}c_k\ \omega^C(z_k^c,t_i)\approx\sum_{k=1}^{N}c_k\ \omega^C_h(z_k^c,t_i),\ i=1,2,\cdots,n,$$ and $$\label{IntSin_1Frac} {}^{C}D_{0^+}^{\alpha}y(t)\Big|_{t=t_i}=\int_{0}^{+\infty}z^{\alpha}\omega^S(z,t_i)\,dz\approx\sum_{k=1}^{N}s_k\ \omega^C(z_k^s,t_i)\approx\sum_{k=1}^{N}s_k\ \omega^S_h(z_k^s,t_i),\ i=1,2,\cdots,n.$$ Next remark provides some important issues to obtain numerical approximations of Caputo fractional derivative of order $0<\alpha<1$. **Remark 2**. *For the first step of the numerical method which concerns with the use of a suitable quadrature rules, it is worthy to point out that various types of the quadrature rules have been proposed to handle the semi-infinite integral of the DR, recently. In the following we list some of them.* *To handle semi-infinite integral, in fact, Yuan and Agrawal proposed the classical Gauss-Laguerre quadrature rule [@Yuan2002]. Then Lu and Hanyga suggested to split the semi-infinite integral into two integrals $[0,c]$ and $[c,+\infty)$ [@LuHanyga2005]. They used the Gauss-Jacobi and shifted Gauss-Laguerre rules to compute the integral over $[0,c]$ and $[c,+\infty)$, respectively. Generalized Gauss-Laguerre and Gauss-Jacobi quadrature rules have been successfully carried out by K. Diethelm in [@Diethelm2008; @Diethelm2009]. Composite Gauss-Jacobi quadrature rule is used by Hinze et al. [@Hinze2019].* *For the second step, which provides some ODE solver, the backward Euler and trapezoidal methods have been suggested (See [@Diethelm2021; @Diethelm2008] for more comments on the ODE solvers).* We also point out that, in our computations, due to the asymptotic behaviors of the functions $\omega^C(z,t)$ and $\omega^S(z,t)$ when $z\to 0$ and $z\to+\infty$, we will use the generalized Gauss-Laguerre formula to approximate the semi-infinite integrals (See Theorem [Theorem 6](#AsymtoticCDRandSDR){reference-type="ref" reference="AsymtoticCDRandSDR"} and Remark [Remark 4](#RemWeight){reference-type="ref" reference="RemWeight"}). This means that for the CDR semi-infinite integral the generalized Gauss-Laguerre with respect to the weight function $w(z)=z^{\alpha-1}e^{-z}$ is carried out. On the other hand, for $k=1,2,\cdots,n$, we write: $$\label{CDRINT} {}^{C}D_{0^+}^{\alpha}y(t)\Big|_{t=t_k}= % \int_{0}^{\infty} \omega^C(z,t_{k})\,dz= \int_{0}^{\infty}z^{\alpha-1} e^{-z}\left[ e^{z}\omega^C(z,t_k)\right]\,dz\approx\sum_{i=1}^{N}w_i^{(\alpha-1)}\ e^{z_i^{(\alpha-1)}}\omega^C(z_i^{(\alpha-1)},t_k),$$ where $z_i^{(\alpha-1)}$ and $w_i^{(\alpha-1)}$ are the Gauss-Laguerre nodes and weights associated with the weight function $w(z)=z^{\alpha-1}e^{-z}$. Now, we need to approximate $\omega^C(z_i^{(\alpha-1)},t_k)$ (by $\omega_h^C(z_i^{(\alpha-1)},t_k)$) numerically. So, the following backward Euler and the trapezoidal methods for [\[SysODE2\]](#SysODE2){reference-type="eqref" reference="SysODE2"} with $z=z_i^{(\alpha-1)},\ i=1,2,\cdots,N$ are suggested as: $$\label{Syatem_11} \begin{cases} x_1^E(z,t_k)=x_1^E(z,t_{k-1})+hx_2^E(z,t_{k-1}),\\ x_2^E(z,t_k)=\frac{1}{1+z^2h^2}\left[x_2^E(z,t_{k-1})-z^2hx_1^E(z,t_{k-1})+\dfrac{2\sin(\tfrac{\pi\alpha}{2})}{\pi} \left(y'(t_k)-y'(t_{k-1})\right)\right], \end{cases},\ k=2,3,...,n,$$ and (for $\ k=2,3,...,n$) $$\label{Syatem_111} \begin{cases} x_1^T(z,t_k)=x_1^T(z,t_{k-1})+\frac{h}{2}\left[x_2^T(z,t_{k-1})+x_2^E(z,t_{k})\right],\\ x_2^T(z,t_k)=\frac{1}{1+\frac{z^2h^2}{4}}\left[\left(1-\frac{z^2h^2}{4}\right)x_2^T(z,t_{k-1})-z^2hx_1^T(z,t_{k-1})+\dfrac{2\sin(\tfrac{\pi\alpha}{2})}{\pi} \left(y'(t_k)-y'(t_{k-1})\right)\right], \end{cases}$$ respectively, where $$\label{InitCDR} x_1(z,t_1)=0,\ x_2(z,t_1)=\frac{2\sin(\tfrac{\pi\alpha}{2})}{\pi} y'(0).$$ Substituting the solutions $x_1^E(z_i^{(\alpha-1)},t_k)$ and $x_1^T(z_i^{(\alpha-1)},t_k)$ as approximations of $\omega_h^C(z_i^{(\alpha-1)},t_k)$ into [\[CDRINT\]](#CDRINT){reference-type="eqref" reference="CDRINT"}, the approximations of the Caputo fractional derivative are obtained. **Remark 3**. *Due to the initial conditions [\[InitCDR\]](#InitCDR){reference-type="eqref" reference="InitCDR"} and to solve systems [\[Syatem_11\]](#Syatem_11){reference-type="eqref" reference="Syatem_11"} and [\[Syatem_111\]](#Syatem_111){reference-type="eqref" reference="Syatem_111"} numerically, we need to approximate $y'(0)$. So, we can use the following approximation formula: $$y'(0)\approx \frac{y(h)-y(0)}{h}.$$* Similarly, for the SDR semi-infinite integral the generalized Gauss-Laguerre with respect to the weight function $w(z)=z^{\alpha}e^{-z}$ is used. This means that, for $k=1,2,\cdots,n$, we have: $$\label{SDRINT} {}^{C}D_{0^+}^{\alpha}y(t)\Big|_{t=t_k}=\int_{0}^{\infty}z^{\alpha} e^{-z}\left[e^{z}\omega^S(z,t_k)\right]\,dz\approx\sum_{i=1}^{N}w_i^{(\alpha)}\ e^{z_i^{(\alpha)}}\omega^S(z_i^{(\alpha)},t_k),$$ where $z_i^{(\alpha)}$ and $w_i^{(\alpha)}$ are the Gauss-Laguerre nodes and weights associated with the weight function $w(z)=z^{\alpha}e^{-z}$. Finally and similar to the previous method, to approximate $\omega^S(z_i^{(\alpha)},t_k)$ (by $\omega^S_h(z_i^{(\alpha)},t_k)$) the following backward Euler method together with the trapezoidal method for [\[SysODE3\]](#SysODE3){reference-type="eqref" reference="SysODE3"} with $z=z_i^{(\alpha)},\ i=1,2,\cdots,N$ are given as: $$\label{Syatem_22} \begin{cases} x_1^E(z,t_k)=x_1^E(z,t_{k-1})+hx_2^E(z,t_{k-1}),\\ x_2^E(z,t_k)=\frac{1}{1+z^2h^2}\left[x_2^E(z,t_{k-1})-z^2hx_1^E(z,t_{k-1})+\dfrac{2\cos(\tfrac{\pi\alpha}{2})}{\pi} \left(y(t_k)-y(t_{k-1})\right)\right], \end{cases},\ k=2,3,...,n,$$ and (for $\ k=2,3,...,n$) $$\label{Syatem_222} \begin{cases} x_1^T(z,t_k)=x_1^T(z,t_{k-1})+\frac{h}{2}\left[x_2^T(z,t_{k-1})+x_2^E(z,t_{k})\right],\\ x_2^T(z,t_k)=\frac{1}{1+\frac{z^2h^2}{4}}\left[\left(1-\frac{z^2h^2}{4}\right)x_2^T(z,t_{k-1})-z^2hx_1^T(z,t_{k-1})+\dfrac{2\sin(\tfrac{\pi\alpha}{2})}{\pi} \left(y(t_k)-y(t_{k-1})\right)\right], \end{cases}$$ respectively, where $$x_1(z,t_1)=0,\ x_2(z,t_1)=0,$$ Now, plugging the obtained solutions $x_1^E(z_i^{(\alpha)},t_k)$ and $x_1^T(z_i^{(\alpha)},t_k)$ as approximations of $\omega^S_h(z_i^{(\alpha)},t_k)$ into [\[SDRINT\]](#SDRINT){reference-type="eqref" reference="SDRINT"}, the approximations of Caputo fractional derivative can be obtained. ## Error analysis of CDR and SDR {#Sec_2.2} The main goal of this subsection is to provide the error analysis of the approximation methods CDR and SDR when the parameters $N$ (the number of integration points in the generalized Gauss-Laguerre formula) and $h$ (the step size of the ODE solvers) vary. Thanks to the fact that the CDR and SDR approximations are constructed from two steps (ODE solver together with the quadrature formula), so, it is natural to take into account both the errors of the generalized Gauss-Laguerre quadrature and the ODE solvers, to obtain the complete error analysis of the introduced methods. To do so, we denote the CDR and SDR approximations of the Caputo fractional derivatives of order $\alpha\in(0,1)$ by: $${}^{C}D_{0^+,C,N,h}^{\alpha}y(t)=\sum_{i=1}^{N}w_i^{(\alpha-1)}\ e^{z_i^{(\alpha-1)}}\omega^C_h(z_i^{(\alpha-1)},t),$$ and $${}^{C}D_{0^+,S,N,h}^{\alpha}y(t)=\sum_{i=1}^{N}w_i^{(\alpha)}\ e^{z_i^{(\alpha)}}\omega^S_h(z_i^{(\alpha)},t),$$ respectively. Now in the next subsections, the error analysis of the generalized Gauss-Laguerre formula together with the ODE solver is provided. ### The contribution of the generalized Gauss-Laguerre formula To obtain the error analysis of the generalized Gauss-Laguerre formula, we need to define: $$\begin{aligned} \label{Resid_CDR} R^{\alpha}_{C,N,h}y(t)&:=&{}^{C}D_{0^+}^{\alpha}y(t)-{}^{C}D_{0^+,C,N,h}^{\alpha}y(t)\nonumber\\ &=&\int_{0}^{+\infty}z^{\alpha-1}\omega^C(z,t)\,dz-\sum_{i=1}^{N}w_i^{(\alpha-1)}\ e^{z_i^{(\alpha-1)}}\omega^C_h(z_i^{(\alpha-1)},t)\nonumber\\ &=&R_{C,N,\alpha-1}^{GGL}\left[ e^{(.)}\omega^C(.,t)\right]+Q_{C,N,\alpha-1}^{GGL}\left[ e^{(.)}E^C_h(.,t)\right], \end{aligned}$$ where $$\label{ErrQuadCDR} R_{C,N,\alpha-1}^{GGL}\left[ e^{(.)}\omega^C(.,t)\right]:=\int_{0}^{+\infty}z^{\alpha-1}\omega^C(z,t)\,dz-\sum_{i=1}^{N}w_i^{(\alpha-1)}\ e^{z_i^{(\alpha-1)}}\omega^C(z_i^{(\alpha-1)},t),$$ is the error of generalized Gauss-Laguerre and $$\label{ErrODECDR} Q_{C,N,\alpha-1}^{GGL}\left[ e^{(.)}E^C_h(.,t)\right]:=\sum_{i=1}^{N}w_i^{(\alpha-1)}\ e^{z_i^{(\alpha-1)}}\left[\omega^C(z_i^{(\alpha-1)},t)-\omega^C_h(z_i^{(\alpha-1)},t)\right],$$ is the error of the ODE solver. Similarly, we denote: $$\begin{aligned} \label{Resid_SDR} R^{\alpha}_{S,N,h}y(t)&:=&{}^{C}D_{0^+}^{\alpha}y(t)-{}^{C}D_{0^+,S,N,h}^{\alpha}y(t)\nonumber\\ &=&\int_{0}^{+\infty}z^{\alpha}\omega^S(z,t)\,dz-\sum_{i=1}^{N}w_i^{(\alpha)}\ e^{z_i^{(\alpha)}}\omega^S_h(z_i^{(\alpha)},t)\nonumber\\ &=&R_{S,N,\alpha}^{GGL}\left[ e^{(.)}\omega^S(.,t)\right]+Q_{S,N,\alpha}^{GGL}\left[ e^{(.)}E^S_h(.,t)\right], \end{aligned}$$ where $$\label{ErrQuadSDR} R_{S,N,\alpha}^{GGL}\left[ e^{(.)}\omega^S(.,t)\right]:=\int_{0}^{+\infty}z^{\alpha}\omega^S(z,t)\,dz-\sum_{i=1}^{N}w_i^{(\alpha)}\ e^{z_i^{(\alpha)}}\omega^S(z_i^{(\alpha)},t),$$ denotes the error of generalized Gauss-Laguerre and $$\label{ErrODESDR} Q_{S,N,\alpha}^{GGL}\left[ e^{(.)}E^S_h(.,t)\right]:=\sum_{i=1}^{N}w_i^{(\alpha)}\ e^{z_i^{(\alpha)}}\left[\omega^S(z_i^{(\alpha)},t)-\omega^S_h(z_i^{(\alpha)},t)\right],$$ used for the error of the ODE solver. In this position, we start to analysis the errors of the generalized Gauss-Laguerre formulae [\[ErrQuadCDR\]](#ErrQuadCDR){reference-type="eqref" reference="ErrQuadCDR"} and [\[ErrQuadSDR\]](#ErrQuadSDR){reference-type="eqref" reference="ErrQuadSDR"}. To reach this aim, we need to have the behaviour of the integrands $\omega^C(z,t)$ and $\omega^S(z,t)$ when $z\to0$ and $z\to\infty$. Here, the symbol $a(v)\sim b(v)$ means that there exist two strictly positive constants $A$ and $B$ such that: $$\left|\frac{a(v)}{b(v)}\right|\in [A, B],$$ as $v$ tends to the indicated limit. **Theorem 6**. *Let $t \in(0, T)$ be fixed and $0<\alpha<1$.* 1. *Assume that there exists some constant $C>0$, such that for all $t\in(0,T)$ we have $|y'(t)|>C$ then functions $\omega^C(.,t)$ and $\omega^S(.,t)$ defined in [\[Kernel_1\]](#Kernel_1){reference-type="eqref" reference="Kernel_1"} and [\[Kernel_2\]](#Kernel_2){reference-type="eqref" reference="Kernel_2"}, respectively behave as: $$\begin{aligned} \label{Asymp_1} &&z^{\alpha-1}\omega^C(z,t)\sim z^{\alpha-1}\ \ \ \text{as}\ z\to 0,\\ && z^{\alpha}\omega^S(z,t)\sim z^{\alpha}\ \ \ \text{as}\ z\to 0. \end{aligned}$$* 2. *Let $y(t)\in C^2[0,T]$ and $y(0)=y'(0)=0$. Assume that $y(t)$ and $y'(t)$ are of exponential order, then we have: $$\begin{aligned} \label{Asymp_2} &&z^{\alpha-1}\omega^C(z,t)\sim z^{\alpha-3}\ \ \ \text{as}\ z\to +\infty. % && \omega^S(z,t)\sim z^{\alpha-2}\ \ \ \text{as}\ z\to +\infty. \end{aligned}$$* 3. *Let $y(t)\in C^1[0,T]$ and $y(t)$ and $y'(t)$ be of exponential order then we have: $$\begin{aligned} \label{Asymp_3} && z^{\alpha}\omega^S(z,t)\sim z^{\alpha-2}\ \ \ \text{as}\ z\to +\infty. \end{aligned}$$* **Proof.* For part (a), using the integration by part, yields: $$\begin{aligned} \int_{0}^{t}\cos((t-\tau)z)y'(\tau)\,d\tau&=&y(\tau)\cos((t-\tau)z)\Bigg]_{\tau=0}^{\tau=t}-z\int_{0}^{t}\sin((t-\tau)z)y(\tau)\,d\tau\\ \\ &=&y(t)-\cos(tz)y(0)-z\int_{0}^{t}\sin((t-\tau)z)y(\tau)\,d\tau. \end{aligned}$$ For fixed $t$, the right side integral remains bounded as $z\to0$, that proves: $$\lim_{z\to0}\int_{0}^{t}\cos((t-\tau)z)y'(\tau)\,d\tau=\lim_{z\to0}\left[y(t)-\cos(tz)y(0)\right]=y(t)-y(0).$$ Substituting the above relation into [\[Kernel_1\]](#Kernel_1){reference-type="eqref" reference="Kernel_1"}, completes the proof.* *For function $\omega^S(z,t)$, we write: $$\begin{aligned} \frac{1}{z}\int_{0}^{t}\sin((t-\tau)z)y'(\tau)\,d\tau&=&\frac{1}{z}y(\tau)\sin((t-\tau)z)\Bigg]_{\tau=0}^{\tau=t}+\int_{0}^{t}\cos((t-\tau)z)y(\tau)\,d\tau\\ \\ &=&-\frac{\sin(tz)}{z}y(0)+\int_{0}^{t}\cos((t-\tau)z)y(\tau)\,d\tau. \end{aligned}$$ Now, we have: $$\begin{aligned} \lim_{z\to0}\frac{1}{z}\int_{0}^{t}\sin((t-\tau)z)y'(\tau)\,d\tau&=&\lim_{z\to0}\left[-\frac{\sin(tz)}{z}y(0)+\int_{0}^{t}\cos((t-\tau)z)y(\tau)\,d\tau\right]\\ &=&-ty(0)+\int_{0}^{t}y(\tau)\,d\tau. \end{aligned}$$ The above relation together with [\[Kernel_2\]](#Kernel_2){reference-type="eqref" reference="Kernel_2"}, concludes the proof.* *For part (b), thanks to the fact that $y(t)$ and $y'(t)$ are continuous and of exponential order and then using the Laplace transform, formally gives: $$\begin{aligned} z^2\mathcal{L}\left\{\int_{0}^{t}\cos((t-\tau)z)y'(\tau)\,d\tau\right\}&=&z^2 \mathcal{L}\left\{\cos(tz)\right\}\mathcal{L}\left\{y'(t)\right\} =z^2\frac{s}{s^2+z^2}\left(s\mathcal{L}\left\{y(t)\right\}-y(0)\right). \end{aligned}$$ Thus, $$\displaystyle z^2\int_{0}^{t}\cos((t-\tau)z)y'(\tau)\,d\tau=\mathcal{L}^{-1}\left\{z^2\frac{s}{s^2+z^2}\left(s\mathcal{L}\left\{y(t)\right\}-y(0)\right)\right\}.$$ Now by taking the limit when $z\to+\infty$, we formally obtain: $$\begin{aligned} \displaystyle \lim_{z\to+\infty}z^2\int_{0}^{t}\cos((t-\tau)z)y'(\tau)\,d\tau&=&\mathcal{L}^{-1}\left\{\lim_{z\to+\infty}\left[z^2\frac{s^2\mathcal{L}\left\{y(t)\right\}}{s^2+z^2}\right]\right\}\\ &=&y''(t). \end{aligned}$$ Substituting the obtained result into [\[Kernel_1\]](#Kernel_1){reference-type="eqref" reference="Kernel_1"}, completes the proof.* *Similarly, we can write: $$\begin{aligned} \displaystyle \lim_{z\to+\infty}z\int_{0}^{t}\sin((t-\tau)z)y'(\tau)\,d\tau&=&\mathcal{L}^{-1}\left\{\lim_{z\to+\infty}\left[z^2\frac{s\mathcal{L}\left\{y(t)\right\}-y(0)}{s^2+z^2}\right]\right\}=y'(t), \end{aligned}$$ Plugging the last relation into [\[Kernel_2\]](#Kernel_2){reference-type="eqref" reference="Kernel_2"}, the proof is concluded. ◻* **Remark 4**. *Let $0<\alpha<1$. Due to Theorem [Theorem 6](#AsymtoticCDRandSDR){reference-type="ref" reference="AsymtoticCDRandSDR"}, we have the following properties:* - *The asymptotic behaviours of $z^{\alpha-1}\omega^C(z,t)$ and $z^{\alpha}\omega^S(z,t)$ when $z\to0$, indicate that the use of generalized Gauss-Laguerre with the weight functions $w(z)=z^{\alpha-1}e^{-z}$ and $w(z)=z^{\alpha}e^{-z}$, respectively, may lead to the smooth integrands at origin.* - *As we see, $z^{\alpha-1}\omega^C(z,t)$ and $z^{\alpha}\omega^S(z,t)$ when $z\to+\infty$ decay as $z^{\alpha-3}$ and $z^{\alpha-2}$, respectively. On the other hand, the exponent of $z$ for each case is always contained in $(-3,-2)$ and $(-2,-1)$, respectively. This fact is sufficient to make sure that the semi-infinite integrals [\[IntCos_1\]](#IntCos_1){reference-type="eqref" reference="IntCos_1"} and [\[IntSin_1\]](#IntSin_1){reference-type="eqref" reference="IntSin_1"} exist.* Now, in what follows, error analysis of the generalized Gauss-Laguerre formula is given. **Theorem 7**. *Let $y(t)$ and $y'(t)$ be of exponential order.* - *(a): For $0<\alpha<1$ and $y\in C^2[0,T]$ such that $y(0)=y'(0)=0$, we have: $$\label{CDEQuadEr} R_{C,N,\alpha-1}^{GGL}\left[ e^{(.)}\omega^C(.,t)\right]=\mathcal{O}(N^{\alpha-2}),$$ for $t\in[0,T]$.* - *(b): For $0<\alpha<1$ and $y\in C^1[0,T]$, we also have: $$\label{SDEQuadEr} R_{S,N,\alpha}^{GGL}\left[ e^{(.)}\omega^S(.,t)\right]=\mathcal{O}(N^{\alpha-1}),$$ for $t\in[0,T]$.* *Proof.* For the proof of this theorem see [@Diethelm2008]. ◻ **Remark 5**. *Theorem [Theorem 7](#ErrorQuad){reference-type="ref" reference="ErrorQuad"} states that for $0<\alpha<1$ the error of the generalized Gauss-Laguerre quadrature rule of the CDR method when $N\to+\infty$ decays faster than the SDR.* ### The contribution of the ODE solver The second part of the error analysis is about the truncation error of the ODE solver. To do this, we first note that, if $z_k^{(\gamma)},\ k=1,2,\cdots,N$ stands for the nodes of the generalized Laguerre integration formula with respect to the weight function $w(x)=x^{\gamma} e^{-x}$, then we have $z_k^{(\gamma)}=4k+2\gamma+6$ [@Shen2011]. To explain more clearly, we consider the following system of first order differential equations: $$\begin{aligned} \label{SysGen} Y'(t)=F(t,Y(t)),\ Y(0)={\bf a},\ \ t\in[0,T],\end{aligned}$$ for which $$Y(t)=\left[\begin{array}{c} y_1(t)\\ y_2(t)\\ \vdots\\ y_m(t) \end{array}\right],\ \ F(t,Y(t))=\left[\begin{array}{c} f_1(t,y_1,\cdots,y_m)\\ f_2(t,y_1,\cdots,y_m)\\ \vdots\\ f_m(t,y_1,\cdots,y_m) \end{array}\right],\ \ {\bf a}=\left[\begin{array}{c} y_1(0)\\ y_2(0)\\ \vdots\\ y_m(0) \end{array}\right],$$ where $F: [0,T]\times \Bbb{R}^m\longrightarrow \Bbb{R}^m$ is continuous in its first variable and satisfies a Lipschitz condition with constant $L$ in its second variable, i.e., for any $t\in[0,T]$ and $Z,W\in \Bbb{R}^m$, we have $$\|F(t, W)-F(t,Z)\|\leq L\|W-Z\|.$$ The convergence of a numerical method applied to Eq. [\[SysGen\]](#SysGen){reference-type="eqref" reference="SysGen"} requires the step size $h$ to satisfy in $Lh<1$. In our case, viz. Eqs. [\[ODE2\]](#ODE2){reference-type="eqref" reference="ODE2"} and [\[ODE3\]](#ODE3){reference-type="eqref" reference="ODE3"}, we have $L=\left(z_k^{(\gamma)}\right)^2,\ k=1,2,\cdots, N$. This may lead to some difficulties from the numerical point of view for sufficiently large $N$ unless the step sizes $h$ are chosen extremely small. For this reason, we always assume that $$h\left(z_N^{(\gamma)}\right)^2<1\Longrightarrow h<\frac{1}{\left(z_N^{(\gamma)}\right)^2}\sim N^{-2}.$$ So, we have the following lemma. **Lemma 2**. *Assume that a A-stable one-step implicit method of order $p$ is used for Eqs. [\[ODE2\]](#ODE2){reference-type="eqref" reference="ODE2"} and [\[ODE3\]](#ODE3){reference-type="eqref" reference="ODE3"}, then there exists a constant $C>0$ such that: $$\begin{aligned} &&\left|E^C_h(z_k^{(\alpha-1)},t)\right|\leq Ch^p e^{3T\displaystyle \left(z_k^{(\alpha-1)}\right)^2 },\ \\ && \left|E^S_h(z_k^{(\alpha)},t)\right|\leq Ch^p e^{3T\displaystyle \left(z_k^{(\alpha)}\right)^2 }, \end{aligned}$$ for $k=1,2,\cdots,N$, sufficiently small $h>0$ and any $t\in[0,T]$.* **Proof.* For the proof of this lemma see [@Diethelm2008]. ◻* **Theorem 8**. *Under the assumptions of the previous Lemma, there exist constants $C_1 > 0,\ C_2>0$ such that: $$\begin{aligned} && \Big|Q_{C,N,\alpha-1}^{GGL}\left[ e^{(.)}E^C_h(.,t)\right]\Big|\leq C_1h^{p}\int_{0}^{4N}e^{3T\displaystyle z^2 }\,dz,\\ &&\Big|Q_{S,N,\alpha}^{GGL}\left[ e^{(.)}E^S_h(.,t)\right]\Big|\leq C_2h^{p}\int_{0}^{4N}e^{3T\displaystyle z^2 }\,dz. \end{aligned}$$* *Proof.* The proof of this theorem is fairly similar to the proof of Theorem 5 of [@Diethelm2008]. ◻ ### The overall error analysis For the reader's convenience, summary of the error analysis is given in the following theorem. **Theorem 9**. *Let $0<\alpha<1$. If a A-stable one-step implicit method of order $p$ with the step size $h<N^{-2}$, (where $N$ is the number of integration points in the generalized Gauss-Laguerre formula) is used for Eqs. [\[ODE2\]](#ODE2){reference-type="eqref" reference="ODE2"} and [\[ODE3\]](#ODE3){reference-type="eqref" reference="ODE3"}, then the overall error analysis of CDR and SDR approximation formulae satisfies:* - *If $y(t)\in C^1[0,T]$, then for $t\in[0,T]$, we have $$\label{OVERERRCDR} \Big|R^{\alpha}_{C,N,h}y(t)\Big|=\mathcal{O}(N^{\alpha-2})+\mathcal{O}(h^p)\int_{0}^{4N}e^{3T\displaystyle z^2 }\,dz.$$* - *If $y(t)\in C^1[0,T]$, then for $t\in[0,T]$, we have $$\label{OVERERRSDR} \Big|R^{\alpha}_{S,N,h}y(t)\Big|=\mathcal{O}(N^{\alpha-1})+\mathcal{O}(h^p)\int_{0}^{4N}e^{3T\displaystyle z^2 }\,dz.$$* *Proof.* The proofs are immediately obtained from Theorems [Theorem 7](#ErrorQuad){reference-type="ref" reference="ErrorQuad"} and [Theorem 8](#BoundODESolver){reference-type="ref" reference="BoundODESolver"}. ◻ To have a good sense and in order to compare the CDR and SDR methods and the Yuan and Agrawal (YA) one, the error analysis of their method is provided here [@Diethelm2008]. **Theorem 10**. *Let $0<\alpha<1$ and $y(t)\in C^1[0,T]$. If a A-stable one-step implicit method of order $p$ with the step size $h<N^{-2}$ is used for Eq. [\[ODE1\]](#ODE1){reference-type="eqref" reference="ODE1"}, then the overall error analysis of YA approximation formula satisfies: $$\label{OVERERRYA} \Big|R^{\alpha}_{N,h}y(t)\Big|=\mathcal{O}(N^{2\alpha-2})+\mathcal{O}(h^p)\int_{0}^{4N}e^{3T\displaystyle z^2 }\,dz,$$ for $t\in[0,T]$.* *Proof.* See Theorem 6 of [@Diethelm2008]. ◻ # Numerical results {#Sec_3} In this position, we proceed to testify the numerical methods with some examples. To make a good comparison, the proposed methods CDR and SDR have compared with the Yuan and Agrawal method (YA). To do so, we first denote $$E_{\infty}(N)=\max_{t\in[a,b]}\left|{}^{C}D_{0^+}^{\alpha}y(t)-{}^{C}D_{0^+,N,h}^{\alpha}y(t)\right|,$$ for the maximum errors obtained by the methods YA, CDR and SDR for fixed $N$ and $t$ varies on the domain $[a,b]$. **Example 1**. *For the first example we consider [@Diethelm2021; @Diethelm2008] $$y(t)=t^{1.6},\ \ t\in[0,3],$$ where $${}^{C}D_{0^+}^{\alpha}y(t)=\frac{\Gamma(2.6)}{\Gamma(2.6-\alpha)}t^{1.6-\alpha},\ \alpha=0.4.$$ We also note that $y\in C^1[0,3]$. Relative errors of the approximation of ${}^{C}D_{0^+}^{\alpha}y(t)$ by the backward Euler method for some values of $n$ with the use of the generalized Gauss-Laguerre quadrature rule with $N=50$-points, obtained from YA, CDR and SDR methods are plotted in Figs. [2](#Fig_1){reference-type="ref" reference="Fig_1"}-[4](#Fig_2){reference-type="ref" reference="Fig_2"}. Also, a comparison of maximum absolute errors of approximations obtained from YA, CDR and SDR methods for some values of $N$ and $n=10^4$ is shown in Fig. [7](#Fig_3){reference-type="ref" reference="Fig_3"}.* *{#Fig_1 width="12cm"}{#Fig_1 width="12cm"}\ * *{#Fig_2 width="12cm"}{#Fig_2 width="12cm"}\ * *{#Fig_3 width="7.cm" height="15cm"}{#Fig_3 width="7.cm" height="15cm"}{#Fig_3 width="7.cm" height="15cm"}\ * **Example 2**. *For the second example, consider the following sufficiently smooth function [@Diethelm2021; @Diethelm2008]: $$y(t)=t^{3},\ \ t\in[0,1],$$ where $${}^{C}D_{0^+}^{\alpha}y(t)=\frac{\Gamma(4)}{\Gamma(4-\alpha)}t^{3-\alpha},\ \alpha=0.6.$$ We also have, $y\in C^{\infty}[0,1]$. Similarly, the relative errors of approximation ${}^{C}D_{0^+}^{\alpha}y(t)$ (with $N=50$) for three methods YA, CDR and SDR are also plotted in Figs. [9](#Fig_4){reference-type="ref" reference="Fig_4"} and [11](#Fig_5){reference-type="ref" reference="Fig_5"}. A comparisons of the maximum errors of approximation ${}^{C}D_{0^+}^{\alpha}y(t)$ (with $N=50$) is shown in Fig. [14](#Fig_6){reference-type="ref" reference="Fig_6"}.* *{#Fig_4 width="12cm"}{#Fig_4 width="12cm"}\ * *{#Fig_5 width="12cm"}{#Fig_5 width="12cm"}\ * *{#Fig_6 width="7cm" height="15cm"}{#Fig_6 width="7cm" height="15cm"}{#Fig_6 width="7cm" height="15cm"}\ * **Example 3**. *For the third example consider the sufficiently smooth and periodic function [@Sugiura2009]: $$y(t)=\sin t, \ t\in[0,1],$$ where $${}^{C}D_{0^+}^{\alpha}y(t)=t^{1-\alpha}\sum_{k=0}^{+\infty}\frac{(-t)^{2k}}{\Gamma(2k+2-\alpha)},\ \alpha=0.5.$$* *As we know, $y\in C^{\infty}[0,1]$. The relative errors obtained by the YA, CDR and SDR methods to approximate ${}^{C}D_{0^+}^{\alpha}y(t)$ for $\alpha=0.5$ and $N=50$ versus some valued of $n$ have been reported in Figs. [16](#Fig_7){reference-type="ref" reference="Fig_7"} and [18](#Fig_8){reference-type="ref" reference="Fig_8"}.* *We also report the maximum errors of the methods for some values of $N$ with $n=10^4$ in Fig. [21](#Fig_9){reference-type="ref" reference="Fig_9"}.* *{#Fig_7 width="12cm"}{#Fig_7 width="12cm"}\ * *{#Fig_8 width="12cm"}{#Fig_8 width="12cm"}\ * *{#Fig_9 width="7cm" height="15cm"}{#Fig_9 width="7cm" height="15cm"}{#Fig_9 width="7cm" height="15cm"}\ * **Example 4**. *For the last example consider the function [@Sugiura2009]: $$y(t)=t^{\frac{\nu}{2}}J_\nu(2\sqrt{t}), \ t\in[0,1], \ \nu=3,$$ where $J_\nu(z)$ is the Bessel function of the first kind. It is easy to show that: $${}^{C}D_{0^+}^{\alpha}y(t)=t^{\frac{\nu-\alpha}{2}}J_{\nu-\alpha}(2\sqrt{t}),\ \alpha=0.5.$$ Due to the fact that (cf. Property 2.3 of [@MR3742072]): $$J_\nu(t)\sim \frac{t^{\nu}}{2^\nu\Gamma(\nu+1)},\ t\to 0,\ \nu\ne-1,-2,\cdots.$$ It is easy to verify that for $\nu=3$, we have $y\in C^3[0,1]$. The relative errors of the approximations of the Caputo fractional derivative of order $\alpha=0.5$ of the function $y(t)$ obtained from three mentioned methods are depicted in Figs. [23](#Fig_10){reference-type="ref" reference="Fig_10"} and [25](#Fig_11){reference-type="ref" reference="Fig_11"}. The maximum errors of the approximations for some values of $N$ with $n=10^4$ are also graphed in Fig. [28](#Fig_12){reference-type="ref" reference="Fig_12"}.* *{#Fig_10 width="12cm"}{#Fig_10 width="12cm"}\ * *{#Fig_11 width="12cm"}{#Fig_11 width="12cm"}\ * *{#Fig_12 width="7cm" height="15cm"}{#Fig_12 width="7cm" height="15cm"}{#Fig_12 width="7cm" height="15cm"}\ * **Remark 6**. *Due to the results obtained from Examples [Example 1](#Ex_1){reference-type="ref" reference="Ex_1"}-[Example 4](#Ex_4){reference-type="ref" reference="Ex_4"}, the following conclusions can be drawn:* - *Numerical experiments show that the first term of the overall errors of the methods CDR, SDR and YA proposed in Theorems [Theorem 9](#OverErrCDRSDR){reference-type="ref" reference="OverErrCDRSDR"} and [Theorem 10](#OverErrYA){reference-type="ref" reference="OverErrYA"} will usually dominate the overall error. This means that we can say that the errors of the methods CDR, SDR and YA decay like $\mathcal{O}(N^{\alpha-2})$, $\mathcal{O}(N^{\alpha-1})$ and $\mathcal{O}(N^{2\alpha-2})$, respectively, where $N$ is the number of quadrature points. So, we don't need to apply these methods with small values of $h$. In other word, it is not necessary to use the step size which satisfies $h<N^{-2}$.* - *As we expected from Theorems [Theorem 9](#OverErrCDRSDR){reference-type="ref" reference="OverErrCDRSDR"} and [Theorem 10](#OverErrYA){reference-type="ref" reference="OverErrYA"}, the convergence rate of the SDR method is (very) slow while the error of the CDR method (for function $y(t)$ for which $y''\in C[a,b]$) decays faster than the YA method.* ## An improvement of SDR method {#Sec_3.1} As we saw in the previous section, the convergence rate of the SDR method is very slow. In fact, the slow convergence of the SDR method comes from the asymptotic behavior of $z^{\alpha}\omega^S(z,t)$ when $z\to0$ and $z\to+\infty$. As it can be seen in Theorems [Theorem 6](#AsymtoticCDRandSDR){reference-type="ref" reference="AsymtoticCDRandSDR"} and [Theorem 7](#ErrorQuad){reference-type="ref" reference="ErrorQuad"}, the convergence rate of the $N$-point generalized Gauss-Laguerre formula for the SDR method applied to Examples [Example 1](#Ex_1){reference-type="ref" reference="Ex_1"} and [Example 2](#Ex_2){reference-type="ref" reference="Ex_2"} is proportional to $\mathcal{O}\left(N^{-0.6}\right)$ and $\mathcal{O}\left(N^{-0.4}\right)$, respectively. To improve this difficulty, we will use a simple change of variable $z=\theta^2$ in Theorem [Theorem 3](#Sin){reference-type="ref" reference="Sin"}. So, we have the following theorem: **Theorem 11**. *(The improved sine diffusive representation (ISDR)). For $0<\alpha<1$, we have $$\label{SDR1} {}^{C}D_{0^+}^{\alpha}y(t)=\frac{4\cos(\tfrac{\pi\alpha}{2})}{\pi}\int_{0}^{\infty}\theta^{2\alpha-1}\left(\int_{0}^{t}\sin\left((t-\tau)\theta^2\right){y'(\tau)}\,d\tau\right)\,d\tau=\int_{0}^{\infty}\theta^{2\alpha-1}\omega^{IS}(\theta,t)\,d\theta,$$ where $$\label{Kernel_2_Imp} \omega^{IS}(\theta,t)=\frac{4\cos(\tfrac{\pi\alpha}{2})}{\pi} \left(\int_{0}^{t}\sin\left((t-\tau)\theta^2\right){y'(\tau)}\,d\tau\right).$$ Also, for a given function $y$ for which its first derivative exists on $[0,T]$, $\omega^{IS}(\theta,t)$ (for fixed $\theta>0$) satisfies the following second-order differential equation: $$\label{ODE3IM} \begin{cases} \displaystyle\frac{\partial^2 \omega^{IS}}{\partial t^2}+\theta^4\omega^{IS}=\frac{4\cos(\tfrac{\pi\alpha}{2})}{\pi}\ \theta^2y'(t),\\ \displaystyle \omega^{IS}(\theta,0)=\frac{\partial}{\partial t}\omega^{IS}(\theta,0)=0. \end{cases}$$* **Proof.* The proof is straightforward. ◻* In the following we present the error analysis of the new improvement of the SDR method which we denote by ISDR. The error analysis of the ISDR is fairly similar to those provided in Section [2.2](#Sec_2.2){reference-type="ref" reference="Sec_2.2"}. So we denote: $$\begin{aligned} \label{Resid_ISDR} R^{\alpha}_{IS,N,h}y(t)&:=&{}^{C}D_{0^+}^{\alpha}y(t)-{}^{C}D_{0^+,IS,N,h}^{\alpha}y(t)\nonumber\\ &=&\int_{0}^{+\infty}z^{2\alpha-1}\omega^{IS}(z,t)\,dz-\sum_{i=1}^{N}w_i^{(2\alpha-1)}\ e^{z_i^{(2\alpha-1)}}\omega^{IS}_h(z_i^{(\alpha)},t)\nonumber\\ &=&R_{IS,N,\alpha}^{GGL}\left[ e^{(.)}\omega^{IS}(.,t)\right]+Q_{IS,N,\alpha}^{GGL}\left[ e^{(.)}E^{IS}_h(.,t)\right], \end{aligned}$$ where $$\label{ErrQuadISDR} R_{IS,N,\alpha}^{GGL}\left[ e^{(.)}\omega^{IS}(.,t)\right]:=\int_{0}^{+\infty}z^{2\alpha-1}\omega^{IS}(z,t)\,dz-\sum_{i=1}^{N}w_i^{(2\alpha-1)}\ e^{z_i^{(2\alpha-1)}}\omega^{IS}(z_i^{(2\alpha-1)},t),$$ denotes the error of generalized Gauss-Laguerre formula with respect to the weight function $w(z)=z^{2\alpha-1}e^{-z}$ and $$\label{ErrODEISDR} Q_{IS,N,\alpha}^{GGL}\left[ e^{(.)}E^{IS}_h(.,t)\right]:=\sum_{i=1}^{N}w_i^{(2\alpha-1)}\ e^{z_i^{(2\alpha-1)}}\left[\omega^{IS}(z_i^{(2\alpha-1)},t)-\omega^{IS}_h(z_i^{(2\alpha-1)},t)\right],$$ used for the error of the ODE solver. In the next theorem the asymptotic behavior of the function $\omega^{IS}(z,t)$ when $z\to 0$ and $z\to+\infty$ is provided. **Theorem 12**. *Let $t \in(0, T)$ be fixed and $0<\alpha<1$.* 1. *Assume that there exists some constant $C>0$, such that for all $t\in(0,T)$ we have $|y'(t)|>C$ then function $\omega^{IS}(.,t)$ defined in [\[Kernel_2\_Imp\]](#Kernel_2_Imp){reference-type="eqref" reference="Kernel_2_Imp"} behave as: $$\label{Asymp_1-imp} z^{2\alpha-1}\omega^{IS}(z,t)\sim z^{2\alpha+1}\ \ \ \text{as}\ z\to 0.$$* 2. *Let $y(t)\in C^1[0,T]$ and $y(t)$ and $y'(t)$ be of exponential order then we have: $$\begin{aligned} \label{Asymp_3_imp} && z^{2\alpha-1}\omega^{IS}(z,t)\sim z^{2\alpha-3}\ \ \ \text{as}\ z\to +\infty. \end{aligned}$$* *Proof.* The proof is fairly similar to the proof of Theorem [Theorem 6](#AsymtoticCDRandSDR){reference-type="ref" reference="AsymtoticCDRandSDR"}. ◻ The next theorem, gives the error bound of the new improvement of the SDR method. **Theorem 13**. *Let $0<\alpha<1$. If a A-stable one-step implicit method of order $p$ with the step size $h<N^{-4}$, (where $N$ is the number of integration points in the generalized Gauss-Laguerre formula) is used for Eq. [\[ODE3IM\]](#ODE3IM){reference-type="eqref" reference="ODE3IM"}, then for $y(t)\in C^1[0,T]$ and $t\in[0,T]$, we have the overall error analysis of ISDR approximation formula:* *$$\label{OVERERRISDR} \Big|R^{\alpha}_{IS,N,h}y(t)\Big|=\mathcal{O}(N^{2\alpha-2})+\mathcal{O}(h^p)\int_{0}^{4N}e^{3T\displaystyle z^4 }\,dz.$$* *Proof.* The proof is obtained by the similar fashion which used for Theorem [Theorem 9](#OverErrCDRSDR){reference-type="ref" reference="OverErrCDRSDR"}. ◻ **Remark 7**. *It is worthy to point out that as we stated in Remark [Theorem 9](#OverErrCDRSDR){reference-type="ref" reference="OverErrCDRSDR"}, in practice, we don't need to use the step size $h$ in such a way $h<N^{-4}$.* **Remark 8**. *Thanks to the overall error of the ISDR presented in Theorem [Theorem 13](#OverErrISDR){reference-type="ref" reference="OverErrISDR"}, we expect that the new improvement method can work like the YA one (See Theorem [Theorem 10](#OverErrYA){reference-type="ref" reference="OverErrYA"}).* **Example 5**. *To show the efficiency and accuracy of the ISDR, we use this method to approximate the Caputo fractional derivative of order $\alpha$ of the following functions: $$\begin{aligned} && y(t)=t^{1.6},\ \alpha=0.4,\ t\in[0,3],\\ && y(t)=t^{3},\ \alpha=0.6,\ t\in[0,1],\\ && y(t)=\sin t,\ \alpha=0.5,\ t\in[0,1],\\ && y(t)=t^{\frac{\nu}{2}}J_{\nu}(2\sqrt{t}),\ \nu=3,\ \alpha=0.5,\ t\in[0,1].\end{aligned}$$* *To make a good comparison, the maximal errors of the approximation methods CDR, SDR, YA and ISDR obtained by the backward Euler method with $n=10^4$ and various values of $N$ of these functions are shown in Figs. [30](#Fig_14){reference-type="ref" reference="Fig_14"} and [32](#Fig_15){reference-type="ref" reference="Fig_15"}.* *It is clearly observed from Figs. [30](#Fig_14){reference-type="ref" reference="Fig_14"} and [32](#Fig_15){reference-type="ref" reference="Fig_15"} that, although, the convergence rate of the ISDR method applied to the functions $y(t)=t^3$ and $y(t)=t^{\frac{\nu}{2}}J_{\nu}(2\sqrt{t})$ with $\nu=3$, is the same as the YA method, but for other functions, the errors of ISDR method decay like the SDR one and thus we have the surprising results. This problem (may) comes from the smoothness of the Caputo fractional derivative of the functions $y(t)=t^{1.6}$ and $y(t)=\sin t$.* *{#Fig_14 width="12cm"} {#Fig_14 width="12cm"}* *{#Fig_15 width="12cm"}{#Fig_15 width="12cm"}\ * # Concluding remarks and future works {#Sec_4} The diffusive representation for the Caputo fractional derivative has very interesting feature from the numerical point of view (See [@Diethelm2009; @Diethelm2008]). This paper presents two new classes of diffusive representations with sine and cosine kernels to approximate the Caputo fractional derivative which were called as the cosine and sine diffusive representations and denoted by CDR and SDR, respectively. The error analysis of the CDR and SDR methods proved in detail. Some numerical examples have also provided to show the efficiency and accuracy of the new methods. Our numerical experiments show that for function $y(t)$ which ${}^{C}D_{0^+}^{\alpha}y(t)\in C^2[0,T]$, in opposite to the SDR method, the CDR one is faster than the Yuan and Agrawal (YA) method. So, in the final part of the paper, a new version of the SDR method (which was denoted by ISDR) is also proposed and verified numerically. The maximal error of the ISDR method for function $y(t)$ which ${}^{C}D_{0^+}^{\alpha}y(t)\in C^2[0,T]$ decays like YA method. The authors believed that the proposed methods will open a new window for researchers to investigate the diffusive representation methods in more and in-depth details. So, in the following, the authors suggest some future works which can be considered in the continuation of this paper. 1. The first suggestion is to propose some new modifications and improvements of the CDR and SDR methods to obtain some fast and accurate numerical methods to approximate the Caputo fractional derivative (See some improvements of the YA method by Diethelm and et al., [@Diethelm2008; @Diethelm2009; @Diethelm2021; @AnewDiff; @Liu2018; @MR3936246; @Birk2010]). 2. As we saw in the previous sections, the CDR and SDR methods have been used to approximate the Caputo fractional derivative of order $\alpha\in(0,1)$. So, our second suggestion is to extend these methods for $\alpha>1$. 3. Our third suggestion is to apply the CDR and SDR methods to solve problem with fractional derivatives such as: 1. Fractional ordinary and partial differential equation. 2. Fractional optimal control and calculus of variation problems. 4. Our last suggestion is to follow the idea of this paper to introduce some new generalizations of the diffusive representation. # Acknowledgment The authors would like to express their special thanks to Professor K. Diethelm for his helpful comments and suggestions on the first version of the current paper. 10 Om Prakash Agrawal. A numerical scheme for initial compliance and creep response of a system. , 36(4):444--451, 2009. Jacques Audounet, Denis Matignon, and Gérard Montseny. Semi-linear diffusive representations for nonlinear fractional differential systems. In *Nonlinear control in the Year 2000*, pages 73--82. Springer London. Daniel Baffet. A Gauss-Jacobi kernel compression scheme for fractional differential equations. , 79(1):227--248, 2019. Maamar Bettayeb and Said Djennoune. Design of sliding mode controllers for nonlinear fractional-order systems via diffusive representation. , 84(2):593--605, 2016. Carolin Birk and Chongmin Song. An improved non-classical method for the solution of fractional differential equations. , 46(5):721--734, 2010. P. Castillo and S. Gómez. On the conservation of fractional nonlinear schrödinger equation's invariants by the local discontinuous galerkin method. , 77(3):1444--1467, 2018. Mehdi Dehghan, Ehsan-Allah Hamedi, and Hassan Khosravian-Arab. A numerical scheme for the solution of a class of fractional variational and optimal control problems using the modified Jacobi polynomials. , 22(6):1547--1559, 2016. Kai Diethelm. An investigation of some nonclassical methods for the numerical approximation of Caputo-type fractional derivatives. , 47(4):361--390, 2008. Kai Diethelm. An improvement of a nonclassical numerical method for the computation of fractional derivatives. , 131(1), 2009. Kai Diethelm. , volume 2004 of *Lecture Notes in Mathematics*. Springer-Verlag, Berlin, 2010. Kai Diethelm. Fast solution methods for fractional differential equations in the modeling of viscoelastic materials. In *2021 9th International Conference on Systems and Control (ICSC)*. IEEE, 2021. Kai Diethelm. A new diffusive representation for fractional derivatives and its application. , 2022. Kai Diethelm, Virginia Kiryakova, Yuri Luchko, J. A. Tenreiro Machado, and Vasily E. Tarasov. Trends, directions for further research, and some open problems of fractional calculus. , 107(4):3245--3270, 2022. Qiang Du, Jiang Yang, and Zhi Zhou. Time-fractional allencahn equations: Analysis and numerical methods. , 85(2), 2020. Neville J. Ford and A. Charles Simpson. The numerical solution of fractional differential equations: Speed versus accuracy. , 26(4):333--346, 2001. Roberto Garrappa. Numerical solution of fractional differential equations: A survey and a software tutorial. , 6(2):16, 2018. Matthias Hinze, André Schmidt, and Remco I. Leine. Numerical solution of fractional-order ordinary differential equations using the reformulated infinite state representation. , 22(5):1321--1350, 2019. Cui--cui Ji, Weizhong Dai, and Zhi--zhong Sun. Numerical method for solving the time-fractional dual-phase-lagging heat conduction equation with the temperature-jump boundary condition. , 75(3):1307--1336, 2017. Hassan Khosravian-Arab, Mehdi Dehghan, and M. R. Eslahchi. Fractional Sturm-Liouville boundary value problems in unbounded domains: theory and applications. , 299:526--560, 2015. Hassan Khosravian-Arab, Mehdi Dehghan, and M. R. Eslahchi. Fractional spectral and pseudo-spectral methods in unbounded domains: theory and applications. , 338:527--566, 2017. Hassan Khosravian-Arab, Mehdi Dehghan, and M. R. Eslahchi. Generalized Bessel functions: theory and their applications. , 40(18):6389--6410, 2017. Hassan Khosravian-Arab and M. R. Eslahchi. Müntz pseudo-spectral method: theory and numerical experiments. , 93:Paper No. 105510, 29, 2021. Hassan Khosravian-Arab and Mohammad Reza Eslahchi. Müntz Sturm-Liouville problems: theory and numerical experiments. , 24(3):775--817, 2021. Anatoly A. Kilbas, Hari M. Srivastava, and Juan J. Trujillo. , volume 204 of *North-Holland Mathematics Studies*. Elsevier Science B.V., Amsterdam, 2006. Siu-Long Lei, Wenfei Wang, Xu Chen, and Deng Ding. A fast preconditioned penalty method for american options pricing under regime-switching tempered fractional diffusion models. , 75(3):1633--1655, 2017. Changpin Li and Min Cai. . Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2020. Changpin Li and Fanhai Zeng. . Chapman and Hall/CRC, 2015. Q. X. Liu, Y. M. Chen, and J. K. Liu. An improved YuanAgrawal method with rapid convergence rate for fractional differential equations. , 63(4):713--723, 2018. Jian-Fei Lu and Andrzej Hanyga. Wave field simulation for heterogeneous porous media with singular memory drag force. , 208(2):651--674, 2005. D Matignon. Diffusive representations for fractional Laplacian: systems theory framework and numerical issues. , T136:014009, 2009. André Schmidt and Lothar Gaul. On a critique of a numerical scheme for the calculation of fractionally damped dynamical systems. , 33(1):99--107, 2006. Ao Shen, Yuxiang Guo, and Qingping Zhang. A novel diffusive representation of fractional calculus to stability and stabilisation of noncommensurate fractional-order nonlinear systems. , 10(1):283--295, 2022. Jie Shen, Tao Tang, and Li-Lian Wang. . Springer Berlin Heidelberg, 2011. Hiroshi Sugiura and Takemitsu Hasegawa. Quadrature rule for abel's equations: Uniformly approximating fractional derivatives. , 223(1):459--468, 2009. C. Trinks and P. Ruge. Treatment of dynamic systems with fractional derivatives without evaluating memory-integrals. , 29(6):471--476, 2002. Lixia Yuan and Om P. Agrawal. A numerical scheme for dynamic systems containing fractional derivatives. , 124(2):321--324, 2002. Mohsen Zayernouri and George Em Karniadakis. Fractional Sturm-Liouville eigen-problems: theory and numerical approximation. , 252:495--517, 2013. Daniel Zwillinger. . Academic Press, 7 edition, 2007. [^1]: Department of Applied Mathematics and Computer Science, Faculty of Mathematics and Statistics, University of Isfahan, 81746-73441 Isfahan, Iran. *E-mail Address:* h.khosravian\@sci.ui.ac.ir. [^2]: Corresponding author. Department of Applied Mathematics, Faculty of Mathematics and Computer Science, Amirkabir University of Technology (Tehran Polytechnic), No.424, Hafez Avenue, Tehran, Iran. *E-mail Address:* mdehghan\@aut.ac.ir, mdehghan.aut\@gmail.com.

arxiv_math

--- abstract: | We consider the Boltzmann equation with the soft potential and angular cutoff. Inspired by the methods from dispersive PDEs, we establish its sharp local well-posedness and ill-posedness in $H^{s}$ Sobolev space. We find the well/ill-posedness separation at regularity $s=\frac{d-1}{2}$, strictly $\frac{1}{2}$-derivative higher than the scaling-invariant index $s=\frac{d-2}{2}$, the usually expected separation point. address: - Department of Mathematics, University of Rochester, Rochester, NY 14627, USA - School of Mathematical Sciences, Peking University, Beijing, 100871, China - School of Mathematical Sciences, Peking University, Beijing, 100871, China author: - Xuwen Chen - Shunlin Shen - Zhifei Zhang bibliography: - references.bib title: Well/Ill-posedness of the Boltzmann Equation with Soft Potential --- # Introduction We consider the Boltzmann equation $$\label{equ:Boltzmann} \left\{ \begin{aligned} \left( \partial_t + v \cdot \nabla_x \right) f (t,x,v) =& Q(f,f),\\ f(0,x,v)=& f_{0}(x,v), \end{aligned} \right.$$ where $f(t,x,v)$ is the distribution function for the particles at time $t\geq 0$, position $x\in \mathbb{R}^{d}$ and velocity $v\in \mathbb{R}^{d}$. The collision operator $Q$ is conventionally split into a gain term and a loss term $$Q(f,g)=Q^{+}(f,g)-Q^{-}(f,g)$$ where the gain term is $$\begin{aligned} Q^{+}(f,g)=&\int_{\mathbb{R}^{d}}\int_{\mathbb{S}^{d-1}} f(v^{\ast })g(u^{\ast}) B(u-v,\omega)dud\omega,\end{aligned}$$ and the loss term is $$\begin{aligned} Q^{-}(f,g)=& \int_{\mathbb{R}^{d}}\int_{\mathbb{S}^{d-1}} f(v)g(u) B(u-v,\omega) du d\omega,\end{aligned}$$ with the relation between the pre-collision and after-collision velocities that $$\begin{aligned} u^{*}=u+\omega\cdot (v-u) \omega,\quad v^{*}=v-\omega\cdot(v-u)\omega.\end{aligned}$$ The Boltzmann collision kernel function $B(u-v,\omega)$ is a non-negative function depending only on the relative velocity $|u-v|$ and the deviation angle $\theta$ through $\cos \theta:=\frac{u-v}{|u-v|}\cdot \omega$. Throughout the paper, we consider $$\begin{aligned} \label{equ:kernel function} B(u-v,\omega)=|u-v|^{\gamma}\textbf{b}(\cos \theta)\end{aligned}$$ under the Grad's angular cutoff assumption $$\begin{aligned} 0\leq \textbf{b}(\cos \theta)\leq C|\cos\theta|.\end{aligned}$$ The different ranges $\gamma<0$, $\gamma=0$, $\gamma>0$ correspond to soft potentials, Maxwellian molecules, and hard potentials, respectively. See also [@cercignani1994mathematical; @cercignani1988boltzmann; @villani2002review] for a more detailed physics background. This collision kernel [\[equ:kernel function\]](#equ:kernel function){reference-type="eqref" reference="equ:kernel function"} comes from an important model case of inverse-power law potentials and there have been a large amount of literature devoted to various problems for this model, such as its hydrodynamics limits which provide a description between the kinetic theory and hydrodynamic equations. For a detailed presentation and the derivation of macroscopic equations from the fundamental laws of physics, see for example [@saint2009hydrodynamcis]. The Cauchy problem for the Boltzmann equation is one of the fundamental mathematical problems in kinetic theory, as it is of vital importance for the physical interpretation and practical application. For instance, in the absence of uniqueness or continuous dependence on the initial condition, numerical calculations and algorithms, even if they can be done, could present puzzling results. Despite the innovative work [@diperna1989cauchy; @gressman2011global] and many nice developments, the well/ill-posedness of the Boltzmann equation remains largely open. So far, there have been many developed methods and techniques for well-posedness, see for example [@alexandre2011global; @alexandre2013local; @ampatzoglou2022global; @arsenio2011global; @duan2017global; @duan2016global; @guo2003classical; @he2017well; @he2023cauchy; @sohinger2014boltzmann; @toscani1988global]. In the recent series of paper [@chen2019local; @chen2019moments; @chen2021small], by taking dispersive techniques on the study of the quantum many-body hierarchy dynamics, especially space-time collapsing/multi-linear estimates techniques (see for instance [@chen2015unconditional; @chen2014derivation; @chen2013rigorous; @chen2016focusing; @chen2016collapsing; @chen2016klainerman; @chen2016correlation; @chen2019derivation; @chen2022quantitative; @chen2023derivation; @herr2016gross; @herr2019unconditional; @kirkpatrick2011derivation; @klainerman2008uniqueness; @sohinger2015rigorous]), T. Chen, Denlinger, and Pavlovi$\acute{c}$ provided a new approach to prove the well-posedness of the Boltzmann equation and suggested the possibility of a systematic study of Boltzmann equation using dispersive tools. With the dispersive techniques, the regularity index for well-posedness, which is usually at least the continuity threshold $s>\frac{d}{2}$, has been relaxed to $s>\frac{d-1}{2}$ for both Maxwellian molecules and hard potentials with cutoff in [@chen2019local]. It is of mathematical and physical interest to prove well-posedness at the optimal regularity. From the scaling point of view, the Boltzmann equation [\[equ:Boltzmann\]](#equ:Boltzmann){reference-type="eqref" reference="equ:Boltzmann"} is invariant under the scaling $$\begin{aligned} f_{\lambda}(t,x,v)=\lambda^{\alpha+(d-1+\gamma)\beta}f(\lambda^{\alpha-\beta}t,\lambda^{\alpha}x,\lambda^{\beta}v),\end{aligned}$$ for any $\alpha$, $\beta\in \mathbb{R}$ and $\lambda>0$. Then in the $L^{2}$ setting, it holds that $$\begin{aligned} \Vert |\nabla_{x}|^{s}|v|^{r}f_{\lambda}\Vert _{L_{xv}^{2}}=\lambda^{^{\alpha+(d-1+\gamma)\beta}}\lambda^{\alpha s-\beta r}\lambda^{-\frac{d}{2}\alpha-\frac{d}{2}\beta} \Vert |\nabla_{x}|^{s}|v|^{r}f\Vert _{L_{xv}^{2}}.\end{aligned}$$ This gives the scaling-critical index $$\begin{aligned} \label{equ:scaling,L2} s=\frac{d-2}{2},\quad r=s+\gamma.\end{aligned}$$ From the past experience of scaling analysis, it is believed that the well/ill-posedness threshold[^1] in $H^{s}$ Sobolev space is $s_{c}=\frac{d-2}{2}$ with $r\geq 0$. Surprisingly, for the 3D constant kernel case, X. Chen and Holmer in [@chen2022well] prove the well/ill-posedness threshold in $H^{s}$ Sobolev space is exactly at regularity $s=\frac{d-1}{2}$, and thus point out the actual optimal regularity for the global well-posedness problem. On the one hand, while there are many well-known progress such as [@christ2003asymptotics; @christ2003ill-posedness; @kenig1996bilinear; @kenig1996quadratic; @kenig2001ill-posedness; @molinet2001ill-posedness; @molinet2002well-posedness; @tzvetkov2006ill-posedness] regarding the study of dispersive equations, the illposedness of the Boltzmann equation remains largely open away from [@chen2022well]. One certainly would like to have the sharp problem resolved for the Boltzmann equations. On the other hand, To initiate a systematic study of a large project including sharp well-posedness, blow-up analysis, regularity criteria, etc, it is of priority to find out the well/ill-posedness separation point. In the paper, moving forward from the special case [@chen2022well], we investigate the general kernel with soft potentials, for which both the sharp well-posedness and ill-posedeness are open. We settle this problem and provide the well/ill-posedness threshold. With the finding of this optimal regularity index, we deal with the sharp small data global well-posedness in another paper [@chen2023sharp].[^2] We start with the connection between the analysis of [\[equ:Boltzmann\]](#equ:Boltzmann){reference-type="eqref" reference="equ:Boltzmann"} and the theory of nonlinear dispersive PDEs. Let $\widetilde{f}(t,x,\xi)$ be the inverse Fourier transform in the velocity variable, that is, $$\begin{aligned} \widetilde{f}(t,x,\xi)=\mathcal{F}_{v\mapsto \xi}^{-1}(f).\end{aligned}$$ Then the linear part of [\[equ:Boltzmann\]](#equ:Boltzmann){reference-type="eqref" reference="equ:Boltzmann"} is changed into the symmetric hyperbolic Schrödinger equation $$\begin{aligned} i\partial_{t}\widetilde{f}+\nabla_{\xi}\cdot \nabla_{x}\widetilde{f}=0,\end{aligned}$$ which, in the nonlinear context, enables the application of Strichartz estimates that $$\begin{aligned} \Vert e^{it\nabla_{\xi}\cdot \nabla_{x}}\widetilde{f}_{0}\Vert _{L_{t}^{q}L_{x\xi}^{p}}\lesssim \Vert \widetilde{f}_{0}\Vert _{L_{x\xi}^{2}},\quad \frac{2}{q}+\frac{2d}{p}=d,\quad q\geq 2,\ d\geq2.\end{aligned}$$ We introduce the Sobolev norms $$\begin{aligned} \Vert \widetilde{f}\Vert _{H_{x}^{s}H_{\xi}^{r}}=\Vert \langle \nabla_{x} \rangle ^{s}\langle \nabla_{\xi} \rangle ^{r}\widetilde{f}\Vert _{L_{x\xi}^{2}}=\Vert \langle \nabla_{x} \rangle ^{s}\langle v \rangle ^{r}f\Vert _{L_{xv}^{2}} =\Vert f\Vert _{L_{v}^{2,r}H_{x}^{s}},\end{aligned}$$ and the Fourier restriction norms (see [@beals1983self; @bourgain1993fourier1; @bourgain1993fourier2; @klainerman1993space; @rauch1982nonlinear]) $$\begin{aligned} \label{equ:fourier restriction norm} \Vert \widetilde{f}\Vert _{X^{s,r,b}}=\Vert \hat{f}(\tau,\eta,v)\langle \tau+\eta\cdot v \rangle ^{b}\langle \eta \rangle ^{s}\langle v \rangle ^{r}\Vert _{L_{\tau,\eta,v}^{2}},\end{aligned}$$ where $\hat{f}(\tau,\eta,v)$ denotes the Fourier transform of $\widetilde{f}(t,x,\xi)$ in $(t,x,\xi)\mapsto (\tau,\eta,v)$, and is thus the Fourier transform of $f(t,x,v)$ itself in only $(t,x)\mapsto (\tau,\eta)$, that is, $$\hat{f}(\tau,\eta,v)=\mathcal{F}(\widetilde{f})=\mathcal{F}_{(t,x)\mapsto(\tau,\eta)}\left( f \right) .$$ It is customary to define their finite time restrictions via $$\begin{aligned} \label{equ:fourier restriction norm,finite time} \Vert \widetilde{f}\Vert _{X_{T}^{s,r,b}}=\inf\left\{ \Vert F\Vert _{X^{s,r,b}}:F|_{[-T,T]}=\widetilde{f} \right\} .\end{aligned}$$ We recall the definition of well-posedness, see for example [@mihaela2023local; @tao2006nonlinear]. **Definition 1**. We say that ([\[equ:Boltzmann\]](#equ:Boltzmann){reference-type="ref" reference="equ:Boltzmann"}) is well-posed in $L_{v}^{2,r}H_{x}^{s}$ if for each $R>0$, there exists a time $T=T(R)>0$, and a set $X$, such that all of the following are satisfied. 1. (Existence and Uniqueness) For each $f_{0}\in L_{v}^{2,r}H_{x}^{s}$ with $\left\Vert f_{0}\right\Vert _{L_{v}^{2,r}H_{x}^{s}}\leqslant R$, there exists a unique solution $f(t,x,v)$ to the integral equation of [\[equ:Boltzmann\]](#equ:Boltzmann){reference-type="eqref" reference="equ:Boltzmann"} in $$\begin{aligned} C([-T,T];L_{v}^{2,r}H_{x}^{s})\bigcap X.\end{aligned}$$ Moreover, $f(t,x,v)\geqslant 0$ if $f_{0}\geqslant 0$.[^3] 2. (Uniform Continuity of the Solution Map)[^4] The map $f_{0}\mapsto f$ is uniform continuous with the $C([-T,T];L_{v}^{2,r}H_{x}^{s})$ norm. Specifically, suppose $f$ and $g$ are two solutions to ([\[equ:Boltzmann\]](#equ:Boltzmann){reference-type="ref" reference="equ:Boltzmann"}) on $[-T,T],$ $\forall \varepsilon >0$, $\exists\ \delta (\varepsilon )$ independent of $f$ or $g$ such that $$\left\Vert f(t)-g(t)\right\Vert _{C([-T,T];{L_{v}^{2,r}H_{x}^{s}} )}<\varepsilon \text{ provided that }\left\Vert f(0)-g(0)\right\Vert _{L_{v}^{2,r}H_{x}^{s}}<\delta (\varepsilon ) . \label{eq:uniform continuity}$$ We take $X$ to be the Fourier restriction norm space $X_{T}^{s,s+\gamma,b}$ defined by [\[equ:fourier restriction norm,finite time\]](#equ:fourier restriction norm,finite time){reference-type="eqref" reference="equ:fourier restriction norm,finite time"} with $b\in (\frac{1}{2},1)$. **Theorem 2** (Main Theorem). *Let $d=2,3$.* 1. *For $s>\frac{d-1}{2}$, $\frac{1-d}{2}\leq \gamma\leq 0$, [\[equ:Boltzmann\]](#equ:Boltzmann){reference-type="eqref" reference="equ:Boltzmann"} is locally well-posed in $L_{v}^{2,s+\gamma}H_{x}^{s}$.* 2. *For $0 \leq s_{0}<\frac{d-1}{2}$, $\frac{1-d}{2}\leq \gamma\leq 0$, $r_{0}=\max\left\{ 0,s_{0}+\gamma \right\}$, [\[equ:Boltzmann\]](#equ:Boltzmann){reference-type="eqref" reference="equ:Boltzmann"} is ill-posed in $L_{v}^{2,r_{0}}H_{x}^{s_{0}}$ in the sense that the data-to-solution map is not uniformly continuous. In particular, for each $M\gg1$, there exists a time sequence $\left\{t_{0}^{M}\right\} _{M}$ such that $$t_{0}^{M}<0, \quad t_{0}^{M}\nearrow 0,$$ and two solutions $f^{M}(t)$, $g^{M}(t)$ in $[t_{0}^{M},0]$ with $$\begin{aligned} \Vert f^{M}(t_{0}^{M})\Vert _{L_{v}^{2,r_{0}}H_{x}^{s_{0}}}\sim \Vert g^{M}(t_{0}^{M})\Vert _{L_{v}^{2,r_{0}}H_{x}^{s_{0}}} \sim 1,\end{aligned}$$ such that they are initially close at $t=t_{0}^{M}$ $$\Vert f^{M}(t_{0}^{M})-g^{M}(t_{0}^{M})\Vert _{L_{v}^{2,r_{0}}H_{x}^{s_{0}}}\leqslant \frac{1}{\ln \ln M}\ll 1,$$ but become fully separated at $t=0$ $$\Vert f^{M}(0)-g^{M}(0)\Vert _{L_{v}^{2,r_{0}}H_{x}^{s_{0}}}\sim 1.$$* Theorem [Theorem 2](#thm:main theorem){reference-type="ref" reference="thm:main theorem"} is the main novelty, which finds the well/ill-posedness threshold, by establishing the sharp local well-posedness, and proving the ill-posedness for the soft potential case. There have been many nice work on the well-posedness part by the energy method which requires higher regularity, see for example [@guo2003classical; @guo2003the; @strain2008exponential; @alexandre2013local]. For both Maxwellian molecules and hard potentials, the regularity index $s>\frac{d-1}{2}$ for well-posedness was achieved in [@chen2019local] without ill-posedness. Our well-posedness result solves the remaining soft potential case. We remark that, as scaling [\[equ:scaling,L2\]](#equ:scaling,L2){reference-type="eqref" reference="equ:scaling,L2"} in $L^{2}$ setting gives the restriction that $s+\gamma\geq 0$, the range $\frac{1-d}{2}\leq \gamma\leq 0$ should be sharp if one seeks the optimal regularity $s>\frac{d-1}{2}$. In addition, the endpoint case $\gamma=-1$ with $d=3$ plays an important role in the derivation of the Boltzmann equation from quantum many-body dynamics in [@chen2023derivationboltzmann], where the collision kernel is composed of part hard sphere and part inverse power potential: $$\begin{aligned} \label{equ:hard,soft,kernel} B(u-v,\omega)=\left( 1_{\left\{ |u-v|\leq 1 \right\} }|u-v|+1_{\left\{ |u-v|\geq 1 \right\} }|u-v|^{-1} \right) \textbf{b}(\frac{u-v}{|u-v|}\cdot \omega),\end{aligned}$$ which also provides yet another physical background to our problem here. Our proof for ill-posedness also works for kernel [\[equ:hard,soft,kernel\]](#equ:hard,soft,kernel){reference-type="eqref" reference="equ:hard,soft,kernel"}. **Corollary 3**. For $d=3$, $0 \leq s_{0}<1$, [\[equ:Boltzmann\]](#equ:Boltzmann){reference-type="eqref" reference="equ:Boltzmann"} is ill-posed in $L_{v}^{2}H_{x}^{s_{0}}$ with the kernel [\[equ:hard,soft,kernel\]](#equ:hard,soft,kernel){reference-type="eqref" reference="equ:hard,soft,kernel"} in the sense that the data-to-solution map is not uniformly continuous. ## Outline of the Paper In Section [2](#section:Well-posedness){reference-type="ref" reference="section:Well-posedness"}, we prove the well-posedeness of [\[equ:Boltzmann\]](#equ:Boltzmann){reference-type="eqref" reference="equ:Boltzmann"}. The bilinear estimates for gain/loss terms are the key step to conclude the well-posedness and the proof highly relies on the techniques from dispersive PDEs. In Section [2.1](#section:Bilinear Estimate for Loss Term){reference-type="ref" reference="section:Bilinear Estimate for Loss Term"}, we appeal to dispersive estimates to prove the loss term bilinear estimate. This can be directly handled because of the factorization of the kernel. In Section [2.2](#section:Bilinear Estimate for Gain Term){reference-type="ref" reference="section:Bilinear Estimate for Gain Term"}, we deal with the gain term, which requires a more subtle analysis due to the complicated partial convolution structure. One important observation is that the energy conservation provides a lower bound estimate for after-collision velocities, which enables the application of the Littlewood-Paley theory and frequency analysis techniques in multi-linear estimates. Then with a convolution type estimate in [@alonso2010estimates], we are able to establish the gain term bilinear estimate with the help of Strichartz estimates in the Fourier restriction norm space. Finally, in Section [2.3](#section:Proof of Well-posedness){reference-type="ref" reference="section:Proof of Well-posedness"}, we complete the proof of well-posedness after our built-up $X^{s,r,b}$ spaces and its related frequency analysis in this context. In Section [3](#section:Ill-posedness){reference-type="ref" reference="section:Ill-posedness"}, we prove the ill-posedness of [\[equ:Boltzmann\]](#equ:Boltzmann){reference-type="eqref" reference="equ:Boltzmann"}. The idea is to construct an approximation solution which has the norm deflation property and then perturb it into an exact solution. We improvise and sharpen the prototype approximation solution found in [@chen2022well]. To overcome the singularities carried by the soft potentials, which were known to be the main difficulties, we introduce a new scaling on the approximation solution, create an elaborate $Z$-norm, which is used to prove a closed estimate for the gain and loss terms, that is, $$\label{equ:closed estimate,z norm} \|Q^\pm (f_1,f_2) \|_{Z} \lesssim \|f_1\|_{Z} \|f_2\|_{Z},$$ and conclude the existence of small corrections. With this new treatment, the extra restriction that $s_{0}>\frac{1}{2}$ in [@chen2022well] can now be removed. In Section [3.1](#equ:Bounds on the approximation solution){reference-type="ref" reference="equ:Bounds on the approximation solution"}, we first construct the approximation solution $f_{\textrm{a}}$ and prove its norm deflation. Then in Section [3.1.1](#section:Discussion on the $L^{1}$-based space and hard potentials){reference-type="ref" reference="section:Discussion on the $L^{1}$-based space and hard potentials"}, we give a discussion on the $L^{1}$-based spaces and the hard potentials case, for which our approximation solution also gives desired bad behaviors.[^5] Therefore, a similar mechanism of norm deflation in different settings is possible and deserves further investigations. In Sections [3.2](#section:$Z$-norm Bounds on the Approximation Solution){reference-type="ref" reference="section:$Z$-norm Bounds on the Approximation Solution"}--[3.4](#section:Bounds on the Correction Term){reference-type="ref" reference="section:Bounds on the Correction Term"}, we introduce the $Z$-norm space and perform a perturbation argument to turn the approximation solution into the exact solution. In Section [3.2](#section:$Z$-norm Bounds on the Approximation Solution){reference-type="ref" reference="section:$Z$-norm Bounds on the Approximation Solution"}, we first provide the $Z$-norm bounds on the approximation solution. Then in Section [3.3](#section:Bounds on the Error Terms){reference-type="ref" reference="section:Bounds on the Error Terms"}, we deal with the error terms and prove the $Z$-norm error estimates. Proving the error estimates, as it includes a large quantity of error terms involving singularities at which we need geometric techniques on the nonlinear interactions between frequencies, is the most intricate part which we treat in Sections [3.3.1](#section:Analysis of term1){reference-type="ref" reference="section:Analysis of term1"}--[3.3.4](#section:Analysis of Q+){reference-type="ref" reference="section:Analysis of Q+"},. After dealing with the error terms, we prove that there is an exact solution which is mostly $f_{\textrm{a}}$ in Section [3.4](#section:Bounds on the Correction Term){reference-type="ref" reference="section:Bounds on the Correction Term"}, and thus conclude the ill-posedness result in Section [3.5](#section:Proof of Illposedness){reference-type="ref" reference="section:Proof of Illposedness"}. After the proof of the main theorem, we put and review some tools in Appendix [4](#section:Sobolev-type and Time-independent Bilinear Estimates){reference-type="ref" reference="section:Sobolev-type and Time-independent Bilinear Estimates"} and the Strichartz estimates in Appendix [5](#section:Strichartz Estimates){reference-type="ref" reference="section:Strichartz Estimates"}. \ **Acknowledgements** The authors would like to thank professors Yan Guo and Tong Yang for helpful discussions about this work. X. Chen was supported in part by NSF grant DMS-2005469 and a Simons fellowship numbered 916862, S. Shen was supported in part by the Postdoctoral Science Foundation of China under Grant 2022M720263, and Z. Zhang was supported in part by NSF of China under Grant 12171010 and 12288101. # Well-posedness {#section:Well-posedness} To conclude the well-posedeness of [\[equ:Boltzmann\]](#equ:Boltzmann){reference-type="eqref" reference="equ:Boltzmann"}, it suffices to prove the following bilinear estimates $$\begin{aligned} \label{equ:lwp,bilinear estimate} \Vert \langle \nabla_{x} \rangle ^{s}\langle v \rangle ^{s+\gamma}Q^{\pm}(f,g)\Vert _{L_{t}^{2}L_{x,v}^{2}}\lesssim \Vert \widetilde{f}\Vert _{X^{s,s+\gamma,b}}\Vert \widetilde{g}\Vert _{X^{s,s+\gamma,b}}.\end{aligned}$$ Note that no $v$-variable Fourier transform of the collision kernel in [\[equ:lwp,bilinear estimate\]](#equ:lwp,bilinear estimate){reference-type="eqref" reference="equ:lwp,bilinear estimate"} is needed if we fully work in the $X^{s,s+\gamma,b}$ space. Here, we will work on the $(x,\xi)$ side and prove [\[equ:lwp,bilinear estimate\]](#equ:lwp,bilinear estimate){reference-type="eqref" reference="equ:lwp,bilinear estimate"} by use of the Fourier transform of the kernel. Taking the inverse $v$-variable Fourier transform on both side of [\[equ:Boltzmann\]](#equ:Boltzmann){reference-type="eqref" reference="equ:Boltzmann"}, we get $$\begin{aligned} i\partial_{t}\widetilde{f}+\nabla_{\xi}\cdot \nabla_{x}\widetilde{f}=i\mathcal{F}_{v\mapsto \xi}^{-1}\left[ Q(f,f) \right] .\end{aligned}$$ By the well-known Bobylev identity in a more general case, see for example [@alexandre2000entropy; @desvillettes2003use], it holds that (up to an unimportant constant) $$\begin{aligned} \mathcal{F}_{v\mapsto \xi}^{-1}\left[ Q^{-}(f,g) \right] (\xi)=& \Vert \textbf{b}\Vert _{L^{1}(\mathbb{S}^{d-1})}\int \frac{\widetilde{f}(\xi-\eta) \widetilde{g}(\eta)}{|\eta|^{d+\gamma}}d\eta,\\ \mathcal{F}_{v\mapsto \xi}^{-1}\left[ Q^{+}(f,g) \right] (\xi)=&\int_{\mathbb{R}^{d}\times \mathbb{S}^{d-1}}\frac{\widetilde{f}(\xi^{+}+\eta)\widetilde{g}(\xi^{-}-\eta)}{|\eta|^{d+\gamma}} \textbf{b}(\frac{\xi}{|\xi|}\cdot \omega)d\eta d\omega,\end{aligned}$$ where $\xi^{+}=\frac{1}{2}\left( \xi+|\xi|\omega \right)$ and $\xi^{-}=\frac{1}{2}\left( \xi-|\xi|\omega \right)$. For convenience, we take the notation that $\widetilde{Q}^{\pm}(\widetilde{f},\widetilde{g})=\mathcal{F}_{v\mapsto \xi}^{-1}\left[ Q^{\pm}(f,g) \right]$. In Sections [2.1](#section:Bilinear Estimate for Loss Term){reference-type="ref" reference="section:Bilinear Estimate for Loss Term"}-[2.2](#section:Bilinear Estimate for Gain Term){reference-type="ref" reference="section:Bilinear Estimate for Gain Term"}, we establish the bilinear estimates for the loss and gain terms respectively. Then in Section [2.3](#section:Proof of Well-posedness){reference-type="ref" reference="section:Proof of Well-posedness"}, we complete the proof of the well-posedness of [\[equ:Boltzmann\]](#equ:Boltzmann){reference-type="eqref" reference="equ:Boltzmann"}. ## Bilinear Estimate for Loss Term {#section:Bilinear Estimate for Loss Term} **Lemma 4**. *For $s>\frac{d-1}{2}$, it holds that $$\begin{aligned} \label{equ:bilinear estimate,Q-,L2} \Vert \langle \nabla_{x} \rangle ^{s}\langle \nabla_{\xi} \rangle ^{s+\gamma}\widetilde{Q}^{-}(\widetilde{f},\widetilde{g})\Vert _{L_{t}^{2+}L_{x\xi}^{2}}\lesssim \Vert \widetilde{f}\Vert _{X^{s,s+\gamma,b}}\Vert \widetilde{g}\Vert _{X^{s,s+\gamma,b}}.\end{aligned}$$* *Proof.* By the fractional Leibniz rule in Lemma [Lemma 23](#lemma:generalized leibniz rule){reference-type="ref" reference="lemma:generalized leibniz rule"}, we have $$\begin{aligned} \left\Vert \widetilde{Q}^{-}(\widetilde{f},\widetilde{g})\right\Vert_{L_{t}^{2+}H_{x}^{s}H_{\xi}^{s+\gamma}} = &\left\Vert \left\langle \nabla_{x}\right\rangle^{s} \int \left\langle \nabla _{\xi}\right\rangle ^{s+\gamma}\widetilde{f}(t,x,\xi-\eta) \frac{\widetilde{g}(t,x,\eta)}{|\eta|^{d+\gamma}}d\eta \right\Vert _{L_{t}^{2+}L_{x\xi}^{2}}\\ \lesssim& \left\Vert \int \big \Vert \langle \nabla_{x} \rangle ^{s} \left\langle \nabla _{\xi}\right\rangle ^{s+\gamma}\widetilde{f}(t,x,\xi-\eta) \big \Vert _{L_{x}^{2}} \Big\Vert \frac{\widetilde{g}(t,x,\eta)}{|\eta|^{d+\gamma}} \Big \Vert _{L_{x}^{\infty}}d\eta \right\Vert _{L_{t}^{2+}L_{\xi }^{2}}\\ &+\left\Vert \int \big \Vert \left\langle \nabla _{\xi}\right\rangle ^{s+\gamma}\widetilde{f}(t,x,\xi-\eta) \big \Vert _{L_{x}^{2d+}} \Big\Vert \frac{\langle \nabla_{x} \rangle ^{s} \widetilde{g}(t,x,\eta)}{|\eta|^{d+\gamma}} \Big \Vert _{L_{x}^{\frac{2d}{d-1}-}}d\eta \right\Vert _{L_{t}^{2+}L_{\xi }^{2}}.\end{aligned}$$ Applying Sobolev inequalities that $W^{s,\frac{2d}{d-1}-}\hookrightarrow L^{\infty}$, $W^{s,2}\hookrightarrow L^{2d+}$ and Young's inequality, $$\begin{aligned} \left\Vert \widetilde{Q}^{-}(\widetilde{f},\widetilde{g})\right\Vert_{L_{t}^{2+}H_{x}^{s}H_{\xi}^{s+\gamma}}\lesssim& \big \Vert \langle \nabla_{x} \rangle ^{s} \left\langle \nabla_{\xi}\right\rangle^{s+\gamma}\widetilde{f}(t,x,\xi) \big \Vert _{L_{t}^{\infty}L_{\xi}^{2}L_{x}^{2}} \Big\Vert \frac{\langle \nabla_{x} \rangle ^{s} \widetilde{g}(t,x,\eta)}{|\eta|^{d+\gamma}} \Big \Vert _{L_{t}^{2+}L_{\eta}^{1}L_{x}^{\frac{2d}{d-1}-}}\notag\\ &+ \big \Vert \left\langle \nabla _{\xi}\right\rangle ^{s+\gamma}\widetilde{f}(t,x,\xi) \big \Vert _{L_{t}^{\infty}L_{\xi}^{2}L_{x}^{2d+}} \Big\Vert \frac{\langle \nabla_{x} \rangle ^{s} \widetilde{g}(t,x,\eta)}{|\eta|^{d+\gamma}} \Big \Vert _{L_{t}^{2+}L_{\eta}^{1}L_{x}^{\frac{2d}{d-1}-}}\notag\\ \lesssim& \big \Vert \langle \nabla_{x} \rangle ^{s} \left\langle \nabla_{\xi}\right\rangle^{s+\gamma}\widetilde{f}(t,x,\xi) \big \Vert _{L_{t}^{\infty}L_{\xi}^{2}L_{x}^{2}} \Big\Vert \frac{\langle \nabla_{x} \rangle ^{s} \widetilde{g}(t,x,\eta)}{|\eta|^{d+\gamma}} \Big \Vert _{L_{t}^{2+}L_{\eta}^{1}L_{x}^{\frac{2d}{d-1}-}}.\label{equ:bilinear estimate,Q-,g}\end{aligned}$$ We are left to deal with the last term on the right hand side of [\[equ:bilinear estimate,Q-,g\]](#equ:bilinear estimate,Q-,g){reference-type="eqref" reference="equ:bilinear estimate,Q-,g"}. Set $$\begin{aligned} G(\eta)=\Big\Vert \langle \nabla_{x} \rangle ^{s} \widetilde{g}(t,x,\eta) \Big \Vert _{L_{x}^{\frac{2d}{d-1}-}}.\end{aligned}$$ Then by Hardy-Littlewood-Sobolev inequality [\[equ:endpoint estimate,hls,L1\]](#equ:endpoint estimate,hls,L1){reference-type="eqref" reference="equ:endpoint estimate,hls,L1"} in Lemma [Lemma 25](#lemma:endpoint estimate,hls){reference-type="ref" reference="lemma:endpoint estimate,hls"} with $\lambda=d+\gamma$, we obtain $$\begin{aligned} \int \frac{G(\eta)}{|\eta|^{d+\gamma}}d\eta\lesssim \Vert G\Vert _{L^{\frac{2d}{d-1}-}}^{\alpha}\Vert G\Vert _{L^{\frac{d}{-\gamma}+}}^{1-\alpha}.\end{aligned}$$ Therefore, we have $$\begin{aligned} & \Big\Vert \frac{\langle \nabla_{x} \rangle ^{s} \widetilde{g}(t,x,\eta)}{|\eta|^{d+\gamma}} \Big \Vert _{L_{t}^{2+}L_{\eta}^{1}L_{x}^{\frac{2d}{d-1}-}}\\ \lesssim& \big \Vert \langle \nabla_{x} \rangle ^{s} \widetilde{g}(t,x,\eta) \big \Vert _{L_{t}^{2+}L_{\eta}^{\frac{2d}{d-1}-}L_{x}^{\frac{2d}{d-1}-}}^{\alpha} \big \Vert \langle \nabla_{x} \rangle ^{s} \widetilde{g}(t,x,\eta) \big \Vert _{L_{t}^{2+}L_{\eta}^{\frac{d}{-\gamma}+}L_{x}^{\frac{2d}{d-1}-}}^{1-\alpha}\\ \leq& \big \Vert \langle \nabla_{x} \rangle ^{s} \widetilde{g}(t,x,\eta) \big \Vert _{L_{t}^{2+}L_{x}^{\frac{2d}{d-1}-}L_{\eta}^{\frac{2d}{d-1}-}}^{\alpha} \big \Vert \langle \nabla_{x} \rangle ^{s} \widetilde{g}(t,x,\eta) \big \Vert _{L_{t}^{2+}L_{x}^{\frac{2d}{d-1}-}L_{\eta}^{\frac{d}{-\gamma}+}}^{1-\alpha}\end{aligned}$$ where in the last inequality we have used the Minkowski inequality. Applying Sobolev inequality that $W^{s+\gamma,\frac{2d}{d-1}-}\hookrightarrow L^{\frac{d}{-\gamma}+}$ and Strichartz estimate [\[equ:strichartz estimate,xsb\]](#equ:strichartz estimate,xsb){reference-type="eqref" reference="equ:strichartz estimate,xsb"}, we arrive at $$\begin{aligned} &\Big\Vert \frac{\langle \nabla_{x} \rangle ^{s} \widetilde{g}(t,x,\eta)}{|\eta|^{d+\gamma}} \Big \Vert _{L_{t}^{2+}L_{\eta}^{1}L_{x}^{\frac{2d}{d-1}-}}\\ \leq& \big \Vert \langle \nabla_{x} \rangle ^{s} \widetilde{g}(t,x,\eta) \big \Vert _{L_{t}^{2+}L_{x}^{\frac{2d}{d-1}-}L_{\eta}^{\frac{2d}{d-1}-}}^{\alpha} \big \Vert \langle \nabla_{\eta} \rangle ^{s+\gamma}\langle \nabla_{x} \rangle ^{s} \widetilde{g}(t,x,\eta) \big \Vert _{L_{t}^{2+}L_{x}^{\frac{2d}{d-1}-}L_{\eta}^{\frac{2d}{d-1}-}}^{1-\alpha}\\ \leq&\big \Vert \langle \nabla_{\eta} \rangle ^{s+\gamma}\langle \nabla_{x} \rangle ^{s} \widetilde{g}(t,x,\eta) \big \Vert _{L_{t}^{2+}L_{x}^{\frac{2d}{d-1}-}L_{\eta}^{\frac{2d}{d-1}-}}\\ \leq& \Vert \widetilde{g}\Vert _{X^{s,s+\gamma,b}}.\end{aligned}$$ Hence, we complete the proof of [\[equ:bilinear estimate,Q-,L2\]](#equ:bilinear estimate,Q-,L2){reference-type="eqref" reference="equ:bilinear estimate,Q-,L2"}. ◻ ## Bilinear Estimate for Gain Term {#section:Bilinear Estimate for Gain Term} Before proving the bilinear estimate for the gain term, we first give a useful lemma as follows. **Lemma 5**. *Let $\frac{1}{p}+\frac{1}{q}=\frac{1}{2}$. $$\begin{aligned} \label{equ:Q+,bilinear estimate} \Big\Vert \int_{\mathbb{S}^{d-1}}\int_{\mathbb{R}^{d}}\frac{\widetilde{f}(\xi^{+}+\eta)\widetilde{g}(\xi^{-}-\eta)}{|\eta|^{d+\gamma}} \textbf{b}(\frac{\xi}{|\xi|}\cdot \omega)d\eta d\omega \Big \Vert _{L_{\xi}^{2}}\lesssim \Vert \widetilde{f}\Vert _{L^{\frac{2pd}{2d-p\gamma}}} \Vert \widetilde{g}\Vert _{L^{\frac{2qd}{2d-q\gamma}}}.\end{aligned}$$ In particular, we have $$\begin{aligned} \Vert \widetilde{Q}^{+}(\widetilde{f},\widetilde{g})\Vert _{L_{\xi}^{2}} \lesssim& \Vert \widetilde{f}\Vert _{L_{\xi}^{2}} \Vert \widetilde{g}\Vert _{L_{\xi}^{\frac{d}{-\gamma}}},\label{equ:Q+,bilinear estimate,PM,f,g}\\ \Vert \widetilde{Q}^{+}(\widetilde{f},\widetilde{g})\Vert _{L_{\xi}^{2}} \lesssim& \Vert \widetilde{f}\Vert _{L_{\xi}^{\frac{d}{-\gamma}}} \Vert \widetilde{g}\Vert _{L_{\xi}^{2}}.\label{equ:Q+,bilinear estimate,PM,g,f}\end{aligned}$$* *Proof.* For the case of Maxwellian molecules, it holds that $$\begin{aligned} \label{equ:Q+,bilinear estimate,constant kernel} \Big\Vert \int_{\mathbb{S}^{d-1}} \widetilde{f}(\xi^{+})\widetilde{g}(\xi^{-}) \textbf{b}(\frac{\xi}{|\xi|}\cdot \omega) d\omega \Big \Vert _{L_{\xi}^{2}}\lesssim \Vert \widetilde{f}\Vert _{L_{\xi}^{p}}\Vert \widetilde{g}\Vert _{L_{\xi}^{q}},\quad \frac{1}{p}+\frac{1}{q}=\frac{1}{2},\end{aligned}$$ which is proved in [@alonso2010estimates Theorem 1]. By Cauchy-Schwarz inequality and then [\[equ:Q+,bilinear estimate,constant kernel\]](#equ:Q+,bilinear estimate,constant kernel){reference-type="eqref" reference="equ:Q+,bilinear estimate,constant kernel"}, we have $$\begin{aligned} \label{equ:Q+,bilinear estimate,proof} &\Big\Vert \int_{\mathbb{S}^{d-1}}\int_{\mathbb{R}^{d}}\frac{\widetilde{f}(\xi^{+}+\eta)\widetilde{g}(\xi^{-}-\eta)}{|\eta|^{d+\gamma}} \textbf{b}(\frac{\xi}{|\xi|}\cdot \omega)d\eta d\omega \Big \Vert _{L_{\xi}^{2}}\\ \leq& \Big\Vert \int_{\mathbb{S}^{d-1}}\left[ \int_{\mathbb{R}^{d}}\frac{|\widetilde{f}(\xi^{+}+\eta)|^{2}}{|\eta|^{d+\gamma}} d\eta \right] ^{\frac{1}{2}} \left[ \int_{\mathbb{R}^{d}}\frac{|\widetilde{g}(\xi^{-}-\eta)|^{2}}{|\eta|^{d+\gamma}}d\eta \right] ^{\frac{1}{2}}\textbf{b}(\frac{\xi}{|\xi|}\cdot \omega) d\omega \Big \Vert _{L_{\xi}^{2}}\notag\\ \lesssim& \Big\Vert \left[ \int_{\mathbb{R}^{d}}\frac{|\widetilde{f}(\xi+\eta)|^{2}}{|\eta|^{d+\gamma}} d\eta \right] ^{\frac{1}{2}} \Big \Vert _{L_{\xi}^{p}} \Big\Vert \left[ \int_{\mathbb{R}^{d}}\frac{|\widetilde{g}(\xi-\eta)|^{2}}{|\eta|^{d+\gamma}}d\eta \right] ^{\frac{1}{2}} \Big \Vert _{L_{\xi}^{q}} \notag\\ =& \Big\Vert \int_{\mathbb{R}^{d}}\frac{|\widetilde{f}(\xi+\eta)|^{2}}{|\eta|^{d+\gamma}} d\eta \Big \Vert ^{\frac{1}{2}}_{L_{\xi}^{\frac{p}{2}}} \Big\Vert \int_{\mathbb{R}^{d}}\frac{|\widetilde{g}(\xi-\eta)|^{2}}{|\eta|^{d+\gamma}}d\eta \Big \Vert ^{\frac{1}{2}}_{L_{\xi}^{\frac{q}{2}}}\notag\\ \lesssim& \Vert \widetilde{f}\Vert _{L^{\frac{2pd}{2d-p\gamma}}} \Vert \widetilde{g}\Vert _{L^{\frac{2qd}{2d-q\gamma}}},\notag\end{aligned}$$ where in the last inequality we have used Hardy-Littlewood-Sobolev inequality [\[equ:hardy-littlewood-sobolev inequality\]](#equ:hardy-littlewood-sobolev inequality){reference-type="eqref" reference="equ:hardy-littlewood-sobolev inequality"}. This completes the proof of [\[equ:Q+,bilinear estimate\]](#equ:Q+,bilinear estimate){reference-type="eqref" reference="equ:Q+,bilinear estimate"}. Then by taking $$(p,q)=(\frac{2d}{d+\gamma},-\frac{2d}{\gamma}),\quad (p,q)=(-\frac{2d}{\gamma},\frac{2d}{d+\gamma}),$$ we immediately obtain [\[equ:Q+,bilinear estimate,PM,f,g\]](#equ:Q+,bilinear estimate,PM,f,g){reference-type="eqref" reference="equ:Q+,bilinear estimate,PM,f,g"} and [\[equ:Q+,bilinear estimate,PM,g,f\]](#equ:Q+,bilinear estimate,PM,g,f){reference-type="eqref" reference="equ:Q+,bilinear estimate,PM,g,f"}. ◻ To prove the bilinear estimate for the gain term, we need a detailed frequency analysis from Littlewood-Paley theory.[^6] Let $\chi(x)$ be a smooth function and satisfy $\chi(x)=1$ for all $|x|\leq 1$ and $\chi(x)=0$ for $|x|\geq 2$. Let $N$ be a dyadic number and set $\varphi_{N}(x)=\chi(\frac{x}{N})-\chi(\frac{x}{2N})$. Define the Littlewood-Paley projector $$\begin{aligned} \widehat{P_{N}u}(\eta)=\varphi_{N}(\eta)\widehat{u}(\eta).\end{aligned}$$ We denote by $P_{N}^{x}$/$P_{M}^{\xi}$ the projector of the $x$-variable and $\xi$-variable respectively. Now, we delve into the analysis of the bilinear estimate. **Lemma 6**. *For $s>\frac{d-1}{2}$, we have $$\begin{aligned} \label{equ:Q+,bilinear estimate,L2} \Vert \langle \nabla_{x} \rangle ^{s}\langle \nabla_{\xi} \rangle ^{s+\gamma}\widetilde{Q}^{+}(\widetilde{f},\widetilde{g})\Vert _{L_{t}^{2}L_{x\xi}^{2}}\lesssim \Vert \widetilde{f}\Vert _{X^{s,s+\gamma,b}}\Vert \widetilde{g}\Vert _{X^{s,s+\gamma,b}}.\end{aligned}$$* *Proof.* By duality, [\[equ:Q+,bilinear estimate,L2\]](#equ:Q+,bilinear estimate,L2){reference-type="eqref" reference="equ:Q+,bilinear estimate,L2"} is equivalent to $$\begin{aligned} \label{equ:Q+,bilinear estimate,L2,equivalent} \int \widetilde{Q}^{+}(\widetilde{f},\widetilde{g}) h dx d\xi dt\lesssim \Vert \widetilde{f}\Vert _{X^{s,s+\gamma,b}}\Vert \widetilde{g}\Vert _{X^{s,s+\gamma,b}} \Vert h\Vert _{L_{t}^{2}H_{\xi}^{-s-\gamma}H_{x}^{-s}} .\end{aligned}$$ We denote by $I$ the integral in [\[equ:Q+,bilinear estimate,L2,equivalent\]](#equ:Q+,bilinear estimate,L2,equivalent){reference-type="eqref" reference="equ:Q+,bilinear estimate,L2,equivalent"}. Inserting a Littlewood-Paley decomposition gives that $$\begin{aligned} I=\sum_{\substack{M,M_{1},M_{2}\\N,N_{1},N_{2}}} I_{M,M_{1},M_{2},N,N_{1},N_{2}}\end{aligned}$$ where $$\begin{aligned} I_{M,M_{1},M_{2},N,N_{1},N_{2}}= \int \widetilde{Q}^{+}(P_{N_{1}}^{x}P_{M_{1}}^{\xi}\widetilde{f},P_{N_{2}}^{x}P_{M_{2}}^{\xi}\widetilde{g}) P_{N}^{x}P_{M}^{\xi}h dx d\xi dt.\end{aligned}$$ Note that $\widetilde{Q}^{+}$ commutes with $P_{N}^{x}$, so this gives the constraint that $N\lesssim \max\left( N_{1},N_{2} \right)$ due to that $$\begin{aligned} \label{equ:property,constraint,projector} P_{N}^{x}(P_{N_{1}}^{x}\widetilde{f}P_{N_{2}}^{x}\widetilde{g})=0, \quad \text{if $N\geq 10\max\left( N_{1},N_{2} \right) $.} \end{aligned}$$ In addition, we observe that such a property [\[equ:property,constraint,projector\]](#equ:property,constraint,projector){reference-type="eqref" reference="equ:property,constraint,projector"} is also hinted in the $\xi$-variable, that is, $$\begin{aligned} \label{equ:property,constraint,projector,v,variable} P_{M}^{\xi}\widetilde{Q}^{+}(P_{M_{1}}^{\xi}\widetilde{f},P_{M_{2}}^{\xi}\widetilde{g})=0, \quad \text{if $M\geq 10\max\left( M_{1},M_{2} \right) $.}\end{aligned}$$ Indeed, notice that $$\begin{aligned} \label{equ:fourier transform,decomposition,Q+} {\mathcal F}_{\xi}\left( P_{M}^{\xi}\widetilde{Q}^{+}(P_{M_{1}}^{\xi}\widetilde{f},P_{M_{2}}^{\xi}\widetilde{g}) \right) =\varphi_{M}(v)\int_{\mathbb{R}^{d}}\int_{\mathbb{S}^{d-1}} (\varphi_{M_{1}}f)(v^{\ast })(\varphi_{M_{2}}g)(u^{\ast}) B(u-v,\omega)dud\omega.\end{aligned}$$ Then from the energy conservation which implies the inequality $|v|^{2}\leq |v^{*}|^{2}+|u^{*}|^{2}$, we have the lower bound that $$\begin{aligned} \label{equ:u*,v*,lower bound} \text{$|u^{*}|\geq \frac{M}{4}$, or $|v^{*}|\geq \frac{M}{4}$}\end{aligned}$$ for all $(u,\omega)\in \mathbb{R}^{d}\times \mathbb{S}^{d-1}$ and $|v|\geq \frac{M}{2}$. Therefore, for $M\geq 10\max\left( M_{1},M_{2} \right)$, the lower bound [\[equ:u\*,v\*,lower bound\]](#equ:u*,v*,lower bound){reference-type="eqref" reference="equ:u*,v*,lower bound"} forces the $v^{*}$-variable or $u^{*}$-variable off their own support set, which makes the integral on the right hand side of [\[equ:fourier transform,decomposition,Q+\]](#equ:fourier transform,decomposition,Q+){reference-type="eqref" reference="equ:fourier transform,decomposition,Q+"} vanish. Hence, this gives the constraint that $M\lesssim \max\left( M_{1},M_{2} \right)$. Now, we divide the sum into four cases as follows Case A. $M_{1}\geq M_{2}$, $N_{1}\geq N_{2}$. Case B. $M_{1}\leq M_{2}$, $N_{1}\geq N_{2}$. Case C. $M_{1}\geq M_{2}$, $N_{1}\leq N_{2}$. Case D. $M_{1}\leq M_{2}$, $N_{1}\leq N_{2}$. We only need to treat Cases A and B, as Cases C and D follow similarly. **Case A. $M_{1}\geq M_{2}$, $N_{1}\geq N_{2}$.** Let $I_{A}$ denote the integral restricted to the Case A. By Cauchy-Schwarz, $$\begin{aligned} I_{A}\lesssim \sum_{\substack{ _{\substack{ M,M_{1}\geq M_{2} \\ N,N_{1}\geq N_{2}}} \\ M_{1}\geqslant M,N_{1}\geqslant N}} \Vert \widetilde{Q}^{+}(P_{N_{1}}^{x}P_{M_{1}}^{\xi}\widetilde{f},P_{N_{2}}^{x}P_{M_{2}}^{\xi}\widetilde{g})\Vert _{L_{t}^{2}L_{x\xi}^{2}} \Vert P_{N}^{x}P_{M}^{\xi}h\Vert _{L_{t}^{2}L_{x\xi}^{2}}.\end{aligned}$$ By using the estimate [\[equ:Q+,bilinear estimate,PM,f,g\]](#equ:Q+,bilinear estimate,PM,f,g){reference-type="eqref" reference="equ:Q+,bilinear estimate,PM,f,g"} in Lemma [\[equ:Q+,bilinear estimate\]](#equ:Q+,bilinear estimate){reference-type="ref" reference="equ:Q+,bilinear estimate"} and then Hölder inequality, we have $$\begin{aligned} I_{A} \lesssim &\sum_{\substack{ _{\substack{ M,M_{1}\geq M_{2} \\ N,N_{1}\geq N_{2}}} \\ M_{1}\geqslant M,N_{1}\geqslant N}} \Big\Vert \Vert P_{N_{1}}^{x}P_{M_{1}}^{\xi }\widetilde{f}\Vert _{L_{\xi}^{2}} \Vert P_{N_{2}}^{x}P_{M_{2}}^{\xi }\widetilde{g}\Vert _{L_{\xi }^{\frac{d}{-\gamma}}} \Big \Vert _{L_{t}^{2}L_{x}^{2}}\Vert P_{N}^{x}P_{M}^{\xi}h\Vert _{L_{t}^{2}L_{x\xi}^{2}} \\ \leq&\sum_{\substack{ _{\substack{ M,M_{1}\geq M_{2} \\ N,N_{1}\geq N_{2}}} \\ M_{1}\geqslant M,N_{1}\geqslant N}} \Vert P_{N_{1}}^{x}P_{M_{1}}^{\xi }\widetilde{f}\Vert _{L_{t}^{\infty}L_{x}^{2}L_{\xi }^{2}} \Vert P_{N_{2}}^{x}P_{M_{2}}^{\xi }\widetilde{g}\Vert _{L_{t}^{2}L_{x}^{\infty}L_{\xi }^{\frac{d}{-\gamma}}} \Vert P_{N}^{x}P_{M}^{\xi}h\Vert _{L_{t}^{2}L_{x\xi}^{2}} .\end{aligned}$$ By using Minkowski inequality, Sobolev inequality that $W^{\frac{d-1}{2}+\gamma,\frac{2d}{d-1}}\hookrightarrow L^{\frac{d}{-\gamma}}$, and Bernstein inequality that $\Vert P_{N_{2}}^{x}\widetilde{g}\Vert _{L_{x}^{\infty}}\lesssim \Vert \langle \nabla_{x} \rangle ^{\frac{d-1}{2}}P_{N_{2}}^{x}\widetilde{g}\Vert _{L_{x}^{\frac{2d}{d-1}}}$, we obtain $$\begin{aligned} I_{A} \lesssim &\sum_{\substack{ _{\substack{ M,M_{1}\geq M_{2} \\ N,N_{1}\geq N_{2}}} \\ M_{1}\geqslant M,N_{1}\geqslant N}} \frac{N^{s}M^{s+\gamma}}{N_{1}^{s}M_{1}^{s+\gamma}} \Vert P_{N_{1}}^{x}P_{M_{1}}^{\xi }\langle \nabla_{x} \rangle ^{s}\langle \nabla _{\xi} \rangle ^{s+\gamma}\widetilde{f}\Vert _{L_{t}^{\infty}L_{x}^{2}L_{\xi}^{2}} \\ &\times \Vert P_{N_{2}}^{x}P_{M_{2}}^{\xi}\langle \nabla_{x} \rangle ^{\frac{d-1}{2}}\langle \nabla_{\xi} \rangle ^{\frac{d-1}{2}+\gamma} \widetilde{g}\Vert _{L_{t}^{2}L_{x}^{\frac{2d}{d-1}}L_{\xi}^{\frac{2d}{d-1}}} \Vert P_{N}^{x}P_{M}^{\xi}\langle \nabla_{x} \rangle ^{-s}\langle \nabla_{\xi} \rangle ^{-s-\gamma}h\Vert _{L_{t}^{2}L_{x\xi}^{2}}\\ \lesssim &\sum_{\substack{ _{\substack{ M,M_{1}\geq M_{2} \\ N,N_{1}\geq N_{2}}} \\ M_{1}\geqslant M,N_{1}\geqslant N}} \frac{N^{s}M^{s+\gamma}}{ N_{1}^{s}M_{1}^{s+\gamma}} \frac{1}{N_{2}^{s-\frac{d-1}{2}}}\frac{1}{M_{2}^{s-\frac{d-1}{2}}} \Vert P_{N}^{x}P_{M}^{\xi}h\Vert _{L_{t}^{2}H_{\xi}^{-s-\gamma}H_{x}^{-s}}\\ &\times \Vert P_{N_{1}}^{x}P_{M_{1}}^{\xi }\left\langle \nabla _{x}\right\rangle^{s}\left\langle \nabla_{\xi}\right\rangle^{s+\gamma}\widetilde{f}\Vert _{L_{t}^{\infty}L_{x}^{2}L_{\xi }^{2}} \Vert P_{N_{2}}^{x}P_{M_{2}}^{\xi}\left\langle \nabla _{x}\right\rangle^{s}\left\langle \nabla _{\xi }\right\rangle ^{s+\gamma}\widetilde{g}\Vert _{L_{t}^{2}L_{x}^{\frac{2d}{d-1}}L_{\xi}^{\frac{2d}{d-1}}},\end{aligned}$$ where in the last inequality we have used Bernstein inequality again. By Strichartz estimate [\[equ:strichartz estimate,xsb\]](#equ:strichartz estimate,xsb){reference-type="eqref" reference="equ:strichartz estimate,xsb"}, $$\begin{aligned} I_{A}\lesssim&\sum_{\substack{ _{\substack{ M,M_{1}\geq M_{2} \\ N,N_{1}\geq N_{2}}} \\ M_{1}\geqslant M,N_{1}\geqslant N}} \frac{N^{s}M^{s+\gamma}}{ N_{1}^{s}M_{1}^{s+\gamma}} \frac{1}{N_{2}^{s-\frac{d-1}{2}}}\frac{1}{M_{2}^{s-\frac{d-1}{2}}} \Vert P_{N}^{x}P_{M}^{\xi}h\Vert _{L_{t}^{2}H_{\xi}^{-s-\gamma}H_{x}^{-s}}\\ &\times \Vert P_{N_{1}}^{x}P_{M_{1}}^{\xi }\widetilde{f}\Vert _{X^{s,s+\gamma,b}}\Vert P_{N_{2}}^{x}P_{M_{2}}^{\xi }\widetilde{g}\Vert _{X^{s,s+\gamma,b}}. \end{aligned}$$ Note that $s>\frac{d-1}{2}$, so we use that $\Vert P_{N_{2}}^{x}P_{M_{2}}^{\xi }\widetilde{g}\Vert _{X^{s,s+\gamma,b}}\lesssim \Vert\widetilde{g}\Vert _{X^{s,s+\gamma,b}}$ and then carry out the $N_{2}$, $M_{2}$ sums to obtain $$\begin{aligned} I_{A}\lesssim& \Vert \widetilde{g}\Vert _{X^{s,s+\gamma,b}}\sum_{\substack{ M_{1}\geq M \\ N_{1}\geq N}} \frac{N^{s}M^{s+\gamma}}{N_{1}^{s}M_{1}^{s+\gamma}}\Vert P_{N_{1}}^{x}P_{M_{1}}^{\xi}\widetilde{f}\Vert _{X^{s,s+\gamma,b}}\Vert P_{N}^{x}P_{M}^{\xi}h\Vert _{L_{t}^{2}H_{\xi}^{-s-\gamma}H_{x}^{-s}}.\end{aligned}$$ By Cauchy-Schwarz in $M$, $M_{1}$, $N$ and $N_{1}$, we have $$\begin{aligned} \label{equ:bilinear estimate,Q+,cauchy-schwarz} I_{A}\lesssim& \Vert \widetilde{g}\Vert _{X^{s,s+\gamma,b}}\left( \sum_{\substack{ M_{1}\geq M \\ N_{1}\geq N}} \frac{N^{s}M^{s+\gamma}}{N_{1}^{s}M_{1}^{s+\gamma}} \Vert P_{N_{1}}^{x}P_{M_{1}}^{\xi}\widetilde{f}\Vert _{X^{s,s+\gamma,b}}^{2} \right) ^{\frac{1}{2}}\\ &\left( \sum_{\substack{ M_{1}\geq M \\ N_{1}\geq N}} \frac{N^{s}M^{s+\gamma}}{N_{1}^{s}M_{1}^{s+\gamma}} \Vert P_{N}^{x}P_{M}^{\xi}h\Vert _{L_{t}^{2}H_{\xi}^{-s-\gamma}H_{x}^{-s}}^{2} \right) ^{\frac{1}{2}} \notag \\ \lesssim &\Vert \widetilde{f}\Vert _{X^{s,s+\gamma,b}}\Vert \widetilde{g}\Vert _{X^{s,s+\gamma,b}} \Vert h\Vert _{L_{t}^{2}H_{\xi}^{-s-\gamma}H_{x}^{-s}},\notag\end{aligned}$$ which completes the proof of [\[equ:Q+,bilinear estimate,L2,equivalent\]](#equ:Q+,bilinear estimate,L2,equivalent){reference-type="eqref" reference="equ:Q+,bilinear estimate,L2,equivalent"} for Case A. **Case B. $M_{1}\leq M_{2}$, $N_{1}\geq N_{2}$.** Following the same way as Case A, we have $$\begin{aligned} I_{B}\lesssim \sum_{\substack{ _{\substack{ M,M_{2}\geq M_{1} \\ N,N_{1}\geq N_{2}}} \\ M_{2}\geqslant M,N_{1}\geqslant N}} \Vert \widetilde{Q}^{+}(P_{N_{1}}^{x}P_{M_{1}}^{\xi}\widetilde{f},P_{N_{2}}^{x}P_{M_{2}}^{\xi}\widetilde{g})\Vert _{L_{t}^{2}L_{x\xi}^{2}} \Vert P_{N}^{x}P_{M}^{\xi}h\Vert _{L_{t}^{2}L_{x\xi}^{2}}.\end{aligned}$$ By using the estimate [\[equ:Q+,bilinear estimate,PM,g,f\]](#equ:Q+,bilinear estimate,PM,g,f){reference-type="eqref" reference="equ:Q+,bilinear estimate,PM,g,f"} in Lemma [\[equ:Q+,bilinear estimate\]](#equ:Q+,bilinear estimate){reference-type="ref" reference="equ:Q+,bilinear estimate"} and then Hölder inequality, we have $$\begin{aligned} I_{B} \lesssim &\sum_{\substack{ _{\substack{ M,M_{2}\geq M_{1} \\ N,N_{1}\geq N_{2}}} \\ M_{2}\geqslant M,N_{1}\geqslant N}} \Big\Vert \Vert P_{N_{1}}^{x}P_{M_{1}}^{\xi }\widetilde{f}\Vert _{L_{\xi}^{\frac{d}{-\gamma}}} \Vert P_{N_{2}}^{x}P_{M_{2}}^{\xi }\widetilde{g}\Vert _{L_{\xi }^{2}} \Big \Vert _{L_{t}^{2}L_{x}^{2}}\Vert P_{N}^{x}P_{M}^{\xi}h\Vert _{L_{t}^{2}L_{x\xi}^{2}} \\ \leq&\sum_{\substack{ _{\substack{ M,M_{2}\geq M_{1} \\ N,N_{1}\geq N_{2}}} \\ M_{2}\geqslant M,N_{1}\geqslant N}} \Vert P_{N_{1}}^{x}P_{M_{1}}^{\xi }\widetilde{f}\Vert _{L_{t}^{2}L_{x}^{\frac{2d}{d-1}}L_{\xi }^{\frac{d}{-\gamma}}} \Vert P_{N_{2}}^{x}P_{M_{2}}^{\xi }\widetilde{g}\Vert _{L_{t}^{\infty}L_{x}^{2d}L_{\xi}^{2}} \Vert P_{N}^{x}P_{M}^{\xi}h\Vert _{L_{t}^{2}L_{x\xi}^{2}} .\end{aligned}$$ By Minkowski inequality, Sobolev inequality that $W^{\frac{d-1}{2}+\gamma,\frac{2d}{d-1}}\hookrightarrow L^{\frac{d}{-\gamma}}$, $W^{\frac{d-1}{2},2}\hookrightarrow L^{2d}$ and Bernstein inequality, we obtain $$\begin{aligned} I_{B} \lesssim &\sum_{\substack{ _{\substack{ M,M_{2}\geq M_{1} \\ N,N_{1}\geq N_{2}}} \\ M_{2}\geqslant M,N_{1}\geqslant N}} \Vert P_{N_{1}}^{x}P_{M_{1}}^{\xi }\langle \nabla_{\xi} \rangle ^{\frac{d-1}{2}+\gamma}\widetilde{f}\Vert _{L_{t}^{2}L_{x}^{\frac{2d}{d-1}}L_{\xi}^{\frac{2d}{d-1}}} \\ &\times \Vert P_{N_{2}}^{x}P_{M_{2}}^{\xi}\langle \nabla_{x} \rangle ^{\frac{d-1}{2}} \widetilde{g}\Vert _{L_{t}^{\infty}L_{x}^{2}L_{\xi}^{2}} \Vert P_{N}^{x}P_{M}^{\xi}h\Vert _{L_{t}^{2}L_{x\xi}^{2}}\\ \lesssim &\sum_{\substack{ _{\substack{ M,M_{2}\geq M_{1} \\ N,N_{1}\geq N_{2}}} \\ M_{2}\geqslant M,N_{1}\geqslant N}} \frac{N^{s}M^{s+\gamma}}{ N_{1}^{s}M_{2}^{s+\gamma}} \frac{1}{N_{2}^{s-\frac{d-1}{2}}}\frac{1}{M_{1}^{s-\frac{d-1}{2}}} \Vert P_{N}^{x}P_{M}^{\xi}h\Vert _{L_{t}^{2}H_{\xi}^{-s-\gamma}H_{x}^{-s}}\\ &\times \Vert P_{N_{1}}^{x}P_{M_{1}}^{\xi }\langle \nabla_{x} \rangle ^{s}\langle \nabla_{\xi} \rangle ^{s+\gamma}\widetilde{f}\Vert _{L_{t}^{2}L_{x}^{\frac{2d}{d-1}}L_{\xi}^{\frac{2d}{d-1}}} \Vert P_{N_{2}}^{x}P_{M_{2}}^{\xi}\langle \nabla _{x} \rangle ^{s}\left\langle \nabla_{\xi }\right\rangle ^{s+\gamma}\widetilde{g}\Vert _{L_{t}^{\infty}L_{x}^{2}L_{\xi }^{2}}.\end{aligned}$$ By Strichartz estimate [\[equ:strichartz estimate,xsb\]](#equ:strichartz estimate,xsb){reference-type="eqref" reference="equ:strichartz estimate,xsb"}, $$\begin{aligned} I_{B}\lesssim&\sum_{\substack{ _{\substack{ M,M_{2}\geq M_{1} \\ N,N_{1}\geq N_{2}}} \\ M_{2}\geqslant M,N_{1}\geqslant N}} \frac{N^{s}M^{s+\gamma}}{ N_{1}^{s}M_{2}^{s+\gamma}} \frac{1}{N_{2}^{s-\frac{d-1}{2}}}\frac{1}{M_{1}^{s-\frac{d-1}{2}}} \Vert P_{N}^{x}P_{M}^{\xi}h\Vert _{L_{t}^{2}H_{\xi}^{-s-\gamma}H_{x}^{-s}}\\ &\times \Vert P_{N_{1}}^{x}P_{M_{1}}^{\xi }\widetilde{f}\Vert _{X^{s,s+\gamma,b}}\Vert P_{N_{2}}^{x}P_{M_{2}}^{\xi }\widetilde{g}\Vert _{X^{s,s+\gamma,b}}. \end{aligned}$$ Note that $s>\frac{d-1}{2}$, so we use that $$\Vert P_{N_{1}}^{x}P_{M_{1}}^{\xi }\widetilde{f}\Vert _{X^{s,s+\gamma,b}}\lesssim \Vert P_{N_{1}}^{x}\widetilde{f}\Vert _{X^{s,s+\gamma,b}},\quad \Vert P_{N_{2}}^{x}P_{M_{2}}^{\xi }\widetilde{g}\Vert _{X^{s,s+\gamma,b}}\lesssim \Vert P_{M_{2}}^{\xi }\widetilde{g}\Vert _{X^{s,s+\gamma,b}},$$ and then carry out the $N_{2}$, $M_{1}$ sums to obtain $$\begin{aligned} I_{B}\lesssim& \sum_{\substack{ M_{2}\geq M \\ N_{1}\geq N}} \frac{N^{s}M^{s+\gamma}}{N_{1}^{s}M_{2}^{s+\gamma}} \Vert P_{N_{1}}^{x}\widetilde{f}\Vert _{X^{s,s+\gamma,b}} \Vert P_{M_{2}}^{\xi }\widetilde{g}\Vert _{X^{s,s+\gamma,b}} \Vert P_{N}^{x}P_{M}^{\xi}h\Vert _{L_{t}^{2}H_{\xi}^{-s-\gamma}H_{x}^{-s}}.\end{aligned}$$ In a similar way to [\[equ:bilinear estimate,Q+,cauchy-schwarz\]](#equ:bilinear estimate,Q+,cauchy-schwarz){reference-type="eqref" reference="equ:bilinear estimate,Q+,cauchy-schwarz"}, we use Cauchy-Schwarz inequality to get $$\begin{aligned} I_{B}\lesssim& \sum_{N_{1}\geq N}\frac{N^{s}}{N_{1}^{s}}\Vert P_{N_{1}}^{x}\widetilde{f}\Vert _{X^{s,s+\gamma,b}}\left( \sum_{M_{2}\geq M}\frac{M^{s+\gamma}}{M_{2}^{s+\gamma}}\Vert P_{M_{2}}^{\xi }\widetilde{g}\Vert _{X^{s,s+\gamma,b}}^{2} \right) ^{\frac{1}{2}}\\ &\times \left( \sum_{M_{2}\geq M}\frac{M^{s+\gamma}}{M_{2}^{s+\gamma}}\Vert P_{N}^{x}P_{M}^{\xi}h\Vert _{L_{t}^{2}H_{\xi}^{-s-\gamma}H_{x}^{-s}}^{2} \right) ^{\frac{1}{2}}\\ \lesssim&\Vert \widetilde{g}\Vert _{X^{s,s+\gamma,b}}\sum_{N_{1}\geq N}\frac{N^{s}}{N_{1}^{s}}\Vert P_{N_{1}}^{x}\widetilde{f}\Vert _{X^{s,s+\gamma,b}}\Vert P_{N}^{x}h\Vert _{L_{t}^{2}H_{\xi}^{-s-\gamma}H_{x}^{-s}}\\ \lesssim& \Vert \widetilde{f}\Vert _{X^{s,s+\gamma,b}}\Vert \widetilde{g}\Vert _{X^{s,s+\gamma,b}} \Vert h\Vert _{L_{t}^{2}H_{\xi}^{-s-\gamma}H_{x}^{-s}}.\end{aligned}$$ Hence, we complete the proof of of [\[equ:Q+,bilinear estimate,L2,equivalent\]](#equ:Q+,bilinear estimate,L2,equivalent){reference-type="eqref" reference="equ:Q+,bilinear estimate,L2,equivalent"} for Case $B$. ◻ ## Well-posedness in Fourier Restriction Norm Space {#section:Proof of Well-posedness} We first recall some standard results on the Fourier restriction norms and Strichartz estimates. **Lemma 7**. *Let $b\in (\frac{1}{2},1)$, $s\in \mathbb{R}$, $r\in \mathbb{R}$, and $\theta(t)$ be a smooth cutoff function. Define $$\begin{aligned} U(t):=e^{it \nabla_{x}\cdot \nabla_{\xi}},\quad D(F):=\int_{0}^{t}U(t-\tau)F(\tau)d\tau.\end{aligned}$$ Then we have $$\begin{aligned} \Vert \widetilde{f}\Vert _{C_{t}^{0}H_{x}^{s}H_{\xi}^{r}}\lesssim& \Vert \widetilde{f}\Vert _{X^{s,r,b}} \label{equ:xsb,infty} ,\\ \Vert \theta(t)S(t)\widetilde{f}_{0}\Vert _{X^{s,r,b}}\lesssim& \Vert \widetilde{f}_{0}\Vert _{H_{x}^{s}H_{\xi}^{r}}\label{equ:xsb,free solution},\\ \Vert \theta(t)D(F)\Vert _{X^{s,r,b}}\lesssim& \Vert F\Vert _{X^{s,r,b-1}}\label{equ:xsb,duhamel estimate},\\ \Vert \widetilde{f}\Vert _{X^{s,r,b-1}}\lesssim&\Vert \widetilde{f}\Vert _{L_{t}^{p}H_{x}^{s}H_{\xi}^{r}},\quad p\in (1,2],\ b\leq \frac{3}{2}-\frac{1}{p},\label{equ:xsb,Lp estimate}\\ \Vert \widetilde{f}\Vert _{L_{t}^{q}L_{x\xi}^{p}}\lesssim& \Vert \widetilde{f}\Vert _{X^{0,0,b}},\quad \frac{2}{q}+\frac{2d}{p}=d,\quad q\geq 2,\ d\geq2.\label{equ:strichartz estimate,xsb}\end{aligned}$$* *Proof.* These type estimates are well-known in the dispersive literatures. The Strichartz estimate [\[equ:strichartz estimate,xsb\]](#equ:strichartz estimate,xsb){reference-type="eqref" reference="equ:strichartz estimate,xsb"} follows from the linear Strichartz estimate [\[equ:strichartz estimate,linear\]](#equ:strichartz estimate,linear){reference-type="eqref" reference="equ:strichartz estimate,linear"} and the transference principle. See for example [@tao2006nonlinear Chapter 2.6]. ◻ We prove the existence, uniqueness, and the Lipschitz continuity of the solution map. The nonnegativity of $f$ follows from the persistence of regularity (as shown in [@chen2019moments; @lu2001on]) by use of the bilinear estimates [\[equ:bilinear estimate,Q-,L2\]](#equ:bilinear estimate,Q-,L2){reference-type="eqref" reference="equ:bilinear estimate,Q-,L2"} and [\[equ:Q+,bilinear estimate,L2\]](#equ:Q+,bilinear estimate,L2){reference-type="eqref" reference="equ:Q+,bilinear estimate,L2"} for the soft potential case. ***Proof of Well-posedness in Theorem $\ref{thm:main theorem}$**.* Let $\theta_{T}(t)=\theta(t/T)$. By estimate [\[equ:xsb,Lp estimate\]](#equ:xsb,Lp estimate){reference-type="eqref" reference="equ:xsb,Lp estimate"}, Hölder inequality, and the bilinear estimates for $Q^{\pm}$, we have $$\begin{aligned} \label{equ:bilinear estimate,Q,xsb} &\Vert \theta_{T}(t)\widetilde{Q}(\widetilde{f},\widetilde{g})\Vert _{X^{s,s+\gamma,b-1}}\\ \lesssim& \Vert \theta_{T}(t)\widetilde{Q}(\widetilde{f},\widetilde{g})\Vert _{L_{t}^{\frac{2}{3-2b}}H_{x}^{s}H_{\xi}^{s+\gamma}}\notag\\ \lesssim& T^{1-b}\Vert \theta_{T}(t)\widetilde{Q}^{+}(\widetilde{f},\widetilde{g})\Vert _{L_{t}^{2}H_{x}^{s}H_{\xi}^{s+\gamma}}+ T^{1-b}\Vert \theta_{T}(t)\widetilde{Q}^{-}(\widetilde{f},\widetilde{g})\Vert _{L_{t}^{2+}H_{x}^{s}H_{\xi}^{s+\gamma}}\notag\\ \lesssim& T^{1-b}\Vert \widetilde{f}\Vert _{X^{s,s+\gamma,b}}\Vert \widetilde{g}\Vert _{X^{s,s+\gamma,b}}.\notag\end{aligned}$$ Let $B=\left\{ \widetilde{f}:\Vert \widetilde{f}\Vert _{X^{s,s+\gamma,b}}\leq R \right\}$ with $R=2C\Vert \widetilde{f}_{0}\Vert _{H_{x}^{s}H_{\xi}^{s+\gamma}}$ and define the nonlinear map $$\begin{aligned} \Phi(\widetilde{f}):=\theta_{T}(t)U(t)\widetilde{f}_{0}+D(\widetilde{f},\widetilde{f}),\end{aligned}$$ where $$\begin{aligned} D(\widetilde{f},\widetilde{f}):=\theta_{T}(t)\int_{0}^{t}U(t-\tau)\theta_{T}(\tau)\widetilde{Q}(\widetilde{f}(\tau),\widetilde{f}(\tau))d\tau.\end{aligned}$$ By estimates [\[equ:xsb,free solution\]](#equ:xsb,free solution){reference-type="eqref" reference="equ:xsb,free solution"}, [\[equ:xsb,duhamel estimate\]](#equ:xsb,duhamel estimate){reference-type="eqref" reference="equ:xsb,duhamel estimate"}, and [\[equ:bilinear estimate,Q,xsb\]](#equ:bilinear estimate,Q,xsb){reference-type="eqref" reference="equ:bilinear estimate,Q,xsb"}, we obtain $$\begin{aligned} \Vert \Phi(\widetilde{f})\Vert _{X^{s,s+\gamma,b}}\leq& \Vert \theta_{T}(t)U(t)\widetilde{f}_{0}\Vert _{X^{s,s+\gamma,b}}+\Vert D(\widetilde{f},\widetilde{f})\Vert _{X^{s,s+\gamma,b}}\\ \leq& C\Vert \widetilde{f}_{0}\Vert _{H_{x}^{s}H_{\xi}^{s+\gamma}}+C\Vert \theta_{T}\widetilde{Q}(\widetilde{f},\widetilde{f})\Vert _{X^{s,s+\gamma,b-1}}\\ \leq& \frac{R}{2}+CT^{1-b}\Vert \widetilde{f}\Vert _{X^{s,s+\gamma,b}}^{2}\\ \leq& R\end{aligned}$$ where in the last inequality we have used that $CT^{1-b}R\leq \frac{1}{2}$. Thus, $\Phi$ maps the set $B$ into itself. In a similar way, for $\widetilde{f}$ and $\widetilde{g}\in B$ we have $$\begin{aligned} \Vert \Phi(\widetilde{f})-\Phi(\widetilde{g})\Vert _{X^{s,s+\gamma,b}}=&\Vert D(\widetilde{f},\widetilde{f})-D(\widetilde{g},\widetilde{g})\Vert _{X^{s,s+\gamma,b}}\\ \leq& C\Vert \theta_{T}\widetilde{Q}(\widetilde{f}-\widetilde{g},\widetilde{f})\Vert _{X^{s,s+\gamma,b-1}}+C\Vert \theta_{T}\widetilde{Q}(\widetilde{g},\widetilde{f}-\widetilde{g})\Vert _{X^{s,s+\gamma,b-1}}\\ \leq& CT^{b-1}\left( \Vert \widetilde{f}\Vert _{X^{s,s+\gamma,b}}+\Vert \widetilde{g}\Vert _{X^{s,s+\gamma,b}} \right) \Vert \widetilde{f}-\widetilde{g}\Vert _{X^{s,s+\gamma,b}}\\ \leq& \frac{1}{2}\Vert \widetilde{f}-\widetilde{g}\Vert _{X^{s,s+\gamma,b}}.\end{aligned}$$ Therefore, $\Phi$ is a contraction mapping in $X^{s,s+\gamma,b}$ and has a unique fixed point $\widetilde{f}$ on the time scale $|T|\sim \langle R \rangle ^{\frac{1}{b-1}}$. Given two initial data $\widetilde{f}_{0}$ and $\widetilde{g}_{0}$, we set $$R_{1}=2\max\left( \Vert \widetilde{f}_{0}\Vert _{H_{x}^{s}H_{\xi}^{s+\gamma}},\Vert \widetilde{g}_{0}\Vert _{H_{x}^{s}H_{\xi}^{s+\gamma}} \right) .$$ Let $\widetilde{f}$, $\widetilde{g}$ be the corresponding unique fixed points. Taking a difference gives that $$\begin{aligned} \widetilde{f}-\widetilde{g}=\theta_{T}(t)U(t)(\widetilde{f}_{0}-\widetilde{g}_{0})+D(\widetilde{f}-\widetilde{g},\widetilde{f})+D(\widetilde{g},\widetilde{f}-\widetilde{g}).\end{aligned}$$ By estimates [\[equ:xsb,free solution\]](#equ:xsb,free solution){reference-type="eqref" reference="equ:xsb,free solution"}, [\[equ:xsb,duhamel estimate\]](#equ:xsb,duhamel estimate){reference-type="eqref" reference="equ:xsb,duhamel estimate"}, and [\[equ:bilinear estimate,Q,xsb\]](#equ:bilinear estimate,Q,xsb){reference-type="eqref" reference="equ:bilinear estimate,Q,xsb"}, we have $$\begin{aligned} \Vert \widetilde{f}-\widetilde{g}\Vert _{X^{s,s+\gamma,b}}\leq& \Vert \theta_{T}(t)U(t)(\widetilde{f}_{0}-\widetilde{g}_{0})\Vert _{X^{s,s+\gamma,b}}+C\Vert \theta_{T}\widetilde{Q}(\widetilde{f}-\widetilde{g},\widetilde{f})\Vert _{X^{s,s+\gamma,b-1}}\\ &+C\Vert \theta_{T}\widetilde{Q}(\widetilde{g},\widetilde{f}-\widetilde{g})\Vert _{X^{s,s+\gamma,b-1}}\\ \leq& C\Vert \widetilde{f}_{0}-\widetilde{g}_{0}\Vert _{H_{x}^{s}H_{\xi}^{s+\gamma}}+CT^{1-b}\left( \Vert \widetilde{f}\Vert _{X^{s,s+\gamma,b}} +\Vert \widetilde{g}\Vert _{X^{s,s+\gamma,b}} \right) \Vert \widetilde{f}-\widetilde{g}\Vert _{X^{s,s+\gamma,b}},\end{aligned}$$ which together with $CT^{1-b}R_{1}\leq \frac{1}{2}$ gives that $$\begin{aligned} \Vert \widetilde{f}-\widetilde{g}\Vert _{X^{s,s+\gamma,b}}\leq 2C\Vert \widetilde{f}_{0}-\widetilde{g}_{0}\Vert _{H_{x}^{s}H_{\xi}^{s+\gamma}}.\end{aligned}$$ The Lipschitz continuity of the data-to-solution map on the time $[-T,T]$ follows from the embedding $X^{s,s+\gamma,b}\hookrightarrow C([-T,T];H_{x}^{s}H_{\xi}^{s+\gamma})$. ◻ # Ill-posedness {#section:Ill-posedness} The idea is to first construct an approximation solution $f_{\mathrm{a}}(t)$ with the norm deflation property that $$\begin{aligned} \label{equ:fa,expected} \Vert f_{\mathrm{a}}(0)\Vert _{L_{v}^{2,r_{0}}H_{x}^{s_{0}}}\ll 1, \quad \Vert f_{\mathrm{a}}(T_{*})\Vert _{L_{v}^{2,r_{0}}H_{x}^{s_{0}}}\gtrsim 1,\end{aligned}$$ with $T_{*}\nearrow 0$, and then use stability theory to perturb the approximation solution into an exact solution. Specifically, from the exact solution $f_{\mathrm{ex}}(t)$ to the Boltzmann equation [\[equ:Boltzmann\]](#equ:Boltzmann){reference-type="eqref" reference="equ:Boltzmann"} $$\left\{ \begin{aligned} &\partial_{t}f_{\mathrm{ex}}+v\cdot \nabla_{x}f_{\mathrm{ex}}=Q(f_{\mathrm{ex}},f_{\mathrm{ex}}),\\ &f_{\mathrm{ex}}(t)=f_{\mathrm{a}}(t)+f_{\mathrm{c}}(t). \end{aligned} \right.$$ we have the equation for the correction term $f_{\mathrm{c}}$ that $$\label{equ:correction term,fc} \left\{ \begin{aligned} \partial_t f_{\mathrm{c}} + v\cdot \nabla_x f_{\mathrm{c}} = & \pm Q^\pm(f_{\mathrm{c}},f_{ \mathrm{a}}) \pm Q^\pm(f_{\mathrm{a}},f_{\mathrm{c}}) \pm Q^\pm(f_{\mathrm{c}},f_{\mathrm{c} })-F_{\text{err}},\\ F_{\mathrm{err}}=&\partial_{t}f_{\mathrm{a}}+v\cdot \nabla_{x}f_{\mathrm{a}}+Q^{-}(f_{\mathrm{a}},f_{\mathrm{a}})-Q^{+}(f_{\mathrm{a}},f_{\mathrm{a}}). \end{aligned} \right.$$ To prove the existence of $f_{\mathrm{c}}$, we work with a $Z$-norm defined by [\[equ:z-norm\]](#equ:z-norm){reference-type="eqref" reference="equ:z-norm"} which is tailored to be stronger than the $L_{v}^{2,r_{0}}H_{x}^{s_{0}}$ norms. For the $Z$-norm, we are able to provide a closed bilinear estimate for the gain and loss terms in Lemma [Lemma 20](#lemma:binlinear estimate){reference-type="ref" reference="lemma:binlinear estimate"}. Additionally, to work on the $Z$-norm space, we provide effective $Z$-norm bounds on the approximation solution $f_{\mathrm{a}}$, which we conclude in Proposition [Proposition 15](#lemma:z-norm bounds on fa){reference-type="ref" reference="lemma:z-norm bounds on fa"}, and then prove the $Z$-norm error estimates on the error term $F_{\mathrm{err}}$ that $$\begin{aligned} \label{equ:error bound,Ferr} \Big\Vert \int_{\tau}^{t}e^{-(t-t_{0})v\cdot \nabla _{x}}F_{\text{err} }(t_{0})\,dt_{0} \Big \Vert _{Z}\ll 1,\end{aligned}$$ which we set up in Proposition [Proposition 16](#lemma:bounds on ferr){reference-type="ref" reference="lemma:bounds on ferr"}. Then by a perturbation argument in Proposition [Proposition 21](#lemma:perturbation){reference-type="ref" reference="lemma:perturbation"}, we prove that the correction term indeed satisfies the smallness property that $$\begin{aligned} \Vert f_{\mathrm{c}}(t)\Vert _{L^{\infty}([T_{*},0];Z)}\ll 1.\end{aligned}$$ Finally in Section [3.5](#section:Proof of Illposedness){reference-type="ref" reference="section:Proof of Illposedness"}, we conclude the ill-posedness results. ## Norm Deflation of the Approximation Solution {#equ:Bounds on the approximation solution} In the section, we get into the analysis of the construction of the approximation solution and its norm deflation property. Following the analysis of a prototype approximation solution in [@chen2022well],[^7] we decompose $$\begin{aligned} f_{\mathrm{a}}(t)=f_{\mathrm{r}}(t)+f_{\mathrm{b}}(t).\end{aligned}$$ On the unit sphere, set $J\sim M^{d-1}N_{2}^{d-1}$ points $\left\{ e_{j} \right\} _{j=1}^{J}$, where the points $e_{j}$ are roughly equally spaced. Let $P_{e_{j}}$ be the orthogonal projection onto the $1D$ subspace spanned by $e_{j}$ and $P_{e_{j}}^{\perp}$ denote the orthogonal projection onto the orthogonal complement space $\left\{ e_{j} \right\} ^{\perp}$. Set $$f_{\mathrm{b}}(t,x,v)=\frac{M^{\frac{d-1}{2}-s}}{N_{2}^{d+\gamma}}\sum_{j=1}^{J}K_{j}(x-vt)I_{j}(v),$$ where $$\begin{aligned} K_{j}(x)=\chi (MP_{e_{j}}^{\perp }x)\chi (\frac{P_{e_{j}}x}{N_{2}}),\quad I_{j}(v)=\chi(MP_{e_{j}}^{\perp }v)\chi (\frac{10P_{e_{j}}(v-N_{2}e_{j})}{N_{2}}).\end{aligned}$$ In fact, $f_{\mathrm{b}}(t,x,v)$ is a linear solution to the transport equation: $$\begin{aligned} \label{equ:fb equation} \partial _{t}f_{\mathrm{b}}+v\cdot \nabla _{x}f_{\mathrm{b}}=0.\end{aligned}$$ Let $f_{\mathrm{r}}(t,x,v)$ be the solution to a drift-free linearized Boltzmann equation: $$\label{equ:fr equation} \partial _{t}f_{\mathrm{r}}(t,x,v)=-f_{\mathrm{r}}(t,x,v)\int \frac{f_{\mathrm{b}}(t,x,u)}{|u-v|^{-\gamma}}\,du=-Q^{-}(f_{\mathrm{r}},f_{\mathrm{b}}),$$ with initial data $f_{\mathrm{r}}(0)=M^{\frac{d}{2}-s}N_{1}^{\frac{d}{2}} \chi (Mx)\chi(N_{1}v)$. Therefore, we write out $$\label{equ:fr} f_{\mathrm{r}}(t,x,v)=M^{\frac{d}{2}-s}N_{1}^{\frac{d}{2}}\exp \left[ -\int_{0}^{t}\int\frac{f_{\mathrm{b}}(\tau,x,u)}{|u-v|^{-\gamma}}\,du\,d\tau\right] \chi (Mx)\chi(N_{1}v).$$ Recall that $0 \leq s_{0}<\frac{d-1}{2}$, $\frac{1-d}{2}\leq \gamma\leq 0$, and $r_{0}=\max\left\{ 0,s_{0}+\gamma \right\}$. In what follows, the parameters are set by $$\begin{aligned} &M\gg 1, \quad N_{1}\geq N_{2}^{10}\geq M^{100},\label{equ:condition,parameters}\\ &s=s_{0}+\frac{\ln\ln \ln M}{\ln M}, \label{equ:condition,s,s0}\\ &T_{*}= -M^{s-\frac{d-1}{2}}(\ln \ln \ln M).\label{equ:T*}\end{aligned}$$ Next, we give the Sobolev norm estimates on $f_{\mathrm{b}}$, $f_{\mathrm{r}}$ and $f_{\mathrm{a}}$. **Lemma 8** (Sobolev norm bounds on $f_{\mathrm{b}}$). *We have for $t\leq 0$, $$\begin{aligned} \label{equ:critical norm estimate for fb} \Vert f_{\mathrm{b}}(t,x,v)\Vert _{L_{v}^{2,r_{0}}H_{x}^{s_{0}}}\lesssim M^{s_{0}-s}N_{2}^{\max\left\{ s_{0},-\gamma \right\} -\frac{d-1}{2}}\leq \frac{1}{\ln \ln M}.\end{aligned}$$* *Proof.* Recall that $$f_{\mathrm{b}}(t,x,v)=\frac{M^{\frac{d-1}{2}-s}}{N_{2}^{d+\gamma}}\sum_{j=1}^{J}K_{j}(x-vt)I_{j}(v).$$ Due to the $v$-support of $f_{\mathrm{b}}$, the weight on $v$-variable produces a factor of $N_{2}^{r_{0}}$. Then expanding $f_{\mathrm{b}}$ gives that $$\begin{aligned} \Vert \langle \nabla_{x} \rangle ^{s_{0}}f_{\mathrm{b}}(t,x,v)\Vert _{L_{v}^{2,r_{0}}L_{x}^{2}}^{2} \lesssim N_{2}^{2r_{0}}\frac{M^{d-1-2s}}{N_{2}^{2d+2\gamma}}\Big\Vert \sum_{j=1}^{J}\langle \nabla_{x} \rangle ^{s_{0}}K_{j}(x-vt)I_{j}(v) \Big \Vert _{L_{v}^{2}L_{x}^{2}}^{2}.\end{aligned}$$ Due to the disjointness of the $v$-support, we have $$\begin{aligned} \Vert \langle \nabla_{x} \rangle ^{s_{0}}f_{\mathrm{b}}(t,x,v)\Vert _{L_{v}^{2,r_{0}}L_{x}^{2}}^{2}&\lesssim N_{2}^{2r_{0}}\frac{M^{d-1-2s}}{N_{2}^{2d+2\gamma}}\sum_{j=1}^{J}\Vert \langle \nabla_{x} \rangle ^{s_{0}}K_{j}(x-vt)I_{j}(v)\Vert _{L_{v}^{2}L_{x}^{2}}^{2}\\ &\lesssim N_{2}^{2r_{0}}\frac{M^{d-1-2s}}{N_{2}^{2d+2\gamma}}\sum_{j=1}^{J}\Vert \langle \nabla_{x} \rangle ^{s_{0}}K_{j}(x)\Vert _{L_{x}^{2}}^{2}\Vert I_{j}(v)\Vert _{L_{v}^{2}}^{2}\\ &\lesssim N_{2}^{2r_{0}}\frac{M^{d-1-2s}}{N_{2}^{2d+2\gamma}}(MN_{2})^{d-1}(M^{2s_{0}+1-d}N_{2})(M^{1-d}N_{2})\\ &= N_{2}^{2r_{0}}\frac{M^{2s_{0}-2s}}{N_{2}^{d-1+2\gamma}},\end{aligned}$$ where in the second-to-last inequality we used that $$\begin{aligned} \Vert \langle \nabla_{x} \rangle ^{s_{0}}K_{j}(x)\Vert _{L_{x}^{2}}^{2}\lesssim &M^{2s_{0}}M^{1-d}N_{2},\quad \Vert I_{j}(v)\Vert _{L_{v}^{2}}^{2}\sim M^{1-d}N_{2}.\end{aligned}$$ Notice that $$\begin{aligned} r_{0}=\max\left\{ 0,s_{0}+\gamma \right\} ,\quad M^{s-s_{0}}=\ln\ln M,\quad \max\left\{ s_{0},-\gamma \right\} \leq \frac{d-1}{2}.\end{aligned}$$ Hence, we complete the proof of [\[equ:critical norm estimate for fb\]](#equ:critical norm estimate for fb){reference-type="eqref" reference="equ:critical norm estimate for fb"}. ◻ Before proceeding to the analysis of $f_{\mathrm{r}}$, we give a useful pointwise bound on $f_{\mathrm{b}}$. **Lemma 9** (Pointwise estimate on $f_{\mathrm{b}}$). *Let $-\frac{1}{4}\leq t\leq 0$ and $$\beta(t,x,v) = \int_{0}^{t} \int \frac{f_{\mathrm{b} }(t_{0},x,u)}{|u-v|^{-\gamma}}\,du dt_0\leq 0.$$ For $k=0,1,2$, we have the pointwise upper bound $$\begin{aligned} \label{equ:upper bound,fr,k} \big|\chi(N_{1}v) \nabla_{x}^{k}\beta(t,x,v)\big |\lesssim |t|M^{k+\frac{d-1}{2}-s}.\end{aligned}$$ For the pointwise lower bound, we have $$\label{equ:upper lower bound,fr} |\chi(N_{1}v)\beta(t,x,v) |\gtrsim |t| M^{\frac{d-1}{2}-s} \chi(N_{1}v),\quad \text{for $|x|\leq M^{-1}$}.$$* *Proof.* For $-\frac{1}{4}\leq t\leq 0$, given the constraints on the $u$-variable, we have $$\begin{aligned} \label{equ:pointwise estimate on fb} &\frac{M^{\frac{d-1}{2}-s}}{N_{2}^{d+\gamma}}\sum_{j=1}^{J}\chi (10MP_{e_{j}}^{\perp }x)\chi (\frac{10P_{e_{j}}x}{N_{2}})\chi (MP_{e_{j}}^{\perp }u)\chi (\frac{10P_{e_{j}}(u-N_{2}e_{j})}{N_{2}}) \\ \leq& f_{\mathrm{b}}(t,x,u)\leq \frac{M^{\frac{d-1}{2}-s}}{N_{2}^{d+\gamma}}\sum_{j=1}^{J}\chi (\frac{% MP_{e_{j}}^{\perp }x}{10})\chi (\frac{P_{e_{j}}x}{10N_{2}})\chi (MP_{e_{j}}^{\perp }u)\chi (\frac{10P_{e_{j}}(u-N_{2}e_{j})}{N_{2}}).\notag\end{aligned}$$ From the $v$-support and $u$-support, we have $|v|\sim N_{1}^{-1}$, $|u|\sim N_{2}$, and hence $|u-v|\sim N_{2}$. Then by using [\[equ:pointwise estimate on fb\]](#equ:pointwise estimate on fb){reference-type="eqref" reference="equ:pointwise estimate on fb"}, we get $$\begin{aligned} \label{equ:pointwise estimate on fb,integral,final} \chi(N_{1}v)\int \frac{f_{\mathrm{b} }(t,x,u)}{|u-v|^{-\gamma}}\,du \sim& N_{2}^{\gamma} \chi(N_{1}v)\int f_{\mathrm{b} }(t,x,u)\,du \\ \sim& N_{2}^{\gamma} \frac{M^{\frac{d-1}{2}-s}}{N_{2}^{d+\gamma}}M^{1-d}N_{2}\chi(N_{1}v)\sum_{j}^{J}\chi (MP_{e_{j}}^{\perp }x)\chi (\frac{P_{e_{j}}x}{N_{2}})\notag\\ =&\frac{M^{\frac{1-d}{2}-s}}{N_{2}^{d-1}}\chi(N_{1}v)\sum_{j}^{J}\chi (MP_{e_{j}}^{\perp }x)\chi (\frac{P_{e_{j}}x}{N_{2}})\notag.\end{aligned}$$ Thus, for the upper bound [\[equ:upper bound,fr,k\]](#equ:upper bound,fr,k){reference-type="eqref" reference="equ:upper bound,fr,k"} with $k=0$, we use that $|J|\sim (MN_{2})^{d-1}$ to obtain $$\begin{aligned} |\chi(N_{1}v) \beta(t,x,v)|\lesssim |t|\frac{M^{\frac{1-d}{2}-s}}{N_{2}^{d-1}} (MN_{2})^{d-1}\chi(N_{1}v)=|t|M^{\frac{d-1}{2}-s}\chi(N_{1}v).\end{aligned}$$ Notice that the upper bounds of estimates [\[equ:pointwise estimate on fb\]](#equ:pointwise estimate on fb){reference-type="eqref" reference="equ:pointwise estimate on fb"} and [\[equ:pointwise estimate on fb,integral,final\]](#equ:pointwise estimate on fb,integral,final){reference-type="eqref" reference="equ:pointwise estimate on fb,integral,final"} remain true with the extra factor $M^{k}$ if $\nabla_x^k$ is applied, for $k\geq 0$. Therefore, we conclude the pointwise upper bound [\[equ:upper bound,fr,k\]](#equ:upper bound,fr,k){reference-type="eqref" reference="equ:upper bound,fr,k"} on $\beta(t,x,v)$ for $k=0,1,2$. For the lower bound [\[equ:upper lower bound,fr\]](#equ:upper lower bound,fr){reference-type="eqref" reference="equ:upper lower bound,fr"}, by noting that $\chi (MP_{e_{j}}^{\perp }x)\chi (\frac{P_{e_{j}}x}{N_{2}})=1$ for $|x|\leq M^{-1}$, we use [\[equ:pointwise estimate on fb,integral,final\]](#equ:pointwise estimate on fb,integral,final){reference-type="eqref" reference="equ:pointwise estimate on fb,integral,final"} again to get $$\begin{aligned} |\chi(N_{1}v) \beta(t,x,v)|=& \chi(N_{1}v)\int_{t}^{0} \int \frac{f_{\mathrm{b} }(t_{0},x,u)}{|u-v|^{-\gamma}}\,du \, dt_0\\ \gtrsim& |t|\frac{M^{\frac{1-d}{2}-s}}{N_{2}^{d-1}} (MN_{2})^{d-1}\chi(N_{1}v) =|t|M^{\frac{d-1}{2}-s}\chi(N_{1}v),\end{aligned}$$ which completes the proof of [\[equ:upper lower bound,fr\]](#equ:upper lower bound,fr){reference-type="eqref" reference="equ:upper lower bound,fr"}. ◻ Now, we are able to give the upper and lower bounds on $f_{\mathrm{r}}$. **Lemma 10** (Sobolev norm bounds on $f_{\rm r}$). *For $-\frac{1}{4} \leq t\leq 0$, we have the upper bound estimate $$\begin{aligned} &\Vert f_{\mathrm{r}}(t)\Vert _{L_{v}^{2,r_{0}}H_{x}^{s_{0}}}\lesssim M^{s_{0}-s} \exp[|t| M^{\frac{d-1}{2}-s}]\langle |t|M^{\frac{d-1}{2}-s} \rangle ,\label{equ:critical norm estimate for fr,upper bound}\end{aligned}$$ and the lower bound estimate $$\begin{aligned} &\Vert f_{\mathrm{r}}(t)\Vert _{L_{v}^{2,r_{0}}H_{x}^{s_{0}}}\gtrsim M^{s_{0}-s} \exp[|t| M^{\frac{d-1}{2}-s}].\label{equ:critical norm estimate for fr,lower bound}\end{aligned}$$ In particular, we have $$\begin{aligned} \Vert f_{\mathrm{r}}(0)\Vert _{L_{v}^{2,r_{0}}H_{x}^{s_{0}}}\lesssim& \frac{1}{\ln\ln M},\label{equ:upper bound,fr,t=0}\\ \Vert f_{\mathrm{r}}(T_{*})\Vert _{L_{v}^{2,r_{0}}H_{x}^{s_{0}}}\gtrsim& 1,\label{equ:lower bound,fr,T}\end{aligned}$$ with $T_{*}= -M^{s-\frac{d-1}{2}}(\ln \ln \ln M)$.* *Proof.* Recall that $$\label{equ:fr,expression} f_{\mathrm{r}}(t,x,v)=M^{\frac{d}{2}-s}N_{1}^{\frac{d}{2}}\exp \left[ -\beta(t,x,v)\right] \chi (Mx)\chi ( N_{1}v).$$ Due to the $v$-support of $f_{\mathrm{r}}$, we can discard the weight on the $v$-variable. For upper bound estimate [\[equ:critical norm estimate for fr,upper bound\]](#equ:critical norm estimate for fr,upper bound){reference-type="eqref" reference="equ:critical norm estimate for fr,upper bound"} on $f_{\mathrm{r}}$, we use the pointwise upper bound [\[equ:upper bound,fr,k\]](#equ:upper bound,fr,k){reference-type="eqref" reference="equ:upper bound,fr,k"} to get $$\begin{aligned} \label{equ:upper bound,fr,sobolev} &\Vert \nabla_{x}f_{\rm{r}}(t)\Vert _{L_{v}^{2,r_{0}}L_{x}^{2}}\\ \leq&M^{1+\frac{d}{2}-s}N_{1}^{\frac{d}{2}}\Vert \exp[-\beta(t,x,v)](\nabla\chi)(Mx)\chi(N_{1}v)\Vert _{L_{v}^{2}L_{x}^{2}}\notag\\ &+M^{\frac{d}{2}-s}N_{1}^{\frac{d}{2}}\Vert \nabla_{x}\beta(t,x,v)\exp[-\beta(t,x,v)]\chi(Mx)\chi(N_{1}v)\Vert _{L_{v}^{2}L_{x}^{2}}\notag\\ \lesssim& M^{1+\frac{d}{2}-s}N_{1}^{\frac{d}{2}}\exp[|t|M^{\frac{d-1}{2}-s}]\Vert (\nabla\chi)(Mx)\Vert _{L_{x}^{2}}\Vert \chi(N_{1}v)\Vert _{L_{v}^{2}}\notag\\ &+M^{\frac{d}{2}-s}N_{1}^{\frac{d}{2}}\langle |t|M^{1+\frac{d-1}{2}-s} \rangle \exp[|t|M^{\frac{d-1}{2}-s}]\Vert \chi(Mx)\Vert _{L_{x}^{2}}\Vert \chi(N_{1}v)\Vert _{L_{v}^{2}}\notag\\ \lesssim& M^{1-s}\exp[|t| M^{\frac{d-1}{2}-s}]\langle |t|M^{\frac{d-1}{2}-s} \rangle .\notag\end{aligned}$$ In the same way, we also have $$\begin{aligned} \Vert f_{\mathrm{r}}(t)\Vert _{L_{v}^{2,r_{0}} L_{x}^{2}}\lesssim M^{-s}\exp[|t| M^{\frac{d-1}{2}-s}].\end{aligned}$$ By the interpolation inequality, we obtain $$\begin{aligned} \Vert f_{\mathrm{r}}(t)\Vert _{L_{v}^{2,r_{0}} H_x^{s_{0}}}\leq& \Vert f_{\mathrm{r}}(t)\Vert _{L_{v}^{2,r_{0}} H_{x}^{1}}^{s_{0}} \Vert f_{\mathrm{r}}(t)\Vert _{L_{v}^{2,r_{0}} L_{x}^{2}}^{1-s_{0}} \lesssim M^{s_{0}-s} \exp[|t| M^{\frac{d-1}{2}-s}]\langle |t|M^{\frac{d-1}{2}-s} \rangle .\end{aligned}$$ For the lower bound estimate [\[equ:critical norm estimate for fr,lower bound\]](#equ:critical norm estimate for fr,lower bound){reference-type="eqref" reference="equ:critical norm estimate for fr,lower bound"} on $f_{\mathrm{r}}$, we use the Sobolev inequality and lower bound estimate [\[equ:upper lower bound,fr\]](#equ:upper lower bound,fr){reference-type="eqref" reference="equ:upper lower bound,fr"} to obtain $$\begin{aligned} &\Vert \langle \nabla_{x} \rangle ^{s_{0}}f_{\mathrm{r}}(t,x,v)\Vert _{L_{v}^{2,r_{0}}L_{x}^{2}}\\ \gtrsim&\Vert f_{\mathrm{r}}(t,x,v)\Vert _{L_{v}^{2,r_{0}}L_{x}^{\frac{2d}{d-2s_{0}}}}\\ \gtrsim&M^{\frac{d}{2}-s}N_{1}^{\frac{d}{2}}\Vert \exp \left[ -\beta(t,x,v)\right] \chi (Mx)\chi ( N_{1}v)\Vert _{L_{v}^{2}L_{x}^{\frac{2d}{d-2s_{0}}}}\\ \gtrsim& M^{\frac{d}{2}-s}N_{1}^{\frac{d}{2}}\exp[|t| M^{\frac{d-1}{2}-s}] \Vert \chi(Mx)\Vert _{L_{x}^{\frac{2d}{d-2s_{0}}}}\Vert \chi(N_{1}v)\Vert _{L_{v}^{2}}\\ \gtrsim& M^{s_{0}-s}\exp[|t| M^{\frac{d-1}{2}-s}].\end{aligned}$$ Hence, we have done the proof of estimate $(\ref{equ:critical norm estimate for fr,lower bound})$. Inserting in $|T_{*}|= M^{s-\frac{d-1}{2}}(\ln \ln \ln M)$ and $M^{s_{0}-s}=\frac{1}{\ln\ln M}$, we have $$\begin{aligned} \Vert f_{\mathrm{r}}(T_{*})\Vert _{L_{v}^{2,r_{0}}H_{x}^{s_{0}}}\gtrsim M^{s_{0}-s}\exp[|T_{*}| M^{\frac{d-1}{2}-s}]\gtrsim 1,\end{aligned}$$ which completes the proof of [\[equ:lower bound,fr,T\]](#equ:lower bound,fr,T){reference-type="eqref" reference="equ:lower bound,fr,T"}. ◻ **Remark 11**. The lower bound estimate [\[equ:critical norm estimate for fr,lower bound\]](#equ:critical norm estimate for fr,lower bound){reference-type="eqref" reference="equ:critical norm estimate for fr,lower bound"} on $f_{\mathrm{r}}(t)$ also holds for the kernel ($d=3$) $$\begin{aligned} B(u-v,\omega)=\left( 1_{\left\{ |u-v|\leq 1 \right\} }|u-v|+1_{\left\{ |u-v|\geq 1 \right\} }|u-v|^{-1} \right) \textbf{b}(\frac{u-v}{|u-v|}\cdot \omega).\end{aligned}$$ Indeed, in the proof of the lower bound estimate [\[equ:upper lower bound,fr\]](#equ:upper lower bound,fr){reference-type="eqref" reference="equ:upper lower bound,fr"}, the term $1_{\left\{ |u-v|\leq 1 \right\} }|u-v|$ would vanish due to that $|u-v|\sim N_{2}\gg 1$. In the end, we conclude the norm deflation property of the approximation solution $f_{\mathrm{a}}$. **Proposition 12** (Norm deflation of $f_{\mathrm{a}}$). Let $T_{*}= -M^{s-\frac{d-1}{2}}(\ln \ln \ln M)$. We have $$\begin{aligned} \Vert f_{\mathrm{a}}(0)\Vert _{L_v^{2,r_{0}} H_x^{s_{0}}}\lesssim& \frac{1}{\ln \ln M}\ll 1,\label{equ:critical norm estimate for fa,t=0}\\ \Vert f_{\mathrm{a}}(T_{*})\Vert _{L_v^{2,r_{0}} H_x^{s_{0}}}\gtrsim& 1.\label{equ:critical norm estimate for fa,t=T}\end{aligned}$$ *Proof.* Since $f_{\mathrm{b}}$ and $f_{\mathrm{r}}$ have disjoint velocity supports, we get $$\begin{aligned} \Vert f_{\mathrm{a}}(t)\Vert _{L_v^{2,r_{0}} H_x^{s_{0}}}\sim \Vert f_{\mathrm{r}}(t)\Vert _{L_v^{2,r_{0}} H_x^{s_{0}}}+\Vert f_{\mathrm{b}}(t)\Vert _{L_v^{2,r_{0}} H_x^{s_{0}}}.\end{aligned}$$ Then by estimate [\[equ:critical norm estimate for fb\]](#equ:critical norm estimate for fb){reference-type="eqref" reference="equ:critical norm estimate for fb"} on $f_{\mathrm{b}}$ in Lemma [Lemma 8](#lemma:Hs,bounds on fb){reference-type="ref" reference="lemma:Hs,bounds on fb"} and estimates [\[equ:upper bound,fr,t=0\]](#equ:upper bound,fr,t=0){reference-type="eqref" reference="equ:upper bound,fr,t=0"}--[\[equ:lower bound,fr,T\]](#equ:lower bound,fr,T){reference-type="eqref" reference="equ:lower bound,fr,T"} on $f_{\mathrm{r}}$ in Lemma [Lemma 10](#lemma:bounds on fr){reference-type="ref" reference="lemma:bounds on fr"}, we arrive at estimates [\[equ:critical norm estimate for fa,t=0\]](#equ:critical norm estimate for fa,t=0){reference-type="eqref" reference="equ:critical norm estimate for fa,t=0"} and [\[equ:critical norm estimate for fa,t=T\]](#equ:critical norm estimate for fa,t=T){reference-type="eqref" reference="equ:critical norm estimate for fa,t=T"}. ◻ ### Discussion on the $L^{1}$-based space and hard potentials {#section:Discussion on the $L^{1}$-based space and hard potentials} The equation [\[equ:Boltzmann\]](#equ:Boltzmann){reference-type="eqref" reference="equ:Boltzmann"} is invariant under the scaling $$\begin{aligned} f_{\lambda}(t,x,,v)=\lambda^{\alpha+(d-1+\gamma)\beta}f(\lambda^{\alpha-\beta}t,\lambda^{\alpha}x,\lambda^{\beta}v),\end{aligned}$$ for any $\alpha$, $\beta\in \mathbb{R}$ and $\lambda>0$. Then $$\begin{aligned} \Vert |\nabla_{x}|^{s}|v|^{r}f_{\lambda}\Vert _{L_{xv}^{1}}=\lambda^{^{\alpha+(d-1+\gamma)\beta}}\lambda^{\alpha s-\beta r}\lambda^{-d\alpha-d\beta} \Vert |\nabla_{x}|^{s}|v|^{r}f\Vert _{L_{xv}^{1}},\end{aligned}$$ which gives the $L^{1}$-based scaling-critical index $$\begin{aligned} s_{1}=d-1,\quad r_{1}=1+\gamma.\end{aligned}$$ In the $L^{1}$ setting, we construct the approximation solution $f_{a,1}=f_{\mathrm{b},1}+f_{\mathrm{r},1}$, where $$\begin{aligned} f_{\mathrm{b},1}(t,x,v)=&\frac{M^{d-1-s}}{N_{2}^{d+2+\gamma}}\sum_{j=1}^{J}K_{j}(x-vt)I_{j}(v),\\ f_{\mathrm{r,1}}(t,x,v)=&M^{d-s}N_{1}^{d}\exp \left[ -\beta(t,x,v)\right] \chi (Mx)\chi ( N_{1}v).\end{aligned}$$ Repeating the proof of estimates [\[equ:critical norm estimate for fb\]](#equ:critical norm estimate for fb){reference-type="eqref" reference="equ:critical norm estimate for fb"} and [\[equ:critical norm estimate for fr,lower bound\]](#equ:critical norm estimate for fr,lower bound){reference-type="eqref" reference="equ:critical norm estimate for fr,lower bound"}, we also have $$\begin{aligned} \Vert \langle \nabla \rangle ^{s_{0}}f_{\mathrm{b},1}\Vert _{L_{v}^{1,r_{1}}L_{x}^{1}}\lesssim& M^{s_{0}-s},\\ \Vert \langle \nabla \rangle ^{s_{0}}f_{\mathrm{r},1}\Vert _{L_{v}^{1,r_{1}}L_{x}^{1}}\gtrsim& M^{s_{0}-s}\exp[|t|N_{2}^{-2}M^{d-1-s}].\end{aligned}$$ If $s_{0}<s_{1}=d-1$, a similar mechanism of norm deflation could be possible in the $L^{1}$ setting. For the hard potential case that $\gamma> 0$, the norm deflation of the approximation solution $f_{\mathrm{a}}(t)$ also holds. But, to perturb it into the exact solution, it requires a much more different work space to prove the error bounds in Proposition [Proposition 16](#lemma:bounds on ferr){reference-type="ref" reference="lemma:bounds on ferr"} and provide a closed estimate in Lemma [Lemma 20](#lemma:binlinear estimate){reference-type="ref" reference="lemma:binlinear estimate"}. We leave the problem for future work. ## $Z$-norm Bounds on the Approximation Solution {#section:$Z$-norm Bounds on the Approximation Solution} To perturb the approximation solution $f_{\mathrm{a}}(t)$ into an exact solution $f_{\mathrm{ex}}(t)$, we need to prove the existence of a small correction term $f_{\mathrm{c}}(t)$. As it satisfies a more complicated equation [\[equ:correction term,fc\]](#equ:correction term,fc){reference-type="eqref" reference="equ:correction term,fc"}, some terms of [\[equ:correction term,fc\]](#equ:correction term,fc){reference-type="eqref" reference="equ:correction term,fc"} cannot be effectively treated using Strichartz estimates like [\[equ:lwp,bilinear estimate\]](#equ:lwp,bilinear estimate){reference-type="eqref" reference="equ:lwp,bilinear estimate"}. Hence, we tailor a $Z$-norm to provide a closed estimate for the gain and loss terms, that is, $$\label{equ:closed z-norm estimate} \|Q^\pm (f_1,f_2) \|_{Z} \lesssim \|f_1\|_{Z} \|f_2\|_{Z},$$ where the $Z$-norm is given by $$\begin{aligned} \label{equ:z-norm} \Vert f(t)\Vert _{Z}=&M^{\frac{d-3}{2}}\Vert \nabla_{x} f(t)\Vert _{L_{v}^{2,r_{0}}L_{x}^{2}} +M^{\frac{d-1}{2}}\Vert f(t)\Vert _{L_{v}^{2,r_{0}}L_{x}^{2}}+N_{2}^{\gamma}\Vert f(t)\Vert _{L_{v}^{1}L_{x}^{\infty}}\\ &+N_{2}^{\frac{2d}{5}+\gamma}\Vert f(t)\Vert _{L_{v}^{\frac{5}{3}}L_{x}^{\infty}} +M^{-1}N_{2}^{\gamma}\Vert \nabla_{x}f(t)\Vert _{L_{v}^{1}L_{x}^{\infty}}+ M^{-1}N_{2}^{\frac{2d}{5}+\gamma}\Vert \nabla_{x}f(t)\Vert _{L_{v}^{\frac{5}{3}}L_{x}^{\infty}}.\notag\end{aligned}$$ The closed estimate [\[equ:closed z-norm estimate\]](#equ:closed z-norm estimate){reference-type="eqref" reference="equ:closed z-norm estimate"} which we will prove in Section [3.4](#section:Bounds on the Correction Term){reference-type="ref" reference="section:Bounds on the Correction Term"} indeed plays a key role in the perturbation argument. In the section, we give $Z$-norm bounds on the approximation solution $f_{\mathrm{a}}=f_{\mathrm{r}}+f_{\mathrm{b}}$, which will be used to control the error term $F_{\mathrm{err}}$. **Lemma 13** ($Z$-norm bounds on $f_{\mathrm{b}}$). *For the $Z$-norm, we have $$\begin{aligned} \label{equ:z-norm estimate for fb} \Vert f_{\mathrm{b}}(t)\Vert _{L^{\infty}([T_{*},0];Z)}\lesssim M^{\frac{d-1}{2}-s}.\end{aligned}$$* *Proof.* **The $M^{\frac{d-3}{2}}\Vert \nabla_{x}\bullet\Vert _{L_{v}^{2,r_{0}}L_{x}^{2}}$ and $M^{\frac{d-1}{2}}\Vert \bullet\Vert _{L_{v}^{2,r_{0}}L_{x}^{2}}$ estimates.** This can be done in the same way as estimate [\[equ:critical norm estimate for fb\]](#equ:critical norm estimate for fb){reference-type="eqref" reference="equ:critical norm estimate for fb"} with the regularity index $s_{0}$ replaced by $1$ and $0$. Therefore, we obtain $$\begin{aligned} \label{equ:z-norm estimate for fb,L2} M^{\frac{d-3}{2}}\Vert \nabla_{x} f_{\mathrm{b}}\Vert _{L_{v}^{2}L_{x}^{2}}\lesssim & M^{\frac{d-1}{2}-s}N_{2}^{\max\left\{ s_{0},-\gamma \right\} -\frac{d-1}{2}}\leq M^{\frac{d-1}{2}-s},\\ M^{\frac{d-1}{2}}\Vert f_{\mathrm{b}}\Vert _{L_{v}^{2}L_{x}^{2}}\lesssim &M^{\frac{d-1}{2}-s}N_{2}^{\max\left\{ s_{0},-\gamma \right\} -\frac{d-1}{2}}\leq M^{\frac{d-1}{2}-s}.\end{aligned}$$ **The $N_{2}^{\gamma}\Vert \bullet \Vert _{L_{v}^{1}L_{x}^{\infty}}$ and $N_{2}^{\frac{2d}{5}+\gamma}\Vert \bullet \Vert _{L_{v}^{\frac{5}{3}}L_{x}^{\infty}}$ estimates.** $$\begin{aligned} \label{equ:z-norm estimate for fb,Lv1} N_{2}^{\gamma}\Vert f_{\mathrm{b}}\Vert _{L_{v}^{1}L_{x}^{\infty}} \lesssim &N_{2}^{\gamma}\frac{M^{\frac{d-1}{2}-s}}{N_{2}^{d+\gamma}} \Big\Vert \sum_{j=1}^{J}\Vert K_{j}(x-vt)\Vert _{L_{x}^{\infty}}I_{j}(v) \Big \Vert _{L_{v}^{1}}\\ \lesssim &\frac{M^{\frac{d-1}{2}-s}}{N_{2}^{d}} \Big\Vert \sum_{j=1}^{J}I_{j}(v) \Big \Vert _{L_{v}^{1}}\notag\\ \lesssim&\frac{M^{\frac{d-1}{2}-s}}{N_{2}^{d}}N_{2}^{d}=M^{\frac{d-1}{2}-s},\notag\end{aligned}$$ where in the last inequality we have used that $$\begin{aligned} \sum_{j=1}^{J}I_{j}(v)\sim 1_{\left\{ \frac{9N_{2}}{10}\leq|v|\leq \frac{11N_{2}}{10} \right\} }(v),\quad \Big\Vert 1_{\left\{ \frac{9N_{2}}{10}\leq|v|\leq \frac{11N_{2}}{10} \right\} }(v) \Big \Vert _{L_{v}^{1}}\lesssim N_{2}^{d}.\end{aligned}$$ In the same way, we also have $$\begin{aligned} N_{2}^{\frac{2d}{5}+\gamma}\Vert f_{\mathrm{b}}\Vert _{L_{v}^{\frac{5}{3}}L_{x}^{\infty}} \lesssim &N_{2}^{\frac{2d}{5}+\gamma}\frac{M^{\frac{d-1}{2}-s}}{N_{2}^{d+\gamma}} \Big\Vert \sum_{j=1}^{J}\Vert K_{j}(x-vt)\Vert _{L_{x}^{\infty}}I_{j}(v) \Big \Vert _{L_{v}^{\frac{5}{3}}}\\ \lesssim &N_{2}^{\frac{2d}{5}}\frac{M^{\frac{d-1}{2}-s}}{N_{2}^{d}} \Big\Vert \sum_{j=1}^{J}I_{j}(v) \Big \Vert _{L_{v}^{\frac{5}{3}}}\\ \lesssim&N_{2}^{\frac{2d}{5}}\frac{M^{\frac{d-1}{2}-s}}{N_{2}^{d}}N_{2}^{\frac{3d}{5}}=M^{\frac{d-1}{2}-s}.\end{aligned}$$ The same bound is obtained for $M^{-1}N_{2}^{\gamma}\Vert \nabla_{x}f_{\mathrm{b}} \Vert _{L_{v}^{1}L_{x}^{\infty}}$ and $M^{-1}N_{2}^{\frac{2d}{5}+\gamma}\Vert \nabla_{x}f_{\mathrm{b}} \Vert _{L_{v}^{\frac{5}{3}}L_{x}^{\infty}}$ with one $x$-derivative producing a factor of $M$. Therefore, we complete the proof of the $Z$-norm estimate [\[equ:z-norm estimate for fb\]](#equ:z-norm estimate for fb){reference-type="eqref" reference="equ:z-norm estimate for fb"}. ◻ **Lemma 14** ($Z$-norm bounds on $f_{\mathrm{r}}$). *For $T_{*}\leq t\leq 0$, we have $$\begin{aligned} \label{equ:z-norm estimate for fr} \Vert f_{\mathrm{r}}(t)\Vert _{Z}\lesssim M^{\frac{d-1}{2}-s} \exp[|t| M^{\frac{d-1}{2}-s}]\langle |t|M^{\frac{d-1}{2}-s} \rangle .\end{aligned}$$ In particular, $$\begin{aligned} \label{equ:z-norm bounds for fr} \Vert f_{\mathrm{r}}(t)\Vert _{L^{\infty}([T_{*},0];Z)}\lesssim M^{\frac{d-1}{2}-s} (\ln \ln M)^{2}.\end{aligned}$$* *Proof.* Recall $$f_{\mathrm{r}}(t,x,v)=M^{\frac{d}{2}-s}N_{1}^{\frac{d}{2}}\exp \left[ -\beta(t,x,v)\right] \chi (Mx)\chi ( N_{1}v).$$ **The $M^{\frac{d-3}{2}}\Vert \nabla_{x}\bullet\Vert _{L_{v}^{2,r_{0}}L_{x}^{2}}$ and $M^{\frac{d-1}{2}}\Vert \bullet\Vert _{L_{v}^{2,r_{0}}L_{x}^{2}}$ estimates.** The weight on $v$-variable plays no role due to the $v$-support set, so we can discard it. By the pointwise upper bound [\[equ:upper bound,fr,k\]](#equ:upper bound,fr,k){reference-type="eqref" reference="equ:upper bound,fr,k"}, we get $$\begin{aligned} &M^{\frac{d-3}{2}}\Vert \nabla_{x}f_{\rm{r}}(t)\Vert _{L_{v}^{2,r_{0}}L_{x}^{2}}\\ \leq&M^{\frac{d-3}{2}}M^{1+\frac{d}{2}-s}N_{1}^{\frac{d}{2}}\Vert \exp[-\beta(t,x,v)](\nabla\chi)(Mx)\chi(N_{1}v)\Vert _{L_{v}^{2}L_{x}^{2}}\\ &+M^{\frac{d-3}{2}}M^{\frac{d}{2}-s}N_{1}^{\frac{d}{2}}\Vert \nabla_{x}\beta(t,x,v)\exp[-\beta(t,x,v)]\chi(Mx)\chi(N_{1}v)\Vert _{L_{v}^{2}L_{x}^{2}}\\ \lesssim& M^{\frac{d-1}{2}}M^{\frac{d}{2}-s}N_{1}^{\frac{d}{2}}\exp[|t|M^{\frac{d-1}{2}-s}]\Vert (\nabla\chi)(Mx)\Vert _{L_{x}^{2}}\Vert \chi(N_{1}v)\Vert _{L_{v}^{2}}\\ &+M^{\frac{d-3}{2}}M^{\frac{d}{2}-s}N_{1}^{\frac{d}{2}}\langle |t|M^{1+\frac{d-1}{2}-s} \rangle \exp[|t|M^{\frac{d-1}{2}-s}]\Vert \chi(Mx)\Vert _{L_{x}^{2}}\Vert \chi(N_{1}v)\Vert _{L_{v}^{2}}\\ \lesssim& M^{\frac{d-1}{2}-s}\exp[|t| M^{\frac{d-1}{2}-s}]\langle |t|M^{\frac{d-1}{2}-s} \rangle .\end{aligned}$$ The $M^{\frac{d-1}{2}}\Vert f_{\mathrm{r}}\Vert _{L_{v}^{2,r_{0}}L_{x}^{2}}$ estimate can be handled in the same way. **The $N_{2}^{\gamma}\Vert \bullet \Vert _{L_{v}^{1}L_{x}^{\infty}}$ and $M^{-1}N_{2}^{\gamma}\Vert \nabla_{x}\bullet\Vert _{L_{v}^{1}L_{x}^{\infty}}$ estimates.** We only need to treat the $M^{-1}N_{2}^{\gamma}\Vert \nabla_{x}\bullet\Vert _{L_{v}^{1}L_{x}^{\infty}}$ norm, as the $N_{2}^{\gamma}\Vert \bullet\Vert _{L_{v}^{1}L_{x}^{\infty}}$ norm can be dealt with in a similar way. We use the pointwise upper bound [\[equ:upper bound,fr,k\]](#equ:upper bound,fr,k){reference-type="eqref" reference="equ:upper bound,fr,k"} to obtain $$\begin{aligned} \label{equ:fr,Z-norm,L1} &M^{-1}N_{2}^{\gamma}\Vert \nabla_{x}f_{\rm{r}}(t)\Vert _{L_{v}^{1}L_{x}^{\infty}}\\ \leq& M^{-1}N_{2}^{\gamma} M^{1+\frac{d}{2}-s}N_{1}^{\frac{d}{2}}\Vert \exp[-\beta(t,x,v)](\nabla\chi)(Mx)\chi(N_{1}v)\Vert _{L_{v}^{1}L_{x}^{\infty}}\notag\\ &+M^{-1}N_{2}^{\gamma}M^{\frac{d}{2}-s}N_{1}^{\frac{d}{2}}\Vert \nabla_{x}\beta(t,x,v)\exp[-\beta(t,x,v)]\chi(Mx)\chi(N_{1}v)\Vert _{L_{v}^{1}L_{x}^{\infty}}\notag\\ \lesssim& M^{-1}N_{2}^{\gamma}M^{1+\frac{d}{2}-s}N_{1}^{\frac{d}{2}}\exp[|t|M^{\frac{d-1}{2}-s}]\Vert (\nabla\chi)(Mx)\Vert _{L_{x}^{\infty}}\Vert \chi(N_{1}v)\Vert _{L_{v}^{1}}\notag\\ &+M^{-1}N_{2}^{\gamma}M^{\frac{d}{2}-s}N_{1}^{\frac{d}{2}}\langle |t|M^{1+\frac{d-1}{2}-s} \rangle \exp[|t|M^{\frac{d-1}{2}-s}]\Vert \chi(Mx)\Vert _{L_{x}^{\infty}}\Vert \chi(N_{1}v)\Vert _{L_{v}^{1}}\notag\\ \lesssim& M^{-1}N_{2}^{\gamma} M^{1+\frac{d}{2}-s}N_{1}^{-\frac{d}{2}}\exp[|t| M^{\frac{d-1}{2}-s}]\langle |t|M^{\frac{d-1}{2}-s} \rangle \notag\\ \lesssim& N_{1}^{-\frac{d}{2}}N_{2}^{\gamma}M^{\frac{d}{2}-s}\exp[|t| M^{\frac{d-1}{2}-s}]\langle |t|M^{\frac{d-1}{2}-s} \rangle .\notag\end{aligned}$$ This bound is enough as it carries the smallness factor $N_{1}^{-\frac{d}{2}}$. **The $N_{2}^{\frac{2d}{5}+\gamma}\Vert \bullet \Vert _{L_{v}^{\frac{5}{3}}L_{x}^{\infty}}$ and $M^{-1}N_{2}^{\frac{2d}{5}+\gamma}\Vert \nabla_{x}\bullet\Vert _{L_{v}^{\frac{5}{3}}L_{x}^{\infty}}$ estimates.** These two norms can be controlled in the same manner as [\[equ:fr,Z-norm,L1\]](#equ:fr,Z-norm,L1){reference-type="eqref" reference="equ:fr,Z-norm,L1"} with the $L_{v}^{1}$ norm replaced by the $L_{v}^{\frac{5}{3}}$ norm. As a result, we also have $$\begin{aligned} N_{2}^{\frac{2d}{5}+\gamma} \Vert f_{\rm{r}}(t)\Vert _{L_{v}^{\frac{5}{3}}L_{x}^{\infty}}\lesssim& N_{1}^{-\frac{d}{10}}N_{2}^{\frac{2d}{5}+\gamma} M^{\frac{d}{2}-s}\exp[|t| M^{\frac{d-1}{2}-s}]\langle |t|M^{\frac{d-1}{2}-s} \rangle ,\label{equ:vfr,L35}\\ M^{-1}N_{2}^{\frac{2d}{5}+\gamma} \Vert \nabla_{x}f_{\rm{r}}(t)\Vert _{L_{v}^{\frac{5}{3}}L_{x}^{\infty}}\lesssim& N_{1}^{-\frac{d}{10}}N_{2}^{\frac{2d}{5}+\gamma}M^{\frac{d}{2}-s}\exp[|t| M^{\frac{d-1}{2}-s}]\langle |t|M^{\frac{d-1}{2}-s} \rangle .\label{equ:vfr,L35,H1}\end{aligned}$$ By the condition [\[equ:condition,parameters\]](#equ:condition,parameters){reference-type="eqref" reference="equ:condition,parameters"} that $N_{1}\geq N_{2}^{10}\geq M^{100}$, it is sufficient to obtain the desired bound. Thus, we complete the proof of [\[equ:z-norm estimate for fr\]](#equ:z-norm estimate for fr){reference-type="eqref" reference="equ:z-norm estimate for fr"}. Inserting in $|T_{*}|=M^{s-\frac{d-1}{2}}(\ln \ln \ln M)$ and $M^{s-s_{0}}=\ln \ln M$, we obtain $$\begin{aligned} \Vert f_{\mathrm{r}}(t)\Vert _{L^{\infty}([T_{*},0];Z)} \lesssim M^{\frac{d-1}{2}-s}(\ln \ln M)^{2},\end{aligned}$$ which completes the proof of [\[equ:z-norm bounds for fr\]](#equ:z-norm bounds for fr){reference-type="eqref" reference="equ:z-norm bounds for fr"}. ◻ To the end, we conclude the $Z$-norm bounds on $f_{\mathrm{a}}=f_{\mathrm{r}}+f_{\mathrm{b}}$. **Proposition 15** ($Z$-norm bounds on $f_{\mathrm{a}}$). For the $Z$-norm, $$\begin{aligned} \label{equ:z-norm estimate for fa} \Vert f_{\mathrm{a}}(t)\Vert _{L^{\infty}([T_{*},0];Z)}\lesssim M^{\frac{d-1}{2}-s} (\ln \ln M)^{2}.\end{aligned}$$ *Proof.* By the triangle inequality, we have $$\begin{aligned} \Vert f_{\mathrm{a}}(t)\Vert _{Z}\lesssim \Vert f_{\mathrm{r}}(t)\Vert _{Z}+\Vert f_{\mathrm{b}}(t)\Vert _{Z}.\end{aligned}$$ Then combining estimate [\[equ:z-norm estimate for fb\]](#equ:z-norm estimate for fb){reference-type="eqref" reference="equ:z-norm estimate for fb"} on $f_{\mathrm{b}}$ and estimate [\[equ:z-norm estimate for fr\]](#equ:z-norm estimate for fr){reference-type="eqref" reference="equ:z-norm estimate for fr"} on $f_{\mathrm{r}}$, we complete the proof of estimate [\[equ:z-norm estimate for fa\]](#equ:z-norm estimate for fa){reference-type="eqref" reference="equ:z-norm estimate for fa"}. ◻ ## $Z$-norm Bounds on the Error Terms {#section:Bounds on the Error Terms} In the section, we give the $Z$-norm bounds on the error term $F_{\mathrm{err}}$. Recall the error term $$\begin{aligned} F_{\text{err}}& =\partial _{t}f_{\mathrm{a}}+v\cdot \nabla _{x}f_{\mathrm{a}}+Q^{-}(f_{\mathrm{a}},f_{\mathrm{a}})-Q^{+}(f_{\mathrm{a}},f_{ \mathrm{a}}) \\ & =v\cdot \nabla _{x}\,f_{\mathrm{r}}-Q^{+}(f_{\mathrm{r}},f_{\mathrm{b}})\mp Q^{\pm}(f_{\mathrm{b}},f_{\mathrm{r}})\mp Q^{\pm}(f_{\mathrm{r}},f_{ \mathrm{r}})\mp Q^{\pm}(f_{\mathrm{b}},f_{\mathrm{b}}),\end{aligned}$$ and thus the estimate on $F_{\mathrm{err}}$ highly relies on the $Z$-norm bounds of $f_{\mathrm{r}}$ and $f_{\mathrm{b}}$. Recall the estimate [\[equ:z-norm estimate for fb,L2\]](#equ:z-norm estimate for fb,L2){reference-type="eqref" reference="equ:z-norm estimate for fb,L2"} in Lemma [Lemma 13](#lemma:bounds on fb){reference-type="ref" reference="lemma:bounds on fb"} that $$\begin{aligned} M^{\frac{d-3}{2}}\Vert \nabla_{x} f_{\mathrm{b}}\Vert _{L_{v}^{2,r_{0}}L_{x}^{2}}\lesssim M^{\frac{d-1}{2}-s}N_{2}^{\max\left\{ s_{0},-\gamma \right\} -\frac{d-1}{2}},\\ M^{\frac{d-1}{2}}\Vert f_{\mathrm{b}}\Vert _{L_{v}^{2,r_{0}}L_{x}^{2}}\lesssim M^{\frac{d-1}{2}-s}N_{2}^{\max\left\{ s_{0},-\gamma \right\} -\frac{d-1}{2}}.\end{aligned}$$ For the case $\gamma\in(\frac{1-d}{2},0]$, the extra smallness comes from the factor $N_{2}^{\max\left\{ s_{0},-\gamma \right\} -\frac{d-1}{2}}$ as we have required that $s_{0}<\frac{d-1}{2}$ and $N_{2}\gg M$. Thus, it is enough to deal with the hardest endpoint case that $\gamma=\frac{1-d}{2}$, in which the $M^{\frac{d-3}{2}}\Vert \nabla_{x}\bullet\Vert _{L_{v}^{2,r_{0}}L_{x}^{2}}$ and $M^{\frac{d-1}{2}}\Vert \bullet\Vert _{L_{v}^{2,r_{0}}L_{x}^{2}}$ norms of $f_{\mathrm{b}}$ are the order of $M^{\frac{d-1}{2}-s}$ and hence would not give any smallness for $s<\frac{d-1}{2}$. Additionally, we only need to prove the $d=3$ case as the $d=2$ case follows from a similar way. In the section, we set $d=3$, $\gamma=-1$ and hence $r_{0}=0$, for which the $Z$-norm is $$\begin{aligned} \Vert f(t)\Vert _{Z}=&\Vert \nabla_{x} f(t)\Vert _{L_{v}^{2}L_{x}^{2}} +M\Vert f(t)\Vert _{L_{v}^{2}L_{x}^{2}}+N_{2}^{-1}\Vert f(t)\Vert _{L_{v}^{1}L_{x}^{\infty}}\\ &+N_{2}^{\frac{1}{5}}\Vert f(t)\Vert _{L_{v}^{\frac{5}{3}}L_{x}^{\infty}} +M^{-1}N_{2}^{-1}\Vert \nabla_{x}f(t)\Vert _{L_{v}^{1}L_{x}^{\infty}}+ M^{-1}N_{2}^{\frac{1}{5}}\Vert \nabla_{x}f(t)\Vert _{L_{v}^{\frac{5}{3}}L_{x}^{\infty}}.\notag\end{aligned}$$ The following is the main result about the $Z$-norm bounds on the error term $F_{\mathrm{err}}$. **Proposition 16** ($Z$-norm bounds on $F_{\text{err}}$). For $T_{*}\leq \tau\leq t \leq 0$, $$\label{equ:Ferr_bound} \Big\Vert \int_{\tau}^{t}e^{-(t-t_{0})v\cdot \nabla _{x}}F_{\text{err} }(t_{0})\,dt_{0} \Big \Vert _{Z}\lesssim M^{-1}.$$ We deal with all of the terms in the following separate sections. In section [3.3.1](#section:Analysis of term1){reference-type="ref" reference="section:Analysis of term1"}, we give estimates on the term $v \cdot \nabla_{x}f_{\mathrm{r}}$. In section [3.3.2](#section:Analysis of term2){reference-type="ref" reference="section:Analysis of term2"}, we handle the bilinear terms which contain $f_{\mathrm{r}}$. Finally, we deal with $Q^{\pm}(f_{\mathrm{b}},f_{\mathrm{b}})$ in sections [3.3.3](#section:Analysis of Q-){reference-type="ref" reference="section:Analysis of Q-"}--[3.3.4](#section:Analysis of Q+){reference-type="ref" reference="section:Analysis of Q+"}, which are the most intricate parts. The estimates are mainly achieved by moving the $t_{0}$ integration to the outside as follows: $$\Big\Vert \int_{\tau }^{t}e^{-(t-t_{0})v\cdot \nabla _{x}}F_{\text{err}% }(t_{0})\,dt_{0} \Big \Vert _{Z}\lesssim |T_{*}|\Vert F_{\text{err}}\Vert _{L_{t}^{\infty }Z}.$$ The only exception is the treatment of the bound on $L_{v}^{1}L_{x}^{\infty }$ of $Q^{\pm }(f_{\mathrm{b}},f_{\mathrm{b}})$, where a substantial gain is captured by carrying out the $t_{0}$ integration first. ### Analysis of $v\cdot \nabla_{x}f_{\mathrm{r}}$ {#section:Analysis of term1} **Lemma 17**. *For $T_{*}\leq \tau\leq t \leq 0$, $$\begin{aligned} \Big\Vert \int_{\tau }^{t}e^{-(t-t_{0})v\cdot \nabla_{x}}(v\cdot \nabla_{x}f_{\mathrm{r}})(t_{0})\,dt_{0} \Big \Vert _{Z}\lesssim M^{-1}.\end{aligned}$$* *Proof.* As we have required that $N_{1}\gg M$ in [\[equ:condition,parameters\]](#equ:condition,parameters){reference-type="eqref" reference="equ:condition,parameters"}, the desired decay bound is achieved provided the upper bound carries the smallness factor $N_{1}^{-\delta}$ for some $\delta>0$. **The $\Vert \nabla_{x}\bullet \Vert_{L_{v}^{2}L_{x}^{2}}$ and $M\Vert \bullet \Vert_{L_{v}^{2}L_{x}^{2}}$ estimates.** It suffices to deal with the $\Vert \nabla_{x}\bullet \Vert_{L_{v}^{2}L_{x}^{2}}$ norm, as the estimate for the $M\Vert \bullet \Vert_{L_{v}^{2}L_{x}^{2}}$ norm follows the same way. Noting that $f_{\mathrm{r}}$ is supported on $\left\{ |v|\lesssim N_{1}^{-1} \right\}$, we have $$\begin{aligned} \label{equ:v term,H1} \Vert \nabla_{x} (v\cdot \nabla_{x} f_{\mathrm{r}})\Vert _{L_{v}^{2}L_{x}^{2}}\lesssim& N_{1}^{-1}\Vert \Delta_{x}f_{\mathrm{r}}\Vert _{L_{v}^{2}L_{x}^{2}} \lesssim N_{1}^{-1}M^{2-s}\exp[|t| M^{1-s}]\langle |t|M^{1-s} \rangle ^{2},\end{aligned}$$ where the last inequality follows from the proof of [\[equ:upper bound,fr,sobolev\]](#equ:upper bound,fr,sobolev){reference-type="eqref" reference="equ:upper bound,fr,sobolev"} with one $x$-derivative producing a factor of $M$. We then insert in $|T_{*}|=M^{s-1}(\ln\ln\ln M)$ to get $$\begin{aligned} &\Big\Vert \nabla_{x} \int_{\tau}^{t}e^{-(t-t_{0})v\cdot \nabla _{x}}(v\cdot \nabla_{x}f)(t_{0})\,dt_{0} \Big \Vert _{L_{v}^{2}L_{x}^{2}}\\ \lesssim& |T_{*}| \sup_{t_{0}\in[T_{*},0]}\Vert \nabla_{x}(v\cdot \nabla_{x} f_{\mathrm{r}})\Vert _{L_{v}^{2}L_{x}^{2}}\\ \lesssim&M^{s-1}(\ln \ln \ln M) N_{1}^{-1}M^{2-s}(\ln \ln M)^{3}\\ \lesssim& N_{1}^{-1}M (\ln \ln M)^{4}.\end{aligned}$$ **The $N_{2}^{-1}\Vert \bullet \Vert _{L_{v}^{1}L_{x}^{\infty}}$ and $M^{-1}N_{2}^{-1}\Vert \nabla_{x}\bullet\Vert _{L_{v}^{1}L_{x}^{\infty}}$ estimates.** We only need to treat the $M^{-1}N_{2}^{-1}\Vert \nabla_{x}\bullet\Vert _{L_{v}^{1}L_{x}^{\infty}}$ norm, as the $N_{2}^{-1}\Vert \bullet\Vert _{L_{v}^{1}L_{x}^{\infty}}$ norm can be dealt with in a similar way. Recalling that $(d=3)$ $$f_{\mathrm{r}}(t,x,v)=M^{\frac{3}{2}-s}N_{1}^{\frac{3}{2}}\exp \left[ -\beta(t,x,v)\right] \chi (Mx)\chi ( N_{1}v),$$ we use the pointwise upper bound [\[equ:upper bound,fr,k\]](#equ:upper bound,fr,k){reference-type="eqref" reference="equ:upper bound,fr,k"} to get $$\begin{aligned} &\Vert \nabla_{x}(v\cdot \nabla_{x}f_{\mathrm{r}})\Vert _{L_{v}^{1}L_{x}^{\infty}}\\ \lesssim &N_{1}^{-1}M^{2+\frac{3}{2}-s}N_{1}^{\frac{3}{2}}\Vert \exp[-\beta(t,x,v)](\nabla^{2} \chi)(Mx)\chi(N_{1}v)\Vert _{L_{v}^{1}L_{x}^{\infty}}\\ &+N_{1}^{-1}M^{1+\frac{3}{2}-s}N_{1}^{\frac{3}{2}}\Vert \nabla_{x}\beta(t,x,v)\exp[-\beta(t,x,v)](\nabla \chi)(Mx)\chi(N_{1}v)\Vert _{L_{v}^{1}L_{x}^{\infty}}\\ &+N_{1}^{-1}M^{\frac{3}{2}-s}N_{1}^{\frac{3}{2}}\Vert \nabla_{x}^{2}\beta(t,x,v)\exp[-\beta(t,x,v)] \chi (Mx)\chi(N_{1}v)\Vert _{L_{v}^{1}L_{x}^{\infty}}\\ &+N_{1}^{-1}M^{\frac{3}{2}-s}N_{1}^{\frac{3}{2}}\Vert |\nabla_{x}\beta(t,x,v)|^{2}\exp[-\beta(t,x,v)] \chi (Mx)\chi(N_{1}v)\Vert _{L_{v}^{1}L_{x}^{\infty}}\\ \lesssim& N_{1}^{-1}N_{1}^{-\frac{3}{2}}M^{2-s}M^{\frac{3}{2}} \exp[M^{1-s}|t|]\langle M^{1-s}|t| \rangle ^{2}.\end{aligned}$$ When multiplied by $|T_{*}|=M^{s-1}(\ln\ln\ln M)$, this gives $$\begin{aligned} \label{equ:vfr,L1} &M^{-1}N_{2}^{-1} \Big\Vert \nabla_{x} \int_{\tau}^{t}e^{-(t-t_{0})v\cdot \nabla _{x}}(v\cdot \nabla_{x}f)(t_{0})\,dt_{0} \Big \Vert _{L_{v}^{1}L_{x}^{\infty}}\\ \lesssim& |T_{*}|N_{1}^{-\frac{5}{2}}N_{2}^{-1}M^{1-s}M^{\frac{3}{2}}\exp[M^{1-s}|T_{*}|]\langle M^{1-s}|T_{*}| \rangle ^{2}\notag\\ \lesssim& N_{1}^{-\frac{5}{2}}N_{2}^{-1}M^{\frac{3}{2}} (\ln \ln M)^{4}. \notag\end{aligned}$$ **The $N_{2}^{\frac{1}{5}}\Vert \bullet \Vert _{L_{v}^{\frac{5}{3}}L_{x}^{\infty}}$ and $M^{-1}N_{2}^{\frac{1}{5}}\Vert \nabla_{x}\bullet\Vert _{L_{v}^{\frac{5}{3}}L_{x}^{\infty}}$ estimates.** These two norms can be estimated in the same manner as [\[equ:vfr,L1\]](#equ:vfr,L1){reference-type="eqref" reference="equ:vfr,L1"} with the $L_{v}^{1}$ norm replaced by the $L_{v}^{\frac{5}{3}}$ norm. Therefore, we also have $$\begin{aligned} &M^{-1}N_{2}^{\frac{1}{5}} \Big\Vert \nabla_{x} \int_{\tau}^{t}e^{-(t-t_{0})v\cdot \nabla _{x}}(v\cdot \nabla_{x}f)(t_{0})\,dt_{0} \Big \Vert _{L_{v}^{\frac{5}{3}}L_{x}^{\infty}}\\ \lesssim& |T_{*}|N_{1}^{-1}N_{1}^{-\frac{3}{10}}N_{2}^{\frac{1}{5}}M^{1-s}M^{\frac{3}{2}}\exp[M^{1-s}|T_{*}|]\langle M^{1-s}|T_{*}| \rangle ^{2}\notag\\ \lesssim& N_{1}^{-1}N_{1}^{-\frac{3}{10}}N_{2}^{\frac{1}{5}}M^{\frac{3}{2}} (\ln \ln M)^{4}\notag\\ \lesssim& N_{1}^{-\frac{11}{10}}M^{\frac{3}{2}} (\ln \ln M)^{4},\notag\end{aligned}$$ where in the last inequality we have used that $N_{1}\geq N_{2}$. ◻ ### Analysis of $Q^{+}(f_{\mathrm{r}},f_{\mathrm{b}})$, $Q^{\pm}(f_{\mathrm{b}},f_{\mathrm{r}})$, and $Q^{\pm}(f_{\mathrm{r}},f_{\mathrm{r}})$ {#section:Analysis of term2} Before getting into the analysis of the terms, we recall some estimates on $f_{\mathrm{b}}$ and $f_{\mathrm{r}}$, which are established in Lemma [Lemma 13](#lemma:bounds on fb){reference-type="ref" reference="lemma:bounds on fb"} and Lemma [Lemma 14](#lemma:z-norm bounds on fr){reference-type="ref" reference="lemma:z-norm bounds on fr"}. That is, $$\begin{aligned} \Vert f_{\mathrm{b}}\Vert _{L^{\infty}([T_{*},0];Z)}\lesssim& M^{1-s}, \label{equ:z-norm,fb,used}\\ \Vert f_{\mathrm{r}}\Vert _{L^{\infty}([T_{*},0];Z)}\lesssim& M^{1-s}(\ln \ln M)^{2},\label{equ:z-norm,fr,used}\\ \Vert f_{\mathrm{r}}\Vert _{L_{v}^{1}L_{x}^{\infty}}^{\frac{1}{6}}\Vert f_{\mathrm{r}}\Vert _{L_{v}^{\frac{5}{3}}L_{x}^{\infty}}^{\frac{5}{6}}\lesssim & N_{1}^{-\frac{1}{2}}M^{\frac{3}{2}-s}(\ln \ln M)^{2},\label{equ:z-norm,lv1,used}\\ M^{-1}\Vert \nabla_{x}f_{\mathrm{r}}\Vert _{L_{v}^{1}L_{x}^{\infty}}^{\frac{1}{6}}\Vert \nabla_{x}f_{\mathrm{r}}\Vert _{L_{v}^{\frac{5}{3}} L_{x}^{\infty}}^{\frac{5}{6}}\lesssim& N_{1}^{-\frac{1}{2}}M^{\frac{3}{2}-s} (\ln \ln M)^{2},\label{equ:z-norm,lv1,H1,used}\end{aligned}$$ where the last two inequalities [\[equ:z-norm,lv1,used\]](#equ:z-norm,lv1,used){reference-type="eqref" reference="equ:z-norm,lv1,used"}--[\[equ:z-norm,lv1,H1,used\]](#equ:z-norm,lv1,H1,used){reference-type="eqref" reference="equ:z-norm,lv1,H1,used"} follow from estimates [\[equ:fr,Z-norm,L1\]](#equ:fr,Z-norm,L1){reference-type="eqref" reference="equ:fr,Z-norm,L1"}--[\[equ:vfr,L35,H1\]](#equ:vfr,L35,H1){reference-type="eqref" reference="equ:vfr,L35,H1"}. In addition, during the proof of the bilinear estimate on $Q^{\pm}$ in Lemma [Lemma 20](#lemma:binlinear estimate){reference-type="ref" reference="lemma:binlinear estimate"} we postpone to Section [3.4](#section:Bounds on the Correction Term){reference-type="ref" reference="section:Bounds on the Correction Term"}, we actually have that $$\begin{aligned} \Vert Q^{-}(f_{\mathrm{b}},f_{\mathrm{r}})\Vert _{Z}\lesssim \Vert f_{\mathrm{b}}\Vert _{Z}\left( \Vert f_{\mathrm{r}}\Vert _{L_{v}^{1}L_{x}^{\infty}}^{\frac{1}{6}}\Vert f_{\mathrm{r}}\Vert _{L_{v}^{\frac{5}{3}}L_{x}^{\infty}}^{\frac{5}{6}} +M^{-1}\Vert \nabla_{x}f_{\mathrm{r}}\Vert _{L_{v}^{1}L_{x}^{\infty}}^{\frac{1}{6}}\Vert \nabla_{x}f_{\mathrm{r}}\Vert _{L_{v}^{\frac{5}{3}}L_{x}^{\infty}}^{\frac{5}{6}} \right) ,\\ \Vert Q^{+}(f_{\mathrm{b}},f_{\mathrm{r}})\Vert _{Z}\lesssim \Vert f_{\mathrm{b}}\Vert _{Z}\left( \Vert f_{\mathrm{r}}\Vert _{L_{v}^{1}L_{x}^{\infty}}^{\frac{1}{6}}\Vert f_{\mathrm{r}}\Vert _{L_{v}^{\frac{5}{3}}L_{x}^{\infty}}^{\frac{5}{6}} +M^{-1}\Vert \nabla_{x}f_{\mathrm{r}}\Vert _{L_{v}^{1}L_{x}^{\infty}}^{\frac{1}{6}}\Vert \nabla_{x}f_{\mathrm{r}}\Vert _{L_{v}^{\frac{5}{3}}L_{x}^{\infty}}^{\frac{5}{6}} \right) ,\\ \Vert Q^{+}(f_{\mathrm{r}},f_{\mathrm{b}})\Vert _{Z}\lesssim \Vert f_{\mathrm{b}}\Vert _{Z}\left( \Vert f_{\mathrm{r}}\Vert _{L_{v}^{1}L_{x}^{\infty}}^{\frac{1}{6}}\Vert f_{\mathrm{r}}\Vert _{L_{v}^{\frac{5}{3}}L_{x}^{\infty}}^{\frac{5}{6}} +M^{-1}\Vert \nabla_{x}f_{\mathrm{r}}\Vert _{L_{v}^{1}L_{x}^{\infty}}^{\frac{1}{6}}\Vert \nabla_{x}f_{\mathrm{r}}\Vert _{L_{v}^{\frac{5}{3}}L_{x}^{\infty}}^{\frac{5}{6}} \right) ,\\ \Vert Q^{\pm}(f_{\mathrm{r}},f_{\mathrm{r}})\Vert _{Z}\lesssim \Vert f_{\mathrm{r}}\Vert _{Z}\left( \Vert f_{\mathrm{r}}\Vert _{L_{v}^{1}L_{x}^{\infty}}^{\frac{1}{6}}\Vert f_{\mathrm{r}}\Vert _{L_{v}^{\frac{5}{3}}L_{x}^{\infty}}^{\frac{5}{6}} +M^{-1}\Vert \nabla_{x}f_{\mathrm{r}}\Vert _{L_{v}^{1}L_{x}^{\infty}}^{\frac{1}{6}}\Vert \nabla_{x}f_{\mathrm{r}}\Vert _{L_{v}^{\frac{5}{3}}L_{x}^{\infty}}^{\frac{5}{6}} \right) .\end{aligned}$$ Note that such an estimate is not possible for $Q^{-}(f_{\mathrm{r}},f_{\mathrm{b}})$, which is not contained in the error terms. Therefore, for $$(sgn,1,2)\in\left\{ (+,r,b),(\pm,b,r),(\pm,r,r) \right\} ,$$ by estimates [\[equ:z-norm,fb,used\]](#equ:z-norm,fb,used){reference-type="eqref" reference="equ:z-norm,fb,used"}--[\[equ:z-norm,lv1,used\]](#equ:z-norm,lv1,used){reference-type="eqref" reference="equ:z-norm,lv1,used"}, we have $$\begin{aligned} &\Big\Vert \int_{\tau }^{t}e^{-(t-t_{0})v\cdot \nabla_{x}}Q^{sgn}(f_{1},f_{2})(t_{0})\,dt_{0} \Big \Vert _{Z}\\ \lesssim& |T_{*}|\left( \Vert f_{\mathrm{r}}\Vert _{Z}+\Vert f_{\mathrm{b}}\Vert _{Z} \right) \left( \Vert f_{\mathrm{r}}\Vert _{L_{v}^{1}L_{x}^{\infty}}^{\frac{1}{6}}\Vert f_{\mathrm{r}}\Vert _{L_{v}^{\frac{5}{3}}L_{x}^{\infty}}^{\frac{5}{6}} +M^{-1}\Vert \nabla_{x}f_{\mathrm{r}}\Vert _{L_{v}^{1}L_{x}^{\infty}}^{\frac{1}{6}}\Vert \nabla_{x}f_{\mathrm{r}}\Vert _{L_{v}^{\frac{5}{3}}L_{x}^{\infty}}^{\frac{5}{6}} \right) \\ \lesssim& |T_{*}|M^{1-s}(\ln \ln M)^{2}N_{1}^{-\frac{1}{2}}M^{\frac{3}{2}-s} (\ln \ln M)^{2}\\ \lesssim& N_{1}^{-\frac{1}{2}}M^{\frac{3}{2}-s}(\ln \ln M)^{5},\end{aligned}$$ where in the last inequality we have inserted in $|T_{*}|=M^{s-1}(\ln\ln\ln M)$. This bound suffices for our goal as it carries the smallness factor $N_{1}^{-\frac{1}{2}}$. ### Analysis of $Q^{-}(f_{\mathrm{b}},f_{\mathrm{b}})$ {#section:Analysis of Q-} **Lemma 18**. *For $T_{*}\leq \tau\leq t \leq 0$, $$\begin{aligned} \Big\Vert \int_{\tau}^{t}e^{-(t-t_{0})v\cdot \nabla _{x}}Q^{-}(f_{\mathrm{b}},f_{\mathrm{b}})(t_{0})\,dt_{0} \Big \Vert _{Z}\lesssim M^{-1}. \end{aligned}$$* *Proof.* As we have required that $N_{2}\gg M$ in [\[equ:condition,parameters\]](#equ:condition,parameters){reference-type="eqref" reference="equ:condition,parameters"}, the desired smallness comes from the factor $N_{2}^{-\delta}$ for some $\delta>0$. As the $x$-derivative, which is put on $f_{\mathrm{b}}$, produces a factor of $M$, it is sufficient to estimate the $L_{v}^{2}L_{x}^{2}$, $L_{v}^{1}L_{x}^{\infty}$ and $L_{v}^{\frac{5}{3}}L_{x}^{\infty}$ norms. **The $M\Vert \bullet \Vert _{L_{v}^{2}L_{x}^{2}}$ estimate.** Note that $$\label{equ:equ:Q-,fb,fb,pointwise,first} M\Big\Vert \int_{\tau}^{t}e^{-(t-t_{0})v\cdot \nabla _{x}}Q^{-}(f_{\mathrm{b}},f_{\mathrm{b}})(t_{0})\,dt_{0} \Big \Vert _{L_{v}^{2}L_{x}^{2}}\lesssim |T_{*}|M\Vert Q^{-}(f_{\mathrm{b}},f_{\mathrm{b}})\Vert _{L_{t}^{\infty }(T_{*},0;L_{v}^{2}L_{x}^{2})}.$$ We only need to control $M\Vert Q^{-}(f_{\mathrm{b}},f_{\mathrm{b}})\Vert _{L_{t}^{\infty }L_{v}^{2}L_{x}^{2}}$. Recall the upper bound [\[equ:pointwise estimate on fb\]](#equ:pointwise estimate on fb){reference-type="eqref" reference="equ:pointwise estimate on fb"} that $$\begin{aligned} \label{equ:pointwise estimate on fb,recall} f_{\mathrm{b}}(t,x,u)\lesssim& \frac{M^{1-s}}{N_{2}^{2}}\sum_{j=1}^{J}\widetilde{K}_{j}(x)I_{j}(u), \end{aligned}$$ where $$\begin{aligned} \widetilde{K}_{j}(x)=\chi (\frac{ MP_{e_{j}}^{\perp }x}{10})\chi (\frac{P_{e_{j}}x}{10N_{2}}),\quad I_{j}(u)=\chi(MP_{e_{j}}^{\perp }u)\chi (\frac{10P_{e_{j}}(u-N_{2}e_{j})}{N_{2}}). \end{aligned}$$ Then we have $$\begin{aligned} \label{equ:Q-,fb,fb,pointwise,two cases} &Q^{-}(f_{\mathrm{b}},f_{\mathrm{b}})(t,x,v)\\ \lesssim &\left( \frac{M^{1-s}}{N_{2}^{2}} \right) ^{2} \left( \sum_{|j-k|\lesssim 1}\widetilde{K}_{j}(x)\widetilde{K}_{k}(x)Q^{-}(I_{j},I_{k})(v)+\sum_{|j-k|\gtrsim 1}\widetilde{K}_{j}(x)\widetilde{K}_{k}(x)Q^{-}(I_{j},I_{k})(v) \right) .\notag\end{aligned}$$ **Case $I$: $|j-k|\lesssim 1$.** For the case that $|j-k|\lesssim 1$, the summands in the double sum $\sum_{k}^{J}\sum_{j}^{J}$ are reduced to $(MN_{2})^{2}$. By Hölder and Hardy-Sobolev-Littlewood inequality [\[equ:endpoint estimate,hls\]](#equ:endpoint estimate,hls){reference-type="eqref" reference="equ:endpoint estimate,hls"}, we obtain $$\begin{aligned} \label{equ:Q-,fb,fb,pointwise,case 1} &\left( \frac{M^{1-s}}{N_{2}^{2}} \right) ^{2}\Big\Vert \sum_{|j-k|\lesssim 1}\widetilde{K}_{j}(x)\widetilde{K}_{k}(x)Q^{-}(I_{j},I_{k})(v) \Big \Vert _{L_{x}^{2}L_{v}^{2}}\\ \lesssim &\left( \frac{M^{1-s}}{N_{2}^{2}} \right) ^{2}\Big\Vert \sum_{j}\widetilde{K}_{j}(x)I_{j}(v) \Big \Vert _{L_{x}^{2}L_{v}^{2}}\Vert \widetilde{K}_{k}(x)\Vert _{L_{x}^{\infty}}\Big\Vert \int \frac{I_{k}(u)}{|u-v|}du \Big \Vert _{L_{v}^{\infty}}\notag\\ \lesssim &\left( \frac{M^{1-s}}{N_{2}^{2}} \right) ^{2}\Big\Vert \sum_{j}\widetilde{K}_{j}(x)I_{j}(v) \Big \Vert _{L_{x}^{2}L_{v}^{2}} \Vert I_{k}\Vert _{L_{v}^{1}}^{\frac{1}{3}}\Vert I_{k}\Vert _{L_{v}^{2}}^{\frac{2}{3}}\notag\\ \lesssim &\left( \frac{M^{1-s}}{N_{2}^{2}} \right) ^{2} (M^{-1}N_{2}^{2})(M^{-2}N_{2})^{\frac{2}{3}}\notag\\ =&M^{-\frac{1}{3}-2s}N_{2}^{-\frac{4}{3}}\notag\end{aligned}$$ where in the second-to-last inequality we have used the disjointness of the $v$-support to get $$\begin{aligned} \Big\Vert \sum_{j}\widetilde{K}_{j}(x)I_{j}(v) \Big \Vert _{L_{x}^{2}L_{v}^{2}}^{2}\lesssim \sum_{j}\Vert \widetilde{K}_{j}(x)\Vert _{L_{x}^{2}}^{2}\Vert I_{j}(v)\Vert _{L_{v}^{2}}^{2}\lesssim (MN_{2})^{2}(M^{-2}N_{2})^{2}=M^{-2}N_{2}^{4}.\end{aligned}$$ **Case $II$: $|j-k|\gtrsim 1$.** For the case that $|j-k|\gtrsim 1$, this implies that $\sin \alpha_{j,k}\gtrsim (MN_{2})^{-1}$, where $\alpha_{j,k}$ denotes the angle between $e_{j}$ and $e_{k}$. Then we have $$\begin{aligned} Q^{-}(I_{j},I_{k})(v)=&I_{j}(v)\int \frac{I_{k}(u)}{|u-v|}du\\ =&I_{j}(v)\int \frac{1}{|u-v|}\chi (MP_{e_{k}}^{\perp }u)\chi (\frac{10P_{e_{k}}(u-N_{2}e_{k})}{N_{2}})du\\ \lesssim &I_{j}(v)\int \frac{1}{|P_{e_{k}}(u-v)|+|P_{e_{k}}^{\perp}(u-v)|}\chi (MP_{e_{k}}^{\perp }u)\chi (\frac{10P_{e_{k}}(u-N_{2}e_{k})}{N_{2}})du.\end{aligned}$$ Due to the $v$-support and $u$-support, we write $$v=ae_{j}+ce_{j}^{\perp},\quad u=be_{k}+de_{k}^{\perp}$$ where $a\sim b\sim N_{2}$ and $c\sim d\sim M^{-1}$. Therefore, this gives $$\begin{aligned} \label{equ:lower bound,ej,ek} |P_{e_{k}}^{\perp}(u-v)|=&|P_{e_{k}}^{\perp}(be_{k}+de_{k}^{\perp}-ae_{j}-ce_{j}^{\perp})|\\ \gtrsim &a|P_{e_{k}}^{\perp}e_{j}|-d-c\notag\\ \gtrsim &N_{2}\sin \alpha_{j,k}-M^{-1}\gtrsim M^{-1}\notag\end{aligned}$$ where in the last inequality we have used that $\sin \alpha_{j,k}\gtrsim (MN_{2})^{-1}$. By the estimate [\[equ:lower bound,ej,ek\]](#equ:lower bound,ej,ek){reference-type="eqref" reference="equ:lower bound,ej,ek"}, we then set $\xi=\langle u,e_{k} \rangle$ to get $$\begin{aligned} \label{equ:estimate,IjIk} I_{j}(v)\int \frac{I_{k}(u)}{|u-v|}du\lesssim &I_{j}(v)\int \frac{1}{|P_{e_{k}}(u-v)|+M^{-1}}\chi (MP_{e_{k}}^{\perp }u)\chi (\frac{10P_{e_{k}}(u-N_{2}e_{k})}{N_{2}})du\\ \lesssim &I_{j}(v)\int \frac{1}{|\xi-\langle e_{k},v \rangle |+M^{-1}}\chi (M\xi^{\perp})\chi (\frac{10(\xi-N_{2})}{N_{2}})d\xi d\xi^{\perp}\notag\\ \lesssim &I_{j}(v)M^{-2}\int\frac{1}{|\xi|+M^{-1}}\chi(\frac{10(\xi+\langle e_{k},v \rangle -N_{2})}{N_{2}})d\xi\notag\\ =&I_{j}(v)M^{-2}N_{2}\int_{-1}^{1}\frac{M}{MN_{2}|\xi|+1}\chi(\frac{10(N_{2}\xi+\langle e_{k},v \rangle -N_{2})}{N_{2}})d\xi\notag\\ \lesssim &I_{j}(v)M^{-2}N_{2}\int_{-1}^{1}\frac{M}{MN_{2}|\xi|+1}d\xi\notag\\ \lesssim &I_{j}(v)\frac{\ln(MN_{2})}{M^{2}}.\notag\end{aligned}$$ Consequently, we arrive at $$\begin{aligned} &\left( \frac{M^{1-s}}{N_{2}^{2}} \right) ^{2}\sum_{|j-k|\gtrsim 1}\widetilde{K}_{j}(x)\widetilde{K}_{k}(x)Q^{-}(I_{j},I_{k})\\ =&\left( \frac{M^{1-s}}{N_{2}^{2}} \right) ^{2}\sum_{|j-k|\gtrsim 1}\widetilde{K}_{j}(x)\widetilde{K}_{k}(x)I_{j}(v)\int \frac{I_{k}(u)}{|u-v|}du\notag\\ \lesssim & \left( \frac{M^{1-s}}{N_{2}^{2}} \right) ^{2}\sum_{j,k}\widetilde{K}_{j}(x)\widetilde{K}_{k}(x)I_{j}(v)\frac{\ln(MN_{2})}{M^{2}}\notag\\ \leq& \left( \frac{M^{1-s}}{N_{2}^{2}} \right) ^{2} \frac{\ln(MN_{2})}{M^{2}} \left( \frac{N_{2}}{|x|+M^{-1}} \right) ^{2}\chi(\frac{x}{N_{2}}) \sum_{j}\widetilde{K}_{j}(x)I_{j}(v)\notag\end{aligned}$$ where in the last inequality we have used that $$\begin{aligned} \label{equ:fb,sum} \sum_{k}\widetilde{K}_{k}(x)=\sum_{k}^{J}\chi (\frac{MP_{e_{k}}^{\perp }x}{10})\chi (\frac{P_{e_{k}}x}{N_{2}})\lesssim\left( \frac{N_{2}}{ |x|+M^{-1}} \right) ^{2}\chi (\frac{x}{N_{2}}).\end{aligned}$$ To see ([\[equ:fb,sum\]](#equ:fb,sum){reference-type="ref" reference="equ:fb,sum"}), we might as well take $x=(0,0,|x|)$ with $M^{-1}\leq |x|\leq N_{2}$. Let $\theta_{j}$ be the angle between $e_{j}$ and $(0,0,1)$. Then, we have $$\begin{aligned} \sum_{j}^{J}\chi (\frac{MP_{e_{j}}^{\perp }x}{10})\chi (\frac{P_{e_{j}}x}{N_{2}})=\sum_{j}^{J}\chi (\frac{M|x|\sin \theta_{j}}{10})=\sum_{j:\sin\theta_{j}\lesssim \frac{1}{|x|M}}1\sim \frac{(MN_{2})^{2}}{(|x|M)^{2}}= \frac{N_{2}^{2}}{|x|^{2}}.\end{aligned}$$ Applying the $L_{v}^{2}L_{x}^{2}$ norm, we have $$\begin{aligned} \label{equ:Q-,fb,fb,pointwise,case 2} &\left( \frac{M^{1-s}}{N_{2}^{2}} \right) ^{2}\Big\Vert \sum_{|j-k|\gtrsim 1}\widetilde{K}_{j}(x)\widetilde{K}_{k}(x)Q^{-}(I_{j},I_{k}) \Big \Vert _{L_{v}^{2}L_{x}^{2}}\\ \lesssim& \left( \frac{M^{1-s}}{N_{2}^{2}} \right) ^{2} \frac{\ln(MN_{2})}{M^{2}} \Big\Vert \sum_{j}\widetilde{K}_{j}(x)I_{j}(v) \Big \Vert _{L_{v}^{2}L_{x}^\infty} \Big\Vert \left( \frac{N_{2}}{|x|+M^{-1}} \right) ^{2}\chi(\frac{x}{N_{2}}) \Big \Vert _{L_{x}^{2}}\notag\\ \lesssim& \left( \frac{M^{1-s}}{N_{2}^{2}} \right) ^{2} \frac{\ln(MN_{2})}{M^{2}} N_{2}^{\frac{3}{2}} (M^{\frac{1}{2}}N_{2}^{2})\notag\\ =& M^{\frac{1}{2}-2s}N_{2}^{-\frac{1}{2}}\ln(MN_{2})\notag\end{aligned}$$ where we have used that $$\begin{aligned} &\Big\Vert \sum_{j}\widetilde{K}_{j}(x)I_{j}(v) \Big \Vert _{L_{v}^{2}L_{x}^\infty}\lesssim \Big\Vert \sum_{j}I_{j}(v) \Big \Vert _{L_{v}^{2}}\sim N_{2}^{\frac{3}{2}},\end{aligned}$$ and $$\begin{aligned} & \Big\Vert \left( \frac{N_{2}}{|x|+M^{-1}} \right) ^{2}\chi(\frac{x}{N_{2}}) \Big \Vert _{L_{x}^{2}}=M^{\frac{1}{2}}N_{2}^{2} \Big\Vert \left( \frac{1}{|x|+1} \right) ^{2}\chi(\frac{x}{MN_{2}}) \Big \Vert _{L_{x}^{2}} \lesssim M^{\frac{1}{2}}N_{2}^{2}.\label{equ:pointwise estimate,fb}\end{aligned}$$ Combining estimates [\[equ:Q-,fb,fb,pointwise,case 1\]](#equ:Q-,fb,fb,pointwise,case 1){reference-type="eqref" reference="equ:Q-,fb,fb,pointwise,case 1"} and [\[equ:Q-,fb,fb,pointwise,case 2\]](#equ:Q-,fb,fb,pointwise,case 2){reference-type="eqref" reference="equ:Q-,fb,fb,pointwise,case 2"} in the two cases, we finally reach $$\begin{aligned} M\Vert Q^{-}(f_{\mathrm{b}},f_{\mathrm{b}})\Vert _{L_{v}^{2}L_{x}^{2}}\lesssim M^{\frac{3}{2}-2s}N_{2}^{-\frac{1}{2}}\ln(MN_{2}).\end{aligned}$$ Together with [\[equ:equ:Q-,fb,fb,pointwise,first\]](#equ:equ:Q-,fb,fb,pointwise,first){reference-type="eqref" reference="equ:equ:Q-,fb,fb,pointwise,first"}, we insert in $|T_{*}|=M^{s-1}(\ln \ln \ln M)$ to obtain $$\begin{aligned} M\Big\Vert \int_{\tau}^{t}e^{-(t-t_{0})v\cdot \nabla _{x}}Q^{-}(f_{\mathrm{b}},f_{\mathrm{b}})(t_{0})\,dt_{0} \Big \Vert _{L_{v}^{2}L_{x}^{2}}\lesssim N_{2}^{-\frac{1}{2}} M^{\frac{1}{2}-s}\ln(MN_{2})(\ln \ln \ln M),\end{aligned}$$ which suffices for our goal. **The $N_{2}^{-1}\Vert \bullet\Vert _{L_{v}^{1}L_{x}^{\infty}}$ estimate.** For convenience, we use the notation $$D^{-}=\int_{\tau}^{t}e^{-(t-t_{0})v\cdot \nabla _{x}}Q^{-}(f_{\mathrm{b}},f_{\mathrm{b}})(t_{0})dt_{0}.$$ From the analysis on $Q^{-}(f_{\mathrm{b}},f_{\mathrm{b}})$ in estimates [\[equ:Q-,fb,fb,pointwise,case 1\]](#equ:Q-,fb,fb,pointwise,case 1){reference-type="eqref" reference="equ:Q-,fb,fb,pointwise,case 1"} and [\[equ:Q-,fb,fb,pointwise,case 2\]](#equ:Q-,fb,fb,pointwise,case 2){reference-type="eqref" reference="equ:Q-,fb,fb,pointwise,case 2"}, we actually get a pointwise estimate on $Q^{-}(f_{\mathrm{b}},f_{\mathrm{b}})$ that $$\begin{aligned} Q^{-}(f_{\mathrm{b}},f_{\mathrm{b}})\lesssim \left( \frac{M^{1-s}}{N_{2}^{2}} \right) ^{2}\sum_{j}\widetilde{K}_{j}(x)I_{j}(v)\left[ \frac{\ln(MN_{2})}{M^{2}} \left( \frac{N_{2}}{|x|+M^{-1}} \right) ^{2}\chi (\frac{x}{N_{2}})+(M^{-2}N_{2})^{\frac{2}{3}} \right] .\end{aligned}$$ Expanding $D^{-}$ gives that $$\begin{aligned} D^{-}=&\int_{\tau}^{t}Q^{-}(f_{\mathrm{b}},f_{\mathrm{b}})(t_{0},x-v(t-t_{0}),v)dt_{0}\\ \lesssim &\int_{\tau}^{t} \left( \frac{M^{1-s}}{N_{2}^{2}} \right) ^{2} \sum_{j}\widetilde{K}_{j}(x-v(t-t_{0}))I_{j}(v) \\ &\times \left[ \frac{\ln(MN_{2})}{M^{2}} \left( \frac{N_{2}}{|x-v(t-t_{0})|+M^{-1}} \right) ^{2}\chi(\frac{x-v(t-t_{0})}{N_{2}})+(M^{-2}N_{2})^{\frac{2}{3}} \right] dt_{0}\\ \lesssim &\frac{M^{-2s}}{N_{2}^{4}}I(v) \left[ \ln (MN_{2}) \int_{T_{*}}^{0} \left( \frac{N_{2}}{ |x-v(t-t_{0})|+M^{-1}} \right) ^{2} \chi (\frac{x-v(t-t_{0})}{N_{2}})dt_{0}+(MN_{2})^{\frac{2}{3}} \right] ,\end{aligned}$$ where in the last inequality we have used that $$\begin{aligned} \sum_{j}\widetilde{K}_{j}(x-v(t-t_{0}))I_{j}(v)\leq \sum_{j}I_{j}(v)=:I(v)\sim 1_{\left\{ \frac{9N_{2}}{10}\leq|v|\leq \frac{11N_{2}}{10} \right\} }(v).\end{aligned}$$ We then deal with the time integral. By change of variable, we have $$\begin{aligned} & I(v)\int_{T_{*}}^{0} \left( \frac{N_{2}}{ |x-v(t-t_{0})|+M^{-1}} \right) ^{2} \chi (\frac{x-v(t-t_{0})}{N_{2}})dt_{0}\\ \leq&I(v)\int_{T_{*}}^{|T_{*}|} \left( \frac{MN_{2}}{|Mx-Mv\sigma|+1} \right) ^{2} \chi (\frac{x-v\sigma}{N_{2}})d\sigma\\ \leq& \frac{(MN_{2})^{2}I(v)}{M|v|}\int_{-M|T_{*}||v|}^{M|T_{*}||v|}\left( \frac{1}{\big | \sigma-M|x| \big|+1} \right) ^{2}d\sigma\\ \lesssim& MN_{2}I(v),\end{aligned}$$ where in the last inequality we have used that $|v|\sim N_{2}$ and $\int \frac{d\tau}{\langle \tau \rangle ^{2}}\lesssim 1$. Hence, after carrying out the 1D $dt_{0}$ integral, we arrive at $$\begin{aligned} \label{equ:Q-,fb,fb,L1,final} N_{2}^{-1}\Vert D^{-}\Vert _{L_{v}^{1}L_{x}^{\infty}}\lesssim&N_{2}^{-1}\frac{M^{-2s}}{N_{2}^{4}} \Vert I(v)\Vert _{L_{v}^{1}} \left[ \ln (MN_{2})MN_{2}+(MN_{2})^{\frac{2}{3}} \right] \\ \lesssim& N_{2}^{-1}\frac{M^{-2s}}{N_{2}^{4}} N_{2}^{3} \ln (MN_{2})MN_{2}\notag\\ =&N_{2}^{-1} M^{1-2s}\ln (MN_{2}).\notag\end{aligned}$$ **The $N_{2}^{\frac{1}{5}}\Vert \bullet\Vert _{L_{v}^{\frac{5}{3}}L_{x}^{\infty}}$ estimate.** By the interpolation inequality, we have $$\begin{aligned} \label{equ:Q-,fb,fb,L53} N_{2}^{\frac{1}{5}}\Vert D^{-}\Vert _{L_{v}^{\frac{5}{3}}L_{x}^{\infty}}\leq \left( N_{2}^{-1}\Vert D^{-}\Vert _{L_{v}^{1}L_{x}^{\infty}} \right) ^{\frac{1}{5}} \left( N_{2}^{\frac{1}{2}}\Vert D^{-}\Vert _{L_{v}^{2}L_{x}^{\infty}} \right) ^{\frac{4}{5}}.\end{aligned}$$ For the $L_{v}^{2}L_{x}^{\infty}$ norm on $D^{-}$, by Hölder inequality, we have $$\begin{aligned} N_{2}^{\frac{1}{2}}\Vert D^{-}\Vert _{L_{v}^{2}L_{x}^{\infty}}\lesssim& N_{2}^{\frac{1}{2}}|T_{*}|\Vert Q^{-}(f_{\mathrm{b}},f_{\mathrm{b}})\Vert _{L_{v}^{2}L_{x}^{\infty}}\\ \lesssim &N_{2}^{\frac{1}{2}}|T_{*}| \Big\Vert \int \frac{f_{\mathrm{b}}(x,u)}{|u-v|}du \Big \Vert _{L_{v}^{\infty}L_{x}^{\infty}} \Vert f_{\mathrm{b}}\Vert _{L_{v}^{2}L_{x}^{\infty}}.\end{aligned}$$ We then use the $L^{\infty}$ estimate [\[equ:endpoint estimate,hls\]](#equ:endpoint estimate,hls){reference-type="eqref" reference="equ:endpoint estimate,hls"} in Lemma [Lemma 25](#lemma:endpoint estimate,hls){reference-type="ref" reference="lemma:endpoint estimate,hls"} and interpolation inequality to get $$\begin{aligned} \Big\Vert \int \frac{f_{\mathrm{b}}(x,u)}{|u-v|}du \Big \Vert _{L_{v}^{\infty}L_{x}^{\infty}}\lesssim \Vert f_{\mathrm{b}}\Vert _{L_{v}^{1}L_{x}^{\infty}}^{\frac{1}{6}}\Vert f_{\mathrm{b}}\Vert _{L_{v}^{\frac{5}{3}}L_{x}^{\infty}}^{\frac{5}{6}}\lesssim N_{2}^{-1}\Vert f_{\mathrm{b}}\Vert _{L_{v}^{1}L_{x}^{\infty}}+N_{2}^{\frac{1}{5}}\Vert f_{\mathrm{b}}\Vert _{L_{v}^{\frac{5}{3}}L_{x}^{\infty}}\leq \Vert f_{\mathrm{b}}\Vert _{Z}.\end{aligned}$$ By the $Z$-norm bound on $f_{\mathrm{b}}$ in Lemma [Lemma 13](#lemma:bounds on fb){reference-type="ref" reference="lemma:bounds on fb"}, we have that $$\Vert f_{\mathrm{b}}\Vert _{Z}\lesssim M^{1-s},\quad N_{2}^{\frac{1}{2}}\Vert f_{\mathrm{b}}\Vert _{L_{v}^{2}L_{x}^{\infty}}\lesssim M^{1-s}.$$ Thus, inserting in $|T_{*}|=M^{s-1}(\ln\ln\ln M)$, we obtain $$\begin{aligned} \label{equ:Q-,fb,fb,L2} N_{2}^{\frac{1}{2}}\Vert D^{-}\Vert _{L_{v}^{2}L_{x}^{\infty}} \lesssim& M^{1-s}(\ln\ln\ln M).\end{aligned}$$ Combining estimates [\[equ:Q-,fb,fb,L1,final\]](#equ:Q-,fb,fb,L1,final){reference-type="eqref" reference="equ:Q-,fb,fb,L1,final"}, [\[equ:Q-,fb,fb,L53\]](#equ:Q-,fb,fb,L53){reference-type="eqref" reference="equ:Q-,fb,fb,L53"} and [\[equ:Q-,fb,fb,L2\]](#equ:Q-,fb,fb,L2){reference-type="eqref" reference="equ:Q-,fb,fb,L2"}, we reach $$\begin{aligned} N_{2}^{\frac{1}{5}}\Vert D^{-}\Vert _{L_{v}^{\frac{5}{3}}L_{x}^{\infty}}\lesssim N_{2}^{-\frac{1}{5}}M^{1-\frac{6}{5}s}\ln (MN_{2})(\ln\ln\ln M).\end{aligned}$$ This bound is enough as it carries the smallness parameter $N_{2}^{-\frac{1}{5}}$. ◻ ### Analysis of $Q^{+}(f_{\mathrm{b}},f_{\mathrm{b}})$ {#section:Analysis of Q+} **Lemma 19**. *For $T_{*}\leq \tau\leq t \leq 0$, $$\begin{aligned} \Big\Vert \int_{\tau}^{t}e^{-(t-t_{0})v\cdot \nabla _{x}}Q^{+}(f_{\mathrm{b}},f_{\mathrm{b}})(t_{0})\,dt_{0} \Big \Vert _{Z}\lesssim M^{-1}. \end{aligned}$$* *Proof.* In a similar way to estimate $Q^{-}(f_{\mathrm{b}},f_{\mathrm{b}})$, we obtain the desired estimate provided the upper bound carries the smallness factor $N_{2}^{-\delta}$ for some $\delta>0$. For convenience, we use the notation $$D^{+}=\int_{\tau}^{t}e^{-(t-t_{0})v\cdot \nabla _{x}}Q^{+}(f_{\mathrm{b}},f_{\mathrm{b}})(t_{0})dt_{0}.$$ The $x$-derivative produces the factor of $M$, so we only need to estimate the $L_{v}^{2}L_{x}^{2}$, $L_{v}^{1}L_{x}^{\infty}$ and $L_{v}^{\frac{5}{3}}L_{x}^{\infty}$ norms. **The $M\Vert \bullet \Vert _{L_{v}^{2}L_{x}^{2}}$ estimate.** We use again the upper bound [\[equ:pointwise estimate on fb\]](#equ:pointwise estimate on fb){reference-type="eqref" reference="equ:pointwise estimate on fb"} that $$\begin{aligned} f_{\mathrm{b}}(t,x,u)\lesssim& \frac{M^{1-s}}{N_{2}^{2}}\sum_{j=1}^{J}\widetilde{K}_{j}(x)I_{j}(u), \end{aligned}$$ where $$\widetilde{K}_{j}(x)=\chi (\frac{ MP_{e_{j}}^{\perp }x}{10})\chi (\frac{P_{e_{j}}x}{10N_{2}}),\quad I_{j}(u)=\chi(MP_{e_{j}}^{\perp }u)\chi (\frac{10P_{e_{j}}(u-N_{2}e_{j})}{N_{2}}).$$ Then we expand $Q^{+}(f_{\mathrm{b}},f_{\mathrm{b}})$ to get $$\begin{aligned} &\Vert Q^{+}(f_{\mathrm{b}},f_{\mathrm{b}})\Vert _{L_{v}^{2}L_{x}^{2}}^{2}\\ \lesssim & \frac{M^{4-4s}}{N_{2}^{8}}\sum_{j_{1},j_{2},j_{3},j_{4}} \int \widetilde{K}_{j_{1}}(x) \widetilde{K}_{j_{2}}(x)\widetilde{K}_{j_{3}}(x)\widetilde{K}_{j_{4}}(x) Q^{+}(I_{j_{1}},I_{j_{2}})(v)Q^{+}(I_{j_{3}},I_{j_{4}})(v) dx dv.\end{aligned}$$ By using that $\widetilde{K}_{j_{2}}(x)\lesssim 1$ and $\widetilde{K}_{j_{4}}(x)\lesssim 1$, we obtain $$\begin{aligned} &\sum_{j_{1},j_{2},j_{3},j_{4}} \int \widetilde{K}_{j_{1}}(x) \widetilde{K}_{j_{2}}(x)\widetilde{K}_{j_{3}}(x)\widetilde{K}_{j_{4}}(x) Q^{+}(I_{j_{1}},I_{j_{2}})(v)Q^{+}(I_{j_{3}},I_{j_{4}})(v) dx dv\\ \lesssim& \sum_{j_{1},j_{2},j_{3},j_{4}} \int \widetilde{K}_{j_{1}}(x) \widetilde{K}_{j_{3}}(x)dx \int Q^{+}(I_{j_{1}},I_{j_{2}})(v)Q^{+}(I_{j_{3}},I_{j_{4}})(v) dv\\ =&\sum_{j_{1},j_{3}} \int \widetilde{K}_{j_{1}}(x) \widetilde{K}_{j_{3}}(x)dx \int Q^{+}(I_{j_{1}},I)(v)Q^{+}(I_{j_{3}},I)(v) dv\end{aligned}$$ where $$\begin{aligned} I(v)=\sum_{j}^{J}I_{j}(v)\sim 1_{\left\{ \frac{9N_{2}}{10}\leq|v|\leq \frac{11N_{2}}{10} \right\} }(v).\end{aligned}$$ By Hölder inequality and bilinear estimate [\[equ:bilinear estimate,Q+\]](#equ:bilinear estimate,Q+){reference-type="eqref" reference="equ:bilinear estimate,Q+"} for $Q^{+}$ in Lemma [Lemma 26](#lemma:bilinear estimate,Q+){reference-type="ref" reference="lemma:bilinear estimate,Q+"}, $$\begin{aligned} \Vert Q^{+}(f_{\mathrm{b}},f_{\mathrm{b}})\Vert _{L_{v}^{2}L_{x}^{2}}^{2}\lesssim& \frac{M^{4-4s}}{N_{2}^{8}}\sum_{j_{1},j_{3}} \int \widetilde{K}_{j_{1}}(x) \widetilde{K}_{j_{3}}(x)dx \Vert Q^{+}(I_{j_{1}},I)\Vert _{L_{v}^{2}} \Vert Q^{+}(I_{j_{3}},I)\Vert _{L_{v}^{2}}\\ \lesssim&\frac{M^{4-4s}}{N_{2}^{8}} \Vert I\Vert _{L_{v}^{3}}^{2} \sum_{j_{1},j_{3}} \int \widetilde{K}_{j_{1}}(x) \widetilde{K}_{j_{3}}(x)dx \Vert I_{j_{1}}\Vert _{L_{v}^{\frac{6}{5}}}\Vert I_{j_{3}}\Vert _{L^{\frac{6}{5}}}.\end{aligned}$$ Using $\Vert I_{j}\Vert _{L_{v}^{\frac{6}{5}}}\lesssim (M^{-2}N_{2})^{\frac{5}{6}}$, $\Vert I\Vert _{L_{v}^{3}}\lesssim N_{2}$, estimates [\[equ:fb,sum\]](#equ:fb,sum){reference-type="eqref" reference="equ:fb,sum"} and [\[equ:pointwise estimate,fb\]](#equ:pointwise estimate,fb){reference-type="eqref" reference="equ:pointwise estimate,fb"} for the sum, we obtain $$\begin{aligned} \Vert Q^{+}(f_{\mathrm{b}},f_{\mathrm{b}})\Vert _{L_{v}^{2}L_{x}^{2}}^{2}\lesssim & \frac{M^{4-4s}}{N_{2}^{8}}(M^{-2}N_{2})^{\frac{5}{3}} N_{2}^{2}\Big\Vert \sum_{j}\widetilde{K}_{j} \Big \Vert _{L_{x}^{2}}^{2}\\ \lesssim &\frac{M^{4-4s}}{N_{2}^{8}}(M^{-2}N_{2})^{\frac{5}{3}} N_{2}^{2} \Big\Vert \left( \frac{N_{2}}{|x|+M^{-1}} \right) ^{2}\chi(\frac{x}{N_{2}}) \Big \Vert _{L_{x}^{2}}^{2}\\ \lesssim &\frac{M^{4-4s}}{N_{2}^{8}}(M^{-2}N_{2})^{\frac{5}{3}} N_{2}^{2}(MN_{2}^{4})\\ =& M^{\frac{5}{3}-4s}N_{2}^{-\frac{1}{3}}.\end{aligned}$$ Thus, we arrive at $$\begin{aligned} \label{equ:Q+,fb,fb,est1,L2} M\Vert Q^{+}(f_{\mathrm{b}},f_{\mathrm{b}})\Vert _{L_{v}^{2}L_{x}^{2}}\lesssim N_{2}^{-\frac{1}{6}} M^{\frac{11}{6}-2s}.\end{aligned}$$ Upon multiplying by the time factor $|T_*|=M^{s-1}(\ln\ln\ln M)$, this yields a desired bound $$\begin{aligned} M\Vert D^{+}\Vert _{L_{v}^{2}L_{x}^{2}}\lesssim& N_{2}^{-\frac{1}{6}} M^{\frac{5}{6}-s}(\ln\ln\ln M).\end{aligned}$$ **The $N_{2}^{-1}\Vert \bullet\Vert _{L_{v}^{1}L_{x}^{\infty}}$ estimate.** Recall that $$\begin{aligned} f_{\mathrm{b}}(t)=&\frac{M^{1-s}}{N_{2}^{2}}\sum_{j=1}^{J}K_{j}(x-vt)I_{j}(v),\end{aligned}$$ where $K_{j}(x)=\chi (MP_{e_{j}}^{\perp }x)\chi (\frac{P_{e_{j}}x}{N_{2}})$, $I_{j}(v)=\chi(MP_{e_{j}}^{\perp }v)\chi (\frac{10P_{e_{j}}(v-N_{2}e_{j})}{N_{2}})$. The gain term is $$Q^{+}(f,g)=\int_{\mathbb{S}^{2}}\int_{\mathbb{R}^{3}}B(u-v,\omega)f(v^{\ast })g(u^{\ast })\,du\,d\omega,$$ with the relationship that $$\begin{aligned} v^{\ast } =&P_{\omega }^{\Vert }u+P_{\omega }^{\bot }v,\quad u^{\ast }=P_{\omega }^{\Vert }v+P_{\omega }^{\bot }u, \\ v =&P_{\omega }^{\bot }v^{\ast }+P_{\omega }^{\Vert }u^{\ast },\quad% u=P_{\omega }^{\Vert }v^{\ast }+P_{\omega }^{\bot }u^{\ast }.\end{aligned}$$ Then, expanding $D^{+}$ gives $$\begin{aligned} D^{+} =&\frac{M^{2-2s}}{N_{2}^{4}}\sum_{k}^{J}\sum_{j}^{J}\int_{\tau}^{t}e^{-(t-t_{0})v\cdot \nabla_{x}}Q^{+}(K_{k}(x-tv)I_{k}(v),K_{j}(x-tv)I_{j}(v))(t_{0})dt_{0}\\ =&\frac{M^{2-2s}}{N_{2}^{4}}\sum_{k}^{J}\sum_{j}^{J}\int_{ \mathbb{S}^{2}} \int_{\mathbb{R}^{3}} B(u-v,\omega) S_{j,k}(t,x,\omega,u^{\ast },v^{\ast})dud\omega,\end{aligned}$$ where $$\begin{aligned} &S_{j,k}(t,x,\omega,u^{\ast },v^{\ast })\\ =&\int_{\tau}^{t} K_{k}(x-v(t-t_{0})-v^{*}t_{0})I_{k}(v^{*}) K_{j}(x-v(t-t_{0})-u^{*}t_{0})I_{j}(u^{*})dt_{0}.\notag\end{aligned}$$ We estimate by $$\begin{aligned} \label{equ:estimate,D+} \left\Vert D^{+}\right\Vert _{L_{v}^{1}L_{x}^{\infty }} \leqslant& \frac{M^{2-2s}}{N_{2}^{4}}\left\Vert \sum_{k}^{J}\sum_{j}^{J}\int_{ \mathbb{S}^{2}} \int_{\mathbb{R}^{3}} B(u-v,\omega) S_{j,k}(t,x,\omega,u^{\ast },v^{\ast })dud\omega \right\Vert _{L_{v}^{1}L_{x}^{\infty }} \\ \leqslant &\frac{M^{2-2s}}{N_{2}^{4}}\sum_{k}^{J}\sum_{j}^{J}\int_{ \mathbb{S}^{2}} \int_{\mathbb{R}^{3}\times \mathbb{R}^{3}} B(u-v,\omega)\left\Vert S_{j,k}(t,x,\omega,u^{\ast },v^{\ast })\right\Vert _{L_{x}^{\infty }} du dv d\omega\notag\\ =&\frac{M^{2-2s}}{N_{2}^{4}}\sum_{k}^{J}\sum_{j}^{J}\int_{ \mathbb{S}^{2}} \int_{\mathbb{R}^{3}\times \mathbb{R}^{3}} B(u-v,\omega) \left\Vert S_{j,k}(t,x,\omega,u^{\ast },v^{\ast })\right\Vert _{L_{x}^{\infty }}du^{*}dv^{*} d\omega \notag\\ \lesssim &\frac{M^{2-2s}}{N_{2}^{4}}\sum_{k}^{J}\sum_{j}^{J}\int du^{\ast }\int dv^{\ast }\int_{\mathbb{S}^{2}}d\omega \frac{1}{|u^{*}-v^{*}|}\left\Vert S_{j,k}(t,x,\omega,u^{\ast },v^{\ast })\right\Vert _{L_{x}^{\infty }}\notag\end{aligned}$$ where in the second-to-last equality we used the change of variable, and in the last inequality we used that $B(u-v,\omega)=|u-v|^{-1}\textbf{b}(\cos \theta)\lesssim |u-v|^{-1}$ and $|u-v|=|u^{*}-v^{*}|$. We note that $$v-u^{\ast }=P_{\omega }^{\bot }(v^{\ast }-u^{\ast }),\quad v-v^{\ast }=-P_{\omega }^{\Vert }(v^{\ast }-u^{\ast }),$$ and hence get $$\begin{aligned} &x-v(t-t_{0})-v^{\ast }t_{0} =x-vt-P_{\omega }^{\Vert }(v^{\ast }-u^{\ast })t_{0},\\ &x-v(t-t_{0})-u^{\ast }t_{0} =x-vt+P_{\omega }^{\bot }(v^{\ast }-u^{\ast })t_{0}.\end{aligned}$$ For fixed $u^{*}$ and $v^{*}$, we get $$\begin{aligned} \label{equ:Sjk,estimate} &S_{j,k}(t,x,\omega,u^{\ast },v^{\ast }) \\ \lesssim &\int_{T_{*}}^{0}\chi \left( MP_{e_{k}}^{\perp }(x-vt-P_{\omega }^{\Vert }(v^{\ast }-u^{\ast })t_{0})\right) \chi (\frac{P_{e_{k}}(x-vt-P_{\omega }^{\Vert }(v^{\ast }-u^{\ast })t_{0})}{N_{2}}) I_{k}(v^{*})\notag\\ &\chi \left( MP_{e_{j}}^{\perp }(x-vt+P_{\omega }^{\bot }(v^{\ast }-u^{\ast })t_{0})\right) \chi (\frac{P_{e_{j}}(x-vt+P_{ \omega }^{\bot }(v^{\ast }-u^{\ast })t_{0})}{N_{2}}) I_{j}(u^{*})dt_{0}\notag\\ \leq &I_{k}(v^{*}) I_{j}(u^{*})E_{k}(t,x,v,\omega,u^{*},v^{*}),\notag\end{aligned}$$ where $$\begin{aligned} E_{k}(t,x,v,\omega,u^{*},v^{*}) =:&\int_{T_{*}}^{0}\chi \left( MP_{e_{k}}^{\perp }(x-vt-P_{\omega }^{\Vert }(v^{\ast }-u^{\ast })t_{0})\right) dt_{0}.\end{aligned}$$ We split into two cases in terms of the angle $\alpha_{j,k}$ between $e_{j}$ and $e_{k}$. **Case $I$: $\alpha_{j,k}\neq 0$.** (In this case, we have that $\sin \alpha_{j,k}\gtrsim \frac{1}{MN_{2}}$.) Now, we get into the analysis of $E_{k}(t,x,v,\omega,u^{*},v^{*})$. First of all, it gives a trial upper bound that $$\begin{aligned} E_{k}(t,x,v,\omega,u^{*},v^{*})\leq |T_{*}|\leq 1.\end{aligned}$$ By the radial symmetry and monotonicity of the cutoff function $\chi$, we obtain $$\begin{aligned} \label{equ:chi,inequality} \int_{\mathbb{R}} \chi(\overrightarrow{n}t_{0}+\overrightarrow{m})dt_{0}\leq \frac{1}{|\overrightarrow{n}|}\int_{\mathbb{R}}\chi(t_{0})dt_{0}.\end{aligned}$$ To see [\[equ:chi,inequality\]](#equ:chi,inequality){reference-type="eqref" reference="equ:chi,inequality"}, without loss of generality, we take $\overrightarrow{n}=(0,0,1)$ and $\overrightarrow{m}=(m_{1},m_{2},m_{3})$ to get $$\begin{aligned} \int_{\mathbb{R}} \chi(\overrightarrow{n}t_{0}+\overrightarrow{m})dt_{0}=&\int_{\mathbb{R}} \chi\left( \sqrt{m_{1}^{2}+m_{2}^{2}+(t_{0}+m)^{2}} \right) dt_{0}\\ \leq & \int_{\mathbb{R}} \chi\left( |t_{0}+m| \right) dt_{0}=\int_{\mathbb{R}} \chi\left( t_{0} \right) dt_{0}.\end{aligned}$$ Thus, by [\[equ:chi,inequality\]](#equ:chi,inequality){reference-type="eqref" reference="equ:chi,inequality"} we arrive at $$\begin{aligned} E_{k}(t,x,v,\omega,u^{*},v^{*}) =&\int_{T_{*}}^{0}\chi \left( MP_{e_{k}}^{\perp }(x-vt-P_{\omega }^{\Vert }(v^{\ast }-u^{\ast })t_{0})\right)dt_{0} \notag\\ \lesssim& \frac{\int \chi(t_{0})dt_{0}}{M|P_{e_{k}}^{\perp}P_{\omega }^{\Vert }(v^{\ast }-u^{\ast })|}\\ \lesssim & \frac{1}{M\sin \phi_{k}} \frac{1}{|P_{\omega }^{\Vert }(v^{\ast }-u^{\ast })|},\notag\end{aligned}$$ where $\phi_{k}$ is the angle between $\omega$ and $e_{k}$. Due to the $v^{*}$-support and $u^{*}$-support, we have $$\begin{aligned} |v^{*}-u^{*}|^{2}\sim |ae_{k}-be_{j}|^{2}=&(a-b)^{2}+2ab(1-\cos \alpha_{j,k})\\ \geq& N_{2}^{2}(1-\cos \alpha_{j,k})\gtrsim N_{2}^{2}(\sin\alpha_{j,k})^{2}.\notag\end{aligned}$$ Let $\theta$ be the angle between $w$ and $v^{*}-u^{*}$. Then we obtain $$\begin{aligned} |P_{\omega }^{\Vert }(v^{\ast }-u^{\ast })|=|v^{\ast }-u^{\ast }|\cos \theta\gtrsim N_{2}\sin \alpha_{j,k} \cos \theta.\end{aligned}$$ Therefore, we get a useful upper bound that $$\begin{aligned} \label{equ:upper bound,Ek} I_{k}(v^{*}) I_{j}(u^{*})E_{k}(t,x,v,\omega,u^{*},v^{*}) \lesssim &\frac{I_{k}(v^{*}) I_{j}(u^{*})}{M N_{2} \sin \phi_{k} \cos \theta \sin \alpha_{j,k}}.\end{aligned}$$ Now, we are able to establish the effective bound on $E_{k}(t,x,v,\omega,u^{*},v^{*})$. Set $$\begin{aligned} A=\left\{ \omega\in \mathbb{S}^{2}:\phi_{k}\leq \frac{1}{MN_{2}} \right\} \bigcup \left\{ \frac{\pi}{2}-\theta\leq \frac{1}{MN_{2}} \right\} ,\end{aligned}$$ and denote by $A^{c}$ the complementary set of $A$. With the trivial bound that $E_{k}\leq 1$ on the set $A$, we have $$\begin{aligned} \int_{\mathbb{S}^{2}} \Vert E_{k}(t,x,v,\omega,u^{*},v^{*})\Vert _{L_{x}^{\infty}}d\omega \leq &\int_{A}1d\omega +\int_{A^{c}}\Vert E_{k}(t,x,v,\omega,u^{*},v^{*})\Vert _{L_{x}^{\infty}}d\omega\notag\\ \lesssim & \frac{1}{(MN_{2})^{2}}+\int_{A^{c}}\Vert E_{k}(t,x,v,\omega,u^{*},v^{*})\Vert _{L_{x}^{\infty}}d\omega.\label{equ:estimate,Ek,Ac}\end{aligned}$$ For the second term on the right hand side of [\[equ:estimate,Ek,Ac\]](#equ:estimate,Ek,Ac){reference-type="eqref" reference="equ:estimate,Ek,Ac"}, by the upper bound [\[equ:upper bound,Ek\]](#equ:upper bound,Ek){reference-type="eqref" reference="equ:upper bound,Ek"}, we get $$\begin{aligned} & I_{k}(v^{*})I_{j}(u^{*})\int_{A^{c}}\Vert E_{k}(t,x,v,\omega,u^{*},v^{*})\Vert _{L_{x}^{\infty}}d\omega\notag\\ \leq&\frac{I_{k}(v^{*})I_{j}(u^{*})}{MN_{2}\sin \alpha_{j,k}}\int_{A^{c}} \frac{1}{\sin \phi_{k}\cos \theta}d\omega\notag\\ \lesssim & \frac{I_{k}(v^{*})I_{j}(u^{*})}{M N_{2}\sin \alpha_{j,k}}\left[ \int_{A^{c}}\frac{1}{(\sin \phi_{k})^{2}}d\omega+\int_{A^{c}}\frac{1}{(\cos \theta)^{2}}d\omega \right] .\label{equ:estimate,Ek,Ac,two terms}\end{aligned}$$ For the last two terms on the right hand side of [\[equ:estimate,Ek,Ac,two terms\]](#equ:estimate,Ek,Ac,two terms){reference-type="eqref" reference="equ:estimate,Ek,Ac,two terms"}, by the rotational symmetry, we might as well to consider $$\begin{aligned} \int_{\mathbb{S}^{2}\bigcap \left\{ |\phi|\geq \frac{1}{MN_{2}} \right\} } \frac{1}{(\sin \phi)^{2}}d\omega,\end{aligned}$$ where $\phi$ is the angle between the $z$-vector $(0,0,1)$ and $\omega$. Using the surface integral formula, $$\begin{aligned} \int_{\mathbb{S}^{2}\bigcap \left\{ |\phi|\geq \frac{1}{MN_{2}} \right\} } \frac{1}{(\sin \phi)^{2}}d\omega\lesssim \int_{|\phi|\geq \frac{1}{MN_{2}}}\frac{|\sin \phi|}{(\sin \phi)^{2}} d\phi\lesssim \int_{|\phi|\geq\frac{1}{MN_{2}}}^{\frac{\pi}{2}} \frac{1}{|\phi|}d\phi \lesssim \ln (MN_{2}).\end{aligned}$$ Together with [\[equ:estimate,Ek,Ac,two terms\]](#equ:estimate,Ek,Ac,two terms){reference-type="eqref" reference="equ:estimate,Ek,Ac,two terms"}, this bound yields $$\begin{aligned} \label{equ:estimate,Ek,Ac,final} I_{k}(v^{*})I_{j}(u^{*})\int_{A^{c}}\Vert E_{k}(t,x,v,\omega,u^{*},v^{*})\Vert _{L_{x}^{\infty}}d\omega \leq \frac{I_{k}(v^{*})I_{j}(u^{*})\ln (MN_{2})}{MN_{2}\sin \alpha_{j,k}}.\end{aligned}$$ Therefore, combining estimates [\[equ:Sjk,estimate\]](#equ:Sjk,estimate){reference-type="eqref" reference="equ:Sjk,estimate"}, [\[equ:estimate,Ek,Ac\]](#equ:estimate,Ek,Ac){reference-type="eqref" reference="equ:estimate,Ek,Ac"} and [\[equ:estimate,Ek,Ac,final\]](#equ:estimate,Ek,Ac,final){reference-type="eqref" reference="equ:estimate,Ek,Ac,final"}, we arrive at $$\begin{aligned} \int_{\mathbb{S}^{2}}\Vert S_{j,k}(t,x,\omega,u^{\ast },v^{\ast })\Vert _{L_{x}^{\infty}}d\omega \lesssim \frac{\ln (MN_{2})}{MN_{2}\sin \alpha_{j,k}}I_{k}(v^{*})I_{j}(u^{*}).\end{aligned}$$ Then, going back to the estimate [\[equ:estimate,D+\]](#equ:estimate,D+){reference-type="eqref" reference="equ:estimate,D+"} on $D^{+}$, we have $$\begin{aligned} \label{equ:D+,Lv1} &\Vert D^{+}\Vert _{L_{v}^{1}L_{x}^{\infty}}\\ \lesssim & \frac{M^{2-2s}}{N_{2}^{4}}\sum_{k\neq j}\int du^{*}\int dv^{*} \int_{\mathbb{S}^{2}} d\omega \frac{1}{|u^{*}-v^{*}|}\Vert S_{j,k}(t,x,\omega,u^{\ast },v^{\ast })\Vert _{L_{x}^{\infty}}\notag\\ \lesssim & \frac{M^{2-2s}}{N_{2}^{4}}\sum_{k\neq j} \frac{\ln (MN_{2})}{MN_{2}\sin \alpha_{j,k}} \int du^{*}\int dv^{*} \frac{1}{|u^{*}-v^{*}|}I_{k}(v^{*})I_{j}(u^{*}).\notag\end{aligned}$$ In this case, since we have that $\sin \alpha_{j,k}\gtrsim \frac{1}{MN_{2}}$, we can use estimate [\[equ:estimate,IjIk\]](#equ:estimate,IjIk){reference-type="eqref" reference="equ:estimate,IjIk"}, which is established in the analysis of $Q^{-}(f_{\mathrm{b}},f_{\mathrm{b}})$, to get $$\begin{aligned} \label{equ:D+,Lv1,IjIk} \int du^{*}\int dv^{*} \frac{1}{|u^{*}-v^{*}|}I_{k}(v^{*})I_{j}(u^{*}) \lesssim&\frac{\ln(MN_{2})}{M^{2}} \int I_{j}(v^{*})dv^{*}=\frac{\ln(MN_{2})}{M^{2}}M^{-2}N_{2}.\end{aligned}$$ Consequently, combining estimates [\[equ:D+,Lv1\]](#equ:D+,Lv1){reference-type="eqref" reference="equ:D+,Lv1"} and [\[equ:D+,Lv1,IjIk\]](#equ:D+,Lv1,IjIk){reference-type="eqref" reference="equ:D+,Lv1,IjIk"}, we arrive at $$\begin{aligned} \label{equ:Q+,fb,fb,est2,final,case1} N_{2}^{-1}\Vert D^{+}\Vert _{L_{v}^{1}L_{x}^{\infty}}\lesssim& N_{2}^{-1}\frac{M^{2-2s}}{N_{2}^{4}}\sum_{k\neq j}^{J} \frac{\ln (MN_{2})}{MN_{2}\sin \alpha_{j,k}} \frac{\ln (MN_{2})}{M^{4}}N_{2}\\ \lesssim&N_{2}^{-1} \frac{M^{2-2s}}{N_{2}^{4}} \frac{\ln (MN_{2})}{MN_{2}} (MN_{2})^{4} \frac{\ln (MN_{2})}{M^{4}}N_{2}\notag\\ =&N_{2}^{-1}M^{1-2s}\left[ \ln (MN_{2}) \right] ^{2},\notag\end{aligned}$$ where in the second-to-last inequality we have used that $$\begin{aligned} \sum_{k\neq j}^{J} \frac{1}{\sin \alpha_{j,k}}=&\sum_{ j}^{J}\sum_{i=1}^{MN_{2}}\sum_{\sin \alpha_{j,k}\sim \frac{i}{MN_{2}}} \frac{1}{\sin \alpha_{j,k}} \\ \lesssim&\sum_{ j}^{J}\sum_{i=1}^{MN_{2}} MN_{2}\sin \alpha_{j,k} \frac{1}{\sin \alpha_{j,k}} \lesssim (MN_{2})^{4}.\end{aligned}$$ This completes the estimate of the $L_{v}^{1}L_{x}^{\infty}$ norm for $D^{+}$. **Case $II$: $\alpha_{j,k}\sim 0$.** (That is, $|j-k|\lesssim 1$.) In this case, the summands in the double sum $\sum_{k}^{J}\sum_{j}^{J}$ are reduced to $(MN_{2})^{2}$, so we only need to use the trivial bound that $$\begin{aligned} \int_{\mathbb{S}^{2}}\Vert S_{j,k}(t,x,\omega,u^{\ast },v^{\ast })\Vert _{L_{x}^{\infty}}d\omega\lesssim I_{k}(v^{*})I_{j}(u^{*}).\end{aligned}$$ Then, with the estimate [\[equ:estimate,D+\]](#equ:estimate,D+){reference-type="eqref" reference="equ:estimate,D+"} on $D^{+}$, we use Hardy-Sobolev-Littlewood inequality [\[equ:hls,integral\]](#equ:hls,integral){reference-type="eqref" reference="equ:hls,integral"} to get $$\begin{aligned} \label{equ:Q+,fb,fb,est2,final,case2} &\Vert D^{+}\Vert _{L_{v}^{1}L_{x}^{\infty}}\\ \lesssim& \frac{M^{2-2s}}{N_{2}^{4}}\sum_{k}^{J}\sum_{j}^{J}\int du^{\ast }\int dv^{\ast }\int_{\mathbb{S}^{2}}d\omega \frac{1}{|u^{*}-v^{*}|}\left\Vert S_{j,k}(t,x,\omega,u^{\ast },v^{\ast })\right\Vert _{L_{x}^{\infty }}\notag\\ \lesssim& \frac{M^{2-2s}}{N_{2}^{4}}\sum_{|j-k|\lesssim 1}^{J} \int du^{*}\int dv^{*} \frac{1}{|u^{*}-v^{*}|}I_{k}(v^{*})I_{j}(u^{*})\notag\\ \lesssim& \frac{M^{2-2s}}{N_{2}^{4}}\sum_{|j-k|\lesssim 1}^{J} \Vert I_{j}\Vert _{L^{\frac{6}{5}}}\Vert I_{k}\Vert _{L^{\frac{6}{5}}}\notag\\ \lesssim& \frac{M^{2-2s}}{N_{2}^{4}} (MN_{2})^{2} (M^{-2}N_{2})^{\frac{5}{3}}\notag\\ \lesssim& M^{\frac{2}{3}-2s} N_{2}^{-\frac{1}{3}}.\notag\end{aligned}$$ Combining estimates [\[equ:Q+,fb,fb,est2,final,case1\]](#equ:Q+,fb,fb,est2,final,case1){reference-type="eqref" reference="equ:Q+,fb,fb,est2,final,case1"} and [\[equ:Q+,fb,fb,est2,final,case2\]](#equ:Q+,fb,fb,est2,final,case2){reference-type="eqref" reference="equ:Q+,fb,fb,est2,final,case2"} in the two cases, we finally reach $$\begin{aligned} \label{equ:Q+,fb,fb,L1,final} N_{2}^{-1}\Vert D^{+}\Vert _{L_{v}^{1}L_{x}^{\infty}}\lesssim M^{1-2s}N_{2}^{-1}\left[ \ln(MN_{2}) \right] ^{2}.\end{aligned}$$ **The $N_{2}^{\frac{1}{5}}\Vert \bullet\Vert _{L_{v}^{\frac{5}{3}}L_{x}^{\infty}}$ estimate.** By the interpolation inequality, we have $$\begin{aligned} \label{equ:Q+,fb,fb,L53} N_{2}^{\frac{1}{5}}\Vert D^{+}\Vert _{L_{v}^{\frac{5}{3}}L_{x}^{\infty}}\leq \left( N_{2}^{-1}\Vert D^{+}\Vert _{L_{v}^{1}L_{x}^{\infty}} \right) ^{\frac{1}{5}} \left( N_{2}^{\frac{1}{2}}\Vert D^{+}\Vert _{L_{v}^{2}L_{x}^{\infty}} \right) ^{\frac{4}{5}}.\end{aligned}$$ For the $L_{v}^{2}L_{x}^{\infty}$ norm, by the bilinear estimate [\[equ:bilinear estimate,Q+\]](#equ:bilinear estimate,Q+){reference-type="eqref" reference="equ:bilinear estimate,Q+"} for $Q^{+}$ in Lemma [Lemma 26](#lemma:bilinear estimate,Q+){reference-type="ref" reference="lemma:bilinear estimate,Q+"}, we have $$\begin{aligned} \label{equ:Q+,fb,fb,L2} N_{2}^{\frac{1}{2}}\Vert D^{+}\Vert _{L_{v}^{2}L_{x}^{\infty}}\lesssim& N_{2}^{\frac{1}{2}}|T_{*}|\Vert Q^{+}(f_{\mathrm{b}},f_{\mathrm{b}})\Vert _{L_{v}^{2}L_{x}^{\infty}}\\ \lesssim &N_{2}^{\frac{1}{2}}|T_{*}| \Vert f_{\mathrm{b}}\Vert _{L_{v}^{\frac{3}{2}}L_{x}^{\infty}} \Vert f_{\mathrm{b}}\Vert _{L_{v}^{2}L_{x}^{\infty}}\notag\\ \lesssim& |T^{*}|M^{1-s}M^{1-s}\notag\\ \lesssim& M^{1-s}(\ln \ln \ln M),\notag\end{aligned}$$ where we have used the bounds on $f_{b}$ established in Lemma [Lemma 13](#lemma:bounds on fb){reference-type="ref" reference="lemma:bounds on fb"} that $$\begin{aligned} \Vert f_{\mathrm{b}}\Vert _{L_{v}^{\frac{3}{2}}L_{x}^{\infty}}\leq & N_{2}^{-1}\Vert f_{\mathrm{b}}\Vert _{L_{v}^{1}L_{x}^{\infty}}+N_{2}^{\frac{1}{5}}\Vert f_{\mathrm{b}}\Vert _{L_{v}^{\frac{5}{3}}L_{x}^{\infty}} \leq \Vert f_{\mathrm{b}}\Vert _{Z}\lesssim M^{1-s},\\ N_{2}^{\frac{1}{2}}\Vert f_{\mathrm{b}}\Vert _{L_{v}^{2}L_{x}^{\infty}}\lesssim &M^{1-s}.\end{aligned}$$ Thus, combining estimates [\[equ:Q+,fb,fb,L1,final\]](#equ:Q+,fb,fb,L1,final){reference-type="eqref" reference="equ:Q+,fb,fb,L1,final"}, [\[equ:Q+,fb,fb,L53\]](#equ:Q+,fb,fb,L53){reference-type="eqref" reference="equ:Q+,fb,fb,L53"} and [\[equ:Q+,fb,fb,L2\]](#equ:Q+,fb,fb,L2){reference-type="eqref" reference="equ:Q+,fb,fb,L2"}, we reach $$\begin{aligned} N_{2}^{\frac{1}{5}}\Vert D^{+}\Vert _{L_{v}^{\frac{5}{3}}L_{x}^{\infty}}\lesssim N_{2}^{-\frac{1}{5}}M^{1-\frac{6}{5}s}(\ln\ln\ln M)\ln (MN_{2}),\end{aligned}$$ which is sufficient for our goal. ◻ ## $Z$-norm Bounds on the Correction Term {#section:Bounds on the Correction Term} Recall the equation [\[equ:correction term,fc\]](#equ:correction term,fc){reference-type="eqref" reference="equ:correction term,fc"} for the correction term $f_{\mathrm{c}}$ that $$\left\{ \begin{aligned} &\partial_t f_{\mathrm{c}} + v\cdot \nabla_x f_{\mathrm{c}} = G, \\ &G= \pm Q^\pm(f_{\mathrm{c}},f_{ \mathrm{a}}) \pm Q^\pm(f_{\mathrm{a}},f_{\mathrm{c}}) \pm Q^\pm(f_{\mathrm{c}},f_{\mathrm{c} })-F_{\text{err}}. \end{aligned} \label{equ:fc} \right.$$ For $T_{*}=-M^{s-\frac{d-1}{2}}(\ln \ln \ln M) \leq t\leq 0$, we are looking for the correction term $f_{\mathrm{c}}(t)$ with $$\begin{aligned} \|f_{\mathrm{c}}(t)\|_{L_v^{2,r_{0}}H_x^{s_{0}}} \lesssim M^{-1/2}.\end{aligned}$$ To achieve it, we apply a perturbation argument and work on the stronger $Z$-norm [\[equ:z-norm\]](#equ:z-norm){reference-type="eqref" reference="equ:z-norm"}. By interpolation inequality, for $d=2,3$, we indeed have $$\begin{aligned} \Vert f\Vert _{L_{v}^{2,r_{0}}H_{x}^{s_{0}}}\leq \Vert f\Vert _{L_{v}^{2,r_{0}}H_{x}^{\frac{d-1}{2}}}\leq & \Vert f\Vert _{L_{v}^{2,r_{0}}L_{x}^{2}}^{\frac{3-d}{2}}\Vert \langle \nabla_{x} \rangle f\Vert _{L_{v}^{2,r_{0}}L_{x}^{2}}^{\frac{d-1}{2}}\\ \leq & M^{\frac{d-1}{2}}\Vert f\Vert _{L_{v}^{2,r_{0}}L_{x}^{2}}+ M^{\frac{d-3}{2}}\Vert \langle \nabla_{x} \rangle f\Vert _{L_{v}^{2,r_{0}}L_{x}^{2}}\leq \Vert f\Vert _{Z}.\end{aligned}$$ Certainly, there are multiple choices of $Z$-norms. As we are fully in the perturbation regime, we expect the correction term $f_{\mathrm{c}}$ to be much smoother and hence we choose the $L_{v}^{2,r_{0}}H_{x}^{1}$ norm. On the other hand, to beat the difficulties caused by singularities of soft potentials, the $L_{v}^{1}L_{x}^{\infty}$ and $L_{v}^{\frac{5}{3}}L_{x}^{\infty}$ norms[^8] are needed as shown in the following estimate [\[equ:z-norm,L1,L53\]](#equ:z-norm,L1,L53){reference-type="eqref" reference="equ:z-norm,L1,L53"}. In the section, we first prove a closed estimate for the loss and gain terms in Lemma [Lemma 20](#lemma:binlinear estimate){reference-type="ref" reference="lemma:binlinear estimate"} and then use it to conclude the existence of small correction term $f_{\mathrm{c}}(t)$ in Proposition [Proposition 21](#lemma:perturbation){reference-type="ref" reference="lemma:perturbation"}. **Lemma 20** (Bilinear $Z$-norm estimates for loss/gain operator $Q^\pm$). *[\[L:Z_bilinear\]]{#L:Z_bilinear label="L:Z_bilinear"} For $f_1$, $f_2$, we have $$\|Q^\pm (f_1,f_2) \|_{Z} \lesssim \|f_1\|_{Z} \|f_2\|_{Z}.$$* *Proof.* We only need to prove that $$\begin{aligned} \label{equ:z-norm,L1,L53} \Vert Q^{\pm}(f_{1},f_{2})\Vert _{Z}\lesssim \Vert f_{1}\Vert _{Z}\left( \Vert f_{2}\Vert _{L_{v}^{1}L_{x}^{\infty}}^{1+\frac{5\gamma}{2d}}\Vert f_{2}\Vert _{L_{v}^{\frac{5}{3}}L_{x}^{\infty}}^{\frac{-5\gamma}{2d}} +M^{-1}\Vert \nabla_{x}f_{2}\Vert _{L_{v}^{1}L_{x}^{\infty}}^{1+\frac{5\gamma}{2d}}\Vert \nabla_{x}f_{2}\Vert _{L_{v}^{\frac{5}{3}}L_{x}^{\infty}}^{\frac{-5\gamma}{2d}} \right) .\end{aligned}$$ since we have that $$\begin{aligned} \Vert f\Vert _{L_{v}^{1}L_{x}^{\infty}}^{1+\frac{5\gamma}{2d}}\Vert f\Vert _{L_{v}^{\frac{5}{3}}L_{x}^{\infty}}^{\frac{-5\gamma}{2d}}\leq& N_{2}^{-1}\Vert f\Vert _{L_{v}^{1}L_{x}^{\infty}} +N_{2}^{\frac{2d}{5}+\gamma}\Vert f\Vert _{L_{v}^{\frac{5}{3}}L_{x}^{\infty}}\leq \Vert f\Vert _{Z},\\ M^{-1}\Vert \nabla_{x} f\Vert _{L_{v}^{1}L_{x}^{\infty}}^{1+\frac{5\gamma}{2d}} \Vert \nabla_{x} f\Vert _{L_{v}^{\frac{5}{3}}L_{x}^{\infty}}^{\frac{-5\gamma}{2d}}\leq& M^{-1}N_{2}^{-1}\Vert \nabla_{x}f\Vert _{L_{v}^{1}L_{x}^{\infty}}+ M^{-1}N_{2}^{\frac{2d}{5}+\gamma}\Vert \nabla_{x}f\Vert _{L_{v}^{\frac{5}{3}}L_{x}^{\infty}}\leq \Vert f\Vert _{Z}.\end{aligned}$$ **The $M^{\frac{d-3}{2}}\Vert \nabla_{x}\bullet \Vert_{L_{v}^{2,r_{0}}L_{x}^{2}}$ and $M^{\frac{d-1}{2}}\Vert \bullet \Vert_{L_{v}^{2,r_{0}}L_{x}^{2}}$ estimates for $Q^{\pm}(f_{1},f_{2})$.** It suffices to deal with $M^{\frac{d-3}{2}}\Vert \nabla_{x}\bullet \Vert_{L_{v}^{2,r_{0}}L_{x}^{2}}$ norm as the $M^{\frac{d-1}{2}}\Vert \bullet \Vert_{L_{v}^{2,r_{0}}L_{x}^{2}}$ norm can be estimated in a similar way. For the estimate on $Q^{-}$, we use Leibniz rule and Hölder inequality to get $$\begin{aligned} &M^{\frac{d-3}{2}}\Vert \nabla_{x}Q^{-}(f_1,f_2)\Vert _{L_{v}^{2,r_{0}}L_{x}^{2}}\\ \leq& M^{\frac{d-3}{2}}\Vert Q^{-}(\nabla_{x}f_1,f_2)\Vert _{L_{v}^{2,r_{0}}L_{x}^{2}}+M^{\frac{d-3}{2}}\Vert Q^{-}(f_1,\nabla_{x}f_2)\Vert _{L_{v}^{2,r_{0}}L_{x}^{2}}\\ \lesssim & M^{\frac{d-3}{2}}\Big\Vert (\nabla_{x}f_{1})(x,v)\int \frac{f_{2}(x,u)}{|u-v|^{-\gamma}} du \Big \Vert _{L_{v}^{2,r_{0}}L_{x}^{2}}+ M^{\frac{d-3}{2}}\Big\Vert f_{1}(x,v)\int \frac{\nabla_{x}f_{2}(x,u)}{|u-v|^{-\gamma}} du \Big \Vert _{L_{v}^{2,r_{0}}L_{x}^{2}}\\ \lesssim& M^{\frac{d-3}{2}}\Vert \nabla_{x}f_{1}\Vert _{L_{v}^{2,r_{0}}L_{x}^{2}}\Big\Vert \int \frac{f_{2}(x,u)}{|u-v|^{-\gamma}}du \Big \Vert _{L_{v,x}^{\infty}}+ M^{\frac{d-3}{2}}\Vert f_{1}\Vert _{L_{v}^{2,r_{0}}L_{x}^{2}} \Big\Vert \int \frac{\nabla_{x}f_{2}(x,u)}{|u-v|^{-\gamma}}du \Big \Vert _{L_{v,x}^{\infty}}.\end{aligned}$$ Then by $L^{\infty}$ estimate [\[equ:endpoint estimate,hls\]](#equ:endpoint estimate,hls){reference-type="eqref" reference="equ:endpoint estimate,hls"} in Lemma [Lemma 25](#lemma:endpoint estimate,hls){reference-type="ref" reference="lemma:endpoint estimate,hls"}, we obtain $$\begin{aligned} &M^{\frac{d-3}{2}}\Vert \nabla_{x}Q^{-}(f_1,f_2)\Vert _{L_{v}^{2,r_{0}}L_{x}^{2}}\\ \lesssim&M^{\frac{d-3}{2}} \Vert \nabla_{x}f_{1}\Vert _{L_{v}^{2,r_{0}}L_{x}^{2}}\Vert f_{2}\Vert _{L_{v}^{1}L_{x}^{\infty}}^{1+\frac{5\gamma}{2d}} \Vert f_{2}\Vert _{L_{v}^{\frac{5}{3}}L_{x}^{\infty}}^{\frac{-5\gamma}{2d}}+ M^{\frac{d-1}{2}}\Vert f_{1}\Vert _{L_{v}^{2,r_{0}}L_{x}^{2}}M^{-1}\Vert \nabla_{x} f_{2}\Vert _{L_{v}^{1}L_{x}^{\infty}}^{1+\frac{5\gamma}{2d}} \Vert \nabla_{x} f_{2}\Vert _{L_{v}^{\frac{5}{3}}L_{x}^{\infty}}^{\frac{-5\gamma}{2d}}\\ \lesssim& \Vert f_{1}\Vert _{Z}\Vert f_{2}\Vert _{Z}.\end{aligned}$$ For the estimate on $Q^{+}$, from the conservation of energy that $|v|^{2}+|u|^{2}=|v^{*}|^{2}+|u^{*}|^{2}$, we use Leibniz rule to get $$\begin{aligned} |\langle v \rangle ^{r_{0}}\nabla_{x}Q^{+}(f_{1},f_{2})|\lesssim & Q^{+}(\langle v \rangle ^{r_{0}}|\nabla_{x}f_{1}|,|f_{2}|)+ Q^{+}(\langle v \rangle ^{r_{0}}|f_{1}|,|\nabla_{x}f_{2}|)\\ &+Q^{+}(|\nabla_{x}f_{1}|,\langle v \rangle ^{r_{0}}|f_{2}|)+Q^{+}(|f_{1}|,\langle v \rangle ^{r_{0}}|\nabla_{x}f_{2}|).\end{aligned}$$ Then by bilinear estimate [\[equ:bilinear estimate,Q+\]](#equ:bilinear estimate,Q+){reference-type="eqref" reference="equ:bilinear estimate,Q+"} on $Q^{+}$, we have $$\begin{aligned} &M^{\frac{d-3}{2}}\Vert \nabla_{x}Q^{+}(f_{1},f_{2})\Vert _{L_{v}^{2,r_{0}}L_{x}^{2}}\\ \lesssim& M^{\frac{d-3}{2}}\Vert \langle v \rangle ^{r_{0}}\nabla_{x}f_{1}\Vert _{L_{v}^{2}L_{x}^{2}} \Vert f_{2}\Vert _{L_{v}^{\frac{d}{d+\gamma}}L_{x}^{\infty}}+ M^{\frac{d-1}{2}}\Vert \langle v \rangle ^{r_{0}}f_{1}\Vert _{L_{v}^{2}L_{x}^{2}}M^{-1}\Vert \nabla_{x} f_{2}\Vert _{L_{v}^{\frac{d}{d+\gamma}}L_{x}^{\infty}}\\ &+M^{-1}\Vert \nabla_{x}f_{1}\Vert _{L_{v}^{\frac{d}{d+\gamma}}L_{x}^{\infty}} M^{\frac{d-1}{2}}\Vert \langle v \rangle ^{r_{0}}f_{2}\Vert _{L_{v}^{2}L_{x}^{2}} +\Vert f_{1}\Vert _{L_{v}^{\frac{d}{d+\gamma}}L_{x}^{\infty}} M^{\frac{d-3}{2}}\Vert \langle v \rangle ^{r_{0}}\nabla_{x} f_{2}\Vert _{L_{v}^{2}L_{x}^{2}}.\end{aligned}$$ By the interpolation inequality that $$\begin{aligned} \Vert f\Vert _{L_{v}^{\frac{d}{d+\gamma}}L_{x}^{\infty}}\leq \Vert f\Vert _{L_{v}^{1}L_{x}^{\infty}}^{1+\frac{5\gamma}{2d}}\Vert f\Vert _{L_{v}^{\frac{5}{3}}L_{x}^{\infty}}^{\frac{-5\gamma}{2d}},\end{aligned}$$ we arrive at $$\begin{aligned} M^{\frac{d-3}{2}}\Vert \nabla_{x}Q^{+}(f_{1},f_{2})\Vert _{L_{v}^{2,r_{0}}L_{x}^{2}}\lesssim \Vert f_{1}\Vert _{Z}\Vert f_{2}\Vert _{Z}.\end{aligned}$$ **The $N_{2}^{\gamma}\Vert \bullet\Vert _{L_{v}^{1}L_{x}^{\infty}}$ and $N_{2}^{\frac{2d}{5}+\gamma}\Vert \bullet\Vert _{L_{v}^{\frac{5}{3}}L_{x}^{\infty}}$ estimates for $Q^{\pm}(f_{1},f_{2})$.** For the estimate on $Q^{-}$ , we use Hölder inequality and the $L^{\infty}$ estimate [\[equ:endpoint estimate,hls\]](#equ:endpoint estimate,hls){reference-type="eqref" reference="equ:endpoint estimate,hls"} to get $$\begin{aligned} \label{equ:Q-,Lv1} N_{2}^{\gamma}\Vert Q^{-}(f_{1},f_{2})\Vert _{L_{v}^{1}L_{x}^{\infty}}\lesssim& N_{2}^{\gamma}\Vert f_{1}\Vert _{L_{v}^{1}L_{x}^{\infty}}\Big\Vert \int \frac{f_{2}(x,u)}{|u-v|}du \Big \Vert _{L_{v}^{\infty}L_{x}^{\infty}}\\ \lesssim&N_{2}^{\gamma}\Vert f_{1}\Vert _{L_{v}^{1}L_{x}^{\infty}}\Vert f_{2}\Vert _{L_{v}^{1}L_{x}^{\infty}}^{1+\frac{5\gamma}{2d}} \Vert f_{2}\Vert _{L_{v}^{\frac{5}{3}}L_{x}^{\infty}}^{\frac{-5\gamma}{2d}}\notag\\ \lesssim& \Vert f_{1}\Vert _{Z}\Vert f_{2}\Vert _{Z}.\notag\end{aligned}$$ In the same way, we also have $$\begin{aligned} N_{2}^{\frac{2d}{5}+\gamma}\Vert Q^{-}(f_{1},f_{2})\Vert _{L_{v}^{\frac{5}{3}}L_{x}^{\infty}} \lesssim& N_{2}^{\frac{2d}{5}+\gamma}\Vert f_{1}\Vert _{L_{v}^{\frac{5}{3}}L_{x}^{\infty}} \Vert f_{2}\Vert _{L_{v}^{1}L_{x}^{\infty}}^{1+\frac{5\gamma}{2d}}\Vert f_{2}\Vert _{L_{v}^{\frac{5}{3}}L_{x}^{\infty}}^{\frac{-5\gamma}{2d}}\\ \lesssim& \Vert f_{1}\Vert _{Z}\Vert f_{2}\Vert _{Z}.\end{aligned}$$ For the estimate on $Q^{+}$, by the bilinear estimate [\[equ:endpoint estimate,Q+,f\]](#equ:endpoint estimate,Q+,f){reference-type="eqref" reference="equ:endpoint estimate,Q+,f"} for $Q^{+}$ in Lemma [Lemma 27](#lemma:endpoint estimate,Q+){reference-type="ref" reference="lemma:endpoint estimate,Q+"}, we have $$\begin{aligned} \label{equ:Q+,Lv1} N_{2}^{\gamma}\Vert Q^{+}(f_{1},f_{2})\Vert _{L_{v}^{1}L_{x}^{\infty}}\lesssim N_{2}^{\gamma}\Vert f_{1}\Vert _{L_{v}^{1}L_{x}^{\infty}}\Vert f_{2}\Vert _{L_{v}^{1}L_{x}^{\infty}}^{1+\frac{5\gamma}{2d}} \Vert f_{2}\Vert _{L_{v}^{\frac{5}{3}}L_{x}^{\infty}}^{\frac{-5\gamma}{2d}} \lesssim \Vert f_{1}\Vert _{Z}\Vert f_{2}\Vert _{Z}.\end{aligned}$$ Similarly, by the bilinear estimate [\[equ:bilinear estimate,Q+\]](#equ:bilinear estimate,Q+){reference-type="eqref" reference="equ:bilinear estimate,Q+"} in Lemma [Lemma 26](#lemma:bilinear estimate,Q+){reference-type="ref" reference="lemma:bilinear estimate,Q+"}, we obtain $$\begin{aligned} N_{2}^{\frac{2d}{5}+\gamma}\Vert Q^{+}(f_{1},f_{2})\Vert _{L_{v}^{\frac{5}{3}}L_{x}^{\infty}}\lesssim N_{2}^{\frac{2d}{5}+\gamma}\Vert f_{1}\Vert _{L_{v}^{\frac{5}{3}}L_{x}^{\infty}}\Vert f_{2}\Vert _{L_{v}^{1}L_{x}^{\infty}}^{1+\frac{5\gamma}{2d}} \Vert f_{2}\Vert _{L_{v}^{\frac{5}{3}}L_{x}^{\infty}}^{\frac{-5\gamma}{2d}} \lesssim \Vert f_{1}\Vert _{Z}\Vert f_{2}\Vert _{Z}.\end{aligned}$$ **The $M^{-1}N_{2}^{\gamma}\Vert \nabla_{x}\bullet\Vert _{L_{v}^{1}L_{x}^{\infty}}$ and $M^{-1}N_{2}^{\frac{2d}{5}+\gamma}\Vert \nabla_{x}\bullet\Vert _{L_{v}^{\frac{5}{3}}L_{x}^{\infty}}$ estimates for $Q^{\pm}(f_{1},f_{2})$.** For the estimate on $Q^{-}$, in a similar way to [\[equ:Q-,Lv1\]](#equ:Q-,Lv1){reference-type="eqref" reference="equ:Q-,Lv1"}, we use the Leibniz rule to get $$\begin{aligned} &M^{-1}N_{2}^{\gamma}\Vert \nabla_{x}Q^{-}(f_{1},f_{2})\Vert _{L_{v}^{1}L_{x}^{\infty}}\\ \leq& M^{-1}N_{2}^{\gamma}\left( \Vert Q^{-}(\nabla_{x}f_{1},f_{2})\Vert _{L_{v}^{1}L_{x}^{\infty}} +\Vert Q^{-}(f_{1},\nabla_{x} f_{2})\Vert _{L_{v}^{1}L_{x}^{\infty}} \right) \\ \lesssim &M^{-1}N_{2}^{\gamma}\Vert \nabla_{x}f_{1}\Vert _{L_{v}^{1}L_{x}^{\infty}} \Vert f_{2}\Vert _{L_{v}^{1}L_{x}^{\infty}}^{1+\frac{5\gamma}{2d}}\Vert f_{2}\Vert _{L_{v}^{\frac{5}{3}}L_{x}^{\infty}}^{\frac{-5\gamma}{2d}}\\ &+ N_{2}^{\gamma}\Vert f_{1}\Vert _{L_{v}^{1}L_{x}^{\infty}} M^{-1}\Vert \nabla_{x}f_{2}\Vert _{L_{v}^{1}L_{x}^{\infty}}^{1+\frac{5\gamma}{2d}}\Vert \nabla_{x}f_{2}\Vert _{L_{v}^{\frac{5}{3}}L_{x}^{\infty}}^{\frac{-5\gamma}{2d}}\\ \lesssim& \Vert f_{1}\Vert _{Z}\Vert f_{2}\Vert _{Z}.\end{aligned}$$ The same also holds for the $M^{-1}N_{2}^{\frac{2d}{5}+\gamma}\Vert \nabla_{x} Q^{-}(f_{1},f_{2})\Vert _{L_{v}^{\frac{5}{3}}L_{x}^{\infty}}$ norm. For the estimate on $Q^{+}$, in a similar way to [\[equ:Q+,Lv1\]](#equ:Q+,Lv1){reference-type="eqref" reference="equ:Q+,Lv1"}, we also have $$\begin{aligned} &M^{-1}N_{2}^{\gamma}\Vert \nabla_{x}Q^{+}(f_{1},f_{2})\Vert _{L_{v}^{1}L_{x}^{\infty}}\\ \leq& M^{-1}N_{2}^{\gamma}\left( \Vert Q^{+}(\nabla_{x}f_{1},f_{2})\Vert _{L_{v}^{1}L_{x}^{\infty}} +\Vert Q^{+}(f_{1},\nabla_{x}f_{2})\Vert _{L_{v}^{1}L_{x}^{\infty}} \right) \\ \lesssim& M^{-1}N_{2}^{\gamma}\Vert \nabla_{x}f_{1}\Vert _{L_{v}^{1}L_{x}^{\infty}} \Vert f_{2}\Vert _{L_{v}^{1}L_{x}^{\infty}}^{1+\frac{5\gamma}{2d}} \Vert f_{2}\Vert _{L_{v}^{\frac{5}{3}}L_{x}^{\infty}}^{\frac{-5\gamma}{2d}}\\ &+ N_{2}^{\gamma}\Vert f_{1}\Vert _{L_{v}^{1}L_{x}^{\infty}} M^{-1}\Vert \nabla_{x}f_{2}\Vert _{L_{v}^{1}L_{x}^{\infty}}^{1+\frac{5\gamma}{2d}} \Vert \nabla_{x}f_{2}\Vert _{L_{v}^{\frac{5}{3}}L_{x}^{\infty}}^{\frac{-5\gamma}{2d}}\\ \lesssim& \Vert f_{1}\Vert _{Z}\Vert f_{2}\Vert _{Z}.\end{aligned}$$ The estimate for the $M^{-1}N_{2}^{\frac{2d}{5}+\gamma}\Vert \nabla_{x} Q^{+}(f_{1},f_{2})\Vert _{L_{v}^{\frac{5}{3}}L_{x}^{\infty}}$ norm follows the same way by using bilinear estimate [\[equ:bilinear estimate,Q+\]](#equ:bilinear estimate,Q+){reference-type="eqref" reference="equ:bilinear estimate,Q+"} in Lemma [Lemma 26](#lemma:bilinear estimate,Q+){reference-type="ref" reference="lemma:bilinear estimate,Q+"}. ◻ Now, we take a perturbation argument to generate the correction term $f_{\mathrm{c}}(t)$ using the $Z$-norm bounds on $f_{\mathrm{a}}$ in Proposition [Proposition 15](#lemma:z-norm bounds on fa){reference-type="ref" reference="lemma:z-norm bounds on fa"} and the $Z$-norm bounds on $F_{\mathrm{err}}$ in Proposition [Proposition 16](#lemma:bounds on ferr){reference-type="ref" reference="lemma:bounds on ferr"}. **Proposition 21**. Suppose that $f_{\mathrm{c}}$ solves [\[equ:fc\]](#equ:fc){reference-type="eqref" reference="equ:fc"} with $f_{\mathrm{c}}(0)=0$. Then for all $t$ such that $$T_{\ast }=- M^{s-\frac{d-1}{2}}(\ln \ln \ln M)\leq t\leq 0,$$ we have the bound $$\Vert f_{\mathrm{c}}(t)\Vert_{Z}\lesssim M^{-1/2}. \label{E:fc_bound2}$$ *Proof.* Let the time interval $T_{\ast }\leq t\leq 0$ be partitioned as $$T_{\ast }=T_{n}<T_{n-1}<T_{n-2}<\cdots <T_{2}<T_{1}<T_{0}=0$$ where $$T_{j}= \frac{- jM^{s-\frac{d-1}{2}}}{\sqrt{\ln M}},\quad n= \sqrt{\ln M} (\ln\ln \ln M).$$ Thus, the length of each time interval $I_{j}=[T_{j+1},T_{j}]$ is $$|I_{j}|= \frac{M^{s-\frac{d-1}{2}}}{\sqrt{\ln M}}.$$ For $t\in I_{j}=[T_{j+1},T_{j}]$, we rewrite the equation [\[equ:fc\]](#equ:fc){reference-type="eqref" reference="equ:fc"} in Duhamel form $$\begin{aligned} f_{\mathrm{c}}(T_{j}+t)=e^{-(t-T_{j})v\cdot \nabla_{x}}f_{\mathrm{c}}(T_{j})+\int_{T_{j}}^{t}e^{-(t-t_{0})v\cdot \nabla_{x}}G(t_{0})dt_{0}\end{aligned}$$ with $f_{\mathrm{c}}(T_{0})=0$. Applying the $Z$-norm, $$\begin{aligned} \Vert f_{\mathrm{c}}\Vert _{L_{I_{j}}^{\infty }Z} \leq& \Vert f_{\mathrm{c}}(T_{j})\Vert _{Z} +\Big\Vert \int_{T_{j}}^{t}e^{-(t-t_{0})v\cdot \nabla_{x}}G(t_{0})dt_{0} \Big \Vert _{L_{I_{j}}^{\infty}Z} \\ \leq& \Vert f_{\mathrm{c}}(T_{j})\Vert _{Z}+ |I_{j}|\Vert Q^{\pm }(f_{\mathrm{c}},f_{\mathrm{a}})\Vert _{L_{I_{j}}^{\infty }Z}+|I_{j}|\Vert Q^{\pm }(f_{\mathrm{a}},f_{\mathrm{c}% })\Vert _{L_{I_{j}}^{\infty }Z}+|I_{j}|\Vert Q^{\pm }(f_{\mathrm{c}},f_{\text{c% }})\Vert _{L_{I_{j}}^{\infty }Z} \\ & +\Big\Vert \int_{T_{j}}^{t}e^{-(t-t_{0})v\cdot \nabla_{x}}F_{\mathrm{err}}(t_{0})dt_{0} \Big \Vert _{L_{I_{j}}^{\infty}Z}. \end{aligned}$$ For these terms on the second line, we apply the bilinear estimate in Lemma [Lemma 20](#lemma:binlinear estimate){reference-type="ref" reference="lemma:binlinear estimate"}, and then the estimate [\[equ:z-norm estimate for fa\]](#equ:z-norm estimate for fa){reference-type="eqref" reference="equ:z-norm estimate for fa"} on $\Vert f_{\mathrm{a}}\Vert _{L_{I_{j}}^{\infty }Z}$ from Lemma [Proposition 15](#lemma:z-norm bounds on fa){reference-type="ref" reference="lemma:z-norm bounds on fa"}. For the $F_{\mathrm{err}}$ term on the last line, we use the estimate [\[equ:Ferr_bound\]](#equ:Ferr_bound){reference-type="eqref" reference="equ:Ferr_bound"} in Proposition [Proposition 16](#lemma:bounds on ferr){reference-type="ref" reference="lemma:bounds on ferr"}. Then we have $$\Vert f_{\mathrm{c}}\Vert _{L_{I_{j}}^{\infty }Z}\leq \Vert f_{\mathrm{c}% }(T_{j})\Vert _{Z}+ \frac{C(\ln \ln M)^{2}}{\sqrt{\ln M}}\Vert f_{\mathrm{c}}\Vert _{L_{I_{j}}^{\infty }Z}+ \frac{CM^{s-\frac{d-1}{2}}}{\sqrt{\ln M}}\Vert f_{\mathrm{c}}\Vert _{L_{I_{j}}^{\infty }Z}^{2}+CM^{-1},$$ where $C$ is some absolute constant. Absorbing the $\Vert f_{\mathrm{c}}\Vert _{L_{I_{j}}^{\infty }Z}$ term on the right gives $$\Vert f_{\mathrm{c}}\Vert _{L_{I_{j}}^{\infty }Z}\leq 2\Vert f_{\mathrm{c} }(T_{j})\Vert _{Z}+2CM^{-1}.$$ Applying this successively for $j=0,1,\ldots$, we obtain $$\Vert f_{\mathrm{c}}\Vert _{L_{I_{j}}^{\infty }Z}\leq (2^{j+1}-1)2CM^{-1}.$$ With $j=n=\sqrt{\ln M}(\ln\ln \ln M)$, we arrive at $$\Vert f_{\mathrm{c}}(T_{\ast })\Vert _{Z}\leq \frac{Ce^{\sqrt{\ln M}\ln \ln M}}{M} = \frac{Ce^{\sqrt{\ln M}\ln \ln M}}{e^{ \ln M}}\leq M^{-1 /2}\ll 1.$$ ◻ ## Proof of Illposedness {#section:Proof of Illposedness} We get into the proof the ill-posedness. ***Proof of Ill-posedness in Theorem $\ref{thm:main theorem}$**.* Let $$f_{\mathrm{ex}}(t)=f_{\mathrm{r}}(t)+f_{\mathrm{b}}(t)+f_{\mathrm{c}}(t),$$ with $f_{\mathrm{c}}(t)$ given in Proposition [Proposition 21](#lemma:perturbation){reference-type="ref" reference="lemma:perturbation"}. By the upper and lower bounds in Lemma [Lemma 10](#lemma:bounds on fr){reference-type="ref" reference="lemma:bounds on fr"} that $$\Vert f_{\mathrm{r}}(0)\Vert _{L_{v}^{2,r_{0}}H_{x}^{s_{0}}}\lesssim \frac{1}{\ln\ln M},\quad \Vert f_{\mathrm{r}}(T_{*})\Vert _{L_{v}^{2,r_{0}}H_{x}^{s_{0}}}\gtrsim 1,\quad$$ we can take $t_{0}\in [T_{*},0]$ such that $\Vert f_{\mathrm{r}}(t_{0})\Vert _{L_{v}^{2,r_{0}}H_{x}^{s_{0}}}=1$. Note that $$\begin{aligned} \Vert f_{\mathrm{r}}\Vert _{Z}\lesssim M^{\frac{d-1}{2}-s} (\ln \ln M)^{2},\quad \Vert v\cdot \nabla_{x}f_{\mathrm{r}}\Vert _{Z}\ll M^{-1},\quad \Vert Q^{\pm}(f_{\mathrm{r}},f_{\mathrm{r}})\Vert _{Z}\ll M^{-1},\end{aligned}$$ which are established in Lemma [Lemma 14](#lemma:z-norm bounds on fr){reference-type="ref" reference="lemma:z-norm bounds on fr"} and Section [3.3](#section:Bounds on the Error Terms){reference-type="ref" reference="section:Bounds on the Error Terms"}. Therefore, by the same perturbation argument in Lemma [Proposition 21](#lemma:perturbation){reference-type="ref" reference="lemma:perturbation"}, we generate an exact solution $g_{\mathrm{ex}}(t)$ to Boltzmann equation $$g_{\mathrm{ex}}(t)=f_{\mathrm{r}}(t_{0})+g_{\mathrm{c}}(t),$$ with $g_{\mathrm{c}}(0)=0$. This gives that $$\left\{ \begin{aligned} &\Vert g_{\mathrm{ex}}(0)\Vert _{L_{v}^{2,r_{0}}H_{x}^{s_{0}}}=\Vert f_{\mathrm{r}}(t_{0})\Vert _{L_{v}^{2,r_{0}}H_{x}^{s_{0}}}=1,\\ &\Vert g_{\mathrm{c}}(t)\Vert _{L^{\infty}([T_{*},0];Z)}\lesssim M^{-\frac{1}{2}}. \end{aligned} \right.$$ Now, we have two solutions with the decompositions $$\left\{ \begin{aligned} f_{\mathrm{ex}}(t)=&f_{\mathrm{r}}(t)+f_{\mathrm{b}}(t)+f_{\mathrm{c}}(t),\\ g_{\mathrm{ex}}(t)=&f_{\mathrm{r}}(t_{0})+g_{c}(t), \end{aligned} \right.$$ which gives $$f_{\mathrm{ex}}(t)-g_{\mathrm{ex}}(t)=(f_{\mathrm{r}}(t)-f_{\mathrm{r}}(t_{0}))+f_{ \mathrm{b}}(t)+f_{\mathrm{c}}(t)-g_{\mathrm{c}}(t).$$ For $t\in [T_{*},0]$, by Lemma [Lemma 8](#lemma:Hs,bounds on fb){reference-type="ref" reference="lemma:Hs,bounds on fb"} and Proposition [Proposition 21](#lemma:perturbation){reference-type="ref" reference="lemma:perturbation"}, we have $$\begin{aligned} &\Vert f_{\mathrm{b}}(t)\Vert _{L_{v}^{2,r_{0}}H_{x}^{s_{0}}}\lesssim M^{s_{0}-s}= \frac{1}{\ln \ln M},\\ &\Vert f_{\mathrm{c}}(t)\Vert _{L_{v}^{2,r_{0}}H_{x}^{s_{0}}}\leq \Vert f_{ \mathrm{c}}(t)\Vert _{Z}\lesssim M^{-\frac{1}{2}},\\ &\Vert g_{\mathrm{c}}(t)\Vert _{L_{v}^{2,r_{0}}H_{x}^{s_{0}}}\leq \Vert g_{\mathrm{c}}(t)\Vert _{Z}\lesssim M^{-\frac{1}{2}}.\end{aligned}$$ Thus, we obtain $$\Vert f_{\mathrm{ex}}(t_{0})-g_{\mathrm{ex}}(t_{0})\Vert _{L_{v}^{2,r_{0}}H_{x}^{s_{0}}}\lesssim \frac{1}{\ln\ln M},$$ and $$\Vert f_{\mathrm{ex}}(0)-g_{\mathrm{ex}}(0)\Vert _{L_{v}^{2,r_{0}}H_{x}^{s_{0}}}\sim \Vert f_{\mathrm{a}}(0)-f_{\mathrm{r} }(t_{0})\Vert _{L_{v}^{2,r_{0}}H_{x}^{s_{0}}}\sim \Vert f_{\mathrm{r} }(t_{0})\Vert _{L_{v}^{2,r_{0}}H_{x}^{s_{0}}}=1,$$ where we have used that $\Vert f_{\mathrm{a}}(0)\Vert _{L_{v}^{2,r_{0}}H_{x}^{s_{0}}}\lesssim \frac{1}{\ln \ln M}$. Hence, we complete the proof. ◻ **Remark 22**. We actually have found an exact solution $f_{\mathrm{ex}}(t)$ which satisfies the norm deflation property. This is the key to conclude the failure of uniform continuity of the data-to-solution map. In the end, we prove Corollary [Corollary 3](#lemma:ill-posedness,kernels){reference-type="ref" reference="lemma:ill-posedness,kernels"}. ***Proof of Corollary $\ref{lemma:ill-posedness,kernels}$**.* Recall the kernel $$\begin{aligned} \label{equ:kernels} B(u-v,\omega)=\left( 1_{\left\{ |u-v|\leq 1 \right\} }|u-v|+1_{\left\{ |u-v|\geq 1 \right\} }|u-v|^{-1} \right) \textbf{b}(\frac{u-v}{|u-v|}\cdot \omega),\end{aligned}$$ and notice the pointwise upper bound estimate $$\begin{aligned} \label{equ:pointwise upper bound,kernels} \left( 1_{\left\{ |u-v|\leq 1 \right\} }|u-v|+1_{\left\{ |u-v|\geq 1 \right\} }|u-v|^{-1} \right) \textbf{b}(\frac{u-v}{|u-v|}\cdot \omega)\leq \frac{1}{|u-v|} \textbf{b}(\frac{u-v}{|u-v|}\cdot \omega).\end{aligned}$$ Therefore, for the kernel $B(u-v,\omega)$ in [\[equ:kernels\]](#equ:kernels){reference-type="eqref" reference="equ:kernels"}, all the same upper bound estimates on $f_{\mathrm{b}}$, $f_{\mathrm{r}}$, $f_{\mathrm{a}}$, $F_{\mathrm{err}}$, and $f_{\mathrm{c}}$ follow from the pointwise upper bound estimate [\[equ:pointwise upper bound,kernels\]](#equ:pointwise upper bound,kernels){reference-type="eqref" reference="equ:pointwise upper bound,kernels"}. The only one lower bound on $f_{\mathrm{r}}$ we need is given in Remark [Remark 11](#remark:lower bound,kernels){reference-type="ref" reference="remark:lower bound,kernels"}. Then by repeating the proof of ill-posedness for the endpoint case $(d,\gamma,r_{0})=(3,-1,0)$ in Theorem [Theorem 2](#thm:main theorem){reference-type="ref" reference="thm:main theorem"}, we complete the proof of Corollary [Corollary 3](#lemma:ill-posedness,kernels){reference-type="ref" reference="lemma:ill-posedness,kernels"}. ◻ # Sobolev-type and Time-independent Bilinear Estimates {#section:Sobolev-type and Time-independent Bilinear Estimates} **Lemma 23** (Fractional Leibniz rule, [@gulisashvili1996exact]). *Suppose $1<r<\infty$, $s\geq 0$ and $\frac{1}{r}=\frac{1}{p_{i}}+\frac{1}{q_{i}}$ with $i=1,2$, $1<q_{1}\leq \infty$, $1<p_{2}\leq \infty$. Then $$\begin{aligned} \Vert \langle \nabla_{x} \rangle ^{s}(fg)\Vert _{L^{r}}\leq C\Vert \langle \nabla_{x} \rangle ^{s}f\Vert _{L^{p_{1}}}\Vert g\Vert _{L^{q_{1}}}+\Vert f\Vert _{L^{p_{2}}} \Vert \langle \nabla_{x} \rangle ^{s}g\Vert _{L^{q_{2}}}\end{aligned}$$ where the constant $C$ depends on all of the parameters.* Next, we present the standard Hardy-Littlewood-Sobolev inequality, which is widely used in our various estimates for the soft potential case. **Lemma 24**. *Let $p>1$, $r>1$ and $-d<\gamma \leq 0$ with $$\frac{1}{p}+\frac{1}{r}=2+\frac{\gamma}{d}.$$ Let $f\in L^{p}(\mathbb{R}^{d})$ and $h\in L^{r}(\mathbb{R}^{d})$, then there exists a constant $C(d, \gamma, p)$, independent of $f$ and $h$, such that $$\begin{aligned} \label{equ:hls,integral} \int_{\mathbb{R}^d} \int_{\mathbb{R}^d} f(x)|x-y|^{\gamma} h(y) \mathrm{d} x \mathrm{d} y\leq C(d, \gamma, p,r)\|f\|_p\|h\|_r.\end{aligned}$$ In particular, for $p>1$, $q>1$ with $$1+\frac{1}{q}+\frac{\gamma}{d}=\frac{1}{p},$$ we also have $$\begin{aligned} \label{equ:hardy-littlewood-sobolev inequality} \Vert f*|\cdot|^{\gamma}\Vert _{L^{q}}\leq C(d, \gamma, p,q)\Vert f\Vert _{L^{p}}.\end{aligned}$$* **Lemma 25** (Endpoint case). *Let $d\geq 2$, $-d<\gamma\leq 0$, and $1 \leq p< \frac{d}{d+\gamma}<q\leq \infty$. Then for $f \in L^p\left(\mathbb{R}^d\right) \cap L^q\left(\mathbb{R}^d\right)$, it holds that $$\begin{aligned} \label{equ:endpoint estimate,hls,L1} \int |x|^{\gamma}|f(x)|dx \lesssim\Vert f\Vert _{L^{p}}^{\frac{\frac{q-1}{q}+\frac{\gamma}{d}}{\frac{1}{p}-\frac{1}{q}}} \Vert f\Vert _{L^{q}}^{\frac{-\frac{\gamma}{d}-\frac{p-1}{p}}{\frac{1}{p}-\frac{1}{q}}}.\end{aligned}$$ In particular, when $\gamma=-1$, $p=1$, and $q>\frac{d}{d-1}$, we have $$\begin{aligned} \label{equ:endpoint estimate,hls} \Big\Vert \int \frac{f(y)}{|x-y|}dy \Big \Vert _{L_{x}^{\infty}} \lesssim\Vert f\Vert _{L^{1}}^{1-\frac{1}{d\left(1-\frac{1}{q}\right)}}\Vert f\Vert _{L^{q}}^{\frac{1}{d\left(1-\frac{1}{q}\right)}}.\end{aligned}$$* *Proof.* The endpoint case is also known. For completeness, we include a proof. We split the integral into two parts and use Hölder inequality to get $$\begin{aligned} \int_{\mathbb{R}^{d}}|x|^{\gamma}|f(x)|dx\leq& \int_{|x|\leq \eta}|x|^{\gamma}|f(x)|dx+\int_{|x|> \eta}|x|^{\gamma}|f(x)|dx\\ \lesssim &\Vert f\Vert _{L^{q}}\eta^{\frac{d}{q'}+\gamma}+\Vert f\Vert _{L^{p}}\eta^{\frac{d}{p'}+\gamma},\end{aligned}$$ where $p'=\frac{p}{p-1}$ and $q'=\frac{q}{q-1}$. Optimizing the choice of $\eta$ gives the desired estimate that $$\begin{aligned} \int_{\mathbb{R}^{d}}|x|^{\gamma}|f(x)|dx \lesssim &\Vert f\Vert _{L^{p}}^{\frac{\frac{q-1}{q}+\frac{\gamma}{d}}{\frac{1}{p}-\frac{1}{q}}} \Vert f\Vert _{L^{q}}^{\frac{-\frac{\gamma}{d}-\frac{p-1}{p}}{\frac{1}{p}-\frac{1}{q}}}.\end{aligned}$$ ◻ The following parts focus on time-independent bilinear estimates for gain/loss terms. **Lemma 26** ([@alonso2010convolution Theorem 2, Corollary 9]). *Let $1<p, q, r<\infty$ and $-d<\gamma \leq 0$ with $$\frac{1}{p}+\frac{1}{q}=1+\frac{\gamma}{d}+\frac{1}{r}.$$ Assume the collision kernel $$B\left(u-v,\omega\right)=\left|u-v\right|^\gamma \textbf{b}(\frac{u-v}{|u-v|}\cdot \omega),$$ with $\textbf{b}(\frac{u-v}{|u-v|}\cdot \omega)$ satisfying Grad's angular cutoff assumption. Then, it holds that $$\begin{aligned} &\left\|Q^{+}(f, g)\right\|_{L^{r}\left(\mathbb{R}^{d}\right)} \leq C\|f\|_{L^{p}\left(\mathbb{R}^{d}\right)}\|g\|_{L^{q}\left(\mathbb{R}^{d}\right)},\label{equ:bilinear estimate,Q+}\\ &\left\|Q^{-}(f, g)\right\|_{L^{r}\left(\mathbb{R}^{d}\right)} \leq C\|f\|_{L^{p}\left(\mathbb{R}^{d}\right)}\|g\|_{L^{q}\left(\mathbb{R}^{d}\right)},\quad p>r.\label{equ:bilinear estimate,Q-}\end{aligned}$$* **Lemma 27** ($L^{1}$ endpoint estimate for $Q^{+}$). *For $\gamma=-1$, we have $$\begin{aligned} \label{equ:endpoint estimate,Q+,f} \Vert Q^{+}(f,g)\Vert _{L^{1}} \leq& \Vert f\Vert _{L^{1}}\Vert g\Vert _{L^1}^{1-\frac{1}{d\left(1-\frac{1}{p}\right)}}\Vert g\Vert _{L^p}^{\frac{1}{d\left(1-\frac{1}{p}\right)}},\end{aligned}$$ $$\begin{aligned} \label{equ:endpoint estimate,Q+,g} \Vert Q^{+}(f,g)\Vert _{L^{1}} \leq& \|f\|_{L^1}^{1-\frac{1}{d\left(1-\frac{1}{p}\right)}}\|f\|_{L^p}^{\frac{1}{d\left(1-\frac{1}{p}\right)}}\Vert g\Vert _{L^{1}}.\end{aligned}$$* *Proof.* By the change of variable, we have $$\begin{aligned} \Vert Q^{+}(f,g)\Vert _{L^{1}} \lesssim& \int_{\mathbb{S}^{d-1}} \int_{\mathbb{R}^{2d}}\frac{|f(u^{*})g(v^{*})|}{|u^{*}-v^{*}|} du dv d\omega\\ =&\int_{\mathbb{S}^{d-1}} \int_{\mathbb{R}^{2d}}\frac{|f(u^{*})g(v^{*})|}{|u^{*}-v^{*}|} du^{*} dv^{*} d\omega\\ \lesssim& \Vert f\Vert _{L^{1}}\Big\Vert \int \frac{|g(v^*)|}{|u^{*}-v^{*}|}dv^{*} \Big \Vert _{L^{\infty}}.\end{aligned}$$ Using the $L^{\infty}$ estimate [\[equ:endpoint estimate,hls\]](#equ:endpoint estimate,hls){reference-type="eqref" reference="equ:endpoint estimate,hls"}, we get $$\begin{aligned} \Vert Q^{+}(f,g)\Vert _{L^{1}}\lesssim \Vert f\Vert _{L^{1}}\Vert g\Vert _{L^1}^{1-\frac{1}{d\left(1-\frac{1}{p}\right)}}\Vert g\Vert _{L^p}^{\frac{1}{d\left(1-\frac{1}{p}\right)}}.\end{aligned}$$ In the same way, we also obtain estimate [\[equ:endpoint estimate,Q+,g\]](#equ:endpoint estimate,Q+,g){reference-type="eqref" reference="equ:endpoint estimate,Q+,g"}. ◻ # Strichartz Estimates {#section:Strichartz Estimates} Recall the abstract Strichartz estimates. **Theorem 28** ([@keel1998endpoint Theorem 1.2]). *Suppose that for each time $t$ we have an operator $U(t)$ such that $$\begin{aligned} \Vert U(t)f\Vert _{L_{x}^{2}}\lesssim& \Vert f\Vert _{L_{x}^{2}},\\ \Vert U(t)(U(s)^{*})f\Vert _{L_{x}^{\infty}}\lesssim& |t-s|^{-\sigma}\Vert f\Vert _{L_{x}^{1}}.\end{aligned}$$ Then it holds that $$\begin{aligned} \label{equ:strichartz,usual one} \Vert U(t)f\Vert _{L_{t}^{q}L_{x}^{p}}\lesssim \Vert f\Vert _{L_{x}^{2}},\end{aligned}$$ for all sharp $\sigma$-admissible exponent pair that $$\begin{aligned} \frac{2}{q}+\frac{2\sigma}{p}=\sigma,\quad q\geq 2,\ \sigma>1.\end{aligned}$$* The symmetric hyperbolic Schrödinger equation is $$\left\{ \begin{aligned} i\partial_{t}\phi+\nabla_{\xi}\cdot \nabla_{x}\phi=&0,\\ \phi(0)=&\phi_{0}. \end{aligned} \right.$$ Note that the linear propagator $U(t)=e^{it\nabla_{\xi}\cdot \nabla_{x}}$ satisfies the energy and dispersive estimates $$\begin{aligned} &\Vert e^{it\nabla_{\xi}\cdot\nabla_{x}}\phi_{0}\Vert _{L_{x\xi}^{2}}\lesssim \Vert \phi_{0}\Vert _{L_{x\xi}^{2}},\\ &\Vert e^{it\nabla_{\xi}\cdot\nabla_{x}}\phi_{0}\Vert _{L_{x\xi}^{\infty}}\lesssim t^{-d}\Vert \phi_{0}\Vert _{L_{x\xi}^{1}}. \end{aligned}$$ Then by Theorem [Theorem 28](#lemma:strichartz estimate,keel-tao){reference-type="ref" reference="lemma:strichartz estimate,keel-tao"}, this gives a Strichartz estimate that $$\begin{aligned} \label{equ:strichartz estimate,linear} \Vert e^{it\nabla_{\xi}\cdot \nabla_{x}}\phi_{0}\Vert _{L_{t}^{q}L_{x\xi}^{p}}\lesssim \Vert \phi_{0}\Vert _{L_{x\xi}^{2}},\quad \frac{2}{q}+\frac{2d}{p}=d,\quad q\geq 2,\ d\geq2.\end{aligned}$$ [^1]: Instead of scaling invariance of equation, the critical regularity for the Boltzmann equation is sometimes believed at $s=\frac{d}{2}$ in the sense that the critical embedding $H^{\frac{d}{2}}\hookrightarrow L^{\infty}$ fails, see for example [@alexandre2013local; @duan2016global; @duan2021global; @duan2018solution]. [^2]: The hard potential case is also interesting and the ill-posedness result remains open. Our approximation solution gives desired bad behaviors for the hard potential. But it needs a totally different work space to generate the exact solution. Hence, we put it for further work. [^3]: If $f_{0}\in L_{x,v}^{1}$, the solution $f(t)$ should also have the $L_{x,v}^{1}$ integrability in terms of the mass conservation law. However, this is not a simple problem. We deal with it in [@chen2023sharp] by using regularity criteria which are beyond the scope of this paper. [^4]: One could replace (c) with the Lipschitz continuity which is usually the case as well. [^5]: It then provides a formal answer to a question raised by K. Nakanishi. [^6]: See [@chen2019derivation; @chen2022unconditional; @CSZ22] for some examples sharing similar critical flavor but carrying completely different structures. [^7]: One could see [@chen2022well Figure 1] for a picture of the approximation solutions there. They look like bullets hitting a rock in [@chen2022well]. Our improvised and refined version is more like needles poking a rock through. [^8]: The index $\frac{5}{3}$ is just one of the multiple choices. We choose it, as it would not yield much more difficulties in the estimates on the approximation solution and error terms.

arxiv_math

--- abstract: | We characterize the maximizers of a functional involving the minimization of the Wasserstein distance between equal volume sets. This functional appears as a repulsive interaction term in some models describing biological membranes. We combine a symmetrization-by-reflection technique with the uniqueness of optimal transport plans to prove that balls are the only maximizers. Further, in one dimension, we provide a sharp quantitative version of this maximality result. address: - Department of Mathematics, University of Toronto, ON, Canada - Scuola Normale Superiore, Pisa, Italy - Department of Mathematics and Applied Mathematics, Virginia Commonwealth University, Richmond VA, United States author: - Almut Burchard - Davide Carazzato - Ihsan Topaloglu bibliography: - references.bib title: Maximizers of nonlocal interactions of Wasserstein Type --- # Introduction {#sec: intro} In this paper we study a variational problem involving the Wasserstein distance between equal volume sets. Specifically, for any $p>1$ we consider the following energy defined on subsets of $\mathbb R^N$: $$\label{eq:wasserstein} \mathcal{W}_p(E)\coloneqq \inf\Big\{W_p(\mathscr{L}^N\mathop{\hbox{\vrule height 7pt width .5pt depth 0pt \vrule height .5pt width 6pt depth 0pt}}\nolimits E,\mathscr{L}^N\mathop{\hbox{\vrule height 7pt width .5pt depth 0pt \vrule height .5pt width 6pt depth 0pt}}\nolimits F)\colon|F|=|E|,|E\cap F|=0\Big\},$$ where $W_p(\mu_1,\mu_2)$ is the $p$-Wasserstein distance between two measures $\mu_1,\mu_2\in\mathsf{M}_+(\mathbb R^N)$ with $\mu_1(\mathbb R^N)=\mu_2(\mathbb R^N)<+\infty$. Here $\mathscr{L}^N$ denotes the Lebesgue measure in $\mathbb R^N$, and for any measurable set $E\subset \mathbb R^N$, we use the notation $|E|=\mathscr{L}^N(E)$. The functional [\[eq:wasserstein\]](#eq:wasserstein){reference-type="eqref" reference="eq:wasserstein"} appears in [@BCL2020], where Buttazzo, Carlier and Laborde investigate the Wasserstein distance between two mutually singular measures for any $p\geq 1$. In particular, given a measure $\mu$ they prove that the infimum is achieved among measures that are singular with respect to $\mu$. They also show that, when the admissible class consists of densities bounded by 1, the optimal solution is given by the characteristic function of a set. In [@BCL2020] the authors also introduce and analyze the perimeter regularization of [\[eq:wasserstein\]](#eq:wasserstein){reference-type="eqref" reference="eq:wasserstein"}. Namely, they consider the problem $$\label{eq:wasserstein-perimeter} \inf \Big\{ P(E) + \lambda W_p({\mathcal{L}^n \mathop{\hbox{\vrule height 7pt width .5pt depth 0pt \vrule height .5pt width 6pt depth 0pt}}\nolimits} E,{\mathcal{L}^n \mathop{\hbox{\vrule height 7pt width .5pt depth 0pt \vrule height .5pt width 6pt depth 0pt}}\nolimits}F) \colon E, F \subset \mathbb R^N, \,|E\cap F|=0, \, |E|=|F|=1 \Big\},$$ and show, for any $\lambda>0$, the existence of minimizers when admissible sets $E$ and $F$ are required to be subsets of a bounded domain $\Omega$. This problem (with $p=1$) is introduced by Peletier and Röger as a simplified model for lipid bilayer membranes where the sets $E$ and $F$ represent the densities of the hydrophobic tails and hydrophilic heads of the two part lipid molecules, respectively [@PR2009; @LPR2014]. The perimeter term accounts for an interfacial energy arising from hydrophobic effects, while the Wasserstein term models the weak bonding between the head and tail particles. When posed over the unbounded space, Buttazzo, Carlier and Laborde prove the existence of minimizers for the problem [\[eq:wasserstein-perimeter\]](#eq:wasserstein-perimeter){reference-type="eqref" reference="eq:wasserstein-perimeter"} in two dimensions. Xia and Zhou [@XZ2021] extend this result to higher dimensions but under the additional assumptions that $\lambda$ is sufficiently small and that $p<n/(n-2)$. Recently, Novack, Venkatraman and the third author [@NTV2023] prove that minimizers to [\[eq:wasserstein-perimeter\]](#eq:wasserstein-perimeter){reference-type="eqref" reference="eq:wasserstein-perimeter"} exist in any dimension *and* for all values of $\lambda>0$ and $p \in [1,\infty)$. Simultaneously, Candau-Tilh and Goldman [@C-TG2022] also obtain the existence of minimizers via an alternative argument and characterize global minimizers in the small $\lambda$ regime. The analysis in [@C-TG2022] and [@NTV2023] show that there is a direct competition between the perimeter and the Wasserstein terms in [\[eq:wasserstein-perimeter\]](#eq:wasserstein-perimeter){reference-type="eqref" reference="eq:wasserstein-perimeter"}. This, also as pointed out by Rupert Frank to the third author, leads to the question whether the functional [\[eq:wasserstein\]](#eq:wasserstein){reference-type="eqref" reference="eq:wasserstein"} is *maximized* when the set $E$ is a ball. We investigate this question in this paper *when* $p>1$. It often happens that we need to relax a functional to exploit some compactness. We denote by $\mathsf{A}_m$ the class of admissible densities with mass $m$ that we use to relax the problem, i.e., $$\mathsf{A}_m\coloneqq \left\{\rho\in L^1(\mathbb R^N)\colon0\leq \rho\leq 1, \int\rho \, dx=m\right\}.$$ We will use the shorthand notation $\mathsf{A}\coloneqq\mathsf{A}_1$ when we deal with probability densities. We define the relaxation of [\[eq:wasserstein\]](#eq:wasserstein){reference-type="eqref" reference="eq:wasserstein"} to densities $\rho$ with $0\leq \rho\leq 1$ as follows: $$\label{eq:wasserstein-densities} \begin{split} \mathcal{W}_p(\rho)\coloneqq \inf\left\{W_p(\rho,\rho')\colon 0 \leq \rho', \ 0\leq \rho+\rho'\leq 1, \int\rho' \, dx=\int\rho \, dx\right\}. \end{split}$$ Our main result in this paper is the following theorem. **Main Theorem 1**. *The only maximizer of [\[eq:wasserstein-densities\]](#eq:wasserstein-densities){reference-type="eqref" reference="eq:wasserstein-densities"} in the class $\mathsf{A}_m$, up to translations, is the characteristic function of a ball $B$ with $|B|=m$.* By [@DMSV2016 Proposition 5.2] in the case $p=2$, and by the same result combined with [@BCL2020 Theorem 3.10] and [@C-TG2022 Proposition 2.1] in the case $p\neq 2$, the expression [\[eq:wasserstein-densities\]](#eq:wasserstein-densities){reference-type="eqref" reference="eq:wasserstein-densities"} extends the definition on sets given in [\[eq:wasserstein\]](#eq:wasserstein){reference-type="eqref" reference="eq:wasserstein"}. By these results, we also have that for any $\rho\in\mathsf{A}_m$ there is a unique density $\eta_{\rho}$ realizing [\[eq:wasserstein-densities\]](#eq:wasserstein-densities){reference-type="eqref" reference="eq:wasserstein-densities"} when $p>1$. Note that, for $p>1$ [@V2003 Theorem 2.44] guarantees that there is only one optimal transport plan $\pi_{\rho}$ between $\rho$ and $\eta_{\rho}$, and it is induced by a map. The class of transport plans, which we will call *admissible plans*, that play a role in the definition of $\mathcal{W}_p(\rho)$ is given by $$\begin{split} \mathsf{AP}_{\rho} \coloneqq \left\{\pi\in\mathsf{M}_+(\mathbb R^N\times \mathbb R^N)\colon (p_1)_{\#}\pi = \rho\mathscr{L}^N, \ \rho\mathscr{L}^N+(p_2)_{\#}\pi\leq \mathscr{L}^N\right\}, \end{split}$$ where $\mathsf{M}(\mathbb R^N)$ denotes the set of signed Borel measures in $\mathbb R^N$, and $\mathsf{M}_+(\mathbb R^N)\subset \mathsf{M}(\mathbb R^N)$ denotes the set of non-negative measures. Here $p_1$ and $p_2$ are the two usual projections from $\mathbb R^N\times \mathbb R^N$ in $\mathbb R^N$. Notice that, thanks to the properties of the push-forward, it is automatically true that the density of $(p_2)_{\#}\pi$ with respect to $\mathscr{L}^N$ belongs to $\mathsf{A}_m$ whenever $\rho\in\mathsf{A}_m$ and $\pi\in \mathsf{AP}_{\rho}$. *Remark 1*. We point out that the energy $\mathcal{W}_p(\rho)$ can be defined whenever we have a metric space with a reference measure (in our case, the euclidean space $\mathbb R^N$ endowed with $\mathscr{L}^N$). If $(X,d)$ is a Polish metric space, and $\gamma\in\mathsf{M}_+(X)$ is a Borel measure, then for any density $\rho\colon X\to[0,1]$ we can define its Wasserstein energy as $$\mathcal{W}_p(\rho) \coloneqq \inf \left\{W_p(\rho\gamma,\rho'\gamma)\colon 0 \leq \rho', \ \rho+\rho'\leq1, \ \int\rho'\,d\gamma = \int\rho \,d\gamma\right\},$$ and the $p$-Wasserstein distance can be defined in any metric space. We continue to denote by $\mathsf{AP}_{\rho}$ the set of admissible plans, i.e. $$\mathsf{AP}_{\rho} = \left\{\pi\in\mathsf{M}_+(X\times X)\colon (p_1)_{\#}\pi = \rho\gamma, \ \rho\gamma+(p_2)_{\#}\pi\leq\gamma\right\}.$$ We cannot expect to have many invariance properties in an abstract setting, but some analytic-flavoured features could be retrieved in wide generality. We will not use this abstract formulation in this paper, with the exception of Proposition [Proposition 14](#prop:maximizer-1D){reference-type="ref" reference="prop:maximizer-1D"} where we consider the space $X=\mathbb R^+$ with a weight. This appears because in Section [3](#sec:maximizer){reference-type="ref" reference="sec:maximizer"} we reduce to radial densities, and it is convenient to look at them as $1$-dimensional densities (a weight pops up because of the coarea formula). ## Plan of the paper {#plan-of-the-paper .unnumbered} In Section [2](#sec:preliminary results){reference-type="ref" reference="sec:preliminary results"} we introduce some preliminary results that are useful for the problem. After recalling briefly some well-known theorems about the existence and uniqueness of the optimal transport map, we introduce some very simple properties of the functional $\mathcal{W}_p$ that were essentially already present in the literature for slightly different problems. In particular, Lemma [Lemma 7](#lem:full-ball){reference-type="ref" reference="lem:full-ball"} is devoted to the saturation of the constraint in a certain region, and Corollary [Corollary 8](#cor:bounded-transport-distance){reference-type="ref" reference="cor:bounded-transport-distance"} provides a uniform control on the transport distance. These two results are quite robust, as they do not require any geometric property of the Euclidean space, but just its metric-measure structure. Lemma [Lemma 10](#lem:weak-continuity-wasserstein){reference-type="ref" reference="lem:weak-continuity-wasserstein"} and Lemma [Lemma 11](#lemma:symmetry-transport){reference-type="ref" reference="lemma:symmetry-transport"} are an original contribution. The first one shows the continuity of the functional $\mathcal{W}_p$ with respect to the weak$*$ convergence (when there is no loss of mass), and it is fundamental to prove the existence of maximizers for $\mathcal{W}_p$. The second one, instead, shows that some symmetries of a density $\rho$ can be inherited by the optimal plan $\pi_{\rho}$ that realizes $\mathcal{W}_p(\rho)$. In Section [3](#sec:maximizer){reference-type="ref" reference="sec:maximizer"} we deal with the maximizers of $\mathcal{W}_p$, whose existence is proved in Proposition [Theorem 13](#thm:existence-maximizers){reference-type="ref" reference="thm:existence-maximizers"} applying the concentration compactness principle. This is a building block also for our successive characterization of the maximizers, since we combine a symmetrization technique and the uniqueness of the optimal transport plan to show that the maximizers have some symmetry. In fact, our plan to characterize them is the following: 1. prove that the segments maximize a 1-dimensional weighted version of $\mathcal{W}_p$, in Proposition [Proposition 14](#prop:maximizer-1D){reference-type="ref" reference="prop:maximizer-1D"}; 2. prove that, if $\rho$ is a given maximizer, then the optimal transport plan realizing $\mathcal{W}_p(\rho)$ is radial. This is contained in Corollary [Corollary 16](#cor:radial-transport-for-maximizers){reference-type="ref" reference="cor:radial-transport-for-maximizers"}, as a consequence of Lemma [Lemma 15](#lemma:non-crossing){reference-type="ref" reference="lemma:non-crossing"}; 3. combine the first two points to show that the maximizers have to be star-shaped sets, and then conclude that the ball is the only possible maximizer thanks to the saturation of the constraint exposed in Lemma [Lemma 7](#lem:full-ball){reference-type="ref" reference="lem:full-ball"}. This is contained in Theorem [Theorem 17](#thm:ball-max){reference-type="ref" reference="thm:ball-max"}, and it is our main contribution. Finally, in Section [4](#sec:quant_ineq){reference-type="ref" reference="sec:quant_ineq"} we prove a quantitative version of this maximality result in one dimension, where we show that the deficit of maximality is controlled from below by the square of an asymmetry given as the $L^1$ distance between the ball and any density. Our inequality is asymptotically sharp, in the sense that the exponent of the asymmetry cannot be lowered. A few days before submitting this paper, we became aware of the independent work by Candau-Tilh, Goldman and Merlet [@C-TGM2023preprint] (posted on arXiv on September 6, 2023) studying the same maximization problem. Their result is more general, as it considers a broader class of cost functions in the transport problem. Our strategy, pursued in Section [3](#sec:maximizer){reference-type="ref" reference="sec:maximizer"}, instead, is more geometric, and we circumvent the need to introduce Kantorovich potentials to deal with the transport problem. ## Notation {#notation .unnumbered} Throughout the paper, with an abuse of notation, we will denote the Wasserstein distance between two disjoint set, $W_p(\mathscr{L}^N\mathop{\hbox{\vrule height 7pt width .5pt depth 0pt \vrule height .5pt width 6pt depth 0pt}}\nolimits E,\mathscr{L}^N\mathop{\hbox{\vrule height 7pt width .5pt depth 0pt \vrule height .5pt width 6pt depth 0pt}}\nolimits F)$, by $W_p(E,F)$. By $B_r(x)$ we will denote the open ball of center $x$ and radius $r$, and we will write $B_r$ for $B_r(0)$. The cube of side length $2l$ centered at the origin will be denoted by $Q_l = [-l,l]^N\subset \mathbb R^N$; hence, $Q_l(x) = x+Q_l$. For $\rho\in\mathsf{A}$ by $\eta_{\rho}$ we will denote any density in $\mathsf{A}$ such that $\mathcal{W}_p(\rho) = W_p(\rho,\eta_{\rho})$. Note that for $p>1$ we have that $\eta_\rho$ is unique (cfr. [@BCL2020 Remark 3.11]). Similarly, for $\rho\in\mathsf{A}$, $\pi_{\rho}$ will denote the optimal plan $\mathcal{W}_p^p(\rho) = \int|x-y|^p\,d\pi_{\rho}(x,y)$, and $T_{\rho}$ is the optimal transport map that induces $\pi_{\rho}$. If we have a density $f$, we will sometimes use the short-hand notation $T_{\#}f$ to denote the push forward of the measure $T_{\#}(f\mathscr{L}^N)$. # Preliminary results {#sec:preliminary results} ## The optimal transport problem We introduce in this section the optimal transport problem. The general theory is well developed, and goes far beyond the needs of this paper. We state some results, and we define the optimal transport problem, just in the setting that we need. The interested reader may find much more general statements, and much deeper developments, in the references that we cite, as well as in other books on the subject. Instead, one of the crucial restrictions that we impose is to work with cost $c(x)=|x|^p$ with $p>1$ and (mostly) in the Euclidean space $\mathbb R^N$. This is necessary when we characterize the maximizers of $\mathcal{W}_p$ since we use some uniqueness result valid for these special cost functions, while some parts of our strategy work also for $p=1$ with a slightly different discussion. The next definitions describe rigorously our framework. A quite general setting for the optimal transport problem is that of Polish metric spaces, that are defined as follows. **Definition 2** (Polish metric space). A metric space $(X,d)$ is *Polish* if it is complete and separable. **Definition 3** (Push forward). Let $(X,d_X)$ and $(Y,d_Y)$ be two Polish metric spaces. Given $f:X\to Y$ a Borel function, and given a measure $\mu\in\mathsf{M}(X)$, the *push forward* of $\mu$ induced by $f$ is a new measure denoted by $f_{\#}\mu$. It is defined as follows: for every $A\subset X$ Borel, we have that $$(f_{\#}\mu)(A) = \mu(f^{-1}(A)).$$ Given $(X,d)$ a Polish metric space, $p>1$ a real exponent, and given $\mu_1,\mu_2\in \mathsf{M}_+(X)$ with $\mu_1(X) = \mu_2(X)<+\infty$, we can consider the optimal transport problem with cost $c(x)=|x|^p$: $$W_p^p(\mu_1,\mu_2) = \inf\left\{\iint_{X\times X} |x-y|^p\, d\pi(x,y): \pi\in\mathsf{M}_+(X\times X): (p_1)_{\#}\pi = \mu_1, (p_2)_{\#}\pi = \mu_2\right\}.$$ It is well known that for every couple of marginals $\mu_1$ and $\mu_2$ the infimum is attained (see [@V2003 Theorem 1.3] for a more general result). In some special cases, there are some structure theorems for the optimal transport plans, i.e. those measures $\pi$ that realize the aforementioned infimum. The following is such a result that holds for strictly convex costs. **Theorem 4**. *[@V2003 Theorem 2.44][\[thm:existence-uniqueness-transport\]]{#thm:existence-uniqueness-transport label="thm:existence-uniqueness-transport"} Let $p>1$ be given, and $\mu_1,\mu_2\in\mathsf{M}_+(\mathbb R^N)$ be two measures with $\mu_1(\mathbb R^N) = \mu_2(\mathbb R^N)<+\infty$. Suppose that $\mu_1\ll \mathscr{L}^N$ and that $W_p(\mu_1,\mu_2)<+\infty$. Then, there is a unique optimal transport plan $\pi$, and it is of the form $$\pi = (\mathrm{Id},T)_{\#}\mu_1,$$ where $T$ denotes the unique optimal transport map.* In Section [3](#sec:maximizer){reference-type="ref" reference="sec:maximizer"} it is crucial to characterize the maximizers in one dimension to later pass to higher dimension. Our task is simplified in one dimension because the transport problem has a very easy solution. **Theorem 5**. *[@V2003 Remarks 2.19][\[thm:transport-1D\]]{#thm:transport-1D label="thm:transport-1D"} Let $p>1$ be given, and let $\mu_1,\mu_2\in\mathsf{M}_+(\mathbb R)$ be two measures with $\mu_1(\mathbb R) = \mu_2(\mathbb R) <+\infty$. If they are non-atomic, then the only optimal transport map realizing $W_p(\mu_1,\mu_2)$ is monotone.* ## Properties of $\mathcal{W}_p$ The most basic fact is the following existence theorem. **Theorem 6**. *[@DMSV2016 Section 5][\[thm:existence-W_p\]]{#thm:existence-W_p label="thm:existence-W_p"} Let $p>1$ be given. For any $m>0$ and for any $\rho\in\mathsf{A}_m$, there exists a unique density, called $\eta_{\rho}\in\mathsf{A}_m$, realizing the infimum in [\[eq:wasserstein-densities\]](#eq:wasserstein-densities){reference-type="eqref" reference="eq:wasserstein-densities"}.* Combining this result with Theorem [\[thm:existence-uniqueness-transport\]](#thm:existence-uniqueness-transport){reference-type="ref" reference="thm:existence-uniqueness-transport"} we obtain the existence and uniqueness of the optimal transport plan $\pi_{\rho}$ and the map inducing it, called $T_{\rho}$, which satisfy $$\mathcal{W}_p^p(\rho) = W_p^p(\rho,\eta_{\rho}) = \int |x-y|^p \, d\pi_{\rho}(x,y) = \int |x-T_{\rho}(x)|^p\rho(x) \, dx.$$ We point out that the objects $\eta_{\rho}$, $\pi_{\rho}$ and $T_{\rho}$ all depend implicitly on $p$. We do not stress that dependence because we suppose $p>1$ to be fixed in the whole paper. One important result contains a geometric property of the optimal plan $\pi_{\rho}$. The proof of the following lemma is purely metric, and concerns mostly the structure of $\eta_{\rho}$, rather than the optimal transport problem that is hidden in $\mathcal{W}_p$. Indeed, we do not exploit the $c$-cyclical monotonicity of the optimal plans. This result is a natural generalization of [@DMSV2016 Lemma 5.1]. **Lemma 7**. *Let $(X,d)$ be a Polish metric space, and let $\gamma\in\mathsf{M}_+(X)$ be a given measure. Let $\rho\colon X\to[0,1]$ be a Borel density. If $\pi$ is an optimal plan to compute $\mathcal{W}_p(\rho)$ and $(x,y)\in{\rm spt}\pi$, then $$\label{eq:full-ball} (p_2)_{\#}\pi = (1-\rho)\gamma \qquad \gamma-\text{a.e. in }B_{|y-x|}(x).$$ Moreover, we have that $(p_2)_{\#}\pi \geq \min\{1-\rho,\rho\}\gamma$.* *Proof.* We start proving that $(p_2)_{\#}\pi$ saturates the constraint in the ball, and the second statement will follow easily. The idea is very simple: if $\pi$ does not saturate the constraint in that ball, then we can lower the energy of $\rho$ adding some mass close to $x$. We define $r=|y-x|$. Let us suppose by contradiction that there exist $\varepsilon,\delta>0$ and a set $E\subset B_{r-4\delta}(x)$ with $\gamma(E)$ strictly positive and finite and such that $$(1-\rho)\gamma-(p_2)_{\#}\pi \geq \varepsilon\gamma\qquad \text{in }E.$$ We take $\mu_1 = (p_1)_{\#}(\pi\mathop{\hbox{\vrule height 7pt width .5pt depth 0pt \vrule height .5pt width 6pt depth 0pt}}\nolimits B_{\delta}(x)\times B_{\delta}(y))$ and $\mu_2 = \varepsilon\gamma\mathop{\hbox{\vrule height 7pt width .5pt depth 0pt \vrule height .5pt width 6pt depth 0pt}}\nolimits E$, and we modify $\pi$ in the following way: we take $0<t<\min\{1,\mu_1(X)/\mu_2(X)\}$, and we take $$\begin{split} \tilde \pi = \pi - t\frac{\mu_2(X)}{\mu_1(X)}\pi\mathop{\hbox{\vrule height 7pt width .5pt depth 0pt \vrule height .5pt width 6pt depth 0pt}}\nolimits(B_{\delta}(x)\times B_{\delta}(y)) + \frac{t}{\mu_1(X)}\mu_1\times\mu_2. \end{split}$$ One can check that $\tilde \pi\in\mathsf{AP}_{\rho}$ thanks to our choice of $t$. Since $\pi$ is an optimal plan to compute $\mathcal{W}_p(\rho)$, we have that $$\begin{split} 0&\leq \int |x-y|^p\,(d\tilde\pi-d\pi) = -t\frac{\mu_2(X)}{\mu_1(X)}\int_{B_{\delta}(x)\times B_{\delta}(y)}|x-y|^p\,d\pi+\frac{t}{\mu_1(X)}\int|x-y|^p\,d\mu_1 \, d\mu_2\\ &\leq -t\frac{\mu_2(X)}{\mu_1(X)}(r-2\delta)^p\mu_1(X)+\frac{t}{\mu_1(X)}(r-4\delta+\delta)^p\mu_1(X)\mu_2(X)\\ &= t\mu_2(X)\left[(r-3\delta)^p-(r-2\delta)^p\right]<0, \end{split}$$ and thus we reach a contradiction. We now address the last inequality. Suppose by contradiction that the opposite inequality holds in a set $E\subset X$ with $\int_E \rho d \gamma>0$. Then, thanks to what we have proved so far, we know that the set $$\label{eq:no-motion} \{x\in E\colon {\rm spt}\pi\cap (\{x\}\times X) = (x,x)\}$$ has full $\gamma$-measure in $E$. In fact, if this was not the case, then we could find $E'\subset E$ with $\gamma(E')>0$ and such that, for every $x\in E'$, there exists $y\in X\setminus \{x\}$ such that $(x,y)\in{\rm spt}\pi$. Then, using [\[eq:full-ball\]](#eq:full-ball){reference-type="eqref" reference="eq:full-ball"} we find an open covering of $E'$ where the contradiction hypothesis is not satisfied, against the definition of $E$. Condition [\[eq:no-motion\]](#eq:no-motion){reference-type="eqref" reference="eq:no-motion"} means that we are not moving mass in $E$, and thus $$(p_2)_{\#}(\pi\mathop{\hbox{\vrule height 7pt width .5pt depth 0pt \vrule height .5pt width 6pt depth 0pt}}\nolimits(E\times X)) = (p_1)_{\#}(\pi\mathop{\hbox{\vrule height 7pt width .5pt depth 0pt \vrule height .5pt width 6pt depth 0pt}}\nolimits(E\times X)) = \rho\hbox{{\large $\chi$}{\Large $_{_{E}}$}}\gamma.$$ This is sufficient to conclude since $(p_2)_{\#}\pi \geq (p_2)_{\#}(\pi\mathop{\hbox{\vrule height 7pt width .5pt depth 0pt \vrule height .5pt width 6pt depth 0pt}}\nolimits(E\times X)) = \rho\hbox{{\large $\chi$}{\Large $_{_{E}}$}}\gamma$, that is incompatible with our contradiction hypothesis. ◻ **Corollary 8**. *Let us consider the functional $\mathcal{W}_p$ on the Euclidean space $\mathbb R^N$ with the usual metric and the Lebesgue measure $\mathscr{L}^N$. There exists a constant $C_N<+\infty$ such that, for any $\rho\in\mathsf{A}_m$ and for any $(x,y)\in {\rm spt}\pi_{\rho}$, we have that $$\label{eq:bounded-transport-distance} |x-y|\leq C_Nm^{\frac{1}{N}},$$ where $\pi_{\rho}$ is any optimal transport plan associated to $\rho$ and $\eta_{\rho}$. Therefore, we also have that $\mathcal{W}_p^p(\rho)\leq Cm^{1+\frac{p}{N}}$.* *Proof.* This is a consequence of Lemma [Lemma 7](#lem:full-ball){reference-type="ref" reference="lem:full-ball"}. In fact, if we fix $r>0$ such that $|B_r|=3m$, then for any $\rho'\in\mathsf{A}_m$ and for any $z\in\mathbb R^N$ we have that $$\int_{B_r(z)}\rho+\rho'\, dx\leq 2m.$$ Therefore, the condition [\[eq:full-ball\]](#eq:full-ball){reference-type="eqref" reference="eq:full-ball"} is not satisfied for any couple of points $(x,y)\in {\rm spt}\pi_{\rho}$ with $|x-y|>r$. This is precisely the required estimate, since we bound the transport distance with a quantity proportional to the radius of a ball with mass $m$. ◻ *Remark 9*. We report here the scaling property of the energy $\mathcal{W}_p$, that is already stated in [@NTV2023 Lemma 2.5] for sets. Let $\rho$ be a density satisfying the constraint $0\leq \rho\leq 1$ and let $t>0$ be a given constant. If we consider $\tilde \rho(x)=\rho(x/t)$, then we have that $\mathcal{W}_p^p(\tilde \rho) = t^{p+N}\mathcal{W}_p^p(\rho)$. In fact, it is sufficient to consider the density $\eta_{\rho}(\cdot/t)$, rescaling appropriately the transport map. **Lemma 10** (Continuity of $\mathcal{W}_p$). *Let $\rho\in\mathsf{A}_m$ be a given density and let $\{\rho_n\}_{n\in\mathbb N}\subset \mathsf{A}_m$ be a sequence such that $\rho_n\overset{\ast}{\rightharpoonup}\rho$. Then, the limit of $\mathcal{W}_p(\rho_n)$ exists and $\mathcal{W}_p(\rho) = \lim_n \mathcal{W}_p(\rho_n)$.* *Proof.* We prove this proposition in two steps. In the first step we establish that for any $p \geq 1$ [\[eq:wasserstein-densities\]](#eq:wasserstein-densities){reference-type="eqref" reference="eq:wasserstein-densities"} is the lower semicontinuous envelope of the functional in [\[eq:wasserstein\]](#eq:wasserstein){reference-type="eqref" reference="eq:wasserstein"} in the class $\mathsf{A}_m$ with respect to the weak-$*$ topology. As a consequence, $\mathcal{W}_p$ is lower semicontinuous in $\mathsf{A}_m$. In the second step we obtain the upper semicontinuity of $\mathcal{W}_p$ in $\mathsf{A}_m$. *Step 1.* Thanks to Remark [Remark 9](#rem:scaling-wasserstein){reference-type="ref" reference="rem:scaling-wasserstein"} we can consider only the case $m=1$. Let $\{E_n\}_{n\in\mathbb N}$ be a sequence of sets with $|E_n|=1$ such that $E_n\overset{\ast}{\rightharpoonup}\rho$ for some $\rho\in\mathsf{A}$, and let us call $\rho_n=\hbox{{\large $\chi$}{\Large $_{_{E_n}}$}}$. Since we preserve the total mass, we know that for any $\varepsilon>0$ there exist $R>0$ and $k\in\mathbb N$ such that $\int_{B_R}\rho_n \,dx>1-\varepsilon$ for every $n>k$. Using Corollary [Corollary 8](#cor:bounded-transport-distance){reference-type="ref" reference="cor:bounded-transport-distance"} we know that the transport distance is uniformly bounded by a constant $C$, and thus $\int_{B_{R+C}}\eta_{\rho_n}\,dx\geq 1-\varepsilon$ for any $n>k$. Therefore, up to a subsequence, we have that also $\eta_{\rho_n}\overset{\ast}{\rightharpoonup}\rho'$ for some density $\rho'$ with $\int \rho' \, dx=1$. It is then easy to see that $\rho+\rho'\leq 1$ almost everywhere, and thus $$\mathcal{W}_p(\rho)\leq W_p(\rho,\rho') \leq \liminf_n W_p(\rho_n,\eta_{\rho_n}) = \mathcal{W}_p(\rho_n),$$ where we used the well-known lower semicontinuity of the Wasserstein distance (it is sufficient to take the weak limit of the optimal transport plans). This proves that the functional in [\[eq:wasserstein-densities\]](#eq:wasserstein-densities){reference-type="eqref" reference="eq:wasserstein-densities"} is smaller than the lower semicontinuous envelope of $\mathcal{W}_p$ with respect to the weak$*$ topology. Next, we will find a sequence that realizes the equality, proving that our definition of $\mathcal{W}_p(\rho)$ in $\mathsf{A}$ *is* the lower semicontinuous envelope of the functional defined in [\[eq:wasserstein\]](#eq:wasserstein){reference-type="eqref" reference="eq:wasserstein"}. Given $\rho\in\mathsf{A}$, for any $n\in\mathbb N$ we consider a partition of $\mathbb R^N$ with a family of cubes $\mathcal{F}_n = \{Q^k_n\}_{k\in\mathbb N}$ with diameter $1/n$. Thanks to the compatibility condition $\rho+\eta_{\rho}\leq 1$, for any $n$ we can find two sets $E_n$ and $F_n$ with $|E_n\cap F_n|=0$ and such that $$|E_n\cap Q^k_n| = \int_{Q^k_n}\rho \,dx,\qquad |F_n\cap Q^k_n|=\int_{Q^k_n}\eta_{\rho}\,dx,\qquad \forall Q^k_n\in\mathcal{F}_n.$$ It is immediate to see that $E_n\overset{\ast}{\rightharpoonup}\rho$ and $F_n\overset{\ast}{\rightharpoonup}\eta_{\rho}$ as $n\to+\infty$. Recalling $m=1$, we also note that $W_p(E_n,\rho) \leq \mathop{\mathrm{diam}}(Q^k_n)$ and $W_p(\eta_{\rho},F_n)\leq \mathop{\mathrm{diam}}(Q^k_n)$. To see this, it is sufficient to consider the (non-optimal) transport plan given by $$\label{eq:plan-lower-semicontinuity} \pi_n = \sum_{k\in\mathbb N} \frac{1}{|E_n\cap Q^k_n|}(\hbox{{\large $\chi$}{\Large $_{_{E_n\cap Q^k_n}}$}}\mathscr{L}^N)\times(\rho\hbox{{\large $\chi$}{\Large $_{_{Q^k_n}}$}}\mathscr{L}^N)\in\mathcal P(\mathbb R^N\times \mathbb R^N),$$ and notice that $|x-y|\leq \mathop{\mathrm{diam}}(Q^k_n)=1/n$ for any $(x,y)\in{\rm spt}\pi_n$. The proof of the inequality for $F_n$ and $\eta_{\rho}$ is analogous, and thus we obtain that $$W_p(E_n,F_n) \leq W_p(E_n,\rho)+W_p(\rho,\eta_{\rho})+W_p(\eta_{\rho},F_n) \leq \frac{2}{n}+W_p(\rho,\eta_{\rho}).$$ This, combined with the first part, shows that $$\mathcal{W}_p(\rho) = \inf_{E_n\overset{\ast}{\rightharpoonup}\rho,|E_n|=m} \liminf_n \mathcal{W}_p(E_n)\qquad \forall \rho\in\mathsf{A}.$$ *Step 2.* We remind that, thanks to Theorem [\[thm:existence-uniqueness-transport\]](#thm:existence-uniqueness-transport){reference-type="ref" reference="thm:existence-uniqueness-transport"}, there exists an optimal transport map for every transport problem that we consider in this paper. Up to taking a subsequence, we can suppose that $\lim_n\mathcal{W}_p(\rho_n)$ exists, and we prove that $\mathcal{W}_p(\rho) = \lim_n\mathcal{W}_p(\rho_n)$. Since we can extract one of such subsequence from any subsequence of $\{\rho_n\}_n$, this guarantees the existence of that limit for the whole sequence. We proceed by contradiction, and we suppose that there exists $\delta>0$ such that $\mathcal{W}_p(\rho)<\lim_n\mathcal{W}_p(\rho_n)-\delta$. The idea is to modify $\eta_{\rho}$ and produce a competitor to compute $\mathcal{W}_p(\rho_n)$, proving that we cannot have a strict inequality. To proceed with this plan we first truncate the densities to guarantee a convergence in Wasserstein distance. Up to taking another subsequence, we can suppose that $\eta_{\rho_n}\overset{\ast}{\rightharpoonup}\rho'$ for some $\rho'\in\mathsf{A}$ with $\rho+\rho'\leq1$ (using the same argument as in Step 1). Since the sequences $\{\rho_n\}_n$ and $\{\eta_{\rho_n}\}_n$ do not lose mass, for any $\varepsilon<1/2$ there exists $\bar n,k_1\in\mathbb N$ such that $$\label{eq:large-cube} \int_{\mathbb R^N\setminus Q_{3k_1}}(\rho_n+\eta_{\rho_n})\, dx<\varepsilon\qquad \forall n>\bar n.$$ We will choose $\varepsilon$ later on in order to make some approximations precise enough to obtain a contradiction out of the strict inequality. Now take $k_2 = \lceil 3/\varepsilon\rceil$, so that $k_2\varepsilon\in[3,3+\varepsilon]$, and we consider the cube $\bar Q = [-k_1k_2\varepsilon,k_1k_2\varepsilon]^N$. It is easy to see that we can partition $\mathbb R^N$ with a family $\mathcal{F} = \{Q^k\}_{k\in\mathbb N}$ of cubes with side length equal to $\varepsilon$ and such that $|Q^k\cap \bar Q|\in\{0,\varepsilon^N\}$ (i.e. $\mathcal{F}$ contains two disjoint subfamilies that partition $\bar Q$ and $\mathbb R^N\setminus\bar Q$). Moreover, it is also possible to find a partition of $\mathbb R^N\setminus\bar Q$ with a family $\tilde{\mathcal{F}}=\{\tilde Q^k\}_{k\in\mathbb N}$ of cubes with side length $k_2\varepsilon$. We will use the first partition to control the cost of an approximation of $\eta_{\rho}$ inside $\bar Q$, where we move mass at short distance. The second one, instead, will be used to estimate the energy carried by the mass outside of that cube (thanks to [\[eq:large-cube\]](#eq:large-cube){reference-type="eqref" reference="eq:large-cube"}, that mass is small). We call $T$ the optimal transport map between $\rho$ and $\eta_{\rho}$, and for any $n$ we define the truncated densities $\tilde\rho_n = \rho_n\hbox{{\large $\chi$}{\Large $_{_{\bar Q}}$}}$. For any $n$ we also take $L_n>0$ such that $\int_{Q_{L_n}}\rho \,dx = \int\tilde\rho_n\,dx$, and we define the densities $\zeta_n\coloneqq \rho\hbox{{\large $\chi$}{\Large $_{_{Q_{L_n}}}$}}$ and $\zeta'_n\coloneqq (T_{\rho})_{\#}\zeta_n$. Since $\rho_n\overset{\ast}{\rightharpoonup}\rho$, then $\tilde\rho_n\overset{\ast}{\rightharpoonup}\rho\hbox{{\large $\chi$}{\Large $_{_{\bar Q}}$}}$ and we can choose the sequence $\{L_n\}_n$ to be bounded. Moreover, we have that $\zeta_n\overset{\ast}{\rightharpoonup}\rho\hbox{{\large $\chi$}{\Large $_{_{\bar Q}}$}}$. Since the supports of the truncated densities are equibounded, then the $p$th-moment of $\zeta_n$ converges, as well as the $p$th-moment of $\tilde\rho_n$, and thus $W_p(\tilde\rho_n,\zeta_n)\to0$ (see e.g. [@V2003 Theorem 7.12]) We take $h^1_n$ any non-negative density such that $\rho_n+h_n^1\leq 1$ and for any $k\in\mathbb N$ $$\int_{Q^k}h^1_n \,dx = \min\left\{\int_{Q^k}\zeta'_n \,dx,\ \int_{Q^k}1-\rho_n\,dx\right\}.$$ Since $\zeta_n' = (T_{\rho})_{\#}\zeta_n$, we can apply Corollary [Corollary 8](#cor:bounded-transport-distance){reference-type="ref" reference="cor:bounded-transport-distance"} and see that ${\rm spt}h^1_n$ is contained in $Q_{L_n+C}$ for any $n$, where $C$ is a constant depending only on $N$. Since $\tilde\rho_n\overset{\ast}{\rightharpoonup}\rho\hbox{{\large $\chi$}{\Large $_{_{\bar Q}}$}}$ and $\zeta'_n\overset{\ast}{\rightharpoonup}(T_{\rho})_{\#}(\rho\hbox{{\large $\chi$}{\Large $_{_{\bar Q}}$}})$, then we have that $\left\lVert h^1_n\right\rVert_1-\left\lVert\zeta'_n\right\rVert_1\to0$ (notice that here only a finite number of cubes in $\mathcal{F}$ play an active role). We choose any non-negative density $h^2_n$ with ${\rm spt}h^2_n\subset 3\bar Q$ and such that $$\rho_n+h^1_n+h^2_n\leq 1\qquad \text{and}\qquad\left\lVert h^1_n+h^2_n\right\rVert_1 = \left\lVert\tilde\rho_n\right\rVert_1,$$ and our candidate to compute $\mathcal{W}_p(\tilde\rho_n)$ will be $\tilde\rho'_n\coloneqq h^1_n+h^2_n$. Observe that, by definition of $h^1_n$ and thanks to the properties of the pus-forward of measures, we have that $\left\lVert h^1_n\right\rVert_1\leq \left\lVert\zeta'_n\right\rVert_1 = \left\lVert\zeta_n\right\rVert_1 = \left\lVert\tilde\rho_n\right\rVert_1$. Thanks to the triangle inequality for the $p$-Wasserstein distance, we have that $$W_p(\tilde \rho_n,\tilde\rho'_n)\leq W_p(\tilde\rho_n,\zeta_n)+W_p(\zeta_n,\zeta_n')+W_p(\zeta_n',\tilde\rho'_n).$$ The first term on the right hand side is going to $0$ because, as we already noticed, the sets ${\rm spt}\tilde\rho_n$ and ${\rm spt}\zeta_n$ are uniformly bounded and these densities are converging to $\rho\hbox{{\large $\chi$}{\Large $_{_{\bar Q}}$}}$. Hence, up to taking $\bar n$ large enough, we can suppose that $W_p(\tilde\rho_n,\zeta_n)<\varepsilon$. Likewise, the last term is controlled by $\varepsilon$, and we use a plan similar to [\[eq:plan-lower-semicontinuity\]](#eq:plan-lower-semicontinuity){reference-type="eqref" reference="eq:plan-lower-semicontinuity"} to show this. We choose a density $\zeta''_n\leq \zeta'_n$ such that $$\int_{Q^k}\zeta''_n \,dx= \int_{Q^k}h^1_n \,dx \qquad \forall k\in\mathbb N,$$ and we consider the plan $$\tilde\pi_n = \sum_{k\in\mathbb N}\frac{1}{\left\lVert h^1_n\hbox{{\large $\chi$}{\Large $_{_{Q^k}}$}}\right\rVert_1}(\zeta''_n\hbox{{\large $\chi$}{\Large $_{_{Q^k}}$}}\mathscr{L}^N)\times(h^1_n\hbox{{\large $\chi$}{\Large $_{_{Q^k}}$}}\mathscr{L}^N)+\frac{1}{\left\lVert h^2_n\right\rVert_1}((\zeta'_n-\zeta''_n)\mathscr{L}^N)\times (h^2_n\mathscr{L}^N),$$ where the sum is intended to run only on the indices for which $h^1_n\hbox{{\large $\chi$}{\Large $_{_{Q^k}}$}}$ is not identically zero. Using $\tilde \pi_n$ as test plan to compute $W_p(\zeta'_n,\tilde\rho'_n)$ we obtain the following upper bound: $$W_p^p(\zeta'_n,\tilde\rho'_n)\leq \int |x-y|^p\,d\tilde\pi_n(x,y) \leq C\varepsilon^p+\mathop{\mathrm{diam}}({\rm spt}h^2_n+\zeta'_n)\left(\left\lVert\zeta'_n\right\rVert_1-\left\lVert h^1_n\right\rVert_1\right)\leq C\varepsilon^p,$$ where we used that the mass of $h^1_n$ remains inside the small cubes with side length $\varepsilon$, and the remaining mass is transported at finite distance in any case (the constant $C$ depends only on $N$ and $p$). The last inequality holds if we take $\bar n$, and thus $n$, large enough, and if we adjust the constant $C$. Adding up the various terms, we conclude that for any $n>\bar n$ there is an optimal transport plan $\pi_n$ for $\tilde\rho_n$ and $\tilde\rho'_n$ such that $$\begin{split} W_p(\tilde\rho_n,\tilde\rho'_n) = \left(\int|x-y|^p\,d\pi_n(x,y)\right)^{\frac{1}{p}}\leq W_p(\zeta_n,\zeta'_n)+C\varepsilon. \end{split}$$ To conclude, we observe that the cubes in $\tilde{\mathcal{F}}$ are so large that we can find a non-negative density $h^3_n$ such that $\rho_n+\tilde\rho'_n+h^3_n\leq 1$ and $$\int_{\tilde Q^k}h^3_n \,dx = \int_{\tilde Q^k}\rho_n\,dx \qquad \forall k\in\mathbb N.$$ Therefore, we consider the plan $\gamma_n$ associated to $\rho_n$ and $\tilde\rho'_n+h^3_n$ defined as $$\gamma_n = \pi_n + \sum_{k\in\mathbb N}\frac{1}{\left\lVert\rho_n\hbox{{\large $\chi$}{\Large $_{_{\tilde Q^k}}$}}\right\rVert_1}(\rho_n\hbox{{\large $\chi$}{\Large $_{_{\tilde Q^k}}$}}\mathscr{L}^N)\times(h^3_n\hbox{{\large $\chi$}{\Large $_{_{\tilde Q^k}}$}}\mathscr{L}^N),$$ again summing only on the cubes with non-trivial measure. This gives the following estimate for $W_p^p(\rho_n,\tilde\rho'_n+h^3_n)$: $$\begin{split} W_p^p(\rho_n,\tilde\rho'_n+h^3_n)&\leq \Big(W_p(\zeta_n,\zeta'_n)+C\varepsilon\Big)^p+C\left\lVert h^3_n\right\rVert_1\\ &\leq \Big(\mathcal{W}_p(\rho)+C\varepsilon\Big)^p+C\left\lVert h^3_n\right\rVert_1\\ &\leq \Big(\mathcal{W}_p(\rho_n)-\delta+C\varepsilon\Big)^p+C\varepsilon. \end{split}$$ Since $\delta>0$ is fixed and since the constant $C$ in that estimate depends only on $N$ and $p$, we can find $\varepsilon$ small enough so that $W_p^p(\rho_n,\tilde\rho'_n+h^3_n)<\mathcal{W}_p^p(\rho_n)$, and this is impossible since $\tilde\rho'_n+h^3_n$ is a competitor in the definition of $\mathcal{W}_p(\rho_n)$. ◻ The next lemma describes particular symmetries of the problem [\[eq:wasserstein-densities\]](#eq:wasserstein-densities){reference-type="eqref" reference="eq:wasserstein-densities"} which are crucial in proving properties of maximizers of $\mathcal{W}_p$ in the next section. **Lemma 11** (Symmetries of the transport problem). *Let $F\colon\mathbb R^N\to\mathbb R^N$ be an isometry and let $\rho\in\mathsf{A}$ be a given density such that $F_{\#}(\rho\mathscr{L}^N) = \rho\mathscr{L}^N$. Then the following hold:* 1. *$F_{\#}(\eta_{\rho}\mathscr{L}^N) = \eta_{\rho}\mathscr{L}^N$ and $\tilde F_{\#}\pi_{\rho} = \pi_{\rho}$, where $\tilde F$ is the map from $\mathbb R^N\times \mathbb R^N$ into itself defined as $\tilde F(x,y) = (F(x),F(y))$.* 2. *If $F$ is a reflection of the form $F(x) = x-2\langle x, \nu \rangle\nu$ for some $\nu\in\mathbb S^{N-1}$, then we have that $$\label{eq:split-hyperplane} \pi_{\rho}\left(\{(x,y)\colon\langle x, \nu \rangle\langle y, \nu \rangle<0\}\right) = 0.$$ In other words, $\pi_{\rho}$ does not transport mass from one side of the reflection hyperplane $\{x\colon\langle x, \nu \rangle=0\}$ to the other.* *Proof.* We recall that the optimal plan $\pi_{\rho}$ is unique (see Theorem [\[thm:existence-uniqueness-transport\]](#thm:existence-uniqueness-transport){reference-type="ref" reference="thm:existence-uniqueness-transport"}). Also, notice that $F_{\#}(\rho\mathscr{L}^N)$ and $F_{\#}(\eta_{\rho}\mathscr{L}^N)$ are absolutely continuous with respect to the Lebesgue measure, and we have that $F_{\#}(\rho\mathscr{L}^N) = (\rho\circ F)\mathscr{L}^N$ and $F_{\#}(\eta_{\rho}\mathscr{L}^N) = (\eta_{\rho}\circ F)\mathscr{L}^N$. Therefore, it is trivial to see that $F_{\#}(\rho\mathscr{L}^N)\in\mathsf{A}$, $F_{\#}(\eta_{\rho}\mathscr{L}^N)\in\mathsf{A}$ and $F_{\#}((\rho+\eta_{\rho})\mathscr{L}^N)\leq \mathscr{L}^N$. It is easy to see that $\tilde \pi_{\rho} = (\tilde F)_{\#}\pi_{\rho}$ is a transport plan associated to $F_{\#}(\rho\mathscr{L}^N)$ and $F_{\#}(\eta_{\rho}\mathscr{L}^N)$: by the properties of the push forward, we have that $(p_1\circ \tilde F)_{\#}\pi_{\rho} = (p_1)_{\#}(\tilde F_{\#}\pi_{\rho})$, and $p_1\circ \tilde F = F\circ p_1$, therefore $(p_1)_{\#}\tilde \pi_{\rho} = F_{\#}(\rho\mathscr{L}^N)$. An analogous property holds for the second projection $p_2$, and thus $\tilde\pi_{\rho}$ has the correct marginals. Then, we consider the plan $(\pi_{\rho}+\tilde \pi_{\rho})/2$, whose marginals are $\rho\mathscr{L}^N$ and $\frac{1}{2}(\eta_{\rho}+\eta_{\rho}\circ F)\mathscr{L}^N$, and we observe that $$\begin{split} \mathcal{W}_p^p(\rho) &\leq \frac{1}{2}\int |x-y|^p\,d\pi_{\rho}(x,y)+\frac{1}{2}\int|x-y|^p \,d\tilde F_{\#}\pi_{\rho}(x,y)\\ &= \frac{1}{2}\int |x-y|^p\,d\pi_{\rho}(x,y)+\frac{1}{2}\int|F(x)-F(y)|^p \,d\pi_{\rho}(x,y) = W_p^p(\rho,\eta_{\rho}), \end{split}$$ where we used that $F$ is an isometry to obtain the last identity. This implies that $\eta_{\rho}\circ F$ is also an optimal density to compute $\mathcal{W}_p(\rho)$. Since there exists a unique density which realizes $\mathcal{W}_p(\rho)$, then $\eta_{\rho}\mathscr{L}^N = F_{\#}(\eta_{\rho}\mathscr{L}^N)$ and $\tilde F_{\#}\pi_{\rho}=\pi_{\rho}$. In order to prove (ii), suppose that $F(x)=x-2\langle x, \nu \rangle\nu$ for some $\nu\in\mathbb S^{N-1}$. From the previous point we know that $\pi_{\rho}$ satisfies $\tilde F_{\#}\pi_{\rho} = \pi_{\rho}$. We want to prove that, whenever [\[eq:split-hyperplane\]](#eq:split-hyperplane){reference-type="eqref" reference="eq:split-hyperplane"} does not hold, we can find a better plan, contradicting the definition of $\pi_{\rho}$. In fact, we consider the plan $$\tilde \pi_{\rho} = \pi_{\rho}\mathop{\hbox{\vrule height 7pt width .5pt depth 0pt \vrule height .5pt width 6pt depth 0pt}}\nolimits(H_1\times H_1)+\pi_{\rho}\mathop{\hbox{\vrule height 7pt width .5pt depth 0pt \vrule height .5pt width 6pt depth 0pt}}\nolimits(H_2\times H_2)+(\mathrm{Id},F)_{\#}(\pi_{\rho}\mathop{\hbox{\vrule height 7pt width .5pt depth 0pt \vrule height .5pt width 6pt depth 0pt}}\nolimits(H_1\times H_2))+(\mathrm{Id},F)_{\#}(\pi_{\rho}\mathop{\hbox{\vrule height 7pt width .5pt depth 0pt \vrule height .5pt width 6pt depth 0pt}}\nolimits(H_2\times H_1)),$$ where $H_1=\{x\colon\langle x, \nu \rangle>0\}$ and $H_2=F(H_1)=\{x\colon\langle x, \nu \rangle<0\}$. We observe that, since $(p_1)_{\#}\pi_{\rho}$ and $(p_2)_{\#}\pi_{\rho}$ are absolutely continuous with respect to Lebesgue measure, then $\pi_{\rho}$ does not give mass to $\partial(H_i\times H_j)$ for any $i,j\in\{1,2\}$. Therefore, $\tilde \pi_{\rho}$ is a probability measure, and the well-known properties of the push-forward operation guarantee that $(p_1)_{\#}\tilde\pi_{\rho} = \rho\mathscr{L}^N$. Since $\pi_{\rho}=\tilde F_{\#}\pi_{\rho}$ and $\tilde F(H_1\times H_2) = H_2\times H_1$, then $\pi_{\rho}\mathop{\hbox{\vrule height 7pt width .5pt depth 0pt \vrule height .5pt width 6pt depth 0pt}}\nolimits(H_1\times H_2) = \tilde F_{\#}(\pi_{\rho}\mathop{\hbox{\vrule height 7pt width .5pt depth 0pt \vrule height .5pt width 6pt depth 0pt}}\nolimits(H_2\times H_1))$. With this observation we arrive to $$\begin{aligned} \left((p_2)_{\#}\tilde\pi_{\rho}\right)\mathop{\hbox{\vrule height 7pt width .5pt depth 0pt \vrule height .5pt width 6pt depth 0pt}}\nolimits H_1 &= (p_2)_{\#}\left(\pi_{\rho}\mathop{\hbox{\vrule height 7pt width .5pt depth 0pt \vrule height .5pt width 6pt depth 0pt}}\nolimits(H_1\times H_1)\right)+(p_2)_{\#}\left((\mathrm{Id},F)_{\#}(\pi_{\rho}\mathop{\hbox{\vrule height 7pt width .5pt depth 0pt \vrule height .5pt width 6pt depth 0pt}}\nolimits(H_1\times H_2))\right)\\ &=(p_2)_{\#}\left(\pi_{\rho}\mathop{\hbox{\vrule height 7pt width .5pt depth 0pt \vrule height .5pt width 6pt depth 0pt}}\nolimits(H_1\times H_1)\right)+(p_2)_{\#}\left((\mathrm{Id},F)_{\#}\tilde F_{\#}(\pi_{\rho}\mathop{\hbox{\vrule height 7pt width .5pt depth 0pt \vrule height .5pt width 6pt depth 0pt}}\nolimits(H_2\times H_1))\right)\\ &=(p_2)_{\#}\left(\pi_{\rho}\mathop{\hbox{\vrule height 7pt width .5pt depth 0pt \vrule height .5pt width 6pt depth 0pt}}\nolimits(H_1\times H_1)\right)+(p_2)_{\#}\left((F,\mathrm{Id})_{\#}(\pi_{\rho}\mathop{\hbox{\vrule height 7pt width .5pt depth 0pt \vrule height .5pt width 6pt depth 0pt}}\nolimits(H_2\times H_1))\right)\\ &=(p_2)_{\#}\left(\pi_{\rho}\mathop{\hbox{\vrule height 7pt width .5pt depth 0pt \vrule height .5pt width 6pt depth 0pt}}\nolimits(H_1\times H_1)\right)+(p_2)_{\#}\left(\pi_{\rho}\mathop{\hbox{\vrule height 7pt width .5pt depth 0pt \vrule height .5pt width 6pt depth 0pt}}\nolimits(H_2\times H_1)\right)\\ &=(p_2)_{\#}(\pi_{\rho}\mathop{\hbox{\vrule height 7pt width .5pt depth 0pt \vrule height .5pt width 6pt depth 0pt}}\nolimits(\mathbb R^N\times H_1)) = ((p_2)_{\#}\pi_{\rho})\mathop{\hbox{\vrule height 7pt width .5pt depth 0pt \vrule height .5pt width 6pt depth 0pt}}\nolimits H_1, \end{aligned}$$ where we used that $(\mathrm{Id},F)\circ \tilde F = (F,\mathrm{Id})$ and the fact that $F$ is an isometry to pass from the second to the third line. Arguing in the same way, one can also see that $((p_2)_{\#}\tilde \pi_{\rho})\mathop{\hbox{\vrule height 7pt width .5pt depth 0pt \vrule height .5pt width 6pt depth 0pt}}\nolimits H_2 = ((p_2)_{\#}\pi_{\rho})\mathop{\hbox{\vrule height 7pt width .5pt depth 0pt \vrule height .5pt width 6pt depth 0pt}}\nolimits H_2$. This is sufficient to say that $\rho\mathscr{L}^N+(p_2)_{\#}(\tilde \pi_{\rho}\mathscr{L}^N)\leq \mathscr{L}^N$, and thus $\tilde \pi_{\rho}\in\mathsf{AP}_{\rho}$. Now we can compare the costs associated to $\tilde \pi_{\rho}$ and $\pi_{\rho}$. Discarding the common terms, we get that $$\label{eq:reflection-inequality} \int |x-y|^p \, d(\tilde \pi_{\rho}-\pi_{\rho}) = \int_{(H_1\times H_2)\cup (H_2\times H_1)} \left(|x-F(y)|^p-|x-y|^p\right)\,d\pi_{\rho}(x,y),$$ and a simple geometric argument shows that the function inside the integral is strictly negative. Therefore, if the domain appearing in the right hand side of [\[eq:reflection-inequality\]](#eq:reflection-inequality){reference-type="eqref" reference="eq:reflection-inequality"} has positive $\pi_{\rho}$ measure, then $\tilde \pi_{\rho}$ is a strictly better competitor to compute $\mathcal{W}_p(\rho)$, in contradiction with the definition of $\pi_{\rho}$. To conclude, we observe that we have just proved that $\pi_{\rho}((H_1\times H_2)\cup (H_2\times H_1))=0$, and this is equivalent to [\[eq:split-hyperplane\]](#eq:split-hyperplane){reference-type="eqref" reference="eq:split-hyperplane"}. ◻ # Maximizer of $\mathcal{W}_p$ {#sec:maximizer} ## Existence of maximizers In this section we first prove the existence of maximizers of the energies [\[eq:wasserstein-densities\]](#eq:wasserstein-densities){reference-type="eqref" reference="eq:wasserstein-densities"} in $\mathsf{A}$ by applying the concentration compactness principle to a maximizing sequence of densities, where we consider them as measures. Even though we consider a maximization problem, our strategy works since $\mathcal{W}_p$ is continuous with respect to the weak$*$ convergence, as shown in Lemma [Lemma 10](#lem:weak-continuity-wasserstein){reference-type="ref" reference="lem:weak-continuity-wasserstein"}. Here we state concentration compactness lemma for measures for the convenience of the reader. **Lemma 12** (Concentration compactness, [@S]). *Let $\mu_n\in\mathcal P(\mathbb R^N)$ be a given sequence of probability measures. Then there exists a subsequence (not relabelled) such that one of the following holds:* 1. **(Compactness)* There exists a sequence of points $x_n\in\mathbb R^N$ such that, for every $\varepsilon>0$, there exists $L>0$ large enough such that $\mu_n(Q_L(x_n))>1-\varepsilon$.* 2. **(Vanishing)* For every $\varepsilon>0$ and every $L>0$ there exists $\bar n\in\mathbb N$ such that $$\mu_n(Q_L(x))<\varepsilon\qquad \forall x\in\mathbb R^N, \forall n>\bar n.$$* 3. **(Dichotomy)* There exist $\lambda \in(0,1)$ and a sequence of points $x_n\in\mathbb R^N$ with the following property: for any $\varepsilon>0$, there exists $L>0$ such that, for any $L'>L$ there exist two non-negative measures $\mu^1_n$ and $\mu^2_n$ that satisfy, for every $n$ large enough, the following conditions $$\begin{split} &\mu^1_n+\mu^2_n\leq \mu_n,\\ &{\rm spt}\mu^1_n\subset Q_L(x_n),\quad {\rm spt}\mu^2_n\subset \mathbb R^N\setminus Q_{L'}(x_n),\\ &\left|\mu^1_n(\mathbb R^N)-\lambda\right|+\left|\mu^2_n(\mathbb R^N)-(1-\lambda)\right|<\varepsilon. \end{split}$$* **Theorem 13**. *Let $p>1$ be fixed. Then there exists a maximizer of $\mathcal{W}_p$ in $\mathsf{A}$.* *Proof.* Let us consider a maximizing sequence $\rho_n\in\mathsf{A}$ with $\mathcal{W}_p(\rho_n)\to \sup_{\rho\in\mathsf{A}}\mathcal{W}_p(\rho)$. Notice that, thanks to Corollary [Corollary 8](#cor:bounded-transport-distance){reference-type="ref" reference="cor:bounded-transport-distance"}, we have that $\sup_{\rho\in\mathsf{A}}\mathcal{W}_p(\rho)\leq C<+\infty$ for some constant $C=C(p,N)$. We are going to apply the concentration compactness lemma to $\mu_n=\rho_n\mathscr{L}^N$, and show that the vanishing and dichotomy phenomena do not happen. Then exploiting the invariance of the energy under translations and Lemma [Lemma 10](#lem:weak-continuity-wasserstein){reference-type="ref" reference="lem:weak-continuity-wasserstein"} we establish the existence of a maximizer. We first exclude the vanishing case. Up to translations, we can suppose that the points $x_n$ appearing in Lemma [Lemma 12](#lemma:concentration-compactness){reference-type="ref" reference="lemma:concentration-compactness"} all coincide with the origin. Suppose by contradiction that, for any $\varepsilon>0$ and any $L>0$ we can find $\bar n\in\mathbb N$ such that $\mu_n(Q_L(x))<(\varepsilon/3)^N$ for every $x\in\mathbb R^N$. Then, we fix a partition $\mathcal{F}=\{Q^k\}_{k\in\mathbb N}$ of $\mathbb R^N$ made of cubes with side length $\varepsilon$. Since by hypothesis $\mu_n(Q^k)<|Q^k|/3$ for every $n>\bar n$ and every $k\in\mathbb N$, then for every $n>\bar n$ there exists $\rho'_n\in\mathsf{A}$ such that $\rho_n+\rho'_n\leq 1$ and $$\int_{Q^k}\rho_n \,dx = \int_{Q^k}\rho'_n\,dx \qquad \forall k\in\mathbb N.$$ Using a transport plan similar to $\pi_n$ defined in [\[eq:plan-lower-semicontinuity\]](#eq:plan-lower-semicontinuity){reference-type="eqref" reference="eq:plan-lower-semicontinuity"}, it is immediate to see that $$\mathcal{W}_p^p(\rho_n) \leq W_p^p(\rho_n,\rho'_n) \leq \mathop{\mathrm{diam}}(Q^k)^p = C_{N,p}\varepsilon^p.$$ If we take $\varepsilon$ sufficiently small, we clearly have that $\rho_n$ is not a maximizing sequence for $\mathcal{W}_p$, arriving to a contradiction. Now we treat the dichotomy case. Suppose for a contradiction that there exists $\lambda\in(0,1)$ such that, for any $\varepsilon>0$ there exist $\bar n\in\mathbb N$, $L>0$ and two sequences of non-negative densities $\rho^1_n$, $\rho^2_n$ that satisfy $$\label{eq:dichotomy-densities} \begin{split} &\rho^1_n+\rho^2_n\leq \rho_n\\ &{\rm spt}\rho^1_n\subset Q_L\quad {\rm spt}\rho^2_n\subset \mathbb R^N\setminus Q_{L+3C_N},\\ &\left|\int\rho^1_n\,dx -\lambda\right|+\left|\int\rho^2_n\,dx-(1-\lambda)\right|<\varepsilon, \end{split}$$ where $C_N$ is the constant appearing in [\[eq:bounded-transport-distance\]](#eq:bounded-transport-distance){reference-type="eqref" reference="eq:bounded-transport-distance"}. Since the distance between ${\rm spt}\rho^1_n$ and ${\rm spt}\rho^2_n$ is larger than $3C_N$, then applying Corollary [Corollary 8](#cor:bounded-transport-distance){reference-type="ref" reference="cor:bounded-transport-distance"} we obtain that $\mathcal{W}_p^p(\rho^1_n+\rho^2_n) = \mathcal{W}_p^p(\rho^1_n)+\mathcal{W}_p^p(\rho^2_n)$. Combining the first and the third conditions in [\[eq:dichotomy-densities\]](#eq:dichotomy-densities){reference-type="eqref" reference="eq:dichotomy-densities"}, we get that $\left\lVert\rho_n-\rho^1_n-\rho^2_n\right\rVert_1<\varepsilon$, and we define $m^1_n = \left\lVert\rho^1_n\right\rVert_1$ and $m^2_n = \left\lVert\rho^2_n\right\rVert_1$. Using this fact, and that $\rho^1_n+\rho^2_n+\eta_{\rho^1_n+\rho^2_n}\leq 1$, we deduce that $$\label{eq:almost-compatible} \int (\eta_{\rho^1_n+\rho^2_n}-(1-\rho_n))_+\,dx\leq \varepsilon.$$ We denote by $T_n$ the optimal transport map to compute $\mathcal{W}_p(\rho^1_n+\rho^2_n)$, and we define $$\zeta_n = \min\{\eta_{\rho^1_n+\rho^2_n},1-\rho_n\},\qquad \tilde\rho_n = (T_n^{-1})_{\#}\zeta_n,$$ so that $\tilde \rho_n$ is an approximation of $\rho^1_n+\rho^2_n$, and it is smaller than that sum. We let $\mathcal{F}=\{Q^k\}_{k\in\mathbb N}$ be a partition of $\mathbb R^N$ made of cubes with side length equal to $3$, and we can find, as we did before, a density $\zeta'_n$ such that $\rho_n+\zeta_n+\zeta'_n\leq 1$ and $$\int_{Q^k}\zeta'_n \,dx= \int_{Q^k}\rho_n-\tilde \rho_n \,dx\qquad \forall k\in\mathbb N.$$ Therefore, we estimate the energy of $\rho_n$ with the plan $$\tilde \pi_n = (\mathrm{Id},T_n)_{\#}\tilde \rho_n + \sum_{k\in\mathbb N} \frac{1}{\left\lVert\zeta'_n\hbox{{\large $\chi$}{\Large $_{_{Q^k}}$}}\right\rVert_1}((\rho_n-\tilde\rho_n)\hbox{{\large $\chi$}{\Large $_{_{Q^k}}$}}\mathscr{L}^N)\times (\zeta'_n\hbox{{\large $\chi$}{\Large $_{_{Q^k}}$}}\mathscr{L}^N).$$ In fact, combining [\[eq:almost-compatible\]](#eq:almost-compatible){reference-type="eqref" reference="eq:almost-compatible"} and the fact that $\left\lVert\rho_n-\rho^1_n-\rho^2_n\right\rVert_1\leq \varepsilon$, we have that $\left\lVert\rho_n-\tilde\rho_n\right\rVert_1\leq 2\varepsilon$, and thus $$\label{eq:dichotomy-estimate} \begin{split} \mathcal{W}_p^p(\rho_n)&\leq\int|x-y|^p\,d\tilde\pi_n \leq \int |T_n(x)-x|^p\tilde\rho_n(x)\,dx+2(\mathop{\mathrm{diam}}Q^k)^p \varepsilon\\ &\leq \mathcal{W}_p^p(\rho^1_n+\rho^2_n)+C_{N,p}\varepsilon\\ &=\mathcal{W}_p^p(\rho^1_n)+\mathcal{W}_p^p(\rho^2_n)+C_{N,p}\varepsilon\\ &\leq \sup\left\{\mathcal{W}_p^p(\rho)\colon\rho\in \mathsf{A}_{m^1_n}\right\}+\sup\left\{\mathcal{W}_p^p(\rho)\colon\rho\in \mathsf{A}_{m^2_n}\right\}+C_{N,p}\varepsilon. \end{split}$$ Using the rescaling exploited in Remark [Remark 9](#rem:scaling-wasserstein){reference-type="ref" reference="rem:scaling-wasserstein"} we see that $$\sup\left\{\mathcal{W}_p^p(\rho)\colon\rho\in\mathsf{A}_m\right\} = m^{1+\frac{p}{N}}\sup\left\{\mathcal{W}_p^p(\rho)\colon\rho\in\mathsf{A}\right\};$$ hence, [\[eq:dichotomy-estimate\]](#eq:dichotomy-estimate){reference-type="eqref" reference="eq:dichotomy-estimate"} implies that $$\mathcal{W}_p^p(\rho_n)\leq C_{N,p}\varepsilon+\left((m^1_n)^{1+\frac{p}{N}}+(m^2_n)^{1+\frac{p}{N}}\right)\sup\left\{\mathcal{W}_p^p(\rho)\colon\rho\in\mathsf{A}\right\}.$$ If $\varepsilon$ is small enough, this is incompatible with the fact that $\lim_n\mathcal{W}_p(\rho_n) = \sup_{\rho\in\mathsf{A}}\mathcal{W}_p(\rho)$. In fact, the function $t\mapsto t^{1+\frac{p}{N}}$ is strictly convex, and if $\varepsilon<\frac12\min\{\lambda,1-\lambda\}$, then $m^1_n$ and $m^2_n$ are far away from $0$. ◻ ## The only maximizer is the ball In the second part of this section we will characterize the maximizers of $\mathcal{W}_p$ over $\mathsf{A}$. In fact, we prove that the only maximizer of $\mathcal{W}_p$ is the characteristic function of a ball (with the correct volume). The intuition behind this result is that, if we have a set, and we create some holes in it (adding some mass somewhere else), we are lowering the energy since the additional mass can be transported at shorter distance. We obtain the main result in several steps: First we study the $1$-dimensional case, possibly with a weight, where the structure of the transport plan is known explicitly. Then, using a symmetrization argument we show that the optimal plan associated to a maximizer has some geometric properties, and, in fact, it is radial. Next, using the $1$-dimensional case, we prove that a maximizer has to be a star-shaped set, and via an optimality argument we deduce that a star-shaped maximizer must actually be a ball. **Proposition 14**. *Let $m>0$ be a given parameter. Let $w\colon(0,+\infty)\to(0,+\infty)$ be a non-decreasing weight and let $I=[0,\ell]$ be the unique segment such that $\int_I w\,dx=m$. For any density $\rho\colon\mathbb R^+\to [0,1]$ with $\int_{\mathbb R^+} \rho w\,dx = m$, we have that $$\label{eq:1D-ineq} \mathcal{W}_p(\hbox{{\large $\chi$}{\Large $_{_{I}}$}})\geq \mathcal{W}_p(\rho),$$ where $\mathcal{W}_p$ is defined in the metric-measure setting with base space $\mathbb R^+$ endowed with the usual distance and reference measure equal to $w\mathscr{L}^1$. Moreover, the equality holds if and only if $\rho = \hbox{{\large $\chi$}{\Large $_{_{I}}$}}$ almost everywhere.* *Proof.* We note that, also in this weighted case, the transport distance is bounded (using again Lemma [Lemma 7](#lem:full-ball){reference-type="ref" reference="lem:full-ball"}), and thus for any density the infimum in the definition of $\mathcal{W}_p$ is achieved thanks to Theorem [\[thm:existence-uniqueness-transport\]](#thm:existence-uniqueness-transport){reference-type="ref" reference="thm:existence-uniqueness-transport"} and Theorem [\[thm:existence-W_p\]](#thm:existence-W_p){reference-type="ref" reference="thm:existence-W_p"}. Therefore, there exists $\eta_{\rho}$ such that $\mathcal{W}_p(\rho) = W_p(\rho\gamma,\eta_{\rho}\gamma)$, where we use the notation $\gamma=w\mathscr{L}^1$. Moreover, since we have an increasing cost, we actually have that $\mathcal{W}_p(\hbox{{\large $\chi$}{\Large $_{_{I}}$}}) = W_p(\hbox{{\large $\chi$}{\Large $_{_{I}}$}}\gamma,\hbox{{\large $\chi$}{\Large $_{_{I'}}$}}\gamma)$, where $I' = [\ell,\ell']$ for some $\ell'>\ell$, and the transport plan is induced by a monotone map $T$ (see Theorem [\[thm:transport-1D\]](#thm:transport-1D){reference-type="ref" reference="thm:transport-1D"}). Now we introduce an auxiliary problem that produces a non-optimal candidate to estimate $\mathcal{W}_p(\rho)$. The advantage of this modified problem is that it enforces a "geometric" constraint that clarifies some arguments. The auxiliary functional, which considers only plans which move mass "forward", is given by $$\mathcal{AW}_p^p(\rho)\coloneqq \inf\left\{\int |x-y|^p\,d\pi(x,y)\colon \pi\in\mathsf{AP}_{\rho}, \ \pi\big(\left\{(x,y)\colon y<x\right\}\big) = 0\right\}.$$ We observe that the infimum is actually a minimum since the additional constraint is closed under weak$*$ convergence. Moreover, applying the standard results for the one dimensional transport problem, we know that the optimal plan is induced by a non-decreasing map. Since we have already observed that $\mathcal{W}_p(\hbox{{\large $\chi$}{\Large $_{_{I}}$}}) = W_p(\hbox{{\large $\chi$}{\Large $_{_{I}}$}}\gamma,\hbox{{\large $\chi$}{\Large $_{_{I'}}$}}\gamma)$, then the monotonicity of the optimal map ensures that $\mathcal{W}_p(\hbox{{\large $\chi$}{\Large $_{_{I}}$}}) = \mathcal{AW}_p(\hbox{{\large $\chi$}{\Large $_{_{I}}$}})$. For a general density $\rho$, instead, we have just the inequality $\mathcal{AW}_p(\rho)\geq \mathcal{W}_p(\rho)$ due to the introduction of the additional constraint. With these observations, we reduce to proving the following (stronger) inequality: $$\mathcal{W}_p(\hbox{{\large $\chi$}{\Large $_{_{I}}$}})\geq \mathcal{AW}_p(\rho),$$ and [\[eq:1D-ineq\]](#eq:1D-ineq){reference-type="eqref" reference="eq:1D-ineq"} simply follows. From now on we denote by $\tilde T_{\rho}$ the transport map appearing when we compute $\mathcal{AW}_p(\rho)$. We define the following "volume" functions with domain $\mathbb R^+$: $$V(x) \coloneqq \int_0^xw(t)\,dt, \qquad V_{\rho}(x) \coloneqq \int_0^x\rho(t)w(t)\,dt.$$ We also denote by $d(v)$ (resp. $d_{\rho}(v)$) the transport distance of the point $V^{-1}(v)$ (resp. $V_{\rho}^{-1}(v)$) when we compute $\mathcal{W}_p(\hbox{{\large $\chi$}{\Large $_{_{I}}$}})$ (resp. $\mathcal{W}_p(\rho)$), i.e. $$d(v)\coloneqq |T(V^{-1}(v))-V^{-1}(v)|,\qquad d_{\rho}(v)\coloneqq |\tilde T_{\rho}(V_{\rho}^{-1}(v))-V_{\rho}^{-1}(v)|.$$ Using the explicit expression of the optimal transport map in $1$D (see for example [@V2003 Remarks 2.19 (iv)]), we have that $$\gamma([V^{-1}(v),V^{-1}(v)+d(v)]) = m\qquad \forall v\in[0,m].$$ One can easily adapt the proof of Lemma [Lemma 7](#lem:full-ball){reference-type="ref" reference="lem:full-ball"} to the auxiliary functional and see that, if $x$ is a Lebesgue point for $\tilde T_{\rho}$ and $r=|\tilde T_{\rho}(x)-x|$, then $(\tilde T_{\rho})_{\#}(\rho\gamma) = (1-\rho)\gamma$ in $[x,x+r]$. Moreover, since $\tilde T_{\rho}$ is non-decreasing, we also have that $$\label{eq:constraint-saturation} (\tilde T_{\rho})_{\#}\left(\rho\gamma\mathop{\hbox{\vrule height 7pt width .5pt depth 0pt \vrule height .5pt width 6pt depth 0pt}}\nolimits[0,x]\right) = (1-\rho)\gamma\qquad \text{in }[x,x+r].$$ We claim that $d_{\rho}\leq d$. In fact, suppose for a contradiction that there exists $v\in(0,m)$ such that $d_{\rho}(v)>d(v)$. Since $\rho\leq1$, then $V^{-1}\leq V_{\rho}^{-1}$, and thus $$\begin{split} \int_0^{V_{\rho}^{-1}(v)+d_{\rho}(v)}\rho(t)w(t)\,dt &\geq \int_{V_{\rho}^{-1}(v)}^{V_{\rho}^{-1}(v)+d_{\rho}(v)}w(t)\,dt \geq \int_{V^{-1}(v)}^{V^{-1}(v)+d_{\rho}(v)}w(t)\,dt\\ & > \int_{V^{-1}(v)}^{V^{-1}(v)+d(v)}w(t)\,dt = \gamma([V^{-1}(v),V^{-1}(v)+d(v)]) = m, \end{split}$$ where we used [\[eq:constraint-saturation\]](#eq:constraint-saturation){reference-type="eqref" reference="eq:constraint-saturation"} applied to $x=V_{\rho}^{-1}(v)$ and $r=d_{\rho}(v)$ to get the first inequality, and the monotonicity of $w$ to obtain the second one. This chain of inequalities of course leads to a contradiction since $m=\int \rho \,d\gamma$. Therefore $d_{\rho}\leq d$. Since $w$ and $\rho w$ are locally bounded in $[0,+\infty)$, then both $V$ and $V_{\rho}$ are locally Lipschitz, and we can apply the fundamental theorem of calculus: using $v=V_{\rho}(x)$ as variable in the computation of $\mathcal{AW}_p(\rho)$ we obtain that $$\begin{split} \mathcal{AW}_p^p(\rho) = \int_{\mathbb R^+} |\tilde T_{\rho}(x)-x|^p \rho(x)w(x)\,dx = \int_0^m d_{\rho}(v)^p\,dv\leq \int_0^md(v)^p\,dv = \mathcal{W}_p^p(\hbox{{\large $\chi$}{\Large $_{_{I}}$}}), \end{split}$$ where the inequality follows from comparison between $d$ and $d_{\rho}$, and this is the desired inequality. Finally, one can notice that the only way to obtain an equality in the previous chain of inequalities is that $\rho= \hbox{{\large $\chi$}{\Large $_{_{I''}}$}}$ for some segment $I''$ and $w$ is constant in ${\rm spt}(\rho+T_{\#}\rho)$. However, if $I''\neq I$, then one can construct a better transport plan moving some mass to the left (this plan should belong to $\mathsf{AP}_{\rho}$, but it is not admissible for the auxiliary problem). Therefore, the equality in [\[eq:1D-ineq\]](#eq:1D-ineq){reference-type="eqref" reference="eq:1D-ineq"} holds only for $\rho=\hbox{{\large $\chi$}{\Large $_{_{I}}$}}$. ◻ **Lemma 15**. *Let $p>1$ be given, and let $\rho\in\mathsf{A}$ be a maximizer of $\mathcal{W}_p$. If $\nu\in\mathbb S^{N-1}$ is such that $$\label{eq:half-direction} \int_{\{x\colon\langle x, \nu \rangle>0\}}\rho \,dx = \int_{\{x\colon\langle x, \nu \rangle<0\}}\rho \,dx = \frac12,$$ then the optimal plan $\pi_{\rho}$ satisfies $$\label{eq:splitting-plan} \pi_{\rho}(\{(x,y)\colon\langle x, \nu \rangle\cdot\langle y, \nu \rangle<0\})=0.$$* *Proof.* The idea is to consider an auxiliary functional, as in the proof of Proposition [Proposition 14](#prop:maximizer-1D){reference-type="ref" reference="prop:maximizer-1D"}, and show that it coincides with $\mathcal{W}_p$ when evaluated at $\rho$ (due to the maximality of this density). This ensures that $\pi_{\rho}$ has some additional structure due to the uniqueness of the optimal plan. We define the auxiliary functional $$\mathcal{AW}_{p}^p(\rho,\nu)\coloneqq \inf\left\{\int |x-y|^p\,d\pi(x,y)\colon\pi\in\mathsf{AP}_{\rho}, \pi(\{(x,y)\colon\langle x, \nu \rangle\cdot\langle y, \nu \rangle<0\})=0\right\}.$$ Loosely speaking, this auxiliary functional uses only plans that do not transport mass across the hyperplane $\{x\colon\langle x, \nu \rangle=0\}$. As before, we are introducing an additional constraint that is closed under weak$*$ convergence, and thus there exists an optimal plan in the definition of $\mathcal{AW}_p(\rho,\nu)$. Clearly, since we are introducing a constraint in the minimization process, we have that $\mathcal{W}_p(\rho)\leq \mathcal{AW}_p(\rho,\nu)$. Let $F(x)= x-2\langle x, \nu \rangle\nu$ be the reflection map, and define the two symmetrizations of $\rho$ with respect to $\nu$: $$\rho_1 = \rho\mathop{\hbox{\vrule height 7pt width .5pt depth 0pt \vrule height .5pt width 6pt depth 0pt}}\nolimits H_1+F_{\#}(\rho\mathop{\hbox{\vrule height 7pt width .5pt depth 0pt \vrule height .5pt width 6pt depth 0pt}}\nolimits H_1), \qquad \rho_2 = \rho\mathop{\hbox{\vrule height 7pt width .5pt depth 0pt \vrule height .5pt width 6pt depth 0pt}}\nolimits H_2+F_{\#}(\rho\mathop{\hbox{\vrule height 7pt width .5pt depth 0pt \vrule height .5pt width 6pt depth 0pt}}\nolimits H_2),$$ where $H_1 = \{x\colon\langle x, \nu \rangle>0\}$ and $H_2 = F(H_1) = \{x\colon\langle x, \nu \rangle<0\}$. We denote by $\bar \pi_{1}$ and $\bar \pi_{2}$ the two optimal plans realizing $\mathcal{AW}_p(\rho_1,\nu)$ and $\mathcal{AW}_p(\rho_2,\nu)$, respectively. We claim that $$\bar \pi = \bar \pi_1\mathop{\hbox{\vrule height 7pt width .5pt depth 0pt \vrule height .5pt width 6pt depth 0pt}}\nolimits(H_1\times H_1)+ \bar \pi_2\mathop{\hbox{\vrule height 7pt width .5pt depth 0pt \vrule height .5pt width 6pt depth 0pt}}\nolimits(H_2\times H_2)$$ realizes $\mathcal{AW}_p(\rho,\nu)$. In fact, $\bar \pi$ is admissible to compute $\mathcal{AW}_p(\rho,\nu)$, and if we find a better candidate $\pi$ to compute $\mathcal{AW}_p(\rho,\nu)$, then we can also construct the following plans that are good candidates to compute $\mathcal{AW}_p(\rho_1,\nu)$ and $\mathcal{AW}_p(\rho_2,\nu)$ respectively: $$\pi_1 = \pi\mathop{\hbox{\vrule height 7pt width .5pt depth 0pt \vrule height .5pt width 6pt depth 0pt}}\nolimits(H_1\times H_1)+\tilde F_{\#}(\pi\mathop{\hbox{\vrule height 7pt width .5pt depth 0pt \vrule height .5pt width 6pt depth 0pt}}\nolimits(H_1\times H_1)),\qquad \pi_2 = \pi\mathop{\hbox{\vrule height 7pt width .5pt depth 0pt \vrule height .5pt width 6pt depth 0pt}}\nolimits(H_2\times H_2)+\tilde F_{\#}(\pi\mathop{\hbox{\vrule height 7pt width .5pt depth 0pt \vrule height .5pt width 6pt depth 0pt}}\nolimits(H_2\times H_2)),$$ where $\tilde F(x,y) = (F(x),F(y))$. Then we observe that $$\begin{gathered} \mathcal{AW}_p^p(\rho_1,\nu) = \int |x-y|^p\,d\bar\pi_1 = 2 \int_{H_1\times H_1} |x-y|^p\,d\bar\pi_1,\\ \mathcal{AW}_p^p(\rho_2,\nu) = \int |x-y|^p\,d\bar\pi_2 = 2 \int_{H_2\times H_2} |x-y|^p\,d\bar\pi_2,\\ \int |x-y|^p\,d\bar \pi = \frac12\left(\mathcal{AW}_p^p(\rho_1,\nu)+\mathcal{AW}_p^p(\rho_2,\nu)\right),\\ \int |x-y|^p\,d \pi = \frac12\left(\int |x-y|^p\,d \pi_1+\int |x-y|^p\,d \pi_2\right). \end{gathered}$$ If $\mathcal{AW}_p^p(\rho,\nu)<\int|x-y|^p\,d\bar\pi$, then at least one between $\pi_1$ and $\pi_2$ is a better competitor for $\mathcal{AW}_p(\rho_1,\nu)$ or $\mathcal{AW}_p(\rho_2,\nu)$, contradicting the definition of $\bar \pi_1$ and $\bar \pi_2$. Therefore, the following conditions hold: $$\label{eq:symmetrization-maximizer} \mathcal{W}_p^p(\rho)\leq \mathcal{AW}_p^p(\rho,\nu) = \frac12\left(\mathcal{AW}_p^p(\rho_1,\nu)+\mathcal{AW}_p^p(\rho_2,\nu)\right) = \frac12\left(\mathcal{W}_p^p(\rho_1)+\mathcal{W}_p^p(\rho_2)\right),$$ where we used the second part of Lemma [Lemma 11](#lemma:symmetry-transport){reference-type="ref" reference="lemma:symmetry-transport"} to obtain the last equality. Since $\rho$ is a maximizer, then [\[eq:symmetrization-maximizer\]](#eq:symmetrization-maximizer){reference-type="eqref" reference="eq:symmetrization-maximizer"} guarantees that $\rho_1$ and $\rho_2$ are also maximizers. This, however, implies that $\mathcal{W}_p(\rho) = \mathcal{AW}_p(\rho,\nu)$. In other words, $\bar \pi$ realizes $\mathcal{W}_p(\rho)$ and satisfies [\[eq:splitting-plan\]](#eq:splitting-plan){reference-type="eqref" reference="eq:splitting-plan"}. Therefore, necessarily, we have that $\pi_{\rho}=\bar \pi$, concluding the proof. ◻ **Corollary 16**. *Let $p>1$ be given, and let $\rho\in\mathsf{A}$ be a maximizer of $\mathcal{W}_p$. Then there exists $x_0\in\mathbb R^N$ such that $\pi_{\rho}$ has the following property: $$\label{eq:radial-plan} \pi_{\rho}\left(\{(x,y)\colon \langle y-x_0, x-x_0 \rangle\neq |y-x_0||x-x_0|\}\right) = 0.$$ That is, $\pi_{\rho}$ is radial with center $x_0$.* *Proof.* By sliding each hyperplane $\{x\colon\langle x, e_i \rangle=0\}$ until it splits the mass of $\rho$ in half, and by taking the intersection of the $N$ hyperplanes, we find a point $x_0\in\mathbb R^N$ such that $$\int_{\{x\colon\langle x-x_0, e_i \rangle>0\}}\rho \,dx = \int_{\{x\colon\langle x-x_0, e_i \rangle<0\}}\rho \,dx = \frac12\qquad \forall i\in\{1,\ldots,N\}.$$ Up to translations, we suppose that $x_0=0$. By [\[eq:symmetrization-maximizer\]](#eq:symmetrization-maximizer){reference-type="eqref" reference="eq:symmetrization-maximizer"} we know that suitable symmetrizations of $\rho$ with respect to the coordinate axes are again maximizers. Iterating this procedure, we obtain a maximizer $\tilde \rho$ taking successive reflections of the sector $$\label{eq:sector-rho} \rho\mathop{\hbox{\vrule height 7pt width .5pt depth 0pt \vrule height .5pt width 6pt depth 0pt}}\nolimits\{x\colon\langle x, e_i \rangle>0\ \forall i=1,\ldots,N\},$$ and the result is a density symmetric with respect to each coordinate direction. The symmetries of $\tilde \rho$ guarantee that $$\int_{\{x\colon\langle x, \nu \rangle>0\}}\tilde{\rho} \,dx = \int_{\{x\colon\langle x, \nu \rangle<0\}}\tilde{\rho} \,dx = \frac12\qquad \forall \nu\in\mathbb S^{N-1}.$$ Hence, applying Lemma [Lemma 15](#lemma:non-crossing){reference-type="ref" reference="lemma:non-crossing"} to $\tilde \rho$ we obtain that $\pi_{\tilde \rho}$ satisfies the splitting condition [\[eq:splitting-plan\]](#eq:splitting-plan){reference-type="eqref" reference="eq:splitting-plan"} for any vector $\nu$. Thus, the condition [\[eq:radial-plan\]](#eq:radial-plan){reference-type="eqref" reference="eq:radial-plan"} holds for $\pi_{\tilde\rho}$. We finally conclude by uniqueness of the optimal plan, as we did in the last part of Lemma [Lemma 15](#lemma:non-crossing){reference-type="ref" reference="lemma:non-crossing"}: we can use the same strategy starting from a different sector in [\[eq:sector-rho\]](#eq:sector-rho){reference-type="eqref" reference="eq:sector-rho"}, defining a different symmetric density $\tilde \rho$. The same conclusion holds for the new optimal plan associated to that density, namely $\pi_{\tilde \rho}$. By uniqueness of the optimal plan, we know that $\pi_{\rho}$ can be obtained gluing together the plans of each sector, and thus also $\pi_{\rho}$ satisfies [\[eq:radial-plan\]](#eq:radial-plan){reference-type="eqref" reference="eq:radial-plan"}. ◻ Now we can state and prove our main result. **Theorem 17**. *Let $p>1$ be given. Then the only maximizer of $\mathcal{W}_p$ in the class $\mathsf{A}$, up to translations, is the characteristic function of $B$ with $|B|=1$.* *Proof.* We prove this result in two steps: in the first one we prove that any maximizer must be the characteristic function of a star-shaped set, while in the second one we exploit the inner-ball condition exposed in Lemma [Lemma 7](#lem:full-ball){reference-type="ref" reference="lem:full-ball"} to see that the length of the rays must be constant. Without loss of generality, we can suppose $N\geq 2$ since the $1$-dimensional case has already been treated in Proposition [Proposition 14](#prop:maximizer-1D){reference-type="ref" reference="prop:maximizer-1D"}. *Step 1.* First we will apply Corollary [Corollary 16](#cor:radial-transport-for-maximizers){reference-type="ref" reference="cor:radial-transport-for-maximizers"} and decompose the transport along rays. Then, exploit the one dimensional result obtained in Proposition [Proposition 14](#prop:maximizer-1D){reference-type="ref" reference="prop:maximizer-1D"} to prove that, along the rays, we see only segments, and this is equivalent to saying that the maximizer is a star-shaped set. Let $\rho$ be any maximizer of $\mathcal{W}_p$ in $\mathsf{A}$. We apply Corollary [Corollary 16](#cor:radial-transport-for-maximizers){reference-type="ref" reference="cor:radial-transport-for-maximizers"} to $\rho$, and we can suppose, without loss of generality, that the point $x_0$ coincides with the origin. Therefore, the optimal plan $\pi_{\rho}$ is induced by a radial map $T_{\rho}$. Since in this proof we do not need to stress the dependence of $\eta_{\rho}$, $\pi_{\rho}$ and $T_{\rho}$ on the density $\rho$, we simplify the notation, and we denote those objects by $\eta$, $\pi$ and $T$, respectively. We decompose every function in radial coordinates, and let $w(r) = r^{N-1}$ denote the coarea factor when we integrate in polar coordinates. For any $\omega\in\mathbb S^{N-1}$ we define the functions $$\rho^{\omega}(r) = \rho(r\omega),\qquad \eta^{\omega}(r) = \eta(r\omega),\qquad T^{\omega}(r) = |T(r\omega)|$$ for every $r\in[0,+\infty)$. We consider them as functions defined (almost everywhere) on the metric-measure space $(X,d,\gamma)$, where $X=\mathbb R^+$, $\gamma=w\mathscr{L}^1$ and $d$ is the usual distance. We claim that, since $T(r\omega) = T^{\omega}(r)\omega$ and $T_{\#}\rho = \eta$, we have $$\label{eq:factorization-transport} (T^{\omega})_{\#}\left(\rho^{\omega}\gamma\right) = \eta^{\omega}\gamma\qquad \text{for a.e. } \omega\in\mathbb S^{N-1}.$$ For any $s>0$ and any $E\subset \mathbb S^{N-1}$ we define the set $F = \{r\omega\colon0\leq r\leq s,\, \omega\in E\}$ and we have that $$\begin{split} \int_Ed\mathscr H^{N-1}_{\omega}\int_0^s\eta^{\omega}\,d\gamma &= \int_Ed\mathscr H^{N-1}_{\omega}\int_0^s \eta(r\omega)r^{N-1}\,dr = \int_F\eta(x)\,dx\\ &=\int_F(T_{\#}\rho)(x)\,dx =\int_{T^{-1}(F)}\rho(x)\,dx\\ &=\int_Ed\mathscr H^{N-1}_{\omega}\int_{(T^{\omega})^{-1}([0,s])}\rho(r\omega)r^{N-1}\,dr\\ &=\int_Ed\mathscr H^{N-1}_{\omega}\int_{(T^{\omega})^{-1}([0,s])}\rho^{\omega}(r)\,d\gamma\\ &=\int_Ed\mathscr H^{N-1}_{\omega}\int_0^s(T^{\omega})_{\#}\rho^{\omega}\,d\gamma. \end{split}$$ Here we used that $T$ is radial to pass from the second to the third line, in combination with the integration in polar coordinates. Since $E$ and $s$ are arbitrary, this proves [\[eq:factorization-transport\]](#eq:factorization-transport){reference-type="eqref" reference="eq:factorization-transport"}. We obtain the result of this first step by applying Proposition [Proposition 14](#prop:maximizer-1D){reference-type="ref" reference="prop:maximizer-1D"} separately for any $\omega\in\mathbb S^{N-1}$. In fact, we can integrate in polar coordinates the transport cost and obtain that $$\begin{aligned} \int |T(x)-x|^p\rho(x)\,dx &= \int_{\mathbb S^{N-1}}\int_0^{+\infty} |T(r\omega)-r\omega|^p\rho(r\omega)r^{N-1}\,dr\,d\omega\\ &= \int_{\mathbb S^{N-1}}\left(\int_0^{+\infty}|T^{\omega}(r)-r|^p\rho^{\omega}(r)w(r)\,dr\right)\,d\omega. \end{aligned}$$ The inner integral in the last expression coincides with the transport cost of $T^{\omega}$ between $\rho^{\omega}\gamma$ and $\eta^{\omega}\gamma$, and since $T$ is the optimal transport map between $\rho$ and $\eta$, then also $T^{\omega}$ must be optimal between $\rho^{\omega}\gamma$ and $\eta^{\omega}\gamma$ for every $\omega\in\mathbb S^{N-1}$. This is properly justified by showing that gluing the optimizers $\omega$-by-$\omega$ we obtain a measurable density. We sketch the proof of this fact in Appendix [5](#sec:appendix){reference-type="ref" reference="sec:appendix"}. Therefore, if we denote by $m(\omega) = \int_{\mathbb R^+} \rho^{\omega}\,d\gamma$, then $$\begin{gathered} \label{eq:radial-optimal-decomposition} \mathcal{W}_p^p(\rho) = \int_{\mathbb S^{N-1}}\mathcal{W}_p^p(\rho^{\omega})\,d\omega \\ \leq \int_{\mathbb S^{N-1}} \sup\left\{\mathcal{W}_p^p(\theta)\colon \theta\colon X\to[0,1],\int_{X}\theta \,d\gamma=m(\omega)\right\}\,d\omega, \end{gathered}$$ where we use the metric-measure definition of $\mathcal{W}_p$ in those integrals (see Remark [Remark 1](#rem:metric-measure-setting){reference-type="ref" reference="rem:metric-measure-setting"}). By Proposition [Proposition 14](#prop:maximizer-1D){reference-type="ref" reference="prop:maximizer-1D"}, for every $\omega\in\mathbb S^{N-1}$, the supremum inside the last integral coincides with $\mathcal{W}_p^p(\hbox{{\large $\chi$}{\Large $_{_{I^{\omega}}}$}})$, where $I^{\omega}\subset X$ is the unique segment of the form $[0,\ell^{\omega}]$ with $\gamma(I^{\omega}) = m(\omega)$. Moreover, the inequality is strict whenever $\rho^{\omega}$ is not equivalent to $\hbox{{\large $\chi$}{\Large $_{_{I^{\omega}}}$}}$. Since the map $\omega\mapsto m(\omega)$ is measurable, we can glue the segments $I^{\omega}$ together and obtain another candidate to compute $\mathcal{W}_p$. The density $\rho$ is a maximizer; hence, for almost every $\omega\in\mathbb S^{N-1}$ the density $\rho^{\omega}$ must be equivalent to $\hbox{{\large $\chi$}{\Large $_{_{I^{\omega}}}$}}$, concluding the proof of the first step. *Step 2.* For any $\omega\in\mathbb S^{N-1}$ we know that $T(\ell^{\omega}\omega) =(T^{\omega}(\ell^{\omega}))\omega$, and Lemma [Lemma 7](#lem:full-ball){reference-type="ref" reference="lem:full-ball"} guarantees that $\eta(x) = 1-\rho(x)$ for every $x\in\mathbb R^N$ such that $|x-\ell^{\omega}\omega|\leq T^{\omega}(\ell^{\omega})-\ell^{\omega}$. Let $\nu\in\mathbb S^{N-1}$ be another unit vector. Note that $T^{\omega}(\ell^{\omega}) = 2^{1/N}\ell^{\omega}$ (see e.g. [@C-TG2022] where the transport map in the case of a ball is given explicitly). Thanks to the inner ball condition, we obtain that $T^{\nu}(\ell^{\nu})$ is larger than $t$ for any $t>0$ such that $|t\nu-\ell^{\omega}\omega|\leq T^{\omega}(\ell^{\omega})-\ell^{\omega}$. In order to simplify the notation we define $c=2^{1/N}$, $r=\ell^{\omega}$ and $s=\ell^{\nu}$. Taking the square of both sides of the inner-ball inequality (see Figure [1](#fig:star-shaped-comparison){reference-type="ref" reference="fig:star-shaped-comparison"} for a geometric intuition of the inner ball condition in this situation), we get that $s\geq t$ for every $t>0$ satisfying $$c^2t^2-2c\langle \nu, \omega \rangle rt+c(2-c)r^2= 0.$$ Solving the above equation in $t$ one gets that $$s\geq \frac{\langle \nu, \omega \rangle+\sqrt{ \langle \nu, \omega \rangle^{2}-c(2-c) }}{c}\,r \, .$$ By the definition of $c$, the expression under the square root is non-negative whenever $\langle \nu, \omega \rangle$ is close enough to $1$ since $c>1$ and $1-c(2-c) = (c-1)^2>0$. Swapping the roles of $\nu$ and $\omega$ we also arrive to the analogous inequality $$r\geq \frac{\langle \nu, \omega \rangle+\sqrt{ \langle \nu, \omega \rangle^{2}-c(2-c) }}{c}\,s \, .$$ {#fig:star-shaped-comparison width="30%"} Combining these two inequalities we can control the difference between $s$ and $r$ in terms of the distance between $\nu$ and $\omega$: $$\begin{aligned} s-r&\geq \frac{r}{c}\left( \langle \nu, \omega \rangle-c+\sqrt{ \langle \nu, \omega \rangle^2-c(2-c) } \right) %&=\frac{r}{c} \frac{(\scal{\nu}{\omega}-c)^{2}-\scal{\nu}\omega^2+2c-c^2}{\scal{\nu}{\omega}-c-\sqrt{ \scal{\nu}{\omega}^2-c(2-c) }}\\ = \frac{2r(1-\langle \nu, \omega \rangle)}{\langle \nu, \omega \rangle-c-\sqrt{ \langle \nu, \omega \rangle^2-c(2-c) }},\\ s-r&\leq \frac{s}{c}\left(c-\langle \nu, \omega \rangle-\sqrt{ \langle \nu, \omega \rangle^2-c(2-c) }\right) %&=\frac{c^2+\scal{\nu}{\omega}^{2}-2c\scal{\nu}{\omega}-\scal{\nu}{\omega}^{2}+2c-c^{2}}{c\left( c-\scal{\nu}{\omega}+\sqrt{ \scal{\nu}{\omega}^{2}-c(2-c) } \right) }s\\ =\frac{2s(1-\langle \nu, \omega \rangle)}{c-\langle \nu, \omega \rangle+\sqrt{ \langle \nu, \omega \rangle^{2}+c(2-c) }}. \end{aligned}$$ By Corollary [Corollary 8](#cor:bounded-transport-distance){reference-type="ref" reference="cor:bounded-transport-distance"} we have that $|T(x)-x|\leq C_N$ for a dimensional constant $C_N$; hence, $r$ and $s$ are also uniformly bounded. Since $2(1-\langle \nu, \omega \rangle) = |\nu-\omega|^2$, we can combine the previous estimates and obtain that $$|\ell^{\nu}-\ell^{\omega}|\leq \tilde C_N |\nu-\omega|^2$$ for any $\nu$ and $\omega$ sufficiently close. This implies that the map $\omega\mapsto \ell^{\omega}$ is $2$-Hölder continuous on the sphere, hence it is constant. This is equivalent to showing that the only maximizer is the ball, and thus the proof is concluded. ◻ # Quantitative inequality in one dimension {#sec:quant_ineq} In this section we prove a quantitative inequality for $\mathcal{W}_p$ in one dimension, so we manage to strengthen the result obtained in Section [3](#sec:maximizer){reference-type="ref" reference="sec:maximizer"} adding a term that measures the displacement of a density $\rho$ respect to the characteristic function of a ball. In order to measure that distance, we consider a version of the Frankel asymmetry that, loosely speaking, is the $L^1$ distance between a density and a ball. This choice is by no means new: for example, the asymmetry was used in the quantitative isoperimetric inequality (cfr. [@FMP2008; @FMP2010]) and in the quantitative Brunn-Minkowski inequality (cfr. [@BJ2017]). See also [@FP2020; @FL2021] for a quantitative inequality involving a functional of Riesz type. **Definition 18**. We define the following quantity, that we will just call asymmetry in the sequel: $$A(\rho)\coloneqq \inf\left\{\left\lVert\rho-\hbox{{\large $\chi$}{\Large $_{_{B_r(x)}}$}}\right\rVert_1\colon x\in\mathbb R^N,|B_r(x)|=1\right\}\qquad \forall \rho\in\mathsf{A}.$$ With this notion, our quantitative inequality reads as the following. **Theorem 19**. *For $N=1$ and $p>1$ fixed, there exists a constant $C_p>0$ such that $$\mathcal{W}_p^p(B)-\mathcal{W}_p^p(\rho)\geq C_pA(\rho)^2\qquad \text{for all } \rho\in\mathsf{A}.$$* *Remark 20*. We point out that the exponent $2$ in our quantitative inequality is sharp, in the sense that the inequality would be false with a smaller exponent for densities with small asymmetry. This can be seen by taking $\rho = \hbox{{\large $\chi$}{\Large $_{_{[-1/2-\varepsilon,-1/2]}}$}}+\hbox{{\large $\chi$}{\Large $_{_{[-1/2+\varepsilon,1/2-\varepsilon]}}$}}+\hbox{{\large $\chi$}{\Large $_{_{[1/2,1/2+\varepsilon]}}$}}$ for $\varepsilon$ small. *Proof.* By definition of asymmetry, $A(\rho)\leq 2$ for every $\rho\in\mathsf{A}$, and without loss of generality we can suppose that $A(\rho)>0$. Up to translations, we can suppose that $$\int_{-\infty}^{0}\rho \,dx = \int_{0}^{+\infty}\rho \,dx = \frac12.$$ As we showed in Proposition [Proposition 14](#prop:maximizer-1D){reference-type="ref" reference="prop:maximizer-1D"}, it is possible to find a transport plan $\bar \pi\in\mathsf{AP}_{\rho}$ such that $|x-y|\leq 1/2$ for any $(x,y)\in{\rm spt}\bar\pi$. Loosely speaking, that transport plan moves mass "away from the origin". Now we want to get a quantitative inequality modifying $\bar\pi$ and finding another plan $\pi\in\mathsf{AP}_{\rho}$ for which the transport distance is again controlled by $1/2$, and moreover $$\label{eq:measure-distance-estimate} \pi\left(\left\{(x,y)\colon|x-y|\leq d\right\}\right)\geq \frac{A(\rho)}{100},\quad \text{where }d\coloneqq \frac12-\frac{A(\rho)}{100}.$$ With this competitor, if $E=\left\{(x,y)\in\mathbb R^N\times\mathbb R^N\colon|x-y|\leq d\right\}$ is the set considered in the previous inequality, we have that $$\begin{split} \mathcal{W}_p^p(\rho)&\leq \int|x-y|^p\,d\pi(x,y) \leq d^p\pi(E)+\frac{1}{2^p}(1-\pi(E))\\ &= \frac{1}{2^p}+\frac{\pi(E)}{2^p}\left[\left(1-\frac{A(\rho)}{50}\right)^p-1\right]\leq \frac{1}{2^p}+\frac{\pi(E)}{2^p}\left(-C_pA(\rho)\right)\\ &= \mathcal{W}_p^p(B)-C_pA(\rho)^2, \end{split}$$ where $C_p$ is a constant depending only on $p$. Therefore, we need to find such a plan $\pi$ to complete the proof. We denote by $\bar T$ the map that induces $\bar \pi$. Let us look at the set $\{x\geq0\}$, and we define $x_R$ as the smallest point that is moved at distance $d$, i.e. $x_R\coloneqq \inf\{x>0\colon\bar T(x)-x>d\}$. Exploiting the same arguments of Proposition [Proposition 14](#prop:maximizer-1D){reference-type="ref" reference="prop:maximizer-1D"}, it can be shown that $\bar T_{\#}(\rho\mathop{\hbox{\vrule height 7pt width .5pt depth 0pt \vrule height .5pt width 6pt depth 0pt}}\nolimits[0,x_R]) = 1-\rho$ in $[x_R,x_R+d]$, and thus $\int_0^{x_R}\rho\,dx +\int_{x_R}^{x_R+d}\rho \,dx \geq d$. Now we explore the different cases that may appear. #### *Case 1* If we have that $\int_0^{x_R}\rho \,dx\geq \frac{A(\rho)}{100}$, then the plan $\bar\pi$ already satisfies [\[eq:measure-distance-estimate\]](#eq:measure-distance-estimate){reference-type="eqref" reference="eq:measure-distance-estimate"} and there is nothing to do. #### *Case 2* Let us suppose that both of the following conditions hold $$\int_0^{x_R}\rho \,dx< \frac{A(\rho)}{100},\qquad \int_0^{x_R}(1-\rho) \,dx> \frac{A(\rho)}{100}.$$ In this case, we take a point $x_R^1>x_R$ such that $\int_0^{x_R^1}\rho \,dx = \frac{A(\rho)}{100}$, and we try to move mass in the opposite direction in the segment $[0,x_R^1]$. This is necessary in order to take into account densities similar to the characteristic function of the union of two intervals: in that case, the optimal map actually moves mass toward the origin (see Figure [2](#fig:segments){reference-type="ref" reference="fig:segments"}). ![Denoting by $\rho = \hbox{{\large $\chi$}{\Large $_{_{[-a-1/2,-a]\cup[a,a+1/2]}}$}}$ for some $a>0$, we represent in this picture the matching induced by a plan realizing every $\mathcal{W}_p(\rho)$. In fact, if $x\in{\rm spt}\rho$ has a given color, it will be mapped to a point with the corresponding dashed color.](two-segments.eps){#fig:segments width="40%"} To do this, we consider a transport plan tailored to $\rho$ and depending on $x_R^1$, and it is obtained again through a minimization process: $$\label{eq:constrained-transport-backward} \min\left\{\int |x-y|^p\,d\pi(x,y)\colon\pi\in\mathsf{AP}_{\rho}, {\rm spt}\pi\subset D\right\},$$ where $D\subset \mathbb R\times\mathbb R$ is the following domain: $$D\coloneqq \{(x,y)\colon x\not\in(0,x_R^1), x\cdot(y-x)\geq 0\}\cup\left([0,x_R^1]\times[0,x_R^1]\right).$$ Observe that, since $\int_0^{x_R^1}\rho \,dx=\frac{A(\rho)}{100}<\int_0^{x_R^1}(1-\rho)\,dx$, then it is possible to find a minimizer $\pi$ of [\[eq:constrained-transport-backward\]](#eq:constrained-transport-backward){reference-type="eqref" reference="eq:constrained-transport-backward"}. Applying again the structure theorem for optimal plans in one dimension, we find a map $T$ that induces an optimal plan. This transport problem is actually decoupled, considering independently $\rho\mathop{\hbox{\vrule height 7pt width .5pt depth 0pt \vrule height .5pt width 6pt depth 0pt}}\nolimits[0,x_R^1]$ and $\rho-(\rho\mathop{\hbox{\vrule height 7pt width .5pt depth 0pt \vrule height .5pt width 6pt depth 0pt}}\nolimits[0,x_R^1])$. Hence, it is possible to adapt [@DMSV2016 Lemma 5.1] separately to both pieces and see that $|T(x)-x|\leq d$ for every $x\in[0,x_R^1]$. In fact, if this is not the case, then $T_{\#}\rho = 1-\rho$ in a segment $I\subset [0,x_R^1]$ longer than $d$. This is impossible since $$d\leq |I| = \int_I (\rho+(1-\rho))\,dx= \int_I(\rho+T_{\#}\rho)\,dx \leq 2\int_0^{x_R^1}\rho \,dx = 2\cdot \frac{A(\rho)}{100}\leq\frac{1}{25},$$ and $d = \frac12-\frac{A(\rho)}{100}>\frac13$. Having this uniform bound on the transport length in $[0,x_R^1]$, then we see that $\pi$ satisfies [\[eq:measure-distance-estimate\]](#eq:measure-distance-estimate){reference-type="eqref" reference="eq:measure-distance-estimate"} because $\int_0^{x_R^1}\rho \,dx= \frac{A(\rho)}{100}$. #### *Case 3* Finally, let us suppose that the following inequalities hold at the same time: $$\int_0^{x_R}\rho \,dx< \frac{A(\rho)}{100},\qquad\int_0^{x_R}(1-\rho)\,dx\leq \frac{A(\rho)}{100}.$$ At this point, we can explore each of the previous cases on the left side of the real line, producing the analogous $x_L = \sup\left\{x<0\colon x-\bar T(x)>d\right\}$. Since in the first two cases we managed to construct the desired $\pi$, we can suppose without loss of generality that we are in Case 3 also on the left side. In other words, the following holds $$\max\left\{\int_0^{x_R}\rho \,dx, \int_0^{x_R}(1-\rho) \,dx, \int_{x_L}^0\rho \,dx, \int_{x_L}^0(1-\rho) \,dx \right\}\leq \frac{A(\rho)}{100}.$$ Combining these information we obtain an estimate on $|x_R-x_L|$: $$x_R-x_L = \int_{x_L}^0(\rho+(1-\rho))\,dx + \int_0^{x_R}(\rho+(1-\rho))\,dx \leq \frac{A(\rho)}{25},$$ and we will see that this is not possible because we can get an inequality for the asymmetry of $\rho$. We repeat here the argument of Proposition [Proposition 14](#prop:maximizer-1D){reference-type="ref" reference="prop:maximizer-1D"}: adapting [@DMSV2016 Lemma 5.1] we obtain that $\bar T_{\#}(\rho\mathop{\hbox{\vrule height 7pt width .5pt depth 0pt \vrule height .5pt width 6pt depth 0pt}}\nolimits[x_L,0]) = 1-\rho$ in $[x_L-d,x_L]$ and $\bar T_{\#}(\rho\mathop{\hbox{\vrule height 7pt width .5pt depth 0pt \vrule height .5pt width 6pt depth 0pt}}\nolimits[0,x_R]) = 1-\rho$ in $[x_R,x_R+d]$, and thus $$\begin{split} \int_{x_L-d}^{x_R+d}\rho \,dx &= \int_{x_L-d}^{x_L}\rho \,dx+\int_{x_L}^0\rho \,dx+\int_0^{x_R}\rho \,dx + \int_{x_R}^{x_R+d}\rho \,dx\\ & \geq \int_{x_L-d}^{x_L}\rho \,dx+ \int_{x_L-d}^{x_L}\bar T_{\#}\rho \,dx +\int_{x_R}^{x_R+d}\bar T_{\#}\rho \,dx+\int_{x_R}^{x_R+d}\rho\,dx=2d. \end{split}$$ This means that $\int_{x_L-d}^{x_R+d}\rho \geq 1-\frac{A(\rho)}{50}$. If $x_R+d-(x_L-d)\leq1$, then by definition of asymmetry $$A(\rho)\leq 2\int_{-\infty}^{x_L-d}\rho\,dx+2\int_{x_R+d}^{+\infty}\rho\,dx\leq \frac{A(\rho)}{25},$$ that is impossible. Hence, we know that $x_L-d+1<x_R+d$. Since we proved that $x_R-x_L\leq \frac{A(\rho)}{25}$, we obtain an inequality always valid in our case: $x_R+d-(x_L-d) = 1-\frac{A(\rho)}{50}+x_R-x_L \leq 1+\frac{A(\rho)}{50}$. Therefore, we have that $$\begin{aligned} A(\rho) &\leq 2\int_{x_L-d}^{x_L-d+1}(1-\rho)\,dx \leq 2\int_{x_L-d}^{x_R+d}(1-\rho)\,dx \leq 2(x_R-x_L+2d)-2\left(1-\frac{A(\rho)}{50}\right)\\ &\leq 2+\frac{A(\rho)}{25}-2+\frac{A(\rho)}{25} = \frac{2}{25}A(\rho), \end{aligned}$$ and thus we reach a contradiction, concluding the last remaining case. ◻ # Sketch of the measurability of the construction in Theorem [Theorem 17](#thm:ball-max){reference-type="ref" reference="thm:ball-max"} {#sec:appendix} In Theorem [Theorem 17](#thm:ball-max){reference-type="ref" reference="thm:ball-max"} we needed to check that the density $$(r,\omega) \mapsto \bar \zeta^{\omega}(r)$$ is measurable, where $\bar \zeta^{\omega}$ satisfies $\mathcal{W}_p(\rho^{\omega}) = W_p(\rho^{\omega},\bar\zeta^{\omega})$. This is necessary to have the representation in [\[eq:radial-optimal-decomposition\]](#eq:radial-optimal-decomposition){reference-type="eqref" reference="eq:radial-optimal-decomposition"}. To do that, we approximate $\rho$ in $L^1$ with densities $\rho_k\in\mathsf{A}$ that are piecewise constant along the sphere. In other words, for every $k$ there exists a partition of the sphere $\mathbb S^{N-1} = \bigcup_j E^k_j$ with sets such that $\mathop{\mathrm{diam}}(E^k_j)+|E^k_j|\leq 1/k$, and such that for every $j$ $$\rho_k(r\omega) = \rho_k(r\omega') \qquad\forall \omega,\omega'\in E^k_j.$$ We construct the following densities: for every $k$ and every $\omega\in\mathbb S^{N-1}$ we take $\zeta$ such that $\mathcal{W}_p(\rho_k^{\omega}) = W_p(\rho_k^{\omega},\zeta)$ (in the metric-measure sense), and we define $$\zeta_k(r,\omega) = \zeta(r).$$ In other words, $\zeta_k^{\omega}$ is the optimal density to compute $\mathcal{W}_p(\rho_k^{\omega})$. This density is measurable since it is piecewise constant along the sphere. Since $\rho_k\to\rho$ in $L^1$, then $\rho_k^{\omega}\to \rho^{\omega}$ in $L^1$ for a.e. $\omega\in\mathbb S^{N-1}$. For this reason, we say that $\zeta_k^{\omega}\to \bar \zeta^{\omega}$ in weak$*$ sense for a.e. $\omega$. To see this, notice that $\zeta_k^{\omega}$ converges to some density $\phi^{\omega}$ because the sequence $\rho_k^{\omega}$ is bounded in $L^{\infty}$, and the transport distance is bounded when the mass of $\rho_k^{\omega}$ is finite, that happens for a.e. $\omega$. By lower semicontinuity of the transport distance we have that $$\mathcal{W}_p(\rho^{\omega}) = W_p(\rho^{\omega},\bar\zeta^{\omega}) \leq W_p(\rho^{\omega},\phi^{\omega}) \leq \liminf_k W_p(\rho_k^{\omega},\zeta_k^{\omega}) = \mathcal{W}_p(\rho^{\omega}),$$ where we used that $\rho^{\omega}+\phi^{\omega}\leq1$ in the first inequality, and the continuity of $\mathcal{W}_p$ with respect the weak$*$ convergence in the last equality. Since the optimal density to compute $\mathcal{W}_p(\rho^{\omega})$ is unique, then $\bar \zeta^{\omega} = \phi^{\omega} = \lim_k \zeta_k^{\omega}$. We finally conclude because $\zeta_k\to \zeta_\infty$ for some $\zeta_\infty$ in weak$*$ sense, and $\zeta_\infty$ is therefore measurable. Moreover, a little argument shows that, whenever $f_k:X\times Y\to\mathbb R$ converges in weak$*$ sense to $f$ ($X$ and $Y$ being reasonable spaces, in our case $X=\mathbb R^+$ and $Y=\mathbb S^{N-1}$), then for almost every $x\in X$ we have that $$f_k\mathop{\hbox{\vrule height 7pt width .5pt depth 0pt \vrule height .5pt width 6pt depth 0pt}}\nolimits(\{x\}\times Y) \overset{\ast}{\rightharpoonup}f\mathop{\hbox{\vrule height 7pt width .5pt depth 0pt \vrule height .5pt width 6pt depth 0pt}}\nolimits(\{x\}\times Y).$$ Hence, for almost every $\omega\in\mathbb S^{N-1}$ we have that $$\zeta_k^{\omega}\overset{\ast}{\rightharpoonup}\zeta_\infty^{\omega},$$ and our previous argument shows also that $$\zeta_k^{\omega}\overset{\ast}{\rightharpoonup}\bar\zeta^{\omega}\qquad \text{for a.e. }\omega\in\mathbb S^{N-1}.$$ Combining these fact, we get that $\bar\zeta = \zeta_\infty$ almost everywhere, and thus $\bar \zeta$ is measurable, as we wanted. ## Acknowledgments {#acknowledgments .unnumbered} D.C. is member of the Istituto Nazionale di Alta Matematica (INdAM), Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA), and is partially supported by the INdAM--GNAMPA 2023 Project *Problemi variazionali per funzionali e operatori non-locali*, codice CUP_E53C22. D.C. wishes to thank Virginia Commonwealth University for the generous hospitality provided during his visit, which marked the beginning of this project. I.T.'s research was partially supported by a Simons Collaboration grant 851065 and an NSF grant DMS 2306962.

arxiv_math

--- abstract: | These are informal lecture notes for a three-hour minicourse on Kac--Moody groups, given at the workshop "Kac--Moody geometry" in July 2023 in Kiel. They provide a concise overview of the book *An introduction to Kac--Moody groups over fields*, EMS Textbooks in Mathematics (2018). They assume a previous familiarity with the (very) basics of Kac--Moody algebras. For readers unfamiliar with the latter topic, short "Prerequisites" notes (referenced within the text) are also freely available. author: - Timothée [Marquis]{.smallcaps}$^*$ bibliography: - overviewKM.bib title: Minicourse on Kac--Moody groups --- [^1] All results mentioned in these notes are contained in the [book](http://dx.doi.org/10.4171/187) "An introduction to Kac--Moody groups over fields". So as to lighten the presentation, no bibliographical details are provided within the text. Instead, attributions of all mentioned results can be found at the end of Chapters 6, 7 and 8, and in Chapter 9 of the book. For readers unfamiliar with Kac--Moody algebras (and/or Coxeter groups, buildings and BN-pairs), short "Prerequisites" notes are also available [here](https://perso.uclouvain.be/timothee.marquis/papers/PrerequisitesKM.pdf). # Setting - Let $A=(a_{ij})_{i,j\in I}$ be a generalised Cartan matrix (GCM). - Let $(\mathfrak{h},\Pi,\Pi^{\vee})$ be a realisation of $A$: $\mathfrak{h}$ is a complex vector space with $\dim_{\mathbb{C}}\mathfrak{h}=|I|+\mathrm{corank}(A)$ and $\Pi=\{\alpha_i \ | \ i\in I\}$ and $\Pi^{\vee}=\{\alpha_i^{\vee} \ | \ i\in I\}$ are linearly independant subsets of $\mathfrak{h}^*$ and $\mathfrak{h}$ respectively such that $\langle\alpha_j,\alpha_i^{\vee}\rangle=a_{ij}$ for all $i,j\in I$. - Let $\mathfrak{g}(A)=\langle \textrm{$e_i,f_i$ ($i\in I$), $\mathfrak{h}$} \ | \ \textrm{(RA1)--(RA4)} \rangle$ be the Kac-Moody algebra of type $A$, where - $[h,h']=0$ for all $h,h'\in\mathfrak{h}$; - $[h,e_i]=\alpha_i(h)e_i$ and $[h,f_i]=-\alpha_i(h)f_i$ for all $h\in\mathfrak{h}$ and $i\in I$; - $[f_i,e_j]=\delta_{ij}\alpha_i^{\vee}$ for all $i,j\in I$; - $(\mathop{\mathrm{ad}}e_i)^{|a_{ij}|+1}e_j=0=(\mathop{\mathrm{ad}}f_i)^{|a_{ij}|+1}f_j$ for all $i,j\in I$ with $i\neq j$. - We have a root space decomposition $$\mathfrak{g}(A)=\mathfrak{h}\oplus\bigoplus_{\alpha\in\Delta}\mathfrak{g}_{\alpha}\quad\textrm{where $\mathfrak{g}_{\alpha}=\{x\in\mathfrak{g}(A) \ | \ [h,x]=\alpha(h)x \ \forall h\in\mathfrak{h}\}$}$$ and a triangular decomposition $$\mathfrak{g}(A)=\mathfrak{n}^-\oplus\mathfrak{h}\oplus\mathfrak{n}^+$$ where $\mathfrak{n}^+$ (resp. $\mathfrak{n}^-$) is the subalgebra generated by the $e_i$ (resp. $f_i$), $i\in I$. We also consider the derived Kac--Moody algebra $$\mathfrak{g}_A=\mathfrak{n}^-\oplus\mathfrak{h}'\oplus\mathfrak{n}^+\quad\textrm{where $\mathfrak{h}'=\sum_{i\in I}\mathbb{C}\alpha_i^{\vee}\subseteq\mathfrak{h}$.}$$ - The root system $\Delta=\{\alpha\in\mathfrak{h}^*\setminus\{0\} \ | \ \mathfrak{g}_{\alpha}\neq\{0\}\}$ decomposes into positive/negative roots: $$\Delta=\Delta^+\sqcup \Delta^-\quad\textrm{where}\quad \Delta^{\pm}=\{\alpha=\pm\sum_{i\in I}n_i\alpha_i\in\Delta \ | \ n_i\in\mathbb{N}\}.$$ For $\alpha$ as above, we write $\mathop{\mathrm{ht}}(\alpha)=\pm\sum_{i\in I}n_i\in\mathbb{Z}$ for its height. - The Weyl group $$W=\langle s_i\colon\thinspace\alpha\mapsto \alpha-\langle\alpha,\alpha_i^{\vee}\rangle \alpha_i \ | \ i\in I\rangle\leq\mathop{\mathrm{GL}}(\mathfrak{h}^*)$$ of $A$ is a Coxeter group, with set of simple reflections $S=\{s_i \ | \ i\in I\}$. - $\Delta$ also splits into the sets of real/imaginary roots: $$\Delta=\Delta^{re}\sqcup\Delta^{im}\quad\textrm{where $\Delta^{re}=W\Pi$ and $\Delta^{im}=\Delta\setminus\Delta^{re}$}$$ and we set $\Delta^{re\pm}=\Delta^{re}\cap\Delta^{\pm}$ and $\Delta^{im\pm}=\Delta^{im}\cap\Delta^{\pm}$. **Example 1**. 1. If $A=\begin{psmallmatrix}2&-1\\ -1&2\end{psmallmatrix}$, then $\mathfrak{g}_A=\mathfrak{g}(A)\cong\mathfrak{sl}_3(\mathbb{C})$. 2. If $A=\begin{psmallmatrix}2&-2\\ -2&2 \end{psmallmatrix}$ with $I=\{0,1\}$, then $\mathfrak{g}_A\cong\mathfrak{sl}_2(\mathbb{C}[t,t^{-1}])\rtimes \mathbb{C}K$ is a one-dimensional (nontrivial) central extension of $\mathfrak{sl}_2(\mathbb{C}[t,t^{-1}])$, with $$e_1=\begin{psmallmatrix}0&1\\ 0&0 \end{psmallmatrix}, \quad f_1=\begin{psmallmatrix}0&0\\ -1&0 \end{psmallmatrix}, \quad \alpha_1^{\vee}=\begin{psmallmatrix}1&0\\ 0&-1 \end{psmallmatrix}$$ and $$e_0=\begin{psmallmatrix}0&0\\ -t&0 \end{psmallmatrix}, \quad f_0=\begin{psmallmatrix}0&t^{-1}\\ 0&0 \end{psmallmatrix}, \quad \alpha_0^{\vee}=-\alpha_1^{\vee}+K.$$ We conclude this section with the Gabber--Kac theorem, justifying why Kac--Moody algebras are generalisations of the *simple* finite-dimensional complex Lie algebras, at least when $A$ is symmetrisable (i.e. $A=DB$ with $D$ a diagonal and $B$ a symmetric matrix). **Theorem 2** (Gabber--Kac). *If $A$ is an indecomposable symmetrisable GCM, then $\mathfrak{g}_A$ is simple modulo its center $\mathcal Z(\mathfrak{g}_A)\subseteq\mathfrak{h}'$.* # The world of Kac--Moody algebras {#section:KMA} Call an (indecomposable) GCM $A$ of *finite type* if $A=DB$ with $B$ positive definite, of *affine type* if $A=DB$ with $B$ positive semidefinite and of corank $1$, and of *indefinite type* otherwise. \|m4cm\|m3cm\|m4cm\|m4cm\| &Finite type&Affine type&Indefinite type\ $\dim\mathfrak{g}_A$ &$<\infty$&$\infty$&$\infty$\ growth as $n\to\infty$ of $\dim\bigoplus_{0<\mathop{\mathrm{ht}}(\alpha)<n}\mathfrak{g}_{\alpha}$&constant&polynomial&exponential\ Known "realisations"&e.g. $\mathfrak{sl}_n(\mathbb{C})$& e.g. $\mathfrak{sl}_n(\mathbb{C}[t,t^{-1}])$ & ???\ Coxeter group $W$& finite (spherical geometry)&affine (Euclidean geometry)& indefinite for $|I|\geq 3$ (hyperbolic-like geometry)\ Roots &$\Delta^{re}=\Delta$ &$\Delta^{re}\subsetneq\Delta$, $\Delta^{im}=\mathbb{Z}_{\neq 0}\delta$& $\Delta^{re}\subsetneq\Delta$, $\Delta^{im}$ "big"\ $\mathfrak{g}_{\alpha}$, $\alpha\in\Delta^{re}$&\ $\mathfrak{g}_{\alpha}$, $\alpha\in\Delta^{im}$&$\varnothing$&$\sup_{\alpha\in\Delta^{im}}\dim\mathfrak{g}_{\alpha}<\infty$&$\sup_{\alpha\in\Delta^{im}}\dim\mathfrak{g}_{\alpha}=\infty$\ **Example 3**. In the notations of Example [Example 1](#example:gA){reference-type="ref" reference="example:gA"}(2): $\Delta^{im}=\mathbb{Z}_{\neq 0}\delta$ with $\delta=\alpha_0+\alpha_1$, and $\mathfrak{g}_{n\delta}=\mathbb{C}\begin{psmallmatrix}t^n&0\\0&-t^n\end{psmallmatrix}$. Similarly, $\Delta^{re}=\{n\delta\pm\alpha_1 \ | \ n\in\mathbb{Z}\}$ with $e_{n\delta+\alpha_1}=\begin{psmallmatrix}0&t^n\\0&0\end{psmallmatrix}$ and $e_{n\delta-\alpha_1}=\begin{psmallmatrix}0&0\\-t^n&0\end{psmallmatrix}$. **Remark 4**. Here is another key difference between real and imaginary roots: if $x\in\mathfrak{g}_{\alpha}$ is nonzero, then the endomorphism $\mathop{\mathrm{ad}}x\in\mathop{\mathrm{End}}(\mathfrak{g}_A)$ is *locally nilpotent* (i.e. for all $y\in\mathfrak{g}_A$ there exists $n=n(y)\in\mathbb{N}$ such that $(\mathop{\mathrm{ad}}x)^ny=0$) if and only if $\alpha\in\Delta^{re}$. # Kac--Moody groups {#section:KMG} There are many objects deserving the name of Kac--Moody group; the most flexible "definition" of a Kac--Moody group is then as follows. **Definition 5**. A **Kac--Moody group** is a group attached to a Kac--Moody algebra. More precisely, in order for a group $G$ to deserve the name of Kac--Moody group of type $A$, it should have an adjoint action $\mathop{\mathrm{Ad}}\colon\thinspace G\to\mathop{\mathrm{Aut}}(\mathfrak{g}_A)$ on $\mathfrak{g}_A$ (or some variation of $\mathfrak{g}_A$, such as a completion) with small, central kernel. The most obvious way to construct a Kac--Moody group would then be to exponentiate the adjoint representation $\mathop{\mathrm{ad}}\colon\thinspace\mathfrak{g}_A\to\mathop{\mathrm{End}}(\mathfrak{g}_A)$. On the other hand, if $\alpha\in\Delta$ and $x\in\mathfrak{g}_{\alpha}$ is nonzero, then the map $$\mathfrak{g}_A\to\mathfrak{g}_A:y\mapsto (\exp\mathop{\mathrm{ad}}x)y:=\sum_{n\geq 0}\tfrac{1}{n!}(\mathop{\mathrm{ad}}x)^ny$$ is a well-defined automorphism of $\mathfrak{g}_A$ if and only if the above sum is always finite, which happens precisely when $\alpha\in\Delta^{re}$ by Remark [Remark 4](#remark:locnilp){reference-type="ref" reference="remark:locnilp"}. This leads us to define the group $$G_A^{\mathop{\mathrm{ad}}}(\mathbb{C}):=\langle \exp\mathop{\mathrm{ad}}x \ | \ x\in\mathfrak{g}_{\alpha}, \ \alpha\in\Delta^{re}\rangle=\langle x_{\alpha}(r)=\exp\mathop{\mathrm{ad}}re_{\alpha} \ | \ r\in\mathbb{C}, \ \alpha\in\Delta^{re}\rangle\leq\mathop{\mathrm{Aut}}(\mathfrak{g}_A),$$ which one can call a **minimal[^2] (adjoint) Kac--Moody group over $\mathbb{C}$**. **Example 6**. In the notations of Example [Example 3](#example:A1tilde1){reference-type="ref" reference="example:A1tilde1"}, we have $G_A^{\mathop{\mathrm{ad}}}(\mathbb{C})=\mathrm{PSL}_2(\mathbb{C}[t,t^{-1}])$ and $x_{\alpha}(r)=I+re_{\alpha}\in G_A^{\mathop{\mathrm{ad}}}(\mathbb{C})$ for all $\alpha\in\Delta^{re}$. **Remark 7**. If $h\in\mathfrak{h}'$ and $y\in\mathfrak{g}_{\beta}$, then $(\exp \mathop{\mathrm{ad}}h)y=e^{\beta(h)}y$ also makes sense, and we can thus also exponentiate the adjoint action of $\mathfrak{h}'$, to get a **torus** $$T:=\langle e^{\mathop{\mathrm{ad}}h}=\exp\mathop{\mathrm{ad}}h \ | \ h\in\mathfrak{h}'\rangle\leq\mathop{\mathrm{Aut}}(\mathfrak{g}_A).$$ But as we will see, $T$ is in fact already contained in $G_A^{\mathop{\mathrm{ad}}}(\mathbb{C})$. **Remark 8**. Let $\lambda\in\mathfrak{h}^*$ be dominant integral, i.e. such that $\lambda(\alpha_i^{\vee})\in\mathbb{N}$ for all $i\in I$. Then the irreducible highest-weight $\mathfrak{g}_A$-module $L(\lambda)$ with highest weight $\lambda$ is integrable: the representation $\pi_{\lambda}\colon\thinspace\mathfrak{g}_A\to\mathop{\mathrm{End}}(L(\lambda))$ is such that each $\pi_{\lambda}(e_{\alpha})$ ($\alpha\in\Delta^{re}$) is locally nilpotent. One can thus define in a same the minimal Kac--Moody group $$G_A^{\pi_{\lambda}}(\mathbb{C}):=\langle \exp\pi_{\lambda}(x) \ | \ x\in\mathfrak{g}_{\alpha}, \ \alpha\in\Delta^{re}\rangle\leq\mathop{\mathrm{Aut}}(L(\lambda)).$$ Since finite-dimensional simple Lie algebras yield groups that can be defined over any field or even (commutative, associative, unital) ring $k$, such as $\mathop{\mathrm{SL}}_n(k)$, we would now like to define Kac--Moody groups over arbitrary fields or even rings. Moreover, we would like to have an intrinsic definition of a Kac--Moody group that does not depend on an ambiant space such as $\mathop{\mathrm{Aut}}(\mathfrak{g}_A)$ or $\mathop{\mathrm{Aut}}(L(\lambda))$. In particular, we would like to understand how the various groups $G_A^{\mathop{\mathrm{ad}}}(\mathbb{C})$ and $G_A^{\pi_{\lambda}}(\mathbb{C})$ constructed above compare to each other. Note that $G_A^{\mathop{\mathrm{ad}}}(\mathbb{C})$ (and $G_A^{\pi_{\lambda}}(\mathbb{C})$) is generated by a torus $T$ and by copies $U_{\alpha}:=x_{\alpha}(\mathbb{C})\cong (\mathbb{C},+)$ of the additive group of $\mathbb{C}$ for each $\alpha\in\Delta^{re}$. One could then define a Kac--Moody group over $k$ as a free product of groups $\mathfrak{U}_{\alpha}(k)\cong (k,+)$ for $\alpha\in\Delta^{re}$ (we then denote by $$x_{\alpha}\colon\thinspace k\to \mathfrak{U}_{\alpha}(k):r\mapsto x_{\alpha}(r)$$ the corresponding isomorphism) and of a torus $T_k\cong (k^{\times})^{|I|}$ (we then write $$T_k=\langle r^{\alpha_i^{\vee}} \ | \ i\in I\rangle\cong (k^{\times})^{|I|},$$ with an injective group morphism $k^{\times}\to T_k:r\mapsto r^{\alpha_i^{\vee}}$ for each $i\in I$), which we quotient out by all the relations we observe between these generators inside $\mathop{\mathrm{Aut}}(\mathfrak{g}_A)$ and $\mathop{\mathrm{Aut}}(L(\lambda))$ (at least over $\mathbb{C}$), with the hope to find sufficiently many such relations to get back the groups $G_A^{\mathop{\mathrm{ad}}}(\mathbb{C})$ and $G_A^{\pi_{\lambda}}(\mathbb{C})$ for $k=\mathbb{C}$. **Definition 9**. Let $k$ be a ring. Define the group $\mathfrak{G}_A(k)=T_k*(\ast_{\alpha\in\Delta^{re}}\mathfrak{U}_{\alpha}(k))/\textrm{(R0)--(R4)}$ where the relations (R0)--(R4) are as follows. For each $i\in I$ and $r\in k^{\times}$, set $\widetilde{s}_i(r):=x_{\alpha_i}(r)x_{-\alpha_i}(r^{-1})x_{\alpha_i}(r)$ and $\widetilde{s}_i:=\widetilde{s}_i(1)$. For all $\alpha,\beta\in\Delta^{re}$, fix a total order on $]\alpha,\beta[_{\mathbb{N}}:=(\mathbb{N}_{>0}\alpha+\mathbb{N}_{>0}\beta)\cap\Delta$. 1. $[x_{\alpha}(t),x_{\beta}(u)]=\prod_{\gamma=i\alpha+j\beta\in ]\alpha,\beta[_{\mathbb{N}}}x_{\gamma}(C^{\alpha\beta}_{ij}t^iu^j)$ for all prenilpotent pairs $\{\alpha,\beta\}\subseteq\Delta^{re}$, where the $C^{\alpha\beta}_{ij}$ are given integers; 2. $r^{\alpha_i^{\vee}}x_{\alpha_j}(t)r^{-\alpha_i^{\vee}}=x_{\alpha_j}(r^{a_{ij}}t)$ for all $r\in k^{\times}$, $i,j\in I$ and $t\in k$; 3. $\widetilde{s}_ir^{\alpha_j^{\vee}}\widetilde{s}_i^{-1}=r^{s_i\alpha_j^{\vee}}$ for all $r\in k^{\times}$ and $i,j\in I$; 4. $r^{\alpha_i^{\vee}}=\widetilde{s}_i^{-1}\widetilde{s}_i(r^{-1})$ for all $r\in k^{\times}$ and $i\in I$; 5. $\widetilde{s}_ix_{\gamma}(t)\widetilde{s_i}^{-1}=x_{s_i\gamma}(t)$ for all $i\in I$, $t\in k$ and $\gamma\in\Delta^{re}$. The group functor $\mathfrak{G}_A\colon\thinspace\mathop{\mathrm{{\mathbb Z}-alg}}\to\mathop{\mathrm{Gr}}$ is called the **constructive Tits functor** of type $A$. For $\mathbb{K}$ a field, the group $\mathfrak{G}_A(\mathbb{K})$ is called the **minimal Kac--Moody group** of (simply connected) type $A$ over $\mathbb{K}$. The relation (R1) says that the torus $T_k$ normalises each **real root group** $\mathfrak{U}_{\alpha}(k)$. The relation (R2) says that the elements $\widetilde{s}_i$ normalise $T_k$ (the notation $s_i\alpha_j^{\vee}$ refers to the dual action of $W\leq\mathop{\mathrm{GL}}(\mathfrak{h}^*)$ on $\mathfrak{h}$). The relation (R3) implies that $T_k$ is already contained in the subgroup generated by the real root groups. The relation (R4) says that the elements $\widetilde{s}_i$ permute (by conjugation) the real root groups according to the corresponding action of the Weyl group $W$ on $\Delta^{re}$. We now explain (R0). Note that the product $\prod_{\gamma=i\alpha+j\beta\in ]\alpha,\beta[_{\mathbb{N}}}x_{\gamma}(C^{\alpha\beta}_{ij}t^iu^j)$ only makes sense if the interval $]\alpha,\beta[_{\mathbb{N}}$ is finite and contained in $\Delta^{re}$. A pair of roots $\{\alpha,\beta\}\in\Delta^{re}$ is **prenilpotent** if there exist $v,w\in W$ such that $\{v\alpha,v\beta\}\subseteq\Delta^{re+}$ and $\{w\alpha,w\beta\}\subseteq\Delta^{re-}$. If $\beta\neq\pm\alpha$, this turns out to be equivalent to requiring that $]\alpha,\beta[_{\mathbb{N}}$ be finite and contained in $\Delta^{re}$. Finally, we explain where the constants $C^{\alpha\beta}_{ij}\in\mathbb{Z}$ come from. Using (R4) and the definition of a prenilpotent pair, we may assume that $\alpha,\beta\in\Delta^{re+}$. We would like to be able to write down the exponential $\exp(re_{\alpha})=\sum_{n\geq 0}r^ne_{\alpha}^n/n!$ for $r\in k$ ($k$ a ring) in a suitable space. A natural candidate would be the enveloping algebra of $\mathfrak{n}^+$, except we need to define a $k$-form of this algebra in which the fractions $1/n!$ make sense, and which also allows for infinite sums. **Definition 10**. Let $\mathcal{U}_{\mathbb{C}}(\mathfrak{g}_A)$ be the universal enveloping algebra of $\mathfrak{g}_A$, and consider its $\mathbb{Z}$-subalgebra $\mathcal{U}^+_{\mathbb{Z}}$ generated by the elements $e_i^n/n!$ for $i\in I$ and $n\in\mathbb{N}$. Then $\mathcal{U}^+_{\mathbb{Z}}$ is a $\mathbb{Z}$-form of $\mathcal{U}_{\mathbb{C}}(\mathfrak{n}^+)$, that is, the canonical map $\mathcal{U}^+_{\mathbb{Z}}\otimes_{\mathbb{Z}}\mathbb{C}\to\mathcal{U}_{\mathbb{C}}(\mathfrak{n}^+)$ is an isomorphism. For a ring $k$, one also defines the $k$-form $\mathcal{U}^+_k:=\mathcal{U}^+_{\mathbb{Z}}\otimes_{\mathbb{Z}}k$. Let $\mathcal{U}^+_{k}=\bigoplus_{\alpha\in Q_+}\mathcal{U}^+_{\alpha k}$ be the $Q_+$-gradation of $\mathcal{U}^+_k$ inherited from the $Q_+$-gradation of $\mathfrak{n}^+$, where $Q_+:=\bigoplus_{i\in I}\mathbb{N}\alpha_i$ (note that the $\mathcal{U}^+_{\alpha k}$ are finite-dimensional $k$-modules). Finally, set $\widehat{\mathcal{U}}^+_k:=\prod_{\alpha\in Q_+}\mathcal{U}^+_{\alpha k}$. For each $\alpha\in\Delta^{re+}$ we have $e_{\alpha}^n/n!\in\mathcal{U}^+_{n\alpha\mathbb{Z}}$, so that for any $r\in k$ the exponential $$\exp(re_{\alpha})=\sum_{n\geq 0}r^ne_{\alpha}^n/n!\in (\widehat{\mathcal{U}}^+_k)^{\times}$$ belongs to the group of invertible elements of $\widehat{\mathcal{U}}^+_k$. The integers $C^{\alpha\beta}_{ij}$ can then be computed from the group commutator $[g,h]:=g^{-1}h^{-1}gh$ of the exponentials $\exp(te_{\alpha})$ and $\exp(ue_{\beta})$ inside $\widehat{\mathcal{U}}^+_k$. **Example 11**. Suppose that the simple roots $\{\alpha_i,\alpha_j\}$ form a subsystem of type $A_2$, that is, the corresponding sub-matrix of $A$ is $\begin{psmallmatrix}2&-1\\ -1&2\end{psmallmatrix}$. We then have $]\alpha_i,\alpha_j[_{\mathbb{N}}=\{\alpha_i+\alpha_j\}$ and $e_{\alpha_i}=e_i$, $e_{\alpha_j}=e_j$ and $e_{\alpha_i+\alpha_j}=[e_i,e_j]$. Using the Serre relations $(\mathop{\mathrm{ad}}e_i)^2e_j=0=(\mathop{\mathrm{ad}}e_j)^2e_i$, one can compute in $\widehat{\mathcal{U}}^+_k$ that $$[\exp(te_{\alpha_i}),\exp(ue_{\alpha_j})]=[\exp(te_i),\exp(ue_j)]=\exp(tu[e_i,e_j])=\exp(tue_{\alpha_i+\alpha_j}),$$ so that $C^{\alpha_i\alpha_j}_{11}=1$. Now that we have explained Definition [Definition 9](#definition:constructiveTF){reference-type="ref" reference="definition:constructiveTF"}, we make two remarks, respectively justifying why $\mathfrak{G}_A(\mathbb{K})$ is the right object for $A$ of finite type, and why it is not too small (the presentation does not collapse) over any ring $k$. **Remark 12**. If $A$ is a Cartan matrix and $\mathbb{K}$ a field, then $\mathfrak{G}_A(\mathbb{K})$ is the Chevalley group of (simply connected) type $A$. **Remark 13**. One can extend $\mathcal{U}^+_{\mathbb{Z}}$ (see Definition [Definition 10](#definition:Zform){reference-type="ref" reference="definition:Zform"}) to a $\mathbb{Z}$-form $\mathcal{U}_{\mathbb{Z}}$ of $\mathcal{U}_{\mathbb{C}}(\mathfrak{g}_A)$, and hence for any ring $k$ define a $k$-form $\mathfrak{g}_{Ak}:=(\mathfrak{g}_A\cap\mathcal{U}_{\mathbb{Z}})\otimes_{\mathbb{Z}}k$ of $\mathfrak{g}_A$. As before, we can then define the group $$G_A^{ad}(k):=\langle \exp\mathop{\mathrm{ad}}re_{\alpha} \ | \ r\in k, \ \alpha\in\Delta^{re}\rangle\leq\mathop{\mathrm{Aut}}(\mathfrak{g}_{Ak}).$$ One easily checks that the relations (R0)--(R4) are satisfied in $G_A^{ad}(k)$ (this is how the relations were found!), and so we have an adjoint action map $$\mathop{\mathrm{Ad}}_k\colon\thinspace\mathfrak{G}_A(k)\twoheadrightarrow G_A^{ad}(k)\leq\mathop{\mathrm{Aut}}(\mathfrak{g}_{Ak}).$$ In particular, $\mathfrak{G}_A(k)$ is "big enough" as it admits $G_A^{ad}(k)$ as a quotient (and the same can be done with the highest weight representations $\pi_{\lambda}$ instead of $\mathop{\mathrm{ad}}$). To justify why $\mathfrak{G}_A(\mathbb{K})$ is not too big either over fields $\mathbb{K}$ (i.e. we found sufficiently many relations so that $\mathop{\mathrm{Ad}}_{\mathbb{K}}$ has small, central kernel), we will need the theory of buildings (see the "Prerequisites" notes). # Buildings If $\mathbb{K}$ is a field, it readily follows from the relations (R0)--(R4) that the real root groups $\mathfrak{U}_{\alpha}(\mathbb{K})$ satisfy the axioms of an RGD system (which is no surprise as RGD systems were defined to fit this picture). **Lemma 14**. *Let $\mathbb{K}$ be a field. Then $(\mathfrak{G}_A(\mathbb{K}),(\mathfrak{U}_{\alpha}(\mathbb{K}))_{\alpha\in\Delta^{re}},T_{\mathbb{K}})$ is an RGD system of type $(W,S)$.* The deeper result of this section, which is purely building-theoretic, is that RGD systems yield (twin) BN-pairs, and hence strongly transitive actions on buildings. **Corollary 15**. *Let $\mathbb{K}$ be a field. Set $$\mathfrak{U}^{\pm}_A(\mathbb{K})=\langle \mathfrak{U}_{\alpha}(\mathbb{K}) \ | \ \alpha\in\Delta^{re\pm}\rangle, \quad B^{\pm}=T_{\mathbb{K}}\ltimes U^{\pm}_A(\mathbb{K})\quad\textrm{and}\quad N:=\langle\widetilde{s}_i, \ T_{\mathbb{K}} \ | \ i\in I\rangle.$$ Then $(B^+,N)$ and $(B^-,N)$ are (twinned, saturated) BN-pairs. In particular, $\mathfrak{G}_A(\mathbb{K})$ acts strongly transitively on the associated buildings $X_{\pm}=X(\mathfrak{G}_A(\mathbb{K}),B^{\pm})$, and we have Bruhat decompositions $\mathfrak{G}_A(\mathbb{K})=\coprod_{w\in W}B^{\pm}wB^{\pm}$.* # Axiomatic At this point, we could already justify why $\mathfrak{G}_A(\mathbb{K})$ is not too big, but before we do so, let us also ask ourselves the question whether this group $\mathfrak{G}_A(\mathbb{K})$ is essentially unique, or in other words whether any group deserving the name of "(minimal) Kac--Moody" in the sense of Definition [Definition 5](#definition:KMG){reference-type="ref" reference="definition:KMG"} is already isomorphic to $\mathfrak{G}_A(\mathbb{K})$. Such deserving candidates should be group functors $\mathfrak{G}\colon\thinspace\mathop{\mathrm{{\mathbb Z}-alg}}\to\mathop{\mathrm{Gr}}$ such that $\mathfrak{G}(\mathbb{C})$ has an adjoint action on $\mathfrak{g}_A$ with small, central kernel. In other words, they should come equipped with group functor morphisms $\varphi_i\colon\thinspace\mathop{\mathrm{SL}}_2\to\mathfrak{G}$ ($i\in I$) and $\eta\colon\thinspace T\to \mathfrak{G}$ (respectively exponentiating the fundamental copies $\mathbb{C}e_i+\mathbb{C}\alpha_i^{\vee}+\mathbb{C}f_i$ of $\mathfrak{sl}_2(\mathbb{C})$ and the Cartan subalgebra $\mathfrak{h}'$), such that 1. there is an adjoint action $\mathop{\mathrm{Ad}}_{\mathbb{C}}\colon\thinspace\mathfrak{G}(\mathbb{C})\to \mathop{\mathrm{Aut}}(\mathfrak{g}_A)$, with central kernel contained in $T_{\mathbb{C}}$, such that $$\mathop{\mathrm{Ad}}_{\mathbb{C}}\varphi_i\begin{psmallmatrix}1 &r\\ 0&1\end{psmallmatrix}=\exp\mathop{\mathrm{ad}}re_i, \quad \mathop{\mathrm{Ad}}_{\mathbb{C}}\varphi_i\begin{psmallmatrix}1 &0\\ -r&1\end{psmallmatrix}=\exp\mathop{\mathrm{ad}}rf_i\quad\textrm{and}\quad \mathop{\mathrm{Ad}}_{\mathbb{C}}(\eta(e^{r\alpha_i^{\vee}}))=\exp\mathop{\mathrm{ad}}r\alpha_i^{\vee}$$ for all $r\in\mathbb{C}$ and $i\in I$. Since we want our group to be "minimal", it should also be generated by the fundamental copies of $\mathop{\mathrm{SL}}_2$ and the torus (at least over fields): 1. If $\mathbb{K}$ is a field, $\mathfrak{G}(\mathbb{K})$ is generated by the $\varphi_i(\mathop{\mathrm{SL}}_2(\mathbb{K}))$ ($i\in I$) and $\eta(T(\mathbb{K}))$. Note that these two axioms are not sufficient: over $\mathbb{C}$, the group $\mathfrak{G}(\mathbb{C})$ is small enough thanks to (KMG5), but for $k\neq\mathbb{C}$, we could take for $\mathfrak{G}(k)$ the free product of the $\varphi_i(\mathop{\mathrm{SL}}_2(k))$ ($i\in I$) and $\eta(T(k))$ without violating (KMG1) or (KMG5). To ensure that $\mathfrak{G}(k)$ is also small enough for $k\neq\mathbb{C}$ (at least over fields), we then impose one last axiom: 1. If $k\to\mathbb{C}$ is an injective morphism from a ring $k$ to $\mathbb{C}$, then the corresponding group morphism $\mathfrak{G}(k)\to\mathfrak{G}(\mathbb{C})$ is also injective. **Definition 16**. A triple $(\mathfrak{G},(\varphi_i)_{i\in I},\eta)$ as above satisfying the axioms (KMG1), (KMG4), (KMG5) is called a **Tits functor** of type $A$. Here is now the desired uniqueness statement. Let $\mathop{\mathrm{\mathbb{G}_a}}\colon\thinspace\mathop{\mathrm{{\mathbb Z}-alg}}\to\mathop{\mathrm{Gr}}:k\mapsto (k,+)$ be the additive group functor, and let $x_{\pm}\colon\thinspace\mathop{\mathrm{\mathbb{G}_a}}\to\mathop{\mathrm{SL}}_2$ be the functors defined by $$x_+(r)=\begin{psmallmatrix}1 &r\\ 0&1\end{psmallmatrix}\quad\textrm{and}\quad x_-(r)=\begin{psmallmatrix}1 &0\\ -r&1\end{psmallmatrix}.$$ **Theorem 17**. *Let $(\mathfrak{G},(\varphi_i)_{i\in I},\eta)$ be a Tits functor of type $A$. Then there is a unique morphism of group functors $\pi\colon\thinspace\mathfrak{G}_A\to\mathfrak{G}$ such that the diagrams $$\xymatrix{ T \ar[r] \ar[rd]_{\eta} & \mathfrak{G}_{A} \ar[d]^{\pi}\\ &\mathfrak{G}}\qquad\quad\textrm{and}\qquad\quad \xymatrix{ \mathop{\mathrm{\mathbb{G}_a}}\ar[r]^{x_{\pm \alpha_i}} \ar[rd]_{x_{\pm}} & \mathfrak{U}_{\pm\alpha_i} \ar[r] & \mathfrak{G}_{A} \ar[d]^{\pi}\\ &\mathop{\mathrm{SL}}_2\ar[r]_{\varphi_i} & \mathfrak{G}}$$ are commutative. Moreover, $\pi_{\mathbb{K}}\colon\thinspace\mathfrak{G}_A(\mathbb{K})\to\mathfrak{G}(\mathbb{K})$ is an isomorphism for any field $\mathbb{K}$ (up to kernel contained in $T_{\mathbb{K}}$), provided $\varphi_i(\mathop{\mathrm{SL}}_2(\mathbb{K}))\not\subseteq\pi_{\mathbb{K}}(\mathfrak{U}^+_A(\mathbb{K}))$ for all $i\in I$.[^3]* *Proof.* Here is a sketch proof of this theorem. 1. By (KMG5), we have the morphism $\pi_{\mathbb{C}}\colon\thinspace\mathfrak{G}_A(\mathbb{C})\to\mathfrak{G}(\mathbb{C})$ (as (KMG5) says that essentially, $\mathfrak{G}(\mathbb{C})$ is just $G^{ad}_A(\mathbb{C})$). 2. By (KMG4), we then get the morphism $\pi_k\colon\thinspace\mathfrak{G}_A(k)\to\mathfrak{G}(k)$ for any subring $k$ of $\mathbb{C}$. 3. Using the functoriality of $\mathfrak{G}_A$ and $\mathfrak{G}$, this then yields the desired morphism $\pi_k$ for any ring $k$. 4. Assume now that $k=\mathbb{K}$ is a field. Then $\pi_{\mathbb{K}}$ is surjective by (KMG1). Moreover, $\pi_{\mathbb{K}}$ maps the RGD system of $\mathfrak{G}_A(\mathbb{K})$ to an RGD system of $\mathfrak{G}(\mathbb{K})$ of the same type[^4]. In particular, the kernel of $$\pi_{\mathbb{K}}\colon\thinspace\mathfrak{G}_A(\mathbb{K})=\coprod_{w\in W}B^+wB^+\to\mathfrak{G}(\mathbb{K})=\coprod_{w\in W}\pi_{\mathbb{K}}(B^+)w\pi_{\mathbb{K}}(B^+)$$ lies in $B^+$. Hence $\mathrm{Ker}(\pi_{\mathbb{K}})\subseteq\bigcap_{g\in G}gB^+g^{-1}=T_{\mathbb{K}}$, as desired (where the last equality follows from the fact that the BN-pair $(B^+,N)$ is saturated).  ◻ **Remark 18**. Note that, as shown by the above proof, we need for the uniqueness statement over fields that the Tits functors be defined over rings and not just fields: otherwise, there would for instance be no way to go from $\mathbb{C}$ to a finite field when trying to construct the morphism $\pi$. We also note that the terminology is a bit confusing, in that the constructive Tits functor $\mathfrak{G}_A$ is (probably) not a Tits functor, as it (probably) does not satisfy (KMG4) in full generality (see also Section [7.1](#subsection:Injectivity){reference-type="ref" reference="subsection:Injectivity"}). As the group functor $\mathfrak{G}=G^{ad}_A$ from Remark [Remark 13](#remark:GAadk){reference-type="ref" reference="remark:GAadk"} trivially satisfies the assumptions of Theorem [Theorem 17](#thm:uniquenessTF){reference-type="ref" reference="thm:uniquenessTF"}[^5], we can now justify why $\mathfrak{G}_A(\mathbb{K})$ is not too big for $\mathbb{K}$ a field. **Corollary 19**. *Let $\mathbb{K}$ be a field. Then the adjoint action map $\mathop{\mathrm{Ad}}_{\mathbb{K}}\colon\thinspace\mathfrak{G}_A(\mathbb{K})\twoheadrightarrow G^{ad}_A(\mathbb{K})\leq\mathop{\mathrm{Aut}}(\mathfrak{g}_{A\mathbb{K}})$ has kernel contained in $T_{\mathbb{K}}$.* Since Kac--Moody algebras of affine type are realised as (twisted) loop algebras over a finite-dimensional simple Lie algebra, Theorem [Theorem 17](#thm:uniquenessTF){reference-type="ref" reference="thm:uniquenessTF"} also allows to identify the corresponding minimal Kac--Moody groups over fields. **Corollary 20**. *Let $A=\begin{psmallmatrix}2&-2\\ -2&2\end{psmallmatrix}$. Then $\mathfrak{G}_A(\mathbb{K})\cong\mathop{\mathrm{SL}}_2(\mathbb{K}[t,t^{-1}])\rtimes\mathbb{K}^{\times}$ for $\mathbb{K}$ a field.* # Maximal Kac--Moody groups Let $A=(a_{ij})_{i,j\in I}$ be a GCM and $\mathbb{K}$ be a field. So far, we have constructed a "minimal" Kac--Moody group $\mathfrak{G}_A(\mathbb{K})$ by exponentiating the real root spaces of the Kac--Moody algebra $\mathfrak{g}_A$, and have seen that $\mathfrak{G}_A(\mathbb{K})$ admits the following action maps with kernel contained in $T_{\mathbb{K}}$: 1. $\mathfrak{G}_A(\mathbb{K})\to\mathop{\mathrm{Aut}}(\mathfrak{g}_{A\mathbb{K}})$; 2. $\mathfrak{G}_A(\mathbb{K})\to\mathop{\mathrm{Aut}}(L_{\mathbb{K}}(\lambda))$ for $\lambda\in\mathfrak{h}^*$ dominant integral (where $L_{\mathbb{K}}(\lambda)$ is a $\mathbb{K}$-form of $L(\lambda)$ defined using $\mathcal{U}_{\mathbb{Z}}$); 3. $\mathfrak{G}_A(\mathbb{K})\to\mathop{\mathrm{Aut}}(X_{\pm})$ where $(X_+,X_-)$ is the twin building of $\mathfrak{G}_A(\mathbb{K})$. Going back to the beginning of Section [3](#section:KMG){reference-type="ref" reference="section:KMG"}, we could try a bit harder to define an "adjoint" Kac--Moody group over $\mathbb{C}$ by also exponentiating the adjoint action of imaginary root spaces on $\mathfrak{g}_A$. The problem was that if $\alpha\in\Delta^{im}$ and $x\in\mathfrak{g}_{\alpha}$ is nonzero, then $(\exp\mathop{\mathrm{ad}}x)y=\sum_{n\geq 0}\tfrac{1}{n!}(\mathop{\mathrm{ad}}x)^ny$ is in general an infinite sum for $y\in\mathfrak{g}_A$. To remedy, this, we could try to complete $\mathfrak{g}_A$ to allow such infinite sums: recall that $$\mathfrak{g}_A=\bigoplus_{\alpha\in\Delta\cup\{0\}}\mathfrak{g}_{\alpha}=\mathfrak{n}^-\oplus\mathfrak{h}'\oplus\mathfrak{n}^+.$$ If we allow for infinite sums of homogeneous elements, say $x=\sum_{\alpha}x_{\alpha}$ and $y=\sum_{\beta}y_{\beta}$ with $x_{\beta},y_{\beta}\in\mathfrak{g}_{\beta}$, then we need to be able to define the Lie bracket $[x,y]$, which for each $\gamma\in\Delta$ has homogeneous component of degree $\gamma$ given by $$[x,y]_{\gamma}=[\sum_{\alpha}x_{\alpha},\sum_{\beta}y_{\beta}]_{\gamma}=\sum_{\alpha+\beta=\gamma}[x_{\alpha},x_{\beta}].$$ But this last sum only makes sense if it is finite, and this prevents us from simultaneously allowing infinitely many nonzero homogeneous components of positive and negative degrees. In other words, we have to choose a direction in which to complete, positive or negative: we set $$\widehat{\mathfrak{g}}_A:=\mathfrak{n}^-\oplus\mathfrak{h}'\oplus\widehat{\mathfrak{n}}^+\quad\textrm{where}\quad \widehat{\mathfrak{n}}^+:=\prod_{\alpha\in\Delta^+}\mathfrak{g}_{\alpha}.$$ We could then define[^6] $$G_{A,ad}^{\max}(\mathbb{C}):=\langle \exp\mathop{\mathrm{ad}}x \ | \ x\in\mathfrak{g}_{\alpha}, \ \alpha\in\Delta^{re}\cup\Delta^{im+}\rangle\leq\mathop{\mathrm{Aut}}(\widehat{\mathfrak{g}}_A).$$ More conceptually, for any field $\mathbb{K}$, we could define a completion of $\mathfrak{G}_A(\mathbb{K})$ inside each of the spaces $\mathop{\mathrm{Aut}}(\mathfrak{g}_{A\mathbb{K}})$, $\mathop{\mathrm{Aut}}(L_{\mathbb{K}}(\lambda))$ and $\mathop{\mathrm{Aut}}(X_+)$. **Definition 21**. We define the completions of (the image of) $\mathfrak{G}_A(\mathbb{K})$ inside each the following spaces, with respect to the topology of uniform convergence on bounded sets[^7]: 1. inside $\mathop{\mathrm{Aut}}(\widehat{\mathfrak{g}}_{A\mathbb{K}})$: this is the **algebraic completion** of $\mathfrak{G}_A(\mathbb{K})$, denoted $\mathfrak{G}_A^{\mathrm{alg}}(\mathbb{K})$; 2. inside $\mathop{\mathrm{Aut}}(L_{\mathbb{K}}(\lambda))$: this is the **representation-theoretic completion** of $\mathfrak{G}_A(\mathbb{K})$, denoted $\mathfrak{G}_A^{\mathrm{rt}\lambda}(\mathbb{K})$; 3. inside $\mathop{\mathrm{Aut}}(X_+)$: this is the **geometric completion** of $\mathfrak{G}_A(\mathbb{K})$, denoted $\mathfrak{G}_A^{\mathrm{geo}}(\mathbb{K})$. The metric on each of the spaces $\widehat{\mathfrak{g}}_{A\mathbb{K}}$, $L_{\mathbb{K}}(\lambda)$ and $X_+$ that give a sense to "bounded" sets (and hence to the topology on their automorphism group) is as follows. For $X_+$, the metric is the chamber distance. Since $\widehat{\mathfrak{g}}_{A\mathbb{K}}$ and $L_{\mathbb{K}}(\lambda)$ are $\mathbb{Z}$-graded vector spaces (for the gradation induced by root height), they are also equiped with a natural metric, in which two vectors are close if their difference is a sum of homogeneous elements of high degree. **Example 22**. If $A=\begin{psmallmatrix}2&-2\\ -2&2\end{psmallmatrix}$, then one can check[^8] that the completions of $\mathfrak{G}_A(\mathbb{K})\cong\mathop{\mathrm{SL}}_2(\mathbb{K}[t,t^{-1}])\rtimes\mathbb{K}^{\times}$ coincide with $\mathop{\mathrm{SL}}_2(\mathbb{K}(\!(t)\!))\rtimes\mathbb{K}^{\times}$. **Example 23**. If $\mathbb{K}=\mathbb{F}_q$ is a finite field, then these completions are totally disconnected locally compact (tdlc) groups, which are locally pro-$p$ where $p=\mathop{\mathrm{char}}(\mathbb{K})$. As for minimal Kac--Moody groups, we would also like to define a more intrinsic completion of $\mathfrak{G}_A(\mathbb{K})$, i.e. one that does not depend on an ambiant space. With the experience of Section [3](#section:KMG){reference-type="ref" reference="section:KMG"}, it seems to be a good idea to look at exponentials $\exp(x)$ of homogeneous elements $x\in\mathfrak{g}_{\alpha k}$ with $\alpha\in\Delta^{+}$ inside $\widehat{\mathcal{U}}^+_{k}$, for each ring $k$. **Proposition 24**. *Let $k$ be a ring. For each $\alpha\in\Delta^+$, let $\mathcal{B}_{\alpha}$ be a $\mathbb{Z}$-basis of $\mathfrak{g}_{\alpha\mathbb{Z}}:=\mathfrak{g}_{\alpha}\cap \mathcal{U}^+_{\mathbb{Z}}$, and fix a total order on $\mathcal{B}=\bigcup_{\alpha\in\Delta^+}\mathcal{B}_{\alpha}$. Set[^9] $$\mathfrak{U}^{ma+}_A(k):=\Big\{\prod_{x\in\mathcal{B}}[\exp](\lambda_x x)\in \widehat{\mathcal{U}}^+_k \ | \ \lambda_x\in k\Big\}\subseteq \widehat{\mathcal{U}}^+_k.$$ Then the following assertions hold:* 1. *$\mathfrak{U}^{ma+}_A(k)$ is a subgroup of $(\widehat{\mathcal{U}}^+_k)^{\times}$. In fact, the group functor $\mathfrak{U}^{ma+}_A\colon\thinspace\mathop{\mathrm{{\mathbb Z}-alg}}\to\mathop{\mathrm{Gr}}$ is even an affine group scheme.* 2. *Each element $g\in\mathfrak{U}^{ma+}_A(k)$ has a unique expression $g=\prod_{x\in\mathcal{B}}[\exp](\lambda_x x)$ with $\lambda_x\in k$.* 3. *We have a group morphism $\mathfrak{U}^+_A(k)\to\mathfrak{U}^{ma+}_A(k):x_{\alpha}(\lambda)\mapsto \exp(\lambda e_{\alpha})$, which is injective if $k$ is a field.* 4. *The sets $$\mathfrak{U}^{ma}_n(k):=\Big\{\prod_{x\in\mathcal{B}, \ \mathop{\mathrm{ht}}(\deg(x))\geq n}[\exp](\lambda_x x)\in \widehat{\mathcal{U}}^+_k \ | \ \lambda_x\in k\Big\}$$ for $n\in\mathbb{N}$ are normal subgroups of $\mathfrak{U}^{ma+}_A(k)$ and form a basis of identity neighbourhoods for a complete Hausdorff group topology on $\mathfrak{U}^{ma+}_A(k)$.* 5. *If $\mathop{\mathrm{char}}\mathbb{K}=0$ or $\mathop{\mathrm{char}}\mathbb{K}>M_A:=\max_{i\neq j}|a_{ij}|$, then $\mathfrak{U}^+_A(\mathbb{K})$ is dense in $\mathfrak{U}^{ma+}_A(\mathbb{K})$.* We then define an intrinsic completion of $\mathfrak{G}_A(k)$ in the spirit of Definition [Definition 9](#definition:constructiveTF){reference-type="ref" reference="definition:constructiveTF"}. **Definition 25**. For $k$ a ring, we define the **scheme-theoretic completion** $\mathfrak{G}_A^{\mathrm{sch}}(k)$ of $\mathfrak{G}_A(k)$ via a presentation $$\mathfrak{G}_A^{\mathrm{sch}}(k)=\mathfrak{G}_A(k)*\mathfrak{U}^{ma+}_A(k)/\textrm{(obvious relations)}.$$ We then have a natural inclusion $\mathfrak{G}_A(\mathbb{K})\hookrightarrow \mathfrak{G}_A^{\mathrm{sch}}(\mathbb{K})$ and $\overline{\mathfrak{G}_A(\mathbb{K})}=\mathfrak{G}^{\mathrm{sch}}_A(\mathbb{K})$ as soon as $\mathop{\mathrm{char}}\mathbb{K}=0$ or $\mathop{\mathrm{char}}\mathbb{K}>M_A$. An analogue of the uniqueness statement for maximal Kac--Moody groups is still out of reach in general, but we nevertheless have the following. **Theorem 26**. 1. *The indentity map on $\mathfrak{G}_A(\mathbb{K})$ induces continuous group morphisms[^10] $$\overline{\mathfrak{G}_A(\mathbb{K})}\to \mathfrak{G}_A^{\mathrm{alg}}(\mathbb{K})\to\mathfrak{G}_A^{\mathrm{rt}\lambda}(\mathbb{K})\to \mathfrak{G}_A^{\mathrm{geo}}(\mathbb{K}).$$* 2. *If $\mathop{\mathrm{char}}\mathbb{K}=0$ and $A$ is symmetrisable, these are isomorphisms of topological groups.* 3. *If $\mathbb{K}=\mathbb{F}_q$ is a finite field, these are surjective, and are isomorphisms if and only if[^11] $\mathfrak{G}_A^{\mathrm{sch}}(\mathbb{K})$ is **GK-simple**, in the sense that the kernel $$Z'(\mathfrak{G}_A^{\mathrm{sch}}(\mathbb{K})):=\bigcap_{g\in \mathfrak{G}_A^{\mathrm{sch}}(\mathbb{K})}g(T_{\mathbb{K}}\mathfrak{U}^{ma+}_A(\mathbb{K}))g^{-1}$$ of the $\mathfrak{G}_A^{\mathrm{sch}}(\mathbb{K})$-action on the building $X_+$ is contained in $T_{\mathbb{K}}$.* **Remark 27**. One can also construct a **(positive) maximal Kac--Moody ind-group scheme[^12]** $\mathfrak{G}_A^{\mathrm{pma}}\colon\thinspace\mathop{\mathrm{{\mathbb Z}-alg}}\to\mathop{\mathrm{Gr}}$ such that 1. there is a natural morphism $\mathfrak{G}_A^{\mathrm{sch}}\to\mathfrak{G}_A^{\mathrm{pma}}$ of group functors which is an isomorphism of topological groups over fields. 2. if $A$ is a Cartan matrix, then $\mathfrak{G}_A^{\mathrm{pma}}$ is the Chevalley--Demazure affine group scheme of type $A$. # Open questions {#section:openpb} We collect here a few open questions pertaining to the foundations of the theory. More details can be found in Chapter 8 and 9 of the book. ## Injectivity of the Tits functor {#subsection:Injectivity} First, as noted in Remark [Remark 18](#remark:CTFTF){reference-type="ref" reference="remark:CTFTF"}, it is unclear how far the constructive Tits functor is from being a Tits functor because of the axiom (KMG4). This leads to the following question. **Question 28**. Given a domain $k$ with field of fractions $\mathbb{K}$, when is the natural map $\mathfrak{G}_A(k)\to\mathfrak{G}_A(\mathbb{K})$ injective? This question can also be stated for arbitrary rings as follows: **Question 29**. Given a ring $k$, when is the natural map $\mathfrak{G}_A(k)\to\mathfrak{G}^{\mathrm{pma}}_A(k)$ injective? For $A$ of finite type, a lot of work has been done in that direction (the keywords being "K2-theory for Chevalley groups"). ## GK-simplicity As we saw, having a uniqueness statement for maximal Kac--Moody groups over fields $\mathbb{K}$ (at least when $\mathop{\mathrm{char}}\mathbb{K}=0$ or $\mathop{\mathrm{char}}\mathbb{K}>M_A$) essentially amounts to establishing that $\mathfrak{G}_A^{\mathrm{sch}}(\mathbb{K})$ is GK-simple. **Question 30**. When is $\mathfrak{G}_A^{\mathrm{sch}}(\mathbb{K})$ GK-simple? Note that when $0<\mathop{\mathrm{char}}\mathbb{K}<M_A$, this is in general false, but the hope is that $\mathfrak{G}_A^{\mathrm{sch}}(\mathbb{K})$ would be GK-simple whenever $\mathop{\mathrm{char}}\mathbb{K}>M_A$. When $\mathop{\mathrm{char}}\mathbb{K}=0$, the group $\mathfrak{G}_A^{\mathrm{sch}}(\mathbb{K})$ is GK-simple if $A$ is symmetrisable by Theorem [Theorem 26](#thm:GKsimplicity){reference-type="ref" reference="thm:GKsimplicity"}(2). For $A$ non-necessarily symmetrisable, this is equivalent to proving the Gabber--Kac simplicity Theorem [Theorem 2](#thm:GKac){reference-type="ref" reference="thm:GKac"} for $A$ (whence the terminology "Gabber--Kac simple" or "GK-simple"). When $\mathbb{K}=\mathbb{F}_q$ is a finite field, GK-simplicity amounts to all the completions of $\mathfrak{G}_A(\mathbb{K})$ being isomorphic (see Theorem [Theorem 26](#thm:GKsimplicity){reference-type="ref" reference="thm:GKsimplicity"}(3)). Beyond aiming for a uniqueness statement for maximal Kac--Moody groups over fields, there is another motivation to study the GK-simplicity of $\mathfrak{G}_A^{\mathrm{sch}}(\mathbb{K})$: it amounts to understand the kernel of the map $$\mathfrak{G}_A^{\mathrm{sch}}(\mathbb{K})\to\mathfrak{G}^{\mathrm{geo}}_A(\mathbb{K})\leq\mathop{\mathrm{Aut}}(X_+),$$ and hence the difference between, on the one hand, the group $\mathfrak{G}^{\mathrm{geo}}_A(\mathbb{K})$ which has the nice property of being simple (see §[7.4](#subsection:simplicity){reference-type="ref" reference="subsection:simplicity"}), and on the other hand the group $\mathfrak{G}_A^{\mathrm{sch}}(\mathbb{K})$ whose fine algebraic structure is much more easily accessed thanks to the connection between $\mathfrak{U}^{ma+}_A(\mathbb{K})$ and $\mathfrak{g}_{A\mathbb{K}}$. In practise, this means that the GK-simplicity question is often in the way when trying to prove fundamental properties of maximal Kac--Moody groups (see also §[7.3](#subsection:linearity){reference-type="ref" reference="subsection:linearity"} and [7.5](#subsection:isompb){reference-type="ref" reference="subsection:isompb"} for illustrations of this). On the other hand, when $0<\mathop{\mathrm{char}}\mathbb{K}<M_A$, all hell seems to break loose (non-GK simplicity, non-density of $\mathfrak{G}_A(\mathbb{K})$ in $\mathfrak{G}_A^{\mathrm{sch}}(\mathbb{K})$, exceptional isomorphisms, etc), and it would be nice to understand what exactly is going on. **Question 31**. What is happening in small characteristic??? ## Linearity {#subsection:linearity} A group $G$ is *linear* if there exists a group morphism $\varphi\colon\thinspace G\to\mathop{\mathrm{GL}}_n(F)$ for some field $F$, with central kernel. If $A$ is of finite or affine type, then the groups $\mathfrak{G}_A(\mathbb{K})$ and $\mathfrak{G}_A^{\mathrm{sch}}(\mathbb{K})$ are linear. On the other hand, as soon as $A$ is of indefinite type, the group $\mathfrak{G}_A(\mathbb{K})$ is not linear (with some exceptions for $\mathbb{K}=\mathbb{F}_q$ and $|I|=2$). The following question, on the other hand, is still open: **Question 32**. Suppose $A$ is of indefinite type. When are the groups $\mathfrak{U}^+_A(\mathbb{K})$ and $\mathfrak{U}^{ma+}_A(\mathbb{K})$ (non-)linear? For instance, it is a long-standing open question whether $\mathfrak{U}^{ma+}_A(\mathbb{F}_q)$ can be linear over a local field $F$. Note that if $\mathfrak{G}_A^{\mathrm{sch}}(\mathbb{F}_q)$ were known to be GK-simple, then this question can be answered (by the negative, for $A$ of indefinite type). ## Simplicity {#subsection:simplicity} Suppose that $A$ is indecomposable. It is known that $\mathfrak{G}_A^{\mathrm{sch}}(\mathbb{K})$ is (abstractly) simple modulo its Gabber--Kac kernel for most fields $\mathbb{K}$ (including fields of characteristic zero and finite fields). For the minimal Kac--Moody group $\mathfrak{G}_A(\mathbb{K})$, things are more complicated: it is known that $\mathfrak{G}_A(\mathbb{F}_q)$ is simple modulo center provided $A$ is not of affine type (and with additional mild assumptions on $|I|$ and $q$), but the following question remains open: **Question 33**. Suppose $A$ is of indefinite indecomposable type. Is $\mathfrak{G}_A(\mathbb{K})$ simple modulo centre when $\mathbb{K}$ is a field of characteristic zero? ## Isomorphism problem {#subsection:isompb} Another natural question is the isomorphism problem for Kac--Moody groups. The minimal Kac--Moody group $\mathfrak{G}_A(\mathbb{K})$ determines $A$ when $\mathbb{K}$ has characteristic zero, but when $\mathbb{K}=\mathbb{F}_q$ is a finite field, there are exceptional isomorphisms and $\mathfrak{G}_A(\mathbb{K})$ turns out to carry very little information about $A$. One can ask a similar question for maximal Kac--Moody groups: **Question 34**. Does the topological group $\mathfrak{G}_A^{\mathrm{sch}}(\mathbb{K})$ determine $A$? As the exceptional isomorphisms between minimal Kac--Moody groups over $\mathbb{F}_q$ extend to topological isomorphisms of the corresponding geometric completions, it is important to consider the scheme-theoretic completion $\mathfrak{G}_A^{\mathrm{sch}}(\mathbb{K})$ rather than another completion of $\mathfrak{G}_A(\mathbb{K})$, as it is the completion that seems to carry the most information about $A$, even over finite fields. Again, partial results to answer the above question can be obtained if we knew the groups $\mathfrak{G}_A^{\mathrm{sch}}(\mathbb{K})$ were GK-simple. ## Characterisations of Kac--Moody groups Finally, as mentioned in Section [2](#section:KMA){reference-type="ref" reference="section:KMA"}, there are no known "realisations" of Kac--Moody algebras (and hence groups) when $A$ is of indefinite type. It would be nice to be able to give examples of Kac--Moody groups of indefinite type, in the same way one can give $\mathop{\mathrm{SL}}_n(\mathbb{K}[t,t^{-1}])$ and $\mathop{\mathrm{SL}}_n(\mathbb{K}(\!(t)\!))$ as examples of Kac--Moody groups of affine type. More importantly, we would like to be able to characterise Kac--Moody groups as "building blocks" of various theories, thereby underlining their fundamental nature. This raises the following question, which we then decline in three sub-questions respectively pertaining to the theories of schemes, locally compact groups, and buildings. **Question 35**. Can we characterise (a class of) Kac--Moody groups without reference to Kac--Moody algebras? It is known that the Chevalley--Demazure affine group schemes (which maximal Kac--Moody ind-group schemes generalise by Remark [Remark 27](#remark:indgroupscheme){reference-type="ref" reference="remark:indgroupscheme"}(2)) are precisely the (split, semisimple) affine group schemes over $\mathbb{Z}$. **Question 36**. Are maximal Kac--Moody ind-group schemes the (*suitable adjectives*) ind-group schemes over $\mathbb{Z}$? The quotient of $\mathfrak{G}_A^{\mathrm{sch}}(\mathbb{F}_q)$ by its Gabber--Kac kernel belongs to the class $\mathscr{S}$ of compactly generated, (topologically) simple, non-discrete tdlc groups. This class $\mathscr{S}$ plays a fundamental role in the general structure theory of locally compact groups (with suitable non-discreteness assumptions). Here are two facts about $\mathscr{S}$[^13]: 1. To each group $G$ in $\mathscr{S}$ one can attach in a canonical way a compact space $\Omega_G$ on which $G$ acts nicely. 2. The only known examples of groups $G$ in $\mathscr{S}$ such that $\Omega_G$ is trivial (i.e. reduced to a point) are simple algebraic groups over local fields and (conjecturally) locally compact Kac--Moody groups (namely, maximal Kac--Moody groups over finite fields). In other words, locally compact Kac--Moody groups are the only known examples (at least conjecturally) of groups $G$ in $\mathscr{S}$ that are non-linear and with trivial $\Omega_G$. **Question 37**. Can we characterise locally compact Kac--Moody groups within the class $\mathscr{S}$? Finally, recall that any minimal Kac--Moody group $\mathfrak{G}_A(\mathbb{K})$ acts on a twin building $(X_+,X_-)$. When $A$ is *$2$-spherical* (i.e. $a_{ij}a_{ji}\leq 3$ for all $i\neq j$), the automorphism group $\mathop{\mathrm{Aut}}(X_+,X_-)$ (that is, the group of automorphisms of $X_+\times X_-$ preserving the twinning) is not "too big" compared to $\mathfrak{G}_A(\mathbb{K})$ (or rather, it is too big if $A$ is not $2$-spherical). **Question 38**. Assume that $A$ is $2$-spherical, with corresponding Weyl group $(W,S)$. Can we characterise minimal Kac--Moody groups $\mathfrak{G}_A(\mathbb{K})$ as automorphism groups of twin buildings of type $(W,S)$ (with suitable extra assumptions) ? The answer to this last question is probably already known.[^14] [^1]: $^*$F.R.S.-FNRS Research associate; supported in part by the FWO and the F.R.S.-FNRS under the EOS programme (project ID 40007542). [^2]: here, "minimal" refers to the fact that we only exponentiate the real root spaces and not the imaginary root spaces. [^3]: *This last condition is a "non-degeneracy condition" that prevents degenerate examples.* [^4]: This is where the non-degeneracy condition is used. [^5]: Technically speaking, one has to replace the group functors $\mathop{\mathrm{SL}}_2$ by the elementary subgroup functors $\mathop{\mathrm{E}}_2$. [^6]: The proper way to do it is in fact to take a closure of that group, see Definition [Definition 21](#definition:completions){reference-type="ref" reference="definition:completions"}. [^7]: The correct definition of these three completions of $\mathfrak{G}_A(\mathbb{K})$ is actually a slight modification of the completions defined here, in order to ensure that the torus $T_{\mathbb{K}}$ of $\mathfrak{G}_A(\mathbb{K})$ injects in each of them. [^8]: up to a difference at the level of the torus. [^9]: *The elements $[\exp](\lambda x)$ are called *twisted exponentials*: as we have seen, when $x\in\mathcal{B}_{\alpha}$ with $\alpha\in\Delta^{re+}$, the divided powers $x^n/n!$ belong to $\mathcal{U}^+_{\mathbb{Z}}$ and hence the exponentials $\exp(\lambda x)=\sum_{n\geq 0}\lambda^nx^n/n!$ make sense in $\widehat{\mathcal{U}}^+_k$. However, this is no longer true if $\alpha\in\Delta^{im+}$: in that case, if $k$ is not of characteristic zero, we have to replace exponentials by these twisted exponentials, which are in some sense the "best possible approximations" of exponentials living inside $\widehat{\mathcal{U}}^+_k$.* [^10]: *There is a subtlety for the second arrow when $0<\mathop{\mathrm{char}}\mathbb{K}<M_A$.* [^11]: *or rather, if and only if $\overline{\mathfrak{G}_A(\mathbb{K})}$ is GK-simple.* [^12]: that is, $\mathfrak{G}_A^{\mathrm{pma}}$ is a group functor which is an inductive limit of affine schemes. [^13]: For more details, we refer to the survey paper "[Non-discrete simple locally compact groups](http://dx.doi.org/10.4171/176-1/15)" by Pierre-Emmanuel Caprace. [^14]: See the paper "[Twin buildings and groups of Kac--Moody type](http://dx.doi.org/10.1017/CBO9780511629259.023)" by Jacques Tits, as well as subsequent work of Bernhard Mühlherr.

arxiv_math

--- abstract: | This paper deals with a variant of the optimal transportation problem. Given $f \in L^1( \mathbb{R}^d, [0,1])$ and a cost function $c \in C(\mathbb{R}^d \times \mathbb{R}^d)$ of the form $c(x,y)=k(y-x)$, we minimise $\smallint c \,d\gamma$ among transport plans $\gamma$ whose first marginal is $f$ and whose second marginal is not prescribed but constrained to be smaller than $1-f$. Denoting by $\Upsilon(f)$ the infimum of this problem, we then consider the maximisation problem $\sup \{\Upsilon(f) : \, \smallint f = m \}$ where $m > 0$ is given. We prove that maximisers exist under general assumptions on $k$, and that for $k$ radial, increasing and coercive these maximisers are the characteristic functions of the balls of volume $m$. **Keywords and phrases.** Optimal transport, dual problem, existence of maximisers. **2020 Mathematics Subject Classification.** 49Q22, 49Q20, 49J35. address: - "J. C-T.: Univ. Lille, CNRS, UMR 8524, Inria - Laboratoire Paul Painlevé, F-59000 Lille" - "M.G.: CMAP, CNRS, École polytechnique, Institut Polytechnique de Paris, 91120 Palaiseau, France" - "B.M.: Univ. Lille, CNRS, UMR 8524, Inria - Laboratoire Paul Painlevé, F-59000 Lille" author: - Jules Candau-Tilh - Michael Goldman - Benoit Merlet bibliography: - bib_can_gol_mer.bib title: An exterior optimal transport problem --- # Introduction In this paper, we study the optimization problems associated with functionals which favour dispersion and are based on some Wasserstein energies. These functionals correspond to the non-local term of the energy studied in [@CanGol]. We denote by $\mathcal{M}_+(\mathbb{R}^d)$ the set of positive Radon measures on $\mathbb{R}^d$. Given a cost function $c$ and $\mu, \nu \in \mathcal{M}_+(\mathbb{R}^d)$, we let $\mathcal{T}_c(\mu, \nu)$ be the $c$-transport cost between $\mu$ and $\nu$ (see Section [2](#StdOT){reference-type="ref" reference="StdOT"} for the exact definition of $\mathcal{T}_c$). Given a measurable set $E \subset \mathbb{R}^d$ with finite volume, we consider the optimisation problem $$\label{Upsfunc} \Upsilon_{\mathrm{set}}(E) := \inf\left\{ \mathcal{T}_c(E, F):F\subset \mathbb{R}^d\text{ Lebesgue measurable}, |F|=|E|, \, |F\cap E| = 0\right\}$$ where we identify $E$ with the restriction of the Lebesgue measure on $E$. Given $m >0$, we introduce the maximisation problem $$\label{maxE} \mathcal{E}_{\mathrm{set}}(m) := \sup_{|E|=m} \Upsilon_{\mathrm{set}}(E).$$ The main goal of the article is to investigate the existence of maximisers for this problem and to characterise these latter.\ If we apply the direct method of the Calculus of Variations, we obtain that, up to extraction, any maximising sequence $E_n$ converges weakly to some function $u_\infty\in L^1(\mathbb{R}^d, [0,1])$. However, there is no guarantee at this point that $u_\infty$ is a characteristic function or has mass $m$. Our strategy is to extend the functional $\Upsilon_{\mathrm{set}}$ as a functional $\Upsilon$ defined on $L^1(\mathbb{R}^d, [0,1])$. Applying the bathtub principle (see Proposition [Proposition 27](#prop_bathtub){reference-type="ref" reference="prop_bathtub"}) to a maximiser of the relaxed problem, we show that the supremum in [\[maxE\]](#maxE){reference-type="eqref" reference="maxE"} is actually reached (see Corollary [Corollary 2](#coro_maxE){reference-type="ref" reference="coro_maxE"}). This relaxation approach is not new: it was successfully applied to several variational problems in the last few years (see for instance [@CiDeNoPo15; @BonKnuRog; @PegSanXia; @BuChoTop]). Given $f \in L^1(\mathbb{R}^d, \, [0,1])$, the set of admissible exterior transport plans is defined as $${\varPi}_f :=\left\{\gamma\in\mathcal{M}_+(\mathbb{R}^d \times \mathbb{R}^d) : \gamma_x = f ,\, \gamma_y \le 1-f\right\}.$$ Here, the measures $f\,dx$ and $(1-f)\,dy$ are identified with their respective densities and $\gamma_x$ and $\gamma_y$ denote respectively the first and second marginals of $\gamma$. We then define the primal problem $$\Upsilon(f) :=\inf \left\{\int c\,d\gamma : \gamma\in {\varPi}_f\right\}.$$ We have $\Upsilon(\chi_E)=\Upsilon_{\mathrm{set}}(E)$ under mild assumptions on $c$ (see Theorem [Theorem 20](#thm_saturation){reference-type="ref" reference="thm_saturation"}). Given $m>0$, our maximisation problem is now $$\mathcal{E}(m):=\sup \left\{ \Upsilon(f) : f \in L^1(\mathbb{R}^d, [0,1]), \,\int f\,dx=m\right\}.$$ By abuse of notation and when no confusion is possible, we refer to the variational problems by the values they attain (*e.g.* we write $\Upsilon_{\mathrm{set}}(E)$ for [\[Upsfunc\]](#Upsfunc){reference-type="eqref" reference="Upsfunc"}). ## Main results The first important result of this article is that maximisers of $\mathcal{E}(m)$ exist whenever $c$ is of the form $c(x,y) = k(y-x)$ for some $k: \mathbb{R}^d \to \mathbb{R}_+$ and satisfies 1. [\[cont\]]{#cont label="cont"} $k \in C(\mathbb{R}^d, \mathbb{R}_+)$, $k(0) = 0$ and $k(x) \to \infty$ as $|x| \to \infty$, 2. [\[cone\]]{#cone label="cone"} $\forall x \not = 0$, $$\limsup_{r \to 0} \frac{1}{ r^d} \big|B_r(x) \cap \{y \in \mathbb{R}^d, \, k(y) < k(x)\} \big| > 0,$$ 3. [\[monot\]]{#monot label="monot"} $\forall \, \sigma \in \mathbb{S}^{d-1}$, $r \mapsto k(r\sigma)$ is increasing on $\mathbb{R}_+$. Notice that $k$ is not assumed to be strictly convex, so that our results hold in cases where the existence of an optimal transport map is not guaranteed. Also observe that all the costs of the form $k(z) = |z|^p$ with $0 < p < \infty$ satisfy the above hypotheses. However, radial symmetry is not required and the costs $k(z)= |z|^p \,h(z/|z|)$ with $h$ positive and Lipschitz continuous on $\mathbb{S}^{d-1}$ are also admissible. **Theorem 1**. *Assume that $c(x,y) = k(y-x)$ for $x,y\in \mathbb{R}^d$ with $k$ satisfying (H[\[cont\]](#cont){reference-type="ref" reference="cont"}),(H[\[cone\]](#cone){reference-type="ref" reference="cone"})&(H[\[monot\]](#monot){reference-type="ref" reference="monot"}). Then, for any $m>0$ the supremum in $\mathcal{E}(m)$ is attained. Moreover, there exists $R_* = R_*(m)$ such that (up to translation) any maximiser is supported in the ball $\overline B_{R_*}$.* Once the existence of maximisers for $\mathcal{E}(m)$ is established, the bathtub principle (see Proposition [Proposition 27](#prop_bathtub){reference-type="ref" reference="prop_bathtub"}) and a saturation result (see Theorem [Theorem 20](#thm_saturation){reference-type="ref" reference="thm_saturation"}) imply that [\[maxE\]](#maxE){reference-type="eqref" reference="maxE"} admits solutions. **Corollary 2**. *Assume that $c$ satisfies the hypotheses of Theorem [Theorem 1](#thm_MP_max){reference-type="ref" reference="thm_MP_max"}. Then, [\[maxE\]](#maxE){reference-type="eqref" reference="maxE"} admits a maximiser and $\mathcal{E}_{\mathrm{set}}(m) = \mathcal{E}(m)$ for any $m>0$.* As a second main result, we establish that if $k$ is furthermore radially symmetric then $\mathcal{E}(m)$ and $\mathcal{E}_{\mathrm{set}}(m)$ are uniquely maximised by balls of volume $m$. **Theorem 3**. *Assume that $c(x,y)= k(|y-x|)$ for some $k \in C(\mathbb{R}_+, \mathbb{R}_+)$ increasing and such that $k(0) = 0$ and $k(x) \to \infty$ as $x \to \infty$. Then, for any $m>0$, the maximisers of $\mathcal{E}(m)$ (and consequently those of $\mathcal{E}_{\mathrm{set}}(m)$) are the balls of volume $m$.* We point out that cost functions satisfying the hypotheses of Theorem [Theorem 3](#thm_ball_unique_max){reference-type="ref" reference="thm_ball_unique_max"} also satisfy hypotheses (H[\[cont\]](#cont){reference-type="ref" reference="cont"}),(H[\[cone\]](#cone){reference-type="ref" reference="cone"})&(H[\[monot\]](#monot){reference-type="ref" reference="monot"}). Let us briefly sketch the proofs of these three results. They all strongly rely on the properties of the dual problem $$\Upsilon^*(f):=\sup\left\{\int \left(f \varphi+ (1-f) \psi \right) \, dx : (\varphi,\psi) \in \Phi \right\},$$ where $$\Phi := \left\{(\varphi, \psi) \in C_b(\mathbb{R}^d) \times C_b(\mathbb{R}^d), \, \psi \leq 0, \, \varphi(x)+\psi(y) \leq c(x,y) \ \forall \, (x,y) \in \mathbb{R}^d \times \mathbb{R}^d \right\}.$$ We establish Theorem [Theorem 1](#thm_MP_max){reference-type="ref" reference="thm_MP_max"} using the direct method of Calculus of Variations. The main difficulty is to establish compactness of maximising sequences. If we refer to the concentration-compactness principle [@PLLconccomp], we have to prove that given a maximising sequence $f_n$, no mass escapes at infinity. To do so we establish two crucial results. The first one is that $m \mapsto \mathcal{E}(m)/m$ is increasing (see Proposition [Proposition 23](#prop_mu_inc){reference-type="ref" reference="prop_mu_inc"}). This implies that $m \mapsto \mathcal{E}(m)$ is strictly superadditive, *i.e.* that for $m>m'>0$, $$\label{stricsubadd} \mathcal{E}(m) + \mathcal{E}(m-m')< \mathcal{E}(m).$$ Notice that this is the counterpart of the strict subbadditivity inequality (also called binding inequality) which is known to provide compactness in minimisation problems, see *e.g.* [@PLLconccomp; @FrankLieb; @frankNam]. Using the dual formulation $\Upsilon^*$ of $\Upsilon$, we obtain the second crucial result for Theorem [Theorem 3](#thm_ball_unique_max){reference-type="ref" reference="thm_ball_unique_max"}: a monotonicity principle on the sum of marginals of maximisers $\gamma$ of $\Upsilon(f)$ (see Corollary [Corollary 22](#coro_monot_f){reference-type="ref" reference="coro_monot_f"}). Combining this and [\[stricsubadd\]](#stricsubadd){reference-type="eqref" reference="stricsubadd"}, we prove that if $f$ is almost maximising then most of its mass must remain in a bounded region (see Proposition [Proposition 25](#prop_strong_tightness){reference-type="ref" reference="prop_strong_tightness"}). This gives tightness of maximising sequences for $\mathcal{E}(m)$. To prove Corollary [Corollary 2](#coro_maxE){reference-type="ref" reference="coro_maxE"}, we consider a maximiser $f$ of $\mathcal{E}(m)$ provided by Theorem [Theorem 1](#thm_MP_max){reference-type="ref" reference="thm_MP_max"} and a pair of potentials $(\varphi, \psi)$ optimal for the dual problem $\Upsilon^*(f)$. Using the definition of $\Upsilon^*$ we see that $f$ is a maximiser of $$\sup\left\{\int \widetilde{f} (\varphi- \psi)\, : 0 \leq \widetilde{f} \,\leq 1, \, \int \widetilde{f}\, = m\right\}.$$ By the bathtub principle, $f = \chi_{\{\varphi-\psi > t\}} + \theta$ for some $t \in \mathbb{R}$ and some $\theta \in L^1(\mathbb{R}^d, [0,1])$ supported in $\{\varphi-\psi = t \}$. Then for any measurable subset $G\subset \{\varphi- \psi = t\}$ with $|G|=\int\theta\,$, the characteristic function of $E:=\{\varphi-\psi > t\}\cup G$ is also a maximiser for $\Upsilon(f)$. By Theorem [Theorem 20](#thm_saturation){reference-type="ref" reference="thm_saturation"} and Corollary [Corollary 21](#coro_unique_g){reference-type="ref" reference="coro_unique_g"} applied to $E$, there exists $F \subset \mathbb{R}^d$ such that any minimiser $\gamma$ of $\Upsilon(\chi_E)$ satisfies $\gamma_y = \chi_F$. This finally implies that $E$ maximises [\[maxE\]](#maxE){reference-type="eqref" reference="maxE"}. Regarding Theorem [Theorem 3](#thm_ball_unique_max){reference-type="ref" reference="thm_ball_unique_max"}, as explained in Section [5](#ballisunique){reference-type="ref" reference="ballisunique"}, we may assume without loss of generality that $m = \omega_d$, the volume of the unit ball. Combining Theorem [Theorem 1](#thm_MP_max){reference-type="ref" reference="thm_MP_max"} and Lemma [Lemma 26](#lem_ctransfglobal){reference-type="ref" reference="lem_ctransfglobal"} yields that $$\label{doublesup} \sup_{\smallint f\, = \omega_d} \left\{ \sup_{(\psi^c, \psi) \in \Phi} \left\{ \int f (\psi^c - \psi)\, + \int \psi\, \right\}\right\},$$ coincides with $\mathcal{E}(m)$ and admits a solution $(f, \psi^c, \psi)$, where $\psi^c$ is the $c$-transform of $\psi$ (see Definition [Definition 5](#def_ctransf){reference-type="ref" reference="def_ctransf"}). To show that balls are maximisers of $\mathcal{E}(m)$, we establish that each term in [\[doublesup\]](#doublesup){reference-type="eqref" reference="doublesup"} is improved by replacing $f$ by $\chi_{B_1}$ and $\psi$ by its symmetric increasing rearrangement $\psi_*$ (see Definition [\[def_sym_rear\]](#def_sym_rear){reference-type="ref" reference="def_sym_rear"}). As $\smallint \psi\, = \smallint \psi_*\,$, the third term in [\[doublesup\]](#doublesup){reference-type="eqref" reference="doublesup"} does not change under rearrangement. Regarding the second term, combining the Hardy-Littlewood inequality (see [@LiebLoss Theorem 3.4]) and the bathtub principle yields (recall that $\psi \leq 0$) $$-\int f \psi\, \leq -\int f^* \psi_*\, \leq -\int \chi_{B_{1}} \psi_*\,,$$ where $f^*$ is the symmetric decreasing rearrangement of $f$ (see Definition [\[def_sym_rear\]](#def_sym_rear){reference-type="ref" reference="def_sym_rear"}). The study of the first term $\smallint f \psi^c\,$ is more involved. Using the Brunn-Minkowski inequality, we obtain the following crucial comparison: $$(\psi^c)^* \leq (\psi_*)^c.$$ Combining this inequality with the Hardy-Littlewood inequality yields $$\label{introHL} \int f \psi^c\, \leq \int f^* (\psi^c)^*\, \leq \int f^* (\psi_*)^c\,.$$ Additionally, as $(\psi_*)^c$ is non-increasing, $\chi_{B_1}$ is a maximiser of $$\label{introbathtub} \sup \left\{\int \widetilde{f} (\psi_*)^c\,: 0 \leq \widetilde f \leq 1, \, \int \widetilde{f}\, = \omega_d \right\},$$ so that $\smallint f^* (\psi_*)^c\, \leq \smallint (\psi_*)^c \chi_{B_{1}}\,$. Lastly, by [\[introHL\]](#introHL){reference-type="eqref" reference="introHL"}, $\smallint f \psi^c\,\leq \smallint \chi_{B_{1}} (\psi_*)^c\,$. This eventually proves that unit balls maximise $\mathcal{E}(\omega_d)$. As for uniqueness, the key property to establish is that $(\psi_*)^c$ is decreasing on $B_{1}$ (see Lemma [Lemma 30](#lem_monotpotball){reference-type="ref" reference="lem_monotpotball"}). Indeed, by [@LiebLoss Theorem 3.4], this implies that $\chi_{B_{1}}$ is the unique maximiser of [\[introbathtub\]](#introbathtub){reference-type="eqref" reference="introbathtub"}. Combining this with the fact that the inequalities in [\[introHL\]](#introHL){reference-type="eqref" reference="introHL"} are now equalities, we obtain that $f^* = \chi_{B_{1}}$, so that $f = \chi_{E}$ for some $E \subset \mathbb{R}^d$. Using the equality case of the Brunn-Minkowski inequality, we then show that (up to a translation) $f = \chi_{B_{1}}$, concluding the proof. ## Motivation In [@BCL2020], the following variational problem was introduced: $$\label{singleOptim} \inf_{|E| = \omega_d} \left\{ P(E) + \alpha \Upsilon_p(E) \right\},$$ where $\alpha >0$ and where $\Upsilon_p$ is the functional $\Upsilon$ defined in [\[Upsfunc\]](#Upsfunc){reference-type="eqref" reference="Upsfunc"} with the cost $c(x,y) = |x-y|^p$. Such a variational problem may be used to model the formation of bi-layer biological membranes (see [@PeRo09; @LuPeRo14]). Existence of minimisers were obtained in the series of work [@BCL2020; @XiaZhou; @NoToVenk; @CanGol]. Notice that [\[singleOptim\]](#singleOptim){reference-type="eqref" reference="singleOptim"} is an isoperimetric problem with a non-local term $\Upsilon_p$. One of the best-known examples of this type of problem is Gamow's liquid drop model for the atomic nucleus. Since the beginning of the 2010s (see [@gamowhist] for an historical perspective), this model has received a lot of attention from the mathematical community, and several versions of it have been studied, see for instance [@KnMu; @golnovruf; @KnMuNov; @GolMerPe]. In this framework, the perimeter term represents the local attractive forces while the repulsive non-local term is given by the Riesz potential $$V_\beta(E) := \int_E\int_E \frac{dxdy}{|x-y|^{d-\beta}},$$ where $\beta \in (0,d)$. A consequence of Riesz's rearrangement inequality is that balls are the volume constrained maximisers of $V_{\beta}$. This illustrates the competition between the perimeter and the Riesz potential. It is thus natural to investigate whether similar properties hold for [\[singleOptim\]](#singleOptim){reference-type="eqref" reference="singleOptim"}. In our case, the proof is much more involved since the rearrangement argument does not seem to work well for the primal problem. We consider instead the dual problem $\Upsilon^*$ and study the (fortunately favourable) interplay between rearrangement and $c-$transforms. As a closing remark, we point out that the functional $\Upsilon$ is a particular case of the optimal partial transport problem studied in [@FigPartial; @DePMSV]. ## Organization of the article The paper is structured as follows. In Section [2](#StdOT){reference-type="ref" reference="StdOT"}, we introduce the notation and review standard facts related to optimal transport in complete separable metric spaces. In Section [3](#Xcompact){reference-type="ref" reference="Xcompact"}, we obtain preliminary results on the functional $\Upsilon$ defined in compact spaces. In Section [4](#trslinv){reference-type="ref" reference="trslinv"}, we establish Theorem [Theorem 1](#thm_MP_max){reference-type="ref" reference="thm_MP_max"}. Eventually, in Section [5](#ballisunique){reference-type="ref" reference="ballisunique"}, we prove Theorem [Theorem 3](#thm_ball_unique_max){reference-type="ref" reference="thm_ball_unique_max"}. # Notation and preliminary results {#StdOT} ## Notation Let $(X, d_X)$ be a Polish space endowed with a positive Radon measure $\lambda$. Given a function $f : X \to \mathbb{R}$, we decompose it as: $$f = f_+ + f_- \quad \text{ with } \quad f_+ := \max(0, f) := 0 \vee f \quad\ \text{and}\ \quad f_- := \min(0, f) := 0 \wedge f.$$ Let us stress that $f_-$ is non-positive, contrary to the classical decomposition of a function into its positive and negative parts. We endow $\mathcal{M}_+(X)$ with the topology of weak-$*$ convergence, that is the topology induced by duality with $C_b(X)$. The convergence of a sequence $\mu_n \in \mathcal{M}_+(X)$ to $\mu \in \mathcal{M}_+$ is written: $\mu_n \stackrel*\rightharpoonup \mu$ as $n \to \infty$. Given a measure $\mu \in \mathcal{M}_+(X)$ and a set $A \subset X$, the restriction of $\mu$ to $A$ is the measure $\mu \hskip 2.5pt{\vrule height7pt width.5pt depth0pt} \hskip-.2pt\vbox{\hrule height.5pt width7pt depth0pt} \, A$ defined as $\mu \hskip 2.5pt{\vrule height7pt width.5pt depth0pt} \hskip-.2pt\vbox{\hrule height.5pt width7pt depth0pt} \, A (B) := \mu (B \cap A)$ for every Borel set $B$ of $X$. The support of $\mu$, denoted by $\mathop{\mathrm{supp}}\mu$, is the closed set defined by $$\mathop{\mathrm{supp}}\mu := \left\{x \in X : \mu(A) > 0 \text{ for all open set A containing } x\right\}.$$ Given $f \in L^1(X, \lambda)$ the support of $f$ is defined as the support of the measure $f d\lambda$ and denoted by $\mathop{\mathrm{supp}}f$. We identify the measure $f d\lambda$ with its density $f$ and write $f_n\stackrel*\rightharpoonup f$ as $n \to \infty$ to signify that $\int f_n\xi\,$ converges to $\int f\xi\,$ for every $\xi \in C_b(X)$. Given a function $f \in L^1_{\mathrm{loc}}(\mathbb{R}^d, \mathbb{R})$, we denote by $\mathrm{Leb}(f)$ the set of its Lebesgue points. Given $x \in \mathbb{R}^d$ and $r >0$, $B_r(x)$ denotes the open ball of radius $r$ centred at $x$, and $B_r$ denotes the open ball of radius $r$ centred at $0$. The closed ball of radius $r$ centred at $x$ is denoted by $\overline{B}_r(x)$. The volume of the unit ball in $\mathbb{R}^d$ is denoted by $\omega_d$. Given two sets $A, B$ of $\mathbb{R}^d$, we define their sum $A+B:= \{a +b, \, a \in A, \, b \in B\}$. The gap between $A$ and $B$ is $d(A,B) := \inf \{|a-b|, \, a \in A, \, b \in B\}$. ## Optimal transport theory In this subsection, we recall some results regarding standard optimal transport theory. Most of the material presented here comes from [@OTAM Chapter 1]. Let $(X, d_X)$ be a complete separable metric space (*i.e. *a Polish space) and let $c : X \times X \to \mathbb{R}$ be measurable. Given $\mu, \nu \in \mathcal{M}_+(X)$ such that $\mu(X) = \nu(X)$, the Kantorovitch problem with marginals $\mu$ and $\nu$ and cost $c$ is $$\label{KP} \mathcal{T}_c(\mu, \nu) := \inf \left\{\int c \, d\gamma : \, \gamma \in {\varPi}(\mu, \nu)\right\},$$ where ${\varPi}(\mu, \nu)$ is the set of transport plans between $\mu$ and $\nu$, *i.e. * $${\varPi}(\mu, \nu) := \left\{ \gamma \in \mathcal{M}_+(X \times X) : \, \gamma_x = \mu, \, \gamma_y = \nu \right\}.$$ Problem [\[KP\]](#KP){reference-type="eqref" reference="KP"} admits a dual formulation given by $$\label{DKP} \mathcal{T}_c^*(\mu, \nu) := \sup \left\{\int \varphi\, d\mu +\int \psi \, d\nu : \, \varphi, \psi \in C_b(X), \, \varphi\oplus \psi \leq c\right\},$$ where the function $\varphi\oplus \psi$ is defined on $X \times X$ by $(\varphi\oplus \psi)(x,y):= \varphi(x)+\psi(y)$. **Theorem 4** (Theorem 1.7 of [@OTAM]). *Let $c : X \times X \to \mathbb{R}$ be lower semi-continuous and bounded from below and let $\mu, \nu \in \mathcal{M}_+(X)$ with $\mu(X) = \nu (X)$. Then [\[KP\]](#KP){reference-type="eqref" reference="KP"} admits a solution and $$\mathcal{T}_c(\mu, \nu) = \mathcal{T}_c^*(\mu, \nu).$$* Using the notion of $c$-transform of a function, the maxima of [\[DKP\]](#DKP){reference-type="eqref" reference="DKP"} can be further characterised. **Definition 5**. Given a function $\xi : X \to \mathbb{R}\cup \{+\infty\}$, we define its $c$-transform (or $c$-conjugate) $\xi^c : X \to \mathbb{R}\cup \{-\infty\}$ by $$\xi^c(y) := \inf_{x \in X} \left\{c(x,y) - \xi(x) \right\}.$$ Denoting $\bar c(y,x) := c(x,y)$, the $\bar c$-transform of $\zeta : X \to \mathbb{R}\cup \{+\infty\}$ is given by $$\zeta^{\bar c}(x) := \inf_{y \in X} \left\{\bar c(y,x) - \zeta(y) \right\}.$$ A function $\psi : X \to \mathbb{R}\cup \{-\infty\}$ is said to be $\bar c$-concave if there exists $\xi :X \to \mathbb{R}\cup \{+\infty\}$ such that $\psi = \xi^c$ (the definition of $c$-concavity is analogous). **Definition 6**. Let $(X, d_X)$ be a metric space and $\omega\in C(\mathbb{R}_+, \mathbb{R}_+)$ be increasing and such that $\omega(0) = 0$. A function $\varphi: X \to \mathbb{R}$ is $\omega$-continuous if for all $x, x' \in X$, $$|\varphi(x) - \varphi(x')| \leq \omega(d_X(x, x')).$$ Similarly, we say that $c : X \times X \to \mathbb{R}$ is $\omega$-continuous if for all $x, x', y, y' \in X$, $$|c(x,y)-c(x', y')| \leq \omega(d_X(x,x')+d_X(y,y')).$$ **Proposition 7**. *Let $\varphi, \psi : X \to \mathbb{R}$ be fixed and assume that $\varphi^c$ and $\psi^{\bar c}$ take real values. The following statements hold:* (i) *If $c$ is $\omega$-continuous, then $\varphi^c$ is also $\omega$-continuous,* (ii) *$\varphi^{c \bar c} \geq \varphi$, and $\varphi^{c \bar c} = \varphi$ if and only if $\varphi$ is $c$-concave,* (iii) *$\varphi^c$ is the largest function $\psi$ compatible with the constraint $\varphi\oplus \psi \leq c$ and $\psi^{\bar c}$ is the largest function $\varphi$ compatible with the constraint $\varphi\oplus \psi \leq c$.* Remark that if $X$ is compact and $\varphi$, $\psi$ and $c$ are bounded then $\varphi^c$ and $\psi^{\bar c}$ take real values. Moreover, if $c$ is continuous, say $\omega$-continuous, the proposition states that $\varphi^c$ and $\psi^{\bar c}$ are $\omega$-continuous. This yields the following existence result for [\[DKP\]](#DKP){reference-type="eqref" reference="DKP"}. **Theorem 8** (Proposition 1.11 of [@OTAM]). *Let $X$ be a compact metric space and $c : X \times X \to \mathbb{R}$ be continuous. Then there exists a solution $(\varphi, \psi)$ to [\[DKP\]](#DKP){reference-type="eqref" reference="DKP"}, where $\varphi$ is $c$-concave and $\psi = \varphi^c$. In particular, $$\mathcal{T}^c_*(\mu, \nu) = \max \left\{\int \varphi\, d\mu + \int \varphi^c \, d\nu :\, \varphi\, \text{ c-concave} \right\}.$$* A pair of functions maximising [\[DKP\]](#DKP){reference-type="eqref" reference="DKP"} is called a pair of Kantorovitch potentials. # Study of $\Upsilon$ in compact metric spaces {#Xcompact} Let $(X, d_X)$ be a compact metric space and let $c : X \times X \to \mathbb{R}$ be a continuous cost function. We endow $(X, d_X)$ with a measure $\lambda \in \mathcal{M}_+(X)$ such that $\lambda(X)>0$ and denote by $L^1(X)$ the set of $\mathbb{R}$-valued functions integrable with respect to $\lambda$. Given $f \in L^1(X)$, we define the set of admissible transport plans $${\varPi}_f :=\left\{\gamma\in\mathcal{M}_+(X\times X) : \gamma_x = f,\, \gamma_y \le 1-f \right\}$$ and the primal problem $$\label{PPXcomp} \Upsilon(f) :=\inf \left\{\int c\,d\gamma : \gamma\in {\varPi}_f\right\}.$$ Notice that ${\varPi}_f$ is empty whenever $f$ does not satisfy $0 \leq f \leq 1$ or when $\smallint f\,d\lambda > \lambda(X)/2$. In the other cases, there exists $g \in L^1(X)$ such that $g\ge0$, $f+g\le1$ and $\smallint g \, d\lambda=\smallint f \,d\lambda$. Thus, $$\gamma:= \frac{1}{\smallint f \, d\lambda}(f \, d\lambda)\otimes (g \,d\lambda)\ \in{\varPi}_f,$$ and ${\varPi}_f$ is not empty. We now fix $0 < m \leq \lambda(X)/2$ and define $$L^1_m := \left\{f \in L^1(X, [0,1]) : \int f\, \leq m \right\}.$$ Given $f \in L^1_m$ and $\varphi, \psi \in C(X)$, we set $$\label{Kf} K_f(\varphi, \psi) := \int \left(f \varphi+ (1-f) \psi \right) \, d\lambda$$ and define the dual problem $$\label{DPXcomp} \Upsilon^*(f):=\sup\left\{K_f(\varphi, \psi) : (\varphi, \psi) \in \Phi \right\},$$ where $$\Phi := \left\{(\varphi, \psi) \in C(X) \times C(X), \, \psi \leq 0, \, \varphi\oplus \psi \leq c \right\}.$$ For the remainder of the section we fix $f \in L^1_m$. As in the classical theory of optimal transport, a simple application of the direct method of Calculus of Variations shows that [\[PPXcomp\]](#PPXcomp){reference-type="eqref" reference="PPXcomp"} admits a minimiser. **Proposition 9**. *Assume that $X$ is a compact metric space and that $c \in C(X \times X, \mathbb{R})$. Then, the infimum in [\[PPXcomp\]](#PPXcomp){reference-type="eqref" reference="PPXcomp"} is a minimum.* **Remark 10**. If we let $f \in L^1_m$, by Proposition [Proposition 9](#prop_exis_gamma_opt){reference-type="ref" reference="prop_exis_gamma_opt"}, there exists $\gamma \in \mathcal{M}_+(X \times X)$ optimal for $\Upsilon(f)$. Notice that $\gamma$ solves the classical optimal transport problem from $f$ towards $g := \gamma_y$ defined by [\[KP\]](#KP){reference-type="eqref" reference="KP"}. Moreover, we have the identity $\Upsilon(f) = \mathcal{T}_c(f,g)$. Let us now show that $\Upsilon^*(f)=\Upsilon(f)$ and that [\[DPXcomp\]](#DPXcomp){reference-type="eqref" reference="DPXcomp"} admits a maximising pair $(\varphi,\psi)$. We first establish that we can reduce the set of competitors for [\[DPXcomp\]](#DPXcomp){reference-type="eqref" reference="DPXcomp"}. To simplify the notation we denote by $\varphi^c_{\,-}$ the function $(\varphi^c)_- := \varphi^c \wedge 0$. **Lemma 11**. *Assume that $X$ is a compact metric space and that $c \in C(X \times X, \mathbb{R})$. Then, there holds $$\label{DPp} \Upsilon^*(f)=\sup\,\{K_f(\psi^{\bar c}, \psi) : \psi = \varphi^c_{\,-} \text{ for some } \varphi\in \Phi' \},$$* *where $$\label{defPhiprime} \Phi' := \left\{\varphi\in C(X), \, \varphi= (\varphi^c_{\,-})^{\bar c}, \; \max \varphi^c \geq 0 \right\}.$$* *Proof.*  \ *Step 1. We can replace $\psi$ by $\varphi^c_{\,-}$ and assume that $\max \varphi^c \geq 0$.* Let $(\varphi,\psi)\in \Phi$. By Proposition [Proposition 7](#prop_ctransf){reference-type="ref" reference="prop_ctransf"} $(iii)$, $\psi\le\varphi^c$, so that $\psi\le\varphi^c\wedge0=\varphi^c_{\,-}$. As $1-f\ge0$, $K_f(\varphi,\varphi^c_{\,-})\ge K_f(\varphi,\psi)$. Therefore, we can restrict the maximisation to the pairs $(\varphi,\varphi^c_{\,-})$ in the supremum [\[DPXcomp\]](#DPXcomp){reference-type="eqref" reference="DPXcomp"}. Now, if $\max \varphi^c=-t<0$ we set $\widetilde \varphi:=\varphi-t$ so that ${\widetilde\varphi}^c=\varphi^c+t$. Consequently, $\max\widetilde\varphi^c=0$ and in particular, ${\widetilde\varphi}^c_{\,-}={\widetilde\varphi}^c$ so that $({\widetilde\varphi},{\widetilde\varphi}^c)\in\Phi$. We then compute $$\begin{aligned} K_f(\widetilde\varphi,\widetilde\varphi^c_{\-}) &=\int f(\widetilde\varphi-\widetilde\varphi^c)\,d\lambda+\int \widetilde\varphi^c\,d\lambda\\ &=\int f(\varphi-\varphi^c)\,d\lambda+\int \varphi^c\,d\lambda + t(\lambda(X) -2m)\\ &=K_f(\varphi,\varphi^c_{\,-})+ t(\lambda(X)-2m). \end{aligned}$$ As $2m \leq \lambda(X)$ we obtain $K_f(\widetilde\varphi,\widetilde\varphi^c_{\,-}) \geq K_f(\varphi,\varphi^c_{\,-})$. Hence $$\Upsilon^*(f)=\sup \left\{K_f(\varphi,\varphi^c_{\,-}) : \varphi\in C(X), \, \max \varphi^c \geq 0 \right\}. \smallskip$$ *Step 2. There holds $\varphi= (\varphi^c_{\,-})^{\bar c}$.* Let us introduce the mapping $P:C(X)\to C(X)$ defined by $P(\varphi):=(\varphi^c_{\,-})^{\bar c}$. For $\varphi\in C(X)$, $\varphi^c\ge\varphi^c_{\,-}$, so that $P(\varphi)=(\varphi^c_{\,-})^{\bar c}\ge \varphi^{c \bar c}$. By Proposition [Proposition 7](#prop_ctransf){reference-type="ref" reference="prop_ctransf"} $(ii)$, $\varphi^{c \bar c} \geq \varphi$, hence $$\label{proof_lem_Phi_1} P(\varphi)\ge \varphi.$$ By Proposition [Proposition 7](#prop_ctransf){reference-type="ref" reference="prop_ctransf"} $(ii)$ again there holds $P(\varphi)^c=(\varphi^c_{\,-})^{\bar cc} \ge \varphi^c_{\,-}$. Taking the negative part yields $$\label{proof_lem_Phi_2} P(\varphi)^c_{\,-} \geq \varphi^c_{\,-}.$$ We deduce from [\[proof_lem_Phi_1\]](#proof_lem_Phi_1){reference-type="eqref" reference="proof_lem_Phi_1"} and [\[proof_lem_Phi_2\]](#proof_lem_Phi_2){reference-type="eqref" reference="proof_lem_Phi_2"} that $$K_f(P(\varphi),P(\varphi)^c_{\,-}) \ge K_f(\varphi,\varphi^c_{\,-}).$$ Now, we observe that if $\max \varphi^c \geq 0$ we also have $\max \varphi^c_{\,-} = 0$ and, by [\[proof_lem_Phi_2\]](#proof_lem_Phi_2){reference-type="eqref" reference="proof_lem_Phi_2"}, $\max P(\varphi)^c_{\,-} = 0$ which implies that $\max {P(\varphi)^c} \geq 0$. Hence, $$\label{proof_lem_Phi_25} \Upsilon^*(f)=\sup \left\{K_f(\widetilde \varphi, \widetilde \varphi^c_{\,-}) : \widetilde\varphi\in C(X), \, \max \widetilde \varphi^c \geq 0,\, \widetilde \varphi=P(\varphi)\text{ for some } \varphi\in C(X) \right\}.$$ To conclude, we show that $P(P(\varphi)) = P(\varphi)$ for any $\varphi\in C(X)$. By [\[proof_lem_Phi_1\]](#proof_lem_Phi_1){reference-type="eqref" reference="proof_lem_Phi_1"}, $P(P(\varphi)) \geq P(\varphi)$. Taking the $\bar c$-transform in [\[proof_lem_Phi_2\]](#proof_lem_Phi_2){reference-type="eqref" reference="proof_lem_Phi_2"} yields $P(P(\varphi))\leq P(\varphi)$ and we have indeed $P(P(\varphi))=P(\varphi)$. Hence we have $\tilde \varphi\in\Phi'$ in [\[proof_lem_Phi_25\]](#proof_lem_Phi_25){reference-type="eqref" reference="proof_lem_Phi_25"} and we get $$\label{proof_lem_Phi_3} \Upsilon^*(f) = \sup\left\{ K_f (\tilde\varphi, \tilde\varphi^c_{\,-}) : \tilde\varphi\in \Phi' \right\}.$$ Finally, by definition $\tilde\varphi= (\tilde\varphi^c_{\,-})^{\bar c}$ for $\tilde\varphi\in \Phi'$ and [\[DPp\]](#DPp){reference-type="eqref" reference="DPp"} follows from [\[proof_lem_Phi_3\]](#proof_lem_Phi_3){reference-type="eqref" reference="proof_lem_Phi_3"} by letting $\psi := \tilde\varphi^c_{\,-}$. ◻ We can now establish that the supremum in [\[DPp\]](#DPp){reference-type="eqref" reference="DPp"} is reached. **Proposition 12**. *Assume that $X$ is a compact metric space and that $c \in C(X \times X, \mathbb{R})$. Then, the set $\Phi'$ is compact in $(C(X), \|\cdot\|_\infty)$ and the suprema in [\[DPp\]](#DPp){reference-type="eqref" reference="DPp"} and [\[DPXcomp\]](#DPXcomp){reference-type="eqref" reference="DPXcomp"} are attained.* *Proof.*  \ Let us show that $\Phi'$ is compact. Let $\varphi_n$ be a sequence in $\Phi'$. The function $c$ is $\omega$-continuous for some modulus of continuity $\omega\in C(\mathbb{R}_+,\mathbb{R}_+)$, so that by Proposition [Proposition 7](#prop_ctransf){reference-type="ref" reference="prop_ctransf"} $(ii)$ for every $n \geq 0$, $\varphi^c_n:=(\varphi_n)^c$ and ${\varphi_n^c}_-:=((\varphi_n)^c)_-$ are $\omega$-continuous. By definition of $\Phi'$, $\varphi_n = ({\varphi_n^c}_-)^{\bar c}$, so that $\varphi_n$ is also $\omega$-continuous for every $n \geq 0$. Let us show that the sequences $\varphi_n$ and ${\varphi_n^c}_-$ are uniformly bounded in $(C(X), \|\cdot\|_\infty)$. We observe that for every $n \geq 0$, $\max \varphi_n^c \geq 0$. Denoting by $x_n$ a point of $X$ such that ${\varphi^c_n}_- (x_n) = 0$, by $\omega$-continuity we have for $x \in X$ and $n \geq 0$, $$-\omega(\mathop{\mathrm{diam}}(X))\leq -\omega(|x-x_n|) \leq {\varphi_n^c}_-(x)-{\varphi_n^c}_-(x_n) = {\varphi_n^c}_-(x) \le 0.$$ Thus the sequence ${\varphi_n^c}_-$ is uniformly bounded in $(C(X), \|\cdot\|_\infty)$. By definition of the $c$-transform $$\min_{X\times X} c -\max_X {\varphi_n^c}_- \le ({\varphi_n^c}_-)^{\bar c} \le \max_{X\times X} c-\min_X {\varphi_n^c}_-.$$ Hence the sequence $\varphi_n$ is also uniformly bounded. By Arzelá-Ascoli's theorem, there exists a pair $(\varphi, \psi) \in C(X) \times C(X)$ such that, up to extraction, $(\varphi_n, {\varphi_n^c}_-)$ converges uniformly to $(\varphi, \psi)$. Let us show that $\varphi\in \Phi'$. By Proposition [Proposition 7](#prop_ctransf){reference-type="ref" reference="prop_ctransf"} $(iii)$ and by uniform convergence $\varphi_n^c \to \varphi^c$ as $n \to \infty$ so that $$\label{ctransfunifconv} {\varphi_n^c}_- \to \varphi^c_{\,-} \quad \text{uniformly as } n \to \infty,$$ which yields $\psi = \varphi^c_{\,-}$. From [\[ctransfunifconv\]](#ctransfunifconv){reference-type="eqref" reference="ctransfunifconv"} and the uniform continuity of $c$, we deduce that $$({\varphi_n^c}_-)^{\bar c} =\varphi_n \to (\varphi^c_{\,-})^{\bar c} \quad \text{ uniformly as } n \to \infty.$$ Since $\varphi_n \to \varphi$ as $n \to \infty$, we obtain $\varphi= (\varphi^c_{\,-})^{\bar c}$. Lastly, by uniform convergence, the fact that $\max \varphi_n^c \geq 0$ for all $n \geq 0$ implies that $\max \varphi^c \geq 0$, so that $\varphi\in \Phi'$. This shows that $\Phi'$ is a compact subset of $(C(X), \|\cdot\|_\infty)$. Let now $\psi_n$ be a maximising sequence for [\[DPp\]](#DPp){reference-type="eqref" reference="DPp"}. For all $n \geq 0$, there exists $\varphi_n \in \Phi'$ such that $\psi_n = {\varphi_n^c}_-$. By compactness of $\Phi'$, $\varphi_n \to \varphi$ as $n \to \infty$ for some $\varphi\in \Phi'$. Setting $\psi = \varphi^c_{\,-}$, we have $\psi_n \to \psi$ and $\psi_n^{\bar c} \to \psi^{\bar c}$ as $n \to \infty$. The functional $K_f$ being continuous with respect to uniform convergence, we obtain $$K_f(\psi^{\bar c}, \psi )=\lim K_f(\psi^{\bar c}_n, \psi_n) = \Upsilon^*(f).$$ This proves that $\psi$ is a maximiser for [\[DPp\]](#DPp){reference-type="eqref" reference="DPp"} and by Lemma [Lemma 11](#lem_Phi){reference-type="ref" reference="lem_Phi"}, $\psi$ also maximises [\[DPXcomp\]](#DPXcomp){reference-type="eqref" reference="DPXcomp"}. ◻ We are now ready to prove that there is no duality gap between [\[PPXcomp\]](#PPXcomp){reference-type="eqref" reference="PPXcomp"} and [\[DPXcomp\]](#DPXcomp){reference-type="eqref" reference="DPXcomp"}. The proof is an adaptation of [@OTAM Section 1.6.3]. **Proposition 13**. *Assume that $X$ is a compact metric space and that $c \in C(X \times X, \mathbb{R})$. Then, $$\Upsilon^*(f)= \Upsilon(f).$$* *Proof.*  \ *Step 1. Definition of $H$ and first properties.* For $p\in C(X\times X)$, we define $$H(p):=-\sup\left\{\int \left(f\varphi+(1-f)\psi \right) \,d\lambda : (\varphi,\psi) \in \Phi_p \right\}$$ where $$\Phi_p := \left\{(\varphi,\psi) \in C(X) \times C(X),\ \psi\le0,\ \varphi\oplus\psi\le c-p \right\}.$$ We first observe that $c-p$ is continuous and bounded from below. Thus, by applying Proposition [Proposition 12](#prop_ex_DPp){reference-type="ref" reference="prop_ex_DPp"} with $c-p$ in place of $c$, we see that the above supremum is a maximum. Let us now show that $H$ is convex. Let $p_0,p_1\in C(X \times X)$ and $\theta\in[0,1]$ and let us set $p:=(1-\theta)p_0+\theta p_1$. We denote by $(\varphi_0,\psi_0)$ and $(\varphi_1,\psi_1)$ two maximising pairs associated with $p_0$ and $p_1$ and set $\varphi:=(1-\theta)\varphi_0+\theta \varphi_1$, $\psi:=(1-\theta)\psi_0+\theta \psi_1$. We see that $(\varphi,\psi)$ is an admissible pair ($\psi \leq 0$ and $\varphi\oplus \psi \leq c-p$), so that $$H(p)\le -\int \left(f\varphi+(1-f)\psi\right)\,d\lambda = (1-\theta)H(p_0) +\theta H(p_1).$$ This proves that $H$ is convex. Next, we establish that $H$ is lower semi-continuous in $(C(X\times X),\|\cdot\|_\infty)$. Let $p_n$ and $p$ be elements of $C(X\times X)$ such that $p_n\to p$ uniformly as $n \to \infty$. The sequence $c-p_n$ is uniformly equi-continuous. Therefore, proceeding as in the proof of Proposition [Proposition 12](#prop_ex_DPp){reference-type="ref" reference="prop_ex_DPp"}, there exists a sequence of uniformly bounded and equi-continuous admissible pairs $(\varphi_n, \psi_n)$ such that $$H(p_n)=-\int \left(f\varphi_n+(1-f)\psi_n\right)\,d\lambda\qquad\text{for every }n \geq 0.$$ We first extract a subsequence $p_{n'}$ such that $\lim_{n'} H(p_{n'})=\liminf_n H(p_n)$. By Arzelà-Ascoli's theorem, there exists $(\varphi, \psi) \in C(X) \times C(X)$ such that $\varphi_{n'}\to\varphi$ and $\psi_{n'}\to\psi$ uniformly as $n'\to \infty$. By pointwise convergence, $\psi\le0$ and $\varphi\oplus\psi\le c-p$. Passing to the limit yields $$H(p)\le -\int \left(f\varphi+(1-f)\psi\right)\,d\lambda = -\lim_{n'} \int \left(f\varphi_{n'}+(1-f)\psi_{n'}\right)\,d\lambda = \liminf_n H(p_n).$$ Thus $H$ is lower semi-continuous. *Step 2. Absence of duality gap.* Since $H$ is convex and lower semi-continuous on the Banach space $(C(X\times X),\|\cdot\|_\infty)$, there holds $H=H^{**}$ on $C(X\times X)$. Here, for a Banach space $\mathcal{X}$ and a function $F:\mathcal{X}\to\mathbb{R}\cup\{+\infty\}$, $F^*$ denotes the Legendre transform of $F$ defined on the topological dual $\mathcal{X}^*$ of $\mathcal{X}$ by $$F^*(x^*):=\sup \left\{ x^*(x)-F(x) : x\in \mathcal{X}\right\}.$$ In particular, $$\label{proof_prop_DP=OP_1} \Upsilon^*(f)=-H(0)=-H^{**}(0) = \inf\{ H^*(\gamma) : \gamma\in \mathcal{M}(X\times X)\}.$$ We now compute $H^*$. Let $\gamma\in \mathcal{M}(X\times X)$. By definition, $$H^*(\gamma)=\sup_{p \in C(X\times X) } \left\{\int p\,d\gamma + \sup_{(\varphi, \psi) \in \Phi_p}\left\{ \int \left(f\varphi+(1-f)\psi\right)\,d\lambda\right\}\right\}.$$ Let us first assume that there exists $q\in C(X\times X,\mathbb{R}_+)$ such that $t:=-\int q\,d\gamma >0$. We set $\varphi=\min c$, $\psi=0$ and $p_n:=-nq$ for $n\ge 1$. We obtain $$H^*(\gamma)\ge n t - m \min c\ \to \infty \qquad \text{ as } n \to \infty.$$ Thus, when computing $H^*(\gamma)$, we may assume that $\gamma \geq 0$. We rewrite $H^*(\gamma)$ as $$\label{Hstar} \begin{aligned} H^*(\gamma) = \int c \, d\gamma + \sup_{p \in C(X \times X)} \sup_{(\varphi, \psi) \in \Phi_p} &\bigg\{\int (p-c + \varphi\oplus \psi) \, d\gamma \\ &+\int \varphi\, d (f \lambda - \gamma_x) + \int \psi \, d((1-f) \lambda - \gamma_y))\bigg\}. \end{aligned}$$ Let us set $$G(\gamma) := \sup\left\{\int \varphi\, d(f \lambda - \gamma_x) + \int \psi \, d((1-f)\lambda - \gamma_y) : (\varphi,\psi) \in C(X) \times C(X),\ \psi\le0 \right\}.$$ On the one hand, given $(\varphi, \psi) \in \Phi_p$ and $\gamma \ge 0$, $$\int (p - c + \varphi\oplus \psi)\,d\gamma \leq 0.$$ Therefore, $H^*(\gamma) \leq \smallint c \, d \gamma + G(\gamma)$. On the other hand, given $(\varphi, \psi)$ admissible for $G(\gamma)$, setting $p = c - \varphi\oplus \psi$ yields the converse inequality thanks to [\[Hstar\]](#Hstar){reference-type="eqref" reference="Hstar"}. Hence $$\label{Hstar2} H^*(\gamma)=\int c\, d\gamma + G(\gamma).$$ Given $\gamma \in \mathcal{M}_+(X \times X)$, we have $G(\gamma) = 0$ if $\gamma \in {\varPi}_f$ and $G(\gamma) = +\infty$ otherwise. Combining this with [\[Hstar2\]](#Hstar2){reference-type="eqref" reference="Hstar2"}, we obtain that for $\gamma \in \mathcal{M}(X \times X)$, $$H^*(\gamma)= \begin{cases} \displaystyle\int c\, d\gamma &\text{if }\gamma\in {\varPi}_f,\\ +\infty&\text{in the other cases.}\end{cases}$$ Taking the infimum with respect to $\gamma\in \mathcal{M}(X\times X)$ and recalling [\[proof_prop_DP=OP_1\]](#proof_prop_DP=OP_1){reference-type="eqref" reference="proof_prop_DP=OP_1"}, we get $$\Upsilon^*(f)= \inf \left\{H^*(\gamma):\gamma\in \mathcal{M}(X\times X)\right\} = \inf \left\{\int c\,d\gamma :\gamma\in{\varPi}_f\right\}=\Upsilon(f),$$ which concludes the proof. ◻ **Remark 14**. There is still no duality gap between [\[PPXcomp\]](#PPXcomp){reference-type="eqref" reference="PPXcomp"} and [\[DPXcomp\]](#DPXcomp){reference-type="eqref" reference="DPXcomp"} if we only assume $c$ to be lower semi-continuous. This result can be obtained by approximating $c$ pointwise from below by a non-decreasing sequence of continuous functions. In the remainder of the section, we focus on the properties of the potentials $(\psi^{\bar c}, \psi)$ maximising [\[DPp\]](#DPp){reference-type="eqref" reference="DPp"}. We first show that the sign of $\psi^{\bar cc}$ enforces constraints on the local values of the marginals of any plan $\gamma$ optimal for $\Upsilon(f)$. **Proposition 15**. *Assume that $X$ is a compact metric space and that $c \in C(X \times X, \mathbb{R})$. Let $(\psi^{\bar c}, \psi)$ be a maximiser of [\[DPXcomp\]](#DPXcomp){reference-type="eqref" reference="DPXcomp"} and let $\gamma$ be a minimiser of [\[PPXcomp\]](#PPXcomp){reference-type="eqref" reference="PPXcomp"}. We set $g := \gamma_y$. Then, $\gamma$ is a minimiser of [\[KP\]](#KP){reference-type="eqref" reference="KP"} with $(\mu, \nu) = (f,g)$ and $(\psi^{ \bar c}, \psi^{\bar c c})$ is a pair of Kantorovitch potentials realising the maximum in [\[DKP\]](#DKP){reference-type="eqref" reference="DKP"}. Moreover, up to $\lambda$-negligible sets, $$\label{psi_KP} f+g\equiv1\quad\text{on }\{\psi^{\bar cc} < 0 \}\qquad\text{and}\qquad g\equiv0\quad\text{on }\{\psi^{\bar cc}>0\}.$$* *Proof.*  \ By Remark [Remark 10](#rem_KP_Ups){reference-type="ref" reference="rem_KP_Ups"}, $\gamma$ realises the minimum in [\[KP\]](#KP){reference-type="eqref" reference="KP"} and $\Upsilon(f)= \mathcal{T}_c(f,g)$. As there is no duality gap in [\[PPXcomp\]](#PPXcomp){reference-type="eqref" reference="PPXcomp"} nor in [\[KP\]](#KP){reference-type="eqref" reference="KP"}, $\Upsilon^*(f) = \mathcal{T}_c^*(f,g)$. Additionally, $(\psi^{\bar c}, \psi^{\bar cc})$ is admissible for $\mathcal{T}_c^*(f,g)$ and $\psi = (\psi^{\bar cc})_-$. Thus $$\label{proof_psi_KP} K_f(\psi^c, \psi)=\int f\psi^{\bar c}\, d\lambda + \int (1-f)(\psi^{\bar cc})_-\, d\lambda = \mathcal{T}_c^*(f,g) \geq \int f \psi^{\bar c}\, d\lambda + \int g \psi^{\bar cc}\, d\lambda.$$ Hence $$\int (1-f-g) (\psi^{\bar cc})_-\, d\lambda \ge \int g (\psi^{\bar cc})_+\,d\lambda.$$ Since $(1-f-g) (\psi^{\bar cc})_-\le0$ and $g (\psi^{\bar cc})_+ \geq 0$, the integrands must vanish $\lambda$-almost everywhere: we deduce [\[psi_KP\]](#psi_KP){reference-type="eqref" reference="psi_KP"}. Additionally, the inequality in [\[proof_psi_KP\]](#proof_psi_KP){reference-type="eqref" reference="proof_psi_KP"} is an equality. Consequently, $(\psi^{\bar c}, \psi^{\bar cc})$ is a pair of Kantorovitch potentials for [\[DKP\]](#DKP){reference-type="eqref" reference="DKP"}. ◻ To end this section, we establish a comparison principle on the potentials maximising [\[DPp\]](#DPp){reference-type="eqref" reference="DPp"}. We say that a set $\Psi \subset C(X)$ admits a minimal (respectively maximal) element for the relation $\leq$ if there exists $\psi_0 \in \Psi$ such that for any $\psi \in \Psi$, $\psi_0 \leq \psi$ (respectively $\psi_0 \geq \psi$). **Proposition 16**. *Assume that $X$ is a compact metric space and that $c \in C(X \times X, \mathbb{R})$. Let $f \in L^1_m$ and let us define $$\Psi_f:=\left\{\psi, \, \psi = \varphi^c_{\,-} \text{ for some } \varphi\in \Phi' \text{ and } K_f(\psi^{\bar c}, \psi)=\Upsilon^*(f)\right\},$$* *where $K_f$ is defined in [\[Kf\]](#Kf){reference-type="eqref" reference="Kf"} and $\Phi'$ in Lemma [Lemma 11](#lem_Phi){reference-type="ref" reference="lem_Phi"}.* *Then:* (i) *$\Psi_f$ admits a maximal element for the relation $\leq$, denoted by $\psi_f$ in the sequel,* (ii) *For $f_1, f_2 \in L^1_m$, there holds $f_1 \leq f_2\implies\psi_{f_1}\geq \psi_{f_2}$.* *Proof.*  \ *Step 1. Sufficient condition and preliminary claim.* Notice that $\Psi_f$ is not empty by Proposition [Proposition 12](#prop_ex_DPp){reference-type="ref" reference="prop_ex_DPp"}. To obtain $(i)$, we prove that the set $$\Phi_f := \left\{\varphi\in \Phi', \, K_f(\varphi, \varphi^c_{\,-}) = \Upsilon^*(f)\right\}$$ admits a minimal element $\varphi_f$ and then $\psi_f := (\varphi^c_f)_-$ is the desired maximal element of $\Psi_f$. Let us make a preliminary observation. **Claim 1**. *Let $f_1, f_2 \in L^1_m$ with $f_1 \leq f_2$ and set $\varphi_i \in \Phi_{f_i}$ for $i \in \{1, 2\}$. Then $\varphi_{\wedge} := \varphi_1 \wedge \varphi_2 \in \Phi_{f_1}$.* Let us first prove that $\varphi_{\wedge} \in \Phi'$. In the sequel we write $\varphi_i^c:=(\varphi_i)^c$ and ${\varphi_i^c}_-:=((\varphi_i)^c)_-$ for $i\in\{1,2,\wedge\}$. We observe that $\varphi_{\wedge} \in C(X)$. By definition of the $c$-transform, we obtain $$\label{ctransfwedgegeq} \varphi_{\wedge}^c = (\varphi_1 \wedge \varphi_2)^c \geq \varphi_1^c \vee \varphi_2^c$$ and $$\label{ctransfwedgeq} (\varphi_1 \vee \varphi_2)^c = \varphi_1^c \wedge \varphi_2^c.$$ Since for $i \in \{1, 2\}$, $\max \varphi_i^c \geq 0$ we have by [\[ctransfwedgegeq\]](#ctransfwedgegeq){reference-type="eqref" reference="ctransfwedgegeq"} that $\max \varphi_{\wedge}^c \geq 0$. We now prove that $({\varphi_{\wedge}^c}_-)^{\bar c} = \varphi_{\wedge}$. We observe that ${\varphi_{\wedge}^c}_- \leq \varphi_{\wedge}^c$ which implies $({\varphi_{\wedge}^c}_-)^{\bar c} \geq \varphi_{\wedge}^{c \bar c}$. By Proposition [Proposition 7](#prop_ctransf){reference-type="ref" reference="prop_ctransf"} $(ii)$, $\varphi_{\wedge}^{c \bar c} \geq \varphi_{\wedge}$ so that $({\varphi_{\wedge}^c}_-)^{\bar c} \geq \varphi_{\wedge}$. Conversely, taking the negative part of [\[ctransfwedgegeq\]](#ctransfwedgegeq){reference-type="eqref" reference="ctransfwedgegeq"}, we have ${\varphi_{\wedge}^c}_- \geq (\varphi_1^c \vee \varphi_2^c)_- = {\varphi_1^c}_- \vee {\varphi_2^c}_-$. Taking the $\bar c$-transform and using [\[ctransfwedgeq\]](#ctransfwedgeq){reference-type="eqref" reference="ctransfwedgeq"} yields $$({\varphi_{\wedge}^c}_-)^{\bar c} \leq ({\varphi_1^c}_- \vee {\varphi_2^c}_-)^{\bar c} = ({\varphi_1^c}_-)^{\bar c} \wedge ({\varphi_2^c}_-)^{\bar c} = \varphi_1 \wedge \varphi_2 = \varphi_{\wedge}.$$ Hence $({\varphi_{\wedge}^c}_-)^{\bar c} = \varphi_{\wedge}$ and $\varphi_{\wedge} \in \Phi'$. We now show that the pair $(\varphi_{\wedge}, {\varphi_{\wedge}^c}_-)$ maximises $\Upsilon^*(f_1)$. We set $$\Delta_K := K_{f_1}(\varphi_{\wedge}, {\varphi_{\wedge}^c}_-)-K_{f_1}(\varphi_1, {\varphi_1^c}_-) = \int f_1 (\varphi_{\wedge}-\varphi_1)\, + \int (1-f_1)({\varphi^c_{\wedge}}_--{\varphi_1^c}_-)\,.$$ By optimality of $\varphi_1$, $\Delta_K \leq 0$. Let us prove the converse inequality. Substituting $f_1 = f_2 + f_1-f_2$ in the definition of $\Delta_K$, we obtain $$\Delta_K=\int f_2 (\varphi_{\wedge} - \varphi_1) \,+\int (1-f_2)({\varphi_{\wedge}^c}_- - {\varphi_1^c}_-)\, +\int (f_2-f_1)(\varphi_1 - \varphi_{\wedge} + {\varphi_{\wedge}^c}_- - {\varphi_1^c}_-)\,.$$ We have $f_2-f_1\ge0$ and $\varphi_1- \varphi_{\wedge} \ge0$. Additionally, $\varphi_{\wedge}^c \geq \varphi_1^c$, so that ${\varphi_{\wedge}^c}_-\ge{\varphi_1^c}_-$. Thus the last integral in $\Delta_K$ is non-negative. Adding and subtracting $f_2 \varphi_2$ in the first integral yields $$\label{lowerboundDeltaK} \Delta_K \geq \int f_2 \varphi_2 \,+ \int f_2 (\varphi_{\wedge}-\varphi_1-\varphi_2)\, +\int (1-f_2)({\varphi_{\wedge}^c}_--{\varphi_1^c}_-)\,.$$ Let us set $\varphi_{\vee} := \varphi_1 \vee \varphi_2$. By optimality of $\varphi_2$, we have $K_{f_2}(\varphi_2,{\varphi_2^c}_-)\ge K_{f_2}(\varphi_{\vee}, {\varphi_{\vee}^c}_-)$, which rewrites as $$\int f_2\varphi_2\, \ge \int f_2 \varphi_{\vee}\, + \int(1-f_2)({\varphi_{\vee}^c}_--{\varphi_2^c}_-)\,.$$ Injecting this inequality in the first term of the right-hand side of [\[lowerboundDeltaK\]](#lowerboundDeltaK){reference-type="eqref" reference="lowerboundDeltaK"} yields $$\label{boundeddeltaK} \Delta_K \geq \int f_2(\varphi_{\wedge} + \varphi_{\vee} -\varphi_1-\varphi_2)\, +\int (1-f_2)({\varphi_{\vee}^c}_-+{\varphi_{\wedge}^c}_--{\varphi_1^c}_--{\varphi_2^c}_-)\,.$$ The integrand in the first integral of [\[boundeddeltaK\]](#boundeddeltaK){reference-type="eqref" reference="boundeddeltaK"} vanishes. Regarding the second term, using [\[ctransfwedgeq\]](#ctransfwedgeq){reference-type="eqref" reference="ctransfwedgeq"} and [\[ctransfwedgegeq\]](#ctransfwedgegeq){reference-type="eqref" reference="ctransfwedgegeq"} we obtain $${\varphi_{\vee}^c}_-+{\varphi_{\wedge}^c}_-\ge {\varphi_1^c}_-\wedge{\varphi_2^c}_-+ {\varphi_1^c}_-\vee{\varphi_2^c}_- ={\varphi_1^c}_-+{\varphi_2^c}_-.$$ Hence the integrand in the second integral is non-negative. We conclude that $\Delta_K \geq 0$ and finally that $\Delta_K=0$ so that the claim is proved. *Step 2. Construction of the minimal element of $\Phi_f$.* By Lemma [Proposition 12](#prop_ex_DPp){reference-type="ref" reference="prop_ex_DPp"}, $\Phi'$ is compact. As $\varphi\mapsto K_f(\varphi, \varphi^c_{\,-})$ is continuous for the norm of uniform convergence, $\Phi_f$ is compact as well. Let $(\varphi_j)_{j \geq 0}$ be a dense subset of $\Phi_f$. For $x \in X$ and $j \geq 0$, we define $\widetilde{\varphi}_j$ and $\varphi_f$ by $$\widetilde \varphi_j(x):=\min(\varphi_0(x),\dots,\varphi_j(x)) \qquad \text{and} \qquad \varphi_f(x):= \inf\{ \varphi(x), \, \varphi\in \Phi_f \}.$$ Using our preliminary claim with $f_1=f_2=f$ recursively, we obtain that for any $j \geq 0$, $\widetilde \varphi_j \in \Phi_f$. As $\Phi_f$ is compact and $\widetilde \varphi_j\to\varphi_f$ pointwise, we obtain that $\widetilde \varphi_j\to\varphi_f$ uniformly and $\varphi_f\in\Phi_f$, so that $\varphi_f$ is the desired minimal element of $\Phi_f$. *Step 3. Conclusion.* Taking $\psi_f:={\varphi_f^c}_-$ proves $(i)$. Let $f_1 \leq f_2$ as given in the statement of $(ii)$. By the previous step, there exist $\varphi_1, \varphi_2$ respective minimal elements for $\Phi_{f_1}$ and $\Phi_{f_2}$ such that $\psi_1 := {\varphi_1^c}_-$ and $\psi_2 :={\varphi_2^c}_-$ are respective maximal elements for $\Psi_{f_1}$ and $\Psi_{f_2}$. By the preliminary claim, $\varphi_1 \wedge \varphi_2 \in \Phi_{f_1}$ and by minimality of $\varphi_1$ we have $\varphi_1 \leq \varphi_1 \wedge \varphi_2$, so that $\varphi_2 \geq \varphi_1$. Hence $\psi_2 \leq \psi_1$. ◻ # Existence of maximisers of [\[MPintro\]](#MPintro){reference-type="eqref" reference="MPintro"} for translation invariant costs in $\mathbb{R}^d$ {#trslinv} We now assume that $X=\mathbb{R}^d$, that $\lambda$ is the Lebesgue measure and that $c(x,y)=k(y-x)$, with $k : \mathbb{R}^d \to \mathbb{R}_+$. We recall the following hypotheses on $k$. 1. $k \in C(\mathbb{R}^d, \mathbb{R}_+)$, $k(0)=0$ and $k(x) \to \infty$ as $|x| \to \infty$, 2. $\forall x \not = 0$, $$\limsup_{r \to 0} \frac{1}{r^d} \left|B_r(x) \cap \{y \in \mathbb{R}^d, \, k(y) < k(x)\} \right| > 0,$$ 3. $\forall \, \sigma \in \mathbb{S}^{d-1}$, $r \mapsto k(r\sigma)$ is increasing on $\mathbb{R}_+$. Notice that under hypotheses (H[\[cont\]](#cont){reference-type="ref" reference="cont"})&(H[\[cone\]](#cone){reference-type="ref" reference="cone"}), there holds $k(x)> 0$ for $x \neq 0$. The primal problem is now defined as $$\label{PPRd} \Upsilon(f) :=\inf \left\{\int c\,d\gamma : \gamma\in {\varPi}_f\right\},$$ where $${\varPi}_f :=\left\{\gamma\in\mathcal{M}_+(\mathbb{R}^d\times \mathbb{R}^d) : \gamma_x = f,\, \gamma_y \le 1-f \right\}.$$ The goal of this section is to prove that for every $m>0$ the energy $$\label{MPintro} \mathcal{E}(m) := \sup \left\{\Upsilon(f) : f \in L^1(\mathbb{R}^d, [0,1]), \int f \, = m \right\}$$ admits a maximiser. ## First properties of $\Upsilon$ and saturation theorem. In this subsection, we collect some properties of the functional $\Upsilon$ defined in $\mathbb{R}^d$ and establish a saturation property (Theorem [Theorem 20](#thm_saturation){reference-type="ref" reference="thm_saturation"}), namely that if $\gamma$ is a minimiser for $\Upsilon(f)$ then $\gamma_y(x) \in \{f(x), 1-f(x)\}$ for almost every $x \in \mathbb{R}^d$. We start by proving that minimisers of [\[PPRd\]](#PPRd){reference-type="eqref" reference="PPRd"} exist. The proof of this result is similar to the proof of [@CanGol Proposition 2.1], but with weaker assumptions on the cost $c$ and in the context of functions taking values in $[0,1]$ rather than in $\{0, 1\}$. **Proposition 17**. *Assume that $k$ satisfies (H[\[cont\]](#cont){reference-type="ref" reference="cont"}). Then, for any $m >0$ and $f \in L^1_m$, the infimum in [\[PPRd\]](#PPRd){reference-type="eqref" reference="PPRd"} is attained. Additionally, given any minimiser $\gamma$ of [\[PPRd\]](#PPRd){reference-type="eqref" reference="PPRd"} we have $\Upsilon(f) = \mathcal{T}_c(f,g)$, where $g := \gamma_y$.* *Lastly, there exists $R = R(m)$ non-decreasing in $m$ such that for any $f \in L^1_m$, $$\label{infrestr} \Upsilon(f)=\min \left\{\int c\, d\gamma :\gamma\in {\varPi}_f, \, \forall \, (x,y) \in \mathop{\mathrm{supp}}\gamma, \, |x-y| \leq R \right\}$$* *and for any minimiser $\gamma$ of [\[PPRd\]](#PPRd){reference-type="eqref" reference="PPRd"}, there holds $|x-y| \leq R$ on $\mathop{\mathrm{supp}}\gamma$.* *Proof.*  \ The strategy of the proof is to first establish [\[infrestr\]](#infrestr){reference-type="eqref" reference="infrestr"} with an infimum in place of the minimum. Then we use this property to derive compactness for [\[PPRd\]](#PPRd){reference-type="eqref" reference="PPRd"}. *Step 1. Restricting the set of competitors for [\[PPRd\]](#PPRd){reference-type="eqref" reference="PPRd"}.* We let $\gamma \in {\varPi}_f$ and set $g := \gamma_y$. We want to build a competitor $\widetilde{\gamma}$ for $\Upsilon(f)$ such that for some $R >0$, $|x-y| \leq R$ for every $(x,y) \in \mathop{\mathrm{supp}}\widetilde{\gamma}$. For $R > 0$, we define $$\Gamma_R := \left\{ (x,y) \in \mathbb{R}^d \times \mathbb{R}^d, \, |x-y| \geq R \right\}.$$ We consider a standard partition of $\mathbb{R}^d$ into cubes $(Q_i)_{i \geq 0}$ with side-length $\rho_1(m) := (3m)^{1/d}$. We define $$I := \{ i \geq 0, \, m_i := \gamma(\Gamma_R \cap (Q_i \times \mathbb{R}^d))>0\},$$ and for $i \in I$, we set $$\gamma_{\text{bad}, i} := \chi_{\Gamma_R \cap (Q_i \times \mathbb{R}^d)} \gamma.$$ As $|Q_i| - \smallint f - \smallint g \geq m \geq m_i$, there exists a positive measure $\mu_i\ll\chi_{Q_i}\lambda$ such that $$\mu_i \leq \chi_{Q_i}(1 - f - g) \qquad \text{and} \qquad \mu_i (\mathbb{R}^d)=\gamma_{\text{bad},i}(\mathbb{R}^d \times \mathbb{R}^d).$$ Denoting by $\theta_i$ the first marginal of $\gamma_{\text{bad},i}$ we set $$\widetilde{\gamma}_i := \chi_{Q_i \times \mathbb{R}^d} \gamma - \gamma_{\text{bad},_i} + \frac{1}{m_i}\theta_i \otimes \mu_i\ \ge0.$$ For $i \in I^c$, we simply define $\widetilde{\gamma}_i := \chi_{(Q_i \times \mathbb{R}^d )} \gamma$. As a consequence, $\widetilde{\gamma} := \sum_{i \geq 0} \widetilde{\gamma_i}$ is a transport plan whose first marginal is $f$ and second marginal $\widetilde{g}$ verifies $\widetilde{g} \leq g \leq 1-f$. By construction, for $R > \sqrt{d}\rho_1(m)$, we have $\widetilde{\gamma}(\Gamma_R) = 0$. Let us now compare the transportation cost of $\gamma$ and $\widetilde{\gamma}$. We compute: $$\begin{aligned} \int c \, d\widetilde{\gamma} -\int c \, d\gamma &= \sum_{i \in I} \int_{Q_i \times \mathbb{R}^d} c \, d\left(\frac{\theta_i \otimes \mu_i}{m_i} - \gamma_{\mathrm{bad},i} \right) \\ &\leq \left(\sum_{i \in I} m_i \right) \left( \max_{z \in \overline{Q}_{\rho_1(m)}} k(z) - \inf_{|z|\geq R} k(z) \right). \end{aligned}$$ Let us set $M := \max\{k(z), \, z \in \overline{Q}_{\rho_1(m)}\}$. By (H[\[cont\]](#cont){reference-type="ref" reference="cont"}), there exists $R> \sqrt{d}\rho_1(m)$ such that if $|z| > R$, then $k(z)>M$. With this choice of $R$ we have $\smallint c \,d\widetilde{\gamma} \leq \smallint c \,d\gamma$. Lastly, whenever $\gamma(\Gamma_R) >0$, $$\label{stricineqgamma} \int c \,d\widetilde{\gamma} < \int c \,d\gamma.$$ *Step 2 : Lower semi-continuity of the transportation cost.* This step is classical. To prove that $\gamma \mapsto \smallint c \,d\gamma$ is lower semi-continuous with respect to weak convergence, we proceed by approximation. Let us assume that $\gamma_n \stackrel*\rightharpoonup \gamma$ as $n \to \infty$. For $j \geq 0$, we define $c_j := c \wedge j$. The sequence $c_j$ is non-decreasing and converges pointwise to $c$. For every $j \geq 0$, $c_j \in C_b(\mathbb{R}^d\times \mathbb{R}^d)$, so that $$\int c_j \, d\gamma = \lim_n \int c_j \, d\gamma_n \leq \liminf_n\int c \, d\gamma_n.$$ By the monotone convergence theorem, $$\int c \, d\gamma = \lim_j \int c_j \, d\gamma \leq \liminf_n\int c \, d\gamma_n,$$ which concludes the second step of the proof. *Step 3. $\Upsilon(f)$ admits a minimiser.* Let $\gamma_n$ be a minimising sequence for $\eqref{PPRd}$. Let us show that the sequence $\gamma_n$ is tight. By the first step, we can assume that there exists $R = R(m)$ such that for any $n \geq 0$ there holds $|x-y| \leq R$ on $\mathop{\mathrm{supp}}\gamma_n$. Now, because $\smallint f \leq m < \infty$, there exists $R' = R'(m)> 0$ such that $\smallint_{\mathbb{R}^d \setminus B_{R'}} f \leq \varepsilon$. Hence, $$\gamma_n \left(\mathbb{R}^d \times \mathbb{R}^d \setminus (B_{R'} \times B_{R+R'})\right)= \gamma_n\left(B_{R'} \times (\mathbb{R}^d \setminus B_{R+R'})\right) + \gamma_n \left((\mathbb{R}^d \setminus B_{R'}) \times \mathbb{R}^d\right) \leq 0 + \varepsilon$$ which proves that the sequence $\gamma_n$ is tight. Together with the second step, this shows that [\[PPRd\]](#PPRd){reference-type="eqref" reference="PPRd"} admits a minimiser. Moreover, by [\[stricineqgamma\]](#stricineqgamma){reference-type="eqref" reference="stricineqgamma"} for any minimiser $\gamma$ of [\[PPRd\]](#PPRd){reference-type="eqref" reference="PPRd"} there holds $|x-y| \leq R$ on $\mathop{\mathrm{supp}}\gamma$. Lastly, setting $g := \gamma_y$ the identity $\Upsilon(f) = \mathcal{T}_c(f,g)$ is immediate. ◻ We now establish some basic properties of the functional $\Upsilon$. The results here are similar to [@CanGol Proposition 2.2 & Lemma 2.4]. **Proposition 18**. *Assume that $k$ satisfies (H[\[cont\]](#cont){reference-type="ref" reference="cont"}). Given $m>0$ and $f_1, f_2 \in L^1_m$ we have:* (i) *If $f_1 + f_2 \leq 1$, then $$\Upsilon(f_1 + f_2) \geq \Upsilon(f_1) + \Upsilon(f_2).$$* *As a consequence, if $f_1 \leq f_2$, then $\Upsilon(f_1) \leq \Upsilon(f_2)$.* (ii) *There exists $R=R(m)$ such that if $d(\mathop{\mathrm{supp}}f_1, \mathop{\mathrm{supp}}f_2) \geq R$, then $$\Upsilon(f_1 + f_2) = \Upsilon(f_1) + \Upsilon(f_2).$$* (iii) *There exists $C=C(m)>0$ such that $$|\Upsilon(f_1)-\Upsilon(f_2)|\le C\|f_1-f_2\|_{L^1}.$$* (iv) *Let $f, f_n\in L^1(\mathbb{R}^d,[0,1])$ be such that the sequence $f_n\lambda$ is tight and $f_n\stackrel*\rightharpoonup f$. Then $\Upsilon(f_n)\to\Upsilon(f)$.* *Proof.*  \ *Step 1. Proof of (i)&(ii)*. To prove $(i)$, we consider a transport plan $\gamma$ optimal for $\Upsilon(f_1 + f_2)$ whose existence is guaranteed by Proposition [Proposition 17](#prop_Ups_max){reference-type="ref" reference="prop_Ups_max"}. We would like to extract from $\gamma$ two plans $\gamma^1$ and $\gamma^2$ admissible for $\Upsilon(f_1)$ and $\Upsilon(f_2)$ respectively. Using the convention $0/0 = 0$, we define $\gamma^1$ and $\gamma^2$ through $$d\gamma^1(x,y) := \frac{f_1(x)}{(f_1 + f_2)(x)} \, d\gamma(x,y) \qquad \text{and} \qquad d\gamma^2(x,y) := \frac{f_2(x)}{(f_1 + f_2)(x)} \, d\gamma(x,y).$$ By construction, $\gamma^1_x = f_1$ and $\gamma^2_x = f_2$. We also have $\gamma^1 \leq \gamma$, so that $$\gamma^1_y \leq \gamma_y \leq 1-(f_1+f_2) \leq 1-f_1.$$ Likewise, $\gamma^2_y \leq 1 - f_2$. Therefore, $\gamma^1$ and $\gamma^2$ are admissible for $\Upsilon(f_1)$ and $\Upsilon(f_2)$ respectively. Moreover $$\Upsilon(f_1) + \Upsilon(f_2) \leq \int c \, d\gamma^1 + \int c\, d\gamma^2 = \int c \, d\gamma = \Upsilon(f_1 + f_2),$$ which is the desired conclusion. To prove $(ii)$, we consider transport plans $\gamma^1$ and $\gamma^2$ which are optimal for $\Upsilon(f_1)$ and $\Upsilon(f_2)$ respectively. We define $g_1 := \gamma^1_y$ and $g_2 := \gamma^2_y$. If we set $\gamma := \gamma^1 + \gamma^2$, we have $\gamma_x = f_1+f_2$ and $\gamma_y = g_1 + g_2$. Moreover, by Proposition [Proposition 17](#prop_Ups_max){reference-type="ref" reference="prop_Ups_max"}, if $d(\mathop{\mathrm{supp}}f_1, \mathop{\mathrm{supp}}f_2) \geq R$ for $R=R(m)$ large enough, then the supports of $g_1$ and $g_2$ are also disjoint. Consequently, $g_1 + g_2 \leq 1 - (f_1 + f_2)$, so that $\gamma$ is admissible for $\Upsilon(f_1+f_2)$ and we have the desired converse inequality $$\Upsilon(f_1 + f_2) \leq \int c \, d(\gamma^1 + \gamma^2) \leq \Upsilon(f_1) + \Upsilon(f_2).$$ *Step 2. Proof of (iii).* Exchanging the roles of $f_1$ and $f_2$, it is enough to prove the estimate $$\label{upperLip} \Upsilon(f_2)-\Upsilon(f_1) \leq C \|f_2-f_1\|_{L^1}.$$ Let $\gamma^1$ be a minimiser of $\Upsilon(f_1)$ and let us set $g_1 := \gamma^1_x$. In the next substeps, we build from $\gamma^1$ an exterior transport plan $\gamma^2$ for $f_2$ with controlled cost. *Step 2.a. Transporting most of $f_1 \wedge f_2$.* Using the convention $0/0= 0$, we define a plan $\gamma'$ by $$d\gamma'(x,y) := \frac{(f_1 \wedge f_2)(x)}{f_1(x)} d\gamma_1(x,y).$$ We set $g' := \gamma'_y$. Notice that $\gamma' \leq \gamma^1$, which implies $g' \leq g_1$. Additionally, $\gamma'_x = f_1 \wedge f_2$, so that $$\label{massg2a} \gamma^1(\mathbb{R}^d \times \mathbb{R}^d) - \gamma'(\mathbb{R}^d \times \mathbb{R}^d) = \int(f_1-f_1\wedge f_2)\,= \int (f_1 - f_2)_+\,.$$ Heuristically, $\gamma'$ corresponds to sending through $\gamma^1$ as much mass from $f_2$ as possible. However, we have to remove some of this mass because the constraint $g' \leq 1 - f_2$ might not hold true everywhere. Let $$u:=(f_2+g'-1)_+$$ and define $\gamma''$ as $$d\gamma''(x,y) := \frac{g'(y)-u(y)}{g'(y)} \, d\gamma'(x,y).$$ We set $f'' := \gamma''_x$ and $g'':=\gamma''_y$. By construction, $g''=g'-u$ so $g''\le1-f_2$ as desired. Since $\gamma'' \leq \gamma'$, we also have $f'' \leq f_1 \wedge f_2 \leq f_2$. Now since $g' \leq g_1 \leq 1 -f_1$ we have $u\le (f_2-f_1)_+$ from which we infer $$\gamma'(\mathbb{R}^d \times \mathbb{R}^d) - \gamma''(\mathbb{R}^d \times \mathbb{R}^d) =\int (g'-g'')\,=\int u\, \leq \int (f_2 - f_1)_+\,.$$ Summing this and [\[massg2a\]](#massg2a){reference-type="eqref" reference="massg2a"} yields $$\label{massg2b} \gamma^1(\mathbb{R}^d \times \mathbb{R}^d) - \gamma''(\mathbb{R}^d \times \mathbb{R}^d) \leq \|f_2-f_1\|_{L^1}.$$ Eventually since $\gamma''\le\gamma'\le\gamma^1$ and $c\ge0$ we have $$\label{boundedg2b} \int c \, d\gamma'' - \int c \, d\gamma^1 \le0.$$ *Step 2.b. Final construction.* We are now ready to build an admissible transport plan $\gamma^2$ for $\Upsilon(f_2)$. Noticing that $f_2 - f''\ge0$ we write $f_2 = f'' + (f_2 - f'')$. By [\[massg2b\]](#massg2b){reference-type="eqref" reference="massg2b"} we have $$\label{totalmassr} \int (f_2 - f'')\, = \int(f_2- f_1)\, + \int(f_1 -f'')\, \leq 2 \|f_2-f_1\|_{L^1}.$$ Arguing as in the proof of Proposition [Proposition 17](#prop_Ups_max){reference-type="ref" reference="prop_Ups_max"}, we can find a function $0\le g''' \leq 1- f_2 - g''$ with $\int g'''\,\le\int(f_2-f'')\,$ and a transport plan $\gamma'''$ between $f_2-f''$ and $g'''$ such that for some $C=C(m)>0$, $$\label{boundedgr} \int c \, d\gamma''' \leq C\int(f_2 - f'')\, \stackrel{\eqref{totalmassr}}{\leq} C \|f_2 - f_1\|_{L^1}.$$ Finally we define $\gamma^2 := \gamma'' + \gamma'''$ which is admissible for $\Upsilon(f_2)$ by construction. Summing [\[boundedg2b\]](#boundedg2b){reference-type="eqref" reference="boundedg2b"} and [\[boundedgr\]](#boundedgr){reference-type="eqref" reference="boundedgr"}, we get $$\Upsilon(f_2)\le\int c\,d\gamma^2\le\int c\, d\gamma_1+C \|f_2 - f_1\|_{L^1}=\Upsilon(f_1)+C \|f_2 - f_1\|_{L^1}.$$ This proves [\[upperLip\]](#upperLip){reference-type="eqref" reference="upperLip"} and thus point (iii). *Step 3. Proof of (iv).* Let $f_n$ and $f_n$ be as in the statement of the proposition. By weak convergence, we have $f_n,f\in L^1_m$ for some $m>0$. Using the Lipschitz continuity of $\Upsilon$ with respect to $L^1$ convergence, we may assume without loss of generality that $f_n$ (and thus also $f$) are supported in $\overline{B}_{R_0}$ for some $R_0>0$. Applying Proposition [Proposition 17](#prop_Ups_max){reference-type="ref" reference="prop_Ups_max"} we get that minimisers of $\Upsilon(f_n)$ and $\Upsilon(f)$ are supported in $\overline{B}_R\times\overline{B}_R$ for some $R>R_0>0$. We may thus restrict these problems to the compact set $\overline{B}_R$. Using Proposition [Proposition 13](#prop_DP=OP){reference-type="ref" reference="prop_DP=OP"} we have $\Upsilon(f_n)=\Upsilon^*(f_n)$ and it is thus enough to prove the continuity of $\Upsilon^*$ with respect to the weak-$*$ topology.\ By Proposition [Proposition 12](#prop_ex_DPp){reference-type="ref" reference="prop_ex_DPp"}, for every $n\ge 0$ there exists a pair of potentials $(\varphi_n, \psi_n)$ maximising $\Upsilon^*(f_n)$. Since for every $n$, $\varphi_n$ belongs to $\Phi'$ (where $\Phi'$ is defined by [\[defPhiprime\]](#defPhiprime){reference-type="eqref" reference="defPhiprime"}) and since this set is compact by Proposition [Proposition 12](#prop_ex_DPp){reference-type="ref" reference="prop_ex_DPp"} we have that a subsequence $\varphi_{n'}$ of $\varphi_n$ converges in $C(\overline{B}_R)$ to some $\varphi\in \Phi'$. Arguing as in the proof of Proposition [Proposition 12](#prop_ex_DPp){reference-type="ref" reference="prop_ex_DPp"} we see that $\psi_{n'}$ also converges to $\psi$ with $(\varphi,\psi)$ admissible for $\Upsilon^*(f)$. By weak-strong convergence we then have $$\limsup\Upsilon^*(f_{n'})= \limsup \int f_{n'} \varphi_{n'} +(1-f_{n'}) \psi_{n'} \,=\int f \varphi+(1-f)\psi\,\le \Upsilon^*(f).$$ Similarly, if $(\varphi,\psi)$ are optimal potentials for $\Upsilon^*(f)$, they are admissible for $\Upsilon^*(f_n)$ and thus $$\liminf\Upsilon^*(f_n)\ge \liminf \int f_n \varphi+(1-f_n) \psi =\int f \varphi+(1-f)\psi= \Upsilon^*(f).$$ We then have $\lim \Upsilon^*(f_{n'})=\Upsilon^*(f)$ and by uniqueness of the limit we see that the extraction was not necessary. This establishes (iv) and ends the proof of the proposition. ◻ The next lemma and theorem state very important saturation properties satisfied by the optimal exterior transport plan. These results extend [@DePMSV Lemma 5.1 & Proposition 5.2] to more general costs $c$. **Lemma 19**. *Assume that $k$ satisfies (H[\[cont\]](#cont){reference-type="ref" reference="cont"})&(H[\[cone\]](#cone){reference-type="ref" reference="cone"}). For $f \in L^1_m$ let $\gamma$ be optimal for $\Upsilon(f)$. Then for every $(x_0,y_0)\in\mathop{\mathrm{supp}}\gamma$ there holds $f+\gamma_y\equiv1$ almost everywhere on the saturation set $$S(x_0, y_0) := \{ y \in \mathbb{R}^d, \, k(y-x_0) < k(y_0-x_0) \}.$$* *Proof.*  \ In the proof we set $g := \gamma_y$ and $h := f+g$. Let $(x_0, y_0)\in\mathop{\mathrm{supp}}\gamma$ and assume without loss of generality that $x_0 = 0$. We suppose by contradiction that there exists $\varepsilon> 0$ such that the set $$S_{\varepsilon} := \{h < 1\} \cap \{y \in \mathbb{R}^d, \, k(y) < k(y_0) - \varepsilon\}$$ has positive Lebesgue measure. Notice that by (H[\[cont\]](#cont){reference-type="ref" reference="cont"}), $k(x) \to \infty$ as $|x|\to \infty$ so that $S_{\varepsilon}$ is bounded. Therefore, $$m_{\varepsilon} := \int_{S_{\varepsilon}} (1-h)\,\ \in (0, \infty).$$ We now exhibit an exterior transport plan $\overline{\gamma}$ whose transportation cost is strictly smaller than the one of $\gamma$. Given $r >0$, we define the measure $\gamma^0 := \gamma \hskip 2.5pt{\vrule height7pt width.5pt depth0pt} \hskip-.2pt\vbox{\hrule height.5pt width7pt depth0pt} \, (B_r \times B_r(y_0))$. As $(0, y_0) \in \mathop{\mathrm{supp}}\gamma$, for every $r> 0$, $$\label{defsup} 0 < \gamma^0(\mathbb{R}^d \times \mathbb{R}^d) \leq \int_{B_r} f\, \leq |B_r|.$$ Thus, by the last inequality in [\[defsup\]](#defsup){reference-type="eqref" reference="defsup"} there exists $r_{\varepsilon} > 0$ such that for every $r \in (0, r_\varepsilon]$, $$\gamma^0(\mathbb{R}^d \times \mathbb{R}^d) = \alpha m_{\varepsilon}$$ for some $0 < \alpha \leq 1$. Let us fix $r \in (0, r_{\varepsilon}]$. We define a competitor $\widetilde\gamma$ for $\Upsilon(f)$ by setting $\widetilde \gamma := \gamma - \gamma^0 + \eta$, where $$\eta := \gamma_x^0 \otimes \frac{1-h}{m_{\varepsilon}}\chi_{S_{\varepsilon}}.$$ By construction, $\widetilde\gamma_x = \gamma_x = f$. We also have $$f+ \widetilde\gamma_y \leq f +g + \alpha(1-h)\chi_{S_{\varepsilon}} \leq h + \alpha(1-h) = 1 - (1-h)(1-\alpha) \leq 1,$$ so that $\widetilde\gamma$ is admissible for $\Upsilon(f)$. We compute $$\int c \, d\widetilde\gamma - \int c \, d\gamma = \int c \, d\eta - \int c \, d\gamma^0 \leq \alpha m_{\varepsilon} \left(\max_{\overline{B}_r \times \overline{S}_{\varepsilon}} c(x,y) - \min_{\overline{B}_r \times \overline{B}_r(y_0)} c(x,y)\right).$$ By continuity of $c$ there exists $r_\varepsilon>0$ such that for $0<r\le r_{\varepsilon}$, $$\max_{\overline{B}_r \times \overline{S}_{\varepsilon}} c(x,y) \leq k( y_0) - \varepsilon/2 \qquad \text{and} \qquad \min_{\overline{B}_r \times \overline{B}_r(y_0)} c(x,y) \geq k( y_0) - \varepsilon/4.$$ Thus for $0<r \leq r_{\varepsilon}$, $$\int c \, d\widetilde\gamma - \int c \, d\gamma \leq -\alpha \varepsilon m_{\varepsilon}/4 <0,$$ which contradicts the fact that $\gamma$ is a minimiser for $\Upsilon(f)$. ◻ **Theorem 20**. *Assume that $k$ satisfies (H[\[cont\]](#cont){reference-type="ref" reference="cont"})&(H[\[cone\]](#cone){reference-type="ref" reference="cone"}). For $f \in L^1_m$, let $\gamma\in{\varPi}_f$ be a minimiser of [\[PPRd\]](#PPRd){reference-type="eqref" reference="PPRd"} and set $g := \gamma_y$. Then, defining $$E:=\{x : \exists\, y \neq x\text{ such that } (x,y) \in \mathop{\mathrm{supp}}\gamma \text{ or } (y,x) \in \mathop{\mathrm{supp}}\gamma\},$$ the set $E$ is Lebesgue measurable and we have the identity $g = (1-f)\chi_E + f\chi_{E^c}$.* *Proof.*  \ *Step 1. A preliminary claim.* We first prove the following. Let $\mu, \nu \in \mathcal{M}_+(\mathbb{R}^d)$ be such that $\mu(\mathbb{R}^d) = \nu(\mathbb{R}^d)$, and let $\gamma \in {\varPi}(\mu, \nu)$. Assume that $\mu$ and $\nu$ are absolutely continuous with respect to the Lebesgue measure. If we define the set $$\mathcal{A}(\gamma) := \{ x : \exists \, y \neq x\text{ such that }(x,y) \in \mathop{\mathrm{supp}}\gamma \},$$ then $\mu \leq \nu$ on $\mathcal{A}(\gamma)^c$. To prove the claim, let us first show that $\mathcal{A}(\gamma)$ is Lebesgue measurable. We define $$\mathcal{D}(\gamma) := \mathop{\mathrm{supp}}\gamma \setminus \{(x,x): \, x \in \mathbb{R}^d\},$$ which is a Borel set of $\mathbb{R}^d \times \mathbb{R}^d$. If we denote by $p_x : X \times X \to X$ the canonical projection on the first variable, we have $$p_x(\mathcal{D}(\gamma)) = \{x \in \mathbb{R}^d: \, \exists y \not = x, \, (x,y) \in \mathop{\mathrm{supp}}\gamma \} = \mathcal{A}(\gamma).$$ We deduce that $\mathcal{A}(\gamma)$ is Lebesgue measurable as the continuous image of a Borel set. We now show that $\mu \le \nu$ on $\mathcal{A}(\gamma)^c$. Let $\phi \in C_c(\mathbb{R}^d, \mathbb{R}_+)$. By definition of $\mathcal{A}(\gamma)$, if $(x,y) \in \mathop{\mathrm{supp}}\gamma$ and $x\in\mathcal{A}(\gamma)^c$ then $x=y$. Therefore $$\begin{aligned} \int_{\mathcal{A}(\gamma)^c} \phi \, d\mu = \int \phi(x) \chi_{\mathcal{A}(\gamma)^c}(x) \, d\gamma(x,y) &= \int \phi(y) \chi_{\mathcal{A}(\gamma)^c}(y) \chi_{\mathcal{A}(\gamma)^c}(x) \, d\gamma(x,y)\\ &\le \int \phi(y) \chi_{\mathcal{A}(\gamma)^c}(y) \, d\gamma(x,y) =\int_{\mathcal{A}(\gamma)^c}\phi \, d\nu, \end{aligned}$$ and the claim is proved. *Step 2. Construction of $E$.* We now consider an optimal exterior transport plan $\gamma$ for $\Upsilon(f)$ and set $g := \gamma_y$, $h := f+g$. By Proposition [Proposition 17](#prop_Ups_max){reference-type="ref" reference="prop_Ups_max"}, $\gamma$ is an optimal transport plan from $f$ to $g$. Let $\overline\gamma$ be the image of $\gamma$ through the map $(x,y) \mapsto (y,x)$ and define $$E := \mathcal{A}(\gamma) \cup \mathcal{A}(\overline\gamma).$$ We have $E^c=\mathcal{A}(\gamma)^c\cap\mathcal{A}(\overline\gamma)^c$ and by the first step there holds $f\le g$ and $g\le f$ almost everywhere on $E^c$. Hence, $$g\chi_{E^c}=f\chi_{E^c}.$$ To conclude the proof, we have to show that $g \equiv 1-f$ on $E$ or equivalently that up to Lebesgue negligible sets $\mathcal{A}(\gamma)$ and $\mathcal{A}(\overline\gamma)$ are included in $\{g = 1-f\}$. On the one hand, if $x_0 \in \mathcal{A}(\gamma)$ there exists $y_0 \neq x_0$ such that $(x_0, y_0) \in \mathop{\mathrm{supp}}\gamma$. By Lemma [Lemma 19](#lem_saturation){reference-type="ref" reference="lem_saturation"}, denoting $$S(x_0, y_0) := \{y \in \mathbb{R}^d, \, k(y-x_0) < k(y_0-x_0)\},$$ we have $g = 1-f$ on $S(x_0, y_0)$. Notice that $S(x_0, y_0)$ is an open set and that $x_0 \in S(x_0, y_0)$ (since for $x \neq 0$, $k(x) >0=k(0)$). Hence $g(x_0) = 1-f(x_0)$ and $\mathcal{A}(\gamma) \subset \{g = 1-f\}$. On the other hand, if $y_0 \in \mathcal{A} (\overline\gamma)$ there exists $x_0 \neq y_0$ such that $(x_0, y_0) \in \mathop{\mathrm{supp}}\gamma$. Let us assume by contradiction that $g(y_0) < 1 - f(y_0)$. Without loss of generality, we can assume that $y_0$ is a point of Lebesgue density one of $\{g < 1 -f\}$. Then $$\lim_{r \to 0} \frac{1}{r^d}\Big|\left\{g = 1-f\right\} \cap B(y_0, r) \Big| = 0.$$ Thus by Lemma [Lemma 19](#lem_saturation){reference-type="ref" reference="lem_saturation"}, $$\lim_{r \to 0} \frac{1}{ r^d}\left|S(x_0, y_0) \cap B(y_0, r) \right| = 0,$$ which contradicts (H[\[cone\]](#cone){reference-type="ref" reference="cone"}) as $y_0 \neq x_0$. Hence $g(y_0) = 1-f(y_0)$ and $\mathcal{A}(\overline\gamma) \subset \{g = 1-f \}$. This concludes the proof of the theorem. ◻ An important corollary is the uniqueness of the second marginal of minimisers of [\[PPRd\]](#PPRd){reference-type="eqref" reference="PPRd"}. **Corollary 21**. *Assume that $k$ satisfies (H[\[cont\]](#cont){reference-type="ref" reference="cont"})&(H[\[cone\]](#cone){reference-type="ref" reference="cone"}). Let $f \in L^1(\mathbb{R}^d)$. Then all minimisers $\gamma$ of $\Upsilon(f)$ have the same second marginal $\gamma_y$.* *Proof.*  \ We let $\gamma, \gamma'$ be two minimisers of [\[PPRd\]](#PPRd){reference-type="eqref" reference="PPRd"} and define $\widetilde \gamma := (\gamma + \gamma')/2$ which also minimises [\[PPRd\]](#PPRd){reference-type="eqref" reference="PPRd"}. We denote $g := \gamma_y$, $g' := \gamma'_y$ and $\widetilde{g} := \widetilde{\gamma}_y$ and introduce the set $$F := \{x \in \mathbb{R}^d: g(x), g'(x), \widetilde{g}(x) \in \{f(x), 1-f(x)\}\}.$$ Assuming by contradiction that $g(x) \not = g'(x)$ for some $x \in F$ , we have $1/2 = \widetilde{g}(x) \in \{f(x), 1-f(x)\}$ so that $f(x) = 1/2$ and $g(x) = g'(x) = 1/2$, which is absurd. Hence $g = g'$ on $F$ and since $F$ is of full measure by Theorem [Theorem 20](#thm_saturation){reference-type="ref" reference="thm_saturation"}, the proof is complete. ◻ ## Preliminary results for the existence of a maximiser of [\[MP\]](#MP){reference-type="eqref" reference="MP"}. We now gather results which, combined with Theorem [Theorem 20](#thm_saturation){reference-type="ref" reference="thm_saturation"}, allow us to prove existence of a maximiser for both $\mathcal{E}_{\textrm{set}}(m)$ and $$\label{MP} \mathcal{E}(m) := \sup \left\{\Upsilon(f) : f \in L^1(\mathbb{R}^d, [0,1]), \int f\, = m \right\}.$$ We first establish a corollary of Theorem [Theorem 20](#thm_saturation){reference-type="ref" reference="thm_saturation"} regarding the monotonicity of the sum of the marginals of solutions to [\[PPRd\]](#PPRd){reference-type="eqref" reference="PPRd"}. **Corollary 22**. *Assume that $k$ satisfies (H[\[cont\]](#cont){reference-type="ref" reference="cont"})&(H[\[cone\]](#cone){reference-type="ref" reference="cone"}). Let $m >0$, let $f_1,f_2 \in L^1_m$ be such that $f_1 \leq f_2$ and let $\gamma^1, \gamma^2$ be respective minimisers of $\Upsilon(f_1)$ and $\Upsilon(f_2)$. Then setting $g_1 := \gamma^1_y$ and $g_2 := \gamma^2_y$, we have $f_1+ g_1 \le f_2+ g_2.$* *Proof.*  \ Let $f_1, f_2 \in L^1_m$ be such that $f_1 \leq f_2$. In the first three steps of the proof, we additionally assume that they are compactly supported. This condition is relaxed in the fourth and final step.\ By Proposition [Proposition 17](#prop_Ups_max){reference-type="ref" reference="prop_Ups_max"}, we can assume that the ambient space is a compact ball $\overline{B}_R$. Let $\gamma^1, \gamma^2$ be minimisers for $\Upsilon(f_1)$ and $\Upsilon(f_2)$ respectively. For $i \in \{1,2\}$ we define $g_i:= \gamma^i_y$, $h_i:=f_i+g_i$ and set $$F:=\{h_1 > h_2\}.$$ We shall prove that $|F|=0$. By Theorem [Theorem 20](#thm_saturation){reference-type="ref" reference="thm_saturation"}, there exists $E_1, E_2 \subset \overline{B}_R$ such that $$h_1 = \chi_{E_1} + 2 f_1 \chi_{E_1^c} \qquad \text{and} \qquad h_2 = \chi_{E_2} + 2f_2 \chi_{E_2^c}.$$ Since $h_2\ge0$, $h_1\le 1$ and $h_2\ge f_2 \geq f_1$ we have $$\label{honF} h_1>0,\qquad h_2=2f_2<1\qquad\text{and}\qquad f_1 < 1\quad \text{on } F.$$ *Step 1. $|E_1^c \cap F|=0$.* By definition of $E_1$ we have $h_1= 2f_1$ on $E_1^c$ and by [\[honF\]](#honF){reference-type="eqref" reference="honF"} we have $h_2 = 2f_2$ on $F$ and since $f_1\le f_2$ we get $h_1\le h_2$ on $E_1^c \cap F$. This contradicts the definition of $F$, hence $E_1^c \cap F =\emptyset$ and in particular $|E_1^c \cap F | = 0$. *Step 2. Intermediate claim.* Let $\psi_1$ be the maximal potential for $\Upsilon^*(f_1)$ given by Proposition [Proposition 16](#prop_pot_monot){reference-type="ref" reference="prop_pot_monot"}. We define $$G := \{\psi_1^{\bar cc} =0\}\cap E_1 \cap F$$ and claim that $|G| = 0$. Let us assume by contradiction that $|G|>0$. First notice that on $F$, $$f_1 +g_1 =h_1> h_2 = 2f_2,$$ so that $$g_1 > 2f_2 - f_1\geq f_2 \geq f_1.$$ Thus $$\label{Gsubg1f1} G\subset E_1 \cap F \subset \{ g_1 > f_1 \}.$$ Now recall that by Theorem [Theorem 20](#thm_saturation){reference-type="ref" reference="thm_saturation"}, $$E_1=\{x : \exists y \neq x\text{ such that } (x,y) \in\mathop{\mathrm{supp}}\gamma^1\text{ or }(y,x)\in\mathop{\mathrm{supp}}\gamma^1\}.$$ Together with [\[Gsubg1f1\]](#Gsubg1f1){reference-type="eqref" reference="Gsubg1f1"} we obtain that for almost every $y_0 \in G$ there exists $x_0 \neq y_0$ with $(x_0, y_0) \in \mathop{\mathrm{supp}}\gamma^1$. Without loss of generality, we assume that $y_0$ is a point of density of $G$ and we set $$S(x_0, y_0) := \left\{y \in \mathbb{R}^d: \, k(y-x_0) < k(y_0 - x_0) \right\}.$$ By (H[\[cone\]](#cone){reference-type="ref" reference="cone"}), we have $|G \cap S(x_0, y_0)|> 0$. Let now $\widetilde{y} \in G \cap S(x_0, y_0)$. By Proposition [Proposition 15](#prop_Ups_OT){reference-type="ref" reference="prop_Ups_OT"}, $(\psi^{\bar c}_1, \psi^{\bar cc}_1)$ forms a pair of Kantorovitch potentials for the optimal transport from $f_1$ to $g_1$. Thus $$\psi^{\bar c}_1 (x_0) + \psi^{\bar cc}_1(y_0) = k(y_0-x_0) \qquad \text{and} \qquad \psi^{\bar c }_1(x_0) + \psi^{\bar cc}_1(\widetilde{y}) \leq k(\widetilde{y}-x_0).$$ However, $y_0, \widetilde{y} \in G$, so that $\psi^{\bar cc}_1(y_0) = \psi^{\bar cc}_1(\widetilde{y}) = 0$, hence $$\psi^{\bar c}_1 (x_0) = k(y_0-x_0) \qquad \text{and} \qquad \psi^{\bar c }_1(x_0) \leq k(\widetilde{y}-x_0).$$ Eventually, as $\widetilde{y} \in S(x_0, y_0)$, we conclude that $$\psi^{\bar c}_1 (x_0) \leq k(\widetilde{y}-x_0) < k(y_0-x_0) = \psi^{\bar c}_1 (x_0),$$ obtaining a contradiction. Thus $|G|=0$, which is the claim. *Step 3. $|E_1 \cap F|=0$.* By Proposition [Proposition 15](#prop_Ups_OT){reference-type="ref" reference="prop_Ups_OT"}, $$\{\psi^{\bar cc}_1 > 0\} \subset \{g_1 = 0\} \quad\text{ so that }\quad \{g_1 > 0\} \subset \{\psi^{\bar cc}_1 \leq 0\}.$$ We observe that $g_1 = 1-f_1$ on $E_1 \cap F$. By [\[honF\]](#honF){reference-type="eqref" reference="honF"}, $E_1 \cap F \subset \{ g_1 > 0\}$ and by the previous step, $E_1 \cap F \subset \{\psi^{\bar cc}_1 \neq 0\}$, hence $\psi_1^{\bar cc} < 0$ almost everywhere on $E_1 \cap F$. Let $\psi_2$ be the maximal potential for $\Upsilon^*(f_2)$ given by Proposition [Proposition 16](#prop_pot_monot){reference-type="ref" reference="prop_pot_monot"}. As $f_1 \leq f_2$, we have $\psi_1 \geq \psi_2$ so that $\psi^{\bar cc}_1 \geq \psi^{\bar cc}_2$. Thus $$\psi^{\bar cc}_2 < 0 \text{ on } E_1 \cap F.$$ By Proposition [Proposition 15](#prop_Ups_OT){reference-type="ref" reference="prop_Ups_OT"} we deduce that $$h_2 = g_2 + f_2 = 1 \text{ on } E_1 \cap F.$$ But since $h_2 < 1$ on $F$ we get that $|E_1 \cap F | = 0$ and with the first step we conclude that $|F| = 0$. *Step 4. Extension to the non-compact case*. Let $f_1,f_2 \in L^1_m$ be such that $f_1 \leq f_2$. For $i \in \{1, 2\}$, we set $f_{i, R} = f_i \chi_{B_R}$, consider $\gamma^i_R$ an optimal exterior transport plan for $\Upsilon(f_{i, R})$ and set $g_{i, R} := (\gamma^i_R)_y$. Applying the previous steps to $f_{1,R}$ and $f_{2, R}$, we obtain $$\label{compactineq} f_{1,R} + g_{1,R} \leq f_{2,R} + g_{2,R}.$$ For $i \in\{1, 2\}$, $f_{i,R}$ $L^1$-converges to $f_i$ as $R \to \infty$. By Proposition [Proposition 18](#prop_basic_ups){reference-type="ref" reference="prop_basic_ups"} $(iii)$, $\Upsilon(f_{i, R}) \to \Upsilon(f_i)$ as $R \to \infty$. Additionally, $\gamma^i_R$ admits a subsequence converging weakly-$*$ to some $\widetilde{\gamma}^i$ admissible for $\Upsilon(f_i)$. By lower semi-continuity of $\gamma \mapsto \smallint c \, d \gamma$ with respect to weak convergence, we get $$\int c \, d\widetilde{\gamma^i} \leq \liminf_R \int c \, d\gamma^i_R = \liminf_R \Upsilon(f_{i,R}) = \Upsilon(f_i).$$ Hence $\widetilde{\gamma}^i$ is optimal for $\Upsilon(f_i)$, so that by Corollary [Corollary 21](#coro_unique_g){reference-type="ref" reference="coro_unique_g"}, $\widetilde{\gamma}^i_y = g_i$. Finally, as $\gamma^i_R\stackrel*\rightharpoonup \widetilde{\gamma}^i$ as $R \to \infty$, $g_{j,R}$ converges in duality with $C_c(\mathbb{R}^d)$ to $g_i$ as $R \to \infty$. Multiplying [\[compactineq\]](#compactineq){reference-type="eqref" reference="compactineq"} by $\phi \in C_c(\mathbb{R}^d, \mathbb{R}_+)$, integrating and passing to the limit we obtain that for any $\phi \in C_c(\mathbb{R}^d, \mathbb{R}_+)$, $$\int (f_1 + g_1) \phi\, \leq \int (f_2+g_2) \phi\,.$$ Hence $f_1 + g_1 \leq f_2 + g_2$ which completes the proof. ◻ We now prove that $\mathcal{E}$ is strictly superadditive. **Proposition 23**. *Assume that $k$ satisfies (H[\[cont\]](#cont){reference-type="ref" reference="cont"})&(H[\[monot\]](#monot){reference-type="ref" reference="monot"}). Let $m \in (0, \infty)$ and define $e(m) :=\mathcal{E}(m)/m$. Then, $e$ is increasing on $(0, \infty)$. As a consequence, given $0 < m' < m$, $$\mathcal{E}(m') + \mathcal{E}(m-m') < \mathcal{E}(m).$$* *Proof.*  \ Let $M>m>0$. We have to establish that $\mathcal{E}(m) < (m/M)\mathcal{E}(M)$. *Step 1. $\mathcal{E}(m) \leq (m/M)\mathcal{E}(M)$.* For $R>0$ we set $$\Gamma_R:=\{(x,y) \in \mathbb{R}^d \times \mathbb{R}^d: |x-y|> R\}.$$ Let $0 \leq \varepsilon< \mathcal{E}(m)/2$ and $f \in L^1_m$ of mass exactly $m$ and such that $\Upsilon(f)\ge \mathcal{E}(m)-\varepsilon$. We denote $\lambda:=(M/m)^{1/d} >1$ and we set $$f_{\lambda}(x):=f(x/\lambda)\qquad\text{for }x\in\mathbb{R}^d,$$ so that $\int f_{\lambda}\,=M$. Let $\gamma^{\lambda}$ be an optimal transport plan for $\Upsilon(f_{\lambda})$. We define a Radon measure $\gamma$ by $$\int \xi(x,y)\, d\gamma(x,y) := \frac{m}{M}\int \xi(x/\lambda,y/\lambda) \,d\gamma^{\lambda}(x,y)\qquad\text{for }\xi\in C_c(\mathbb{R}^d \times \mathbb{R}^d).$$ Observe that $\gamma$ is admissible for $\Upsilon(f)$. By Proposition [Proposition 17](#prop_Ups_max){reference-type="ref" reference="prop_Ups_max"}, there exists $R_{\lambda} = R_{\lambda}(M)$ such that $\gamma^{\lambda}(\Gamma_{R_{\lambda}})=0$. Setting $R := R_{\lambda}/\lambda$, we then have $$\label{defR} \gamma(\Gamma_R) = \frac{m}{M} \gamma^{\lambda}(\Gamma_{R_{\lambda}}) = 0.$$ Let us define $$\kappa(r):=\min \left\{ k(z)-k(z/\lambda) :r \leq |z|\leq R_{\lambda} \right\}.$$ As $\lambda > 1$ we have by (H[\[monot\]](#monot){reference-type="ref" reference="monot"}) that $\kappa(r) > 0$ for $0 <r < R_{\lambda}$. Additionally, $k(z/\lambda) \leq k(z)$ for any $z \in \mathbb{R}^d$. Consequently, for any $0 <r < R_{\lambda}$, $$\begin{aligned} \Upsilon(f) &\le\int k(y-x)\,d \gamma(x,y)\\ &= \frac{m}{M}\int k\left(\dfrac{y-x}\lambda\right)\,d\gamma^{\lambda}(x,y)\\ &= \frac{m}{M} \int \left[ k\left(\dfrac{y-x}\lambda\right)-k(y-x) \right] d\gamma^{\lambda}(x,y) + \frac{m}{M} \int k(y-x) \, d \gamma^{\lambda}(x,y) \\ &\le \frac{m}{M} \int_{\Gamma_r} \left[k\left(\dfrac{y-x}\lambda\right)-k(y-x) \right] d\gamma^{\lambda}(x,y) + \frac{m}{M}\mathcal{E}(M). \end{aligned}$$ In the integral over $\Gamma_r$, the term in brackets is smaller than $-\kappa(r)$. Hence, for every $0 < r < R_{\lambda}$ $$\label{prf_prop_mu_inc_1} \mathcal{E}(m)-\varepsilon\le \Upsilon(f) \leq \frac{m}{M}\mathcal{E}(M)- \frac{m}{M}\kappa(r)\gamma^{\lambda}(\Gamma_r).$$ At this point we can send $\varepsilon$ to 0 and deduce that $\mathcal{E}(m) \leq (m/M)\mathcal{E}(M)$. However we need to establish a strict inequality. For this we prove in the next step that there exist $r^*,\delta>0$ not depending on $\varepsilon$ or $f$ such that $\gamma^\lambda(\Gamma_{r^*})\ge \delta$. *Step 2. Conclusion.* For $r\ge 0$, we set $$\overline{k}(r):=\max\{k(z):|z|\le r\}.$$ This function is increasing, continuous and there holds $\overline{k}(0)=0$. Notice that using a ball of mass $m$ as a candidate for the energy $\mathcal{E}(m)$, we see that $\mathcal{E}(m) > 0$ for any $m>0$. Let us fix $0 < r^* < R$ such that $$\label{prf_prop_mu_inc_2} m\overline{k}(r^*)\le \mathcal{E}(m)/4.$$ By [\[defR\]](#defR){reference-type="eqref" reference="defR"} we have $|x-y| \leq R$ for $(x,y) \in \mathop{\mathrm{supp}}\gamma$ and by definition $|x-y|\leq r^*$ for $(x,y)\not\in\Gamma_{r^*}$. We deduce $$\begin{aligned} \frac{\mathcal{E}(m)}{2} <\Upsilon(f) = \int_{\Gamma_{r^*}} c \, d\gamma + \int_{\Gamma_{r^*}^c} c \, d\gamma &\le\gamma(\Gamma_{r^*})\overline{k}(R)+\left(m-\gamma(\Gamma_{r^*})\right)\overline{k}({r^*})\\ &\stackrel{\eqref{prf_prop_mu_inc_2}}{\le}\gamma(\Gamma_{r^*})\overline{k}(R)+\left(m-\gamma(\Gamma_{r^*})\right)\frac{\mathcal{E}(m)}{4m}. \end{aligned}$$ This implies $$\gamma(\Gamma_{r^*}) \left(\overline{k}(R) - \frac{\mathcal{E}(m)}{4m}\right) > \frac{\mathcal{E}(m)}{4}.$$ Thus $4m\overline{k}(R) > \mathcal{E}(m)$ and $$\frac{m}{M}\gamma^{\lambda}(\Gamma_{\lambda r^*})=\gamma(\Gamma_{r^*})\ge\frac{m\mathcal{E}(m)}{4m\overline{k}(R)-\mathcal{E}(m)}=:m^*>0.$$ Plugging this in [\[prf_prop_mu_inc_1\]](#prf_prop_mu_inc_1){reference-type="eqref" reference="prf_prop_mu_inc_1"} with $r= \lambda r^*<R_\lambda$ we obtain $$\mathcal{E}(m)-\varepsilon\le\frac{m}{M}\mathcal{E}(M)-m^*\kappa(r^*).$$ Since $\varepsilon\in[0,\mathcal{E}(m)/2]$ is arbitrary and $m^*\kappa(r^*)>0$, this proves the proposition. ◻ We close this subsection with a lemma establishing that if a function $f$ nearly maximises $\mathcal{E}(m)$ for some $m>0$ then there exists a cube which is at least half filled by $f$. **Lemma 24**. *Let $m>0$. There exists a non-decreasing function $r_0 : m \mapsto r_0(m)$ such that for $m>0$ and $f \in L^1_m$ with $\Upsilon(f)\ge \mathcal{E}(m)/2$, there exists a cube $Q_0$ of side-length $r_0(m)$ such that: $$\int_{Q_0}f\,\ge \frac{|Q_0|}2.$$* *Proof.*  \ Let $r_0>0$ to be fixed later and assume by contradiction that there exists a partition $\mathcal{Q}$ of $\mathbb{R}^d$ in cubes with side-length $r_0$ such that for every $Q\in\mathcal{Q}$, $$\int_Qf\,<\frac{|Q|}{2}.$$ The strategy to get a contradiction from this hypothesis is to build an exterior transport plan for $f$ with too small transport cost. Let $Q\in\mathcal{Q}$. Since $\smallint_Q (1-f)\,\ge\smallint_Qf\,$ there exists a function $g_Q\ge0$ supported in $Q$ such that $\smallint g_Q\,=\smallint_Qf\,$ and $f \chi_Q + g_Q \leq 1$. We then set $$\gamma_Q:= f \chi_Q \otimes \frac{g_Q}{\int_Q f\,} \qquad \text{and} \qquad \gamma := \sum_{Q \in \mathcal{Q}} \gamma_Q.$$ Notice that $\gamma$ is a valid competitor for $\Upsilon(f)$. Next for $R>0$, we define $$\overline{k}(R) := \max \{ k(x), \, |x| \le R\}.$$ We compute: $$\begin{gathered} \label{E(m)<kbar} 0 < \frac{\mathcal{E}(m)}{2} \leq \Upsilon(f) \leq \sum_{Q \in \mathcal{Q}} \int_{Q \times Q} k(y-x) \, d \gamma_Q\\ \le \overline{k}\left(\sqrt{d}r_0\right) \sum_{Q \in \mathcal{Q}} \int_Q f\, = \overline{k}\left(\sqrt{d}r_0\right)\int f\, \le \overline{k}\left(\sqrt{d}r_0\right) m. \end{gathered}$$ Remarking that $\overline{k}: \mathbb{R}_+ \to \mathbb{R}_+$ is continuous at $0$, increasing and with $\overline{k}(0) = 0$, we set $$r_0 := \max \left\{ r>0 : \overline{k}\left(\sqrt{d}r\right) \leq \frac{\mathcal{E}(m)}{4m}\right\}\ >0$$ and obtain a contradiction with [\[E(m)\<kbar\]](#E(m)<kbar){reference-type="eqref" reference="E(m)<kbar"}. This concludes the proof. ◻ ## Existence of a maximiser for [\[MP\]](#MP){reference-type="eqref" reference="MP"} In the following subsection, we assume that (H[\[cont\]](#cont){reference-type="ref" reference="cont"}),(H[\[cone\]](#cone){reference-type="ref" reference="cone"})&(H[\[monot\]](#monot){reference-type="ref" reference="monot"}) hold and prove the existence of maximisers for [\[MP\]](#MP){reference-type="eqref" reference="MP"}. We only have to prove that maximising sequences for $\mathcal{E}(m)$ are tight. However our result is more precise. We obtain that if $f$ nearly maximises $\mathcal{E}(m)$ then almost all its mass concentrates in a closed ball with radius $R_*=R_*(m)$. In the limit, maximisers are supported in such balls. **Proposition 25**. *Let $m > 0$. There exist $R_* = R_*(m)>0$, $\varepsilon_0 = \varepsilon_0(m)>0$ non-decreasing in $m$ with the following property. Let $0 < \varepsilon\le \varepsilon_0$ and let $f \in L^1_m$ such that $\smallint f = m$ and $\Upsilon(f) \geq \mathcal{E}(m) - \varepsilon$, then up to a translation there holds $$\int_{\mathbb{R}^d\setminus B_{R_*}}f\,\le\frac{2m}{\mathcal{E}(m)}\varepsilon.$$* *Proof.*  \ *Outline of the proof.*\ (Step 1) We start by using Lemma [Lemma 24](#lem_concentration){reference-type="ref" reference="lem_concentration"} to get a collection $\mathcal{Q}_0$ of cubes $Q$ of side-length $r_0=r_0(m)$ such that $\int_Q f+g\,\ge|Q|/2$. We denote $\Omega_0:=\cup\mathcal{Q}_0$. We also consider the set $\Omega$ obtained by thickening $\Omega_0$ by adding the cubes closer than some distance $R=R(m)$. The real $R$ is chosen so that no mass of $f\chi_{\Omega_0}$ is sent outside $\Omega$ by any optimal exterior transport plan of $f$.\ (Step 2) We build an exterior transport plan for $f$ whose cost is very close to $\Upsilon(f \chi_\Omega)$.\ (Step 3) Next, we show that $\Omega$ concentrates almost all the mass of $f$. Using the strict superadditivity of $m \mapsto \mathcal{E}(m)$ and the previous step, we deduce that $m_\Omega:=\smallint f \chi_\Omega$ is close to $m$.\ (Step 4) Eventually, we show that the distance between cubes in $\mathcal{Q}_0$ is uniformly bounded. As the cardinal of $\mathcal{Q}_0$ is also bounded, we conclude that the diameter of $\Omega$ is bounded by a distance only depending on $m$. *Step 1. Construction of a collection of cubes on which $\smallint_Q(f+g)\,\ge|Q|/2$.* Let $m>0$ and $f$ as in the statement of the proposition and assume that $\Upsilon(f)\ge \mathcal{E}(m)/2$ so that $$\label{defeps} \varepsilon:=\mathcal{E}(m)-\Upsilon(f)\le\mathcal{E}(m)/2.$$ Let $\gamma$ be a minimiser for $\Upsilon(f)$ and let us set $g := \gamma_y$. Let $r_0$ and $Q_0$ be given by Lemma [Lemma 24](#lem_concentration){reference-type="ref" reference="lem_concentration"}. We denote by $\hat{\mathcal{Q}}$ the regular partition of $\mathbb{R}^d$ into cubes of side-length $r_0$ such that $Q_0 \in \hat{\mathcal{Q}}$. For $j \geq 0$ to be fixed later, we set $r_j:=2^{-j}r_0$. Considering the partition $\mathcal{Q}$ of $\mathbb{R}^d$ into cubes of side-length $r_j$ obtained by refining $\hat{\mathcal{Q}}$, we define $\mathcal{Q}_0$ as the subset formed by the elements $Q\in\mathcal{Q}$ such that $$\int_Q (f+g) \,\ge\frac{|Q|}4.$$ We remark that $\mathcal{Q}_0$ is not empty since $$\int_Q(f+g)\,\ge\int_Qf\,\ge\frac{|Q|}2$$ for at least one of the $2^j$ sub-cubes of $Q_0$ in the partition $\mathcal{Q}$. Let us define $\Omega_0:=\cup\mathcal{Q}_0$. By Proposition [Proposition 17](#prop_Ups_max){reference-type="ref" reference="prop_Ups_max"}, there exists $R = R(m)$ such that $|x-y| \leq R$ on $\mathop{\mathrm{supp}}\gamma$. We denote by $\mathcal{Q}_R$ the collection of cubes $Q \in\mathcal{Q}$ such that $d(Q, \Omega_0)\le R$, and by $\Omega$ their union. By construction, there holds $\gamma(\Omega_0 \times\Omega^c)=0$. We now define $$f_\Omega:= f \chi_\Omega, \qquad \text{ and }\qquad m_\Omega:=\int f_\Omega\,,$$ and we let $\gamma_\Omega$ be an optimal exterior transport plan for $f_\Omega$, that is $\gamma\in\varPi_{f_\Omega}$ with $\int c\,d\gamma_\Omega= \Upsilon(f_\Omega)$. By Proposition [Proposition 17](#prop_Ups_max){reference-type="ref" reference="prop_Ups_max"} again, we have (since $m_\Omega\le m$) that $$\label{gammaOmega} \gamma_\Omega(\Omega_0 \times \Omega^c) = 0.$$ *Step 2. Building a transport plan for $f$ whose cost is close to $\Upsilon(f_\Omega)$.* In this step we modify $\gamma_\Omega$ to build an exterior transport plan $\gamma$ for $f$ with a cost close to $\Upsilon(f_\Omega)$. More precisely, we require that for some constant $C = C(r_j)>0$ with $C(r_j)\to 0$ as $r_j\to 0$, $$\int c \, d \gamma - \int c \, d\gamma_\Omega\leq C(m-m_\Omega).$$ The proof is a refinement of the proof of the Lipschitz continuity of $\Upsilon$, see Proposition [Proposition 18](#prop_basic_ups){reference-type="ref" reference="prop_basic_ups"} (iii). In the following we define successively the plans $\gamma^0$, $\gamma^1$, $\gamma^2$, $\gamma^3$ which satisfy in particular $$\mathop{\mathrm{supp}}\gamma^0\subset\overline\Omega\times\overline\Omega,\qquad \mathop{\mathrm{supp}}\gamma^1\subset\overline\Omega\times\overline{\Omega^c},\qquad \mathop{\mathrm{supp}}\gamma^2\subset\overline{\Omega\setminus\Omega_0}\times\overline{\Omega\setminus\Omega_0},\qquad \mathop{\mathrm{supp}}\gamma^3\subset\overline{\Omega^c}\times\overline{\Omega^c}.$$ First we set $\gamma^0:=\gamma_\Omega \hskip 2.5pt{\vrule height7pt width.5pt depth0pt} \hskip-.2pt\vbox{\hrule height.5pt width7pt depth0pt} \, \Omega\times\Omega$ and denote $f^0:=\gamma^0_x$, $g^0:=\gamma^0_y$. We build the three remaining plans in the following substeps. These constructions will satisfy $$(\gamma^1+\gamma^2)_x=f_\Omega-\gamma^0_x=f\chi_\Omega-f^0\qquad\text{and}\qquad\gamma^3_x=f-f_\Omega=f\chi_{\Omega^c}.$$ We will set eventually $\widehat\gamma:=\gamma^0+\gamma^1+\gamma^2+\gamma^3$ which will be an optimal transport plan for $f$. The difficulty is to preserve the constraint $f+\widehat\gamma_y\le1$ while controlling the cost. *Step 2.a. Construction of $\gamma^1$.* Let us denote $\gamma_\Omega^1:=\gamma_\Omega \hskip 2.5pt{\vrule height7pt width.5pt depth0pt} \hskip-.2pt\vbox{\hrule height.5pt width7pt depth0pt} \, \Omega\times\Omega^c$, $f_\Omega^1:=(\gamma_\Omega^1)_x$ and $g_\Omega^1:=(\gamma_\Omega^1)_y=\chi_{\Omega^c}g_\Omega$. We can not rule out the possibility that $f+g_\Omega>1$ in some part of $\Omega^c$ so that we cannot set $\gamma^1=\gamma_\Omega^1$. However, we will transport as much as possible mass through $\gamma_\Omega^1$. Let us define $$u:=(f+g_\Omega-1)_+,$$ which corresponds to the excess mass transported through $\gamma^1_\Omega$. Using the convention $0/0=0$, we define $\gamma^1$ by $$d\gamma^1(x,y):=\frac{g_\Omega(y)-u(y)}{g_\Omega(y)}d\gamma^1_\Omega(x,y).$$ At this point, we have $$\label{estim_cost_0+1} \int c\,d(\gamma^0+\gamma^1)\le \int c\, d\gamma_\Omega=\Upsilon(f_\Omega).$$ Moreover setting $f^1:=\gamma^1_x$ and $g^1:=\gamma^1_y$, there holds $\mathop{\mathrm{supp}}g^1\subset\overline{\Omega^c}$. Notice that since $f_\Omega\le f$, by Corollary [Corollary 22](#coro_monot_f){reference-type="ref" reference="coro_monot_f"} we have $f_\Omega+g_\Omega\le f+g$, so that $g_\Omega\le f+ g$ in $\Omega^c$ which implies $g^1\le f+g$. Thus $$\label{intQf+Q1} \int_Q (f+g^1)\,\le\int_Q (2f+g)\,<\frac{|Q|}2,\qquad\text{for every }Q\in \mathcal{Q}\setminus\mathcal{Q}_R,$$ where we used the definition of $\mathcal{Q}_0$ and the fact that $[\mathcal{Q}\setminus\mathcal{Q}_R] \cap \mathcal{Q}_0=\emptyset$.\ Let us compute for later use the mass from $\Omega$ that still requires to be transported. By construction $$\label{massleft} \int_\Omega(f -f^0-f^1) \,=\int\, d(\gamma^1_\Omega-\gamma^1)=\int\frac{u(y)}{g_\Omega(y)}\,d\gamma_\Omega^1(x,y) =\int_{\Omega^c}(f+g_\Omega-1)_+\,\le\int_{\Omega^c}f\,.$$ *Step 2.b. Construction of $\gamma^2$.* We now define $$\gamma^2_\Omega:= \gamma^1_\Omega- \gamma^1 = \gamma_\Omega-\gamma^0 - \gamma^1.$$ Notice that by [\[gammaOmega\]](#gammaOmega){reference-type="eqref" reference="gammaOmega"}, $\gamma^1_\Omega(\Omega_0 \times \Omega^c) = 0$, so that $$\mathop{\mathrm{supp}}\gamma^2_\Omega\subset\overline{\Omega\setminus\Omega_0}\times\overline{\Omega^c}.$$ In particular, $f^2:=f_\Omega-f^0-f^1$ is supported in $\overline{\Omega\setminus\Omega_0}$. Let $Q\in\mathcal{Q}_R\setminus\mathcal{Q}_0$. Since $g^0\le g_\Omega$ and $f=f_\Omega$ on $Q$, using Corollary [Corollary 22](#coro_monot_f){reference-type="ref" reference="coro_monot_f"} again we see that $$\int_Q(f+g^0)\,\le\int_Q(f_\Omega+g_\Omega)\,\le\int_Q(f+g)\,\le\frac{|Q|}4.$$ Therefore for such $Q$ there exists a function $g^2_Q:Q\to\mathbb{R}_+$ such that $f+g^0+g^2_Q\le1$ and $\int g^2_Q\,=\int_Qf^2\,$. Defining $$\gamma^2_Q:=\frac1{\int_Qf^2\,}\left[\chi_Q f^2\right]\otimes g^2_Q\quad\text{ for }Q\in\mathcal{Q}_R\setminus\mathcal{Q}_0\quad\text{ and then } \quad\gamma^2:=\sum_{Q\in\mathcal{Q}_R\setminus\mathcal{Q}_0} \gamma^2_Q,$$ we have $\gamma^2_x=f^2$ and $g^2:=\gamma^2_y=\sum_Q g^2_Q$. Hence $f+g^0+g^2\le 1$ and $$\int c\,d\gamma^2 \le\left(\int f^2\,\right)\overline{k}\left(\sqrt dr_j\right),$$ where as in the proof of Proposition [Proposition 23](#prop_mu_inc){reference-type="ref" reference="prop_mu_inc"} we denote $\overline{k}(r):=\max\{k(x):|x|\le r\}$.\ By construction $f^2=f_\Omega-f^0-f^1$, so by [\[massleft\]](#massleft){reference-type="eqref" reference="massleft"} there holds $\int f^2\,\le m-m_\Omega$ which leads to the cost estimate $$\label{estim_cost_2} \int c\,d\gamma^2\le(m-m_\Omega)\overline{k}\left(\sqrt d r_j\right).$$ *Step 2.c. Construction of $\gamma^3$.* We still have to transport the mass corresponding to $\chi_{\Omega^c}f$. For every $Q\in\mathcal{Q}\setminus\mathcal{Q}_R$ we have $\int_Qf\,\le\int_Q(f+g)\,\le|Q|/4$, therefore, in view of [\[intQf+Q1\]](#intQf+Q1){reference-type="eqref" reference="intQf+Q1"}, there exists a function $g^3_Q:Q\to\mathbb{R}_+$ such that $\int g^3_Q\,=\int_Q f\,$ and $f+g^1+g^3_Q\le1$. As in the previous step, we define $$\gamma^3_Q:=\frac1{\int_Qf\,}\left[\chi_Q f\right]\otimes g^3_Q \qquad\text{and}\qquad \gamma^3:=\sum_Q \gamma^3_Q.$$ By construction, $(\gamma^3)_x=\chi_{\Omega^c}f$ and denoting $g^3:=(\gamma^3)_y$, we have $\mathop{\mathrm{supp}}g^3\subset\overline{\Omega^c}$ as well as $f+g^1+g^3\le1$. Moreover $$\label{estim_cost_3} \int c\,d\gamma^3\le\left(\int_{\Omega^c}f\,\right)\overline{k}\left(\sqrt dr_j\right)=(m-m_\Omega)\overline{k}\left(\sqrt dr_j\right).$$ *Step 2.d. Conclusion : definition and properties of $\widehat\gamma$.* Eventually, we set $\widehat\gamma:=\gamma^0+\gamma^1+\gamma^2+\gamma^3$ and $\widehat g:=\widehat\gamma_y$. There holds $\widehat\gamma_x=f^0+f^1+f^2+f^3=f$ and $f+\widehat g\le1$ so that $\widehat\gamma$ is an admissible exterior transport plan for $f$. Besides, collecting the estimates [\[estim_cost_0+1\]](#estim_cost_0+1){reference-type="eqref" reference="estim_cost_0+1"},[\[estim_cost_2\]](#estim_cost_2){reference-type="eqref" reference="estim_cost_2"}&[\[estim_cost_3\]](#estim_cost_3){reference-type="eqref" reference="estim_cost_3"} we get $$\label{Upsf<UpsfOmega+} \Upsilon(f)\le\int c\, d\widehat\gamma\le\Upsilon(f_\Omega)+2(m-m_\Omega)\overline{k}\left(\sqrt dr_j\right).$$ *Step 3. We show that $m - m_\Omega\leq C(m) \varepsilon$ (recall the definition [\[defeps\]](#defeps){reference-type="eqref" reference="defeps"} of $\varepsilon$).* As $\Upsilon(f_\Omega) \leq \mathcal{E}(m_\Omega)$ and $\mathcal{E}(m) - \varepsilon=\Upsilon(f)$, [\[Upsf\<UpsfOmega+\]](#Upsf<UpsfOmega+){reference-type="eqref" reference="Upsf<UpsfOmega+"} yields $$\mathcal{E}(m)-\varepsilon\le \mathcal{E}(m_\Omega)+ 2 \overline{k}\left(\sqrt{d} r_j\right)(m-m_\Omega).$$ Additionally by Proposition [Proposition 23](#prop_mu_inc){reference-type="ref" reference="prop_mu_inc"}, $\mathcal{E}(m_\Omega) \leq \frac{m_\Omega}{m} \mathcal{E}(m)$. Hence $$\left(\frac{\mathcal{E}(m)}{m}- 2 \overline{k}\left(\sqrt{d} r_j\right)\right)(m-m_\Omega)\le \varepsilon.$$ By continuity of $k$, $\overline{k}(\sqrt{d} r_j) \to 0$ as $r_j \to 0$. Recalling that $r_j = 2^{-j} r_0$, we fix $j\ge0$ as the first integer such that $\overline{k}(\sqrt{d} r_j) \le \mathcal{E}(m)/4m$ (notice that $j$ does not depend on $\varepsilon$). Therefore $$\label{prf_lem_tight_3} m-m_\Omega\le \dfrac{2m\varepsilon}{\mathcal{E}(m)}.$$ This yields $$\label{prf_lem_tight_35} \int_{\mathbb{R}^d \setminus\Omega}f \,=\int f\, - \int f_\Omega\, = m-m_\Omega\le\dfrac{2m \varepsilon}{\mathcal{E}(m)}.$$ For future use, let us also notice that injecting [\[prf_lem_tight_3\]](#prf_lem_tight_3){reference-type="eqref" reference="prf_lem_tight_3"} into [\[Upsf\<UpsfOmega+\]](#Upsf<UpsfOmega+){reference-type="eqref" reference="Upsf<UpsfOmega+"} we obtain $$\label{energyf} \mathcal{E}(m) - \varepsilon\leq \Upsilon(f) \leq \Upsilon(f_\Omega) + \varepsilon.$$ *Step 4 : Bounding the diameter of $\Omega$.* We finally prove that $\Omega$ is uniformly bounded which would conclude the proof. For $Q_-$, $Q_+\in\mathcal{Q}_0$, we write $Q_-\sim Q_+$ if there exists a finite chain $$\label{prf_lem_tight_4} Q_-=Q_0,Q_1,\dots,Q_n=Q_+$$ such that $Q_i\in\mathcal{Q}_0$ and $d(Q_{i-1},Q_i)\le 4R +\sqrt{d}r_j$ for $1\le i\le n$. This defines an equivalence relation. Let us show that there exists only one equivalence class. We assume by contradiction that there exist at least two equivalence classes, and we let $\mathcal{C}^1$ be one of these classes and $\mathcal{C}^2$ be the union of the remaining classes. For $i \in \{1, 2\}$, we then define $\Omega^i$ to be the reunion of the cubes $Q$ such that $d(Q, \mathcal{C}^i)\le R$. By construction, $d(\Omega_1,\Omega_2)>2R$. Recalling that $\Omega$ is the reunion of the cubes $Q$ such that $d(Q, \Omega_0) \leq R$, we have $\Omega^1\cup\Omega^2=\Omega$. For $i \in \{1, 2\}$, we set $f_\Omega^i:= f_\Omega\chi_{\Omega^i}$ and $m_\Omega^i=\int f_\Omega^i\,$. We have $m_\Omega^1+m_\Omega^2=m_\Omega\le m$ and $m_\Omega^1,m_\Omega^2\ge 2^{-jd}|Q_0|/4=2^{-jd-2}|Q_0|$. Additionally, by Proposition [Proposition 18](#prop_basic_ups){reference-type="ref" reference="prop_basic_ups"} $(ii)$, $$\Upsilon(f_\Omega)=\Upsilon(f^1_\Omega)+\Upsilon(f^2_\Omega)\le \mathcal{E}(m^1_\Omega)+\mathcal{E}(m^2_\Omega).$$ Injecting this inequality into [\[energyf\]](#energyf){reference-type="eqref" reference="energyf"} yields $$\mathcal{E}(m) - \varepsilon\leq \Upsilon(f) \leq \Upsilon(f_\Omega) +\varepsilon\leq \mathcal{E}(m^1_\Omega)+\mathcal{E}(m^2_\Omega)+\varepsilon.$$ By definition of $e$ this rewrites as $$\label{ineqmu} m e(m) \leq m^1_\Omega e(m^1_\Omega)+m^2_\Omega e(m^2_\Omega)+2\varepsilon.$$ As $m^1_\Omega+m^2_\Omega\le m$ and for $i \in \{1, 2\}$, $m^i_\Omega\ge2^{-jd-2}|Q_0|$, we have $m^i_\Omega\leq m - 2^{-jd-2}|Q_0|$. Recall that by Proposition [Proposition 23](#prop_mu_inc){reference-type="ref" reference="prop_mu_inc"}, $e$ is increasing, so that $e(m^i_\Omega) \leq e(m-2^{-jd}m_0)$. Hence $$m^1_\Omega e(m^1_\Omega)+m^2_\Omega e(m^2_\Omega) \leq m e\left(m-2^{-jd}m_0\right).$$ With [\[ineqmu\]](#ineqmu){reference-type="eqref" reference="ineqmu"}, we obtain $$m e (m) \leq me\left(m-2^{-jd}m_0\right) + 2\varepsilon,$$ which is absurd for $\varepsilon$ small enough because $e$ is increasing. It follows that for $\varepsilon>0$ small enough the relation $\sim$ has a single class. Recall that for all $Q \in \mathcal{Q}_0$, $\smallint_Q (f+g)\, \,\geq 2^{-jd-2}|Q_0|$. Thus the maximal length of a chain in [\[prf_lem_tight_4\]](#prf_lem_tight_4){reference-type="eqref" reference="prf_lem_tight_4"} without any repetition is bounded by $N := \lfloor 2^{jd+3}m/|Q_0|\rfloor$. Therefore, the diameter of $\Omega$ is bounded by $(4R +2\sqrt{d}r_j)(N+1)$ with $r_j$ and $N$ only depending on $m$, the dimension $d$ and the cost $c$. Together with [\[prf_lem_tight_35\]](#prf_lem_tight_35){reference-type="eqref" reference="prf_lem_tight_35"} this proves the proposition. ◻ We can now apply the direct method of Calculus of Variations to establish the existence of a maximiser for [\[MP\]](#MP){reference-type="eqref" reference="MP"}. *Proof of Theorem [Theorem 1](#thm_MP_max){reference-type="ref" reference="thm_MP_max"}.*    Let $f_n$ be a maximising sequence for [\[MP\]](#MP){reference-type="eqref" reference="MP"} and let $R_* = R_*(m)$ be given by Proposition [Proposition 25](#prop_strong_tightness){reference-type="ref" reference="prop_strong_tightness"} so that up to translation, $$\int_{\mathbb{R}^d \setminus B_{R_*}} f_n\, \to 0 \quad \text{as} \quad n \to \infty.$$ Therefore, $f_n\lambda$ is a tight sequence of $\mathcal{M}_+(\mathbb{R}^d)$ and up to extraction of a subsequence it converges weakly-$*$ to $f\lambda$ where $f$ is admissible for [\[MP\]](#MP){reference-type="eqref" reference="MP"}. By Proposition [Proposition 18](#prop_basic_ups){reference-type="ref" reference="prop_basic_ups"} $(iv)$, $$\Upsilon(f) = \lim \Upsilon(f_n) = \mathcal{E}(m),$$ so that $f$ is a maximiser for $\mathcal{E}(m)$. Let now $f$ be any maximiser of $\mathcal{E}(m)$. Applying Proposition [Proposition 25](#prop_strong_tightness){reference-type="ref" reference="prop_strong_tightness"} to $f$ we have that up to a translation $\mathop{\mathrm{supp}}f \subset \overline B_{R_*}$. This concludes the proof. ◻ Let us show that when $f$ is compactly supported there exist Kantorovitch potentials for the problem [\[PPRd\]](#PPRd){reference-type="eqref" reference="PPRd"} (this is the situation of interest as we have just established that the maximisers of $\mathcal{E}(m)$ are compactly supported in $\mathbb{R}^d$). **Lemma 26**. *Let $m >0$ and assume that $f \in L^1_m$ is compactly supported. Let $R = R(m)$ be given by Proposition [Proposition 17](#prop_Ups_max){reference-type="ref" reference="prop_Ups_max"} such that all maximisers $\gamma$ of $\Upsilon(f)$ are supported in $X:= \mathop{\mathrm{supp}}f + \overline{B}_R$. Then, there exists a pair $(\varphi, \psi) \in C_c(\mathbb{R}^d) \times C_c(\mathbb{R}^d)$ optimal for $\Upsilon^*(f)$. Additionally, $\varphi= \psi^c$, $\psi = \varphi^c_{\,-}$ and both $\varphi$ and $\psi$ are compactly supported in $X$.* *Proof.* Let us introduce $\tilde c := c_{|X \times X}$ which is a continuous cost function on the compact set $X$. By Proposition [Proposition 12](#prop_ex_DPp){reference-type="ref" reference="prop_ex_DPp"}, there exists $\widetilde \psi \in C(X)$ with $\widetilde \psi = ({\widetilde \psi}^{\tilde c \tilde c})_-$ such that $\Upsilon^*(f) = K_f({\widetilde \psi}^{\tilde c}, \widetilde \psi)$. By Proposition [Proposition 15](#prop_Ups_OT){reference-type="ref" reference="prop_Ups_OT"}, $$\{{\widetilde \psi}^{\tilde c \tilde c} < 0 \} \subset \{f+g=1\}\subset X.$$ Combining this with $\widetilde \psi = ({\widetilde \psi}^{\tilde c \tilde c})_-$ and $f+g=0$ on $\partial X$ , we get $\widetilde \psi=0$ on $\partial X$. We extend the potentials on $\mathbb{R}^d$ by setting $$\psi := \begin{cases} \widetilde \psi &\text{in} \quad X,\\ 0 &\text{in} \quad X^c, \end{cases} \qquad \text{and for }x\in\mathbb{R}^d,\quad \varphi(x) := \psi^c(x) = \inf\{c(x,y) - \psi(y) : \, y \in \mathbb{R}^d \}.$$ We now show that the pair $(\varphi, \psi)$ satisfies the conclusion of the lemma. Observe that $\psi$ is continuous and supported in $X$ and that $\psi \leq 0$. Hence $\varphi\geq 0$. Moreover, for $x \in\mathbb{R}^d\setminus\mathop{\mathrm{supp}}\psi$, $$\varphi(x) \leq c(x,x) - \psi(x) = 0,$$ so that $\varphi$ is also supported in $X$.\ Next, for $x \in X$, $$\varphi(x) = \min \left( \min_{y \in \mathbb{R}^d\setminus X} c(x,y), \ \min_{y \in X} \{c(x,y) - \widetilde \psi(y) \} \right).$$ Let $x \in X$. For $y \in \mathbb{R}^d\setminus X$, there exists $\widetilde y$ in the intersection of the segment $[x,y]$ with $\partial X$. By continuity, $\widetilde \psi(\widetilde y) = 0= \widetilde\psi(y)$ and moreover by (H[\[monot\]](#monot){reference-type="ref" reference="monot"}), $c(x, \widetilde y) \leq c(x, y)$ so that $c(x, \widetilde y) \le c(x,y)- \widetilde \psi(y)$. We deduce that for $x \in X$ the above formula simplifies as $$\varphi(x) = \min_{y \in X} \{c(x,y) - \widetilde \psi(y) \} = {\widetilde \psi}^{\tilde c}(x).$$ This proves that $\varphi$ is continuous and that $K_f(\varphi, \psi) = K_f({\widetilde \psi}^{\tilde c}, \widetilde \psi) = \Upsilon^*(f)$. Moreover, using the same argument as above, we have $\varphi^c(y) = 0$ for $y \not \in X$. For $y \in X$, $$\varphi^c(y) = \min \left( \min_{x \in \mathbb{R}^d\setminus X} c(x,y), \ \min_{x \in X} \{c(x,y) - {\widetilde \psi}^{\tilde c}(y) \} \right) = \min_{x \in X} \{c(x,y) - {\widetilde \psi}^{\tilde c}(y) \} = {\widetilde \psi}^{\tilde c \tilde c}(y).$$ We deduce $\varphi^c_{\,-} = 0 = \psi$ in $X^c$, and $\varphi^c_{\,-} = (\psi^{\tilde c \tilde c})_- = \widetilde \psi$ in $X$. Thus $\varphi^c_{\,-} = \psi$ everywhere. This ends the proof of the lemma. ◻ Let us now recall a variant of the bathtub principle, see [@LiebLoss Theorem 1.14]. **Proposition 27**. *Let $\xi : \mathbb{R}^d \to \mathbb{R}_+$ be measurable and such that for all $t \ge 0$, $|\{ \xi > t\}| < \infty$. Given $m >0$, let $$t := \inf \{ s \ge 0, \, |\{ \xi > s \}| \leq m\}.$$ Then, the maximisers of $$\sup_{\widetilde{f}} \left\{\int \widetilde{f} \xi \, : \widetilde{f} \in L^1(\mathbb{R}^d), \, 0 \leq \widetilde{f} \leq 1, \, \int \widetilde{f}\, =m\right\}$$ are the functions $f := \chi_{\{\xi > t \}} + \theta$, where $\theta \in L^1(\mathbb{R}^d, [0,1])$ is supported in $\{\xi = t\}$ and satisfies $$\int \theta\,= m-|\{\xi>t\}|.$$* We are now ready to establish Corollary [Corollary 2](#coro_maxE){reference-type="ref" reference="coro_maxE"}. *Proof of Corollary [Corollary 2](#coro_maxE){reference-type="ref" reference="coro_maxE"}.*    By Theorem [Theorem 1](#thm_MP_max){reference-type="ref" reference="thm_MP_max"}, the optimisation problem [\[MP\]](#MP){reference-type="eqref" reference="MP"} admits a compactly supported solution $f$. Let $(\varphi, \psi) \in C_c(\mathbb{R}^d)\times C_c(\mathbb{R}^d)$ be an optimal pair for $\Upsilon^*(f)$ provided by Lemma [Lemma 26](#lem_ctransfglobal){reference-type="ref" reference="lem_ctransfglobal"}, so that $$\Upsilon^*(f)= \int f(\varphi- \psi)\, + \int \psi\,.$$ We see that $f$ is a maximiser of: $$\sup\left\{ \int \widetilde{f}(\varphi- \psi)\,: \widetilde{f} \in L^1(\mathbb{R}^d), \, 0 \leq \widetilde{f} \leq 1, \, \int \widetilde{f}\, = m\right\}.$$ Let us set $\xi := \varphi- \psi\ge 0$. By Proposition [Proposition 27](#prop_bathtub){reference-type="ref" reference="prop_bathtub"} there exists $t \ge 0$ and $\theta \in L^1(\mathbb{R}^d, [0,1])$ supported in $\{\xi =t \}$ such that $f = \chi_{\{\xi> t \}} + \theta$. Notice in particular that since $\theta\in [0,1]$, we have $$|\{\xi=t\}|\ge \int \theta \, = m - |\{\xi> t \}|.$$ and there exist measurable subsets $G\subset \{\xi = t\}$ with $|G|=m- |\{\xi> t \}|$. For any such set, setting $$\bar f := \chi_{\{\xi> t \}} + \chi_G,$$ we have $\Upsilon^*(\bar f) = \Upsilon^*(f)$ and $\bar f$ is also a maximiser of [\[MP\]](#MP){reference-type="eqref" reference="MP"}. Since $\bar f$ is a characteristic function, by Theorem [Theorem 20](#thm_saturation){reference-type="ref" reference="thm_saturation"} and Corollary [Corollary 21](#coro_unique_g){reference-type="ref" reference="coro_unique_g"}, there exists $F \subset \mathbb{R}^d$ such that any minimiser $\gamma$ of $\Upsilon({\bar f})$ satisfies $\gamma_y = \chi_F$. Setting $E := \{\xi> t \} \cup G$, we deduce that $$\Upsilon_{\mathrm{set}}(E) =\Upsilon^*(\bar f) =\Upsilon(\bar f) = \Upsilon(f)$$ so that $\mathcal{E}(m) = \mathcal{E}_{\mathrm{set}}(m)$, which concludes the proof. ◻ # Maximisers of [\[MP\]](#MP){reference-type="eqref" reference="MP"} are characteristic functions of balls {#ballisunique} In this section we prove Theorem [Theorem 3](#thm_ball_unique_max){reference-type="ref" reference="thm_ball_unique_max"}. We assume that $c(x,y)=k(|y-x|)$ with $k\in C(\mathbb{R}_+,\mathbb{R}_+)$ increasing and coercive and with $k(0)=0$. In particular, we have now $c = \bar c$, so that the operations of $c$-transform and $\bar c$-transform coincide. Also notice that by Theorem [Theorem 20](#thm_saturation){reference-type="ref" reference="thm_saturation"}, if $f = \chi_E$ for some Lebesgue measurable set $E$ then $\Upsilon_{\mathrm{set}}(E) = \Upsilon(\chi_E)$. By abuse of notation, we write $\Upsilon(E)$ for $\Upsilon(\chi_E)$. Since the class of costs that we consider is invariant by scaling we assume without loss of generality that $m = \omega_d$. We now recall the definition of symmetric rearrangement of functions with constant sign (see [@LiebLoss Chapter 3] for more details on symmetric rearrangements). **Definition 28**.  [\[def_sym_rear\]]{#def_sym_rear label="def_sym_rear"} (i) Given a measurable set $A \subset \mathbb{R}^d$, we define the symmetric rearrangement of $A$ as the open ball $A^*$ centred at the origin and of volume $|A|$. (ii) Let $\varphi: \mathbb{R}^d \to \mathbb{R}_+$ be measurable and such that for every $t\geq 0$, $|\{\varphi> t\}| <\infty$. Its symmetric decreasing rearrangement is defined by $$\varphi^*(x) := \int_{\mathbb{R}_+} \chi_{\{ \varphi> t \}^*} (x) \, dt.$$ (iii) Let $\psi : \mathbb{R}^d \to \mathbb{R}_-$ be measurable and such that for every $t\leq 0$, $|\{\psi < t\}| < \infty$. Its symmetric increasing rearrangement is defined by $$\psi_*(x) :=-(-\psi)^*(x)= - \int_{\mathbb{R}_-} \chi_{\{ \psi < t \}^*} (x) \, dt.$$ The following lemma recalls some basic properties of the symmetric increasing rearrangement $\psi_*$ of a non-positive function $\psi$. All these properties but the continuity of $\psi_*$ follow immediately from the definition. The fact that continuity is preserved by symmetric rearrangement is well-known but we have no reference for this at hand. We provide a short proof for the reader's convenience. **Lemma 29**. *Let $\psi : \mathbb{R}^d \to \mathbb{R}_-$ be as in Definition [\[def_sym_rear\]](#def_sym_rear){reference-type="ref" reference="def_sym_rear"}. Then, $\psi_*$ is non-positive, radial, non-decreasing, and for any $t \leq 0$, $\{\psi_*<t\}=\{\psi<t\}^*$. Besides, if $\psi$ is supported in a compact set of diameter bounded by $2R >0$ then $\psi_*$ is supported in $\overline B_R$. If moreover $\psi$ is continuous then $\psi_*$ is also continuous.* *Proof of the last point.* Let $\psi\in C_c(\mathbb{R}^d,\mathbb{R}_-)$. First, as the strict sublevels sets $\{\psi_* < t\}$ are the open balls $\{\psi < t\}^*$, $\psi_*$ is upper semi-continuous (note that this is true even when $\psi$ is not continuous). Let us now establish that $\psi_*$ is lower semi-continuous, *i.e. *that for any $t \leq 0$, $\{ \psi_* \leq t \}$ is closed. We first notice that $\{\psi_* \leq 0 \} = \mathbb{R}^d$ is closed. Given $t < 0$, let $t_n<0$ be a decreasing sequence converging to $t$. Observe that if for some $n \geq 0$, $\{\psi < t_n\} = \emptyset$, then $\{\psi_* \le t\} = \emptyset$ is closed. Next, we assume that for every $n \geq 0$, $$\label{sublvlpsi} \{\psi < t_n \} \neq \emptyset.$$ We denote by $R_n$ the radius of the ball $\{\psi_* < t_n\}$. Notice that the sequence $R_n$ is non-increasing and bounded by $0$, so that $R_n$ converges to some $R \geq 0$. Let us show that the sequence $R_n$ is decreasing. By contradiction, we assume that $R_n = R_{n+1}$ for some $n \geq 0$. Then $\{\psi < t_n\}^* = \{\psi < t_{n+1}\}^*$ and $|\{t_{n+1} \leq \psi < t_n\}| = 0$. Using [\[sublvlpsi\]](#sublvlpsi){reference-type="eqref" reference="sublvlpsi"} and the fact that $\psi$ is compactly supported, there exists $x$ such that $\psi(x)<t_{n+1}$ and $y$ such that $\psi(y)>t_n$. Thus by continuity of $\psi$ there exists $z$ such that $\psi(z) = (t_{n+1} + t_n)/2$. By continuity of $\psi$ again, there exists $\eta >0$ such that $B_\eta(z) \subset \{t_{n+1} < \psi < t_n\}$, contradicting the fact that $|\{t_{n+1} \leq \psi < t_n\}| = 0$. As a conclusion, the sequence $R_n$ is decreasing and $$\{\psi_* \leq t \} = \bigcap_{n \geq 0} \{\psi_* < t_n\} = \bigcap_{n \geq 0} B_{R_n} = \overline{B}_{R}.$$ Hence $\psi_*$ is lower semi-continuous and therefore continuous. ◻ To prove Theorem [Theorem 3](#thm_ball_unique_max){reference-type="ref" reference="thm_ball_unique_max"}, we need a last lemma about final lemma characterising optimal potentials of $\Upsilon^*(\chi_{B_1})$. **Lemma 30**. *Let $(\psi^c, \psi)$ be a pair of optimal potentials for $\Upsilon^*(\chi_{B_1})$ such that $\psi$ is radially symmetric and non-decreasing. Then $\psi^c$ is radially symmetric and non-increasing. Besides, $\psi^c$ is radially decreasing on $B_{1}$.* *Proof.* Combining the facts that $k$ is continuous, that $k(r) \to \infty$ as $r \to \infty$ and that $\psi$ is bounded, we see that for any $x \in \mathbb{R}^d$, $$\psi^c(x) = \min \{ k(|y-x|) - \psi(y) : y \in \mathbb{R}^d \}.$$ As $\psi$ is radially symmetric non-decreasing and $k$ is increasing, we easily see that $$\label{altinfdef} \psi^c(x) = \min \{k(|y-x|) - \psi(y) : y, \, \exists \lambda \geq 1, \, y = \lambda x\},$$ which in turn implies that $\psi^c$ is radially symmetric. From now on, for radial functions $\zeta:\mathbb{R}^d\to\mathbb{R}$, we make the abuse of notation $\zeta(r)=\zeta(r\sigma)$ for $r\ge0$ where $\sigma$ is some fixed element of $\mathbb{S}^{d-1}$. With this convention [\[altinfdef\]](#altinfdef){reference-type="eqref" reference="altinfdef"} reads $$\label{altinfdef_bis} \psi^c(r) = \min_{s\ge r} k(s-r) - \psi(s).$$ Let us prove that $\psi^c$ is non-decreasing. Let $0 \le r_1 \le r_2$. By [\[altinfdef_bis\]](#altinfdef_bis){reference-type="eqref" reference="altinfdef_bis"}, there exists $r\ge r_1$ such that $$\label{psicx1} \psi^c(r_1) = k(r-r_1) - \psi(r).$$ If $r \leq r_2$, we use $\psi^c(r_2)\le k(0) -\psi(r_2)=-\psi(r_2)$ and deduce from [\[psicx1\]](#psicx1){reference-type="eqref" reference="psicx1"} and the fact that $\psi$ is non-decreasing that $$\psi^c(r_2) - \psi^c(r_1) \leq \psi(r) - \psi(r_2) -k(r-r_1) \leq 0.$$ If $r > r_2$, we use $\psi^c(r_2)\le k(r-r_2)-\psi(r)$ to get $$\psi^c(r_2) - \psi^c(r_1) \leq k(r-r_2) - k(r-r_1) \leq 0,$$ because $r_1\le r_2<r$ and $k$ is increasing. In both cases $\psi^c(r_2) - \psi^c(r_1) \leq 0$. Hence $\psi^c$ is non-increasing on $\mathbb{R}^d$. We finally prove that $\psi^c$ is decreasing on $B_{1}$. Let $0 < r_1 < r_2 < 1$. Given $\gamma$ a minimiser for $\Upsilon(B_{1})$, there exists $y\in \mathbb{R}^d\setminus B_1$ such that $(r_1 \sigma, y) \in \mathop{\mathrm{supp}}\gamma$. By Proposition [Proposition 15](#prop_Ups_OT){reference-type="ref" reference="prop_Ups_OT"}, $\gamma$ is an optimal transport plan between $f$ and $g := \gamma_y$ and $(\psi^c, \psi^{cc})$ is a pair of Kantorovitch potentials for the transport between $f$ and $g$. Therefore, $$\label{y1min} \psi^c(r_1) + \psi^{cc}(|y|) = k(|y-r_1 \sigma|).$$ Let us prove by contradiction that $y\in [1,+\infty)\sigma$. Assume it is not and let $y' := |y|\sigma$. Recalling that $|y|\ge1>r_1$, we have $|y'-r_1 \sigma| = |y|-r_1 < |y - r_1 \sigma|$ and since $k$ is increasing we deduce $$k(|y'-r_1 \sigma|) < k(|y - r_1 \sigma|).$$ Then, by definition of $\psi^{cc}$ and taking into account that it is radially symmetric we get $$\psi^c(r_1) + \psi^{cc}(|y|) \leq k(|y'-r_1 \sigma|)< k(|y - r_1 \sigma|)$$ which contradicts [\[y1min\]](#y1min){reference-type="eqref" reference="y1min"}. Therefore, $y = r \sigma$ for some $r \geq 1$. By definition of the $c$-transform, $$\label{defKP} \psi^c(r_2) + \psi^{cc}(r) \leq k(r-r_2).$$ Subtracting [\[y1min\]](#y1min){reference-type="eqref" reference="y1min"} to [\[defKP\]](#defKP){reference-type="eqref" reference="defKP"}, we obtain $$\psi^c(r_2) - \psi^c(r_1) \leq k(r-r_2) - k(r-r_1) < 0,$$ where we used $r_1 < r_2 <1\le r$. This shows that $\psi^c$ is decreasing on $B_{1}$. ◻ We are now ready to prove Theorem [Theorem 3](#thm_ball_unique_max){reference-type="ref" reference="thm_ball_unique_max"}. *Proof of Theorem [Theorem 3](#thm_ball_unique_max){reference-type="ref" reference="thm_ball_unique_max"}.*    *Part I : Unit balls are maximisers of $\mathcal{E}(\omega_d)$.* By Theorem [Theorem 1](#thm_MP_max){reference-type="ref" reference="thm_MP_max"}, there exists a compactly supported maximiser $f$ for [\[MP\]](#MP){reference-type="eqref" reference="MP"} with $m = \omega_d$. By Lemma [Lemma 26](#lem_ctransfglobal){reference-type="ref" reference="lem_ctransfglobal"}, there exists an optimal pair $(\psi^c, \psi) \in C_c(\mathbb{R}^d) \times C_c(\mathbb{R}^d)$ for problem $\Upsilon^*(f)$ such that $\psi = (\psi^{cc})_-$. *Step 1. We build a radially symmetric maximiser for [\[MP\]](#MP){reference-type="eqref" reference="MP"}.* Let $\psi_*$ be the symmetric increasing rearrangement of $\psi$. By Lemma [Lemma 29](#lem_psi_star){reference-type="ref" reference="lem_psi_star"}, as $\psi \in C_c(\mathbb{R}^d)$, we also have $\psi_* \in C_c(\mathbb{R}^d)$. We denote by $\psi_*^{\,c}$ the function $(\psi_*)^c$. By definition, $\psi_*^{\,c} \oplus \psi_* \leq c$. Proceeding as in the proof of Lemma [Lemma 26](#lem_ctransfglobal){reference-type="ref" reference="lem_ctransfglobal"}, we obtain $\psi_*^{\,c} \in C_c(\mathbb{R}^d)$. Thus $(\psi_*^{\,c}, \psi_*)$ is admissible for $\Upsilon^*(B_{1})$. Notice that $(f, \psi)$ solves the double supremum problem (recall the definition [\[Kf\]](#Kf){reference-type="eqref" reference="Kf"} of $K_f$) $$\sup_f \sup_{\psi \in C_c(\mathbb{R}^d)} \left\{K_f(\psi^c, \psi) : 0\leq f \leq 1, \, \int f \,= \omega_d, \, \psi \leq 0 \right\}.$$ Hence $$\mathcal{E}(\omega_d) = K_f(\psi^c, \psi) = \int f (\psi^c - \psi) \,+\int \psi\, \geq \Upsilon^*(B_{1}) \geq K_{\chi_{B_{1}}}(\psi_*^{\,c}, \psi_*) = \int_{B_{1}} (\psi_*^{\,c}-\psi_*)\, + \int \psi_*\,.$$ In the remainder of this step, we establish the converse inequality $$\label{converseineq} K_f(\psi^c, \psi) \leq K_{\chi_{B_{1}}}(\psi_*^{\,c}, \psi_*),$$ so that $B_1$ is a maximiser of $\mathcal{E}(\omega_d)$ and the first part of Theorem [Theorem 3](#thm_ball_unique_max){reference-type="ref" reference="thm_ball_unique_max"} is proved. Notice that [\[converseineq\]](#converseineq){reference-type="eqref" reference="converseineq"} also implies that $(\psi_*^{\,c}, \psi_*)$ is a pair of optimal potentials for $\Upsilon^*(B_{1})$. To establish [\[converseineq\]](#converseineq){reference-type="eqref" reference="converseineq"}, we first notice that by construction $\smallint \psi = \smallint \psi^*$ so that we only need to prove $$\label{toprovereduced} \int f (\psi^c - \psi)\, \leq \int_{B_{1}} (\psi_*^{\,c}-\psi_*)\,.$$ In Step 2 below we establish the inequality $$\label{psistarccstar} (\psi^c)^* \leq (\psi_*)^c = \psi_*^{\,c},$$ where $(\psi^c)^*$ denotes the symmetric decreasing rearrangement of $\psi^c$. Admitting that [\[psistarccstar\]](#psistarccstar){reference-type="eqref" reference="psistarccstar"} holds we deduce [\[toprovereduced\]](#toprovereduced){reference-type="eqref" reference="toprovereduced"} as follows. Since $f$ is non-negative and compactly supported we have by the Hardy-Littlewood inequality (see [@LiebLoss Theorem 3.4]) $$\label{ineqHL} -\int f \psi \,\leq - \int f^* \psi_* \,\qquad \text{and} \qquad \int f \psi^c\, \leq \int f^* (\psi^c)^*\, \stackrel{\eqref{psistarccstar}}{\leq} \int f^* \psi_*^{\,c}\,.$$ Using that $-\psi_*$ and $\psi_*^{\,c}$ are radially symmetric and non-increasing, we may appeal to Proposition [Proposition 27](#prop_bathtub){reference-type="ref" reference="prop_bathtub"} and conclude that separately, $$\label{lefttoprove} - \int f \psi \,\leq - \int \chi_{B_1} \psi_*\, \qquad \textrm{and } \qquad \int f \psi^c\, \leq \int \chi_{B_1}\psi_*^{\,c}\,.$$ Summing these inequalities gives [\[toprovereduced\]](#toprovereduced){reference-type="eqref" reference="toprovereduced"} and thus [\[converseineq\]](#converseineq){reference-type="eqref" reference="converseineq"}. This proves that $\chi_{B_1}$ is a maximiser for $\mathcal{E}(\omega_d)$ and then that $B_1$ is a maximiser for $\mathcal{E}_{\textrm{set}}(\omega_d)$. *Step 2. Proof of [\[psistarccstar\]](#psistarccstar){reference-type="eqref" reference="psistarccstar"}.* As $\psi_*^{\,c}$ and $(\psi^c)^*$ are both continuous radially symmetric functions, to prove [\[psistarccstar\]](#psistarccstar){reference-type="eqref" reference="psistarccstar"} it is sufficient to establish that for any $t> 0$, $\{(\psi^c)^* >t \} \subset \{\psi_*^{\,c} >t \}$, *i.e. *that $$\label{rearr} |\{(\psi^c)^* > t\}| = |\{\psi^c > t\}| \leq |\{\psi_*^{\,c} > t\}| .$$ Recall that as $\psi \in C_c(\mathbb{R}^d)$ and $k \in C(\mathbb{R}_+, \mathbb{R}_+)$ with $k(x) \to \infty$ as $x \to \infty$, for any $x \in \mathbb{R}^d$ the function $k(|y-x|) - \psi(y)$ admits a minimum on $\mathbb{R}^d$. Thus for any $x \in \mathbb{R}^d$ the infimum defining $\psi^c(x)$ (see Definition [Definition 5](#def_ctransf){reference-type="ref" reference="def_ctransf"}) is reached. Recalling that $k$ is also radially symmetric and increasing, we obtain $$\begin{aligned} \{\psi^c > t\}&=\{x \in \mathbb{R}^d : \min \{k(|y-x|)-\psi(y) : y \in \mathbb{R}^d\} > t \}\\ &=\{x\in \mathbb{R}^d :-\psi > t-k(r) \text{ on } \overline{B}_r(x) \ \forall r\geq 0\}\\ &=\bigcap_{r\geq 0} \{x\in \mathbb{R}^d :-\psi > t-k(r) \text{ on } \overline{B}_r(x)\}\\ &=\bigcap_{r\geq 0} \{-\psi > t-k(r)\}_r, \end{aligned}$$ where for $\Omega\subset \mathbb{R}^d$ and $r\geq 0$, $\Omega_r$ is defined as $\Omega_r:=\{x\in \Omega:d(x, \mathbb{R}^d\setminus\Omega) > r\}$. In particular, $$\label{infboundpsic} |\{\psi^c> t\} |\le \inf_{r\geq 0} |\{-\psi > t-k(r)\}_r|.$$ We observe that $\{- \psi > t - k(r)\}$ is an open set for any $t>0$ and $r\geq 0$. We also notice that [\[infboundpsic\]](#infboundpsic){reference-type="eqref" reference="infboundpsic"} holds for all $\psi \in C_c(\mathbb{R}^d)$. In particular, it holds for $\psi_*$. Moreover, as $\psi_*$ is radially non-decreasing by construction, the sets $\{-\psi_*> t-k(r)\}_r$ are open balls centred at the origin and we have in fact $$|\{\psi_*^{\,c} > t\} |= \inf_{r\geq 0} |\{-\psi_* > t-k(r)\}_r|.$$ Let us now prove the following claim. **Claim 2**. *Let $s>0$ and $V> 0$.* (i) *If $V>\omega_d s^d$ then, among open sets $\Omega\subset \mathbb{R}^d$ of volume $V$, $|\Omega_s|$ is maximal if and only if $\Omega$ is a ball.* (ii) *If $V\le\omega_d s^d$ then $|\Omega_s|=0$ for any set of volume $V$.* Let $V>0$ and $s>0$ and let $\Omega\subset \mathbb{R}^d$ be an open set. We assume without loss of generality that $|\Omega| = V$ and $|\Omega_s| > 0$. Notice that we always have $\Omega_s + B_s \subset \Omega$ (but the converse inclusion may fail). By the Brunn-Minkowski inequality (see for instance [@gardnerbrunnmink]) applied to $\Omega_s$ and $B_s$, there holds $$\label{brunnmink} V^{1/d}=|\Omega|^{1/d} \geq |\Omega_s + B_s|^{1/d} \geq |\Omega_s|^{1/d} + |B_s|^{1/d}.$$ If $\Omega_s$ is a ball, then $\Omega$ is a ball of volume $V$, $\Omega= \Omega_s + B_s$ and we have equality in [\[brunnmink\]](#brunnmink){reference-type="eqref" reference="brunnmink"}. Conversely if we have equality in [\[brunnmink\]](#brunnmink){reference-type="eqref" reference="brunnmink"}, by the equality case of the Brunn-Minkowski inequality and the fact that $s>0$, $\Omega_s$ is a ball and $|\Omega| = |\Omega_s +B_s|$, so that $\Omega$ is a ball. This proves the first part of the claim. Regarding the second part, we assume that $|\Omega|\le\omega_d s^d$ and (by contradiction) that $|\Omega_s|>0$. The above reasoning applies and we have $\Omega=\Omega_s+B_s$ so that $|\Omega_s|>0$ implies $|\Omega|>|B_s|=\omega_d s^d$ and we get a contradiction. This proves the claim. By definition, $\{-\psi > t-k(r)\}$ and $\{-\psi_* > t-k(r)\}$ have the same volume. As a consequence of the claim, for any $t > 0$ and $r > 0$, $$\label{inegrearr} |\{- \psi > t - k(r)\}_r | \leq | \{- \psi_* > t - k(r) \}_r|.$$ Notice that the previous inequality is an equality if $r=0$, as $\Omega_0 = \Omega$ for any open set $\Omega$. Taking the infimum on $r \geq 0$ yields $$\label{ineqvol} |\{\psi^c> t\} |\le \inf_{r \geq 0} |\{-\psi > t-k(r)\}_r| \leq \inf_{r\geq 0} |\{-\psi_* > t-k(r)\}_r| = |\{\psi_*^{\,c} > t\} |.$$ This proves [\[rearr\]](#rearr){reference-type="eqref" reference="rearr"} which in turn implies [\[psistarccstar\]](#psistarccstar){reference-type="eqref" reference="psistarccstar"}. *Part II : Unit balls are the unique maximisers of $\mathcal{E}(\omega_d)$.* *Step 1. Proof of $f=\chi_{\{\psi^c > \psi_*^{\,c}(1)\}}$ (exploiting the equality case in the bathtub principle).* We now show that any maximiser $f$ is of the form $\chi_{\{\psi^c > \psi_*^{\,c}(1)\}}$. By Lemma [Lemma 30](#lem_monotpotball){reference-type="ref" reference="lem_monotpotball"}, $\psi_*^{\,c}$ is radially decreasing on $B_{1}$ and non-increasing on $\mathbb{R}^d$. Thus $\chi_{B_{1}}$ is the only function maximising $$\sup_{\widetilde{f}} \left\{\int \widetilde{f} \psi_*^{\,c} \,: 0 \leq \widetilde{f} \leq 1, \, \int \widetilde{f}\,= \omega_d \right\}.$$ As $\Upsilon^*(\chi_{B_{1}}) = \mathcal{E}(\omega_d)$, the inequalities in [\[ineqHL\]](#ineqHL){reference-type="eqref" reference="ineqHL"} and [\[lefttoprove\]](#lefttoprove){reference-type="eqref" reference="lefttoprove"} are in fact equalities (and [\[psistarccstar\]](#psistarccstar){reference-type="eqref" reference="psistarccstar"} is also an equality in $B_1$). Namely, there hold $$(\psi^c)^* = \psi_*^{\,c},\ \qquad -\int f \psi \,= - \int f^* \psi_*\qquad\text{ and }\qquad \int f \psi^c\, =\int f^* \psi_*^{\,c}\,.$$ This leads to $$\int f \psi^c \,= \int f^* (\psi^c)^* \,= \int f^* \psi_*^{\,c} \,= \int_{B_{1}}\psi_*^{\,c}\,,$$ and $f$ is a maximiser of $$\sup_{\widetilde{f}}\left\{ \int \widetilde{f} \psi^c\, : 0 \leq \widetilde{f} \leq 1, \, \int \widetilde{f}\,= \omega_d\right\}.$$ Let us now prove that $|\{\psi^c>\psi_*^{\,c}(1)\}|=\omega_d$ (which with Proposition [Proposition 27](#prop_bathtub){reference-type="ref" reference="prop_bathtub"} yields $f=\chi_{\{\psi^c > \psi_*^{\,c}(1)\}}$). Since $(\psi^c)^*=\psi_*^{\,c}$ in $B_1$, and $\psi_*^{\,c}$ is decreasing in $B_1$ by Lemma [Lemma 30](#lem_monotpotball){reference-type="ref" reference="lem_monotpotball"}, there holds for $t\ge \psi^c_*(1)$, $$\label{equalvol} |\{\psi^c >t \}| =|\{(\psi^c)^* >t \}|= |\{\psi_*^{\,c} >t \}|.$$ Using this for $t=\psi_*^{\,c}(1)$ we get $|\{\psi^c>\psi_*^{\,c}(1)\}|=\omega_d$ and we conclude with Proposition [Proposition 27](#prop_bathtub){reference-type="ref" reference="prop_bathtub"} that $f=\chi_{\{\psi^c > \psi_*^{\,c}(1)\}}$. *Step 2. We prove that $\{\psi^c >t \}$ is a ball for $t>\psi^c_*(1)$ (exploiting the equality case in the Brunn-Minkowski inequality).*\ *Step 2.a.* We fix $t> \psi^c_*(1)$. Combining [\[equalvol\]](#equalvol){reference-type="eqref" reference="equalvol"} and [\[ineqvol\]](#ineqvol){reference-type="eqref" reference="ineqvol"}, we get that $$\label{egrearr} |\{\psi^c > t \}| = \inf_{r \geq 0} |\{-\psi > t-k(r)\}_r| = \inf_{r \geq 0} |\{-\psi_* > t-k(r)\}_r|= |\{\psi_*^{\,c} > t \}|.$$ The following claim is established in Step 2.b below. **Claim 3**. *There exists $r_*=r_*(t)>0$ such that $$|\{\psi_*^{\,c} > t \}| = \inf_{r \geq 0} |\{-\psi_* > t-k(r)\}_r| = |\{-\psi_*> t-k(r_*)\}_{r_*}.$$* Provisionally assuming the claim let us prove that $\{\psi^c >t \}$ is a ball.\ We assume without loss of generality that $|\{\psi_*^{\,c}>t\}|>0$ (otherwise $|\{\psi^c >t \}|\le |\{\psi_*^{\,c} >t \}|=0$ by [\[inegrearr\]](#inegrearr){reference-type="eqref" reference="inegrearr"} and the open set $\{\psi^c >t \}$ is empty). Next, the claim, [\[egrearr\]](#egrearr){reference-type="eqref" reference="egrearr"} and [\[inegrearr\]](#inegrearr){reference-type="eqref" reference="inegrearr"} yield that $r_*(t)$ also minimises $\inf_{r \geq 0} |\{-\psi > t-k(r)\}_r|$. Thus by [\[egrearr\]](#egrearr){reference-type="eqref" reference="egrearr"}, $\{-\psi_* > t-k(r_*(t))\}_{r_*(t)}$ is a ball of positive volume. As $r_*(t)>0$, by the equality case of the claim of Part I, Step 2, the set $\{-\psi > t-k(r_*(t))\}_{r_*(t)}$ is also a ball. As $\{\psi^c > t \} \subset \{-\psi > t-k(r_*(t))\}_{r_*(t)}$, by [\[egrearr\]](#egrearr){reference-type="eqref" reference="egrearr"} the inclusion is actually an equality. Hence $\{\psi^c >t \}$ is a ball. *Step 2.b. Proof of the claim.* We first show that there exist $0<r_t<R_t<\infty$ such that $$\label{reducr} \inf_{r \geq 0} |\{-\psi_* > t-k(r)\}_r| = \inf_{r_t \leq r \leq R_t } |\{-\psi_* > t-k(r)\}_r|.$$ We start with the upper bound on $r$. By (H[\[cont\]](#cont){reference-type="ref" reference="cont"})&(H[\[monot\]](#monot){reference-type="ref" reference="monot"}), there exists $R_t$ such that $k(R_t) = t+1$. Hence, if $r > R_t$, $\{-\psi_* >t - k(r) \} = \mathbb{R}^d$. We can thus only consider the radii $r\le R_t$. We now prove the lower bound on $r$. Recall that $t> \psi_*^{\,c}(1)$ and that $\psi_*^{\,c}$ is decreasing in $B_1$. Therefore there exists $R_*(t)<1$ such that $$\{\psi_*^{\,c}>t\}=B_{R_*(t)}.$$ We set $r_t:= \frac{1-R_*(t)}{2}$ and claim that [\[reducr\]](#reducr){reference-type="eqref" reference="reducr"} holds for this value. To ease notation, let us set for $r>0$ $$S_r:=\{-\psi_* > t-k(r)\}_r=\{x\in \mathbb{R}^d :-\psi_* > t-k(r) \text{ on } \overline{B}_r(x)\}.$$ We also define $\overline{R}:=\frac{1+R_*(t)}{2}$. In order to prove [\[reducr\]](#reducr){reference-type="eqref" reference="reducr"} it is enough to show that $$\label{psiSr} \{\psi_*^{\,c}>t\}=\cap_{r\ge r_t} S_r.$$ Recalling that the sets $S_r$ are centred balls and that $B_{R_*(t)}\subset B_{\overline{R}}$, we have $$\{\psi_*^{\,c}>t\}=\cap_{r\ge 0} (S_r\cap B_{\overline{R}}).$$ We now claim that $$\cap_{ r\ge r_t} (S_r \cap B_{\overline{R}}) \subset \cap_{r<r_t} (S_r \cap B_{\overline{R}}),$$ which is equivalent to $$\label{cupcup} \cup_{r<r_t} (S_r^c\cap B_{\overline{R}})\subset \cup_{ r\ge r_t} (S^c_r \cap B_{\overline{R}}).$$ To prove this let $x\in S_r^c\cap B_{\overline{R}}$ for some $r<r_t$. By definition of $S_r^c$, $$\min_{y\in \overline{B}_r(x)} k(r)-\psi_*(y)\le t.$$ In particular since $k$ is increasing, there exists $y\in \overline{B}_r(x)$ such that $$k(|x-y|)-\psi^*(y)\le t.$$ As $x\in B_{\overline{R}}\subset B_1$, and $(\psi_*^{\,c},\psi_*^{cc})$ are Kantorovitch potentials for the external transport minimising $\Upsilon(B_1)$ (see Proposition [Proposition 15](#prop_Ups_OT){reference-type="ref" reference="prop_Ups_OT"}) there exists $z\in B_1^c$ such that $\psi_*^{cc}(z)=\psi_*(z)$ (by [\[psi_KP\]](#psi_KP){reference-type="eqref" reference="psi_KP"}) and $$\psi_*^{\,c}(x)=k(x-z)-\psi_*(z)=\min_y k(|x-y|)-\psi_*(y)\le t.$$ Since $z\in B_1^c$ and $x\in B_{\overline{R}}$ we have $$r'=|z-x|\ge 1-\overline{R}=\frac{1-R_*(t)}{2}=r_t$$ and thus $$\min_{z\in \overline{B}_{r'}(x)} k(r')-\psi_*(z)\le t$$ so that $x\in S_{r'}$. This shows [\[cupcup\]](#cupcup){reference-type="eqref" reference="cupcup"} which implies $$\{\psi_*^{\,c}>t\}=\cap_{ r\ge r_t} (S_r \cap B_{\overline{R}}).$$ Eventually, we must have $S_r \subset B_{\overline{R}}$ for some $r\ge r_t$ (otherwise $\{\psi_*^{\,c}>t\}=B_{\overline{R}}$ which is absurd). This concludes the proof of [\[psiSr\]](#psiSr){reference-type="eqref" reference="psiSr"} and thus of [\[reducr\]](#reducr){reference-type="eqref" reference="reducr"}. Next, setting $$L(r):=|\{-\psi_* > t-k(r)\}_r|,$$ we still have to establish that the infimum of $L$ over $[r_t,R_t]$ is reached. For this we establish that $L$ is lower semi-continuous (together with [\[reducr\]](#reducr){reference-type="eqref" reference="reducr"} this will conclude the proof of the existence of $r_*(t)>0$ minimising $L$ over $\mathbb{R}_+$). We start by noticing that, $r \mapsto |\{-\psi_* > r \}|$ is lower semi-continuous on $\mathbb{R}_+$. Let us denote by $\rho_t(r)$ the radius of the ball $\{-\psi_* > t - k(r) \}$. As $k$ is continuous, the function $r \mapsto \rho_t(r)$ is also lower semi-continuous. Finally, as $L(r) = \omega_d[(\rho_t(r) - r)_+]^d$, $L$ is lower semi-continuous as well. This ends the proof of the claim. *Step 3. Conclusion.* Let now $t_n$ be a decreasing sequence converging to $\psi_*^{\,c}(1)$. We have $$\label{sequenceballpsi} \{\psi^c >\psi_*^{\,c}(1)\} = \bigcup_{n \geq 0} \{\psi^c > t_n \} \qquad \text{and} \qquad \{\psi_*^{\,c} > \psi_*^{\,c}(1)\} = \bigcup_{n \geq 0} \{\psi_*^{\,c} > t_n \} = B_{1}.$$ By [\[egrearr\]](#egrearr){reference-type="eqref" reference="egrearr"}, for every $n \geq 0$, $\{\psi^c > t_n \} = B_{r_n}(z_n)$, where $r_n$ is the radius of $\{\psi_*^{\,c} > t_n\}$ and $z_n \in \mathbb{R}^d$. Since $t_n$ is decreasing the sequence $B_{r_n}(z_n)$ is non-decreasing. Moreover, by [\[sequenceballpsi\]](#sequenceballpsi){reference-type="eqref" reference="sequenceballpsi"} $r_n \to 1$ as $n \to \infty$. Hence there exists $z \in \mathbb{R}^d$ such that $\chi_{B_{r_n}} \to \chi_{B_1}(z)$ monotonically in $L^1(\mathbb{R}^d)$ as $n \to \infty$. Eventually [\[sequenceballpsi\]](#sequenceballpsi){reference-type="eqref" reference="sequenceballpsi"} implies that $\{\psi^c >\psi_*^{\,c}(1) \} = B_{1}(z)$. Consequently, $f = \chi_{\{\psi^c > \psi_*^{\,c}(1)\}} = \chi_{B_{1}(z)}$. This concludes the proof of the fact that balls are the unique maximisers to [\[MP\]](#MP){reference-type="eqref" reference="MP"}. ◻

arxiv_math

--- abstract: | The Collatz conjecture implies that an iterated function sequence under a certain linear operator, beginning with a certain complex valued function, must converge to a certain complex function. author: - | Kerry M. Soileau\ kerry at kerrysoileau dot com title: A Necessary Condition on the Collatz Conjecture --- **Keywords** Collatz, iteration. # Background The Collatz Conjecture is named for the mathematician Lothar Collatz, who introduced it in 1937.[@O'Connor] J. Lagarias provided a useful survey of the subject.[@Lagarias] P. Erdős remarked that "Mathematics may not be ready for such problems.\"[@Guy] By 2020, the conjecture had been verified by computer for all starting values up to $2^{68}.$[@Barina] # Introduction Let $C(m)$ be the Collatz function given by $$C(n)= \begin{cases} \frac{n}{2},& \text{for } n \text{ even}\\ 3 n+1,& \text{for } n \text{ odd} \end{cases}$$ for $n=1,2,3,\dots$ Suppose $\left\{ a_n \right \}_{n=1}^\infty \subset \mathbb{C}$ with $\sum \limits_{n=1}^\infty |a_n|<\infty.$ Let the operator $L_n$ be given by $$L_{0}\left(\sum \limits_{n=1}^\infty a_n e^{n i t} \right) \equiv \sum \limits_{n=1}^\infty a_n e^{{n i t}},$$ $$L_{1}\left(\sum \limits_{n=1}^\infty a_n e^{n i t} \right) \equiv \sum \limits_{n=1}^\infty a_n e^{{C(n) i t}},$$ and in general $$L_{n+1} \equiv L_1 \circ L_n \text{ for }n=1,2,3,\cdots.$$ Note that in particular if $f(t) \equiv \sum \limits_{n=1}^\infty a_n e^{{C(n) i t}},$ its absolute convergence implies $$\begin{gathered} L_{1}\left(f(t) \right) =\sum \limits_{n=1}^\infty a_n e^{{C(n) i t}} =\sum \limits_{m=1}^\infty a_{2 m} e^{C(2 m) i t}+\sum \limits_{m=0}^\infty a_{2 m+1} e^{C(2 m+1) i t}\\ =\sum \limits_{m=1}^\infty a_{2 m} e^{m i t}+\sum \limits_{m=0}^\infty a_{2 m+1} e^{(6m+4) i t}\\ =\frac{1}{2} \left(f\left(\frac{t}{2}\right)+f\left(\frac{t}{2}+\pi\right)+ e^{i t}\left(f(3 t)-f(3 t + \pi)\right) \right)\end{gathered}$$ In the following we consider the case in which $a_n=c^n$ for some $c \in \mathbb{C}$ in the open unit disk and $n=1,2,3,\cdots.$ Suppose $f(t) \equiv \sum \limits_{n=0}^\infty c^n e^{i n t}$ with $|c|<1,$ then $f(t)=\frac{1}{1- c e^{i t}}$ and $$\begin{gathered} L_1(f(t)) =\sum \limits_{n=0}^\infty c^n e^{{C(n) i t}}=1+c e^{4 i t}+c^2 e^{i t}+c^3 e^{10 i t}+c^4 e^{2 i t}+c^5 e^{16 i t}\\ +c^6 e^{3 i t}+c^7 e^{22 i t}+c^8 e^{4 i t}+c^9 e^{28 i t}+c^{10} e^{5 i t}+c^{11} e^{34 i t}+c^{12} e^{6 i t}+c^{13} e^{40 i t} \\ +c^{14} e^{7 i t}+c^{15} e^{46 i t}+c^{16} e^{8 i t}+c^{17} e^{52 i t}+c^{18} e^{9 i t}+c^{19} e^{58 i t}+c^{20} e^{10 i t} +\cdots\\ =\frac{1}{2} \left(f\left(\frac{t}{2}\right)+f\left(\frac{t}{2}+\pi\right)+ e^{i t}\left(f(3 t)-f(3 t + \pi)\right) \right)\\ =\frac{c e^{i t} \left(c^3 \left(-e^{6 i t}\right)-c^2 e^{4 i t}+c+e^{3 i t}\right)}{c^4 e^{7 i t}-c^2 e^{i t} \left(1+e^{5 i t}\right)+1}\end{gathered}$$ Continuing, we get $$\begin{split} L^2(f(t)) \equiv L_1(L_1(f(t)))\\ =\frac{c e^{i t} \left(c^7 \left(-e^{6 i t}\right)-c^6 e^{5 i t}-c^5 e^{4 i t}-c^4 e^{2 i t}+c^3+c^2 e^{4 i t}+c e^{3 i t}+e^{i t}\right)}{c^8 e^{7 i t}-c^4 e^{i t} \left(1+e^{5 i t}\right)+1} \end{split}$$ and so on. # Averaging Lemma **Lemma 1**. *Suppose $a_1,a_2,a_3,\cdots$ is a sequence of complex numbers that eventually repeats with period $P>0.$ By this we mean that there exists some integer $K \geqslant 1$ such that $a_{n+P}=a_n$ for all $n\geqslant K,$ and $K$ is the least integer with this property. Then $$\lim_{n \to \infty} \frac{1}{n}\sum \limits_{i=1}^n a_i = \frac{1}{P}\sum \limits_{i=K}^{K+P-1} a_i,$$ which is the average of the terms in the repeated subsequence.* *Let $Q=\min \limits_{K \leqslant j \leqslant K+P-1} \left|\sum \limits_{i=K}^j a_i \right|$ and $R=\max \limits_{K \leqslant j \leqslant K+P-1} \left|\sum \limits_{i=K}^j a_i \right|.$ Let $V=\sum \limits_{i=K}^{K+P-1} a_i$ and $W=\sum \limits_{i=1}^{K-1} a_i.$ Clearly $W$ is the sum of the sequence terms appearing before the repetition starts (if any), $V$ is the contribution due to each repetition subsequence, and $Q$ and $R$ bound the running total contribution of a repetition sequence. If we define the running mean $M_n \equiv \frac{1}{n}\sum \limits_{i=1}^n a_i,$ then for $n \geqslant K+P$ we have $$\begin{gathered} n M_n =\sum \limits_{i=1}^{K-1} a_i + \sum \limits_{i=K}^{\Bigl\lfloor \frac{n-K+1}{P} \Bigl\rfloor P+K-1} a_i + \sum \limits_{i=\Bigl\lfloor \frac{n-K+1}{P} \Bigl\rfloor P+K}^n a_i\\ =W + \Bigl\lfloor \frac{n-K+1}{P} \Bigl\rfloor V+ \sum \limits_{i=\Bigl\lfloor \frac{n-K+1}{P} \Bigl\rfloor P+K}^n a_i \end{gathered}$$ Note that $$Q \leqslant \left|\sum \limits_{i=\Bigl\lfloor \frac{n-K+1}{P} \Bigl\rfloor P+K}^n a_i \right| \leqslant R,$$ whence $$Q \leqslant \left|n M_n -W - \Bigl\lfloor \frac{n-K+1}{P} \Bigl\rfloor V \right| \leqslant R,$$ and thus $$\frac{Q}{n} \leqslant \left|M_n - \frac{W}{n} - \frac{1}{n}\Bigl\lfloor \frac{n-K+1}{P} \Bigl\rfloor V \right| \leqslant \frac{R}{n}.$$ Taking limits as $n \to \infty,$ we get $$0 \leqslant \lim_{n \to \infty}\left|M_n - \frac{W}{n} - \frac{1}{n}\Bigl\lfloor \frac{n-K+1}{P} \Bigl\rfloor V \right| \leqslant 0,$$ i.e. $$\lim_{n \to \infty}\left|M_n - \frac{W}{n} - \frac{1}{n}\Bigl\lfloor \frac{n-K+1}{P} \Bigl\rfloor V \right|=0.$$ Thus $$\lim_{n \to \infty}\left|M_n - 0 - \frac{1}{P}\sum \limits_{i=K}^{K+P-1} a_i \right|=0,$$ whence $$\lim_{n \to \infty} M_n = \frac{1}{P}\sum \limits_{i=K}^{K+P-1} a_i.$$* # Convergence Lemma **Lemma 2**. *If for all $n=1,2,3,\cdots$ and $j=1,2,3,\cdots,$* 1. *$a_{n,j} \in \mathbb{C}$,* 2. *$\lim \limits_{n \to \infty} a_{n,j} =0$ and* 3. *there exist some $K, \rho>0$ such that $|a_{n,j}| \leqslant K \rho^{\,j}<\infty,$* *then $$\sum \limits_{j=1}^\infty a_{n,j} z^j$$ converges to $0$ for $|z|<\frac{1}{\rho}.$* We define $f_n(z) \equiv \sum \limits_{j=1}^\infty a_{n,j} z^j.$ Note that $\inf \limits_k \sup \limits_{j \geqslant k} |a_{n,j}|^{\frac{1}{j}} \leqslant \inf \limits_k \sup \limits_{j \geqslant k} (K \rho^{\,j})^{\frac{1}{j}} =\inf \limits_k \sup \limits_{j \geqslant k} (K^{\frac{1}{j}} \rho)=\rho,$ so the radius of convergence of each $\sum \limits_{j=1}^\infty a_{n,j} z^j$ is at least $\frac{1}{\rho},$ and thus $f_n(z)$ is well-defined for $|z|<\frac{1}{\rho}.$ Fix $|z_0|<\frac{1}{\rho}.$ We claim that $\lim \limits_{n \to \infty} f_n(z_0)=0.$ Indeed, fix $\epsilon>0.$ For each $m=1,2,3,\cdots,$ we define $A_{n,m}(z_0)=\sum \limits_{j=1}^m a_{n,j} z_0^j$ and $B_{n,m}(z_0)=\sum \limits_{j=m+1}^\infty a_{n,j} z_0^j.$ Clearly $f_n(z_0)=A_{n,m}(z_0)+B_{n,m}(z_0).$ Let $m_0$ be the smallest positive integer satisfying $m_0 > \frac {\ln \frac{\epsilon }{2 K}+\ln (1-\rho |z_0|)} {\ln \rho+\ln |z_0|} -1 .$ Then $|B_{n,m_0}(z_0)|=\Big|\sum \limits_{j=m_0+1}^\infty a_{n,j} z_0^j\Big|\leqslant \sum \limits_{j=m_0+1}^\infty |a_{n,j}| |z_0|^j\leqslant \sum \limits_{j=m_0+1}^\infty K \rho^{\,j} |z_0|^j=\frac{K |\rho z_0|^{m_0+1}}{1-\rho |z_0|}<\frac{\epsilon}{2},$ the last inequality due to our assumption on $m_0.$ Thus $|B_{n,m_0}(z_0)|<\frac{\epsilon}{2}$ for every $n=1,2,3,\cdots.$ Next, since $\lim \limits_{n \to \infty} a_{n,j}=0$ there exist integers $N_j>0$ such that for each $j=1,2,\cdots,m_0$ we have $|a_{n,j}|<\frac{\epsilon}{2(m_0+1)}$ for $n \geqslant N_j.$ By taking $N=\max \limits_{1 \leqslant j \leqslant m_0} N_j,$ we have found an $N$ such that $n \geqslant N$ implies $|a_{n,j}|<\frac{\epsilon}{2(m_0+1)}$ for $j=1,2,\cdots,m_0.$ Now note that for such $n$ we have $|A_{n,m_0}(z_0)|=\Big|\sum \limits_{j=1}^{m_0} a_{n,j} z_0^j\Big| \leqslant \sum \limits_{j=1}^{m_0} |a_{n,j}| |z_0|^j < \sum \limits_{j=1}^{m_0} |a_{n,j}| < \sum \limits_{j=1}^{m_0} \frac{\epsilon}{2(m_0+1)}=\frac{\epsilon}{2}.$ Finally, $|f_n(z_0)|=|A_{n,m_0}(z_0)+B_{n,m_0}(z_0)| \leqslant |A_{n,m_0}(z_0)|+|B_{n,m_0}(z_0)|<\frac{\epsilon}{2}.$ This implies $\lim \limits_{n \to \infty} \sum \limits_{j=1}^\infty a_{n,j} z^j=0$ for any $|z|<1,$ as desired. # Necessary Condition Let $|c|<1.$ We define $$f_n(\theta) \equiv L_1(f_{n-1}(\theta))$$ for $n=1,2,\cdots,$ where $f_0(\theta) \equiv 1+c e^{i \theta}+c^2 e^{2 i \theta}+ \cdots = \frac{1}{1-c\, e^{i \theta}}.$ This implies $$f_n(\theta)=\sum \limits_{r=1}^\infty c^r e^{C^n(r) i \theta}$$ where $C^0(m) \equiv m$ and $C^k(m) \equiv C(C^{k-1}(m))$ for $m=1,2,3,\cdots.$ Next, we now define the $n^{th}$ mean: $$M_n(\theta)=\frac{1}{n} \sum \limits_{j=1}^{n} f_j(\theta)$$ for $n=1,2,3,\cdots.$ Finally we define $$g_{n,m}(\theta) \equiv \frac{1}{n} \sum \limits_{j=1}^{n} e^{C^j(m)i\theta}$$ for $n=1,2,3,\cdots$ and $m=1,2,3,\cdots.$ We then have $$M_n(\theta)=\sum \limits_{m=1}^\infty c^m g_{n,m}(\theta)$$ **Proposition 3**. *If every Collatz sequence eventually reaches $1,$ then for each $m=1,2,3,\cdots,$ $$\lim \limits_{n \to \infty} g_{n,m}(\theta)=\frac{1}{3}\left( e^{i\theta}+e^{2 i\theta}+e^{4 i\theta} \right).$$* Fix $m \in \{1,2,3,\cdots\}$ and $\theta \in [0,2\pi).$ Then the sequence $$\left\{ e^{C^n(m)i\theta} \right\}_{n=0}^\infty$$ eventually repeats with period $3$ and repeating subsequence $\left\{ e^{4i\theta},e^{2i\theta},e^{i\theta} \right\}.$ By Lemma [Lemma 1](#averaginglemma){reference-type="ref" reference="averaginglemma"}, it follows that $\lim \limits_{n \to \infty} g_{n,m}(\theta)=\frac{1}{3}\left( e^{i\theta}+e^{2 i\theta}+e^{4 i\theta} \right).$ **Theorem 4**. *If every Collatz sequence eventually reaches $1,$ then for each $\theta \in [0,2 \pi)$ we have $$\lim \limits_{n \to \infty} M_n(\theta)=1+\frac{1}{3}\left( e^{i\theta}+e^{2 i\theta}+e^{4 i\theta} \right) \frac{c}{1-c}$$* *Let $$\lim \limits_{n \to \infty} M_k(\theta)=\lim \limits_{n \to \infty}\sum \limits_{m=1}^\infty c^m g_{n,m}(\theta).$$ Note that $$\begin{gathered} \left| g_{n,m}(\theta)- \frac{1}{3}\left( e^{i\theta}+e^{2 i\theta}+e^{4 i\theta} \right)\right|\\ =\left| \frac{1}{n} \sum \limits_{j=1}^{n} e^{C^j(m)i\theta}-\frac{1}{3}\left( e^{i\theta}+e^{2 i\theta}+e^{4 i\theta} \right)\right|\\ \leqslant \left| \frac{1}{n} \sum \limits_{j=1}^{n} e^{C^j(m)i\theta}\right|+\left|-\frac{1}{3}\left( e^{i\theta}+e^{2 i\theta}+e^{4 i\theta} \right)\right|\\ \leqslant \frac{1}{n} \sum \limits_{j=1}^{n} \left|e^{C^j(m)i\theta}\right|+\frac{1}{3}\left| e^{i\theta}+e^{2 i\theta}+e^{4 i\theta} \right|\\ \leqslant 1 +1=2<\infty\\\end{gathered}$$ for all $m=1,2,3,\cdots$ and $m=1,2,3\cdots.$* *Now we invoke Lemma [Lemma 2](#convergencelemma){reference-type="ref" reference="convergencelemma"} to infer that $$\lim \limits_{n \to \infty}\sum \limits_{m=1}^\infty c^m \left( g_{n,m}(\theta)- \frac{1}{3}\left( e^{i\theta}+e^{2 i\theta}+e^{4 i\theta} \right) \right)=0,$$ i.e. $$\begin{gathered} \lim \limits_{n \to \infty}\sum \limits_{m=1}^\infty c^m g_{n,m}(\theta) =\sum \limits_{m=1}^\infty c^m \frac{1}{3}\left( e^{i\theta}+e^{2 i\theta}+e^{4 i\theta} \right)\\ =\frac{1}{3}\left( e^{i\theta}+e^{2 i\theta}+e^{4 i\theta} \right) \frac{c}{1-c},\end{gathered}$$ whence $$\lim \limits_{n \to \infty} M_n(\theta) =\frac{1}{3}\left( e^{i\theta}+e^{2 i\theta}+e^{4 i\theta} \right) \frac{c}{1-c},$$ as desired.* 9 O'Connor, J.J.; Robertson, E.F. (2006). \"Lothar Collatz\". St Andrews University School of Mathematics and Statistics, Scotland. Lagarias, Jeffrey C. (1985). \"The 3x + 1 problem and its generalizations\". The American Mathematical Monthly. 92 (1): 323. Guy, Richard K. (2004). \"\"E16: The 3x+1 problem\"\". Unsolved Problems in Number Theory (3rd ed.). Springer-Verlag. pp. 3306. Barina, David (2020). \"Convergence verification of the Collatz problem\". The Journal of Supercomputing. 77 (3): 26812688.

arxiv_math

--- abstract: | We consider a compact, polynomially convex, regular set $K \subset \mathbb{C}$ and a sequence $(p_n)_{n=1}^\infty$ of polynomials with uniformly bounded zeros and such that $\lim_{n \to \infty}\|p_n\|_K^{1/({\rm deg} \ p_n)}={\rm cap}(K)$, where ${\rm cap}(K)$ is the logarithmic capacity of $K$. Taking an arbitrary sequence $(d_k)_{k=1}^\infty$ of integers greater than $1$ we prove that there exists a nonempty set $\mathcal{K}[(p_{d_k})_{k=1}^\infty]$, depending only on the sequence $(p_{d_k})_{k=1}^\infty$, such that for any compact polynomially convex regular set $E$ the preimages $(p_{d_k}\circ...\circ p_{d_1})^{-1}(E)$ converge in Klimek's metric to $\mathcal{K}[(p_{d_k})_{k=1}^\infty]$. We call the set $\mathcal{K}[(p_{d_k})_{k=1}^\infty]$ the non-autonomous filled Julia set generated by the polynomial sequence $(p_{d_k})_{k=1}^\infty$. Our toy example is generated by $t_n=\frac{1}{2^{n-1}}T_n,\ n\in\{1,2,...\}$, associated with $K=[-1,1]$, where $T_n$ is the classical Chebyshev polynomial of degree $n$. author: - | Marta Kosek\ Institute of Mathematics\ Faculty of Mathematics and Computer Science\ Jagiellonian University\ Łojasiewicza 6, 30-348 Kraków, POLAND\ E-mail: Marta.Kosek\@im.uj.edu.pl\ - | Małgorzata Stawiska\ American Mathematical Society--Mathematical Reviews\ 416 Fourth St., Ann Arbor, Michigan, USA\ E-mail: stawiska\@umich.edu date: title: Non-autonomous Julia sets for sequences of polynomials satisfying Kalmár-Walsh theorem --- [^1] [^2] # Introduction Behavior of iterates $p^n:=p^{\circ n}$ of a polynomial $p: \mathbb{C} \longrightarrow \mathbb{C}$ has been intensely studied since the nineteenth century. For an introduction to the subject see e.g. [@CGbook]. Much more recent (and much less advanced) is the study of sequences of maps of the type $p_n\circ...\circ p_1$ where each $p_n$, $n\in\{1,2,...\},$ is a polynomial, but not necessarily the same one. Under the name of \"generalized iteration\", the works [@Al], [@BB03] and [@B97] deal with such sequences when the coefficients of underlying polynomials satisfy some conditions. In this article we extend the dynamical study to sequences of polynomials without any concrete assumptions on their coefficients. We work instead with sequences of polynomials associated with a (fixed) compact set in the complex plane that have been widely studied in complex approximation and interpolation theory, often in connection with logarithmic potential theory ([@Walsh], [@BBCL], [@CSZ4], [@BE]), namely sequences satisfying the Kalmár-Walsh condition. More specifically, let $K \subset \mathbb{C}$ be compact and regular (i.e., the Green function of $K$ exists and is continuous), and let $(p_n)_{n=1}^{\infty}$ be a sequence of polynomials such that ${\rm deg} \ p_n=n$. By the *Kalmár-Walsh condition* we mean the following equality: $$\label{eq: KWnormy} \lim_{n\to \infty}\|p_n\|_K^{1/n}={\rm cap}(K),$$ where ${\rm cap}(K)$ is the logarithmic capacity of $K$. See subsection [2.2](#subsection: KW){reference-type="ref" reference="subsection: KW"} for examples of such sequences of polynomials. Here is the example motivating our study of the behavior of compositions $p_n\circ...\circ p_1$, where $(p_n)_{n=1}^\infty$ is a sequence of polynomials satisfying ([\[eq: KWnormy\]](#eq: KWnormy){reference-type="ref" reference="eq: KWnormy"}). *Main Example 1*. Recall that the classical Chebyshev polynomials satisfy $T_{d_1}\circ T_{d_2} =T_{d_2}\circ T_{d_1}=T_{d_1d_2}$. All zeros of $T_d$ belong to the segment $K=[-1,1]$. The polynomial $t_d=\frac{1}{2^{d-1}}T_d$ is the minimal polynomial of degree $d$ on $[-1,1]$, so $$\lim_{d \to \infty}\|t_d\|_K^{1/d}= {\rm cap}([-1,1]) =\frac12 %1/2$$ and $$\lim_{d \to \infty}\frac{1}{d}\log|t_d(z)|=g_{[-1,1]}(z) -\log 2$$ locally uniformly in $\mathbb{C}\setminus [-1,1]$. Here $g_{[-1,1]}$ is the complex Green function of the segment $[-1,1]$. In consequence $\lim_{d \to \infty}\frac{1}{d}\log|T_d(z)|=g_{[-1,1]}(z)$ locally uniformly in $\mathbb{C}\setminus [-1,1]$. For a sequence of integers $(d_n)_{n=1}^\infty$ not smaller than 2 we get $T_{d_n} \circ ...\circ T_{d_1}=T_{d_n...d_1}$, hence $(T_{d_ n} \circ ...\circ T_{d_1})_{n=1}^\infty$ is a subsequence of the sequence of all classical Chebyshev polynomials with increasing degrees and $$\lim_{n \to \infty}\frac{1}{d_n...d_1}\log \left|(T_{d_n}\circ...\circ T_{d_1}) (z)\right|=g_{[-1,1]}(z)$$ locally uniformly in $\mathbb{C}\setminus [-1,1]$. By a more dynamical approach it was shown in [@Ko21] that $$\lim_{n \to \infty}\frac{1}{d_n...d_1}\log^+ \left|(T_{d_n}\circ...\circ T_{d_1})(z)\right|=g_{[-1,1]}(z)$$ for any $z\in\mathbb{C}$. Moreover, the convergence is uniform in the whole complex plane. The function $\frac{1}{d_n...d_1}\log^+ \left|(T_{d_n}\circ...\circ T_{d_1})\right|$ is the Green function of the preimage of the unit disk, $g_{(T_{d_n}\circ...\circ T_{d_1})^{-1}\left(%\mathbb{D} \overline{\mathbb{D}}(0,1)\right)}$, and the convergence of functions can be thought of as the convergence in Klimek's metric of the sets $(T_{d_n}\circ...\circ T_{d_1})^{-1}\left(%\mathbb{D} \overline{\mathbb{D}}(0,1)\right)$ to $[-1,1]$. It is natural to ask whether the convergence still holds if $[-1,1]$ is replaced by a (fixed) general compact, polynomially convex regular set $K$, $(T_n)_{n=1}^\infty$ is replaced by a sequence $(p_n)_{n=1}^\infty$ of polynomials associated with $K$ and $%\mathbb{D} \overline{\mathbb{D}}(0,1)$ is replaced by an arbitrary compact, polynomially convex regular set $E$ (not necessarily equal to $K$). We show that this is indeed the case when the polynomials $p_n$ satisfy ([\[eq: KWnormy\]](#eq: KWnormy){reference-type="ref" reference="eq: KWnormy"}) and their zeros are uniformly bounded (we call such polynomials *KW polynomials*). Our main result is the following (see Theorem [Theorem 35](#thm:zbiornieautonomiczny){reference-type="ref" reference="thm:zbiornieautonomiczny"}): **Main Theorem 1**. *Let $K\subset \mathbb{C}$ be a regular polynomially convex compact set and let $(p_n)_{n=1}^\infty$ be a sequence of KW polynomials associated with $K$. Fix now a sequence $(d_k)_{k=1}^\infty$ of integers greater than $1$. Then, for every compact, polynomially convex and regular set $E$, the sequence $$\left((p_{d_k}\circ...\circ p_{d_1})^{-1}(E)\right)_{k=1}^\infty$$ is convergent in Klimek's metric to the limit ${\mathcal K}[(p_{d_k})_{k=1}^\infty]$, independent of $E$.* The set ${\mathcal K}[(p_{d_k})_{k=1}^\infty]$ is the non-autonomous Julia set generated by the sequence $(p_{d_k})_{k=1}^\infty$. We establish basic properties of such Julia sets, showing that they are nonempty, compact, polynomially convex and regular. Our methods are primarily based on the approach developed in [@KK03] (see also [@KK18]). We further study in some more detail the non-autonomous filled Julia set generated by $t_d=\frac{1}{2^{d-1}}T_d,\ d\in\{1,2,...\},$ and the connections with the theory of best uniform approximation by polynomials on a compact set in $\mathbb{C}$.\ We include a good amount of background material in our presentation to make it as self-contained as possible. Section 2 contains preliminaries from logarithmic potential theory and complex approximation theory, with quite a detailed discussion of Kalmár-Walsh theorem. Proposition [Proposition 17](#prop: tnzez){reference-type="ref" reference="prop: tnzez"} and Corollary [Corollary 18](#cor:przeciwobrazkola){reference-type="ref" reference="cor:przeciwobrazkola"} sharpen some previously available results, while Proposition [Proposition 20](#prop: characterization){reference-type="ref" reference="prop: characterization"} seems to be new. More new material appears in Section 3, including the definition of a non-autonomous filled Julia set, the proof of its existence and properties. The approach using Klimek's metric is explained. In Section 4 we explore the case of Chebyshev polynomials on a compact set.\ # The Green function, the logarithmic capacity and their approximations ## The Green function and related notions We will study certain sequences of polynomials associated with a compact subset $K$ of $\mathbb{C}$, namely polynomials occurring in approximating the Green function (with pole at infinity) of $K$, provided such a function exists and is continuous. Let us first define the Green function and the logarithmic capacity of a set (mostly following [@STo Section II.4]). **Definition 1**. Let $D \subset \mathbb{C}$ be an unbounded domain. Consider a function $g:D \longrightarrow \mathbb{R}$ with the following properties: 1. $g_D$ is harmonic and positive in $D$; 2. $g_D(z)$ tends to $0$ as $z \to \partial D$; 3. $g_D(z)-\log |z|$ tends to a finite number $\gamma$ as $z \to \infty$. If such a function $g_D$ exists, it is called the *Green function* of $D$ with pole at infinity. It can be proved that if the Green function exists, it is unique.\ Fix a compact subset $K$ of $\mathbb{C}$. Define the polynomially convex hull of $K$ as $$\widehat{K}:=\left\{z \in \mathbb{C}:\; \forall p \mbox{ polynomial }\; |p(z)|\leq \|p\|_K\right\},$$ where $\|p\|_K:=\sup_{z\in K}|p(z)|$. Let $D_\infty^K$ denote the unbounded component of $\mathbb{C}\setminus K$. We have $\widehat{K}=\mathbb{C}\setminus D_\infty^K$. The set $K$ is called *polynomially convex* if $K=\widehat{K}$. Note that polynomial convexity of $K$ is equivalent to the connectedness of $\mathbb{C}\setminus K$.\ **Definition 2**. Let $K\subset \mathbb{C}$ be such a compact set that the Green function $g_{D_\infty^K}$ of $D_\infty^K$ with pole at infinity exists. We say then that $K$ is *regular* and define *the Green function $g_K:\mathbb{C}\longrightarrow \mathbb{R}$ of $K$* via formula $$g_K(z)=\begin{cases} g_{D_\infty^K}(z), & \mbox{ if } z\in {D_\infty^K}\\ 0, & \text{ if } z\in \widehat{K} \end{cases}.$$ The number $\exp(-\gamma)$, where $\gamma$ is the limit from the last item of Definition [Definition 1](#defin: Greenfunction){reference-type="ref" reference="defin: Greenfunction"}, will be then called the *logarithmic capacity* of $K$ and denoted by ${\rm cap}(K)$. Note that if $K$ is regular, then $g_K=g_{\widehat{K}}$ and it is a continuous function in view of Definitions [Definition 1](#defin: Greenfunction){reference-type="ref" reference="defin: Greenfunction"} and [Definition 2](#def:g_K){reference-type="ref" reference="def:g_K"}. Moreover $g_K$ is subharmonic in $\mathbb{C}$.\ *Example 3*. Let $a \in \mathbb{C}$, $R>0$ and let $K=\overline{\mathbb{D}}(a,R):=\{z: |z-a|\leq R\}$. Then $g_K(z)=\log^+(|z-a|/R):=\max\{0, \log (|z-a|/R)\}$ and ${\rm cap}(K)=R$. *Example 4*. Let $K=[-1,1]$. Then $g_K(z)=\log\left|z+\sqrt{z^2-1}\right|$, with the square root branch defined in $\mathbb{C}\setminus (-\infty,0)$ so that $\sqrt{1}=1$, and ${\rm cap}(K)=1/2$. *Example 5*. Fix an $R>1$. Let $K=E_R$ be the ellipse with foci $-1,+1$ and semiaxes $a=\frac{1}{2}\bigl(R+\frac{1}{R}\bigr), \ b= \frac{1}{2}\bigl(R-\frac{1}{R}\bigr)$. Then $g_K(z)=\log^+\left( {\left|z+\sqrt{z^2-1}\right|}/{R}\right)$ and ${\rm cap}(K)=(a+b)/2=R/2$. It follows from Definitions [Definition 1](#defin: Greenfunction){reference-type="ref" reference="defin: Greenfunction"} and [Definition 2](#def:g_K){reference-type="ref" reference="def:g_K"} that $\exists C\in\mathbb{R}\; \forall z\in\mathbb{C}:\; g_K(z) \leq C+\log^+|z|$. The class of all functions subharmonic in $\mathbb{C}$ and satisfying such an inequality -- the Lelong class -- is denoted by $\mathcal{L}$. A theorem of Siciak (see [@Klimekbook Theorem 5.6.1]) characterizing the class $\mathcal{L}$ says that for a function $u \in \mathcal{L}$ there exists a sequence $(p_n)_{n=1}^\infty$ of complex polynomials in $\mathbb{C}$ such that $\forall n\geq 1: {\rm deg }\ p_n \leq n$ and $u=(\limsup_{n \to \infty}\frac{1}{n}\log |p_n|)^*$. Here $^*$ denotes the upper semicontinuous regularization of a function, $v^*(x)=\limsup_{y \to x}v(y)$. We will consider such sequences of polynomials for $g_K$.\ The following estimates will be used later: **Lemma 6**. *Let $K\subset \mathbb{C}$ be compact and regular. Then $$\exists M>0:\quad |z|\geq M\quad \Longrightarrow\quad 2<g_K(z)+\log {\rm cap}(K) < 1+\log|z|.$$* *Proof.* In view of Definitions [Definition 1](#defin: Greenfunction){reference-type="ref" reference="defin: Greenfunction"} and [Definition 2](#def:g_K){reference-type="ref" reference="def:g_K"}, $$\lim_{z\rightarrow \infty} \left(g_K(z)-\log|z|+\log {\rm cap}(K)\right)= 0,$$ so there exists $\widetilde{M}>0$, such that $$|z|\geq \widetilde{M} \quad \Longrightarrow \quad -1+\log|z|<g_K(z)+\log {\rm cap}(K)< 1+\log|z|.$$ It suffices now to take $M:=\max\left(\widetilde{M},e^3+1\right).$ ◻ **Definition 7**. Let $\varepsilon>0$ and let $K\subset \mathbb{C}$ be compact and regular. Then the $\varepsilon$-sublevel set of $g_K$ (also called the *$\varepsilon$-augmentation* of $K$) is $K_\varepsilon:=\left\{z\in \mathbb{C}:\; g_K(z)\leq \varepsilon\right\}$. If in addition $K$ is polynomially convex, the family $\{K_\varepsilon\}_{\varepsilon >0}$ forms a neighbourhood base of the set $K$ in $\mathbb{C}$ as shown in [@Kl95 Corollary 1]. It was proved by M. Mazurek (published in [@Sic1981 Proposition 5.11]) that $$\label{e:Mazurek} g_{K_\varepsilon}=\max(0,g_K-\varepsilon).$$ This result implies the following properties of the sublevel sets: **Proposition 8** (cf. [@BCKS Proposition 2.3]). *Let $K$ be a regular compact subset of $\mathbb{C}$. Then:\ $(i)$ For every $\varepsilon >0$ the set $K_\varepsilon$ is polynomially convex.\ $(ii)$ For every $\varepsilon >0$ the set $K_\varepsilon$ is regular.\ $(iii)$ $K_{\varepsilon+\sigma}=(K_{\varepsilon})_{\sigma}$ for every $\varepsilon, \sigma >0$.\ $(iv)$ ${\rm cap} (K_\varepsilon)=\exp{(-\gamma+\varepsilon)}$.* *Example 9*. From Examples [Example 4](#ex: odcGreen){reference-type="ref" reference="ex: odcGreen"} and [Example 5](#ex: elipsyGreen){reference-type="ref" reference="ex: elipsyGreen"} it follows that for the segment $K=[-1,1]$ the $\varepsilon$-sublevel set of $g_K$ is the (filled) ellipse with foci $-1,+1$ and semiaxes $a=\frac{1}{2}(e^\varepsilon + e^{-\varepsilon}), \ b= \frac{1}{2}(e^\varepsilon - e^{-\varepsilon})$. The well known Bernstein-Walsh inequality ([@Walsh Lemma in Section 4.6]; for a new proof see also [@Sch]) states that for every regular compact set $K\subset \mathbb{C}$ and for any polynomial $f$ of degree $n$ $$%\label{e:BW} \forall z\in\mathbb{C}: \quad \frac{|f(z)|}{\|f\|_K}\leq \exp(ng_K(z)).$$ In particular $$\label{e:BW} \forall z\in\mathbb{C}: \qquad |f(z)|^{1/n}\leq \|f\|_K^{1/n} \exp(g_K(z)).$$ Recall also the following polynomial transformation formula: Let $f$ be a polynomial of degree $n \geq 1$. Then $$\label{e:Greenaprzeciwobrazu} g_{f^{-1}(K)}=\frac1n g_K\circ f.$$ In particular, for any polynomial $f$ of degree $n \geq 1$ the preimage under $f$ of a regular set, e.g., a closed disk, a line segment or an ellipse, is a regular set. ## Sequences of KW polynomials -- definitions and examples {#subsection: KW} Before formulating our main definition let us recall the known notion of Chebyshev polynomials. **Definition 10**. Let $E \subset \mathbb{C}$ be a compact set. A monic polynomial $t_n$ of degree $n \geq 1$, $t_n(z)=z^n+a_{n-1}z^{n-1}+...+a_1z+a_0$, is called the *$n$th Chebyshev polynomial* (or the *$n$th minimal polynomial*) on $E$ if $\|t_n\|_E \leq \|q\|_E$ for any monic polynomial $q$ of degree $n$. Let us recall a well known example of Chebyshev polynomials (cf. Main Example from the Introduction). *Example 11*. Let $E=[-1,1]$ and $n\geq 1$. Then $t_n:=\frac{1}{2^{n-1}}T_n$ is the $n$th minimal polynomial on $E$, where $T_n$ satisfies the formula $T_n\left(\frac{z+ z^{-1}}{2}\right)=\frac{z^n+z^{-n}}{2}$. The polynomials $T_n$ are called the *classical Chebyshev polynomials*. The following fact is due to Fejér [@Fejer22], but can be also found e.g. in [@Leja47]. **Lemma 12**. *All zeros of the Chebyshev polynomials on a compact set $E$ lie in the convex hull $\rm conv(E)$.* The main definition of our article follows. **Definition 13**. Let $K \subset \mathbb{C}$ be a regular polynomially convex compact set. Consider a sequence of polynomials $(p_n)_{n=1}^\infty$ such that $$p_n:\mathbb{C}\ni z \longmapsto \big(z-\zeta_1^{(n)}\big)...\big(z-\zeta_n^{(n)}\big)\in\mathbb{C},\quad n\in\{1,2,...\}.$$ Thus all $p_n$ are monic polynomials and $\forall n: \; \deg p_n=n.$ We say that $(p_n)_{n=1}^\infty$ is a *sequence of KW polynomials* associated with $K$ if the set $\bigcup_{n \in \mathbb{N}}\left\{\zeta_1^{(n)},...,\zeta_n^{(n)}\right\}$ is bounded and $$\label{eq: normy} \lim\limits_{n\to\infty} \|p_n\|_K^{1/n} = {\rm cap}(K).$$ Recall that ([\[eq: normy\]](#eq: normy){reference-type="ref" reference="eq: normy"}) is the Kalmár-Walsh condition ([\[eq: KWnormy\]](#eq: KWnormy){reference-type="ref" reference="eq: KWnormy"}) from the Introduction. Before we comment extensively on this definition, let us note that Chebyshev polynomials (Definition [Definition 10](#def:nthCheb){reference-type="ref" reference="def:nthCheb"}) on a compact polynomially convex regular set $K$ provide examples of polynomials satisfying Definition [Definition 13](#def:KWpolynomials){reference-type="ref" reference="def:KWpolynomials"}. We observe the following fact: **Corollary 14**. *If $K\subset \mathbb{C}$ is compact, polynomially convex and regular then the sequence of Chebyshev polynomials on $K$ is a KW sequence associated with $K$.* *Proof.* The condition on the zeros of the polynomials follows from Lemma [Lemma 12](#lemma: zeros){reference-type="ref" reference="lemma: zeros"}. Moreover, in [@Fekete] it was proved that ([\[eq: normy\]](#eq: normy){reference-type="ref" reference="eq: normy"}) holds for the sequence of Chebyshev polynomials on a compact set. ◻ The letters KW in Definition [Definition 13](#def:KWpolynomials){reference-type="ref" reference="def:KWpolynomials"} refer to a very well known theorem. Namely, if $\bigcup_{n \in \mathbb{N}}\left\{\zeta_1^{(n)},...,\zeta_n^{(n)}\right\}\subset K$, then the classical result of Kalmár and Walsh (see [@Walsh Section 7.3, Theorem 3 and Section 7.4, Theorem 4] or [@BBCL Theorem 1.4 and Theorem 1.5]; cf. [@Gaier II.2.B, Theorem 1 and Lemma 1]) says that ([\[eq: normy\]](#eq: normy){reference-type="ref" reference="eq: normy"}) is equivalent to the fact, that $$\label{eq: zbieznosc} \lim\limits_{n\to\infty} |p_n(z)|^{1/n} = {\rm cap}(K)\cdot \exp(g_K(z))$$ uniformly on compact subsets of $\mathbb{C}\setminus K$, as well as to the following statement: for any function $f$ holomorphic in a neighbourhood of $K$ the sequence of its interpolation polynomials with nodes $\zeta^{(n)}$ converge uniformly on $K$ to $f$. Some examples of sequences $(p_n)_{n=1}^\infty$ such that $\bigcup_{n \in \mathbb{N}}\left\{\zeta_1^{(n)},...,\zeta_n^{(n)}\right\}\subset K$ and ([\[eq: normy\]](#eq: normy){reference-type="ref" reference="eq: normy"}) is satisfied can be found in [@Gaier II.2]. Thus more examples of KW polynomials are provided.\ Let us comment more on the situation when the Kalmár-Walsh condition ([\[eq: normy\]](#eq: normy){reference-type="ref" reference="eq: normy"}) is equivalent to ([\[eq: zbieznosc\]](#eq: zbieznosc){reference-type="ref" reference="eq: zbieznosc"}) or in other words to $$\label{eq: KWgreen} \lim_{n \to \infty}\frac{1}{n}\log|p_n(z)|=g_K(z)+{\rm cap}(K).$$ The work [@BBCL], investigates, among others, conditions sufficient for ([\[eq: normy\]](#eq: normy){reference-type="ref" reference="eq: normy"}) (see [@BBCL Theorem 1.5]), under the assumptions that $K$ is polynomially convex and that the zeros of all $p_n$ lie in $K$. The equivalence between ([\[eq: normy\]](#eq: normy){reference-type="ref" reference="eq: normy"}) and ([\[eq: zbieznosc\]](#eq: zbieznosc){reference-type="ref" reference="eq: zbieznosc"}) is stated there as part of [@BBCL Theorem 1.4]). For the proof the authors refer the reader to [@Walsh], where the topic is treated in [@Walsh Section 7.3, Theorem 3 and Section 7.4, Theorem 4]. The assumptions in [@Walsh] are that $K$ is a compact, regular and polynomially convex set and that the zeros of $p_n$ do not have accumulation points in $\mathbb{C}\setminus K$ (in particular, they do not have to lie all in $K$). The convergence in ([\[eq: KWgreen\]](#eq: KWgreen){reference-type="ref" reference="eq: KWgreen"}) is then shown to hold uniformly on compact subsets in $\mathbb{C}\setminus K$.\ The equivalence between ([\[eq: normy\]](#eq: normy){reference-type="ref" reference="eq: normy"}) and ([\[eq: KWgreen\]](#eq: KWgreen){reference-type="ref" reference="eq: KWgreen"}), with uniform convergence on compact subsets of $\mathbb{C}\setminus K$ in ([\[eq: KWgreen\]](#eq: KWgreen){reference-type="ref" reference="eq: KWgreen"}), is also explicitly proved in [@Gaier II.2.B, Theorem 1 and Lemma 1], under the assumption that $K$ is a compact subset of $\mathbb{C}$ whose complement $\mathbb{C}\setminus K$ is a simply connected domain. Moreover, all zeros of all $p_n$ are assumed to lie in $K$.\ A proof of implication $$(\ref{eq: normy}) \Longrightarrow (\ref{eq: KWgreen})$$ in [@Leja47] for Chebyshev (minimal) polynomials on a compact set $K$ is easily generalized to any polynomial sequence satisfying ([\[eq: normy\]](#eq: normy){reference-type="ref" reference="eq: normy"}). The zeros of $p_n$ are assumed to be uniformly bounded but are allowed to have limit points in the unbounded component of $\mathbb{C}\setminus K$ (as may be the case for Chebyshev polynomials). Let $Z'$ be the set of all limit points of zeros of $p_n$ in the unbounded component of $\mathbb{C}\setminus K$. Then, if ([\[eq: normy\]](#eq: normy){reference-type="ref" reference="eq: normy"}) holds, ([\[eq: KWgreen\]](#eq: KWgreen){reference-type="ref" reference="eq: KWgreen"}) holds uniformly on compact subsets of $\mathbb{C}\setminus (K\cup Z')$. In [@Leja47] there is also an example of a set $K$ and a sequence of polynomials $p_n$ satisfying ([\[eq: KWnormy\]](#eq: KWnormy){reference-type="ref" reference="eq: KWnormy"}) for which ([\[eq: KWgreen\]](#eq: KWgreen){reference-type="ref" reference="eq: KWgreen"}) may fail at a point $z \in Z'$. There is no treatment of bounded components of $\mathbb{C}\setminus K$, so there seems to be no loss in assuming that $K$ is polynomially convex.\ Finally, we are assuming that the set of the zeros of all $p_n$ is bounded. It can be shown that this assumption is not too restrictive. Namely, a polynomial sequence $(p_n)_{n=1}^\infty$ such that ([\[eq: normy\]](#eq: normy){reference-type="ref" reference="eq: normy"}) holds can be replaced by a polynomial sequence $(q_n)_{n=1}^\infty$ such that ([\[eq: normy\]](#eq: normy){reference-type="ref" reference="eq: normy"}) holds for $q_n$ and the zeros of $q_n$ are uniformly bounded. More precisely, the following proposition is true: **Proposition 15** ([@FW57 Theorem 12]). *Let $K \subset \mathbb{C}$ be a compact set with $\tau ={\rm cap}(K)>0$ and let $\Gamma$ be a Jordan curve such that $K \subset {\rm int} \widehat{\Gamma}$. Suppose that $(p_n)_{n=2}^\infty$ is a sequence of polynomials such that $p_n(z)=z^n+a_{n-1}^{(n)}z^{n-1}+...+a_0^{(n)}$ and $\limsup_{n \to \infty} \|p_n\|_K^{1/n} \leq \tau$. Let us decompose $p_n$ as $p_n(z)=q_{n-\sigma}(z)r_\sigma(z)$, where $r_\sigma(z)=z^\sigma+...$ is a polynomial whose zeros are precisely the zeros of $p_n$ not belonging to $\widehat{\Gamma}$ (or $r_\sigma \equiv 1$ if there are no such zeros).* *Then we have: $\sigma=\sigma(n)=o(n)$, $\lim_{n \to \infty}\|q_{n-\sigma}\|_K^{1/(n-\sigma)}=\tau$ and $\lim_{n \to \infty}\frac{1}{n-\sigma}\log |q_{n-\sigma}(z)|=\log \tau +g_K(z)$ uniformly on any compact subset of $\mathbb{C} \setminus \widehat{\Gamma}$. Moreover, $\lim_{n \to \infty}\|r_\sigma\|_K^{1/n}=1$.* ## Properties of sequences of KW polynomials We will now establish some properties which will be crucial in the further investigation. We start with a lemma valid for all polynomials of degree at least 2. **Lemma 16**. *If $P:\mathbb{C}\longrightarrow \mathbb{C}$ is a polynomial of degree at least $2$, then $$\label{e:theta} \forall \theta>1 \ \exists R=R(\theta)>0: \quad |z|\geq R\quad \Longrightarrow \quad |P(z)|\geq \theta |z|.$$* *Proof.* Observe that $$\lim_{z\rightarrow \infty}\frac{|P(z)|}{|z|}=\infty,$$ since $\deg P\geq 2$. The relation ([\[e:theta\]](#e:theta){reference-type="ref" reference="e:theta"}) follows. ◻ A property of KW polynomials now follows. **Proposition 17**. *Let $K\subset \mathbb{C}$ be a regular polynomially convex compact set. Let further $(p_n)_{n=1}^\infty$ be a sequence of KW polynomials associated with $K$. Then $$\exists R>0\ \forall n\geq 2:\quad |z|\geq R\quad \Longrightarrow \quad |p_n(z)|\geq e|z|.$$* *Proof.* By Definition [Definition 13](#def:KWpolynomials){reference-type="ref" reference="def:KWpolynomials"} there exists such $r>0$ that all zeros of all $p_n$ lie in $\mathbb{D}(0,r) \supset K$. Put $\gamma:=\log{\rm cap}(K).$ Let $M>0$ be as in Lemma [Lemma 6](#lem:Greennierownosci){reference-type="ref" reference="lem:Greennierownosci"} and fix $R_0> \max(M,r)$. We have $\min\{g_K(z)+\gamma:\; |z|=R_0\}>2$ by Lemma [Lemma 6](#lem:Greennierownosci){reference-type="ref" reference="lem:Greennierownosci"}. As $n \to \infty$, since $(\ref{eq: normy}) \Longrightarrow (\ref{eq: KWgreen})$, $$\frac1n\log|p_n(z)|-\frac1n\log|z|\longrightarrow g_K(z)+\gamma$$ uniformly on compact subsets of $\mathbb{C}\setminus \overline{\mathbb{D}}(0,r)$, so we can choose an integer $N=N(R_0)\geq 2$ such that $$\label{e:varepsilon} \forall n\geq N :\quad \frac1n\log|p_n(z)| -\frac1n\log|z| >1,$$ provided $|z|=R_0$. Since all zeros of the polynomials $p_n$ are contained in $\mathbb{D}(0,r) \varsubsetneq \overline{\mathbb{D}}(0,R_0)$, we may apply the Minimum Principle for harmonic functions, which implies that ([\[e:varepsilon\]](#e:varepsilon){reference-type="ref" reference="e:varepsilon"}) is satisfied for all $z\in \mathbb{C}$ such that $|z|\geq R_0$. Therefore $$\forall n\geq N :\quad |z|\geq R_0\quad \Longrightarrow \quad |p_n(z)| \geq e|z|.$$ Now, by Lemma [Lemma 16](#lem: theta){reference-type="ref" reference="lem: theta"} $$\forall j\in\{2,3,...,N-1\}\ \exists R_j>0:\quad |z|\geq R_j\quad \Longrightarrow \quad |p_j(z)|\geq e|z|.$$ It suffices to take $R:=\max\{R_j: j\in\{0,2,3,...,N-1\}\}.$ ◻ **Corollary 18**. *If $K\subset \mathbb{C}$ is a regular polynomially convex compact set and $(p_n)_{n=1}^\infty$ is a sequence of KW polynomials associated with $K$, then $$\exists \varrho>0\; \forall R\geq \varrho\; \forall n\geq 2:\quad p_n^{-1}\left(\overline{\mathbb{D}}(0,R)\right)\subset \overline{\mathbb{D}}(0,R).$$* *Proof.* It follows from the previous proposition that $$\exists \varrho>0\ \forall R>\varrho \ \forall n\geq 2:\quad |z|\geq R\quad \Longrightarrow \quad |p_n(z)|\geq e|z|,$$ since $|z|\geq R\Longrightarrow |z|\geq \varrho.$ In particular for a fixed $R>\varrho$ $$|z|\geq R\quad \Longrightarrow\quad |p_n(z)|>R.$$ ◻ *Remark 19*. The Proposition [Proposition 17](#prop: tnzez){reference-type="ref" reference="prop: tnzez"} and the Corollary [Corollary 18](#cor:przeciwobrazkola){reference-type="ref" reference="cor:przeciwobrazkola"} have some similarities to [@CHPP Proposition 3.3], where a different asymptotic condition on polynomials was assumed. We do not know of any examples of sequences of polynomials satisfying that condition which would not be KW polynomials. At the end of this section we want to note the following characterization of a compact polynomially convex set via KW polynomials associated with it. Recall that set $Z'$, defined in the proposition below, appeared also in the previous subsection (in the context of the convergence in ([\[eq: KWgreen\]](#eq: KWgreen){reference-type="ref" reference="eq: KWgreen"})). **Proposition 20**. *Let $K\subset \mathbb{C}$ be a regular polynomially convex compact set and let $(p_n)_{n=1}^\infty$ be a sequence of KW polynomials associated with $K$. Let $Z'$ be a set of the limit points of the set of the zeros of all $p_n$ contained in $\mathbb{C}\setminus K$. If $Z'=\emptyset$, then $K=\bigcap_{n \geq 1}Z_n,$ where $$Z_n =\{z \in \mathbb{C}: |p_n(z)| \leq \| p_n\|_K\}, \quad n\in\{1,2,...\}.$$* *Proof.* The inclusion $K \subset \bigcap_{n \geq 1}Z_n$ follows from polynomial convexity of $K$. Assume now that $z\in \bigcap_{n \geq 1}Z_n\setminus K$. By the definition of $Z_n$ $$\forall n\in\{1,2,...\}:\quad |p_n(z)|^{1/n}\leq \|p_n\|_K^{1/n}.$$ By ([\[eq: normy\]](#eq: normy){reference-type="ref" reference="eq: normy"}) and its consequence ([\[eq: zbieznosc\]](#eq: zbieznosc){reference-type="ref" reference="eq: zbieznosc"}), taking the limit as $n \to \infty$ we see that ${\rm cap}(K)\cdot \exp(g_K(z)) \leq {\rm cap}(K)$, hence $g_K(z)\leq 0$. This is a contradiction with the polynomial convexity of $K$. ◻ # Polynomial Julia type sets ## Autonomous Julia sets **Definition 21**. Let $P:\mathbb{C}\longrightarrow \mathbb{C}$ be a polynomial of degree $d\geq 2$. The *(autonomous) filled Julia set* of $P$ is $$\begin{aligned} {\mathcal K}[P]:=&\ \{z\in\mathbb{C}:\; (P^n(z))_{n=1}^\infty \text{ is bounded}\}.\end{aligned}$$ It is well known that the filled Julia set is nonempty, compact, perfect, totally invariant under $P$ (i.e., $P({\mathcal K}[P])=P^{-1}({\mathcal K}[P])={\mathcal K}[P]$) and moreover $$\begin{aligned} \label{e:infty} {\mathcal K}[P]:= \mathbb{C}\setminus \left\{z\in\mathbb{C}:\: \lim_{n\rightarrow \infty} P^n(z)= \infty\right\}.\end{aligned}$$ For more information, see [@CGbook]. *Example 22*. $$\forall n\geq 2:\quad {\mathcal K}[z\longmapsto z^n]=\overline{\mathbb{D}}(0,1)\quad \text{ and } \quad {\mathcal K}[T_n]=[-1,1],$$ where $T_n$ is the $n$th classical Chebyshev polynomial for $n\geq 2$. Note that polynomials of degree 1 could be included in Definition [Definition 21](#def:filledJulia){reference-type="ref" reference="def:filledJulia"}. However, then for $z\longmapsto z$, we would obtain the whole complex plane and for $z\longmapsto z+1$ the empty set as the filled Julia sets, so we would loose two nice properties of Julia sets. This is why the definition is restricted to polynomials of degree at least 2.\ Let us define the notion of escape radius of a polynomial, following [@KK18 page 53] (a slightly different definition was proposed in [@Douady]). **Definition 23**. An *escape radius* for a polynomial $P:\mathbb{C}\longrightarrow \mathbb{C}$ is a number $R>0$ with the following property $$|z|>R\quad \Longrightarrow \quad \lim_{n\rightarrow \infty} P^n(z)= \infty.$$ **Lemma 24**. *$R>0$ is an escape radius for a polynomial $P$ if and only if ${\mathcal K}[P]\subset \overline{\mathbb{D}}(0,R).$ In particular, if $R$ is an escape radius for $P$ and $\varrho>R$, then $\varrho$ is an escape radius for $P$ too.* *Proof.* This follows from the definition and from ([\[e:infty\]](#e:infty){reference-type="ref" reference="e:infty"}). ◻ **Corollary 25**. *If $R>0$ is an escape radius for a polynomial $P$, then $${\mathcal K}[P]=\bigcap_{n=1}^{\infty}P^{-n}\left(\overline{\mathbb{D}}(0,R)\right).$$* *Proof.* Let $R>0$ be such that ${\mathcal K}[P]\subset \overline{\mathbb{D}}(0,R)$. Then for every $n \geq 1$ we have $\mathcal{K}[P]=P^{-n}(\mathcal{K}[P]) \subset P^{-n}(\overline{\mathbb{D}}(0,R))$. Conversely, if $z \in P^{-n}\left(\overline{\mathbb{D}}(0,R)\right)$ for every $n \geq 1$, then $(P^n(z))_{n=1}^{\infty} \subset \overline{\mathbb{D}}(0,R)$, hence $z \in {\mathcal K}[P]$. ◻ A sufficient condition for a positive number to be an escape radius for a polynomial was given in Lemma [Lemma 16](#lem: theta){reference-type="ref" reference="lem: theta"}. ## Non-autonomous Julia sets We will consider now a generalization of the autonomous filled Julia set. As in Definition [Definition 21](#def:filledJulia){reference-type="ref" reference="def:filledJulia"} here also usually only polynomials of degree greater than 1 are considered (cf. e.g. [@B97], [@KK18]), however since we are interested in sequences of KW polynomials, we will allow one polynomial to be of degree 1. **Definition 26**. Let $(d_n)_{n=1}^\infty$ be a sequence of integers such that $d_1\geq 1$ and $\forall n\geq 2: d_n\geq 2.$ Let $(p_n)_{n=1}^\infty$ be a sequence of polynomials with $\deg p_n=d_n$. We define the *(non-autonomous) filled Julia set* of the sequence $(p_n)_{n=1}^\infty$ to be $$\begin{aligned} {\mathcal K}[(p_n)_{n=1}^\infty]:=&\ \{z\in\mathbb{C}:\; ((p_n\circ\dots\circ p_1)(z))_{n=1}^\infty \text{ is bounded}\}.\end{aligned}$$ *Remark 27*. It is straightforward that in the situation from the definition $${\mathcal K}[(p_n)_{n=1}^\infty]=p_1^{-1}\left({\mathcal K}[(p_n)_{n=2}^\infty]\right).$$ In particular if $p_1:\mathbb{C}\ni z \longmapsto z\in\mathbb{C}$, then ${\mathcal K}[(p_n)_{n=1}^\infty]={\mathcal K}[(p_n)_{n=2}^\infty]$. Note that it follows from Definition [Definition 26](#def:nonautonomous){reference-type="ref" reference="def:nonautonomous"} that $${\mathcal K}[(p_n)_{n=1}^\infty]=\bigcup_{r\in \mathbb{N}}\bigcap_{n\geq 1}(p_n\circ...\circ p_1)^{-1}\left(\overline{\mathbb{D}}(0,r)\right),$$ hence ${\mathcal K}[(p_n)_{n=1}^\infty]$ is of $F_\sigma$-type. A non-autonomous filled Julia set may be finite, which is impossible for an autonomous one. As shown in [@B97], if we take $p_n:\mathbb{C}\ni z \longmapsto n^{2^n}z^2\in\mathbb{C}, n\geq 1$, then ${\mathcal K}[(p_n)_{n=1}^\infty]=\{0\}$. Similarly, if we just exchange the first polynomial taking $q_1:\mathbb{C}\ni z\longmapsto z^2-1\in\mathbb{C}$ and $\forall n\geq 2: q_n:=p_n$, then ${\mathcal K}[(q_n)_{n=1}^\infty]=\{-1,1\}$. If $(p_n)_{n=1}^\infty$ in Definition [Definition 26](#def:nonautonomous){reference-type="ref" reference="def:nonautonomous"} above is periodic (i.e., there exists an $m$ such that $p_{m+i}=p_i$ for every $i$), then we obtain the autonomous filled Julia set (see Definition [Definition 21](#def:filledJulia){reference-type="ref" reference="def:filledJulia"}) of the composition of the polynomials $p_1,...,p_m$. In particular for a constant sequence we also get an autonomous Julia set. The interesting case is when the sequence is not periodic, e.g., when each polynomial in the sequence has a different degree. *Example 28*. If $(d_n)_{n=1}^\infty$ is a sequence of integers not smaller than 2 and $p_n:\mathbb{C}\ni z \longmapsto z^{d_n}$, then ${\mathcal K}[(p_n)_{n=1}^\infty]=\overline{\mathbb{D}}(0,1)$. Furthermore ${\mathcal K}[(T_{d_n})_{n=1}^\infty]=[-1,1]$. This follows from Example [Example 22](#ex: filled){reference-type="ref" reference="ex: filled"} and the composition formulae $p_n\circ p_k=p_{nk}$ and $T_n\circ T_k=T_{nk}$. In particular ${\mathcal K}[(z\longmapsto z^n)_{n=1}^\infty]=\overline{\mathbb{D}}(0,1)$ and ${\mathcal K}[(T_n)_{n=1}^\infty]=[-1,1]$. The following example is a consequence of the previous one and of Remark [Remark 27](#rem:pierwszy){reference-type="ref" reference="rem:pierwszy"}. *Example 29*. For any non-constant polynomial $P:\mathbb{C}\longrightarrow \mathbb{C}$, the sets $P^{-1}\left(\overline{\mathbb{D}}(0,1\right))$ and $P^{-1}\left([-1,1]\right)$ are (non-autonomous) filled Julia sets. The authors of [@BB03] define non-autonomous Julia sets for special sequences of polynomials. Namely, they define a class ${\mathcal B}$ of sequences $(f_n)_{n=1}^\infty$, where $$f_n(z)=\sum_{j=1}^{d_n} a_{n,j}z^j$$ and $d_n\geq2$ for any $n$. The sequences in this class satisfy some conditions, in particular 1. there is a constant $A\geq 0$ such that $|a_{n,j}|\leq A|a_{n,d_n}|$ for $j\in\{0,...,d_n\}$ and all integers $n$. However, this condition does not hold for some important sequences of polynomials. *Remark 30*. The sequence of the Chebyshev polynomials on $[-1,1]$ does not belong to the class $\mathcal{B}$. *Proof.* Recall that $T_n$ is the classical Chebyshev polynomial of degree $n$ and the $n$th Chebyshev polynomial on $[-1,1]$ is $\frac{1}{2^{n-1}}T_n$ (see Example [Example 11](#ex:Td){reference-type="ref" reference="ex:Td"}). We may write $T_n(z)=2^{n-1}z^n+a_{n-1}^{(n)}z^{n-1}+...+a_0^{(n)}$. Note that $a_{n-1}^{(n)}=0$ for every $n \geq 1$. First we will prove that $a_{n-2}^{(n)}=-n2^{n-3}$ for $n\in\{1,2,...\}$. Indeed, this is true for $T_1(z)=z, \ T_2(z)=2z^2-1, \ T_3(z)=4z^3-3z$. To argue by induction, let $n$ be such that $a_{k-2}^{(k)}=-k2^{k-3}$ for $k\in\{2,3,...,n\}$. From the recurrence formula $T_{n+1}(z)=2zT_n(z)-T_{n-1}(z)$ (valid for all $n \geq 1$, if we set $T_0(z)\equiv 1$) we can express $T_{n+1}(z)$ as $$\begin{aligned} 2z\left(2^{n-1}z^n+(-n2^{n-3})z^{n-2}+a_{n-3}^{(n)}z^{n-3}+...+a_0^{(n)}\right)+\\-\left(2^{n-2}z^{n-1}+(-(n-1))2^{n-4}z^{n-3}+a_{n-4}^{(n-1)}z^{n-4}+...+a_0^{(n-1)}\right).\end{aligned}$$ Then $a_{n-1}^{(n+1)}=-n2^{n-2}-2^{n-2}=-(n+1)2^{n-2}$, as claimed. Thus for the polynomial $\frac{1}{2^{n-1}}T_n$ the coefficient corresponding to $z^{n-2}$ is $-n/4$. Hence condition $(P2)$ does not hold and $\left(\frac{1}{2^{n-1}}T_n\right)_{n=1}^\infty\notin \mathcal{B}.$ ◻ For the explicit expression of coefficients of $T_n$ corresponding to the powers of the variable $z$ see [@Pasz75 T1.2 (40)].\ Remark [Remark 30](#rem:notinB){reference-type="ref" reference="rem:notinB"} shows that the approach from [@BB03] cannot be used in the study of non-autonomous Julia sets for the sequences of Chebyshev polynomials on compact sets. Another approach was proposed in [@KK18]. There all polynomials from the sequence $(p_n)_{n=1}^\infty$ have to have a common escape radius $R$ such that $$\label{e:suppn}\sup_{n}\|p_n\|_{\overline{\mathbb{D}}(0,R)}<\infty.$$ In this case ${\mathcal K}[(p_n)_{n=1}^\infty]$ is compact, non-empty and has some better properties too. Let us however note that no proof is provided in [@KK18] and one has to follow ideas from [@KK03], which are presented in quite a specific case of polynomials of the same degree. Note first that not all sequences have the property of common escape radius. For instance there is no common escape radius for the sequence $(z\longmapsto z^2-n)_{n=1}^\infty$. One can namely check that the smallest escape radius for polynomial $z\longmapsto z^2-n$ is $\frac12+\sqrt{\frac14+n}$ (see also Example [Example 48](#ex:escapec){reference-type="ref" reference="ex:escapec"}). On the other hand, even if there is a common escape radius, the condition ([\[e:suppn\]](#e:suppn){reference-type="ref" reference="e:suppn"}) does not have to be satisfied, which can be seen in the following example. If $p_1:\mathbb{C}\ni z \longmapsto z^2-2\in\mathbb{C}$, then ${\mathcal K}[p_1]=[-2,2]$, therefore in view of Lemma [Lemma 24](#lem:escaperadius){reference-type="ref" reference="lem:escaperadius"} the smallest escape radius for $p_1$ is 2. Take now $p_n:\mathbb{C}\ni z\longmapsto z^n\in\mathbb{C}$ for $n\geq 2$. It is obvious that $R$ is a common escape radius for thus defined sequence $(p_n)_{n=1}^\infty$ if and only if $R\geq 2$. However $\sup_n\|p_n\|_{\overline{\mathbb{D}}(0,2)}\geq \sup_n2^n=\infty$. It might be hence difficult to follow the idea from [@KK18]. Moreover, in the case of polynomials of different degrees one has to totally rewrite the proof from [@KK03]. Therefore we prefer to present our whole route, for the completeness of the article, without referring to that case. The following theorem is our first result about the non-autonomous Julia set defined with use of a sequence of KW polynomials. **Theorem 31**. *Let $K\subset \mathbb{C}$ be a regular polynomially convex compact set and $(p_n)_{n=1}^\infty$ be a sequence of KW polynomials associated with $K$. If $(d_k)_{k=1}^\infty$ is a sequence of integers not smaller than $2$, then ${\mathcal K}[(p_{d_k})_{k=1}^\infty]$ is nonempty and compact.* *Moreover ${\mathcal K}[(p_{n})_{n=1}^\infty]$ is nonempty and compact too.* *Proof.* Let $R$ be as in Proposition [Proposition 17](#prop: tnzez){reference-type="ref" reference="prop: tnzez"}. We have $$|z|\geq R\quad \Longrightarrow \quad |(p_{d_k}\circ...\circ p_{d_1})(z)|\geq e^k|z|\longrightarrow \infty, \text{ if } k\rightarrow \infty.$$ Therefore ${\mathcal K}[(p_{d_k})_{k=1}^\infty]=\bigcap_{k=1}^{\infty}(p_{d_k}\circ...\circ p_{d_1})^{-1}\left( \overline{\mathbb{D}}(0,R)\right)$. It follows that ${\mathcal K}[(p_{n})_{n=2}^\infty]$ is nonempty and compact too. For the additional assertion it suffices to use Remark [Remark 27](#rem:pierwszy){reference-type="ref" reference="rem:pierwszy"}. ◻ ## Klimek's metric In our further investigation we will need some results for the space with Klimek's metric. We use the following notation $$\begin{aligned} {\mathcal R}&={{\mathcal R}}(\mathbb{C})=\\&:= \{K\subset \mathbb{C}:\; K \text{ is compact,} \text{ regular and polynomially convex}\}.\end{aligned}$$ For $E, F\in {{\mathcal R}}$ Klimek defined in [@Kl95] their distance $$\label{e:Gamma} \Gamma (E,F):=\sup_{z\in\mathbb{C}}|g_E(z)-g_F(z)|=\max\left(\sup_{z\in E} g_F(z) , \sup_{z\in F} g_E(z) \right)$$ and showed that $({{\mathcal R}},\Gamma)$ is a complete metric space. Note that a sequence $(E_n)_{n=1}^\infty$ is convergent to $F$ in $({{\mathcal R}},\Gamma)$ if and only if $g_{E_n}\rightrightarrows g_F$, i.e. the function sequence $(g_{E_n})_{n=1}^\infty$ is uniformly convergent to $g_F$ in the whole complex plane.\ Fix now a polynomial $P$ of degree $d\geq 1$ and consider the following mapping: $$\label{e:A_P} A_P: {\mathcal R}\ni K\longmapsto P^{-1}(K)\in {\mathcal R}.$$ ([\[e:Greenaprzeciwobrazu\]](#e:Greenaprzeciwobrazu){reference-type="ref" reference="e:Greenaprzeciwobrazu"}) yields that this mapping is an isometry if $d=1$ (since $f$ is bijective) and a contraction with contraction ratio $1/d$ if $d\geq 2$. In the latter case, since $({\mathcal R},\Gamma)$ is a complete metric space, by Banach Contraction Principle the mapping $A_P$ has a unique fixed point. This fixed point is the above defined (Definition [Definition 21](#def:filledJulia){reference-type="ref" reference="def:filledJulia"}) filled Julia set ${\mathcal K}[P]$ (see [@Kl95]). In particular ${\mathcal K}[P]\in {\mathcal R}$. Moreover, by the classical proof of Banach Contraction Principle $$\forall E\in{\mathcal R}:\quad \lim_{n\rightarrow \infty} P^{-n}(E)=\lim_{n\rightarrow \infty}(A_P)^n(E)= {\mathcal K}[P].$$ Once again using ([\[e:Greenaprzeciwobrazu\]](#e:Greenaprzeciwobrazu){reference-type="ref" reference="e:Greenaprzeciwobrazu"}) we deduce that $$\forall E\in{\mathcal R}:\quad \frac1{d^n}g_E\circ P^n \rightrightarrows g_{{\mathcal K}[P]}.$$ Now we would like to use the Klimek metric in our case of KW polynomials. The following proposition is an important step in proving that the compact sets ${\mathcal K}[(p_{d_k})_{k=1}^\infty]$ and ${\mathcal K}[(p_{n})_{n=1}^\infty]$ obtained in Theorem [Theorem 31](#thm:JuliadlaKW){reference-type="ref" reference="thm:JuliadlaKW"} are regular and polynomially convex. **Proposition 32**. *Let $K\subset \mathbb{C}$ be a regular polynomially convex compact set and $(p_n)_{n=1}^\infty$ be a sequence of KW polynomials associated with $K$. If $E\in{\mathcal R}$, then $$\exists C>0\ \forall n\geq 1: \quad \Gamma(E,p_n^{-1}(E))\leq C.$$* *Proof.* Fix $E\in{\mathcal R}$. Take $\varrho$ from Corollary [Corollary 18](#cor:przeciwobrazkola){reference-type="ref" reference="cor:przeciwobrazkola"}, fix $R\geq \varrho$ big enough to satisfy $E\subset \overline{\mathbb{D}}(0,R)$. By Corollary [Corollary 18](#cor:przeciwobrazkola){reference-type="ref" reference="cor:przeciwobrazkola"} $$p_n^{-1}(E)\subset p_n^{-1}\left(\overline{\mathbb{D}}(0,R)\right)\subset \overline{\mathbb{D}}(0,R) \quad \text{ for } n\geq 2.$$ Hence $$\label{e:C_1} \forall n\geq 2:\qquad \sup_{z\in p_n^{-1}(E)}g_E(z)\leq \sup_{z\in \overline{\mathbb{D}}(0,R)} g_E(z)=:C_1.$$ Note that $C_1$ is a non-negative number and does not depend on $n$. By the properties of Green's function $C_2:=\max_{z\in\overline{\mathbb{D}}(0,R)}g_K(z)$ is well defined and non-negative. By assumption $(p_n)_{n=1}^\infty$ satisfies ([\[eq: normy\]](#eq: normy){reference-type="ref" reference="eq: normy"}). Therefore there exists $C_3>1$ such that $\forall n\geq 1:\quad \|p_n\|_K^{1/n}\leq C_3$. Hence by ([\[e:BW\]](#e:BW){reference-type="ref" reference="e:BW"}) $$\label{e:szacowaniezBW} \forall n\geq 1 \ \forall z\in E : \quad |p_n(z)|\leq C_4^n,$$ where $C_4=C_3\exp(C_2)>1$. In view of ([\[e:Greenaprzeciwobrazu\]](#e:Greenaprzeciwobrazu){reference-type="ref" reference="e:Greenaprzeciwobrazu"}) we have $$\begin{aligned} \nonumber \forall n\geq 1: \quad \sup_{z\in E}g_{p_n^{-1}(E)}(z)&=\sup_{z\in E} \frac1ng_E(p_n(z))\leq\\&\leq \sup_{z\in \overline{\mathbb{D}}(0,C_4^n)}\frac1ng_E(z)=\sup_{z\in \partial\overline{\mathbb{D}}(0,C_4^n)}\frac1ng_E(z), \label{e:szacowanieGreenaprzeciwobrazu}\end{aligned}$$ and the last equality follows from the Maximum Principle. By Lemma [Lemma 6](#lem:Greennierownosci){reference-type="ref" reference="lem:Greennierownosci"} there exists $N_1> 1$ such that for $n\geq N_1$ $$\forall z\in \partial\overline{\mathbb{D}}(0,C_4^n):\quad g_E(z)\leq n\log C_4-\log {\rm cap}(E)+1.$$ And furthermore there exists $N_2\geq N_1$ such that for $n\geq N_2$ $$\forall z\in \partial\overline{\mathbb{D}}(0,C_4^n):\quad \frac1ng_E(z)\leq 2+\log C_4.$$ Combining that with ([\[e:szacowanieGreenaprzeciwobrazu\]](#e:szacowanieGreenaprzeciwobrazu){reference-type="ref" reference="e:szacowanieGreenaprzeciwobrazu"}) gives $$\begin{aligned} \label{e:C_5} \forall n\geq N_2: \quad \sup_{z\in E}g_{p_n^{-1}(E)}(z)\leq C_5:=2+\log C_4.\end{aligned}$$ Put also $C_6:=\max\left\{\Gamma(E,p_n^{-1}(E)): n\in\{1,...,N_2-1\}\right\}.$ We see that $$\forall n\geq 1: \quad \Gamma(E,p_n^{-1}(E))\leq \max\{C_1,C_5,C_6\}.$$ ◻ Some properties of the filled Julia set of a polynomial of degree at least 2 (see Definition [Definition 21](#def:filledJulia){reference-type="ref" reference="def:filledJulia"}) followed from the Banach Contraction Principle. Now we need a generalization of this result. **Theorem 33** (Enhanced version of Banach's Contraction Principle, [@KK03 Lemma 4.5]). *Let $(X, \rho)$ be a complete metric space and let $(H_n)_{n=1}^\infty$ be a sequence of contractions of X with contraction ratios not greater than $L<1$. If $$\forall x\in X:\quad \sup_{n\geq 1}\rho(H_n(x),x)<\infty,$$ then there exists a unique point $c\in X$ such that the sequence $(H_1\circ...\circ H_n)_{n=1}^\infty$ converges pointwise to $c$.* Let us also quote the following result. **Proposition 34** ([@Kl01 Proposition 1]). *Let $P_n:\mathbb{C}\longrightarrow \mathbb{C}$ be a polynomial of degree $d_n\geq 2$ for $n\in\{1,2,...\}$. Let $E\in{\mathcal R}$ and define $E_n:=(P_n\circ...\circ P_1)^{-1}(E)$ for $n\in\{1,2,...\}$. If $$\label{e:sum} \sum_{n=1}^\infty \frac{\Gamma(P_{n+1}^{-1}(E), E)}{d_1d_2\cdots d_n}<\infty,$$ then the sequence $(E_n)_{n=1}^\infty$ is convergent in $({\mathcal R},\Gamma)$ to a set $F$. Any other choice of $\widetilde{E}\in {\mathcal R}$ for which $(\ref{e:sum})$ is satisfied, results in the same limit $F$. If we assume that $P_n^{-1}(E)\subset E$ for all $n$, then the sequence $(E_n)_{n=1}^\infty$ is decreasing and $$%\label{e:product} F=\bigcap_{n\geq 1} E_n=\{z\in E: \; (P_n\circ...\circ P_1)(z)\in E \text{ for all }n\geq 1\}.$$* ## Julia sets of sequences of KW polynomials We will now apply the results from the previous subsection to a sequence of contractions of the type defined in ([\[e:A_P\]](#e:A_P){reference-type="ref" reference="e:A_P"}). **Theorem 35**. *Let $K\subset \mathbb{C}$ be a regular polynomially convex compact set and $(p_n)_{n=1}^\infty$ be a sequence of KW polynomials associated with $K$. Fix now a sequence $(d_k)_{k=1}^\infty$ of integers greater than $1$. Then the sequence $$\left(A_{p_{d_1}}\circ...\circ A_{p_{d_k}}\right)_{n=1}^\infty$$ converges pointwise in ($\mathcal{R},\Gamma)$ to a constant mapping with the value ${\mathcal K}[(p_{d_k})_{k=1}^\infty]$.* *In particular ${\mathcal K}[(p_{d_k})_{k=1}^\infty]$ and ${\mathcal K}[(p_n)_{n=1}^\infty]$ are polynomially convex and regular.* *Proof.* By Theorem [Theorem 33](#th: EnhancedBCP){reference-type="ref" reference="th: EnhancedBCP"} and Proposition [Proposition 32](#prop:GammaEiprzeciwobrazu){reference-type="ref" reference="prop:GammaEiprzeciwobrazu"}, for every $E \in \mathcal{R}$ the sequence $$\left(A_{p_{d_1}}\circ...\circ A_{p_{d_k}}(E)\right)_{n=1}^\infty$$ is convergent to the same set $F\in{\mathcal R}$. Take $\varrho>0$ from Corollary [Corollary 18](#cor:przeciwobrazkola){reference-type="ref" reference="cor:przeciwobrazkola"}. Proposition [Proposition 34](#p:nagoya){reference-type="ref" reference="p:nagoya"} yields $$\forall R\geq \varrho:\quad F= \bigcap_{k=1}^\infty (p_{d_k}\circ...\circ p_{d_1})^{-1}\left(\overline{\mathbb{D}}(0,R)\right) .$$ Hence $F={\mathcal K}[(p_{d_k})_{k=1}^\infty]$ by Theorem [Theorem 31](#thm:JuliadlaKW){reference-type="ref" reference="thm:JuliadlaKW"}. In order to get the assertion for ${\mathcal K}[(p_n)_{n=1}^\infty]$ it suffices to use Remark [Remark 27](#rem:pierwszy){reference-type="ref" reference="rem:pierwszy"} to ${\mathcal K}[(p_{n})_{n=2}^\infty]$. ◻ **Corollary 36**. *Let $K\subset \mathbb{C}$ be a regular polynomially convex compact set and $(p_n)_{n=1}^\infty$ be a sequence of KW polynomials associated with $K$. Fix now a sequence $(d_k)_{k=1}^\infty$ of integers greater than 1. Then $$\forall E\in {\mathcal R}:\quad g_{(p_{d_n}\circ...\circ p_{d_1})^{-1}(E)}\rightrightarrows g_{{\mathcal K}\left[(p_{d_n})_{n=1}^\infty\right]}.$$* *In particular the function sequence $$\left(\frac1{d_n\cdot...\cdot d_1}\log^+\left|p_{d_n}\circ...\circ p_{d_1}\right|\right)_{n=1}^\infty$$ is uniformly convergent in $\mathbb{C}$.* *Proof.* It follows directly from Theorem [Theorem 35](#thm:zbiornieautonomiczny){reference-type="ref" reference="thm:zbiornieautonomiczny"}, Definition [Definition 26](#def:nonautonomous){reference-type="ref" reference="def:nonautonomous"} of the filled Julia set ${\mathcal K}\left[(p_{d_n})_{n=1}^\infty\right]$ and of Klimek's metric $\Gamma$. The last assertion is a consequence of ([\[e:Greenaprzeciwobrazu\]](#e:Greenaprzeciwobrazu){reference-type="ref" reference="e:Greenaprzeciwobrazu"}) and the formula for the Green function of the unit disk (cf. Example [Example 3](#ex: koloGreen){reference-type="ref" reference="ex: koloGreen"}). ◻ We will now consider ${\mathcal K}[(p_n)_{n=1}^\infty]$. **Corollary 37**. *Let $K\subset \mathbb{C}$ be a regular polynomially convex compact set and $(p_n)_{n=1}^\infty$ be a sequence of KW polynomials associated with $K$. Then the function sequence $$\left(\frac1{n!}\log^+\left|p_n\circ...\circ p_1\right|\right)_{n=1}^\infty$$ is uniformly convergent in $\mathbb{C}$ to $g_{{\mathcal K}[(p_n)_{n=1}^\infty]}$.* *Proof.* By the previous corollary $$g_{(p_n\circ...\circ p_2)^{-1}\left(\overline{\mathbb{D}}(0,1)\right)}\rightrightarrows g_{{\mathcal K}[(p_n)_{n=2}^\infty]}.$$ Recall the formula ([\[e:Greenaprzeciwobrazu\]](#e:Greenaprzeciwobrazu){reference-type="ref" reference="e:Greenaprzeciwobrazu"}) and note that $p_1:\mathbb{C}\longrightarrow \mathbb{C}$ is bijective. Since $\deg p_1=1$, we have $$\begin{aligned} \frac1{n!}\log^+\left|p_n\circ...\circ p_1\right|&=g_{(p_n\circ...\circ p_1)^{-1}\left(\overline{\mathbb{D}}(0,1)\right)}=\\&=g_{(p_n\circ...\circ p_2)^{-1}\left(\overline{\mathbb{D}}(0,1)\right)}\circ p_1\rightrightarrows g_{{\mathcal K}[(p_n)_{n=2}^\infty]}\circ p_1=\\&\qquad \qquad =g_{p_1^{-1}\left({\mathcal K}[(p_n)_{n=2}^\infty]\right)}=g_{{\mathcal K}[(p_n)_{n=1}^\infty]}. \end{aligned}$$ ◻ The following approximation of the non-autonomous Julia set by the autonomous Julia sets of compositions can be easily shown (cf. [@AKK Proposition 5]). **Corollary 38**. *Let $K\subset \mathbb{C}$ be a regular polynomially convex compact set and $(p_n)_{n=1}^\infty$ be a sequence of KW polynomials associated with $K$. Fix now a sequence $(d_k)_{k=1}^\infty$ of integers greater than $1$. Then $$\lim_{k\rightarrow \infty} \Gamma\left({\mathcal K}[p_{d_k}\circ...\circ p_{d_1}], {\mathcal K}[(p_{d_k})_{k=1}^\infty]\right)=0.$$* We recall now another result due to Klimek. **Theorem 39** ([@Kl95 Corollary 5]). *$$\forall E,F\in {\mathcal R}:\quad \left|\log{\rm cap} (E)-\log{\rm cap}(F)\right|\leq \Gamma(E,F).$$ In particular the logarithmic capacity is continuous on $({\mathcal R},\Gamma)$.* Recall that ${\rm cap}\left(\overline{\mathbb{D}}(0,1)\right)=1$, moreover because of ([\[e:Greenaprzeciwobrazu\]](#e:Greenaprzeciwobrazu){reference-type="ref" reference="e:Greenaprzeciwobrazu"}) we have $$\label{e:capprzeciwobrazukola} {\rm cap}\left(f^{-1}\left(\overline{\mathbb{D}}(0,1)\right)\right)=1$$ for any non constant monic polynomial $f$ too. **Corollary 40**. *Let $K\subset \mathbb{C}$ be a regular polynomially convex compact set and $(p_n)_{n=1}^\infty$ be a sequence of KW polynomials associated with $K$. Let $(d_k)_{k=1}^\infty$ be a sequence of integers greater than $1$.* *Then ${\rm cap}\left({\mathcal K}[(p_{d_k})_{k=1}^\infty]\right)= {\rm cap}\left({\mathcal K}[(p_{n})_{n=1}^\infty]\right)=1$.* *Proof.* By Theorem [Theorem 35](#thm:zbiornieautonomiczny){reference-type="ref" reference="thm:zbiornieautonomiczny"} $$\left(p_{d_{n}}\circ...\circ p_{d_1}\right)^{-1}\left(\overline{\mathbb{D}}(0,1)\right)\longrightarrow {\mathcal K}\left[(p_{d_{k}})_{k=1}^\infty\right]\quad (n\rightarrow \infty)$$ with respect to $\Gamma$. The assertion follows from ([\[e:capprzeciwobrazukola\]](#e:capprzeciwobrazukola){reference-type="ref" reference="e:capprzeciwobrazukola"}) and Theorem [Theorem 39](#th:Klimekcap){reference-type="ref" reference="th:Klimekcap"}. ◻ At the end of this subsection we give some information about our toy case of Chebyshev polynomials on $[-1,1]$. *Example 41*. The following pictures (prepared by Maciej Klimek) show approximations of ${\mathcal K}[(t_n)_{n=1}^\infty]$, where $(t_n)_{n=1}^\infty$ is the sequence of minimal polynomials on $[-1,1]$ (see Example [Example 11](#ex:Td){reference-type="ref" reference="ex:Td"}). The sets depicted here are, from left to right: - $(t_8\circ...\circ t_2\circ t_1)^{-1}([-1,1]\times [-0.0005,0.0005])$, (which is used as an approximation of $(t_8\circ...\circ t_2\circ t_1)^{-1}([-1,1])$), - $(t_5\circ...\circ t_2\circ t_1)^{-1}\left(\overline{\mathbb{D}}(0,1)\right)$, - $(t_{100}\circ...\circ t_2\circ t_1)^{-1}\left(\overline{\mathbb{D}}(0,1)\right)$. {width="4 cm"}{width="4 cm"}{width="4 cm"} Observe some simple geometric properties of the set ${\mathcal K}[(t_n)_{n=1}^\infty]$.\ [Property 1:]{.ul} $I=[-1,1] \subset {\mathcal K}[(t_n)_{n=1}^\infty]$.\ In particular, ${\mathcal K}[(t_n)_{n=1}^\infty] \cap \{z: {\rm Im}z=0\}\neq \emptyset$. *Proof.* The set $I$ is totally invariant under every polynomial $T_n$. Hence $t_1(I)=T_1(I)=I$, $t_2(I)=(1/2)T_2(I)=[-1/2,1/2] \subset I$, $t_n(I)=(1/2^{n-1})T_n(I)=[-1/2^{n-1},1/2^{n-1}] \subset I$, and $(t_n\circ...\circ t_2\circ t_1)(I) \subset I$ for every $n \geq 1$. ◻ [Property 2:]{.ul} If $z \in {\mathcal K}[(t_n)_{n=1}^\infty]$, then $-z \in {\mathcal K}[(t_n)_{n=1}^\infty]$ and $\bar{z} \in {\mathcal K}[(t_n)_{n=1}^\infty]$. *Proof.* We have $t_1(z)=z$ and $t_2(-z)=t_2(z)$ for every $z \in \mathbb{C}$. Hence $(t_n\circ...\circ t_2\circ t_1)(-z)=(t_n\circ...\circ t_2\circ t_1)(z)$ for every $n \geq 2$ and every $z \in \mathbb{C}$. Moreover, all $t_n$ have real coefficients, so $(t_n\circ...\circ t_1)(\overline{z})=\overline{(t_n\circ...\circ t_1)(z)}$ for every $n \geq 2$ and every $z \in \mathbb{C}$. It follows that each of the sequences $\left((t_n\circ...\circ t_1)(\overline{z})\right)_{n=1}^\infty$ and $\left((t_n\circ...\circ t_1)(-z)\right)_{n=1}^\infty$ is bounded if and only if $\left((t_n\circ...\circ t_1)(z)\right)_{n=1}^\infty$ is. ◻ [Property 3:]{.ul} $\exists R\geq 2:\ {\mathcal K}[(t_n)_{n=1}^\infty]\subset E_R$, where $E_R$ is the filled ellipse with foci $\pm 1$ and semiaxes $a_R=\frac{1}{2}(R+\frac{1}{R}), \ b_R=\frac{1}{2}(R-\frac{1}{R})$. *Proof.* When $R >1$, such (filled) ellipses are sublevel sets (see Definition [Definition 7](#def:sublevel){reference-type="ref" reference="def:sublevel"} and Example [Example 9](#ex:elipsy){reference-type="ref" reference="ex:elipsy"}) of the Green function $g_I$, which tends to infinity as $|z| \to \infty$. Hence we have $\mathbb{C}=\bigcup_{R>1}E_R$, and, by compactness, $\exists R>1: {\mathcal K}[(t_n)_{n=2}^\infty] \subset E_R$. The capacity of $E_R$ is $(a_R+b_R)/2=R/2$ (see Example [Example 5](#ex: elipsyGreen){reference-type="ref" reference="ex: elipsyGreen"}). By monotonicity of capacity and Corollary [Corollary 40](#cor:1){reference-type="ref" reference="cor:1"}, we need $R \geq 2$ for the inclusion ${\mathcal K}[(t_n)_{n=2}^\infty]\subset E_R$. ◻ [Property 4:]{.ul} ${\mathcal K}[(t_n)_{n=1}^\infty]\subset\!\!\!\!\!/ \ E_2$. *Proof.* Note that $E_2$ has the major semiaxis $a_2=5/4$ and the minor semiaxis $b_2=3/4$. We need to find a point in ${\mathcal K}[(t_n)_{n=1}^\infty]\setminus E_2$. Let us start with the following properties of the polynomials $t_n$, $n\in\{1,2,...\}$: 1. $t_n(-x)=(-1)^nt_n(x), \ x \in \mathbb{R}$; 2. $t_n$ is increasing in the interval $(1,+\infty)$; 3. $\max_{z \in E_2}|t_n(z)|=|t_n(5/4)|=|t_n(-5/4)|=1+2^{-2n} \leq 5/4$. \(a\) and (b) are known. To prove (c), let us first compute $\max_{w \in E_2}|T_n(w)|$ (cf. [@Faber], [@Gaier]). Recall that the classical Chebyshev polynomials satisfy the relation $$T_n \left(\frac{z+z^{-1}}{2}\right)=\frac{z^n+z^{-n}}{2},\qquad n\in\{1,2,...\}.$$ For $z=2e^{i\theta}$ with $\theta \in [0,2\pi)$ we thus have $$\begin{aligned} T_n \left(\frac{z+z^{-1}}{2}\right)&=\frac{2^ne^{in\theta}+2^{-n}e^{-in\theta}}{2}\\&=\frac{1}{2}\left((2^n+2^{-n})\cos n\theta +i(2^n-2^{-n})\sin n\theta\right).\end{aligned}$$ Then$$\left|T_n \left(\frac{z+z^{-1}}{2}\right)\right|^2=\frac{1}{4}\left(2^{2n}+2\cos 2n\theta +2^{-2n}\right)$$ achieves its maximum when $\theta =0$ or $\theta=\pi$. Checking values for the corresponding $z=2$ or $z=-2$ we get $$\max_{w \in E_2}\left|T_n(w)\right|=\left|T_n\left(\frac{2+2^{-1}}2\right)\right|=\left|T_n\left(-\frac{2+2^{-1}}2\right)\right|=2^{n-1}+2^{-(n-1)}.$$ Hence (c) is proved. Observe now that Property 1 together with (a), (b) and (c) implies that $[-5/4,5/4] \subset {\mathcal K}[(t_n)_{n=1}^\infty]$. Indeed, for every $n \geq 1$ we have $$t_n\left([-5/4,5/4])=t_n([-5/4,-1]\cup[-1,1]\cup[1,5/4]\right) \subset [-5/4,5/4],$$ consequently $$(t_n\circ...\circ t_1)([-5/4,5/4])\subset [-5/4,5/4]$$ and we get the inclusion $[-5/4,5/4] \subset {\mathcal K}[(t_n)_{n=1}^\infty]$. Consider the point $z_0=4i/5$, which does not belong to $E_2$ (since the minor semiaxis of $E_2$ is $b_2=3/4 < 4/5$). Now, $$t_2(z_0)=z_0^2-1/2=-57/50 \in (-5/4,-1),$$ hence, for every $n \geq 2$ we have $(t_n \circ ...\circ t_2\circ t_1)(z_0) \in [-5/4,5/4]$, and so $z_0 \in {\mathcal K}[(t_n)_{n=1}^\infty]$. ◻ # Chebyshev polynomials on compact sets, revisited ## More notions and examples We will use Definition [Definition 10](#def:nthCheb){reference-type="ref" reference="def:nthCheb"} of minimal polynomials. It is known that for a fixed infinite compact set $E$ and for each $n$ the $n$th Chebyshev polynomial on $E$ is unique (see e.g. [@O Chapter II. Theorem 7] or [@CSZ4 page 2]). Few explicit examples of Chebyshev polynomials are known. We already have Example [Example 11](#ex:Td){reference-type="ref" reference="ex:Td"}. Recall also the following: *Example 42*. Fix an $R>0$. Then $t_n(z)=z^n$ is the $n$th Chebyshev polynomial for $\{z\in\mathbb{C}:\ |z|=R\}$ as well as for $\overline{\mathbb{D}}(0,R):=\{z\in\mathbb{C}: |z|\leq R\}$. *Example 43*. The polynomial $t_n:=\frac{1}{2^{n-1}}T_n$ from Example [Example 11](#ex:Td){reference-type="ref" reference="ex:Td"} is also the $n$th Chebyshev polynomial on any ellipse $E$ with foci $-1,+1$ (see [@Faber]). Recall that these ellipses are level sets of the Green function (cf. Definition [Definition 7](#def:sublevel){reference-type="ref" reference="def:sublevel"}) of the segment $[-1, 1]$ (cf. Example [Example 9](#ex:elipsy){reference-type="ref" reference="ex:elipsy"}). *Example 44*. Let $E \subset \mathbb{C}$ be a compact set, let $p$ be the $n$th Chebyshev polynomial on $E$ and let $f$ be an arbitrary polynomial of degree $m \geq 1$. Then $p\circ f$ is the $(m\cdot n)$th Chebyshev polynomial on $f^{-1}(E)$ (see [@OPZ] or [@BC00]). **Definition 45** ([@OPZ Definition 1]). Let $E \subset \mathbb{C}$ be a compact set. The closed disc $\overline{\mathbb{D}}_C:=\overline{\mathbb{D}}(a,r_C)$ of the smallest radius which contains $E$ is called the *Chebyshev disc* of $E$. Its center $a$ is called the *Chebyshev center* of $E$; and its radius $r_C$ is called the *Chebyshev radius* of $E$. **Lemma 46**. *The point $a \in \mathbb{C}$ is the Chebyshev center of a compact set $E \subset \mathbb{C}$ if and only if $t_1(z)=z-a$ is the first Chebyshev polynomial on $E$.* *Proof.* By Definitions [Definition 10](#def:nthCheb){reference-type="ref" reference="def:nthCheb"} and [Definition 45](#def: chebradius){reference-type="ref" reference="def: chebradius"}, if $a \in \mathbb{C}$ is the Chebyshev center of $E$, then $\|t_1\|_E=r_C$ and conversely, if $t_1(z)=z-a$ is the first Chebyshev polynomial on $E$, then $\overline{\mathbb{D}}(a, \|z-a\|_E)$ is the Chebyshev disc of $E$. ◻ Let us now point out a relation between an escape radius (recall Definition [Definition 23](#def:escape){reference-type="ref" reference="def:escape"}) and the Chebyshev radius for a filled Julia set. **Corollary 47**. *If $P:\mathbb{C}\longrightarrow \mathbb{C}$ is a polynomial of degree $d\geq 2$ and $\overline{\mathbb{D}}(a,r_C)$ is the Chebyshev center of ${\mathcal K}[P]$, then $r_C+|a|$ is an escape radius for $P$.* *Proof.* By Definition [Definition 45](#def: chebradius){reference-type="ref" reference="def: chebradius"} we have ${\mathcal K}[P]\subset \overline{\mathbb{D}}(a,r_C)\subset \overline{\mathbb{D}}(0,r_C+|a|)$. It suffices to apply Lemma [Lemma 24](#lem:escaperadius){reference-type="ref" reference="lem:escaperadius"}. ◻ *Example 48*. Consider now the special case $P_c:\mathbb{C}\ni z\longmapsto z^2+c\in\mathbb{C}$ for $c\in\mathbb{C}$. Note that the Chebyshev center of ${\mathcal K}[P_c]$ is 0, since the set ${\mathcal K}[P_c]$ is symmetric with respect to 0. If additionally $c\in (-\infty,0]$, then the Chebyshev radius of ${\mathcal K}[P_c]$ is $$\label{e:rCPa} r_C=\frac12+\sqrt{\frac14-c}$$ (cf. [@beardon]). Observe that in this case every escape radius of $P_c$ is not smaller than $\frac12+\sqrt{\frac14-c}.$ Indeed, Corollary [Corollary 47](#cor:Chebradiusescaperadius){reference-type="ref" reference="cor:Chebradiusescaperadius"} yields that $r_C$ given in ([\[e:rCPa\]](#e:rCPa){reference-type="ref" reference="e:rCPa"}) is an escape radius for $P_c$. Definition [Definition 45](#def: chebradius){reference-type="ref" reference="def: chebradius"} and Lemma [Lemma 24](#lem:escaperadius){reference-type="ref" reference="lem:escaperadius"} give the assertion. We will show now the uniform convergence of a function sequence defined with use of Chebyshev polynomials (cf. Main Example from the Introduction and [@Ko21]). **Corollary 49**. *Let $K\subset \mathbb{C}$ be a regular polynomially convex compact set. If $(t_n)_{n=1}^\infty$ is the sequence of Chebyshev polynomials on $K$, then the function sequence $$\left(\frac1{n!}\log^+\left|t_{ n}\circ...\circ t_{ 1}\right|\right)_{n=1}^\infty$$ is uniformly convergent in $\mathbb{C}$.* *Proof.* We apply Proposition [Corollary 14](#cor:czebyszewKW){reference-type="ref" reference="cor:czebyszewKW"} and Corollary [Corollary 37](#cor:p1...pn){reference-type="ref" reference="cor:p1...pn"} to the sequence $(t_n)_{n=1}^\infty$. ◻ ## Chebyshev polynomials on Julia sets In either case of the Example [Example 22](#ex: filled){reference-type="ref" reference="ex: filled"}, the $n$th Chebyshev polynomial for the filled Julia set coincides with the polynomial of degree $n$ generating the Julia set. Sequences of Chebyshev polynomials on Julia sets were determined in [@BGH83] (for quadratic polynomials) and [@KB]. Chebyshev polynomials on level sets of Green functions for Julia sets were studied in [@St96], [@XQ09] and [@XQ10].\ In [@Al] Chebyshev polynomials on non-autonomous Julia sets of sequences from class ${\mathcal B}$ (defined in [@BB03]) are discussed. Recall that in view of Remark [Remark 30](#rem:notinB){reference-type="ref" reference="rem:notinB"} we cannot apply the results from [@Al] directly to our case. We can however prove an analogue of the main theorem there. **Theorem 50**. *Let $K\subset \mathbb{C}$ be a regular polynomially convex compact set and let $(p_n)_{n=1}^\infty$ be a sequence of KW polynomials associated with K. If $(d_k)_{k=1}^\infty$ is a sequence of integers not smaller than $2$, then $\forall k\geq 1\ \exists \tau_k\in\mathbb{C}: \; p_{d_k}\circ... \circ p_{d_1}-\tau_k$ is the $d_k\cdot...\cdot d_1$-th Chebyshev polynomial on ${\mathcal K}[(p_{d_k})_{k=1}^\infty].$* *Moreover, $\forall n\geq 2\ \exists \pi_n\in\mathbb{C}: \; p_n\circ... \circ p_1-\pi_n$ is the $n!$-th Chebyshev polynomial on ${\mathcal K}[(p_n)_{n=1}^\infty].$* *Proof.* Let $R$ be as in Proposition [Proposition 17](#prop: tnzez){reference-type="ref" reference="prop: tnzez"}. We follow the lines of the proof of [@Al Theorem 4] with this $R$ for $(p_{d_k})_{k=1}^\infty$ and $(p_n)_{n=2}^\infty$. We obtain the assertion for ${\mathcal K}[(p_{d_k})_{k=1}^\infty]$, which yields also that $p_n\circ...\circ p_2-\tau_n$ is the $n!$-th Chebyshev polynomial on ${\mathcal K}[(p_n)_{n=2}^\infty]$. To finish the proof it suffices to use Remark [Remark 27](#rem:pierwszy){reference-type="ref" reference="rem:pierwszy"} and Example [Example 44](#ex:przeciwobraz){reference-type="ref" reference="ex:przeciwobraz"}. ◻ *Example 51*. Let $t_n=\frac{1}{2^{n-1}}T_n$, where $T_n$ is the classical Chebyshev polynomial, $n\in\{1,2,...\}$. The Chebyshev polynomial of degree 1 on ${\mathcal K}[(t_n)_{n=1}^\infty]$ is $p(z)=z$. *Proof.* Property 2 in Example [Example 41](#ex:rysunki){reference-type="ref" reference="ex:rysunki"} shows that ${\mathcal K}[(t_n)_{n=1}^\infty]$ is symmetric with respect to 0. Hence its Chebyshev center is 0. It is enough now to apply Lemma [Lemma 46](#l:Chebcenter0){reference-type="ref" reference="l:Chebcenter0"}. ◻ **Acknowledgements 1**. Both authors thankfully acknowledge their participation in Thematic Research Programme "Modern holomorphic dynamics and related fields", Excellence Initiative -- Research University programme at the University of Warsaw (a mini-semester in spring 2023). The paper was partially written thanks to the support from the programme. The second named author extends her thanks to the Faculty of Mathematics, Informatics and Mechanics of the University of Warsaw for supporting her participation in the thematic semester "Dynamical Systems. Topological, smooth and holomorphic dynamics, ergodic theory, fractals" in Stefan Banach International Mathematical Center at the Institute of Mathematics of the Polish Academy of Sciences in Warsaw (part of "Simons Semesters in Banach Center: 2020s vision.") and in the conference "Complex dynamics: connections to other fields", at the University of Warsaw Conference Center in Chęciny, Poland, as well as to the Chair of Approximation, Institute of Mathematics, Jagiellonian University, Kraków, for hosting her in March -- June 2023 during her study leave from the American Mathematical Society. We would also like to thank Maciej Klimek for preparing the figures for us. BBCL A. Alghamdi, M. Klimek, M. Kosek, *Attractors of compactly generated semigroups of regular polynomial mappings*, Complexity 2018 (2018), Article ID 5698021, 11 pp. G. Alpan, *Chebyshev polynomials on generalized Julia sets*, Comput. Methods Funct. Theory 16 (2016), 387-393. M. F. Barnsley, J. S. Geronimo, A. N. 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Qiu, *Chebyshev polynomials on Julia sets and equipotential curves* (Chinese. English summary), Acta Math. Sin., Chin. Ser. 53 (2010), No. 2, 323-328. [^1]: 2020 *Mathematics Subject Classification*: Primary 37F10; Secondary 30C15, 30E10, 31A15 [^2]: *Key words and phrases*: Julia sets, polynomials, Green function, Kalmár-Walsh theorem, Chebyshev polynomials.

arxiv_math

--- abstract: | For a group acting on a hyperbolic space, we set up an algorithm in the group algebra showing that ideals generated by few elements are free, where few is a function of the minimal displacement of the action, and derive algebraic, geometric, and topological consequences. author: - Grigori Avramidi and Thomas Delzant bibliography: - trees.bib title: Group rings and hyperbolic geometry --- # Introduction Let $G$ be a group and $\mathbb{K}$ a field. A natural problem is to study relations between the group $G$ and its group algebra $\mathbb K[G]$. For instance, in 1953 Fox suggested that > "*It seems reasonable to conjecture that a group ring $\mathbb Z[G]$ can not have divisors of zero unless $G$ has elements of finite order; this seems to be not an easy question.*"([@fox], p.557). This was also conjectured by Kaplansky in [@kaplansky] and by Higman in his (unpublished) thesis (see [@sandling]). It is equivalent to the statement that ideals generated by one element are free modules. In the early 60's, Cohn [@cohnfir; @cohnicm; @cohnbook] investigated rings in which ideals generated by any number of elements are free as $\mathbb K[G]$-modules (calling them *free ideal rings*, or *firs* for short), and showed that group algebras of free groups have this property. Soon after, Stallings proved his celebrated result on ends of groups [@stallings] which implies no other group algebras do. **Theorem 1** (Cohn [@cohnfir]+(a consequence of) Stallings [@stallings]). *The group $G$ is a free group if and only if all ideals in the group algebra $\mathbb K[G]$ are free as submodules.* A related question is to describe, for a ring $R$, the automorphism group $\mathrm{GL}_n(R)$ of a free $R$-module. Denote by $\text{GE}_n(R)$ the subgroup generated by elementary and diagonal matrices. Cohn also showed that for group algebras of free groups this is the entire automorphism group. **Theorem 2** (Cohn [@cohngl]). *For every $n$, $\mathrm{GE}_n(\mathbb K[F])=\mathrm{GL}_n(\mathbb K[F])$.* Bass [@bass] used these two theorems to show projective modules over the integral group ring of a free group, $\mathbb Z[F]$, are free. This algebraic result has a striking topological consequence: **Corollary 3** (3.3. in [@wall]). *Any two-dimensional complex with free fundamental group is homotopy equivalent to a wedge of circles and $2$-spheres.* Our goal is to establish similar results for groups $G$ acting on hyperbolic spaces. Earlier steps in this direction were taken in [@delzant] and [@avramidi], where it was proved that ideals in $\mathbb K[G]$ generated by one, respectively two elements are free if the minimum displacement of the action is large enough. Going further, we show in this paper that ideals generated by $n$ elements are free if ($\mathcal{H}_{n, \delta}$): *$G$ acts on a $\delta$-hyperbolic space $\mathcal{H}$ with displacement greater than $(2n+11)^2\delta$* and---under the same hypothesis---describe the automorphism groups of free, rank $n$ modules. **Theorem 4**. *Assume the group $G$ satisfies $\mathcal H_{n,\delta}$. Then* 1. *Every $n$-generated ideal in $\mathbb K[G]$ is a free $\mathbb K[G]$-module, and* 2. *$\mathrm{GE}_n(\mathbb K[G])=\mathrm{GL}_n(\mathbb K[G]).$* **Example 1**. *If $G$ is the fundamental group of a compact riemannian manifold of curvature $\leq -1$ acting on its universal cover, then the minimal displacement is twice the injectivity radius of the manifold and the hyperbolicity constant is $\delta=\log 2$. So, if the injectivity radius is greater than $(2n+11)^2(\log 2)/2$ then $G$ satisfies $\mathcal H_{n,\delta}$. If $G$ is any residually finite hyperbolic group, then it has a finite index subgroup satisfying $\mathcal H_{n, \delta}$. The free product of any two groups satisifying ${\mathcal{H}_{n, \delta}}$ also satisfies it.* ## Applications {#applications .unnumbered} To augment the description of $\mathrm{GL}_n$, one needs to understand diagonal matrices, which amounts to describing the units in $\mathbb K[G]$. This was already done in [@delzant], where it was shown (assuming minimal displacement is greater than $4\delta$) that all units are trivial, i.e. have the form $\lambda g$ for $\lambda\in\mathbb K^*,g\in G$. In particular, for a finite field $\mathbb K$ and finitely generated group $G$ the unit group of $\mathbb K[G]$ is finitely generated. Combining this with our theorem, we obtain an analogous result for $\mathrm{GL}_n$. **Theorem 5**. *Assume $G$ satisfies $\mathcal H_{n,\delta}$. If the field $\mathbb K$ is finite and the group $G$ is finitely generated, then $\mathrm{GL}_n(\mathbb K[G])$ is finitely generated.* In a different direction, a geometric consequence of our theorem is a lower bound for critical points of Morse functions of a given index on essential manifolds. **Theorem 6**. *Let $X^d$ be a closed $d$-manifold, $G$ a group that satisfies $\mathcal H_{n,\delta}$ and $BG$ its classifying space. If there is a continuous map $f:X^d\rightarrow BG$ with $f_*[X]\not=0$ in $H_d(BG;\mathbb K)$ then for each $0<k<d$, a Morse function on $X^d$ has at least $n+1$ critical points of index $k$.* The theorem applies---for instance---if the classifying space $BG$ is a closed, riemannian manifold of curvature $\leq -1$ and injectivity radius greater than $(2n+11)^2$ and $X=BG$, or more generally if $X$ has a map of non-zero degree to such a $BG$. Using Bass's local-to-global method, we also obtain the following version of Corollary [Corollary 3](#standard){reference-type="ref" reference="standard"} for some $2$-dimensional hyperbolic groups (e.g. very high genus surface groups). **Theorem 7**. *Assume that $G$ satisfies $\mathcal H_{n,\delta}$, and that there is an aspherical $2$-complex $Y$ with fundamental group $G$. Then every presentation $2$-complex for $G$ with less than $n+1$ relations is homotopy equivalent to $$Y\vee S^2\vee\dots\vee S^2.$$* It is known ([@cohenlyndon]) that one relator groups have geometric dimenension at most two. In our setting of groups acting with large displacement on hyperbolic spaces, we can show that "few relator" groups have cohomological dimension $\leq 2$. **Theorem 8**. *An $n$-relator group satisfying $\mathcal H_{n,\delta}$ has cohomological dimension $\leq 2$.* Finally, let us mention a consequence that can be thought of either as a generalization of Theorem [Theorem 8](#fewrelator){reference-type="ref" reference="fewrelator"} to higher dimensions or as a generalization of (the $X=BG$ case of[^1]) Theorem [Theorem 6](#manifold){reference-type="ref" reference="manifold"} to hyperbolic groups. **Theorem 9**. *Assume that the group $G$ satisfies $\mathcal H_{n,\delta}$ and has cohomological dimension $d$. Then, every aspherical complex with fundamental group $G$ has more than $n$ cells in each dimension $0<k<d$.* ## Algorithms {#algorithms .unnumbered} The proof of the first part of Theorem [Theorem 4](#mainfirtheorem){reference-type="ref" reference="mainfirtheorem"} is based on an algorithm which can be seen as a geometric version of the euclidean algorithm. To describe it, let us first sketch the approach to the "only if" direction of Theorem [Theorem 1](#ffir){reference-type="ref" reference="ffir"} given in Cohn's book [@cohnbook]. It consists of two distinct steps: - Fix a basis for the free group $F$, let $F_+$ be the (monoid of) non-negative words in this basis and denote by $\mathbb K[F_+]\subset\mathbb K[F]$ the subring of the group algebra generated by these words. Cohn first shows that ideals in $\mathbb K[F_+]$ are free. (Corollary 2.5.2 and Theorem 2.4.6) - He then gives a localization procedure for passing from $\mathbb K[F_+]$ to $\mathbb K[F]$ that preserves the free ideal property. (Corollary 7.11.8) Cohn measures size via filtrations. Recall that a *filtration* on a ring $R$ is a map $|\cdot| : R\rightarrow \mathbb{N}\cup\{-\infty\}$ such that $|0|=-\infty$, $|1| = 0$, and for all $x,y\in R$ $|x-y|\leqslant\max(|x|,|y|),$ $|xy|\leqslant|x| + |y|.$ Given a generating set for a group $G$, the group algebra $\mathbb K[G]$ has a natural filtration by word length $l(g)$ of elements $g$ in $G$: $$\left|\sum_{g\in G}\lambda^g\cdot g\right|:=\max_{\lambda^g\not=0}l(g).$$ On the ring $\mathbb K[F_+]$ of positive words in the free group, this filtration satisfies $|xy|=|x|+|y|$. The relevant notion of dependence says that some linear combination has smaller size than the maximum dictated by its terms: A family $\xi_1, \dots, \xi_n$ in $R$ is *$|\cdot|$-dependent* if there is a non-zero $(\alpha_1, \ldots, \alpha_n)\in R^n$ such that $$\left|\sum_i \alpha_i \xi_i\right|<\max_i\left|\alpha_i\xi_i\right|.$$ **Example 2**. *This is the case if the family is linearly dependent in the usual sense.* The first step is accomplished via the following algorithmic theorem. **Theorem 10** (Cohn [@cohnbook]). *Let $|\cdot|$ be the word length filtration of $\mathbb K[F_+]$. If $\xi_1,\dots,\xi_n$ in $\mathbb K[F_+]$ is a $|\cdot|$-dependent family then, up to reordering, there exist $\beta_2,\dots,\beta_n$ in $\mathbb K[F_+]$ such that $$\left|\xi_1 + \sum_{i = 2}^n \beta_i \xi_i\right| < | \xi_1|.$$* For a finitely generated ideal in $\mathbb K[F_+]$, one can repeatedly apply this theorem to decrease the size of members of a finite generating set until one arrives at a generating set that does not satisfy any dependence relations and hence forms a basis, showing the ideal is free. Doing a bit more work, Cohn uses this theorem to show infinitely generated ideals in $\mathbb K[F_+]$ are free, as well (Thm. 2.4.6). Finally, Cohn's localization proceedure shows the same is true for $\mathbb K[F]$. In [@hogangeloni], Hog-Angeloni gave a beautiful, geometric version of this algorithm that applies to the entire group algera $\mathbb K[F]$. It can be used to bypass the localization step if one is interested exclusively in finitely generated ideals. Hog-Angeloni's proof goes by looking at the action of $F$ on its Cayley graph, which is a tree $\mathcal T$. It uses the same notion of dependence as Cohn's proof but the natural notion of size of group ring elements relevant to her argument is the *diameter* $$\mathrm{diam}\left(\sum_{g\in G}\lambda^g\cdot g\right):=\max_{\lambda^g\not=0\not=\lambda^h}l(g^{-1}h).$$ **Remark 1**. *Note that $|\xi|=0$ means $\xi \in\mathbb{K}^*$, while $\mathrm{diam} (\xi) = 0$ means $\xi=\lambda g$ for a non-zero $\lambda\in \mathbb{K}^*$ and group element $g \in F$. The diameter is invariant by left translation, while $|\cdot|$ is not.* **Theorem 11** (Hog-Angeloni [@hogangeloni]). *Let $|\cdot|$ be the word length filtration of $\mathbb K[F]$. If $\xi_1,\dots,\xi_n$ in $\mathbb K[F]$ is a $|\cdot|$-dependent family then, up to reordering, there exist $\beta_2,\dots,\beta_n$ in $\mathbb K[F]$ such that $$\mathrm{diam}\left(\xi_1+\sum_{i=2}^n\beta_i\xi_i\right)<\mathrm{diam}(\xi_1).$$* An attentive reader of [@hogangeloni] can check that Hog-Angeloni does not use that the group is free nor that the tree is a Cayley graph, but only the fact that the group acts freely on a tree. We build on her geometric approach, replacing $F$ acting on the tree $\mathcal T$ by a group $G$ that acts on a hyperbolic space $\mathcal H$ with large minimum displacement. In our approach we will use the natural filtration on $\mathbb K[G]$ obtained from the action of $G$ on the hyperbolic space $\mathcal H$. **Notation 1**. *Let $\mathcal H$ be a geodesic metric space and $o$ a basepoint (or origin). Denote by $|p-q|$ the distance between two points in $\mathcal H$, and by $|p|=|p-o|$ the distance to the origin. If $\mathcal X\subset\mathcal H$ is a finite subset then $|\mathcal X|=\max_{p\in\mathcal H}|p|$ is called its absolute value and $\mathrm{diam}(\mathcal X)=\max_{p,q\in\mathcal X}|p-q|$ its diameter.* *If a group $G$ acts isometrically on $\mathcal H$ and $\xi=\sum_{g\in G}\lambda^g\cdot g$ is an element in the group algebra, denote by $\mathcal X=\{g\cdot o\mid \lambda^g\not=0\}$ the orbit of the basepoint under group elements appearing with non-zero coefficient in $\xi$ (i.e. under group elements in the algebraic support of $\xi$) and call it the geometric support of $\xi$. The diameter $\mathrm{diam}(\xi)$ and absolute value $|\xi|$ are $\mathrm{diam}(\xi)=\mathrm{diam}(\mathcal X)$ and $|\xi|=|\mathcal X|$. By convention, $\mathrm{diam} (0) = | 0 | = - \infty$.* *The minimal displacement of the action of $G$ on $\mathcal H$ is $\min_{g\in G-\{1\}, p\in\mathcal H}|g\cdot p-p|$.* Now, we can state our hyperbolic version of Hog-Angeloni's theorem. **Theorem 12**. *Set $\delta_n=(n^2+10n)\delta$. Let the group $G$ act on a $\delta$-hyperbolic space $\mathcal H$ with minimum displacement $>4\delta_n+(10+2n)\delta$. If $\xi_1,\dots,\xi_n\in\mathbb K[G]$ and there is a non-zero $(\alpha_1,\dots,\alpha_n)\in\mathbb K[G]^n$ such that $$\left|\sum_i\alpha_i\xi_i\right|<\max_i|\alpha_i\xi_i|-\delta_n,$$ then, up to re-ordering, there exist $\beta_2,\dots,\beta_n$ in $\mathbb K[G]$ such that $$\mathrm{diam} \left(\xi_1+\sum_{i = 2}^n \beta_i \xi_i\right)<\mathrm{diam}(\xi_1)-\delta.$$* **Remark 2**. *Theorem [Theorem 12](#intromaintheorem){reference-type="ref" reference="intromaintheorem"} will be used to replace the euclidean algorithm in the classical study of ideals, submodules of free modules, and matrices over euclidean rings. This is the key tool needed to obtain the geometric and algebraic applications (Theorems [Theorem 4](#mainfirtheorem){reference-type="ref" reference="mainfirtheorem"} through [Theorem 9](#cellbound){reference-type="ref" reference="cellbound"}) as we shall see in the last section.* ## Acknowledgements {#acknowledgements .unnumbered} We would like to thank Misha Gromov for his interest in this work and for mentioning the possibility of extending our results to essential manifolds. G.A. would like to thank the Max Planck Institut für Mathematik for its hospitality and financial support. # Hyperbolic preliminaries[\[hypsection\]]{#hypsection label="hypsection"} Fix a $\delta$-hyperbolic metric space $\mathcal H$ with basepoint $o$. See [@gromovhyperbolic],[@cdp], or [@bhbook] for background on hyperbolicity. ### Radii and centers {#radii-and-centers .unnumbered} Let $\mathcal X$ be a bounded subset in the metric space $\mathcal H$. Denote by $$r(\mathcal X):=\inf\{r\mid\mbox{there is } c \mbox{ for which } B(c,r)\supset\mathcal X\}$$ the infimum of radii of closed balls containing $\mathcal X$. We call it the *radius* of $\mathcal X$. If $\mathcal H$ is a proper metric space, then this infimum is realized and there is a closed ball of radius $r(\mathcal X)$ containing $\mathcal X$. In general, we only have for any positive $\epsilon$ a closed ball $B(c,r(\mathcal X)+\epsilon)$ containing $\mathcal X$. We call such a $c$ an $\epsilon$-center of $\mathcal X$. **Remark 3**. *If $\mathcal H$ is a proper, complete $\mathrm{CAT}(-1)$ space, then there is a unique $0$-center.* ### Gromov products {#gromov-products .unnumbered} Recall ([@gromovhyperbolic; @cdp; @bhbook]) the definition of the Gromov product $$\left<p,q\right>_r:={1\over 2}(|p-r|+|q-r|-|p-q|).$$ To simplify notation, recall that the distance from $p$ to the origin $o$ is denoted by $|p|$ and set $\left<p,q\right>:=\left<p,q\right>_o$. First, we give an estimate for the Gromov product of an $\epsilon$-center of $\mathcal X$ with a point of $\mathcal X$ that follows directly from the triangle inequality. **Lemma 13**. *Let $p$ be a point in a set $\mathcal X$ with $\epsilon$-center $c$ and radius $r$. Then $$|c|\geq\left<p,c\right>\geq{|\mathcal X|+|p|\over 2}-r-\epsilon.$$* *Proof.* The triangle inequality $|c|\geq|p|-|c-p|$ implies $$\begin{aligned} |c|&\geq&{1\over 2}\left(|c|+|p|-|c-p|\right)\\ &\geq&|p|-|c-p|\\ &\geq&|p|-r-\epsilon.\end{aligned}$$ This proves the left inequality and, since it is true for all $p\in\mathcal X$, shows $|c|\geq|\mathcal X|-r-\epsilon$. Plugging it back into the formula for Gromov product give the right inequality: $$\begin{aligned} \left<p,c\right>&=&{1\over 2}(|c|+|p|-|p-c|)\\ &\geq&{1\over 2}((|\mathcal X|-r-\epsilon)+|p|-(r+\epsilon)).\end{aligned}$$ ◻ ### Thin triangles and projections in hyperbolic spaces {#thin-triangles-and-projections-in-hyperbolic-spaces .unnumbered} In a $\delta$-hyperbolic space, any geodesic triangle is $\delta$-thin. This means that if $[r,p,q]$ is a triangle then two (oriented) geodesics $[r,p]$ and $[r,q]$ parametrized by arc length $p(t),q(t)$ remain $\delta$-close (i.e. $|p(t)-q(t)|\leq\delta$) until $t=\left<p,q\right>_r$. We will repeatedly use the following consequence of this: Given a segment $[a,b]$ and a point $x\in\mathcal H$, the *projection of $x$ to $[a,b]$* is the point $x'$ on $[a,b]$ such that $|a-x'|=\left<b,x\right>_a$.[^2] Then, for points $y_1$ in the initial subsegment $[a,x']$ of $[a,b]$ we have a $\delta$-converse to the triangle inequality $$|x-y_1|+|y_1-a|\leq|x-a|+\delta,$$ while for points $y_2$ in the subsegment $[x',b]$ we have $$|x-y_2|+|y_2-b|\leq|x-b|+\delta.$$ ### Diameter vs radius {#diameter-vs-radius .unnumbered} First, we note that in a $\delta$-hyperbolic space---just like in a tree---the diameter and radius are closely related. More precisely, **Lemma 14**. *Let $[a,b]$ be a diameter realizing segment of a finite set $\mathcal X$ with radius $r$ and let $m$ be the midpoint of $[a,b]$. Then $\mathcal X\subset B\left(m,{|a-b|\over 2}+\delta\right)$. In particular, $m$ is a $\delta$-center of $\mathcal X$ and $${|a-b|\over 2}\leq r\leq {|a-b|\over 2}+\delta.$$* *Proof.* The left inequality holds in any geodesic space. For the right one, let $p$ be a point in $\mathcal X$ and assume (without loss of generality) that its projection to $[a,b]$ is on $[m,b]$. Then $$\begin{aligned} {|a-b|\over 2}+|m-p|&=&|a-m|+|m-p|\\ &\leq&|a-p|+\delta\\ &\leq&|a-b|+\delta,\end{aligned}$$ whence the result. ◻ ### Center vs midpoint {#center-vs-midpoint .unnumbered} Next, we observe that an $\epsilon$-center is $(2\delta+\epsilon)$-close to the midpoint of any diameter realizing segment. **Lemma 15**. *Let $m$ be the midpoint of a diameter $[a,b]$ of a finite set $\mathcal X$ with $\epsilon$-center $c$. Then $$|c-m|\leq \epsilon+2\delta.$$* *Proof.* Without loss of generality, the projection of $c$ to $[a,b]$ is on $[m,b]$, so $$\begin{aligned} |c-m|+{|a-b|\over 2}&=&|c-m|+|m-a|\\ &\leq&|c-a|+\delta\\ &\leq&r+\epsilon+\delta\end{aligned}$$ implies $|c-m|\leq r-{|a-b|\over 2}+\epsilon+\delta\leq\epsilon+ 2\delta$. ◻ In particular, any two $\epsilon$-centers of a finite set are $(4\delta+2\epsilon)$-close to each other and the midpoints of any two diameters are $3\delta$-close to each other. ### An equivariant choice of centers {#an-equivariant-choice-of-centers .unnumbered} **Corollary 16**. *Suppose $G$ acts on a $\delta$-hyperbolic space $\mathcal H$ with displacement greater than $3\delta$. Then $G$ acts freely on the collection of all finite subsets of $\mathcal H$.* *Proof.* Let $\mathcal X$ be a finite subset and $m$ the midpoint of a diameter of $\mathcal X$. If $g\mathcal X=\mathcal X$ for some $g\in G$ then $m$ and $gm$ are $\delta$-centers by Lemma [Lemma 14](#diamvrad){reference-type="ref" reference="diamvrad"} and hence $3\delta$-close by Lemma [Lemma 15](#centervsmidpoint){reference-type="ref" reference="centervsmidpoint"}. Therefore, $g=1$. ◻ Therefore, as long as the displacement is greater than $3\delta$ we can choose for each finite subset $\mathcal X\subset \mathcal H$ an $\epsilon$-center $c(\mathcal X)$ such that $c(g\mathcal X)=gc(\mathcal X)$ for each $g\in G$. ### Distance from origin to center {#distance-from-origin-to-center .unnumbered} Now, we estimate the distance from the origin to an $\epsilon$-center of a set $\mathcal X$ in terms of its radius and $|\mathcal X|$. **Lemma 17**. *Let $\mathcal X$ be a finite set with $\epsilon$-center $c$. Then $$|\mathcal X|-r-\epsilon\leq|c|\leq |\mathcal X|-r+4\delta+\epsilon$$* *Proof.* The left inequality already appeared in Lemma [Lemma 13](#fellowtravel1){reference-type="ref" reference="fellowtravel1"}. For the right one, we again pick a diameter realizing segment $[a,b]$ and let $m$ be its midpoint. Without loss of generality, the projection of $o$ to $[a,b]$ is on $[m,b]$ and we have $$\begin{aligned} |c|&\leq&|m|+|m-c|\\ &\leq&(|a|-|a-m|+\delta)+2\delta+\epsilon\\ &\leq& |\mathcal X|-{|a-b|\over 2}+3\delta+\epsilon.\end{aligned}$$ Therefore $$|c|+r\leq|c|+\left({|a-b|\over 2}+\delta\right)\leq |\mathcal X|+4\delta+\epsilon$$ which proves the desired inequality. ◻ ### Diameter of intersection of two balls {#diameter-of-intersection-of-two-balls .unnumbered} Another property of $\delta$-hyperbolic spaces we will need is a bound on the diameter of the intersection of two balls. **Lemma 18**. *In a $\delta$-hyperbolic space $\mathcal H$, the diameter of the intersection of two closed balls $B(c_1,r_1)\cap B(c_2,r_2)$ is bounded above by $r_1+r_2-|c_1-c_2|+2\delta.$* *Proof.* Let $[a,b]$ be a segment realizing the diameter of the intersection of balls, and let $a',b'$ be the two projections of $a, b$ on $[c_1, c_2]$. We assume that $a'$ is on the left of $b'$ on this segment. Using $a'\in[c_1,b']$, hyperbolicity and the fact that $b \in B (c_1, r_1)$ $$\begin{aligned} | b - b' | + |b'-a'|+|a'-c_1|&=&|b-b'|+|b'-c_1|\\ &\leqslant&\delta+| b - c_1| \\ &\leqslant&\delta+r_1.\end{aligned}$$ Similarly, $b'\in[a',c_2]$, hyperbolicity and the fact that $a\in B(c_2,r_2)$ implies $$\begin{aligned} |a-a'|+|a'-b'|+|b'-c_2|&=&| a - a' | + | a' - c_2 | \\ &\leqslant& \delta + | a - c_2|\\ &\leqslant& \delta+r_2.\end{aligned}$$ Adding these two inequalities together and using the triangle inequality on the left gives $$|a-b|+|c_1-c_2|\leq 2\delta+r_1+r_2.$$ Since $|a-b|$ is the diameter, this finishes the proof. ◻ ### Gromov product inequality {#gromov-product-inequality .unnumbered} Finally, we recall a key inequality that will be used to estimate distances: **Lemma 19** ([@gromovhyperbolic] &6 page 155 or [@cdp] 8.2 page 91). *For a sequence of $2^k+1$ points $x_0,\dots,x_{2^k}$, we have $$\left<x_0,x_{2^k}\right>\geq\min_i\left<x_i,x_{i+1}\right>-k\delta.$$* # Extremal graphs[\[extremalsection\]]{#extremalsection label="extremalsection"} Let $V$ be a finite set and suppose we have a family[^3] of bounded sets $Y=(\mathcal Y_v)_{v\in V}$. Let $\mu$ be a fixed parameter. A point $p\in\cup_{v\in V}\mathcal Y_v$ satisfying $|p|\geq |\cup_{v\in V}\mathcal Y_v|-\mu$ is called a *$\mu$-extremal point* (of $Y$). A $0$-extremal point will also be called an *extremal point*. ## The graph $\Gamma_{\mu}(Y)$ Let us first define the *$\mu$-extremal graph $\Gamma_{\mu}(Y)$* of the family $Y$. The vertices of $\Gamma_{\mu}(Y)$ are the indices $v\in V$ such that $\mathcal Y_v$ contains a $\mu$-extremal point (of $Y$), and there is an edge between $v$ and $w$ for each $\mu$-extremal point in $\mathcal Y_v\cap\mathcal Y_w$. ### Distance between centers {#distance-between-centers .unnumbered} For each vertex $v\in V$ we let $r_v$ be radius of the set $\mathcal Y_v$ and choose an $\epsilon$-center $c_v$. We estimate the distance between two centers $c_v$ and $c_w$ in terms of the distance between the vertices $v$ and $w$ in the graph $\Gamma_{\mu}(Y)$. **Lemma 20**. *For subset $W\subset V$ denote $r_{\max(W)}=\max_{v\in W}r_v$.* 1. *If $v$ and $w$ are connected by a path $P$ of length $m$ in $\Gamma_\mu(Y)$, then $$|c_v-c_w|\leqslant (r_{\max(P)}-r_v)+(r_{\max(P)}-r_w)+2\mu+(8+2\lceil\log_2(2m)\rceil)\delta+4\epsilon.$$* 2. *If $v$ and $w$ are adjacent vertices in $\Gamma_{\mu}(Y)$, then we have $$|c_v-c_w|\leq|r_v-r_w|+2\mu+10\delta+4\epsilon.$$* *Proof.* The path $v=v_0,e_0,v_1,e_1,\dots,e_{m-1},v_m=w$ in the graph gives an alternating sequence of centers and $\mu$-extremal points $c_{v_0},p_{e_0},\dots,p_{e_{m-1}},c_{v_m}$. The Gromov product inequality and Lemma [Lemma 13](#fellowtravel1){reference-type="ref" reference="fellowtravel1"} gives $$\begin{aligned} \left<c_v,c_w\right>&\geq&\min_{v_i} (d-\mu-r_{v_i}-\epsilon)-\lceil\log_2(2m)\rceil\delta.\end{aligned}$$ Using this and the inequality $|c_v|\leq d-r_v+4\delta+\epsilon$ obtained in Lemma [Lemma 17](#distancetocenter){reference-type="ref" reference="distancetocenter"} we get $$|c_v|-\left<c_v,c_w\right>\leq (\max_{v_i}r_{v_i})-r_v+\mu+(4+\lceil\log_2(2m)\rceil)\delta+2\epsilon,$$ and a similar inequality for $w$ in place of $v$. Since $$|c_v-c_w|=(|c_v|-\left<c_v,c_w\right>)+(|c_w|-\left<c_v,c_w\right>),$$ we obtain the first inequality. For adjacent vertices $v$ and $w$, we have $m=1$ and $2\max\{r_v,r_w\}-r_v-r_w=|r_v-r_w|$, which gives the second inequality. ◻ ### Colors {#colors .unnumbered} Consider a partition $V=V_1\sqcup\dots\sqcup V_n$ into $n$ subsets called *colors* such that any two vertices of the same color have the same radius (if $v,w\in V_i$ then $r_v=r_w$). Denote by $r_i$ the radius of a set of color $i$. ### Bounding the diameter of $\Gamma_{\mu}(Y)$ {#bounding-the-diameter-of-gamma_muy .unnumbered} Next, we will give an upper bound on the diameter of a component of an extremal graph in terms of the number $n$ of colors. The result will be proved by induction on the number of colors $n$. In fact, we prove something stronger, namely an upper bound on the length of an embedded path in the graph. **Proposition 21**. *Let $V=V_1\sqcup\dots\sqcup V_n$ be a finite set partitioned into $n$ colors and let $Y=(\mathcal Y_v)_{v\in V}$ be a collection of bounded subsets of $\mathcal H$ such that for each $i$ and any two different $v,w\in V_i$ we have* - *$r_v=r_w$ and* - *$|c_v-c_w|>2\mu+(10+2n)\delta+4\epsilon$.* *Then every embedded path in $\Gamma_{\mu}(Y)$ has length at most $2^n-2$.* *Proof.* We argue by contradiction. If there is an embedded path in $\Gamma_{\mu}(Y)$ of length more than $2^n-2$, then there is an embedded path of length precisely $2^n-1$. Let $P_n$ be such a path. It has $2^n$ vertices. Reorder the list of colors so that $$r_1\geq\dots\geq r_n.$$ Let $P_k$ be the sequence of vertices obtained by throwing out from $P_n$ all vertices of colors $\{k+1,\dots,n\}$. The sequence $P_k$ is colored by the set $\{1,\dots,k\}$. We prove by reverse induction (starting with base case $k=n$ and going down to $k=1$) that - $P_k$ has at least $2^k$ vertices, and - any two consecutive vertices in $P_k$ have different colors. Note that consecutive vertices of $P_n$ have different colors by Lemma [Lemma 20](#centerdistance){reference-type="ref" reference="centerdistance"}.2 and the second hypothesis. So $P_n$ satisfies the two properties above, verifying the base case. To prove the induction step, suppose we know the statement for $P_{k+1}$. Since it has at least $2^{k+1}$ vertices an no two consecutive vertices have the same color, at most $\lceil|P_{k+1}|/2\rceil$ have the color $r_{k+1}$, so that $P_k$ has at least $2^k$ vertices. Suppose two consecutive vertices $v,w$ in the sequence $P_k$ have the same color. Then $r_v=r_w$ and the two vertices are connected by a path (of length at most $2^n$) in $P_n$ through colors $\{k+1,\dots,n\}$ in which all vertices have radii $\leq r_v=r_w$ by our choice of ordering. Therefore, Lemma [Lemma 20](#centerdistance){reference-type="ref" reference="centerdistance"}.1 implies: $$|c_v-c_w|\leq 2\mu+(8+2(n+1))\delta+4\epsilon.$$ This contradicts the second hypothesis of our proposition. So, we have shown that $v$ and $w$ have different colors, completing the induction step. We have shown $P_1$ has at least two vertices and consecutive ones have different colors. This is absurd since all vertices of $P_1$ have the same color. So, we arrive at a contradiction to our initial assumption, and conclude that all embedded paths in $\Gamma_{\mu}(Y)$ have length $\leq 2^n-2$. ◻ As a consequence, we see that there is a unique vertex of largest color in each component, and a similar statement for other colors with suitably large radii. Recall that $r_1\geq\dots\geq r_n$ so that $r_{\max(V)}=r_1$. **Corollary 22** (Uniqueness). *In the sitation of Prop. [Proposition 21](#pathbound){reference-type="ref" reference="pathbound"}, if for any pair of distinct vertices of the same color $v,w\in V_i$ we have $$|c_v-c_w|>2(r_{1}-r_i)+2\mu+(10+2n)\delta+4\epsilon$$ then there is at most one vertex of color $i$ in each component of $\Gamma_{\mu}(Y)$.* *Proof.* Assume $v$ and $w$ are in the same component of $\Gamma_{\mu}(Y)$. Then, by Proposition [Proposition 21](#pathbound){reference-type="ref" reference="pathbound"}, they are connected by an embedded path $P$ of length at most $2^{n}-2$. Therefore, Lemma [Lemma 20](#centerdistance){reference-type="ref" reference="centerdistance"}.1 implies that $|c_v-c_w|\leq 2(r_{1}-r_i)+2\mu+(10+2n)\delta+4\epsilon$, contradicting the hypothesis. So, $v$ and $w$ must be in different components. ◻ ## The graph $\Gamma_{\mu}(\Xi)$ We will apply Proposition [Proposition 21](#pathbound){reference-type="ref" reference="pathbound"} and Corollary [Corollary 22](#uniqueness){reference-type="ref" reference="uniqueness"} in the following situation. Let $\Xi=(\xi_v)_{v\in V}$ in $\mathbb K[G]$ be a finite family of group ring elements, such that no two are $\mathbb K$-scalar multiples of each other. Recall that the (geometric) support of $\xi_v=\sum_{g\in G}\xi_v^g\cdot g$ is the set $\mathcal X_v=\{g\cdot o\mid \xi_v^g\not=0\}$. Let $X=(\mathcal X_v)_{v\in V}$ be the family of these supports, and set $\Gamma_{\mu}(\Xi):=\Gamma_{\mu}(X)$. In this situation, there is a canonical partition of the index set $V$ into colors $V=V_1\sqcup\dots\sqcup V_n$ defined as follows. We declare that $v$ and $w$ have the same color if and only if there is $\lambda\in\mathbb K^*$ and $g\in G$ such that $\xi_v=\lambda g\xi_w$. Note that if $v$ and $w$ have the same color then $g\mathcal X_v=\mathcal X_w$ and hence $r_v=r_w$. Let $r_i$ be the radius of elements with color $i$. **Corollary 23** (Uniform diameter bound+Uniqueness). *Suppose that $G$ acts on a $\delta$-hyperbolic space $\mathcal H$ with displacement greater than $2\mu+(10+2n)\delta$. Let $\Xi=(\xi_v)_{v\in V}$ in $\mathbb K[G]$ be a family of group ring elements consisting of $n$ colors, and such that no two are $\mathbb K$-scalar multiples of each other. Then,* - *Each component of $\Gamma_{\mu}(\Xi)$ has diameter at most $2^n-2$.* - *If the minimal displacement is greater than $2(r_{1}-r_i)+2\mu+(10+2n)\delta$, then there is at most one vertex of color $i$ in each component of $\Gamma_{\mu}(\Xi)$.* *Proof.* For two distinct elements $v,w\in V_i$, since $\xi_v$ is not a $\mathbb K$-scalar multiple of $\xi_w$ there is a non-trivial $g\in G$ such that $\lambda g\xi_v=\xi_w$. So, we have $r_v=r_w$. Since the displacement is greater than $3\delta$ we can, by Corollary [Corollary 16](#equivcenter){reference-type="ref" reference="equivcenter"}, pick $\epsilon$-centers equivariantly, so that $gc_v=c_w$. Then $|c_v-c_w|=|c_v-gc_v|>2\mu+(10+2n)\delta+4\epsilon$ for small enough $\epsilon$. So, Proposition [Proposition 21](#pathbound){reference-type="ref" reference="pathbound"} and Corollary [Corollary 22](#uniqueness){reference-type="ref" reference="uniqueness"} apply. ◻ ### Components of $\Gamma_{\mu}(\Xi)$ and weak relations {#components-of-gamma_muxi-and-weak-relations .unnumbered} A finite collection of group ring elements $(\xi_v)_{v\in V}$ defines a *$\mu$-relation* if $$\label{murelation} \left|\sum_{v\in V}\xi_v\right|<\max_{v\in V}|\xi_v|-\mu,$$ The components of $\Gamma_{\mu}(\Xi)$ help us keep track of such $\mu$-relations. **Lemma 24**. *If the family $\Xi=(\xi_v)_{v\in V}$ define a $\mu$-relation, then every connected component $C$ of the graph $\Gamma_{\mu}(\Xi)$ that contains a vertex of $\Gamma_0(\Xi)$ defines its own $\mu$-relation $|\sum_{v\in C}\xi_v|<\max_{v\in C}|\xi_v|-\mu$.* *Proof.* Set $d=\max_{v\in V}|\xi_v|$. Denote the complement of $C$ by $D=\Gamma_{\mu}-C$. Since the $(\xi_v)_{v\in V}$ define a $\mu$-relation, we have $$\left|\sum _{v\in C}\xi_v+\sum_{v\in D}\xi_v\right|<d-\mu.$$ Since $C$ is a component of $\Gamma_{\mu}(\Xi)$, the supports of $\sum_{v\in C}\xi_v$ and $\sum_{v\in D}\xi_v$ have no $\mu$-extremal points (that is, points $p$ with $|p|\geq d-\mu$) in common. Therefore, we must have $$\left|\sum _{v\in C}\xi_v\right|<d-\mu.$$ Our choice of $C$ implies that $d=\max_{v\in C}|\xi_v|$, proving the lemma. ◻ # Proof of Theorem [Theorem 12](#intromaintheorem){reference-type="ref" reference="intromaintheorem"} {#proof-of-theorem-intromaintheorem} ## Restatement of the main theorem [\[restate\]]{#restate label="restate"} We will now restate Theorem [Theorem 12](#intromaintheorem){reference-type="ref" reference="intromaintheorem"} in terms of colors and $\mu$-relations. We assume that $\xi_1,\dots,\xi_n$ are elements in the group algebra $\mathbb{K} [G]$ so that there exist $\alpha_1, \ldots, \alpha_n$ in $\mathbb{K} [G]$, not all zero, satisfying $| \sum \alpha_i \xi_i | < \max_i | \alpha_i \xi_i | - \delta_n$, where $\delta_n =(n^2+10n)\delta$. Recall that two elements $\xi_1$ and $\xi_2$ of $\mathbb{K} [G]$ have the same color if there exists a trivial unit $\lambda g$ in $\mathbb{K} [G]$ such that $\xi_1 - \lambda g \xi_2 = 0$. Certainly if two of the $\xi_i$ have the same color, the conclusion of the theorem follows. So, from now on, we assume that the $\xi_i$ have different colors, so that the set $\{ 1, \ldots, n \}$ can be used as the set of colors. For each $\alpha_i = \sum_{g \in G} \alpha^g_i \cdot g$, we let $A_i = \{g \in G \mid \alpha^g_i \neq 0\}$ be its (algebraic) support in the group $G$. The group ring elements $(\alpha^g_i g \xi_i)_{g \in A_i, 1 \leq i \leq n}$ are distinct, no two of them are scalar multiples of each other, and for $i \neq j$ they do not have the same color. We rename them $(\xi_v)_{v \in V}$ which is a collection of elements in $\mathbb{K} [G]$ having $n$ colors. With this notation, the hypothesis $| \sum \alpha_i \xi_i | < \max_i | \alpha_i \xi_i | - \delta_n$ implies that: $$\left| \sum_{v \in V} \xi_v \right| < \max_{v \in V} \left| \xi_v \right| - \delta_n.$$ Recall that in this situation we say that the family $(\xi_v)_{v \in V}$ defines a $\delta_n$-relation. Under this hypothesis, we will show that there exists a vertex $v_*$ in $V$ and a subset $S \subset V$ made of elements of different colors from the color of $v_*$ such that $\mathrm{diam} (\xi_{v_*} + \sum_{v \in S} \xi_v) < \mathrm{diam} (\xi_{v_*}) - \delta$. **Remark 4**. *This conclusion is slightly stronger than that of Theorem [Theorem 12](#intromaintheorem){reference-type="ref" reference="intromaintheorem"}. It implies in addition that---after reordering and replacing $\xi_1$ by an element of the same color---the $\beta_i$ have the form $\beta_i=\sum_{g\in B_i}\alpha_i^g\cdot g$, where $B_i$ is a subset of $A_i$.* Summing up, Theorem [Theorem 12](#intromaintheorem){reference-type="ref" reference="intromaintheorem"} will follow from **Theorem 25**. *Set $\delta_n=(n^2+10n)\delta$. Suppose $G$ acts on a $\delta$-hyperbolic space $\mathcal H$ with minimal displacement $\rho>4\delta_n+(10+2n)\delta$. Let $\Xi=(\xi_v)_{v\in V}$ be a finite collection of elements in $\mathbb K[G]$ consisting of $n$ colors, and with no two elements being scalar multiples of each other. If $(\xi_v)_{v\in V}$ satisfies a $\delta_n$-relation, then there is a subset $\{v_*\}\sqcup S$ of $V$ such that no element of $S$ has the same color as $v_*$ and $$\mathrm{diam}\left(\xi_{v_*}+\sum_{v\in S}\xi_v\right)< \mathrm{diam}(\xi_{v_*})-\delta.$$* **Remark 5**. *The set $S\sqcup\{v_*\}$ will be constructed as the set of vertices of a component of $\Gamma_{\delta_k}(\Xi)$ for some $1\leq k\leq n$.* **Remark 6**. *The constant $\delta_n$ is obtained inductively as follows. Set $\delta_0=0$ and define $\delta_{k}=\delta_{k-1}+(2k+9)\delta$. Then $\delta_n/\delta=\sum_{k=1}^n(9+2k)=9n+(n+1)n=n^2+10n$. Note also that the minimal displacement condition is $${\rho\over\delta}>4n^2+42n+10=(2n+10.5)^2-10.5^2+10.$$ So, $\rho>(2n+11)^2\delta$ is enough to obtain the conclusion of the theorem.* ## The case of trees[\[hogangeloniproof\]]{#hogangeloniproof label="hogangeloniproof"} We keep the notations of the previous paragraph unchanged, but assume that the space $\mathcal{H}$ is a tree. This is the case $(\delta = 0)$ treated by Hog-Angeloni in [@hogangeloni]. We slightly rephrase her proof. Recall that $\Gamma_0 = \Gamma_0 (\Xi)$ is the graph whose vertices are the elements $v$ in $V$ such that if $\mathcal X_v$ is the support of $\xi_v$, then $\mathcal X_v$ contains an extremal point, i.e. $| \xi_v | = d = \max_{v \in V} | \xi_v |$, and there is an edge between $v$ and $w$ for each extremal point in $\mathcal X_v\cap\mathcal X_w$. The colors $\{ 1, \ldots, n \}$ are organized so that the $r_i$ (the radii of the $\mathcal X_i$) are in decreasing order: $r_1 \geq \cdots \geq r_n$. Let $\hat{\Gamma}_0$ be a component of $\Gamma_0$ containing a vertex $v_*$ with color $1$, the color of largest radius. By Lemma [Lemma 24](#relationcomponents){reference-type="ref" reference="relationcomponents"}, this graph defines a $0$-relation $$\left| \sum_{v \in \hat{\Gamma}_0} \xi_v \right| < \max_{v \in \hat{\Gamma}_0} | \xi_v | .$$ By Corollary [Corollary 23](#diamcor){reference-type="ref" reference="diamcor"}, $v_*$ is the unique vertex in $\hat{\Gamma}_0$ of color $1$. So, removing $v_*$ from $\hat\Gamma_0$ and setting $S = \hat{\Gamma}_0 - \{v_*\}$, we see that to conclude the proof of the theorem we need to estimate the diameter of the group ring element $\sum_{v\in\hat\Gamma_0}\xi_v=\xi_{v_*}+\sum_{v\in S}\xi_v$. Note that for any two adjacent vertices $v$ and $w$ in $\hat{\Gamma}_0$, both centers $c_v$ and $c_w$ lie on a geodesic $[o, p]$ from the origin to an extremal point. The oriented geodesics $[o, c_v]$ and $[o, c_w]$ coincide along a segment of length $\min \{|c_v |, |c_w |\} \geq d - r_1 = | c_{v_*} | .$ By following a path from $v_*$ to $v$ in the graph $\hat{\Gamma}_0$ and noting that $r_1 \geq r_w$ for every vertex $w$ along that path, we conclude that $c_{v_*}$ belongs to every geodesic $[o, c_v]$, and $$|c_{v_*} - c_v | = r_1 - r_v .$$ For every $v,$ the support of $\xi_v$ is contained in the closed ball $B (c_{v_*}, r_1)$, so the same is true for the support of $\sum_{v \in\hat\Gamma_0} \xi_v$. Moreover, because of the $0$-relation, this support has no $0$-extremal points, i.e. it is contained in the ball $B (o, d-\alpha)$ for some $\alpha > 0$. So, applying Lemma [Lemma 18](#twoballs){reference-type="ref" reference="twoballs"}, we see that the the diameter of the support of $\sum_{v\in\hat\Gamma_0}\xi_v$ is at most $$d - \alpha + r_1 - |o - c_{v_*} | = 2 r_1 = \mathrm{diam} (\xi_{v_*}) - \alpha .$$ This finishes the proof. **Remark 7**. *In this result, we do not use that $\mathcal{H}$ is a combinatorial tree. It might be an $\mathbb{R}$-tree with a free action of a surface group, for instance. But, for proving that the algorithm ends in a finite number of steps, we need that the diameter decreases by a given value, say $1$.* ## Proof of Theorem [Theorem 25](#maintheorem){reference-type="ref" reference="maintheorem"} for hyperbolic spaces[\[proofsubsection\]]{#proofsubsection label="proofsubsection"} {#proof-of-theorem-maintheorem-for-hyperbolic-spacesproofsubsection} In order to prove Theorem [Theorem 25](#maintheorem){reference-type="ref" reference="maintheorem"}, we keep the notations of [\[restate\]](#restate){reference-type="ref" reference="restate"} unchanged and proceed by induction on the number $n$ of colors. Recall that $\delta_0=0$ and $\delta_n=\delta_{n-1}+(2n+9)\delta$. Let us fix some notation. We are given a collection of elements $\Xi=(\xi_v)_{v\in V}$ defining a $\delta_n$-relation. Denote $d=\max_{v\in V}|\xi_v|$. We abreviate $\Gamma_{\mu}=\Gamma_{\mu}(\Xi)$, so that vertices of $\Gamma_{\mu}$ are those $v\in V$ for which $|\xi_v|\geq d-\mu$. ## Base case {#base-case .unnumbered} If there is one color, then $\delta_1=11\delta$ and the minimum displacement is greater than $4\delta_1+12\delta$. So, Corollary [Corollary 23](#diamcor){reference-type="ref" reference="diamcor"} applies and implies that connected components of $\Gamma_{\delta_1}$ are points, and there are no $\delta_1$-relations. **Remark 8**. *This case implies, in particular, that $\mathbb K[G]$ has no zero divisors. (The zero-divisor relation $0=\alpha\xi=\sum_{g\in G}\alpha^gg\xi$ is a $\delta_1$-relation defined by a finite collection of elements of a single color.)* ## Inductive step {#inductive-step .unnumbered} Suppose now that we know the theorem for $n-1$ colors and want to prove it for $n$. By the inductive hypothesis, we may assume that $\Xi$ satisfies the following "minimality" condition: - No subset $W\subset V$ consisting of fewer than $n$ colors defines a $\delta_{n-1}$-relation. Since we will consider different values of $\mu$, it is useful to keep in mind that for $\mu<\mu'$ the graph $\Gamma_{\mu}$ is a subgraph of $\Gamma_{\mu'}$. In particular, for a vertex $v_0\in\Gamma_0$ (corresponding to an element $\xi_{v_0}$ whose support contains an extremal point) we have inclusions $$v_0\in\Gamma_0\subset\Gamma_{\delta_{n-1}}\subset\Gamma_{\delta_n}.$$ Denote by $\hat\Gamma_{\delta_{n-1}}$ and $\hat\Gamma_{\delta_n}$ the connected components of $\Gamma_{\delta_{n-1}}$ and $\Gamma_{\delta_n}$ containing the vertex $v_0$. Since $\Xi$ defines a $\delta_n$-relation (and $\delta_n>\delta_{n-1}$) Lemma [Lemma 24](#relationcomponents){reference-type="ref" reference="relationcomponents"} implies that $\hat{\Gamma}_{\delta_{n-1}}$ defines a $\delta_{n-1}$-relation, so minimality implies $\hat\Gamma_{\delta_{n-1}}$ contains all $n$ colors. ### Large and small colors {#large-and-small-colors .unnumbered} Order the colors in decreasing order, $r_1\geq\dots\geq r_n$, and let $k$ be the largest integer for which $r_k\geq r_1-\delta_n$. We call $\{1,\dots,k\}$ the large colors and the rest small colors. The minimal displacement $> 4\delta_n+(10+2n)\delta$ assumption implies, by Corollary [Corollary 23](#diamcor){reference-type="ref" reference="diamcor"}.2, that each large color appears exactly once in the connected graph $\hat\Gamma_{\delta_n}$. Thus, we can identify the set of large colors $\{1,\dots,k\}$ with a set of $k$ vertices $\{v_1,\dots,v_k\}$ in $\hat\Gamma_{\delta_n}$. Since each color appears in $\hat\Gamma_{\delta_{n-1}}$, we conclude $$\{v_1,\dots,v_k\}\subset\hat\Gamma_{\delta_{n-1}}\subset\hat\Gamma_{\delta_n}.$$ ### Bounding the diameter of support {#bounding-the-diameter-of-support .unnumbered} Recall that the family $\Xi$ defines a $\delta_n$-relation, so the support of $\sum_{v\in\hat\Gamma_{\delta_n}}\xi_v$ is contained in the ball $B(o,d-\delta_n)$. Let $c^*$ be the point in $[o,c_{v_1}]$ such that $|c^*|=d-\delta_{n-1}-r_1-(n+1)\delta$. In order to conclude the proof of Theorem [Theorem 25](#maintheorem){reference-type="ref" reference="maintheorem"}, we shall prove that the ball $B(c^*,r_1+(n+3)\delta)$ also contains this support: **Lemma 26**. *For $w\in\hat\Gamma_{\delta_n}$ and a point $p\in\mathcal X_w$ that is not $\delta_n$-extremal we have $$|c^*-p|\leq r_1+(n+3)\delta.$$* Given this, Lemma [Lemma 18](#twoballs){reference-type="ref" reference="twoballs"} implies the diameter of the support of $\sum_{v\in\hat \Gamma_{\delta_n}}\xi_v$ is $$\begin{aligned} &\leq&(d-\delta_n)+(r_1+(n+3)\delta)-(d-r_1-\delta_{n-1}-(n+1)\delta)+2\delta\\ &=&(2r_1-2\delta)-\delta_n+\delta_{n-1}+(2n+8)\delta\\ &\leq&\mathrm{diam}(\xi_{v_1})-\delta,\end{aligned}$$ and since $v_1$ is the unique vertex of color $1$ in $\hat\Gamma_{\delta_n}$, this establishes the theorem. In order to prove Lemma [Lemma 26](#radiuslemma){reference-type="ref" reference="radiuslemma"} we first need to discuss the positions of centers of the sets $\mathcal X_v$. ### Centers {#centers .unnumbered} **Lemma 27**. *If $v$ and $w$ are adjacent vertices in $\hat\Gamma_{\delta_n}$, then $$\left<c_v,c_w\right>\geq d-{\delta_n+\delta_{n-1}\over 2}-r_1-\delta.$$* *Proof.* If $v$ and $w$ are adjacent in $\hat\Gamma_{\delta_n}$, then the sets $\mathcal X_v$ and $\mathcal X_w$ contain a common $\delta_n$-extremal point $p$. Since $p$ is $\delta_n$-extremal, Lemma [Lemma 13](#fellowtravel1){reference-type="ref" reference="fellowtravel1"} implies $$\left<p,c_v\right>\geq{|\mathcal X_v|+|p|\over 2}-r_v\geq{|\mathcal X_v|+d-\delta_n\over 2}-r_v.$$ If the color of $v$ is small, then $r_v\leq r_1-\delta_n$ and $|\mathcal X_v|\geq d-\delta_n$ so that $\left<p,c_v\right>\geq d-r_1$. If the color of $v$ is large, then $v\in\hat\Gamma_{\delta_{n-1}}$ so $|\mathcal X_v|\geq d-\delta_{n-1}$ and $-r_v\geq-r_1$, implying that $\left<p,c_v\right>\geq{d-\delta_{n-1}+d-\delta_n\over 2}-r_1$. In either case, we get the inequality $$\left<p,c_v\right>\geq d-{\delta_n+\delta_{n-1}\over 2}-r_1,$$ and also the same inequality with $v$ replaced by $w$. This implies the claimed inequality since, by definition of $\delta$-hyperbolicity, $\left<c_v,c_w\right>\geq\min(\left<c_v,p\right>,\left<p,c_w\right>)-\delta.$ ◻ Putting this together with the $2^n$-diameter bound for the connected graph $\hat\Gamma_{\delta_n}$ (Corollary [Corollary 23](#diamcor){reference-type="ref" reference="diamcor"}.1) and using the Gromov product inequality (Lemma [Lemma 19](#gromovproductinequality){reference-type="ref" reference="gromovproductinequality"}) gives for any pair of vertices $v,w\in\hat\Gamma_{\delta_n}$ $$\label{centerproduct} \left<c_v,c_w\right>\geq d-{\delta_n+\delta_{n-1}\over 2}-r_1-(n+1)\delta.$$ Now, for any $w\in\hat\Gamma_{\delta_n}$, pick $c'_w\in [o,c_w]$ such that $|c'_w|=d-{\delta_n+\delta_{n-1}\over 2}-r_1-(n+1)\delta$. Note that by ($\ref{centerproduct}$) all these points $c'_w$ are $\delta$-close to each other. The next lemma records this and summarizes how these points compare to the centers $c_w$ and the basepoint $c^*$ we chose earlier. **Lemma 28**. *For any vertices $v,w$ in $\hat\Gamma_{\delta_n}$ we have $$\begin{aligned} |c'_v-c'_w|&\leq&\delta,\\ |c^*-c'_{v_1}|&=&{\delta_n-\delta_{n-1}\over 2},\\ |c_w-c_w'|&\geq&{\delta_n-\delta_{n-1}\over 2}+(n+1)\delta.\end{aligned}$$* *Proof.* The first inequality follows from ([\[centerproduct\]](#centerproduct){reference-type="ref" reference="centerproduct"}) and hyperbolicity. The equality follows directly from the definition of $c^*$ as the point on $[o,c_{v_1}]$ such that $|c^*|=d-r_1-\delta_{n-1}-(n+1)\delta$. For the last inequality, note that $$|c_w|\geq d-\delta_{n-1}-r_1$$ (either $w$ is small, in which case $|c_w|\geq d-\delta_{n}-r_v\geq d-r_1$ or $w\in\hat\Gamma_{\delta_{n-1}}$ and then $|c_w|\geq d-\delta_{n-1}-r_v\geq d-\delta_{n-1}-r_1)$ and the inequality follows from this by subtracting $|c_w'|$ from both sides. ◻ Now we can prove Lemma [Lemma 26](#radiuslemma){reference-type="ref" reference="radiuslemma"} and thus finish the proof of the theorem. ### Proof of Lemma [Lemma 26](#radiuslemma){reference-type="ref" reference="radiuslemma"} {#proof-of-lemma-radiuslemma .unnumbered} Let $p'$ be the projection of $p$ onto $[o,c_w]$. There are two cases. - If $p'\in[o,c_w']$ then $${\delta_n-\delta_{n-1}\over 2}+|c_w'-p|\leq |c_w-c'_w|+|c_w'-p|\stackrel{h}\leq |c_w-p|+\delta\leq r_1+\delta$$ - If $p'\in[c_w',c_w]$ then, since $p$ is not $\delta_n$-extremal, $$\left(d-r_1-{\delta_n+\delta_{n-1}\over 2}-(n+1)\delta\right)+|c_w'-p|=|c'_w|+|c_w'-p|\stackrel{h}\leq|p|+\delta\leq d-\delta_n+\delta.$$ **Remark 9**. *The inequalities following from hyperbolicity are denoted $\stackrel{h}\leq$ for emphasis.* Rearranging, we see that in either case we have obtained the inequality $$|c_w'-p|\leq r_1 -{\delta_{n}-\delta_{n-1}\over 2} +(n+2)\delta.$$ Therefore, $$\begin{aligned} |c^*-p|&\leq&|c^*-c'_{v_1}|+|c'_{v_1}-c'_{w}|+|c'_w-p|,\\ &\leq&\left({\delta_n-\delta_{n-1}\over 2}\right)+\delta+\left(r_1-{\delta_n-\delta_{n-1}\over 2}+(n+2)\delta\right),\\ &=&r_1+(n+3)\delta. \end{aligned}$$ which finishes the proof. # Applications We now apply the algorithm. The main hypothesis in this section is: ($\mathcal{H}_{n, \delta}$): *$G$ acts on a $\delta$-hyperbolic space $\mathcal{H}$ with displacement greater than $(2n+11)^2\delta$.* Let $\mathrm{E}_n (\mathbb{K} [G])$ be the subgroup of elementary matrices in $\mathrm{GL}_n (\mathbb{K} [G])$. We begin with a linear algebraic lemma. **Lemma 29**. *Suppose the group $G$ satisfies $\mathcal H_{n,\delta}$. Let $\xi=(\xi_1,\dots,\xi_n)\in\mathbb K[G]^n$.* 1. *If the coordinates of $\xi$ are linearly dependent over $\mathbb K[G]$, then the $\mathrm{E}_n(\mathbb K[G])$-orbit of $\xi$ contains a vector which has at least one coordinate equal to $0$.* 2. *If there is $\alpha=(\alpha_1,\dots,\alpha_n)\in\mathbb K[G]^n$ satisfying $\alpha\cdot\xi=\sum\alpha_i\xi_i=1$, then the $\mathrm{E}_n(\mathbb K[G])$-orbit of $\xi$ contains $(\lambda g,0,\dots,0)$ for some $\lambda\in\mathbb K^*$ and $g\in G$.* *Proof.* 0. Pick a vector $\xi'$ in the $\mathrm{E}_n(\mathbb K[G])$-orbit of $\xi$ that minimizes the sum of diameters of its coordinates. The coordinates of $\xi'$ are still linearly dependent. If none of them are zero, then Theorem [Theorem 12](#intromaintheorem){reference-type="ref" reference="intromaintheorem"} would let us reduce the sum of diameters, contradicting the minimality assumption. 1\. We argue the same way. Pick a vector $\xi'=\xi U$ in the $\mathrm{E}_n(\mathbb K[G])$-orbit of $\xi$ that minimizes the sum of diameters of *nonzero* coordinates. Then $$1=\alpha\cdot\xi=\alpha U^{-t}\cdot \xi U=\alpha'\cdot\xi'.$$ Pick $i$ with $\alpha_i'\xi_i'\not=0$. If $|\alpha_i'\xi_i'|>0$ then we can apply Theorem [Theorem 12](#intromaintheorem){reference-type="ref" reference="intromaintheorem"} to reduce the sum of diameters of nonzero coordinates of $\xi'$, contradicting minimality. So, $|\alpha'_i\xi'_i|=0$, i.e. $\xi_i'$ is a unit. By [@delzant], $\mathbb K[G]$ only has trivial units, so $\xi_i'=\lambda g$ for some $\lambda\in\mathbb K^*$ and $g\in G$. The conclusion follows by applying elementary transformations to $\xi'$. ◻ ## Freeness This enables the study of finitely generated submodules of free modules. **Theorem 30**. *Assume the group $G$ satisfies ${\mathcal{H}_{n, \delta}}$.* 1. *Every $n$-generated ideal in $\mathbb{K} [G]$ is a free $\mathbb{K}[G]$-module.* 2. *Every $n$-generated submodule of a free $\mathbb{K} [G]$-module is a free $\mathbb{K} [G]$-module.* *Proof.* 1. Suppose we have shown that ideals generated by fewer than $n$ elements are free, and let $\mathcal{I}$ be an ideal in $\mathbb{K} [G]$ generated $n$ elements $\xi_1,\dots,\xi_n$. Consider the map $$\begin{aligned} \mathbb K[G]^n&{\rightarrow}&\mathbb{K}[G],\\ (\alpha_1, \ldots, \alpha_n) & \mapsto & \alpha_1 \xi_1+\ldots +\alpha_n\xi_n.\end{aligned}$$ If this map is injective, then it provides an isomorphism from the free module $\mathbb K[G]^n$ to the ideal $\mathcal I$. If it is not injective, then there is a non-trivial relation $\alpha_1\xi_1 + \ldots + \alpha_n \xi_n = 0$. In other words the family $\xi_1,\dots,\xi_n$ is linearly dependent. By Lemma [Lemma 29](#elementary){reference-type="ref" reference="elementary"}.0 we can do permutations and elementary transformations to replace $\xi_1,\dots,\xi_n$ by a generating set for $\mathcal I$ consisting of $n-1$ elements. Therefore $\mathcal I$ is free. 2\. Suppose $M \subset \mathbb{K} [G]^m$ is an $n$-generated submodule. Note that $M \otimes_{\mathbb{K} [G]} \mathbb{K}$ is a finite dimensional $\mathbb{K}$-vector space of some dimension $d\leq n$. We argue by induction on $d$. If $M \neq 0$ there is a projection to a factor $p : \mathbb{K} [G]^m\rightarrow \mathbb{K} [G]$ such that $p (M)$ is non-trivial. Since $p (M)$ is an $n$-generated ideal, it is free as a $\mathbb{K} [G]$ module by part 1. It follows that the module $M$ maps onto a non-zero free $\mathbb{K} [G]$-module and, a fortiori, onto $\mathbb{K} [G]$. Therefore, the module $M$ splits as $M \cong M' \oplus \mathbb{K} [G]$. Note that $\dim_{\mathbb{K}} (M\otimes_{\mathbb{K} [G]} \mathbb{K}) - 1 = \dim_{\mathbb{K}} (M' \otimes_{\mathbb{K} [G]} \mathbb{K})$, and the result follows. ◻ From this theorem and Stallings result on groups with infinitely many ends, we can also deduce: **Corollary 31**. *Under the same hypotheses, every $n$ generated subgroup of the group $G$ is free.* *Proof.* Let $\mathbb{K}$ be any field, for instance $\mathbb{K}=\mathbb{F}_2$. Suppose $(g_1, \ldots, g_n) = H<G$ is an $n$-generated subgroup. Then its augmentation ideal $(g_1 - 1, \ldots, g_n - 1)$ is a free ideal in $\mathbb{K} [H]$ by Theorem [Theorem 30](#freemodules){reference-type="ref" reference="freemodules"}.1. Clearly the group $H$ is torsion-free, so by 3.14 of [@dicksdunwoody], the group $H$ is a free group. ◻ **Remark 10**. *For the convenience of the reader, let us recall the argument of Dicks and Dunwoody ([@dicksdunwoody]). The main ingredient behind the passage from ideals to subgroups is Stallings theorem on ends ([@stallings]). If the augmentation ideal $\mathcal{I}$ is free, we have a free resolution $0\rightarrow \mathcal{I} \rightarrow \mathbb{F}_2 [G] \rightarrow\mathbb{F}_2 \rightarrow 0$, so that $H^1 (G, \mathbb{F}_2 [G]) \neq 0$: the group $G$ has several ends. As the group $G$ is torsion free, Stallings theorem implies that either it is infinite cyclic or that it splits as a free product $G = G_1 \ast G_2$. By Grushko's theorem, the groups $G_1, G_2$ have smaller rank, and one can conclude by induction. This argument was used by Stallings to prove that a group of cohomological dimension one is a free group.* **Remark 11**. *This result---large displacement ($\geq \rho$) implies that every $n$-generated subgroup is free---is not new. It has been stated by Gromov [@gromovhyperbolic], and proofs have been given by Arzhantseva [@arzhantseva] and Kapovich-Weidmann [@kapovichweidmann]. We can express the known quantitative results as follows. Let $N_{fr}(G)$ be the largest number $n$ such that $n$-generated subgroups of $G$ are free, and set $$N\left(\rho/\delta\right):=\min_{G}N_{fr}(G),$$ where the minimum is over all groups $G$ for which there is a $\delta$-hyperbolic space $\mathcal H$ and a $G$ action on $\mathcal H$ with minimal displacement $\geq \rho$. The best known estimate for $N$---due to Gromov [@gromovexpanders] p.763---is $N(\rho/\delta)\geq 10^{- 6} \frac{\rho /\delta}{\log(\rho/\delta)}$, which is a much better bound than ours, which is $N(\rho/\delta)\geq{1\over 2}\sqrt{\rho/\delta}-6$. Conjecturally ([@gromovexpanders]), $N(\rho/\delta)\geq (1+\varepsilon)^{\rho/\delta}$ for a universal positive constant $\varepsilon$.* ## The group $\text{GL}_n (\mathbb{K} [G])$ Recall that $\text{GE}_n (\mathbb{K} [G])$ is the subgroup of $\text{GL}_n (\mathbb{K} [G])$ generated by elementary and diagonal matrices. **Theorem 32**. *Assume $G$ satisfies ${\mathcal{H}_{n, \delta}}$. Then $$\mathrm{GL}_n (\mathbb{K} [G])=\mathrm{GE}_n (\mathbb{K} [G]).$$* *Proof.* We copy the usual proof of the fact that $\text{GL}_n (\mathbb{Z})$ is generated by elementary matrices and diagonal matrices with entries $\pm 1$. Let $X = (\xi_{ij})$ be in $\text{GL}_n (\mathbb{K} [G])$, and choose $A=(\alpha_{ij})$ in $\text{GL}_{_n} (\mathbb{K} [G])$ such that $AX = 1$. As $\sum_{i = 1}^n \alpha_{1 i} \xi_{i 1} = 1$, we can apply Lemma [Lemma 29](#elementary){reference-type="ref" reference="elementary"}.1, and deduce that there is a matrix $U$ in $\text{GE}_n (\mathbb{K} [G])$ such that the first row of $XU$ is $(u,0,\dots,0)$ for some unit $u\in\mathbb K[G]$. Left multiplying $XU$ by a product of elementary matrices, say $V \in \text{E}_n (\mathbb{K} [G])$, we obtain a matrix $VXU$ of the form $$\left(\begin{array}{cc} u & 0\\ 0 & Y \end{array}\right),$$ where $Y$ is a matrix in $\text{GL}_{n - 1} (\mathbb{K}[G])$. So, the theorem follows by induction on $n$. ◻ **Theorem 33**. *Assume $\mathcal H_{n,\delta}$. If $\mathbb K$ is finite and $G$ is finitely generated, then $\mathrm{GL}_n (\mathbb{K} [G])$ is finitely generated.* *Proof.* Recall that, because of the large displacement assumption (greater than $4\delta$ is enough), all units in $\mathbb K[G]$ are trivial by [@delzant]. Let $e_1,\dots,e_n$ be the standard basis for $\mathbb K[G]^n$. The group $\text{GE}_n(\mathbb K[G])$ is generated by the finitely many elementary transformations of the form $(e_i\mapsto e_i+e_j)$ and multiplication of basis elements by units of the form $\lambda g$, where $\lambda$ is in $\mathbb K^*$ and $g$ is in a generating set for $G$. So, the corollary follows from the previous theorem. ◻ **Remark 12**. *The proof only uses that $\mathbb K^*$ is finitely generated. However, no example of an infinite field with finitely generated $\mathbb K^*$ is known.* Theorems [Theorem 30](#freemodules){reference-type="ref" reference="freemodules"}.1 and [Theorem 32](#gltheorem){reference-type="ref" reference="gltheorem"} together establish Theorem [Theorem 4](#mainfirtheorem){reference-type="ref" reference="mainfirtheorem"} from the introduction. ## Submodules of free $\mathbb{Z} [G]$-modules Until now we've been studying the group algebra $\mathbb{K} [G]$ with coefficients in a field $\mathbb{K}$. Our next goal is to extend freeness results to the integral group ring $\mathbb{Z} [G]$. We follow the method Bass used in [@bass] to show projective $\mathbb Z[F]$-modules are free, but with two differences. First, the article [@bass] studies group rings with coefficients in a principal ideal domain. As we have no applications in this degree of generality, we restrict ourselves to the ring $\mathbb{Z}$. Second, the main hypothesis of [@bass] is that $M$ is a projective module. Instead, we use - The module $M$ embeds as a submodule of a free module $\oplus\mathbb{Z} [G]$ such the quotient $(\oplus \mathbb{Z}[G])/M$ is torsion free as an abelian group. This condition is probably known to specialists but hard to locate in the literature. It is a weakening of the more familiar "projective". Indeed, if $M$ is a projective module, there exists a module $N$ such that $M \oplus N = \oplus \mathbb{Z} [G]$ is a free module, in particular the quotient $N$ is torsion free. It is useful for obtaining topological consequences thanks to the following important example. **Example 3**. *Let $X$ be a finite cell complex with fundamental group $G$, $\widetilde{X}$ its universal cover and $C_{*} := C_{*} (\widetilde{X}; \mathbb{Z})$ its cellular chain complex. Then the kernels of each boundary map $\partial : C_k \rightarrow C_{k - 1}$ and of the augmentation map $C_0 \rightarrow \mathbb{Z}$ all satisfy condition $(\star)$, as their quotients are submodules of the free modules $C_{k - 1}$ and of $\mathbb{Z}$, respectively (see the proof of Theorem [Theorem 37](#cdim){reference-type="ref" reference="cdim"}). In particular the augmentation ideal of the group $G$, the relation module of a generating set for $G$, and the second homotopy module of a presentation $2$-complex for $G$ all satisfy $(\star)$.* Here is the main theorem of this subsection. **Theorem 34**. *Suppose the group $G$ satisfies $\mathcal{H}_{n, \delta}$. Let $M$ be a $n$-generated submodule of a free module $\oplus \mathbb{Z}G$ such that $(\oplus \mathbb{Z}G)/ M$ is torsion free as an abelian group. Then $M$ is free.* **Remark 13**. *Some assumption such as $(\star)$ is necessary to establish freeness: the ideal in $\mathbb{Z} [t, t^{- 1}]$ generated by $u = 2$ and $v = t - 1$ is not free, as it satisfies the relation $(t - 1) u - 2 v = 0$ and is not generated by a single element.* To ellucidate the role of $(\star)$ in the proof, we recall some basics on abelian groups. ### On abelian groups and torsion {#on-abelian-groups-and-torsion .unnumbered} For an abelian group $A$, let $A_{\mathbb Q}:=A\otimes_{\mathbb Z}\mathbb Q$ be its *rationalization* and $A_p:=A\otimes_{\mathbb Z}\mathbb F_p$ its *mod $p$ reduction*. An inclusion of abelian groups induces homomorphisms of ratinalizations and mod $p$ reductions. The later may no longer be an inclusion. In our proof, condition ($\star$) will be relevant because it implies injectivity of the induced map on mod $p$ reductions via the last part of the following lemma which summarizes basic properties of rationalizations and mod $p$ reductions that we will need. **Lemma 35**. *For any abelian group $A$,* - *the mod $p$ reduction can be expressed as $A_p=A\otimes_{\mathbb Z}\mathbb F_p\cong A/pA$.* - *If $A$ is torsion-free, then the natural map $A\rightarrow A_{\mathbb Q}$ is an embedding.* *Let $A\hookrightarrow B$ be an embedding of abelian groups. Then* - *the induced map of rationalizations $A_{\mathbb Q}\rightarrow B_{\mathbb Q}$ is injective, and* - *if $B/A$ has no $p$-torsion then the induced map $A_p\rightarrow B_p$ is injective.* *Proof.* The failure of tensor products to preserve injectivity is measured by Tor as appying $A\otimes_{\mathbb Z}-$ to an exact sequence of abelian groups $0\rightarrow B\rightarrow C\rightarrow D\rightarrow 0$ gives the exact sequence $$\mathrm{Tor}(A,D)\rightarrow A\otimes_{\mathbb Z}B\rightarrow A\otimes_{\mathbb Z}C\rightarrow A\otimes_{\mathbb Z}D\rightarrow 0.$$ The basic properties of Tor we will use can be found in 3A.5 of [@hatcherbook]. For the first point, apply $A\otimes_{\mathbb Z}-$ to the exact sequence $0\rightarrow\mathbb Z\stackrel p\rightarrow\mathbb Z\rightarrow\mathbb F_p\rightarrow 0$. For the second, apply it to the exact sequence $0\rightarrow\mathbb Z\rightarrow\mathbb Q\rightarrow\mathbb Q/\mathbb Z\rightarrow 0$ and note that the tor term $\mathrm{Tor}(A,\mathbb Q/\mathbb Z)$ vanishes because $A$ is torsion-free. For the third point, apply $-\otimes_{\mathbb Z}\mathbb Q$ to $0\rightarrow A\rightarrow B\rightarrow B/A\rightarrow 0$ and note that $\mathrm{Tor}(B/A,\mathbb Q)$ vanishes beecause $\mathbb Q$ is torsion-free. For the forth point, apply $-\otimes_{\mathbb Z}\mathbb F_p$ to the same sequence and note that the tor term $\mathrm{Tor}(B/A,\mathbb F_p)=\ker(B/A\stackrel{p}\rightarrow B/A)$ vanishes because $B/A$ has no $p$-torsion. ◻ ### A 'local-to-global' principle for rings {#a-local-to-global-principle-for-rings .unnumbered} We can now prove Theorem [Theorem 34](#freezmodules){reference-type="ref" reference="freezmodules"}. Note that $\mathbb Z[G]$ is torsion-free as an abelian group, so any submodule of $\oplus\mathbb Z[G]$ is, as well. Moreover, the rank of a finitely generated free $\mathbb K[G]$-module is the dimension of the vector space of coinvariants $\mathbb K[G]^m\otimes_{\mathbb K[G]}\mathbb K=\mathbb K^m$. So, Theorem [Theorem 34](#freezmodules){reference-type="ref" reference="freezmodules"} is a consequence of Theorem [Theorem 30](#freemodules){reference-type="ref" reference="freemodules"}.2, Theorem [Theorem 32](#gltheorem){reference-type="ref" reference="gltheorem"} and the $R=\mathbb Z[G]$ case of the following purely ring theoretic proposition. **Proposition 36**. *Suppose $R$ is a ring satisfying the following properties:* - *every $n$-generated submodule of $\oplus R_{\mathbb Q}$ is free of unique rank[^4],* *and for all primes $p$* - *every $n$-generated submodule of $\oplus R_p$ is free of unique rank, and* - *for $m\leqslant n-1$ we have $\mathrm{GL}_m(R_p)=\mathrm{GE}_m(R_p)$.* *If $M$ is an $n$-generated submodule of $\oplus R$ such that both $M$ and $(\oplus R)/M$ are torsion-free abelian groups, then $M$ is a free $R$-module.* *Proof.* Since $M$ is torsion-free, the map $M\rightarrow M_{\mathbb Q}$ is an embedding. So, we can think of $M$ as a subgroup of $M_{\mathbb Q}$. Let $x_1, \dots, x_n$ generate the $R$-module $M$. By Lemma [Lemma 35](#abgroup){reference-type="ref" reference="abgroup"}.3, $M_{\mathbb{Q}}$ is a submodule of the free $R_{\mathbb Q}$-module $\oplus R_{\mathbb Q}$. So, by our ($\mathbb{Q}$) hypothesis, it is a free $R_{\mathbb Q}$-module. If this module is of rank $n$ (not $\leqslant n - 1$), then the family $x_1, \ldots, x_n$ is an $R_{\mathbb Q}$-basis for $M_{\mathbb Q}$. Therefore, it is an $R$-basis for $M$, and hence $M$ is free of rank $n$, and we are done. Otherwise, $M_{\mathbb{Q}}$ is of rank $m < n$. Let $y_1,\ldots, y_m$ be an $R_{\mathbb Q}$-basis for $M_{\mathbb Q}$. Clearing denominators, we may assume that each $y_i$ is in $M$. Then, the $y_i$ generate a free, rank $m$ $R$-submodule $Y:=\left<y_1,\dots,y_m\right>$ of $M$. Next, since the $y_i$ form an $R_{\mathbb Q}$-basis for $M$, each $x_i$ can expressed as a $R_{\mathbb Q}$-linear combination of the $y_i$. We can clear denominators in these expressions and find a single positive integer $k$ such that---for all $j$---$kx_j$ is a $R$-linear combination of the $y_i$. In summary we have obtained a free $R$-module $Y$, a positive number $k$, and inclusions $$kM\subset Y\subset M.$$ Let $k\geq 1$ be the smallest number such that there is a free $R$-module $Y$ of rank $m$ with $kM\subset Y\subset M$. If $k=1$, then we are done since then $M=Y$ is a free $R$-module. So, towards a contradiction, suppose $k>1$. We will find a free, rank $m$ $R$-module $Y'$ and $1\leq k'<k$ such that $k'M\subset Y'\subset M$. To that end, pick a prime $p$ dividing $k$ and let $f:Y_p\rightarrow M_p$ be the mod $p$ reduction of the second inclusion above. By Lemma [Lemma 35](#abgroup){reference-type="ref" reference="abgroup"}.4 and the hypothesis that $(\oplus R)/M$ is torsion-free as an abelian group, the module $M_p$ embeds in the free module $\oplus R_p$. Since $M_p$ is generated by $n$-elements, our hypothesis ($p_0$) shows that it is free. The module $f (Y_p)$ is a submodule of $M_p$, and is generated by $m<n$ elements, so it is again free by $(p_0)$. Thus, we have a splitting $$Y_p \cong\ker(f) \oplus\mathrm{im} (f) .$$ Note that the left hand side is also a free $R_p$-module, since it the mod $p$ reduction of the free $R$-module $Y$. The splitting shows that $\ker (f)$ is an $m$-generated submodule of a free $R_p$-module, so it is also free. Therefore, we can pick an $R_p$-basis $z_1,\dots,z_k,z_{k+1},\dots,z_m$ for the module $Y_p$ so that the first $k$ elements are a basis for the kernel of $f$. By our ($p_1$) hypothesis, there exists a matrix $U$ in $\mathrm{E}_m (R_p)$ transforming the mod $p$ reduction of the family $\{y_i\}$ to the family $\{u_i z_i\}$ where the $u_i$ are units in $R_p$. As a matrix in $\mathrm{E}_m (R_p)$ is a product of elementary matrices, we can lift $U$ to $\mathrm{E}_m (R)$ transforming the family $\{y_i\}$ to a family $\{y'_i\}$ such that the reduction mod $p$ of $y'_i$ is $u_i z_i$. Let $P:=\left<y_1',\dots,y_k'\right>$ and $Q:=\left<y_{k+1}',\dots,y_m'\right>$, so that $$Y=P\oplus Q,$$ and on mod $p$ reductions - $P_p\rightarrow M_p$ is the zero map, i.e. $P\subset pM$, while - $Q_p\rightarrow M_p$ is injective, i.e. $pQ=Q\cap pM$. For $x\in M$, the inclusion $kM\subset Y=P\oplus Q$ lets us write $kx=a+b$ where $a\in P$ and $b\in Q$. Since $P\subset pM$, we have[^5] ${a\over p}\in {1\over p}P\subset M$ and hence $b=p({k\over p}x-{a\over p})\in Q\cap pM=pQ$. Therefore, ${b\over p}\in Q$. Since ${k\over p}x={a\over p}+{b\over p}$ we have obtained the inclusions $${k\over p}M\subset {1\over p}P\oplus Q\subset M.$$ Since ${1\over p}P\oplus Q$ is a free $R$-module of rank $m$ (with basis $\{{y_1'\over p},\dots,{y_k'\over p},y'_{k+1},\dots,y_m'\}$) we arrive at a contradiction to the minimality of $Y$. ◻ **Remark 14**. *One may ask whether there is a local-to-global argument taking as input $\mathrm{GL}_m(R_{\mathbb K})=\mathrm{GE}_m(R_{\mathbb K})$ for all $m\leq n$ and all fields $\mathbb K$ that leads to $\mathrm{GL}_n(R)=\mathrm{GE}_n(R)$. This is not the case. Indeed, the polynomial ring $R=\mathbb Z[t]$ satisfies the hypothesis for all $n$ and all $\mathbb K$, but the matrix $$\left( \begin{array}{cc} 4&1+2t\\ 1-2t&-t^2 \end{array} \right)\in \mathrm{GL}_2(\mathbb Z[t])$$ is not in $\mathrm{GE}_2(\mathbb Z[t])$ ([@cohngl], p.30). In fact, $\mathrm{GL}_2(\mathbb Z[t])/\mathrm{GE}_2(\mathbb Z[t])$ is quite large ([@krsticmccool]). However, the above example goes away if we invert $t$, and [@abramenko] conjectures that $\mathrm{GL}_2(\mathbb Z[t,t^{-1}])=\mathrm{GE}_2(\mathbb Z[t,t^{-1}])$.* ## Chain complexes, cell decompositions, and Morse theory Recall that the cohomological dimension of the group $G$, denoted $\mathrm{cd}(G)$, is the minimal length of a free $\mathbb Z[G]$-resolution of $\mathbb Z$ (see [@brownbook]). **Theorem 37**. *Assume that the group $G$ satisfies $\mathcal{H}_{n, \delta}$.* 1. *For every free $\mathbb Z[G]$-resolution $C_*\rightarrow\mathbb Z$ and each $0<k<\mathrm{cd}(G)$ we have $$\mathrm{rank}_{\mathbb Z[G]}(C_k)>n.$$* 2. *(Theorem [Theorem 9](#cellbound){reference-type="ref" reference="cellbound"} in the introduction) Every aspherical cell complex with fundamental group $G$ has more than $n$ cells of each dimension $0 < k < \mathrm{cd} (G)$.* *Proof.* First, we prove the algebraic part. Suppose rank$(C_k)\leqslant n$. Then the module of boundaries, $B_{k-1}:= \partial (C_k)\subset C_{k - 1}$ is an $n$-generated submodule of a free module. Since $C_{\ast}$ is a resolution, $\ker \left( C_{k - 1} \overset{\partial}{\rightarrow} C_{k - 2} \right) = \partial (C_k)$, so $C_{k - 1} / \partial (C_k)$ injects into $C_{k - 2}$, which is either a free module (if $k \geq 2$) or $\mathbb{Z}$ (if $k = 1$). In either case, $C_{k - 1} / \partial (C_k)$ is torsion-free as an abelian group. Hence, $B_{k-1}$ satisfies $(\star)$ and we conclude from Theorem [Theorem 34](#freezmodules){reference-type="ref" reference="freezmodules"} that it is free. So, we obtain a new free resolution $$0 \rightarrow B_{k-1} \rightarrow C_{k - 1}\rightarrow \ldots \rightarrow C_0 \rightarrow \mathbb{Z} \rightarrow 0$$ of length $k$. Thus $\mathrm{cd} (G) \leq k$. This proves 1. Now, suppose $X$ is an aspherical cell complex with fundamental group $G$. Let $C_* (\widetilde{X} ; \mathbb{Z})$ be the cellular chain complex of the universal cover of $X$. The augmented complex $C_*(\widetilde X;\mathbb Z)\rightarrow \mathbb{Z}$ is a free $\mathbb{Z} [G]$ resolution of $\mathbb{Z}$ ([@brownbook], Prop I. 4.1). Applying part 1 to this resolution gives part 2. ◻ ### Essential maps {#essential-maps .unnumbered} If $\mathbb{K}$ is a field, the same result is true with $\mathbb Z$ replaced by $\mathbb K$ and cohomological dimension by $\mathbb K$-cohomological dimension ($=$ minimal length of a free $\mathbb K[G]$-resolution of $\mathbb K$), and easier to prove as we don't need Theorem [Theorem 34](#freezmodules){reference-type="ref" reference="freezmodules"}, but only Theorem [Theorem 30](#freemodules){reference-type="ref" reference="freemodules"}.2. In fact, in the setting of $\mathbb K$-cohomological dimension we have the following more general result suggested by a question of Gromov. Let $G$ be a classifying space for the group $G$. For a cell complex $X$, we say that a map $X\rightarrow BG$ is *$d$-essential (with local coefficients)* if there is a $\mathbb K[G]$-module $V$ such that the induced map $H^{d}(BG;V)\rightarrow H^{d}(X;V)$ is non-zero. For the sake of brevity, we will omit "with local coefficients" from now on. We will call a $d$-manifold $X$ *essential* if the map $X\rightarrow B\pi_1(X)$ is $d$-essential. **Example 4**. *Any map of non-zero degree from a closed manifold to a closed aspherical manifold of dimension $d$ is $d$-essential.* **Theorem 38**. *Assume that the group $G$ satisfies $\mathcal H_{n,\delta}$. If $X\rightarrow BG$ is a $d$-essential map, then $X$ has more than $n$ cells in each dimension $0<k<d$.* *Proof.* Suppose not. Let $\hat X$ be the $G$-cover of $X$ induced by the map $f:X\rightarrow BG$. Note that the image of the map $f_*\circ\partial:C_k(\hat X;\mathbb K)\rightarrow C_{k-1}(EG;\mathbb K)$ is an $n$-generated submodule of a free module, hence it is free by Theorem [Theorem 30](#freemodules){reference-type="ref" reference="freemodules"}.2. So, if $i$ denotes the inclusion $\partial C_k\hookrightarrow C_{k-1}$, then map $f_*\circ i:\partial C_k(\hat X;\mathbb K)\rightarrow C_{k-1}(EG;\mathbb K)$ lifts to $f':\partial C_k(\hat X;\mathbb K)\rightarrow C_k(EG;\mathbb K)$ satisfying $f_*\circ i =\partial\circ f'$. In summary, we have a commutative diagram $$\begin{array}{cccccccc} \dots\rightarrow & C_{k+1}(\hat X;\mathbb K)&\rightarrow &C_k(\hat X\;\mathbb K)&\stackrel{\partial}\rightarrow &C_{k-1}(\hat X;\mathbb K)&\rightarrow\dots\rightarrow &C_0(\hat X;\mathbb K)\\ &\downarrow&&\downarrow\partial&&||\hspace{0.5cm}&&||\\ \dots\rightarrow & 0&\rightarrow &\partial C_k(\hat X;\mathbb K)&\stackrel{i}\hookrightarrow &C_{k-1}(\hat X;\mathbb K)&\rightarrow\dots\rightarrow &C_0(\hat X;\mathbb K)\\ &\downarrow&&\downarrow f'&&\downarrow f_*&&\downarrow\\ \dots\rightarrow & C_{k+1}(EG;\mathbb K)&\rightarrow &C_k(EG;\mathbb K)&\stackrel{\partial}\rightarrow &C_{k-1}(EG;\mathbb K)&\rightarrow\dots\rightarrow &C_0(EG;\mathbb K). \end{array}$$ This shows that the chain map $C_*(\hat X;\mathbb K)\rightarrow C_*(EG;\mathbb K)$ factors through a chain complex $T_*$ (the middle row in the above diagram) which has no terms in degree $d>k$. Therefore, the composition $$H^{d}(BG;V)\rightarrow H^{d}(T_*;V)\rightarrow H^{d}(X;V)$$ is the zero map, contradicting the hypothesis that $X\rightarrow BG$ is $d$-essential. ◻ Here is a special case, where we suppose that $X^d$ is a closed $d$-manifold. **Theorem 39**. *Let $X^d$ be a $d$-manifold, $G$ a group that satisfies $\mathcal H_{n,\delta}$ and $BG$ its classifying space. If there is a continuous map $f:X^d\rightarrow BG$ with $f_*[X]\not=0$ in $H_d(BG;\mathbb K)$ then for each $0<k<d$, a Morse function on $X^d$ has at least $n+1$ critical points of index $k$.* **Remark 15**. *Since the fundamental group of a hyperbolic manifold is not free, the case of critical points of index $1$ or $d-1$ follows from the aforementioned theorem of Arzhantseva, Gromov, Kapovich-Weidmann ([@arzhantseva; @gromovexpanders; @kapovichweidmann]). For hyperbolic manifolds of dimension three, a much better bound is given by [@bachman].* ## Dimensions, few relator groups, and $2$-complexes Concerning few relator groups, we get the following. **Theorem 40**. *An $n$-relator group satisfying $\mathcal{H}_{n, \delta}$ has cohomological dimension $\leq 2$.* *Proof.* An $n$-relator group is the fundamental group of an aspherical cell complex with $n$ $2$-cells (and cells of higher dimension). If such a group satisfies $\mathcal{H}_{n, \delta}$ then, by Theorem [Theorem 37](#cdim){reference-type="ref" reference="cdim"}.2, it has cohomological dimension $\leq 2$. ◻ **Question 41**. *Does every $n$-relator group satisfying $\mathcal{H}_{n,\delta}$ have geometric dimension $\leq 2$?* **Remark 16**. *This question is an instance of a problem raised by Eilenberg and Ganea in [@eilenbergganea], asking whether there is an example of a group whose geometric and cohomological dimensions differ:* > *"*We do not know whether these exceptional cases are actually present. The inequality dim $\Pi<$ cat $\Pi$ (cases A and B) is equivalent with the assertion that $\Pi$ is one-dimensional but not free. The problem of the existence of such a group has equivalent formulations in terms of group extensions and also in terms of properties of the integral group ring $\Lambda=\mathbb Z[\Pi]$. Similarly, the inequality cat $\Pi<$ geom. dim $\Pi$ (cases B and C) is related to properties of the ring $\Lambda$. For instance, if it can be shown that a direct summand of a free $\Lambda$-module is free, then the equality cat $\Pi$=geom. dim $\Pi$ follows.*" ([@eilenbergganea], p. 517-518).* *In [@eilenbergganea] "dim" is the cohomological dimension and there is also an intermediate dimension "cat" that was later shown to be equivalent to cohomological dimension by Stallings [@stallings]. Cases A and B refer to the hypothetical situation ($1=\dim \Pi<$ cat $\Pi=2$) which is now known to not occur. Case C refers to the potential situation ($2=$ dim $\Pi=$ cat $\Pi<$ geom. dim $\Pi=3$). The conjecture that case C does not occur, either, is nowadays referred to as the Eilenberg-Ganea conjecture. A proof of the reduction of this conjecture to a question about group rings claimed in the last sentence of the quote may help with Question [Question 41](#hypeg){reference-type="ref" reference="hypeg"}, but we do not know how to establish such a reduction, even with the additional assumption $\mathrm{GL}_n(\Lambda)=\mathrm{GE}_n(\Lambda)$ for all $n$.* Concerning the topology of presentation complexes, we get the following. **Theorem 42**. *Assume that the group $G$ satisfies $\mathcal{H}_{n, \delta}$.* 1. *Every presentation $2$-complex for $G$ with $n$ relations (or less) has a free $\pi_2$.* 2. *(Theorem [Theorem 7](#hypstandard){reference-type="ref" reference="hypstandard"} from introduction) Assume further that the group $G$ has geometric dimension two, and let $Y$ be an aspherical $2$-complex with fundamental group $G$. Then every presentation $2$-complex with less than $n+1$ relations is standard, i.e. has the same homotopy type as $$Y \vee S^2 \vee \ldots \vee S^2.$$* *Proof.* If $X$ is an $n$-relator presentation $2$-complex for such a group and $C_{\ast} = C_{\ast} (\tilde{X} ; \mathbb{Z})$ is the chain complex of its universal cover, then $\partial (C_2)$ is an $n$-generated submodule of $C_1$ satisfying $(\star)$. Hence it is a free module (by Theorem [Theorem 34](#freezmodules){reference-type="ref" reference="freezmodules"}), and we have a splitting $C_2 = \pi_2 (X) \oplus \partial (C_2)$. By projecting the generators of $C_2$ to $\pi_2 (X)$, we see that $\pi_2 (X)$ is generated by $n$ elements. Moreover, $\pi_2 (X)$ satisfies $(\star)$ (in fact, it is projective) so $\pi_2 (X)$ is free by Theorem [Theorem 34](#freezmodules){reference-type="ref" reference="freezmodules"}, i.e. $\pi_2 (X) \cong\mathbb{Z} [G]^m$ for some $m$. This proves 1. If $Y$ is an aspherical $2$-complex $Y$ with fundamental group $G$, we can construct a homotopy equivalence $f : Y \vee S_1^2 \vee \ldots \vee S^2_m \rightarrow X$ as follows. First, construct a map realizing the $\pi_1$-isomorphism $Y \rightarrow X$. This can be done since $Y$ is a $2$-complex. Second, let the $(S^2_i)_{1 \leqslant i \leqslant m}$ represent a $\mathbb{Z} [G]$-basis of $\pi_2 (X)$. Then the resulting map $f$ is a $\pi_1$-isomorphism and a homology isomorphism of universal covers, hence a homotopy equivalence. This proves 2. ◻ [^1]: See also Theorem [Theorem 38](#essential){reference-type="ref" reference="essential"}. [^2]: This is also the point such that $|b-x'|=\left<a,x\right>_b$, since $\left<a,x\right>_b+\left<b,x\right>_a=|a-b|$. [^3]: Repetitions in $Y=(\mathcal Y_v)_{v\in V}$ are allowed, i.e. we may have $\mathcal Y_v=\mathcal Y_w$ even if $v\not=w$. [^4]: *A finitely generated $R$-module is free of unique rank if it is isomorphic to $R^m$ for a unique $m$.* [^5]: For an element $a$ in a torsion-free abelian group, if there is an element $a_0$ such that $a=pa_0$, then there is a unique such element. We call this element ${a\over p}$.

arxiv_math

--- abstract: | We show there exist representations of each maximal compact subgroup $K$ of the $p$-adic group $G=\mathrm{SL}(2,F)$, attached to each nilpotent coadjoint orbit, such that every irreducible representation of $G$, upon restriction to a suitable subgroup of $K$, is a sum of these five representations in the Grothendieck group. This is a representation-theoretic analogue of the analytic local character expansion due to Harish-Chandra and Howe. Moreover, we show for general connected reductive groups that the wave front set of many irreducible positive-depth representations of $G$ are completely determined by the *nilpotent support* of their unrefined minimal $K$-types. address: Department of Mathematics and Statistics, University of Ottawa, Canada K1N 6N5 author: - Monica Nevins title: "The local character expansion as branching rules: nilpotent cones and the case of $\\mathrm{SL}(2)$" --- [^1] # Introduction The distribution character of an admissible representation of a $p$-adic group can be expressed, in a neighbourhood of the identity, as a linear combination of Fourier transforms of the finitely many nilpotent orbital integrals in the dual of the Lie algebra. This remarkable theorem, known as the Harish-Chandra--Howe local character expansion, has many variations (such as expansions on neighbourhoods of other semisimple elements, or expansions in terms of other collections of orbital integrals [@KimMurnaghan2003; @KimMurnaghan2006; @Spice2018]) and many applications (such as determining the Gel'fand--Kirillov dimension of a representation, or relating to conjectural classifications such as the orbit method, or the local Langlands correspondence [@BarbaschMoy1997; @CiubotaruMason-BrownOkada2021; @CiubotaruMason-BrownOkada2022; @JiangLiuZhang2022]). Though it is primarily considered in characteristic zero, it also holds when the characteristic is sufficiently large and a suitable substitute for the exponential map exists [@CluckersGordonHalupczok2014]. In this paper, we propose an interpretation of the local character expansion as a statement in the Grothendieck group of representations of a maximal compact open subgroup, upon restriction to a subgroup of suitable depth, and prove this for the case that $G=SL(2,F)$, where $F$ is a local nonarchimedean field of residual characteristic at least $3$. In particular, we construct for each nilpotent orbit $\mathcal{O}$ of $G$ in the dual of its Lie algebra $\mathfrak{g}^*$ a (highly reducible) representation $\tau_x(\mathcal{O})$ of each maximal compact open subgroup $G_x$ with the following property. **Theorem 1**. *Let $\pi$ be an irreducible admissible representation of $G=\mathrm{SL}(2,F)$ of depth $r\geq 0$ and let $x$ be a vertex in the building of $G$. Then there exist integers $c_{x,\mathcal{O}}(\pi)$ such that in the Grothendieck group of representations we have $$\label{LCEbrules} \mathrm{Res}^G_{G_{x,r+}}\pi = \sum_{\mathcal{O}} c_{x,\mathcal{O}}(\pi) \mathrm{Res}^{G_x}_{G_{x,r+}}\tau_{x}(\mathcal{O})$$ where $G_{x,r+}$ is the Moy--Prasad filtration subgroup of $G_x$ of depth $r+$, and the sum is over all nilpotent orbits in $\mathfrak{g}^*$.* Moreover, the coefficients corresponding to the regular nilpotent orbits in this expansion are nonnegative integers and agree with those of the Harish-Chandra--Howe local character expansion (subject to suitable normalizations). Note that while inherently expressing the same local nature of representations, our statement holds with fewer restrictions on $F$ than does the local character expansion, because it does not depend on the existence of a $G$-equivariant map, such as the exponential or a Cayley transform, from the Lie algebra to the group. Now suppose $G$ is a general connected reductive group. It is reasonable to conjecture that a decomposition of the form [\[LCEbrules\]](#LCEbrules){reference-type="eqref" reference="LCEbrules"} should hold for an irreducible representation $\pi$ of $G$, and we develop some theory towards this end in Section [3](#S:Nilpotent){reference-type="ref" reference="S:Nilpotent"}, as follows. The set of maximal orbits appearing in the local character expansion for an admissible representation $\pi$ is denoted $\mathcal{WF}(\pi)$; the closure of the union of these orbits is the wave front set of $\pi$. For depth-zero representations $\pi$, Barbasch and Moy [@BarbaschMoy1997] proved that $\mathcal{WF}(\pi)$ is determined by the depth-zero components of the restriction of $\pi$ to various maximal compact subgroups, through the theory of Gel'fand--Graev representations. For a positive-depth representation with minimal $K$-type $\Gamma$ (in the sense of Moy and Prasad [@MoyPrasad1994]), we should instead infer $\mathcal{WF}(\pi)$ from the *nilpotent support* $\mathrm{Nil}(\Gamma)$ (Definition [Definition 4](#D:nilpotentsupport){reference-type="ref" reference="D:nilpotentsupport"}) of $\Gamma$. This definition, of independent interest, depends strongly on the classification of nilpotent orbits using Bruhat--Tits theory [@BarbaschMoy1997; @DeBackerNilpotent2002]. In fact, in Proposition [Proposition 6](#P:cone=nil){reference-type="ref" reference="P:cone=nil"} we show that the algebraic notion of nilpotent support can be characterized as the set of nonzero nilpotent orbits appearing in the asymptotic cone on $\Gamma$, as defined in [@AdamsVogan2021]. In Theorem [Theorem 7](#T:nil=wf){reference-type="ref" reference="T:nil=wf"} (proof due to Fiona Murnaghan), we prove that $\mathcal{WF}(\pi)$ is the set of maximal orbits of $\mathrm{Nil}(\Gamma)$ whenever the $\Gamma$-asymptotic expansion [@KimMurnaghan2003] reduces to a single term. This last result is similar to recent work of Ciubotaru and Okada, who show that the depth-$r$ components of the restriction to certain compact open subgroups determine the wave front set of $\pi$ [@CiubotaruOkada2023]. The idea of the nilpotent support is also central to [@CiubotaruOkada2023], where they develop it using, among other things, the geometry of the associated finite reductive group. Now again suppose that $G=\mathrm{SL}(2,F)$. Our result gives a second characterization of $\mathcal{WF}(\pi)$: it can be entirely determined from the *non-typical* representations occurring in the restriction of $\pi$ to a maximal compact open subgroup, for $\pi$ of any depth. That is, the asymptotic decomposition of $\mathrm{Res}_{G_x}\pi$ unfolds exactly as the representations $\tau_x(\mathcal{O})$ for $\mathcal{O}\in \mathcal{WF}(\pi)$. For the case of a positive-depth representation $\pi$, our main theorem is stated in Theorem [Theorem 21](#T:posdepth2){reference-type="ref" reference="T:posdepth2"}, with the explicit values of the constant coefficient given in Proposition [Proposition 23](#P:npi){reference-type="ref" reference="P:npi"}. To prove the theorem, we first show that the restriction of $\pi$ to a maximal compact subgroup can be expressed entirely in terms of twists of the pair $(\Gamma,\chi)$ used in the construction of $\pi$ (Theorem [Theorem 18](#T:posdepth){reference-type="ref" reference="T:posdepth"}), using results from [@Nevins2005; @Nevins2013]. Here, $\chi$ is a character of a torus $T=\mathrm{Cent}_G(\Gamma)$ that is realized by $\Gamma \in \mathfrak{g}^*$, and the realization of the irreducible components of the restriction is framed in terms of a generalization (Proposition [Proposition 14](#P:Gxreps){reference-type="ref" reference="P:Gxreps"}) of a construction due to Shalika in his thesis. From this characterization, and a key technical result (Lemma [Lemma 15](#L:Shalikaequiv){reference-type="ref" reference="L:Shalikaequiv"}), it follows that the expansion [\[LCEbrules\]](#LCEbrules){reference-type="eqref" reference="LCEbrules"} exists and has leading terms corresponding to the nilpotent support of $\Gamma$. Since $\Gamma$ represents a minimal $K$-type of $\pi$ in the sense of Moy and Prasad [@MoyPrasad1994], we independently recover from Theorem [Theorem 7](#T:nil=wf){reference-type="ref" reference="T:nil=wf"} that the maximal orbits in $\mathrm{Nil}(\Gamma)$ coincide with $\mathcal{WF}(\pi)$. For representations of depth zero, the principal technical difficulties lie in matching the depth-zero components with nilpotent orbits, particularly in the case of the twelve "exceptional" representations: the reducible principal series, the principal series composed of the trivial and the Steinberg representation, and the four special supercuspidal representations. Once these are addressed, Theorem [Theorem 28](#T:zerodepth2){reference-type="ref" reference="T:zerodepth2"} follows by carefully extracting the necessary branching rules from [@Nevins2005; @Nevins2013]. Again, the orbits in $\mathcal{WF}(\pi)$ are obtained both from the depth-zero components (via [@BarbaschMoy1997]) and from the asymptotic development of the branching rules. At two crucial junctures we use information that is currently only known for $G=\mathrm{SL}(2,F)$ and a handful of other small rank groups: one is the explicit calculation of the asymptotic cone on any semisimple element of $\mathfrak{g}^*$ (Section [4](#S:sl2nil){reference-type="ref" reference="S:sl2nil"}); the other is the full knowledge of the representation theory of the maximal compact subgroups of $G$ (Section [5](#S:Representations){reference-type="ref" reference="S:Representations"}). While the former seems a tractable and interesting question in general, the latter is quite daunting: it is not expected that we will achieve a classification of the representations of maximal compact open subgroups of $p$-adic reductive groups. Note that a full classification is not necessary to prove the theorem: what is needed is a construction of an appropriate representation of $G_x$ attached to each nilpotent orbit, and we offer indications as to how this might be done, in Section [5.2](#S:gx){reference-type="ref" reference="S:gx"}. Attaching representations of $G$ to nilpotent orbits to reflect the wave front set has been an active area of research for some time. For example, in [@Savin1996], Savin establishes the result for minimal representations of groups of type $D$ and $E$. For groups of type $A_n$, there are many results available that may suffice in some cases, including the interesting work of Patel and Singla [@PatelSingla2022] about regular generic representations. Our work suggests a variation on this theme. There are many interesting applications and open directions left to pursue. Evidently an immediate goal is to establish a result like [\[LCEbrules\]](#LCEbrules){reference-type="eqref" reference="LCEbrules"} for a large class of groups. It may also be fruitful to build representations of the groups $G_{x,0+}$ directly, rather than to construct representations of $G_{x,0}$; this has the advantage of avoiding the difficulties inherent at depth zero. It may also allow for a more uniform treatment of all points $x$ of the building; in this paper, we consider only vertices, and the union of all $G_{x,r+}$ as $x$ runs over vertices is not equal to $G_{r+}$ in general. In another direction: the $\Gamma$-asymptotic expansions of [@KimMurnaghan2003; @KimMurnaghan2006] describe the character of a positive-depth representation in a larger neighbourhood than does the local character expansion, by incorporating a minimal $K$-type $\Gamma$. Then Theorem [Theorem 18](#T:posdepth){reference-type="ref" reference="T:posdepth"} can be interpreted as analogously formulating these expansions in terms of branching rules. It would be interesting to explore this idea further. The paper is organized as follows. We set our notation in Section [2](#S:Notation){reference-type="ref" reference="S:Notation"} and then present some background on the local character expansion that provide the motivation and context for our results. In Section [3](#S:Nilpotent){reference-type="ref" reference="S:Nilpotent"} we consider a general connected reductive group $G$. We define the nilpotent support of an element $\Gamma$ of $\mathfrak{g}^*$, show it defines the asymptotic cone of $\Gamma$, and relate this to the wave front set via the theory of $\Gamma$-asymptotic expansions. We then specialize to $G=\mathrm{SL}(2,F)$. In Section [4](#S:sl2nil){reference-type="ref" reference="S:sl2nil"} we characterize the nilpotent cones $\mathrm{Nil}(\Gamma)$ in many ways (Proposition [Proposition 9](#P:asymptotic_orbit_invariance){reference-type="ref" reference="P:asymptotic_orbit_invariance"}) and compute them explicitly. In Section [5](#S:Representations){reference-type="ref" reference="S:Representations"} we recall the construction of certain irreducible representations of $\mathrm{SL}(2,\mathcal{R})$ by Shalika in his 1966 thesis [@Shalika1966], and then rephrase it using Bruhat--Tits theory and derive some consequences. This allows us to define, for each vertex $x\in\mathcal{B}(G)$, each nilpotent orbit $\mathcal{O}\subset \mathfrak{g}^*$, and each central character $\zeta$ a representation $\tau_x(\mathcal{O},\zeta)$ of $G_x$. We prove our main theorems for representations of positive depth in Section [6](#S:posdepth){reference-type="ref" reference="S:posdepth"} and for representations of depth zero in Section [7](#S:depthzero){reference-type="ref" reference="S:depthzero"}. We conclude with two brief applications of Theorem [Theorem 1](#maintheorem){reference-type="ref" reference="maintheorem"} in Section [8](#S:applications){reference-type="ref" reference="S:applications"}: an explicit formula for the functions $\hat{\mu}_\mathcal{O}$ in terms of the trace character of the representation $\tau_x(\mathcal{O})$ of the compact group $G_x$; and an explicit polynomial expression for $\dim(\pi^{G_{x,2n}})$, whose existence is predicted by the local character expansion. ## Acknowledgements {#acknowledgements .unnumbered} This work was instigated by a question posed to the author by David Vogan and has benefitted enormously from many conversations with him in the online research community on Representation Theory and Noncommutative Geometry sponsored by the American Institute of Mathematics. The approach to $\mathrm{Nil}(\Gamma)$ given here was signficantly refined through conversations with Fiona Murnaghan and Loren Spice. This work progressed over a period of visits to many colleagues, and benefitted from their comments and interest: Vincent Sécherre, Laboratoire de Mathématiques de Versailles, Université Paris-Saclay; Anne-Marie Aubert, Institut de Mathématiques de Jussieu-Paris Rive Gauche, Université de Paris/Sorbonne Université and Jessica Fintzen, Universität Bonn. It is a true pleasure to thank all of these generous people. # Notation and background {#S:Notation} Let $F$ be a local nonarchimedean field of residual characteristic $p\neq 2$, with integer ring $\mathcal{R}$, maximal ideal $\mathcal{P}$ and residue field $\mathfrak{f}$ of cardinality $q$. We impose additional hypotheses on $p$ in Section [2.2](#SS:prestrict){reference-type="ref" reference="SS:prestrict"}, below. Fix once and for all an additive character $\psi$ of $F$ that is trivial on $\mathcal{P}$ and nontrivial on $\mathcal{R}$. Fix a uniformizer $\varpi$ and normalize the valuation on $F$ (and any extension thereof) by $\mathrm{val}(\varpi)=1$. We write $\mathrm{val}(0):=\infty$. Let $\mathbf{G}$ denote a connected reductive algebraic group defined over $F$ whose group of $F$-rational points is denoted $G$; we use $\mathfrak{g}=\mathrm{Lie}(\mathbf{G})(F)$ to denote its Lie algebra over $F$. We simplify notation by refering to tori, Borel subgroups and parabolic subgroups of $G$ when we mean the $F$-points of such algebraic $F$-subgroups of $\mathbf{G}$, and denote them in roman font. Let $G^{\mathrm{reg}}$, respectively $\mathfrak{g}^{\mathrm{reg}}$, denote the set of regular elements of $G$, respectively $\mathfrak{g}$. The group $G$ acts on $\mathfrak{g}$ via the adjoint action $\mathop{Ad}$ and on its dual $\mathfrak{g}^*$ via the coadjoint action $\mathop{Ad}^*$; we abbreviate these by both $g\cdot X$ or ${}^gX$ for $g\in G$ and $X$ in $\mathfrak{g}$ or $\mathfrak{g}^*$. Similarly, if $H$ is a subgroup of $G$ we write ${}^gH$ for the group $gHg^{-1}$. An element $X\in\mathfrak{g}^*$ or $\mathfrak{g}$ is called *semisimple* (or *almost stable*) if its $G$-orbit is closed. We define $X\in \mathfrak{g}^*$ or $\mathfrak{g}$ to be *nilpotent* if there exists an $F$-rational one-parameter subgroup $\lambda \in X_*(\mathbf{G})$ such that $\lim_{t\to 0}{}^{\lambda(t)}X = 0$. By [@AdlerDeBacker2002 §2.5], this is equivalent to a more usual definition that the closure of the coadjoint orbit in the rational topology contains $0$. We say the one-parameter subgroup $\lambda$ is *adapted* to $X$ [@DeBackerNilpotent2002 Definition 4.5.6], if ${}^{\lambda(t)}X =t^2X$. We write $\mathcal{N}^*$ for the set of nilpotent elements of $\mathfrak{g}^*$ and $\mathscr{O}(0)$ for the (finite) set of $G$-orbits in $\mathcal{N}^*$. We sometimes specify a group of matrices merely by the sets in which its entries lie; in this case, that the resulting subgroup is the intersection of this set with $G$ is understood. We write $\lceil t \rceil = \min\{n\in \mathbb{Z}\mid n\geq t\}$ and $\lfloor t \rfloor = \max\{n\in \mathbb{Z}\mid n\leq t\}$. Write $\mathrm{Cent}_G(S)$ for the centralizer in $G$ of the element or set $S$. We may write $[\sigma]$ for the trace character of a representation $\sigma$ of a finite or compact group. The trivial representation is denoted ${\bf 1}$, and the characteristic function of a subset $S$ is denoted ${\bf 1}_S$. ## The Bruhat--Tits building and Moy--Prasad filtration subgroups Let $\mathcal{B}(G)=\mathcal{B}(\mathbf{G},F)$ denote the (enlarged) Bruhat-Tits building of $G$; then to each $x\in \mathcal{B}(G)$ we associate its stabilizer $G_x$, which is a compact subgroup of $G$ containing the parahoric subgroup $G_{x,0}$. These admit a Moy-Prasad filtration by normal subgroups $G_{x,r}$ with $r\in \mathbb{R}_{\geq 0}$ defined relative to the valuation on $F$. We briefly recap the definition; for a careful and detailed summary, see for example [@FintzenMP2021 §2]. To define $G_{x,r}$, choose an apartment $\mathcal{A}\subset \mathcal{B}(G)$ containing $x$; this is the affine space over $X_*(T)\otimes_\mathbb{Z}\mathbb{R}$ for some maximal split torus $T$ of $G$ and we write $\mathcal{A}= \mathcal{A}(G,T)$. Let $\Phi=\Phi(G,T)$ denote the corresponding root system and $\Psi$ the set of affine roots, viewed as functions on $\mathcal{A}$. For each root $\alpha\in \Phi$, let $U_\alpha$ denote the corresponding root subgroup. The affine roots $\psi$ with gradient $\alpha$ define a filtration of $U_\alpha$ by compact open subgroups $U_\psi$. Let $C=\mathrm{Cent}_G(T)$; it contains a parahoric subgroup $C_0$, and a filtration by compact open normal subgroups $C_r$, $r>0$, that is independent of the point $x\in \mathcal{A}$. Then for any $r\geq 0$ we define compact open subgroups $$G_{x,r} = \langle C_r, U_\psi \mid \psi\in \Psi, \psi(x)\geq r\rangle;$$ if $r=0$ this is the parahoric subgroup and for $r>0$ it is a Moy--Prasad filtration subgroup of $G_{x,0}$. It is independent of the choice of apartment containing $x$. The Moy--Prasad filtration is $G$-equivariant; for example ${}^gG_{x,r}=G_{gx,r}$ for all $x\in \mathcal{B}(G)$ and $r\geq 0$. Similarly, the Lie algebra $\mathfrak{g}$ admits a filtration $\mathfrak{g}_{x,r}$ by $\mathcal{R}$-modules indexed by $r\in \mathbb{R}$, as follows. Let $\mathfrak{t}$ denote the Lie algebra of $T$, $\mathfrak{c}$ its centralizer in $\mathfrak{g}$ and for each $\alpha \in \Phi$, let $\mathfrak{g}_\alpha$ denote the corresponding root subspace. These subspaces admit filtrations by $\mathcal{R}$-submodules $\mathfrak{c}_r$ with $r\in \mathbb{R}$ and $\mathfrak{g}_{\psi}$ for $\psi\in \Psi$, respectively, such that $$\label{E:MPfiltration} \mathfrak{g}_{x,r} = \mathfrak{c}_r \oplus \bigoplus_{\alpha} \mathfrak{g}_{\alpha,x,r},$$ where $\mathfrak{g}_{\alpha,x,r}$ is the union of the $\mathcal{R}$-submodules $\mathfrak{g}_\psi$ such that $\psi\in \Psi$, the gradient of $\psi$ is $\alpha$ and $\psi(x)\geq r$. We write $$G_{x,r+}=\bigcup_{s>r}G_{x,s}, \quad \text{and} \quad \mathfrak{g}_{x,r+} = \bigcup_{s>r} \mathfrak{g}_{x,s}.$$ By [@Adler1998 §1.6], there exists a mock exponential map $e=e_x\colon \mathfrak{g}_{x,0+}\to G_{x,0+}$ that induces the Moy--Prasad isomorphism $\mathfrak{g}_{x,r}/\mathfrak{g}_{x,2r} \cong G_{x,r}/G_{x,2r}$ for any $r>0$ (among other desirable properties). Writing $\langle X,Y\rangle$ for the natural pairing of $X\in \mathfrak{g}^*$ with $Y\in \mathfrak{g}$, the Moy-Prasad filtration on the dual of the Lie algebra is defined by $\mathfrak{g}^*_{x,r} = \{X\in \mathfrak{g}^*\mid \forall Y\in \mathfrak{g}_{x,(-r)+}, \langle X,Y\rangle\in \mathcal{P}\}$. We again define $\mathfrak{g}^*_{x,r+} = \cup_{s>r}\mathfrak{g}^*_{x,s}$. Finally, for any $r\geq 0$ we define $G$-stable subsets $$G_{r} = \bigcup_{x\in \mathcal{B}(G)}G_{x,r} \quad \text{and} \quad G_{r+} = \bigcup_{x\in \mathcal{B}(G)}G_{x,r+}.$$ For any real number $r$ we do the same to define $\mathfrak{g}_r$ and $\mathfrak{g}_{r+}$. If $(\pi,V)$ is an irreducible admissible representation of $G$, then its depth is defined as the least real number $r\geq 0$ such that there exists $x\in\mathcal{B}(G)$ for which $V^{G_{x,r+}} \neq \{0\}$. We define the depth of a smooth irreducible representation $\rho$ of $G_x$, for fixed $x$, in the same way; this is equivalent to the least $r\geq 0$ for which $\rho$ factors through $G_x/G_{x,r+}$. ## Restrictions on $p$ {#SS:prestrict} We impose the restriction that $G$ splits over a tamely ramified extension of $F$ and that $p$ does not divide the order of the absolute Weyl group of $\mathbf{G}$. One of the main results of [@Fintzen2021] is that this is sufficient to ensure that all irreducible admissible representations are tame. Combining [@Fintzen2021 Lemma 2.2, Table 1] and [@AdlerRoche2000 §1], one sees that the hypotheses of [@Adler1998 Hypothesis 2.1.1] or [@AdlerRoche2000 Prop 4.1] hold, so that there is a non-degenerate $G$-invariant bilinear form on $\mathfrak{g}$ under which $\mathfrak{g}^*_{x,r}$ and $\mathfrak{g}_{x,r}$ are identified for all $x$ and $r$. For $\mathbf{G}=\mathrm{SL}(2)$ and $p\neq 2$ we may take the trace form, and define for each $\dot{X}\in \mathfrak{g}$ the element $X\in\mathfrak{g}^*$ by $\langle X,\cdot\rangle =\mathrm{tr}(\dot{X}\cdot)$. We also impose the hypotheses of [@DeBackerNilpotent2002 §4] to obtain the classification of nilpotent orbits; this requires the use of $\mathfrak{sl}_2(F)$ triples over the residue field as well as some properties of a mock exponential map. By recent work of Stewart and Thomas, [@StewartThomas2017] the former condition is satisfied for $p>h$, where $h$ is the Coxeter number of $G$. To satisfy all hypotheses for $G=\mathrm{SL}(2,F)$, it suffices to take $p\geq 3$. In contrast, to state the local character expansion, which relates a function on the group to one on the Lie algebra, one needs a $G$-equivariant map $\mathfrak{g}_{0+}\to G_{0+}$ satisfying [@DeBackerHomogeneity2002 Hypothesis 3.2.1]. Such a map, which we'll simply denote $\exp$, can exist in large positive characteristic (see, for example, the discussion in [@CluckersGordonHalupczok2014 §2]); in characteristic zero, [@DeBackerReeder2009 Lemma B.0.3] gives an effective lower bound on $p$. For $G=\mathrm{SL}(2,F)$, this entails in characteristic zero that $p>e+1$ where $e$ is the ramification index of $F$ over $\mathbb{Q}_p$, for example. ## The local character expansion As detailed in the expanded notes [@HarishChandra1999], Harish-Chandra proved in the 1970s that the distribution character of an irreducible admissible representation $\pi$ of $G$, which is given on $f\in C_c^\infty(G)$ by $$\Theta_\pi(f) = \mathrm{tr}\int f(g)\pi(g) \, dg,$$ is well-defined and representable by a function, which we also denote $\Theta_\pi$, that is locally integrable on $G$ and locally constant on the set $G^\mathrm{reg}$ of regular semisimple elements of $G$ (see [@AdlerKorman2007 §13] and the discussion therein). Similarly, to each coadjoint orbit $\mathcal{O}\subset \mathfrak{g}^*$ we associate its orbital integral, given on $f\in C_c^\infty(\mathfrak{g}^*)$ by $$\label{E:orbitalintegral} \mu_\mathcal{O}(f) = \int_{\mathcal{O}} f(X) \, d\mu_\mathcal{O}(X)$$ where $d\mu_\mathcal{O}$ is a Radon measure [@RangaRao1972]. Relative to $\psi$, the fixed additive character of $F$, the Fourier transform of $f\in C_c^\infty(\mathfrak{g})$ is a function $\hat{f} \in C_c^\infty(\mathfrak{g}^*)$. The Fourier transform of the orbital integral $\mu_\mathcal{O}$ is the distribution given on $f\in C_c^\infty(\mathfrak{g})$ by $\widehat{\mu_\mathcal{O}}(f) = \mu_\mathcal{O}(\hat{f})$. Then $\widehat{\mu_\mathcal{O}}$ is representable by a locally integrable function on $\mathfrak{g}$ that is locally constant on $\mathfrak{g}^{\mathrm{reg}}$ [@HarishChandra1999 Theorem 4.4]. We set $\mathfrak{g}^{\mathrm{reg}}_{r+}:=\cup_{x\in \mathcal{B}(G)}\mathfrak{g}_{x,r+}\cap \mathfrak{g}^{\mathrm{reg}}$. The local character expansion expresses that these finitely many functions $\widehat{\mu_\mathcal{O}}$, for $\mathcal{O}\in \mathscr{O}(0)$, form a basis, in a neighbourhood of $0$, for the space of locally integrable $G$-invariant functions that are locally constant on $\mathfrak{g}^{\mathrm{reg}}$. The nature of the expansion was first proven for $G=\mathrm{GL}(n,F)$ in characteristic $0$ by Roger Howe [@Howe1974] and then in the generality of connected reductive groups in characteristic zero by Harish-Chandra [@HarishChandra1999]. Cluckers, Gordon and Halupczok proved its validity in large positive characteristic in [@CluckersGordonHalupczok2014]; Adler and Korman proved an analogous result for expansions centered at other semisimple elements in [@AdlerKorman2007]. The precise domain on which the local character expansion holds was conjectured by Hales, Moy and Prasad [@MoyPrasad1994] and proven in [@Waldspurger1995] for a large class of groups and by [@DeBackerHomogeneity2002] in the following generality. **Theorem 2** (The Local Character Expansion). *If $\pi$ is an irreducible admissible representation of $G$ of depth $r$, then there exist unique $c_\mathcal{O}(\pi)\in \mathbb{C}$ such that for all $X\in \mathfrak{g}^{\mathrm{reg}}_{r+}$ we have $$\label{E:LCE} \Theta_\pi(\exp(X)) = \sum_{\mathcal{O}\in \mathscr{O}(0)} c_\mathcal{O}(\pi) \widehat{\mu_\mathcal{O}}(X).$$* We denote by $\mathcal{WF}(\pi)$ the set of *maximal* nilpotent orbits $\mathcal{O}$ such that $c_\mathcal{O}(\pi)\neq 0$, where the ordering is taken in the local topology; this is the set denoted $\mathrm{WF}^{\mathrm{rat}}(\pi)$ in [@Tsai2023]. In [@Heifetz1985], Heifetz defined and developed the analytic notion of the *wave front set* of a representation of a $p$-adic group, in analogy with the work of Howe [@Howe1979] in the real case. In [@Przebinda1990], Przebinda proved that the wave front set coincides with the support of the right side of [\[E:LCE\]](#E:LCE){reference-type="eqref" reference="E:LCE"}, which is the closure of the union of these orbits. Recent work of Tsai [@Tsai2022; @Tsai2023] has shown that the orbits of $\mathcal{WF}(\pi)$ may fail to be stably conjugate. Finally, note that for $G=\mathrm{GL}(n,F)$, Howe proved that for each $\mathcal{O}\in \mathscr{O}(0)$, there is a corresponding parabolic subgroup $P$ such that in a neighbourhood $O$ of $0\in \mathfrak{g}$ we have $$\widehat{\mu_\mathcal{O}}|_O =\Theta_\pi\circ \exp|_O$$ where $\pi = \mathrm{Ind}_P^G \bf{1}$ [@Howe1974 Lemma 5]. In the same vein, for $\mathrm{SL}(2,F)$, the functions $\widehat{\mu_\mathcal{O}}$ are almost equal to the characters of special unipotent representations (see [\[E:specialchar\]](#E:specialchar){reference-type="eqref" reference="E:specialchar"}). We cannot expect such equalities in general as, for example, for classical groups nonspecial orbits cannot occur in $\mathcal{WF}(\pi)$ for any $\pi$ [@Moeglin1996 Thm 1.4]). The main goal in this paper is to propose an example of a weaker form of Howe's theorem, based on representations of a maximal compact open subgroup, that one may hope can hold true in general. # Nilpotent orbits and nilpotent support {#S:Nilpotent} In this section, $G$ is an arbitrary connected reductive group, subject to the hypothesis on $p$ of Section [2.2](#SS:prestrict){reference-type="ref" reference="SS:prestrict"}. We define the (local) nilpotent support of an element of $\mathfrak{g}^*$, and relate this both to the asymptotic cone and to the wave front set of a representation of positive depth. ## Degenerate cosets and nilpotent orbits In [@AdlerDeBacker2002 §3], Adler and DeBacker generalize ideas of Moy and Prasad to establish, for connected reductive groups, that for all $r\in \mathbb{R}$ $$\mathfrak{g}^*_r = \bigcap_{x\in \mathcal{B}(G)}(\mathfrak{g}^*_{x,r}+\mathcal{N}^*),$$ where $\mathfrak{g}^*_r := \bigcup_{x\in \mathcal{B}(G)} \mathfrak{g}^*_{x,r}$. They further show that $$\mathcal{N}^* = \bigcap_{r\in \mathbb{R}} \mathfrak{g}_r^*.$$ Given $x\in \mathcal{B}(G)$ and $X\in \mathfrak{g}^*\setminus \{0\}$, the *depth of $X$ at $x$* is the unique value $t = d_x(X)$ such that $X\in \mathfrak{g}^*_{x,t}\setminus \mathfrak{g}^*_{x,t+}$. When $X$ is not nilpotent, they prove that the *depth of $X$*, given by $$d(X)=\max\{d_x(X)\mid x\in \mathcal{B}(G)\} = \max\{r\mid X\in \mathfrak{g}^*_r\}$$ is well-defined and rational. For $X$ nilpotent, we set $d(X)=\infty$. Depth is $G$-invariant. For semisimple $\Gamma\in \mathfrak{g}^*$, let $T\subset \mathrm{Cent}_G(\Gamma)$ be a maximal torus with associated absolute root system $\Phi(G,T)$. Then $\Gamma$ is called *good* if for all $\alpha \in \Phi(G,T)$, we have $\mathrm{val}(\Gamma(d\alpha^\vee(1)))\in \{d(\Gamma),\infty\}$. By [@KimMurnaghan2003 Thm 2.3.1], if $\Gamma$ is good then the set of points $x\in \mathcal{B}(G)$ at which $d_x(\Gamma)$ attains its maximum value $d(\Gamma)$ is exactly $\mathcal{B}(\mathrm{Cent}_G(\Gamma))\subset \mathcal{B}(G)$. For any $\Gamma \in \mathfrak{g}^*$ set $d=d_x(\Gamma)$. The coset $\Gamma+\mathfrak{g}^*_{x,d+}$ is called *degenerate* if it contains a nilpotent element $X\in \mathcal{N}^*$. From the relations above it follows that this happens if and only if $d < d(\Gamma)$. In [@DeBackerNilpotent2002 §5], DeBacker proves that the set of nilpotent $G$-orbits meeting a degenerate coset $\Gamma+\mathfrak{g}^*_{x,d+}$ has a unique minimal element with respect to the (rational) closure relation on orbits, which we'll denote $\mathcal{O}(\Gamma,x)$. This generalizes a result of Barbasch and Moy [@BarbaschMoy1997 Prop 3.1.6] for $d=0$, which was integral to their determination of the wave front set of a depth zero representation. To classify nilpotent orbits in this way, DeBacker proceeds as follows. Identify $\mathfrak{g}$ and $\mathfrak{g}^*$. Given a nilpotent element $X\in \mathfrak{g}$, complete $X$ to an $\mathfrak{sl}_2(F)$ triple $(X,H,Y)$. Choose $r\in \mathbb{R}$ and create the building set $$\mathcal{B}_r(X,H,Y) = \{x\in \mathcal{B}(G) \mid X \in \mathfrak{g}_{x,r}, \ H \in \mathfrak{g}_{x,0}, \ Y\in \mathfrak{g}_{x,-r}\};$$ he proves this set is a nonempty, closed, convex subset of $\mathcal{B}(G)$ with the property that for all $x\in \mathcal{B}_r(X,H,Y)$ we have $\mathcal{O}(X,x)=G\cdot X$. **Remark 3**. Note that for each $g\in G$, we have $\mathcal{B}_r({}^gX,{}^gH,{}^gY)={}^g\mathcal{B}_r(X,H,Y)$, and for fixed $X$ the union of these need not cover $\mathcal{B}(G)$. Moreover, if $\mu$ is a one-parameter subgroup adapted to this triple, then by [@DeBackerNilpotent2002 Remark 5.1.5], $\mathcal{B}_r(X,H,Y)=\mathcal{B}_0(X,H,Y)+\frac{r}{2}\mu$, where this sum is taken in any apartment in $\mathcal{B}(C_G(\mu))$. It follows that (if the rank of $G$ is greater than $1$) there exist orbits $\mathcal{O}$ (such as ones for which $\mathcal{B}_r(X,H,Y)$ is a point) for which there exist $y\in \mathcal{B}(G)$ such that $\mathcal{O}\neq \mathcal{O}(X,y)$ for any $X\in \mathcal{O}$. For example, in $\mathrm{Sp}(4,F)$, the principal nilpotent orbits are only obtained along certain lines emanating from vertices. ## Nilpotent support and nilpotent cones We now explore different ways to understand the asymptotic nilpotent support of a general element $\Gamma\in \mathfrak{g}^*$ and show their equivalence. **Definition 4**. Let $\Gamma\in\mathfrak{g}^*$. If $x\in \mathcal{B}(G)$, then the *local nilpotent support at $x$* of $\Gamma$ is $$\mathrm{Nil}_x(\Gamma) = \{ \mathcal{O}({}^g\Gamma,x) \mid g\in G, d_x(g\cdot \Gamma)<d(\Gamma)\},$$ which is the set of nilpotent orbits defined by degenerate cosets at $x$ of elements of the $G$-orbit of $\Gamma$. On the other hand, the *nilpotent support* of $\Gamma$ is $$\mathrm{Nil}(\Gamma) = \{ \mathcal{O}(\Gamma,x) \mid x\in \mathcal{B}(G), d_x(\Gamma)<d(\Gamma)\},$$ the set of nilpotent orbits corresponding to any (nontrivial) degenerate coset of $\Gamma$. Note that if $\Gamma$ is nilpotent, then $\mathrm{Nil}(\Gamma) = G\cdot \Gamma$. More generally, for any $g\in G$, we have $d_x(\Gamma)=d_{gx}({}^g\Gamma)$ and ${}^g(\Gamma+\mathfrak{g}^*_{x,d+})= {}^g \Gamma + \mathfrak{g}^*_{gx,d+}$. Thus $\mathcal{O}(\Gamma,x)=\mathcal{O}({}^g\Gamma,gx)$ and $$\mathrm{Nil}(\Gamma)= \bigcup_{x\in \mathcal{B}(G)}\mathrm{Nil}_x(\Gamma),$$ that is, the nilpotent support is the union of the local nilpotent supports, and $\mathrm{Nil}(\Gamma)$ is an invariant of the $G$-orbit of $\Gamma$. One may alternately restrict this union to one over the points in a fundamental domain for the action of $G$ on $\mathcal{B}(G)$. By Remark [Remark 3](#rem:coverage){reference-type="ref" reference="rem:coverage"}, when the rank of $G$ is greater than $1$, not all nilpotent orbits will occur as some $\mathcal{O}(\Gamma,x)$ for a given point $x\in \mathcal{B}(G)$, so $\mathrm{Nil}_x(\Gamma)\neq \mathrm{Nil}(\Gamma)$ in general. Even when these sets are equal, as for $\mathrm{SL}(2,F)$ (see Proposition [Proposition 9](#P:asymptotic_orbit_invariance){reference-type="ref" reference="P:asymptotic_orbit_invariance"}), they are interesting subsets of the nilpotent cone (see Lemma [Lemma 10](#L:NilGamma){reference-type="ref" reference="L:NilGamma"}). On the other hand, the asymptotic cone on an element $\Gamma$ is defined in [@AdamsVogan2021 Def 3.9] analytically as follows. **Definition 5**. Let $\Gamma\in \mathfrak{g}^*$. The *asymptotic cone* on $\Gamma$ is the set $$\mathop{Cone}(\Gamma) = \{X\in \mathfrak{g}^* \mid \exists \varepsilon_i \to 0, \varepsilon_i \in F^\times, \exists g_i\in G, \lim_{i\to\infty}\varepsilon_i^2 {\mathop{Ad}}^*(g_i)\Gamma = X\}.$$ This is a closed, nonempty union of nilpotent orbits of $G$ on $\mathfrak{g}^*$. **Proposition 6**. *Let $\Gamma\in \mathfrak{g}^*$. Then the nonzero $G$-orbits occurring in the asymptotic cone of $\Gamma$ are those in its nilpotent support, that is, $$\mathop{Cone}(\Gamma) = \bigcup_{\mathcal{O}\in \mathrm{Nil}(\Gamma)} \mathcal{O}\cup \{0\}.$$* *Proof.* It suffices to prove this result for $\Gamma \in\mathfrak{g}$, where we may apply the theory of $\mathfrak{sl}_2(F)$ triples. Let $\Gamma \in \mathfrak{g}$ have depth $r\leq \infty$ and let $\mathcal{O}\in \mathrm{Nil}(\Gamma)$. Then there exists $x\in \mathcal{B}(G)$ and $d<r$ such that $d_x(\Gamma)=d$ and $\mathcal{O}=\mathcal{O}(\Gamma,x)$. Choose a representative $$X\in \mathcal{O}(\Gamma,x) \cap (\Gamma+\mathfrak{g}_{x,d+}).$$ Choose an $\mathfrak{sl}_2(F)$ triple $(X,H,Y)$ and the corresponding one-parameter subgroup $\mu$ adapted to $X$. By [@DeBackerNilpotent2002 Lemma 5.2.1], we have $$X + \mathfrak{g}_{x,d+} = \mathop{Ad}(G_{x,0+})(X+C_{\mathfrak{g}_{x,d+}}(Y)).$$ Therefore there exist $g \in G_{x,0+}$ and $C\in C_{\mathfrak{g}_{x,d+}}(Y)$ for which $$\Gamma = \mathop{Ad}(g^{-1})(X+C).$$ Note that $C_{\mathfrak{g}}(Y)$ is spanned by the lowest weight vectors of $\mathrm{ad}(H)$, so we may decompose $C=\sum_{i\leq 0}C_i$ where $\mathop{Ad}(\mu(t))C_i = t^iC_i$ for all $t\in F^\times$. Similarly, for all $t\in F^\times$ we have $\mathop{Ad}(\mu(t))X = t^2X$. Therefore $$\lim_{t\to 0} t^2\mathop{Ad}(\mu(t^{-1})g)\Gamma = \lim_{t\to 0} t^2\mathop{Ad}(\mu(t^{-1}))(X+C)= X$$ so $X\in \mathop{Cone}(\Gamma)$. Since $\mathop{Cone}(\Gamma)$ is $G$-invariant, we deduce $\mathcal{O}\subset \mathop{Cone}(\Gamma)$. Conversely, let $X\in \mathop{Cone}(\Gamma)$ be nonzero, so that there exists a sequence of elements $\varepsilon_i \in F^\times$, with $\varepsilon_i\to 0$, and a sequence of elements $g_i \in G$, such that $$\lim_{i\to \infty} \varepsilon_i^2 \mathop{Ad}(g_i)\Gamma = X.$$ Complete $X$ to an $\mathfrak{sl}_2(F)$ triple $(X,H,Y)$ and choose a point $x\in \mathcal{B}_0(X,H,Y)$. Since the given sequence converges to $X$, it enters the neighbourhood $X+\mathfrak{g}_{x,0+}$ so we may choose $i\in \mathbb{N}$ such that $$\varepsilon_i^2\mathop{Ad}(g_i)\Gamma \in X + \mathfrak{g}_{x,0+}.$$ It follows that $\mathop{Ad}(g_i)\Gamma \in \varepsilon_i^{-2}X + \mathfrak{g}_{x,-2\mathrm{val}(\varepsilon_i)+}$, a nontrivial degenerate coset of depth $-2\mathrm{val}(\varepsilon_i)$. Since $(\varepsilon_i^{-2}X, H, \varepsilon_i^2Y)$ is again an $\mathfrak{sl}_2(F)$ triple and $\mathcal{B}_{-2\mathrm{val}(\varepsilon_i)}(\varepsilon_i^{-2}X, H, \varepsilon_i^2Y) = \mathcal{B}_0(X,H,Y)$, we infer that the minimal nilpotent orbit meeting this coset is $\mathop{Ad}(G)(\varepsilon_i^{-2}X) = \mathop{Ad}(G)X$. Thus $\mathop{Ad}(G)X=\mathcal{O}({}^{g_i}\Gamma,x)\in \mathrm{Nil}_x(\Gamma)\subset \mathrm{Nil}(\Gamma)$, as required. ◻ ## Connection with the wave front set of a positive depth representation Suppose now that $\pi$ is an irreducible admissible representation of $G$ of depth $r$ with good minimal $K$-type $\Gamma$ of depth $-r$ (in the sense of [@KimMurnaghan2003 Def 2.4.3, 2.4.6]). Then, under suitable hypotheses (that are satisfied if $F$ has characteristic zero and the exponential map converges on $\mathfrak{g}_{0+}$), Kim and Murnaghan prove a version of the local character expansion that is valid on the strictly larger neighbourhood $\mathfrak{g}^{\mathrm{reg}}_r$. The $\Gamma$-asymptotic expansion [@KimMurnaghan2003 Thm 5.3] asserts that there exist complex coefficients $c_{\mathcal{O}'}(\pi)$ such that for any $X\in \mathfrak{g}^{\mathrm{reg}}_r$ we have $$\label{E:KM} \Theta_\pi(\exp(X)) = \sum_{\mathcal{O}' \in \mathscr{O}(\Gamma)} c_{\mathcal{O}'}(\pi) \widehat{\mu_{\mathcal{O}'}}(X),$$ where $\mathscr{O}(\Gamma)$ denotes the set of $G$-orbits in $\mathfrak{g}^*$ with $\Gamma$ in their closure, and for $\mathcal{O}'\in \mathscr{O}(\Gamma)$, $\widehat{\mu_{\mathcal{O}'}}$ denotes the Fourier transform of the corresponding orbital integral [\[E:orbitalintegral\]](#E:orbitalintegral){reference-type="eqref" reference="E:orbitalintegral"}. This yields a special case of interest: that of the expansion [\[E:KM\]](#E:KM){reference-type="eqref" reference="E:KM"} having a single nonzero term $c_{\mathcal{O}'}(\pi)\widehat{\mu_{\mathcal{O}'}}$ corresponding to $\mathcal{O}'=G\cdot \Gamma$. We claim this happens, for example, when $G'=\mathrm{Cent}_G(\Gamma)$ is compact-mod-centre, such as when $\Gamma$ is a regular element. Namely, let $\mathfrak{g}'$ denote the Lie algebra of $G'$. Then the set $\mathscr{O}(\Gamma)$ indexing the sum in [\[E:KM\]](#E:KM){reference-type="eqref" reference="E:KM"} is in bijective correspondence with the set of nilpotent $G'$-orbits in $(\mathfrak{g}')^*$, which is the singleton $\{G\cdot\Gamma\}$ under this hypothesis. **Theorem 7**. *Let $\pi$ be an irreducible representation of $G$ of depth $r>0$, and let $\Gamma \in \mathfrak{g}^*$ be a good minimal $K$-type of $\pi$ such that $\pi$ admits a $\Gamma$-asymptotic expansion. Suppose further that this expansion has a unique nonzero term, corresponding to the Fourier transform of the orbital integral corresponding to $\Gamma$ itself. Then $\mathcal{WF}(\pi)$ coincides with the maximal elements of $\mathrm{Nil}(\Gamma)$; that is, the asymptotic cone on $\Gamma$ is the wave front set of $\pi$.* The following proof was communicated to me by Fiona Murnaghan. *Proof.* We are given that on $\mathfrak{g}^{\mathrm{reg}}\cap\mathfrak{g}_r$, $\Theta_\pi\circ \exp = t\widehat{\mu_{G\cdot\Gamma}}$ for some nonzero scalar $t$. Applying the inverse Fourier transform to the local character expansion [\[E:LCE\]](#E:LCE){reference-type="eqref" reference="E:LCE"} of $\Theta_\pi$ (which is valid on the smaller set $\mathfrak{g}^{\mathrm{reg}}\cap\mathfrak{g}_{r+}$) we may write the equality of distributions $$\label{E:germ} t\mu_{G\cdot \Gamma}=\sum_{\mathcal{O}\in \mathscr{O}(0)} c_{\mathcal{O}}(\pi) \mu_\mathcal{O}$$ which by [@DeBackerHomogeneity2002 Cor 3.4.6] holds in particular for all compactly supported functions on $\mathfrak{g}^*/\mathfrak{g}^*_{x,-r}$, for any $x \in \mathcal{B}(G)$. So let $x \in \mathcal{B}(G)$ and let $d$ be such that $\mathfrak{g}_{x,d+}^*\supset \mathfrak{g}_{x,-r}^*$. Given a nonzero coset $\xi\in \mathfrak{g}^*_{x,d}/\mathfrak{g}^*_{x,d+}$ let ${\bf 1}_\xi$ denote the characteristic function of this subset of $\mathfrak{g}^*$. Note that if $X\in \xi\cap \mathcal{O}$ for some (not necessarily nilpotent) $G$-orbit $\mathcal{O}$, then this intersection contains the open set $G_{x,0+}\cdot X$ as well. Thus we have $$\label{*} \mu_\mathcal{O}({\bf 1}_\xi) = 0 \iff \xi\cap\mathcal{O}=\emptyset.$$ Now suppose that $\mathcal{O}\in \mathscr{O}(0)$, and choose $x\in \mathcal{B}(G)$ and $\xi=X+\mathfrak{g}^*_{x,d+}$ with $\mathfrak{g}^*_{x,d+}\supset \mathfrak{g}^*_{x,-r}$ with the property that $\mathcal{O}= \mathcal{O}(X,x)$. The minimality of $\mathcal{O}(X,x)$ proven by DeBacker implies that any nilpotent orbit $\mathcal{O}'$ meeting $\xi$ (or equivalently, by [\[\*\]](#*){reference-type="eqref" reference="*"}, satisfying $\mu_{\mathcal{O}'}({\bf 1}_\xi)\neq 0$) must contain $\mathcal{O}$ in its closure. Suppose first that $\mathcal{O}$ is not in the wave front set $\cup_{\mathcal{O}' \in \mathcal{WF}(\pi)}\overline{\mathcal{O}'}$ of $\pi$. Let $\mathcal{O}'\in \mathscr{O}(0)$ be such that $c_{\mathcal{O}'}(\pi)\neq 0$; then $\mathcal{O}'$ is in the wave front set, so $\mathcal{O}\not\subset\overline{\mathcal{O}'}$. This implies by the preceding paragraph that $\mu_{\mathcal{O}'}({\bf 1}_\xi) = 0$. As this holds for all such $\mathcal{O}'$, we conclude from [\[E:germ\]](#E:germ){reference-type="eqref" reference="E:germ"} that $\mu_{G\cdot\Gamma}({\bf 1}_\xi)=0$, whence by [\[\*\]](#*){reference-type="eqref" reference="*"} we have $\xi\cap G\cdot \Gamma = \emptyset$, and thus $\mathcal{O}\notin \mathrm{Nil}(\Gamma)$. It follows that every $\mathcal{O}\in \mathrm{Nil}(\Gamma)$ lies in the wave front set of $\pi$. Now suppose $\mathcal{O}\in \mathcal{WF}(\pi)$; that is, it is maximal among nilpotent orbits with nonzero coefficient in [\[E:germ\]](#E:germ){reference-type="eqref" reference="E:germ"}. Thus the preceding argument implies $\mu_{\mathcal{O}'}({\bf 1}_\xi)=0$ for all $\mathcal{O}'\neq \mathcal{O}$ in the wave front set. Therefore [\[E:germ\]](#E:germ){reference-type="eqref" reference="E:germ"} yields $t\mu_{G\cdot \Gamma}({\bf 1}_\xi)=c_{\mathcal{O}}(\pi) \mu_{\mathcal{O}}({\bf 1}_\xi) \neq 0$, so by [\[\*\]](#*){reference-type="eqref" reference="*"}, $\xi$ must meet $G\cdot \Gamma$ and thus $\mathcal{O}\in \mathrm{Nil}(\Gamma)$. Hence, the maximal elements of $\mathrm{Nil}(\Gamma)$ coincide with $\mathcal{WF}(\pi)$. ◻ In fact, the key to the proof is that the maximal nilpotent orbits occuring in the Shalika germ expansion of $\mu_{G\cdot \Gamma}$ are the maximal orbits of $\mathrm{Nil}(\Gamma)$. In [@CiubotaruOkada2023], Ciubotaru and Okada obtain a similar result directly, by analysing the asymptotic nilpotent cone of the characters of $G_{x,r}/G_{x,r+}$ appearing in $\pi^{G_{x,r+}}$. **Remark 8**. One might ask if Theorem [Theorem 7](#T:nil=wf){reference-type="ref" reference="T:nil=wf"} could be extended to show that $\mathcal{WF}(\pi)$ is the union of the nilpotent supports of the maximal orbits occurring in the $\Gamma$-asymptotic expansion [\[E:KM\]](#E:KM){reference-type="eqref" reference="E:KM"}. The answer is expected to be negative. In the supercuspidal case, the key result is [@Spice2021 Cor 10.2.3(1)], which implies that this latter set of orbits (in $\mathscr{O}(\Gamma)$) corresponds exactly to $\mathcal{WF}(\pi^0)$ (in $\mathscr{O}(0)$ for $G^0=\mathrm{Cent}_G(\Gamma)^\circ$), where $\pi^0$ is the associated depth-zero supercuspidal representation of $G^0$. Cheng-Chiang Tsai[^2] has constructed explicit examples of supercuspidal representations where the wave front set does not follow such a pleasant inductive structure. In effect, one expects that when substituting Shalika germ expansions into the $\Gamma$-asymptotic expansion, cancellations among coefficients may occur. While the proof of Theorem [Theorem 7](#T:nil=wf){reference-type="ref" reference="T:nil=wf"} entails some additional hypotheses on $F$, a consequence of the main theorem of Section [6](#S:posdepth){reference-type="ref" reference="S:posdepth"} is that, for $G=\mathrm{SL}(2,F)$, the conclusion of the theorem holds whenever the characteristic and residual characteristic of $F$ are not $2$. # Nilpotent orbits and nilpotent cones of $G=\mathrm{SL}(2,F)$ {#S:sl2nil} For the rest of this paper we suppose that $\mathbf{G}=\mathrm{SL}(2)$ and $\mathfrak{g}= \mathfrak{sl}(2,F)$. In this section, we derive some additional properties of the nilpotent support of an element $\Gamma \in \mathfrak{g}^*$. We identify $\mathfrak{g}$ and $\mathfrak{g}^*$ with the trace form. There are five nilpotent orbits: the zero orbit, and four two-dimensional principal (or regular) orbits that are in bijection with the rational square classes $F^\times/(F^\times)^2$. Representatives of these five orbits in $\mathfrak{g}$ are $$\label{E:nilpotentorbits} \dot{X}_u=\left[ \begin{matrix} 0&u\\0&0 \end{matrix} \right]$$ where $u$ runs over the set $\{0,1,\varepsilon,\varpi,\varepsilon\varpi\}$ modulo $(F^\times)^2$ and $\varepsilon\in \mathcal{R}^\times$ is a fixed nonsquare. For each $u$, write $\mathcal{O}_u$ for the orbit in $\mathfrak{g}^*$ corresponding to $\dot{X}_u$. The following proposition relaxes the conditions for identifying the orbits in the nilpotent support of an element $\Gamma$. **Proposition 9**. *Let $\mathfrak{g}=\mathfrak{sl}_2(F)$ and $\Gamma \in \mathfrak{g}^*\setminus\{0\}$. Set $r=d(\Gamma)\in \mathbb{R}\cup \{\infty\}$. Then* (a) *every $\mathcal{O}\in\mathrm{Nil}(\Gamma)$ meets $\Gamma+\mathfrak{g}^*_{x,r}$ for some $x\in \mathcal{B}(G)$ such that $d_x(\Gamma)<r$;* (b) *for each $x\in\mathcal{B}(G)$ such that $d_x(\Gamma)<r$, if $\Gamma+\mathfrak{g}^*_{x,r}$ meets a nilpotent orbit $\mathcal{O}$, then $\mathcal{O}\in\mathrm{Nil}(\Gamma)$;* (c) *for each $x\in \mathcal{B}(G)$, $\mathrm{Nil}(\Gamma) = \{ \mathcal{O}({}^g\Gamma, x) \mid g\in G\}=\mathrm{Nil}_x(\Gamma)$, that is, every nonzero nilpotent orbit in $\mathop{Cone}(\Gamma)$ appears in the local nilpotent support at every $x$.* *Proof.* The first two statements use that there are no closure relations between the principal orbits of $\mathfrak{sl}_2(F)$, and so the uniqueness of the minimal nilpotent orbit meeting any degenerate coset implies that any nontrivial degenerate coset meets only one nilpotent orbit. For (a), suppose $\mathcal{O}\in \mathrm{Nil}(\Gamma)$; then $\mathcal{O}= \mathcal{O}(\Gamma,x)$ for some $x\in \mathcal{B}(G)$, implying $d_x(\Gamma)<r$. Since $\Gamma\in \mathfrak{g}^*_{r}\subset \mathfrak{g}^*_{x,r}+\mathcal{N}^*$, the set $\Gamma+\mathfrak{g}^*_{x,r}$ contains a (nonzero) nilpotent element $Y$. Since $Y\in \Gamma+\mathfrak{g}^*_{x,r}\subset \Gamma+\mathfrak{g}^*_{x,d_x(\Gamma)}$, it lies in $\mathcal{O}$, so $\mathcal{O}$ meets the smaller coset, as required. For (b), note that if $d_x(\Gamma)<r$ then $0\notin \Gamma+\mathfrak{g}^*_{x,r}\subset \Gamma+\mathfrak{g}^*_{x,d(\Gamma)+}$; any nilpotent orbit meeting the smaller set meets the larger one, and thus by uniqueness this orbit is $\mathcal{O}(\Gamma,x)\in\mathrm{Nil}(\Gamma)$. To prove (c), let $x\in \mathcal{B}(G)$ and let $\mathrm{Nil}_{x}(\Gamma)$ be the local nilpotent support of $\Gamma$ at $x$; we have already noted that $\mathrm{Nil}_x(\Gamma)\subset \mathrm{Nil}(\Gamma)$. The reverse inclusion follows from the one-dimensionality of $\mathcal{B}(G)$. Let $\mathcal{O}\in \mathrm{Nil}(\Gamma)$; then $\mathcal{O}= \mathcal{O}(\Gamma,y)$ for some $y\in \mathcal{B}(G)$. Let $S$ be a split torus with associated root system $\Phi(G,S)=\{\pm \alpha\}$ such that $y\in \mathcal{A}(G,S)$. Set $d=d_y(\Gamma)$ and let $\dot{\Gamma}\in \mathfrak{g}$ correspond to $\Gamma$ via the trace form. Choose $\dot{X}\in \mathcal{O}$ such that $\dot{\Gamma}\in \dot{X}+\mathfrak{g}_{y,d+}$. Conjugating both $\dot{\Gamma}$ and $\dot{X}$ by $G_y$ as necessary we may assume $\dot{X}\in \mathfrak{g}_\alpha$. Relative to the pinning of a fixed base point, we have the decomposition of $\mathcal{R}$-modules $$\mathfrak{g}_{y,d} = \mathfrak{g}_{-\alpha, d+\alpha(y)}\oplus \mathfrak{s}_d \oplus \mathfrak{g}_{\alpha,d-\alpha(y)}.$$ Let $\alpha^\vee$ denote the positive coroot, and choose $g\in G$ so that $gx\in \mathcal{A}(G,S)$ and $gx=y-\ell\alpha^\vee$ for some $\ell\geq 0$. Therefore if $d' = d-2\ell$ then $\mathfrak{g}_{y,d}\subset \mathfrak{g}_{gx,d'}$. Since $\dot{X}\in \mathfrak{g}_{\alpha,d-\alpha(y)}\setminus \mathfrak{g}_{\alpha,(d-\alpha(y))+}$ and $d'-\alpha(gx)=d-\alpha(y)$, we conclude $d_{gx}(\dot{\Gamma})=d_{gx}(\dot{X})=d'$ and $\dot{\Gamma}-\dot{X}\in \mathfrak{g}_{gx,d'+}$. By uniqueness, we infer that $\mathcal{O}= \mathcal{O}(\dot{\Gamma},gx)=\mathcal{O}({}^{g^{-1}} \dot{\Gamma},x) \in \mathrm{Nil}_x(\dot{\Gamma})$, yielding the result. ◻ We next determine $\mathrm{Nil}(\Gamma)$ explicitly, for any $\Gamma \in \mathfrak{g}=\mathfrak{sl}_2(F)$ (identified with its dual via the trace form). There is nothing to do if $\Gamma$ is nilpotent. If $\Gamma \neq 0$ is semisimple, then it is $G$-conjugate to a matrix of the form $$\label{E:form} \dot{X}(u,v)=\left[ \begin{matrix} 0&u\\v&0 \end{matrix} \right],$$ for some $u,v\in F^\times$. Its centralizer is a maximal torus. There is one $G$-conjugacy class of split torus, represented by any diagonal element, and two classes of unramified anisotropic tori, represented by $\dot{X}(1,\varepsilon)\in \mathfrak{g}$ and $\dot{X}(\varpi^{-1},\varepsilon\varpi)\in\mathfrak{g}$, respectively. The classes of ramified tori are represented by $\dot{X}(1,t)\in \mathfrak{g}$ with $t\in \{\varpi, \varepsilon\varpi, \varepsilon^2\varpi, \varepsilon^3\varpi\}$, noting that if $-\varepsilon\in F^2$ then there are only two classes. We can now describe the nilpotent support of each such element, using the parametrization given in [\[E:nilpotentorbits\]](#E:nilpotentorbits){reference-type="eqref" reference="E:nilpotentorbits"}. **Lemma 10**. *Let $G=\mathrm{SL}(2,F)$ and $\Gamma\in \mathfrak{g}\setminus\{0\}$ semisimple. If $\Gamma$ splits over $F$, then $$\mathrm{Nil}(\Gamma)=\{\mathcal{O}_1, \mathcal{O}_\varepsilon, \mathcal{O}_\varpi, \mathcal{O}_{\varepsilon\varpi}\}.$$ Otherwise, $\Gamma$ is conjugate to $\dot{X}(u,v)$ for some $u,v \in F^\times$, and splits over $E= F[\sqrt{uv}]$. Let $\mathrm{Norm}_{E/F}(E^\times)/(F^\times)^2$ be represented by $\{1,\gamma\}$. Then $u$ and $v$ are uniquely defined mod $\mathrm{Norm}_{E/F}(E^\times)$ and $$\mathrm{Nil}(\Gamma) = \{\mathcal{O}_u, \mathcal{O}_{u\gamma}\}.$$* *Proof.* By Proposition [Proposition 9](#P:asymptotic_orbit_invariance){reference-type="ref" reference="P:asymptotic_orbit_invariance"}, we may fix the choice $x=x_0\in\mathcal{B}(G)$ to be the vertex such that $\mathfrak{g}_{x,r}$ is the set of traceless $2\times 2$ matrices with entries in $\mathcal{P}^{\lceil r \rceil}$, and replace $\Gamma$ by any $G$-conjugate. First suppose $\Gamma=\textrm{diag}(a,-a)$ with $\mathrm{val}(a)=r$. Let $u \in F^\times$ and note that if $g_u=\left[ \begin{smallmatrix} 1&-\frac 12a^{-1}u\\0&1 \end{smallmatrix} \right]\in G$ then ${}^{g_u}\Gamma = \left[ \begin{smallmatrix} a&u\\0&-a \end{smallmatrix} \right].$ Therefore, for any $u$ such that $\mathrm{val}(u)=d<r$, we have ${}^{g_u} \Gamma \in \dot{X}_u+\mathfrak{g}_{x,d+}$. Thus $\mathrm{Nil}(\Gamma)$ contains every nonzero nilpotent orbit. Now suppose $\Gamma=\dot{X}(u,v)$ for some $u,v\in F^\times$ such that $uv\notin (F^\times)^2$ and set $E=F[\sqrt{uv}]$. We calculate directly that the upper triangular entry of any $G$-conjugate of $\Gamma$ takes the form $$u'=a^2 u - b^2 v = u(a^2-b^2vu^{-1}) \in u\mathrm{Norm}_{E/F}(E^\times) %= -v(b^2-a^2uv^{-1}).$$ for some $a,b\in F$, not both zero, from which it follows that $\mathrm{Nil}(\Gamma)\subset \{\mathcal{O}_u,\mathcal{O}_{u\gamma}\}$. For the reverse inclusion, first note that $\dot{X}(u,v)$ is $G$-conjugate to $\dot{X}(u\varpi^{-2n},v\varpi^{2n})$ for all $n\in \mathbb{Z}$ and for $n$ sufficiently large $\dot{X}(u\varpi^{-2n},v\varpi^{2n})-\dot{X}_{u\varpi^{-2n}}\in \mathfrak{g}_{x,r}$. Thus $\mathcal{O}_u\in \mathrm{Nil}(\Gamma)$. Now note that when $E$ is ramified, we may take $\gamma = -uv$ so $\mathcal{O}_{u\gamma}=\mathcal{O}_{-v}$; since $\dot{X}(u,v)$ is $G$-conjugate to $\dot{X}(-v,-u)$ we are done by the preceding. If $E$ is unramified, we have instead $\gamma=uv$, whence $\mathcal{O}_{u\gamma}=\mathcal{O}_{v}$. As $-1$ is a norm, we may choose $\alpha,\beta\in F$ such that $-1 = \beta^2-\alpha^2uv^{-1}$; then $g = \left[ \begin{smallmatrix} \alpha & \beta \\ \beta & \alpha uv^{-1} \end{smallmatrix} \right] \in G$ satisfies ${}^g\dot{X}(u,v) = \dot{X}(v,u)$, and again by the preceding we may conclude $\mathcal{O}_v \in \mathrm{Nil}(\Gamma)$. ◻ # Representations of $G_x$ associated to nilpotent orbits {#S:Representations} ## Shalika's representations of $\mathrm{SL}(2,\mathcal{R})$ {#SS:Shalika} In his thesis, Shalika constructed all irreducible representations of $K=\mathrm{SL}(2,\mathcal{R})$. In this section we recap his explicit construction for the so-called ramified case, which attaches an irreducible representation of $K$ to certain $K$-orbits in $\mathfrak{g}^*$; we'll then provide a coordinate-free generalization more suited to our needs in the next section. Let $S$ be the diagonal split torus, $B$ the upper triangular Borel subgroup and $U$ its unipotent radical. We use a subscript $0$ to indicate their intersections with $K$: $S_0=S\cap K$, $B_0=B\cap K$ and $U_0=U\cap K$. Let $x_0\in \mathcal{A}(G,S)$ be such that $K=G_{x_0}$ and $z_0$ the barycentre of the positive alcove adjacent to $x_0$ (relative to $B$). Let $d$ be a positive integer. Choose $u \in \mathcal{P}^{-d}\setminus \mathcal{P}^{-d+1}$ and nonzero $v\in \mathcal{P}^{-d+1}$ and consider the anti-diagonal matrix $\dot{X}:= \dot{X}(u,v)\in \mathfrak{g}_{x_0,-d}$ of [\[E:form\]](#E:form){reference-type="eqref" reference="E:form"}. Identify this with the element $X\in \mathfrak{g}^*_{x_0,-d}$ by the rule $X(Z)=\mathrm{tr}(\dot{X}Z)$ for all $Z\in\mathfrak{g}$. If $v=0$ then $X$ is nilpotent and its centralizer $C_K(X)$ in $K$ coincides with $ZU_0$, where $Z=\{\pm I\}$. Otherwise, $X$ is semisimple and $C_K(X)$ is a torus. Note that every $X\in \mathfrak{g}^*_{x_0,-d}$ that represents a degenerate coset is $K$-conjugate to one of this form. Define an open subgroup of $K$ by $$J_{d} = \left[ \begin{matrix} 1+\mathcal{P}^{\lceil d/2 \rceil}&\mathcal{P}^{\lceil d/2 \rceil} \\ \mathcal{P}^{\lceil (d+1)/2\rceil} & 1+\mathcal{P}^{\lceil d/2 \rceil} \end{matrix} \right] \cap K.$$ It is straightforward to verify that $X$ gives a well-defined character $\eta_X$ of $J_d$, trivial on $G_{x_0,d+}$, by the rule $$\label{E:formula} \eta_X(g)=\psi(\mathrm{tr}(\dot{X}(g-I))).$$ This character is trivial on $K_{d+}$ and depends only on the classes $u+\mathcal{P}^{\lceil (-d+1)/2\rceil}$ and $v+\mathcal{P}^{\lceil -d/2 \rceil}$. For any choice of character $\theta$ of $C_K(X)$ agreeing with $\eta_X(g)$ on $C_K(X)\cap J_d$, write $\eta(X,\theta)$ for the resulting extension to a character of $C_K(X)J_d$. Then Shalika proves the following result with an intricate elementary argument [@Shalika1966 Thm 4.2.1, Thm 4.2.5, §4.3]. **Proposition 11**. *Let $X\in \mathfrak{g}^*_{x_0,-d}$ be as above, and $\theta$ any character of its centralizer $C_K(X)$ agreeing with $\eta_X$ on $C_K(X)\cap J_d$. Then the representation $$\mathcal{S}_{x_0}(X,\theta) = \mathrm{Ind}_{C_K(X)J_d}^K \eta(X,\theta)$$ is irreducible and independent (up to equivalence) of the choice of representative in the $K$-orbit of $\dot{X}(u+\mathcal{P}^{\lfloor(d+1)/2\rfloor},v+\mathcal{P}^{\lceil (d+1)/2\rceil})$. It is of degree $\frac12 q^{d-1}(q^2-1)$ and of depth $d$, meaning it is nontrivial on $K_d$ but trivial on $K_{d+}$.* ## Irreducible representations of $G_x$ parametrized by degenerate cosets at $x$ {#S:gx} Our goal in this section is to give a coordinate-free interpretation of Shalika's construction that allows us to unambiguously attach representations of $G_x$ to any degenerate coset of negative depth. Note that $\mathrm{GL}(2,F)$ acts on $\mathcal{B}(G)$, and all vertices are conjugate under this action. This conjugacy does not in general preserve the $\mathrm{SL}(2,F)$-orbit of $\Gamma$ or $X$. *Example 1*. Let $x_0, z_0$ be as in Section [5.1](#SS:Shalika){reference-type="ref" reference="SS:Shalika"} and $x_1$ the other vertex of the chamber containing $x_0$ in its closure. The element $\omega = \left[ \begin{smallmatrix} 0 & 1 \\ \varpi & 0 \end{smallmatrix} \right]$ used in [@Nevins2005] is an affine reflection such that $\omega\cdot x_0 = x_1$, and ${}^\omega \dot{X}(u,v) = \dot{X}(\varpi^{-1}v, \varpi u)$. Thus in particular in the case of nilpotent orbits, where $\dot{X}(0,1)\sim \dot{X}(-1,0)$, we have ${}^\omega \mathcal{O}_1 = \mathcal{O}_{-\varpi}$. On the other hand, the element $\eta = \left[ \begin{smallmatrix} 1 & 0\\0&\varpi \end{smallmatrix} \right]$ used in [@Nevins2013] is a translation such that $\eta\cdot x_0 = x_1$, but now ${}^\eta \dot{X}(u,v) = \dot{X}(\varpi^{-1}u,\varpi v)$ so ${}^\eta \mathcal{O}_1 = \mathcal{O}_{\varpi}$ instead. We begin by showing that any degenerate coset determines a chamber of $\mathcal{B}(G)$ adjacent to $x$. **Lemma 12**. *Let $G=\mathrm{SL}(2,F)$. Let $x\in \mathcal{B}(G)$ be any vertex and let $\Gamma \in \mathfrak{g}^*_{x,-d}\setminus \mathfrak{g}^*_{x,-d+}$ represent a degenerate coset for some $d>0$. Then there exists a unique chamber $\mathcal{C}=\mathcal{C}_\Gamma$ of $\mathcal{B}(G)$ adjacent to $x$, independent of the choice of representative of $\Gamma+\mathfrak{g}^*_{x,-d+}$, such that for any $z\in \mathcal{C}$ we have $\Gamma \in \mathfrak{g}^*_{x,-d}\cap\mathfrak{g}^*_{z,-d+}$. Moreover, we have $\mathrm{Cent}_{G_x}(\Gamma)=\mathrm{Cent}_{G_z}(\Gamma)$.* *Proof.* Uniqueness is immediate: given $z'$ in any other chamber adjacent to $x$, the geodesic from $z$ to $z'$ contains $x$; hence $\mathfrak{g}^*_{z,-d+}\cap \mathfrak{g}^*_{z',-d+}\subset \mathfrak{g}^*_{x,-d+}$, so does not contain $\Gamma$. Identify $\Gamma$ with an element $\dot{\Gamma}\in \mathfrak{g}_{x,-d}$ via the trace form. Choose a nilpotent element $\dot{X}\in \dot{\Gamma}+\mathfrak{g}_{x,-d+}$. By [@DeBackerNilpotent2002 §5], we may complete $\dot{X}$ to an $\mathfrak{sl}_2(F)$-triple $\{\dot{X}, \dot{H}\in \mathfrak{g}_{x,0}, \dot{Y}\in \mathfrak{g}_{x,d}\}$ and find a split torus $S$ and corresponding apartment $\mathcal{A}(G,S)$ containing $x$, such that if $\Phi(G,S)=\{\pm \alpha\}$, then $\dot{X}\in \mathfrak{g}_{\alpha}$ and $\dot{Y}\in \mathfrak{g}_{-\alpha}$. Let $\mathcal{C}$ be the positive alcove adjacent to $x$ in this apartment. Note that we have $\mathrm{Cent}_{\mathfrak{g}}(\dot{Y}) = \mathfrak{g}_{-\alpha}$. From [@DeBackerNilpotent2002 Lemma 5.2.1] we know that $$\dot{X}+\mathfrak{g}_{x,-d+} = {}^{G_{x,0+}}\left(\dot{X} + \mathrm{Cent}_{\mathfrak{g}_{x,-d+}}(\dot{Y}) \right);$$ thus there exists $g\in G_{x,0+}$ such that $\dot{\Gamma}\in {}^g(\dot{X}+ \mathfrak{g}_{-\alpha}\cap \mathfrak{g}_{x,-d+})$. Since $G_{x,0+}$ fixes $C$ and the coset $\dot{\Gamma}+\mathfrak{g}_{x,-d+}$, we may without loss of generality replace the Lie triple and torus of the preceding paragraph with their $g$-conjugate, so that we have $\dot{\Gamma} \in \dot{X}+\mathfrak{g}_{-\alpha}\cap \mathfrak{g}_{x,-d+}$. For any $z\in C$ we have $0 < \alpha(z-x) < 1$; thus since $\alpha(x), d\in \mathbb{Z}$ we may conclude $$\mathfrak{g}_\alpha \cap \mathfrak{g}_{x,-d} = \mathfrak{g}_\alpha \cap \mathfrak{g}_{z,-d+}. \quad \text{and} \quad \mathfrak{g}_{-\alpha}\cap \mathfrak{g}_{x,-d+} = \mathfrak{g}_{-\alpha}\cap \mathfrak{g}_{z,-d+}.$$ Since $\dot{\Gamma}$ lies in the sum of these two spaces we have $\dot{\Gamma}\in \mathfrak{g}_{z,-d+}$, whence $\Gamma \in \mathfrak{g}^*_{x,-d}\cap \mathfrak{g}^*_{z,-d+}$. Finally, note that $\mathrm{Cent}_{G}(\dot{X})=U_\alpha$ and $U_\alpha\cap G_x=U_\alpha\cap G_z$. Since $\dot{\Gamma}\in\dot{X}+\mathfrak{g}_{x,-d+}$, we have $\mathrm{Cent}_{G_x}(\dot{\Gamma}) \subset \mathrm{Cent}_{G_x}(\dot{X})G_{x,0+}=\mathrm{Cent}_{G_z}(\dot{X})G_{x,0+}\subset G_z$. ◻ **Definition 13**. Let $d=-d_x(\Gamma)$ be such that $\Gamma+\mathfrak{g}^*_{x,-d+}$ is a degenerate coset. Let $z$ be the barycentre of the associated alcove $\mathcal{C}_\Gamma$. Define the subgroup $$\label{E:defJ} J_{x,\Gamma}= \begin{cases} G_{x,d/2} & \text{if $d=d_x(\Gamma)$ is odd};\\ G_{z,d/2} & \text{if $d$ is even}. \end{cases}$$ Note that when $x=x_0$ and $z=z_0$ we have $J_{x,\Gamma}=J_d$. Since $G_{x,n+}\subseteq G_{z,n} \subseteq G_{x,n}$ for any integer $n$, it follows directly that for all $d$, we have $$G_{x,d/2+}\subseteq J_{x,\Gamma} \subseteq G_{x,d/2}.$$ Since $\Gamma \in \mathfrak{g}^*_{x,-d}\cap \mathfrak{g}^*_{z,-d+}$, it defines a character $\eta_{\Gamma}$ of $J_{x,\Gamma}$ that is trivial on $G_{x,d+}$ via the corresponding Moy--Prasad isomorphism. The character depends only on the coset $\Gamma + \mathfrak{g}_{x,-d/2}^*$ if $d$ is odd and on $\Gamma+\mathfrak{g}^*_{z,-d/2+}$ otherwise. Moreover, since $\mathrm{Cent}_{G_x}(\Gamma) = \mathrm{Cent}_{G_z}(\Gamma)$ we deduce directly that $J_{x,\Gamma}$ is normalized by $C_x(\Gamma):=\mathrm{Cent}_{G_x}(\Gamma)$. Thus, for any character $\theta$ of $C_x(\Gamma)$ coinciding with $\eta_{\Gamma}$ on the intersection of their domains there is a unique extension $\eta(\Gamma,\theta)$ of $\eta_\Gamma$ to $C_x(\Gamma)J_{x,\Gamma}$. Define $$\mathcal{S}_x(\Gamma,\theta) = \mathrm{Ind}_{C_x(\Gamma)J_{x,\Gamma}}^{G_x} \eta(\Gamma,\theta).$$ **Proposition 14**. *Suppose $\Gamma$ represents a degenerate coset at a vertex $x\in \mathcal{B}(G)$ and $-d=d_x(\Gamma)<0$. Suppose $\theta$ is a character of the centralizer $C_x(\Gamma)$ of $\Gamma$ in $G_x$ defining a character $\eta(\Gamma,\theta)$ of $C_x(\Gamma)J_{x,\Gamma}$. Then* (a) *$\mathcal{S}_x(\Gamma,\theta)$ is an irreducible representation of $G_x$ of depth $d$ and degree $\frac12 q^{d-1}(q^2-1)$;* (b) *$\mathcal{S}_x(\Gamma,\theta) \cong \mathcal{S}_x(\Gamma',\theta')$ if and only if there exists $g\in G_x$ such that $\eta(\Gamma,\theta)={}^g\eta(\Gamma',\theta')$; and* (c) *for any $\nu\in \mathrm{GL}(2,F)$ we have $$\label{Eq:GL2conj} {}^\nu \mathcal{S}_{x}(\Gamma,\theta) \cong \mathcal{S}_{\nu\cdot x}({}^\nu\Gamma, {}^\nu \theta).$$* *Proof.* When $x=x_0$ and $\Gamma\in \mathfrak{g}^*$ corresponds to some $\dot{X}(u,v)\in\mathfrak{g}_{x_0,-d}\setminus \mathfrak{g}_{x_0,-d+}$, then this construction coincides with Shalika's. If $g\in G_{x}$, then ${}^gC_{x}(\Gamma) = C_{x}({}^g\Gamma)$ and ${}^gJ_{x,\Gamma} = J_{x,{}^g\Gamma}$, so we obtain the invariance of $\mathcal{S}_x(\Gamma,\theta)$ under $G_x$-conjugacy and the choice of representative of the appropriate coset of $\Gamma$. More generally, for any $\nu\in \mathrm{GL}(2,F)$ such that $\nu\cdot x_0 = x$, we have ${}^\nu(\mathfrak{g}^*_{x_0,d})=\mathfrak{g}^*_{x,d}$, ${}^\nu C_{{x_0}}(\Gamma) = C_{x}({}^\nu\Gamma)$ and ${}^\nu J_{x_0,\Gamma} = J_{x,{}^\nu\Gamma}$. Thus $${}^\nu \mathcal{S}_{x_0}(\Gamma,\theta) \cong \mathcal{S}_{x}({}^\nu\Gamma, {}^\nu \theta),$$ where we have identified a $\nu$-conjugate of a representation of $G_{x_0}$ with a representation of $G_x$ under the group isomorphism ${}^\nu G_{x_0}\cong G_x$. Since $\mathrm{GL}(2,F)$ acts transitively on the set of vertices of $\mathcal{B}(\mathrm{SL}(2,F))$, the rest of the statements follow from Proposition [Proposition 11](#P:Shalika){reference-type="ref" reference="P:Shalika"}. ◻ The simple nature of the representations $\mathcal{S}_x(\Gamma,\theta)$ is revealed as follows. **Lemma 15**. *Suppose $x$ is a vertex of $\mathcal{B}(G)$ and $\Gamma_1, \Gamma_2 \in \mathfrak{g}^*_{x,-d}$ represent nonzero but degenerate cosets of $\mathfrak{g}^*_{x,-d}/\mathfrak{g}^*_{x,-d+}$ for some $d>0$. Suppose $s \in \mathbb{R}$ satisfies $\Gamma_1\in \Gamma_2 + \mathfrak{g}^*_{x,-s}$. Then for any choice of characters $\theta_i$ of $C_x(\Gamma_i)$ such that the characters $\eta(\Gamma_i,\theta_i)$ agree upon restriction to $C_x(\Gamma_i)J\cap G_{x,s+}$ for $i\in \{1,2\}$, we have $$\label{E:restrictionsame} \mathrm{Res}_{G_{x,s+}}\mathcal{S}_x(\Gamma_1,\theta_1)\cong \mathrm{Res}_{G_{x,s+}}\mathcal{S}_x(\Gamma_2,\theta_2).$$ In particular, if $s\geq d/2$ then [\[E:restrictionsame\]](#E:restrictionsame){reference-type="eqref" reference="E:restrictionsame"} holds independent of $\theta_i$.* *Proof.* For any $\Gamma_i$, the two representations have the same degree $\frac12q^{d-1}(q^2-1)$ and the same depth $d$. If $s \geq d$ then both sides are $1$-isotypic of the same degree hence equivalent. Suppose $s<d$. Since $\Gamma_1\in \Gamma_2 + \mathfrak{g}^*_{x,-s}$, we have $C_x(\Gamma_1) \subset C_x(\Gamma_2)G_{x,d-s}$. Since $\Gamma_1\in \Gamma_2+\mathfrak{g}^*_{x,-d+}$, Lemma [Lemma 12](#L:CGamma){reference-type="ref" reference="L:CGamma"} yields $J_{x,\Gamma_1}=J_{x,\Gamma_2}$; let us denote this group $J$. Thus $\eta_{\Gamma_i}$ for $i\in \{1,2\}$ are characters of $J$ that agree on $J\cap G_{x,s+}$. If $s\geq d/2$ then $G_{x,s+}\subset J$, and so $\mathrm{Res}_{G_{x,s+}\cap C_x(\Gamma_i)J}\; \eta(\Gamma_i,\theta_i)=\eta_{\Gamma_i}$ is independent of $\theta_i$. Mackey theory thus yields the decomposition $$\label{E:etai} \mathrm{Res}_{G_{x,s+}}\mathcal{S}_x(\Gamma_i,\theta_i) \cong \bigoplus_{\gamma \in G_{x}/C_x(\Gamma_i)J} {}^\gamma\eta_{\Gamma_i}|_{G_{x,s+}}.$$ Each $\gamma \in C_x(\Gamma_i)G_{x,d-s}/C_x(\Gamma_i)J$ fixes the character $\eta_{\Gamma_i}|_{G_{x,s+}}$. The elements $\gamma' \in G_x/C_x(\Gamma_1)G_{x,d-s}=G_x/C_x(\Gamma_2)G_{x,d-s}$ parametrize the orbit of the coset $\Gamma_1+\mathfrak{g}^*_{x,-s} = \Gamma_2+\mathfrak{g}^*_{x,-s}$. Thus [\[E:etai\]](#E:etai){reference-type="eqref" reference="E:etai"} gives the same sum of characters for $i\in \{1,2\}$. If instead $s<d/2$, then $G_{x,d-s}\subseteq J$ so $C_x(\Gamma_1)J=C_x(\Gamma_2)J$. Since $J\subseteq G_{x,s+}$, the double coset space $G_{x,s+}\backslash G_{x} /C_x(\Gamma_i) J$ is now equal to $G_x/C_x(\Gamma_i)G_{x,s+}$, and is independent of $i$. So again by Mackey theory we have $$\begin{aligned} \mathrm{Res}_{G_{x,s+}}\mathcal{S}_x(\Gamma_i,\theta_i) &= \bigoplus_{\gamma \in G_x/C_x(\Gamma_i)G_{x,s+}} \mathrm{Ind}_{G_{x,s+}\cap {}^\gamma(C_x(\Gamma_i) J)}^{G_{x,s+}} {}^\gamma(\eta(\Gamma_i,\theta_i))\\ &=\bigoplus_{\gamma \in G_x/C_x(\Gamma_i)G_{x,s+}}{}^\gamma\left(\mathrm{Ind}_{G_{x,s+}\cap C_x(\Gamma_i) J}^{G_{x,s+}} \eta(\Gamma_i,\theta_i))\right).\end{aligned}$$ When the restriction of $\eta(\Gamma_i,\theta_i)$ to $G_{x,s+}\cap C_x(\Gamma_1) J=G_{x,s+}\cap C_x(\Gamma_2) J$ is independent of $i$, we infer [\[E:restrictionsame\]](#E:restrictionsame){reference-type="eqref" reference="E:restrictionsame"}. ◻ ## Representations attached to nilpotent orbits {#attach} Let $X\in \mathcal{N}^*\setminus\{0\}$ and let $\lambda$ be a corresponding adapted one-parameter subgroup, whose centralizer in $G$ is a maximal split torus $S$. In fact, $S$ is generated by $S_0$ and $\lambda(\varpi)$, and $\mathrm{Cent}_G(X)=ZU$ where $Z$ is the center of $G$ and $B=SU$ is a Borel subgroup. For any vertex $x\in\mathcal{B}(G)$, applying the Cartan decomposition yields $$\label{E:GGxorbit} \mathcal{O}= G\cdot X = \bigsqcup_{n\in \mathbb{Z}}G_x \cdot (\lambda(\varpi)^n\cdot X) = \bigsqcup_{n\in \mathbb{Z}}G_x \cdot (\varpi^{2n} X)$$ as the decomposition of the $G$-orbit of $X$ into disjoint $G_x$-orbits. **Proposition 16**. *Let $x$ be a vertex in $\mathcal{B}(\mathbf{G},F)$, $\mathcal{O}$ a nonzero nilpotent $G$-orbit in $\mathfrak{g}^*$ and $\zeta$ a character of $Z$. Let $\{X_{-d}\mid d_x(X)=-d<0\}$ be any set of representatives of the $G_x$-orbits in $\mathcal{O}\setminus \mathfrak{g}^*_{x,0}$. Then the *representation of $G_x$ attached to $\mathcal{O}$ with central character $\zeta$*, given by $$\label{E:deftau} \tau_x(\mathcal{O},\zeta) = \bigoplus_{d>0} \mathcal{S}_x(X_{-d},\zeta),$$ is independent of choices up to $G_x$-equivalence.* *Proof.* The chamber $\mathcal{C}_X$ associated to $(X_{-d},x)$ by Lemma [Lemma 12](#L:CGamma){reference-type="ref" reference="L:CGamma"} defines an Iwahori subgroup of $G_x$ with pro-p unipotent radical $U_x$; by construction, we have $C_{x}(X) = ZU_x$. Since $\eta_X$ is trivial on $ZU_x\cap J_{x,X}$, the character $\zeta$ of $ZU_x$ defined by $\zeta(zu)=\zeta(z)$ for all $z\in Z$ and $u\in U_x$ extends $\eta_X$. Thus Proposition [Proposition 14](#P:Gxreps){reference-type="ref" reference="P:Gxreps"} applies. ◻ It follows from [\[E:GGxorbit\]](#E:GGxorbit){reference-type="eqref" reference="E:GGxorbit"} that the parity of $d_x(Y)$, for any $Y\in \mathcal{O}$, is an invariant of the $G$-orbit, and we call this the *parity depth of $\mathcal{O}$ at $x$*. Therefore the depths $d$ of the components of $\tau_x(\mathcal{O},\zeta)$ all have parity equal to the parity depth of $\mathcal{O}$ at $x$. Note that the restriction of $\tau_x(\mathcal{O},\zeta)$ to any subgroup of $G_{x,0+}$ is independent of the choice of $\zeta$, so we may drop $\zeta$ from the notation in such cases. As needed, we associate to the zero nilpotent orbit the trivial representation of $G_x$, and denote it $\tau_x(\{0\})$. # The case of positive-depth representations of $\mathrm{SL}(2,F)$ {#S:posdepth} We briefly recap the classification of irreducible representations $\pi = \pi(\chi,\Gamma)$ of $\mathrm{SL}(2,F)$ of positive depth, then establish that their explicit branching to a maximal compact open subgroup $G_x$ can be described as twists of the datum $(\chi,\Gamma)$ defining $\pi$. This allows us to state and prove our main theorem in this case, and to explicitly compute the constant terms that arise. ## Representation of $\mathrm{SL}(2,F)$ of positive depth The classification of the irreducible admissible representations of $G=\mathrm{SL}(2,F)$ was given in Shalika's 1966 thesis [@Shalika1966], and Sally and Shalika obtained many explicit results on their characters in several papers, including [@SallyShalika1984]. An excellent overview is given [@AdlerDeBackerSallySpice2011]. In this section we address the positive-depth supercuspidal representations, using the parametrization of Adler and Yu [@Adler1998; @Yu2001; @FintzenFixYu2021]. Because the tori in $\mathrm{SL}(2,F)$ are one-dimensional, the correcting twist to this construction given by Fintzen, Kaletha and Spice in [@FintzenKalethaSpice2021 Definition 3.1] is trivial in this case. **Proposition 17**. *The isomorphism classes of irreducible representations of $\mathrm{SL}(2,F)$ of positive depth $r$ are parametrized by the $G$-conjugacy classes of pairs $(T,\chi)$, where $T$ is a maximal torus of $G$ and $\chi$ is a character of $T$ of depth $r$.* This is equivalent to the well-known classification of the irreducible positive depth representations of $\mathrm{SL}(2,F)$ in terms of unrefined minimal $K$-types. To construct the representations explicitly we first recall some facts about the maximal tori and their characters. Let $T$ be a maximal torus of $G$ and let $\chi$ be a character of $T$ of depth $r>0$. The building $\mathcal{B}(T)$ of $T$ embeds into $\mathcal{B}(G)$ as the apartment $\mathcal{A}(G,T)$ if $T$ is split and as a single point $\{x_T\}$ otherwise, which is a vertex if $T$ is unramified and the midpoint of a chamber if $T$ is ramified. It follows that the depth $r$ is an integer if $T$ splits over an unramified extension and an element of $\frac12 + \mathbb{Z}$ otherwise. To each pair $(T,\chi)$ we associate an element $\Gamma$ as follows. If $\mathfrak{t}$ denotes the Lie algebra of $T$, then via the Moy-Prasad isomorphism $e\colon \mathfrak{t}_{r/2+}/\mathfrak{t}_{r+} \to T_{r/2+}/T_{r+}$ there exists a nonzero element $\Gamma =\Gamma_\pi \in \mathfrak{t}^*_{-r}$, uniquely defined modulo $\mathfrak{t}^*_{-r/2}$, such that $$\chi(t) = \psi(\Gamma(e^{-1}(t))).$$ We identify $\Gamma$ with an element of $\mathfrak{g}^*$ that is zero on the $T$-invariant complement of $\mathfrak{t}$ in $\mathfrak{g}$. Then $\Gamma\in \mathfrak{g}^*_{x,-r}$ for any $x\in \mathcal{B}(T)$ and we recover $T$ as $\mathrm{Cent}_G(\Gamma)$. Moreover, $\Gamma$ thus defines a character of $G_{x,r}/G_{x,r+}\cong \mathfrak{g}_{x,r}/\mathfrak{g}_{x,r+}$, and following the work of Moy and Prasad, the pair $(G_{x,r},\Gamma)$ is called an *unrefined minimal $K$-type*. *Proof of Proposition [Proposition 17](#P:posdepthpara){reference-type="ref" reference="P:posdepthpara"}.* The classification is known; we only wish to construct the representations $\pi = \pi(T,\chi)$ variously as follows. If $T$ is a split torus, then choose a Borel subgroup $B=TN$ of $G$ containing $T$ and extend $\chi$ trivially across $N$ to a character of $B$. Set $$\mathrm{Ind}_{TN}^G(\chi) = \{ f\colon G\to \mathbb{C}\mid f(tng)=\chi(t)\nu(t)f(g) \forall t\in T, n\in N, g\in G\}$$ where $\nu$ is the square root of the modular character and is given on $T\cong F^\times$ by the $p$-adic norm. Then $\pi(T,\chi)=\mathrm{Ind}_B^G(\chi)$ is an irreducible principal series representation. If $T$ is anisotropic, with associated point $x_T\in\mathcal{B}(G)$, then we first extend $\chi$ to a character of $TG_{x_T,r/2+}$, by setting $$\chi(tg) = \chi(t) \psi(\Gamma(e^{-1}(g)))$$ where $e \colon \mathfrak{g}_{x_T,r/2+}/\mathfrak{g}_{x_T,r+} \to G_{x,r/2+}/G_{x,r+}$ is the Moy-Prasad isomorphism. When $G_{x_T,r/2}\neq G_{x_T,r/2+}$ (which will happen only if $T$ is unramified and $r\in 2\mathbb{Z}$), we take a certain Weil-Heisenberg lift of $\chi|_{G_{x_T,r}}$ to form a $q$-dimensional representation $\omega$ of $T\ltimes G_{x_T,r/2}$, and set $\kappa(tg) = \chi(t)\omega(t,g)$. Then $\pi(T,\chi) = \textrm{c-}\mathrm{Ind}_{TG_{x_T,r/2}}^G \kappa$ is an irreducible supercuspidal representation. ◻ Given $\pi=\pi(T,\chi)$, we let $\Gamma = \Gamma_\pi$ denote a choice of element in $\mathfrak{g}^*$ realizing the character $\chi$, as preceded the proof. Then since $T=\mathrm{Cent}_G(\Gamma)$ we may also say that $(\chi,\Gamma)$ is the data defining $\pi$. ## Branching rules obtained as twists of the inducing datum We begin by proving that the branching rules obtained in [@Nevins2005 Theorem 7.4] and [@Nevins2013 Theorem 6.2] are in fact constructable from twists of the data $(\chi,\Gamma)$. **Theorem 18**. *Let $\pi=\pi(T,\chi)$ be an irreducible representation of $G$ of depth $r>0$. Let $\Gamma=\Gamma_\pi\in \mathfrak{g}^*$ realize $\chi$ as above. Then for any vertex $x\in\mathcal{B}(G)$ we have $$\label{E:basicdecomp} \mathrm{Res}_{G_{x}}\pi = \pi^{G_{x,r+}} \oplus \bigoplus_{g\in [G_x\backslash G/\mathrm{Cent}(\Gamma)]^{\text{deg}}} \mathcal{S}_x({}^g\Gamma, {}^g\chi)$$ where $[G_x\backslash G/\mathrm{Cent}(\Gamma)]^{\text{deg}}$ denotes a parameter set for the $G_x$-orbits in $G\cdot \Gamma$ that do not meet $\mathfrak{g}^*_{x,-r}$, that is, such that the coset ${}^g\Gamma + \mathfrak{g}^*_{x,d_x({}^g\Gamma)+}$ is degenerate.* *Proof.* We begin with the case that $T=\mathrm{Cent}(\Gamma)$ is anisotropic. Set $y=x_T$ and let $\pi(T,\chi) =\textrm{c-}\mathrm{Ind}_{TG_{y,r/2}}^G \kappa$ be the corresponding supercuspidal representation. First suppose that we are in the special case that $x=x_0$ and $y \in \overline{\mathcal{C}}$, the closure of the fundamental alcove, which was the case considered in [@Nevins2013]. By [@Nevins2013 Prop 4.4], the double coset space $G_x\backslash G/TG_{y,r/2}$ that arises in the Mackey decomposition $$\mathrm{Res}_{G_x}\pi = \bigoplus_{g \in G_x\backslash G/TG_{y,r/2}}\mathrm{Ind}_{G_x\cap {}^g(TG_{y,r/2})}^{G_x} {}^g\kappa$$ is independent of $r$ and is given by $G_x\backslash G/T$. Since $T=\mathrm{Cent}_G(\Gamma)$, this latter space parametrizes the $G_x$-orbits in the $G$-orbit of $\Gamma$ in $\mathfrak{g}^*$. By [@Nevins2013 Thm 6.1], each of these Mackey components is irreducible. Now since $\Gamma$ has depth $-r$ and depth is $G$-invariant, ${}^g\Gamma$ meets $\mathfrak{g}^*_{x,-r}$ if and only if $d_x({}^g\Gamma)=-r$. Since $\mathrm{Cent}_G(\Gamma)=T$, this happens if and only if $gx = x_T= y$ in which case the corresponding Mackey component has depth $r$ and so lies in $\pi^{G_{x,r+}}$. Note that $\pi^{G_{x,r+}} \neq \{0\}$ thus arises only when $T$ is an unramified torus attached a vertex $y$ in the same $G$-conjugacy class as $x$. When $gx\neq y$, then by in [@Nevins2013 Thm 6.2],[^3] the corresponding Mackey component satisfies $$\mathrm{Ind}_{G_{x}\cap {}^g(TG_{y,r/2})}^{G_x} {}^g\kappa \cong \mathcal{S}_{x}({}^g\Gamma, {}^g\chi),$$ as required, yielding [\[E:basicdecomp\]](#E:basicdecomp){reference-type="eqref" reference="E:basicdecomp"} for the fundamental case. Now suppose that $x\in \mathcal{B}(G)$ is an arbitrary vertex. Then there exists $k\in \mathrm{GL}(2,F)$ such that $k x = x_0$. Choose $h\in \mathrm{SL}(2,F)$ such that $hy\in k^{-1} \overline{\mathcal{C}}$. Then via the identification ${}^kG_{x}= G_{x_0}$, and then the isomorphism ${}^h\pi\cong \pi$, we may write $$\mathrm{Res}_{G_{x_0}}{}^{kh}\pi = \mathrm{Res}_{G_x}{}^h\pi \cong \mathrm{Res}_{G_x}\pi.$$ Even when $kh\notin \mathrm{SL}(2,F)$, the data defining the representation ${}^{kh}\pi$ is simply $({}^{kh} T,{}^{kh}\chi,{}^{kh}\Gamma, khy)$. We may therefore apply the decomposition [\[E:basicdecomp\]](#E:basicdecomp){reference-type="eqref" reference="E:basicdecomp"} to $\mathrm{Res}_{G_{x_0}}{}^{kh}\pi$. Since ${}^kG_{x,r+}=G_{x_0,r+}$, we have $({}^{kh}\pi)^{G_{x_0,r+}}=({}^h\pi)^{G_{x,r+}}$. Moreover, for any $g$ defining a $G_{x_0}$-orbit of ${}^{kh}\Gamma$ that does not meet $\mathfrak{g}_{x_0,-r}^*$, we have $$S_{x_0}({}^{gkh}\Gamma,{}^{gkh}\chi)%=S_{x_0}({}^{kk^{-1}gk}\Gamma,{}^{kk^{-1}gk}\chi) ={}^k(S_{x_0}({}^{(k^{-1}gk) h}\Gamma,{}^{(k^{-1}gk)h}\chi))= S_x({}^{g'h}\Gamma,{}^{g'h}\chi),$$ where $g' = k^{-1}gk$; then $g'h$ defines a $G_x$-orbit of $\Gamma$ that does not meet $\mathfrak{g}_{x,-r}^*$, showing the index sets correspond. We now consider the case that $T$ is a split torus, so that $\pi=\pi(T,\chi)=\mathrm{Ind}_B^G\chi$ for some Borel subgroup $B=TU$ containing $T$ having $U$ as its unipotent radical. Since $G=G_xB$, there is a unique (highly reducible) Mackey component in this case. Instead, in the special case that $x\in \{x_0,x_1\}$ and $T=S$, the decomposition of $\mathrm{Res}_{G_x}\pi$ into irreducibles is found in [@Nevins2005] by explicitly decomposing the $G_x$-subrepresentations $\pi^{G_{x,n}}$ as $n\to \infty$. We need to show that this decomposition is in fact of the form [\[E:basicdecomp\]](#E:basicdecomp){reference-type="eqref" reference="E:basicdecomp"}. We begin with $x=x_0$ and $T=S$. First note that as $\pi$ has depth $r$ at $x$, the subrepresentation $\pi^{G_{x,r+}}$ is nonzero, and in fact is irreducible by [@Nevins2005 Prop 4.4] (where this space is denoted $V^{K_m}$, for $m=r+1$ the conductor of $\chi$). By depth we know that ${}^g\Gamma \in \mathfrak{g}_{x,-r}^*$ if and only if ${}^gT \subset G_x$, so the $G_x$-orbits in $G\cdot \Gamma$ meeting $\mathfrak{g}_{x,-r}^*$ correspond to the trivial double coset of $G_{x}\backslash G/T$. The irreducible representations of $G_{x}$ of depth greater than $r$ appearing in $\mathrm{Res}_{G_x}\pi$ are classified in [@Nevins2005 Thm 7.4]. The notation in that paper relates to ours as follows. Identify $\mathfrak{g}$ and $\mathfrak{g}^*$ via the trace form. The conductor $n$ is $d+1$, and for any $u\in \mathcal{R}^\times, v\in \mathcal{P}$ we have $$\label{NotationNevins2005} \mathcal{D}_n(\rho, \dot{X}(u,v)) := \mathcal{S}_{x}(\varpi^{-d}\dot{X}(u,v), \rho).$$ If $\dot{\Gamma}$ is the diagonal matrix $\textrm{diag}(a,-a)\in \mathfrak{g}_{x,-r}$, set $\gamma_0 = a\varpi^{d}$, $\gamma_1 = a\varepsilon^{-1}\varpi^{d}$ and write $$g_i= \left[ \begin{matrix} 1& -\frac 12 \gamma_i^{-1}\\ \gamma_i & \frac 12 \end{matrix} \right].$$ Following the notation in *loc. cit.*, define $Y_0 = \dot{X}(1,\gamma_0^2)$ and $Y_1 = \dot{X}(\varepsilon,\varepsilon\gamma_1^2)$; then $$\label{E:Y0Y1} \varpi^{-d}Y_0%:=\smat{0&1\\\gamma_0^2 &0} = \varpi^{-d}\dot{X}(1,\gamma_0^2) ={}^{g_0}\Gamma \quad \text{and}\quad \varpi^{-d}Y_1 %:= \smat{0&\ep \\ \ep \gamma_1^2 & 0} = \ep \varpi^{-d}\dot{X}(1,\gamma_1^2) ={}^{g_1}\Gamma.$$ It follows that $\rho_i:={}^{g_i}\chi$ is a character of $C_{{x}}(Y_i)=C_{{x}}({}^{g_i}\Gamma)$ extending $\eta_{{}^{g_i}\Gamma}$. Then, in our notation, [@Nevins2005 Thm 7.4] asserts that for each integer $d>r$, $\mathrm{Res}_{G_x}\pi$ has two irreducible components of depth $d$, denoted $W_{d-1}^\pm$, and these are explicitly given by $$\label{E:psdecomp} W_{d-1}^+\oplus W_{d-1}^-\cong \mathcal{S}_{x}({}^{g_0}\Gamma, {}^{g_0}\chi) \oplus \mathcal{S}_{x}({}^{g_1}\Gamma, {}^{g_1}\chi).$$ Noting the factorization $$\left[ \begin{matrix} 1 & -\frac 12\gamma^{-1}\\ \gamma & \frac 12 \end{matrix} \right] = \left[ \begin{matrix} 1&0\\\gamma&1 \end{matrix} \right]\left[ \begin{matrix} 1 & -\frac 12 \gamma^{-1}\\0&1 \end{matrix} \right],$$ we see that as $\gamma$ runs over the distinct square classes in $\gamma\in\mathcal{P}^{d}\setminus \mathcal{P}^{d+1}$, for all $d>0$, we obtain representatives of the distinct nontrivial cosets of $G_{x}\backslash G/T$, which is the index set $[G_x\backslash G/\mathrm{Cent}(\Gamma)]^{\text{deg}}$. The same result is obtained for $x=x_1$ in [@Nevins2005 Cor 4.6, Thm 7.4] via conjugation by $\omega = \left[ \begin{smallmatrix} 0 & 1 \\ \varpi & 0 \end{smallmatrix} \right] \in \mathrm{GL}(2,F)$, as $\omega x_0=x_1$. For arbitrary $x$, we proceed as in the previous case, this time choosing $k, h\in \mathrm{SL}(2,F)$ for which $kx \in \{x_0,x_1\}$ and for which ${}^hT = S$. ◻ ## Main theorem for representations of positive depth Before stating the next theorem, we require a short lemma about filtrations of tori. **Lemma 19**. *Let $T$ be a maximal torus of $G=\mathrm{SL}(2,F)$, let $x\in \mathcal{B}(G)$. Let $\dot{X}\in \mathfrak{g}$ be nilpotent and denote its centralizer in $G_x$ by $U_x$. Then for any $\ell\in \mathbb{Z}_+$, we have $$T \cap U_xG_{x,\ell} \subseteq ZT_{\ell}.$$* *Proof.* As in the proof of Lemma [Lemma 12](#L:CGamma){reference-type="ref" reference="L:CGamma"}, we may choose an apartment $\mathcal{A}(G,S)\subset\mathcal{B}(G)$ containing $x$ with root system $\Phi = \{\pm \alpha\}$ such that $\dot{X} \in \mathfrak{g}_\alpha$; then $U_x = (G_x\cap U_{\alpha})Z$. Given $t\in T\cap U_xG_{x,\ell}$, write $t=uzg$ with $z\in Z$, $u\in G_x\cap U_{\alpha}$ and $g\in G_{x,\ell}$. Then $z^{-1}t \in (G_{x,\ell}\cap U_{-\alpha})S_\ell(G_x\cap U_{\alpha})$, so its trace is that of an element of $S_\ell$. Since for any torus $T$ of $G$ and any $\ell>0$, we have $t'\in T_\ell$ if and only if the trace of $t'$ lies in $2+\mathcal{P}^{2\ell}$, we conclude that $t\in T_\ell$. ◻ **Lemma 20**. *Let $T=\mathrm{Cent}_{G}(\Gamma)$, where $\Gamma$ is a semisimple element of depth $-r$. Suppose that at $x\in\mathcal{B}(G)$ we have $d_{x}(\Gamma)=-d<-r$. Then $T\cap G_x = ZT_{d-r}$, so that $T\cap G_{x,\ell} = T_{d-r+\ell}$ for any $\ell \geq 0$.* *Proof.* Let $\mathfrak{t}$ be the Lie algebra of $T$. The hypotheses imply that $d_x(\varpi^k\Gamma)=d(\varpi^k\Gamma)-(d-r)$ for any $k\in \mathbb{Z}$. Thus since $\mathfrak{t}$ is one-dimensional, for any element $\dot{X}\in \mathfrak{t}_\ell\setminus \mathfrak{t}_{\ell+}$ we have $d_x(\dot{X}) = \ell-(d-r)$, yielding $\mathfrak{t}\cap \mathfrak{g}_{x,\ell} = \mathfrak{t}_{d-r+\ell}$. Passage to the group yields the desired result, where at depth zero, we observe that $Z\subset T\cap G_x$ for all $x$. ◻ **Theorem 21**. *Let $\pi = \pi(T,\chi)$ be an irreducible representation of $G=\mathrm{SL}(2,F)$ of depth $r>0$ and let $\Gamma = \Gamma_\pi \in \mathfrak{g}^*$. Then for each maximal compact $G_x$, there is an integer $n_{x}(\pi)$ such that in the Grothendieck group of representations we have $$\label{E:posdepththm} \mathrm{Res}_{G_{x,r+}}\pi \cong n_{x}(\pi){\bf 1}+ \sum_{\mathcal{O}\in \mathrm{Nil}(\Gamma)} \mathrm{Res}_{G_{x,r+}}\tau_{x}(\mathcal{O}).$$ That is, up to some copies of the trivial representation, the representation $\pi$ is locally completely determined by the nilpotent support of $\Gamma$.* *Proof.* The restriction to $G_{x,r+}$ will be trivial on any $G_x$-representations of depth less than or equal to $r$, so our first step is to match components of depth $d>r$ in $\mathrm{Res}_{G_x}\pi$ and in $\sum_{\mathcal{O}\in \mathrm{Nil}(\Gamma)}\tau_x(\mathcal{O})$. Note that the restriction to $G_{x,r+}$ is independent of the choice of central character $\zeta$ so it is omitted from the notation. Theorem [Theorem 18](#T:posdepth){reference-type="ref" reference="T:posdepth"} gives the decomposition [\[E:basicdecomp\]](#E:basicdecomp){reference-type="eqref" reference="E:basicdecomp"} of the left side: the components of depth greater than $r$ are parametrized by the degenerate $G_x$-orbits of $\Gamma$ at $x$. From [\[E:deftau\]](#E:deftau){reference-type="eqref" reference="E:deftau"} we infer the decomposition of the right side: the components are parametrized by nilpotent $G_x$-orbits in $\mathcal{O}\setminus \mathfrak{g}_{x,0}^*$ for each $\mathcal{O}\in \mathrm{Nil}(\Gamma)$. By Proposition [Proposition 9](#P:asymptotic_orbit_invariance){reference-type="ref" reference="P:asymptotic_orbit_invariance"}, each degenerate coset $\xi = {}^g\Gamma+\mathfrak{g}^*_{x,-d+}$, where $d=-d_x({}^g\Gamma)$, is represented by a nilpotent element $X\in \mathcal{O}({}^g\Gamma,x)$ such that moreover ${}^g\Gamma-X\in \mathfrak{g}^*_{x,-r}$. The $G_x$-orbit of $\xi$ determines the $G_x$-orbit of $X$ and by definition $G\cdot X\in \mathrm{Nil}(\Gamma)$. Thus for each $d>r$ there is a one-to-one correspondence between the $G_x$-orbits in $G\cdot \Gamma$ whose depth at $x$ is $-d$ and the $G_x$-orbits in $\mathrm{Nil}(\Gamma)$ whose depth at $x$ is $-d$. To complete the proof we need to show the corresponding representations are isomorphic upon restriction to $G_{x,r+}$. Let $\zeta = \chi|_Z$ be the central character of $\pi$. If $r<d\leq 2\lfloor r \rfloor + 1$ then applying Lemma [Lemma 15](#L:Shalikaequiv){reference-type="ref" reference="L:Shalikaequiv"} to the pair $\Gamma_1 = {}^g\Gamma$ and $\Gamma_2=X$, with $s=r$ gives $\mathrm{Res}_{G_{x,r+}}\mathcal{S}_x({}^g\Gamma,{}^g\chi) \cong \mathrm{Res}_{G_{x,r+}}\mathcal{S}_x(X,\zeta)$ as required. If $d>2r$, then we have a stronger result. Lemma [Lemma 20](#L:toriintersection2){reference-type="ref" reference="L:toriintersection2"} implies that $\mathrm{Cent}_{G_x}({}^g\Gamma) = Z\;{}^gT_{d-r}\subseteq Z\;{}^gT_{r+}$, and thus ${}^g\chi$ is given on this subgroup by the central character $\zeta$. Since the chambers attached to ${}^g\Gamma$ and to $X$ by Lemma [Lemma 12](#L:CGamma){reference-type="ref" reference="L:CGamma"} coincide, we have $J:=J_{x,{}^g\Gamma}=J_{x,X}$. Since ${}^g\Gamma-X\in \mathfrak{g}^*_{x,-r}\subset \mathfrak{g}^*_{x,-d/2+}$, we have $\eta_{{}^g\Gamma}=\eta_X$ as characters of $J$. Moreover, since ${}^g\Gamma \in X+\mathfrak{g}^*_{x,-r}$, we have $C_{G_x}({}^g\Gamma) \subseteq C_{G_x}(X)G_{x,d-r} \subseteq C_{G_x}(X)J$. Therefore $\eta({}^g\Gamma,{}^g\chi)=\eta(X,\zeta)$ as characters of this common group so that $\mathcal{S}_x({}^g\Gamma,{}^g\chi) = \mathcal{S}_x(X,\zeta)$ as representations of $G_x$. ◻ In the course of the proof we established that the components of depth $d>2r$ occurring in $\mathrm{Res}_{G_x}\pi$ coincide as representations of $G_x$, not just as representations of $G_{x,r+}$, with the components of depth $d>2r$ in $\sum_{\mathcal{O}\in \mathrm{Nil}(\Gamma)} \tau_{x}(\mathcal{O},\zeta)$, where $\zeta$ is the central character of $\pi$. This was proven case by case in [@Nevins2005 Rem 7.5] and [@Nevins2013 Prop 7.6]. **Remark 22**. Comparing [\[E:posdepththm\]](#E:posdepththm){reference-type="eqref" reference="E:posdepththm"} with the known values of $\mathcal{WF}(\pi)$ from [@DeBackerSally2000 Tables 1--4] reveals that $\mathrm{Nil}(\Gamma)=\mathcal{WF}(\pi)$ in all cases (*cf* Theorem [Theorem 7](#T:nil=wf){reference-type="ref" reference="T:nil=wf"}). Moreover, with the standard normalization chosen in [@MoeglinWaldspurger1987 I.8], the coefficients of the leading terms of the local character expansion agree with those of [\[E:posdepththm\]](#E:posdepththm){reference-type="eqref" reference="E:posdepththm"}; namely $c_\mathcal{O}(\pi)=1$ for all $\mathcal{O}\in \mathcal{WF}(\pi)$. Thus Theorem [Theorem 21](#T:posdepth2){reference-type="ref" reference="T:posdepth2"} is a representation-theoretic analogue of the analytic local character expansion. On the other hand, the constant term $n_{x}(\pi)$ of the decomposition [\[E:posdepththm\]](#E:posdepththm){reference-type="eqref" reference="E:posdepththm"} does not (and could not) agree with the constant term $c_{0}(\pi)$ of the local character expansion. For one, $n_x(\pi)\in \mathbb{Z}$ whereas $c_0(\pi)$ may be half-integral; see Table [5](#Table:c0){reference-type="ref" reference="Table:c0"}. For another, $n_x(\pi)$ depends on the dimension of $\pi^{G_{x,r+}}$, which may vary based on the $G$-conjugacy class of the vertex $x\in \mathcal{B}(G)$. Let us compute the constant terms $n_x(\pi)$ explicitly. **Proposition 23**. *Let $\pi=\pi(T,\chi)$ be an irreducible representation of depth $r>0$ as in Theorem [Theorem 18](#T:posdepth){reference-type="ref" reference="T:posdepth"}. Then for each vertex $x\in \mathcal{B}(G)$, the dimension of the subspace of $G_{x,r+}$-fixed vectors, as well as the value of the coefficient $n_{x}(\pi)$ appearing in [\[E:posdepththm\]](#E:posdepththm){reference-type="eqref" reference="E:posdepththm"} are as given in Table [1](#T:c0x){reference-type="ref" reference="T:c0x"}.* -------------------------------- ------------------------ ------------------------- ----------------------------- -------------------------------------- Type of torus $T$ split unramified, $x_T\sim x$ unramified, $x_T\not\sim x$ ramified depth $r$ $r\in \mathbb{Z}_{>0}$ $r\in \mathbb{Z}_{>0}$ $r\in \mathbb{Z}_{>0}$ $r\in \frac12 + \mathbb{Z}_{\geq 0}$ $\dim(\pi(T,\chi)^{G_{x,r+}})$ $(q+1)q^r$ $(q-1)q^r$ $0$ $0$ $n_{x}(\pi)$ $q+1$ $q-q^r$ if $r$ is even $1-q^r$ if $r$ is even $\displaystyle (1-q^{r-1/2})(q+1)/2$ $1-q^r$ if $r$ is odd $q-q^r$ if $r$ is odd -------------------------------- ------------------------ ------------------------- ----------------------------- -------------------------------------- : The values of $n_{x}(\pi)$ appearing in [\[E:posdepththm\]](#E:posdepththm){reference-type="eqref" reference="E:posdepththm"} for each irreducible representation of $\mathrm{SL}(2,F)$ of depth $r>0$. *Proof.* Let $\pi=\pi(T,\chi)$ have depth $r>0$. From Theorem [Theorem 18](#T:posdepth){reference-type="ref" reference="T:posdepth"} we have the equality $$\label{E:valgxr} n_{x}(\pi)=\dim(\pi^{G_{x,r+}}) - \sum_{\mathcal{O}\in \mathrm{Nil}(\Gamma)}\dim(\tau_x(\mathcal{O}))^{G_{x,r+}}.$$ Let us first compute $\dim(\pi^{G_{x,r+}})$ in each case. If $\pi$ is a principal series representation and $B$ is a Borel subgroup containing $T$, then $\pi^{G_{x,r+}} = \mathrm{Ind}_{(B\cap G_x)G_{x,r+}}^{G_x}\chi$, whence $\dim(\pi^{G_{x,r+}}) = \vert G_x/(B\cap G_x)G_{x,r+}\vert = (q+1)q^r$. If $\pi=\pi(T,\chi)=\textrm{c-}\mathrm{Ind}_{TG_{x_T,r/2}}^G\kappa$ is an unramified supercuspidal representation such that some $G$-conjugate of $T$ is contained in $G_x$, then (replacing $T$ and $\pi$ by this conjugate) we have $x_T=x$ and $\pi^{G_{x,r+}} = \mathrm{Ind}_{TG_{x,r/2}}^{G_x} \kappa(\chi)$. It follows from a calculation in [@Nevins2013 Prop 4.8] that independently of the parity of $r$ we have $\dim(\pi^{G_{x,r+}})=(q-1)q^r$. On the other hand, if $T$ is not conjugate to a torus contained in $G_x$, then $\mathrm{Res}_{G_x}\pi(T,\chi)$ has no components of depth $r$ and $\dim(\pi^{G_{x,r+}})=0$. Similarly, if $\pi =\pi(T,\chi)$ is a ramified supercuspidal representation, then its depth is half-integral, whence for a vertex $x$ we have $G_{x,r}=G_{x,r+}$, and thus by definition of depth $\pi^{G_{x,r+}}=\{0\}$. On the other hand, the space of $G_{x,r+}$-fixed vectors of $\tau_x(\mathcal{O})$ is exactly the sum of its irreducible components of depth $d\leq r$. These have total dimension $\frac12q^{d-1}(q^2-1)$ and correspond to the $G_x$-orbits of $\mathcal{O}$ whose depths $-d$ at $x$ satisfy $-r\leq -d\leq -1$. Thus if the parity depth of $\mathcal{O}$ at $x$ is even then $$\dim(\tau_x(\mathcal{O})^{G_{x,r+}}) = %\sum_{e=1}^{k} \dim(\Sh_{x}(\varpi^{-2e}X,\zeta))= \sum_{e=1}^{\lfloor r/2 \rfloor} \frac12q^{2e-1}(q^2-1) %=\frac12(q^2-1)q^2q^{-1}\frac{(q^2)^k-1}{q^2-1} = \frac12q(q^{2\lfloor r/2\rfloor}-1).$$ whereas if it is odd we have $$\dim(\tau_{x}(\mathcal{O})^{G_{x,r+}}) = %\sum_{e=1}^{\ell} \dim(\Sh_{x}(\varpi^{-2e}X,\zeta))= \sum_{e=1}^{\lceil r/2 \rceil} \frac12q^{2e-2}(q^2-1) %=\frac12(q^2-1)\frac{(q^2)^{k}-1}{q^2-1} = \frac12(q^{2\lceil r/2 \rceil}-1).$$ Note that $\dim(\tau_{x}(\mathcal{O})\oplus \tau_{x}(\mathcal{O}'))^{G_{x,r+}}= \frac12(q+1)(q^{\lfloor r\rfloor} -1)$ when the $G_x$-orbits in $\mathcal{O}$ and $\mathcal{O}'$ have opposite parity depths at $x$. Consequently, we can compute $n_{x}(\pi)$ using [\[E:valgxr\]](#E:valgxr){reference-type="eqref" reference="E:valgxr"} and the explicit determination of $\mathrm{Nil}(\Gamma)$ in Lemma [Lemma 10](#L:NilGamma){reference-type="ref" reference="L:NilGamma"}, as follows. If $\pi$ is a principal series representation, then $T$ is a torus and $\Gamma$ is split. Thus all principal nilpotent orbits occur, yielding $$n_{x}(\pi) = (q+1)q^r - 2\left(\frac12(q+1)(q^r-1)\right) = q+1.$$ If $\pi$ is a supercuspidal representation corresponding to a ramified torus, then $\mathrm{Nil}(\Gamma)$ consists of two nonzero orbits which will be of opposite parity at any vertex $x$, and $\lfloor r \rfloor = r-\frac12$. Thus we simply have $n_{x}(\pi)=0-\frac12(q+1)(q^{r-1/2}-1)$. On the other hand, if $T$ is unramified, then $\mathrm{Nil}(\Gamma)$ consists of two nonzero orbits and at any vertex $x$, the parity of the depths of elements of these orbits is that of $d_{x}(\Gamma)$. Thus if $x$ is $G$-conjugate to $x_T$, the orbits that occur have the same parity as $-r=d_{x_T}(\Gamma)$, so $$n_{x}(\pi) = \begin{cases} (q-1)q^r - 2\left(\frac12q(q^{r}-1)\right) = q-q^r & \text{if $r$ is even;}\\ (q-1)q^r - 2\left(\frac12(q^{r+1}-1)\right) = 1-q^r & \text{if $r$ is odd.} \end{cases}$$ Finally, if $x$ is not $G$-conjugate to $x_T$, then $d_x(\Gamma)$ and $d_{x_T}(\Gamma)$ have opposite parity. Thus we conclude $$n_{x}(\pi) = \begin{cases} 0 - 2\left(\frac12(q^{r}-1)\right) = 1-q^r & \text{if $r$ is even;}\\ 0 - 2\left(\frac12q(q^{r-1}-1)\right) = q-q^r & \text{if $r$ is odd.} \end{cases}$$ ◻ # The case of depth-zero representations of $\mathrm{SL}(2,F)$ {#S:depthzero} To establish our result for depth-zero representations of $\mathrm{SL}(2,F)$, we apply a result by Barbash and Moy relating the wave front set of $\pi$ to that of $\pi^{G_{x,0+}}$, viewed as a representation of $\mathrm{SL}(2,\mathfrak{f})\cong G_{x,0}/G_{x,0+}$ (Proposition [Proposition 25](#P:BM){reference-type="ref" reference="P:BM"}). We begin by recalling the representation theory of $\mathrm{SL}(2,\mathfrak{f})$ and then the classification of depth-zero representations of $\mathrm{SL}(2,F)$. ## Representations of $\mathrm{SL}(2,\mathfrak{f})$ This theory is well-known and is beautifully recapped in [@DigneMichel1991 §15]. Let $\mathsf{G}= \mathrm{SL}(2,\mathfrak{f})$, $\mathsf{T}$ a maximal torus of $\mathsf{G}$ and $\overline{\chi}$ a character of $\mathsf{T}$ (which is assumed to be nontrivial if $\mathsf{T}$ is anisotropic). The irreducible representations of $\mathsf{G}$ are parametrized by these pairs $(\mathsf{T},\overline{\chi})$ as follows. If $\mathsf{T}$ is split and $\overline{\chi}^2\neq {\bf 1}$ then $\sigma(\mathsf{T},\overline{\chi})$ is an irreducible principal series representation; if $\mathsf{T}$ is anisotropic and $\overline{\chi}^2\neq {\bf 1}$ then $\sigma(\mathsf{T},\overline{\chi})$ is a (Deligne--Lusztig) cuspidal representation. If $\mathsf{T}$ is split and $\overline{\chi}={\bf 1}$ then $\sigma(\mathsf{T},\overline{\chi}) = {\bf 1}\oplus \overline{\mathrm{St}}$ where $\overline{\mathrm{St}}$ denotes the Steinberg representation of $\mathsf{G}$. For either $\mathsf{T}$, when $\overline{\chi}$ is a strictly quadratic character, we obtain two irreducible representations $\sigma^u(\mathsf{T},\overline{\chi})$ for $u\in \{1,\varepsilon\}$ (as the components of the restriction $\sigma(\mathsf{T},\overline{\chi})$ to $\mathrm{SL}(2,\mathfrak{f})$ of a corresponding (irreducible) representation of $\mathrm{GL}(2,\mathfrak{f})$). They are distinguished by the theory of Gel'fand--Graev representations, as follows. Let $X\in \mathfrak{g}(\mathfrak{f})^*\setminus\{0\}$ be nilpotent, and identify this with a nilpotent element $\dot{X}\in \mathfrak{g}(\mathfrak{f})$. Complete this to an $\mathfrak{sl}(2,\mathfrak{f})$ triple $\{\dot{Y},\dot{H},\dot{X}\}$ and let $\mathfrak{u}(\mathfrak{f}) = \mathfrak{f}\dot{Y}$. Then $X$ defines a character of $\mathsf{U}=\exp(\mathfrak{u}(\mathfrak{f}))$ by $\psi_X(\exp(W)) = \psi(X(W))$ for all $W\in \mathfrak{u}(\mathfrak{f})$. Then the representation of $\mathrm{SL}(2,\mathfrak{f})$ given by $$\label{E:GGrep} \gamma_{\mathcal{O}} = \mathrm{Ind}_{\mathsf{U}}^\mathsf{G}\psi_X$$ depends (up to equivalence) only on the nonzero orbit $\mathcal{O}=\mathsf{G}\cdot X$, and is called the Gel'fand--Graev representation of $\mathsf{G}$ associated to $\mathcal{O}$. Contrary to convention, we parametrize our nonzero nilpotent orbits by upper triangular matrices $\dot{X}_u \in \mathfrak{g}(\mathfrak{f})$ as in [\[E:nilpotentorbits\]](#E:nilpotentorbits){reference-type="eqref" reference="E:nilpotentorbits"}, where $u \in \mathfrak{f}^\times/(\mathfrak{f}^\times)^2 \sim \{1,\varepsilon\}$. With respect to this choice, one can compute directly that the character $[\gamma_{\mathcal{O}_u}]$ of the Gel'fand--Graev representation associated to $\mathcal{O}=\mathsf{G}\cdot X_u$ is given by $$[\gamma_{\mathcal{O}_u}](g) = \begin{cases} q^2-1 & \text{if $g=I$};\\ 2\sigma_{-us} = 2\sum_{t \in (\mathfrak{f}^\times)^2}\psi(-ust) & \text{if $g \sim \left[ \begin{smallmatrix} 1&s\\0&1 \end{smallmatrix} \right]$};\\ 0 & \text{otherwise}. \end{cases}$$ By [@DigneMichel1991 Thm 14.30], the decomposition into irreducible subrepresentations of $\gamma_{\mathcal{O}_u}$ is multiplicity-free. Using character tables it is straightforward to see that it contains all irreducible principal series representations, all Deligne--Lusztig cuspidal representations, the Steinberg representation, and exactly one from each pair of representations arising from quadratic characters. Our parametrization is therefore as follows: for $u\in \mathfrak{f}^\times/(\mathfrak{f}^\times)^2$, and a quadratic character $\overline{\chi}$ of $\mathsf{T}$, let $\sigma^{u}(\mathsf{T},\overline{\chi})$ denote the component of $\sigma(\mathsf{T},\overline{\chi})$ occuring in $\gamma_{\mathcal{O}_u}$. In the notation of [@DigneMichel1991 §15], the characters of these representations are labeled $\overline{\chi}^\pm$ where $[\sigma^{-1}(\mathsf{T},\overline{\chi})] = \overline{\chi}^+$ and $[\sigma^{-\varepsilon}(\mathsf{T},\overline{\chi})] = \overline{\chi}^-$. ## Depth-zero representations of $\mathrm{SL}(2,F)$ {#SS:SL2F} Now let $G=\mathrm{SL}(2,F)$ and let $\chi$ be a depth-zero character of a maximal split or unramified torus. Assume $\chi$ is nontrivial if the torus is nonsplit. There are two nonconjugate choices $T_1,T_2$ for an unramified anisotropic torus. If $x_0$ and $x_1$ are the vertices of the standard alcove, as before, then we can choose representatives $T^i$ of the conjugacy classes of maximal tori such that $T^i\subset G_{x_i}$, for $i\in \{0,1\}$. Then $\mathsf{T}_i = T^i_0/T^i_{0+}$ is a maximal anisotropic torus of $G_{x_i,0}/G_{x_i,0+}=:\mathsf{G}_i \cong \mathrm{SL}(2,\mathfrak{f})$. Let $T$ denote the split torus corresponding to the standard apartment and set $\mathsf{T}=T_0/T_{0+}$, which is a maximal split torus of both $\mathsf{G}_1$ and $\mathsf{G}_2$. In each case, the character $\chi$ factors to a character $\overline{\chi}$ of the quotient. In the nonsplit case, for each $i\in \{0,1\}$ inflate the representation $\sigma(\mathsf{T}_i,\overline{\chi})$ of $\mathsf{G}_i$ to a representation of $G_{x_i}$ and define $\pi(T^i,\chi) = \textrm{c-}\mathrm{Ind}_{G_{x_i}}^G \sigma(\mathsf{T}_i,\overline{\chi})$ when $\chi^2\neq {\bf 1}$. When $\chi^2={\bf 1}$ set $\pi^u(T^i,\chi) = \textrm{c-}\mathrm{Ind}_{G_{x_i}}^G \sigma^u(\mathsf{T}_i,\overline{\chi})$ for $u\in \{1,\varepsilon\}$. These representations are supercuspidal and irreducible; the latter four were called the special representations. For $\eta = \left[ \begin{smallmatrix} 1&0\\0&\varpi \end{smallmatrix} \right] \in \mathrm{GL}(2,F)$ we have ${}^\eta \pi^*(T^0,\chi)=\pi^*(T^1,{}^\eta\chi)$, where $*$ indicates that this applies both to the special and nonspecial representations. It follows from [@Nevins2013 Thm 5.3] that for any vertex $x=gx_i$, with $g\in G$, the depth-zero component $\pi^*(T^i,\chi)^{G_{x,0+}}$ is the inflation to $G_x$ of ${}^g\sigma^*(\mathsf{T}_i,\overline{\chi})$, but $\pi^*(T^i,\chi)^{G_{x,0+}}=\{0\}$ if $x$ is not in the $G$-orbit of $x_i$. If $T$ is split, contained in a Borel subgroup $B$, then $\pi(T,\chi) = \mathrm{Ind}_B^G\chi$ is again in the principal series. It is immediate to see that for any vertex $x$, $\pi(T,\chi)^{G_{x,0+}} \cong \sigma(\mathsf{T},\overline{\chi})$ under the isomorphism $G_{x,0}/G_{x,0+}\cong \mathrm{SL}(2,\mathfrak{f})$. In fact, $\pi(T,\chi)$ is irreducible if and only if $\sigma(\mathsf{T},\overline{\chi})$ is; its factors in the remaining cases are as follows. When $\chi \in \{\nu,\nu^{-1}\}$, its Jordan-Hölder factors are the trivial representation and the Steinberg representation $\mathrm{St}$, and we have $\mathrm{St}^{G_{x,0+}} = \overline{\mathrm{St}}$ and ${\bf 1}^{G_{x,0+}} = {\bf 1}$. When $\chi$ is quadratic, it is the sign character $\mathsf{sgn}_\tau$ corresponding to the extension $E=F[\sqrt{\tau}]$. As described in [@Nevins2005 §8], there is in this case a realization of $\pi(T,\chi)$ on the space $L^2(F^\times)$ such that its two irreducible summands $H^\tau_i$, with $i\in \{+,-\}$, are the functions supported $N_i^\tau=\{u\in F^\times\mid i\mathsf{sgn}_\tau(u)=1\}$, respectively. Note that the cases $-1\in (F^\times)^2$ and $-1\notin (F^\times)^2$ thus need to be considered separately while computing the components of depth zero, but in fact the result may be stated uniformly, as follows. **Proposition 24**. *For each $\tau\in \{\varepsilon,-\varpi,-\varepsilon\varpi\}$ and $i\in \{0,1\}$, the $\mathsf{G}_i$-representations $(H_\pm^\tau)^{G_{x_i,0+}}$ are irreducible and their isomorphism classes are given in Table [2](#Table:H+){reference-type="ref" reference="Table:H+"}.* $\pi$ $H^\varepsilon_{+}$ $H^\varepsilon_{-}$ $H^{-\varpi}_{+}$ $H^{-\varpi}_{-}$ $H^{-\varepsilon\varpi}_{+}$ $H^{-\varepsilon\varpi}_{-}$ -------------------- -------------------------- -------------------------- --------------------------------------- ------------------------------------------------- ------------------------------------------------- ------------------------------------------------- $\pi^{G_{x_0,0+}}$ $\overline{\mathrm{St}}$ ${\bf 1}$ $\sigma^{1}(\mathsf{T},\mathsf{sgn})$ $\sigma^{\varepsilon}(\mathsf{T},\mathsf{sgn})$ $\sigma^{1}(\mathsf{T},\mathsf{sgn})$ $\sigma^{\varepsilon}(\mathsf{T},\mathsf{sgn})$ $\pi^{G_{x_1,0+}}$ ${\bf 1}$ $\overline{\mathrm{St}}$ $\sigma^{1}(\mathsf{T},\mathsf{sgn})$ $\sigma^{\varepsilon}(\mathsf{T},\mathsf{sgn})$ $\sigma^{\varepsilon}(\mathsf{T},\mathsf{sgn})$ $\sigma^{1}(\mathsf{T},\mathsf{sgn})$ : The isomorphism classes of the depth-zero representations of $G_{x_i}$ occuring in the restriction to $G_{x_i}$ of the decomposable principal series. *Proof.* Without loss of generality we may assume $x_0,x_1 \in \mathcal{A}(G,T)$. Since $\overline{\mathsf{sgn}_\varepsilon} = {\bf 1}$ and $\overline{\mathsf{sgn}_{-\varpi}}=\overline{\mathsf{sgn}_{-\varepsilon\varpi}}=\mathsf{sgn}$, the unique quadratic character of $\mathsf{T}$, we immediately have the relation $$(H^\tau_{+})^{G_{x,0+}} \oplus (H^\tau_{-})^{G_{x,0+}} \cong %\sigma(\sT,\overline{\chi}) = \sigma^1(\mathsf{T},\overline{\chi})\oplus \sigma^\varepsilon(\mathsf{T},\overline{\chi}),$$ for any vertex $x\in \mathcal{A}(G,T)$. The restriction to $G_{x_0,0+}$ of these components was determined via character computations in [@Nevins2005 Thm 9.1], where (in the notation of that paper, of [@DigneMichel1991 §15], and ours, respectively), $\Xi^+_{\mathsf{sgn}}=\chi^-_{\alpha_0}=[\sigma^{-\varepsilon}(\mathsf{T},\mathsf{sgn})]$ and $\Xi^-_{\mathsf{sgn}}=\chi^+_{\alpha_0}=[\sigma^{-1}(\mathsf{T},\mathsf{sgn})]$. The negative signs used in our parametrizations simplify the statement of the result, yielding the first row of the table. For the restriction to $G_{x_1,0+}$, the proof of [@Nevins2005 Cor 9.3] showed that twisting $\pi(T,\mathsf{sgn}_\tau)$ by $\omega=\left[ \begin{smallmatrix} 0&1\\\varpi&0 \end{smallmatrix} \right] \in \mathrm{GL}(2,F)$ preserves $H_{\pm}^\tau$ when $\mathsf{sgn}_\tau(-\varpi)=1$ and interchanges them otherwise. Applying this twist to $\pi^{G_{x_0,0+}}$ yields $\pi^{G_{x_1,0+}}$ and sends a representation $\sigma$ of $G_{x_0}$ to the representation ${}^\omega \sigma$ of $G_{x_1}$. Twisting by $\omega$ sends the inflation of the representation $\sigma^u(\mathsf{T},\overline{\chi})$ of $G_{x_0}$ to the inflation of the representation $\sigma^{-u}(\mathsf{T},\overline{\chi})$ of $G_{x_1}$, since it takes $\mathcal{O}_u$ to $\mathcal{O}_{-u\varpi}$, which in turn determines the choice of Gel'fand--Graev representation $\gamma_{\mathcal{O}_u}$.[^4] A careful accounting of signs yields the second row of the table. ◻ For convenience, we list the isomorphism class of $\pi^{G_{x,0+}}$ for the remaining irreducible representations $\pi$ in Table [3](#Table:depthzerocomponents){reference-type="ref" reference="Table:depthzerocomponents"}. $T$ $\pi$ $\pi^{G_{x,0+}}$ ------------------ ----------------------- ----------------------------------------- -------------------- $T$ split $\pi=\pi(T,\chi)$ $\sigma(\mathsf{T}, \overline{\chi})$ $\pi=\mathrm{St}$ ${}^g\overline{\mathrm{St}}$ $T^i$ unramified $\pi = \pi(T^i,\chi)$ $\sigma(\mathsf{T}_i, \overline{\chi})$ if $x \sim x_i$ $i\in \{0,1\}$ $\{0\}$ if $x\not\sim x_i$ : The depth-zero representations of $G_{x}$ occuring in the restriction to $G_x$ of the irreducible principal series, Steinberg and supercuspidal representations, for any vertex $x\in \mathcal{B}(G)$. ## Wave front sets The wave front set is determined with the following result that is based on [@BarbaschMoy1997 Thm 4.5]. **Proposition 25**. *Let $\pi$ be an irreducible representation of depth zero of $\mathrm{SL}(2,F)$. Suppose $\mathrm{char}(F)=0$ and $p>3e+1$, where $e$ is the absolute ramification index of $F$ over $\mathbb{Q}_p$. Then we have $$\mathcal{WF}(\pi) = \{ \mathcal{O}\in \mathscr{O}(0)\mid \exists x \text{ a vertex of }\mathcal{B}(G), \sigma \in \pi^{G_{x,0+}} \text{such that $\sigma$ occurs in $\gamma_\mathcal{O}$}\},$$ where $\gamma_\mathcal{O}$ is the Gel'fand--Graev representation [\[E:GGrep\]](#E:GGrep){reference-type="eqref" reference="E:GGrep"} of $\mathsf{G}_x \cong \mathrm{SL}(2,\mathfrak{f})$.* *Proof.* The hypotheses imply that $\exp$ converges on $\mathfrak{g}_{0+}$ and that the local character expansion holds. In [@BarbaschMoy1997], Barbasch and Moy used (generalized) Gel'fand--Graev characters as test functions to determine the wave front set of $\pi$. For each nilpotent orbit $\mathcal{O}$ that is represented by a depth-zero coset at some $x$, let $[\gamma_\mathcal{O}]$ denote the lift to $G_{x,0}$ of the character of the corresponding Gel'fand--Graev representation of $\mathsf{G}_x = G_{x,0}/G_{x,0+}$, viewed as a function on $G$. It is supported on the subset $G_{0+}\cap G_{x,0}$ of topologically unipotent elements. Let $f_{x,\mathcal{O}}$ be the function on $\mathfrak{g}$, with support in $\mathfrak{g}_{0+}\cap\mathfrak{g}_{x,0}$, that is given by $f_{x,\mathcal{O}} = [\gamma_\mathcal{O}]\circ \exp$. Then they show that $\widehat{\mu_{\mathcal{O}'}}(f_{x,\mathcal{O}})=0$ if $\mathcal{O}\not\subset \overline{\mathcal{O}'}$ and is nonzero if $\mathcal{O}=\mathcal{O}'$. Thus $\Theta_\pi([\gamma_\mathcal{O}]) =0$ for all $\mathcal{O}$ that do not meet the wave front set of $\pi$ and is nonzero when $\mathcal{O}\in \mathcal{WF}(\pi)$. For any irreducible representation $\sigma$ of $\mathsf{G}_x$, let $m(\sigma,\pi)$ denote the multiplicity of (the inflation of) $\sigma$ in $\pi^{G_{x,0+}}$ and $m(\sigma,\gamma_\mathcal{O})$ the multiplicity of $\sigma$ in $\gamma_\mathcal{O}$. Then [@BarbaschMoy1997 Thm 4.5(4)] becomes $$\Theta_\pi([\gamma_{\mathcal{O}}]) = \sum_{\sigma}m(\sigma,\pi)m(\sigma,\gamma_\mathcal{O}),$$ whence our result for the case of $\mathrm{SL}(2,F)$. ◻ **Corollary 26**. *Under the hypothesis of Proposition [Proposition 25](#P:BM){reference-type="ref" reference="P:BM"}, the wave front sets corresponding to the depth-zero representations of $\mathrm{SL}(2,F)$ are as given in Table [4](#Table:WF){reference-type="ref" reference="Table:WF"}.* Representation $\pi$ $\mathcal{WF}(\pi)$ -------------------------------------------- --------------------------- ------------------------------------- ---------------------------------------------- $\pi(T,\chi)$ irreducible principal series $\{1,\varepsilon,\varpi,\varepsilon\varpi\}$ $\pi(T_0,\chi)$ irreducible supercuspidal $\{1,\varepsilon\}$ $\pi(T_1,\chi)$ irreducible supercuspidal $\{\varpi,\varepsilon\varpi\}$ $\pi$ $\mathcal{WF}(\pi)$ $\pi$ $\mathcal{WF}(\pi)$ ${\bf 1}$ $\{0\}$ $\mathrm{St}$ $\{1,\varepsilon,\varpi,\varepsilon\varpi\}$ $H^\varepsilon_+$ $\{1,\varepsilon\}$ $H^\varepsilon_{-}$ $\{\varpi,\varepsilon\varpi\}$ $H^{-\varpi}_+$ $\{1,\varpi\}$ $H^{-\varpi}_{-}$ $\{\varepsilon,\varepsilon\varpi\}$ $H^{-\varepsilon\varpi}_+$ $\{1,\varepsilon\varpi\}$ $H^{-\varepsilon\varpi}_{-}$ $\{\varepsilon,\varpi\}$ $\pi^1(T_0,\mathsf{sgn})$ $\{1\}$ $\pi^\varepsilon(T_0,\mathsf{sgn})$ $\{\varepsilon\}$ $\pi^1(T_1,\mathsf{sgn})$ $\{\varpi\}$ $\pi^\varepsilon(T_1,\mathsf{sgn})$ $\{\varepsilon\varpi\}$ : For each irreducible depth-zero representation of $\mathrm{SL}(2,F)$, we list under the heading $\mathcal{WF}(\pi)$ the set of elements $u \in \{0,1,\varepsilon,\varpi,\varepsilon\varpi\}$ such that $\mathcal{O}_u\in \mathcal{WF}(\pi)$. *Proof.* For $u\in \{1,\varepsilon\}$ and $j \in \{0,1\}$, the nilpotent orbit $\mathcal{O}_{u\varpi^j}$ is represented by a depth-zero coset at $x_i$ if and only if $i=j$, and in this case it corresponds to the nilpotent orbit in the quotient $\mathfrak{g}_{x_i,0}/\mathfrak{g}_{x_i,0+}\cong \mathfrak{sl}(2,\mathfrak{f})$ under $\mathsf{G}_i\cong \mathrm{SL}(2,\mathfrak{f})$ that we denoted $\mathcal{O}_u$. Therefore the Gel'fand--Graev representations $\gamma_\mathcal{O}$ referred to in Proposition [Proposition 25](#P:BM){reference-type="ref" reference="P:BM"} are $\gamma_{\mathcal{O}_1}$ and $\gamma_{\mathcal{O}_\varepsilon}$ for $x=x_0$, and $\gamma_{\mathcal{O}_{\varpi}}$ and $\gamma_{\mathcal{O}_{\varepsilon\varpi}}$ for $x=x_1$. By conjugacy, these two vertices suffice. The decomposition of $\pi^{G_{x,0+}}$ for $x\in \{x_0,x_1\}$ was given in Tables [2](#Table:H+){reference-type="ref" reference="Table:H+"} and [3](#Table:depthzerocomponents){reference-type="ref" reference="Table:depthzerocomponents"} for all irreducible depth-zero representations $\pi$, and matching these with the decomposition of the Gel'fand--Graev representations of the corresponding groups $\mathsf{G}_i$ yields Table [4](#Table:WF){reference-type="ref" reference="Table:WF"}. ◻ In light of Proposition [Proposition 25](#P:BM){reference-type="ref" reference="P:BM"}, we may define $\mathcal{WF}(\pi)$ by Table [4](#Table:WF){reference-type="ref" reference="Table:WF"}, even over fields where Proposition [Proposition 25](#P:BM){reference-type="ref" reference="P:BM"} does not apply. Theorem [Theorem 28](#T:zerodepth2){reference-type="ref" reference="T:zerodepth2"} below expresses that, just as in the positive-depth case, this is consistent (for all fields with residual characteristic different from $2$). **Corollary 27**. *For each depth-zero irreducible representation $\pi$ of $\mathrm{SL}(2,F)$ there exists an element $\Gamma\in \mathfrak{g}_{x,0}^*$, for some $x\in \mathcal{B}(G)$, such that $\mathcal{WF}(\pi)=\mathrm{Nil}(\Gamma)$.* Existence follows immediately from Table [4](#Table:WF){reference-type="ref" reference="Table:WF"} and Lemma [Lemma 10](#L:NilGamma){reference-type="ref" reference="L:NilGamma"}, though the elements $\Gamma$ for which $\mathcal{WF}(\pi)=\mathrm{Nil}(\Gamma)$ do not correspond to minimal $K$-types for $\pi$ (as these latter are not realized by elements on the Lie algebra). However, on an *ad-hoc* basis, we can make this association of $\pi$ with $\Gamma$ more explicit, as follows. For $T$ unramified or split, and $x\in \mathcal{B}(T)\subset \mathcal{B}(G)$, we can in the same spirit attach to any *regular* $\pi(T,\chi)$ (in the sense of Kaletha [@Kaletha2019 Prop 3.4.27]) any element $\Gamma\in \mathfrak{g}_{x,0}^*$ whose centralizer in $G$ is $T$. The same holds for $\pi=\mathrm{St}$, whereas we associate $\Gamma=0$ to ${\bf 1}$. When $\pi^u(T^i,\mathsf{sgn})$ is a special representation (for some $u\in \{1,\varepsilon\}$ and $i\in \{0,1\}$) then it is a supercuspidal unipotent representation and $\Gamma$ is chosen to be a nilpotent element in the lift to $\mathfrak{g}_{x_i,0}^*$ of the nilpotent orbit corresponding to $\sigma^u(\mathsf{T}_i,\mathsf{sgn})$. When $\pi \in \{H^\tau_{\pm} \mid \tau \in \{\varepsilon,\varpi,\varepsilon\varpi\}\}$, $\Gamma$ is a choice of element of an anisotropic torus $T$ that splits over $F[\sqrt{\tau}]$. However, while the orbit of $\Gamma$ satisfies $\mathrm{Nil}(\Gamma)=\mathcal{WF}(\pi)$, its centralizer in this case need not correspond to $\pi$: when $-1\in (F^\times)^2$ and $\tau \in \{\varpi,\varepsilon\varpi\}$, the centralizer may be one of two possible tori $T = \mathrm{Cent}_G(\Gamma)$ up to conjugacy, and neither one is expressly associated to $\pi$. We conclude this section with our main result. **Theorem 28**. *Let $\pi$ be an irreducible representation of $G$ of depth zero with central character $\zeta$. For any vertex $x\in \mathcal{B}(G)$, we have $$\label{E:desired} \mathrm{Res}_{G_{x}}\pi \cong \pi^{G_{x,0+}} \oplus \bigoplus_{\mathcal{O}\in \mathcal{WF}(\pi)}\tau_x(\mathcal{O},\zeta),$$ where $\mathcal{WF}(\pi)$ is as in Table [4](#Table:WF){reference-type="ref" reference="Table:WF"}. It follows that $\mathrm{Res}_{G_{x,0+}}\pi$ takes the form of [\[E:posdepththm\]](#E:posdepththm){reference-type="eqref" reference="E:posdepththm"} with constant coefficient $n_x(\pi)=\dim(\pi^{G_{x,0+}})$.* *Proof.* The decomposition will follow from the main results of [@Nevins2005; @Nevins2013] as in the proof of Theorem [Theorem 18](#T:posdepth){reference-type="ref" reference="T:posdepth"}. Let $\pi$ be a depth-zero representation of $G$ with central character $\zeta$, and let $x\in \{x_0,x_1\}$. For irreducible depth-zero principal series, one has $\gamma_0=\gamma_1=0$ in [@Nevins2005 Thm 7.4]. Matching notation as is [\[NotationNevins2005\]](#NotationNevins2005){reference-type="eqref" reference="NotationNevins2005"}, we conclude that $\mathcal{S}_{x}(X_{u\varpi^{-d}},\zeta)$ occurs in $\mathrm{Res}_{G_{x}}\pi$, for $u\in \{1,\varepsilon\}$, for each $d>0$, and that these exhaust the irreducible summands. Therefore the summands can be regrouped as the sum of $\tau_{x}(\mathcal{O},\zeta)$, as defined in [\[E:deftau\]](#E:deftau){reference-type="eqref" reference="E:deftau"}, over all regular nilpotent orbits, as required. As the positive-depth summands of $\mathrm{Res}_{G_x}\pi$ are identical for all depth-zero irreducible principal series, the case of $\pi=\mathrm{St}$ follows since $\mathrm{Res}_{G_x}{\bf 1}$ has no positive-depth components. The results of [@Nevins2005 Thm 9.1, Thm 9.2] yield $$\mathrm{Res}_{G_{x_0}}H^\tau_+ = \begin{cases} \mathrm{St}\oplus \bigoplus_{d>0}(\mathcal{S}_{x_0}(\varpi^{-2d}X_1, \zeta) \oplus \mathcal{S}_{x_0}(\varpi^{-2d}X_\varepsilon,\zeta)) & \text{if $\tau=\varepsilon$};\\ \sigma^1(\mathsf{T},\mathsf{sgn}) \oplus \bigoplus_{d>0}\mathcal{S}_{x_0}(\varpi^{-d}X_1, \zeta) & \text{if $\tau=-\varpi$};\\ \sigma^1(\mathsf{T},\mathsf{sgn}) \oplus \bigoplus_{d>0}(\mathcal{S}_{x_0}(\varpi^{-2d}X_1, \zeta) \oplus \mathcal{S}_{x_0}(\varpi^{-2d+1}X_\varepsilon,\zeta))& \text{if $\tau=-\varepsilon\varpi$.} \end{cases}$$ Regrouping the positive-depth summands yields the decomposition $$\mathrm{Res}_{G_{x_0}}H^\tau_+ = \begin{cases} \mathrm{St}\oplus \tau_{x_0}(\mathcal{O}_1,\zeta) \oplus \tau_{x_0}(\mathcal{O}_\varepsilon, \zeta)& \text{if $\tau=\varepsilon$};\\ \sigma^1(\mathsf{T},\mathsf{sgn}) \oplus \tau_{x_0}(\mathcal{O}_1,\zeta) \oplus \tau_{x_0}(\mathcal{O}_{\varpi},\zeta) & \text{if $\tau=-\varpi$};\\ \sigma^1(\mathsf{T},\mathsf{sgn}) \oplus \tau_{x_0}(\mathcal{O}_1,\zeta) \oplus \tau_{x_0}(\mathcal{O}_{\varepsilon\varpi},\zeta) & \text{if $\tau=-\varepsilon\varpi$,} \end{cases}$$ which coincides with the wave front set computed in Table [2](#Table:H+){reference-type="ref" reference="Table:H+"}. Since the positive-depth summands of $H^\tau_+\oplus H^\tau_-$ form $\bigoplus_{\mathcal{O}\in \mathscr{O}(0)\setminus \{0\}}\tau_{x_0}(\mathcal{O},\zeta)$, and the wave front sets of these representations are complementary, this yields the result for $\mathrm{Res}_{G_{x_0}}H^\tau_-$ as well. To determine $\mathrm{Res}_{G_{x_1}}\pi$ we proceed as in the proof of Proposition [Proposition 24](#P:pigx0+){reference-type="ref" reference="P:pigx0+"}. Conjugation by $\omega$ interchanges the components of the principal series *except* when: $\tau=-\varpi$ and $-1\in (F^\times)^2$; or $\tau=-\varepsilon\varpi$ and $-1\notin (F^\times)^2$. Since the depth-zero components were computed in Proposition [Proposition 24](#P:pigx0+){reference-type="ref" reference="P:pigx0+"} and ${}^\omega\tau_{x_0}(\mathcal{O}_u,\zeta) =\tau_{x_1}(\mathcal{O}_{-u\varpi},\zeta)$, we deduce that $$\mathrm{Res}_{G_{x_1}}H^\tau_- = \begin{cases} \mathrm{St}\oplus \tau_{x_1}(\mathcal{O}_\varpi,\zeta) \oplus \tau_{x_1}(\mathcal{O}_{\varepsilon\varpi}, \zeta)& \text{if $\tau=\varepsilon$};\\ \sigma^{-1}(\mathsf{T},\mathsf{sgn}) \oplus \tau_{x_1}(\mathcal{O}_{-\varpi},\zeta) \oplus \tau_{x_1}(\mathcal{O}_{-\varpi^2},\zeta) & \text{if $\tau=-\varpi$ and $-1\notin (F^\times)^2$};\\ \sigma^{-1}(\mathsf{T},\mathsf{sgn}) \oplus \tau_{x_1}(\mathcal{O}_{-\varpi},\zeta) \oplus \tau_{x_1}(\mathcal{O}_{-\varepsilon\varpi^2},\zeta) & \text{if $\tau=-\varepsilon\varpi$ and $-1\in (F^\times)^2$;}\\ \sigma^{-\varepsilon}(\mathsf{T},\mathsf{sgn}) \oplus \tau_{x_1}(\mathcal{O}_{-\varepsilon\varpi},\zeta) \oplus \tau_{x_1}(\mathcal{O}_{-\varepsilon\varpi^2},\zeta) & \text{if $\tau=-\varpi$ and $-1\in (F^\times)^2$};\\ \sigma^{-\varepsilon}(\mathsf{T},\mathsf{sgn}) \oplus \tau_{x_1}(\mathcal{O}_{-\varepsilon\varpi},\zeta) \oplus \tau_{x_1}(\mathcal{O}_{-\varpi^2},\zeta) & \text{if $\tau=-\varepsilon\varpi$ and $-1\notin (F^\times)^2$;} \end{cases}$$ Thus, in any case, the nilpotent orbits arising in $\mathrm{Res}_{G_{x_1}}H^\varepsilon_-$ are $\{\mathcal{O}_\varpi, \mathcal{O}_{\varepsilon\varpi}\}$; those arising in $\mathrm{Res}_{G_{x_1}}H^{-\varpi}_-$ are $\{\mathcal{O}_\varepsilon, \mathcal{O}_{\varepsilon\varpi}\}$; and those arising in $\mathrm{Res}_{G_{x_1}}H^{-\varepsilon\varpi}_-$ are $\{\mathcal{O}_{\varpi},\mathcal{O}_{\varepsilon}\}$, which again is consistent with Table [4](#Table:WF){reference-type="ref" reference="Table:WF"}, as required. Now suppose that $\pi_i = \textrm{c-}\mathrm{Ind}_{G_i}^G\sigma$ is a supercuspidal representation. We use [@Nevins2013 Cor 5.2, Thm 5.3], where $\eta=\left[ \begin{smallmatrix} 1&0\\0&\varpi \end{smallmatrix} \right]$, $\sigma_0^+$ corresponds to our $\sigma^{-1}(T^0,\chi)$, and $\sigma_0^-$ is our $\sigma^{-\varepsilon}(T^0,\chi)$. Since ${}^\eta\mathcal{O}_u=\mathcal{O}_{u\varpi}$, the $G_{x_1}$-representation ${}^\eta(\sigma_0^+)$ is the inflation of $\sigma^{-1}(T^1,\chi)$ to $G_{x_1}$. We thus infer the decompositions $$\mathrm{Res}_{G_{x_0}}\pi = \begin{cases} \sigma \oplus \bigoplus_{t>0}\left(\mathcal{S}_{x_0}(-\varpi^{-2t}X_1,\zeta) \oplus \mathcal{S}_{x_0}(-\varpi^{-2t}X_\varepsilon,\zeta)\right) & \text{if $\pi$ nonspecial and $i=0$;}\\ \bigoplus_{t>0}\left(\mathcal{S}_{x_0}(-\varpi^{-2t+1}X_1,\zeta) \oplus \mathcal{S}_{x_0}(-\varpi^{-2t+1}X_\varepsilon,\zeta)\right) & \text{if $\pi$ nonspecial and $i=1$;}\\ \sigma \oplus \bigoplus_{t>0} \mathcal{S}_{x_0}(-\varpi^{-2t}X_u, \zeta) & \text{if $\pi = \pi^{-u}(T^0,\chi)$ is special and $i=0$;}\\ \bigoplus_{t>0} \mathcal{S}_{x_0}(-\varpi^{-2t+1}X_u, \zeta) & \text{if $\pi = \pi^{-u}(T^1,\chi)$ is special and $i=1$.} \end{cases}$$ Comparing with Table [4](#Table:WF){reference-type="ref" reference="Table:WF"}, we conclude that [\[E:desired\]](#E:desired){reference-type="eqref" reference="E:desired"} holds for $\mathrm{Res}_{G_{x_0}}\pi_i$ in each case. The result for general $x$ now follows as in the proof of Theorem [Theorem 18](#T:posdepth){reference-type="ref" reference="T:posdepth"}. Finally, the values $n_x(\pi)=\dim(\pi^{G_{x,0+}})$ can be deduced from Tables [2](#Table:H+){reference-type="ref" reference="Table:H+"} and [3](#Table:depthzerocomponents){reference-type="ref" reference="Table:depthzerocomponents"}: it is $q+1$ for irreducible principal series, $q-1$ for Deligne-Lusztig cuspidal representations, $q$ for $\overline{\mathrm{St}}$, $(q-1)/2$ for the special unipotent representations and $(q+1)/2$ for the components of the reducible principal series. ◻ # Applications {#S:applications} ## The Fourier transform of a nilpotent orbital integral {#S:FT} As a first application, we derive a formula for the Fourier transform of a nilpotent orbital integral in any open set of the form $\mathfrak{g}_{x,0+}$ in terms of the trace characters of the representations $\tau_x(\mathcal{O},\zeta)$. **Proposition 29**. *Let $x\in \mathcal{B}(G)$ be a vertex. Let $[\tau_x(\mathcal{O})]$ denote the restriction to $G^{\mathrm{reg}}_{x,0+}$ of the trace character of the representation $\tau_x(\mathcal{O},\zeta)$, for either choice of central character $\zeta$. Assume $\exp$ converges on $\mathfrak{g}_{x,0+}$. Then for each nonzero nilpotent orbit $\mathcal{O}$ and $X\in \mathfrak{g}^{\mathrm{reg}}_{x,0+}$ we have $$\widehat{\mu_\mathcal{O}}(X) = \begin{cases} %1 & \text{if $\cO=\{0\}$;}\\ q/2 + [\tau_x(\mathcal{O})](\exp X) &\text{if $\mathcal{O}$ has even parity depth at $x$;}\\%$\cO \cap \g_{x,0}\setminus \g_{x,0+} \neq \emptyset$}\\ 1/2 + [\tau_x(\mathcal{O})](\exp X) & \text{if $\mathcal{O}$ has odd parity depth at $x$}. \end{cases}$$ As $x$ ranges over the vertices of $\mathcal{B}(G)$, these expressions determine the function $\widehat{\mu_\mathcal{O}}$ on $\mathfrak{g}^{\mathrm{reg}}_{1/2+}$.* *Proof.* Let $\Theta_\pi$ denote the character of the depth-$r$ representation $\pi$. We assume the functions $\widehat{\mu}_{\mathcal{O}}$ are normalized as in [@MoeglinWaldspurger1987], so that the coefficients $c_\mathcal{O}$ corresponding to $\mathcal{O}\in \mathcal{WF}(\pi)$ in the local character expansion of $\Theta_\pi\circ \exp$ are all equal to $1$. Thus on $\mathfrak{g}^{\mathrm{reg}}_{x,r+}$ we have $$\Theta_\pi\circ \exp = c_0(\pi) + \sum_{\mathcal{O}\in \mathcal{WF}(\pi)}\hat{\mu}_\mathcal{O}.$$ The constant coefficients for supercuspidal representations are given in Table [5](#Table:c0){reference-type="ref" reference="Table:c0"}, following, for example, [@DeBackerSally2000 Tables 1--4]. For (irreducible components of) principal series, the constant term of the local character expansion is trivial, except in the case of the trivial and Steinberg representations, which have constant terms $1$ and $-1$, respectively. ------------------------------------------------------------- ------------------------------------ Representation of $\mathrm{SL}(2,F)$ coefficient $c_0$ of $\mu_{\{0\}}$ of depth $r\geq 0$ in local character expansion $\pi(T,\chi)$, $T$ unramified $-q^r$ $\pi^u(T^i,\chi)$, $i\in \{0,1\}$, $u\in \{1,\varepsilon\}$ $-1/2$ $\pi(T,\chi)$, $T$ ramified $q^{r-1/2}(q+1)/2$ $\mathrm{St}$, Steinberg representation $-1$ ------------------------------------------------------------- ------------------------------------ : Values of the constant term in the local character expansion of supercuspidal and Steinberg representations of $\mathrm{SL}(2,F)$. Theorem [Theorem 28](#T:zerodepth2){reference-type="ref" reference="T:zerodepth2"}, on the other hand, gives a formula for the character of any irreducible depth-zero representation on $G_{x,0+}$. Matching these for the special unipotent representations $\pi = \pi^u(T,\chi)$ yields the given formula. It is moreover direct to verify the consistency of this expression across the local character expansions of all irreducible representations, including those of positive depth (on $G_{x,r+}$ as in Theorem [Theorem 21](#T:posdepth2){reference-type="ref" reference="T:posdepth2"}). Finally, we note that for $G=\mathrm{SL}(2,F)$ we have $\mathfrak{g}_{1/2+} = G\dot (\mathfrak{g}_{x_0,0+} \cup \mathfrak{g}_{x_1,0+}) \subsetneq \mathfrak{g}_{0+}$, which limits the $G$-domain on which the formulas hold. ◻ Much more explicit formulae for the functions $\widehat{\mu}_\mathcal{O}$ have been computed for the group $\mathrm{SL}(2,F)$ in [@Assem1994; @DeBackerSally2000] among others. They have also noted that, under the exponential map, the characters of the five representations $\{1, \pi^u(T^i,\chi)\mid u\in \{1,\varepsilon\}, i\in \{0,1\}\}$ form another basis for the span of the functions $\widehat{\mu}_\mathcal{O}$. In fact the special representations have local character expansions of the form $$\label{E:specialchar} \Theta_\pi(\exp(X))=\widehat{\mu_\mathcal{O}}(X) - 1/2,$$ for the single corresponding orbit $\mathcal{O}$, and this holds on the strictly larger set $\mathfrak{g}^{\mathrm{reg}}_{0+}$. An advantage to Proposition [Proposition 29](#C:orbint){reference-type="ref" reference="C:orbint"} is the simplicity and explicitness of the construction, which uses no more than a vertex and a representative of the orbit as input. In this, it recalls some of the original formulae for these Fourier transforms of nilpotent orbital integrals in [@HarishChandra1999]. ## Computing the polynomial $\dim(\pi^{G_{x,2n}})$ This answers a question posed to me by Marie-France Vignéras. If $\pi$ is an irreducible representation of $G$, the local character expansion implies that $\dim(\pi^{G_{x,2n}})$ is expressible as a polynomial in $q$, as described in [@BarbaschMoy1997 §5.1]. Here we can obtain this polynomial as a corollary of Theorem [Theorem 21](#T:posdepth2){reference-type="ref" reference="T:posdepth2"} and [Theorem 28](#T:zerodepth2){reference-type="ref" reference="T:zerodepth2"}, using the explicit values computed in Proposition [Proposition 23](#P:npi){reference-type="ref" reference="P:npi"}. **Corollary 30**. *Let $\pi$ be an irreducible representation of $G=\mathrm{SL}(2,F)$ of depth $r$. Then for each integer $n>0$, we have $$\dim(\pi^{G_{x,2n}}) = \begin{cases} q^{2n}+q^{2n-1} & \text{if $\pi$ is an irreducible principal series},\\ q^{2n-1}-q^r & \text{if $\pi$ is supercuspidal nonspecial, from a vertex $\sim x$},\\ q^{2n}-q^r & \text{if $\pi$ is supercuspidal nonspecial, from a vertex $\not\sim x$},\\ \frac12(q+1)(q^{2n-1}-q^{r-\frac12}) & \text{if $\pi$ is supercuspidal, from a nonvertex}. \end{cases}$$ On the other hand, if $\pi_s = H^\varepsilon_{s}$ then $\dim(\pi^{G_{x,2n}}) =q^{2n-1}$ when the parity depth at $x$ of the orbits in $\mathcal{WF}(\pi_s)$ is even, and equals $q^{2n}$ otherwise; and if $\pi=\mathrm{St}$, then $\dim(\pi^{G_{x,2n}})=q^{2n}+q^{2n-1}-1$. In all other cases, $\dim(\pi^{G_{x,2n}})$ is exactly half of that of a corresponding nonspecial representation.* CMBO22 Jeffrey D. Adler and Stephen DeBacker, *Some applications of Bruhat-Tits theory to harmonic analysis on the Lie algebra of a reductive $p$-adic group*, Michigan Math. 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[^1]: Supported by NSERC Discovery grant RGPIN-2020-05020. [^2]: private correspondence and forthcoming work [^3]: Errata to [@Nevins2013 Thm 6.2]: the decomposition in case $y=1$ is missing the term corresponding to the double coset representative $\mathsf{e}^{\eta}$. [^4]: This calculation was neglected in the proof of [@Nevins2005 Cor 9.3], yielding an incorrect statement for the depth-zero components.

arxiv_math

--- abstract: | Let $M$ be an open (complete and non-compact) manifold with $\mathrm{Ric}\ge 0$ and escape rate not $1/2$. It is known that under these conditions, the fundamental group $\pi_1(M)$ has a finitely generated torsion-free nilpotent subgroup $\mathcal{N}$ of finite index, as long as $\pi_1(M)$ is an infinite group. We show that the nilpotency step of $\mathcal{N}$ must be reflected in the asymptotic geometry of the universal cover $\widetilde{M}$, in terms of the Hausdorff dimension of an isometric $\mathbb{R}$-orbit: there exist an asymptotic cone $(Y,y)$ of $\widetilde{M}$ and a closed $\mathbb{R}$-subgroup $L$ of the isometry group of $Y$ such that its orbit $Ly$ has Hausdorff dimension at least the nilpotency step of $\mathcal{N}$. This resolves a question raised by Wei and the author (see [@PW_ex Remark 1.7] and [@Pan_cone Conjecture 0.2]) and extends previous results on virtual abelianness by the author [@Pan_es0; @Pan_esgap; @Pan_cone]. address: Department of Mathematics, University of California, Santa Cruz, California, US. author: - Jiayin Pan title: | Nonnegative Ricci curvature, nilpotency,\ and Hausdorff dimension --- plus 1pt # Introduction Collapsed Ricci limit spaces in general may admit isometric orbits whose Hausdorff dimension exceeds their topological dimension. The first examples with this feature are constructed by Wei and the author [@PW_ex] as the asymptotic cone of the universal cover of an open (complete and non-compact) manifold $M$ with $\mathrm{Ric}\ge 0$ and $\pi_1(M)=\mathbb{Z}$. More precisely, it is the equivariant Gromov-Hausdorff limit of $$(r_i^{-1}\widetilde{M},\tilde{p},\pi_1(M,p))\overset{GH}\longrightarrow (Y,y,L),$$ where $r_i\to\infty$ and $(\widetilde{M},\tilde{p})$ is the universal cover of $(M,p)$. In the limit space, $L$ is a closed $\mathbb{R}$-subgroup of $\mathrm{Isom}(Y)$ and the orbit $Ly$ has Hausdorff dimension $1+\alpha$, where $\alpha\ge 0$ can be any large number by choosing a suitable metric and dimension of $M$. In the same paper, we asked whether the (non-abelian) nilpotency of $\pi_1(M)$ implies the existence of some asymptotic $\mathbb{R}$-orbit of large Hausdorff dimension [@PW_ex Remark 1.7]. This question was later formalized by the author in [@Pan_cone Conjecture 0.2], relating the nilpotency step to Hausdorff dimension. More precisely, for any open manifold $M$ with $\mathrm{Ric}\ge 0$ and escape rate not $1/2$, given that $\pi_1(M)$ contains a torsion-free nilpotent subgroup of nilpotency step $l$, is it true that $\dim_{\mathcal{H}}(Ly)\ge l$ for some asymptotic cone $(Y,y)$ of $\widetilde{M}$ and some closed $\mathbb{R}$-subgroup $L$ of $\mathrm{Isom}(Y)$? Before proceeding further, we give more background on this problem. Let $M$ be an open manifold with $\mathrm{Ric}\ge 0$. By the work of Kapovitch-Wilking [@KW], $\pi_1(M)$ contains a nilpotent subgroup of index at most $C(n)$, a constant only depending on $n$. Also see [@Mil; @Gro_poly]. In the other direction, by the work of Wei [@Wei] and Wilking [@Wilk], it is known that any finitely generated virtually nilpotent can be realized as the fundamental group of some open manifold with $\mathrm{Ric}\ge 0$. This is distinct from open manifolds with nonnegative sectional curvature, whose fundamental groups are always virtually abelian [@CG_soul]. Therefore, it is natural to investigate on what additional conditions $\pi_1(M)$ is virtually abelian for nonnegative Ricci curvature; or equivalently, we can ask how virtual abelianness or nilpotency of $\pi_1(M)$ is related to the geometry of $M$. The author has studied this question in [@Pan_es0; @Pan_esgap; @Pan_cone]. [@Pan_es0] introduced a geometric quantity, the escape rate $E(M,p)$, which measures how fast representing geodesic loops escape from bounded sets. The escape rate takes value within $[0,1/2]$. It is known that $E(M,p)<1/2$ implies the finite generation of $\pi_1(M)$ by Sormani's halfway lemma [@Sor]. In [@Pan_esgap], we proved that if the escape rate of $M^n$ is smaller than some universal constant $\epsilon(n)$, then $\pi_1(M)$ is virtually abelian. In [@Pan_cone], we proved that if the escape rate of $M$ is not $1/2$ and the universal cover is (metric) conic at infinity, then $\pi_1(M)$ is virtually abelian. The proofs in both [@Pan_esgap] and [@Pan_cone] relate the equivariant asymptotic geometry to the structure of $\pi_1(M)$. Given the above results, it is naturally to further study the equivariant asymptotic geometry without the smallness of escape rate and without the (metric) conic structure at infinity. Such understanding should have implications on virtual abelianness or nilpotency of $\pi_1(M)$. A close look at Wei's examples of open manifolds with $\mathrm{Ric}\ge 0$ and torsion-free nilpotent fundamental groups [@Wei] indicates that the nilpotency step is reflected in a lower bound of the Hausdorff dimension of isometric $\mathbb{R}$-orbits. See Section [3](#sec_exmp){reference-type="ref" reference="sec_exmp"} for these motivating examples. This problem of nilpotency step and Hausdorff dimension is also related to the structure of Carnot groups. Let $\Gamma$ be a finitely generated virtually nilpotent group with nilpotency step $l$. Any finite generating set $S$ of $\Gamma$ defines a word length metric $d_S$ on $\Gamma$. The asymptotic structure of $(\Gamma,d_S)$ was studied by Gromov [@Gro_poly] and Pansu [@Pansu]. For any sequence $r_i\to\infty$, Gromov-Hausdorff convergence holds: $$(r_i^{-1}\Gamma,e,d_S)\overset{GH}\longrightarrow (G,e,d).$$ The unique limit space $(G,d)$ is a Carnot group, that is, a simply connected stratified nilpotent Lie group $G$ with nilpotency step $l$ and a distance $d$ induced by a left-invariant subFinsler metric. Moreover, $\dim_{\mathcal{H}}(Ly)=l$ for any one-parameter subgroup $L$ in $\zeta_{l-1}(G)$, the last nontrivial subgroup in the lower central series (see Definition [Definition 6](#def_nilstep){reference-type="ref" reference="def_nilstep"}). This structure also applies to closed manifolds. For a closed Riemannian manifold $(M,g)$ with a virtually nilpotent fundamental group $\Gamma$, although $g$ cannot have nonnegative Ricci curvature when $\mathrm{step}(\Gamma)\ge 2$, the blow-down sequence of the universal cover $(r_i^{-1}\widetilde{M},\tilde{p},\tilde{g})$ actually converges in the Gromov-Hausdorff topology to a limit space $(G,e,d)$ as described above. Therefore, we can view the proposed problem as an extension of part of the Carnot group structure from closed manifolds to open ones. The main result of this paper confirms this conjecture about nilpotency step and Hausdorff dimension [@Pan_cone Conjecture 0.2], thus resolves the question raised in [@PW_ex Remark 1.7]. **Theorem A 1**. *Let $(M,p)$ be an open $n$-manifold with $\mathrm{Ric}\ge 0$ and $E(M,p)\not=1/2$. Let $\mathcal{N}$ be a torsion-free nilpotent subgroup of $\pi_1(M,p)$ with nilpotency step $l$ and finite index. Then there exists an asymptotic cone $(Y,y)$ of $\widetilde{M}$, the universal cover of $M$, and a closed $\mathbb{R}$-subgroup $L$ of $\mathrm{Isom}(Y)$ such that $\dim_{\mathcal{H}}(Ly)\ge l$.* The condition $E(M,p)\not= 1/2$ guarantees that $\pi_1(M)$ is finitely generated. Thus we can always choose a torsion-free nilpotent subgroup $\mathcal{N}$ of $\pi_1(M)$ with finite index as long as $\pi_1(M)$ is infinite. Moreover, its nilpotency step $l$ is independent of the choice of $\mathcal{N}$. Hence we can naturally define the nilpotency step of $\pi_1(M)$ as $\mathrm{step}(\pi_1(M))=l$. The inequality in Theorem A is sharp. In fact, we have examples of open manifolds such that the equality holds on every asymptotic cone of $\widetilde{M}$ (see Section [3.2](#subsec_exmp_min){reference-type="ref" reference="subsec_exmp_min"}). The $\mathbb{R}$-subgroup $L$ in Theorem A indeed comes from $\langle \gamma \rangle$-action on $\widetilde{M}$, where $\gamma$ belongs to $\zeta_{l-1}(N)$. In terms of the asymptotic limit of $\mathcal{N}$-action, we can show that its asymptotic orbit $Gy$ always has a connected and simply connected nilpotent Lie group structure (Proposition [Proposition 43](#topol_dim){reference-type="ref" reference="topol_dim"}). We emphasize that this nilpotent structure on $Gy$ could be abelian when $\mathrm{step}(\mathcal{N})>1$ (see Remark [Remark 17](#rem_abel_limit){reference-type="ref" reference="rem_abel_limit"} or [@Pan_cone Appendix A]). In other words, the (non-abelian) nilpotency structure of $\pi_1(M)$ may not be preserved in the algebraic aspect of asymptotic limits. However, according to Theorem A, it must be reflected in the metric aspects of asymptotic limits. It is unclear to the author whether the condition $E(M,p)\not=1/2$ can be replaced by the finite generation of $\pi_1(M)$, mainly due to the lack of examples with $\mathrm{Ric}\ge 0$, finitely generated $\pi_1(M)$, and $E(M,p)=1/2$. If such examples do exist, then we don't expect their equivariant asymptotic geometry to have good structures. Also see the recent examples by Bruè-Naber-Semola [@BNS] with infinitely generated $\pi_1(M)$ (thus $E(M,p)=1/2$); in particular, one may refer to [@BNS Section 2.2.4], where the equivariant asymptotic geometry of their examples are described. As applications of Theorem A, we can use the asymptotic geometry of $\widetilde{M}$ to control the nilpotency step of $\pi_1(M)$. For convenience, we write $\Omega(\widetilde{M})$ as the set of all asymptotic cones of $\widetilde{M}$ and define $$\mathcal{D}_\infty(\widetilde{M})=\sup\{\dim_{\mathcal{H}}(Ly)|(Y,y)\in\Omega(\widetilde{M}), \text{ $L$ is a closed $\mathbb{R}$-subgroup of $\mathrm{Isom}(Y)$} \}.$$ Using this quantity, we can reformulate Theorem A as follows. **Corollary B 1**. *Let $(M,p)$ be an open $n$-manifold with $\mathrm{Ric}\ge 0$ and $E(M,p)\not= 1/2$. Then $$\mathrm{step}(\pi_1(M))\le \mathcal{D}_\infty(\widetilde{M}).$$ In particular, if $\mathcal{D}_\infty(\widetilde{M})<2$, then $\pi_1(M)$ is virtually abelian.* Corollary B generalizes the main result in [@Pan_cone], where the universal cover $\widetilde{M}$ is assumed to be (metric) conic at infinity. When an asymptotic cone $(Y,y)$ is a metric cone with vertex $y$, the orbit $Ly$ always has Hausdorff dimension $1$ for any closed $\mathbb{R}$-subgroup $L$ of $\mathrm{Isom}(Y)$ (see proof of Corollary [Corollary 65](#cor_cone){reference-type="ref" reference="cor_cone"} for details); in other words, $\mathcal{D}_\infty(\widetilde{M})=1$ if $\widetilde{M}$ is conic at infinity. Then it follows from Corollary B that $\pi_1(M)$ is virtually abelian. Besides metric cones, Corollary B applies to other asymptotic cones that are not covered by previous results. For instance, following the methods in [@PW_ex] and [@Pan_cone Appendix A], we can construct the Grushin halfspace below as the asymptotic cone of the universal cover of some open manifold with $\mathrm{Ric}\ge 0$ (also see [@DHPW Remark 3.9], where the metric is clarified as a Grushin-type almost Riemannian metric). Given $0\le \alpha_1\le...\le \alpha_k$, we define an incomplete Riemannian metric $g$ on $\mathbb{R}^k \times (0,\infty)$ by $$g=dr^2 + \sum_{j=1}^k r^{-2\alpha_j}dx_j^2.$$ By taking its metric completion, $g$ defines a distance $d$ on $Y=\mathbb{R}^k \times [0,\infty)$. We denote this Grushin halfspace $(Y,0,d)$ by $\mathbb{G}^+(\alpha_1,...,\alpha_k)$. Each $x_j$-curve through $0\in Y$ is the orbit of some isometric $\mathbb{R}$-action with Hausdorff dimension $1+\alpha_j$. Suppose that every asymptotic cone of $\widetilde{M}$ is isometric to a Grushin halfplane $\mathbb{G}^+(\alpha_1,..,\alpha_k)$ for some $0\le \alpha_1\le...\le \alpha_k$, then by Corollary B, $$\mathrm{step}(\pi_1(M))\le 1+\alpha_k$$ for some $Y\in \Omega(\widetilde{M})$; in particular, if $\alpha_k<1$ holds for all $Y\in \Omega(\widetilde{M})$, then $\pi_1(M)$ is virtually abelian. Corollary B also extends the main result in [@Pan_esgap] about small escape rate and virtual abelianness. In fact, when $E(M,p)\le \epsilon$, we can show that the orbit of all asymptotic $\mathbb{R}$-orbits are close to $1$. See Proposition [Proposition 66](#cor_small_escape){reference-type="ref" reference="cor_small_escape"} for the precise statement. **Outline of the proof.** To illustrate our approach to Theorem A, we break its proof into two parts, as Proposition C(1) and C(2) below. We write $\Omega(\widetilde{M},\langle\gamma \rangle)$ as the set of equivariant asymptotic cones of $(\widetilde{M},\langle\gamma \rangle)$, where $\gamma\in \pi_1(M,p)$ and $\langle \gamma \rangle$ is the subgroup generated by $\gamma$. **Proposition C 1**. *Let $(M,p)$ be an open $n$-manifold with $\mathrm{Ric}\ge 0$ and $E(M,p)\not= 1/2$. Let $\mathcal{N}$ be a torsion-free nilpotent subgroup of $\pi_1(M,p)$ with finite index and let $l$ be the nilpotency step of $\mathcal{N}$. Then the followings holds for any $\gamma\in \zeta_{l-1}(\mathcal{N})-\{\mathrm{id}\}$.\ (1) For every $(Y,y,H)\in\Omega(\widetilde{M},\langle\gamma \rangle)$, the orbit $Hy$ is homeomorphic to $\mathbb{R}$.\ (2) There exists $(Y,y,H)\in\Omega(\widetilde{M},\langle\gamma \rangle)$ such that $\dim_{\mathcal{H}}(Hy)\ge l$.* Below are the major steps in the proof of Proposition C(1):\ $E(M,p)\not=1/2$;\ $\Rightarrow$ Equivariant GH distance gaps between different types of possible equivariant asymptotic cones of $(\widetilde{M},\mathcal{N})$ (Propositions [Proposition 25](#eGH_gap_1){reference-type="ref" reference="eGH_gap_1"} and [Proposition 26](#eGH_gap_2){reference-type="ref" reference="eGH_gap_2"});\ $\Rightarrow$ There is an integer $k_0$ such that for every $(Y,y,G)\in \Omega(\widetilde{M},\mathcal{N})$, the orbit $Gy$ has a natural simply connected nilpotent Lie group structure of dimension $k_0$ (Proposition [Proposition 43](#topol_dim){reference-type="ref" reference="topol_dim"});\ $\Rightarrow$ Proposition C(1). Though some intermediate results (for example, Propositions [Proposition 43](#topol_dim){reference-type="ref" reference="topol_dim"} and C(1)) have their counterparts in [@Pan_cone], where $\widetilde{M}$ is assumed to be conic at infinity, many new techniques are developed in this paper to study the equivariant asymptotic geometry without the metric cone structures. Compared to the conic at infinity case, the new difficulties are mainly due to the lack of structure results on singular sets with large Hausdorff dimension. When an asymptotic cone $(Y,y)$ is a metric cone with vertex $y$, it follows from Cheeger-Colding splitting theorem [@CC96] and the cone structure that the orbit $Gy$ of an isometric $G$-action must stay in an Euclidean factor of $Y$. This fact about metric cones is used extensively in [@Pan_cone]. In contrast, in the proof of Theorem A here, we are essentially considering orbits of large Hausdorff dimension in collapsed Ricci limits. Such examples are first constructed in [@PW_ex] and no general structure results are known so far. Among the new techniques, we highlight the proof of distance gaps (Section [4](#sec_egh_gap){reference-type="ref" reference="sec_egh_gap"}). We develop convergence of tunnels (continuous curves in the orbit) and apply the large fiber lemma (Lemma [Lemma 27](#large_fiber){reference-type="ref" reference="large_fiber"}) from topological dimension theory to derive equivariant GH distance gaps between equivariant asymptotic cones of different types. We shall describe more about this novel method in Section [4.2](#subsec_egh_idea){reference-type="ref" reference="subsec_egh_idea"}. Proposition [Proposition 43](#topol_dim){reference-type="ref" reference="topol_dim"} depends on the above mentioned distance gaps and a critical rescaling argument. This kind of arguments was first developed in [@Pan_eu] and applied in different contexts to study the equivariant asymptotic geometry [@Pan_al_stable; @Pan_es0; @Pan_esgap; @Pan_cone]. The proof of Proposition C(2) relies on many structure results of spaces in $\Omega(\widetilde{M},\langle \gamma\rangle)$, including Proposition C(1). Below we give some indications how Hausdorff dimension of $Hy$ and nilpotency step are related in the proof of Proposition C(2). For $\mathcal{N}$ and $\gamma$ in Proposition C, it is known that any word length metric $d_S$ on $\mathcal{N}$ satisfies $$C_1\cdot b^{1/l} \le d_S(\gamma^b \tilde{p},\tilde{p})\le C_2\cdot b^{1/l}$$ In fact, this two-side inequality implies $\dim_{\mathcal{H}}(Ly)=l$ in a Carnot group $G$, where $L$ is the asymptotic limit of $\langle \gamma \rangle$. For an isometric-$\mathcal{N}$ orbit on $\widetilde{M}$, the nilpotency step only offers an upper bound: $$d(\gamma^b \tilde{p},\tilde{p})\le C\cdot b^{1/l}$$ for all $b\in\mathbb{Z}_+$ (see Corollary [Corollary 10](#orbit_length_upper){reference-type="ref" reference="orbit_length_upper"}). We approach Proposition C(2) by contradiction. We define $$\mathcal{D}:=\sup\{\dim_{\mathcal{H}}(Hy)|(Y,y,H)\in\Omega(\widetilde{M},\langle\gamma \rangle)\}$$ and seek a contradiction if $\mathcal{D}<l$. We remark that $\mathcal{D}$ as a supremum indeed can be obtained (Proposition [Proposition 61](#dim_sup){reference-type="ref" reference="dim_sup"}). The idea is to relate the Hausdorff dimension to a lower bound for orbit length: for each $s>\mathcal{D}$, there is a constant $C'$ such that $$d(\gamma^b \tilde{p},\tilde{p})\ge C'\cdot b^{1/s}$$ for all $b\in \mathbb{Z}_+$ large (Proposition [Proposition 64](#orbit_length_lower){reference-type="ref" reference="orbit_length_lower"}). Then a contradiction to the orbit length upper bound would arise if $\mathcal{D}<l$. To close the introduction, we point out that the proof of Theorem A naturally extends to nilpotent isometric actions on open manifolds with $\mathrm{Ric}\ge 0$ as well. **Theorem D 1**. *Let $M$ be an open $n$-manifold with $\mathrm{Ric}\ge 0$ and let $\mathcal{N}$ be a simply connected nilpotent Lie group with nilpotency step $l$. Suppose that $\mathcal{N}$ acts effectively and isometrically on $M$ and $(M,p,G)$ has escape rate less than $1/2$ (see Definition [Definition 16](#defn_es_rate_general){reference-type="ref" reference="defn_es_rate_general"}). Then there is an equivariant asymptotic cone $(Y,y,G)$ of $(M,\mathcal{N})$ and a closed $\mathbb{R}$-subgroup $L$ of $G$ such that $\dim_{\mathcal{H}}(Ly)\ge l$.* **Acknowledgments.** The author is partially supported by National Science Foundation DMS-2304698 and Simons Foundation Travel Support for Mathematicians. A part of the paper was drafted when the author was supported by Fields Postdoctoral Fellowship from the Fields Institute. The author would like to thank Vitali Kapovitch and Guofang Wei for helpful discussions during the preparation of this paper. # Preliminaries ## Equivariant Gromov-Hausdorff convergence Throughout the paper, we use a tuple $(Y,y,G)$ to denote a pointed complete length metric space $(Y,y)$ with a closed subgroup $G$ of the isometry group $\mathrm{Isom}(Y)$. We recall the basics of (pointed) equivariant Gromov-Hausdorff convergence from [@Fu; @FY]. For $R>0$, we put $$G(R)=\{g\in G\ |\ d(gy,y)\le R \}.$$ **Definition 1**. [@Fu; @FY] Let $(Y,y,G)$ and $(Z,z,H)$ be two spaces. We say that $$d_{GH}((Y,y,G),(Z,z,H))\le \epsilon,$$ where $\epsilon>0$, if there are $\epsilon$-approximation maps $(f,\psi,\phi)$, that is, $$f:B_{1/\epsilon}(y)\to Z,\quad \psi: G(1/\epsilon) \to H(1/\epsilon), \quad \phi:H(1/\epsilon) \to G(1/\epsilon)$$ with the following properties:\ (1) $f(y)=z$;\ (2) the $\epsilon$-neighborhood $f(B_{1/\epsilon}(y))$ contains $B_{1/\epsilon}(z)$;\ (3) $|d(f(x),f(x'))-d(x,x')|\le\epsilon$ for all $x,x'\in B_{1/\epsilon}(y)$;\ (4) if $g\in G(1/\epsilon)$ and $x,gx\in B_{1/\epsilon}(y)$, then $d(f(gx),\psi(g)f(x))\le\epsilon$;\ (5) if $h\in H(1/\epsilon)$ and $x,\phi(h)x\in B_{1/\epsilon}(y)$, then $d(f(\phi(h)x),h f(x))\le\epsilon$. **Theorem 2**. *[@Fu; @FY] Let $$(Y_i,y_i)\overset{GH}\longrightarrow (Z,z)$$ be a Gromov-Hausdorff convergence sequence and let $G_i$ be closed subgroups of $\mathrm{Isom}(Y_i)$. Then the followings hold.\ (1) After passing to a subsequence, we have an equivariant Gromov-Hausdorff convergence sequence $$(Y_i,y_i,G_i)\overset{GH} \longrightarrow (Z,z,H),$$ where $H$ is closed subgroup of $\mathrm{Isom}(Z)$.\ (2) The sequence of quotient spaces converges $$(Y_i/G_i,\bar{y}_i)\overset{GH}\longrightarrow (Z/H,\bar{z}).$$* Later in Sections [5.2](#subsec_one_para_orbit){reference-type="ref" reference="subsec_one_para_orbit"} and [5.3](#subsec_pf_c1){reference-type="ref" reference="subsec_pf_c1"}, we shall use the convergence of symmetric subsets $S_i\subseteq G_i$. Recall that a subset $S$ in a group $G$ is *symmetric*, if $e\in S$ and $g^{-1}\in S$ for every $g\in S$. **Definition 3**. Let $$(Y_i,y_i,G_i)\overset{GH}\longrightarrow (Z,z,H)$$ be an equivariant Gromov-Hausdorff convergent sequence and let $S_i$ be a sequence of closed symmetric subsets in $G_i$. We say that $$(Y_i,y_i,S_i)\overset{GH}\longrightarrow (Z,z,S),$$ for a closed symmetric subset $S$ of $H$, if\ (1) for every $h\in S$, there is a sequence $g_i\in S_i$ converging to $h$,\ (2) every convergent sequence $g_i \in S_i$ has the limit $h$ in $S$. A corresponding precompactness result follows directly from the proof of [@FY Proposition 3.6]. **Proposition 4**. *Let $$(Y_i,y_i,G_i)\overset{GH}\longrightarrow (Z,z,H)$$ be an equivariant Gromov-Hausdorff convergent sequence and let $S_i$ be a sequence of closed symmetric subsets in $G_i$. Then after passing to a subsequence, we have convergence $$(Y_i,y_i,S_i)\overset{GH}\longrightarrow (Z,z,S),$$ where $S$ is a closed symmetric subset of $H$.* Let $M$ be an open manifold with $\mathrm{Ric}\ge 0$. Let $\widetilde{M}$ be its Riemannian universal cover and let $\Gamma=\pi_1(M,p)$. For a sequence $r_i\to\infty$, after passing to a subsequence, we have convergence $$(r_i^{-1}\widetilde{M},p,\Gamma)\overset{GH}\longrightarrow (Y,y,G).$$ The limit space $(Y,y,G)$ is called an *equivariant asymptotic cone* of $(\widetilde{M},\Gamma)$. By the work of Colding-Naber [@CN], $G$ is a Lie group. In general, the above limit space $(Y,y,G)$ is not unique and may depend on the choice of $r_i\to\infty$. We denote $\Omega(\widetilde{M},\Gamma)$ as the set of all equivariant asymptotic cones of $(\widetilde{M},\Gamma)$. **Proposition 5**. *The set $\Omega(\widetilde{M},\Gamma)$ is compact and connected in the pointed equivariant Gromov-Hausdorff topology.* See [@Pan_eu Proposition 2.1] for a proof of Proposition [Proposition 5](#cpt_cnt){reference-type="ref" reference="cpt_cnt"}. ## Nilpotent groups We recall some basics results about nilpotent groups. **Definition 6**. A group $\mathcal{N}$ is *nilpotent*, if the following central series terminates $$\mathcal{N}=\zeta_0(\mathcal{N})\triangleright \zeta_1(\mathcal{N}) \triangleright ... \triangleright \zeta_l(\mathcal{N})=\{\mathrm{id}\},$$ where $\zeta_{j+1}(\mathcal{N})=[\mathcal{N},\zeta_j(\mathcal{N})]$. We define the *nilpotency step or class* of $\mathcal{N}$, denoted by $\mathrm{step}(\mathcal{N})$, as the smallest integer $l$ such that $\zeta_l(\mathcal{N})=\{\mathrm{id}\}$. It is clear that $\zeta_{l-1}(\mathcal{N})$ is contained in $Z(\mathcal{N})$, the center of $\mathcal{N}$. The following result is standard for finitely generated nilpotent groups; see [@KM]. **Proposition 7**. *Let $\Gamma$ be a finitely generated nilpotent group. Then\ (1) $\Gamma$ has a torsion-free nilpotent subgroup $\mathcal{N}$ of finite index;\ (2) if $\mathcal{N'}$ is another torsion-free nilpotent subgroup of finite index, then $\mathrm{step}(\mathcal{N})=\mathrm{step}(\mathcal{N'})$.* By Proposition [Proposition 7](#torsion_free_index){reference-type="ref" reference="torsion_free_index"}, we can naturally define nilpotency steps for finitely generated virtually nilpotent groups. **Definition 8**. Let $\Gamma$ be a finitely generated virtually nilpotent group. We define $$\mathrm{step}(\Gamma):=\mathrm{step}(\mathcal{N}),$$ where $\mathcal{N}$ is a torsion-free nilpotent subgroup of $\Gamma$ with finite index. The growth rate of finitely generated nilpotent groups are well understood by the work of Bass [@Bass] and Guivarc'h [@Guiv]. In particular, the following word length estimate holds. **Proposition 9**. *[@Bass; @Guiv] Let $\mathcal{N}$ be a finitely generated torsion-free nilpotent group of nilpotency step $l$ and let $\gamma$ be an element in $\zeta_{l-1}(\mathcal{N})-\{\mathrm{id}\}$. Let $S$ be a finite generating set of $\mathcal{N}$ and let $d_S$ be the corresponding word length metric on $\mathcal{N}$. Then there are positive constants $C_1$ and $C_2$ such that $$C_1\cdot b^{1/l} \le d_S(\gamma^b,e)\le C_2\cdot b^{1/l}$$ for all $b\in \mathbb{Z}_+$.* Proposition [Proposition 9](#word_length_estimate){reference-type="ref" reference="word_length_estimate"} immediately implies an upper bound on the orbit length for $\mathcal{N}$-action on a metric space. **Corollary 10**. *Let $\mathcal{N}$ and $\gamma$ as in Proposition [Proposition 9](#word_length_estimate){reference-type="ref" reference="word_length_estimate"}. Suppose that $\mathcal{N}$ acts freely and discretely on a metric space $(X,d)$ by isometries. Then for every $x\in X$, there is a constant $C$ such that $$d(\gamma^b x,x)\le C\cdot b^{1/l}$$ for all $b\in \mathbb{Z}_+$.* *Proof.* Let $S$ be a finite generating set of $\mathcal{N}$ and let $$R:=\max_{g\in S} d(g x,x).$$ By Proposition [Proposition 9](#word_length_estimate){reference-type="ref" reference="word_length_estimate"}, we can express $\gamma^b$ by at most $C_2b^{1/l}$ many elements in $S$. Thus by triangle inequality, it is clear that $$d(\gamma^b x,x)\le R\cdot C_2b^{1/l}.$$ ◻ In general, the lower bound in Proposition [Proposition 9](#word_length_estimate){reference-type="ref" reference="word_length_estimate"} cannot be transferred to orbit length on a non-compact metric space $X$. Lastly, below are some results about nilpotent Lie groups. See [@HN Section 11.2] for references. **Lemma 11**. *Let $G$ be a connected nilpotent Lie group. Then any maximal torus of $G$ is central in $G$; in particular, $G$ has a unique maximal torus.* **Lemma 12**. *Let $G$ be a connected nilpotent Lie group. Then the exponential map $\exp:\mathrm{Lie}(G)\to G$ is a smooth covering map. If in addition that $G$ is simply connected, then $\exp$ is a diffeomorphism.* For an element $g\not= e$ in a connected and simply connected nilpotent Lie group $G$, according to Lemma [Lemma 12](#nil_exp){reference-type="ref" reference="nil_exp"}, $g$ uniquely determines a parameter subgroup $\{\exp(tv)| t\in\mathbb{R} \}$ of $G$, where $v \in \mathrm{Lie}(G)$ such that $\exp(v)=g$. ## Escape rate [@Pan_es0] introduces the notion of escape rate by comparing the size of representing geodesic loops to their lengths. Let $M$ be an open manifold with an infinite $\pi_1(M)$. For every element $\gamma\in \pi_1(M,p)$, we draw a representing loop $c_\gamma$ at $p$ such that it has the minimal length among all loops at $p$ in the homotopy class $\gamma$. $c_\gamma$ is always a geodesic loop at $p$. We denote $$|\gamma|=d(\gamma \tilde{p},\tilde{p})=\mathrm{length}(c_\gamma).$$ The escape rate of $(M,p)$ is defined as $$E(M,p)=\limsup_{|\gamma|\to\infty} \dfrac{d_H(p,c_\gamma)}{|\gamma|},$$ where $d_H(p,c_\gamma)$ is the Hausdorff distance between $c_\gamma$ and $p$ (in other words, the smallest radius $R$ such that the closed ball $\overline{B_R(p)}$ covers $c_\gamma$.) If $\pi_1(M)$ is finite, then we set $E(M,p)=0$ as a convention. It is clear that $E(M,p)$ takes its value in $[0,1/2]$. In [@Sor], Sormani proved that if $\pi_1(M,p)$ is not finitely generated, then there is a sequence of element $\gamma_i$ (the short generators) such that their representing loops $c_{\gamma_i}$ are minimal up to halfway. Therefore, in the terminology of escape rate, we have **Theorem 13**. *[@Sor][\[escape_onehalf\]]{#escape_onehalf label="escape_onehalf"} Let $M$ be an open manifold with $E(M,p)<1/2$, then $\pi_1(M)$ is finitely generated.* It is unclear to the author whether the converse of Theorem [\[escape_onehalf\]](#escape_onehalf){reference-type="ref" reference="escape_onehalf"} holds for nonnegative Ricci curvature. Throughout the paper, we always assume that $M$ is an open (non-compact and complete) Riemannian manifold with $\mathrm{Ric}\ge 0$ and $E(M,p)\not=1/2$; in particular, $\pi_1(M)$ is finitely generated. When $\pi_1(M)$ is an infinite group, by [@Mil; @Gro_poly] and Proposition [Proposition 7](#torsion_free_index){reference-type="ref" reference="torsion_free_index"}, $\pi_1(M)$ has a finitely generated torsion-free nilpotent subgroup $\mathcal{N}$ of finite index. We gather some elementary lemmas from [@Pan_cone] about the basics of escape rate. **Lemma 14**. *[@Pan_cone Lemma 1.5][\[escape_index\]]{#escape_index label="escape_index"} Let $(M,p)$ be an open manifold with $\mathrm{Ric}\ge 0$ and let $F: (\hat{M},\hat{p})\to (M,p)$ be a finite cover. Then $E(\hat{M},\hat{p})\le E(M,p)$.* In practice, we choose a torsion-free nilpotent subgroup $\mathcal{N}$ of $\pi_1(M,p)$ with finite index and set $\hat{M}=\widetilde{M}/\mathcal{N}$ as a finite cover of $M$. By Lemma [\[escape_index\]](#escape_index){reference-type="ref" reference="escape_index"}, we have $$E(\hat{M},\hat{p})\le E(M,p)<1/2,\quad \pi_1(\hat{M},\hat{p})=\mathcal{N}.$$ Therefore, without loss of generality, we can replace $(M,p)$ by $(\hat{M},\hat{p})$ and then assume that $\pi_1(M,p)=\mathcal{N}$ is a finitely generated torsion-free nilpotent group. **Lemma 15**. *[@Pan_cone Lemma 2.1 and Proposition 2.2][\[midpt_orb\]]{#midpt_orb label="midpt_orb"} Let $(Y,y,G)\in\Omega(\widetilde{M},\Gamma)$. For any point $gy\in Gy$ that is not $y$, there is a minimal geodesic $\sigma$ from $y$ to $gy$ and an orbit point $g'y\in Gy$ such that $$d(m,g'y)\le E\cdot d(y,gy),$$ where $m$ is the midpoint of $\sigma$. As a consequence, the orbit $Gy$ is connected for all $(Y,y,G)\in \Omega(\widetilde{M},\Gamma)$.* The inequality in [@Pan_cone Lemma 2.1] states $d(m,g'y)<(1/2)\cdot d(y,gy)$. However, inspecting its proof, it is clear that a stronger inequality as stated above holds. Lastly, we mention that the notion of escape rate and its properties can be naturally extended to group actions on non-compact length metric spaces. **Definition 16**. Let $(X,p)$ be a complete non-compact length metric spaces and let $G$ be a group that acts effectively and isometrically on $X$. For each $g\in G$, we denote $c_g$ as a minimal geodesic from $p$ to $g\cdot p$ and we put $$|g|=d(p,g\cdot p)=\mathrm{length}(c_g).$$ We define $E(X,p,G)$, the *escape rate* of $(X,p,g)$, by $$E(X,p,G):=\limsup_{|g|\to \infty} \dfrac{R(c_g)}{|g|},$$ where $R(c_g)$ is the infimum of all $R>0$ such that $c_g$ is contained in the $R$-tubular neighborhood of $G\cdot p$. If the orbit $G\cdot p$ is bounded, then we set $E(X,p,G)=0$ as a convention. # Motivating examples {#sec_exmp} ## Wei's examples {#subsec_exmp_wei} To better motivate Theorem A, we have a brief review of Wei's construction [@Wei] of open manifolds with $\mathrm{Ric}>0$ and torsion-free nilpotent fundamental groups. We explain the choice of warping functions and its relation to Hausdorff dimension of asymptotic orbits. Also see [@PW_ex], [@Pan_es0 Appendix B], and [@Pan_cone Appendix A]. For simplicity, we use the discrete Heisenberg $3$-group $$\Gamma=\left\{ \begin{pmatrix} 1 & a & c\\ 0 & 1 & b\\ 0 & 0 & 1 \end{pmatrix} \bigg| a,b,c\in\mathbb{Z} \right\}\subseteq \left\{ \begin{pmatrix} 1 & a & c\\ 0 & 1 & b\\ 0 & 0 & 1 \end{pmatrix} \bigg| a,b,c\in\mathbb{R} \right\}=:\widetilde{N}.$$ $\Gamma$ is torsion-free nilpotent with nilpotency step $2$. Let $\mathfrak{n}$ be the Lie algebra of $\widetilde{N}$. $\mathfrak{n}$ has a basis $$X_0=\begin{pmatrix} 0 & 1 & 0\\ 0 & 0 & 0\\ 0 & 0 & 0 \end{pmatrix}, \quad X_1=\begin{pmatrix} 0 & 0 & 0\\ 0 & 0 & 1\\ 0 & 0 & 0 \end{pmatrix}, \quad X_2=\begin{pmatrix} 0 & 0 & 1\\ 0 & 0 & 0\\ 0 & 0 & 0 \end{pmatrix}$$ with $[X_0,X_1]=X_2$ as the only nontrivial Lie bracket relation. Let $$h_0(r)=h_1(r)=(1+r^2)^{-\alpha},\quad h_2(r)=(1+r^2)^{-\beta},$$ where $\alpha,\beta>0$ to be specified later. We define a family of inner products with parameter $r\in [0,\infty)$ on $\mathfrak{n}$ by $$||X_j||_r:=h_j(r)$$ and setting $X_i$ and $X_j$ orthogonal when $i\not= j$. This naturally defines a family of left-invariant metrics $g_r$ on $\widetilde{N}$. Note $$||[X_0,X_1]||=||X_2||=(1+r^2)^{-\beta}=(1+r^2)^{-\beta+2\alpha}||X_0||_r||X_1||_r.$$ It follows from basic curvature formulas for left-invariant metrics on Lie group that $$|\mathrm{sec}(g_r)|\le C\cdot (1+r^2)^{-2\beta+4\alpha},$$ where $C$ is a constant. Therefore, for $g_r$ to be almost flat as $r\to\infty$, we should require $\beta>2\alpha$. Next, we define a doubly warped product $$\widetilde{M}=[0,\infty) \times_{f(r)} S^{k-1} \times (\widetilde{N},g_r), \quad g= dr^2 + f(r)^2 ds_{k-1}^2 + g_r,$$ where $f(r)=r(1+r^2)^{-1/4}$. $\widetilde{M}$ is diffeomorphic to $\mathbb{R}^k \times \widetilde{N}$. Following a similar computation as in [@Wei; @BW], for $g$ to have $\mathrm{Ric}>0$ when $k$ is sufficiently large, the term $|\mathrm{Ric}(g_r)|$ should be comparable to or smaller than $(1+r^2)^{-1}$; equivalently, we require that $$\beta\ge 2\alpha+1/2.$$ To construct $M$, noting that $\widetilde{N}$ naturally acts on $\widetilde{M}$ freely by isometries, we can take the quotient Riemannian manifold $M=\widetilde{M}/\Gamma$. Then it is clear that $M$ satisfies $\mathrm{Ric}>0$ and $\pi_1(M)=\Gamma$. We check the Hausdorff dimension of asymptotic orbits for the above example. Let $\gamma_j=\exp(X_j)\in \Gamma$ and let $\tilde{p}=(0,\star,\mathrm{id})\in \widetilde{M}$. We use $|\cdot|$ to denote the displacement of an isometry at $\tilde{p}$. Following the length estimate in [@PW_ex Section 1.2], as $b\to\infty$ we have $$|\gamma_0^b|=|\gamma_1^b|\sim b^{\frac{1}{1+2\alpha}},\quad |\gamma_2^{(b^2)}|=|[\gamma_0^b,\gamma_1^b]|\le C\cdot b^{\frac{1}{1+2\alpha}}.$$ $|\gamma_2^b|$ can also be estimated by $h_2(r)$. Since $$\dfrac{1}{1+2\beta}\le \dfrac{1}{2(1+2\alpha)},$$ we have $$|\gamma_2^b|\sim b^{\frac{1}{1+2\beta}}.$$ After blowing-down $$(r_i^{-1}\widetilde{M},\tilde{p},\langle \gamma_j \rangle,\Gamma)\overset{GH}\longrightarrow (Y,y,H_j,G),$$ each $H_j$ is a closed $\mathbb{R}$-subgroup of $G$. It follows from the length estimate that the asymptotic orbits have Hausdorff dimension $$\dim_{\mathcal{H}}(H_0y)=\dim_{\mathcal{H}}(H_1 y)=1+2\alpha,\quad \dim_{\mathcal{H}}(H_2 y)=1+2\beta >2.$$ *Remark 17*. As observed in [@Pan_cone Appendix A], we also mention that the algebraic structure of $G$ depends on $\beta$. In fact, if $\beta=2\alpha+1/2$, then $G$ is isomorphic to the $3$-dimensional Heisenberg group $\widetilde{N}$; if $\beta>2\alpha+1/2$, then $G$ is isomorphic to the abelian group $\mathbb{R}^3$. ## Examples with minimal Hausdorff dimension {#subsec_exmp_min} In this subsection, we give examples of open manifolds $M$ with $\mathrm{Ric}>0$, $E(M,p)\not=1/2$, and the following properties:\ (1) $\pi_1(M,p)=\mathcal{N}$ is a torsion-free nilpotent group of nilpotency step $l\ge 2$;\ (2) for every $(Y,y,H)\in \Omega(\widetilde{M},\langle \gamma \rangle )$, where $\gamma \in \zeta_{l-1}(\mathcal{N})-\{\mathrm{id}\}$, $H$ is a closed $\mathbb{R}$-subgroup of $\mathrm{Isom}(Y)$ and the orbit $Hy$ has Hausdorff dimension exactly $l$. These examples show that the inequality in Proposition C(2) is sharp. The construction is a slight modification of Wei's examples [@Wei]. We remark that the choice of functions in [@Wei] always yield examples with $\dim_{\mathcal{H}}(Hy)> l$, a strict inequality, for all $(Y,y,H)\in \Omega(\widetilde{M},\langle \gamma \rangle )$. Hence some modifications are required. Let $\mathfrak{n}$ be $(l+1)$-dimensional filiform Lie algebra spanned by $\{X_0,X_1,...,X_l\}$ with only nontrivial Lie bracket relations $$[X_0,X_j]=X_{j+1} \quad \text{for } j=1,...,l-1.$$ $\mathfrak{n}$ is nilpotent with nilpotency step $l$. Let $\widetilde{N}$ be the simply connected nilpotent Lie group with Lie algebra $\mathfrak{n}$. Let $$h_0(r)=h_1(r)=1+(1+r^2)^{-1/4},\quad h_j(r)=(1+r^2)^{-(j-1)/2} \text{ for } j=2,...,l.$$ Note that $h_0$ and $h_1$ decays to $1$ as $r\to\infty$; this is the main difference compared to [@Wei]. Then similar to the construction in the last section, we define a family of inner products on $\mathfrak{n}$ by $||X_j||_r:=h_j(r)$ and setting $\{X_j\}$ orthogonal to each other, thus define a family of left-invariant Riemannian metrics $\{g_r\}$ on $\widetilde{N}$. Given the Lie bracket relations and the choice of $h_j$, we have $$||[X_0,X_j]||_r=||X_{j+1}||_r=h_{j+1}(r) \le (1+r^2)^{-1/2} ||X_0||_r||X_j||_r,$$ where $j=1,...,l-1$. Hence $g_r$ satisfies $$|\mathrm{Ric}(g_r)|\le C\cdot (1+r^2)^{-1}.$$ Then we similarly define a doubly warped product $$\widetilde{M}=[0,\infty) \times_{f(r)} S^{k-1} \times (\widetilde{N},g_r), \quad g= dr^2 + f(r)^2 ds_{k-1}^2 + g_r,$$ where $f(r)=r(1+r^2)^{-1/4}$. Then $g$ satisfies $\mathrm{Ric}(g)> 0$ when $k$ is sufficiently large. Let $\gamma= \exp(X_l) \in \zeta_{l-1}(\widetilde{N})$. we choose a lattice $\mathcal{N}$ in $\widetilde{N}$ such that $\zeta_{l-1}(\mathcal{N})=\langle \gamma \rangle$ and set $M$ as the quotient manifold $\widetilde{M}/\mathcal{N}$. Then $M$ is our desired example. In fact, at $\tilde{p}=(0,\star,e)\in \widetilde{M}$, by the choice of $\{h_j\}$ and the estimate in [@PW_ex], we can similarly show that $$|\gamma^b| \sim b^{1/l}.$$ Therefore, $\dim_{\mathcal{H}}(Hy)=l$ for any $(Y,y,H)\in \Omega(\widetilde{M},\langle \gamma \rangle)$. # Equivariant Gromov-Hausdorff distance gaps {#sec_egh_gap} The main goal of this section is to establish equivariant Gromov-Hausdorff distance gaps between different types of equivariant asymptotic cones of $(\widetilde{M},\mathcal{N})$ (Propositions [Proposition 25](#eGH_gap_1){reference-type="ref" reference="eGH_gap_1"} and [Proposition 26](#eGH_gap_2){reference-type="ref" reference="eGH_gap_2"}). In subsection [4.1](#subsec_egh_tunnel){reference-type="ref" reference="subsec_egh_tunnel"}, we introduce the notion of tunnels, that is, continuous curves inside the orbit $Gy$. Using the condition $E(M,p)<1/2$, we study controls on the size of tunnels in the asymptotic cones and the convergence of tunnels. In subsection [4.2](#subsec_egh_idea){reference-type="ref" reference="subsec_egh_idea"}, we give the statements of distance gaps and a rough idea of the proof by using the convergence of tunnels and large fiber lemma. Subsection [4.3](#subsec_egh_adapt){reference-type="ref" reference="subsec_egh_adapt"} is a technical part that we introduce some tools so that we can carry out the rough idea in general nilpotent group actions; in particular, we introduce the notion of adapted bases and adapted maps. We prove the distance gaps Propositions [Proposition 25](#eGH_gap_1){reference-type="ref" reference="eGH_gap_1"} and [Proposition 26](#eGH_gap_2){reference-type="ref" reference="eGH_gap_2"} in subsection [4.4](#subsec_egh_proof){reference-type="ref" reference="subsec_egh_proof"}. ## Tunnels {#subsec_egh_tunnel} In this subsection, we use $E$ exclusively to denote the value of $E(M,p)<1/2$. **Definition 18**. Let $(Y,y,G)$ be a space. A *tunnel* is a continuous path $\sigma:[0,1] \to Gy$. We say that $(Y,y,G)$ is *$C$-tunneled* for some constant $C\in [1,\infty)$, if for every orbit point $gy\in Gy$ with $d(gy,y)=:d$, there is a tunnel $\sigma$ from $y$ to $gy$ that is contained in $\overline{B_{Cd}}(y)$. *Remark 19*. We remark that, in general, it is possible that every nontrivial curve in $Gy$ is has infinite length. For example, as constructed in [@PW_ex] and clarified in [@DHPW Remark 3.9], the Grushin halfplane $\mathbb{G}(\alpha)$, or more generally the Grushin halfplane $\mathbb{G}(\alpha_1,...,\alpha_k)$ mentioned in the introduction, are asymptotic cones of open manifolds with $\mathrm{Ric}\ge 0$. The orbit $Gy$ is exactly $\mathbb{R}^k \times \{0\}$ under the $\mathbb{R}^k \times [0,\infty)$-coordinate of $\mathbb{G}(\alpha_1,...,\alpha_k)$. When all $\alpha_i$ are positive, every nontrivial curve in $Gy$ is not rectifiable. Therefore, we use size, instead of length, to measure tunnels in Definition [Definition 18](#def_tunnel){reference-type="ref" reference="def_tunnel"}. **Proposition 20**. *Given $E\in[0,1/2)$, there is a constant $C_0(E)$ such that the following holds.* *Let $M$ be an open manifold with $\mathrm{Ric}\ge 0$ and $E(M,p)=E$. Then any $(Y,y,G)\in\Omega(\widetilde{M},\Gamma)$ is $C_0(E)$-tunneled.* *Proof.* We put $$C_0(E)=\sum_{j=1}^\infty \left(E+1/2\right)^j<\infty.$$ Let $gy\in Gy-\{y\}$ with $d:=d(y,gy)$. We use Lemma [\[midpt_orb\]](#midpt_orb){reference-type="ref" reference="midpt_orb"} to construct a desired tunnel $\sigma$ from $y$ to $gy$ as follows. We define $\sigma(0)=y$ and $\sigma(1)=gy$. By Lemma [\[midpt_orb\]](#midpt_orb){reference-type="ref" reference="midpt_orb"}, there is an orbit point $g_{(1,1)}y\in Gy$ such that $$d(g_{(1,1)}y,y)\le \left(E+1/2\right)d,\quad d(g_{(1,1)}y,gy)\le \left(E+1/2\right)d.$$ We define $\sigma(1/2)=g_{(1,1)}y$. Next, we set $$P_j=\{k/2^j|k=0,1,...,2^j\}.$$ Inductively, suppose that we have defined $\sigma$ on $P_j$ with $\sigma(k/2^j)=\sigma(g_{(k,j)}y)$ for some $g_{(k,j)}\in G$ such that $$d(g_{(k-1,j)}y,g_{(k,j)}y)\le \left(E+1/2\right)^{j}d.$$ for all $k=1,...,2^j$. Then we use Lemma [\[midpt_orb\]](#midpt_orb){reference-type="ref" reference="midpt_orb"} again to define $\sigma$ on $P_{j+1}$ as follows. When $k$ is even, then $k/2^{j+1}\in P_j$; hence $\sigma(k/2^{j+1})$ has been defined in the previous steps. When $k$ is odd, we assign $\sigma(k/2^{j+1})$ as an orbit point $g_{(k,j+1)}y$ with $$d(g_{(k,j+1)}y,g_{(\frac{k-1}{2},j)}y)\le \left(E+1/2\right)d(g_{(\frac{k-1}{2},j)}y,g_{(\frac{k+1}{2},j)}y)\le \left(E+1/2\right)^{j+1}d,$$ $$d(g_{(k,j+1)}y,g_{(\frac{k+1}{2},j)}y)\le \left(E+1/2\right)^{j+1}d.$$ So far, we have defined $\sigma$ on $\cup_j P_j$ with image in $Gy$. We show that $\sigma$ is uniform continuous on $\cup_j P_j$. In fact, for any $\epsilon>0$, let $J\in\mathbb{N}$ large with $$\sum_{j=J}^\infty (1/2+E)^j\le\epsilon/2,$$ and we choose $\delta=1/2^{J+1}$. Then for any $t_1,t_2\in[0,1]$ with $|t_1-t_2|\le \delta$, there is some $k_0/2^{J}\in P_{J}$ such that $$\max\{|k_0/2^{J}-t_1|,|k_0/2^{J}-t_2|\}|\le 1/2^J.$$ Then by construction of $\sigma$, $$\begin{aligned} d(\sigma(t_1),\sigma(t_2))\le& d(\sigma(t_1),\sigma(k_0/2^{J}))+d(\sigma(k_0/2^{J}),\sigma(t_2))\\ \le & 2\sum_{j=J}^\infty (1/2+E)^j\le \epsilon. \end{aligned}$$ This verifies that $\sigma$ is uniform continuous on $\cup_j P_j$, thus $\sigma$ extends to a continuous path from $y$ to $gy$ in $Gy$. Lastly, observe that by construction, $$d(y,\sigma(t))\le \sum_{j=1}^\infty (1/2+E)^{j}d = C_0(E)d$$ for all $t\in[0,1]$. Therefore, the image of $\sigma$ is contained in $\overline{B_{C_0(E)d}}(y)$. This proves that $(Y,y,G)$ is $C_0(E)$-tunneled. ◻ *Remark 21*. Regarding Proposition [Proposition 20](#C_tunnel){reference-type="ref" reference="C_tunnel"}, we mention that a stronger result holds: $(Y,y,G)$ is $3$-tunneled. Hence the size of the tunnel actually can be uniformly controlled regardless of the value of $E$, as long as $E\not=1/2$. Because Proposition [Proposition 20](#C_tunnel){reference-type="ref" reference="C_tunnel"} is sufficient for this paper, we omit the proof of this stronger result. **Lemma 22**. *Let $(Y_i,y_i,G_i)$ be a sequence of spaces such that $$(Y_i,y_i,G_i)\overset{GH}\longrightarrow (Z,z,H).$$ Suppose that there is $C_0$ such that $(Y_i,y_i,G_i)$ is $C_0$-tunneled for all $i$. Then for any tunnel $\sigma:[0,1]\to Hz$ from $z$, there is a sequence of tunnels $\sigma_i:[0,1]\to G_iy_i$ from $y_i$ such that $\sigma_i$ converges uniformly to $\sigma$.* *Proof.* Let $$\epsilon_i=d_{GH}((Y_i,y_i,G_i),(Z,z,H))\to 0.$$ We shall construct $\sigma_i$ on $(Y_i,y_i,G_i)$ for each large $i$. By the uniform continuity of $\sigma$, we can choose a large integer $N$ such that $$\mathrm{diam}(\sigma|_{[(j-1)/N,j/N]})\le\epsilon_i$$ for all $j=1,2,..,N$. Let $z_j=\sigma(j/\mathcal{N})\in Hz$. For each $j$, we choose $y_{i,j}\in G_iy_i$ that is $\epsilon_i$-close to $z_j$; for $j=0$, we use $y_{i,0}=y$. By triangle inequality, it is clear that $$d(y_{i,j},y_{i,j+1})\le 3\epsilon$$ for all $j$. Next, for two adjacent $y_{i,j}$ and $y_{i,j+1}$, we join them by a tunnel $$\sigma_{i,j}:[(j-1)/N,j/N]\to G_iy_i$$ such that it is contained in $\overline{B_{3C_0\epsilon}}(y_{i,j})$. Let $\sigma_i:[0,1]\to Gy$ be the concatenation of all $\sigma_{i,j}$, where $j=1,...,N$. For any $t\in [0,1]$, let $j\in\{1,...,N\}$ such that $t\in[(j-1)/N,j/N]$, then by construction we have $$\begin{aligned} d(\sigma_i(t),\sigma(t))&\le d(\sigma_i(t),y_{i,j})+d(y_{i,j},z_j)+d(z_j,\sigma(t))\\ &\le 3C_0\epsilon_i + \epsilon_i + \epsilon_i =(3C_0+2)\epsilon_i\to 0. \end{aligned}$$ ◻ Let $G$ be a Lie group. We use $G_0$ to denote the identity component subgroup of $G$. In a space $(Y,y,G)$, where $G$ is a Lie group, if $Gy$ is connected, then $Gy=G_0y$; consequently, every orbit point $z\in Gy$ can be represented as $gy$ for some $g\in G_0$. **Lemma 23**. *Let $$(Y_i,y_i,G_i)\overset{GH}\longrightarrow (Z,z,H)$$ be a convergent sequence such that each $(Y_i,y_i,G_i)$ is $C_0$-tunneled. Suppose that a sequence of orbit points $g_iy_i\in G_iy_i$ converges to a limit orbit point $gz\in Hz$, where $g_i\in (G_i)_0$ and $g\in H_0$. Then after passing to a subsequence if necessary, $g_i$ converges to $gh\in H_0$ with $h\in \mathrm{Iso}(z,H_0)$, the isotropy subgroup of $H_0$ at $z$.* *Proof.* Let $d=d(z,gz)$. Passing to a subsequence, we have convergence $$(Y_i,y_i,G_i,g_i)\overset{GH}\longrightarrow (Z,z,H,g_\infty),$$ where $g_\infty \in H$. We claim that $g_\infty \in H_0$. We argue by contradiction. Suppose that $g_\infty\not\in H_0$, then there is a point $q\in Z$ such that the orbit $Hq$ has multiple components and $g_\infty q$ is not in the component containing $q$. On $(Y_i,y_i,G_i)$, let $\sigma_i:[0,1]\to G_iy_i$ be a tunnel from $y_i$ to $g_iy_i$ such that $\mathrm{im}\sigma_i \subseteq \overline{B_{2C_0d}}(y_i)$. Let $\widetilde{\sigma_i}:[0,1]\to (G_i)_0$ be a continuous path from $\mathrm{id}$ to $g_i$ such that $\sigma_i(t)=\widetilde{\sigma_i}(t)\cdot y_i$. Let $q_i\in Y_i$ converging to $q$. We consider the continuous path $\tau_i(t):=\widetilde{\sigma_i}(t)\cdot q_i$ in $G_iq_i$ from $q_i$ to $g_iq_i$. Then $$d(\tau_i(t),y_i)\le d(\widetilde{\sigma_i}(t) q_i,\widetilde{\sigma_i}(t) y_i)+d(\widetilde{\sigma_i}(t) y_i,y_i)\le d(q_i,y_i)+2C_0d.$$ We write $l=d(q,y)$. After passing to a subsequence, connected and closed subsets $\mathrm{im}\tau_i$ converges to a limit connected and closed subset $S\subseteq \overline{B_{l+2C_0d}(y)}\cap Hq$. Since $S$ contain both $q$ and $g_\infty q$, we conclude that $q$ and $g_\infty q$ belong to the same connected component of $Hq$. This proves the claim. We have $g_i\to g_\infty$ and $g_iy_i\to gz$, thus $g_\infty z=gz$. Let $h=g^{-1}g_\infty$, then $h$ fixes $z$. Together with $g\in H_0$ and the claim $g_\infty\in H_0$, we conclude that $h\in \mathrm{Iso}(z,H_0)$. ◻ ## Statements of distance gaps and a rough idea of the proof {#subsec_egh_idea} As explained in Lemma [\[escape_index\]](#escape_index){reference-type="ref" reference="escape_index"}, without loss generality, we assume that $\pi_1(M,p)=\mathcal{N}$ is a finitely generated torsion-free nilpotent group. Then for any $(Y,y,G)\in\Omega(\widetilde{M},\mathcal{N})$, $G$ is a nilpotent Lie group. **Definition 24**. Let $(Y,y,G)$ be a space, where $G$ is a nilpotent Lie group. Let $T$ be the maximal torus of $G_0$. We say that $(Y,y,G)$ is of type $(k,d)$, if $$\dim G - \dim T =k,\quad \mathrm{diam}(Ty)=d.$$ Using Definition [Definition 24](#def_type_kd){reference-type="ref" reference="def_type_kd"}, we state the equivariant Gromov-Hausdorff distance gaps. **Proposition 25**. *There is a constant $\delta_1=\delta_1(\widetilde{M},\mathcal{N})>0$ such that the following holds.* *Let $(Y,y,G)$ and $(Y',y',G')\in\Omega(\widetilde{M},\mathcal{N})$ of type $(k,d)$ and $(k',d')$, respectively. Suppose that $k<k'$ and $d=0$, then $$d_{GH}((Y,y,G),(Y',y',G'))\ge \delta_1.$$* **Proposition 26**. *There is a constant $\delta_2=\delta_2(\widetilde{M},\mathcal{N})>0$ such that the following holds.* *Let $(Y,y,G)$ and $(Y',y',G')\in\Omega(\widetilde{M},\mathcal{N})$ of type $(k,d)$ and $(k',d')$, respectively. Suppose that $k=k'$, $d\le 1$, and $d'\ge 10$, then $$d_{GH}((Y,y,G),(Y',y',G'))\ge \delta_2.$$* Besides using tunnels, a key ingredient in the proof is the large fiber lemma from topological dimension theory. **Lemma 27** (Large Fiber Lemma). *Let $F:[0,1]^{k+1}\to \mathbb{R}^k$ be a continuous map. Then there are $a,b\in [0,1]^{k+1}$ such that $F(a)=F(b)$ and $|a-b|\ge 1$.* *Remark 28*. The large fiber lemma is a corollary of the Lebesgue covering lemma in topological dimension theory (see [@Guth Section 6]). It also follows from the Borsuk-Ulam theorem in algebraic topology (see [@Hatcher Corollary 2B.7]): we take a $k$-sphere of radius $1/2$ in $[0,1]^{k+1}$, then by Borsuk-Ulam theorem, there exists a pair of antipodal points on the sphere such that they have the same image under $F$. To illustrate how the large fiber lemma and the $C$-tunneled property can be applied to prove equivariant Gromov-Hausdorff distance gaps between two spaces, we rule out the following scenario. Suppose that there is a sequence $$(Y_i,y_i,G_i)\overset{GH}\longrightarrow (Z,z,H)$$ such that\ (1) for each $i$, $(Y_i,y_i,G_i)$ is $C_0$-tunneled and $G_i$ is isomorphic $\mathbb{R}^k$;\ (2) $H$ is isomorphic to $\mathbb{R}^{k+1}$. Let $\{e_1,...,e_{k+1}\}\subseteq H$ be an $\mathbb{R}$-basis of $H=\mathbb{R}^{k+1}$. Note that $H$-action does not have isotropy subgroups at $z$ (otherwise, $H$ would have a nontrivial compact subgroup). For each $j=1,...,k+1$, we write $\{te_j\}_{t\in\mathbb{R}}$ as the one-parameter subgroup through $e_j$. We consider an embedding $$F:[0,1]^{k+1} \to Hz, \quad (t_1,...,t_{k+1})\mapsto \prod_{j=1}^{k+1} (t_je_j)\cdot z.$$ $\sigma_j(t):=(te_j)z$, where $t\in[0,1]$, gives a tunnel in $Hz$ from $z$ to $e_jz$. We apply Lemma [Lemma 22](#tunnel_conv){reference-type="ref" reference="tunnel_conv"} to construct tunnels $\sigma_{i,j}:[0,1]\to G_iy_i$ from $y_i$ that converges uniformly to $\sigma_j$ as $i\to\infty$. Because $G_i$-action does not have isotropy subgroups at $y_i$, each $\sigma_{i,j}$ uniquely defines a continuous curve $\widetilde{\sigma_{i,j}}:[0,1]\to G_i$ such that $\widetilde{\sigma_{i,j}}(t)\cdot y_i=\sigma_{i,j}(t)$. This allows us to define a continuous map $$F_i:[0,1]^{k+1} \to G_iy_i,\quad (t_1,...,t_{k+1})\mapsto \prod_{j=1}^{k+1} \widetilde{\sigma_{i,j}}(t_j) \cdot y_i.$$ By construction, it is not difficult to show that $F_i$ converges uniformly to $F$. Since $G_iy_i$ is homeomorphic to $\mathbb{R}^k$, we can apply the large fiber lemma to $F_i$. It follows that there are $a_i,b_i\in [0,1]^{k+1}$ such that $F_i(a_i)=F_i(b_i)$ and $|a_i-b_i|\ge 1$. Passing to a subsequence, we have limit points $a',b'\in [0,1]^{k+1}$ such that $F(a')=F(b')$ and $|a'-b'|\ge 1$ by the uniform convergence of $F_i$ to $F$. However, this contradicts the injectivity of $F$. *Remark 29*. We remark that in general, it is possible for a sequence of $\mathbb{R}^k$-actions converges to a limit $\mathbb{R}^{k+1}$-action, as shown in the example below. Hence the $C$-tunneled property is crucial here. We consider a sequence of $\mathbb{R}$-actions on the standard Euclidean space $\mathbb{R}^3$ as follows. Set $y=(0,0,0)$ as our base point. For each $i\in\mathbb{Z}_+$, $G_i=\mathbb{R}$ acts on $\mathbb{R}^3$ by rotating $xy$-plane by angle $2\pi t$ with center $(i,0,0)$ while translating along $z$-axis by $t/i$, where $t\in\mathbb{R}$. Then $$(\mathbb{R}^3,p,G_i)\overset{GH}\longrightarrow (\mathbb{R}^3,p,\mathbb{R}^2).$$ The limit $\mathbb{R}^2$-action are translations in $yz$-plane. Note that these spaces $(\mathbb{R}^3,p,G_i)$ are not $C$-tunneled for a uniform $C$. ## Adapted bases and adapted maps {#subsec_egh_adapt} In general, the group actions involved are nilpotent and may have nontrivial torus subgroups that move the base point. Hence we need more preparations to carry out the strategy in subsection [4.2](#subsec_egh_idea){reference-type="ref" reference="subsec_egh_idea"}. For convenience, we define $$\begin{aligned} \Omega_Q(\widetilde{M},\mathcal{N})=\{(Y/H,\bar{y},G/H)|& (Y,y,G)\in\Omega(\widetilde{M},\mathcal{N}),\\ & H \text{ is a closed normal subgroup of }G \}.\end{aligned}$$ Note that $\Omega_Q(\widetilde{M},\mathcal{N})$ includes $\Omega(\widetilde{M},\mathcal{N})$ since we can take $H=\{\mathrm{id}\}$. **Definition 30**. We say that a space $(Y,y,G)$ is *good*, if the followings hold:\ (1) $(Y,y)\in \Omega_Q(\widetilde{M},\mathcal{N})$;\ (2) $G\subseteq \mathrm{Isom}(Y)$ is closed nilpotent subgroup;\ (3) $(Y,y,G)$ is $C_0$-tunneled, where $C_0$ is the constant in Proposition [Proposition 20](#C_tunnel){reference-type="ref" reference="C_tunnel"};\ (4) The isotropy subgroup of $G$ at $y$ is finite;\ (5) $G_0$, the identity component subgroup of $G$, is simply connected. **Lemma 31**. *Let $(Y,y,G)\in \Omega(\widetilde{M},\mathcal{N})$ and let $T$ be the maximal torus subgroup of $G_0$. Then the quotient space $(Y/T,\bar{y},G/T)$ is good in the sense of Definition [Definition 30](#def_good){reference-type="ref" reference="def_good"}.* *Proof.* We first remark that by Lemma [Lemma 11](#max_torus){reference-type="ref" reference="max_torus"} $T$ is normal in $G$, thus the quotient group $G/T$ is defined. It is clear that (1,2) in Definition [Definition 30](#def_good){reference-type="ref" reference="def_good"} are fulfilled. (4,5) are also straightforward since $(G/T)_0=G_0/T$ does not have any nontrivial torus subgroup. It remains to show (3). Let $\pi:Y\to Y/T$ be the quotient map. Since $\pi$ maps $Gy$ to $(G/T)\bar{y}$ and $Gy$ is connected by Lemma [\[midpt_orb\]](#midpt_orb){reference-type="ref" reference="midpt_orb"}, we see that $(G/T)\bar{y}$ is also connected. For any orbit point $\bar{z}\in (G/T)\bar{y}$, because $(G/T)\bar{y}$ is connected, we can write $\bar{z}=\bar{g}\bar{y}$, where $\bar{g}\in (G/T)_0=G_0/T$. We choose $g\in G_0$ such that $g$ projects to $\bar{g}$ and $$d_Y(gy,y)=d_{Y/T}(\bar{g}\bar{y},\bar{y})=:d.$$ By Lemma [Proposition 20](#C_tunnel){reference-type="ref" reference="C_tunnel"}, there is a tunnel $\sigma:[0,1]\to Gy$ from $y$ to $gy$ that is contained in $\overline{B_{C_0d}}(y)$. Then $\pi\circ\sigma$ is a desired tunnel from $\bar{y}$ to $\bar{g}\bar{y}$ in $(G/T)\bar{y}$. ◻ We first construct adapted bases and maps for good spaces in the sense of Definition [Definition 30](#def_good){reference-type="ref" reference="def_good"}. In this case, the maximal torus of $G_0$ is trivial. **Definition 32**. Let $(Y,y,G)$ be a space, where $G_0$ is a simply connected nilpotent Lie group. Let $$G_0=\zeta_0(G_0)\triangleright \zeta_1(G_0) \triangleright ...\triangleright \zeta_{l-1}(G_0)\triangleright \zeta_l(G_0)=\{\mathrm{id}\}$$ be the lower central series of $G_0$, where $\zeta_{l-1}(G_0)\not=\{\mathrm{id}\}$. We say an element $e_1\in G_0$ is *initial*, if $e_1\in \zeta_{l-1}(G_0)$ and $d(e_1y,y)=1$. Note that every one-parameter subgroup of $\zeta_{l-1}(G_0)$ has an unbounded orbit at $y$. Thus the initial element defined above always exists. This initial element $e_1$ is the first element of an adapted basis $\{e_1,...,e_k\}$ with respect to $(Y,y,G)$, where $k$ is the dimension of $G$. We choose the remaining elements by induction. Let $H_1$ be the unique one-parameter subgroup through $e_1$ (see Lemma [Lemma 12](#nil_exp){reference-type="ref" reference="nil_exp"}). By construction, $H_1$ is normal in $G_0$. We choose $\bar{e}_2\in G_0/H_1$ as the initial element in $(Y/H_1,\bar{y},G_0/H_1)$. Let $e_2\in G_0$ such that $e_2$ projects to $\bar{e}_2\in G_0/H_1$ and $$d_Y(e_2 y,y)=d_{Y/H_1}(\bar{e}_2\bar{y},\bar{y})=1.$$ Note that because $\bar{e}_2$ belongs to the last nontrivial subgroup of the lower central series of $G_0/H_1$, we have $$[v,v_2]\in \mathrm{span}_\mathbb{R} \{v_1\}$$ for all $v\in \mathrm{Lie}(G_0)$, where $v_j\in \mathrm{Lie}(G_0)$ such that $\exp(v_j)=e_j$. Inductively, we choose $\{e_1,...,e_k\}$ such that\ (1) $[v,v_{j+1}]\in \mathrm{span}_\mathbb{R} \{v_1,...,v_{j}\}$ for all $j=1,2,...,k-1$, where $v_j\in \mathrm{Lie}(G_0)$ such that $\exp(v_j)=e_j$;\ (2) $\bar{e}_{j+1}$ is an initial element of $(Y/H_j,\bar{y},G_0/H_j)$, where $H_j$ is the Lie subgroup with Lie algebra as $\mathrm{span}\{v_1,...,v_j\}$, and $e_{j+1}\in G_0$ such that $e_{j+1}$ projects to $\bar{e}_{j+1}$ and $$d_Y(e_{j+1} y,y)=d_{Y/H_j}(\bar{e}_{j+1}\bar{y},\bar{y})=1.$$ **Definition 33**. Let $(Y,y,G)$ be a good space in the sense of Definition [Definition 30](#def_good){reference-type="ref" reference="def_good"}. We call the above constructed $\{e_1,...,e_k\}\subseteq G_0$ an *adapted basis* with respect to $(Y,y,G)$, where $k$ is the dimension of $G$. As a convention, $\prod$ means a product multiplying on the left $\prod_{j=1}^k g_j=g_k...g_2g_1.$ **Definition 34**. Let $(Y,y,G)$ be a good space in the sense of Definition [Definition 30](#def_good){reference-type="ref" reference="def_good"}. Let $\mathcal{E}=\{e_1,...,e_k\}\subseteq G_0$ be an adapted basis with respect to $(Y,y,G)$, where $k$ is the dimension of $G$. We define an *adapted map* for $\mathcal{E}$: $$F:[0,1]^k \to Gy,\quad (t_1,...,t_k)\mapsto \prod_{j=1}^k (t_je_j) \cdot y,$$ where $te_j$ denotes the elements on the unique one-parameter subgroup through $e_j$. **Lemma 35**. *Let $(Y,y,G)$ be a good space in the sense of Definition [Definition 30](#def_good){reference-type="ref" reference="def_good"}. Let $F:[0,1]^k \to Gy$ be an adapted map for an adapted basis $\mathcal{E}$ as in Definition [Definition 34](#def_adapted_map_good){reference-type="ref" reference="def_adapted_map_good"}. Then $F$ is a continuous injection.* *Proof.* Note that $G_0$-action is free at $y$; otherwise $G_0$ would have a compact torus subgroup as the isotropy subgroup at $y$. Because $G_0$-action is continuous and free at $y$, it suffices to show that the map $$\widetilde{F}:[0,1]^k \to G_0,\quad (t_1,...,t_k)\mapsto \prod_{j=1}^k (t_je_j)$$ is a continuous injection. It is clear that $\widetilde{F}$ is continuous. We prove its injectivity by induction on the nilpotency step of $G_0$. When $G_0$ is abelian, it is clear that $\widetilde{F}$ is injective. Assuming that $\widetilde{F}$ is injective when $G_0$ has nilpotency step $\le l$, we consider the case that $G_0$ has nilpotency step $l+1$. For each $j$, let $v_j\in \mathrm{Lie}(G_0)$ such that $\exp(v_j)=e_j$. Suppose that $$\prod_{j=1}^k \exp(t_jv_j) = \prod_{j=1}^k \exp(s_jv_j).$$ By the construction of the adapted basis, there is an integer $m\in [1,k)$ such that $\zeta_{l-1}(G_0)=\exp(V_m)$, where $V_m$ is the span of $\{v_1,...,v_m\}$. The quotient group $G_0/\zeta_{l-1}(G_0)$ has nilpotency step $l$. Let $\bar{v_j}$, where $j>m$, be the projection of $v_j$ in the quotient Lie algebra $\mathrm{Lie}(G_0)/V_m=\mathrm{Lie}(G_0/\zeta_{l-1}(G_0))$. Then in $G_0/\zeta_{l-1}(G_0)$ we have $$\prod_{j=m+1}^k \exp(t_j\bar{v_j})=\prod_{j=m+1}^k \exp(s_j\bar{v_j}).$$ By the induction assumption, we have $t_j=s_j$ for all $j>m$. By Lemma [Lemma 12](#nil_exp){reference-type="ref" reference="nil_exp"}, there is an element $Z\in \mathrm{Lie}(G_0)$ such that $$\exp(Z)=\prod_{j=m+1}^k \exp(t_j {v_j}).$$ Because $v_1,..,v_m$ are in the center of $\mathrm{Lie}(G_0)$, by the Baker--Campbell--Hausdorff formula, it follows that $$\exp\left(\sum_{j=1}^m t_jv_j+Z\right)=\exp\left(\sum_{j=1}^m s_jv_j+Z\right).$$ By Lemma [Lemma 12](#nil_exp){reference-type="ref" reference="nil_exp"} again, we see that $$\sum_{j=1}^m t_jv_j+Z=\sum_{j=1}^m s_jv_j+Z.$$ We conclude that $t_j=s_j$ also holds for $j=1,...,m$. This completes the inductive step and thus $\widetilde{F}$ is injective. ◻ In general, we will also consider spaces that do not satisfy Definition [Definition 30](#def_good){reference-type="ref" reference="def_good"}. There are mainly two cases, corresponding to Propositions [Proposition 25](#eGH_gap_1){reference-type="ref" reference="eGH_gap_1"} and [Proposition 26](#eGH_gap_2){reference-type="ref" reference="eGH_gap_2"} respectively. We shall similarly construct adapted bases and adapted maps in each case. For Proposition [Proposition 25](#eGH_gap_1){reference-type="ref" reference="eGH_gap_1"}, we consider a space $(Z,z,H)\in\Omega_Q(\widetilde{M},\mathcal{N})$ of type $(k,d)$. Let $T_H$ be the maximal torus subgroup of $H_0$. The quotient space $(Z/T_H,\bar{z},H/T_H)$ is a good space in the sense of Definition [Definition 30](#def_good){reference-type="ref" reference="def_good"} by Lemma [Lemma 31](#lem_quo_tori_good){reference-type="ref" reference="lem_quo_tori_good"}. Thus we can follow Definitions [Definition 33](#def_adapted_basis_good){reference-type="ref" reference="def_adapted_basis_good"} and [Definition 34](#def_adapted_map_good){reference-type="ref" reference="def_adapted_map_good"} to construct an adapted basis $\{\bar{e}_1,...,\bar{e}_k\}\subseteq (H/T_H)_0=H_0/T_H$. For each $j=1,...,k$, we choose $e_j\in H_0$ such that $e_j$ projects to $\bar{e_j}\in H/T_H$ and $$d_Y(e_jz,z)=d_{Y/T_H} (\bar{e_j}\bar{z},\bar{z})=1.$$ **Definition 36**. Let $(Z,z,H)\in\Omega_Q(\widetilde{M},\mathcal{N})$ be a space of type $(k,d)$. We call the above constructed $\mathcal{E}=\{e_1,...e_k\}\subseteq H_0$ an *adapted basis* with respect to $(Z,z,H)$. To further construct an adapted map, for each $e_k$, we choose a parameter subgroup $\tau_j:\mathbb{R}\to H_0$ such that $\tau_j(1)=e_j$; note that the choice of $\tau_j$ may not be unique. We define an *adapted map* for $\mathcal{E}$ as follows: $$F:[0,1]^k\to Hz,\quad (t_1,...,t_k)\mapsto \prod_{j=1}^k \tau_j(t_j)\cdot z.$$ **Lemma 37**. *Let $(Z,z,H)\in\Omega_Q(\widetilde{M},\mathcal{N})$ be a space of type $(k,d)$ and let $T_H$ be the maximal torus subgroup of $H$. Let $\mathcal{E}=\{e_1,...e_k\}\subseteq H_0$ be an adapted basis with respect to $(Z,z,H,T_H)$ and let $F:[0,1]^k\to Hz$ an adapted map for $\mathcal{E}$. Then $F$ is a continuous injection.* *Proof.* The continuity of $F$ is clear. We prove its injectivity. Recall that $\mathcal{E}=\{e_1,...,e_k\}$ is the lift of an adapted basis $\overline{\mathcal{E}}=\{\bar{e_1},...,\bar{e_k}\}$ with respect to the quotient space $(Z/T_H,\bar{z},H/T_H)$. Note that a one-parameter subgroup $\tau_j$ through $e_j$ projects to the unique one-parameter subgroup through $\bar{e_j}$. Let $\pi:Z\to Z/T_H$ be the quotient map. By construction, $$(\pi\circ F)(t_1,...,t_k)=\prod_{j=1}^k \pi\circ\tau_j(t_j)\cdot \bar{z}=\prod_{j=1}^k t_j \bar{e}_j \cdot\bar{z}.$$ Thus $\pi\circ F$ is the adapted map for $\overline{\mathcal{E}}$. By Lemma [Lemma 35](#adapted_map_inj_good){reference-type="ref" reference="adapted_map_inj_good"}, $\pi\circ F$ is injective, thus $F$ is injective as well. ◻ Next, we consider the scenario for Proposition [Proposition 26](#eGH_gap_2){reference-type="ref" reference="eGH_gap_2"}. Let $(Z,z,H)\in\Omega(\widetilde{M},\mathcal{N})$ be a space of type $(k,d)$ with $d\ge 5$. Let $T_H$ be the maximal torus subgroup of $H_0$ and let $\{e_1,...,e_k\}$ be an adapted basis with respect to $(Z,z,H)$ as constructed in Definition [Definition 36](#def_adapted_ver1){reference-type="ref" reference="def_adapted_ver1"}. For the sake of a dimensional argument, we need an additional element from $T_H$. We choose an element $e_0\in T_H$ such that\ (1) $d({e_0}{z},{z})=1$,\ (2) there is a piece of one-parameter subgroup $\tau_0$ from $\mathrm{id}$ to $e_0$ such that $\tau_0|_{(0,1]}$ is outside $\mathrm{Iso}(z,H_0)$. **Definition 38**. Let $(Z,z,H)\in\Omega_Q(\widetilde{M},\mathcal{N})$ be a space of type $(k,d)$, where $d\ge 5$, and let $T_H$ be the normal torus subgroup of $H$. We call the above constructed $\{e_0,e_1...,e_k\}\subseteq H_0$ an *adapted basis* with respect to $(Y,y,H,T_H)$. Next, we construct an adapted map. For $e_0$, we have already chosen a piece of one-parameter subgroup $\tau_0$ from $\mathrm{id}$ to $e_0$. For each $e_j$, where $j=1,...,k$, let $\tau_j$ be a piece of a one-parameter subgroup from $\mathrm{id}$ to $e_j$. We define an *adapted map* for $\mathcal{E}$: $$F:[0,1]^{k+1}\to Hz,\quad (t_0,t_1,...,t_k)\mapsto \prod_{j=0}^k \tau_j(t_j)\cdot z.$$ **Lemma 39**. *Let $(Z,z,H)\in\Omega_Q(\widetilde{M},\mathcal{N})$ be a space of type $(k,d)$, where $d\ge 5$. Let $\mathcal{E}=\{e_0,e_1,...e_k\}\subseteq H_0$ be an adapted basis with respect to $(Z,z,H,T_H)$ and let $F:[0,1]^{k+1}\to Hz$ be an adapted map for $\mathcal{E}$. Then $F$ is a continuous injection.* *Proof.* It is clear that $F$ is continuous. Suppose that $$F(t_0,t_1,...,t_k)=F(t'_0,t'_1,...,t'_k).$$ Let $\pi: Z\to Z/T_H$ be the quotient map. By the proof of Lemma [Lemma 37](#adapted_map_inj_ver1){reference-type="ref" reference="adapted_map_inj_ver1"}, $\pi\circ F$ is an adapted map and thus is injective. This shows that $t'_j=t_j$ for $j=1,...,k$. Now we have $$\left(\prod_{j=1}^k \tau_j(t_j)\right)\tau_0(t_0)z=\left(\prod_{j=1}^k \tau_j(t_j)\right)\tau_0(t'_0)z.$$ Thus $\tau_0(t_0)z=\tau_0(t'_0)z$. Because $\tau_0$ is constructed from a one-parameter subgroup, we have $\tau_0(t_0-t'_0)z=z$. Recall that $\tau|_{(0,1]}$ is outside $\mathrm{Iso}(z,H_0)$. Hence we must have $t_0=t'_0$. ◻ To complete this subsection, we use the properties of tunnels (Lemmas [Lemma 22](#tunnel_conv){reference-type="ref" reference="tunnel_conv"} and [Lemma 23](#tunnel_by_isotropy){reference-type="ref" reference="tunnel_by_isotropy"}) to construct maps converging uniformly to an adapted map. **Lemma 40**. *Let $$(Y_i,y_i,G_i)\overset{GH}\longrightarrow(Z,z,H)$$ be a convergent sequence of spaces in $\Omega_Q(\widetilde{M},\mathcal{N})$. Suppose that\ (1) each $(Y_i,y_i,G_i)$ is good in the sense of Definition [Definition 30](#def_good){reference-type="ref" reference="def_good"};\ (2) on the limit space $(Z,z,H)$, there is an adapted map $F$ defined in either Definition [Definition 36](#def_adapted_ver1){reference-type="ref" reference="def_adapted_ver1"} or [Definition 38](#def_adapted_ver2){reference-type="ref" reference="def_adapted_ver2"} with domain $[0,1]^k$ or $[0,1]^{k+1}$, respectively.\ Then there is a sequence of continuous maps $$F_i:[0,1]^k \text{ or } [0,1]^{k+1} \to G_iy_i\subseteq Y_i$$ that converges uniformly to $F$.* *Proof.* Before starting the proof, we remark that the limit space $(Z,z,H)$ may not be good in the sense of Definition [Definition 30](#def_good){reference-type="ref" reference="def_good"}. Let $J=\{1,...,k\}$ or $\{0,1,...,k\}$. We use $\vv{t}$ to denote $$\vv{t}=(t_1,...,t_k)\in [0,1]^k\ \text{ or }\ \vv{t}=(t_0,t_1,...,t_k)\in [0,1]^{k+1}.$$ In both Definitions [Definition 36](#def_adapted_ver1){reference-type="ref" reference="def_adapted_ver1"} and [Definition 38](#def_adapted_ver2){reference-type="ref" reference="def_adapted_ver2"}, the adapted map has the form $$F:[0,1]^k \text{ or } [0,1]^{k+1} \to Hz,\quad \vv{t} \mapsto \prod_{j\in J} \tau_j(t_j)\cdot z,$$ where $\tau_j:[0,1]\to H_0$ is a piece of one-parameter subgroup from $\mathrm{id}$ to $e_j$, an element in the adapted basis $\mathcal{E}$. For each $j\in J$, let $\sigma_j(t)=\tau_j(t)\cdot z$, which is a tunnel from $z$ to $e_jz$. By Lemma [Lemma 22](#tunnel_conv){reference-type="ref" reference="tunnel_conv"}, there is a sequence of tunnels $$\sigma_{i,j}:[0,1]\to G_iy_i$$ from $y_i$ that converges uniformly to $\sigma_j$ as $i\to\infty$. Because each $(Y_i,y_i,G_i)$ is good, $(G_i)_0$ acts freely at $y_i$. Hence $\sigma_{i,j}$ uniquely determines a continuous path $$\widetilde{\sigma_{i,j}}:[0,1]\to (G_i)_0$$ such that $\widetilde{\sigma_{i,j}}(t) \cdot y_i=\sigma_{i,j}(t)$. We construct $F_i$ as $$F_i:[0,1]^k \text{ or } [0,1]^{k+1} \to G_iy_i,\quad \vv{t} \mapsto \prod_{j\in J} \widetilde{\sigma_{i,j}}(t_j) \cdot y_i.$$ We prove that $F_i$ converges uniformly to $F$. It suffices to show that for every convergent sequence $$(\vv{t})_i=(t_{i,j})_{j\in J}\to \vv{t}=(t_j)_{j\in J},$$ it holds that $F_i((\vv{t})_i)\to F(\vv{t})$ as $i\to\infty$, that is, $$\prod_{j\in J} \widetilde{\sigma_{i,j}}(t_j)\cdot y_i \to \prod_{j\in J} \tau_j(t_j)\cdot z$$ given $t_{i,j}\to t_j$ for each $j\in J$. By construction of $\widetilde{\sigma_{i,j}}$, we have $\widetilde{\sigma_{i,j}}(t_j) y_i$ converges to $\tau_j(t_j)z$. After passing to a subsequence, we assume that for each $j\in J$, $\widetilde{\sigma_{i,j}}(t_j)$ converges to some element in $H$ as $i\to\infty$. By Lemma [Lemma 23](#tunnel_by_isotropy){reference-type="ref" reference="tunnel_by_isotropy"}, we have $$\widetilde{\sigma_{i,j}}(t_j)\overset{GH}\to \tau_j(t_j)h_j,$$ where $h_j\in \mathrm{Iso}(z,H_0)$. The compact subgroup $\mathrm{Iso}(z,H_0)$ must be contained in the maximal torus subgroup of $H_0$, thus each $h_j$ is central in $H_0$ by Lemma [Lemma 11](#max_torus){reference-type="ref" reference="max_torus"}. It follows that $$\prod_{j\in J} \widetilde{\sigma_{i,j}}(t_j)\cdot y_i \to \prod_{j\in J} (\tau_j(t_j)h_j) \cdot z =\prod_{j\in J} \tau_j(t_j) \cdot \prod_{j\in J} h_j \cdot z= \prod_{j\in J} \tau_j(t_j)\cdot z.$$ This verifies the uniform convergence of $F_i$ to $F$. ◻ ## Proof of the distance gaps {#subsec_egh_proof} We prove Propositions [Proposition 25](#eGH_gap_1){reference-type="ref" reference="eGH_gap_1"} and [Proposition 26](#eGH_gap_2){reference-type="ref" reference="eGH_gap_2"} in this subsection. **Lemma 41**. *Let $(Y_i,y_i,G_i)$ be a sequence of spaces in $\Omega(\widetilde{M},\mathcal{N})$ and let $T_i$ is the maximal torus subgroup of $G_i$. Suppose that there is $D>0$ such that $\mathrm{diam}(T_iy_i)\le D$ for all $i$ and $$(Y_i,y_i,G_i,T_i)\overset{GH}\longrightarrow (Z,z,H,K).$$ Then $$(Y_i/T_i,\bar{y}_i,G_i/T_i)\overset{GH}\longrightarrow (Z/K,\bar{z},H/K).$$* *Proof.* The proof is standard by approximation maps. We give some details below for readers' convenience. Let $$\epsilon_i=10\cdot d_{GH}((Y_i,y_i,G_i),(Z,z,H))\to 0.$$ It is clear that $\mathrm{diam}(Kz)\le D$. When $1/\epsilon_i \gg D$, we have a tuple of $\epsilon_i$-approximation maps $(f_i,\psi_i,\phi_i)$, that is, $$f_i:B_{1/\epsilon_i}(y_i)\to Z,\quad \psi_i: G_i(1/\epsilon_i) \to H(1/\epsilon_i), \quad \phi_i:H(1/\epsilon_i) \to G_i(1/\epsilon_i)$$ with the properties (1)-(5) in Definition [Definition 1](#defn_FY){reference-type="ref" reference="defn_FY"} and\ (6) $\psi_i(T_i)\subseteq K$, $\phi_i(K)\subseteq T_i$.\ By Theorem [Theorem 2](#eqGH_FY){reference-type="ref" reference="eqGH_FY"}(2), we have $$(Y_i/T_i,\bar{y_i})\overset{GH}\longrightarrow (Z/K,\bar{z}).$$ Moreover, the approximation map $\bar{f}_i$ from $B_{1/(5\epsilon_i)}(\bar{y}_i)\subseteq Y_i/T_i$ to $Z/K$ can be chosen as an quotient of $f_i$; more precisely, for each $\bar{x}\in B_{1/(5\epsilon_i)}(\bar{y}_i)$, we define $$\bar{f}_i(\bar{x}):=\overline{f_i(x)}\in Z/K,$$ where $x\in B_{1/(5\epsilon_i)}(y)$ is a point projecting to $\bar{x}\in Y_i/T_i$. Let $\bar{g}\in \frac{G_i}{T_i}(\frac{1}{5\epsilon_i})$, then there are $t_1,t_2\in T_i$ and $g\in G_i$ projecting to $\bar{g}$ such that $$d(t_1g y_i,t_2 y_i)=d(Tgy,Ty)=d(\bar{g}\bar{y}_i,\bar{y}_i)\le \frac{1}{5\epsilon_i}.$$ Thus $$d(gy_i,y_i)\le d(gy_i,t_1 g y_i)+d(t_1 g y_i,t_2y_i)+d(t_2 y_i,y_i) \le D+\frac{1}{5\epsilon_i}+D.$$ We define $$\bar{\psi}_i: \dfrac{G_i}{T_i}\left(\frac{1}{5\epsilon_i}\right) \to \dfrac{H}{K},\quad \bar{g} \mapsto \overline{\psi_i(g)}.$$ We estimate $$\begin{aligned} d(\overline{\psi_i(g)}\bar{z},\bar{z})&=d(K\psi_i(g)z,Kz)\\ &= d(k_1\psi_i(g)z,k_2z) \ \ \text{for some } k_1,k_2\in K \\ &\le d(\psi_i(g)z,z)+d(k_1z,z)+d(k_2z,z)\\ &\le d(\psi_i(g)z,gy_i)+d(gy_i,y_i)+d(y_i,z)+2D\\ &\le \epsilon_i+ \left[2D + 1/(5\epsilon_i)\right] +\epsilon_i + 2D\\ &\le 1/(10\epsilon_i). \end{aligned}$$ Thus $\mathrm{im}(\bar{\psi}_i)\subseteq \frac{H}{K}(\frac{1}{10\epsilon_i})$. For any $g\in \frac{G_i}{T_i}(\frac{1}{5\epsilon_i})$ and $\bar{x},\bar{g}\bar{x}\in B_{1/\epsilon_i}(\bar{y})\subset Y/T$, $$\begin{aligned} d(\bar{f}_i(\bar{g}\bar{x}),\bar{\psi}_i\bar{f}_i(\bar{x}))& = d(\overline{f_i(gx)},\overline{\psi_i(g)}\cdot \overline{f_i(x)}) \\ & =d(K\cdot f_i(gx),K\cdot \psi_i(g)\cdot f_i(x))\\ & \le d(f_i(gx),\psi_i(g)\cdot f_i(x))\\ & \le \epsilon_i. \end{aligned}$$ Similarly, we can construct $$\bar{\phi}_i: \frac{H}{K}\left(\dfrac{1}{5\epsilon_i}\right) \to \dfrac{G_i}{T_i},\quad \bar{h} \mapsto \overline{\phi_i(h)}$$ with the desired estimates. Therefore, $(\bar{f}_i,\bar{\psi}_i,\bar{\phi}_i)$ gives $(10\epsilon_i)$-approximation maps between $(Y_i/T_i,\bar{y}_i,G_i/T_i)$ and $(Z/K,\bar{z},H/K)$. This completes the proof. ◻ **Lemma 42**. *Let $$(Y_i,y_i,G_i)\overset{GH}\longrightarrow (Z,z,H)$$ be a convergent sequence of spaces in $\Omega(\widetilde{M},\mathcal{N})$. Let $(k_i,d_i)$ be the type of $(Y_i,y_i,G_i)$ and let $(k_\infty,d_\infty)$ be the type of $(Z,z,H)$.\ (1) Suppose that $k_i\ge k$ for all $i$, then $k_\infty\ge k$.\ (2) Suppose that $k_i=k$ and $d_i\ge 10$ for all $i$, then either $k_\infty>k$ holds, or $k_\infty=k$ and $d_\infty\ge 10$ hold.* *Proof.* (1) For each $i$, let $\{e_{i,1},...,e_{i,k_i}\}$ be an adapted basis for $(Y_i,y_i,G_i)$ as defined in Definition [Definition 36](#def_adapted_ver1){reference-type="ref" reference="def_adapted_ver1"}, where $k_i\ge k$. Let $T_i$ be the maximal torus subgroup of $G_i$ and let $L_{i,j}$ be the subgroup $$L_{i,j}=\langle T_i,\mathbb{R}e_{i,1},...,\mathbb{R}e_{i,j} \rangle,$$ where $j=1,...,k$. We remark that although the one-parameter subgroup $\mathbb{R}e_{i,j}$ is not unique, the above defined subgroup $L_{i,j}$ is uniquely defined and independent of the choice of the one-parameter subgroup through $e_j$. We consider the convergence $$(Y_i,y_i,G_i,T_i,L_{i,j})\overset{GH}\longrightarrow (Z,z,H,T_\infty,L_{\infty,j}).$$ Let $(l_j,c_j)$ be the type of $(Z,z,L_{\infty,j})$. We prove $l_j\ge j$ by induction on $j$, then (1) follows by setting $j=k$. Let $j=1$. Note that the quotient group $L_{i,1}/T_i$ is a closed $\mathbb{R}$-subgroup of $\mathrm{Isom}(Y_i/T_i)$. Thus for each $\delta>0$, there is an element $g_{i}(\delta) \in L_{i,1}-T_i$ such that $$\delta= d(g_i(\delta)\cdot T_iy_i,T_i y_i)=d(g_i(\delta)\cdot y_i,T_iy_i).$$ Passing this property to the limit, then for any $\delta>0$, we have an element $g_\infty(\delta)\in L_{\infty,1}-T_\infty$ such that $$\delta= d(g_\infty(\delta)\cdot z,T_\infty z).$$ In particular, there is a closed $\mathbb{R}$-subgroup of $L_{\infty,1}$ that is outside $T_\infty$. Thus $l_1\ge 1$. Suppose that the statement holds for $j$. We consider $j+1$ next. The argument is similar to the case $j=1$. Since $L_{i,j+1}/L_{i,j}$ is a closed $\mathbb{R}$-subgroup of $\mathrm{Isom}(Y_i/L_{i,j})$, for each $\delta>0$ there is an element $g_{i}(\delta) \in L_{i,j+1}-L_{i,j}$ such that $$\delta= d(g_i(\delta)\cdot y_i,L_{i,j} y_i).$$ We pass this property to the limit. Hence there is $g_\infty(\delta)\in L_{\infty,j+1}-L_{\infty,j}$ such that $$\delta= d(g_\infty(\delta)\cdot z,L_{\infty,j} z).$$ This shows that there is a closed $\mathbb{R}$-subgroup of $L_{\infty,j+1}$ that is outside $L_{\infty,j}$. Together with the inductive assumption, we conclude that $l_{j+1}\ge j+1$. This completes the induction. \(2\) Assuming that $k_i=k$, $d_i\ge 10$ for all $i$, and $k_\infty=k$, we shall prove that $d_\infty\ge 10$. We follow the same notations as in the proof of (1); in particular, we have $$(Y_i,y_i,G_i,T_i,L_{i,j})\overset{GH}\longrightarrow (Z,z,H,T_\infty,L_{\infty,j}).$$ We set $L_{\infty,0}:=T_\infty$ for convenience. From the proof of (1), we know that for each $j=1,...,k$, there is a closed $\mathbb{R}$-subgroup in $L_{\infty,j}-L_{\infty,j-1}$. This implies that $T_\infty$ must be compact; otherwise, $T_\infty$ would contain a closed $\mathbb{R}$-subgroup and thus $k_\infty \ge k+1$, which violates with $k_\infty=k$. It follows that there is $D>0$ such that $d_i\le D$ for all $i$. Thus $$d_\infty = \lim_{i\to\infty} d_i \ge 10.$$ ◻ We are ready to prove the distance gaps. *Proof of Propositions [Proposition 25](#eGH_gap_1){reference-type="ref" reference="eGH_gap_1"} and [Proposition 26](#eGH_gap_2){reference-type="ref" reference="eGH_gap_2"}.* The proofs of these two statements are similar. We argue by contradiction to prove them. For Proposition [Proposition 25](#eGH_gap_1){reference-type="ref" reference="eGH_gap_1"}, suppose that there are two sequences $\{(Y_i,y_i,G_i)\}_i$ and $\{(Y'_i,y'_i,G'_i)\}_i$ of spaces in $\Omega(\widetilde{M},\mathcal{N})$ with the conditions below:\ (1) each $(Y_i,y_i,G_i)$ has type $(k,0)$;\ (2) each $(Y'_i,y'_i,G'_i)$ has type $(k',d'_i)$ with $k'>k$;\ (3) $d_{GH}((Y_i,y_i,G_i),(Y'_i,y'_i,G'_i))\to 0$ as $i\to\infty$.\ After passing to some subsequences, we can assume that two sequences converge to the same limit $(Z,z,H)\in \Omega(\widetilde{M},\mathcal{N})$. We write $(k_\infty,d_\infty)$ as the type of $(Z,z,H)$. Condition (2) above and Lemma [Lemma 42](#dist_gap_lem){reference-type="ref" reference="dist_gap_lem"}(1) imply $k_\infty\ge k'>k$. Let $T_i$ be the maximal torus subgroup of $G_i$ and let $K\subseteq H$ be its limit. Since $T_i$ fixes $y_i$, $K$ must fix $z$ as well. Thus the quotient space $(Z/K,\bar{z},H/K)$ also has type $(k_\infty,d_\infty)$. By Lemma [Lemma 41](#mod_max_torus){reference-type="ref" reference="mod_max_torus"}, we have convergence $$(Y_i/T_i,\bar{y}_i,G_i/T_i)\overset{GH}\longrightarrow (Z/T_\infty,\bar{z},H/T_\infty).$$ Let $F:[0,1]^{k_\infty} \to (H/T_\infty)\bar{z}$ be an adapted map constructed in Definition [Definition 36](#def_adapted_ver1){reference-type="ref" reference="def_adapted_ver1"}. $F$ is continuous and injective by Lemma [Lemma 37](#adapted_map_inj_ver1){reference-type="ref" reference="adapted_map_inj_ver1"}. According to Lemma [Lemma 40](#conv_to_adapted_map){reference-type="ref" reference="conv_to_adapted_map"}, there is a sequence of continuous maps $F_i:[0,1]^{k_\infty}\to (G_i/T_i)\bar{y}_i$ converges uniformly to $F$. Note that each $(G_i/T_i)\bar{y}_i$ is homeomorphic to $\mathbb{R}^k$ with $k<k_\infty$. By large fiber lemma, there are two sequences $\{a_i\}$ and $\{b_i\}$ in $[0,1]^{k_\infty}$ such that $$F_i(a_i)=F_i(b_i),\quad |a_i-b_i|\ge 1.$$ By the uniform convergence of $F_i$ to $F$, we can find $a,b\in [0,1]^{k_\infty}$ such that $$F(a)=F(b),\quad |a-b|\ge 1.$$ This contradicts the injectivity of $F$. For Proposition [Proposition 26](#eGH_gap_2){reference-type="ref" reference="eGH_gap_2"}, suppose that we have contradicting convergent sequences $\{(Y_i,y_i,G_i)\}_i$ and $\{(Y'_i,y'_i,G'_i)\}_i$ such that\ (1) each $(Y_i,y_i,G_i)$ has type $(k,d_i)$ with $d_i\le 1$;\ (2) each $(Y'_i,y'_i,G'_i)$ has type $(k,d'_i)$ with $d'_i\ge 10$;\ (3) $d_{GH}((Y_i,y_i,G_i),(Y'_i,y'_i,G'_i))\to 0$ as $i\to\infty$.\ After passing to some subsequences, we let $(Z,z,H)$ be their common limit, whose type is denoted as $(k_\infty,d_\infty)$. By Condition (2) above and Lemma [Lemma 42](#dist_gap_lem){reference-type="ref" reference="dist_gap_lem"}(2), either\ *Case I.* $k_\infty>k$, or\ *Case II.* $k_\infty=k$ and $d_\infty\ge 10$.\ Let $T_i$ be the maximal torus subgroup of $G_i$ and let $T_\infty\subseteq H$ be its limit. By Lemma [Lemma 41](#mod_max_torus){reference-type="ref" reference="mod_max_torus"}, we have convergence $$(Y_i/T_i,\bar{y}_i,G_i/T_i)\overset{GH}\longrightarrow (Z/T_\infty,\bar{z},H/T_\infty).$$ In Case I, we construct an adapted map $F:[0,1]^{k_\infty} \to (H/T_\infty)\bar{z}$ as in Definition [Definition 36](#def_adapted_ver1){reference-type="ref" reference="def_adapted_ver1"}. Then we use Lemma [Lemma 40](#conv_to_adapted_map){reference-type="ref" reference="conv_to_adapted_map"} to obtain $F_i:[0,1]^{k_\infty}\to (G_i/T_i)\bar{y}_i$ that converges uniformly to $F$, where $(G_i/T_i)\bar{y}_i$ is homeomorphic to $\mathbb{R}^k$ with $k<k_\infty$. Then a similar contradiction follows from the large fiber lemma and injectivity of $F$. In Case II, because $\mathrm{diam}(T_iy_i)\le 1$ for all $i$, the limit space $(Z/T_\infty,\bar{z},H/T_\infty)$ has type $(k,\bar{d})$ with $\bar{d}\ge 9$. Then we construct an adapted map $F:[0,1]^{k+1} \to (H/T_\infty)\bar{z}$ as in Definition [Definition 38](#def_adapted_ver2){reference-type="ref" reference="def_adapted_ver2"}. A similar contradiction arises since the targets of the approximated maps $F_i:[0,1]^{k+1}\to (G_i/T_i)\bar{y}_i$ are homeomorphic to $\mathbb{R}^k$. ◻ # Asymptotic orbits of $\langle\gamma\rangle$-action {#sec_cyclic} This section studies the geometry of spaces in $\Omega(\widetilde{M},\mathcal{N})$ and $\Omega(\widetilde{M},\langle \gamma \rangle)$, where $\gamma \in \zeta_{l-1}(\mathcal{N})$. One of the main goals of this section is Proposition C(1), which states that $Hy$ is homeomorphic to $\mathbb{R}$ for all $(Y,y,H)\in \Omega(\widetilde{M},\langle \gamma \rangle)$. Some understandings of $(Y,y,G)\in \Omega(\widetilde{M},\mathcal{N})$ are required beforehand to prove Proposition C(1). In subsection [5.1](#subsec_whole_orbit){reference-type="ref" reference="subsec_whole_orbit"}, we show that every $(Y,y,G)\in\Omega (\widetilde{M},\mathcal{N})$ is of type $(k_0,0)$ for some uniform $k_0$ (Proposition [Proposition 43](#topol_dim){reference-type="ref" reference="topol_dim"}). As mentioned in the introduction, the proof of Proposition [Proposition 43](#topol_dim){reference-type="ref" reference="topol_dim"} uses the distance gaps in Section [4](#sec_egh_gap){reference-type="ref" reference="sec_egh_gap"} and a critical rescaling argument. Subsection [5.2](#subsec_one_para_orbit){reference-type="ref" reference="subsec_one_para_orbit"} studies the one-parameter orbits of $Gy$ and makes the preparation for Proposition C(1). Lastly, we prove Proposition C(1) in subsection [5.3](#subsec_pf_c1){reference-type="ref" reference="subsec_pf_c1"}. The proof of Proposition C(1) follow a similar strategy as [@Pan_cone Section 3]: suppose that the statement fails, then we would find a suitable rescaling such that its limit space violates Proposition [Proposition 43](#topol_dim){reference-type="ref" reference="topol_dim"}. ## Uniform type of asymptotic orbits {#subsec_whole_orbit} **Proposition 43**. *Let $(M,p)$ be an open $n$-manifold with $\mathrm{Ric}\ge 0$ and $E(M,p)\not=1/2$. Let $\mathcal{N}$ be a torsion-free nilpotent subgroup of finite index. Then there is an integer $k_0$ such that every $(Y,y,G)\in\Omega (\widetilde{M},\mathcal{N})$ is of type $(k_0,0)$. Consequently,\ (1) the orbit $Gy$ has a natural simply connected nilpotent group structure of dimension $k_0$;\ (2) any compact subgroup of $Z(G)$ fixes $y$;\ (3) $G$ has at most finitely many components.* *Remark 44*. We mention that by using structure result of nilpotent groups, Proposition [Proposition 43](#topol_dim){reference-type="ref" reference="topol_dim"}(2) can be further improved: any compact subgroup of $G$ fixes $y$. The statement in its current form is sufficient for the rest of the paper. We first prove a weaker statement without a uniform $k_0$. **Lemma 45**. *Under the assumptions of Proposition [Proposition 43](#topol_dim){reference-type="ref" reference="topol_dim"}, let $(Y,y,G)\in \Omega(\widetilde{M},\mathcal{N})$. Then $(Y,y,G)$ is of type $(k,0)$ for some integer $k$.* *Proof.* We argue by contradiction. Suppose that the statement fails. Then we can choose $(Y,y,G)\in\Omega(\widetilde{M},\mathcal{N})$ with type $(k_0,d)$ such that\ (1) $d>0$, and\ (2) if another space $(Y',y',G')\in\Omega(\widetilde{M},\mathcal{N})$ is of type $(k',d')$ with $d'>0$, then $k'\ge k_0$. Let $$(Y_1,y_1,G_1)=(10d^{-1}Y,y,G),\quad (Y_2,y_2,G_2)=(d^{-1}Y,y,G)$$ be two spaces in $\Omega(\widetilde{M},\mathcal{N})$. It is clear that they are of type $(k_0,10)$ and $(k_0,1)$, respectively. Let $r_i,s_i\to\infty$ such that $$(r_i^{-1}\widetilde{M},\tilde{p},\mathcal{N})\overset{GH}\longrightarrow (Y_1,y_1,G_1),\quad (s_i^{-1}\widetilde{M},\tilde{p},\mathcal{N})\overset{GH}\longrightarrow (Y_2,y_2,G_2).$$ Passing to some subsequence, we can assume that $t_i:=r_i/s_i\to\infty$. Let $$(N_i,q_i,\Gamma_i)=(r_i^{-1}\widetilde{M},\tilde{p},\mathcal{N}),$$ then $$(N_i,q_i,\Gamma_i)\overset{GH}\longrightarrow (Y_1,y_1,G_1),\quad (t_iN_i,q_i,\Gamma_i)\overset{GH}\longrightarrow (Y_2,y_2,G_2).$$ Let $\delta=\min\{\delta_1,\delta_2\}>0$, where $\delta_1$ and $\delta_2$ are the constants in Propositions [Proposition 25](#eGH_gap_1){reference-type="ref" reference="eGH_gap_1"} and [Proposition 26](#eGH_gap_2){reference-type="ref" reference="eGH_gap_2"}, respectively. For each $i$, we define a set of scales $$\begin{aligned} L_i=\{ l\ge 1|&d_{GH}((lN_i,q_i,\Gamma),(W,w,H))\le \delta/10 \text{ for some} \\ &(W,w,H)\in\Omega(\widetilde{M},\mathcal{N}) \text{ such that } (W,w,H) \text{ has}\\ &\text{type $(k,d)$ with } k<k_0, \text{ or with } k=k_0 \text{ and } d\le 1 \}. \end{aligned}$$ Recall that $(Y_2,y_2,G_2)$ is of type $(k_0,1)$, thus $t_i\in L_i$ for all $i$ large. We choose $l_i\in L_i$ with $\inf L_i\le l_i \le \inf L_i+1/i$. **Claim 1:** $l_i\to\infty$. Suppose that $l_i\to l_\infty <\infty$ for a subsequence, then $$(l_iN_i,q_i,\Gamma_i)\overset{GH}\longrightarrow (l_\infty Y_1,y_1,G_1).$$ Since $l_i\in L_i$, for each $i$ there is some $(W_i,w_i,H_i)$ with the properties in the definition of $L_i$ such that $$d_{GH}((l_iN_i,q_i,\Gamma_i),(W_i,w_i,H_i))\le \delta/10.$$ Hence for $i$ large, $$d_{GH}((W_i,w_i,H_i),(l_\infty Y_1,y_1,G_1))\le \delta/2,$$ where $(l_\infty Y_1,y_1,G_1)$ is of type $(k_0,10l_\infty)$ with $10l_\infty\ge 10$. Let $(k_i,d_i)$ be the type of $(W_i,w_i,H_i)$. If $k_i<k_0$, then $d_i=0$ by our choice of $k_0$, and the above Gromov-Hausdorff distance estimate cannot hold due to Proposition [Proposition 25](#eGH_gap_1){reference-type="ref" reference="eGH_gap_1"} and the choice of $\delta$. If $k_i=k_0$ and $d_i\le 1$, then it also leads to a contradiction due to Proposition [Proposition 26](#eGH_gap_2){reference-type="ref" reference="eGH_gap_2"}. Therefore, $l_i\to\infty$. Next, we consider the convergence $$(l_iN_i,q_i,\Gamma_i)\overset{GH}\longrightarrow (Y',y',G')\in\Omega(\widetilde{M},\mathcal{N}).$$ Let $(k',d')$ be the type of $(Y',y',G')$. **Claim 2:** $k'\le k_0$; moreover, $d'< 10$ when $k'=k_0$. For each $i$, there is some $(W_i,w_i,H_i)$ with the properties in the definition of $L_i$ and $$d_{GH}((l_iN_i,q_i,\Gamma_i),(W_i,w_i,H_i))\le \delta/10.$$ It follows that for $i$ large, $$d_{GH}((W_i,w_i,H_i),(Y',y',G'))\le\delta/2.$$ If $k'>k_0$, then we end in a contradiction to Proposition [Proposition 25](#eGH_gap_1){reference-type="ref" reference="eGH_gap_1"}. If $k'=k_0$ and $d'>10$, then this contradicts with Proposition [Proposition 26](#eGH_gap_2){reference-type="ref" reference="eGH_gap_2"}. This proves Claim 2. By Claim 2 and our choice of $k_0$, $(Y',y',G')$ has type $(k',d')$ with one of the following cases:\ *Case 1:* $k'<k_0$ and $d'=0$;\ *Case 2:* $k'=k_0$ and $d'< 10$.\ For Case 1, we consider the convergence sequence $$(\frac{1}{2}l_iN_i,q_i,\Gamma_i)\overset{GH}\longrightarrow (\frac{1}{2}Y',y',G')\in\Omega(\widetilde{M},\mathcal{N}).$$ The limit space $(\frac{1}{2}Y',y',G')$ is of type $(k',0)$, where $k'<k_0$. This implies that $l_i/2\in L_i$, a contradiction to $\inf L_i\le l_i\le \inf L_i+1/i$. For Case 2, we consider $$(\frac{1}{10}l_iN_i,q_i,\Gamma_i)\overset{GH}\longrightarrow (\frac{1}{10}Y',y',G')\in\Omega(\widetilde{M},\mathcal{N}),$$ where $(\frac{1}{10}Y',y',G')$ is of type $(k',d'/10)$ with $d'/10< 1$. We result in $l_i/10\in L_i$ for $i$ large and thus a desired contradiction. With all possibilities of $(Y',y',G')$ being ruled out, we complete the proof. ◻ Next, we prove Proposition [Proposition 43](#topol_dim){reference-type="ref" reference="topol_dim"}. *Proof of Proposition [Proposition 43](#topol_dim){reference-type="ref" reference="topol_dim"}.* Let $(Y,y,G)$ and $(Y',y',G')$ in $\Omega(\widetilde{M},\mathcal{N})$ having type $(k,0)$ and $(k',0)$, respectively. We show that $k= k'$. Let $\delta_1$ be the constant in Proposition [Proposition 25](#eGH_gap_1){reference-type="ref" reference="eGH_gap_1"} and let $\epsilon=\delta_1/2$. Because the set $\Omega(\widetilde{M},\mathcal{N})$ is connected in the pointed equivariant Gromov-Hausdorff topology (Proposition [Proposition 5](#cpt_cnt){reference-type="ref" reference="cpt_cnt"}), for the above $\epsilon>0$, there is a chain of elements $\{(W_j,w_j,H_j)\}_{j=1}^J$ in $\Omega(\widetilde{M},\mathcal{N})$ such that $$(W_1,w_1,H_1)=(Y,y,G),\quad (W_J,w_J,H_J)=(Y',y',G'),$$ and $$d_{GH}((W_j,w_j,H_j),(W_{j+1},w_{j+1},H_{j+1}))\le \epsilon$$ for all $j=1,...,J-1$. Applying Lemma [Lemma 45](#width_zero){reference-type="ref" reference="width_zero"}, we know that each $(W_j,w_j,H_j)$ is of type $(k_j,0)$ for some $k_j$. By Proposition [Proposition 25](#eGH_gap_1){reference-type="ref" reference="eGH_gap_1"} and our choice of $\epsilon$, all $k_j$ must be the same. In particular, we conclude $k=k'$. Now we proceed to prove the consequences (1)-(3). To prove (1), let $T$ be the maximal torus subgroup of $G_0$. Because $(Y,y,G)$ is of type $(k_0,0)$, $T$-action fixes $y$. Thus the orbit $Gy=G_0y$ can be naturally identified with the quotient group $G_0/T$, which is a simply connected nilpotent Lie group of dimension $k_0$. Next, we prove (2). Let $K$ be a compact subgroup of $Z(G)$. If $K$ is contained in a torus subgroup of $G$, then it is clearly that $Ky=y$ because $(Y,y,G)$ is of type $(k,0)$. In general, suppose that (2) fails, then we can find a finite cyclic subgroup $\langle h \rangle \subseteq K$ such that $hy\not= y$. Let $g\in G_0$ such that $hy=gy$. Note that $g$ is outside the maximal torus $T$ of $G_0$ because $T$ fixes $y$; in particular, $\langle g \rangle y$ is unbounded in $Y$. On the other hand, because $hy=gy$ and $h\in Z(G)$, we have $h^k y=g^k y$ for all $k\in \mathbb{Z}$. Thus the set $\{g^k y| k\in\mathbb{Z}\}$ is bounded because $\langle h \rangle$ is a finite group. A contradiction. Lastly, we prove (3). Suppose that $G$ has infinitely many connected components. We consider the continuous map $$A: G \to Gy, \quad g \mapsto gy.$$ Let $\mathcal{C}$ be any connected component of $G$. Because the orbit $Gy$ is connected, we have $A(\mathcal{C})=Gy$. Thus we can choose $g_{\mathcal{C}}\in \mathcal{C}$ such that $g_{\mathcal{C}}\cdot y=y$. By hypothesis, this gives a set of infinitely many elements $$\mathcal{F}=\{g_\mathcal{C}\ |\ \mathcal{C} \text{ is a connected component of }G\}$$ that fixes $y$. Since the isotropy subgroup of $G$ at $y$ is compact, we can find a convergence subsequence from $\mathcal{F}$. This contradicts with the fact that elements in $\mathcal{F}$ belong to distinct connected components of $G$. ◻ *Remark 46*. For any convergent sequence $$(r_i^{-1}\widetilde{M},\tilde{p},\mathcal{N})\overset{GH}\longrightarrow (Y,y,G),$$ it is clear that $$\mathrm{step}(Gy)\le \mathrm{step}(G) \le \mathrm{step}(\mathcal{N})=\mathrm{step}(\pi_1(M)),$$ where the nilpotent structure of $Gy$ comes from Proposition [Proposition 43](#topol_dim){reference-type="ref" reference="topol_dim"}(1). ## One-parameter orbits {#subsec_one_para_orbit} **Definition 47**. Let $(Y,y,G)\in\Omega(\widetilde{M},\mathcal{N})$. Given an orbit point $z\in Gy-\{y\}$, we define the *one-parameter orbit* through $z$ as the one-parameter subgroup of $Gy$ through $z$, where the simply connected nilpotent Lie group structure of $Gy$ is given in Proposition [Proposition 43](#topol_dim){reference-type="ref" reference="topol_dim"}(1). By Lemma [Lemma 12](#nil_exp){reference-type="ref" reference="nil_exp"}, this is well-defined. We denote this one-parameter orbit through $z$ as $\mathbb{R}z$, and the points on it as $tz$, where $t\in\mathbb{R}$. As indicated in its name, a one-parameter orbit is indeed the orbit of a one-parameter subgroup. More precisely, we have the description below. **Lemma 48**. *Let $(Y,y,G)\in\Omega(\widetilde{M},\mathcal{N})$ and let $z\in Gy-\{y\}$ be an orbit point. We write $z=gy$, where $g\in G_0$. Let $\sigma:\mathbb{R}\to G_0$ be one-parameter subgroup with $\sigma(1)=g$. Then $\sigma(t)y=tz$ for all $t\in\mathbb{R}$.* *Proof.* Let $T$ be the maximal torus subgroup of $G_0$. We naturally identify the orbit $Gy$ with the quotient group $G_0/T$ as in the proof of Proposition [Proposition 43](#topol_dim){reference-type="ref" reference="topol_dim"}(1). Let $\pi:G_0 \to G_0/T$ be the quotient map and let $\bar{g}=\pi(g)$. Because $G_0/T$ is connected and simply connected, by Lemma [Lemma 12](#nil_exp){reference-type="ref" reference="nil_exp"}, there is a unique one-parameter subgroup $\bar{\sigma}:\mathbb{R}\to G_0/T$ such that $\bar{\sigma}(1)=\bar{g}$. Since $\pi$ is a group homomorphism, we have $$\sigma(t)y=\pi\circ \sigma(t)=\bar{\sigma}(t).$$ Because the choice of $\bar{g}\in G_0/T$ and $\bar{\sigma}$ are uniquely determined by the orbit point $z=gy$, the result follows. ◻ **Corollary 49**. *Let $(Y,y,G)\in\Omega(\widetilde{M},\mathcal{N})$ and let $z\in Gy-\{y\}$ be an orbit point. Let $g\in G_0$ such that $z=gy$. Then $g\cdot (tz)=(1+t)z\in \mathbb{R}z$ for all $t\in\mathbb{R}$.* *Proof.* Let $\sigma:\mathbb{R}\to G_0$ be a one-parameter subgroup of $G_0$ with $\sigma(1)=g$. By Lemma [Lemma 48](#one_para_orbit){reference-type="ref" reference="one_para_orbit"}, for any $t\in\mathbb{R}$, we have $$g\cdot (tz)=g\cdot \sigma(t)y=\sigma(1+t)y=(1+t)z.$$ The result follows. ◻ Given $\gamma\in \mathcal{N}-\{\mathrm{id}\}$ and $k\in\mathbb{Z}$, we denote $$S(\gamma,k):=\{\gamma^m\ |\ m=0,\pm1,...,\pm k\},$$ which is a symmetric subset of $\langle \gamma \rangle$. Recall that $\mathcal{N}$ is torsion-free, thus $\gamma$ has infinite order. Below, we will use the notion of convergence of symmetric subsets; see Definition [Definition 3](#def_conv_symsubset){reference-type="ref" reference="def_conv_symsubset"}. **Lemma 50**. *Let $\gamma\in \mathcal{N}-\{\mathrm{id}\}$. We consider the convergence $$(r_i^{-1}\widetilde{M},\tilde{p},\gamma)\overset{GH}\longrightarrow (Y,y,g).$$ Then there is a non-decreasing sequence $\{t_i\}$ in $\mathbb{Z}_+$ such that after passing to a subsequence, we have $$(r_i^{-1}\widetilde{M},\tilde{p},\gamma^{t_i},S(\gamma,t_i))\overset{GH}\longrightarrow (Y,y,\mathrm{id},S)$$ with $Sy=y$.* *Proof.* Note that $g$ fixes the base point $y$. If $g$ has finite order, we let $t\in \mathbb{Z}_+$ such that $g^t=\mathrm{id}$. Then it holds that $$(r_i^{-1}\widetilde{M},\tilde{p},\gamma^{t},S(\gamma,t))\overset{GH}\longrightarrow (Y,y,\mathrm{id},S)$$ with $S$ fixing $y$. It remains to consider the case that $g$ has infinite order. Since $g$ fixes $y$, $\langle g \rangle$ is a precompact subgroup of the isotropy subgroup at $y$. Let $t_j\to \infty$ be a sequence such that $g^{t_j}\to \mathrm{id}$ as $j\to\infty$. By a standard diagonal argument, we can choose a subsequence and $t_{i(j)}\to\infty$ such that $$(r_{i(j)}^{-1}\widetilde{M},\tilde{p},\gamma^{t_{i(j)}})\overset{GH}\longrightarrow (Y,y,\mathrm{id}).$$ Also, by construction, the limit of $S(\gamma,t_{i(j)})$ fixes $y$ with respect to the above convergence. ◻ **Lemma 51**. *Let $\gamma\in \mathcal{N}-\{\mathrm{id}\}$ and $r_i\to\infty$. We choose $t_i$ as in Lemma [Lemma 50](#reduction_id){reference-type="ref" reference="reduction_id"} and $$k_i:=\min\{l \in\mathbb{Z}_+\ |\ d(\gamma^{t_i l} \tilde{p},\tilde{p}) \ge r_i\}\to\infty.$$ Passing to a subsequence, we consider the convergence $$(r_i^{-1}\widetilde{M},\tilde{p},\gamma^{t_i k_i},S(\gamma^{t_i},k_i),\langle\gamma \rangle,\mathcal{N})\overset{GH}\longrightarrow (Y,y,g,A,H,G).$$ We denote $z=gy$. Then\ (1) $A$ is connected, in particular $A\subseteq G_0$;\ (2) $Ay$ contains the set $\{tz|t\in[0,1]\}$;\ (3) $Hy$ contains the one-parameter orbit $\mathbb{R}z$; moreover, there is a one-parameter subgroup $L$ of $H_0$ such that $Ly=\mathbb{R}$z.* *Proof.* (1) First note that by the choice of $k_i$ and the construction of $A$, it is clear that $Ay\subseteq \overline{B_1}(y)$. We argue by contradiction to prove that $A$ is connected. Suppose otherwise, then there is a point $w\in Y$ such that $Aw$ has multiple connected components. Let $\alpha w\in Aw$ be a point outside the connected component of $Aw$ that contains $w$. We choose a sequence of point $q_i\in\widetilde{M}$ and a sequence of integers $m_i$ within $[-k_i,k_i]$ such that $$(r_i^{-1}\widetilde{M},\tilde{p},q_i,\gamma^{t_im_i})\overset{GH}\longrightarrow (Y,y,w,\alpha).$$ Replacing $\alpha$ by $\alpha^{-1}$ if necessary, we can assume that $m_i>0$. Recall that $t_i$ is chosen in Lemma [Lemma 50](#reduction_id){reference-type="ref" reference="reduction_id"} such that $\gamma^{t_i}$ converges to identity under the convergence; in particular, we have $$r_i^{-1}d(\gamma^{t_i}q_i,q_i)\to 0$$ as $i\to\infty$. Since $\alpha$ moves $w$, we must have $m_i\to\infty$. Let $\epsilon\ll d(w,\alpha w)$. For each sufficiently large $i$, we choose an integer $s_i$ within $[1,m_i]$ such that $$\epsilon/2 \le r_i^{-1}d(\gamma^{t_is_i}q_i,q_i) \le \epsilon.$$ We consider the symmetric subset $$T_i=\langle \gamma^{t_is_i} \rangle \cap S(\gamma^{t_i}, m_i)$$ and its convergence $$(r_i^{-1}\widetilde{M},\tilde{p},q_i,T_i)\overset{GH}\longrightarrow (Y,y,q,T).$$ By construction and the compactness of $Tw\subseteq Aw \subseteq \overline{B_{1+2D}}(y)$, where $D=d(w,y)$, we can find an $\epsilon$-chain $\{y_0=y,y_1,....,y_N=\alpha w\}$ in $Tw\cup \{\alpha w\}\subseteq Aw$ such that $$d(y_j,y_{j+1})\le\epsilon$$ for all $j$. Because the small $\epsilon>0$ is arbitrary, this shows that $\alpha w$ belongs to the same connected component of $Aw$ as $w$; a contradiction. \(2\) Note that by the choice of $k_i$, it holds that $$r_i \le d(\gamma^{t_i k_i}\tilde{p},\tilde{p})\le r_i+d(\gamma^{t_i}\tilde{p},\tilde{p})$$ with $r_i^{-1}d(\gamma^{t_i}\tilde{p},\tilde{p})\to 0$. Thus $d(gy,y)=1$. By (1), we have $g\in A \subseteq G_0$. Below we write $\delta_i=\gamma^{t_i}$ for convenience. Let $b\ge 2$ be an integer. Due to the choice of $k_i$, we have $$d(\delta_i^{\lceil k_i/b \rceil}\tilde{p},\tilde{p})<r_i,$$ where $\lceil\cdot\rceil$ is the ceiling function. After passing to a subsequence, we can assume that $$b\cdot \lceil k_i/b\rceil=k_i+b_0$$ for all $i$, where $b_0\in\{0,1,...,b-1\}$, and the convergence $$(r_i^{-1}\widetilde{M},\tilde{p},\delta_i^{\lceil k_i/b \rceil},\delta_i^{b_0},\delta_i^{b\cdot \lceil k_i/b\rceil})\overset{GH}\longrightarrow (Y,y,\beta,\mathrm{id},g).$$ By construction, we have $\beta^b=g;$ moreover, $\beta\in A\subseteq G_0$. Let $\sigma:\mathbb{R}\to G_0$ be a one-parameter subgroup of $G_0$ with $\sigma(1)=\beta$. Then $$\eta:\mathbb{R}\to G_0\quad t\mapsto \sigma(bt)$$ defines a one-parameter subgroup of $G_0$ with $\eta(1)=\sigma(b)=g$. By Lemma [Lemma 48](#one_para_orbit){reference-type="ref" reference="one_para_orbit"}, we have $$\sigma(bt)y=\eta(t)y=tz$$ for all $t\in\mathbb{R}$, where $z=gy$. We set $t=m/b$, where $m\in \{0,\pm1,...,\pm (b-1)\}$. Then we have $$\frac{m}{b}z=\sigma(m)y=\beta^m y.$$ Note that $\beta^m$ is the limit of $\delta_i^{m\lceil k_i/b \rceil}\in S(\delta_i,k_i)$. Thus $\frac{m}{b}z \in Ay$ for all $b\ge 2$ and all $m\in \{0,\pm1,...,\pm (b-1)\}$. We conclude that $Ay$ must contain $\{tz|t\in[0,1]\}$. \(3\) Using the notations in the proof of (2), we choose $t=m/b$, where $m\in\mathbb{Z}$. Then similarly, we have $$\frac{m}{b}z=\beta^m y.$$ Since $\langle \beta\rangle \in H$, we conclude that the orbit $Hy$ contains the points $$\left\{\frac{m}{b}z|m\in\mathbb{Z}\right\}.$$ Because the integer $b\ge 2$ is arbitrary, $Hy$ must contain $\mathbb{R}z$. Lastly, we show that $\mathbb{R}z=Ly$ for a one-parameter subgroup $L$ of $H_0$. In fact, let $\mathcal{C}$ be the connected component of $Hy$ that contains $y$. Note that $$\mathbb{R}z\subseteq \mathcal{C} \subseteq H_0y.$$ Now the result follows from Lemma [Lemma 48](#one_para_orbit){reference-type="ref" reference="one_para_orbit"}. ◻ *Remark 52*. In the context of Lemma [Lemma 51](#limit_one_para){reference-type="ref" reference="limit_one_para"}, if a sequence $n_i$ satisfies $n_i\gg {t_i k_i}$, then with respect to the convergence $$(r_i^{-1}\widetilde{M},\tilde{p},S(\gamma,n_i))\overset{GH}\longrightarrow (Y,y,B),$$ $By$ must contain $\mathbb{R}z$. In fact, from the proof of Lemma [Lemma 51](#limit_one_para){reference-type="ref" reference="limit_one_para"}(2), we have $$(r_i^{-1}\widetilde{M},\tilde{p},\delta_i^{\lceil k_i/b \rceil})\overset{GH}\longrightarrow (Y,y,\beta),\quad \frac{m}{b}z=\beta^m y,$$ where $b\in\mathbb{Z}\cap [2,\infty)$ and $m\in\mathbb{Z}$. Together with $n_i\gg t_ik_i$, $\frac{m}{b}z$ is the limit of $\delta_i^{m\lceil k_i/b \rceil}\cdot \tilde{p}\in S(\gamma,n_i)\cdot \tilde{p}$. It follows that $By$ contains $\mathbb{R}z$. ## Proof of Proposition C(1) {#subsec_pf_c1} We prove Proposition C(1) in this subsection. For convenience, we restate it as follows by using the terminology from Section [5.2](#subsec_one_para_orbit){reference-type="ref" reference="subsec_one_para_orbit"}. **Proposition 53**. *Let $(M,p)$ be an open $n$-manifold with $\mathrm{Ric}\ge 0$ and $E(M,p)\not= 1/2$. Let $\mathcal{N}$ be a torsion-free nilpotent subgroup of $\pi_1(M,p)$ with finite index and let $l$ be the nilpotency step of $\mathcal{N}$.* *Then for every $\gamma\in \zeta_{l-1}(\mathcal{N})-\{\mathrm{id}\}$ and every $(Y,y,H,G)\in \Omega(\widetilde{M},\langle \gamma \rangle,\mathcal{N})$, the orbit $Hy$ is exactly the one-parameter orbit $\mathbb{R}z$, where $z\in Hy-\{y\}$. Moreover, the orbit $Hy$ can be represented by $\{\sigma(t)y|t\in \mathbb{R}\}$, where $\sigma:\mathbb{R}\to H_0 \subseteq Z(G)$ is a one-parameter subgroup through $g$ with $gy\in Hy$.* Two corollaries below follow directly from Proposition [Proposition 43](#topol_dim){reference-type="ref" reference="topol_dim"}(2). **Corollary 54**. *Let $(Y,y,H)\in\Omega(\widetilde{M},\langle \gamma \rangle)$ and let $h_1,h_2\in H$. Suppose that there is an integer $m\ge 2$ such that $h_1^m y=h_2^m y$, then $h_1y=h_2y$.* *Proof.* Let $r_i\to\infty$ such that $$(r_i^{-1}\widetilde{M},\tilde{p},\langle \gamma \rangle, \mathcal{N})\overset{GH}\longrightarrow (Y,y,H,G).$$ Since $\gamma\in Z(\mathcal{N})$, we have $H\subseteq Z(G)$. We first claim that if an element $h\in H$ satisfies $h^m y=y$ for some integer $m\ge 2$, then $hy=y$. In fact, let $H\subseteq G$ be the closure of the subgroup generated by $h$. By assumption, its orbit $Hy$ consists of at most $m$ many elements. In particular, $H\subseteq Z(G)$ is compact. By Proposition [Proposition 43](#topol_dim){reference-type="ref" reference="topol_dim"}(2), we conclude $Hy=y$. Now let $h_1, h_2\in H \subseteq Z(G)$ such that $h_1^my=h_2^m y$. We have $$(h_2^{-1}h_1)^m y=h_2^{-m} h_1^m y=y.$$ It follows from the Claim that $h_2^{-1}h_1y=y$. ◻ **Corollary 55**. *Let $(Y,y,H)\in\Omega(\widetilde{M},\langle \gamma \rangle)$ and let $S$ be a closed symmetric subset of $H$. Suppose that the set $Sy=\{sy|s\in S\}$ satisfies the following:\ (1) $Sy$ is closed under multiplication, that is, if $s_1,s_2\in S$, then $s_1s_2y\in Sy$;\ (2) $Sy$ is bounded.\ Then $Sy=\{y\}$.* *Proof.* Let $L={\langle S \rangle}$, the subgroup generated by $S$. By assumptions, $Ly=Sy$ is bounded. Thus the closure $\overline{L}$ is a compact subgroup of $H$. It follows from $H\subseteq Z(G)$ and Proposition [Proposition 43](#topol_dim){reference-type="ref" reference="topol_dim"}(2) that $\overline{L}$ must fix $y$. Thus $Sy=Ly=\{y\}$. ◻ We are ready to prove Proposition [Proposition 53](#c1){reference-type="ref" reference="c1"}. *Proof of Proposition [Proposition 53](#c1){reference-type="ref" reference="c1"}.* From Lemma [Lemma 51](#limit_one_para){reference-type="ref" reference="limit_one_para"}, we know that the limit orbit $Hy$ from $$(r_i^{-1}\widetilde{M},\tilde{p},\langle \gamma \rangle,\mathcal{N})\overset{GH}\longrightarrow (Y,y,H,G)$$ contains a one-parameter orbit $\mathbb{R}z$, where $z=gy$ is constructed in Lemma [Lemma 51](#limit_one_para){reference-type="ref" reference="limit_one_para"}. Below, we continue to use the notations from Lemma [Lemma 51](#limit_one_para){reference-type="ref" reference="limit_one_para"}; in particular, we have $$(r_i^{-1}\widetilde{M},\tilde{p}),\gamma^{t_i k_i},S(\gamma^{t_i},k_i))\overset{GH}\longrightarrow (Y,y,g,A)$$ with $$g\in A\subseteq G_0,\quad d(gy,y)=1,\quad Ay\subseteq \overline{B_1}(y),$$ where the sequences $t_i$ and $k_i$ are described in Lemmas [Lemma 50](#reduction_id){reference-type="ref" reference="reduction_id"} and [Lemma 51](#limit_one_para){reference-type="ref" reference="limit_one_para"}. We argue by contradiction to show that $Hy=\mathbb{R}z$. Suppose that there is $h\in H$ such that $hy\not\in \mathbb{R}z$. **Claim 1:** Without lose of generality, we can assume that $d(hy,\mathbb{R}z)\ge 2$. The element $h$ may not be in $G_0$. However, because $G$ has at most finitely many components (see Proposition [Proposition 43](#topol_dim){reference-type="ref" reference="topol_dim"}(3)), we can find a power $n\in\mathbb{Z}_+$ such that $h^n\in G_0$. For this $h^n$, we still have the property that $h^n y \notin \mathbb{R}z$. In fact, suppose that $h^n y\in \mathbb{R}z$. By Lemma [Lemma 51](#limit_one_para){reference-type="ref" reference="limit_one_para"}(3), we have a one parameter subgroup $\eta:\mathbb{R}\to H_0\subseteq Z(G)$ such that $\eta(1)=g_0\in H_0$ and $g_0y=h^n y$. Then by Lemma [Corollary 54](#orbit_power_back){reference-type="ref" reference="orbit_power_back"}, we see that $hy=\eta(1/n)y\in\mathbb{R}z$; a contradiction. Now that we have $h^n\in H\cap G_0$ with $h^n y\notin \mathbb{R}z$; next, we show that $d((h^n)^m y,\mathbb{R}z)$ is unbounded as $m\to \infty$. In fact, let $\sigma:\mathbb{R}\to G_0$ be a one-parameter subgroup such that $\sigma(1)y=z$. Let $\overline{L}$ be the subgroup generated by elements in $\sigma$ and $T$, the maximal torus subgroup of $G_0$. Because $T$ is central in $G_0$, each element $\overline{L}$ can be expressed as $\sigma(t)\cdot \theta$, where $\theta\in T$. Moreover, $\overline{L}y=\mathbb{R}z$ because $T$ fixes $y$ according to Proposition [Proposition 43](#topol_dim){reference-type="ref" reference="topol_dim"}; in particular, $h^n\notin \overline{L}$. By construction, the quotient group $G_0/\overline{L}$ is a connected and simply connected nilpotent Lie group and $G_0/\overline{L}$ acts on the quotient space $(Y/\overline{L},\bar{y})$. Let $q:G_0\to G_0/\overline{L}$ be the quotient homomorphism, then $q(h^n)$ generates a discrete $\mathbb{Z}$-subgroup in $G_0/\overline{L}$. Thus $d(q(h^n)^m\bar{y},\bar{y})$ is unbounded as $m\to\infty$. As a consequence, $d((h^n)^m y,\mathbb{R}z)$ is unbounded as $m\to \infty$. To this end, we choose $m$ such that $d((h^n)^m y,\mathbb{R}z)\ge 2$. Replacing $h$ by $h^{nm}$, we complete Claim 1. Let $m_i\in\mathbb{Z}$ such that $$(r_i^{-1}\widetilde{M},\tilde{p},\gamma^{m_i})\overset{GH}\longrightarrow (Y,y,h).$$ Replacing $h$ by $h^{-1}$ if necessary, we can assume that $m_i>0$. **Claim 2:** $m_i \gg t_i k_i$. By $d(hy,y)\ge 2$ and the choice of $k_i$, we clearly have $m_i>t_ik_i$. To prove Claim 2, suppose that $m_i/(t_ik_i)\to C<\infty$ for a subsequence, then we can write $$m_i=\lfloor C \rfloor t_ik_i +o_i,$$ where $\lfloor \cdot \rfloor$ is the floor function and $o_i\in \mathbb{Z}\cap [0,t_ik_i]$. Passing to a subsequence, we have convergence $$(r_i^{-1}\widetilde{M},\tilde{p},\gamma^{\lfloor C \rfloor t_ik_i},\gamma^{o_i},\gamma^{m_i})\overset{GH}\longrightarrow (Y,y,g^{\lfloor C \rfloor},\delta,h),$$ where $\delta\in A$. Since $g\in G_0$, by Corollary [Corollary 49](#one_para_trans){reference-type="ref" reference="one_para_trans"}, we have $$g^{\lfloor C \rfloor}\cdot(tz)=(\lfloor C \rfloor+t) z\in\mathbb{R}z$$ for all $t\in\mathbb{R}$. Consequently, $$d(hy,\mathbb{R}z)=d(g^{\lfloor C \rfloor}\delta y,\mathbb{R}z)=d(\delta y,\mathbb{R}z)\le 1;$$ A contradiction to $d(hy,\mathbb{R}z)\ge 2$. This proves Claim 2. For each $i$, we define $$d_i=\max\{d(\gamma^k\tilde{p},\tilde{p})\ |\ k\in \mathbb{Z}\cap [t_ik_i,m_i] \}.$$ **Claim 3:** $d_i \gg r_i$. It is clear that $d_i\ge r_i$. Suppose that $d_i/r_i\to C<\infty$ for a subsequence. Then we consider the convergence $$(d_i^{-1}\widetilde{M},\tilde{p},S(\gamma,m_i))\overset{GH}\longrightarrow (C^{-1}Y,y,B).$$ By the proof of Lemma [Lemma 51](#limit_one_para){reference-type="ref" reference="limit_one_para"} and $m_i\gg t_ik_i$, $By$ must contain $\mathbb{R}z$ (see Remark [Remark 52](#rem_sym_one_para){reference-type="ref" reference="rem_sym_one_para"}). Hence $By$ is unbounded. On the other hand, by the choice of $d_i$, we should have $By\subseteq \overline{B_1}(y)$; a contradiction. This proves Claim 3. Next, we consider the blow-down under $d_i\to\infty$: $$(d_i^{-1}\widetilde{M},\tilde{p},\gamma^{m_i},S(\gamma,m_i),\langle \gamma \rangle)\overset{GH}\longrightarrow (Y',y',h',B',H').$$ By the choice of $d_i$, we have $B'y'\subseteq \overline{B_1}(y')$. Also, note that Claim 3 implies that $h'y'=y'$. **Claim 4:** $B'y'$ is closed under multiplication. Let $\beta_1,\beta_2\in B'$. We shall show $\beta_1\beta_2y'\in B'y'$. We choose $b_{1,i},b_{2,i}\in \mathbb{Z}\cap[-m_i,m_i]$ such that $$(d_i^{-1}\widetilde{M},\tilde{p},\gamma^{b_{1,i}},\gamma^{b_{2,i}})\overset{GH}\longrightarrow (Y',y',\beta_1,\beta_2).$$ Then $\beta_1\beta_2$ is the limit of $\gamma^{b_{1,i}+b_{2,i}}$. If $b_{1,i}+b_{2,i}\in [-m_i,m_i]$, then $\beta_1\beta_2\in B'$ clearly holds. If not, we write $$b_{1,i}+b_{2,i}=\pm m_i + o_i,$$ where $o_i\in \mathbb{Z}\cap[-m_i,m_i]$. Passing to a subsequence if necessary, we have $$(d_i^{-1}\widetilde{M},\tilde{p},\gamma^{m_i},\gamma^{o_i})\overset{GH}\longrightarrow (Y',y',h',\beta_0),$$ where $\beta_0\in B'$. Then with respect to the blow-down sequence of $\widetilde{M}$ by $d_i^{-1}$, we have $$\beta_1\beta_2y'=\lim \gamma^{o_i}\gamma^{\pm m_i} \tilde{p}=\beta_0(h')^{\pm 1}y'=\beta_0 y'\in B'y'.$$ This proves Claim 4. Lastly, by Claim 4 and Corollary [Corollary 55](#bdd_sym_fix){reference-type="ref" reference="bdd_sym_fix"}, we conclude that $B'y'=\{y'\}$. On the other hand, the choice of $d_i$ implies that $d_H(B'y',y')=1$. This contradiction shows that $Hy=\mathbb{R}z$ and thus completes the proof of Proposition [Proposition 53](#c1){reference-type="ref" reference="c1"}. ◻ We complete this section by proving a distance control on the one-parameter orbit. **Lemma 56**. *Under the assumptions of Proposition C, there is a constant $C_1=C_1(\widetilde{M},\gamma)$ such that for any $(Y,y,H)\in\Omega(\widetilde{M},\langle\gamma \rangle)$ and any orbit point $z\in Hy-\{y\}$, we have $$d(tz,y)\le C_1\cdot d(z,y)$$ for all $t\in [0,1]$.* *Proof.* Scaling $(Y,y,H)$ by a constant, we may assume that $d(z,y)=1$. Recall that by Proposition [Proposition 53](#c1){reference-type="ref" reference="c1"} we can choose $h\in H_0$ such that $z=hy$. We argue by contradiction and suppose that there are contradicting sequences $(Y_j,y_j,H_j)\in \Omega(\widetilde{M},\langle \gamma \rangle)$ and $h_j\in (H_j)_0$ with $d(h_jy_j,y_j)=1$ but $$R_j:=\max_{t\in [0,1]} d(tz_j,y_j)\to\infty$$ as $j\to\infty$, where $z_j=h_jy_j\in Y_j$. For each $j$, we choose a one-parameter subgroup of $(H_j)_0$ through $h_j$ and use $t h_j$ to denote elements in the subgroup, where $t\in\mathbb{R}$. By Lemma [Lemma 48](#one_para_orbit){reference-type="ref" reference="one_para_orbit"}, $(th_j)y_j=tz_j$ for all $t\in\mathbb{R}$. We consider symmetric subsets $$S_j=\{ th_j |t\in[-1,1]\}$$ and the convergence $$(R_j^{-1}Y_j,y_j,H_j,S_j)\overset{GH}\longrightarrow (Y',y',H',S'),$$ where $(Y',y',H')\in\Omega(\widetilde{M},\langle\gamma\rangle)$. Since $R_j\to\infty$ and $d(h_jy_j,y_j)=1$, we have $h_jy_j\overset{GH}\to y'$ and $S'y'\subseteq \bar{B}_1(y')$ with respect to the above convergence. **Claim:** The set $S'y'$ is closed under multiplication. The proof is similar to Claim 4 in the proof of Proposition [Proposition 53](#c1){reference-type="ref" reference="c1"}. Let $\beta_1,\beta_2\in S'$ and let $b_{j,1},b_{j,2}\in [-1,1]$ such that $$(R_j^{-1}Y_j,y_j,b_{j,1}h_j,b_{j,2}h_j)\overset{GH}\longrightarrow (Y',y',\beta_1,\beta_2).$$ If $b_{j,1}+b_{j,2}\in[-1,1]$, then clearly $\beta_1\beta_2\in S'$. If not, we write $$b_{j,1}+b_{j,2}=\pm 1 + o_j,$$ where $o_j\in [-1,1]$. In a convergent subsequence, we have $$(R_j^{-1}Y_j,y_j,h_j,o_jh_j)\overset{GH}\longrightarrow (Y',y',h',\beta_0)$$ with $\beta_0\in S'$. Then with respect to this convergence, $$\beta_1\beta_2y'=\lim_{j\to\infty} (o_jh_j)\cdot(\pm h_j) y_j=\beta_0y'\in S'y'.$$ This proves the Claim. Together with Corollary [Corollary 55](#bdd_sym_fix){reference-type="ref" reference="bdd_sym_fix"}, we conclude $S'y'=y'$. On the other hand, by the construction of $S_j$ and $R_j$, the limit $S'y'$ should have a point with distance $1$ to $y'$. A contradiction. ◻ *Remark 57*. Let $z\in Hy \subseteq Gy$ as in Lemma [Lemma 56](#boomer_control){reference-type="ref" reference="boomer_control"} and let $d=d(z,y)$. Recall that by Proposition [Proposition 20](#C_tunnel){reference-type="ref" reference="C_tunnel"}, there is a tunnel $\sigma:[0,1]\to Gy$ from $y$ to $z$ that is contained in $\overline{B_{C_0 d}}(y)$. Lemma [Lemma 56](#boomer_control){reference-type="ref" reference="boomer_control"} shows that we can follow a specific tunnel, the one-parameter orbit, such that it is contained in $\overline{B_{C_1 d}}(y)$. # Hausdorff dimension and orbit distance estimates {#sec_haus} This section studies the Hausdorff dimension of the orbit $Hy$ in $(Y,y,H)\in \Omega(\widetilde{M},\langle \gamma \rangle)$ and proves Proposition C(2). In subsection [6.1](#subsec_haus_one_para){reference-type="ref" reference="subsec_haus_one_para"}, we prove distance controls on the orbit $Hy$ and show that the supremum of $\dim_{\mathcal{H}}(Hy)$ among all $(Y,y,H)\in \Omega(\widetilde{M},\langle \gamma \rangle)$ can be obtained (Proposition [Proposition 61](#dim_sup){reference-type="ref" reference="dim_sup"}). In subsection [6.2](#subsec_pf_c2){reference-type="ref" reference="subsec_pf_c2"}, we relate the Hausdorff dimension of $Hy$ to a lower bound on the orbit length (Proposition [Proposition 64](#orbit_length_lower){reference-type="ref" reference="orbit_length_lower"}) and then complete the proof of Proposition C(2). Lastly, we have a short subsection [6.3](#subsec_recover){reference-type="ref" reference="subsec_recover"} that relates Proposition C(2) to previous results on virtual abelianness [@Pan_esgap; @Pan_cone]. ## Hausdorff dimension of one-parameter orbits {#subsec_haus_one_para} We fix an element $\gamma\in \zeta_{l-1}(\mathcal{N})-\{\mathrm{id}\}$. From Proposition C(1), we know that for all $(Y,y,H)\in\Omega(\widetilde{M},\langle\gamma \rangle)$, the orbit $Hy$ is homeomorphic to $\mathbb{R}$. For each $(Y,y,H)\in\Omega(\widetilde{M},\langle\gamma \rangle)$, we choose an orbit point $z\in Hy$ with $d(z,y)=1$. The choice of such a point $z$ may not be unique since the orbit $Hy$ may cross $\partial B_1(y)$ multiple times. By Lemma [Lemma 56](#boomer_control){reference-type="ref" reference="boomer_control"}, we always have distance control $$d(tz,y)\le C_1$$ for all $t\in[0,1]$. For convenience, we denote $$\Omega(\widetilde{M},\langle \gamma \rangle,1)=\{(Y,y,H,z)|(Y,y,H)\in\Omega(\widetilde{M},\langle \gamma \rangle),z\in Hy\cap \partial B_1(y)\}.$$ Given $(Y,y,H,z)\in \Omega(\widetilde{M},\langle \gamma \rangle,1)$, for each $L\in\mathbb{Z}_+$, we define $$\mathcal{O}^L_{(Y,y,H,z)}=\{tz\ |\ t\in[0,1/L]\}\subseteq Hy.$$ If the space $(Y,y,H,z)$ is clear, we shall write $\mathcal{O}^L$ for simplicity. **Lemma 58**. *Let $C_1=C_1(\widetilde{M},\gamma)$ be the constant in Lemma [Lemma 56](#boomer_control){reference-type="ref" reference="boomer_control"}. Then the followings hold for all $(Y,y,H,z)\in \Omega(\widetilde{M},\langle\gamma \rangle,1)$ and all $L\in\mathbb{Z}_+$:\ (1) $\mathrm{diam}(\mathcal{O}^L)\le 2C_1^3L^{-1/n}$;\ (2) $(L+1)\cdot\mathrm{diam}(\mathcal{O}^{L+1})^s\le L\cdot \mathrm{diam}(\mathcal{O}^{L})^s+ C_1^s$, where $s\ge 1$.* *Proof.* (1) We write $r=d(\frac{1}{L}z,y)$. First note that by Lemma [Lemma 56](#boomer_control){reference-type="ref" reference="boomer_control"}, the points $\{\frac{j}{L}z\}_{j=0}^{L}$ are all contained in $\overline{B_{C_1}}(y)$. We claim that the points $\{\frac{j}{L}z\}_{j=0}^{L}$ are pairwise $C_1^{-1}r$-disjoint. In fact, suppose that there are $j_1<j_2$ in $\{0,1,...,L\}$ with $$d\left(\frac{j_1}{L}z,\frac{j_2}{L}z\right)< C_1^{-1}r.$$ Since $\mathbb{R}z$ is represented by the orbit of a one-parameter subgroup at $y$, we have $$d\left(\frac{j_1-j_2}{L}z,y\right)< C_1^{-1}r.$$ However, by Lemma [Lemma 56](#boomer_control){reference-type="ref" reference="boomer_control"}, $$r=d((1/L)z,y)\le C_1\cdot d\left(\frac{j_1-j_2}{L}z,y\right)<r.$$ A contradiction. This verifies the claim. Next, by a standard packing argument with respect to a limit renormalized measure on $Y$, we have $$L\le \left(\dfrac{C_1}{C_1^{-1}r/2}\right)^n=(2C_1^2)^{n}\cdot r^{-n}.$$ Thus $$\mathrm{diam}(\mathcal{O}^L)\le C_1\cdot r\le 2 C_1^{3}\cdot L^{-1/n}.$$ (2) It is clear that $$\mathrm{diam}(\mathcal{O}^{L+1})\le \mathrm{diam}(\mathcal{O}^{L})$$ because $\mathcal{O}^{L+1} \subseteq \mathcal{O}^{L}$. Thus $$\begin{aligned} (L+1)\cdot\mathrm{diam}(\mathcal{O}^{L+1})^s \le& L\cdot \mathrm{diam}(\mathcal{O}^{L})^s + \mathrm{diam}(\mathcal{O}^{L+1})^s\\ \le& L\cdot \mathrm{diam}(\mathcal{O}^{L})^s+C_1^s, \end{aligned}$$ where the last inequality holds because $\mathrm{diam}(\mathcal{O}^{L+1})\le C_1$. ◻ Recall that for a metric space $(X,d)$, we have definition $$\mathcal{H}^s_\delta(X)=\inf \left\{\sum_{j=1}^\infty r_j^s\ \bigg|\ X\subseteq \cup_{j=1}^\infty B_j, \text{ where each $B_j$ has diameter } r_j\le\delta\right\},$$ then $s$-dimensional Hausdorff measure and Hausdorff dimension of $X$ are defined by $$\mathcal{H}^s(X)=\lim_{\delta\to 0} \mathcal{H}^s_\delta(X),$$ $$\dim_{\mathcal{H}}(X)=\inf \{s>0| \mathcal{H}^s(X)=0\}=\sup \{s>0| \mathcal{H}^s(X)=\infty\}.$$ When $X$ is compact, we can use finite covers $\{B_j\}$ instead of countable ones to define $\mathcal{H}_\delta^s(X)$. Next, we use equal partitions of $\mathcal{O}^1$ to give an alternative way to calculate its Hausdorff dimension. **Definition 59**. We define $$\mathcal{E}^s(\mathcal{O}^1)=\liminf\limits_{L\to\infty} L\cdot \mathrm{diam}(\mathcal{O}^L)^s,$$ where $L$ takes values in $\mathbb{Z}_+$. **Lemma 60**. *Given $s\ge 1$, there is a constant $C_2=C_2(\widetilde{M},\gamma,s)>1$ such that $$\mathcal{H}^s(\mathcal{O}^1_{(Y,y,H,z)})\le\mathcal{E}^s(\mathcal{O}^1_{(Y,y,H,z)})\le C_2\cdot \mathcal{H}^s(\mathcal{O}^1_{(Y,y,H,z)})$$ holds for any $(Y,y,H,z)\in\Omega(\widetilde{M},\langle\gamma\rangle,1)$. As a consequence, $$\dim_{\mathcal{H}}(Hy)=\inf \{s>0| \mathcal{E}^s(\mathcal{O}^1)=0\}=\sup \{s>0| \mathcal{E}^s(\mathcal{O}^1)=\infty \}.$$* *Proof.* We first show that $\mathcal{H}^s(\mathcal{O}^1)\le \mathcal{E}^s(\mathcal{O}^1)$, which is straightforward. Let $L\in\mathbb{Z}_+$. Note that $\mathcal{O}^L$ has diameter at most $2C_1^3L^{-1/n}$ by Lemma [Lemma 58](#partition_dist_estimate){reference-type="ref" reference="partition_dist_estimate"}(1). For each $j=1,...,L$, we define $$\mathcal{B}_j=\{tz\ |\ t\in [(j-1)/L,j/L]\}.$$ Then $\{\mathcal{B}_j\}_{j=1}^L$ covers $\mathcal{O}^1$ and each $\mathcal{B}_j$ has diameter at most $2C_1^3 L^{-1/n}$. This shows that for $\delta=2C_1^3L^{-1/n}$, we have $$\mathcal{H}^s_\delta(\mathcal{O}^1) \le \sum_{j=1}^{L} \mathrm{diam}(\mathcal{B}_j)^s= L\cdot \mathrm{diam}(\mathcal{O}^L)^s.$$ Let $L\to\infty$, then $\delta\to 0$ and we conclude that $\mathcal{H}^s(\mathcal{O}^1)\le \mathcal{E}^s(\mathcal{O}^1)$. Next, we prove that $\mathcal{E}^s(\mathcal{O}^1)\le C_2\cdot \mathcal{H}^s(\mathcal{O}^1)$ for some constant $C_2(\widetilde{M},\gamma,s)$. Let $\delta>0$. Let $\{B_j\}_{j=1}^J$ be a cover of $\mathcal{O}^1$ such that each $B_j$ is contained in $\mathcal{O}^1$, has diameter $r_j\le \delta$, and $$\sum_{j=1}^J r_j^s \le 2 \cdot\mathcal{H}^s_\delta(\mathcal{O}^1).$$ By replacing $B_j$ by its closure, without loss of generality, we can assume that all $B_j$ are closed. For each $j$, let $B'_j\subseteq \mathcal{O}^1$ be the smallest connected closed subset that contains $B_j$; in other words, we put $$\alpha_j=\min\{t|tz\in B_j\},\quad \beta_j=\max\{t|tz\in B_j\},$$ and $$B'_j=\{tz|t\in[\alpha_j,\beta_j]\}.$$ Note that by Lemma [Lemma 56](#boomer_control){reference-type="ref" reference="boomer_control"}, $$\mathrm{diam}(B'_j)\le C_1\cdot d(\alpha_j z,\beta_j z)\le C_1r_j.$$ We further modify the cover $\{B'_j\}_{j=1}^{J}$ to a cover $\{D'_j\}_{j\in I}$ in the following way, where the index set $I$ has cardinality at most $J$ and the adjacent $D'_j$ and $D'_{j+1}$ have exactly one common point. We define the index set $$I=\{j=1,...,J | \text{$B'_j$ is not contained in $B'_m$ for any $m\not= j$}\}.$$ Clearly, $\{B'_j\}_{j\in I}$ still covers $\mathcal{O}^1$. We rearrange $\{B'_j\}_{j\in I}$ by the order of their left endpoints and then relabel the sets in $\{B'_j\}_{j\in I}$ to $\{D_j\}_{j=1}^{|I|}$. Let $$\alpha'_j=\min\{t|tz\in D_j\},\quad \beta'_j=\max\{t|tz\in D_j\};$$ then $\alpha'_j< \alpha'_{j+1}$, $\beta'_j<\beta'_{j+1}$ for all $j=1,...,|I|-1$ and $\beta'_{|I|}=1$. Now we define $$D'_j=\{tz|t\in[\alpha'_j,\alpha'_{j+1}]\}$$ for each $j=1,...,|I|-1$ and define the last one $D'_{|I|}=D_{|I|}$. By construction, $\{D'_j\}_{j=1}^{|I|}$ covers $\mathcal{O}^1$; moreover, each $D'_j$ is not a single point and the adjacent two share exactly one common point. Let $d_j=\mathrm{diam}(D'_j)$. By construction, each $D'_j$ is contained in $D_{j}$, and each $D_j$ is indeed the relabel of some unique $B'_{m}$, thus $$\sum_{j=1}^{|I|} d_j^s \le \sum_{j=1}^{J}\mathrm{diam}(B'_j)^s\le C_1^s\cdot \sum_{j=1}^{J} r_j^s\le 2C_1^s \cdot\mathcal{H}^s_\delta(\mathcal{O}^1).$$ For each $j= 1,...,|I|$, we put $$\tau_j=\max\{t|tz\in D'_j\}-\min\{t|tz\in D'_j\}.$$ Because two adjacent $D'_j$ and $D'_{j+1}$ share exactly one common point, we see that $\sum_{j=1}^{|I|}\tau_j=1$. Let $j_0\in\{1,...,|I|\}$ such that $$\dfrac{d_{j_0}^s}{\tau_{j_0}}=\min_{j\in I} \dfrac{d_j^s}{\tau_j}=:\rho.$$ We choose $L\in\mathbb{Z}_+$ with $1\le L\tau_{j_0}\le 2$. Then $$\begin{aligned} &L\cdot \mathrm{diam}(\mathcal{O}^L)^s \le L\cdot \mathrm{diam}(D'_{j_0})^s =L\cdot \rho \tau_{j_0}\\ \le&\ 2\rho=2\rho\sum_{j=1}^{|I|} \tau_j \le 2\sum_{j=1}^{|I|} d_j^s \le 4C_1^s \cdot\mathcal{H}^s_\delta(\mathcal{O}^1). \end{aligned}$$ By triangle inequality, $$L\cdot \mathrm{diam}(\mathcal{O}^L)\ge L\cdot d((1/L)z,y)\ge d(z,y)=1.$$ Thus $$L\ge \dfrac{1}{\mathrm{diam}(\mathcal{O}^L)}\ge \dfrac{1}{\mathrm{diam}(D'_{j_0})}\ge \dfrac{1}{C_1\delta}.$$ This means that the above chosen $L\to\infty$ as $\delta\to 0$. Let $\delta\to 0$; we conclude that $$\mathcal{E}^s(\mathcal{O}^1)\le 4C_1^s\cdot \mathcal{H}^s(\mathcal{O}^1).$$ ◻ **Proposition 61**. *$\mathcal{D}:=\sup\{\dim_{\mathcal{H}}(Hy)|(Y,y,H)\in\Omega(\widetilde{M},\langle\gamma \rangle)\}$ can be obtained.* *Proof.* If $\mathcal{D}=1$, then $\dim_{\mathcal{H}}(Hy)=1$ for all $(Y,y,H)\in\Omega(\widetilde{M},\langle\gamma\rangle)$ and the result holds trivially. Below, we assume that $\mathcal{D}>1$. Let $\{(Y_j,y_j,H_j,z_j)\}_j$ be a sequence of spaces in $\Omega(\widetilde{M},\langle\gamma \rangle,1)$ such that $$\lim_{j\to\infty} \dim_{\mathcal{H}}(H_jy_j)\to \mathcal{D},$$ Let $s\in(1,\mathcal{D})$, then $s<\dim_{\mathcal{H}}(H_jy_j)$ for all $j$ large. By Lemma [Lemma 60](#Hdim_equal_partition){reference-type="ref" reference="Hdim_equal_partition"}, this implies that $$L\cdot \mathrm{diam}(\mathcal{O}^L_{(Y_j,y_j,H_j,z_j)})^s \to \infty$$ as $L\to\infty$. Together with Lemma [Lemma 58](#partition_dist_estimate){reference-type="ref" reference="partition_dist_estimate"}(2), for each $j\in\mathbb{N}$, there is an integer $L_j$ such that\ (1) $L\cdot \mathrm{diam}(\mathcal{O}^L_{(Y_j,y_j,H_j,z_j)})^s \ge 2^j$ for all $L\ge L_j$, and\ (2) $L_j\cdot \mathrm{diam}(\mathcal{O}^{L_j}_{(Y_j,y_j,H_j,z_j)})^s\in [2^j,2^j+C_1^s].$\ Let $r_j=d((1/L_j)z_j,y_j)$. By Lemma [Lemma 58](#partition_dist_estimate){reference-type="ref" reference="partition_dist_estimate"}(1), $r_j\to 0$ as $j\to \infty$. Passing to a subsequence if necessary, we consider $$(r_j^{-1}Y_j,y_j,H_j,(1/L_j)z_j)\overset{GH}\longrightarrow (Y',y',H',z')\in \Omega(\widetilde{M},\langle\gamma \rangle,1).$$ Let $K\in \mathbb{Z}_+$. We estimate that $$\begin{aligned} r_j^{-1}\cdot\mathrm{diam}(\mathcal{O}^{KL_j}_{(Y_j,y_j,H_j,z_j)})\ge & \dfrac{\mathrm{diam}(\mathcal{O}^{KL_j}_{(Y_j,y_j,H_j,z_j)})}{\mathrm{diam}(\mathcal{O}^{L_j}_{(Y_j,y_j,H_j,z_j)})}\\ \ge & \dfrac{\left(\dfrac{2^j}{KL_j}\right)^{1/s}}{\left(\dfrac{2^j+C_1^s}{L_j}\right)^{1/s}} \to \left(\dfrac{1}{K}\right)^{1/s} \end{aligned}$$ as $j\to \infty$. Note that for any $K\in \mathbb{Z}_+$, it holds the convergence $$(r_j^{-1}Y_j,y_j,\mathcal{O}^{KL_j}_{(Y_j,y_j,H_j,z_j)})\overset{GH}\longrightarrow (Y',y',\mathcal{O}^K_{(Y',y',H',z')}).$$ Thus $$K\cdot \mathrm{diam}(\mathcal{O}^K_{(Y',y',H',z')})^s\ge K \cdot \dfrac{1}{K}=1$$ for any integer $K\ge 1$. This shows that $\dim_{\mathcal{H}}(H'y')\ge s$ by Lemma [Lemma 60](#Hdim_equal_partition){reference-type="ref" reference="Hdim_equal_partition"}. Since $s$ is arbitrarily chosen within $(1,\mathcal{D})$, it follows that $\dim_{\mathcal{H}}(H'y')=\mathcal{D}$. ◻ ## Proof of Proposition C(2) {#subsec_pf_c2} We continue to use the notation $\mathcal{D}$ as in Proposition [Proposition 61](#dim_sup){reference-type="ref" reference="dim_sup"}. **Lemma 62**. *Let $s>\mathcal{D}$ and $\epsilon>0$. Then there is a constant $L_0=L_0(\epsilon,s,\widetilde{M},\gamma)$ such that for all $(Y,y,H,z)\in\Omega(\widetilde{M},\langle \gamma \rangle,1)$, there exists an integer $2\le L\le L_0$ with $$L\cdot \mathrm{diam}(\mathcal{O}^L_{(Y,y,H,z)})^s\le\epsilon.$$* *Proof.* We argue by contradiction. Suppose that for each integer $L_j=j$, there is some space $(Y_j,y_j,H_j,z_j)\in \Omega(\widetilde{M},\langle \gamma \rangle,1)$ such that $$L\cdot \mathrm{diam}(\mathcal{O}^L_{(Y_j,y_j,H_j,z_j)})^s > \epsilon$$ for all $2\le L\le L_j$. After passing to a subsequence, we consider $$(Y_j,y_j,H_j,z_j)\overset{GH}\longrightarrow (Y',y',H',z').$$ For any integer $L\ge 2$, we observe that $$L\cdot\mathrm{diam}(\mathcal{O}^L_{(Y',y',H',z')})^s=\lim\limits_{j\to\infty} L\cdot \mathrm{diam}(\mathcal{O}^L_{(Y_j,y_j,H_j,z_j)})^s\ge \epsilon.$$ Thus $\dim_{\mathcal{H}}(H'y')\ge s$ by Lemma [Lemma 60](#Hdim_equal_partition){reference-type="ref" reference="Hdim_equal_partition"}, which is a contradiction to $s>\mathcal{D}$. ◻ Below we write $|\gamma|=d(\gamma \tilde{p},\tilde{p})$ for convenience. Next, we transfer Lemma [Lemma 62](#uniform_partition){reference-type="ref" reference="uniform_partition"} to a distance estimate of $\langle\gamma\rangle$-action on $\widetilde{M}$. **Lemma 63**. *Let $s>\mathcal{D}$. Then there are constants $L'=L'(s,\widetilde{M},\gamma)$ and $R=R(s,\widetilde{M},\gamma)$ such that the following holds.* *For any $\gamma^b\in\langle\gamma\rangle$ with $|\gamma^b|\ge R$, where $b\in\mathbb{Z}_+$, there exists an integer $2\le L\le L'$ such that $$|\gamma^b|\ge L^{1/s}\cdot |\gamma^{\lceil b/L \rceil}|,$$ where $\lceil \cdot \rceil$ means the ceiling function.* *Proof.* We choose $L'(s,\widetilde{M},\gamma)=L_0(\frac{1}{2},s,\widetilde{M},\gamma)$, the constant in Lemma [Lemma 62](#uniform_partition){reference-type="ref" reference="uniform_partition"}. We argue by contradiction to prove the statement. Suppose that there is a sequence $b_i\to\infty$ such that $$|\gamma^{b_i}|\le L^{1/s}\cdot |\gamma^{\lceil b_i/L\rceil}|$$ for all $L=2,...,L'.$ Let $r_i=|\gamma^{b_i}|\to\infty$, we consider $$(r_i^{-1}\widetilde{M},\tilde{p},\langle\gamma\rangle,\gamma^{b_i})\overset{GH}\longrightarrow (Y,y,H,g).$$ It is clear that $g\in H$ satisfies $d(gy,y)=1$. We put $z=gy$. We claim that for each integer $L\in\mathbb{Z}_+$, we have convergence $$(r_i^{-1}\widetilde{M},\tilde{p},\gamma^{\lceil b_i/L \rceil}\tilde{p}) \overset{GH}\to (Y,y,(1/L)z).$$ In fact, we can write $$\lceil b_i/L \rceil\cdot L=b_i+o_i,$$ where $o_i\in \{0,1,...,L-1\}$. After passing to a subsequence, we have $$(r_i^{-1}\widetilde{M},\tilde{p},\gamma^{\lceil b_i/L \rceil},\gamma^{o_i})\overset{GH}\longrightarrow (Y,y,h,\delta)$$ with $h^L=g \delta$; moreover, $\delta y=y$ because each $o_i$ is at most $L-1$. Thus we have $$h^L y= g\delta y= gy=z.$$ By Lemma [Lemma 51](#limit_one_para){reference-type="ref" reference="limit_one_para"}(3), we have a one-parameter subgroup $\sigma:\mathbb{R}\to H_0$ with $\sigma(1)=g'$ and $g'y=z$. Then by Corollary [Corollary 54](#orbit_power_back){reference-type="ref" reference="orbit_power_back"} and the fact that $h^L y=\sigma(1) y$, we conclude that $$hy=\sigma(1/L)y=(1/L)z.$$ This proves the Claim. From the above Claim and the hypothesis, we have $$\mathrm{diam}(\mathcal{O}^L_{(Y,y,H,z)})\ge d((1/L)z,y)=\lim\limits_{i\to\infty} \dfrac{d(\gamma^{\lceil b_i/L \rceil}\tilde{p},\tilde{p})}{d(\gamma^{b_i}\tilde{p},\tilde{p})}\ge \left(\dfrac{1}{L}\right)^{1/s}$$ for all $L\in\{2,...,L'\}$. On the other hand, by the choice $L'=L_0(\frac{1}{2},s,\widetilde{M},\gamma)$ and Lemma [Lemma 62](#uniform_partition){reference-type="ref" reference="uniform_partition"}, $$\mathrm{diam}(\mathcal{O}^L_{(Y,y,H,z)})\le \left(\dfrac{1}{2L}\right)^{1/s}$$ for some $L\in\{2,...,L'\}$. A contradiction. ◻ Then we use Lemma [Lemma 63](#length_growth_estimate){reference-type="ref" reference="length_growth_estimate"} repeatedly to derive a lower bound for $|\gamma^b|$. **Proposition 64**. *Let $s>\mathcal{D}$. Then there is a constant $C_3=C_3(s,\widetilde{M},\gamma)$ such that $$|\gamma^b|\ge C_3\cdot b^{1/s}$$ for all $b\in \mathbb{Z}_+$ large.* *Proof of Proposition C(2).* Let $s>\mathcal{D}$ and let $P_0$ be a large integer such that $$|\gamma^b|\ge R(s,\widetilde{M},\gamma)$$ holds for all $b\ge P_0$, where $R(s,\widetilde{M},\gamma)$ is the constant in Lemma [Lemma 63](#length_growth_estimate){reference-type="ref" reference="length_growth_estimate"}. Let $b>P_0$. By Lemma [Lemma 63](#length_growth_estimate){reference-type="ref" reference="length_growth_estimate"}, there is some integer $L_1\in\{2,...,L'\}$ such that $$|\gamma^b|\ge L_1^{1/s}\cdot |\gamma^{\lceil b/L_1 \rceil}|.$$ If $\lceil b/L_1 \rceil\le P_0$, then we stop here. Otherwise, we apply Lemma [Lemma 63](#length_growth_estimate){reference-type="ref" reference="length_growth_estimate"} again to find some integer $L_2\in \{2,...,L'\}$ such that $$|\gamma^b|\ge L_1^{1/s}\cdot |\gamma^{\lceil b/L_1 \rceil}|\ge (L_1L_2)^{1/s}\cdot |\gamma^{\lceil\lceil b/L_1 \rceil/L_2\rceil}|.$$ Repeating this process, we eventually obtain $$|\gamma^b|\ge \left( \prod_j L_j \right)^{1/s} \cdot |\gamma^{\lceil...\lceil b/L_1\rceil/L_2.../L_k\rceil}|\ge \left( \prod_j L_j \right)^{1/s} \cdot r_0,$$ where $\lceil...\lceil b/L_1\rceil/L_2.../L_k\rceil\le P_0$ and $r_0=\min_{m\in \mathbb{Z}}|\gamma^m|$. Note that $$\dfrac{b}{\prod_j L_j}\le \lceil...\lceil b/L_1\rceil/L_2.../L_k\rceil \le P_0.$$ It follows that $$|\gamma^b|\ge \left(\dfrac{b}{P_0}\right)^{1/s}\cdot r_0=C_3\cdot b^{1/s},$$ where $C_3=r_0/P_0^{1/s}$. ◻ As indicated in the introduction, Proposition C(2) follows immediately from Proposition [Proposition 64](#orbit_length_lower){reference-type="ref" reference="orbit_length_lower"} and Corollary [Corollary 10](#orbit_length_upper){reference-type="ref" reference="orbit_length_upper"}. *Proof of Proposition C(2).* Let $s>\mathcal{D}$. By Proposition [Proposition 64](#orbit_length_lower){reference-type="ref" reference="orbit_length_lower"}, we have a lower bound $$|\gamma^b|\ge C_3\cdot b^{1/l}$$ for all $b\in \mathbb{Z}_+$ large. On the other hand, by Corollary [Corollary 10](#orbit_length_upper){reference-type="ref" reference="orbit_length_upper"}, we have an upper bound $$|\gamma^b|\le C_4\cdot b^{1/l}$$ for all $b\in\mathbb{Z}_+.$ Combining these two inequalities together, we obtain $$C_3\cdot b^{1/s}\le C_4\cdot b^{1/l}$$ for all $b$ large. We conclude that $s\ge l$. Recall that $s>\mathcal{D}$ is arbitrary, thus $\mathcal{D}\ge l$. Lastly, by Proposition [Proposition 61](#dim_sup){reference-type="ref" reference="dim_sup"}, there exists $(Y,y,H)\in\Omega(\widetilde{M},\langle \gamma \rangle)$ such that $$\dim_{\mathcal{H}}(Hy)=\mathcal{D}\ge l.$$ This completes the proof. ◻ ## Relations to previous results on virtual abelianness {#subsec_recover} As indicated in the introduction, Proposition C(2) immediately recovers the main result on metric cones and virtual abelianness in [@Pan_cone]. Recall that an open manifold with $\mathrm{Ric}\ge 0$ is *conic at infinity*, if every asymptotic cone $(Y,y)$ is a metric cone with vertex $y$. **Corollary 65**. *Let $(M,p)$ be an open $n$-manifold with $\mathrm{Ric}\ge 0$ and $E(M,p)\not=1/2$. If its Riemannian universal cover is conic at infinity, then $\pi_1(M)$ is virtually abelian.* *Proof.* We shall show that $$\dim_{\mathcal{H}}(Ly)=1$$ for any $(Y,y)\in \Omega(\widetilde{M})$ and any closed $\mathbb{R}$-subgroup $L$ of $\mathrm{Isom}(Y)$. Then by Proposition C(2), we have $$\mathrm{step}(\pi_1(M))\le 1,$$ that is, $\pi_1(M)$ is virtually abelian (see Definition [Definition 8](#def_virnilstep){reference-type="ref" reference="def_virnilstep"}). The verification of $\dim_{\mathcal{H}}(Ly)=1$ is standard given the metric cone structure. We give some details below for readers' convenience. Let $(Y,y)$ be any asymptotic cone of $\widetilde{M}$. By assumption, $Y$ is a metric cone with vertex $y$. By Cheeger-Colding splitting theorem, $(Y,y)$ splits isometrically as $$(\mathbb{R}^k \times C(Z),(0,z)),$$ where $k\in\mathbb{N}\cap [0,n]$, $C(Z)$ is a metric cone without lines, and $z$ is the unique vertex of $C(Z)$. Moreover, its isometry group also splits as a product $$\mathrm{Isom}(Y)=\mathrm{Isom}(\mathbb{R}^k) \times \mathrm{Isom} (C(Z)).$$ Because $z$ is the unique vertex of $C(Z)$, any isometry of $C(Z)$ must fix $z$. As a consequence, $$g\cdot y=g\cdot (0,z)\in \mathbb{R}^k \times \{z\}$$ for any isometry $g$ of $Y$. In particular, we have the orbit $$Ly \subseteq \mathbb{R}^k \times \{z\}$$ for any closed $\mathbb{R}$-subgroup $L$ of $\mathrm{Isom}(Y)$. Thus we can view $Ly$ as a $C^1$-curve in the Euclidean factor $\mathbb{R}^k \times \{z\}$. Consequently, $Ly$ must have Hausdorff dimension $1$. ◻ Proposition C(2) also extends the main result on small escape rate and virtual abelianness in [@Pan_esgap]. To explain this, we prove a proposition below, which is based on the results in this paper and [@Pan_esgap]. We continue to use the notation $\mathcal{D}$ from Proposition [Proposition 61](#dim_sup){reference-type="ref" reference="dim_sup"}. **Proposition 66**. *Let $M$ be an open $n$-manifold with $\mathrm{Ric}\ge 0$ and $E(M,p)\le \epsilon$, where $\epsilon>0$ is a small number. Then $$\mathcal{D}\le 1+\delta(\epsilon|n),$$ where $\delta(\epsilon|n)\to 0$ as $\epsilon\to 0$.* *Proof.* Let $\delta>0$. Suppose that $\mathcal{D}=1+\delta$. We set $s=1+\delta/2$. Following the proof of Proposition [Proposition 61](#dim_sup){reference-type="ref" reference="dim_sup"}, we can find a space $(Y,y,H,z)\in \Omega(\widetilde{M},\langle \gamma \rangle,1)$ such that $$K\cdot \mathrm{diam}(\mathcal{O}^K)^s\ge 1$$ for any integer $K\ge 1$. On the other hand, using the small escape rate condition, it follows from [@Pan_esgap Theorem 0.1 and Lemma 4.6] that $$d_{GH}((Y,y,H,z),(\mathbb{R}^k\times X,(0,x),\mathbb{R},(1,x)))\le \Psi(\epsilon|n),$$ where $\times$ means a metric product and the group $\mathbb{R}$ acts as translations in $\mathbb{R}^k \times X$. For a fixed $\delta>0$ and $s=1+\delta/2$, we can choose a large integer $K$ such that $$2^s K^{1-s} \le 1/2.$$ When $\epsilon$ is so small that $\Psi(\epsilon|n)\le 1/K$, we have $$1 \le K\cdot \mathrm{diam}(\mathcal{O}^K_{(Y,y,H,z)})^s \le K\cdot (\frac{1}{K}+\Psi)^s\le 2^s K^{1-s} \le 1/2.$$ This clear contradiction shows that $\delta \to 0$ as $\epsilon \to 0$. ◻ Combining Proposition [Proposition 66](#cor_small_escape){reference-type="ref" reference="cor_small_escape"} and Proposition C(2), it is clear that $\mathrm{step}(\pi_1(M))\ge 2$ implies $E(M,p)>\epsilon(n)$ for some universal constant $\epsilon(n)$, which is the main result in [@Pan_esgap]. 10 H. Bass. *The degree of polynomial growth of finitely generated nilpotent groups.* Proc. London Math. Soc. (3)25, 603--614, 1972. I. Belegradek and G. Wei. *Metrics of positive Ricci curvature on vector bundles over nilmanifolds*. Geom. Funct. Anal., 12, 56-72, 2002. E. Bruè, A. Naber, and D. Semola. *Fundamental groups and the Milnor conjecture.* arXiv:2303.15347. J. Cheeger and T. H. Colding. *Lower bounds on Ricci curvature and the almost rigidity of warped products.* Ann. of Math.(2) 144, no.1, 189-237, 1996. J. Cheeger and T. H. Colding. *On the structure of spaces with Ricci curvature bounded from below. I.* J. Differential Geom., 46, no.3, 406-480, 1997. J. Cheeger and D. Gromoll. *On the structure of complete manifolds of nonnegative curvature.* Ann. of Math. (2), 96(3):413-443, 1972. T. H. Colding and A. Naber. *Sharp Hölder continuity of tangent cones for spaces with a lower Ricci curvature bound and applications.* Ann. of Math. (2), 176:1173-1229, 2012. X. Dai, S. Honda, J. Pan, and G. Wei. *Singular Weyl's law with Ricci curvature bounded.* To appear in Trans. Amer. Math. Soc. Ser. B, arXiv:2208.13962. K. Fukaya. *Theory of convergence for Riemannian orbifolds.* Japan J. Math. 12, 121-160, 1986. Y. Guivarc'h. *Croissance polynomiale et périodes des fonctions harmoniques.* Bull. Soc. Math. France 101, 333--379, 1973. K. Fukaya and T. Yamaguchi. *The fundamental groups of almost nonnegatively curved manifolds.* Ann. of Math(2), 136, no.2, 253-333, 1992. M. Gromov. *Groups of polynomial growth and expanding maps.* Publications mathematiques I.H.É.S. 53(1981), 53-75. L. Guth. *Metaphors in systolic geometry.* in Proceedings of the International Congress of Mathematics, Vol. II. Hindustan Book Agency, New Delhi, 745--768, 2010. A. Hatcher. *Algebraic Topology.* Cambridge University Press, Cambridge, 2002. J. Hilgert and K. H. Neeb. *Structure and geometry of Lie groups.* Springer Monogr. Math. Springer, New York, 2012. M. I. Kargapolov and J. I. Merzljakov. *Fundamentals of the theory of groups*. Springer-Verlag, New York-Berlin, 1979. V. Kapovitch and B. Wilking. *Structure of fundamental groups of manifolds of Ricci curvature bounded below.* arXiv:1105.5955. J. Milnor. *A note on curvature and the fundamental group.* J. Differential Geom., 2:1-7, 1968. J. Pan. *Nonnegative Ricci curvature, stability at infinity, and finite generation of fundamental groups.* Geom. & Topol., 23, no. 6:3203--3231, 2019. J. Pan. *Nonnegative Ricci curvature, almost stability at infinity, and structure generation of fundamental groups.* arXiv:1809.10220. J. Pan. *On the escape rate of geodesic loops in an open manifold with nonnegative Ricci curvature.* Geom. & Topol., 25, no.2:1059-1085, 2021. J. Pan. *Nonnegative Ricci curvature and escape rate gap.* J. Reine Angew. Math., 782, 175-196, 2022. J. Pan. *Nonnegative Ricci curvature, metric cones, and virtual abelianness.* To appear in Geom. & Topol., arXiv:2201.07852. J. Pan and G. 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arxiv_math

--- abstract: | We study the perturbed Sobolev space $H^{1,r}_\alpha$, $r \in (1,\infty),$ associated with singular perturbation $\Delta_\alpha$ of Laplace operator in Euclidean space of dimension $2.$ The main results give the possibility to extend the $L^2$ theory of perturbed Sobolev space to the $L^r$ case. When $r \in (2,\infty)$ we have appropriate representation of the functions in $H^{1,r}_\alpha$ in regular and singular part. An application to local well - posedness of the NLS associated with this singular perturbation in the mass critical and mass supercritical cases is established too. address: - Department of Mathematics, University of Pisa, Largo Bruno Pontecorvo 5, I - 56127 Pisa, Italy - Faculty of Science and Engineering, Waseda University, 3-4-1, Okubo, Shinjuku-ku, Tokyo 169-8555, Japan - Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev Str., Block 8, Sofia, 1113, Bulgaria - Department of Mathematics, University of Pisa, Largo Bruno Pontecorvo 5, I - 56127 Pisa, Italy author: - Vladimir Georgiev - Mario Rastrelli bibliography: - SING_PERT_GR.bib title: Sobolev spaces for singular perturbation of Laplace operator --- # Introduction and basic definitions We study the singular-perturbed Laplacian $-\Delta_{\alpha}$, $\alpha\in \mathbb{R}$, on $L^2(\mathbb{R}^2)$, that is a delta-like perturbation of the free Laplacian in $\mathbb{R}^2$. This operator describes a zero-range interaction between particles or the presence of an impurity. The parameter $\alpha$ expresses, in suitable units, the inverse scattering length of the interaction supported at $x_0=0$ and for $\alpha=\infty$ it is the classical Laplacian on $L^2.$ We mention that $-\Delta_{\alpha}$ is a non-trivial self-adjoint extension on $L^2$ of the symmetric operator $$-\Delta|_{C^\infty_0(\mathbb{R}^2\setminus\{0\})}.$$ This is today a well-known one parameter class of operators, since the first rigorous attempt [@BF61] by Berezin and Faddeev in 1961, the seminal work [@AH81] by Albeverio and Høegh-Krohn in 1981, and subsequent characterizations by many others authors. In dimension $d=2,3$, the explicit characterization of the domain $\mathcal{D}(-\Delta_\alpha)$ is well-known and consists of functions that can be decomposed in a $H^2$ function plus a singular one, the Green function. The similarities with the Sobolev spaces gave the name of the perturbed Sobolev space $H^2_\alpha=\mathcal{D}(-\Delta_\alpha)$. In [@GMS18] there is a focus on the operator $(-\Delta_\alpha)^{s/2},$ with $s\in (0,2),$ that provides the definition of the singular-perturbed Sobolev Spaces $H^s_\alpha=\mathcal{D}((-\Delta_\alpha)^{s/2})$. These spaces allow the study of PDEs like nonlinear Schrödinger equations (NLSE) with a point interaction, thanks to the develop of tools like Strichartz estimates [@DMSY18; @IS17]. One of the basic motivation of our work is to develop some missing tools of harmonic analysis needed to study dispersive equations associated with $-\Delta_{\alpha}$. For example, the classical Nonlinear Schrödinger equation (NLSE) $$\label{eq.NLSEfr} ( i \partial_t + \Delta) u =\mu u|u|^{p-1}, \ p >1, \mu = \pm 1,$$ is a well studied object and we can immediately refer to the book of T. Cazenave [@c] and list the following basic tools used to obtain local and global well - posedness for [\[eq.NLSEfr\]](#eq.NLSEfr){reference-type="eqref" reference="eq.NLSEfr"}: 1. Strichartz estimates for the linear Schrödinger group $e^{it \Delta}$ using $L^q(0,T)L^r(\mathbb{R}^2)$ spaces with admissible couples $(q,r);$ 2. systematic use of the Sobolev spaces $W^{1,p}(\mathbb{R}^2) = H^{1,p}(\mathbb{R}^2)$ and application of Strichartz estimates using these spaces. As a typical example we can recall the following estimate of the composition operator $$\label{eq.co100} \|u|u|^{p-1}\|_{H^{1,r}(\mathbb{R}^2)} \lesssim \|u\|^p_{H^1(\mathbb{R}^2)}$$ for $r<2$ and $r$ sufficiently close to $2.$ While Strichartz estimates for the Schrödinger group $e^{it \Delta_\alpha}$ are obtained recently by establishing the existence and completeness of the wave operators in $L^r$( see [@DMSY18], [@CMY19], [@CMY19b]), to our knowledge there is a lack of results on the definition and properties of Sobolev spaces $H^{1,r}_\alpha(\mathbb{R}^2)$ associated with the perturbed operator $\Delta_\alpha.$ Let us mention that the necessity to have $L^p$ version of classical Sobolev spaces $H^{1,p}$ is dicussed in [@GMS22], [@GM22Ob], [@FGI22]. Our Theorems [Theorem 3](#t.2.1){reference-type="ref" reference="t.2.1"} and [Theorem 4](#t.2.2){reference-type="ref" reference="t.2.2"} will show a new description of $H^{1,p}_\alpha$ in dimension $d=2$, moving the focus also into $L^p$ spaces. This is crucial because it allows to gain new Strichartz estimates that involve the spaces $H^{1,p}_\alpha$ and the energy space $H^1_\alpha$. Now the contraction argument is available and we can give a new proof of local well - posedness of the following NLSE: $$\label{eq.NLSE} ( i \partial_t + \Delta_\alpha) u =\mu u|u|^{p-1}, \ p >1, \mu = \pm 1.$$ The key estimate we use in this local existence result is the following variant of [\[eq.co100\]](#eq.co100){reference-type="eqref" reference="eq.co100"} $$\label{eq.co119} \|u|u|^{p-1}\|_{H^{1,r}_\alpha(\mathbb{R}^2)} \lesssim \|u\|^p_{H^1_\alpha(\mathbb{R}^2)}$$ for $r<2$ and $r$ sufficiently close to $2.$ ## Overview on existing results First we give a short description on results treating the Strichartz estimates. for $-\Delta_\alpha$ in dimension $d=2.$ In the case of dimension $d=2$ and when we have $N$ singularities, the existence and completeness on $L^2(\mathbb R^2)$ of wave operators $W_{\alpha,N}^\pm$ is well-known thanks to Kato-Birman-Rosenblum theorem [@KATO66]. Using this result, Cornean, Michelangeli and Yajima in [@CMY19; @CMY19b] defined a condition of regularity for the singularities and proved that, under this condition (always fulfilled for $N=1$, that is our case), the wave operators are bounded in $L^p$ for every $p$. As a corollary they obtained the $L^{p^\prime}-L^{p}$ estimates without weights for every $p\in [2,\infty)$ and, immediately after, the Strichartz estimates: $$\begin{aligned} & \left\| e^{i t \Delta_\alpha} P_{ac}f \right\|_{L^q(\mathbb{R}_t)L^p(\mathbb{R}^2)} \lesssim \|f\|_{L^2}, \\ & \left\|\int_0^t e^{i(t-\tau)\Delta_\alpha} P_{ac}F(\tau) d\tau\right\|_{L^q(\mathbb{R}_t)L^p(\mathbb{R}^2)} \lesssim \left\| F\right\|_{L^{{s}'}(\mathbb{R}_t)L^{{r}'}(\mathbb{R}^2) }, \end{aligned}$$ where the couples $(p,q)$ and $(s,r)$ are 2-dimensional Strichartz exponents, i.e. $$\frac{1}{p}+\frac{1}{q}=\frac{1}{2},\ \ 2<q\leq \infty.$$ For $N=1$ and the explicit structure of the absolutely continuous subspace for $-\Delta_\alpha$ allows to generalize the above inequalities local in time, without the orthogonal projection. These results, with the perturbed Sobolev spaces develop, led to an in intense research in PDEs. For the case $d=3$ we refer to [@DPT06] and [@IS17]. In [@DMSY18] Dell'Antonio, Michelangeli, Scandone and Yajima proved that the wave operators associated to the pair $(-\Delta_\alpha,-\Delta)$, defined by the following strong limits, $$W_{\alpha,N}^\pm=\lim_{t\to \pm\infty}e^{it\Delta_{\alpha,N}}e^{-it\Delta}$$ exist, they are complete in $L^2(\mathbb{R}^3)$ and are bounded on $L^q(\mathbb{R}^3)$ for $1<q<3$. As a consequence, the weighted estimate $$\label{eq.APT} \|w^{-1}e^{it\Delta_{\alpha,N}}P_{ac}f\|_{L^\infty(\mathbb{R}^3)}\leq {C}{t^{-\frac{3}{2}}}\|wf\|_{L^1(\mathbb{R}^3)}$$ from [@DPT06] can be extended in the $L^p-L^q$ setting with $$\label{eq.IS} \|e^{it\Delta_{\alpha,N}}P_{ac}f\|_{L^{p}(\mathbb{R}^3)}\leq {C}{t^{-\frac{3}{2}\left(\frac{1}{p^\prime}-\frac{1}{p}\right)}}\|f\|_{L^{p^\prime}(\mathbb{R}^3)}\ \ \mbox{for} \ \ p\in[2,3).$$ The local well - posedness is a necessary element in the study of standing waves. Let us give a brief overview on the results in this direction. The existence of standing waves for the 2d Hartree type equation with point interaction is studied in [@GMS22b] $$i\partial_tu=-\Delta_\alpha u +(w*|u|^2)u,$$ where $w$ is a real-valued measurable function. The existence and stability of standing waves for power nonlinearity is studied in [@FGI22] in all possible cases: mass critical, mass subcritical and subcritical ones. In alternative way in the mass subcritical case the ground states can be obtained by looking for constraint minimization of the energy. This approach is developed in [@ABCT]. The 3d case can be seen in [@ABCT22]. Each of the works treating existence and stability/instability of standing waves need appropriate local well posedness in energy space. In all of them this is dome by appropriate modification of Cazenave approach by using a compactness argument. It is not difficult (using only Strichartz estimates to establish local existence and uniqueness of a solution in $C([0,T]; L^2)$ in the mass sub-critical case (see in Theorem [Theorem 5](#t.le19){reference-type="ref" reference="t.le19"} below). The main novelty is the prove of the local existence and uniqueness of the solution of [\[eq.NLSE\]](#eq.NLSE){reference-type="eqref" reference="eq.NLSE"} in $C([0,T]; H^1_\alpha)$, for the mass critical and super critical cases $(p\geq 3)$, by using a classical contraction argument. The precise statement is given in Theorem [Theorem 11](#t.lems8){reference-type="ref" reference="t.lems8"}. ## Heuristic introduction of $\Delta_\alpha$ An important feature of the family $-\Delta_{\alpha}$, $\alpha\in \mathbb{R}$, is the following explicit formula for the resolvent, valid for sufficiently large $\omega > 0$. $$\label{eq:res_formula} (-\Delta_\alpha+\omega)^{-1} f\;=\;(-\Delta+\omega)^{-1}f + \frac{1}{\beta_{\alpha}(\omega) }\mathbb{G}_\omega \langle f, \mathbb{G}_\omega \rangle .$$ Here $\beta_\alpha(\omega) = \alpha + c(\omega),$ where $c(\omega)$ is associated with the asymptotics of $\mathbb{G}_\omega(x)$ near the origin. Identity [\[eq:res_formula\]](#eq:res_formula){reference-type="eqref" reference="eq:res_formula"} says that the resolvent of $-\Delta_\alpha$ is a rank-one perturbation of the free resolvent. Formal substitution $$\phi =(-\Delta_\alpha+\omega)^{-1} f,\ \ g= (-\Delta+\omega)^{-1}f$$ gives $$\label{eq.dom96} \phi = g + \frac{1}{\beta_{\alpha}(\omega) }\mathbb{G}_\omega \langle f, \mathbb{G}_\omega \rangle = g+ \frac{1}{\beta_{\alpha}(\omega) }\mathbb{G}_\omega \langle g, (\omega-\Delta) \mathbb{G}_\omega \rangle.$$ Our choice of the singular perturbation is determined by the requirement $$\label{eq.gom01} (\omega-\Delta) \mathbb{G}_\omega = \delta.$$ Hence, $$\label{eq:defGlambda} \mathbb{G}_\omega(x)\;= (2\pi)^{-1} K_{0}(\sqrt{\omega} |x|),$$ where $K_{0}$ is modified Bessel function of order zero. The constant $\beta_{\alpha}(\omega)$ in [\[eq:res_formula\]](#eq:res_formula){reference-type="eqref" reference="eq:res_formula"} shall be determined by the special asymptotics of $K_0$ near the origin. We shall turn to this choice later on. Therefore if we assume formally that $\omega - \Delta_\alpha$ is a self -adjoint positive operator (on $L^2$ for example and its resolvent is rank one perturbation of type [\[eq:res_formula\]](#eq:res_formula){reference-type="eqref" reference="eq:res_formula"} with $\mathbb{G}_\omega$ determined by [\[eq.gom01\]](#eq.gom01){reference-type="eqref" reference="eq.gom01"}, then we have $$\label{eq.dom107} \mathcal{D}(-\Delta_{\alpha}) = \left\{ \phi \in L^2; \phi = g+ \frac{1}{\beta_{\alpha}(\omega) } g(0) \mathbb{G}_\omega, g \in H^2 \right\}.$$ The space in [\[eq.dom107\]](#eq.dom107){reference-type="eqref" reference="eq.dom107"} shall be denoted by $H^2_\alpha(\mathbb{R}^2).$ Since this is a linear space we can write $$\label{eq.dom116} H^2_\alpha(\mathbb{R}^2) = \mathcal{D}(-\Delta_{\alpha}) = \left\{ \phi \in L^2; \phi = \beta_{\alpha}(\omega) g + g(0) \mathbb{G}_\omega, g \in H^2 \right\}.$$ Taking $g\in H^2$ such that $g(0)=1,$ we see that $G_\omega$ is in the domain of $\Delta_\alpha$ if and only if $$\beta_\alpha(\omega) = \alpha + c(\omega) =0.$$ In this case, using [\[eq.dom116\]](#eq.dom116){reference-type="eqref" reference="eq.dom116"} and taking $$\phi = \beta_{\alpha}(\omega) g + g(0) \mathbb{G}_\omega = \mathbb{G}_\omega.$$ we see that $$(\omega-\Delta_\alpha) \mathbb{G}_\omega =0.$$ To close our heuristic description of $\Delta_\alpha$ let us note that [\[eq.dom96\]](#eq.dom96){reference-type="eqref" reference="eq.dom96"} implies $$(\omega- \Delta_\alpha) \phi = f = (\omega-\Delta)g.$$ ## Precise definitions The above observation enables one to construct a rigorous definition of $\mathcal{D}(-\Delta_\alpha)$ and its action. Namely, $$\begin{gathered} \label{eq:op_dom}\mathcal{D}(-\Delta_\alpha)\;=\;\Big\{\phi\in L^2(\mathbb{R}^2)\,\Big|\,\phi=g_{\alpha,\omega}+\frac{ g_{\alpha,\omega}(0)}{\beta_\alpha (\omega)}\,\mathbb{G}_\omega\textrm{ with }g_{\alpha,\omega}\in H^2(\mathbb{R}^2)\Big\},\\ \label{eq:opaction} (-\Delta_\alpha+\omega)\,\phi\;=\;(-\Delta+\omega)\,g_\omega\,,\end{gathered}$$ where $\omega>0$ is an arbitrarily fixed constant, $$\label{eq.2d1} \beta_\alpha(\omega):= \alpha + c(\omega), \ c(\omega)= \frac{\gamma}{2\pi} + \frac{1}{2\pi} \ln \left( \frac{\sqrt{\omega}}{2} \right),$$ $\gamma$ denoting Euler-Mascheroni constant, Owing to [\[eq:defGlambda\]](#eq:defGlambda){reference-type="eqref" reference="eq:defGlambda"}, we have the explicit formula $$\label{eq:defGlambda47} \mathbb{G}_\omega(x)\;= (2\pi)^{-1} K_{0}(\sqrt{\omega} |x|).$$ The results in [@AH81] guarantee the following statements: - we have the relation $$\label{eq.h26} \mathcal{D}(-\Delta_\alpha) = H^2_\alpha(\mathbb{R}^2) = \left\{ \phi \in L^2; \phi = (\omega-\Delta_\alpha)^{-1}f, f \in L^2 \right\};$$ - the operator $\Delta_\alpha$ is self-adjoint, its spectrum consist of absolutely continuous part $(-\infty,0]$ and it has point eigenvalue at $\omega_0$ determined by $\alpha + c(\omega_0)=0,$ i.e. $$\label{eq.pi1} \omega_0= 4 e^{-4\pi\alpha - 2\gamma};$$ - the domain and the action are independent of the choice of $\omega;$ - the resolvent identity [\[eq:res_formula\]](#eq:res_formula){reference-type="eqref" reference="eq:res_formula"} holds. Summarizing, we can define the perturbed Sobolev space $H^2_\alpha(\mathbb{R}^2)=\mathcal{D}(-\Delta_\alpha)$ and moreover $(\omega-\Delta_\alpha)^{1/2}$ is well defined. In particular, as in the case of classical Sobolev spaces, we can define $H^1_\alpha(\mathbb{R}^2)=\mathcal{D}((\omega-\Delta_\alpha)^{1/2})$. The space $H^1_\alpha(\mathbb{R}^2)$ is defined explicitly in [@MOS-2018FP] as follows $$\label{eq.defh1a81} H^1_\alpha(\mathbb{R}^2) = \left\{\phi = g + c \mathbb{G}_\omega, g \in H^1, c \in \mathbb{C}\right\}$$ and the corresponding norm is $$\label{eq.ns77} \|\phi\|_{H^1_\alpha}^2 = \|g\|_{H^1}^2 + |c|^2.$$ ## $L^p$ extension of the resolvent First we define $H^{2,p}_\alpha$ by relation similar to [\[eq:op_dom\]](#eq:op_dom){reference-type="eqref" reference="eq:op_dom"}. $$\begin{gathered} \label{eq:op_dom63}H^{2,p}_\alpha\;=\;\Big\{\phi\in L^p(\mathbb{R}^2)\,\Big|\,\phi=g_{\alpha,\omega}+\frac{ g_{\alpha,\omega}(0)}{\beta_\alpha (\omega)}\,\mathbb{G}_\omega\textrm{ with }g_{\alpha,\omega}\in H^{2,p}(\mathbb{R}^2)\Big\},\\ \label{eq:opaction67} (-\Delta_\alpha+\omega)\,\phi\;=\;(-\Delta+\omega)\,g_\omega\,,\end{gathered}$$ where $\omega > \omega_0 = 4 e^{-4\pi\alpha - 2\gamma}$ with $\omega_0$ being the unique eigenvalue of $\Delta_\alpha.$ Using the resolvent relation [\[eq:res_formula\]](#eq:res_formula){reference-type="eqref" reference="eq:res_formula"} one can easily obtain the following. **Lemma 1**. *If $\omega > \omega_0 = 4 e^{-4\pi\alpha - 2\gamma},$ the operator $$(\omega - \Delta_\alpha)^{-1}$$ can be extended as a closed operator on $L^p.$* Next, we define the spaces $H^{1,p}_\alpha$ for $p \in (1,\infty).$ $$\label{d} H^{1,p}_\alpha(\mathbb{R}^2) = \left\{\phi \in L^p; \phi = (\omega - \Delta_\alpha)^{-1/2}f, f \in L^p \right\} .$$ It is not difficult to check the following **Lemma 2**. *For any $q \in (2,\infty)$ there the space $H^2_\alpha$ is dense in $H^{1,q}.$* *Proof.* It is sufficient to use the definition of perturbed Sobolev space, the Sobolev embedding of Lemma [Lemma 10](#l.sem1){reference-type="ref" reference="l.sem1"} so we can conclude that $H^2_\alpha$ is embedded in $H^{1,q}_\alpha.$ The density property follows from the fact that any $f \in L^q$ can be approximated by $$f_\varepsilon = (\omega-\varepsilon \Delta_\alpha)^{-1} f$$ as $\varepsilon \to 0.$ Therefore the Yosida approximation completes the proof. ◻ The key questions we discuss are: 1. The space $H^2_\alpha$ is dense in $H^{1,p}_\alpha$ for $p \in (1,\infty)$? 2. Can we extend the characterization of $H^{1,p}_\alpha$ in a way similar to the relation $$\phi=g+\frac{ g(0)}{\beta_\alpha (\omega)}\,\mathbb{G}_\omega$$ used in [\[eq:op_dom63\]](#eq:op_dom63){reference-type="eqref" reference="eq:op_dom63"}, when $p>2$? 3. Can we say that $H^{1,p}$ and $H^{1,p}_\alpha$ coincide for $p \in (1,2)$ ? # Main results Our main result is quite similar to the $L^2$ case studied in [@GMS18]. First of all we have the representation of the operator $(\omega - \Delta_\alpha)^{-1/2}$ used in the definition of $H^{1,p}_\alpha.$ $$\label{eq.Darr} \begin{aligned} (\omega - \Delta_\alpha)^{-1/2}f = (\omega - \Delta)^{-1/2}f + \frac{1}{\pi}\int_0^\infty t^{-1/2} \mathbb{G}_{\omega+t}(|x|) \langle \mathbb{G}_{\omega+t}, f \rangle \frac{dt}{\beta_\alpha(\omega+t)} \end{aligned}$$ To justify it we use the relation (4.7) in [@GMS18] and we can write $$(\omega-\Delta_\alpha)^{-1/2} = \frac{1}{\pi}\int_0^\infty t^{-1/2} (\omega+t-\Delta_\alpha)^{-1} dt$$ Using the relation [\[eq:res_formula\]](#eq:res_formula){reference-type="eqref" reference="eq:res_formula"} we get [\[eq.Darr\]](#eq.Darr){reference-type="eqref" reference="eq.Darr"}. Our main canonical representation of this operator is closely connected with the following asymptotic representation of $\mathbb{G}_\omega(r)$. $$\label{eq.decomposition} \mathbb{G}_\omega(r) = \varphi_0(\sqrt{\omega}r) + R(\sqrt{\omega}r),$$ where $$\label{eq.vp08} \varphi_0(r) = - (2\pi)^{-1}\left( \log \left( r/2\right) + \gamma \right) \phi(r),$$ $\phi$ is a smooth non - negative function, such that $$\phi(r) = \left\{ \begin{aligned} & 1, \ \ \mbox{if} \ \ r<1;\\ & 0, \ \ \mbox{if} \ \ r>2. \end{aligned}\right.$$ Our first result is the following. **Theorem 3**. *We have the following properties:* - *if $p>2$ and $f \in L^p,$ then there exists a unique $g \in H^{1,p}(\mathbb{R}^2)$ so that $$(\omega - \Delta_\alpha)^{-1/2}f = g + \varphi_0(\sqrt{\omega}r) C(f),$$ where $C(f)$ is the linear functional defined by $$C(f) = \frac{1}{\pi}\int_0^\infty t^{-1/2} \langle \mathbb{G}_{\omega+t}, f \rangle \frac{dt}{\beta_\alpha(\omega+t)}$$ that is bounded functional on $L^p:$* - *if $p \in (1,2),$ then $$(\omega - \Delta_\alpha)^{-1/2}f \in H^{1,p}(\mathbb{R}^2)$$ and we have the estimate $$\label{eq.stimaellittica} \|(\omega-\Delta)^{1/2} (\omega-\Delta_\alpha)^{-1/2}f\|_{L^p} \lesssim \|f\|_{L^p}.$$* By using the approximation $$(\omega-\Delta_\alpha)^{-1/2}f = \lim_{\varepsilon \to 0} (\omega-\Delta_\alpha)^{-1/2} f_\varepsilon, \ \ f_\varepsilon = \omega^{1/2} (\omega-\varepsilon \Delta_\alpha)^{-1/2}f$$ we obtain the following. **Theorem 4**. *If $p>2$ and $f \in L^p,$ then there exists a unique $g \in H^{1,p}(\mathbb{R}^2)$ so that $$\label{eq.rappresentazione} (\omega - \Delta_\alpha)^{-1/2}f =g+\frac{ g(0)}{\beta_\alpha (\omega)}\,\mathbb{G}_\omega.$$* Next we turn to an application of the above results. We consider the following NLS associated with $\Delta_\alpha$ $$\label{eq.CP81} \begin{aligned} &( i \partial_t + \Delta_\alpha) u =\mu u|u|^{p-1}, \ p >1, \mu = \pm 1.\\ & u(0) = u_0 \in H^1_\alpha(\mathbb{R}^2). \end{aligned}$$ Formally, the conservation of mass and energy are associated with the relations $$\|u(t)\|^2_{L^{2}(\mathbb{R}^2)} = \|u(0)\|^2_{L^{2}(\mathbb{R}^2)} , \ \ E(t) = E(0),$$ where $$\begin{aligned} & E(t) = \frac{1}{2} \langle -\Delta_\alpha u(t), u(t) \rangle_{L^2} + \frac{\mu}{p+1} \|u(t)\|^{p+1}_{L^{p+1}(\mathbb{R}^2)} = \\ & =\frac{1}{2} \left\| (\omega-\Delta_\alpha)^{1/2} u(t)\right\|^2_{L^2} - \frac{\omega}{2} \|u(t)\|^2_{L^{2}(\mathbb{R}^2)} + \frac{\mu}{p+1} \|u(t)\|^{p+1}_{L^{p+1}(\mathbb{R}^2)}. \end{aligned}$$ First we consider the mass subcritical case $p \in (1,3)$ and we state the following local existence result in $L^2.$ **Theorem 5**. *For any $p \in (1,3)$ and any $R>0$ there exists $T=T(R,p)>0$ so that for any $$u_0 \in B_{L^2}(R) = \left\{ \phi \in L^2; \|\phi\|_{L^2} \leq R \right\}$$ there exists a unique solution $$u \in C([0,T]; L^2)$$ to the integral equation $$\label{eq.ie30} u = e^{it \Delta_\alpha} u_0 -i \int_0^t e^{i(t-\tau)\Delta_\alpha} u(\tau)|u(\tau)|^{p-1} d\tau.$$ associated to [\[eq.CP81\]](#eq.CP81){reference-type="eqref" reference="eq.CP81"}.* **Remark 1**. *Using a rescaling argument (see Section [7](#sec.as8){reference-type="ref" reference="sec.as8"}) in the mass subcritical case and assuming initial data in $H^1_\alpha$ one can prove the conservation of energy and global existence result.* **Remark 2**. *In the mass critical $(p=3)$ and mass super critical $(p>3)$ cases we can obtain local existence result in $H^1_\alpha.$ See Theorem below* # Characterization of $H^{1,p}_\alpha$ We start with the well-known fact that $H^{1,p}(\mathbb{R}^2)$ has as a norm (see section 1.3.1 and the identity i) in section 1.4.1 in [@YSY10]) $$\label{eq.s1p3} \|u\|_{H^{1,p}} = \|u\|_{W^{1,p}} = \sum_{|\alpha|\leq 1}\|\partial_x^\alpha u\|_{L^p} , \ \ 1 < p < \infty.$$ Another equivalent norm is the following one $$\|u\|_{H^{1,p}} = \|(1-\Delta)^{1/2}u\|_{L^p} , \ \ 1 < p < \infty.$$ **Remark 3**. *Using the norm [\[eq.s1p3\]](#eq.s1p3){reference-type="eqref" reference="eq.s1p3"}, we obtain the following property: the functions $\varphi_0$ defined in [\[eq.vp08\]](#eq.vp08){reference-type="eqref" reference="eq.vp08"} as well as the function $\mathbb{G}_\omega$ defined in [\[eq.decomposition\]](#eq.decomposition){reference-type="eqref" reference="eq.decomposition"} are in $H^{1,p}(\mathbb{R}^2)$ if and only if $p<2.$* *Proof of Theorem [Theorem 3](#t.2.1){reference-type="ref" reference="t.2.1"}.* Using the rescaling argument of Section [6](#sec.res2){reference-type="ref" reference="sec.res2"}, we can assume $\omega =1$ and $\alpha$ is so large that the unique eigenvalue $4 e^{-4\pi\alpha - 2\gamma}$ determined in [\[eq.pi1\]](#eq.pi1){reference-type="eqref" reference="eq.pi1"} is in the interval $(0,1).$ We start from the case $p>2$. Thanks to the decomposition of $\mathbb{G}_{1+t}$ in [\[eq.decomposition\]](#eq.decomposition){reference-type="eqref" reference="eq.decomposition"}, we can write $$\label{eq.cpg160} (1-\Delta_\alpha)^{-1/2}f=g+ \frac{1}{\pi}\varphi_0(r)\int_0^\infty t^{-1/2} \langle \mathbb{G}_{1+t}, f \rangle \frac{dt}{\beta_\alpha(1+t)},$$ with $$\label{eq.def g} g=(1 - \Delta)^{-1/2}f+\Gamma(f)+\Gamma_0(f),$$ where $$\label{eq.Ga16} \begin{aligned} & \Gamma (f) (x) = \int_0^\infty t^{-1/2} R( \sqrt{t+1} |x|) \left(\int_{\mathbb{R}^2} \mathbb{G}_{1+t}( |y|) \overline{f(y)} dy\right) dt \end{aligned}$$ and $$\label{eq.Ga022} \begin{aligned} & \Gamma_0 (f) (x) = \int_0^\infty t^{-1/2} [\varphi_0( \sqrt{t+1} |x|) - \varphi_0(|x|)] \left(\int_{\mathbb{R}^2} \mathbb{G}_{1+t}( |y|) \overline{f(y)} dy\right) dt. \end{aligned}$$ Thanks to Lemmas [Lemma 7](#l.gamma){reference-type="ref" reference="l.gamma"} and [Lemma 8](#l.gamma0){reference-type="ref" reference="l.gamma0"}, in the Section [4](#s.lp){reference-type="ref" reference="s.lp"} we will prove that $\Gamma(f),\Gamma_1(f)\in H^{1,p}$. We note in particular that $\varphi_0(r)\notin H^{1,p}$, for $p>2$ because $|\partial_x^\alpha \varphi_0(r)|\sim 1/r$ near zero. If $1<p<2$, we have instead $$(1-\Delta_\alpha)^{-1/2}f=(1 - \Delta)^{-1/2}f+\Gamma(f)+\Gamma_1(f),$$ with $$\begin{aligned} & \Gamma_1 (f) (x) = \int_0^\infty t^{-1/2} \varphi_0( \sqrt{t+1} |x|) \left(\int_{\mathbb{R}^2} \mathbb{G}_{1+t}( |y|) \overline{f(y)} dy\right) dt. \end{aligned}$$ Again in Section [4](#s.lp){reference-type="ref" reference="s.lp"}, Lemmas [Lemma 7](#l.gamma){reference-type="ref" reference="l.gamma"} and [Lemma 9](#l.gamma1){reference-type="ref" reference="l.gamma1"} will give that $\Gamma(f),\Gamma_1(f)\in H^{1,p}$ and moreover [\[eq.lll86\]](#eq.lll86){reference-type="eqref" reference="eq.lll86"} and [\[eq.gamma1.2\]](#eq.gamma1.2){reference-type="eqref" reference="eq.gamma1.2"}, with an elliptic estimate give $$\|(1-\Delta_\alpha)^{-1/2}f\|_{H^{1,p}}\lesssim\|f\|_{L^p},$$ that is equivalent to [\[eq.stimaellittica\]](#eq.stimaellittica){reference-type="eqref" reference="eq.stimaellittica"}. ◻ *Proof of Theorem [Theorem 4](#t.2.2){reference-type="ref" reference="t.2.2"}.* We start with the uniqueness. It follows by contradiction argument. So let us assume that there exist $g_1$ , $g_2\in H^{1,p}$ such that $$(\omega - \Delta_\alpha)^{-1/2}f =g_1+\frac{ g_1(0)}{\beta_\alpha (\omega)}\,\mathbb{G}_\omega$$ and $$(\omega - \Delta_\alpha)^{-1/2}f =g_2+\frac{ g_2(0)}{\beta_\alpha (\omega)}\,\mathbb{G}_\omega.$$ From these two equations we obtain that $$g_1-g_2=\frac{ g_2(0)-g_1(0)}{\beta_\alpha (\omega)}\,\mathbb{G}_\omega.$$ We remember that $\mathbb{G}_\omega\notin H^{1,p}$ for $p>2$, because its first derivative behaves like $1/r$ for $r$ near zero, so $g_2(0)-g_1(0)=0$ and $g_1=g_2$. Our next step is to prove that for any $\phi = (1-\Delta_\alpha)^{-1/2}f \in H^{1,p}_\alpha$ we have $$\label{eq.dec522} \phi=\Lambda(f)+C(f)\mathbb{G}_1,$$ where $$\Lambda: L^p \to H^{1,p}$$ is a linear bounded operator and $C(f)$ is a bounded functional on $L^p.$ As before we can assume $\omega=1$. We have the relations [\[eq.def g\]](#eq.def g){reference-type="eqref" reference="eq.def g"} for $p >2.$ So we can write $$\label{eq.cpg1515} (1-\Delta_\alpha)^{-1/2}f= \tilde{g} + C(f)\varphi_0(r),$$ where $$\tilde{g}=(1 - \Delta)^{-1/2}f+\Gamma(f)+\Gamma_0(f).$$ Here $\Gamma$ and $\Gamma_0$ are $L^p-H^{1,p}$ continuous and $C(f)$ is a $L^p$ bounded functional. Recall that $G_1(r) = \varphi_0(r)+R(r)$ due to [\[eq.decomposition\]](#eq.decomposition){reference-type="eqref" reference="eq.decomposition"} and $R(|x|)$ is in $H^{1,p}.$ Hence defining $$g= \tilde{g} - C(f)R = (1 - \Delta)^{-1/2}f+\Gamma(f)+\Gamma_0(f)-C(f)R = \Lambda(f),$$ we arrive at [\[eq.dec522\]](#eq.dec522){reference-type="eqref" reference="eq.dec522"}. Now we can prove [\[eq.rappresentazione\]](#eq.rappresentazione){reference-type="eqref" reference="eq.rappresentazione"} by using [\[eq.dec522\]](#eq.dec522){reference-type="eqref" reference="eq.dec522"} and density argument. Let $\phi=(1-\Delta_\alpha)^{-1/2}f$ with $f \in L^q$ Then we can approximate $f$ by $f_n \in H^{1,p}_\alpha$ so that $f_n\to f$ in $L^p$. On one hand, $$\phi_n = (1-\Delta_\alpha)^{-1/2}f_n \in H^2_\alpha$$ so $$\phi_n = g_n + g_n(0) \frac{\mathbb{G}_1}{\beta_1(\alpha)}$$ On the other hand , from [\[eq.dec522\]](#eq.dec522){reference-type="eqref" reference="eq.dec522"} we can deduce $$\label{eq.dec548} \phi_n=\Lambda(f_n)+C(f_n)\mathbb{G}_1,$$ so the uniqueness observation discussed above implies $$g_n = \Lambda(f_n), \ C(f_n) = \frac{g_n(0)}{\beta_1(\alpha)} .$$ and after taking the limit we get $$\phi = g + g(0) \frac{\mathbb{G}_1}{\beta_1(\alpha)}.$$ This completes the proof. ◻ # $L^p$ estimates of the operators $\Gamma$ {#s.lp} The canonical representation [\[eq.Darr\]](#eq.Darr){reference-type="eqref" reference="eq.Darr"} shows that we have to consider (after rescaling) the term $$\label{eq.mop2} \frac{1}{\pi}\int_0^\infty t^{-1/2} \mathbb{G}_{1+t}(|x|) \langle \mathbb{G}_{1+t}, f \rangle \frac{dt}{\beta_\alpha(1+t)}.$$ Then [\[eq:defGlambda\]](#eq:defGlambda){reference-type="eqref" reference="eq:defGlambda"} and asymptotics of Appendix [7](#sec.as8){reference-type="ref" reference="sec.as8"} imply that $$\label{eq:fgl10a} \mathbb{G}_{1+t}(x)\;= \varphi_0( \sqrt{1+t}|x|) +R(\sqrt{1+t}|x|), \ \ t>0,$$ where $\varphi_0(r)$ is smooth in $(0,\infty)$ and satisfies $$\left\{ \begin{aligned} & \mathrm{supp} \ \varphi_0 (r) \subset \{ r \leq 2 \}, \\ & \varphi_0(r)= - (2\pi)^{-1}\left( \log \left( r/2\right) + \gamma \right) , \ \ r \leq 1 ,\\ & | \partial_r^k \varphi_0(r)| \lesssim r^{-k} \log^{1-k}(2/r)\ \ \ k=0,1, \end{aligned}\right.$$ while the remainder $R$ is represented by two terms localised near $0$ and $\infty$ respectively. More precisely, we have $$R(r) = R_{small}(r) + R_{large}(r)$$ where $$\label{eq.pss39} \left\{ \begin{aligned} & \mathrm{supp} \ R_{small}(r) \subset \{ r \leq 2 \}, \\ & | \partial_r^k R_{small}(r)| \lesssim r^{2-k} \log(2/r) , r \leq 1, \ k=0,1, \\ \end{aligned}\right.$$ and $$\label{eq.pss47} \left\{ \begin{aligned} & \mathrm{supp} \ R_{large} (r) \subset \{ r \geq 1/\sqrt 2 \}, \\ & R_{large}(r)\in L^q((0,\infty), rdr), \ \ \forall q \in (1,\infty), \\ & \exists \delta>0, \ \ \mbox{so that} \ \ |R^\prime_{large}(\sigma)| \leq e^{-\delta \sigma} , \forall \sigma \geq 1/\sqrt 2. \end{aligned}\right.$$ We have the following **Lemma 6**. *If $\mathrm{arg} \lambda \in (\varepsilon, \pi)$ and $|\lambda| >0,$ then for any $p \in (1,\infty)$ we have $$\label{eq.SSE9} \left\{ \begin{aligned} & \sum_{|\alpha|=m} \|\partial_x^\alpha R(\lambda |x|)\|_{H^{k,p}(\mathbb{R}^2_x)} \lesssim |\lambda|^{k+m-2/p}, k,m=0,1, \\ & \|\varphi_0(\lambda |x|)\|_{L^p(\mathbb{R}^2_x)} \lesssim |\lambda|^{-2/p},\\ & \|G_{\lambda^2}(x)\|_{L^p(\mathbb{R}^2_x)} \lesssim |\lambda|^{-2/p}. \end{aligned}\right.$$ Moreover for $p \in [1,2)$ we have $$\label{eq.phi.2} \|\varphi_0(\lambda |x|)\|_{H^{1,p}(\mathbb{R}^2_x)} \lesssim |\lambda|^{1-2/p}.$$* The term [\[eq.mop2\]](#eq.mop2){reference-type="eqref" reference="eq.mop2"} suggests to consider the operator $$\begin{aligned} & \Gamma (f) (x) = \int_0^\infty t^{-1/2} R( \sqrt{t+1} |x|) \left(\int_{\mathbb{R}^2} \mathbb{G}_{1+t}( |y|) \overline{f(y)} dy\right) dt. \end{aligned}$$ Note that for simplicity we do not put the factor $\beta_1(\alpha)$ in denominator, since this factor is bounded from below. **Lemma 7**. *For any $p \in (1,\infty)$ the operator $\Gamma$ maps $L^{p}(\mathbb{R}^2)$ into $H^{1, p}(\mathbb{R}^2).$* *Proof.* It is sufficient to prove $$\label{eq.lll82} \|\Gamma(f)\|_{L^{q}} \lesssim \|f\|_{L^q}$$ and $$\label{eq.lll86} \sum_{|\alpha|=1}\|\partial_x^\alpha \Gamma(f)\|_{L^{q,\infty}} \lesssim \|f\|_{L^q}$$ and then apply Marcinkiewicz interpolation theorem. Using [\[eq.SSE9\]](#eq.SSE9){reference-type="eqref" reference="eq.SSE9"} we have $$\label{eq.11192} \begin{aligned} & |\Gamma(f)(x)| \lesssim \int_0^\infty t^{-1/2} R( \sqrt{t+1} |x|) \left\| \mathbb{G}_{1+t}( |y|) \right\|_{L^{q^\prime}_y} dt \|f\|_{L^q} \lesssim \\ & \int_0^\infty t^{-1/2} R( \sqrt{t+1} |x|) (1+t)^{-1/q^\prime} dt \|f\|_{L^q} \end{aligned}$$ so $$\begin{aligned} &\|\Gamma(f)\|_{L^{q}} \lesssim \int_0^\infty t^{-1/2} \| R( \sqrt{t+1} |x|)\|_{L^q} (1+t)^{-1/q^\prime} dt \|f\|_{L^q} \\ & \lesssim \left( \int_0^\infty t^{-1/2} (1+t)^{-1} dt \right) \ \|f\|_{L^q} \end{aligned}$$ and we have [\[eq.lll82\]](#eq.lll82){reference-type="eqref" reference="eq.lll82"}. The proof of [\[eq.lll86\]](#eq.lll86){reference-type="eqref" reference="eq.lll86"} is more delicate. We have the decomposition $$\Gamma (f) (x) = \Gamma_{small} (f) (x) + \Gamma_{large} (f) (x),$$ where $$\Gamma_{small} (f) (x)= \int_0^\infty t^{-1/2} R_{small}( \sqrt{t+1} |x|) \left(\int_{\mathbb{R}^2} \mathbb{G}_{1+t}( |y|) \overline{f(y)} dy\right) dt,$$ $$\Gamma_{large} (f) (x)= \int_0^\infty t^{-1/2} R_{large}( \sqrt{t+1} |x|) \left(\int_{\mathbb{R}^2} \mathbb{G}_{1+t}( |y|) \overline{f(y)} dy\right) dt$$ As in [\[eq.11192\]](#eq.11192){reference-type="eqref" reference="eq.11192"} we have $$\label{eq.111317} \left\{\begin{aligned} & \sum_{|\alpha|=1} |\partial_x^\alpha\Gamma_{small}(f)(x)| \lesssim \int_0^\infty t^{-1/2} \sum_{|\alpha|=1} \left|\partial_x^\alpha R_{small}( \sqrt{t+1} |x|) \right| (1+t)^{-1/q^\prime} dt \|f\|_{L^q}, \\ & \sum_{|\alpha|=1} |\partial_x^\alpha\Gamma_{large}(f)(x)| \lesssim \int_0^\infty t^{-1/2} \sum_{|\alpha|=1} \left|\partial_x^\alpha R_{large}( \sqrt{t+1} |x|) \right| (1+t)^{-1/q^\prime} dt \|f\|_{L^q}. \end{aligned}\right.$$ First we shall estimate $$\sum_{|\alpha|=1} |\partial_x^\alpha\Gamma_{small}(f)(x)|.$$ Using the support assumption in [\[eq.pss39\]](#eq.pss39){reference-type="eqref" reference="eq.pss39"} and the estimate $\left|\partial_r R_{small}( r )\right| \lesssim 1$ stated in [\[eq.pss39\]](#eq.pss39){reference-type="eqref" reference="eq.pss39"}, we obtain $$\begin{aligned} & \int_0^\infty t^{-1/2} \sum_{|\alpha|=1} \left| \partial_x^\alpha \left( R_{small}( \sqrt{t+1} |x|) \right)\right| \ (1+t)^{-1/q^\prime} dt \\ & \lesssim \int_0^{4/|x|^2} t^{-1/2} (1+t)^{1/2} (1+t)^{-1/q^\prime} dt . \end{aligned}$$ Now we have two possibilities: $|x|$ bounded, say $|x| \leq 2$ and then the other case is $|x| >2.$ If $|x|\leq 2,$ then $$\begin{aligned} & \int_0^{4/|x|^2} t^{-1/2} (1+t)^{1/2} (1+t)^{-1/q^\prime} dt \lesssim \\ & \lesssim \int_0^{1/4} t^{-1/2} dt+ \int_{1/4}^{4/|x|^2} t^{-1/2} (1+t)^{-1/2+1/q} dt\\ &\lesssim 1 + \int_{1/4}^{4/|x|^2} t^{-1+1/q} dt \lesssim |x|^{-2/q} \end{aligned}$$ If $|x|>2,$ then $r= \sqrt{1+t}|x|$ is outside the support of $R_{small}(r)$ so we can conclude $$\begin{aligned} & \int_0^\infty t^{-1/2} \sum_{|\alpha|=1} \left| \partial_x^\alpha \left( R_{small}( \sqrt{t+1} |x|) \right)\right| \ (1+t)^{-1/q^\prime} dt \lesssim |x|^{-2/q} \end{aligned}$$ and we arrive at $$\label{eq.sme52} \begin{aligned} \sum_{|\alpha|=1} |\partial_x^\alpha\Gamma_{small}(f)(x)| \lesssim |x|^{-2/q} \|f\|_{L^q} \end{aligned}$$ Next we shall estimate $$\sum_{|\alpha|=1} |\partial_x^\alpha\Gamma_{large}(f)(x)|.$$ For the purpose we set $$Q(r) = \sum_{|\alpha|=1} \left|\partial_x^\alpha R_{large}( |x|)\right|, r=|x|,$$ Then [\[eq.111317\]](#eq.111317){reference-type="eqref" reference="eq.111317"} shows that we need to evaluate $$\left|\int_0^\infty t^{-1/2} (1+t)^{1/2} Q( \sqrt{t+1} |x|) \ (1+t)^{-1/q^\prime} dt \right|.$$ In the case $|x|\leq 1/\sqrt{2}$ we use the support assumption of $Q(r)$ and we see that integration domain for $t$ is determined by $$t > \frac{1}{4r^2}-1 \geq \frac{1}{8r^2}\geq 1 .$$ Hence $$\begin{aligned} & \int_0^\infty t^{-1/2} (1+t)^{1/2} Q( \sqrt{t+1} |x|) \ (1+t)^{-1/q^\prime} dt \\ & \lesssim \int_{1/8r^2}^\infty t^{-1/2} (1+t)^{1/2} Q( \sqrt{t+1} |x|) \ (1+t)^{-1/q^\prime} dt . \end{aligned}$$ We have also $$\int_{1/8r^2}^\infty t^{-1/2} (1+t)^{1/2} Q( \sqrt{t+1} |x|) \ (1+t)^{-1/q^\prime} dt \lesssim$$ $$\int_{1/8r^2}^\infty (1+t)^{-1+1/q} Q( \sqrt{t+1} |x|) dt$$ Now we make change of variables $\sigma=\sqrt{t+1} |x|$ and from $$t > \frac{1}{8r^2}$$ we find $$\sigma^2 = |x|^2 + t|x|^2 \geq |x|^2 + \frac{1}8 \geq \frac{1}8$$ so we find $$\int_{1/8r^2}^\infty (1+t)^{-1+1/q} Q( \sqrt{t+1} |x|) dt \lesssim$$ $$\int_{1/2\sqrt{2}}^\infty \sigma^{-2+2/q} |x|^{2-2/q} Q( \sigma) \ \frac{2\sigma d\sigma}{|x|^2} .$$ Applying the assumption [\[eq.pss47\]](#eq.pss47){reference-type="eqref" reference="eq.pss47"}, we see that $$\int_0^\infty \sigma^{-1+2/q}| \partial_\sigma R_{large}(\sigma)| d\sigma \lesssim 1$$ and we find $$\int_{1/8r^2}^\infty (1+t)^{-1+1/q} Q( \sqrt{t+1} |x|) dt \lesssim |x|^{-2/q}.$$ Hence we have $$\label{eq.lae87} \sum_{|\alpha|=1} |\partial_x^\alpha\Gamma_{large}(f)(x)|\lesssim |x|^{-2/q} \|f\|_{L^q}, \ \ |x| \leq 1/\sqrt{2}.$$ We turn to the case $|x| > 1/\sqrt 2.$ Then the assumption $$\exists \delta>0, \ \ \mbox{so that} \ \ Q(\sigma)= |R^\prime_{large}(\sigma)| \leq e^{-\delta \sigma} , \forall \sigma \geq 1/\sqrt2$$ from [\[eq.pss47\]](#eq.pss47){reference-type="eqref" reference="eq.pss47"} guarantees that $$Q( \sqrt{t+1} |x|) \leq e^{-\delta |x| \sqrt{1+t}} \lesssim e^{-\delta \sqrt{1+t}/4} e^{-\delta |x|/2}, \ \ \forall |x| >1/\sqrt2, t \geq 0.$$ Therefore, we have $$\sum_{|\alpha|=1} |\partial_x^\alpha\Gamma_{large}(f)(x)| \lesssim \int_0^\infty t^{-1/2} (1+t)^{-1/2+ q} \sum_{|\alpha|=1} Q( \sqrt{t+1} |x|) dt \|f\|_{L^q} \lesssim e^{-\delta |x|/2} \|f\|_{L^q},$$ so we arrive at $$\label{eq.lae02} \sum_{|\alpha|=1} |\partial_x^\alpha\Gamma_{large}(f)(x)|\lesssim |x|^{-2/q} \|f\|_{L^q}, \ \ |x| > 1/\sqrt2.$$ From [\[eq.sme52\]](#eq.sme52){reference-type="eqref" reference="eq.sme52"}, [\[eq.lae87\]](#eq.lae87){reference-type="eqref" reference="eq.lae87"} and [\[eq.lae02\]](#eq.lae02){reference-type="eqref" reference="eq.lae02"} we conclude that [\[eq.lll86\]](#eq.lll86){reference-type="eqref" reference="eq.lll86"} is true. ◻ Next we consider the operator $$\begin{aligned} & \Gamma_0 (f) (x) = \int_0^\infty t^{-1/2} [\varphi_0( \sqrt{t+1} |x|) - \varphi_0(|x|)] \left(\int_{\mathbb{R}^2} \mathbb{G}_{1+t}( |y|) \overline{f(y)} dy\right) dt. \end{aligned}$$ **Lemma 8**. *We have the estimates $$\label{eq.ll1423} \|\Gamma_0(f)\|_{L^{q}} \lesssim \|f\|_{L^q}$$ and $$\label{eq.lll427} \sum_{|\alpha|=1}\|\partial_x^\alpha \Gamma_0(f)\|_{L^{q,\infty}} \lesssim \|f\|_{L^q}$$* *Proof.* The proof of [\[eq.ll1423\]](#eq.ll1423){reference-type="eqref" reference="eq.ll1423"} is the same as the proof of [\[eq.lll82\]](#eq.lll82){reference-type="eqref" reference="eq.lll82"}. To prove [\[eq.lll427\]](#eq.lll427){reference-type="eqref" reference="eq.lll427"} we use the inequality $$\sum_{|\alpha|=1} \left| \partial_x^\alpha [\varphi_0( \sqrt{t+1} |x|) - \varphi_0(|x|)] \right| \lesssim \left| \sqrt{1+t}\varphi_0^\prime( |x| \sqrt{1+t}) - \varphi_0^\prime(|x|) \right|,$$ We lose no generality assuming $$\varphi_0(r)= - (2\pi)^{-1}\left( \log \left( r/2\right) + \gamma \right) \phi(r),$$ where $\phi(r)$ is smooth, $\phi(r)=1, 0 < r \leq 1$ and $\phi(r)=0, r>2.$ Then $$-(2\pi)\partial_r \left( \varphi_0(r\sqrt{1+t}) - \varphi_0(r) \right)= \frac{1}{ r}\left(\phi(r\sqrt{1+t})-\phi(r) \right) +$$ $$+\left( \log \left( r/2\right) + \log \left( \sqrt{1+t}\right) +\gamma \right) \sqrt{1+t} \phi^\prime(r\sqrt{1+t}) - \left( \log \left( r/2\right) + \gamma \right) \phi^\prime(r).$$ In the case, when $\delta < r < 2$ with $\delta>0$ small we can conclude that $$1+t \leq 4/r^2$$ implies $t$ is bounded, so $$\partial_r \left( \varphi_0(r\sqrt{1+t}) - \varphi_0(r) \right) = O(1), \ \delta < r < 2, \ 1+t \leq 4/r^2.$$ Further for $q>2,$ $\delta < r < 2$ and $1+t \geq 4/r^2$ we have $$\partial_r \left( \varphi_0(r\sqrt{1+t}) - \varphi_0(r) \right) = \frac{1}{2\pi r}\phi(r) + O(1)$$ Now we can follow the proof of [\[eq.sme52\]](#eq.sme52){reference-type="eqref" reference="eq.sme52"} so that $$\sum_{|\alpha|=1}|\partial_x^\alpha\Gamma_0 (f) (x)| \lesssim r^{-1}\int_{2/r^2}^\infty t^{-1/2} (1+t)^{-1+1/q} dt\ \|f\|_{L^q} +$$ $$+\int_0^{2/r^2} t^{-1/2} (1+t)^{-1+1/q} dt \ \|f\|_{L^q} \lesssim \|f\|_{L^q}, \ \ \delta < r < 2$$ provided $$\label{eq.q265} q > 2.$$ Next we turn to the case $0 < r < \delta.$ Then we can write $$\frac{1}{ r}\left(\phi(r\sqrt{1+t})-\phi(r) \right) \lesssim r^{-1} \mathds{1}_{r^{-2} \leq t} .$$ Further, we have $$\left| \left( \log \left( r/2\right) + \log \left( \sqrt{1+t}\right) +\gamma \right) \sqrt{1+t} \phi^\prime(r\sqrt{1+t}) \right|=$$ $$\frac{1}{r}\left| \left( \log \left(\sqrt{1+t} r/2\right) +\gamma \right) r\sqrt{1+t} \phi^\prime(r\sqrt{1+t}) \right| \lesssim r^{-1} \mathds{1}_{r^{-2} \sim t} .$$ Finally, $$\left|\left( \log \left( r/2\right) + \gamma \right) \phi^\prime(r)\right| =0,$$ provided $\delta$ is small. Hence we arrive at $$\sum_{|\alpha|=1}|\partial_x^\alpha\Gamma_0 (f) (x)| \lesssim r^{-1}\int_{1/(r^2)}^\infty t^{-1/2} (1+t)^{-1+1/q} dt\ \|f\|_{L^q} \lesssim r^{-2/q}\|f\|_{L^q}$$ provided $r<\delta$ and [\[eq.q265\]](#eq.q265){reference-type="eqref" reference="eq.q265"} holds. ◻ We finally consider the operator $$\begin{aligned} & \Gamma_1 (f) (x) = \int_0^\infty t^{-1/2} \varphi_0( \sqrt{t+1} |x|) \left(\int_{\mathbb{R}^2} \mathbb{G}_{1+t}( |y|) \overline{f(y)} dy\right) dt, \end{aligned}$$ and we have the following Lemma. **Lemma 9**. *For any $p \in (1,2)$ the operator $\Gamma_1$ maps $L^{p}(\mathbb{R}^2)$ into $H^{1, p}(\mathbb{R}^2).$ In particular we have the following estimates: $$\label{eq.gamma1} \|\Gamma_1(f)\|_{L^{p}} \lesssim \|f\|_{L^p},$$ and $$\label{eq.gamma1.2} \|\Gamma_1(f)\|_{H^{1,p}} \lesssim \|f\|_{L^p}.$$* *Proof.* The proof of [\[eq.gamma1\]](#eq.gamma1){reference-type="eqref" reference="eq.gamma1"} is similar to [\[eq.lll82\]](#eq.lll82){reference-type="eqref" reference="eq.lll82"}. The proof of [\[eq.gamma1.2\]](#eq.gamma1.2){reference-type="eqref" reference="eq.gamma1.2"} follows from [\[eq.phi.2\]](#eq.phi.2){reference-type="eqref" reference="eq.phi.2"}. ◻ # Sobolev embedding and local well posedness {#section.5} In this section we consider the Cauchy problem [\[eq.CP81\]](#eq.CP81){reference-type="eqref" reference="eq.CP81"} and we shall give alternative proof of the local existence result established in Theorem B.1 in [@FGI22]. Our first step is the following Sobolev inequality. **Lemma 10**. *For any $q \in (2,\infty)$ there is a constant $C=c(q)>0$ so that for any $\phi \in H^1_\alpha$ we have $\phi \in L^q$ and $$\|\phi\|_{L^q(\mathbb{R}^2)} \leq C \|\phi\|_{H^1_\alpha}.$$* *Proof.* We know from [\[eq.defh1a81\]](#eq.defh1a81){reference-type="eqref" reference="eq.defh1a81"} that $$\phi = g + c_* \mathbb{G}_\omega, g \in H^1.$$ Since the classical Sobolev embedding implies $$\|g\|_{L^q(\mathbb{R}^2)} \leq C \|g\|_{H^1_\alpha}$$ moreover $G_\omega \in L^q.$ Hence, $$\|\phi\|_{L^q} \lesssim \|g\|_{H^1_\alpha} + |c_*| \sim \|\phi\|_{H^1_\alpha}.$$ ◻ Further we recall the Strichartz estimates for $\Delta_\alpha$ that are obtained in [@CMY19; @CMY19b]. $$\label{eq.str710} \begin{aligned} & \left\| e^{i t \Delta_\alpha} f \right\|_{L^q(0,T)L^r} \lesssim \|f\|_{L^2}, \\ & \left\|\int_0^t e^{i(t-\tau)\Delta_\alpha} F(\tau) d\tau\right\|_{L^q(0,T)L^r} \lesssim \left\| F\right\|_{L^{\tilde{q}^\prime}(0,T)L^ {\tilde{r}^\prime}}, \end{aligned}$$ provided $$\label{eq.adm18} \frac{1}{q} + \frac{1}{r} = \frac{1}{2}, \ q \in (2, \infty], \ \frac{1}{\tilde{q}} + \frac{1}{\tilde{r}} = \frac{1}{2}, \ \tilde{q} \in (2, \infty]$$ It is easy to obtain local well - posedness of the problem [\[eq.CP81\]](#eq.CP81){reference-type="eqref" reference="eq.CP81"} in the mass subcritical case. *Proof of Theorem [Theorem 5](#t.le19){reference-type="ref" reference="t.le19"}.* Consider the operator $$\label{eq.kop37} \mathfrak{K} (u) = e^{it \Delta_\alpha} u_0 -i \int_0^t e^{i(t-\tau)\Delta_\alpha} u(\tau)|u(\tau)|^{p-1} d\tau$$ and define the Banach space $$L^\infty(0,T)L^2 \cap L^{\tilde{q}^\prime}(0,T) L^{\tilde{r}^\prime}$$ with Applying the Strichartz estimate with $$\label{tqr46} \tilde{r} =2 ,\ \tilde{q}= \infty ,$$ we get $$\left\| \mathfrak{K}(u)\right\|_{L^q(0,T)L^r} \lesssim \|u_0\|_{L^2} + \left\| u|u|^{p-1}\right\|_{L^{1}(0,T)L^{2} } \lesssim \|u_0\|_{L^2} + \left\|u\right\|^p_{L^{p}(0,T)L^{2p} } .$$ so we can choose $$\begin{aligned} & r=2p, \\ & q = \frac{2p}{p-1}. \end{aligned}$$ so that $(q,r)$ is admissible couple satisfying [\[eq.adm18\]](#eq.adm18){reference-type="eqref" reference="eq.adm18"}. Now we need $$\left\|u\right\|_{L^{p}(0,T)L^{2p} } \lesssim T^{\alpha} \left\|u\right\|_{L^{q}(0,T)L^{2p} }$$ with $\alpha=(3-p)/(2p)>0$ and this can be done if and only if $$p < q = \frac{2p}{p-1}$$ that is $p<3.$ The estimate $$\label{eq.sest60} \left\| \mathfrak{K}(u)\right\|_{L^{2p/(p-1)}(0,T)L^{2p}} \lesssim \|u_0\|_{L^2} + T^{p\alpha} \left\| \mathfrak{K}(u)\right\|^p_{L^{2p/(p-1)}(0,T)L^{2p}}$$ shows that $\mathfrak{K}$ maps $$\label{eq.2R62} \left\{ u \in L^{2p/(p-1)}(0,T)L^{2p}; \left\| u\right\|_{L^{2p/(p-1)}(0,T)L^{2p}} \leq 2R \right\}$$ provided $u_0 \in B_{L^2}(R)$ and $T=T(R,p)$ is sufficiently small. In a similar way we deduce $$\left\| \mathfrak{K}(u)-\mathfrak{K}(\tilde{u})\right\|_{L^{2p/(p-1)}(0,T)L^{2p}} \lesssim \frac{1}{2} \left\| u-\tilde{u}\right\|_{L^{2p/(p-1)}(0,T)L^{2p}}$$ so $\mathfrak{K}$ is a contraction in [\[eq.2R62\]](#eq.2R62){reference-type="eqref" reference="eq.2R62"}. Observing that the estimate [\[eq.sest60\]](#eq.sest60){reference-type="eqref" reference="eq.sest60"} and Strichartz estimates imply $$\label{eq.sest75} \left\| \mathfrak{K}(u)\right\|_{L^{q}(0,T)L^{r}} \lesssim \|u_0\|_{L^2} + T^{p\alpha} \left\| u\right\|^p_{L^{2p/(p-1)}(0,T)L^{2p}}$$ for any admissible couple, we complete the proof. ◻ Our Theorem [Theorem 3](#t.2.1){reference-type="ref" reference="t.2.1"} guarantees the more general Strichartz estimates $$\label{eq.SH123} \begin{aligned} & \left\| e^{i t \Delta_\alpha} f \right\|_{L^q(0,T)H^{1,r}_\alpha} \lesssim \|f\|_{H^1_\alpha}, \\ & \left\|\int_0^t e^{i(t-\tau)\Delta_\alpha} F(\tau) d\tau\right\|_{L^q(0,T)H^{1,r}_\alpha} \lesssim \left\| F\right\|_{L^{\tilde{q}^\prime}(0,T)H^{1,{\tilde{r}^\prime}}_\alpha } \end{aligned}$$ Our next local existence result treats the case $p \geq 3$. **Theorem 11**. *For any $p\geq 3$ and any $R>0$ there exists $T=T(R,p)>0$ so that for any $$u_0 \in B(R) = \left\{ \phi \in H^1_\alpha; \|\phi\|_{H^1_\alpha} \leq R \right\}$$ there exists a unique solution $$u \in C([0,T]; H^1_\alpha)$$ to the integral equation $$u = e^{it \Delta_\alpha} u_0 -i \int_0^t e^{i(t-\tau)\Delta_\alpha} u(\tau)|u(\tau)|^{p-1} d\tau.$$ associated to [\[eq.CP81\]](#eq.CP81){reference-type="eqref" reference="eq.CP81"}.* *Proof.* Consider the operator $$\mathfrak{K} (u) = e^{it \Delta_\alpha} u_0 -i \int_0^t e^{i(t-\tau)\Delta_\alpha} u(\tau)|u(\tau)|^{p-1} d\tau.$$ Further, we define the Banach space $\mathcal{B}=L^\infty(0,T)H^1_\alpha$ and the corresponding ball of radius $R$ $$B_{\mathcal{B}} = \left\{u \in \mathcal{B} ; \|u\|_{\mathcal{B}} \leq R \right\}.$$ Applying the Strichartz estimate [\[eq.SH123\]](#eq.SH123){reference-type="eqref" reference="eq.SH123"}, we find $$\left\| \mathfrak{K}(u)\right\|_{L^q(0,T)H^1_\alpha} \lesssim \|u_0\|_{H^1_\alpha} + \left\| u|u|^{p-1}\right\|_{L^{\tilde{q}^\prime}(0,T)H^{1, \tilde{r}^\prime}_\alpha } .$$ Now we choose $$\label{eq.rq} \tilde{r} = \frac{2-\varepsilon}{1-\varepsilon},\ \tilde{q}= \frac{4-2\varepsilon}{\varepsilon}$$ so that $$\label{eq.rqprime} \tilde{r}^\prime = 2-\varepsilon,\ \tilde{q}^\prime= \frac{\varepsilon}{3\varepsilon-4} .$$ Since $\tilde{r}^\prime <2,$ we see that Theorem [Theorem 3](#t.2.1){reference-type="ref" reference="t.2.1"} implies $$\left\| u|u|^{p-1}\right\|_{H^{1,{\tilde{r}^\prime}}_\alpha } \sim \left\| u|u|^{p-1}\right\|_{H^{1,{\tilde{r}^\prime}} } \sim \left\| u|u|^{p-1}\right\|_{L^{\tilde{r}^\prime} } + \left\| \nabla u|u|^{p-1}\right\|_{L^{\tilde{r}^\prime} }$$ Now we use the fact that $u \in H^1_\alpha$ $$u= g + c_*G_\omega$$ and we can continue the estimates as follows $$\left\| |\nabla u||u|^{p-1}\right\|_{L^{\tilde{r}^\prime} } \leq \left\| |\nabla g||u|^{p-1}\right\|_{L^{\tilde{r}^\prime} } + |c_*|\left\| |\nabla \mathbb{G}_\omega||u|^{p-1}\right\|_{L^{\tilde{r}^\prime} }.$$ Then we estimate each of the terms in the right side and find $$\left\| |\nabla g||u|^{p-1}\right\|_{L^{\tilde{r}^\prime} }\lesssim\|\nabla g\|_{L^2}\|u\|^{(p-1)}_{L^{\frac{2(2-\varepsilon)}{\varepsilon}(p-1)}}\lesssim \|\nabla g\|_{L^2}\|u\|^{p-1}_{H^1_\alpha},$$ $$\left\| |\nabla \mathbb{G}_\omega||u|^{p-1}\right\|_{L^{\tilde{r}^\prime} }\lesssim\|\nabla \mathbb{G}_\omega\|_{L^{2-\frac{\varepsilon}{2}}}\|u\|_{L^{\frac{(2-\varepsilon)(4-\varepsilon)}{\varepsilon}(p-1)}}^{(p-1)}\lesssim \|u\|^{p-1}_{H^1_\alpha}.$$ Hence $$\left\| |\nabla u||u|^{p-1}\right\|_{L^{\tilde{r}^\prime} } \lesssim (\left\| \|\nabla g\|_{L^2}+ |c_*| \right) \|u\|^{p-1}_{H^1_\alpha}$$ and via [\[eq.ns77\]](#eq.ns77){reference-type="eqref" reference="eq.ns77"} we get $$\left\| |\nabla u||u|^{p-1}\right\|_{L^{\tilde{r}^\prime} } \lesssim \|u\|^{p}_{H^1_\alpha}.$$ So $$\left\| \mathfrak{K}(u)\right\|_{L^q(0,T)H^{1,r}_\alpha} \lesssim \|u_0\|_{H^1_\alpha} + T^{p/\tilde{q}'}\|u\|^{p}_{L^\infty(0,T) H^1_\alpha}$$ In a similar way we deduce $$\left\| \mathfrak{K}(u)-\mathfrak{K}(\tilde{u})\right\|_{L^{\infty}(0,T) H^1_\alpha} \lesssim \frac{1}{2} \left\| u-\tilde{u}\right\|_{L^\infty(0,T) H^1_\alpha}$$ so $\mathfrak{K}$ is a contraction in $L^\infty(0,T)H^1_\alpha.$ ◻ # Rescaling {#sec.res2} We recall the rescaling argument from section 5 in [@FGI22]. Let $$\phi(x) = g(x) + \frac{g(0)}{\beta_\alpha(\omega)} \mathbb{G}_\omega(x) \in \mathcal D(\Delta_\alpha)$$ so that $$(\omega -\Delta_\alpha) \phi = \Phi$$ and $$\begin{aligned} & \tilde{\phi} (x) = \phi \left( \frac{x}{\sqrt{\omega}}\right), \\ & \tilde{\Phi} (x) = \Phi \left( \frac{x}{\sqrt{\omega}}\right), \\ &\tilde{g} (x) = g \left( \frac{x}{\sqrt{\omega}}\right),\\ & \tilde{\alpha} = \alpha + \frac{1}{4\pi} \ln (\omega). \end{aligned}$$ It is easy to deduce $$\begin{aligned} & \beta_\alpha(\omega) = \beta_{\tilde{\alpha}}(1), \\ & \tilde{\phi} (x) = \tilde{g} (x) + \frac{\tilde{g}(0)}{\beta_{\tilde{\alpha}}(1)} \mathbb{G}_1(x) \end{aligned}$$ and moreover $$\label{eq.rescalingDelta} \widetilde{(\omega-\Delta_\alpha) \phi} = \omega (1-\Delta_{\tilde{\alpha}}) \tilde{\phi}.$$ Applying the spectral theorem we find $$\label{eq.rescalingDelta0.5} \widetilde{(\omega-\Delta_\alpha)^{s/2}\phi} = \omega^{s/2} (1-\Delta_{\tilde{\alpha}})^{s/2} \tilde{\phi}, \ \forall s \in [0,2].$$ We turn to the rescaling of the linear Schrödinger equation: $$(i\partial_t+\Delta_\alpha)u=0.$$ With the change of variable $$y=\frac{x}{\sqrt{\omega}}, \ \ s=\frac{t}{\omega},$$ it is easy to see that $$u\left(\frac{x}{\sqrt{\omega}},\frac{t}{\omega}\right)=e^{it\Delta_{\tilde\alpha}}\tilde{u_0}.$$ This change, works also for the NLS $$(i\partial_t+\Delta_\alpha)u= \mu u|u|^{p-1}.$$ We consider the rescaling $S_\lambda(u)(t,x) = \lambda^{-1} u \left(\frac{t}{\lambda^2}, \frac{x}{\lambda} \right)$ we have that $$\|S_\lambda(u)(t)\|^2_{L^{2}(\mathbb{R}^2)} = \|u(t)\|^2_{L^{2}(\mathbb{R}^2)}$$ and $$\lambda^{-1}(\omega-\Delta_\alpha)u(\frac{t}{\lambda^2},\frac{x}{\lambda})=\lambda(\omega-\Delta_{\alpha+\frac{1}{2\pi}\ln(\lambda)})(u(\frac{t}{\lambda^2},\frac{x}{\lambda})),$$ that gives $$\|S_\lambda(u)(t)\|^2_{L^2(\mathbb{R}^2)} = \|u(t)\|^2_{L^2(\mathbb{R}^2)}$$ and $$\begin{aligned} E(S_\lambda(u))=&\frac{1}{2}\int_{\mathbb{R}^2} (\omega-\Delta_\alpha) S_\lambda(u)(t) \overline{S_\lambda(u)(t)} -\frac{\omega}{2}\|S_\lambda (u)(t)\|^{2}_{L^{2}(\mathbb{R}^2)}+ \frac{\mu}{p+1} \|S_\lambda (u)(t)\|^{p+1}_{L^{p+1}(\mathbb{R}^2)}=\\ &=\lambda^{-2}\frac{1}{2}\|(\omega-\Delta_{\alpha^*})^{1/2}u(t)\|_{L^2}^2-\lambda^{-2}\frac{\omega}{2}\|u(t)\|^{2}_{L^{2}(\mathbb{R}^2)}+\lambda^{1-p}\frac{\mu}{p+1} \| u(t)\|^{p+1}_{L^{p+1}(\mathbb{R}^2)}, \end{aligned}$$ where $\alpha^*=\alpha+\frac{1}{2\pi}\ln(\lambda)$. Note that $\alpha^* > \alpha$ if $\lambda>1.$ Now it is clear that the mass critical case is defined by $\lambda^{-2} = \lambda^{1-p}$ with $\lambda >1$ so mass critical case is $p=3.$ # Asymptotics {#sec.as8} The function $K_0 ( \sqrt{z}|x|)$ has the asymptotics $$K_0( \sqrt{z}r) = - \log \left(r\right) - c(\omega) + O(|\sqrt{z}|r), \ \ \ c(\omega)= \log\left(\frac{\sqrt{\omega}}{2} \right)+\gamma$$ where $$r|\sqrt{z}| \leq 1, \ \ z \in \mathbb{C}, \mathrm{arg}(z) \in (-\pi, \pi).$$ This follows from the following asymptotic expansion (see (38), p.9 in [@BE]) $$\begin{aligned} & K_0(z)=-I_0(z) \log \left(\frac{z}{2}\right)+ \sum_{m=0}^{\infty} \left(\frac{z}{2}\right)^{2 m} \frac{\psi(m+1)}{ \left[(m !)^2\right]}= \\ = & - \log \left(\frac{z}{2}\right) -\gamma + O(\log(1/|z|) |z|^2), \ \ |z|\leq 1, \end{aligned}$$ where $\psi(z)=\Gamma^\prime(z)/\Gamma(z)$ and $\gamma$ is the Euler-Mascheroni constant. We have also $$\begin{aligned} & K_0^\prime(z)=-\frac{2}{z} + O(\log(1/|z|) |z|), \ \ |z|\leq 1, \\ & K_0^{\prime\prime}(z)=\frac{2}{z} + O(\log(1/|z|) ), \ \ |z|\leq 1, \end{aligned}$$ We have the following asymptotic expansion valid if $|z| \to \infty$ and $\mathrm{arg} z \in (-\pi, \pi)$ (see relation (20), section 7.23 in [@W95]) $$\label{eq.Bf7a23} K_\nu(z) = \left( \frac{\pi}{2} \right)^{1/2} e^{- (\log|z| +\mathrm{i}\mathrm{arg} z)/2} e^{-z} \left( 1+ O(|z|^{-1}) \right).$$

arxiv_math

--- abstract: | We use a recent formalism of quantum geodesics in noncommutative geometry to construct geodesic flow on the infinite chain $\cdots\bullet$--$\bullet$--$\bullet\cdots$. We find that noncommutative effects due to the discretisation of the line cause an initially real geodesic flow amplitude $\psi$ (for which the density is $|\psi|^2$) to become complex. This has been noted also for other quantum geometries and suggests that the complex nature of the wave function in quantum mechanics (and the interference effects that follow) may have its origin in a quantum/discrete nature of spacetime at the Planck scale. address: | Department of Mathematics, Bay Campus, Swansea University, SA1 8EN, UK\ Queen Mary University of London, School of Mathematical Sciences, London E1 4NS, UK\ author: - Edwin Beggs and Shahn Majid title: Quantum geodesic flow on the integer lattice line --- # Introduction In recent works[@Beg:geo; @BegMa:geo; @BegMa:cur], we have introduced a radically new way of thinking about geodesics even on a classical manifold $M$ (which then extends to noncommutative or 'quantum' Riemannian geometry). The idea is to think of not one geodesic at a time but a flow of geodesics much like in fluid dynamics where each particle moves along a geodesic. The tangent vectors to these geodesics form a geodesic-time dependent vector field $X(s)$, where $s$ is geodesic time, and it turns out that these obey a simple *geodesic velocity equation* $$\label{claveleq} \dot X+\nabla_XX=0$$ where dot is with respect to $s$. Next, instead of an actual (evolving) fluid particle density $\rho(s)$, we have an evolving wave function $\psi(s)$ or amplitude with $|\psi(s)|^2=\rho(s)$, and we solve for $\psi$ by the *amplitude flow* equation $$\label{claamp} \dot \psi+ \psi \kappa + X(\mathrm{d}\psi)=0,\quad \kappa=\frac{1}{2}{\rm div}(X).$$ If one considers bump functions, then classically this reproduces a bump travelling with velocity $X(s)$ evaluated at the bump, i.e. a classical geodesic as expected. If $\psi$ is real-valued then this is not really different from working with $\rho$, i.e. a fluid flow language. But when $\psi$ is complex-valued then there are possible interference effects as normally associated with quantum mechanics. Indeed, we make a quantum mechanics-like interpretation with respect to geodesic time $s$ and ([\[claamp\]](#claamp){reference-type="ref" reference="claamp"}) in the role of Schroedinger's equation. - This way of thinking of geodesics rips apart the usual concept of geodesics as points with tangent vectors and *puts them back in reverse order*, now starting with a velocity field and only afterwards introducing particle densities. - Geodesic time $s$ is no longer arclength or proper time of just one particle but the time experienced by an observer who watches all the particles evolving together. It provides a new coordinate for the physical system. - If $M$ is spacetime then $\psi(x,t)$ is (an evolving) spacetime amplitude with $|\psi(x,t)|^2$ the probability density to find a particle at $(x,t)$ at time $s$, which is unfamiliar. However, if the system is $t$-independent then one can focus on wave functions of a fixed frequency in $t$, then the spatial evolution looks more like quantum mechanics with respect to $s$ cf. [@BegMa:geo]. - The Ricci curvature $R_{\mu\nu}$ enters naturally via the *convected derivative* ${D\over D s} f={\partial\over\partial s} + X(\mathrm{d}f)$ of a function (the rate of change seen moving with the fluid flow). Then [@BegMa:cur] $${D\over Ds}{\rm div}(X)=-(\nabla_\mu X^\nu)(\nabla_\nu X^\mu) - R_{\mu\nu}X^\mu X^\nu$$ This new way of thinking about particles in GR is more field-theoretic (starting with the velocity field $X$ as fundamental) but takes getting used. Here, quantum geodesics on classical Minkowski spacetime and on the bicrossproduct model quantum spacetime $[x,t]=\mathrm{i}\lambda_P x_i$ in [@MaRue] (where $\lambda_P$ is the Planck scale) are studied in [@LiuMa]. One of the results there is that a 'point' in spacetime modelled as bump with width $\sigma$ moving with respect to $s$ gets a correction to its effective velocity of order $(\lambda_P/\sigma)^2$, i.e. there can be no such thing as a truly point particle *due to quantum gravity effects* if these are modelled by quantum spacetime. Another phenomenon in [@LiuMa], which we support now with new and direct calculations on the integer lattice, is that an initially real $\psi$ *does not stay real* when the geometry is quantum. Quantum geometry here means that we replace $C^\infty(M)$ by a possibly noncommutative $*$-algebra $A$. We replace differential forms on $M$ by an $A$-bimodule $\Omega^1$ etc., see below. We also replace the measure on $M$ with respect to which the probabilistic interpretation of $\psi$ is defined by a reference *state* or positive linear functional $\int:A\to \mathbb{C}$. In this case, $\psi(s)\in A$ is an element of a $*$-algebra and $|\psi(s)|^2$ is replaced by the positive operator $\psi(s)^*\psi(s)\ge 0$ but nevertheless behaves like a probability density when used to compute expectation values via $\int$. A reality condition on the geodesic velocity field $X$ ensures that the flow is unitary in the sense of preserving $\int \psi^*\psi$. As well as being applicable to actual quantum mechanics (where the usual Schroedinger equation can be seen as a certain quantum geodesic flow)[@BegMa:geo] and to relativistic quantum mechanics, the theory also applies to discrete geometry such as the lattice line in the present work. The algebra of functions $A$ is still commutative, namely functions on the vertices, but the space of differentials $\Omega^1$ is intrinsically noncommutative, being spanned as a vector space by the arrows $\{x\to y\}$ of the graph. Here, the product on either side by a function $f$ on the vertex set is $$f.(x\to y)=f(x)x\to y\quad \ne\quad f(y)x\to y=(x\to y).f;$$ once we allow such noncommutativity between differentials and functions then discrete geometry sits very naturally as a special case of noncommutative differential geometry. Section [2](#secpre){reference-type="ref" reference="secpre"} of the paper outlines the tools of *quantum Riemannian geometry* (QRG) as developed in works of the authors and others, see the text [@BegMa], used to write down the quantum versions of ([\[claveleq\]](#claveleq){reference-type="ref" reference="claveleq"}) and ([\[claamp\]](#claamp){reference-type="ref" reference="claamp"}). Section [3](#secZ){reference-type="ref" reference="secZ"} then shows how the theory works on the integer line $\mathbb{Z}$, for which the unique $*$-preserving QLC was found in [@ArgMa1]. We show by solving the flow equations that, even for the flat lattice where all edge lengths are the same, if $\psi$ starts off real then it evolves into a complex wave function due to the noncommutative nature of the differential geometry of $\mathbb{Z}$. This is also known for geodesics on $A=M_2(\mathbb{C})$ in [@BegMa:cur], but here it arises out of the discreteness. This suggests that the set-up of quantum mechanics with complex wave functions could itself have its origin in quantum gravity leading to spacetime being better modelled[@Sny; @Ma:pla] as discreteness/noncommutative at the Planck scale. We note that the methods of QRG used here are very different from the approach to noncommutative geometry of Connes[@Con] and others coming out of operator theory and cyclic cohomology. The more constructive QRG approach grew out of [@BegMa:rie; @BegMa:gra] and more broadly including experience with quantum groups. It was applied in Physics to models of quantum gravity and Beckenstein-Hawking radiation in [@Ma:sq; @Ma:haw; @ArgMa2; @LirMa] and to Kaluza-Klein models for elementary particles in [@ArgMa4; @LiuMa2]. Here [@ArgMa4] also provides an introduction for Physicists to the QRG formalism. Quantum geodesics, however, link back to operator theory via the KSGNS construction and KK-theory as a motivation in [@Beg:geo], suggesting a new direction for convergence of these approaches. # Recap of the formalism {#secpre} We recap elements of QRG from [@BegMa] and then of quantum geodesics from [@Beg:geo; @BegMa:geo; @BegMa:cur]. ## Quantum Riemannian Geometry We start with a $*$-algebra $A$, which will play the role of our 'coordinate algebra' but could be noncommutative. We require this to have a differential structure in the form of an $A$-bimodule $\Omega^1$ of differential forms equipped with a map $\mathrm{d}:A\to \Omega^1$ obeying the Leibniz rule $$\mathrm{d}(ab)=a\mathrm{d}b+ (\mathrm{d}a)b$$ and such that $\Omega^1$ is spanned by elements of the form $a\mathrm{d}b$ for $a,b\in A$. This can be extended to a full differential graded 'exterior algebra' $\Omega$ although not uniquely (there is a unique maximal extension). We assume that at least $\Omega^2$ has been specified and that $*$ extends at least to $\Omega^1$ as a graded-involution (i.e., with a minus sign on swapping odd degrees) and commutes with $\mathrm{d}$. Next, a metric means for us an element $\mathfrak{g}\in\Omega^1\mathop{{\otimes}}_A\Omega^1$ which is invertible in the sense of a bimodule map $(\ ,\ ):\Omega^1\mathop{{\otimes}}_A\Omega^1\to A$ obeyng the usual requirements as inverse to $\mathfrak g$, which forces $\mathfrak g$ to be central[@BegMa:gra]. There are ways to go beyond, for example starting with $(\ ,\ )$ or with a metric as a map from $\Omega^1$ to vector fields, as in the hermitian version of the theory[@BegMa Chap. 8.5]. Next, a QLC or *quantum Levi-Civita connection* is a left bimodule connection $(\nabla,\sigma)$ on $\Omega^1$ which is metric compatible and torsion free in the sense in the sense $$\label{nablag} \nabla \mathfrak{g}:=(\nabla\mathop{{\otimes}}\mathrm{id}+(\sigma\mathop{{\otimes}}\mathrm{id})(\mathrm{id}\mathop{{\otimes}}\nabla))\mathfrak g=0,\quad T_\nabla:= \wedge\nabla-\mathrm{d}=0$$ for a left bimodule connection $\nabla:\Omega^1\to \Omega^1\mathop{{\otimes}}_A\Omega^1$ characterised by[@DVM; @Mou] $$\nabla(a.\omega)=\mathrm{d}a\mathop{{\otimes}}\omega+ a.\nabla\omega,\quad \nabla(\omega.a)=\sigma(\omega\mathop{{\otimes}}\mathrm{d}a)+(\nabla\omega).a,$$ where the 'generalised braiding' $\sigma:\Omega^1\mathop{{\otimes}}_A\Omega^1\to \Omega^1\mathop{{\otimes}}_A\Omega^1$ is assumed to exist and is uniquely determined by the second equation. The thinking behind these formulae is that the left factor in the output of $\nabla$ would classically evaluate against a vector field to provide a covariant derivative along it. There is an analogous theory with right bimodule connections where the right hand factor of the output would be evaluated against a vector field to recover classical geometry. In terms of $(\ ,\ )$, metric compatibility amounts to $$\label{metnabla} \mathrm{d}(\omega,\eta)=(\mathrm{id}\mathop{{\otimes}}(\ ,\ ))(\nabla\omega\mathop{{\otimes}}\eta+(\sigma\mathop{{\otimes}}\mathrm{id})(\omega\mathop{{\otimes}}\nabla\eta)),$$ see [@BegMa Lemma 8.4] for the equivalence with ([\[nablag\]](#nablag){reference-type="ref" reference="nablag"}). We also require 'reality' of the metric and for $\nabla$ to be $*$-preserving[@BegMa:rie; @BegMa] in the sense $$\mathfrak{g}^\dagger=\mathfrak{g},\quad \sigma\,\dag\,\nabla(\xi^*)=\nabla\xi,\quad \dagger:={\rm flip}(*\mathop{{\otimes}}*).$$ Finally, we will need $\mathfrak{X}={}_A\hom(\Omega^1,A)$, the space of left quantum vector fields defined as left $A$-module maps $X: \Omega^1\to A$. This has a bimodule structure $$(a.X.b)(\omega)= (X(\omega.a))b$$ for all $\omega\in \Omega^1$ and $a,b\in A$, and inherits from $\nabla$ a dual right connection $$\nabla_\mathfrak{X}:\mathfrak{X}\to \mathfrak{X}\mathop{{\otimes}}_A\Omega^1,\quad \sigma_\mathfrak{X}: \Omega^1\mathop{{\otimes}}_A\mathfrak{X}\to \mathfrak{X}\mathop{{\otimes}}_A\Omega^1.$$ characterised by $$\mathrm{d}(\mathrm{ev}(\omega\mathop{{\otimes}}X))=(\mathrm{id}\mathop{{\otimes}}\mathrm{ev})(\nabla \omega \mathop{{\otimes}}X)+(\mathrm{ev}\mathop{{\otimes}}\mathrm{id})(\omega\mathop{{\otimes}}\nabla_\mathfrak{X}X)$$ for all $\omega\in \Omega^1$ and $X\in \mathfrak{X}$. We use this to define the divergence of a vector field by $${\rm div}:=\mathrm{ev}\circ\sigma_\mathfrak{X}^{-1}\nabla_\mathfrak{X},\quad \mathrm{ev}:\Omega^1\mathop{{\otimes}}_A\mathfrak{X}\to A,\quad \mathrm{ev}(\omega\mathop{{\otimes}}_A X)=X(\omega)$$ If $\nabla$ is metric compatible then so is $\nabla_\mathfrak{X}$ in a suitable sense and then one can prove that $${\rm div}(\mathfrak{g}_2(\omega))=(\ ,\ )\nabla \omega;\quad \mathfrak{g}_2:\Omega^1\to \mathfrak{X},\quad \mathfrak{g}_2(\omega)=(\ ,\omega)$$ for all $\omega\in \Omega^1$, as would be usual in GR (i.e., one can compute divergences as the codifferential of the 1-form corresponding via the metric to a vector field). The bottom line is, all the basic tools of Riemannian Geometry extend to this set-up provided one is rather careful. ## Quantum geodesics {#secgeo} We are now ready for quantum geodesics. Although the theory is more general, we will focus on 'wave functions' in $\psi\in E=C^\infty(\mathbb{R}, A)$ where $s\in\mathbb{R}$ will be the 'geodesic time' parameter. We assume that $A$ is equipped with enough structure for such functions (or some variant of them) to make sense. Likewise, we let $X_s$ be a time-dependent left vector field on $A$ and $\kappa_s$ another time-dependent element of $A$. These obey the *geodesic velocity equations* if $$\label{veleqX} \dot X_s +[X_s,\kappa_s]+(\mathrm{id}\mathop{{\otimes}}X_s)\nabla_\mathfrak{X}(X_s)=0$$ where dot means differential with respect to $s$. Next, we require $\int: A\to \mathbb{C}$ to be a non-degenerate positive linear functional (so we can think of it as a probability measure if normalisable so that $\int 1=1$) and define ${\rm div}_{\int}(X)$ of a vector field by $$\label{divint} \int a\, {\rm div}_{\int}(X)+ \int X(\mathrm{d}a)=0$$ for all $a\in A$. We require that the geodesic velocity field and $\kappa$ at each $s$ to obey the *unitarity conditions* $$\label{unitarity} \int \kappa^* a+a\kappa+X(\mathrm{d}a)=0,\quad \int X(\omega^*)=\int X(\omega)^*$$ for all $a\in A$ and all $\omega\in\Omega^1$. Note that if the second of ([\[unitarity\]](#unitarity){reference-type="ref" reference="unitarity"}) applies then one says that $X$ is *real with respect to $\int$* and then we can canonically solve the first equation in the pair by $$\kappa={1\over 2}{\rm div}_{\int}(X),$$ see after [@BegMa:cur Def. 4.5]. It is not automatic that if $X_s$ is initially real with respect to $\int$ that it necessarily remains so under the geodesic velocity equation, we have to impose this as a further *improved auxiliary equation* obtained as the difference between ([\[veleqX\]](#veleqX){reference-type="ref" reference="veleqX"}) and its conjugate under the reality assumption. Given this geodesic velocity vector field, we then require the *amplitude flow equation* $$\dot\psi= -\psi\kappa_s-X_s(\mathrm{d}\psi),$$ where $\mathrm{d}$ acts on $\psi(s)\in A$ and dot is with respect to $s$ as before. The above conditions ensure that $\int\psi^*\psi$ is constant in time, which is needed for a probabilistic interpretation. Finally, the above works for any $\int$ but in Riemannian geometry we would normally use a particular measure defined by $\sqrt{|g|}$ characterised as vanishing on a total divergence. We do the same and say that $\int$ is *divergence compatible* (with $\nabla$) if $$\int{\rm div} X=0$$ for all $X\in \mathfrak{X}$. This is equivalent to saying that ${\rm div}_{\int}={\rm div}$. If it is also (as it will be in our case) a trace so that $\int (ab)=\int (ba)$ then as a special case of the theory in [@BegMa:cur], $\mathfrak{X}$ acquires a $*$ operation characterised by $$X^*(\omega)=\big(\mathrm{ev}\circ\sigma_\mathfrak{X}^{-1}(X\mathop{{\otimes}}\omega^*)\big)^*$$ for all $\omega\in\Omega^1$ and such that our previous $X$ being real with respect to $\int$ now appears as $X^*=X$. # Quantum geodesics on $\mathbb{Z}$ {#secZ} We illustrate the above machinery on the integer lattice. Here $A=C(\mathbb{Z})$ denotes all (or some class of) functions on the vertices of the lattice line labelled by integers $\mathbb{Z}$. These are viewed as a graph with arrows in both directions $i\to i+1$ and $i\leftarrow i+1$ for all $i$. In fact this is a Cayley graph and we can formally define $e^+$ as the sum of all increasing arrows and $e^-$ as the sum of all decreasing arrows. We do not need to worry about this sum and can just take $e^\pm$ as, by definition, a basis of $\Omega^1$ over $A$, i.e. every 1-form is of the form $\omega=\omega_+ e^++ \omega_- e^i$ for $\omega_\pm\in A$. We just need not know that the bimodule commutation relations and exterior derivative are $$e^\pm f= R_\pm(f) e^\pm,\quad \mathrm{d}f= (\partial_\pm f)e^\pm;\quad R_\pm(f)(i)=f(i\pm 1),\quad \partial_\pm=R_\pm-\mathrm{id}.$$ So the $R_\pm$ are just lattice shift up and down and $\partial_\pm$ are just the usual lattice finite difference. A metric $\mathfrak{g}\in\Omega^1\mathop{{\otimes}}_A\Omega^1$ is then just an element of the form $$\mathfrak{g}= g_+ e^+\mathop{{\otimes}}e^- + g_- e^-\mathop{{\otimes}}e^+;\quad g_\pm \in \mathbb{R}\setminus\{0\}$$ where the other terms are not allowed as the metric has to be central. The inverse is $$(e^+,e^-)={1\over R_+(g_-)},\quad (e^-,e^+)={1\over R_-(g_+)}.$$ These functions correspond to square-lengths attached to the arrows as $g_\pm(i)=g_{i\to i\pm 1}$ and it is natural to suppose that the metric is 'edge symmetric' i.e. independent of the arrow direction[@Ma:sq], which amounts to $g_-=R_-(g_+)$. In this case we have only one independent function $g:=g_+$ with $g_i$ being the square length of the link $i$--$i+1$, $$\cdots-\bullet_{i-1} {\buildrel g_{i-1}\over -}\bullet_i{\buildrel g_i\over -}\bullet_{i+1}{\buildrel g_{i+1}\over-}\bullet_{i+2}-\cdots.$$ So edge symmetric metrics correspond exactly to what you would expect for a metric on a lattice. We also let $$\rho_\pm=R_\pm({g_\pm\over g_\mp}),\quad \rho:=\rho_+$$ as 'ratio derivative' of the metric function $g$. Then [@ArgMa1] showed that there is a unique ($*$-preserving, left) QLC $$\nabla e^\pm=(1-\rho_\pm) e^\pm\mathop{{\otimes}}e^\pm,\quad \sigma(e^\pm\mathop{{\otimes}}e^\pm)=\rho_\pm e^\pm\mathop{{\otimes}}e^\pm ,\quad \sigma(e^\pm\mathop{{\otimes}}e^\mp)=e^\mp\mathop{{\otimes}}e^\pm.$$ The exterior algebra here is supposed to be the standard one where $e^\pm$ are Grassmann and anticommute with each other. If we write $\nabla e^a=-\Gamma^a{}_{bc}e^b\mathop{{\otimes}}e^c$ where indices run over $\pm$ then these same Christoffel symbols define $\nabla_\mathfrak{X}$ on a dual basis $f_a$ by $$\nabla_\mathfrak{X}f_a=f_b\mathop{{\otimes}}\Gamma^a{}_{bc} e^c.$$ This applies when there is a basis of 1-forms, as is the case of any Cayley graph (where the vertices form a group and the arrows are right translation by a fixed set of generators). Now, in our case $\Gamma$ has only two nonzero entries namely $$\Gamma^\pm{}_{\pm\pm}=\rho_\pm-1$$ hence $$\nabla_\mathfrak{X}f_\pm = f_\pm\mathop{{\otimes}}(\rho_\pm-1) e^\pm,\quad \sigma_\mathfrak{X}(e^\pm\mathop{{\otimes}}f_\pm)= f_\pm\mathop{{\otimes}}\rho_\pm e^\pm,\quad \sigma_\mathfrak{X}(e^\pm\mathop{{\otimes}}f_\mp)= f_\mp\mathop{{\otimes}}e^\pm.$$ Using the commutation rules, $$e_\pm f= R_\pm(f) e^\pm,\quad f f_\pm= f_\pm R_\pm(f)$$ we check this is consistent with being a right connection, $$\begin{aligned} \nabla_\mathfrak{X}(f f_\pm)&=\sigma_\mathfrak{X}(\mathrm{d}f\mathop{{\otimes}}f_\pm)+ f f_\pm \mathop{{\otimes}}(\rho_\pm-1) e^\pm=(\partial_a f)\sigma_\mathfrak{X}(e^a\mathop{{\otimes}}f_\pm)+ f_\pm \mathop{{\otimes}}(\rho_\pm-1)e^\pm f\\ &=(\partial_\mp f) f_\pm \mathop{{\otimes}}e^\mp + f_\pm \mathop{{\otimes}}\rho_\pm e^\pm \partial_\pm f+ f_\pm \mathop{{\otimes}}(\rho_\pm-1) e^\pm f\\ &= f_\pm \mathop{{\otimes}}(\partial_\mp R_\pm f)e^\mp +f_\pm \mathop{{\otimes}}e^\pm \partial_\pm f+ f_\pm \mathop{{\otimes}}(\rho_\pm-1) e^\pm \partial_\pm f+ f_\pm \mathop{{\otimes}}(\rho_\pm-1) e^\pm f\\ &= f_\pm \mathop{{\otimes}}(\partial_a R_\pm f)e^a+ f_\pm \mathop{{\otimes}}(\rho_\pm-1) e^\pm R_\pm f= \nabla_\mathfrak{X}(f_\pm R_\pm f)\end{aligned}$$ as required. Then $$\sigma_\mathfrak{X}^{-1}(f_\pm\mathop{{\otimes}}e^\pm)={1\over R_\mp(\rho_\pm)}e^\pm\mathop{{\otimes}}f_\pm,\quad \sigma_\mathfrak{X}^{-1}(f_\pm\mathop{{\otimes}}e^\mp)=e^\mp\mathop{{\otimes}}f_\pm,$$ $$\begin{aligned} {\rm div}(f_\pm)&=\mathrm{ev}(\sigma_\mathfrak{X}^{-1}\nabla_\mathfrak{X}f_\pm)=\sigma_\mathfrak{X}^{-1}(f_\pm(\rho_\pm-1)\mathop{{\otimes}}e^\pm)=R_\mp(\rho_\pm-1)\mathrm{ev}(\sigma_\mathfrak{X}^{-1}(f_\pm\mathop{{\otimes}}e^\pm))\\ &=R_\mp(\rho_\pm-1)\mathrm{ev}({1\over R_\mp(\rho_\pm)}e^\pm\mathop{{\otimes}}f_\pm)= 1- { g_\mp\over g_\pm}\end{aligned}$$ **Lemma 1**. *$\int f=\sum_i f\mu$ defined by measure $\mu$ is divergence-compatible iff $\rho$ is constant and ${R_\pm(\mu)\over\mu}=\rho^{\pm 1}$. Here, one may take $\mu=g$.* *Proof.* For the divergence compatibility, we need for all $f$, $$\sum \mu f {\rm div} f_\pm = - \sum \mu f_\pm(\mathrm{d}f)= -\sum\mu f_\pm((\partial_a f)e^a)=-\sum\mu \partial_\pm f=\sum\mu f-\sum R_\mp(\mu)f$$ which using $R_\pm(\mu)=\mu (1- {\rm div}(f_\mp))$, needs $$R_\pm(\mu)=\mu { g_\pm\over g_\mp};\quad R_+(\mu)=\mu {g\over R_- (g)},\quad R_-(\mu)=\mu {R_- (g)\over g}$$ which requires for a solution a constraint on $g$ $$R_+(g) R_-(g)= g^2,\quad g(i)=\left({ g(1)\over g(0)}\right)^i g(0).$$ We see that the condition on $g$ amounts to $\rho(i)=\rho(0)=g(1)/g(0)$, i.e. to $\rho$ a constant (this is equivalent geometrically to $\nabla$ having zero curvature) as well as $$\rho_\pm=\rho^{\pm 1}$$ The simplest solution for $\mu$ is then $\mu=g$. $\hfill\square$ Proceeding for $\int$ divergence-compatible, $e_\pm^*=-e_\mp$ implies $$f_\pm^*= -f_\mp {1\over R_{\mp}(\rho_\pm)}= - f_\mp {g_\mp\over g_\pm},\quad f_+^*=-{f_-\over\rho},\quad f_-^*=-\rho f_+.$$ Hence, writing $X=f_+ X^++ f_- X^-$, this is real with respect to our $*$ on vector fields iff $$\label{XrealZ} X^\pm{}^*=-\rho_\pm R_\pm(X^\mp)=-\rho^{\pm 1}R_\pm (X_\mp).$$ Next, from the above, we have $$\label{kappaZ} \kappa={1\over 2}{\rm div}(X)={1\over 2}((\mathrm{id}- {R_-\over \rho})X^++ ((\mathrm{id}-\rho R_+)X^-),$$ which is real. We put this into the velocity equation ([\[veleqX\]](#veleqX){reference-type="ref" reference="veleqX"}), which for $\nabla_\mathfrak{X}$ above can be written as $$\begin{aligned} \label{veleqZa} \dot X^+ &= \partial_+(\kappa)X^+ + (1-\rho)X^+ X^+ -\rho\partial_+(X^+)X^+- \partial_-(X^+)X^- \\ \label{veleqZb} \dot X^- &= \partial_-(\kappa)X^- + (1-\rho^{-1})X^-X^--\rho^{-1}\partial_-(X^-)X^-- \partial_+(X^-)X^+\end{aligned}$$ where $$\partial_+\kappa={1\over 2}((1-\rho R_+)\partial_+ X^-+ (\partial_++ {1\over\rho}\partial_-)X^+),\quad \partial_-\kappa={1\over 2}((1-{1\over \rho} R_-)\partial_- X^++ (\partial_-+\rho\partial_+)X^-).$$ Now, applying $*$ to ([\[veleqZb\]](#veleqZb){reference-type="ref" reference="veleqZb"}) and subtracting from ([\[veleqZa\]](#veleqZa){reference-type="ref" reference="veleqZa"}) gives $$((1-\rho)({1\over \rho^2}R_-- R_+) X^+)X^++((\rho\partial_++\rho-1)\partial_+ X^-)X^++(\partial_- X^+)X^-+ \rho (\partial_+ X^+)R_+^2(X^-)=0$$ which simplifies to the 'improved auxiliary equation' $$\label{auxZ} (\partial_-+\rho\partial_+)(X^+R_+X^-)=(1-\rho)((R_+-{R_-\over \rho^2})X^+)X^+$$ where $X^+R_+X^-$ is real and $\partial_-+\rho\partial^+$ if $\rho=1$ would be the usual finite different Laplacian $\Delta_\mathbb{Z}$ (but is not the QRG Laplacian which would be $\Delta =-{1+\rho\over h}\Delta_\mathbb{Z}$.) Note that if we take the \* of the 2nd velocity equation and compare with the first, we obtain instead $$(\partial_++{1\over \rho}\partial_-)(X^-R_- X^+)=(1-{1\over\rho})(( R_--\rho^2 R_+)X^-)X^-$$ which is equivalent to $*$ applied to ([\[auxZ\]](#auxZ){reference-type="ref" reference="auxZ"}). The latter when $\rho\ne 1$ amounts to the implicit condition $$(L X^+)X^+=\rho^2 R_+((L X^-)X^-);\quad L=R_+-{1\over \rho^2}R_-.$$ and is needed precisely so that any initially real $X$ (in our sense ([\[XrealZ\]](#XrealZ){reference-type="ref" reference="XrealZ"})) stays real during the evolution, which in turn is needed for unitarity ([\[unitarity\]](#unitarity){reference-type="ref" reference="unitarity"}). Once we have found the geodesic velocity field, it only remains to solve for the amplitude flow itself. This means we solved for $\psi_s\in C(\mathbb{Z})$ such that $\nabla_E\psi=0$, which is $$\label{ampZ} \dot \psi = -X_s(\mathrm{d}\psi )-\psi \kappa_s= - (f_a X^a)((\partial_b \psi)e^b)- \psi \kappa=-(\partial_a\psi)X^a-\psi\kappa.$$ ## Geodesics for the constant metric $\rho=1$ on $\mathbb{Z}$ Here real $X$ means $X^+{}^*=-R_+(X^-)$ and $X^-{}^*=-R_-(X^+)$ and $$\begin{aligned} \kappa&=-{1\over 2}(\partial_-X^++ \partial_+X^-)\\ 2\partial_+ \kappa&=-( R_+-1)(R_--1)X^+-\partial_+^2 X^-=\Delta_\mathbb{Z}X^+ - \partial_+^2 X^-.\end{aligned}$$ The geodesic velocity eqn is $$\dot X^+=(\partial_+ \kappa)X^+-(\partial_+ X^+)X^+-(\partial_- X^+)X^-,\quad \dot X^-=(\partial_-\kappa)X^--(\partial_-X^-)X^--(\partial_+X^-)X^+.$$ Applying $*$ to the second and comparing to the first gives $$(\partial_+^2X^-)X^++(\partial_- X^+)X^-+(\partial_+X^+)R_+^2 X^-=0$$ which simplifies to $$\Delta_\mathbb{Z}(X^+ R_+ X^-)=0,$$ where $X^+R_+X^-$ is real. Solving the first velocity equation and this 'improved auxiliary equation' for real $X$ is equivalent to solving both velocity equations for real $X$. We first look at this latter moduli of solutions of the auxiliary equation as the subspace on which we will be solving the 1st velocity equation. Here among real non-negative functions $$\ker\Delta_\mathbb{Z}\cup\{{\rm non-negative}\}=\mathbb{R}_{\ge 0}1$$ (Here a general zero mode of $\Delta_\mathbb{Z}$ is $Y(i)=i \alpha - (i-1)\beta$ for $\alpha=Y(1),\beta=Y(0)$ but we need $\alpha\ge\beta\ge0$ for $Y$ to be positive for positive $i$ and $\beta\ge\alpha\ge0$ for $Y$ to be positive for negative $i$.) We therefore can parameterize members of this aux moduli space as $$X^+=r e^{\mathrm{i}\theta},\quad X^-=-R_-(re^{-\mathrm{i}\theta});\quad r\in \mathbb{R}_{\ge 0},\quad \theta\in \mathbb{C}(\mathbb{Z}).$$ All of this applies at each geodesic time $s$ and we now consider $r,\theta$ as functions of $s$ and solve the first vel equation which in terms of $X:=X^+,r$ looks like $$\dot X={1\over 2}(\Delta_\mathbb{Z}(X+ {r^2\over X}))X- (\partial_+ X+ \partial_-({r^2\over X}))X+ \partial_- r^2={1\over 2}\left((R_--R_+)(X-{r^2\over X})\right)X$$ which, in terms of $r,\theta$, is $$\dot\theta= r(R_--R_+) (\sin(\theta)), \quad \dot r=0$$ for the real and imaginary parts. Thus, $r$ becomes a constant parameter and any initial $\theta_0$ at $s=0$ evolves according to the equation shown for the angle phase $\theta$. For example, if we start at a sampled Gaussian peaked at $i=50$ then the flow is plotted (smoothly interpolated in $i$ for visual clarity) in Figure [\[XsolZ\]](#XsolZ){reference-type="ref" reference="XsolZ"}(a). (By contrast, if we start at $\theta_0$ constant then this remains constant for all time.) The divergence for our form of $X^\pm$ is $$\kappa= -{1\over 2}\partial_-(X+{r^2\over X})=-r\partial_-\cos(\theta)$$ and plotted in Figure [\[XsolZ\]](#XsolZ){reference-type="ref" reference="XsolZ"}(b). $$\includegraphics[scale=.9]{XsolZ.pdf}$$ Finally, the amplitude flow is $$\dot\psi = - r(\partial_+\psi)e^{\mathrm{i}\theta}+ r(\partial_-\psi)R_-( e^{-\mathrm{i}\theta})-\psi \kappa.$$ An example is shown in Figure [\[psiZ\]](#psiZ){reference-type="ref" reference="psiZ"}. We see that it acquires an imaginary component and becomes more wavelike with time. The imaginary component appears sooner if we start nearer (or on top of) the initial bump in $\theta$. Thus we see literally how a real bump function approximating a point in classical geometry evolves over geodesic time into a complex wave packet. $$\includegraphics[scale=0.8]{psiZ.pdf}$$ ## Geodesics for generic metrics on $\mathbb{Z}$ For a generic metric on $\mathbb{Z}$, $\int$ defined by $\mu$ won't be divergence-compatible. But we can still define ${\rm div}_{\int}$ and require our vector field $X$ to be real with respect to $\int$. Here $$\int X(\mathrm{d}f)=\int (\partial_\pm f) X(e^\pm)=\sum_i \mu (R_\pm f-f)X^\pm=\sum_i f (R_\mp (\mu X^\pm)- \mu X^\pm)$$ (sum over the $\pm$ understood). For this to equal $-\int f {\rm div}_{\int}(X)$ for all $f$ as in ([\[divint\]](#divint){reference-type="ref" reference="divint"}), we need $$\label{divintZ} {\rm div}_{\int}(X)= -{1\over\mu}(\partial_+(\mu X^+)+\partial_-(\mu X^-)).$$ Next, $X$ is real with respect to $\int$ if the second of ([\[unitarity\]](#unitarity){reference-type="ref" reference="unitarity"}) holds, which means $$\int X((fe^\pm)^*)=\int X(e^\pm{}^* f^*)=-\int R_\mp(f^*) X(e^\mp )=-\int R_\mp(f^*) X^\mp=-\sum_i f^* R_\pm(\mu X^\mp)$$ has to equal $$\int X(fe^\pm)^*=\int f^*X^\pm{}^*=\sum_i f^* \mu X^\pm{}^*$$ for all $f$. This gives us reality with respect to $\int$ as $$\label{XrealintZ} X^\pm{}^*=-{R_\pm(\mu X^\mp)\over\mu}.$$ We impose this on $X$ at all times $s$ and let $\kappa={1\over 2}{\rm div}_{\int}(X)$ so that the flow is unitary. One can check that $\kappa$ is then real. The velocity equation ([\[veleqX\]](#veleqX){reference-type="ref" reference="veleqX"}) still gives ([\[veleqZa\]](#veleqZa){reference-type="ref" reference="veleqZa"})-([\[veleqZb\]](#veleqZb){reference-type="ref" reference="veleqZb"}) but with $\rho^{-1}$ in the latter replaced by $\rho_-$ since the metric is now arbitrary. Comparison of one half of these velocity equations with the conjugate of the other half to give the improved auxiliary equation is now more complicated. One can also add a driving force term for maximal generality[@BegMa:gra]. 9 J. Argota-Quiroz and S. Majid, Quantum gravity on polygons and $\mathbb{R}\times \mathbb{Z}_n$ FLRW model, Class. Quant. Grav. 37 (2020) 245001 J. Argota-Quiroz and S. Majid, Fuzzy and discrete black hole models, Class. Quant. Grav. 38 (2021) 145020 J. Argota-Quiroz and S. Majid, Quantum gravity on finite spacetimes and dynamical mass, Corfu Summer Institute 2021: School and Workshops on Elementary Particle Physics and Gravity, PoS (2022) 210 (41pp) E.J. Beggs, Noncommutative geodesics and the KSGNS construction, J. Geom. Phys. 158 (2020) 103851 E.J. Beggs and S. Majid, *Quantum Riemannian Geometry*, Grundlehren der mathematischen Wissenschaften, Vol. 355, Springer (2020) 809pp E.J. Beggs and S. Majid, \*-compatible connections in noncommutative Riemannian geometry, J. Geom. Phys. 61 (2011) 95--124 E.J. Beggs and S. Majid, Quantum geodesics in quantum mechanics, arXiv:1912.13376 (math-ph) E.J. Beggs and S. Majid, Gravity induced by quantum spacetime, Class. Quant. Grav. 31 (2014) 035020 E. Beggs and S. Majid, Quantum geodesic flows and curvature, Lett. Math. Phys. (2023) 113:73 E.J. Beggs and S. Majid, Quantum geodesics flows on graphs, preprint in preparation. A. Connes, Noncommutative Geometry, Academic Press, Inc., San Diego, CA, 1994 M. Dubois-Violette and P.W. Michor, Connections on central bimodules in noncommutative differential geometry, J. Geom. Phys. 20 (1996) 218--232 C. Liu and S. Majid, Quantum geodesics on quantum Minkowski spacetime, J. Phys. A 55 (2022) 424003 C. Liu and S. Majid, Quantum Kaluza-Klein theory with $M_2(\mathbb{C})$, J. High Energ. Phys. (2023) E. Lira-Torres and S. Majid, Quantum gravity and Riemannian geometry on the fuzzy sphere, Lett. Math. Phys. (2021) 111:29 S. Majid, Hopf algebras for physics at the Planck scale, Class. Quant. Grav. 5 (1988) 1587--1607 S. Majid, Quantum gravity on a square graph, Class. Quant. Grav 36 (2019) 245009 S. Majid, Noncommutative Riemannian geometry of graphs, J. Geom. Phys. 69 (2013) 74--93 S. Majid, Quantum Riemannian geometry and particle creation on the integer line, Class. Quant. Grav 36 (2019) 135011 S. Majid and H. Ruegg, Bicrossproduct structure of the $\kappa$-Poincare group and non-commutative geometry, Phys. Lett. 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arxiv_math

--- abstract: | A blocker of $123$-avoiding permutation matrices refers to the set of zeros contained within an $n\times n$ $123$-forcing matrix. Recently, Brualdi and Cao provided a characterization of all minimal blockers, which are blockers with a cardinality of $n$. Building upon their work, a new type of blocker, flag-shaped blockers, which can be seen as a generalization of the $L$-shaped blockers defined by Brualdi and Cao, are introduced. It is demonstrated that all flag-shaped blockers are minimum blockers. The possible cardinalities of flag-shaped blockers are also determined, and the dimensions of subpolytopes that are defined by flag-shaped blockers are examined. author: - Megan Bennett$^\ast$ - Lei Cao$^{\ast\ast}$ title: Flag-Shaped Blockers of 123-Avoiding Permutation Matrices --- **Key words.** 123-pattern, 123-avoiding permutation matrices, blockers **AMS subject classifications.** 05A05, 15A03, 15B99, 52A20. # Introduction Let $n$ be a positive integer and let ${\mathcal P}_n$ be the set of $n\times n$ permutation matrices corresponding to the set ${\mathcal S}_n$ of permutations of $\{1,2,\ldots,n\}$. A permutation $\sigma$ of $\{1,2,\ldots,n\}$ *contains a $123$-pattern* provided it contains an increasing subsequence of length $3$ and, otherwise, is *$123$-avoiding*. In terms of the $n\times n$ permutation matrix $P$ corresponding to $\sigma$, $P$ contains a $123$-pattern provided the $3\times 3$ identity matrix $I_3$ is a submatrix of $P$. If $A$ is an $n\times n$ $(0,1)$-matrix, then $A$ is *$123$-forcing* provided every permutation matrix $P\le A$ (entrywise order) contains a $123$-pattern; the matrix $A$ thus *blocks* all $123$-avoiding permutations in that every $123$-avoiding permutation matrix has at least one $1$ in a position of a $0$ of $A$. The set of zeros contained in a $123$-forcing matrix $A$ is called a blocker of $123$-avoiding permutation matrices. A blocker $\mathcal{B}$ is *minimum* if removing any element from $\mathcal{B}$ makes it no longer a blocker. If there does not exist a blocker $\mathcal{D}$ such that the cardinality of $\mathcal{D}$ is less than the cardinality of $\mathcal{B},$ then $\mathcal{B}$ is *minimal*. Thus, all minimal blockers are minimum. $123$-forcing matrices and blockers of $123$-avoiding permutation matrices have been previously investigated in [@BC; @BC2; @BC1]. Denote the $n\times n$ matrix of all $1'$s as $J_n$. We define the cyclic-Hankel decomposition of $J_n$ into $n$ permutation matrices by starting with row $1$ and cyclically permuting it as for circulant matrices but in a right-to-left fashion, rather than the left-to-right fashion. We call these permutation matrices cyclic-Hankel permutation matrices or cyclic-Hankel diagonals. The cyclic-Hankel decomposition is illustrated for $n=6$ using letters $a, b, c, d, e, f$ below: $$\left[\begin{array}{c|c|c|c|c|c} a&b&c&d&e&f\\ \hline b&c&d&e&f&a\\ \hline c&d&e&f&a&b\\ \hline d&e&f&a&b&c\\ \hline e&f&a&b&c&d\\ \hline f&a&b&c&d&e\end{array}\right].$$ **Lemma 1** (Lemma 2.2 in [@BC]). *The number of $0$'s in an $n\times n$ $123$-forcing $(0,1)$-matrix is at least $n$. An $n\times n$ $123$-forcing $(0,1)$-matrix with exactly n $0$'s contains exactly one $0$ from each cyclic-Hankel permutation matrix.* One particular importance of all blockers with exact $n$ zeros is the $L$-shaped blockers, a set of $n$ adjacent positions denoted $L_n(s,r)$, where $s$ corresponds to the width of the blocker, $r$ corresponds to the height, and $r+s=n+1$. Clearly, an $n\times n$ $(0,1)$-matrix $A$ with a row or column of all $0$'s is $123$-forcing, since there do not exist any permutation matrices $P\le A$. If the zeros are not contained in one row or one column, $L$-shaped blockers must contain either the $(1,n)$ or $(n,1)$ position. In [@BC], it was shown that all blockers with exact $n$ zeros can be obtained from $L$-shaped blockers by shifting some zeros along the cyclic-Hankel diagonals. The following examples illustrate these results. **Example 2**. *[\[ex:123avoid\]]{#ex:123avoid label="ex:123avoid"} For $n=6$, one possible $L$-shaped blocker is $$L_6(4,3)=\left[\begin{array}{c|c|c|c|c|c} 1&1&0&0&0&0\\ \hline 1&1&1&1&1&0\\ \hline 1 &1&1&1&1&0\\ \hline 1&1&1&1&1&1\\ \hline 1 &1&1&1&1&1\\ \hline 1 &1&1&1&1&1\end{array}\right].$$* *Every permutation matrix $P\le L_6(4,3)$ contains one of the two $1$'s from row $1$, one of the three $1$'s from column $6$, and then necessarily one of the $1$'s from the $2\times 3$ submatrix formed by rows $2$ and $3$, and columns $3, 4$, and $5$, as given by the Frobenius-König theorem [@FK1933], thereby resulting in a $123$-pattern. Thus $A$ is a $123$-forcing matrix; equivalently, $A$ blocks all $6\times 6$ $123$-avoiding permutation matrices. Another example of a ${123}$-forcing matrix with 6 $0$'s obtained from $L_6(4,3)$ by shifting some zeros along the cyclic-Hankel diagonals is $$\left[\begin{array}{c|c|c|c|c|c} 1&1&1&0&0&0\\ \hline 1&0&1&1&1&0 \\ \hline 1&1&1&1&1&1 \\ \hline 1&1&1&1&0&1 \\ \hline 1&1&1&1&1&1 \\ \hline 1&1&1&1&1&1 \end{array}\right].$$* The main purpose of this paper is to characterize $123$-forcing matrices (equivalently, blockers of $123$-avoiding permutation matrices) $A$ with $0$'s in certain shapes. We now briefly summarize the content of this paper. In Section 2, we define flag-shaped blockers of $123$-avoiding permutation matrices and show that they are minimum blockers. In Section 3, we give all possible cardinalities (number of zeros) of an $n\times n$ flag-shaped blocker of $123$-avoiding permutation matrices and propose a conjecture that any $n\times n$ minimum blocker of $123$-avoiding permutation matrices contains at least $n$ zeros and at most $rs$ zeros, where $|r-s|\leq 1$ and $r+s=n+1.$ In Section 4, we explore the dimensions of the subpolytopes of a $123$-avoiding polytope determined by a flag-shaped blocker. # Defining Flag-Shaped Blockers {#Section 2} A flag-shaped blocker of an $n \times n$ matrix is a set of adjacent positions, denoted $B_n(m,t)$, in the following form. The blocker is composed of two parts, the \"pole,\" shown in green, and the \"flag,\" shown in red. The \"pole\" is a $(n-t)\times 1$ submatrix contained in the $m^{th}$ column, where $1\leq m\leq n.$ The \"flag\" is an $(n-m+1)\times t$ submatrix with consecutive rows and columns contained in row $1$ to row $n-m+1$ and column $m-t$ to column $m-1,$ where $t$ is the number of unoccupied rows below the blocker. We require the height of the pole to be greater or equal to the height of the flag, hence $0 \leq t \leq m-1$. Note that $B_n(m,t)$ has cardinality $n+t(n-m).$ Since the $123$-avoiding property is preserved when we take the transpose, take the Hankel transpose, or rotate the matrix by $180^{\circ},$ a flag-shaped blocker is still a blocker if it is reflected across the main or Hankel diagonal or rotated $180^{\circ}$ about the center of the matrix. Thus, without loss of generality, we will only consider the case where flag-shaped blockers are oriented in the same manner as the figure above, unless stated otherwise. Interestingly, these flag-shaped blockers can be considered generalizations of $L$-shaped blockers, which are introduced in [@BC2], as an $L$-shaped blocker with $m=n$ is a flag-shaped blocker. Furthermore, the flag-shaped blockers $B_n(m,m-1)$ are special types of blockers given by the Frobenius-König theorem. Thus, $L$-shaped blockers and certain cases of Frobenius-König blockers are both special cases of flag-shaped blockers. Starting with Lemma [Lemma 3](#Lemma2.1){reference-type="ref" reference="Lemma2.1"}, we first prove that $B_n(m,t),$ the set of positions arranged in a flag-shape, are blockers, then we show they are minimum blockers in Lemma [Lemma 4](#minimumR1){reference-type="ref" reference="minimumR1"} and Theorem [Theorem 5](#Theorem2.3){reference-type="ref" reference="Theorem2.3"}. In the following proofs and examples, we will utilize the cyclic-Hankel decomposition with red positions denoting blocker positions, unless specified otherwise. **Lemma 3**. *Let $Q$ be a set of positions of an $n \times n$ matrix. If $Q$ is a flag-shaped blocker $B_n(m,t)$ with cardinality $n+t(n-m)$, where $0 \leq t \leq m-1$ and $1 \leq m \leq n$, then $Q$ is a blocker of all $n \times n$ $123$-avoiding permutation matrices.* *Proof.* Without loss of generality, we consider a $10 \times 10$ matrix with the flag-shaped blocker $B_n(7,3)$ for illustrative purposes, though the general proof follows in the same manner. We know that if there exists a $123$-avoiding permutation matrix that does not intersect $Q$, then it must contain one of the positions in yellow submatrix in the below matrix. $$\left[\begin{array}{c|c|c|c|c|c|c|c|c|c} a&b&c&\cellcolor{red!20}d&\cellcolor{red!20}e&\cellcolor{red!20}f&\cellcolor{red!20}g&h&i&j\\ \hline b&c&d&\cellcolor{red!20}e&\cellcolor{red!20}f&\cellcolor{red!20}g&\cellcolor{red!20}h&i&j&a\\ \hline c&d&e&\cellcolor{red!20}f&\cellcolor{red!20}g&\cellcolor{red!20}h&\cellcolor{red!20}i&j&a&b\\ \hline d&e&f&\cellcolor{red!20}g&\cellcolor{red!20}h&\cellcolor{red!20}i&\cellcolor{red!20}j&a&b&c\\ \hline e&f&g&\cellcolor{green!20}h&\cellcolor{green!20}i&\cellcolor{green!20}j&\cellcolor{red!20}a&b&c&d\\ \hline f&g&h&\cellcolor{green!20}i&\cellcolor{green!20}j&\cellcolor{green!20}a&\cellcolor{red!20}b&c&d&e\\ \hline g&h&i&\cellcolor{green!20}j&\cellcolor{green!20}a&\cellcolor{green!20}b&\cellcolor{red!20}c&d&e&f\\ \hline h&i&j&a&b&c&\cellcolor{yellow!20}d&e&f&g\\ \hline i&j&a&b&c&d&\cellcolor{yellow!20}e&f&g&h\\ \hline j&a&b&c&d&e&\cellcolor{yellow!20}f&g&h&i \end{array}\right]$$ We also know that at least one of the green positions in the above figure must be used to construct the permutation matrix, as the set of green and red positions forms a blocker given by the Frobenius-König theorem, which intersects every permutation matrix. The yellow submatrix in the upper right corner shown in the figure below will always contain more rows, $n-m+1$, than columns, $n-m$. Thus, it is necessary that we use at least one position from the green submatrix in the upper left corner in the figure below to construct the permutation matrix. $$\left[\begin{array}{c|c|c|c|c|c|c|c|c|c} \cellcolor{green!20}a&\cellcolor{green!20}b&\cellcolor{green!20}c&\cellcolor{red!20}d&\cellcolor{red!20}e&\cellcolor{red!20}f&\cellcolor{red!20}g&\cellcolor{yellow!20}h&\cellcolor{yellow!20}i&\cellcolor{yellow!20}j\\ \hline \cellcolor{green!20}b&\cellcolor{green!20}c&\cellcolor{green!20}d&\cellcolor{red!20}e&\cellcolor{red!20}f&\cellcolor{red!20}g&\cellcolor{red!20}h&\cellcolor{yellow!20}i&\cellcolor{yellow!20}j&\cellcolor{yellow!20}a\\ \hline \cellcolor{green!20}c&\cellcolor{green!20}d&\cellcolor{green!20}e&\cellcolor{red!20}f&\cellcolor{red!20}g&\cellcolor{red!20}h&\cellcolor{red!20}i&\cellcolor{yellow!20}j&\cellcolor{yellow!20}a&\cellcolor{yellow!20}b\\ \hline \cellcolor{green!20}d&\cellcolor{green!20}e&\cellcolor{green!20}f&\cellcolor{red!20}g&\cellcolor{red!20}h&\cellcolor{red!20}i&\cellcolor{red!20}j&\cellcolor{yellow!20}a&\cellcolor{yellow!20}b&\cellcolor{yellow!20}c\\ \hline e&f&g&h&i&j&\cellcolor{red!20}a&b&c&d\\ \hline f&g&h&i&j&a&\cellcolor{red!20}b&c&d&e\\ \hline g&h&i&j&a&b&\cellcolor{red!20}c&d&e&f\\ \hline h&i&j&a&b&c&d&e&f&g\\ \hline i&j&a&b&c&d&e&f&g&h\\ \hline j&a&b&c&d&e&f&g&h&i \end{array}\right]$$ However, this means the permutation matrix constructed cannot be $123$-avoiding without intersecting the blocker, as at least one element from each of the yellow submatrices below must be utilized when constructing a permutation matrix that does not intersect $Q$. Thus, $Q$ is a blocker. $$\left[\begin{array}{c|c|c|c|c|c|c|c|c|c} \cellcolor{yellow!20}a&\cellcolor{yellow!20}b&\cellcolor{yellow!20}c&\cellcolor{red!20}d&\cellcolor{red!20}e&\cellcolor{red!20}f&\cellcolor{red!20}g&h&i&j\\ \hline \cellcolor{yellow!20}b&\cellcolor{yellow!20}c&\cellcolor{yellow!20}d&\cellcolor{red!20}e&\cellcolor{red!20}f&\cellcolor{red!20}g&\cellcolor{red!20}h&i&j&a\\ \hline \cellcolor{yellow!20}c&\cellcolor{yellow!20}d&\cellcolor{yellow!20}e&\cellcolor{red!20}f&\cellcolor{red!20}g&\cellcolor{red!20}h&\cellcolor{red!20}i&j&a&b\\ \hline \cellcolor{yellow!20}d&\cellcolor{yellow!20}e&\cellcolor{yellow!20}f&\cellcolor{red!20}g&\cellcolor{red!20}h&\cellcolor{red!20}i&\cellcolor{red!20}j&a&b&c\\ \hline e&f&g&\cellcolor{yellow!20}h&\cellcolor{yellow!20}i&\cellcolor{yellow!20}j&\cellcolor{red!20}a&b&c&d\\ \hline f&g&h&\cellcolor{yellow!20}i&\cellcolor{yellow!20}j&\cellcolor{yellow!20}a&\cellcolor{red!20}b&c&d&e\\ \hline g&h&i&\cellcolor{yellow!20}j&\cellcolor{yellow!20}a&\cellcolor{yellow!20}b&\cellcolor{red!20}c&d&e&f\\ \hline h&i&j&a&b&c&\cellcolor{yellow!20}d&e&f&g\\ \hline i&j&a&b&c&d&\cellcolor{yellow!20}e&f&g&h\\ \hline j&a&b&c&d&e&\cellcolor{yellow!20}f&g&h&i \end{array}\right]$$ ◻ We now show that flag-shaped blockers $B_n(m,t)$ are minimum blockers. **Lemma 4**. *For any element $b$ in the first row of a flag-shaped blocker $B_n(m,t)$ of all $n \times n$ $123$-avoiding permutation matrices, there exists at least one $123$-avoiding permutation matrix that intersects the blocker at most once at $b$.* *Proof.* Without loss of generality, we consider a $10 \times 10$ matrix with the flag-shaped blocker $B_n(7,3)$ for illustrative purposes, though the general proof follows in the same manner. Recall that if a blocker is minimum, then removing any position will no longer make it a blocker. To prove that all the blocker positions in the first row are necessary, we need to show that a $123$-avoiding permutation matrix can be constructed using any of the positions of the blocker in the first row. We start by considering the first $m-t-1$ columns of the matrix, the first three columns in the matrix below. We can begin constructing the $123$-avoiding permutation matrix that only intersects the blocker in row 1 by utilizing the highest positions, going from northeast to southwest and starting at row 2. These are the three green positions in the figure below. $$\left[\begin{array}{c|c|c||c|c|c|c|c|c|c} a&b&c&\cellcolor{red!20}d&\cellcolor{red!20}e&\cellcolor{red!20}f&\cellcolor{red!20}g&h&i&j\\ \hline b&c&\cellcolor{green!20}d&\cellcolor{red!20}e&\cellcolor{red!20}f&\cellcolor{red!20}g&\cellcolor{red!20}h&i&j&a\\ \hline c&\cellcolor{green!20}d&e&\cellcolor{red!20}f&\cellcolor{red!20}g&\cellcolor{red!20}h&\cellcolor{red!20}i&j&a&b\\ \hline \cellcolor{green!20}d&e&f&\cellcolor{red!20}g&\cellcolor{red!20}h&\cellcolor{red!20}i&\cellcolor{red!20}j&a&b&c\\ \hline e&f&g&h&i&j&\cellcolor{red!20}a&b&c&d\\ \hline f&g&h&i&j&a&\cellcolor{red!20}b&c&d&e\\ \hline g&h&i&j&a&b&\cellcolor{red!20}c&d&e&f\\ \hline h&i&j&a&b&c&d&e&f&g\\ \hline i&j&a&b&c&d&e&f&g&h\\ \hline j&a&b&c&d&e&f&g&h&i \end{array}\right]$$ Next, we will consider the last $n-m$ columns. We can use positions from these columns to construct the $123$-avoiding permutation matrix, starting with the most northeastern position in the row below the lowest position used in the previous step. $$\left[\begin{array}{c|c|c|c|c|c|c||c|c|c} a&b&c&\cellcolor{red!20}d&\cellcolor{red!20}e&\cellcolor{red!20}f&\cellcolor{red!20}g&h&i&j\\ \hline b&c&\cellcolor{green!20}d&\cellcolor{red!20}e&\cellcolor{red!20}f&\cellcolor{red!20}g&\cellcolor{red!20}h&i&j&a\\ \hline c&\cellcolor{green!20}d&e&\cellcolor{red!20}f&\cellcolor{red!20}g&\cellcolor{red!20}h&\cellcolor{red!20}i&j&a&b\\ \hline \cellcolor{green!20}d&e&f&\cellcolor{red!20}g&\cellcolor{red!20}h&\cellcolor{red!20}i&\cellcolor{red!20}j&a&b&c\\ \hline e&f&g&h&i&j&\cellcolor{red!20}a&b&c&\cellcolor{green!20}d\\ \hline f&g&h&i&j&a&\cellcolor{red!20}b&c&\cellcolor{green!20}d&e\\ \hline g&h&i&j&a&b&\cellcolor{red!20}c&\cellcolor{green!20}d&e&f\\ \hline h&i&j&a&b&c&d&e&f&g\\ \hline i&j&a&b&c&d&e&f&g&h\\ \hline j&a&b&c&d&e&f&g&h&i \end{array}\right]$$ Exactly $n-t-1$ out of $n$ columns have been accounted for, and elements in these columns that we use to construct the $123$-avoiding permutation matrix will never intersect the blocker since there are no positions of the blocker in these columns. This implies that there are exactly $n-t-1$ rows in the matrix that pose no issues when constructing the permutation matrix. Now, all that is left to consider is row 1 and the last $t$ rows of the matrix, for a total of $t+1$ rows. By definition, the last $t$ rows contain no positions of the blocker, so it is always possible to use $t$ positions from northeast to southwest for the portion of the $123$-avoiding permutation matrix in these last $t$ rows. So far, the $123$-avoiding permutation matrix being constructed is disjoint from the blocker. However, the last element of the permutation matrix must come from row 1. This element will intersect the blocker, so there exists a permutation matrix that intersects the blocker at most once, as shown in the following examples, where the yellow positions represent the intersections of the blocker and the $123$-avoiding permutation matrix. $$\left[\begin{array}{c|c|c|c|c|c|c|c|c|c} a&b&c&\cellcolor{red!20}d&\cellcolor{red!20}e&\cellcolor{red!20}f&\cellcolor{yellow!20}g&h&i&j\\ \hline b&c&\cellcolor{green!20}d&\cellcolor{red!20}e&\cellcolor{red!20}f&\cellcolor{red!20}g&\cellcolor{red!20}h&i&j&a\\ \hline c&\cellcolor{green!20}d&e&\cellcolor{red!20}f&\cellcolor{red!20}g&\cellcolor{red!20}h&\cellcolor{red!20}i&j&a&b\\ \hline \cellcolor{green!20}d&e&f&\cellcolor{red!20}g&\cellcolor{red!20}h&\cellcolor{red!20}i&\cellcolor{red!20}j&a&b&c\\ \hline e&f&g&h&i&j&\cellcolor{red!20}a&b&c&\cellcolor{green!20}d\\ \hline f&g&h&i&j&a&\cellcolor{red!20}b&c&\cellcolor{green!20}d&e\\ \hline g&h&i&j&a&b&\cellcolor{red!20}c&\cellcolor{green!20}d&e&f\\ \hline h&i&j&a&b&\cellcolor{green!20}c&d&e&f&g\\ \hline i&j&a&b&\cellcolor{green!20}c&d&e&f&g&h\\ \hline j&a&b&\cellcolor{green!20}c&d&e&f&g&h&i \end{array}\right] \ \ or \ \ \left[\begin{array}{c|c|c|c|c|c|c|c|c|c} a&b&c&\cellcolor{red!20}d&\cellcolor{yellow!20}e&\cellcolor{red!20}f&\cellcolor{red!20}g&h&i&j\\ \hline b&c&\cellcolor{green!20}d&\cellcolor{red!20}e&\cellcolor{red!20}f&\cellcolor{red!20}g&\cellcolor{red!20}h&i&j&a\\ \hline c&\cellcolor{green!20}d&e&\cellcolor{red!20}f&\cellcolor{red!20}g&\cellcolor{red!20}h&\cellcolor{red!20}i&j&a&b\\ \hline \cellcolor{green!20}d&e&f&\cellcolor{red!20}g&\cellcolor{red!20}h&\cellcolor{red!20}i&\cellcolor{red!20}j&a&b&c\\ \hline e&f&g&h&i&j&\cellcolor{red!20}a&b&c&\cellcolor{green!20}d\\ \hline f&g&h&i&j&a&\cellcolor{red!20}b&c&\cellcolor{green!20}d&e\\ \hline g&h&i&j&a&b&\cellcolor{red!20}c&\cellcolor{green!20}d&e&f\\ \hline h&i&j&a&b&c&\cellcolor{green!20}d&e&f&g\\ \hline i&j&a&b&c&\cellcolor{green!20}d&e&f&g&h\\ \hline j&a&b&\cellcolor{green!20}c&d&e&f&g&h&i \end{array}\right]$$ Thus, for any element $b$ in the flag-shaped blocker that resides in the first row of an $n \times n$ matrix, there exists at least one $123$-avoiding permutation matrix that intersects the blocker at most once at $b$. By definition, this means that every element of the blocker in the first row of the matrix is a necessary position of the blocker. ◻ **Theorem 5**. *Let $Q$ be a set of positions of an $n \times n$ matrix. If $Q$ is a flag-shaped blocker $B_n(m,t)$ with cardinality $n+t(n-m)$, where $0 \leq t \leq m-1$ and $1 \leq m \leq n$, then $Q$ is a minimum blocker of all $n \times n$ $123$-avoiding permutation matrices.* *Proof.* Without loss of generality, we consider a $10 \times 10$ matrix with the flag-shaped blocker $B_n(7,3)$ for illustrative purposes, though the general proof follows in the same manner. By Lemma [Lemma 4](#minimumR1){reference-type="ref" reference="minimumR1"}, every element of $Q$ in the first row of the matrix is necessary for the minimum blocker, so we can consider the following matrix. $$\left[\begin{array}{c|c|c|c|c|c|c|c|c||c} a&b&c&\cellcolor{red!20}d&\cellcolor{red!20}e&\cellcolor{red!20}f&\cellcolor{red!20}g&h&i&\cellcolor{green!20}j\\ \hline \hline b&c&d&\cellcolor{red!20}e&\cellcolor{red!20}f&\cellcolor{red!20}g&\cellcolor{red!20}h&i&j&a\\ \hline c&d&e&\cellcolor{red!20}f&\cellcolor{red!20}g&\cellcolor{red!20}h&\cellcolor{red!20}i&j&a&b\\ \hline d&e&f&\cellcolor{red!20}g&\cellcolor{red!20}h&\cellcolor{red!20}i&\cellcolor{red!20}j&a&b&c\\ \hline e&f&g&h&i&j&\cellcolor{red!20}a&b&c&d\\ \hline f&g&h&i&j&a&\cellcolor{red!20}b&c&d&e\\ \hline g&h&i&j&a&b&\cellcolor{red!20}c&d&e&f\\ \hline h&i&j&a&b&c&d&e&f&g\\ \hline i&j&a&b&c&d&e&f&g&h\\ \hline j&a&b&c&d&e&f&g&h&i \end{array}\right]$$ We can use the green $j$ to attempt to construct a $123$-avoiding permutation matrix, allowing us to focus on the $9 \times 9$ submatrix at the lower left corner. In this submatrix, we have another flag-shaped blocker. By repeating the above steps, we can show that all the blocker elements in row 2 must be included in the minimum blocker. This process of reducing the size of the flag-shaped blocker can be repeated until we are left with an $L$-shaped minimal blocker, all positions of which are necessary. Thus, all positions of $Q$ are necessary. Since removing any element of $Q$ makes it no longer a blocker, $Q$ is a minimum blocker. ◻ # The Cardinality of Flag-Shaped Blockers Flag-shaped blockers have a cardinality of $n$ only when $m=n$ or $m=1$. More generally, the cardinality of a flag-shaped minimum blocker is given by the expression $n+t(n-m)$, where $0 \leq t \leq m-1$ and $1 \leq m \leq n$. With this expression, we are able to determine which cardinalities can and cannot be achieved by a flag-shaped blocker. **Theorem 6**. *Let $n,m,$ and $t$ be nonnegative integers, such that $0 \leq t \leq m-1$ and $1 \leq m \leq n$, and let $p:=n+t(n-m)$, where $n \leq p \leq n+(\lceil \frac{n}{2} \rceil -1)(n - \lceil \frac{n}{2} \rceil)$. There exists a flag-shaped blocker $B_n(m,t)$ with cardinality $p$ if and only if $p-n \leq m-1$ or if $p-n$ is a composite number.* *Proof.* $\Rightarrow$ Suppose there exists a flag-shaped blocker with cardinality $p$. If $p-n$ is a prime number greater than $m-1$, then this implies that $t(n-m)$ is also prime and greater than $m-1$. However, $t(n-m)$ can only be prime if either $t=1$ or $n-m=1$. The requirement that $t(n-m) > m-1$ prevents either case from occurring. Thus, $p-n=t(n-m)$ cannot be prime, a contradiction. This implies that if there exists a flag-shaped blocker with cardinality $p$, then either $p-n \leq m-1$ is true or $p-n$ is a composite number. $\Leftarrow$ For $n-m=1$, $p=n+t$ and $0 \leq t \leq m-1,$ so $$n \leq p \leq n+m-1$$ $$0 \leq p-n \leq m-1,$$ and all conditions on $p, n, m,$ and $t$ for a flag-shaped blocker, as defined in Section [2](#Section 2){reference-type="ref" reference="Section 2"}, are met. This means that when $p-n \leq m-1$, there does exist a flag-shaped blocker with cardinality $p$. Now, suppose that $p-n$ is a composite number. If $n-m \geq 2$ and $t \geq 2$, then $p-n = t(n-m)$ must be a composite number such that $1 \leq m \leq n$ and $0 \leq t \leq m-1$. All conditions on $p, n, m,$ and $t$ for a flag-shaped blocker are satisfied, so there does exist a flag-shaped blocker with cardinality $p$. ◻ Since we know the range of cardinalities for flag-shaped blockers, we briefly expand our view to consider all minimum blockers, not simply flag-shaped blockers. Because all minimal blockers are minimum blockers, the lower bound for the cardinality of a minimum blocker of an $n \times n$ matrix is $n$, as each letter of the Hankel-cyclic decomposition must appear at least once in the blocker. Determining the upper bound is more complicated, since the shapes of all minimum blockers have not yet been fully characterized. Nonetheless, we provide the following conjecture for the upper bound of the cardinality of all minimum blockers and show that flag-shaped blockers can achieve this upper bound. **Conjecture 7**. *The upper bound for the cardinality of a minimum blocker of all $n \times n$ $123$-avoiding permutation matrices is $r \times s$, where $|r-s| \leq 1$ and $r + s = n + 1$.* There exist minimum flag-shaped blockers with exactly $r \times s$ positions. These can occur when an adjacent set of positions forming a blocker given by the Frobenius-König theorem, such that $|r-s| \leq 1$, is placed at the northwest or southeast corner of a matrix, since these blockers are special cases of flag-shaped blockers, as described in Section [2](#Section 2){reference-type="ref" reference="Section 2"}. The upper bound can also be expressed in terms of only $m$ and $n$ by denoting $m=\lceil \frac{n}{2} \rceil$ and $t=m-1=\lceil \frac{n}{2} \rceil -1$ for a maximum cardinality of $n+(\lceil \frac{n}{2} \rceil -1)(n - \lceil \frac{n}{2} \rceil)$. **Example 8**. *Minimum blockers of $10 \times 10$ matrices with cardinality $r \times s =30$.* *$$\left[\begin{array}{c|c|c|c|c|c|c|c|c|c} \cellcolor{red!20}a&\cellcolor{red!20}b&\cellcolor{red!20}c&\cellcolor{red!20}d&\cellcolor{red!20}e&\cellcolor{red!20}f&g&h&i&j\\ \hline \cellcolor{red!20}b&\cellcolor{red!20}c&\cellcolor{red!20}d&\cellcolor{red!20}e&\cellcolor{red!20}f&\cellcolor{red!20}g&h&i&j&a\\ \hline \cellcolor{red!20}c&\cellcolor{red!20}d&\cellcolor{red!20}e&\cellcolor{red!20}f&\cellcolor{red!20}g&\cellcolor{red!20}h&i&j&a&b\\ \hline \cellcolor{red!20}d&\cellcolor{red!20}e&\cellcolor{red!20}f&\cellcolor{red!20}g&\cellcolor{red!20}h&\cellcolor{red!20}i&j&a&b&c\\ \hline \cellcolor{red!20}e&\cellcolor{red!20}f&\cellcolor{red!20}g&\cellcolor{red!20}h&\cellcolor{red!20}i&\cellcolor{red!20}j&a&b&c&d\\ \hline f&g&h&i&j&a&b&c&d&e\\ \hline g&h&i&j&a&b&c&d&e&f\\ \hline h&i&j&a&b&c&d&e&f&g\\ \hline i&j&a&b&c&d&e&f&g&h\\ \hline j&a&b&c&d&e&f&g&h&i \end{array}\right] \ \ and \ \ \left[\begin{array}{c|c|c|c|c|c|c|c|c|c} a&b&c&d&e&f&g&h&i&j\\ \hline b&c&d&e&f&g&h&i&j&a\\ \hline c&d&e&f&g&h&i&j&a&b\\ \hline d&e&f&g&h&i&j&a&b&c\\ \hline e&f&g&h&i&\cellcolor{red!20}j&\cellcolor{red!20}a&\cellcolor{red!20}b&\cellcolor{red!20}c&\cellcolor{red!20}d\\ \hline f&g&h&i&j&\cellcolor{red!20}a&\cellcolor{red!20}b&\cellcolor{red!20}c&\cellcolor{red!20}d&\cellcolor{red!20}e\\ \hline g&h&i&j&a&\cellcolor{red!20}b&\cellcolor{red!20}c&\cellcolor{red!20}d&\cellcolor{red!20}e&\cellcolor{red!20}f\\ \hline h&i&j&a&b&\cellcolor{red!20}c&\cellcolor{red!20}d&\cellcolor{red!20}e&\cellcolor{red!20}f&\cellcolor{red!20}g\\ \hline i&j&a&b&c&\cellcolor{red!20}d&\cellcolor{red!20}e&\cellcolor{red!20}f&\cellcolor{red!20}g&\cellcolor{red!20}h\\ \hline j&a&b&c&d&\cellcolor{red!20}e&\cellcolor{red!20}f&\cellcolor{red!20}g&\cellcolor{red!20}h&\cellcolor{red!20}i \end{array}\right]$$ $~\vbox{\hrule\hbox{% \vrule height1.3ex\hskip 0.8ex\vrule}\hrule }$* Furthermore, we are able to determine which cardinalities are unable to be obtained by flag-shaped blockers by using the sieve of Eratosthenes to find all values of $p-n$ that are prime, where $m-1 < p-n \leq rs-n$ and $|r-s| \leq 1$. The last thing we will prove in this section is that Conjecture [Conjecture 7](#conjecture1){reference-type="ref" reference="conjecture1"} is true for the $n=3$ case. That is, we can show that the upper bound of a minimum flag-shaped blocker of a $3 \times 3$ matrix is $r \times s = 2 \times 2 = 4$. *Proof.* Let $A$ be a $3\times 3$ $(0,1)$-matrix avoiding $123$-permutation. If the permanent of $A$ is $0,$ then $A$ contains a rectangular $r\times s$ zero submatrix with $r+s=4.$ According to Frobenius-König theorem, any zero not in the $r\times s$ submatrix is not necessary, so a minimum blocker of $A$ at most contains four zeros when $r=s=2.$ If the permanent of $A$ is nonzero, then $A$ contains three $1$'s on the diagonal. $$\left[\begin{array}{c|c|c} 1&\phantom{0}&\phantom{0} \\ \hline \phantom{0}&1&\phantom{0} \\ \hline \phantom{0}&\phantom{0}&1 \end{array}\right]$$ If there is a pair of $1'$s symmetric with respect to the diagonal, we can construct a $123$-avoiding permutation matrix by replacing the two $1'$s on the diagonal by the pair of $1'$s. $$\left[\begin{array}{c|c|c} 1&\phantom{0}&\cellcolor{red!20}\phantom{0} \\ \hline \phantom{0}&1&\phantom{0} \\ \hline \cellcolor{red!20}\phantom{0}&\phantom{0}&1 \end{array}\right] \Rightarrow \left[\begin{array}{c|c|c} &\phantom{0}&\cellcolor{red!20}1 \\ \hline \phantom{0}&1&\phantom{0} \\ \hline \cellcolor{red!20}1&\phantom{0}& \end{array}\right]$$ There are three pairs of elements and we just need to block one element in each pair, so a minimum blocker has cardinality $3.$ ◻ # The Polytope Generated by 123-Avoiding Permutation Matrices One of our motivations for studying blockers of $123$-avoiding permutation matrices is to gain a better understanding of the polytope generated by $123$-avoiding permutation matrices, denoted $\Omega_n(\overline{123})$, whose dimension is $(n-1)^2$. In [@BC3], Brualdi and Cao show that each minimal blocker of $n\times n$ $123$-avoiding permutation matrices determines a facet of $\Omega_n(\overline{123}),$ and a face of $\Omega_n(\overline{123})$ lives in dimension $(n-1)^2-1.$ We seek to learn more about the dimension of the faces of $\Omega_n(\overline{123})$ determined by flag-shaped blocker. The dimension of a face of $\Omega_n(\overline{123})$ determined by a minimum blocker is equivalent to the number of linearly independent $123$-avoiding permutation matrices that intersect the blocker exactly once, and no more. We present an inductive argument to find a more precise upper bound for the dimension. **Lemma 9**. *Given a flag-shaped minimum blocker $B_n(m,t)$ of all $n \times n$ $123$-avoiding permutation matrices with $m=n-1$, there is a $t \times 1$ submatrix of adjacent positions containing the $(n,n)$ position that cannot be used to construct a $123$-avoiding permutation matrix that intersects the blocker exactly once.* *Proof.* Consider an $n \times n$ matrix containing a flag-shaped blocker such that $m=n-1$. Our claim is that the elements at the intersection of column $n$ and the last $t$ rows cannot be used to construct any $123$-avoiding permutation matrix that intersects the blocker exactly once. Suppose we use any position from the intersection of column $n$ and the last $t$ rows to attempt to construct a $123$-avoiding permutation matrix that intersects the blocker exactly once. Now, consider the submatrix formed by deleting the column and row in which this position resides. We must utilize $t-1$ additional positions from the last $t-1$ rows of the submatrix to attempt to construct a $123$-avoiding permutation matrix. The specific positions we use are not important in this case, so long as they do not form a $123$-pattern with the first position we chose from column $n$. We can now delete the last $t-1$ rows from the submatrix. Notice that the new submatrix is not square, as it has $t-1$ more columns than rows. No matter which $t-1$ columns we delete to form an $(n-t) \times (n-t)$ submatrix, we can never construct a $12$-avoiding permutation matrix in this submatrix without intersecting the blocker more than once. This is due to the fact that the remaining flag portion of the blocker will always form at least an $(n-m+1) \times 2$ submatrix with at least two blocker positions on each full diagonal of the $(n-t) \times (n-1)$ submatrix. Thus, this $(n-t) \times (n-1)$ submatrix must contain a $12$-pattern. However, this $12$-pattern paired with the position from column $n$ forms a $123$-pattern. Thus, we cannot construct a $123$-avoiding permutation matrix using any position from the intersection of column $n$ and the last $t$ rows. ◻ **Theorem 10**. *Given a flag-shaped minimum blocker $B_n(m,t)$ of all $n \times n$ $123$-avoiding permutation matrices, there is a $t \times (n-m)$ submatrix of adjacent positions containing the $(n,n)$ position that cannot be used to construct a $123$-avoiding permutation matrix that intersects the blocker exactly once.* *Proof.* Define $p:=n-m$. Lemma [Lemma 9](#Lemma18){reference-type="ref" reference="Lemma18"} describes the case where $p=1$. Now, suppose that when $2\leq p\leq k$, there are $pt$ positions that cannot be used to construct a $123$-avoiding permutation matrix that intersects the blocker exactly once. Suppose $p=k+1$. Consider any of the $t$ positions of an $n \times n$ matrix in the intersection of the $n$th column and the last $t$ rows. In order to construct a $123$-avoiding permutation matrix using any of these positions, a necessary condition is that we can construct a $123$-avoiding permutation matrix in the submatrix obtained by deleting the row and column the position that intersects the blocker only once resides in. However, the $(n-1) \times (n-1)$ submatrix we obtain contains a flag-shaped blocker (perhaps with more blocker positions than necessary) with $p=k$. Then by the induction hypothesis, there are $kt$ positions we cannot use to construct a $123$-avoiding permutation matrix in the $(n-1) \times (n-1)$ submatrix that intersects the blocker only once. We also know there are $t$ positions from the intersection of the last column and the last $t$ rows that we cannot use to construct a $123$-avoiding permutation matrix. Thus, there are a total of $kt+t=(k+1)t$ positions we cannot use to construct a $123$-avoiding permutation matrix that intersects the blocker only once. Then by mathematical induction, we have shown that there are $pt=(n-m)t$ positions of an $n\times n$ matrix that we cannot use. Thus, there is a $t \times (n-m)$ submatrix at the lower right corner of the matrix from which we cannot use any positions to construct a $123$-avoiding permutation matrix that intersects the blocker exactly once. ◻ **Example 11**. *To illustrate Theorem [Theorem 10](#Theorem19){reference-type="ref" reference="Theorem19"}, consider the flag-shaped blocker $B_{10}(8,3)$. We are unable to construct a $123$-avoiding permutation matrix that intersects the blocker once, and no more, using any of the yellow positions.* *$$\left[\begin{array}{c|c|c|c|c|c|c|c|c|c} a&b&c&d&\cellcolor{red!20}e&\cellcolor{red!20}f&\cellcolor{red!20}g&\cellcolor{red!20}h&i&j\\ \hline b&c&d&e&\cellcolor{red!20}f&\cellcolor{red!20}g&\cellcolor{red!20}h&\cellcolor{red!20}i&j&a\\ \hline c&d&e&f&\cellcolor{red!20}g&\cellcolor{red!20}h&\cellcolor{red!20}i&\cellcolor{red!20}j&a&b\\ \hline d&e&f&g&h&i&j&\cellcolor{red!20}a&b&c\\ \hline e&f&g&h&i&j&a&\cellcolor{red!20}b&c&d\\ \hline f&g&h&i&j&a&b&\cellcolor{red!20}c&d&e\\ \hline g&h&i&j&a&b&c&\cellcolor{red!20}d&e&f\\ \hline h&i&j&a&b&c&d&e&\cellcolor{yellow!20}f&\cellcolor{yellow!20}g\\ \hline i&j&a&b&c&d&e&f&\cellcolor{yellow!20}g&\cellcolor{yellow!20}h\\ \hline j&a&b&c&d&e&f&g&\cellcolor{yellow!20}h&\cellcolor{yellow!20}i \end{array}\right]$$* *The permutation matrix consisting of all $i$'s is the only $123$-avoiding permutation matrix using the yellow $i$. Clearly, this intersects the blocker twice. Thus, we can focus on just the yellow $f$ or one of the yellow $g$'s or $h$'s to illustrate the issue that arises when attempting to use any of these positions to form a $123$-avoiding permutation matrix. Without loss of generality, we will consider the $h$ in the last column.* *We can delete column $10$ and row $9$ since we do not have to use another position from either in our construction of a $123$-avoiding permutation matrix. We also know that we will use one position from row $8$ and one from row $10$ for the permutation matrix, so we can focus on the $7 \times 9$ submatrix formed by deleting the last $t=3$ rows and the last column.* *$$\left[\begin{array}{c|c|c|c|c|c|c|c|c||c} a&b&c&d&\cellcolor{red!20}e&\cellcolor{red!20}f&\cellcolor{red!20}g&\cellcolor{red!20}h&i&j\\ \hline b&c&d&e&\cellcolor{red!20}f&\cellcolor{red!20}g&\cellcolor{red!20}h&\cellcolor{red!20}i&j&a\\ \hline c&d&e&f&\cellcolor{red!20}g&\cellcolor{red!20}h&\cellcolor{red!20}i&\cellcolor{red!20}j&a&b\\ \hline d&e&f&g&h&i&j&\cellcolor{red!20}a&b&c\\ \hline e&f&g&h&i&j&a&\cellcolor{red!20}b&c&d\\ \hline f&g&h&i&j&a&b&\cellcolor{red!20}c&d&e\\ \hline g&h&i&j&a&b&c&\cellcolor{red!20}d&e&f\\ \hline \hline h&i&j&a&b&c&d&e&f&g\\ \hline i&j&a&b&c&d&e&f&g&\cellcolor{green!20}h\\ \hline j&a&b&c&d&e&f&g&h&i \end{array}\right]$$* *We can delete the two columns in which the positions we use from rows $8$ and $10$ reside. However, notice that there are no two columns we can delete to form a $7 \times 7$ submatrix in which we can construct a $12$-avoiding permutation matrix that intersects the blocker exactly once. For example, suppose we utilize the $e$ in row $8$ and the $f$ in row $10$ for the permutation matrix. The submatrix formed by deleting all rows and columns that already contain a position of the $123$-avoiding permutation matrix we are attempting to construct follows.* *$$\left[\begin{array}{c|c|c|c|c|c|c} a&b&c&d&\cellcolor{red!20}e&\cellcolor{red!20}f&g\\ \hline b&c&d&e&\cellcolor{red!20}f&\cellcolor{red!20}g&a\\ \hline c&d&e&f&\cellcolor{red!20}g&\cellcolor{red!20}a&b\\ \hline d&e&f&g&a&b&c\\ \hline e&f&g&a&b&c&d\\ \hline f&g&a&b&c&d&e\\ \hline g&a&b&c&d&e&f \end{array}\right]$$* *The $h$ from row $9$ of the original matrix is completely below and to the right of this $7 \times 7$ submatrix, so the only way we can complete the $n \times n$ $123$-avoiding permutation matrix is to construct a $12$-avoiding permutation matrix in this submatrix. However, this is only possible if we take the Hankel diagonal of the submatrix, but we intersect the blocker twice by doing so. Thus, it is impossible to construct a $123$-avoiding permutation matrix using the $h$ in the $(9,10)$ position without intersecting the blocker more than once. The same process can be used to show that none of the remaining positions in the $t \times (n-m)$ submatrix at the lower right corner of the matrix can be used to construct a $123$-avoiding permutation matrix intersecting the blocker only once. $~\vbox{\hrule\hbox{% \vrule height1.3ex\hskip 0.8ex\vrule}\hrule }$* Using Theorem [Theorem 10](#Theorem19){reference-type="ref" reference="Theorem19"}, we can determine the upper bound for the dimension of a face of $\Omega_n(\overline{123})$. **Theorem 12**. *A flag-shaped minimum blocker $B_n(m,t)$ of all $n \times n$ $123$-avoiding permutation matrices determines a face of $\Omega_n(\overline{123})$ with dimensions at most $(n-1)^2+1-t(n-m)$.* *Proof.* Suppose there are $a$ linearly independent permutation matrices that intersect the blocker exactly once and that do not use any positions from the $t \times (n-m)$ submatrix at the lower right corner below and to the right of the flag-shaped blocker. It is possible to find $t(n-m)$ linearly independent permutation matrices (which do not intersect the blocker exactly once according to Theorem [Theorem 10](#Theorem19){reference-type="ref" reference="Theorem19"}), each of which uses a unique position of the $t \times (n-1)$ submatrix. These $t(n-m)$ permutation matrices are linearly independent from one another and from the $a$ linearly independent permutation matrices found earlier. Thus, we have a total of $a+t(n-m)$ linearly independent permutation matrices, and this total cannot exceed the total number of linearly independent permutation matrices of an $n \times n$ matrix, which is $(n-1)^2+1$. Therefore, the number of linearly independent permutation matrices that intersects the blocker exactly once, given by $a$, is $$a \leq (n-1)^2+1 - t(n-m).$$ ◻ We have shown that there are at least $t(n-m)$ positions of an $n \times n$ matrix that we cannot use to construct a $123$-avoiding permutation matrix that intersects a minimum flag-shaped blocker exactly once. It is important to note that not all flag-shaped blockers will define a face of $\Omega_n(\overline{123})$ with dimension $(n-1)^2+1-t(n-m)$. However, there exist flag-shaped blockers that do define a face with precisely this dimension. Before describing these blockers, we first note an important aspect of of row and column blockers. **Lemma 13**. *For each minimum blocker of all $n \times n$ $123$-avoiding permutation matrices that are composed of an entire row or column of an $n \times n$ matrix, there exist $(n-1)^2+1$ linearly independent $123$-avoiding permutation matrices that intersect the blocker exactly once each.* *Proof.* Every $n \times n$ permutation matrix must intersect a row or column blocker exactly once. The polytope $\Omega_n(\overline{123})$ lives in dimension $(n-1)^2+1$, so there must exist the same number of linearly independent $123$-avoiding permutation matrices. ◻ **Theorem 14**. *A rectangular flag-shaped blocker $B_n(m,m-1)$ of all $n \times n$ $123$-avoiding permutation matrices determines a face of $\Omega_n(\overline{123})$ with the maximum dimension $(n-1)^2+1-t(n-m)$.* *Proof.* We use induction on $p:=n-m$, starting by showing that when $p=1$, it is possible to find $(n-1)^2+1-t \cdot 1$ linearly independent $123$-avoiding permutation matrices that intersect the blocker exactly once. For illustrative purposes, we consider a $10 \times 10$ matrix with the flag-shaped blocker $B_n(9,8)$ while describing the general proof. Additionally, notice that $t=m-1$ for all rectangular flag-shaped blockers, and $m=n-1$ when $p=1$, so we are looking to construct $(n-1)^2+1-(n-2)\cdot 1$ linearly independent $123$-avoiding permutation matrices. Using the $(1,n)$ position, there are $(n-2)^2+1$ linearly independent $123$-avoiding permutation matrices that intersect the blocker exactly once. This is because the $(n-1) \times (n-1)$ submatrix obtained by deleting the first row and last column contains $[(n-1)-1]^2+1$ such permutation matrices according to Lemma [Lemma 13](#Lemma12){reference-type="ref" reference="Lemma12"}. $$\left[\begin{array}{c|c|c|c|c|c|c|c|c||c} \cellcolor{red!20}a&\cellcolor{red!20}b&\cellcolor{red!20}c&\cellcolor{red!20}d&\cellcolor{red!20}e&\cellcolor{red!20}f&\cellcolor{red!20}g&\cellcolor{red!20}h&\cellcolor{red!20}i&\cellcolor{green!20}j\\ \hline \hline \cellcolor{red!20}b&\cellcolor{red!20}c&\cellcolor{red!20}d&\cellcolor{red!20}e&\cellcolor{red!20}f&\cellcolor{red!20}g&\cellcolor{red!20}h&\cellcolor{red!20}i&\cellcolor{red!20}j&a\\ \hline c&d&e&f&g&h&i&j&a&b\\ \hline d&e&f&g&h&i&j&a&b&c\\ \hline e&f&g&h&i&j&a&b&c&d\\ \hline f&g&h&i&j&a&b&c&d&e\\ \hline g&h&i&j&a&b&c&d&e&f\\ \hline h&i&j&a&b&c&d&e&f&g\\ \hline i&j&a&b&c&d&e&f&g&h\\ \hline j&a&b&c&d&e&f&g&h&i \end{array}\right]$$ Using the $(2,n)$ position, there are an additional $n-1$ linearly independent $123$-avoiding permutation matrices that intersect the blocker exactly once, which are obtained using the $n-1$ blocker positions in the first row. The $123$-avoiding permutation matrices corresponding to each position will contain a unique position of the matrix, making them linearly independent from one another. For example, consider the $123$-avoiding permutation matrix obtained by using the yellow $e$ from the blocker in the example below. $$\left[\begin{array}{c|c|c|c|c|c|c|c|c|c} \cellcolor{red!20}a&\cellcolor{red!20}b&\cellcolor{red!20}c&\cellcolor{red!20}d&\cellcolor{yellow!20}e&\cellcolor{red!20}f&\cellcolor{red!20}g&\cellcolor{red!20}h&\cellcolor{red!20}i&j\\ \hline \cellcolor{red!20}b&\cellcolor{red!20}c&\cellcolor{red!20}d&\cellcolor{red!20}e&\cellcolor{red!20}f&\cellcolor{red!20}g&\cellcolor{red!20}h&\cellcolor{red!20}i&\cellcolor{red!20}j&\cellcolor{green!20}a\\ \hline c&d&e&f&g&h&i&j&\cellcolor{green!20}a&b\\ \hline d&e&f&g&h&i&j&\cellcolor{green!20}a&b&c\\ \hline e&f&g&h&i&j&\cellcolor{green!20}a&b&c&d\\ \hline f&g&h&i&j&\cellcolor{green!20}a&b&c&d&e\\ \hline g&h&i&\cellcolor{green!20}j&a&b&c&d&e&f\\ \hline h&i&\cellcolor{green!20}j&a&b&c&d&e&f&g\\ \hline i&\cellcolor{green!20}j&a&b&c&d&e&f&g&h\\ \hline \cellcolor{green!20}j&a&b&c&d&e&f&g&h&i \end{array}\right]$$ Thus, in total we have $$(n-2)^2+1 + n-1 = (n-1)^2+1-(n-2)\cdot 1$$ linearly independent $123$-avoiding permutation matrices that intersect the blocker exactly once, as desired. We move on to the induction step. Suppose that when $p=k$, where $2 \leq k \leq n-3$, the corresponding rectangular flag-shaped blocker achieves the maximum dimension $$(n-1)^2+1-(m-1)(k)=(n-1)^2+1-(m-1)(n-m-1),$$ since $t=m-1$ and $k=n-m-1$ for these blockers. We will show that if $p=k+1$, the rectangular flag-shaped blocker achieves the maximum dimension $$(n-1)^2+1-(m-1)(k+1)=(n-1)^2+1-(m-1)(n-m).$$ For illustrative purposes, we consider a $10 \times 10$ matrix with the flag-shaped blocker $B_n(5,4)$ while describing the general proof. Using the inductive hypothesis, it is possible to find $$[(n-1)-1]^2+1-(m-1)[(n-1)-m] = (n-2)^2+1-(m-1)(n-m-1)$$ linearly independent $123$-avoiding permutation matrices that intersect the blocker exactly once using the $(1,n)$ position. $$\left[\begin{array}{c|c|c|c|c|c|c|c|c||c} \cellcolor{red!20}a&\cellcolor{red!20}b&\cellcolor{red!20}c&\cellcolor{red!20}d&\cellcolor{red!20}e&f&g&h&i&\cellcolor{green!20}j\\ \hline \hline \cellcolor{red!20}b&\cellcolor{red!20}c&\cellcolor{red!20}d&\cellcolor{red!20}e&\cellcolor{red!20}f&g&h&i&j&a\\ \hline \cellcolor{red!20}c&\cellcolor{red!20}d&\cellcolor{red!20}e&\cellcolor{red!20}f&\cellcolor{red!20}g&h&i&j&a&b\\ \hline \cellcolor{red!20}d&\cellcolor{red!20}e&\cellcolor{red!20}f&\cellcolor{red!20}g&\cellcolor{red!20}h&i&j&a&b&c\\ \hline \cellcolor{red!20}e&\cellcolor{red!20}f&\cellcolor{red!20}g&\cellcolor{red!20}h&\cellcolor{red!20}i&j&a&b&c&d\\ \hline \cellcolor{red!20}f&\cellcolor{red!20}g&\cellcolor{red!20}h&\cellcolor{red!20}i&\cellcolor{red!20}j&a&b&c&d&e\\ \hline g&h&i&j&a&b&c&d&e&f\\ \hline h&i&j&a&b&c&d&e&f&g\\ \hline i&j&a&b&c&d&e&f&g&h\\ \hline j&a&b&c&d&e&f&g&h&i \end{array}\right]$$ Using the $(2,n)$ position and the blocker positions in the first row, we can find $m$ additional linearly independent $123$-avoiding permutation matrices that intersect the blocker exactly once and that use a unique position of the matrix. For example, consider such a permutation matrix utilizing the yellow $d$ from the first row of the below matrix. $$\left[\begin{array}{c|c|c|c|c|c|c|c|c|c} \cellcolor{red!20}a&\cellcolor{red!20}b&\cellcolor{red!20}c&\cellcolor{yellow!20}d&\cellcolor{red!20}e&f&g&h&i&j\\ \hline \cellcolor{red!20}b&\cellcolor{red!20}c&\cellcolor{red!20}d&\cellcolor{red!20}e&\cellcolor{red!20}f&g&h&i&j&\cellcolor{green!20}a\\ \hline \cellcolor{red!20}c&\cellcolor{red!20}d&\cellcolor{red!20}e&\cellcolor{red!20}f&\cellcolor{red!20}g&h&i&j&\cellcolor{green!20}a&b\\ \hline \cellcolor{red!20}d&\cellcolor{red!20}e&\cellcolor{red!20}f&\cellcolor{red!20}g&\cellcolor{red!20}h&i&j&\cellcolor{green!20}a&b&c\\ \hline \cellcolor{red!20}e&\cellcolor{red!20}f&\cellcolor{red!20}g&\cellcolor{red!20}h&\cellcolor{red!20}i&j&\cellcolor{green!20}a&b&c&d\\ \hline \cellcolor{red!20}f&\cellcolor{red!20}g&\cellcolor{red!20}h&\cellcolor{red!20}i&\cellcolor{red!20}j&\cellcolor{green!20}a&b&c&d&e\\ \hline g&h&i&j&\cellcolor{green!20}a&b&c&d&e&f\\ \hline h&i&\cellcolor{green!20}j&a&b&c&d&e&f&g\\ \hline i&\cellcolor{green!20}j&a&b&c&d&e&f&g&h\\ \hline \cellcolor{green!20}j&a&b&c&d&e&f&g&h&i \end{array}\right]$$ Furthermore, the $n-m-1$ unused positions to the right of the blocker in the first row can also be used to construct linearly independent $123$-avoiding permutation matrices that intersect the blocker exactly once and that use a unique position of the matrix. For example, consider such a permutation matrix utilizing the green $g$ from the first row below, where the yellow represents the intersection of the blocker and the $123$-avoiding permutation matrix. $$\left[\begin{array}{c|c|c|c|c|c|c|c|c|c} \cellcolor{red!20}a&\cellcolor{red!20}b&\cellcolor{red!20}c&\cellcolor{red!20}d&\cellcolor{red!20}e&f&\cellcolor{green!20}g&h&i&j\\ \hline \cellcolor{red!20}b&\cellcolor{red!20}c&\cellcolor{red!20}d&\cellcolor{red!20}e&\cellcolor{red!20}f&g&h&i&j&\cellcolor{green!20}a\\ \hline \cellcolor{red!20}c&\cellcolor{red!20}d&\cellcolor{red!20}e&\cellcolor{red!20}f&\cellcolor{red!20}g&h&i&j&\cellcolor{green!20}a&b\\ \hline \cellcolor{red!20}d&\cellcolor{red!20}e&\cellcolor{red!20}f&\cellcolor{red!20}g&\cellcolor{red!20}h&i&j&\cellcolor{green!20}a&b&c\\ \hline \cellcolor{red!20}e&\cellcolor{red!20}f&\cellcolor{red!20}g&\cellcolor{red!20}h&\cellcolor{red!20}i&\cellcolor{green!20}j&a&b&c&d\\ \hline \cellcolor{red!20}f&\cellcolor{red!20}g&\cellcolor{red!20}h&\cellcolor{red!20}i&\cellcolor{yellow!20}j&a&b&c&d&e\\ \hline g&h&i&\cellcolor{green!20}j&a&b&c&d&e&f\\ \hline h&i&\cellcolor{green!20}j&a&b&c&d&e&f&g\\ \hline i&\cellcolor{green!20}j&a&b&c&d&e&f&g&h\\ \hline \cellcolor{green!20}j&a&b&c&d&e&f&g&h&i \end{array}\right]$$ Similarly, there are $n-t-2=n-m-1$ positions in the intersection of the last column and row $3$ through row $n-m+1$ that can be used to construct linearly independent $123$-avoiding permutation matrices that intersect the blocker exactly once and that use a unique position of the matrix. For example, consider such a permutation matrix utilizing the green $c$ from the last column below. $$\left[\begin{array}{c|c|c|c|c|c|c|c|c|c} \cellcolor{red!20}a&\cellcolor{red!20}b&\cellcolor{red!20}c&\cellcolor{red!20}d&\cellcolor{red!20}e&f&g&h&\cellcolor{green!20}i&j\\ \hline \cellcolor{red!20}b&\cellcolor{red!20}c&\cellcolor{red!20}d&\cellcolor{red!20}e&\cellcolor{red!20}f&g&h&\cellcolor{green!20}i&j&a\\ \hline \cellcolor{red!20}c&\cellcolor{red!20}d&\cellcolor{red!20}e&\cellcolor{red!20}f&\cellcolor{red!20}g&h&\cellcolor{green!20}i&j&a&b\\ \hline \cellcolor{red!20}d&\cellcolor{red!20}e&\cellcolor{red!20}f&\cellcolor{red!20}g&\cellcolor{red!20}h&i&j&a&b&\cellcolor{green!20}c\\ \hline \cellcolor{red!20}e&\cellcolor{red!20}f&\cellcolor{red!20}g&\cellcolor{red!20}h&\cellcolor{red!20}i&\cellcolor{green!20}j&a&b&c&d\\ \hline \cellcolor{red!20}f&\cellcolor{red!20}g&\cellcolor{red!20}h&\cellcolor{red!20}i&\cellcolor{yellow!20}j&a&b&c&d&e\\ \hline g&h&i&\cellcolor{green!20}j&a&b&c&d&e&f\\ \hline h&i&\cellcolor{green!20}j&a&b&c&d&e&f&g\\ \hline i&\cellcolor{green!20}j&a&b&c&d&e&f&g&h\\ \hline \cellcolor{green!20}j&a&b&c&d&e&f&g&h&i \end{array}\right]$$ Thus, in total we have found $$(n-2)^2+1-(m-1)(n-m-1) + m + 2(n-m-1) = (n-1)^2+1-(m-1)(n-m)$$ linearly independent $123$-avoiding permutation matrices that intersect the blocker exactly once, as desired. ◻ While Theorem [Theorem 12](#Theorem11){reference-type="ref" reference="Theorem11"} presents the upper bound of the dimension of a face of $\Omega_n(\overline{123})$ as determined by a flag-shaped minimum blocker, it does not establish the lower bound. We propose one lower bound, though we first consider a theorem regarding $L$-shaped blockers, a special type of flag-shaped blocker. **Theorem 15** (Theorem 3.7 in [@BC3]). *A minimum blocker of all $n \times n$ $123$-avoiding permutation matrices determines a facet of the polytope $\Omega_n(\overline{123})$ whose extreme points are the $n \times n$ $123$-avoiding permutation matrices that intersect the blocker exactly once.* **Corollary 16**. *For $L$-shaped minimum blockers of all $n \times n$ $123$-avoiding permutation matrices, there exist $(n-1)^2$ linearly independent $n \times n$ $123$-avoiding permutation matrices that each contain exactly one blocker position.* We can utilize Corollary [Corollary 16](#Lemma 14){reference-type="ref" reference="Lemma 14"} to determine a lower bound for the dimension of a face of $\Omega_n(\overline{123})$. **Theorem 17**. *A flag-shaped minimum blocker $B_n(m,t)$ of all $n \times n$ $123$-avoiding permutation matrices determines a face of $\Omega_n(\overline{123})$ with dimension at least $(n-1)^2+1-(t+2)(n-m)$.* *Proof.* Inducting on $p:=n-m$, we first show that when $p=1$, it is possible to find $(n-1)^2+1-(t+2)$ linearly independent $123$-avoiding permutation matrices that intersect the blocker exactly once. For illustrative purposes, we consider a $10 \times 10$ matrix with the flag-shaped blocker $B_n(9,3)$ while describing the general proof. Using the $(1,n)$ position, there are $(n-2)^2$ linearly independent $123$-avoiding permutation matrices that intersect the blocker exactly once. This is because the $(n-1) \times (n-1)$ submatrix obtained by deleting the first row and last column contains $[(n-1)-1]^2$ such permutation matrices according to Lemma [Corollary 16](#Lemma 14){reference-type="ref" reference="Lemma 14"}. $$\left[\begin{array}{c|c|c|c|c|c|c|c|c||c} a&b&c&d&e&\cellcolor{red!20}f&\cellcolor{red!20}g&\cellcolor{red!20}h&\cellcolor{red!20}i&\cellcolor{green!20}j\\ \hline \hline b&c&d&e&f&\cellcolor{red!20}g&\cellcolor{red!20}h&\cellcolor{red!20}i&\cellcolor{red!20}j&a\\ \hline c&d&e&f&g&h&i&j&\cellcolor{red!20}a&b\\ \hline d&e&f&g&h&i&j&a&\cellcolor{red!20}b&c\\ \hline e&f&g&h&i&j&a&b&\cellcolor{red!20}c&d\\ \hline f&g&h&i&j&a&b&c&\cellcolor{red!20}d&e\\ \hline g&h&i&j&a&b&c&d&\cellcolor{red!20}e&f\\ \hline h&i&j&a&b&c&d&e&f&g\\ \hline i&j&a&b&c&d&e&f&g&h\\ \hline j&a&b&c&d&e&f&g&h&i \end{array}\right]$$ Using $(2,n)$ in combination with the first $m-t-1$ positions from row $1$, in addition to $(n-t, n)$ in combination with the blocker positions in row $1$, we have an additional $m=n-1$ linearly independent $123$-avoiding permutation matrices that intersect the blocker once and that use a unique position of the matrix. Consider the following two matrices as examples, where yellow positions represent the intersection of the permutation matrix with the blocker. $$\left[\begin{array}{c|c|c|c|c|c|c|c|c|c} a&b&\cellcolor{green!20}c&d&e&\cellcolor{red!20}f&\cellcolor{red!20}g&\cellcolor{red!20}h&\cellcolor{red!20}i&j\\ \hline b&c&d&e&f&\cellcolor{red!20}g&\cellcolor{red!20}h&\cellcolor{red!20}i&\cellcolor{red!20}j&\cellcolor{green!20}a\\ \hline c&d&e&f&g&h&i&j&\cellcolor{yellow!20}a&b\\ \hline d&e&f&g&h&i&j&\cellcolor{green!20}a&\cellcolor{red!20}b&c\\ \hline e&f&g&h&i&j&\cellcolor{green!20}a&b&\cellcolor{red!20}c&d\\ \hline f&g&h&i&j&\cellcolor{green!20}a&b&c&\cellcolor{red!20}d&e\\ \hline g&h&i&j&\cellcolor{green!20}a&b&c&d&\cellcolor{red!20}e&f\\ \hline h&i&j&\cellcolor{green!20}a&b&c&d&e&f&g\\ \hline i&\cellcolor{green!20}j&a&b&c&d&e&f&g&h\\ \hline \cellcolor{green!20}j&a&b&c&d&e&f&g&h&i \end{array}\right] \text{ \ \ and \ \ } \left[\begin{array}{c|c|c|c|c|c|c|c|c|c} a&b&c&d&e&\cellcolor{red!20}f&\cellcolor{red!20}g&\cellcolor{yellow!20}h&\cellcolor{red!20}i&j\\ \hline b&c&d&e&\cellcolor{green!20}f&\cellcolor{red!20}g&\cellcolor{red!20}h&\cellcolor{red!20}i&\cellcolor{red!20}j&a\\ \hline c&d&e&\cellcolor{green!20}f&g&h&i&j&\cellcolor{red!20}a&b\\ \hline d&e&\cellcolor{green!20}f&g&h&i&j&a&\cellcolor{red!20}b&c\\ \hline e&\cellcolor{green!20}f&g&h&i&j&a&b&\cellcolor{red!20}c&d\\ \hline \cellcolor{green!20}f&g&h&i&j&a&b&c&\cellcolor{red!20}d&e\\ \hline g&h&i&j&a&b&c&d&\cellcolor{red!20}e&\cellcolor{green!20}f\\ \hline h&i&j&a&b&c&d&e&\cellcolor{green!20}f&g\\ \hline i&j&a&b&c&d&\cellcolor{green!20}e&f&g&h\\ \hline j&a&b&c&d&\cellcolor{green!20}e&f&g&h&i \end{array}\right]$$ Additionally, we can construct $n-t-3$ more linearly independent $123$-avoiding permutation matrices that intersect the blocker only once by utilizing the positions at the intersection of column $n$ and rows $3$ through $n-t-1$. Consider the following example. $$\left[\begin{array}{c|c|c|c|c|c|c|c|c|c} a&b&c&d&e&\cellcolor{red!20}f&\cellcolor{red!20}g&\cellcolor{red!20}h&\cellcolor{yellow!20}i&j\\ \hline b&c&d&e&\cellcolor{green!20}f&\cellcolor{red!20}g&\cellcolor{red!20}h&\cellcolor{red!20}i&\cellcolor{red!20}j&a\\ \hline c&d&e&\cellcolor{green!20}f&g&h&i&j&\cellcolor{red!20}a&b\\ \hline d&e&f&g&h&i&j&a&\cellcolor{red!20}b&\cellcolor{green!20}c\\ \hline e&f&\cellcolor{green!20}g&h&i&j&a&b&\cellcolor{red!20}c&d\\ \hline f&\cellcolor{green!20}g&h&i&j&a&b&c&\cellcolor{red!20}d&e\\ \hline \cellcolor{green!20}g&h&i&j&a&b&c&d&\cellcolor{red!20}e&f\\ \hline h&i&j&a&b&c&d&\cellcolor{green!20}e&f&g\\ \hline i&j&a&b&c&d&\cellcolor{green!20}e&f&g&h\\ \hline j&a&b&c&d&\cellcolor{green!20}e&f&g&h&i \end{array}\right]$$ In total we have $$(n-2)^2 + n-1 + n-t-3= (n-1)^2 +1 - (t+2)$$ linearly independent $123$-avoiding permutation matrices that intersect the blocker exactly once, as desired. Moving onto the induction step, suppose that when $p=k$, where $2 \leq k \leq n-3$, the corresponding flag-shaped blocker lives in dimension at least $$(n-1)^2+1-(t+2)(k).$$ We will show that if $p=k+1$, the flag-shaped blocker at least achieves the minimum dimension $$(n-1)^2+1-(t+2)(k+1).$$ For illustrative purposes, we consider a $10 \times 10$ matrix with the flag-shaped blocker $B_n(7,2)$ while describing the general proof. Using the inductive hypothesis, it is possible to find $$[(n-1)-1)]^2+1-(t+2)(k) = (n-2)^2+1-(t+2)(n-m)$$ linearly independent $123$-avoiding permutation matrices that intersect the blocker exactly once using the $(1,n)$ position. $$\left[\begin{array}{c|c|c|c|c|c|c|c|c||c} a&b&c&d&\cellcolor{red!20}e&\cellcolor{red!20}f&\cellcolor{red!20}g&h&i&\cellcolor{green!20}j\\ \hline \hline b&c&d&e&\cellcolor{red!20}f&\cellcolor{red!20}g&\cellcolor{red!20}h&i&j&a\\ \hline c&d&e&f&\cellcolor{red!20}g&\cellcolor{red!20}h&\cellcolor{red!20}i&j&a&b\\ \hline d&e&f&g&\cellcolor{red!20}h&\cellcolor{red!20}i&\cellcolor{red!20}j&a&b&c\\ \hline e&f&g&h&i&j&\cellcolor{red!20}a&b&c&d\\ \hline f&g&h&i&j&a&\cellcolor{red!20}b&c&d&e\\ \hline g&h&i&j&a&b&\cellcolor{red!20}c&d&e&f\\ \hline h&i&j&a&b&c&\cellcolor{red!20}d&e&f&g\\ \hline i&j&a&b&c&d&e&f&g&h\\ \hline j&a&b&c&d&e&f&g&h&i \end{array}\right]$$ Utilizing the first $m-t-1$ positions of row $1$ in conjunction with the $(2,n)$ position, we can construct $m-t-1$ additional linearly independent $123$-avoiding permutation matrices intersecting the blocker only once and containing a unique position of the matrix. Consider the below example using $c$ from row $1$ along with $a$ from row $2$. $$\left[\begin{array}{c|c|c|c|c|c|c|c|c|c} a&b&\cellcolor{green!20}c&d&\cellcolor{red!20}e&\cellcolor{red!20}f&\cellcolor{red!20}g&h&i&j\\ \hline b&c&d&e&\cellcolor{red!20}f&\cellcolor{red!20}g&\cellcolor{red!20}h&i&j&\cellcolor{green!20}a\\ \hline c&d&e&f&\cellcolor{red!20}g&\cellcolor{red!20}h&\cellcolor{red!20}i&j&\cellcolor{green!20}a&b\\ \hline d&e&f&g&\cellcolor{red!20}h&\cellcolor{red!20}i&\cellcolor{red!20}j&\cellcolor{green!20}a&b&c\\ \hline e&f&g&h&i&j&\cellcolor{yellow!20}a&b&c&d\\ \hline f&g&h&i&j&\cellcolor{green!20}a&\cellcolor{red!20}b&c&d&e\\ \hline g&h&i&j&\cellcolor{green!20}a&b&\cellcolor{red!20}c&d&e&f\\ \hline h&i&j&\cellcolor{green!20}a&b&c&\cellcolor{red!20}d&e&f&g\\ \hline i&\cellcolor{green!20}j&a&b&c&d&e&f&g&h\\ \hline \cellcolor{green!20}j&a&b&c&d&e&f&g&h&i \end{array}\right]$$ Furthermore, we can use the row $1$ blocker positions along with the $(m-t+1,n)$ position to obtain $t+1$ additional linearly independent $123$-avoiding permutation matrices that intersect the blocker once each. Consider such a permutation matrix using the $f$ from the first row and the $e$ from the last column below. $$\left[\begin{array}{c|c|c|c|c|c|c|c|c|c} a&b&c&d&\cellcolor{red!20}e&\cellcolor{yellow!20}f&\cellcolor{red!20}g&h&i&j\\ \hline b&c&d&\cellcolor{green!20}e&\cellcolor{red!20}f&\cellcolor{red!20}g&\cellcolor{red!20}h&i&j&a\\ \hline c&d&\cellcolor{green!20}e&f&\cellcolor{red!20}g&\cellcolor{red!20}h&\cellcolor{red!20}i&j&a&b\\ \hline d&\cellcolor{green!20}e&f&g&\cellcolor{red!20}h&\cellcolor{red!20}i&\cellcolor{red!20}j&a&b&c\\ \hline \cellcolor{green!20}e&f&g&h&i&j&\cellcolor{red!20}a&b&c&d\\ \hline f&g&h&i&j&a&\cellcolor{red!20}b&c&d&\cellcolor{green!20}e\\ \hline g&h&i&j&a&b&\cellcolor{red!20}c&d&\cellcolor{green!20}e&f\\ \hline h&i&j&a&b&c&\cellcolor{red!20}d&\cellcolor{green!20}e&f&g\\ \hline i&j&a&b&c&d&\cellcolor{green!20}e&f&g&h\\ \hline j&a&b&c&\cellcolor{green!20}d&e&f&g&h&i \end{array}\right]$$ Using the $m+1$ through $n-2$ positions from row one, we can obtain $k-2=n-m-2$ additional linearly independent $123$-avoiding permutation matrices that intersect the blocker once and that use a unique position. Consider the below example. $$\left[\begin{array}{c|c|c|c|c|c|c|c|c|c} a&b&c&d&\cellcolor{red!20}e&\cellcolor{red!20}f&\cellcolor{red!20}g&\cellcolor{green!20}h&i&j\\ \hline b&c&d&e&\cellcolor{red!20}f&\cellcolor{red!20}g&\cellcolor{red!20}h&i&j&\cellcolor{green!20}a\\ \hline c&d&e&f&\cellcolor{red!20}g&\cellcolor{red!20}h&\cellcolor{red!20}i&j&\cellcolor{green!20}a&b\\ \hline d&e&f&g&\cellcolor{red!20}h&\cellcolor{red!20}i&\cellcolor{yellow!20}j&a&b&c\\ \hline e&f&g&h&i&\cellcolor{green!20}j&\cellcolor{red!20}a&b&c&d\\ \hline f&g&h&i&\cellcolor{green!20}j&a&\cellcolor{red!20}b&c&d&e\\ \hline g&h&i&\cellcolor{green!20}j&a&b&\cellcolor{red!20}c&d&e&f\\ \hline h&i&\cellcolor{green!20}j&a&b&c&\cellcolor{red!20}d&e&f&g\\ \hline i&\cellcolor{green!20}j&a&b&c&d&e&f&g&h\\ \hline \cellcolor{green!20}j&a&b&c&d&e&f&g&h&i \end{array}\right]$$ Thus far, we have used $3$ positions from column $n$ in our constructions. There are $n-t-3$ additional positions in this column that we may utilize, each of which will result in a permutation matrix that contains a unique position of the matrix. Consider two such examples below. $$\left[\begin{array}{c|c|c|c|c|c|c|c|c|c} a&b&c&d&\cellcolor{red!20}e&\cellcolor{red!20}f&\cellcolor{red!20}g&h&\cellcolor{green!20}i&j\\ \hline b&c&d&e&\cellcolor{red!20}f&\cellcolor{red!20}g&\cellcolor{red!20}h&\cellcolor{green!20}i&j&a\\ \hline c&d&e&f&\cellcolor{yellow!20}g&\cellcolor{red!20}h&\cellcolor{red!20}i&j&a&b\\ \hline d&e&f&\cellcolor{green!20}g&\cellcolor{red!20}h&\cellcolor{red!20}i&\cellcolor{red!20}j&a&b&c\\ \hline e&f&\cellcolor{green!20}g&h&i&j&\cellcolor{red!20}a&b&c&d\\ \hline f&\cellcolor{green!20}g&h&i&j&a&\cellcolor{red!20}b&c&d&e\\ \hline \cellcolor{green!20}g&h&i&j&a&b&\cellcolor{red!20}c&d&e&f\\ \hline h&i&j&a&b&c&\cellcolor{red!20}d&e&f&\cellcolor{green!20}g\\ \hline i&j&a&b&c&d&\cellcolor{green!20}e&f&g&h\\ \hline j&a&b&c&d&\cellcolor{green!20}e&f&g&h&i \end{array}\right] \text{ \ \ and \ \ } \left[\begin{array}{c|c|c|c|c|c|c|c|c|c} a&b&c&d&\cellcolor{red!20}e&\cellcolor{red!20}f&\cellcolor{red!20}g&h&\cellcolor{green!20}i&j\\ \hline b&c&d&e&\cellcolor{red!20}f&\cellcolor{red!20}g&\cellcolor{red!20}h&\cellcolor{green!20}i&j&a\\ \hline c&d&e&f&\cellcolor{red!20}g&\cellcolor{red!20}h&\cellcolor{yellow!20}i&j&a&b\\ \hline d&e&f&g&\cellcolor{red!20}h&\cellcolor{red!20}i&\cellcolor{red!20}j&a&b&\cellcolor{green!20}c\\ \hline e&f&g&h&i&\cellcolor{green!20}j&\cellcolor{red!20}a&b&c&d\\ \hline f&g&h&i&\cellcolor{green!20}j&a&\cellcolor{red!20}b&c&d&e\\ \hline g&h&i&\cellcolor{green!20}j&a&b&\cellcolor{red!20}c&d&e&f\\ \hline h&i&\cellcolor{green!20}j&a&b&c&\cellcolor{red!20}d&e&f&g\\ \hline i&\cellcolor{green!20}j&a&b&c&d&e&f&g&h\\ \hline \cellcolor{green!20}j&a&b&c&d&e&f&g&h&i \end{array}\right]$$ After adding and rewriting, in total we have constructed $(n-1)^2+1-(t+2)(k+1)$ linearly independent $123$-avoiding permutation matrices that intersect the blocker exactly once, as desired. ◻ Our results relating to the upper and lower bound for the dimension of a face of $\Omega_n(\overline{123})$ provide insight into the geometric properties of the polytope, though there is still work to be done in terms of more precisely determining the dimension of the faces of $\Omega_n(\overline{123})$. 1 Brualdi, R.A., & Cao, L. (2021). Pattern-avoiding $(0,1)$-matrices and bases of permutation matrices. *Discrete Appl. Math.*, 304, 196-211. Brualdi, R.A., & Cao, L. (2022). Blockers of pattern avoiding permutation matrices. *Australasian J. Combin.*, 83(2), 274-303. Brualdi, R.A., & Cao, L. (2023). $123$-Forcing matrices. *Australasian J. Combin.*, 86(1), 169-186. Brualdi, R.A., & Cao, L. (2023). $123$-Avoiding doubly stochastic matrices. *Linear Algebra Appl.*, 1-33. König, D. (1933). Über trennende knotenpunkte in graphen (nebst anwendundungen auf determinanten und matrizen). *Sectio Scientiarum Mathematicarum (Szeged)*, 6, 155-179.

arxiv_math

--- abstract: | We establish the existence of multiple solutions for a nonlinear problem of critical type. The problem considered is fractional in nature, since it is obtained by the superposition of $(s,p)$-fractional Laplacians of different orders. The results obtained are new even in the case of the sum of two different fractional $p$-Laplacians, or the sum of a fractional $p$-Laplacian and a classical $p$-Laplacian, but our framework is general enough to address also the sum of finitely, or even infinitely many, operators. In fact, we can also consider the superposition of a continuum of operators, modulated by a general signed measure on the fractional exponents. When this measure is not positive, the contributions of the individual operators to the whole superposition operator is allowed to change sign. In this situation, our structural assumption is that the positive measure on the higher fractional exponents dominates the rest of the signed measure. address: - "Serena Dipierro: Department of Mathematics and Statistics, The University of Western Australia, 35 Stirling Highway, Crawley, Perth, WA 6009, Australia" - "Kanishka Perera: Department of Mathematical Sciences, Florida Institute of Technology, 150 W University Blvd, Melbourne, FL 32901-6975, USA" - "Caterina Sportelli: Department of Mathematics and Statistics, The University of Western Australia, 35 Stirling Highway, Crawley, Perth, WA 6009, Australia" - "Enrico Valdinoci: Department of Mathematics and Statistics, The University of Western Australia, 35 Stirling Highway, Crawley, Perth, WA 6009, Australia" author: - Serena Dipierro - Kanishka Perera - Caterina Sportelli - Enrico Valdinoci title: | An existence theory\ for nonlinear superposition operators\ of mixed fractional order --- # Introduction ## Nonlinear nonlocal operators and existence theory {#AB1} The goal of this article is to construct solutions for an operator obtained through the superposition of possibly nonlinear, fractional operators. The results obtained will be very general, but they are also new in a number of specific interesting cases which will be obtained as a simple byproduct of our comprehensive approach. We consider two finite (Borel) measures $\mu^+$ and $\mu^-$ on the interval of fractional exponents $[0,1]$ satisfying, for some $\overline s\in ({{ 0 }}, 1]$ and $\gamma\geqslant 0$, that $$\label{mu00} {\mu^+}([\overline s, 1])>0,$$ $$\label{mu3} {\mu^-}_{\big|_{[\overline s, 1]}}=0$$ and $$\label{mu2} \mu^-\big([{{ 0 }}, \overline s]\big)\leqslant\gamma\, \mu^+\big([\overline s, 1]\big).$$ We consider the signed measure $$\mu:=\mu^+-\mu^-$$ and the nonlinear fractional operator $$A_{\mu,p}u:=\int_{[0, 1]} (-\Delta)_p^{s} u \, d\mu(s).$$ Notice that the above integration may occur with "variable signs", since $\mu$ is a signed measure with possibly negative components induced by the measure $\mu^-$: however, conditions [\[mu00\]](#mu00){reference-type="eqref" reference="mu00"} and [\[mu3\]](#mu3){reference-type="eqref" reference="mu3"} say that these negative components do not affect the higher values of fractional exponents and, in fact, by condition [\[mu2\]](#mu2){reference-type="eqref" reference="mu2"}, the measure of these higher fractional exponents suitably dominate the negative contribution. This setting has been introduced in [@CATERINA] for the case of linear fractional operators (corresponding to $p=2$). Here, we can also allow a nonlinear nonlocal structure of the operator, hence we assume that $p\in(1,+\infty)$. In our setting, when $s\in(0,1)$ the fractional $p$-Laplacian $(- \Delta)_p^s$ is the nonlinear nonlocal operator defined on smooth functions by $$(- \Delta)_p^s\, u(x) := 2c_{N,s,p} \,\lim_{\varepsilon\searrow 0} \int_{\mathbb{R}^N \setminus B_\varepsilon(x)} \frac{|u(x) - u(y)|^{p-2}\, (u(x) - u(y))}{|x - y|^{N+sp}}\, dy.$$ See e.g. [@MR3593528; @MR3861716] and the references therein for more information on the fractional $p$-Laplace operator. The exact value of the positive normalizing constant $c_{N,s,p}$ is not important[^1] for us, except for allowing a consistent setting for the limit cases, namely that $$\begin{aligned} &&\lim_{s\searrow0}(- \Delta)_p^s\, u=(- \Delta)_p^0\, u:=u\\{\mbox{and}}\qquad&& \lim_{s\nearrow1}(- \Delta)_p^s\, u=(- \Delta)_p^1\, u:=-\Delta_pu=-{\operatorname{div}}(|\nabla u|^{p-2}\nabla u).\end{aligned}$$ To develop our analysis, we consider a bounded open set $\Omega\subset\mathbb{R}^N$ and rely on the spectral theory for the operator $A_{\mu,p}$. For this, one looks at the values of $\lambda$ allowing for nontrivial solutions of the equation $$\label{EUGE} \begin{cases}A_{\mu,p}u=\lambda |u|^{p-2}u &{\mbox{ in }}\Omega,\\u=0&{\mbox{ in }}\mathbb{R}^N\setminus\Omega.\end{cases}$$ The cohomological index theory by Fadell and Rabinowitz (put forth in [@MR0478189] and whose essential ingredients will also be recalled in the forthcoming Section [1.2](#AB2){reference-type="ref" reference="AB2"}) addresses the Dirichlet eigenvalue theory related to equation [\[EUGE\]](#EUGE){reference-type="eqref" reference="EUGE"}, provides the existence of a sequence $\lambda_l$ of eigenvalues, with $0<\lambda_1\leqslant\lambda_2\leqslant\dots$, and will constitute an important building block for our existence theory. Given the superposition nature of the operator, there is another fractional exponent which may play a critical role. To appreciate this, we observe that, by assumption [\[mu00\]](#mu00){reference-type="eqref" reference="mu00"}, there exists $s_\sharp\in [\overline s, 1]$ such that $$\label{scritico} \mu^+ ([s_\sharp, 1])> 0.$$ We will see below that the exponent $s_\sharp$ plays also the role of a critical exponent (and we remark that, while some arbitrariness is allowed in the choice of $s_\sharp$ here above, the results obtained will be stronger if one picks $s_\sharp$ "as large as possible" but still fulfilling [\[scritico\]](#scritico){reference-type="eqref" reference="scritico"}). Specifically, we will look at the equation $$\label{mainp} \left\{\begin{aligned} A_{\mu,p}u& = \lambda\, |u|^{p-2}\, u + |u|^{p_{s_\sharp}^* - 2}\, u && \text{in } \Omega,\\ u & = 0 && \text{in } \mathbb{R}^N \setminus \Omega. \end{aligned}\right.$$ Notice that if a nontrivial solution $u$ exists, then $-u$ is a nontrivial solutions as well (hence, nontrivial solutions go automatically in pairs). The exponent $p_{s_\sharp}^*$ is the fractional critical exponent related to the fractional parameter $s_\sharp$ for which [\[scritico\]](#scritico){reference-type="eqref" reference="scritico"} holds true, namely $$p_{s_\sharp}^*:=\frac{Np}{N-s_\sharp p}\,,$$ under the condition[^2] that $N>p$. We also introduce the notion of Sobolev constant suitable for our framework. Namely, we set $$[u]_{s,p}:=\begin{dcases} \|u\|_{L^p(\mathbb{R}^N)} & {\mbox{ if }}s=0,\\ \displaystyle\left( c_{N,s,p}\iint_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^p}{|x-y|^{N+sp}}\,dx\,dy\right)^{1/p} & {\mbox{ if }}s\in(0,1),\\ \|\nabla u\|_{L^p(\mathbb{R}^N)} & {\mbox{ if }}s=1. \end{dcases}$$ Thanks to the normalizing constant $c_{N,s,p}$ we have that $$\lim_{s\searrow0}[u]_{s,p}=[u]_{0,p}\qquad{\mbox{and}}\qquad \lim_{s\nearrow1}[u]_{s,p}=[u]_{1,p}.$$ The Sobolev constant that we consider is thus defined as $$\label{DESOCO} {\mathcal S(p)}:=\inf \int_{[0,1]}[u]_{s,p}^p \,d\mu^+(s),$$ where the infimum is taken over all the measurable functions with $\|u\|_{L^{p^*_{s_\sharp}}(\mathbb{R}^N)}=1$. With this, our main result goes as follows: **Theorem 1**. *Let $\mu=\mu^+-\mu^-$ with $\mu^+$ and $\mu^-$ satisfying [\[mu00\]](#mu00){reference-type="eqref" reference="mu00"}, [\[mu3\]](#mu3){reference-type="eqref" reference="mu3"} and [\[mu2\]](#mu2){reference-type="eqref" reference="mu2"}. Let $s_\sharp$ be the exponent given by [\[scritico\]](#scritico){reference-type="eqref" reference="scritico"}.* *Assume that $$\label{LLP1} \lambda_l - \frac{\mathcal S(p)}{|\Omega|^{(s_\sharp p)/N}} < \lambda < \lambda_l = \dots = \lambda_{l+m-1}$$ for some $l, m \geqslant 1$.* *Then, there exists $\gamma_0>0$ depending only on $N$, $\Omega$, $p$, $s_\sharp$, $\mu$, $\lambda$ and $l$ such that if $\gamma\in[0,\gamma_0]$, problem [\[mainp\]](#mainp){reference-type="eqref" reference="mainp"} has $m$ distinct pairs of nontrivial solutions $\pm u^\lambda_1,\dots,\pm u^\lambda_m$.* As an immediate consequence of this result, taking $m:=1$ here above, we obtain: **Corollary 2**. *Let $\mu=\mu^+-\mu^-$ with $\mu^+$ and $\mu^-$ satisfying [\[mu00\]](#mu00){reference-type="eqref" reference="mu00"}, [\[mu3\]](#mu3){reference-type="eqref" reference="mu3"} and [\[mu2\]](#mu2){reference-type="eqref" reference="mu2"}. Let $s_\sharp$ be the exponent given by [\[scritico\]](#scritico){reference-type="eqref" reference="scritico"}.* *Assume that, for some $l\geqslant 1$, $$\lambda_l - \frac{\mathcal S(p)}{|\Omega|^{(s_\sharp p)/N}} < \lambda < \lambda_l .$$* *Then, there exists $\gamma_0>0$ depending only on $N$, $\Omega$, $p$, $s_\sharp$, $\mu$, $\lambda$ and $l$ such that if $\gamma\in[0,\gamma_0]$, problem [\[mainp\]](#mainp){reference-type="eqref" reference="mainp"} has a nontrivial solution (and therefore a pair of nontrivial solutions, one equal to the other up to a minus sign).* Taking $\mu$ as the Dirac measure at $1$, Theorem [Theorem 1](#mainp1){reference-type="ref" reference="mainp1"} recovers the multiplicity result stated in [@MR3469053 Theorem 1.1], see the forthcoming Corollary [Corollary 22](#C1){reference-type="ref" reference="C1"}. Also, taking $\mu$ as the Dirac measure at some $s\in(0,1)$, Theorem [Theorem 1](#mainp1){reference-type="ref" reference="mainp1"} recovers [@PSY Theorem 1.1] see Corollary [Corollary 23](#C2){reference-type="ref" reference="C2"}. Theorem [Theorem 1](#mainp1){reference-type="ref" reference="mainp1"} is new and very general, in view of the structure of the superposition operator $A_{\mu,p}$. Indeed, to the best of our knowledge, several special cases of interest of Theorem [Theorem 1](#mainp1){reference-type="ref" reference="mainp1"} happen to be new. In particular: - Let $s\in [0, 1)$ and set $\mu:= \delta_1 +\delta_s$, where $\delta_1$ and $\delta_s$ denote the Dirac measures centered at the points $1$ and $s$, respectively. With this choice, the operator $A_{\mu,p}$ boils down to $-\Delta_p + (-\Delta)_p^s$. We will deal with this case in Corollary [Corollary 24](#C3){reference-type="ref" reference="C3"}. As far as we know, this result is new, even when $p=2$ (for this case, we refer the interested reader to [@arXiv.2209.07502] for some existence results). - Let $s\in[0,1)$ and $\mu:= \delta_1 -\alpha\delta_s$, where $\alpha$ is a small positive constant. This choice corresponds to the operator $-\Delta_p-\alpha(-\Delta)^s_p$. Notice that with this choice the measure $\mu$ changes sign and the second term in the operator has the "wrong" sign. This case is new and will be treated in Corollary [Corollary 25](#C5){reference-type="ref" reference="C5"}. - Let $1\geqslant s_0 > s_1> s_2 >\dots \geqslant 0$ and $$\mu:= \sum_{k=0}^{+\infty} c_k \delta_{s_k},\qquad{\mbox{where }}\, \sum_{k=0}^{+\infty} c_k\in(0,+\infty),$$ with - either $c_0>0$ and $c_k\geqslant 0$ for all $k\geqslant 1$, - or $$\begin{aligned} & & c_k>0\ \text{ for all } k\in\{0,\dots, \overline k\} \text{ and } \sum_{k=\overline k +1}^{+\infty} c_k \leqslant\gamma \sum_{k=0}^{\overline k} c_k,\\ &&{\mbox{for some~$\overline k\in\mathbb{N}$ and~$\gamma\geqslant 0$.}}\end{aligned}$$ These two cases of convergent series of Dirac measures are new and will be deal with in Corollaries [Corollary 26](#serie1){reference-type="ref" reference="serie1"} and [Corollary 27](#serie2){reference-type="ref" reference="serie2"}. - Let $f$ be a measurable and non identically zero function and consider a continuous superposition of fractional $p$-Laplacian of the form $$\int_0^1 f(s) (-\Delta)_p^s \, u \, ds.$$ This case is also new in the literature and will be treated in Corollary [Corollary 28](#function){reference-type="ref" reference="function"}. We will discuss in detail all these cases in Section [6](#sec-app){reference-type="ref" reference="sec-app"}. An interesting feature of the operators considered here is that not only we can deal with nonlinear operators and infinitely many (possibly uncountably many) fractional operators at the same time, but also that some of these operators may have the "wrong sign" (provided that there is a "dominant part" given by the operators with higher fractional order). Indeed, we think that the possibility of dealing with complicated operators having some components with the "wrong sign" is particularly intriguing in view of applications in biology and population dynamics. For example, in light of the Lévy flight foraging hypothesis, it is often appropriate to model animal dispersal as a superposition of (possibly fractional) operators of different order, corresponding to different individuals of a given population adopting different foraging strategies (see e.g. [@MR4249816]). In this sense, the possibility of including also operators with the "wrong sign" provides a simple way to model individuals which, rather than diffusing to search for food, tend to concentrate together (e.g., for social or mating reason), exhibiting patterns induced by a retrograde fractional heat equation. ## An abstract formulation {#AB2} It will now be convenient to cast the framework described in Section [1.1](#AB1){reference-type="ref" reference="AB1"} into an "abstract" setting. The advantage of this procedure is to develop all the methodology at the level of a suitable functional analysis, relying on general topological methods, which will then provide the proof of the main result in Theorem [Theorem 1](#mainp1){reference-type="ref" reference="mainp1"} as a simple byproduct. To this end, we consider $$\label{UNICO} {\mbox{a uniformly convex Banach space~$W$,}}$$ with its norm $\left\|\, \cdot\,\right\|_{}$. The dual space will be denoted by $W^*$, endowed with the dual norm $\left\|\, \cdot\,\right\|_{}^*$. The duality pairing will be denoted by $\left(\cdot,\cdot\right)_{}$. In this setting, one says that $f \in C(W,W^*)$ is a potential operator if there is a functional $F \in C^1(W,\mathbb{R})$, called a potential for $f$, such that $F' = f$. We consider $$\label{UNICO2} {\mbox{two potential operators~$A_p\, , B_p\, \in C(W,W^*)$}}$$ satisfying the following assumptions: 1. the operator $A_p\,$ is $(p - 1)$-homogeneous and odd for some $p \in (1,+\infty)$, i.e., for all $u \in W$ and $t \in\mathbb{R}$, we have that $$A_p\, (tu) = |t|^{p-2}\, t\, A_p\, u,$$ 2. for all $u, v \in W$, we have that $$\left(A_p\, u,v\right)_{} \leqslant\left\|u\right\|_{}^{p-1} \left\|v\right\|_{}$$ and equality holds if and only if $\alpha u = \beta v$ for some $\alpha, \beta \geqslant 0$, with $\alpha^2+\beta^2\ne0$, 3. the operator $B_p\,$ is $(p - 1)$-homogeneous and odd, i.e., for all $u \in W$ and $t \in\mathbb{R}$, $$B_p\, (tu) = |t|^{p-2}\, t\, B_p\, u,$$ 4. for all $u \in W \setminus \left\{0\right\}$ we have that $$\left(B_p\, u,u\right)_{} > 0,$$ and for all $u, v \in W$ we have that $$\left(B_p\, u,v\right)_{} \leqslant\left(B_p\, u,u\right)_{}^{(p-1)/p} \left(B_p\, v,v\right)_{}^{1/p},$$ 5. $B_p\,$ is a compact operator. We will develop the theory in this abstract framework and then we will reduce it to the case of interest in Section [1.1](#AB1){reference-type="ref" reference="AB1"} by taking $$\label{2wePAL:a}A_pu:=\int_{[0, 1]} (-\Delta)_p^{s} u \, d\mu^+(s)\qquad{\mbox{ and }}\qquad B_p u:= |u|^{p- 2}\, u .$$ We define $$\label{SETT1} I_p(u) := \frac{1}{p} \left(A_p\, u,u\right)_{}\qquad{\mbox{and}} \qquad J_p(u) := \frac{1}{p} \left(B_p\, u,u\right)_{}.$$ The interest of these objects is that $$\label{ARETRH} \begin{split}&{\mbox{$I_p$ and~$J_p$ are the potentials of~$A_p\, $ and~$B_p\, $,}}\\&{\mbox{satisfying~$I_p(0) = 0$ and~$J_p(0) = 0$, respectively,}}\end{split}$$ see [@MR4293883 Proposition 3.1]. We also observe that, as a consequence of assumption $(A_2)$, for all $u \in W$ we have that $$\left(A_p\, u,u\right)_{} = \left\|u\right\|_{}^p$$ and therefore $$\label{SETT2} I_p(u) = \frac{1}{p} \left\|u\right\|_{}^p.$$ The reason for introducing hypotheses $(A_1)$, $(A_2)$, $(B_1)$, $(B_2)$ and $(B_3)$ is that, under the above assumptions, it is known that the nonlinear eigenvalue problem $$\label{2.888} A_p\, u = \lambda B_p\, u \quad \text{in } W^*$$ possesses an unbounded sequence of eigenvalues $\lambda_1\leqslant\lambda_2\leqslant\dots$ and that $\lambda_1>0$, see [@MR1998432], and also [@MR2640827 Theorem 4.6] and [@MR4293883 Theorem 1.3] for full details on this topic. Now we consider $$\label{2.888-0} {\mbox{a potential operator~$f \in C(W,W^*)$.}}$$ We assume that $$\label{2.888-01} {\mbox{$f$ is odd (i.e. $f(-u)=-f(u)$)}}$$ and that the potential $F$ of $f$ is normalized such that $F(0) = 0$. We also consider $$\label{lppotop} {\mbox{a potential operator~$L_p\in C(W, W^*)$}}$$ such that 1. the operator $L_p$ is $(p - 1)$-homogeneous and odd for some $p \in (1,+\infty)$, i.e., for all $u \in W$ and $t \in\mathbb{R}$, we have that $$L_p(tu) = |t|^{p-2}\, t\, L_p(u).$$ Additionally, we define $$N_p(u):=\frac1p(L_pu,u).$$ In this setting, $N_p$ is the potential of $L_p$. Given $\lambda > 0$, our goal is now to study the equation $$\label{2.10} A_p\, u = \lambda B_p\, u + L_p u + f(u) \quad \text{in } W^*.$$ This abstract formulation will then be reduced to the concrete case showcased in Section [1.1](#AB1){reference-type="ref" reference="AB1"} through the choice $$L_pu:=\int_{[0, 1]} (-\Delta)_p^{s} u \, d\mu^-(s)$$ and $f(u):= |u|^{p_{s_\sharp}^* - 2}\, u$ (together with the setting in [\[2wePAL:a\]](#2wePAL:a){reference-type="eqref" reference="2wePAL:a"}). To study [\[2.10\]](#2.10){reference-type="eqref" reference="2.10"} in its more general formulation, it is convenient to define, for all $u \in W$, $$\label{EDE} E(u) := I_p(u) - N_p(u) - \lambda J_p(u) - F(u) .$$ The convenience of this definition is that, in our setting, $E$ will play the role of the variational functional associated with equation [\[2.10\]](#2.10){reference-type="eqref" reference="2.10"}. We suppose that the following structural assumptions are satisfied: 1. $F(u) = \text{o}(\left\|u\right\|_{}^p)$ as $u \to 0$, 2. there exist $\beta \in( 0,+\infty)$ and $q \in( p,+\infty)$ such that, for all $u \in W$, $$F(u) \geqslant\dfrac{\beta}{q}\, \big(p\, J_p(u)\big)^{q/p},$$ 3. there exists $c^* \in( 0,+\infty)$ such that the functional $E$ satisfies the Palais-Smale condition $(\text{PS})_{c}$ for all $c \in (0,c^*)$, 4. there exists $\eta\in (0, 1)$ such that $$0\leqslant N_p(u)\leqslant\eta\, I_p(u).$$ Our main result in this framework[^3] goes as follows: **Theorem 3**. *Suppose that $$\label{2.11} \lambda_l - \beta^{p/q} \left(\frac{pqc^*}{q - p}\right)^{1 - p/q} < \lambda < \lambda_l = \dots = \lambda_{l+m-1}$$ for some $l, m \geqslant 1$.* *Then, there exists $\eta_0>0$, depending only on $N$, $\Omega$, $p$, $\mu$, $\lambda$ and $l$, such that if $\eta\in[0,\eta_0]$, then equation [\[2.10\]](#2.10){reference-type="eqref" reference="2.10"} has $m$ distinct pairs of nontrivial solutions $\pm u^\lambda_1,\dots,\pm u^\lambda_m$.* An immediate consequence of this result occurs by taking $m:=1$ here above. In this way, we obtain: **Corollary 4**. *Suppose that, for some $l \geqslant 1$, $$\lambda_l - \beta^{p/q} \left(\frac{pqc^*}{q - p}\right)^{1 - p/q} < \lambda < \lambda_l .$$* *Then, there exists $\eta_0>0$, depending only on $N$, $\Omega$, $p$, $\mu$, $\lambda$ and $l$, such that if $\eta\in[0,\eta_0]$, then equation [\[2.10\]](#2.10){reference-type="eqref" reference="2.10"} has a nontrivial solution (and therefore a pair of nontrivial solutions, one equal to the other up to a minus sign).* The rest of this paper is organized as follows. In Section [2](#ANasqdf){reference-type="ref" reference="ANasqdf"} we deal with the abstract formulation and prove Theorem [Theorem 3](#Theorem 2.9){reference-type="ref" reference="Theorem 2.9"}. Then, we focus on the proof of Theorem [Theorem 1](#mainp1){reference-type="ref" reference="mainp1"}. For this, in Section [3](#SOBOLEV){reference-type="ref" reference="SOBOLEV"} we present some uniform inequalities of Sobolev type, of independent interest, and in Section [4](#BaUPS){reference-type="ref" reference="BaUPS"} we introduce a suitable functional setting to deal with the problem under consideration. The proof of Theorem [Theorem 1](#mainp1){reference-type="ref" reference="mainp1"} is completed in Section [5](#BaUPS-2){reference-type="ref" reference="BaUPS-2"}, by checking that the hypotheses of the abstract result in Theorem [Theorem 3](#Theorem 2.9){reference-type="ref" reference="Theorem 2.9"} are fulfilled. In Section [6](#sec-app){reference-type="ref" reference="sec-app"} we "specialize\" the superposition operator $A_{\mu,p}$ and we employ Theorem [Theorem 1](#mainp1){reference-type="ref" reference="mainp1"} to provide new multiplicity results for many cases of interest. # Proof of Theorem [Theorem 3](#Theorem 2.9){reference-type="ref" reference="Theorem 2.9"} {#ANasqdf} Given $r > 0$, we consider the sphere of radius $r$ in $W$, namely $$S_r := \left\{u \in W : \left\|u\right\|_{} = r\right\}.$$ We also use the short notation $S:=S_1$. In view of [\[SETT1\]](#SETT1){reference-type="eqref" reference="SETT1"} and [\[SETT2\]](#SETT2){reference-type="eqref" reference="SETT2"}, we have that $$S = \left\{u \in W \,:\, I_p(u) = \frac{1}{p}\right\}.$$ For every $u\in S$, we let $$\label{2.8} \Psi(u) := \frac{1}{p\, J_p(u)} .$$ It is known that critical points of $\Psi$ in $S$ are related to solutions of the eigenvalue problem in [\[2.888\]](#2.888){reference-type="eqref" reference="2.888"}, see [@MR2640827 Theorem 4.6] and [@MR4293883 Theorem 1.3], and, in particular, that $$\label{2.9} \lambda_1 = \inf_{u \in S}\, \Psi(u) > 0.$$ To prove Theorem [Theorem 3](#Theorem 2.9){reference-type="ref" reference="Theorem 2.9"}, we will rely on an abstract critical point theory, detailed in [@MR3469053], based on the $\mathbb{Z}_2$-cohomological index introduced by Fadell and Rabinowitz in [@MR0478189] and denoted here by $i(\cdot)$. To this end, we say that a subset $A$ of $W$ is symmetric if $x\in A$ if and only if $-x\in A$. When $E$ is an even functional satisfying $(\mathcal F_3)$, the following result holds true: **Theorem 5** ([@MR3469053 Theorem 2.2]). *Let $A_0$ and $B_0$ be symmetric subsets of the unit sphere $S$ such that $A_0$ is compact and $B_0$ is closed.* *Assume that, for some integers $l, m \geqslant 1$, we have that $$i(A_0) \geqslant l + m - 1 \qquad{\mbox{and}}\qquad i(S \setminus B_0) \leqslant l - 1.$$* *Let $r\in(0,+\infty)$ and $R\in(r,+\infty)$. We define $$\label{LAUa}\begin{split} &A := \left\{Ru : u \in A_0\right\},\\ &B: = \left\{ru : u \in B_0\right\}\\ {\mbox{and }}\quad&X := \left\{tu : u \in A,\, 0 \leqslant t \leqslant 1\right\}.\end{split}$$* *We assume that $$\label{2.7} \sup_{u \in A}\, E(u) \leqslant 0 < \inf_{u \in B}\, E(u) \qquad{\mbox{and}}\qquad \sup_{u \in X}\, E(u) < c^*.$$* *Then, $E$ has $m$ distinct pairs of critical points.* Given $a \in \mathbb{R}$, we consider the sublevel and superlevel sets of $\Psi$, as defined in [\[2.8\]](#2.8){reference-type="eqref" reference="2.8"}, namely we set $$\Psi^a := \left\{u \in S : \Psi(u) \leqslant a\right\}\qquad{\mbox{and}}\qquad\Psi_a = \left\{u \in S : \Psi(u) \geqslant a\right\}.$$ With this notation, we have: **Theorem 6** ([@MR2640827 Theorem 4.6] and [@MR4293883 Theorem 1.3]). *Assume that $l\geqslant 2$ and that $\lambda_{l-1} < \lambda_l$.* *Then, $$i(\Psi^{\lambda_{l-1}}) = i(S \setminus \Psi_{\lambda_l}) = l - 1$$ and $\Psi^{\lambda_{l-1}}$ has a compact symmetric subset of index $l - 1$.* With these preliminary results, we can now complete the proof of Theorem [Theorem 3](#Theorem 2.9){reference-type="ref" reference="Theorem 2.9"}, by arguing as follows: *Proof of Theorem [Theorem 3](#Theorem 2.9){reference-type="ref" reference="Theorem 2.9"}.* Regarding condition [\[2.11\]](#2.11){reference-type="eqref" reference="2.11"}, we can take $l$ as small as possible and $m$ as large as possible satisfying $\lambda_l = \dots = \lambda_{l+m-1}$. Therefore, we may assume that $$\label{REDU1} \lambda_{l+m-1} < \lambda_{l+m}$$ and (unless $l=1$) that $$\label{REDU2} \lambda_{l-1} < \lambda_l.$$ In view of [\[REDU1\]](#REDU1){reference-type="eqref" reference="REDU1"}, we can use Theorem [Theorem 6](#Theorem 2.7){reference-type="ref" reference="Theorem 2.7"} (with $l$ replaced here by $l+m$, which is greater than or equal to $2$) and deduce that $\Psi^{\lambda_{l+m-1}}$ possesses a compact symmetric subset $C$ of index $l + m - 1$. We now take $A_0 := C$ and $B_0 := \Psi_{\lambda_l}$ and we observe that, by construction, $$i(A_0) = l + m - 1.$$ We claim that $$\label{INPRO} i(S \setminus B_0) = l - 1.$$ To prove this, we distinguish two cases, either $l=1$ or $l\geqslant 2$. If $l = 1$, then we can use [\[2.9\]](#2.9){reference-type="eqref" reference="2.9"} and deduce that $B_0 = S$, which gives that $i(S \setminus B_0) = 0$, thus establishing [\[INPRO\]](#INPRO){reference-type="eqref" reference="INPRO"} in this case. If instead $l \geqslant 2$, we can use [\[REDU2\]](#REDU2){reference-type="eqref" reference="REDU2"}, which in turn allows us to Theorem [Theorem 6](#Theorem 2.7){reference-type="ref" reference="Theorem 2.7"} and infer that $i(S \setminus \Psi_{\lambda_l}) = l - 1$, which proves [\[INPRO\]](#INPRO){reference-type="eqref" reference="INPRO"} in this case as well. Now we recall [\[2.8\]](#2.8){reference-type="eqref" reference="2.8"} and we observe that, for all $u \in S$ and $t \geqslant 0$, $$\label{2.12} E(tu) = t^p\, \big(I_p(u) -N_p(u) - \lambda J_p(u)\big) - F(tu) = \frac{t^p}{p} \left(1 -pN_p(u) - \frac{\lambda}{\Psi(u)}\right) - F(tu).$$ We also pick $r\in(0,+\infty)$ and $R\in(r,+\infty)$ and we define $A$, $B$, and $X$ as in [\[LAUa\]](#LAUa){reference-type="eqref" reference="LAUa"}. Our goal is to find $r$ and $R$ such that condition [\[2.7\]](#2.7){reference-type="eqref" reference="2.7"} is satisfied. To this end, we recall [\[2.8\]](#2.8){reference-type="eqref" reference="2.8"} and $(\mathcal F_2)$, to see that, for all $u \in A_0 \subset \Psi^{\lambda_{l+m-1}} = \Psi^{\lambda_l}$, $$F(Ru) \geqslant\dfrac{\beta R^q}{q}\, \big(p\, J_p(u)\big)^{q/p} = \dfrac{\beta R^q}{q\, \Psi^{q/p}(u)} \geqslant\dfrac{\beta R^q}{q\, \lambda_l^{q/p}}.$$ This, together with [\[2.12\]](#2.12){reference-type="eqref" reference="2.12"}, gives that $$\label{2.13} E(Ru) \leqslant\frac{R^p}{p} \left(1 - \frac{\lambda}{\lambda_l}\right) - \dfrac{\beta R^q}{q\, \lambda_l^{q/p}}.$$ Regarding $(\mathcal F_2)$, we stress that $\beta > 0$ and $q > p$. As a result, the first inequality in [\[2.7\]](#2.7){reference-type="eqref" reference="2.7"} holds true as a consequence of [\[2.13\]](#2.13){reference-type="eqref" reference="2.13"} as long as $R$ is sufficiently large. Now, to complete the check of the validity of [\[2.7\]](#2.7){reference-type="eqref" reference="2.7"}, we recall that $F(ru) = \text{o}(r^p)$, thanks to $(\mathcal F_1)$. Hence, we exploit [\[2.12\]](#2.12){reference-type="eqref" reference="2.12"}, $(\mathcal F_1)$ and $(\mathcal N_1)$ to see that, for all $u \in B_0 = \Psi_{\lambda_l}$, $$E(ru) \geqslant\frac{r^p}{p} \left(1-\eta - \frac{\lambda}{\lambda_l} + \text{o}(1)\right) \text{ as } r \to 0 .$$ Since $\lambda < \lambda_l$, it follows from this that the second inequality in [\[2.7\]](#2.7){reference-type="eqref" reference="2.7"} also holds if $r$ and $\eta$ are sufficiently small. Furthermore, we employ [\[2.13\]](#2.13){reference-type="eqref" reference="2.13"} to see that, for all $u \in A$ and $0 \leqslant t \leqslant 1$, $$\label{2.14} E(tu) \leqslant\frac{t^p R^p}{p} \left(1 - \frac{\lambda}{\lambda_l}\right) - \dfrac{\beta t^q R^q}{q\, \lambda_l^{q/p}} = \frac{s^p}{p}\, (\lambda_l - \lambda) - \dfrac{\beta s^q}{q},$$ where we set $s = tR/\lambda_l^{1/p}$. Looking at the maximum attained by the last expression in [\[2.14\]](#2.14){reference-type="eqref" reference="2.14"} over all $s \geqslant 0$ and using [\[2.11\]](#2.11){reference-type="eqref" reference="2.11"}, we obtain that $$\sup_{u \in X}\, E(u) \leqslant\left(\frac{1}{p} - \frac{1}{q}\right)\!\left(\frac{\lambda_l - \lambda}{\beta^{p/q}}\right)^{q/(q - p)} < c^*.$$ This gives that condition [\[2.7\]](#2.7){reference-type="eqref" reference="2.7"} is fulfilled in this case. Hence, we can use Theorem [Theorem 5](#Theorem 2.6){reference-type="ref" reference="Theorem 2.6"} and thus obtain $m$ distinct pairs of nontrivial critical points of $E$, as desired. ◻ # Uniform Sobolev embeddings {#SOBOLEV} Here we discuss some uniform embedding of Sobolev type, whose interest may possibly go even beyond the specific goals of this paper (and, in relation to this, it is a pleasure to thank Oscar Domı́nguez and Petru Mironescu for sharing information about the state of the art on the fractional Sobolev embeddings). **Proposition 7**. *Let $\Omega$ be a bounded, open subset of $\mathbb{R}^N$ and $p\in [1,N)$.* *Then, there exists $C_0=C_0(N,\Omega,p)>0$ such that, for every $s\in[0,1]$ and every measurable function $u:\mathbb{R}^N\to\mathbb{R}$ with $u=0$ a.e. in $\mathbb{R}^N\setminus\Omega$, one has that $$\|u\|_{L^p(\mathbb{R}^N)}\leqslant C_0\,[u]_{s,p}.$$* *Proof.* Suppose not. Then, there exist sequences $s_k\in[0,1]$ and $u_k:\mathbb{R}^N\to\mathbb{R}$ with $u_k=0$ a.e. in $\mathbb{R}^N\setminus\Omega$ and such that $$\label{NAR} \|u_k\|_{L^p(\mathbb{R}^N)}^p\geqslant k\,[u_k]_{s_k,p}^p.$$ Since $[u_k]_{0,p}=\|u_k\|_{L^p(\mathbb{R}^N)}$, we have that $s_k\ne0$. Also, from the classical Sobolev-Poincaré Inequality, we have that $s_k\ne1$. Up to replacing $u_k$ by $u_k/\|u_k\|_{L^p(\mathbb{R}^N)}$, we can assume that $$\label{Normaliz} \|u_k\|_{L^p(\mathbb{R}^N)}=1.$$ We also take a cube $Q\subset\mathbb{R}^N$ sufficiently large such that $\Omega\subset Q$. In this way, if $$c_k:=\frac1{|Q|}\int_Q u_k(x)\,dx,$$ we deduce from the Hölder Inequality and [\[Normaliz\]](#Normaliz){reference-type="eqref" reference="Normaliz"} that $$|c_k|\leqslant\frac1{|Q|}\int_Q |u_k(x)|\,dx\leqslant\frac{\| u_k\|_{L^p(Q)}}{|Q|^{1/p}}\leqslant\frac{1}{|Q|^{1/p}}$$ and therefore, up to a subsequence, $c_k$ converges to some $c\in\mathbb{R}$ as $k\to+\infty$. Up to replacing $u_k$ with $-u_k$, we may also suppose that $c\geqslant 0$. Additionally, we have that $$\label{AJSMr} [u_k]_{s_k,p}^p\geqslant c_0\,s_k(1-s_k)\iint_{\mathbb{R}^{2N}}\frac{|u_k(x)-u_k(y)|^p}{|x-y|^{N+s_kp}}\,dx\,dy$$ for some $c_0=c_0(N,p)>0$. Accordingly, if $R>0$ is large enough such that $Q\subset B_R$, using that $u=0$ a.e. outside $B_R$, recalling [\[Normaliz\]](#Normaliz){reference-type="eqref" reference="Normaliz"} we find that, if $s_k\in(0,1)$, $$\label{NARR2}\begin{split}& [u_k]_{s_k,p}^p\geqslant c_0\,s_k(1-s_k)\iint_{\Omega\times(\mathbb{R}^N\setminus B_R)}\frac{|u_k(x)|^p}{|x-y|^{N+s_kp}}\,dx\,dy\\&\qquad\qquad\qquad\geqslant c_0\,s_k(1-s_k)\,\|u_k\|_{L^p(\Omega)}^p \int_{\mathbb{R}^N\setminus B_{2R}}\frac{dz}{|z|^{N+s_kp}}=c_1\,(1-s_k),\end{split}$$ with $c_1=c_1(N,\Omega,p)>0$. By [\[NAR\]](#NAR){reference-type="eqref" reference="NAR"}, [\[Normaliz\]](#Normaliz){reference-type="eqref" reference="Normaliz"} and [\[NARR2\]](#NARR2){reference-type="eqref" reference="NARR2"}, it follows that $$\begin{aligned} \frac1k\geqslant[u_k]_{s_k,p}^p\geqslant c_1\,(1-s_k),\end{aligned}$$ therefore $s_k\to1$ as $k\to+\infty$. Now we recall[^4] Theorem 1 in [@MR1945278], according to which $$\iint_{Q\times Q}\frac{|u_k(x)-u_k(y)|^p}{|x-y|^{N+s_kp}}\,dx\,dy\geqslant c_2 \frac{(N-s_k p)^{p-1}}{1-s_k}\| u_k-c_k\|^p_{L^{Np/(N-s_kp)}(Q)},$$ for some $c_2=c_2(N,\Omega)>0$. From this, [\[NAR\]](#NAR){reference-type="eqref" reference="NAR"}, [\[Normaliz\]](#Normaliz){reference-type="eqref" reference="Normaliz"} and [\[AJSMr\]](#AJSMr){reference-type="eqref" reference="AJSMr"}, we arrive at $$\begin{aligned} && \frac1k\geqslant[u_k]_{s_k,p}^p\geqslant c_0\,s_k(1-s_k)\iint_{\mathbb{R}^{2N}}\frac{|u_k(x)-u_k(y)|^p}{|x-y|^{N+s_kp}}\,dx\,dy\\&&\qquad\quad\geqslant c_0\,c_2\,(N-s_k p)^{p-1}\,s_k\| u_k-c_k\|^p_{L^{Np/(N-s_kp)}(Q)}.\end{aligned}$$ Hence, using the Hölder Inequality with exponents $\frac{N}{N-s_kp}$ and $\frac{N}{s_kp}$, $$\begin{aligned} && \frac1k\geqslant\frac{c_0\,c_2\,(N-s_k p)^{p-1}\,s_k}{|Q|^{s_kp/N}}\,\| u_k-c_k\|^p_{L^{p}(Q)}.\end{aligned}$$ As a consequence, taking the limit as $k\to+\infty$, we obtain that $$(N-p)^{p-1}\,\lim_{k\to+\infty}\| u_k-c_k\|^p_{L^{p}(Q)}=0,$$ giving that $u_k-c_k\to0$ in $L^p(\Omega)$. Thus, for all $\rho>0$ and $x_0\in\mathbb{R}^N$, we define $$m_{k,x_0,\rho}:=\frac{1}{|B_\rho|}\int_{B_\rho(x_0)} u_k(x)\,dx=\frac{1}{|B_\rho|}\int_{B_\rho(x_0)\cap \Omega} u_k(x)\,dx$$ and we have that $$\begin{aligned} && \lim_{k\to+\infty} \left|m_{k,x_0,\rho}-\frac{c\,|B_\rho(x_0)\cap\Omega|}{|B_\rho|}\right|= \frac{1}{|B_\rho|}\lim_{k\to+\infty} \left|\,\int_{B_\rho(x_0)\cap\Omega}\big( u_k(x)-c\big)\,dx\right|\\&&\qquad\leqslant \frac{1}{|B_\rho|}\lim_{k\to+\infty} \left[\,\int_{B_\rho(x_0)\cap\Omega}\big| u_k(x)-c_k\big|\,dx+ \int_{B_\rho(x_0)\cap\Omega}\big| c_k-c\big|\,dx \right]=0.\end{aligned}$$ Consequently, $$\begin{aligned} && \lim_{k\to+\infty} \|u_k-m_{k,x_0,\rho}\|_{L^p(B_\rho(x_0))}^p\\&&\qquad= \lim_{k\to+\infty} \left(\,\int_{B_\rho(x_0)\cap\Omega} \big|u_k(x)-m_{k,x_0,\rho}\big|^p\,dx+ \int_{B_\rho(x_0)\setminus\Omega} |m_{k,x_0,\rho}|^p\,dx\right)\\&&\qquad= \int_{B_\rho(x_0)\cap\Omega} \left|c-\frac{c\,|B_\rho(x_0)\cap\Omega|}{|B_\rho|}\right|^p\,dx+ \int_{B_\rho(x_0)\setminus\Omega} \frac{c^p\,|B_\rho(x_0)\cap\Omega|^p}{|B_\rho|^p}\,dx\\&&\qquad=c^p\, \left(\frac{|B_\rho(x_0)\setminus\Omega|^p\, |B_\rho(x_0)\cap\Omega|}{|B_\rho|^p}+ \frac{|B_\rho(x_0)\cap\Omega|^p\, |B_\rho(x_0)\setminus\Omega|}{|B_\rho|^p}\right).\end{aligned}$$ Furthermore, by [@MR4225499 Theorem 2.4], $$\sup_{{x_0\in\mathbb{R}^N}\atop{\rho>0}}\rho^{-s_k}\|u_k-m_{k,x_0,\rho}\|_{L^p(B_\rho(x_0))} \leqslant\widehat{C}\,[u_k]_{s_k,p},$$ for some $\widehat{C}=\widehat{C}(N,p)>0$. From these observations, we obtain that, for every $\rho>0$ and $x_0\in\mathbb{R}^N$, $$\begin{aligned} && \frac{c}\rho\, \left(\frac{|B_\rho(x_0)\setminus\Omega|^p\, |B_\rho(x_0)\cap\Omega|}{|B_\rho|^p}+ \frac{|B_\rho(x_0)\cap\Omega|^p\, |B_\rho(x_0)\setminus\Omega|}{|B_\rho|^p}\right)^{1/p}\\&&\qquad= \lim_{k\to+\infty}\frac1\rho\,\|u_k-m_{k,x_0,\rho}\|_{L^p(B_\rho(x_0))}= \lim_{k\to+\infty} \rho^{-s_k}\|u_k-m_{k,x_0,\rho}\|_{L^p(B_\rho(x_0))}\\&&\qquad \leqslant\widehat{C}\,\lim_{k\to+\infty}[u_k]_{s_k,p}\leqslant\widehat{C}\,\lim_{k\to+\infty}\frac1{k^{1/p}}=0.\end{aligned}$$ This gives that $c=0$, and accordingly that $u_k\to0$ in $L^p(\Omega)$, but this is in contradiction with [\[Normaliz\]](#Normaliz){reference-type="eqref" reference="Normaliz"}. ◻ **Theorem 8**. *Let $\Omega$ be a bounded, open subset of $\mathbb{R}^N$ and $p\in (1,N)$.* *Then, there exists $C=C(N,\Omega,p)>0$ such that, for every $s$, $S\in[0,1]$ with $s\leqslant S$ and every measurable function $u:\mathbb{R}^N\to\mathbb{R}$ with $u=0$ a.e. in $\mathbb{R}^N\setminus\Omega$, one has that $$[u]_{s,p}\leqslant C\,[u]_{S,p}.$$* *Proof.* Let us first suppose that $S=1$. In this case, we can also assume that $\|\nabla u\|_{L^p(\mathbb{R}^N)}<+\infty$, otherwise the desired result is obvious. Also, if $S=1$ and $s=1$, the desired result is obvious, and if $S=1$ and $s=0$, then the desired result follows from Proposition [Proposition 7](#PRBMI){reference-type="ref" reference="PRBMI"}. If $S=1$ and $s\in(0,1)$, then we argue as follows. For all $x$, $y\in\mathbb{R}^N$ with $|x-y|<1$ we have that $$\begin{aligned} |u(x)-u(y)|\leqslant|x-y|\int_0^1 |\nabla u(tx+(1-t)y)|\,dt\end{aligned}$$ and thus, using the substitutions $z:=x-y$ and $w:=tx+(1-t)y$, and noticing that $dz\,dw=dx\,dy$, $$\begin{aligned} &&\iint_{\mathbb{R}^{2N}\cap\{|x-y|<1\}}\frac{|u(x)-u(y)|^p}{|x-y|^{N+sp}}\,dx\,dy \leqslant\int_{\mathbb{R}^{2N}\cap\{|z|<1\}} |z|^{p(1-s)-N}\left(\int_0^1 |\nabla u(w)|\,dt\right)^p\,dz\,dw \\&&\qquad\qquad\leqslant\frac{C_0}{1-s}\|\nabla u\|_{L^p(\mathbb{R}^N)}^p,\end{aligned}$$ for some $C_0=C_0(N,p)>0$. Consequently, $$\begin{aligned} && \iint_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^p}{|x-y|^{N+sp}}\,dx\,dy\\ &&\qquad\leqslant\frac{C_0}{1-s}\|\nabla u\|_{L^p(\mathbb{R}^N)}^p +\iint_{\mathbb{R}^{2N}\cap\{|x-y|\geqslant 1\}}\frac{(|u(x)|+|u(y)|)^p}{|x-y|^{N+sp}}\,dx\,dy\\ &&\qquad\leqslant\frac{C_0}{1-s}\|\nabla u\|_{L^p(\mathbb{R}^N)}^p+\frac{C_1}{s}\| u\|_{L^p(\mathbb{R}^N)}^p,\end{aligned}$$ for some $C_1=C_1(N,p)>0$. This and the classical Sobolev-Poincaré Inequality yield that $$\iint_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^p}{|x-y|^{N+sp}}\,dx\,dy\leqslant C_2\left( \frac{1}{1-s}+\frac1s\right)\|\nabla u\|_{L^p(\mathbb{R}^N)}^p$$ for some $C_2=C_2(N,\Omega,p)>0$ and the desired result follows. Now we suppose $S\ne1$. If $s=0$, the desired result follows from Proposition [Proposition 7](#PRBMI){reference-type="ref" reference="PRBMI"}. If instead $s\in(0,1)$, we use [@MR4525724 Theorem 3.9] with $\alpha:=1$ and we find that $$\begin{aligned} && \|u\|_{L^p(\mathbb{R}^N)}+\left(\min\{s,1-s\} \iint_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^p}{|x-y|^{N+sp}}\,dx\,dy\right)^{1/p} \\&&\qquad\leqslant C_3\,\left(\|u\|_{L^p(\mathbb{R}^N)}+\left(\min\{S,1-S\} \iint_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^p}{|x-y|^{N+Sp}}\,dx\,dy \right)^{1/p}\right),\end{aligned}$$ for some $C_3=C_3(N,p)>0$, and therefore $$_{s,p}\leqslant\|u\|_{L^p(\mathbb{R}^N)}+[u]_{s,p}\leqslant C_4\,\Big(\|u\|_{L^p(\mathbb{R}^N)}+[u]_{S,p}\Big),$$ for some $C_4=C_4(N,p)>0$. This and Proposition [Proposition 7](#PRBMI){reference-type="ref" reference="PRBMI"} give the desired result. ◻ # Functional setting towards the proof of Theorem [Theorem 1](#mainp1){reference-type="ref" reference="mainp1"} {#BaUPS} Here we construct suitable functional spaces which are helpful to prove Theorem [Theorem 1](#mainp1){reference-type="ref" reference="mainp1"}. For this, given a measure $\mu^+$ satisfying [\[mu00\]](#mu00){reference-type="eqref" reference="mu00"}, we set $$\rho_p(u):=\left( \,\int_{[0,1]}[u]_{s,p}^p \,d\mu^+(s)\right)^{1/p}$$ and define $\mathcal{X}_p(\Omega)$ as the set of measurable functions $u:\mathbb{R}^N\to\mathbb{R}$ such that $u=0$ in $\mathbb{R}^N\setminus\Omega$ and $\rho_p(u)<+\infty$. In this setting, we can "reabsorb" the negative part of the signed measure $\mu$: **Proposition 9**. *Let $p\in (1, N)$ and assume that [\[mu3\]](#mu3){reference-type="eqref" reference="mu3"} and [\[mu2\]](#mu2){reference-type="eqref" reference="mu2"} hold.* *Then, there exists $c_0=c_0(N,\Omega, p)>0$ such that, for any $u\in\mathcal{X}_p(\Omega)$, we have $$\int_{[{{ 0 }}, \overline s]} [u]_{s, p}^p \, d\mu^- (s) \leqslant c_0\,\gamma \int_{[\overline s, 1]} [u]^p_{s, p} \, d\mu(s).$$* *Proof.* We observe that if $\mu^+([\overline s,1])=0$, condition [\[mu2\]](#mu2){reference-type="eqref" reference="mu2"} implies that $\mu^-([0,\overline s])=0$ and then the result is proved. Thus, for the remaining part of this proof we assume that $\mu^+([\overline s,1])>0$. By using Theorem [Theorem 8](#SOBOLEVEMBEDDING){reference-type="ref" reference="SOBOLEVEMBEDDING"} with $s$ and $S:=\overline s$ we get, $$[u]_{s, p}\leqslant C(N,\Omega,p) [u]_{\overline s, p}.$$ Furthermore, employing Theorem [Theorem 8](#SOBOLEVEMBEDDING){reference-type="ref" reference="SOBOLEVEMBEDDING"} with $s:=\overline s$ and $S:= s$, we have $$[u]_{\overline s, p}\leqslant C(N,\Omega,p) [u]_{s, p}.$$ Hence, by using the previous inequalities together with the assumptions [\[mu3\]](#mu3){reference-type="eqref" reference="mu3"} and [\[mu2\]](#mu2){reference-type="eqref" reference="mu2"}, we obtain that $$\begin{aligned} && \int_{[{{{0}}}, \overline s]} [u]_{s, p}^p \, d\mu^-(s) \leqslant C^p(N,\Omega,p) \int_{[{{{0}}}, \overline s]} [u]_{\overline s, p}^p \, d\mu^-(s) = C^p(N,\Omega,p)\, [u]_{\overline s, p}^p \,\mu^-([{{{0}}}, \overline s]) \\ &&\qquad \leqslant C^p(N,\Omega,p)\gamma \, [u]_{\overline s, p}^p \,\mu^+([ \overline s,1]) = C^p(N,\Omega,p)\gamma \int_{[\overline s,1]} [u]_{\overline s, p}^p \, d\mu^+(s)\\ &&\qquad\leqslant C^{2p}(N,\Omega,p)\gamma \int_{[\overline s,1]} [u]_{ s, p}^p \, d\mu^+(s) = C^{2p}(N,\Omega,p)\gamma \int_{[\overline s,1]} [u]_{ s, p}^p \, d\mu(s),\end{aligned}$$ which gives the desired result with $c_0:= C^{2p}(N,\Omega,p)$. ◻ We notice that $$\label{complete00} {\mbox{$\mathcal X_p(\Omega)$ is complete}}$$ (the case $p=2$ being proved in [@CATERINA], the general case being similar). Also, we have that: **Lemma 10**. *$\mathcal X_p(\Omega)$ is a uniformly convex space.* *Proof.* We need to prove that for every $\varepsilon\in(0,2]$ there exists $\delta>0$ such that if $u$, $v\in{\mathcal{X}}_p(\Omega)$ are such that $\rho_p(u)=\rho_p(v)=1$ and $\rho_p(u-v)\geqslant\varepsilon$, then $\rho_p(u+v)\leqslant 2-\delta$. To this end, we modify some classical approaches (see e.g. [@MR748950; @MR3859645] and the references therein). The details are not completely obvious and go as follows. Let us first deal with the case $p\in[2,+\infty)$. In this case, for all $a$, $b\in\mathbb{R}$, we have that $$\label{PPlas} |a+b|^p+|a-b|^p\leqslant 2^{p-1}\big( |a|^p+|b|^p\big),$$ see [@MR1501880 Theorem 2]. Hence, we use [\[PPlas\]](#PPlas){reference-type="eqref" reference="PPlas"} with $$a:=u(x)-u(y)\qquad{\mbox{and}}\qquad b:=v(x)-v(y)$$ and we find that $$\begin{aligned} ^p &=&\int_{[0,1]}\left( c_{N,s,p}\iint_{\mathbb{R}^{2N}}\frac{|u(x)+v(x)-u(y)-v(y)|^p}{|x-y|^{N+sp}}\,dx\,dy\right)\,d\mu^+(s) \\&\leqslant& 2^{p-1}\Bigg[\; \int_{[0,1]}\left( c_{N,s,p}\iint_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^p}{|x-y|^{N+sp}}\,dx\,dy\right)\,d\mu^+(s)\\&&\qquad\qquad + \int_{[0,1]}\left( c_{N,s,p}\iint_{\mathbb{R}^{2N}}\frac{|v(x)-v(y)|^p}{|x-y|^{N+sp}}\,dx\,dy\right)\,d\mu^+(s) \Bigg]\\&&\qquad -\int_{[0,1]}\left( c_{N,s,p}\iint_{\mathbb{R}^{2N}}\frac{|u(x)-v(x)-u(y)+v(y)|^p}{|x-y|^{N+sp}}\,dx\,dy\right)\,d\mu^+(s)\\ &=& 2^{p-1}\Big[ [\rho_p(u)]^p+[\rho_p(v)]^p \Big]-[\rho_p(u-v)]^p\\ &\leqslant& 2^{p}-\varepsilon^p\\&=&(2-\delta)^p,\end{aligned}$$ with $$\delta=\delta(\varepsilon):=2-\big(2^p-\varepsilon^p\big)^{1/p}.$$ We stress that $\delta$ is strictly increasing in $\varepsilon\in(0,2]$, therefore $\delta>\delta(0)=0$, and the proof of the uniform convexity in this case is thereby complete. If instead $p\in(1,2)$, we use that, for every $\Phi$, $\Psi\in L^p(\mathbb{R}^{2N})$, $$\label{ShbsnTA}\begin{split}& \left\| \frac{|\Phi|+|\Psi|}2\right\|_{L^p(\mathbb{R}^{2N})}^{2-p} \left( \frac{\|\Phi\|_{L^p(\mathbb{R}^{2N})}^p+\|\Psi\|_{L^p(\mathbb{R}^{2N})}^p}2- \left\| \frac{\Phi+\Psi}2\right\|_{L^p(\mathbb{R}^{2N})}^p\right)\\&\qquad\qquad\geqslant\frac{p(p-1)}8 \|\Phi-\Psi\|_{L^p(\mathbb{R}^{2N})}^2,\end{split}$$ see [@MR748950 Theorem 1]. We choose $$\Phi(x,y,s):=c_{N, s, p}^{\frac1p}\,\frac{u(x)-u(y)}{|x-y|^{\frac{N+sp}p}}\qquad{\mbox{and}}\qquad \Psi(x,y,s):=c_{N, s, p}^{\frac1p}\,\frac{v(x)-v(y)}{|x-y|^{\frac{N+sp}p}}.$$ In this way, $$\begin{aligned} &&\|\Phi\|_{L^p(\mathbb{R}^{2N}\times [0, 1])}=\rho_p(u)=1,\\ &&\|\Psi\|_{L^p(\mathbb{R}^{2N}\times [0, 1])}=\rho_p(v)=1,\\ && \|\Phi+\Psi\|_{L^p(\mathbb{R}^{2N}\times [0, 1])}=\rho_p(u+v),\\ && \|\Phi-\Psi\|_{L^p(\mathbb{R}^{2N}\times [0, 1])}=\rho_p(u-v)\geqslant\varepsilon\\{\mbox{and }}&& \| |\Phi|+|\Psi|\|_{L^p(\mathbb{R}^{2N}\times [0, 1])}\leqslant\| \Phi\|_{L^p(\mathbb{R}^{2N}\times [0, 1])}+\|\Psi\|_{L^p(\mathbb{R}^{2N}\times [0, 1])}=\rho_p(u)+\rho_p(v)=2.\end{aligned}$$ Therefore, in light of [\[ShbsnTA\]](#ShbsnTA){reference-type="eqref" reference="ShbsnTA"}, $$1-\frac{[\rho_p(u+v)]^p}{2^p}\geqslant\frac{p(p-1)}8 \varepsilon^2$$ and, as a result, $$\begin{aligned} \rho_p(u+v) \leqslant 2\left(1-\frac{\varepsilon^2p(p-1)}8 \right)^{1/p}=2-\delta,\end{aligned}$$ with $$\delta=\delta(\varepsilon):=2\left[1-\left(1-\frac{\varepsilon^2p(p-1)}8 \right)^{1/p}\right].$$ We stress that $\varepsilon^2p(p-1)<8$ in this case, whence $\delta$ is strictly increasing in $\varepsilon\in(0,2]$. This entails that $\delta>\delta(0)=0$, thus completing the proof of the uniform convexity property. ◻ The setting of solutions that we consider here is the one induced by this functional framework, namely: **Definition 11**. A weak solution of problem [\[mainp\]](#mainp){reference-type="eqref" reference="mainp"} is a function $u \in \mathcal{X}_p(\Omega)$ such that[^5] $$\begin{split} &\int_{[{{{0}}}, 1]}\left(\;\iint_{\mathbb{R}^{2N}} \frac{c_{N,s,p}\,|u(x) - u(y)|^{p-2}\, (u(x) - u(y))\, (v(x) - v(y))}{|x - y|^{N+sp}}\, dx dy\right) d\mu^+(s)\\[5pt] &-\int_{[{{{0}}}, \overline s]}\left(\;\iint_{\mathbb{R}^{2N}} \frac{c_{N,s,p}\,|u(x) - u(y)|^{p-2}\, (u(x) - u(y))\, (v(x) - v(y))}{|x - y|^{N+sp}}\, dx dy\right) d\mu^-(s)\\[5pt] &= \lambda \int_\Omega |u|^{p-2}\, uv\, dx + \int_\Omega |u|^{p_{s_\sharp}^* - 2}\, uv\, dx, \end{split}$$ for all $v \in \mathcal{X}_p(\Omega)$. Solutions of problem [\[mainp\]](#mainp){reference-type="eqref" reference="mainp"} coincide with the critical points of the functional $E: \mathcal X_p(\Omega)\to\mathbb{R}$ given by $$\label{funp} E_p(u) =\frac1p [\rho_p(u)]^p -\frac1p\int_{[{{{0}}}, \overline s]} [u]_{s, p}^p \, d\mu^-(s) - \frac{\lambda}{p} \int_\Omega |u|^p \, dx - \frac{1}{p_{s_\sharp}^*} \int_\Omega |u|^{p_{s_\sharp}^*}\, dx.$$ # Proof of Theorem [Theorem 1](#mainp1){reference-type="ref" reference="mainp1"} {#BaUPS-2} We now prove Theorem [Theorem 1](#mainp1){reference-type="ref" reference="mainp1"}. The strategy is to reduce ourselves to the abstract setting introduced in Section [1.2](#AB2){reference-type="ref" reference="AB2"} and exploit Theorem [Theorem 3](#Theorem 2.9){reference-type="ref" reference="Theorem 2.9"}. To this end, as already anticipated in Section [1.2](#AB2){reference-type="ref" reference="AB2"}, we choose $$\label{SETTING}\begin{split} &W:={\mathcal{X}}_p(\Omega),\qquad A_pu:=\int_{[0, 1]} (-\Delta)_p^{s} u \, d\mu^+(s),\qquad B_pu:=|u|^{p-2}u,\\& L_pu:=\int_{[0, 1]} (-\Delta)_p^{s} u \, d\mu^-(s) \qquad{\mbox{and}}\qquad f(u):= |u|^{p_{s_\sharp}^* - 2}\, u . \end{split}$$ Our goal is now to systematically check that the abstract hypotheses requested in Section [1.2](#AB2){reference-type="ref" reference="AB2"} are fulfilled. Specifically, we need to check [\[UNICO\]](#UNICO){reference-type="eqref" reference="UNICO"}, [\[UNICO2\]](#UNICO2){reference-type="eqref" reference="UNICO2"}, [\[2.888-0\]](#2.888-0){reference-type="eqref" reference="2.888-0"}, [\[2.888-01\]](#2.888-01){reference-type="eqref" reference="2.888-01"}, [\[lppotop\]](#lppotop){reference-type="eqref" reference="lppotop"}, as well as the structural conditions $(A_1)$, $(A_2)$, $(B_1)$, $(B_2)$, $(B_3)$, $(L_1)$, $(\mathcal F_1)$, $(\mathcal F_2)$, $(\mathcal F_3)$ and $(\mathcal N_1)$. Let us proceed in order. First of all, by [\[complete00\]](#complete00){reference-type="eqref" reference="complete00"} and Lemma [Lemma 10](#UNCONCVE){reference-type="ref" reference="UNCONCVE"}, we have that $\mathcal X_p(\Omega)$ is a uniformly convex Banach space, which is the claim in [\[UNICO\]](#UNICO){reference-type="eqref" reference="UNICO"}. Regarding the claim in [\[UNICO2\]](#UNICO2){reference-type="eqref" reference="UNICO2"}, we first need to interpret $A_p$ and $B_p$ as operators from $W$ to its dual, namely we rephrase [\[SETTING\]](#SETTING){reference-type="eqref" reference="SETTING"}, given $u$, $v\in{\mathcal{X}}_p(\Omega)$, as $$\begin{aligned} && ( A_pu,v)=\int_{\mathbb{R}^N}\left(\;\int_{[0, 1]} (-\Delta)_p^{s} u(x)\,v(x) \, d\mu^+(s)\right)\,dx\\{\mbox{and }}&&(B_pu,v)=\int_{\mathbb{R}^N} |u(x)|^{p-2}u(x)\,v(x)\,dx.\end{aligned}$$ We observe that: **Lemma 12**. *$A_p$ is continuous from $W$ to $W^*$.* *Proof.* We notice that $$\label{hdsu3y259676tgrhueh}\begin{split} &\int_{\mathbb{R}^N} (-\Delta)_p^{s} u(x)\,v(x)\,dx\\ =\;& 2c_{N,s,p} \,\lim_{\varepsilon\searrow 0} \int_{\mathbb{R}^N}\left[\; \int_{\mathbb{R}^N \setminus B_\varepsilon(x)} \frac{|u(x) - u(y)|^{p-2}\, (u(x) - u(y)) v(x)}{|x - y|^{N+sp}}\, dy\right]\,dx \\=\;& {c_{N,s,p}} \,\iint_{\mathbb{R}^{2N}} \frac{|u(x) - u(y)|^{p-2}\, (u(x) - u(y))( v(x)-v(y))}{|x - y|^{N+sp}}\, dx\,dy. \end{split}$$ The Hölder Inequality with exponents $\frac{p}{p-1}$ and $p$ applied to the functions $$\label{2Aalsdx029oiefkvHO-bn} \frac{c_{N,s,p}^{(p-1)/p}\,|u(x) - u(y)|^{p-2}\, (u(x) - u(y))}{|x - y|^{(N+sp)(p-1)/p}} \qquad{\mbox{ and }}\qquad \frac{c_{N,s,p}^{1/p}\,( v(x)-v(y))}{|x - y|^{(N+sp)/p}}$$ returns that $$\label{2Aalsdx029oiefkvHO}\begin{split} |(A_pu,v)|&= \left|\;\;\iint_{[0,1]\times\mathbb{R}^N} (-\Delta)_p^{s} u(x)\,v(x)\,d\mu^+(s)\,dx\right|\\ %%%%%%&\le \iiint_{[0,1]\times\R^{2N}} \frac{c_{N,s,p}\,|u(x) - u(y)|^{p-1}| v(x)-v(y)|}{|x - y|^{N+sp}}\, d\mu^+(s)\,dx\,dy\\ &\leqslant \left[\;\; \iiint_{[0,1]\times\mathbb{R}^{2N}} c_{N,s,p}\frac{|u(x) - u(y)|^p}{|x - y|^{N+sp}}\,d\mu^+(s)\, dx\,dy\right]^{(p-1)/p}\\&\qquad\qquad\times \left[\;\;\iiint_{[0,1]\times\mathbb{R}^{2N}} c_{N,s,p}\frac{| v(x)-v(y)|^p}{|x - y|^{N+sp}}\,d\mu^+(s)\, dx\,dy\right]^{1/p} \\&=\left(\;\int_{[0,1]} [u]_{s,p}^p\,d\mu^+(s)\right)^{(p-1)/p}\left(\;\int_{[0,1]} [v]_{s,p}^p\,d\mu^+(s)\right)^{1/p} \\&=[\rho_p(u)]^{p-1}\,\rho_p(v). \end{split}$$ This gives that $$\label{Aalsdx029oiefkv} |(A_pu,v)|\leqslant[\rho_p(u)]^{p-1}\,\rho_p(v),$$ which yields the desired result. ◻ Moreover, we have that: **Lemma 13**. *$B_p$ is continuous from $W$ to $W^*$.* *Proof.* Using the Hölder Inequality with exponents $\frac{p}{p-1}$ and $p$, we have that $$\begin{aligned} |( B_pu,v)|\leqslant\int_{\Omega} |u(x)|^{p-1}\,|v(x)|\,dx\leqslant\|u\|_{L^p(\mathbb{R}^N)}^{p-1}\,\|v\|_{L^p(\mathbb{R}^N)}.\end{aligned}$$ Consequently, by Proposition [Proposition 7](#PRBMI){reference-type="ref" reference="PRBMI"}, for all $s\in[0,1]$, $$|( B_pu,v)|\leqslant\widetilde C\,[u]_{s,p}^{p-1}\,[v]_{s,p},$$ for some $\widetilde C=\widetilde C(N,\Omega,p)>0$. Integrating with respect to $\mu^+$ and using again the Hölder Inequality with exponents $\frac{p}{p-1}$ and $p$ we deduce that $$|(B_pu,v)|\leqslant\widehat C\,[\rho_p(u)]^{p-1}\,\rho_p(v),$$ for some $\widehat C=\widehat C(N,\Omega,p,\mu)>0$, which establishes the desired result. ◻ By Lemmata [Lemma 12](#APC){reference-type="ref" reference="APC"} and [Lemma 13](#BPC){reference-type="ref" reference="BPC"}, combined with the general result in [\[ARETRH\]](#ARETRH){reference-type="eqref" reference="ARETRH"}, it follows that $A_p$ and $B_p$ are potential operators, and we have thus checked condition [\[UNICO2\]](#UNICO2){reference-type="eqref" reference="UNICO2"}. As for [\[2.888-0\]](#2.888-0){reference-type="eqref" reference="2.888-0"}, again we have to interpret the definition of $f$ in [\[SETTING\]](#SETTING){reference-type="eqref" reference="SETTING"} as an operator from $W$ to its dual, namely, for all $u$, $v\in\mathcal{X}_p(\Omega)$, $$(f(u),v)=\int_{\mathbb{R}^N} |u(x)|^{p_{s_\sharp}^* - 2}\, u(x)\,v(x)\,dx.$$ In this setting, we have that **Lemma 14**. *$f$ is continuous from $W$ to $W^*$.* *Proof.* By the Hölder Inequality with exponents ${p_{s_\sharp}^* }/({p_{s_\sharp}^* -1})$ and ${p_{s_\sharp}^*}$, we have that $$\begin{aligned} \big|( f(u),v)\big|\leqslant\int_{\mathbb{R}^N} |u(x)|^{p_{s_\sharp}^* -1}\, |v(x)|\,dx\leqslant \|u\|^{p_{s_\sharp}^*-1}_{L^{p_{s_\sharp}^*}(\mathbb{R}^N)}\,\|v\|_{L^{p_{s_\sharp}^*}(\mathbb{R}^N)}.\end{aligned}$$ This and the fractional Sobolev embedding (see e.g. [@MR2944369]) gives that $$\big|( f(u),v)\big|\leqslant C_\star\,[u]^{p-1}_{s_\sharp,p} \,[v]_{s_\sharp,p},$$ for some $C_\star=C_\star(N,p,s_\sharp)>0$. Hence, by Theorem [Theorem 8](#SOBOLEVEMBEDDING){reference-type="ref" reference="SOBOLEVEMBEDDING"}, for all $s\in[s_\sharp,1]$, $$\big|( f(u),v)\big|\leqslant C_\sharp\,[u]^{p-1}_{ s,p} \,[v]_{s,p},$$ for some $C_\sharp=C_\sharp(N,\Omega, p,s_\sharp)>0$. This and the Hölder inequality with exponents $\frac{p}{p-1}$ and $p$ give that $$\begin{aligned} && \mu^+\big([s_\sharp,1]\big) \big|( f(u),v)\big|\leqslant C_\sharp\,\int_{[s_\sharp,1]}[u]^{p-1}_{ s,p} \,[v]_{s,p}\,d\mu^+(s)\\&&\qquad\leqslant C_\sharp\,\left(\; \int_{[s_\sharp,1]}[u]^{p}_{ s,p} \,d\mu^+(s)\right)^{(p-1)/p}\left(\;\int_{[s_\sharp,1]} [v]_{s,p}^p\,d\mu^+(s) \right)^{1/p}\leqslant C_\sharp\,[\rho_p(u)]^{p-1}\,\rho_p(v),\end{aligned}$$ which gives the desired result. ◻ **Corollary 15**. *$f$ is a potential operator.* *Proof.* We know by Lemma [Lemma 14](#fcot){reference-type="ref" reference="fcot"} that $f\in C(W,W^*)$. A potential of $f$ is given by $$\label{POTEF}F(u):= \frac1{p_{s_\sharp}^*} \int_{\mathbb{R}^N} |u(x)|^{p_{s_\sharp}^* }\,dx.\qedhere$$ ◻ In light of Corollary [\[2.888-0C\]](#2.888-0C){reference-type="eqref" reference="2.888-0C"}, we have checked condition [\[2.888-0\]](#2.888-0){reference-type="eqref" reference="2.888-0"}, as desired. The fact that $f$ is odd, as requested in [\[2.888-01\]](#2.888-01){reference-type="eqref" reference="2.888-01"} is obvious in our case. As for the assumption on $L_p$ in [\[lppotop\]](#lppotop){reference-type="eqref" reference="lppotop"}, we have to interpret the definition of $L_p$ in [\[SETTING\]](#SETTING){reference-type="eqref" reference="SETTING"} as an operator from $W$ to its dual, namely, for all $u$, $v\in\mathcal{X}_p(\Omega)$, $$(L_p(u),v)=\int_{\mathbb{R}^N}\left(\;\int_{[0, 1]} (-\Delta)_p^{s} u(x)\,v(x) \, d\mu^-(s)\right)\,dx.$$ In this case, we have that **Lemma 16**. *$L_p$ is continuous from $W$ to $W^*$.* *Proof.* We use [\[hdsu3y259676tgrhueh\]](#hdsu3y259676tgrhueh){reference-type="eqref" reference="hdsu3y259676tgrhueh"} and argue as in [\[2Aalsdx029oiefkvHO\]](#2Aalsdx029oiefkvHO){reference-type="eqref" reference="2Aalsdx029oiefkvHO"} with $A_p$ replaced by $L_p$ and $\mu^+$ replaced by $\mu^-$ to conclude that $$|(L_pu,v)|\leqslant[\rho_p(u)]^{p-1}\,\rho_p(v),$$ which yields the desired result. ◻ **Corollary 17**. *$L_p$ is a potential operator.* *Proof.* Lemma [Lemma 16](#Lpcont){reference-type="ref" reference="Lpcont"} informs us that $L_p\in C(W,W^*)$. Moreover, a potential of $L_p$ is given by $$\label{Npippoepluto} N_p(u):=\frac1p(L_pu,u)=\frac1p \int_{\mathbb{R}^N}\left(\;\int_{[0, 1]} (-\Delta)_p^{s} u(x)\,u(x) \, d\mu^-(s)\right)\,dx.\qedhere$$ ◻ Lemmata [Lemma 16](#Lpcont){reference-type="ref" reference="Lpcont"} and [Corollary 17](#Lppotope000){reference-type="ref" reference="Lppotope000"} establish that the assumption in [\[lppotop\]](#lppotop){reference-type="eqref" reference="lppotop"} is satisfied. The homogeneity required in $(A_1)$ is obvious. In relation to $(A_2)$, for all $u, v \in {\mathcal{X}}_p(\Omega)$, the bound on $\left(A_p\, u,v\right)_{}$ is a consequence of [\[Aalsdx029oiefkv\]](#Aalsdx029oiefkv){reference-type="eqref" reference="Aalsdx029oiefkv"}. In addition, equality holds true in [\[Aalsdx029oiefkv\]](#Aalsdx029oiefkv){reference-type="eqref" reference="Aalsdx029oiefkv"} if and only if it holds in [\[2Aalsdx029oiefkvHO\]](#2Aalsdx029oiefkvHO){reference-type="eqref" reference="2Aalsdx029oiefkvHO"}, and so if and only if the $\frac{p}{p-1}$ power of the absolute value of the first function in [\[2Aalsdx029oiefkvHO-bn\]](#2Aalsdx029oiefkvHO-bn){reference-type="eqref" reference="2Aalsdx029oiefkvHO-bn"} is proportional to the $p$ power of the absolute value of the second function in [\[2Aalsdx029oiefkvHO-bn\]](#2Aalsdx029oiefkvHO-bn){reference-type="eqref" reference="2Aalsdx029oiefkvHO-bn"} (and with a consistent sign if we want $\left(A_p\, u,v\right)_{}\geqslant 0$), i.e. if and only if there exist $\alpha$, $\beta\geqslant 0$, with $\alpha^2+\beta^2\ne0$, such that $$\alpha\, (u(x) - u(y))= \beta\, (v(x)-v(y)).$$ Choosing $y\in\mathbb{R}^n\setminus\Omega$ (in which case $u(y)=v(y)=0$), we see that this condition is equivalent to $\alpha u = \beta v$. These observations give that $(A_2)$ holds true. The homogeneity and the odd property in $(B_1)$ is obvious, and $(B_2)$ follows from $$\label{NASDL} (B_pu,u)=\int_{\mathbb{R}^N} |u(x)|^{p}\,dx=\|u\|_{L^p(\mathbb{R}^N)}^p>0,$$ unless $u$ vanishes identically, and $$\begin{aligned} (B_pu,v)\leqslant\int_{\mathbb{R}^N} |u(x)|^{p-1}\,|v(x)|\,dx\leqslant\|u\|_{L^p(\mathbb{R}^N)}^{p-1}\,\|v\|_{L^p(\mathbb{R}^N)}=(B_pu,u)^{(p-1)/p}\,(B_pv,v)^{1/p}.\end{aligned}$$ With respect to the compactness property requested in $(B_3)$, it is a consequence of the following result: **Lemma 18**. *Let $u_n$ be a bounded sequence in ${\mathcal{X}}_p(\Omega)$. Then, the sequence $U_n:=B_pu_n$ is precompact in the dual of ${\mathcal{X}}_p(\Omega)$.* *Proof.* We exploit Theorem [Theorem 8](#SOBOLEVEMBEDDING){reference-type="ref" reference="SOBOLEVEMBEDDING"} to see that $$\begin{aligned} \int_{[\overline s,1]}[u_n]_{s,p}^p \,d\mu^+(s) \geqslant\frac1{C(N,\Omega,p)}\int_{[\overline s,1]}[u_n]_{\overline s,p}^p \,d\mu^+(s)=\frac{\mu^+([\overline s,1])}{C(N,\Omega,p)}[u_n]_{\overline s,p}^p .\end{aligned}$$ In light of the assumption on $\mu$ in [\[mu00\]](#mu00){reference-type="eqref" reference="mu00"}, this implies that $[u_n]_{\overline s,p}$ is uniformly bounded in $n$, and therefore by the fractional compact embedding (see e.g. [@MR2944369 Corollary 7.2]) we obtain that, up to a subsequence, $$\label{fhwerutgvbhsd879999999999962-945} {\mbox{$u_n$ converges to some~$u\in L^p(\Omega)$.}}$$ Thus, we set $U:=B_pu$ and we have that, for all $v\in{\mathcal{X}}_p(\Omega)$, $$( U_n,v)-(U,v)=\int_{\mathbb{R}^N} \Big(|u_n(x)|^{p-2}u_n(x)-|u(x)|^{p-2}u(x)\Big)\,v(x)\,dx.$$ By the Hölder Inequality with exponents $\frac{p}{p-1}$ and $p$ and Theorem [Theorem 8](#SOBOLEVEMBEDDING){reference-type="ref" reference="SOBOLEVEMBEDDING"}, we obtain that $$\label{sweo0p7i8u6t453rkjtrg}\begin{split} |( U_n,v)-(U,v)|\leqslant\;& \left(\;\int_{\mathbb{R}^N} \Big|\,|u_n(x)|^{p-2}u_n(x)-|u(x)|^{p-2}u(x)\Big|^{\frac{p}{p-1}}\,dx\right)^{\frac{p-1}{p}} \|v\|_{L^p(\mathbb{R}^N)}\\ \leqslant\;& C \left(\;\int_{\mathbb{R}^N} \Big|\,|u_n(x)|^{p-2}u_n(x)-|u(x)|^{p-2}u(x)\Big|^{\frac{p}{p-1}}\,dx\right)^{\frac{p-1}{p}} \rho_p(v), \end{split}$$ where $C=C(N, \Omega, p,\mu^+)>0$. We point out that if $p=2$ the desired result follows from this and [\[fhwerutgvbhsd879999999999962-945\]](#fhwerutgvbhsd879999999999962-945){reference-type="eqref" reference="fhwerutgvbhsd879999999999962-945"}. If instead $p\neq2$ some more algebraic manipulations are needed to establish the desired convergence result. Hence, from now on, we suppose that $p\neq2$. When $p>2$, we utilize the following inequality $$\label{dkoejtiohnytkvfxdls} \big|\,|a|^{p-2}a-|b|^{p-2}b\,\big|\leqslant(p-1) \big( |a|^{p-2} +|b|^{p-2}\big) |a-b|,$$ which holds true for all $a$, $b\in\mathbb{R}$. Indeed, we suppose that $a<b$ and we have that $$\begin{aligned} && \big|\,|a|^{p-2}a-|b|^{p-2}b\,\big|=\left|\int_a^b (p-1) |t|^{p-2}\,dt \right|\leqslant(p-1) \big( |a|^{p-2} +|b|^{p-2}\big) |a-b|,\end{aligned}$$ which is [\[dkoejtiohnytkvfxdls\]](#dkoejtiohnytkvfxdls){reference-type="eqref" reference="dkoejtiohnytkvfxdls"}. Hence, using formula [\[dkoejtiohnytkvfxdls\]](#dkoejtiohnytkvfxdls){reference-type="eqref" reference="dkoejtiohnytkvfxdls"} with $a:=u_n(x)$ and $b:=u(x)$, we conclude that $$\label{djweioguoe9786543e}\begin{split} &\int_{\mathbb{R}^N} \Big|\,|u_n(x)|^{p-2}u_n(x)-|u(x)|^{p-2}u(x)\Big|^{\frac{p}{p-1}}\,dx \\ \leqslant\;& C\int_{\mathbb{R}^N} \big( |u_n(x)|^{p-2} +|u(x)|^{p-2}\big)^{\frac{p}{p-1}} | u_n(x) -u(x)|^{\frac{p}{p-1}}\,dx \end{split}$$ for some $C=C(p)>0$. In this case, we have that $p-1>1$ and therefore we can use the Hölder Inequality with exponents $\frac{p-1}{p-2}$ and $p-1$ and obtain from [\[djweioguoe9786543e\]](#djweioguoe9786543e){reference-type="eqref" reference="djweioguoe9786543e"} that $$\begin{split} &\int_{\mathbb{R}^N} \Big|\,|u_n(x)|^{p-2}u_n(x)-|u(x)|^{p-2}u(x)\Big|^{\frac{p}{p-1}}\,dx \\ \leqslant\;& C\left(\;\int_{\mathbb{R}^N} \big( |u_n(x)|^{p-2} +|u(x)|^{p-2}\big)^{\frac{p}{p-2}}\,dx\right)^{\frac{p-2}{p-1}} \left(\;\int_{\mathbb{R}^N} | u_n(x) -u(x)|^{p}\,dx\right)^{\frac{1}{p-1}} \\ \leqslant\;& C\left(\;\int_{\mathbb{R}^N} \big( |u_n(x)|^{p} +|u(x)|^{p}\big)\,dx\right)^{\frac{p-2}{p-1}} \left(\;\int_{\mathbb{R}^N} | u_n(x) -u(x)|^{p}\,dx\right)^{\frac{1}{p-1}}, \end{split}$$ up to renaming $C$. The desired result in the case $p>2$ then follows from this, [\[fhwerutgvbhsd879999999999962-945\]](#fhwerutgvbhsd879999999999962-945){reference-type="eqref" reference="fhwerutgvbhsd879999999999962-945"} and [\[sweo0p7i8u6t453rkjtrg\]](#sweo0p7i8u6t453rkjtrg){reference-type="eqref" reference="sweo0p7i8u6t453rkjtrg"}. If instead $p\in(1,2)$, we first observe that, for all $t\in\mathbb{R}\setminus\{0\}$, $$\label{pso2} \frac{\big| |1+t|^{p-2}(1+t)-1\big|}{|t|^{p-1}}\leqslant C_\star,$$ for some $C_\star>0$, depending only on $p$. To check this, one can define $\phi(t)$ as the left hand side of [\[pso2\]](#pso2){reference-type="eqref" reference="pso2"} and notice that $$\lim_{t\to\pm\infty}\phi(t)=1.$$ Moreover, by L'Hôpital's Rule, since $p<2$, $$\lim_{t\to0}\phi(t)=\lim_{t\to0} \frac{\big| (1+t)^{p-1}-1\big|}{|t|^{p-1}}=\lim_{t\to0} \big( {\rm sign}(t)\big)^p\; \frac{ (1+t)^{p-1}-1}{t^{p-1}} =\pm\lim_{t\to0} \frac{ (1+t)^{p-2}}{t^{p-2}}=0.$$ These observations establish [\[pso2\]](#pso2){reference-type="eqref" reference="pso2"}. Moreover, in this case, for all $a$, $b\in\mathbb{R}$, $$\label{pso23} \big|\,|a|^{p-2}a-|b|^{p-2}b\,\big|\leqslant C_\star \,|a-b|^{p-1}.$$ Indeed, if $a=b$, or $a=0$, or $b=0$ this is obvious, hence we can suppose that $a\ne b$, that $a\ne0$ and that $b\ne0$. Also, we can assume that $a$ and $b$ have the same sign, because if, say, $a>0>b$, then $$\big|\,|a|^{p-2}a-|b|^{p-2}b\,\big|= a^{p-1}+|b|^{p-1}\leqslant(a+|b|)^{p-1}+(a+|b|)^{p-1} =2(a-b)^{p-1}=2|a-b|^{p-1}$$ and we are done. Hence, up to swapping the signs of $a$ and $b$, which would not change the desired claim, we can assume, without loss of generality, that $a$, $b>0$. Accordingly, we can define $$t:=\frac{a}{b}-1.$$ Notice that $t\ne0$ and then, in view of [\[pso2\]](#pso2){reference-type="eqref" reference="pso2"}, $$\begin{aligned} C_\star\geqslant\frac{\big| |1+t|^{p-2}(1+t)-1\big|}{|t|^{p-1}} = \frac{\big| (a/b)^{p-1}-1\big|}{|(a/b)-1|^{p-1}}= \frac{\big| a^{p-1}-b^{p-1}\big|}{|a-b|^{p-1}},\end{aligned}$$ which establishes [\[pso23\]](#pso23){reference-type="eqref" reference="pso23"}. As a consequence, owing to [\[sweo0p7i8u6t453rkjtrg\]](#sweo0p7i8u6t453rkjtrg){reference-type="eqref" reference="sweo0p7i8u6t453rkjtrg"} and [\[pso23\]](#pso23){reference-type="eqref" reference="pso23"}, $$\begin{aligned} |( U_n,v)-(U,v)|&\leqslant& C\,C_\star\,\left(\;\int_{\mathbb{R}^N} |u_n(x)-u(x)|^p\,dx\right)^{\frac{p-1}{p}} \rho_p(v).\end{aligned}$$ This and the convergence of $u_n$ in $L^p(\Omega)$ prove the desired result. ◻ The homogeneity assumption in $(L_1)$ is also obvious. With respect to the validity of $(\mathcal F_1)$, we note that the potential $F$ is given by [\[POTEF\]](#POTEF){reference-type="eqref" reference="POTEF"} and satisfies $$\label{FDCP} |F(u)|=\frac1{p_{s_\sharp}^*}\|u\|^{p_{s_\sharp}^* }_{L^{p_{s_\sharp}^* }(\mathbb{R}^N)}.$$ Besides, from the fractional Sobolev embedding (see e.g. [@MR2944369 Theorem 6.5]) and Theorem [Theorem 8](#SOBOLEVEMBEDDING){reference-type="ref" reference="SOBOLEVEMBEDDING"}, for all $s\in[s_\sharp,1]$, $$\|u\|_{L^{p_{s_\sharp}^* }(\mathbb{R}^N)}\leqslant C_1\,[u]_{s_\sharp,p}\leqslant C_2\,[u]_{s,p},$$ for suitable $C_1=C_1(N,s_\sharp,\Omega,p)>0$ and $C_2=C_2(N,s_\sharp,\Omega,p)>0$. Integrating over $s$ we find that $$\mu^+\big([s_\sharp,1]\big)\|u\|_{L^{p_{s_\sharp}^* }(\mathbb{R}^N)}^p\leqslant C_2^p\int_{[s_\sharp,1]}[u]_{s,p}^p\,d\mu^+(s)\leqslant \big[ C_2\,\rho_p(u)\big]^p.$$ This and [\[FDCP\]](#FDCP){reference-type="eqref" reference="FDCP"} give that $$|F(u)|\leqslant C_3\,[\rho(u)]^{p_{s_\sharp}^*/p},$$ with $C_3=C_3(N,s_\sharp,\Omega,p,\mu)>0$ and , since $p_{s_\sharp}^*>p$, we see that $(\mathcal F_1)$ is satisfied in our setting. In regard to $(\mathcal F_2)$, one uses [\[SETT1\]](#SETT1){reference-type="eqref" reference="SETT1"}, [\[POTEF\]](#POTEF){reference-type="eqref" reference="POTEF"} and [\[NASDL\]](#NASDL){reference-type="eqref" reference="NASDL"} and sees that $$\begin{aligned} F(u)-\frac\beta{q}\big(pJ_p(u)\big)^{q/p}&=&\frac1{p_{s_\sharp}^*}\; \int_{\mathbb{R}^N} |u(x)|^{p_{s_\sharp}^* }\,dx-\frac{\beta}q\big( \left(B_p\, u,u\right)_{}\big)^{q/p}\\ &=&\frac1{p_{s_\sharp}^*}\;\int_{\Omega} |u(x)|^{p_{s_\sharp}^* }\,dx- \frac{\beta}q\left(\int_{\Omega} |u(x)|^{p}\,dx\right)^{q/p}.\end{aligned}$$ Hence, choosing $$\label{plutoepaper} q:=p_{s_\sharp}^* >p \qquad {\mbox{and}}\qquad \beta:=\frac1{|\Omega|^{(p_{s_\sharp}^* -p)/p }},$$ the Hölder Inequality with exponents $p_{s_\sharp}^* /p$ and $p_{s_\sharp}^* /(p_{s_\sharp}^* -p)$ gives that $$F(u)-\frac\beta{q}\big(pJ_p(u)\big)^{q/p}\geqslant\frac1{p_{s_\sharp}^*}\; \int_{\Omega} |u(x)|^{p_{s_\sharp}^* }\,dx- \frac{\beta\,|\Omega|^{(p_{s_\sharp}^* -p)/p }}{p_{s_\sharp}^* } \int_{\Omega} |u(x)|^{p_{s_\sharp}^*}\,dx=0 ,$$ establishing $(\mathcal F_2)$. Referring to $(\mathcal F_3)$, we first point out the following weak convergence result: **Lemma 19**. *Let $u_n$ be a bounded sequence in $\mathcal X_p(\Omega)$.* *Then, there exists $u\in\mathcal X_p(\Omega)$ such that, for any $v\in\mathcal X_p(\Omega)$, we have $$\label{Vconv} \begin{split} &\lim_{n\to+\infty}\int_{[{{ 0 }}, 1]} \left(\;\iint_{\mathbb{R}^{2N}} \frac{|u_n(x)-u_n(y)|^{p-2}(u_n(x)-u_n(y)) (v(x)-v(y))}{|x-y|^{N+sp}} \, dxdy\right)\, d\mu^\pm(s)\\ &\qquad=\int_{[{{ 0 }}, 1]} \left(\;\iint_{\mathbb{R}^{2N}} \frac{|u(x) - u(y)|^{p-2}(u(x)-u(y)) (v(x)-v(y))}{|x-y|^{N+sp}} \, dxdy\right)\, d\mu^\pm(s) . \end{split}$$* *Proof.* On the one hand, by [\[complete00\]](#complete00){reference-type="eqref" reference="complete00"} and Lemma [Lemma 10](#UNCONCVE){reference-type="ref" reference="UNCONCVE"}, we have that $\mathcal X_p(\Omega)$ is complete and uniformly convex. On the other hand, the Milman-Pettis Theorem (see e.g. [@MR100215]) states that every uniformly convex Banach space is reflexive. Therefore, $\mathcal X_p(\Omega)$ is reflexive (and the same holds if one replaces $\mu^+$ by $\mu^-$). Hence, by Kakutani's Theorem we know that the closed balls of ${\mathcal{X}}_p(\Omega)$ are compact in the weak topology, which yields the desired result. ◻ Moreover, to establish the threshold $c^*$ for the condition $(\mathcal F_3)$ to hold, we will also need the following Brézis--Lieb type result (see [@MR699419] for the original statement): **Lemma 20**. *Let $u_n$ be a bounded sequence in $\mathcal X_p(\Omega)$. Suppose that $u_n$ converges to some $u$ a.e. in $\mathbb{R}^N$ as $n\to +\infty$.* *Then, $$\label{djieow34LLLtyb5o4yo3tu3493pp} \int_{[{{ 0 }}, 1]} [u]^p_{s, p}\, d\mu^{\pm}(s) = \lim_{n\to +\infty}\left(\;\int_{[{{ 0 }}, 1]} [u_n]^p_{s, p}\, d\mu^{\pm}(s) - \int_{[{{ 0 }}, 1]} [u_n -u]^p_{s, p}\, d\mu^{\pm}(s) \right).$$* *Proof.* The aim is to apply [@MR1817225 Theorem 1.9] to the measure space given by $(\mathbb{R}^{2N}\times [0, 1], \,dx\,dy\, d\mu^+(s))$ and the functions $$f_n(x, y, s):= c_{N, s, p}^{1/p} \frac{u_n(x) -u_n(y)}{|x-y|^{\frac{N+sp}{p}}}\qquad\text{ and }\qquad f(x, y, s):= c_{N, s, p}^{1/p} \frac{u(x) -u(y)}{|x-y|^{\frac{N+sp}{p}}}.$$ We observe that, since $u_n$ is a bounded sequence in $\mathcal X_p(\Omega)$, there exists $C>0$, independent of $n$, such that $$\begin{aligned} && C\geqslant\int_{[0,1]}[u_n]_{s,p}^p \,d\mu^+(s) =\int_{[0,1]} c_{N, s, p}\iint_{\mathbb{R}^{2N}} \frac{|u_n(x) -u_n(y)|^p}{|x-y|^{N+sp} }\,dx\,dy\,d\mu^+(s) \\&&\qquad\qquad = \iiint_{[0,1]\times\mathbb{R}^{2N}} |f_n(x, y, s)|^p\,dx\,dy\,d\mu^+(s) ,\end{aligned}$$ which implies that $f_n$ is uniformly bounded in $L^p\big( \mathbb{R}^{2N}\times [0, 1], \,dx\,dy\, d\mu^+(s)\big)$. Moreover, since $u_n$ converges to $u$ a.e. in $\mathbb{R}^N$ as $n\to +\infty$, we have that $f_n$ converges to $f$ a.e. in $\mathbb{R}^{2N}\times[0,1]$ as $n\to +\infty$. Hence, the assumptions in [@MR1817225 Theorem 1.9] are fulfilled, and therefore $$\begin{aligned} && \int_{[{{ 0 }}, 1]} [u]^p_{s, p}\, d\mu^{+}(s) = \iiint_{[0,1]\times\mathbb{R}^{2N}} |f(x, y, s)|^p\,dx\,dy\,d\mu^+(s)\\& =& \lim_{n\to +\infty}\left(\;\;\iiint_{[0,1]\times\mathbb{R}^{2N}} |f_n(x, y, s)|^p\,dx\,dy\,d\mu^+(s) - \iiint_{[0,1]\times\mathbb{R}^{2N}} |f(x, y, s)-f_n(x,y,s)|^p\,dx\,dy\,d\mu^+(s) \right)\\& =& \lim_{n\to +\infty}\left(\;\int_{[{{ 0 }}, 1]} [u_n]^p_{s, p}\, d\mu^{+}(s) - \int_{[{{ 0 }}, 1]} [u_n -u]^p_{s, p}\, d\mu^{+}(s) \right).\end{aligned}$$ This establishes the desired result for $\mu^+$. We now focus on proving the claim in [\[djieow34LLLtyb5o4yo3tu3493pp\]](#djieow34LLLtyb5o4yo3tu3493pp){reference-type="eqref" reference="djieow34LLLtyb5o4yo3tu3493pp"} for $\mu^-$. For this, we use [\[mu3\]](#mu3){reference-type="eqref" reference="mu3"} and Proposition [Proposition 9](#absorb){reference-type="ref" reference="absorb"} to see that $$\begin{aligned} && \int_{[{{ 0 }}, 1]} [u_n]_{s, p}^p \, d\mu^- (s)= \int_{[{{ 0 }}, \overline s]} [u_n]_{s, p}^p \, d\mu^- (s) \leqslant c_0(N,\Omega, p) \,\gamma\int_{[\overline s, 1]} [u_n]^p_{s, p} \, d\mu(s) \\&&\qquad\qquad \leqslant c_0(N,\Omega, p) \,\gamma\int_{[{{ 0 }}, 1]} [u_n]^p_{s, p}\, d\mu^+ (s).\end{aligned}$$ This and the fact that $u_n$ is bounded in ${\mathcal{X}}_p(\Omega)$ give that the quantity $$\int_{[{{ 0 }}, 1]} [u_n]_{s, p}^p \, d\mu^- (s)$$ is uniformly bounded in $n$. Therefore, one can repeat the same argument as above, replacing $\mu^+$ with $\mu^-$ and obtain the desired result in [\[djieow34LLLtyb5o4yo3tu3493pp\]](#djieow34LLLtyb5o4yo3tu3493pp){reference-type="eqref" reference="djieow34LLLtyb5o4yo3tu3493pp"}. ◻ To check $(\mathcal F_3)$, we also point out that: **Proposition 21**. *Let $\theta_0\in(0,1)$ and $$\label{topolino} c^* := \frac{s_\sharp}{N} \, \Big( (1-\theta_0)\mathcal S(p)\Big)^{N/s_\sharp p}.$$* *Then, there exists $\gamma_0>0$, depending on $N$, $\Omega$, $p$, $s_\sharp$ and $\theta_0$, such that if $\gamma\in[0,\gamma_0]$ and $c\in(0,c^*)$, then the functional in [\[funp\]](#funp){reference-type="eqref" reference="funp"} satisfies the $(\text{PS})_{c}$ condition.* *Proof.* Take $c\in(0, c^*)$ and let $u_n \subset \mathcal{X}_p(\Omega)$ be a sequence verifying $$\label{PS1}\begin{split}\lim_{n\to+\infty} E(u_n) =\,&\lim_{n\to+\infty} \frac1p [\rho_p(u_n)]^p -\frac1p\int_{[{{{0}}}, \overline s]} [u_n]_{s, p}^p \, d\mu^-(s) - \frac{\lambda}{p} \int_\Omega |u_n|^p \, dx - \frac{1}{p_{s_\sharp}^*} \int_\Omega |u_n|^{p_{s_\sharp}^*}\, dx\\ =\,& c\end{split}$$ and $$\label{PS2} \begin{split}&\lim_{n\to+\infty} \sup_{v \in {\mathcal{X}}_p(\Omega)} ( dE (u_n), v) \\= \,&\lim_{n\to+\infty} \sup_{v \in {\mathcal{X}}_p(\Omega)}\int_{[{{{0}}}, 1]}\left(\; c_{N,s,p}\iint_{\mathbb{R}^{2N}} \frac{|u_n(x) - u_n(y)|^{p-2}\, (u_n(x) - u_n(y))\, (v(x) - v(y))}{|x - y|^{N+sp}}\, dx dy\right) d\mu^+(s)\\ &\qquad-\int_{[{{{0}}}, \overline s]}\left(\;c_{N,s,p}\iint_{\mathbb{R}^{2N}} \frac{|u_n(x) - u_n(y)|^{p-2}\, (u_n(x) - u_n(y))\, (v(x) - v(y))}{|x - y|^{N+sp}}\, dx dy\right) d\mu^-(s)\\ &\qquad- \lambda \int_\Omega |u_n|^{p-2}\, u_n v\, dx - \int_\Omega |u_n|^{p_{s_\sharp}^* - 2}\, u_n v\, dx \\=\,& 0. \end{split}$$ Testing [\[PS2\]](#PS2){reference-type="eqref" reference="PS2"} with $v:=-u_n$, $$\begin{split}0\leqslant\,&\lim_{n\to+\infty} \int_{[{{{0}}}, 1]}\left(\;c_{N,s,p}\iint_{\mathbb{R}^{2N}} \frac{|u_n(x) - u_n(y)|^{p}}{|x - y|^{N+sp}}\, dx dy\right) d\mu^+(s)\\ &\qquad-\int_{[{{{0}}}, \overline s]}\left(\;c_{N,s,p}\iint_{\mathbb{R}^{2N}} \frac{|u_n(x) - u_n(y)|^{p}}{|x - y|^{N+sp}}\, dx dy\right) d\mu^-(s)\\ &\qquad- \lambda \int_\Omega |u_n|^{p}\, dx - \int_\Omega |u_n|^{p_{s_\sharp}^* }\, dx\\ =\,&\lim_{n\to+\infty} p\,E(u_n)+\left(\frac{p}{p^*_{s_\sharp}}-1 \right)\int_\Omega |u_n|^{p_{s_\sharp}^* }\, dx, \end{split}$$ and then, by [\[PS1\]](#PS1){reference-type="eqref" reference="PS1"}, $$\label{boh}\lim_{n\to+\infty} \left(\frac{1}{p}-\frac{1}{p^*_{s_\sharp}}\right) \|u_n\|_{L^{p^*_{s_\sharp}}(\mathbb{R}^N)}^{p^*_{s_\sharp}} \leqslant c .$$ Furthermore, from Proposition [Proposition 9](#absorb){reference-type="ref" reference="absorb"} we deduce that $$\int_{[{{{0}}}, \overline s]} [u_n]_{s, p}^p \, d\mu^-(s) \leqslant c_0 \gamma \int_{[\overline s,1]} [u_n]_{ s, p}^p \, d\mu^+(s) \leqslant c_0 \gamma[\rho_p(u_n)]^p .$$ Hence, $$\label{PKSD-LPAIKSEASCMC} [\rho_p(u_n)]^p -\int_{[{{{0}}}, \overline s]} [u_n]_{s, p}^p \, d\mu^-(s)\geqslant \Big(1-C^{2p}(N,\Omega,p)\gamma\Big) [\rho_p(u_n)]^p.$$ From this and [\[PS1\]](#PS1){reference-type="eqref" reference="PS1"}, we obtain that $\rho_p(u_n)$ is bounded uniformly in $n$, provided that $1-C^{2p}(N,\Omega,p)\gamma>0$. Accordingly, in view of Lemma [Lemma 19](#Vitali){reference-type="ref" reference="Vitali"}, there exists $u\in\mathcal{X}_p(\Omega)$ such that, up to subsequences, $$\begin{aligned} {2}\nonumber & u_n\rightharpoonup u && \text{ in } \mathcal{X}_p(\Omega),\\\label{4.7} & u_n\to u && \text{ in } L^r(\Omega) \text{ for any } r\in [1, p^*_{s_\sharp}),\\\nonumber & u_n\to u && \text{ a.e. in } \Omega.\end{aligned}$$ It remains to prove that $u_n \to u$ in $\mathcal{X}_p(\Omega)$ as $n\to+\infty$. For this, we set $\widetilde{u}_n :=u_n - u$. By [@MR699419 Theorem 1] we have that $$\label{L32} \|u\|^{p^*_{s_\sharp}}_{L^{p^*_{s_\sharp}}(\mathbb{R}^N)} = \lim_{n\to+\infty}\|u_n\|^{p^*_{s_\sharp}}_{L^{p^*_{s_\sharp}}(\mathbb{R}^N)} - \|\widetilde{u}_n\|_{L^{p^*_{s_\sharp}}(\mathbb{R}^N)}^{p^*_{s_\sharp}}.$$ Moreover, by Lemma [Lemma 20](#Brezis-Lieb){reference-type="ref" reference="Brezis-Lieb"} we get that $$\label{L32BIS} \int_{[{{{0}}}, 1]} [u]^p_{s, p}\, d\mu^\pm(s)= \lim_{n\to+\infty}\int_{[{{{0}}}, 1]} [u_n]^p_{s, p}\, d\mu^\pm(s)- \int_{[{{{0}}}, 1]} [\widetilde u_n]^p_{s, p}\, d\mu^\pm(s) .$$ Hence, testing Definition [Definition 11](#wsol){reference-type="ref" reference="wsol"} with $v:=u$, $$\label{test1} [\rho_p (u)]^p -\int_{[{{{0}}}, \overline s]} [u]^p_{s, p}\, d\mu^-(s) = \lambda \|u\|^p_{L^p(\mathbb{R}^N)} + \|u\|_{L^{p^*_{s_\sharp}}(\mathbb{R}^N)}^{p^*_{s_\sharp}}.$$ Similarly, testing identity [\[PS2\]](#PS2){reference-type="eqref" reference="PS2"} with $v:=\pm u_n$, $$\lim_{n\to+\infty} [\rho_p (u_n)]^p -\int_{[{{{0}}}, \overline s]} [u_n]^p_{s, p}\, d\mu^-(s) - \lambda \|u_n\|^p_{L^p(\mathbb{R}^N)} - \|u_n\|_{L^{p^*_{s_\sharp}}(\mathbb{R}^N)}^{p^*_{s_\sharp}} =0.$$ This and [\[4.7\]](#4.7){reference-type="eqref" reference="4.7"} give that $$\label{L32BIS03plrf-73874} \lim_{n\to+\infty} [\rho_p (u_n)]^p -\int_{[{{{0}}}, \overline s]} [u_n]^p_{s, p}\, d\mu^-(s) - \|u_n\|_{L^{p^*_{s_\sharp}}(\mathbb{R}^N)}^{p^*_{s_\sharp}} =\lambda \|u\|^p_{L^p(\mathbb{R}^N)}.$$ We compare this with [\[test1\]](#test1){reference-type="eqref" reference="test1"} and we see that $$\lim_{n\to+\infty} [\rho_p (u_n)]^p -\int_{[{{{0}}}, \overline s]} [u_n]^p_{s, p}\, d\mu^-(s) - \|u_n\|_{L^{p^*_{s_\sharp}}(\mathbb{R}^N)}^{p^*_{s_\sharp}} =[\rho_p (u)]^p -\int_{[{{{0}}}, \overline s]} [u]^p_{s, p}\, d\mu^-(s)-\|u\|_{L^{p^*_{s_\sharp}}(\mathbb{R}^N)}^{p^*_{s_\sharp}}.$$ Hence, in light of [\[L32BIS\]](#L32BIS){reference-type="eqref" reference="L32BIS"}, $$\lim_{n\to+\infty} [\rho_p (\widetilde u_n)]^p -\int_{[{{{0}}}, \overline s]} [\widetilde u_n]^p_{s, p}\, d\mu^-(s) - \|u_n\|_{L^{p^*_{s_\sharp}}(\mathbb{R}^N)}^{p^*_{s_\sharp}} = -\|u\|_{L^{p^*_{s_\sharp}}(\mathbb{R}^N)}^{p^*_{s_\sharp}}.$$ Accordingly, by [\[L32\]](#L32){reference-type="eqref" reference="L32"}, $$\label{NBSFVCECGNAS} \lim_{n\to+\infty} [\rho_p (\widetilde u_n)]^p -\int_{[{{{0}}}, \overline s]} [\widetilde u_n]^p_{s, p}\, d\mu^-(s) - \|\widetilde u_n\|_{L^{p^*_{s_\sharp}}(\mathbb{R}^N)}^{p^*_{s_\sharp}} = 0.$$ Combining this and the definition of the Sobolev constant in [\[DESOCO\]](#DESOCO){reference-type="eqref" reference="DESOCO"}, $$\lim_{n\to+\infty} [\rho_p (\widetilde u_n)]^p -\int_{[{{{0}}}, \overline s]} [\widetilde u_n]^p_{s, p}\, d\mu^-(s) -\left(\frac1{{\mathcal S(p)}}\int_{[{{{0}}},1]} [\widetilde u_n]^p_{s, p}\, d\mu^+(s)\right)^{p^*_{s_\sharp}/p} \leqslant 0,$$ that is $$\lim_{n\to+\infty} [\rho_p (\widetilde u_n)]^p -\int_{[{{{0}}}, \overline s]} [\widetilde u_n]^p_{s, p}\, d\mu^-(s) -\frac{[\rho_p (\widetilde u_n)]^{p^*_{s_\sharp}}}{({\mathcal S(p)})^{p^*_{s_\sharp}/p}} \leqslant 0.$$ We now recall [\[PKSD-LPAIKSEASCMC\]](#PKSD-LPAIKSEASCMC){reference-type="eqref" reference="PKSD-LPAIKSEASCMC"}, applied here to $\widetilde u_n$ instead of $u_n$, and we conclude that $$\lim_{n\to+\infty} \Big(1-C^{2p}(N,\Omega,p)\gamma\Big)[\rho_p(\widetilde u_n)]^p -\frac{[\rho_p (\widetilde u_n)]^{p^*_{s_\sharp}}}{({\mathcal S(p)})^{p^*_{s_\sharp}/p}} \leqslant 0,$$ or equivalently $$\label{moOmdRIJSmdfGOIANMSnao0lol} \lim_{n\to+\infty} [\rho_p(\widetilde u_n)]^p \left[ \Big(1-C^{2p}(N,\Omega,p)\gamma\Big){({\mathcal S(p)})^{p^*_{s_\sharp}/p}} -{[\rho_p (\widetilde u_n)]^{p^*_{s_\sharp}-p}}\right]\leqslant 0.$$ Now we reconsider [\[PS1\]](#PS1){reference-type="eqref" reference="PS1"} in the light of [\[L32BIS03plrf-73874\]](#L32BIS03plrf-73874){reference-type="eqref" reference="L32BIS03plrf-73874"} and we see that $$\begin{aligned} c&=&\lim_{n\to+\infty} \frac1p [\rho_p(u_n)]^p -\frac1p\int_{[{{{0}}}, \overline s]} [u_n]_{s, p}^p \, d\mu^-(s) - \frac{\lambda}{p} \int_\Omega |u_n|^p \, dx - \frac{1}{p_{s_\sharp}^*} \int_\Omega |u_n|^{p_{s_\sharp}^*}\, dx\\&=&\lim_{n\to+\infty} \left(\frac1p-\frac1{p^*_{s_\sharp}}\right)\left( [\rho_p(u_n)]^p -\int_{[{{{0}}}, \overline s]} [u_n]_{s, p}^p \, d\mu^-(s) \right)- \frac{\lambda}{p} \int_\Omega |u_n|^p \, dx + \frac\lambda{p^*_{s_\sharp}}\int_\Omega|u|^p\,dx.\end{aligned}$$ This and the strong convergence in [\[4.7\]](#4.7){reference-type="eqref" reference="4.7"} yield that $$\begin{aligned} c&=&\lim_{n\to+\infty} \frac{s_\sharp}N\left( [\rho_p(u_n)]^p -\int_{[{{{0}}}, \overline s]} [u_n]_{s, p}^p \, d\mu^-(s) \right)- \frac{\lambda \,s_\sharp}{N} \int_\Omega |u|^p \, dx.\end{aligned}$$ From this and [\[L32BIS\]](#L32BIS){reference-type="eqref" reference="L32BIS"} we arrive at $$\begin{aligned} \frac{cN}{s_\sharp}&=&\lim_{n\to+\infty} [\rho_p(u)]^p+[\rho_p(\widetilde u_n)]^p -\int_{[{{{0}}}, \overline s]} [u]_{s, p}^p \, d\mu^-(s)-\int_{[{{{0}}}, \overline s]} [\widetilde u_n]_{s, p}^p \, d\mu^-(s)- \lambda\int_\Omega |u|^p \, dx.\end{aligned}$$ As a consequence, by [\[test1\]](#test1){reference-type="eqref" reference="test1"}, $$\begin{aligned} \frac{cN}{s_\sharp}&=&\lim_{n\to+\infty} [\rho_p(\widetilde u_n)]^p -\int_{[{{{0}}}, \overline s]} [\widetilde u_n]_{s, p}^p \, d\mu^-(s) +\int_\Omega |u|^{p^*_{s_\sharp}} \, dx.\end{aligned}$$ Hence, recalling [\[PKSD-LPAIKSEASCMC\]](#PKSD-LPAIKSEASCMC){reference-type="eqref" reference="PKSD-LPAIKSEASCMC"} (used here for $\widetilde u_n$ instead of $u_n$), $$\begin{aligned} \frac{cN}{s_\sharp}&\geqslant&\lim_{n\to+\infty}\Big(1-C^{2p}(N,\Omega,p)\gamma\Big) [\rho_p(\widetilde u_n)]^p +\int_\Omega |u|^{p^*_{s_\sharp}} \, dx\\ &\geqslant&\lim_{n\to+\infty}\Big(1-C^{2p}(N,\Omega,p)\gamma\Big) [\rho_p(\widetilde u_n)]^p.\end{aligned}$$ This and [\[topolino\]](#topolino){reference-type="eqref" reference="topolino"} tell us that $$\begin{aligned} \Big( (1-\theta_0)\mathcal S(p)\Big)^{N/s_\sharp p}= \frac{c^* N}{s_\sharp}> \frac{cN}{s_\sharp}\geqslant\lim_{n\to+\infty}\Big(1-C^{2p}(N,\Omega,p)\gamma\Big) [\rho_p(\widetilde u_n)]^p\end{aligned}$$ and consequently $$\begin{aligned} &&\liminf_{n\to+\infty}\Big(1-C^{2p}(N,\Omega,p)\gamma\Big){({\mathcal S(p)})^{p^*_{s_\sharp}/p}} -{[\rho_p (\widetilde u_n)]^{p^*_{s_\sharp}-p}}\\&&\qquad=\Big(1-C^{2p}(N,\Omega,p)\gamma\Big){({\mathcal S(p)})^{N/(N-s_\sharp p)}}-\limsup_{n\to+\infty} [\rho_p (\widetilde u_n)]^{{s_\sharp}p^2/(N-{s_\sharp}p)}\\ &&\qquad\geqslant \Big(1-C^{2p}(N,\Omega,p)\gamma\Big){({\mathcal S(p)})^{N/(N-s_\sharp p)}}- \left(\frac{\Big( (1-\theta_0)\mathcal S(p)\Big)^{N/s_\sharp p}}{1-C^{2p}(N,\Omega,p)\gamma }\right)^{{s_\sharp}p/(N-{s_\sharp}p)}\\&&\qquad= \left[ 1-C^{2p}(N,\Omega,p)\gamma- \left(\frac{ (1-\theta_0)^{N/s_\sharp p}}{1-C^{2p}(N,\Omega,p)\gamma }\right)^{{s_\sharp}p/(N-{s_\sharp}p)}\right] {({\mathcal S(p)})^{N/(N-s_\sharp p)}}.\end{aligned}$$ Thus, since $$\lim_{\gamma\searrow0} 1-C^{2p}(N,\Omega,p)\gamma- \left(\frac{ (1-\theta_0)^{N/s_\sharp p}}{1-C^{2p}(N,\Omega,p)\gamma }\right)^{{s_\sharp}p/(N-{s_\sharp}p)}= 1-(1-\theta_0)^{N/(N-{s_\sharp}p)}>0,$$ we infer that $$\liminf_{n\to+\infty}\Big(1-C^{2p}(N,\Omega,p)\gamma\Big){({\mathcal S(p)})^{p^*_{s_\sharp}/p}} -{[\rho_p (\widetilde u_n)]^{p^*_{s_\sharp}-p}}>0,$$ as long as $\gamma$ is sufficiently small. Combining this information with [\[moOmdRIJSmdfGOIANMSnao0lol\]](#moOmdRIJSmdfGOIANMSnao0lol){reference-type="eqref" reference="moOmdRIJSmdfGOIANMSnao0lol"}, we gather that $$\lim_{n\to+\infty} \rho_p(\widetilde u_n)=0,$$ and thus $u_n \to u$ in $\mathcal{X}_p(\Omega)$ as $n\to+\infty$. ◻ Moreover, as it concerns $(\mathcal N_1)$, in view of [\[Npippoepluto\]](#Npippoepluto){reference-type="eqref" reference="Npippoepluto"} and Theorem [Theorem 8](#SOBOLEVEMBEDDING){reference-type="ref" reference="SOBOLEVEMBEDDING"} we have that $$N_p(u)=\frac1p\int_{[{{{0}}}, \overline s]} [u]_{s, p}^p \, d\mu^-(s)\leqslant \frac{C(N,\Omega,p)\,\mu^-\big( {[{{{0}}}, \overline s]}\big)}p\,[u]^p_{\overline s,p}.$$ Hence, recalling [\[mu2\]](#mu2){reference-type="eqref" reference="mu2"} and using again Theorem [Theorem 8](#SOBOLEVEMBEDDING){reference-type="ref" reference="SOBOLEVEMBEDDING"}, $$N_p(u)\leqslant\frac{C(N,\Omega,p)\,\gamma\,\mu^+\big([\overline s, 1]\big)}p\,[u]^p_{\overline s,p} \leqslant\frac{\big( C(N,\Omega,p)\big)^2\,\gamma}p\,\int_{[\overline s, 1]}[u]^p_{ s,p}\,d\mu^+(s) .$$ As a result, we can take $$\eta:=\big( C(N,\Omega,p)\big)^2\,\gamma$$ and notice that, if $\gamma$ is sufficiently small, possibly in dependence of $N$, $\Omega$ and $p$, then $\eta\in(0,1)$. In this case, $$N_p(u)\leqslant \frac{\eta}p \,[\rho_p(u)]^p=\eta \,I_p(u),$$ and $(\mathcal N_1)$ is thereby established. In this way we have: *Proof of Theorem [Theorem 1](#mainp1){reference-type="ref" reference="mainp1"}.* We have just checked that all the abstract assumptions in Section [1.2](#AB2){reference-type="ref" reference="AB2"} are fulfilled under the hypotheses of Theorem [Theorem 1](#mainp1){reference-type="ref" reference="mainp1"}. We can thereby exploit Theorem [Theorem 3](#Theorem 2.9){reference-type="ref" reference="Theorem 2.9"}, with the choices of $q$, $\beta$ and $c^*$ as in [\[plutoepaper\]](#plutoepaper){reference-type="eqref" reference="plutoepaper"} and [\[topolino\]](#topolino){reference-type="eqref" reference="topolino"}, which in turn implies the thesis of Theorem [Theorem 1](#mainp1){reference-type="ref" reference="mainp1"} as a particular case. To make this argument work, we have to check that one can choose $\theta_0\in(0,1)$ in [\[topolino\]](#topolino){reference-type="eqref" reference="topolino"} so to satisfy [\[2.11\]](#2.11){reference-type="eqref" reference="2.11"}. To this end, we observe that [\[2.11\]](#2.11){reference-type="eqref" reference="2.11"} in this specific case boils down to $$\lambda_l - \left(\frac1{|\Omega|^{(p_{s_\sharp}^* -p)/p }}\right)^{p/p^*_{s_\sharp}} \left(\frac{p\,p^*_{s_\sharp}\,c^*}{p^*_{s_\sharp} - p}\right)^{1 - p/p^*_{s_\sharp}} < \lambda < \lambda_l ,$$ that is $$\lambda_l - \left(\frac{N\,c^*}{|\Omega|\, {s_\sharp} }\right)^{{s_\sharp}p/N} < \lambda < \lambda_l ,$$ which in turn, recalling [\[topolino\]](#topolino){reference-type="eqref" reference="topolino"}, reduces to $$\lambda_l - \frac{(1-\theta_0)\mathcal S(p)}{|\Omega|^{{s_\sharp}p/N}} < \lambda < \lambda_l .$$ This condition is fulfilled thanks to the structural assumption in [\[LLP1\]](#LLP1){reference-type="eqref" reference="LLP1"}, which allows us to define $$\theta_0:= \frac12\left(1-\frac{|\Omega|^{{s_\sharp}p/N}}{\mathcal S(p)}(\lambda_l-\lambda)\right)\in(0,1).\qedhere$$ ◻ # Applications {#sec-app} The overall versatility of the measure $\mu$ that we treat in this paper allows us to treat a variety of operators. In this section we focus our attention on illustrating some examples that stem from the particular choice of the measure $\mu$. We point out that most of the multiplicity results provided here are new and they enrich the already existing literature. To start with, we showcase multiplicity results for the classical $p$-Laplacian and the fractional $p$-Laplacian. Indeed, with specific choices of the measure $\mu$, Theorem [Theorem 1](#mainp1){reference-type="ref" reference="mainp1"} recovers [@MR3469053 Theorem 1.1] and [@PSY Theorem 1.1]). **Corollary 22**. *Let $p\in (1, N)$, let $p^*:= Np/(N-p)$ be the classical critical Sobolev exponent and $S_p$ the classical best Sobolev constant. Suppose that, for some $l$, $m \geqslant 1$, $$\lambda_l - \frac{S_{p}}{|\Omega|^{p/N}} < \lambda < \lambda_l = \dots = \lambda_{l+m-1}.$$* *Then, the following problem $$\left\{\begin{aligned} -\Delta_p \, u & = \lambda |u|^{p-2} u + |u|^{p^* - 2}\, u && \text{in } \Omega,\\ u & = 0 && \text{on } \partial\Omega\end{aligned}\right.$$ has $m$ distinct pairs of nontrivial solutions $\pm u^\lambda_1,\dots,\pm u^\lambda_m$.* *Proof.* Let $\mu:= \delta_1$ be the Dirac measure centered at the point $1$. Notice that $\mu$ verifies conditions [\[mu00\]](#mu00){reference-type="eqref" reference="mu00"}, [\[mu3\]](#mu3){reference-type="eqref" reference="mu3"} and [\[mu2\]](#mu2){reference-type="eqref" reference="mu2"} with $\overline s:=1$ and $\gamma=0$. Furthermore, we can take $s_\sharp:=1$, so that ${\mathcal{S}(p)}$ defined in [\[DESOCO\]](#DESOCO){reference-type="eqref" reference="DESOCO"} boils down to the classical Sobolev constant $S_p$. The desired result now follows from Theorem [Theorem 1](#mainp1){reference-type="ref" reference="mainp1"}. ◻ **Corollary 23**. *Let $s\in ({{ 0 }}, 1)$ and $p\in (1, N)$. Let $p_s^* := Np/(N-sp)$ be the fractional critical Sobolev exponent and let $S_p(s)$ be the fractional best Sobolev constant. Suppose that, for some $l$, $m \geqslant 1$, $$\label{lambda-pslap} \lambda_l - \frac{S_{p}(s)}{|\Omega|^{sp/N}} < \lambda < \lambda_l = \dots = \lambda_{l+m-1}.$$* *Then, the following problem $$\left\{\begin{aligned} (-\Delta)^s_p \, u & = \lambda |u|^{p-2} u + |u|^{p_s^* - 2}\, u && \text{in } \Omega,\\ u & = 0 && \text{in } \mathbb{R}^N \setminus \Omega\end{aligned}\right.$$ has $m$ distinct pairs of nontrivial solutions $\pm u^\lambda_1,\dots,\pm u^\lambda_m$.* *Proof.* In this case we take $\mu:= \delta_s$, being $\delta_s$ the Dirac measure centered at $s$. With this choice, $\mu$ satisfies the conditions [\[mu00\]](#mu00){reference-type="eqref" reference="mu00"}, [\[mu3\]](#mu3){reference-type="eqref" reference="mu3"} and [\[mu2\]](#mu2){reference-type="eqref" reference="mu2"} with $\overline s:=s$ and $\gamma=0$. Moreover, here we can take $s_\sharp:=s$, and then the desired result follows from Theorem [Theorem 1](#mainp1){reference-type="ref" reference="mainp1"}. ◻ The next result provides the existence of multiple solutions for the mixed operator $-\Delta_p + (-\Delta)_p^s$. To our best knowledge, this result is new even for the case $p=2$: **Corollary 24**. *Let $s\in [{{ 0 }}, 1)$ and $p\in (1, N)$. Let $p^* = Np/(N-p)$ be the classical critical Sobolev exponent and let $S_p$ be the best Sobolev constant. Suppose that, for some $l$, $m \geqslant 1$, $$\lambda_l - \frac{S_{p}}{|\Omega|^{p/N}} < \lambda < \lambda_l = \dots = \lambda_{l+m-1}.$$* *Then, the following problem $$\left\{\begin{aligned} -\Delta_p \, u + (-\Delta)_p^s \, u & = \lambda |u|^{p-2} u + |u|^{p^* - 2}\, u && \text{in } \Omega,\\ u & = 0 && \text{in } \mathbb{R}^N \setminus \Omega\end{aligned}\right.$$ has $m$ distinct pairs of nontrivial solutions $\pm u^\lambda_1,\dots,\pm u^\lambda_m$.* *Proof.* Let $\delta_1$ and $\delta_s$ denote the Dirac measures centered at $1$ and $s$, respectively. We define $\mu:=\delta_1 + \delta_s$. Here we take $\overline s:=1$, $\gamma=0$ and $s_\sharp:=1$ and we observe that conditions [\[mu00\]](#mu00){reference-type="eqref" reference="mu00"}, [\[mu3\]](#mu3){reference-type="eqref" reference="mu3"} and [\[mu2\]](#mu2){reference-type="eqref" reference="mu2"} are fulfilled. Thus, the desired result follows from Theorem [Theorem 1](#mainp1){reference-type="ref" reference="mainp1"}. ◻ An interesting scenario turns out when the measure $\mu$ changes sign. It means, for example, that the operator is allowed to include a small term with the "wrong" sign. Again, to our best knowledge, the existing literature lacks of a result of this kind. **Corollary 25**. *Let $s\in [{{ 0 }}, 1)$,  $p\in (1, N)$ and $\alpha\in\mathbb{R}$. Denote by $p^* := Np/(N-p)$ the classical critical Sobolev exponent and by $S_p$ the classical Sobolev constant. Suppose that, for some $l$, $m \geqslant 1$, $$\lambda_l - \frac{S_{p}}{|\Omega|^{p/N}} < \lambda < \lambda_l = \dots = \lambda_{l+m-1} .$$* *Then, there exists $\alpha_0>0$, depending only on $N$, $\Omega$, $p$, $s$, $\lambda$ and $l$ such that if $\alpha\leqslant\alpha_0$, then the following problem $$\left\{\begin{aligned} -\Delta_p \, u - \alpha(-\Delta)_p^s \, u & = \lambda |u|^{p-2} + |u|^{p^* - 2}\, u && \text{in } \Omega,\\ u & = 0 && \text{in } \mathbb{R}^N \setminus \Omega \end{aligned}\right.$$ has $m$ distinct pairs of nontrivial solutions $\pm u^\lambda_1,\dots,\pm u^\lambda_m$.* *Proof.* We define $\mu:= \delta_1 -\alpha\delta_s$ and take $\overline s:=1$ and $s_\sharp:=1$. In this way, conditions [\[mu00\]](#mu00){reference-type="eqref" reference="mu00"} and [\[mu3\]](#mu3){reference-type="eqref" reference="mu3"} are satisfied. Furthermore, $$\begin{aligned} \mu^-\big([{{ 0 }}, \overline s]\big)\leqslant\max\{0,\alpha\} = \max\{0,\alpha\}\mu^+\big([\overline s, 1]\big),\end{aligned}$$ which gives that condition [\[mu2\]](#mu2){reference-type="eqref" reference="mu2"} holds true taking $\gamma:=\max\{0,\alpha\}$. Hence, we obtain the desired result as an application of Theorem [Theorem 1](#mainp1){reference-type="ref" reference="mainp1"}. ◻ Our setting allows us to choose $\mu$ as a convergent series of Dirac measures. To be more precise, the next two results hold true. **Corollary 26**. *Let  $p\in (1, N)$ and $1\geqslant s_0 > s_1> s_2 >\dots \geqslant 0$. Consider the operator $$\sum_{k=0}^{+\infty} c_k (-\Delta)_p^{s_k} \qquad{\mbox{with }} \,\sum_{k=0}^{+\infty} c_k \in (0, +\infty).$$ Suppose that $c_0>0$ and $c_k\geqslant 0$ for all $k\geqslant 1$.* *Denote by $p^*_{s_0} := Np/(N-ps_0)$ the fractional critical Sobolev exponent and by $S_p(s_0)$ the fractional Sobolev constant corresponding to the exponent $s_0$. Assume that, for some $l$, $m \geqslant 1$, $$\lambda_l - \frac{S_{p}(s_0)}{|\Omega|^{s_0 p/N}} < \lambda < \lambda_l = \dots = \lambda_{l+m-1}.$$* *Then, the following problem $$\label{serie} \left\{\begin{aligned} \sum_{k=0}^{+\infty} c_k (-\Delta)_p^{s_k} u & = \lambda |u|^{p-2} u + |u|^{p_{s_0}^* - 2}\, u && \text{in } \Omega,\\ u & = 0 && \text{in } \mathbb{R}^N \setminus \Omega \end{aligned}\right.$$ has $m$ distinct pairs of nontrivial solutions $\pm u^\lambda_1,\dots,\pm u^\lambda_m$.* *Proof.* Here we set $$\mu:=\sum_{k=0}^{+\infty} c_k \,\delta_{s_k} ,$$ where $\delta_{s_k}$ denote the Dirac measures centered at each $s_k$. In this case, we can take $\overline s:=s_0$ and $s_\sharp:=s_0$, and notice that conditions [\[mu00\]](#mu00){reference-type="eqref" reference="mu00"}, [\[mu3\]](#mu3){reference-type="eqref" reference="mu3"} and [\[mu2\]](#mu2){reference-type="eqref" reference="mu2"} are satisfied (with $\gamma=0$). Hence, the desired result is a byproduct of Theorem [Theorem 1](#mainp1){reference-type="ref" reference="mainp1"}. ◻ **Corollary 27**. *Let  $p\in (1, N)$ and $1\geqslant s_0 > s_1> s_2 >\dots \geqslant 0$. Consider the operator $$\sum_{k=0}^{+\infty} c_k (-\Delta)_p^{s_k} \qquad{\mbox{with }} \,\sum_{k=0}^{+\infty} c_k \in (0, +\infty).$$ Assume that there exist $\gamma\geqslant 0$ and $\overline k\in\mathbb{N}$ such that $$\label{<gamma} c_k>0\ \text{ for all } k\in\{0,\dots, \overline k\}\quad \text{ and }\quad \sum_{k=\overline k +1}^{+\infty} c_k \leqslant\gamma \sum_{k=0}^{\overline k} c_k.$$* *Denote by $p^*_{s_0} := Np/(N-ps_0)$ the fractional critical Sobolev exponent and by $S_p(s_0)$ the fractional Sobolev constant corresponding to the exponent ${s_0}$. Assume that, for some $l$, $m \geqslant 1$, $$\lambda_l - \frac{S_{p}(s_0)}{|\Omega|^{s_0 p/N}} < \lambda < \lambda_l = \dots = \lambda_{l+m-1}$$ holds.* *Then, there exists $\gamma_0>0$ depending on $N$, $\Omega$, $p$, $s_k$, $c_k$, $\lambda$ and $l$ such that if $\gamma\in[0,\gamma_0]$, problem [\[serie\]](#serie){reference-type="eqref" reference="serie"} has $m$ distinct pairs of nontrivial solutions $\pm u^\lambda_1,\dots,\pm u^\lambda_m$.* *Proof.* We set once again $$\mu:=\sum_{k=0}^{+\infty} c_k \,\delta_{s_k}$$ where $\delta_{s_k}$ denote the Dirac measures centered at each $s_k$. In this case, condition [\[\<gamma\]](#<gamma){reference-type="eqref" reference="<gamma"} assures that the assumptions [\[mu00\]](#mu00){reference-type="eqref" reference="mu00"}, [\[mu3\]](#mu3){reference-type="eqref" reference="mu3"} and [\[mu2\]](#mu2){reference-type="eqref" reference="mu2"} on the measure $\mu$ are fulfilled with $\overline s:=s_{\overline k}$. Thus, taking $s_\sharp:=s_0$ we derive the desired result from Theorem [Theorem 1](#mainp1){reference-type="ref" reference="mainp1"}. ◻ We point out that another interesting result comes from the continuous superposition of fractional operators of $p$-Laplacian type. To the best of our knowledge, also this result happens to be new: **Corollary 28**. *Let $s_\sharp\in ({{ 0 }}, 1)$, $\gamma\geqslant 0$ and $f$ be a measurable and non identically zero function satisfying $$\label{fun-int}\begin{split} & {\mbox{$f\geqslant 0$ in~$(s_\sharp,1)$,}}\\& \int_{s_\sharp}^1 f(s) \,ds >0 \\ {\mbox{and }}\qquad& \int_0^{s_\sharp} \max\{0,-f(s)\} \,ds \leqslant\gamma \int_{s_\sharp}^1 f(s) \,ds . \end{split}$$* *Denote by $p_{s_\sharp}^* := Np/(N-ps_\sharp)$ the fractional critical Sobolev exponent and by $S_p(s_\sharp)$ the best Sobolev constant corresponding to the exponent $s_\sharp$. Assume that, for some $l$, $m \geqslant 1$, $$\lambda_l - \frac{S_{p}(s_\sharp)}{|\Omega|^{s_\sharp p/N}} < \lambda < \lambda_l = \dots = \lambda_{l+m-1}.$$* *Then, there exists $\gamma_0>0$, depending only on $N$, $\Omega$, $p$, $s_\sharp$, $f$, $\lambda$ and $l$, such that if $\gamma\in[0,\gamma_0]$, then problem $$\left\{\begin{aligned} \int_0^1 f(s) (-\Delta)_p^s \, u \, ds & = \lambda |u|^{p-2} u + |u|^{p_{s_\sharp}^* - 2}\, u && \text{in } \Omega,\\ u & = 0 && \text{in } \mathbb{R}^N \setminus \Omega, \end{aligned}\right.$$ has $m$ distinct pairs of nontrivial solutions $\pm u^\lambda_1,\dots,\pm u^\lambda_m$.* *Proof.* In this case, the operator $A_{\mu, p}$ takes the form $$\int_0^1 f(s) (-\Delta)_p^s \, u \, ds,$$ which means that we are taking $d\mu(s) = f(s)\,ds$. Moreover, here $s_\sharp$ acts as the critical fractional Sobolev exponent. Owing to the conditions stated in [\[fun-int\]](#fun-int){reference-type="eqref" reference="fun-int"}, we have that [\[mu00\]](#mu00){reference-type="eqref" reference="mu00"}, [\[mu3\]](#mu3){reference-type="eqref" reference="mu3"} and [\[mu2\]](#mu2){reference-type="eqref" reference="mu2"} are fulfilled with $\overline s:=s_\sharp$. Hence, the desired result follows from Theorem [Theorem 1](#mainp1){reference-type="ref" reference="mainp1"}. ◻ # Acknowledgements {#acknowledgements .unnumbered} SD and EV are members of the Australian Mathematical Society (AustMS). EV is supported by the Australian Laureate Fellowship FL190100081 "Minimal surfaces, free boundaries and partial differential equations". CS is member of INdAM-GNAMPA. This work was partially completed while KP was visiting the Department of Mathematics and Statistics at the University of Western Australia, and he is grateful for the hospitality of the host department. His visit to the UWA was supported by the Simons Foundation Award 962241 "Local and nonlocal variational problems with lack of compactness". 99999 [^1]: However, one can take $$c_{N,s,p}:=\frac{s\,2^{2s-1}\,\Gamma\left(\frac{ps+p+N-2}{2}\right)}{\pi^{N/2}\,\Gamma(1-s)},$$ see e.g. [@MR3473114 page 130] and the references therein. [^2]: From now on, this condition will always be assumed, to allow us to write critical exponents in our setting; this condition can also be relaxed by taking $N>s_\star p$, being $s_\star$ the supremum of the support of $\mu$. [^3]: As a technical observation, we stress that the eigenvalues that appear in the forthcoming Theorem [Theorem 3](#Theorem 2.9){reference-type="ref" reference="Theorem 2.9"} are defined using the cohomological index. It is not clear that this theorem is true for the standard sequence of eigenvalues defined using the genus since the proof of Theorem [Theorem 5](#Theorem 2.6){reference-type="ref" reference="Theorem 2.6"} is based on the piercing property of the cohomological index and the genus does not have this property. [^4]: We think that there is a very small typo in [@MR1945278 Theorem 1]. Namely, the condition $sp<1$ there should read $sp$ less than the dimension, thus allowing the use of this result in our context. See also [@MR1940355 formula (1)]. [^5]: We stress that expressions such as $$\int_{[{{{0}}}, 1]}\left(\;\iint_{\mathbb{R}^{2N}} \frac{c_{N,s,p}\,|u(x) - u(y)|^{p-2}\, (u(x) - u(y))\, (v(x) - v(y))}{|x - y|^{N+sp}}\, dx dy\right) d\mu(s)$$ are a slight abuse of notation. To be precise, one should write instead $$\begin{aligned} && \int_{({{{0}}}, 1)}\left(\;\iint_{\mathbb{R}^{2N}} \frac{c_{N,s,p}\,|u(x) - u(y)|^{p-2}\, (u(x) - u(y))\, (v(x) - v(y))}{|x - y|^{N+sp}}\, dx dy\right) d\mu (s)\\ &&\qquad\qquad+ \mu(0)\int_{\mathbb{R}^N} u(x)\,v(x)\,dx+\mu(1)\int_{\mathbb{R}^N} \nabla u(x)\cdot\nabla v(x)\,dx.\end{aligned}$$ For the sake of shortness, however, we will accept the above abuse of notation whenever typographically convenient.

arxiv_math

--- abstract: | We introduce operator theory on the pentablock. A commuting triple of Hilbert space operators $(A,S,P)$ is said to be a *$\mathbb{P}$-contraction* if the closed pentablock $\overline{\mathbb P}$ is a spectral set for $(A,S,P)$, where $$\mathbb{P}:=\left\{(a_{21}, \mbox{tr}(A_0), \mbox{det}(A_0))\ : \ A_0=[a_{ij}]_{2 \times 2} \; \; \& \;\; \|A_0\| <1 \right\} \subseteq \mathbb{C}^3.$$ A commuting triple of normal operators $(A, S, P)$ acting on a Hilbert space is said to be a *$\mathbb P$-unitary* if the Taylor-joint spectrum $\sigma_T(A, S, P)$ of $(A, S, P)$ is contained in the distinguished boundary $b\mathbb{P}$ of $\overline{\mathbb{P}}$. Also, $(A, S , P)$ is called a *$\mathbb P$-isometry* if it is the restriction of a $\mathbb P$-unitary $(\hat A, \hat S, \hat P)$ to a joint invariant subspace of $\hat A, \hat S, \hat P$. We find several characterizations for the $\mathbb P$-unitaries and $\mathbb P$-isometries. We show that every $\mathbb P$-isometry admits a Wold type decomposition that splits it into a direct sum of a $\mathbb P$-unitary and a pure $\mathbb P$-isometry. Moving one step ahead we show that every $\mathbb P$-contraction $(A,S,P)$ possesses a canonical decomposition that orthogonally decomposes $(A,S,P)$ into a $\mathbb P$-unitary and a completely non-unitary $\mathbb P$-contraction. We find a necessary and sufficient condition such that a $\mathbb P$-contraction $(A, S, P)$ dilates to a $\mathbb P$-isometry $(X, T, V)$ with $V$ being the minimal isometric dilation of $P$. Then we show an explicit construction of such a conditional dilation. A commuting tuple of Hilbert space operators $(T_1, \dots , T_n)$ having the closed unit ball $\overline{\mathbb B}_n$ as a spectral set is called a $\mathbb B_n$-*contraction* and a commuting pair $(S,P)$ having the closed symmetrized bidisc $\overline{\mathbb G_2} \,(=\Gamma)$ as a spectral set is called a $\Gamma$-*contraction*, where $$\begin{aligned} \mathbb B_n & =\{ (w_1,\dots , w_n)\in \mathbb{C}^n \; : \, |w_1|^2+ \dots + |w_n|^2 <1 \}, \\ \mathbb G_2 & = \{ (z_1+z_2,z_1z_2) \in \mathbb{C}^2 \;: \; |z_1|<1, \, |z_2|<1 \}. \end{aligned}$$ We characterize isometries and unitaries associated with $\mathbb B_n$. Then we present an analogous canonical decomposition for a $\mathbb B_n$-contraction. Also, we prove that if $(A,S,P)$ is a $\mathbb P$-contraction then $(A,S\slash 2)$ is a $\mathbb B_2$-contraction and $(S,P)$ is a $\Gamma$-contraction but the converse does not hold. We show interplay between operator theory on the three domains $\mathbb P, \mathbb B_2$ and $\mathbb G_2$. address: - Mathematics Department, Indian Institute of Technology Bombay, Powai, Mumbai - 400076, India. - Mathematics Department, Indian Institute of Technology Bombay, Powai, Mumbai-400076, India. author: - Sourav Pal and Nitin Tomar title: Operators associated with the pentablock and their relations with biball and symmetrized bidisc --- # Introduction {#Intro} Throughout the paper, all operators are bounded linear operators acting on complex Hilbert spaces. We define Taylor joint spectrum, spectral set, distinguished boundary and rational dilation in Section [2](#basic){reference-type="ref" reference="basic"}. In this article, we introduce operator theory on the pentablock $\mathbb P$, a domain related to a special case of $\mu$-synthesis, which is defined by $$\mathbb{P}:=\left\{(a_{21}, \mbox{tr}(A_0), \mbox{det}(A_0))\ : \ A_0=[a_{ij}]_{2 \times 2}, \|A_0\| <1 \right\} \subseteq \mathbb{C}^3.$$ Also, we study operators having the closed polyball $\overline{\mathbb{B}}_n$ as a spectral set and then connect the operator theory of the three domains: $\mathbb{P}, \mathbb{B}_2$ and the symmetrized bidisc $\mathbb G_2$, where $$\begin{aligned} \mathbb B_n & =\{ (w_1,\dots , w_n)\in \mathbb{C}^n \; : \, |w_1|^2+ \dots + |w_n|^2 <1 \}, \\ \mathbb G_2 & = \{ (z_1+z_2,z_1z_2) \in \mathbb{C}^2 \;: \; |z_1|<1, \, |z_2|<1 \}. \end{aligned}$$ In [@Agler], Agler, Lykova and Young introduced the pentablock to study a special case of $\mu$-synthesis. The $\mu$-synthesis is a part of the theory of robust control of systems comprising interconnected electronic devices whose outputs are linearly dependent on the inputs. Given a linear subspace $E$ of $\mathcal M_n(\mathbb C)$, the space of all $n \times n$ complex matrices, the functional $$\mu_E(A_0):= (\text{inf} \{ \|Y \|: Y\in E \text{ and } (I-A_0Y) \text{ is singular } \})^{-1}, \; A_0\in \mathcal M_n(\mathbb C),$$ is called a *structured singular value*, where the linear subspace $E$ is referred to as the *structure*. If $E=\mathcal M_n(\mathbb C)$, then $\mu_E (A_0)$ is equal to the operator norm $\|A_0\|$, while if $E$ is the space of all scalar multiples of the identity matrix, then $\mu_E(A_0)$ is the spectral radius $r(A_0)$. For any linear subspace $E$ of $\mathcal M_n(\mathbb C)$ that contains the identity matrix $I$, $r(A_0)\leq \mu_E(A_0) \leq \|A_0\|$. We refer to the pioneering work of Doyle [@Doyle] on control-theory and motivation behind considering $\mu_E$. Also, for further details on this topic an interested reader can see [@Francis]. Given distinct points $\alpha_1, \dots , \alpha_d \in \mathbb{D}$, the open unit disk in the complex-plane $\mathbb{C}$, and matrices $B_1, \dots , B_d \in \mathcal M_n(\mathbb{C})$, the aim of $\mu$-synthesis is to find an analytic function $F:\,\mathbb{D}\rightarrow \mathcal M_n(\mathbb{C})$ with $\mu_E(F(\lambda))<1$ for all $\lambda \in \mathbb D$ such that $F$ interpolates the given data, i.e. $F(\alpha_i)=B_i$ for $1\leq i \leq d$. The pentablock arises naturally in connection with a special case of $\mu$-synthesis. Indeed, if $$E=\bigg\{ \begin{pmatrix} \lambda & \mu \\ 0 & \lambda \end{pmatrix}\,: \, \lambda , \mu \in \mathbb{C}\bigg\},$$ then $\mu_E(A_0)<1$ for $A_0=[a_{ij}]_{2 \times 2}$ if and only if $(a_{21}, \mbox{tr}(A_0), \mbox{det}(A_0)) \in \mathbb{P}$. Thus, the function $$\pi: \,A_0=[a_{ij}] \mapsto (a_{21}, \mbox{tr}(A_0), \mbox{det}(A_0))$$ maps the $\mu_E$ unit ball onto the pentablock. However, Agler, Lykova and Young refined this result in [@Agler] and showed that the pentablock is the image under $\pi$ of the norm unit ball $\mathbb{B}_{\|.\|}$ in $\mathcal M_2(\mathbb{C})$, which is strictly smaller than the $\mu_E$ unit ball. The pentablock has attracted considerable attentions in recent past from complex geometric and function theoretic aspects [@Alsheri; @Jindal; @Kosinski; @Su; @SuII; @Zapalowski]. In this article, we initiate operator theory on the pentablock. Thus, our primary object of study in this paper is a commuting operator triple that has the closed pentablock as a spectral set. **Definition 1**. A commuting triple of operators $(A,S,P)$ is said to be a $\mathbb{P}$-*contraction* if $\overline{\mathbb{P}}$ is a spectral set for $(A,S,P)$, that is, the Taylor joint spectrum $\sigma_T(A,S,P) \subseteq \overline{\mathbb{P}}$ and von Neumann's inequality holds, i.e., $$\|f(A,S,P) \| \leq \sup_{(z_1,z_2,z_2)\in \overline{\mathbb{P}}} \; |f(z_1,z_2,z_3)| = \|f\|_{\infty, \, \overline{\mathbb{P}}\,},$$ for every rational function $f=p \slash q$, where $p,q$ are polynomials in $\mathbb{C}[z_1,z_2,z_3]$ with $q$ having no zeros in $\overline{\mathbb{P}}$. Similarly, a $\mathbb{B}_n$-*contraction* is a commuting $n$-tuple of operators having $\overline{\mathbb{B}}_n$ as a spectral set and we call $(S,P)$ a $\Gamma$-*contraction* if $\overline{\mathbb{G}}_2\,(=\Gamma)$ is a spectral set for the commuting operator pair $(S,P)$. Unitaries, isometries and co-isometries are special classes of contractions. A unitary is a normal operator having its spectrum on the unit circle $\mathbb{T}$. An isometry is the restriction of a unitary to an invariant subspace and a co-isometry is the adjoint of an isometry. In an analogous manner, we define unitary, isometry and co-isometry associated with the pentablock. **Definition 2**. Let $A,S,P$ be commuting operators acting on a Hilbert space $\mathcal{H}$. Then the triple $(A,S,P)$ is called - a $\mathbb{P}$-*unitary* if $A,S,P$ are normal operators and the Taylor joint spectrum $\sigma_T(A,S,P)$ lies in the distinguished boundary of the pentablock ; - a $\mathbb{P}$-*isometry* if there is a Hilbert space $\mathcal K \supseteq \mathcal{H}$ and a $\mathbb{P}$-unitary $(\hat A, \hat S, \hat P)$ on $\mathcal K$ such that $\mathcal{H}$ is a joint invariant subspace for $\hat A, \hat S, \hat P$ and $\hat A|_{\mathcal{H}}=A, \hat S|_{\mathcal{H}}= S$ and $\hat P|_{\mathcal{H}}=P$ ; - a $\mathbb{P}$-*co-isometry* if $(A^*,S^*,P^*)$ is a $\mathbb{P}$-isometry. Similarly, one defines unitary, isometry and co-isometry for the classes of $\overline{\mathbb{B}}_n$-contractions and $\Gamma$-contractions. Moreover, an isometry (on $\mathcal{H}$) associated with a domain is called *pure* if there is no nonzero proper joint reducing subspace of the isometry on which it acts like a unitary associated with the domain. Amongst the central theorems that constitute the foundation of the one-variable operator theory, the following two results are remarkable: the first is due to von Neumann [@vN], which states that an operator is a contraction if and only if the closed unit disc $\overline{\mathbb{D}}$ is a spectral set for it and the second is Sz.-Nagy's celebrated theorem [@Nagy], which asserts that an operator $T$ on a Hilbert space $\mathcal{H}$ is a contraction if and only if it dilates to a unitary $U$ acting on a Hilbert space $\mathcal K \supseteq \mathcal{H}$, i.e. $T^n = P_{\mathcal{H}}U^n|_{\mathcal{H}}$ for every positive integer $n$, where $P_{\mathcal{H}}$ is the orthogonal projection of $\mathcal K$ onto $\mathcal{H}$. Moreover, such a dilation is called *minimal* if $$\mathcal K = \overline{Span}\; \{ U^nh\,:\, h \in \mathcal{H}, \, n \in \mathbb{Z}\}.$$ Thus, von Neumann's famous result compels to realize a contraction as an operator having $\overline{\mathbb{D}}$ as a spectral set and Sz.-Nagy's dilation theorem paves a way to dilate such an operator to a normal operator having its spectrum on the boundary of $\overline{\mathbb{D}}$. Taking cue from such inspiring classical concepts, one considers operators having other domains as spectral sets and in the same spirit studies if they dilate to normal operators associated with the boundary of the domain. In this article, we first focus on $\mathbb{P}$-unitaries and $\mathbb{P}$-isometries; characterize them in several different ways and decipher their structures in Sections [5](#P-uni){reference-type="ref" reference="P-uni"} & [6](#P-iso){reference-type="ref" reference="P-iso"}. We show that every $\mathbb{P}$-isometry admits a Wold decomposition that splits it into two orthogonal parts of which one is a $\mathbb{P}$-unitary and the other is a pure $\mathbb{P}$-isometry. This is parallel to the Wold decomposition of an isometry into a unitary and a pure isometry. Also, more generally every contraction orthogonally decomposes into a unitary and a completely non-unitary contraction. A completely non-unitary contraction is a contraction that does not have a unitary part. Such a decomposition is called the *canonical decomposition* of a contraction, see Chapter I of [@Nagy] for details. In Section [7](#decomp){reference-type="ref" reference="decomp"}, we show that such a canonical decomposition is possible for a $\mathbb{P}$-contraction. In Section [4](#Polyball){reference-type="ref" reference="Polyball"}, we establish analogues of these results for $\mathbb{B}_n$-contractions. Operators having the (closed) pentablock as a spectral set have close connections with the operators associated with the biball and the symmetrized bidisc. Indeed, in Section [3](#Prelims){reference-type="ref" reference="Prelims"} we show that if $(A,S,P)$ is a $\mathbb{P}$-contraction then $(A, S \slash 2)$ is a $\mathbb{B}_2$-contraction and $(S,P)$ is a $\Gamma$-contraction. However, a converse to this result does not hold and we show it by a counter example. Naturally, operators associated with the symmetrzed bidisc come into the picture while studying $\mathbb{P}$-contractions. Operator theoretic aspects of the symmetrize bidisc have rich literature, e.g. [@AglerII; @AglerVII; @Bhattacharyya; @Tirtha-Sourav1]. An interested reader may also see the references therein. In [@AglerII], Agler and Young profoundly established the success of rational dilation on the symmetrized bidisc and in [@Bhattacharyya], Bhattacharyya, Pal and Shyam Roy explicitly constructed such a dilation. In Section [8](#dilation){reference-type="ref" reference="dilation"}, we mention this dilation theorem. Note that it suffices to find an isometric dilation to a commuting operator tuple associated with a domain, because, by definition every isometry associated with a domain extends to a unitary with respect to the same domain. The explicit $\Gamma$-isometric dilation of a $\Gamma$-contraction from [@Bhattacharyya] motivates us to construct explicitly an isometric dilation for a $\mathbb{P}$-contraction under certain conditions. Still it is unknown if every $\mathbb{P}$-contraction dilates to a $\mathbb{P}$-isometry. The fact that the component $P$ of a $\mathbb{P}$-contraction $(A,S,P)$ is a contraction leads to the possibility of a $\mathbb{P}$-isometric dilation $(X,T,V)$ of $(A,S,P)$, when $V$ is the minimal isometric dilation of $P$. We capitalize this idea in Section [8](#dilation){reference-type="ref" reference="dilation"}. In Theorem [Theorem 61](#thm:main-dilation){reference-type="ref" reference="thm:main-dilation"}, we find a necessary and sufficient condition such that a $\mathbb{P}$-contraction $(A,S,P)$ dilates to a pentablock isometry $(X,T,V)$, where $V$ is the minimal dilation of $P$. Then we explicitly construct such a conditional $\mathbb{P}$-isometric dilation in Theorem [Theorem 62](#lem6.9){reference-type="ref" reference="lem6.9"}. The following operator equation in $Z$ associated with a $\mathbb{P}$-contraction $(A,S,P)$ plays an important role in these dilation theorems: $$I-A^*A-\frac{1}{4}S^*S=D_P\big(Z^*Z+\frac{1}{4}FF^*\big)D_P,$$ where $D_P=(I-P^*P)^{1\slash 2}$ and $F$ satisfies $S-S^*P=D_PFD_P$. In Section [8](#dilation){reference-type="ref" reference="dilation"}, we also find a necessary and sufficient condition such that the above operator equation has a solution and prove that such a solution when exists is unique. At every stage of this paper, we explore and find interaction of a $\mathbb{P}$-contraction with $\mathbb{B}$-contractions and $\Gamma$-contractions. # Preliminaries {#basic} In this Section, we recall from the literature a few basic facts that are necessary in the contex of the results of this paper. We begin with the definition of the Taylor joint spectrum of a tuple of commuting operators. ## The Taylor joint spectrum Let $E$ be the exterior algebra on $n$ generators $\nu_1, \dotsc, \nu_n$ with identity $\nu_0 \equiv 1$. $\Lambda$ is the algebra of forms in $\nu_1, \dotsc, \nu_n$ with complex coefficients, subject to the property $\nu_i\nu_j+\nu_j\nu_i=0 \ (1 \leq i,j \leq n).$ Let $\Delta_i: \Lambda \to \Lambda$ denote the operator, given by $\Delta_i\xi=\xi \ (\xi \in \Lambda, 1 \leq i \leq n).$ If we declare $\{\nu_{i_1} \dotsc \nu_{i_k} : \ 1 \leq i_1 < \dotsc < i_k \leq n \}$ to be an orthogonal basis, the exterior algebra $E$ becomes a Hilbert space, admitting an orthogonal decomposition $\Lambda=\overset{n}{\underset{k=1}{\oplus}} \Lambda^k$ where dim$\Lambda^k=\begin{pmatrix} n\\ k\\ \end{pmatrix}.$ Thus, each $\xi \in \Lambda$ admits a unique orthogonal decomposition $\xi=\nu_i\xi'+\xi''$, where $\xi'$ and $\xi''$ have no $\nu_i$ contribution. It then follows that $\Delta_i^*\xi=\xi',$ and we have that each $\Delta_i$ is a partial isometry, satisfying $\Delta_i^*\Delta_j+\Delta_j^*\Delta_i=\delta_{ij}$. Let $Y$ be a normed space and $\underline{A}=(A_1, \dotsc, A_n)$ be a commuting $n$-tuple of bounded operators on $Y$ and set $\Lambda(Y)=Y\otimes_\mathbb{C}\Lambda.$ We define $D_{\underline{A}}: \Lambda(Y) \to \Lambda(Y)$ by $D_{\underline{A}}=\overset{n}{\underset{i=1}{\sum}}A_i \otimes \Delta_i.$ Then it is easy to see that $D_{\underline{A}}^2=0,$ so $Ran D_{\underline{A}} \subset KerD_{\underline{A}}.$ The commuting $n$-tuple is said to be *non-singular* on $Y$ if $RanD_{\underline{A}} = KerD_{\underline{A}}.$ **Definition 3**. The Taylor joint spectrum of $\underline{A}$ on $Y$ is the set $\sigma_T(\underline{A}, Y)=\{\mu=(\mu_1, \dotsc, \mu_n) \in \mathbb{C}^n \ : \ \ \underline{A}-\mu \ \mbox{is singular}\}.$ $$ For a further reading on Taylor joint spectrum, reader is referred to Taylor's works [@Taylor] and [@TaylorII]. ## The distinguished boundary For a compact subset $X$ of $\mathbb{C}^n$, let $A(X)$ be the algebra of continuous complex-valued functions on $X$ that are holomorphic in the interior of $X$. A *boundary* for $X$ is a closed subset $C$ of $X$ such that every function in $A(X)$ attains its maximum modulus on $C$. It follows from the theory of uniform algebras that the intersection of all the boundaries of $X$ is also a boundary of $X$ and it is the smallest among all boundaries. This is called the *distinguished boundary* of $X$ and is denoted by $bX$. For a bounded domain $\Omega \subset \mathbb{C}^n$, we denote by $b\Omega$ the distinguished boundary of $\overline{\Omega}$ and for the sake of simplicity we call it the distinguished boundary of $\Omega$. ## Spectral set, complete spectral set and rational dilation Let $X$ be a compact subset of $\mathbb{C}^n$ and $\mathcal R (X)$ be the algebra of rational functions $p\slash q$, where $p,q \in \mathbb{C}[z_1, \dots , z_n]$ such that $q$ does not have any zeros in $X$. Let $\underline{A}=(A_1, \dotsc, A_n)$ be a commuting tuple of operators acting on a Hilbert space $\mathcal{H}$. Then $X$ is said to be a *spectral set* for $\underline{A}$ if the Taylor joint spectrum of $\underline{A}$ is contained in $X$ and von Neumann's inequality holds for any $g \in \mathcal R(X)$, i.e. $$\|g(\underline{A})\| \leq \sup_{x \in X} \;|g(x)|=\|g\|_{\infty, \, X},$$ where $g(\underline{A})=p(\underline{A})q(\underline{A})^{-1}$ when $g=p \slash q$. Also, $X$ is said to be a *complete spectral set* if for any $g=[g_{ij}]_{m\times m}$, where each $g_{ij}\in \mathcal R(X)$, we have $$\|g(\underline{A})\|=\|\,[g_{ij}(\underline{A})]_{m\times m} \,\| \leq \sup_{x \in X} \;\| \, [g_{ij}(x)]_{m\times m} \,\|.$$ A commuting $n$-tuple of operators $\underline{A}$ having $X$ as a spectral set, is said to have a *rational dilation* or *normal b$X$-dilation* if there exist a Hilbert space $\mathcal{K}$, an isometry $V: \mathcal{H} \to \mathcal{K}$ and a commuting $n$-tuple of normal operators $\underline{B}=(B_1, \dots , B_n)$ on $\mathcal{K}$ with $\sigma_T(\underline{B}) \subseteq bX$ such that $g(\underline{A})=V^*g(\underline{B})V \quad \text{ for all }\; g \in \mathcal R(X)$  . In other words, $g(\underline{A})=P_\mathcal{H}g(\underline{B})|_\mathcal{H}$ for every $g \in \mathcal R(X)$ when $\mathcal{H}$ is realized as a closed subspace of $\mathcal{K}$. # The $\mathbb{P}$-contractions, $\mathbb{B}_2$-contractions and $\Gamma$-contractions {#Prelims} Recall that a $\mathbb{P}$-contraction is a commtuing operator triple that has the closed pentablock $\overline{\mathbb{P}}$ as a spectral set. Similarly, for a commuting operator pair if the closed biball $\overline{\mathbb{B}}_2$ or the closed symmetrized bidisc $\Gamma$ is a spectral set, then it is called a $\mathbb{B}_2$-contraction or a $\Gamma$-contraction respectively. In this Section, we present a few basic results on $\mathbb{P}$-contractions and explore their interactions with $\mathbb{B}_2$-contractions and $\Gamma$-contractions. We begin with an elementary proposition which ensures that the Taylor spectrum condition can be dropped from the definition of spectral set if the underlying compact set is polynomially convex. We give a short proof to this result here for the convenience of a reader. However, a proof can also be found in the literature. **Proposition 4**. *A polynomially convex set $X \subset \mathbb{C}^n$ is a spectral set for a commuting tuple of operators $(T_1,\dots, ,T_n)$ if and only if $$\label{eq-new1} \|p(T_1, \dots , T_n)\| \leq \|p\|_{\infty,\, X}$$ for every polynomial $p$ in $\mathbb{C}[z_1, \dots , z_n]$.* *Proof.* If $X$ is a spectral set for $(T_1, \dots, T_n)$ then the desired von Neumann's inequality ([\[eq-new1\]](#eq-new1){reference-type="ref" reference="eq-new1"}) holds for all polynomials and this is a part of the definition of spectral set. So, we prove the converse here. Suppose von Neumann's inequality ([\[eq-new1\]](#eq-new1){reference-type="ref" reference="eq-new1"}) holds for all polynomials in $\mathbb{C}[z_1, \dots, z_n]$. If the Taylor spectrum $\sigma_T(T_1,\dots, T_n)$ is not contained in $X$, then there is a point say $(w_1, \dots, w_n)$ in $\sigma_T(T_1, \dots , T_n)$ which is not in $X$. Since $X$ is polynomially convex, there is a polynomial $q$ such that $|q(z_1, \dots ,z_n)| > \|q\|_{\infty, X}$. The spectral mapping theorem for polynomials yields that $$\sigma_T(q(T_1, \dots , T_n))=\{q(z_1,\dots , z_n) \ : \ (z_1, \dots , z_n) \in \sigma_T(T_1, \dots , T_n) \},$$ and it follows that the spectral radius of $q(T_1, \dots, T_n)$, i.e. $r(q(T_1, \dots, T_n))$ is bigger than $\|q\|_{\infty,\, X}$. Then, $\|q(T_1, \dots , T_n)\| \geq r(q(T_1, \dots, T_n)) > \|q\|_{\infty, \, X}$, contradicting the fact that $X$ is a spectral set for $(T_1, \dots, T_n)$. Therefore, $\sigma_T(T_1, \dots, T_n) \subseteq X$. Since $X$ is polynomially convex, by Oka-Weil theorem (Theorem 5.1 in [@Gamelin]), any rational function with singularities outside $X$ can be uniformly approximated over $X$ by a sequence of polynomials from $\mathbb{C}[z_1, \dots, z_n]$. In fact such polynomial approximation is possible for any function that is holomorphic in a neighbourhood of $X$. So, for any rational function $f$ with singularities off $X$, there is a sequence of holomorphic polynomials $\{p_n\}$ such that $\|f-p_n\|_{ \infty, \,X} \to 0$ as $n \to \infty$. Then $p_n(T_1, \dots , T_n) \to f(T_1, \dots, T_n)$ in operator norm as $n \to \infty$. Since each $p_n$ satisfies ([\[eq-new1\]](#eq-new1){reference-type="ref" reference="eq-new1"}), we have that $$\|f(T_1, \dots , T_n)\| = \lim_{n \to \infty}\|p_n(T_1, \dots, T_n)\| \leq \|f\|_{\infty,\, X}.$$ This completes the proof. ◻ Interestingly, the Taylor spectrum condition in the definition of spectral set suffices to produce the von Neumann's inequality once we deal with commuting normal operators as the following proposition explains. **Proposition 5**. *A compact subset $X$ of $\mathbb{C}^n$ is a spectral set for a commuting tuple of normal operators $\underline{N}=(N_1, \dots , N_n)$ if and only if $\sigma_T(N_1, \dots , N_n) \subseteq X$.* *Proof.* If $X$ is a spectral set of $\underline{N}$, then it follows from the definition of spectral set that $\sigma_T(\underline{N})$ is a subset of $X$. Conversely, assume that $\sigma_T(\underline{N}) \subseteq X$ and take any $f=p\slash q \in \mathcal{R}(X)$, the rational algebra on $X$. Then $f(\underline{N})=p(\underline{N})q(\underline{N})^{-1}$ and so, $f(\underline{N})$ is a normal operator. Since the norm of a normal operator is equal to its spectral radius, we have $$\begin{split} \|f(\underline{N})\| =\sup\{|\lambda| : \lambda \in \sigma_T(f(\underline{N})) \} &=\sup\{|\lambda| : \lambda \in f(\sigma_T(\underline{N})) \} \quad [\text{Spectral mapping theorem}]\\ &=\sup\{|f(x)| : x \in \sigma_T(\underline{N}) \}\\ &\leq \sup\{|f(x)| : x \in X \},\\ \end{split}$$ where the last inequality follows from the fact that $\sigma_T(\underline{N}) \subseteq X$. Therefore, $X$ is a spectral set for $\underline{N}$ and the proof is complete. ◻ Since the closed pentablock $\overline{\mathbb{P}}$ is polynomially convex (see [@Agler]), we have the following lemmas as a consequence of Proposition [Proposition 4](#basicprop:01){reference-type="ref" reference="basicprop:01"}. **Lemma 6**. *A commuting triple of operators $(A, S, P)$ is a $\mathbb{P}$-contraction if and only if $$\label{eq} \|f(A, S, P)\| \leq \|f\|_{\infty, \overline{\mathbb{P}}}$$ for every polynomial $f$ in $\mathbb{C}[z_1,z_2,z_3]$.* We obtain the following result by an application of Lemma [Lemma 6](#lem2.2){reference-type="ref" reference="lem2.2"}. **Lemma 7**. *Let $(A,S,P)$ on a Hilbert space $\mathcal{H}$ be a $\mathbb{P}$-contraction. Then* - *$(A^*,S^*,P^*)$ is a $\mathbb{P}$-contraction ;* - *$(A|_{\mathcal{L}}, S|_{\mathcal{L}} , P|_{\mathcal{L}})$ is a $\mathbb{P}$-contraction for any joint invariant subspace $\mathcal L \subseteq \mathcal{H}$ of $A,S,P$.* *Proof.* $(i)$ Given a polynomial $f(z_1, z_2, z_3)=\underset{0 \leq i,j,k \leq m}{\sum}a_{ijk}z_1^iz_2^jz_3^k$, we define another polynomial $$\hat{f}(z_1, z_2, z_3)=\underset{0 \leq i,j,k \leq m}{\sum}\overline{a_{ijk}}z_1^iz_2^jz_3^k.$$ For any $\mathbb{P}$-contraction $(A, S, P)$, we have $$\|f(A^*, S^*, P^*)\| =\|\hat{f}(A, S, P)^*\|=\|\hat{f}(A, S, P)\| \leq \|\hat{f}\|_{\infty,\; \overline{\mathbb{P}}}.$$ Let $(a, s, p) \in \overline{\mathbb{P}}$ be a point at which $\hat f$ attains its maximum modulus value. Since for every $(z_1, z_2, z_3) \in \overline{\mathbb{P}}$, the conjugate triple $(\overline{z_1}, \overline{z_2}, \overline{z_3}) \in \overline{\mathbb{P}}$, we get that $$\|\hat f\|_{\infty, \overline{\mathbb{P}}}=|\hat f(A, S, P)|=|f(\overline{a}, \overline{s}, \overline{p})| \leq \|f\|_{\infty,\; \overline{\mathbb{P}}}.$$ Consequently, it follows that $$\|f(A^*, S^*, P^*)\| \leq \|f\|_{\infty,\; \overline{\mathbb{P}}},$$ for every polynomial $f$ in three variables. Hence, $(A^*, S^*, P^*)$ is a $\mathbb{P}$-contraction if $(A, S, P)$ is a $\mathbb{P}$-contraction. $(ii)$. Let $\mathcal{L}$ be a joint invariant subspace of a $\mathbb{P}$-contraction $(A, S, P)$ and let $f$ is any polynomial in three variables. Then $$\|f(A|_{\mathcal{L}}, S|_{\mathcal{L}} , P|_{\mathcal{L}})\|= \|f(A, S, P)|_{\mathcal{L}}\| \leq \|f(A, S, P)\| \leq \|f\|_{\infty,\; \overline{\mathbb{P}}}$$ and the proof is complete. ◻ Now we move forward to explore relations of $\mathbb{P}$ with the biball $\mathbb{B}_2$ and the symmetrized bidisc $\mathbb{G}_2$, which result in interplay between $\mathbb{P}$-contractions, $\mathbb{B}_2$-contractions and $\Gamma$-contractions. We start with a couple of useful results from the literature, of which the first is due to Agler and Young [@AglerVII]. **Theorem 8** ([@AglerVII], Theorems 2.2 & 2.6). *Let $S, P$ be commuting operators on a Hilbert space $\mathcal{H}$. Then* 1. *$(S, P)$ is a $\Gamma$-unitary if and only if $S=S^*P$, $P$ is unitary and $\|S\| \leq 2$.* 2. *$(S, P)$ is a $\Gamma$-isometry if and only if $S=S^*P$, $P$ is isometry and $\|S\| \leq 2$.* The next result is due to Agler, Lykova nad Young from [@Agler] which characterizes the points in $\overline{\mathbb{P}}$ in several ways. **Theorem 9** ([@Agler], Theorem 5.3). *Let $$(s,p)=(\beta+\overline{\beta}p, p)=(\lambda_1+\lambda_2, \lambda_1\lambda_2) \in \Gamma$$ where $\lambda_1, \lambda_2 \in \overline{\mathbb{D}}$ and $|\beta| \leq 1$. If $|p|=1$, then $\beta=\frac{1}{2}s$. Let $a \in \mathbb{C}$. The following statements are equivalent:* 1. *$(a, s, p) \in \overline{\mathbb{P}}$;* 2. *$|a| \leq \bigg|1-\frac{\frac{1}{2}s\overline{\beta}}{1+\sqrt{1-|\beta|^2}} \bigg|$;* 3. *$|a| \leq \frac{1}{2}|1-\overline{\lambda}_2\lambda_1|+\frac{1}{2}(1-|\lambda_1|^2)^{\frac{1}{2}}(1-|\lambda_2|^2)^{\frac{1}{2}}$;* 4. *there exists $A_0 \in M_2(\mathbb{C})$ such that $\|A_0\| \leq 1$ and $\pi(A_0)=(a, s, p)$.* It is evident from the above theorem that if $(a, s, p)$ is an element in $\overline{\mathbb{P}}$, then $(s, p) \in \Gamma$. Also, the same holds if we consider $\mathbb{P}$ and $\mathbb{G}$ (see [@Agler]). Indeed, if $(a, s, p) \in \overline{\mathbb{P}}$, then there is a $2 \times 2$ contraction $A_0=[a_{ij}]$ such that $a_{21}=a, \quad tr(A_0)=s \quad \text{and} \quad det(A_0)=p$. Therefore, $(s,p) \in \Gamma$ as we have from [@AglerII] that $\Gamma=\{(tr(A_0), det(A_0)) \in \mathbb{C}^2 \ : \ A_0 \in M_2(\mathbb{C}), \ \|A_0\| \leq 1 \}$. So, we have the following lemma which can also be found in [@Agler]. **Lemma 10**. *If $(a,s,p) \in \overline{\mathbb{P}}$ (or $\in \mathbb{P}$), then $(s,P) \in \Gamma$ (or $\in \mathbb{G}$).* This scalar result naturally extends to the following operator theoretic version. **Proposition 11**. *If $(A, S, P)$ is a $\mathbb{P}$-contraction, then $(S, P)$ is a $\Gamma$-contraction.* *Proof.* Let $(a, s, p) \in \overline{\mathbb{P}}$ be any point. By Theorem [Theorem 9](#thm2.1){reference-type="ref" reference="thm2.1"}, there is a $2 \times 2$ matrix $A_0=[a_{ij}]$ with $\|A_0\| \leq 1$ such that $\pi(A_0)=(a, s, p)$. Evidently, $|a|=|a_{21}| \leq 1$ and $(s, p)=(tr(A_0), det(A_0))$. Thus, $a \in \overline{\mathbb{D}}$ and $(s, p) \in \Gamma$. Therefore, we have that $\overline{\mathbb{P}}\subseteq \overline{\mathbb{D}}\times \Gamma$. The set $\Gamma$ is polynomially convex and thus it follows from Proposition [Proposition 4](#basicprop:01){reference-type="ref" reference="basicprop:01"} that $(S, P)$ is a $\Gamma$-contraction if and only if $$\|g(S, P)\| \leq \|g\|_{\infty,\; \Gamma}=\sup\{|g(z_2, z_3)| \ : \ (z_2, z_3) \in \Gamma \}$$ for every holomorphic polynomial $g$ in two variables. For a polynomial $g \in \mathbb{C}[]z_1,z_2$, let us set $f(z_1, z_2, z_3)=g(z_2, z_3)$. Using the fact that $\overline{\mathbb{P}} \subseteq \overline{\mathbb{D}} \times \Gamma$, we have $$\begin{split} \|g(S, P)\|=\|f(A, S, P)\| & \leq \sup\{|f(z_1, z_2, z_3)| \ : \ (z_1, z_2, z_3) \in \overline{\mathbb{P}} \}\\ & \leq \sup\{|f(z_1, z_2, z_3)| \ : \ z_1 \in \overline{\mathbb{D}}, \ (z_2, z_3) \in \Gamma \}\\ &=\sup\{|g(z_2, z_3)| \ : \ (z_2, z_3) \in \Gamma \}.\\ \end{split}$$ Therefore, $(S, P)$ is a $\Gamma$-contraction. ◻ It is not difficult to see that if $(a, s, p) \in \overline{\mathbb{P}}$ and $\alpha \in \overline{\mathbb{D}}$, then $(\alpha a, \alpha s, \alpha^2 p) \in \overline{\mathbb{P}}$ and thus *$\overline{\mathbb{P}}$ is $(1,1,2)$-quasi-balanced*. See Section 6 in [@Agler] for a detailed proof of this. The next proposition generalizes this result to $\mathbb{P}$-contractions. **Proposition 12**. *Let $(A, S, P)$ be a $\mathbb{P}$-contraction on a Hilbert space $\mathcal{H}$. Then $(\alpha A, \alpha S, \alpha^2 P)$ is a $\mathbb{P}$-contraction for every $\alpha \in \overline{\mathbb{D}}$.* *Proof.* We have that $\overline{\mathbb{P}}$ is $(1, 1, 2)$-quasi-balanced. So, for any $\alpha \in \overline{\mathbb{D}}$, the map $f_\alpha : \overline{\mathbb{P}}\to \overline{\mathbb{P}}$ defined by $f_\alpha(a, s, p)=(\alpha a, \alpha s, \alpha^2 p)$ is analytic. For any holomorphic polynomial $g$ in $3$-variables, we have that $$\begin{split} \|g(\alpha A, \alpha S, \alpha^2 P)\|& =\|g\circ f_{\alpha}(A, S, P)\| \\ & \leq \|g\circ f_\alpha\|_{\infty, \overline{\mathbb{P}}}\\ & =\sup\left\{|g(\alpha a, \alpha s, \alpha^2 p)| \ : \ (a, s, p) \in \overline{\mathbb{P}}\right\}\\ & \leq \|g\|_{\infty, \overline{\mathbb{P}}} \ . \end{split}$$ It follows from Lemma [Lemma 6](#lem2.2){reference-type="ref" reference="lem2.2"} that $(\alpha A, \alpha S, \alpha^2 P)$ is a $\mathbb{P}$-contraction. ◻ It is evident from Theorem [Theorem 9](#thm2.1){reference-type="ref" reference="thm2.1"} that $(\alpha a, s, p) \in \overline{\mathbb{P}}$ for every $(a, s, p) \in \overline{\mathbb{P}}$ and $\alpha \in \overline{\mathbb{D}}$. Thus, by an application of a similar idea as in the previous proposition, we arrive at the following result. **Proposition 13**. *Let $(A, S, P)$ be a $\mathbb{P}$-contraction and $\alpha \in \overline{\mathbb{D}}$. Then $(\alpha A, S, P)$ is a $\mathbb{P}$-contraction.* *Proof.* For any $\alpha \in \overline{\mathbb{D}}$, the map $\phi_\alpha : \overline{\mathbb{P}}\to \overline{\mathbb{P}}$ defined by $\phi_\alpha(a, s, p)=(\alpha a, s, p)$ is analytic. For any polynomial $q(z_1,z_2,z_3)$, we have $$\begin{split} \|q(\alpha A, S, P)\| =\|g\circ \phi_{\alpha}(A, S, P)\| \leq \|q\circ \phi_\alpha\|_{\infty, \overline{\mathbb{P}}} =\sup\left\{|q(\alpha a, s, p)| \ : \ (a, s, p) \in \overline{\mathbb{P}}\right\} \leq \|q\|_{\infty, \overline{\mathbb{P}}} \ . \end{split}$$ Using Lemma [Lemma 6](#lem2.2){reference-type="ref" reference="lem2.2"}, we get that $(\alpha A, S, P)$ is a $\mathbb{P}$-contraction. ◻ The following observation provides a way to construct $\mathbb{P}$-contraction from a given $\Gamma$-contraction. **Proposition 14**. *$(S, P)$ is a $\Gamma$-contraction if and only if $(0, S, P)$ is a $\mathbb{P}$-contraction.* *Proof.* Assume that $(S, P)$ is a $\Gamma$-contraction. Let $f$ be a holomorphic polynomial in $3$-variables and let $g(z_2, z_3)=f(0, z_2, z_3)$. Then $$\begin{split} \|f(0, S, P)\| & =\|g(S, P)\|\\ & \leq \sup\{|g(z_2, z_3)| \ : \ (z_2, z_3) \in \Gamma \} \qquad \text{[since $\Gamma$ is a spectral set for $(S, P)$]}\\ & =\sup \{|f(z_1, z_2, z_3)| \ : \ z_1=0, \ (z_2, z_3) \in \Gamma \} \\ & \leq \sup \{|f(z_1, z_2, z_3)| \ : \ (z_1, z_2, z_3) \in \overline{\mathbb{P}}\}. \\ \end{split}$$ The last inequality follows from the fact that for any $(s, p) \in \Gamma$ the point $(0, s, p) \in \overline{\mathbb{P}}$ and this is a consequence of Theorem [Theorem 9](#thm2.1){reference-type="ref" reference="thm2.1"}. It follows from Lemma [Lemma 6](#lem2.2){reference-type="ref" reference="lem2.2"} that $(0, S, P)$ is a $\mathbb{P}$-contraction. The converse part follows from Proposition [Proposition 11](#lem2.3){reference-type="ref" reference="lem2.3"}. ◻ **Proposition 15**. *A pair $(T, T')$ of operators acting on a Hilbert space $\mathcal{H}$ is a commuting pair of contractions if and only if $(T, 0, T')$ is a $\mathbb{P}$-contraction on $\mathcal{H}$.* *Proof.* We have shown in the proof of Proposition [Proposition 11](#lem2.3){reference-type="ref" reference="lem2.3"} that $\overline{\mathbb{P}}\subseteq \overline{\mathbb{D}}\times \Gamma$. Since $\Gamma \subseteq 2\overline{\mathbb{D}}\times \overline{\mathbb{D}}$, we have that $\overline{\mathbb{P}}\subseteq \overline{\mathbb{D}}\times 2\overline{\mathbb{D}}\times \overline{\mathbb{D}}$. Let $f_1(z_1, z_2, z_3)=z_1$ and $f_3(z_1, z_2, z_3)=z_3$. So, if $(T, 0, T')$ is a $\mathbb{P}$-contraction, then for $j=1,3$ we have $$\begin{split} \|f_j(T, 0, T')\| & \leq \sup\{|f_j(z_1, z_2, z_3)| \ : \ (z_1, z_2, z_3) \in \overline{\mathbb{P}}\}\\ & \leq \sup\{|f_j(z_1, z_2, z_3)| \ : \ (z_1, z_2, z_3) \in \overline{\mathbb{D}}\times 2 \overline{\mathbb{D}}\times \overline{\mathbb{D}}\}\\ & \leq 1. \end{split}$$ Therefore, $\|T\|, \|T'\| \leq 1$. Conversely, let us assume that $(T, T')$ is a commuting pair of contractions. Then it follows from Ando's inequality [@Nagy] that $$\label{Ando} \|p(T, T')\| \leq \|p\|_{\infty, \overline{\mathbb{D}}^2},$$ for every holomorphic polynomial $p$ in $2$-variables. An application of Theorem [Theorem 9](#thm2.1){reference-type="ref" reference="thm2.1"} gives $\overline{\mathbb{D}}\times \{0\} \times \overline{\mathbb{D}}\subseteq \overline{\mathbb{P}}$. Let $f$ be a holomorphic polynomial in $3$-variables and let $g(z, w)=f(z, 0, w)$. Since $\overline{\mathbb{D}}$ is a spectral set for $T$, we have that $$\begin{split} \|f(T, 0, T')\| & =\|g(T, T')\|\\ & \leq \sup\{|g(z_1, z_3)| \ : \ z_1, z_3 \in \overline{\mathbb{D}}\} \quad \text{[by (\ref{Ando})]}\\ & =\sup \{|f(z_1, z_2, z_3)| \ : \ (z_1, 0, z_3) \in \overline{\mathbb{D}}\times \{0\} \times \overline{\mathbb{D}}\} \\ & \leq \sup \{|f(z_1, z_2, z_3)| \ : \ (z_1, z_2, z_3) \in \overline{\mathbb{P}}\}. \\ \end{split}$$ It follows from Lemma [Lemma 6](#lem2.2){reference-type="ref" reference="lem2.2"} that $(T, 0, T')$ is a $\mathbb{P}$-contraction. ◻ We now show interplay between the pentablock and the Euclidean unit ball $\mathbb{B}_2$ in $\mathbb{C}^2$. **Lemma 16**. *If $(a, s, p) \in \overline{\mathbb{P}}$, then $\left(a, s\slash 2 \right) \in \overline{\mathbb{B}}_2$.* *Proof.* Let $(a, s, p) \in \overline{\mathbb{P}}$. Then it follows from Theorem [Theorem 9](#thm2.1){reference-type="ref" reference="thm2.1"} that there is a $2 \times 2$ matrix $A_0=[a_{ij}]$ such that $\|A_0\| \leq 1$ and $a_{21}=a, \quad a_{11}+a_{22}=s \quad \text{and} \quad a_{11}a_{22}-a_{12}a_{21}=p$. Let us assume that $|a_{11}| \leq |a_{22}|$. Also, let $e_1=(1, 0)$ and $e_2=(0, 1)$ in $\mathbb{C}^2$. Then $$\begin{split} |a|^2+\frac{|s|^2}{4} & =|a_{21}|^2+\frac{1}{4}|a_{11}+a_{22}|^2 \\ &=|a_{21}|^2+\frac{1}{4}\left(|a_{11}|^2+|a_{22}|^2+2Re \ (\overline{a}_{11}a_{22})\right)\\ &\leq |a_{21}|^2+\frac{1}{4}\left(|a_{11}|^2+|a_{22}|^2+ 2|a_{11}a_{22}|\right)\\ &\leq |a_{21}|^2+|a_{22}|^2\\ &=\|A^*e_2 \|^2 \\ & \leq 1.\\ \end{split}$$ Similarly, one can prove that $|a|^2 + \frac{1}{4}|s|^2 \leq 1$ when $|a_{22}| \leq |a_{11}|$. The proof is complete. ◻ As expected, this result has an operator theoretic extension, which is given below. **Proposition 17**. *If $(A, S, P)$ is a $\mathbb{P}$-contraction, then $\left(A, \ S \slash 2\right)$ is a $\mathbb{B}_2$-contraction.* *Proof.* By Lemma [Lemma 16](#lem2.10){reference-type="ref" reference="lem2.10"}, the map $f: \overline{\mathbb{P}}\to \overline{\mathbb{B}}_2$ defined by $f(a, s, p)=(a, s\slash 2)$ is well-defined and analytic on $\overline{\mathbb{P}}$. Now for any $g\in \mathbb{C}[z_1,z_2]$, we have $$\begin{split} \|g(A, S\slash 2)\|& =\|g\circ f(A, S, P)\| \\ & \leq \|g\circ f\|_{\infty, \overline{\mathbb{P}}}\\ & =\sup\left\{|g(a, s \slash 2)| \ : \ (a, s, p) \in \overline{\mathbb{P}}\right\}\\ & \leq \sup\left\{|g(a, s \slash 2)| \ : \ (a, s\slash 2) \in \overline{\mathbb{B}}_2\right\}\\ & = \|g\|_{\infty, \overline{\mathbb{B}}_2} \ . \end{split}$$ Since $\overline{\mathbb{B}}_2$ is a polynomially convex set, Proposition [Proposition 4](#basicprop:01){reference-type="ref" reference="basicprop:01"} shows that $\overline{\mathbb{B}}_2$ is a spectral set for $(A, S \slash 2)$. ◻ Putting together everything we obtain the following theorem, which is a main result of this Section. **Theorem 18**. *If $(A, S, P)$ is a $\mathbb{P}$-contraction, then* 1. *$(A, S\slash 2)$ is a $\mathbb{B}_2$-contraction;* 2. *$(S, P)$ is a $\Gamma$-contraction;* 3. *$(A, P)$ is a commuting pair of contractions.* The converse of Theorem [Theorem 18](#lem2.12){reference-type="ref" reference="lem2.12"} is not true. Indeed, below we show that there exist $a, s$ and $p$ in $\overline{\mathbb{D}}$ such that $(a, s\slash 2) \in \overline{\mathbb{B}}_2, (s, p) \in \Gamma$ but $(a, s, p) \notin \overline{\mathbb{P}}$. **Example 19**. Let $\lambda_1=1$ and $\lambda_2=0$ in $\overline{\mathbb{D}}$. Then it follows from the definition of $\Gamma$ that $(s, p)=(\lambda_1+\lambda_2, \lambda_1\lambda_2)=(1, 0) \in \Gamma$. Let $(a, s, p)=\left(\sqrt{3} \slash 2, 1, 0 \right)$. Then obviously $|a|^2+\frac{1}{4}|s|^2=1$. Thus, $(a, s \slash 2) \in \overline{\mathbb{B}}_2$. Let if possible, $(a, s, p) \in \overline{\mathbb{P}}$. Then, by Theorem [Theorem 9](#thm2.1){reference-type="ref" reference="thm2.1"}, we must have $$\frac{\sqrt{3}}{2}=|a| \leq \frac{1}{2}|1-\overline{\lambda}_2\lambda_1|+\frac{1}{2}(1-|\lambda_1|^2)^{\frac{1}{2}}(1-|\lambda_2|^2)^{\frac{1}{2}}= \frac{1}{2},$$ which is a contradiction. Thus, $(a, s, p) \notin \overline{\mathbb{P}}$. One naturally asks if there is $p \in \overline{\mathbb{D}}$ such that $(a, s\slash 2) \in \overline{\mathbb{B}}_2$ implies that $(a, s, p) \in \overline{\mathbb{P}}$. Indeed, existence of such a $p$ is guaranteed by the next lemma. **Lemma 20**. *$(a, s\slash 2) \in \overline{\mathbb{B}}_2$ if and only if there exists $p \in \mathbb{T}$ such that $(a, s, p) \in \overline{\mathbb{P}}$.* *Proof.* Let $(a, s\slash 2) \in \overline{\mathbb{B}}_2$. Then $|a| \leq 1$ and $|s| \leq 2$. If $s=0$, then it follows from Theorem [Theorem 9](#thm2.1){reference-type="ref" reference="thm2.1"} that $(a, 0, p) \in \overline{\mathbb{P}}$ for any $p \in \overline{\mathbb{D}}$. If $s \ne 0$, we choose $p=s\slash \overline{s}$. Then $(s, p) \in b\Gamma$ by Theorem [Theorem 8](#Gamma_uni){reference-type="ref" reference="Gamma_uni"}, as $|p|=1, s=\overline{s}p$ and $|s| \leq 2$. Furthermore, $(a, s \slash 2) \in \overline{\mathbb{B}}_2$, we have that $$\begin{split} |a| \leq \bigg|1-\frac{\frac{1}{4}s^2}{1+\sqrt{1-\frac{1}{4}|s|^2}} \bigg|=\sqrt{1-\frac{1}{4}|s|^2}. \end{split}$$ Hence, by Theorem [Theorem 9](#thm2.1){reference-type="ref" reference="thm2.1"}, $(a, s, p) \in \overline{\mathbb{P}}$. The converse part follows from Lemma [Lemma 16](#lem2.10){reference-type="ref" reference="lem2.10"}. ◻ Lemma [Lemma 20](#basiclem:03){reference-type="ref" reference="basiclem:03"} can be extended to the class of $\mathbb{P}$-contractions consisting of normal operators. Such a $\mathbb{P}$-contraction is called a *normal $\mathbb{P}$-contraction*. To obtain the desired conclusion, we use the polar decomposition of normal operators. The *polar decomposition theorem* (Theorem 12.35 in [@Rudin]) states that if $N$ is a normal operator on a Hilbert space $\mathcal{H}$, then there exists a unitary operator $U$ on $\mathcal{H}$ such that $U$ commutes with any operator that commutes with $N$ and $N=U(N^*N)^{1 \slash 2}=(N^*N)^{1 \slash 2}U$. Set $p(\lambda)=|\lambda|, \ u(\lambda)=\frac{\lambda}{|\lambda|}$ if $\lambda \ne 0$ and $u(0)=1$. Then $p$ and $u$ are bounded Borel functions on $\sigma(N)$. Define $P=p(N)$ and $U=u(N)$. Since $u\overline{u}=1, UU^*=U^*U=I$. Since $\lambda=u(\lambda)p(\lambda)$, the conclusion $N=UP$ follows from the Borel functional calculus. Furthermore, $P$ is a positive operator that satisfies $\|Px\|=\|Nx\|$ for every $x \in \mathcal{H}$ and so, $P=(N^*N)^{1\slash 2}$. If $T \in \mathcal{B}(\mathcal{H})$ commutes with both $N$ and $N^*$, then $T$ commutes with $u(N)=U$. **Lemma 21**. *Let $(A, S\slash 2)$ be a $\mathbb{B}_2$-contraction consisting of normal operators acting on a space $\mathcal{H}$. Then there exists a unitary $P$ on $\mathcal{H}$ such that $(S, P)$ is a $\Gamma$-unitary and $(A, S, P)$ is a normal $\mathbb{P}$-contraction.* *Proof.* It follows from given hypothesis that $S$ is normal and $\|S\| \leq 2$. Now, from the above discussion we have that there is a unitary $U$ on $\mathcal{H}$ such that $S=(S^*S)^{1\slash 2}U=U(S^*S)^{1\slash 2}$ and $U$ commutes with any operator that commutes with $S$. Hence, $U$ commutes with $B$. Set $P=U^2$. Then $(A, S, P)$ is a triple of commuting normal operators. Moreover $S^*P=(S^*S)^{1\slash 2}U^*U^2=(S^*S)^{1\slash 2}U=S$ and $\|S\| \leq 2$. Thus, $(S, P)$ is a $\Gamma$-contraction with $P$ being unitary and so, $(S, P)$ is a $\Gamma$-unitary by the virtue of Theorem [Theorem 8](#Gamma_uni){reference-type="ref" reference="Gamma_uni"}. Let $(a, s, p)$ in $\sigma_T(A, S, P)$ and let $f(z_1, z_2, z_3)=(z_1, z_2\slash 2)$. Then the spectral mapping theorem yields that $$(a, s\slash 2)=f(a, s, p) \in f(\sigma_T(A, S, P))=\sigma_T(f(A, S, P))=\sigma_T(A, S\slash 2).$$ By Proposition [Proposition 17](#prop2.11){reference-type="ref" reference="prop2.11"}, $(a, s\slash 2) \in \overline{\mathbb{B}}_2$. From the projection property of Taylor spectrum, it follows that $p \in \sigma(P)$ and so, $|p|=1$ since $P$ is a unitary. Furthermore, for $\beta=\frac{s}{2}$, we have $$\bigg|1-\frac{\frac{1}{2}s\overline{\beta}}{1+\sqrt{1-|\beta|^2}} \bigg|=\bigg|1-\frac{\frac{1}{4}s^2}{1+\sqrt{1-\frac{1}{4}|s|^2}} \bigg|=\sqrt{1-\frac{1}{4}|s|^2} \geq |a|,$$ where the last inequality follows from the fact that $(a, s\slash 2) \in \overline{\mathbb{B}}_2$. By Theorem [Theorem 9](#thm2.1){reference-type="ref" reference="thm2.1"}, $(a, s, p) \in \overline{\mathbb{P}}$. Consequently, $(A, S, P)$ is a commuting triple of normal operators so that $\sigma_T(A, S, P) \subseteq \overline{\mathbb{P}}$. Hence, $(A, S, P)$ is a normal $\mathbb{P}$-contraction by Proposition [Proposition 5](#basicprop:02){reference-type="ref" reference="basicprop:02"}. ◻ We have seen that if $(A, S, 0)$ is a $\mathbb{P}$-contraction, then $(A, S \slash 2)$ is a $\mathbb{B}_2$-contraction but the converse does not hold. In this sense, every $\mathbb{P}$-contraction gives rise to a $\mathbb{B}_2$-contraction. It is interesting to explore if one can obtain a $\mathbb{P}$-contraction from a $\mathbb{B}_2$-contraction. The subsequent results provide an answer to this question. **Lemma 22**. *Let $(a, s) \in \overline{\mathbb{B}}_2$. Then $(a, s, 0) \in \overline{\mathbb{P}}$.* *Proof.* The proof follows from Theorem [Theorem 9](#thm2.1){reference-type="ref" reference="thm2.1"}. If $(a, s) \in \overline{\mathbb{B}}_2$, then $b, s \in \overline{\mathbb{D}}$ and so, $(s, 0)=(\lambda_1+\lambda_2, \lambda_1\lambda_2)$ for $\lambda_1=s$ and $\lambda_2=0$. Thus, $(s, 0) \in \Gamma$. By Theorem [Theorem 9](#thm2.1){reference-type="ref" reference="thm2.1"}, $(a, s, 0) \in \overline{\mathbb{P}}$ if and only if the following inequality holds. $$|a| \leq \frac{1}{2}|1-\overline{\lambda}_2\lambda_1|+\frac{1}{2}(1-|\lambda_1|^2)^{\frac{1}{2}}(1-|\lambda_2|^2)^{\frac{1}{2}}=\frac{1}{2}\bigg(1+\sqrt{1-|s|^2}\bigg).$$ Since $|a| \leq \sqrt{1-|s|^2}$, the above inequality holds and so, $(a, s, 0) \in \overline{\mathbb{P}}$. ◻ **Proposition 23**. *Let $(A, S)$ be a $\mathbb{B}_2$-contraction. Then $(A, S, 0)$ is a $\mathbb{P}$-contraction.* *Proof.* Let $f \in \mathbb{C}[z_1,z_2,z_3]$ be arbitrary polynomial and let $g(z_1, z_2)=f(z_1, z_2, 0)$. Then $$\begin{split} \|f(A, S, 0)\| & =\|g(S, P)\|\\ & \leq \sup\{|g(z_1, z_2)| \ : \ (z_1, z_2) \in \overline{\mathbb{B}}_2\} \quad \text{($\because \overline{\mathbb{B}}_2$ is a spectral set for $(B, S)$)}\\ & =\sup \{|f(z_1, z_2, z_3)| \ : \ (z_1, z_2, z_3) \in \overline{\mathbb{B}}_2\times \{0\} \} \\ & \leq \sup \{|f(z_1, z_2, z_3)| \ : \ (z_1, z_2, z_3) \in \overline{\mathbb{P}}\}, \\ \end{split}$$ where, the last inequality follows from Lemma [Lemma 22](#lem2.13){reference-type="ref" reference="lem2.13"}. Consequently, Lemma [Lemma 6](#lem2.2){reference-type="ref" reference="lem2.2"} shows that $(A, S, 0)$ is a $\mathbb{P}$-contraction. ◻ The converse to Proposition [Proposition 23](#prop2.14){reference-type="ref" reference="prop2.14"} is not even true for scalars. For example, take $(a, s)=(1\slash 2, 1)$. Then we have that $|a|^2+|s|^2>1$. Since $(s, 0) \in \Gamma$, Theorem [Theorem 9](#thm2.1){reference-type="ref" reference="thm2.1"} yields that $(a, s, 0) \in \overline{\mathbb{P}}$. Thus there exist scalars $a$ and $s$ so that $(a, s, 0) \in \overline{\mathbb{P}}$ but $(a, s) \notin \overline{\mathbb{B}}_2$. # Operator theory on the unit ball in $\mathbb{C}^n$ {#Polyball} In the previous Section, we have witnessed some interplay between the closed unit ball $\overline{\mathbb{B}}_2$ and $\overline{\mathbb{P}}$. Indeed, if $(A, S, P)$ is a $\mathbb{P}$-contraction, then $(A, S\slash 2)$ is a $\mathbb{B}_2$-contraction. Also, if $(A, S)$ is a $\mathbb{B}_2$-contraction, then $(A, S, 0)$ is a $\mathbb{P}$-contraction. It motivates us to study the operator theoretic aspects of the unit ball $\mathbb{B}^n$ in $\mathbb{C}^n$, where $$\mathbb{B}_n=\left\{(z_1, \dotsc, z_n) \in \mathbb{C}^n \ : \ |z_1|^2+\dotsc+|z_n|^2<1 \right\}.$$ We shall develop some useful operator theory on $\overline{\mathbb{B}}_n$. A commuting tuple of operators $(T_1, \dots , T_n)$ for which $\overline{\mathbb{B}}_n$ is a spectral set is called a $\mathbb{B}_n$-contraction. Contractions have special classes like unitary, isometry, completely non-unitary contraction etc. As we have mentioned in Section 1 that a contraction is an operator for which $\overline{\mathbb{D}}$ is a spectral set. A unitary is a normal operator having its spectrum on the unit circle $\mathbb{T}$ and an isometry is the restriction of a unitary to an invariant subspace. We shall define $\mathbb{B}_n$-unitary and $\mathbb{B}_n$-isometry in an analogous manner. **Definition 24**. Let $\underline{T}=(T_1, \dotsc, T_n)$ be a tuple of commuting operators acting on a Hilbert space $\mathcal{H}$. We say that $\underline{T}$ is 1. a *$\mathbb{B}_n$-unitary* if $T_1, \dotsc, T_n$ are normal operators and $\sigma_T(\underline{T}) \subseteq b \mathbb{B}_n$, where $b\mathbb{B}_n$ is the distinguished boundary of $\mathbb{B}_n$; 2. a *$\mathbb{B}_n$-isometry* if there exist a Hilbert space $\mathcal{K} \supseteq \mathcal{H}$ and a $\mathbb{B}_n$-unitary $(U_1, \dotsc, U_n)$ on $\mathcal{K}$ such that $\mathcal{H}$ is a joint invariant subspace of $U_1, \dots , U_n$ and $T_j=U_j|_\mathcal{H}$ for $j=1, \dotsc, n$. ; 3. *completely non-unitary $\mathbb{B}_n$-contraction* or simply a *c.n.u. $\mathbb{B}_n$-contraction* if $\underline{T}$ is a $\mathbb{B}_n$-contraction and there is no closed joint reducing subspace of $\underline{T}$, on which $\underline{T}$ acts as a $\mathbb{B}_n$-unitary. Note that $\overline{\mathbb{B}}_n$ is a convex compact set and hence is polynomially convex. An interesting fact about $\mathbb{B}_n$ is that its topological boundary $\partial \mathbb{B}_n$ and distinguished boundary $b\mathbb{B}_n$ coincide unlike the polydisc $\mathbb{D}^n$. Needless to mention that $$\partial\mathbb{B}_n=\left\{(z_1, \dotsc, z_n) \in \mathbb{C}^n \ : \ |z_1|^2+\dotsc+|z_n|^2 =1 \right\}.$$ The fact that $\partial \mathbb{B}_n=b\mathbb{B}_n$ is explained in [@Mackey] (see Example 4.10 in [@Mackey] and the discussion thereafter). For the sake of completeness we include a proof here. **Lemma 25**. *For any $n\geq 1$, $b \mathbb{B}_n=\partial \mathbb{B}_n$.* *Proof.* It follows from the maximum-modulus principle that $b\mathbb{B}_n$ is a subset of $\partial \mathbb{B}_n$. Conversely, let $w=(w_1, \dotsc, w_n) \in \partial \mathbb{B}_n$. Consider the function $f_w: \overline{\mathbb{B}}_n \to \mathbb{C}$ defined by $f_w(z)=\langle z, w\rangle=z_1\overline{w}_1+\dotsc + z_n\overline{w}_n$ for all $z=(z_1, \dotsc, z_n)$ in $\overline{\mathbb{B}}_n$. Evidently $f_w$ is holomorphic on $\overline{\mathbb{B}}_n$ and by an application of Cauchy-Schwarz inequality we get $|f_w(z)| \leq 1$ and $|f_w(e^{i\theta}w)|=1$ for all $z$ $\overline{\mathbb{B}}_n$ and for all $\theta \in \mathbb{R}$. Let $z=(z_1, \dotsc, z_n) \in \overline{\mathbb{B}}_n$ be such that $|f_w(z)|=1$. Again. by Cauchy-Schwarz inequality we have $1=|\langle z, w \rangle| \leq \ \|z\| \cdot \|w\| \leq 1$ which happens if and only if $z=\alpha w$ for some $\alpha \in \mathbb{T}$. Therefore, the maximum-modulus of $f_w$ occurs only at points of the form $e^{i\theta}w$ for $\theta \in \mathbb{R}$. It then follows from the definition of distinguished boundary that there is some $\theta \in \mathbb{R}$ such that $e^{i\theta}w \in b\mathbb{B}_n$. An application of Corollary 3.2 in [@KosinskiII] yields that the automorphism $z \mapsto e^{-i\theta}z$ of $\mathbb{B}_n$ maps $b\mathbb{B}_n$ onto itself. Since $e^{i\theta}w\in b\mathbb{B}_n$, we have that $w \in b\mathbb{B}_n$. Thus, $b\mathbb{B}_n=\partial \mathbb{B}_n$. ◻ The works of Arveson, Eschmeier and Athavale [@ArvesonIII; @EschmeierII; @AthavaleII] show that the spherical contractions naturally occur in the study of operators associated with the unit ball. Before proceeding further, we recall the definition of this class along with its special subclasses from the literature. **Definition 26**. A commuting tuple $\underline{T}=(T_1, \dotsc, T_n)$ of operators acting on a Hilbert space $\mathcal{H}$ is said to be 1. a *spherical contraction* if $T_1^*T_1 + \dotsc + T_n^*T_n \leq I_\mathcal{H}$; 2. a *spherical unitary* if each $T_j$ is normal and $T_1^*T_1 + \dotsc + T_n^*T_n = I_\mathcal{H}$; 3. a *spherical isometry* if $T_1^*T_1 + \dotsc + T_n^*T_n =I_\mathcal{H}$; 4. a *row contraction* if $(T_1^*, \dotsc, T_n^*)$ is a spherical contraction. Not every spherical contraction or row contraction is a $\mathbb{B}_2$-contraction. We recall Arveson's example from [@ArvesonIII] in this context. **Example 27**. Consider the pair of multiplication operator $(M_{z_1}, M_{z_2})$ on the *Drury-Arveson space* $H^2_2$, where $H_2^2$ is the reproducing kernel Hilbert space with the kernel $$k(z,w)=\frac{1}{1-\langle z, w\rangle} \quad (z, w \in \mathbb{B}_2).$$ It follows from Corollary 2 in [@ArvesonIII] that $(M_{z_1}, M_{z_2})$ satisfies $M_{z_1}M_{z_1}^*+M_{z_2}M_{z_2}^* \leq I$ but does not form a $\mathbb{B}_2$-contraction as explained in [@ArvesonIII]. Thus, $(M_{z_1}^*, M_{z_2}^*)$ is a spherical contraction which is not a $\mathbb{B}_2$-contraction which is same as saying that the row contraction $(M_{z_1},M_{z_2})$ is not a $\mathbb{B}_2$-contraction. However, we shall see below that $\mathbb{B}_n$-contractions and spherical contractions agree at the level of unitaries and isometries. **Theorem 28**. *Let $\underline{U}=(U_1, \dotsc, U_n)$ be a tuple of commuting operators acting on $\mathcal{H}$. Then $\underline{U}$ is a $\mathbb{B}_n$-unitary if and only if $\underline U$ is a spherical unitary.* *Proof.* Let $\underline{U}$ be a $\mathbb{B}_n$-unitary. Then each $U_j$ is normal and $\sigma_T(\underline{U}) \subseteq b \mathbb{B}_n$. The continuous functional calculus $$\phi: C(\sigma_T(\underline{U})) \to \mathcal{B}(\mathcal{H}), \quad 1 \mapsto I_\mathcal{H} \quad \text{and} \quad z_j \mapsto U_j \quad (j=1, \dotsc, n)$$ defines an algebra homomorphism. Therefore, $U_1^*U_1+\dotsc+U_n^*U_n=I_\mathcal{H}$ as the coordinate functions satisfy $|z_1|^2+\dotsc+|z_n|^2=1$ on $\sigma_T(\underline{U})$. Thus, $\underline U$ is a spherical unitary. Conversely, let $\underline{U}$ be a tuple of commuting normal operators that satisfies $U_1^*U_1+\dotsc + U_n^*U_n=I_\mathcal{H}$. Let $w=(w_1, \dotsc, w_n) \in \sigma_T(\underline{U})$. The function given by $f(z_1, \dotsc, z_n)=|z_1|^2+\dotsc+|z_n|^2$ is in $C(\sigma_T(\underline{U}))$. The continuous functional caluclus gives $f(U_1, \dotsc, U_n)=U_1^*U_1+\dotsc + U_n^*U_n=I_\mathcal{H}.$ It follows from spectral mapping theorem that $\{1\ =\sigma_T(f(\underline{U}))=f(\sigma_T(\underline{U})).$ Since $w\in \sigma_T(\underline{U})$, we have that $f(w)=1$ and so, $w \in b \mathbb{B}_n$. Consequently, $\sigma_T(\underline{U}) \subseteq \partial \mathbb{B}_n$ which yields that $\underline{U}$ is a $\mathbb{B}_n$-unitary. The proof is complete. ◻ Now we are going to prove that a $\mathbb{B}_n$-isometry is nothing but a spherical isometry and vice-versa. In this connection, recall that a *subnormal tuple* is a tuple $(T_1, \dotsc, T_n)$ of commuting operators that admits a simultaneous normal extension. The following result due to Athavale [@AthavaleIII], which was later proved independently by Arveson [@ArvesonIII], will be useful in this context. **Lemma 29** ([@ArvesonIII], Corollary 1). *Let $T_1, \dotsc, T_n$ be a set of commuting operators on a Hilbert space $\mathcal{H}$ such that $T_1^*T_1+\dotsc+T_n^*T_n=I_\mathcal{H}$. Then $(T_1, \dotsc, T_n)$ is a subnormal tuple.* **Theorem 30**. *Let $\underline{V}=(V_1, \dotsc, V_n)$ be a tuple of commuting operators acting on $\mathcal{H}$. Then $\underline{V}$ is a $\mathbb{B}_n$ -isometry if and only if $\underline{V}$ is a spherical isometry, i.e. $V_1^*V_1+\dotsc + V_n^*V_n=I_\mathcal{H}.$* *Proof.* Let $\underline{V}$ be a $\mathbb{B}_n$-isometry acting on $\mathcal{H}$. Then there exist a Hilbert space $\mathcal{K} \supseteq \mathcal{H}$ and a $\mathbb{B}_n$-unitary $\underline{U}=(U_1, \dotsc, U_n)$ on $\mathcal{K}$ such that $\mathcal{H}$ is invariant under each $V_j$ and $V_j=U_j|_\mathcal{H}$. Then for any $x \in \mathcal{H}$, we have $\|V_1x\|^2+\dotsc \|V_nx\|^2= \|U_1x\|^2+ \dotsc \|U_nx\|^2=\|x\|^2$, which is equivalent to saying that $V_1^*V_1+\dotsc + V_n^*V_n=I_\mathcal{H}$. Conversely, let $\underline{V}$ be a spherical isometry. Lemma [Lemma 29](#subnormal){reference-type="ref" reference="subnormal"} yields that there exists a commuting tuple $\underline{N}=(N_1, \dotsc, N_n)$ of normal operators acting on some Hilbert space $\mathcal{K}$ containing $\mathcal{H}$ such that $(N_1|_\mathcal{H}, \dotsc, N_n|_\mathcal{H})=(V_1, \dotsc, V_n)$. Without loss of generality, we assume that $\underline{N}$ on $\mathcal{K}$ is the *minimal normal extension* of $\underline{V}$ and the space $\mathcal{K}$ is given by $$\label{eqn3.1} \mathcal{K}=\overline{span}\{N_1^{*j_1}\dotsc N_n^{*j_n}h \ : \ j_1, \dotsc, j_n \in \mathbb{N}\cup \{0\} , \ \ h \in \mathcal{H}\}.$$ For any $h \in \mathcal{H}$, a straightforward computation yields that $$\begin{split} &\left\|\left(I-\overset{n}{\underset{j=1}{\sum}}N_j^*N_j\right)h\right\|^2\\ &=\left(\|h\|^2-\overset{n}{\underset{j=1}{\sum}}\|V_jh\|^2\right)-\left(\|V_1h\|^2-\overset{n}{\underset{j=1}{\sum}}\|V_jV_1h\|^2\right)-\dots -\left(\|V_nh\|^2-\overset{n}{\underset{j=1}{\sum}}\|V_jV_nh\|^2\right)\\ &=0, \end{split}$$ where the last equality follows from the fact $\overset{n}{\underset{j=1}{\sum}}\|V_jx\|^2=\|x\|^2$ for every $\ x \in \mathcal{H}$. Thus, $I-N_1^*N_1-\dotsc -N_n^*N_n=0$ on $\mathcal{H}$. Since $N_1, \dotsc, N_n$ are commuting normal operators, it follows from ([\[eqn3.1\]](#eqn3.1){reference-type="ref" reference="eqn3.1"}) that $I-N_1^*N_1-\dotsc -N_n^*N_n=0$ on $\mathcal{K}$. Hence, by Proposition [Theorem 28](#prop3.2){reference-type="ref" reference="prop3.2"}, $\underline{N}$ is a $\mathbb{B}_n$-unitary and consequently $\underline{V}$ is a $\mathbb{B}_n$-isometry. ◻ One of the most important results in one variable operator theory is the canonical decomposition of a contraction (see CH-I of [@Nagy]), which states that for every contraction $T$ on a Hilbert space $\mathcal{H}$, the space $\mathcal{H}$ admits a unique orthogonal decomposition $\mathcal{H}=\mathcal{H}_1 \oplus \mathcal{H}_2$ into joint reducing subspaces of $T$ such that $T|_{\mathcal{H}_1}$ is a unitary and $T|_{\mathcal{H}_2}$ is a completely non-unitary contraction. Below we show that a $\mathbb{B}_n$-contraction admits an analogous orthogonal decomposition into a $\mathbb{B}_n$-unitary and a completely non-unitary $\mathbb{B}_n$-contraction. For the ease of computations, we introduce the following notations : for a tuple of commuting operators $\underline{T}=(T_1, \dotsc, T_n)$ on space $\mathcal{H}$ and for $\alpha=(\alpha_1, \dotsc, \alpha_n) \in \mathbb{N}^n$, we write $\underline{T}^\alpha=T_1^{\alpha_1} \dotsc T_n^{\alpha_n}$ and $\underline{T}^{*\alpha}=T_1^{*\alpha_1} \dotsc T_n^{*\alpha_n}$. Also, by *normal tuple*, we mean a tuple of commuting normal operators. **Theorem 31** (**Canonical decomposition**). *Let $\underline{T}=(T_1, \dotsc, T_n)$ be a $\mathbb{B}_n$-contraction on a Hilbert space $\mathcal{H}$. Then there is an orthogonal decomposition of $\mathcal{H}$ into joint reducing subspaces $\mathcal{H}_u$ and $\mathcal{H}_c$ of $\underline T$ such that* 1. *$(T_1|_{\mathcal{H}_u}, \dotsc, T_n|_{\mathcal{H}_u})$ is a $\mathbb{B}_n$-unitary;* 2. *$(T_1|_{\mathcal{H}_c}, \dotsc, T_n|_{\mathcal{H}_c})$ is a completely non-unitary $\mathbb{B}_n$-contraction.* *Proof.* It follows from Corollary 4.2 in [@Eschmeier] that the closed linear subspace given by $$\mathcal{H}_0=\bigcap_{\alpha \in \mathbb{N}^n}\bigcap_{ \beta \in \mathbb{N}^n}Ker\bigg[\underline{T}^\alpha\underline{T}^{*\beta}-\underline{T}^{*\beta}\underline{T}^{\alpha}\bigg].$$ is a *maximal* joint reducing subspace for $\underline{T}$ on which $\underline{T}$ acts as a normal tuple. By *maximality*, we mean that if there is a joint reducing subspace $\mathcal{L}$ of $\underline{T}$ such that each $T_j|_{\mathcal{L}}$ is normal, then $\mathcal{L} \subseteq \mathcal{H}_0$. Define $$\mathcal{H}_u:=\{h \in \mathcal{H}_0 \ : \ \|T_1h\|^2+\dotsc+\|T_nh\|^2=\|h\|^2 \}.$$ Then, $\mathcal{H}_u=\left\{h \in \mathcal{H}_0 \ : \ (I-T_1^*T_1\dotsc-T_n^*T_n)h=0 \right\}$. We show that $\mathcal{H}_u$ is a joint reducing subspace of $\underline{T}$. Let $h \in \mathcal{H}_u$ and let $1 \leq j \leq n$. Since $\mathcal{H}_0$ is a joint reducing subspace, we have that $T_jh \in \mathcal{H}_0$ and $T_j^*h \in \mathcal{H}_0$. Since $(T_1|_{\mathcal{H}_0}, \dotsc, T_n|_{\mathcal{H}_0})$ is a commuting normal tuple and $(I-T_1^*T_1\dotsc-T_n^*T_n)h=0$, we have $(I-T_1^*T_1\dotsc-T_n^*T_n)T_jh=T_j(I-T_1^*T_1\dotsc-T_n^*T_n)h=0$. We used the fact that if $N_1, N_2$ are commuting normal operators, then $N_1^*N_2=N_2^*N_1$. Therefore, $T_jh \in \mathcal{H}_u$ and similarly $T_j^*h \in \mathcal{H}_u$. Thus, $\mathcal{H}_u$ is a joint reducing subspace of $\underline{T}$ and $(T_1|_{\mathcal{H}_u}, \dotsc, T_n|_{\mathcal{H}_u})$ is a normal tuple satisfying $\|T_1h\|^2+\dotsc \|T_nh\|^2=\|h\|^2 \quad \text{for all} \ h \in \mathcal{H}_u$. It follows from Theorem [Theorem 28](#prop3.2){reference-type="ref" reference="prop3.2"} that $(T_1|_\mathcal{H_u}, \dotsc, T_n|_{\mathcal{H}_u})$ is a $\mathbb{P}$-unitary on $\mathcal{H}_u$. Setting $\mathcal{H}_c=\mathcal{H}\ominus \mathcal{H}_u,$ we see that $\mathcal{H}_c$ is a joint reducing subspace of $\underline{T}$. Let $\mathcal{L} \subseteq \mathcal{H}_c$ be a joint reducing subspace of $\underline{T}$ such that $(T_1|_\mathcal{L}, \dotsc, T_n|_\mathcal{L})$ is a $\mathbb{P}$-unitary. Thus $(T_1|_\mathcal{L}, \dotsc, T_n|_\mathcal{L})$ is a normal tuple and the maximality of $\mathcal{H}_0$ implies that $\mathcal{L} \subseteq \mathcal{H}_0$. By Theorem [Theorem 28](#prop3.2){reference-type="ref" reference="prop3.2"}, we have that $\|T_1h\|^2+\dotsc \|T_nh\|^2=\|h\|^2 \quad \text{for all} \ h \in \mathcal{L}$. Hence, $\mathcal{L} \subseteq \mathcal{H}_u$. Putting everything together, we have that $\mathcal{L} \subset \mathcal{H}_u \cap \mathcal{H}_c=\{0\}$ and so, $\mathcal{L}=\{0\}$. Thus, $(T_1|_\mathcal{L}, \dotsc, T_n|_\mathcal{L})$ is a completely non-unitary $\mathbb{B}_n$-contraction and the proof is complete. ◻ # The pentablock unitaries {#P-uni} Recall that a $\mathbb{P}$-unitary is a normal $\mathbb{P}$-contraction whose Taylor joint spectrum lies in the distinguished boundary $b\mathbb{P}$ of the pentablock. In this Section, we find several characterization for the $\mathbb{P}$-unitaries and find their interplay with $\mathbb{B}_2$-unitaries and $\Gamma$-unitaries. First we collect from the existing literature [@Agler; @Jindal], similar various characterizations for the points in the distinguished boundary of the pentablock $b\mathbb{P}$. **Theorem 32**. *For $(a, s, p) \in \mathbb{C}^3$, the following are equivalent:* 1. *$(a, s, p) \in b\mathbb{P}$;* 2. *$(s, p) \in b\Gamma, |a|=\sqrt{1-\frac{1}{4}|s|^2}$ ;* 3. *There is a unitary matrix $U=[u_{ij}]_{2 \times 2}$ such that $u_{11}=u_{22}$ & $(a, s, p)=(u_{21}, tr(U), det(U)).$* In other words, we have the following description for the points in the distinguished boundary $b \mathbb{P}$: $$b\mathbb{P}=\bigg\{(a, s, p) \in \mathbb{C}^3 \ : \ |a|=\sqrt{1-\frac{1}{4}|s|^2}, \ (s, p) \in b\Gamma \bigg\}\,.$$ Interestingly, each of the above characterizations for a point in $b\mathbb{P}$ gives rise to a characterization of a $\mathbb{P}$-unitary. Also, we have other characterizations in terms of $\mathbb{B}_2$-unitaries and $\Gamma$-unitaries as shown below. **Theorem 33**. *Let $\underline{N}=(N_1, N_2, N_3)$ be a commuting triple of bounded linear operators. Then the following are equivalent:* 1. *$\underline{N}$ is a $\mathbb{P}$-unitary ;* 2. *$N_1^*$ is subnormal, $(N_2, N_3)$ is a $\Gamma$-unitary and $N_1^*N_1=I-\frac{1}{4}N_2^*N_2$ ;* 3. *$(N_2, N_3)$ is a $\Gamma$-unitary and $N_1^*N_1=I-\frac{1}{4}N_2^*N_2$ and $N_1N_1^*=I-\frac{1}{4}N_2N_2^*$ ;* 4. *$(N_1, N_2\slash 2)$ is a $\mathbb{B}_2$-unitary and $(N_2, N_3)$ is a $\Gamma$-unitary ;* 5. *There is a $2 \times 2$ unitary block matrix $U=[U_{ij}]$, where $U_{ij}$ are commuting normal operators, such that $U_{11}=U_{22}$ and $\underline{N}=(U_{21}, U_{11}+U_{22}, U_{11}U_{22}-U_{12}U_{21}).$* *Proof.* $(1) \implies (2):$ By definition $N_1, N_2, N_3$ are commuting normal operators and $\sigma_T(N_1,N_2,N_3) \subseteq b\mathbb{P}$. By the spectral mapping theorem, $\sigma_T(N_2, N_3)=P_{2,3}\sigma_T(\underline{N})$ where $P_{2,3}$ is the projection onto the second and third coordinates. It follows from Theorem [Theorem 32](#thm:distP){reference-type="ref" reference="thm:distP"} and the projection property of the Taylor-joint spectrum that $\sigma_T(N_2, N_3)\subseteq b\Gamma$. Hence, $(N_2, N_3)$ is a $\Gamma$-unitary. Again, the commutative $C^*$-algebra generated by the commuting normal operators $N_1, N_2, N_3$ is isometrically isomorphic to the $C(\sigma_T(\underline{N}))$ via the continuous functional calculus. The continuous functional takes the coordinate function $z_i$ to $N_i$ for $i=1,2,3$. The coordinate functions satisfy $|z_1|^2=1-\frac{1}{4}|z_2|^2$ on $b\mathbb{P}$ and hence on $\sigma_T(\underline{N})$. Thus, $N_1^*N_1=I-\frac{1}{4}N_2^*N_2$. $(2) \implies (1):$ From the hypothesis that $N_1^*N_1=I-\frac{1}{4}N_2^*N_2$ and Lemma [Lemma 29](#subnormal){reference-type="ref" reference="subnormal"}, it follows that $N_1$ is subnormal. Thus, $N_1^*, N_1$ are subnormal operators and consequently $N_1$ is a normal operator. Let $(a, s, p) \in \sigma_T(\underline{N})$. It follows from the projection property of Taylor-joint spectrum that $(s, p) \in \sigma_T(N_2, N_3)$. Since $(N_2, N_3)$ is a $\Gamma$-unitary, we have that $(s, p)\in b\Gamma$. The function $$f(z_1, z_2, z_3)=|z_1|^2-\bigg(1-\frac{|z_2|^2}{4}\bigg),$$ is continuous on $\sigma_T(\underline{N})$. Then it follows from the continuous functional calculus that $$f(N_1, N_2, N_3)=N_1^*N_1-\bigg(I-\frac{1}{4}N_2^*N_2\bigg)=0.$$ Now spectral mapping theorem gives $\{0\}=\sigma_T(f(\underline{N}))=f(\sigma_T(\underline{N})).$ Since $(a, s, p) \in \sigma_T(\underline{N})$, we have that $f(a, s, p)=0$ and the desired conclusion follows. $(2) \implies (3):$ Since $N_1^*N_1+\frac{1}{4}N_2^*N_2=I$, Lemma [Lemma 29](#subnormal){reference-type="ref" reference="subnormal"} yields that $N_1$ is a normal operator. Hence, $N_1N_1^*=N_1^*N_1=I-\frac{1}{4}N_2^*N_2.$ $(3) \implies (2):$ This is obvious. $(3) \iff (4):$ Follows from Theorem [Theorem 28](#prop3.2){reference-type="ref" reference="prop3.2"}. $(2) \implies (5):$ Set $U=\begin{bmatrix} \frac{1}{2}N_2 & -N_1^*N_3\\ N_1 & \frac{1}{2}N_2 \end{bmatrix}.$ Since $N_1, N_2, N_3$ are commuting normal operators, $U_{ij}$ are commuting normal operators. We show that $U$ is a unitary matrix. Since $N_3$ is unitary and $N_2^*N_3=N_2$ as $(N_2, N_3)$ is a $\Gamma$-unitary, we have $$\begin{split} UU^*=\begin{bmatrix} \frac{1}{2}N_2 & -N_1^*N_3\\ \\ N_1 & \frac{1}{2}N_2 \end{bmatrix} \begin{bmatrix} \frac{1}{2}N_2^* & N_1^*\\ \\ -N_1N_3^* & \frac{1}{2}N_2^* \end{bmatrix} &=\begin{bmatrix} \frac{1}{4}N_2N_2^*+ N_1^*N_3N_1N_3^* & \frac{1}{2}N_2N_1^*-\frac{1}{2}N_1^*N_3N_2^*\\ \\ \frac{1}{2}N_1N_2^*-\frac{1}{2}N_2N_1N_3^* & N_1N_1^*+\frac{1}{4}N_2N_2^* \end{bmatrix}\\ \\ &=\begin{bmatrix} \frac{1}{4}N_2^*N_2+ N_1^*N_1N_3^*N_3 & \frac{1}{2}N_1^*N_2-\frac{1}{2}N_1^*N_2^*N_3\\ \\ \frac{1}{2}N_1N_2^*-\frac{1}{2}N_1N_2N_3^* & N_1^*N_1+\frac{1}{4}N_2^*N_2 \end{bmatrix}\\ \\ &=\begin{bmatrix} \frac{1}{4}N_2^*N_2+ N_1^*N_1 & \frac{1}{2}N_1^*N_2-\frac{1}{2}N_1^*N_2\\ \\ \frac{1}{2}N_1N_2^*-\frac{1}{2}N_1N_2^* & \frac{1}{4}N_2^*N_2+ N_1^*N_1 \end{bmatrix}=\begin{bmatrix} I & 0 \\ 0 & I\\ \end{bmatrix}. \end{split}$$ Similarly, we can show that $U^*U=I$. Now $U_{21}=N_1$ and $U_{11}+U_{22}=N_2$. It only remains to show that $U_{11}U_{22}-U_{12}U_{21}=N_3$ which we prove using the fact that $N_2=N_2^*N_3$ in the following way. $$U_{11}U_{22}-U_{12}U_{21}=\frac{1}{4}N_2^2+N_1^*N_1N_3=\frac{1}{4}N_2^*N_2N_3+N_1^*N_1N_3=\bigg(\frac{1}{4}N_2^*N_2+N_1^*N_1\bigg)N_3=N_3.$$ $(5) \implies (2):$ Let $U$ be a $2 \times 2$ unitary block matrix $[U_{ij}]$ where $U_{ij}$ are commuting normal operators such that $U_{11}=U_{22}$ and $\underline{N}=(U_{21}, U_{11}+U_{22}, U_{11}U_{22}-U_{12}U_{21}).$ It is evident that $\|N_2\| \leq \|U_{11}\|+\|U_{22}\| \leq 2\|U\|=2.$ The condition $U^*U=I$ gives the following set of equations. $$\label{eq1} U_{11}^*U_{11}+U_{21}^*U_{21}=I, \quad U_{12}^*U_{12}+U_{22}^*U_{22}=I,$$ $$\label{eq2} U_{12}^*U_{11}+U_{22}^*U_{21}=0, \quad U_{11}^*U_{12}+U_{21}^*U_{22}=0.$$ Again, $UU^*=I$ provides the following equations. $$\label{eq3} U_{11}U_{11}^*+U_{12}U_{12}^*=I, \quad U_{21}U_{21}^*+U_{22}U_{22}^*=I,$$ $$\label{eq4} U_{21}U_{11}^*+U_{22}U_{12}^*=0, \quad U_{11}U_{21}^*+U_{12}U_{22}^*=0 .$$ Using the above equations, we have the following. $$\begin{split} N_2^*N_3&=(U_{11}^*+U_{22}^*)(U_{11}U_{22}-U_{12}U_{21})\\ &=(U_{11}^*U_{11})U_{22}-U_{11}^*U_{12}U_{21} +(U_{22}^*U_{22})U_{11}-U_{22}^*U_{12}U_{21}\\ & =(I-U_{21}^*U_{21})U_{22}-U_{11}^*U_{12}U_{21} +(I-U_{12}^*U_{12})U_{11}-U_{22}^*U_{12}U_{21} \quad [\text{ by } {(\ref{eq1})}] \ \\ &=U_{22}-(U_{22}U_{21}^*)U_{21} -U_{11}^*U_{12}U_{21}+U_{11}-(U_{12}^*U_{11})U_{12}- U_{22}^*U_{12}U_{21}\\ &=(U_{11}+U_{22})+(U_{11}^*U_{12})U_{21} -U_{11}^*U_{12}U_{21}+(U_{22}^*U_{12})U_{12}- U_{22}^*U_{12}U_{21} \ \quad [\text{ by } {(\ref{eq2})}]\\ &=N_2.\\ \end{split}$$ We show that $N_3$ is unitary. Since $N_3$ is normal, it suffices to show that $N_3^*N_3=I$. $$\begin{split} N_3^*N_3&=(U_{11}^*U_{22}^*-U_{12}^*U_{21}^*)(U_{11}U_{22}-U_{12}U_{21})\\ &=U_{11}^*U_{11}U_{22}^*U_{22}-(U_{11}^*U_{12})U_{22}^*U_{21}-(U_{21}^*U_{11})U_{21}^*U_{22}+U_{12}^*U_{21}^*U_{12}U_{21}\\ & =U_{11}^*U_{11}U_{22}^*U_{22}+(U_{21}^*U_{22})U_{22}^*U_{21}+(U_{22}^*U_{21})U_{21}^*U_{22}+U_{12}^*U_{21}^*U_{12}U_{21} \quad [\text{ by } (\ref{eq2})]\\ &=(U_{11}^*U_{11}+U_{21}^*U_{21})U_{22}^*U_{22}+(U_{22}^*U_{22}+U_{12}^*U_{12})U_{21}^*U_{21}\\ & =U_{22}^*U_{22}+U_{21}^*U_{21} \quad [\text{ by } (\ref{eq1})]\\ & =I. \quad [\text{ by } (\ref{eq3})] \end{split}$$ Hence, $(N_2, N_3)$ is a commuting pair of normal operators such that $\|N_2\| \leq 2, N_2^*N_3=N_2$ and $N_3$ is a unitary. Therefore, $(N_2,N_3)$ is a $\Gamma$-unitary. Since $U_{11}=U_{22}$, we have $$\begin{split} I-\frac{1}{4}N_2^*N_2=I-\frac{1}{4}(U_{11}^*+U_{22}^*)(U_{11}+U_{22}) =I-U_{22}^*U_{22} =U_{21}^*U_{21} =N_1^*N_1,\\ \end{split}$$ where the second last equality follows from ([\[eq3\]](#eq3){reference-type="ref" reference="eq3"}). The proof is now complete. ◻ Note that the assumption that $N_1^*$ is subnormal in Theorem [Theorem 33](#P_unitary){reference-type="ref" reference="P_unitary"} cannot be dropped. Also, unlike operators associated with the symmetrized bidisc and tetrablock (see Theorem 2.5 in [@Bhattacharyya] and Theorem 5.4 in [@Bhattacharyya-01] respectively), it is not always true that a $\mathbb{P}$-unitary is a commuting triple $(N_1,N_2,N_3)$ which is a $\mathbb{P}$-contraction and $N_3$ is a unitary. The following example explains all these together. **Example 34**. Consider the commuting triple of subnormal operators $\underline{N}=(N_1, N_2, N_3)=(T_z, \ 0, \ -I)$ on $\ell^2(\mathbb{N})$, where $T_z$ is the unilateral shift on $\ell^2(\mathbb{N})$. Then 1. $(0, -I)$ is a $\Gamma$-unitary since $\sigma_T(0, -I)=\{(0, -1)\} \subset b\Gamma$ and; 2. $N_1^*N_1=T_z^*T_z=I$ which is same as $N_1^*N_1+\frac{1}{4}N_2^*N_2=I$. Hence, $\underline{N}$ is a commuting triple such that $(N_2, N_3)$ is a $\Gamma$-unitary and $N_1^*N_1=I-\frac{1}{4}N_2^*N_2$ but $N_1^*$ is not subnormal. Thus, $\underline{N}$ is not a $\mathbb{P}$-unitary as $N_1$ is not normal. However, it is true that $(N_1,N_2,N_3)$ is a $\mathbb{P}$-contraction, in fact is a $\mathbb{P}$-isometry (which follows from Theorem [Theorem 46](#P_isometry){reference-type="ref" reference="P_isometry"}) with $N_3$ being a unitary. Note that the tetrablock is a domain in $\mathbb{C}^3$ that appeared in [@Abouhajar] to study another special case of $\mu$-synthesis. The following example shows that one cannot drop the hypothesis $(N_1, N_2\slash 2)$ being a $\mathbb{B}_2$-unitary in Theorem [Theorem 33](#P_unitary){reference-type="ref" reference="P_unitary"}. Indeed, we show that there exists $(a, s, p) \in \overline{\mathbb{P}}$ such that $(s, p) \in b\Gamma$ but $(a, s, p) \notin b\mathbb{P}$. **Example 35**. Let $(a, s, p)=(0, 0, 1).$ Then $(s, p) \in b\Gamma, \quad (a, s, p) \in \overline{\mathbb{P}}$ and $|a|^2+\frac{1}{4}|s|^2 \ne 1$. Thus $(a, s, p) \notin b\mathbb{P}$. Moreover, it shows that $$b\mathbb{P}\ne \{(a, s, p) \in \overline{\mathbb{P}}\ : \ |p|=1 \}.$$ The next corollary is an immediate consequence of Theorem [Theorem 8](#Gamma_uni){reference-type="ref" reference="Gamma_uni"} and Theorem [Theorem 33](#P_unitary){reference-type="ref" reference="P_unitary"}. **Corollary 36**. *$(U_1, U_2)$ is a pair of commuting unitaries if and only if $(U_1, 0, U_2)$ is a $\mathbb{P}$-unitary.* One can easily construct a $\mathbb{P}$-unitary from a given $\Gamma$-unitary in the following way. **Example 37**. Let $(N_2, N_3)$ be a $\Gamma$-unitary on a Hilbert space $\mathcal{H}$. It follows from the definition of $\Gamma$-contraction that $\frac{1}{2}N_2$ is a contraction. Therefore, we can consider its defect operator which is $D_{N_2\slash 2}=(I-\frac{1}{4}N_2^*N_2)^{1\slash 2}.$ Since $(N_2, N_3)$ are commuting normal operators, it immediately follows that $D_{N_2\slash 2}$ commutes with $N_1$ and $N_2$. Therefore, $(D_{N_2\slash 2}, N_2, N_3)$ is a triple of commuting normal operators such that $(N_2, N_3)$ is a $\Gamma$-unitary and $D_{N_2\slash 2}^2+\frac{1}{4}N_2^*N_2=I.$ Thus, it follows from Theorem [Theorem 33](#P_unitary){reference-type="ref" reference="P_unitary"} that $(D_{N_2\slash 2}, N_2, N_3)$ is a $\Gamma$-unitary on $\mathcal{H}$. We shall use the polar decomposition for normal operators to show that the aforementioned example serves as a prototype of a $\mathbb{P}$-unitary. The proof of the next corollary follows from the polar decomposition theorem, Theorem [Theorem 33](#P_unitary){reference-type="ref" reference="P_unitary"} and Example [Example 37](#eg3.5){reference-type="ref" reference="eg3.5"}. **Theorem 38**. *A commuting triple of operators $\underline{N}=(N_1, N_2, N_3)$ acting on a Hilbert space $\mathcal{H}$ is a $\mathbb{P}$-unitary if and only $(N_2, N_3)$ is a $\Gamma$-unitary and there is a unitary $U$ on $\mathcal{H}$ such that $U$ commutes with $N_2, N_3$ and $N_1=UD_{N_2\slash 2}=D_{N_2\slash 2}U.$* *Proof.* Let $(N_2, N_3)$ be a $\Gamma$-unitary and let $U$ be a unitary on $\mathcal{H}$ that commutes with $N_2$ and $N_3$. Then $U$ commutes with $N_2^*$ due to Fuglede's theorem and so, $$U\left(I-\frac{1}{4}N_2^*N_2\right)=\left(I-\frac{1}{4}N_2^*N_2\right)U.$$ Consequently, the continuous functional calculus for normal operators yields that $U$ commutes with $D_{N_2\slash 2}$. If we take $N_1=UD_{N_2\slash 2}$, then $N_1$ commutes with $N_2$ and $N_3$ since $U$ and $D_{N_2\slash 2}$ commute with $N_2, N_3$. Also, we have $$N_1^*N_1+\frac{1}{4}N_2^*N_2=U^*UD_{N_2\slash2}^2+\frac{1}{4}N_2^*N_2=I.$$ Thus, it follows from Theorem [Theorem 33](#P_unitary){reference-type="ref" reference="P_unitary"} that $(N_1, N_2, N_3)$ is a $\mathbb{P}$-unitary. To see the converse, let $(N_1, N_2, N_3)$ be a $\mathbb{P}$-unitary. By Theorem [Theorem 33](#P_unitary){reference-type="ref" reference="P_unitary"}, $(N_2, N_3)$ is a $\Gamma$-unitary. Moreover, $N_1^*N_1=I-\frac{1}{4}N_2^*N_2=D_{N_2\slash2}^2$ and thus $(N_1^*N_1)^{1\slash 2}=D_{N_2\slash 2}$. It follows from polar decomposition theorem (see the discussion after Lemma [Lemma 20](#basiclem:03){reference-type="ref" reference="basiclem:03"}) that there is a unitary $U$ on $\mathcal{H}$ which commutes with $N_2, N_3$ and $N_1=U(N_1^*N_1)^{1\slash 2}=(N_1^*N_1)^{1\slash 2}U.$ Consequently, $N_1=UD_{N_2\slash 2}=D_{N_2\slash 2}U$ and the proof is complete. ◻ We conclude the section with the following sufficient condition for a $\mathbb{P}$-unitary. Recall that a commuting tuple of operators $(T_1, \dotsc, T_n)$ is said to be *doubly commuting* if $T_iT_j^*=T_j^*T_i$ for all $i \ne j$. **Proposition 39**. *Let $(A, S, P)$ be a doubly commuting $\mathbb{P}$-contraction on $\mathbb{C}^2$ such that $\sigma_T(A, S, P)\subseteq b\mathbb{P}$ and let $(A, S\slash 2)$ be a spherical contraction. Then $(A, S, P)$ is a $\mathbb{P}$-unitary.* *Proof.* With respect to a fixed orthonormal basis, we can write $(A, S, P)$ in the following way: $$A=\begin{bmatrix} a_{11} & a_{12}\\ 0 & a_{22} \end{bmatrix}, \quad S=\begin{bmatrix} s_{11} & s_{12}\\ 0 & s_{22} \end{bmatrix} \quad \text{and} \quad P=\begin{bmatrix} p_{11} & p_{12}\\ 0 & p_{22} \end{bmatrix}.$$ Note that $\sigma_T(A, S, P)=\left\{(a_{11}, s_{11}, p_{11}), (a_{22}, s_{22}, p_{22})\right\} \subset b\mathbb{P}.$ By Theorem [Theorem 33](#P_unitary){reference-type="ref" reference="P_unitary"}, we have $$\label{doublycommuting} |p_{11}|=|p_{22}|=1 \quad \text{and} \quad |a_{11}|^2+\frac{1}{4}|s_{11}|^2=1=|a_{22}|^2+\frac{1}{4}|s_{22}|^2.$$ It follows from Lemma [Theorem 18](#lem2.12){reference-type="ref" reference="lem2.12"} that $P$ is a contraction and thus, we have $$0 \leq I-P^*P=\begin{bmatrix} 1-|p_{11}|^2 & -\overline{p_{11}}p_{12} \\ -p_{11}\overline{p_{12}} & 1-|p_{22}|^2-|p_{12}|^2 \end{bmatrix}=\begin{bmatrix} 0 & -\overline{p_{11}}p_{12} \\ -p_{11}\overline{p_{12}} & -|p_{12}|^2 \end{bmatrix}.$$ So, we have $p_{12}=0$ and thus $P$ is a normal operator such that $P^*P=I$. Therefore, $P$ is unitary and $(S, P)$ is a $\Gamma$-contraction which yields that $(S, P)$ is a $\Gamma$-unitary. It follows from Theorem [Theorem 9](#thm2.1){reference-type="ref" reference="thm2.1"} that $S-S^*P=0$. A straight forward calculation gives the following: $$S-S^*P=\begin{bmatrix} s_{11}-\overline{s_{11}}p_{11} & s_{12}\\ \overline{s_{12}}p_{11} & s_{22}-\overline{s_{22}}p_{22} \end{bmatrix}$$ Hence, $S-S^*P=0$ gives that $s_{12}=0$. Putting everything together, we have that $$A=\begin{bmatrix} a_{11} & a_{12}\\ 0 & a_{22} \end{bmatrix}, \quad S=\begin{bmatrix} s_{11} & 0\\ 0 & s_{22} \end{bmatrix} \quad \text{and} \quad P=\begin{bmatrix} p_{11} & 0\\ 0 & p_{22} \end{bmatrix} \quad (|p_{11}|=|p_{22}|=1).$$ If $a_{12}=0$, then $(A, S, P)$ is a normal $\mathbb{P}$-contraction with $\sigma_T(A, S, P) \subset b\mathbb{P}$ and hence, $(A, S, P)$ is a $\mathbb{P}$-unitary. Let us assume that $a_{12} \ne 0$. Now, we use the fact that $A$ doubly commutes with $S$ and $P$. A routine computation yields that $$AS^*-S^*A=\begin{bmatrix} 0 & a_{12}(\overline{s_{22}}-\overline{s}_{11})\\ 0 & 0 \end{bmatrix} \quad \text{and} \quad AP^*-P^*A=\begin{bmatrix} 0 & a_{12}(\overline{p_{22}}-\overline{p}_{11})\\ 0 & 0 \end{bmatrix}.$$ Thus, $p_{11}=p_{22}$ and $s_{11}=s_{22}$. Now, using the hypothesis that $I-A^*A-\frac{1}{4}S^*S \geq 0$, we have $$0 \leq I-A^*A-\frac{1}{4}S^*S=\begin{bmatrix} 1-|a_{11}|^2-\frac{1}{4}|s_{11}|^2 & -\overline{a_{11}}a_{12} \\ -a_{11}\overline{a_{12}} & 1-|a_{22}|^2-\frac{1}{4}|s_{11}|^2-|a_{12}|^2\\ \end{bmatrix} =\begin{bmatrix} 0 & -\overline{a_{11}}a_{12} \\ -a_{11}\overline{a_{12}} & -|a_{12}|^2\\ \end{bmatrix},$$ where the last inequality follows from ([\[doublycommuting\]](#doublycommuting){reference-type="ref" reference="doublycommuting"}). The positive semi-definiteness of $I-A^*A-\frac{1}{4}S^*S$ implies that its trace $-|a_{12}|^2 \geq 0$ and hence, $a_{12}=0$. This is a contradiction. Hence, $a_{12}=0$ and the proof is complete. ◻ # The pentablock isometries {#P-iso} In this Section, we decipher the structure of a $\mathbb{P}$-isometry and find different characterization for them. A $\mathbb{P}$-isometry is the restriction of a $\mathbb{P}$-unitary $(A,S,P)$ to a joint invariant subspace of $A,S$ and $P$. Thus, a $\mathbb{P}$-isometry is a subnormal triple. Note that a tuple of commuting operators $\underline{T}=(T_1, \dotsc, T_m)$ acting on a Hilbert space $\mathcal{H}$ is said to be *subnormal* if there is a Hilbert space $\mathcal{K} \supseteq \mathcal{H}$ and a commuting tuple of normal operators $\underline{N}=(N_1, \dotsc, N_m)$ of commuting normal operators in $\mathcal{B}(\mathcal{K})$ such that $\mathcal{H}$ is invariant under $N_1, \dotsc, N_m$ and $N_j|_{\mathcal{H}}=T_j$ for each $j=1, \dots , m$. The tuple $\underline{N}$ is said to be a *normal extension* of $\underline{T}$. It follows from the theory of subnormal operators (see [@Lubin; @Athavale]) that every subnormal tuple admits a minimal normal extension to the space $$\mathcal{K}=\overline{span}\left\{N_1^{*k_1} \dotsc N_m^{*k_m}h \ : \ h \in \mathcal{H} \ \& \ \ k_1, \dotsc, k_m \in \mathbb{N}\cup \{0\} \right\},$$ and a minimal normal extension is unique upto unitary equivalence. We invoke the following results on subnormal operators to prove our theorems of this Section. **Lemma 40** ([@Bram], Theorem 8). *If $S$ is subnormal and $T$ is normal such that $ST=TS$, then $S$ and $T$ have a commuting normal extension.* **Lemma 41** ([@Abrahamese], pp. 173). *If $S$ and $T$ are commuting subnormal operators, then $S$ and $T$ have a commuting normal extension if $\sigma(T)$ is finitely connected and the spectrum of minimal normal extension of $T$ is contained in the topological boundary of $\sigma(T)$.* **Lemma 42** ([@Athavale], Proposition 0). *Let $(S_1, \dotsc, S_n)$ be a commuting $n$-tuple of contractions acting on the space $\mathcal{H}$. Then TFAE:* 1. *There is a commuting $n$-tuple $(N_1, \dotsc, N_n)$ of normal operators on the space $\mathcal{K} \supseteq \mathcal{H}$ such that $S_i=N_i|_\mathcal{H}, \ i=1, \dotsc, n$.* 2. *For every non-negative integers $k_1, \dotsc, k_n$, we have $$\underset{\substack{0 \leq p_i \leq k_i\\ 1 \leq i \leq n}}{\sum}(-1)^{p_1+\dotsc+p_n}\binom{k_1}{p_1}\dotsc \binom{k_n}{p_n}S_1^{*p_1}\dotsc S_n^{*p_n}S_1^{p_1}\dotsc S_n^{p_n} \geq 0.$$* **Lemma 43** ([@Lubin], Corollary 1). *Let $\underline{S}=(S_1, \dotsc, S_n)$ be a subnormal tuple and let $\underline{N}=(N_1, \dotsc, N_n)$ be the minimal normal extension of $\underline{S}$. Then each $N_i$ is unitarily equivalent to the minimal normal extension of $S_i$.* **Lemma 44** ([@Lubin], Corollary 2). *Let $\underline{S}=(S_1, \dotsc, S_n)$ be a subnormal tuple and let $\underline{N}=(N_1, \dotsc, N_n)$ be the minimal normal extension of $\underline{S}$. Then for any $n$-variable polynomial $p, p(\underline{N})$ is unitarily equivalent to the minimal normal extension of $p(\underline{S})$.* Also, we recall from the literature the following theorem, which gives characterizations of $\Gamma$-unitaries, appeared in parts in [@AglerVII] and [@Bhattacharyya]. We shall use this theorem below. **Theorem 45**. *Let $(S, P)$ be a pair of commuting operators on a Hilbert space $\mathcal{H}$. Then, the following statements are equivalent:* 1. *$(S, P)$ is a $\Gamma$-unitary;* 2. *there exists commuting unitary operators $U_1$ and $U_2$ on $\mathcal{H}$ such that $$S=U_1+U_2, \quad P=U_1U_2;$$* 3. *$P$ is unitary, $S=S^*P$ and $r(S) \leq 2$, where $r(S)$ is the spectral radius of $S$;* 4. *$(S, P)$ is a $\Gamma$-contraction and $P$ is unitary;* 5. *$P$ is a unitary and $S=U+U^*P$ for unitary $U$ commuting with $P$.* Our first main result of this Section is the following. **Theorem 46**. *Let $(V_1, V_2, V_3)$ be a commuting triple of operators acting on the Hilbert space $\mathcal{H}$. Then the following are equivalent.* 1. *$(V_1, V_2, V_3)$ is a $\mathbb{P}$-isometry;* 2. *$(V_1, V_2\slash 2)$ is a $\mathbb{B}_2$-isometry and $(V_2, V_3)$ is a $\Gamma$-isometry.* *Proof.* $(1) \implies (2):$ Let $(V_1,V_2,V_3)$ on $\mathcal{H}$ be a $\mathbb{P}$-isometry. Then there is a pentblock unitary $(U_1, U_2, U_3)$ on a Hilbert space $\mathcal{K} \supseteq \mathcal{H}$ such that $\mathcal{H}$ is a joint invariant subspace and $(V_1, V_2, V_3)=(U_1|_{\mathcal{H}}, U_2|_{\mathcal{H}} ,U_3|_{\mathcal{H}})$. Theorem [Theorem 33](#P_unitary){reference-type="ref" reference="P_unitary"} yields that $U_1$ is normal and $(U_2, U_3)$ is a $\mathbb{P}$-unitary and as a consequence we have that $V_1$ is subnormal and $(V_2, V_3)$ is a $\Gamma$-isometry. Since each $V_j$ is the restriction of $U_j$ to $\mathcal{H}$, we have that $V_j^*V_j=P_\mathcal{H}U_j^*U_j|_\mathcal{H}$ for $j=1, 2, 3.$ Therefore, it follows from Theorem [Theorem 33](#P_unitary){reference-type="ref" reference="P_unitary"} that $$I-\frac{1}{4}V_2^*V_2-V_1^*V_1=P_{\mathcal{H}}\bigg(I-\frac{1}{4}U_2^*U_2-U_1^*U_1\bigg)\bigg|_\mathcal{H}=0,$$ and consequently $(V_1, V_2\slash 2)$ is a $\mathbb{B}_2$-isometry by Theorem [Theorem 30](#prop3.5){reference-type="ref" reference="prop3.5"}. $(2) \implies (1):$ We first show that $(V_1, V_2, V_3)$ has a simultaneous normal extension which is same as showing that $(V_1, V_2\slash 2, V_3)$ has a simultaneous normal extension. For the ease of writing, we denote $(S_1,S_2,S_3)=(V_1, V_2\slash 2,V_3).$ Since $S_1, S_2, S_3$ are all contractions, Theorem [Lemma 42](#Athavale){reference-type="ref" reference="Athavale"} yields that there is a commuting triple $(U_1, U_2, U_3)$ of normal operators on a space $\mathcal{K}$ such that $\mathcal{H}$ is a joint invariant subspace of $\mathcal{K}$ and $S_i=N_i|_\mathcal{H}$ for $i=1,2,3$ if and only if the following operators are non-negative. 1. $\Delta_i=\underset{0 \leq p_i \leq k_i}{\sum}(-1)^{p_i}\binom{k_i}{p_i}S_i^{*p_i}S_i^{p_i}$ for $k_i \in \mathbb{N} , i=1,2,3;$ 2. $\Delta_{ij}= \underset{\substack{0 \leq p_i \leq k_i\\ 0 \leq p_j \leq k_j}}{\sum}(-1)^{p_i+p_j}\binom{k_i}{p_i} \binom{k_j}{p_j}S_i^{*p_i}S_j^{*p_j}S_i^{p_i} S_j^{p_j}$ for $k_i , k_j \in \mathbb{N}$, $i,j=1,2,3$ and $i\ne j;$ 3. $\Delta_{123}= \underset{\substack{0 \leq p_i \leq k_i\\ 1 \leq i \leq 3}}{\sum}(-1)^{p_1+p_2+p_3}\binom{k_1}{p_1}\binom{k_2}{p_2} \binom{k_3}{p_3}S_1^{*p_1}S_2^{*p_2} S_3^{*p_3}S_1^{p_1}S_2^{p_2} S_3^{p_3}$ for $k_1, k_2, k_3 \in \mathbb{N}$. We prove the non-negativity of each of the operators defined above. 1. Since $S_2$ and $S_3$ are subnormal operators, it follows again from Theorem [Lemma 42](#Athavale){reference-type="ref" reference="Athavale"} that $\Delta_2, \Delta_3 \geq 0$. Note that $S_1^*S_1+S_2^*S_2=V_1^*V_1+\frac{1}{4}V_2^*V_2=I.$ By Lemma [Lemma 29](#subnormal){reference-type="ref" reference="subnormal"}, we have that $S_1$ is subnormal and so $\Delta_1 \geq 0$. 2. Again by Theorem [Lemma 29](#subnormal){reference-type="ref" reference="subnormal"}, we have that $(S_1, S_2)$ is a subnormal pair. Thus $\Delta_{12} \geq 0$. Another application of Theorem [Lemma 42](#Athavale){reference-type="ref" reference="Athavale"} gives $\Delta_{23} \geq 0$ because, $(V_2, V_3)$ is a $\Gamma$-isometry and hence admits a simultaneous commuting normal extension which is also true for $(S_2, S_3)$. We now prove that $\Delta_{13} \geq 0$. Since $V_3$ is an isometry, $V_3$ can either be a unitary or has a non-zero shift part. $$\ $$\ *Case I:* Let $S_3=V_3$ be a unitary. Then $(S_1, S_3)$ is a commuting pair of operators such that $S_1$ is subnormal and $S_3$ is normal. By Lemma [Lemma 40](#Bram){reference-type="ref" reference="Bram"}, $S_1$ and $S_3$ have a simultaneous commuting normal extension. $$\ $$\ *Case II:* Suppose that $S_3=V_3$ has a non-zero shift part. In this case $\sigma(S_3)=\overline{\mathbb{D}}$. The minimal normal extension, say, $N_3$ of $S_3$ is a unitary and hence, we have $\sigma(N_3) \subseteq \mathbb{T}=\partial \mathbb{D}=\partial \sigma(S_3)$. Lemma [Lemma 41](#Abrahamese){reference-type="ref" reference="Abrahamese"} yields that $S_1$ and $S_3$ have a simultaneous commuting normal extension. In either case, $(S_1, S_3)$ admits a simultaneous commuting normal extension and so, $\Delta_{13} \geq 0$ which follows from Theorem [Lemma 42](#Athavale){reference-type="ref" reference="Athavale"}. 3. It is only remaining to show that $\Delta_{123} \geq 0$. In the computation of $\Delta_{123}$ we use the fact that $S_3^{*p}S_3^p=I$ for every $p \geq 0$, which happens because $V_3$ is an isometry. So, we have the following. $$\begin{split} \Delta_{123}&= \underset{\substack{0 \leq p_i \leq k_i\\ 1 \leq i \leq 3}}{\sum}(-1)^{p_1+p_2+p_3}\binom{k_1}{p_1}\binom{k_2}{p_2} \binom{k_3}{p_3}S_1^{*p_1}S_2^{*p_2} S_1^{p_1}S_2^{p_2} \\ &=\bigg[\overset{k_3}{\underset{p_3=0}{\sum}}(-1)^{p_3}\binom{k_3}{p_3}\bigg]\; \bigg[\;\overset{k_1}{\underset{p_1=0}{\sum}}\overset{k_2}{\underset{p_2=0}{\sum}}(-1)^{p_1+p_2}\binom{k_1}{p_1} \binom{k_2}{p_2}S_1^{*p_1}S_2^{*p_2}S_1^{p_1}S_2^{p_2}\bigg]\\ &=\overset{k_1}{\underset{p_1=0}{\sum}}\overset{k_2}{\underset{p_2=0}{\sum}}(-1)^{p_1+p_2}\binom{k_1}{p_1} \binom{k_2}{p_2}S_1^{*p_1}S_2^{*p_2}S_1^{p_1}S_2^{p_2},\\ \end{split}$$ where, the last equality again uses the fact that $\overset{k_3}{\underset{p_3=0}{\sum}}(-1)^{p_3}\binom{k_3}{p_3}=1$ since $k_3 \ne 0$. Now, the non-negativity of $\Delta_{12}$ gives $\Delta_{123} \geq 0$. Thus, combining everything together, we see that there is a commuting triple $(U_1, U_2, U_3)$ of normal operators on a space $\mathcal{K}$ such that $\mathcal{H}$ is a joint invariant subspace of $\mathcal{K}$ and $(V_1, V_2, V_3)=(U_1|_\mathcal{H}, U_2|_\mathcal{H}, U_3|_\mathcal{H}).$ Without loss of generality, we assume that $(U_1, U_2, U_3)$ on $\mathcal{K}$ is the minimal normal extension of the triple $(V_1, V_2, V_3)$ and the space $\mathcal{K}$ is given by $$\overline{\mbox{span}}\{U_1^{*j_1}U_2^{*j_2} U_3^{*j_3}h \ | \ j_1, j_2, j_3 \geq 0, \ h \in \mathcal{H}\}.$$ We claim that $(U_1, U_2, U_3)$ on $\mathcal{K}$ is a $\mathbb{P}$-unitary. It follows from Lemma [Lemma 43](#Lubin){reference-type="ref" reference="Lubin"} that each $U_i$ on $\mathcal{K}$ is unitarily equivalent to the minimal normal extension of $V_i$ for $i=1,2,3$. We prove that $(U_2, U_3)$ is a $\Gamma$-unitary. By Theorem [Theorem 45](#G_unitary){reference-type="ref" reference="G_unitary"}, it suffices to show that $(U_2, U_3)$ is a $\Gamma$-contraction and $U_3$ is a unitary. Note that $U_3$ is a unitary by being the normal extension of the isometry $V_3$. Let $g$ be a holomorphic polynomial in $2$-variables. Let $f(z_1, z_2, z_3)=g(z_2, z_3)$. It follows from Lemma [Lemma 44](#Lubin2){reference-type="ref" reference="Lubin2"} that $f(U_1, U_2, U_3)$ is unitarily equivalent to the minimal normal extension, say, $\widetilde{N}$ of $\widetilde{S}=f(V_1, V_2, V_3)$. Bram proved [@Bram] that a subnormal operator satisfies the spectral inclusion relation, that is $\sigma(\widetilde{N}) \subseteq \sigma(\widetilde{S})$. Since $\widetilde N$ is normal, we have that $$\begin{split} \|\widetilde{N}\|=\sup\{|\lambda| \ : \ \lambda \in \sigma(\widetilde{N})\} \leq \sup\{|\lambda| \ : \ \lambda \in \sigma(\widetilde{S})\} & = \sup\{|\lambda| \ : \ \lambda \in \sigma(f(V_1, V_2, V_3))\}\\ & = \sup\{|\lambda| \ : \ \lambda \in \sigma(g(V_2, V_3))\}\\ & = \sup\{|\lambda| \ : \ \lambda \in g(\sigma_T(V_2, V_3))\}\\ & \leq \sup\{|\lambda| \ : \ \lambda \in g(\Gamma)\}\\ &=\|g\|_{\infty, \Gamma}. \end{split}$$ Since $g(U_2, U_3)=f(U_1, U_2, U_3)$ and $f(U_1,U_2,U_3)$ is unitarily equivalent to $\widetilde{N}$, we must have $$\|g(U_2, U_3)\| =\|\widetilde{N}\| \leq \|g\|_{\infty, \Gamma}.$$ Proposition [Proposition 4](#basicprop:01){reference-type="ref" reference="basicprop:01"} yields that $(U_2, U_3)$ is a $\Gamma$-contraction. Thus, $(U_2, U_3)$ is a $\Gamma$-unitary. We now show that $(U_1,U_2)$ is a $\mathbb{B}_2$-unitary, that is $U_1^*U_1+\frac{1}{4}U_2^*U_2-I=0.$ Let $h \in \mathcal{H}$. Then $$\begin{split} & \|(U_1^*U_1+\frac{1}{4}U_2^*U_2-I)h\|^2 = \\ & \bigg(\|U_1^2h\|^2+\frac{1}{4}\|U_1U_2h\|^2-\|U_1h\|^2\bigg) +\bigg(\|U_1U_2h\|^2+\frac{1}{4}\|U_2^2h\|^2-\|U_2h\|^2\bigg) +\bigg(\|U_1h\|^2+\frac{1}{4}\|U_2h\|^2-\|h\|^2\bigg)\\ &=\bigg(\|V_1^2h\|^2+\frac{1}{4}\|V_1V_2h\|^2-\|V_1h\|^2\bigg) +\bigg(\|V_1V_2h\|^2+\frac{1}{4}\|V_2^2h\|^2-\|V_2h\|^2\bigg) +\bigg(\|V_1h\|^2+\frac{1}{4}\|V_2h\|^2-\|h\|^2\bigg)\\ & \rightline{$[\because V_i=U_i|_{\mathcal{H}}]$} \\ & =0,\\ \end{split}$$ where, the last equality holds because $$\|V_1h\|^2+\frac{1}{4}\|V_2h\|^2-\|h\|^2=\langle (V_1^*V_1-I+\frac{1}{4}V_2^*V_2)h, h \rangle=0,$$ for every $h \in \mathcal{H}$. Therefore, $(U_1^*U_1+\frac{1}{4}U_2^*U_2-I)h=0$ for every $h \in \mathcal{H}$. From the definition of $\mathcal{K}$, it follows that $U_1^*U_1+\frac{1}{4}U_2^*U_2-I=0$ on $\mathcal{K}$. Theorem [Theorem 33](#P_unitary){reference-type="ref" reference="P_unitary"} yields that $(U_1, U_2, U_3)$ on $\mathcal{K}$ is a $\mathbb{P}$-unitary. Hence, $(V_1, V_2, V_3)$ is a $\mathbb{P}$-isometry by being the restriction of the $\mathbb{P}$-unitary $(U_1, U_2, U_3)$ to that joint invariant subspace $\mathcal{H}$. The proof is now complete. ◻ The following two results are direct consequences of Theorem [Theorem 8](#Gamma_uni){reference-type="ref" reference="Gamma_uni"} and Theorem [Theorem 46](#P_isometry){reference-type="ref" reference="P_isometry"}. **Corollary 47**. *$(V_1, V_2)$ is a pair of commuting isometries if and only if $(V_1, 0, V_2)$ is a $\mathbb{P}$-isometry.* **Corollary 48**. *Let $\underline{N}=(N_1, N_2, N_3)$ be a commuting triple of bounded linear operators. Then $\underline{N}$ is a $\mathbb{P}$-unitary if and only if both $(N_1, N_2, N_3)$ and $(N_1^*, N_2^*, N_3^*)$ are $\mathbb{P}$-isometries.* *Proof.* The necessary condition follows from Theorem [Theorem 33](#P_unitary){reference-type="ref" reference="P_unitary"} and Theorem [Theorem 46](#P_isometry){reference-type="ref" reference="P_isometry"}. Let us assume that $(N_1, N_2, N_3)$ and $(N_1^*, N_2^*, N_3^*)$ are $\mathbb{P}$-isometries. In particular, each $N_j$ and $N_j^*$ are subnormal operators and so, each $N_j$ is normal. Therefore, $(N_1, N_2, N_3)$ is a commuting triple of normal operators such that $N_1^*N_1+\frac{1}{4}N_2^*N_2=I$ and $N_3^*N_3=N_3N_3^*=I$. Theorem [Theorem 8](#Gamma_uni){reference-type="ref" reference="Gamma_uni"} and Theorem [Theorem 33](#P_unitary){reference-type="ref" reference="P_unitary"} yield that $(N_1, N_2, N_3)$ is a $\mathbb{P}$-unitary. ◻ Our next theorem is an analogue of the Wold decomposition for an $\mathbb{P}$-isometry. Indeed, we show that a $\mathbb{P}$-contraction orthogonally decomposes into a $\mathbb{P}$-unitary and a pure $\mathbb{P}$-isometry. Before that we state a result due to Agler and Young from [@AglerVII], which will be useful. **Theorem 49** ([@AglerVII], Theorem 2.6). *Let $(S,P)$ be a a $\Gamma$-isometry and $\mathcal{H}=\mathcal{H}_1 \oplus \mathcal{H}_2$ be the Wold decomposition of the isometry $P$ into its unitary part $P|_{\mathcal{H}_1}$ and the shift part $P|_{\mathcal{H}_2}$. Then $\mathcal{H}_1$ and $\mathcal{H} _2$ are reducing subspaces for $S$ such that $(S|_{\mathcal{H}_1}, P|_{\mathcal{H}_1})$ is $\Gamma$-unitary and $(S|_{\mathcal{H}_2}, P|_{\mathcal{H}_2})$ is a pure $\Gamma$-isometry i.e. $(S|_{\mathcal{H}_2}, P|_{\mathcal{H}_2})$ is a pure $\Gamma$-isometry and $P|_{\mathcal{H}_2}$ is a shift.* **Theorem 50**. ***(Wold decomposition for a $\mathbb{P}$-isometry).**[\[Wold\]]{#Wold label="Wold"} Let $(V_1, V_2, V_3)$ be a $\mathbb{P}$-isometry on a Hilbert space $\mathcal{H}$. Then, there is a unique orthogonal decomposition $\mathcal{H}=\mathcal{H}_u \oplus \mathcal{H}_c$ such that $\mathcal{H}_u$ and $\mathcal{H}_c$ are reducing subspaces of $V_1, V_2, V_3$ and that $(V_1|_{\mathcal{H}_u}, V_2|_{\mathcal{H}_u},V_3|_{\mathcal{H}_u})$ is a $\mathbb{P}$-unitary and $(V_1|_{\mathcal{H}_c}, V_2|_{\mathcal{H}_c},V_3|_{\mathcal{H}_c})$ is a pure $\mathbb{P}$-isometry.* *Proof.* Let $V_3=P_3 \oplus Q_3$ with respect to the orthogonal decomposition $\mathcal{H}=\mathcal{H}_1 \oplus \mathcal{H}_2$ be the Wold decomposition of the isometry $V_3$ such that $P_3$ on $\mathcal{H}_1$ is a unitary and $Q_3$ on $\mathcal{H}_2$ is a pure isometry i.e. a unilateral shift. It follows from Theorem [Theorem 46](#P_isometry){reference-type="ref" reference="P_isometry"} that $(V_2, V_3)$ is a $\Gamma$-isometry. Therefore, Theorem [Theorem 49](#Wold_Gamma_Isometry){reference-type="ref" reference="Wold_Gamma_Isometry"} yields that $\mathcal{H}_1, \mathcal{H}_2$ are reducing subspaces for $V_2$ such that $(V_2|_{\mathcal{H}_1}, V_3|_{\mathcal{H}_1})$ is $\Gamma$-unitary and $(V_2|_{\mathcal{H}_2}, V_3|_{\mathcal{H}_2})$ is a pure $\Gamma$-isometry. Thus, if $V_2|_{\mathcal{H}_1}=P_2$ and $V_2|_{\mathcal{H}_2}=Q_2$, then with respect to the decomposition $\mathcal{H}=\mathcal{H}_1 \oplus \mathcal{H}_2$, $$\quad V_2= \begin{bmatrix} P_2 & 0\\ 0 & Q_2\\ \end{bmatrix}, \quad V_3=\begin{bmatrix} P_3 & 0\\ 0 & Q_3\\ \end{bmatrix}.$$ Suppose $$V_1=\begin{bmatrix} P_{1} & A_{12}\\ A_{21} & Q_{1}\\ \end{bmatrix}, \quad \text{with respect to } \mathcal{H}=\mathcal{H}_1 \oplus \mathcal{H}_2.$$ By the commutativity of $V_1$ with $V_3$, we have that $A_{12}Q_3=P_3A_{12}$ and $A_{21}P_3=Q_3A_{21}$. It is well-known that (see Lemma 2.13 in [@Bhattacharyya]) that no non zero operator can have such intertwining relation since $P_3$ is a unitary and $Q_3$ is a shift. Consequently, $A_{12}=A_{21}=0$. Thus, with respect to the decomposition $\mathcal{H}=\mathcal{H}_1 \oplus \mathcal{H}_2$, we have $$V_1=\begin{bmatrix} P_1 & 0\\ 0 & Q_1\\ \end{bmatrix}.$$ Evidently, $P_{1}$ and ${Q_{1}}$ are contractions. Thus, we have that the commuting triple $(P_1, P_2, P_3)$ on $\mathcal{H}_1$ is a $\mathbb{P}$-isometry such that $(P_2, P_3)$ is a $\Gamma$-unitary and $P_1$ is a subnormal contraction. We further decompose the space $\mathcal{H}_1$. It follows from Lemma 3.1 in [@Morrel] that the space $$\mathcal{H}_u=\overset{\infty}{\underset{j=0}{\bigcap}}\text{Ker}\bigg(P_1^{*j}P_1^j-P_1^jP_1^{*j}\bigg) \subseteq \mathcal{H}_1,$$ is a reducing subspace for $P_1$ on which $P_1$ acts a normal operator. Since $P_2$ and $P_3$ are normal operators that commute with $P_1$, Fuglede's theorem [@Fuglede] yields that $P_2^*$ and $P_3^*$ also commute with $P_1$. Consequently, $P_2$ and $P_3$ doubly commute with $(P_1^{*j}P_1^j-P_1^jP_1^{*j})$ for every $j \geq 0$. Thus $\mathcal{H}_u$ is a reducing subspace for $P_2$ and $P_3$ as well. With respect to the orthogonal decomposition $\mathcal{H}_1=\mathcal{H}_u \oplus \mathcal{H}_{0}$, let the block matrix form of each $P_i$ be given by $$P_1=\begin{bmatrix} U_1 & 0\\ 0 & B_1\\ \end{bmatrix}, \quad P_2=\begin{bmatrix} U_2 & 0\\ 0 & B_2\\ \end{bmatrix} \quad \text{and} \quad P_3=\begin{bmatrix} U_3 & 0\\ 0 & B_3\\ \end{bmatrix}.$$ Since $U_1, U_2, U_3$ are restrictions of $V_1, V_2, V_3$ respectively to the common reducing subspace $\mathcal{H}_u$, therefore, $U_1^*U_1=I-\frac{1}{4}U_2^*U_2.$ Hence, it follows from Theorem [Theorem 33](#P_unitary){reference-type="ref" reference="P_unitary"} that the commuting triple $(U_1, U_2, U_3)$ of normal operators acting on $\mathcal{H}_u$ is indeed a $\mathbb{P}$-unitary. With respect to the decomposition of the whole space $\mathcal{H}=\mathcal{H}_u \oplus \mathcal{H}_0 \oplus \mathcal{H}_2$, the block matrix form of each $V_i$ is given by $$V_1=\begin{bmatrix} U_1 & 0 & 0\\ 0 & B_1 & 0\\ 0 & 0 & Q_1 \end{bmatrix} , \quad V_2=\begin{bmatrix} U_2 & 0 & 0\\ 0 & B_2 & 0\\ 0 & 0 & Q_2 \end{bmatrix}, \quad V_3=\begin{bmatrix} U_3 & 0 & 0\\ 0 & B_3 & 0\\ 0 & 0 & Q_3 \end{bmatrix},$$ which we re-write as $$V_1=\begin{bmatrix} U_1 & 0 \\ 0 & S_1 \end{bmatrix} , \quad V_2=\begin{bmatrix} U_2 & 0 \\ 0 & S_2 \end{bmatrix}, \quad V_3=\begin{bmatrix} U_3 & 0 \\ 0 & S_3 \end{bmatrix},$$ with respect to the decomposition $\mathcal{H}=\mathcal{H}_u\oplus \mathcal{H}_c$, where, $\mathcal{H}_c=\mathcal{H}_0 \oplus \mathcal{H}_2$. Denoting $$(V_1|_{\mathcal{H}_c}, V_2|_{\mathcal{H}_c}, V_3|_{\mathcal{H}_c})=(S_1, S_2, S_3),$$ we now show that $(S_1,S_2,S_3)$ is a pure $\mathbb{P}$-isometry. If possible, let there be a closed joint reducing subspace say, $\mathcal{L}$ of $\mathcal{H}_c$ on which $(S_1, S_2, S_3)$ acts as a $\mathbb{P}$-unitary. Theorem [Theorem 33](#P_unitary){reference-type="ref" reference="P_unitary"} shows that $(S_2, S_3)$ is a $\Gamma$-unitary and hence $S_3$ is a unitary. Consequently, $\mathcal{L} \subseteq \mathcal{H}_1$. On the subspace $\mathcal{L}$, the contraction $P_1$ acts as a normal operator because, $\mathcal{L} \subseteq \mathcal{H}_1$ implying that $S_1|_{\mathcal{L}}=V_1|_{\mathcal{L}}=P_1|_{\mathcal{L}}$. Since $\mathcal{H}_u$ is the maximal closed subspace of $\mathcal{H}_1$ which reduces $P_1$ and on which $P_1$ acts as a normal, therefore, $\mathcal{L} \subseteq \mathcal{H}_u$. Hence, $\mathcal{L} \subseteq \mathcal{H}_u \cap \mathcal{H}_c=\{0\}$. This also shows that any closed joint reducing subspace of $\mathcal{H}$ on which $(V_1, V_2, V_3)$ acts as a $\mathbb{P}$-unitary must be contained in $\mathcal{H}_u$ and in this sense, $\mathcal{H}_u$ is maximal. We now prove the uniqueness of the decomposition. Let $\mathcal{H}=\mathcal{L}_u \oplus \mathcal{L}_c$ be an arbitrary decomposition of $\mathcal{H}$ with the properties in the statement of the theorem. The maximality of $\mathcal{H}_u$ implies that $\mathcal{L}_u \subseteq \mathcal{H}_u$. The spaces $\mathcal{H}_u$ and $\mathcal{L}_u$ reduce each $V_i$, therefore, the same is true for $\mathcal{H}_u \ominus \mathcal{L}_u$ and $(V_1, V_2, V_3)|_{\mathcal{H}_u \ominus \mathcal{L}_u}$ is a $\mathbb{P}$-unitary. Since $\mathcal{H}_u \ominus \mathcal{L}_u \subseteq \mathcal{H} \ominus \mathcal{L}_u=\mathcal{L}_c$ and since $(V_1, V_2, V_3)$ is a pure $\mathbb{P}$-isometry on $\mathcal{L}_c$, we have that $\mathcal{H}_u \ominus \mathcal{L}_u=\{0\}$. This shows that $\mathcal{H}_u =\mathcal{L}_u$ and the desired conclusion follows. ◻ Note that it is not necessary that a $\mathbb{P}$-isometry is a commuting triple $(A,S,P)$ that is a $\mathbb{P}$-contraction with $P$ being an isometry unlike the isometries associated with the symmetrized bidisc and tetrablock, (see Theorem 2.14 in [@Bhattacharyya] and Theorem 5.7 in [@Bhattacharyya-01] respectively). The following example shows this. **Example 51**. We recall Example [Example 34](#exm:imp){reference-type="ref" reference="exm:imp"} first. It follows from Theorem [Theorem 46](#P_isometry){reference-type="ref" reference="P_isometry"} that the commuting triple of subnormal operators $\underline{N}=(N_1, N_2, N_3)=(T_z, \ 0, \ -I)$ on $\ell^2(\mathbb{N})$, where $T_z$ is the unilateral shift on $\ell^2(\mathbb{N})$, is a $\mathbb{P}$-isometry but not a $\mathbb{P}$-unitary. Thus, its adjoint $(T_z^*,0,-I)$ is a $\mathbb{P}$-contraction by Lemma [Lemma 7](#basiclem:01){reference-type="ref" reference="basiclem:01"} whose last component, that is $-I$ is an isometry. However, it follows from Theorem [Theorem 46](#P_isometry){reference-type="ref" reference="P_isometry"} that $(T_z^*,0,-I)$ is not a $\mathbb{P}$-isometry. A $\mathbb{P}$-isometry with its last component being a pure isometry, plays major role in determining the structure of a $\mathbb{P}$-isometry. Indeed, in the proof of Theorem [\[Wold\]](#Wold){reference-type="ref" reference="Wold"}, the last component of the $\mathbb{P}$-isometry $(Q_1,Q_2,Q_3)=(V_1|_{\mathcal{H}_2}, V_2|_{\mathcal{H}_2}, V_3|_{\mathcal{H}_2})$ was a pure isometry. We conclude this Section by producing a concrete operator model for a $\mathbb{P}$-isometry whose last component is a pure isometry. For this we need to mention the highly efficient machinery called the fundamental operator of a $\Gamma$-contraction. It was proved in [@Bhattacharyya] that to every $\Gamma$-contraction $(S,P)$ there is a unique operator $F \in \mathcal B(\mathcal D_P)$ with numerical radius $\omega(F) \leq 1$ such that $$\label{eqn:funda-01} S-S^*P=D_PFD_P,$$ where $D_P=(I-P^*P)^{\frac{1}{2}}$ and $\mathcal D_P=\overline{Ran}\, D_P$. Indeed, a major role in the operator theory of the symmetrized bidisc is played by this unique operator. For this reason $F$ was named the *fundamental operator* of the $\Gamma$-contraction $(S,P)$. **Theorem 52**. *Let $(V_1, V_2, V_3)$ be a commuting triple of operators on a Hilbert space $\mathcal{H}$. If $V_3$ is a pure isometry, then there is a unitary operator $U: \mathcal{H} \to H^2(\mathcal{D}_{V_3^*})$ and a partial isometry $V$ on $\mathcal{H}$ such that $$V_1=VU^*D_{\frac{1}{2}T_\phi}U, \quad V_2=U^*T_\phi U , \quad \text{and} \quad V_3=U^*T_z U,$$ where $\phi(z)=F_*^*+F_*z$ and $F_*$ is the fundamental operator of $(V_2^*, V_3^*)$.* *Proof.* It follows from Theorem [Theorem 46](#P_isometry){reference-type="ref" reference="P_isometry"} that $(V_2, V_3)$ is a $\Gamma$-isometry. Since $V_3$ is a pure isometry, Theorem 2.16 in [@PalI] yields that there is a unitary operator $U: \mathcal{H} \to H^2(\mathcal{D}_{V_3^*})$ such that $$V_2=U^*T_\phi U \quad \text{and} \quad V_3=U^*T_z U, \quad \phi(z)=F_*^*+F_*z,$$ $F_*$ being the fundamental operator of $(V_2^*, V_3^*)$. Again by Theorem [Theorem 46](#P_isometry){reference-type="ref" reference="P_isometry"}, we have $V_1^*V_1=I-\frac{1}{4}V_2^*V_2$ and hence $V_1^*V_1=U^*\left(I-\frac{1}{4}T_\phi^*T_\phi\right)U.$ It follows by iteration that $$(V_1^*V_1)^n=U^*\left(I-\frac{1}{4}T_\phi^*T_\phi\right)^nU \quad \text{for} \quad n=0,1 ,2, \dotsc.$$ Consequently, we have that $p(V_1^*V_1)=U^*p\left(I-\frac{1}{4}T_\phi^*T_\phi\right)U$ for every polynomial $p(\lambda)=a_0+a_1\lambda+ \dotsc + a_n\lambda^n$. Choose a sequence of polynomials $p_m(\lambda)$ which tends to the function $\lambda^{1\slash 2}$ on the interval $0 \leq \lambda \leq 1$. The sequence of operators $p_m(T)$ converges then converges to $T^{1\slash 2}$ in operator norm. Therefore, $(V_1^*V_1)^{1\slash 2}=U^*\left(I-\frac{1}{4}T_\phi^*T_\phi\right)^{1\slash 2}U.$ Recall that for every $T \in \mathcal{B}(\mathcal{H})$, there is a partial isometry $V$ on $\mathcal{H}$ such that $T=V(T^*T)^{1 \slash 2}$. Therefore, there is a partial isometry $V$ on $\mathcal{H}$ such that $$V_1=V(V_1^*V_1)^{1\slash 2}=VU^*\left(I-\frac{1}{4}T_\phi^*T_\phi\right)^{1\slash 2}U=VU^*D_{\frac{1}{2}T_\phi}U$$ and this completes the proof. ◻ # Canonical decomposition of a $\mathbb{P}$-contraction {#decomp} As we have mentioned in Section [4](#Polyball){reference-type="ref" reference="Polyball"} (see the discussion before theorem [Theorem 31](#thm:decomp-Ball){reference-type="ref" reference="thm:decomp-Ball"}) that every contraction $T$ acting on a Hilbert space $\mathcal{H}$ admits a canonical decomposition $T_1\oplus T_2$ with respect to $\mathcal{H}=\mathcal{H}_1 \oplus \mathcal{H}_2$, where $T_1$ is a unitary and $T_2$ is a completely non-unitary contraction. The maximal reducing subspace $\mathcal{H}_1$ on which $T$ acts as a unitary is given by $$\begin{split} \mathcal{H}_1 =\{h \in \mathcal{H}: \|T^nh\|=\|h\|=\|T^{*n}h\|, \ n=1,2, \dotsc \} = \underset{n \in \mathbb{Z}}{\bigcap} Ker D_{T(n)},\\ \end{split}$$ where, $$D_{T(n)}= \left\{ \begin{array}{ll} (I-T^{*n}T^n)^{1\slash 2} & n \geq 0 \\ (I-T^{|n|}T^{*|n|})^{1\slash 2} & n <0 \,.\\ \end{array} \right.$$ An analogous result holds for a pair of doubly commuting contractions as the following result shows. **Theorem 53** ([@Pal-II], Theorem 4.2). *For a pair of doubly commuting contractions $P,Q$ acting on a Hilbert space $\mathcal{H}$, if $Q=Q_1 \oplus Q_2$ is the canonical decomposition of $Q$ with respect to the orthogonal decomposition $\mathcal{H}=\mathcal{H}_1 \oplus \mathcal{H}_2$, then $\mathcal{H}_1, \mathcal{H}_2$ are reducing subspaces for $P$.* In [@AglerVII], Agler and Young proved an analogue of canonical decomposition for a $\Gamma$-contraction $(S, P)$. Interestingly, such a decomposition of $(S, P)$ corresponds to the canonical decomposition of the contraction $P$ as the following theorem shows. **Theorem 54** ([@AglerVII], Theorem 2.8). *Let $(S, P)$ be a $\Gamma$-contraction on a Hilbert space $\mathcal{H}$. Let $\mathcal{H}_1$ be the maximal subspace of $\mathcal{H}$ which reduces $P$ and on which $P$ is unitary. Let $\mathcal{H}_2=\mathcal{H} \ominus \mathcal{H}_1$. Then $\mathcal{H}_1, \mathcal{H}_2$ reduces $S, (S|_{\mathcal{H}_1}, P|_{\mathcal{H}_1})$ is a $\Gamma$-unitary and $(S|_{\mathcal{H}_2}, P|_{\mathcal{H}_2})$ is a $\Gamma$-contraction for which $P|_{\mathcal{H}_2}$ is a completely non-unitary contraction.* Here we present a canonical decomposition of a $\mathbb{P}$-contraction. Indeed, we show that every $\mathbb{P}$-contraction admits an orthogonal decomposition into a $\mathbb{P}$-unitary and a completely non-unitary $\mathbb{P}$-contraction. We divide our proof into two parts. First we prove the result for a normal $\mathbb{P}$-contraction. **Proposition 55**. *Let $(A, S, P)$ be a normal $\mathbb{P}$-contraction on a Hilbert space $\mathcal{H}$. Then there is an orthogonal decomposition $\mathcal{H}=\mathcal{H}_u\oplus \mathcal{H}_c$ into joint reducing subspaces $\mathcal{H}_u, \mathcal{H}_c$ of $A, S, P$ such that $(A|_{\mathcal{H}_u}, S|_{\mathcal{H}_u}, P|_{\mathcal{H}_u})$ is a $\mathbb{P}$-unitary and $(A|_{\mathcal{H}_c}, S|_{\mathcal{H}_c}, P|_{\mathcal{H}_c})$ is a completely non-unitary $\mathbb{P}$-contraction. Moreover, $\mathcal{H}_u$ is the maximal closed joint reducing subspace of $\mathcal{H}$ on which $(A, S, P)$ acts as a $\mathbb{P}$-unitary.* *Proof.* It follows from Lemma [Proposition 11](#lem2.3){reference-type="ref" reference="lem2.3"} that $(S, P)$ on $\mathcal{H}$ is a $\Gamma$-contraction and thus $P$ and $S\slash 2$ are contractions. Let $\mathcal{H}=\mathcal{H}_1\oplus \mathcal{H}_2$ be the canonical decomposition of $P$. An application of Lemma [Theorem 53](#lem5.2){reference-type="ref" reference="lem5.2"} yields that $\mathcal{H}_1, \mathcal{H}_2$ are reducing subspaces for $A$ and $S$. Let $$A=\begin{bmatrix} A_1 & 0 \\ 0 & A_2 \end{bmatrix}, \quad S=\begin{bmatrix} S_1 & 0 \\ 0 & S_2 \end{bmatrix} \quad \text{and} \quad P=\begin{bmatrix} P_1 & 0 \\ 0 & P_2 \end{bmatrix}$$ with respect to the decomposition $\mathcal{H}=\mathcal{H}_1\oplus \mathcal{H}_2$, so that $P_1$ is unitary and $P_2$ is completely non-unitary. Theorem [Theorem 54](#thm5.3){reference-type="ref" reference="thm5.3"} yields that $(S_1, P_1)$ is a $\Gamma$-unitary on $\mathcal{H}_1$. We now further decompose $\mathcal{H}_1$ into an orthogonal sum of two joint reducing subspaces, say, $\mathcal{H}_1=\mathcal{H}_{11} \oplus \mathcal{H}_{12}$ so that $(A_1|_{\mathcal{H}_{11}}, S_1|_{\mathcal{H}_{11}}, P_1|_{\mathcal{H}_{11}})$ is a $\mathbb{P}$-unitary. To do this, Theorem [Theorem 33](#P_unitary){reference-type="ref" reference="P_unitary"} implies that we must have $A_1^*A_1=I-\frac{1}{4}S_1^*S_1$ on $\ \mathcal{H}_{11}$. Indeed, we take $$\mathcal{H}_{11}= Ker\bigg[I-A_1^{*}A_1-\frac{1}{4}S_1^*S_1 \bigg] \subseteq \mathcal{H}_1.$$ Since $A_1, S_1, P_1$ are commuting normal operators, it follows that $A_1, S_1, P_1$ doubly commutes with the operator $(I-A_1^{*}A_1-\frac{1}{4}S_1^*S_1)$. Therefore, $\mathcal{H}_{11}$ reduces $A_1, S_1, P_1$ and hence, $A, S, P$. For any $x \in \mathcal{H}_{11}$, we have $(I-A_1^*A_1-\frac{1}{4}S_1^*S_1)x=0.$ Then by Theorem [Theorem 33](#P_unitary){reference-type="ref" reference="P_unitary"}, $(A_1|_{\mathcal{H}_{11}}, S_1|_{\mathcal{H}_{11}}, P_1|_{\mathcal{H}_{11}})$ is a $\mathbb{P}$-unitary. Setting $\mathcal{H}_{u}=\mathcal{H}_{11}$ and $\mathcal{H}_c=\mathcal{H}\ominus \mathcal{H}_{11}$, it is immediate that $\mathcal{H}_c$ reduces $A, S, P$. We show that $(A_1|_{\mathcal{H}_c}, S_1|_{\mathcal{H}_c}, P_1|_{\mathcal{H}_c})$ is a completely non-unitary $\mathbb{P}$-contraction. Assume that there is a closed joint reducing subspace, say, $\mathcal{L}$ of $\mathcal{H}$ on which $(A, S, P)$ acts as a $\mathbb{P}$-unitary. Theorem [Theorem 33](#P_unitary){reference-type="ref" reference="P_unitary"} implies that $(S, P)$ is a $\Gamma$-unitary and hence $P$ is a unitary. Consequently, $\mathcal{L} \subseteq \mathcal{H}_1$. On the subspace $\mathcal{L}$, the triple $(A, S, P)$ acts as $\mathbb{P}$-unitary. Thus, Theorem [Theorem 33](#P_unitary){reference-type="ref" reference="P_unitary"} yields that $A^*A-\frac{1}{4}S^*S-I=0$ on $\ \mathcal{L}$. Consequently, $\mathcal{L} \subseteq \mathcal{H}_u$. Hence, $\mathcal{H}_u$ is the maximal closed joint reducing subspace of $\mathcal{H}$ restricted to which $(A, S, P)$ acts as a $\mathbb{P}$-unitary. Let $\mathcal{L}$ be a closed joint reducing subspace of $\mathcal{H}_c$ on which $(A, S, P)$ acts as a $\mathbb{P}$-unitary. Since $\mathcal{H}_u$ is a maximal such subspace, $\mathcal{L} \subseteq \mathcal{H}_u$. Hence, $$\mathcal{L} \subseteq \mathcal{H}_u \cap \mathcal{H}_c=\{0\}$$ and so, $(A|_{\mathcal{H}_c}, S|_{\mathcal{H}_c}, P|_{\mathcal{H}_c})$ is a completely non-unitary $\mathbb{P}$-contraction. The proof is now complete. ◻ Now we are going to present the main theorem of this Section, the canonical decomposition of a $\mathbb{P}$-contraction. We shall follow the same notations as in Section [4](#Polyball){reference-type="ref" reference="Polyball"}, that is to say for a commuting operator tuple $\underline{T}=(T_1, \dotsc, T_n)$ and for $\alpha=(\alpha_1, \dotsc, \alpha_n) \in \mathbb{N}^n$, we write $\underline{T}^\alpha=T_1^{\alpha_1} \dotsc T_n^{\alpha_n}$ and $\underline{T}^{*\alpha}=T_1^{*\alpha_1} \dotsc T_n^{*\alpha_n}.$ **Theorem 56**. ***(Canonical decomposition of a $\mathbb{P}$-contraction).** Let $(A, S, P)$ be a $\mathbb{P}$-contraction on a Hilbert space $\mathcal{H}$. Then $\mathcal{H}$ admits an orthogonal decomposition $\mathcal{H}=\mathcal{H}_u\oplus \mathcal{H}_c$ into joint reducing subspaces $\mathcal{H}_u, \mathcal{H}_c$ of $A, S, P$ such that $(A|_{\mathcal{H}_u}, S|_{\mathcal{H}_u}, P|_{\mathcal{H}_u})$ is a $\mathbb{P}$-unitary and $(A|_{\mathcal{H}_c}, S|_{\mathcal{H}_c}, P|_{\mathcal{H}_c})$ is a completely non-unitary $\mathbb{P}$-contraction.* *Proof.* Let $\underline{T}=(A, S, P)$ and let $$\mathcal{H}_0=\bigcap_{\alpha \in \mathbb{N}^3}\bigcap_{ \beta \in \mathbb{N}^3}Ker\bigg[\underline{T}^\alpha\underline{T}^{*\beta}-\underline{T}^{*\beta}\underline{T}^{\alpha}\bigg].$$ It follows from Eschmeier's work (see Corollary 4.2 in [@Eschmeier]) that $\mathcal{H}_0$ is the largest joint reducing subspace of $A,S,P$ on which $(A, S, P)$ acts as a commuting triple of normal operators. Let $(A_0, \ S_0, \ P_0)=(A|_{\mathcal{H}_0}, S|_{\mathcal{H}_0}, P|_{\mathcal{H}_0})$ which is a $\mathbb{P}$-contraction consisting of normal operators. Theorem [Proposition 55](#normal){reference-type="ref" reference="normal"} yields that $\mathcal{H}_0$ admits an orthogonal decomposition $\mathcal{H}_0=\mathcal{H}_u\oplus \mathcal{H}_{0c}$ such that $\mathcal{H}_u$ and $\mathcal{H}_c$ reduce $A_0, S_0, P_0$ and hence $A, S, P$. Moreover, $(A|_{\mathcal{H}_u}, S|_{\mathcal{H}_u}, P|_{\mathcal{H}_u})$ is a $\mathbb{P}$-unitary and $(A|_{\mathcal{H}_{0c}}, S|_{\mathcal{H}_{0c}}, P|_{\mathcal{H}_{0c}})$ is a completely non-unitary $\mathbb{P}$-contraction. Let $\mathcal{H}_c=\mathcal{H}\ominus\mathcal{H}_u.$ Let if possible, there is a non-zero closed joint reducing subspace $\mathcal{L}$ of $\mathcal{H}_u$ on which $(A, S, P)$ acts as a $\mathbb{P}$-unitary. In particular, $(A, S, P)$ acts as a commuting triple of normal operators on $\mathcal{H}_u$ and so, $\mathcal{L} \subseteq \mathcal{H}_0$. Since $\mathcal{H}_u$ is the maximal joint reducing subspace of $\mathcal{H}_0$ restricted to which $(A, S, P)$ is a $\mathbb{P}$-unitary, we have that $\mathcal{L} \subseteq \mathcal{H}_u$. Therefore, $\mathcal{L} \subseteq \mathcal{H}_u \cap \mathcal{H}_c=\{0\}$. Hence, $(A|_{\mathcal{H}_c}, S|_{\mathcal{H}_c}, P|_{\mathcal{H}_c})$ is a completely non-unitary $\mathbb{P}$-contraction and proof is complete. ◻ # Dilation of a $\mathbb{P}$-contraction {#dilation} In this Section, we find a necessary and sufficient condition such that a $\mathbb{P}$-contraction $(A,S,P)$ admits a $\mathbb{P}$-isometric dilation on the minimal dilation space of the contraction $P$ and then explicitly construct such a dilation. Note that the existence of a $\mathbb{P}$-siometric dilation guarantees the existence of a $\mathbb{P}$-unitary dilation as every $\mathbb{P}$-isometry extends to a $\mathbb{P}$-unitary. However, our result does not ensure the success of rational dilation on the pentablock. Again, the pentablock is a polynomially convex domain. So, the Oka-Weil theorem (see CH-7 of [@Oka-Weil]) yields that the algebra of polynomials is dense in the rational algebra $\mathcal R(\overline{\mathbb{P}})$. Also, rational dilation for a $\mathbb{P}$-contraction is just a $\mathbb{P}$-unitary dilation of it. So, the definition of rational dilation (as in Section [2](#basic){reference-type="ref" reference="basic"}) can we simplified using the polynomials or more precisely the monomials only. Below we define $\mathbb{P}$-isometric and $\mathbb{P}$-unitary dilations of a $\mathbb{P}$-contraction. **Definition 57**. Let $(A, S, P)$ be a $\mathbb{P}$-contraction on $\mathcal{H}$. A $\mathbb{P}$-isometry (or $\mathbb{P}$-unitary) $(X, T, V)$ acting on $\mathcal{K} \supseteq \mathcal{H}$ is said to be a *$\mathbb{P}$-isometric dilation* (or a $\mathbb{P}$-*unitary dilation*) of $(A, S, P)$ if $$A^{i}S^{j}P^{k}=P_\mathcal{H}X^{i}T^{j}V^{k}|_\mathcal{H}, \quad \text{for all } \ \ i, j, k \in \mathbb{N} \cup \{0\}.$$ Moreover, such a $\mathbb{P}$-isometric dilation is called *minimal* if $$\mathcal{K}=\overline{\text{span}}\{X^iT^jV^kh \ : \ h \in \mathcal{H} \ \text{and} \ i, j, k \in \mathbb{N} \cup \{0\} \}.$$ However, the *minimality* of a $\mathbb{P}$-unitary dilation demands $i,j,k$ to vary over the set of integers $\mathbb{Z}$. We begin with a few preparatory results associated with $\mathbb{P}$-contractions. **Proposition 58**. *If a $\mathbb{P}$-contraction $(A, S, P)$ defined on $\mathcal{H}$ has a $\mathbb{P}$-isometric dilation, then it has a minimal $\mathbb{P}$-isometric dilation.* *Proof.* Let $(X, T, V)$ on $\mathcal{K} \supseteq \mathcal{H}$ be a $\mathbb{P}$-isometric dilation of $(A, S, P)$. Let $\mathcal{K}_0$ be the space defined as $$\mathcal{K}_0= \overline{\text{span}}\{X^iT^jV^kh \ : \ h \in \mathcal{H} \ \text{and} \ i, j, k \in \mathbb{N} \cup \{0\} \}.$$ It is easy to see that $\mathcal{K}_0$ is invariant under $X^{i}, T^j$ and $V^k$, for any non-negative integers $i, j, k$. Therefore, if we denote the restrictions of $X, T, V$ to the common invariant subspace $\mathcal{K}_0$ by $X_1, T_1, V_1$ respectively, we get $X_1^iy=X^iy$, $T_1^jy=T^jy$, and $V_1^ky=V^ky$ for all $y \in \mathcal{K}_0$. Hence $$\mathcal{K}_0= \overline{\text{span}}\{X_1^iT_1^jV_1^kh \ : \ h \in \mathcal{H} \ \text{and} \ i, j, k \in \mathbb{N} \cup \{0\} \}.$$ Therefore, for any non-negative integers $i, j$ and $k$, we have that $P_\mathcal{H}(X_1^iT_1^jV_1^k)h=A^iS^jP^kh$, for all $h \in \mathcal{H}$. Since $(X, T, V)$ on $\mathcal{K}$ is a $\mathbb{P}$-isometry, it follows from the definition that there is a larger space $\mathcal{K}$ containing $\mathcal{H}$ and a $\mathbb{P}$-unitary $(U_1, U_2, U_3)$ on $\mathcal{K}$ such that $\mathcal{H}$ is a common invariant subspace of $\mathcal{K}$ and $(X, T, V)=(U_1|_{\mathcal{H}}, U_2|_{\mathcal{H}}, U_3|_{\mathcal{H}}).$ Since $\mathcal{K}_0$ is a subspace of $\mathcal{H}$ which is invariant under $X, T$ and $V$, we have that $$(X_1, T_1, V_1)=(X|_{\mathcal{K}_0}, T|_{\mathcal{K}_0}, V|_{\mathcal{K_0}})=(U_1|_{\mathcal{K}_0}, U_2|_{\mathcal{K}_0}, U_3|_{\mathcal{K_0}}).$$ Therefore, $(X_1, T_1, V_1)$ on $\mathcal{K}_0$ is a minimal $\mathbb{P}$-isometric dilation of $(A, S, P)$. ◻ **Proposition 59**. *Let $(X, T, V)$ on $\mathcal{K}$ be an $\mathbb{P}$-isometric dilation of a $\mathbb{P}$-contraction $(A, S, P)$ on $\mathcal{H}$. If $(X, T, V)$ is minimal, then $(A^*, S^*, P^*)$ is a $\mathbb{P}$-isometric extension of $(X^*, T^*, V^*)$.* *Proof.* We first prove that $AP_\mathcal{H}=P_\mathcal{H}X, SP_\mathcal{H}=P_\mathcal{H}T$ and $PP_\mathcal{H}=P_\mathcal{H}V$. Clearly $$\mathcal{K}=\overline{\text{span}}\{X^{i}T^{j}V^{k}h \ : \ h \in \mathcal{H} \ \text{and} \ i, j, k \in \mathbb{N} \cup \{0\} \}.$$ Now for $h \in \mathcal{H}$, we have $$AP_\mathcal{H}(X^{i}T^{j}V^{k}h)=A(A^{i}S^{j}P^{k}h)=A^{i+1}S^{j}P^{k}h=P_\mathcal{H}(X^{i+1}T^{j}V^{k}h)=P_\mathcal{H}X(X^{i}T^{j}V^{k}h).$$ Thus we get that $AP_\mathcal{H}=P_\mathcal{H}X$ and similarly, we can show that $SP_\mathcal{H}=P_\mathcal{H}T$ and $PP_\mathcal{H}=P_\mathcal{H}V$. Also for $h \in \mathcal{H}$ and $k \in \mathcal{K}$, we have $$\langle A^*h, k \rangle =\langle P_\mathcal{H}A^*h, k \rangle =\langle A^*h, P_\mathcal{H}k \rangle =\langle h, AP_\mathcal{H}k \rangle =\langle h, P_\mathcal{H}Xk \rangle =\langle X^*h, k \rangle.$$ Hence, $A^*=X^*|_\mathcal{H}$ and similarly $S^*=T^*|_\mathcal{H}$ and $P^*=V^*|_\mathcal{H}$. The proof is complete. ◻ We have explained in Section [3](#Prelims){reference-type="ref" reference="Prelims"} the connection between $\mathbb{P}$-contractions and the operator theory on the symmetrized bidisc. Indeed, Proposition [Proposition 11](#lem2.3){reference-type="ref" reference="lem2.3"} shows that if $(A,S,P)$ is a $\mathbb{P}$-contraction then $(S,P)$ is a $\Gamma$-contraction. For this reason, the success of rational dilation on $\Gamma$ (see [@AglerII; @Bhattacharyya]) will play a major role in the dilation of a $\mathbb{P}$-contraction. In [@Bhattacharyya], an explicit $\Gamma$-isometric dilation was constructed for any $\Gamma$-contraction. Below we mention this dilation theorem from [@Bhattacharyya]. **Theorem 60** ([@Bhattacharyya], Theorem 4.3). *Let $(S,P)$ be a $\Gamma$-contraction on a Hilbert space $\mathcal{H}$. Let $F$ be the fundamental operator of $(S,P)$, that is, unique solution of the operator equation $S-S^*P=D_PXD_P$ as in $(\ref{eqn:funda-01})$. Consider the operators $T_F, V_0$ defined on $\mathcal{H}\bigoplus \ell^2(\mathcal{D}_P)$ by $$\begin{split} & T_F(x_0, x_1, x_2, \dotsc)=(Sh_0, F^*D_Ph_0+Fh_1, F^*h_1+Fh_2, F^*h_2+Fh_1, \dotsc)\\ & V_0(x_0, x_1, x_2, \dotsc)=(Ph_0, D_Ph_0,h_1, h_2, \dotsc). \end{split}$$ Then* 1. *$(T_F, V_0)$ is a $\Gamma$-isometric dilation of $(S,P)$.* 2. *If $(\widehat{T}, V_0)$ on $\mathcal H \oplus l^2(\mathcal D_P)$ is a $\Gamma$-isometric dilation of $(S, P)$, then $\widehat{T}=T_F$.* 3. *If $(T, V)$ is a $\Gamma$-isometric dilation of $(S, P)$ where $V$ is a minimal isometric dilation of $P$, then $(T, V)$ is unitarily equivalent to $(T_F, V_0)$.* It is evident from the definition that with respect to the decomposition $\mathcal H \oplus l^2(\mathcal D_P) = \mathcal H \oplus \mathcal D_P \oplus \mathcal D_P \oplus \dots$, the operators $T_F, V_0$ have the following form: $$T_F= \begin{bmatrix} S & 0 & 0 & 0 & \dotsc \\ F^*D_{P} & F & 0 & 0 & \dotsc \\ 0 & F^* & F & 0 & \dotsc\\ 0 & 0 & F^* & F & \dotsc\\ \dotsc & \dotsc & \dotsc & \dotsc & \dotsc \\ \end{bmatrix}, \quad V_0= \begin{bmatrix} P & 0 & 0 & 0 & \dotsc \\ D_{P} & 0 & 0 & 0 & \dotsc \\ 0 & I & 0 & 0 & \dotsc\\ 0 & 0 & I & 0 & \dotsc\\ \dotsc & \dotsc & \dotsc & \dotsc & \dotsc \\ \end{bmatrix}.$$ it is evident that if $(X, T, V)$ is a $\mathbb{P}$-isometric dilation of a $\mathbb{P}$-contraction $(A, S, P)$, then $(T,V)$ is a $\Gamma$-isometric dilation of the $\Gamma$-contraction $(S,P)$. Again, Theorem [Theorem 60](#thm6.2){reference-type="ref" reference="thm6.2"} tells us that if $V$ is the minimal isometric dilation of $P$, then $(T, V)$ is unitarily equivalent to $(T_F, V_0)$. Taking cue from this, we find a necessary and sufficient conditions such that $(A, S, P)$ dilates to a $\mathbb{P}$-isometry $(X, T_F, V_0)$ acting on the space $\mathcal{H} \oplus \ell^2(\mathcal{D}_P)$. **Theorem 61**. *Let $(A, S, P)$ be a $\mathbb{P}$-contraction on $\mathcal{H}$. Then $(A, S, P)$ admits a $\mathbb{P}$-isometric dilation $(X,T,V)$ with $V$ being the minimal isometric dilation of $P$ if and only if there exist sequences $(X_{n1})_{n=2}^\infty$ and $(X_n)_{n=2}^\infty$ of operators acting on $\mathcal{H}$ and $\mathcal{D}_P$ respectively such that the following hold:* 1. *$X_{n1}=X_{n+1, 1}P+X_{n+1}D_P \quad$ for $n=2,3, \dotsc$ ,\ * 2. *$X_{21}P+X_2D_P=D_PA$,\ * 3. *$X_{21}S+X_2F^*D_P=F^*D_PA+FX_{21}$,\ * 4. *$X_{n1}S+X_nF^*D_P=F^*X_{n-1, 1}+FX_{n1} \quad$ for $n=3,4,\dotsc$,\ * 5. *$X_2F=FX_2$,\ * 6. *$X_nF+X_{n-1}F^*=F^*X_{n-1}+FX_n \quad$ for $n=3,4,\dotsc$,\ * 7. *$I-A^*A-\frac{1}{4}S^*S=\overset{\infty}{\underset{n=2}{\sum}}X_{n1}^*X_{n1}+\frac{1}{4}D_PFF^*D_P$,\ * 8. *$\overset{\infty}{\underset{n=2}{\sum}}X_{n}^*X_{n}=I-\frac{1}{4}(F^*F+FF^*)$,\ * 9. *$\overset{\infty}{\underset{n=2}{\sum}}X_{n}^*X_{n+k,1}=0=\overset{\infty}{\underset{n=2}{\sum}}X_{n+k+1}^*X_{n} \quad$ for $k=1,2, \dotsc$,\ * 10. *$\overset{\infty}{\underset{n=2}{\sum}}X_{n1}^*X_{n}+\frac{1}{4}D_PF^2=0=\overset{\infty}{\underset{n=2}{\sum}}X_{n+1}^*X_{n}+\frac{1}{4}F^2$.* *Proof.* Suppose that a $\mathbb{P}$-contraction $(A, S, P)$ acting on a Hilbert space $\mathcal{H}$ dilates to a $\mathbb{P}$-isometry $(X, T, V)$ on a Hilbert spaces $\mathcal{K}$ with $V$ being the minimal isometric dilation of $P$. Since the minimal isometric dilation of a contraction is unique upto a unitary, without loss of generality let us assume that $V$ is the Schäffer's minimal isometric of $P$, that is $$V= \begin{bmatrix} P & 0 & 0 & 0 & \dotsc \\ D_{P} & 0 & 0 & 0 & \dotsc \\ 0 & I & 0 & 0 & \dotsc\\ 0 & 0 & I & 0 & \dotsc\\ \dotsc & \dotsc & \dotsc & \dotsc & \dotsc \\ \end{bmatrix}$$ acting on the space $\mathcal K =\mathcal{H}\oplus \mathcal D_P \oplus \mathcal D_P \oplus \dots$ . Then $V=\begin{bmatrix} P & 0\\ C_3 & E_3 \end{bmatrix}$ with respect to the decomposition $\mathcal{H} \oplus \ell^2(\mathcal{D}_P)$ of $\mathcal{K}$, where, $$C_3=\begin{bmatrix} D_P\\ 0 \\ 0\\ \dotsc \end{bmatrix} \ : \ \mathcal{H} \to \mathcal{D}_P \oplus \mathcal{D}_P \oplus \mathcal{D}_P \oplus \dotsc \quad \& \quad E_3= \begin{bmatrix} 0 & 0 & 0 & \dotsc \\ I & 0 & 0 & \dotsc\\ 0 & I & 0 & \dotsc\\ \dotsc & \dotsc & \dotsc & \dotsc \\ \end{bmatrix} \ \ \text{on} \ \mathcal{D}_P\oplus \mathcal{D}_P \oplus \mathcal{D}_P \oplus \dotsc .$$ Using the $2 \times 2$ block matrix form of $V$ and the fact that $X$ and $T$ commute with $V$, it follows from straightforward computation that with respect to the decomposition $\mathcal{K}=\mathcal{H} \oplus \ell^2(\mathcal{D}_P)$, the operators $X$ and $T$ have the following operator matrix forms: $$X=\begin{bmatrix} A & 0\\ C_1 & E_1 \end{bmatrix} \quad \& \quad T=\begin{bmatrix} S & 0\\ C_2 & E_2 \end{bmatrix},$$ for some $C_i$ and $E_i$ and $1\leq i \leq 2$. It then follows from Theorem [Theorem 60](#thm6.2){reference-type="ref" reference="thm6.2"} that $$T=T_F= \begin{bmatrix} S & 0 & 0 & 0 & \dotsc \\ F^*D_{P} & F & 0 & 0 & \dotsc \\ 0 & F^* & F & 0 & \dotsc\\ 0 & 0 & F^* & F & \dotsc\\ \dotsc & \dotsc & \dotsc & \dotsc & \dotsc \\ \end{bmatrix} \quad \text{and} \quad V=V_0= \begin{bmatrix} P & 0 & 0 & 0 & \dotsc \\ D_{P} & 0 & 0 & 0 & \dotsc \\ 0 & I & 0 & 0 & \dotsc\\ 0 & 0 & I & 0 & \dotsc\\ \dotsc & \dotsc & \dotsc & \dotsc & \dotsc \\ \end{bmatrix},$$ with respect to the decomposition $\mathcal{K} = \mathcal{H} \oplus \mathcal{D}_P\oplus \mathcal{D}_P\oplus \dotsc$. Suppose that with respect to the same decomposition of $\mathcal{K}, X$ has the operator matrix form given by $$X= \begin{bmatrix} A & 0 & 0 & 0 & \dotsc \\ X_{21} & X_{22} & X_{23} & X_{24} & \dotsc \\ X_{31} & X_{32} & X_{33} & X_{34} & \dotsc \\ X_{41} & X_{42} & X_{43} & X_{44} & \dotsc \\ \dotsc & \dotsc & \dotsc & \dotsc & \dotsc \\ \end{bmatrix}.$$ Some routine but laborious calculations yield the following: $$XV_0= \begin{bmatrix} AP & 0 & 0 & 0 & \dotsc \\ X_{21}P+X_{22}D_P & X_{23} & X_{24} & X_{25} & \dotsc \\ X_{31}P+X_{32}D_P & X_{33} & X_{34} & X_{35} & \dotsc\\ X_{41}P+X_{42}D_P & X_{43} & X_{44} & X_{45} & \dotsc\\ \dotsc & \dotsc & \dotsc & \dotsc & \dotsc \\ \end{bmatrix} \quad \text{and} \quad V_0X= \begin{bmatrix} PA & 0 & 0 & 0 & \dotsc \\ D_{P}A & 0 & 0 & 0 & \dotsc \\ X_{21} & X_{22} & X_{23} & X_{24} & \dotsc \\ X_{31} & X_{32} & X_{33} & X_{34} & \dotsc \\ \dotsc & \dotsc & \dotsc & \dotsc & \dotsc \\ \end{bmatrix}.$$ This shows that $X$ and $V_0$ commute if and only if 1. $X_{ij}=0 \ $ for all $2 \leq i<j$, 2. $X_{ii}=X_{i+k, i+k} \ $ for all $i \geq 2$ and $k \in \mathbb{N}$, 3. $X_{21}P+X_{22}D_P=D_PA$, 4. $X_{n1}=X_{n+1, 1}P+X_{n+1, 2}D_P$ for all $n \geq 2$. Hence, the operator matrix form of $X$ with respect to the decomposition of $\mathcal{K}=\mathcal{H} \oplus \mathcal{D}_P\oplus \mathcal{D}_P\oplus \dotsc$ takes the form $$\label{eqnX} X= \begin{bmatrix} A & 0 & 0 & 0 & \dotsc \\ X_{21} & X_2 & 0 & 0 & \dotsc \\ X_{31} & X_3 & X_2 & 0 & \dotsc \\ X_{41} & X_4 & X_3 & X_2 & \dotsc \\ \dotsc & \dotsc & \dotsc & \dotsc & \dotsc \\ \end{bmatrix}\ ,,$$ where $$\label{eqn6.2} X_{21}P+X_2D_P=D_PA \quad \text{and} \quad X_{n1}=X_{n+1, 1}P+X_{n+1}D_P, \quad n=2,3, \dotsc.$$ Again, straightforward computations show that $$XT_F= \begin{bmatrix} AS & 0 & 0 & 0 & \dotsc \\ X_{21}S+X_2F^*D_P & X_2F & 0 & 0 & \dotsc \\ X_{31}S+X_3F^*D_P & X_3F+X_2F^* & X_2F & 0 & \dotsc \\ X_{41}S+X_4F^*D_P & X_4F+X_3F^* & X_3F+X_2F^* & X_2F & \dotsc \\ X_{51}S+X_5F^*D_P & X_5F+X_4F^* & X_4F+X_3F^* & X_3F+X_2F^* & \dotsc \\ \dotsc & \dotsc & \dotsc & \dotsc & \dotsc \\ \end{bmatrix}$$ and $$T_FX= \begin{bmatrix} SA & 0 & 0 & 0 & \dotsc \\ F^*D_PA+FX_{21} & FX_2 & 0 & 0 & \dotsc \\ F^*X_{21}+FX_{31} & F^*X_2+FX_3 & FX_2 & 0 & \dotsc \\ F^*X_{31}+FX_{41} & F^*X_3+FX_4 & F^*X_2+FX_3 & FX_2 & \dotsc \\ F^*X_{41}+FX_{51} & F^*X_4+FX_5 & F^*X_3+FX_4 & F^*X_2+FX_3 & \dotsc \\ \dotsc & \dotsc & \dotsc & \dotsc & \dotsc \\ \end{bmatrix}.$$ Therefore, $X$ and $T_F$ commutes if and only if $$\label{eqn6.3} \left.\begin{split} & (a) \quad X_{21}S+X_2F^*D_P=F^*D_PA+FX_{21}, \\ & (b) \quad X_{n1}S+X_nF^*D_P=F^*X_{n-1, 1}+FX_{n1} \quad \text{for} \quad n=3,4,\dotsc, \\ & (c) \quad X_2F=FX_2 \quad \text{and} \\ & (d) \quad X_nF+X_{n-1}F^*=F^*X_{n-1}+FX_n \quad \text{for} \quad n=3,4,\dotsc. \end{split}\right\}$$ Again, a sequence of routine computations yield $$X^*X= \begin{bmatrix} A^*A+\overset{\infty}{\underset{n=2}{\sum}}X_{n1}^*X_{n1} & \overset{\infty}{\underset{n=2}{\sum}}X_{n1}^*X_{n} & \overset{\infty}{\underset{n=2}{\sum}}X_{n+1, 1}^*X_{n} & \overset{\infty}{\underset{n=2}{\sum}}X_{n+2, 1}^*X_{n} & \dotsc \\ \\ \overset{\infty}{\underset{n=2}{\sum}}X_{n}^*X_{n1} & \overset{\infty}{\underset{n=2}{\sum}}X_{n}^*X_{n} & \overset{\infty}{\underset{n=2}{\sum}}X_{n+1}^*X_{n} & \overset{\infty}{\underset{n=2}{\sum}}X_{n+2}^*X_{n} & \dotsc \\ \\ \overset{\infty}{\underset{n=2}{\sum}}X_{n}^*X_{n+1, 1} & \overset{\infty}{\underset{n=2}{\sum}}X_{n}^*X_{n+1} & \overset{\infty}{\underset{n=2}{\sum}}X_{n}^*X_{n} & \overset{\infty}{\underset{n=2}{\sum}}X_{n+1}^*X_{n} & \dotsc \\ \\ \overset{\infty}{\underset{n=2}{\sum}}X_{n}^*X_{n+2, 1} & \overset{\infty}{\underset{n=2}{\sum}}X_{n}^*X_{n+2} & \overset{\infty}{\underset{n=2}{\sum}}X_{n}^*X_{n+1} & \overset{\infty}{\underset{n=2}{\sum}}X_{n}^*X_{n} & \dotsc \\ \dotsc & \dotsc & \dotsc & \dotsc & \dotsc \\ \end{bmatrix}$$ and $$T_F^*T_F= \begin{bmatrix} S^*S+D_PFF^*D_P & D_PF^2 & 0 & 0 & \dotsc \\ F^{*2}D_{P} & F^*F+FF^* & F^2 & 0 & \dotsc \\ 0 & F^{*2} & F^*F+FF^* & F^2 & \dotsc\\ 0 & 0 & F^{*2} & F^*F+FF^* & \dotsc\\ \dotsc & \dotsc & \dotsc & \dotsc & \dotsc \\ \end{bmatrix}.$$ Hence, $X^*X+\frac{1}{4}T_F^*T_F= I$ if and only if $$\label{eqn6.4} \left.\begin{split} & (a) \quad I-A^*A-\frac{1}{4}S^*S=\overset{\infty}{\underset{n=2}{\sum}}X_{n1}^*X_{n1}+\frac{1}{4}D_PFF^*D_P\\ & (b) \quad \overset{\infty}{\underset{n=2}{\sum}}X_{n}^*X_{n}=I-\frac{1}{4}(F^*F+FF^*)\\ & (c) \quad \overset{\infty}{\underset{n=2}{\sum}}X_{n}^*X_{n+k,1}=0=\overset{\infty}{\underset{n=2}{\sum}}X_{n+k+1}^*X_{n} \quad \text{for} \ k=1,2, \dotsc\\ & (d) \quad \overset{\infty}{\underset{n=2}{\sum}}X_{n1}^*X_{n}+\frac{1}{4}D_PF^2=0=\overset{\infty}{\underset{n=2}{\sum}}X_{n+1}^*X_{n}+\frac{1}{4}F^2.\\ \end{split}\right\}$$ Hence, the necessary part follows from equations ([\[eqn6.2\]](#eqn6.2){reference-type="ref" reference="eqn6.2"}) -- ([\[eqn6.4\]](#eqn6.4){reference-type="ref" reference="eqn6.4"}).\ Conversely, let us assume that the operator equations given in the statement of the theorem hold. Set $$X= \begin{bmatrix} A & 0 & 0 & 0 & \dotsc \\ X_{21} & X_2 & 0 & 0 & \dotsc \\ X_{31} & X_3 & X_2 & 0 & \dotsc \\ X_{41} & X_4 & X_3 & X_2 & \dotsc \\ \dotsc & \dotsc & \dotsc & \dotsc & \dotsc \\ \end{bmatrix}, \; T_F= \begin{bmatrix} S & 0 & 0 & 0 & \dotsc \\ F^*D_{P} & F & 0 & 0 & \dotsc \\ 0 & F^* & F & 0 & \dotsc\\ 0 & 0 & F^* & F & \dotsc\\ \dotsc & \dotsc & \dotsc & \dotsc & \dotsc \\ \end{bmatrix}, \; V_0= \begin{bmatrix} P & 0 & 0 & 0 & \dotsc \\ D_{P} & 0 & 0 & 0 & \dotsc \\ 0 & I & 0 & 0 & \dotsc\\ 0 & 0 & I & 0 & \dotsc\\ \dotsc & \dotsc & \dotsc & \dotsc & \dotsc \\ \end{bmatrix}$$ on the space $\mathcal{H} \oplus \mathcal{D}_P\oplus \mathcal{D}_P\oplus \dotsc=\mathcal{H} \oplus \ell^2(\mathcal{D}_P)$. It follows from Theorem [Theorem 60](#thm6.2){reference-type="ref" reference="thm6.2"} that $(T_F, V_0)$ is a $\Gamma$-isometry on $\mathcal{H} \oplus \ell^2(\mathcal{D}_P)$. Again, using the same computations for equations ([\[eqn6.2\]](#eqn6.2){reference-type="ref" reference="eqn6.2"}) -- ([\[eqn6.4\]](#eqn6.4){reference-type="ref" reference="eqn6.4"}), we get that $X$ commutes with both $T_F$ as well as with $V_0$ and $X^*X+\frac{1}{4}T_F^*T_F=I.$ Consequently, Theorem [Theorem 46](#P_isometry){reference-type="ref" reference="P_isometry"} yields that $(X, T_F, V_0)$ is a $\mathbb{P}$-isometry on $\mathcal{H} \oplus \ell^2(\mathcal{D}_P)$. Evidently, $A^*=X^*|_\mathcal{H}$, $S^*=T_F^*|_\mathcal{H}$ and $P^*=V_0^*|_\mathcal{H}$ and hence $(X,T,V)$ dilates $(A,S,P)$. The proof is now complete. ◻ The conditions in Theorem [Theorem 61](#thm:main-dilation){reference-type="ref" reference="thm:main-dilation"} can be a bit relaxed if want a dilation of a special kind. Indeed, we shall see below that under seven of the conditions as in theorem [Theorem 61](#thm:main-dilation){reference-type="ref" reference="thm:main-dilation"}, we can exhibit a particular $\mathbb{P}$-isometric dilation of a $\mathbb{P}$-contraction $(A,S,P)$ on the minimal dilation space of $P$. However, Theorem [Theorem 61](#thm:main-dilation){reference-type="ref" reference="thm:main-dilation"} provides the general case which can come only in the presence of all ten conditions. **Theorem 62**. *Let $(A, S, P)$ be a $\mathbb{P}$-contraction on a Hilbert space $\mathcal{H}$. if there are two operators $F_1, F_2\in \mathcal B(\mathcal{D}_P)$ satisfying the following* *$$\begin{split} & 1. \ \ F_2D_PP+F_1D_P=D_PA, \\ & 3. \ \ F_1F=FF_1 , \\ & 5. \ \ F_2F+F_1F^*=FF_2+F^*F_1 , \\ & 7. \ \ I-A^*A-\frac{1}{4}S^*S=D_P\bigg(F_2^*F_2+\frac{1}{4}FF^*\bigg)D_P ,\\ \end{split}$$* *$$\label{eqn7} \begin{split} & 2. \ \ F_2F^*=F^*F_2 , \\ & 4. \ \ F_2^*F_1+F^2 \slash 4=0, \\ & 6. \ \ F_1^*F_1+F_2^*F_2=I-\frac{1}{4}(F^*F+FF^*),\\ \end{split}$$* *then $(X,T_F,V_0)$ on $\mathcal H \oplus \ell^2(\mathcal{D}_P)$ is a minimal $\mathbb{P}$-isometric dilation of $(A, S, P)$, where $$X= \begin{bmatrix} A & 0 & 0 & 0 & \dotsc \\ F_2D_{P} & F_1 & 0 & 0 & \dotsc \\ 0 & F_2 & F_1 & 0 & \dotsc\\ 0 & 0 & F_2 & F_1 & \dotsc\\ \dotsc & \dotsc & \dotsc & \dotsc & \dotsc \\ \end{bmatrix}, \; T_F= \begin{bmatrix} S & 0 & 0 & 0 & \dotsc \\ F^*D_{P} & F & 0 & 0 & \dotsc \\ 0 & F^* & F & 0 & \dotsc\\ 0 & 0 & F^* & F & \dotsc\\ \dotsc & \dotsc & \dotsc & \dotsc & \dotsc \\ \end{bmatrix}, \; V_0= \begin{bmatrix} P & 0 & 0 & 0 & \dotsc \\ D_{P} & 0 & 0 & 0 & \dotsc \\ 0 & I & 0 & 0 & \dotsc\\ 0 & 0 & I & 0 & \dotsc\\ \dotsc & \dotsc & \dotsc & \dotsc & \dotsc \\ \end{bmatrix}.$$* *Proof.* The minimality is immediate once we prove that $(X,T_F,V_0)$ is a $\mathbb{P}$-isometric dilation of $(A,S,P)$, because, $V_0$ acting on $\mathcal{H}\oplus \ell^2(\mathcal D_P)$ is the minimal isometric dilation of $P$. If we put $$X_2=F_1, \quad X_3=F_2, \quad X_{21}=F_2D_P \quad \text{and} \quad X_{n1}=0=X_{n+1} \quad \text{for} \ n \geq 3$$ in Theorem [Theorem 61](#thm:main-dilation){reference-type="ref" reference="thm:main-dilation"}, then the conditions (1) and (9) in Theorem [Theorem 61](#thm:main-dilation){reference-type="ref" reference="thm:main-dilation"} become redundant and the operator $X$ takes the block-matrix form as in the statement of this theorem. Thus, to ensure that $(X, T_F,V_0)$ is a $\mathbb{P}$-isometric dilation of $(A,S,P)$ in view of Theorem [Theorem 61](#thm:main-dilation){reference-type="ref" reference="thm:main-dilation"}, it suffices to prove $$F_2D_PS+F_1F^*D_P=F^*D_PA+FF_2D_P\,,$$ because, the other conditions are the hypotheses of this theorem. We deduce the above condition from the identities $(1), (4)$ and $(5)$ in ([\[eqn7\]](#eqn7){reference-type="ref" reference="eqn7"}). Note that the fundamental operator $F$ of a $\Gamma$-contraction $(S, P)$ satisfies $$\label{eqn6.7'} D_PS=FD_P+F^*D_PP.$$ See the last section of [@Bhattacharyya] for a proof. Let $G=D_PS-FD_P-F^*D_PP$. Then $G:\mathcal{H} \to \mathcal{D}_P$ satisfies the following: $$\begin{split} D_PG=D_P^2S-D_PFD_P-D_PF^*D_PP=(I-P^*P)S-(S-S^*P)-(S^*-P^*S)P=0. \end{split}$$ Now, $\langle Gx, D_Py\rangle=\langle D_PGx, y\rangle=0$ for all $x, y \in \mathcal{H}$, which implies that $G=0$. Now multiplying both sides of $F_2D_PP+F_1D_P=D_PA$ by $F^*$ both sides, we have $$\begin{split} F^*D_PA& = F^*F_2D_PP+F^*F_1D_P\\ &=F_2F^*D_PP+F^*F_1D_P \qquad [\text{ by condition-(2)}]\\ &=F_2D_PS-F_2FD_P+F^*F_1D_P \quad [\text{ by } \ (\ref{eqn6.7'})] \\ &= F_2D_PS-(F_2F-F^*F_1)D_P \\ &=F_2D_PS-(FF_2-F_1F^*)D_P. \quad [\text{ by condition-(5) of } (\ref{eqn7})]\\ \end{split}$$ Thus, we have that $F_2D_PS+F_1F^*D_P=F^*D_PA+FF_2D_P$ and this completes the proof. ◻ **Remark 63**. The conditional dilations as in Theorems [Theorem 61](#thm:main-dilation){reference-type="ref" reference="thm:main-dilation"} & [Theorem 62](#lem6.9){reference-type="ref" reference="lem6.9"} determine a class of $\mathbb{P}$-contractions $(A,S,P)$ that dilate to $\mathbb{P}$-isometries on the minimal isometric dilation space for $P$. However, there are limitations these theorems mainly because the concerned dilation space, i.e. $\mathcal{H}\oplus \ell^2(\mathcal D_P)$ is too small. Below we provide examples to show that Theorems [Theorem 61](#thm:main-dilation){reference-type="ref" reference="thm:main-dilation"} & [Theorem 62](#lem6.9){reference-type="ref" reference="lem6.9"} provide dilations to nontrivial classes of $\mathbb{P}$-contractions and also at the same time they are not applicable for some $\mathbb{P}$-contractions. 1. Let $T$ be a contraction acting on $\mathcal{H}$ such that $D_TT=0$. Then the $\mathbb{P}$-contraction $(A, S, P)=(T, 0, 0)$ admits a $\mathbb{P}$-isometric dilation $(X, T_F, V)$, where $V$ is a minimal isometric dilation space of $P$. Indeed, it follows from Theorem [Theorem 62](#lem6.9){reference-type="ref" reference="lem6.9"} that if there exist $Y$ and $Z$ in $\mathcal{B}(\mathcal{D}_P)$ such that the operator equations in ([\[eqn7\]](#eqn7){reference-type="ref" reference="eqn7"}) hold, then the desired conclusion follows. Here $F=0, D_P=I$ and so $\mathcal{D}_P=\mathcal{H}$. A straightforward computation shows that for $(Y, Z)=(T, D_T)$, the operator equations in ([\[eqn7\]](#eqn7){reference-type="ref" reference="eqn7"}) admit a solution. 2. Proposition [Proposition 15](#prop2.8){reference-type="ref" reference="prop2.8"} yields that $(I, 0, T)$ is a $\mathbb{P}$-contraction for any contraction $T$. Then the $\mathbb{P}$-contraction $(A, S, P)=(I, 0, T)$ admits a $\mathbb{P}$-isometric dilation $(X, T_F, V)$, where $V$ is a minimal isometric dilation space of $P$. The choice of $F_1=I$ and $F_2=0$ in $\mathcal{B}(\mathcal{D}_P)$ gives a solution to ([\[eqn7\]](#eqn7){reference-type="ref" reference="eqn7"}) and the rest follows from Theorem [Theorem 62](#lem6.9){reference-type="ref" reference="lem6.9"}. 3. On the other hand, $(A,S,P)=(0,0,I)$ is a $\mathbb{P}$-contraction as it is a commuting normal triple with $\sigma_T(A,S,P)=\{ (0,0,1) \} \subset \overline{\mathbb{P}}$. Now, Since $(S, P) =(0,I)$ on $\mathbb{C}^2$ is a $\Gamma$-isometry (in fact a $\Gamma$-unitary), it follows that the minimal isometric dilation space of $P$ is $\mathbb{C}^2$ itself. Note that $(0,0,I)$ is not a $\mathbb{P}$-isometry as the first two component i.e., $(0,0)$ is not a $\mathbb B_2$-isometry. If $(0, 0, I)$ were to dilate to a $\mathbb{P}$-isometry $(X,T,V)$ on $\mathbb{C}^2$ with $V$ being the minimal dilation of $I$, then $P_{\mathbb{C}^2}X|_{\mathbb{C}^2}=X=0$, $P_{\mathbb{C}^2}T|_{\mathbb{C}^2}=T=0$ and $P_{\mathbb{C}^2}V|_{\mathbb{C}^2}=V=I$, but Theorem [Theorem 33](#P_unitary){reference-type="ref" reference="P_unitary"} yields that $X^*X+\frac{1}{4}T^*T=I$, which is a contradiction. Hence, $(0,0,I)$ do not dilate to a $\mathbb{P}$-isometry on the minimal isometric dilation space of the last component. If we move out of the territory of the minimal isometric dilation of the last component as in Theorems [Theorem 61](#thm:main-dilation){reference-type="ref" reference="thm:main-dilation"} & [Theorem 62](#lem6.9){reference-type="ref" reference="lem6.9"}, we can find $\mathbb{P}$-isometric dilation for some $\mathbb{P}$-contractions as shown below. **Proposition 64**. *Every $\mathbb{P}$-contraction of the form $(T_1, 0, T_2)$ admits a $\mathbb{P}$-isometric dilation.* *Proof.* It follows from Proposition [Proposition 15](#prop2.8){reference-type="ref" reference="prop2.8"} that $(T_1, T_2)$ is a commuting pair of contractions if and only if $(T_1,0, T_2)$ is a $\mathbb{P}$-contraction. Let $(T_1, 0, T_2)$ be a $\mathbb{P}$-contraction acting on a Hilbert space $\mathcal{H}$. A famous result due to Ando (see Chapter-I of [@Nagy]) yields that $(T_1,T_2)$ dilates to a pair of commuting isometries $(V_1, V_2)$. By Corollary [Corollary 47](#V_1, 0, V_2){reference-type="ref" reference="V_1, 0, V_2"}, we have that $(V_1, 0, V_2)$ is a $\mathbb{P}$-isometry. Evidently, $(V_1, 0, V_2)$ is a $\mathbb{P}$-isometric dilation of $(T_1, 0, T_2)$. ◻ A major role in the $\mathbb{P}$-isometric dilation of Theorem [Theorem 62](#lem6.9){reference-type="ref" reference="lem6.9"} is played by the existence of a solution to the operator equation $$\label{eqn:dilation-01} I-A^*A-\frac{1}{4}S^*S=D_P\big(Z^*Z+\frac{1}{4}FF^*\big)D_P.$$ Indeed, it is evident from Theorem [Theorem 62](#lem6.9){reference-type="ref" reference="lem6.9"} that if ([\[eqn:dilation-01\]](#eqn:dilation-01){reference-type="ref" reference="eqn:dilation-01"}) has a solution $Z=F_2$ satisfying $I-\dfrac{1}{4}(F^*F+FF^*)-F_2^*F_2 \geq 0$, then it confirms the existence of $F_1$ and the rest boils down to $F_1,F_2$ satisfying the other identities. For this reason, we put special emphasis on Equation-([\[eqn:dilation-01\]](#eqn:dilation-01){reference-type="ref" reference="eqn:dilation-01"}). In other words, we seek a solution $X \in \mathcal (\mathcal{D}_P)$ to the operator equation $$\label{eqn6.8} I-A^*A-\frac{1}{4}S^*S=D_PXD_P.$$ such that $X=F_2^*F_2+\frac{1}{4}FF^*.$ Moreover, if there is a solution to Equation-([\[eqn6.8\]](#eqn6.8){reference-type="ref" reference="eqn6.8"}), then $X\geq 0$ and consequently $D_PXD_P \geq 0$ which implies that $I-A^*A-\frac{1}{4}S^*S\geq 0$. Then we have $$\langle XD_Ph, D_Ph \rangle = \langle D_PXD_Ph, h \rangle= \langle (I-A^*A-S^*S \slash 4)h,h \rangle \leq \langle h , h\rangle=\|h\|^2$$ for every $h \in \mathcal{H}$ and hence it is necessary that $\omega(X) \leq 1$. In this connection let us recall an important result associated with the numerical radius. **Lemma 65** ([@Bhattacharyya], Lemma 2.9). *The numerical radius of an operator $T$ is not greater than one if and only if Re $\beta T \leq I$ for all complex numbers $\beta$ of unit modulus.* It follows from the above lemma that $\omega\left( Z^*Z+\frac{1}{4}FF^* \right) \leq 1$ if and only if $Re \ \beta\left(Z^*Z+\frac{1}{4}FF^*\right) \leq I$ for all $\beta \in \mathbb{T}$. This is equivalent to saying that $Z^*Z+\frac{1}{4}FF^* \leq I$ as $Z^*Z+\frac{1}{4}FF^*$ is self-adjoint. In order to solve the operator equations in Theorem [Theorem 62](#lem6.9){reference-type="ref" reference="lem6.9"}, we must have $$I-F_2^*F_2-\frac{1}{4}FF^*=F_1^*F_1+\frac{1}{4}F^*F \geq 0.$$ Thus, to obtain solutions that fit in with the system of equations in Theorem [Theorem 62](#lem6.9){reference-type="ref" reference="lem6.9"}, we have to find $X \geq 0$ in $\mathcal{B}(\mathcal{D}_P)$ with $\omega(X) \leq 1$ such that ([\[eqn6.8\]](#eqn6.8){reference-type="ref" reference="eqn6.8"}) is satisfied. Our next results characterizes the class of $\mathbb{P}$-contractions for which $(\ref{eqn6.8})$ has a solution with the desired properties. The proof of this result requires the following lemma. **Lemma 66** ([@Bhattacharyya], Lemma 4.1). *Let $\Sigma$ and $D$ be two bounded operators on $\mathcal{H}$. Then $$DD^* \geq Re (e^{i\theta}\Sigma) \quad \text{for all} \ \theta \in \mathbb{R}$$ if and only if there is $X \in \mathcal{B}(\mathcal{D}_*)$ with numerical radius of $X$ not greater than $1$ such that $\Sigma=DXD^*$, where $\mathcal{D}_*=\overline{Ran}(D^*)$.* **Theorem 67**. *Let $(A, S, P)$ be a $\mathbb{P}$-contraction on $\mathcal{H}$. Then there is a unique solution $X \in \mathcal{B}(\mathcal{D}_P)$ with $\omega(X) \leq 1$ to the operator equation ([\[eqn6.8\]](#eqn6.8){reference-type="ref" reference="eqn6.8"}), i.e. $I-A^*A-\frac{1}{4}S^*S=D_PXD_P$ if and only if $$\label{eqn6.10} \pm \left(I-A^*A-\frac{1}{4}S^*S\right) \leq D_P^2.$$ Moreover, if such a solution $X$ exists, then $X \geq 0 \ \text{if and only if} \ \text{$(A, S\slash 2)$ is a spherical contraction}.$* *Proof.* Let $\Sigma=I-A^*A-\frac{1}{4}S^*S$ and $D=D_P$. Then, it follows from Lemma [Lemma 66](#lem6.11){reference-type="ref" reference="lem6.11"} that there is an operator $X \in \mathcal{B}(\mathcal{D}_P)$ with $\omega(X) \leq 1$ such that $I-A^*A-\frac{1}{4}S^*S=D_PXD_P$ if and only if $$\begin{split} 0 & \leq D_P^2-Re(e^{i\theta} \Sigma)=D_P^2-\Sigma \ Re(e^{i\theta})=D_P^2-\cos\theta \ \Sigma \quad \text{for all} \ \theta \in \mathbb{R}. \end{split}$$ We show that $D_P^2 \geq \cos \theta \Sigma$ for all $\theta \in \mathbb{R}$ if and only if $D_P^2 \geq \pm \ \Sigma$. The necessary part is obvious. we prove the converse. Let $D_P^2 \geq \pm \ \Sigma$. Since $\Sigma$ is a self-adjoint operator, we have that $\langle \Sigma x, x \rangle \in \mathbb{R}$ for every $x \in \mathcal{H}$. Take any $\theta \in \mathbb{R}$ and $x \in \mathcal{H}$. We consider two different cases here depending on whether $\langle \Sigma x, x \rangle$ is positive or negative. If $\langle \Sigma x, x \rangle \geq 0$, then $\cos \theta \langle \Sigma x, x \rangle \leq \langle \Sigma x, x \rangle \leq \langle D_P^2x, x \rangle.$ Also, if $\langle \Sigma x, x \rangle \leq 0$, then $\cos \theta \langle \Sigma x, x \rangle \leq -\langle \Sigma x, x \rangle \leq \langle D_P^2x, x \rangle.$ In either case, we have that $\langle (D_P^2-\cos \theta \Sigma) x, x \rangle \geq 0$. Thus, $D_P^2 \geq \cos\theta \ \Sigma$ for all $\theta \in \mathbb{R}$. Thus, there is a solution $X \in \mathcal{B}(\mathcal{D}_P)$ with $\omega(X) \leq 1$ to the operator equation $I-A^*A-\frac{1}{4}S^*S=D_PXD_P$ if and only if $D_P^2 \geq \pm\Sigma$ which is equivalent to saying that $\pm \left(I-A^*A-\frac{1}{4}S^*S\right) \leq D_P^2$. For the uniqueness part, let there be two such solutions $X_1$ and $X_2$. Then $D_P\widehat{X}D_P=0$, where $\widehat{X}=X_1-X_2 \in \mathcal{B}(\mathcal{D}_P)$. Then, for all $x, y \in \mathcal{H}$, we have $\langle \widehat{X}D_Px, D_Py\rangle=\langle D_P\widehat{X}D_Px, y \rangle =0 ,$ which shows that $\widehat{X}=0$. Hence, $X_1=X_2$. Let us assume that there is an operator $X \in \mathcal{B}(\mathcal{D}_P)$ such that Equation-([\[eqn6.8\]](#eqn6.8){reference-type="ref" reference="eqn6.8"}) holds. For any $x \in \mathcal{H}$, we have $$\langle XD_Px, D_Px \rangle=\langle D_PXD_Px, x \rangle=\left\langle \left(I-A^*A-\frac{1}{4}S^*S\right)x, x \right \rangle$$ which shows that $I-A^*A-\frac{1}{4}S^*S \geq 0$ if and only if $X \geq 0$. The proof is now complete. ◻ Note that Equation-([\[eqn6.10\]](#eqn6.10){reference-type="ref" reference="eqn6.10"}) does not hold for all $\mathbb{P}$-contractions. The scalar version of Equation-([\[eqn6.10\]](#eqn6.10){reference-type="ref" reference="eqn6.10"}) is given by $$\pm \left(1-|a|^2-\frac{1}{4}|s|^2\right) \leq 1-|p|^2.$$ Now $(a, s, p)=(0, 0, 1)$, which in $\overline{\mathbb{P}}$, does not satisfy the above inequality. Also, $I-A^*A-\frac{1}{4}S^*S \geq 0$ if and only if $(A,S \slash 2)$ is a spherical contraction. Again, for every $\mathbb{P}$-contraction $(A,S,P)$ we have that $(A, S \slash 2)$ is a $\mathbb{B}_2$-contraction. Thus, we are in search of $\mathbb{B}_2$-contractions that are spherical contractions. In the special case when $(A,S,P)$ is a subnormal $\mathbb{P}$-contraction, i.e., a $\mathbb{P}$-contraction that admits an extension to a normal $\mathbb{P}$-contraction, we have success by an application of an elegant result due to Athavale, [@AthavaleII]. **Lemma 68**. *A subnormal $\mathbb{P}$-contraction $(A, S, P)$ satisfies $I-A^*A-\frac{1}{4}S^*S \geq 0$.* *Proof.* Let $(A, S, P)$ be a subnormal $\mathbb{P}$-contraction. It follows from Proposition [Proposition 17](#prop2.11){reference-type="ref" reference="prop2.11"} that $\sigma_T(A, S\slash 2) \subseteq \overline{\mathbb{B}}_2$. Now Theorem 5.2 in [@AthavaleII] yields that $I-A^*A-\frac{1}{4}S^*S \geq 0.$ ◻ It is never easy to determine the success or failure of rational dilation on a domain. Rational dilation succeeds on the bidisc $\mathbb D^2$ and on the symmetrized bidisc $\mathbb G_2$ (see [@AglerII; @Bhattacharyya]), but it is unclear at this point if it succeeds on the pentablock. 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arxiv_math

--- abstract: | We introduce the notion of clique number of a tournament and investigate its relation with the dichromatic number. In particular, it permits defining $\mathop{\mathrm{\overrightarrow{\chi}}}$-bounded classes of tournaments, which is the paper's main topic. author: - Pierre Aboulker - Guillaume Aubian - Pierre Charbit - Raul Lopes bibliography: - refs.bib title: Clique number of tournaments --- # Introduction In this paper, we only consider *graphs* or *directed graphs* (*digraphs* in short) with no loops, no parallel edges or arcs nor anti-parallel arcs (in particular our digraphs contain no cycle of length $2$). Given an undirected graph $G$, we denote by $\omega(G)$ the size of a maximum clique of $G$ and by $\chi(G)$ its chromatic number. Given a digraph $G$, we denote by $\mathop{\mathrm{\overrightarrow{\chi}}}(G)$ its *dichromatic number*, that is the minimum integer $k$ such that the set of vertices of $G$ can be partitioned into $k$ acyclic subdigraphs. Relations between the chromatic number and the clique number of a graph have been studied for decades in structural graph theory. The goal of this paper is to introduce a notion of clique number for digraphs, that would be a lower bound for the dichromatic number as in the undirected case, and start to investigate its relation with the dichromatic number. Given a digraph $D$, and a total order $\prec$ on $V(D)$, we denote $D^{\prec}$ the (undirected) graph with vertex set $V(D)$ and edge $uv$ if $u\prec v$ and $vu\in A(D)$. We call it the *backedge graph* of $D$ with respect to $\prec$. It is straightforward that every independent set of $D^{\prec}$ induces an acyclic digraph. As a consequence, we have that $\mathop{\mathrm{\overrightarrow{\chi}}}(D) \le \chi(D^{\prec})$. Conversely, by taking an ordering built from a $\mathop{\mathrm{\overrightarrow{\chi}}}(D)$-dicolouring, that is taking colour classes one after the other, and ordering each colour class in a topological ordering, we get that: $$\mathop{\mathrm{\overrightarrow{\chi}}}(D) = \min \, \big\{ \chi(D^{\prec}) : \mbox{$\prec$ is a total order of $V(D)$} \big\}$$ This gives an alternative definition for the dichromatic number, which naturally leads to the following definition of the *clique number of a digraph*[^1] $$\mathop{\mathrm{\overrightarrow{\omega}}}(D) = \min \,\big\{ \omega(D^{\prec}) : \mbox{$\prec$ is a total order on $V(D)$} \big\}$$ We point out that although this is the first time (up to our knowledge), that this definition formally appears, the idea of looking at the clique number of ordered digraphs is not new, and in the nice survey [@NSS23], Nguyen, Scott and Seymour study (amongst other things) the clique number of backedge graphs of tournaments, so the idea was clearly in their minds. Obviously, since $\omega(G)\leq \chi(G)$ for any graph $G$, we also have $\mathop{\mathrm{\overrightarrow{\omega}}}(D)\leq \mathop{\mathrm{\overrightarrow{\chi}}}(D)$ for any digraph $D$. In the context of graphs, since there are families of graphs with clique number $2$ but an arbitrarily large chromatic number, there has been in the past decades a very important amount of work dedicated to the study of so-called $\chi$-bounded classes of graphs, that is classes for which $\chi$ is bounded above by a function of $\omega$. See [@SS20] for a survey on $\chi$-boundedness. Analogously, we say that a class of digraphs is *$\mathop{\mathrm{\overrightarrow{\chi}}}$-bounded* if there exists $f$ such that for every digraph $D \in \mathcal C$, $\mathop{\mathrm{\overrightarrow{\chi}}}(D) \leq f(\mathop{\mathrm{\overrightarrow{\omega}}}(D))$. The object of the paper is to give first results and conjectures about clique number of tournaments and $\mathop{\mathrm{\overrightarrow{\chi}}}$-bounded classes of tournaments. We briefly discuss clique number of general digraphs in Section [6](#sec:conclusion){reference-type="ref" reference="sec:conclusion"}. Note that another definition of $\mathop{\mathrm{\overrightarrow{\chi}}}$-boundedness is given in [@ACN21] where the clique number of a digraph $D$ is defined as the maximum size of a transitive tournament contained in $D$. (More precisely, it is defined as the size of a maximum clique of the underlying of $D$, but since an orientation of the complete graphs on $2^k$ vertices contains $TT_k$, if a class of oriented graphs is $\mathop{\mathrm{\overrightarrow{\chi}}}$-bounded for one notion, it is also $\mathop{\mathrm{\overrightarrow{\chi}}}$-bounded for the other). Such a definition does not give a lower bound on the dichromatic number, which is the reason why we were looking for another definition. The next section is devoted to notations and definitions used throughout the paper. Section [3](#sec:chibound){reference-type="ref" reference="sec:chibound"} establishes some first properties of clique number, and explores the connections between the clique number of a tournament and the clique number of its backedge graphs. We then endeavour to extend standard results on $\chi$-boundedness to tournaments. In subsection [3.2](#sec:subst_S_k){reference-type="ref" reference="sec:subst_S_k"}, we describe a simple family having arbitrarily large clique number and prove that $\mathop{\mathrm{\overrightarrow{\chi}}}$-boundedness of tournaments is preserved by substitution (and a similar result for some classes of digraphs). We then discuss in subsection [3.3](#sec:twinwidth){reference-type="ref" reference="sec:twinwidth"} if, as in the undirected case, classes of tournaments of bounded twin-width are $\mathop{\mathrm{\overrightarrow{\chi}}}$-bounded. A fruitful discussion when studying tournaments involves examining the class of tournaments not containing a given tournament $T$, and deciding which $T$ will ensure that this class has a given property. For example, choices of $T$ guaranteeing a small dichromatic number [@hero] (such $T$ are called heroes), a small domination number [@CKLST18] or a small twin-width [@GT22] have been studied before. Section [4](#sec:Hfree){reference-type="ref" reference="sec:Hfree"} is devoted to this for the property of being $\mathop{\mathrm{\overrightarrow{\chi}}}$-bounded. In Section [4.1](#sec:gentlemen_heroes){reference-type="ref" reference="sec:gentlemen_heroes"} we study gentlemen, which are tournaments such that the clique number of tournaments not containing them is bounded, and prove that gentlemen are the same as heroes. In Subsection [4.2](#sec:gyarfas_sumner){reference-type="ref" reference="sec:gyarfas_sumner"}, we propose an analogue of Gyárfás-Summner Conjecture for tournaments, proving multiple results supporting this conjecture. We then link $\mathop{\mathrm{\overrightarrow{\chi}}}$-binding tournaments to the famous Erdős-Hajnal property and to the $BIG \Rightarrow BIG$ conjecture in Subsection [4.3](#sec:erdos_hajnal){reference-type="ref" reference="sec:erdos_hajnal"}. Eventually, in Section [5](#sec:cluster){reference-type="ref" reference="sec:cluster"}, we discuss local to global results for clique number, trying to adapt and generalize results of Harutyunyan, Le, Thomassé and Wu in [@HLTW19] about dichromatic number and domination number. # Definitions and Notations Definitions and notations of this paper that are not explained in this section follow from classical textbooks such as [@BG18], [@BM08] or [@D05]. Given two disjoint sets of vertices $X, Y$ of a digraph $D$, we write $X \Rightarrow Y$ to say that for every $x \in X$ and for every $y \in Y$, $xy \in A(D)$, and we write $X \rightarrow Y$ to say that every arc with one end in $X$ and the other one in $Y$ is oriented from $X$ to $Y$ (but some vertices of $X$ might be non-adjacent to some vertices of $Y$). When $X=\{x\}$ we write $x \Rightarrow Y$ and $x \rightarrow Y$. We also use the symbol $\Rightarrow$ to denote a composition operation on digraphs: for two digraphs $D_1$ and $D_2$, $D_1\Rightarrow D_2$ is the digraph obtained from the disjoint union of $D_1$ and $D_2$ by adding all arcs from $V(D_1)$ to $V(D_2)$. A *dominating set* of a digraph $D$ is a set of vertices $X$ such that $N^+[X] = V(D)$. The *dominating number* $\mathop{\mathrm{dom}}(D)$ of $D$ is the size of a smallest dominating set of $D$. A *tournament* is an orientation of a complete graph. A *transitive tournament* is an acyclic tournament and we denote by $TT_n$ the unique acyclic tournament on $n$ vertices. Given three tournaments $T_1,T_2,T_3$, we denote by $\Delta(T_1,T_2,T_3)$ the tournament obtained from disjoint copies of $T_1,T_2,T_3$ by adding arcs in such a way that $T_1\Rightarrow T_2$, $T_2\Rightarrow T_3$ and $T_3\Rightarrow T_1$. If one or more of the tournaments $T_i$ is a transitive tournament $TT_k$, we simplify the notation by using its size $k$ instead of writing $TT_k$ in the $\Delta$ construction: for example, $\Delta(1,k, T)$ corresponds to $\Delta(TT_1,TT_k,T)$ and $\Delta(1,1,1)$ is simply the directed triangle, which we also denote by $C_3$. A *class* of graphs (resp. digraphs) is a collection of graphs (resp. digraphs) that is closed under induced subgraphs, meaning that if $G$ belongs to the collection, then any induced subgraph of $G$ also belongs to the collection. Given a collection $\mathcal{C}$ the *hereditary closure* of $\mathcal{C}$ is the class of all induced subgraphs of elements of $\mathcal{C}$. Given a set of digraphs $\mathcal H$, we say that a digraph $G$ is *$\mathcal H$-free* if it contains no member of $\mathcal H$ as an induced subgraph and denote by $Forb(\mathcal H)$ the class of $\mathcal H$-free digraphs. We write $Forb(F_1, \dots, F_k)$ instead of $Forb(\{F_1, \dots, F_k\})$ for simplicity. Because of the definition of $\mathop{\mathrm{\overrightarrow{\omega}}}$, this paper is very often concerned with total orders on the vertices of a graph or a tournament. For a graph or tournament $T$ we denote by $\mathfrak S(T)$ the set of total orderings of $V(T)$. Given a graph or tournament with a total ordering $\prec$ of its vertex set $V$ and two disjoint subsets $A, B$ of $V$, we write $A \prec B$ to say that for every $a \in A$ and every $b \in B$, $a\prec b$. For a digraph or tournament with a total ordering $\prec$ of its vertex set $V$, an arc $uv$ such that $u\prec v$ is called *forward*, and otherwise it is called *backward*. Recall that given a tournament $T$, and a total order $\prec$ on $V(T)$, the backedge graph $T^{\prec}$ of $T$ with respect to $\prec$ is the (undirected) graph with vertex set $V(T)$ and edges $uv$ if $u\prec v$ and $vu\in A(T)$ (i.e. $vu$ is backward). An ordering $\prec$ such that $\omega(T^{\prec})= \mathop{\mathrm{\overrightarrow{\omega}}}(T)$ (resp. $\mathop{\mathrm{\overrightarrow{\chi}}}(T^{\prec}) = \mathop{\mathrm{\overrightarrow{\chi}}}(T)$) is called an *$\mathop{\mathrm{\overrightarrow{\omega}}}$-ordering* of $T$ (resp. $\mathop{\mathrm{\overrightarrow{\chi}}}$-ordering). We denote by $\mathfrak S_{\mathop{\mathrm{\overrightarrow{\omega}}}}(T)$ the set of $\mathop{\mathrm{\overrightarrow{\omega}}}$-orderings and by $\mathfrak S_{\mathop{\mathrm{\overrightarrow{\chi}}}}(T)$ the set of $\mathop{\mathrm{\overrightarrow{\chi}}}$-orderings. # $\chi$-bounded classes of tournaments {#sec:chibound} ## First properties about $\mathop{\mathrm{\overrightarrow{\chi}}}$ and $\mathop{\mathrm{\overrightarrow{\omega}}}$ {#sec:first_prop} We begin with an easy fact relating the clique number of a digraph and the clique number of its strong components. The clique number of a digraph is equal to the maximum clique number of its strong components. *Proof.* Assume a digraph $D$ has two strong components $A$ and $B$ such that $A \rightarrow B$ and let us prove that $\mathop{\mathrm{\overrightarrow{\omega}}}(D) = \max (\mathop{\mathrm{\overrightarrow{\omega}}}(A), \mathop{\mathrm{\overrightarrow{\omega}}}(B))$. It is clear that $\mathop{\mathrm{\overrightarrow{\omega}}}(D) \geq \max (\mathop{\mathrm{\overrightarrow{\omega}}}(A), \mathop{\mathrm{\overrightarrow{\omega}}}(B))$. Let $\prec_A$ (resp. $\prec_B$) be an $\mathop{\mathrm{\overrightarrow{\omega}}}$-ordering of $A$ (resp. of $B$). Then the ordering of $D$ obtained from $\prec_A$ and $\prec_B$ by setting, for every $a \in V(A)$ and $b \in V(B)$, $b \prec a$ satisfies $\omega(D^{\prec}) = \max (\mathop{\mathrm{\overrightarrow{\omega}}}(A), \mathop{\mathrm{\overrightarrow{\omega}}}(B))$. ◻ As observed in the introduction, the definition of $\mathop{\mathrm{\overrightarrow{\omega}}}$ immediately implies that $\mathop{\mathrm{\overrightarrow{\omega}}}(T) \leq \mathop{\mathrm{\overrightarrow{\chi}}}(T)$ for any tournament $T$. The following proves a relation with the domination number. In Section [5](#sec:cluster){reference-type="ref" reference="sec:cluster"} we will expose some other results linking $\mathop{\mathrm{dom}}$, $\mathop{\mathrm{\overrightarrow{\chi}}}$ and $\mathop{\mathrm{\overrightarrow{\omega}}}$ [\[prop:dom\<omega\]]{#prop:dom<omega label="prop:dom<omega"} For every tournament $T$, $\mathop{\mathrm{dom}}(T) \leq \mathop{\mathrm{\overrightarrow{\omega}}}(T) \leq \mathop{\mathrm{\overrightarrow{\chi}}}(T)$. *Proof.* We already know that the second inequality holds. Let $T$ be a tournament, set $V(T) = \{v_1, \dots, v_n\}$ and assume that $v_1 \prec v_2 \prec \dots \prec v_n$ is an $\mathop{\mathrm{\overrightarrow{\omega}}}$ ordering of $T$. Greedily construct a dominating set $X$ of $T$ as follows: $v_1 \in X$, and if $v_i$ is the last vertex added to $X$, add $v_j$ to $X$ where $j$ is minimum such that there is an arc from $v_j$ to every vertex of $X$. Then $X$ is a dominating set of $T$, and it induces a clique in the backedge graph of $T$ defined by an $\mathop{\mathrm{\overrightarrow{\omega}}}$-ordering. So $\mathop{\mathrm{dom}}(T) \leq \mathop{\mathrm{\overrightarrow{\omega}}}(T)$. ◻  \ In [@NSS23], the following fundamental inequality is proved (the second inequality is trivial by definition). We give a proof anyway, to make the paper self-contained and familiarize the reader with the notations. Moreover our prove is presented slightly differently then the one in [@NSS23]. [\[eq:bounddic\]]{#eq:bounddic label="eq:bounddic"} For any tournament $T$ and ordering $\prec$ of $V(T)$. $$\frac{\chi(T^{\prec})}{\omega(T^{\prec})} \leq \mathop{\mathrm{\overrightarrow{\chi}}}(T) \leq \chi(T^{\prec})$$ *Proof.* Let $T$ be a tournament and $\prec$ an ordering of $V(T)$. Let $X \subseteq V(T)$ such that $T[X]$ is a transitive tournament. To prove that $\chi(T^{\prec}) \leq \omega(T^{\prec}) \mathop{\mathrm{\overrightarrow{\chi}}}(T)$, it suffices to prove that $\chi(T^{\prec}[X]) \leq \omega(T^{\prec})$. Let $\varphi : X \to \mathbb{N}$ be such that $\varphi(x)$ is the number of vertices of a longest $\prec$-decreasing path in $T^{\prec}[X]$ finishing in $x$. We claim that $\varphi$ is a $\omega(T^{\prec})$-colouring of $T^{\prec}$. Let $u,v \in X$ with $u\prec v$ and $uv \in E(T^{\prec})$. Then $\varphi(u) \geq \varphi(v) + 1$, so $\varphi$ is a colouring of $T^{\prec}[X]$. If $x_1,x_2,x_3 \in X$, $x_3 \prec x_2 \prec x_1$ and $x_1x_2, x_2x_3 \in E(T^{\prec})$, then $x_1x_2, x_2x_3 \in A(T)$ and thus, since $T[X]$ is a transitive tournament, $x_1x_3 \in A(T)$, i.e. $x_1x_3 \in E(T^{\prec})$. This implies that the vertices of a $\prec$-decreasing path in $T^{\prec}[X]$ induces a clique of $T^{\prec}$. So for every vertex $x \in X$, $\varphi(x) \leq \omega(T^{\prec})$. ◻ Observe that for an aribtrary order $\prec$, $\omega(T^{\prec})$ and $\chi(T^{\prec})$ can be arbitrarily larger than $\mathop{\mathrm{\overrightarrow{\omega}}}(T)$ or $\mathop{\mathrm{\overrightarrow{\chi}}}(T)$. For example, there is an ordering $\prec$ of $TT_n$ such that $\omega(T^{\prec}) = \chi(T^{\prec})=n$, while $\mathop{\mathrm{\overrightarrow{\omega}}}(TT_n) = \mathop{\mathrm{\overrightarrow{\chi}}}(TT_n) = 1$. However, an $\mathop{\mathrm{\overrightarrow{\omega}}}$-ordering always provides a good approximation of $\mathop{\mathrm{\overrightarrow{\chi}}}$ in the following sense: For every tournament $T$ and every $\mathop{\mathrm{\overrightarrow{\omega}}}$-ordering $\prec$ we have: $$\mathop{\mathrm{\overrightarrow{\chi}}}(T) \leq \chi(T^{\prec}) \leq \mathop{\mathrm{\overrightarrow{\chi}}}(T)^2$$ *Proof.* Let $T$ be a tournament and $\prec$ an $\mathop{\mathrm{\overrightarrow{\omega}}}$-ordering of $T$. By Theorem [\[eq:bounddic\]](#eq:bounddic){reference-type="ref" reference="eq:bounddic"}, we have that $\chi(T^\prec) \leq \omega(G^\prec) \mathop{\mathrm{\overrightarrow{\chi}}}(T)$. But since $\omega(G^\prec) = \mathop{\mathrm{\overrightarrow{\omega}}}(T)$ and $\mathop{\mathrm{\overrightarrow{\omega}}}(T) \leq \mathop{\mathrm{\overrightarrow{\chi}}}(T)$, we get that: $$\mathop{\mathrm{\overrightarrow{\chi}}}(T) \leq \chi(T^{\prec}) \leq \mathop{\mathrm{\overrightarrow{\chi}}}(T)^2$$ ◻ It is natural to ask if the following stronger form of the above property holds (we have no reason to believe it does, but we could not find a counter-example). Is it true that for every tournament $T$, there exists $\prec \in \mathfrak S(T)$ such that $\prec$ is both a $\mathop{\mathrm{\overrightarrow{\omega}}}$-ordering and a $\mathop{\mathrm{\overrightarrow{\chi}}}$-ordering? Given a class of tournaments $\mathcal{T}$, let us denote by $\mathcal{T}^\prec$ the class of all backedge graphs of tournaments in $\mathcal{T}$ : $$\mathcal T^{\prec} = \{T^{\prec}: T \in \mathcal T, \prec \in \mathfrak S(T)\}$$ A natural question is whether $\mathcal{T}$ is $\mathop{\mathrm{\overrightarrow{\chi}}}$-bounded has to do with the fact that $\mathcal{T}^\prec$ is $\chi$-bounded in the usual sense for undirected graphs. And one can ask the same question for the more restricted class of \"optimal\" backedge graphs $\mathcal T^{\prec_{\mathop{\mathrm{\overrightarrow{\omega}}}}}$ : $$\mathcal T^{\prec_{\mathop{\mathrm{\overrightarrow{\omega}}}}} = \{T^{\prec}: T \in \mathcal T, \prec \in \mathfrak S_{\mathop{\mathrm{\overrightarrow{\omega}}}}(T)\}$$ The following theorem answers these questions. [\[thm2:equivchi-boundedness\]]{#thm2:equivchi-boundedness label="thm2:equivchi-boundedness"} Let $\mathcal T$ be a class of tournaments. The following properties are equivalent: 1. $\mathcal T$ is $\mathop{\mathrm{\overrightarrow{\chi}}}$-bounded. 2. $\mathcal T^{\prec}$ is $\chi$-bounded. 3. $\mathcal T^{\prec_{\mathop{\mathrm{\overrightarrow{\omega}}}}}$ is $\chi$-bounded. *Proof.* $(i) \Rightarrow(ii)$: let $f$ be a function such that any tournament $T \in \mathcal T$ satisfies $\mathop{\mathrm{\overrightarrow{\chi}}}(T) \leq f(\mathop{\mathrm{\overrightarrow{\omega}}}(T))$. Now for any tournament $T \in \mathcal T$ and $\prec \in \mathfrak S(T)$. $$\begin{aligned} \chi(T^{\prec}) &\leq \omega(T^{\prec}) \mathop{\mathrm{\overrightarrow{\chi}}}(T)& \text {by~Theorem \ref{eq:bounddic}}\\ &\leq \omega(T^{\prec}) f(\mathop{\mathrm{\overrightarrow{\omega}}}(T))& \text{ by $(i)$}\\ &\leq \omega(T^{\prec}) f(\omega(T^{\prec}))& \end{aligned}$$ $(ii) \Rightarrow(iii)$ is trivial since $\mathcal T^{\prec_{\mathop{\mathrm{\overrightarrow{\omega}}}}}\subset \mathcal T^{\prec}$ $(iii) \Rightarrow(i)$: let $g$ be a function such that for every $T \in \mathcal T$ and for every $\prec \in \mathfrak S_{\mathop{\mathrm{\overrightarrow{\omega}}}}(T)$, $\chi(T^{\prec}) \leq g(\omega^{\prec}(T))$. Now for any $T \in \mathcal T$ and $\prec \in \mathfrak S_{\mathop{\mathrm{\overrightarrow{\omega}}}}(T)$. $$\mathop{\mathrm{\overrightarrow{\chi}}}(T) \leq \chi(T^{\prec}) \leq g(\omega(T^{\prec})) = g(\mathop{\mathrm{\overrightarrow{\omega}}}(T)) \\ $$ ◻ In a converse manner, given a class of undirected graphs $\mathcal C$, one can define $\mathcal T[\mathcal C]$ as the set of tournaments $T$ that admits a backedge graph belonging to $\mathcal C$. This operation has already been used to construct classes of tournaments with nice properties, see for example [@NSS23]. More formally: $$\mathcal T[\mathcal C] = \{T: T \mbox{ is a tournament such that there exists} \prec \in \mathfrak S(T) \mbox{ with } T^{\prec} \in \mathcal C \}$$ It is natural to wonder if, given a $\chi$-bounded class of undirected graph $\mathcal C$, the class of tournaments $\mathcal T[\mathcal C]$ is $\mathop{\mathrm{\overrightarrow{\chi}}}$-bounded. It turns out that the answer is no as we explain now. Given a graph $G$, denote by $G^c$ be the complement of $G$, that is the graph $\big(V(G), \binom{V(G)}{2} \setminus E(G)\big)$, and set $\mathcal C^{c} = \{G^c: G \in \mathcal C\}$. Let $\mathcal C$ be a class of undirected graphs. Then $\mathcal C^c \subseteq \mathcal T[\mathcal C]^{\prec}$. *Proof.* Let $G \in \mathcal \mathcal C$. Let $T$ and $\prec$ be such that $T^{\prec} =G$. Denoting $\prec_r$ the reverse ordering of $\prec$, we get that $T^{\prec_r} = G^c \in \mathcal T[\mathcal C]^{\prec}$. Hence, $\mathcal C^c \subseteq \mathcal T[\mathcal C]$. ◻ Let now $\mathcal C$ be the class of graphs with stability number at most $2$. Then for every $G \in \mathcal C$, $\chi(G) \leq |V(G)| \leq \omega(G) + \omega(G)^2$ (by Ramsey theory), so $\mathcal C$ is $\chi$-bounded. But $\mathcal T[\mathcal C]^{\prec}$ contains $\mathcal C^c$, the class of triangle-free graphs and is thus not $\chi$-bounded. By Theorem [\[thm2:equivchi-boundedness\]](#thm2:equivchi-boundedness){reference-type="ref" reference="thm2:equivchi-boundedness"}, this implies that $\mathcal T[\mathcal C]$ is not $\mathop{\mathrm{\overrightarrow{\chi}}}$-bounded. ## Substitution and a class of tournaments with unbounded $\mathop{\mathrm{\overrightarrow{\omega}}}$ {#sec:subst_S_k} We start this section by giving a simple and natural construction of tournaments with arbitrarily large clique number. Let $\widetilde S_1 = TT_1$ and inductively, for $n \geq 1$, let $\widetilde S_n=\Delta(\widetilde S_{n-1}, \widetilde S_{n-1}, \widetilde S_{n-1})$. [\[lem:Stilde_omega\]]{#lem:Stilde_omega label="lem:Stilde_omega"} For any integer $n$, $\mathop{\mathrm{\overrightarrow{\omega}}}(\widetilde S_n)\geq n$ *Proof.* It is clear that $\mathop{\mathrm{\overrightarrow{\omega}}}(\widetilde S_1) = 1$. Let $n \geq 2$ and let $A,B,C$ be the partition of $V(\widetilde S_n)$ such that each set induces a copy of $\widetilde S_{n-1}$ and $A\Rightarrow B\Rightarrow C \Rightarrow A$. Let $\prec$ be an $\mathop{\mathrm{\overrightarrow{\omega}}}$-ordering of $\widetilde S_n$. Let $a$ be the first vertex of $\prec$ and assume without loss of generality that $a \in A$. By definition of $\mathop{\mathrm{\overrightarrow{\omega}}}$, there exists a clique in the backedge graph of $T[C]=\widetilde S_{n-1}$ with respect to $\prec$ that is of size at least $\mathop{\mathrm{\overrightarrow{\omega}}}(\widetilde S_{n-1})$, which, together with vertex $a$, form a clique of $\widetilde S_n^{\prec}$ of size $1+\mathop{\mathrm{\overrightarrow{\omega}}}(\widetilde S_{n-1}) \leq \mathop{\mathrm{\overrightarrow{\omega}}}(\widetilde S_n)$, the desired bound. ◻ It is also easy to see that $\mathop{\mathrm{\overrightarrow{\chi}}}(\widetilde S_n)\leq 2\mathop{\mathrm{\overrightarrow{\chi}}}(\widetilde S_{n-1})$ since one can colour two copies of $\widetilde S_{n-1}$ with $\mathop{\mathrm{\overrightarrow{\chi}}}(\widetilde S_{n-1})$ colours, and then use $\widetilde S_{n-1}$ new colours for the third copy. So $\mathop{\mathrm{\overrightarrow{\chi}}}(S_n)\leq 2^n \leq 2^{\mathop{\mathrm{\overrightarrow{\omega}}}(\widetilde S_n)}$ (in fact one can improve this to $3/2$ instead of $2$). One can wonder whether this inequality is still true for any $T$ in the hereditary closure of the family $\{\widetilde S_n \mid n\in \mathbf{N}\}$, that is for $T$ being a subtournament of some $\widetilde S_n$. It is in fact not obvious that this hereditary closure is even $\chi$-bounded (a subtournament of $\widetilde S_n$ can have clique number much smaller than $n$). Our next Theorem [\[thm:stable_by_subst\]](#thm:stable_by_subst){reference-type="ref" reference="thm:stable_by_subst"} implies that $\mathop{\mathrm{\overrightarrow{\chi}}}(T)\leq 9^{\mathop{\mathrm{\overrightarrow{\omega}}}(T)}$ for every tournament $T$ in this class. This is due to the fact that this class can also be defined through the operation of *substitution*, defined below. Given two digraphs $D_1$ and $D_2$ with disjoint vertex sets, a vertex $u \in V(D_1)$, and a digraph $D$, we say that $D$ is obtained by *substituting* $D_2$ for $u$ in $D_1$ provided that the following holds: - $V(D) = (V(D_1) \setminus u) \cup V(D_2)$, - $D[V(D_1) \setminus u] = D_1 \setminus u$, - $D[V(D_2)] = D_2$, - for all $v \in V(D_1) \setminus u$ if $vu \in A(D_1)$ (resp. $uv \in A(D_1)$, resp. $u$ and $v$ are non-adjacent in $D_1$), then $V(H_1) \Rightarrow v$ (resp. $v \Rightarrow V(D_2)$, resp. there is no arcs between $v$ and $V(D_2)$) in $D$. Given a class of digraphs $\mathcal{C}$ we define $\mathcal{C}^{subst}$ to be the closure of $\mathcal{C}$ under substitution.\ It is a well-known easy-to-prove fact that, if a class of (undirected) graphs is $\chi$-bounded, so is $\mathcal C^{subst}$. The first item of the following theorem proves the analogue for classes of tournaments. The second item applies to classes of digraphs (instead of tournaments) but only in the case of families of digraphs whose underlying graphs have bounded chromatic number. In that case we get a better $\mathop{\mathrm{\overrightarrow{\chi}}}$-binding function. [\[thm:stable_by_subst\]]{#thm:stable_by_subst label="thm:stable_by_subst"} Let $\mathcal{T}$ be a class of digraphs. 1. If $\mathcal{T}$ is a class of tournaments $\mathop{\mathrm{\overrightarrow{\chi}}}$-bounded by a function $f$, then $\mathcal{T}^{subst}$ is $\mathop{\mathrm{\overrightarrow{\chi}}}$-bounded by function $g(w) = (3wf(w))^w$. 2. If there exists $K$ such that for every digraph $D$ in $\mathcal{T}$ the underlying graph of $D$ has chromatic number at most $K$, then $\mathcal{T}^{subst}$ is $\mathop{\mathrm{\overrightarrow{\chi}}}$-bounded by function $g(w) = (3K)^w$. *Proof.* We will prove the two statements simultaneously, as the difference only occurs at one particular point of the proof. We want to prove that for every $T \in \mathcal T^{subst}$, $\mathop{\mathrm{\overrightarrow{\chi}}}(T) \leq g(\mathop{\mathrm{\overrightarrow{\omega}}}(T))$. Assume by contradiction that the result does not hold and let $T \in \mathcal T^{subst}$ be a counter-example minimizing $\mathop{\mathrm{\overrightarrow{\omega}}}(T) + |V(T)|$. We refer to this by saying "by minimality of $T$". Define $w:=\mathop{\mathrm{\overrightarrow{\omega}}}(T)$ and observe that $w\geq 2$ since if $\mathop{\mathrm{\overrightarrow{\omega}}}(T)=1$, then $T$ is acylic, so $\mathop{\mathrm{\overrightarrow{\chi}}}(T)=1$ and the result holds. Since $T \in \mathcal{T}^{subst}$, $T$ can be constructed from a digraph $X \in \mathcal{T}$, with $|V(X)| \geq 2$, by substituting each vertex $v \in V(X)$ by a digraph $T_v \in \mathcal{T}^{subst}$. For any set of vertices $Y \subseteq V(X)$, we define $T_Y = T[\cup_{y \in Y} V(T_y)]$. We call a vertex $v \in X$ *big* if $$\mathop{\mathrm{\overrightarrow{\chi}}}(T_v) \geq 2g(w-1)+3$$ and *small* otherwise. We call $B$ the set of big vertices and $S$ the set of small vertices (so $V(X) = B \cup S$). Let $\prec$ be an $\mathop{\mathrm{\overrightarrow{\omega}}}$-ordering of $T$. We are going to construct, in two steps, a dicolouring of $T$ using at most $g(w)$ colours. First we colour the vertices of $T_S$ with at most $g(w)$ colours in such a way that for every small vertex $s$ the digraph $T_s$ is properly dicoloured, and with the additional property that if $xy$ is a backward arc of $T$ such that there is two distinct small vertices $s$ and $s'$ such that $x\in T_s$ and $y\in T_{s'}$, then $x$ and $y$ receive distinct colours. Secondly we use induction to colour the vertices of $T_B$ and argue that no directed cycle is monochromatic. For the first step, we distinguish between the two cases of the theorem's statement. For both cases, we set $\phi_s$ to be a dicolouring of $T_s$ using at most $2g(w-1)+2$ colours (such a dicolouring exists by definition of $S$). - In case $1$ we observe that since $X \in \mathcal T$ and $\mathop{\mathrm{\overrightarrow{\omega}}}(X) \leq \mathop{\mathrm{\overrightarrow{\omega}}}(T) = w$, we have $\mathop{\mathrm{\overrightarrow{\chi}}}(X) \leq f(w)$ and thus there exists a colouring $\phi$ of $X[S]$ using at most $f(w)$ colours. We colour the vertices of $T_S$ by a Cartesian product of colourings: for each $s \in S$, each vertex $x\in T_s$ receives the colour $(\phi(s),\phi_s(x))$. It is clear that this yields a dicolouring of $T_S$ using at most $f(w)(2g(w-1)+2)$ distinct colours. Now, by Theorem [\[eq:bounddic\]](#eq:bounddic){reference-type="ref" reference="eq:bounddic"}, we have: $$\chi(T_S^{\prec}) \leq \omega(T_S^{\prec}) \mathop{\mathrm{\overrightarrow{\chi}}}(T_S) \leq wf(w)(2g(w-1) + 2) \leq 3wf(w)g(w-1) \leq g(w)$$ A colouring of $T_S^{\prec}$ with at most $g(w)$ gives a dicolouring of $T_S$ such that no backward arc is monochromatic, which implies the property described in the previous paragraph. - In case $2$, we apply a similar approach. Define $H$ to be the undirected graph with vertex set $S$ and with an edge $ss'$ if $ss'$ is an arc of $X$ and there exists $x\in T_s$ and $y\in T_{s'}$ with $y\prec x$. Observe that $H$ is a subgraph of the underlying graph of $X[S]$ so by assumption there exists a colouring $\phi$ of $H$ using at most $K$ colours. We colour the vertices of $T_S$ again by a Cartesian product of colourings: each vertex $x\in T_s$ receives the colour $(\phi(s),\phi_s(x))$. This colouring uses at most $K(2g(w-1)+2)\leq g(w)$ colours and satisfies the desired property: no $T_s$ contains a monochromatic directed cycle (because of $\phi_s$) and if $xy$ is a backward arc of $T$ such that there is two distinct small vertices $s$ and $s'$ such that $x\in T_s$ and $y\in T_{s'}$, then $ss' \in E(H)$ and thus $\phi(s)\neq \phi(s')$ which implies that $x$ and $y$ receive distinct colours as desired. For the second step, we first observe that for each big vertex $b \in B$, the digraph $T_b$ has strictly less vertices then $T$ (because $X$ has at least two vertices), so by minimality of $T$ we have $\mathop{\mathrm{\overrightarrow{\chi}}}(T_b)\leq g(\mathop{\mathrm{\overrightarrow{\omega}}}(T_b))\leq g(w)$. Moreover since $b$ is big, $\mathop{\mathrm{\overrightarrow{\chi}}}(T_b) > g(w-1)$ , so by minimality of $T$ we also get $\mathop{\mathrm{\overrightarrow{\omega}}}(T_b) = w$. For each $b \in B$, and each $x \in T_b$, we define the two following digraphs: $$T_b[\prec x] = T[\{u \in V(T_b): u \prec x\}] \text{ and } T_b[x \prec] = T[\{u \in V(T_b): x \prec u\}]$$ and, since $\mathop{\mathrm{\overrightarrow{\chi}}}(T_b) \geq 2g(w-1) + 3$, there is a vertex $m_b \in V(T_b)$ such that $$\mathop{\mathrm{\overrightarrow{\chi}}}(T_b[\prec m_b]) \geq g(w-1) + 1 \text{ and } \mathop{\mathrm{\overrightarrow{\chi}}}(T_b[m_b \prec]) \geq g(w-1) + 1$$ Hence, by minimality of $T$, for every $b \in B$: $$\mathop{\mathrm{\overrightarrow{\omega}}}(T_b[\prec m_b]) \geq w \text{ and } \mathop{\mathrm{\overrightarrow{\omega}}}(T_b[m_b \prec]) \geq w$$ Let $b \in B$. We claim that the inequalities above imply that if $m_b$ is incident to a backward arc, then the other extremity of the arc is in $T_b$. There are two symmetric cases (depending on whether $m_b$ is the tail or the head of the backward arc), so let us assume by contradiction that $xm_b$ is an backward arc and $x\in T_x$, $x\neq b$. Then because of the substitution $xm_b\in A(T)$ implies that $xy\in A(T)$ for any $y\in T_b$. In particular $xy\in A(T)$ for any $y\in T_b[\prec m_b]$. Since $\mathop{\mathrm{\overrightarrow{\omega}}}( T_b[\prec m_b]) \geq w$, a clique of size $w$ in $T_b[\prec m_b]^{\prec}$ together with $y$ form a clique of size $w+1$ in $T^{\prec}$, a contradiction. The argument is exactly the same if $m_bx$ is an arc with $x\prec m_b$ because $\mathop{\mathrm{\overrightarrow{\omega}}}(T_b[m_b \prec])$ also equals $w$. We are now ready to conclude. Consider the colouring of $V(T_S)$ obtained in the first step of the proof and extend it to $V(T)$ by assigning to the vertices of each $T_b$ for $b$ big a valid dicoulouring using at most $g(w)$ colours (remember that we showed that $\mathop{\mathrm{\overrightarrow{\chi}}}(T_b)\leq g(w)$). We claim that this defines a valid dicolouring of $T$. Assume by contradiction that there exists a monochromatic directed cycle and let $C$ be a minimal such cycle. Since for a fixed $x\in V(X)$, any two vertices in $V(T_x)$ share the same adjacency relation with each vertex of $V(T) \setminus V(T_x)$, the minimality of $C$ implies that $C$ contains at most one vertex of $V(T_x)$ for any $x \in V(X)$, for otherwise $C$ would be entirely included into some $T_x$, which is not possible since the colouring is valid dicolouring on each digraph $T_x$. Now define $C'$ to be the directed cycle obtained from $C$ by replacing each vertex belonging to $V(T_b)$ for some $b \in B$ by the vertex $m_b$. Since $C'$ is a directed cycle, it must contain some backward arc, i.e. some arc $uv$ with $v\prec u$. Since $u$ and $v$ do not belong to the same $T_x$, and because of the property of the vertices $m_b$ proven in the previous paragraph, both vertices $u$ and $v$ belong to $T_S$. But then they are vertices of $C$ and the arc $uv$ is thus monochromatic and backward, which contradicts the property of the colouring of $T_S$ established in the first step of the proof. ◻ Note that the hereditary closure of $\{\widetilde S_n, n\in\mathbb N\}$ mentioned earlier is easily seen to be exactly $\{TT_1,TT_2,\overrightarrow{C_{3}}\}^{subst}$. Therefore the first item implies $\mathop{\mathrm{\overrightarrow{\chi}}}(T)\leq 9^{\mathop{\mathrm{\overrightarrow{\omega}}}(T)}$ for any $T$ which is a subtournament of some $\widetilde S_n$. We do not know if this class is $\mathop{\mathrm{\overrightarrow{\chi}}}$-bounded by a polynomial function. One can prove for example that that the order of magnitude of $\mathop{\mathrm{\overrightarrow{\chi}}}(\widetilde S_n)$ is $(3/2)^n$ but it could be that the lower bound $\mathop{\mathrm{\overrightarrow{\omega}}}(\widetilde S_n)\geq n$ given by Lemma [\[lem:Stilde_omega\]](#lem:Stilde_omega){reference-type="ref" reference="lem:Stilde_omega"} is far from tight. Chudnovsky, Penev, Scott and Trotignon [@CPST13] proved that if a class of graphs is $\chi$-bounded by a polynomial function, so is its closure under substitution. Could it be that the same holds for tournaments? Is it true that if a class of tournaments $\mathcal T$ is polynomially $\mathop{\mathrm{\overrightarrow{\chi}}}$-bounded, then so is $\mathcal T^{subst}$. Before closing this section on substitutions we mention here another sequence of tournaments belonging to $\{TT_1,TT_2,\overrightarrow{C_{3}}\}^{subst}$ that will be of use in the proof of Theorem [\[thm:gentlemen\]](#thm:gentlemen){reference-type="ref" reference="thm:gentlemen"}. Let $S_1 = TT_1$ and inductively, for $n \geq 2$, let $S_n=\Delta(1, S_{n-1}, S_{n-1})$. It is easy to observe that $\mathop{\mathrm{\overrightarrow{\chi}}}(S_n)=n$. Since $S_n$ is obviously a subtournament of $\widetilde S_n$, we have therefore $\mathop{\mathrm{\overrightarrow{\omega}}}(S_n)\geq \log_9(n)$. Again it could be that this logarithm is not necessary. It is clear that $\mathop{\mathrm{\overrightarrow{\omega}}}(S_1) = 1$, $\mathop{\mathrm{\overrightarrow{\omega}}}(S_2) = 2$ and it is not hard (but a bit laborious) to prove that $\mathop{\mathrm{\overrightarrow{\omega}}}(S_3) = 2$ and $\mathop{\mathrm{\overrightarrow{\omega}}}(S_4) = 3$. The clique number of $S_k$ for $k \geq 5$ is not known but we doubt that one can compute an exact formula for it. ## Do tournaments with bounded twin-width are $\mathop{\mathrm{\overrightarrow{\chi}}}$-bounded? {#sec:twinwidth} *Twin-width* is a parameter introduced in [@tww1] measuring the complexity of a binary structure. We refer to [@GT22] for the definitions of the twin-width of a graph, an ordered graph and a tournament. Given a graph or digraph $G$ and a total order $\prec$ on $V(G)$, we denote by $tww(G)$ the twin-width of $G$ and by $tww(G, \prec)$ the twin-width of the ordered graph (digraph) $(G, \prec)$. Classes of graphs of bounded twin-width have been shown to be $\chi$-bounded [@tww3; @PS22], and even polynomially $\chi$-bounded [@BT23]. For every $k \geq 1$, the class of undirected graphs with twin-width at most $k$ is polynomially $\chi$-bounded. Observe that for every integer $k \geq 2$, $S_k$ has twin-width $1$ and dichromatic number $k$, so tournaments with bounded twin-width can have arbitrarily large dichromatic number. [\[conj:tww_chi_bounded\]]{#conj:tww_chi_bounded label="conj:tww_chi_bounded"} Let $k \geq 1$. The class of tournaments with twin-width at most $k$ is $\mathop{\mathrm{\overrightarrow{\chi}}}$-bounded. Given a tournament $T$ and an ordering $\prec$ of $V(T)$, we denote by $(T, \prec)$ the ordered tournament with ordering $\prec$, and by $(T^{\prec}, \prec)$ the ordered backedge graph with ordering $\prec$. The following conjecture implies Conjecture [\[conj:tww_chi_bounded\]](#conj:tww_chi_bounded){reference-type="ref" reference="conj:tww_chi_bounded"}. [\[conj:good_ordering_tww_omega\]]{#conj:good_ordering_tww_omega label="conj:good_ordering_tww_omega"} There exists a function $f$, such that for every tournament $T$, there exists an ordering $\prec^*$ of $V(T)$ such that: $$\omega(T^{\prec^*}) \leq f( \mathop{\mathrm{\overrightarrow{\omega}}}(T)) \quad \text{ and } \quad tww(T, \prec^*) \leq f(tww(T))$$ Conjecture [\[conj:good_ordering_tww_omega\]](#conj:good_ordering_tww_omega){reference-type="ref" reference="conj:good_ordering_tww_omega"} implies Conjecture [\[conj:tww_chi_bounded\]](#conj:tww_chi_bounded){reference-type="ref" reference="conj:tww_chi_bounded"}. *Proof.* Let $\mathcal T_k$ the class of tournaments with twin-width at most $k$. For each $T \in \mathcal T_k$, we associate an ordering $\prec_T^*$ given by Conjecture [\[conj:good_ordering_tww_omega\]](#conj:good_ordering_tww_omega){reference-type="ref" reference="conj:good_ordering_tww_omega"}. For every $T \in \mathcal T_k$, we have (the two first inequalities are true for every tournament and any ordering, the last one comes from the property of $\prec_T^*$): $$\label{eq:tww} tww(T^{\prec_T^*}) \leq tww(T^{\prec_T^*}, \prec_T^*) \leq tww(T, \prec_T^*) \leq f(k)$$ Hence, the class of undirected graphs $\mathcal C_k=\{T^{\prec_T^*}\mid T \in \mathcal T_k\}$ has bounded twin-width, and is thus $\chi$-bounded by a polynomial function $g$. Let $T \in \mathcal T_k$. We have: $T^{\prec_T^*} \in \mathcal C_k$, $tww(T^{\prec_T^*}) \leq f(k)$ by [\[eq:tww\]](#eq:tww){reference-type="eqref" reference="eq:tww"}, $\omega(T^{\prec_T^*}) \leq f(\mathop{\mathrm{\overrightarrow{\omega}}}(T))$ by the choice of $\prec_T^*$. Hence, $$\chi(T^{\prec_T^*}) \leq g(f(\mathop{\mathrm{\overrightarrow{\omega}}}(T)))$$ and thus $\mathop{\mathrm{\overrightarrow{\chi}}}(T) \leq g(f(\mathop{\mathrm{\overrightarrow{\omega}}}(T))$. ◻ Geniet and Thomassé [@GT22] introduced a particular ordering of tournaments called *BST-ordering*. Informally, a $BST$-ordering of a tournament $T$ is based on a rooted binary search tree on vertex set $V(T)$ with the property that for every vertex $x$, the left child of $x$ and its descendent are in $N^-(x)$, and the right child of $x$ and its descendent are in $N^+(x)$, and the order is the left-to-right order defined by this tree. See [@GT22] Section 4 for a formal definition. They prove that $BST$-orderings give an approximation of the twin-width in the following sense: There exists a function $f$ such that, for every tournament $T$ and any $BST$-ordering $\prec$ of $T$, we have: $$tww(T, \prec) \leq f(tww(T))$$ Hence, $BST$-orderings are natural candidates for the ordering of Conjecture [\[conj:good_ordering_tww_omega\]](#conj:good_ordering_tww_omega){reference-type="ref" reference="conj:good_ordering_tww_omega"}. There exists a function $f$ such that, for every tournament $T$, there exists a $BST$-ordering $\prec$ of $T$ such: $$\omega(T^{\prec}) \leq f(\mathop{\mathrm{\overrightarrow{\omega}}}(T))$$ # Classes of tournaments defined by forbidding a single tournament {#sec:Hfree} In this section, we will investigate the classes defined by forbidding a single tournament. Our main question will be to understand which tournaments $H$ are such that $\mathop{\mathrm{Forb}}(H)$ is $\chi$-bounded. Such a tournament is said to be *$\mathop{\mathrm{\overrightarrow{\chi}}}$-binding*. ## Gentlemen are the same as heroes {#sec:gentlemen_heroes} The most trivial case of $\chi$-bounding function is a constant function. A tournament $H$ is a *hero* if there exists an integer $c_H$ such that every $H$-free tournament $T$ has dichromatic number at most $c_H$. In a seminal paper, Berger, Choromanski, Chudnovsky, Fox, Loebl, Scott, Seymour and Thomassé [@hero] characterized heroes: [\[thm:heroes\]]{#thm:heroes label="thm:heroes"} A tournament $H$ is a hero if and only if: - $H=TT_1$, or - $H=H_1 \Rightarrow H_2$, where $H_1$ and $H_2$ are heroes in tournaments, or - $H=\Delta(1, k, H_1)$ or $H = \Delta(1, H_1, k)$, where $k\geq 1$ and $H_1$ is a hero in tournaments. Similarly, we say that a tournament $H$ is a *gentleman* if there exists a number $c_H$ such that every $H$-free tournament has clique number at most $c_H$. Since $\mathop{\mathrm{\overrightarrow{\omega}}}(T)\leq \mathop{\mathrm{\overrightarrow{\chi}}}(T)$ for any tournament $T$, heroes are gentlemen. We prove that the converse is also true. In [@NSS23], Nguyen, Scott and Seymour introduce a class of tournaments called *crossing tournaments* (it is the class $\mathcal T[\mathcal C]$ where $\mathcal C$ is the class of circle graphs). They prove that crossing tournaments are $S_3$-free and prove a result (see 12.4 in [@NSS23]) that can be translated in our language by saying that they have arbitrarily large clique number. It is a key ingredient in the proof of the following theorem. [\[thm:gentlemen\]]{#thm:gentlemen label="thm:gentlemen"} Gentlemen and heroes are the same. *Proof.* For every tournament $T$, $\mathop{\mathrm{\overrightarrow{\omega}}}(T) \leq \mathop{\mathrm{\overrightarrow{\chi}}}(T)$, thus it is clear that all heroes are gentlemen. Let us now prove that all gentlemen are heroes. Suppose there exists a gentleman $H$ that is not a hero, and let it be chosen so as to minimize $|V(H)|$. Since all subtournaments of a gentleman are gentlemen (because tournaments not containing a subtournament of $H$ do not contain $H$ and thus have bounded clique number), every subtournament of $H$ is a hero by minimality of $V(H)$. Consider the sequence of tournaments $S_n$ defined at the end of subsection [3.2](#sec:subst_S_k){reference-type="ref" reference="sec:subst_S_k"}. Since this sequence has unbounded clique number and $H$ is a gentlemen, there exists an integer $k$ such that $H$ is a subtournament of $S_k$. This implies that either $H = A \Rightarrow B$ or $H = \Delta(1,A,B)$ for some tournaments $A$ and $B$. Since $A$ and $B$ are two strict subtournaments of $H$, they are heroes by minimality of $H$. Thus $H \neq A \Rightarrow B$ for $H$ would be a hero by Theorem [\[thm:heroes\]](#thm:heroes){reference-type="ref" reference="thm:heroes"}. Thus $H = \Delta(1,A,B)$. But we know that $S_3$ is not a gentleman since crossing tournaments are $S_3$-free and can have arbitrarily large clique number. Thus $H$ does not contain $S_3 = \Delta(1, \vec C_3 , \vec C_3)$. This implies that one of $A$ or $B$ does not contain $\vec C_3$ and thus either $A$ or $B$ is a transitive tournament, which implies $H$ is a hero, a contradiction. ◻ ## Gyárfás-Sumner Conjecture for tournaments {#sec:gyarfas_sumner} We propose the following analogue of the celebrated Gyárfás-Sumner Conjecture [@G75; @S81] that states that a graph $F$ is $\chi$-binding if and only if $F$ is a forest (where, as in the directed case, a graph $F$ is $\chi$-binding if the class of graphs not containing $F$ as an induced subgraph is $\chi$-bounded). [\[conj:GS_tournaments\]]{#conj:GS_tournaments label="conj:GS_tournaments"} A tournament $H$ is $\mathop{\mathrm{\overrightarrow{\chi}}}$-binding if and only if $H$ has a backedge graph which is a forest. Despite the link between $\mathop{\mathrm{\overrightarrow{\chi}}}$-bounded classes of tournaments and $\chi$-bounded classes of graphs given by Theorem [\[thm2:equivchi-boundedness\]](#thm2:equivchi-boundedness){reference-type="ref" reference="thm2:equivchi-boundedness"}, we were not able to prove that Gyárfás-Sumner Conjecture applies or is applied by Conjecture [\[conj:GS_tournaments\]](#conj:GS_tournaments){reference-type="ref" reference="conj:GS_tournaments"}. We now believe that the two conjectures are independent, but we would be very happy if a bridge between them was shown. To support the conjecture, we prove that : - the \"only if\" part is true (Theorem [\[thm:only_if_part_GS\]](#thm:only_if_part_GS){reference-type="ref" reference="thm:only_if_part_GS"}), - it is enough to prove it for trees instead of forests (Proposition [\[lem:GS_for_trees\]](#lem:GS_for_trees){reference-type="ref" reference="lem:GS_for_trees"}), - if it holds for a tournament $T$ then it holds for the tournaments obtained by reversing every arc of $T$ (Proposition [\[lem:reverse\]](#lem:reverse){reference-type="ref" reference="lem:reverse"}), - if it holds for two tournaments $H_1$ and $H_2$ then it holds for the tournament $H_1 \Rightarrow H_2$ (Theorem [\[thm:GS_H=\>H\]](#thm:GS_H=>H){reference-type="ref" reference="thm:GS_H=>H"}), A *star* is a tree that has at most one non-leaf vertex. We also prove that heroes admit a backedge graph that is a disjoint union of stars. See Proposition [\[prop:herostar\]](#prop:herostar){reference-type="ref" reference="prop:herostar"}. At the end of the section we also discuss the case of tournaments $T$ that admits a backedge graph that is a matching. [\[thm:only_if_part_GS\]]{#thm:only_if_part_GS label="thm:only_if_part_GS"} Let $H$ be a tournament. If $H$ is $\mathop{\mathrm{\overrightarrow{\chi}}}$-binding, then $H$ admits an ordering whose backedge graph is a forest. *Proof.* Let $H$ be a tournament that does not admit an ordering whose backedge graph is a forest. Let $\mathcal C$ be the class of undirected graphs with girth at least $|V(H)|+1$, and let $\mathcal T[\mathcal C]$ be the class of tournaments that admit a graph of $\mathcal C$ as a backedge graph. Let $T \in \mathcal T[\mathcal C]$ and let $X \subseteq V(T)$ such that $|X| = |V(H)|$. $T$ admits an ordering such that the backedge graph has girth at least $|V(H)|+1$, so $T[X]$ admits an ordering for which the backedge graph is a forest. So $T[X] \neq H$. This proves that tournaments in $\mathcal T[\mathcal C]$ are $H$-free. Since for every $T \in \mathcal T[\mathcal C]$, a backedge graph of $T$ has girth $|V(H)| + 1 \geq 4$, we have $\mathop{\mathrm{\overrightarrow{\omega}}}(T) \leq 2$. By a celebrated result of Erdős [@E59], for every integer $k$, there exists $G \in \mathcal C$ such that $\chi(G) \geq k$. Let $T \in \mathcal T[\mathcal C]$ such that $T$ admits an ordering $\prec$ such that $T^{\prec}=G$. By Theorem [\[eq:bounddic\]](#eq:bounddic){reference-type="ref" reference="eq:bounddic"}, $\mathop{\mathrm{\overrightarrow{\chi}}}(T) \geq \chi(T^{ \prec }) / \omega(T^{\prec}) = k/2$. This proves that there are $H$-free tournaments with clique number $2$ and arbitrarily large dichromatic number, i.e. $H$ is not $\mathop{\mathrm{\overrightarrow{\chi}}}$-binding. ◻ Let $T$ be a tournament admitting a forest as a backedge graph. We claim that $T$ also admits a tree as a backedge graph. Indeed, let $\prec$ be an ordering of $V(T)$ such that $T^{\prec}$ is a forest and among such ordering, assume it minimizes the number of connected components of $T^{\prec}$. We claim that $T^{\prec}$ is a tree. Assume for contradiction that it is not. Let $v$ be the smallest vertex not in the same connected component of $T^{\prec}$ as the first vertex, and let $u$ be the vertex preceding $v$ in $\prec$. Then the backedge graph resulting from switching $u$ and $v$ in the ordering is obtained from $T^{\prec}$ by adding the edges $uv$ and thus has one less connected component than $T^{\prec}$, a contradiction. We thus have the following: [\[lem:GS_for_trees\]]{#lem:GS_for_trees label="lem:GS_for_trees"} It is enough to prove Conjecture [\[conj:GS_tournaments\]](#conj:GS_tournaments){reference-type="ref" reference="conj:GS_tournaments"} for tournaments that admit a tree as a backedge graph. As mentioned before, heroes are by definition $\mathop{\mathrm{\overrightarrow{\chi}}}$-binding tournaments, and by Theorem [\[thm:only_if_part_GS\]](#thm:only_if_part_GS){reference-type="ref" reference="thm:only_if_part_GS"} they admit a backedge graph that is a forest. The following proposition proves that for heroes, this backedge graphs is a star forest. [\[prop:herostar\]]{#prop:herostar label="prop:herostar"} If $H$ is a hero, then $H$ admits a backedge graph that is a disjoint union of stars. *Proof.* We prove this using the inductive construction of heroes given by Theorem [\[thm:heroes\]](#thm:heroes){reference-type="ref" reference="thm:heroes"}. It is true for $TT_1$ so we need to maintain this property if $H=H_1\Rightarrow H_2$ and if $H=\Delta(1,k,H_1)$. If $H=H_1\Rightarrow H_2$, consider orderings $\prec_i$ of $H_i$ given by the induction, and simply construct the order on $V(H)$ in which all vertices of $H_1$ are placed before those of $H_2$ (respecting $\prec_1$ and $\prec_2$). This adds no new back arc, so the back edge graph is the union of those of $H_1$ and $H_2$, so we have our result. If $H=\Delta(1,k,H_1)$, then consider the ordering $\prec$ of $H_i$ given by the induction, and construct the order on $V(H)$ obtained by placing the vertices of $TT_k$ first so that all arcs go forward, then the vertices of $H_1$ in the order $\prec$, and finally the vertex $x$ corresponding to the \"1\" in $\Delta(1,k,H_1)$. The only new back arcs are the one from $x$ to the vertices of the $TT_k$, which produce a star, so we again get our desired result. ◻ Given a tournament $T$, the reverse $T_r$ of $T$ is the tournament obtained from $T$ by reversing the direction of every arc. [\[lem:reverse\]]{#lem:reverse label="lem:reverse"} If $H$ is $\mathop{\mathrm{\overrightarrow{\chi}}}$-binding, then the reverse $H_r$ of $H$ is also $\mathop{\mathrm{\overrightarrow{\chi}}}$-binding. *Proof.* Observe that, for every tournament $H$ and every ordering $\prec$ of $H$, $H^{\prec} = H_r^{\prec_r}$. So $\mathop{\mathrm{Forb}}(H)^{\prec} = \mathop{\mathrm{Forb}}(H_r)^{\prec}$ and the result holds by Theorem [\[thm2:equivchi-boundedness\]](#thm2:equivchi-boundedness){reference-type="ref" reference="thm2:equivchi-boundedness"}. ◻ In order to prove our next theorem, we will use the following very nice result of Le, Harutyunyan, Thomassé and Wu that states that if the outneighbourhood of each vertex of a tournament has bounded dichromatic number, so does the whole tournament. We will discuss more of these kinds of "local to global" properties in Section [5](#sec:cluster){reference-type="ref" reference="sec:cluster"}. [\[thm:local_to_global\]]{#thm:local_to_global label="thm:local_to_global"} There exists a function $\lambda_1$ such that, for every integer $t$, if $T$ is a tournament such that for every $v \in V(T)$, $\mathop{\mathrm{\overrightarrow{\chi}}}(T[N^+(v)]) \leq t$, then $\mathop{\mathrm{\overrightarrow{\chi}}}(T) \leq \lambda_1(t)$. Following [@KN23], we define the *neighbourhood $N(uv)$ of an arc $uv$* to be $N^-(u) \cap N^+(v)$, that is the set of vertices forming a directed triangle with $uv$. [\[thm:GS_H=\>H\]]{#thm:GS_H=>H label="thm:GS_H=>H"} If $H_1$ and $H_2$ are two $\mathop{\mathrm{\overrightarrow{\chi}}}$-binding tournaments, then $H_1 \Rightarrow H_2$ is also $\mathop{\mathrm{\overrightarrow{\chi}}}$-binding. *Proof.* For $i=1,2$, let $f_i$ be a $\mathop{\mathrm{\overrightarrow{\chi}}}$-binding function for the class of $H_i$-free tournaments. Set $f=\max(f_1, f_2)$ and $h=\max(|H_1|, |H_2|)$. We are going to prove that the class of $(H_1 \Rightarrow H_2)$-free tournaments is $\mathop{\mathrm{\overrightarrow{\chi}}}$-bounded by a function $g$ that will be fixed at the end of the proof. The result is trivial for tournaments with clique number $1$. We proceed by induction on $\mathop{\mathrm{\overrightarrow{\omega}}}$. Let $k \geq 1$. Assume the result holds for tournaments with clique number at most $k$ and let $T$ be a $(H_1 \Rightarrow H_2)$-free tournament with clique number $k+1$. Let $\prec$ be an $\mathop{\mathrm{\overrightarrow{\omega}}}$-ordering of $T$. We call an arc $uv$ heavy if $\mathop{\mathrm{\overrightarrow{\chi}}}(T[N(uv)]) \geq 2g(k) + 1$, and light otherwise. **Claim 1**. *If $uv$ is a heavy arc, then $v \prec u$* Assume for contradiction that $u \prec v$. Let $(L, M, R)$ be the partition of $N(uv)$ such that $L \prec u \prec M \prec v \prec R$. Since $L \cup M \prec v$ and $v \Rightarrow L \cup M$, $\mathop{\mathrm{\overrightarrow{\omega}}}(T[L \cup M]) \leq k$. Since $u \prec R$ and $R \Rightarrow u$, $\mathop{\mathrm{\overrightarrow{\omega}}}(T[R]) \leq k$. Hence, by induction, $\mathop{\mathrm{\overrightarrow{\chi}}}(T[N(uv)]) \leq 2g(k)$, a contradiction with the fact that $uv$ is heavy. $\square$ **Claim 2**. *Every vertex $u$ satisfies $\min(\mathop{\mathrm{\overrightarrow{\chi}}}(T[N^-(u)]), \mathop{\mathrm{\overrightarrow{\chi}}}(T[N^+(u)]) \leq f(k+1) + g(k) + 2hg(k)$.* Let $u \in V(T)$. Set $X_h^+ = \{v \mid uv \text{ is heavy}\}$, $X_h^- = \{v \mid vu \text{ is heavy}\}$, $X_{\ell}^+ = \{v \mid uv \text{ is light}\}$ and $X_{\ell}^- = \{v \mid vu \text{ is light}\}$. Note that $N^+(u) = X_h^+ \cup X_{\ell}^+$ and $N^-(u) = X_h^- \cup X_{\ell}^-$ By claim [Claim 1](#clm:uv_heavy_arc){reference-type="ref" reference="clm:uv_heavy_arc"}, $X_h^+ \prec u \prec X_h^-$, and since $X_h^- \Rightarrow u \Rightarrow X_h^+$, we have $\mathop{\mathrm{\overrightarrow{\omega}}}(T[X_h^+ \cup X_h^-]) \leq k$ and thus, by induction, $\mathop{\mathrm{\overrightarrow{\chi}}}(T[X_h^+]) \leq g(k)$ and $\mathop{\mathrm{\overrightarrow{\chi}}}(T[X_h^-]) \leq g(k)$. Now, if $\min(\mathop{\mathrm{\overrightarrow{\chi}}}(T[X_{\ell}^+]), \mathop{\mathrm{\overrightarrow{\chi}}}(T[X_{\ell}^-]))\leq f(k+1) + 2hg(k)$, we are done. Assume first that $\mathop{\mathrm{\overrightarrow{\chi}}}(T[X_{\ell}^+]) > f(k+1) + 2hg(k)$. Then there exists $V_2 \subseteq X_{\ell}^+$ such that $T[V_2]=H_2$. Let $X \subseteq X_{\ell}^-$ such that $X \Rightarrow V_2$. Then $X$ is $H_1$-free and thus $\mathop{\mathrm{\overrightarrow{\chi}}}(T[X]) \leq f(k+1)$. For each $w \in X_{\ell}^- \setminus X$, there exists $v \in V_2$ such that $w \in N(uv)$. Since for each $v \in V_2$, $uv$ is light, i.e. $\mathop{\mathrm{\overrightarrow{\chi}}}(T[N(uv)]) \leq 2g(k)$, we have $\mathop{\mathrm{\overrightarrow{\chi}}}(T[X_{\ell}^- \setminus X]) \leq 2hg(k)$. Hence $\mathop{\mathrm{\overrightarrow{\chi}}}(T[N^-(u)]) \leq g(k) + f(k+1) + 2hg(k)$. If $\mathop{\mathrm{\overrightarrow{\chi}}}(T[X_{\ell}^-]) > f(k+1) + 2hg(k)$, we similarly get that $\mathop{\mathrm{\overrightarrow{\chi}}}(T[N^+(u)]) \leq g(k) + f(k+1) + 2hg(k)$. $\square$ By claim [Claim 2](#clm:vertices_are_small){reference-type="ref" reference="clm:vertices_are_small"}, we can partition $V(T)$ into two sets $V_1$ and $V_2$ such that for every $v \in V_1$, $\mathop{\mathrm{\overrightarrow{\chi}}}(T[N^+(v)]) \leq f(k+1) + g(k) + 2hg(k)$ and for every $v \in V_2$, $\mathop{\mathrm{\overrightarrow{\chi}}}(T[N^-(v)]) \leq f(k+1) + g(k) + 2hg(k)$. Hence, by Theorem [\[thm:local_to_global\]](#thm:local_to_global){reference-type="ref" reference="thm:local_to_global"}, $\mathop{\mathrm{\overrightarrow{\chi}}}(V_i) \leq \gamma(f(k+1) + g(k) + 2hg(k))$. Hence, the function $g=2\gamma(f(k+1) + g(k) + 2hg(k))$ is a $\mathop{\mathrm{\overrightarrow{\chi}}}$-binding function for the class of $H_1 \Rightarrow H_2$-free tournaments. ◻ For the last part of this section we need to consider *ordered graphs*, that is pairs $(G,\prec)$, where $G$ is an undirected graph, and $\prec \in \mathfrak S(G)$ is an order on its vertices. We say that an ordered graph $(H,\prec_H)$ is an induced ordered subgraph of an ordered graph $(G,\prec_G)$ if there exists an injective mapping $\phi:V(H)\to V(G)$ such that for all vertices $x,y$ of $V(H)$, $x \prec_H y$ if and only if $\phi(x) \prec_G \phi(y)$ and $xy\in E(H)$ if an only if $\phi(x)\phi(y)\in E(G)$. Given that definition, and for a given class $\mathcal{O}$ of ordered graphs, one can define $\mathop{\mathrm{Forb}}(\mathcal{O})$ as the set of ordered graphs that do not contain any member of $\mathcal{O}$ as an induced ordered subgraph. We say that a class of ordered graphs $\mathcal{O}$ is $\chi$-bounded if the set of graphs $G$ such that there exists $\prec$ with $(G,\prec)\in \mathcal{O}$ is $\chi$-bounded (we simply ignore the orderings here). Let $\mathcal T$ be a class of tournaments. Recall that $\mathcal T^{\prec}$ is the set of graphs that are backedge graphs of a tournament in $\mathcal T$. We now define the ordered version of it as follows: $$\mathcal T_o^{\prec} = \{(T^{\prec}, \prec): T \in \mathcal T, \prec\in \mathfrak S(T)\}$$ Note that, given a tournament $T$, $\{T\}^{\prec}$ is the set of backedge graphs of $T$ and $\{T\}^{\prec}_o$ the set of ordered backedge graphs of $T$. The following is an ordered analogue of Theorem [\[thm2:equivchi-boundedness\]](#thm2:equivchi-boundedness){reference-type="ref" reference="thm2:equivchi-boundedness"}. [\[prop:tournaments_orderedgraphs\]]{#prop:tournaments_orderedgraphs label="prop:tournaments_orderedgraphs"} Let $T$ be a tournament. The class of tournaments $Forb(T)$ is $\mathop{\mathrm{\overrightarrow{\chi}}}$-bounded if and only if the class of ordered undirected graphs $Forb(\{T\}^{\prec}_o)$ is $\chi$-bounded. *Proof.* Assume first that $Forb(T)$ is $\mathop{\mathrm{\overrightarrow{\chi}}}$-bounding by a function $f$. Let $(G, \prec_G) \in Forb(\{T\}^{\prec}_o)$. Let $T$ be the tournament on the same vertex set as $G$, such that $(T^{\prec_G}, \prec_G) = (G, \prec_G)$ (i.e. $T$ is obtained from $(G, \prec_G)$ by oriented edges of $G$ from right to left, and all other edges from left to right). Since $(G, \prec_G) \in Forb(\{T\}^{\prec}_o)$, $T \in Forb(T)$ and thus $\mathop{\mathrm{\overrightarrow{\chi}}}(T) \leq f(\mathop{\mathrm{\overrightarrow{\omega}}}(T)) \leq f(\omega(G))$ (because $\mathop{\mathrm{\overrightarrow{\omega}}}(T) \leq \omega(G)$). We have the first inequality by Theorem [\[eq:bounddic\]](#eq:bounddic){reference-type="ref" reference="eq:bounddic"} $$\chi(G) = \chi(T^{\prec}) \leq \mathop{\mathrm{\overrightarrow{\chi}}}(T) \omega(T^{\prec}) \leq f(\omega(G))\omega(G)$$ which proves that $Forb(\{T\}^{\prec}_o)$ is $\chi$-bounded. Assume now that $Forb(\{T\}^{\prec}_o)$ is $\chi$-bounded by a function $f$. Let $T \in Forb(T)$ and let $\prec$ be an $\mathop{\mathrm{\overrightarrow{\omega}}}$-ordering of $T$. Then $T^{\prec} \in Forb(\{T\}^{\prec}_o)$. Hence: $\mathop{\mathrm{\overrightarrow{\chi}}}(T) \leq \chi(T^{\prec}) \leq f(\omega(T^{\prec})) = f(\mathop{\mathrm{\overrightarrow{\omega}}}(T))$ ◻ Briański, Davies and Walczak [@BDW22] studied for which ordered graphs $(G, \prec_G)$, $Forb((G, \prec_G))$ is $\chi$-bounded, and claimed in a personal communications to have proven that excluding any ordered matching yields a $\chi$-bounded class. [\[conj:bartosz\]]{#conj:bartosz label="conj:bartosz"} Let $(M,\prec)$ be an ordered graph with maximum degree $1$. Then the class of $(M,\prec)$-free ordered graphs is $\chi$-bounded. By Property [\[prop:tournaments_orderedgraphs\]](#prop:tournaments_orderedgraphs){reference-type="ref" reference="prop:tournaments_orderedgraphs"}, we have: [\[conj:ordered_matching\]]{#conj:ordered_matching label="conj:ordered_matching"} If Conjecture [\[conj:bartosz\]](#conj:bartosz){reference-type="ref" reference="conj:bartosz"} holds, then any tournament that admits a backedge graph of maximum degree $1$ is $\mathop{\mathrm{\overrightarrow{\chi}}}$-binding. For the usual undirected version of Gyárfás-Sumner conjecture, one of the first non trivial cases that was proven (by Gyárfás, see [@Gya87]) concerns the class of graphs that do not contain a path of fixed length as an induced subgraph. An analogue for tournaments could be the tournament $T\hspace{-0.1em}P_n$ on $n$ vertices obtained from the transitive tournament $TT_n$ by reversing the direction of each arc of the unique Hamiltonian path $v_1v_2\ldots v_n$. Now consider $\prec$ to be the ordering $v_2 \prec v_1 \prec v_4 \prec v_3 \prec \ldots \prec v_{2p} \prec v_{2p-1} \prec \ldots \prec v_n \prec v_{n-1}$ (assuming for simplicity that $n$ is even, otherwise we end with $v_n$). It is easy to observe that the backedge graph of $T\hspace{-0.1em}P_n$ with respect to $\prec$ has maximum degree $1$. Hence, by  [\[prop:tournaments_orderedgraphs\]](#prop:tournaments_orderedgraphs){reference-type="ref" reference="prop:tournaments_orderedgraphs"}, we have: [\[lem:GS_holds_for_Pk\]]{#lem:GS_holds_for_Pk label="lem:GS_holds_for_Pk"} If Conjecture [\[conj:bartosz\]](#conj:bartosz){reference-type="ref" reference="conj:bartosz"} holds, then for every $k \geq 1$, $T\hspace{-0.1em}P_k$ is $\mathop{\mathrm{\overrightarrow{\chi}}}$-binding. ## Relation with the Erdős-Hajnal property and the $BIG \Rightarrow BIG$ conjecture {#sec:erdos_hajnal} A tournament $H$ has the *Erdős-Hajnal property* if there exists an integer $c$ such that for every $H$-free tournaments $T$, $T$ contains a transitive tournament on $|T|^{c}$ vertices. It was proven in [@alon2001ramsey] that the famous Erdős-Hajnal conjecture on undirected graphs is equivalent to the conjecture saying that every tournament has the Erdős-Hajnal property. If $H$ is a polynomially $\mathop{\mathrm{\overrightarrow{\chi}}}$-binding tournament, then $H$ has the Erdős-Hajnal property. *Proof.* Let $H$ be a tournament and $c$ an integer such that for every $H$-free tournaments $T$, $\mathop{\mathrm{\overrightarrow{\chi}}}(T) \leq \mathop{\mathrm{\overrightarrow{\omega}}}(T)^c$. Let us prove that $H$ has the Erdős-Hajnal property (for the constant $\frac{1}{1+c}$). Let $T$ be an $H$-free tournament. If $\mathop{\mathrm{\overrightarrow{\omega}}}(T) \geq n^{\frac{1}{1+c}}$, then $T$ contains a transitive tournament of size $n^{\frac{1}{1+c}}$ and we are done. So assume that $\mathop{\mathrm{\overrightarrow{\omega}}}(T) \leq n^{\frac{1}{1+c}}$. Then $\mathop{\mathrm{\overrightarrow{\chi}}}(T) \leq n^{\frac{c}{1+c}}$ and thus $T$ contains a transitive tournament on $\frac{n}{n^{\frac{c}{1+c}}}=n^{\frac{1}{1+c}}$ vertices. ◻ A tournament $H$ has the *strong Erdős-Hajnal property* if there exists a number $c$ such that all $H$-free tournaments contain two disjoint set of vertices $A$ and $B$ such that $A \Rightarrow B$ and $|A|,|B| \geq c|H|$. It can be shown that if $H$ has the strong Erdős-Hajnal then it has the Erdős-Hajnal property, and that not every tournament has the strong Erdős-Hajnal property (the Paley tournament on $7$ vertices being an example). Heroes have the strong Erdős-Hajnal property since bounded dichromatic number implies a transitive tournament of linear size, and thus a directed cut of linear size. In [@Chud24], the authors prove that every tournament that has the strong Erdős-Hajnal property admits a backedge graph that is a forest and conjecture that the converse is true : [\[conj:SEH_BEF\]]{#conj:SEH_BEF label="conj:SEH_BEF"} A tournament $H$ has the strong Erdős-Hajnal property if and only if it has a backedge graph that is a forest. Note that in view of our conjecture [\[conj:GS_tournaments\]](#conj:GS_tournaments){reference-type="ref" reference="conj:GS_tournaments"}, it could be that being chi-bounding is the same has having the strong Erdős-Hajnal property. We are not able to prove any inclusion yet.\ We say that a class of tournaments $\mathcal T$ has the *$BIG \Rightarrow BIG$ property* if there exists a function $f$ such that, for every $T \in \mathcal T$, if $\mathop{\mathrm{\overrightarrow{\chi}}}(T) \geq f(t)$, then $T$ contains two disjoint subtournaments $A$ and $B$ such that $\mathop{\mathrm{\overrightarrow{\chi}}}(A), \mathop{\mathrm{\overrightarrow{\chi}}}(B) \geq t$ and $A \Rightarrow B$. In [@NSS23], the following beautiful conjecture is proposed: The class of all tournaments has the $BIG \Rightarrow BIG$ property. Nguyen, Scott and Seymour proved in [@elzahar] that the $BIG \Rightarrow BIG$ Conjecture implies the Erdős-El-Zahar conjecture, which states that there exists a function $f$ such that, for every integer $c$, every graph $G$ with $\chi(G) \geq f(\omega(G), c)$ contains two disjoint subgraphs $A$ and $B$ such that $\chi(A), \chi(B) \geq c$ and there is no edge between $A$ and $B$. Klingelhöfer and Newman [@KN23] showed recently the other direction, that is the Erdős-El-Zahar conjecture implies the $BIG \Rightarrow BIG$ Conjecture. To prove it, they first prove the following beautiful theorem. Given an oriented graph $G$, we denote by $\alpha(G)$ the size of a maximum independent set of $G$. [\[thm:arc_local_to_global_alpha\]]{#thm:arc_local_to_global_alpha label="thm:arc_local_to_global_alpha"} There exists a function $\lambda$ such that, for every integer $t$, if $G$ is an oriented graph such that for every $a \in A(T)$, $\mathop{\mathrm{\overrightarrow{\chi}}}(G[N^+(a)]) \leq t$, then $\mathop{\mathrm{\overrightarrow{\chi}}}(G) \leq \lambda(t, \alpha(G))$. Applying the exact same method as Klingelhöfer and Newman used to prove that the Erdős-El-Zahar conjecture implies the $BIG \Rightarrow BIG$ Conjecture, we can prove the following. If $H$ is a $\mathop{\mathrm{\overrightarrow{\chi}}}$-binding tournament, then the class of $H$-free tournaments has the $BIG \Rightarrow BIG$ property *Proof.* Let $\ell$ be a function defined inductively as follows: $\ell(1) = 1$ and for every $t \geq 1$, $\ell(t+1)=(t+1)+\binom{t+1}{2} \ell(t)$. Given a tournament $T$, we say that a subtournament $X$ of $T$ is a $t$-cluster if $\mathop{\mathrm{\overrightarrow{\chi}}}(X) \geq t$ and $|X| \leq \ell(t)$ Let $H$ be a $\mathop{\mathrm{\overrightarrow{\chi}}}$-binding tournament and $c$ be an integer. Let $T$ be an $H$-free tournament such that $T$ does not contain two disjoint subtournaments $A$ and $B$ such that $A \Rightarrow B$ and $\mathop{\mathrm{\overrightarrow{\chi}}}(A),\mathop{\mathrm{\overrightarrow{\chi}}}(B) \geq c$. We want to prove that the dichromatic number of $T$ is bounded by a function of $c$. We first prove a weaker statement : we prove by induction on $t$ that if $T$ contains no $t$-cluster for some $t$ and no two disjoint subtournaments $A$ and $B$ such that $A \Rightarrow B$ and $\mathop{\mathrm{\overrightarrow{\chi}}}(A),\mathop{\mathrm{\overrightarrow{\chi}}}(B) \geq c$, then $\mathop{\mathrm{\overrightarrow{\chi}}}(T)$ is bounded (by a function of $c$ and $t$). Since a $1$-cluster is a vertex, the result trivially holds for $t=1$. Now assume it holds for $t <2c$, and let us prove it for $t+1$. So assume $T$ has no $(t+1)$-cluster, and say that an arc $a$ is *heavy* if $N(a)$ contains a $t$-cluster, and it is *light* otherwise (we recall that if $xy$ is an arc, $N(xy)$ denotes the set of vertices $z$ such that $yz\in A$ and $zx\in A$). Let $T_h$ be the oriented graph induced by the heavy arcs, and $T_{\ell}$ the oriented graphs induced by the light arcs. We first claim that the underlying graph of $T_h$ has maximum clique at most $t$. Assume by contradiction there exists $K$ of size $t+1$ inducing a tournament in $T_{h}$. For every arc $a$ with both endvertices in $K$, $a$ is heavy so there exists $C_a$ a $t$-cluster included in $N(a)$. Let $X$ be the subtournament of $T$ induced by the union of $K$ and all such sets $C_a$. The number of its vertices is at most $(t+1)+\binom{t+1}{2}l(t)=l(t+1)$. If $X$ admits a dicolouring with at most $t$ colours then there must be two vertices $x,y$ in $K$ that get the same colour (because $K$ has size $t+1$), but then this colour cannot appear in $C_{xy}$ (for it would create a monochromatic $C_3$), which contradicts the fact that $C_{xy}$ has dichromatic number at least $t$. Hence $\mathop{\mathrm{\overrightarrow{\chi}}}(X)\geq t+1$ and so $X$ is a $(t+1)$-cluster which contradicts our hypothesis. Hence we are proven our claim, which can be stated as $\alpha(T_{\ell}) \leq t$. By induction, since for every light arc $a$, $N(a)$ contains no $t$-cluster, we have that $\mathop{\mathrm{\overrightarrow{\chi}}}(N^+(a))$ is bounded for every light arc $a$. Now, by Theorem [\[thm:arc_local_to_global_alpha\]](#thm:arc_local_to_global_alpha){reference-type="ref" reference="thm:arc_local_to_global_alpha"}, $\mathop{\mathrm{\overrightarrow{\chi}}}(T_{\ell})$ is also bounded, say by $k$. Let $(S_1, \dots, S_k)$ be a dicolouring of $T_{\ell}$, i.e. $T_{\ell}[S_i]$ is acylic for $i=1, \dots, k$. Then, for $i=1, \dots, k$, there is an ordering $\prec_i$ of $S_i$ such that all backward arcs of $(T[S_i], \prec_i)$ are heavy. Hence $\mathop{\mathrm{\overrightarrow{\omega}}}(T[S_i]) \leq t$, and since $T[S_i]$ is $H$-free and $H$ is $\mathop{\mathrm{\overrightarrow{\chi}}}$-binding, $\mathop{\mathrm{\overrightarrow{\chi}}}(T[S_i])$ is bounded, which implies that $\mathop{\mathrm{\overrightarrow{\chi}}}(T)$ is also bounded. We can now conclude. Either $T$ contains no $2c$-cluster and we win by what precedes, or $T$ contains a $2c$-cluster $X$. Partition $V(T) \setminus V(X)$ with respect to their adjacency to $X$. This gives a partition of $V(T) \setminus X$ into at most $2^{|X|} \leq 2^{\ell(2c)}$ parts. Assume by contradiction that one of these parts, let it be $A$, has dichromatic number at least $c$. Call $B^+$ (resp. $B^-$) the subset of $X$ such that $A \Rightarrow B^+$ (resp $B^- \Rightarrow A$). Since $\mathop{\mathrm{\overrightarrow{\chi}}}(X) \geq 2c$, one of $B^+$ or $B^-$ has dichromatic number of at least $c$, and we get a contradiction with the assumption on $T$. So every such part $A$ has dichromatic number at most $c$ and hence $\mathop{\mathrm{\overrightarrow{\chi}}}(T)$ is bounded by $\ell(2c)+2^{\ell(2c)}c$. ◻ # Local to Global - Links with domination number {#sec:cluster} Informally, given a digraph parameter $\gamma$, a $\gamma$-cluster of a tournament $T$ is a subtournament $X$ of $T$ of bounded size with large $\gamma$. In this section, we investigate for which parameters $\gamma_1$ and $\gamma_2$ we have that, for all tournaments $T$ with sufficiently large $\gamma_1$, $T$ has a $\gamma_2$-cluster. We say that *large $\gamma_1$ implies a $\gamma_2$-cluster* if there exists two functions $f$ and $\ell$ such that for every integer $k$, if $\gamma_1(T) \geq k$, then $T$ contains a subtournament $X$ such that $\gamma_1(X) \geq k$ and $|X| \leq \ell(k)$. We review what is known on this topic and propose some new conjectures. Such property was first studied by Thomassé, Le, Harutyunyan and Wu in [@HLWT19]: [\[thm:dom_chi_cluster\]]{#thm:dom_chi_cluster label="thm:dom_chi_cluster"} There exists two functions $f$ and $\ell$ such that, for every integer $k$, every tournament $T$ with $\mathop{\mathrm{dom}}(T) \geq f(k)$ contains a subtournament $X$ with $|X| \leq \ell(k)$ and $\mathop{\mathrm{\overrightarrow{\chi}}}(X) \geq k$. In the same paper, Thomassé, Le, Harutyunyan and Wu conjectured the following: [\[conj:dom_dom_cluster\]]{#conj:dom_dom_cluster label="conj:dom_dom_cluster"} There exist two functions $f$ and $\ell$ such that, for every integer $k$, every tournament $T$ with $\mathop{\mathrm{dom}}(T) \geq f(k)$ contains a subtournament $X$ with $|X| \leq \ell(k)$ and $\mathop{\mathrm{dom}}(X) \geq k$. Since $\mathop{\mathrm{dom}}(T) \leq \mathop{\mathrm{\overrightarrow{\omega}}}(T) \leq \mathop{\mathrm{\overrightarrow{\chi}}}(T)$, the following is stronger than Theorem [\[thm:dom_chi_cluster\]](#thm:dom_chi_cluster){reference-type="ref" reference="thm:dom_chi_cluster"} but weaker than Conjecture [\[conj:dom_dom_cluster\]](#conj:dom_dom_cluster){reference-type="ref" reference="conj:dom_dom_cluster"}. [\[conj:dom_omega_cluster\]]{#conj:dom_omega_cluster label="conj:dom_omega_cluster"} There exist two functions $f$ and $\ell$ such that, for every integer $k$, every tournament $T$ with $\mathop{\mathrm{dom}}(T) \geq f(k)$ contains a subtournament $X$ with $|X| \leq \ell(k)$ and $\mathop{\mathrm{\overrightarrow{\omega}}}(X) \geq k$. A tournament $R$ is a *rebel* if tournaments not containing $R$ have bounded domination number. A tournament is a poset tournament if it admits a backedge graph that is a comparability graph. Chudnovsky, Kim, Liu, Seymour and Thomassé proved in [@CKLST18] that every rebel is a poset tournament, but the converse remains open. [\[conj:rebel\]]{#conj:rebel label="conj:rebel"} Every poset tournament is a rebel. In particular, the $S_k$ defined in Section [3.2](#sec:subst_S_k){reference-type="ref" reference="sec:subst_S_k"} are posets tournaments, and have arbitrarily large clique number (see Corollary [\[coro:S_k\]](#coro:S_k){reference-type="ref" reference="coro:S_k"}). Hence, if one can prove that, for every integer $k$, $S_k$ is a rebel, then Conjecture [\[conj:dom_omega_cluster\]](#conj:dom_omega_cluster){reference-type="ref" reference="conj:dom_omega_cluster"} holds. More generally: Conjecture [\[conj:rebel\]](#conj:rebel){reference-type="ref" reference="conj:rebel"} implies Conjecture [\[conj:dom_omega_cluster\]](#conj:dom_omega_cluster){reference-type="ref" reference="conj:dom_omega_cluster"}. Thomassé, Le, Harutyunyan and Wu applied Theorem [\[thm:dom_chi_cluster\]](#thm:dom_chi_cluster){reference-type="ref" reference="thm:dom_chi_cluster"} to prove Theorem [\[thm:local_to_global\]](#thm:local_to_global){reference-type="ref" reference="thm:local_to_global"}, which can be seen as a local to global theorem about dichromatic number. The analogue of Theorem [\[thm:local_to_global\]](#thm:local_to_global){reference-type="ref" reference="thm:local_to_global"} for clique number is the following conjecture. [\[conj:localtoglobalomega\]]{#conj:localtoglobalomega label="conj:localtoglobalomega"} There exists a function $g$ such that, for every integer $t$, if $T$ is a tournament such that for every $v \in V(T)$, $\mathop{\mathrm{\overrightarrow{\omega}}}(N^+(v)) \leq t$, then $\mathop{\mathrm{\overrightarrow{\omega}}}(T) \leq g(t)$. The analogue of Theorem [\[thm:dom_chi_cluster\]](#thm:dom_chi_cluster){reference-type="ref" reference="thm:dom_chi_cluster"} for clique number is Conjecture [\[conj:dom_omega_cluster\]](#conj:dom_omega_cluster){reference-type="ref" reference="conj:dom_omega_cluster"} and indeed we have the following implication. Conjecture [\[conj:dom_omega_cluster\]](#conj:dom_omega_cluster){reference-type="ref" reference="conj:dom_omega_cluster"} implies Conjecture [\[conj:localtoglobalomega\]](#conj:localtoglobalomega){reference-type="ref" reference="conj:localtoglobalomega"} *Proof.* Let $T$ be a tournament and $t \in \mathbb{N}$ such that for every vertex $v \in V(T)$, $\mathop{\mathrm{\overrightarrow{\omega}}}(N^+(v)) \leq t$. Let $f$ and $\ell$ be the functions given by Conjecture [\[conj:dom_omega_cluster\]](#conj:dom_omega_cluster){reference-type="ref" reference="conj:dom_omega_cluster"}. We will prove that $\mathop{\mathrm{\overrightarrow{\omega}}}(T) \leq \max(tf(t+1), t\ell(t+1))$. If $\mathop{\mathrm{dom}}(T) < f(t+1)$, then, since $\mathop{\mathrm{\overrightarrow{\omega}}}(N^+(v)) \leq t$ for every $v \in V(T)$, we have $\mathop{\mathrm{\overrightarrow{\omega}}}(T) < tf(t+1)$. So we may assume that $\mathop{\mathrm{dom}}(T) \geq f(t+1)$ and thus $T$ has a subtournament $X$ such that $|X| \leq \ell(t+1)$ and $\mathop{\mathrm{\overrightarrow{\omega}}}(A) \geq t+1$. Hence, since for every $v \in V(T)$, $\mathop{\mathrm{\overrightarrow{\omega}}}(N^+(v)) \leq t$, we have $A \subsetneq N^+(v)$. So $A$ is a dominating set of $T$, and thus $\mathop{\mathrm{\overrightarrow{\omega}}}(T) \leq t|A| \leq t\ell(t+1)$ ◻ Since $\mathop{\mathrm{dom}}(T) \leq \mathop{\mathrm{\overrightarrow{\omega}}}(T)$ for every tournament $T$, the following conjecture implies Conjecture [\[conj:dom_omega_cluster\]](#conj:dom_omega_cluster){reference-type="ref" reference="conj:dom_omega_cluster"} and would give a natural property of the clique number of tournaments. [\[conj:omega_omega_cluster\]]{#conj:omega_omega_cluster label="conj:omega_omega_cluster"} There exists two functions $f$ and $\ell$ such that, for every integer $k$, every tournament $T$ with $\mathop{\mathrm{\overrightarrow{\omega}}}(T) \geq f(k)$ contains a subtournament $X$ with $|X| \leq \ell(k)$ and $\mathop{\mathrm{\overrightarrow{\omega}}}(X) \geq k$. We believe (or maybe only hope) that the above conjecture is true, and actually we were not even able to disprove the following stronger form of it, where $f$ is taken to be the identity: [\[question:super_omega_cluster\]]{#question:super_omega_cluster label="question:super_omega_cluster"} Is there a function $\ell$ such that, for every tournament $T$, if $\mathop{\mathrm{\overrightarrow{\omega}}}(T) \geq k$, then $T$ has a subtournament $A$ such that $|A| \leq \ell(k)$ and $\mathop{\mathrm{\overrightarrow{\omega}}}(A) \geq k$. Let us say that a tournament $T$ is $k$-$\mathop{\mathrm{\overrightarrow{\omega}}}$-critical if $\mathop{\mathrm{\overrightarrow{\omega}}}(T) = k$ and for every $v \in V(T)$, $\mathop{\mathrm{\overrightarrow{\omega}}}(T-v) = k-1$. Observe that the only $1$-$\mathop{\mathrm{\overrightarrow{\omega}}}$-critical tournament if the one vertex tournament, and the only $2$-$\mathop{\mathrm{\overrightarrow{\omega}}}$-critical tournament is $\vec C_3$. [\[conj:critique\]]{#conj:critique label="conj:critique"} For every integer $k \geq 3$, there is an infinite number of $k$-$\mathop{\mathrm{\overrightarrow{\omega}}}$-critical tournaments. Observe that if Conjecture [\[conj:critique\]](#conj:critique){reference-type="ref" reference="conj:critique"} is true (resp. false), then it answers to the negative (resp. to the positive) to Question [\[question:super_omega_cluster\]](#question:super_omega_cluster){reference-type="ref" reference="question:super_omega_cluster"}. It is also open if large $\mathop{\mathrm{\overrightarrow{\omega}}}$ implies a $\mathop{\mathrm{\overrightarrow{\chi}}}$-cluster (resp. a $dom$-cluster). On the negative side, Thomassé, Le, Harutyunyan and Wu proved the following: For every integer $K, \ell$, there exists a tournament $T$ such that $\mathop{\mathrm{\overrightarrow{\chi}}}(T) \geq K$, and all subtournaments $X$ of $T$ on at most $\ell$ vertices are $2$-dicolourable. # Conclusion and future direction {#sec:conclusion} We did not give much thought yet to the clique number of digraphs. On this matter, it is to be noted that Theorem [\[eq:bounddic\]](#eq:bounddic){reference-type="ref" reference="eq:bounddic"} does not hold for digraphs, which implies that many of the results that we proved for tournaments cannot be proved as easily for digraphs. Actually, even if our definition of clique number makes at first glance as much sense applied to tournaments or digraphs, we strongly believe that the notion will be very fruitful on tournaments, while on digraphs it is less clear for the moment. On the positive side, we think that Theorem [\[thm:stable_by_subst\]](#thm:stable_by_subst){reference-type="ref" reference="thm:stable_by_subst"} can be generalised to classes of digraphs. [\[conj:closed_subst\]]{#conj:closed_subst label="conj:closed_subst"} If a class of digraphs $\mathcal C$ is $\mathop{\mathrm{\overrightarrow{\chi}}}$-bounded, then so is its closure under substitution. Note that, while Theorem [\[eq:bounddic\]](#eq:bounddic){reference-type="ref" reference="eq:bounddic"} does not hold for digraphs, a similar weaker bound exists for digraphs of bounded independence number. We denote by $\alpha(D)$ the size of a maximum independent set in $D$ and by $R(i,j)$ the smallest integer such that every graph on $R(i,j)$ vertices contains either a clique of size $j$, or an independent set of size $j$. $R(i,j)$ exists for all integers $i,j$ by Ramsey Theorem. [\[eq:bounddic_alpha\]]{#eq:bounddic_alpha label="eq:bounddic_alpha"} For any digraph $D$ and ordering $\prec$ of $V(D)$, we have: $$\frac{\chi(D^{\prec})}{R(\omega(D^{\prec})+1, \alpha(D)+1)} \leq \mathop{\mathrm{\overrightarrow{\chi}}}(D) \leq \chi(D^{\prec})$$ *Proof.* Let $D$ be a digraph and $\prec$ an ordering of $V(D)$. Set $\omega = \omega(D^{\prec})$ and $\alpha = \alpha(D)$. Let $X \subseteq V(D)$ such that $D[X]$ is acyclic. To prove that $\chi(D^{\prec}) \leq R(\omega+1,\alpha+1) \mathop{\mathrm{\overrightarrow{\chi}}}(D)$, it suffices to prove that $\chi(D^{\prec}[X]) \leq R(\omega+1,\alpha+1)$. Let $\varphi : X \to \mathbb{N}$ be such that $\varphi(x)$ is the number of vertices of a longest $\prec$-decreasing path in $D^{\prec}[X]$ finishing in $x$. We claim that $\varphi$ is a $R(\omega+1,\alpha+1)$-colouring of $D^{\prec}$. Let $u,v \in X$ with $u\prec v$ and $uv \in E(D^{\prec})$. Then $\varphi(u) \geq \varphi(v) + 1$, so $\varphi$ is a colouring of $D^{\prec}[X]$. Suppose for contradiction that $\varphi$ uses more than $R(\omega+1,\alpha+1)$ colours. Then there is a $\prec$-decreasing path $P$ of size $R(\omega+1,\alpha+1) + 1$. Note that since $D[V(P)]$ is acyclic, every arc of $D[V(P)]$ corresponds to an edge of $D^{\prec}[V(P)]$ (i.e. for every $x,y \in V(P)$, if $xy \in A(D)$, then $y \prec x$ and thus $xy \in E(D^{\prec})$). Hence, $\alpha(D^{\prec}[V(P)] = \alpha(D[V(P)]) \leq \alpha$. Now, since $|V(P)| > R(\omega+1,\alpha+1)$ and $\alpha(D^{\prec}[V(P)]) \leq \alpha$, we get that $\omega(D^{\prec}[V(P)]) \geq \omega + 1$, a contradiction. ◻ Using the above inequation, most results of the paper can be adapted to classes of tournaments with bounded independence number. An obvious topic that we did not investigate is the complexity of computing the clique number. Nguyen, Scott and Seymour ask in Section 9 of [@NSS23] if deciding if a tournament has bounded clique number is in co-NP. ### Acknowledgement {#acknowledgement .unnumbered} This research was partially supported by the ANR project DAGDigDec (JCJC) ANR-21-CE48-0012 and by the group Casino/ENS Chair on Algorithmics and Machine Learning. [^1]: This definition was introduced during a discussion between the authors and Stéphan Thomassé in Sète during the [fifth ANR Digraphs meeting](https://project.inria.fr/anrdigraphs/meetings/).

arxiv_math

--- abstract: | We obtain exact conditions for global weak solutions of the problem $$\left\{ \begin{aligned} & u_t - \sum_{|\alpha| = m} \partial^\alpha a_\alpha (x, t, u) \ge f (|u|) \quad \mbox{in } {\mathbb R}_+^{n+1} = {\mathbb R}^n \times (0, \infty), \\ & u (x, 0) = u_0 (x) \ge 0, \end{aligned} \right.$$ to be identically zero, where $m$ and $n$ are positive integers, $a_\alpha$ and $f$ are some functions, and $u_0 \in L_{1, loc} ({\mathbb R}^n)$. address: - Department of Differential Equations, Faculty of Mechanics and Mathematics, Moscow Lomonosov State University, Vorobyovy Gory, Moscow, 119992 Russia. Center of Nonlinear Problems of Mathematical Physics, RUDN University, Miklukho-Maklaya str. 6, Moscow, 117198 Russia. - Center of Nonlinear Problems of Mathematical Physics, RUDN University, Miklukho-Maklaya str. 6, Moscow, 117198 Russia. author: - A.A. Kon'kov - A.E. Shishkov title: On blow-up conditions for nonlinear higher order evolution inequalities --- # Introduction We study the problem $$\left\{ \begin{aligned} & u_t - \sum_{|\alpha| = m} \partial^\alpha a_\alpha (x, t, u) \ge f (|u|) \quad \mbox{in } {\mathbb R}_+^{n+1}, \\ & u (x, 0) = u_0 (x) \ge 0, \end{aligned} \right. \label{1.1}$$ where ${\mathbb R}_+^{n+1} = {\mathbb R}^n \times (0, \infty)$, $m, n \ge 1$, and $a_\alpha$ are Caratheodory functions such that $$|a_\alpha (x, t, \zeta)| \le A |\zeta|^p, \quad |\alpha| = m,$$ with some constants $A > 0$ and $p > 0$ for almost all $(x, t) \in {\mathbb R}_+^{n+1}$ and for all $\zeta \in {\mathbb R}$. As is customary, by $\alpha$ we mean a multi-index $\alpha = {(\alpha_1, \ldots, \alpha_n)}$. In so doing, $|\alpha| = \alpha_1 + \ldots + \alpha_n$ and $\partial^\alpha = {\partial^{|\alpha|} / (\partial_{x_1}^{\alpha_1} \ldots \partial_{x_n}^{\alpha_n})}.$ Let us denote by $B_r$ the open ball in ${\mathbb R}^n$ of radius $r > 0$ and center at zero. Also let $Q_r = B_r \times (0, r^m)$. We say that a function $u$ belongs to $L_{1, loc} ({\mathbb R}^n)$ if $u \in L_1 (B_r)$ for any real number $r > 0$. In its turn, $u \in L_{1, loc} ({\mathbb R}^{n+1}_+)$ if $u \in L_{1, loc} (Q_r)$ for any real number $r > 0$. Throughout the paper, it is assumed that $u_0 \in L_{1, loc} ({\mathbb R}^n)$ and, moreover, $f (\zeta)$ and $f (\zeta^{1/p})$ are non-decreasing convex functions on the interval $[0, \infty)$ with $f (\zeta) > 0$ for all $\zeta \in (0, \infty)$. A function $u \in {L_{1, loc} ({\mathbb R}_+^{n+1})} \cap {L_{p, loc} ({\mathbb R}_+^{n+1})}$ is called a global weak solution of [\[1.1\]](#1.1){reference-type="eqref" reference="1.1"} if ${f (|u|)} \in {L_{1, loc} ({\mathbb R}^{n+1}_+)}$ and $$\begin{aligned} & - \int_{ {\mathbb R}_+^{n+1} } u \varphi_t dx dt - \int_{ {\mathbb R}_+^{n+1} } \sum_{|\alpha| = m} (- 1)^m a_\alpha (x, t, u) \partial^\alpha \varphi dx dt \nonumber \\ & \qquad \ge \int_{ {\mathbb R}^n } u_0 (x) \varphi dx + \int_{ {\mathbb R}_+^{n+1} } f (|u|) \varphi dx dt \label{1.2}\end{aligned}$$ for any non-negative function $\varphi \in C_0^\infty ({\mathbb R}^{n+1})$. Analogously, a function $u \in {L_{1, loc} ({\mathbb R}_+^{n+1})} \cap {L_{p, loc} ({\mathbb R}_+^{n+1})}$ is a global weak solution of the problem $$\left\{ \begin{aligned} & u_t - \sum_{|\alpha| = m} \partial^\alpha a_\alpha (x, t, u) = f (|u|) \quad \mbox{in } {\mathbb R}_+^{n+1}, \\ & u (x, 0) = u_0 (x) \ge 0, \end{aligned} \right. \label{1.3}$$ if ${f (|u|)} \in {L_{1, loc} ({\mathbb R}^{n+1}_+)}$ and $$\begin{aligned} & - \int_{ {\mathbb R}_+^{n+1} } u \varphi_t dx dt - \int_{ {\mathbb R}_+^{n+1} } \sum_{|\alpha| = m} (- 1)^m a_\alpha (x, t, u) \partial^\alpha \varphi dx dt \\ & \qquad = \int_{ {\mathbb R}^n } u_0 (x) \varphi dx + \int_{ {\mathbb R}_+^{n+1} } f (|u|) \varphi dx dt\end{aligned}$$ for any $\varphi \in C_0^\infty ({\mathbb R}^{n+1})$. It is obvious that every global weak solution of [\[1.3\]](#1.3){reference-type="eqref" reference="1.3"} is also a global weak solution of [\[1.1\]](#1.1){reference-type="eqref" reference="1.1"}. In our paper, we obtain conditions guaranteeing that any global weak solution of [\[1.1\]](#1.1){reference-type="eqref" reference="1.1"} is trivial, i.e. this solution is equal to zero almost everywhere in ${\mathbb R}_+^{n+1}$. The absence of non-trivial solutions of differential equations and inequalities or, in other words, the blow-up phenomenon was studied by many authors \[1--19\]. In most cases, these studies were limited to the power nonlinearity $f (t) = t^\lambda$ or dealt with second-order differential operators in the space variables. Stationary higher order inequalities with arbitrary nonlinearity were considered in [@Nonlinearity]. However, for higher order evolution inequalities with arbitrary nonlinearity, blow-up conditions remained unknown. Theorems [Theorem 1](#T2.1){reference-type="ref" reference="T2.1"}--[Theorem 3](#T2.3){reference-type="ref" reference="T2.3"} given below are intended to correct this shortcoming. As a corollary, these theorems imply blow-up conditions for solutions of evolution inequalities with power nonlinearity obtained earlier in papers [@EGKPCR; @EGKPMZ; @F; @H; @MPbook] and, in particular, the well-known Fujita--Hayakawa condition (see Example [Example 1](#E2.1){reference-type="ref" reference="E2.1"}). Also note that we impose no ellipticity conditions on the coefficients $a_\alpha$ of the differential operator. Thus, our results can be applied to both parabolic and so-called anti-parabolic inequalities. We need the following notation. For $$\int_1^\infty \frac{ d\zeta }{ f (\zeta) } < \infty \label{1.4}$$ and $$\int_1^\infty \frac{ \zeta^{p / m - 1} d\zeta }{ f^{1 / m} (\zeta) } < \infty, \label{1.5}$$ we put $$F (s) = \left( \int_s^\infty \frac{ d\zeta }{ f (\zeta) } \right)^{1 / m} + \int_s^\infty \frac{ \zeta^{p / m - 1} d\zeta }{ f^{1 / m} (\zeta) }.$$ # Main results **Theorem 1**. *Let $p \ge 1$, conditions [\[1.4\]](#1.4){reference-type="eqref" reference="1.4"} and [\[1.5\]](#1.5){reference-type="eqref" reference="1.5"} be valid and $$\liminf_{s \to +0} s^{1 / n} F (s) < \infty. \label{T2.1.1}$$ Then any global weak solution of [\[1.1\]](#1.1){reference-type="eqref" reference="1.1"} is trivial.* **Theorem 2**. *Let $0 < p < 1$, conditions [\[1.4\]](#1.4){reference-type="eqref" reference="1.4"} and [\[1.5\]](#1.5){reference-type="eqref" reference="1.5"} be valid and $$\liminf_{s \to +0} s^{p / n} F (s) < \infty. \label{T2.2.1}$$ Then any global weak solution of [\[1.1\]](#1.1){reference-type="eqref" reference="1.1"} is trivial.* **Theorem 3**. *Suppose that $$\int_0^\infty \frac{ \zeta^{p / m - 1} d\zeta }{ f^{1 / m} (\zeta) } < \infty \label{T2.3.1}$$ and $$\int_0^\infty \frac{ d\zeta }{ f (\zeta) } < \infty. \label{T2.3.2}$$ Then [\[1.1\]](#1.1){reference-type="eqref" reference="1.1"} has no global weak solutions.* *Remark 1*. According to [@meIzv Lemma 2.3], we have $$\left( \int_s^\infty \frac{ d\zeta }{ f (2 \zeta) } \right)^{1 / m} \le C \int_s^\infty \frac{ \zeta^{1 / m - 1} d\zeta }{ f^{1 / m} (\zeta) }$$ for all $s \in (0, \infty)$, where the constant $C > 0$ depends only on $m$. Thus, in the case of $p = 1$, condition [\[T2.1.1\]](#T2.1.1){reference-type="eqref" reference="T2.1.1"} in Theorem [Theorem 1](#T2.1){reference-type="ref" reference="T2.1"} can be replaced by $$\liminf_{s \to +0} s^{1 / n} \int_s^\infty \frac{ \zeta^{1 / m - 1} d\zeta }{ f^{1 / m} (\zeta) } < \infty.$$ In this case, condition [\[T2.3.2\]](#T2.3.2){reference-type="eqref" reference="T2.3.2"} in Theorem [Theorem 3](#T2.3){reference-type="ref" reference="T2.3"} can obviously be omitted. Theorems [Theorem 1](#T2.1){reference-type="ref" reference="T2.1"}--[Theorem 3](#T2.3){reference-type="ref" reference="T2.3"} are proved in Section [3](#proof){reference-type="ref" reference="proof"}. Now, we give some examples. *Example 1*. Consider the problem $$\left\{ \begin{aligned} & u_t = \Delta^{m/2} u^p + |u|^\lambda \quad \mbox{in } {\mathbb R}_+^{n+1}, \\ & u (x, 0) = u_0 (x) \ge 0, \end{aligned} \right. \label{E2.1.1}$$ where $\lambda$ is a real number and $m$ is an even positive integer. If $p \ge 1$, then in accordance with Theorem [Theorem 1](#T2.1){reference-type="ref" reference="T2.1"}, the inequality $$p < \lambda \le 1 + \frac{m}{n} \label{E2.1.2}$$ implies that any global weak solution of [\[E2.1.1\]](#E2.1.1){reference-type="eqref" reference="E2.1.1"} is trivial. Now assume that $0 < p < 1$. By Theorem [Theorem 2](#T2.2){reference-type="ref" reference="T2.2"}, if $$1 < \lambda \le \left( 1 + \frac{m}{n} \right) p, \label{E2.1.3}$$ then any global weak solution of [\[E2.1.1\]](#E2.1.1){reference-type="eqref" reference="E2.1.1"} is trivial. We note that [\[E2.1.2\]](#E2.1.2){reference-type="eqref" reference="E2.1.2"} and [\[E2.1.3\]](#E2.1.3){reference-type="eqref" reference="E2.1.3"} coincide with similar conditions given in [@EGKPCR; @EGKPMZ; @MPbook]. In so doing, for $p = 1$ and $m = 2$, condition [\[E2.1.2\]](#E2.1.2){reference-type="eqref" reference="E2.1.2"} coincides with the well-known Fujita--Hayakawa condition [@F; @H]. *Example 2*. Let us examine the critical exponent $\lambda = p$ in [\[E2.1.2\]](#E2.1.2){reference-type="eqref" reference="E2.1.2"}. Namely, consider the problem $$\left\{ \begin{aligned} & u_t = \Delta^{m/2} u^p + |u|^p \log^\nu (2 + |u|) \quad \mbox{in } {\mathbb R}_+^{n+1}, \\ & u (x, 0) = u_0 (x) \ge 0, \end{aligned} \right. \label{E2.2.1}$$ where $p \ge 1$, $m$ is an even positive integer, and $\nu$ is a real number. Applying Theorem [Theorem 1](#T2.1){reference-type="ref" reference="T2.1"}, we obtain that, for $$m < \nu \quad \mbox{and} \quad p \le 1 + \frac{m}{n},$$ any global weak solution of [\[E2.2.1\]](#E2.2.1){reference-type="eqref" reference="E2.2.1"} is trivial. *Example 3*. Now we examine the critical exponent $\lambda = 1$ in [\[E2.1.3\]](#E2.1.3){reference-type="eqref" reference="E2.1.3"}. Consider the problem $$\left\{ \begin{aligned} & u_t = \Delta^{m/2} u^p + |u| \log^\nu (2 + |u|) \quad \mbox{in } {\mathbb R}_+^{n+1}, \\ & u (x, 0) = u_0 (x) \ge 0, \end{aligned} \right. \label{E2.3.1}$$ where $0 < p < 1$, $m$ is an even positive integer, and $\nu$ is a real number. By Theorem [Theorem 2](#T2.2){reference-type="ref" reference="T2.2"}, if $$1 < \nu \quad \mbox{and} \quad 1 \le \left( 1 + \frac{m}{n} \right) p,$$ then any global weak solution of [\[E2.3.1\]](#E2.3.1){reference-type="eqref" reference="E2.3.1"} is trivial. # Proof of Theorems [Theorem 1](#T2.1){reference-type="ref" reference="T2.1"}--[Theorem 3](#T2.3){reference-type="ref" reference="T2.3"} {#proof} In this section, we assume that $u$ is a global weak solution of [\[1.1\]](#1.1){reference-type="eqref" reference="1.1"}. We also denote by $C$ and $k$ various positive constants that depend only on $A$, $m$, $n$, and $p$. **Lemma 1**. *Let $R > 0$ be a real number. Then $$\begin{aligned} & \frac{1}{r_2^m - r_1^m} \int_{r_1^m}^{r_2^m} \int_{B_{r_2}} |u| dx dt + \frac{1}{(r_2 - r_1)^m} \int_0^{r_2^m} \int_{B_{r_2} \setminus B_{r_1}} |u|^p dx dt \nonumber \\ & \qquad {} \ge C \int_{Q_{r_1}} f (|u|) dx dt \label{L3.1.1}\end{aligned}$$ for all real numbers $R \le r_1 < r_2 \le 2 R$.* *Proof.* Take a non-decreasing function $\varphi_0 \in C^\infty ({\mathbb R})$ such that $$\left. \varphi_0 \right|_{ (- \infty, 0] } = 0 \quad \mbox{and} \quad \left. \varphi_0 \right|_{ [1, \infty) } = 1.$$ We put $$\varphi (x, t) = \varphi_0 \left( \frac{r_2 - |x|}{r_2 - r_1} \right) \varphi_0 \left( \frac{r_2^m - t}{r_2^m - r_1^m} \right). \label{PL3.1.1}$$ It is easy to see that $$\left| \int_{ {\mathbb R}_+^{n+1} } u \varphi_t dx dt \right| \le \frac{C}{r_2^m - r_1^m} \int_{r_1^m}^{r_2^m} \int_{B_{r_2}} |u| dx dt,$$ $$\left| \int_{ {\mathbb R}_+^{n+1} } \sum_{|\alpha| = m} (- 1)^m a_\alpha (x, t, u) \partial^\alpha \varphi dx dt \right| \le \frac{C}{(r_2 - r_1)^m} \int_0^{r_2^m} \int_{B_{r_2} \setminus B_{r_1}} |u|^p dx dt,$$ and $$\int_{ {\mathbb R}_+^{n+1} } f (|u|) \varphi dx dt \ge \int_0^{r_1^m} \int_{B_{r_1}} f (|u|) dx dt = \int_{Q_{r_1}} f (|u|) dx dt.$$ Thus, taking [\[PL3.1.1\]](#PL3.1.1){reference-type="eqref" reference="PL3.1.1"} as a test function in [\[1.2\]](#1.2){reference-type="eqref" reference="1.2"}, we readily obtain [\[L3.1.1\]](#L3.1.1){reference-type="eqref" reference="L3.1.1"}. ◻ **Lemma 2**. *Let $p \ge 1$, then for all real numbers $R > 0$ and $R \le r_1 < r_2 \le 2 R$ at least one of the following two inequalities is valid: $$I (r_2) - I (r_1) \ge C (r_2 - r_1)^p R^{(m - 1) p} f^p (I^{1 / p} (r_1)), \label{L3.2.1}$$ $$I (r_2) - I (r_1) \ge C (r_2 - r_1)^m f (I^{1 / p} (r_1)), \label{L3.2.2}$$ where $$I (r) = \frac{ 1 }{ \mathop{\rm mes}\nolimits Q_{2 R} } \int_{Q_r} |u|^p dx dt, \quad R < r < 2 R. \label{L3.2.3}$$* *Proof.* At first, assume that $$\frac{1}{r_2^m - r_1^m} \int_{r_1^m}^{r_2^m} \int_{B_{r_2}} |u| dx dt \ge \frac{1}{(r_2 - r_1)^m} \int_0^{r_2^m} \int_{B_{r_2} \setminus B_{r_1}} |u|^p dx dt. \label{PL3.2.1}$$ Then Lemma [Lemma 1](#L3.1){reference-type="ref" reference="L3.1"} implies the estimate $$\frac{1}{r_2^m - r_1^m} \int_{r_1^m}^{r_2^m} \int_{B_{r_2}} |u| dx dt \ge C \int_{Q_{r_1}} f (|u|) dx dt.$$ Since $$\int_{ Q_{r_2} \setminus Q_{r_1} } |u| dx dt \ge \int_{r_1^m}^{r_2^m} \int_{B_{r_2}} |u| dx dt,$$ this yields $$\int_{ Q_{r_2} \setminus Q_{r_1} } |u| dx dt \ge C (r_2^m - r_1^m) \int_{Q_{r_1}} f (|u|) dx dt. \label{PL3.2.5}$$ Evaluating the left-hand side of the last expression by Hölder's inequality $$(\mathop{\rm mes}\nolimits Q_{r_2} \setminus Q_{r_1})^{(p - 1) / p} \left( \int_{ Q_{r_2} \setminus Q_{r_1} } |u|^p dx dt \right)^{1 / p} \ge \int_{ Q_{r_2} \setminus Q_{r_1} } |u| dx dt,$$ we obtain $$\left( \int_{ Q_{r_2} \setminus Q_{r_1} } |u|^p dx dt \right)^{1 / p} \ge \frac{ C (r_2^m - r_1^m) }{ (\mathop{\rm mes}\nolimits Q_{r_2} \setminus Q_{r_1})^{(p - 1) / p} } \int_{Q_{r_1}} f (|u|) dx dt,$$ whence it follows that $$\begin{aligned} \frac{ 1 }{ \mathop{\rm mes}\nolimits Q_{2 R} } \int_{ Q_{r_2} \setminus Q_{r_1} } |u|^p dx dt \ge {} & C (r_2^m - r_1^m)^p \left( \frac{ \mathop{\rm mes}\nolimits Q_{2 R} }{ \mathop{\rm mes}\nolimits Q_{r_2} \setminus Q_{r_1} } \right)^{p - 1} \nonumber \\ & {} \times \left( \frac{1}{\mathop{\rm mes}\nolimits Q_{2 R}} \int_{Q_{r_1}} f (|u|) dx dt \right)^p. \label{PL3.2.2}\end{aligned}$$ Since $f (\zeta^{1/p})$ is a non-decreasing convex function, we have $$\begin{aligned} \frac{1}{\mathop{\rm mes}\nolimits Q_{2 R}} \int_{Q_{r_1}} f (|u|) dx dt & \ge \frac{ 1 }{ 2^{n + m} \mathop{\rm mes}\nolimits Q_{r_1} } \int_{Q_{r_1}} f (|u|) dx dt \nonumber \\ & \ge \frac{ 1 }{ 2^{n + m} } f \left( \left( \frac{ 1 }{ \mathop{\rm mes}\nolimits Q_{r_1} } \int_{Q_{r_1}} |u|^p dx dt \right)^{1 / p} \right) \nonumber \\ & \ge \frac{ 1 }{ 2^{n + m} } f \left( \left( \frac{ 1 }{ \mathop{\rm mes}\nolimits Q_{2 R} } \int_{Q_{r_1}} |u|^p dx dt \right)^{1 / p} \right). \label{PL3.2.3}\end{aligned}$$ In view of [\[PL3.2.2\]](#PL3.2.2){reference-type="eqref" reference="PL3.2.2"}, this implies the estimate $$\begin{aligned} \frac{ 1 }{ \mathop{\rm mes}\nolimits Q_{2 R} } \int_{ Q_{r_2} \setminus Q_{r_1} } |u|^p dx dt \ge {} & C (r_2^m - r_1^m)^p \left( \frac{ \mathop{\rm mes}\nolimits Q_{2 R} }{ \mathop{\rm mes}\nolimits Q_{r_2} \setminus Q_{r_1} } \right)^{p - 1} \\ & {} \times f^p \left( \left( \frac{ 1 }{ \mathop{\rm mes}\nolimits Q_{2 R} } \int_{Q_{r_1}} |u|^p dx dt \right)^{1 / p} \right),\end{aligned}$$ combining which with the evident inequalities $$\left( \frac{ \mathop{\rm mes}\nolimits Q_{2 R} }{ \mathop{\rm mes}\nolimits Q_{r_2} \setminus Q_{r_1} } \right)^{p - 1} \ge 1$$ and $$r_2^m - r_1^m \ge (r_2 - r_1) R^{m - 1}, \label{PL3.2.4}$$ we arrive at [\[L3.2.1\]](#L3.2.1){reference-type="eqref" reference="L3.2.1"}. Now, let [\[PL3.2.1\]](#PL3.2.1){reference-type="eqref" reference="PL3.2.1"} not hold. In this case, Lemma [Lemma 1](#L3.1){reference-type="ref" reference="L3.1"} implies that $$\frac{1}{(r_2 - r_1)^m} \int_0^{r_2^m} \int_{B_{r_2} \setminus B_{r_1}} |u|^p dx dt \ge C \int_{Q_{r_1}} f (|u|) dx dt, \label{PL3.2.6}$$ whence in accordance with [\[PL3.2.3\]](#PL3.2.3){reference-type="eqref" reference="PL3.2.3"} and the inequality $$\int_{Q_{r_2} \setminus Q_{r_1}} |u|^p dx dt \ge \int_0^{r_2^m} \int_{B_{r_2} \setminus B_{r_1}} |u|^p dx dt \label{PL3.2.7}$$ we obtain $$\frac{ 1 }{ \mathop{\rm mes}\nolimits Q_{2 R} } \int_{Q_{r_2} \setminus Q_{r_1}} |u|^p dx dt \ge C (r_2 - r_1)^m f \left( \left( \frac{ 1 }{ \mathop{\rm mes}\nolimits Q_{2 R} } \int_{Q_{r_1}} |u|^p dx dt \right)^{1 / p} \right).$$ To complete the proof, it remains to note that the last expression is equivalent to [\[L3.2.2\]](#L3.2.2){reference-type="eqref" reference="L3.2.2"}. ◻ **Lemma 3**. *Let $0 < p < 1$, then for all real numbers $R > 0$ and $R \le r_1 < r_2 \le 2 R$ at least one of the following two inequalities is valid: $$J (r_2) - J (r_1) \ge C (r_2 - r_1) R^{m - 1} f (J (r_1)), \label{L3.3.1}$$ $$J (r_2) - J (r_1) \ge C (r_2 - r_1)^{m / p} f^{1 / p} (J (r_1)), \label{L3.3.2}$$ where $$J (r) = \frac{ 1 }{ \mathop{\rm mes}\nolimits Q_{2 R} } \int_{Q_r} |u| dx dt, \quad R < r < 2 R. \label{L3.3.3}$$* *Proof.* At first, assume that [\[PL3.2.1\]](#PL3.2.1){reference-type="eqref" reference="PL3.2.1"} is valid. In this case, repeating the arguments given in the proof of Lemma [Lemma 2](#L3.2){reference-type="ref" reference="L3.2"}, we conclude that [\[PL3.2.5\]](#PL3.2.5){reference-type="eqref" reference="PL3.2.5"} holds. At the same time, since $f$ is a non-decreasing convex function, we have $$\begin{aligned} \frac{1}{\mathop{\rm mes}\nolimits Q_{2 R}} \int_{Q_{r_1}} f (|u|) dx dt & \ge \frac{ 1 }{ 2^{n + m} \mathop{\rm mes}\nolimits Q_{r_1} } \int_{Q_{r_1}} f (|u|) dx dt \nonumber \\ & \ge \frac{ 1 }{ 2^{n + m} } f \left( \frac{ 1 }{ \mathop{\rm mes}\nolimits Q_{r_1} } \int_{Q_{r_1}} |u| dx dt \right) \nonumber \\ & \ge \frac{ 1 }{ 2^{n + m} } f \left( \frac{ 1 }{ \mathop{\rm mes}\nolimits Q_{2 R} } \int_{Q_{r_1}} |u| dx dt \right). \label{PL3.3.1}\end{aligned}$$ Thus, [\[PL3.2.5\]](#PL3.2.5){reference-type="eqref" reference="PL3.2.5"} implies the estimate $$\frac{1}{\mathop{\rm mes}\nolimits Q_{2 R}} \int_{ Q_{r_2} \setminus Q_{r_1} } |u| dx dt \ge C (r_2^m - r_1^m) f \left( \frac{ 1 }{ \mathop{\rm mes}\nolimits Q_{2 R} } \int_{Q_{r_1}} |u| dx dt \right),$$ whence in accordance with [\[PL3.2.4\]](#PL3.2.4){reference-type="eqref" reference="PL3.2.4"} inequality [\[L3.3.1\]](#L3.3.1){reference-type="eqref" reference="L3.3.1"} follows at once. In its turn, if [\[PL3.2.1\]](#PL3.2.1){reference-type="eqref" reference="PL3.2.1"} is not valid, then Lemma [Lemma 1](#L3.1){reference-type="ref" reference="L3.1"} implies [\[PL3.2.6\]](#PL3.2.6){reference-type="eqref" reference="PL3.2.6"}, whence in accordance with [\[PL3.2.7\]](#PL3.2.7){reference-type="eqref" reference="PL3.2.7"} and [\[PL3.3.1\]](#PL3.3.1){reference-type="eqref" reference="PL3.3.1"} we obtain $$\frac{1}{\mathop{\rm mes}\nolimits Q_{2 R}} \int_{Q_{r_2} \setminus Q_{r_1}} |u|^p dx dt \ge C (r_2 - r_1)^m f \left( \frac{ 1 }{ \mathop{\rm mes}\nolimits Q_{2 R} } \int_{Q_{r_1}} |u| dx dt \right). \label{PL3.3.2}$$ By Hölder's inequality, $$\int_{ Q_{r_2} \setminus Q_{r_1} } |u|^p dx dt \le (\mathop{\rm mes}\nolimits Q_{r_2} \setminus Q_{r_1})^{1 - p} \left( \int_{ Q_{r_2} \setminus Q_{r_1} } |u| dx dt \right)^p;$$ therefore, [\[PL3.3.2\]](#PL3.3.2){reference-type="eqref" reference="PL3.3.2"} allows us to assert that $$\begin{aligned} \frac{1}{\mathop{\rm mes}\nolimits Q_{2 R}} \int_{Q_{r_2} \setminus Q_{r_1}} |u| dx dt \ge {} & C (r_2 - r_1)^{m / p} \left( \frac{ \mathop{\rm mes}\nolimits Q_{2 R} }{ \mathop{\rm mes}\nolimits Q_{r_2} \setminus Q_{r_1} } \right)^{1 / p - 1} \\ & {} \times f^{1 / p} \left( \frac{ 1 }{ \mathop{\rm mes}\nolimits Q_{2 R} } \int_{Q_{r_1}} |u| dx dt \right).\end{aligned}$$ Since $$\left( \frac{ \mathop{\rm mes}\nolimits Q_{2 R} }{ \mathop{\rm mes}\nolimits Q_{r_2} \setminus Q_{r_1} } \right)^{1 / p - 1} \ge 1,$$ this readily implies [\[L3.3.2\]](#L3.3.2){reference-type="eqref" reference="L3.3.2"}. ◻ **Lemma 4**. *Let $p \ge 1$ and, moreover, [\[1.4\]](#1.4){reference-type="eqref" reference="1.4"} and [\[1.5\]](#1.5){reference-type="eqref" reference="1.5"} be valid. Then $$F \left( \left( \frac{ k }{ R^{n + m} } \int_{Q_R} |u|^p dx dt \right)^{1 / p} \right) \ge C R, \label{L3.4.1}$$ for any real number $R > 0$ such that $\mathop{\rm mes}\nolimits \{ (x, t) \in Q_R : u (x, t) \ne 0 \} > 0.$* *Proof.* Take the minimal positive integer $l$ such that $2^l I (R) \ge I (2 R)$, where the function $I$ is defined by [\[L3.2.3\]](#L3.2.3){reference-type="eqref" reference="L3.2.3"}. Let us construct a finite sequence of real numbers $r_i$, $i = 0, \ldots, l$, putting $r_0 = R$, $$r_i = \sup \{ r \in (r_{i - 1}, 2 R) : I (r) \le 2 I (r_{i - 1}) \}, \quad 0 < i \le l - 1,$$ and $r_l = 2 R$. It can easily be seen that $$I (r_i) = 2 I (r_{i-1}) \quad \mbox{for all } 0 < i \le l - 1 \quad \mbox{and} \quad I (r_l) \le 2 I (r_{l-1}). \label{PL3.4.6}$$ According to Lemma [Lemma 2](#L3.2){reference-type="ref" reference="L3.2"}, for any integer $0 < i \le l$ at least one of the following two inequalities is valid: $$I (r_i) - I (r_{i-1}) \ge C (r_i - r_{i-1})^p R^{(m - 1) p} f^p (I^{1 / p} (r_{i-1})), \label{PL3.4.1}$$ $$I (r_i) - I (r_{i-1}) \ge C (r_i - r_{i-1})^m f (I^{1 / p} (r_{i-1})). \label{PL3.4.2}$$ We denote by $\Xi_1$ the set of all integers $0 < i \le l$ for which [\[PL3.4.1\]](#PL3.4.1){reference-type="eqref" reference="PL3.4.1"} holds. Also let $\Xi_2$ be the set of all other integers $0 < i \le l$. If $i \in \Xi_1$, then [\[PL3.4.6\]](#PL3.4.6){reference-type="eqref" reference="PL3.4.6"} and [\[PL3.4.1\]](#PL3.4.1){reference-type="eqref" reference="PL3.4.1"} imply that $$\frac{ I^{1 / p} (r_{i-1}) }{ f (I^{1 / p} (r_{i-1})) } \ge C (r_i - r_{i-1}) R^{m-1},$$ whence in accordance with the evident inequality $$\int_{ I (r_{i-1}) }^{ 2 I (r_{i-1}) } \frac{ \zeta^{1 / p - 1} d\zeta }{ f (2^{- 1 / p} \zeta^{1 / p}) } \ge \frac{ 2^{1 / p - 1} I^{1 / p} (r_{i-1}) }{ f (I^{1 / p} (r_{i-1})) }$$ we obtain $$\int_{ I (r_{i-1}) }^{ 2 I (r_{i-1}) } \frac{ \zeta^{1 / p - 1} d\zeta }{ f (2^{- 1 / p} \zeta^{1 / p}) } \ge C (r_i - r_{i-1}) R^{m-1}, \label{PL3.4.3}$$ In its turn, if $i \in \Xi_2$, then [\[PL3.4.6\]](#PL3.4.6){reference-type="eqref" reference="PL3.4.6"} and [\[PL3.4.2\]](#PL3.4.2){reference-type="eqref" reference="PL3.4.2"} allow us to assert that $$\frac{ I^{1 / m} (r_{i-1}) }{ f^{1 / m} (I^{1 / p} (r_{i-1})) } \ge C (r_i - r_{i-1}).$$ Since $$\int_{ I (r_{i-1}) }^{ 2 I (r_{i-1}) } \frac{ \zeta^{1 / m - 1} d\zeta }{ f^{1 / m} (2^{- 1 / p} \zeta^{1 / p}) } \ge \frac{ 2^{1 / m - 1} I^{1 / m} (r_{i-1}) }{ f^{1 / m} (I^{1 / p} (r_{i-1})) },$$ this readily implies the estimate $$\int_{ I (r_{i-1}) }^{ 2 I (r_{i-1}) } \frac{ \zeta^{1 / m - 1} d\zeta }{ f^{1 / m} (2^{- 1 / p} \zeta^{1 / p}) } \ge C (r_i - r_{i-1}). \label{PL3.4.4}$$ At first, we assume that $$\sum_{i \in \Xi_1} (r_i - r_{i-1}) \ge \frac{R}{2}. \label{PL3.4.5}$$ Summing [\[PL3.4.3\]](#PL3.4.3){reference-type="eqref" reference="PL3.4.3"} over all $i \in \Xi_1$, we obtain $$\int_{ I (R) }^\infty \frac{ \zeta^{1 / p - 1} d\zeta }{ f (2^{- 1 / p} \zeta^{1 / p}) } \ge C R^m.$$ By the change of variable $\xi = 2^{- 1 / p} \zeta^{1/p}$, the last expression is reduced to the form $$\int_{ (I (R) / 2)^{1 / p} }^\infty \frac{ d\xi }{ f (\xi) } \ge C R^m.$$ This, in its turn, implies [\[L3.4.1\]](#L3.4.1){reference-type="eqref" reference="L3.4.1"}. Now let [\[PL3.4.5\]](#PL3.4.5){reference-type="eqref" reference="PL3.4.5"} not hold. Then it is obvious that $$\sum_{i \in \Xi_2} (r_i - r_{i-1}) \ge \frac{R}{2}; \label{PL3.4.7}$$ therefore, summing [\[PL3.4.4\]](#PL3.4.4){reference-type="eqref" reference="PL3.4.4"} over all $i \in \Xi_2$, we have $$\int_{ I (R) }^\infty \frac{ \zeta^{1 / m - 1} d\zeta }{ f^{1 / m} (2^{- 1 / p} \zeta^{1 / p}) } \ge C R.$$ By the change of variable $\xi = 2^{- 1 / p} \zeta^{1/p}$, this can be transformed to the inequality $$\int_{ (I (R) / 2)^{1 / p} }^\infty \frac{ \xi^{p / m - 1} d\xi }{ f^{1 / m} (\xi) } \ge C R,$$ whence [\[L3.4.1\]](#L3.4.1){reference-type="eqref" reference="L3.4.1"} follows again. The proof is completed. ◻ **Lemma 5**. *Let $0 < p < 1$ and, moreover, [\[1.4\]](#1.4){reference-type="eqref" reference="1.4"} and [\[1.5\]](#1.5){reference-type="eqref" reference="1.5"} be valid. Then $$F \left( \frac{ k }{ R^{n + m} } \int_{Q_R} |u| dx dt \right) \ge C R \label{L3.5.1}$$ for any real number $R > 0$ such that $\mathop{\rm mes}\nolimits \{ (x, t) \in Q_R : u (x, t) \ne 0 \} > 0.$* *Proof.* Take the minimal positive integer $l$ such that $2^l J (R) \ge J (2 R)$, where the function $J$ is defined by [\[L3.3.3\]](#L3.3.3){reference-type="eqref" reference="L3.3.3"}. Let us consider the sequence of real numbers $r_i$, $i = 0, \ldots, l$, constructed in the proof of Lemma [Lemma 4](#L3.4){reference-type="ref" reference="L3.4"} with $I$ replaced by $J$. We obviously have $$J (r_i) = 2 J (r_{i-1}) \quad \mbox{for all } 0 < i \le l - 1 \quad \mbox{and} \quad J (r_l) \le 2 J (r_{l-1}). \label{PL3.5.1}$$ In view of Lemma [Lemma 3](#L3.3){reference-type="ref" reference="L3.3"}, for any integer $0 < i \le l$ at least one of the following two inequalities is valid: $$J (r_i) - J (r_{i-1}) \ge C (r_i - r_{i-1}) R^{m - 1} f (J (r_{i-1})), \label{PL3.5.2}$$ $$J (r_i) - J (r_{i-1}) \ge C (r_i - r_{i-1})^{m / p} f^{1 / p} (J (r_{i-1})). \label{PL3.5.3}$$ Let us denote by $\Xi_1$ the set of integers $0 < i \le l$ such that [\[PL3.5.2\]](#PL3.5.2){reference-type="eqref" reference="PL3.5.2"} holds and let $\Xi_2$ be the set of all other positive integers $0 < i \le l$. For any $i \in \Xi_1$, it follows from [\[PL3.5.1\]](#PL3.5.1){reference-type="eqref" reference="PL3.5.1"} and [\[PL3.5.2\]](#PL3.5.2){reference-type="eqref" reference="PL3.5.2"} that $$\frac{ J (r_{i-1}) }{ f (J (r_{i-1})) } \ge C (r_i - r_{i-1}) R^{m - 1}.$$ Combining this with the evident inequality $$\int_{ J (r_{i-1}) }^{ 2 J (r_{i-1}) } \frac{ d\zeta }{ f (\zeta / 2) } \ge \frac{ J (r_{i-1}) }{ f (J (r_{i-1})) },$$ we obtain $$\int_{ J (r_{i-1}) }^{ 2 J (r_{i-1}) } \frac{ d\zeta }{ f (\zeta / 2) } \ge C (r_i - r_{i-1}) R^{m - 1}. \label{PL3.5.4}$$ In its turn, for any $i \in \Xi_2$ in accordance with [\[PL3.5.1\]](#PL3.5.1){reference-type="eqref" reference="PL3.5.1"} and [\[PL3.5.3\]](#PL3.5.3){reference-type="eqref" reference="PL3.5.3"} we have $$\frac{ J^{p / m} (r_{i-1}) }{ f^{1 / m} (J (r_{i-1})) } \ge C (r_i - r_{i-1}).$$ In view of the estimate $$\int_{ J (r_{i-1}) }^{ 2 J (r_{i-1}) } \frac{ \zeta^{p / m - 1} d\zeta }{ f^{1 / m} (\zeta / 2) } \ge \frac{ 2^{p / m - 1} J^{p / m} (r_{i-1}) }{ f^{1 / m} (J (r_{i-1})) },$$ this implies that $$\int_{ J (r_{i-1}) }^{ 2 J (r_{i-1}) } \frac{ \zeta^{p / m - 1} d\zeta }{ f^{1 / m} (\zeta / 2) } \ge C (r_i - r_{i-1}). \label{PL3.5.5}$$ At first, let [\[PL3.4.5\]](#PL3.4.5){reference-type="eqref" reference="PL3.4.5"} be fulfilled. Summing [\[PL3.5.4\]](#PL3.5.4){reference-type="eqref" reference="PL3.5.4"} over all $i \in \Xi_1$, we obtain $$\int_{ J (R) }^\infty \frac{ d\zeta }{ f (\zeta / 2) } \ge C R^m.$$ After the change of variable $\xi = \zeta / 2$, the last expression takes the form $$\int_{ J (R) / 2 }^\infty \frac{ d\xi }{ f (\xi) } \ge C R^m,$$ whence [\[L3.5.1\]](#L3.5.1){reference-type="eqref" reference="L3.5.1"} immediately follows. Now, assume that [\[PL3.4.5\]](#PL3.4.5){reference-type="eqref" reference="PL3.4.5"} is not valid. In this case, [\[PL3.4.7\]](#PL3.4.7){reference-type="eqref" reference="PL3.4.7"} holds and, summing [\[PL3.5.5\]](#PL3.5.5){reference-type="eqref" reference="PL3.5.5"} over all $i \in \Xi_2$, we arrive at the inequality $$\int_{ J (R) }^\infty \frac{ \zeta^{p / m - 1} d\zeta }{ f^{1 / m} (\zeta / 2) } \ge C R$$ which, by the change of variable $\xi = \zeta / 2$, can be transformed to $$\int_{ J (R) / 2 }^\infty \frac{ \xi^{p / m - 1} d\xi }{ f^{1 / m} (\xi) } \ge C R.$$ This again implies [\[L3.5.1\]](#L3.5.1){reference-type="eqref" reference="L3.5.1"}. ◻ *Proof of Theorem $\ref{T2.1}$.* Assume the converse. Let problem [\[1.1\]](#1.1){reference-type="eqref" reference="1.1"} have a non-trivial global weak solution. Lemma [Lemma 1](#L3.1){reference-type="ref" reference="L3.1"} with $r_1 = R / 2$ and $r_2 = R$, implies the estimate $$\frac{ 1 }{ R^m } \int_{ Q_R \setminus Q_{R / 2} } |u| dx dt + \frac{ 1 }{ R^m } \int_{ Q_R \setminus Q_{R / 2} } |u|^p dx dt \ge C \int_{Q_{R / 2}} f (|u|) dx dt \label{PT2.1.5}$$ for all real numbers $R > 0$, whence in accordance with Hölder's inequality $$(\mathop{\rm mes}\nolimits Q_R \setminus Q_{R / 2})^{(p - 1) / p} \left( \int_{ Q_R \setminus Q_{R / 2} } |u|^p dx dt \right)^{1 / p} \ge \int_{ Q_R \setminus Q_{R / 2} } |u| dx dt$$ it follows that $$\left( \frac{ 1 }{ R^{n + m} } \int_{ Q_R \setminus Q_{R / 2} } |u|^p dx dt \right)^{1 / p} + \frac{ 1 }{ R^{n + m} } \int_{ Q_R \setminus Q_{R / 2} } |u|^p dx dt \ge \frac{C}{R^n} \int_{Q_{R / 2}} f (|u|) dx dt \label{PT2.1.1}$$ for all real numbers $R > 0$. At the same time, by Lemma [Lemma 4](#L3.4){reference-type="ref" reference="L3.4"}, we have $$\left( \frac{ k }{ R^{n + m} } \int_{Q_R} |u|^p dx dt \right)^{1 / p} \le F^{-1} (C R) \label{PT2.1.2}$$ for all sufficiently large real numbers $R > 0$, where $F^{-1}$ is the inverse function to $F$. Let us note that $F (s) \to \infty$ as $s \to +0$; otherwise [\[L3.4.1\]](#L3.4.1){reference-type="eqref" reference="L3.4.1"} cannot be satisfied for all real numbers $R > 0$ in a neighborhood of infinity. Hence, $f (0) = 0$ and, moreover, $f$ and $F$ are one-to-one continuous maps of the interval $(0, \infty)$ into itself with $F^{-1} (s) \to 0$ as $s \to \infty$. By [\[PT2.1.2\]](#PT2.1.2){reference-type="eqref" reference="PT2.1.2"}, we have $$\lim_{R \to \infty} \frac{ 1 }{ R^{n + m} } \int_{Q_R} |u|^p dx dt = 0;$$ therefore, [\[PT2.1.1\]](#PT2.1.1){reference-type="eqref" reference="PT2.1.1"} implies the estimate $$\left( \frac{ 1 }{ R^{n + m} } \int_{ Q_R \setminus Q_{R / 2} } |u|^p dx dt \right)^{1 / p} \ge \frac{C}{R^n} \int_{Q_{R / 2}} f (|u|) dx dt \label{PT2.1.3}$$ for all sufficiently large $R > 0$. It can be seen that condition [\[T2.1.1\]](#T2.1.1){reference-type="eqref" reference="T2.1.1"} is equivalent to $$\liminf_{R \to \infty} R^n F^{-1} (R) < \infty.$$ Thus, taking into account [\[PT2.1.2\]](#PT2.1.2){reference-type="eqref" reference="PT2.1.2"} and [\[PT2.1.3\]](#PT2.1.3){reference-type="eqref" reference="PT2.1.3"}, we obtain $$\int_{ {\mathbb R}_+^{n+1} } f (|u|) dx dt < \infty. \label{PT2.1.4}$$ Since $f (\zeta^{1/p})$ is a convex function, we have $$\begin{aligned} & \frac{ 1 }{ \mathop{\rm mes}\nolimits(Q_R \setminus Q_{R / 2}) } \int_{ Q_r \setminus Q_{R / 2} } f (|u|) dx dt \\ & \qquad {} \ge f \left( \left( \frac{ 1 }{ \mathop{\rm mes}\nolimits(Q_r \setminus Q_{R / 2}) } \int_{ Q_R \setminus Q_{R / 2} } |u|^p dx dt \right)^{1 / p} \right)\end{aligned}$$ for all real numbers $R > 0$. This, in its turn, yields $$\begin{aligned} & f^{-1} \left( \frac{ 1 }{ \mathop{\rm mes}\nolimits(Q_R \setminus Q_{R / 2}) } \int_{ Q_R \setminus Q_{R / 2} } f (|u|) dx dt \right) \\ & \qquad {} \ge \left( \frac{ 1 }{ \mathop{\rm mes}\nolimits(Q_R \setminus Q_{R / 2}) } \int_{ Q_R \setminus Q_{R / 2} } |u|^p dx dt \right)^{1 / p}\end{aligned}$$ for all real numbers $R > 0$, where $f^{-1}$ is the inverse function to $f$. Combining the last inequality with [\[PT2.1.3\]](#PT2.1.3){reference-type="eqref" reference="PT2.1.3"}, we obtain $$f^{-1} \left( \frac{ 1 }{ \mathop{\rm mes}\nolimits(Q_R \setminus Q_{R / 2}) } \int_{ Q_R \setminus Q_{R / 2} } f (|u|) dx dt \right) \ge \frac{ C }{ R^n } \int_{ Q_{R / 2} } f (|u|) dx dt$$ for all sufficiently large $R > 0$, whence it follows that $$\frac{ h (R) }{ f^{n / (n + m)} (h (R)) } \left( \int_{ Q_R \setminus Q_{R / 2} } f (|u|) dx dt \right)^{n / (n + m)} \ge C \int_{ Q_{R / 2} } f (|u|) dx dt \label{PT2.1.6}$$ for all sufficiently large $R > 0$, where $$h (R) = f^{-1} \left( \frac{ 1 }{ \mathop{\rm mes}\nolimits(Q_R \setminus Q_{R / 2}) } \int_{ Q_R \setminus Q_{R / 2} } f (|u|) dx dt \right). \label{PT2.1.9}$$ Let us show that $$\liminf_{R \to \infty} \frac{ h (R) }{ f^{n / (n + m)} (h (R)) } < \infty. \label{PT2.1.7}$$ We have $$F (s) \ge \left( \int_s^{2 s} \frac{ d\zeta }{ f (\zeta) } \right)^{1 / m} \ge \left( \frac{ s }{ f (2 s) } \right)^{1 / m}$$ for all real numbers $s > 0$. By [\[T2.1.1\]](#T2.1.1){reference-type="eqref" reference="T2.1.1"}, this yields $$\liminf_{s \to +0} \frac{ s^{1 / n + 1 / m} }{ f^{1 / m} (s) } < \infty$$ or, in other words, $$\liminf_{s \to +0} \frac{ s }{ f^{n / (n + m)} (s) } < \infty. \label{PT2.1.8}$$ We note that $h$ is a continuous non-negative function. This function cannot identically vanish in any neighborhood of infinity; otherwise, in view of [\[PT2.1.3\]](#PT2.1.3){reference-type="eqref" reference="PT2.1.3"}, we would have $u = 0$ almost everywhere in ${\mathbb R}_+^{n+1}$ contrary to our assumption. In so doing, according to [\[PT2.1.4\]](#PT2.1.4){reference-type="eqref" reference="PT2.1.4"}, one can assert that $h (R) \to 0$ as $R \to \infty$. Thus, [\[PT2.1.8\]](#PT2.1.8){reference-type="eqref" reference="PT2.1.8"} implies [\[PT2.1.7\]](#PT2.1.7){reference-type="eqref" reference="PT2.1.7"}; therefore, [\[PT2.1.6\]](#PT2.1.6){reference-type="eqref" reference="PT2.1.6"} leads to the relation $$\limsup_{R \to \infty} \left( \int_{ Q_R \setminus Q_{R / 2} } f (|u|) dx dt \right)^{n / (n + m)} \ge C \int_{ {\mathbb R}_+^{n+1} } f (|u|) dx dt. \label{PT2.1.10}$$ In accordance with [\[PT2.1.4\]](#PT2.1.4){reference-type="eqref" reference="PT2.1.4"} the left-hand side of the last expression is equal to zero. This immediately implies that $u = 0$ almost everywhere in ${\mathbb R}_+^{n+1}$. The resulting contradiction completes the proof. ◻ *Proof of Theorem $\ref{T2.2}$.* Arguing by contradiction, we assume that [\[1.1\]](#1.1){reference-type="eqref" reference="1.1"} has a non-trivial global weak solution. By Lemma [Lemma 1](#L3.1){reference-type="ref" reference="L3.1"}, this solution satisfies [\[PT2.1.5\]](#PT2.1.5){reference-type="eqref" reference="PT2.1.5"} for all real numbers $R > 0$. Combining [\[PT2.1.5\]](#PT2.1.5){reference-type="eqref" reference="PT2.1.5"} with the estimate $$(\mathop{\rm mes}\nolimits Q_R \setminus Q_{R / 2})^{1 - p} \left( \int_{ Q_R \setminus Q_{R / 2} } |u| dx dt \right)^p \ge \int_{ Q_R \setminus Q_{R / 2} } |u|^p dx dt$$ which follows from Hölder's inequality, we obtain $$\frac{ 1 }{ R^{n + m} } \int_{ Q_R \setminus Q_{R / 2} } |u| dx dt + \left( \frac{ 1 }{ R^{n + m} } \int_{ Q_R \setminus Q_{R / 2} } |u| dx dt \right)^p \ge \frac{C}{R^n} \int_{Q_{R / 2}} f (|u|) dx dt \label{PT2.2.1}$$ for all real numbers $R > 0$. In so doing, from Lemma [Lemma 5](#L3.5){reference-type="ref" reference="L3.5"}, it follows that $$\frac{ k }{ R^{n + m} } \int_{Q_R} |u| dx dt \le F^{-1} (C R) \label{PT2.2.2}$$ for all sufficiently large real numbers $R > 0$, where $F^{-1}$ is the inverse function to $F$. Since [\[L3.5.1\]](#L3.5.1){reference-type="eqref" reference="L3.5.1"} is valid all real numbers $R > 0$ in a neighborhood of infinity, one can assert that $F (s) \to \infty$ as $s \to +0$. Consequently, $f (0) = 0$ and, moreover, $F$ and $f$ are one-to-one continuous maps of the interval $(0, \infty)$ into itself with $F^{-1} (s) \to 0$ as $s \to \infty$. In view of [\[PT2.2.2\]](#PT2.2.2){reference-type="eqref" reference="PT2.2.2"}, we have $$\lim_{R \to \infty} \frac{ 1 }{ R^{n + m} } \int_{Q_R} |u| dx dt = 0;$$ therefore, [\[PT2.2.1\]](#PT2.2.1){reference-type="eqref" reference="PT2.2.1"} implies the inequality $$\left( \frac{ 1 }{ R^{n + m} } \int_{ Q_R \setminus Q_{R / 2} } |u| dx dt \right)^p \ge \frac{C}{R^n} \int_{Q_{R / 2}} f (|u|) dx dt \label{PT2.2.3}$$ for all sufficiently large $R > 0$. It does not present any particular problem to verify that [\[T2.2.1\]](#T2.2.1){reference-type="eqref" reference="T2.2.1"} is equivalent to $$\liminf_{R \to \infty} R^{n / p} F^{-1} (R) < \infty.$$ Thus, taking into account [\[PT2.2.2\]](#PT2.2.2){reference-type="eqref" reference="PT2.2.2"} and [\[PT2.2.3\]](#PT2.2.3){reference-type="eqref" reference="PT2.2.3"}, we arrive at [\[PT2.1.4\]](#PT2.1.4){reference-type="eqref" reference="PT2.1.4"}. From the convexity condition of the function $f$, it follows that $$\frac{ 1 }{ \mathop{\rm mes}\nolimits(Q_R \setminus Q_{R / 2}) } \int_{ Q_r \setminus Q_{R / 2} } f (|u|) dx dt \ge f \left( \frac{ 1 }{ \mathop{\rm mes}\nolimits(Q_r \setminus Q_{R / 2}) } \int_{ Q_R \setminus Q_{R / 2} } |u| dx dt \right)$$ for all real numbers $R > 0$. Hence, $$f^{-1} \left( \frac{ 1 }{ \mathop{\rm mes}\nolimits(Q_R \setminus Q_{R / 2}) } \int_{ Q_R \setminus Q_{R / 2} } f (|u|) dx dt \right) \ge \frac{ 1 }{ \mathop{\rm mes}\nolimits(Q_R \setminus Q_{R / 2}) } \int_{ Q_R \setminus Q_{R / 2} } |u| dx dt$$ for all real numbers $R > 0$, where $f^{-1}$ is the inverse function to $f$. Combining the last estimate with [\[PT2.2.3\]](#PT2.2.3){reference-type="eqref" reference="PT2.2.3"}, we have $$f^{-1} \left( \frac{ 1 }{ \mathop{\rm mes}\nolimits(Q_R \setminus Q_{R / 2}) } \int_{ Q_R \setminus Q_{R / 2} } f (|u|) dx dt \right) \ge \frac{ C }{ R^{n / p} } \left( \int_{ Q_{R / 2} } f (|u|) dx dt \right)^{1 / p}$$ for all sufficiently large $R > 0$. In its turn, this implies the inequality $$\frac{ h^p (R) }{ f^{n / (n + m)} (h (R)) } \left( \int_{ Q_R \setminus Q_{R / 2} } f (|u|) dx dt \right)^{n / (n + m)} \ge C \int_{ Q_{R / 2} } f (|u|) dx dt \label{PT2.2.4}$$ for all sufficiently large $R > 0$, where the function $h$ is defined by [\[PT2.1.9\]](#PT2.1.9){reference-type="eqref" reference="PT2.1.9"}. Let us establish the validity of the relation $$\liminf_{R \to \infty} \frac{ h^p (R) }{ f^{n / (n + m)} (h (R)) } < \infty. \label{PT2.2.5}$$ Indeed, it can be seen that $$F (s) \ge \int_s^{2 s} \frac{ \zeta^{p / m - 1} d\zeta }{ f^{1 / m} (\zeta) } \ge \frac{ 2^{p / m - 1} s^{p / m} }{ f^{1 / m} (2 s) }$$ for all real numbers $s > 0$, whence in accordance with [\[T2.2.1\]](#T2.2.1){reference-type="eqref" reference="T2.2.1"} we obtain $$\liminf_{s \to +0} \frac{ s^{p / n + p / m} }{ f^{1 / m} (s) } < \infty$$ or, in other words, $$%\begin{equation} \liminf_{s \to +0} \frac{ s^p }{ f^{n / (n + m)} (s) } < \infty. % \label{PT2.2.6} %\end{equation}$$ The last inequality immediately leads us to [\[PT2.2.5\]](#PT2.2.5){reference-type="eqref" reference="PT2.2.5"}, because $h$ is a non-negative continuous function such that $h (R) \to 0$ as $R \to \infty$ and, in addition, $h$ does not vanish identically in any neighborhood of infinity. Combining [\[PT2.2.4\]](#PT2.2.4){reference-type="eqref" reference="PT2.2.4"} and [\[PT2.2.5\]](#PT2.2.5){reference-type="eqref" reference="PT2.2.5"}, we obviously obtain [\[PT2.1.10\]](#PT2.1.10){reference-type="eqref" reference="PT2.1.10"}. As in the proof of Theorem [Theorem 1](#T2.1){reference-type="ref" reference="T2.1"}, this contradicts [\[PT2.1.4\]](#PT2.1.4){reference-type="eqref" reference="PT2.1.4"} and our assumption that $u$ is not identically zero in ${\mathbb R}_+^n$. The proof is completed. ◻ *Proof of Theorem $\ref{T2.3}$.* We consider the case of $p \ge 1$. For $0 < p < 1$, the proof is exactly the same with Lemma [Lemma 4](#L3.4){reference-type="ref" reference="L3.4"} replaced by Lemma [Lemma 5](#L3.5){reference-type="ref" reference="L3.5"}. Since $f$ is a non-decreasing convex function, it follows from [\[T2.3.2\]](#T2.3.2){reference-type="eqref" reference="T2.3.2"} that $f (0) > 0$. In this case one can assert that $\mathop{\rm mes}\nolimits \{ (x, t) \in Q_R : u (x, t) \ne 0 \} > 0$ for any real number $R > 0$. Thus, according to Lemma [Lemma 4](#L3.4){reference-type="ref" reference="L3.4"}, for any real number $R > 0$ inequality [\[L3.4.1\]](#L3.4.1){reference-type="eqref" reference="L3.4.1"} holds. This contradicts [\[T2.3.1\]](#T2.3.1){reference-type="eqref" reference="T2.3.1"} and [\[T2.3.2\]](#T2.3.2){reference-type="eqref" reference="T2.3.2"}. ◻ 100 Egorov, Yu.V., Galaktionov, V.A., Kondratiev, V.A., Pohozaev, S.I.: On the necessary conditions of global existence of solutions to a quasilinear inequality in the half-space. C. R. Acad. Sci. Paris, Sér. 1: Math. **330**, 93--98 (2000). Filippucci, R., Ghergu, M.: Fujita type results for quasilinear parabolic inequalities with nonlocal terms. Discrete Contin. Dyn. Syst. **42**(4), 1817--1833 (2022). Fujita, H.: On the blowing up of solutions of the Cauchy problem for $u_t = \Delta u + u^{1 + \alpha}$. J. Fac. Sci. Univ. Tokyo Sect. IA Math. **13**, 109--124 (1966). Galaktionov, V.A., Egorov, Yu.V., Kondratiev, V.A., Pohozaev, S.I.: Conditions for the existence of solutions to a quasilinear inequality in half-space. Math. Notes volume **67**, 119--121 (2000). Galaktionov, V.A., Shishkov, A.E.: Higher-order quasilinear paprabolic equations with singular initial data. Comm. Cont. Math. **8**(3), 1--24 (2006). Gladkov, A.L.: Unique solvability of the Cauchy problem for certain quasilinear pseudoparabolic equations. Math. Notes **60**, 264--268 (1996). Hayakawa, K.: On nonexistence of global solutions of some semilinear parabolic differential equations. Proc. Japan Acad. **49**, 503--505 (1973). Keller, J.B.: On solution of $\Delta u = f(u)$. Comm. Pure. Appl. Math. **10**, 503--510 (1957). Kondratiev, V.A., Veron, L.: Asymptotic behavior of solutions of some nonlinear parabolic or elliptic equations. Asympt. Anal. **14**, 117--156 (1997). Kon'kov, A.A.: On the asymptotic behaviour of solutions of nonlinear parabolic equations. Proc. Royal Soc. Edinburgh **136**, 365--384 (2006). Kon'kov, A.A., Shishkov, A.E.: Generalization of the Keller-Osserman theorem for higher order differential inequalities. Nonlinearity **32**, 3012--3022 (2019). Kon'kov, A.A.: On solutions of non-autonomous ordinary differential equations. Izv. Math. **65**, 285--327 (2001). Mitidieri, E., Pohozaev, S.I.: A priori estimates and blow-up of solutions to nonlinear partial differential equations and inequalities. Proc. V.A. Steklov Inst. Math. **234**, 3--383 (2001). Osserman, R.: On the inequality $\Delta u \ge f(u)$. Pacific J. Math. **7**, 1641--1647 (1957). Palmieri, A., Takamura, H.: A blow-up result for a Nakao-type weakly coupled system with nonlinearities of derivative-type. Math. Ann. **387**, 111-132 (2023). Pohozaev, S.I., Tesei, A.: Blow-up of nonnegative solutions to quasilinear parabolic inequalities. Atti Accad. Naz. Lincei. Cl. Sci. Fis. Mat. Natur. Rend. Lincei. Ser. 9. **11**, 99--109 (2000). Pucci, P., Serrin, J.: Global nonexistence for abstract evolution equations with positive initial energy. J. Diff. Equat. **150**(1), 203--214 (1998). Samarskii, A. A., Galaktionov, V. A., Kurdyumov, S.P. Mikhailov, A. P.: Blow-Up in Quasilinear Parabolic Equations. De Gruyter, Berlin, New York (1995). Shishkov, A., Veron, L.: The balance between diffusion and absorption in semilinear parabolic equations. Rend. Lincei Mat. Appl. **18**, 59--96 (2007).

arxiv_math

--- abstract: | The relative distance between eigenvalues of the compression of a not necessarily semibounded self-adjoint operator to a closed subspace and some of the eigenvalues of the original operator in a gap of the essential spectrum is considered. It is shown that this distance depends on the maximal angles between pairs of associated subspaces. This generalises results by Drmač in \[Linear Algebra Appl. **244** (1996), 155--163\] from matrices to not necessarily (semi)bounded operators. address: A. Seelmann, Technische Universität Dortmund, Fakultät für Mathematik, D-44221 Dortmund, Germany author: - Albrecht Seelmann title: Relative residual bounds for eigenvalues in gaps of the essential spectrum --- # Introduction and main results {#sec:intro} Let $H$ be a not necessarily semibounded self-adjoint operator in a Hilbert space ${\mathcal H}$ with bounded inverse. We denote by $\lambda_j \in (0,\infty)$ the $j$-th positive eigenvalue of $H$ below $\inf\mathop{\mathrm{\sigma}}_\mathrm{ess}(H) \cap (0,\infty)$, in increasing order and counting multiplicities, provided that this eigenvalue exists. Let ${\mathcal U}$ be a finite dimensional subspace of $\mathop{\mathrm{Dom}}(H)$, and write $P_{\mathcal U}$ for the orthogonal projection onto ${\mathcal U}$. In the Hilbert space ${\mathcal U}$ we then consider the compression $M$ of $H$ to ${\mathcal U}$, that is, the self-adjoint operator $$M = P_{\mathcal U}H|_{\mathcal U} \colon {\mathcal U}\to {\mathcal U} ,$$ with eigenvalues $$\mu_1 \leq \dots \leq \mu_m ,\quad m = \dim{\mathcal U} .$$ Under suitable additional assumptions on ${\mathcal U}$, one expects at least some of the eigenvalues of $M$ to be close to certain eigenvalues of $H$ in a relative sense; cf. [@Drm96]. In order to make this precise, consider the finite dimensional subspaces $${\mathcal V}= \mathop{\mathrm{Ran}}H|_{\mathcal U} \quad\text{ and }\quad {\mathcal W}= \mathop{\mathrm{Ran}}H^{-1}|_{\mathcal U} ,$$ and denote by $P$ the (in general non-orthogonal) projection in ${\mathcal H}$ onto ${\mathcal V}$ along the orthogonal complement ${\mathcal W}^\perp$ of ${\mathcal W}$; it will be established in Lemma [Lemma 10](#lem:projection){reference-type="ref" reference="lem:projection"} below that $P$ always exists and is given by $P = HP_{\mathcal U}H^{-1}$. The main result of this note now generalises Theorem 3 in [@Drm96] from matrices to the current setting of (unbounded) operators $H$. **Theorem 1**. *Let $H$, $\lambda_j$, ${\mathcal U}$, $M$, $\mu_k$, and $P$ be as above, and suppose that $\eta := \norm{ P_{\mathcal U}- P } < 1$. Then:* 1. *$M$ is invertible.* 2. *If numbers $m_0, m_1 \in \mathbb{N}$ with $m_0 \leq m_1 \leq \dim{\mathcal U}$ satisfy $\mu_{m_0} > 0$ and $\mu_{m_1} < (1-\eta)d$, where $d := \inf(\mathop{\mathrm{\sigma}}_\mathrm{ess}(H) \cap (0,\infty)) \in (0,\infty]$, then $H$ has at least $m_1 - m_0 + 1$ positive eigenvalues below $d$, counting multiplicities, and there are indices $j_{m_0} < \dots < j_{m_1}$ with $$\label{eq:relBound} \frac{\abs{\lambda_{j_k} - \mu_k}}{\lambda_{j_k}} \leq \eta \quad\text{ for all }\ m_0 \leq k \leq m_1 .$$* Roughly speaking, Theorem [Theorem 1](#thm:genFinite){reference-type="ref" reference="thm:genFinite"} states that if $\eta < 1$, then *small enough* positive eigenvalues of $M$ can be matched to certain positive eigenvalues of $H$ with a suitable relative bound. Here, small enough refers to being well below a threshold close to the bottom of the positive essential spectrum of $H$, cf. parts (1) and (2) of Remark [Remark 2](#rem:genFinite){reference-type="ref" reference="rem:genFinite"} below. As in [@Drm96; @Gru06; @GN12], the proof of Theorem [Theorem 1](#thm:genFinite){reference-type="ref" reference="thm:genFinite"} relies on perturbing $H$ into its diagonal part with respect to the decomposition $\mathop{\mathrm{Ran}}P_{\mathcal U}\oplus \mathop{\mathrm{Ran}}(I-P_{\mathcal U})$, which is reduced by ${\mathcal U}$ with corresponding part $M$, see Section [3](#sec:geomRelBounds){reference-type="ref" reference="sec:geomRelBounds"} below. Note also that the subspace ${\mathcal U}$ is invariant (and then, in fact, reducing) for $H$ if and only if ${\mathcal V}\subset {\mathcal U}$. In this case, one even has ${\mathcal V}= {\mathcal U}= {\mathcal W}$ and, therefore, $P = P_{\mathcal U}$, see Lemma [Lemma 8](#lem:invariant){reference-type="ref" reference="lem:invariant"} below. In this respect, the norm of the difference $P_{\mathcal U}- P$ can be regarded as an appropriate measure for how far ${\mathcal U}$ is off from being an invariant subspace for $H$. Also, if $H = H^{-1}$, then we have ${\mathcal V}= {\mathcal W}$ and, thus, $P = P_{\mathcal V}= P_{\mathcal W}$. *Remark 2*. (1) If $H$ has no positive essential spectrum at all, that is, if $d = \infty$, then the condition $\mu_{m_1} < (1-\eta)d$ in part (b) of Theorem [Theorem 1](#thm:genFinite){reference-type="ref" reference="thm:genFinite"} is automatically satisfied and *all* positive eigenvalues of $M$ can be matched to some positive eigenvalues of $H$, provided that $\eta < 1$. \(2\) It is worth to note that the bound [\[eq:relBound\]](#eq:relBound){reference-type="eqref" reference="eq:relBound"} together with $\mu_k < (1-\eta)d$ indeed entails $\lambda_{j_k} < d$. In this regard, it is a priori not possible to obtain in Theorem [Theorem 1](#thm:genFinite){reference-type="ref" reference="thm:genFinite"} analogous statements for eigenvalues $\mu_k \geq (1-\eta)d$. In fact, $H$ may not even have correspondingly many positive eigenvalues below $d$. \(3\) As already mentioned in [@GV06 Remark 2.3], a bound of the form [\[eq:relBound\]](#eq:relBound){reference-type="eqref" reference="eq:relBound"} also yields the relative bound $$\frac{\abs{\lambda_{j_k} - \mu_k}}{\mu_k} = \frac{\frac{\abs{\lambda_{j_k} - \mu_k}}{\lambda_{j_k}}}{1-\frac{\lambda_{j_k} - \mu_k}{\lambda_{j_k}}} \leq \frac{\eta}{1-\eta} \quad\text{ for all }\ m_0 \leq k \leq m_1 .$$ \(4\) Upon replacing $H$ and $M$ by $-H$ and $-M$, respectively, one gets the analogous statement of Theorem [Theorem 1](#thm:genFinite){reference-type="ref" reference="thm:genFinite"} for negative eigenvalues in the gap of the essential spectrum. \(5\) Similar statements regarding eigenvalues in gaps of the essential spectrum not containing zero are also possible (while still keeping the requirement of bounded invertibility of $H$), but this then requires a stronger assumption on the norm $\norm{P_{\mathcal U}- P}$ depending on the gap under consideration, see Remark [Remark 15](#rem:gap){reference-type="ref" reference="rem:gap"} below. The latter can, of course, formally be avoided with a suitable spectral shift of $H$ (and $M$), but this then also affects the subspaces ${\mathcal V}$ and ${\mathcal W}$ and, thus, the projection $P$. Let us now compare Theorem [Theorem 1](#thm:genFinite){reference-type="ref" reference="thm:genFinite"} to [@Drm96 Theorem 3] and comment on other related results in the literature. *Remark 3*. (1) If $J \colon {\mathcal K}\to {\mathcal H}$ is an isometry from some Hilbert space ${\mathcal K}$ with range ${\mathcal U}$, then the operator $M$ is unitarily equivalent to $J^*HJ$. In this sense, the above setting is consistent with the framework of [@Drm96]. \(2\) It is easily seen that $P_{\mathcal U}- P = (P_{\mathcal U}- P_{\mathcal U}^\perp)(P_{\mathcal U}(I-P) + P_{\mathcal U}^\perp P)$, where $P_{\mathcal U}- P_{\mathcal U}^\perp$ is unitary; cf. the proof of Lemma [Lemma 12](#lem:reprHoffH){reference-type="ref" reference="lem:reprHoffH"} below. In particular, we have $\norm{P_{\mathcal U}- P} = \norm{P_{\mathcal U}(I-P) + P_{\mathcal U}^\perp P}$. Taking into account parts (1) of this remark and of Remark [Remark 2](#rem:genFinite){reference-type="ref" reference="rem:genFinite"}, Theorem [Theorem 1](#thm:genFinite){reference-type="ref" reference="thm:genFinite"} therefore indeed contains [@Drm96 Theorem 3] as a special case and, thus, generalises it from matrices to (possibly unbounded) operators $H$. \(3\) To the best of the author's knowledge, Theorem [Theorem 1](#thm:genFinite){reference-type="ref" reference="thm:genFinite"} is the first result of this kind applicable for gaps in the essential spectrum of not necessarily semibounded operators $H$. By contrast, for nonnegative operators $H$ stronger results have been obtained in [@GV06; @GV07] for eigenvalues below the essential spectrum. In particular, [@GV06 Theorem 2.2] allows to consider subspaces ${\mathcal U}$ in the *form domain* of $H$ and provides a stronger relative bound already in the case of matrices considered earlier in [@Drm96], cf. [@Drm96 Example 10]. The following result gives a geometric bound on the norm of the difference $P_{\mathcal U}- P$ in terms of the maximal angles between the pairs of subspaces $({\mathcal U},{\mathcal V})$, $({\mathcal U},{\mathcal W})$, and $({\mathcal V},{\mathcal W})$. In this regard, it recovers Proposition 5 in [@Drm96] in the current setting. Recall that the *maximal angle* $\theta({\mathcal M},{\mathcal N})$ between two closed subspaces ${\mathcal M}$ and ${\mathcal N}$ of ${\mathcal H}$ can be defined as $$\theta({\mathcal M},{\mathcal N}) = \arcsin( \norm{P_{\mathcal M}- P_{\mathcal N}} ),$$ see, e.g., [@AM13 Definition 2.1]. **Theorem 4**. *Let ${\mathcal U}$, ${\mathcal V}$, ${\mathcal W}$, and $P$ be as in Theorem [Theorem 1](#thm:genFinite){reference-type="ref" reference="thm:genFinite"}. Then $$\norm{P_{\mathcal U}- P} \leq \min\bigl\{ \sin\theta({\mathcal U},{\mathcal V}) , \sin\theta({\mathcal U},{\mathcal W}) \bigr\} + \tan\theta({\mathcal V},{\mathcal W}) .$$* The rest of this note is organised as follows: Section [2](#sec:relBounds){reference-type="ref" reference="sec:relBounds"} presents a general perturbation result that addresses relative bounds for eigenvalues in gaps of the essential spectrum. In essence, it reproduces a result from [@Ves08] in an operator framework, but is proved here in an alternative way using the variational principle from [@DES00; @DES23]. Section [3](#sec:geomRelBounds){reference-type="ref" reference="sec:geomRelBounds"} then adds a geometric component in terms of the projections $P_{\mathcal U}$, $P_{\mathcal V}$, $P_{\mathcal W}$, and $P$ that allows to infer from the general result in Section [2](#sec:relBounds){reference-type="ref" reference="sec:relBounds"} the core result of this note, Theorem [Theorem 13](#thm:gen){reference-type="ref" reference="thm:gen"}. The latter includes Theorem [Theorem 1](#thm:genFinite){reference-type="ref" reference="thm:genFinite"} as a particular case, while allowing the subspace ${\mathcal U}$ to have infinite dimension. A likewise more general version of Theorem [Theorem 4](#thm:angleFinite){reference-type="ref" reference="thm:angleFinite"}, Theorem [Theorem 18](#thm:angle){reference-type="ref" reference="thm:angle"}, is also proved in that section utilizing known results on maximal angles between closed subspaces. # Relative bounds for eigenvalues {#sec:relBounds} In this section we prove a general residual bound for eigenvalues in gaps of the essential spectrum of self-adjoint operators, which lays the foundation for the proof of Theorem [Theorem 1](#thm:genFinite){reference-type="ref" reference="thm:genFinite"}. The corresponding result essentially reproduces [@Ves08 Theorem 4.13] in the particular case of an operator framework; see also [@VS93 Theorem 3.16] for the matrix case. For a self-adjoint operator $T$, we denote by $\mathsf{E}_T$ the projection-valued spectral measure for $T$, and for $\gamma \in \mathbb{R}$ we write $\lambda_{\gamma,j}(T) = \lambda_{j}(T|_{\mathop{\mathrm{Ran}}\mathsf{E}_T((\gamma,\infty))}) \geq \gamma$, $j \in \mathbb{N}$, $j \leq \dim \mathop{\mathrm{Ran}}\mathsf{E}_T((\gamma,\infty))$, for the $j$-th standard variational value of the lower semibounded part $T|_{\mathop{\mathrm{Ran}}\mathsf{E}((\gamma,\infty))}$ of $T$. It agrees with the $j$-th eigenvalue of $T|_{\mathop{\mathrm{Ran}}\mathsf{E}((\gamma,\infty))}$ below its essential spectrum, in nondecreasing order and counting multiplicities, if this eigenvalue exists, and otherwise equals the bottom of the essential spectrum of $T|_{\mathop{\mathrm{Ran}}\mathsf{E}((\gamma,\infty))}$. In fact, if $\mathop{\mathrm{Ran}}\mathsf{E}((\gamma,\infty))$ is infinite dimensional, then $\lambda_{\gamma,j}(T) \to \inf(\mathop{\mathrm{\sigma}}_\mathrm{ess}(T) \cap (\gamma,\infty)) \in [\gamma,\infty]$ as $j \to \infty$. Let $A$ be self-adjoint, and let $V$ be symmetric with $\mathop{\mathrm{Dom}}(V) \supset \mathop{\mathrm{Dom}}(A)$. Suppose that for some constants $a \in \mathbb{R}$, $b \in [0,1)$ the operator $A_1 := a + b\abs{A}$ is nonnegative and that $\norm{ Vx } \leq \norm{ A_1 x }$ for all $x \in \mathop{\mathrm{Dom}}(A)$. In particular, this gives $\norm{ Vx } \leq \abs{a} \norm{ x } + b \norm{ Ax }$ for all $x \in \mathop{\mathrm{Dom}}(A)$, so that $B := A + V$ is self-adjoint on $\mathop{\mathrm{Dom}}(B) = \mathop{\mathrm{Dom}}(A)$ by the well-known Kato-Rellich theorem. The following result is used in Section [3](#sec:geomRelBounds){reference-type="ref" reference="sec:geomRelBounds"} below only in the particular case where $a = 0$. However, the more general case of $a \in \mathbb{R}$ does not require much more efforts and is more in line with the mentioned guiding statement from [@Ves08]. **Proposition 5**. *Let the interval $(\alpha , \beta)$ with $\beta - \alpha > 2a + b(\abs{\alpha} + \abs{\beta})$ be in the resolvent set of $A$. Then:* 1. *The interval $( \alpha + b\abs{\alpha} + a , \beta - b\abs{\beta} - a )$ belongs to the resolvent set of $B = A + V$.* 2. *The subspace $\mathop{\mathrm{Ran}}\mathsf{E}_A((\alpha,\infty))$ has finite dimension if and only if $\mathop{\mathrm{Ran}}\mathsf{E}_B((\alpha+b\abs{\alpha}+a,\infty))$ has finite dimension, and in this case $\dim\mathop{\mathrm{Ran}}\mathsf{E}_A((\alpha,\infty)) = \dim\mathop{\mathrm{Ran}}\mathsf{E}_B((\alpha+b\abs{\alpha}+a,\infty))$ holds.* 3. *We have $$\abs{ \lambda_{\alpha,j}(A) - \lambda_{\alpha+b\abs{\alpha}+a,j}(B) } \leq a + b\abs{\lambda_{\alpha,j}(A)}$$ for all $j \in \mathbb{N}$ with $j \leq \dim\mathop{\mathrm{Ran}}\mathsf{E}_A((\alpha,\infty))$.* 4. *With $d := \inf( \mathop{\mathrm{\sigma}}_\mathrm{ess}(A) \cap (\alpha,\infty) ) \in [\beta , \infty]$ we have $$d - b\abs{d} - a \leq \inf\bigl( \mathop{\mathrm{\sigma}}_\mathrm{ess}(B) \cap (\alpha+b\abs{\alpha}+a,\infty) \bigr) \leq d + b\abs{d} + a ,$$ where the lower and upper bounds are interpreted as $\infty$ if $d = \infty$. In particular, the spectral part $\mathop{\mathrm{\sigma}}(B) \cap (\alpha+b\abs{\alpha}+a,\infty)$ is purely discrete if $\mathop{\mathrm{\sigma}}(A) \cap (\alpha,\infty)$ is purely discrete.* For the convenience of the reader, a proof of Proposition [Proposition 5](#prop:eig){reference-type="ref" reference="prop:eig"} is presented below. Other than the approach in [@Ves08], which was based on analyticity properties, this proof alternatively relies on the minimax principle from [@DES00; @DES23] for eigenvalues in gaps of the essential spectrum. The following proposition formulates a variant of this result tailored to the current situation; cf. also [@SST20; @DES]. **Proposition 6** ([@DES23 Theorem 1]). *Let $T$ be self-adjoint, and let $\Lambda$ be an orthogonal projection in the same Hilbert space such that $\mathop{\mathrm{Dom}}(T)$ is invariant for $\Lambda$. With ${\mathcal D}_+ := \mathop{\mathrm{Dom}}(T) \cap \mathop{\mathrm{Ran}}\Lambda$ and ${\mathcal D}_- := \mathop{\mathrm{Dom}}(T) \cap \mathop{\mathrm{Ran}}(I-\Lambda)$, suppose that $$\label{eq:gapCondition} \nu := \sup_{\substack{x_- \in {\mathcal D}_-\\ \norm{x_-} = 1}} \langle x_- , Tx_- \rangle < \inf_{\substack{x_+ \in {\mathcal D}_+\\ \norm{x_+} = 1}} \langle x_+ , Tx_+ \rangle .$$ Then, $$\label{eq:minimax} \lambda_{\nu,j}(T) = \inf_{\substack{{\mathfrak M}\subset {\mathcal D}_+\\ \dim{\mathfrak M}= j}} \sup_{\substack{x \in {\mathfrak M}\oplus {\mathcal D}_-\\ \norm{x} = 1}} \langle x , Tx \rangle$$ for $j \in \mathbb{N}$ with $j \leq \dim\mathop{\mathrm{Ran}}\Lambda$, and these describe all variational values of the lower semibounded part $T|_{\mathop{\mathrm{Ran}}\mathsf{E}_T((\nu,\infty))}$ of $T$.* *Remark 7*. The inequality [\[eq:gapCondition\]](#eq:gapCondition){reference-type="eqref" reference="eq:gapCondition"} is usually called a *gap condition* for $T$. In [@DES00; @DES23], the right-hand side of [\[eq:gapCondition\]](#eq:gapCondition){reference-type="eqref" reference="eq:gapCondition"} is replaced by the possibly larger term $$\inf_{x_+ \in {\mathcal D}_+ \setminus \{0\}} \sup_{x_- \in {\mathcal D}_-} \frac{\langle x_+ + x_- , T(x_+ + x_-) \rangle}{\norm{x_+ + x_-}^2},$$ which agrees with the right-hand side of [\[eq:minimax\]](#eq:minimax){reference-type="eqref" reference="eq:minimax"} for $j = 1$. In particular, the condition formulated by [\[eq:gapCondition\]](#eq:gapCondition){reference-type="eqref" reference="eq:gapCondition"} is stricter than the corresponding one in [@DES00; @DES23]. However, it is exactly [\[eq:gapCondition\]](#eq:gapCondition){reference-type="eqref" reference="eq:gapCondition"} that is verified in the proof of Proposition [Proposition 5](#prop:eig){reference-type="ref" reference="prop:eig"} below. An implicit part of Proposition [Proposition 6](#prop:DES){reference-type="ref" reference="prop:DES"} is that under the hypotheses the subspace $\mathop{\mathrm{Ran}}\mathsf{E}_T((\nu,\infty))$ has finite dimension if and only if $\mathop{\mathrm{Ran}}\Lambda$ has, and, in this case, the two subspaces have the same dimension. Moreover, the interval $(\nu,\lambda_{\nu,1}(T))$ belongs to the resolvent set of $T$ and, in particular, so does the interval $(\nu,\nu')$, where $\nu'$ denotes the right-hind side of [\[eq:gapCondition\]](#eq:gapCondition){reference-type="eqref" reference="eq:gapCondition"}, cf. Remark [Remark 7](#rem:DES){reference-type="ref" reference="rem:DES"}. With this is mind, we are ready to prove Proposition [Proposition 5](#prop:eig){reference-type="ref" reference="prop:eig"}. *Proof of Proposition [Proposition 5](#prop:eig){reference-type="ref" reference="prop:eig"}.* We follow the general strategy of the proof of [@VS93 Theorem 3.16]. Since by hypothesis $A_1$ is self-adjoint and nonnegative and $V$ is symmetric with $\mathop{\mathrm{Dom}}(V) \supset \mathop{\mathrm{Dom}}(A_1) = \mathop{\mathrm{Dom}}(A)$ and $\norm{ Vx } \leq \norm{ A_1x }$ for all $x \in \mathop{\mathrm{Dom}}(A)$, it follows from Löwner's theorem, see, e.g., [@Kato95 Theorem V.4.12], that $$\abs{ \langle x , Vx \rangle } \leq \langle x , A_1 x \rangle \quad\text{ for all }\ x \in \mathop{\mathrm{Dom}}(A) .$$ As a consequence, we have $$\label{eq:monotonicity} A - A_1 \leq B \leq A + A_1$$ in the sense of quadratic forms, where $\mathop{\mathrm{Dom}}(A \pm A_1) = \mathop{\mathrm{Dom}}(A) = \mathop{\mathrm{Dom}}(B)$. In the notation of Proposition [Proposition 6](#prop:DES){reference-type="ref" reference="prop:DES"}, we take $\Lambda = \mathsf{E}_A((\alpha,\infty)) = \mathsf{E}_A([\beta,\infty))$ and $${\mathcal D}_+ = \mathop{\mathrm{Dom}}(A) \cap \mathop{\mathrm{Ran}}\mathsf{E}_A([\beta,\infty)) ,\quad {\mathcal D}_- = \mathop{\mathrm{Dom}}(A) \cap \mathop{\mathrm{Ran}}\mathsf{E}_A((-\infty,\alpha]) .$$ Define $f_\pm \colon \mathbb{R}\to \mathbb{R}$ by $f_\pm(t) = t \pm (a + b\abs{t})$, which both are continuous, bijective, and strictly increasing. Taking into account that $A \pm A_1 = f_\pm(A)$ by functional calculus, we then have $$\langle x, (A-A_1)x \rangle \geq f_-(\beta)\norm{x}^2 \quad\text{ for }\quad x \in {\mathcal D}_+$$ and $$\langle x, (A+A_1)x \rangle \leq f_+(\alpha)\norm{x}^2 \quad\text{ for }\quad x \in {\mathcal D}_- .$$ Moreover, the hypothesis on $\alpha$ and $\beta$ guarantees that $f_-(\beta) > f_+(\alpha)$. In light of [\[eq:monotonicity\]](#eq:monotonicity){reference-type="eqref" reference="eq:monotonicity"}, for each of the choices $T \in \{A\pm A_1,B\}$ the gap condition [\[eq:gapCondition\]](#eq:gapCondition){reference-type="eqref" reference="eq:gapCondition"} is therefore satisfied with $$\sup_{\substack{x_-\in{\mathcal D}_-\\\norm{x_-}=1}} \langle x_- , Tx_- \rangle \leq f_+(\alpha) < f_-(\beta) \leq \inf_{\substack{x_+\in{\mathcal D}_+\\\norm{x_+}=1}} \langle x_+ , Tx_+ \rangle ,$$ so that Proposition [Proposition 6](#prop:DES){reference-type="ref" reference="prop:DES"} can be applied for all three choices. In particular, the interval $(f_+(\alpha),f_-(\beta))$ belongs to the resolvent set of all three operators $A\pm A_1$ and $B$. With $T = B$, this proves parts (a) and (b) of the claim. Furthermore, with $\gamma = f_+(\alpha)$ we obtain from [\[eq:monotonicity\]](#eq:monotonicity){reference-type="eqref" reference="eq:monotonicity"} and the representation of the variational values in [\[eq:minimax\]](#eq:minimax){reference-type="eqref" reference="eq:minimax"} that $$\label{eq:monotonicityMinimax} \lambda_{\gamma,j}(A-A_1) \leq \lambda_{\gamma,j}(B) \leq \lambda_{\gamma,j}(A+A_1)$$ for all $j \in \mathbb{N}$, $j \leq \dim\mathop{\mathrm{Ran}}\mathsf{E}_A((\alpha,\infty))$. Here, we also have the representation $\lambda_{\gamma,j}(A - A_1) = \lambda_{f_-(\alpha),j}(f_-(A))$ since also the interval $(f_-(\alpha),f_-(\beta))$ belongs to the resolvent set of $f_-(A)$ by the spectral mapping theorem, as well as trivially $\lambda_{\gamma,j}(A + A_1) = \lambda_{f_+(\alpha),j}(f_+(A))$. In turn, again by the spectral mapping theorem, we have $\lambda_{f_\pm(\alpha),j}(f_\pm(A)) = f_\pm(\lambda_{\alpha,j}(A))$, so that we arrive at the representations $\lambda_{\gamma,j}(A\pm A_1) = f_\pm(\lambda_{\alpha,j}(A))$. Plugging the latter into [\[eq:monotonicityMinimax\]](#eq:monotonicityMinimax){reference-type="eqref" reference="eq:monotonicityMinimax"} gives $$\label{eq:boundEig} \lambda_{\alpha,j}(A) - (a+b\abs{\lambda_{\alpha,j}(A)}) \leq \lambda_{\gamma,j}(B) \leq \lambda_{\alpha,j}(A) + (a+b\abs{\lambda_{\alpha,j}(A)})$$ for all $j \in \mathbb{N}$ with $j \leq \dim\mathop{\mathrm{Ran}}\mathsf{E}_A((\alpha,\infty))$, which proves part (c) of the claim. It remains to show part (d). If $\mathop{\mathrm{Ran}}\mathsf{E}_A((\alpha,\infty))$ has finite dimension, then also $\mathop{\mathrm{Ran}}\mathsf{E}_B((f_+(\alpha),\infty))$ has finite dimension by part (b) and there is nothing to prove. So, suppose that $\mathop{\mathrm{Ran}}\mathsf{E}_A((\alpha,\infty))$, and hence also $\mathop{\mathrm{Ran}}\mathsf{E}_B((\gamma,\infty))$, is infinite dimensional. The claim of part (d) then follows by taking in [\[eq:boundEig\]](#eq:boundEig){reference-type="eqref" reference="eq:boundEig"} the limit as $j \to \infty$. This completes the proof. ◻ # Geometric residual bounds and proof of main results {#sec:geomRelBounds} A large part of the considerations in this section also works under more general assumptions than the ones from Section [1](#sec:intro){reference-type="ref" reference="sec:intro"}. With this in mind, let $H$ and $\lambda_j$ be as in Section [1](#sec:intro){reference-type="ref" reference="sec:intro"}, and let ${\mathcal U}$ be a (not necessarily finite dimensional) closed subspace such that $\mathop{\mathrm{Dom}}(H)$ is invariant for the orthogonal projection $P_{\mathcal U}$ onto ${\mathcal U}$; this obviously includes the case where ${\mathcal U}$ is just a finite dimensional subspace of $\mathop{\mathrm{Dom}}(H)$ as in Section 1. Let $M$ be the compression of $H$ to ${\mathcal U}$, that is, $$\label{eq:defM} M = P_{\mathcal U}H|_{\mathcal U} \quad\text{ with }\ \mathop{\mathrm{Dom}}(M) = \mathop{\mathrm{Dom}}(H) \cap {\mathcal U}\subset {\mathcal U} ,$$ as an operator in the Hilbert space ${\mathcal U}$. Finally, denote by ${\mathcal V}$ and ${\mathcal W}$ the closed subspaces $${\mathcal V}= \mathop{\mathrm{Ran}}H|_{\mathcal U} \quad\text{ and }\quad {\mathcal W}= \overline{\mathop{\mathrm{Ran}}H^{-1}|_{\mathcal U}} .$$ We begin with the following elementary, essentially well-known lemma. **Lemma 8**. 1. *$\mathop{\mathrm{Dom}}(H)$ is invariant also for $P_{\mathcal U}^\perp = I - P_{\mathcal U}$.* 2. *$\mathop{\mathrm{Dom}}(H)$ splits as $$\mathop{\mathrm{Dom}}(H) = ( \mathop{\mathrm{Dom}}(H) \cap {\mathcal U}) \oplus ( \mathop{\mathrm{Dom}}(H) \cap {\mathcal U}^\perp ).$$* 3. *$M$ is densely defined in ${\mathcal U}$.* 4. *If ${\mathcal U}$ is invariant for $H$, then ${\mathcal V}= {\mathcal W}= {\mathcal U}$.* *Proof.* (a) is clear, and (b) follows immediately from (a) and the identity $I = P_{\mathcal U}+ P_{\mathcal U}^\perp$. For part (c), let $u \in {\mathcal U}$. Since $H$ is densely defined, we may choose a sequence $(x_k)$ in $\mathop{\mathrm{Dom}}(H)$ that converges to $u$. Taking into account that $P_{\mathcal U}$ is bounded, the sequence $(u_k)$ with $u_k = P_{\mathcal U}x_k \in \mathop{\mathrm{Dom}}(M)$ then converges to $P_{\mathcal U}u = u$ in ${\mathcal U}$, which proves the claim. Finally, for part (d), suppose that ${\mathcal U}$ is invariant for $H$, that is, ${\mathcal V}\subset {\mathcal U}$. In view of part (c), a standard argument then shows that also ${\mathcal U}^\perp$ is invariant for $H$. Now, let $y \in {\mathcal U}$. By part (b), we may decompose $x := H^{-1}y \in \mathop{\mathrm{Dom}}(H)$ as $x = u + v$ with $u \in \mathop{\mathrm{Dom}}(H) \cap {\mathcal U}$ and $v \in \mathop{\mathrm{Dom}}(H) \cap {\mathcal U}^\perp$. Then, we have $Hu + Hv = Hx = y \in {\mathcal U}$, which by $Hu \in {\mathcal U}$ and $Hv \in {\mathcal U}^\perp$ implies that $Hv = 0$, so that $v = 0$ because $H$ is invertible. We conclude that $H^{-1}y = u \in {\mathcal U}$ and $y = H u \in {\mathcal V}$. Since $y \in {\mathcal U}$ was arbitrary and taking into account that ${\mathcal U}$ is closed, the former yields ${\mathcal W}\subset {\mathcal U}$, and the latter implies ${\mathcal U}\subset {\mathcal V}$, that is, ${\mathcal U}= {\mathcal V}$. In order to show the remaining inclusion ${\mathcal U}\subset {\mathcal W}$, we observe that the invariance of ${\mathcal U}$ for $H$ implies that $\mathop{\mathrm{Dom}}(H) \cap {\mathcal U}\subset \mathop{\mathrm{Ran}}H^{-1}|_{\mathcal U}\subset {\mathcal W}$. In view of part (c) and the closedness of ${\mathcal W}$, this shows that indeed ${\mathcal U}\subset {\mathcal W}$, which completes the proof. ◻ *Remark 9*. The above reasoning for part (d) of Lemma [Lemma 8](#lem:invariant){reference-type="ref" reference="lem:invariant"} is essentially contained, at least in part, in the proof of Lemma 2.1 in [@MSS16]; cf. also Remark 2.3 and Lemma 2.4 in [@TW14]. The next lemma proves the existence of the (not necessarily orthogonal) projection onto ${\mathcal V}$ along ${\mathcal W}^\perp$ by providing an explicit representation in terms of $H$ and $P_{\mathcal U}$. **Lemma 10**. *The operator $P = HP_{\mathcal U}H^{-1}$ is the projection onto ${\mathcal V}$ along ${\mathcal W}^\perp$, that is, $P$ is bounded with $P^2 = P$ and satisfies $\mathop{\mathrm{Ran}}P = {\mathcal V}$ and $\mathop{\mathrm{Ker}}P = {\mathcal W}^\perp$.* *Proof.* Observe that $P = HP_{\mathcal U}H^{-1}$ is closed and everywhere defined, hence bounded by the closed graph theorem. It is then obvious that also $P^2 = P$. Finally, we have the identities $\mathop{\mathrm{Ran}}P = \mathop{\mathrm{Ran}}( H P_{\mathcal U}|_{\mathop{\mathrm{Dom}}(H)} ) = {\mathcal V}$ as well as $\mathop{\mathrm{Ker}}P = \mathop{\mathrm{Ker}}(P_{\mathcal U}H^{-1}) = (\mathop{\mathrm{Ran}}( H^{-1} P_{\mathcal U}))^\perp = {\mathcal W}^\perp$. ◻ *Remark 11*. More generally, if $L$ is a closed densely defined operator with bounded inverse such that $\mathop{\mathrm{Dom}}(L)$ is invariant for $P_{\mathcal U}$, then $LP_{\mathcal U}L^{-1}$ is the projection onto $\mathop{\mathrm{Ran}}L|_{\mathcal U}$ along $\mathop{\mathrm{Ker}}(P_{\mathcal U}L^{-1}) = (\mathop{\mathrm{Ran}}(L^{-*}|_{\mathcal U}))^\perp$. In light of the domain splitting in part (b) of Lemma [Lemma 8](#lem:invariant){reference-type="ref" reference="lem:invariant"}, we may define the diagonal and off-diagonal parts of $H$ with respect to ${\mathcal U}\oplus {\mathcal U}^\perp$ as $$H_{\mathrm{diag}}= P_{\mathcal U}H P_{\mathcal U}+ P_{\mathcal U}^\perp H P_{\mathcal U}^\perp,\quad H_{\mathrm{off}}= P_{\mathcal U}H P_{\mathcal U}^\perp + P_{\mathcal U}^\perp H P_{\mathcal U}$$ with $\mathop{\mathrm{Dom}}(H_{\mathrm{diag}}) = \mathop{\mathrm{Dom}}(H) = \mathop{\mathrm{Dom}}(H_{\mathrm{off}})$; cf. also [@Gru06; @GN12]. In particular, we have the operator identity $$H = H_{\mathrm{diag}}+ H_{\mathrm{off}} .$$ Clearly, the subspace ${\mathcal U}$ reduces $H_{\mathrm{diag}}$ in the sense that $H_{\mathrm{diag}}$ is the direct sum of operators defined in ${\mathcal U}$ and ${\mathcal U}^\perp$, respectively, and $M$ is the part of $H_{\mathrm{diag}}$ associated to ${\mathcal U}$. We now aim to apply Proposition [Proposition 5](#prop:eig){reference-type="ref" reference="prop:eig"} from the previous section with $A = H$ and $V = -H_{\mathrm{off}}$, so that $A + V = H_{\mathrm{diag}}$. To this end, we make the following elementary observation. **Lemma 12**. *We have $$\label{eq:reprHoffH} H_{\mathrm{off}}H^{-1} = (P_{\mathcal U}- P_{\mathcal U}^\perp) (P_{\mathcal U}- P)$$ with $P$ as in Lemma [Lemma 10](#lem:projection){reference-type="ref" reference="lem:projection"}.* *Proof.* We calculate $$\begin{aligned} H_{\mathrm{off}}H^{-1} &= P_{\mathcal U}H P_{\mathcal U}^\perp H^{-1} + P_{\mathcal U}^\perp H P_{\mathcal U}H^{-1} = P_{\mathcal U}(I - P) + P_{\mathcal U}^\perp P\\ &= (P_{\mathcal U}- P_{\mathcal U}^\perp) (P_{\mathcal U}- P).\qedhere \end{aligned}$$ ◻ Note that the factor $P_{\mathcal U}- P_{\mathcal U}^\perp$ on the right-hand side of [\[eq:reprHoffH\]](#eq:reprHoffH){reference-type="eqref" reference="eq:reprHoffH"} is self-adjoint and unitary and can therefore be ignored when it comes to estimating $H_{\mathrm{off}}H^{-1}$ in norm. With this in mind, we are now able to formulate and prove the core result of this note. Here, the particular case where ${\mathcal U}$ has finite dimension agrees with Theorem [Theorem 1](#thm:genFinite){reference-type="ref" reference="thm:genFinite"}. **Theorem 13**. *Suppose that $\eta := \norm{ P_{\mathcal U}- P } < 1$ with $P = HP_{\mathcal U}H^{-1}$ as in Lemma [Lemma 10](#lem:projection){reference-type="ref" reference="lem:projection"}.* 1. *The operator $M$ in [\[eq:defM\]](#eq:defM){reference-type="eqref" reference="eq:defM"} is self-adjoint and has a bounded inverse.* 2. *With $d := \inf(\mathop{\mathrm{\sigma}}_\mathrm{ess}(H) \cap (0,\infty)) \in (0,\infty]$ we have $$\inf\bigl( \mathop{\mathrm{\sigma}}_\mathrm{ess}(M) \cap (0,\infty) \bigr) \geq (1-\eta)d .$$* 3. *Denote by $(\mu_k)_{k \in J}$ with $J \subset \mathbb{N}$ the (finite or infinite) collection of eigenvalues of $M$ in the interval $(0,(1-\eta)d)$, in increasing order and counting multiplicities. Then, there is a family of indices $j_k \in \mathbb{N}$, $k \in J$, strictly increasing in $k$, such that for each $k \in J$ we have $$\frac{\abs{\lambda_{j_k} - \mu_k}}{\lambda_{j_k}} \leq \eta .$$* *Proof.* In view of Lemma [Lemma 12](#lem:reprHoffH){reference-type="ref" reference="lem:reprHoffH"}, we have $\norm{H_{\mathrm{off}}H^{-1}} = \norm{P_{\mathcal U}-P} = \eta < 1$. In particular, this gives $$\norm{H_{\mathrm{off}}x} \leq \eta\norm{Hx} = \eta\norm{\abs{H}x}$$ for all $x \in \mathop{\mathrm{Dom}}(H)$. In the notation of Section [2](#sec:relBounds){reference-type="ref" reference="sec:relBounds"}, we may therefore take $A = H$ and $V = -H_{\mathrm{off}}$ with $a = 0$ and $b = \eta \in [0,1)$. Moreover, since $H$ has a bounded inverse, there are numbers $\alpha,\beta \in \mathbb{R}$ with $\alpha < 0 < \beta$ such that the interval $(\alpha,\beta)$ belongs to the resolvent set of $H$; in particular, we have $$d = \inf(\mathop{\mathrm{\sigma}}_\mathrm{ess}(H) \cap (0,\infty)) = \inf(\mathop{\mathrm{\sigma}}_\mathrm{ess}(H) \cap (\alpha,\infty)) \geq \beta .$$ We observe that $2a + b(\abs{\alpha}+\abs{\beta}) = \eta(\beta-\alpha) < \beta - \alpha$, so that the hypotheses of Proposition [Proposition 5](#prop:eig){reference-type="ref" reference="prop:eig"} are satisfied. We conclude that $H_{\mathrm{diag}}= H - H_{\mathrm{off}}$ is self-adjoint and that the interval $((1-\eta)\alpha,(1-\eta)\beta)$ belongs to its resolvent set; in particular, $H_{\mathrm{diag}}$ has a bounded inverse. Moreover, part (d) of Proposition [Proposition 5](#prop:eig){reference-type="ref" reference="prop:eig"} gives $$\inf(\mathop{\mathrm{\sigma}}_\mathrm{ess}(H_{\mathrm{diag}}) \cap ((1-\eta)\alpha,\infty)) \geq (1-\eta)d .$$ Since ${\mathcal U}$ reduces $H_{\mathrm{diag}}$ and $M$ is the part of $H_{\mathrm{diag}}$ associated to ${\mathcal U}$, this proves (a) and (b). Taking into account that each $\lambda_{\alpha,j}(H)$ is positive, it follows from part (c) of Proposition [Proposition 5](#prop:eig){reference-type="ref" reference="prop:eig"} that $$\label{eq:eigHHdiag} \frac{\abs{\lambda_{\alpha,j}(H) - \lambda_{(1-\eta)\alpha,j}(H_{\mathrm{diag}})}}{\lambda_{\alpha,j}(H)} \leq \eta$$ for all $j \in \mathbb{N}$ with $j \leq \dim\mathop{\mathrm{Ran}}\mathsf{E}_H((\alpha,\infty))$. In particular, this implies that $\lambda_{\alpha,j}(H) < d$ if $\lambda_{(1-\eta)\alpha,j}(H_{\mathrm{diag}}) < (1-\eta)d$. Now, by definition of the $\mu_k$ there are indices $j_k$ with $\lambda_{(1-\eta)\alpha,j_k}(H_{\mathrm{diag}}) = \mu_k \in (0,(1-\eta)d)$ for all $k \in J$. Thus, $\lambda_{\alpha,j_k}(H) < d$ is the $j_k$-th positive eigenvalue of $H$ below $d$, that is, $\lambda_{\alpha,j_k}(H) = \lambda_{j_k}$. Together with [\[eq:eigHHdiag\]](#eq:eigHHdiag){reference-type="eqref" reference="eq:eigHHdiag"}, this shows part (c) and, hence, completes the proof of the theorem. ◻ Let us collect some useful observations regarding part (b) of Theorem [Theorem 13](#thm:gen){reference-type="ref" reference="thm:gen"}. *Remark 14*. (1) The proof of Theorem [Theorem 13](#thm:gen){reference-type="ref" reference="thm:gen"} gives $$(1-\eta)d \leq \inf(\mathop{\mathrm{\sigma}}_\mathrm{ess}(H_{\mathrm{diag}}) \cap ((1-\eta)\alpha,\infty)) \leq \inf(\mathop{\mathrm{\sigma}}_\mathrm{ess}(M) \cap ((1-\eta)\alpha,\infty)) ,$$ and either inequality may a priori be strict. Thus, eigenvalues of $M$ that are larger than (or equal to) $(1-\eta)d$ are not necessarily accessible via the variational values $\lambda_{(1-\eta)\alpha,j}(H_{\mathrm{diag}})$, and even for those that are accessible, we can no longer guarantee that the corresponding variational values $\lambda_{\alpha,j}(H)$ for $H$ are smaller than $d$. The latter may therefore not correspond to eigenvalues of $H$. \(2\) If ${\mathcal U}$ has finite dimension, then $$\inf(\mathop{\mathrm{\sigma}}_\mathrm{ess}(H_{\mathrm{diag}}) \cap ((1-\eta)\alpha,\infty)) = \inf(\mathop{\mathrm{\sigma}}_\mathrm{ess}(H) \cap (\alpha,\infty)) = d .$$ Indeed, in this case $\mathop{\mathrm{Ran}}(P_{\mathcal U}-P)$ has finite dimension and, consequently, in view of Lemma [Lemma 12](#lem:reprHoffH){reference-type="ref" reference="lem:reprHoffH"}, $H_{\mathrm{diag}}^{-1} - H^{-1} = H_{\mathrm{diag}}^{-1}H_{\mathrm{off}}H^{-1}$ is compact. Hence, $\mathop{\mathrm{\sigma}}_\mathrm{ess}(H_{\mathrm{diag}}) = \mathop{\mathrm{\sigma}}_\mathrm{ess}(H)$, see, e.g., [@Kato95 Theorem IV.5.35]. \(3\) Although $\inf(\mathop{\mathrm{\sigma}}_\mathrm{ess}(H_{\mathrm{diag}}) \cap ((1-\eta)\alpha,\infty)) \leq (1+\eta)d$ by part (d) of Proposition [Proposition 5](#prop:eig){reference-type="ref" reference="prop:eig"}, the term $\inf(\mathop{\mathrm{\sigma}}_\mathrm{ess}(M) \cap ((1-\eta)\alpha,\infty))$ might a priori be a lot larger, for instance if $H_{\mathrm{diag}}$ has positive essential spectrum but $M$ does not. In view of part (2) of this remark, this is the case, in particular, if $H$ has positive essential spectrum and ${\mathcal U}$ has finite dimension. The following remark addresses an extension of Theorem [Theorem 13](#thm:gen){reference-type="ref" reference="thm:gen"} to gaps of the essential spectrum of $H$ that do not contain zero. *Remark 15*. The general form of Proposition [Proposition 5](#prop:eig){reference-type="ref" reference="prop:eig"} allows to obtain also similar statements as in Theorem [Theorem 13](#thm:gen){reference-type="ref" reference="thm:gen"} for eigenvalues in gaps of the essential spectrum not containing zero. More precisely, instead of the interval $(\alpha,\beta)$ in the proof of Theorem [Theorem 13](#thm:gen){reference-type="ref" reference="thm:gen"}, we may consider any interval $(\tilde{\alpha},\tilde{\beta})$ belonging to the resolvent set of $H$ such that $\eta = \norm{P_{\mathcal U}- P}$ satisfies the (stronger) condition $$\eta < \frac{\tilde{\beta}-\tilde{\alpha}}{\abs{\tilde{\alpha}}+\abs{\tilde{\beta}}} .$$ The terms $(1-\eta)\alpha$, $(1-\eta)\beta$, and $(1-\eta)d$ in the proof then just have to be replaced by $\tilde{\alpha} + \eta\abs{\tilde{\alpha}}$, $\tilde{\beta} - \eta\abs{\tilde{\beta}}$, and $\tilde{d} - \eta\abs{\tilde{d}}$, respectively, where $\tilde{d}$ is given by $\tilde{d} = \inf(\mathop{\mathrm{\sigma}}_\mathrm{ess}(H) \cap (\tilde{\alpha},\infty)) \geq \tilde{\beta}$. Theorem [Theorem 13](#thm:gen){reference-type="ref" reference="thm:gen"} relies on the crucial condition $\norm{ P_{\mathcal U}- P } < 1$, so let us now address how the norm of $P_{\mathcal U}- P$ can be estimated. To this end, we may choose one of the alternative decompositions $$\label{eq:projDecomp} P_{\mathcal U}- P = ( P_{\mathcal U}- P_{\mathcal V}) + ( P_{\mathcal V}- P ) = ( P_{\mathcal U}- P_{\mathcal W}) + ( P_{\mathcal W}- P) .$$ Here, the terms $P_{\mathcal U}- P_{\mathcal V}$ and $P_{\mathcal U}- P_{\mathcal W}$ correspond to sines of the operator angles associated to the pairs of subspaces $({\mathcal U},{\mathcal V})$ and $({\mathcal U},{\mathcal W})$, respectively. More precisely, $$\label{eq:opAngle} \abs{P_{\mathcal U}- P_{\mathcal V}} = \sin\Theta({\mathcal U},{\mathcal V}) \quad\text{ and }\quad \abs{P_{\mathcal U}- P_{\mathcal W}} = \sin\Theta({\mathcal U},{\mathcal W}),$$ where $\Theta(\cdot,\cdot)$ denotes the operator angle associated with the respective subspaces, see, e.g., [@Seel14 Section 2] and the references cited therein for a discussion. In particular, the maximal angle introduced in Section [1](#sec:intro){reference-type="ref" reference="sec:intro"} satisfies $\theta(\cdot,\cdot) = \norm{\Theta(\cdot,\cdot)}$. In order to address the other two terms, $P_{\mathcal V}- P$ and $P_{\mathcal W}- P$, we make the following considerations: Since $\mathop{\mathrm{Ran}}(I_{\mathcal H}- P) = \mathop{\mathrm{Ker}}P = {\mathcal W}^\perp$, we obtain from $P + (I-P) = I$ that $P_{\mathcal W}P = P_{\mathcal W}$. Hence, the projection $P$ can be represented with respect to the orthogonal decomposition ${\mathcal W}\oplus {\mathcal W}^\perp$ as the $2\times 2$ block operator matrix $$\label{eq:P} P = \begin{pmatrix} I_{\mathcal W}& 0\\ X & 0 \end{pmatrix}$$ with $X := P_{\mathcal W}^\perp P|_{\mathcal W}$, interpreted as an operator from ${\mathcal W}$ to ${\mathcal W}^\perp$. In particular, ${\mathcal V}= \mathop{\mathrm{Ran}}P$ admits the graph subspace representation $$\label{eq:V} {\mathcal V}= \{ f \oplus Xf \colon f \in {\mathcal W}\}.$$ Recall from [@KMM03:181 Corollary 3.4 and Remark 3.6] that consequently we have $\norm{ P_{\mathcal W}- P_{\mathcal V}} < 1$ and that $X$ corresponds to the tangent of the operator angle associated to the subspaces ${\mathcal W}$ and ${\mathcal V}$, more precisely $$\label{eq:reprX} \begin{pmatrix} \abs{X} & 0\\ 0 & \abs{X^*} \end{pmatrix} = \tan\Theta({\mathcal W},{\mathcal V}) .$$ Moreover, we have $$\label{eq:PV} P_{\mathcal V}= U P_{\mathcal W}U^*,$$ where $U$ is the unitary operator given by the $2\times 2$ block operator matrix $$\label{eq:U} U = \begin{pmatrix} (I_{\mathcal W}+ X^*X)^{-1/2} & -X^*(I_{{\mathcal W}^\perp} + XX^*)^{-1/2}\\ X(I_{\mathcal W}+ X^*X)^{-1/2} & (I_{{\mathcal W}^\perp} + XX^*)^{-1/2} \end{pmatrix} .$$ A broader discussion on the operator angle and graph subspace representations can be found, for instance, in [@SeelDiss Sections 1.3 and 1.5] and the references cited therein. *Remark 16*. The inequality $\norm{ P_{\mathcal W}- P_{\mathcal V}} < 1$ can alternatively also be verified as follows: Since the projection $P$ onto ${\mathcal V}$ along ${\mathcal W}^\perp$ exists by Lemma [Lemma 10](#lem:projection){reference-type="ref" reference="lem:projection"}, Proposition 1.6 in [@BS10] yields that $\norm{ P_{\mathcal V}P_{\mathcal W}^\perp } < 1$. Taking into account that $P^*$ is the projection onto ${\mathcal W}$ along ${\mathcal V}^\perp$, we obtain in the same way that $\norm{ P_{\mathcal V}^\perp P_{\mathcal W}} = \norm{ P_{\mathcal W}P_{\mathcal V}^\perp } < 1$. Using $\norm{ P_{\mathcal W}- P_{\mathcal V}} = \max\{ \norm{P_{\mathcal V}P_{\mathcal W}^\perp} , \norm{P_{\mathcal V}^\perp P_{\mathcal W}} \}$, see, e.g., [@AG93 Section 34], this gives $\norm{ P_{\mathcal W}- P_{\mathcal V}} < 1$. **Lemma 17**. *With $X = P_{\mathcal W}^\perp P|_{\mathcal W}\colon {\mathcal W}\to {\mathcal W}^\perp$ and $U$ as in [\[eq:U\]](#eq:U){reference-type="eqref" reference="eq:U"} we have $$P_{\mathcal W}- P = \begin{pmatrix} 0 & 0\\ -X & 0 \end{pmatrix}$$ and $$P_{\mathcal V}- P = U \begin{pmatrix} 0 & X^*\\ 0 & 0 \end{pmatrix} U^*.$$* *Proof.* The representation for $P_{\mathcal W}- P$ follows directly from [\[eq:P\]](#eq:P){reference-type="eqref" reference="eq:P"}. Moreover, using the identity $X^*(I_{{\mathcal W}^\perp} + XX^*)^{-1/2} = (I_{\mathcal W}+ X^*X)^{-1/2}X^*$, the representation for $P_{\mathcal V}- P$ is verified from [\[eq:P\]](#eq:P){reference-type="eqref" reference="eq:P"}, [\[eq:PV\]](#eq:PV){reference-type="eqref" reference="eq:PV"}, and [\[eq:U\]](#eq:U){reference-type="eqref" reference="eq:U"} by plain multiplication of $2\times 2$ block operator matrices. ◻ We now arrive at the following result, the particular case of which where ${\mathcal U}$ has finite dimension agrees with Theorem [Theorem 4](#thm:angleFinite){reference-type="ref" reference="thm:angleFinite"}. **Theorem 18**. *We have $$\norm{P_{\mathcal U}- P} \leq \min\bigl\{ \sin\theta({\mathcal U},{\mathcal V}) , \sin\theta({\mathcal U},{\mathcal W}) \bigr\} + \tan\theta({\mathcal V},{\mathcal W}) .$$* *Proof.* From Lemma [Lemma 17](#lem:annular){reference-type="ref" reference="lem:annular"} and [\[eq:reprX\]](#eq:reprX){reference-type="eqref" reference="eq:reprX"} we obtain that $$\norm{ P_{\mathcal W}- P } = \norm{ P_{\mathcal V}- P } = \norm{ X } = \tan \theta({\mathcal V}, {\mathcal W}) ,$$ where for the last equality we used that $\norm{ \tan\Theta({\mathcal W}, {\mathcal V}) } = \tan \theta({\mathcal V}, {\mathcal W})$. Combining the latter with [\[eq:projDecomp\]](#eq:projDecomp){reference-type="eqref" reference="eq:projDecomp"} and [\[eq:opAngle\]](#eq:opAngle){reference-type="eqref" reference="eq:opAngle"} gives $$\begin{aligned} \norm{P_{\mathcal U}- P} &\leq \min\bigl\{ \norm{P_{\mathcal U}-P_{\mathcal V}} , \norm{P_{\mathcal U}-P_{\mathcal W}} \bigr\} + \norm{X}\\ &= \min\bigl\{ \sin\theta({\mathcal U},{\mathcal V}) , \sin\theta({\mathcal U},{\mathcal W}) \bigr\} + \tan\theta({\mathcal V},{\mathcal W}) , \end{aligned}$$ which proves the claim. ◻ # Acknowledgements {#acknowledgements .unnumbered} The author is grateful to Zlatko Drmač and Ivan Veselić for suggesting this research direction. He also thanks Krešimir Veselić for a helpful communication. \[10\] N. I. Akhiezer, I. M. Glazman, *Theory of Linear Operators in Hilbert Space*, Dover Publications, New York, 1993. 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A geometric approach*, In: Advances in Differential Equations and Mathematical Physics (Birmingham, AL, 2002), Contemp. Math., vol. 327, Amer. Math. Soc., Providence, RI, 2003, pp. 181--198. A. Seelmann, *Notes on the $\sin2\Theta$ theorem*, Integral Equations Operator Theory **79** (2014), 579--597. A. Seelmann, *Perturbation theory for spectral subspaces*, Dissertation, Johannes Gutenberg-Universität Mainz, 2014. K. Veselić, I. Slapničar, *Floating-point perturbations of Hermitian Matrices*, Linear Algebra Appl. **195** (1993), 81--116. K. A. Makarov, S. Schmitz, A. Seelmann, *On invariant graph subspaces*, Integral Equations Operator Theory **85** (2016), 399--425. L. Schimmer, J. P. Solovej, S. Tokus, *Friedrichs extension and min-max principle for operators with a gap*, Ann. Henri Poincaré **21** (2020), 327--357. C. Tretter, C. Wyss, *Dichotomous Hamiltonians with unbounded entries and solutions of Riccati equations*, J. Evol. Equ. **14** (2014), 121--153. 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arxiv_math

--- abstract: | We demonstrate how a generic automated theorem prover can be applied to establish the non-orderability of groups. Our approach incorporates various tools such as positive cones, torsions, generalised torsions and cofinal elements. author: - Alexei Lisitsa - Zipei Nie - Alexei Vernitski bibliography: - ref.bib title: Automated reasoning for proving non-orderability of groups --- # Introduction The study of orderable groups has a rich history [@kokorin1974fully; @mura1977orderable; @kopytov1996right; @glass1999partially] among group theorists. In the last decade of the 20th century, researchers gradually recognised the significance of orderability in topology, and it has since remained an active and thriving area of research. The three most extensively studied variants of group orderability are left order, bi-order, and (left-invariant) circular order. In this paper, we will also study the existence of bi-invariant circular orders. We consider the following algorithmic problem. [\[Problem\]]{#Problem label="Problem"} Given a group $G$ with presentation $\langle S|R \rangle$ and a type of group orderability, determine whether $G$ is orderable. In general, Problem [\[Problem\]](#Problem){reference-type="ref" reference="Problem"} has been proven to be undecidable for any type of orderability by [@baumslag2012algorithms Theorem 3.3]. Despite its undecidability, the problem remains highly intriguing when considering specific groups that arise in topology. Notably, the braid groups [@short1999orderings; @rolfsen1998braids], the mapping class groups [@rourke2000order; @hyde2019group], the fundamental groups of $3$-manifolds [@boyer2005orderable; @boyer2013spaces; @juhasz2015survey; @ito2016alexander], and lattices in Lie groups [@witte1994arithmetic] are among the groups where Problem [\[Problem\]](#Problem){reference-type="ref" reference="Problem"} holds great significance. We focus on the non-orderability aspect of Problem [\[Problem\]](#Problem){reference-type="ref" reference="Problem"}, which involves finding a contradiction assuming the existence of an order. We believe that, by providing simple proofs demonstrating the non-orderability of a group of interest, one may deepen our understanding and reveal intriguing topological structures. Most previous automated proofs of non-orderability are variants of the algorithm described in [@calegari2003laminations Section 8]. This algorithm relies on a short-lex automatic structure to tackle the word problem and seeks to identify contradictions within the positive cone of a left order. Later, Dunfield [@dunfield2020floer] enhanced this algorithm for the fundamental group of a finite-volume hyperbolic $3$-manifold by solving the word problem through an $SL_2(\mathbb{C})$ representation. We present a methodology for establishing non-orderability using generic automated theorem proving instead of specialized algorithms. In contrast to previous approaches, our method offers a unified framework capable of handling all variants of orderability without any assumptions about the group. The flexibility of the imposed assumptions makes it easier to discover new proofs and results in non-orderability. We provide many examples to illustrate our methodology, ranging in difficulty. A particularly interesting one is Example [\[ex:Hyde\]](#ex:Hyde){reference-type="ref" reference="ex:Hyde"}, where we provide an alternative proof of the non-left-orderability [@hyde2019group] of $\mathrm{Homeo}(D,\partial D)$, the group of homeomorphisms of the disk fixing the boundary, using the concept of left absolutely cofinal elements; see also [@triestino2021james]. ## Main results and organisation of the paper In Section [2](#sec:FP){reference-type="ref" reference="sec:FP"}, we present the first principles approach. As its name suggests, we directly break down the original problem into axioms in first-order logic, and then prove the non-orderability through the automatically-derived contradiction. To derive sufficient axioms from the group presentation, a necessary technical step is to identify inequalities using an automated finite model finder. When compared to specific algorithms, like the one described in [@calegari2003laminations Section 8], our approach offers a significant advantage: the ability to readily modify input axioms. We have noticed that the axiom of connectedness consumes excessive computational resources. Therefore, in Subsection [3.1](#subsec:weakened){reference-type="ref" reference="subsec:weakened"}, we introduce the weakened theory approach, in which we substitute the axiom of connectedness with weaker assumptions. In Subsection [3.2](#subsec:positive_cone){reference-type="ref" reference="subsec:positive_cone"}, we discuss a further optimisation of our approach: the positive cone translation. This method has been used in the literature for establishing non-left-orderability and non-bi-orderability. We extend its applicability to left-invariant or bi-invariant circular orders. From the perspective of automated reasoning, the positive cone translation results in an equiconsistent theory with predicates of smaller arities, thereby enhancing the efficiency of our reasoning process. In Section [4](#sec:torsions){reference-type="ref" reference="sec:torsions"}, we establish the equivalence between an element $g\in G$ being a torsion or a generalised torsion and the inconsistency of the corresponding weakened theory with respect to the pair $(e,g)$. One may compare this result with the well-known fact that the existence of a nontrivial torsion (resp. generalised torsion) implies non-left-orderability (resp. non-bi-orderability). Furthermore, we extend this analysis to bi-invariant circular orders. If the axiom of cyclicity is dropped, and the axiom of connectedness is weakened with respect to the triple $(e, fgf^{-1}, g)$, then the theory for bi-invariant circular orders is inconsistent if and only if $f^{-1}$ is in the monoid generated by $f$ and the centraliser of $g$. In Section [5](#sec:convexity){reference-type="ref" reference="sec:convexity"}, we further study the strength of the axiom of cyclicity for bi-invariant circular orders. For this purpose, we develop the theory of left relatively convex subgroups defined by Antolın, Dicks, and Sunic [@antolin2015left]. Unlike previous works such as [@kopytov1996right], their definition does not impose restrictions on the left-orderability of the ambient group. We extend some properties of relatively convex subgroups of left-orderable groups to general cases. 1. The equivalence (a) $\Leftrightarrow$ (c) in Proposition [\[prop-rel-convex-1\]](#prop-rel-convex-1){reference-type="ref" reference="prop-rel-convex-1"} generalises the usual definition of relatively convex subgroups of left-orderable groups. 2. The equivalence (a) $\Leftrightarrow$ (b) in Proposition [\[prop-rel-convex-2\]](#prop-rel-convex-2){reference-type="ref" reference="prop-rel-convex-2"} generalises [@clay2010space Theorem 1.4.10]. 3. Proposition [\[convex-intersection\]](#convex-intersection){reference-type="ref" reference="convex-intersection"} generalises [@kopytov1996right Proposition 5.1.10], establishing the closure property under arbitrary intersection. Using the theory of left relatively convex subgroups, we demonstrate that a relation on the group $G$ satisfying all axioms except cyclicity exists if and only if the centraliser of each subset of $G$ is left relatively convex in $G$. In Section [6](#sec:cofinality){reference-type="ref" reference="sec:cofinality"}, we introduce the concept of the left relatively convex subgroup closure, denoted as $\mathrm{cl}(A)$, for a subset $A$. This closure is defined as the intersection of all left relatively convex subgroups containing $A$. Additionally, we define the left absolute cofinal subgroup as a subgroup $H$ for which $\mathrm{cl}(H)=G$. By the equivalence (a) $\Leftrightarrow$ (c) established in Proposition [\[prop-equiv-cofinal\]](#prop-equiv-cofinal){reference-type="ref" reference="prop-equiv-cofinal"}, a subgroup is left absolutely cofinal if and only if it is cofinal with respect to every left total preorder. Based on the search for left absolute cofinal cyclic subgroups, we introduce a methodology for establishing the non-left-orderability of every nontrivial quotient using automated theorem proving. Our approach formalises a common practice in establishing the non-left-orderability, which involves tracking fixed points in the dynamical realisation. We believe this formalisation could also be used in other scenarios, including the development of fast algorithms such as the one detailed in [@calegari2003laminations Section 8], as well as in human proofs. Propositions [\[prop:main\]](#prop:main){reference-type="ref" reference="prop:main"}, [\[prop:weak\]](#prop:weak){reference-type="ref" reference="prop:weak"}, [\[weak-cbo\]](#weak-cbo){reference-type="ref" reference="weak-cbo"}, [\[symmetry\]](#symmetry){reference-type="ref" reference="symmetry"}, [\[prop:positive_cone\]](#prop:positive_cone){reference-type="ref" reference="prop:positive_cone"}, [\[prop:positive_cone_C\]](#prop:positive_cone_C){reference-type="ref" reference="prop:positive_cone_C"}, [\[prop-torsion\]](#prop-torsion){reference-type="ref" reference="prop-torsion"}, [\[prop-generalised-torsion\]](#prop-generalised-torsion){reference-type="ref" reference="prop-generalised-torsion"}, [\[prop-CBO-monoid\]](#prop-CBO-monoid){reference-type="ref" reference="prop-CBO-monoid"}, [\[prop-cl\]](#prop-cl){reference-type="ref" reference="prop-cl"}, [\[prop-fixed-point\]](#prop-fixed-point){reference-type="ref" reference="prop-fixed-point"} establish the group theoretic implications of the consistency or inconsistency of various theories. The utilisation of an automated theorem prover like Prover9 enables the automatic detection of contradictions in a first-order theory, thereby implying corresponding group theoretic properties. We provide numerous examples at the end of each section to illustrate our methodology. In Section [7](#sec:tasks){reference-type="ref" reference="sec:tasks"}, we elaborate our approach for performing automated reasoning tasks in these examples using Prover9 and Mace4. Furthermore, we provide a summary of the performance of these automated theorem provers when executing these tasks. # Non-orderability from first principles {#sec:FP} In this section, we demonstrate a methodology for reducing the non-orderability of a group $G$ with presentation $\langle S|R\rangle$ to automated theorem proving tasks. Specifically, we address the following variant of Problem [\[Problem\]](#Problem){reference-type="ref" reference="Problem"}. [\[Problem\'\]]{#Problem' label="Problem'"} Given a group $G$ with presentation $\langle S|R \rangle$ and a type of group orderability, provide an automated proof of the non-orderability of $G$. To achieve this goal, we first extract the assumptions of our problem setting in first-order logic. Then we utilise two tools for experimental evaluation: Prover9 and Mace4 [@prover9-mace4]. Prover9 is a widely used automated theorem prover that operates on first-order logic. It takes a set of logical axioms as input and attempts to find a proof of a conjecture. And the finite model finder Mace4 is designed to search for finite models that satisfy a given set of first-order logical formulas. ## Group axioms {#subsec:group_axioms} A group is a set together with an associative binary operation $\cdot$, such that there exists an identity element, and every element has an inverse. In a first-order logic perspective, we consider the groups as models for the following standard system of axioms $\mathbf{Gr}$ in a vocabulary consisting a binary functional symbol $\cdot$ for group multiplication, a unary functional symbol $'$ (in postfix notation) for group inverse operation, and a constant $e$ for the identity element in the group: 1. $\forall x \forall y \forall z((x \cdot y) \cdot z = x \cdot (y \cdot z))$,*(associativity)* 2. $\forall x(x \cdot e = e \cdot x = x)$,*(identity element)* 3. $\forall x(x'\cdot x = x \cdot x' = e)$.*(inverse element)* For the group with presentation $\langle S|R\rangle$, we encode every generator in $S$ as an additional constant, and every relation in $R$ as an additional equational axiom, in a standard way. We denote this system of axioms by $\mathbf{Ax}_R$. In addition to the group $G$ presented by $\langle S|R\rangle$, any quotient group of $G$ is also a model of $\mathbf{Gr}\cup \mathbf{Ax}_R$. To establish non-orderability, it is crucial to distinguish between $G$ and its quotient groups. Therefore, we need to include an additional situation-dependent set $\mathfrak{S}$ of true statements for the group $G$. In practice, $\mathfrak{S}$ can be taken as a set of inequalities in $G$. ## Order axioms {#subsec:order_axioms} By definition, a *left order* on a group $G$ is a linear order $<$ on $G$ that is invariant under left multiplication, and a *bi-order* is one that is invariant under left and right multiplication. Thus the left order and bi-order satisfy the following axioms $\mathbf{AxL}$ for linear orders: 1. $\forall x (\neg (x < x))$, *(irreflexivity)* 2. $\forall x \forall y \forall z((x < y) \land (y < z) \to (x < z))$, *(transitivity)* 3. $\forall x \forall y ((x=y) \lor (x < y)\lor(y<x))$. *(connectedness)* In addition, the left order satisfies the left-invariance axiom $\mathbf{OrdL}$: 1. $\forall x \forall y \forall z ((x<y) \to (z\cdot x < z\cdot y))$. *(left-invariance)* And the bi-order satisfies the bi-invariance axiom $\mathbf{OrdB}$: 1. $\forall x \forall y \forall z \forall u ((x<y) \to ((z\cdot x)\cdot u < (z\cdot y)\cdot u))$. *(bi-invariance)* By definition, a *cyclic order* on a set is a ternary relation $C(\cdot,\cdot,\cdot)$ satisfying the following axioms $\mathbf{AxC}$: 1. $\forall x \forall y \forall z(C(x,y,z) \to C(y,z,x))$, *(cyclicity)* 2. $\forall x \forall y (\neg C(x, y, y))$, *(irreflexivity)* 3. $\forall x \forall y\forall z \forall u (C(x,y,z)\land C(x,z,u) \to C(x,y,u))$, *(transitivity)* 4. $\forall x \forall y\forall z ((x=y)\lor(y=z)\lor(z=x)\lor C(x,y,z)\lor C(x,z,y))$. *(connectedness)* When referring to a *circular order* on a group $G$, it is conventionally understood as a cyclic order on the elements of $G$ that is invariant under left multiplication. So the circular order also satisfies the left-invariance axiom $\mathbf{OrdCL}$: 1. $\forall x \forall y \forall z\forall u (C(x,y,z) \to C(u\cdot x ,u\cdot y,u\cdot z))$. *(left-invariance)* And the *bi-invariant circular order* satisfies the bi-invariance axiom $\mathbf{OrdCB}$: 1. $\forall x \forall y \forall z \forall u\forall v (C(x,y,z) \to C((u\cdot x)\cdot v ,(u\cdot y)\cdot v,(u\cdot z)\cdot v))$. *(bi-invariance)* ## The first principles approach If a group is orderable, then with the corresponding order it would constitute a model for the first-order theory with applicable axioms from Subsection [2.1](#subsec:group_axioms){reference-type="ref" reference="subsec:group_axioms"} and Subsection [2.2](#subsec:order_axioms){reference-type="ref" reference="subsec:order_axioms"}, which in turn would entail that the theory is consistent. In other words, we have the following proposition. [\[prop:main\]]{#prop:main label="prop:main"} Let $G$ be a group with presentation $\langle S|R\rangle$. Let $\mathfrak{S}$ be a set of true statements for $G$. Then the following statements hold: 1. If $\mathbf{Gr}\cup \mathbf{Ax}_R\cup \mathbf{AxL}\cup \mathbf{OrdL}\cup \mathfrak{S}$ is inconsistent, then $G$ is not left-orderable. 2. If $\mathbf{Gr}\cup \mathbf{Ax}_R\cup\mathbf{AxL}\cup \mathbf{OrdB}\cup \mathfrak{S}$ is inconsistent, then $G$ is not bi-orderable. 3. If $\mathbf{Gr}\cup \mathbf{Ax}_R\cup \mathbf{AxC}\cup \mathbf{OrdCL}\cup \mathfrak{S}$ is inconsistent, then $G$ is not circularly orderable. 4. If $\mathbf{Gr}\cup \mathbf{Ax}_R\cup\mathbf{AxC}\cup \mathbf{OrdCB}\cup \mathfrak{S}$ is inconsistent, then $G$ does not admit a bi-invariant circular order. According to Proposition [\[prop:main\]](#prop:main){reference-type="ref" reference="prop:main"}, given the group presentation, to establish non-orderability of the presented group, one can apply an automated theorem prover in the first-order logic to derive a contradiction from the corresponding theory. Let us outline some important observations: 1. The proposed approach is not fully automatic. It requires a set of true statements $\mathfrak{S}$ for the group $G$ to be provided before attempting a proof. In the next subsection, we will discuss how to establish such statements using a finite model finder. 2. The proposed approach is limited to establishing the non-orderability of groups. While many groups are orderable, proving it involves second-order reasoning that includes the quantifier "there exists an order". Alternatively, it could be handled by inductive reasoning. Both directions of automation appear to be promising areas for further research, but they are beyond the scope of this paper. 3. An application of the automated reasoning to orderability can be found in [@wehrung2021]. In that work, finite model building was used to establish the orderability of finite monoids. However, this approach cannot be applied to show the left-orderability and bi-orderability of infinite groups. ## Finite models as a source of true statements As we have already noticed, the proposed methodology is not complete and is not fully automated. The choice of a set of true statements (inequalities) $\mathfrak{S}$ in a group of interest remains a crucial and, generally, creative step. In applications one may use any known equationally expressible property of the group, such as non-commutativity. We propose here a partial automation of the search for true statements using an automated reasoning technique, *finite model finding*. For a $G$ presented by $\langle S|R\rangle$, a model of $\mathbf{Gr}\cup \mathbf{Ax}_R$ can be viewed as a quotient group of $G$. Hence any inequality $t_{1} \neq t_{2}$ among ground terms $t_{1}$ and $t_{2}$ which holds true in a model of $\mathbf{Gr}\cup \mathbf{Ax}_R$ also holds true in $G$. The proposed approach then works as follows: for a group presentation $\langle S|R\rangle$, we search for finite models of $\mathbf{Gr}\cup \mathbf{Ax}_R$ using an automated finite model finder tool, such as Mace4. If a finite model $G'$ is discovered, we can take any subset of ground inequalities true in $G'$ as a set of true statements $\mathfrak{S}$ in $G$. If a group of interest $G$ does not have nontrivial finite quotients, then the finite model finder will never find a useful model. Hence this approach is incomplete. Empirically, it has been effective in many, though not all, of our experiments. ## Examples Now we show how the proposed first principles approach works on some simple examples. The fundamental group of Klein bottle has a presentation $$\langle a, b\; |\; a^{-1}ba = b^{-1} \rangle.$$ This group is known to be left-orderable, but not bi-orderable [@boyer2005orderable]. One can prove that the group is not bi-orderable by noticing that one has to have both $b < e$ and $e < b$, which is impossible. This argument has to be complemented by a proof of the fact that $b\neq e$. If $b=e$ in the group, then $a$ becomes a generator. Thus it suffices to prove that the Klein bottle group is not cyclic. If we want to apply automated reasoning, the corresponding theory $\mathbf{Gr}\cup \mathbf{Ax}_R\cup\mathbf{AxL}\cup \mathbf{OrdB}$ can be formulated as follows in the syntax of Prover9: % Gr % Ax_R (x * y) * z = x * (y * z). (a' * b) * a = b'. x * e = x. e * x = x. x' * x = e. x * x' = e. % AxL % OrdB - L(x,x). L(x,y) -> L((z*x)*u,(z*y)*u). L(x,y) & L(y,z) -> L(x,z). (x=y) | L(x,y) | L(y,x). One can notice that it is impossible to prove contradiction from such a theory because it is consistent and has a one-element model, namely, the trivial group. In order to get a contradiction, one needs to add some true statements in the group ($\mathfrak{S}$ in Proposition [\[prop:main\]](#prop:main){reference-type="ref" reference="prop:main"}) to the theory. In this example, $b\neq e$ is sufficient. To prove $b\neq e$ in the Klein bottle group automatically, we use the finite model building technique introduced in the previous subsection. When asked if the theory $\mathbf{Gr}\cup \mathbf{Ax}_{R} \cup \{b\neq e\}$ has a model (Task 1.1), the model builder Mace4 produces a model of size $2$; see Table [2](#tab:summary2){reference-type="ref" reference="tab:summary2"}. Therefore, the inequality $b\neq e$ is confirmed in the Klein bottle group. We can use Prover9 to prove contradiction (Task 1.2) from the theory $\mathbf{Gr}\cup \mathbf{Ax}_R\cup\mathbf{AxL}\cup \mathbf{OrdB}\cup\{b\neq e\}$. Note that Prover9 rediscovers the human-authored proof given above, finding a contradiction after deriving both $L(b,e)$ and $L(e,b)$.  [\[ex:sl2\]]{#ex:sl2 label="ex:sl2"} Special linear group $SL_{2}(\mathbb{Z})$ has a presentation $$\langle a,b \;|\; a^{4}=e, (ba)^{3} = b^{2} \rangle$$ and is known to be non-left-orderable because $a$ is a nontrivial torsion. We prove this fact automatically. First we validate that both $a\neq e$ and $b\neq e$ hold true by finding models (Task 2.1) for the theory $\mathbf{Gr}\cup \mathbf{Ax}_R\cup \{a\neq e, b\neq e\}$ using Mace4. Then by adding $\{a\neq e, b\neq e\}$ to the theory $\mathbf{Gr}\cup \mathbf{Ax}_R\cup \mathbf{AxL}\cup \mathbf{OrdL}$, we derive contradiction (Task 2.2) using Prover9. [\[ex:Fibonacci\]]{#ex:Fibonacci label="ex:Fibonacci"} The $n$-th Fibonacci group $F(2,n)$ ($n\ge 2$) has a presentation $$\langle a_0,\ldots, a_{n-1} \;|\; a_i a_{i+1}= a_{i+2} \mbox{ for } i=0,\ldots, n-1\rangle.$$ When $n$ is odd, this group contains a nontrivial torsion by [@bardakov2003generalization Proposition 3.1], which implies the non-left-orderability. When $n$ is even, this group is the fundamental group of a cyclic branched cover of the figure-eight knot by [@hilden1992arithmeticity Theorem 1], and is not left-orderable by [@dkabkowski2005non Theorem 2]. While our methods cannot automatically verify the non-left-orderability for every positive integer $n\ge 2$, we can apply them to some relatively large integers. These instances could serve as inspiration for mathematicians to establish non-left-orderability in general cases. Suppose that $n=12$. Define the set of inequalities $\mathfrak{S}$ by $$\mathfrak{S}:=\{a_i\neq e : i=0,\ldots,n-1\}.$$ Using Mace4, we prove that $\mathbf{Gr}\cup \mathbf{Ax}_R\cup \mathfrak{S}$ is consistent (Task 3.1). Using Prover9, we prove that $\mathbf{Gr}\cup \mathbf{Ax}_R\cup \mathbf{AxL}\cup \mathbf{OrdL}\cup \mathfrak{S}$ is inconsistent (Task 3.2). Thus the Fibonacci group $F(2,12)$ is not left-orderable. When $n=11$, the inconsistency of $\mathbf{Gr}\cup \mathbf{Ax}_R\cup \mathbf{AxL}\cup \mathbf{OrdL}\cup \mathfrak{S}$ can still be verified (Task 3.3) using Prover9. However, it is difficult to find a finite model of $\mathbf{Gr}\cup \mathbf{Ax}_R\cup\mathfrak{S}$ using Mace4. To deal with the computational challenge, one approach is to manually deduce these inequalities based on the fact that $F(2,11)$ is infinite [@chalk1998fibonacci]. [\[ex:B3\]]{#ex:B3 label="ex:B3"} The braid group $B_3$ on three strands is isomorphic to the knot group of the trefoil knot $T_{2,3}$, and have a presentation $$\langle a,b\;|\; aba=bab \rangle.$$ This group does not admit bi-invariant circular orders, according to [@ba2023knot Corollary 8.8]. An alternative proof follows from the left-orderability (thus they are torsion-free) and the non-bi-orderability of braid groups, as well as knot groups of nontrivial torus knots. According to [@zheleva1976cyclically Proposition 3], a torsion-free group admits a bi-invariant circular order if and only if it is bi-orderable. To prove this fact automatically, we define the set of inequalities $\mathfrak{S}$ by $$\mathfrak{S}:=\{a\neq e,b\neq e,a \neq b)\}.$$ Using Mace4, we prove that $\mathbf{Gr}\cup \mathbf{Ax}_R\cup \mathfrak{S}$ is consistent (Task 4.1). Using Prover9, we prove that $\mathbf{Gr}\cup \mathbf{Ax}_R\cup \mathbf{AxC}\cup \mathbf{OrdCB}\cup \mathfrak{S}$ is inconsistent (Task 4.2). Thus the group $B_3$ does not admit bi-invariant circular orders. [\[ex:D7\]]{#ex:D7 label="ex:D7"} To illustrate how our approach can be applied to circular non-orderability, consider the dihedral group $D_{7}$ of order $14$ given by the presentation $$\langle a, b\; |\; a^{7} =e, b^{2} =e, bab=a^{-1} \rangle.$$ This group is finite and non-cyclic, thus by [@zheleva1976cyclically Theorem 1], it is not circularly orderable. We prove this fact directly using automated reasoning. Define the set of inequalities $\mathfrak{S}$ by $$\mathfrak{S}:= \{a\neq e, b\neq e, a\neq b\}.$$ Using Mace4, we prove that $\mathbf{Gr}\cup \mathbf{Ax}_R\cup \mathfrak{S}$ is consistent (Task 5.1). Using Prover9, we prove that $\mathbf{Gr}\cup \mathbf{Ax}_R\cup \mathbf{AxC}\cup \mathbf{OrdCL}\cup \mathfrak{S}$ is inconsistent (Task 5.2). Thus the group $D_{7}$ is not circularly orderable. Note that this example already takes much longer to compute than the previous ones; see Table [1](#tab:summary1){reference-type="ref" reference="tab:summary1"}. To improve the applicability of our approach and handle more complicated examples, we present some methods in the subsequent section to deal with the computational challenge. # Weakened theories and positive cones ## Weakened theories {#subsec:weakened} Establishing the non-orderability using automated theorem provers is inherently incomplete methodology. It may fail for various reasons: 1. If the orderability status of a group is unknown, it may turned out to be orderable. In the case it is not possible to derive contradiction. 2. The group may turned out to be non-orderable, but a supplied set of true statements is not sufficient to derive contradiction. 3. The supplied set of true statements may be sufficient to derive a contradiction, but it takes too long to find a proof automatically. In practice, it is difficult to distinguish between these alternatives. Hence to improve utility of the methodology one needs to consider possible optimisations to improve efficiency of the proof search. One of possible optimisation is based on using weaker theory to derive contradictions. For example, in a typical derivation of contradiction in the'first principle approach the axiom 1. $\forall x \forall y ((x=y) \lor (x < y)\lor(y<x))$ *(connectedness)* in $\mathbf{AxL}$ can be used with a supplied inequality $t_{1} \neq t_{2}$ to derive $(t_{1} < t_{2}) \lor (t_{2} < t_{1}).$ So, the search space for proof can potentially be reduced by removing this axiom and replacing supplied inequality $t_{1} \neq t_{2}$ with $(t_{1} < t_{2}) \lor (t_{2} < t_{1}).$ In summary, we have the following proposition. [\[prop:weak\]]{#prop:weak label="prop:weak"} Let $G$ be a group with presentation $\langle S|R\rangle$. Let $\mathfrak{S}$ be a set of true statements for $G$. Let $\mathfrak{S}^{<}$ be a set of formulas obtained by replacing all inequalities $t_{1} \neq t_{2}$ from $\mathfrak{S}$ by the corresponding formulas $(t_{1} < t_{2}) \lor (t_{2} < t_{1})$. Let $\mathbf{AxL}'$ denote $\mathbf{AxL}$ minus the axiom of connectedness. Then the following statements hold: 1. If $\mathbf{Gr}\cup \mathbf{Ax}_R\cup \mathbf{AxL}'\cup \mathbf{OrdL}\cup \mathfrak{S}^<$ is inconsistent, then $G$ is not left-orderable. 2. If $\mathbf{Gr}\cup \mathbf{Ax}_R\cup\mathbf{AxL}' \cup \mathbf{OrdB}\cup \mathfrak{S}^<$ is inconsistent, then $G$ is not bi-orderable. One can derive similar results for circular orders and bi-invariant circular orders. [\[weak-cbo\]]{#weak-cbo label="weak-cbo"} Let $G$ be a group with presentation $\langle S|R\rangle$. Let $\mathfrak{S}$ be a set of true statements for $G$. Let $\mathfrak{S}^{C}$ be a set of formulas obtained by replacing each triple of inequalities $\{ t_{1} \neq t_{2}, t_{2} \neq t_{3}), t_{3} \neq t_{1}\}$ from $\mathfrak{S}$ by the corresponding formula $C(t_1, t_2,t_3) \lor C(t_1,t_3,t_2)$. Let $\mathbf{AxC}'$ denote $\mathbf{AxC}$ minus the axiom of connectedness. Then the following statements hold: 1. If $\mathbf{Gr}\cup \mathbf{Ax}_R\cup \mathbf{AxC}'\cup \mathbf{OrdCL}\cup \mathfrak{S}^C$ is inconsistent, then $G$ is not circularly orderable. 2. If $\mathbf{Gr}\cup \mathbf{Ax}_R\cup\mathbf{AxC}' \cup \mathbf{OrdCB}\cup \mathfrak{S}^C$ is inconsistent, then $G$ does not admit a bi-invariant circular order. To improve the efficiency further, we can strengthen the assumption $(t_{1} < t_{2}) \lor (t_{2} < t_{1})$ in $\mathfrak{S}^<$ to $t_{1} < t_{2}$, or strengthen the assumption $C(t_1, t_2,t_3) \lor C(t_1,t_3,t_2)$ in $\mathfrak{S}^C$ to $C(t_1, t_2,t_3)$. By imposing stronger assumptions, we are testing the existence of an order satisfying extra formulas. A contradiction of a strengthened theory leads to a partial result on non-orderability. However, the first strengthening is free by symmetry. [\[symmetry\]]{#symmetry label="symmetry"} If we strengthen one formula of form $(t_{1} < t_{2}) \lor (t_{2} < t_{1})$ in $\mathfrak{S}^<$ to $t_{1} < t_{2}$, or of form $C(t_1, t_2,t_3) \lor C(t_1,t_3,t_2)$ in $\mathfrak{S}^C$ to $C(t_1, t_2,t_3)$, then the conclusions in Proposition [\[prop:weak\]](#prop:weak){reference-type="ref" reference="prop:weak"} and Proposition [\[weak-cbo\]](#weak-cbo){reference-type="ref" reference="weak-cbo"} still hold. *Proof.* For each linear order $<$, we say $x<_{op}y$ if and only if $y<x$. Then all axioms in $\mathbf{AxL}$, $\mathbf{OrdL}$, and $\mathbf{OrdB}$ hold invariant when replacing $<$ with $<_{op}$. Thus if $<$ is a left order (resp., a bi-order), then $<_{op}$ is also a left order (resp., a bi-order). Either $<$ or $<_{op}$ satisfies the formula $t_1<t_2$. For a cyclic order $C$, we say $C_{op}(x,y,z)$ if and only if $C(x,z,y)$. Then all axioms in $\mathbf{AxC}$, $\mathbf{OrdCL}$, and $\mathbf{OrdCB}$ hold invariant when replacing $C$ with $C_{op}$. Thus if $<$ is a circular order (resp., a bi-invariant circular order), then $<_{op}$ is also a circular order (resp., a bi-invariant circular order). Either $C$ or $C_{op}$ satisfies the formula $C(t_1, t_2,t_3)$. ◻ ## Positive cones {#subsec:positive_cone} In this subsection, we incorporate a well-known technique in the theory of ordered groups into our methodology: the positive cone technique. A positive cone of an order $<$ on the group $G$ is defined as the set of all positive elements. In other words, it is the set $\{x\in G: e < x\}$. This concept is particularly useful for left orders and bi-orders, as the positive cone determines the left order or the bi-order. According to the left-invariance, we can show that $x<y$ if and only if $x^{-1}y$ is in the positive cone. By translating the axioms for orders to the corresponding axioms for positive cones, we can gain a computational advantage, as it reduces a binary predicate to a unary one. The positive cone of a left order or a bi-order satisfies the following axioms, which we denote by $\mathbf{AxPL}$: 1. $\neg P(e)$, *(irreflexivity)* 2. $\forall x \forall y (P(x) \land P(y) \to P(x\cdot y)$), *(closure)* 3. $\forall x ((x=e) \lor P(x)\lor P(x'))$. *(connectedness)* Additionally, the positive cone of a bi-order satisfies the conjugacy invariance axiom $\mathbf{PB}$: 1. $\forall x \forall y (P(x) \to P((y\cdot x)\cdot y' ) )$. *(conjugacy invariance)* We can generalise the positive cone method to circular orders. We define the positive cone of a circular order $C$ on the group $G$ as the set $\{(x,y)\in G: C(e,x,y)\}$. Because $C(x,y,z)$ if and only if $(x^{-1}y,x^{-1}z)$ is in the positive cone, the positive cone determines the circular order. Thus we can translate the axioms for circular orders and bi-invariant circular orders to the axioms for their positive cones as follows. The positive cone of a circular order satisfies the following axioms, which we denote by $\mathbf{AxPCL}$: 1. $\forall x \forall y (P(x,y) \to P(x'\cdot y,x'))$, *(cyclicity)* 2. $\forall x (\neg P(x, x))$, *(irreflexivity)* 3. $\forall x \forall y\forall z (P(x,y)\land P(y,z) \to P(x,z))$, *(transitivity)* 4. $\forall x \forall y ((e=x)\lor(e=y)\lor(x=y)\lor P(x,y)\lor P(y,x))$. *(connectedness)* Additionally, the positive cone of a bi-invariant circular order satisfies the conjugacy invariance axiom $\mathbf{PCB}$: 1. $\forall x \forall y\forall z (P(x,y) \to P((z \cdot x)\cdot z' ),(z \cdot y)\cdot z' )$. *(conjugacy invariance)* We encourage the readers to verify the following lemmas. [\[lem:1\]]{#lem:1 label="lem:1"} Assume the axioms in $\mathbf{Gr}$ holds for binary function $\cdot$, unary function $'$ and constant $e$. Let $<$ be a binary predicate satisfying $\mathbf{OrdL}$ or $\mathbf{OrdB}$. Define a unary predicate $P$ by $P(x)$ if and only if $e<x$. Then 1. the irreflexivity axiom in $\mathbf{AxL}$ implies the irreflexivity axiom in $\mathbf{AxPL}$; 2. the transitivity axiom in $\mathbf{AxL}$ implies the closure axiom in $\mathbf{AxPL}$; 3. the connectedness axiom in $\mathbf{AxL}$ implies the connectedness axiom in $\mathbf{AxPL}$; 4. the axiom $\mathbf{OrdB}$ implies the axiom $\mathbf{PB}$. [\[lem:2\]]{#lem:2 label="lem:2"} Assume the axioms in $\mathbf{Gr}$ holds for binary function $\cdot$, unary function $'$ and constant $e$. Let $P$ be a unary predicate. Define a binary predicate $<$ by $x<y$ if and only if $P(x' \cdot y)$. Then 1. the axiom $\mathbf{OrdL}$ holds; 2. the irreflexivity axiom in $\mathbf{AxPL}$ implies the irreflexivity axiom in $\mathbf{AxL}$; 3. the closure axiom in $\mathbf{AxPL}$ implies the transitivity axiom in $\mathbf{AxL}$; 4. the connectedness axiom in $\mathbf{AxPL}$ implies the connectedness axiom in $\mathbf{AxL}$; 5. the axiom $\mathbf{PB}$ implies the axiom $\mathbf{OrdB}$. [\[lem:3\]]{#lem:3 label="lem:3"} Assume the axioms in $\mathbf{Gr}$ holds for binary function $\cdot$, unary function $'$ and constant $e$. Let $C$ be a ternary predicate satisfying $\mathbf{OrdCL}$ or $\mathbf{OrdCB}$. Define a binary predicate $P$ by $P(x,y)$ if and only if $C(e,x,y)$. Then 1. the cyclicity axiom in $\mathbf{AxC}$ implies the cyclicity axiom in $\mathbf{AxPCL}$; 2. the irreflexivity axiom in $\mathbf{AxC}$ implies the irreflexivity axiom in $\mathbf{AxPCL}$; 3. the transitivity axiom in $\mathbf{AxC}$ implies the transitivity axiom in $\mathbf{AxPCL}$; 4. the connectedness axiom in $\mathbf{AxC}$ implies the connectedness axiom in $\mathbf{AxPCL}$; 5. the axiom $\mathbf{OrdCB}$ implies the axiom $\mathbf{PCB}$. [\[lem:4\]]{#lem:4 label="lem:4"} Assume the axioms in $\mathbf{Gr}$ holds for binary function $\cdot$, unary function $'$ and constant $e$. Let $P$ be a binary predicate. Define a ternary predicate $C$ by $C(x,y,z)$ if and only if $P(x'\cdot y,x'\cdot z)$. Then 1. the axiom $\mathbf{OrdCL}$ holds; 2. the cyclicity axiom in $\mathbf{AxPCL}$ implies the cyclicity axiom in $\mathbf{AxC}$; 3. the irreflexivity axiom in $\mathbf{AxPCL}$ implies the irreflexivity axiom in $\mathbf{AxC}$; 4. the transitivity axiom in $\mathbf{AxPCL}$ implies the transitivity axiom in $\mathbf{AxC}$; 5. the connectedness axiom in $\mathbf{AxPCL}$ implies the connectedness axiom in $\mathbf{AxC}$; 6. the axiom $\mathbf{PCB}$ implies the axiom $\mathbf{OrdCB}$. By replacing $\mathbf{AxL}\cup\mathbf{OrdL}$ with $\mathbf{AxPL}$, replacing $\mathbf{AxL}\cup\mathbf{OrdB}$ with $\mathbf{AxPL}\cup \mathbf{PB}$, replacing $\mathbf{AxC}\cup \mathbf{OrdCL}$ with $\mathbf{AxPCL}$, and replacing $\mathbf{AxC}\cup \mathbf{OrdCB}$ with $\mathbf{AxPCL}\cup \mathbf{PCB}$, we can translate Proposition [\[prop:main\]](#prop:main){reference-type="ref" reference="prop:main"}, Proposition [\[prop:weak\]](#prop:weak){reference-type="ref" reference="prop:weak"}, Proposition [\[weak-cbo\]](#weak-cbo){reference-type="ref" reference="weak-cbo"}, and Proposition [\[symmetry\]](#symmetry){reference-type="ref" reference="symmetry"} into positive cone forms. We prove that these positive cone translations lead to *equiconsistent* theories. [\[prop:positive_cone\]]{#prop:positive_cone label="prop:positive_cone"} Let $s_i$, $t_i$ $(i=0,1,\ldots,k)$ be ground terms. Let $\mathfrak{S}$ denote the set of inequalities $$\{s_i\neq t_i: i=0,1,\ldots,k\}.$$ Let $\mathfrak{S}^<$ denote the set of axioms $$\{s_0< t_0\}\cup\{(s_i<t_i)\lor (t_i<s_i): i=1,\ldots, k\}.$$ Let $\mathfrak{S}^P$ denote the set of axioms $$\{P(s_0' \cdot t_0)\}\cup\{P(s_i'\cdot t_i)\lor P(t_i'\cdot s_i): i=1,\ldots, k\}.$$ Let $\mathbf{AxPL}'$ (resp. $\mathbf{AxL}'$) denote $\mathbf{AxPL}$ (resp. $\mathbf{AxL}$) minus the axiom of connectedness. Then 1. $\mathbf{Gr}\cup \mathbf{Ax}_R\cup \mathbf{AxL}\cup \mathbf{OrdL}\cup \mathfrak{S}$ is consistent if and only if $\mathbf{Gr}\cup \mathbf{Ax}_R\cup \mathbf{AxPL}\cup \mathfrak{S}$ is consistent; 2. $\mathbf{Gr}\cup \mathbf{Ax}_R\cup\mathbf{AxL}\cup \mathbf{OrdB}\cup \mathfrak{S}$ is consistent if and only if $\mathbf{Gr}\cup \mathbf{Ax}_R\cup \mathbf{AxPL}\cup \mathbf{PB}\cup \mathfrak{S}$ is consistent; 3. $\mathbf{Gr}\cup \mathbf{Ax}_R\cup \mathbf{AxL}'\cup \mathbf{OrdL}\cup \mathfrak{S}^<$ is consistent if and only if $\mathbf{Gr}\cup \mathbf{Ax}_R\cup \mathbf{AxPL}'\cup \mathfrak{S}^P$ is consistent; 4. $\mathbf{Gr}\cup \mathbf{Ax}_R\cup\mathbf{AxL}' \cup \mathbf{OrdB}\cup \mathfrak{S}^<$ is consistent if and only if $\mathbf{Gr}\cup \mathbf{Ax}_R\cup \mathbf{AxPL}'\cup \mathbf{PB}\cup \mathfrak{S}^P$ is consistent. *Proof.* If a theory $T_1$ involving $<$ (in either of the four cases) is consistent, then there is a model $\mathfrak{M}_1$ of it. We extend $\mathfrak{M}_1$ by the interpretation of $P$, defined as $P(x)$ if and only if $e<x$, then remove the interpretation of $<$. Then by Lemma [\[lem:1\]](#lem:1){reference-type="ref" reference="lem:1"}, the resulting model $\mathfrak{M}_2$ is a model of the corresponding theory $T_2$ involving $P$. Conversely, if a theory $T_2$ involving $P$ is consistent, then there is a model $\mathfrak{M}_2$ of it. We interpret $<$ by $x<y$ if and only if $P(x'\cdot y)$ and then remove the interpretation of $P$. Then by Lemma [\[lem:2\]](#lem:2){reference-type="ref" reference="lem:2"}, the resulting model $\mathfrak{M}_1$ is a model of the corresponding theory $T_1$ involving $<$. ◻ Similarly, by Lemma [\[lem:3\]](#lem:3){reference-type="ref" reference="lem:3"} and Lemma [\[lem:4\]](#lem:4){reference-type="ref" reference="lem:4"}, we have the following proposition. [\[prop:positive_cone_C\]]{#prop:positive_cone_C label="prop:positive_cone_C"} Let $r_i$, $s_i$, $t_i$ $(i=1,\ldots,k)$ be ground terms. Let $\mathfrak{S}$ denote the set of inequalities $$\{r_i\neq s_i,s_i\neq t_i, t_i\neq r_i:i=1,\ldots,k\}.$$ Let $\mathfrak{S}^C$ denote the set of axioms $$\{C(r_0,s_0,t_0\}\cup\{C(r_i,s_i,t_i)\lor C(r_i,s_i,t_i): i=1,\ldots, k\}.$$ Let $\mathfrak{S}^P$ denote the set of axioms $$\{P(r_0'\cdot s_0,r_0' \cdot t_0)\}\cup\{P(r_i'\cdot s_i,r_i' \cdot t_i)\lor P(r_i' \cdot t_i,r_i'\cdot s_i): i=1,\ldots, k\}.$$ Let $\mathbf{AxPCL}'$ (resp. $\mathbf{AxC}'$) denote $\mathbf{AxPCL}$ (resp. $\mathbf{AxC}$) minus the axiom of connectedness. Then 1. $\mathbf{Gr}\cup \mathbf{Ax}_R\cup \mathbf{AxC}\cup \mathbf{OrdCL}\cup \mathfrak{S}$ is consistent if and only if $\mathbf{Gr}\cup \mathbf{Ax}_R\cup \mathbf{AxPCL}\cup \mathfrak{S}$ is consistent; 2. $\mathbf{Gr}\cup \mathbf{Ax}_R\cup\mathbf{AxC}\cup \mathbf{OrdCB}\cup \mathfrak{S}$ is consistent if and only if $\mathbf{Gr}\cup \mathbf{Ax}_R\cup \mathbf{AxPCL}\cup \mathbf{PCB}\cup \mathfrak{S}$ is consistent; 3. $\mathbf{Gr}\cup \mathbf{Ax}_R\cup \mathbf{AxC}' \cup \mathbf{OrdCL}\cup \mathfrak{S}^C$ is consistent if and only if $\mathbf{Gr}\cup \mathbf{Ax}_R\cup \mathbf{AxPCL}'\cup \mathfrak{S}^P$ is consistent; 4. $\mathbf{Gr}\cup \mathbf{Ax}_R\cup\mathbf{AxC}' \cup \mathbf{OrdCB}\cup \mathfrak{S}^C$ is consistent if and only if $\mathbf{Gr}\cup \mathbf{Ax}_R\cup \mathbf{AxPCL}'\cup \mathbf{PCB}\cup \mathfrak{S}^P$ is consistent. Proposition [\[prop:positive_cone\]](#prop:positive_cone){reference-type="ref" reference="prop:positive_cone"} and Proposition [\[prop:positive_cone_C\]](#prop:positive_cone_C){reference-type="ref" reference="prop:positive_cone_C"} offer an alternative approach to demonstrating the inconsistency of the theories in Proposition [\[prop:main\]](#prop:main){reference-type="ref" reference="prop:main"} and Proposition [\[symmetry\]](#symmetry){reference-type="ref" reference="symmetry"} through automated reasoning, which usually enhances efficiency. It is worth noting that the performance of the automated theorem prover Prover9 on a single task is highly unpredictable. For further optimisations, one may fine tune the proof search strategy or attempt alternative presentations of the same group. ## Examples In Example [\[ex:D7\]](#ex:D7){reference-type="ref" reference="ex:D7"}, we established the non-circular-orderability of the dihedral group $D_{7}$ using automated reasoning. However, the process of deducing the inconsistency of $\mathbf{Gr}\cup \mathbf{Ax}_R\cup \mathbf{AxC}\cup\mathbf{OrdCL}\cup \mathfrak{S}$ with Prover9 is time-consuming. By the statement (a) in Proposition [\[prop:positive_cone_C\]](#prop:positive_cone_C){reference-type="ref" reference="prop:positive_cone_C"}, we can alternatively demonstrate the inconsistency of $\mathbf{Gr}\cup \mathbf{Ax}_R \cup \mathbf{AxPCL}\cup \mathfrak{S}$ (Task 6.1). The positive cone translation significantly reduces the time required; see Table [1](#tab:summary1){reference-type="ref" reference="tab:summary1"}. [\[ex:sl2-weak\]]{#ex:sl2-weak label="ex:sl2-weak"} Consider the special linear group $SL_2(\mathbb{Z})$ with the presentation as shown in Example [\[ex:sl2\]](#ex:sl2){reference-type="ref" reference="ex:sl2"}. By [@giraudet2018first Theorem 5.10], in a group with a bi-invariant circular order, the torsion part is central. Since $a$ is a torsion element in $SL_2(\mathbb{Z})$ with $ab\neq ba$, it follows that $SL_2(\mathbb{Z})$ does not admit a bi-invariant circular order. To prove this fact automatically, we first establish the inequalities $e\neq ab$, $e\neq ba$, $ab\neq ba$ by building a model (Task 7.1) of $\mathbf{Gr}\cup \mathbf{Ax}_R\cup \{e\neq a\cdot b,e\neq b\cdot a,a\cdot b\neq b\cdot a\}$ by Mace4. Then we prove that $\mathbf{Gr}\cup \mathbf{Ax}_R\cup \mathbf{AxPCL}'\cup \mathbf{PCB}\cup\{P(a\cdot b,b\cdot a)\}$ is inconsistent (Task 7.2) by Prover9. Thus by Proposition [\[weak-cbo\]](#weak-cbo){reference-type="ref" reference="weak-cbo"}, Proposition [\[symmetry\]](#symmetry){reference-type="ref" reference="symmetry"} and the statement (d) in Proposition [\[prop:positive_cone_C\]](#prop:positive_cone_C){reference-type="ref" reference="prop:positive_cone_C"}, the special linear group $SL_2(\mathbb{Z})$ does not admit a bi-invariant circular order. [\[ex:Poincare\]]{#ex:Poincare label="ex:Poincare"} The fundamental group of the Poincaré homology sphere has a presentation $$\langle a,b\;|\; (ab)^{2} = a^{3}, a^{3} = b^{5} \rangle.$$ This group is finite (of order $120$), hence it is not left-orderable. To prove this fact automatically, we check that $a\neq e$ by finding a model (Task 8.1) of $\mathbf{Gr}\cup \mathbf{Ax}_R\cup \{e\neq a\}$ using Mace4. Then we prove that $\mathbf{Gr}\cup\mathbf{Ax}_R\cup \mathbf{AxPL}'\cup\{P(a)\}$ is inconsistent (Task 8.2) by Prover9. Thus by Proposition [\[prop:weak\]](#prop:weak){reference-type="ref" reference="prop:weak"}, Proposition [\[symmetry\]](#symmetry){reference-type="ref" reference="symmetry"} and the statement (a) in Proposition [\[prop:positive_cone\]](#prop:positive_cone){reference-type="ref" reference="prop:positive_cone"}, the fundamental group of the Poincaré homology sphere is not left-orderable. [\[ex:5_2\]]{#ex:5_2 label="ex:5_2"} The knot group of the knot $5_{2}$ has a presentation $$\langle a,b \;|\; b^{2}a^{2}b^{2} = a b^{3}a\rangle.$$ This group is known to be left-orderable, as all knot groups, and not bi-orderable, according to [@chiswell2015residual page 5] and [@naylor2016generalized Theorem 7]. For the same reason as in Example [\[ex:B3\]](#ex:B3){reference-type="ref" reference="ex:B3"}, this group does not admit a bi-invariant circular order. We first prove the non-bi-orderability using automated reasoning. A model for $\mathbf{Gr}\cup \mathbf{Ax}_{R} \cup \{b\cdot a \neq (a\cdot b)\cdot b\}$ can be found (Task 9.1) using Mace4. Thus $ba \neq ab^{2}$ holds true in this group. We can verify that $\mathbf{Gr}\cup \mathbf{Ax}_R\cup\mathbf{AxPL}' \cup \mathbf{PB}\cup \{P((b\cdot a)'\cdot ((a\cdot b)\cdot b))\}$ is inconsistent (Task 9.2) by Prover9. By Proposition [\[prop:weak\]](#prop:weak){reference-type="ref" reference="prop:weak"}, Proposition [\[symmetry\]](#symmetry){reference-type="ref" reference="symmetry"} and the statement (d) in Proposition [\[prop:positive_cone\]](#prop:positive_cone){reference-type="ref" reference="prop:positive_cone"}, the knot group of $5_{2}$ is not bi-orderable. Alternatively, we may use the inequality $ab \neq ba$ in the automated proof. (Task 9.3 and Task 9.4) Now we show how to prove the non-existence of a bi-invariant circular order using automated reasoning. First, we prove $e\neq ab$, $e\neq ba$ and $ab\neq ba$ by building a model (Task 9.5) of $\mathbf{Gr}\cup \mathbf{Ax}_R\cup\{e\neq a\cdot b, e\neq b\cdot a, a\cdot b\neq b\cdot a\}$ using Mace4. Then we prove the inconsistency of $\mathbf{Gr}\cup \mathbf{Ax}_R\cup\mathbf{AxPCL}'\cup \mathbf{PCB}\cup\{P(a\cdot b,b\cdot a)\}$ (Task 9.6) using Prover9. By Proposition [\[weak-cbo\]](#weak-cbo){reference-type="ref" reference="weak-cbo"}, Proposition [\[symmetry\]](#symmetry){reference-type="ref" reference="symmetry"} and the statement (d) in Proposition [\[prop:positive_cone_C\]](#prop:positive_cone_C){reference-type="ref" reference="prop:positive_cone_C"}, the knot group of $5_{2}$ does not admit a bi-invariant circular order. Alternatively, we may use the inequalities $e\neq ba$, $e\neq ab^2$ and $ba\neq ab^2$ in the automated proof. (Task 9.7 and Task 9.8) [\[ex:weeks\]]{#ex:weeks label="ex:weeks"} The fundamental group of the Weeks manifold has a presentation $$\langle a,b\;|\; a^{2}b^{2}a^{2} = ba^{-1}b,b^2a^{2}b^{2} = ab^{-1}a \rangle.$$ According to [@calegari2003laminations Theorem 9.2], this group is not circularly orderable. While our approach is not sophisticated enough to provide an automated proof of the non-circular-orderability in reasonable time, we can prove two weaker properties using automated reasoning: the non-left-orderability and the absence of bi-invariant circular orders. By building a finite model (Task 10.1) of $\mathbf{Gr}\cup \mathbf{Ax}_R \cup \{ a\neq e\}$ using Mace4, we obtain that $a\neq e$. By deriving a contradiction (Task 10.2) of $\mathbf{Gr}\cup\mathbf{Ax}_R\cup\mathbf{AxPL}\cup\{a\neq e\}$ using Prover9, it follows from Proposition [\[prop:main\]](#prop:main){reference-type="ref" reference="prop:main"} and the statement (a) in Proposition [\[prop:positive_cone\]](#prop:positive_cone){reference-type="ref" reference="prop:positive_cone"} that, the Weeks manifold group is not left-orderable. In order to prove the non-existence of bi-invariant circular orders automatically, we need to confirm three inequalities $e\neq ab$, $e\neq ba$ and $ab\neq ba$. The first two inequalities can be verified by building a finite model (Task 10.3) of $\mathbf{Gr}\cup\mathbf{Ax}_R\cup\{e\neq a\cdot b, e\neq b\cdot a\}$ using Mace4. However, the last inequality $ab\neq ba$ turns out to be more challenging. One way to prove $ab\neq ba$ in the Weeks manifold group is through an $SL_2(\mathbb{C})$-representation as described on [@chinburg2001arithmetic page 24]. Alternatively, we can argue that the equality $ab=ba$ implies that the group of interest is isomorphic to the product of two cyclic groups of order $5$, hence it is finite and non-cyclic. According to [@zheleva1976cyclically Theorem 1], in such a case, it is not circularly orderable. We can prove that the theory $\mathbf{Gr}\cup\mathbf{Ax}_R\cup \mathbf{AxPCL}'\cup \mathbf{PCB}\cup\{P(a\cdot b, b\cdot a)\}$ is inconsistent (Task 10.4) using Prover9. Thus by Proposition [\[weak-cbo\]](#weak-cbo){reference-type="ref" reference="weak-cbo"}, Proposition [\[symmetry\]](#symmetry){reference-type="ref" reference="symmetry"} and the statement (d) in Proposition [\[prop:positive_cone_C\]](#prop:positive_cone_C){reference-type="ref" reference="prop:positive_cone_C"}, the Weeks manifold group does not admit a bi-invariant circular order. # Torsions, generalised torsions, and more {#sec:torsions} ## Torsions and generalised torsions In this subsection, we show that, if $\mathfrak{S}$ contains a single inequality $t_1\neq t_2$, then the methods of establishing the non-left-orderability (resp. non-bi-orderability) via the weakened theory described in Subsection [3.1](#subsec:weakened){reference-type="ref" reference="subsec:weakened"} is essentially detecting whether $t_1^{-1} t_2$ represents a torsion (resp. generalised torsion) in the group presented by $\langle S|R\rangle$. Note that a nontrivial torsion (resp. generalised torsion) is a well-known obstruction to left-orderability (resp. bi-orderability); see [@clay2016ordered Proposition 1.3 and Problem 1.22] for example. First, we present the definition of torsion and establish the desired equivalence. A group element $x \in G$ is called a torsion if there exists a positive integer $n$ such that $x^{n} =e$ where $e$ is the identity element of the group $G$. [\[prop-torsion\]]{#prop-torsion label="prop-torsion"} Let $G$ be a group with presentation $\langle S|R\rangle$. Let $t$ be a ground term representing a group element in $G$. Then $\mathbf{Gr}\cup\mathbf{Ax}_R\cup\mathbf{AxPL}'\cup\{P(t)\}$ is inconsistent if and only if $t$ represents a torsion in $G$. *Proof.* We first prove the "if" part. If $t$ represents a torsion in $G$, then there exists a positive integer $n$ such that $t^n=e$ holds assuming the axioms $\mathbf{Gr}$ and $\mathbf{Ax}_R$, where the ground term $t^n$ is recursively defined by $t^1=t$ and $t^n=t^{n-1}\cdot t$ for $n\ge 2$. The closure axiom in $\mathbf{AxPL}'$ and the axiom $P(t)$ implies that $P(t^n)$ for every positive integer $n$ inductively, hence we have $P(e)$, which contradicts to the irreflexivity axiom in $\mathbf{AxPL}'$. Therefore $\mathbf{Gr}\cup\mathbf{Ax}_R\cup\mathbf{AxPL}'\cup\{P(t)\}$ is inconsistent if $t$ represents a torsion in $G$. Next, we prove the "only if" part. Suppose that $t$ represents a non-torsion element $\bar{t}$ in $G$. For any $x\in G$, let $P(x)$ be the proposition that $\bar{t}^n=x$ in $G$ for some positive integer $n$. We prove that the group $G$ together with the predicate $P$ constitutes a model for $\mathbf{Gr}\cup\mathbf{Ax}_R\cup\mathbf{AxPL}'\cup\{P(t)\}$: 1. The axioms in $\mathbf{Gr}\cup Ax_R$ are satisfied because $G$ is the group with presentation $\langle S|R\rangle$. 2. If $P(e)$ holds, then there exists a positive integer $n$ such that $\bar{t}^n=e$, which contradicts to the assumption that $\bar{t}$ is not a torsion. Thus the irreflexivity axiom in $\mathbf{AxPL}'$ is satisfied. 3. If $P(x)$ and $P(y)$ holds for $x,y\in G$, then there exist positive integers $m$ and $n$ such that $\bar{t}^m=x$ and $\bar{t}^{n}=y$, so we have $\bar{t}^{m+n}=x$ and therefore $P(xy)$ holds. Thus the transitivity axiom in $\mathbf{AxPL}'$ is satisfied. 4. The axiom $P(t)$ is satisfied by the definition of $P$. Therefore the theory $\mathbf{Gr}\cup\mathbf{Ax}_R\cup\mathbf{AxPL}'\cup\{P(t)\}$ is consistent if $t$ represents a non-torsion element in $G$. ◻ Then, we present the definition of generalised torsion and prove the analogous statement to Proposition [\[prop-torsion\]](#prop-torsion){reference-type="ref" reference="prop-torsion"}. For convenience, we consider the identity element $e$ as a generalised torsion. A group element $x \in G$ is called a generalised torsion if there exist $y_{1}, \ldots, y_{n} \in G$ such that $$(y_{1}xy^{-1}_{1})(y_{2}xy^{-1}_{2})\cdots (y_{n}xy^{-1}_{n})=e,$$ where $e$ is the identity element of of the group $G$. [\[prop-generalised-torsion\]]{#prop-generalised-torsion label="prop-generalised-torsion"} Let $G$ be a group with presentation $\langle S|R\rangle$. Let $t$ be a ground term representing a group element in $G$. Then $\mathbf{Gr}\cup\mathbf{Ax}_R\cup\mathbf{AxPL}'\cup \mathbf{PB}\cup\{P(t)\}$ is inconsistent if and only if $t$ represents a generalised torsion in $G$. *Proof.* We first prove the "if" part. If $t$ represents a generalised torsion $\bar{t}$ in $G$, then there exist $y_{1}, \ldots, y_{n} \in G$ such that $$(y_{1}\bar{t} y^{-1}_{1})(y_{2}\bar{t} y^{-1}_{2})\cdots (y_{n}\bar{t} y^{-1}_{n})=e.$$ By the definition of $\langle S|R \rangle$, there exist ground terms $t_1, \ldots, t_n$, such that the product of $(t_i\cdot \bar{t})\cdot t_i'$ $(i=1,\ldots, n)$ equals to $e$, assuming the axioms $\mathbf{Gr}$ and $\mathbf{Ax}_R$. The conjugacy invariance axiom $\mathbf{PB}$ and $P(t)$ implies that $P((t_i\cdot \bar{t})\cdot t_i')$ for each $i=1,\ldots, n$. The closure axiom in $\mathbf{AxPL}'$ implies $P(e)$ by induction, which contradicts to the irreflexivity axiom in $\mathbf{AxPL}'$. Therefore $\mathbf{Gr}\cup\mathbf{Ax}_R\cup\mathbf{AxPL}'\cup \mathbf{PB}\cup\{P(t)\}$ is inconsistent if $t$ represents a generalised torsion in $G$. Next, we prove the "only if" part. Suppose that $t$ represents a element $\bar{t}$ in $G$ that is not a generalised torsion. For any $x\in G$, let $P(x)$ be the proposition that $$(y_1\bar{t}y_1^{-1})(y_2 \bar{t} y_2^{-1})\cdots(y_n \bar{t} y_n^{-1})=x$$ in $G$ for some $y_1,\ldots, y_n\in G$. We prove that the group $G$ together with the predicate $P$ constitutes a model for $\mathbf{Gr}\cup\mathbf{Ax}_R\cup\mathbf{AxPL}'\cup \mathbf{PB}\cup\{P(t)\}$: 1. The axioms in $\mathbf{Gr}\cup Ax_R$ are satisfied because $G$ is the group with presentation $\langle S|R\rangle$. 2. Since $\bar{t}$ is not a generalised torsion, we have $\neg P(e)$. Thus the irreflexivity axiom in $\mathbf{AxPL}'$ is satisfied. 3. If $P(x)$ and $P(y)$ holds for $x,y\in G$, then there exist $y_1,\ldots, y_m\in G$ and $z_1,\ldots, z_n\in G$ such that $$(y_1\bar{t}y_1^{-1})(y_2 \bar{t} y_2^{-1})\cdots(y_m \bar{t} y_m^{-1})=x$$ and $$(z_1\bar{t}z_1^{-1})(z_2 \bar{t} z_2^{-1})\cdots(z_m \bar{t} z_n^{-1})=y,$$ so we have $$(y_1\bar{t}y_1^{-1})(y_2 \bar{t} y_2^{-1})\cdots(y_m \bar{t} y_m^{-1})(z_1\bar{t}z_1^{-1})(z_2 \bar{t} z_2^{-1})\cdots(z_m \bar{t} z_n^{-1})=xy$$ and therefore $P(xy)$ holds. Thus the transitivity axiom in $\mathbf{AxPL}'$ is satisfied. 4. If $P(x)$ holds for $x\in G$, then there exist $y_1,\ldots, y_n\in G$ such that $$(y_1\bar{t}y_1^{-1})(y_2 \bar{t} y_2^{-1})\cdots(y_n \bar{t} y_n^{-1})=x.$$ Then for any $y\in G$ we have $$((y y_1)\bar{t}(y y_1)^{-1})((y y_2) \bar{t} (y y_2)^{-1})\cdots((y y_n) \bar{t} (y y_n)^{-1})=y x y^{-1}.$$ Thus the conjugacy invariant axiom $\mathbf{PB}$ is satisfied. 5. The axiom $P(t)$ is satisfied by the definition of $P$. Therefore the theory $\mathbf{Gr}\cup\mathbf{Ax}_R\cup\mathbf{AxPL}'\cup\mathbf{PB}\cup\{P(t)\}$ is consistent if $t$ does not represent a generalised torsion in $G$. ◻ ## Analogous statement for bi-invariant circular orders In this subsection, we establish an analogous statement for bi-invariant circular orders. We give an equivalent condition for the inconsistency of a theory where the axiom of connectedness is weakened as described in Subsection [3.1](#subsec:weakened){reference-type="ref" reference="subsec:weakened"} and the axiom of cyclicity is removed.  [\[prop-CBO-monoid\]]{#prop-CBO-monoid label="prop-CBO-monoid"} Let $G$ be a group with presentation $\langle S|R\rangle$. Let $t_1$ and $t_2$ be ground terms representing the group elements $\bar{t}_1$ and $\bar{t}_2$ in $G$ respectively. Let $\overline{\mathbf{AxPCL}}'$ denote $\mathbf{AxPCL}$ minus the axioms of cyclicity and connectedness. Then $\mathbf{Gr}\cup\mathbf{Ax}_R\cup\overline{\mathbf{AxPCL}}'\cup \mathbf{PCB}\cup\{P((t_2 \cdot t_1)\cdot t_2',t_1)\}$ is inconsistent if and only if $\bar{t}_2^{-1}$ is in the monoid generated by $\bar{t}_2$ and the centraliser of $\bar{t}_1$. *Proof.* We first prove the "if" part. If $\bar{t}_2^{-1}$ is in the monoid generated by $\bar{t}_1$, $\bar{t}_1^{-1}$, and $\bar{t}_2$, then there exist a nonnegative integer $n$ and some ground terms $s_0=t_2, s_1,s_2,\ldots s_n$, such that $s_i$ ($i=1,2,\ldots,n$) is either $s_{i-1}\cdot t_2$ or $s_{i-1}\cdot t$ where $t$ and $t_1$ represent commutative elements in $G$, and that $s_n = e$ assuming $\mathbf{Gr}$ and $\mathbf{Ax}_R$. We prove $P((s_i\cdot t_1)\cdot s_i', t_1)$ ($i=0,1,\ldots, n$) inductively. For $i=0$, the statement $P((s_0\cdot t_1)\cdot s_0', t_1)$ holds true by assumption. Suppose that the statement $P((s_{i-1}\cdot t_1)\cdot s_{i-1}', t_1)$ holds true for some $i\in\{0,1,\ldots,n-1\}$. If $s_i= s_{i-1}\cdot t_2$, then we have $$(s_{i}\cdot t_1)\cdot s_{i}'=(s_{i-1}\cdot ((t_2\cdot t_1)\cdot t_2'))\cdot s_{i-1}'$$ assuming $\mathbf{Gr}$. By the conjugacy invariance axiom $\mathbf{PCB}$ and $P((t_2\cdot t_1)\cdot t_2', t_1)$, we have $P((s_{i}\cdot t_1)\cdot s_{i}', ((s_{i-1}\cdot t_1)\cdot s_{i-1}'))$. By the transitivity axiom in $\overline{\mathbf{AxPCL}}'$ and the inductive hypothesis, the statement $P((s_{i}\cdot t_1)\cdot s_{i}', t_1)$ holds true in this case. If $s_i= s_{i-1}\cdot t$ where $t$ and $t_1$ represent commutative elements in $G$, then we have $$(s_{i}\cdot t_1)\cdot s_{i}'=(s_{i-1}\cdot t_1)\cdot s_{i-1}'$$ assuming $\mathbf{Gr}$, thus $P((s_{i}\cdot t_1)\cdot s_{i}', t_1)$ also holds true. By taking $i=n$, we have $P(t_1,t_1)$, which contradicts to the irreflexivity axiom in $\overline{\mathbf{AxPCL}}'$. Therefore $\mathbf{Gr}\cup\mathbf{Ax}_R\cup\overline{\mathbf{AxPCL}}'\cup \mathbf{PCB}\cup\{P((t_2 \cdot t_1)\cdot t_2',t_1)\}$ is inconsistent if $\bar{t}_2^{-1}$ is in the monoid generated by $\bar{t}_2$ and the centraliser of $\bar{t}_1$. Next, we prove the "only if" part. Suppose that $\bar{t}_2^{-1}$ is not in the monoid generated by $\bar{t}_2$ and the centraliser of $\bar{t}_1$. For any $x, y\in G$, let $P(x, y)$ be the proposition that there exist $z_1, z_2\in G$, such that: 1. $x= z_1 z_2 \bar{t}_2 \bar{t}_1 \bar{t}_2^{-1} z_2^{-1} z_1^{-1}$, 2. $y=z_1 \bar{t}_1 z_1^{-1}$, and 3. $z_2$ is in the monoid generated by $\bar{t}_2$ and the centraliser of $\bar{t}_1$. We prove that the group $G$ together with the predicate $P$ constitutes a model for $\mathbf{Gr}\cup \mathbf{Ax}_R \cup \overline{\mathbf{AxPCL}}' \cup \mathbf{PCB}\cap\{P((t_2\cdot t_1) \cdot t'_2,t_1)\}$: 1. The axioms in $\mathbf{Gr}\cup Ax_R$ are satisfied because $G$ is the group with presentation $\langle S|R\rangle$. 2. If $P(x,x)$ holds for some $x\in G$, then there exists $z_2$ in the monoid generated by $\bar{t}_2$ and the centraliser of $\bar{t}_1$, such that $$z_2 \bar{t}_2 \bar{t}_1 \bar{t}_2^{-1} z_2^{-1} = z_1^{-1} x z_1=\bar{t}_1.$$ In this case, the element $\bar{t}_2^{-1} z_2^{-1}$ is in the centraliser of $\bar{t_1}$. Since a monoid is closed under multiplication by definition, the element $\bar{t}_2^{-1}$ is in the monoid generated by $\bar{t}_2$ and the centraliser of $\bar{t}_1$, which contradicts to the assumption. Thus the the irreflexivity axiom in $\overline{\mathbf{AxPCL}}'$ is satisfied. 3. If $P(x,y)$ and $P(y,z)$ holds for $x,y\in G$, then there exist $z_1,z_2,z_3,z_4\in G$, such that $$x=z_1 z_2 \bar{t}_2\bar{t}_1\bar{t}_2^{-1} z_2^{-1} z_1^{-1},$$ $$y=z_1 \bar{t}_1 z_1^{-1} = z_3 z_4 \bar{t}_2\bar{t}_1\bar{t}_2^{-1} z_4^{-1} z_3^{-1},$$ $$z=z_3 \bar{t}_1 z_3^{-1},$$ and $z_2$ and $z_4$ are in the monoid generated by $\bar{t}_2$ and the centraliser of $\bar{t}_1$. By the second equality, the element $\bar{t}_2^{-1} z_4^{-1} z_3^{-1} z_1$ is in the centraliser of $\bar{t}_1$. Since a monoid is closed under multiplication by definition, the element $$z_3^{-1} z_1 z_2 =z_4 \bar{t}_2(\bar{t}_2^{-1} z_4^{-1} z_3^{-1} z_1)z_2$$ is in the monoid generated by $\bar{t}_2$ and the centraliser of $\bar{t}_1$. The elements $z_3, z_3^{-1} z_1 z_2 \in G$ satisfy the conditions in the definition of $P(x,z)$, hence $P(x,z)$ holds true. Thus the transitivity axiom in $\overline{\mathbf{AxPCL}}'$ is satisfied. 4. If $P(x,y)$ holds for some $x,y\in G$, then there exist $z_1, z_2\in G$ such that the conditions in the definition of $P(x,y)$ are satisfied. Then $z z_1, z_2 \in G$ satisfy the conditions in the definition of $P(zxz^{-1},y)$, hence $P(zxz^{-1},y)$. Thus the conjugacy invariance axiom $\mathbf{PCB}$ is satisfied. 5. The axiom $P((t_2 \cdot t_1)\cdot t_2', t_1)$ is satisfied by taking $z_1=z_2=e$ in the definition of $P$. Therefore the theory $\mathbf{Gr}\cup \mathbf{Ax}_R \cup \overline{\mathbf{AxPCL}}' \cup \mathbf{PCB}\cap\{P((t_2\cdot t_1) \cdot t'_2,t_1)\}$ is consistent if $\bar{t}_2^{-1}$ is not in the monoid generated by $\bar{t}_2$ and the centraliser of $\bar{t}_1$. ◻ ## Examples Consider the Fibonacci group $F(2,n)$ $(n\ge 2)$ with the presentation a shown in Example [\[ex:Fibonacci\]](#ex:Fibonacci){reference-type="ref" reference="ex:Fibonacci"}. According to [@motegi2017generalized Theorem 5.2], the element $a_0$ is a generalised torsion. To prove this fact automatically when $n=11$ or $n=12$, we can verify the inconsistency (Task 11.1 and Task 11.2) of $\mathbf{Gr}\cup \mathbf{Ax}_R\cup\mathbf{AxPL}'\cup \mathbf{PB}\cup\{P(a_0)\}$ using Prover9, and then apply Proposition [\[prop-generalised-torsion\]](#prop-generalised-torsion){reference-type="ref" reference="prop-generalised-torsion"}. In Example [\[ex:sl2-weak\]](#ex:sl2-weak){reference-type="ref" reference="ex:sl2-weak"}, we established the absence of bi-invariant circular orders via weakened theories. In fact, the axiom of cyclicity is redundant in the automated proof. By Prover9, we can prove that $\mathbf{Gr}\cup \mathbf{Ax}_R\cup \overline{\mathbf{AxPCL}}'\cup \mathbf{PCB}\cup\{P(a\cdot b,b\cdot a)\}$ is inconsistent (Task 12.1). Thus by Proposition [\[prop-CBO-monoid\]](#prop-CBO-monoid){reference-type="ref" reference="prop-CBO-monoid"}, the element $a^{-1}$ is in the monoid generated by $a$ and the centraliser of $ba$. Alternatively, this condition follows from the relation $a^{-1}=a^3$. In Example [\[ex:Poincare\]](#ex:Poincare){reference-type="ref" reference="ex:Poincare"}, we established the non-left-orderability of the fundamental group of the Poincaré homology sphere via weakened theories. By Proposition [\[prop-torsion\]](#prop-torsion){reference-type="ref" reference="prop-torsion"}, the inconsistency of the weakened theory implies that $a$ is a torsion. Alternatively, it follows from that this group is finite. In Example [\[ex:5_2\]](#ex:5_2){reference-type="ref" reference="ex:5_2"}, we established the non-bi-orderability of the knot group of $5_2$ via weakened theories in two ways (the inequalities $ba\neq ab^2$ and $ba\neq ab$). By Proposition [\[prop-generalised-torsion\]](#prop-generalised-torsion){reference-type="ref" reference="prop-generalised-torsion"}, it follows that $a^{-1}b^{-1}ab^2$ and $b^{-1}a^{-1}ba$ are generalised torsions. Note that the latter one was previously discovered in [@naylor2016generalized Theorem 7]. In Example [\[ex:weeks\]](#ex:weeks){reference-type="ref" reference="ex:weeks"}, we established the absence of bi-invariant circular orders via weakened theories. In this example, the axiom of cyclicity is also redundant. We can verify the inconsistency of $\mathbf{Gr}\cup \mathbf{Ax}_R\cup \overline{\mathbf{AxPCL}}'\cup \mathbf{PCB}\cup\{P(a\cdot b,b\cdot a)\}$ (Task 15.1) using Prover9. Thus by Proposition [\[prop-CBO-monoid\]](#prop-CBO-monoid){reference-type="ref" reference="prop-CBO-monoid"}, the element $a^{-1}$ is in the monoid generated by $a$ and the centraliser of $ba$. Alternatively, this condition follows from the relations $b^{-1} = a(ba)^{-1}$ and $a^{-1}= b^{-3} a b^{-1} a^3b^{-1}$. # Relative convexity and strength of cyclicity axiom {#sec:convexity} In this section, we further explore the consistency of the theory regarding bi-invariant circular orders when the axiom of cyclicity is removed. We do not weaken the axiom of connectedness. To state our criterion, we introduce the concept of left relative convexity. In contrast to traditional usage (relative convexity of left-orderable groups), we do not restrict the ambient group to a left-orderable group. Thus we adopt the following definition introduced by Antolın, Dicks, and Sunic [@antolin2015left]. The proof of equivalence of definitions could be found in [@antolin2021space Lemma 2.1]. [\[def-rel-convex\]]{#def-rel-convex label="def-rel-convex"} Let $G$ be a group and $H$ be a subgroup of $G$. We say $H$ is left relatively convex in $G$ when any of the following equivalent conditions hold. 1. There exists a $G$-invariant order on the left $G$-set $G/H$. 2. There exists a subsemigroup $P$ of $G$ such that $P\sqcup H\sqcup P^{-1}$ is a partition of $G$, and $H P H\subseteq P$. A total preorder $\le$ on a group $G$ is called a *left total preorder* if it is invariant under multiplication. A subset $S$ of $G$ is called *convex* relative to the left total preorder $\le$ if $x,z\in S$ and $x\le y\le z$ implies $y\in S$. Based on the condition (b) in Definition [\[def-rel-convex\]](#def-rel-convex){reference-type="ref" reference="def-rel-convex"}, we establish the following equivalent conditions for a subgroup to be left relatively convex. By the equivalence (a) $\Leftrightarrow$ (c) in the next proposition, the definition of left relative convexity serves as a left total preorder adaptation of the concept of relative convexity in left-orderable groups. [\[prop-rel-convex-1\]]{#prop-rel-convex-1 label="prop-rel-convex-1"} Let $G$ be a group and $H$ be a subgroup of $G$. The following statements are equivalent. 1. The subgroup $H$ is left relatively convex in $G$. 2. The subgroup $H$ is the residue group $\{x\in G: e\le x\le e\}$ of some left total preorder $\le$ on $G$. 3. The subgroup $H$ is convex relative to some left total preorder $\le$ on $G$. *Proof.* For the implication (a) $\Rightarrow$ (b), let $P$ be the subsemigroup described in the condition (b) in Definition [\[def-rel-convex\]](#def-rel-convex){reference-type="ref" reference="def-rel-convex"}. Define the binary relation $\le$ on $G$ by $x\le y$ if and only if $x^{-1}y\in P\cup H$. Then we can check that $\le$ is a left total preorder on $G$: the reflexivity follows from $e\in H$, the transitivity follows from $PP \cup PH\cup HP \subseteq P$ and $HH\subseteq H$, the totality follows from $$(P\cup H)\cup(P\cup H)^{-1} = P\cup H\cup P^{-1}=G,$$ and the left-invariance follows from the definition of $\le$. Now we prove that the residue group of $\le$ is $H$. The inequality $e\le x\le e$ holds if and only if both $x$ and $x^{-1}$ are in $P\cup H$, which holds if and only if $$x\in (P\cup H)\cap(P\cup H)^{-1} = (P\cap P^{-1})\cup H=H.$$ The implication (b) $\Rightarrow$ (c) holds because the residue group is always convex by definition. For the implication (c) $\Rightarrow$ (a), suppose that $\le$ is a left total preorder on $G$ relative to which $H$ is convex. Let $P$ denote the set $\{x\in G\setminus H: e\le x\}$. We prove that $P$ satisfies the condition (b) in Definition [\[def-rel-convex\]](#def-rel-convex){reference-type="ref" reference="def-rel-convex"}. 1. For any $x,y\in P$, by the transitivity and the left-invariance of $\le$, we have $e\le x\le xy$. Then by the convexity of $\le$, we have $xy\in G\setminus H$, hence we have $xy\in P$. Therefore $P$ is a subsemigroup. 2. Because $P$ is a subsemigroup, $H$ is a subgroup, and $P\cap H=\emptyset$, the subsets $P$, $H$, and $P^{-1}$ are disjoint. 3. For any $x\in G\setminus H$, by the totality and the left-invariance of $\le$, either $e\le x$ or $e\le x^{-1}$ holds. Since $H$ is a subgroup, either $x\in P$ or $x\in P^{-1}$ holds. Thus we have $G=P\sqcup H\sqcup P^{-1}$. 4. Suppose that $x\in H$ and $y\in P$. If $xy\in H$, then $y=x^{-1}(xy)\in P\cap H$. If $xy\in P^{-1}$, then $x=(xy)y^{-1}\in H\cap P^{-1}$. Because $G=P\sqcup H\sqcup P^{-1}$, we have $xy\in P$. Thus we have $HP\subseteq P$. For the same reason, we have $PH\subseteq P$. Therefore $HPH\subseteq PH\subseteq P$.  ◻ Next, we establish some equivalent statements for the condition (a) in Definition [\[def-rel-convex\]](#def-rel-convex){reference-type="ref" reference="def-rel-convex"}. Note that the equivalence (a) $\Leftrightarrow$ (b) in the following proposition generalises [@clay2010space Theorem 1.4.10]. [\[prop-rel-convex-2\]]{#prop-rel-convex-2 label="prop-rel-convex-2"} Let $G$ be a group and $H$ be a subgroup of $G$. The following statements are equivalent. 1. The subgroup $H$ is left relatively convex in $G$. 2. The subgroup $H$ is a kernel of an order preserving $G$-action on some totally ordered set $(\Omega,<)$. 3. For any finite set of elements $g_1,\ldots, g_n\in G\setminus H$, there exist $\varepsilon_1,\ldots,\varepsilon_n\in\{-1,1\}$ such that the subsemigroup generated by $H g_1^{\varepsilon_1}H,\ldots,Hg_n^{\varepsilon_n}H$ does not contain the identity element $e$. 4. For any finite set of elements $g_1,\ldots, g_n\in G\setminus H$, there exist $\varepsilon_1,\ldots,\varepsilon_n\in\{-1,1\}$ such that the subsemigroup generated by $g_1^{\varepsilon_1},\ldots,g_n^{\varepsilon_n}$ has empty intersection with $H$. *Proof.* For the implication (a) $\Rightarrow$ (b), suppose that $H$ is left relatively convex in $G$. By the condition (a) in Definition [\[def-rel-convex\]](#def-rel-convex){reference-type="ref" reference="def-rel-convex"}, there exists a $G$-invariant order $<$ on the left $G$-set $G/H$. And the subgroup $H$ is the kernel of the order preserving $G$-action on the totally ordered set $(G/H,<)$. The implication (b) $\Rightarrow$ (c) was proved by Tararin; see [@kopytov1996right Proposition 5.1.5]. The implication (c) $\Rightarrow$ (d) holds because $H$ is a subgroup: if $h\in H$ is in the subsemigroup generated by $g_1^{\varepsilon_1},\ldots, g_n^{\varepsilon_n}$, then $e$ is in the subsemigroup generated by $g_1^{\varepsilon_1},\ldots, g_n^{\varepsilon_n}$ and $h^{-1} g_1^{\varepsilon_1},\ldots, h^{-1} g_n^{\varepsilon_n}$. The implication (d) $\Rightarrow$ (a) can be deduced from [@chiswell1993soluble Lemma 8]; see also [@glass1999partially Lemma 2.2.3]. ◻ Finally, we prove that the set of left relatively convex subgroups is closed under arbitrary intersection, generalising [@kopytov1996right Proposition 5.1.10], the same property for relatively convex subgroups of a left-orderable group. [\[convex-intersection\]]{#convex-intersection label="convex-intersection"} The intersection of left relatively convex subgroups of $G$ is left relatively convex in $G$. *Proof.* Let $\left\{H_\alpha: \alpha \in I\right\}$ be a family of left relatively convex subgroups of $G$. By the implication (a) $\Rightarrow$ (b) in Proposition [\[prop-rel-convex-2\]](#prop-rel-convex-2){reference-type="ref" reference="prop-rel-convex-2"}, there exist totally ordered sets $(\Omega_\alpha,<_\alpha)$ ($\alpha\in I$) such that $H_\alpha$ is the kernel of an order preserving $G$-action on $(\Omega_\alpha,<_\alpha)$. Let $\Omega$ denote the disjoint union of all $\Omega_\alpha$ ($\alpha\in I$). We choose an arbitrary total order $<_{\mbox{index}}$ on $I$, then define $x<y$ for some $x\in \Omega_\alpha$ and $y\in \Omega_\beta$ if and only if either $\alpha<_{\mbox{index}}\beta$, or $\alpha=\beta$ and $x<_\alpha y$. In this way, the binary relation $<$ be a total order on $\Omega$ such that each inclusion map $\Omega_\alpha\hookrightarrow \Omega$ is order preserving. Therefore, the natural $G$-action on $\Omega$ is order preserving. Since the kernel of this group action is $\bigcap_{\alpha \in I} H_\alpha$, by the implication (b) $\Rightarrow$ (a) in Proposition [\[prop-rel-convex-2\]](#prop-rel-convex-2){reference-type="ref" reference="prop-rel-convex-2"}, the statement holds true. ◻ Now we state the criterion for the existence of a predicate satisfying all axioms for a bi-invariant circular order, except for the axiom of cyclicity. Let $\overline{\mathbf{AxPCL}}$ denote $\mathbf{AxPCL}$ minus the axiom of cyclicity. There exists a binary relation $P(\cdot,\cdot)$ on the group $G$ satisfying the axioms in $\overline{\mathbf{AxPCL}}\cup\mathbf{PCB}$ if and only if the centraliser of each subset of $G$ is left relatively convex in $G$. *Proof.* We first prove the "only if" part. Suppose that there exists a binary relation $P(\cdot,\cdot)$ on $G$ satisfying the axioms in $\overline{\mathbf{AxPCL}}\cup\mathbf{PCB}$. For each $x\in G$, we define a binary relation $\le_x$ on $G$ by $y\le_x z$ if and only if either $y^{-1}z$ commutes with $x$, or $P(yxy^{-1},zxz^{-1})$ holds true. We prove that $\le_x$ is a left total preorder on $G$ with the residue being the centraliser of $x$. 1. The reflexivity follows from that $e$ commutes with $x$. 2. Suppose that $y\le_x z$ and $z\le_x u$. If both $y^{-1}z$ and $z^{-1} u$ commute with $x$, then $y^{-1} u$ also commutes with $x$. We consider four scenarios. If $y^{-1}z$ commutes with $x$ and $P(zxz^{-1},uxu^{-1})$ holds, then by $zxz^{-1}= yxy^{-1}$, we have $P(yxy^{-1},uxu^{-1})$. If $P(yxy^{-1},zxz^{-1})$ holds and $z^{-1}u$ commutes with $x$ then by $zxz^{-1}= uxu^{-1}$, we also have $P(yxy^{-1},uxu^{-1})$. If both $P(yxy^{-1},zxz^{-1})$ and $P(zxz^{-1},uxu^{-1})$ hold true, then by the transitivity axiom in $\overline{\mathbf{AxPCL}}$, we have $P(yxy^{-1},uxu^{-1})$. In either way, we have $z\le_x u$. Thus we proved the transitivity of $\le_x$. 3. The totality follows from the connectedness axiom in $\overline{\mathbf{AxPCL}}$. 4. The left-invariance follows from the conjugacy invariance axiom $\mathbf{PCB}$. 5. If $y$ is in the centraliser of $x$, then so is $y^{-1}$. By definition, we have $e\le_x y\le_x e$. So the centraliser of $x$ is a subgroup of the residue of $\le_x$. 6. If $y$ is not in the centraliser of $x$, then neither is $y^{-1}$. If we also have $e\le_x y \le_x e$, then by definition, we have $P(x,yxy^{-1})$ and $P(yxy^{-1},x)$. By the transitivity axiom in $\overline{\mathbf{AxPCL}}$, we have $P(x,x)$, which contradicts to the irreflexivity axiom in $\overline{\mathbf{AxPCL}}$. Thus the residue of $\le_x$ is a subgroup of the centraliser of $x$. By the implication (b) $\Rightarrow$ (a) in Proposition [\[prop-rel-convex-1\]](#prop-rel-convex-1){reference-type="ref" reference="prop-rel-convex-1"}, the centraliser of any group element is left relatively convex in $G$. By Proposition [\[convex-intersection\]](#convex-intersection){reference-type="ref" reference="convex-intersection"}, the centraliser of any subset is left relatively convex in $G$. Then we prove the "if" part. Suppose that the centraliser of any $x\in G$ is left relatively convex in $G$. Let $\mathrm{Con}(G)$ denote the set of conjugacy classes of $G$. For each conjugacy class $\gamma\in \mathrm{Con}(G)$, select an group element $x_\gamma$ in $\gamma$. By the implication (a) $\Rightarrow$ (b) in Proposition [\[prop-rel-convex-1\]](#prop-rel-convex-1){reference-type="ref" reference="prop-rel-convex-1"}, there exists a left total preorder $\le_\gamma$ on $G$ such that the residue group of $\le_\gamma$ is the centraliser of $x_\gamma$. We choose an arbitrary total order $<_{\mbox{index}}$ on $\mathrm{Con}(G)$, then define $P(y,z)$ for some $y\in \gamma_1$ and $z\in \gamma_2$ ($\gamma_1, \gamma_2\in \mathrm{Con}$) if and only if either $\gamma_1<_{\mbox{index}}\gamma_2$, or $\gamma_1=\gamma_2$ and the following four conditions hold for some pair $(y_0, z_0)\in G\times G$: 1. $y= y_0 x_{\gamma_1} y_0^{-1}$. 2. $z= z_0 x_{\gamma_1} z_0^{-1}$. 3. $y_0\le_{\gamma_1} z_0$ holds. 4. $z_0\le_{\gamma_1} y_0$ does not hold. We prove that the binary relation $P(\cdot,\cdot)$ satisfies the axioms in $\overline{\mathbf{AxPCL}}\cup\mathbf{PCB}$. 1. We first prove the irreflexivity axiom in $\overline{\mathbf{AxPCL}}$. Suppose that $P(y,y)$ for some $y\in G$. By definition, there exists $(y_0,z_0)\in G\times G$ satisfying the four conditions above. Let $\gamma$ denote the conjugacy class of $y$, then by $y= y_0 x_{\gamma} y_0^{-1}=z_0 x_{\gamma} z_0^{-1}$, the element $y_0^{-1} z_0$ is in the centraliser of $x_\gamma$. In this case, we have $z_0\le_{\gamma_1} y_0$, which contradicts to the fourth condition. 2. Then we prove the transitivity axiom in $\overline{\mathbf{AxPCL}}$. Suppose that both $P(y,z)$ and $P(z,u)$ hold true. Let $\gamma_1, \gamma_2,\gamma_3$ denote the conjugacy classes of $y,z,u$ respectively. By definition, we have $\gamma_1\le_{\mbox{index}}\gamma_2$ and $\gamma_2\le_{\mbox{index}}\gamma_3$. By the transitivity of $<_{\mbox{index}}$, we have $\gamma_1\le_{\mbox{index}}\gamma_3$. If $\gamma_1<_{\mbox{index}}\gamma_3$, then $P(y,u)$ holds true. Otherwise, we have $\gamma_1=\gamma_2=\gamma_3$, and there exist pairs $(y_0,z_0)$ and $(y_1,z_1)\in G\times G$ satisfying the four conditions for $(y,z)$ and $(z,u)$ respectively. We prove that the pair $(y_0,z_1)$ satisfies the four conditions for $(y,u)$. The first two conditions follow from corresponding conditions for $(y,z)$ and $(z,u)$. By $z= z_0 x_{\gamma_1} z_0^{-1}= y_1x_{\gamma_1} y_1^{-1}$, the element $y_1^{-1} z_0$ is in the centraliser of $x_{\gamma_1}$. Thus we have $z_0\le_{\gamma_1} y_1$. By the transitivity of $\le_{\gamma_1}$, we have $y_0 \le_{\gamma_1} z_0\le_{\gamma_1} y_1\le_{\gamma_1} z_1$. Thus the third condition is satisfied. If $z_1\le_{\gamma_1}y_0$ holds, then by the transitivity of $\le_{\gamma_1}$, we have $z_0\le_{\gamma_1} y_1\le_{\gamma_1} z_1\le_{\gamma_1} y_0$, which contradicts to the corresponding conditions for $(y,z)$. Therefore, we have $P(y,u)$ if both $P(y,z)$ and $P(z,u)$ hold true, 3. Now we prove the connectedness axiom in $\overline{\mathbf{AxPCL}}$. Suppose that $y$ and $z$ are two distinct group elements. Let $\gamma_1$ and $\gamma_2$ denote the conjugacy classes of $y$ and $z$ respectively. If $\gamma_1\neq \gamma_2$, then by the connectedness of $<_{\mbox{index}}$, we have either $\gamma_1<_{\mbox{index}}\gamma_2$ or $\gamma_2<_{\mbox{index}}\gamma_2$, thus we have either $P(y,z)$ or $P(z,y)$. Otherwise, suppose that $\gamma_1= \gamma_2$, then there exists $(y_0,z_0)\in G\times G$ such that $y=y_0 x_{\gamma_1}y_0^{-1}$ and $z=z_0 x_{\gamma_1}z_0^{-1}$. By the totality of $\le_{\gamma_1}$, we have either $y_0\le_{\gamma_1} z_0$ or $z_0\le_{\gamma_1} y_0$. By $y\neq z$, the element $y_0^{-1}z_0$ is not in the centraliser of $x_{\gamma_1}$, thus only one of $y_0\le_{\gamma_1} z_0$ and $z_0\le_{\gamma_1} y_0$ holds. 4. Finally we prove the conjugacy invariance axiom $\mathbf{PCB}$. Suppose that $P(y,z)$ holds true. Let $\gamma_1$ and $\gamma_2$ denote the conjugacy classes of $y$ and $z$ respectively. Then by definition, either $\gamma_1<_{\mbox{index}} \gamma_2$, or $\gamma_1=\gamma_2$ and there exists $(y_0,z_0)\in G\times G$ satisfying the four conditions for $(y,z)$. For any $u\in G$, we have $uyu^{-1}\in\gamma_1$ and $uzu^{-1}\in \gamma_2$. If $\gamma_1<_{\mbox{index}} \gamma_2$, then we have $P(uyu^{-1},uzu^{-1})$. Otherwise, we prove that $(u y_0,u z_0)\in G\times G$ satisfies the four conditions for $(uyu^{-1}, u z u^{-1})$. The first two conditions follow from corresponding conditions for $(y,z)$. The last two conditions follow from corresponding conditions for $(y,z)$ and the left-invariance of $\le_{\gamma_1}$. Therefore $P(y,z)$ implies $P(uyu^{-1},uzu^{-1})$.  ◻ # Absolute cofinality and non-left-orderability {#sec:cofinality} In this section, we introduce a methodology to integrate the fixed point method for non-left-orderability into automated reasoning. The term "fixed point method" originates from the dynamic realisation of a left order. However, instead of introducing the dynamic realisation, we opt for the concept of cofinal elements for convenience, which essentially yields the same proofs. We define the left absolute cofinality as a dual concept of the left relative convexity. This makes our definition slightly different from traditional usage. To begin with, we introduce the concept of the left relatively convex subgroup closure and integrate it into automated reasoning. ## Relatively convex subgroup closure By Proposition [\[convex-intersection\]](#convex-intersection){reference-type="ref" reference="convex-intersection"}, the left relatively convex subgroups form a Moore collection. So we can define a natural closure operator $\mathrm{cl}$ based on this family. The following definition extends the definition of the relatively convex subgroup closure in [@longobardi2000right] to arbitrary groups. [\[def-closure\]]{#def-closure label="def-closure"} For a subset $A$ in a group $G$, we define the left relatively convex subgroup closure $\mathrm{cl}(A)$ as the intersection of all left relatively convex subgroups $H$ with $A\subseteq H \subseteq G$. By Proposition [\[convex-intersection\]](#convex-intersection){reference-type="ref" reference="convex-intersection"}, a subset $A\subseteq G$ is a left relatively convex subgroup of $G$ if and only if $\mathrm{cl}(A)=A$. [\[prop-equiv-cofinal\]]{#prop-equiv-cofinal label="prop-equiv-cofinal"} Let $A$ be a subset of the group $G$, and $g\in G$ be a group element. Let $\langle A\rangle$ denote the subgroup generated by $A$. The following statements are equivalent. 1. The element $g$ is in the left relatively convex subgroup closure $\mathrm{cl}(A)$. 2. For any subsemigroup $S$ with $A\cup A^{-1}\subseteq S\subseteq G$ and $S\cup S^{-1}=G$, we have $g\in S$. 3. For any left total preorder $\le$ on $G$, there exists $x\in \langle A\rangle$ such that $g\le x$. *Proof.* First we prove the implication (a) $\Rightarrow$ (b). Suppose that $S$ is a subsemigroup $S$ with $A\cup A^{-1}\subseteq S\subseteq G$ and $S\cup S^{-1}=G$. Let $P$ denote the set $S\setminus S^{-1}$. By $S\cup S^{-1}=G$, the group $G$ can be written as the disjoint union $P\sqcup (S\cap S^{-1})\sqcup P^{-1}$. We first prove that $P$ is a subsemigroup. For any $x,y\in P$, we have $xy\in S$ by the closure of $S$ under multiplication. If we have $xy\in S^{-1}$, then $y^{-1}=(xy)^{-1}x$ is in $S$ by the closure of $S$ under multiplication, which contradicts to the assumption that $y\in P$. Now we prove that $(S\cap S^{-1})P(S\cap S^{-1})\subseteq P$. On the one hand, since $P\subseteq S$, by the closure of $S$ under multiplication, we have $(S\cap S^{-1})P(S\cap S^{-1})\subseteq S$. On the other hand, if there exist $x,z\in S\cap S^{-1}$ and $y\in P$ such that $xyz\in S^{-1}$, then $y^{-1}=z(xyz)^{-1}x$ is in $S$ by the closure of $S$ under multiplication, which contradicts to the assumption that $y\in P$. So we have $(S\cap S^{-1})P(S\cap S^{-1})\subseteq P$ by the definition of $P$. Therefore $S\cap S^{-1}$ is left relatively convex by the condition (b) in Definition [\[def-rel-convex\]](#def-rel-convex){reference-type="ref" reference="def-rel-convex"}. By the assumption that $g\in\mathrm{cl}(A)$, we have $$g\in\mathrm{cl}(A)\subseteq \mathrm{cl}(S\cap S^{-1})= S\cap S^{-1} \subseteq S.$$ Then we prove the implication (b) $\Rightarrow$ (c). For any left total preorder $\le$ on $G$, consider the set $$S:=\{x\in G: \mbox{for all } a_1 \in \langle A\rangle\mbox{ there exists } a_2\in \langle A\rangle \mbox{ such that } x a_1\le a_2\}.$$ We prove that $S$ satisfies the condition described in (b). First, for any $x\in A\cup A^{-1}$ and any $a_1\in \langle A\rangle$, we can choose $a_2 = x a_1 \in \langle A\rangle$. Thus by the reflexivity of $\le$, we have $A\cup A^{-1}\subseteq M$. Second, if $x, y\in S$, then for all $a_1\in \langle A\rangle$ there exists $a_2\in \langle A\rangle$ such that $y a_1\le a_2$, and also there exists $a_3 \in \langle A\rangle$ such that $x a_2 \le a_3$. By the left-invariance and the transitivity of $\le$, we have $xya_1\le x a_2\le a_3$. Thus we have $xy\in S$ by the definition of $S$. In other words, $S$ is a subsemigroup. Finally, for any $x\in G$, if both $x\not\in S$ and $x^{-1}\not \in S$ hold true, then there exists $a_1\in \langle A\rangle$ such that $\lnot(x a_1\le a_2)$ for all $a_2\in \langle A\rangle$, and there exists $a_3\in \langle A\rangle$ such that $\lnot(x^{-1} a_3\le a_4)$ for all $a_4\in \langle A\rangle$. In particular, we have $\lnot(x a_1\le a_3)$ and $\lnot(x^{-1} a_3\le a_1)$. By the left-invariance of $\le$, neither $x a_1\le a_3$ nor $a_3\le x a_1$ holds true, which contradicts to the totality of $\le$. Thus we have $S\cup S^{-1}=G$. Suppose that (b) holds, then we have $g\in S$. By taking $a_1=e$ in the definition of $S$, there exists $x\in \langle A\rangle$ such that $g\le x$. Finally we prove the implication (c) $\Rightarrow$ (a). Suppose that $H$ is an arbitrary left relatively convex subgroup with $A\subseteq H\subseteq G$. Since $H$ is a subgroup, we have $\langle A\rangle \subseteq H$. By the implication (a) $\Rightarrow$ (b) in Proposition [\[prop-rel-convex-1\]](#prop-rel-convex-1){reference-type="ref" reference="prop-rel-convex-1"}, there exists a left total preorder $\le$ on $G$ with the residue being $H$. Suppose that (c) holds, then there exists $x\in \langle A\rangle$ such that $g\le x$. Define $x\le_{op} y$ if and only if $y\le x$, then we can check that $\le_{op}$ is also a left total preorder. By the statement (c), there exists $y\in \langle A\rangle$ such that $g\le_{op} y$. Because $e\le y\le g\le x\le e$, we have $g\in H$. By the arbitrariness of $H$, we have $g\in \mathrm{cl}(A)$. ◻ Now we present a methodology for proving $g_{k+1}\in \mathrm{cl}(\{g_1,\ldots, g_k\})$ in a group $G$ using generic automated theorem proving based on the implication (b) $\Rightarrow$ (a) in Proposition [\[prop-equiv-cofinal\]](#prop-equiv-cofinal){reference-type="ref" reference="prop-equiv-cofinal"}. For convenience, let $\mathbf{CC}$ denote the following axioms for a unary predicate $P(\cdot)$: 1. $\forall x \forall y (P(x)\land P(y)\to P(x\cdot y))$, *(closure)* 2. $\forall x (P(x)\lor P(x'))$. *(connectedness)* [\[prop-cl\]]{#prop-cl label="prop-cl"} Let $G$ be a group with presentation $\langle S|R\rangle$. Let $t_1, \ldots, t_k, t_{k+1}$ be ground terms representing group elements $\bar{t}_1, \ldots, \bar{t}_k, \bar{t}_{k+1}$ in $G$ respectively. Then the theory $\mathbf{Gr}\cup \mathbf{Ax}_R\cup \mathbf{CC}\cup \{P(t_i)\land P(t_i'): i=1,\ldots, k\} \cup \{\lnot P(t_{k+1})\}$ is inconsistent if and only if $\bar{t}_{k+1}$ is in the left relatively convex subgroup closure $\mathrm{cl}(\{\bar{t}_1,\ldots,\bar{t}_k\})$. *Proof.* We first prove the "only if" part. Assume that $\bar{t}_{k+1}$ is not in the left relatively convex subgroup closure $\mathrm{cl}(\{\bar{t}_1,\ldots,\bar{t}_k\})$. By the implication (b) $\Rightarrow$ (a) in Proposition [\[prop-equiv-cofinal\]](#prop-equiv-cofinal){reference-type="ref" reference="prop-equiv-cofinal"}, there exists a subsemigroup $S_+\subseteq G$ with $\bar{t}_i,\bar{t}_i^{-1}\in S_+$ ($1\le i\le k$), $S_+\cup S_+^{-1}=G$, and $\bar{t}_{k+1}\not\in S_+$. Define the unary predicate $P$ by $P(t)$ if and only if $t$ represents an element in $S_+$. Then we can check that the group $G$ together with the predicate $P$ constitutes a model for the theory $\mathbf{Gr}\cup \mathbf{Ax}_R\cup\mathbf{CC}\cup \{P(t_i)\land P(t_i'): i=1,\ldots, k\} \cup \{\lnot P(t_{k+1})\}$. Therefore it is consistent if $\bar{t}_{k+1}$ is not in $\mathrm{cl}(\{\bar{t}_1,\ldots,\bar{t}_k\})$. Then we prove the "if" part. Assume that the theory $\mathbf{Gr}\cup \mathbf{Ax}_R\cup \mathbf{CC}\cup \{P(t_i)\land P(t_i'): i=1,\ldots, k\} \cup \{\lnot P(t_{k+1})\}$ is consistent, then there is a model $\mathfrak{M}$ of it. Let $S_+\subseteq G$ denote the subset $$S_+:=\{x\in G: x\mbox{ is represented by a ground term }t\mbox{ such that } P(t) \mbox{ in } \mathfrak{M}\}$$ Then $S_+$ has the following properties. 1. For any $x,y\in S_+$, there exist ground terms $t_1, t_2$ representing $x,y$ respectively such that $P(t_1)$ and $P(t_2)$ hold true in $\mathfrak{M}$. By the closure axiom in $\mathbf{CC}$, $P(t_1\cdot t_2)$ holds true in $\mathfrak{M}$. Since $xy$ is represented by $t_1\cdot t_2$, we have $xy\in S_+$ for any $x,y\in S_+$. Thus $S_+$ is a subsemigroup. 2. For each $1\le i\le k$, the elements $\bar{t}_i$ and $\bar{t}_i^{-1}$ in $G$ are represented by the ground terms $t_i$ and $t_i'$ respectively. Thus by $P(t_i)\land P(t_i')$, we have $\bar{t}_i, \bar{t}_i^{-1}\in S_+$. 3. For any $x\in G$, let $t$ be a ground term representing $x$. Then $x^{-1}$ is represented by the ground term $t'$. By the connectedness axiom in $\mathbf{CC}$, either $P(t)$ or $P(t')$ holds true in $\mathfrak{M}$. Thus we have $S_+\cup S_+^{-1}=G$. For any ground term $s$ representing $\bar{t}_{k+1}$ in $G$, by the definition of $\langle S|R\rangle$, the axioms $\mathbf{Gr}\cup \mathbf{Ax}_R$ imply that $s=t_{k+1}$ in $\mathfrak{M}$. By the axiom $\neg P(t_{k+1})$, we have $\neg P(s)$ in $\mathfrak{M}$. Thus we have $\bar{t}_{k+1}\not\in S_+$. By the implication (a) $\Rightarrow$ (b) in Proposition [\[prop-equiv-cofinal\]](#prop-equiv-cofinal){reference-type="ref" reference="prop-equiv-cofinal"}, the element $\bar{t}_{k+1}$ is not in the left relatively convex subgroup closure $\mathrm{cl}(\{\bar{t}_1,\ldots, \bar{t}_{k}\})$. ◻ ## Absolute cofinality We say a subgroup $H\subseteq G$ is *left absolutely cofinal* if the left relatively convex subgroup closure $\mathrm{cl}(H)$ equals to $G$. By the equivalence (a) $\Leftrightarrow$ (c) in Proposition [\[prop-equiv-cofinal\]](#prop-equiv-cofinal){reference-type="ref" reference="prop-equiv-cofinal"}, a subgroup $H\subseteq G$ is left absolutely cofinal if and only if it is cofinal[^1] with respect to every left total preorder $\le$ on $G$. It follows from the condition (a) in Definition [\[def-rel-convex\]](#def-rel-convex){reference-type="ref" reference="def-rel-convex"} that the trivial subgroup $\{e\}$ is left relatively convex if and only if $G$ is left-orderable. Now we establish the left absolutely cofinal counterpart of this fact. [\[proposition-e-cofinal\]]{#proposition-e-cofinal label="proposition-e-cofinal"} The trivial subgroup $\{e\}$ is not left absolutely cofinal in the group $G$ if and only if $G$ admits a nontrivial left-orderable quotient. *Proof.* We first prove the "only if" part. If $e$ is not left absolutely cofinal in the group $G$, then there exists a proper subgroup $H$ which is left relatively convex in $G$. By the implication (a) $\Rightarrow$ (d) in Proposition [\[prop-rel-convex-2\]](#prop-rel-convex-2){reference-type="ref" reference="prop-rel-convex-2"}, for any finite set of elements $g_1,\ldots, g_n\in G\setminus H$, there exist $\varepsilon_1,\ldots,\varepsilon_n\in\{-1,1\}$ such that the subsemigroup generated by $g_1^{\varepsilon_1},\ldots,g_n^{\varepsilon_n}$ has empty intersection with $H$. By [@glass1999partially Lemma 2.2.3], the quotient group $G/\mathrm{core}(H)$ is left-orderable, where $\mathrm{core}(H):=\cap\{g H g^{-1}:g\in G\}$ is the largest normal subgroup of $G$ contained in $H$. Since $H$ is proper, the quotient group $G/\mathrm{core}(H)$ is nontrivial. Then we prove the "if" part. Suppose that $N$ is a proper normal subgroup of $G$ such that $G/N$ is left-orderable. The $G$-action on $G/N$ preserves the left orders, so by the condition (a) in Definition [\[def-rel-convex\]](#def-rel-convex){reference-type="ref" reference="def-rel-convex"}, the subgroup $N$ is left relatively convex in $G$. Therefore $e$ is not left absolutely cofinal in $G$. ◻ It is worth noting that the condition that a group does not admit any nontrivial left-orderable quotients holds significant importance in topology. According to the L-space conjecture, it is conjectured that a closed connected $3$-manifold is an L-space if and only if its fundamental group does not admit any nontrivial left-orderable quotient. If we replace this condition with the non-left-orderability, we would have to assume the manifold is at least irreducible. This would only make things more complicated; compare [@boyer2022order Theorem 1.9] to [@boyer2022order Corollary 1.10] for example. While one may establish the non-left-orderability by proving $\mathrm{cl}(e)=G$ through successive applications of Proposition [\[prop-cl\]](#prop-cl){reference-type="ref" reference="prop-cl"}, this method is not much different from the positive cone formalisation presented in Proposition [\[prop:positive_cone\]](#prop:positive_cone){reference-type="ref" reference="prop:positive_cone"} and Proposition [\[prop:positive_cone_C\]](#prop:positive_cone_C){reference-type="ref" reference="prop:positive_cone_C"}. In the remainder of this subsection, we introduce an alternative approach that simplifies the computation. It is well-known that the absolute cofinality of a cyclic subgroup $\langle g \rangle$ implies that every conjugate of $g$ has the same sign with respect to any given left order on $G$; see [@conrad1959right Property 3.1] or [@boyer2017foliations Lemma 4.6] for example. In order to generalise this fact to left total preorders, we prove the following statement. [\[prop-isolated\]]{#prop-isolated label="prop-isolated"} Let $\le$ be a left total preorder on the group $G$. If $e \le x^n$ for some $x\in G$ and some positive integer $n$, then $e\le x$. *Proof.* By the connectedness of $\le$, we have either $e\le x$ or $x\le e$. In the second case, we have $$e\le x^n\le x^{n-1}\le \cdots \le x$$ by the transitivity and the left-invariance of $\le$. Therefore, in either case, we have $e\le x$. ◻ Now we generalise [@boyer2017foliations Lemma 4.6] to left total preorders. [\[prop-cofinal-conjugation\]]{#prop-cofinal-conjugation label="prop-cofinal-conjugation"} Let $\le$ be a left total preorder on the group $G$. Let $x\in G$ be an element with $e\le x$ and $\langle x\rangle$ being left absolutely cofinal. Then we have $e\le y x y^{-1}$ for every $y\in G$. *Proof.* By the connectedness of $\le$, we have either $y\le e$ or $e\le y$. We consider two scenarios separately. First, suppose that $y\le e$. Then by the cofinality of $\langle x\rangle$ with respect to $\le$, there exists an integer $n$ such that $y^{-1}\le x^n$. Because $$y^{-1}\le x^n\le x^{n+1}\le x^{n+2}\le \cdots,$$ we can assume that $n$ is positive. And we have $$e \le yx^n \le yx^n y^{-1}=(y x y^{-1})^n.$$ By Proposition [\[prop-isolated\]](#prop-isolated){reference-type="ref" reference="prop-isolated"}, we have $e\le yxy^{-1}$. Then, suppose instead that $e \le y$. Define the binary relation $\le_{op}$ by $x\le_{op}y$ if and only if $y\le x$, then we can check that $\le_{op}$ is also a left total order on $G$. By the cofinality of $\langle x \rangle$ with respect to $\le_{op}$, there exists an integer $n$ such that $y^{-1}\le_{op} x^{-n}$. Because $$\cdots\le x^{-n-2}\le x^{-n-1}\le x^{-n}\le y^{-1},$$ we can also assume that $n$ is positive. And we have $$e\le y \le y x^n y^{-1}= (yxy^{-1})^n.$$ By Proposition [\[prop-isolated\]](#prop-isolated){reference-type="ref" reference="prop-isolated"}, we have $e\le yxy^{-1}$. ◻ A subsemigroup $A$ of the group $G$ is called *isolated* if $x^n\in A$ for some positive integer $n$ implies $x\in A$ for every $x\in G$. It is called *normal* if $xAx^{-1}\subseteq A$ for every $x\in G$. The following statement is a left total preorder adaptation of the technique developed in [@nie20201 Section 3]. [\[prop-stage-2\]]{#prop-stage-2 label="prop-stage-2"} Let $g_0, g_1, \ldots, g_k\in G$. Suppose that $\langle g_i\rangle$ is left absolutely cofinal in $G$ for each $0\le i\le k$, and that $\{e\}$ is not left absolutely cofinal in $G$. Then there exists an isolated normal subsemigroup $S$ such that: 1. for each $1\le i\le k$, either $g_i\in S$ or $g_i^{-1}\in S$ holds; 2. the element $g_0$ is in $S$; 3. the identity element $e$ is not in $S$. *Proof.* Since $\{e\}$ is not left absolutely cofinal in $G$, there exists a proper subgroup $H$ which is left relatively cofinal in $G$. By the implication (a) $\Rightarrow$ (b) in Proposition [\[prop-rel-convex-1\]](#prop-rel-convex-1){reference-type="ref" reference="prop-rel-convex-1"}, there exists a left total preorder $\le$ on $G$ with the residue being $H$. We assume $e\le g_0$ without loss of generality, because otherwise we can replace $\le$ with $\le_{op}$, where $x\le_{op} y$ if and only if $y\le x$. For any $g\in G$, we define the binary relation $\le_{g}$ by $x \le_g y$ if and only if $x g \le y g$. Then the reflexivity, transitivity, totality, and left-invariance of $\le$ imply the same properties for $\le_g$ respectively. Therefore $\le_g$ is a left total order on $G$. Moreover, the residue group of $\le_g$ is $g^{-1}Hg$. Let $P_g$ denote the set $\{x\in G\setminus g^{-1} H g: e\le_g x\}$. Then by the proof of the implication (b) $\Rightarrow$ (a) in Proposition [\[prop-rel-convex-1\]](#prop-rel-convex-1){reference-type="ref" reference="prop-rel-convex-1"}, the set $P_g$ is a subsemigroup of $G$ such that $P_g\sqcup g^{-1} H g\sqcup P_g^{-1}$ is a partition of $G$, and $H P_g H\subseteq P_g$. We prove that the subset $S:= \bigcap_{g\in G} P_g$ satisfies the desired conditions. 1. $S$ is a subsemigroup, because every $P_g$ is a subsemigroup. 2. If $x^n\in S$ for some $x\in G$ and some positive integer $n$, then we have $e\le_g x^n$ and $x^n\not\in g^{-1} H g$ for every $g\in G$. By Proposition [\[prop-isolated\]](#prop-isolated){reference-type="ref" reference="prop-isolated"}, we have $e\le_g x$ for every $g\in G$. Since $g^{-1}H g$ is a subgroup, we have $x\not\in g^{-1} H g$ for any $g\in G$. Thus we have $x\in S$. In other words, $S$ is isolated. 3. By the equation $x P_g x^{-1}= P_{xg}$, the subsemigroup $S$ is normal. 4. For each $0\le i\le k$, since $\langle g_i\rangle$ is left absolutely cofinal in $G$, by the implication (b) $\Rightarrow$ (a) in Proposition [\[prop-rel-convex-1\]](#prop-rel-convex-1){reference-type="ref" reference="prop-rel-convex-1"}, we have $g_i\not \in g^{-1}H g$ for any $g\in G$. By $g_i\not\in H$, there exist exponents $\varepsilon_i\in\{-1,1\}$ such that $g_i^{\varepsilon_i}\in P_e$. By $e\le g_0$, we have $\varepsilon_0=1$. Since $g^{-1}H g$ is a subgroup, we have $g_i^{\varepsilon_i}\not \in g^{-1}H g$ for any $g\in G$. Since $\langle g_i^{\varepsilon_i}\rangle=\langle g_i\rangle$ is left absolutely cofinal in $G$, by Proposition [\[prop-cofinal-conjugation\]](#prop-cofinal-conjugation){reference-type="ref" reference="prop-cofinal-conjugation"}, we have $e\le_g g_i^{\varepsilon_i}$ for any $g\in G$. Therefore, we have $g_0\in S$, and $g_i^{\varepsilon_i}\in S$ for each $1\le i \le k$. 5. Since $e\in g^{-1}H g$ for every $g\in G$, we have $e\not\in S$.  ◻ Now we present our methodology to integrate the fixed point method for non-left-orderability into automated reasoning. We suppose that the left absolute cofinality was checked through successive uses of Proposition [\[prop-cl\]](#prop-cl){reference-type="ref" reference="prop-cl"}. Then we may use Proposition [\[prop-fixed-point\]](#prop-fixed-point){reference-type="ref" reference="prop-fixed-point"} below to establish the non-left-orderability. For each positive integer $m$ and each ground term $t$, recursively define the ground term $t^m$ by $t^1=t$ and $t^m=t^{m-1}\cdot t$ for $m\ge 2$. Let $M$ be a set of positive integers. Let $\mathbf{Isolated}(M)$ denote the set of axioms for a unary predicate $P(\cdot)$, containing the following axiom for each $m\in M$: 1. $\forall x (P(x^m)\to P(x))$. *($m$-isolation)* [\[prop-fixed-point\]]{#prop-fixed-point label="prop-fixed-point"} Let $G$ be a group with presentation $\langle S|R \rangle$. Let $t_0, t_1,\ldots,t_k$ be ground terms representing group elements $\bar{t}_0,\bar{t}_1,\ldots, \bar{t}_k$ in $G$ respectively. Suppose that $\langle \bar{t}_i\rangle$ is left absolutely cofinal in $G$ for each $0\le i\le k$. Let $\mathbf{AxPL}'$ denote $\mathbf{AxPL}$ minus the axiom of connectedness. Let $M$ be a set of positive integers. If the theory $\mathbf{Gr}\cup\mathbf{Ax}_R\cup\mathbf{AxPL}' \cup \mathbf{PB}\cup \mathbf{Isolated}(M)\cup \{P(t_0)\} \cup\{P(t_i)\lor P(t_i'): i=1,\ldots, k\}$ is inconsistent, then $G$ does not admit nontrivial left-orderable quotients. *Proof.* If $G$ admits a nontrivial left-orderable quotient, then by Proposition [\[proposition-e-cofinal\]](#proposition-e-cofinal){reference-type="ref" reference="proposition-e-cofinal"}, the trivial subgroup $\{e\}$ is not left absolutely cofinal in $G$. Then by Proposition [\[prop-stage-2\]](#prop-stage-2){reference-type="ref" reference="prop-stage-2"}, there exists an isolated normal subsemigroup $S_+$ such that: 1. for each $1\le i\le k$, either $\bar{t}_i\in S_+$ or $\bar{t}_i^{-1}\in S_+$ holds; 2. the element $\bar{t}_0$ is in $S_+$; 3. the identity element $e$ is not in $S_+$. Define the unary predicate $P$ by $P(t)$ if and only if $t$ represents an element in $S_+$. Then we can check that the group $G$ together with the predicate $P$ constitutes a model for the theory $\mathbf{Gr}\cup\mathbf{Ax}_R\cup\mathbf{AxPL}' \cup \mathbf{PB}\cup \mathbf{Isolated}(M)\cup\{P(t_0)\}\cup \{P(t_i)\lor P(t_i'): i=1,\ldots, k\}$. Therefore the theory is consistent if $G$ admits a nontrivial left-orderable quotient. ◻ Notice that the set of axioms $\{P(t_0)\} \cup\{P(t_i)\lor P(t_i'): i=1,\ldots, k\}$ in Proposition [\[prop-fixed-point\]](#prop-fixed-point){reference-type="ref" reference="prop-fixed-point"} has the same form as $\mathfrak{S}^P$ defined in Proposition [\[prop:positive_cone\]](#prop:positive_cone){reference-type="ref" reference="prop:positive_cone"}. Thus, for convenience, we name this set as $\mathfrak{S}^P$ with respect to the pairs $(e, t_i)$ ($i=0,1,\ldots k$). ## Examples Consider the Weeks manifold group with the presentation as shown in Example [\[ex:weeks\]](#ex:weeks){reference-type="ref" reference="ex:weeks"}. We prove that this group does not admit nontrivial left-orderable quotients using automated reasoning. First, we prove that $\mathbf{Gr}\cup \mathbf{Ax}_R\cup\mathbf{CC}\cup\{P(a)\land P(a'), \neg P(b)\}$ is inconsistent (Task 16.1) using Prover9. By Proposition [\[prop-cl\]](#prop-cl){reference-type="ref" reference="prop-cl"}, we have $b\in\mathrm{cl}(a)$. Thus $\langle a\rangle$ is left absolutely cofinal. By symmetry, the subgroup $\langle b\rangle$ is also left absolutely cofinal. Consider the set $\mathfrak{S}^P$ with respect to the pairs $(e,a)$ and $(e,b)$. We prove that $\mathbf{Gr}\cup\mathbf{Ax}_R\cup\mathbf{AxPL}' \cup \mathbf{PB}\cup \mathfrak{S}^P$ is inconsistent (Task 16.2) using Prover9. By Proposition [\[prop-fixed-point\]](#prop-fixed-point){reference-type="ref" reference="prop-fixed-point"}, the group of interest does not admit nontrivial left-orderable quotients. The non-left-orderability follows from the nontriviality. Consider the fundamental group of the fourfold branched cover of the two-bridge knot $K_{[6,-6]}$. According to [@gordon2014taut Theorem 1.9], this group is not left-orderable. We consider the following presentation of the group: $$\langle a_i,b_i \;|\; a_i^3b_i=a_{i+1}^3, a_i b_i^3= b_{i-1}^3 \mbox{ for } i=0,1,2,3\rangle,$$ where the indices are taken modulo $4$. We can derive contradictions (Task 17.1 and Task 17.2) from the theories $\mathbf{Gr}\cup \mathbf{Ax}_R\cup \mathbf{CC}\cup\{P(a_0)\land P(a_0'),\neg P(b_0)\}$ and $\mathbf{Gr}\cup \mathbf{Ax}_R\cup \mathbf{CC}\cup\{P(b_0)\land P(b_0'),\neg P(a_1)\}$ using Prover9. By Proposition [\[prop-cl\]](#prop-cl){reference-type="ref" reference="prop-cl"}, we obtain $b_0\in\mathrm{cl}(a_0)$ and $a_1\in\mathrm{cl}(b_0)$. By symmetry, we have $b_i\in \mathrm{cl}(a_i)$ and $a_{i+1}\in\mathrm{cl}(b_i)$ for each $i=0,1,2,3$. Therefore, each $\langle a_i\rangle$ ($i=0,1,2,3$) is left absolutely cofinal. Consider the set $\mathfrak{S}^P$ with respect to the pairs $(e,a_i)$ ($i=0,1,2,3$). We prove that the theory $\mathbf{Gr}\cup \mathbf{Ax}_R \cup \mathbf{AxPL}'\cup \mathbf{PB}\cup \mathfrak{S}^P$ is inconsistent (Task 17.3) using Prover9. By Proposition [\[prop-fixed-point\]](#prop-fixed-point){reference-type="ref" reference="prop-fixed-point"}, the group does not admit nontrivial left-orderable quotients. The non-left-orderability follows from the nontriviality, which can be verified by building a model (Task 17.4) of $\mathbf{Gr}\cup \mathbf{Ax}_R\cup \{e\neq a_0\}$ using Mace4. [\[ex:Hyde\]]{#ex:Hyde label="ex:Hyde"} Hyde [@hyde2019group] proved that the group $\mathrm{Homeo}(D,\partial D)$ of homeomorphisms of the disc that fix the boundary is not left-orderable. He constructed a subgroup $H$ generated by six elements $a,b,c_0,d_0, c_1, d_1$, corresponding to $\alpha^{-1},\beta^{-1},\gamma, \delta, \gamma^{\eta}, \delta^{\eta}$ on [@hyde2019group page 4]. Let the subgroups $H_0, H_1$ be generated by $a,b,c_0,d_0$ and $a,b,c_1,d_1$ in $H$. Then he proved that, for any left order on $H_0$ (resp. $H_1$), we have $\max(a,a^{-1})<\max(b,b^{-1})$ (resp. $\max(a,a^{-1})>\max(b,b^{-1})$). In this example, we establish the non-left-orderability through automated reasoning. This proof is different from Hyde's proof and the one in [@triestino2021james]. The elements $a,b,c_0,d_0, c_1, d_1$ satisfy the following relations: $ab=ba$, $c_0b=bc_0$, $d_0 b=b d_0$, $c_0 a^3 c_0 = a^3$, $d_0 a^3 d_0 = a^3$, $(d_0^{-1} c_0 d_0 a)^6 =a^6 b^{36}$, $c_1a=ac_1$, $d_1a=ad_1$, $c_1 b^3 c_1 = b^3$, $d_1 b^3 d_1= b^3$, $(d_1^{-1} c_1 d_1 b)^6 =a^{36} b^{6}$. Note that the terms containing $a^{36}$ or $b^{36}$ pose a significant computational challenge for the automated prover. To address this issue, our first task is to manually simplify these relations. Define the set of group elements $T=\{t_0,t_1,t_2,t_3,t_4,t_5,t_6\}$ in $H$ by $t_0=a^3$, $t_1=b^3$, $t_2=t_1^{-2} d_0^{-1} c_0 d_0$, $t_3= t_0^{-2} d_1^{-1} c_1 d_1$, $t_4= t_2 a$, $t_5=t_3 b$, $t_6=t_4^3$, $t_7=t_5^3$. Let $R_0$ be the set of the following relations: $t_0=a^3$, $t_0 t_1=t_1 t_0$, $c_0 t_1 = t_1 c_0$, $d_0 t_1=t_1 d_0$, $c_0 t_0=t_0 c_0^{-1}$, $d_0 t_0=t_0d_0^{-1}$, $t_1^2 t_2 =d_0^{-1} c_0 d_0$, $t_4=t_2 a$, $t_6=t_4^3$, $t_6^2=t_0^2$. Similarly, let $R_1$ be the set of the following relations: $t_1=b^3$, $t_1 t_0=t_0 t_1$, $c_1 t_0=t_0 c_1$, $d_1 t_0=t_0 d_1$, $c_1 t_1=t_1 c_1^{-1}$, $d_1 t_1=t_1 d_1^{-1}$, $t_0^2 t_3=d_1^{-1} c_1 d_1$, $t_5= t_3 b$, $t_7=t_5^3$, $t_7^2=t_1^2$. Then we can check that $R_0$ and $R_1$ are satisfied in $H$. Note that the sets $R_0$ and $R_1$ are symmetric: $R_1$ can be obtained by replacing $a,c_0,d_0,t_0,t_1,t_2,t_4,t_6$ in $R_0$ with $b,c_1,d_1,t_1,t_0,t_3,t_5,t_7$ respectively. Let $G$ be the group with presentation $$\langle\{a,b,c_0,d_0,c_1,d_1\}\cup T\;|\;R_0\cup R_1\rangle,$$ then $H$ is a quotient group of $G$. By the $\mathrm{Homeo}(D,\partial D)$-representation [@hyde2019group] of $H$, we conclude that $H$ is nontrivial. We are going to prove that $G$ does not admit nontrivial left-orderable quotients, which implies that $H$ is not left-orderable. For each $t\in\{a,c_0,d_0, t_1\}$, we check that $\mathbf{Gr}\cup\mathbf{Ax}_{R_0}\cup \mathbf{CC}\cup\{P(t_0)\land P(t_0'), \neg P(t)\}$ is inconsistent (Task 18.1, Task 18.2, Task 18.3 and Task 18.4) using Prover9. By Proposition [\[prop-cl\]](#prop-cl){reference-type="ref" reference="prop-cl"}, we obtain that $a,c_0,d_0,t_1\in \mathrm{cl}(t_0)$ in $G$. By symmetry, we have $b,c_1,d_1,t_0\in \mathrm{cl}(t_1)$. Since $G$ is generated by $a,b,c_0,d_0,c_1,d_1$, we have $\mathrm{cl}(t_0)=\mathrm{cl}(t_1)=G$. In other words, $\langle t_0\rangle$ and $\langle t_1 \rangle$ are left absolutely cofinal in $G$. We verify that the theories $\mathbf{Gr}\cup\mathbf{Ax}_{R_0}\cup \mathbf{CC}\cup\{P(t_0\cdot t_1)\land P((t_0\cdot t_1)'), \neg P(t_1)\}$ and $\mathbf{Gr}\cup\mathbf{Ax}_{R_0}\cup \mathbf{CC}\cup\{P(t_0\cdot t_1')\land P((t_0\cdot t_1')'), \neg P(t_1)\}$ are inconsistent (Task 18.5 and Task 18.6) using Prover9. By Proposition [\[prop-cl\]](#prop-cl){reference-type="ref" reference="prop-cl"}, we obtain $t_1 \in \mathrm{cl}(t_0 t_1)$ and $t_1\in \mathrm{cl}(t_0t_1^{-1})$, which implies that $\langle t_0 t_1\rangle$ and $\langle t_0 t_1^{-1}\rangle$ are left absolutely cofinal in $G$. Consider the set $\mathfrak{S}^P$ with respect to the pairs $(e,t_0)$, $(e, t_1)$, $(e, t_0\cdot t_1)$ and $(e, t_0\cdot t_1')$. In order to apply Proposition [\[prop-fixed-point\]](#prop-fixed-point){reference-type="ref" reference="prop-fixed-point"}, we would like to prove the inconsistency of $\mathbf{Gr}\cup \mathbf{Ax}_{R_0}\cup \mathbf{Ax}_{R_1}\cup \mathbf{AxPL}'\cup \mathbf{PB}\cup \mathbf{Isolated}(\{2\})\cup \mathfrak{S}^P$. However, this task is computationally challenging for Prover9, so we break the task into two cases according to the comparison between $\max(a,a^{-1})$ and $\max(b,b^{-1})$. We verify that $\mathbf{Gr}\cup\mathbf{Ax}_{R_0}\cup\mathbf{AxPL}'\cup \mathbf{PB}\cup\mathbf{Isolated}(\{2\})\cup \mathfrak{S}^P\cup\{ P(t_0'\cdot t_1')\lor P(t_0'\cdot t_1)\}$ and $\mathbf{Gr}\cup\mathbf{Ax}_{R_1}\cup\mathbf{AxPL}'\cup \mathbf{PB}\cup\mathbf{Isolated}(\{2\})\cup \mathfrak{S}^P\cup\{\neg(P(t_0'\cdot t_1')\lor P(t_0'\cdot t_1))\}$ are inconsistent (Task 18.7 and Task 18.8) by Prover9. Since either $P(t_0'\cdot t_1')\lor P(t_0'\cdot t_1)$ or $\neg(P(t_0'\cdot t_1')\lor P(t_0'\cdot t_1))$ holds, the theory $\mathbf{Gr}\cup \mathbf{Ax}_{R_0}\cup \mathbf{Ax}_{R_1}\cup \mathbf{AxPL}'\cup \mathbf{PB}\cup \mathbf{Isolated}(\{2\})\cup \mathfrak{S}^P$ is inconsistent. By Proposition [\[prop-fixed-point\]](#prop-fixed-point){reference-type="ref" reference="prop-fixed-point"}, the group $G$ does not admit nontrivial left-orderable quotients. # Automated reasoning tasks {#sec:tasks} We have applied our methodology to many groups given by their presentations. In these examples, numerous automated reasoning tasks are performed either by Prover9 to derive a contradiction or by Mace4 to find a finite model. We always use Knuth-Bendix ordering when performing Prover9 tasks. We provide a Python 3 script on [@zenodo_dataset] that generates input files for Prover9 and Mace4, and runs them by calling the automated theorem provers. This script comes with following predefined axiom sets in Prover9/Mace4 format: $\mathbf{Gr}$, $\mathbf{AxL}$, $\mathbf{OrdL}$, $\mathbf{OrdB}$, $\mathbf{AxC}$, $\mathbf{OrdCL}$, $\mathbf{OrdCB}$, $\mathbf{AxPL}$, $\mathbf{PB}$, $\mathbf{AxPCL}$, $\mathbf{PCB}$, $\mathbf{CC}$, and $\mathbf{Isolated}(M)$. It also generates axiom sets $\mathbf{Ax}_R$, $\mathfrak{S}$, and $\mathfrak{S}^P$ from given pairs or triples. For each automated reasoning task described in this paper, we create a task specifying the program name (`prover9` or `mace4`) and the selected axiom sets, and then execute this task. To run the Python program, one need to install Prover9 and Mace4 [@prover9-mace4], and set the value of `BIN_LOCATION` to the location of the binary files. The output of this program consists of the standard inputs and outputs of the automated theorem provers, which can also be accessed on [@zenodo_dataset]. We execute our code using Python 3.9.12 and version LADR-Dec-2007 of Prover9 and Mace4, running on hardware with an AMD Ryzen 7 4800HS processor operating at 2.90 GHz, equipped with 16.0 GB of RAM, and operating on Windows 11 Home. Table [1](#tab:summary1){reference-type="ref" reference="tab:summary1"} summarises the time spent by Prover9 in seconds. **Prover9 tasks** **User CPU time** **System CPU time** **Wall clock time** ------------------- ------------------- --------------------- --------------------- Task 1.2 0.00 0.00 0 Task 2.2 0.00 0.00 0 Task 3.2 4.83 0.05 31 Task 3.3 1.78 0.08 31 Task 4.2 27.77 0.16 38 Task 5.2 973.94 16.66 1756 Task 6.1 39.84 0.06 44 Task 7.2 0.00 0.00 0 Task 8.2 0.00 0.00 0 Task 9.2 15.31 0.00 19 Task 9.4 25.72 0.03 32 Task 9.6 0.80 0.00 10 Task 9.8 6.73 0.05 40 Task 10.2 59.33 0.05 66 Task 10.4 0.00 0.03 0 Task 11.1 10.42 0.01 13 Task 11.2 3.41 0.00 7 Task 12.1 0.00 0.00 0 Task 15.1 0.00 0.00 1 Task 16.1 2.06 0.00 5 Task 16.2 0.00 0.00 1 Task 17.1 6.61 0.08 21 Task 17.2 5.34 0.00 20 Task 17.3 0.75 0.01 9 Task 18.1 0.00 0.00 0 Task 18.2 0.00 0.00 0 Task 18.3 0.00 0.00 0 Task 18.4 0.16 0.00 2 Task 18.5 1.81 0.06 12 Task 18.6 0.64 0.00 8 Task 18.7 71.31 4.23 245 Task 18.8 5.33 0.03 18 : Time spent by Prover9, measured in seconds. All of our Mace4 tasks can be solved in less than one second, except for Task 8.1, which takes 139 seconds to solve. Table [2](#tab:summary2){reference-type="ref" reference="tab:summary2"} summarises the sizes of the finite models found by Mace4. **Mace4 tasks** **model size** **Mace4 tasks** **model size** ----------------- ---------------- ----------------- ---------------- Task 1.1 2 Task 9.1 14 Task 2.1 2 Task 9.3 14 Task 3.1 4 Task 9.5 14 Task 4.1 6 Task 9.7 14 Task 5.1 14 Task 10.1 5 Task 7.1 6 Task 10.3 5 Task 8.1 60 Task 17.4 5 : Sizes of finite models found by Mace4. ### Acknowledgement {#acknowledgement .unnumbered} The work of first and third named authors was supported by the Leverhulme Trust Research Project Grant RPG-2019-313. Part of the work was done when the second author visited Institut des Hautes Études Scientifiques. [^1]: A subset $B\subseteq A$ of a preordered set $(A,\le)$ is called *cofinal* if for every $a\in A$ there exists $b\in B$ such that $a\le b$.

arxiv_math

--- author: - Farnaz Ghanbari - Samreena title: | Collapsing of Mean Curvature Flow of Hypersurfaces\ to Complex Submanifolds --- **Abstract.** In this paper, we produce explicit examples of mean curvature flow of $(2m-1)$-dimensional submanifolds which converge to $(2m-2)$-dimensional submanifolds at a finite time. These examples are a special class of hyperspheres in $\mathbb{C}^{m}$ with a $U(m)$-invariant Kähler metrics. We first discuss the mean curvature flow problem and then investigate the type of singularities for them. # Introduction Mean curvature flow is a well-known geometric evolution equation for hypersurfaces in which each point moves with a velocity given by the mean curvature vector. If the hypersurface is compact, the short time existence and uniqueness of the mean curvature flow are well-known. In general, it is very hard to find an exact solution of mean curvature flow problem. In fact there are very few explicit examples. Round spheres in Euclidean space are non trivial examples of evolving hypersurface under mean curvature flow which concentrically shrink inward until they collapse at a finite time to a single point. Another instance would be the marriage ring that under mean curvature flow shrinks to a circle. A round cylinder also remains round and finally converges to a line. Mean curvature flow develops singularities if the second fundamental forms of the time dependent immersions become unbounded. It is well-known that mean curvature flow of any closed manifold in Euclidean space develops singularities at a finite time. The mean curvature flow has first been investigated by Brakke in 1978 [@Brakke]. Later Huisken [@Huisken:; @1984] showed that any closed convex hypersurface in Euclidean space shrinks to a round point at a finite time. He then proved [@Huisken:1986] that the same holds for hypersurfaces in general Riemannian manifolds satisfying a strong convexity condition which takes into account the geometry of the ambient space. Brakke used geometric measure theory, but Huisken employed a more classical differential geometric approach. In order to describe singularities of the flow, Osher-Sethian introduced a level-set formulation for the mean curvature flow which was investigated later by Evans-Spruck ([@Evans], [@Evans1], [@Evans2], [@Evans3]) and Chen-Giga-Goto [@Chen] in details. Ilmanen revealed in [@Ilmanen] the relation between the level-set formulation and the geometric measure theory approach. In this paper, we consider a class of canonical hyperspheres in $\mathbb{C}^{m}$. We will make an important assumption about the symmetry group. Namely, we will require that the Kähler metric on $\mathbb{C}^m\setminus {0}$ has $U(m)$ as the group of isometries. We study the mean curvature flow problem for hyperspheres in $Bl_{0}\mathbb{C}^m$ which reduce to an ordinary differential equation due to invariance of the metric and mean curvature under isometries. In general, it is not easy to compute the second fundamental form to investigate the singularities and also their types. We have computed all the principal curvatures and observed that near the exceptional divisor, all the principal curvatures vanish except for one direction which goes to infinity. By knowing the principal curvatures, we can compute the mean curvature and also the square of the norm of the second fundamental form. There is a known example, Burns metric on $Bl_{0}\mathbb{C}^2$, for which we will study the mean curvature flow problem in Section [5](#sec:5){reference-type="ref" reference="sec:5"} and show the exact time of singularity. The rest of the paper is organized as follows. Section [2](#sec:2){reference-type="ref" reference="sec:2"} is devoted to definitions, some well-known theorems and some results that will be used throughout the work. Section [3](#sec:3){reference-type="ref" reference="sec:3"} focuses on the blow up of $\mathbb{C}^m$ at the origin. We discuss the condition when a $U(m)$-invariant metric on $\mathbb{C}^m\setminus {0}$ can extend to the blow up of $\mathbb{C}^m$ at the origin. In Section [4](#sec:4){reference-type="ref" reference="sec:4"}, we state and prove a proposition on computing principal curvatures on special cases that leads to the proof of our main theorem. Finally Section [5](#sec:5){reference-type="ref" reference="sec:5"} is dedicated to the mean curvature flow for our setting and some examples. # Preliminaries {#sec:2} In this section, we recall some basic notions and fix some notations used throughout this paper. **Definition 1**. *Let $F_{0}: \Sigma^{m} \longrightarrow M^{m+1}$ be a smooth immersion of an $m$-dimensional manifold. The mean curvature flow of $F_{0}$ is a family of smooth immersions $F_{t}: \Sigma \longrightarrow M^{m+1}$ for $t\in [0,T)$ such that setting $F(p,t)=F_{t}(p)$ the map $F: \Sigma \times [0,T) : \Sigma^{m} \longrightarrow M^{m+1}$ is a smooth solution of the following system of PDE's* *$$\label{eq:in:introduction 1} \begin{cases} \frac{\partial}{\partial{t}} F(p,t)=H(p,t)n(p,t),\\ F(p,0)=F_{0}(p) \,, \end{cases}$$ where $H(p,t)$ and $n(p,t)$ are respectively the mean curvature and the unit normal of the hypersurface $F_{t}$ at the point $p\in \Sigma$.* Usually the Riemannian manifold $M$ is called the ambient manifold and the parameter t is considered as time. Minimal submanifolds, i.e. submanifolds with zero mean curvature everywhere, are the stationary solutions of this flow. There are two important propositions in the Euclidean case. We use them in the proof of our main theorem on mean curvature flow problem [@Mantegazza]. The propositions are as follows. **Proposition 2**. *If the second fundamental form is bounded in the interval $[0,T)$ with $T< +\infty$, then all its covariant derivatives are also bounded.* **Proposition 3**. *If the second fundamental form is bounded in the interval $[0,T)$ with $T< +\infty$, then $T$ cannot be a singular time for the mean curvature flow of a compact hypersurface $F: \Sigma \times[0,T) \longrightarrow \mathbb{R}^{n+1}$.* From these two propositions, we have the following Remark. **Remark 4**. *The above estimate can be found independent of $T$ and also independent of initial data.* One of the most important problems in studying the mean curvature flow is to understand the possible singularities the flow goes through. We introduce the notion of singularity in mean curvature flow and their types in the following. **Definition 5**. *If the second fundamental form $|A|^2$ blows up at $t\longrightarrow T$, then we call $T$ a singular time of the flow.* **Definition 6**. *We say that the flow is developing a type $I$ singularity at time $T$ if there exists a constant $C>1$ such that we have the upper bound $$max_{p\in \Sigma}|A(p,t)|^2 \leq \frac{C}{T-t}.$$ Otherwise, we say it is a type $II$ singularity.* # Kähler Metrics on the Blow Up of $\mathbb{C}^m$ at the Origin {#sec:3} We consider the blow up of $\mathbb{C}^m$ at the origin and denote it by $Bl_0\mathbb{C}^m$. It is defined as following: $$Bl_0\mathbb{C}^m = \{((z_1,z_2,\dots,z_m),[t_1,t_2,\dots,t_{m}]) \in \mathbb{C}^m \times \mathbb{C}P^{m-1} : z_it_j-z_jt_i=0\} \subset\mathbb{C}^m \times \mathbb{C}P^{m-1}\,.$$ There is a natural projection map $\pi_1 : Bl_0\mathbb{C}^m \rightarrow{\mathbb{C}^m}$ defined by $$\pi_1((z_1,z_2,\dots,z_m),[t_1,t_2,\dots,z_{m}])=(z_1,z_2,\dots,z_m)\,.$$ The inverse image $\pi^{-1}_1(p)$ of $p \in \mathbb{C}^{m}$ is a line passing the point $p$. The **exceptional divisor** $E$ is defined as the inverse image of the origin i.e., $\pi^{-1}(0)= \mathbb{C}P^{m-1}.$ Moreover the map $\pi_{1}$ can be restricted to a biholomorphism $$\pi_1: Bl_0\mathbb{C}^m \setminus E \rightarrow{ \mathbb{C}^m\setminus 0}.$$ A system of charts that covers the exceptional divisor is given as follows: for every $i=1,2,\dots,m$, $$U_{i} = \{((z_1,z_2,\dots,z_m),[t_1,t_2,\dots,z_{m}]): t_i\not=0, z_j=z_it_j \}\,.$$ The coordinate map $\Phi_i:U_{i} \rightarrow{}\mathbb{C}^m$ is defined as $$((z_1,z_2,\dots,z_m),[t_1,t_2,\dots,t_{m}])\to \left(z_i, \frac{t_1}{t_i},\dots,\frac{t_{i-1}}{t_i},\frac{t_{i+1}}{t_i},\dots,\frac{t_m}{t_i}\right),$$ with inverse map $\Phi_{i}^{-1}:\mathbb{C}^m\rightarrow{U_{i}}$ $$\label{coordinate} (z_1,z_2,\dots,z_m)\to ((z_1z_{i},z_{i} z_{2},\dots,z_{i},\dots,z_{i}z_{m}),[z_{1},\dots,z_{i-1},1,z_{i+1},\dots,z_{m}]).$$ For every $i=1,2,\dots,m,$ the chart $U_{i}$ intersects the exceptional divisor $E$: $$E \cap U_{i}=\{ z_{i}=0\}\,.$$ We now take the smooth $(1,1)$-form on $\mathbb{C}^m\setminus 0$ given by $$\omega = \sqrt{-1} \partial \bar{\partial}\log (S),$$ where $S= \sum^m_{i=1}|z_i|^2$. The pull back of the smooth form $\omega = \sqrt{-1} \partial \bar{\partial}\log (S)$ on $\mathbb{C}^m\setminus{0}$ extends to the Fubini Study metric on the exceptional divisor $E=\mathbb{C}P^{m-1}$. The pull back $\pi^{*}_{1}\omega$ is given in local coordinates ([\[coordinate\]](#coordinate){reference-type="ref" reference="coordinate"}) by, $$\begin{aligned} \label{ metric on E} \pi^{*}_{1}\omega &=\partial \bar{\partial}\log(|z_i|^2(|z_{1}^2|+|z_2|^2\dots+|z_{i-1}|^2+1+|z_{i+1}|^2+\dots+|z_m|^2)\nonumber\\ &= \partial \bar{\partial}\log(|z_{1}|^2+|z_2|^2\dots+|z_{i-1}|^2+1+|z_{i+1}|^2+\dots+|z_m|^2).\end{aligned}$$ Clearly ([\[ metric on E\]](# metric on E){reference-type="ref" reference=" metric on E"}) is the Fubini Study metric on the exceptional divisor $E$ in homogeneous coordinates $[z_{1},\dots,z_{i-1},1,z_{i+1},\dots,z_{m}].$ Let $g:\mathbb{C}^m\to \mathbb{R}$ be a smooth function that depends on $S=\sum^m_{i=1}|z_i|^2$. Then the smooth form $$\label{K\"ahler: form} \omega =\sqrt{-1}\partial \bar{\partial}f(S)= \sqrt{-1}\partial \bar{\partial}( \log S+g(S))\$$ gives Kähler metric on $\mathbb{C}^m\setminus \{0\}$ if and only if $\frac{1}{S} +g_{S}>0$ and $g_{S}+Sg_{SS}>0$. The next proposition explains when the Kähler form [\[K\\\"ahler: form\]](#K\"ahler: form){reference-type="eqref" reference="K\\\"ahler: form"} on $\mathbb{C}^m\setminus{0}$ can be extended to $Bl_0\mathbb{C}^m$. **Proposition 7**. *The smooth form $\omega =\sqrt{-1} \partial \bar{\partial}( \log S+g(S))$ on $\mathbb{C}^m\setminus \{0\}$ extends to Kähler metric on $Bl_0\mathbb{C}^m$ if and only if $g_{S}(0)>0$, $\frac{1}{S} +g_{S}>0$ and $g_{S}+Sg_{SS}>0$.* *Proof.* For the sake of simplicity, we only prove the case when $m=2.$ The general case follows from the same argument. Given the projection map $$\pi_{1} : Bl_0\mathbb{C}^2\rightarrow{}\mathbb{C}^2 ,$$ on the chart $U_{1}$ we have $S=|z_{1}|^2(1+|z_{2}|^2)$ and $E \cap U_{1}=\{z_1=0 \}\,.$ The pull back of the Kähler metric ([\[K\\\"ahler: form\]](#K\"ahler: form){reference-type="ref" reference="K\\\"ahler: form"}) to $Bl_0\mathbb{C}^2$ is given in coordinates ([\[coordinate\]](#coordinate){reference-type="ref" reference="coordinate"}) by $$\pi_{1}^*\omega= \begin{bmatrix}(1 + |z_2|^2)(g_{S}+ S g_{SS})& z_1 \bar{z_2}(g_{S}+ S g_{S S})\\ z_2 \bar{z_1}(g_{S}+ S g_{SS}) & |z_1|^2 (g_{S} + |z_1|^2|z_2|^2 g_{SS})+ \frac{1}{1+|z_2|^2} \end{bmatrix}.$$ The restriction of $\pi_{1}^*\omega$ to the exceptional divisor $E$ is: $$\pi_{1}^*\omega|_{E} = \begin{bmatrix}(1 + |z_2|^2)g_{S}(0)& 0\\ 0& \frac{1}{1+|z_2|^2} \end{bmatrix}.$$ Clearly $\pi_{1}^*\omega|_{E}$ is positive definite if and only if $g_{S}(0)>0$. In the same way on $U_{2}$, the pull back $\pi_{1}^*\omega$ where $$\pi_{1}^*\omega = \begin{bmatrix}\frac{1}{1+|z_1|^2} + |z_2|^2 (g_{S} + |z_1|^2|z_2|^2g_{SS})& z_1 \bar{z_2}(g_{S}+ S g_{SS})\\ z_2 \bar{z_1}(g_{S}+ S g_{SS})& (1+|z_1|^2 )(g_{S}+ S g_{SS}) \end{bmatrix}$$ can be restricted to the exceptional divisor as follows: $$\pi_{1}^*\omega|_{E} = \begin{bmatrix}\frac{1}{1+|z_1|^2} & 0\\ 0& (1+|z_1|^2 )g_{S} \end{bmatrix}$$ $\pi_{1}^*\omega|_{E}$ is positive definite if and only if $g_{S}(0) > 0$. ◻ 0◻ **Remark 8**. *If $g_{S}(0)=0$, then $\pi_{1}^*\omega|_{E}$ defines a metric only along the exceptional divisor. Therefore the condition $g_{S}(0)\neq 0$ guarantees the non degeneracy of the metric orthogonal to the exceptional divisor. The other two conditions $\frac{1}{S} +g_{S}>0$ and $g_{S}+Sg_{SS}>0$ are considered because $\omega$ must be a Kähler metric on $\mathbb{C}^m\setminus{0.}$* # Principal Curvatures of Hyperspheres {#sec:4} In this section, we compute the second fundamental form for hyperspheres under special conditions. In order to investigate the mean curvature flow for our examples, we need to know the principal curvatures which are the eigenvalues of the second fundamental form. Let $\Sigma$ be an $d$-dimensional smooth submanifold in an $d+1$-dimensional manifold $M$ and $g$ be the Riemannian metric on $M$ with Levi Civita connection $\nabla$. **Definition 9**. *The second fundamental form of $\Sigma$ is defined by $$\label{eq:coefficient :of:2nd form} \Pi_{n}(X,X)=g\left(\nabla_{X}(X),n\right)\,,$$ where $X \in T_{p}M$ and $n \in (T_{p}\Sigma)^{\perp}.$* **Lemma 10**. *Suppose $X$ and $n$ are local vector fields on M such that* - *$||X||^2_{g}$ and $||n||^2_{g}$ are constants ,* - *for all $p \in \Sigma$, $X(p) \in T_{p}\Sigma$ and $n(p) \in (T_{p}\Sigma)^{\perp}$,* *then $$\Pi_{n}(X,X)=-g([X,n],X).$$* *Proof.* $$\begin{aligned} \Pi_{n}(X,X)& =g\left(\nabla_{X}(X),\eta\right) = -g(\nabla_{X}(n),X ) = -g([X,n]+\nabla_n(X),X)\\ &= -g([X,n],X) +\frac{1}{2} n(||X||_{g}) =-g([X,n],X). \end{aligned}$$ ◻ 0◻ In the next proposition, we state and prove the main result of this section and calculate the second fundamental from for hyperspheres with some particular assumptions. **Proposition 11**. *Suppose that $g_{0}$ and $g$ are Euclidean and Riemannian metrics on M respectively. Let $e_{1},...,e_{d+1}$ be orthonormal local vector fields for $M$ with respect to $g_{0}$ i.e., $g_{0}(e_{i},e_{j})=\delta_{ij}$. $\Sigma \subset M$ is an $m$-dimensional submanifold such that for each $p \in \Sigma$ we have $e_{d+1}(p) \perp T_{p}\Sigma$. Let $n=e_{d+1}$ and $A,\eta,\mu$ be local functions on $M$ such that their restrictions on $\Sigma$ are constants. We have the following conditions:* - *$g(e_{d+1}, e_{d+1}) = A^{2}$ , $g(e_{d}, e_{d})= \mu^{2}$,* - *$g(e_{i},e_{i})=\eta^{2}$ if $1 \leq i \leq d-1$* - *$g(e_{i},e_{j})=0$ $\forall i \neq j$* - *$[e_{d},e_{d+1}] \in \mathbb{R}<e_{d},e_{d+1}>.$* *Now if $\Pi_{\Sigma}(g_{0})=\tau g_{0}$ for some $\tau \in \mathbb{R}$, then* *$$\Pi_{g}(e_{i},e_{j})= \begin{bmatrix} (\eta^{2}A^{-1}\tau+ \eta A^{-1}\nabla_{n}\eta)I_{m-1}& 0\\ 0 & \mu^{2}A^{-1}\tau + \mu A^{-1}\nabla_{n}\mu \end{bmatrix}.$$* *Proof.* We fix some notations which will be used in the proof. Let $[n,e_i]= \sum_{j=1}^{d+1} a_{ij}e_{j}$ for $1\leq i\leq d$. Then, - $a_{ii}= g_{0}([n,e_{i}],e_{i})= -g_{0}([e_i,n],e_{i}) = \Pi_{g_{0}}(e_{1},e_{1}) = \tau$ - $0=2\Pi_{g_{0}}(e_i,e_{j}) =g_{0} \nonumber([e_{i},n],e_{j})+g_{0}([e_{j},n],e_{i}) =a_{ij}+a_{ji}.$ Notice that $\{\eta^{-1}e_{1},...,\eta^{-1}e_{d-1} ,\mu^{-1}e_{d},A^{-1}n \}$ is an orthonormal frame for the metric $g$. We prove the proposition in the following steps. **Step 1**: By the Lemma [Lemma 10](#2nd fundamental form){reference-type="ref" reference="2nd fundamental form"} we have $$\begin{aligned} \Pi(\eta^{-1}e_{1},\eta^{-1}e_{1})&=-g([\eta^{-1}e_{1}, A^{-1}n],\eta^{-1}e_{1} )\\ &= -g(\eta^{-1} A^{-1}[e_1,n]-A^{-1}\nabla_{n}(\eta^{-1}e_1), \eta^{-1}e_{1} )\\ &= -\eta^{-1} A^{-1}g([e_1,n], \eta^{-1}e_{1} ) + \eta^{-1}A^{-1}\nabla_{n}(\eta^{-1}) g(e_1, e_{1} )\\ &=-\eta^{-2} A^{-1}g([e_1,n], e_{1} ) + A^{-1} \eta \nabla_{n}(\eta^{-1})\\ &= -\eta^{-2} A^{-1}g([e_1,n], e_{1} ) - A^{-1} \eta^{-1} \nabla_{n}(\eta)\\ &=- A^{-1}(\tau + \eta^{-1}\nabla _{n}\eta)\end{aligned}$$ where we employed the property of Lie bracket and the fact that $\nabla_{e_{1}}A^{-1}=0$ on $\Sigma$. For the last step we use the following relation: $$g([e_1,n], e_{1})= a_{11} g_{11}=\tau \eta^2.$$ **Step 2**: The same calculation shows that $$\Pi(\eta^{-1}e_{i},\eta^{-1}e_{i}) = A^{-1}(\tau + \eta^{-1}\nabla _{n}\eta),$$ if $1 \leq i \leq d-1$, $$\Pi(\mu^{-1}e_{d},\mu^{-1}e_{d}) = g([\mu^{-1} e_{d}, A^{-1}n], \mu^{-1}e_{d}).$$ Similar to the step 1 we have: $$\Pi(\mu^{-1}e_{d},\mu^{-1}e_{d}) = A^{-1}(\tau + \mu^{-1}\nabla _{n}\mu).$$ Now similar to the last calculations we get $\Pi(e_{i},e_{j})=0$ for each $1 \leq i < j\leq d-1$. In the next step we show that $\Pi (\eta^{-1}e_{1},\mu^{-1}e_{d})=0$. **Step 3**: $$2\Pi(\eta^{-1}e_{1}, \mu^{-1}e_{d})= g([\eta ^{-1}e_{1},n], \mu^{-1}e_{d})+ g([\mu^{-1}e_{d},n], \eta^{-1}e_{1})$$ We have: $$[\eta^{-1}e_{1},n]= \eta^{-1}[e_{1},n]- \nabla_{n}\eta^{-1}e_{1}= \eta^{-1}\Sigma a_{ij}e_{j} - \nabla_{n}\eta^{-1}e_{1},$$ and $$[\mu^{-1}e_{d},n]= \mu^{-1}[e_{d},n]- \nabla_{n}\mu^{-1}e_{d}=\mu^{-1}\Sigma a_{dj}e_{j} - \nabla_{n}\mu^{-1}e_{d}.$$ $$\begin{aligned} 2\Pi(\eta^{-1}e_{1}, \mu^{-1}e_{d})&= \mu^{-1}\eta^{-1}g([e_{1},n],e_{d})- \mu^{-1}\nabla_{n}\eta^{-1}g(e_{1},e_{d}) +\mu^{-1}\eta^{-1}g([e_{d},n],e_{1})\\ &\hspace{.4cm}- \eta^{-1}\nabla_{n}\mu^{-1}g(e_{d},e_{1})\\ &=\mu^{-1}\eta^{-1}g([e_{1},n],e_{d})+ g([e_{d},n],e_{1}))\\ &=\mu^{-1}\eta^{-1} (\Sigma a_{ij}e_{j},e_{d})+ g(\Sigma a_{dj}e_{j},e_{1}))\\ &=\mu^{-1}\eta^{-1}(a_{1d} g(e_{d},e_{d})+ a_{d1}g(e_{1},e_{1}))\\ &= \mu^{-1}\eta^{-1}a_{d1}(-g(e_{d},e_{d})+ g(e_{1},e_{1})).\end{aligned}$$ Since $[e_{d},n] \in span<e_{d},n>$, then $a_{d1}=...=a_{d-1}=0$. We thus conclude that $$\Pi (\eta^{-1}e_{1},\mu^{-1}e_{d})=0.$$ ◻ 0◻ We consider $\mathbb{C}^{m}\setminus{0}\,$ with Kähler metric $g= \partial \overline {\partial} f(S)=(f_{S}\delta_{ij}+f_{SS}\bar{z_{i}}z_{j})dz_{i} \wedge d\bar{z_{j}}$, $\Sigma=\{(z_1,z_2,\dots,z_m) \in \mathbb{C}^m: S=R^2=|z_1|^2+ |z_2|^2+\dots+|z_m|^2 \} \subset \mathbb{C}^m$, the normal vector $n$ and $J(n)=in$ and moreover an orthonormal basis $e_{1},...,e_{2m-2}$ for $<n,J(n)>^{\perp}$. Let $e_{2m-1}=J(n)$ and $e_{2m}=n$ , the metric g is written by: $$\begin{bmatrix} f_{S}I_{2m-2}& 0\\ 0 & (f_{S}+f_{SS}S)I_{2} \end{bmatrix}.$$ **Theorem 12**. *The principal curvatures of the family $\Sigma_{S}^{2m-1} \subset\mathbb{C}^{m}\setminus{0}$ with a $U(m)$-invariant Kähler metric $\omega = \sqrt{-1} \partial \overline {\partial} f(S)$ are as follows:* *$$\lambda_{1}= \lambda_{2}=...= \lambda_{2m-2} =- \frac{ \sqrt{f_{S}+f_{SS}S}}{f_{S}\sqrt{S}}, \hspace{.1cm} \lambda_{2m-1}=- \frac{f_{S}+ 3Sf_{SS}+ S^{2}f_{SSS}}{(f_{S}+f_{SS}S)^{\frac{3}{2}} \sqrt{S}}.$$* *where $S=\Sigma_{i=1}^{m}|z_{i}|^2$.* *Proof.* In the setting of the Proposition [Proposition 11](#2nd fundamental form 1){reference-type="ref" reference="2nd fundamental form 1"}, we have $\Sigma= S^{2m-1}(r)$ and $M=\mathbb{C}^{m}\setminus{0}$. Furthermore, we have $A^2=\mu^2=f_{S}+f_{SS}S$ and $\eta^{2}=f_{S}$. Additionally we get $\eta^{-1}\nabla_{n}\eta= \frac{S}{f_{S}}f_{SS}$ and $\mu^{-1}\nabla_{n}\mu= \frac{\sqrt{S}}{\mu^{2}}(2f_{SS}+ Sf_{SSS})$. Now by computing $g^{-1}\Pi(g)$, we obtain the following principal curvatures: $$\lambda_{1}=...=\lambda_{2m-2}=- \frac{\sqrt{f_{S}+Sf_{SS}}}{f_{S}\sqrt{S}}, \hspace{.1cm}\lambda_{2m-1}=- \frac{(f_{S}+3Sf_{SS}+S^{2}f_{SSS})}{(f_{S}+f_{SS}S)^{\frac{3}{2}} \sqrt{S}}.$$ ◻ 0◻ # Mean Curvature Flow {#sec:5} In this section, we prove our main theorem presenting the mean curvature flow with initial data given by a special class of hyperspheres in $\mathbb{C}^{m}$ with a $U(m)$-invariant Kähler metrics. To do so, we first compute the mean curvature which is the sum of the eigenvalues of the second fundamental form. **Theorem 13**. *The mean curvature of the family $\Sigma_{S}^{2m-1} \subset \mathbb{C}^{m}\setminus{0}$ with $U(m)$-invariant Kähler metric $\omega=\sqrt{-1}\partial \bar{\partial}f(S)$ is given as follows: $$H(S)=\frac{-1}{(2m-1)(f_{S}+Sf_{SS})^{\frac{3}{2}}\sqrt{S}f_{S}}((2m-2)(f_{S}+Sf_{SS})^2+f_{S}(f_{SSS}S^2+3Sf_{SS}+f_{S})).$$* In the following two lemmas, we compute the square of the second fundamental form to see whether the mean curvature flow contains singularity or not. **Lemma 14**. *Let $A$ be the second fundamental form of the family of $\Sigma_{S}^{2m-1} \subset \mathbb{C}^{m} \setminus {0}$ with $U(m)$-invariant Kähler metric $\omega = \sqrt{-1} \partial \overline {\partial} f(S)$. Then the square of its norm, $|A|^{2}$ is as follows: $$\frac{(2n-2)(f_{S}+f_{SS}S)^{4}+ f_{S}^{2} (f_{S}+ 3Sf_{SS}+ S^{2}f_{SSS})^{2}}{f_{S}^{2} (f_{S}+f_{SS}S)^{3} S}$$* *Proof.* The principal curvatures for the hyperspheres are: $$\lambda_{1}= \lambda_{2}=...= \lambda_{2m-2} =- \frac{ \sqrt{f_{S}+f_{SS}S}}{f_{S}\sqrt{S}}, \hspace{.1cm} \lambda_{2m-1}=- \frac{f_{S}+ 3Sf_{SS}+ S^{2}f_{SSS}}{(f_{S}+f_{SS}S)^{\frac{3}{2}} \sqrt{S}}.$$ Now we can compute $|A|^{2}$ as following: $|A|^{2} = \lambda_{1}^{2}+ \lambda_{2}^{2}+ ...+ \lambda_{2m-1}^{2} = (2m-2)\lambda_{1}^{2}+ \lambda_{2m-1}^{2}$ $$= \frac{(2m-2)(f_{S}+f_{SS}S)^{4}+ f_{S}^{2} (f_{S}+ 3Sf_{SS}+ S^{2}f_{SSS})^{2}}{f_{S}^{2} (f_{S}+f_{SS}S)^{3} S}\,.$$ ◻ 0◻ **Lemma 15**. *For each $g$ with the following conditions, $$g_{S}(0) > 0,\hspace{.1cm}\frac{1}{S} + g_{S} > 0,\hspace{.1cm} \text{and } g_{S} +Sg_{SS} > 0,$$* *$|A|^{2}$ blows up only at $S=0$.* *Proof.* $$|A|^{2} = \frac{(2n-2)(g_{S}+g_{SS}S)^{4}+ (\frac{1}{S}+g_{S})^{2} (g_{S}+ 3Sg_{SS}+ S^{2}g_{SSS})^{2}}{(\frac{1}{S}+g_{S})^{2} (g_{S}+g_{SS}S)^{3} S}$$ We know that $g$ is a smooth function and does not blow up. When $S=0$, the numerator is always positive by the above conditions of $g$. Thus the singularity only happens when $S=0.$ 0◻ ◻ Now we prove the main result of our work in the next theorem in which we investigate the mean curvature flow for our setting. **Theorem 16**. *Consider $\mathbb{C}^{m} \setminus {0}$ with a $U(m)$-invariant Kähler metric $\omega = \sqrt{-1} \partial \overline {\partial} f(S)$ where $f(S)= \log S + g(S)$ and $g$ is an analytic function with the following conditions: $$g_{S}(0) > 0,\hspace{.1cm}\frac{1}{S} + g_{S} > 0,\hspace{.1cm} \text{and } g_{S} +Sg_{SS} > 0.$$ There exists $\epsilon > 0$ such that if $R_{o} < \epsilon$, we can choose one hypersphere with radius $R_{0}$ in such a way that the mean curvature flow with initial condition $\Sigma_{R(0)} = \Sigma_{R_{0}}$ converges to the exceptional divisor at a finite time and we have a singularity of Type I.* *Proof.* The mean curvature flow problem for the hyperspheres $\Sigma_{S}$ is the following ordinary differential equation (ODE): $$\frac{ dR(t)}{dt}= H(R(t)).$$ We can choose $\epsilon > 0$ such that if we start the flow with the initial data $R(0)=R_{0}<\epsilon$, the mean curvature does not vanish and is negative. In the previous lemma we observe that there is only one singularity at $R(t)=0$. Therefore, the time of singularity $(T_{sing})$ happens whenever $R(t)=0$. This means that if the flow starts at $t=0$, then $|A|^{2}$ is bounded for all $t \in [0,T_{sing})$. We can write the mean curvature flow problem as $\frac{ dR(t)}{dt}= \frac{1}{R^\alpha(t)}K(R(t))$ for some $\alpha >0$, where $K(R(t))$ is an analytic function without singularity and its Taylor series near $R(t)=0$ is as follows: $K(R(t)) = \sum_{n=0}^{\infty} \frac{K^{n}(0)}{n!} R^{n}(t)$. We thus get $R^{\alpha} (t) \frac{ dR(t)}{dt}=\sum_{n=0}^{\infty} \frac{K^{n}(0)}{n!} R^{n}(t)$. By applying integral on both sides, we get $\frac{1}{\alpha+1} R^{\alpha+1}(t)= K(0)t + \sum_{n=1}^{\infty} \frac{K^{n}(0)}{(n+1)!} R^{n+1}(t) + C$ for some constant C. Moreover, with initial condition $R(0)=R_{0}$ we have $C=\frac{R_{0}^{\alpha+1}}{\alpha+1} - \sum_{n=1}^{\infty} \frac{K^{n}(0)}{(n+1)!} R_{0}^{n+1}$. Further, we have the singularity only at $R(t)=0$. Hence $$T_{sing} = \frac{1}{K(0)} ( \sum_{n=1}^{\infty} \frac{K^{n}(0)}{(n+1)!} R_{0}^{n+1} - \frac{R_{0}^{\alpha+1}}{\alpha+1} ).$$ We can easily conclude that the time of singularity is finite and we can employ the Propositions [Proposition 2](#Pro:1){reference-type="ref" reference="Pro:1"} and [Proposition 3](#Pro:2){reference-type="ref" reference="Pro:2"}. Since these Propositions are well-known local theorems, we can apply them in Riemannian case. Therefore, we can conclude that the flow does not stop (i.e., keep restarting) and converges to $R(t)=0$ which is the exceptional divisor in $Bl_{0}\mathbb{C}^{m}$. Moreover, We can write the square of the second fundamental form as $|A|^{2}=\frac{W(R(t))}{R^{2}(t)}$, where $W(R(t))$ is an analytic function without singularity. Its Taylor series then near $R(t)=0$ is $W(R(t)) = \sum_{n=0}^{\infty} \frac{W^{n}(0)}{n!} R^{n}(t)$. Clearly we have $$|A|^{2}= \frac{W^{0}(0)}{R^{2}(t)}+ \frac{W^{1}(0)}{R(t)}+\frac{W^{2}(0)}{2}+ \sum_{n=3}^{\infty} \frac{W^{n}(0)}{n!} R^{n-2}(t).$$ Therefore we get $$\lim_{t \to\ T_{sing}}(T_{sing}-t)|A|^{2}= \lim_{t \to\ T_{sing}}(T_{sing}-t)\frac{W^{0}(0)}{R^{2}(t)}+\lim_{t \to\ T_{sing}}(T_{sing}-t)\frac{W^{1}(0)}{R(t)}$$ $$+\lim_{t \to\ T_{sing}}(T_{sing}-t)\frac{W^{2}(0)}{2} +\lim_{t \to\ T_{sing}}(T_{sing}-t)\sum_{n=3}^{\infty} \frac{W^{n}(0)}{n!} R^{n-2}(t).$$ $R(t)$ goes to zero as $t$ goes to $T_{sing}$, so we can easily check that $$\lim_{t \to\ T_{sing}}(T_{sing}-t)\frac{W^{2}(0)}{2}=\lim_{t \to\ T_{sing}}(T_{sing}-t)\sum_{n=3}^{\infty} \frac{W^{n}(0)}{n!} R^{n-2}(t)=0.$$ Since $\frac{ dR(t)}{dt}= H(R(t))$, we have $R^{\prime}(T_{sing})= \frac{ dR(t)}{dt}|_{t=T_{sing}}=H(R(T_{sing}))=H(0)=\infty.$ By using Hopital method we can conclude that $$\lim_{t \to\ T_{sing}}(T_{sing}-t)\frac{W^{1}(0)}{R(t)}=\lim_{t \to\ T_{sing}}\frac{-W^{1}(0)}{R^{\prime}(t)}<\infty .$$ We can also compute that $\lim_{t \to\ T_{sing}}R(t)H(R(t)) \neq 0$. Again by using Hopital method we can easily see that $$\lim_{t \to\ T_{sing}}(T_{sing}-t)\frac{W^{0}(0)}{R^{2}(t)}< \infty.$$ Consequently, $$\lim_{t \to\ T_{sing}}(T_{sing}-t)max|A|^2 < \infty.$$ The singularity is thus Type I. ◻ 0◻ The assumption of analyticity in the above Theorem is not restrictive. Many interesting Kähler metrics are analytic. For example, as proved by Hopf and Morrey constant scalar curvature Kähler metrics satisfy this hyphothesis [@Morrey]. **Remark 17**. *We can observe that when $S(t) \to 0$, then $\lambda_{1}=...=\lambda_{2m-2} \to 0$ and $\lambda_{2m-1}\to \infty$. This means that when $S(t) \to 0$, one of the principal directions collapses and the hypersphere converges to the exceptional divisor, which is holomorphic submanifold of $Bl_{0}\mathbb{C}^{m}$. Since holomorphic submanifolds of complex manifolds are minimal, so one would naturally expect that the principal curvature vanishes there.* In some examples we can estimate $\epsilon$ as $+\infty$ including the Burns metric. Example [Example 18](#eg:1){reference-type="ref" reference="eg:1"} provides an instance of the mean curvature flow problem for the Burns metric. **Example 18**. *Consider $Bl_{0}\mathbb{C}^{2}$ with the Burns metric given by $\omega =\sqrt{-1} \partial \overline {\partial} (\log(S)+ S)$. We can choose an arbitrary hypersphere $\Sigma_{R_{0}}$ as initial condition for the mean curvature flow. The mean curvature flow of the hypersphere converges to $S^{2}$ at a finite time and we have the singularity of Type I.* *Proof.* The mean curvature flow problem for the hyperspheres $\Sigma_{S}$ is the following ODE: $$\frac{ dR(t)}{dt}= H(R(t)).$$ Now the principal curvatures of $\Sigma_{S}$ with Burns metric are: $$\lambda_{1}=\lambda_{2}= \frac{-R}{(R^{2}+1)}, \hspace{.2cm} \lambda_{3}=\frac{-1}{R}.$$ Moreover, the mean curvature of these families and $|A|^{2}$ are given by $$H(R(t))= \frac{-1}{3}\frac{3R^{2}(t)+1}{R(t)(R^{2}(t)+1)}, \hspace{.1cm}|A|^{2}= \frac{2R^{4}(t)+(R^{2}(t)+1)^{2}}{R^{2}(t)(R^{2}(t)+1)^{2}}.$$ Therefore the mean curvature problem is equivalent to: $$\frac{ dR(t)}{dt} = \frac{-1}{3}\frac{3R^{2}(t)+1}{R(t)(R^{2}(t)+1)}.$$ The solution of the equation with initial data $R(0)=R_{0}$ would be $$\frac{R^{2}(t)}{2} + \frac{1}{3} \log(3R^{2}(t)+1)=-t+c.$$ $|A|^{2}$ blows up only when $R(t)=0$. With the initial condition $R(0)=R_{0}$ we get the time of singularity as following: $$T_{sing}= \frac{R_{0}^{2}}{2} + \frac{1}{3} \log(3R_{0}^{2}+1).$$ The time of singularity is finite and the flow exists for all $t \in [0 , T_{sing})$. We can also check that there exists a positive constant $C$ such that $|A|^{2} < \frac{C}{|T_{sing}-t|}$. The singularity is thus Type I. ◻ 0◻ # Acknowledgement {#acknowledgement .unnumbered} We thank Professor Claudio Arezzo for many valuable discussions and comments about this work. His insightful feedback brought our work to a higher level. We are also greatly indebted with Professor Reza Seyyedali. He also contributed to improve our paper by kindly providing several comments on this paper. 999 K. Brakke -- "The motion of a surface by its mean curvature", Princeton University Press. (1978). Y.-G. Chen, Y. Giga, S. Goto -- "Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations", J. Differential Geom. 33 (1991), p. 749--786, announcement in Proc. Japan. Acad. Ser. A 65 (1985) 207--210. L. Evans, J. Spruck -- "Motion of level sets by mean curvature iv.", J. Geom. Anal. 5 (1995), p. 77-- 114. L. Evans, J. Spruck -- "Motion of level sets by mean curvature iii.", J. Geom. Anal. 2 (1992), p. 121-- 150. L. Evans, J. Spruck -- "Motion of level sets by mean curvature ii.", Trans. Amer. Math. Soc. 330 (1992), p. 321--332. L. Evans, J. Spruck -- "Motion of level sets by mean curvature i.", J. Differential Geom. 33 (1991), p. 635--681. G. Huisken -- "Contracting convex hypersurfaces in Riemannian manifolds by their mean curvature", Invent. Math. 84 (1986), 462-480. G. Huisken -- "Flow by mean curvature of convex surfaces into spheres", J. Differential Geom. 20 (1984), p. 237--266. T. Ilmanen -- "Convergence of the allen-cahn equation to brakke's motion by mean curvature", J. Differential Geom. 38 (1993), p. 417--461. C. Mantegazza -- "Lecture Notes on Mean Curvature Flow", Springer Science & Business Media, 290 (2011). C.B. Morrey -- "On the analyticity of the solutions of analytic non-linear elliptic systems of partial differential equations. I. Analyticity in the interior", American Journal of Mathematics. (1958). X. Zhu -- "Lectures on mean curvature flows", American Mathematical Soc. 32 (2002). - Tarbiat Modares University Tehran and ICTP Trieste, farnazghanbari\@modares.ac.ir - SISSA Trieste, samreena01\@gmail.com

arxiv_math

--- abstract: | Let $\sigma(n)$ be the sum of the positive divisors of $n$. A number $n$ is said to be 2-near perfect if $\sigma(n) = 2n +d_1 +d_2$, where $d_1$ and $d_2$ are distinct positive divisors of $n$. We give a complete description of those $n$ which are 2-near perfect and of the form $n=2^k p^i$ where $p$ is prime and $i \in \{1,2\}$. We also prove related results under the additional restriction where $d_1d_2=n$. author: - | **Vedant Aryan**\ varyan24\@students.hopkins.edu\ **Dev Madhavani**\ devmadhavani\@college.harvard.edu\ **Savan Parikh**\ savan.parikh\@yale.edu\ **Ingrid Slattery**\ ingrid.slattery\@yale.edu\ **Joshua Zelinsky**\ jzelinsky\@hopkins.edu\ title: On 2-Near Perfect Numbers --- # Introduction A *perfect number* is a positive integer that is equal to the sum of its proper positive divisors. Equivalently, a perfect number is an $n$ such that $\sigma(n)=2n$, where $\sigma(n)$ is the sum of all positive divisors of $n$. Perfect numbers have been studied since antiquity. The idea of perfect numbers has been generalized in a variety of ways. A classic generalization is the notion of a *multiply perfect number*, defined as a number $n$ which satisfies $\sigma(n)=mn$ for some other integer $m$. Sierpiñski[@Sierpinski] introduced the term *pseudoperfect* number to mean a number $2n$ such that $n$ is the sum of some subset of its divisors. Pollack and Shevelev [@PS] separated the pseudoperfect numbers into separate types by introducing the idea of a $s$-near perfect number. A number $n$ is $s$-near perfect if $2n$ is the sum of all its positive divisors excepting $s$ of them. For example, while 12 is not perfect, it is 1-near perfect, since $1+2+3+6+12=2(12)$. Here the divisor which has been removed from the set is 4. We will refer to divisors removed from the set as *omitted divisors*. Note that any pseudoperfect number is $s$-near perfect for some $s$, and one can think of perfect numbers as $0$-near perfect numbers. In some sense, multiply perfect numbers are a multiplicative generalization of perfect numbers, while $s$-near perfect numbers are a more additive generalization. A number $n$ is said to be abundant if it satisfies $\sigma(n) > 2n$. If $n$ is $s$-near perfect for some $s>0$, then $n$ must be abundant. However, it is possible for a number to be abundant while not being $s$-near perfect for any $s$. An example is 70, where $\sigma(70)=144$. Numbers which are abundant but not $s$-near perfect for any $s$ are said to be *weird*. A classic open problem is whether there are any odd weird numbers. In addition to the more general notion of $s$-near perfect numbers, Pollack and Shevelev [@PS] also used the term near perfect number to mean 1-near perfect number. Another classic open problem is whether there is any $n$ such that $\sigma(n) = 2n+1$. Such numbers are called quasiperfect numbers. Note that any quasiperfect number is a $1$-near perfect number with omitted divisor $1$. Pollack and Shevelev constructed the following three distinct families of 1-near perfect numbers: 1. $2^{t-1}(2^t-2^k-1)$ where $2^t-2^k-1$ is prime. Here $2^k$ is the omitted divisor. 2. $2^{2p-1}(2^p-1)$ where $2^p-1$ is prime. Here $2^p(2^p-1)$ is the omitted divisor. 3. $2^{p-1}(2^p-1)^2$ where $2^p-1$ is prime. Here $2^p-1$ is the omitted divisor. Subsequent work by Ren and Chen [@RC] showed that all near perfects with two distinct prime factors must be either 40, or one of the the three families above. The only known near perfect odd number is $173369889=(3^4)(7^2)(11^2)(19^2).$ Tang, Ma, and Feng [@TMF] showed that this is the only odd near perfect number with four or fewer distinct prime divisors. Cohen, Cordwell, Epstein, Kwan, Lott, and Miller proved general asymptotics for $s$-near perfect numbers for $s \geq 4.$ Li and Liao [@LL] classified all even near perfects of the form $2^a p_1p_2$ where $p_1$ and $p_2$ are distinct primes. The main results of this paper are twofold. First, we give a complete description of $2$-near perfects of the form $2^kp$ or $2^kp^2$ where $p$ is prime. Second, we use these characterizations to introduce a closely related notion of strongly 2-near perfect numbers, and give a characterization of those of the form $2^kp$. In particular, we have the following two main results. **Theorem 1**. *Assume $n$ is a $2$-near perfect number with omitted divisors $d_1$ and $d_2$. Assume further that $n=2^k p$ where $p$ is prime and $k$ is a positive integer. Then one must have, without loss of generality, one of four situations. [\[classification of 2 near perfect of form power of two times a prime\]]{#classification of 2 near perfect of form power of two times a prime label="classification of 2 near perfect of form power of two times a prime"}* 1. *$p=2^k-1$. Here we have $d_1=1$ and $d_2=p$.* 2. *$p=2^{k+1} -2^a -2^b-1$ for some $a, b \in \mathbb{N}$. Here $d_1=2^a$ and $d_2=2^b$.* 3. *$p=\frac{2^{k+1}-2^a-1}{1+2^b}$ for some $a, b \in \mathbb{N}$. Here $d_1 = 2^a$ and $d_2 = 2^bp$.* 4. *$p=\frac{2^{k+1}-1}{1+2^a+2^b}$ for some $a, b \in \mathbb{N}$. Here $d_1 =2^ap$ and $d_2=2^bp$.* **Theorem 2**. *Assume that $n$ is a $2$-near perfect number with omitted divisors $d_1$ and $d_2$. Assume further that $n=2^kp^2$ where $p$ is prime. Then $n \in \{18, 36,200\}$. [\[Second main result power of 2 times square of a prime\]]{#Second main result power of 2 times square of a prime label="Second main result power of 2 times square of a prime"}* We recall the following basic facts about $\sigma(n)$ will be useful throughout: **Lemma 3**. *The function $\sigma(n)$ has the following properties: [\[basic sigma properties\]]{#basic sigma properties label="basic sigma properties"}* 1. *$\sigma(n)$ is multiplicative. That is, $\sigma(ab) =\sigma(a)\sigma(b)$ whenever $a$ and $b$ are relatively prime.* 2. *For a prime $p$, $\sigma(p^k) = p^k + p^{k-1} \cdots +1 = \frac{p^{k+1}-1}{p-1}.$* # Proof of Theorem [\[classification of 2 near perfect of form power of two times a prime\]](#classification of 2 near perfect of form power of two times a prime){reference-type="ref" reference="classification of 2 near perfect of form power of two times a prime"}. Let us now prove Theorem [\[classification of 2 near perfect of form power of two times a prime\]](#classification of 2 near perfect of form power of two times a prime){reference-type="ref" reference="classification of 2 near perfect of form power of two times a prime"}. *Proof.* Assume we have a 2-near perfect number of the form $n=2^{k}p$ with two omitted divisors $d_1, d_2$, $d_1\neq d_2$ and odd prime $p$. Because $n$ is near perfect, we have that: $$\sigma(n)=2n+d_1+d_2.$$ Using Lemma [\[basic sigma properties\]](#basic sigma properties){reference-type="ref" reference="basic sigma properties"}, we then have: $$\sigma(n)=\sigma(2^{k}p)=(2^{k+1}-1)(p+1)$$ So, setting equation (1) equal to equation (2), we have: $$(2^{k+1}-1)(p+1)=2n+d_1+d_2=2^{k+1}p+d_1+d_2,$$ and hence $$p=2^{k+1}-1-d_1-d_2. \label{p equals in 2 to k p equation}$$ Because $p$ is odd, we have that $2^{k+1}-1-d_1-d_2$ is odd. Since $2^{k+1}-1$ is always odd, we have that $-(d_1+d_2)$ must be even. If $-(d_1+d_2)$ is even, $d_1$ and $d_2$ must be of the same parity. We thus need to consider two situations: where $d_1, d_2$ are both odd, and where they are both even. We will call the first situation Case 1, and we shall separate the second situation, Case 2, into three separate subcases without loss of generality.\ \ Case 1: In this case, $d_1, d_2$ are both odd.\ \ The only odd divisors of $n$ are $1$ and $p$, so we can, without loss of generality, set $d_1=1$ and $d_2=p$ to find: $$p=2^{k+1}-1-1-p,$$ and hence $$p=2^{k}-1.$$ Thus, our first family of 2-near perfect numbers correspond to Mersenne primes and have the form $2^k(2^k-1)$ (twice an even perfect number).\ We now consider the situation where where $d_1, d_2$ are both even. We shall break this down into three subcases, depending on the types of values for $d_1$ and $d_2$. Case 2.1: In this case we have $d_1=2^a, d_2=2^b,$ where $0<a<b\leq k$\ We can use in our definitions of $d_1$ and $d_2$ in Equation (3) to find: $$p=2^{k+1}-2^a-2^b-1$$ This is our second family of 2-near perfect numbers.\ \ Case 2.2: In this case, $d_1, d_2$ are both even, and $d_1=2^a, d_2=2^{b}p,$ and $a, b \in (0, k]$\ \ We use a similar strategy, and plug in our definitions of $d_1,d_2$ into Equation [\[p equals in 2 to k p equation\]](#p equals in 2 to k p equation){reference-type="ref" reference="p equals in 2 to k p equation"}: $$p=2^{k+1}-2^a-2^{b}p-1,$$ and so $$p(1+2^b)=2^{k+1}-2^a-1,$$ which becomes $$p=\frac{2^{k+1}-2^a-1}{1+2^b}.$$ This is our third family of 2-near perfect numbers.\ \ Case 2.3: In this case, $d_1, d_2$ are both even and we have $d_1=2^{a}p, d_2=2^{b}p,$ and $0 < a < b \leq k$\ \ Using the same technique, equation (3) tells us: $$p=2^{k+1}-1-2^{a}p-2^{b}p,$$ and hence, $$p(1+2^a+2^b)=2^{k+1}-1,$$ which implies that $$p=\frac{2^{k+1}-1}{1+2^a+2^b}.$$ This is the fourth and final family of 2-near perfect numbers. ◻ # Proof of Theorem [\[Second main result power of 2 times square of a prime\]](#Second main result power of 2 times square of a prime){reference-type="ref" reference="Second main result power of 2 times square of a prime"} One major technique we will use is what we call the *discriminant sandwich* method: we show that a given Diophantine equation has only a restricted set of possible solutions. We do so by showing that the equation is a quadratic equation in one variable, and thus in order to have integer valued solutions, the discriminant must be a perfect square. However, we will show that the discriminant must, except in a limited set of cases, be shown to be strictly between two consecutive perfect squares, and thus aside from those situations, we have no solution. Discriminant sandwiching will be used extensively in what follows. **Lemma 4**. *Let $a$ and $k$ be positive integers such that $D= 2^{2k+2}+2^{k+2}-2^{a+2}-7$. If $0 \leq a < k$ and $D$ is a perfect square, then $k=a=1$. [\[Case I Lemma 1\]]{#Case I Lemma 1 label="Case I Lemma 1"}* *Proof.* Let us assume that $D$ is a perfect square. For all $a$, note that $$2^{2k+2}+2^{k+2}-2^{a+2}-7 < 2^{2k+2} +2^{k+2}+2 = (2^{k+1}+1)^2$$ Thus, if we have $$D = 2^{2k+2}+2^{k+2}-2^{a+2}-7 > 2^{2k+2} = (2^{k+1})^2$$ then the quantity in question cannot be a perfect square because it is sandwiched between two consecutive perfect squares. So we must have that $$2^{2k+2}+2^{k+2}-2^{a+2}-7 \leq 2^{2k+2}$$ and therefore $$2^{k+2}-2^{a+2} \leq 7. \label{Case I Lemma inequality}$$ If $k > a \geq 1$, then from Equation [\[Case I Lemma inequality\]](#Case I Lemma inequality){reference-type="ref" reference="Case I Lemma inequality"} we have that $2^{k+2}-2^{k+1} \leq 2^{k+2}-2^{a+2} \leq 7$. Thus, $$2^{k+2}-2^{k+1} = 2^{k+1} \leq 7,$$ which implies that $k \leq 1$. However, given the conditions for this case, no solutions are possible.\ \ Now, consider the case when $k=a\geq 1$. In this case, we have $$D=2^{2k+2}+2^{k+2}-2^{k+2}-7=2^{2k+2}-7.$$ Given this, note that: $$2^{2k+2}-7<2^{2k+2}=(2^{k+1})^2$$ Using the same logic as earlier, we see that if $$D>(2^{k+1}-1)^2,$$ then $D$ will be sandwiched between two consecutive perfect squares, and thus will not be a square itself. Thus, we can assume: $$2^{2k+2}-7\leq (2^{k+1}-1)^2=2^{2k+2}-2^{k+2}+1,$$ which implies that $k\leq 1$. The bounds for this case require $k\geq 1$, so the only solution possible is $(a,k)=(1,1)$. ◻ Essentially, Lemma [\[Case I Lemma 1\]](#Case I Lemma 1){reference-type="ref" reference="Case I Lemma 1"} is the sandwiching part of the discriminant sandwich we will use in the proof of Proposition [\[n =18 situation\]](#n =18 situation){reference-type="ref" reference="n =18 situation"} below. **Lemma 5**. *Let $b$ and $k$ be non-negative integers and $p$ be an odd number such that $$(2^{k+1}-1)(p^2+p+1)= 2^{k+1}p^2 + 2^bp +1. \label{Copy of Equation for Case III, j=1, general form for Lemma 1}$$ then $p|2^k-1$, and $p+1|2^b-2$. [\[first Lemma for Case III, a=0, j=1\]]{#first Lemma for Case III, a=0, j=1 label="first Lemma for Case III, a=0, j=1"}* *Proof.* Assume one has a solution to Equation [\[Copy of Equation for Case III, j=1, general form for Lemma 1\]](#Copy of Equation for Case III, j=1, general form for Lemma 1){reference-type="ref" reference="Copy of Equation for Case III, j=1, general form for Lemma 1"}. Then, if we take the equation modulo $p$, we get that $$2^{k+1}-1 \equiv 1 \mod p,$$ and hence $p|2^{k+1}-2= 2(2^{k}-1)$. Since $p$ is odd, we have $p|2^{k}-1$. To prove the second half, observe that we can rewrite our initial equation as $$2^{k+1}(p+1)=2^bp +p^2 + p +2,$$ which implies that $p+1|2^bp +p^2 + p +2$. Hence $p+1|2^bp+p^2-p$, and so $p+1| p(2^b+p-1)$. Since $p$ and $p+1$ are relatively prime, we have then that $p+1|2^b+p-1$ Finally, we take away another multiple of $p+1$ to get $p+1|2^b-2$. ◻ **Lemma 6**. *The equation $$(2^{k+1}-1)(p^2+p+1)= 2^{k+1}p^2 + 2^bp +1 \label{Copy of Equation for Case III, j=1, general form for Lemma 3}$$ has no solutions where $p$ is odd, and $2 \leq b \leq k-1$. [\[Lemma for j=1, a=0, b \<k-1 no solutions\]]{#Lemma for j=1, a=0, b <k-1 no solutions label="Lemma for j=1, a=0, b <k-1 no solutions"}* *Proof.* Assume we have a solution to the equation. From Lemma [\[first Lemma for Case III, a=0, j=1\]](#first Lemma for Case III, a=0, j=1){reference-type="ref" reference="first Lemma for Case III, a=0, j=1"}, we choose an integer $x$ such that $x(p+1)=2^b-2$. Note that $x$ must be odd since $p+1$ is even, and $2^b-2$ is not divisible by 4. We then have $2^b=xp+x +2$. When we substitute this back into Equation [\[Copy of Equation for Case III, j=1, general form for Lemma 3\]](#Copy of Equation for Case III, j=1, general form for Lemma 3){reference-type="ref" reference="Copy of Equation for Case III, j=1, general form for Lemma 3"}, and solve for $2^{k+1}$, we get that $2^{k+1}=xp+p+2$. By taking the difference of these two expressions, we obtain: $$2^{k+1} -2^b = (xp+p+2) -(xp+x +2) = p-x.$$ Because $b \leq k-1$, $2^{k+1} -2^b > 2^k$, we have $p-x > 2^k$, and hence $p > 2^k+1$. But this contradicts Lemma [\[first Lemma for Case III, a=0, j=1\]](#first Lemma for Case III, a=0, j=1){reference-type="ref" reference="first Lemma for Case III, a=0, j=1"}, since we must have $p|2^k-1$. ◻ **Lemma 7**. *Let $n$ be a 2-near perfect number of the form $n=2^kp^2$ where $p$ is an odd prime. Assume further that the omitted divisors of $n$ are $d_1$ and $d_2$. Then, we have $$\label{fundamental equation for 2-near perfect of form power 2 times square of a prime} d_1+d_2=-p^2+(2^{k+1}-1)p+(2^{k+1}-1),$$ and $d_1$ and $d_2$ are of opposite parity. [\[d1 and d2 opposite parities in 2-near of form power 2 times square of prime \]]{#d1 and d2 opposite parities in 2-near of form power 2 times square of prime label="d1 and d2 opposite parities in 2-near of form power 2 times square of prime "}* *Proof.* Assume as given. Then we have $$\sigma(n)-d_1-d_2=2n$$ $$\sigma(2^{k}p^2)-d_1-d_2=2(2^{k}p^2)=2^{k+1}p^2$$ $$(2^{k+1}-1)(p^2+p+1)-d_1-d_2=2^{k+1}p^2$$ $$d_1+d_2=-2^{k+1}p^2+(2^{k+1}-1)(p^2+p+1).$$ which is equivalent to Equation [\[fundamental equation for 2-near perfect of form power 2 times square of a prime\]](#fundamental equation for 2-near perfect of form power 2 times square of a prime){reference-type="ref" reference="fundamental equation for 2-near perfect of form power 2 times square of a prime"}. Since the right-hand side of Equation [\[fundamental equation for 2-near perfect of form power 2 times square of a prime\]](#fundamental equation for 2-near perfect of form power 2 times square of a prime){reference-type="ref" reference="fundamental equation for 2-near perfect of form power 2 times square of a prime"} is odd, $d_1$ and $d_2$ must be of opposite parity. ◻ Let's look at the possible divisors of $n$, assuming $n =2^kp^2$. Every possible divisor can be of one of three types. Type I divisors are powers of 2, that is $d=2^a$ for some $0 \leq a \leq k$. Type II divisors are $p$ or $p^2$. That is $d=p^m$ where $m \in \{1,2\}$. Type III divisors are of the form $d= 2^{b}p^j$ where $0< b \leq k$ and $j \in \{1,2\}$. We may then, without loss of generality, break down our situation into the following six cases depending on all the possible combinations of omitted divisor types, $d_1$ and $d_2$. Case $d_1$ $d_2$ ------ ------- ------- 1 I I 2 I II 3 I III 4 II II 5 II III 6 III III We will handle each of these six cases separately. But before we do, we will observe that Cases 4 and 6 are both trivial since they require that both $d_1$ and $d_2$ of the same parity, which contradicts Lemma [\[d1 and d2 opposite parities in 2-near of form power 2 times square of prime \]](#d1 and d2 opposite parities in 2-near of form power 2 times square of prime ){reference-type="ref" reference="d1 and d2 opposite parities in 2-near of form power 2 times square of prime "}. We thus only consider Cases 1, 2, 3, and 5. **Proposition 8**. *If $n$ is a 2-near perfect number of the form $n=2^kp^2$, where $p$ is an odd prime, with omitted divisors of Case 1 form, then $n=18$, and the omitted divisors are $1$ and $2$. [\[n =18 situation\]]{#n =18 situation label="n =18 situation"}* *Proof.* Assume we are in Case 1. In this case, both $d_1$ and $d_2$ must be distinct powers of 2. Since $d_1 +d_2$ is odd, one of the omitted divisors must be odd (and hence equal to 1). Without loss of generality, we will set $d_1 =1$, and $d_2=2^a$ where $1 \leq a \leq k$. Putting this into Equation [\[fundamental equation for 2-near perfect of form power 2 times square of a prime\]](#fundamental equation for 2-near perfect of form power 2 times square of a prime){reference-type="ref" reference="fundamental equation for 2-near perfect of form power 2 times square of a prime"}, we get that $$p^2-(2^{k+1}-1)p-(2^{k+1}-2^a-2)=0. \label{Case I equation form}$$ Equation [\[Case I equation form\]](#Case I equation form){reference-type="ref" reference="Case I equation form"} is a quadratic equation in $p$. Thus, in order to have a solution, its discriminant, defined as $$D=2^{2k+2}+2^{k+2}-2^{a+2}-7$$ must be a perfect square. From Lemma [\[Case I Lemma 1\]](#Case I Lemma 1){reference-type="ref" reference="Case I Lemma 1"}, $D$ is only a perfect square if $k=a=1$. In this case, Equation [\[Case I equation form\]](#Case I equation form){reference-type="ref" reference="Case I equation form"} becomes just $p^2-3p=0$. Thus, one must have $p=3$, and so $n=18$ with $d_1=1, d_2 = 2$. One can verify this result: $2(18)=\sigma(18)-(1+2)$. ◻ **Proposition 9**. *There are no 2-near perfect numbers of the form $2^kp^2$ with omitted divisors of the Case 2.* *Proof.* We will apply the discriminant sandwich method to this situation. Again we are working with $n=2^kp^2$, but we now have omitted divisors of the following form $d_1=2^a$, $d_2=p^m$ where $a \in (0,k]$ and $m \in [1,2].$ We will break Case 2 down into two subcases, depending on whether $m=1$ or $m=2$.\ We first consider the situation where $m=1$. Applying this to Equation [\[fundamental equation for 2-near perfect of form power 2 times square of a prime\]](#fundamental equation for 2-near perfect of form power 2 times square of a prime){reference-type="ref" reference="fundamental equation for 2-near perfect of form power 2 times square of a prime"}, we obtain $$0=p^2-(2^{k+1}-2)p-(2^{k+1}-2^a-1). \label{Case II fundamental for m=1}$$ Equation [\[Case II fundamental for m=1\]](#Case II fundamental for m=1){reference-type="ref" reference="Case II fundamental for m=1"} has an even discriminant $D$, which means that if $D$ is a perfect square, it must be divisible by 4. Thus, we can define $$D' = \frac{D}{4}=2^{2k} -2^a,$$ and just as well assume that $D'$ is a perfect square. Note that $D'$ is still even, so we can skip over checks against odd squares. We have that $2^{2k} -2^a < (2^k)^2$, and so $$2^{2k} -2^a \le (2^k-2)^2.$$ With a little algebra we then obtain that $$2^{k+2}-2^a\leq 4$$ Without loss of generalization, let's say $k=a+m, m\in [0,k), a>0$. We then have that $$2^a(2^{m+2}-1) =2^{a+m+2}-2^{a}\leq 4.$$ It is evident after plugging in the minimum values for $a$ and $m$ that no solution exists. We thus have shown that when $m=1$, no solution exists.\ \ We now consider the case when $m=2.$ We then obtain from Equation [\[fundamental equation for 2-near perfect of form power 2 times square of a prime\]](#fundamental equation for 2-near perfect of form power 2 times square of a prime){reference-type="ref" reference="fundamental equation for 2-near perfect of form power 2 times square of a prime"}, $$0=2p^2-(2^{k+1}-1)p-2^{k+1}+2^a+1. \label{Case II m=2, form of fundamental equation}$$ We then need that the discriminant D, defined as $$D=2^{2k+2}-2^{k+2}+2^{k+4}-2^{a+3}-7.$$ is a perfect square. We thus must have $$D<(2^{k+1}+3)^2.$$ Since $D$ is odd, it cannot be equal to the next smallest square, which is even. So, $$D \leq (2^{k+1}+1)^2.$$ We thus have $$2^{2k+2}-2^{k+2}+2^{k+4}-2^{a+3}-7\le 2^{2k+2}+2^{k+2}+1,$$ which implies that $$2^k-2^a \le 1.$$ Thus, we can only have a solution when $k=a$ or we have $k=1$ and $a=0$. However, since $d_1$ and $d_2$ must be of opposite parity, we cannot have $a=0$. Thus, we need consider only the case when $k=a$. Our expression for $D$ simplifies so that we have $D=2^{2k+2} +2^{k+2}-7$. But this quantity cannot be a perfect square since $$(2^{k+1})^2 < 2^{2k+2} +2^{k+2}-7 < (2^{k+1}+1)^2,$$ and so $D$ is again sandwiched between two consecutive perfect squares. Thus, there are no solutions for Case 2 when $m=2$. Since no solutions exist for all possible cases for $m$, Proposition 9 is proven. ◻ **Lemma 10**. *If $p$ is an odd number such that $$(2^{k+1}-1)(p^2+p+1)=2^{k+1}p^2+2^b p^2 + 1 \label{Equation for Case III, j=2, general form first copy}$$ where $k$ and $b$ are positive integers, then $p|2^{k}-1$. [\[p divides Case III lemma\]]{#p divides Case III lemma label="p divides Case III lemma"}* *Proof.* Assume one has a solution to Equation [\[Equation for Case III, j=2, general form first copy\]](#Equation for Case III, j=2, general form first copy){reference-type="ref" reference="Equation for Case III, j=2, general form first copy"}. Then, if we take the equation modulo $p$, we get that $$2^{k+1}-1 \equiv 1 \mod p,$$ and hence $p|2^{k+1}-2= 2(2^{k}-1)$. Since $p$ is odd, we have $p|2^{k}-1$. ◻ **Lemma 11**. *If $p$ is an odd number which is a solution to Equation [\[Equation for Case III, j=2, general form first copy\]](#Equation for Case III, j=2, general form first copy){reference-type="ref" reference="Equation for Case III, j=2, general form first copy"}, then $p+1|2^b+2$[\[p+1 divides Case III lemma\]]{#p+1 divides Case III lemma label="p+1 divides Case III lemma"}.* *Proof.* Assume one has a solution to Equation [\[Equation for Case III, j=2, general form first copy\]](#Equation for Case III, j=2, general form first copy){reference-type="ref" reference="Equation for Case III, j=2, general form first copy"}. We can rewrite this as $$2^{k+1}(p+1)=(2^b+1)p^2+p+2$$ which implies that $$p+1|(2^b+1)p^2+p+2,$$ and hence $$p+1|(2^b+1)p^2-p = p((2^b+1)p-1).$$ Since $p$ and $p+1$ are relatively prime, $$p+1|(2^b+1)p-1,$$ and so by similar logic, $$p+1|2^b+2.$$ ◻ Note that Lemma [\[p+1 divides Case III lemma\]](#p+1 divides Case III lemma){reference-type="ref" reference="p+1 divides Case III lemma"} is distinct from Lemma [\[first Lemma for Case III, a=0, j=1\]](#first Lemma for Case III, a=0, j=1){reference-type="ref" reference="first Lemma for Case III, a=0, j=1"}, since the equations needed are different, and one has a $+2$, and the other a $-2$ on the right-hand side. **Proposition 12**. *Let $n =2^kp^2$ be a 2-near perfect number with omitted divisors of the Case 3 form with omitted divisors $1$ and $2^b p^2$. Then $n=36$, and our omitted divisors are $1$ and $18.$* *Proof.* Assume as given. So from Equation [\[fundamental equation for 2-near perfect of form power 2 times square of a prime\]](#fundamental equation for 2-near perfect of form power 2 times square of a prime){reference-type="ref" reference="fundamental equation for 2-near perfect of form power 2 times square of a prime"}, we have some $b$ such $b \leq k$, and $p$ prime such that $$(2^{k+1}-1)(p^2+p+1)=2^{k+1}p^2+2^b p^2 + 1\label{Case III p=2 equation}.$$ By Lemma [\[p divides Case III lemma\]](#p divides Case III lemma){reference-type="ref" reference="p divides Case III lemma"} and [\[p+1 divides Case III lemma\]](#p+1 divides Case III lemma){reference-type="ref" reference="p+1 divides Case III lemma"}, we have $$p|2^{k}-1$$ and $$p+1|2^b+2.$$ Thus, there is a positive integer $z$ such that $$z(p+1)=2^b+2,$$ and $$z(p+1)-2=2^b.$$ If we take Equation [\[Case III p=2 equation\]](#Case III p=2 equation){reference-type="ref" reference="Case III p=2 equation"} modulo $2^b$ we also get that $2^b|p^2+p+2$. We also have $$z(p+1)-2|p^2+p+z(p+1)$$ $$z(p+1)-2|p(p+1)+z(p+1)$$ $$z(p+1)-2|(p+1)(p+z). \label{z(p+1)-2 weaker divis}$$ Let $Q$ be some integer such that $Q|z(p+1)-2$ and $Q|(p+1)$. Then Q will divide any linear combination of those terms. Thus we have $$Q|z(p+1)-2-z(p+1) =-2.$$ Thus, the only possible common factors of $z(p+1)-2$ and $(p+1)$ are 1 and 2. Hence Equation [\[z(p+1)-2 weaker divis\]](#z(p+1)-2 weaker divis){reference-type="ref" reference="z(p+1)-2 weaker divis"} may be strengthened to: $$z(p+1)-2|2(p+z) \label{z division equation}$$ Therefore, we know that $$z(p+1)-2 \le 2(p+z),$$ which implies that $$z \le \frac{2p+2}{p-1} = 2 +\frac{4}{p-1}.$$ Since $p\geq 3$, we have $z \leq 3$, and hence have only three cases, $z=1$, $z=2$ or $z=3$. One can easily check that if $p=3$ then the only case which leads to integer values is when $z=1$ and $b=1$. Here $n=36$, and our omitted divisors are $d_1=1, d_2=18$. Thus, we may assume that $p>3$, which implies $z=1$ or $z=2$. However, if $b=1$, we get a contradiction if $p> 3$. Thus, may assume that $b>1$ which forces $z$ to be odd, and so $z=1$. We have from Equation [\[z division equation\]](#z division equation){reference-type="ref" reference="z division equation"}, $p-1 | 2(p+1)$. So there is some $m$ such that $m(p-1) = 2(p+1)$. If $m=1$ then we get a negative value for $p$ and if $m=2$, we immediately get a contradiction. So we may assume that $m \geq 3$. If $m=3$, then we have $3(p-1)=2(p+1)$, which yields $p=5$ which quickly leads to a contradiction. We thus must have $m \geq 4$. However, if $4(p-1) \leq 2(p+1)$, then one must have $p=3$, but we are in the situation where $p>3$. Thus, our only possibility is when $n=36$. ◻ **Proposition 13**. *If $n$ is a 2-near perfect number of the form $n=2^kp^2$, where $p$ is an odd prime, with omitted divisors of Case V form, then $n=200$.* *Proof.* Assume we have such an $n$, with omitted divisors $d_1$ and $d_2$. Then, without loss of generality, we may assume that $d_1=p^j$ for $j\in[1,2]$ and $d_2=2^bp^g$ for some $g\in[1,2]$, and $1\leq b\leq k$. We may break this down into four cases: Case $d_1$ $d_2$ ------ ------- ---------- 1 $p$ $2^bp$ 2 $p$ $2^bp^2$ 3 $p^2$ $2^bp$ 4 $p^2$ $2^bp^2$ Case 1 and Case 2 can both be handled by the discriminant sandwich technique. For Case 1, Equation [\[fundamental equation for 2-near perfect of form power 2 times square of a prime\]](#fundamental equation for 2-near perfect of form power 2 times square of a prime){reference-type="ref" reference="fundamental equation for 2-near perfect of form power 2 times square of a prime"} becomes $$p+2^bp=-p^2+(2^{k+1}-1)p+(2^{k+1}-1) . \label{V-1}$$ The relevant discriminant from Equation [\[V-1\]](#V-1){reference-type="ref" reference="V-1"} is $$D= 2^{2k+2} -2^{k+b+2} +2^{2b} +2^{b+2}.$$ We have then $$(2^{k+1}-2^b)^2 < D< (2^{k+1}-2^b+1)^2,$$ so $D$ cannot be a perfect square. Thus the equation has no solutions.\ Now, consider Case 2. In this case Equation [\[fundamental equation for 2-near perfect of form power 2 times square of a prime\]](#fundamental equation for 2-near perfect of form power 2 times square of a prime){reference-type="ref" reference="fundamental equation for 2-near perfect of form power 2 times square of a prime"} becomes $$p+2^bp^2=-p^2+(2^{k+1}-1)p+(2^{k+1}-1). \label{V-2}$$ The relevant discriminant from Equation [\[V-2\]](#V-2){reference-type="ref" reference="V-2"} is $$D = 2^{2k+2} +2^{k+b+2} +2^{b+2}.$$ We have that $$(2^{k+1}+2^{k+b})^2< D < (2^{k+1}+2^{k+b}+1)^2.$$ Therefore, $D$ cannot be a perfect square. We now consider Case 3, where $d_1=p^2$ and $d_2=2^bp^2$. Equation [\[fundamental equation for 2-near perfect of form power 2 times square of a prime\]](#fundamental equation for 2-near perfect of form power 2 times square of a prime){reference-type="ref" reference="fundamental equation for 2-near perfect of form power 2 times square of a prime"} then becomes: $$p^2+2^bp=-p^2+(2^{k+1}-1)p+2^{k+1}-1,$$ and hence $$-2p^2+(2^{k+1}-2^b-1)p+2^{k+1}-1=0. \label{Case V-3}$$ This has a corresponding discriminant value given by $$D=x^2-2x(2^b-3)+2^{2b}+2^{b+1}-7, \label{other D value eq needing a label}$$ where $x=2^{k+1}$. We note that if $b=1$, then we get that either $p=-1$ or $p=\frac{2^{k+1}-1}{2}$, neither of which is a prime. Thus, we may assume that $b> 1$. Since $b > 1,$ we have $$2^{b+3}>16,$$ which implies that $$2^{b+1}-7>-6\cdot 2^b+9.$$ We then obtain that $$x^2-2x(2^b-3)+2^{2b}+2^{b+1}-7>x^2-2x(2^b-3)+2^{2b}-6\cdot 2^b+9,$$ which implies that $$D>x^2-2x(2^b-3)+(2^b-3)^2 = (x-(2^b-3))^2.$$ If we have a solution to our original equation, $D$ must be a perfect square. Equation [\[other D value eq needing a label\]](#other D value eq needing a label){reference-type="ref" reference="other D value eq needing a label"} also shows that $D$ must be odd. Thus, we cannot have $D= (x-(2^b-2))^2,$ and thus we have $$D \geq (x-(2^b-1))^2 \label{Lower bound for D}$$ $$3\cdot 2^b\geq 2^{k+1}+8,$$ $$4\cdot 2^b> 2^{k+1}. \label{4 2 to b ineq}$$ Inequality [\[4 2 to b ineq\]](#4 2 to b ineq){reference-type="ref" reference="4 2 to b ineq"} implies that $b > k-1$. Since we have that $b \leq k$, and $b$ is a natural number, we conclude that $b=k$. We thus may replace $k$ with $b$ in Equation [\[Case V-3\]](#Case V-3){reference-type="ref" reference="Case V-3"} to obtain $$p^2+2^bp=-p^2+(2^{b+1}-1)p+2^{b+1}-1,$$ which is equivalent to $$2^b(p+2)= 2p^2+p+1.$$ Thus, we have $p+2|2p^2+p+1$. We then have: $$p+2\vert (2p^2+p+1) + (3-2p)(p+2) = 7$$ Since $p+2|7$, we must have $p=5$. We then can solve to get that $b=k=3$. This yields $n=(2^3)(5^2)=200$, which is in fact a 2-near perfect number of the desired form. Here our omitted divisors are $25$ and $40$. We now turn our attention to Case 4. That is, we have $d_1=p^2$ and $d_2=2^bp^2$. In this situation Equation [\[fundamental equation for 2-near perfect of form power 2 times square of a prime\]](#fundamental equation for 2-near perfect of form power 2 times square of a prime){reference-type="ref" reference="fundamental equation for 2-near perfect of form power 2 times square of a prime"} becomes $$(2^{k+1}-1)(p^2+p+1)-2^{k+1}p^2=p^2+2^bp^2,$$ which can be rewritten as $$2^{k+1}(p+1)=2p^2+2^bp^2+p+1. \label{Case V-4}$$ Therefore, we have: $$(p+1)\vert(2p^2+2^bp^2+p+1),$$ $$(p+1)\vert(2p^2+2^bp^2+p+1)-(p+1)=2p^2+2^bp^2,$$ and thus $$(p+1)\vert p^2(2+2^b).$$ Since $p+1$ and $p^2$ are relatively prime this implies $$(p+1)\vert(2+2^b).$$ Thus, there exists a positive integer $z$ such that $z(p+1)=2+2^b,$ and hence $$2^b= z(p+1)-2.\label{2 to be the in terms of p and z}$$ We note that $p+1$ is even and the only way which $2+2^b$ can be divisible by 4 is if $b=1$ (which does not lead to a solution). Thus, $z$ is odd. If we take Equation [\[Case V-4\]](#Case V-4){reference-type="ref" reference="Case V-4"} modulo $2^b$, we obtain that $$2^b|2p^2+p+1. \label{V-4 2 power divisible function of p}$$ We note that Equation [\[V-4 2 power divisible function of p\]](#V-4 2 power divisible function of p){reference-type="ref" reference="V-4 2 power divisible function of p"} allows one to obtain a finite set of possible values $b$ for any given fixed choice of $p$, and then use each to solve for $k$. We may thus with only a small amount of effort verify that we must have $p > 23$. We may combine equation [\[V-4 2 power divisible function of p\]](#V-4 2 power divisible function of p){reference-type="ref" reference="V-4 2 power divisible function of p"} with Equation [\[2 to be the in terms of p and z\]](#2 to be the in terms of p and z){reference-type="ref" reference="2 to be the in terms of p and z"} to obtain: $$z(p+1)-2\vert2p^2+p+1.$$ We then have: $$zp+z-2\vert2p^2+p+1$$ $$zp+z-2\vert2z(2p^2+p+1)-(4p-2)(zp+z-2)=4z+8p-4$$ and so $$zp+z-2\vert4(z+2p-1).\label{V-4 divisility before ineq}$$ Consider now the situation where we have equality in the relationship in Equation [\[V-4 divisility before ineq\]](#V-4 divisility before ineq){reference-type="ref" reference="V-4 divisility before ineq"}. Then we have $$zp+z-2=4(z+2p-1),$$ which is equivalent to $z(p+1)=8p+2$. Thus, $$p+1|8p+2,$$ and hence $$p+1|6p.$$ Since $p+1$ and $p$ are relatively prime, this forces us to have $p+1|6$, and hence we must have $p=5$, which we can verify does not work. Thus, we must have some integer $m \geq 2$ such that $$m(zp+z-2)=4(z+2p-1),$$ and thus we have $$zp +z-2 \leq 2(z+2p-1),$$ which is equivalent to $$z \leq\frac{4p}{p-1}. \label{Final z inequality}$$ We have that $p \geq 7$, and thus, Inequality [\[Final z inequality\]](#Final z inequality){reference-type="ref" reference="Final z inequality"} implies that $z < 6$. Since $z$ is odd, we must then have $z=1$, $z=3$ or $z=5$. If we have $z=1$, then Equation [\[V-4 divisility before ineq\]](#V-4 divisility before ineq){reference-type="ref" reference="V-4 divisility before ineq"} implies that $p-1\vert8p.$ But since $p-1$ is relatively prime to $p$, we must have $p-1\vert8,$ which is impossible since we know that $p>23$.\ \ Using similar logic, for $z=3$, we obtain from Equation [\[V-4 divisility before ineq\]](#V-4 divisility before ineq){reference-type="ref" reference="V-4 divisility before ineq"} that $$3p+1\vert4(2p+2)=8(p+1)$$ $$3p+1\vert8p+8-8(3p+1)=-16p.$$ Thus, $3p+1$ is relatively prime to $p$, so we must have that $$3p+1\vert16.$$ But, once again, we have $p>23$, so there cannot be any solution.\ Finally, for $z=5$, we obtain, from Equation [\[V-4 divisility before ineq\]](#V-4 divisility before ineq){reference-type="ref" reference="V-4 divisility before ineq"}, that $$5p+3\vert4(2p+4)=8p+16,$$ and thus $$5p+3\vert3(8p+16)=24p+48$$ $$5p+3\vert24p+48-16(5p+3)=-56p.$$ Since $p\neq3$, $5p+3$ and $p$ are relatively prime, so we must have $$5p+3\vert56.$$ But, once again this contradicts that $p>23$. ◻ Thus, we have completed the proof of Theorem [\[Second main result power of 2 times square of a prime\]](#Second main result power of 2 times square of a prime){reference-type="ref" reference="Second main result power of 2 times square of a prime"}. # Strongly 2 near perfect numbers A slightly different way of defining a number to be pseudoperfect is to say that a number $n$ is pseudoperfect if there is a set $S$ which is a subset of the positive divisors of $n$ such that the sum of the elements in $S$ sums to $2n$. The last author and Tim McCormack [@MZ] studied what they called strongly pseudoperfect numbers. A number $n$ is said to be strongly pseudoperfect if there is a subset $S$ of divisors of $n$ where the sum of the elements sums to $2n$, and where we also have the property that $d \in S$ if and only if $\frac{n}{d} \in S$. It is natural to combine the notion of 2-near perfect and strongly pseudoperfect as follows: We say that a number $n$ is strongly 2-near perfect if $n$ is strongly pseudoperfect and also 2-near perfect. Note that this is equivalent to $n$ having a divisor $d$ such that $$\sigma(n)-d -\frac{n}{d}= 2n.$$ The following table gives all seven strongly 2-near perfect numbers less than one million: -------- ------------- ------- ------- $n$ $\sigma(n)$ $d_1$ $d_2$ 156 392 2 78 352 756 8 44 6832 15376 4 1708 60976 122512 148 412 91648 184140 128 716 152812 306432 302 506 260865 539136 15 17391 -------- ------------- ------- ------- In this section, we will give a description of all strongly 2-near perfect numbers $n$ of the form $n=2^kp$ for a prime $p$. **Lemma 14**. *If $n$ is a strong 2-near perfect number of the form $2^kp$ for some odd prime $p$ and natural number $k$, then $p=\frac{2^{k+1}-2^a-1}{1+2^{k-a}}$. [\[Strong near perfect only arise from family 3\]]{#Strong near perfect only arise from family 3 label="Strong near perfect only arise from family 3"}* *Proof.* Assume we have a strong 2-near perfect number. By looking at our four families of numbers which arise from Theorem [\[classification of 2 near perfect of form power of two times a prime\]](#classification of 2 near perfect of form power of two times a prime){reference-type="ref" reference="classification of 2 near perfect of form power of two times a prime"}, we can see that only numbers in the third family might possibly be strongly 2-near perfect. In the first family, the product of omitted divisors $d_1d_2$ is odd, so one cannot have $d_1d_2=n$ since $n$ is even. In the second family, we have $d_1d_2$ is a power of 2, and thus is not $n$. In our fourth family, we have $p^2|d_1d_2$ so $d_1d_2 \neq n$. Thus, we may assume that we have a number arising from the third family. In that situation, from $d_1d_2=n$ we get that $a+b=k$, from which the result follows. ◻ **Lemma 15**. *Assume that $n$ is a strong $2$-near perfect number of the form $n=2^kp$ with $p=\frac{2^{k+1}-2^a-1}{1+2^{k-a}}$. Then $k <2a$. [\[k \< 2a lemma\]]{#k < 2a lemma label="k < 2a lemma"}* *Proof.* Assume that $n$ is a strong $2$-near perfect number of the form $n=2^kp$ with $p=\frac{2^{k+1}-2^a-1}{1+2^{k-a}}$, and that $k \geq 2a$. Thus, we have $$1+2^{k-a}|2^{k+1}-2^a-1,$$ which implies that $$1+2^{k-a}|2^{k+1}-2^a-1 +(1+2^{k-a}) = 2^{k+1} -2^a + 2^{k-a} = 2^a(2^{k+1-a} + 2^{k-2a}-1).$$ Since $k \geq 2a$, $2^{k+1-a} + 2^{k-2a}-1$ is a positive integer. We also have that $(1+2^{k-a}, 2^a)=1$, so we have then that $$1+2^{k-a}|2^{k+1-a} + 2^{k-2a}-1 \label{Right before we break down into two cases}.$$ Consider the situation where $k=2a$. Then Equation [\[Right before we break down into two cases\]](#Right before we break down into two cases){reference-type="ref" reference="Right before we break down into two cases"}, becomes $$1+2^{a}|2^{a+1},$$ which has no solutions. So, we may assume that $k> 2a$. We have from Equation [\[Right before we break down into two cases\]](#Right before we break down into two cases){reference-type="ref" reference="Right before we break down into two cases"} that there is some $m$ such that $$\label{m form} m(1+2^{k-a}) = 2^{k+1-a} + 2^{k-2a}-1 .$$ Note that the right-hand side of the equation is odd, so $m$ must be odd. If $m=1$ then we have $$(1+2^{k-a}) = 2^{k+1-a} + 2^{k-2a}-1,$$ which implies that $$\label{m =1}1 + 2^{k-a-1} = 2^{k-a} -2^{k-2a-1}.$$ The left-hand side of Equation [\[m =1\]](#m =1){reference-type="ref" reference="m =1"} is odd, and the only way for the right-hand side to be odd is if $k-2a-1=0$. The only solution of this system of equations is when $k=3$ and $a=1$, which forces $p=\frac{13}{5}$ which is not an integer. Thus, we have $m \neq 1$, and so $m \geq 3$. We thus have $3(1 + 2^{k-a-1}) \leq 2^{k-a} -2^{k-2a-1}$, which is impossible. ◻ **Proposition 16**. *Assume that $n$ is a strong $2$-near perfect number of the form $n=2^kp$ with $p=\frac{2^{k+1}-2^a-1}{1+2^{k-a}}$. Then $k =a+2$, and the omitted divisors are $d_1 =2^{a}$ and $d_2 =4p$, with $p=\frac{2^{a+3}-2^a-1}{5}$, and $a \equiv 3$ (mod 4). [\[description of strongly 2 near perfect of form power of 2 times a prime\]]{#description of strongly 2 near perfect of form power of 2 times a prime label="description of strongly 2 near perfect of form power of 2 times a prime"}* First, we need to prove the following lemma. **Lemma 17**. *If $2^b+1|3(2^a)+1$, then $b=2$. [\[lemma about 2 b+1\|3(2 a)+1\]]{#lemma about 2 b+1|3(2 a)+1 label="lemma about 2 b+1|3(2 a)+1"}* *Proof.* Assume that $2^b+1|3(2^a)+1$. We thus have for some positive integer $m$, $$m(2^b+1)=3(2^a)+1 \label{m form for 2 powers lemma}.$$ Notice that $3(2^a)+1$ is never divisible by 3, and thus $b$ must be even, and $m$ cannot be divisible by 3. If $m=1$ then we have $2^b+1=3(2^a)+1$ which would imply we would have $3|2^b$, which cannot happen. Thus, we may assume that $m \geq 5$. This implies that $a >b$. Set $a=bq +r$ where $0 \leq r < b$. We have $$(3)(2^{qb+r} +1) = (3)((2^b)^q 2^r +1) \equiv 3((-1)^q 2^r +1) \pmod {2^b+1} \equiv 0 \label{q congruence I} .$$ Now, we separate into two cases, depending on whether $q$ is even or odd. If $q$ is even, then Equation [\[q congruence I\]](#q congruence I){reference-type="ref" reference="q congruence I"} yields that $$3(2^r)+1 \equiv 0 \pmod {2^b+1},$$ and so $$2^b +1 |3 (2^r) +1 \label{consequence case even of q congruence}$$ Equation [\[consequence case even of q congruence\]](#consequence case even of q congruence){reference-type="ref" reference="consequence case even of q congruence"} implies that $r \geq b-1$. We thus have $r=b-1$. Thus, $$2^b +1 | 3(2^{b-1}) +1 = 2^{b} +1 +2^{b-1}$$ which is impossible. We now consider the case where $q$ is odd. Then Equation [\[q congruence I\]](#q congruence I){reference-type="ref" reference="q congruence I"} implies that $$-3(2^r)+1 \equiv 0 \pmod {2^b+1}.$$ and thus, $$2^b +1| 3(2^r)-1,$$ which similarly leads to a contradiction, unless $b=2$ and $r=1$. ◻ One might wonder if Lemma [\[lemma about 2 b+1\|3(2 a)+1\]](#lemma about 2 b+1|3(2 a)+1){reference-type="ref" reference="lemma about 2 b+1|3(2 a)+1"} can be strengthened to conclude that if $p$ is a prime where $p|2^b+1$ for some even $b$ and $p|3(2^a) +1$ for some $a$, then one must have $p=5$. However, this is in fact not true. In particular, note that $29|2^{14}+1$, but it is also true that $29|3(2^9)+1.$ We now prove Proposition [\[description of strongly 2 near perfect of form power of 2 times a prime\]](#description of strongly 2 near perfect of form power of 2 times a prime){reference-type="ref" reference="description of strongly 2 near perfect of form power of 2 times a prime"}. *Proof.* Assume that $n$ is a strongly $2$-near perfect number of the form $n=2^kp$ with $p=\frac{2^{k+1}-2^a-1}{1+2^{k-a}}$. Our proof is complete if we can show that we must have $k=a+2$. If we have $k=a+b$, then this is the same as $2^b +1|2^{a+b+1} -2^a -1$, which implies that $2^b +1|2^{a+b+1} -2^a +2^b$. We have $$2^b+1 |2^{a+b+1} -2^a -1 -2^{a+1}(1+2^b) = -3(2^a)-1$$ and so $2^b +1|3(2^a)+1$, which allows us to apply Lemma [\[lemma about 2 b+1\|3(2 a)+1\]](#lemma about 2 b+1|3(2 a)+1){reference-type="ref" reference="lemma about 2 b+1|3(2 a)+1"}, to conclude that $b=2$, and the rest follows simply from noting that all 2-near perfect of this form are in Case 3. ◻ We list below the first few values of $a$ where $\frac{2^{a+3}-2^a-1}{5}$ is prime, and its corresponding prime $p$, each of which corresponds to a strong 2-near perfect number of the form $2^{a+2} p$. We do not include the last two primes as they are too big to fit on one line. ------ ------------------------------------------------ $a$ $p$ 3 11 7 179 19 734003 27 187904819 31 3006477107 39 769658139443 151 3996293539576687666963200714458586381871690547 1 99 -- 451 -- ------ ------------------------------------------------ Standard heuristic arguments suggest that there should be infinitely many primes of the form $\frac{2^{a+3}-2^a-1}{5}$. # Open problems One obvious direction is try to extend the classification of 2-near perfect numbers to classify all of the form $2^kp^m$ where $m\geq3$. **Conjecture 18**. *There are only finitely many 2-near perfect numbers of the form $2^kp^m$ where $m\geq2$.* A slightly weaker conjecture is the following. **Conjecture 19**. *For any fixed $m \geq2$, there are only finitely many 2-near perfect numbers of the form $2^kp^m$.* Another direction to go in is to change the signs in the relationship $\sigma(n)=2n+d_1 +d_2$. The two other options are $\sigma(n)=2n-d_1 -d_2$ and $\sigma(n)=2n+d_1 -d_2$. It seems likely that the main method used in this paper, including the discriminant sandwich would be successful for the first of these two situations, but the situation with mixed signs on the divisors may be more difficult. # Acknowledgements This paper was written as part of the Hopkins School Mathematics Seminar 2022-2023. Steven J. Miller suggested the problem of changing signs as discussed in the final section. 99 P. Cohen, K. Cordwell, A. Epstein, C. Kwan, A. Lott, S. J. Miller On near perfect numbers, Acta Arithmetica 194 (2020), 341-366 Y. Li and Q. Liao, A class of new near perfect numbers. Journal of the Korean Mathematical Society, Volume, 52 Issue 4, pg. 751-763 (2015). T. McCormack, J. Zelinsky, Weighted Versions of the Arithmetic-Geometric Mean and Zaremba's Function, to appear. P. Pollack, V. Shevelev, On perfect and near perfect numbers, J. Number Theory 132, No. 12, 3037-3046 (2012). W. Sierpiñski, Sur les nombres pseudoparfaits, Matematiĉki Vesnik, Vol. 2 (17), No. 33, pg. 212-213 (1965). X. Ren, Y. Chen, On near perfect numbers with two distinct prime factors, Bull. Aust. Math. Soc. 88, No. 3, 520-524 (2013). M. Tang, X. Ma, M. Feng, On near perfect numbers, Colloq. Math. 144, No. 2, 157-188 (2016)

arxiv_math

--- abstract: | The control of relaxation-type systems of ordinary differential equations is investigated using the Hamilton-Jacobi-Bellman equation. It is shown that their solution also allows for an asymptotic expansion in the relaxation parameter. For the first-order expansion, its solution converges toward the solution to a Hamilton--Jacobi--Bellman equation for a reduced control problem. The considered systems are motivated by semi-discretization of kinetic and hyperbolic partial differential equations and several examples are presented. address: - | Institut für Geometrie und Praktische Mathematik\ RWTH Aachen Univeristy\ 52062 Aachen\ Germany. - Institut für Mathematik, RTG "Energy, Entropy, and Dissipative Dynamics", RWTH Aachen University,52062, Aachen, Germany author: - Michael Herty - Hicham Kouhkouh bibliography: - bibliography.bib title: Relaxation and asymptotic expansion of controlled stiff differential equations --- [^1] [^2] [^3] # Introduction We are interested in systems of differential equations of the form $$\label{eq: sys intro 1} \dot{z}(s) = f(z(s),s) + \frac{1}{\varepsilon} \, g(z(s),s)$$ where $\varepsilon>0$ is the stiffness parameter, representing for example a discretization of a system of relaxation-type partial differential equations (PDE), see e.g. Section [5](#Sec: Application){reference-type="ref" reference="Sec: Application"} for some selected examples. Such systems have been studied intensively in the context of numerical methods, see e.g. [@MR2029975; @MR2231946] for implicit--explicit discretization methods, and in particular, in the PDE context where the above form arises e.g. in semi-Lagrangian approximations to hyperbolic and kinetic transport equations [@MR3698447; @MR2970736; @MR1619910; @MR3109810; @MR3202241]. Many approaches focus on analytical [@MR1693210; @MR3267352] or numerical aspects of the previous relaxation--type system [\[eq: sys intro 1\]](#eq: sys intro 1){reference-type="eqref" reference="eq: sys intro 1"}. To the best of our knowledge, only a few recent results on the corresponding control problems [\[eq: cost\]](#eq: cost){reference-type="eqref" reference="eq: cost"}-[\[eq: CSP intro 1\]](#eq: CSP intro 1){reference-type="eqref" reference="eq: CSP intro 1"} exist, see [@MR3921027; @MR4456458; @MR4471483; @MR2891921; @MR2825372]. The latter publications focus on (high--order) numerical discretization of an optimal control problem governed by using Pontryagin's maximum principle. In particular, the discussion of order conditions of the numerical schemes has been a recent focus of the discussion, see e.g. [@MR1804658; @MR3054355; @MR3072232]. Contrary to those approaches, we focus on a formulation using the Hamilton--Jacobi--Bellman equation. Our goal is twofold: first, we seek the limiting differential equation of [\[eq: sys intro 1\]](#eq: sys intro 1){reference-type="eqref" reference="eq: sys intro 1"} when $\varepsilon\to 0$, then we study this limit in the context of optimal control after embedding the latter in a family of parameterized differential equations. In particular, we aim at providing an asymptotic expansion in $\varepsilon$ of the value function [\[eq: value intro 1\]](#eq: value intro 1){reference-type="eqref" reference="eq: value intro 1"} of such optimal control problem. Introducing a new variable $y(\cdot)$ whose dynamics captures the *fast* part in [\[eq: sys intro 1\]](#eq: sys intro 1){reference-type="eqref" reference="eq: sys intro 1"}, that is $$\label{eq: fast intro 1} \dot{y}(s) = \frac{1}{\varepsilon} \, g(z(s),s),$$ together with the dynamics of $z(\cdot)$, we obtain the following equivalent system of differential equations $$\begin{aligned} \dot{z}(s) & = f(z(s),s) + \dot{y}(s)\\ \varepsilon\, \dot{y}(s) &= g(z(s),s) \end{aligned}$$ which takes the form $$\label{eq: SP intro 1} \begin{aligned} \dot{\mathbf{z}}(s) & = F(\mathbf{z}(s), \mathbf{y}(s), s) \\ \varepsilon\, \dot{\mathbf{y}}(s) &= G(\mathbf{z}(s), \mathbf{y}(s), s). \end{aligned}$$ Indeed, we may set $\mathbf{z} := z-y$, $\mathbf{y} := y$, and then $F(\mathbf{z},\mathbf{y},s) := f(\mathbf{z}+\mathbf{y},s)$ and $G(\mathbf{z},\mathbf{y},s) := g(\mathbf{z}+\mathbf{y},s)$. The formulation [\[eq: SP intro 1\]](#eq: SP intro 1){reference-type="eqref" reference="eq: SP intro 1"} has the advantage of being *singularly perturbed* in time only. Such a structure may be easier to study, also when the dynamics are subject to optimal control as in [\[eq: CSP intro 1\]](#eq: CSP intro 1){reference-type="eqref" reference="eq: CSP intro 1"}, and it benefits from a huge literature, e.g. the books [@bensoussan2011asymptotic; @dontchev2006well; @kokotovic1999singular], the papers from dynamical systems viewpoint [@grammel2004nonlinear; @grammel1997averaging; @artstein1997tracking; @artstein2000value; @gaitsgory1992suboptimization] or from PDE viewpoint [@terrone2011limiting; @alvarez2008multiscale], to name but a few. We also refer to the thesis [@terrone2008singular; @kouhkouh22phd] and the references therein. To this end, we introduce the cost functional $$\label{eq: cost} J(\mathbf{z}) = \int_{0}^{t}\ell(\mathbf{z}(s))\,\mbox{\rm d}s + \phi(\mathbf{z}(t))$$ where the dynamics now includes controls $\alpha(\cdot), \beta(\cdot)$ such that $$\label{eq: CSP intro 1} \begin{aligned} \dot{\mathbf{z}}(s) & = F(\mathbf{z}(s), \mathbf{y}(s), \alpha(s), s), \quad \mathbf{z}(0) = \mathbf{z}_0 \\ \varepsilon \, \dot{\mathbf{y}}(s) &= G(\mathbf{z}(s), \mathbf{y}(s), \beta(s), s), \quad \mathbf{y}(0)=\mathbf{y}_0. \end{aligned}$$ The value function is then $$\label{eq: value intro 1} V^{\varepsilon}(\mathbf{z}_0, \mathbf{y}_0, t) = \inf\limits_{ (\alpha,\beta) } \; J(\mathbf{z}), \quad \text{ s.t.: } \quad \eqref{eq: CSP intro 1}.$$ We are interested in the limit of $V^\varepsilon$ when $\varepsilon \to 0$, as well as in an asymptotic expansion such that $$V^{\varepsilon} = V_{0} + \varepsilon\, V_{1} + O(\varepsilon^2).$$ In the second part, we expand the analysis and consider in equation [\[eq: sys intro 1\]](#eq: sys intro 1){reference-type="eqref" reference="eq: sys intro 1"} a function $g(z,t) = g^{\varepsilon}(z,t)$ depending on $\varepsilon$ such that $$\label{eq: g eps} g^{\varepsilon}(z,t) = g_{0}(z,t) + \varepsilon\, g_{1}(z,t) + O(\varepsilon^{2}).$$ The goal thereafter will be to analyse the impact of $g_{0}$ and $g_{1}$ on the asymptotic expansion of the value function defined in [\[eq: value intro 1\]](#eq: value intro 1){reference-type="eqref" reference="eq: value intro 1"}. An important aspect of the present work is to restrict the analysis to the simple, yet important case where the limiting problems are obtained by setting $\varepsilon =0$ and solving for $\mathbf{y}$ the equation $G(\mathbf{z}(s),\mathbf{y}(s),\beta(s),s)=0$. This is known as the *reduced* system. In general, an *averaged* system is obtained. The latter is discussed in the Appendix [7](#appendix){reference-type="ref" reference="appendix"}. # Singular perturbations for linear optimal control problems {#sec: linear} Denote by $z$ and $y$ the slow and fast variables respectively, subject to the following system of singularly perturbed and linearly controlled ODEs $$\label{CSP} \begin{aligned} \dot{z}(s) & = A_{1}(z)y + B_{1}(z)\alpha + C_{1}(z),& z(0) = z_{0}\in \mathds{R}^{m}&\\ %f(z(s),y(s), \alpha(s)) \varepsilon\dot{y}(s) & = A_{2}(z)y + B_{2}(z)\beta + C_{2}(z),& y(0) = y_{0}\in\mathds{R}^{n},& %g(z(s),y(s),\alpha(s)) \end{aligned}$$ where $s\in[0,T]$, $T=1,$ $\varepsilon>0$ is the small parameter, $A_{i}(z), B_{i}(z), C_{i}(z), i=1,2$ are matrices functions such that for some $\lambda$ $$\label{assumption fast} \left\| e^{A_{2}(z)t} \right\| \leq e^{-\lambda t},\quad \lambda>0,\; \text{loc. unif. in } z,$$ and the admissible controls $\alpha(\cdot)$ and $\beta(\cdot)$ are measurable functions with values in $\Omega_A$ and $\Omega_B$ respectively. These sets are assumed to be both, compact and convex subsets of $\mathds{R}^{p}$ and $\mathds{R}^{q}$ respectively. The control parameters $\alpha,\beta$ are chosen so as to minimize the cost functional $$\label{SP} \tag{SP} G_{\varepsilon}:=\inf \; \Phi(z(1))$$ subject to the singularly perturbed and controlled trajectories [\[CSP\]](#CSP){reference-type="eqref" reference="CSP"} with $s\in [0,T]$. Next, we set $\varepsilon=0$ in [\[CSP\]](#CSP){reference-type="eqref" reference="CSP"} and solve the static equation, the formal limit $\varepsilon=0$ to the previous system: $$\label{static} 0 = A_{2}(z)y + B_{2}(z)\beta + C_{2}(z)$$ whose root $y=\psi(z,\beta)$ is then substituted in the first equation in [\[CSP\]](#CSP){reference-type="eqref" reference="CSP"}, yielding what we shall refer to as the reduced dynamics $$\label{reduced} \dot{\bar{z}}(s) = A_{1}(\bar{z})\psi(\bar{z},\beta) + B_{1}(\bar{z})\alpha + C_{1}(\bar{z}), \quad \bar{z}(0) = z_{0}\in \mathds{R}^{m}.$$ This controlled ODE is associated with the minimization of the cost functional $$\label{R} \tag{R} \bar{G}:=\inf \; \Phi(\bar{z}(1))$$ We are now ready to state some results and definitions from [@gaitsgory1992suboptimization] on the limiting problem when $\varepsilon\to 0$. **Definition 1**. *Given a trajectory $x(\cdot)$, an admissible control $(\alpha_{x(\cdot)},\beta_{x(\cdot)})$ is referred to as [$x(\cdot)$-approximating]{.ul} if it generates in [\[CSP\]](#CSP){reference-type="eqref" reference="CSP"} a trajectory $z_{\varepsilon}(\cdot)$ satisfying $$\max\limits_{s\in[0,1]} \|z_{\varepsilon}(s) - x(s)\| \leq \mu(\varepsilon) \to 0 \text{ as } \varepsilon\to 0.$$* Note that in this definition, the control $(\alpha_{x(\cdot)},\beta_{x(\cdot)})$ is not necessarily the control of the admissible trajectory $x(\cdot)$. Moreover, $\alpha$ and $\beta$ do not need to be necessarily different. **Definition 2**. *We say that the problem [\[SP\]](#SP){reference-type="eqref" reference="SP"} is [approximated by]{.ul} the problem [\[R\]](#R){reference-type="eqref" reference="R"} if $G_{\varepsilon}\to \bar{G}$ as $\varepsilon\to 0$ and if, corresponding to any admissible trajectory $\bar{z}_{\nu}(t)$ of the reduced system [\[reduced\]](#reduced){reference-type="eqref" reference="reduced"} such that $\Phi(\bar{z}_{\nu}(1))\leq \bar{G}+\nu$, $\nu>0$, there exists $\bar{z}_{\nu}(\cdot)$-approximating control providing in the problem [\[SP\]](#SP){reference-type="eqref" reference="SP"} the value of the functional differing from the optimal one by $\nu+\kappa(\varepsilon)$, where $\kappa(\varepsilon)\to 0$ as $\varepsilon\to 0$.* **Theorem 1**. *Under the above-mentioned assumptions, the problem [\[SP\]](#SP){reference-type="eqref" reference="SP"} is approximated by the problem [\[R\]](#R){reference-type="eqref" reference="R"}.* For the proof we refer to Theorem 4.3 and Example 4.3 in [@gaitsgory1992suboptimization]. *Remark 1*. Assumption [\[assumption fast\]](#assumption fast){reference-type="eqref" reference="assumption fast"} guarantees that all admissible trajectories $\{z(s),y(s)\}$ of [\[CSP\]](#CSP){reference-type="eqref" reference="CSP"} remain in a compact subset of $\mathds{R}^{m}\times \mathds{R}^{n}$ for all $s\in [0,1]$ and for all initial positions $(z_{0}, y_{0})$ chosen in a compact subset $\mathds{R}^{m}\times \mathds{R}^{n}$. We assume without loss of generality that $y$ remains in the $n$-dimensional torus $\mathds{T}^{n}$, and the dynamics [\[CSP\]](#CSP){reference-type="eqref" reference="CSP"} is periodic in $y$. The convergence stated in Theorem [Theorem 1](#thm: conv G){reference-type="ref" reference="thm: conv G"} is further investigated in the next section where we seek how it translates at the level of the HJB equation and the value function. # On the HJB equation and the reduction method {#sec: hjb} In this section, we analyse how the reduction [\[reduced\]](#reduced){reference-type="eqref" reference="reduced"} extends to the HJB equation associated with the control problems. Consider a finite horizon control problem whose cost functional is $$\label{cost} J(z) := \int_{0}^{t} \ell(z(s)) \,\mbox{\rm d}s + \phi(z(t))$$ where $\ell,\phi$ are real continuous functions. The value function is $$\label{value eps} V^{\varepsilon}(z,y,t) := \inf\limits_{\alpha(\cdot),\beta(\cdot)}\, J(z), \quad \text{ s.t.: } \; \eqref{CSP}$$ and $\alpha(\cdot),\beta(\cdot) \in L^{\infty}([0,+\infty))$ with values in the compact and convex sets $\Omega_A$ and $\Omega_B$ respectively, subsets of $\mathds{R}^{p}$ and $\mathds{R}^{q}$. *Remark 2*. As it is well-known, such a control problem can be reformulated as the problem [\[SP\]](#SP){reference-type="eqref" reference="SP"} which is of Mayer type. To do so, it suffices to add a new *slow* variable $x(\cdot)\in \mathds{R}$ whose dynamics is governed by $$\dot{x}(s) = \ell(z(s)), \quad x(0) = 0.$$ Then the objective function would be to minimize: $\inf\; x(t) + \phi(z(t))$. The time horizon being finite, it can easily be parameterized to $s \in [0,1]$. Here and in what follows, we assume the following holds. **Standing assumptions** - The dynamics $y$ lives in an $n$-dimensional torus $\mathds{T}^{n}$, and all the functions are periodic in $y$; see Remark [Remark 1](#rem periodic){reference-type="ref" reference="rem periodic"}. - The matrices $A_{i}, B_{i}, C_{i}, i=1,2$ are locally Lipschitz functions with at most a linear growth in $z$. - The functions $\ell, \phi$ are continuous and locally bounded, satisfying for some modulus $\omega$ $$|\varphi(z_{1}) - \varphi(z_{2})| \leq \omega(|z_{1}-z_{2}|,R), \,\forall\, |z_{1}|,|z_{2}|\leq R,\; \forall\, R>0,\quad \varphi=\ell, \phi.$$ - The stability condition [\[assumption fast\]](#assumption fast){reference-type="eqref" reference="assumption fast"} holds. - The matrix $B_{2}$ has full rank. - The control parameters (wherever they are mentioned in the sequel) take values in sufficiently large compact and convex sets. The last two assumptions, together with Remark [Remark 1](#rem periodic){reference-type="ref" reference="rem periodic"} ensure the following controllability condition is satisfied $$\label{strong controllability} \begin{aligned} & \text{for any } z \text{ fixed}, \text{ there exists } \, r(z)>0 \, \text{ such that: } \\ & B\big(0,r(z)\big) \subseteq \overline{\text{co}}\big\{ A_{2}(z)y + B_{2}(z)\beta + C_{2}(z)\, : \beta \in \Omega_B\big\}, \; \forall y\in \mathds{T}^{n}, \end{aligned}$$ where $B\big(0,r(z)\big)$ is the ball centered in $0$ with radius $r(z)$, and $\overline{\text{co}}$ refers to the closed convex hull. Indeed, as the dynamics $y$ remain in a compact set, and since the matrix $B_2$ has full rank, the control can span a set that is large enough to contain a ball around $0$. This condition is needed for the proof of Theorem [Theorem 2](#thm conv){reference-type="ref" reference="thm conv"}. By standard results on viscosity solutions for Hamilton-Jacobi-Bellman (HJB) equation (see e.g. [@bardi2008optimal Chapter III]), the value function is such that $V^{\varepsilon}\in C(\mathds{R}^{m}\times\mathds{R}^{n}\times [0,T])$ for $T<\infty$, and it is the unique viscosity solution to the HJB equation $$\label{HJB} \left\{ \; \begin{aligned} & \partial_{t} V^{\varepsilon} + H\left(z,y,D_{z}V^{\varepsilon},\frac{1}{\varepsilon}D_{y}V^{\varepsilon}\right) = \ell(z),\quad \text{in } \mathds{R}^{m}\times\mathds{R}^{n}\times (0,T)\\ & V^{\varepsilon}(z,y,0) = \phi(z), \text{ in } \mathds{R}^{m}\times\mathds{R}^{n} \end{aligned} \right.$$ where the Hamiltonian $H$ is $$\begin{aligned} %\label{H} & H(z,y,p,q)\\ & = \sup\limits_{\alpha\in \Omega_{A}, \beta\in \Omega_{B}}\{-p\cdot (A_{1}(z)y + B_{1}(z)\alpha + C_{1}(z)) - q\cdot (A_{2}(z)y + B_{2}(z)\beta + C_{2}(z))\}\\ & = -(A_{1}(z)y + C_{1}(z))\cdot p + \sup\limits_{\alpha\in \Omega_A}\big\{ -p\cdot B_{1}(z)\alpha \big \} \\ & \quad \quad \quad - (A_{2}(z)y + C_{2}(z))\cdot q + \sup\limits_{\beta\in \Omega_B}\big\{ -q\cdot B_{2}(z)\beta \big \}. \end{aligned}$$ In the sequel, and for the sake of simplicity of notation, we drop the dependence of $A_{i},B_{i},C_{i},i=1,2$ on the variable $z$. ## The First--Order Asymptotic Expansion {#sec: first order asymptot} Given the optimal control problem [\[value eps\]](#value eps){reference-type="eqref" reference="value eps"}, we are interested in the limit as $\varepsilon\to 0$ of both the value function $V^{\varepsilon}$ (at the level of the PDE problem [\[HJB\]](#HJB){reference-type="eqref" reference="HJB"}) and the underlying controlled and singularly perturbed trajectories [\[CSP\]](#CSP){reference-type="eqref" reference="CSP"}. In [@homocp] (see also Chapter I in [@kouhkouh22phd]), this convergence is proved using *Limit Occupational Measure Sets* (LOMS); see Appendix [7](#appendix){reference-type="ref" reference="appendix"}. These measures are essential in the construction of the limiting optimal control problem and they have been used already in [@gaitsgory1992suboptimization] and references therein. Using the results in [@homocp] (see also [@kouhkouh22phd Chapter I]), we show in Theorem [Theorem 2](#thm conv){reference-type="ref" reference="thm conv"} below that the solution $V^{\varepsilon}$ to [\[HJB\]](#HJB){reference-type="eqref" reference="HJB"} converges locally uniformly to $V_{0}$ unique continuous viscosity solution in $\mathds{R}^{m}\times (0,T)$ of $$\label{HJB 0} \left\{\; \begin{aligned} & \partial_{t} V_{0} - C_{1}(z)\cdot\partial_{z}V_{0} + \sup\limits_{\alpha\in \Omega_A}\{-\partial_{z}V_{0}\cdot B_{1}(z)\alpha \} + \bar{\lambda}_{1}(z,\partial_{z}V_{0}) = \ell(z) \\ & V_{0}(z,0) = \phi(z), \text{ in } \mathds{R}^{m} \end{aligned} \right.$$ where, for fixed $z$ and $p:=\partial_{z}V_{0}(z)$, we have $\bar{\lambda}_{1}(z,p)$ a constant in $y$ for which there exists a continuous viscosity solution $V_{1}(y)$ to the so-called *cell problem* $$\label{cell} \begin{aligned} - (A_{2}y + C_{2})\cdot \partial_{y}V_{1}+\sup\limits_{\beta\in \Omega_B}\{ \; -\partial_{y}V_{1}\cdot B_{2}\beta \} - p^{\top}A_{1}y = \bar{\lambda}_{1}(z,p). \end{aligned}$$ In this case, one would have the following first order asymptotic expansion $$\label{expansion 1} V^{\varepsilon}(t,z,y) = V_{0}(t,z) + \varepsilon V_{1}(y) + O(\varepsilon^{2})$$ where $V_{0}$ solves [\[HJB 0\]](#HJB 0){reference-type="eqref" reference="HJB 0"}. More precisely, given the optimal control problem [\[value eps\]](#value eps){reference-type="eqref" reference="value eps"} where the dynamics $(y(\cdot), z(\cdot))$ in [\[CSP\]](#CSP){reference-type="eqref" reference="CSP"} satisfies the standing assumptions, with $z(\cdot)\in \mathds{T}^{n}$ as explained in Remark [Remark 1](#rem periodic){reference-type="ref" reference="rem periodic"}, we have the following results. **Lemma 1**. *Let $z,p$ be fixed in $\mathds{R}^{m}$. Under the general assumptions, there exists a unique $\bar{\lambda}_{1}(z,p)$ such that [\[cell\]](#cell){reference-type="eqref" reference="cell"} has a continuous periodic viscosity solution $V_{1}(\cdot)$.* *Proof.* This is a particular case of [@alvarez2010ergodicity Theorem 6.2] where it is assumed that the fast dynamics in our case is *small time controllable*. But this is ensured by the controllability assumption for linear systems, see [@bardi2008optimal Theorem IV.1.9, p. 231]. This proves the existence of $(\bar{\lambda}_{1}, V_{1}(\cdot))$. ◻ **Theorem 2**. *As $\varepsilon\to 0$, the sequence $V^{\varepsilon}$ of solutions of [\[HJB\]](#HJB){reference-type="eqref" reference="HJB"} converges locally uniformly on $\mathds{R}^{m}\times (0,+\infty)$ to $V_{0}$ solution to $$\label{HJB eff} \left\{\, \begin{aligned} & \partial_{t} V_{0} + \bar{H}\left(z,D_{z}V_{0}\right) = \ell(z),\quad \text{in } \mathds{R}^{m}\times (0,T)\\ & V_{0}(z,0) = \phi(z), \text{ in } \mathds{R}^{m} \end{aligned} \right.$$ where the limit (effective) Hamiltonian is $$\begin{aligned} %\label{H} \bar{H}(z,p) &= \sup\limits_{\alpha\in \Omega_{A},\beta\in \Omega_B}\left\{-p\cdot (A_{1}(z)\psi(z,\beta) + B_{1}(z)\alpha + C_{1}(z)) \right\} \end{aligned}$$ where $\psi(z,\beta)$ solves (for $y$) the static equation [\[static\]](#static){reference-type="eqref" reference="static"}. Moreover $V_{0}$ is the value function of the limit (effective) optimal control problem $$\label{value eff} V_{0}(z,t) := \inf\limits_{\alpha(\cdot),\beta(\cdot)}\, J(z), \quad \text{ s.t.: } \; \eqref{reduced}$$ where $J$ is as defined in [\[cost\]](#cost){reference-type="eqref" reference="cost"}.* The previous result shows the convergence for $\varepsilon\to 0$ of the solution to the HJB. Applying the previous result and with the notation in [\[HJB 0\]](#HJB 0){reference-type="eqref" reference="HJB 0"}, we have $$\bar{H}(z,p) = -p\cdot C_{1}(z) + \sup\limits_{\alpha\in \Omega_A} \{-p\cdot B_{1}(z)\alpha \} + \bar{\lambda}_{1}(z,p) %-\ell(z)$$ and $$\bar{\lambda}_{1}(z,p) = \sup\limits_{\beta\in \Omega_B} \{-p \cdot A_{1}(s)\psi(z,\beta) \}.$$ Furthermore, using [\[expansion 1\]](#expansion 1){reference-type="eqref" reference="expansion 1"}, we obtain an asymptotic expansion of a closed-loop optimal control. *Proof.* The standing assumptions guarantee the controllability condition [\[strong controllability\]](#strong controllability){reference-type="eqref" reference="strong controllability"}. We can then apply Theorem [@homocp Theorem 3.1] (or [@kouhkouh22phd Theorem 1.3.1])which ensures the value function $V^{\varepsilon}$ converges locally uniformly on $\mathds{R}^{m}\times(0,+\infty)$ to $\widetilde{V}$ solution to $$\left\{ \; \begin{aligned} & \partial_{t}\widetilde{V} + \widetilde{H}(z,D_{z}\widetilde{V}) = \ell(z),\; \text{ in } \mathds{R}^{m}\times(0,T)\\ & \widetilde{V}(z,0) = \phi(z),\; \text{ in } \mathds{R}^{m} \end{aligned} \right.$$ where $$\begin{aligned} %\label{H} \widetilde{H}(z,p) &= \sup\limits_{\alpha\in \Omega_{A},\mu\in \mathfrak{L}(z)}%\left\{ -p\cdot \int_{\mathds{T}^{n}\times \Omega_{B}}(A_{1}(z)y + B_{1}(z)\alpha + C_{1}(z)) \, \text{d}\mu(y,\beta) \end{aligned}$$ and $\mathfrak{L}(z)$ is the *Limit Occupational Measure Set* (LOMS; see the definition in the Appendix [7](#appendix){reference-type="ref" reference="appendix"}). It turns out, however, that in the setting of section §[2](#sec: linear){reference-type="ref" reference="sec: linear"}, the LOMS coincides with the reduced dynamics as was proven in [@gaitsgory1992suboptimization Example 3.4]. Indeed, let $\bar{\beta}\in \Omega_B$ be arbitrarily fixed, then using [@gaitsgory1992suboptimization eq. (3.21)], the dynamics [\[CSP\]](#CSP){reference-type="eqref" reference="CSP"} at the limit $\varepsilon\to 0$ becomes $$\dot{\tilde{z}} = A_{1}(\tilde{z})\psi(\tilde{z},\bar{\beta}) + B_{1}(\tilde{z}) \alpha + C_{1}(\tilde{z})$$ where $\psi(\tilde{z},\bar{\beta})$ solves [\[static\]](#static){reference-type="eqref" reference="static"}, that is $$\psi(\tilde{z},\bar{\beta}) = - A_{2}^{-1}B_{2}\bar{\beta} - A_{2}^{-1}C_{2}$$ and is unique (see [@gaitsgory1992suboptimization Lemma 3.1 (ii)]). This leads to the reduced dynamics [\[reduced\]](#reduced){reference-type="eqref" reference="reduced"}, which can be equivalently represented as $$\dot{\tilde{z}} = \int_{\mathds{T}^{n}\times \Omega_B} \big[ A_{1}(\tilde{z})y + B_{1}(\tilde{z})\alpha + C_{1}(\tilde{z})\,\big]\, \delta_{\psi(\tilde{z},\beta)}(\mbox{\rm d}y) \otimes \delta_{\bar{\beta}}(\mbox{\rm d}\beta).$$ Therefore any $\mu \in \mathfrak{L}(z)$ corresponds to $\delta_{\psi(\tilde{z},\bar{\beta})}(\text{d}y)\!\otimes \!\delta_{\bar{\beta}}(\text{d}\beta)$, where $\psi(z,\bar{\beta})$ solves [\[static\]](#static){reference-type="eqref" reference="static"} for some $\bar{\beta}\in \Omega_B$, and so the supremum in the definition of $\widetilde{H}$ is taken over $\alpha\in \Omega_A$ and $\bar{\beta}\in \Omega_B$ which ultimately yields to $\widetilde{H}=\Bar{H}$. ◻ ## Asymptotic Expansion for Second--Order Approximation The next term in the asymptotic expansion can be formally obtained by making the following **Ansatz.** $V^{\varepsilon}(z,y,t) = V_{0}(z,t;\varepsilon) + \varepsilon\,V_{1}(y;\varepsilon) + \varepsilon^{2}\,V_{2}(y) + O(\varepsilon^{3})$ \ Using this ansatz in the PDE in [\[HJB\]](#HJB){reference-type="eqref" reference="HJB"} yields $$\begin{aligned} & 0= \partial_{t} V_{0} - C_{1}\cdot\partial_{z}V_{0} + \sup\limits_{\alpha\in \Omega_A}\{-\partial_{z}V_{0}\cdot B_{1}\alpha \} -\ell(z) - A_{1}y\cdot\partial_{z}V_{0} \\ & \quad \quad - (A_{2}y + C_{2})\cdot\partial_{y}V_{1} + \sup\limits_{\beta\in \Omega_B}\{ \; -\partial_{y}V_{1}\cdot B_{1}\beta -\varepsilon(A_{2}y + C_{2} + B_{2}\beta)\cdot\partial_{y}V_{2} \; \}. \end{aligned}$$ We identify the terms depending on the fast variable $y$ according to the powers of $\varepsilon$. This implies having pairs $(\lambda_{1},V_{1}(\cdot))$ and $(\lambda_{2}, V_{2}(\cdot))$ where $\lambda_{1},\lambda_{2}$ are constant in $y$ and $V_{1},V_{2}$ are functions of $y$ for fixed $z$, $\partial_{z}V_{0}=:p$ and $\beta$, such that - The PDE problem solved by $(\lambda_{2}, V_{2}(\cdot))$ is $$\label{HJB 2} -(A_{2}y + C_{2} + B_{2}\beta)\cdot\partial_{y}V_{2} = \lambda_{2}(z,\beta), \quad \text{for } y\in \mathds{T}^{n}$$ - The PDE problem solved by $(\lambda_{1}, V_{1}(\cdot))$ is $$\label{HJB 1} \begin{aligned} & - A_{1} y \cdot p - (A_{2}y + C_{2})\cdot\partial_{y}V_{1} \\ & \quad \quad \quad + \sup\limits_{\beta\in \Omega_B}\{ \; -\partial_{y}V_{1}\cdot B_{1}\beta +\varepsilon \lambda_{2}(z,\beta) \} = \lambda_{1}(z,p,\varepsilon), \quad \text{for } y\in \mathds{T}^{n} \end{aligned}$$ Then the PDE in [\[HJB\]](#HJB){reference-type="eqref" reference="HJB"}, complemented with the same initial condition, becomes $$\label{HJB eps} \partial_{t} V_{0} - C_{1}(z)\cdot\partial_{z}V_{0} + \sup\limits_{\alpha\in \Omega_A}\{-\partial_{z}V_{0}\cdot B_{1}(z)\alpha \} + \lambda_{1}(z,\partial_{z}V_{0},\varepsilon) = \ell(z).$$ The results of the previous section are not straightforwardly applicable in this situation. This is mainly due to the presence of $\beta$ in both PDEs [\[HJB 2\]](#HJB 2){reference-type="eqref" reference="HJB 2"} and [\[HJB 1\]](#HJB 1){reference-type="eqref" reference="HJB 1"}. Indeed, in order to obtain an asymptotic expansion of $V^{\varepsilon}$, one could start first by solving [\[HJB 2\]](#HJB 2){reference-type="eqref" reference="HJB 2"} for $\beta$ fixed, then plugging $\lambda_{2}$ in [\[HJB 1\]](#HJB 1){reference-type="eqref" reference="HJB 1"} and solving the latter PDE (provided $\lambda_{2}$ is continuous in $\beta$). The next step would be to find $\beta^{*}$ for which the $\sup$ is obtained and solve again [\[HJB 2\]](#HJB 2){reference-type="eqref" reference="HJB 2"} with $\beta^{*}$. Then the resulting function $V_{2}$ is the next term in the asymptotic expansion $$\label{expansion 2} V^{\varepsilon}(t,z,y) = {V}_{0}(t,z;\varepsilon) + \varepsilon {V}_{1}(y;\varepsilon) + \varepsilon^{2}V_{2}(y) + O(\varepsilon^{3})$$ where $V_{0}$ solves [\[HJB eps\]](#HJB eps){reference-type="eqref" reference="HJB eps"} and $V_{1}$ solves [\[HJB 1\]](#HJB 1){reference-type="eqref" reference="HJB 1"}.\ To that must be added the other difficulty arising from the ambiguity of the dependence of $V_{0}(t,z;\varepsilon), V_{1}(z;\varepsilon)$ on $\varepsilon$ (see [\[HJB 1\]](#HJB 1){reference-type="eqref" reference="HJB 1"}-[\[HJB eps\]](#HJB eps){reference-type="eqref" reference="HJB eps"}). Note that even if there was no control $\beta$ in the slow variables, one would have $\lambda_{1}(z,p,\varepsilon)= \bar{\lambda}_{1}(z,p) + \varepsilon\lambda_{2}(z)$ where $\bar{\lambda}_{1}(z,p)$ is the constant obtained in [\[cell\]](#cell){reference-type="eqref" reference="cell"}, and $V_{0}, V_{1}$ will still depend on $\varepsilon$ in an unknown fashion. To bypass these difficulties (mainly, the presence of $\beta$ in [\[HJB 2\]](#HJB 2){reference-type="eqref" reference="HJB 2"} and the dependence on $\varepsilon$ in [\[HJB 1\]](#HJB 1){reference-type="eqref" reference="HJB 1"}), and still keep track of $\varepsilon$ in the asymptotic expansion, we will consider a multiscale approach. This is the object of the next section. # A Multi-Scale System {#sec: multiscale sys} ## The Three-Scale Linear Optimal Control {#sec: linear 2} For a general discussion, we denote by $z,y$ and $x$ the variables in the macro-, meso-, and micro-scale regime respectively, subject to the following system of singularly perturbed and controlled ODEs $$\label{CSP 2} \begin{aligned} \dot{z}(s) & = A_{0}(z)x + A_{1}(z)y + B_{1}(z)\alpha + C_{1}(z),& z(0) = z_{0}\in \mathds{R}^{m},&\\ %f(z(s),y(s), \alpha(s)) \varepsilon\dot{y}(s) & = A_{2}(z)y + B_{2}(z)\beta + C_{2}(z),& y(0) = y_{0}\in\mathds{R}^{n},&\\ %g(z(s),y(s),\alpha(s)) \varepsilon^{2}\dot{x}(s) & = A_{3}(z)x + B_{3}(z)\gamma + C_{3}(z),& x(0) = x_{0}\in\mathds{R}^{\ell},& \end{aligned}$$ where $s\in[0,1]$, $\varepsilon>0$ is a small parameter, $A_{0}(z), A_{i}(z), B_{i}(z), C_{i}(z), i=1,2,3$ are matrices functions satisfying the general assumptions and, moreover, $x(\cdot)$ satisfy the stability and controllability assumptions (as it is assumed for $y(\cdot)$). We recall that $y,x$ live on the torus $\mathds{T}^{n}$ and $\mathds{T}^{\ell}$ respectively and that all functions if depending on $y$ or $x$ are periodic in the latter (see Remark [Remark 1](#rem periodic){reference-type="ref" reference="rem periodic"}). The admissible controls $\alpha(\cdot)$, $\beta(\cdot)$ and $\gamma(\cdot)$ are measurable functions with values in $\Omega_A$, $\Omega_B$ and $\Omega_\Gamma$ respectively, all compact and convex subsets of $\mathds{R}^{p}$, $\mathds{R}^{q}$ and $\mathds{R}^{r}$ respectively. They are chosen as to minimize the cost functional $$\label{SP 2} \tag{SP} \inf \; \int_{0}^{1} \ell(z(s))\,\text{d}s + \phi(z(1))$$ subject to the singularly perturbed and controlled trajectories [\[CSP 2\]](#CSP 2){reference-type="eqref" reference="CSP 2"} with $s\in [0,1]$. The corresponding HJB equation takes the form $$\label{HJB multi} \left\{ \; \begin{aligned} & \partial_{t} V^{\varepsilon} + H\left(z,y,x,D_{z}V^{\varepsilon},\frac{1}{\varepsilon}D_{y}V^{\varepsilon}, \frac{1}{\varepsilon^{2}}D_{x}V^{\varepsilon}\right) = \ell(z),\quad \text{in } \mathds{R}^{m}\times\mathds{R}^{n}\times \mathds{R}^{\ell}\times (0,1)\\ & V^{\varepsilon}(z,y,x,0) = \phi(z), \text{ in } \mathds{R}^{m}\times\mathds{R}^{n}\times \mathds{R}^{\ell} \end{aligned} \right.$$ where the Hamiltonian $H$ is $$\begin{aligned} %\label{H} & H(z,y,x,p,q,r)\\ & = \sup\limits_{\alpha\in \Omega_{A},\beta\in \Omega_{B}, \gamma \in \Omega_{\Gamma}}\big\{-p\cdot (A_{0}(z)x+A_{1}(z)y + B_{1}(z)\alpha + C_{1}(z))\\ & \quad \quad \quad \quad \quad \quad - q\cdot (A_{2}(z)y + B_{2}(z)\beta + C_{2}(z)) - r \cdot(A_{3}(z)x + B_{3}(z)\gamma + C_{3}(z))\big\}\\ & = -(A_{0}(z)x + A_{1}(z)y + C_{1}(z))\cdot p + \sup\limits_{\alpha\in \Omega_A}\big\{ -p\cdot B_{1}(z)\alpha \big \} \quad \leftarrow \text{ contribution of macro-scale}\\ & \quad \quad \quad - (A_{2}(z)y + C_{2}(z))\cdot q + \sup\limits_{\beta\in \Omega_B}\big\{ -q\cdot B_{2}(z)\beta \big \}\quad \quad \quad \leftarrow \text{ contribution of meso-scale}\\ & \quad \quad \quad \quad \quad - (A_{3}(z)x + C_{3}(z))\cdot r + \sup\limits_{\gamma\in \Omega_\Gamma}\big\{ -r\cdot B_{3}(z)\gamma \big \} \quad \leftarrow \text{ contribution of micro-scale} \end{aligned}$$ To obtain the asymptotic expansion of the value function, we consider the asymptotic expansion **Ansatz:** $V^{\varepsilon}(t,z,y,x) = V_{0}(t,z) + \varepsilon V_{1}(y) + \varepsilon^{2} V_{2}(x) + O(\varepsilon^{3})$. This formally leads to the the HJB equations on the different scales: - *(The micro-scale)* we freeze $\bar{z}, \bar{y},\bar{p}:=\partial_{z}V_{0}(\bar{z}), \bar{q}:=\partial_{y}V_{1}(\bar{y})$ and solve the PDE whose unknown is the pair $(\lambda_{2}, V_{2}(\cdot))$ where $\lambda_{2}$ is a constant in $x$ (depending on $\bar{z}$ and on $\bar{p}$) and $V_{2}(\cdot)$ is a function of $x$ such that $$\label{micro} -(A_{3}(\bar{z})x+C_{3}(\bar{z}))\cdot \partial_{x}V_{2}(x) + \sup\limits_{\gamma\in \Omega_\Gamma}\{-\partial_{x}V_{2}(x)\cdot B_{3}(\bar{z})\gamma\} - A_{0}(\bar{z})x\cdot \bar{p} = \lambda_{2},\quad \forall\, x.$$ - *(The meso-scale)* we freeze $\bar{z},\bar{p}:=\partial_{z}V_{0}(\bar{z})$ and solve the PDE whose unknown is the pair $(\lambda_{1}, V_{1}(\cdot))$ where $\lambda_{1}$ is a constant in $y$ (depending on $\bar{z}$ and on $\bar{p}$) and $V_{1}(\cdot)$ is a function of $y$ such that $$\label{meso} - (A_{2}(\bar{z})y + C_{2}(\bar{z}))\cdot \partial_{y}V_{1}(y) + \sup\limits_{\beta\in \Omega_B}\big\{ -\partial_{y}V_{1}(y)\cdot B_{2}(\bar{z})\beta \big\} -A_{1}(\bar{z})y\cdot \bar{p}= \lambda_{1},\quad \forall\, y$$ analogue to [\[cell\]](#cell){reference-type="eqref" reference="cell"}. - *(The macro-scale)* we solve the PDE whose unknown is the function $V_{0}$ and is such that $$\label{eff HJB multi} \left\{ \; \begin{aligned} & \partial_{t}V_{0}(t,z) - C_{1}(z)\cdot \partial_{z}V_{0}(t,z) + \sup\limits_{\alpha\in \Omega_A}\big\{ -\partial_{z}V_{0}(t,z)\cdot B_{1}\alpha \big\}\\ & \quad \quad \quad \quad \quad \quad \quad + \lambda_{1}(z,\partial_{z}V_{0}(t,z)) + \lambda_{2}(z,\partial_{z}V_{0}(t,z)) = \ell(z),\quad \text{in } \mathds{R}^{m}\times (0,1)\\ & V_{0}(z,0) = \phi(z), \text{ in } \mathds{R}^{m} \end{aligned} \right.$$ This is the effective (limit) PDE problem obtained when $\varepsilon\to 0$ in [\[HJB multi\]](#HJB multi){reference-type="eqref" reference="HJB multi"} (compare with [\[HJB 0\]](#HJB 0){reference-type="eqref" reference="HJB 0"}). Summarizing, this requires $(\lambda_{2},V_{2})$, $(\lambda_{1},V_{1})$ and $V_{0}$ solving the latter three PDEs, and prove the convergence of $V^{\varepsilon}$ to $V_{0}(t,z)$ as $\varepsilon\to 0$ in order to justify the ansatz above. For a proof of convergence, we consider a cascaded approach as described in [@alvarez2008multiscale §4.1] (see also [@alvarez2007multiscale]). It goes as follows: 1. First, consider the dynamics of $x$ alone, being the *fast* one (while keeping frozen the dynamics of $z,y$). Then, we pass to the limit and recover a system made of the dynamics of $z$ and $y$ as in the above section §[2](#sec: linear){reference-type="ref" reference="sec: linear"}. 2. Then, we freeze $z$ and consider $y$ to be the *fast* dynamics. So we can pass to the limit again as in §[3.1](#sec: first order asymptot){reference-type="ref" reference="sec: first order asymptot"}. Noting that the dynamics of (the microscopic variable) $x$ satisfies the same assumptions as of the one of (the mesoscopic variable) $y$, the results of the previous section apply so we can pass to the limit in the step (i) and recover a mesoscopic limit problem whose two-scaled variables are now $(z,y)$. The latter falls again in the framework of the two-scaled optimal control problem discussed previously. Thus we can repeat the same procedure to finally get the limit (macroscopic, or effective) problem whose solution is $V_{0}$ and the convergence of $V^{\varepsilon}$ to $V_{0}(t,z)$ is a consequence of [@alvarez2008multiscale Theorem 4.1]. **Theorem 3**. *Under the general assumptions, the following statements hold.* 1. *Let $z,y,p,q$ be fixed. There exists a unique $\lambda_{2}$ for which there exists a continuous periodic viscosity solution $V_{2}(\cdot)$ to [\[micro\]](#micro){reference-type="eqref" reference="micro"}.* 2. *Let $z,p$ be fixed. There exists a unique $\lambda_{1}$ for which there exists a continuous periodic viscosity solution $V_{1}(\cdot)$ to [\[meso\]](#meso){reference-type="eqref" reference="meso"}.* 3. *As $\varepsilon\to 0$, the sequence $V^{\varepsilon}$ of solutions of [\[HJB multi\]](#HJB multi){reference-type="eqref" reference="HJB multi"} converges locally uniformly on $\mathds{R}^{m}\times (0, 1]$ to [\[eff HJB multi\]](#eff HJB multi){reference-type="eqref" reference="eff HJB multi"}.* *Proof.* The first and second statements are the same as Lemma [Lemma 1](#lem: corrector){reference-type="ref" reference="lem: corrector"}. The last statement is analog to Theorem [Theorem 2](#thm conv){reference-type="ref" reference="thm conv"} and is a consequence of [@alvarez2008multiscale Theorem 4.1]. The convergence results (of the dynamics and the value function) hold for all the initial positions of the dynamics $y,x$, and uniformly on compact sets for $z$. This is ensured by the stability assumption. ◻ # Applications and Examples {#Sec: Application} ## Jin-Xin Two Scale Relaxation We start by recalling the relaxation due to Jin and Xin [@MR1322811; @MR1793199; @MR1693210]. The original PDE problem being $$\label{original} \left\{\quad \begin{aligned} & \partial_{t}u + \partial_{x} (\mathcal{F}(u)) = 0\\ & u(x,0)=u_{0}(x). \end{aligned} \right.$$ A new variable $v$ is introduced and yields the following system of PDEs, for $a>0$, $$\label{relax JX} \left\{\quad \begin{aligned} & \partial_{t}u + \partial_{x} v = 0\\ & \partial_{t}v + a\,\partial_{x} u = -\frac{1}{\varepsilon}(v-\mathcal{F}(u))\\ & u(x,0)=u_{0}(x),\; v(x,0)=\mathcal{F}(u_{0}(x)). \end{aligned} \right.$$ This is motivated by the small relaxation limit ($\varepsilon\to 0$) for which one recovers a local equilibrium $v=\mathcal{F}(u)$ and hence the original PDE problem. In order to study this convergence, we rewrite this relaxation by introduce another variable $\omega$ subject to the PDE $$\left\{\quad \begin{aligned} & \partial_{t}\omega = -\frac{1}{\varepsilon}(v-\mathcal{F}(u))\\ & \omega(x,0)=\omega_{0}(x), \end{aligned} \right.$$ where $\omega_0$ will later be discussed. Then we define $\nu := v-\omega$ and write the system of PDEs satisfied by $(u,\nu,\omega)$ that is $$\label{eq: u nu oemga} \left\{\quad \begin{aligned} & \partial_{t}u = - \partial_{x} \nu - \partial_{x}\omega, \quad && u(x,0)=u_{0}(x) \\ & \partial_{t} \nu = - a\,\partial_{x} u , \quad && \nu(x,0)=\mathcal{F}(u_{0}(x)) - \omega_{0}(x)\\ & \partial_{t}\omega = -\frac{1}{\varepsilon}( \nu-\mathcal{F}(u) +\omega), \quad && \omega(x,0) = \omega_{0}(x). \end{aligned} \right.$$ The system satisfied by $(u,\nu,\omega)$ has the advantage of being *singularly perturbed* in time only. This structure is easier to study, at least in the finite-dimensional setting which we discuss in the sequel. Note moreover that the system [\[eq: u nu oemga\]](#eq: u nu oemga){reference-type="eqref" reference="eq: u nu oemga"} still enjoy hyperbolicity. Indeed, we have $$\label{eq operator A} \partial_{t} \begin{pmatrix} u\\ \nu \\ \omega \end{pmatrix} + \mathcal{A} \, \partial_{x} \begin{pmatrix} u\\ \nu \\ \omega \end{pmatrix} = S(u,\nu,\omega)$$ where $$\label{eq: A and S} \mathcal{A} = \begin{pmatrix} 0 & 1 & 1 \\ a & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} \quad \text{ and } \quad S(u,\nu,\omega) = \begin{pmatrix} 0 \\ 0 \\ -\frac{1}{\varepsilon}( \nu-\mathcal{F}(u) +\omega) \\ \end{pmatrix}$$ and $\mathcal{A}$ has the eigenvalues $\{0,\, \sqrt{a}, -\sqrt{a}\}$ corresponding to the eigenvectors $(0, \, -1, \, 1)^{\top}$, $(a^{-1/2}, \, 1, \, 0)^{\top}$ and $(- a^{-1/2}, \, 1, \, 0)^{\top}$ respectively. It can then be expressed such that $\mathcal{A} = T\Lambda T^{-1}$ where $$T = \begin{pmatrix} 0 & \sqrt{a} & -\sqrt{a}\\ -1 & 1 & 1 \\ 1 & 0 & 0 \end{pmatrix}, \quad \Lambda = \begin{pmatrix} 0 & 0 & 0 \\ 0 & \sqrt{a} & 0\\ 0 & 0 & -\sqrt{a} \end{pmatrix}, \quad T^{-1} = \begin{pmatrix} 0 & 0 & 1\\ \sqrt{a}/2 & 1/2 & 1/2\\ -\sqrt{a}/2 & 1/2 & 1/2 \end{pmatrix}.$$ We can then write [\[eq operator A\]](#eq operator A){reference-type="eqref" reference="eq operator A"} in the form $$\label{eq operator A 2} \partial_{t} T^{-1} \begin{pmatrix} u\\ \nu \\ \omega \end{pmatrix} + \Lambda \, \partial_{x} T^{-1} \begin{pmatrix} u\\ \nu \\ \omega \end{pmatrix} = T^{-1}S(u,\nu,\omega).$$ Setting $$\xi = \begin{pmatrix} \xi_{1}\\ \xi_{2} \\ \xi_{3} \end{pmatrix} := T^{-1} \begin{pmatrix} u\\ \nu \\ \omega \end{pmatrix} = \begin{pmatrix} \omega\\ \frac{\sqrt{a}}{2} u + \frac{1}{2}(\nu + \omega)\\ -\frac{\sqrt{a}}{2} + \frac{1}{2}(\nu + \omega) \end{pmatrix} \quad \text{ and } \quad \bar{S}(\xi) := T^{-1} S(u,\nu,\omega)$$ yields the following equivalent system to [\[eq operator A\]](#eq operator A){reference-type="eqref" reference="eq operator A"} $$\left\{\quad \begin{aligned} & \partial_{t} \xi_{1} = \bar{S}_{1}(\xi)\\ & \partial_{t} \xi_{2} + \sqrt{a}\, \partial_{x}\,\xi_{2} = \bar{S}_{2}(\xi)\\ & \partial_{t} \xi_{3} - \sqrt{a}\, \partial_{x}\,\xi_{3} = \bar{S}_{3}(\xi). \end{aligned} \right.$$ We now consider a semi-discretization in space of the PDE system [\[eq: u nu oemga\]](#eq: u nu oemga){reference-type="eqref" reference="eq: u nu oemga"} satisfied by $(u,\nu,\omega)$. We introduce $\mathbf{u}_{\varepsilon}(\cdot), \mathbf{v}_{\varepsilon}(\cdot), \mathbf{w}_{\varepsilon}(\cdot)\in \mathds{R}^{m}$ satisfying the system of ODEs, for $s\in [0,T]$ and $T<+\infty$ $$%\label{relax omega 2} \left\{\quad \begin{aligned} & \dot{\mathbf{u}}_{\varepsilon}(s) = -D\mathbf{v}_{\varepsilon}(s) -D\mathbf{w}_{\varepsilon}(s), \quad && \mathbf{u}_{\varepsilon}(0) = \mathbf{u}_0\in \mathds{R}^{m} \\ & \dot{\mathbf{v}}_{\varepsilon}(s) = -a\, D \mathbf{u}_{\varepsilon}(s), \quad && \mathbf{v}_{\varepsilon}(0)=\mathbf{v}_0\in \mathds{R}^{m} \\ & \dot{\mathbf{w}}_{\varepsilon}(s) = -\frac{1}{\varepsilon}\big[ \mathbf{v}_{\varepsilon}(s)-\mathcal{F}(\mathbf{u}_{\varepsilon}(s)) +\mathbf{w}_{\varepsilon}(s)\big], \quad && \mathbf{w}_{\varepsilon}(0) = \mathbf{w}_0\in \mathds{R}^{m}. \end{aligned} \right.$$ We denote by $D$, e.g. first-order finite-volume spatial discretization of the transport operators $\partial_x$. Its specific structure is for the following considerations not relevant. The $i$th component of vectors $\mathbf{u}_{\varepsilon}(\cdot), \mathbf{v}_{\varepsilon}(\cdot), \mathbf{w}_{\varepsilon}(\cdot)\in \mathds{R}^{m}$ are the values of the solution $(u,v,w)$ at the cell centers $x_i = i \Delta x$ for some spatial grid $\Delta x>0.$ For details on the discretization we refer to [@MR1322811]. The previous system falls within the general theory of *singular perturbation*, or singularly perturbed system of ODEs for which one distinguishes between the *slow* dynamics (here $\mathbf{u}_{\varepsilon}$ and $\mathbf{v}_{\varepsilon}$) and the *fast* dynamics (here $\mathbf{w}_{\varepsilon}$). Furthermore, we introduce control parameters $\alpha(s)\in \Omega_A$ and $\beta(s)\in \Omega_B$ with $\Omega_{A}\subset \mathds{R}^{p}, \Omega_{B} \subset \mathds{R}^{q}$ are compact and convex and $p,q>0$, as follows $$\label{ODE eps app} \left\{\;%\quad \begin{aligned} & \dot{\mathbf{u}}_{\varepsilon}(s) = -D\mathbf{v}_{\varepsilon}(s) -D\mathbf{w}_{\varepsilon}(s) + \mathds{H}(\mathbf{u}_{\varepsilon}(s))\alpha(s), && \mathbf{u}_{\varepsilon}(0) = \mathbf{u}_0 \\ & \dot{\mathbf{v}}_{\varepsilon}(s) = -a\, D \mathbf{u}_{\varepsilon}(s) , && \mathbf{v}_{\varepsilon}(0)=\mathbf{v}_0 \\ & \dot{\mathbf{w}}_{\varepsilon}(s) = -\frac{1}{\varepsilon}\big[ \mathbf{v}_{\varepsilon}(s)-\mathcal{F}(\mathbf{u}_{\varepsilon}(s)) - \mathds{G}(\mathbf{u}_{\varepsilon}(s))\beta(s) +\mathbf{w}_{\varepsilon}(s)\big], && \mathbf{w}_{\varepsilon}(0) = \mathbf{w}_0, \end{aligned} \right.$$ where $\mathbf{u}_0,\mathbf{v}_0,\mathbf{w}_0\in \mathds{R}^{m}$, $\mathds{H}$ and $\mathds{G}$ are matrices functions which multiply the control parameters $\alpha,\beta$. *Remark 3*. In the notation of [\[eq operator A\]](#eq operator A){reference-type="eqref" reference="eq operator A"}, this corresponds to adding $\mathds{H}(u)\alpha$ and $\frac{1}{\varepsilon}\mathds{G}(u)\beta$ to the first and third entries of $S(u,\nu,\omega)$ in [\[eq: A and S\]](#eq: A and S){reference-type="eqref" reference="eq: A and S"} respectively. These control parameters will be subject to minimizing a given cost function $$\label{eq: value eps JX} \tag{SP.1} \inf \; \Phi(\mathbf{u}_{\varepsilon}(T)),\quad \text{s.t. } \; \eqref{ODE eps app}$$ where we assume the cost $\Phi$ to be continuous. The limit $\varepsilon\to 0$ yields the reduced control problem $$\label{eq: value eff JX} \tag{R.1} \inf \; \Phi(\mathbf{u}(T)),\quad \text{s.t. } \; \eqref{ODE eff app}$$ where the reduced dynamics is $$\label{ODE eff app} \left\{\; \begin{aligned} & \dot{\mathbf{u}}(s) + D\big[\mathcal{F}(\mathbf{u}(s))+\mathds{G}(\mathbf{u}(s))\beta(s)\big] = \mathds{H}(\mathbf{u}(s))\alpha(s),\quad s\in (0,T]\\ & \mathbf{u}(0) = u_{0}\in\mathds{R}^{m} \end{aligned} \right.$$ together with $$\label{eq: v eff 1} \dot{\mathbf{v}}_{\varepsilon}(s) = -a\, D \mathbf{u}_{\varepsilon}(s) , \quad \quad \mathbf{v}_{\varepsilon}(0)=\mathbf{v}_0.$$ But since in the control problem, only the function $\mathbf{u}$ is involved, then we omit the dynamics for $\mathbf{v}$.\ The ODE [\[ODE eff app\]](#ODE eff app){reference-type="eqref" reference="ODE eff app"} is obtained by first solving for $\mathbf{w}_{\varepsilon}$ the algebraic equation $$0= \mathbf{v}_{\varepsilon}(s)-\mathcal{F}(\mathbf{u}_{\varepsilon}(s)) - \mathds{G}(\mathbf{u}_{\varepsilon}(s))\beta(s) +\mathbf{w}_{\varepsilon}(s),$$ then substituting the value of $\mathbf{w}_{\varepsilon}$ in the dynamics for $\mathbf{u}_{\varepsilon}$. Note that [\[ODE eff app\]](#ODE eff app){reference-type="eqref" reference="ODE eff app"} is the discretization of $$\label{eq: original pde} \left\{\quad \begin{aligned} & \partial_{t}u(x,t) + \partial_{x}[\mathcal{F}(u(x,t)) + \mathds{G}(u(x,t))\beta(x,t)] = \mathds{H}(u(x,t))\alpha(x,t)\\ & u(x,0)=u_{0}(x), \end{aligned} \right.$$ which is the controlled version of the original PDE problem [\[original\]](#original){reference-type="eqref" reference="original"} with the additional control terms arising from $\mathds{G}$ and $\mathds{H}$. The limit as $\varepsilon\to 0$ is summarized in the following result whose proof is a direct application of Theorem [Theorem 1](#thm: conv G){reference-type="ref" reference="thm: conv G"}. **Corollary 1**. *As $\varepsilon\to 0$, the problem [\[eq: value eps JX\]](#eq: value eps JX){reference-type="eqref" reference="eq: value eps JX"} is approximated by [\[eq: value eff JX\]](#eq: value eff JX){reference-type="eqref" reference="eq: value eff JX"} in the sense of Definition [Definition 2](#def:approx){reference-type="ref" reference="def:approx"}. In particular, the controlled system [\[ODE eps app\]](#ODE eps app){reference-type="eqref" reference="ODE eps app"} with any given initial conditions converges to [\[ODE eff app\]](#ODE eff app){reference-type="eqref" reference="ODE eff app"} locally uniformly on any finite time interval, and we have $\inf\, \Phi(\mathbf{u}_{\varepsilon}(T)) \to \inf \, \Phi(\mathbf{u}(T))$.* Moreover, the convergence in the above corollary extends to the HJB equations whose solutions are the value functions of [\[eq: value eps JX\]](#eq: value eps JX){reference-type="eqref" reference="eq: value eps JX"} and [\[eq: value eff JX\]](#eq: value eff JX){reference-type="eqref" reference="eq: value eff JX"} respectively, in the sense of Theorem [Theorem 2](#thm conv){reference-type="ref" reference="thm conv"}. ## Jin-Xin Three--Scale Relaxation {#sec: JX 3 app} In this section, we would like to go one step further in the analysis by considering a multi-scale approximation of [\[original\]](#original){reference-type="eqref" reference="original"} and [\[relax JX\]](#relax JX){reference-type="eqref" reference="relax JX"}. Assuming a decomposition $$\label{eq: F decomp} \mathcal{F}(u) \approx F_{0}(u) + \varepsilon\mathcal{F}_{1}(u),$$ the following relaxation system for $a,b>0$ is obtained $$\label{relax JX 2} \left\{\quad \begin{aligned} & \partial_{t}u + \partial_{x} v = 0\\ & \partial_{t}v + a\,\partial_{x} u = -\frac{1}{\varepsilon}\big[ v-(\mathcal{F}_{0}(u) + \varepsilon \omega)\big]\\ & \partial_{t}\omega + b\,\partial_{x} u = -\frac{1}{\varepsilon^{2}}\big[\omega-\mathcal{F}_{1}(u)\big]\\ & u(x,0)=u_{0}(x),\; v(x,0)=\mathcal{F}_{0}(u_{0}(x)), \; \omega(x,0) = \mathcal{F}_{1}(u_{0}(x)). \end{aligned} \right.$$ This system exhibits three scales: the first PDE is in the *macro-scale*, the second one is in the *meso-scale*, the third one is in the *micro-scale*. In the latter PDE, the local equilibrium is reached for $\omega = \mathcal{F}_{1}(u)$ and yields the term $F_{0}(u) + \varepsilon\mathcal{F}_{1}(u)$ in the second PDE. Our goal now is to capture the effect of $\mathcal{F}_{0}$ and $\mathcal{F}_{1}$ from [\[eq: F decomp\]](#eq: F decomp){reference-type="eqref" reference="eq: F decomp"} on the value function corresponding to the control problem of the finite-dimensional (discrete) version of [\[relax JX 2\]](#relax JX 2){reference-type="eqref" reference="relax JX 2"}. Following the same idea as in the previous subsection, we introduce new variables $p(x,t),q(x,t)$ whose time derivatives correspond to the r.h.s. of the second and third equations. So starting from $$\partial_{t}v + a\,\partial_{x} u = -\frac{1}{\varepsilon}\big[ v-(\mathcal{F}_{0}(u) + \varepsilon \omega)\big] = -\frac{1}{\varepsilon}\big[ v -\mathcal{F}_{0}(u) \big] + \omega$$ we set $$\partial_{t} p = -\frac{1}{\varepsilon}\big[ v -\mathcal{F}_{0}(u) \big]$$ then $$\partial_{t}v + a\,\partial_{x} u = \partial_{t} p + \omega$$ which is $$\partial_{t}[v-p] = - a\,\partial_{x} u + \omega.$$ Similarly, we start by setting $$\partial_{t} q = -\frac{1}{\varepsilon^{2}}\big[\omega-\mathcal{F}_{1}(u)\big],$$ then we get $$\partial_{t}[\omega-q] =- b\,\partial_{x} u.$$ Ultimately, using $\nu := v-p$ and $\text{w} := \omega-q$ yields the following three--scale system $$\label{eq: multi scale} \left\{\quad \begin{aligned} & \partial_{t}u = - \partial_{x} \nu - \partial_{x}p, \quad && u(x,0)=u_{0}(x) \\ & \partial_{t} \nu = - a\,\partial_{x} u + \text{w} + q, \quad && \nu(x,0)= \mathcal{F}_{0}(u_{0}(x)) - p_{0}(x)\\ & \partial_{t}\text{w} = -b\, \partial_{x} u, \quad && \text{w}(x,0) = \mathcal{F}_{1}(u_{0}(x)) - q_{0}(x)\\ & \partial_{t}p = -\frac{1}{\varepsilon}(\nu - \mathcal{F}_{0}(u) + p), \quad && p(x,0) = p_{0}(x)\\ & \partial_{t}q = -\frac{1}{\varepsilon^{2}}(\text{w} - \mathcal{F}_{1}(u) + q), \quad && q(x,0) = q_{0}(x). \end{aligned} \right.$$ This system enjoys hyperbolicity as it can be expressed such that $$\label{eq operator A tilde} \partial_{t} \begin{pmatrix} u \\\nu \\ \text{w} \\ p \\ q \end{pmatrix} + \widetilde{\mathcal{A}}\; \partial_{x} \begin{pmatrix} u \\ \nu \\ \text{w} \\ p \\ q \end{pmatrix} = \widetilde{S}(u,\nu,\text{w}, p, q)$$ where $$\widetilde{\mathcal{A}} = \begin{pmatrix} 0 & 1 & 0 & 1 & 0 \\ a & 0 & 0 & 0 & 0\\ b & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{pmatrix} \quad \text{ and } \quad \widetilde{S}(u,\nu,\text{w},p , q) = \begin{pmatrix} 0 \\ \text{w} + q \\ 0\\ -\frac{1}{\varepsilon}(\nu - \mathcal{F}_{0}(u) + p) \\ -\frac{1}{\varepsilon^{2}}(\omega - \mathcal{F}_{1}(u) + q) \end{pmatrix}$$ and $\widetilde{A}$ has the eigenvalues $\{0, 0, 0, \sqrt{a}, -\sqrt{a}\}$ corresponding to the eigenvectors $(0,0,1,0,0)^{\top}$, $(0,-1,0, 1,0)^{\top}$, $(0,0,0,0,1)^{\top}$, $(\sqrt{a}/b, a/b,1,0,0)^{\top}$ and $(-\sqrt{a}/b, a/b,1,0,0)^{\top}$ respectively. It can then be expressed such that $\widetilde{A} = \widetilde{T} \widetilde{\Lambda} \widetilde{T}^{-1}$ where we skip the details of $\tilde{T}$. The matrix $\widetilde{\Lambda}$ is diagonal with three diagonal entries being zero, $\pm \sqrt{a}$. Then, we write [\[eq operator A tilde\]](#eq operator A tilde){reference-type="eqref" reference="eq operator A tilde"} in the form (compare with [\[eq operator A 2\]](#eq operator A 2){reference-type="eqref" reference="eq operator A 2"}) $$\label{eq operator A tilde 2} \partial_{t} \widetilde{T}^{-1} \begin{pmatrix} u\\ \nu \\ \text{w} \\ p \\ q \end{pmatrix} + \widetilde{\Lambda} \, \partial_{x} \widetilde{T}^{-1} \begin{pmatrix} u \\ \nu \\ \text{w} \\ p \\ q \end{pmatrix} = \widetilde{T}^{-1}\widetilde{S}(u,\nu,\text{w},p,q).$$ Setting $$\widetilde{\xi} = \begin{pmatrix} \tilde{\xi}_{1} \\ \tilde{\xi}_{2} \\ \tilde{\xi}_{3} \\ \tilde{\xi}_{4} \\ \tilde{\xi}_{5} \end{pmatrix} := \widetilde{T}^{-1} \begin{pmatrix} u \\ \nu \\ \text{w} \\ p \\ q \end{pmatrix} = \begin{pmatrix} \frac{-b}{a}(\nu +p) + \text{w} \\ p \\ q \\ \frac{b}{2 \sqrt{a}}u + \frac{b}{2a} (\nu + p)\\ \frac{-b}{2 \sqrt{a}}u + \frac{b}{2 a}(\nu + p) \end{pmatrix}$$ yileds the following equivalent system to [\[eq operator A tilde\]](#eq operator A tilde){reference-type="eqref" reference="eq operator A tilde"} $$\partial_{t} \widetilde{\xi} + \widetilde{\Lambda} \, \partial_{x}\widetilde{\xi} = \widetilde{T}^{-1} \widetilde{S}(u,\nu,\text{w},p,q).$$ The finite-dimensional version of [\[eq: multi scale\]](#eq: multi scale){reference-type="eqref" reference="eq: multi scale"} is the following system of ODEs to which we have added three control parameters, $\alpha(s)\in \Omega_A$, $\beta(s)\in \Omega_B$ and $\gamma(s)\in \Omega_\Gamma$, where $\Omega_{A},\Omega_{B}, \Omega_{\Gamma}$ are three compact and convex sets $$\label{relax 2 eps} \left\{ \begin{aligned} & \dot{\mathbf{u}}_{\varepsilon}(s) = -D\mathbf{v}_{\varepsilon}(s) -D\mathbf{p}_{\varepsilon}(s) + \mathds{H}(\mathbf{u}_{\varepsilon}(s))\alpha(s), \; && \mathbf{u}_{\varepsilon}(0) = \mathbf{u}_0\in \mathds{R}^{m} \\ & \dot{\mathbf{v}}_{\varepsilon}(s) = -a\, D \mathbf{u}_{\varepsilon}(s) + \mathbf{w}_{\varepsilon} + \mathbf{q}_{\varepsilon}, \; && \mathbf{v}_{\varepsilon}(0)=\mathbf{v}_0\in \mathds{R}^{m} \\ & \dot{\mathbf{w}}_{\varepsilon}(s) = - b \, D \mathbf{u}_{\varepsilon}(s), \; && \mathbf{w}_{\varepsilon}(0)=\mathbf{w}_0\in \mathds{R}^{m} \\ & \dot{\mathbf{p}}_{\varepsilon}(s) = -\frac{1}{\varepsilon}\big[ \mathbf{v}_{\varepsilon}(s)-\mathcal{F}_{0}(\mathbf{u}_{\varepsilon}(s)) - \mathds{G}(\mathbf{u}_{\varepsilon}(s))\beta(s) +\mathbf{p}_{\varepsilon}(s)\big], \; && \mathbf{p}_{\varepsilon}(0) = \mathbf{p}_0\in \mathds{R}^{m}\\ & \dot{\mathbf{q}}_{\varepsilon}(s) = -\frac{1}{\varepsilon^{2}}\big[ \mathbf{w}_{\varepsilon}(s)-\mathcal{F}_{1}(\mathbf{u}_{\varepsilon}(s)) - \mathds{K}(\mathbf{u}_{\varepsilon}(s))\gamma(s)+\mathbf{q}_{\varepsilon}(s)\big], \; && \mathbf{q}_{\varepsilon}(0) = \mathbf{q}_0\in \mathds{R}^{m}. \end{aligned} \right.$$ The limiting (effective) system, as $\varepsilon\to 0$, is again given by [\[ODE eff app\]](#ODE eff app){reference-type="eqref" reference="ODE eff app"}. The system [\[relax 2 eps\]](#relax 2 eps){reference-type="eqref" reference="relax 2 eps"} falls within the framework of section [4](#sec: multiscale sys){reference-type="ref" reference="sec: multiscale sys"} since it can be written in the form [\[CSP 2\]](#CSP 2){reference-type="eqref" reference="CSP 2"} as follows with the dimensions $m=3d$, $n=d$, $\ell=d$, where $d$ is the space dimension (discretization). The variables are $z=[\textbf{u}_{\varepsilon}^{\top},\textbf{v}_{\varepsilon}^{\top},\textbf{w}_{\varepsilon}^{\top}]^{\top}$, $y=\textbf{p}_{\varepsilon}$, $x=\textbf{q}_{\varepsilon}$. Let $0_{d}, 1_{d}$ be the $d$-dimensional column vectors whose entries are all $0$ and $1$ respectively. Let also $a,b>0$ be constant parameters and $D$ the discretization matrix of transport operator as before. Then, we set $$\begin{aligned} & A_{0} = \begin{pmatrix} 0_{d} \\ 1_{d} \\ 0_{d} \end{pmatrix}, \; A_{1} = \begin{pmatrix} -D \\ 0_{d} \\ 0_{d} \end{pmatrix}, \; B_{1} = \begin{pmatrix} \mathds{H}(\mathbf{u}_{\varepsilon})\\ 0_{d} \\ 0_{d} \end{pmatrix},\; C_{1} = \begin{pmatrix} 0_{d} & -D & 0_{d} \\ -a\,D & 0_{d} & 1_{d} \\ -b\,D & 0_{d} & 0_{d} \end{pmatrix}\!\! \begin{pmatrix} \mathbf{u}_{\varepsilon} \\ \mathbf{v}_{\varepsilon} \\ \mathbf{w}_{\varepsilon} \end{pmatrix},\\ & A_{2} = -1_{d}, \quad B_{2} = \mathds{G}(\mathbf{u}_{\varepsilon}),\quad C_{2} = \mathcal{F}_{0}(\mathbf{u}_{\varepsilon}) + \begin{pmatrix} 0_{d}^{\top} & -1_{d}^{\top} & 0_{d}^{\top} \end{pmatrix}\!\! \begin{pmatrix} \mathbf{u}_{\varepsilon} \\ \mathbf{v}_{\varepsilon} \\ \mathbf{w}_{\varepsilon} \end{pmatrix},\\ & A_{3} = -1_{d}, \quad B_{3} = \mathds{K}(\mathbf{u}_{\varepsilon}), \quad C_{3} = \mathcal{F}_{1}(\mathbf{u}_{\varepsilon}) + \begin{pmatrix} 0_{d}^{\top} & 0_{d}^{\top} & -1_{d}^{\top} \end{pmatrix}\!\! \begin{pmatrix} \mathbf{u}_{\varepsilon} \\ \mathbf{v}_{\varepsilon} \\ \mathbf{w}_{\varepsilon} \end{pmatrix}. \end{aligned}$$ Note that the multiscale approach yields at the limit $\varepsilon\to 0$ the same effective ODE system [\[ODE eff app\]](#ODE eff app){reference-type="eqref" reference="ODE eff app"}, the difference being in the dynamics of $\mathbf{v}$ (compare with [\[eq: v eff 1\]](#eq: v eff 1){reference-type="eqref" reference="eq: v eff 1"}) $$\label{eq: v eff 2} \dot{\mathbf{v}}(s) = - a\, D\mathbf{u}(s) + \mathcal{F}_{1}(\mathbf{u}(s)) + \mathds{K}(\mathbf{u}(s))\gamma(s), \quad \quad \mathbf{v}(0) = \mathbf{v}_{0}.$$ and the new variable $\mathbf{w}$ governed by $$\label{eq: w eff 2} \dot{\mathbf{w}}(s) = - b \, D \mathbf{u}(s),\quad \quad \mathbf{w}(0) = \mathbf{w}_{0}.$$ However, the benefit of using three (or more) scales appears in the asymptotic expansion of the value function which keeps track of the higher-order terms in [\[eq: F decomp\]](#eq: F decomp){reference-type="eqref" reference="eq: F decomp"}. Thus, we make a corollary of Theorem [Theorem 3](#thm: conv multi){reference-type="ref" reference="thm: conv multi"} on the convergence at the level of the HJB equations. Let $\ell, \phi$ be as in the standing assumptions of §[3](#sec: hjb){reference-type="ref" reference="sec: hjb"}. Consider the optimal control problem $$\label{SP JX multi} \tag{SP.2} \mathbf{V}^{\varepsilon}(\mathbf{u}_{0}, \mathbf{v}_{0}, \mathbf{w}_{0}, \mathbf{p}_{0}, \mathbf{q}_{0}) = \inf \; \int_{0}^{1} \ell(\mathbf{u}_{\varepsilon}(s))\,\text{d}s + \phi(z(1)), \; \text{ s.t.: } \; \eqref{relax 2 eps} \text{ with } s\in [0,1].$$ And consider the corresponding reduced problem $$\label{SP JX multi R} \tag{R.2} \mathbf{V}_{0}(\mathbf{u}_{0}) = \inf \; \int_{0}^{1} \ell(\mathbf{u}(s))\,\text{d}s + \phi(z(1)), \; \text{ s.t.: } \; \eqref{ODE eff app} \text{ with } s\in [0,1].$$ *Remark 4*. Observe that $\ell,\phi$ being dependent on $\mathbf{u}$ is a choice we made for simplicity only. Indeed the previous theoretical results apply for control problems where the cost functional depends on the slow variables (therein denoted by $z(\cdot)$) and which are in [\[relax 2 eps\]](#relax 2 eps){reference-type="eqref" reference="relax 2 eps"} the variables $\mathbf{u}_{\varepsilon}, \mathbf{v}_{\varepsilon},\mathbf{w}_{\varepsilon}$. Hence, we could also consider $\ell,\phi$ as functions of $\mathbf{u},\mathbf{v},\mathbf{w}$ in which case, the dynamics [\[ODE eff app\]](#ODE eff app){reference-type="eqref" reference="ODE eff app"} would be complemented with [\[eq: v eff 2\]](#eq: v eff 2){reference-type="eqref" reference="eq: v eff 2"} and [\[eq: w eff 2\]](#eq: w eff 2){reference-type="eqref" reference="eq: w eff 2"} and the value function would be $\mathbf{V}_{0}(\mathbf{u}_{0}, \mathbf{v}_{0}, \mathbf{w}_{0})$. A consequence of Theorem [Theorem 3](#thm: conv multi){reference-type="ref" reference="thm: conv multi"} is the following. **Corollary 2**. *The value function $\mathbf{V}^{\varepsilon}$ of [\[SP JX multi\]](#SP JX multi){reference-type="eqref" reference="SP JX multi"} converges locally uniformly on $\big(\mathds{R}^{m}\big)^{5}\times (0,1]$ to the value function $\mathbf{V}_{0}$ of [\[SP JX multi R\]](#SP JX multi R){reference-type="eqref" reference="SP JX multi R"}.* In other words, the above corollary states that relying on the control of the singularly perturbed dynamics to approximate the control of [\[ODE eff app\]](#ODE eff app){reference-type="eqref" reference="ODE eff app"} is sufficient. This control is a discretization of [\[eq: original pde\]](#eq: original pde){reference-type="eqref" reference="eq: original pde"}. Moreover $\mathbf{V}^{\varepsilon}$ admits the asymptotic expansion described in §[4.1](#sec: linear 2){reference-type="ref" reference="sec: linear 2"} where the dominant term is $\mathbf{V}_{0}$. ## Goldstein-Taylor Two--Scale Model {#sec: GT 2} We consider the Goldstein--Taylor model in the formulation of [@albi2014asymptotic]. The latter describes the time evolution of two-particle densities $f^{+}(x,t)$ and $f^{-}(x,t)$, with $x\in \Omega\subset \mathds{R}$ and $t\in \mathds{R}^{+}$, where $f^{+}(x,t)\,$ (respectively $f^{-}(x,t)$) denotes the density of particles at time $t>0$ traveling along a straight line with velocity $+c$ (respectively $-c$). The particle changes with rate $\sigma$ the direction. The differential model can be written as $$\begin{aligned} f^{+}_{t} + c f^{+}_{x} & = \sigma (f^{-} - f^{+}) \\ f^{-}_{t} - c f^{-}_{x} & = \sigma (f^{+} - f^{-}). \end{aligned}$$ Introducing the macroscopic variables $$\rho = f^{+} + f^{-},\quad \quad j = c (f^{+} - f^{-})$$ we obtain the equivalent form $$\left\{\; \begin{aligned} & \partial_{t}\rho + \partial_{x} j = 0,\\ & \partial_{t} j + c^{2}\partial_{x}\rho = 2\sigma j. \end{aligned} \right.$$ Setting $c^{2} = 2\sigma = 1/\varepsilon$ where $\varepsilon>0$ is the relaxation parameter, yields $$\label{eq: pde GT} \left\{\; \begin{aligned} & \partial_{t} \rho = - \partial_{x} j \\ & \partial_{t}j = \frac{1}{\varepsilon}\left( - j -\partial_{x}\rho \right) \end{aligned} \right.$$ which has the desired singularly perturbed structure. When $\varepsilon\to 0$, we obtain $$\label{eq: pde GT 2} j(x,t) = - \partial_{x}\rho(x,t).$$ The previous results can be applied after using a semi--discretization in space. This leads to a similar structure as in the previous section. Given two controls $\alpha,\beta$ (not necessarily different) and two matrices $\mathds{H},\mathds{G}$ depending on $\bm{\rho}$, the singularly perturbed discrete dynamics is $$\label{eq: relax rho j} \left\{\; \begin{aligned} & \dot{\bm{\rho}}_{\varepsilon}(s) = -D\bm{J}_{\varepsilon}(s) + \mathds{H}(\bm{\rho}_{\varepsilon}(s))\alpha(s), && \bm{\rho}_{\varepsilon}(0) = \bm{\rho}_{0} \in \mathds{R}^{m}\\ &\dot{\bm{J}}_{\varepsilon}(s) = - \frac{1}{\varepsilon}[D \bm{\rho}_{\varepsilon}(s) + \bm{J}_{\varepsilon}(s) - \mathds{G}(\bm{\rho}_{\varepsilon}(s))\beta(s)], && \bm{J}_{\varepsilon}(0) = \bm{J}_{0}\in \mathds{R}^{m} \end{aligned} \right.$$ and the corresponding reduced dynamics is $$\label{eq: rho j} \dot{\bm{\rho}}(s) + D [-D\bm{\rho}(s) + \mathds{G}(\bm{\rho}_{\varepsilon}(s))\beta(s)] = \mathds{H}(\bm{\rho}_{\varepsilon}(s))\alpha(s) , \quad \bm{\rho}(0) = \bm{\rho}_{0} \in \mathds{R}^{m}.$$ The latter dynamics corresponds to the discretization of the controlled PDE $$\left\{\; \begin{aligned} & \partial_{t}\rho(t,x) + \partial_{x}[-\partial_{x}\rho(t,x) + \mathds{G}(\rho(t,x))\beta(t,x) ]= \mathds{H}(\rho(t,x))\alpha(t,x),\\ & \rho(0,x) = \rho_{0}(x), \quad (t,x)\in (0,+\infty)\times \mathds{R}. \end{aligned} \right.$$ In the absence of controls, this becomes the heat equation $$\label{eq: heat} \partial_{t}\rho(t,x) = \partial_{xx}\rho(t,x), \quad \rho(0,x) = \rho_{0}(x),\quad (t,x)\in (0,+\infty)\times \mathds{R}$$ which is also what one gets when substituting [\[eq: pde GT 2\]](#eq: pde GT 2){reference-type="eqref" reference="eq: pde GT 2"} in the first PDE of [\[eq: pde GT\]](#eq: pde GT){reference-type="eqref" reference="eq: pde GT"}. Let us introduce the two optimal control problems $$\label{eq: value eps GT} \tag{SP.3} \inf \; \Phi(\bm{\rho}_{\varepsilon}(T)),\quad \text{s.t. } \; \eqref{eq: relax rho j}$$ and $$\label{eq: value eff GT} \tag{R.3} \inf \; \Phi(\bm{\rho}_{\varepsilon}(T)),\quad \text{s.t. } \; \eqref{eq: rho j}$$ The following corollary is a direct consequence of Theorem [Theorem 1](#thm: conv G){reference-type="ref" reference="thm: conv G"}. **Corollary 3**. *As $\varepsilon\to 0$, the problem [\[eq: value eps GT\]](#eq: value eps GT){reference-type="eqref" reference="eq: value eps GT"} is approximated by [\[eq: value eff GT\]](#eq: value eff GT){reference-type="eqref" reference="eq: value eff GT"} in the sense of Definition [Definition 2](#def:approx){reference-type="ref" reference="def:approx"}. In particular, the controlled system [\[eq: relax rho j\]](#eq: relax rho j){reference-type="eqref" reference="eq: relax rho j"} with any given initial conditions converges to [\[eq: rho j\]](#eq: rho j){reference-type="eqref" reference="eq: rho j"} locally uniformly on any finite time interval, and we have $\inf\, \Phi(\bm{\rho}_{\varepsilon}(T)) \to \inf \, \Phi(\bm{\rho}(T))$.* ## Goldstein-Taylor Three--Scale Model {#sec: GT 3} Recalling the model in the previous section §[5.3](#sec: GT 2){reference-type="ref" reference="sec: GT 2"}, we would like now to get a three--scale approximation. To do so, we suppose we have an additional term $\mathcal{F}_{1}(\rho)$ of order $\varepsilon$ in [\[eq: heat\]](#eq: heat){reference-type="eqref" reference="eq: heat"} $$%\label{eq: heat eps} \partial_{t}\rho(t,x) + \partial_{x}\big[-\partial_{x}\rho(t,x) + \varepsilon \, \mathcal{F}_{1}(\rho(t,x))\big] = 0.$$ The three--scale relaxation becomes, for some $a,b>0$ fixed, $$\left\{\; \begin{aligned} & \partial_{t}\rho = -\partial_{x}j\\ & \partial_{t} j + a\, \partial_{x} \rho = - \frac{1}{\varepsilon}\big[ j - ( -\partial_{x}\rho +\varepsilon w) \big]\\ & \partial_{t} w + b\, \partial_{x}\rho = -\frac{1}{\varepsilon^{2}}\big[ w - \mathcal{F}_{1}(\rho) \big]. \end{aligned} \right.$$ Repeating what we have done in the beginning of §[5.2](#sec: JX 3 app){reference-type="ref" reference="sec: JX 3 app"}, we get $$\label{with ab} \left\{\; \begin{aligned} & \partial_{t} \rho &=& -\partial_{x}[j-p] - \partial_{x}p\\ & \partial_{t}[j-p] &=& \; - a \, \partial_{x}\rho + [w-q] + q\\ & \partial_{t} [w-q] &=&\; - b\, \partial_{x}\rho \\ & \partial_{t} p &=& -\frac{1}{\varepsilon}\big( [j-p] +\partial_{x}\rho + p\big)\\ & \partial_{t} q &=& - \frac{1}{\varepsilon^{2}}\big([w-q] - \mathcal{F}_{1}(\rho) + q\big). \end{aligned} \right.$$ Note if we choose $a=b=0$, the latter system simplifies as follows $$\label{without ab} \left\{\; \begin{aligned} & \partial_{t} \rho &=& -\partial_{x}[j-p] - \partial_{x}p\\ & \partial_{t}[j-p] &=& \; w\\ & \partial_{t} p &=& -\frac{1}{\varepsilon}\big( [j-p] +\partial_{x}\rho + p\big)\\ & \partial_{t} w &=& - \frac{1}{\varepsilon^{2}}\big(w - \mathcal{F}_{1}(\rho) \big). \end{aligned} \right.$$ For the sake of generality, we let $a,b\geq 0$. In particular, they are allowed to be null, contrary to Jin-Xin relaxation in the previous sections. Let us consider the semi-discretization in space of the latter PDE system by introducing $(\bm{\rho}_{\varepsilon}, \bm{J}_{\varepsilon}, \bm{w}_{\varepsilon}, \bm{p}_{\varepsilon}, \bm{q}_{\varepsilon})$ the discretization of $(\rho, [j-p], [w-q], p, q)$ respectively, and letting $D$ be for example a first order finite-volume spatial discretization of the transport operator $\partial_{x}$. We shall also introduce three control parameters (not necessarily different) $\alpha,\beta,\gamma$ and three matrices $\mathds{H},\mathds{G},\mathds{K}$ functions of $\bm{\rho}_{\varepsilon}$. Then one gets the system of controlled and singularly perturbed ODEs $$\label{relax 3 eps} \left\{\; \begin{aligned} \dot{\bm{\rho}}_{\varepsilon}(s) & = -D \bm{J}_{\varepsilon}(s) - D\bm{p}_{\varepsilon}(s) + \mathds{H}(\bm{\rho}_{\varepsilon}(s))\alpha(s), && \bm{\rho}_{\varepsilon}(0) = \bm{\rho}_{0}\in \mathds{R}^{m}\\ \dot{\bm{J}}_{\varepsilon}(s) & =\; -a \, D\bm{\rho}_{\varepsilon}(s) + \bm{w}_{\varepsilon}(s) + \bm{q}_{\varepsilon}(s), && \bm{J}_{\varepsilon}(0) = \bm{J}_{0}\in \mathds{R}^{m}\\ \dot{\bm{w}}_{\varepsilon}(s) & =\; -b\, D \bm{\rho}_{\varepsilon}(s), && \bm{w}_{\varepsilon}(0) = \bm{w}_{0}\in \mathds{R}^{m}\\ \dot{\bm{p}}_{\varepsilon}(s) & = -\frac{1}{\varepsilon}\big[\bm{p}_{\varepsilon}(s) + \bm{J}_{\varepsilon}(s) + D \bm{\rho}_{\varepsilon}(s) - \mathds{G}(\bm{\rho}_{\varepsilon}(s))\beta(s)\big], && \bm{p}_{\varepsilon}(0) = \bm{p}_{0}\in \mathds{R}^{m}\\ \dot{\bm{q}}_{\varepsilon}(s) & = -\frac{1}{\varepsilon^{2}} \big[\bm{q}_{\varepsilon}(s) + \bm{w}_{\varepsilon}(s) - \mathcal{F}_{1}(\bm{\rho}_{\varepsilon}(s)) - \mathds{K}(\bm{\rho}_{\varepsilon}(s))\gamma(s)\big], &&\bm{q}_{\varepsilon}(0)= \bm{q}_{0} \in \mathds{R}^{m}. \end{aligned} \right.$$ For simplicity, we shall consider a control problem whose running cost and final costs are functions of $\bm{\rho}_{\varepsilon}$ only, the general case being discussed in Remark [Remark 4](#rem: cost general){reference-type="ref" reference="rem: cost general"}. $$\label{SP GT multi} \tag{SP.4} \mathbf{V}^{\varepsilon}(\bm{\rho}_{0}, \bm{J}_{0}, \bm{w}_{0}, \bm{p}_{0}, \bm{q}_{0}) = \inf \; \int_{0}^{1} \ell(\bm{\rho}_{\varepsilon}(s))\,\text{d}s + \phi(z(1)), \; \text{ s.t.: } \; \eqref{relax 3 eps} \text{ with } s\in [0,1].$$ And consider the corresponding reduced problem $$\label{SP GT multi R} \tag{R.4} \mathbf{V}_{0}(\bm{\rho}_{0}) = \inf \; \int_{0}^{1} \ell(\bm{\rho}(s))\,\text{d}s + \phi(z(1)), \; \text{ s.t.: } \; \eqref{eq: rho j} \text{ with } s\in [0,1].$$ A consequence of Theorem [Theorem 3](#thm: conv multi){reference-type="ref" reference="thm: conv multi"} is the following. **Corollary 4**. *The value function $\mathbf{V}^{\varepsilon}$ of [\[SP GT multi\]](#SP GT multi){reference-type="eqref" reference="SP GT multi"} converges locally uniformly on $\big(\mathds{R}^{m}\big)^{5}\times (0,1]$ to the value function $\mathbf{V}_{0}$ of [\[SP GT multi R\]](#SP GT multi R){reference-type="eqref" reference="SP GT multi R"}.* ## Shallow Water and Inviscid Burger's Equation Let us recall the system as described in [@pareschi2005implicit §6.1] (see also [@jin1995runge]) for shallow water flow $$\left\{\; \begin{aligned} & \partial_{t} h + \partial_{x}(hv) = 0\\ & \partial_{t}(hv) + \partial_{x}\left(h + \frac{1}{2}h^{2}\right) = \frac{1}{\varepsilon}h\left(\frac{h}{2} - v\right) \end{aligned} \right.$$ where $h$ is the water with respect to the bottom and $hv$ the flux. The zero relaxation limit of this model is given by the inviscid Burgers equation. Choosing $g$ such that $\partial_{t} g = \frac{1}{\varepsilon}\left(\frac{h^{2}}{2} - hv\right)$ and letting $f:= hv-g$ yield the system $$\left\{\; \begin{aligned} & \partial_{t} h = - \partial_{x} f -\partial_{x} g\\ & \partial_{t} f = - \partial_{x}\left(h + \frac{h^{2}}{2}\right) \\ & \partial_{t} g = -\frac{1}{\varepsilon}\left(g+f -\frac{h^{2}}{2}\right). \end{aligned} \right.$$ Hence, we recover the desired singularly perturbed structure. In the finite dimensional case, it becomes after introducing two control parameters $\alpha,\beta$ $$\label{eq: relax WB} \left\{\; \begin{aligned} & \dot{\bm{h}}_{\varepsilon}(s) = -D \bm{f}_{\varepsilon}(s) - D \bm{w}_{\varepsilon}(s) + \mathds{H}(\bm{h}_{\varepsilon}(s))\alpha(s), && \bm{h}_{\varepsilon}(0) = \bm{h}_{0} \in \mathds{R}^{m}\\ & \dot{\bm{f}}_{\varepsilon}(s) = -D \bm{h}_{\varepsilon}(s) - \frac{1}{2}D (\bm{h}_{\varepsilon}(s))^{2}, && \bm{f}_{\varepsilon}(0) = \bm{f}_{0} \in \mathds{R}^{m}\\ & \dot{\bm{g}}_{\varepsilon}(s) = -\frac{1}{\varepsilon}\left[\bm{g}_{\varepsilon}(s) + \bm{f}_{\varepsilon}(s) - \frac{1}{2}(\bm{h}_{\varepsilon}(s))^{2} - \mathds{G}(\bm{h}_{\varepsilon}(s))\beta(s)\right], && \bm{w}_{\varepsilon}(0) = \bm{w}_{0}\in \mathds{R}^{m} \end{aligned} \right.$$ Here, $((\bm{h}_{\varepsilon}(s))^{2})$ is the vector whose entries are the square of the entries of $\bm{h}_{\varepsilon}(s)$. The control parameters can for example refer to the bottom profile. At the limit $\varepsilon \to 0$, one gets $$\label{eq: eff WB} \dot{\bm{h}}(s) = -\frac{1}{2}D(\bm{h}(s))^{2} - D\mathds{G}(\bm{h}(s))\beta(s) + \mathds{H}(\bm{h}(s))\alpha(s)$$ which is as expected the discretization of the controlled inviscid Burgers equation $$\partial_{t}h(t,x) + \frac{1}{2}\partial_{x}\big[h^{2}(t,x) + \mathds{G}(h(t,x))\beta(t,x)\big] = \mathds{H}(h(t,x))\alpha(t,x).$$ Let us introduce the two optimal control problems $$\label{eq: value eps WB} \tag{SP.5} \inf \; \Phi(\bm{h}_{\varepsilon}(T)),\quad \text{s.t. } \; \eqref{eq: relax WB}$$ and $$\label{eq: value eff WB} \tag{R.5} \inf \; \Phi(\bm{h}_{\varepsilon}(T)),\quad \text{s.t. } \; \eqref{eq: eff WB}$$ The following corollary is a direct consequence of Theorem [Theorem 1](#thm: conv G){reference-type="ref" reference="thm: conv G"}. **Corollary 5**. *As $\varepsilon\to 0$, the problem [\[eq: value eps WB\]](#eq: value eps WB){reference-type="eqref" reference="eq: value eps WB"} is approximated by [\[eq: value eff WB\]](#eq: value eff WB){reference-type="eqref" reference="eq: value eff WB"} in the sense of Definition [Definition 2](#def:approx){reference-type="ref" reference="def:approx"}. In particular, the controlled system [\[eq: relax WB\]](#eq: relax WB){reference-type="eqref" reference="eq: relax WB"} with any given initial conditions converges to [\[eq: eff WB\]](#eq: eff WB){reference-type="eqref" reference="eq: eff WB"} locally uniformly on any finite time interval, and we have $\inf\, \Phi(\bm{h}_{\varepsilon}(T)) \to \inf \, \Phi(\bm{h}(T))$.* The three--scale approximation follows the computations in §[5.2](#sec: JX 3 app){reference-type="ref" reference="sec: JX 3 app"} and in §[5.4](#sec: GT 3){reference-type="ref" reference="sec: GT 3"}. Starting from $$\left\{\; \begin{aligned} & \partial_{t}h + \partial_{x}\left[ hv + \varepsilon \mathcal{F}_{1}(h) \right] = 0\\ & \partial_{t}(hv) + \partial_{x}\left(h + \frac{1}{2}h^{2}\right) = \frac{1}{\varepsilon}\left(\frac{1}{2}h^{2}-hv\right), \end{aligned} \right.$$ we set $$\partial_{t}g = \frac{1}{\varepsilon}\left(\frac{1}{2}h^{2} - hv\right)$$ which yields $$\partial_{t}[hv -g] = -\partial_{x}\left(h+\frac{1}{2}h^{2}\right).$$ We define $f:= hv - g$ and get the following $$\left\{\; \begin{aligned} & \partial_{t} h =- \partial_{x}(f+g + \varepsilon\mathcal{F}_{1}(h)) \\ & \partial_{t} f = -\partial_{x}\left(h + \frac{1}{2}h^{2}\right)\\ & \partial_{t} g = -\frac{1}{\varepsilon}\left( g+f -\frac{1}{2}h^{2}\right). \end{aligned} \right.$$ Let us introduce $p$ and $q$ such that $$\begin{aligned} & \partial_{t} h = -\partial_{x} k\\ & \partial_{t} k = -\frac{1}{\varepsilon}\big[ k- (f+g+\varepsilon q) \big] \\%= \partial_{t}p + \mathcal{F}_{1}(h)\\ & \partial_{t} p = - \frac{1}{\varepsilon}\big[ k - (f+g) \big]\\ & \partial_{t} q = -\frac{1}{\varepsilon^{2}}[q - \mathcal{F}_{1}(h)]. \end{aligned}$$ This yields $$\left\{\; \begin{aligned} & \partial_{t} h = -\partial_{x}[k-p] - \partial_{x}p\\ & \partial_{t} [k-p] = q\\ & \partial_{t} p = - \frac{1}{\varepsilon}\big[ p + [k-p] - (f+g) \big]\\ & \partial_{t} q = -\frac{1}{\varepsilon^{2}}[q - \mathcal{F}_{1}(h)] \end{aligned} \right.$$ to which we add the two equations for $f,g$ $$\left\{\; \begin{aligned} & \partial_{t} f = -\partial_{x}\left(h + \frac{1}{2}h^{2}\right)\\ & \partial_{t} g = -\frac{1}{\varepsilon}\left( g+f -\frac{1}{2}h^{2}\right). \end{aligned} \right.$$ Note here we did not add the terms in Jin-Xin relaxation (i.e. we took $a=b=0$ in [\[relax JX 2\]](#relax JX 2){reference-type="eqref" reference="relax JX 2"}). Therefore we got a system analogue to [\[without ab\]](#without ab){reference-type="eqref" reference="without ab"}. But we could also consider the additional terms with $a,b>0$, and get a system analogue to [\[with ab\]](#with ab){reference-type="eqref" reference="with ab"}. The discrete version can be expressed as follows. Let $(\bm{h}, \bm{f}, \bm{g}, \bm{k}, \bm{p}, \bm{q})$ be the discretization of $(h, f, g, [k-p], p, q)$ respectively. Then one gets $$\label{SP WB multi} \left\{\; \begin{aligned} & \dot{\bm{h}}_{\varepsilon}(s) = - D \bm{k}_{\varepsilon}(s) - D \bm{p}_{\varepsilon}(s) + \mathds{H}(\bm{h}_{\varepsilon}(s))\alpha(s) , && \, \bm{h}_{\varepsilon}(0) = \bm{h}_{0}\\ & \dot{\bm{f}}_{\varepsilon}(s) = - D\bm{h}_{\varepsilon}(s) - \frac{1}{2}D (\bm{h}_{\varepsilon}(s))^{2} , && \, \bm{f}_{\varepsilon}(0) = \bm{f}_{0}\\ & \dot{\bm{k}}_{\varepsilon}(s) = \bm{q}_{\varepsilon}(s), && \, \bm{k}_{\varepsilon}(0) = \bm{k}_{0}\\ & \dot{\bm{g}}_{\varepsilon}(s) = -\frac{1}{\varepsilon}\left[ \bm{g}_{\varepsilon}(s) + \bm{f}_{\varepsilon}(s) - \frac{1}{2}(\bm{h}_{\varepsilon}(s))^{2} -\mathds{G}(\bm{h}_{\varepsilon}(s))\beta(s) \right], && \, \bm{g}_{\varepsilon}(0) = \bm{g}_{0}\\ & \dot{\bm{p}}_{\varepsilon}(s) = -\frac{1}{\varepsilon}\left[ \bm{p}_{\varepsilon}(s) + \bm{k}_{\varepsilon}(s) - \bm{f}_{\varepsilon}(s) - \bm{g}_{\varepsilon}(s) \right], && \, \bm{p}_{\varepsilon}(0) = \bm{p}_{0}\\ & \dot{\bm{q}}_{\varepsilon}(s) = -\frac{1}{\varepsilon^{2}}\left[\bm{q}_{\varepsilon}(s) - \mathcal{F}_{1}(\bm{h}_{\varepsilon}(s)) - \mathds{K}(\bm{h}_{\varepsilon}(s))\alpha(s)\right], && \, \bm{q}_{\varepsilon}(0) = \bm{q}_{0} \end{aligned} \right.$$ where we have chosen to add three control parameters $\alpha,\beta,\gamma$, although all the ODEs can be controlled as described in section §[4](#sec: multiscale sys){reference-type="ref" reference="sec: multiscale sys"}. At the limit $\varepsilon\to 0$, we obtain [\[eq: eff WB\]](#eq: eff WB){reference-type="eqref" reference="eq: eff WB"}. The analogue of Corollary [Corollary 2](#cor JX 3){reference-type="ref" reference="cor JX 3"} and Corollary [Corollary 4](#cor GT 3){reference-type="ref" reference="cor GT 3"} is the following. **Corollary 6**. *The value function corresponding to the control of [\[SP WB multi\]](#SP WB multi){reference-type="eqref" reference="SP WB multi"} converges locally uniformly on $\big(\mathds{R}^{m}\big)^{6}\times (0,1]$ to the value function corresponding to the control of [\[eq: eff WB\]](#eq: eff WB){reference-type="eqref" reference="eq: eff WB"}.* ## Second-Order Traffic Flow Models As stated in [@pareschi2005implicit §6.2], we recall the model in [@aw2000resurrection] for vehicular traffic $$\left\{\; \begin{aligned} & \partial_{t} \rho + \partial_{x}(\rho v) = 0\\ & \partial_{t}(\rho \omega) + \partial_{x}(v\rho \omega) = A \frac{\rho}{\varepsilon}\big(V(\rho) - v\big)\\ & \omega = v - P(\rho) \end{aligned} \right.$$ where $\rho$ is the density of vehicles subject to the first (continuity) equation, and complemented with an additional velocity equation for the mass flux variations due to the road conditions in front of the driver. Here $P(\rho)$ is a given function describing the anticipation of road conditions in front of the drivers and $V(\rho)$ describes the dependence of the velocity with respect to the density for an equilibrium situation. The parameter $\varepsilon$ is the relaxation time and $A>0$ is a positive constant.\ When the relaxation time goes to zero, we obtain the Lighthill-Whitham [@whitham1974linear] model $$\label{eq: LW} \partial_{t} \rho + \partial_{x}\big( \rho V(\rho) \big) = 0.$$ We introduce a new variable $f$ subject to the PDE $$\partial_{t} f = A\frac{\rho}{\varepsilon}\big(V(\rho) - v\big)$$ and define $g := \rho \omega -f$. Then we obtain the system $$\label{eq: traffic flow SP} \left\{\; \begin{aligned} & \partial_{t} \rho + v(\partial_{x}\rho) + \rho (\partial_{x} v) = 0 \\ & \partial_{t} g + (\partial_{x}v) g + v (\partial_{x} g) + (\partial_{x}v)f + v(\partial_{x}f) = 0\\ & \partial_{t} f = A\frac{1}{\varepsilon}\big(\rho V(\rho) + \rho P(\rho) - f - g \big)%\\ %& v = \frac{1}{\rho}(g+f) - P(\rho),\quad \text{if } \rho\neq 0, \text{ otherwise } ... \end{aligned} \right.$$ and $v$ solves the equation $\rho v = f+g - \rho P(\rho)$. The model [\[eq: traffic flow SP\]](#eq: traffic flow SP){reference-type="eqref" reference="eq: traffic flow SP"} falls within our setting and its discretized version is the following system of ODEs to which we added two control parameters $\alpha$ and $\beta$ $$\label{TF eps} \left\{\; \begin{aligned} \dot{\bm{\rho}}_{\varepsilon}(s) & = -\bm{v}_{\varepsilon}(s) D\bm{\rho}_{\varepsilon}(s) - \bm{\rho}_{\varepsilon}(s) D\bm{v}_{\varepsilon}(s) + \mathds{H}(\bm{\rho}_{\varepsilon}(s))\alpha(s)\\ \dot{\bm{g}}_{\varepsilon}(s) & = -\big(\bm{f}_{\varepsilon}(s) + \bm{g}_{\varepsilon}(s)\big) D\bm{v}_{\varepsilon}(s) -\bm{v}_{\varepsilon}(s) D\big(\bm{f}_{\varepsilon}(s) +\bm{g}_{\varepsilon}(s)\big) \\ \dot{\bm{f}}_{\varepsilon}(s) & = -\frac{1}{\varepsilon}A\big[ \bm{f}_{\varepsilon}(s) + \bm{g}_{\varepsilon}(s) - \bm{\rho}_{\varepsilon}(s) V(\bm{\rho}_{\varepsilon}(s)) - \bm{\rho}_{\varepsilon}(s) P(\bm{\rho}_{\varepsilon}(s))-\mathds{G}(\bm{\rho}_{\varepsilon}(s))\beta(s)\big] \end{aligned} \right.$$ together with the equation defining $\bm{v}_{\varepsilon}(s)$ $$\label{TF v} \bm{\rho}_{\varepsilon}(s) \bm{v}_{\varepsilon}(s) = \bm{f}_{\varepsilon}(s) + \bm{g}_{\varepsilon}(s) - \bm{\rho}_{\varepsilon}(s)P(\bm{\rho}_{\varepsilon}(s)),$$ here the multiplication of two vectors (e.g., in $\bm{\rho}_{\varepsilon}(s) \bm{v}_{\varepsilon}(s)$) is understood component-wise. As before, we denoted by $D$ a discretization matrix. An example of control could also be $V(\bm{\rho}_{\varepsilon}) = \beta$. In this case, the dynamics for $\bm{f}_{\varepsilon}$ becomes $$\dot{\bm{f}}_{\varepsilon}(s) = -\frac{1}{\varepsilon}A\big[ \bm{f}_{\varepsilon}(s) + \bm{g}_{\varepsilon}(s) - \bm{\rho}_{\varepsilon}(s) \beta(s) - \bm{\rho}_{\varepsilon}(s) P(\bm{\rho}_{\varepsilon}(s))\big].$$ When $\varepsilon\to 0$, the local equilibrium is $$\bm{f}(s) + \bm{g}(s) = \bm{\rho}(s) \big[V(\bm{\rho}(s)) + P(\bm{\rho}(s))\big]+\mathds{G}(\bm{\rho}(s))\beta(s).$$ When substituted in [\[TF v\]](#TF v){reference-type="eqref" reference="TF v"}, one gets $$\begin{aligned} \bm{\rho}(s) \bm{v}(s) = \bm{\rho}(s) V(\bm{\rho}(s)) +\mathds{G}(\bm{\rho}(s))\beta(s). \end{aligned}$$ We can choose the matrix $\mathds{G}(\bm{\rho}(s))$ to be zero matrix when $\bm{\rho}(s) =0$. Then, we can write, whenever $\bm{\rho}(s) \neq 0$ $$\begin{aligned} \bm{v}(s) = V(\bm{\rho}(s)) +(\bm{\rho}(s))^{-1}\mathds{G}(\bm{\rho}(s))\beta(s), \end{aligned}$$ the inverse $(\bm{\rho}(s))^{-1}$ being understood component-wise. Therefore, the first equation in [\[TF eps\]](#TF eps){reference-type="eqref" reference="TF eps"} becomes $$\label{eq: rho TF} \begin{aligned} \dot{\bm{\rho}}(s) = & -V(\bm{\rho}(s)) D\bm{\rho}(s) - \bm{\rho}(s) DV(\bm{\rho}(s)) \\ & \quad \quad -\left[(\bm{\rho}(s))^{-1}\mathds{G}(\bm{\rho}(s))\beta(s)\right] D\bm{\rho}(s) - \bm{\rho}(s) D\left[(\bm{\rho}(s))^{-1}\mathds{G}(\bm{\rho}(s))\beta(s)\right]\\ & \quad \quad \quad \quad \quad + \mathds{H}(\bm{\rho}(s))\alpha(s). \end{aligned}$$ This corresponds to the discretization of the controlled PDE $$\partial_{t}\rho + \partial_{x} \big[\rho(t,x)V(\rho(t,x)) + \mathds{G}(\rho(t,x))\beta(t,x)\big] = \mathds{H}(\rho(s))\alpha(t,x),$$ which reduces to [\[eq: LW\]](#eq: LW){reference-type="eqref" reference="eq: LW"} in the absence of controls. Let us introduce the two optimal control problems $$\label{eq: value eps TF} \tag{SP.6} \inf \; \Phi(\bm{\rho}_{\varepsilon}(T)),\quad \text{s.t. } \; \eqref{TF eps}-\eqref{TF v}$$ and $$\label{eq: value eff TF} \tag{R.6} \inf \; \Phi(\bm{\rho}_{\varepsilon}(T)),\quad \text{s.t. } \; \eqref{eq: rho TF}$$ The following corollary is a direct consequence of Theorem [Theorem 1](#thm: conv G){reference-type="ref" reference="thm: conv G"}. **Corollary 7**. *As $\varepsilon\to 0$, the problem [\[eq: value eps TF\]](#eq: value eps TF){reference-type="eqref" reference="eq: value eps TF"} is approximated by [\[eq: value eff TF\]](#eq: value eff TF){reference-type="eqref" reference="eq: value eff TF"} in the sense of Definition [Definition 2](#def:approx){reference-type="ref" reference="def:approx"}. In particular, the controlled system [\[TF eps\]](#TF eps){reference-type="eqref" reference="TF eps"}-[\[TF v\]](#TF v){reference-type="eqref" reference="TF v"} with any given initial conditions converges to [\[eq: rho TF\]](#eq: rho TF){reference-type="eqref" reference="eq: rho TF"} locally uniformly on any finite time interval, and we have $\inf\, \Phi(\bm{\rho}_{\varepsilon}(T)) \to \inf \, \Phi(\bm{\rho}(T))$.* ## Granular Gases We shall consider the model in [@pareschi2005implicit §6.3] for a granular gas [@jenkins1985grad; @toscani2004kinetic] given by $$\left\{ \; \begin{aligned} & \partial_{t}\rho + \partial_{x}(\rho u) = 0 \\ & \partial_{t}(\rho u) + \partial_{x}(\rho u^2 + p) = \rho g \\ & \partial_{t}\left( \frac{1}{2}\rho u^2 + \frac{3}{2}\rho T \right) + \partial_{x}\left( \frac{1}{2}\rho u^3 + \frac{3}{2} u\rho T + pu \right) = - \frac{(1-e^2)}{\varepsilon} G(\rho) \rho^{2} T^{3/2} \end{aligned} \right.$$ where $e$ is the coefficient of restitution, $g$ the acceleration due to gravity, $\varepsilon$ a relaxation time, $p$ is the pressure given by $$p = \rho T (1+2(1+e)G(\rho))$$ and $G(\rho)$ is the statistical correlation function. We introduce new variables $$\label{eq: v w} v=\rho u, \quad \text{ and } \quad \text{w} = \frac{1}{2}\rho u^2 + \frac{3}{2}\rho T,$$ and define $\varphi$ such that $$\partial_{t}\varphi = - \frac{(1-e^2)}{\varepsilon} G(\rho) \rho^{2} T^{3/2}.$$ Then, setting $$\label{eq: omega} \omega = \text{w} - \varphi$$ yields the system $$\left\{ \; \begin{aligned} & \partial_{t}\rho + \partial_{x}v = 0\\ & \partial_{t} v + \partial_{x}(uv + p) = \rho g\\ & \partial_{t} \omega + \partial_{x}(u \omega + up +u\varphi) = 0\\ & \partial_{t}\varphi = - \frac{(1-e^2)}{\varepsilon} G(\rho) \rho^{2} T^{3/2}. \end{aligned} \right.$$ From the definition of $\omega$ and $\text{w}$ above, we can write $\rho T = \frac{2}{3}(\omega + \varphi) - \frac{1}{3}v u$, then we substitute the latter quantity in $\rho^{2}T^{3/2} = (\rho T)\rho T^{1/2}$ and obtain the system $$\left\{ \; \begin{aligned} & \partial_{t}\rho + \partial_{x}v = 0\\ & \partial_{t} v + \partial_{x}(uv + p) = \rho g\\ & \partial_{t} \omega + \partial_{x}(u \omega + up +u\varphi) = 0\\ & \partial_{t}\varphi = \frac{1}{\varepsilon}\,\frac{2(1-e^2)}{3} G(\rho) \rho T^{1/2}\left( -\omega - \varphi + \frac{1}{2}vu\right) \end{aligned} \right.$$ whose structure now complies with our singularly perturbed model. Indeed, the discrete and controlled version of the latter system of PDEs is the following system of ODEs where the control parameters are $\alpha,\beta,\gamma$ $$\label{eq: granular relax} \left\{\, \begin{aligned} \dot{\bm{\rho}}_{\varepsilon}(s) & = -D \bm{v}_{\varepsilon}(s) + \mathds{H}(\bm{\rho}_{\varepsilon}(s))\alpha(s), && \bm{\rho}_{\varepsilon}(0) = \bm{\rho}_{0}\\ \dot{\bm{v}}_{\varepsilon}(s) & = \bm{\rho}_{\varepsilon}(s)g - \bm{u}_{\varepsilon}(s)D \bm{v}_{\varepsilon}(s) - \bm{v}_{\varepsilon}(s)D\bm{u}_{\varepsilon}(s) - D \bm{p}_{\varepsilon}(s)+ \mathds{K}(\bm{\rho}_{\varepsilon}(s))\gamma(s),&& \bm{v}_{\varepsilon}(0) = \bm{v}_{0}\\ \dot{\bm{w}}_{\varepsilon}(s) & = -\bm{u}_{\varepsilon}(s) D\big[\bm{w}_{\varepsilon}(s) + \bm{p}_{\varepsilon}(s) + \bm{\varphi}_{\varepsilon}(s)\big] && \bm{w}_{\varepsilon}(0) = \bm{w}_{0}\\ & \quad \quad \quad - \big[\bm{w}_{\varepsilon}(s) + \bm{p}_{\varepsilon}(s) + \bm{\varphi}_{\varepsilon}(s)\big] D \bm{u}_{\varepsilon}(s), && \\ \dot{\bm{\varphi}}_{\varepsilon}(s) & = -\frac{1}{\varepsilon} M(\bm{\rho}_{\varepsilon}(s))\left[ \bm{\varphi}_{\varepsilon}(s) + \bm{w}_{\varepsilon}(s) - \frac{1}{2}\bm{u}_{\varepsilon}(s)\bm{v}_{\varepsilon}(s) - \mathds{G}(\bm{\rho}_{\varepsilon}(s))\beta(s)\right] , && \bm{\varphi}_{\varepsilon}(0) = \bm{\varphi}_{0} \end{aligned} \right.$$ where $M(\bm{\rho}_{\varepsilon}(s)):=\frac{2(1-e^2)}{3} G(\bm{\rho}_{\varepsilon}(s) )\bm{\rho}_{\varepsilon}(s) T^{1/2}$. When $\varepsilon\to 0$, we obtain $$\bm{\varphi}(s) + \bm{w}(s) = \frac{1}{2}\bm{u}(s)\bm{v}(s) + \mathds{G}(\bm{\rho}(s))\beta(s).$$ In particular, we have $\dot{\bm{\varphi}}(s) = 0$, so recalling [\[eq: omega\]](#eq: omega){reference-type="eqref" reference="eq: omega"}, we get $$\dot{\textbf{w}}(s) = \dot{\bm{w}}(s) + \dot{\bm{\varphi}}(s) = \dot{\bm{w}}(s)$$ thus, from [\[eq: v w\]](#eq: v w){reference-type="eqref" reference="eq: v w"}, we have $$\label{eq: granular 1} \begin{aligned} \frac{\text{d}}{\text{d}t} \left[\frac{1}{2} \bm{\rho}(s)\bm{u}(s)^{2} + \frac{3}{2}\bm{\rho}(s)T\right] & = -\bm{u}(s) D\big[\frac{1}{2}\bm{u}(s)\bm{v}(s) + \mathds{G}(\bm{\rho}(s))\beta(s) + \bm{p}(s)\big] \\ & \quad \quad \quad - \big[\frac{1}{2}\bm{u}(s)\bm{v}(s) + \mathds{G}(\bm{\rho}(s))\beta(s) + \bm{p}(s)\big] D \bm{u}(s), \end{aligned}$$ which complements $$\label{eq: granular 2} \left\{\, \begin{aligned} \dot{\bm{\rho}}(s) & = -D \bm{v}(s) + \mathds{H}(\bm{\rho}(s))\alpha(s), \\ \dot{\bm{v}}(s) & = \bm{\rho}(s)g - \bm{u}(s)D \bm{v}(s) - \bm{v}(s)D\bm{u}(s) - D \bm{p}(s) + \mathds{K}(\bm{\rho}(s))\gamma(s). \end{aligned} \right.$$ Recalling again [\[eq: v w\]](#eq: v w){reference-type="eqref" reference="eq: v w"}, the latter system of ODEs corresponds to the system of PDEs $$\left\{\, \begin{aligned} & \partial_{t} \rho + \partial_{x}\big[ \rho u \big] = \mathds{H}(\rho)\alpha\\ & \partial_{t}\big(\rho u\big) + \partial_{x}\big(u^{2}\rho + p\big) = \rho g + \mathds{K}(\rho)\gamma\\ & \partial_{t}\left(\frac{1}{2}\rho u^{2} + \frac{3}{2}\rho T\right) +\partial_{x}\left[ \frac{1}{2}u^{3}\rho + up + u\,\mathds{G}(\rho)\beta\right] = 0 \end{aligned} \right.$$ Let us introduce the two optimal control problems $$\label{eq: value eps GG} \tag{SP.6} \inf \; \Phi(\bm{\rho}_{\varepsilon}(T)),\quad \text{s.t. } \; \eqref{eq: granular relax}$$ and $$\label{eq: value eff GG} \tag{R.6} \inf \; \Phi(\bm{\rho}_{\varepsilon}(T)),\quad \text{s.t. } \; \eqref{eq: granular 2}-\eqref{eq: granular 1}$$ The following corollary is a direct consequence of Theorem [Theorem 1](#thm: conv G){reference-type="ref" reference="thm: conv G"}. **Corollary 8**. *As $\varepsilon\to 0$, the problem [\[eq: value eps GG\]](#eq: value eps GG){reference-type="eqref" reference="eq: value eps GG"} is approximated by [\[eq: value eff GG\]](#eq: value eff GG){reference-type="eqref" reference="eq: value eff GG"} in the sense of Definition [Definition 2](#def:approx){reference-type="ref" reference="def:approx"}. In particular, the controlled system [\[eq: granular relax\]](#eq: granular relax){reference-type="eqref" reference="eq: granular relax"} with any given initial conditions converges to [\[eq: granular 2\]](#eq: granular 2){reference-type="eqref" reference="eq: granular 2"}-[\[eq: granular 1\]](#eq: granular 1){reference-type="eqref" reference="eq: granular 1"} locally uniformly on any finite time interval, and we have $\inf\, \Phi(\bm{\rho}_{\varepsilon}(T)) \to \inf \, \Phi(\bm{\rho}(T))$.* # Summary The object of the present manuscript is twofold. First, we provide a new formulation for stiff differential equations based on singular perturbations. The convergence of the value functions with respect to the stiffness parameter is shown. An asymptotic expansion of the value function corresponding to the control problem is formulated. This is useful in particular when designing feedback controls depending on the gradient of the value function. The second contribution of the manuscript is the higher order approximation of controlled stiff differential equations. Consequently, this leads to a higher order corrections of the value function. Even so our main motivation has been the Jin-Xin relaxation system, we show also for various other examples the asymptotic expansion of feedback controls. For Jin-Xin we also expanded the system to allow for higher--order formulation and approximation. # Limit Occupational Measure Sets {#appendix} Denote by $\lambda\{I\}$ the Lebesgue measure of the interval $I$, and let $y_{z}$ follow the same dynamics as $y$ in [\[CSP\]](#CSP){reference-type="eqref" reference="CSP"} but where $\varepsilon=1$ and $z$ is frozen (constant). We define the *occupational measures* (see [@gaitsgory1992suboptimization; @homocp; @gaitsgory1999limit; @kouhkouh22phd]) as $$\varphi_{z}^{(y_{0},\beta, T)}(Q) := \frac{1}{T}\lambda\{s \in [0,T] \, | \, (y_{z}(s),\beta(s)) \in Q\}$$ and denote the union of such measures over all admissible controls $$\Phi(z,T,y_{0}) := \bigcup_{\beta}\{\varphi_{z}^{(y_{0},\beta, T)}\}.$$ Then for all $z\in Z,\, y_{0}\in \mathds{T}^{n}$ where $Z$ is a compact set (see remark [Remark 1](#rem periodic){reference-type="ref" reference="rem periodic"}), the limit in the Hausdorff metric as $T\to +\infty$ is $$\label{LOMS} \lim\limits_{T\to +\infty}\Phi(z,T,y_{0}) = \mathfrak{L}(z),\quad \forall\, z\in Z,\, y_{0}\in \mathds{T}^{n}$$ called the *limit occupational measure set* (LOMS) whose existence is discussed in [@gaitsgory1999limit] and references therein. Also define the set $$V_{f}(z) := \left\{ v\, \bigg|\, v = \int_{\mathds{T}^{n}\times \Omega_{B}} f(z,y,\alpha, \beta)\,\varphi(\mbox{\rm d}y \times \mbox{\rm d}\beta),\; \varphi \in \mathfrak{L}(z),\; \alpha\in \Omega_{A} \right\}.$$ where $f$ here denotes the dynamics of the slow variable $z$, that is $f(z,y,\alpha, \beta) = A_{1}(z)y + B_{1}(z)\alpha + C_{1}(z)$. A result from [@gaitsgory1999limit] (see also [@gaitsgory1992suboptimization]) ensures that there exists a function $\mu(\varepsilon)$, $$\lim\limits_{\varepsilon\to 0} \, \mu(\varepsilon)=0,$$ such that corresponding to any trajectory $(z_{\varepsilon}(\cdot),y_{\varepsilon}(\cdot))$ of [\[CSP\]](#CSP){reference-type="eqref" reference="CSP"}, there exists a solution $\tilde{z}(\cdot)$ of $$\label{limit sys} \dot{\tilde{z}}(s) \in V_{f}(\tilde{z}(s)),\quad \tilde{z}(0)=z_{0}$$ such that $$\max\limits_{s\in[0,T]}\|z_{\varepsilon}(s) - \tilde{z}(s)\| \leq \mu(\varepsilon).$$ Conversely, corresponding to any solution of [\[limit sys\]](#limit sys){reference-type="eqref" reference="limit sys"} there exists a trajectory of [\[CSP\]](#CSP){reference-type="eqref" reference="CSP"} such that the same inequality is satisfied. Concerning the HJB equation, when $\varepsilon\to 0$, the *effective* (limit) PDE problem obtained in [@homocp Theorem 3.1] is $$\left\{ \; \begin{aligned} & \partial_{t}\widetilde{v} + \widetilde{H}(z,D_{z}\widetilde{v}) = \ell(z),\; \text{ in } \mathds{R}^{m}\times(0,T)\\ & \widetilde{v}(z,0) = \phi(z),\; \text{ in } \mathds{R}^{m} \end{aligned} \right.$$ for $T\in (0,+\infty)$, where the *effective* Hamiltonian (compare with $\widetilde{H}$ in the proof of Theorem [Theorem 2](#thm conv){reference-type="ref" reference="thm conv"}) is $$\label{eff Ham} \widetilde{H}(z,p) = \sup\limits_{\alpha\in \Omega_{A}, \mu \in \mathfrak{L}(z)}\{ - p\cdot \widetilde{f}(z,\alpha,\mu) \}$$ and the *effective* dynamics is $$\widetilde{f}(z, \alpha,\mu) := \int_{\mathds{T}^{n}\times \Omega_{B}} f(z,y,\alpha,\beta)\,\mu(\mbox{\rm d}y\times \mbox{\rm d}\beta).%,\quad \widetilde{\ell}(z,\mu) := \int \ell(z,\alpha)\,\mu(\dd z\times \dd \alpha).$$ Moreover, $\widetilde{v}$ is the value function of the (*effective*) optimal control problem $$\label{averaged ocp} \begin{aligned} \widetilde{v}(z,t) =& \inf\limits_{\alpha,\mu}\, \int_{0}^{t}{\ell}(\tilde{z}(s)\,\mbox{\rm d}s + \phi(\tilde{z}(t))\\ & \; \text{ subject to: } \dot{\tilde{z}}(s) = \widetilde{f}(\tilde{z}(s), \alpha(s),\mu(s)),\quad \tilde{z}(0) = z_{0},\\ &\quad \quad\quad \; \text{ and } \quad \mu(s) \in \mathfrak{L}(\tilde{z}(s)), \quad \alpha(s)\in \Omega_{A}. \end{aligned}$$ The dynamics can be written as a differential inclusion $$\dot{\Tilde{z}}(s) \in \widetilde{f}(\tilde{z}(s), \alpha(s), \mathfrak{L}(\tilde{z}(s))) %= V_{f}(\tilde{z}(s))$$ which is also [\[limit sys\]](#limit sys){reference-type="eqref" reference="limit sys"}. This is reminiscent to relaxed optimal controls. In our setting, the LOMS reduces to a singleton as stated in the proof of Theorem [Theorem 2](#thm conv){reference-type="ref" reference="thm conv"}. Hence, the value function $V_{0}$ in Theorem [Theorem 2](#thm conv){reference-type="ref" reference="thm conv"} coincides with $\widetilde{v}$ above, and the limiting dynamics $\bar{z}(\cdot)$ in [\[reduced\]](#reduced){reference-type="eqref" reference="reduced"} coincides with $\tilde{z}(\cdot)$. [^1]: [^2]: The authors thank the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) for the financial support through 320021702/GRK2326, 333849990/IRTG-2379, B04, B05 and B06 of 442047500/SFB1481, HE5386/19-2,22-1,23-1,25-1,26-1,27-1 and under Germany's Excellence Strategy EXC-2023 Internet of Production 390621612 and under the Excellence Strategy of the Federal Government and the Länder. Support through the EU DATAHYKING is also acknowledged. [^3]:

arxiv_math

--- abstract: | This manuscript is a continuation of the program started in [@defeo_federico_swiech] of studying optimal control problems for stochastic delay differential equations using the theory of viscosity solutions of the associated Hamilton-Jacobi-Bellman equation. We show how to use the partial $C^{1,\alpha}$-regularity of the value function obtained in [@defeo_federico_swiech] to obtain optimal feedback controls. The main result of the paper is a verification theorem which provides a sufficient condition for optimality using the value function. We then discuss its applicability to the construction of optimal feedback controls. The procedure is based on several layers of approximations. We use inf-convolutions and we adapt partial convolution approximations originally introduced by P.L. Lions for equations with bounded terms. Minimal regularity of the coefficients of the problem is assumed, however we have to assume that the value function is semiconvex in a space defined by a weaker norm. We provide an application to stochastic optimal advertising. address: - "F. De Feo: Department of Mathematics, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy" - "A. Święch: School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA" author: - Filippo De Feo - Andrzej Święch title: "Optimal control of stochastic delay differential equations: Optimal feedback controls" --- **Mathematics Subject Classification (2020):** 49L25, 93E20, 49K45, 60H15, 49L20, 35R15, 49L12, 49N35, 34K50 **Keywords and phrases:** stochastic optimal control, Hamilton-Jacobi-Bellman equation, optimal synthesis, verification theorem, viscosity solution, stochastic delay differential equation # Introduction In this manuscript we continue the program started in [@defeo_federico_swiech] of studying optimal control problems for stochastic delay differential equations (SDDE) using the approach of the dynamic programming principle. In this approach an original optimal control problem for a stochastic delay differential equation is rewritten as an optimal control problem without delay for a stochastic partial differential equation which is viewed as a stochastic differential equation in an infinite dimensional Hilbert space $X=\mathbb R^n\times L^2([-d,0];\mathbb R^n)$, where $d$ is the maximum delay time. In this setup $\mathbb R^n$ (the "present" space) is the space where we keep track of the current state of the controlled random variable and $L^2([-d,0];\mathbb R^n)$ (the "past" space) is where we keep track of the relevant past part. We can then investigate this new equivalent problem using the dynamic programming approach and study the value function $V$ of the problem and the associated Hamilton-Jacobi-Bellman (HJB) equation. It was proved in [@defeo_federico_swiech] that the value function is the unique viscosity solution of the HJB equation in $X$ in the sense of the definition presented in [@fgs_book]. Moreover it was proved in [@defeo_federico_swiech] that the value function $V$ is such that for every $x_1\in L^2([-d,0];\mathbb R^n)$, $V(\cdot,x_1)\in C^{1,\alpha}_{\rm loc}(\mathbb R^n)$. The procedure of rewriting the optimal control problems for SDDE in a Hilbert space as well as the main results of [@defeo_federico_swiech] regarding the value function and the HJB equation are recalled here in Section [3](#sec:prelim){reference-type="ref" reference="sec:prelim"}. The goal of this paper is to use the HJB equation and the partial regularity result for the value function to explore how they can be helpful in the construction of optimal feedback controls. It is standard to construct and optimal feedback control from the HJB equation if the value function is smooth (see [@FS; @yong_zhou] or [@fgs_book] in infinite dimension). Here, under the assumption that the diffusion does not depend on the controls, the partial $C^{1,\alpha}$-regularity result allows to construct a candidate for an optimal feedback map. However the value function is not regular enough, we cannot even write Itô's formula for $V$, some terms in the equation are not well defined, and thus we cannot follow a standard argument. Instead we employ an approximation procedure involving several layers of approximations. We first use inf-convolutions and then we adopt approximations based on partial convolutions which were originally introduced in [@lions-infdim1] and which we adapt here to equations with unbounded terms. We then construct regular functions approximating $V$ which are viscosity supersolutions of slightly perturbed HJB equations. The new functions are regular enough so that they satisfy Itô's formula and all terms in the perturbed equations are satisfied pointwise. We can then work with the approximating functions and HJB equations and the candidate optimal feedback map to show passing to the limit that the optimal feedback map indeed allows to define an optimal feedback control. We work under minimal regularity assumptions on the coefficients of the problem, however we have to assume that the value function is so called $|\cdot|_{-1}$ semiconvex (see Section [4](#sec:approximation via inf-convolutions){reference-type="ref" reference="sec:approximation via inf-convolutions"}). The main result of the paper is a verification theorem, i.e. Theorem [Theorem 32](#th:verification){reference-type="ref" reference="th:verification"}, which provides a sufficient condition for optimality using the value function $V$. We then discuss its applicability to the construction of optimal feedback controls, i.e. Corollary [Corollary 33](#cor:closed_loop){reference-type="ref" reference="cor:closed_loop"}. To complete this we show that the value functions in the weak formulation of optimal control problem using reference probability spaces and so called generalized reference probability spaces (see Section [2](#sec:formul){reference-type="ref" reference="sec:formul"} for definitions) are the same. Finally we provide an application to stochastic optimal advertising. We remark that the current paper as well as [@defeo_federico_swiech] only consider the case where the delay occurs in the state variable. A detailed description of existing research into optimal control problems for SDDE using the Hilbert space approach and dynamic programming/HJB techniques can be found in [@defeo_federico_swiech]. We refer the readers to this paper. Here we just briefly recall relevant literature. Deterministic and stochastic optimal control problems for delay differential equations have been studied using mild solutions, mild solutions in $L^2$ spaces and solutions using backward stochastic differential equations in [@gozzi_masiero_2017; @gozzi_masiero_2017_b; @gozzi_masiero_2022; @masiero_2022; @gozzi_marinelli_2004; @fuhrman_tessitore_masiero_2010]. Classical explicit solutions were employed in [@fabbri_gozzi_2008; @bambi_fabbri_gozzi; @(2012); @bambi_digirolami_federico_gozzi; @(2017); @biffis_gozzi_prosdocini_2020; @biagini_gozzi_zanella_2022; @djehiche_gozzi_zanco_zanella_2022]. An approach based on regular solutions and convex regularization techniques was used in [@faggian_2005; @faggian_2008; @faggian_gozzi_2010]. Viscosity solutions were first used to deal with deterministic optimal control problems in [@goldys_1; @goldys_2; @tacconi; @carlier_tahraoui_2010]. Paper [@defeo_federico_swiech] was the first to study optimal control problems for SDDE by the notion of the so-called $B$-continuous viscosity solutions in a Hilbert space. See also [@deFeo_2023] where stochastic optimal control problems with delays including delays in the control were studied via $B$-continuous viscosity solutions in a Hilbert space. For the approach using path-dependent viscosity solutions see, e.g., [@bayraktar_keller_2018; @bayraktar_keller_2022; @cosso_federico_gozzi_rosestolato_touzi; @ekren_2014; @ekren_2016; @ekren_2016b; @ren_touzi_zhang] and the references therein. Connections between path-dependent viscosity solutions and $B$-continuous viscosity solutions are shown in [@ren_rosestolato]. Other approaches using appropriately defined viscosity solutions in spaces of right-continuous functions and continuous functions were considered to investigate the stochastic case in [@zhou_2018; @zhou_2019; @zhou_2021]. Classical verification theorems for stochastic optimal control problems assume smoothness of the value function and uses the associated HJB equation to obtain sufficient and necessary conditions for optimality. For finite dimensional problems such results can be found in [@FS; @yong_zhou] and for problems in an infinite dimensional Hilbert space corresponding formulations are in [@fgs_book], Sections 2.5.1 and 2.5.2. When the value function is not $C^2$ verification theorems become complicated. A viscosity solution version of the verification theorem is in [@yong_zhou Chapter 5] and full proofs are in [@gozzi_swiech_zhou_2005] and [@gozzi_swiech_zhou_2010]. It is very rare for infinite dimensional problems that the value function is regular enough to apply the smooth verification theorem. Nevertheless some results exist. For deterministic problems using viscosity solution framework, we refer for instance to [@LY Chapter 6], [@can-fr; @fgs_2008]. In the stochastic case versions of the result from [@yong_zhou] appeared recently in [@stannat_wessels_2021; @ChenLu]. Hence, to the best of our knowledge, Theorem [Theorem 32](#th:verification){reference-type="ref" reference="th:verification"} is the first verification theorem in the context of viscosity solutions for a stochastic optimal control problem with delays. Verification theorem and optimal synthesis results using other frameworks are discussed in [@fgs_book]: for mild solutions in Section 4.8; for solutions in $L^2$ spaces in Section 5.5, see also [@federico_gozzi_2018]; for solutions using backward stochastic differential equations in Sections 6.5, 6.6 and 6.10. Viscosity solutions handle second order HJB equations which are fully nonlinear and degenerate, where good regularity results for solutions are hard to get. Hence there are very few results about construction of optimal feedback controls. Optimal feedback controls for deterministic optimal control problems coming from controlled differential delay equations were constructed in [@goldys_2; @tacconi]. Recently optimal feedback controls were constructed in [@mayorga_swiech_2022] for a special class of stochastic optimal control problems with bounded evolution in a Hilbert space coming from a mean field control problem, for which the HJB equation was semilinear, had bounded terms and the value function was $C^{1,1}$ in the state variable. The plan of the manuscript is the following. In Section [2](#sec:formul){reference-type="ref" reference="sec:formul"} we introduce the optimal control problems for SDDE and the main assumptions. In Section [3](#sec:prelim){reference-type="ref" reference="sec:prelim"} we recall the results from [@defeo_federico_swiech] which are the basis for the current paper. Section [4](#sec:approximation via inf-convolutions){reference-type="ref" reference="sec:approximation via inf-convolutions"} deals with the first approximation of the value function $V$ by inf-convolutions. It is proved there that the inf-convolution of the value function is a viscosity supersolution of a perturbed HJB equation. In Section [5](#sec:dynkin_formula){reference-type="ref" reference="sec:dynkin_formula"} we introduce a modified class of functions $\mathcal D$ from [@lions-infdim1] and show that they satisfy Dynkin's formula. In Section [6](#sec:lions_approx){reference-type="ref" reference="sec:lions_approx"} we further perturb the inf-convolutions of $V$ by partial convolutions to obtain functions from the class $\mathcal D$ and which are viscosity supersolutions of another perturbed HJB equations. Section [7](#sec:verthm-optfeed){reference-type="ref" reference="sec:verthm-optfeed"} contains the proof of the verification theorem and construction of an optimal feedback control. Finally, in Section [8](#sec:example){reference-type="ref" reference="sec:example"}, we present an application to stochastic optimal advertising. # The optimal control problem: Setup and assumptions {#sec:formul} We denote by $M^{m \times n}$ the space of real valued $m \times n$-matrices and we denote by $|\cdot|$ the Euclidean norm in $\mathbb{R}^{n}$ as well as the norm of elements of $M^{m \times n}$ regarded as linear operators from $\mathbb{R}^{m}$ to $\mathbb{R}^{n}$. We will write $x \cdot y$ for the inner product in $\mathbb R^n$. We consider the standard Lebesgue space $L^2:=L^2([-d,0];\mathbb{R}^n)$ of square integrable functions from $[-d,0]$ to $\mathbb{R}^{n}$. We denote by $\langle \cdot,\cdot\rangle_{L^{2}}$ the inner product in $L^2$ and by $|\cdot|_{L^{2}}$ the norm. We also consider the standard Sobolev space $W^{1,2}:=W^{1,2}([-d,0]; \mathbb{R}^n)$ of functions in $L^{2}$ admitting weak derivative in $L^{2}$, endowed with the inner product $\langle f,g\rangle_{W^{1,2}}:= \langle f,g\rangle_{L^{2}}+\langle f',g'\rangle_{L^{2}}$ and norm $|f|_{W^{1,2}}:=(|f|^2_{L^{2}}+|f'|^2_{L^{2}})^{\frac{1}{2}}$. We use the setup of [@defeo_federico_swiech]. We say that $\tau=(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\geq 0}, \mathbb{P}, W)$ is a generalized reference probability space if $(\Omega,\mathcal{F},\mathbb{P})$ is a complete probability space, $W=(W(t))_{t\ge 0}$ is a standard $\mathbb{R}^q$-valued Wiener process and $(\mathcal{F}_t)_{t\geq 0}$ is a filtration satisfying the usual conditions, i.e. it is right-continuous and complete (see [@defeo_federico_swiech Definition 1.100]). A generalized reference probability space $\tau=(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\geq 0}, \mathbb{P}, W)$ is called a reference probability space if in addition $W(0)=0$ and $(\mathcal{F}_t)_{t\geq 0}$ is the augmented filtration generated by $W$ (see [@defeo_federico_swiech Definition 2.7]). We consider the following controlled stochastic differential delay equation (SDDE) $$\label{eq:SDDE} \begin{cases} dy(t) = \displaystyle b_0 \left ( y(t),\int_{-d}^0 a_1(\xi)y(t+\xi)\,d\xi ,u(t) \right) dt \displaystyle+ \sigma_0 \left (y(t),\int_{-d}^0 a_2(\xi)y(t+\xi)\,d\xi \right)\, dW(t),\quad t>0,\\ y(0)=x_0, \quad y(\xi)=x_1(\xi)\; \quad \forall \xi\in[-d,0), \end{cases}$$ where $d>0$ is the maximum delay and: (i) $x_0 \in \mathbb{R}^n$, $x_1 \in L^2([-d,0]; \mathbb{R}^n)$ are the initial conditions; (ii) $b_0 \colon \mathbb{R}^n \times \mathbb{R}^h \times U \to \mathbb{R}^n$, $\sigma_0 \colon \mathbb{R}^n \times \mathbb{R}^h \to M^{n\times q}$; (iii) $a_{i}:[-d,0]\to M^{h \times n}$ for $i=1,2$ and if $a_{i}^{j}$ is the $j$-th row of $a_i(\cdot)$, for $j=1,...,h$, then $a_i^{j}\in W^{1,2}$ and $a_i^{j}(-d)=0$. The precise assumptions on $b_0,\sigma_0$ will be given later. We consider the following infinite horizon optimal control problem. Given $x=(x_0,x_1)\in \mathbb{R}^{n}\times L^2$, we define a cost functional of the form $$\label{eq:obj-origbis} J(x;u(\cdot)) = \mathbb E\left[\int_0^{\infty} e^{-\rho t} l(y^{x,u}(t),u(t)) dt \right],$$ where $\rho>0$ is the discount factor, $l \colon \mathbb{R}^n \times U \to \mathbb{R}$ is the running cost and $U \subset \mathbb{R}^{p}$. For every reference probability space $\tau$ we consider the set of control processes $$\label{eq:controlsrps} \mathcal{U}_\tau=\{u(\cdot): \Omega\times [0,+\infty )\to U: \ u(\cdot) \ \mbox{is} \ \mathcal{F}_t\mbox{-progressively measurable} \}.$$ We define $$\label{eq:controlsrps1} \mathcal{U}=\bigcup_{\tau}\mathcal{U}_\tau,$$ where the union is taken over all reference probability spaces $\tau$. The goal is to minimize $J(x;u(\cdot))$ over all $u(\cdot)\in \mathcal{U}$. This is a standard setup of a stochastic optimal control problem (see [@yong_zhou; @fgs_book]) used to apply the dynamic programming approach. We remark (see e.g. [@fgs_book], Section 2.3.2) that $$\label{eq:equivrefprsp} \inf_{u(\cdot)\in \mathcal{ U}} J(x;u(\cdot))=\inf_{u(\cdot)\in \mathcal{U}_\tau} J(x;u(\cdot))$$ for every reference probability space $\tau$ so the optimal control problem is in fact independent of the choice of a reference probability space. We will also consider the optimal control problem using generalized reference probability spaces. For every generalized reference probability space $\tau$ the set of control processes are defined as in [\[eq:controlsrps\]](#eq:controlsrps){reference-type="eqref" reference="eq:controlsrps"} and are denoted by $\mathcal{\overline U}_\tau$ and we then define $$\mathcal{ \overline U}=\bigcup_{\tau}\mathcal{\overline U}_\tau,$$ where the union is taken over all generalized reference probability spaces $\tau$. We will assume the following conditions. **Assumption 1**. *The functions $b_0,\sigma_0$ are continuous and such that there exist constants $L,C>0$ such that, for every $x,x_1,x_2 \in \mathbb R^n,z,z_1,z_2 \in \mathbb{R}^{m}$ and every $u \in U$, $$\begin{aligned} & |b_0(x,z,u)| \leq C(1+|x|+|z|), \\ & |\sigma_0(x,z)| \leq C(1+|x|+|z|),\\ & |b_0(x_2,z_2,u)-b_0(x_1,z_1,u)| \leq L (|x_2-x_1|+|z_2-z_1|), \\ & |\sigma_0(x_2,z_2)-\sigma_0(x_1,z_1)| \leq L(|x_2-x_1|+|z_2-z_1|). \end{aligned}$$* Under Assumption [Assumption 1](#hp:state){reference-type="ref" reference="hp:state"}, by [@revuz Theorem IX.2.1], for each initial datum $x:=(x_0,x_1)\in \mathbb{R}^{n}\times L^2([-d,0];\mathbb R^n)$ and each control $u(\cdot)\in\ \mathcal{\overline U}$, there exists a unique (up to indistinguishability) strong solution to [\[eq:SDDE\]](#eq:SDDE){reference-type="eqref" reference="eq:SDDE"} and this solution admits a version with continuous paths that we denote by $y^{x;u}$. The proof that the assumptions of [@revuz Theorem IX.2.1] are satisfied can be found in [@tankov_federico Proposition 2.5]. **Assumption 2**. *$l \colon \mathbb{R}^n \times U \to \mathbb{R}$ is continuous and is such that the following hold.* - *There exist constants $K>0$, such that $$\label{eq:growth_l} |l(z,u)| \leq K(1+|z|) \ \ \ \forall y\in \mathbb{R}^n, \ \forall u \in U.$$* - *There exists $L>0$ such that $$\label{eq:regularity_l} |l(z,u)-l(z',u)| \leq L |z-z'| \quad \forall z,z' \in \mathbb{R}^n, u \in U.$$* In order for the cost functional to be well defined and continuous we will later assume (see Assumption [Assumption 9](#hp:discount){reference-type="ref" reference="hp:discount"}) that the discount factor $\rho>0$ is sufficiently large. Throughout the paper we will write $C>0,\omega, \omega_R$ to indicate, respectively, a constant, a modulus continuity, and a local modulus of continuity, which may change from place to place if the precise dependence on other data is not important. # Preliminary results {#sec:prelim} ## The equivalent infinite dimensional Markovian representation {#subsec:infinite_dimensional_framework} The optimal control problem we study is not Markovian due to the delay. As in [@defeo_federico_swiech] in order to regain Markovianity and approach the problem by dynamic programming, following a well-known procedure, see [@delfour Part II, Chapter 4] for deterministic delay equations and [@choj78], [@DZ14], [@goldys_1] for the stochastic case, we reformulate the state equation by lifting it to an infinite-dimensional space. We define $X := \mathbb{R}^n \times L^2$. An element $x\in X$ is a couple $x= (x_0,x_1)$, where $x_0 \in \mathbb{R}^n$, $x_{1}\in L^{2}$; sometimes, we will write $x=\begin{bmatrix} x_0\\ x_1 \end{bmatrix}.$ The space $X$ is a Hilbert space when endowed with the inner product $$\begin{aligned} \langle x,z\rangle_{X}& := x_0\cdot z_0 + \langle x_1,z_1 \rangle_{L^2}= x_0 z_0 + \int_{-d}^0 x_1(\xi)\cdot z_1(\xi) \,d\xi, \ \ \ x=(x_{0},x_{1}), \ z=(z_{0},z_{1})\in X.\end{aligned}$$ The induced norm, denoted by $|\cdot|_X$, is then $$|x|_{X} = \left( |x_0|^2 + \int_{-d}^0 |x_1(\xi)|_{L^{2}}^2\,d\xi\right)^{1/2}, \ \ \ x=(x_0,x_1) \in X.$$ For $R>0$, we denote $$B_R:=\{x \in X: |x|_{X} < R\}, \ \ \ B_R^0:=\{x_0 \in \mathbb R^n: |x_0| < R\}, \ \ \ B_R^1:=\{x_1 \in L^2[-d,0]: |x_1|_{L^{2}} < R\},$$ to be the open balls of radius $R$ in $X$, $\mathbb R^n,$ and $L^2$, respectively. We denote by $\mathcal{L}(X)$ the space of bounded linear operators from $X$ to $X$, endowed with the operator norm $$|L|_{\mathcal{L}(X)}=\sup_{|x|_{X}=1} |Lx|_{X}.$$ An operator $L \in \mathcal{L}(X)$ can be seen as $$Lx=\begin{bmatrix} L_{00} & L_{01}\\ L_{10} & L_{11} \end{bmatrix}\begin{bmatrix} x_0\\ x_1 \end{bmatrix}, \quad x=(x_0,x_1) \in X,$$ where $L_{00} \colon \mathbb R^n \to \mathbb R^n$, $L_{01} \colon L^2 \to \mathbb R^n$, $L_{10} \colon \mathbb R^n \to L^2$, $L_{00} \colon L^2 \to L^2$ are bounded linear operators. Moreover, given two separable Hilbert spaces $(H, \langle \cdot , \cdot \rangle_H), (K, \langle \cdot , \cdot \rangle_K)$, we denote by $\mathcal{L}_1(H,K)$ the space of trace-class operators endowed with the norm $$|L|_{\mathcal{L}_1(K,H)}=\inf \left \{\sum_{i \in \mathbb N} |a_i|_K |b_i|_{H}: Lx=\sum_{i \in \mathbb N} b_i \langle a_i,x \rangle, a_i \in K, b_i \in H , \forall i \in \mathbb N \right \}.$$ We also denote by $\mathcal{L}_2(H,K)$ the space of Hilbert-Schmidt operators from $H$ to $K$ endowed with the norm $$|L|_{\mathcal{L}_2(H,K)}=(\operatorname{Tr}(L^*L))^{1/2}.$$ When $H=K$ we simply write $\mathcal{L}_1(H)$, $\mathcal{L}_2(H)$. We denote by $S(H)$ the space of self-adjoint operators in $\mathcal{L}(H)$. If $Y,Z\in S(H)$, we write $Y\geq Z$ if $\langle Yx,x \rangle\leq \langle Zx,x \rangle$ for every $x \in H$. We now recall from [@defeo_federico_swiech Section 3]) how to rewrite the state equation [\[eq:SDDE\]](#eq:SDDE){reference-type="eqref" reference="eq:SDDE"} in the space $X$. In order to be consistent with [@defeo_federico_swiech] we use the same notation as in [@defeo_federico_swiech Section 3]. We define the linear unbounded operator $\tilde A \colon D(\tilde A) \subset X \to X$ by $$\label{eq:tildeA} \tilde A x= \begin{bmatrix} -x_0\\ x_1' \end{bmatrix}, \quad D(\tilde A)= \left\{ x=(x_0,x_1) \in X: x_1 \in W^{1,2}, \ x_1(0)=x_0\right\}.$$ **Proposition 3**. *([@defeo_federico_swiech Proposition 3.2])[\[prop:Amaxdiss\]]{#prop:Amaxdiss label="prop:Amaxdiss"} The operator $\tilde{A}$ defined in [\[eq:tildeA\]](#eq:tildeA){reference-type="eqref" reference="eq:tildeA"} is maximal dissipative.* Hence the operator $\tilde A$ is the generator of the so called delay semigroup $e^{t \tilde A}$ which is a strongly continuous semigroup of contractions on $X$.\ We define $\tilde b \colon X \times U \to X$ (with a small abuse of notation for $\tilde b_0(x,u)$) by $$\begin{aligned} \tilde b(x,u) & = \begin{bmatrix} \tilde b_0 \left ( x,u \right)\\ 0 \end{bmatrix} = \begin{bmatrix} b_0 \left ( x_0,\int_{-d}^0 a_1(\xi)x_1(\xi)\,d\xi ,u \right)+x_0\\ 0 \end{bmatrix}, \quad x=(x_0,x_1) \in X, \ u \in U\end{aligned}$$ and $\sigma \colon X \to \mathcal{L}(\mathbb{R}^q,X)$ (with a small abuse of notation for $\sigma_0(x)$) by $$\sigma(x)w=\begin{bmatrix} \sigma_0 ( x) w\\ 0 \end{bmatrix}= \begin{bmatrix} \sigma_0 \left ( x_0,\int_{-d}^0 a_2(\xi)x_1(\xi)\,d\xi \right)w\\ 0 \end{bmatrix},\quad x=(x_0,x_1) \in X, \ w \in \mathbb{R}^q.$$ We point out that we write $\tilde A$, $\tilde b$ to emphasize that these are translated versions of $A$, $b_0$. We also want to be consistent with the notation used in [@defeo_federico_swiech Section 3]). Given $x\in X$ and a control process $u(\cdot)\in\mathcal{\overline U}$, we consider the following infinite-dimensional stochastic differential equation $$\label{eq:abstract_dissipative_operator} dY(t) = [\tilde A Y(t)+\tilde b(Y(t),u(t))]dt + \sigma(Y(t))\,dW(t) \quad \forall t \geq 0 , \quad Y(0) = x.$$ As in [@defeo_federico_swiech Section 3] there exists a unique mild solution to [\[eq:abstract_dissipative_operator\]](#eq:abstract_dissipative_operator){reference-type="eqref" reference="eq:abstract_dissipative_operator"}, that is an $X$-valued progressively measurable stochastic process $Y$ satisfying $$\begin{aligned} Y(t)=e^{\tilde A t}x+ \int_0^t e^{\tilde A(t-s)}\tilde b(Y(s),u(s)) ds+ \int_0^t e^{\tilde A(t-s)}\sigma(Y(s)) dW(s), \ \ \ \forall t\geq 0.\end{aligned}$$ The infinite dimensional stochastic differential equation [\[eq:abstract_dissipative_operator\]](#eq:abstract_dissipative_operator){reference-type="eqref" reference="eq:abstract_dissipative_operator"} is linked to [\[eq:SDDE\]](#eq:SDDE){reference-type="eqref" reference="eq:SDDE"} by the following result, see [@tankov_federico Theorem 3.4] (cf. also the original result in the linear case [@choj78]). **Proposition 4**. *Let Assumption [Assumption 1](#hp:state){reference-type="ref" reference="hp:state"} hold. Given $x\in X$ and $u(\cdot)\in\mathcal{U}$, let $y^{x,u}$ be the unique strong solution to [\[eq:SDDE\]](#eq:SDDE){reference-type="eqref" reference="eq:SDDE"} and let $Y^{x,u}$ be the unique mild solution to [\[eq:abstract_dissipative_operator\]](#eq:abstract_dissipative_operator){reference-type="eqref" reference="eq:abstract_dissipative_operator"}. Then $$Y^{x,u}(t)=(y^{x,u}(t),y^{x,u}(t+\cdot)|_{[-d,0]}), \ \ \ \forall t\geq 0.$$* Proposition [\[prop:equiv\]](#prop:equiv){reference-type="eqref" reference="prop:equiv"} provides a Markovian reformulation of the optimal control problem in the Hilbert space $X$. Indeed, the cost functional [\[eq:obj-origbis\]](#eq:obj-origbis){reference-type="eqref" reference="eq:obj-origbis"} can be rewritten in $X$ as $$\label{ex1jbisbis} J(x;u(\cdot)) = \mathbb E \left[ \int_0^{\infty} e^{-\rho t}[L(Y^{x,u}(t),u(t))]dt \right],$$ where $L : X \times U\to\mathbb{R}$ is defined by $$L(x,u) := l(x_0,u), \quad x=(x_0,x_1) \in X, \ u \in U.$$ The value function $V \colon X \rightarrow \mathbb R$ for the optimal control problem in the reference probability space formulation is defined by $$V(x) := \inf_{u(\cdot)\in\mathcal{U}} J(x;u(\cdot)).$$ We also define the value function $\overline V \colon X \rightarrow \mathbb R$ for the optimal control problem in the generalized reference probability space formulation $$\overline V(x) := \inf_{u(\cdot)\in\mathcal{\tilde U}} J(x;u(\cdot)).$$ We will later see in Proposition [Proposition 29](#prop:V_bar_equal_V){reference-type="ref" reference="prop:V_bar_equal_V"} that, under proper conditions, $\overline V=V.$ ## Operator $B$ and space $X_{-1}$. {#sec:operator_B} In this subsection, following [@defeo_federico_swiech Section 3], we introduce the operator $B$ and the so-called weak $B$-condition for $\tilde{A}$. First note that, as in [@defeo_federico_swiech Section 3], the adjoint operator $\tilde{A}^{*}:D(\tilde A^*)\subset X\to X$ is given by $$\begin{aligned} \tilde A^* x = \begin{bmatrix} x_1(0)-x_0\\ -x_1' \end{bmatrix}, \quad D(\tilde A^*) = \left\{ x=(x_0,x_1) \in X: x_1 \in W^{1,2}([-d,0];\mathbb{R}^n), \ x_1(-d)=0\right\}.\end{aligned}$$ **Definition 5**. *(See [@fgs_book Definition 3.9]) Let $B \in \mathcal{L}(X)$. We say that $\tilde{A}$ satisfies the weak $B$-condition if the following hold:* (i) *$B$ is strictly positive, i.e. $\langle Bx,x \rangle_{X} >0$ for every $x \neq 0$;* (ii) *$B$ is self-adjoint;* (iii) *$\tilde{A}^* B \in \mathcal{L}(X)$;* (iv) *There exists $C_0 \geq 0$ such that $\langle \tilde{A}^* Bx,x \rangle_{X} \leq C_0 \langle Bx,x \rangle_{X},$ $\forall x \in X.$* Let $\tilde{A}^{-1}$ be the inverse of the operator $\tilde{A}$. As in [@defeo_federico_swiech Section 3] its explicit expression is given by $$\label{eq:tilde_A_inverse} \tilde{A}^{-1}x=\left(-x_0, -x_0 -\int_\cdot^0 x_1(\xi)d\xi\right), \quad \forall x=(x_0,x_1)\in X.$$ Notice that $\tilde A^{-1} \in \mathcal{L}(X)$. Moreover, since $\tilde A^{-1}$ is continuous as an operator from $X$ to $W^{1,2}$, and the embedding $W^{1,2} \hookrightarrow L^{2}$ is compact, $\tilde A^{-1}: L^2\to L^2$ is compact. Define now $$\label{def:B} B:=(\tilde{A}^{-1})^*\tilde{A}^{-1}=(\tilde{A}^*)^{-1}\tilde{A}^{-1}\in\mathcal{L}(X).$$ $B$ is compact by the compactness of $\tilde A^{-1}$. **Proposition 6**. *( [@defeo_federico_swiech Proposition 3.4])[\[prop:weak_b\]]{#prop:weak_b label="prop:weak_b"} Let $B$ be defined by [\[def:B\]](#def:B){reference-type="eqref" reference="def:B"}. Then $\tilde{A}$ satisfies the weak $B$-condition with $C_0=0$.* Observe that if we write $$\begin{aligned} \label{eq:representation_B} B x= \begin{bmatrix} B_{00} & B_{01} \\ B_{10} & B_{11} \end{bmatrix} \begin{bmatrix} x_0 \\ x_1 \end{bmatrix}, \ \ \ \ x=(x_0,x_1)\in X, \end{aligned}$$ by the strict positivity of $B$, $B_{00}\in M^{n\times n}$ is strictly positive and $B_{11}$ is strictly positive as an operator from $L^2$ to $L^2$. Moreover, since $B$ is strictly positive and self-adjoint, the operator $B^{1/2}\in \mathcal{L}(X)$ is well defined, self-adjoint and strictly positive. We introduce the $|\cdot |_{-1}$-norm on $X$ by $$\begin{aligned} \label{eq:properties_norm_-1} |x|_{-1}^2&=\langle B^{1/2}x, B^{1/2}x\rangle_{X}= \langle Bx,x\rangle_{X}=\langle (\tilde{A}^{-1})^*\tilde{A}^{-1}x,x\rangle_{X}=\langle \tilde{A}^{-1}x,\tilde{A}^{-1}x\rangle_{X}= |\tilde A^{-1}x|^{2}_{X} \quad \forall x \in X. \end{aligned}$$ We define $$X_{-1}:= \ \mbox{ the completion of $X$ under} \ |\cdot|_{-1},$$ which is a Hilbert space endowed with the inner product $$\langle x,y\rangle_{-1}:=\langle B^{1/2}x,B^{1/2}y\rangle_{X}= \langle Bx,y\rangle_{X}= \langle \tilde{A}^{-1}x,\tilde A^{-1}y\rangle_{X}.$$ Notice that $|x|_{-1}\leq |\tilde A^{-1}|_{\mathcal L(X)}|x|_X$; in particular, we have $(X,|\cdot|) \hookrightarrow (X_{-1},|\cdot|_{-1})$. Moreover, strict positivity of $B$ ensures that the operator $B^{1 /2}$ can be extended to an isometry $B^{1 /2} \colon (X_{- 1},|\cdot|_{-1}) \to (X,|\cdot|_{X}).$ By [\[eq:properties_norm\_-1\]](#eq:properties_norm_-1){reference-type="eqref" reference="eq:properties_norm_-1"} and an application of [@DZ14 Proposition B.1], we have $\mbox{Range}(B^{1/2})=\mbox{Range}((\tilde A^{-1})^*)$. Since $\mbox{Range}((\tilde A^{-1})^*)=D(\tilde A^*)$, we have $$\begin{aligned} \label{eq:R(B^1/2)=D(A^*)} \mbox{Range}\big(B^{1/2}\big)=D(\tilde A^*).\end{aligned}$$ By [\[eq:R(B\^1/2)=D(A\^\*)\]](#eq:R(B^1/2)=D(A^*)){reference-type="eqref" reference="eq:R(B^1/2)=D(A^*)"}, the operator $\tilde A^* B^{1/2}$ is well defined on the whole space $X$. Moreover, since $\tilde A^*$ is closed and $B^{1/2}\in\mathcal{L}(X)$, $\tilde A^* B^{1/2}$ is a closed operator. Thus, by the closed graph theorem, we have $$\label{eq:A*B^1/2} \tilde A^* B^{1/2} \in \mathcal{L}(X).$$ By [\[eq:tilde_A\_inverse\]](#eq:tilde_A_inverse){reference-type="eqref" reference="eq:tilde_A_inverse"}, we immediately notice that $$\label{eq:|x_0|_leq_|x|} |x_0|\leq |x|_{-1}, \quad \forall x=(x_0,x_1)\in X.$$ Since $B$ is a compact, self-adjoint and strictly positive operator on $X$, by the spectral theorem $B$ admits a set of eigenvalues $\{\lambda_i \}_{i \in \mathbb{N}} \subset (0, +\infty)$ such that $\lambda_{i}\to 0^{+}$ and a corresponding set $\{f_i \}_{i \in \mathbb{N}} \subset X$ of eigenvectors forming an orthonormal basis of $X$. By taking $\{e_i \}_{i \in \mathbb{N}}$ defined by $e_i:=\frac{1 } {\sqrt \lambda_i} f_i$, we then get an orthonormal basis of $X_{-1}$. We set $X^N := \mbox{span} \{f_1,...f_N\}= \mbox{span} \{e_1,...e_N\}$ for $N\geq 1$, and let $P_N \colon X \to X$ be the orthogonal projection onto $X_N$ and $Q_N := I - P_N$. Since $\{e_i \}_{i \in \mathbb{N}}$ is an orthogonal basis of $X_{-1}$, the projections $P_N,Q_N$ extend to orthogonal projections in $X_{-1}$ and we will use the same symbols to denote them. We notice that $$\label{eq:BP_BQ} B P_N =P_N B, \quad B Q_N =Q_N B.$$ Therefore, since $|BQ_N|_{\mathcal L(X)}=|Q_NB|_{\mathcal L(X)}$ and $B$ is compact, we get $$\label{eq:norm_BQ_N_to_zero} \lim_{N \to \infty }|BQ_N|_{\mathcal L(X)} =0.$$ ## Estimates for the state equation and the value function {#subsec:estimates} In this subsection we recall from [@defeo_federico_swiech] estimates for solutions of the state equation, the cost functional and the value function. These results will be needed in the paper. **Lemma 7**. *([@defeo_federico_swiech Lemma 4.1]) Let Assumptions [Assumption 1](#hp:state){reference-type="ref" reference="hp:state"} and [Assumption 2](#hp:cost){reference-type="ref" reference="hp:cost"} hold. There exists $C>0$ and a local modulus of continuity $\omega$ such that the following hold true for every $x,y \in X, u \in U$:* *$$\begin{aligned} & |\tilde b(x, u)-\tilde b(y, u)|_X \leq C|x-y|_{-1} \label{eq:b_lip}, \\ & \langle \tilde b(x, u)-\tilde b(y, u), B(x-y)\rangle_{X} \leq C|x-y|_{-1}^{2}, \label{eq:b_B_property} \\ & |\tilde b(x, u)| \leq C(1+|x|_{X}) , \label{eq:b_sublinear} \\ %& |\sigma(y,u)-\sigma(x,u)|_{\mathcal{L}_{2}(X)} \leq C |x-y|,\label{eq:G_lipschitz} \\ & |\sigma(y)-\sigma(x)|_{\mathcal{L}_{2}(X)} \leq C |x-y|_{-1}, \label{eq:G_lipschitz_norm_B} \\ & |\sigma(x)|_{\mathcal{L}_{2}} \leq C (1+|x|_{X}), \label{eq:G_bounded} \\ %&|L(x, u)-L(y, u)| \leq C|x-y|, \label{eq:L_unif_cont} \\ &|L(x, u)-L(y, u)| \leq C|x-y|_{-1}, \label{eq:L_unif_cont_norm_B} \\ & |L(x, u)| \leq C\left(1+|x|_X\right). \label{eq:L_growth} \end{aligned}$$* *Moreover, $$\label{eq:trace_goes_to_zero} \lim_{N \to \infty} \sup_{u \in U} \operatorname{Tr}\left[\sigma(x) \sigma(x)^{*} B Q_{N}\right]=0, \ \ \ \forall x\in X.$$* Set $$\rho_0 =\max \left \{C+\frac{ C^2}{2}, C+\frac{ C^2|B|_{\mathcal{L}(X)}}{2} \right \} ,$$ where $C$ is the constant from [\[eq:b_B\_property\]](#eq:b_B_property){reference-type="eqref" reference="eq:b_B_property"} and [\[eq:G_lipschitz_norm_B\]](#eq:G_lipschitz_norm_B){reference-type="eqref" reference="eq:G_lipschitz_norm_B"}. **Proposition 8**. *([@fgs_book Proposition 3.24]) Let Assumption [Assumption 1](#hp:state){reference-type="ref" reference="hp:state"} hold and let $\lambda>\rho_0$. Let $Y(t)$ be the mild solution of [\[eq:abstract_dissipative_operator\]](#eq:abstract_dissipative_operator){reference-type="eqref" reference="eq:abstract_dissipative_operator"} with initial datum $x \in X$ and control $u(\cdot) \in \mathcal U$. Then there exists $C_\lambda>0$ such that $$\mathbb{E}\left[|Y(t)|_X\right] \leq C_\lambda\left(1+|x|_X\right)e^{\lambda t}, \quad \forall t \geq 0.$$* We remark that only $\lambda>C+\frac{ C^2}{2}$ is needed in Proposition [Proposition 8](#prop:expect_Y(s)){reference-type="ref" reference="prop:expect_Y(s)"}. The second restriction for $\rho_0$ is necessary to obtain Proposition [Proposition 12](#prop:V_continuous_norm_-1){reference-type="ref" reference="prop:V_continuous_norm_-1"}. We need the following assumption. **Assumption 9**. *$\rho >\rho_{0}.$* **Proposition 10**. *([@defeo_federico_swiech Proposition 4.4].)[\[prop:growth_V\]]{#prop:growth_V label="prop:growth_V"} Let Assumptions [Assumption 1](#hp:state){reference-type="ref" reference="hp:state"}, [Assumption 2](#hp:cost){reference-type="ref" reference="hp:cost"}, and [Assumption 9](#hp:discount){reference-type="ref" reference="hp:discount"} hold. There exists $\bar C>0$ such that $$|J(x;u(\cdot))|\leq \bar C(1+|x|_X) \quad \forall x \in X, \ \forall u(\cdot) \in \mathcal{U}.$$ Hence, $$|V(x)|\leq \bar C(1+|x|_X), \ \ \ \forall x \in X.$$* We now recall the notion of $B$-continuity (see [@fgs_book Definition 3.4]). **Definition 11**. *Let $B \in\mathcal{L}(X)$ be a strictly positive self-adjoint operator. A function $u: X \rightarrow \mathbb{R}$ is said to be $B$-upper semicontinuous (respectively, $B$-lower semicontinuous) if, for any sequence $\left\{ x_{n}\right\}_{n \in \mathbb{N}}\subset X$ such that $x_{n} \rightharpoonup x \in X$ and $B x_{n} \rightarrow B x$ as $n \rightarrow \infty$, we have $$\limsup _{n \rightarrow \infty} u\left( x_{n}\right) \leq u(x) \ \ \ \mbox{ (respectively, } \ \liminf _{n \rightarrow \infty} u\left( x_{n}\right) \geq u(x)).$$ A function $u: X \rightarrow \mathbb{R}$ is said to be $B$-continuous if it is both $B$-upper semicontinuous and $B$-lower semicontinuous.* We remark that, since the operator $B$ defined in [\[def:B\]](#def:B){reference-type="eqref" reference="def:B"} is compact, in our case $B$-upper/lower semicontinuity is equivalent to the weak sequential upper/lower semicontinuity, respectively. The next proposition is proved in [@defeo_federico_swiech Example 6.2]. **Proposition 12**. *Let Assumptions [Assumption 1](#hp:state){reference-type="ref" reference="hp:state"}, [Assumption 2](#hp:cost){reference-type="ref" reference="hp:cost"}, and [Assumption 9](#hp:discount){reference-type="ref" reference="hp:discount"} hold. There exists $K>0$ such that $$\label{Va-1} |V(x)-V(y)| \leq K|x-y|_{-1} , \ \ \ \forall x,y \in X.$$ Hence $V$ is $B$-continuous and thus weakly sequentially continuous.* ## HJB equation: Viscosity solutions {#subsec:viscosity} In this subsection we recall the characterization of $V$ as the unique $B$-continuous viscosity solution to the associated HJB equation that was obtained in [@defeo_federico_swiech]. Given $v \in C^1(X)$, we denote by $D v(x)$ its Fréchet derivative at $x \in X$ and we write $$\begin{aligned} Dv(x)= \begin{bmatrix} D_{x_0}v(x) \\ D_{x_1}v(x) \end{bmatrix},\end{aligned}$$ where $D_{x_0}v(x), D_{x_1}v(x)$ are the partial Fréchet derivatives. For $v \in C^2(X)$, we denote by $D^2 v(x)$ its second order Fréchet derivative at $x \in X$ which we will often write as $$D^2v(x)= \begin{bmatrix} D^2_{x_0^2}v(x) & D^2_{x_0x_1}v(x) \\ D^2_{x_1x_0}v(x) & D^2_{x_1^2}v(x) \end{bmatrix}.$$ We define the Hamiltonian function $H:X\times X\times S(X)\to \mathbb{R}$ by $$\begin{aligned} \label{eq:hamiltonian_synthesis} H(x,p,Z) & = H(x,p) - {1\over 2} \mathop{\mathrm{Tr}}\nolimits\left [ \sigma \left ( x\right) \sigma \left ( x \right) ^* Z \right] \nonumber \\ &=\tilde H \left (x,p_0 \right) - {1\over 2} \mathop{\mathrm{Tr}}\nolimits\left [ \sigma_0 \left ( x_0,\int_{-d}^0 a_2(\xi)x_1(\xi)\,d\xi \right)\sigma_0 \left ( x_0,\int_{-d}^0 a_2(\xi)x_1(\xi)\,d\xi\right)^T Z_{00} \right] \nonumber \\ &=:\tilde H \left (x,p_0 ,Z_{00} \right),\end{aligned}$$ where $$\begin{aligned} \label{eq:hamiltonian_synthesis_no_sigma} H(x,p)=\tilde H \left (x,p_0 \right)&=\sup_{u\in U} \Bigg \{ - \tilde b_0\left ( x ,u \right) \cdot D_{x_0}v(x) - l(x_0,u) \Bigg \} \nonumber \\ &= - x_0 \cdot p_0 + \sup_{u\in U} \Bigg \{ - b_0\left ( x_0,\int_{-d}^0 a_1(\xi)x_1(\xi)\,d\xi ,u \right) \cdot p_0 - l(x_0,u) \Bigg \}. \end{aligned}$$ The Hamiltonian $H$ satisfies the following properties. **Lemma 13**. *([@fgs_book Theorem 3.75]) Let Assumptions [Assumption 1](#hp:state){reference-type="ref" reference="hp:state"} and [Assumption 2](#hp:cost){reference-type="ref" reference="hp:cost"} hold.* (i) *$H$ is uniformly continuous on bounded subsets of $X \times X \times S(X)$.* (ii) *For every $x,p \in X$ and every $Y,Z \in S(X)$ such that $Z \leq Y$, we have $$\label{eq:H_decreasing} H(x,p,Y)\leq H(x,p,Z).$$* (iii) *For every $x,p \in X$ and every $R>0$, we have $$\begin{aligned} \label{eq:H_lambda_BQ_N} \lim_{N \to \infty} \sup \Big \{ |H(x,p,Z+\lambda BQ_N)-H(x,p,Z)|: \ |Z_{00}|\leq R, \ |\lambda| \leq R \Big \}=0.\end{aligned}$$* (iv) *There exists a modulus of continuity $C>0$ such that $$\begin{aligned} \label{eq:H_norm_-1} H \left (z,\frac{B(z-y)}{\varepsilon},Z \right )-H \left (y,\frac{B(z-y)}{\varepsilon},Y \right ) \geq -C\left (|z-y|_{-1}\left(1+\frac{|z-y|_{-1}}{\varepsilon}\right) \right) \end{aligned}$$ for every $\varepsilon>0$, $y,z \in X$, and $Y,Z \in \mathcal{S}(X)$ satisfying $$Y=P_N Y P_N \quad Z=P_N Z P_N$$ and $$\begin{aligned} \frac{3}{\varepsilon}\left(\begin{array}{cc}B P_{N} & 0 \\ 0 & B P_{N}\end{array}\right) \leq\left(\begin{array}{cc}Y & 0 \\ 0 & -Z\end{array}\right) \leq \frac{3}{\varepsilon}\left(\begin{array}{cc}B P_{N} & -B P_{N} \\ -B P_{N} & B P_{N}\end{array}\right).\end{aligned}$$* (v) *If $C>0$ is the constant in [\[eq:b_sublinear\]](#eq:b_sublinear){reference-type="eqref" reference="eq:b_sublinear"} and [\[eq:G_bounded\]](#eq:G_bounded){reference-type="eqref" reference="eq:G_bounded"}, then, for every $x \in X, p, q \in X, Y,Z \in \mathcal{S}(X)$, $$\begin{aligned} \label{eq:Hamiltonian_local_lip} | H(x, p+q,Y+Z)-H(x, p, Y)| \leq C\left(1+|x|_X\right)|q_0|_X+\frac{1}{2}C^2\left(1+|x|_X\right)^{2}|Z_{00}|.\end{aligned}$$* The Hamilton-Jacobi-Bellman (HJB) equation associated with the optimal control problem is the infinite dimensional PDE $$\label{eq:HJB} %\left\{\begin{array}{l} \rho v(x) - \langle\tilde A x,Dv(x)\rangle + H(x,Dv(x),D^2v(x))=0, \quad x \in X. %0 \leq t \leq T\\[8pt] %v(T) = \varphi, %end{array}\right.$$ We recall the definition of $B$-continuous viscosity solution from [@fgs_book]. **Definition 14**. - *$\phi \colon X \to \mathbb R$ is a regular test function if* *$$\begin{aligned} \phi \in \Phi := \{ \phi \in C^2(X): \phi \textit{ is weakly sequentially lower semicontinuous and } \\ \phi, D\phi , D^2\phi , A^* D\phi \textit{ are uniformly continuous on }X\};\end{aligned}$$* - *$g \colon X \to \mathbb R$ is a radial test function if $$\begin{aligned} g \in \mathcal G:= \{ g \in C^2(X): g(x)=g_0(|x|_X) \textit{ for some } g_0 \in C^2([0,\infty)) \textit{ non-decreasing}, g_{0}'(0)=0 \}.\end{aligned}$$* Note that, if $g\in\mathcal{G}$, we have $$\begin{aligned} \label{eq:gradient_radial} D g(x)=\left\{\begin{array}{l} g_0^{\prime}(|x|_{X}) \frac{x}{|x|_{X}}, \quad \ \ \ \mbox{if} \ x \neq 0, \\ 0, \quad\quad \ \ \ \ \ \ \ \ \ \ \ \ \,\,\, \ \mbox{if} \ x=0. \end{array}\right.\end{aligned}$$ We say that a function is locally bounded if it is bounded on bounded subsets of $X$. **Definition 15**. (i) *A locally bounded weakly sequentially upper semicontinuous function $v:X\to\mathbb{R}$ is a viscosity subsolution of [\[eq:HJB\]](#eq:HJB){reference-type="eqref" reference="eq:HJB"} if, whenever $v-\phi-g$ has a local maximum at $x \in X$ for $\phi \in \Phi, g \in \mathcal G$, then $$\rho v(x) - \langle x,\tilde A^* D\phi(x)\rangle_{X} + H(x,D\phi(x)+Dg(x) ,D^2\phi(x)+D^2g(x))\leq 0.$$* (ii) *A locally bounded weakly sequentially lower semicontinuous function $v:X\to\mathbb{R}$ is a viscosity supersolution of [\[eq:HJB\]](#eq:HJB){reference-type="eqref" reference="eq:HJB"} if, whenever $v+\phi+g$ has a local minimum at $x \in X$ for $\phi \in \Phi$, $g \in \mathcal G$, then $$\rho v(x) + \langle x,\tilde A^* D\phi(x)\rangle_{X} + H(x,-D\phi(x)-Dg(x) ,-D^2\phi(x)-D^2g(x))\geq 0.$$* (iii) *A viscosity solution of [\[eq:HJB\]](#eq:HJB){reference-type="eqref" reference="eq:HJB"} is a function $v:X\to\mathbb{R}$ which is both a viscosity subsolution and a viscosity supersolution of [\[eq:HJB\]](#eq:HJB){reference-type="eqref" reference="eq:HJB"}.* Define $\mathcal{S}:=\{u \colon X \to \mathbb{R}: \exists k\geq 0 \ \mbox{satisfying \eqref{eq:k_set_uniqueness} and } \tilde C\geq 0 \,\mbox{such that}\, |u(x)|\leq \tilde C (1+|x|_X^k)\},$ where $$\label{eq:k_set_uniqueness} \begin{cases} k<\frac{\rho}{C+\frac{1}{2} C^{2}}, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \quad \mbox{ if } \ \frac{\rho}{C+\frac{1}{2} C^{2}} \leq 2, \\ C k+\frac{1}{2} C^{2} k(k-1)<\rho, \quad \mbox{ if } \frac{\rho}{C+\frac{1}{2} C^{2}}>2, \end{cases}$$ and $C$ is the constant appearing in [\[eq:b_sublinear\]](#eq:b_sublinear){reference-type="eqref" reference="eq:b_sublinear"} and [\[eq:G_bounded\]](#eq:G_bounded){reference-type="eqref" reference="eq:G_bounded"}. It was proved in [@defeo_federico_swiech] that $V$ is the unique viscosity solution of [\[eq:HJB\]](#eq:HJB){reference-type="eqref" reference="eq:HJB"} in $\mathcal{S}$. **Theorem 16**. *([@defeo_federico_swiech Theorem 5.4])[\[th:existence_uniqueness_viscosity_infinite\]]{#th:existence_uniqueness_viscosity_infinite label="th:existence_uniqueness_viscosity_infinite"} Let Assumptions [Assumption 1](#hp:state){reference-type="ref" reference="hp:state"}, [Assumption 2](#hp:cost){reference-type="ref" reference="hp:cost"}, and [Assumption 9](#hp:discount){reference-type="ref" reference="hp:discount"} hold. The value function $V$ is the unique viscosity solution of [\[eq:HJB\]](#eq:HJB){reference-type="eqref" reference="eq:HJB"} in the set $\mathcal{S}$.* ## Partial regularity of $V$ {#subsec:regularity} In this subsection we recall the partial regularity result for $V$ with respect to the $x_0$-variable which was obtained in [@defeo_federico_swiech]. **Assumption 17**. *For every $R>0$ there exists $\lambda_{R} >0$ such that $$\sigma_0(x) \sigma_0(x)^T\geq \lambda_{R} I, \quad \forall x\,\ \mbox{such that}\, \ |x|_{X} \leq R.$$* For every $\bar x_1 \in L^2$ we define $$V^{\bar x_1}(x_0):=V(x_0,\bar x_1), \quad \forall x_0 \in \mathbb R^n.$$ **Theorem 18**. *([@defeo_federico_swiech Theorem 6.5])[\[th:C1alpha\]]{#th:C1alpha label="th:C1alpha"} Let Assumptions [Assumption 1](#hp:state){reference-type="ref" reference="hp:state"}, [Assumption 2](#hp:cost){reference-type="ref" reference="hp:cost"}, [Assumption 9](#hp:discount){reference-type="ref" reference="hp:discount"}, and [Assumption 17](#hp:uniform_ellipticity){reference-type="ref" reference="hp:uniform_ellipticity"} hold. For every $p>n$ and every fixed $\bar x_1 \in L^2$, we have $V^{\bar x_1}\in W^{2,p}_{\rm loc}(\mathbb R^n)$; thus, by Sobolev embedding, $V^{\bar x_1}\in C^{1,\alpha}_{\rm loc}({\mathbb{R}}^n)$ for all $0<\alpha<1$. Moreover, for every $R>0$, there exists $C_{R}>0$ such that $$|V^{\bar x_1}|_{W^{2,p}(B_{R})}\leq C_R, \ \ \ \ \forall \bar x_{1}\ \mbox{such that}\, \ |\bar x_1|_{L^{2}}\leq R.$$ Finally, $D_{x_0}V$ is continuous with respect to the $|\cdot |_{-1}$ norm on bounded sets of $X$. In particular, $D_{x_0}V$ is continuous in $X$.* # Approximations by inf-convolutions {#sec:approximation via inf-convolutions} In this section we begin the process of approximating the value function $V$ by more regular functions. The first step is to use an appropriately defined inf-convolution $V_\epsilon$ of $V$ and prove that $V_\epsilon$ is a viscosity super-solution of a perturbed HJB equation. To do this we need one more assumption about $V$. We extend $V$ to the function $\tilde V:X_{-1}\to\mathbb{R}$ which then also satisfies [\[Va-1\]](#Va-1){reference-type="eqref" reference="Va-1"} for all $x,y\in X_{-1}$. **Assumption 19**. *$V$ is $|\cdot|_{-1}$-semiconvex, i.e. there exists $C\geq 0$ (called a semiconvexity constant) such that $V(x)+C|x|^2_{-1}$ is convex.* The $|\cdot|_{-1}$-semiconvexity is equivalent to the requirement that here exists $C\geq 0$ such that $$\label{eq:-1semiconvex} \lambda V(x) +(1-\lambda) V(y)- V(\lambda x + (1-\lambda)y) \geq -C|x|^2_{-1}\quad \forall \lambda \in [0,1], x,y \in X.$$ If $V$ satisfies [\[eq:-1semiconvex\]](#eq:-1semiconvex){reference-type="eqref" reference="eq:-1semiconvex"} then the function $\tilde V$ is semiconvex in $X_{-1}$ and it satisfies [\[eq:-1semiconvex\]](#eq:-1semiconvex){reference-type="eqref" reference="eq:-1semiconvex"} for all $x,y\in X_{-1}$. We say that a function $f$ is $|\cdot|_{-1}$-semiconcave if there is $C\geq 0$ such that $f(x)-C|x|^2_{-1}$ is concave. It is rather well known that the semiconcavity of $V$ can be obtained under standard hypotheses on the data, e.g. see [@yong_zhou], [@defeo_swiech_wessels] for the finite and the infinite-dimensional cases respectively. Here instead we require the $|\cdot|_{-1}$- semiconvexity of $V$. We provide two examples where $V$ is convex in the spirit of [@goldys_1], [@goldys_2], where the authors prove the concavity of $V$ for a maximization problem (which corresponds to the convexity for a minimization problem considered here). We remark that in the approximation procedure of Sections [4](#sec:approximation via inf-convolutions){reference-type="ref" reference="sec:approximation via inf-convolutions"} and [6](#sec:lions_approx){reference-type="ref" reference="sec:lions_approx"}, $V$ can be replaced by any viscosity supersolution of [\[eq:HJB\]](#eq:HJB){reference-type="eqref" reference="eq:HJB"} which satisfies [\[Va-1\]](#Va-1){reference-type="eqref" reference="Va-1"}, the regularity properties of Theorem [\[th:C1alpha\]](#th:C1alpha){reference-type="ref" reference="th:C1alpha"} and Assumption [Assumption 19](#hp:convexity){reference-type="ref" reference="hp:convexity"}. **Example 20**. *We assume that $n=h=q=1$ for simplicity but this argument can be extended to bigger dimensions. Let Assumptions [Assumption 1](#hp:state){reference-type="ref" reference="hp:state"} and [Assumption 2](#hp:cost){reference-type="ref" reference="hp:cost"} hold. Assume that $U$ is convex, $a_1(\xi)\geq 0$ for every $\xi \in [-d,0]$, $b_0(x,z,u)$ is jointly concave and non-decreasing with respect to the second variable, $\sigma_0(x,z,u)=\sigma_0>0$ and that $l(x,u)$ is jointly convex and non-increasing with respect to $x \in \mathbb{R}$ for fixed $u \in U$. We show that under these hypotheses $V$ is convex.* *Indeed, let $x, \bar x \in X$, $\lambda \in [0,1]$, $\epsilon>0$ and $u^\epsilon(\cdot), \bar u^\epsilon(\cdot)$ be $\epsilon$-optimal controls for the initial conditions $x, \bar x$ respectively. By [\[eq:equivrefprsp\]](#eq:equivrefprsp){reference-type="eqref" reference="eq:equivrefprsp"} we can assume that the control processes are defined on the same reference probability space. Denote by $x(t), \bar x(t)$ the solutions of [\[eq:SDDE\]](#eq:SDDE){reference-type="eqref" reference="eq:SDDE"} with initial state $x, \bar x$ and controls $u^\epsilon(\cdot), \bar u^\epsilon(\cdot)$ respectively. Moreover, set $x_\lambda=\lambda x + (1-\lambda) \bar x$, $u^\epsilon_\lambda(\cdot)=\lambda u^\epsilon(\cdot) + (1-\lambda) \bar u^\epsilon(\cdot)$ and let $x(t;x_\lambda, u^\epsilon_\lambda(\cdot))$ be the solution of [\[eq:SDDE\]](#eq:SDDE){reference-type="eqref" reference="eq:SDDE"} with the initial state $x_\lambda$ and control $u^\epsilon_\lambda(\cdot)$. Finally, set $x_\lambda(t)=\lambda x(t)+ (1-\lambda) \bar x(t)$. Note that since [\[eq:SDDE\]](#eq:SDDE){reference-type="eqref" reference="eq:SDDE"} is not linear $x(t,x_\lambda, u^\epsilon_\lambda(\cdot)) \neq x_\lambda(t)$ in general. First, since $b_0(x,y,u)$ is jointly concave, we have* *$$\begin{aligned} dx_\lambda(t)& = \lambda dx(t)+(1-\lambda) d\bar x(t)\\ &=\left [\lambda b_0 \left ( x(t),\int_{-d}^0 a_1(\xi)x(t+\xi)\,d\xi ,u^\epsilon(t) \right) +(1-\lambda) b_0 \left ( \bar x(t),\int_{-d}^0 a_1(\xi)\bar x(t+\xi)\,d\xi ,\bar u^\epsilon(t) \right) \right] dt+ \sigma_0 dW(t)\\ & \leq b_0 \left ( x_\lambda(t),\int_{-d}^0 a_1(\xi)x_\lambda(t+\xi)\,d\xi ,u^\epsilon_\lambda(t) \right) dt+ \sigma_0 dW(t).\end{aligned}$$* *Regarding $x(t;x_\lambda, u_\lambda^\epsilon(\cdot))$, of course we have $$\begin{aligned} d x(t;x_\lambda, u_\lambda^\epsilon(\cdot)) = b_0 \left ( x(t;x_\lambda, u_\lambda^\epsilon(\cdot)),\int_{-d}^0 a_1(\xi)x(t+\xi;x_\lambda, u_\lambda^\epsilon(\cdot))d\xi ,u^\epsilon_\lambda(t) \right) dt+ \sigma_0 dW(t).\end{aligned}$$ Thus, by the comparison lemma, Lemma [Lemma 35](#lemma:comparison_sdde){reference-type="ref" reference="lemma:comparison_sdde"}, we have $$\label{eq:x_lambda leq x} x_\lambda(t) \leq x(t;x_\lambda, u_\lambda^\epsilon(\cdot)) \quad \forall t \geq 0.$$ Finally, by [\[eq:x_lambda leq x\]](#eq:x_lambda leq x){reference-type="eqref" reference="eq:x_lambda leq x"}, the fact that $l(\cdot,u)$ is non-increasing, the joint convexity of $l(x,u)$ and since $u^\epsilon(\cdot), \bar u^\epsilon(\cdot)$ are $\epsilon$-optimal controls for the initial states $x, \bar x$ respectively, we have $$\begin{aligned} V(x_\lambda) \leq J(x_\lambda; u^\epsilon_\lambda)& = \mathbb E\left[ \int_0^{+\infty} e^{-\rho t}l(x(t;x_\lambda, u_\lambda^\epsilon(\cdot)),u^\epsilon_\lambda(t))dt \right] \leq \mathbb E\left[ \int_0^{+\infty} e^{-\rho t}l(x_\lambda(t),u_\lambda^\epsilon(t))dt \right]\\ & \leq \mathbb E\left[ \int_0^{+\infty} e^{-\rho t}[\lambda l(x(t),u^\epsilon(t))+(1-\lambda) l(\bar x(t),\bar u^\epsilon(t))]dt \right] \\ & = \lambda J(x;u^\epsilon(\cdot)) +(1-\lambda) J(\bar x; \bar u^\epsilon(\cdot)) \leq \lambda V(x)+ (1-\lambda) V(\bar x) + \epsilon\end{aligned}$$ so that by letting $\epsilon \to 0$ we obtain the convexity of $V$.* **Example 21**. *Again, we assume that $n=h=q=1$ for simplicity and that Assumptions [Assumption 1](#hp:state){reference-type="ref" reference="hp:state"} and [Assumption 2](#hp:cost){reference-type="ref" reference="hp:cost"} hold. Assume that $U$ is convex, $b_0(x,y,u)$ is linear in $x,y,u$, $\sigma_0(x,y,u)=\sigma_0>0$ and that $l(x,u)$ is jointly convex (here we do not require that $l(x,u)$ is non-increasing in $x$ for a fixed $u$). Then $V$ is convex.* *As in the previous example let $x, \bar x \in X$, $\lambda \in [0,1]$, $\epsilon>0$ and $u^\epsilon(\cdot), \bar u^\epsilon(\cdot)$ be $\epsilon$-optimal controls for the initial conditions $x, \bar x$ respectively (defined on the same reference probability space). We use $x(t), \bar x(t), x_\lambda, x_\lambda(t),u_\lambda^\epsilon(\cdot)$, $x(t;x_\lambda, u_\lambda^\epsilon(\cdot))$ to denote the same objects as in Example [Example 20](#ex:convex1){reference-type="ref" reference="ex:convex1"}. Note that since [\[eq:SDDE\]](#eq:SDDE){reference-type="eqref" reference="eq:SDDE"} is now linear, $$\label{eq:x=x_lambda} x(t;x_\lambda, u_\lambda^\epsilon(\cdot))=x_\lambda(t):=\lambda x(t)+ (1-\lambda) \bar x(t)$$ By [\[eq:x=x_lambda\]](#eq:x=x_lambda){reference-type="eqref" reference="eq:x=x_lambda"} we have $$\begin{aligned} V(x_\lambda) \leq J(x_\lambda; u_\lambda^\epsilon(\cdot))&= \mathbb E\left[ \int_0^{+\infty} e^{-\rho t}l(x(t;x_\lambda, u^\epsilon_\lambda(\cdot)),u^\epsilon_\lambda(t))dt \right] = \mathbb E \left[ \int_0^{+\infty} e^{-\rho t}l(x_\lambda(t),u_\lambda^\epsilon(t))dt \right]\end{aligned}$$ and hence, proceeding as in the previous example, we obtain the convexity of $V.$* Let $\epsilon>0$. We define by $\tilde V_{\epsilon}$ the inf-convolution of $\tilde V$, $$\tilde V_{\epsilon}(x):=\inf_{y \in X_{-1}}\left [ \tilde V(y)+\frac{1}{2\epsilon}|x-y|^2_{-1}\right]=\inf_{y \in X}\left [V(y)+\frac{1}{2\epsilon}|x-y|^2_{-1}\right]=-\sup_{y \in X}\left [- V(y)-\frac{1}{2\epsilon}|x-y|^2_{-1}\right].$$ The function $\tilde V_{\epsilon}$ restricted to $X$ will be denoted by $V_{\epsilon}$. We have the following result. **Lemma 22**. *Let Assumptions [Assumption 1](#hp:state){reference-type="ref" reference="hp:state"}, [Assumption 2](#hp:cost){reference-type="ref" reference="hp:cost"}, [Assumption 9](#hp:discount){reference-type="ref" reference="hp:discount"} and [Assumption 19](#hp:convexity){reference-type="ref" reference="hp:convexity"} hold.* (i) *$\tilde V_\epsilon$ satisfies [\[Va-1\]](#Va-1){reference-type="eqref" reference="Va-1"}, it is semiconcave in $X_{-1}$ and, if $\epsilon$ is small enough, it is semiconvex in $X_{-1}$ with a semiconvexity constant $C$ independent of $\epsilon$.* (ii) *$\tilde V_\epsilon \in C^{1,1}(X_{-1})$. It follows that $V_\epsilon \in C^{1,1}(X)$ with $$\begin{aligned} \label{eq:gradient_w_gradient_w_-1} DV_\epsilon(x)=B D_{-1} \tilde V_\epsilon(x) \in D(\tilde A^*)\end{aligned}$$ for every $x \in X$, where $D_{-1}\tilde V_\epsilon$ denotes the Frechet derivative of $\tilde V_\epsilon$ in $X_{-1}$. Moreover, for every $x_1 \in L^2[-d,0]$, $D^2_{x_0}V_\epsilon(x)$ exists for a.e. $x_0 \in \mathbb R^n$ with $x=(x_0,x_1)$.* (iii) *We have $$\begin{aligned} \label{eq:convergence_inf_convolution} \lim_{\epsilon \to 0} V_\epsilon = V \quad \textit{uniformly on}\,\,X,\quad \lim_{\epsilon \to 0} D_{x_0}V_\epsilon = D_{x_0}V \quad \textit{uniformly on } B_R \subset X \quad \forall R>0.\end{aligned}$$* *Proof.* $$ (i) This is a standard result, e.g. see [@lasry]. (ii) It follows from [@lasry] that $\tilde V_\epsilon \in C^{1,1}(X_{-1})$. It is then easy to see that $V_\epsilon \in C^{1,1}(X)$ with $$DV_\epsilon(x)=B D_{-1} \tilde V_\epsilon(x)$$ for every $x \in X$. Since $D_{-1} \tilde V_\epsilon(\bar x) \in X_{-1}$, $B^{1/2}D_{-1} \tilde V_\epsilon(\bar x) \in X$ so that $D V_\epsilon( x)=BD_{-1} \tilde V_\epsilon( x)=B^{1/2} B^{1/2}D_{-1} \tilde V_\epsilon( x)\in R(B^{1/2})= D(\tilde A^*)$, where the last equality follows by [\[eq:R(B\^1/2)=D(A\^\*)\]](#eq:R(B^1/2)=D(A^*)){reference-type="eqref" reference="eq:R(B^1/2)=D(A^*)"}. Finally note that by Alexandrov's theorem, if $\bar x_1 \in L^2([-d,0],\mathbb{R}^n)$ is fixed, $D^2_{x_0}V_\epsilon(x_0,\bar x_1)$ exists for a.e. $x_0 \in \mathbb{R}^n$. (iii) The uniform convergence of $V_\epsilon$ follows by standard theory since $V$ is Lipschitz with respect to the $| \cdot|_{-1}$ norm. We now show that $D_{x_0}V_\epsilon(x) \xrightarrow{\epsilon \to 0} D_{x_0}V(x)$ uniformly on $B_R \subset X$ for every $R>0.$ We will assume without loss of generality that $V$ and $V_\epsilon$ are convex. Fix $R>0$ and by contradiction assume that $D_{x_0}V_\epsilon(x)$ does not converge uniformly to $D_{x_0}V(x)$ on $B_R$. Then there exist $c>0$, $\{(x_0^\epsilon,x_1^\epsilon)\} \subset B_R$, such that if we set $p_0^\epsilon :=D_{x_0}V_\epsilon(x_0^\epsilon,x_1^\epsilon)$, $\bar p_0^\epsilon :=D_{x_0}V(x_0^\epsilon,x_1^\epsilon)$, we have $|p_0^\epsilon-\bar p_0^\epsilon|\geq c.$ Let $q_0^\epsilon$ be such that $|q_0^\epsilon|=1$ and $|p_0^\epsilon-\bar p_0^\epsilon|=(p_0^\epsilon-\bar p_0^\epsilon)\cdot q_0^\epsilon$. Denote $a_\epsilon:=|V_\epsilon-V|_{L^\infty(B_{R+1})}$ and let $0 \leq t \leq 1$. Since $V_\epsilon$ is convex, we have $$V(x_0^\epsilon+t q_0^\epsilon,x_1^\epsilon) \geq V_\epsilon(x_0^\epsilon+t q_0^\epsilon,x_1^\epsilon) - a_\epsilon \geq V_\epsilon(x_0^\epsilon,x_1^\epsilon) + t p_0^\epsilon \cdot q_0^\epsilon -a_\epsilon \geq V(x_0^\epsilon,x_1^\epsilon) + t p_0^\epsilon \cdot q_0^\epsilon - 2 a_\epsilon.$$ On the other hand, as $V^{\bar x_1} \in C^{1,\alpha}_{\rm loc}(\mathbb R^n)$ and has locally uniform $C^{1,\alpha}$-norm for $\bar x_1 \in L^2,$ $|\bar x_1| \leq R$, there exists $C=C_R>0$ such that $$V(x_0^\epsilon+t q_0^\epsilon,x_1^\epsilon) \leq V(x_0^\epsilon,x_1^\epsilon) + t \bar p_0^\epsilon \cdot q_0^\epsilon+ Ct^{1+\alpha}.$$ Putting together these two inequalities we obtain $$ct \leq t (p_0^\epsilon-\bar p_0^\epsilon) \cdot q_0^\epsilon \leq Ct^{1+\alpha} +2a_\epsilon.$$ Letting $\epsilon \to 0$ (note that $a_\epsilon \to 0$ since $V_\epsilon \to V$ uniformly), we thus have $$ct\leq Ct^{1+\alpha}$$ which is impossible for small $t$ as $\alpha>0$.  ◻ We now prove the main result of this section. **Proposition 23**. *Let Assumptions [Assumption 1](#hp:state){reference-type="ref" reference="hp:state"}, [Assumption 2](#hp:cost){reference-type="ref" reference="hp:cost"}, [Assumption 9](#hp:discount){reference-type="ref" reference="hp:discount"} and [Assumption 19](#hp:convexity){reference-type="ref" reference="hp:convexity"} hold. There exists $\gamma(\epsilon) \geq 0$, $\gamma(\epsilon) \to 0$ as $\epsilon \to 0$ such that $V_\epsilon$ is a viscosity supersolution of $$\label{pertHJB-gamma-eps} \rho V_\epsilon(x)-\langle \tilde A^*DV_\epsilon(x),x \rangle_X+\tilde H \left (x,D_{x_0}V_\epsilon(x) ,D^2_{x_0^2}V_\epsilon(x) \right) =-\gamma(\epsilon), \quad x \in X.$$ In fact the viscosity supersolution property of $V_\epsilon$ holds in the following stronger sense: if $x \in X$ is a local minimum of $V_\epsilon + \phi + g$ for test functions $\phi \in \Phi, g \in \mathcal G$, then $Dg(x)\in D(\tilde A^*)$ and $$\begin{aligned} \rho V_\epsilon(x)& + \langle x, \tilde A^*(D \phi ( x) + D g(\bar x)) \rangle_X + H \left ( x, -D \phi (x) - D g(x), -D^2 \phi (x) - D^2 g(x) \right) \geq -\gamma(\epsilon). \end{aligned}$$* *Proof.* **Step 1.** We fix $\epsilon>0$. Let $x_0 \in X$ be a local minimum of $V_\epsilon + \phi + g$. We remark that $\phi, g$ may depend on $\epsilon$ and $x_0$. Assume without loss of generality that the minimum is global and strict and $w_\epsilon+\phi+g \to \infty$ as $|x|\to \infty$ (see [@fgs_book], Lemma 3.37). We can also assume that $$\begin{aligned} \label{eq:w_eps_+phi+g(x_0)=0} V_\epsilon(x_0)+\phi(x_0)+g(x_0)=0\end{aligned}$$ and $\phi$ is bounded. If $\delta>0$, we can then find $R>0$ such that for every $x,y \in X$ with $|x|_X,|y|_X \geq R-1$, $$\begin{aligned} \label{eq:proof_inf_convolution_approx_bound} -V(y)-\frac{1}{2 \epsilon}|x-y|_{-1}^2-\delta |y|_X^2-\phi(x)-g(x) \leq -V(x_0)-\delta |x_0|^2-\phi(x_0)-g(x_0)-1. \end{aligned}$$ Define now $$\bar\Phi(x,y) := -V(y)-\frac{1}{2 \epsilon}|x-y|_{-1}^2-\delta |y|_X^2-\phi(x)-g(x)$$ and observe that $\bar \Phi$ is weakly sequentially upper semicontinuous on $X\times X$. Indeed, since $V$ is Lipschitz continuous with respect to the $|\cdot|_{-1}$ norm and $B$ is compact, it is weakly sequentially continuous and the same holds for $|\cdot|_{-1}^2$. Then the weak sequential upper semicontinuity of $-\bar \Phi$ follows as $\phi \in \Phi$ is a regular test function and $g \in \mathcal G$ is a radial test function which is weakly sequentially lower semicontinuous. Therefore, we can find $\bar x, \bar y \in B_R$ such that $$\begin{aligned} \bar \Phi(\bar x, \bar y)=\sup_{|x|_X,|y|_X \leq R} \bar \Phi(x,y)=\sup_{x,y \in X} \bar \Phi(x,y),\end{aligned}$$ where the last equality follows by [\[eq:proof_inf_convolution_approx_bound\]](#eq:proof_inf_convolution_approx_bound){reference-type="eqref" reference="eq:proof_inf_convolution_approx_bound"}. We now set $$\begin{gathered} m_{\epsilon, \delta}=\sup_{x,y \in X} \left\{-V(y)-\frac{1}{2 \epsilon}|x-y|_{-1}^2-\delta |y|_X^2-\phi(x)-g(x)\right\} =\bar \Phi(\bar x,\bar y) \end{gathered}$$ and note that $$\begin{aligned} \label{eq:lim_phi(bar_x,bar_y)} \lim_{\delta \to 0} m_{\epsilon, \delta}& = \lim_{R \to \infty} \sup_{x \in X}\sup_{y \in B_R} \left\{-V(y)-\frac{1}{2 \epsilon}|x-y|_{-1}^2-\phi(x)-g(x) \right\} \nonumber \\ & = \sup_{x \in X} \left\{-V_\epsilon(x)-\phi(x)-g(x) \right\} = -V_\epsilon(x_0)-\phi(x_0)-g(x_0)=0.\end{aligned}$$ Moreover by the definition of $w_\epsilon$ we have $$\begin{aligned} m_{\epsilon, \delta} & =\bar \Phi(\bar x, \bar y) \leq -V(\bar y)-\frac{1}{2 \epsilon}|\bar x- \bar y|_{-1}^2-\phi(\bar x)-g(\bar x)\leq -V_\epsilon(\bar x)-\phi(\bar x)-g(\bar x).\end{aligned}$$ Letting $\delta \to 0$ and using [\[eq:lim_phi(bar_x,bar_y)\]](#eq:lim_phi(bar_x,bar_y)){reference-type="eqref" reference="eq:lim_phi(bar_x,bar_y)"} we have $$0=-V_\epsilon(x_0)-\phi(x_0)-g(x_0) \leq \lim_{\delta \to 0} -V_\epsilon(\bar x)-\phi(\bar x)-g(\bar x),$$ so that as $x_0$ is a strict maximum for $-V_\epsilon-\phi-g$, we must have $$\begin{aligned} \label{eq:bar_x_=_x_0} \lim_{\delta \to 0}\bar x=x_0.\end{aligned}$$ Moreover, since $m_{\epsilon, \delta}=-V(\bar y)-\frac{1}{2 \epsilon}|\bar x-\bar y|_{-1}^2-\delta |\bar y|_X^2-\phi(\bar x)-g(\bar x)$, so that $m_{\epsilon, \delta}+ \frac{\delta }{2} |\bar y|_X^2=-V(\bar y)-\frac{1}{2 \epsilon}|\bar x-\bar y|_{-1}^2-\frac{\delta}{2} |\bar y|_X^2-\phi(\bar x)-g(\bar x) \leq m_{\epsilon, \delta/2}$, letting $\delta \to 0$ we obtain $$\begin{aligned} \label{eq:delta_bary_squared_to_zero} \lim_{\delta \to 0}\delta |\bar y|_X^2 =0.\end{aligned}$$ By [\[eq:lim_phi(bar_x,bar_y)\]](#eq:lim_phi(bar_x,bar_y)){reference-type="eqref" reference="eq:lim_phi(bar_x,bar_y)"} we have $$\begin{aligned} V(\bar y)+\frac{1}{2 \epsilon}|\bar x-\bar y|_{-1}^2+\delta |\bar y|_X^2+\phi(\bar x)+g(\bar x) = - \Phi(\bar x, \bar y) \leq \omega (\delta, \epsilon),\end{aligned}$$ for some local modulus $\omega(\cdot,\epsilon)$ (depending on $\phi, g, x_0$). Since $V_\epsilon+\phi+g$ has a minimum at $x_0$ then $0 \leq V_\epsilon+\phi+g$ so that $-V(\bar x) \leq -V_\epsilon(\bar x) \leq \phi(\bar x)+ g(\bar x)$. Then, inserting this inequality in the previous one and since $\delta>0$, we have $$V(\bar y)-V(\bar x)+\frac{1}{2 \epsilon}|\bar x-\bar y|_{-1}^2 \leq \omega (\delta, \epsilon)$$ so that, by [\[Va-1\]](#Va-1){reference-type="eqref" reference="Va-1"}, $$-K|\bar x-\bar y|_{-1}+\frac{1}{2 \epsilon}|\bar x-\bar y|_{-1}^2 \leq \omega (\delta, \epsilon)$$ and thus $$\limsup_{\delta \to 0} \left [-K|\bar x-\bar y|_{-1}+\frac{1}{2 \epsilon}|\bar x-\bar y|_{-1}^2 \right] \leq 0$$ for every $\epsilon>0$. Therefore, for every $\epsilon>0$, we have $$\label{eq:proof_barx-bary_1} \limsup_{\delta \to 0} |\bar x-\bar y|_{-1}\leq 2K \epsilon.$$ **Step 2.** Define $$\begin{aligned} & u_1(x)=-\phi(x)-g(x)-\frac{\left\langle B Q_{N}(\bar{x}-\bar{y}), x\right\rangle_X}{\epsilon}-\frac{\left|Q_{N}(x-\bar{x})\right|_{-1}^{2}}{\epsilon} +\frac{\left|Q_{N}(\bar{x}-\bar{y})\right|_{-1}^{2}}{2 \epsilon} \\ & v_1(y)=V(y)+\delta|y|_X^2-\frac{\left\langle B Q_{N}(\bar{x}-\bar{y}), y\right\rangle_X}{\epsilon}+\frac{\left|Q_{N}(y-\bar{y})\right|_{-1}^{2}}{\epsilon}\end{aligned}$$ and notice that $$|x-y|_{-1}^{2}=\left|P_{N}(x-y)\right|_{-1}^{2}+\left|Q_{N}(x-y)\right|_{-1}^{2}$$ and $$\begin{aligned} \left|Q_{N}(x-y)\right|_{-1}^{2} \leq 2\left\langle B Q_{N}(\bar{x}-\bar{y}), x-y\right\rangle_X &+2\left|Q_{N}(x-\bar{x})\right|_{-1}^{2} +2\left|Q_{N}(y-\bar{y})\right|_{-1}^{2}-\left|Q_{N}(\bar{x}-\bar{y})\right|_{-1}^{2} \end{aligned}$$ with equality at $\bar x, \bar y$. Thus, since $(\bar x, \bar y)$ is a global maximum for $\bar \Phi$, $$u_{1}(x)-v_{1}(y)-\frac{1}{2 \epsilon}\left|P_{N}(x-y)\right|_{-1}^{2}$$ has a strict global maximum over $X \times X$ at $(\bar x,\bar y)$.\ Denote $\bar x_N=P_N \bar x, \bar y_N=P_N \bar y$. By [@fgs_book Corollary 3.28] there exist test functions $\{ \varphi_{k}\},\{ \psi_{k} \}_k \subset \Phi$ and points $\{(x_{k}, y_{k})\}_k \subset X \times X$ such that $u_{1}(x)-\varphi_{k}(x)$ has a maximum at $x_{k}$, $v_{1}(y)-\psi_{k}(y)$ has a minimum at $y_{k}$ and such that $$\left(x_{k}, D \varphi_{k}\left(x_{k}\right), D^{2} \varphi_{k}\left(x_{k}\right)\right) \stackrel{k \rightarrow \infty}{\longrightarrow}\left(\bar{x}, \frac{B\left(\bar{x}_{N}-\bar{y}_{N}\right)}{\epsilon}, X_{N}\right)$$ in $X \times X_{2} \times \mathcal{L}\left(X_{-1}, X_{1}\right)$ $$\left(y_{k}, D \psi_{k}\left(y_{k}\right), D^{2} \psi_{k}\left(y_{k}\right)\right) \stackrel{k \rightarrow \infty}{\longrightarrow}\left(\bar{y}, \frac{B\left(\bar{x}_{N}-\bar{y}_{N}\right)}{\epsilon}, Y_{N}\right)$$ in $X \times X_{2} \times \mathcal{L}\left(X_{-1}, X_{1}\right)$ where $X_N=P_N X_N P_N, Y_N=P_N Y_N P_N$ and $$\begin{aligned} \label{eq:maximum_principle_2_derivatives} \frac{3}{\epsilon}\left(\begin{array}{cc}B P_{N} & 0 \\ 0 & B P_{N}\end{array}\right) \leq\left(\begin{array}{cc}X_N & 0 \\ 0 & -Y_N\end{array}\right) \leq \frac{3}{\epsilon}\left(\begin{array}{cc}B P_{N} & -B P_{N} \\ -B P_{N} & B P_{N}\end{array}\right).\end{aligned}$$ Since $v_{1}(y)-\psi_{k}(y)$ has a minimum at $y_{k}$, by defining $$\begin{aligned} &\bar \phi(y)=-\psi_k(y)-\frac{\left\langle B Q_{N}(\bar{x}-\bar{y}), y\right\rangle_X}{\epsilon}+\frac{\left|Q_{N}(y-\bar{y})\right|_{-1}^{2}}{\epsilon}, \quad \quad \bar g(y)=\delta |y|_X^2,\end{aligned}$$ $V+\bar \phi + \bar g$ has a minimum at $y_k$. Since $V$ is a viscosity supersolution of the HJB equation, we have $$\begin{aligned} \rho V(y_k)+\langle y_k, \tilde A^*D \bar \phi (y_k) \rangle_X + H(y_k,-D \bar \phi (y_k)-D \bar g (y_k),-D^2 \bar \phi (y_k)-D^2 \bar g (y_k)) \geq 0.\end{aligned}$$ Note that $D \left|Q_{N}(y-\bar{y})\right|_{-1}^{2}=2B Q_N (y-\bar{y})$. Therefore, $$\begin{aligned} &D \bar \phi(y_k) =-D \psi_k(y_k)-\frac{1}{\epsilon}B Q_N(\bar x- \bar y)+ 2BQ_N (y_k-\bar y)\\ & \quad \xrightarrow[]{k \to \infty} -\frac{1}{\epsilon}B(\bar x_N-\bar y_N)-\frac{1}{\epsilon}B Q_N(\bar x- \bar y) = -\frac{1}{\epsilon}BP_N(\bar x -\bar y)-\frac{1}{\epsilon}B Q_N(\bar x- \bar y)= -\frac{1}{\epsilon}B(\bar x-\bar y)\\\end{aligned}$$ and $D^2 \bar \phi(y_k) =-D^2 \psi_k(y_k)+\frac{2}{\epsilon}BQ_N \xrightarrow[]{k \to \infty} -Y_N + \frac{2}{\epsilon} BQ_N,$ so that letting $k \to \infty$ we have $$\begin{aligned} \label{eq:proof_supersolution_ineq} 0 \geq -\rho V(\bar y)+\frac{1}{\epsilon} \langle \bar y,\tilde A^* B(\bar x- \bar y) \rangle_X - H \left (\bar y,\frac{1}{\epsilon}B(\bar x- \bar y)-2 \delta \bar y,Y_N - \frac{2}{\epsilon}BQ_N-2 \delta I \right). \end{aligned}$$ Since $u_{1}(x)-\varphi_{k}(x)$ has a maximum at $x_k$, we have $$\begin{aligned} D(u_{1}(x_k)-\varphi_{k}(x_k))& =-D \phi (x_k) -Dg(x_k) -\frac{1}{\epsilon}BQ_N(\bar x - \bar y)-2 BQ_N(x_k-\bar x) -D\varphi_k(x_k) = 0\end{aligned}$$ and $$\begin{aligned} D^2(u_{1}(x_k)-\varphi_{k}(x_k))& =-D^2 \phi (x_k) -D^2g(x_k)-\frac{2}{\epsilon}BQ_N-D^2\varphi_k(x_k) \leq 0,\end{aligned}$$ so that, letting $k \to \infty$ above, we obtain $$\begin{aligned} \label{eq:proof_gradient_equal_zero} -D \phi (\bar x) -Dg( \bar x) =\frac{1}{\epsilon}B(\bar x - \bar y), \quad \quad -D^2 \phi (\bar x) -D^2g(\bar x)\leq \frac{2}{\epsilon}BQ_N + X_N.\end{aligned}$$ We now note that by [\[eq:proof_gradient_equal_zero\]](#eq:proof_gradient_equal_zero){reference-type="eqref" reference="eq:proof_gradient_equal_zero"} and [\[eq:gradient_radial\]](#eq:gradient_radial){reference-type="eqref" reference="eq:gradient_radial"}, $$\begin{aligned} \label{eq:gradient_g} Dg( \bar x)=\frac{h_0'(|\bar x|_X)} {|\bar x|_X }\bar x \in D(\tilde A^*)\end{aligned}$$ so that $\bar x \in D(\tilde A^*)$.\ **Step 3.** By [\[eq:proof_supersolution_ineq\]](#eq:proof_supersolution_ineq){reference-type="eqref" reference="eq:proof_supersolution_ineq"}, [\[eq:proof_gradient_equal_zero\]](#eq:proof_gradient_equal_zero){reference-type="eqref" reference="eq:proof_gradient_equal_zero"}, [\[eq:gradient_g\]](#eq:gradient_g){reference-type="eqref" reference="eq:gradient_g"}, the structure conditions [\[eq:H_decreasing\]](#eq:H_decreasing){reference-type="eqref" reference="eq:H_decreasing"}, [\[eq:H_lambda_BQ_N\]](#eq:H_lambda_BQ_N){reference-type="eqref" reference="eq:H_lambda_BQ_N"}, [\[eq:H_norm\_-1\]](#eq:H_norm_-1){reference-type="eqref" reference="eq:H_norm_-1"}, [\[eq:Hamiltonian_local_lip\]](#eq:Hamiltonian_local_lip){reference-type="eqref" reference="eq:Hamiltonian_local_lip"}, the weak $B$-condition with $C_0=0$ and [\[Va-1\]](#Va-1){reference-type="eqref" reference="Va-1"}, we have $$\begin{aligned} \rho& V_\epsilon(\bar x)+ \langle \bar x, \tilde A^*(D \phi (\bar x) + D g(\bar x)) \rangle_X + H \left ( \bar x, -D \phi (\bar x) - D g(\bar x), -D^2 \phi (\bar x) - D^2 g(\bar x) \right) \\ & =\rho V_\epsilon(\bar x) -\frac{1}{\epsilon} \langle \bar x, \tilde A^*B(\bar x - \bar y) \rangle_X + H \left ( \bar x, \frac{1}{\epsilon} B(\bar x - \bar y), -D^2 \phi (\bar x) - D^2 g(\bar x) \right) \\ & \geq \rho V_\epsilon(\bar x) -\frac{1}{\epsilon} \langle \bar x, \tilde A^*B(\bar x - \bar y) \rangle_X + H \left ( \bar x, \frac{1}{\epsilon} B(\bar x - \bar y), -D^2 \phi (\bar x) - D^2 g(\bar x) \right) \\ & \quad - \rho V(\bar y)+\frac{1}{\epsilon} \langle \bar y,\tilde A^* B(\bar x- \bar y) \rangle_X - H \left (\bar y,\frac{1}{\epsilon}B(\bar x- \bar y)-2 \delta \bar y,Y_N - \frac{2}{\epsilon}BQ_N-2 \delta I \right) \\ & \geq \rho V_\epsilon(\bar x) -\frac{1}{\epsilon} \langle \bar x-\bar y, \tilde A^*B(\bar x - \bar y) \rangle_X + H \left ( \bar x, \frac{1}{\epsilon} B(\bar x - \bar y), \frac{2}{\epsilon}BQ_N + X_N \right) \\ & \quad -\rho V(\bar y)- H \left (\bar y,\frac{1}{\epsilon}B(\bar x- \bar y),Y_N - \frac{2}{\epsilon}BQ_N \right) -\delta C (1+|\bar y|_X^2)\\ & = \rho V_\epsilon(\bar x) -\frac{1}{\epsilon} \langle \bar x-\bar y, \tilde A^*B(\bar x - \bar y) \rangle_X + H \left ( \bar x, \frac{1}{\epsilon} B(\bar x - \bar y), X_N \right) \\ & \quad -\rho V(\bar y)- H \left (\bar y,\frac{1}{\epsilon}B(\bar x- \bar y),Y_N \right) -\omega_{\epsilon}(1/N) -\delta C (1+|\bar y|_X^2)\\ & \geq \rho V_\epsilon(\bar x)-\rho V(\bar y) -\frac{1}{\epsilon} \langle \bar x-\bar y, \tilde A^*B(\bar x - \bar y) \rangle_X -C\left (|\bar x- \bar y|_{-1}\left(1+\frac{|\bar x- \bar y|_{-1}}{\epsilon}\right) \right) -\omega_{\epsilon}(1/N) -\delta C (1+|\bar y|_X^2)\\ & \geq \rho V_\epsilon(\bar x)-\rho V (\bar x) +\rho V (\bar x) -\rho V(\bar y) -C\left (|\bar x- \bar y|_{-1}\left(1+\frac{|\bar x- \bar y|_{-1}}{\epsilon}\right) \right) -\omega_{\epsilon}(1/N)-\delta C (1+|\bar y|_X^2)\\ & \geq \rho V_\epsilon(\bar x)-\rho V (\bar x) -\rho K|\bar x-\bar y|_{-1}-C\left (|\bar x- \bar y|_{-1}\left(1+\frac{|\bar x- \bar y|_{-1}}{\epsilon}\right) \right)-\omega_{\epsilon}(1/N) -\delta C (1+|\bar y|_X^2),\end{aligned}$$ where $\omega_{\epsilon}$ is a modulus coming from [\[eq:H_lambda_BQ_N\]](#eq:H_lambda_BQ_N){reference-type="eqref" reference="eq:H_lambda_BQ_N"}. We now let first $N \to \infty$ and then take $\limsup_{\delta \to 0}$, so that by [\[eq:bar_x\_=\_x_0\]](#eq:bar_x_=_x_0){reference-type="eqref" reference="eq:bar_x_=_x_0"}, [\[eq:delta_bary_squared_to_zero\]](#eq:delta_bary_squared_to_zero){reference-type="eqref" reference="eq:delta_bary_squared_to_zero"}, [\[eq:proof_barx-bary_1\]](#eq:proof_barx-bary_1){reference-type="eqref" reference="eq:proof_barx-bary_1"} and [\[eq:convergence_inf_convolution\]](#eq:convergence_inf_convolution){reference-type="eqref" reference="eq:convergence_inf_convolution"} we have $$\begin{aligned} \rho V_\epsilon(x_0)& + \langle \bar x, \tilde A^*(D \phi (\bar x) + D g(\bar x)) \rangle_X + H \left ( x_0, -D \phi (x_0) - D g(x_0), -D^2 \phi (x_0) - D^2 g(x_0) \right)\\ & \geq \rho V_\epsilon(x_0)-\rho V (x_0) - C_1 \epsilon \geq -\nu(\epsilon) - C_1 \epsilon =: -\gamma(\epsilon), \end{aligned}$$ where $\gamma(\epsilon) \to 0$ as $\epsilon \to 0$. We emphasize that the modulus $\nu$ and the constant $C_1$, and hence $\gamma$, are independent of $\phi, g, x_0$. Thus we proved that $V_\epsilon$ is a viscosity solution of [\[pertHJB-gamma-eps\]](#pertHJB-gamma-eps){reference-type="eqref" reference="pertHJB-gamma-eps"} in the stronger sense of Proposition [Proposition 23](#prop:inf_conv_subsolution_perturbed){reference-type="ref" reference="prop:inf_conv_subsolution_perturbed"}. ◻ # Non-smooth Dynkin's Formula {#sec:dynkin_formula} In Sections [5](#sec:dynkin_formula){reference-type="ref" reference="sec:dynkin_formula"} and [6](#sec:lions_approx){reference-type="ref" reference="sec:lions_approx"} we follow the setup and technique introduced in [@lions-infdim1] and modify them to accommodate an equation with an unbounded term. We define the space $\mathcal D$ by $$\mathcal D=\Big \{ \phi \in C^{1,1}(X_{-1}) : \eqref{eq:second_derivative_space_D} \textit{ exists and is uniformly continuous on } X_{-1}\},$$ where $$\label{eq:second_derivative_space_D} \lim_{t \to 0} \frac 1 t \langle D_{-1} \phi(x+tk)-D_{-1} \phi(x),h \rangle_{-1} ,\quad \forall x,h,k \in X_{-1}.$$ Functions in the space $\mathcal D$ possess second order derivatives in some sense. Note that for $\phi \in \mathcal D$, since $D \phi=BD_{-1}\phi$, we have $$\label{eq:second_derivative_space_D_derivatives_in_X} \lim_{t \to 0} \frac 1 t \langle D \phi(x+tk)-D \phi(x),h \rangle_X=\lim_{t \to 0} \frac 1 t \langle D_{-1} \phi(x+tk)-D_{-1} \phi(x),h \rangle_{-1}, \quad \forall x,k \in X_{-1}, h \in X$$ and it is uniformly continuous with respect to $x \in X_{-1}$. Moreover, we have $$\label{eq:second_derivative_space_D_derivatives_in_X_-1_bilinear_form} \lim_{t \to 0} \frac 1 t \langle D_{-1} \phi(x+tk)-D_{-1} \phi(x),h \rangle_{-1} = \langle A(x)k,h \rangle_{-1} ,\quad \forall h,k \in X_{-1}$$ and the limit is uniformly continuous with respect to $x \in X_{-1}$. Here $A(x)$ are bounded, linear, self-adjoint operators on $X_{-1}$ such that $|A(x)|_{\mathcal L(X_{-1})} \leq L_\phi,$ where $L_\phi$ is the Lipschitz constant of $D_{-1} \phi$. We will denote $A(x)=\tilde D_{-1}^2\phi(x)$. We point out that $\tilde D_{-1}^2\phi(x)$ is not the Fréchet or the Gateaux derivative. It is a sort of a weak Gateaux second order derivative in $X_{-1}$ in the sense that $$\frac 1 t \left( D_{-1} \phi(x+tk)-D_{-1} \phi(x) \right) \stackrel{X_{-1}}{\rightharpoonup} \tilde D_{-1}^2\phi(x)k, \quad \forall x, k \in X_{-1},$$ where $\stackrel{X_{-1}}{\rightharpoonup}$ is the weak convergence in $X_{-1}$. Denoting $\tilde D^2\phi(x)=B\tilde D_{-1}^2\phi(x)$, by [\[eq:second_derivative_space_D\_derivatives_in_X\]](#eq:second_derivative_space_D_derivatives_in_X){reference-type="eqref" reference="eq:second_derivative_space_D_derivatives_in_X"} we have $$\label{eq:second_derivative_space_D_derivatives_in_X_bilinear_form} \lim_{t \to 0} \frac 1 t \langle D \phi(x+tk)-D \phi(x),h \rangle_X= \langle \tilde D^2\phi(x) k,h \rangle_X ,\quad \forall k,h \in X$$ and the right-hand side of [\[eq:second_derivative_space_D\_derivatives_in_X\_bilinear_form\]](#eq:second_derivative_space_D_derivatives_in_X_bilinear_form){reference-type="eqref" reference="eq:second_derivative_space_D_derivatives_in_X_bilinear_form"} is uniformly continuous in $x\in X$. Here, $\tilde D^2\phi(x)$ is a bounded, linear, self-adjoint operator on $X$. We have $|\tilde D^2\phi(x)|_{\mathcal L(X)} \leq C_\phi$ and $$\frac 1 t \left(D \phi(x+tk)-D \phi(x) \right) \rightharpoonup \tilde D^2\phi(x)k, \quad \forall x, k \in X.$$ Hence the quantities $$\label{eq:second_derivative_space_D_unif_continuous} \begin{aligned} & \langle \tilde D_{-1}^2\phi(x)k,h \rangle_{-1} \quad \forall h,k \in X_{-1},\quad \quad \langle \tilde D^2\phi(x) k,h \rangle_X \quad \forall k,h \in X \end{aligned}$$ are uniformly continuous with respect to $x \in X_{-1}$ and $x \in X$ respectively. **Remark 24**. *We note that [\[eq:second_derivative_space_D\]](#eq:second_derivative_space_D){reference-type="eqref" reference="eq:second_derivative_space_D"} can be replaced by the following condition: $\partial_{ij} \phi$ exists and is uniformly continuous on $X_{-1}$ for every $i,j \in \mathbb N$. Here $\partial_i$ indicates the partial derivative with respect to $e_i$ (in $X_{-1}$), where $\{e_i\}_{i \in \mathbb N}$ is the basis of $X_{-1}$ defined in Subsection [3.2](#sec:operator_B){reference-type="ref" reference="sec:operator_B"}.* We denote $\tilde D^2_{x_0^2}\phi(x):=P_{x_0}\tilde D^2 \phi(x)P_{x_0}$, where $P_{x_0}$ is the orthogonal projection in $X$ onto the $\mathbb R^n$ component. **Lemma 25**. *Let $\phi \in \mathcal D$ and consider its restriction to $X$. Then, for every $\bar x_1 \in L^2$, we have $\phi^{\bar x_1}:=\phi(\cdot,\bar x_1) \in C^2(\mathbb R^n)$, hence $D^2_{x_0^2}\phi(x)=\tilde D^2_{x_0^2}\phi(x)$. Moreover $D^2_{x_0^2}\phi(x)$ is uniformly continuous on $X$.* *Proof.* Let $\{v_i \}$ be an orthonormal basis of $\mathbb R^n$. Fixing any $i,j \leq n$ and considering [\[eq:second_derivative_space_D\_derivatives_in_X\_bilinear_form\]](#eq:second_derivative_space_D_derivatives_in_X_bilinear_form){reference-type="eqref" reference="eq:second_derivative_space_D_derivatives_in_X_bilinear_form"} with $k=(v_i,0), h=(v_j,0)$ we have that $\frac{\partial^2 }{\partial x_i x_j} \phi^{\bar x_1}$ exists and it is uniformly continuous on $X$. Then, by the fact that in finite dimensional spaces the continuity of all second order partial derivatives implies $C^2$, we have $\phi^{\bar x_1}:=\phi(\cdot,\bar x_1) \in C^2(\mathbb R^n)$ and $D^2_{x_0^2}\phi(x)=\tilde D^2_{x_0^2}\phi(x)$. The uniform continuity in $X$ follows. ◻ Thanks to this lemma we will denote $$\tilde D^2\phi(x)k=\begin{bmatrix} D^2_{x_0^2}\phi(x) & \tilde D^2_{x_0 x_1}\phi(x)\\ \tilde D^2_{x_1 x_0}\phi(x) &\tilde D^2_{x_1^2}\phi(x) \end{bmatrix}\begin{bmatrix}k_0\\ k_1\end{bmatrix}, \quad \forall x=(x_0,x_1), k=(k_0,k_1) \in X.$$ We now prove Dynkin's formula for functions of the form $e^{-\rho t} \phi(x),$ where $\phi \in \mathcal D$. The formula could be extended to more general functions, but we restrict ourselves to functions $e^{-\rho t} \phi(x)$ since only such functions will be used in the proof of the Verification Theorem. **Lemma 26** (Dynkin's formula). *Let Assumptions [Assumption 1](#hp:state){reference-type="ref" reference="hp:state"}, [Assumption 9](#hp:discount){reference-type="ref" reference="hp:discount"} hold. Fix any initial datum $x \in X$, any control $u(\cdot) \in \mathcal {\overline U}$ and denote by $Y(t)$ the solution of the state equation [\[eq:abstract_dissipative_operator\]](#eq:abstract_dissipative_operator){reference-type="eqref" reference="eq:abstract_dissipative_operator"}. Let $R,T>0$ and define $\tau^R:=\inf \{s \in[0, T]:|Y(s)|_X>R\}.$ Then, for every $\phi \in \mathcal D$, for every $0 \leq t \leq T$* *$$\begin{aligned} \mathbb E \left[ e^{-\rho (t \wedge \tau^R)} \phi(Y(t \wedge \tau^R)) \right] &=\phi( x)+\mathbb E \int_0^{t \wedge \tau^R} \Big [- \rho e^{-\rho s} \phi(Y(s)) + e^{-\rho s} \langle Y(s), \tilde A^* D \phi(Y(s)) \rangle_X\\ & \quad +e^{-\rho s } \langle \tilde b(Y(s),u(s)), D \phi(Y(s)) \rangle_X + \frac 1 2 e^{-\rho s} \mathop{\mathrm{Tr}}\nolimits\left ( \sigma(Y(s))\sigma(Y(s))^* \tilde D^2\phi(Y(s)) \right ) \Big ] ds\\ &=\phi( x)+\mathbb E \int_0^{t \wedge \tau^R} \Big [- \rho e^{-\rho s} \phi(Y(s)) + e^{-\rho s} \langle Y(s), \tilde A^* D \phi(Y(s)) \rangle_X\\ & \quad +e^{-\rho s } \tilde b_0(Y(s),u(s)) \cdot D_{x_0} \phi(Y(s)) + \frac 1 2 e^{-\rho s} \mathop{\mathrm{Tr}}\nolimits\left ( \sigma_0(Y(s))\sigma_0(Y(s))^T D^2_{x_0^2}\phi(Y(s)) \right ) \Big ] ds.\end{aligned}$$* *Proof.* Let $R_N:=(N I-\tilde A)^{-1}$ be the resolvent operator of $\tilde A$ for $N \in \mathbb N$ and let $\tilde A_N=N \tilde AR_N$ be the Yosida approximation of $\tilde A$. Denote by $Y^N$ the solution of the state equation with $\tilde A_N$ in place of $\tilde A$, that is $$dY^N(s) = [\tilde A_N Y^N(s)+\tilde b(Y^N(s),u(s))]dt + \sigma(Y^N(s))\,dW(s), \quad Y^N(0)= x.$$ By standard theory, e.g. [@fgs_book Proposition 1.132], we have $$\begin{aligned} \label{eq:proof_conv_expect_YN-Y} \lim_{N \to \infty} \mathbb E \left[ \sup_{s \in [0,T]} \left |Y^N(s)-Y(s) \right|_X^2 \right]=0, \quad \forall T>0.\end{aligned}$$ We define $\tau_N^R:=\inf \left\{s \in[0, T]:\left|Y^N(s)\right|_X>R\right\}.$ By [\[eq:proof_conv_expect_YN-Y\]](#eq:proof_conv_expect_YN-Y){reference-type="eqref" reference="eq:proof_conv_expect_YN-Y"}, up to a subsequence, we have $\sup_{s \in [0,T]} \left |Y^N(s)-Y(s) \right|_X \to 0$ a.s. so that $\lim_{N \to \infty} \tau_N^R=\tau^R \quad a.s. \quad \forall R>0 .$ Since $\tilde A_N \in \mathcal L(X)$ we can apply the non-smooth Ito's formula from [@lions-infdim1 Lemma III.2] which holds for equations with bounded terms and for $\phi$, $D\phi, \tilde D^2\phi$ bounded. Note that in our case $\Phi, D\phi, D^2\phi$ are only bounded on bounded sets of $X$, however, since we are using the stopping time $\tau_N^R$, this is enough in order to apply [@lions-infdim1 Lemma III.2]. (Observe that in [@lions-infdim1] what we call $\tilde D^2 \phi$ is denoted by $D^2 \phi$, see page 246 there.) Therefore we have $$\begin{aligned} \label{eq:proof_ito_projected} \mathbb E \left[ e^{-\rho (t \wedge \tau_N^R)} \phi(Y^N(t)) \right] \nonumber & =\phi( x)+\mathbb E \int_0^{t \wedge \tau_N^R} \Big [- \rho e^{-\rho s} \phi(Y^N(s)) + e^{-\rho s} \langle \tilde A_N Y^N(s), D \phi(Y^N(s)) \rangle_X \nonumber \\ & \quad +e^{-\rho s } \langle \tilde b(Y^N(s),u(s)), D \phi(Y^N(s)) \rangle_X + \frac 1 2 e^{-\rho s} \mathop{\mathrm{Tr}}\nolimits\left ( \sigma(Y^N(s))\sigma(Y^N(s))^* \tilde D^2\phi(Y^N(s)) \right ) \Big ] ds \nonumber \\ &=\phi( x)+\mathbb E \int_0^{t \wedge \tau_N^R} \Big [- \rho e^{-\rho s} \phi(Y^N(s)) + e^{-\rho s} \langle Y^N(s), (\tilde A_N)^* D \phi(Y^N(s)) \rangle_X \nonumber \\ & \quad +e^{-\rho s } \tilde b_0(Y^N(s),u(s)) \cdot D_{x_0} \phi(Y^N(s)) + \frac 1 2 e^{-\rho s} \mathop{\mathrm{Tr}}\nolimits\left ( \sigma_0(Y^N(s))\sigma_0(Y^N(s))^T D^2_{x_0^2}\phi(Y^N(s)) \right ) \Big ] ds.\end{aligned}$$ We now prove that $$\begin{aligned} \lim_{N \to \infty}|(\tilde A_N)^* D \phi(Y^N(s))-\tilde A^* D \phi(Y(s))|_X =0 \quad a.s. \,\, \forall s \geq 0.\end{aligned}$$ Indeed we have $$\begin{aligned} |(\tilde A_N)^* D \phi(Y^N(s))-\tilde A^* D \phi(Y(s))|_X &\leq |(\tilde A_N)^*[D \phi(Y^N(s))- D \phi(Y(s))]|_X+|(\tilde A_N)^* D \phi(Y(s))-\tilde A^* D \phi(Y(s))|_X.\end{aligned}$$ Consider the first term. We first note that by [\[eq:A\*B\^1/2\]](#eq:A*B^1/2){reference-type="eqref" reference="eq:A*B^1/2"} and the fact that $|(R_N)^* |_{\mathcal L(X)} =|R_N |_{\mathcal L(X)}\leq 1/N$ (as $\tilde A$ is maximal dissipative), we have $|(\tilde A_N)^*B^{1/2}|_{\mathcal L(X)}=N| (R_N)^*\tilde A^* B^{1/2}|_{\mathcal L(X)} \leq N |R_N |_{\mathcal L(X)} |\tilde A^* B^{1/2}|_{\mathcal L(X)} \leq C$. Hence, since $D\phi=BD_{-1}\phi$ for $\phi \in C^{1,1} ( X_{-1})$, $$\begin{aligned} |(\tilde A_N)^*[D \phi(Y^N(s))- D \phi(Y(s))]|_X&=|(\tilde A_N)^*B^{1/2}[B^{1/2}D_{-1} \phi(Y^N(s))- B^{1/2}D_{-1} \phi(Y(s))]|_X\\ &\leq |(\tilde A_N)^*B^{1/2} |_{\mathcal L(X)} |D_{-1} \phi(Y^N(s))- D_{-1} \phi(Y(s))|_{-1}\\ & \leq C |D_{-1} \phi(Y^N(s))- D_{-1} \phi(Y(s))|_{-1} \leq C | Y^N(s)- Y(s)|_{-1} \\ & \leq C | Y^N(s)- Y(s)|_X \xrightarrow[]{N \to \infty} 0, \quad a.s.\,\,\forall s \geq 0.\end{aligned}$$ For the second term, we note again that $D \phi(Y(s))=BD_{-1} \phi(Y(s)) \in R(B^{1/2})= D(\tilde A^*)$ for every $s \leq T$, so that by the fundamental property of Yosida approximations $$\begin{aligned} \lim_{N \to \infty}|\tilde A_N^* D \phi(Y(s))-\tilde A^* D \phi(Y(s))|_X=0, \quad a.s.\,\,\forall s \geq 0,\end{aligned}$$ and we have the claim. We also have $|(\tilde A_N)^*D \phi(x)|_X\leq |\tilde A^*D \phi(x)|_X\leq C_R$ for $|x|_X\leq R$. The lemma now follows by letting $N \to \infty$ in [\[eq:proof_ito_projected\]](#eq:proof_ito_projected){reference-type="eqref" reference="eq:proof_ito_projected"} and using the dominated convergence theorem (note the stopping time $\tau_N^R$). ◻ # Second approximation in the space $\mathcal D$ {#sec:lions_approx} In this section we use a regularization procedure inspired by [@lions-infdim1] to produce functions $V_\epsilon^\eta \in \mathcal D$ approximating $V_\epsilon,$ which are almost classical supersolutions of perturbed HJB equations. Since $V_\epsilon^\eta \in \mathcal D$, we will then be able to use Dynkin's formula in order to solve the optimal control problem. **Lemma 27**. *Let Assumptions [Assumption 1](#hp:state){reference-type="ref" reference="hp:state"} and [Assumption 2](#hp:cost){reference-type="ref" reference="hp:cost"} hold. Let $z \in C^{1,1}(X_{-1})$ be such that $D_{-1}z$ is bounded on $X_{-1}$. Suppose $z$ is a viscosity supersolution of $$\begin{aligned} \rho z(x)-\langle \tilde A^*Dz(x),x \rangle_X+\tilde H \left (x,D_{x_0}z(x),D^2_{x^2_0}z(x) \right) =-\gamma \quad \mbox{in}\,\,X.\end{aligned}$$ Then for every $\eta>0$ there exist $z^\eta \in \mathcal D$ such that $$\label{eq:convergence_convolution} |z-z^\eta|\leq C \eta, \quad |D_{-1}z-D_{-1}z^\eta|_{-1}\leq C \eta$$ for some $C>0$ (independent of $\eta$) and such that for every $R>0$, $z^\eta$ is a viscosity supersolution of $$\begin{aligned} \rho z^\eta(x)-\langle \tilde A^*Dz^\eta(x),x \rangle_X+\tilde H \left (x,D_{x_0}z^\eta(x), D^2_{x_0^2}z^\eta(x) \right) \geq -\gamma - \omega_R(\eta) \quad \mbox{in}\,\, B_R\end{aligned}$$ for some local moduli of continuity $\omega_R$.* *Proof.* The proof extends the ideas of [@lions-infdim1 Proof of Lemma IV.1] to the case of HJB equations with unbounded operators. We point out that in this section we use notation which is different from the one used elsewhere. We take the orthonormal basis $\{e_i \}$ of $X_{-1}$ defined in Subsection [3.2](#sec:operator_B){reference-type="ref" reference="sec:operator_B"}, where $e_i \in X$ for $i\in\mathbb N$, and we identify $X_{-1}$ with $l^2(\mathbb N)$ or equivalently with $\mathbb R^k \times X^{k,\perp}$ for $k \in \mathbb N$, where recall from Subsection [3.2](#sec:operator_B){reference-type="ref" reference="sec:operator_B"} $$X^k=\mbox{span}(f_1,...f_k)=\mbox{span}(e_1,...e_k).$$ Hence for $x \in X_{-1}$ we write $x=(x^1,x^2,...)=(x^1...,x^k,x')=(x_k,x')$ where $x_k=(x^1,...,x^k) \in X^k \sim \mathbb R^k$, $x'=(x^{k+1},x^{k+2},...) \in X^{k,\perp}$. Here $\{x_i\}$ represent the coordinates of $x \in X_{-1}$ with respect to the orthonormal basis $\{ e_i\}$ of $X_{-1}$. Since $X \subset X_{-1}$, with this notation any element in $x \in X$ will also be denoted by $x=(x_k,x')$. We remark that this notation should not be confused with the notation $x=(x_0,x_1) \in X$ which is used in the rest of the paper, so in general we have $x_k \neq x_0$ and $x' \neq x_1$. We also point out that sometimes we will use $N$ instead of $k$, i.e. $x=(x_N,x')$. Consider a standard mollifier function $\rho \in C^\infty(\mathbb R)$ with $\mbox{supp}(\rho) \subset [-1,1]$, $\rho \geq 0$, $\int_\mathbb{R} \rho dx=1$. Let $\eta>0$. We define $$\begin{aligned} &z^\eta (x)=\lim_{k \to \infty} z^{\eta,k}(x), \quad \quad z^{\eta,k}(x)=\int_{\mathbb R^k} z(y^1,...,y^k,x') \prod_{i=1}^k \rho_{\eta_i}(x^i-y^i) dy^1...dy^k,\end{aligned}$$ for every $x=(x^1,x^2,...)=(x^1,...,x^k,x')=(x_k,x') \in X_{-1}$ with $x_k \in \mathbb R^k \sim X^k, x' \in X^{k,\perp}$, $\eta_i:=\eta \sqrt{\lambda_i}/2^{i}$, $\rho_h(x)=(1/h ) \rho(x/h)$ for every $h >0$. Recall that the $\lambda_i$ are the eigenvalues of the operator $B$, see Subection [3.2](#sec:operator_B){reference-type="ref" reference="sec:operator_B"}. Note that $$\label{eq:sum_eta_i} \sum_{i=1}^\infty \eta_i =c\eta.$$ **Step 1:** We prove that $z^\eta$ is well defined, $z^\eta \in \mathcal D$ and it is close to $z$. We first claim that $$\begin{aligned} \label{eq:proof_convolution_cauchy} \sup_{X_{-1}} |z-z^{\eta,1}|\leq C \eta_1, \quad \sup_{X_{-1}} |z^{\eta,k+1}-z^{\eta,k} |\leq C \eta_{k+1},\quad k\in\mathbb N. \end{aligned}$$ Indeed, since $z$ is Lipschitz in $X_{-1}$, we have $$\begin{aligned} |z(x)-z^{\eta,1}(x)|& = \Bigg | \int_\mathbb{R} \left [ z(x^1,x')- z(y^1,x') \right ] \rho_{\eta_1}(x^1-y^1) dy^1 \Bigg | \leq C \int_\mathbb{R} |x^1-y^1| \rho_{\eta_1}(x^1-y^1) dy^1 \leq C \eta_1 \end{aligned}$$ and for $k \in \mathbb N$ $$\begin{aligned} |z^{\eta,k+1}(x)-z^{\eta,k}(x)| = & \Bigg | \int_{\mathbb{R}^k} \prod_{i=1}^k \rho_{\eta_i}(x^i-y^i) \Bigg [ \int_\mathbb{R} z(y^1,...,y^k,y^{k+1},x') \rho_{\eta_{k+1}}(x^{k+1}-y^{k+1}) dy^{k+1} \\ & - z(y^1,...,y^k, x^{k+1},x') \Bigg ] dy^1...dy^k \Bigg |\\ \leq & \int_{\mathbb{R}^k} \prod_{i=1}^k \rho_{\eta_i}(x^i-y^i) \int_\mathbb{R} \big |z(y^1,...,y^k,y^{k+1},x') - z(y^1,...,y^k, x^{k+1},x') \big | \\ & \times \rho_{\eta_{k+1}}(x^{k+1}-y^{k+1}) dy^{k+1} dy^1...dy^k \leq C \eta_{k+1} \end{aligned}$$ We recall that $D_{-1}z^{\eta,k}=(D_{-1}z)^{\eta,k}$, i.e. the derivative of the convolution is the convolution of the derivative. Then, since $D_{-1}z$ is Lipschitz in $X_{-1}$, with a similar calculation (with $D_{-1}z$ in place of $z$ inside the integrals and these are now meant in the Bochner sense) we obtain $$\begin{aligned} \label{eq:proof_derivative_convolution_cauchy} \sup_{X_{-1}} |D_{-1}z-D_{-1}z^{\eta,1}|_{-1} \leq C \eta_1, \quad \sup_{X_{-1}} |D_{-1}z^{\eta,k+1}-D_{-1}z^{\eta,k} |_{-1}\leq C \eta_{k+1}, \quad k\in\mathbb N.\end{aligned}$$ Now, since $D_{-1}z$ is Lipshitz, if $x=(x_k,x')\in X_{-1}$, $\partial_{ij} z(x)$ exists for $i,j\leq k$ for a.e. $x_k$ and there exists $C>0$ such that $|\partial_{ij} z| \leq C$ (and $C>0$ is independent of $i,j,k$). Thus, for every $i,j \leq k$, we have $\partial_{ij}z^{\eta,k}=(\partial_{ij}z)^{\eta,k}$ so that $$\begin{aligned} \label{eq:proof_second_derivative_bounded} \sup_{X_{-1}} |\partial_{ij} z^{\eta,k}| \leq C \quad \forall i,j \leq k.\end{aligned}$$ Next we show that $$\begin{aligned} \label{eq:proof_second_derivative_convergence} \sup_{X_{-1}} |\partial_{ij} z^{\eta,k+1}-\partial_{ij} z^{\eta,k}| \leq C \frac{\eta_{k+1}}{\eta_i \eta_j}, \quad \forall i,j \leq k.\end{aligned}$$ Indeed, note that $\partial_{i } \rho_{\eta_i}(x_i)=1/ \eta_i c_{\eta_i}(x_i)$ where $c_{\eta_i}(x_i)=1/ \eta_i \rho' (x_i / \eta_i)$. Assume $i \neq j$ (the case $i=j$ is treated in a similar way). Since $z$ is Lipschitz, we have $$\begin{aligned} |\partial_{ij }z^{\eta,k+1}(x)-& \partial_{ij } z^{\eta,k}(x)| = \Bigg | \int_{\mathbb{R}^k} \partial_{i } \rho_{\eta_i}(x_i-y_i) \partial_{j }\rho_{\eta_j}(x_j-y_j) \prod_{h=1, h \neq i,j}^k \rho_{\eta_h}(x_h-y_h) \times \\ & \qquad\qquad\quad \Bigg [ \int_\mathbb{R} z(y^1,...,y^k,y^{k+1},x') \rho_{\eta_{k+1}}(x^{k+1}-y^{k+1}) dy^{k+1} - z(y^1,...,y^k, x^{k+1},x') \Bigg ] dy^1...dy^k \Bigg |\\ \leq & \frac{1}{\eta_i \eta_j} \int_{\mathbb{R}^k} \prod_{h=1, h \neq i,j}^k \rho_{\eta_h}(x^h-y^h) c_{\eta_i}(x_i) c_{\eta_j}(x_j) \int_\mathbb{R} \big |z(y^1,...,y^k,y^{k+1},x') - z(y^1,...,y^k, x^{k+1},x') \big | \\ & \qquad\qquad\qquad\qquad \times \rho_{\eta_{k+1}}(x^{k+1}-y^{k+1}) dy^{k+1} dy^1...dy^k \leq C \frac{\eta_{k+1}}{\eta_i \eta_j}. \end{aligned}$$ Now observe that $D_{-1} \partial_{ij}z^{\eta,k} =\partial_{ij} D_{-1}z^{\eta,k}$ for every $i,j \leq k$. Then, a similar calculation as that done to prove [\[eq:proof_second_derivative_convergence\]](#eq:proof_second_derivative_convergence){reference-type="eqref" reference="eq:proof_second_derivative_convergence"} (with $D_{-1}z$ in place of $z$ inside the integrals), since $D_{-1}z$ is Lipschitz in $X_{-1}$, we obtain $$\begin{aligned} \label{eq:proof_third_derivative_convergence} \sup_{X_{-1}} |D_{-1} \partial_{ij} z^{\eta,k+1}-D_{-1} \partial_{ij} z^{\eta,k}|_{-1}=\sup_{X_{-1}} |\partial_{ij} D_{-1} z^{\eta,k+1}-\partial_{ij} D_{-1} z^{\eta,k}|_{-1} \leq C \frac{\eta_{k+1}}{\eta_i \eta_j}, \quad \forall i,j \leq k.\end{aligned}$$ Finally, since $z$ is Lipschitz in $X_{-1}$ and thus $D_{-1}z$ is bounded, using a similar calculation as that to prove [\[eq:proof_third_derivative_convergence\]](#eq:proof_third_derivative_convergence){reference-type="eqref" reference="eq:proof_third_derivative_convergence"}, we get $$\begin{aligned} \label{eq:proof_third_derivative_bounded} \sup_{X_{-1}}| D_{-1} \partial_{ij} z^{\eta,k}|_{-1}= \sup_{X_{-1}}| \partial_{ij} D_{-1}z^{\eta,k}|_{-1} \leq \frac{C}{\eta_i \eta_j} , \quad \forall i,j \leq k. \end{aligned}$$ Set $g_k=z^{\eta,k}-z^{\eta,1}$. Note that by [\[eq:sum_eta_i\]](#eq:sum_eta_i){reference-type="eqref" reference="eq:sum_eta_i"} and [\[eq:proof_convolution_cauchy\]](#eq:proof_convolution_cauchy){reference-type="eqref" reference="eq:proof_convolution_cauchy"}, we have $\{g_k\} \subset C^{1,1}_B (X_{-1})$, where $C^{1,1}_B (X_{-1})$ is the subspace of functions in $C^{1,1} (X_{-1})$ which are bounded and have bounded derivatives. Moreover, $g^{k+1}-g^k=z^{\eta,k+1}-z^{\eta,k}$, so that by [\[eq:sum_eta_i\]](#eq:sum_eta_i){reference-type="eqref" reference="eq:sum_eta_i"}, [\[eq:proof_convolution_cauchy\]](#eq:proof_convolution_cauchy){reference-type="eqref" reference="eq:proof_convolution_cauchy"}, [\[eq:proof_derivative_convolution_cauchy\]](#eq:proof_derivative_convolution_cauchy){reference-type="eqref" reference="eq:proof_derivative_convolution_cauchy"}, $\{g_k \}$ is a Cauchy sequence in $C^{1}_b (X_{-1})$, where $C^{1}_b (X_{-1})$ is the subspace of functions in $C^{1} (X_{-1})$ which are bounded and have bounded derivatives. Then $g_k \to g$ in $C^{1}_b (X_{-1})$ as $k\to \infty$ to a function $g \in C^{1}_b (X_{-1})$ of the form $g=z^{\eta}-z^{\eta,1}$ for some $z^{\eta} \in C^{1} (X_{-1})$. This implies that $$\label{eq:proof_convergence_convolution} \lim_{k \to \infty} \sup_{X_{-1}} |z^{\eta,k} - z^{\eta}| =0, \quad \lim_{k \to \infty} \sup_{X_{-1}} |D_{-1}z^{\eta,k} - D_{-1}z^{\eta}|_{-1} =0.$$ Note that since $z, D_{-1}z$ are Lipschitz in $X_{-1}$, $z^{\eta,k},D_{-1}z^{\eta,k}$ are families of Lipschitz functions with respect to $|\cdot |_{-1}$ with Lipschitz constants of $z, D_{-1}z$ (so independent of $\eta,k$). Thus, letting $k \to \infty$, we derive that $z^\eta \in C^{1,1}(X_{-1})$ and $z^\eta, D_{-1}z^\eta$ are Lipschitz with respect to $|\cdot |_{-1}$ with Lipschitz constants independent of $\eta$. Now, by [\[eq:proof_convolution_cauchy\]](#eq:proof_convolution_cauchy){reference-type="eqref" reference="eq:proof_convolution_cauchy"}, [\[eq:proof_derivative_convolution_cauchy\]](#eq:proof_derivative_convolution_cauchy){reference-type="eqref" reference="eq:proof_derivative_convolution_cauchy"}, [\[eq:proof_convergence_convolution\]](#eq:proof_convergence_convolution){reference-type="eqref" reference="eq:proof_convergence_convolution"}, we have $$\begin{aligned} |z^{\eta}(x)-z(x)|\leq |z^{\eta} (x)- z^{\eta,k+1}(x)|+ \sum_{i=1}^{k} |z^{\eta,i+1}(x)-z^{\eta,i}(x)| + |z^{\eta,1}(x)-z(x)| \leq \omega_\eta \left (\frac 1 k \right) +C \sum_{i=0}^{k} \eta_{i+1} \end{aligned}$$ and $$\begin{aligned} |D_{-1}z^{\eta}(x)-D_{-1}z(x)|_{-1} &\leq |D_{-1}z^{\eta} (x)- D_{-1}z^{\eta,k+1}(x)|_{-1}\\ &+ \sum_{i=1}^{k} |D_{-1}z^{\eta,i+1}(x)-D_{-1}z^{\eta,i}(x)|_{-1} + |D_{-1}z^{\eta,1}(x)-D_{-1}z(x)|_{-1}\\ & \leq \omega_\eta \left (\frac 1 k \right) +C \sum_{i=0}^{k} \eta_{i+1} \end{aligned}$$ for some modulus $\omega_\eta$, so that, by letting $k\to \infty$ and recalling [\[eq:sum_eta_i\]](#eq:sum_eta_i){reference-type="eqref" reference="eq:sum_eta_i"}, we obtain [\[eq:convergence_convolution\]](#eq:convergence_convolution){reference-type="eqref" reference="eq:convergence_convolution"}. Proceeding in a similar way, by [\[eq:proof_second_derivative_bounded\]](#eq:proof_second_derivative_bounded){reference-type="eqref" reference="eq:proof_second_derivative_bounded"}, [\[eq:proof_second_derivative_convergence\]](#eq:proof_second_derivative_convergence){reference-type="eqref" reference="eq:proof_second_derivative_convergence"}, [\[eq:proof_third_derivative_convergence\]](#eq:proof_third_derivative_convergence){reference-type="eqref" reference="eq:proof_third_derivative_convergence"}, [\[eq:proof_third_derivative_bounded\]](#eq:proof_third_derivative_bounded){reference-type="eqref" reference="eq:proof_third_derivative_bounded"}, we have $$\label{eq:proof_second_derivative_convergence_to_z_eta} \lim_{k \to \infty} \sup_{X_{-1}} |\partial_{ij}z^{\eta,k} - \partial_{ij} z^\eta| =0, \quad \lim_{k \to \infty} \sup_{X_{-1}} |D_{-1} \partial_{ij}z^{\eta,k} - D_{-1}\partial_{ij} z^\eta|_{-1} =0, \quad \forall i,j \in \mathbb N.$$ Note that, since $z, D_{-1}z$ are Lipschitz in $X_{-1}$, by the properties of convolutions we have that for every $i,j$, $\partial_{ij}z^{\eta,k}, D_{-1} \partial_{ij}z^{\eta,k}=\partial_{ij}D_{-1} z^{\eta,k}$ are families of Lipschitz functions with respect to the $|\cdot|_{-1}$ norm with Lipschitz constants independent of $\eta,k$. Letting $k \to \infty$, it follows that for every $i,j$ the functions $\partial_{ij} z^\eta \in C^{1,1}(X_{-1})$ and $\partial_{ij} z^\eta$ are Lipschitz with respect to the $|\cdot|_{-1}$ norm with a Lipschitz constant independent of $\eta$. Moreover, note that by [\[eq:proof_second_derivative_bounded\]](#eq:proof_second_derivative_bounded){reference-type="eqref" reference="eq:proof_second_derivative_bounded"} we have $|\partial_{ij}z^\eta| \leq C$ for a constant $C$ independent of $i,j$. To conclude that $z^\eta \in \mathcal D$, we have to check that for every $k,h \in X_{-1}$ the limit $$\label{eq:z_eta_second_order_derivative} \lim_{t \to 0} \frac {1} {|t|} \langle D_{-1}z^\eta (x+tk)-D_{-1}z^\eta(x) ,h \rangle_{-1}$$ exists and is uniformly continuous in $X_{-1}$. Fix $k,h \in X_{-1}$, $n>0$ and set $k_n=(k^1,...k^n,0,0...), h_n=(h^1,...h^n,0,0...) \in X_{-1}$. Let $x\in X_{-1}$. We denote by $A_n(x)$ the operator from $X^n$ to $X^n$ given by the matrix $(\partial_{ij}z^\eta(x))_{i,j}$. We extend it to $X_{-1}$ by setting $T_n(x)=P_nA_n(x)P_n$. We have $|T_n(x)|_{\mathcal L(X_{-1})}\leq C$ for all $n$. We have for $n>m$ $$\begin{split} |\langle T_n(x)h,k\rangle_{-1}&-\langle T_m(x)h,k\rangle_{-1}|=|\langle A_n(x)P_n h,P_nk\rangle_{-1}-\langle A_n(x)P_mh,P_mk\rangle_{-1}| \\ &\leq |\langle A_n(x)(P_n-P_m) h,P_nk\rangle_{-1}|+|\langle A_n(x)P_mh,(P_n-P_m)k\rangle_{-1}|\to 0\quad\mbox{as}\,\,m\to\infty. \end{split}$$ Therefore the sequence $\{\langle T_n(x)h,k\rangle_{-1}\}$ is a Cauchy sequence and thus $T_n(x)h$ converges weakly in $X_{-1}$ to an element of $X_{-1}$ which we denote by $\tilde D_{-1}^2z^\eta(x)h$. It is easy to see that such defined $\tilde D_{-1}^2z^\eta(x)$ is a linear, bounded and self-adjoint operator on $X_{-1}$. We fix $\delta>0$ and let $n_0$ be such that for $n\geq n_0$ we have $$\label{eq:A(x)n} \Bigg | \langle \tilde D_{-1}^2z^\eta(x)h,k\rangle_{-1}-\sum_{i,j=1}^n \partial_{ij}z^\eta(x)h_i k_j \Bigg | \leq \frac{\delta}{3}.$$ We now estimate $$\begin{aligned} \label{eq:conv-n-a} \Bigg | \frac {1} {|t|} \langle D_{-1}z^\eta (x+tk)&-D_{-1}z^\eta(x) ,h \rangle_{-1} - \sum_{i,j=1}^n \partial_{ij}z^\eta(x)h^i k^j \Bigg | \\ & \leq \frac {1} {|t|} \Bigg | \langle D_{-1} z^\eta (x+tk)-D_{-1} z^\eta(x) ,h \rangle_{-1} - \langle D_{-1} z^\eta (x+tk_n)-D_{-1} z^\eta(x) ,h_n \rangle_{-1} \Bigg |\nonumber\\ & \quad + \Bigg | \frac {1} {|t|} \langle D_{-1} z^\eta (x+tk_n)-D_{-1} z^\eta (x) ,h_n \rangle_{-1} - \sum_{i,j=1}^n \partial_{ij}z^\eta(x)h^i k^j \Bigg | \nonumber\\ & =: I_1(x,n,t) +I_2(x,n,t).\nonumber\end{aligned}$$ By the Lipschitzianity of $D_{-1} z$ we have $$\begin{aligned} I_1(x,n,t) & \leq \frac {1} {|t|} \Big | \langle D_{-1} z^\eta (x+tk)-D_{-1} z^\eta(x) ,h-h_n \rangle_{-1} \Big | + \frac {1} {|t|} \Big | \langle D_{-1} z^\eta (x+tk)-D_{-1} z^\eta(x+tk_n) ,h_n \rangle_{-1} \Big |\\ & \leq C |k|_{-1} |h-h_n|_{-1}+ C |k-k_n|_{-1} |h|_{-1}.\end{aligned}$$ Regarding $I_2$, since $s\mapsto \langle D_{-1} z^\eta (x+sk_n),h_n \rangle_{-1}$ is smooth, by the mean value theorem there is $\tilde t \in \mathbb R, |\tilde t| \leq |t|$ such that $$\begin{aligned} I_2(x,n,t)& = \left | \sum_{i,j=1}^n \partial_{ij}z^\eta(x+\tilde t k^n)h^i k^j - \sum_{i,j=1}^n \partial_{ij}z^\eta(x)h^i k^j \right | \leq C_n |k|_{-1}^2 |h|_{-1} |t|,\end{aligned}$$ where the inequality follows by the Lipschitzianity of $\partial_{ij}z^\eta$ for every $i,j \in \mathbb N.$ We can now find $\overline n>n_0$ such that $I_1(x,\overline n,t)<\delta /3$. Then we choose $|t|$ small enough such that $I_2(x,\overline n,t) <\delta/3$, so that for such $t$ we have $$\Bigg |\frac {1} {|t|} \langle D_{-1}z^\eta (x+tk)-D_{-1}z^\eta(x) ,h \rangle_{-1}- \langle \tilde D_{-1}^2z^\eta(x)h,k\rangle_{-1}\Bigg |<\delta.$$ Therefore the limit in [\[eq:z_eta_second_order_derivative\]](#eq:z_eta_second_order_derivative){reference-type="eqref" reference="eq:z_eta_second_order_derivative"} exists and is equal to $\langle \tilde D_{-1}^2z^\eta(x)h,k\rangle_{-1}$. To prove that the latter expression is uniformly continuous in $X_{-1}$, we send $t\to 0$ in [\[eq:conv-n-a\]](#eq:conv-n-a){reference-type="eqref" reference="eq:conv-n-a"} to obtain $$\Bigg |\langle \tilde D_{-1}^2z^\eta(x)h,k\rangle_{-1}- \sum_{i,j=1}^n \partial_{ij}z^\eta(x)h^i k^j \Bigg |\leq C |k|_{-1} |h-h_n|_{-1}+ C |k-k_n|_{-1} |h|_{-1}$$ which shows that $\langle \tilde D_{-1}^2z^\eta(x)h,k\rangle_{-1}$ is the uniform limit of uniformly continuous functions in $X_{-1}$. **Step 2:** Let $N \in\mathbb N$. We will prove that for every fixed $\bar x' \in X^{N,\perp}$ such that $(0,\bar x') \in X$, the function $z_{\bar x'}:=z(\cdot,\bar x')$ is a viscosity supersolution of a certain HJB equation on $\mathbb{R}^N$. We recall that we use the notation $x=(x_N,x') \in X$ defined at the beginning of the proof. Let $\bar x_N \in X^N$ be a minimum of $z_{\bar x'}+\Psi(\cdot)=z(\cdot, \bar x')+\Psi(\cdot)$ for $\Psi \in C^2(\mathbb R^N)$. (This means that if $x=\sum_i x^i e_i$ then $\Psi(x)=\Psi(x_N)=\Psi(x^1,...,x^N)$.) Then, for every $x=(x_N,x') \in X$, using $z \in C^{1,1}(X_{-1})$ and Young's inequality, for any $\delta>0$ $$\begin{aligned} z(x_N,x')+\Psi(x_N) =& z(x_N,x')-z(x_N,\bar x')- \langle D_{-1}z(x_N,\bar x'),(0, x'-\bar x') \rangle_{-1} \\ & \quad +z(x_N,\bar x')+\Psi(x_N)+\langle D_{-1}z(x_N,\bar x'),(0, x'-\bar x') \rangle_{-1} \\ & \geq -\frac C 2 |(0,x'-\bar x')|_{-1}^2+z(x_N,\bar x')+\Psi(x_N)+\langle D_{-1}z(\bar x_N,\bar x'),(0, x'-\bar x') \rangle_{-1}\\ & \quad -C |(x_N-\bar x_N,0)|_{-1}|(0,x'-\bar x')|_{-1} \\ & \geq z(\bar x_N,\bar x')+\Psi(\bar x_N)+\langle D_{-1}z(\bar x_N,\bar x'),(0, x'-\bar x') \rangle_{-1}\\ &\quad - \frac \delta 2 |(x_N-\bar x_N,0)|_{-1}^2- \frac C 2 \left (1+ \frac 1 \delta \right )|(0,x'-\bar x')|_{-1}^2.\end{aligned}$$ This implies that $z+\overline \Psi$ has a minimum at $\bar x=(\bar x_N,\bar x')$, where $$\overline \Psi(x):=\Psi(x_N) -\langle D_{-1}z(\bar x_N,\bar x'),(0, x'-\bar x') \rangle_{-1}+ \frac \delta 2 |(x_N-\bar x_N,0)|_{-1}^2+ \frac C 2 \left (1+ \frac 1 \delta \right )|(0,x'-\bar x')|_{-1}^2.$$ We notice that since $x=(x_N,x')\in X$, we can write $$\overline \Psi(x)=\Psi(x_N) -\langle BD_{-1}z(\bar x),(0, x'-\bar x') \rangle_X+ \frac \delta 2 |(x_N-\bar x_N,0)|_{-1}^2+ \frac C 2 \left (1+ \frac 1 \delta \right )|(0,x'-\bar x')|_{-1}^2.$$ Hence, we have $D\overline \Psi( x)=D\Psi(x_N)-Q_NBD_{-1}z(x) + \delta BP_N (x-\bar x) +C \left (1+ \frac 1 \delta \right ) BQ_N(x-\bar x),$ so that $$\begin{aligned} &D\overline \Psi(\bar x)=D\Psi(\bar x_N)-Q_NBD_{-1}z(\bar x), \quad D^2 \overline \Psi(\bar x)=D^2\Psi(\bar x_N)+\delta BP_N + C \left (1+ \frac 1 \delta \right )BQ_N.\end{aligned}$$ Here $D\overline \Psi(x), D^2\Psi(x), D\Psi( x_N), D^2\Psi( x_N)$ denote the standard Fréchet derivatives in $X$. Recalling that $D\Psi( x_N)=P_N D\Psi( x_N) ,D^2\Psi( x_N)=P_N D^2\Psi( x_N) P_N$, $D\Psi( x_N)$ and $D^2\Psi( x_N)$ are also first and second order derivatives of $\Psi$ as a function on $X^N\sim \mathbb R^N$, where $X^N$ is considered as a subspace of $X$. Since $z$ is a viscosity supersolution, we now have $$\begin{aligned} \rho z(\bar x)& + \langle \tilde A^*D \Psi(\bar x_N),\bar x \rangle_X -\langle \tilde A^*Q_NBD_{-1}z(\bar x),\bar x\rangle_X \\ & + H \Big (\bar x,-D \Psi(\bar x_N) +Q_NBD_{-1}z(\bar x),-D^2\Psi(\bar x_N)-\delta BP_N - C \left (1+ \frac 1 \delta \right )BQ_N \Big ) \ \geq - \gamma,\end{aligned}$$ i.e. $$\begin{aligned} \rho z(\bar x)& + \langle \tilde A^*D \Psi(\bar x_N),\bar x \rangle_X-\langle \tilde A^* Q_NBD_{-1}z(\bar x),\bar x\rangle_X\\ & + \tilde H \Big (\bar x,-D_{x_0} \Psi(\bar x_N) +(Q_NBD_{-1}z(\bar x))_0,-D_{x_0^2}^2\Psi(\bar x_N)-\delta (BP_N)_{00} - C \left (1+ \frac 1 \delta \right )(BQ_N)_{00} \Big) \geq - \gamma.\end{aligned}$$ By [\[eq:A\*B\^1/2\]](#eq:A*B^1/2){reference-type="eqref" reference="eq:A*B^1/2"} we can write $\langle \tilde A^* Q_NBD_{-1}z(\bar x),\bar x\rangle_X=\langle \tilde A^*B^{1/2} Q_NB^{1/2}D_{-1}z(\bar x), \bar x\rangle_X$ so that $$\begin{aligned} \rho z(\bar x)& + \langle \tilde A^*D \Psi(\bar x_N),\bar x \rangle_X\\ & \quad + \tilde H \Big (\bar x,-D_{x_0} \Psi(\bar x_N) +(Q_NBD_{-1}z(\bar x))_0,-D_{x_0^2}^2\Psi(\bar x_N)-\delta (BP_N)_{00} - C \left (1+ \frac 1 \delta \right )(BQ_N)_{00} \Big)\\ & \geq - \gamma - C \left |Q_NB^{1/2}D_{-1}z(\bar x)\right | |\bar x|.\end{aligned}$$ By definition of $\tilde H$ we now have $$\begin{aligned} \rho z(\bar x)&+ \langle \tilde A^*D \Psi(\bar x_N),\bar x \rangle_X + \tilde H(\bar x,-D_{x_0} \Psi(\bar x_N) ,-D_{x_0^2}^2\Psi(\bar x_N) )\\ & \geq - \gamma - C \left |Q_NB^{1/2}D_{-1}z(\bar x)\right | |\bar x| + \bar x_0 \cdot (Q_NBD_{-1}z(\bar x))_0 \\ &\quad - \sup_{u\in U} \Bigg \{ - b_0\left (\bar x_0,\int_{-d}^0 a_1(\xi)\bar x_1(\xi)\,d\xi ,u \right) \cdot (Q_NBD_{-1}z(\bar x))_0 \Bigg \} \\ &\quad + {1\over 2} \mathop{\mathrm{Tr}}\nolimits\left [ \sigma_0 \left (\bar x_0,\int_{-d}^0 a_2(\xi)\bar x_1(\xi)\,d\xi \right)\sigma_0 \left ( \bar x_0,\int_{-d}^0 a_2(\xi)\bar x_1(\xi)\,d\xi\right)^T \left (-\delta BP_N -C \left (1+ \frac 1 \delta \right )BQ_N \right)_{00} \right]\\ & \geq - \gamma - C \left |Q_NB^{1/2}D_{-1}z(\bar x)\right | |\bar x| + \bar x_0 \cdot (Q_NBD_{-1}z(\bar x))_0 \\ & \quad - C \left (1+ \frac 1 \delta \right ) \mathop{\mathrm{Tr}}\nolimits\left [ \sigma_0 \left (\bar x_0,\int_{-d}^0 a_2(\xi)\bar x_1(\xi)\,d\xi \right)\sigma_0 \left ( \bar x_0,\int_{-d}^0 a_2(\xi)\bar x_1(\xi)\,d\xi\right)^T (BQ_N)_{00} \right] - C_R \delta \\ &=: -\gamma +f_N(\bar x)- C_R \delta.\end{aligned}$$ We point out that for every fixed $\delta>0$, the functions $f_N$ are continuous, locally uniformly (in $N$) bounded in $X$ and for every $x\in X, f_N(x)\to 0$ as $N\to \infty$. Thus we have shown that for every fixed $x' \in X^{N,\perp}$ such that $(0, x') \in X$ the $x_N$-function $z_{x'}:=z(\cdot,x')$ is a viscosity supersolution of $$\begin{aligned} \rho z_{x'}(x_N )&- \langle \tilde A^*D z_{x'}(x_N ),(x_N ,x') \rangle_X + \tilde H( (x_N ,x') ,D_{x_0} z_{x'}(x_N ) ,D_{x_0^2}^2z_{x'}(x_N ) ) = -\gamma - C_R \delta +f_N(x_N , x'), \end{aligned}$$ that is $$\begin{aligned} \label{eq:proof_eq_z(,y)} \rho z_{x'}(\cdot )&- \langle \tilde A^*D z_{x'}(x_N ),(x_N ,x') \rangle_X + \tilde H( (x_N ,x') ,D_{x_0} z_{x'}(x_N ) ) -\frac{1}{2} \mathop{\mathrm{Tr}}\nolimits\Bigg [ \sigma_0 \left ((x_N ,x')_0,\int_{-d}^0 a_2(\xi)(x_N ,x')_1(\xi)\,d\xi \right) \nonumber \\ & \qquad\quad \times \sigma_0 \left ( (x_N ,x')_0,\int_{-d}^0 a_2(\xi)(x_N ,x')_1(\xi)\,d\xi\right)^T D_{x_0^2}^{2} z_{x'}(x_N ) \Bigg] = -\gamma - C_R \delta +f_N(x_N , x'). \end{aligned}$$ Since this is an equation on $\mathbb R^N$, $z_{x'} \in C^{1,1}(\mathbb R^N)$, $\tilde A^*D z_{x'}$ is continuous and all the terms above are well defined, the left-hand side of [\[eq:proof_eq_z(,y)\]](#eq:proof_eq_z(,y)){reference-type="eqref" reference="eq:proof_eq_z(,y)"} is greater than or equal to the right-hand side for a.e. $x_N \in \mathbb R^N$. **Step 3:** Let $R>0$ and let $x' \in X_{N}^\perp$ be such that $(0,x') \in X, |x'|_X\leq R$ and consider the $x_N$-function $z_{x'}^{\eta,k}:=z^{\eta,k}(\cdot,x')$, $k\leq N$. We will show that this function satisfies a perturbed HJB equation. Applying the $X_k$-convolution to both sides of [\[eq:proof_eq_z(,y)\]](#eq:proof_eq_z(,y)){reference-type="eqref" reference="eq:proof_eq_z(,y)"} we have $$\begin{aligned} \rho z_{x'}^{\eta,k}(x_N)&+ \int_{\mathbb R^k} \Bigg \{ - \langle \tilde A^*D z_{x'}(\hat x^1, ...\hat x^k, x^{k+1},...,x^N),(\hat x^1, ...\hat x^k, x^{k+1},...,x^N,x') \rangle_X \nonumber \\ & \quad + \tilde H( (\hat x^1, ...\hat x^k, x^{k+1},...,x^N,x') ,D_{x_0} z_{x'}(\hat x^1, ...\hat x^k, x^{k+1},...,x^N) ) \nonumber \\ & \quad -1/2 \mathop{\mathrm{Tr}}\nolimits\Big [ \sigma_0 \left ((\hat x^1, ...\hat x^k, x^{k+1},...,x^N,x')_0,\int_{-d}^0 a_2(\xi)(\hat x^1, ...\hat x^k, x^{k+1},...,x^N,x')_1(\xi)\,d\xi \right) \nonumber \\ & \quad \times \sigma_0 \left ( (\hat x^1, ...\hat x^k, x^{k+1},...,x^N,x')_0,\int_{-d}^0 a_2(\xi)(\hat x^1, ...\hat x^k, x^{k+1},...,x^N,x')_1(\xi)\,d\xi\right)^* \nonumber \\ &\quad \times D_{x_0^2}^{2} z_{x'}(\hat x^1, ...\hat x^k, x^{k+1},...,x^N) \Big ] \Bigg \} \prod_{i=1}^k \rho_{\eta_i}(x^i-\hat x^i) d\hat x^1... d\hat x^k \nonumber \\ & \geq -\gamma -C_R \delta +\int_{\mathbb R^k} f_N(\hat x^1, ...\hat x^k, x^{k+1},...,x^N, x') \prod_{i=1}^k \rho_{\eta_i}(x^i-\hat x^i) d\hat x^1... d\hat x^k \end{aligned}$$ for a.e. $x_N \in \mathbb R^N$. Note that, since ${\rm supp}( \rho_{\eta_i} )\subset [-\eta_i,\eta_i]$, we are effectively integrating only with respect to $(\hat {x}^1,...\hat x^k) \in X_k \sim \mathbb R^k$ such that $|\hat x^i-x^i| \leq \eta_i=\eta \sqrt{\lambda_i}/2^{i}$. Then for such $\hat x_i$, recalling that $e_i=f_i/ \sqrt{\lambda_i}$ with $\{f_i\}$ being an orthonormal basis of $X$ and by setting $\hat x=(\hat x^1, ...\hat x^k, x^{k+1},...,x^N,x')$, $x=(x_N,x')=(x^1, ... x^k, x^{k+1},...,x^N,x'),$ we have $|\hat x-x|_X^2=\sum_{i \in \mathbb N} |\hat x^i-x^i|^2 /\lambda_i \leq \eta^2.$ Moreover, since $|\tilde A^*D z_{x'}(x)|_X=|\tilde A^*B^{1/2}B^{1/2}P_ND_{-1} z(x)|_X\leq C|D_{-1} z(x)|_{-1}\leq C$. It then follows that $|\langle \tilde A^*D z_{x'}(\hat x^1, ...\hat x^k, x^{k+1},...,x^N),x \rangle_X - \langle \tilde A^*D z_{x'}(\hat x^1, ...\hat x^k, x^{k+1},...,x^N),\hat x \rangle_X|\leq C_R \eta$ for some constant $C_R$ if in addition $|x_N|_X\leq R$. Arguing similarly for the other terms, by Lemma [Lemma 13](#lemma:properties_H){reference-type="ref" reference="lemma:properties_H"} and the Lipschitzianity of $Dz$ as a map from $X_{-1}$ to $X$, we have that there is a modulus $\omega_R$ such that $$\begin{aligned} \rho z_{x'}^{\eta,k}(x_N)&+ \int_{\mathbb R^k} \Bigg \{ - \langle \tilde A^*D z_{x'}(\hat x^1, ...\hat x^k, x^{k+1},...,x^N),(x_N,x') \rangle_x + \tilde H( (x_N,x') ,D_{x_0} z_{x'}(x_N) ) \nonumber \\ &\quad -1/2 \mathop{\mathrm{Tr}}\nolimits\Big [ \sigma_0 \left ((x_N,x')_0,\int_{-d}^0 a_2(\xi)(x_N,x')_1(\xi)\,d\xi \right) \sigma_0 \left ( (x_N,x')_0,\int_{-d}^0 a_2(\xi)(x_N,x')_1(\xi)\,d\xi\right)^T \nonumber \\ & \quad \times D_{x_0^2}^{2} z_{x'}(\hat x^1, ...\hat x^k, x^{k+1},...,x^N) \Big ] \Bigg \} \prod_{i=1}^k \rho_{\eta_i}(x^i-\hat x^i) d\hat x^1... d\hat x^k \nonumber \\ & \geq -\gamma -C_R \delta -\int_{\mathbb R^k} f_N(\hat x^1, ...\hat x^k, x^{k+1},...,x^N, x') \prod_{i=1}^k \rho_{\eta_i}(x^i-\hat x^i) d\hat x^1... d\hat x^k - \omega_R( \eta)\end{aligned}$$ for a.e. $x_N \in \mathbb R^N$ such that $|x_N|\leq R$. We now notice that by [\[eq:proof_derivative_convolution_cauchy\]](#eq:proof_derivative_convolution_cauchy){reference-type="eqref" reference="eq:proof_derivative_convolution_cauchy"}, for every $\in X$ $$\begin{aligned} |D z^{\eta,k}(x)-D z(x)|& =|BD_{-1} z^{\eta,k}(x)-BD_{-1} z(x)|\leq C |D_{-1} z^{\eta,k}(x)-D_{-1} z(x)|_{-1}\\ & \leq C \sum_{i=2}^{k} |D_{-1}z^{\eta,i}(x)-D_{-1}z^{\eta,i-1}(x)|_{-1} + C_R |D_{-1}z^{\eta,1}(x)-D_{-1}z(x)|_{-1} \leq C \eta.\end{aligned}$$ Using this inequality and the fact that the convolutions of derivatives are the derivatives of the convolutions, we now have $$\begin{aligned} \rho & z_{x'}^{\eta,k}(x_N)- \langle \tilde A^*D z^{\eta,k}_{x'}(x_N),(x_N,x') \rangle_X + \tilde H( (x_N,x') ,D_{x_0} z^{\eta,k}_{x'}(x_N) ) \nonumber \\ & \quad -1/2 \mathop{\mathrm{Tr}}\nolimits\Big [ \sigma_0 \left ((x_N,x')_0,\int_{-d}^0 a_2(\xi)(x_N,x')_1(\xi)\,d\xi \right) \sigma_0 \left ( (x_N,x')_0,\int_{-d}^0 a_2(\xi)(x_N,x')_1(\xi)\,d\xi\right)^T D_{x_0^2}^{2} z_{x'}^{\eta,k}(x_N) \Big ] \nonumber \\ & \geq -\gamma -C_R \delta -\int_{\mathbb R^k} f_N(\hat x^1, ...\hat x^k, x^{k+1},...,x^N, x') \prod_{i=1}^k \rho_{\eta_i}(x^i-\hat x^i) d\hat x^1... d\hat x^k - \omega_R (\eta)\end{aligned}$$ for some modulus $\omega_R$. Therefore we proved that if $x' \in X_{N}^\perp$ is such that $(0,x') \in X, |x'|_X\leq R$, then $$\begin{aligned} \label{eq:proof_eq_z_eta_k(,y)} \rho z_{x'}^{\eta,k}(x_N )&- \langle \tilde A^*Dz_{x'}^{\eta,k}(x_N),(x_N,x') \rangle_X + \tilde H( (x_N,x'),D_{x_0} z_{x'}^{\eta,k}(x_N) ,D^{2}_{x_0^2} z_{x'}^{\eta,k}(x_N) ) \nonumber \\ & \geq -\gamma -C_R \delta - \omega_R(\eta) -\int_{\mathbb R^k} f_N(\hat x^1, ...\hat x^k, x^{k+1},...,x^N, x') \prod_{i=1}^k \rho_{\eta_i}(x^i-\hat x^i) d\hat x^1... d\hat x^k \end{aligned}$$ for a.e. $x_N \in \mathbb R^N$ such that $|x_N|\leq R$. Observe that since $z_{x'}^{\eta,k} \in C^{1,1}(\mathbb R^N)$, it is well known that the inequality holds in the viscosity sense in $\mathbb R^N$ (see e.g. [@lions_1983 Theorem I.2]), that is $z_{x'}^{\eta,k}$ is a viscosity supersolution of [\[eq:proof_eq_z\_eta_k(,y)\]](#eq:proof_eq_z_eta_k(,y)){reference-type="eqref" reference="eq:proof_eq_z_eta_k(,y)"}. **Step 4:** We let $N \to \infty$ and prove that $z^{\eta,k}$ is a viscosity supersolution of a perturbed HJB equation on $X$.\ Let $\bar x=(\bar x_N,\bar x') \in X$ be a minimum of $z^{\eta,k}+\Psi$ for a test function $\Psi=\phi+g$ (defined on $X$) with $\phi \in \Phi$ and $g \in \mathcal G$. Then $\bar x_N$ is a minimum for $z_{\bar x'}^{\eta,k}(\cdot)+\Psi(\cdot,\bar x')=z^{\eta,k}(\cdot,\bar x')+\Psi(\cdot,\bar x')$. Since $z^{\eta,k}\in C^{1,1}(X_{-1}), D\Psi(\bar x)=Dz^{\eta,k}(\bar x)=BD_{-1}z^{\eta,k}(\bar x)\in D(\tilde A^*)$. Moreover, we have $$D\Psi(\bar x_N,\bar x')=P_ND\Psi(\bar x)=P_NDz^{\eta,k}(\bar x)=BP_ND_{-1}z^{\eta,k}(\bar x)\quad\mbox{and}\quad D^2\Psi(\bar x_N,\bar x')=P_ND^2\Psi(\bar x)P_N.$$ Thus, as $|P_ND_{-1}z^{\eta,k}(\bar x)|_{-1}\to 0$, $$\label{eq:convatilde} \tilde A^*D\Psi(\bar x_N,\bar x')=\tilde A^*P_ND\Psi(\bar x)\tilde A^*B^{\frac{1}{2}}B^{\frac{1}{2}}P_ND_{-1}z^{\eta,k}(\bar x)\to \tilde A^*BD_{-1}z^{\eta,k}(\bar x)=\tilde A^*D\Psi(\bar x).$$ Since $z_{\bar x'}^{\eta,k}$ is a viscosity supersolution of [\[eq:proof_eq_z\_eta_k(,y)\]](#eq:proof_eq_z_eta_k(,y)){reference-type="eqref" reference="eq:proof_eq_z_eta_k(,y)"}, we have $$\begin{aligned} \rho z^{\eta,k}(\bar x_N, \bar x' )&+ \langle \tilde A^*D \Psi (\bar x_N,\bar x'),\bar x \rangle_X + \tilde H( \bar x,-D_{x_0} \Psi (\bar x_N,\bar x') ,-D_{x_0^2}^2 \Psi(\bar x_N,\bar x') ) \nonumber \\ & \geq -\gamma -C_R \delta - \omega_R(\eta) -\int_{\mathbb R^k} f_N(\hat x_1, ...\hat x_k, \bar x_{k+1},...\bar x_N, \bar x') \prod_{i=1}^k \rho_{\eta_i}(\bar x_i-\hat x_i) d\hat x_1... d\hat x_k.\end{aligned}$$ We recall that $D_{x_0}\Psi(\bar x_N,\bar x')=P_{x_0}P_ND\Psi(\bar x)$ and $D_{x_0^2}^2 \Psi(\bar x_N,\bar x')=P_{x_0}P_ND^2\Psi(\bar x)P_NP_{x_0}$. Therefore, letting $N \to \infty$, using [\[eq:convatilde\]](#eq:convatilde){reference-type="eqref" reference="eq:convatilde"}, the fact that $f_N(x) \to 0$ and the dominated convergence theorem, we obtain $$\begin{aligned} \rho z^{\eta,k}(\bar x )&+ \langle \tilde A^*D \Psi (\bar x),\bar x \rangle_X + \tilde H( \bar x,-D_{x_0} \Psi (\bar x) ,-D_{x_0^2}^2 \Psi(\bar x) ) \geq -\gamma -C_R \delta - \omega_R(\eta) .\end{aligned}$$ We can now let $\delta \to 0$ to get $$\begin{aligned} \rho z^{\eta,k}(\bar x )&+ \langle \tilde A^*D \Psi (\bar x),\bar x \rangle_X + \tilde H( \bar x,-D_{x_0} \Psi (\bar x) ,-D_{x_0^2}^2 \Psi(\bar x) ) \geq -\gamma - \omega_R(\eta). \end{aligned}$$ In particular, we proved that for every $k$, the function $z^{\eta,k}$ is a viscosity supersolution in $X$ of $$\begin{aligned} \label{eq:proof_z_eta_k_viscosity_supersolution} \rho z^{\eta,k}(x )&- \langle \tilde A^*D z^{\eta,k}, x \rangle_X + \tilde H( x,D_{x_0} z^{\eta,k} (x) ,D_{x_0^2}^2 z^{\eta,k}(x) ) = -\gamma - \omega_R(\eta) .\end{aligned}$$ **Step 5:** We use consistency of viscosity solutions to obtain that $z^\eta$ is a viscosity supersolution of a perturbed HJB equation. Since for every $k$, the function $z^{\eta,k}$ is a viscosity supersolution of [\[eq:proof_z\_eta_k\_viscosity_supersolution\]](#eq:proof_z_eta_k_viscosity_supersolution){reference-type="eqref" reference="eq:proof_z_eta_k_viscosity_supersolution"} and, by [\[eq:proof_convergence_convolution\]](#eq:proof_convergence_convolution){reference-type="eqref" reference="eq:proof_convergence_convolution"}, $z^{\eta,k}$ converges uniformly to $z^\eta$, it follows from consistency of viscosity solutions, [@fgs_book Theorem 3.41], that $z^{\eta}$ is a viscosity supersolution of $$\begin{aligned} \rho z^{\eta}(x )&- \langle \tilde A^*D z^{\eta}, x \rangle_X + \tilde H( x,D_{x_0} z^{\eta} (x) ,D_{x_0^2}^2 z^{\eta}(x) ) = -\gamma - \omega_R(\eta).\end{aligned}$$ This completes the proof of the lemma. ◻ Applying Lemma [Lemma 27](#lemma:z_eta){reference-type="ref" reference="lemma:z_eta"} to $z=\tilde V_\epsilon$, we obtain that for every $\eta>0$ there exist $\tilde V_\epsilon^\eta \in \mathcal D$ such that $$\label{eq:convergence_convolution_w_eps_eta} |\tilde V_\epsilon-\tilde V_\epsilon^\eta|\leq C_\epsilon \eta , \quad |D_{-1}\tilde V_\epsilon-D_{-1}\tilde V_\epsilon^\eta|_{-1}\leq C_\epsilon \eta$$ for some $C_\epsilon>0$ (independent of $\eta$) and such that $\tilde V_\epsilon^\eta$ is a viscosity supersolution of $$\begin{aligned} \label{eq:w_eps_eta_supersolution} \rho \tilde V_\epsilon^\eta(x)-\langle \tilde A^*D\tilde V_\epsilon^\eta(x),x \rangle_X +\tilde H \left (x,D_{x_0}\tilde V_\epsilon^\eta(x),D^2_{x_0^2}\tilde V_\epsilon^\eta(x) \right) = -\gamma(\epsilon) - \omega_{R,\epsilon}(\eta), \quad \forall x \in B_R.\end{aligned}$$ Since $\tilde V_\epsilon^\eta \in \mathcal D$ all terms appearing in [\[eq:w_eps_eta_supersolution\]](#eq:w_eps_eta_supersolution){reference-type="eqref" reference="eq:w_eps_eta_supersolution"} are well defined. Thus we will prove that $\tilde V_\epsilon^\eta$ satisfies [\[eq:w_eps_eta_supersolution\]](#eq:w_eps_eta_supersolution){reference-type="eqref" reference="eq:w_eps_eta_supersolution"} pointwise as inequality. **Lemma 28**. *Let Assumptions [Assumption 1](#hp:state){reference-type="ref" reference="hp:state"}, [Assumption 2](#hp:cost){reference-type="ref" reference="hp:cost"}, [Assumption 9](#hp:discount){reference-type="ref" reference="hp:discount"} and [Assumption 19](#hp:convexity){reference-type="ref" reference="hp:convexity"} hold. For every $R>0$ we have $$\begin{aligned} \rho \tilde V_\epsilon^\eta(x)-\langle \tilde A^*D\tilde V_\epsilon^\eta(x),x \rangle_X +\tilde H \left (x,D_{x_0}\tilde V_\epsilon^\eta(x), D^2_{x_0^2}\tilde V_\epsilon^\eta(x) \right) \geq -\gamma(\epsilon) - \omega_{R,\epsilon}(\eta) \quad \forall x \in B_R.\end{aligned}$$* *Proof.* We go back to the standard notation from Subsection [3.1](#subsec:infinite_dimensional_framework){reference-type="ref" reference="subsec:infinite_dimensional_framework"}, that is $x=(x_0,x_1)$ means $x_0\in\mathbb R^n,x_1\in L^2$. Fix $\bar x=(\bar x_0, \bar x_1) \in B_R \subset X$. Since $\tilde V_\epsilon^\eta \in C^{1,1}(X_{-1})$, using Young's inequality, for every $0 <\delta < 1$ we have $$\begin{aligned} \tilde V_\epsilon^\eta(x_0,x_1)-\tilde V_\epsilon^\eta(x_0,\bar x_1) & \geq \langle D_{-1}\tilde V_\epsilon^\eta(x_0,\bar x_1),(0,x_1-\bar x_1) \rangle_{-1} - C |(0,x_1-\bar x_1)|_{-1}^2 \\ & \geq \langle D_{-1} \tilde V_\epsilon^\eta(\bar x),(0,x_1-\bar x_1) \rangle_{-1} -C |(x_0-\bar x_0,0)|_{-1}|(0,x_1-\bar x_1)|_{-1} - C |(0,x_1-\bar x_1)|_{-1}^2 \\ & \geq \langle D_{-1} \tilde V_\epsilon^\eta(\bar x),(0,x_1-\bar x_1) \rangle_{-1} -\delta |(x_0-\bar x_0,0)|_{-1}^2-\frac{C}{\delta}|(0,x_1-\bar x_1)|_{-1}^2 \\ & = \langle D\tilde V_\epsilon^\eta(\bar x),(0,x_1-\bar x_1) \rangle_X -\delta |(x_0-\bar x_0,0)|_{-1}^2-\frac{C}{\delta}|(0,x_1-\bar x_1)|_{-1}^2\\ & = \langle D_{x_1}\tilde V_\epsilon^\eta(\bar x),x_1-\bar x_1 \rangle_{L^2} -\delta |(x_0-\bar x_0,0)|_{-1}^2 -\frac{C}{\delta}|(0,x_1-\bar x_1)|_{-1}^2\end{aligned}$$ for every $x=(x_0,x_1) \in X$. Since $\tilde w_\epsilon^\eta \in \mathcal D$, by Lemma [Lemma 25](#lemma:D^2_x_0^2phi_in_D){reference-type="ref" reference="lemma:D^2_x_0^2phi_in_D"}, using the second order Taylor expansion with respect to $x_0$, we have $$\begin{aligned} \tilde V_\epsilon^\eta(x_0,\bar x_1)- \tilde V_\epsilon^\eta(\bar x_0,\bar x_1) \geq D_{x_0}\tilde V_\epsilon^\eta(\bar x)\cdot(x_0-\bar x_0)+\frac{1}{2}D^2_{x_0^2}\tilde V_\epsilon^\eta(\bar x)(x_0-\bar x_0) \cdot (x_0-\bar x_0) - \delta |(x_0-\bar x_0,0)|_{-1}^2\end{aligned}$$ when $|(x_0-\bar x_0,0)|_{-1}^2$ is small (and we used that the norms $|(\cdot,0)|$ iand $|(\cdot,0)|_{-1}$ are equivalent on $\mathbb{R}^n$). Adding the last two inequalities we now get $$\begin{aligned} \tilde V_\epsilon^\eta(x)&-\tilde V_\epsilon^\eta(\bar x) \geq D_{x_0}\tilde V_\epsilon^\eta(\bar x)\cdot(x_0-\bar x_0) +\langle D_{x_1}\tilde V_\epsilon^\eta(\bar x),x_1-\bar x_1 \rangle_{L^2} +\frac{1}{2}D^2_{x_0^2}\tilde V_\epsilon^\eta(\bar x)(x_0-\bar x_0)\cdot (x_0-\bar x_0) \\ & \qquad\qquad\qquad\qquad\qquad\qquad -2\delta |(x_0-\bar x_0,0)|_{-1}^2-\frac{C}{\delta}|(0,x_1-\bar x_1)|_{-1}^2 \\ & = \langle D\tilde V_\epsilon^\eta(\bar x), x-\bar x \rangle_X +\frac{1}{2}D^2_{x_0^2}\tilde V_\epsilon^\eta(\bar x)(x_0-\bar x_0)\cdot (x_0-\bar x_0) -2\delta |(x_0-\bar x_0,0)|_{-1}^2-\frac{C}{\delta}|(0,x_1-\bar x_1)|_{-1}^2.\end{aligned}$$ Defining $\phi$ by $$\phi(x)=-\tilde V_\epsilon^\eta(\bar x) - \langle D\tilde V_\epsilon^\eta(\bar x), x-\bar x \rangle_X -\frac{1}{2}D^2_{x^2_0}\tilde V_\epsilon^\eta(\bar x)(x_0-\bar x_0)\cdot (x_0-\bar x_0) +2\delta |(x_0-\bar x_0,0)|_{-1}^2+\frac{C}{\delta}|(0,x_1-\bar x_1)|_{-1}^2$$ we have $\tilde V_\epsilon^\eta(x) +\phi(x) \geq 0$ when $|(x_0-\bar x_0,0)|_{-1}^2$ is small so that $\bar x$ is a local minimum for $\tilde V_\epsilon^\eta+\phi$. We notice that $\phi \in \Phi$ since $D\tilde V_\epsilon^\eta(\bar x) \in D(\tilde A^*)$ (recall that $\tilde V_\epsilon^\eta \in C^{1}(X_{-1})$). We also observe that $D |(x_0-\bar x_0,0)|_{-1}^2=2B(x_0-\bar x_0,0)$, $\frac{\partial^2}{\partial^2 x_0^2} |(x_0-\bar x_0,0)|_{-1}^2=2B_{00}$, so that $$-D \phi(\bar x)=D\tilde V_\epsilon^\eta(\bar x), \quad -D^2_{x^2_0}\phi (\bar x)=D^2_{x_0^2}\tilde V_\epsilon^\eta(\bar x)-4\delta B_{00},$$ where $B_{00}$ was defined in [\[eq:representation_B\]](#eq:representation_B){reference-type="eqref" reference="eq:representation_B"}.\ Since $\bar x$ is a local minimum of $\tilde V_\epsilon^\eta+\phi$ and $w_\epsilon$ is a viscosity supersolution of [\[eq:w_eps_eta_supersolution\]](#eq:w_eps_eta_supersolution){reference-type="eqref" reference="eq:w_eps_eta_supersolution"}, we thus have $$\begin{aligned} \rho \tilde V_\epsilon^\eta(\bar x)& + \langle \bar x, \tilde A^*D \phi (\bar x) \rangle_X + \tilde H \left ( \bar x, -D_{x_0}\phi (\bar x) , -D^2_{x^2_0}\phi(\bar x) \right) \geq -\gamma(\epsilon)-\omega_{R,\epsilon}(\eta), \end{aligned}$$ from which by [\[eq:hamiltonian_synthesis\]](#eq:hamiltonian_synthesis){reference-type="eqref" reference="eq:hamiltonian_synthesis"}, we obtain $$\begin{aligned} \rho \tilde V_\epsilon^\eta(\bar x) & -\langle \bar x, \tilde A^*D\tilde V_\epsilon^\eta(\bar x) \rangle_X+\tilde H \left (\bar x,D_{x_0}\tilde V_\epsilon^\eta(\bar x),D^2_{x^2_0}\tilde V_\epsilon^\eta(\bar x) \right)\\ & \geq -\gamma(\epsilon) -\omega_{R,\epsilon}(\eta) - 4\delta\mathop{\mathrm{Tr}}\nolimits\left [ \sigma_0 \left ( \bar x_0,\int_{-d}^0 a_1(\xi)\bar x_1(\xi)\,d\xi \right)\sigma_0 \left ( \bar x_0,\int_{-d}^0 a_1(\xi)\bar x_1(\xi)\,d\xi\right)^T B_{00} \right].\end{aligned}$$ The result follows by letting $\delta\to 0$. ◻ # Verification Theorem and Optimal Synthesis {#sec:verthm-optfeed} In this section we prove a Verification Theorem and construct an optimal feedback control for our problem. We start by proving the following proposition. **Proposition 29**. *Let Assumptions [Assumption 1](#hp:state){reference-type="ref" reference="hp:state"}, [Assumption 2](#hp:cost){reference-type="ref" reference="hp:cost"}, [Assumption 9](#hp:discount){reference-type="ref" reference="hp:discount"} and [Assumption 19](#hp:convexity){reference-type="ref" reference="hp:convexity"} hold. There exists $\overline \rho\geq \rho_0$ such that if $\rho> \overline \rho$ then $\overline V(x)=V(x) \quad \forall x \in X.$* *Proof.* Since $\mathcal{\overline U } \supset \mathcal U$ we immediately have $\overline V \leq V$. Hence we are left to prove $$\label{eq:tilde_V_leq} V \leq \overline V.$$ We divide the proof of this fact into several parts. \(i\) For every $\xi >0$ we define $L^\xi\colon X \times U \to \mathbb R, \tilde b_0^\xi \colon X \times U \to \mathbb R^{n}, \sigma_0^\xi\colon X \to M^{n \times q }$ by $$\begin{aligned} &L^\xi(x,u)=\sup_{z \in X} \inf_{y \in X} \left[ L(y,u) + \frac{1}{2 \xi} |z-y|_{-1}^2- \frac 1 \xi |z-x|_{-1}^2 \right]\\ &(\tilde b_{0}^\xi)_i(x,u)=\sup_{z \in X} \inf_{y \in X} \left[ \tilde (b_{0})_i(y,u) + \frac{1}{2 \xi} |z-y|_{-1}^2- \frac 1 \xi |z-x|_{-1}^2 \right], \quad i=1,...,n,\\ &(\sigma_{0}^\xi)_{ij}(x)=\sup_{z \in X} \inf_{y \in X} \left[ (\sigma_{0})_{ij}(y) + \frac{1}{2 \xi} |z-y|_{-1}^2- \frac 1 \xi |z-x|_{-1}^2 \right] \quad i=1,...,n, j=1,...,q.\end{aligned}$$ The functions $L(\cdot,u),(\tilde b_0)_i(\cdot,u), (\sigma_0)_{ij}$ are Lipschitz in the $|\cdot|_{-1}$ norm. By [@lasry] we have $L^\xi(\cdot,u), (\tilde b_{0}^\xi)_i(\cdot,u), (\sigma_{0}^\xi)_{ij} \in C^{1,1}(X_{-1})$ (where by density of $X \subset X_{-1}$ we have extended $L^\xi, \tilde b_{0}^\xi, \sigma_{0}^\xi$ to $X_{-1}$). Moreover $L^\xi(\cdot,u),(\tilde b_0^\xi )_i (\cdot,u) , (\sigma_0^\xi)_{ij}$ are Lipschitz in the $|\cdot|_{-1}$ norm (with Lipschitz constants independent of $\xi,u$). In fact the Lipschitz constants of $L^\xi(\cdot,u),(\tilde b_0^\xi )_i (\cdot,u) , (\sigma_0^\xi)_{ij}$ are the same as those of $L(\cdot,u),(\tilde b_0)_i(\cdot,u), (\sigma_0)_{ij}$. Finally $$\label{eq:unif_convercenge_L_xi,b_xi,sigma_xi} L^\xi \xrightarrow{\xi \to 0} L, \quad b_{0}^\xi \xrightarrow{\xi \to 0} \tilde b_{0} \quad \textit{uniformly in } X \times U, \quad \sigma_{0}^\xi \xrightarrow{\xi \to 0} \sigma_{0}\quad \textit{uniformly in } X.$$ Define $$\tilde b^\xi(x,u)=[\tilde b_0^\xi(x,u),0]^T, \quad \sigma^\xi(x)w=\begin{bmatrix} \sigma_0^\xi(x) w, 0 \end{bmatrix}^T \quad \forall x \in X, \ w \in \mathbb{R}^q.$$ Now, for every $x \in X, u(\cdot) \in \mathcal U$ we consider approximating optimal control problems (in the reference probability space formulation) with state equations $$\label{eq:state_eq_approximated_Y_xi} dY^\xi(t)=\tilde AY^\xi(t) dt + \tilde b^\xi(Y^\xi(t),u(t))dt+\sigma^\xi(Y^\xi(t))dW_t, \quad Y^\xi(0)=x,$$ and cost functionals and value functions $$J^\xi(x;u(\cdot))=\mathbb E \int_0^\infty e^{-\rho t} L^\xi(Y^\xi(t),u(t)) dt,\quad V^\xi(x)=\inf_{u(\cdot) \in \mathcal{U}} J^\xi(x;u(\cdot)).$$ Moreover, $V^\xi$ satisfies [\[Va-1\]](#Va-1){reference-type="eqref" reference="Va-1"} and Theorem [\[th:existence_uniqueness_viscosity_infinite\]](#th:existence_uniqueness_viscosity_infinite){reference-type="ref" reference="th:existence_uniqueness_viscosity_infinite"} holds for $V^\xi$, that is it is a unique viscosity solution of the HJB equation [\[eq:HJB\]](#eq:HJB){reference-type="eqref" reference="eq:HJB"} with $b,\sigma, L$ replaced by $b^\xi, \sigma^\xi, L^\xi$. \(ii\) Denoting by $Y(t)$ the solution of [\[eq:abstract_dissipative_operator\]](#eq:abstract_dissipative_operator){reference-type="eqref" reference="eq:abstract_dissipative_operator"}, we prove that there exist a modulus of continuity $\omega$ and a constant $\lambda >0$ (both independent of $x,u(\cdot)$) such that $$\label{eq:E_norm_-1_Y_xi-Y} \mathbb E |Y^\xi(t)-Y(t)|_{-1}^2 \leq t \omega(\xi)e^{\lambda t} \quad \forall t \geq 0, \xi >0.$$ Indeed, since for $\xi>0$ $$d(Y^\xi-Y)(t)=\tilde A(Y^\xi-Y)(t) dt +\left [\tilde b^\xi(Y^\xi(t),u(t))-\tilde b(Y(t),u(t)) \right] dt+\left [ \sigma^\xi(Y^\xi(t))-\sigma(Y(t)) \right ]dW(t), \quad (Y^\xi-Y)(0)=x,$$ by Ito's formula [@fgs_book Proposition 1.165] we have $$\begin{aligned} \mathbb E |Y^\xi(t)-Y(t)|_{-1}^2 & = 2 \int_0^t \mathbb E \langle \tilde A^*B (Y^\xi(s)-Y(s)), Y^\xi(s)-Y(s) \rangle_X + \mathbb E \langle B(Y^\xi(s)-Y(s)), \tilde b^\xi(Y^\xi(s),u(s))-\tilde b(Y(s),u(s)) \rangle_X ds\\ & \quad + \int_0^t \mathbb E \operatorname{Tr}\left[ \left ( \sigma^\xi(Y^\xi(s))-\tilde \sigma (Y(s)) \right ) \left ( \sigma^\xi(Y^\xi(s))-\tilde \sigma (Y(s)) \right )^* B \right ]ds \\ & \leq 2 \int_0^t\mathbb E \langle B(Y^\xi(s)-Y(s)), \tilde b^\xi(Y^\xi(s),u(s))-\tilde b(Y(s),u(s)) \rangle_X ds\\ & \quad + \int_0^t \mathbb E \operatorname{Tr}\left[ \left ( \sigma^\xi(Y^\xi(s),u(s))-\tilde \sigma (Y(s),u(s)) \right ) \left ( \sigma^\xi(Y^\xi(s))-\tilde \sigma (Y(s)) \right )^* B \right ]ds,\end{aligned}$$ where the inequality follows using the weak $B$-condition with $C_0=0$ (i.e. Proposition [\[prop:weak_b\]](#prop:weak_b){reference-type="ref" reference="prop:weak_b"}). Consider the first term on the right-hand-side. Using [\[eq:b_B\_property\]](#eq:b_B_property){reference-type="eqref" reference="eq:b_B_property"}, [\[eq:unif_convercenge_L\_xi,b_xi,sigma_xi\]](#eq:unif_convercenge_L_xi,b_xi,sigma_xi){reference-type="eqref" reference="eq:unif_convercenge_L_xi,b_xi,sigma_xi"} and the uniform convergence of $\tilde b_0^\xi$, we have $$\begin{aligned} \int_0^t & \mathbb E \langle B(Y^\xi(s)-Y(s)), \tilde b^\xi(Y^\xi(s),u(s))-\tilde b(Y(s),u(s)) \rangle_X ds\\ & = \int_0^t\mathbb E \langle B(Y^\xi(s)-Y(s)), \tilde b^\xi(Y^\xi(s),u(s))-\tilde b(Y^\xi(s),u(s)) \rangle_X ds +\int_0^t\mathbb E \langle B(Y^\xi(s)-Y(s)), \tilde b(Y^\xi(s),u(s))-\tilde b(Y(s),u(s)) \rangle_X ds\\ & \leq C \int_0^t \left [ \mathbb E |Y^\xi(s)-Y(s)|_{-1}^2 + \left |\tilde b_0^\xi(Y^\xi(s),u(s))-\tilde b_0(Y^\xi(s),u(s)) \right |^2 \right]ds + C \int_0^t \mathbb E |Y^\xi(s)-Y(s)|_{-1}^2ds\\ & \leq C \int_0^t \mathbb E |Y^\xi(s)-Y(s)|_{-1}^2ds + t\omega (\xi)\end{aligned}$$ for some $C>0$ and a modulus of continuity $\omega$ independent of $x,u(\cdot).$ An analogous inequality is obtained similarly for the second term on the right-hand side. Hence we have $$\begin{aligned} \mathbb E |Y^\xi(t)-Y(t)|_{-1}^2 \leq C \int_0^t \mathbb E |Y^\xi(s)-Y(s)|_{-1}^2ds + t\omega (\xi)\end{aligned}$$ and by Gronwall's lemma we obtain [\[eq:E_norm\_-1_Y\_xi-Y\]](#eq:E_norm_-1_Y_xi-Y){reference-type="eqref" reference="eq:E_norm_-1_Y_xi-Y"} for some $\lambda>0$.\ (iii) We can now prove that $$\label{eq:uniform_convergence_V_xi} V^\xi \xrightarrow{\xi \to 0} V \quad \textit{uniformly in } X.$$ Indeed, fix $\rho> \max(\rho_0,\frac{\lambda}{2})$ and let $x \in X, u(\cdot) \in \mathcal U$. By [\[eq:unif_convercenge_L\_xi,b_xi,sigma_xi\]](#eq:unif_convercenge_L_xi,b_xi,sigma_xi){reference-type="eqref" reference="eq:unif_convercenge_L_xi,b_xi,sigma_xi"}, [\[eq:L_unif_cont_norm_B\]](#eq:L_unif_cont_norm_B){reference-type="eqref" reference="eq:L_unif_cont_norm_B"} and [\[eq:E_norm\_-1_Y\_xi-Y\]](#eq:E_norm_-1_Y_xi-Y){reference-type="eqref" reference="eq:E_norm_-1_Y_xi-Y"} we have $$\begin{aligned} |J^\xi(x;u(\cdot))-J(x;u(\cdot))| & \leq \int_0^\infty e^{-\rho t} \mathbb E |L^\xi(Y^\xi(t),u(t))-L(Y^\xi(t),u(t))| dt + \int_0^\infty e^{-\rho t} \mathbb E |L(Y^\xi(t),u(t))-L(Y(t),u(t))| dt\\ & \leq \int_0^\infty e^{-\rho t} \omega_1(\xi) dt +C \int_0^\infty e^{-\rho t} \mathbb E |Y^\xi(t)-Y(t)|_{-1}dt \leq \omega_1(\xi) +\omega_2(\xi) \int_0^\infty t e^{-(\rho -\frac{\lambda}{2})t} dt \leq \omega(\xi).\end{aligned}$$ for some modulus of continuity $\omega$ independent of $x,u(\cdot).$ This implies [\[eq:uniform_convergence_V\_xi\]](#eq:uniform_convergence_V_xi){reference-type="eqref" reference="eq:uniform_convergence_V_xi"}. \(iv\) We prove that there exists $\bar \rho\geq \max(\rho_0,\frac{\lambda}{2})$ such that for every $\rho> \bar \rho$, $V^\xi$ is $|\cdot|_{-1}$-semiconcave for every $\xi >0$. Fix $\xi >0$. It is enough to show that there exists $\bar \rho>0$ such that $\forall \rho> \bar\rho$ $J^\xi(\cdot, u(\cdot))$ is $|\cdot|_{-1}$-semiconcave with a semiconcavity constant independent of $u(\cdot)$. Indeed let $x, \bar x \in X$, $u(\cdot) \in \mathcal U$, $\lambda \in [0,1]$. Denote by $Y(t), \bar Y(t)$ the solutions of [\[eq:state_eq_approximated_Y\_xi\]](#eq:state_eq_approximated_Y_xi){reference-type="eqref" reference="eq:state_eq_approximated_Y_xi"} with initial state $x, \bar x$ respectively and control $u(\cdot).$ Moreover, set $x_\lambda=\lambda x + (1-\lambda) \bar x$ and let $Y_\lambda(t)$ be the solution of [\[eq:state_eq_approximated_Y\_xi\]](#eq:state_eq_approximated_Y_xi){reference-type="eqref" reference="eq:state_eq_approximated_Y_xi"} with initial state $x_\lambda$ and control $u(\cdot)$. Finally, set $Y^\lambda(t)=\lambda Y(t)+ (1-\lambda) \bar Y(t)$. Then, by the $|\cdot|_{-1}$-Lipschitzianity (uniformly in $u, \xi$) and the $|\cdot|_{-1}$-semiconcavity of $L^\xi(\cdot,u)$ (with a semiconcavity constant uniform in $u$), we have $$\begin{aligned} \lambda J^\xi(x; u(\cdot))&+ (1-\lambda)J^\xi(\bar x;u(\cdot)) - J^\xi(x_\lambda;u(\cdot))\\ &=\int_0^\infty e^{-\rho t}\mathbb E \left [ \lambda L^\xi(Y(t),u(t))+(1-\lambda ) L^\xi(\bar Y(t),u(t)) - L^\xi(Y^\lambda(t), u(t))\right ] dt\\ &\quad + \int_0^\infty e^{-\rho t}\mathbb E \left [L^\xi(Y^\lambda(t), u(t))- L^\xi(Y_\lambda(t), u(t)\right ] dt\\ & \leq C_\xi\lambda (1-\lambda) \int_0^\infty e^{-\rho t} \mathbb E |Y(t)-\bar Y(t)|^2_{-1}dt + C\int_0^\infty e^{-\rho t} \mathbb E |Y^\lambda(t) - Y_\lambda(t))|_{-1} dt.\end{aligned}$$ for some $C_\xi,C>0$ (independent of $u(\cdot)$).\ By [@defeo_swiech_wessels Lemmas 5.3, 5.8] there exist constants $C,\tilde \rho>0$ (both independent of $\xi,u(\cdot)$), and $C_\xi$ (independent of $u(\cdot)$) such that $$\mathbb E |Y(t)-\bar Y(t)|^2_{-1} \leq C e^{\tilde \rho t } |\bar x- x|_{-1}^2 , \quad \mathbb E |Y^\lambda(t) - Y_\lambda(t))|_{-1} \leq C_\xi \lambda (1-\lambda) e^{\tilde \rho t } |\bar x- x|_{-1}^2.$$ By inserting these inequalities in the previous one, for every $\rho> \bar \rho:=\tilde \rho=\max(\tilde\rho,\rho_0,\frac{\lambda}{2})$ $$\begin{aligned} \lambda J^\xi(x, u(\cdot))&+ (1-\lambda)J^\xi(\bar x, u(\cdot)) - J^\xi(x_\lambda,u(\cdot)) \leq C_\xi \lambda (1-\lambda) |\bar x- x|_{-1}^2,\end{aligned}$$ which yields the claim. Hence $(iv)$ follows. \(v\) We now show [\[eq:tilde_V\_leq\]](#eq:tilde_V_leq){reference-type="eqref" reference="eq:tilde_V_leq"}. Following Section [4](#sec:approximation via inf-convolutions){reference-type="ref" reference="sec:approximation via inf-convolutions"} we extend $V^\xi$ to the function $\tilde V^\xi$ on $X_{-1}$, which then satisfies [\[Va-1\]](#Va-1){reference-type="eqref" reference="Va-1"} on $X_{-1}$ and is semiconcave in $X_{-1}$. We now fix $\xi>0$ and for every $\epsilon>0$ consider the sup-convolution $(\tilde V^\xi)^\epsilon$ of $\tilde V^\xi$, that is $$(\tilde V^\xi)^{\epsilon}(x):=\sup_{y \in X_{-1}}\left [ \tilde V^\xi(y)-\frac{1}{2\epsilon}|x-y|^2_{-1}\right].$$ We denote the restriction of $(\tilde V^\xi)^{\epsilon}$ to $X$ by $(V^\xi)^{\epsilon}$. Similarly to Lemma [Lemma 22](#lem:c11){reference-type="ref" reference="lem:c11"}, if $\epsilon$ is small enough, $(\tilde V^\xi)^{\epsilon}\in C^{1,1}(X_{-1})$ and $$(V^\xi)^{\epsilon} \xrightarrow{\epsilon \to 0} V^\xi \quad \textit{uniformly.}$$ By repeating the procedure from Sections [4](#sec:approximation via inf-convolutions){reference-type="ref" reference="sec:approximation via inf-convolutions"} and [6](#sec:lions_approx){reference-type="ref" reference="sec:lions_approx"} (Proposition [Proposition 23](#prop:inf_conv_subsolution_perturbed){reference-type="ref" reference="prop:inf_conv_subsolution_perturbed"}, Lemma [Lemma 27](#lemma:z_eta){reference-type="ref" reference="lemma:z_eta"}, Lemma [Lemma 28](#lemma:w_eta_eps_classical_supersol_perturbed_HJB){reference-type="ref" reference="lemma:w_eta_eps_classical_supersol_perturbed_HJB"} with adjustments since we are now dealing with sup-convolutions and viscosity subsolutions, but the proofs are the same), we obtain that for every $\eta>0$ there exist $(\tilde V^\xi)^\epsilon_\eta \in \mathcal D$ such that $$|(\tilde V^\xi)^\epsilon-(\tilde V^\xi)^\epsilon_\eta|\leq C^\xi_\epsilon \eta , \quad |D_{-1}(\tilde V^\xi)^\epsilon-D_{-1}(\tilde V^\xi)^\epsilon_\eta|_{-1}\leq C^\xi_\epsilon \eta$$ for some $C^\xi_\epsilon>0$ (independent of $\eta$) and such that for every $R>0$ $$\begin{aligned} \label{eq:V_xi_eta_eps_subsolution_perturbed_HJB} \rho (\tilde V^\xi)^\epsilon_\eta(x)-\langle \tilde A^*D(\tilde V^\xi)^\epsilon_\eta(x),x \rangle_X +\tilde H \left (x,D_{x_0}(\tilde V^\xi)^\epsilon_\eta(x), D^2_{x_0^2}(\tilde V^\xi)^\epsilon_\eta(x) \right) \leq \gamma^\xi(\epsilon) + \omega^\xi_{R,\epsilon}(\eta) \quad \forall x \in B_R.\end{aligned}$$ We now fix $x \in X,u(\cdot) \in \overline U$ and denote by $Y(t)$ the solution of the state equation with initial state $x$ and control $u(\cdot)$. Let $R,t>0$ and define $\tau^R = \inf \{s \in[0, t]:|Y(s)|_X>R\}.$ Since $(\tilde V^\xi)_\eta^\epsilon \in \mathcal D$, we can apply Lemma [Lemma 26](#lemma:ito){reference-type="ref" reference="lemma:ito"} to $\phi=(\tilde V^\xi)_\eta^\epsilon$ to get $$\begin{aligned} (\tilde V^\xi)_\eta^\epsilon( x) & =\mathbb E \left[ e^{-\rho (t\wedge \tau^R)} (\tilde V^\xi)_\eta^\epsilon(Y(t \wedge \tau^R )) \right] +\mathbb E \int_0^{t \wedge \tau^R} e^{-\rho s} \Big [ \rho (\tilde V^\xi)_\eta^\epsilon(Y(s)) - \langle Y(s), \tilde A^* D (\tilde V^\xi)_\eta^\epsilon(Y(s)) \rangle_X \\ & \quad - \tilde b_0(Y(s),u(s)) \cdot D_{x_0} (\tilde V^\xi)_\eta^\epsilon(Y(s)) - \frac 1 2 \mathop{\mathrm{Tr}}\nolimits\left ( \sigma_0(Y(s))\sigma_0(Y(s))^T D^2_{x_0^2}(\tilde V^\xi)_\eta^\epsilon(Y(s)) \right ) \Big ] ds\\ & =\mathbb E \left[ e^{-\rho (t\wedge \tau^R)} (\tilde V^\xi)_\eta^\epsilon(Y(t \wedge \tau^R )) \right] +\mathbb E \int_0^{t \wedge \tau^R} e^{-\rho s} l(Y(s),u(s)) ds\\ &\,\, +\mathbb E \int_0^{t \wedge \tau^R} e^{-\rho s} \Big [ \rho (\tilde V^\xi)_\eta^\epsilon(Y(s)) - \langle Y(s), \tilde A^* D (\tilde V^\xi)_\eta^\epsilon(Y(s)) \rangle_X - \tilde b_0(Y(s),u(s)) \cdot D_{x_0} (\tilde V^\xi)_\eta^\epsilon(Y(s)) -l(Y(s),u(s)) \\ &\quad \quad \quad \quad \quad \quad - \frac 1 2 \mathop{\mathrm{Tr}}\nolimits\left ( \sigma_0(Y(s))\sigma_0(Y(s))^T D^2_{x_0^2}(\tilde V^\xi)_\eta^\epsilon(Y(s)) \right ) \Big ] ds\\ & \leq \mathbb E \left[ e^{-\rho (t\wedge \tau^R)} (\tilde V^\xi)_\eta^\epsilon(Y(t \wedge \tau^R )) \right] +\mathbb E \int_0^{t \wedge \tau^R} e^{-\rho s} l(Y(s),u(s)) ds\\ &\,\, +\mathbb E \int_0^{t \wedge \tau^R} e^{-\rho s} \Big [ \rho (\tilde V^\xi)_\eta^\epsilon(Y(s)) - \langle Y(s), \tilde A^* D (\tilde V^\xi)_\eta^\epsilon(Y(s)) \rangle_X +\tilde H\left (Y(s), D_{x_0} (\tilde V^\xi)_\eta^\epsilon(Y(s)) , D^2_{x_0^2}(\tilde V^\xi)_\eta^\epsilon(Y(s)) \right ) ds\\ & \leq \mathbb E \left[ e^{-\rho (t\wedge \tau^R)} (\tilde V^\xi)_\eta^\epsilon(Y(t \wedge \tau^R )) \right] +\mathbb E \int_0^{t \wedge \tau^R} e^{-\rho s} l(Y(s),u(s)) ds+\gamma^\xi(\epsilon) + \omega^\xi_{R,\epsilon}(\eta),\end{aligned}$$ where the first and the second inequalities follow by definition of $\tilde H$ and [\[eq:V_xi_eta_eps_subsolution_perturbed_HJB\]](#eq:V_xi_eta_eps_subsolution_perturbed_HJB){reference-type="eqref" reference="eq:V_xi_eta_eps_subsolution_perturbed_HJB"} respectively. Letting (in this order) $\eta \to 0, \epsilon \to 0$, $R \to \infty$ and $t \to \infty$ we obtain $V^\xi(x) \leq J(x;u(\cdot))$. Finally, letting $\xi \to 0$, we get $V(x) \leq J(x;u(\cdot))$ and hence $$V( x)\leq \inf_{u \in \overline U} J(x;u(\cdot)) = \overline V(x) \quad \forall x \in X.$$ This concludes the proof of [\[eq:tilde_V\_leq\]](#eq:tilde_V_leq){reference-type="eqref" reference="eq:tilde_V_leq"} so that $\overline V=V$. ◻ We now strengthen Assumption [Assumption 9](#hp:discount){reference-type="ref" reference="hp:discount"} by **Assumption 30**. *$\rho >\bar \rho$, where $\bar\rho$ is from Proposition [Proposition 29](#prop:V_bar_equal_V){reference-type="ref" reference="prop:V_bar_equal_V"}.* We make an additional assumption about the Hamiltonian. **Assumption 31**. *We assume that the supremum in [\[eq:hamiltonian_synthesis_no_sigma\]](#eq:hamiltonian_synthesis_no_sigma){reference-type="eqref" reference="eq:hamiltonian_synthesis_no_sigma"} is a maximum, i.e. $$\begin{aligned} \tilde H \left (x,p_0 \right)&= \max_{u\in U} \bigg \{ - \tilde b_0\left ( x ,u \right) \cdot p_0 - l(x_0,u) \bigg \} \\&= - x_0 \cdot p_0 + \max_{u\in U} \Bigg \{ - b_0\left ( x_0,\int_{-d}^0 a_1(\xi)x_1(\xi)\,d\xi ,u \right) \cdot p_0 - l(x_0,u) \Bigg \}. \end{aligned}$$* **Theorem 32** (Verification). *Let Assumptions [Assumption 1](#hp:state){reference-type="ref" reference="hp:state"}, [Assumption 2](#hp:cost){reference-type="ref" reference="hp:cost"}, [Assumption 17](#hp:uniform_ellipticity){reference-type="ref" reference="hp:uniform_ellipticity"}, [Assumption 19](#hp:convexity){reference-type="ref" reference="hp:convexity"}, [Assumption 30](#hp:discount2){reference-type="ref" reference="hp:discount2"}, [Assumption 31](#hp:feedback_map_well-defined){reference-type="ref" reference="hp:feedback_map_well-defined"} hold. Let $x \in X$ and $u^*(\cdot) \in \mathcal {\overline U}$ be an admissible control. Denote by $Y^*(s)$ the solution of [\[eq:abstract_dissipative_operator\]](#eq:abstract_dissipative_operator){reference-type="eqref" reference="eq:abstract_dissipative_operator"} with $u(\cdot)=u^*(\cdot)$. Assume that $$u^*(s)\in {\rm argmax}_{u\in U} \Bigg \{ - b_0\left ( Y_0^*(s),\int_{-d}^0 a_1(\xi)Y_1^*(s)(\xi)\,d\xi ,u \right) \cdot D_{x_0}V(Y^*(s)) - l(Y_0^*(s),u) \Bigg \},$$ $\mathbb P -$a.s. for a.e. $s \geq 0.$ Then $u^*(\cdot)$ is optimal for $x$.* *Proof.* Let $R,t>0$ and define $\tau^R := \inf \{s \in[0, t]:|Y^*(s)|_X>R\}.$ Since $\tilde V^\eta_\epsilon \in \mathcal D$ we can apply Lemma [Lemma 26](#lemma:ito){reference-type="ref" reference="lemma:ito"} to $\phi=\tilde V^\eta_\epsilon$ to get $$\begin{aligned} \tilde V^\eta_\epsilon( x) & =\mathbb E \left[ e^{-\rho (t\wedge \tau^R)} \tilde V^\eta_\epsilon(Y^*(t \wedge \tau^R )) \right] +\mathbb E \int_0^{t \wedge \tau^R} e^{-\rho s} \Big [ \rho \tilde V^\eta_\epsilon(Y^*(s)) - \langle Y^*(s), \tilde A^* D \tilde V^\eta_\epsilon(Y^*(s)) \rangle_X \\ & \quad - \tilde b_0(Y^*(s),u^*(s)) \cdot D_{x_0} \tilde V^\eta_\epsilon(Y^*(s)) - \frac 1 2 \mathop{\mathrm{Tr}}\nolimits\left ( \sigma_0(Y^*(s))\sigma_0(Y^*(s))^T D^2_{x_0^2}\tilde V^\eta_\epsilon(Y^*(s)) \right ) \Big ] ds\\ & =\mathbb E \left[ e^{-\rho (t\wedge \tau^R)} \tilde V^\eta_\epsilon(Y^*(t \wedge \tau^R )) \right] +\mathbb E \int_0^{t \wedge \tau^R} e^{-\rho s} l(Y^*(s),u^*(s)) ds\\ &\quad +\mathbb E \int_0^{t \wedge \tau^R} e^{-\rho s} \Big [ \rho \tilde V^\eta_\epsilon(Y^*(s)) - \langle Y^*(s), \tilde A^* D \tilde V^\eta_\epsilon(Y^*(s)) \rangle_X - \tilde b_0(Y^*(s),u^*(s)) \cdot D_{x_0} \tilde V^\eta_\epsilon(Y^*(s)) -l(Y^*(s),u^*(s)) \\ &\quad \quad \quad \quad \quad \quad - \frac 1 2 \mathop{\mathrm{Tr}}\nolimits\left ( \sigma_0(Y^*(s))\sigma_0(Y^*(s))^T D^2_{x_0^2}\tilde V^\eta_\epsilon(Y^*(s)) \right ) \Big ] ds.\end{aligned}$$ Note that, by [\[eq:convergence_inf_convolution\]](#eq:convergence_inf_convolution){reference-type="eqref" reference="eq:convergence_inf_convolution"}, [\[eq:convergence_convolution_w\_eps_eta\]](#eq:convergence_convolution_w_eps_eta){reference-type="eqref" reference="eq:convergence_convolution_w_eps_eta"}, we have $|D_{x_0} \tilde V^\eta_\epsilon(Y^*(s)) - D_{x_0} V(Y^*(s)) |_X \leq |D_{x_0} \tilde V^\eta_\epsilon(Y^*(s)) - D_{x_0} \tilde V_\epsilon(Y^*(s)) |_X + |D_{x_0} \tilde V_\epsilon(Y^*(s)) - D_{x_0} V(Y^*(s)) |_X \leq C_\epsilon \eta + \omega_R(\epsilon)$. Note also that thanks to $\tau_R$, we have $|\tilde b_0(Y^*(s),u^*(s))| \leq C_R$. From these facts, the definition of $u^*(\cdot)$, [\[eq:Hamiltonian_local_lip\]](#eq:Hamiltonian_local_lip){reference-type="eqref" reference="eq:Hamiltonian_local_lip"} and Lemma [Lemma 28](#lemma:w_eta_eps_classical_supersol_perturbed_HJB){reference-type="ref" reference="lemma:w_eta_eps_classical_supersol_perturbed_HJB"}, we have $$\begin{aligned} \tilde V^\eta_\epsilon( x) & \geq \mathbb E \left[ e^{-\rho (t\wedge \tau^R)} \tilde V^\eta_\epsilon(Y^*(t \wedge \tau^R )) \right] +\mathbb E \int_0^{t \wedge \tau^R} e^{-\rho s} l(Y^*(s),u^*(s)) ds\\ &\quad +\mathbb E \int_0^{t \wedge \tau^R} e^{-\rho s} \Big [ \rho \tilde V^\eta_\epsilon(Y^*(s)) - \langle Y^*(s), \tilde A^* D \tilde V^\eta_\epsilon(Y^*(s)) \rangle_X - \tilde b_0(Y^*(s),u^*(s)) \cdot D_{x_0} V(Y^*(s)) -l(Y^*(s),u^*(s)) \\ &\quad \quad \quad \quad \quad \quad - \frac 1 2 \mathop{\mathrm{Tr}}\nolimits\left ( \sigma_0(Y^*(s),u^*(s))\sigma_0(Y^*(s))^T D^2_{x_0^2}\tilde V^\eta_\epsilon(Y^*(s)) \right )-C_{R,\epsilon} \eta - \omega_R(\epsilon) \Big ] ds \\ & = \mathbb E \left[ e^{-\rho (t\wedge \tau^R)} \tilde V^\eta_\epsilon(Y^*(t \wedge \tau^R )) \right] +\mathbb E \int_0^{t \wedge \tau^R} e^{-\rho s} l(Y^*(s),u^*(s)) ds\\ &\quad +\mathbb E \int_0^{t \wedge \tau^R} e^{-\rho s} \Big [ \rho \tilde V^\eta_\epsilon(Y^*(s)) - \langle Y^*(s), \tilde A^* D \tilde V^\eta_\epsilon(Y^*(s))\rangle_X +\tilde H\left (Y^*(s),D_{x_0}V(Y^*(s) ) \right) \\ &\quad \quad \quad \quad \quad \quad - \frac 1 2 \mathop{\mathrm{Tr}}\nolimits\left ( \sigma_0(Y^*(s))\sigma_0(Y^*(s))^T D^2_{x_0^2}\tilde V^\eta_\epsilon(Y^*(s)) \right )-C_{R,\epsilon} \eta - \omega_R(\epsilon) \Big ] ds \\ & \geq \mathbb E \left[ e^{-\rho (t\wedge \tau^R)} \tilde V^\eta_\epsilon(Y^*(t \wedge \tau^R )) \right] +\mathbb E \int_0^{t \wedge \tau^R} e^{-\rho s} l(Y^*(s),u^*(s)) ds\\ &\quad +\mathbb E \int_0^{t \wedge \tau^R} e^{-\rho s} \Big [ \rho \tilde V^\eta_\epsilon(Y^*(s)) - \langle Y^*(s), \tilde A^* D \tilde V^\eta_\epsilon(Y^*(s))\rangle_X +\tilde H\left (Y^*(s),D_{x_0}\tilde V^\eta_\epsilon((Y^*(s) )) \right) \\&\quad \quad \quad \quad \quad \quad - \frac 1 2 \mathop{\mathrm{Tr}}\nolimits\left ( \sigma_0(Y^*(s))\sigma_0(Y^*(s))^T D^2_{x_0^2}\tilde V^\eta_\epsilon(Y^*(s)) \right )-C_{R,\epsilon} \eta - \omega_R(\epsilon) \Big ] ds \\ & \geq \mathbb E \left[ e^{-\rho (t\wedge \tau^R)} \tilde V^\eta_\epsilon(Y^*(t \wedge \tau^R )) \right]+\mathbb E \int_0^{t \wedge \tau^R} e^{-\rho s} l(Y^*(s),u^*(s)) ds -\omega_{R,\epsilon}( \eta) - \omega_R(\epsilon),\end{aligned}$$ where the constants $C_{R,\epsilon}$ and moduli $\omega_R(\epsilon)$ might have changed from line to line. Letting first $\eta \to 0$ and then $\epsilon \to 0$, by [\[eq:convergence_inf_convolution\]](#eq:convergence_inf_convolution){reference-type="eqref" reference="eq:convergence_inf_convolution"}, [\[eq:convergence_convolution_w\_eps_eta\]](#eq:convergence_convolution_w_eps_eta){reference-type="eqref" reference="eq:convergence_convolution_w_eps_eta"}, we get $$\begin{aligned} V(x) \geq \mathbb E \left[ e^{-\rho (t\wedge \tau^R)} V(Y^*(t \wedge \tau^R )) \right]+\mathbb E \int_0^{t\wedge \tau^R} e^{-\rho s} l(Y^*(s),u^*(s)) ds.\end{aligned}$$ We now send first $R \to \infty$ and then $t \to \infty$. Recalling Proposition [Proposition 29](#prop:V_bar_equal_V){reference-type="ref" reference="prop:V_bar_equal_V"}, we have $$\begin{aligned} \overline V(x)=V( x) \geq \mathbb E \int_0^\infty e^{-\rho s} l(Y^*(s),u^*(s)) ds= J(x;u^*(\cdot)),\end{aligned}$$ from which we obtain the optimality of $u^*(\cdot).$ ◻ We now construct an optimal feedback control. Define the multivalued map $\Psi \colon X \to \mathcal P (U)$, $\forall x \in X$, by $$\begin{aligned} \label{eq:multivalued_function_PSI} \Psi(x):&={\rm argmax}_{u \in U} \tilde H(x,D_{x_0}V(x))={\rm argmax}_{u \in U} \Bigg \{ - b_0\left ( x_0,\int_{-d}^0 a_1(\xi)x_1(\xi)\,d\xi ,u \right) \cdot D_{x_0}V(x) - l(x_0,u) \Bigg \} \neq \emptyset. \end{aligned}$$ **Corollary 33**. *Let the assumptions of Theorem [Theorem 32](#th:verification){reference-type="ref" reference="th:verification"} be satisfied. Assume that $\Psi$ has a measurable selection $\psi$ such that the closed loop equation $$\label{eq:closed_loop} dY(t) = [\tilde A Y(t)+\tilde b(Y(t),\psi(Y(t)) ] dt + \sigma(Y(t)) \,dW(t), \quad Y(0) = x \in X,$$ admits a weak mild solution $Y^\psi(t)$ (e.g. see [@fgs_book Definition 1.121]) in some generalized reference probability space $\tau$. If we set $u^\psi(\cdot) := \psi(Y^\psi (\cdot)),$ then the pair $(u^\psi(\cdot),Y^\psi(\cdot))$ is optimal.* *Proof.* First note that $u^\psi(\cdot) \in \mathcal{ \overline U}$. Hence $Y^\psi(\cdot)$ is the unique mild solution to [\[eq:abstract_dissipative_operator\]](#eq:abstract_dissipative_operator){reference-type="eqref" reference="eq:abstract_dissipative_operator"} (in the strong probabilistic sense now in the generalized reference probability space $\tau$). Now note that by construction, $u^\psi$ satisfies for every $s \geq 0$ $$u^\psi(Y^\psi(s))\in {\rm argmax}_{u\in U} \Bigg \{ - b_0\left ( Y^\psi_0(s),\int_{-d}^0 a_1(\xi)Y_1^\psi(s)(\xi)\,d\xi ,u \right) \cdot D_{x_0}V(Y^\psi(s)) - l(Y_0^\psi(s),u) \Bigg \} .$$ Hence, by the Verification Theorem [Theorem 32](#th:verification){reference-type="ref" reference="th:verification"}, we conclude that the pair $(u^\psi(\cdot),Y^\psi(\cdot))$ is optimal. ◻ We remark that we had to relax the class of admissible controls since the feedback map $\psi$ is not regular enough to guarantee existence of mild solutions of the closed loop equation in a reference probability space. **Remark 34**. *As mentioned in the introduction, verification theorems and optimal feedback laws for stochastic optimal control problems with delays have been studied using mild solutions in $L^2$ spaces or BSDE's. Indeed, in the former a linear structure of the state equation and appropriate conditions ensuring the existence of an invariant measure are assumed, e.g. see [@fgs_book Section 5.6]. In the latter some regularity of the coefficients is assumed (e.g. differentiability and $\sigma_0$ having a bounded inverse), e.g. see [@fgs_book Section 5.6]. Moreover, both approaches can handle pointwise delays when these appear in a linear way in the state equation (e.g. when the state equation is of the form $dy(t)=a y(t-d)+ \cdots$ for some $a \in M^{n \times n}$). Here we work under different assumptions. Indeed, apart from standard conditions, we assume the $|\cdot |_{-1}$-semiconvexity of the value function $V$ and a local non-degeneracy of $\sigma_0$.* # Application to stochastic optimal advertising {#sec:example} The following problem is taken from [@defeo_federico_swiech Section 7], [@gozzi_marinelli_2004 Section 4]. The model for the dynamics of the stock of advertising goodwill $y(s)$ of a product is given by the following controlled 1-dimensional SDDE ($n=h=k=1$) $$\begin{cases} dy(t) = \left[ a_0 y(t)+\int_{-d}^0 a_1(\xi)y(t+\xi)\,d\xi +c_0 u(t) \right] dt + \sigma_0 \, dW(t),\\ y(0)=x_0, \quad y(\xi)=x_1(\xi)\; \quad \forall \xi\in[-d,0), \end{cases}$$ where $d>0$, the control process $u(s)$ models the intensity of advertising spending and $W$ is a real-valued Brownian motion, and (i) $a_0 \leq 0$ is a constant factor of image deterioration in absence of advertising; (ii) $c_0 > 0$ is a constant advertising effectiveness factor; (iii) $a_1 \leq 0$ is a given deterministic function satisfying the assumptions used in the previous sections which represents the distribution of the forgetting time; (iv) $\sigma_0>0$ represents the uncertainty in the model; (v) $x_0 \in \mathbb R$ is the level of goodwill at the beginning of the advertising campaign; (vi) $x_1 \in L^2([-d,0];\mathbb R)$ is the history of the goodwill level. We use the same setup of the stochastic optimal control problem as in Section [2](#sec:formul){reference-type="ref" reference="sec:formul"}. The control set is $U= [0,\bar u]$ for some $\bar u>0$. The optimization problem is $$\inf_{u(\cdot) \in \tilde{\mathcal U}}\mathbb{E} \left[\int_0^\infty e^{-\rho s} l(y(s),u(s)) d s\right],$$ where $\rho >0$ is a discount factor, $l(x,u)=h(u)-g(x)$, with a continuous and strictly convex cost function $h \colon [0,\bar u] \rightarrow \mathbb R$ and a continuous and concave utility function $g \colon \mathbb R \rightarrow \mathbb R$, which satisfy Assumption [Assumption 2](#hp:cost){reference-type="ref" reference="hp:cost"}. Moreover we assume that $g$ is strictly increasing; $h \in C^0([0,\bar u]) \cap C^1([0,\bar u))$; $h$ is strictly increasing and $h(0)=0$; $h'(0)=0$, $\lim_{u \to \bar u} h'(u)=\infty.$ Setting $$b_0 \left ( x_0,\int_{-d}^0 a_1(\xi)x_1(\xi)\,d\xi ,u\right):=a_0 x_0+\int_{-d}^0 a_1(\xi)x_1(\xi)\,d\xi +c_0 u,$$ we are then in the setting of Section [2](#sec:formul){reference-type="ref" reference="sec:formul"}. Therefore, using the infinite dimensional framework of Section [3](#sec:prelim){reference-type="ref" reference="sec:prelim"}, we can use Theorem [\[th:existence_uniqueness_viscosity_infinite\]](#th:existence_uniqueness_viscosity_infinite){reference-type="ref" reference="th:existence_uniqueness_viscosity_infinite"} to characterize the value function $V$ as the unique viscosity solution to [\[eq:HJB\]](#eq:HJB){reference-type="eqref" reference="eq:HJB"}, and Theorem [\[th:C1alpha\]](#th:C1alpha){reference-type="ref" reference="th:C1alpha"} to obtain partial regularity of $V$. Moreover $V$ is convex as the assumptions of Example [Example 21](#ex:convex2){reference-type="ref" reference="ex:convex2"} are satisfied. We want to construct an optimal feedback for the optimization problem. Note that $$\tilde b(x,u)=\begin{bmatrix} b_0 \left ( x_0,\int_{-d}^0 a_1(\xi)x_1(\xi)\,d\xi ,u\right) +x_0 \\ 0 \end{bmatrix} = \begin{bmatrix} (a_0+1) x_0+\int_{-d}^0 a_1(\xi)x_1(\xi)\,d\xi \\ 0 \end{bmatrix}+ \begin{bmatrix} c_0 u\\ 0 \end{bmatrix} =: D x + E u,$$ for every $x=(x_0,x_1) \in X, u \in [0,\overline u].$ Hence [\[eq:abstract_dissipative_operator\]](#eq:abstract_dissipative_operator){reference-type="eqref" reference="eq:abstract_dissipative_operator"} becomes $$\label{eq:abstract_eq_example} d Y(t) = [(\tilde A+D) Y(t)+E u(t) ] dt + \sigma \,dW(t), \quad Y(0) = x \in X.$$ Notice that, since $h$ is strictly convex, then its continuous derivative $h' \colon [0,\bar u) \to [0,\infty)$ is strictly increasing. Hence it is invertible ant its inverse $(h')^{-1} \colon [0,\infty) \to [0,\bar u)$ is continuous. Then $\Psi$ defined by [\[eq:multivalued_function_PSI\]](#eq:multivalued_function_PSI){reference-type="eqref" reference="eq:multivalued_function_PSI"} becomes $$\begin{aligned} \Psi(x)&=argmax_{u \in U} \Bigg \{ - \left (a_0 x_0+\int_{-d}^0 a_1(\xi)x_1(\xi)\,d\xi +c_0 u \right) D_{x_0}V(x) - h(u)+g(x) \Bigg \} \\ & = argmax_{u \in U} \Bigg \{ - c_0 u D_{x_0}V(x) - h(u) \Bigg \}= \begin{cases} & (h')^{-1} \left( c_0 D_{x_0}V(x) \right) \quad \textit{if } D_{x_0}V(x)<0,\\ &0 \quad \quad \quad \quad \quad \quad \quad \quad \quad \textit{if } D_{x_0}V(x) \geq 0, \end{cases} \end{aligned}$$ where we have also used the fact that $c_0>0$ and that for fixed $x$ the argument of the argmax is a linear term $\theta(u)=-c_0u DV_{x_0}(x)$ perturbed by a strictly concave term $\nu(u)=-h(u)$ which is strictly decreasing and such that $\nu(0)=0, \nu'(0)=0$ Hence we can see $\Psi$ as a (single-valued) map, i.e. $\Psi \colon X \rightarrow U$ (so that the measurable selection is trivially $\psi=\Psi$). Moreover, since $D_{x_0}V$ is continuous, we have that $\Psi$ is continuous on $X.$ Next we study the solutions of the closed loop equation $$\label{eq:closed_loop_eq_example} d Y(t) = [(\tilde A+D) Y(t)+ E\Psi(Y(t)) ] dt + \sigma \,dW(t), \quad Y(0) = x \in X,$$ Fix any generalized probability space $\nu=( \Omega, {\mathcal F}, {\mathcal F}_t, \mathbb P, W_t)$ and denote by $Y(t)$ the unique mild solution of the (uncontrolled) equation $$d Y(t) = (\tilde A + D) Y(t) dt + \sigma \,dW(t), \quad Y(0) = x \in X.$$ Denote by $\sigma^{-1} \colon \sigma (\mathbb R) \subset X \to \mathbb R$ the inverse of the operator $\sigma \in \mathcal L(\mathbb R,X)$ and set $$\phi \colon [0,\infty) \rightarrow \mathbb R, \quad \phi(t):=\sigma^{-1} E\Psi(Y(t)) = \frac {c_0} {\sigma_0}\Psi(Y(t)) .$$ Since $|\Psi| \leq \overline u$, we have that $\phi$ is bounded so that [@DZ14 Proposition 10.17 (i)] holds. This means that we can apply Girsanov Theorem [@DZ14 Theorem 10.14] to get the existence of a probability $\overline {\mathbb P}$ on $\Omega$ under which $$\overline W_t:=-\int_0^t \phi(s) ds + W_t=-\sigma^{-1} \int_0^t E\Psi(Y(s)) ds + W_t$$ is a Wiener process. It follows that $Y(t)$ is a mild solution of [\[eq:closed_loop_eq_example\]](#eq:closed_loop_eq_example){reference-type="eqref" reference="eq:closed_loop_eq_example"} in the generalized reference probability space $\overline \nu:=( \Omega, {\mathcal F}, {\mathcal F}_t, \overline {\mathbb P}, \overline W_t)$. Hence $Y(t)$ is a weak mild solution of [\[eq:closed_loop_eq_example\]](#eq:closed_loop_eq_example){reference-type="eqref" reference="eq:closed_loop_eq_example"}. Then we can apply Corollary [Corollary 33](#cor:closed_loop){reference-type="ref" reference="cor:closed_loop"} to get the optimality of $(u^\Psi(\cdot),Y^\Psi(\cdot))$ with $Y^\Psi(t)=Y(t)$, $u^\Psi(\cdot) := \Psi(Y^\Psi (\cdot))$. Optimal feedback laws for this problem can also be constructed using the approaches of mild solutions in $L^2$ spaces or BSDE's, e.g. see [@gozzi_marinelli_2004 Section 4], [@fgs_book Chapter 6]. # Comparison for SDDE's We prove a comparison result for a class of SDDE. In particular we generalize the deterministic result [@goldys_1 Lemma 2.8] to the stochastic case with additive noise and under a more general drift $b_0$. We assume that $n=h=q=1$ for simplicity. Fix a reference probability space $\tau=(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\geq 0}, \mathbb{P}, W)$. Let $x=(x_0,x_1) \in X, u(\cdot ) \in {\mathcal U}_\tau$ and consider the following SDDE with additive noise $$\label{eq:ap1} \begin{cases} dy(t) = \displaystyle b_0 \left ( y(t),\int_{-d}^0 a_1(\xi)y(t+\xi)\,d\xi ,u(t) \right) dt \displaystyle+ \sigma_0 dW(t),\\ y(0)=x_0, \quad y(\xi)=x_1(\xi)\; \quad \forall \xi\in[-d,0). \end{cases}$$ **Lemma 35**. *Let the standing assumptions of Section [2](#sec:formul){reference-type="ref" reference="sec:formul"} be satisfied and let $a_1(\xi) \geq 0$ for $\xi\in [-d,0]$. Let $\sigma_0(x,z,u)=\sigma_0>0$, $b_0$ satisfy Assumption [Assumption 1](#hp:state){reference-type="ref" reference="hp:state"} and moreover assume that $b_0(x,z,u)$ is non-decreasing with respect to the second variable $z$. Let $y(t)$ be the solution of [\[eq:ap1\]](#eq:ap1){reference-type="eqref" reference="eq:ap1"} and let $x(t)$ satisfy* *$$\label{eq:comparison_differential_ineq} \begin{cases} dx(t)=F_tdt+\sigma_0 dW(t) \leq \displaystyle b_0 \left ( x(t),\int_{-d}^0 a_1(\xi)x(t+\xi)\,d\xi ,u(t) \right) dt \displaystyle+ \sigma_0 dW(t),\\ x(0)\leq x_0, \quad x(\xi)\leq x_1(\xi)\; \quad \forall \xi\in[-d,0), \end{cases}$$* *where $F_t$ is a locally integrable adapted process. Then $x(t)\leq y(t)$ for a.e. $\omega \in \Omega,$ $\forall t \geq 0.$* *Proof.* By [\[eq:ap1\]](#eq:ap1){reference-type="eqref" reference="eq:ap1"} and [\[eq:comparison_differential_ineq\]](#eq:comparison_differential_ineq){reference-type="eqref" reference="eq:comparison_differential_ineq"}, for a.e. $\omega \in \Omega$, $\forall t \geq 0$ $$\begin{aligned} x(t)-y(t)& =x(0)-x_0+ \int_0^t \left[ F_s - b_0 \left ( y(s),\int_{-d}^0 a_1(\xi)y(s+\xi)\,d\xi ,u(s) \right) \right] ds\\ & \leq x(0)-x_0 + \int_0^t\left[ b_0 \left ( x(s),\int_{-d}^0 a_1(\xi)x(s+\xi)\,d\xi ,u(s) \right)-b_0 \left ( y(s),\int_{-d}^0 a_1(\xi)y(s+\xi)\,d\xi ,u(s) \right) \right] ds.\end{aligned}$$ Hence the process $(x-y)(t)=x(t)-y(t)$ is differentiable for a.e. $\omega \in \Omega$, for a.e. $t \geq 0$ with $$\begin{aligned} \label{eq:comparison_inequality_(x-y)'} (x-y)'(t)& =F_t- b_0 \left ( x(t),\int_{-d}^0 a_1(\xi)x(t+\xi)\,d\xi ,u(t) \right) \nonumber\\ & \leq b_0 \left ( x(t),\int_{-d}^0 a_1(\xi)x(t+\xi)\,d\xi ,u(t) \right)-b_0 \left ( y(t),\int_{-d}^0 a_1(\xi)y(t+\xi)\,d\xi ,u(t) \right).\end{aligned}$$ We define the process $h(t)=[x(t)-y(t)]^+=\max[x(t)-y(t),0] \geq 0$ and let $\overline a=\sup_{\xi \in [-d,0]}a_1(\xi)$. We show by contradiction that $h(t)=0$ for a.e. $\omega \in \Omega$, $\forall t \geq 0.$ Hence let $\tau>0$ and define the random variable $M=\sup_{t\in [0,\tau]}h(t) \geq 0$.\ By contradiction suppose $\exists G\in \mathcal F$ with $\mathbb P(G)>0$ such that $M(\omega)>0$ $\forall \omega \in G$. By monotonicity of $b_0$ with respect to the second variable and since $a_1 \geq 0$, we have for a.e. $\omega \in G$, $\forall t \in[0, \tau]$ $$\begin{aligned} \label{eq:monotonocity_comparison} b_0 \left ( x(t),\int_{-d}^0 a_1(\xi)x(t+\xi)\,d\xi ,u(t) \right) \leq b_0 \left ( x(t),\int_{-d}^0 a_1(\xi)y(t+\xi)\,d\xi + \overline a M d ,u(t) \right).\end{aligned}$$ Define, for $n \in \mathbb{N}$, $$\begin{aligned} \label{eq:properties_phi_N_comparison_sdde} \varphi_n(x):= \begin{cases}0, & \quad \forall x \leq 0, \\ n x^2, & \quad \forall x \in(0,1 / 2 n] \\ x-1 / 4 n, & \quad \forall x>1 / 2 n .\end{cases}\end{aligned}$$ The sequence $\left \{ \varphi_n\right\} _{n \in \mathbb{N}} \subset C^1( \mathbb{R})$ is such that $$\begin{aligned} & \varphi_n(x)=\varphi_n^{\prime}(x)=0 \quad \forall x \in(-\infty, 0], n \in \mathbb{N}, \quad \quad 0 \leq \varphi_n^{\prime}(x) \leq 1 \quad \forall x \in \mathbb{R}, N \in \mathbb{N}, \\ & \varphi_n(x) \rightarrow x^{+} \quad \text { uniformly on } x \in \mathbb{R}, \quad \quad \quad \quad \quad \varphi_n^{\prime}(x) \rightarrow 1\quad \forall x \in(0,+\infty) . \end{aligned}$$ Noticing that $\varphi_n\left(x(0)-x_0\right) = 0$ as $x(0) \leq x_0$ and using [\[eq:comparison_inequality\_(x-y)\'\]](#eq:comparison_inequality_(x-y)'){reference-type="eqref" reference="eq:comparison_inequality_(x-y)'"}, [\[eq:monotonocity_comparison\]](#eq:monotonocity_comparison){reference-type="eqref" reference="eq:monotonocity_comparison"}, we have for a.e. $\omega \in G$, $\forall t \in[0, \tau]$ $$\begin{aligned} \varphi_n(x(t)&-y(t)) =\varphi_n\left(x(0)-x_0\right)+\int_0^t \varphi_n^{\prime}(x(s)-y(s))\left(x-y\right)'(s) d s \\ & \leq \int_0^t \varphi_n^{\prime}(x(s)-y(s)) \left[ b_0\left(x(s), \int_{-d}^0 a_1(\xi) x(s+\xi) d \xi,u(s)\right)-b_0\left(y(s), \int_{-d}^0 a_1(\xi) y(s+\xi) d \xi,u(s)\right) \right ] d s \\ & \leq \int_0^t \varphi_n^{\prime}(x(s)-y(s)) \left[ b_0\left(x(s), \int_{-d}^0 a_1(\xi) y(s+\xi) d \xi + \overline a M d ,u(s)\right)-b_0\left(y(s), \int_{-d}^0 a_1(\xi) y(s+\xi) d \xi,u(s)\right) \right ] d s \\ & \leq L \int_0^t \varphi_n^{\prime}(x(s)-y(s))\left[|x(s)-y(s)|+ \bar{a} M d\right] ds. \end{aligned}$$ Recalling [\[eq:properties_phi_N\_comparison_sdde\]](#eq:properties_phi_N_comparison_sdde){reference-type="eqref" reference="eq:properties_phi_N_comparison_sdde"}, since $\varphi_n^{\prime}(x(s)-y(s)) =0$ when $x(s)-y(s)\leq 0$, we have $\varphi_n^{\prime}(x(s)-y(s))|x(s)-y(s)|\leq h(s)$, where we have used also the fact that $\varphi_n' \leq 1$. Hence, using again $\varphi_n' \leq 1$, we have for a.e. $\omega \in G$, $\forall t \in[0, \tau]$ $$\begin{aligned} \varphi_n(x(t)-y(t))& \leq L \int_0^t h(s) d s +t L \overline a M d . \end{aligned}$$ Letting $n \to \infty$, for a.e. $\omega \in G$, $\forall t \in[0, \tau]$ $$\begin{aligned} & h(t) \leq L \int_0^t h(s) d s +\tau L \overline a M d . \end{aligned}$$ Therefore, by Gronwall's lemma and recalling that $h(t) \geq 0$, we have for a.e. $\omega \in G$, $\forall t \in[0, \tau]$ $$0 \leq h(t) \leq \tau L \overline a M d \cdot e^{L t} \leq \tau L \overline a M d \cdot e^{L \tau} \quad \forall t \leq \tau.$$ Choosing $\tau$ such that $\tau L \overline a d \cdot e^{L \tau} \leq 1/2$, we have for a.e. $\omega \in G$, $\forall t \in[0, \tau]$ $$0 \leq h(t) \leq M/2.$$ Since we assumed $M( \omega)=\sup_{t\in [0,\tau]}h(t)(\omega)>0$ $\forall \omega \in G$, this is a contradiction by the definition of $M$. 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arxiv_math

--- abstract: | The main objects of this paper include some degenerate and nonlocal elliptic operators which naturally arise in the conformal invariant theory of Poincaré-Einstein manifolds. These operators generally reflect the correspondence between the Riemannian geometry of a complete Poincaré-Einstein manifold and the conformal geometry of its associated conformal infinity. In this setting, we develop the quantitative differentiation theory that includes quantitative stratification for the singular set and Minkowski type estimates for the (quantitatively) stratified singular sets. All these, together with a new $\epsilon$-regularity result for degenerate/singular elliptic operators on Poincaré-Einstein manifolds, lead to uniform Hausdorff measure estimates for the singular sets. Furthermore, the main results in this paper provide a delicate synergy between the geometry of Poincaré-Einstein manifolds and the elliptic theory of associated degenerate elliptic operators. address: - Department of Mathematics, Fordham University, Bronx, NY, USA, 10458 - Department of Mathematics, Johns Hopkins University, Baltimore, MD, USA, 21218 - Department of Mathematics, Princeton University, Princeton, NJ, USA, 08544 author: - Xumin Jiang - Yannick Sire - Ruobing Zhang bibliography: - PE.bib title: The singular sets of degenerate and nonlocal elliptic equations on Poincaré-Einstein manifolds --- [^1] # Introduction The main purpose of this paper is to establish a series of new dimension bounds and geometric measure estimates for the singular sets of a typical class of degenerate and nonlocal elliptic equations in the context of Poincaré-Einstein manifolds. These estimates are new even in the Euclidean model case. First, let us recall the backgrounds and major strategies of several aspects employed in our studies. ## Background and setup A fundamental theme in the analysis of elliptic PDEs is to understand the structure, particularly the geometric profiles, of their solutions. It is a crucial to study the structure of the singular set of these solutions. The implicit function theorem and Sard's theorem provide a qualitative characterization for smooth functions: for a smooth function $f$, its level set $\Sigma$ is a smooth hypersurface except at the critical set of $f$, which has measure zero. In other words, the critical points contribute to the topological and geometric complexity of $\Sigma$, and consequently, of $f$ itself. To obtain more refined geometric information, it is necessary to investigate the structure of the critical set or singular set. In the context of the solutions of elliptic PDEs, the very first step is to consider the model case, namely the Laplace equation and harmonic functions. It is well known that every harmonic on a Euclidean ball must be analytic. Moreover, if a Euclidean harmonic function is homogeneous, then it must be a *homogeneous harmonic polynomial*. From the geometric point of view, any homogeneous polynomial is invariant under rescaling at the origin which is also called a *Euclidean cone* in the literature. Since the critical set of a homogeneous harmonic polynomial is well understood, in the study of general elliptic PDEs, a common strategy is to formulate *effective cone structure* of solutions on small scales. That is, one hopes to understand in what sense solutions can be approximated by Euclidean cones, i.e., homogeneous harmonic polynomials. This natural philosophy has endless descendants in different disciplines of geometry and analysis, such as geometric measure theory, minimal submanifolds, harmonic mappings, mean curvature flow, the metric geometry of Ricci curvature, etc. In the specific context of elliptic PDEs, significant progress has been made in recent decades, with many fundamental works focusing on the measure-theoretic properties of the singular set; see [@DF; @Lin; @Han; @HHL; @Hardt-N; @Han-Lin-harmonic-maps; @Cheeger-Naber-Valtorta; @naber-valtorta]. It is worth mentioning that groundbreaking results have been obtained recently in the studies of Yau's Conjecture and Nadirashvili's Conjecture, which bounds the Hausdorff measure of the nodal sets of eigenfunctions and harmonic functions on Riemannian manifolds; see [@LM; @logu1; @logu2]. In this paper, we are mainly concerned with the singular sets of the solutions for a class of degenerate and nonlocal elliptic equations defined in the context of Poincaré-Einstein manifolds. Before presenting our main results, it is worth introducing the geometric background and motivations of this geometric setting. Extensive studies of Poincaré-Einstein manifolds originally come from the AdS/CFT correspondence in theoretic physics which relates the (Anti-de-Sitter) Riemannian geometry of a complete Einstein manifold $(X^{n + 1}, g_+)$ to the conformal (field) theory of the conformal infinity $(M^n , [h])$, where $M^n$ is the topological boundary of $X^{n + 1}$. This physical philosophy has continuous impacts on conformal geometry so that studying Poincaré-Einstein manifolds has become a very active research direction. On the mathematical side, this topic dates back to the ambient metric construction by Fefferman-Graham in 1985 ([@FG-conformal-invariants; @Fefferman-Graham-ambient]), which successfully produced conformal invariants and conformally covariant elliptic operators. In this setting, a typical class of conformally covariant elliptic operators $P_{2\gamma}$ constructed on the conformal manifolds $(M^n, [h])$ are nowadays called *fractional GJMS operators* with principal symbol equal to the principal symbol of $(-\Delta)^{\gamma}$ on $\mathbb{R}^n$; see [@CG; @Case-Chang; @Chang-Yang] for a synthetic background. The operators $P_{2\gamma}$ depend not only on the conformal geometry of $(M^n, [h])$ but also on the Riemannian geometry of $(X^{n+1}, g_+)$: they exhibit nonlocal nature for generic values of $\gamma$. Regarding the construction and localization of nonlocal GJMS operators, there are two different approaches. First, in Graham-Zworski's work [@GZ], the operators $P_{2\gamma}$ were discovered to have intimate connections with the geometric scattering theory on the asymptotically hyperbolic Poincaré-Einstein manifolds. Roughly speaking, the GJMS operators defined on the conformal infinity $(M^n,[h])$ can be realized as certain "Dirichlet-to-Neumann mapping\", via a scattering operator coming from a non-degenerate Poisson equation defined on the Poincaré-Einstein filling $(X^{n + 1}, g_+)$. A powerful advantage of this framework is that conformally covariant operators and conformal invariants of different orders can be treated in a uniform way when $\gamma\in(0,\frac{n}{2})$. The second approach to constructing GJMS operators $P_{2\gamma}$ of low orders $\gamma \in (0,1)$ originated in Caffarelli-Silvestre's work [@CS]. The main construction of $P_{2\gamma}$ is realized as a "Dirichlet-to-Neumann mapping\" via a degenerate Laplace equation defined on a conformal compactification $(\overline{X^{n + 1}}, \bar{g})$ which degenerates precisely on the boundary $M^n$. This approach translates the difficulty in studying the behaviors of non-degenerate operators near the infinity of a complete manifold $X^{n + 1}$ to investigating degenerate operators defined on a compact manifold $\overline{X^{n + 1}}$ with boundary $M^n$. Recognizing the equivalence between these two constructions and formulating higher order generalizations represents a major progress in the geometric direction; see [@CG] and [@Case-Chang] for the details. To introduce the main results of the paper, we now set up the geometric background. Given any integer $n\geq 2$, let $(X^{n + 1}, g_+)$ be a complete asymptotically hyperbolic Poincaré-Einstein manifold such that $\mathop{\mathrm{Ric}}_{g_+} \equiv - n g_+$. Let $(M^n, [h])$ be the conformal infinity of $(X^{n + 1}, g_+)$: $M^n$ is the topological boundary of $X^{n + 1}$ and $g_+$ admits a compactification $\bar{g}= \rho^2 g_+$ on the compact $\overline{X^{n + 1}}$ with boundary $M^n$ so that $\bar{g}|_{M^n} = \bar{g}|_{\rho = 0} = h$. We refer the readers to Section [2.1](#ss:PE-metrics){reference-type="ref" reference="ss:PE-metrics"} for detailed definitions of the Poincaré-Einstein manifold. As in the aforementioned works [@GZ; @CG], on the conformal infinity $M^n$, for any $\gamma\in(0,\frac{n}{2})$, there is a family of formally self-adjoint pseudo differential operators, $P_{2\gamma}$, called fractional GJMS operators. We restrict ourselves to the case $\gamma \in (0,1)$. The main reason is that we are using monotonicity methods which are less handy when considering $\gamma >1$. A function $f$ is called $\gamma$-harmonic on $B_1(p)\subset M^n$ if $P_{2\gamma}(f) = 0$ on $B_1(p)$. We also denote $$\begin{aligned} \label{e:zero-critical-singular} \mathcal{Z}(f) \equiv \{x\in M^n: f(x) = 0\}, \quad \mathcal{C}(f) \equiv \{x\in M^n: |\nabla f|(x) = 0\}, \quad \mathcal{S}(f) \equiv \mathcal{Z}(f) \cap \mathcal{C}(f). \end{aligned}$$ Our primary interest is to estimate the size of the zero/critical/singular sets in [\[e:zero-critical-singular\]](#e:zero-critical-singular){reference-type="eqref" reference="e:zero-critical-singular"}. Similar to the second-order elliptic case in [@Lin; @HHL; @Cheeger-Naber-Valtorta; @naber-valtorta], the size of those sets essentially rely on the "degree\" of the solutions, more precisely, on the frequency of the solutions. In our setting, to make sense of the frequency of a $\gamma$-harmonic function, one needs to first localize the operator $P_{2\gamma}$. As we mentioned, Chang and González managed to localize the operators $P_{2\gamma}$ via the Caffarelli-Silvestre type extension; see [@CG]. For our purpose, we will choose a canonical conformal compactification, and we specify Fefferman-Graham's metric $\bar{g}\equiv \varrho^2 g_+ = e^{2 w}g_+$ in this paper, where $w$ satisfies the equation $-\Delta_{g_+} w = n$ on $X^{n + 1}$. By [@CG] (see also Proposition [Proposition 18](#p:extension-problem-in-FG-metric){reference-type="ref" reference="p:extension-problem-in-FG-metric"}), a function $f\in C^{\infty}(M^n)$ that is $\gamma$-harmonic on $B_1(p)\subset M^n$ is always associated to the boundary value problem, called *Caffarelli-Silvestre type extension*, $$\begin{aligned} \label{e:harmonic-extension-PE} \begin{cases} -\mathop{\mathrm{div}}_{\bar{g}}(\varrho^{\mathfrak{a}} \nabla_{\bar{g}} U) + C_{n,\gamma}\varrho^{\mathfrak{a}} R_{\bar{g}} U = 0 & \text{in}\ \overline{X^{n+1}}, \\ U = f & \text{on}\ M^n = \{\varrho= 0\}, \\ P_{2\gamma} f = 0 & \text{on}\ B_1(p) \subset M^n, \end{cases}\end{aligned}$$ where $R_{\bar{g}}$ is the scalar curvature of $\bar{g}$, $\mathfrak{a}\equiv 1 - 2\gamma \in (-1,1)$, and $C_{n,\gamma} > 0$ is a constant depending only on $n$ and $\gamma$. We remark that the equation is *degenerate* when $\mathfrak{a}\in (0,1)$, and *singular* when $\mathfrak{a}\in (-1,0)$. For the simplicity of the notion, we just call this equation a degenerate equation. In this case, we also have $$\begin{aligned} P_{2\gamma} f = \frac{2^{2\gamma}\cdot\Gamma(\gamma)}{2\gamma\cdot\Gamma(-\gamma)} \lim\limits_{\varrho\to 0}\varrho^{\mathfrak{a}} \frac{\partial U}{\partial\varrho}, \quad \gamma\in(0,1) ;\end{aligned}$$ see Section [2.2](#ss:GJMS){reference-type="ref" reference="ss:GJMS"} for details. Moreover, the extension $U$ is unique in the space $H^{1,\mathfrak{a}}(\overline{X^{n + 1}})$. In particular, if the (partial) compactified metric $\bar{g}$ is the Euclidean metric on $\mathbb{R}_+^{n + 1}$, then [\[e:harmonic-extension-PE\]](#e:harmonic-extension-PE){reference-type="eqref" reference="e:harmonic-extension-PE"} is precisely reduced to Caffarelli-Silvestre's original harmonic extension. These operators were first studied in the pioneering works [@fks; @fkj; @fjk2], where the basic theory related to De Giorgi-Nash-Moser estimates, classical functional weighted inequalities, and elliptic measure is considered. In this paper, when we refer to a $\gamma$-harmonic function $f$, we always associate it with a pair of functions $(U, f)$ for some $U\in H^{1,\mathfrak{a}}(\overline{X^{n + 1}})$ that satisfies [\[e:harmonic-extension-PE\]](#e:harmonic-extension-PE){reference-type="eqref" reference="e:harmonic-extension-PE"}. Now that the zero and singular sets of $f$ have been defined in [\[e:zero-critical-singular\]](#e:zero-critical-singular){reference-type="eqref" reference="e:zero-critical-singular"}, for the extension $U$, we define $$\begin{aligned} \label{e:U-zero-critical-singular} \mathcal{Z}(U) \equiv \{x_+ \in M^n: U(x_+) = 0\}, \quad \mathcal{C}(U) \equiv \{x_+ \in M^n: |\nabla U|(x_+) = 0\}, \quad \mathcal{S}(U) \equiv \mathcal{Z}(U) \cap \mathcal{C}(U), \end{aligned}$$ which are the **restrictions** of the zero/critical/singular sets of $U$ in $\overline{X^{n + 1}}$ on the boundary $M^n$. To define the frequency, it is convenient to double the compact manifold $(\overline{X^{n + 1}}, \bar{g})$ along the boundary $(M^n, h)$, which gives a closed manifold $\mathfrak{X}^{n + 1} \equiv \overline{X^{n + 1}}\bigcup\limits_{M^n}\overline{X^{n + 1}}$. The doubled metric is still denoted as $\bar{g}$; see Section [2](#s:preliminaries){reference-type="ref" reference="s:preliminaries"} for the regularity of the doubled metric. Now for any $x \in B_{1/2}(p) \subset M^n$ and for any $r\in (0,1/2)$, the *generalized Almgren's frequency* $\mathcal{N}_f (x, r)$ is defined as: $$\begin{aligned} \mathcal{N}_f (x, r) \equiv \frac{ \displaystyle{r\int_{\mathcal{B}_r(x)} \varrho^{\mathfrak{a}} \left(|\nabla_{\bar{g}} U|^2 + C_{n,\gamma} R_{\bar{g}} U^2\right) \mathop{\mathrm{dvol}}_{\bar{g}} } }{\displaystyle{ \int_{\partial\mathcal{B}_r(x)} \varrho^{\mathfrak{a}} U^2 d\sigma_{\bar{g}}} },\end{aligned}$$ where $U$ is the extension of $f$ given in [\[e:harmonic-extension-PE\]](#e:harmonic-extension-PE){reference-type="eqref" reference="e:harmonic-extension-PE"} and $\mathcal{B}_r(x)$ is a metric ball in the doubled space $\mathfrak{X}^{n + 1}$. We also define another quantity, called the *normalized Dirichlet energy* of $f$: $$\begin{aligned} \mathcal{E}_f (x, r) \equiv \frac{ \displaystyle{r\int_{\mathcal{B}_r(x)} \varrho^{\mathfrak{a}} |\nabla_{\bar{g}} U|^2 \mathop{\mathrm{dvol}}_{\bar{g}} } }{\displaystyle{ \int_{\partial\mathcal{B}_r(x)} \varrho^{\mathfrak{a}} U^2 d\sigma_{\bar{g}}} }. \end{aligned}$$ Similarly, given any $x_+ \in \mathcal{B}_{1/2}(p_+)\cap M^n$ and $r\in (0,1/2)$, we define $$\begin{aligned} \mathcal{E}_U (x_+, r) \equiv \frac{ \displaystyle{r\int_{\mathcal{B}_r(x_+)} \varrho^{\mathfrak{a}} |\nabla_{\bar{g}} U|^2 \mathop{\mathrm{dvol}}_{\bar{g}} } }{\displaystyle{ \int_{\partial\mathcal{B}_r(x_+)} \varrho^{\mathfrak{a}} U^2 d\sigma_{\bar{g}}} }. \end{aligned}$$ The relation between the above frequency and normalized Dirichlet energy will be discussed in Sections [4.1](#ss:almost-monotonicity){reference-type="ref" reference="ss:almost-monotonicity"} and [4.2](#ss:quantitative-symmetry-PE){reference-type="ref" reference="ss:quantitative-symmetry-PE"}. How their large-scale information passes to smaller scales will be also discussed there. ## Main results {#ss:main-results} Our main results consist of a series of dimension bounds and geometric measure estimates for the nodal and singular sets of both $U$ and $f$. In the following, for any $p\in M^n$, we denote $p_+ \equiv \iota(p)$, where $\iota:M^n\hookrightarrow \mathfrak{X}^{n + 1}$ is the canonical inclusion. The first result is a Hausdorff measure estimate on the zero set. **Theorem 1** (Nodal set estimate). *Let $f\in C^{\infty}(M^n)$ be $\gamma$-harmonic on $B_2(p) \subset M^n$ for some $p\in M^n$. For any $\Lambda > 0$, there exists a constant $C = C(\Lambda, n, \gamma, \bar{g}) > 0$ such that if $\mathcal{E}_f(p, 2) \leq \Lambda$, then $$\begin{aligned} \begin{split} \mathcal{H}^n (\mathcal{Z}(U) \cap B_1(p)) \leq &\ C, \\ \mathcal{H}^{n - 1}(\mathcal{Z}(f) \cap B_1(p)) \leq &\ C, \end{split} \end{aligned}$$ where $B_1(p)\subset M^n$ is the unit ball.* The next two theorems provide effective Minkowski-type estimates for the singular sets of a $\gamma$-harmonic function $f$ and their Caffarelli-Silvestre type extension $U$. Note that the two singular sets $\mathcal{S}(U)$ and $\mathcal{S}(f)$ exhibit very different behaviors. Such phenomena are particularly intriguing in the non-local setting which distinguishes itself from the second-order elliptic case. Therefore, we state these estimates separately. **Theorem 2** (Minkowski-type estimate for extensions). *Fix a point $p\in M^n$. Let $f\in C^{\infty}(M^n)$ be $\gamma$-harmonic on $B_2(p) \subset M^n$ and let $U$ be its extension in $\overline{\mathfrak{X}^{n + 1}}$ defined by [\[e:harmonic-extension-PE\]](#e:harmonic-extension-PE){reference-type="eqref" reference="e:harmonic-extension-PE"}. For any $\Lambda > 0$ and $\tau \in (0, 1)$, there exists $C = C(\tau, \Lambda, n, \gamma, \bar{g}) > 0$ such that if $\mathcal{E}_f(p, 2) \leq \Lambda$, then $$\begin{aligned} \mathop{\mathrm{Vol}}_{\bar{g}}(T_r(\mathcal{S}(U)) \cap B_1(p)) \leq C \cdot r^{2 - \tau}, \end{aligned}$$ where $T_r(A)$ is the $r$-tubular neighborhood of $A$ in $\overline{X^{n + 1}}$. As a consequence, the following Minkowski dimension bound holds, $$\begin{aligned} \dim_{\mathop{\mathrm{Min}}}(\mathcal{S}(U)\cap B_1(p)) \leq n - 1. \end{aligned}$$* Different from $\mathcal{S}(U)$, the singular set $\mathcal{S}(f)$ of a $\gamma$-harmonic function $f$ on $M^n$ has a more complicated structure. To understand the subtlety of the structure of $\mathcal{S}(f)$, we define $$\begin{aligned} \underline{\mathcal{S}}(f) \equiv \left\{x\in \mathcal{S}(f): \text{the tangent map} \ T_x(f)\ \text{of}\ f\ \text{at}\ x\ \text{is a harmonic homogeneous polynomial}\right\}, \end{aligned}$$ and we denote $\mathfrak{S}(f) \equiv \mathcal{S}(f) \setminus \underline{\mathcal{S}}(f)$. **Theorem 3** (Minkowski type estimate on the boundary). *Fix a point $p\in M^n$ Let $f\in C^{\infty}(M^n)$ be $\gamma$-harmonic on $B_2(p) \subset M^n$. For any $\Lambda > 0$ and $\tau \in (0, 1)$, there exists $C = C(\tau, \Lambda, n, \gamma, \bar{g}) > 0$ such that if $\mathcal{E}_f(p, 2) \leq \Lambda$, then $$\begin{aligned} \mathop{\mathrm{Vol}}_{\bar{g}}(T_r(\underline{\mathcal{S}}(f)) \cap B_1(p)) \leq C \cdot r^{2 - \tau} \quad \text{and} \quad \mathop{\mathrm{Vol}}_{\bar{g}}(T_r(\mathfrak{S}(f)) \cap B_1(p)) \leq C \cdot r^{1 - \tau},\end{aligned}$$ As a consequence, $$\begin{aligned} \dim_{\mathop{\mathrm{Min}}}(\underline{\mathcal{S}}(f)\cap B_1(p)) \leq n - 2\quad \text{and} \quad \dim_{\mathop{\mathrm{Min}}}(\mathfrak{S}(f)\cap B_1(p)) \leq n - 1. \end{aligned}$$* We remark that the Minkowski-type estimates in the case of second-order uniformly elliptic have been established by Cheeger-Naber-Valtorta [@Cheeger-Naber-Valtorta] within the framework of quantitative stratification of the singular set. We will adopt this new technique in the setting of degenerate and nonlocal elliptic equations. The next theorem gives the Hausdorff measure estimate for the singular set, which requires higher regularity of the background metric. In the theorem below, we will add an extra geometric assumption. **Theorem 4** (Hausdorff measure estimates). *Given $n\geq 2$, let $(X^{n + 1}, g_+)$ be a Poincaré-Einstein manifold with conformal infinity $(M^n, [h])$. Assume that $[h]$ is obstruction flat when $n$ is even. Let $f$ be $\gamma$-harmonic on $B_2(p) \subset M^n$ and let $U$ be the Caffarelli-Silvestre type extension in [\[e:harmonic-extension-PE\]](#e:harmonic-extension-PE){reference-type="eqref" reference="e:harmonic-extension-PE"}. For any $\Lambda > 0$, there exists $C = C(\Lambda, n, \gamma, g_+, h) > 0$ such that if $\mathcal{E}_f(p, 2) \leq \Lambda$, then* 1. *the singular set of $U$ yields $\mathcal{H}^{n - 1}(\mathcal{S}(U) \cap B_1(p)) \leq C$.* 2. *the singular set of $f$ yields $\mathcal{H}^{n - 2}(\underline{\mathcal{S}}(f) \cap B_1(p)) \leq C$ and $\mathcal{H}^{n - 1}(\mathfrak{S}(f) \cap B_1(p)) \leq C$.* **Remark 5**. The hyperbolic space $(X^{n + 1}, g_+) \equiv (\mathbb{H}^{n + 1}, y^{-2}(dy^2 + g_{\mathbb{R}^n}))$ has a natural (partial) conformal compactification $\overline{X^{n + 1}} \equiv \mathbb{R}_+^{n + 1}$ equipped with the Euclidean metric. This serves as the blow-up model of the general Poincaré-Einstein setting. In this special case, the first named author has proven, in [@STT], the Hausdorff measure estimate for the nodal set (special case of Theorem [Theorem 1](#t: nodal_set_theo){reference-type="ref" reference="t: nodal_set_theo"}) and a weaker dimension bound on the singular set, i.e., the Hausdorff dimension upper bound. Theorem [Theorem 2](#t:volume-estimate-extension){reference-type="ref" reference="t:volume-estimate-extension"} and Theorem [Theorem 3](#t:volume-estimate-boundary){reference-type="ref" reference="t:volume-estimate-boundary"} are much stronger than the previous Hausdorff dimension estimates in [@STT] even in the Euclidean case. We also notice that all the estimates in Theorems [Theorem 2](#t:volume-estimate-extension){reference-type="ref" reference="t:volume-estimate-extension"}-[Theorem 4](#t:Hausdorff-measure-estimate-(U,f)){reference-type="ref" reference="t:Hausdorff-measure-estimate-(U,f)"} can be strengthened to the critical sets $\mathcal{C}(U)$ and $\mathcal{C}(f)$. An important purpose of the present work is to understand the influence of the interior of the Poincaré-Einstein manifold on the geometry of $\gamma$-harmonic functions on the conformal infinity. As previously mentioned, this heavily relies on a suitable compactification. However, many of the techniques we use in the present work, and in particular the *quantitative stratification* (see the next section for a brief introduction to it), can also be used to give a fine description of the nodal set and its singular part away from the boundary at infinity (see \[Section 4,[@STT])\] for such a study in the Euclidean case). In this case, the operator is of course uniformly elliptic (with a constant degenerating with the distance to the infinity) and the previous results are already known in this case. It is however important to mention that in the same vein, one could investigate the following problem, which is *not* the one we consider but much related. Consider the doubled manifold $\mathfrak{X}^{n + 1}$ and the following equation posed in $\mathcal{B}_1(x_+) \subset \mathfrak{X}^{n + 1}$ $$\label{another_model} \mathop{\mathrm{div}}_{\bar{g}}(\varrho^{\mathfrak{a}} \nabla_{\bar{g}} U)=0,$$ where $\varrho$ has the obvious meaning on $\mathfrak{X}^{n + 1}$ (comparable to the signed distance close enough to the set $\varrho=0$). Notice that our purpose goes somehow in the opposite direction: we consider first a $\gamma$-harmonic function, extend it inside $X^{n + 1}$ and then double the manifold and reflect accordingly (evenly along the normal direction) the previous equation. In the Euclidean model case, a thorough analysis of the nodal sets on the characteristic manifold of solutions of [\[another_model\]](#another_model){reference-type="eqref" reference="another_model"} has been investigated in [@STT], which relies heavily on the oddness/evenness with respect to the doubling $\mathfrak{X}^{n + 1}$ of a general solution of the PDE [\[another_model\]](#another_model){reference-type="eqref" reference="another_model"}. In line with our results, one can also stratify the singular set of the *trace* of $U$ on $\varrho= 0$ and refine the decomposition of the singular set by means of frequencies $\geq 2$ and the symmetry of the associated tangent maps. We would like to point out that the quantitative stratification we implement in the present work would also lead to improvements of the relevant results in [@STT], leading to volume estimates. Of further interest is that, as noted in the previous discussion, the global (nonlocal) effects are also present in this latter investigation. They appear indirectly in the investigation of the singular set of traces of solutions on the *characteristic* manifold $\varrho=0$. ## Outline of the proof and organization of the paper In this subsection, we will explain the main technical ingredients and new analytic inputs in the proof of the main results. We will also outline the interplay of these techniques from different areas for proceeding the main arguments. As well understood in the literature, size estimates for the nodal/singular sets of elliptic equations fundamentally rely on the *unique continuation property* of the solutions. This property is based on sufficient regularity in the ellipticity coefficients of the equations. Notice that Lipschitz continuity of the coefficients is weakest regularity assumption that one can impose, as shown in [@Plis], where it is demonstrated that the unique continuation property fails under Hölder continuous coefficients. In our specific context of Poincaré-Einstein manifold $(X^{n + 1}, g_+)$, it becomes imperative to select an appropriate conformal compactification with sufficient regularity such that the solutions of our interested problem [\[e:harmonic-extension-PE\]](#e:harmonic-extension-PE){reference-type="eqref" reference="e:harmonic-extension-PE"} satisfy the unique continuation. Indeed, the adapted compactified metric constructed in [@CG; @Case-Chang] has only Hölder regularity up to the boundary when the order index of $P_{2\gamma}$ satisfies $\gamma \in (0,1/2)$, which cannot be chosen for our purpose. In the present paper, we will always specify the Fefferman-Graham compacitification the regularity of which will be discussed in detail in Section [2.1](#ss:PE-metrics){reference-type="ref" reference="ss:PE-metrics"}. The entire strategy of analyzing the singular set is built upon the *conic structure analysis* at small scales. The framework has been well understood and widely applied in numerous areas in geometric analysis. In our specific context, to estimate the size of the singular set $\mathcal{S}(f)$ or $\mathcal{S}(U)$ for the solutions of [\[e:harmonic-extension-PE\]](#e:harmonic-extension-PE){reference-type="eqref" reference="e:harmonic-extension-PE"}, one seeks for appropriate stratification of the singular set. *Classical stratification* relies on the symmetry of the tangent map of a solution at a point. For example, the tangent map of a Euclidean harmonic function $f$ can be understood as the leading term, which is a homogeneous polynomial $P$, in its Taylor expansion at a given point, while the *degree of symmetry*, denoted as $k$, of this polynomial refers to the maximal dimension of a hyperplane $\mathcal{L}\subset \mathbb{R}^n$ for which $P$ is invariant. To obtain the correct dimension bound $\dim(\mathcal{S}(f)) \leq n - 2$, one needs to establish the following for $\mathcal{S}^k(f)$: 1. $\mathcal{S}(f) \subset \mathcal{S}^{n - 2}(f)$, namely any $(n - 1)$-symmetric point is not a critical point of $f$; 2. $\dim(\mathcal{S}^k(f)) \leq k$ for any $k \leq n - 2$. Item (1) follows from a simple fact that any single-variable harmonic function must be linear, while item (2) is the key ingredient and relies on certain iterative blow-up arguments in classical works; see [@Lin; @Han; @HHL] for second order elliptic PDEs and see [@STT] for the Euclidean model setting in our specific context. This methodology in the metric geometry of the Ricci curvature has led to a series of fundamentally important works in the field. Pioneering works in this area include [@ChC; @ChC1; @ChC2; @ChC3; @ChCT; @Cheeger-elliptic]. All these works, along with the aforementioned works based on classical stratification contribute to estimating the Hausdorff dimension of the singular set. However, Hausdorff dimension bound is somewhat weak for analysis purposes since a set with *a low (even zero) Hausdorff dimension* can be arbitrarily dense. More effective results are provided by Minkowski-type estimates, which not only bound the interested set itself but also its tubular neighborhood of that set. For example, the set of rational numbers $\mathbb{Q}^n$ as a dense subset of $\mathbb{R}^n$ has a Hausdorff dimension of $0$, but its Minkowski dimension is equal to $n$. The fundamental tools used to prove Minkowski-type estimates are based on the *theory of quantitative differentiation*. This powerful theory was suggested by Cheeger-Kleiner-Naor in [@CKN], and its geometric formulation was further developed by Cheeger and Naber in the context of Ricci curvature; see [@ChNa-quantitative]. This development has resulted in a rather widely applicable machinery for establishing strong estimates on the singular set and curvatures on non-collapsing Einstein manifolds and their limits. The methodology of quantitative differentiation theory has also been successfully applied in other geometric analytic contexts and in the analysis of nonlinear PDEs. Furthermore, the fundamental tools in this theory have inspired resolutions of numerous long-standing conjectures in the field; see [@ChNa-codim-4; @JN; @CJN]. Building on this new framework, the authors obtained a series of effective estimates in the case of second-order uniform elliptic PDEs in [@Cheeger-Naber-Valtorta; @naber-valtorta]. In this paper, our work aims to establish the quantitative differentiation theory for degenerate and nonlocal elliptic PDEs on Poincaré-Einstein manifolds, where the regularity analysis is much more intricate compared to the classical Euclidean case. We will adopt Cheeger-Naber-Valtorta's general framework [@Cheeger-Naber-Valtorta] to formulate quantitative stratification for the singular set of the solutions of [\[e:harmonic-extension-PE\]](#e:harmonic-extension-PE){reference-type="eqref" reference="e:harmonic-extension-PE"}. This fundamental framework will be combined with a careful analysis on the regularity of the underlying Poincaré-Einstein manifolds. The foundation of such analysis is the *almost monotonicity formula* for the generalized Almgren's frequency in the Poincaré-Einstein setting, as stated in Theorem [Theorem 69](#t:almost-monotonicity){reference-type="ref" reference="t:almost-monotonicity"}. This (almost) monotonicity enables us to obtain the conic structure (homogeneous polynomials) on all but finitely many scales, which leads us to prove the Minkowski type estimates for the singular set. In the next stage, in order to obtain a more refined estimate of the Hausdorff measure for the singular set, one approach is to prove a strong $\epsilon$-regularity result. This result was originally established by Han, Hardt, and Lin in [@HHL theorem 4.1] in the context of second-order uniformly elliptic PDEs. Essentially, this $\epsilon$-regularity theorem states that if a solution of an elliptic PDE is sufficiently close to a homogeneous harmonic polynomial, then their singular sets exhibit similar behaviors. To prove such an $\epsilon$-regularity in our specific setting, we need to obtain a precise characterization of the tangent maps, which are *homogeneous hypergeometric polynomials* (see Section [3.5](#ss:epsilon-regularity){reference-type="ref" reference="ss:epsilon-regularity"}). Additionally, the $\epsilon$-regularity in [@HHL] requires very high regularity of the ellipticity coefficients, which leads us to carry out delicate analysis on the compactification of the Poincaré-Einstein metric. Specifically, when the metric under consideration does not have sufficiently high regularity, we are still able to appropriately adjust the original solution to a more regular one. Our new observation for this aspect is included in Proposition [Proposition 85](#p:higher-order-approximation-PE){reference-type="ref" reference="p:higher-order-approximation-PE"}. Taking into account these two technical points, our $\epsilon$-regularity result (Theorem [Theorem 84](#t:eps-reg-PE){reference-type="ref" reference="t:eps-reg-PE"}) is essentially novel and allows for lower regularity requirements compared to those in [@HHL], making it of independent interest. The remaining sections of the paper are organized as follows. Section 2 provides several preliminary materials, including the basics of Poincaré-Einstein manifolds and fractional GJMS operators. It also covers some fundamental regularity properties of the PDEs that are being studied. Section 3 focuses on the quantitative stratification of the singular set and the main estimates in the model case. Specifically, it considers the scenario where the Poincaré-Einstein manifold $(X^{n + 1}, g_+)$ is hyperbolic and its (partial) compactification is Euclidean. In Section 4, the main results are proven in their full generality, significantly extending beyond the model case discussed in Section 3. **Notations and conventions.** For convenience, we list several notations which are frequently used in the later sections. - For any $p\in M^n$, let us denote $p_+ \equiv \iota(p)$ for the inclusion $\iota: M^n \to \overline{X^{n + 1}}$ or $\iota: M^n \to \mathfrak{X}^{n+1}$. - Given $n\geq 2$, we often denote $\mathfrak{m}\equiv n + 1$. - $\mathcal{B}_r(x)$ denotes a ball in $\overline{X}^{n+1}$ or $\mathfrak{X}^{n + 1}$, while $B_r(x)$ denotes a ball in $M^n$. - Given $x\in M^n$, $\mathcal{B}_r^+(x)\equiv\mathcal{B}_r(x) \cap \overline{X^{n+1}}$ denotes the intersection between a ball and the compactified manifold $\overline{X^{n+1}}$. - $\rho^2 g_+$ denotes a general conformal compactification, and $\varrho^2 g_+$ denotes Fefferman-Graham's compactification. ## Acknowledgements The authors would like to express their gratitude to Qing Han and Fang-Hua Lin for their interest in this work and for their valuable comments. The third named author would also like to thank Fang Wang for the insightful discussions on the regularity of the Poincaré-Einstein metrics, and he is indebted to Jeffrey Case and Alice Chang for valuable communications on the fractional GJMS operators. # Preliminaries {#s:preliminaries} ## Poincaré-Einstein metrics and some regularity results {#ss:PE-metrics} **Definition 6**. A complete Riemannian manifold $(X^{n+1},g_+)$ is called asymptotically hyperbolic Poincaré-Einstein if $g_+$ satisfies $\mathop{\mathrm{Ric}}_{g_+} \equiv - n g_+$ and admits a conformal compactification in the following sense. 1. $X^{n+1}$ is diffeomorphic to the interior of a compact manifold $\overline{X^{n + 1}}$ with boundary $M^n$; 2. there is a smooth defining function $\rho:\overline{X^{n + 1}} \to [0,\infty)$ such that $$\begin{aligned} \begin{cases} \rho > 0 & \text{in}\ X^{n+1}, \\ \rho = 0 & \text{on}\ M^n, \\ |\nabla\rho| = 1 & \text{on}\ M^n. \end{cases} \end{aligned}$$ In this case, the conformal manifold $(M^n, [h])$ is called the conformal infinity, where $h \equiv \rho^2 g_+|_{\rho = 0}$. On a Poincaré-Einstein manifold, the existence of the geodesic defining function near the boundary is well known in the literature now; see, e.g., [@Lee-spectrum] or [@Gra2000]. **Lemma 7** (Geodesic defining function). *Let $(M^n,[h])$ be the conformal infinity, then for every $h_0\in[h]$ there exists a unique defining function $y$ in a neighborhood of $M^n$ such that* *$(1)$ $y^2g_+|_{y=0}=h_0$,* *$(2)$ there exists $\epsilon>0$ such that $|\nabla y|_{y^2g_+}\equiv 1$ on $M^n\times [0,\epsilon)$.* On $M^n\times (0, \epsilon),$ the metric splits as $g_+ = \frac{dy^2+ g_y}{y^2},$ where $g_y$ satisfies $g_y|_{y=0} = h$. Moreover, in [@Gra2000], the compactified metric $g_y$ yields an expansion in terms of the geodesic defining function. If $n$ is odd, $$\begin{aligned} \label{eq-gy1} g_y = g^{(0)} + \frac{g^{(2)}}{2}y^2 +\ldots + (\text{even powers}) + g^{(n-1)}y^{n-1} + g^{(n)}y^{n}+\ldots,\end{aligned}$$ where $g^{(2i)}$ depends only on $h$ for any $2i \leq n - 1$, and *the global term* $g^{(n)}$ depends on both $g_+$ and $h$. If $n$ is even, $g_y$ has an expansion of the form $$\begin{aligned} \label{eq-gy2} g_y = g^{(0)} + \frac{g^{(2)}}{2}y^2 +\ldots + (\text{even powers}) + g^{(n)}y^{n} + \omega \cdot y^n \log y + \ldots,\end{aligned}$$ where $g^{(n)}$ is also a global term which depends on both $h$ and $g_+$, and $\omega$ depends only on $h$ and satisfies $\mathop{\mathrm{Tr}}_h(\omega)=0$. For $n=2$, $\omega=0$, and $g_y$ is smooth on $M^n \times [0, \epsilon)$ in this case. It was proved by Biquard that $h$ and $g^{(n)}$ completely determine the compactified metric and hence $g_+$; see [@Biquard]. For our analysis, we need a compactified metric $\bar{g}$ which has sufficiently high regularity and provides explicit geometric properties when interacting with the scattering operators introduced in Section [2.2](#ss:GJMS){reference-type="ref" reference="ss:GJMS"}. For this purpose, we choose the compactification constructed by Fefferman-Graham in [@FG]. Let $(X^{n + 1}, g_+)$ be an asymptotically hyperbolic Poincaré-Einstein manifold and let $(M^n, h)$ be its conformal infinity. Fefferman-Graham proved **Lemma 8** ([@FG]). *For any representative $h$ on the conformal infinity $(M^n, [h])$, there exists a conformal compactification $\bar{g}= \varrho^2 g_+ \equiv e^{2w} g_+$, called the *Fefferman-Graham compactification*, that satisfies $$\begin{aligned} -\Delta_{g_+} w = n \quad \text{on}\ X^{n + 1}, \end{aligned}$$ and $\bar{g}|_{M^n} = h$. Near the boundary, $w = \log y +O(y^\epsilon)$ for some $\epsilon \in (0, 1)$.* Now we discuss the expansion of $w$ in the geodesic defining function $y$ near the conformal infinity $M^n$. It was proved in [@FG theorem 3.1] that $w$ has a formal expansion $$\begin{aligned} \label{e:w-expansion} w = \log y + \mathscr{A}+ \mathscr{B}y^n\log y +O(y^n),\end{aligned}$$ where $\mathscr{A}, \mathscr{B}\in C^\infty(\overline{X^{n + 1}})$ and $\mathscr{A}=O(y^2)$. It is similar to the expansion of $g_y$ in [\[eq-gy1\]](#eq-gy1){reference-type="eqref" reference="eq-gy1"}-[\[eq-gy2\]](#eq-gy2){reference-type="eqref" reference="eq-gy2"} that $\mathscr{A}\mod O(y^n)$ is even in $y$. So it follows that $\varrho$ yields $$\begin{aligned} \label{e:vr-expansion} \varrho= e^w = y\exp\left(\mathscr{A}+ \mathscr{B}y^n\log y +O(y^n)\right).\end{aligned}$$ Note that $\mathscr{B}\equiv 0$ when $n$ is odd, which implies that both $w$ and $\bar{g}$ are $C^{\infty}$ up to the boundary. In general, $\varrho\in C^{n - 1, \alpha}(\overline{X^{n + 1}})$ and hence $\bar g$ is $C^{n-1,\alpha}$ up to boundary. The appearance of $\mathscr{B}$ obstructs $\bar{g}$ to be $C^{\infty}$ up to the boundary. In summary, we have the following helpful result on the regularity of Fefferman-Graham's compactified metric is as follows. Note that a stronger result was proved in [@wang-zhou-2]. **Proposition 9**. *Given $n \geq 2$, let $(X^{n + 1}, g_+)$ be a Poincaré-Einstein manifold with $\mathop{\mathrm{Ric}}_{g_+} \equiv - n g_+$ such that it admits a conformal compactification which is $C^{k,\alpha}$ up to the boundary for some $k \geq 3$ and $\alpha\in(0,1)$. Denote by $\bar{g}$ the Fefferman-Graham compactification. Then the following holds:* 1. *if $n$ is odd, then $\bar{g}$ is $C^{k , \beta}$ up to the boundary for any $\beta\in(0,\alpha)$;* 2. *if $n$ is even and $n\geq 4$, then $\bar{g}$ is $C^{n - 1 , \beta}$ up to the boundary for any $\beta\in(0,1)$.* **Remark 10**. In the simplest case $n = 2$, any Poincaré-Einstein manifold $(X^3, g_+)$ is hyperbolic and its conformal infinity $(M^2,[h])$ is locally conformally flat. This implies that $(X^3, g_+)$ admits a conformal compactification which is $C^{\infty}$ up to the boundary. In this case, the Fefferman-Graham compactified metric is also $C^{\infty}$ up to the boundary. One can see from item (2) of Proposition [Proposition 9](#p:general-regularity-FG){reference-type="ref" reference="p:general-regularity-FG"} that the regularity of $\bar{g}$ does not exceed $C^n$ in general when $n$ is even. A fundamental reason is that the regularity of $\bar{g}$ is determined by some geometric invariant, called the *obstruction tensor*, of the conformal infinity; see [@Graham-Hirachi] for the definition of the obstruction tensor in this context. If we assume the conformal infinity is *obstruction flat*, the following proposition follows from a regularity result due to Graham-Hirachi; see [@Graham-Hirachi theorem 2.2]. **Proposition 11**. *Let $n\geq 4$ be even, and let $(X^{n + 1}, g_+)$ be a Poincaré-Einstein manifold with $\mathop{\mathrm{Ric}}_{g_+} \equiv - n g_+$ such that it admits a conformal compactification which is $C^{k,\alpha}$ up to the boundary for some $k \geq 3$ and $\alpha\in(0,1)$. Assume that the conformal infinity $(M^n, [h])$ is obstruction flat. Then the Fefferman-Graham compactified metric $\bar{g}$ is $C^{k,\beta}$ for any $\beta \in (0,\alpha)$.* **Remark 12**. In the case when $n \geq 4$ is even, the term $\mathscr{B}$ in the expansion [\[e:w-expansion\]](#e:w-expansion){reference-type="eqref" reference="e:w-expansion"} is vanishing if and only if the obstruction tensor of $h$ is vanishing on $M^n$, and $\varrho$ is smooth up to the boundary. We finally recall the following well-known result. **Lemma 13**. *Let $(X^{n + 1}, g_+)$ be a Poincaré-Einstein manifold with a conformal infinity $(M^n, [h])$. Let $\bar{g}$ be the Fefferman-Graham compactification of $(X^{n + 1}, g_+)$. Then $(M^n, h)$ is totally geodesic in $(\overline{X^{n + 1}}, \bar{g})$.* The previous lemma allows us to double the manifold $M^n$ inside $X^{n + 1}$ and consider a suitable *even* extension of solutions of the PDEs under consideration. This is crucial for regularity purposes as discussed in Section [2.3](#ss:regularity-of-solutions){reference-type="ref" reference="ss:regularity-of-solutions"}. The whole package will be used in Section [4](#s:results-on-PE){reference-type="ref" reference="s:results-on-PE"}. ## Scattering and GJMS operators on the Poincaré-Einstein manifolds {#ss:GJMS} This subsection will summary background materials regarding the fractional GJMS operators defined on the conformal infinity $(M^n, h)$ of a Poincaré-Einstein manifold $(X^{n + 1}, g_+)$. Let us consider the Poisson equation $$\begin{aligned} (\Delta_{g_+} + s(n-s)) u = 0, \quad s\in\mathbb{C},\label{e:Poisson}\end{aligned}$$ on $X^{n+1}$. The interaction between conformal geometry of $(M^n, h)$ and scattering operators defined in this setting is systematically in [@GZ]. Let us first describe the structure of the solutions of [\[e:Poisson\]](#e:Poisson){reference-type="eqref" reference="e:Poisson"}. It is pointed out in [@GZ] that for any $s\in\mathbb{C}$ that satisfies $s(n-s)\not\in\sigma_{pp}(-\Delta_{g_+})$, equation [\[e:Poisson\]](#e:Poisson){reference-type="eqref" reference="e:Poisson"} admits a generalized eigenfunction $$\begin{aligned} \label{eq-u-asymp} u = F y^{n-s} + G y^s, \quad F,G\in C^{\infty}(X^{n+1}). \end{aligned}$$ Here when $n$ is odd or $n=2$, $F, G\in C^{\infty}(\overline{X^{n + 1}})$; when $n\geq 4$ is even, $F, G \in C^{n-1, \alpha}(\overline{X^{n + 1}})$ for $\alpha \in (0,1)$. In fact, when $n\geq 4$ even and $h$ is *not obtruction-free*, the terms involving $y^n\log y$ appear in the expansion of $F$ and $G$ with respect to $y$. The scattering operator $S(s)$, by definition, is the following Dirichlet-to-Neumann operator: if $f\equiv F|_{y=0}$, then $$\begin{aligned} S(s) f \equiv G|_{y=0}. \end{aligned}$$ It was proved in [@GZ] that, as conformally covariant pseudo-differential operators, $S(s)$ gives a meromorphic family with respect to $\{s\in\mathbb{C}:\mathop{\mathrm{Re}}(s)>\frac{n}{2}\}$ with simple poles in $\frac{n}{2}+\mathbb{Z}_+\subset \mathbb{R}$. Now for any $\gamma\in(0,\frac{n}{2})$, the *fractional GJMS operator $P_{2\gamma}$* is defined by $$\begin{aligned} P_{2\gamma}(f) \equiv P_{2\gamma}[h,g_+](f)\equiv 2^{2\gamma}\frac{\Gamma(\gamma)}{\Gamma(-\gamma)}S\left(\frac{n}{2}+\gamma\right)(f).\label{regular-fractional-operator}\end{aligned}$$ In this case, the nonlocal curvature $Q_{2\gamma}$ of order $2\gamma$ is defined as $Q_{2\gamma} \equiv (\frac{n - 2\gamma}{2})^{-1}P_{2\gamma}(1)$. Observe that the function $\Gamma(\gamma)$ cancels the simple poles of the scattering operator $S(\frac{n}{2}+\gamma)$ so that we define $$\begin{aligned} P_n \equiv 2^n \frac{\Gamma(\frac{n}{2})}{\Gamma(-\frac{n}{2})}\mathop{\mathrm{Res}}_{s=n}S(s),\end{aligned}$$ and $Q_n \equiv \lim\limits_{\gamma\to n} Q_{2\gamma} \equiv (\frac{n - 2\gamma}{2})^{-1}P_{2\gamma}(1)$. Therefore, $P_{2\gamma}$ is continuous in the range $\gamma\in(0,\frac{n}{2}]$. There are two important special cases: when $\gamma = 1$, the operator $P_2$ coincides with the conformal Laplacian $$\begin{aligned} \mathscr{L}_h \equiv - \Delta_h + \frac{n-2}{4(n-1)} R_h;\end{aligned}$$ when $\gamma = 2$, the operator $P_4$ coincides with the Paneitz operator of order $4$. The pseudo-differential operator $P_{2\gamma}$ has an important conformal covariance property in the following sense: if $\gamma\in (0,\frac{n}{2})$ and $\tilde{h} \equiv v^{\frac{4}{n - 2\gamma}}h$, then $$\begin{aligned} P_{2\gamma}[\tilde{h}, g_+](u) = v^{-\frac{n + 2\gamma}{n - 2\gamma}} P_{2\gamma}[h,g_+](uv); \end{aligned}$$ if $\gamma = \frac{n}{2}$ and $\tilde{h} \equiv e^{2w} h$, then $$\begin{aligned} e^{nw} \widetilde{Q}_n = Q_n + P_n(w).\end{aligned}$$ This gives a uniform way in studying the conformal geometry involving the operator $P_{2\gamma}$ and makes $P_{2\gamma}$ play a crucial role in the conformal invariance theory. How the operators $P_{2\gamma}$ reveal the geometry and topology of the underlying space still needs further exploration. The third named author initiated studies of the topological aspect of the operator $P_{2\gamma}$ and the associated nonlocal curvature $Q_{2\gamma}$; see [@Zhang-Kleinian] and [@Chen-Zhang-rigidity] for topological and isometric rigidity/classification results in this direction. **Remark 14**. We emphasize that, throughout this section, we will assume that $\lambda_1(-\Delta_{g^+}) > \frac{n^2}{4} - \gamma^2$ which guarantees $s = \frac{n}{2} + \gamma$ satisfies $s(n-s)\not\in \sigma_{pp}(-\Delta_{g_+})$ . A sufficient condition for this is that the non-negativity of the Yamabe constant of $(M^n,h)$, i.e., $\mathcal{Y}(M^n, [h]) \geq 0$. In fact, Lee proved that in this case $\lambda_1(-\Delta_{g_+}) = \frac{n^2}{4}$; see [@Lee-spectrum]. Let us discuss a different formulation of the fractional GJMS operator. First let us recall an example in the model case. **Example 15** (Fractional Laplacian in the hyperbolic case). Let $(\mathbb{H}^{n + 1},g_{_1})$ be the hyperbolic space which admits a partial compactification $(\mathbb{R}^{n + 1}, g_0)$, where $g_{-1} = y^{-2}(dy^2 + g_{\mathbb{R}^n})$ is the hyperbolic metric and $g_0 = dy^2 + g_{\mathbb{R}^n}$ is the Euclidean metric on the upper half space $\mathbb{R}_+^{n + 1}$. One can show that $P_{2\gamma}[g_0, g_{-1}] = (-\Delta_{g_0})^{\gamma}$, where $$\begin{aligned} \left((-\Delta_{g_0})^{\gamma}f\right)(x) \equiv C_{n,\gamma} \cdot \mathop{\mathrm{P.V.}}\int_{\mathbb{R}^n}\frac{f(x) - f(\xi)}{|x - \xi|^{n + 2\gamma}}d\xi, \end{aligned}$$ where $C_{n,\gamma}$ is some constant depending only on $n$ and $\gamma$. By solving an extension problem, Caffarelli and Silvestre have introduced in [@CS] an equivalent definition of $(-\Delta_{g_0})^{\gamma}$ when $\gamma\in(0,1)$. For a function $f: \mathbb{R}^n \to \mathbb{R}$, one can construct the extension $U\in H^{1,\mathfrak{a}}(\mathbb{R}_+^{n+1})$ as the unique solution of the following boundary value problem $$\begin{aligned} \begin{cases} \mathop{\mathrm{div}}_{g_0}(y^{\mathfrak{a}} \nabla_{g_0} U) = 0 & \text{in} \ \mathbb{R}_+^{n+1},\\ U(0,\cdot) = f & \text{on} \ \mathbb{R}^n \equiv \mathbb{R}_+^{n+1}\cap \{y = 0\}, \end{cases} \end{aligned}$$ where $\mathfrak{a}\equiv 1 - 2\gamma \in (-1, 1)$ and $$\begin{aligned} H^{1,\mathfrak{a}}(\mathbb{R}_+^{n + 1}) \equiv \overline{C^{\infty}(\mathbb{R}_+^{n + 1})}^{\|\cdot\|_{H^{1,\mathfrak{a}}}},\quad \|U\|_{H^{1,\mathfrak{a}}(\mathbb{R}_+^{n+1})}\equiv \left(\int_{\mathbb{R}_+^{n+1}}y^{\mathfrak{a}} U^2 \mathop{\mathrm{dvol}}_{g_0}+\int_{\mathbb{R}_+^{n+1}}y^{\mathfrak{a}} |\nabla_{g_0} U|^2 \mathop{\mathrm{dvol}}_{g_0} \right)^{\frac{1}{2}}.\end{aligned}$$ Caffarelli and Silvestre proved that $$\begin{aligned} (-\Delta_{g_0})^{\gamma} f = \frac{2^{2\gamma}\cdot\Gamma(\gamma)}{2\gamma\cdot\Gamma(-\gamma)}\lim\limits_{y\to 0} y^{\mathfrak{a}}\frac{\partial U}{\partial y}. \end{aligned}$$ In the general case, Chang and González obtained in [@CG] a similar formulation for the fractional GJMS operator $P_{2\gamma}$. We fix a positive number $\gamma \in (0,1)$. Let $(X^{n+1}, g_+)$ be a Poincaré-Einstein manifold with conformal infinity $(M^n, h)$. Let us take a conformal compactification $(\overline{X^{n + 1}}, \rho^2 g_+)$ of $(X^{n+1}, g_+)$ for some defining function $\rho$ such that $\rho^2 g_+|_{M^n} = h$. Chang-González proved an equivalence relation: the equation $$\begin{aligned} -\Delta_{g_+}u - s(n - s)u = 0\quad \text{in} \ (X^{n + 1}, g_+) \end{aligned}$$ is equivalent to $$\begin{aligned} \label{e:equivalent-to-harmonic-extension} \begin{split} -\mathop{\mathrm{div}}_{\bar{g}}(\rho^{\mathfrak{a}}\nabla_{\bar{g}} U) + E(\rho) U = 0, \text{ in } X^{n + 1}, \\ \bar{g}\equiv \rho^2 g_+, \quad U \equiv \rho^{s - n}u, \end{split} \end{aligned}$$ where $\mathfrak{a}\equiv 1 - 2\gamma$, and $$\begin{aligned} E(\rho) \equiv - \Delta_{\bar{g}}(\rho^{\frac{\mathfrak{a}}{2}})\rho^{\frac{\mathfrak{a}}{2}} + \left(\gamma^2 - \frac{1}{4}\right)\rho^{-2 + \mathfrak{a}} + \frac{n-1}{4n}R_{\bar{g}}\rho^{\mathfrak{a}}. \label{e:equation-of-E} \end{aligned}$$ Furthermore, in [@Case-Chang], there is an improvement of [@Case-Chang theorems 4.3 and 4.7]. Let $y$ be the geodesic defining function corresponding to $h$ and taking a defining function $\rho$ for $M^n$ that satisfies $$\begin{aligned} \rho = y + \Phi y^{1+2\gamma} + o(y^{1+2\gamma}) \label{e:assumption-on-vr}\end{aligned}$$ for some smooth function $\Phi \in C^\infty(M^n)$. Given $f\in C^\infty(M^n)$, let $U$ be the solution of the boundary value problem $$\begin{aligned} \label{e:U-extension-problem-PE} \begin{cases} -\mathop{\mathrm{div}}_{\bar{g}}(\rho^{\mathfrak{a}}\nabla U) + E(\rho) U = 0, &\text{ in } X^{n+1}, \\ U = f &\text{ on } M^n. \end{cases}\end{aligned}$$ Then $$\begin{aligned} P_{2\gamma} f = \frac{d_\gamma}{2\gamma}\lim_{\rho\rightarrow 0} \rho^\mathfrak{a}\frac{\partial U}{\partial \rho} + \frac{n-2\gamma}{2}d_\gamma \Phi f \label{e:expression-of-GJMS-lower-regularity}\end{aligned}$$ for $d_\gamma \equiv 2^{2\gamma} \frac{\Gamma(\gamma)}{\Gamma(-\gamma)}$. In the following lemma, we will compute the term [\[e:equation-of-E\]](#e:equation-of-E){reference-type="eqref" reference="e:equation-of-E"} when $\bar{g}= \varrho^2 g_+$ is the Fefferman-Graham compactified metric. **Lemma 16**. *Let $w$ be the solution of $-\Delta_{g_+} w = n$ that satisfies the asymptotics $w = \log y + O(y^{\epsilon})$ for some $\epsilon > 0$. With respect to Fefferman-Graham's compactification, $$\begin{aligned} \bar{g} \equiv \varrho^2 g_+, \quad \varrho\equiv e^w,\end{aligned}$$ we have that $$\begin{aligned} E(\varrho) = C_{n,\gamma} \cdot \varrho^{\mathfrak{a}} \cdot R_{\bar{g}}. \end{aligned}$$ If $n=2$ or $n$ is odd, then $R_{\bar g}\in C^\infty(\overline{X^{n + 1}})$ is smooth. If $n\geq 4$ is even, then $R_{\bar g}\in C^{n-3,\alpha}(\bar X^{n+1})$.* *Proof.* It is straightforward to check that $$\begin{aligned} \begin{split} -(d-1) g_+= \mathop{\mathrm{Ric}}_{g_+} = &\ \mathop{\mathrm{Ric}}_{\bar{g}} + (d - 2) \varrho^{-1}\nabla_{\bar{g}}^2\varrho+ (\varrho^{-1}\Delta_{\bar{g}}\varrho- (d - 1) \varrho^{-2}|\nabla_{\bar{g}}\varrho|^2) \bar{g}, \\ -d(d-1)= R_{g_+} = &\ \varrho^2 \left(R_{\bar{g}} + (2d-2)\varrho^{-1}\Delta_{\bar{g}}\varrho- d(d-1)\varrho^{-2}|\nabla_{\bar{g}}\varrho|^2\right),. \end{split}\end{aligned}$$ where $d = n + 1$. Applying $-\Delta_{g_+} w = n$, we have that $$\begin{aligned} & R_{\bar{g}} = (d-1)(d-2)\varrho^{-2}(1 - |\nabla_{\bar{g}}\varrho|_{\bar{\rho}}^2), \\ & \mathop{\mathrm{Ric}}_{\bar{g}} = -(d-2)\varrho^{-1}\nabla_{\bar{g}}^2\varrho. \end{aligned}$$ It turns out that $R_{\bar{g}} = -(d - 2) \varrho^{-1}\Delta_{\bar{g}}\varrho$. Now $$\begin{aligned} \Delta_{\bar{g}}(\varrho^{\frac{\mathfrak{a}}{2}})\varrho^{\frac{\mathfrak{a}}{2}} = \frac{\mathfrak{a}}{2}\varrho^{\mathfrak{a}- 1}\Delta_{\bar{g}} \varrho+ \frac{\mathfrak{a}}{2}\left(\frac{\mathfrak{a}}{2} -1 \right)\varrho^{\mathfrak{a}- 2}|\nabla_{\bar{g}}\varrho|^2, \end{aligned}$$ and hence $$\begin{aligned} E(\varrho) = &\ - \frac{\mathfrak{a}}{2}\varrho^{\mathfrak{a}- 1}\Delta_{\bar{g}}\varrho+ \left(\frac{\gamma^2}{4} - 1\right)\cdot\varrho^{\mathfrak{a}- 2}(1 - |\nabla_{\bar{g}}\varrho|^2) + \frac{n-1}{4n} R_{\bar{g}}\varrho^{\mathfrak{a}} \nonumber\\ = &\ \varrho^{\mathfrak{a}}\left(\frac{\mathfrak{a}}{2}\cdot\frac{1}{d-2} + \left(\frac{\gamma^2}{4} - 1\right)\cdot\frac{1}{(d-1)(d-2)} + \frac{n - 1}{4n}\right) R_{\bar{g}} \nonumber\\ = & \ C_{n,\gamma} \cdot \varrho^{\mathfrak{a}} \cdot R_{\bar{g}},\end{aligned}$$ where $$\begin{aligned} C_{n,\gamma} = \frac{n^2 -3 - 4n\gamma + \gamma^2}{4n(n-1)}. \end{aligned}$$ The proof of the lemma is complete. ◻ We end this subsection by mentioning a lemma regarding the simplified expression of $P_{2\gamma}$ when $\bar{g}= \varrho^2 g_+$ is Fefferman-Graham's compactified metric. **Lemma 17**. *Let $(X^{n + 1}, g_+)$ be a Poincaré-Einstein manifold and let $(M^n, h)$ be its conformal infinity. If $\bar{g}= \varrho^2 g_+$ is Fefferman-Graham's compactified metric, then $$\begin{aligned} P_{2\gamma} f = \frac{2^{2\gamma}\cdot\Gamma(\gamma)}{2\gamma\cdot\Gamma(-\gamma)} \lim_{\varrho\rightarrow 0} \varrho^\mathfrak{a}\frac{\partial U}{\partial \varrho}, \quad f\in C^{\infty}(M^n),\end{aligned}$$ where $U$ is the unique solution in [\[e:U-extension-problem-PE\]](#e:U-extension-problem-PE){reference-type="eqref" reference="e:U-extension-problem-PE"}.* *Proof.* By [\[e:vr-expansion\]](#e:vr-expansion){reference-type="eqref" reference="e:vr-expansion"}, $\varrho= y + O(y^3)$ which implies the term $\Phi$ in [\[e:assumption-on-vr\]](#e:assumption-on-vr){reference-type="eqref" reference="e:assumption-on-vr"} is vanishing. Using [\[e:expression-of-GJMS-lower-regularity\]](#e:expression-of-GJMS-lower-regularity){reference-type="eqref" reference="e:expression-of-GJMS-lower-regularity"}, we have that $$\begin{aligned} \label{eqe-Pf} P_{2\gamma} f = \frac{2^{2\gamma}\cdot\Gamma(\gamma)}{2\gamma\cdot\Gamma(-\gamma)} \lim_{\varrho\rightarrow 0} \varrho^\mathfrak{a}\frac{\partial U}{\partial \varrho},\end{aligned}$$ which completes the proof. ◻ Summarizing the above discussion, we have the following proposition which is a combination of Lemma [Lemma 16](#l:E){reference-type="ref" reference="l:E"} and Lemma [Lemma 17](#l:GJMS-operator-regular-metric){reference-type="ref" reference="l:GJMS-operator-regular-metric"}. **Proposition 18**. *Let $(X^{n + 1}, g_+)$ be a Poincaré-Einstein manifold and let $(M^n, h)$ be its conformal infinity. Let $\bar{g}= \varrho^2 g_+$ be Fefferman-Graham's compactified metric associated to $h$. Given a smooth function $f \in C^{\infty}(M^n)$, let $U\in H^{1,\mathfrak{a}}(X^{n + 1})$ be the unique solution of $$\begin{aligned} \begin{cases} -\mathop{\mathrm{div}}_{\bar{g}}(\varrho^{\mathfrak{a}} \nabla_{\bar{g}} U) + \varrho^{\mathfrak{a}} \mathcal{J}_{\bar{g}} U = 0, & \text{in}\ X^{n+1} \\ U = f, & \text{on}\ M^n, \end{cases}\end{aligned}$$ where $\mathcal{J}_{\bar{g}} \equiv C_{n,\gamma} R_{\bar{g}}$, $C_{n,\gamma}$ is the constant in Lemma [Lemma 16](#l:E){reference-type="ref" reference="l:E"}, and $$\begin{aligned} H^{1,\mathfrak{a}}(X^{n + 1}) \equiv \overline{C^{\infty}(X^{n + 1})}^{\|\cdot\|_{H^{1,\mathfrak{a}}}},\quad \|U\|_{H^{1,\mathfrak{a}}(X^{n+1})}\equiv \left(\int_{X^{n+1}}\varrho^{\mathfrak{a}} U^2 \mathop{\mathrm{dvol}}_{\bar{g}}+\int_{X^{n + 1}}\varrho^{\mathfrak{a}} |\nabla_{\bar{g}} U|^2 \mathop{\mathrm{dvol}}_{\bar{g}} \right)^{\frac{1}{2}}.\end{aligned}$$ Then $$\begin{aligned} P_{2\gamma} f = \frac{2^{2\gamma}\cdot\Gamma(\gamma)}{2\gamma\cdot\Gamma(-\gamma)} \lim_{\varrho\rightarrow 0} \varrho^\mathfrak{a}\frac{\partial U}{\partial \varrho}.\end{aligned}$$* ## Degenerate/singular equations and regularity {#ss:regularity-of-solutions} In this section, we clarify the regularity properties for the (weak) solutions $f\in C^{\infty}(M^n)$ of $P_{2\gamma} f = 0$ and the solutions $U\in H^{1,\mathfrak{a}}(X^{n + 1})$ of corresponding extension problem [\[e:U-extension-problem-PE\]](#e:U-extension-problem-PE){reference-type="eqref" reference="e:U-extension-problem-PE"} in $(X^{n + 1}, \bar{g})$. Since we will always specify Fefferman-Graham's metric as our conformal compactification throughout this paper, the simplified equation in the extension problem and the expression of the fractional GJMS operator are given by Proposition [Proposition 18](#p:extension-problem-in-FG-metric){reference-type="ref" reference="p:extension-problem-in-FG-metric"}. Notice first that we consider the range of parameters $\gamma \in (0,1)$, which amounts $\mathfrak{a}=1-2\gamma \in (-1,1)$. The aim of this section is to recall the regularity results needed for our purposes. In the model case when $(X^{n + 1}, g_+)$ is the hyperbolic space, a general regularity theory on the partial compactification $(\mathbb{R}_+^{n + 1}, g_0)$ of $(X^{n + 1}, g_+)$ has been developed in [@STV1; @STV2] for a larger class of parameters $\mathfrak{a}\in (-1, \infty)$. In our particular case, we are only considering the range $\mathfrak{a}\in (-1,1)$. Several of the arguments can then be simplified because of the special properties of the weight described below. By the very construction of the defining function $\varrho$ in Lemma [Lemma 8](#l:Fefferman-Graham){reference-type="ref" reference="l:Fefferman-Graham"}, the following property holds: there is a sufficiently small constant $r_0 = r_0(n, \bar{g}) >0$ such that $$\label{mu.def} \varrho(x)\sim \left\{ \begin{array}{cl} \text{dist}_{\bar g}(x, M^n) & \quad \text{when} \quad \text{dist}_{\bar g}(x, M^n) < r_0,\\ 1 & \quad \text{when} \quad \text{dist}_{\bar g}(x, M^n) > 2r_0. \end{array} \right.$$ It is then easy to check that the function $\varrho^\mathfrak{a}$ is an $A_2$-Muckenhoupt weight (see e.g. [@dyda]). We recall that a function $w \in L^1_{loc} (\mathbb R^{n+1})$ is called an $A_2$ weight if the following holds: $$\begin{aligned} \sup_{B \subset \mathbb R^{n+1}} \left( \fint_B w \mathop{\mathrm{dvol}}_{g_0} \right) \left( \fint_B w^{-1} \mathop{\mathrm{dvol}}_{g_0}\right) <\infty,\end{aligned}$$ whenever $B$ is a ball. In a series of important papers [@fks; @fkj; @fjk2], Fabes, Kenig, Jerison and Serapioni developed the basic theory of equations in divergence form whose main coefficients are $A_2$ functions. Let $\mathfrak{m}\equiv n + 1$ For our purpose, we need to double the compactified manifold $(\overline{X^{\mathfrak{m}}}, \bar{g})$ along the totally geodesic boundary $(M^n, h)$, which gives a closed manifold $\mathfrak{X}^{\mathfrak{m}} \equiv \overline{X^{\mathfrak{m}}}\bigcup\limits_{M^n}\overline{X^{\mathfrak{m}}}$ equipped with a $C^{n-1, \alpha}$-Riemannian metric (still denoted as $\bar{g}$). Since we are interested in local results, we first choose a domain $\Omega$ in $\mathfrak{X}^{\mathfrak{m}}$ and denote now $\varrho$ the previous special defining function where $\text{dist}_{\bar g}$ denotes the *signed* distance to $M^n$. For any $\mathfrak{a}\in (-1,1)$, we define the weighted Sobolev space $H^{1,\mathfrak{a}}(\Omega)$ as the closure of $C^\infty(\overline\Omega)$ with respect to the norm $$\begin{aligned} \|u\|_{H^{1,\mathfrak{a}}(\Omega)} \equiv \left(\int_{\Omega}\varrho^\mathfrak{a}u^2\, \mathop{\mathrm{dvol}}_{\bar g}+\int_{\Omega}\varrho^\mathfrak{a}|\nabla_{\bar{g}} u|^2\, \mathop{\mathrm{dvol}}_{\bar g}\right)^{\frac{1}{2}}.\end{aligned}$$ By the results in [@fks], it is known the space $\|u\|_{H^{1,\mathfrak{a}}(\Omega)}$ is a Hilbert space (for its usual norm). We record here the following Sobolev inequality (which follows from the proof of a more general one in [@fks], see also [@STV1]). **Lemma 19** (Sobolev inequality). *Let $\mathfrak{a}\in (-1,1)$ and $u\in C^1_c(\Omega)$. Then there exists a constant $c(\mathfrak{m}, \Omega)$ such that $$\label{sobo>-1} \left(\int_{\Omega}\varrho^\mathfrak{a}|u|^{2^*(\mathfrak{a})} \mathop{\mathrm{dvol}}_{\bar g}\right)^{2/2^*(a)}\leq c(\mathfrak{m}, \Omega)\int_{\Omega}\varrho^\mathfrak{a}|\nabla_{\bar{g}} u|^2\mathop{\mathrm{dvol}}_{\bar g},$$ where the optimal embedding exponent is $$\label{2*a} 2^*(\mathfrak{a})\equiv \frac{2(n+1+\mathfrak{a}^+)}{n+\mathfrak{a}^+-1}.$$ When $n=1$ and $\mathfrak{a}^+>0$ the same inequality holds. When $n=1$ and $\mathfrak{a}^+=0$ then the embedding holds in any weighted $L^p(\Omega,\varrho^\mathfrak{a}\mathrm{d}z)$ for $p>1$.* We now describe the regularity properties of the solutions under consideration. The next lemmas are straightforward adaptations of the results in [@fks; @STV1]. We will always consider an energy solution of $$\begin{aligned} \label{eq-U-e} -\mathop{\mathrm{div}}_{\bar{g}}(\varrho^{\mathfrak{a}} \nabla_{\bar{g}} U) + \varrho^{\mathfrak{a}} \mathcal{J}_{\bar{g}} U = 0 \quad \text{ in some geodesic ball}\ \mathcal{B}_r(x_+) \subset \mathfrak{X}^{\mathfrak{m}}, \end{aligned}$$ where $\mathcal{J}_{\bar{g}} \equiv C_{n,\gamma} R_{\bar{g}}$. The following lemma follows from classical integration by parts. **Lemma 20** (Caccioppoli type inequality). *Assume that $U$ solves [\[eq-U-e\]](#eq-U-e){reference-type="eqref" reference="eq-U-e"}. Let $r \in (0, 1]$, $\beta>1$ and $\eta \in C_0^\infty(\mathcal{B}_r(x_+))$. Then there exists a constant $C = C(r, \mathfrak{a}, \bar{g}) > 0$ such that $$\begin{aligned} \int_{\mathcal{B}_r(x_+)}\varrho^{\mathfrak{a}} |\nabla_{\bar{g}} (\eta U^\frac{\beta}{2})|^2 \mathop{\mathrm{dvol}}_{\bar{g}} \leq C\int_{\mathcal{B}_r(x_+)} \varrho^{\mathfrak{a}} (|\nabla_{\bar{g}} \eta|^2 +|\mathcal{J}_{\bar{g}}| \eta^2)U^2 \mathop{\mathrm{dvol}}_{\bar{g}}. \end{aligned}$$* Iterating the previous lemma, one gets classically the following upper bound by Moser's argument using the previous Sobolev inequality . **Lemma 21** ($L^\infty$ bounds). *Assume that $U$ solves [\[eq-U-e\]](#eq-U-e){reference-type="eqref" reference="eq-U-e"}. Then for $r\in (0,1)$, there exists a constant $C = C(r, \mathfrak{a}, n, \bar{g}) > 0$ such that $$\begin{aligned} \|U\|_{L^\infty(\mathcal{B}_r(x_+))} \leq C\|U\|_{L^{2,\mathfrak{a}}(\mathcal{B}_1(x_+))}. \end{aligned}$$* The previous lemma allows to consider the term $\varrho^{\mathfrak{a}} \mathcal{J}_{\bar{g}} U$ as the r.h.s. of the equation [\[eq-U-e\]](#eq-U-e){reference-type="eqref" reference="eq-U-e"} and apply directly the arguments in [@STV1] to deduce that **Lemma 22**. *Let $x_+\in M^n$. Assume that $U$ is an even solution to [\[eq-U-e\]](#eq-U-e){reference-type="eqref" reference="eq-U-e"} under the doubling $\mathfrak{X}^{\mathfrak{m}}$ such that $P_{2\gamma} f =0$ where $f$ is the trace of $U$ on $M^n$. Then for $r\in (0,1)$, there exists a constant $C = C(r, \mathfrak{a}, n, \bar{g}) > 0$ such that $$\begin{aligned} \|U\|_{C^{1,\alpha}(\mathcal{B}_r(x_+))} \leq C\|U\|_{L^{2,\mathfrak{a}}(\mathcal{B}_1(x_+))}. \end{aligned}$$* *Proof.* We follow the argument in \[[@STV1], Theorem 1.2\] for instance. By the previous lemma, we know that $U$ solves locally $$\begin{aligned} \label{eq-U-temp} -\mathop{\mathrm{div}}_{\bar g} (\varrho^\mathfrak{a}\nabla_{\bar g} U) = \varrho^\mathfrak{a}\zeta \text{ in } \mathcal{B}_1(x_+), \end{aligned}$$ where $\zeta =- \mathcal{J}_{\bar{g}} U \in L^p(\mathcal{B}_1(x_+))$ for any $p>1$. Now since $U$ is assumed to be even w.r.t. the doubling $\mathfrak{X}^{\mathfrak{m}}$, we are in the *local* situation of [@STV1] and the result follows from applying the techniques to prove Theorem 1.2 there. Notice that our metric enjoys some generic $C^{\mathfrak{m}-2}$ regularity, which is of course enough to derive the desired $C^{1,\alpha}$ bound. ◻ **Remark 23**. In the previous lemma, the symmetry assumption is necessary as pointed out in [@STV1]. In our special case since the right hand side involves $U$ and the *doubled metric* $\bar g$, which is also symmetric. The following lemma is a summary of the previous discussion and provides additional regularity depending on the smoothness of the compactification. **Lemma 24**. *Let $U$ be an even solution to [\[eq-U-e\]](#eq-U-e){reference-type="eqref" reference="eq-U-e"} under the doubling $\mathfrak{X}^{\mathfrak{m}}$ such that $P_{2\gamma} f =0$ where $f$ is the trace of $U$ on $M^n$. Then, $$\begin{aligned} \label{reg1} \|U\|_{C^{n - 1,\alpha}(\mathcal{B}_{2/3}(x_+))}\leq C(n, \mathfrak{a}, \bar{g})\|U\|_{L^{2,\mathfrak{a}}(\mathcal{B}_1(x_+))}.\end{aligned}$$ Furthermore,* 1. *if $n = 2$, or $n \geq 4$ is even and the conformal infinity $(M^n,[h])$ is obstruction flat, then for any $k\in \mathbb{N}_0$ and $\alpha \in (0,1)$, $$\begin{aligned} \label{reg2} \|U\|_{C^{k,\alpha}(\mathcal{B}_{1/2}(x_+))}\leq C(k, \alpha, n, \mathfrak{a}, \bar{g}) \|U\|_{L^{2,\mathfrak{a}}(\mathcal{B}_1(x_+))};\end{aligned}$$* 2. *if $n$ is odd, then for any $k\in\mathbb{N}_0$ and $\alpha \in (0,1)$, $$\begin{aligned} \label{reg3} \|U\|_{C^{k,\alpha}(\mathcal{B}_{1/2}(x_+))}\leq C(k, \alpha, n, \mathfrak{a}, \bar{g})\|U\|_{L^{2,\mathfrak{a}}(\mathcal{B}_1(x_+))}.\end{aligned}$$* *Proof.* Notice first that since the equation under consideration is posed in a ball of the doubled manifold $\mathfrak{X}^{\mathfrak{m}}$, all the estimates are *interior* estimates and in particular hold across the characteristic manifold. By Lemma [Lemma 22](#lem-eps-C1a){reference-type="ref" reference="lem-eps-C1a"}, we already know that $U$ is $C^{1,\alpha}$ for some $\alpha \in (0,1)$. More concretely, invoking [\[eq-gy1\]](#eq-gy1){reference-type="eqref" reference="eq-gy1"} and Proposition [Proposition 9](#p:general-regularity-FG){reference-type="ref" reference="p:general-regularity-FG"}, it can be observed that the even extension of the metric $\bar g$ onto $\mathcal{B}_1(x_+)$ is locally $C^{n - 1,\beta}$ but is not $C^n$ in general and the scalar curvature term $\mathcal{J}_{\bar{g}}$ is then locally $C^{n-3,\beta}$. To get the desired higher estimates under our standing geometric assumptions, we use the bootstrap argument in \[[@STV1], Section 6\] noticing that in our case of $\mathfrak{a}\in (-1,1)$, the \"dual\" operator, which is associated to the parameter $-\mathfrak{a}$ is also of the form $L_{-\mathfrak{a}}$ with $-\mathfrak{a}\in (-1,1)$. Therefore since $U$ satisfies locally $$\begin{aligned} -\mathop{\mathrm{div}}_{\bar{g}}(\varrho^{\mathfrak{a}} \nabla_{\bar{g}} U) =- \varrho^{\mathfrak{a}} \mathcal{J}_{\bar{g}} U, \end{aligned}$$ taking derivatives in the tangential and normal variables respectively, and using the Schauder estimates in Theorem 7.9 in [@STV1] (the lemmas before this theorem allow to deal with the normal derivatives solely) leads to the desired estimate [\[reg1\]](#reg1){reference-type="eqref" reference="reg1"}. The other two estimates [\[reg2\]](#reg2){reference-type="eqref" reference="reg2"}-[\[reg3\]](#reg3){reference-type="eqref" reference="reg3"} are obtained in exactly the same way taking higher derivatives. ◻ **Remark 25**. It is important to notice for later purposes that Lemma [Lemma 24](#eq-C1a-U){reference-type="ref" reference="eq-C1a-U"} provides generically *finite* smoothness of the solution $U$. In view of the $\epsilon-$regularity we will need later, this technical aspect requires an adaptation of stability results. # Singular set in the model case {#model_section} In this section, we focus on the model case. Let $(X^{n+1}, g_+)\equiv (\mathbb{H}^{n+1}, g_{\mathbb{H}^{n+1}})$ be the hyperbolic space, where $$\begin{aligned} g_+ \equiv \frac{dx^2 + dy^2}{y^2},\quad y>0, \quad x\equiv (x_1,\ldots, x_n)\in \mathbb{R}^n\end{aligned}$$ is the hyperbolic metric on $\mathbb{H}^{n+1}$ with $\sec_{g_+} \equiv -1$. Now $(X^{n+1},g_+)$ can be *partially compactified* to the upper half space $\mathbb{R}_+^{n+1}\equiv \{(x,y)| y \geq 0,\ x\in\mathbb{R}^n \}$ equipped with the Euclidean metric $g_{\mathbb{R}_+^{n+1}} \equiv dx^2 + dy^2$. Throughout this section, we will consider the fractional Laplacian $(-\Delta)^{\gamma}$ defined on the conformal infinity $\mathbb{R}^n$, where $\gamma\in(0,1)$. Now let $f:\mathbb{R}^n \to \mathbb{R}$ be a smooth function in the Schwarz space $\mathcal{S}(\mathbb{R}^n)$. Then there exists a unique Caffarelli-Silvestre type extension $U\in H^{1,\mathfrak{a}}(\mathbb{R}_+^{n+1})$ of $f$ that satisfies $$\begin{aligned} \label{e:U-extension} \begin{cases} \mathop{\mathrm{div}}(y^{\mathfrak{a}}\nabla U) = 0 & \text{in}\ \mathbb{R}_+^{n+1}, \\ U(x,0) = f(x) & \text{on} \ \mathbb{R}^n, \end{cases} \end{aligned}$$ where $\mathfrak{a}\equiv 1 - 2\gamma\in(-1,1)$. Then $$\begin{aligned} \label{eq-Delta-f-Uy} (-\Delta)^{\gamma} f \equiv \frac{2^{2\gamma}}{2\gamma}\cdot \frac{\Gamma(\gamma)}{\Gamma(-\gamma)}\cdot \lim\limits_{y\to 0}y^{\mathfrak{a}}\frac{\partial U}{\partial y}. \end{aligned}$$ We are interested in a $\gamma$-harmonic function $f\in \mathcal{S}(\mathbb{R}^n)$ that satisfies $(-\Delta)^{\gamma} f = 0$ on $B_1(\bm{0})\subset \mathbb{R}^n$. ## The quantitative symmetry and quantitative stratification This subsection summarizes some definitions and examples. To begin with, let us recall some basic definitions given in [@Cheeger-Naber-Valtorta]. **Definition 26**. Let $B_1(\bm{0})\subset \mathbb{R}$. A smooth function $u: B_1(\bm{0})\to \mathbb{R}$ is called *nondegenerate* if for every $x$ some derivative of some order is nonzero. **Example 27**. It is immediate to see that any non-constant analytic function is nondegenerate. For nondegenerate functions, it is essential to define the tangent map to study delicate properties on small scales. **Definition 28** (Tangent map on $\mathbb{R}^n$). Let $B_1(\bm{0})\subset\mathbb{R}^n$. Let $u:B_1(\bm{0})\to \mathbb{R}$ be a smooth nondegenerate function and $r>0$. Then for every $x\in B_{1-r}(\bm{0})$, we define $$\begin{aligned} T_{x,r} u(\xi) \equiv \frac{u(x + r \xi) - u(x)}{ {\displaystyle \left(\fint_{\partial B_1(\bm{0})}(u(x + r \zeta) - u(x))^2d\sigma(\zeta)\right)^{\frac{1}{2}} } }, \quad \xi \in B_1(\bm{0})\subset \mathbb{R}^n. \end{aligned}$$ If the denominator vanishes, we set $T_{x,r}u=\infty$. We also define $$\begin{aligned} T_{x,0} u(\xi) = T_x u(\xi) = \lim\limits_{r\to 0} T_{x,r} u(\xi). \end{aligned}$$ We remark that if $u$ is smooth and nondegenerate at $x$, then a unique tangent map $T_x u$ exists. Moreover, up to rescaling, $T_x u$ is the leading order polynomial of the Taylor expansion of $u-u(x)$ at $x$. To define the symmetry of a function, we need the following notion. **Definition 29**. A polynomial $P$ is said to be *homogeneous at $x\in \mathbb{R}^n$* if for some $d\in\mathbb{N}_0$, $$\begin{aligned} P(y) = \sum\limits_{|\beta| = d}A_{\beta}(y - x)^{\beta}, \end{aligned}$$ where $\beta$ is a multi-index and $|\beta|= \sum\limits_{i}\beta_i=d$. **Definition 30** ($k$-symmetry). Let $u:\mathbb{R}^n \to \mathbb{R}$ be a continuous function. - $u$ is called $0$-symmetric at the origin $\bm{0}\in\mathbb{R}^n$ if $u$ is a homogeneous polynomial at the origin $\bm{0}\in \mathbb{R}^n$. - $u$ is called $k$-symmetric at the origin $\bm{0}\in\mathbb{R}^n$ if $u$ is called $0$-symmetric at the origin and there exists a $k$-dimensional subspace $V \subset \mathbb{R}^n$ such that $$\begin{aligned} u(x + y) = u(x), \quad \forall x\in \mathbb{R}^n,\ y\in V. \end{aligned}$$ **Definition 31** (First-order stratification). Given a smooth nondegenerate function $u:B_1(\bm{0})\to \mathbb{R}$ we define the $k^{th}$-singular stratum of $u$ by $$\begin{aligned} \mathcal{S}^k(u) \equiv \{x\in B_1(\bm{0}): T_xu\ \text{is not}\ (k+1)\text{-symmetric}\}. \end{aligned}$$ **Example 32**. Here are some basic examples. - Any homogeneous polynomial of degree $1$, i.e., a linear function, on $\mathbb{R}^n$ is $(n-1)$-symmetric. - Let $u$ be a nondegenerate function on $\mathbb{R}^n$. Then for almost every point $x$, the tangent map $T_x u$ is linear, and hence $\dim_{\mathcal{H}}(\mathcal{S}^{n-1}) = n$ and $\mathcal{S} = \mathcal{S}^{n-2}$. The *quantitative differentiation theory* in our context relies on the quantitative stratification of the singular set which provides a precise characterization of the function on definite scales. **Definition 33** (Quantitative symmetry on $\mathbb{R}^n$). Let $B_1(\bm{0})$ be the unit ball in $\mathbb{R}^n$, and let $u: B_1(\bm{0})\to\mathbb{R}$ be a continuous function. We say $u$ is $(k,\eta,s)$-symmetric at $x$ if there exists a $k$-symmetric polynomial $P$ such that $\fint_{\partial B_1(\bm{0})}|P|^2 d\sigma_{\partial B_1(\bm{0})}= 1$ and $$\begin{aligned} \fint_{B_1(\bm{0})}|T_{x,s} u - P|^2 \mathop{\mathrm{dvol}}_{\mathbb{R}^n} < \eta. \end{aligned}$$ In our interested context of $\mathbb{R}^{n + 1}\equiv\{(x,y): x\in\mathbb{R}^n, \ y\in \mathbb{R}\}$, we will also consider how a function $U: \mathcal{B}_1(\bm{0}_+)\to \mathbb{R}$ behaves near the hypersurface $\mathbb{R}^n \equiv\{y = 0\} \subset \mathbb{R}^{n+1}$, where $\mathcal{B}_1(\bm{0}_+)\subset \mathbb{R}^{n+1}$ is the unit ball centered at the origin $\bm{0}_+ \in\mathbb{R}^{n+1}$. For our purpose, the weighted measure $|y|^{\mathfrak{a}}\mathop{\mathrm{dvol}}_{\mathbb{R}^{n + 1}}$ on $\mathbb{R}^{n+1}$ will be frequently used. Now we can define the tangent map and the quantitative symmetry similarly. **Definition 34** (Tangent map on $\mathbb{R}^{n+1}$). Let $\mathcal{B}_1(\bm{0}_+)$ be the unit ball in $\mathbb{R}^{n+1}$. Let $U:\mathcal{B}_1(\bm{0}_+)\to \mathbb{R}$ be a smooth nondegenerate function and $r>0$. Then for every $x_+ \equiv (x, 0) \in \mathcal{B}_{1-r}(\bm{0}_+)$, we define $$\begin{aligned} \mathcal{T}_{x_+,r} U(\xi) \equiv \frac{U(x_+ + r \xi) - U(x_+)}{ {\displaystyle \left(\frac{1}{r^{\mathfrak{a}}}\fint_{\partial\mathcal{B}_1(\bm{0}_+)} |y(x_+ + r\zeta)|^{\mathfrak{a}} \cdot \left(U(x_+ + r \zeta) - U(x_+)\right)^2 d\sigma_{\partial\mathcal{B}_1(\bm{0}_+)}(\zeta)\right)^{\frac{1}{2}} } }, \quad \xi \in \mathcal{B}_1(\bm{0}_+). \end{aligned}$$ If the denominator vanishes, we set $\mathcal{T}_{x_+,r}U =\infty$. We also define $$\begin{aligned} \mathcal{T}_{x_+,0} U(\xi) = \mathcal{T}_{x_+} U(\xi) = \lim\limits_{r\to 0} \mathcal{T}_{x_+,r} U(\xi). \end{aligned}$$ **Definition 35** (Quantitative symmetry on $\mathbb{R}^{n+1}$). A continuous function $U: \mathcal{B}_1(\bm{0}_+)\to\mathbb{R}$ is said to be $(k,\eta,s ,\mathfrak{a})$-symmetric at $x_+\equiv(x,0)\in\mathbb{R}^{n+1}$ if there exists a $k$-symmetric polynomial $P$ such that $\frac{1}{s^{\mathfrak{a}}}\fint_{\partial\mathcal{B}_1(\bm{0}_+)}|y|^{\mathfrak{a}} \cdot |P|^2 d\sigma_{\partial\mathcal{B}_1(\bm{0}_+)} = 1$ and $$\begin{aligned} \frac{1}{s^{\mathfrak{a}}}\fint_{\mathcal{B}_1(\bm{0}_+)}|y|^{\mathfrak{a}} \cdot |\mathcal{T}_{x_+,s} U - P|^2 \mathop{\mathrm{dvol}}_{\mathbb{R}^n} < \eta. \end{aligned}$$ Now let us formulate the quantitative stratification of a continuous function $U:\mathcal{B}_1(\bm{0}_+)\to \mathbb{R}$. **Definition 36** (Quantitative singular strata). Let us fix $\mathfrak{a}\in (-1,1)$, and let $U:\mathcal{B}_1(\bm{0}_+)\to \mathbb{R}$ and $f: B_1(\bm{0})\to \mathbb{R}$ be continuous functions on $\mathbb{R}^{n + 1}$ and $\mathbb{R}^n$, respectively. Then we define the $(k,\eta,r)$-quantitative singular stratum of $U$ by $$\begin{aligned} \mathcal{S}_{\eta,r}^k(U) \equiv &\ \left\{x\in B_1(\bm{0}) \subset \mathbb{R}^n: U\ \text{is not} \ (k+1,\eta,s, \mathfrak{a})\text{-symmetric at}\ x\ \text{for all}\ s \geq r \right\}, \\ \mathcal{S}_{\eta,r}^k(f) \equiv &\ \left\{x\in B_1(\bm{0}) \subset \mathbb{R}^n: f\ \text{is not} \ (k+1,\eta,s)\text{-symmetric at}\ x\ \text{for all}\ s \geq r \right\}.\end{aligned}$$ **Remark 37**. Immediately, we have the following monotonicity, $$\begin{aligned} \mathcal{S}_{\eta, r}^k(U) \subset \mathcal{S}_{\eta', r'}^{k'}(U) \quad \text{if}\ k\leq k',\ \eta'\leq \eta, \ r\leq r'. \end{aligned}$$ Moreover, we can recover the standard stratification by $$\begin{aligned} \mathcal{S}^k(U) = \bigcup\limits_{\eta > 0}\bigcap\limits_{r > 0} \mathcal{S}_{\eta, r}^k(U). \end{aligned}$$ Here, similar to [\[e:U-zero-critical-singular\]](#e:U-zero-critical-singular){reference-type="eqref" reference="e:U-zero-critical-singular"}, each classical stratum $\mathcal{S}^k(U)$ is defined as the restriction $$\begin{aligned} \mathcal{S}^k(U) \equiv \{x\in B_1(\bm{0}) \cap \mathbb{R}^n : T_x U\ \text{is not}\ (k+1)\text{-symmetric}\}. \end{aligned}$$ Similar relations hold for $\mathcal{S}_{\eta, r}^k(f)$ as well. ## Monotonicity and quantitative rigidity {#ss:quantitative-rigidity-Euclidean} For our purpose, we will consider the extension of a $\gamma$-harmonic function in the *entire* Euclidean space $\mathbb{R}^{n+1}$. For any smooth function $f\in \mathcal{S}(\mathbb{R}^n)$ that is harmonic on $B_1(\bm{0})$, there exists a unique symmetric extension of $f$ (denoted as $U$) in $\mathbb{R}^{n+1}$ satisfying $$\begin{aligned} \label{e:symmetric-extension} \begin{cases} \mathop{\mathrm{div}}(y^{\mathfrak{a}}\nabla U) = 0 & \text{in}\ \mathbb{R}^{n+1}, \\ U(x, -y) = U(x,y) & \text{in} \ \mathbb{R}^{n+1}, \\ U(x,0) = f(x) & \text{on} \ \mathbb{R}^n, \\ (-\Delta)^{\gamma} f = 0 & \text{on} \ B_1(\bm{0}). \end{cases} \end{aligned}$$ **Example 38**. Let $x_1$ be the first component of $x$. A special solution to [\[e:symmetric-extension\]](#e:symmetric-extension){reference-type="eqref" reference="e:symmetric-extension"} when $f(x)=x_1^2$ is $$\begin{aligned} \label{eq-U-special} U(x, y) = x_1^2 -\frac{1}{1+\mathfrak{a}}y^2.\end{aligned}$$ While $y^{1-\mathfrak{a}}$ is a homogeneous solution to $\mathop{\mathrm{div}}(y^{\mathfrak{a}}\nabla U) = 0$, by [\[eq-Delta-f-Uy\]](#eq-Delta-f-Uy){reference-type="eqref" reference="eq-Delta-f-Uy"}, $(-\Delta)^{\gamma} f =0$ implies that the term $y^{1-\mathfrak{a}}$ does not appear in [\[eq-U-special\]](#eq-U-special){reference-type="eqref" reference="eq-U-special"}. Notice that $U$ and $f$ here do not satisfy the decaying conditions: $f\in \mathcal{S}(\mathbb{R}^n)$ and $U\in H^{1,\mathfrak{a}}(\mathbb{R}^{n + 1})$. But they can be served as local blow-up models. One can define Almgren's frequency function for the symmetric extension $U\in H^{1,\mathfrak{a}}(\mathbb{R}^{n+1})$. Let us fix some notations. For any $x\in\mathbb{R}^n$, let us denote $x_+\equiv (x, 0)\in\mathbb{R}^{n+1}$ and $$\begin{aligned} \mathcal{B}_r(x_+)\equiv \{w_+\in \mathbb{R}^{n+1}: |w_+ - x_+|\leq r\},\quad \partial\mathcal{B}_r(x_+)\equiv \{w_+\in \mathbb{R}^{n+1}: |w_+ - x_+| = r\}. \end{aligned}$$ Immediately, $\mathcal{B}_r(x_+)\cap \mathbb{R}^n = B_r(x)$ and $\partial\mathcal{B}_r(x_+)\cap \mathbb{R}^n = \partial B_r(x)$. Then we define $$\begin{aligned} \begin{split} E_U(x_+, r) & \equiv \frac{1}{r^{n+\mathfrak{a}- 1}}\int_{\mathcal{B}_r(x_+)}|y|^{\mathfrak{a}}\cdot |\nabla U|^2 \mathop{\mathrm{dvol}}_{\mathbb{R}^{n+1}}, \\ H_U(x_+, r) & \equiv \frac{1}{r^{n+\mathfrak{a}}}\int_{\partial\mathcal{B}_r(x_+)}|y|^{\mathfrak{a}}\cdot U^2 d\sigma. \end{split} \end{aligned}$$ Then the Almgren's frequency function $N_U(x_+, r)$ is defined to be $$\begin{aligned} N_U(x_+, r) \equiv \frac{E_U(x_+, r)}{H_U(x_+, r)} = \frac{\displaystyle {r \int_{\mathcal{B}_r(x_+)}|y|^{\mathfrak{a}}\cdot |\nabla U|^2 \mathop{\mathrm{dvol}}_{\mathbb{R}^{n+1}}}}{{\displaystyle \int_{\partial\mathcal{B}_r(x_+)}|y|^{\mathfrak{a}}\cdot U^2 d\sigma}}.\end{aligned}$$ The following monotonicity result is due to Caffarelli-Silvestre [@CS theorem 6.1]. **Lemma 39**. *Let $U$ be the symmetric extension of $f$ that solves [\[e:symmetric-extension\]](#e:symmetric-extension){reference-type="eqref" reference="e:symmetric-extension"} in $\mathcal{B}_1(\bm{0}_+)\subset \mathbb{R}^{n+1}$ such that $(-\Delta)^{\gamma} f = 0$ on $B_1(\bm{0})\subset \mathbb{R}^n$. Then for any $x_+\in B_1(\bm{0})$, the frequency function $N_U(x_+, r)$ is monotone nondecreasing in $r\in (0,1 - |x_+|)$. Moreover, if $$\begin{aligned} N_U(x_+,r_1) = N_U(x_+, r_2)\quad \text{for some}\ 0\leq r_1 < r_2 < 1-|x_+|, \end{aligned}$$ then $U$ is a homogeneous polynomial of degree $d = N_U(x,r)$.* Since we are interested in the structure of the critical set of a function (not only the singular set), we will normalize Almgren's frequency as follows. If $U$ is not constant, let us define $$\begin{aligned} \mathcal{N}_U(x_+,r) \equiv \frac{\displaystyle {r\cdot\int_{\mathcal{B}_r(x_+)}|y|^{\mathfrak{a}}\cdot |\nabla U|^2 \mathop{\mathrm{dvol}}_{\mathbb{R}^{n+1}}}}{{\displaystyle \int_{\partial\mathcal{B}_r(x_+)}|y|^{\mathfrak{a}}\cdot (U-U(x_+))^2 d\sigma}}.\end{aligned}$$ When there is no ambiguity, for a $\gamma$-harmonic function $f$, we will denote by $$\begin{aligned} \mathcal{N}(x, r) = \mathcal{N}_f(x, r) = \mathcal{N}_U(x_+,r)\end{aligned}$$ the Almgren's frequency of $f$ at $x\in \mathbb{R}^n$. Similarly, we have the following monotonicity. **Lemma 40** (Monotonicity and rigidity). *Let $U$ be unique symmetric extension of a non-constant $\gamma$-harmonic function $f$ that solves [\[e:symmetric-extension\]](#e:symmetric-extension){reference-type="eqref" reference="e:symmetric-extension"} in $\mathcal{B}_1(\bm{0}_+)\subset \mathbb{R}^{n+1}$ such that $(-\Delta)^{\gamma} f = 0$ on $B_1(\bm{0})\subset \mathbb{R}^n$. Then the following properties hold.* 1. *For any $x\in B_1(\bm{0})$, the frequency function $\mathcal{N}_f(x, r)$ of $f$ at $x_+$ is monotone nondecreasing with respect to $r\in (0,1 - |x|)$.* 2. *If for some $0\leq r_1 < r_2$, $\mathcal{N}_f (x, r_1) = \mathcal{N}_f (x, r_2)$, then $U - U(x_+)$ is a homogeneous polynomial of degree $d = \mathcal{N}_f (x,r)$ centered at $x_+ = (x, 0)\in \mathbb{R}^{n + 1}$.* Next, we will prove a uniform boundedness result that controls the frequency from a large scale to any smaller scales. **Lemma 41** (Uniform bound on frequency). *Let $f \in \mathcal{S}(\mathbb{R}^n)$ be a non-constant function that is $\gamma$-harmonic on $B_1(\bm{0})\subset \mathbb{R}^n$. Let $U$ be its unique symmetric extension such that $\mathcal{N}_f(\bm{0},1)\leq \Lambda$. Then for each $\tau\in(0,1)$, there exists $C=C(n,\Lambda, \tau, \gamma)>0$ such that for each $x\in B_{\tau}(\bm{0})$ and $r\leq \frac{2}{3}(1-\tau)$, we have $$\begin{aligned} \mathcal{N}_f(x,r) \leq C. \end{aligned}$$* *Proof.* There is no harm to only prove the lemma for $\tau = \frac{1}{4}$ and $s = \frac{1}{2}$. We also assume that $u(\bm{0}) = 0$ such that $N(0,r) = \mathcal{N}(0, r)$ for all $r\leq 1$. Let $x_+ \equiv (x, 0)\in \mathbb{R}^{n + 1}$. By the definition of the normalized frequency, $$\begin{aligned} \mathcal{N}_f(x, 1/2) = \mathcal{N}_U(x_+,1/2) \equiv \frac{\displaystyle {\int_{\mathcal{B}_{1/2}(x_+)}|y|^{\mathfrak{a}}\cdot |\nabla U|^2 \mathop{\mathrm{dvol}}_{\mathbb{R}^{n+1}}}}{{\displaystyle 2\int_{\partial\mathcal{B}_{1/2}(x_+)}|y|^{\mathfrak{a}}\cdot (U-U(x_+))^2 d\sigma}} \label{e:1/2-frequency}.\end{aligned}$$ It follows from standard computations that $$\begin{aligned} U(x_+)\cdot \int_{\partial\mathcal{B}_{1/2}(x_+)}|y|^{\mathfrak{a}} d\sigma = \int_{\partial\mathcal{B}_{1/2}(x_+)}|y|^{\mathfrak{a}} \cdot U d \sigma.\end{aligned}$$ Then for any $0<r\leq \frac{1}{2}$, $$\begin{aligned} \begin{split} & \int_{\partial\mathcal{B}_r(x_+)}|y|^{\mathfrak{a}}\cdot (U-U(x_+))^2 d\sigma \nonumber\\ = \ & \int_{\partial\mathcal{B}_r(x_+)}|y|^{\mathfrak{a}} U^2 d\sigma - \left(\int_{\partial\mathcal{B}_r(x_+)}|y|^{\mathfrak{a}} d\sigma\right)|U(x_+)|^2 \nonumber\\ = \ & r^{n+\mathfrak{a}}\cdot \left(H(x_+ , r) - \left(\int_{\partial\mathcal{B}_1(\bm{0})}|\Theta|^{\mathfrak{a}}d\sigma \right) |U(x_+)|^2 \right)\geq 0, \end{split}\end{aligned}$$ where $y \equiv r \cdot \Theta$ and $\Theta$ is a spherical harmonic on the unit sphere. Immediately we have that $$\begin{aligned} \left(\int_{\partial\mathcal{B}_1(\bm{0})}|\Theta|^{\mathfrak{a}}d\sigma \right) |U(x_+)|^2 \leq H(x_+, r),\label{e:H-bound-scale-r}\end{aligned}$$ and the integral in the denominator of [\[e:1/2-frequency\]](#e:1/2-frequency){reference-type="eqref" reference="e:1/2-frequency"} becomes $$\begin{aligned} \int_{\partial\mathcal{B}_{1/2}(x_+)}|y|^{\mathfrak{a}}\cdot (U-U(x_+))^2 d\sigma = &\ \left(\frac{1}{2}\right)^{n+\mathfrak{a}}\left(H(x_+, 1/2) - \left(\int_{\partial\mathcal{B}_1(\bm{0})}|\Theta|^{\mathfrak{a}}d\sigma \right) |U(x_+)|^2 \right).\end{aligned}$$ Now we choose $r=1/3$ in [\[e:H-bound-scale-r\]](#e:H-bound-scale-r){reference-type="eqref" reference="e:H-bound-scale-r"} so that $$\begin{aligned} \int_{\partial\mathcal{B}_{1/2}(x_+)}|y|^{\mathfrak{a}}\cdot (U-U(x_+))^2 d\sigma \geq \left(\frac{1}{2}\right)^{n+\mathfrak{a}}\Big((H(x_+, 1/2) - H(x_+, 1/3)\Big).\label{e:denominator} \end{aligned}$$ To estimate the right-hand side of [\[e:denominator\]](#e:denominator){reference-type="eqref" reference="e:denominator"}, let us apply the formula $$\begin{aligned} \frac{d}{dr}\log H(x_+, r) = \frac{2N(x_+ , r)}{r}, \end{aligned}$$ which implies that $$\begin{aligned} \frac{H(x_+, 1/3)}{H(x_+, 1/2)} = \exp\left(-2\int_{1/3}^{1/2}\frac{N(x_+, r)}{r}dr\right) \leq \exp\left(2N(x_+,1/3)\log(2/3)\right) = \left(2/3\right)^{2N(x_+, 1/3)}. \end{aligned}$$ Plugging the above into [\[e:denominator\]](#e:denominator){reference-type="eqref" reference="e:denominator"}, $$\begin{aligned} \int_{\partial\mathcal{B}_{1/2}(x_+)}|y|^{\mathfrak{a}}\cdot (U-U(x_+))^2 d\sigma \geq \left(\frac{1}{2}\right)^{n+\mathfrak{a}}\left(1-\left(2/3\right)^{2N(x_+, 1/3)}\right) H(x_+, 1/2). \end{aligned}$$ Therefore, $$\begin{aligned} \mathcal{N}_f (x, 1/2) = \mathcal{N}_U (x_+, 1/2) \leq N(x_+, 1/2) \cdot \left(1-\left(2/3\right)^{2N(x_+, 1/3)}\right)^{-1}\leq C(n,\Lambda),\end{aligned}$$ where we used the fact $N(x_+, 1/3) \geq 1$. The proof of the lemma is done. ◻ The rigidity part of Lemma [Lemma 40](#l:monotonicity){reference-type="ref" reference="l:monotonicity"} has a quantitative version. **Proposition 42** (Quantitative symmetry). *For any $\epsilon>0$, $n\geq 2$, $\gamma\in(0,1)$, $\Lambda>0$, there exists a uniform constant $\delta=\delta(\epsilon, n, \gamma, \Lambda)>0$ such that the following holds. Let $f$ be $\gamma$-harmonic on $B_s(\bm{0})\subset \mathbb{R}^n$ with $\mathcal{N}_f(0,s) = \mathcal{N}_U(\bm{0}_+,s)\leq \Lambda$, where $U$ is the unique symmetric extension of $f$ that solves [\[e:symmetric-extension\]](#e:symmetric-extension){reference-type="eqref" reference="e:symmetric-extension"} in $\mathbb{R}^{n+1}$. If $$\begin{aligned} \mathcal{N}_U(\bm{0}_+, s) - \mathcal{N}_U(\bm{0}_+, s\gamma) < \delta, \end{aligned}$$ then $U$ is $(0,\epsilon, s, \mathfrak{a})$-symmetric at the origin $\bm{0}_+\subset \mathbb{R}^{n+1}$.* *Proof.* We will prove it by contradiction. Suppose there exist positive constants $\epsilon_0>0$, $\gamma > 0$, and a sequence of $U_j$ such that $\mathcal{N}_{U_j}(\bm{0}_+,1) \leq \Lambda$, $\mathcal{N}_{U_j}(\bm{0}_+, 1) - \mathcal{N}_{U_j}(\bm{0}_+, \gamma) < \delta_j \to 0$, but $U_j$ is not $(0,\epsilon_0,1, \mathfrak{a})$-symmetric at $\bm{0}_+\in\mathbb{R}^{n+1}$. Without loss of generality, one can assume $$\begin{aligned} U_j(\bm{0}_+) = \bm{0}_+\quad \text{and} \quad \fint_{\partial\mathcal{B}_1(\bm{0}_+)}|y|^{\mathfrak{a}} \cdot U_j^2 d\sigma = 1. \end{aligned}$$ That is, $\mathcal{T}_{\bm{0}_+,1} U_j = U_j$. Then $U_j$ converges in $H^{1,\mathfrak{a}}(\mathcal{B}_1(\bm{0}_+))\cap C_{\mathop{\mathrm{loc}}}^1(\mathcal{B}_1(\bm{0}_+))$ to some function $U_{\infty}$. Moreover, by the weighted Sobolev trace inequality, we have that $$\begin{aligned} \fint_{\partial\mathcal{B}_1(\bm{0}_+)} |y|^{\mathfrak{a}} \cdot U_{\infty}^2 d\sigma = 1. \end{aligned}$$ Since $\mathcal{N}_{U_{\infty}}(\bm{0}_+, 1) - \mathcal{N}_{U_{\infty}}(\bm{0}_+, \gamma) = 0$, by Lemma [Lemma 40](#l:monotonicity){reference-type="ref" reference="l:monotonicity"}, $U_{\infty}$ is a homogeneous polynomial. The convergence $\| U_j - U_{\infty}\|_{H^{1,\mathfrak{a}}(\mathcal{B}_1(\bm{0}_+))} \to 0$ particularly contradicts the assumption that $U_j$ is not $(0,\epsilon_0,1,\mathfrak{a})$-symmetric at the origin $\bm{0}_+ \in \mathbb{R}^{n + 1}$. ◻ **Definition 43** (Good and bad scales). Let us fix $\epsilon>0$, $n\geq 2$, $\gamma\in(0,1)$ and let us take $r_j \equiv \gamma^j$ for any $j\in\mathbb{Z}_+$. Given a smooth function $U$ defined on $\mathcal{B}_1(\bm{0}_+)$ with $x_+\equiv(x,0)\in \mathcal{B}_1(\bm{0}_+)$, a scale $r_j$ is called a *good scale* at $x$ if $U$ is $(0,\epsilon, r_j, \mathfrak{a})$-symmetric at $x$, and a scale $r_j$ is said to be a *bad scale* at $x_+$ if $U$ is not $(0,\epsilon, r_j, \mathfrak{a})$-symmetric at $x_+$, **Corollary 44** (Uniform control on the number of bad scales). *For any $\epsilon>0$, $n\geq 2$, $\gamma\in(0,1)$, $\Lambda>0$, there exists a uniform constant $Q_0=Q_0(\epsilon, n, \gamma, \Lambda, \mathfrak{a})>0$ such that the following holds. Let $U$ be a solution of [\[e:symmetric-extension\]](#e:symmetric-extension){reference-type="eqref" reference="e:symmetric-extension"} with $\mathcal{N}_U(\bm{0}_+, 1)\leq \Lambda$. Then for any $\epsilon > 0$ and $x_+\equiv(x,0)\in \mathcal{B}_{1/2}(\bm{0}_+)$, there are at most $Q_0$ bad scales at $x_+$.* *Proof.* By Lemma [Lemma 41](#l:frequency-control-large-to-small){reference-type="ref" reference="l:frequency-control-large-to-small"}, for any $x_+ = (x,0)\in B_{1/2}(\bm{0}_+)$, we have $\mathcal{N}_U (x_+,1/2) \leq C(n,\Lambda, \tau)$. Then one can obtain $$\begin{aligned} C(n,\Lambda, \tau) \geq \mathcal{N}_U (x_+, 1/2) = \sum\limits_{j = 1}^{\infty} (\mathcal{N}_U (x_+, r_j) - \mathcal{N}_U (x_+, r_{j + 1})). \end{aligned}$$ For any $\epsilon > 0$, let $\delta = \delta (\epsilon, n , \gamma, \Lambda) > 0$ be the constant in Proposition [Proposition 42](#p:quantitative-symmetry){reference-type="ref" reference="p:quantitative-symmetry"}. Applying Lemma [Lemma 40](#l:monotonicity){reference-type="ref" reference="l:monotonicity"}, there are finitely many scales $r_j$ of number bounded by $Q_0\equiv C(n,\Lambda, \tau) / \delta$, for which $$\begin{aligned} \mathcal{N}_U (x_+, r_j) - \mathcal{N}_U (x_+, r_{j + 1}) > \delta. \end{aligned}$$ Applying Proposition [Proposition 42](#p:quantitative-symmetry){reference-type="ref" reference="p:quantitative-symmetry"}, one can see that all other scales are good scales. ◻ ## Quantitative cone-splitting {#ss:quantitative-cone-splitting} We start with a general *cone splitting principle*. The computations are standard now; see also [@Cheeger-Naber-Valtorta proposition 2.11]. **Lemma 45** (Cone-splitting principle). *Let $P:\mathbb{R}^n\to \mathbb{R}$ be a harmonic polynomial of degree $d$, homogeneous with respect to the origin. Assume that $P$ is symmetric with respect to a $k$-dimensional vector space $V\subset \mathbb{R}^n$. Then* 1. *$P$ is a linear function if and only if $P$ is $(n-1)$-symmetric.* 2. *If $P$ is not $(n-1)$-symmetric, and $P$ is also $0$-symmetric with respect to $z\not\in V$, then $P$ is $(k+1)$-symmetric with respect to the vector space $\mathop{\mathrm{Span}}(V,z)$.* **Remark 46**. The main property of cone splitting given in item (2) holds for general homogeneous polynomials. *Proof.* Item (1) is obvious, so we omit the proof. Let us prove item (2). Since $P$ is homogeneous with respect to the origin, $$\begin{aligned} \langle \nabla P (x), x \rangle = P_r (x)\langle\nabla r, x\rangle = d\cdot P(x). \end{aligned}$$ Also, $P$ is symmetric with respect to some $z\not\in V$, which implies $$\begin{aligned} \langle \nabla P , x - z\rangle = d \cdot P(x).\end{aligned}$$ Therefore, for any $x \in \mathbb{R}^n$, we have $\langle \nabla P (x), z \rangle = 0$, which implies that $P$ is constant along the line connecting $\bm{0}$ and $z$. The proof is done. ◻ The following theorem gives a quantitative version of the cone splitting. **Theorem 47** (Quantitative cone splitting). *For any fixed $n\geq 2$, $\epsilon>0$, $\rho>0$, $r\in (0,1)$, $k\in\{0,1,\ldots, n-2\}$, $\Lambda>0$, there exists a positive constant $\underline{\delta}=\underline{\delta}(n,\epsilon, \rho, \Lambda) > 0$ such that the following holds. Let $U$ be the unique solution of [\[e:U-extension\]](#e:U-extension){reference-type="eqref" reference="e:U-extension"} with $\mathcal{N}_U(\bm{0}_+,1)\leq \Lambda$. If* 1. *$U$ is $(k, \underline{\delta}, r , \mathfrak{a})$-symmetric at $\bm{0}_+$ with respect to a $k$-dimensional vector space $V$,* 2. *$U$ is $(0, \underline{\delta}, r, \mathfrak{a})$-symmetric for some $z\in \mathcal{B}_r(\bm{0}_+)\setminus \mathcal{B}_{\rho}(V)$,* *then $U$ is also $(k+1, \epsilon , 1 , \mathfrak{a})$-symmetric at $\bm{0}_+$.* **Remark 48**. Assuming a weaker symmetry on a smaller ball and a cone symmetry with respect to a distinct point of definite distance, this theorem gives a stronger symmetry on a larger ball. *Proof.* We will prove it by contradiction. Suppose there exist positive constants $\epsilon_0>0$, $\rho_0>0$, $r_0>0$, $\Lambda_0$, and a positive integer $0\leq k \leq n-2$ such that for a sequence $\underline{\delta}_j\to 0$ and a sequence of solution $U_j$, the following properties hold. 1. $U_j$ is $(k_0, \underline{\delta}_j, r_0, \mathfrak{a})$-symmetric at $\bm{0}_+$ with respect to a $k_0$-dimensional vector space $V_j$. 2. $U_j$ is $(0, \underline{\delta}_j , r_0, \mathfrak{a})$-symmetric for some point $z_j\in B_{r_0}(\bm{0}_+)\setminus B_{\rho_0}(V_j)$. 3. $U_j$ is not $(k_0+1, \epsilon_0, 1, \mathfrak{a})$-symmetric at $\bm{0}_+$. Without loss of generality, we can assume that $U_j(\bm{0}_+) = \bm{0}_+$ and $\fint_{\partial\mathcal{B}_1(\bm{0}_+)}|y|^{\mathfrak{a}}\cdot|U_j|^2d\sigma =1$. Since $\mathcal{N}_{U_j}(\bm{0}_+, 1) \leq \Lambda_0$, we have that $$\begin{aligned} \int_{\mathcal{B}_1(\bm{0}_+)}|y|^{\mathfrak{a}}\cdot|\nabla U_j|^2\mathop{\mathrm{dvol}}_{\mathbb{R}^{n+1}} \leq \Lambda_0. \end{aligned}$$ Therefore, $U_j$ converges in $H^{1,\mathfrak{a}}(\mathcal{B}_1(\bm{0}_+))\cap C^1(\mathcal{B}_1(\bm{0}_+))$ to $U_{\infty}$. We also have the convergence $V_j \to V_{\infty}$ and $x_j \to x_{\infty} \not\in V_{\infty}$ for some vector space $V_{\infty}$ with $\dim V_{\infty} = k$. By item (1)', there exists a sequence of $k_0$-symmetric homogeneous polynomial $P_j$ such that $$\begin{aligned} \fint_{\mathcal{B}_1(\bm{0}_+)}|y|^{\mathfrak{a}}\cdot |\mathcal{T}_{\bm{0}_+,r_0}U_j - P_j|^2 \mathop{\mathrm{dvol}}_{\mathbb{R}^{n+1}} \to 0 \end{aligned}$$ Notice that by the uniform doubling property, $$\begin{aligned} r_0^{-\mathfrak{a}} \cdot \fint_{\partial\mathcal{B}_1(\bm{0}_+)} |y(\zeta)|^{\mathfrak{a}} \cdot U_{\infty}(r_0\cdot \zeta)^2 d\sigma(\zeta) = \fint_{\partial\mathcal{B}_{r_0}(\bm{0}_+)}|y|^{\mathfrak{a}} \cdot U_{\infty}^2 d\sigma \geq r_0^{2\Lambda} > 0.\end{aligned}$$ Therefore, $\mathcal{T}_{\bm{0}_+,r_0}U_j$ converges to some $k_0$-symmetric homogeneous polynomial $P_{\infty}$ with $\deg(P_{\infty}) = d$. By item (2)', $P_{\infty}$ is symmetric for some point $z_{\infty}\in \mathcal{B}_{r_0}(\bm{0}_+)\setminus \mathcal{B}_{\rho_0}(V_{\infty})$. Applying the cone-splitting principle in Lemma [Lemma 45](#l:cone-splitting-principle){reference-type="ref" reference="l:cone-splitting-principle"}, $P_{\infty}$ is $(k_0+1)$-symmetric at $\bm{0}_+$. To finish the proof, we only need to prove the claim that $P_{\infty} = U_{\infty}$, which gives the desired contradiction. Indeed, if the claim is true, by the normalization condition, we have $U_j = \mathcal{T}_{\bm{0}_+, 1} U_j$. Then $$\begin{aligned} \fint_{\mathcal{B}_1(\bm{0}_+)} |y|^{\mathfrak{a}} \cdot |\mathcal{T}_{\bm{0}_+, 1} U_j - P_{\infty}|^2 \mathop{\mathrm{dvol}}_{\mathbb{R}^{n+1}} \to 0 \end{aligned}$$ so that the desired contradiction arises. Now let us prove the claim. It follows from the convergence that $$\begin{aligned} P_{\infty}(\xi) = \mathcal{T}_{\bm{0}_+,r_0}U_{\infty}(\xi) = \frac{ \displaystyle{U_{\infty}(r_0\cdot \xi)} }{ \displaystyle{ r_0^{\mathfrak{a}} \cdot \fint_{\partial\mathcal{B}_1(\bm{0}_+)} |y(\zeta)|^{\mathfrak{a}} \cdot U_{\infty}(r_0\cdot \zeta)^2 d\sigma(\zeta)} },\end{aligned}$$ which implies that $U_{\infty}$ is homogeneous. Then we have that $$\begin{aligned} r_0^{-\mathfrak{a}} \cdot \fint_{\partial\mathcal{B}_1(\bm{0}_+)} |y(\zeta)|^{\mathfrak{a}} \cdot U_{\infty}(r_0\cdot \zeta)^2 d\sigma(\zeta) = \fint_{\partial\mathcal{B}_{r_0}(\bm{0}_+)}|y|^{\mathfrak{a}} \cdot U_{\infty}^2 d\sigma = r_0^d,\end{aligned}$$ where the last equality follows from the homogeneity of $U_{\infty}$ and the normalization $$\begin{aligned} \fint_{\partial\mathcal{B}_1(\bm{0}_+)}|y|^{\mathfrak{a}}\cdot|U_{\infty}|^2d\sigma =1.\end{aligned}$$ Finally, we obtain that $$\begin{aligned} P_{\infty}(\xi) = \frac{U_{\infty}(r_0\cdot \xi)}{r_0^d} = \frac{U_{\infty}(r_0\cdot \xi)}{r_0^d} = U_{\infty}(\xi),\quad \xi \in \mathcal{B}_1(\bm{0}_+),\end{aligned}$$ which completes the proof. ◻ Using induction, one can obtain the following corollary, which plays a crucial role in our covering arguments. **Corollary 49**. *For any fixed $n\geq 2$, $\epsilon>0$, $\rho>0$, $r\in (0,1)$, $k\in\{0,1,\ldots, n-2\}$, $\Lambda>0$, there exists a positive constant $\delta=\delta(n,\epsilon, \rho, \Lambda) > 0$ such that the following holds. Let $U$ be the unique solution of [\[e:symmetric-extension\]](#e:symmetric-extension){reference-type="eqref" reference="e:symmetric-extension"} with $\mathcal{N}_U(\bm{0}_+, 1)\leq \Lambda$. If* 1. *$U$ is $(0,\delta,r,\mathfrak{a})$-symmetric at $\bm{0}_+$;* 2. *for any vector space $V$ of dimension $\leq k$, there exists some point $z\in \mathcal{B}_r(\bm{0}_+)\setminus B_{\rho}(V)$ such that $U$ is $(0,\delta,r,\mathfrak{a})$-symmetric at $z$,* *then $U$ is also $(k+1,\epsilon,1,\mathfrak{a})$-symmetric at $\bm{0}_+$.* *Proof.* Let $\underline{\delta}\equiv\underline{\delta}(n,\epsilon,\rho,\Lambda)>0$ be the constant in Theorem [Theorem 47](#t:quantitative-cone-splitting){reference-type="ref" reference="t:quantitative-cone-splitting"}. For any $1 \leq i \leq n-1$, let us define the composite function $$\begin{aligned} \underline{\delta}^{(n-1-i)} \equiv \underbrace{\underline{\delta}\circ \underline{\delta}\circ \ldots \circ \underline{\delta}(n,\epsilon, \rho,\Lambda)}_{i\ \text{factors}}.\end{aligned}$$ For example, $\underline{\delta}^{(n-2)}= \underline{\delta}(n,\underline{\delta}, \rho, \Lambda)$. It is obvious that the following monotonicity holds, $$\begin{aligned} \underline{\delta}^{(0)} < \underline{\delta}^{(1)} < \ldots < \underline{\delta}^{(n-2)} < \underline{\delta}^{(n-1)} \equiv \epsilon.\end{aligned}$$ Let us choose $\delta\equiv \underline{\delta}^{(0)}$ (with $n$ factors in the composition). Let $U$ be a function that satisfies properties (1) and (2) for some fixed integer $0\leq k \leq n-2$. Since $U$ is $(0,\delta,r,\mathfrak{a})$-symmetric at $\bm{0}_+$, by applying property (2), there exists a largest integer $1\leq k_0 \leq n-2$ such that $U$ is $(\ell, \underline{\delta}^{(\ell)}, 1,\mathfrak{a})$-symmetric for any $1\leq \ell \leq k_0$. In fact, the existence is given by applying Theorem [Theorem 47](#t:quantitative-cone-splitting){reference-type="ref" reference="t:quantitative-cone-splitting"} to the case $k=0$. We observe that $k_0 \geq k$. Indeed, suppose $k_0 < k$ so that $U$ is $(k_0, \underline{\delta}^{(k_0)},1,\mathfrak{a})$-symmetric at $\bm{0}_+$. In other words, $U$ is $(k_0, \underline{\delta}^{(k_0)}, 1 , \mathfrak{a})$-symmetric at $\bm{0}_+$ with respect to a $k_0$-dimensional vector space $V$ with $k_0 < k$. Applying property (2) and Theorem [Theorem 47](#t:quantitative-cone-splitting){reference-type="ref" reference="t:quantitative-cone-splitting"}, $U$ is $(k_0 + 1, \underline{\delta}^{(k_0+1)}, 1, \mathfrak{a})$-symmetric at $\bm{0}_+$, which contradicts the maximality of $k_0$. Since $k_0 \geq k$, we particularly have that $U$ is $(k, \underline{\delta}^{(k)}, 1, \mathfrak{a})$-symmetric at $\bm{0}_+$. Applying property (2) and Theorem [Theorem 47](#t:quantitative-cone-splitting){reference-type="ref" reference="t:quantitative-cone-splitting"} again, $U$ is in fact $(k+1, \underline{\delta}^{(k+1)}, 1, \mathfrak{a})$-symmetric at $\bm{0}_+$, where $\underline{\delta}^{(k+1)} < \underline{\delta}^{(n-1)} = \epsilon$. The proof is complete. ◻ ## Volume estimate and frequency decomposition {#ss:volume-estimate} Let $f$ be $\gamma$-harmonic and let $U$ be the unique symmetric extension of $f$ given by [\[e:symmetric-extension\]](#e:symmetric-extension){reference-type="eqref" reference="e:symmetric-extension"}. In this subsection, for both $U$ and $f$, we will prove some uniform volume estimates for small tubular neighborhoods of the quantitative strata. We adopt the general formulation of the estimate and strategy of proof from [@Cheeger-Naber-Valtorta]. The main result of this section is as follows. **Theorem 50**. *For every $j\in\mathbb{N}$, $\epsilon>0$, $k\leq \mathfrak{m}-2$, there exist some positive constants $0<\mu(\mathfrak{m},\epsilon,\Lambda)<1$ and $C(\mathfrak{m},\Lambda, \epsilon)>0$ such that the following property holds. Let $U: \mathcal{B}_1(\bm{0}_+)\to \mathbb{R}$ be the unique symmetric extension with $\mathcal{N}_U(\bm{0}_+, 1)\leq \Lambda$. Then $$\begin{aligned} \mathop{\mathrm{Vol}}(B_{\mu^j}(\mathcal{S}_{\epsilon,\mu^j}^k(U))\cap B_{1/2}(\bm{0})) \leq C \cdot (\mu^j)^{\mathfrak{m}- k -\epsilon}. \end{aligned}$$* Since we have mentioned in Remark [Remark 37](#r:inclusion){reference-type="ref" reference="r:inclusion"} that $$\begin{aligned} \mathcal{S}^k(U) = \bigcup\limits_{\epsilon > 0}\bigcap\limits_{r > 0} \mathcal{S}_{\epsilon, r}^k(U), \end{aligned}$$ we also notice that $\mathcal{S}(U) = \mathcal{S}^{\mathfrak{m}- 2}(U)$, where $\mathcal{S}(U)$ is restriction of the singular set of $U$ on $\mathbb{R}^n$. To obtain the dimension estimates for the singular set, we need the following lemma. **Lemma 51**. *Let $f: B_1(\bm{0})\to \mathbb{R}$ be $\gamma$-harmonic and let $U$ be the unique solution of [\[e:symmetric-extension\]](#e:symmetric-extension){reference-type="eqref" reference="e:symmetric-extension"} such that $\mathcal{N}_U(x_+,r) \leq \Lambda$ with $x_+ \equiv (x, 0)\in \mathbb{R}^{n + 1}$. Then for every $\epsilon > 0$, $k\in\mathbb{N}$, and $\alpha\in(0,1)$, there exists $\eta = \eta (\epsilon, n, k, \Lambda, \mathfrak{a}) > 0$ such that if $U$ is $(\mathfrak{m}- 1, \eta, r, \mathfrak{a})$-symmetric at $x_+$, then $$\begin{aligned} \|\mathcal{T}_{x_+,r} U - L \|_{C^{1,\alpha}(\mathcal{B}_{1/2}(\bm{0}_+))} < \epsilon, \end{aligned}$$ where $L$ is a linear polynomial with $\fint_{\partial\mathcal{B}_1(\bm{0}_+)}|L|^2d\sigma = 1$. In particular, choosing any $\epsilon < \epsilon_0 \equiv |\nabla L|/3$, we have $r_{x_+} \geq r$ if $U$ is $(\mathfrak{m}- 1, \epsilon, r_{x_+} , \mathfrak{a})$ at $x_+$.* *Proof.* The proof is standard and follows from a usual contradiction compactness argument, so we omit the details. ◻ Then Theorem [Theorem 50](#t:volume-estimate-model){reference-type="ref" reference="t:volume-estimate-model"} immediately implies dimension estimates on the quantitative singular strata. **Corollary 52**. *Let $U$ be the unique solution of [\[e:symmetric-extension\]](#e:symmetric-extension){reference-type="eqref" reference="e:symmetric-extension"}. Under the same assumptions as Theorem [Theorem 50](#t:volume-estimate-model){reference-type="ref" reference="t:volume-estimate-model"}, we have $$\begin{aligned} \dim_{\mathcal{H}}(\mathcal{C}(U) \cap B_{1/2}(\bm{0}) ) \leq \dim_{\mathop{\mathrm{Min}}}(\mathcal{C}(U) \cap B_{1/2}(\bm{0}) ) \leq \mathfrak{m}-2. \label{e:singular-set-dimension-of-U} \end{aligned}$$* *Proof.* Applying Lemma [Lemma 51](#l:almost-(m-1)-symmetric){reference-type="ref" reference="l:almost-(m-1)-symmetric"}, we have that for sufficiently small $\epsilon > 0$, $$\begin{aligned} \mathcal{C}(U) \subset \mathcal{S}_{\epsilon, r}^{\mathfrak{m}- 2}(U).\end{aligned}$$ Then applying Theorem [Theorem 50](#t:volume-estimate-model){reference-type="ref" reference="t:volume-estimate-model"} and the conclusion follows. ◻ Now we proceed to prove Theorem [Theorem 50](#t:volume-estimate-model){reference-type="ref" reference="t:volume-estimate-model"}. The proof follows from the following effective covering result, Proposition [Proposition 53](#p:effective-covering){reference-type="ref" reference="p:effective-covering"}. Cheeger-Naber-Valtorta proved it in the second-order elliptic case; see [@Cheeger-Naber-Valtorta]. **Proposition 53** (Effective covering). *We fix a constant $\mu\in(0,1)$. For every $\epsilon>0$, $n \geq 2$ and $\Lambda>0$, there exist uniform constants $C_0=C_0(n)$, $C_1=C_1(n)$ and $D=D(n,\epsilon,\Lambda)$ such that the following properties hold for every $j\in\mathbb{N}$.* 1. *$\mathcal{S}_{\epsilon,\mu^j}^k(U) \cap \mathcal{B}_{1/2}(\bm{0}_+)$ is contained in the union of at most $j^D$ nonempty open sets $\mathfrak{A}_{\epsilon,\mu^j}^k$.* 2. *Each set $\mathfrak{A}_{\epsilon,\mu^j}^k$ is the union of at most $(C_1\mu^{-n-1})^D\cdot (C_0\mu^{-k})^{j-D}$ balls of radius $\mu^j$.* Let us make the following notations. $$\begin{aligned} \mathcal{E}(U, x, r) = \inf\{\eta\geq 0| \ U \ \text{is}\ (0,\eta, r)-\text{symmetric at}\ x\}. \end{aligned}$$ The points in $\mathcal{B}_1(\bm{0}_+) \cap \mathbb{R}^n$ are divided in *good points* and *bad points* in terms of the quantitative symmetry: $$\begin{aligned} \mathfrak{G}_{r,\epsilon}(U)\equiv\{x\in \mathcal{B}_{1/2}(\bm{0}_+) \cap \mathbb{R}^n | \mathcal{E}(U, x, r) < \epsilon\}, \\ \mathfrak{H}_{r,\epsilon}(U)\equiv\{x\in \mathcal{B}_{1/2}(\bm{0}_+) \cap \mathbb{R}^n | \mathcal{E}(U, x, r) \geq \epsilon\}.\end{aligned}$$ Let us denote by $\overline{T}^j\equiv(\overline{T}_1^j, \ldots, \overline{T}_{\ell}^j, \ldots ,\overline{T}_j^j)$ a $j$-tuple, where $\overline{T}_{\ell}^j\in\{0,1\}$ for $1\leq \ell \leq j$. The norm $|\overline{T}^j|$ of a $j$-tuple $\overline{T}^j$ is defined to be $\sum\limits_{\ell=1}^j \overline{T}_{\ell}^j$. For fixed $\mu>0$, $\epsilon>0$ and $j\in\mathbb{Z}_+$, we use the map $T^j$ to assign each point $x\in \mathcal{B}_1(\bm{0}_+) \cap \mathbb{R}^n$ to a $j$-tuple $(T_1^j(x), \ldots, T_j^j(x))$ such that for each $1\leq \ell \leq j$, $$\begin{aligned} T_{\ell}^j(x)\equiv \begin{cases} 1, & x\in \mathfrak{H}_{\mu^{\ell},\epsilon}(U), \\ 0, & x\in \mathfrak{G}_{\mu^{\ell},\epsilon}(U). \end{cases} \end{aligned}$$ Then we label all the points in $\mathcal{B}_{1/2}(\bm{0}_+) \cap \mathbb{R}^n$ by $j$-tuples: for each $j$-tuple $\overline{T}^j$ as described above, we define $$\begin{aligned} E(\overline{T}^j) \equiv \{x\in \mathcal{B}_{1/2}(\bm{0}_+) \cap \mathbb{R}^n | T^j(x) = \overline{T}^j\}.\end{aligned}$$ The proof of Proposition [Proposition 53](#p:effective-covering){reference-type="ref" reference="p:effective-covering"} follows from the key lemmas below. **Lemma 54**. *There exists $D=D(\epsilon, \mu, \Lambda, n, \mathfrak{a}) > 0$ such that $E(\overline{T}^j) = \emptyset$ if $|\overline{T}^j| \geq D$.* *Proof.* This lemma follows from the uniform estimate on the number of bad scales, which is given by Corollary [Corollary 44](#c:controlling-bad-scales){reference-type="ref" reference="c:controlling-bad-scales"}. ◻ **Lemma 55**. *Given a positive integer $j\in\mathbb{Z}_+$ and a $j$-tuple $\overline{T}^j = (\overline{T}_1^j,\ldots, \overline{T}_j^j)$ with $\overline{T}_{\ell}^j\in\{0,1\}$. For every $1 \leq \ell \leq j$, the set $$\begin{aligned} \mathcal{A}_{\ell}\equiv \mathcal{S}_{\epsilon, \mu^j}^k(U) \cap \mathcal{B}_{\mu^{\ell-1}}(x) \cap E(\overline{T}^j)\end{aligned}$$ admits the following effective coverings.* 1. *If $\overline{T}_{\ell}^j = 1$, then $\mathcal{A}_{\ell}$ can be covered by $C_1(n)\cdot\mu^{-n-1}$ balls centered in $\mathcal{A}_{\ell}$ of radius $\mu^{\ell}$.* 2. *If $\overline{T}_{\ell}^j = 0$, then $\mathcal{A}_{\ell}$ can be covered by $C_0(n)\cdot\mu^{-k}$ balls centered in $\mathcal{A}_{\ell}$ of radius $\mu^{\ell}$.* *Proof.* The proof of item (1) is trivial. We will prove item (2). We claim that $\mathcal{A}_{\ell} \subset B_{\frac{\mu^{\ell}}{10}}(V^k) \cap \mathcal{B}_{\mu^{\ell - 1}}(x)$ for some vector space $V^k$ of dimension $k$. If the claim is true, then the conclusion immediately follows since $B_{\frac{\mu^{\ell}}{10}}(V^k) \cap \mathcal{B}_{\mu^{\ell - 1}}(x)$ can be covered by $C_0(n)\cdot \mu^{-k}$ balls of radius $\mu^{\ell}$. Now let us prove the claim. Suppose the inclusion is not true, that is, there exists a point $z\in \mathcal{A}_{\ell}$ that satisfies $z \in \mathcal{B}_{\mu^{\ell - 1}}(x)\setminus B_{\frac{\mu^{\ell}}{10}}(V^k)$. It follows from Corollary [Corollary 49](#c:inductive-splitting){reference-type="ref" reference="c:inductive-splitting"} that $U$ is $(k+1, \epsilon, \mu^{\ell - 1})$-symmetric at $z$ which contradicts the fact $z\in \mathcal{A}_{\ell} \subset \mathcal{S}_{\epsilon, \mu^j}^k(U)$. This completes the proof of the claim. ◻ ## A new $\epsilon$-regularity result {#ss:epsilon-regularity} The main result of this subsection is an $\epsilon$-regularity result (Theorem [Theorem 57](#t:eps-regularity-smooth-approximation){reference-type="ref" reference="t:eps-regularity-smooth-approximation"}). This result is a key ingredient to prove the Hausdorff measure estimate in Section [3.6](#ss:Hausdorff-measure){reference-type="ref" reference="ss:Hausdorff-measure"}. We will start with a technical lemma in the two-dimensional case, which shows that the origin is an isolated critical point of our interested model polynomial. **Lemma 56**. *Let $P$ be a non-constant homogeneous polynomial that solves [\[e:symmetric-extension\]](#e:symmetric-extension){reference-type="eqref" reference="e:symmetric-extension"} when $n=1$. Let us write $\nabla P = (P_x, P_y)$. Then $\bm{0}_+\equiv(0,0)\in\mathbb{R}^2$ is an isolated solution of $\nabla P = \bm{0}_+$.* *Proof.* Let $k=\deg(P)$. There are two cases to analyze. **Case (1):** $k$ is even. By [@STT proposition 4.13], the polynomial $P(x,y)$ yields the form $$\begin{aligned} P(x,y) = \Upsilon_e(k,\mathfrak{a}) \cdot \Phi\left(-\frac{k}{2}, \frac{1-k-\mathfrak{a}}{2} , \frac{1}{2} , - \frac{x^2}{y^2} \right) y^k,\end{aligned}$$ where $\Phi(\alpha,\beta,\gamma, z)$ is the hypergeometric function and the normalization constant $\Upsilon_e(k,\mathfrak{a})>0$ is chosen such that $\fint_{\partial\mathcal{B}_1(\bm{0}_+)}y^{\mathfrak{a}} \cdot |P|^2 d\sigma = 1$. Computing the gradient of $P$, we have $$\begin{aligned} P_x &\ = -2 \Upsilon_e \cdot xy^{k-2}\cdot \frac{d\Phi}{dz}, \\ P_y &\ = 2 \Upsilon_e \cdot x^2 y^{k-3} \cdot \frac{d\Phi}{dz} + k \Upsilon_e \cdot y^{k-1} \cdot \Phi. \end{aligned}$$ Let denote by $Z(P_x)$ and $Z(P_y)$ the zero sets of $P_x$ and $P_y$, respectively. It is evident that both $Z(P_x)$ and $Z(P_y)$ are unions of lines passing through the origin of $\mathbb{R}^2$. We will prove that $\bm{0}_+\in\mathbb{R}^2$ is the only zero of $\nabla P$. First, it is not hard to see that $$\begin{aligned} \{(x,y)\in\mathbb{R}^2 | x y = 0\} \cap \{(x,y) \in \mathbb{R}^2| \nabla P(x,y) = 0\} = (0,0).\end{aligned}$$ Indeed, since $P$ is homogeneous and it is even about $\{y=0\}$, $P$ can be written as $$\begin{aligned} P(x,y) = a_0 x^k + \sum\limits_{i=1}^{k-1} a_i x^{k - 2i} y^{2i} + a_k y^k.\end{aligned}$$ Then $x=0$ and $P_y(x,y) = 0$ force $y=0$. Similarly, $y=0$ and $P_x(x,y) = 0$ force $x=0$. Next, we consider the case $xy \neq 0$. Let $|\nabla P| = 0$ along a line $\{y = \tau x\}$ for some $\tau\neq 0$. Then we have $$\begin{aligned} \Phi(-\tau^2) = \frac{d\Phi}{dz}\Big|_{z=-\tau^2} = 0.\end{aligned}$$ Since $\Phi$ satisfies $$\begin{aligned} z(1-z)\frac{d^2 \Phi}{dz^2} +(\gamma - (\alpha + \beta + 1)z) \frac{d\Phi}{dz} - \alpha\beta \Phi = 0, \end{aligned}$$ we have $\frac{d^2\Phi}{dz^2}\big|_{z = -\tau^2} = 0$. By simple induction argument, we have that $\frac{d^m \Phi}{dz^m}\big|_{z = -\tau^2} = 0$ for any $m\in\mathbb{Z}_+$. This implies that $|\nabla^m P| = 0$ along $\{y = \tau x\}$ for any $m\in\mathbb{N}_0$, and thus $P\equiv 0$ on $\mathbb{R}^2$. Contradiction. **Case (2):** $k$ is odd. The polynomial $P$ in this case satisfies $$\begin{aligned} P(x,y) = \Upsilon_o(k,\mathfrak{a}) \cdot \Phi\left(\frac{1-k}{2}, \frac{2-k-\mathfrak{a}}{2} , \frac{3}{2} , - \frac{x^2}{y^2} \right) x y^{k-1},\end{aligned}$$ the normalization constant $\Upsilon_e(k,\mathfrak{a})>0$ is chosen such that $\fint_{\partial\mathcal{B}_1(\bm{0}_+)}|P|^2 d\sigma = 1$. Computing the gradient of $P$, we have $$\begin{aligned} P_x &\ = -2 \Upsilon_o \cdot x^2 y^{k-3}\cdot \frac{d\Phi}{dz} + \Upsilon_o \cdot y^{k-1} \cdot \Phi , \\ P_y &\ = 2 \Upsilon_e \cdot x^3 y^{k-4} \cdot \frac{d\Phi}{dz} + (k - 1) \Upsilon_e \cdot xy^{k-2} \cdot \Phi. \end{aligned}$$ Using the similar arguments as in Case (1), we have that $(0,0)$ is the only zero of $\nabla P$. ◻ **Theorem 57**. *Let $P$ be a non-constant $(\mathfrak{m}-2)$-symmetric homogeneous polynomial of degree $d$ that solves [\[e:symmetric-extension\]](#e:symmetric-extension){reference-type="eqref" reference="e:symmetric-extension"}. Then there exist positive constants $\delta> 0$ and $r > 0$ such that for any $u \in C^{2d^2}(\mathcal{B}_1(\bm{0}_+))$ if $$\begin{aligned} |u - P|_{C^{2d^2}(\mathcal{B}_1(\bm{0}_+))} < \delta, \end{aligned}$$ then $\mathcal{H}^{\mathfrak{m}- 2}(\mathcal{C}(u) \cap \mathcal{B}_r(\bm{0}_+)) \leq C(\mathfrak{m}) (d - 1)^2 r^{\mathfrak{m}-2}$.* The proof of the proposition requires an important stability result in [@HHL theorem 4.1]. **Theorem 58** ([@HHL]). *Let $P:\mathbb{R}^n \to \mathbb{R}^n$ be a mapping with $f(\bm{0}) = \bm{0}$ so that each component is a homogeneous polynomial of degree $d$, where $\bm{0}\in\mathbb{R}^n$. Assume that $P$ can be extended to a holomorphic map from $\mathbb{C}^n$ to $\mathbb{C}^n$ and that the origin of $\mathbb{C}^n$ is an isolated zero. Then there exist positive constants $\delta_0>0$, $k\in\mathbb{Z}_+$, $r>0$ depending on $P$, such that if $U \in C^k(B_1(\bm{0}), \mathbb{R}^n)$ satisfies $\| U - P \|_{C^{2d^n}(B_1(\bm{0}))} < \delta_0$, then $\#\{\mathcal{C}(U) \cap B_r(\bm{0})\} \leq d^n$.* *Proof of Theorem [Theorem 57](#t:eps-regularity-smooth-approximation){reference-type="ref" reference="t:eps-regularity-smooth-approximation"}.* Let $P = P(x, y)$ and let us denote $x_1 = x$, $x_2 = y$, and $\bm{z} = (x_1 , x_2)$. Since the partial derivatives $P_{x_1}$ and $P_{x_2}$ are homogeneous polynomials, one has the decompositions $$\begin{aligned} P_{x_1}(\bm{z}) = & \ \left(\prod\limits_{k = 1}^{d_1} \langle \nu_k^1 , \bm{z}\rangle\right)\cdot\left(\prod\limits_{\mu = 1}^{q_1} Q_{\mu}^1(\bm{z})\right), \\ P_{x_2}(\bm{z}) = & \ \left(\prod\limits_{k = 1}^{d_2} \langle \nu_k^2 , \bm{z}\rangle\right)\cdot\left(\prod\limits_{\mu = 1}^{q_2} Q_{\mu}^2(\bm{z})\right). \end{aligned}$$ Here each $\nu_k^{\alpha}$ is a vector in $\mathbb{R}_{x_1,x_2}^2$ and each $Q_{\ell}^j$ is a quadratic form with the only zero $\bm{z} = 0$. Now restricted onto the plane $\mathbb{R}_{x_1,x_2}^2$, applying Lemma [Lemma 56](#l:hypergeometric-polynomial-critical-point){reference-type="ref" reference="l:hypergeometric-polynomial-critical-point"}, we find that $\nabla P = (P_{x_1}, P_{x_2})$ is vanishing only at $\bm{z} = 0$. Therefore, the vectors $\nu_k^{\alpha}$ and $\nu_{\ell}^{\beta}$ are linearly independent unless $(k, \alpha) = (\ell, \beta)$. Given the expression of $\nabla P$ as above, we will prove the following claim. **Claim.** There exists an open dense subset $\mathfrak{T}\subset SO(\mathfrak{m})$ such that the following holds: for any rotation $\mathscr{T}= (A_{ij})_{1\leq i,j\leq \mathfrak{m}} \in \mathfrak{T}$ with $\bm{x} = \mathscr{T}\bm{w}\in\mathbb{R}^{\mathfrak{m}}$, in terms of the new coordinate system $\bm{w}=(w_1,\ldots, w_{\mathfrak{m}})$, for every coordinate plane $$\begin{aligned} \mathcal{L}_{ij}\equiv \{\bm{w}=(w_1,\ldots, w_{\mathfrak{m}})\in\mathbb{R}^{\mathfrak{m}}: w_{\ell} = 0\ \text{if} \ \ell\neq i \ \text{and}\ \ell \neq j\}, \end{aligned}$$ the only zero of the restriction $(\nabla P)|_{\mathcal{L}_{ij}}$ on the plane $\mathcal{L}_{ij}$ is the origin. To prove the claim, let us take a rotation $\mathscr{T}= (A_{ij})_{1\leq i,j\leq \mathfrak{m}} \in SO(\mathfrak{m})$ with $\bm{x} = \mathscr{T}\bm{w}\in\mathbb{R}^{\mathfrak{m}}$. Then the partial derivative $P_{w_i}$ with respect to the new coordinates $\bm{w}=(w_1,\ldots, w_{\mathfrak{m}})$ is given by, $$\begin{aligned} P_{w_i}(w) = &\ (D_{x_1}P) \cdot \frac{\partial x_1}{\partial w_i} + (D_{x_2}P) \cdot \frac{\partial x_2}{\partial w_i}\nonumber\\ = & \ A_{1i}\left(\left(\prod\limits_{k = 1}^{d_1}\sum\limits_{\ell = 1}^{\mathfrak{m}}\langle\nu_k^1, \zeta_{\ell} \rangle w_{\ell}\right)\cdot\prod\limits_{\mu = 1}^{q_1}(\mathscr{R}\bm{w})^T\mathscr{Q}_{\mu}^1 (\mathscr{R}\bm{w})\right) \\ & \ + A_{2i}\left(\left(\prod\limits_{k = 1}^{d_2}\sum\limits_{\ell = 1}^{\mathfrak{m}}\langle\nu_k^2, \zeta_{\ell} \rangle w_{\ell}\right)\cdot\prod\limits_{\mu = 1}^{q_2}(\mathscr{R}\bm{w})^T\mathscr{Q}_{\mu}^2 (\mathscr{R}\bm{w})\right),\end{aligned}$$ where $\mathscr{R} \equiv \begin{bmatrix} A_{11} & \ldots & A_{1{\mathfrak{m}}} \\ A_{21} & \ldots & A_{2{\mathfrak{m}}} \end{bmatrix}$ is the first two rows of $(A_{ij})$, $\zeta_{\ell}$ is its $\ell^{th}$ column vector $\begin{bmatrix} A_{1\ell} \\ A_{2\ell} \end{bmatrix}$, $\mathscr{Q}_j^{\alpha}$ $(\alpha=1,2)$ is the matrix of the quadratic form $Q_j^{\alpha}$. Restricted on the coordinate plane $\mathcal{L}_{ij}$, we have $$\begin{aligned} \nabla P|_{\mathcal{L}_{ij}} = \mathscr{R}_{ij} \begin{bmatrix} \left(\prod\limits_{k = 1}^{d_1} \left(\langle\nu_k^1, \zeta_i \rangle w_i + \langle\nu_k^1, \zeta_j \rangle w_j\right)\right)\cdot\prod\limits_{j = 1}^{q_1}(\mathscr{R}_{ij}\bm{w}_{ij})^T\mathscr{Q}_j^1 (\mathscr{R}_{ij}\bm{w}_{ij}) \\ \left(\prod\limits_{k = 1}^{d_2}\left(\langle\nu_k^2, \zeta_i \rangle w_i + \langle\nu_k^2, \zeta_j \rangle w_j\right)\right)\cdot\prod\limits_{j = 1}^{q_2}(\mathscr{R}_{ij}\bm{w}_{ij})^T\mathscr{Q}_j^2 (\mathscr{R}_{ij}\bm{w}_{ij}) \end{bmatrix},\end{aligned}$$ where $\bm{w}_{ij} = (0,\ldots, w_i, \ldots, w_j, \ldots, 0 )$ and $$\begin{aligned} \mathscr{R}_{ij} \equiv \begin{bmatrix} A_{1i} & A_{2i} \\ A_{1j} & A_{2j} \end{bmatrix}.\label{e:submatrix-R_{ij}}\end{aligned}$$ Now the subset $\mathfrak{T}\subset \mathop{\mathrm{SO}}(\mathfrak{m})$ is defined as: a rotation matrix $\mathscr{T} = (A_{ij})_{1\leq i,j\leq \mathfrak{m}}$ is contained in $\mathfrak{T}$ if each sub-matrix $\mathscr{R}_{ij}$, as in [\[e:submatrix-R\_{ij}\]](#e:submatrix-R_{ij}){reference-type="eqref" reference="e:submatrix-R_{ij}"}, is nonsingular. Then $\mathfrak{T}\subset \mathop{\mathrm{SO}}(\mathfrak{m})$ is an open subset. Moreover, $\dim(\mathop{\mathrm{SO}}(\mathfrak{m})\setminus \mathfrak{T}) = \frac{\mathfrak{m}(\mathfrak{m}- 1)}{2} - 1$. Therefore, $\mathfrak{T}\subset \mathop{\mathrm{SO}}(\mathfrak{m})$ is an open dense subset. Next, we will show that, for every $\mathscr{T}\in \mathfrak{T}$ with $\bm{x} = \mathscr{T}\bm{w}$ and for every $\bm{w}$-coordinate plane $\mathcal{L}_{ij}$, the only zero of $\nabla P|_{\mathcal{L}_{ij}}$ is the origin. Indeed, by the definition of $\mathfrak{T}$, the matrix $\mathscr{R}_{ij}$ is always nonsingular, so $\nabla P|_{\mathcal{L}_{ij}}$ if and only if $$\begin{aligned} \begin{split} \left(\prod\limits_{k = 1}^{d_1} \left(\langle\nu_k^1, \zeta_i \rangle w_i + \langle\nu_k^1, \zeta_j \rangle w_j\right)\right)\cdot\prod\limits_{j = 1}^{q_1}(\mathscr{R}_{ij}\bm{w}_{ij})^T\mathscr{Q}_j^1 (\mathscr{R}_{ij}\bm{w}_{ij}) = & \ 0, \\ \left(\prod\limits_{k = 1}^{d_2}\left(\langle\nu_k^2, \zeta_i \rangle w_i + \langle\nu_k^2, \zeta_j \rangle w_j\right)\right)\cdot\prod\limits_{j = 1}^{q_2}(\mathscr{R}_{ij}\bm{w}_{ij})^T\mathscr{Q}_j^2 (\mathscr{R}_{ij}\bm{w}_{ij}) = & \ 0. \end{split}\label{e:product-vanishing}\end{aligned}$$ Since $\mathscr{Q}_j^{\alpha}$ is nonsingular, for any $\alpha=1,2$ and $1\leq j\leq q_{\alpha}$, $(\mathscr{R}_{ij}\bm{w}_{ij})^T\mathscr{Q}_j^{\alpha} (\mathscr{R}_{ij}\bm{w}_{ij}) = 0$ if and only if $\bm{w}_{ij} = 0$. It follows that [\[e:product-vanishing\]](#e:product-vanishing){reference-type="eqref" reference="e:product-vanishing"} if and only if $$\begin{aligned} \prod\limits_{k = 1}^{d_1} \left(\langle\nu_k^1, \zeta_i \rangle w_i + \langle\nu_k^1, \zeta_j \rangle w_j\right) = \prod\limits_{k = 1}^{d_2}\left(\langle\nu_k^2, \zeta_i \rangle w_i + \langle\nu_k^2, \zeta_j \rangle w_j \right)= 0. \end{aligned}$$ Since $(\nu_k^1, \nu_{\ell}^2)$ is pair of linearly independent vectors for any $k\neq \ell$, and $(\zeta_i, \zeta_j)$ is also a pair of linearly independent vectors for any $i\neq j$, simple calculations in linear algebra imply that $w_i = w_j = 0$. This completes the proof of the claim. Now we are ready to prove the Hausdorff measure estimate. First, let us denote $$\begin{aligned} DP(i,j) \equiv (D_i P, D_j P)|_{\mathcal{L}_{ij}}.\end{aligned}$$ By the claim, the origin $\bm{0}^{\mathfrak{m}} \in \mathbb{C}^{\mathfrak{m}}$ is also the only zero of the complex extension of $DP(i,j)$. Let $u\in C^{2d^2}(\mathcal{B}_1(\bm{0}_+))$ satisfy $|u - P|_{C^{2d^2}(\mathcal{B}_1(\bm{0}_+))} < \delta$ for some sufficiently small $\delta > 0$ such that the following holds: for any $p \in \mathcal{B}_1(\bm{0}_+)$, $$\begin{aligned} |D_pu(i, j) - DP(i, j)|_{C^{2(d-1)^2}(\mathcal{B}_1(\bm{0}_+))} < \delta_0,\quad \forall\ 1\leq i,j\leq \mathfrak{m}, \end{aligned}$$ where $D_pu(i, j) \equiv (D_i u, D_ju)|_{\mathcal{L}_{ij}(p)}$ and $\mathcal{L}_{ij}(p) \equiv p + \mathcal{L}_{ij}$ is a translation of the plane $\mathcal{L}_{ij}$. Then applying Theorem [Theorem 58](#t:HHL-stability){reference-type="ref" reference="t:HHL-stability"}, there exists some small $r > 0$ such that for all $p$, $$\begin{aligned} \#\left((\nabla u)^{-1}(\bm{0}_+)\cap \mathcal{L}_{ij}(p) \cap \mathcal{B}_r(\bm{0}_+)\right) \leq \#\left((D_pu(i, j))^{-1}(\bm{0}^2)\cap \mathcal{B}_r(\bm{0}_+)\right) \leq (d - 1)^2. \end{aligned}$$ To estimate the Hausdorff measure of $(\nabla u)^{-1}(\bm{0}_+) \cap B_r(\bm{0}_+)$, let us take the natural projection $\pi_{ij}:\mathbb{R}^{\mathfrak{m}} \to \mathbb{R}^{\mathfrak{m}- 2}$ by $$\begin{aligned} \pi_{ij}(\bm{x}) \equiv (x_1,\ldots,\hat{x}_i,\ldots, \hat{x}_j, \ldots x_{\mathfrak{m}}) \in\mathbb{R}^{n-2}, \end{aligned}$$ where $x_i$ and $x_j$ are deleted from $\bm{x}= (x_1,\ldots, x_{\mathfrak{m}}) \in \mathbb{R}^n$. Then for every $q\in B_r(\bm{0}^{\mathfrak{m}- 2}) \subset \mathbb{R}^{\mathfrak{m}- 2}$, we have that $$\begin{aligned} \#\left((\nabla u)^{-1}(\bm{0}_+)\cap (\pi_{ij})^{-1}(q) \cap \mathcal{B}_r(\bm{0}_+)\right) \leq (d - 1)^2. \end{aligned}$$ Therefore, applying the area formula (see [@Federer 3.2.22] or [@Han-Lin theorem 1.2.10]), $$\begin{aligned} &\ \mathcal{H}^{\mathfrak{m}- 2}\left((\nabla u)^{-1}(\bm{0}_+) \cap \mathcal{B}_r(\bm{0}_+)\right) \\ \leq &\ \sum\limits_{1\leq i < j \leq \mathfrak{m}} \int_{B_r(\bm{0}^{\mathfrak{m}- 2})} \#\left((\nabla u)^{-1}(\bm{0}_+)\cap (\pi_{ij})^{-1}(q) \cap \mathcal{B}_r(\bm{0}_+)\right) d\mathcal{H}^{\mathfrak{m}- 2}(q) \nonumber\\ \leq & \ C(\mathfrak{m}) (d - 1)^2 r^{\mathfrak{m}-2}, \end{aligned}$$ which completes the proof of the proposition. ◻ **Theorem 59** ($\epsilon$-regularity). *Let $U$ solve [\[e:symmetric-extension\]](#e:symmetric-extension){reference-type="eqref" reference="e:symmetric-extension"} and satisfy $\mathcal{N}_U(\bm{0}_+ , 1) \leq \Lambda$. Then there exist constants $\epsilon(\mathfrak{m},\mathfrak{a},\Lambda) > 0$ and $\bar{r}(\mathfrak{m},\mathfrak{a},\Lambda) > 0$ such that if there exists a homogeneous polynomial $P$ with $(\mathfrak{m}- 2)$-symmetry such that $$\begin{aligned} \fint_{\partial\mathcal{B}_1(\bm{0}_+)} |y|^\mathfrak{a}P^2 d\sigma = 1\quad \text{and} \quad \|\mathcal{T}_{\bm{0}_+,1}U - P\|_{L^2(\partial\mathcal{B}_1(\bm{0}_+))} \leq \epsilon, \end{aligned}$$ then for all $r\leq \bar{r}$, $$\begin{aligned} \mathcal{H}^{\mathfrak{m}- 2}(\mathcal{C}(U) \cap \mathcal{B}_r(\bm{0}_+)) \leq C(n,\Lambda) r^{\mathfrak{m}- 2}, \end{aligned}$$ where $\mathcal{C}(U)\equiv \{\bm{x}\in \mathbb{R}^n: |\nabla U|(\bm{x}) = 0\} \subset \mathbb{R}^n$ is the restriction of the critical set of $U$ on the boundary $\mathbb{R}^n$.* *Proof.* It follows from Theorem [Theorem 57](#t:eps-regularity-smooth-approximation){reference-type="ref" reference="t:eps-regularity-smooth-approximation"}. ◻ ## Hausdorff measure estimates for the critical sets {#ss:Hausdorff-measure} In this subsection, we will prove the following uniform Hausdorff measure estimate for the critical set of the solutions of [\[e:symmetric-extension\]](#e:symmetric-extension){reference-type="eqref" reference="e:symmetric-extension"}. **Theorem 60**. *Let $U$ solve [\[e:symmetric-extension\]](#e:symmetric-extension){reference-type="eqref" reference="e:symmetric-extension"} with $\mathcal{N}_U(\bm{0}_+,1)\leq \Lambda$. There exists a constant $C=C(\mathfrak{m},\Lambda,\mathfrak{a})>0$ such that $$\begin{aligned} \mathcal{H}^{\mathfrak{m}- 2}(\mathcal{C}(U) \cap \mathcal{B}_{1/2}(\bm{0}_+)) \leq C(\mathfrak{m},\Lambda,\mathfrak{a}), \end{aligned}$$ where $\mathcal{C}(U)\equiv \{\bm{x}\in \mathbb{R}^n: |\nabla U|(\bm{x}) = 0\} \subset \mathbb{R}^n$ is the restriction of the critical set of $U$ on the boundary $\mathbb{R}^n$.* *Proof.* First, Lemma [Lemma 51](#l:almost-(m-1)-symmetric){reference-type="ref" reference="l:almost-(m-1)-symmetric"} implies that there exists some $\epsilon>0$ such that $$\begin{aligned} \mathcal{C}(U)\subset \mathcal{S}_{\epsilon, 1}^{\mathfrak{m}-2}(U).\end{aligned}$$ Then let us write $$\begin{aligned} \mathcal{C}(U) \cap \mathcal{B}_{1/2}(\bm{0}_+) = \mathcal{C}(U) \cap \Big( \left(\mathcal{S}_{\epsilon, 1}^{\mathfrak{m}-2}(U) \setminus \mathcal{S}_{\epsilon, 1}^{\mathfrak{m}-3}(U)\right) \cup \mathcal{S}_{\epsilon, 1}^{\mathfrak{m}-3}(U)\Big) \cap \mathcal{B}_{1/2}(\bm{0}_+). \label{e:CU-symmetry}\end{aligned}$$ By definition, one can obtain the following dyadic decomposition of $\mathcal{S}_{\epsilon, 1}^{\mathfrak{m}-3}(U)$. $$\begin{aligned} \mathcal{S}_{\epsilon, 1}^{\mathfrak{m}-3}(U) = \left( \bigcup\limits_{j=1}^{\infty} \left(\left( \mathcal{S}_{\epsilon, \mu^j}^{\mathfrak{m}-2}(U) \setminus \mathcal{S}_{\epsilon, \mu^j}^{\mathfrak{m}-3}(U) \right) \bigcap \mathcal{S}_{\epsilon, \mu^{j-1}}^{\mathfrak{m}-3}(U) \right)\right) \bigcup \left(\bigcap\limits_{j=1}^{\infty} \mathcal{S}_{\epsilon, \mu^j}^{\mathfrak{m}-3}(U) \right). \label{e:(m-3)-singular-stratum}\end{aligned}$$ Now let us define $$\begin{aligned} &\mathcal{C}^{(0)}(U) = \mathcal{C}(U) \cap \left(\mathcal{S}_{\epsilon, 1}^{\mathfrak{m}-2}(U) \setminus \mathcal{S}_{\epsilon, 1}^{\mathfrak{m}-3}(U)\right) \cap \mathcal{B}_{1/2}(\bm{0}_+), \\ &\mathcal{C}^{(j)}(U) = \mathcal{C}(U) \cap \left(\mathcal{S}_{\epsilon, \mu^j}^{\mathfrak{m}-2}(U) \setminus \mathcal{S}_{\epsilon, \mu^j}^{\mathfrak{m}-3}(U)\right) \cap \mathcal{S}_{\epsilon, \mu^{j-1}}^{\mathfrak{m}-3}\cap \mathcal{B}_{1/2}(\bm{0}_+), \quad j\in\mathbb{Z}_+.\end{aligned}$$ Combining [\[e:CU-symmetry\]](#e:CU-symmetry){reference-type="eqref" reference="e:CU-symmetry"} and [\[e:(m-3)-singular-stratum\]](#e:(m-3)-singular-stratum){reference-type="eqref" reference="e:(m-3)-singular-stratum"}, $\mathcal{C}(U)$ has a further decomposition, $$\begin{aligned} \mathcal{C}(U) \cap \mathcal{B}_{1/2}(\bm{0}_+) = \left( \bigcup\limits_{j=0}^{\infty} \mathcal{C}^{(j)}(U)\right) \bigcup \left(\mathcal{C}(U)\bigcap\limits_{j=1}^{\infty}\mathcal{S}_{\epsilon,\mu^j}^{\mathfrak{m}-3}(U) \right).\end{aligned}$$ If follows from Theorem [Theorem 50](#t:volume-estimate-model){reference-type="ref" reference="t:volume-estimate-model"} that $$\begin{aligned} \mathcal{H}^{\mathfrak{m}-2}\left( \mathcal{C}(U)\bigcap\limits_{j=1}^{\infty}\mathcal{S}_{\epsilon,\mu^j}^{\mathfrak{m}-3}(U) \right) = 0. \end{aligned}$$ Indeed, for any $j$, the singular stratum $\mathcal{S}_{\epsilon,\mu^j}^{\mathfrak{m}-3}(U)$ is covered by $C(\mathfrak{m},\Lambda,\epsilon)(\mu^j)^{3-\mathfrak{m}-\epsilon}$ balls of radius $\mu^j$ and centered at $x_{\alpha}\in \mathcal{S}_{\epsilon,\mu^j}^{\mathfrak{m}-3}(U)$. Then $$\begin{aligned} \mathcal{H}^{\mathfrak{m}-2}\left( \mathcal{C}(U)\bigcap\limits_{j=1}^{\infty}\mathcal{S}_{\epsilon,\mu^j}^{\mathfrak{m}-3}(U) \right) \leq &\ \mathcal{H}^{\mathfrak{m}-2}(\mathcal{S}_{\epsilon,\mu^j}^{\mathfrak{m}-3}(U)) \nonumber\\ \leq &\ C(\mathfrak{m},\Lambda,\epsilon) (\mu^j)^{\mathfrak{m}-2} \cdot (\mu^j)^{3-\mathfrak{m}-\epsilon} \nonumber\\ = &\ C(\mathfrak{m},\Lambda,\epsilon) (\mu^j)^{1-\epsilon}.\end{aligned}$$ Therefore, $$\begin{aligned} \mathcal{H}^{\mathfrak{m}-2}\left( \mathcal{C}(U)\bigcap\limits_{j=1}^{\infty}\mathcal{S}_{\epsilon,\mu^j}^{\mathfrak{m}-3}(U) \right) = 0. \end{aligned}$$ Then we will prove that for any $k\geq 0$, $$\begin{aligned} \mathcal{H}^{\mathfrak{m}-2} \left( \bigcup\limits_{j=0}^k \mathcal{C}^{(j)}(U)\right) \leq C(\Lambda, \mathfrak{m},\epsilon) \sum\limits_{j=0}^k \mu^{(1-\epsilon)j}. \label{e:measure-union-of-Cj}\end{aligned}$$ By Theorem [Theorem 59](#t:eps-reg-Euclidean){reference-type="ref" reference="t:eps-reg-Euclidean"}, we find that the estimate holds for $k = 0$. Next, we will prove [\[e:measure-union-of-Cj\]](#e:measure-union-of-Cj){reference-type="eqref" reference="e:measure-union-of-Cj"} in the general case. Let us choose a covering consisting of finite balls, $$\begin{aligned} \mathcal{D}(\mu^j\cdot \bar{r}) \equiv \left\{\mathcal{B}_{\mu^j \cdot \bar{r}}(x_{\alpha}) | \text{ for some } x_{\alpha}\in \mathcal{C}^{(j)}(U)\right\} \end{aligned}$$ of $\mathcal{C}^{(k)}(U)$ such that for any $\alpha \neq \beta$, $$\begin{aligned} \mathcal{B}_{\frac{\mu^j \cdot \bar{r}}{10}}(x_{\alpha}) \cap \mathcal{B}_{\frac{\mu^j \cdot \bar{r}}{10}}(x_{\beta}) = \emptyset.\end{aligned}$$ Then by Theorem [Theorem 50](#t:volume-estimate-model){reference-type="ref" reference="t:volume-estimate-model"}, we have that $\#\mathcal{D}(\mu^k\cdot \bar{r}) \leq C(\mathfrak{m}, \Lambda, \epsilon) \cdot (\mu^j \cdot \bar{r})^{(3-\mathfrak{m}-\epsilon)}$. By the definition of $\mathcal{C}^{(j)}$, for each $x_{\alpha}$, there exists a scale $r\in [\mu^j, \mu^{j-1}]$ such that for some normalized homogeneous polynomial $P$ of two variables, we have that $$\begin{aligned} \int_{\partial\mathcal{B}_1(\bm{0}_+)} |y|^{\mathfrak{a}} (\mathcal{T}_{\bm{x}_{\alpha}, r}(U) - P)^2 \mathop{\mathrm{dvol}}_{\partial\mathcal{B}_1(\bm{0}_+)} < \epsilon. \end{aligned}$$ Since $U$ is a solution of the boundary value problem [\[e:symmetric-extension\]](#e:symmetric-extension){reference-type="eqref" reference="e:symmetric-extension"}, using a simple compactness & contradiction argument, one can replace $P$ with a homogeneous polynomial that solves [\[e:symmetric-extension\]](#e:symmetric-extension){reference-type="eqref" reference="e:symmetric-extension"}. Applying Theorem [Theorem 59](#t:eps-reg-Euclidean){reference-type="ref" reference="t:eps-reg-Euclidean"}, we have that $$\begin{aligned} \mathcal{H}^{\mathfrak{m}-2}(\mathcal{C}(U) \cap B_{\mu^j\cdot \bar{r}}(x_{\alpha})) \leq C(\mathfrak{m},\Lambda) \cdot (\mu^j\cdot \bar{r})^{\mathfrak{m}-2}.\end{aligned}$$ Therefore, for any $j\in\mathbb{Z}_+$, $$\begin{aligned} \mathcal{H}^{\mathfrak{m}-2}(\mathcal{C}^{(j)}((U))) \leq C(\Lambda, \mathfrak{m}, \epsilon) \cdot \mu^{(1-\epsilon)j}, \end{aligned}$$ which completes the proof. ◻ ## Singular set on the boundary {#ss:boundary-estimate} This subsection considers the structure of the singular set of a function $f$ that is $\gamma$-harmonic on $B_2(\bm{0})\subset \mathbb{R}^n$. As stated in Theorem [Theorem 3](#t:volume-estimate-boundary){reference-type="ref" reference="t:volume-estimate-boundary"} and Theorem [Theorem 83](#t:Hausdorff-measure-estimate-PE){reference-type="ref" reference="t:Hausdorff-measure-estimate-PE"}, the full singular set $\mathcal{S}(f)$ does not satisfy a codimension-$2$ estimate on $\mathbb{R}^n$ due to the nonlocal nature of the fractional GJMS operator $P_{2\gamma}$. In the example presented in Example [Example 38](#ex-special-soln-U){reference-type="ref" reference="ex-special-soln-U"}, we have $$\begin{aligned} \mathcal{S}(f) = \{x \in \mathbb{R}^n : x_1 = 0\},\end{aligned}$$ and the dimension of $\mathcal{S}(f)$ is $\dim(\mathcal{S}(f)) = n-1$. In the Euclidean model case, this phenomenon has been observed in [@STT]. As in [@STT Section 8], the points in $\mathcal{S}(f)$ are divided into two classes depending upon the "local nature\" of the tangent map of $f$ at $x\in \mathcal{S}(f)$. Let us recall that a codimension-$2$ estimate for $\mathcal{S}(U)$ in $\overline{\mathbb{R}^{\mathfrak{m}}}$ with $\mathfrak{m}= n + 1$ is based on a simple fact: if a single variable function $U$ is smooth on $\overline{\mathbb{R}^{\mathfrak{m}}}$ and satisfies $\mathop{\mathrm{div}}(y^{\mathfrak{a}}\nabla U) = 0$ in $\mathbb{R}^{\mathfrak{m}}$, then $U$ must be a linear function on $\mathbb{R}^n$ and independent of $y$. As a consequence, $U$ does not have any critical point unless it is constantly vanishing. Motivated by this, we introduce the following notions. **Definition 61** (Horizontal singular strata). Given a smooth nondegenerate function $f:B_1(\bm{0})\to \mathbb{R}$ we define the $k^{th}$-horizontal singular stratum of $f$ by $$\begin{aligned} \underline{\mathcal{S}}^k (f) \equiv \{x\in B_1(\bm{0}): T_x (f) \ \text{satisfies} \ \Delta_{\mathbb{R}^n} T_x (f) = 0\ \text{and is not}\ (k+1)\text{-symmetric}\},\end{aligned}$$ for any $0 \leq k \leq n - 1$. We also define the horizontal part of the critical set $\mathcal{C}(f)$, $$\begin{aligned} \underline{\mathcal{C}}(f) \equiv \{x\in \mathcal{C}(f)\cap B_1(\bm{0}): T_x (f) \ \text{satisfies} \ \Delta_{\mathbb{R}^n} T_x (f) = 0\}.\end{aligned}$$ Let us denote the complement $\mathfrak{C}(f) \equiv \mathcal{C}(f) \setminus \underline{\mathcal{C}}(f)$ and $\mathfrak{C}^k(f) \equiv \mathcal{S}^k(f) \setminus \underline{\mathcal{S}}^k(f)$. For the singular $\mathcal{S}(f)$ of $f$, we define $$\begin{aligned} \underline{\mathcal{S}}(f) \equiv \mathcal{S}(f) \cap \underline{\mathcal{C}}(f) \quad \text{and} \quad \mathfrak{S}(f) \equiv \mathcal{S}(f) \setminus \underline{\mathcal{S}}(f). \end{aligned}$$ For fixed parameters $r, \epsilon\in (0,1)$ and $0 \leq k \leq n - 1$, one can define the quantitative singular strata $\underline{\mathcal{S}}_{r,\epsilon}^k(f)$ and $\mathfrak{S}_{r,\epsilon}^k(f)$ of the critical/singular set of $f$ in a similar way. Based on the above definition, we have a simple observation. If a tangent map $T_x(f)$ is $(n - 1)$-symmetric and satisfies $\Delta_{\mathbb{R}^n}T_x (f) = 0$, then $T_x(f)$ must be linear, which implies $x \not\in \mathcal{C}(f)$. This tells us $\underline{\mathcal{C}}(f) = \underline{\mathcal{S}}^{n - 2}(f)$ but in general $\mathfrak{C}(f) \setminus \mathfrak{C}^{n - 2}(f) \neq \emptyset$. Therefore, one should establish codimension-$2$ estimates for the *horizontal singular set* instead of the whole singular set. We give a quick lemma relating the quantitative symmetry of function on $\mathbb{R}^{n+1}$ and its restriction on $\mathbb{R}^n$ which enables us to understand the structure of the singular set of $f$ from $U$. **Lemma 62**. *Given a smooth function $f$ that is $\gamma$-harmonic on $B_2(\bm{0})$, let $U$ be its Caffarelli-Silvestre type extension that solves [\[e:U-extension\]](#e:U-extension){reference-type="eqref" reference="e:U-extension"}. For every $\epsilon>0$, there exists $\delta=\delta(n,\epsilon,\Lambda, \mathfrak{a})>0$ such that if $U$ is $(k,\delta,s,\mathfrak{a})$-symmetric at $x_+\equiv (x, 0)\in \mathcal{B}_1(\bm{0}_+)$, then $f$ is $(k - 1, \epsilon, s)$-symmetric at $x\in \mathbb{R}^n$.* *Proof.* We will prove it by contradiction. Suppose there exists a constant $\epsilon_0>0$, a sequence $\delta_j\to 0$, and a sequence of solutions $U_j$ such that $U_j$ is $(k,\delta_j,s,\mathfrak{a})$-symmetric at $\bm{x}_j = (x_j, 0) \in \mathbb{R}^{n+1}$ but $f_j$ is not $(k - 1,\epsilon_0, s)$-symmetric at $x_j$. Without loss of generality, we can assume that $U_j$ satisfies $$\begin{aligned} U_j(\bm{x}_j) = 0 \quad \text{and} \quad \frac{1}{s^{\mathfrak{a}}}\fint_{\partial\mathcal{B}_1(\bm{0}_+)}|y(\bm{x}_j + s\xi)|^{\mathfrak{a}} \cdot U_j(\bm{x}_j + s \xi)^2d\sigma(\xi) = 1.\end{aligned}$$ By the definition of $(k,\delta_j,s,\mathfrak{a})$-symmetry, we have that $$\begin{aligned} \frac{1}{s^{\mathfrak{a}}}\fint_{\mathcal{B}_1(\bm{x}_j)}|y|^{\mathfrak{a}} \cdot |\mathcal{T}_{\bm{x}_j,s} U_j - P_j|^2 \mathop{\mathrm{dvol}}_{\mathbb{R}^{n+1}} < \delta_j \to 0 \end{aligned}$$ for a sequence of $k$-symmetric normalized homogeneous polynomial $P_j$ with degree uniformly bounded by a constant depending upon $\Lambda$. Letting $j\to \infty$, we have that $\mathcal{T}_{\bm{x}_j, s} U_j$ converges to $\mathcal{T}_{\bm{x}_{\infty}, s} U_{\infty}\equiv P_{\infty}$, which is a $k$-symmetric polynomial. By the normalization condition, we have $U_{\infty} \equiv P_{\infty}$. Let $\underline{P}_{\infty}\equiv P_{\infty}|_{\mathbb{R}^n}$ be the restriction of $P_{\infty}$ on $\mathbb{R}^n$, which is apparently $(k-1)$-symmetric. Then the above implies that $$\begin{aligned} \frac{1}{s^{\mathfrak{a}}}\fint_{B_1(\bm{0})}|T_{x_j, s}f_j - \underline{P}_{\infty}|^2 \mathop{\mathrm{dvol}}_{\mathbb{R}^n} \to 0.\end{aligned}$$ The desired contradiction arises. ◻ Then one can develop the same quantitative symmetry and quantitative cone splittings as in Section [3.3](#ss:quantitative-cone-splitting){reference-type="ref" reference="ss:quantitative-cone-splitting"} for the horizontal singular strata. These technical preliminaries lead to the following result. **Theorem 63**. *For every $n \geq 2$, $\gamma \in (0, 1)$, $\Lambda > 0$, and $\epsilon>0$, there exist $0<\mu(n,\epsilon, \gamma, \Lambda)<1$ and $C(n,\Lambda, \gamma, \epsilon)>0$ such that the following property holds. If $f \in C^{\infty}(\mathbb{R}^n)$ is $\gamma$-harmonic on $B_1(\bm{0})\subset \mathbb{R}^n$ and satisfies $\mathcal{N}_f(\bm{0}, 1)\leq \Lambda$, then for any $j\in \mathbb{Z}_+$, $$\begin{aligned} \mathop{\mathrm{Vol}}(T_{\mu^j}(\underline{\mathcal{S}}_{\epsilon,\mu^j}^k(f))\cap B_{1/2}(\bm{0})) \leq C \cdot (\mu^j)^{n - k -\epsilon}, \quad 0\leq k \leq n -2, \\ \mathop{\mathrm{Vol}}(T_{\mu^j}(\mathfrak{S}_{\epsilon,\mu^j}^k(f))\cap B_{1/2}(\bm{0})) \leq C \cdot (\mu^j)^{n - k -\epsilon}, \quad 0\leq k \leq n - 1,\end{aligned}$$ where $T_r(A)$ is the $r$-tubular neighborhood of $A\subset \mathbb{R}^n$. In particular, $$\begin{aligned} \dim_{\mathop{\mathrm{Min}}}(\underline{\mathcal{C}}(f)\cap B_{1/2}(\bm{0})) \leq n - 2\quad \text{and} \quad \dim_{\mathop{\mathrm{Min}}}(\mathfrak{C}(f)\cap B_{1/2}(\bm{0})) \leq n - 1. \end{aligned}$$* **Remark 64**. In fact, one can obtain the same estimates for the critical set of $f$. The proof of the theorem follows along the the same lines as the covering arguments in Section [3.4](#ss:volume-estimate){reference-type="ref" reference="ss:volume-estimate"}, so we omit it. We end this section by introducing the Hausdorff measure estimate for the critical set $\mathcal{C}(f)$ for a $\gamma$-harmonic function $f$. **Theorem 65**. *Given $n\geq 2$ and $\gamma \in (0,1)$, let $f\in C^{\infty}(\mathbb{R}^n)$ be $\gamma$-harmonic on $B_1(\bm{0})$. For any $\Lambda > 0$, there exists $C = C(\Lambda, n, \gamma) > 0$ such that if $\mathcal{N}_f(\bm{0}, 1) \leq \Lambda$, then $\mathcal{H}^{n - 2}(\underline{\mathcal{C}}(f) \cap B_{1/2}(\bm{0})) \leq C$ and $\mathcal{H}^{n - 1}(\mathfrak{C}(f) \cap B_{1/2}(\bm{0})) \leq C$.* # Singular set in general Poincaré-Einstein manifolds {#s:results-on-PE} This section is devoted to the proof of our main results in the general Poincaré-Einstein case. Given $n\geq 2$, denote $\mathfrak{m}\equiv n + 1$ and let $(X^{\mathfrak{m}}, g_+)$ be a complete Poincaré-Einstein manifold with conformal infinity $(M^n, h)$. We choose the Fefferman-Graham compactification $\bar{g}= \varrho^2 g_+ = e^{2w} g_+$ of $(X^{\mathfrak{m}}, g_+)$ as in Lemma [Lemma 16](#l:E){reference-type="ref" reference="l:E"} such that $-\Delta_{g_+} w = n$ and $\bar{g}|_{M^n} = h$. Throughout this section, we make some fundamental assumptions on the regularity of the Poincaré-Einstein metric $g_+$, that we precisely state below. 1. $(X^{\mathfrak{m}}, g_+)$ admits a $C^{\infty}$ conformal compactification up to the boundary $M^n$. 2. $(M^n,[h])$ has nonnegative Yamabe invariant, i.e., $\mathcal{Y}(M^n,[h]) \geq 0$. This particularly implies $\lambda_1(-\Delta_{g_+}) = \frac{n^2}{4}$ so that one can define fractional GJMS operators $P_{2\gamma}$ on $M^n$ for all $\gamma\in(0,1)$; see Remark [Remark 14](#r:bottom-of-spectrum){reference-type="ref" reference="r:bottom-of-spectrum"}. 3. $(M^n, h)$ is obstruction flat when $n$ is even. Under these assumptions, by Propositions [Proposition 9](#p:general-regularity-FG){reference-type="ref" reference="p:general-regularity-FG"} and [Proposition 11](#p:regularity-obstruction-free){reference-type="ref" reference="p:regularity-obstruction-free"}, the Fefferman-Graham compactified metric $\bar{g}$ is $C^{\infty}$ up to the boundary $M^n$. To avoid unnecessary technical complications, there is no harm to make the following assumption on the compactified space $(\overline{X^{\mathfrak{m}}}, \bar{g})$. 1. **(4).** There exists $\iota_0>0$ such that $\mathop{\mathrm{Inj}}_{\bar{g}}(x_+)\geq \iota_0 > 0$ for any $x_+\in \overline{X^{\mathfrak{m}}}$. 2. **(5).** *The boundary injectivity radius* $\mathop{\mathrm{Inj}}_{\partial}(M^n) \geq 2$, namely the *normal exponential map* is a diffeomorphism within the tubular neighborhood $$\begin{aligned} T_2(M^n)\equiv \{p\in \overline{X^{\mathfrak{m}}}: d_{\bar{g}}(p, M^n) \leq 2\}.\end{aligned}$$ In particular, $T_2(M^n)\equiv \{p\in \overline{X^{\mathfrak{m}}}: d_{\bar{g}}(p, M^n) \leq 2\}$ of $M^n$ is always diffeomorphic to $M^n \times [0,2]$. Indeed, one can always achieve this by a finite rescaling. For our purpose, we need to double the compactified manifold $(\overline{X^{\mathfrak{m}}}, \bar{g})$ along the totally geodesic boundary $(M^n, h)$, which gives a closed manifold $\mathfrak{X}^{\mathfrak{m}} \equiv \overline{X^{\mathfrak{m}}}\bigcup\limits_{M^n}\overline{X^{\mathfrak{m}}}$ equipped with a $C^{\mathfrak{m}-2,1}$-Riemannian metric (still denoted as $\bar{g}$): $\bar{g}$ fails to be smooth only when crossing $M^n$. Now let us take a smooth function $f\in C^{\infty}(M^n)$ that satisfies $P_{2\gamma}(f) = 0$ for some $\gamma \in (0,1)$. Let $U\in H^{1,\mathfrak{a}}(X^{\mathfrak{m}})$ be the *even extension* of $f$ that solves $$\begin{aligned} \label{e:CS-extension-on-PE} \begin{cases} - \mathop{\mathrm{div}}_{\bar{g}}(\varrho^{\mathfrak{a}}\nabla_{\bar{g}} U) + \varrho^{\mathfrak{a}} \mathcal{J}_{\bar{g}} U = 0 & \text{in}\ \mathfrak{X}^{\mathfrak{m}}, \\ U = f, & \text{on}\ M^n, \\ P_{2\gamma} f = 0 & \text{on} \ B_1(p)\subset M^n, \end{cases}\end{aligned}$$ where $\mathfrak{a}\equiv 1 - 2 \gamma$ and $\mathcal{J}_{\bar{g}}\equiv C_{n,\gamma}R_{\bar{g}}$. Lemma [Lemma 24](#eq-C1a-U){reference-type="ref" reference="eq-C1a-U"} implies that the solution $U$ yields $U\in C^{\mathfrak{m}-2,\alpha}(\overline{X^{\mathfrak{m}}})$ for some $\alpha\in(0,1)$. ## Frequency and almost monotonicity {#ss:almost-monotonicity} This subsection is to develop an almost monotonicity formula for the generalized Almgren's frequency associated to the solution $U$ in [\[e:CS-extension-on-PE\]](#e:CS-extension-on-PE){reference-type="eqref" reference="e:CS-extension-on-PE"}. To begin with, for any $x_+\in M^n$ and $r\in (0,1)$, let us define $$\begin{aligned} H_U(x_+, r) & \equiv \int_{\partial \mathcal{B}_r(x_+)} \varrho^{\mathfrak{a}} U^2 d\sigma_{\bar{g}},\label{eq-H}\\ I_U(x_+, r) & \equiv \int_{\mathcal{B}_r(x_+)} \varrho^{\mathfrak{a}} (|\nabla_g U|^2 + \mathcal{J}_{\bar{g}} U^2) \mathop{\mathrm{dvol}}_{\bar{g}}.\label{eq-I}\end{aligned}$$ Define the *generalized Almgren's frequency* of $U$ by $$\begin{aligned} \mathcal{N}_U(x_+, r) = \frac{rI_U(x_+, r)}{H_U(x_+, r)},\end{aligned}$$ if $H_U(x_+,r) \neq 0$. For the convenience of our later computations, we also define the weighted Dirichlet energy $$\begin{aligned} D_U(x_+, r) & \equiv \int_{\mathcal{B}_r(x_+)} \varrho^{\mathfrak{a}} |\nabla_g U|^2 \mathop{\mathrm{dvol}}_{\bar{g}}.\end{aligned}$$ For our later computations, we introduce some notations on the geodesic polar coordinates. Fix a point $x_+\in M^n$ and let $r_0 \in(0, \mathop{\mathrm{Inj}}_h(x_+)/10)$ be a small number. Consider the geodesic polar coordinate system $(r,\Theta)$ on $\mathcal{B}_{r_0}(x_+)$ defined by the exponential map at $x_+$, $$\begin{aligned} \exp_{x_+}: B_{r_0}(0^{n + 1}) \to \mathcal{B}_{r_0}(x_+),\end{aligned}$$ where $B_{r_0}(0^{n + 1})\subset \mathbb{R}^{n+1}$. As $M^n\subset (\overline{X^{\mathfrak{m}}}, \bar{g})$ is totally geodesic, one can also choose normal coordinates $\{x^1, \cdots, x^{n}, x^{n+1}\}$ at $x_+$ such that $$\begin{aligned} \label{eq-norm-n+1} \{x^{n+1}=0\}\subseteq M, \quad r^2 = (x^1)^2+\cdots + (x^{n+1})^2.\end{aligned}$$ We can define $\frac{\partial}{\partial r}$ using the polar coordinates. We have $$\begin{aligned} \frac{\partial}{\partial r} =\frac{x^i}{r}\frac{\partial }{\partial x^i} = \nabla_{\bar g}r.\end{aligned}$$ Under polar coordinates, $$\begin{aligned} \bar g = dr^2 + r^2 b_{ij}(r,\Theta) d\theta^i d\theta^j,\end{aligned}$$ where $b_{ij} =\delta_{ij}$ at $x_+.$ **Lemma 66**. *Let $\varrho$ be the function defined in [\[e:vr-expansion\]](#e:vr-expansion){reference-type="eqref" reference="e:vr-expansion"}, and let $r$ be the geodesic radial coordinate centered at $x_+ \in M^n$. Then there exists a small positive constant $r_0 = r_0(\mathfrak{m}, \iota_0) > 0$ such that the following holds, for $r\in (0, r_0)$, $$\begin{aligned} \label{eq-rho-a-r} \frac{\partial \varrho^{\mathfrak{a}}}{\partial r} = \frac{\mathfrak{a}\varrho^{\mathfrak{a}}}{r}(1 + O(r)).\end{aligned}$$ In particular, for $r\in (0, r_0)$, $$\begin{aligned} \frac{\partial \varrho^{\mathfrak{a}}}{\partial r} \geq 0.\end{aligned}$$* *Proof.* First, we need to estimate $\frac{\partial y}{\partial r}$. For $\alpha = 1,\cdots, n$, we have $\frac{\partial y}{\partial x^\alpha} =0$ on $\{y=0\}$, which implies $$\begin{aligned} \frac{\partial y}{\partial x^\alpha} = O(y).\end{aligned}$$ Then, we have $$\begin{aligned} \frac{\partial y}{\partial r} = \frac{x^{n+1}}{r} \frac{\partial y}{\partial x^{n+1}} + \frac{x^{\alpha}}{r} \frac{\partial y}{\partial x^{\alpha}} = \frac{x^{n+1}}{r} \frac{\partial y}{\partial x^{n+1}} +O(y).\end{aligned}$$ Notice that $dy = dx^{n+1}$ at $x_+$. By Proposition [Proposition 9](#p:general-regularity-FG){reference-type="ref" reference="p:general-regularity-FG"}, $\exp_{x_+}$ is a local diffeomorphism, which gives $$\begin{aligned} dy = (1+O(r))dx^{n+1} + \sum_{\alpha =1}^n O_\alpha(y) dx^\alpha,\end{aligned}$$ which implies $$\begin{aligned} \label{eq-dydxn+1} \frac{\partial y}{\partial x^{n+1}} = dy \left(\frac{\partial}{\partial x^{n+1}}\right) = 1 + O(r).\end{aligned}$$ So we obtain $$\begin{aligned} \label{eq-DyDr} \frac{\partial y}{\partial r} = \frac{x^{n+1}}{r} +O(x^{n+1}) = \frac{y}{r} +O(y).\end{aligned}$$ Here we used the fact that $\frac{x^{n+1}}{y}=1+O(r)$ near $x_+$. Now we compute $\frac{\partial \varrho^{\mathfrak{a}}}{\partial r}$. For $\varrho=\varrho$, applying [\[e:vr-expansion\]](#e:vr-expansion){reference-type="eqref" reference="e:vr-expansion"}, [\[eq-dydxn+1\]](#eq-dydxn+1){reference-type="eqref" reference="eq-dydxn+1"} and [\[eq-DyDr\]](#eq-DyDr){reference-type="eqref" reference="eq-DyDr"}, we have $$\begin{aligned} \varrho&= y + O(y^{3}),\label{eq-rho-y-relation}\\ \varrho_{, n+1} &= 1 + O(r),\label{eq-rho-y-1}\\ \varrho_{,\alpha} &= O(y),\label{eq-rho-y-2}\end{aligned}$$ where $\varrho_{,\alpha}$ denotes the derivative of $\varrho$ with respect to $x^\alpha$ for $\alpha = 1,\cdots, n$. Thus, we find $$\begin{aligned} \label{eq-rho-a-r-1} \frac{\partial \varrho^{\mathfrak{a}}}{\partial r} = \mathfrak{a}\varrho^{\mathfrak{a}-1} \left(\varrho_{, n+1} (x^{n+1})_{,r} + \varrho_{, \alpha} (x^{\alpha})_{,r}\right) = \mathfrak{a}\varrho^{\mathfrak{a}-1} \left(\frac{y}{r} + O(y)\right) = \frac{\mathfrak{a}\varrho^{\mathfrak{a}}}{r}(1 + O(r)).\end{aligned}$$ Then, combining [\[eq-rho-y-relation\]](#eq-rho-y-relation){reference-type="eqref" reference="eq-rho-y-relation"} and [\[eq-rho-a-r-1\]](#eq-rho-a-r-1){reference-type="eqref" reference="eq-rho-a-r-1"}, we conclude the lemma. ◻ To establish the almost monotonicity formula for the generalized frequency $\mathcal{N}_U(x_+, r)$, one needs the following radial dilation in a given geodesic polar coordinate system $(\mathcal{O}_{x_+}, (s,\Theta))$ around a point $x_+\in M^n$, where $s \in (0, r_0)$ for some constant $r_0 = r_0(\iota_0, \mathfrak{m}) > 0$. For any sufficiently small $t \in (0, r_0)$, we define the quantitative tangent map (radial blow-up function) $U_{x_+, t} = \mathcal{T}_{x_+, t}(U) : \mathcal{B}_1(x_+, \bar{g}_t) \to \mathbb{R}$ as follows $$\begin{aligned} U_{x_+, t}(x) = U_{x_+, t}(s,\Theta) \equiv \frac{\displaystyle{ U(ts, \Theta) } }{\displaystyle{\left(\frac{1}{t^{\mathfrak{a}}}\fint_{\partial\mathcal{B}_1(x_+, \bar{g}_t)}\varrho^{\mathfrak{a}} U^2 d\sigma_t\right)^{\frac{1}{2}}}},\label{e:blow-up-function-U}\end{aligned}$$ where $\bar{g}_t(s,\Theta) = \bar{g}(ts, \Theta)$ and $d\sigma_t\equiv d\sigma_{\bar{g}_t}$. The following lemma will be used in proving the almost monotonicity formula, and the proof of the lemma follows from straightforward computations. **Lemma 67**. *For any given $t\in(0,1)$, let $U_{x_+, t}$ be the quantitative tangent map of $U$ at $x_+$ defined by [\[e:blow-up-function-U\]](#e:blow-up-function-U){reference-type="eqref" reference="e:blow-up-function-U"}. We define $$\begin{aligned} & D(U_{x_+, t}, x_+, r, \bar{g}_t) \equiv \int_{\mathcal{B}_r^t(x_+)} \varrho^{\mathfrak{a}} |\nabla_{\bar{g}_t} U_{x_+, t}|^2 \mathop{\mathrm{dvol}}_{\bar{g}_t}, \\ & I(U_{x_+, t}, x_+, r, \bar{g}_t) \equiv \int_{\mathcal{B}_r^t(x_+)} \varrho^{\mathfrak{a}}\left( |\nabla_{\bar{g}_t} U_{x_+, t}|^2 + t^2 \cdot \mathcal{J}_{\bar{g}} U_{x_+, t}^2 \right) \mathop{\mathrm{dvol}}_{\bar{g}_t}, \\ & H(U_{x_+, t}, x_+, r, \bar{g}_t) \equiv \int_{\partial\mathcal{B}_r^t(x_+)} \varrho^{\mathfrak{a}} U_{x_+, t}^2 d\sigma_t, \\ & \mathcal{N}(U_{x_+, t}, x_+, r, \bar{g}_t) \equiv \frac{r I(U_{x_+, t}, x_+, r, \bar{g}_t)}{H(U_{x_+, t}, x_+, r, \bar{g}_t)}, \end{aligned}$$ where $\mathcal{B}_r^t(x_+) \equiv \mathcal{B}_r(x_+, \bar{g}_t)$. Then the following holds: $$\begin{aligned} \begin{split} & D(U_{x_+, t}, x_+, t^{-1} r, \bar{g}_t) = t^{2 - \mathfrak{m}} \omega^{-2} D(U, x_+, r, \bar{g}) = t^{2 - \mathfrak{m}} \omega^{-2} D_U(x_+, r), \\ & I(U_{x_+, t}, x_+, t^{-1} r, \bar{g}_t) = t^{2 - \mathfrak{m}} \omega^{-2} I(U, x_+, r, \bar{g}) = t^{2 - \mathfrak{m}} \omega^{-2} I_U(x_+, r), \\ & H(U_{x_+, t}, x_+, t^{-1} r, \bar{g}_t) = t^{1 - \mathfrak{m}}\omega^{-2} H(U, x_+, r, \bar{g}) = t^{1 - \mathfrak{m}} \omega^{-2} H_U(x_+, r), \\ & \mathcal{N}(U_{x_+, t}, x_+, t^{-1}r, \bar{g}_t) = \mathcal{N}(U, x_+, r, \bar{g}) = \mathcal{N}_U(x_+, r), \end{split} \end{aligned}$$ where $\omega \equiv \left(\frac{1}{t^{\mathfrak{a}}}\fint_{\partial\mathcal{B}_1^t(x_+)}\varrho^{\mathfrak{a}} U^2 d\sigma_t\right)^{\frac{1}{2}}$ and $\bar{g}_t(s,\Theta) = \bar{g}(t\cdot s, \Theta)$.* **Remark 68**. In the above notation, actually it holds that $\mathcal{J}_{\bar{g}_t} = t^2 \mathcal{J}_{\bar{g}}$. The main result in this subsection is the following almost monotonicity theorem for generalized Almgren's frequency. **Theorem 69** (Almost monotonicity). *Let $U$ be an even solution of the boundary value problem [\[e:CS-extension-on-PE\]](#e:CS-extension-on-PE){reference-type="eqref" reference="e:CS-extension-on-PE"}. There exist $C=C(\mathfrak{m}, \mathfrak{a}, \bar{g}) >0$, $r_0 = r_0(\mathfrak{m}, \mathfrak{a}, \bar{g}) >0$, $\epsilon_0 = \epsilon_0(\mathfrak{m}, \mathfrak{a}, \bar{g}) > 0$, and $\tau_0 = \tau_0 (\mathfrak{m}, \mathfrak{a}, \bar{g}) > 0$ such that either one of the following holds for any $x_+\in M^n$:* 1. *$|U_{x_+, t}| \geq \tau_0 > 0$ on $\mathcal{B}_{1/2}(x_+, \bar{g}_t)$ for some $t\in(0, r_0]$;* 2. *$\mathcal{N}_U(x_+,t) > \epsilon_0$ for any $t\in(0,r_0]$, and $e^{C t}\mathcal{N}_U(x_+, t)$ is non-decreasing in $t \in (0, r_0]$.* **Remark 70**. By Proposition [Proposition 74](#p:eps-non-vanishing){reference-type="ref" reference="p:eps-non-vanishing"}, one can find that item (1) happens when $\mathcal{N}_U(x_+,t) \leq \epsilon_0$ for any $t\in(0,r_0]$. In this case, $x_+$ is a good point and stays away from any quantitative singular stratum. In the proof, we will only consider the case of item (2). To prove this theorem, we need a series of preliminary results. **Lemma 71**. *Let $r\in (0,2)$ and $x_+\in M^n$ such that $\mathcal{B}_r(x_+) \subset \mathfrak{X}^{\mathfrak{m}}$ has a smooth boundary. There exists a constant $C>0$ such that for all $u\in H^{1,\mathfrak{a}}(\mathcal{B}_r(x_+))$, $$\begin{aligned} \label{eq-ET} \int_{\mathcal{B}_r(x_+)} \varrho^{\mathfrak{a}} u^2 \mathop{\mathrm{dvol}}_{\bar g} \leq C\left(r^2\int_{\mathcal{B}_r(x_+)} \varrho^\mathfrak{a}|\nabla_{\bar g} u|^2 \mathop{\mathrm{dvol}}_{\bar g} +\ r \int_{\partial \mathcal{B}_r(x_+)} \varrho^\mathfrak{a}U^2 d\sigma_{\bar{g}}\right). \end{aligned}$$* This inequality is standard in the literature; see [@Hebey Chapter 10] for the unweighted version on a compact manifold with boundary. *Proof.* The inequality is scale invariant so that one can assume $r = 1$. We first prove that there exists some constant $\Lambda_0 >0$ such that for all $U \in H^{1,\mathfrak{a}}_0 (\mathcal{B}_1(x_+))$, $$\begin{aligned} \label{eq-Poincare}\int_{\mathcal{B}_1(x_+)} \varrho^{\mathfrak{a}} u^2 \mathop{\mathrm{dvol}}_{\bar g} \leq \Lambda_0 \int_{\mathcal{B}_1(x_+)} \varrho^\mathfrak{a}|\nabla_{\bar g} u|^2 \mathop{\mathrm{dvol}}_{\bar g} . \end{aligned}$$ If it does not hold, then for any $i\in \mathbb{N}$, there is $u_i \in H_0^{1,\mathfrak{a}}(\mathcal{B}_1(x_+))$ such that $$\begin{aligned} \int_{\mathcal{B}_1(x_+)}\varrho^{\mathfrak{a}} |\nabla_{\bar g} u_i|^2 \mathop{\mathrm{dvol}}_{\bar g} < i^{-1}, \,\, \int_{\mathcal{B}_1(x_+)}\varrho^{\mathfrak{a}} u_i^2 \mathop{\mathrm{dvol}}_{\bar g} =1.\end{aligned}$$ Then $u_i \rightarrow u_{\infty}$ weakly in $H^{1, \mathfrak{a}}_0(\mathcal{B}_1(x_+))$. By the compact embedding $H^{1, \mathfrak{a}}_0(\mathcal{B}_1(x_+)) \hookrightarrow L^{2,\mathfrak{a}}(\mathcal{B}_1(x_+))$, we have $$\begin{aligned} \int_{\mathcal{B}_1(x_+)}\varrho^{\mathfrak{a}} |\nabla_{\bar g} u_{\infty}|^2 \mathop{\mathrm{dvol}}_{\bar g} =0, \,\, \int_{\mathcal{B}_1(x_+)}\varrho^{\mathfrak{a}} u_{\infty}^2 \mathop{\mathrm{dvol}}_{\bar g} =1.\end{aligned}$$ Then $u_{\infty}$ must be a nonzero constant, which contradicts $u_{\infty} \in H^{1,\mathfrak{a}}_0(\mathcal{B}_1(x_+))$. Next, we prove [\[eq-ET\]](#eq-ET){reference-type="eqref" reference="eq-ET"}. Suppose it is not true, i.e., there exists contradiction sequences $A_j \to \infty$ and $u_j\in H^{1,\mathfrak{a}}(\mathcal{B}_1(x_+))$ such that $$\begin{aligned} \int_{\mathcal{B}_1(x_+)} \varrho^{\mathfrak{a}} u_j^2 \mathop{\mathrm{dvol}}_{\bar g} > A_j \left(\int_{\mathcal{B}_1(x_+)} \varrho^\mathfrak{a}|\nabla_{\bar g} u_j|^2 \mathop{\mathrm{dvol}}_{\bar g} + \int_{\partial \mathcal{B}_1(x_+)} \varrho^\mathfrak{a}u_j^2 d\sigma_{\bar{g}}\right).\end{aligned}$$ Without loss of generality, we assume that $$\begin{aligned} \label{eq-Ua} \int_{\partial \mathcal{B}_1(x_+)} \varrho^\mathfrak{a}u_j^2 d\sigma_{\bar{g}}+ \int_{\mathcal{B}_1(x_+)} \varrho^{\mathfrak{a}} u_j^2 \mathop{\mathrm{dvol}}_{\bar g}=1.\end{aligned}$$ Then $$\begin{aligned} \label{eq-DUa} \int_{\mathcal{B}_1(x_+)} \varrho^\mathfrak{a}|\nabla_{\bar g} u_j|^2 \mathop{\mathrm{dvol}}_{\bar g} < A_j^{-1}, \end{aligned}$$ which implies that $\|u_j\|_{H^{1,\mathfrak{a}}(\mathcal{B}_1(x_+))}$ is uniformly bounded. Now letting $j\rightarrow \infty$, $u_j \rightarrow u_{\infty}$ weakly in $H^{1,\mathfrak{a}}(\mathcal{B}_1(x_+))$ and strongly in $L^{2, \mathfrak{a}}(\mathcal{B}_1(x_+))$. By [\[eq-Ua\]](#eq-Ua){reference-type="eqref" reference="eq-Ua"} and [\[eq-DUa\]](#eq-DUa){reference-type="eqref" reference="eq-DUa"}, $$\begin{aligned} \label{eq-Ubar} \int_{\mathcal{B}_1(x_+)}\varrho^{\mathfrak{a}} u_{\infty}^2\mathop{\mathrm{dvol}}_{\bar g}=1, \,\, \int_{\mathcal{B}_1(x_+)} \varrho^\mathfrak{a}|\nabla_{\bar g} u_{\infty}|^2 \mathop{\mathrm{dvol}}_{\bar g} = 0. \end{aligned}$$ In addition, $\int_{\partial \mathcal{B}_1(x_+)} \varrho^\mathfrak{a}u_j^2 \mathop{\mathrm{dvol}}_{\bar g} \rightarrow 0$, which implies that $u_{\infty} \in H^{1,\mathfrak{a}}_0(\mathcal{B}_1(x_+))$. Then [\[eq-Ubar\]](#eq-Ubar){reference-type="eqref" reference="eq-Ubar"} contradicts [\[eq-Poincare\]](#eq-Poincare){reference-type="eqref" reference="eq-Poincare"} which completes the proof of the proposition. ◻ The above inequality has a quick corollary on the non-vanishing of $H_U(x_+, r)$ which implies that the generalized frequency $\mathcal{N}_U(x_+, r)$ is well-defined for a non-trivial solution of [\[e:CS-extension-on-PE\]](#e:CS-extension-on-PE){reference-type="eqref" reference="e:CS-extension-on-PE"}. **Corollary 72**. *Let $U \in H^{1, \mathfrak{a}}(\mathcal{B}_1(x_+))$ be a non-trivial solution of [\[e:CS-extension-on-PE\]](#e:CS-extension-on-PE){reference-type="eqref" reference="e:CS-extension-on-PE"}. There exists $r_0 = r_0 (\mathfrak{m}, \mathfrak{a}, \bar{g}) > 0$ such that $H_U(x_+, r) \neq 0$ for any $r\in (0, r_0)$.* *Proof.* Assume that $H_U(x_+, r) =0$ for some $r \in (0, r_0)$, so $U \equiv 0$ on $\partial\mathcal{B}_r(x_+)$. By the divergence theorem, $I_U(r,x_+) =0$. So it follows that $$\begin{aligned} D_U(x_+,r) \leq C_0(\mathfrak{m}, \mathfrak{a}, \bar{g}) \int_{\mathcal{B}_r(x_+)}\varrho^{\mathfrak{a}} U^2 \mathop{\mathrm{dvol}}_{g}\leq C_0'(\mathfrak{m}, \mathfrak{a}, \bar{g}) \left(r^2 D_U(x_+,r)+ r H_U(x_+,r)\right), \end{aligned}$$ where we used Lemma [Lemma 71](#l:L2-estimate-large-scale){reference-type="ref" reference="l:L2-estimate-large-scale"} for the last inequality. If $r$ is chosen such that $r^2 < \frac{1}{2C_0'(\mathfrak{m},\mathfrak{a}, \bar{g})}$, then $D_U(x_+, r) \equiv 0$. Therefore, $U \equiv 0$ on $\mathcal{B}_r(x_+)$. Applying standard unique continuation, we obtain a contradiction to the nontriviality assumption on $U$. ◻ **Lemma 73**. *For any $\Lambda > 0$, there exist $r_0 = r_0 (\Lambda, \mathfrak{m}, \mathfrak{a}, \bar{g}) > 0$ and $D_0 = D_0(\Lambda, \mathfrak{m}, \mathfrak{a}, \bar{g}) > 0$ such that if a solution $U$ of [\[e:CS-extension-on-PE\]](#e:CS-extension-on-PE){reference-type="eqref" reference="e:CS-extension-on-PE"} satisfies $\mathcal{N}_U(x,2r)\leq \Lambda$ for some $x\in M^n$ and $r\in (0,r_0/10)$, then the following holds:* 1. *$\mathcal{N}_U(x, s) \leq 2\Lambda$ for any $s \in (r/2,2r)$;* 2. *$H_U(x, 2r) \leq D_0 \cdot H_U(x, r)$.* *Proof.* We will prove item (1) by contradiction. Suppose there does not exist such a constant $r_0 > 0$. That is, one can find a sequence of solutions $U_j$ of [\[e:CS-extension-on-PE\]](#e:CS-extension-on-PE){reference-type="eqref" reference="e:CS-extension-on-PE"} and sequences of numbers $r_j \to 0$ and $s_j \in (r_j/2, 2r_j)$ such that $$\begin{aligned} \mathcal{N}_U(x, 2r_j) \leq \Lambda \quad \text{and} \quad \mathcal{N}_U(x, s_j) > 2\Lambda.\end{aligned}$$ Let us take rescaled sequences $\widetilde{U}_j(s,\Theta)\equiv U_j(r_j s, \Theta)$ and $g_j(s,\Theta) \equiv g(r_js,\Theta)$ for any sufficiently large $j$. Then by scale invariance, $$\begin{aligned} \mathcal{N}(\widetilde{U}_j,x, 2, g_j) \leq \Lambda \quad \text{and} \quad \mathcal{N}(\widetilde{U}_j,x, s_j/r_j, g_j) > 2\Lambda.\end{aligned}$$ We also assume that $U_j$ is normalized by $H(\widetilde{U}_j,x,2, g_j) = 1$. Letting $j\to \infty$, passing to a subsequence, we have $g_j$ converges to the Euclidean metric $g_0$ on $\mathbb{R}^{\mathfrak{m}}$, $\lim\limits_{j\to\infty} s_j/r_j = t_0 \in [1/2, 2]$ and $$\begin{aligned} \widetilde{U}_j \xrightarrow{H^{1,\mathfrak{a}}(\mathcal{B}_2(x))} \widetilde{U}_{\infty}\in H^{1,\mathfrak{a}}(\mathcal{B}_2(0^{\mathfrak{m}})) \end{aligned}$$ such that $$\begin{aligned} \mathcal{N}(\widetilde{U}_{\infty}, 0^{\mathfrak{m}}, 2, g_0) \leq \Lambda \quad \text{and} \quad \mathcal{N}(\widetilde{U}_{\infty}, 0^{\mathfrak{m}},t_0, g_0) > 2\Lambda.\end{aligned}$$ But this contradicts to Lemma [Lemma 39](#l:Euclidean-monotonicity){reference-type="ref" reference="l:Euclidean-monotonicity"}, which completes the proof of item (1). Now we are ready to prove the doubling property in item (2). In geodesic polar coordinates, we rewrite $H_U(x, s)$ as $$\begin{aligned} H_U(x, s) = s^{\mathfrak{m}-1} \int_{\partial \mathcal{B}_1}\varrho^{\mathfrak{a}}(s,\Theta) U^2 (s, \Theta) \sqrt{b(s, \Theta)} d\Theta.\end{aligned}$$ For fixed center $x$, let us denote $H_U(s) \equiv H_U(s,x)$. Applying [\[eq-rho-a-r\]](#eq-rho-a-r){reference-type="eqref" reference="eq-rho-a-r"}, we obtain $$\begin{aligned} H_U'(s) = \left(\frac{\mathfrak{m}- 1 + \mathfrak{a}}{s}+O(1)\right)H_U(s) +\int_{\partial \mathcal{B}_s} \varrho^{\mathfrak{a}}(\partial_n \log\sqrt{b})U^2 d\sigma_{\bar{g}}+ 2\int_{\partial \mathcal{B}_s} \varrho^{\mathfrak{a}} U (\partial_n U) d\sigma_{\bar{g}},\end{aligned}$$ which implies $$\begin{aligned} \label{eq-H'} \frac{H_U'(s)}{H_U(s)} = \frac{\mathfrak{m}- 1 + \mathfrak{a}}{s}+O(1) + 2\ \cdot \frac{\displaystyle{\int_{\partial \mathcal{B}_s} \varrho^{\mathfrak{a}} U (\partial_n U) d\sigma_{\bar{g}}}}{\displaystyle{\int_{\partial \mathcal{B}_r} \varrho^{\mathfrak{a}} U^2 d\sigma_{\bar{g}}}} = \frac{\mathfrak{m}- 1 + \mathfrak{a}}{s}+O(1) + \frac{2\mathcal{N}_U(s)}{s}.\end{aligned}$$ Here $O(1)$ denotes a function of $s$ and $\Theta$, which is bounded in absolute value by a constant $C$. Therefore, $$\begin{aligned} \frac{d}{ds} \left( \log\frac{H_U(s)}{s^{\mathfrak{m}- 1 + \mathfrak{a}}}\right) = O(1) + \frac{2\mathcal{N}_U(s)}{s}.\label{e:derivative-normalized-log-H}\end{aligned}$$ By item (1), $\mathcal{N}_U(s) \leq 2\Lambda$ for any $s\in [r,2r]$. Then integrating [\[e:derivative-normalized-log-H\]](#e:derivative-normalized-log-H){reference-type="eqref" reference="e:derivative-normalized-log-H"}, we have $$\begin{aligned} H_U(2r) \leq D_0\cdot H_U(r)\end{aligned}$$ for some constant $D_0 = D_0(\Lambda, \mathfrak{m}, \mathfrak{a}, \bar{g})$, which completes the proof of item (2). ◻ Next, we prove an $\epsilon$-regularity result for small frequency. **Proposition 74**. *Let $U$ be a non-trivial even solution of [\[e:CS-extension-on-PE\]](#e:CS-extension-on-PE){reference-type="eqref" reference="e:CS-extension-on-PE"}. For every $\eta > 0$, there exist small constants $r_0 = r_0(\eta, \mathfrak{m}, \mathfrak{a}, \bar{g}) >0$, $\epsilon_0 = \epsilon_0(\eta, \mathfrak{m}, \mathfrak{a}, \bar{g})> 0$, and $\tau_0 = \tau_0(\eta, \mathfrak{m}, \mathfrak{a}, \bar{g}) >0$ such that if $$\begin{aligned} \mathcal{N}_U(x_+, r) \leq \epsilon_0 \quad \text{or} \quad \frac{r D_U(x_+, r)}{H_U(x_+, r)} \leq \epsilon_0\label{e:small-frequency}\end{aligned}$$ for some $x_+\in \mathcal{B}_{1/2}(p_+)$ and $r \in (0, r_0]$, then for any $0 \leq k \leq \mathfrak{m}- 1$, $$\begin{aligned} |U_{x_+, r} - \tau_0|_{C^k(\mathcal{B}_{1/2}(x_+, \bar{g}_r))} \leq \eta.\end{aligned}$$ In particular, $U$ is nowhere vanishing in $\mathcal{B}_{r/2}(x_+)$.* **Remark 75**. This proposition tells us that a point at which the frequency is sufficiently small must be away from any quantitative singular stratum. In other words, such points must be "good points\" which are not in our considerations. Therefore, in the following analysis, there is no harm to assume the definite lower bound $\mathcal{N}_U(x_+, r) > \epsilon_0$ for any of our interested point $x_+$ in a quantitative singular stratum. *Proof of Proposition [Proposition 74](#p:eps-non-vanishing){reference-type="ref" reference="p:eps-non-vanishing"}.* We will only give a proof in the first case of [\[e:small-frequency\]](#e:small-frequency){reference-type="eqref" reference="e:small-frequency"} since the proof in the other case is the same. We will prove the proposition by contradiction. Suppose no such constants $r_0$, $\epsilon_0$, and $\tau_0 > 0$ exist. That is, there exists a sequence of solutions $U_j$ of [\[e:CS-extension-on-PE\]](#e:CS-extension-on-PE){reference-type="eqref" reference="e:CS-extension-on-PE"} that serves as a contradiction sequence such that $$\begin{aligned} \mathcal{N}_{U_j}(x_j, r_j)\leq \epsilon_j\end{aligned}$$ for $r_j \to 0$ and $\epsilon_j \to 0$, but $\mathscr{U}_j\equiv U_{x_j, r_j}$ is not $C^k$-close to any constant for some $0\leq k\leq \mathfrak{m}- 2$. By scale invariance, we have $$\begin{aligned} \mathcal{N}_j(x_j, 1) \equiv \mathcal{N}_{\mathscr{U}_j}(x_j, 1, \bar{g}_j) \leq \epsilon_j,\end{aligned}$$ where $\bar{g}_j (s,\Theta) \equiv \bar{g}(r_j s, \Theta)$ in a local geodesic polar coordinate system $(s,\Theta)$ centered at $x_j$. Since the quantitative tangent map $\mathscr{U}_j$ satisfies the normalization $$\begin{aligned} \label{eq-H=1} H(\mathscr{U}_j, x_j, 1, \bar{g}_j) = 1 \ \text{and}\ D(\mathscr{U}_j, x_j, 1, \bar{g}_j) + r_j^2 \int_{\mathcal{B}_1(x_j, \bar{g}_j)}\mathcal{J}_{\bar{g}} \mathscr{U}_j^2\mathop{\mathrm{dvol}}_{\bar{g}_j}\leq \epsilon_j,\end{aligned}$$ applying Lemma [Lemma 71](#l:L2-estimate-large-scale){reference-type="ref" reference="l:L2-estimate-large-scale"}, we have $$\begin{aligned} D(\mathscr{U}_j, x_j, 1, \bar{g}_j) \leq \epsilon_j + Cr_j^2(D(\mathscr{U}_j, x_j, 1, \bar{g}_j) + H(\mathscr{U}_j, x_j, 1, \bar{g}_j)).\end{aligned}$$ Therefore, $D(\mathscr{U}_j, x_j, 1, \bar{g}_j) \leq C(r_j^2 + \epsilon_j)$. Applying Lemma [Lemma 71](#l:L2-estimate-large-scale){reference-type="ref" reference="l:L2-estimate-large-scale"}, we have $$\begin{aligned} \int_{\mathcal{B}_1(x_j, \bar{g}_j)}\varrho^{\mathfrak{a}}\mathscr{U}_j^2d\sigma_{\bar{g}_j}\leq C(\mathfrak{m}, \mathfrak{a}, \bar{g}).\end{aligned}$$ Letting $j\to \infty$, it holds that $\bar{g}_j$ converges to the Euclidean metric $g_0$, $x_j$ converges to some point in the Euclidean space, and $\mathscr{U}_j$ converges to $\mathscr{U}_{\infty}$ that satisfies $$\begin{aligned} \int_{\mathcal{B}_1(0^{\mathfrak{m}})}|y|^{\mathfrak{a}} |\nabla_{g_0} \mathscr{U}_{\infty}|^2 \mathop{\mathrm{dvol}}_{g_0} = 0.\end{aligned}$$ By the doubling property in Lemma [Lemma 73](#l:doubling-fixed-scale){reference-type="ref" reference="l:doubling-fixed-scale"}, there exists $D_0 = D_0(\mathfrak{m}, \mathfrak{a}, \bar{g}) > 0$ such that $$\begin{aligned} \int_{\partial\mathcal{B}_{1/2}(x_+, \bar{g}_j)}\varrho^\mathfrak{a}\mathscr{U}_j ^2 d\sigma_{\bar{g}_j}\geq D_0^{-1}.\end{aligned}$$ Then one can conclude that $\mathscr{U}_{\infty} \equiv \tau_0$ on $\mathcal{B}_1(0^{\mathfrak{m}})\subset \mathbb{R}^{\mathfrak{m}}$ for some uniform constant $\tau_0 \equiv \tau_0(\mathfrak{m}, \mathfrak{a}, \bar{g}) > 0$. So the desired contradiction arises by using elliptic regularity, which completes the proof. ◻ **Corollary 76**. *There exist $C=C(\mathfrak{m}, \mathfrak{a}, \bar{g}) >0$ and $r_0 = r_0(\mathfrak{m}, \mathfrak{a}, \bar{g}) >0$ such that if $U$ is not $(\mathfrak{m}, 10^{-6}, t, r_0)$-symmetric at $x_+$ for some $t\in (0,r_0)$, then $$\begin{aligned} (1 - Ct^2) \int_{\mathcal{B}_1^t(x_+)}\varrho^{\mathfrak{a}} |\nabla_{\bar{g}_t} \mathscr{U}|^2\mathop{\mathrm{dvol}}_{\bar{g}_t} \leq I(\mathscr{U},x_+, 1, \bar{g}_t) \leq (1 + Ct^2) \int_{\mathcal{B}_1^t(x_+)}\varrho^{\mathfrak{a}} |\nabla_{\bar{g}_t} \mathscr{U}|^2\mathop{\mathrm{dvol}}_{\bar{g}_t},\end{aligned}$$ where $\mathcal{B}_1^t(x_+)\equiv \mathcal{B}_1(x_+, \bar{g}_t)$ and $\mathscr{U}\equiv U_{t,x+}$.* *Proof.* It suffices to estimate the $0^{\text{th}}$-order term in $$\begin{aligned} I(\mathscr{U},x_+, 1, \bar{g}_t)\equiv \int_{\mathcal{B}_1^t(x_+)}\varrho^{\mathfrak{a}}(|\nabla_{\bar{g}_t}\mathscr{U}|^2 + t^2\mathcal{J}_{\bar{g}}\mathscr{U}^2)\mathop{\mathrm{dvol}}_{\bar{g}_t}.\end{aligned}$$ By Lemma [Lemma 71](#l:L2-estimate-large-scale){reference-type="ref" reference="l:L2-estimate-large-scale"}, $$\begin{aligned} \left|\int_{\mathcal{B}_1^t(x_+)} \varrho^{\mathfrak{a}}\mathcal{J}_{\bar{g}}\mathscr{U}^2\mathop{\mathrm{dvol}}_{\bar{g}_t}\right| \leq C(\mathfrak{m}, \mathfrak{a}, \bar{g}) \left(\int_{\mathcal{B}_1^t(x_+)}\varrho^{\mathfrak{a}}|\nabla_{\bar{g}_t}\mathscr{U}|^2\mathop{\mathrm{dvol}}_{\bar{g}_t} + \int_{\partial\mathcal{B}_1^t(x_+)}\varrho^{\mathfrak{a}} \mathscr{U}^2d\sigma_t\right).\end{aligned}$$ Since $U$ is not $(\mathfrak{m}, 10^{-6}, t, r_0)$-symmetric at $x_+$, applying Proposition [Proposition 74](#p:eps-non-vanishing){reference-type="ref" reference="p:eps-non-vanishing"}, we have $\mathcal{N}(\mathscr{U}, x_+, 1, \bar{g}_t) > \epsilon_0$ with $\epsilon_0(\mathfrak{m},\mathfrak{a}, \bar{g}) > 0$ the uniform constant in Proposition [Proposition 74](#p:eps-non-vanishing){reference-type="ref" reference="p:eps-non-vanishing"}, which gives $$\begin{aligned} \left|\int_{\mathcal{B}_1^t(x_+)} \varrho^{\mathfrak{a}}\mathcal{J}_{\bar{g}}\mathscr{U}^2\mathop{\mathrm{dvol}}_{\bar{g}_t}\right| \leq C(\mathfrak{m}, \mathfrak{a}, \bar{g}) (1 + \epsilon_0^{-1}) \int_{\mathcal{B}_1^t(x_+)}\varrho^{\mathfrak{a}}|\nabla_{\bar{g}_t}\mathscr{U}|^2\mathop{\mathrm{dvol}}_{\bar{g}_t}, \end{aligned}$$ which completes the proof. ◻ Now we are ready to complete the proof of Theorem [Theorem 69](#t:almost-monotonicity){reference-type="ref" reference="t:almost-monotonicity"}. *Proof of Theorem [Theorem 69](#t:almost-monotonicity){reference-type="ref" reference="t:almost-monotonicity"}.* Recalling Remark [Remark 70](#r:alternative){reference-type="ref" reference="r:alternative"} and Proposition [Proposition 74](#p:eps-non-vanishing){reference-type="ref" reference="p:eps-non-vanishing"}, we will only consider the case of item (2) of the Theorem [Theorem 69](#t:almost-monotonicity){reference-type="ref" reference="t:almost-monotonicity"}. To prove the theorem, we will first carry out some reduction by rescaling. Theorem [Theorem 69](#t:almost-monotonicity){reference-type="ref" reference="t:almost-monotonicity"} is equivalent to $$\begin{aligned} \frac{\mathcal{N}_U'(x_+, t)}{\mathcal{N}_U(x_+, t)} \geq - C, \ \quad \forall t\in(0,r_0), \label{e:derivative-of-log-N-bounded-below}\end{aligned}$$where $\mathcal{N}_U'(x_+, t)=\frac{d}{dt}\mathcal{N}_U(x_+, t)$. For fixed $t > 0$ and $x_+\in M^n$, let $U(x_+, r)$ be the blow-up function of $U$. Let us also denote $$\begin{aligned} \mathcal{N}_t(r) \equiv \mathcal{N}(U(x_+, r), x_+, r, \bar{g}_t).\end{aligned}$$ By Lemma [Lemma 67](#l:frequency-rescaling){reference-type="ref" reference="l:frequency-rescaling"}, the derivative bound [\[e:derivative-of-log-N-bounded-below\]](#e:derivative-of-log-N-bounded-below){reference-type="eqref" reference="e:derivative-of-log-N-bounded-below"} is equivalent to $$\begin{aligned} \frac{\mathcal{N}_t'(1)}{\mathcal{N}_t(1)} \geq - C t, \end{aligned}$$where $\mathcal{N}_t'(1) = \left.\frac{d}{dr}\right|_{r = 1}\mathcal{N}(U(x_+, r), x_+, r, \bar{g}_t)$. Throughout the proof, we will use the simplified notation $\mathscr{U}\equiv U(x_+, r)$. The first step is to estimate the term $H_t(1)$. $$\begin{aligned} H_t(r) = \int_{\partial\mathcal{B}_r^t(x_+)}\varrho^{\mathfrak{a}}\mathscr{U}^2 d\sigma_t = r^{\mathfrak{m}- 1}\int_{\partial\widetilde{\mathcal{B}}_1^t(x_+)}\varrho^{\mathfrak{a}}\mathscr{U}^2 d\sigma_{t,r},\end{aligned}$$ where $\mathcal{B}_r^t(x_+)\equiv \mathcal{B}_r(x_+, \bar{g}_t)$, $\widetilde{\mathcal{B}}_1^t(x_+) \equiv \mathcal{B}_1(x_+, r^{-2}\bar{g}_t)$, and $d\sigma_{t,r}\equiv d\sigma_{r^{-2}\bar{g}_t}$. So it follows that $$\begin{aligned} \begin{split} H_t'(1) = & \ (\mathfrak{m}- 1) H_t(1) + 2 \int_{\partial\widetilde{\mathcal{B}}_1^t(x_+)}\varrho^{\mathfrak{a}}\mathscr{U}\cdot (\partial_r\mathscr{U}) d\tilde{\sigma}_{t, r} + \int_{\partial\widetilde{\mathcal{B}}_1^t(x_+)} (\partial_r\varrho^{\mathfrak{a}}) \cdot \mathscr{U}^2 d\tilde{\sigma}_{t, r} \\ & \ + \int_{\partial\widetilde{\mathcal{B}}_1^t(x_+)}\varrho^{\mathfrak{a}}\mathscr{U}^2 \left(\partial_r\log\sqrt{\widetilde{G}_{t,r}}\right) d\tilde{\sigma}_{t, r}. \end{split}\end{aligned}$$ First we estimate the third and the last term in $H_t'(1)$. For the third term, applying Lemma [Lemma 66](#lem-rho-a){reference-type="ref" reference="lem-rho-a"}, we have that $$\begin{aligned} \int_{\partial\widetilde{\mathcal{B}}_1^t(x_+)} \frac{\partial\varrho^{\mathfrak{a}}}{\partial r}\mathscr{U}^2 d\tilde{\sigma}_{t, r} = (\mathfrak{a}+ O(t))\cdot H_t(1). \end{aligned}$$ For the last term, since $\sqrt{\widetilde{G}_{t,r}} = t^{\mathfrak{m}- 1}\sqrt{B(tr,\Theta)}$ with $B(tr,\Theta)\equiv\det(b_{ij}(tr,\Theta))$, taking the derivative, $$\begin{aligned} \left.\frac{\partial}{\partial r}\right|_{r = 1} \log \sqrt{\widetilde{G}_{t,r}} = \frac{t}{2}\cdot \left.\frac{\partial}{\partial r}\right|_{r = 1} \log B.\end{aligned}$$ Then there exists a constant $C > 0$ such that $$\begin{aligned} \left| \int_{\partial\widetilde{\mathcal{B}}_1^t(x_+)}\varrho^{\mathfrak{a}}\mathscr{U}^2 \left(\partial_r\log\sqrt{\widetilde{G}_{t,r}}\right) d\tilde{\sigma}_{t, r} \right| \leq C \cdot t \cdot H_t(1). \end{aligned}$$ Combining the above, $$\begin{aligned} \left|H_t'(1) - (\mathfrak{m}- 1 + \mathfrak{a}) H_t(1) -2 \int_{\partial\mathcal{B}_1^t(x_+)} \varrho^{\mathfrak{a}}\mathscr{U}\cdot \mathscr{U}_n d\sigma_t \right| \leq O(t) \cdot H_t(1),\end{aligned}$$ which implies $$\begin{aligned} \label{e:derivative-log-H} \left|\frac{H_t'(1)}{H_t(1)} - (\mathfrak{m}- 1 + \mathfrak{a}) -2 \frac{\displaystyle{\int_{\partial\mathcal{B}_1^t(x_+)} \varrho^{\mathfrak{a}}\mathscr{U}\cdot \mathscr{U}_n d\sigma_t}}{\displaystyle{ \int_{\partial\mathcal{B}_1^t(x_+)} \varrho^{\mathfrak{a}}\mathscr{U}^2 d\sigma_t }} \right| \leq O(t). \end{aligned}$$ In the next step, we estimate the derivative of $I_t$. Let us split the integral $$\begin{aligned} \label{e:derivative-I_t} I_t'(r) = \int_{\partial\mathcal{B}_r^t}\varrho^{\mathfrak{a}}|\nabla_{\bar{g}_t}\mathscr{U}|^2 d\sigma_t+ t^2 \int_{\partial\mathcal{B}_r^t} \varrho^{\mathfrak{a}} \cdot \mathcal{J} \cdot\mathscr{U}^2 d\sigma_t\equiv \mathcal{I}_D + \mathcal{I}_Q.\end{aligned}$$ We pick a normal vector field $X \equiv s\nabla_{\bar{g}_t} s$, where $s(x) \equiv d_{\bar{g}_t}(x,x_+)$ is the distance to $x_+$ in terms of $\bar{g}_t$. Then we have $$\begin{aligned} \begin{split} \mathcal{I}_D(r) = &\ \frac{1}{r}\int_{\partial\mathcal{B}_r^t}\varrho^{\mathfrak{a}}|\nabla_{\bar{g}_t}\mathscr{U}|^2 \langle X, r^{-1}X\rangle d\sigma_t \nonumber\\ = & \ \frac{1}{r}\int_{\mathcal{B}_r^t}\mathop{\mathrm{div}}\left(\varrho^{\mathfrak{a}}|\nabla_{\bar{g}_t}\mathscr{U}|^2 X \right)\mathop{\mathrm{dvol}}_{\bar{g}_t} \nonumber\\ = & \ \frac{1}{r}\int_{\mathcal{B}_r^t}\left(\varrho^{\mathfrak{a}}|\nabla_{\bar{g}_t}\mathscr{U}|^2 \mathop{\mathrm{div}}(X) \right) \mathop{\mathrm{dvol}}_{\bar{g}_t} + \frac{1}{r}\int_{\mathcal{B}_r^t}\langle\nabla_{\bar{g}_t}\varrho^{\mathfrak{a}}, X\rangle|\nabla_{\bar{g}_t}\mathscr{U}|^2\mathop{\mathrm{dvol}}_{\bar{g}_t} \nonumber\\ & \ + \frac{1}{r} \int_{\mathcal{B}_r^t}\varrho^{\mathfrak{a}} \langle\nabla_{\bar{g}_t}|\nabla_{\bar{g}_t}\mathscr{U}|^2, X\rangle \mathop{\mathrm{dvol}}_{\bar{g}_t}. \end{split}\end{aligned}$$ By Lemma [Lemma 66](#lem-rho-a){reference-type="ref" reference="lem-rho-a"}, the second term in $\mathcal{I}_D(r)$ yields the bound $$\begin{aligned} \begin{split} \frac{1}{r}\int_{\mathcal{B}_r^t}\langle\nabla_{\bar{g}_t}\varrho^{\mathfrak{a}}, X \rangle|\nabla_{\bar{g}_t}\mathscr{U}|^2\mathop{\mathrm{dvol}}_{\bar{g}_t} = & \ \int_{\mathcal{B}_r^t} (\partial_r\varrho^{\mathfrak{a}})|\nabla_{\bar{g}_t}\mathscr{U}|^2 \mathop{\mathrm{dvol}}_{\bar{g}_t} \nonumber\\ = & \ \left(\frac{\mathfrak{a}}{r} + O(rt)\right)\int_{\mathcal{B}_r^t} \varrho^{\mathfrak{a}}|\nabla_{\bar{g}_t}\mathscr{U}|^2 \mathop{\mathrm{dvol}}_{\bar{g}_t}.\end{split}\end{aligned}$$ Now let us analyze the last term. By local computations, $$\begin{aligned} \begin{split} \langle\nabla_{\bar{g}_t}\langle\nabla_{\bar{g}_t} \mathscr{U}, X\rangle,\nabla_{\bar{g}_t} \mathscr{U}\rangle = &\ (\nabla_{\bar{g}_t} \mathscr{U})\langle\nabla_{\bar{g}_t} \mathscr{U}, X\rangle \\ = & \ \frac{1}{2}\langle\nabla_{\bar{g}_t} |\nabla_{\bar{g}_t} \mathscr{U}|^2, X\rangle + \langle \nabla_{\bar{g}_t} \mathscr{U}, (\nabla_{\bar{g}_t})_{\nabla_{\bar{g}_t} \mathscr{U}}X\rangle \\ = &\ \frac{1}{2}\langle\nabla_{\bar{g}_t}|\nabla_{\bar{g}_t} \mathscr{U}|^2, X\rangle + (\mathscr{U}_r)^2 + r(\nabla_{\bar{g}_t}^2 r)(\nabla_{\bar{g}_t} \mathscr{U},\nabla_{\bar{g}_t} \mathscr{U}) \\ = &\ \frac{1}{2}\langle\nabla_{\bar{g}_t}|\nabla_{\bar{g}_t} \mathscr{U}|^2, X\rangle + (1 + O(rt)) |\nabla_{\bar{g}_t} \mathscr{U}|^2, \end{split}\end{aligned}$$ where we used the local estimate $$\begin{aligned} |(X^i)_j - \delta_{ij}| \leq Crt. \end{aligned}$$ Therefore, $$\begin{aligned} \label{e:crossing} \begin{split} \frac{1}{r} \int_{\mathcal{B}_r^t}\varrho^{\mathfrak{a}} \langle\nabla_{\bar{g}_t}|\nabla_{\bar{g}_t} \mathscr{U}|^2, X\rangle \mathop{\mathrm{dvol}}_{\bar{g}_t} = \frac{2}{r} \int_{\mathcal{B}_r^t}\varrho^{\mathfrak{a}}\left(\langle\nabla_{\bar{g}_t}\langle\nabla_{\bar{g}_t} \mathscr{U}, X\rangle,\nabla_{\bar{g}_t} \mathscr{U}\rangle- (1 + O(rt)) |\nabla_{\bar{g}_t} \mathscr{U}|^2\right)\mathop{\mathrm{dvol}}_{\bar{g}_t}. \end{split} \end{aligned}$$ Since $$\begin{aligned} \mathop{\mathrm{div}}(\varrho^{\mathfrak{a}} \langle\nabla_{\bar{g}_t} \mathscr{U}, X\rangle \nabla_{\bar{g}_t} \mathscr{U}) = \varrho^{\mathfrak{a}} \langle\nabla_{\bar{g}_t}\langle\nabla_{\bar{g}_t} \mathscr{U}, X\rangle,\nabla_{\bar{g}_t} \mathscr{U}\rangle + t^2\cdot \varrho^{\mathfrak{a}} \cdot \mathcal{J}\mathscr{U}\langle\nabla_{\bar{g}_t} \mathscr{U}, X\rangle.\end{aligned}$$ Applying divergence theorem to the first term in [\[e:crossing\]](#e:crossing){reference-type="eqref" reference="e:crossing"}, we have $$\begin{aligned} \frac{2}{r} \int_{\mathcal{B}_r^t}\varrho^{\mathfrak{a}} \langle\nabla_{\bar{g}_t}\langle\nabla_{\bar{g}_t} \mathscr{U}, X\rangle,\nabla_{\bar{g}_t} \mathscr{U}\rangle \mathop{\mathrm{dvol}}_{\bar{g}_t} = \frac{2}{r}\int_{\partial\mathcal{B}_r^t} \varrho^{\mathfrak{a}} \mathscr{U}_n^2 d\sigma_t- \frac{2\cdot t^2}{r}\int_{\mathcal{B}_r^t}\varrho^{\mathfrak{a}} \cdot \mathcal{J}\mathscr{U}\langle\nabla_{\bar{g}_t} \mathscr{U}, X\rangle \mathop{\mathrm{dvol}}_{\bar{g}_t},\end{aligned}$$ where $\mathscr{U}_n \equiv \partial\mathscr{U}/\partial n$ and $\partial/\partial n$ is the outward unit normal. Combining the above computations, $$\begin{aligned} \left|\mathcal{I}_D(1) - (\mathfrak{m}- 2 + \mathfrak{a}) \int_{\mathcal{B}_1^t}\varrho^{\mathfrak{a}}|\nabla_{\bar{g}_t} \mathscr{U}|^2 \mathop{\mathrm{dvol}}_{\bar{g}_t} - 2\int_{\partial\mathcal{B}_1^t}\varrho^{\mathfrak{a}} \mathscr{U}_n^2 d\sigma_t\right|\leq Ct\int_{\mathcal{B}_1^t}\varrho^{\mathfrak{a}}|\nabla_{\bar{g}_t} \mathscr{U}|^2 \mathop{\mathrm{dvol}}_{\bar{g}_t} .\end{aligned}$$ Finally, Corollary [Corollary 76](#c:approximate-Dirichlet-energy){reference-type="ref" reference="c:approximate-Dirichlet-energy"} leads to $$\begin{aligned} \left|\mathcal{I}_D(1) - (\mathfrak{m}- 2 + \mathfrak{a}) I_t(1) - 2\int_{\partial\mathcal{B}_1^t}\varrho^{\mathfrak{a}} \mathscr{U}_n^2d\sigma_t\right|\leq Ct\cdot I_t(1). \label{e:I_D-bound}\end{aligned}$$ Now we estimate the term $\mathcal{I}_Q$. By Proposition [Proposition 74](#p:eps-non-vanishing){reference-type="ref" reference="p:eps-non-vanishing"}, $H_t(r)\leq \frac{r I_t(r)}{\epsilon_0}$, which implies that $$\begin{aligned} \label{e:I_Q-bound} \mathcal{I}_Q(1) \leq Ct^2 H_t(1) \leq \frac{Ct^2}{\epsilon_0} I_t(1) .\end{aligned}$$ Combining [\[e:derivative-I_t\]](#e:derivative-I_t){reference-type="eqref" reference="e:derivative-I_t"}, [\[e:I_D-bound\]](#e:I_D-bound){reference-type="eqref" reference="e:I_D-bound"}, [\[e:I_Q-bound\]](#e:I_Q-bound){reference-type="eqref" reference="e:I_Q-bound"}, we have $$\begin{aligned} \left|I_t'(1) - (\mathfrak{m}- 2 + \mathfrak{a}) I_t(1) - 2\int_{\partial\mathcal{B}_1^t}\varrho^{\mathfrak{a}} \mathscr{U}_n^2d\sigma_t\right|\leq Ct\cdot I_t(1). \end{aligned}$$ Notice that, by the divergence theorem, $$\begin{aligned} I_t(1) = \int_{\partial\mathcal{B}_1^t} \varrho^{\mathfrak{a}} \mathscr{U}\mathscr{U}_n d\sigma_t,\end{aligned}$$ which implies $$\begin{aligned} \label{e:derivative-log-I} \left|\frac{I_t'(1)}{I_t(1)} - (\mathfrak{m}- 2 + \mathfrak{a}) - 2\frac{\displaystyle{\int_{\partial\mathcal{B}_1^t}\varrho^{\mathfrak{a}} \mathscr{U}_n^2d\sigma_t}}{\displaystyle{\int_{\partial\mathcal{B}_1^t} \varrho^{\mathfrak{a}} \mathscr{U}\mathscr{U}_n d\sigma_t}}\right|\leq Ct. \end{aligned}$$ Finally, it follows from [\[e:derivative-log-I\]](#e:derivative-log-I){reference-type="eqref" reference="e:derivative-log-I"} and [\[e:derivative-log-H\]](#e:derivative-log-H){reference-type="eqref" reference="e:derivative-log-H"} that $$\begin{aligned} \begin{split} \frac{\mathcal{N}_t'(1)}{\mathcal{N}_t(1)} = 1 + \frac{I_t'(1)}{I_t(1)} - \frac{H_t'(1)}{H_t(1)} \geq & \ 2\left(\frac{\displaystyle{\int_{\partial\mathcal{B}_1^t}\varrho^{\mathfrak{a}}\mathscr{U}_n^2d\sigma_t}}{\displaystyle{\int_{\partial\mathcal{B}_1^t}\varrho^{\mathfrak{a}}\mathscr{U}\mathscr{U}_nd\sigma_t}} -\frac{\displaystyle{\int_{\partial\mathcal{B}_1^t}\varrho^{\mathfrak{a}}\mathscr{U}\mathscr{U}_nd\sigma_t}}{\displaystyle{\int_{\partial\mathcal{B}_1^t}\varrho^{\mathfrak{a}} \mathscr{U}^2d\sigma_t}}\right) - C t \\ \geq & \ -Ct. \end{split}\end{aligned}$$ This completes the proof. ◻ ## Quantitative symmetry and quantitative cone-splitting {#ss:quantitative-symmetry-PE} In this subsection, we will look at the behaviors of the solutions of [\[e:CS-extension-on-PE\]](#e:CS-extension-on-PE){reference-type="eqref" reference="e:CS-extension-on-PE"} on small scales in a quantitative way. Mainly, for a solution $U$ of the boundary value problem [\[e:CS-extension-on-PE\]](#e:CS-extension-on-PE){reference-type="eqref" reference="e:CS-extension-on-PE"}, we will summarize several quantitative rigidity results which are used to prove the uniform volume estimates for the quantitative strata of the singular set in Section [4.3](#ss:main-results-PE){reference-type="ref" reference="ss:main-results-PE"}. Let us begin with the following proposition which shows how the energy boundedness property [\[e:N(0,1)-bounded-by-Lambda\]](#e:N(0,1)-bounded-by-Lambda){reference-type="eqref" reference="e:N(0,1)-bounded-by-Lambda"} on a large scale can pass to any smaller scales. This is quite important in the implementation of the quantitative analysis of the singular set. **Proposition 77** (Uniform boundedness). *Let $p\in M^n$ and assume that $\mathcal{B}_1(p_+) \subset \overline{X^{\mathfrak{m}}}$ has a smooth boundary. Let $U: \mathfrak{X}^{\mathfrak{m}}\to \mathbb{R}$ be an even solution of [\[e:CS-extension-on-PE\]](#e:CS-extension-on-PE){reference-type="eqref" reference="e:CS-extension-on-PE"}. Given any $\Lambda > 0$, there exist constants $N_0 = N_0 (\Lambda, \mathfrak{m}, \mathfrak{a}, \bar{g}) > 0$ and $r_0 = r_0 (\Lambda, \mathfrak{m}, \mathfrak{a}, \bar{g}) > 0$ such that if $$\begin{aligned} \frac{\displaystyle{\int_{\mathcal{B}_1(p_+)}\varrho^{\mathfrak{a}}|\nabla_{\bar{g}}U|^2 \mathop{\mathrm{dvol}}_{\bar{g}}}}{\displaystyle{\int_{\partial\mathcal{B}_1(p_+)}\varrho^{\mathfrak{a}}U^2 d\sigma_{\bar{g}}}} \leq \Lambda, \label{e:N(0,1)-bounded-by-Lambda}\end{aligned}$$ then for every $x \in \mathcal{B}_{1/2}(p_+)$ and $r \in (0, r_0)$, $$\begin{aligned} \mathcal{N}_U(x, r) \leq N_0.\end{aligned}$$* *Proof.* Since the condition and the conclusion of proposition are scale-invariant in $U$, we assume that $$\begin{aligned} \int_{\mathcal{B}_1(p_+)}\varrho^{\mathfrak{a}}|\nabla_{\bar{g}}U|^2 \mathop{\mathrm{dvol}}_{\bar{g}} \leq \Lambda, \quad \int_{\partial\mathcal{B}_1(p_+)}\varrho^{\mathfrak{a}}U^2 d\sigma_{\bar{g}}= 1.\label{e:unit-ball-U-normalization} \end{aligned}$$ Then Lemma [Lemma 71](#l:L2-estimate-large-scale){reference-type="ref" reference="l:L2-estimate-large-scale"} implies that $\|U\|_{H^{1,\mathfrak{a}}(\mathcal{B}_1(p_+))} \leq C(\Lambda,\mathfrak{m},\mathfrak{a}, \bar{g})$. It suffices to show that there exists $r_0 = r_0(\Lambda, \mathfrak{m}, \mathfrak{a}, \bar{g}) > 0$ satisfying the following property: for any $r\in(0, r_0]$, there exists $\delta = \delta (r, \Lambda, \mathfrak{m}, \mathfrak{a}, \bar{g}) > 0$ such that for every $x\in \mathcal{B}_{1/2}(p_+)$, $$\begin{aligned} \int_{\partial\mathcal{B}_r(x)} \varrho^{\mathfrak{a}} U^2 \mathop{\mathrm{dvol}}_{\bar{g}} > \delta. \label{e:positive-lower-bound-of-boundary-L2}\end{aligned}$$ Indeed, if [\[e:positive-lower-bound-of-boundary-L2\]](#e:positive-lower-bound-of-boundary-L2){reference-type="eqref" reference="e:positive-lower-bound-of-boundary-L2"} is proved, then we can choose $\delta_0 \equiv \delta(r_0,\Lambda, \mathfrak{m}, \mathfrak{a}, \bar{g}))$ so that $$\begin{aligned} \begin{split} \mathcal{N}_U(x, r_0) = &\ \frac{\displaystyle{r_0\int_{\mathcal{B}_{r_0}(x)}\varrho^{\mathfrak{a}}(|\nabla_{\bar{g}}U|^2 + \mathcal{J}_{\bar{g}} U^2) \mathop{\mathrm{dvol}}_{\bar{g}}}}{\displaystyle{\int_{\partial\mathcal{B}_{r_0}(x)} \varrho^{\mathfrak{a}} U^2 d\sigma_{\bar{g}}}} \\ \leq \ &\delta_0^{-1}r_0 \cdot \left(1 + C_{\mathfrak{m}, \gamma}\sup\limits_{\overline{X^{\mathfrak{m}}}}|R_{\bar{g}}|\right)\cdot \left(\int_{\mathcal{B}_1(p_+)}\varrho^{\mathfrak{a}} (|\nabla_{\bar{g}}U|^2 + U^2) \mathop{\mathrm{dvol}}_{\bar{g}}\right). \end{split}\end{aligned}$$ Applying Lemma [Lemma 71](#l:L2-estimate-large-scale){reference-type="ref" reference="l:L2-estimate-large-scale"}, $$\begin{aligned} \begin{split} \mathcal{N}_U(x, r_0) \leq &\ C_1(\Lambda, \mathfrak{m}, \mathfrak{a}, \bar{g}) \cdot \left( \int_{\mathcal{B}_1(p_+)}\varrho^{\mathfrak{a}}|\nabla_{\bar{g}}U|^2 \mathop{\mathrm{dvol}}_{\bar{g}} + \int_{\partial\mathcal{B}_1(p_+)} \varrho^{\mathfrak{a}} U^2 d\sigma_{\bar{g}}\right) \\ \leq & \ C_1(\Lambda, \mathfrak{m}, \mathfrak{a}, \bar{g}) \cdot (\Lambda+1). \end{split}\end{aligned}$$ By Theorem [Theorem 69](#t:almost-monotonicity){reference-type="ref" reference="t:almost-monotonicity"}, $\mathcal{N}_U(x,r) \leq e^{C_2(\Lambda, \mathfrak{m}, \mathfrak{a}, \bar{g})r_0}(\Lambda + 1)$ for any $r\in(0,r_0)$, which proves the uniform boundedness. Now let us prove [\[e:positive-lower-bound-of-boundary-L2\]](#e:positive-lower-bound-of-boundary-L2){reference-type="eqref" reference="e:positive-lower-bound-of-boundary-L2"}. The proof follows from a standard compactness/contradiction argument. Suppose that for some sufficiently small $r > 0$, there exist contradiction sequences: a sequence of solutions $U_j$ of [\[e:CS-extension-on-PE\]](#e:CS-extension-on-PE){reference-type="eqref" reference="e:CS-extension-on-PE"}, a sequence of points $x_j \in \mathcal{B}_{1/2}(p_+)$, and a sequence $\delta_j \to 0$ such that $\|U_j\|_{H^{1,\mathfrak{a}}(\mathcal{B}_1(p_+))} \leq C$ and $H_U(x_j,r)\leq \delta_j \to 0$. This implies that, passing to a subsequence, $U_j$ converges to $U_{\infty}$ weakly in the $H^{1,\mathfrak{a}}(\mathcal{B}_1(p_+))$-topology and $$\begin{aligned} \label{eq-U-2-P0} \int_{\partial\mathcal{B}_{r_0}(x_{\infty})}\varrho^{\mathfrak{a}} U_{\infty}^2 d\sigma_{\bar{g}} = 0,\end{aligned}$$ where $x_j\to x_{\infty} \in \mathcal{B}_{1/2}(p_+)$. Therefore, $U_{\infty} \equiv 0$ on $\partial\mathcal{B}_r(x_{\infty})$. Since $U_{\infty}$ solves [\[e:CS-extension-on-PE\]](#e:CS-extension-on-PE){reference-type="eqref" reference="e:CS-extension-on-PE"} and $U_{\infty}\in C^{n,\alpha}(\mathcal{B}_1(p_+))$, it follows that $$\begin{aligned} \int_{\mathcal{B}_r(x_{\infty})} \varrho^{\mathfrak{a}}(|\nabla_{\bar{g}} U_{\infty}|^2 + \mathcal{J}_{\bar{g}} U_{\infty}^2 ) \mathop{\mathrm{dvol}}_{\bar{g}} = \int_{\partial\mathcal{B}_r(x_{\infty})} \varrho^{\mathfrak{a}} \cdot U_{\infty} \cdot \frac{\partial U_{\infty}}{\partial n}d\sigma_{\bar{g}}= 0, \end{aligned}$$ which leads to $$\begin{aligned} D_U(x_{\infty}, r) \leq C(\mathfrak{m},\mathfrak{a},\bar{g})\int_{\mathcal{B}_r(x_{\infty})} \varrho^{\mathfrak{a}} U^2 \mathop{\mathrm{dvol}}_{\bar{g}} = C(\mathfrak{m},\mathfrak{a},\bar{g})\left(r H_U(x_{\infty}, r) + r^2D_U(x_{\infty}, r)\right). \end{aligned}$$ If $r > 0$ is chosen such that $C(\mathfrak{m}, \mathfrak{a}, \bar{g}) r^2 < \frac{1}{2}$, then $D_U(x_{\infty}, r) \equiv 0$, and hence $U_{\infty} \equiv 0$ on $\mathcal{B}_r(x_{\infty})$. Notice that $u_{\infty} = \varrho^{n-s} U_{\infty}$ satisfies $$\begin{aligned} \Delta_{g_+} u_{\infty} + s(n-s) u_{\infty} = 0, \quad s = \frac{n}{2} + \gamma,\end{aligned}$$ in an open subset $\mathcal{O}\subset X^{\mathfrak{m}}$. Here $\mathcal{O}$ is the conformal image of $\mathcal{B}_1(p_+)$ in terms of $g_+ = \varrho^{-2} \bar{g}$. By the standard unique continuation, $u_{\infty} \equiv 0$ on $\mathcal{O}$, which implies $U_{\infty}\equiv 0$ on $\mathcal{B}_1(p_+)$. But this contradicts the normalization condition [\[e:unit-ball-U-normalization\]](#e:unit-ball-U-normalization){reference-type="eqref" reference="e:unit-ball-U-normalization"}, which completes the proof of [\[e:positive-lower-bound-of-boundary-L2\]](#e:positive-lower-bound-of-boundary-L2){reference-type="eqref" reference="e:positive-lower-bound-of-boundary-L2"}. ◻ Next, we will list a series of quantitative rigidity results which replace the corresponding results in Sections [3.2](#ss:quantitative-rigidity-Euclidean){reference-type="ref" reference="ss:quantitative-rigidity-Euclidean"} and [3.3](#ss:quantitative-cone-splitting){reference-type="ref" reference="ss:quantitative-cone-splitting"}. The quantitative rigidity below can be proved by standard contradiction and compactness arguments, which is a replacement of Proposition [Proposition 42](#p:quantitative-symmetry){reference-type="ref" reference="p:quantitative-symmetry"} in the general Poincaré-Einstein manifold. **Proposition 78** (Quantitative rigidity). *For fixed $\mathfrak{a}\in (-1,1)$, let $U$ be the unique solution to [\[e:CS-extension-on-PE\]](#e:CS-extension-on-PE){reference-type="eqref" reference="e:CS-extension-on-PE"}. Given $\epsilon > 0$, $\gamma \in (0,1)$, and $\Lambda > 0$, there exist $\delta = \delta(\epsilon,\Lambda, \gamma, n, \mathfrak{a}, g_+, h) > 0$, $r_c = r_c(\epsilon,\Lambda,\gamma, n, \mathfrak{a}, g_+, h) > 0$, and $C = C(\epsilon,\Lambda, \gamma, n,\mathfrak{a}, g_+, h) > 0$ such that if $$\begin{aligned} \frac{\displaystyle{\int_{\mathcal{B}_1(p_+)}\varrho^{\mathfrak{a}}|\nabla_{\bar{g}}U|^2 \mathop{\mathrm{dvol}}_{\bar{g}}}}{\displaystyle{\int_{\partial\mathcal{B}_1(p_+)}\varrho^{\mathfrak{a}}U^2 \mathop{\mathrm{dvol}}_{\bar{g}} }} \leq \Lambda, \end{aligned}$$ and for $s \in (0, r_c)$, $$\begin{aligned} \mathcal{N}(x, s) - \mathcal{N}(x, s\gamma) < \delta, \end{aligned}$$ then $U$ is $(0, \epsilon , s , \mathfrak{a})$-symmetric at $x \in B_{1/2}(p)$.* Applying the almost monotonicity in Theorem [Theorem 69](#t:almost-monotonicity){reference-type="ref" reference="t:almost-monotonicity"} and contradiction arguments, we have the following *inductive cone-splitting principle* which replaces Corollary [Corollary 49](#c:inductive-splitting){reference-type="ref" reference="c:inductive-splitting"}. **Proposition 79** (Inductive cone-splitting). *For any fixed $n\geq 2$, $\epsilon>0$, $\gamma \in (0,1)$, $\rho>0$, $r\in (0,1)$, $k\in\{0,1,\ldots, n-2\}$, $\Lambda>0$, there exist $\delta=\delta(\epsilon, n, \rho, \Lambda, \mathfrak{a}) > 0$ and $r_{\#} = r_{\#}(\epsilon, n, \rho, \Lambda, \mathfrak{a}) > 0$ such that the following holds. Let $U$ be the unique solution to [\[e:CS-extension-on-PE\]](#e:CS-extension-on-PE){reference-type="eqref" reference="e:CS-extension-on-PE"} with $$\begin{aligned} \frac{\displaystyle{\int_{\mathcal{B}_1(p_+)}\varrho^{\mathfrak{a}}|\nabla_{\bar{g}}U|^2 \mathop{\mathrm{dvol}}_{\bar{g}}}}{\displaystyle{\int_{\partial\mathcal{B}_1(p_+)}\varrho^{\mathfrak{a}}U^2 \mathop{\mathrm{dvol}}_{\bar{g}} }} \leq \Lambda, \end{aligned}$$ If for some $s\in (0,r_{\#})$ and $x \in B_{1/2}(p)$,* 1. *$U$ is $(0,\delta, s\gamma ,\mathfrak{a})$-symmetric at $x$,* 2. *for any vector space $V$ of dimension $\leq k$, there exists some point $z\in B_r(x)\setminus B_{\rho}(V)$ such that $U$ is $(0,\delta,s\gamma,\mathfrak{a})$-symmetric at $z$,* *then $U$ is also $(k+1,\epsilon,s,\mathfrak{a})$-symmetric at $x$.* The following lemma is used in estimating the dimension and measure of the singular set which is a replacement of Lemma [Lemma 51](#l:almost-(m-1)-symmetric){reference-type="ref" reference="l:almost-(m-1)-symmetric"}. **Lemma 80**. *Let $U$ be the unique even solution of [\[e:CS-extension-on-PE\]](#e:CS-extension-on-PE){reference-type="eqref" reference="e:CS-extension-on-PE"} such that $$\begin{aligned} \frac{\displaystyle{\int_{\mathcal{B}_1(p_+)}\varrho^{\mathfrak{a}}|\nabla_{\bar{g}}U|^2 \mathop{\mathrm{dvol}}_{\bar{g}}}}{\displaystyle{\int_{\partial\mathcal{B}_1(p_+)}\varrho^{\mathfrak{a}}U^2 \mathop{\mathrm{dvol}}_{\bar{g}} }} \leq \Lambda. \end{aligned}$$ Then for every $\epsilon > 0$, $k\in\mathbb{N}$, and $\alpha\in(0,1)$, there exists $\eta = \eta (\epsilon, n, k, \Lambda, \mathfrak{a}) > 0$ such that if $U$ is $(\mathfrak{m}- 1, \eta, r, \mathfrak{a})$-symmetric at $p_+$, then $$\begin{aligned} \|\mathcal{T}_{p_+,r} U - L \|_{C^{1,\alpha}(\mathcal{B}_{1/2}(\bm{0}_+))} < \epsilon, \end{aligned}$$ where $L$ is a linear polynomial with $\frac{1}{r^{\mathfrak{a}}}\fint_{\partial\mathcal{B}_1(\bm{0}_+)}|L|^2 d\sigma_0 = 1$. In particular, choosing any positive number $\epsilon < \epsilon_0 \equiv |\nabla L|/3$, we have $r_{p_+} \geq r$ if $U$ is $(\mathfrak{m}- 1, \epsilon, r_{p_+} , \mathfrak{a})$ at $p_+$.* ## Volume and measure estimates for the singular set {#ss:main-results-PE} In this subsection, we prove the main results of this paper. To start with, we recall the equations together with basic settings of the underlying Poincaré-Einstein manifolds. For any given $n\geq 2$, let $\mathfrak{m}\equiv n + 1$ and let $(X^{\mathfrak{m}}, g_+)$ be a complete Poincaré-Einstein manifold with $\mathop{\mathrm{Ric}}_{g_+} \equiv - n g_+$. Assume that a closed manifold $M^n$ equipped with a conformal class $[h]$ is the conformal infinity of $(X^{\mathfrak{m}},g_+)$. We choose Fefferman-Graham's compactification $\bar{g}= \varrho^2 g_+ = e^{2w} g_+$ of $(X^{\mathfrak{m}}, g_+)$ as described previously in Section [2.1](#ss:PE-metrics){reference-type="ref" reference="ss:PE-metrics"} such that $-\Delta_{g_+} w = n$ on $X^{\mathfrak{m}}$ and $\bar{g}|_{M^n} = h$. In addition, we also assume that $\bar{g}$ satisfies Assumptions (1) - (4) on $\overline{X^{\mathfrak{m}}}$ as introduced at the beginning of Section [4](#s:results-on-PE){reference-type="ref" reference="s:results-on-PE"}. We also denote by $\mathfrak{X}^{\mathfrak{m}} \equiv \overline{X^{\mathfrak{m}}}\bigcup\limits_{M^n} \overline{X^{\mathfrak{m}}}$ the doubling of $(\overline{X^{\mathfrak{m}}},\bar{g})$ along the totally geodesic boundary $(M^n, h)$. In the following, we will collect our main results in the above setting. For any fixed $\gamma\in(0,1)$, let $f \in C^{\infty}(M^n)$ satisfy $P_{2\gamma}(f) = 0$ on $B_1(p) \subset M^n$, where $P_{2\gamma}$ is the fractional GJMS-operator. Let $\mathfrak{a}\equiv 1 - 2\gamma$ and let $U\in H^{1,\mathfrak{a}}(\mathfrak{X}^{\mathfrak{m}})$ be an even solution of the following boundary value problem: $$\begin{aligned} \label{e:PE-boundary-value-problem} \begin{cases} -\mathop{\mathrm{div}}_{\bar{g}}(\varrho^{\mathfrak{a}} \nabla_{\bar{g}} U) + \varrho^{\mathfrak{a}} \mathcal{J}_{\bar{g}} U = 0 & \text{in} \ \mathfrak{X}^{\mathfrak{m}} \\ U = f & \text{on} \ M^n,\\ P_{2\gamma} f = 0 &\text{on}\ B_1(p) \subset M^n, \end{cases} \end{aligned}$$ where $\mathcal{J}_{\bar{g}}\equiv C_{n,\gamma}R_{\bar{g}}$. **Theorem 81**. *For any $j\in\mathbb{N}$, $\epsilon>0$, $\Lambda > 0$, $k\leq \mathfrak{m}-2$, and $\mathfrak{a}\in (-1,1)$, there exist positive constants $\mu = \mu(\epsilon,\mathfrak{m},\Lambda, \mathfrak{a}, \bar{g})\in(0,1)$ and $C = C(\epsilon,\mathfrak{m},\Lambda, \mathfrak{a}, \bar{g})>0$ such that the following property holds. Let $U$ be the unique even solution of [\[e:PE-boundary-value-problem\]](#e:PE-boundary-value-problem){reference-type="eqref" reference="e:PE-boundary-value-problem"} with $$\begin{aligned} \frac{\displaystyle{\int_{\mathcal{B}_1(p_+)}\varrho^{\mathfrak{a}}|\nabla_{\bar{g}}U|^2 \mathop{\mathrm{dvol}}_{\bar{g}}}}{\displaystyle{\int_{\partial\mathcal{B}_1(p_+)}\varrho^{\mathfrak{a}}U^2 \mathop{\mathrm{dvol}}_{\bar{g}} }} \leq \Lambda. \end{aligned}$$ Then $$\begin{aligned} \mathop{\mathrm{Vol}}_{\bar{g}}(B_{\mu^j}(\mathcal{S}_{\epsilon,\mu^j}^k(U))\cap B_{1/2}(p)) \leq C \cdot (\mu^j)^{\mathfrak{m}- k -\epsilon}. \end{aligned}$$* *Proof.* The strategy of proof is the same as that of Theorem [Proposition 53](#p:effective-covering){reference-type="ref" reference="p:effective-covering"}. Here we only mention the technical difference. In the proof of Theorem [Proposition 53](#p:effective-covering){reference-type="ref" reference="p:effective-covering"}, the uniform boundedness estimate in Lemma [Lemma 41](#l:frequency-control-large-to-small){reference-type="ref" reference="l:frequency-control-large-to-small"} is replaced by Proposition [Proposition 77](#p:frequency-bound-U-PE){reference-type="ref" reference="p:frequency-bound-U-PE"}, the quantitative rigidity in Proposition [Proposition 42](#p:quantitative-symmetry){reference-type="ref" reference="p:quantitative-symmetry"} is replaced by Proposition [Proposition 78](#p:quantitative-symmetry-U-PE){reference-type="ref" reference="p:quantitative-symmetry-U-PE"}, and the inductive splitting property in Corollary [Corollary 49](#c:inductive-splitting){reference-type="ref" reference="c:inductive-splitting"} is replaced by Proposition [Proposition 79](#p:inductive-cone-splitting-PE){reference-type="ref" reference="p:inductive-cone-splitting-PE"}. ◻ Next, we will exhibit two results on the Hausdorff measure estimates. Regarding the $(\mathfrak{m}- 1)$-dimensional Hausdorff measure estimate on the zero set of the solution $U$. We have the following. **Theorem 82** (Hausdorff measure of the zero set). *For $p\in M^n$, let $U$ be the unique even solution of [\[e:PE-boundary-value-problem\]](#e:PE-boundary-value-problem){reference-type="eqref" reference="e:PE-boundary-value-problem"} that satisfies $$\begin{aligned} \frac{\displaystyle{\int_{\mathcal{B}_1(p_+)}\varrho^{\mathfrak{a}}|\nabla_{\bar{g}}U|^2 \mathop{\mathrm{dvol}}_{\bar{g}}}}{\displaystyle{\int_{\partial\mathcal{B}_1(p_+)}\varrho^{\mathfrak{a}}U^2 \mathop{\mathrm{dvol}}_{\bar{g}} }} \leq \Lambda. \label{e:bounded-energy-large-scale} \end{aligned}$$ There exists a constant $C=C(\mathfrak{m},\Lambda,\mathfrak{a},\bar{g})>0$ such that $$\begin{aligned} \mathcal{H}^{\mathfrak{m}- 1}(\mathcal{Z}(U) \cap B_{1/2}(p)) \leq C. \end{aligned}$$* The uniformly elliptic version of Theorem [Theorem 82](#t:zero-set-PE){reference-type="ref" reference="t:zero-set-PE"} was proved in [@HL-geometric-measure; @Lin]. The main technical tools employed there include the uniform doubling property (guaranteed by the almost monotonicity of frequency), compactness of solutions with bounded energy, and harmonic approximation. In our setting, all these tools are available. Since the main strategy and arguments are the same, we omit the proof of Theorem [Theorem 82](#t:zero-set-PE){reference-type="ref" reference="t:zero-set-PE"}. The last result is the $(\mathfrak{m}- 2)$-dimensional Hausdorff measure estimate for the singular set of the solution $U$ of [\[e:PE-boundary-value-problem\]](#e:PE-boundary-value-problem){reference-type="eqref" reference="e:PE-boundary-value-problem"}. **Theorem 83** (Hausdorff measure estimate). *For $p\in M^n$, let $U$ be the unique solution of [\[e:PE-boundary-value-problem\]](#e:PE-boundary-value-problem){reference-type="eqref" reference="e:PE-boundary-value-problem"} with $$\begin{aligned} \frac{\displaystyle{\int_{\mathcal{B}_1(p_+)}\varrho^{\mathfrak{a}}|\nabla_{\bar{g}}U|^2 \mathop{\mathrm{dvol}}_{\bar{g}}}}{\displaystyle{\int_{\partial\mathcal{B}_1(p_+)}\varrho^{\mathfrak{a}}U^2 \mathop{\mathrm{dvol}}_{\bar{g}} }} \leq \Lambda. \end{aligned}$$ There exists a constant $C=C(\mathfrak{m},\Lambda,\mathfrak{a},\bar{g})>0$ such that $$\begin{aligned} \mathcal{H}^{\mathfrak{m}- 2}(\mathcal{S}(U) \cap B_{1/2}(p)) \leq C. \end{aligned}$$* The proof of Theorem [Theorem 83](#t:Hausdorff-measure-estimate-PE){reference-type="ref" reference="t:Hausdorff-measure-estimate-PE"} is the same as Theorem [Theorem 60](#t:Hausdorff-measure-estimate-Euclidean){reference-type="ref" reference="t:Hausdorff-measure-estimate-Euclidean"}. The main step is to establish the following $\epsilon$-regularity theorem which replaces Theorem [Theorem 59](#t:eps-reg-Euclidean){reference-type="ref" reference="t:eps-reg-Euclidean"}. **Theorem 84** ($\epsilon$-regularity). *For $p\in M^n$, let $U$ be the unique solution of [\[e:PE-boundary-value-problem\]](#e:PE-boundary-value-problem){reference-type="eqref" reference="e:PE-boundary-value-problem"} that satisfies [\[e:bounded-energy-large-scale\]](#e:bounded-energy-large-scale){reference-type="eqref" reference="e:bounded-energy-large-scale"}. Then there exist constants $\epsilon(\mathfrak{m},\mathfrak{a},\Lambda,\bar{g}) > 0$ and $\bar{r}(\mathfrak{m},\mathfrak{a},\Lambda,\bar{g}) > 0$ such that if there exists a normalized homogeneous polynomial $P$ with $(\mathfrak{m}- 2)$-symmetries such that $$\begin{aligned} \fint_{\partial\mathcal{B}_1(\bm{0}_+)} \varrho^{\mathfrak{a}} P^2 d\sigma_{\bar{g}}= 1\quad \text{and} \quad \int_{\partial\mathcal{B}_1(\bm{0}_+)}\varrho^{\mathfrak{a}}(U_{x_+, r} - P)^2 d\sigma_{\bar{g}}\leq \epsilon, \label{e:L2-approximation-PE}\end{aligned}$$ for some $x_+ \in B_{1/2}(p)$, then for all $r\leq \bar{r}$, $$\begin{aligned} \mathcal{H}^{\mathfrak{m}- 2}(\mathcal{S}(U) \cap B_r(x_+)) \leq C(\mathfrak{m},\mathfrak{a},\Lambda,\bar{g}) r^{\mathfrak{m}- 2}. \end{aligned}$$* To prove Theorem [Theorem 84](#t:eps-reg-PE){reference-type="ref" reference="t:eps-reg-PE"}, it is the same as Theorem [Theorem 59](#t:eps-reg-Euclidean){reference-type="ref" reference="t:eps-reg-Euclidean"} that one needs to upgrade the approximation [\[e:L2-approximation-PE\]](#e:L2-approximation-PE){reference-type="eqref" reference="e:L2-approximation-PE"} and establish a higher order estimate. That is, **Proposition 85** (Higher order approximation). *Let $U$ be the unique solution of [\[e:PE-boundary-value-problem\]](#e:PE-boundary-value-problem){reference-type="eqref" reference="e:PE-boundary-value-problem"} that satisfies [\[e:bounded-energy-large-scale\]](#e:bounded-energy-large-scale){reference-type="eqref" reference="e:bounded-energy-large-scale"}. For every $\epsilon > 0$, there exists $\delta (\epsilon, \mathfrak{m},\mathfrak{a},\Lambda,\bar{g}) > 0$ and $r_0 (\epsilon, \mathfrak{m},\mathfrak{a},\Lambda,\bar{g}) > 0$ such that if there exists a normalized homogeneous polynomial $P$ with $(\mathfrak{m}- 2)$-symmetries that satisfies $$\begin{aligned} \fint_{\partial\mathcal{B}_1(\bm{0}_+)} \varrho^{\mathfrak{a}} P^2 d\sigma_{\bar{g}}= 1\quad \text{and} \quad \int_{\partial\mathcal{B}_1(\bm{0}_+)}\varrho^{\mathfrak{a}}(U_{x_+, r} - P)^2 d\sigma_{\bar{g}}\leq \epsilon, \end{aligned}$$ for some $x_+ \in \mathcal{Z}(U) \cap \mathcal{B}_{1/2}(p_+)$ and $r\in(0,r_0)$, then there exists a function $\widetilde{U}\in C^{2\mathfrak{m}^2}(\mathcal{B}_{2r/3}(x_+))$ such that $\widetilde{U}\equiv U$ in $\overline{\mathcal{B}_{2r/3}^+(x_+)}$ and $$\begin{aligned} \|\widetilde{U}_{x_+, r} - P\|_{C^{2\mathfrak{m}^2}(\mathcal{B}_{2/3}(\bm{0}_+))} < \epsilon. \label{e:extension-estimate} \end{aligned}$$* **Remark 86**. Note that the doubling metric $\bar{g}$ on $\mathfrak{X}^{\mathfrak{m}}$ is only $C^{\mathfrak{m}- 2,1}$ in general. Especially, when $n$ is odd and the *global term* in the normal expansion of $\bar{g}$ is non-vanishing, the $C^{\mathfrak{m}- 2,1}$-smoothness of $\bar{g}$ is optimal. Since a key tool, theorem [Theorem 58](#t:HHL-stability){reference-type="ref" reference="t:HHL-stability"}, in proving Theorem [Theorem 84](#t:eps-reg-PE){reference-type="ref" reference="t:eps-reg-PE"} requires a very high order approximation of $U$ (up to $2\mathfrak{m}^2$), [\[e:L2-approximation-PE\]](#e:L2-approximation-PE){reference-type="eqref" reference="e:L2-approximation-PE"} cannot be upgraded to this by directly using elliptic estimates. Proposition [Proposition 85](#p:higher-order-approximation-PE){reference-type="ref" reference="p:higher-order-approximation-PE"} constructs a sufficiently regular replacement of $U$ which overcomes insufficient regularity of the metric $\bar{g}$. Once this proposition is proved, the rest of the proof of Theorem [Theorem 84](#t:eps-reg-PE){reference-type="ref" reference="t:eps-reg-PE"} is identical to Theorem [Theorem 59](#t:eps-reg-Euclidean){reference-type="ref" reference="t:eps-reg-Euclidean"}. *Proof of Proposition [Proposition 85](#p:higher-order-approximation-PE){reference-type="ref" reference="p:higher-order-approximation-PE"}.* To construct the extension $\widetilde{U}$, let us first analyze the regularity of $U$ in the upper half space. In this proposition, the estimate is required to pass to a small scale $r_0 > 0$ for carrying out elliptic estimates. Indeed, one needs to pass to a small scale on which the elliptic operator is sufficiently close to the model operator in the Euclidean case. In order to avoid unnecessary technical complications, we just set $r_0 = 1$ and assume that the distortion between our interested elliptic operator and the (degenerate) Euclidean Laplacian is sufficiently small. First, applying elliptic regularity, for any $k\in\mathbb{Z}_+$, there exists $C_k = C_k (\mathfrak{m}, \mathfrak{a}, \bar{g}) > 0$ such that $$\begin{aligned} \|U - P\|_{C^k \left( \overline{\mathcal{B}_{2/3}^+(\bm{0}_+)} \right)} < C_k\delta. \label{e:derivative-estimate-order-k}\end{aligned}$$ Notice that, by the standard compactness argument, the homogeneous polynomial $P$ can be chosen as a solution in the model case of [\[e:PE-boundary-value-problem\]](#e:PE-boundary-value-problem){reference-type="eqref" reference="e:PE-boundary-value-problem"}, $P$ is even in $y$. Moreover, there exists a uniform constant $N_0 = N_0(\Lambda, \mathfrak{m}, \mathfrak{a}, \bar{g}) > 0$ such that $d_0 \leq N_0$ since the frequency of $P$ is uniformly bounded due to the compactness. Let us pick a sufficiently large integer $k_0\in\mathbb{Z}_+$ (to be fixed later) which is determined by the structure of the Taylor expansions of $U$ and $P$. For any $y < \frac{2}{3}$, applying Taylor's theorem in the $y$-direction, we have that $$\begin{aligned} U = U_{k_0} + R_{k_0} \equiv \sum\limits_{\ell = 0}^{k_0} c_{\ell} y^{\ell} + R_{k_0},\quad R_{k_0} = \frac{U^{(k_0+1)}(\xi y, \cdot)}{(k_0 + 1)!}\cdot y^{k_0 + 1}, \label{e:U-expansion-in-y}\end{aligned}$$ where each $c_{\ell} = c_{\ell}(\cdot)$ is independent of $y$, $0< \xi < 1$, and $U^{(k_0+1)}$ denotes the $(k_0+1)^{\text{th}}$-derivative of $U$ in the $y$ direction. For the homogeneous polynomial $P$, we write $P = \sum\limits_{\substack{\ell = 0, \\ \ell \ \text{is even}}}^{m_0} p_{\ell} y^{\ell}$, where $m_0\leq d_0$ is some even integer and every $p_{\ell} = p_{\ell}(\cdot)$ is a polynomial in the directions transversal to the $y$-direction. Next, we write $$\begin{aligned} U - P = (U_{k_0} - P) + R_{k_0} = \sum\limits_{0\leq \ell\leq k_0}\hat{c}_{\ell} y^{\ell} + \widehat{R}_{k_0},\end{aligned}$$ where $\hat{c}_{\ell} \equiv c_{\ell} - p_{\ell} = \frac{(U-P)^{(\ell)}(0, \cdot)}{\ell!}$ and $\widehat{R}_{k_0}$ is the remainder in Taylor's theorem. Now we are ready to choose the truncation parameter $k_0$. For any fixed $\epsilon > 0$, let us choose $$\begin{aligned} k_0 \equiv \max\left\{2\mathfrak{m}^2, N_0 + 1, \frac{\log\epsilon}{\log(2/3)}\right\}.\end{aligned}$$ Immediately, $y^{k_0 + 1} \leq (2/3)^{k_0 + 1} < \epsilon$. For such an integer $k_0$, one can further choose a sufficiently small $\delta = \delta(\epsilon, \mathfrak{m}, \mathfrak{a}, \bar{g}) > 0$ such that if [\[e:L2-approximation-PE\]](#e:L2-approximation-PE){reference-type="eqref" reference="e:L2-approximation-PE"} holds, then $$\begin{aligned} \|\nabla^{\nu} (U - P)\|_{C^0(\mathcal{B}_{2/3}^+(\bm{0}_+))} < \frac{\epsilon}{2 (k_0 + 1)}, \quad \forall \ 0\leq \nu\leq 2\mathfrak{m}^2 + k_0 + 1. \label{e:fixed-derivative-error}\end{aligned}$$ We also make some simple observations. 1. Since $\deg(P)$ is even, it follows that $\hat{c}_{\ell} = c_{\ell}$ for any odd number $0\leq \ell \leq k_0$. 2. By our choice, $k_0 > N_0 \geq d_0$, which gives $\widehat{R}_{k_0} \equiv R_{k_0}$. By observation (2) and [\[e:fixed-derivative-error\]](#e:fixed-derivative-error){reference-type="eqref" reference="e:fixed-derivative-error"}, $$\begin{aligned} \|R_{k_0}\|_{C^{2\mathfrak{m}^2}(\mathcal{B}_{2/3}^+(\bm{0}_+))} = \|\widehat{R}_{k_0}\|_{C^{2\mathfrak{m}^2}(\mathcal{B}_{2/3}^+(\bm{0}_+))}\leq \frac{\|\nabla^{2\mathfrak{m}^2 + k_0 + 1} (U - P)\|_{C^0(\mathcal{B}_{2/3}^+(\bm{0}_+))}}{(k_0 + 1)!}\cdot y^{k_0 + 1}. \label{e:remainder-bound}\end{aligned}$$ So the remainder yields $$\begin{aligned} \|R_{k_0}\|_{C^{2\mathfrak{m}^2}(\mathcal{B}_{2/3}^+(\bm{0}_+))} < \frac{\epsilon^2}{10}.\end{aligned}$$ Finally, we define $$\begin{aligned} \widetilde{U}\equiv \begin{cases} U, & y \geq 0, \\ & \\ \sum\limits_{\substack{0\leq \ell\leq k_0, \\ \ell \ \text{is odd}}}(-c_{\ell}) y^{\ell} +\sum\limits_{\substack{0\leq \ell\leq k_0, \\ \ell \ \text{is even}}} c_{\ell} y^{\ell} + R_{k_0}, & y < 0. \end{cases} \end{aligned}$$ Clearly, $\widetilde{U}\in C^{2\mathfrak{m}^2}(\mathcal{B}_1(\bm{0}_+))$. The rest of the proof is to verify the approximation [\[e:extension-estimate\]](#e:extension-estimate){reference-type="eqref" reference="e:extension-estimate"} in the lower half space. In fact, one can write $$\begin{aligned} \widetilde{U}- P = \sum\limits_{\substack{0\leq \ell\leq k_0, \\ \ell \ \text{is odd}}}(-\hat{c}_{\ell}) y^{\ell} +\sum\limits_{\substack{0\leq \ell\leq k_0, \\ \ell \ \text{is even}}} \hat{c}_{\ell} y^{\ell} + \widehat{R}_{k_0} \quad y < 0. \end{aligned}$$ Then $$\begin{aligned} \begin{split} \|\widetilde{U}- P\|_{C^{2\mathfrak{m}^2}(\mathcal{B}_{2/3}^-(\bm{0}_+))} \leq &\ ( k_0 + 1 ) \cdot \max\limits_{0\leq \nu k_0}\|\nabla^{\nu} (U - P)\|_{C^0(\mathcal{B}_{2/3}^+(\bm{0}_+))} + \|R_{k_0}\|_{C^{2\mathfrak{m}^2}(\mathcal{B}_{2/3}^+(\bm{0}_+))} \\ \leq &\ \frac{\epsilon}{2} + \frac{\epsilon^2}{10} < \epsilon. \end{split} \end{aligned}$$ Here we used [\[e:fixed-derivative-error\]](#e:fixed-derivative-error){reference-type="eqref" reference="e:fixed-derivative-error"} and [\[e:remainder-bound\]](#e:remainder-bound){reference-type="eqref" reference="e:remainder-bound"}. The proof is complete. ◻ For the Minkowski type estimates and Hausdorff measure estimates for $\mathcal{S}(f)$ on the boundary, one needs to split $\mathcal{S}(f)$ in a horizontal part and a mixed part as in Section [3.7](#ss:boundary-estimate){reference-type="ref" reference="ss:boundary-estimate"} The main results have been stated in Section [1.2](#ss:main-results){reference-type="ref" reference="ss:main-results"}, while the technical refinements can be stated in a similar way as the relevant results in Section [3.7](#ss:boundary-estimate){reference-type="ref" reference="ss:boundary-estimate"}. **Remark 87**. The techniques used to prove the previous theorems allow actually to prove a rectifiability result for the singular set $\mathcal S(U)$ and its stratification $\mathcal S^k(U)$. More precisely, one can prove that the following decomposition holds $$\begin{aligned} \mathcal S^k(U)= \bigcup_{j=1}^\infty{E_j},\end{aligned}$$ for some sets $E_j$. Those sets can be shown to be closed, using Monneau's type monotonicity formulae for instance; see [@STT]. The previous characterization of the each singular stratum comes from a growth lemma together with a non-degeneracy lemma. The idea is that singular strata can be described also as the blow-ups having homogeneity $k$. This part follows from the Almgren's frequency function. The classification and behaviour of blow-ups is then done by the introduction of a suitable Monneau monotonicity formula originally introduced for the obstacle problem; see [@monneau]. Therefore as in [@GarofaloPetrosyan Theorem 1.3.8], the rectifiability of each stratum follows from Whitney's extension theorem and the implicit function theorem; see also [@STT Theorem 7.7]. This approach describes just another way of stratifying the singular set. [^1]: YS is partially supported by NSF Grant DMS-2154219; RZ is supported by NSF Grants DMS-1906265 and DMS-2304818.

arxiv_math

--- abstract: | In this paper we continue the investigation of a real number object, i.e., an object representing the real numbers, in categories of relations. Our axiomatization is based on a relation algebraic version of Tarski's axioms of the real numbers. It was already shown that the addition of such an object forms a dense, linear ordered abelian group. In the current paper we will focus on the least-upper-bound property of such an object. author: - "Michael Winter[^1]" title: | Relational Algebraic Approach to the Real Numbers\ The Least-Upper-Bound Property --- # Introduction In [@PartI] the notion of a real number object in Heyting categories with relational powers was introduced as an abstract version of the real numbers in a suitable category of relations. The axioms of such an object are based on Tarski's axioms of the real numbers. Due to the existence of relational powers it was possible to formulate relation-algebraic versions of Tarski's second order axioms in a purely equational style. In addition, since Heyting algebras do not provide Boolean complements the theory of a real number object is complement-free and all results so far were shown without the usage of Boolean complements. Our main motivation for avoiding Boolean complements is that the results transfer immediately to so-called $L$-fuzzy relations, i.e., to relations that use a Heyting algebra $L$ as truth values instead of the Boolean truth values `true` and `false`. This makes it possible to transfer real number objects and their properties also to the fuzzy case. The investigation of a real number object is an important study since categories of relations are used to specify, implement, and verify programs. Usually objects of the category represent types, and relations represent programs of the given programming language and/or properties thereof. The current stream of investigation continued in the current paper allows to utilize the real numbers in such an environment. In this paper we will focus on the least-upper-bound property of a real number object. This property is essential for showing the Archimedean property of the additive group which is planned as future work. # Mathematical Preliminaries In this chapter we want to introduce the mathematical notions used in this paper. We start with the theory of of Heyting algebras as a suitable abstract theory of relations. ## Heyting Categories In this section we want to recall some basic notions from categories and allegories [@Freyd; @Book; @Fuzzy5]. First, we are going to introduce Heyting categories as an extension of division allegories defined in [@Freyd], i.e., a Heyting category is a division allegory where the lattice of relations between two objects is a Heyting algebra instead of just a distributive lattice. Heyting categories are also a version of Dedekind categories introduced in [@oli80; @oli95] without the requirement of completeness of lattice of relations between two objects. We will write $R:A\to B$ to indicate that a morphism $R$ of a category ${\cal C}$ has source $A$ and target $B$. Composition and the identity morphism are denoted by $;$ and ${\mathbb I}_A$, respectively. Please note that composition has to be read from left to right, i.e., $Q;R$ means first $Q$ and then $R$. A Heyting category ${\cal R}$ is a category satisfying the following: 1. For all objects $A$ and $B$ the collection ${\cal R}[A,B]$ is a Heyting algebra. Meet, join, relative pseudo-complement, the induced ordering, the least and the greatest element are denoted by $\sqcap,\sqcup,\to,\sqsubseteq,\Bot_{AB},\Top_{AB}$, respectively. 2. $Q;\Bot_{BC}=\Bot_{AC}$ for all relations $Q:A\to B$. 3. There is a monotone operation ${\rule{0pt}{6pt}}^{\smallsmile}$ (called converse) mapping a relation $Q:A\to B$ to ${Q}^{\smallsmile}:B\to A$ such that for all relations $Q:A\to B$ and $R:B\to C$ the following holds: ${(Q;R)}^{\smallsmile}={R}^{\smallsmile};{Q}^{\smallsmile}$ and ${({Q}^{\smallsmile})}^{\smallsmile}=Q$. 4. For all relations $Q:A\to B, R:B\to C$ and $S:A\to C$ the modular inclusion $(Q;R)\sqcap S\sqsubseteq Q;(R\sqcap({Q}^{\smallsmile};S))$ holds. 5. For all relations $R:B\to C$ and $S:A\to C$ there is a relation $S/R:A\to B$ (called the right residual of $S$ and $R$) such that for all $X:A\to B$ the following holds: $X;R\sqsubseteq S\iff X\sqsubseteq S/R.$ Throughout the paper we will use the axioms and some basics facts such as monotonicity of the operations without mentioning. If we define the left residual $Q\backslash R:B\to C$ of two relations $Q:A\to B$ and $R:A\to C$ by $Q\backslash R:={({R}^{\smallsmile}/{Q}^{\smallsmile})}^{\smallsmile}$ we immediately obtain $X\sqsubseteq Q\backslash R$ iff $Q;X\sqsubseteq R$. Using both residuals we define the symmetric quotient as ${\rm syQ}(Q,R)=(Q\backslash R)\sqcap({Q}^{\smallsmile}/{R}^{\smallsmile})$. This construction is characterized by $X\sqsubseteq{\rm syQ}(Q,R)$ iff $Q;X\sqsubseteq R$ and $R;{X}^{\smallsmile}\sqsubseteq Q$. As usual we define the pseudo-complement $Q^\star$ of a relation $Q:A\to B$ by $Q^\star=Q\to\Bot_{AB}$. Notice that the operation $^\star$ is antitone and that we have $Q^{\star\star\star}=Q^\star$, $\Bot_{AB}^\star=\Top_{AB}$, $\Top_{AB}^\star=\Bot_{AB}$, $(Q\sqcup R)^\star=Q^\star\sqcap R^\star$ and $(Q\sqcap R)^{\star\star}=Q^{\star\star}\sqcap R^{\star\star}$. Following usual conventions we call a relation $Q:A\to B$ complemented iff there is a relations $R:A\to B$ with $Q\sqcap R=\Bot_{AB}$ and $Q\sqcup R=\Top_{AB}$ and regular iff $Q^{\star\star}=Q$. The following lemma relates these notions. [\[LemCompReg\]]{#LemCompReg label="LemCompReg"} Suppose $Q:A\to B$ has a complement $R:A\to B$. Then we have 1. $Q^\star=R$ and $R^\star=Q$, 2. $Q$ and $R$ are regular. *Proof.* 1. From $Q\sqcap R=\Bot_{AB}$ we immediately conclude $R\sqsubseteq Q^\star$. For the converse inclusion consider 2 R &= R(QQ\^)\ &= (RQ)(RQ\^)\ &= RQ\^, && $R$ complement of $Q$ which shows that $Q^\star\sqsubseteq R$. By switching the role of $Q$ and $R$ we obtain the second equation. 2. The first property follows from $Q^{\star\star}=R^\star=Q$ using 1. The second property is shown analogously.0◻  ◻ Now, we provide a version of the so-called Schröder equivalences known to be valid if each lattice of relations is in fact a Boolean algebra in the more general context of Heyting categories. [\[Lem:Schroeder\]]{#Lem:Schroeder label="Lem:Schroeder"} Let $Q:A\to B$, $R:B\to C$ and $S:A\to C$ be relations. The we have $$Q;R\sqsubseteq S^\star \iff {Q}^{\smallsmile};S\sqsubseteq R^\star \iff S;{R}^{\smallsmile}\sqsubseteq Q^\star.$$ *Proof.* We only show $\Rightarrow$ of the first equivalence. All other implications follow analogously. Suppose $Q;R\sqsubseteq S^\star$. Then we have $Q;R\sqcap S=\Bot_{AC}$. We obtain ${Q}^{\smallsmile};S\sqcap R\sqsubseteq {Q}^{\smallsmile};(S\sqcap Q;R)=\Bot_{BC}$, which immediately implies ${Q}^{\smallsmile};S\sqsubseteq R^\star$.0◻ ◻ The Schröder equivalences allow to replace certain residuals by a combination of negations and compositions as the next lemma shows. [\[Lem:ResNeg\]]{#Lem:ResNeg label="Lem:ResNeg"} Suppose $Q:A\to B$, $R:A\to C$, and $S:B\to C$ are relations. Then we have 1. ${{Q}^{\smallsmile}}^\star={{Q^\star}}^{\smallsmile}$, 2. $R^\star/S=(R^{\star\star};{S}^{\smallsmile})^\star$, 3. $Q\backslash R^\star=({Q}^{\smallsmile};R^{\star\star})^\star$. *Proof.* 1. From ${{Q^\star}}^{\smallsmile}\sqcap{Q}^{\smallsmile}={(Q^\star\sqcap Q)}^{\smallsmile}=\Bot_{BA}$ we obtain the inclusion $\sqsupseteq$. For the converse inclusion compute ${{{{Q}^{\smallsmile}}^\star}}^{\smallsmile}\sqcap Q={({{Q}^{\smallsmile}}^\star\sqcap{Q}^{\smallsmile})}^{\smallsmile}=\Bot_{AB}$. This implies ${{{{Q}^{\smallsmile}}^\star}}^{\smallsmile}\sqsubseteq Q^\star$, and, hence, ${{Q}^{\smallsmile}}^\star\sqsubseteq{{Q^\star}}^{\smallsmile}$. 2. We have 2 (R\^/S)R\^;S\^ &((R\^/S);SR\^);S\^\ &(R\^R\^);S\^\ &= \_BA, which immediately implies $R^\star/S\sqsubseteq(R^{\star\star};{S}^{\smallsmile})^\star$. For the converse inclusion we apply Lemma [\[Lem:Schroeder\]](#Lem:Schroeder){reference-type="ref" reference="Lem:Schroeder"} to the inclusion $R^{\star\star};{S}^{\smallsmile}\sqsubseteq(R^{\star\star};{S}^{\smallsmile})^{\star\star}$ and obtain $(R^{\star\star};{S}^{\smallsmile})^\star;S\sqsubseteq R^{\star\star\star}=R^\star$. This implies $(R^{\star\star};{S}^{\smallsmile})^\star\sqsubseteq R^\star/S$. 3. We immediately compute 2 Q\\R\^ &= (R\^\^/Q\^)\^\ &= (R\^\^/Q\^)\^ && by 1.\ &= (R\^\^;Q)\^\^ && by 2.\ &= (R\^\^;Q)\^\^&& by 1.\ &= (Q\^;R\^\^\^)\^\ &= (Q\^;R\^)\^&& by 1.  ◻ The following lemma summarizes some basic properties that will be used throughout the paper. A proof can be found in [@Schmidt; @RedBible; @Winter; @Book]. [\[Lem:Basics\]]{#Lem:Basics label="Lem:Basics"} Let $Q:A\to B$, $R:A\to C$ and $S:C\to D$ be relations, and $i,j:A\to A$ be partial identities, i.e., $i,j\sqsubseteq{\mathbb I}_A$. Then we have 1. ${i}^{\smallsmile}=i$, 2. $i;j=i\sqcap j$ and $i;i=i$, 3. $(Q;\Top_{BC}\sqcap R);S=Q;\Top_{BD}\sqcap R;S$. The domain of a relation $R:A\to B$, i.e., the set of elements that are related to at least one other element, can be be defined as ${\rm dom}(R)={\mathbb I}_A\sqcap R;{R}^{\smallsmile}$. Please note that we have ${\rm dom}(R);R=R$ which we will use throughout the paper without mentioning. A relation $Q:A\to B$ is called univalent (or partial function) iff ${Q}^{\smallsmile};Q\sqsubseteq{\mathbb I}_B$, total iff ${\mathbb I}_A\sqsubseteq Q;{Q}^{\smallsmile}$, injective iff ${Q}^{\smallsmile}$ is univalent, surjective iff ${Q}^{\smallsmile}$ is total, a map iff $Q$ is total and univalent. The following lemma states some basic properties of univalent relations and maps. Again, a proof can be found in [@Schmidt; @RedBible; @Winter; @Book]. [\[Lem:Maps\]]{#Lem:Maps label="Lem:Maps"} Let $f:A\to B$ be a mapping, $g:B\to A$ univalent, $Q:C\to A$, $R:C\to B$, $S:A\to C$ and $T,U:A\to D$. Then we have 1. $Q;f\sqsubseteq R$ iff $Q\sqsubseteq R;{f}^{\smallsmile}$, 2. $(Q;{g}^{\smallsmile}\sqcap R);g=Q\sqcap R;g$, 3. $g;(T\sqcap U)=g;T\sqcap f;U$. The following lemma lists some important properties of the residuals and the symmetric quotient that are needed throughout the paper. A proof can be found in [@Freyd; @Schmidt; @RedBible; @Book]. [\[Lem:ResBasics\]]{#Lem:ResBasics label="Lem:ResBasics"} Suppose $Q:A\to B$, $R:A\to C$, and $S:B\to C$ are relations, and $f:D\to B$ is a map. Then we have 1. $Q;(Q\backslash R)\sqsubseteq R$ and $(R/S);S\sqsubseteq R$, 2. $f;(Q\backslash R)=Q;{f}^{\smallsmile}\backslash R$ and $(R/S);{f}^{\smallsmile}=R/f;S$, 3. ${{\rm syQ}(Q,R)}^{\smallsmile}={\rm syQ}(R,Q)$, 4. $f;{\rm syQ}(Q,R)={\rm syQ}(Q;{f}^{\smallsmile},R)$. For a singleton set $\{*\}$ and concrete relations we obviously have ${\mathbb I}_{\{*\}}=\Top_{\{*\}\{*\}}$. Furthermore, for any set $A$ the relation $\Top_{A\{*\}}$ is actually a map. The first property together with the totality in the second property also characterize singleton sets up to isomorphism. Therefore, we define a unit $1$ as an abstract version of a singleton set by ${\mathbb I}_1=\Top_{11}$ and $\Top_{A1}$ is total for every object $A$. Considering concrete relation a map $p:1\to A$, i.e., a relation that maps $*$ to one element $a$ in $A$, can be identified with the element $a$. Therefore we call a map $p:1\to A$ a point (of $A$). Another important concept is the notion of a relational product, i.e., an abstract version of the Cartesian product of sets. The object $A\times B$ is characterized by the projection relations $\pi:A\times B\to A$ and $\rho:A\times B\to B$ satisfying $${\pi}^{\smallsmile};\pi\sqsubseteq{\mathbb I}_A,~~~ {\rho}^{\smallsmile};\rho\sqsubseteq{\mathbb I}_B,~~~ \pi;{\pi}^{\smallsmile}\sqcap\rho;{\rho}^{\smallsmile}={\mathbb I}_{A\times B},~~~ {\pi}^{\smallsmile};\rho=\Top_{AB}.$$ Given relational products we will use the following abbreviations 2 QR &= Q;\^R;\^,\ QS &= ;Q; S,\ QT &= ;Q;\^;T;\^=Q;\^T;\^=;Q;T, and obtain the following properties [@SchmidtWinter]. [\[Lem:Products\]]{#Lem:Products label="Lem:Products"} If all relational products exist, then we have 1. ${(Q\olessthan R)}^{\smallsmile}={Q}^{\smallsmile}\ogreaterthan{R}^{\smallsmile}$ and ${(Q\ogreaterthan S)}^{\smallsmile}={Q}^{\smallsmile}\olessthan{S}^{\smallsmile}$, 2. If $R$ is total, then $(Q\olessthan R);\pi=Q$ and if $Q$ is total, then $(Q\olessthan R);\rho=R$, 3. If $S$ is surjective, then ${\pi}^{\smallsmile};(Q\ogreaterthan S)=Q$ and if $Q$ is surjective, then ${\rho}^{\smallsmile};(Q\ogreaterthan S)=S$, 4. If $f$ is univalent, then $f;(Q\olessthan R)=f;Q\olessthan f;R$ and if $g$ is injective, then $(Q\ogreaterthan S);g=Q;g\ogreaterthan S;g$, 5. $(Q\olessthan R);(T\ogreaterthan U)=Q;T\sqcap R;U$, 6. $(Q\olessthan R);(T\otimes V)=Q;T\olessthan R;V$ and $(Q\otimes X);(T\ogreaterthan U)=Q;T\ogreaterthan X;U$, 7. $Q;{\pi}^{\smallsmile}\ogreaterthan(R\olessthan S)=(Q\ogreaterthan R)\olessthan\rho;S$. We also use the following two bijective mappings ${\rm assoc}:A\times (B\times C)\to (A\times B)\times C$ and ${\rm swap}:A\times B\to B\times A$ defined by 2 assoc&= ;\^;\^;;\^;\^;;\^= (I\_A);= \^;\^(\^\_C),\ swap&= ;\^;\^= = \^\^. The following properties have been shown in [@Winter22-1]. [\[Lem:AssocSwap\]]{#Lem:AssocSwap label="Lem:AssocSwap"} 1. ${{\rm swap}}^{\smallsmile}={\rm swap}$. 2. $(Q\olessthan R);{\rm swap}=R\olessthan Q$ and ${\rm swap};(Q\ogreaterthan S)=S\ogreaterthan Q$. 3. ${\rm swap};(Q\otimes T)=(T\otimes Q);{\rm swap}$. 4. $(U\olessthan(Q\olessthan R));{\rm assoc}=(U\olessthan Q)\olessthan R$ and ${\rm assoc};((Q\ogreaterthan S)\ogreaterthan V)=Q\ogreaterthan (S\ogreaterthan V)$. 5. ${\rm assoc};((Q\otimes T)\otimes X)=(Q\otimes (T\otimes X));{\rm assoc}$. With the maps above we are now ready to define an abelian group within a Heyting category. A quadruple $(A,e,f,n)$ in a Heyting category ${\cal R}$ is called an abelian group iff $A$ is an object, $e:1\to A$ is a point, and $f:A\times A\to A$ and $n:A\to A$ are maps satisfying: 1. $f$ is associative, i.e., $({\mathbb I}_A\otimes f);f={\rm assoc};(f\otimes{\mathbb I}_A);f$, 2. $e$ is the neutral element of $f$, i.e., $({\mathbb I}_A\olessthan\Top_{A1};e);f={\mathbb I}_A$, 3. $n$ is the complement map, i.e., $({\mathbb I}_A\olessthan n);f=\Top_{A1};e$, 4. $f$ is commutative. i.e., ${\rm swap};f=f$. The next lemma lists some basic properties of abelian groups. [\[Lem:GroupProps\]]{#Lem:GroupProps label="Lem:GroupProps"} Let $(A,e,f,n)$ be an abelian group. The we have 1. ${{\rm assoc}}^{\smallsmile};({\mathbb I}_A\otimes f);f=(f\otimes{\mathbb I}_A);f$, 2. $(\Top_{A1};e\olessthan{\mathbb I}_A);f={\mathbb I}_A$ and $(n\olessthan {\mathbb I}_A);f=\Top_{A1};e$, 3. If $g,h:A\to A$ are maps with $(g\olessthan h);f=\Top_{A1};e$, then $g;n=h$, 4. $(n\otimes n);f;n = f$. *Proof.* 1. This follows immediately from the fact that ${\rm assoc}$ is a bijective map. 2. Both equations follow immediately form the commutativity of $f$. 3. We immediately compute 2 g;n &= g;n;(I\_A\_A1;e);f && $e$ is neutral\ &= (g;n\_A1;e);f && Lemma [\[Lem:Basics\]](#Lem:Basics){reference-type="ref" reference="Lem:Basics"}(3)\ &= (g;n(gh);f);f && assumption\ &= (g;n(gh));(I\_Af);f && Lemma [\[Lem:Products\]](#Lem:Products){reference-type="ref" reference="Lem:Products"}(6)\ &= (g;n(gh));assoc;(f_A);f && $f$ associative\ &= ((g;ng)h);(f_A);f && Lemma [\[Lem:AssocSwap\]](#Lem:AssocSwap){reference-type="ref" reference="Lem:AssocSwap"}(4)\ &= (g;(n_A)h);(f_A);f && Lemma [\[Lem:Maps\]](#Lem:Maps){reference-type="ref" reference="Lem:Maps"}(3)\ &= (g;(n_A);fh);f && Lemma [\[Lem:Products\]](#Lem:Products){reference-type="ref" reference="Lem:Products"}(6)\ &= (g;\_A1;eh);f && by 2.\ &= (\_A1;eh);f && $g$ total\ &= h;(\_A1;e_A);f && Lemma [\[Lem:Basics\]](#Lem:Basics){reference-type="ref" reference="Lem:Basics"}(3)\ &= h. && by 2. 4. First of all, the relation $b=({\rm swap}\otimes{\mathbb I}_{A\times A});{\rm assoc};({{\rm assoc}}^{\smallsmile};{\rm swap}\otimes{\mathbb I}_A);{{\rm assoc}}^{\smallsmile}:(A\times A)\times (A\times A)\to (A\times A)\times (A\times A)$ is a bijective map because ${\rm swap}$ and ${\rm assoc}$ are. Furthermore, we have 2\ &= ((;n;n)\_AA);(swap\_AA);assoc\ &= ((;n;n);swap\_AA);assoc&& Lemma [\[Lem:Products\]](#Lem:Products){reference-type="ref" reference="Lem:Products"}(6)\ &= ((;n;n)\_AA);assoc&& Lemma [\[Lem:AssocSwap\]](#Lem:AssocSwap){reference-type="ref" reference="Lem:AssocSwap"}(2)\ &= ((;n;n)());assoc\ &= ((;n;n))&& Lemma [\[Lem:AssocSwap\]](#Lem:AssocSwap){reference-type="ref" reference="Lem:AssocSwap"}(4)\ \ &= (((;n;n));assoc\^;swap);assoc\^ && Lemma [\[Lem:Products\]](#Lem:Products){reference-type="ref" reference="Lem:Products"}(6)\ &= ((;n(;n));swap);assoc\^ && Lemma [\[Lem:AssocSwap\]](#Lem:AssocSwap){reference-type="ref" reference="Lem:AssocSwap"}(4)\ &= (((;n);n));assoc\^ && Lemma [\[Lem:AssocSwap\]](#Lem:AssocSwap){reference-type="ref" reference="Lem:AssocSwap"}(2)\ &= (;n)(;n) && Lemma [\[Lem:AssocSwap\]](#Lem:AssocSwap){reference-type="ref" reference="Lem:AssocSwap"}(4)\ &= ;(n_A);(n_A) && Lemma [\[Lem:Maps\]](#Lem:Maps){reference-type="ref" reference="Lem:Maps"}(3)\ &= (n_A)(n_A), i,e., $((n\otimes n)\olessthan{\mathbb I}_{A\times A});b=(n\olessthan{\mathbb I}_A)\otimes(n\olessthan{\mathbb I}_A)$. 2\ &= assoc\^;(swap;ff);f && Lemma [\[Lem:Products\]](#Lem:Products){reference-type="ref" reference="Lem:Products"}(6)\ &= assoc\^;(ff);f && $f$ commutative\ &= assoc\^;(f\_AA);(I\_Af);f && Lemma [\[Lem:Products\]](#Lem:Products){reference-type="ref" reference="Lem:Products"}(6)\ &= assoc\^;(f(I\_A_A));(I\_Af);f && I\_A_A=I\_AA\ &= ((f_A)\_A);assoc\^;(I\_Af);f && Lemma [\[Lem:AssocSwap\]](#Lem:AssocSwap){reference-type="ref" reference="Lem:AssocSwap"}(5)\ &= ((f_A)\_A);(f_A);f && by 1.\ &= ((f_A);f_A);f && Lemma [\[Lem:Products\]](#Lem:Products){reference-type="ref" reference="Lem:Products"}(6)\ \ &= assoc;(swap;assoc;(f_A);f_A);f && Lemma [\[Lem:Products\]](#Lem:Products){reference-type="ref" reference="Lem:Products"}(6)\ &= assoc;(swap;(I\_Af);f_A);f && $f$ associative\ &= assoc;((f_A);swap;f_A);f && Lemma [\[Lem:AssocSwap\]](#Lem:AssocSwap){reference-type="ref" reference="Lem:AssocSwap"}(3)\ &= assoc;((f_A);f_A);f && $f$ commutative\ &= (f(I\_A_A));assoc;(f_A);f && Lemma [\[Lem:AssocSwap\]](#Lem:AssocSwap){reference-type="ref" reference="Lem:AssocSwap"}(5)\ &= (f\_AA);assoc;(f_A);f && I\_A_A=I\_AA\ &= (f\_AA);(I\_Af);f && $f$ associative\ &= (ff);f, && Lemma [\[Lem:Products\]](#Lem:Products){reference-type="ref" reference="Lem:Products"}(6) i.e., ${b}^{\smallsmile};(f\otimes f);f=(f\otimes f);f$. Together we conclude 2 ((nn);ff);f &= ((nn)\_AA);(ff);f && Lemma [\[Lem:Products\]](#Lem:Products){reference-type="ref" reference="Lem:Products"}(6)\ &= ((nn)\_AA);b;b\^;(ff);f && $b$ bijective\ &= ((n_A)(n_A));(ff);f && see above\ &= ((n_A);f(n_A);f);f && Lemma [\[Lem:Products\]](#Lem:Products){reference-type="ref" reference="Lem:Products"}(6)\ &= (\_A1;e\_A1;e);f && by 2.\ &= \_A1;e;(I\_A\_A1;e);f && Lemma [\[Lem:Basics\]](#Lem:Basics){reference-type="ref" reference="Lem:Basics"}(3)\ &= \_A1;e. && $e$ neutral From 3. we obtain $(n\otimes n);f;n=f$.0◻  ◻ A relation $C:X\to X$ is called transitive if $C;C\sqsubseteq C$, dense if $C\sqsubseteq C;C$, asymmetric if $C\sqcap{C}^{\smallsmile}=\Bot_{XX}$, a strict-order if $C$ is transitive and asymmetric, a linear strict-order if $C$ is a strict-order and ${\mathbb I}_X\sqcup C\sqcup{C}^{\smallsmile}=\Top_{XX}$. Given a strict-order $C:X\to X$ we define its associate ordering $E={\mathbb I}_X\sqcup C$. It is easy to verify that $E$ is an ordering, i.e., $E$ is reflexive ${\mathbb I}_X\sqsubseteq E$, $E$ is transitive, and $E$ is antisymmetric $E\sqcap{E}^{\smallsmile}={\mathbb I}_X$. The next lemma verifies that a linear strict-order does always have a complement. [\[Lem:StrictComp\]]{#Lem:StrictComp label="Lem:StrictComp"} If $C:A\to A$ is a linear strict-order and $E$ its associated ordering, then we have 1. $C\sqcap{\mathbb I}_A=\Bot_{AA}$, 2. $C$ is complemented with complement ${E}^{\smallsmile}$, 3. $C^\star={E}^{\smallsmile}$ and $E^\star={C}^{\smallsmile}$, 4. $C$ and $E$ are regular. *Proof.* 1. We have 2 C_A &= (C_A)(C_A)\ &= (C_A)\^ && Lemma [\[Lem:Basics\]](#Lem:Basics){reference-type="ref" reference="Lem:Basics"}(1)\ &= C_A\^\ &= \_AA. && $C$ linear 2. We have $C\sqcup{E}^{\smallsmile}=C\sqcup{C}^{\smallsmile}\sqcup{\mathbb I}_A=\Top_{AA}$ because $C$ is linear. On the other hand, the asymmetry and (1) implies $C\sqcap{E}^{\smallsmile}=C\sqcap({C}^{\smallsmile}\sqcup{\mathbb I}_A)=(C\sqcap{C}^{\smallsmile})\sqcup(C\sqcap{\mathbb I}_A)=\Bot_{AA}$. Property 3. and 4. follow immediately from 2. and Lemma [\[LemCompReg\]](#LemCompReg){reference-type="ref" reference="LemCompReg"}.0◻ ◻ If $E$ is an order we define 2 ubd\_E(R) &:= R\^\\E,\ lbd\_E(R) &:= R\^\\E\^,\ lub\_E(R) &:= ubd\_E(R)\_E(ubd\_E(R)),\ glb\_E(R) &:= lbd\_E(R)\_E(lbd\_E(R)). For concrete relations the construction ${\rm ubd}_{E}(R)$ relates $b$ with the upper bounds of the set of elements related to $b$ in $R$, i.e., the image $\{a\mid (b,a)\in R\}$ of $b$ in $R$. Similarly, ${\rm lbd}_{E}(R)$ computes the lower bounds. Finally, ${\rm lub}_{E}(R)$ (resp. ${\rm glb}_{E}(R)$) maps $b$ to the least upper (greatest lower) bound of the image of $b$ in $R$. [\[Lem:BoundLemma\]]{#Lem:BoundLemma label="Lem:BoundLemma"} Let $E:A\to A$ be an ordering, and $X:B\to A$ a relation. Then we have ${\rm ubd}_{E}(X);E={\rm ubd}_{E}(X)$ and ${\rm lbd}_{E}(X);{E}^{\smallsmile}={\rm lbd}_{E}(X)$. *Proof.* We only show the first property since the second can be shown analogously. We have ${X}^{\smallsmile};{\rm ubd}_{E}(X);E={X}^{\smallsmile};({X}^{\smallsmile}\backslash E);E\sqsubseteq E;E\sqsubseteq E$ since $E$ is transitive. This implies ${\rm ubd}_{E}(X);E\sqsubseteq{X}^{\smallsmile}\backslash E={\rm ubd}_{E}(X)$. The converse inclusion follows from the reflexivity of $E$.0◻ ◻ If we consider a concrete linear strict-order and an element $a$ strictly below the least upper bound of a set $M$, then there is an element $b$ in $M$ that is already strictly greater than $a$. We want to show this property in arbitrary Heyting categories. It turns out that we can only show this property if we use double-negation in the conclusion. [\[Lem:DownClosed\]]{#Lem:DownClosed label="Lem:DownClosed"} If $C:A\to A$ is a linear strict-order and $X:B\to A$ a relation, then ${\rm lub}_{E}(X);{C}^{\smallsmile}\sqsubseteq(X;{C}^{\smallsmile})^{\star\star}$. *Proof.* First of all, we have 2 lub\_E(X)\^;(X;C\^)\^ &= lub\_E(X)\^;(X;C\^\^)\^&& Lemma [\[Lem:StrictComp\]](#Lem:StrictComp){reference-type="ref" reference="Lem:StrictComp"}(4)\ &= lub\_E(X)\^;(X\^\\C\^\^) && Lemma [\[Lem:ResNeg\]](#Lem:ResNeg){reference-type="ref" reference="Lem:ResNeg"}(3)\ &= lub\_E(X)\^;(X\^\\E) && Lemma [\[Lem:StrictComp\]](#Lem:StrictComp){reference-type="ref" reference="Lem:StrictComp"}(3)\ &= lub\_E(X)\^;ubd\_E(X)\ &\^;ubd\_E(X)\ &= (E/ubd\_E(X));ubd\_E(X)\ &E && Lemma [\[Lem:ResBasics\]](#Lem:ResBasics){reference-type="ref" reference="Lem:ResBasics"}(1)\ &= C\^\^. && Lemma [\[Lem:StrictComp\]](#Lem:StrictComp){reference-type="ref" reference="Lem:StrictComp"}(3) This implies ${\rm lub}_{E}(X);{C}^{\smallsmile}\sqsubseteq(X;{C}^{\smallsmile})^{\star\star}$ by using the Schröder equivalences (Lemma [\[Lem:Schroeder\]](#Lem:Schroeder){reference-type="ref" reference="Lem:Schroeder"}).0◻ ◻ The following example demonstrates that we cannot remove the double-negation on the right-hand side of the inclusion of the previous lemma. For this consider the 3-element chain $0,u,1$ with $0$ and $1$ as smallest resp. greatest element. It is well-known that the finite matrices with coefficients from a Heyting algebra form a Heyting category. Now consider the following relations: $$C=\left(\begin{array}{cc}0 & 1\\ 0 & 0\end{array}\right),~~~~X=\left(\begin{array}{cc}0 & u\end{array}\right).$$ $C$ is a linear strict-order and we have $$\begin{array}{l@{~~~~}l@{~~~~}l} {\rm ubd}_{E}(X)=\left(\begin{array}{cc}0 & 1\end{array}\right), & {\rm lbd}_{E}({\rm ubd}_{E}(X))=\left(\begin{array}{cc}1 & 1\end{array}\right), & {\rm lub}_{E}(X)=\left(\begin{array}{cc}0 & 1\end{array}\right),\\ {\rm lub}_{E}(X);{C}^{\smallsmile}=\left(\begin{array}{cc}1 & 0\end{array}\right), & X;{C}^{\smallsmile}=\left(\begin{array}{cc}u & 0\end{array}\right), & (X;{C}^{\smallsmile})^{\star\star}=\left(\begin{array}{cc}1 & 0\end{array}\right). \end{array}$$ The relation algebraic version of a power set is given by a so-called relational (or direct) power. An object ${\mathcal P}(A)$ together with a relation $\varepsilon:A\to{\mathcal P}(A)$ is called a relational (or direct) power of $A$ iff $${\rm syQ}(\varepsilon,\varepsilon)={\mathbb I}_{{\mathcal P}(A)}~~~~\mbox{and}~~~~{\rm syQ}(Q,\varepsilon)\mbox{ is total for every }Q:A\to B.$$ Please note that ${\rm syQ}({R}^{\smallsmile},\varepsilon)$ is a map for every relation $R:B\to A$. In fact, this construction is the existential image of $R$, i.e., $x$ is mapped by ${\rm syQ}({R}^{\smallsmile},\varepsilon)$ to the set $\{y\mid (x,y)\in R\}$ for concrete relations. Furthermore, we have the following [@Schmidt; @RedBible; @Winter; @Book]. [\[Lem:PowerBasic\]]{#Lem:PowerBasic label="Lem:PowerBasic"} Suppose $R:A\to B$ is a relation. Then we have ${\rm syQ}({R}^{\smallsmile},\varepsilon);{\varepsilon}^{\smallsmile}=R$. The fourth axiom of Tarski's axiomatization of the real numbers requires for all non-empty subsets $X,Y$ of the real numbers with $x<y$ for every $x\in X$ and $y\in Y$ the existence of an element $z$with $x\leq z$ and $z\leq y$ for all $x\in X$ and $y\in Y$. Our original definition [@PartI] of a real number object used the relational power from above which includes the empty set in the case of concrete relations. This is not correct but, fortunately, the results presented in [@PartI] did not rely on Axiom 4, i.e., Axiom 4 was never used. In the current paper we fix this mistake and switch to the non-empty relational power that corresponds to the set of non-empty subsets in the case of concrete relations. An object ${\mathcal P}_{\!ne}(A)$ together with a relation $\epsilon:A\to{\mathcal P}_{\!ne}(A)$ is called a non-empty relational power of $A$ iff $${\rm syQ}(\epsilon,\epsilon)={\mathbb I}_{{\mathcal P}_{\!ne}(A)}~~~~\mbox{and}~~~~{\rm dom}({\rm syQ}(Q,\epsilon))={\rm dom}({Q}^{\smallsmile})\mbox{ for every }Q:A\to B.$$ We want to show that non-empty relational powers exists if relational powers and splitting exists. Given a partial equivalence relation $X:A\to A$, i.e., $X$ is symmetric (${X}^{\smallsmile}=X$) and transitive, an object $B$ together with a relation $R:B\to A$ is called a splitting of $X$ iff $R;{R}^{\smallsmile}={\mathbb I}_B$ and ${R}^{\smallsmile};R=X$. Intuitively, the object $B$ consists of all (existing) equivalence classes of $X$ and $R$ relates such an equivalence class with its elements. Please note that requiring the existence of splittings is not really an additional assumption since every Heyting category can be fully embedded into a Heyting category with all splittings [@Freyd; @Winter]. If $i:C\to{\mathbb P}(A)$ splits ${\rm dom}({\varepsilon}^{\smallsmile})$, then $C$ together with $\varepsilon;{i}^{\smallsmile}$ is a non-empty relational power. *Proof.* First of all, we have 2 syQ(;i\^,;i\^) &= i;syQ(,);i\^ && Lemma [\[Lem:ResBasics\]](#Lem:ResBasics){reference-type="ref" reference="Lem:ResBasics"}(4)\ &= i;i\^ && definition $\varepsilon$\ &= I\_C. && definition $i$ Now, suppose $Q:A\to B$. Then we have 2\ &= I\_B\^;Q\ &= I\_B(Q,);\^;;syQ(,Q) && Lemma [\[Lem:PowerBasic\]](#Lem:PowerBasic){reference-type="ref" reference="Lem:PowerBasic"}\ &= I\_B(Q,);\^;;syQ(Q,)\^ && Lemma [\[Lem:ResBasics\]](#Lem:ResBasics){reference-type="ref" reference="Lem:ResBasics"}(3)\ &= I\_B(Q,);syQ(Q,)\^(Q,);\^;;syQ(Q,)\^ && ${\rm syQ}(Q,\varepsilon)$ total\ &= I\_B(Q,);(I\^;);syQ(Q,)\^ && Lemma [\[Lem:Maps\]](#Lem:Maps){reference-type="ref" reference="Lem:Maps"}(3)\ &= I\_B(Q,);i\^;i;syQ(Q,)\^ && definition $i$\ &= I\_B(Q,;i\^);syQ(Q,;i\^)\^ && Lemma [\[Lem:ResBasics\]](#Lem:ResBasics){reference-type="ref" reference="Lem:ResBasics"}(3)\ &= dom(syQ(Q,;i\^)).  ◻ Last but not least, we want to show Lemma [\[Lem:PowerBasic\]](#Lem:PowerBasic){reference-type="ref" reference="Lem:PowerBasic"} also for non-empty powers. [\[Lem:TPowerBasic\]]{#Lem:TPowerBasic label="Lem:TPowerBasic"} Suppose $R:A\to B$ is a relation. Then we have ${\rm syQ}({R}^{\smallsmile},\epsilon);{\epsilon}^{\smallsmile}=R$. *Proof.* First of all, we have ${\rm syQ}({R}^{\smallsmile},\epsilon);{\epsilon}^{\smallsmile}\sqsubseteq(R/{\epsilon}^{\smallsmile}) ;{\epsilon}^{\smallsmile}\sqsubseteq R$. 2 R &= dom(R);R\ &= dom(syQ(R\^,));R && definition $\epsilon$\ &(R\^,);syQ(R\^,)\^;R\ &= syQ(R\^,);syQ(,R\^);R && Lemma [\[Lem:ResBasics\]](#Lem:ResBasics){reference-type="ref" reference="Lem:ResBasics"}(3)\ &(R\^,);(\^/R);R\ &(R\^,);\^.  ◻ # Real Number Object In this section we want to recall the results of [@PartI]. We start with Tarski's axioms as they were stated in [@Tarski]. His axioms are based on the language ${\mathbb R},<,+,1$: Axiom 1: : If $x\ne y$, then $x<y$ or $y<x$. Axiom 2: : If $x<y$, then $y\nless x$. Axiom 3: : If $x<z$, then there is a $y$ such that $x<y$ and $y<z$. Axiom 4: : If $\emptyset\ne X\subseteq{\mathbb R}$ and $\emptyset\ne Y\subseteq{\mathbb R}$ so that for every $x\in X$ and every $y\in Y$ we have $x<y$, then there is a $z$ so that for all $x\in X$ and $y\in Y$ we have $x\leq z$ and $z\leq y$ ($x\leq y$ shorthand for $x<y$ or $x=y$). Axiom 5: : $x+(y+z)=(x+z)+y$. Axiom 6: : For every $x$ and $y$ there is a $z$ such that $x=y+z$. Axiom 7: : If $x+z<y+t$, then $x<y$ or $z<t$. Axiom 8: : $1\in{\mathbb R}$. Axiom 9: : $1<1+1$. The axioms above can be translated into the language of relations leading the definition below. Please note that we added Axiom 0 that states that ${\rm add}$ is a map explicitly since we are dealing with relations rather than functions. [\[Def:RealNumberObject\]]{#Def:RealNumberObject label="Def:RealNumberObject"} An object ${\mathbb R}$ together with three relations ${\i}:1\to{\mathbb R}$, $C:{\mathbb R}\to{\mathbb R}$ and ${\rm add}:{\mathbb R}\times {\mathbb R}\to{\mathbb R}$ is called a real number object if the following holds: 0. ${\rm add}$ is a map. 1. ${\mathbb I}_{\mathbb R}\sqcup C\sqcup{C}^{\smallsmile}=\Top_{\mathbb{R R}}$. 2. $C\sqcap{C}^{\smallsmile}=\Bot_{\mathbb{R R}}$. 3. $C\sqsubseteq C;C$. 4. $\epsilon\backslash (C/{\epsilon}^{\smallsmile})\sqsubseteq(\epsilon\backslash (C\sqcup{\mathbb I}_{\mathbb R}));{(\epsilon\backslash {(C\sqcup{\mathbb I}_{\mathbb R})}^{\smallsmile})}^{\smallsmile}$. 5. $({\mathbb I}_{\mathbb R}\otimes{\rm add});{\rm add}=({\mathbb I}_{\mathbb R}\otimes{\rm swap});{\rm assoc};({\rm add}\otimes{\mathbb I}_{\mathbb R});{\rm add}$. 6. ${\pi}^{\smallsmile};{\rm add}=\Top_{\mathbb{R R}}$. 7. ${\rm add};C;{{\rm add}}^{\smallsmile}\sqsubseteq\pi;C;{\pi}^{\smallsmile}\sqcup\rho;C;{\rho}^{\smallsmile}$. 8. ${\i}$ is a map, i.e., a point. 9. ${\i}\sqsubseteq {\i};({\mathbb I}_{\mathbb R}\olessthan{\mathbb I}_{\mathbb R});{\rm add};{C}^{\smallsmile}$. First we define abstract versions of the number $0$ and of the negation operation on the real numbers by $0=\Top_{1{\mathbb R}};({{\rm add}}^{\smallsmile}\sqcap{\pi}^{\smallsmile});\rho$ and ${\rm neg}={\pi}^{\smallsmile};({\rm add};{Z}^{\smallsmile}\sqcap\rho)$. The first main result of [@PartI] is the following theorem. [\[Th:AddGroup\]]{#Th:AddGroup label="Th:AddGroup"} The quadruple $({\mathbb R},0,{\rm add},{\rm neg})$ is an abelian group. The second main result is concerned with the strict-order $C$. The relation $C:{\mathbb R}\to {\mathbb R}$ is a dense strict linear order. Last but not least, the final result of [@PartI] addresses the monotonicity of ${\rm add}$. [\[Th:AddMono\]]{#Th:AddMono label="Th:AddMono"} We have the following 1. ${\rm add}$ is strictly monotone in each parameter, i.e., $({\mathbb I}_{\mathbb R}\otimes C);{\rm add}\sqsubseteq{\rm add};C$ and $(C\otimes{\mathbb I}_{\mathbb R});{\rm add}\sqsubseteq{\rm add};C$, 2. ${\rm add}$ is strictly monotone. i.e., $(C\otimes C);{\rm add}\sqsubseteq{\rm add};C$, 3. ${\rm add}$ is monotone, i.e., $(E\otimes E);{\rm add}\sqsubseteq{\rm add};E$. # Least-Upper-Bound Property In this final section we want to show the least-upper-bound property of a real number object. But first we will show that adding a constant to a number and subtracting the same constant are inverse operations. Suppose $p:1\to{\mathbb R}$ is a point. Then $({\mathbb I}_{\mathbb R}\olessthan \Top_{{\mathbb R}1};p);{\rm add}$ is strictly monotone and a bijective map with ${(({\mathbb I}_{\mathbb R}\olessthan \Top_{{\mathbb R}1};p);{\rm add})}^{\smallsmile}=({\mathbb I}_{\mathbb R}\olessthan \Top_{{\mathbb R}1};p;{\rm neg});{\rm add}$. *Proof.* First of all, $({\mathbb I}_{\mathbb R}\olessthan \Top_{{\mathbb R}1};p);{\rm add}$ is a map because $p$ and ${\rm add}$ are. Now we show $({\mathbb I}_{\mathbb R}\olessthan \Top_{{\mathbb R}1};p);{\rm add};({\mathbb I}_{\mathbb R}\olessthan \Top_{{\mathbb R}1};p;{\rm neg});{\rm add}={\mathbb I}_{\mathbb R}$ by computing 2\ &= ((I\_R\_R1;p);add\_R1;p;neg);add&& Lemma [\[Lem:Basics\]](#Lem:Basics){reference-type="ref" reference="Lem:Basics"}(3)\ &= ((I\_R\_R1;p)\_R1;p;neg);(add\_R);add&& Lemma [\[Lem:Products\]](#Lem:Products){reference-type="ref" reference="Lem:Products"}(6)\ &= ((I\_R\_R1;p)\_R1;p;neg);assoc\^;(I\_R);add&& Lemma [\[Lem:GroupProps\]](#Lem:GroupProps){reference-type="ref" reference="Lem:GroupProps"}(1)\ &= ((I\_R(\_R1;p\_R1;p;neg));(I\_R);add&& Lemma [\[Lem:AssocSwap\]](#Lem:AssocSwap){reference-type="ref" reference="Lem:AssocSwap"}(3)\ &= ((I\_R\_R1;p;(I\_R));(I\_R);add&& Lemma [\[Lem:Maps\]](#Lem:Maps){reference-type="ref" reference="Lem:Maps"}(3)\ &= ((I\_R\_R1;p;(I\_R);add);add&& Lemma [\[Lem:Products\]](#Lem:Products){reference-type="ref" reference="Lem:Products"}(6)\ &= ((I\_R\_R1;p;\_R1;0);add&& Theorem [\[Th:AddGroup\]](#Th:AddGroup){reference-type="ref" reference="Th:AddGroup"}\ &= ((I\_R\_R1;\_11;0);add&& $p$ total\ &= ((I\_R\_R1;0);add&& $\Top_{11}={\mathbb I}_1$\ &= I\_R. && Theorem [\[Th:AddGroup\]](#Th:AddGroup){reference-type="ref" reference="Th:AddGroup"} The property $({\mathbb I}_{\mathbb R}\olessthan \Top_{{\mathbb R}1};p;{\rm neg});{\rm add};({\mathbb I}_{\mathbb R}\olessthan \Top_{{\mathbb R}1};p);{\rm add}={\mathbb I}_{\mathbb R}$ can be shown analogously. Using Lemma [\[Lem:Basics\]](#Lem:Basics){reference-type="ref" reference="Lem:Basics"}(1) the two properties above imply $({\mathbb I}_{\mathbb R}\olessthan \Top_{{\mathbb R}1};p);{\rm add}\sqsubseteq{(({\mathbb I}_{\mathbb R}\olessthan \Top_{{\mathbb R}1};p;{\rm neg});{\rm add})}^{\smallsmile}$ and $({\mathbb I}_{\mathbb R}\olessthan \Top_{{\mathbb R}1};p;{\rm neg});{\rm add}\sqsubseteq{(({\mathbb I}_{\mathbb R}\olessthan \Top_{{\mathbb R}1};p);{\rm add})}^{\smallsmile}$, and, hence, ${(({\mathbb I}_{\mathbb R}\olessthan \Top_{{\mathbb R}1};p);{\rm add})}^{\smallsmile}=({\mathbb I}_{\mathbb R}\olessthan \Top_{{\mathbb R}1};p;{\rm neg});{\rm add}$ and that $(({\mathbb I}_{\mathbb R}\olessthan \Top_{{\mathbb R}1};p);{\rm add}$ is bijective. It remains to show that $({\mathbb I}_{\mathbb R}\olessthan \Top_{{\mathbb R}1};p);{\rm add}$ is strictly monotone. The computation 2 C;(I\_R\_R1;p);add &= (C\_R1;p);add&& Lemma [\[Lem:Basics\]](#Lem:Basics){reference-type="ref" reference="Lem:Basics"}(3)\ &= (I\_R\_R1;p);(C\_R);add&& Lemma [\[Lem:Products\]](#Lem:Products){reference-type="ref" reference="Lem:Products"}(6)\ &(I\_R\_R1;p);add;C. && Theorem [\[Th:AddMono\]](#Th:AddMono){reference-type="ref" reference="Th:AddMono"}(1) shows that property.0◻ ◻ We now use ${\i}$ for $p$ and define ${\rm succ}=({\mathbb I}_{\mathbb R}\olessthan \Top_{{\mathbb R}1};{\i});{\rm add}$ and ${\rm prec}=({\mathbb I}_{\mathbb R}\olessthan\Top_{{\mathbb R}1};{\i};{\rm neg});{\rm add}$. From the previous lemma we obtain that ${\rm succ}$ as well as ${\rm prec}$ are monotone, strictly monotone, bijective, and ${{\rm succ}}^{\smallsmile}={\rm prec}$. [\[Lem:Props\]]{#Lem:Props label="Lem:Props"} We have 1. ${\rm succ};{\rm neg}={\rm neg};{\rm prec}$, 2. $0\sqsubseteq{\i};{C}^{\smallsmile}$, 3. ${\rm succ}\sqsubseteq C$ and ${\rm prec}\sqsubseteq{C}^{\smallsmile}$, 4. $C$ is total and surjective. *Proof.* 1. From the computation 2 neg;prec;neg &= neg;(I\_R\_R1;ı;neg);add;neg\ &= (neg\_R1;ı;neg);add;neg&& Lemma [\[Lem:Basics\]](#Lem:Basics){reference-type="ref" reference="Lem:Basics"}(3)\ &= (I\_R\_R1;ı);(neg);add;neg&& Lemma [\[Lem:Products\]](#Lem:Products){reference-type="ref" reference="Lem:Products"}(6)\ &= (I\_R\_R1;ı);add&& Lemma [\[Lem:GroupProps\]](#Lem:GroupProps){reference-type="ref" reference="Lem:GroupProps"}(4)\ &= succ we immediately obtain the assertion since ${\rm neg}$ is a bijective map. 2. First of all, from the fact that ${\i}$ is total we obtain ${\i};\Top_{{\mathbb R}1}=\Top_{11}={\mathbb I}_1$. Now we compute 2 0 &= ı;\_R1;0 && see above\ &= ı;(I\_R);add&& Theorem [\[Th:AddGroup\]](#Th:AddGroup){reference-type="ref" reference="Th:AddGroup"}\ &= (ı;neg);add&& Lemma [\[Lem:Maps\]](#Lem:Maps){reference-type="ref" reference="Lem:Maps"}(3)\ &= (ı\_11;ı;neg);add&& \_11=I\_1\ &= ı;(I\_R\_R1;ı;neg);add&& Lemma [\[Lem:Basics\]](#Lem:Basics){reference-type="ref" reference="Lem:Basics"}(3)\ &= ı;prec\ &;(I\_R\_R);add;C\^;prec&& Axiom 9\ &= (ı);add;C\^;prec&& Lemma [\[Lem:Maps\]](#Lem:Maps){reference-type="ref" reference="Lem:Maps"}(3)\ &= (ı\_11ı);add;C\^;prec&& \_11=I\_1\ &= ı;(I\_R\_R1ı);add;C\^;prec&& Lemma [\[Lem:Basics\]](#Lem:Basics){reference-type="ref" reference="Lem:Basics"}(3)\ &= ı;succ;C\^;prec\ &= ı;succ;prec;C\^ && ${\rm prec}$ strictly monotone\ &= ı;C\^. && ${\rm succ}$ and ${\rm prec}$ inverse 3. We obtain 2 I\_R &= (I\_R\_R1;0);add&& Theorem [\[Th:AddGroup\]](#Th:AddGroup){reference-type="ref" reference="Th:AddGroup"}\ &(I\_R\_R1;ı;C\^);add&& by 2.\ &= (I\_R\_R1;ı);(I\_R\^);add&& Lemma [\[Lem:Products\]](#Lem:Products){reference-type="ref" reference="Lem:Products"}(6)\ &(I\_R\_R1;ı);add;C\^ && ${\rm add}$ strictly monotone\ &=succ;C\^ from which we conclude ${\rm succ}\sqsubseteq C;{{\rm succ}}^{\smallsmile};{\rm succ}\sqsubseteq C$ since ${\rm succ}$ is univalent. The second inclusion follows from the first by ${\rm prec}={{\rm succ}}^{\smallsmile}\sqsubseteq{C}^{\smallsmile}$. 4. Both properties follow immediately form 3. because ${\rm succ}$and ${\rm prec}$ are total.0◻  ◻ The following lemma will be needed in the proof of the least-upper-bound property. [\[Lem:XCProps\]]{#Lem:XCProps label="Lem:XCProps"} Suppose $X:A\to{\mathbb R}$. Then we have 1. ${\rm dom}(X)={\rm dom}(X;{C}^{\smallsmile})$, 2. ${\rm ubd}_{E}(X)={\rm ubd}_{E}(X;{C}^{\smallsmile})$. *Proof.* 1. The inclusion $\sqsupseteq$ follows from 2 dom(X;C\^) &= I\_RX;C\^;C;X\^\ &= I\_R(X);X;C\^;C;X\^\ &(X);(dom(X)\^X;C\^;C;X\^)\ &(X);dom(X)\^\ &= dom(X), && Lemma [\[Lem:Basics\]](#Lem:Basics){reference-type="ref" reference="Lem:Basics"}(1&2) and the opposite inclusion from 2 dom(X) &= I\_RX;X\^\ &= I\_R(XX;C\^;C);X\^ && Lemma [\[Lem:Props\]](#Lem:Props){reference-type="ref" reference="Lem:Props"}(4)\ &\_R(I\_RX;C\^;C;X\^);X;X\^\ &= I\_R(X;C\^);X;X\^\ &(X;C\^);(dom(X;C\^)\^X;X\^)\ &(X;C\^);dom(X;C\^)\^\ &= dom(X;C\^). && Lemma [\[Lem:Basics\]](#Lem:Basics){reference-type="ref" reference="Lem:Basics"}(1&2) 2. First of all, we have 2 X\^;ubd\_E(X;C\^)\^ &\^;ubd\_E(X;C\^)\^;C\^ && $C$ transitive\  &\^;(C;X\^;ubd\_E(X;C\^)\^)\ &= C\^;(C;X\^;(C;X\^\\E)\^)\ &\^;(E\^) && Lemma [\[Lem:ResBasics\]](#Lem:ResBasics){reference-type="ref" reference="Lem:ResBasics"}(1)\ &= C\^;\_RR && Lemma [\[Lem:StrictComp\]](#Lem:StrictComp){reference-type="ref" reference="Lem:StrictComp"}(2)\ &= \_RR. This implies 2\ &= X\^;ubd\_E(X;C\^)(C\^E) && $C$ linear\ &= (X\^;ubd\_E(X;C\^)\^)(X\^;ubd\_E(X;C\^)E)\ &= X\^;ubd\_E(X;C\^)E, && see above i.e., ${X}^{\smallsmile};{\rm ubd}_{E}(X;{C}^{\smallsmile})\sqsubseteq E$. We conclude ${\rm ubd}_{E}(X;{C}^{\smallsmile})\sqsubseteq{X}^{\smallsmile}\backslash E={\rm ubd}_{E}(X)$. For the opposite inclusion consider 2 C;X\^;ubd\_E(X) &= C;X\^;(X\^\\E)\ &C;E && Lemma [\[Lem:ResBasics\]](#Lem:ResBasics){reference-type="ref" reference="Lem:ResBasics"}(1)\ &= C;CC\ &= C && $C$ transitive\ &E from which we conclude ${\rm ubd}_{E}(X)\sqsubseteq C;{X}^{\smallsmile}\backslash E={\rm ubd}_{E}(X;{C}^{\smallsmile})$.0◻  ◻ Now, we are ready to show the least-upper-bound property. A relation $X:A\to {\mathbb R}$ can be seen a a collection of subsets of ${\mathbb R}$ indexed by $A$, i.e., every $a\in A$ is related to its image under $X$. The element $a$ is in the domain of $X$ iff its image is non-empty. Therefore, the relation ${\rm dom}(X)\sqcap{\rm dom}({\rm ubd}_{E}(X))$ relates an element $a$ to itself iff its image and the upper bound of its image are not empty. The least-upper-bound property now states that least upper bound for such a set exists, i.e., that ${\rm dom}(X)\sqcap{\rm dom}({\rm ubd}_{E}(X))\sqsubseteq{\rm lub}_{E}(X)$. This is our main theorem of the paper. [\[Th:LUP\]]{#Th:LUP label="Th:LUP"} For every relation $X:A\to{\mathbb R}$ we have ${\rm dom}(X)\sqcap{\rm dom}({\rm ubd}_{E}(X))\sqsubseteq{\rm dom}({\rm lub}_{E}(X))$. *Proof.* First of all, we have 2\ &= (syQ(C;X\^,);\^)\^syQ(ubd\_E(X)\^,));\^\ &= C;X\^;ubd\_E(X) && Lemma [\[Lem:TPowerBasic\]](#Lem:TPowerBasic){reference-type="ref" reference="Lem:TPowerBasic"}\ &= C;X\^;(X\^\\E)\ &C;E && Lemma [\[Lem:ResBasics\]](#Lem:ResBasics){reference-type="ref" reference="Lem:ResBasics"}(1)\ &= C;(C\_R)\ &= C;CC\ &= C && $C$ transitive which immediately implies ${{\rm syQ}(C;{X}^{\smallsmile},\epsilon)}^{\smallsmile};{\rm syQ}({{\rm ubd}_{E}(X)}^{\smallsmile},\epsilon)\sqsubseteq\varepsilon\backslash (C/{\varepsilon}^{\smallsmile})$. We obtain 2\ &= dom(X);dom(ubd\_E(X)) && Lemma [\[Lem:Basics\]](#Lem:Basics){reference-type="ref" reference="Lem:Basics"}(2)\ &= dom(X;C\^);dom(ubd\_E(X)) && Lemma [\[Lem:XCProps\]](#Lem:XCProps){reference-type="ref" reference="Lem:XCProps"}(1)\ &= dom(syQ(C;X\^,));dom(syQ(ubd\_E(X)\^,)) && Definition $\epsilon$\ &(C;X\^,);syQ(C;X\^,)\^;syQ(ubd\_E(X)\^,);syQ(ubd\_E(X)\^,)\^\ &(C;X\^,);(\\(C/\^));syQ(ubd\_E(X)\^,)\^ && see above\ &(C;X\^,);(\\E);(\\E\^)\^;syQ(ubd\_E(X)\^,)\^ && Axiom 4\ &(;syQ(C;X\^,)\^\\E);(;syQ(ubd\_E(X)\^,)\^\\E\^)\^ && Lemma [\[Lem:ResBasics\]](#Lem:ResBasics){reference-type="ref" reference="Lem:ResBasics"}(2)\ &= ((X;C\^)\^\\E);(ubd\_E(X)\^\\E\^)\^ && Lemma [\[Lem:TPowerBasic\]](#Lem:TPowerBasic){reference-type="ref" reference="Lem:TPowerBasic"}\ &= ubd\_E(X;C\^);lbd\_E(ubd\_E(X))\^ && Definition\ &= ubd\_E(X);lbd\_E(ubd\_E(X))\^. && Lemma [\[Lem:XCProps\]](#Lem:XCProps){reference-type="ref" reference="Lem:XCProps"}(2) This immediately implies 2\ &= dom(X)(ubd\_E(X))\^ && Lemma [\[Lem:Basics\]](#Lem:Basics){reference-type="ref" reference="Lem:Basics"}(1)\ &\_E(X);lbd\_E(ubd\_E(X))\^\_E(ubd\_E(X));ubd\_E(X)\^ && see above\ &(ubd\_E(X)\_E(ubd\_E(X));ubd\_E(X)\^;lbd\_E(ubd\_E(X)));lbd\_E(ubd\_E(X))\^\ &= (ubd\_E(X)\_E(ubd\_E(X));ubd\_E(X)\^;(ubd\_E(X)\^\\E\^));lbd\_E(ubd\_E(X))\^\ &(ubd\_E(X)\_E(ubd\_E(X));E\^);lbd\_E(ubd\_E(X))\^ && Lemma [\[Lem:ResBasics\]](#Lem:ResBasics){reference-type="ref" reference="Lem:ResBasics"}(1)\ &= (ubd\_E(X)\_E(ubd\_E(X)));lbd\_E(ubd\_E(X))\^ && Lemma [\[Lem:BoundLemma\]](#Lem:BoundLemma){reference-type="ref" reference="Lem:BoundLemma"}\ &= lub\_E(X);lbd\_E(ubd\_E(X))\^ and 2\ &= dom(X)(ubd\_E(X))\^ && Lemma [\[Lem:Basics\]](#Lem:Basics){reference-type="ref" reference="Lem:Basics"}(1)\ &\_E(X);lbd\_E(ubd\_E(X))\^\_E(ubd\_E(X));ubd\_E(X)\^ && see above\ &(ubd\_E(X);lbd\_E(ubd\_E(X))\^;ubd\_E(X)\_E(ubd\_E(X)));ubd\_E(X)\^\ &= (ubd\_E(X);(ubd\_E(X)\^;(ubd\_E(X)\^\\E\^))\^\_E(ubd\_E(X)));ubd\_E(X)\^\ &(ubd\_E(X);E\_E(ubd\_E(X)));ubd\_E(X)\^ && Lemma [\[Lem:ResBasics\]](#Lem:ResBasics){reference-type="ref" reference="Lem:ResBasics"}(1)\ &= (ubd\_E(X)\_E(ubd\_E(X)));ubd\_E(X)\^ && Lemma [\[Lem:BoundLemma\]](#Lem:BoundLemma){reference-type="ref" reference="Lem:BoundLemma"}\ &= lub\_E(X);ubd\_E(X)\^. Together we obtain 2\ &\_E(X);ubd\_E(X)\^\_E(X);lbd\_E(ubd\_E(X))\^ && see above\ &= lub\_E(X);(ubd\_E(X)\^\^) && Lemma [\[Lem:Maps\]](#Lem:Maps){reference-type="ref" reference="Lem:Maps"}(3)\ &= lub\_E(X);lub\_E(X)\^ which immediately implies ${\rm dom}(X)\sqcap{\rm dom}({\rm ubd}_{E}(X))\sqsubseteq{\rm dom}({\rm lub}_{E}(X))$ since ${\rm dom}(X)\sqcap{\rm dom}({\rm ubd}_{E}(X))\sqsubseteq{\mathbb I}_A$.0◻ ◻ One would expect that the inclusion of the previous theorem is in fact an equation. However, for arbitrary Heyting categories we need an additional regularity condition. If $X:A\to{\mathbb R}$ with $X;{C}^{\smallsmile}$ regular, then we have ${\rm dom}(X)\sqcap{\rm dom}({\rm ubd}_{E}(X))={\rm dom}({\rm lub}_{E}(X))$. *Proof.* By Theorem [\[Th:LUP\]](#Th:LUP){reference-type="ref" reference="Th:LUP"} it is sufficient to show ${\rm dom}({\rm lub}_{E}(X))\sqsubseteq{\rm dom}(X)\sqcap{\rm dom}({\rm ubd}_{E}(X))$. First of all, we have obviously have ${\rm dom}({\rm lub}_{E}(X))\sqsubseteq{\rm dom}({\rm ubd}_{E}(X))$. Furthermore we have 2 dom(lub\_E(X)) &= dom(lub\_E(X);C\^) && Lemma [\[Lem:XCProps\]](#Lem:XCProps){reference-type="ref" reference="Lem:XCProps"}(1)\ &((X;C\^)\^) && Lemma [\[Lem:DownClosed\]](#Lem:DownClosed){reference-type="ref" reference="Lem:DownClosed"}\ &= dom(X;C\^) && assumption\ &= dom(X). && Lemma [\[Lem:XCProps\]](#Lem:XCProps){reference-type="ref" reference="Lem:XCProps"}(1)  ◻ # Conclusion and Future Work The current paper has shown the least-upper-bound property for a real number object in a Heyting category. This is the first step for showing that this additive group is Archimedean. For a next step one first has to define the the operation of summing up $n$ copies of an element $a$, i.e., a map ${\mathbb N}\times{\mathbb R}\to{\mathbb R}$. This requires either an external object of the natural numbers or to identify the natural numbers within the real number object. Another paper will concentrate on the multiplicative group of a real number object. The definition of the multiplication operation requires the Archimedean property and shows that the multiplication of natural number has a unique extension in the real numbers. Last but not least, we would like to study the topology induced by the order structure on a real number object using the relation algebraic approach to topological spaces [@SchmidtWinter]. 99 Freyd P., Scedrov A.: Categories, allegories. North-Holland Mathematical Library Vol. 39, North-Holland, Amsterdam (1990). Olivier J.P., Serrato D.: Catégories de Dedekind. Morphismes dans les Catégories de Schröder. C.R. Acad. Sci. Paris 290 (1980) 939-941. Olivier J.P., Serrato D.: Squares and Rectangles in Relational Categories - Three Cases: Semilattice, Distributive lattice and Boolean Non-unitary. Fuzzy sets and systems 72 (1995), 167-178. Schmidt G., Ströhlein T.: Relations and graphs. Discrete mathematics for computer scientists. EATCS Monographs on Theoretical Computer Science, Springer, Berlin (1993). Schmidt G.: Relational mathematics. Encyplopedia of Mathematics and Its Applications Vol. 132, Cambridge University Press, Cambridge (2011). Schmidt G., Winter M.: Relational Topology. LNM 2208 (2018). Tarski A.: Introduction to Logic. Oxford University Press (1941). Winter M.: Strukturtheorie heterogener Relationenalgebren mit Anwendung auf Nichtdetermismus in Programmiersprachen. Dissertationsverlag NG Kopierladen GmbH, München (1998) Winter M.: Goguen categories -- A categorical approach to $L$-fuzzy relations. Trends in Logic Vol. 25, Springer, Berlin (2007). Winter M.: Arrow Categories. Fuzzy Sets and Systems 160, 2893-2909 (2009). Winter M.: Fixed Point Operators in Heyting Categories. Part I - Internal Fixed Point Theorem and Calculus. (submitted to Journal of Pure and Applied Algebra, 2022). Winter M.: Relational Algebraic Approach to the Real Numbers - The Additive Group. Relational and Algebraic Methods in Computer Science, 20th International Conference, RAMiCS 2023, LNCS 13896, 274-292 (2023) [^1]: The author gratefully acknowledges support from the Natural Sciences and Engineering Research Council of Canada (283267).

arxiv_math

--- abstract: | We classify twistings of Grothendieck's differential operators on a smooth variety $X$ in prime characteristic $p$. We prove isomorphism classes of twistings are in bijection with $H^2(X,\mathbb{Z}_p(1))$, the degree 2, weight 1 syntomic cohomology of $X$. We also discuss the relationship between twistings of crystalline and Grothendieck differential operators. Twistings in mixed characteristic are also analyzed. address: | University of Wisconsin-Madison\ Madison\ WI author: - Joshua Mundinger bibliography: - twist-references.bib date: October 9, 2023 title: Twisting the Infinitesimal Site --- # Introduction Beilinson and Bernstein defined and studied rings of twisted differential operators in connection with their study of representations of semisimple Lie algebras [@bb93]. If $\mathrm{Diff}_X$ is the ring of differential operators on a smooth space $X$, then a ring of twisted differential operators on $X$ is essentially a filtered ring $D = \cup_i D_{\leq i}$ such that $$\mathop{\mathrm{gr}}D \cong \mathop{\mathrm{gr}}\mathrm{Diff}_X$$ as Poisson algebras. If $X$ is a holomorphic manifold or complex algebraic variety, then such rings $D$ are generated in degree 1 and are thus classified by the extension $$0 \to \mathcal O_X \to D_{\leq 1} \to T_X \to 0$$ of sheaves of Lie algebras. The set of isomorphism classes of such extensions is thus an abelian group isomorphic to $H^2(X,\Omega_X^{\geq 1})$, which coincides with the Hodge filtration subspace $F^1H_{dR}^2(X)$ when $X$ is a compact Kähler manifold or a proper complex algebraic variety [@bb93 Lemma 2.1.6]. In this paper, we study twisted differential operators when $X \to S$ is a smooth morphism of schemes over a nonzero characteristic base $S$. When $S$ is not of characteristic zero, then there are multiple inequivalent notions of differential operators on $X$. Two main rings of interest are the *crystalline differential operators* $\mathcal D^{crys}_X$ and the *Grothendieck differential operators* $\mathrm{Diff}_X$. The ring of crystalline differential operators $\mathcal D^{crys}_X$ is the enveloping algebra of the tangent sheaf $T_{X/S}$, with associated graded $\mathop{\mathrm{Sym}}_{\mathcal O_X} T_{X/S}$; the ring of Grothendieck differential operators, or "full" ring of differential operators, $\mathrm{Diff}_X$ includes divided powers of partial derivatives, and has associated graded $\Gamma_{\mathcal O_X} T_{X/S}$, the divided symmetric power algebra. There is a filtered map $$\mathcal D^{crys}_X \to \mathrm{Diff}_X,$$ inducing the canonical map $\mathop{\mathrm{Sym}}_{\mathcal O_X} T_{X/S} \to \Gamma_{\mathcal O_X} T_{X/S}$ on associated graded. In positive characteristic, this map is neither injective nor surjective. We study twistings $D$ of Grothendieck's differential operators $\mathrm{Diff}_X$. These algebras $D$ are not generated in degree 1 and so are not determined by the extension class of $D_{\leq 1}$. Instead, such algebras are classified by a $p$-adic analogue of $F^1 H^2_{dR}(X)$. Given a prime $p$, the *weight 1 syntomic cohomology* of a scheme $X$ is by definition $$R\Gamma(X, \mathbb Z_p(1)) = R\Gamma(X,\mathbb G_m)^\wedge_p[-1],$$ the derived $p$-completion of $R\Gamma(X,\mathbb G_m)[-1]$. **Theorem 1**. *Let $k$ be a perfect field of characteristic $p$ and $X \to \mathop{\mathrm{Spec}}k$ a smooth variety. Then isomorphism classes of rings of twisted Grothendieck differential operators are in bijection with $$H^2(X, \mathbb Z_p(1)),$$ the degree 2, weight 1 syntomic cohomology of $X$.* The group $H^2(X,\mathbb Z_p(1))$ can be expressed in terms of the Picard and Brauer groups via the short exact sequence $$\begin{tikzcd} 0 & {\mathop{\mathrm{Pic}}(X)^\wedge_p} & {H^2(X,\mathbb Z_p(1))} & {T_p\mathop{\mathrm{Br}}(X)} & 0 \arrow[from=1-1, to=1-2] \arrow[from=1-2, to=1-3] \arrow[from=1-3, to=1-4] \arrow[from=1-4, to=1-5] \end{tikzcd},$$ where $T_p(-)$ is the $p$-adic Tate module and $\mathop{\mathrm{Pic}}(X)^\wedge_p$ is the classical $p$-adic completion of the Picard group. The first map is induced by the Chern class $c_1: \mathop{\mathrm{Pic}}(X) \to H^2(X,\mathbb Z_p(1))$ [@bhattlurie22 §7]. In terms of twisted differential operators, this assignment sends a line bundle $\mathcal L$ to $\mathrm{Diff}_{\mathcal L}$, the ring of differential operators on $\mathcal L$. If $k$ is algebraically closed, then the syntomic cohomology group $H^2(X,\mathbb Z_p(1))$ may be computed via crystalline cohomology as $$H^2(X, \mathbb Z_p(1)) \cong H^2(X, W\Omega^{\geq 1}_X)^{\varphi= p},$$ see §3.2.1 below. This isomorphism transforms the syntomic Chern class to the crystalline Chern class, of interest in the Tate conjecture. If $X$ is a variety over a field $k$, then $H^2(X,\Omega_X^{\geq 1})$ is a $k$-vector space, while $H^2(X,\mathbb Z_p(1))$ is a $\mathbb Z_p$-module regardless of $k$. Here is the intuition why twisted differential operators should be controlled by a $\mathbb Z_p$-module and not a vector space. Fix an invertible function $f$; if $f^\lambda$ may be differentiated, then conjugation by $f^\lambda$ defines an automorphism of $\mathrm{Diff}_X$. The function $f^\lambda$ satisfies the differential equation $(d - \lambda d\log(f))f^\lambda = 0$, which makes sense for any parameter $\lambda$ in the base field. However, over a positive characteristic field $k$, we must specify not only the first derivative but all divided power derivatives. If $\partial$ is a derivation admitting divided powers $\{\partial^{(i)}\}_{i \geq 1}$, then Faà di Bruno's formula gives $$\partial^{(i)}(f^\lambda) = \sum \binom{m}{m_1,\ldots, m_i} \binom{\lambda}{m} f^{\lambda - m} \prod_{j=1}^i \left( \partial^{(j)} f\right)^{m_j},$$ where the sum runs over all $i$-tuples $(m_1,\ldots, m_i)$ such that $\sum_j j m_j = i$, while $m = \sum_j m_j$. If we wish to form $\partial^{(i)}f^\lambda$ for all $i \geq 0$, then $\binom{\lambda}{m}$ must make sense for all $m \geq 0$, that is, the parameter $\lambda$ must be a $k$-point of a free binomial ring on one generator. The set of residue characteristic $p$ points of the free binomial ring on one generator is exactly $\mathbb Z_p$ [@elliott06]. The same considerations explain why T. Bitoun's $b$-function in positive characteristic is a locally constant function on $\mathbb Z_p$ [@bitoun18]. One might ask whether the map $\mathcal D^{crys}_X \to \mathrm{Diff}_X$ can be twisted. Given an algebra of twisted Grothendieck differential operators $D$ on smooth $X \to \mathop{\mathrm{Spec}}k$, we construct in §[4.2](#subsection: compatibility){reference-type="ref" reference="subsection: compatibility"} a Frobenius-split twisting $\widetilde{\mathcal D}^{crys}$ of $\mathcal D^{crys}_X$ depending on $D$ and a map $\widetilde{\mathcal D}^{crys} \to D$ such that the kernel is generated by the ideal of the zero section of $T^*X = \mathop{\mathrm{Spec}}Z(\widetilde{\mathcal D}^{crys})$. We also study a mixed characteristic version of this problem, when $S = \mathop{\mathrm{Spec}}W_m(k)$ for finite $m$. If $X \to \mathop{\mathrm{Spec}}W_m(k)$ is smooth and of finite type, then in Theorem [Theorem 21](#theorem: mixed-characteristic-syntomic-comparison){reference-type="ref" reference="theorem: mixed-characteristic-syntomic-comparison"} we construct a map from $H^2(X,\mathbb Z_p(1))$ to isomorphism classes of twisted differential operators on $X$. This map may or may not be an isomorphism, depending on the action of Frobenius on $R\Gamma(X_0,\mathcal O_{X_0})$, where $X_0$ is the special fiber of $X$. We caution the reader that our twisted differential operators are in the sense of Beilinson-Bernstein, and do not involve $q$-difference operators as in Gros-Le Stum-Quirós [@groslestumquiros22]. ## Acknowledgments {#acknowledgments .unnumbered} The author thanks Victor Ginzburg for his encouragement of this project. The author is grateful to Daniel Bragg, Luc Illusie, Akhil Mathew, and Vadim Vologodsky for useful conversations and communications. Ben Antieau hosted stimulating workshops at Northwestern University which considerably advanced the progress of this work. The appendix was written through discussions with Dima Arinkin. The author was partially supported by NSF Graduate Research Fellowship DGE 1746045. # Twistings and twisted differential operators ## Algebras of twisted differential operators {#subsection: tdo} Let $S$ be an affine scheme and $X \to S$ a morphism of schemes. Undecorated products of schemes over $S$ are understood to be products over $S$. **Definition 2**. [@bb93 §1.1.4] A differential algebra on $X/S$ is a sheaf of associative algebras $D$ on $X$ equipped with a morphism $\iota: \mathcal O_X \to D$ such that the image of $\mathcal O_S$ is central and such that as an $\mathcal O_X$-bimodule, $D$ is supported on the diagonal of $X \times X$. A differential algebra on $X/S$ is equipped with a canonical filtration $D_{\leq -1 } = 0, D_{\leq i} = \{Q \in D \mid [Q,f] \in D_{\leq i-1} \text{ for all }f \in \mathcal O_X\}$. Since $D$ is supported on the diagonal in $X \times X$, this filtration is complete. The first example of a differential algebra is Grothendieck's ring of differential operators $\mathrm{Diff}_X$, defined as follows: $\mathrm{Diff}_{X,\leq i}$ consists of $\mathcal O_S$-linear operators $Q: \mathcal O_X \to \mathcal O_X$ such that for all $f_0,f_1,\ldots, f_i \in \mathcal O_X$, the iterated commutator $[\ldots[[Q,f_0],f_1],\ldots,f_i]$ vanishes. If $\mathcal O_S$ contains $\mathbb Q$, then $\mathrm{Diff}_X$ is generated by operators of degree at most 1; if $\mathcal O_S$ contains $\mathbb F_p$, then $\mathrm{Diff}_X$ is not Noetherian, and contains operators such as $(p^r)!^{-1} (d/dt)^{p^r}$. If $X/S$ is smooth, then the associated graded ring $\mathop{\mathrm{gr}}\mathrm{Diff}_X$ is isomorphic to the divided power symmetric algebra $\Gamma_{\mathcal O_X} T_{X/S}$ on the tangent sheaf $T_{X/S}$ [@berthelotcrystalline 2.6 Proposition]. We are interested in twisted versions of Grothendieck's differential operators $\mathrm{Diff}_X$. Definition [Definition 3](#defn: tdo){reference-type="ref" reference="defn: tdo"} below modifies Beilinson and Bernstein's definition of twisted differential operators [@bb93 §2] to deal with divided powers. If $D$ is a differential algebra on $X/S$ and $f \in \mathcal O_X$, then the commutator with $f$ sends $D_{\leq \ast}$ into $D_{\leq \ast - 1}$ and thus defines $[-,f] :\mathop{\mathrm{gr}}_\ast D \to \mathop{\mathrm{gr}}_{\ast - 1} D$. The assignment $f \mapsto [-,f]$ satisfies the Leibniz rule. If $f_1\in \mathcal O_X$ and $f_2 \in \mathcal O_X$, then $f_1$ and $f_2$ commute, so $[-,f_1]$ and $[-,f_2]$ are commuting operators on $\mathop{\mathrm{gr}}D$. Thus we obtain a pairing $$\mathop{\mathrm{Sym}}^i \Omega^1_{X/S} \times \mathop{\mathrm{gr}}_\ast D \to \mathop{\mathrm{gr}}_{\ast - i} D.$$ **Definition 3**. If $X/S$ is smooth, then an *algebra of twisted Grothendieck differential operators* on $X/S$, or a *Grothendieck tdo* on $X/S$, is a differential algebra $D$ such that $\iota: \mathcal O_X \to D_{\leq 0}$ is an isomorphism and the bilinear map $$\mathop{\mathrm{Sym}}^i \Omega^1_{X/S} \times \mathop{\mathrm{gr}}_i D \to \mathop{\mathrm{gr}}_0 D \cong \mathcal O_X$$ defined by $$\langle df_1\cdots df_i, Q \rangle =[[[Q,f_1],\ldots ],f_i]$$ is a perfect pairing. **Remark 4**. For the rest of this paper, we will refer to Grothendieck tdo's simply as tdo's. Now assume $X/S$ is smooth. If $D$ is a tdo, then $\mathop{\mathrm{gr}}D$ is the divided power symmetric algebra on $T_{X/S}$. $\mathrm{Diff}_X$ is a tdo. The category of tdo's on $X$ is a groupoid. If $u: U \to X$ is a smooth morphism, then pullback of left $\mathcal O$-modules sends differential algebras on $X$ to differential algebras on $U$ [@bb93 .§1.5]; if $U \to X$ is étale, then $u^* \Omega^1_{X/S} = \Omega^1_{U/S}$, so the pullback of a tdo is a tdo. Hence tdo's form a 1-stack $TDO(X/S)$ on $X_{\acute{e}t}$. **Example 5**. If $\mathcal L$ is a line bundle on $X$, then $\mathrm{Diff}_{\mathcal L}$ is a tdo on $X$. ## The stacky approach {#subsection: stacky approach} Gaitsgory and Rozenblyum developed a general theory of twistings in order to correctly formulate twisted $D$-modules on a stack in characteristic zero using the de Rham stack [@gaitsgoryrozenblyum14]. In nonzero characteristic, different flavors of $D$-modules correspond to different variants of the de Rham stack. Grothendieck's differential operators correspond to the infinitesimal site $(X/S)_{inf}$ associated to $X/S$, defined below. We now recall Gaitsgory and Rozenblyum's formulations of twists [@gaitsgoryrozenblyum14 §6.6] and prove that twistings of the infinitesimal site are equivalent to the tdo's defined in §[2.1](#subsection: tdo){reference-type="ref" reference="subsection: tdo"}. **Definition 6**. Given a morphism $X\to S$ of schemes, the infinitesimal prestack $X_{inf}: \mathrm{Aff}_{/S} \to \mathrm{Set}$ is defined by $$X_{inf}(Z) = X(Z_{red}).$$ There is a natural map $\rho: X \to X_{inf}$ over $S$. **Definition 7**. A *twisting of the infinitesimal site* of $X$ is a $\mathbb G_m$-gerbe on $X_{inf}$ equipped with a trivialization of its pullback along $\rho: X \to X_{inf}$. The Čech nerve $X_\bullet$ of $X \to X_{inf}$ is the formal completion of $X^{\times \bullet + 1}$ along the diagonal $X \to X^{\times \bullet + 1}$. **Proposition 8**. *[@gaitsgoryrozenblyum14 Lemma 1.2.4] [\[lemma: cech-nerve\]]{#lemma: cech-nerve label="lemma: cech-nerve"} If $X/S$ is formally smooth, then $X_{inf} \simeq |X_\bullet|$ in $\infty$-groupoid-valued prestacks over $S$.* The proposition means that we may regard the $\infty$-category of abelian sheaves on $X_{inf}$ as the $\infty$-category of abelian sheaves on $X$ equipped with descent data along $X_\bullet$. **Proposition 9**. *[@gaitsgoryrozenblyum14 §6.6.4] If $X/S$ is smooth, then the groupoid of algebras of twisted differential operators on $X$ is equivalent to the groupoid of twistings of $X \to X_{inf}$.* *Proof.* By Lemma [\[lemma: cech-nerve\]](#lemma: cech-nerve){reference-type="ref" reference="lemma: cech-nerve"}, $|X_\bullet| \simeq X_{inf}$. A $\mathbb G_m$ gerbe on $X_\bullet$ with a trivialization of its pullback along $X \to X_\bullet$ has two different trivializations upon pullback to $\widehat{X \times X}$; hence we obtain a line bundle $D$ on $\widehat{X \times X}$. The pullback of $D$ along $X \to \widehat{X \times X}$ is trivialized by a bimodule homomorphism $\iota: \mathcal O_X \cong D_{\leq 0} \subseteq D$. The face maps along the diagrams $$X_3 \to X_2 \to X_0 = \widehat{X \times X}$$ define an associative product on $D$; the degeneracy and face maps imply $\iota$ is a morphism of associative rings. Since $D$ is a line bundle on $\widehat{X \times X}$, the commutator pairings $\mathop{\mathrm{Sym}}^i\Omega^1_{X/S} \times \mathop{\mathrm{gr}}_i D \to \mathcal O_X$ are isomorphisms. Conversely, reversing the arguments above shows a tdo $D$ defines a line bundle on $X_0$ along with associated coherence data on $X_\bullet$, descending to a gerbe on $|X_\bullet|$. ◻ **Corollary 10**. *[@gaitsgoryrozenblyum14 §6.5] [\[corollary: twistings-classified-by-fiber\]]{#corollary: twistings-classified-by-fiber label="corollary: twistings-classified-by-fiber"} The groupoid of twistings of the infinitesimal site is controlled by $$\tau^{\leq 2} \mathop{\mathrm{fiber}}(R\Gamma(X_{inf},\mathbb G_m) \to R\Gamma(X,\mathbb G_m)).$$* *Proof.* For a prestack $Y$, we have $$Maps(Y, B^2\mathbb G_m) \simeq \tau^{\leq 0}(R\Gamma(Y,\mathbb G_m)[2]).$$ Given a map $Y \to Y'$, the space of $\mathbb G_m$-gerbes on $Y'$ equipped with a trivialization of its pullback to $Y$ is the homotopy fiber of $$Maps(Y',B^2\mathbb G_m) \to Maps(Y, B^2\mathbb G_m).\qedhere$$ ◻ If $S$ is of characterstic zero, then Gaitsgory and Rozenblyum calculated the groupoid of twistings as follows: $R\Gamma(X_{inf},\mathbb G_m)$ is computed by the logarithmic de Rham complex $d\log: \mathcal O_X^\times \to \Omega^{\geq 1}_X$. Hence the desired fiber is exactly $R\Gamma(X,\Omega_X^{\geq 1})$, recovering the classical answer of Beilinson and Bernstein [@bb93 Lemma 2.1.6]. To compute the groupoid of twistings in nonzero characteristic, we must compute $R\Gamma(X_{inf},\mathbb G_m)$. # F-divided structures and syntomic cohomology Fix a perfect field $k$ of characteristic $p > 0$. It is a well-known theorem of Katz that if $X/\mathop{\mathrm{Spec}}k$ is smooth, then modules over $\mathrm{Diff}_X$ are equivalent to $F$-divided sheaves of $\mathcal O_X$-modules [@gieseker75 Theorem 1.3]. This is an extension of Cartier's classical theorem on descent along the Frobenius. Ogus later applied this idea in order to compute cohomology of coherent sheaves on the infinitesimal site in terms of crystalline cohomology [@ogus75]. Berthelot extended Katz's theorem to smooth $X \to \mathop{\mathrm{Spec}}W_m(k)$ admitting a lift of Frobenius [@berthelot12]. In this section, we apply these ideas to compute $R\Gamma(X_{inf},\mathbb G_m)$ and thus compute the groupoid of twistings of the infinitesimal site. From now on $S = \mathop{\mathrm{Spec}}W_m(k)$ for $1 \leq m < \infty$. ## Stratifications via a lift of Frobenius Assume for this subsection that $X \to S$ is smooth. Let $X^{(r)}$ be the $r$-fold Frobenius twist of $X$, defined to be the pullback of $X$ along the Frobenius automorphism of $W_m(k)$. We further assume that $X$ has a lift $\varphi: X \to X^{(r)}$ of the relative Frobenius. The method suggested in [@ogus75 (4.12) Remark] for calculating $R\Gamma(X_{inf},-)$ is that the infinitesimal groupoid $X \times_{X_{inf}} X$ is the colimit of the Čech nerves of $X\to X^{(r)}$, in other words, that as $\infty$-prestacks, $X_{inf} = \mathop{\mathrm{\varinjlim}}_r X^{(r)}$. To treat the entire infinitesimal groupoid $X \times_{ X_{inf}} X$ and not just the formal neighborhood of the diagonal in $X \times X$, we must generalize Berthelot's calculations in [@berthelot12 §1] slightly. We begin by recalling the main lemma of [@berthelot12]. Let $A$ be a commutative ring with an ideal $I \subseteq A$. Given such, define $I^{(i)} = (a^{p^i} : a \in I)$ and $\widetilde{I^{(i)}} = I^{(i)} + p I^{(i-1)} + p^2 I^{(i-2)} + + \cdots + p^{i-1} I + p^i A$. **Lemma 11**. *[@berthelot12 Lemma 1.3] [\[lemma: berthelot-pth-power\]]{#lemma: berthelot-pth-power label="lemma: berthelot-pth-power"} If $a \in \widetilde{I^{(i)}}$, then $a^p \in \widetilde{I^{(i+1)}}$.* **Lemma 12**. *[@berthelot12 Lemma 1.4] Let $\mathcal I_r$ be the ideal of the closed immersion $X^{(r)} \to \left(X^{(r)}\right)^{\times n}$. Then $(\varphi^{\times n})^i (\mathcal I_r) \subseteq \widetilde{\mathcal I^{(i)}_{r-i}}$.* *Proof.* Berthelot proved this in case $n=2$ using Lemma [\[lemma: berthelot-pth-power\]](#lemma: berthelot-pth-power){reference-type="ref" reference="lemma: berthelot-pth-power"}. The general case follows since the ideal of $X^{(r)} \to \left(X^{(r)}\right)^{\times n}$ is generated by the the ideals of "set two coordinates equal." ◻ **Corollary 13**. *Let $\mathcal I$ be the ideal of the closed immersion $X \to X^{\times n}$. Then the sequences of ideals $\{\mathcal I^N\}_{N\geq 1}$ and $\{(\varphi^{\times n})^r \mathcal I^{(r)})\}_{r\geq 1}$ are locally cofinal.* *Proof.* Since $X \to S$ is locally of finite type, the ideal $\mathcal I$ is locally finitely generated, and thus the powers $\{\mathcal I^N\}_{N \geq 1}$ and symbolic powers $\{\mathcal I^{(i)}\}_{i \geq 1}$ are locally cofinal. Since $p^m = 0$ on $S$, we have $\varphi^{\times r}(\mathcal I_r) \subseteq \tilde{\mathcal I^{(r)}} \subseteq \mathcal I^{(r-m)}$ for $r \geq m$. Conversely, if $a \in I$, then $a^p \equiv \varphi(a) \mod p$, so by Dwork's Lemma $a^{p^r} \equiv (\varphi^{r-m+1}(a))^{p^{m-1}} = \varphi^{r-m+1}(a^{p^{m-1}})\mod p^m$, so $\mathcal I^{(r)} \subseteq ((\varphi^{\times n})^{r-m+1} \mathcal I^{(r-m+1)})$. Thus the Frobenius pullbacks $\{(\varphi^{\times n})^r\mathcal I^{(r)}\}_{r \geq 1}$ and symbolic powers $\{\mathcal I^{(i)}\}_{i \geq 1}$ of $\mathcal I$ are cofinal. ◻ **Proposition 14**. *If $X\to S = \mathop{\mathrm{Spec}}W_m(k)$ is a smooth morphism of finite type with a lift of Frobenius $\varphi$, then $$R\Gamma(X_{inf},\mathbb G_m) \simeq R\varprojlim_\varphi R\Gamma(X^{(r)},\mathbb G_m).$$* *Proof.* Let $(X/X^{(r)})_\bullet$ be the Čech nerve of $\varphi^r: X \to X^{(r)}$. By Corollary [Corollary 13](#corollary: cofinal-chains){reference-type="ref" reference="corollary: cofinal-chains"}, $(X/X_{inf})_\bullet$ is the the degreewise colimit of $(X/X^{(r)})_\bullet$. Since cohomology commutes with inverse limts, we have $R\Gamma(X_{inf},\mathbb G_m) = R\varprojlim R\Gamma((X/X^{(r)})_\bullet, \mathbb G_m)$. The functor $R\Gamma(-,\mathbb G_m)$ is an fppf sheaf by Grothendieck's version of Hilbert 90, so we may identify the preceding inverse limit with $R\varprojlim_\varphi R\Gamma(X^{(r)},\mathbb G_m)$. ◻ **Remark 15**. This method provides a proof of Ogus' result that for smooth and proper $X /k$, the cohomology of the infinitesimal site with respect to the structure sheaf is $H^\bullet(X_{inf}, \mathcal O) \cong H^\bullet(X,\mathcal O)^s$, the subspace of $H^\bullet(X,\mathcal O)$ on which the Frobenius acts as an isomorphism [@ogus75]. Indeed, more generally we obtain $$R\Gamma(X_{inf},\mathcal O_{X_{inf}}) \simeq R\varprojlim_\varphi R\Gamma(X, \mathcal O_X).$$ By Lemma [Lemma 39](#lemma: tate-module-sequence){reference-type="ref" reference="lemma: tate-module-sequence"}, this reduces to Ogus' formula if the cohomology groups $H^i(X,\mathcal O_X)$ are finite-dimensional. We recall the derived completion of an object $A$ with respect to an endomorphism $a$ is the homotopy inverse limit $$A^\wedge_a = R\varprojlim_n \mathop{\mathrm{cone}}(\begin{tikzcd} A & A \arrow["{a^n}", from=1-1, to=1-2] \end{tikzcd}),$$ and thus $A^\wedge_a$ is the cone of $(R\varprojlim_a A) \to A$; see Appendix [5](#appendix: completion){reference-type="ref" reference="appendix: completion"}. **Corollary 16**. *If $X\to S = \mathop{\mathrm{Spec}}W_n(k)$ is smooth and of finite type with a lift of Frobenius $\varphi$, then $$\mathop{\mathrm{fiber}}(R\Gamma(X_{inf},\mathbb G_m) \to R\Gamma(X,\mathbb G_m)) \simeq R\Gamma(X,\mathbb G_m)^\wedge_\varphi[-1].$$* *Proof.* Follows from Proposition [Proposition 14](#proposition: derived-inverse-limit){reference-type="ref" reference="proposition: derived-inverse-limit"} and definitions. ◻ ## Syntomic cohomology in prime characteristic **Definition 17**. Let $X$ be a scheme. The syntomic cohomology of $X$ in weight one is defined by $$R\Gamma(X,\mathbb Z_p(1)) = R\Gamma(X,\mathbb G_m)^{\wedge}_p[-1].$$ In characteristic $p$, pullback by Frobenius on $\mathbb G_m$ is multiplication by $p$. Thus $\varphi$-adic completion is $p$-adic completion: **Corollary 18**. *If $X \to \mathop{\mathrm{Spec}}k$ is a smooth variety, then the groupoid of TDO's on $X$ is controlled by $$\tau_{\leq 2} R\Gamma(X,\mathbb Z_p(1)).$$* *Proof.* Follows from Corollaries [\[corollary: twistings-classified-by-fiber\]](#corollary: twistings-classified-by-fiber){reference-type="ref" reference="corollary: twistings-classified-by-fiber"}, [Corollary 16](#corollary: phi-adic-completion){reference-type="ref" reference="corollary: phi-adic-completion"}, and Definition [Definition 17](#defn: syntomic-cohomology){reference-type="ref" reference="defn: syntomic-cohomology"}. ◻ Suppose the Néron-Severi group $\mathop{\mathrm{NS}}(X) = \mathop{\mathrm{Pic}}(X)/\mathop{\mathrm{Pic}}^0(X)$ is a finitely generated abelian group, which holds if $X \to \mathop{\mathrm{Spec}}k$ is proper by the Theorem of the Base [@SPEC-theorem-of-the-base]. As $\mathop{\mathrm{Pic}}^0(X)$ is $p$-divisible, the $p$-adic completion of $\mathop{\mathrm{Pic}}(X)$ is identified with $\mathop{\mathrm{NS}}(X) \otimes \mathbb Z_p$. Lemma [Lemma 39](#lemma: tate-module-sequence){reference-type="ref" reference="lemma: tate-module-sequence"} gives a short exact sequence $$%\label{eq: ses-for-Zp1} 0 \to \mathop{\mathrm{NS}}(X)\otimes \mathbb Z_p \to H^2(X,\mathbb Z_p(1)) \to T_p\mathop{\mathrm{Br}}(X) \to 0,$$ which appears in [@illusie79 (5.8.5)]. If $k$ is algebraically closed, Illusie also showed that $R\Gamma(X,\mathbb Z_p(1))$ may be computed using the crystalline cohomology of $X$. Let $W \Omega_X$ be the de Rham-Witt complex of $X/k$ and $W \Omega_X^{\geq 1} \subseteq W\Omega_X$ be the stupid truncation above degree 1. Let $F': W \Omega^{\geq 1} \to W\Omega^{\geq 1}$ be the crystalline Frobenius divided by $p$. Then there is a long exact sequence $$\begin{tikzcd}[cramped,column sep=small] \cdots & {H^\bullet(X, \mathbb Z_p(1))} & {H^\bullet(X, W\Omega^{\geq 1}_X)} & {H^\bullet(X, W\Omega^{\geq 1}_X)} & \cdots \arrow[from=1-1, to=1-2] \arrow[from=1-2, to=1-3] \arrow["{1 - F'}", from=1-3, to=1-4] \arrow[from=1-4, to=1-5] \end{tikzcd}$$ [@illusie79 p. 627, (5.5.2)]. Using that $k$ is closed under Artin-Schreier extensions, in low degrees the above sequence splits into short exact sequences $$\begin{tikzcd}[cramped,column sep=small] 0 & {H^i(X, \mathbb Z_p(1))} & {H^i(X, W\Omega^{\geq 1}_X)} & {H^i(X, W\Omega^{\geq 1}_X)} & 0 \arrow[from=1-1, to=1-2] \arrow[from=1-2, to=1-3] \arrow["{1 - F'}", from=1-3, to=1-4] \arrow[from=1-4, to=1-5] \end{tikzcd}$$ for $i = 1,2$ [@illusie79 p. 629, §5.8]. **Example 19**. Suppose $k$ is algebraically closed and $X/k$ is an ordinary K3 surface. Then $H^2(X,\mathbb Z_p(1)) \cong \mathbb Z_p^{20}$, so $T_p\mathop{\mathrm{Br}}(X) \cong \mathbb Z_p^{20 -\rho}$ where $\rho = \mathop{\mathrm{rank}}\mathop{\mathrm{NS}}(X)$ is the Picard rank of $X$ [@illusie79 p. 653, §7.2]. The Picard rank is often much less than 20; for example, the theorem of M. Noether implies that the Picard rank of a very general fourfold in $\mathbb P^3$ is $1$ [@sga-7-2 Exposé XIX]. Note that $X$ being very general requires such $X$ to be defined over a transcendental extension of $\overline{\mathbb F_p}$, as the Picard rank of a K3 over $\overline{\mathbb F_p}$ is even by the Tate conjecture. **Example 20** (Zariski-locally trivial tdo's). A tdo $D$ is *Zariski-locally trivial* if there is a Zariski cover $U_\alpha \subset X$ such that $D|_{U_\alpha} \cong \mathrm{Diff}_{U_\alpha}$. By [@milne-etale Corollary IV.2.6], if $X \to \mathop{\mathrm{Spec}}k$ is smooth, then $\mathop{\mathrm{Br}}(X) \to \mathop{\mathrm{Br}}(K(X))$ is injective, so the image of $[D]$ in $T_p\mathop{\mathrm{Br}}(X)$ must be zero. Thus, $D$ lies in the image of $\mathop{\mathrm{Pic}}(X)^\wedge_p$. Again assuming that $\mathop{\mathrm{NS}}(X)$ is finitely generated, every element of $\mathop{\mathrm{Pic}}(X)^\wedge_p \cong \mathop{\mathrm{NS}}(X) \otimes \mathbb Z_p$ is Zariski-locally trivial, as we may choose a cover of $X$ which simultaneously trivializes a finite generating set for $\mathop{\mathrm{NS}}(X)$. For example, if $k$ is algebraically closed, Zariski-locally trivial tdo's on $\mathbb P^n$ are classified by $\mathop{\mathrm{NS}}(\mathbb P^n) \otimes \mathbb Z_p \cong \mathbb Z_p$, which was found by D.P. Faurot using explicit Čech cocycles [@faurot94]. As $T_p\mathop{\mathrm{Br}}(\mathbb P^n) = T_p\mathop{\mathrm{Br}}(k) = 0$, all tdo's on $\mathbb P^n$ are Zariski-locally trivial. ## Comparison with syntomic cohomology in mixed characteristic Consider now a smooth morphism $X \to \mathop{\mathrm{Spec}}W_m(k)$ of finite type, and let $X_0$ be the reduction of $X$ modulo $p$. If $X$ admits a lift of Frobenius, then Corollary [Corollary 16](#corollary: phi-adic-completion){reference-type="ref" reference="corollary: phi-adic-completion"} describes the groupoid of tdo's on $X$. In mixed characteristic, the Frobenius on $\mathbb G_m$ no longer agrees with multiplication by $p$, but they may be compared. We show in Theorem [Theorem 21](#theorem: mixed-characteristic-syntomic-comparison){reference-type="ref" reference="theorem: mixed-characteristic-syntomic-comparison"} that there is a morphism from $\tau_{\leq 2}R\Gamma(X,\mathbb Z_p(1))$ to the complex controlling tdo's on $X$, which may or may not be an isomorphism, depending on action of Frobenius on $R\Gamma(X_0,\mathcal O_{X_0})$. **Theorem 21**. *Suppose that $X \to \mathop{\mathrm{Spec}}W_m(k)$ is a smooth morphism of finite type where $p = \mathop{\mathrm{char}}k > 2$.* (i) *The mapping fiber $$\mathop{\mathrm{fiber}}(R\Gamma(X_{inf},\mathbb G_m) \to R\Gamma(X,\mathbb G_m))$$ is derived $p$-complete, so that there is an essentially unique morphism $R\Gamma(X,\mathbb Z_p(1)) \to \mathop{\mathrm{fiber}}(R\Gamma(X_{inf},\mathbb G_m) \to R\Gamma(X,\mathbb G_m))$ fitting into a factorization $$\begin{tikzcd}[cramped] {R\Gamma(X,\mathbb G_m)[-1]} \\ {R\Gamma(X,\mathbb Z_p(1))} & {\mathop{\mathrm{fiber}}(R\Gamma(X_{inf},\mathbb G_m) \to R\Gamma(X,\mathbb G_m))} \arrow[from=1-1, to=2-2] \arrow[from=1-1, to=2-1] \arrow[dashed, from=2-1, to=2-2] \end{tikzcd}.$$* (ii) *Suppose that pullback by the relative Frobenius $\varphi$ of $X_0$ induces an isomorphism $$\varphi^*: H^i(X_0,\mathcal O_{X_0}) \to H^i(X_0,\mathcal O_{X_0})$$ for $0 \leq i \leq 3$. Then reduction modulo $p$ defines an equivalence $$\begin{tikzcd}[cramped] {\tau_{\leq 2} \mathop{\mathrm{fiber}}(R\Gamma(X_{inf},\mathbb G_m) \to R\Gamma(X,\mathbb G_m))} \\ {\tau_{\leq 2} \mathop{\mathrm{fiber}}(R\Gamma((X_0)_{inf},\mathbb G_m) \to R\Gamma(X_0,\mathbb G_m))} \arrow["\sim"', from=1-1, to=2-1] \end{tikzcd},$$ so that tdo's on $X$ are classified by $\tau_{\leq 2} R\Gamma(X_0,\mathbb Z_p(1))$.* (iii) *Suppose that pullback by the relative Frobenius $\varphi$ of $X_0$ is zero on $H^i(X_0,\mathcal O_{X_0})$ for $0 \leq i \leq 3$. Then the morphism from i. is an equivalence, so that tdo's on $X$ are classified by $\tau_{\leq 2} R\Gamma(X, \mathbb Z_p(1))$.* *Proof.* Let $Z$ be any scheme over $W_m(k)$, and let $Z_1$ be its reduction modulo $p$. Then there is a short exact sequence of sheaves $$\label{eq: exponential-triangle} \begin{tikzcd} 0 & {p\mathcal O_Z} & {\mathcal O_Z^\times} & {\mathcal O_{Z_1}^\times} & 0 \arrow[from=1-1, to=1-2] \arrow["\exp", from=1-2, to=1-3] \arrow[from=1-3, to=1-4] \arrow[from=1-4, to=1-5] \end{tikzcd},$$ where $\exp: p\mathcal O_Z \cong 1 + p\mathcal O_Z$ makes sense because the $p$-adic exponential has radius of convergence $p^{-1/(p-1)} > p^{-1}$ for $p > 2$. Applying this short exact sequence to the Čech nerve $X_\bullet$ of $X \to X_{inf}$ and the corresponding reduction $(X_0)_\bullet$ of $X_0 \to (X_0)_{inf}$ gives a diagram $$\label{eq: exponential-inf-diagram} % https://q.uiver.app/#q=WzAsOCxbMCwwLCJSXFxHYW1tYShYX3tpbmZ9LHBcXE9PX3tYX3tpbmZ9fSkiXSxbMCwxLCJSXFxHYW1tYShYLHBcXE9PX1gpIl0sWzEsMCwiUlxcR2FtbWEoWF97aW5mfSxcXG1hdGhiYiBHX20pIl0sWzEsMSwiUlxcR2FtbWEoWCxcXG1hdGhiYiBHX20pIl0sWzIsMSwiUlxcR2FtbWEoWF8wLFxcbWF0aGJiIEdfbSkiXSxbMiwwLCJSXFxHYW1tYSgoWF8wKV97aW5mfSxcXG1hdGhiYiBHX20pIl0sWzMsMF0sWzMsMV0sWzAsMl0sWzAsMV0sWzIsM10sWzEsM10sWzMsNF0sWzIsNV0sWzUsNF0sWzUsNiwiKzEiXSxbNCw3LCIrMSJdXQ== \begin{tikzcd}[cramped, column sep = scriptsize] {R\Gamma(X_{inf},p\mathcal O_{X_{inf}})} & {R\Gamma(X_{inf},\mathbb G_m)} & {R\Gamma((X_0)_{inf},\mathbb G_m)} & {} \\ {R\Gamma(X,p\mathcal O_X)} & {R\Gamma(X,\mathbb G_m)} & {R\Gamma(X_0,\mathbb G_m)} & {} \arrow[from=1-1, to=1-2] \arrow[from=1-1, to=2-1] \arrow[from=1-2, to=2-2] \arrow[from=2-1, to=2-2] \arrow[from=2-2, to=2-3] \arrow[from=1-2, to=1-3] \arrow[from=1-3, to=2-3] \arrow["{+1}", from=1-3, to=1-4] \arrow["{+1}", from=2-3, to=2-4] \end{tikzcd}$$ of distinguished triangles. Now $p^{m-1}$ acts as zero on the left column, and derived completion is triangulated, so the fiber of the left column is derived $p$-complete. The fiber of $R\Gamma((X_0)_{inf},\mathbb G_m) \to R\Gamma(X_0, \mathbb G_m)$ is derived $p$-complete by Corollary [Corollary 16](#corollary: phi-adic-completion){reference-type="ref" reference="corollary: phi-adic-completion"}. Again since derived completion is triangulated, we conclude the fiber of the middle column is derived $p$-complete. The universal property of derived completion (Corollary [Corollary 38](#corollary: completion-left-adjoint){reference-type="ref" reference="corollary: completion-left-adjoint"}) gives the desired factorization through $R\Gamma(X, \mathbb Z_p(1))$, completing proof of (i). Now suppose we are in the situation of (ii). Taking $\tau_{\leq 2}$ of [\[eq: exponential-inf-diagram\]](#eq: exponential-inf-diagram){reference-type="eqref" reference="eq: exponential-inf-diagram"}, it suffices to show that $$\tau_{\leq 2} \left(R\Gamma(X_{inf},p\mathcal O_{X_{inf}})\to R\Gamma(X,p\mathcal O_X)\right)$$ is an equivalence. By taking the $p$-adic filtration, it suffices to show $$\tau_{\leq 2}\left(R\Gamma((X_0)_{inf},\mathcal O_{(X_0)_{inf}}) \to R\Gamma(X_0, \mathcal O_{X_0})\right)$$ is an equivalence, that is, $\tau_{\leq 2} (R\Gamma(X_0,\mathcal O_{X_0})^\wedge_\varphi ) \simeq 0$. If $\varphi^*$ is an isomorphism on $H^i(X_0,\mathcal O_{X_0})$ for $0 \leq i \leq 3$, then Lemma [Lemma 39](#lemma: tate-module-sequence){reference-type="ref" reference="lemma: tate-module-sequence"} shows $H^i(R\Gamma(X_0,\mathcal O_{X_0})^\wedge_\varphi) = 0$ for $0 \leq i \leq 2$, as desired. Now suppose we are in the situation of (iii). Taking $p$-adic completion of [\[eq: exponential-triangle\]](#eq: exponential-triangle){reference-type="eqref" reference="eq: exponential-triangle"} where $Z = X$ gives a triangle $$R\Gamma(X, p\mathcal O_X)^\wedge_p[-1] \to R\Gamma(X,\mathbb Z_p(1)) \to R\Gamma(X_0,\mathbb Z_p(1)) \to^{+1}.$$ Comparing this triangle to [\[eq: exponential-inf-diagram\]](#eq: exponential-inf-diagram){reference-type="eqref" reference="eq: exponential-inf-diagram"} and taking the $p$-adic filtration on $p\mathcal O_X$ implies that we must show $$\tau_{\leq 2} (R\Gamma(X_0,\mathcal O_{X_0})^\wedge_p) \to \tau_{\leq 2} (R\Gamma(X_0, \mathcal O_{X_0})^\wedge_\varphi)$$ is an equivalence. The $k$-vector space $R\Gamma(X_0,\mathcal O_{X_0})$ is $p$-complete, while the hypothesis that $\varphi^*: H^i(X_0,\mathcal O_{X_0}) \to H^i(X_0,\mathcal O_{X_0})$ is zero for $0 \leq i \leq 3$ implies $$\tau_{\leq 2} R\Gamma(X_0,\mathcal O_{X_0}) \to \tau_{\leq 2} (R\Gamma(X_0,\mathcal O_{X_0})^\wedge_\varphi)$$ is an equivalence by Lemma [Lemma 39](#lemma: tate-module-sequence){reference-type="ref" reference="lemma: tate-module-sequence"}. ◻ **Remark 22**. If $X \to \mathop{\mathrm{Spec}}W_m(k)$ is proper, then the hypothesis on $H^3(X_0,\mathcal O_{X_0})$ in Theorem [Theorem 21](#theorem: mixed-characteristic-syntomic-comparison){reference-type="ref" reference="theorem: mixed-characteristic-syntomic-comparison"}, parts (ii) and (iii) above may be dropped, as the Tate module of an endomorphism of a finite-dimensional vector space is zero. **Example 23**. Lemma [Lemma 39](#lemma: tate-module-sequence){reference-type="ref" reference="lemma: tate-module-sequence"} furnishes us with a map $$\mathop{\mathrm{Pic}}(X)^\wedge_p \to H^2(X,\mathbb Z_p(1)) \to \{\text{tdo's on }X\}/\cong.$$ If $\mathcal L_r$ is a $p$-adic Cauchy sequence in $\mathop{\mathrm{Pic}}(X)$, then the associated tdo is constructed as follows: $\mathrm{Diff}_{\leq i}(\mathcal L_r)$ is independent of $r$ for $r$ large enough, as differential operators of order at most $i$ commute with $p^r$th powers for $r$ large enough. Setting $D_{\leq i}$ to be the stable value of $\mathrm{Diff}_{\leq i}(\mathcal L_r)$ defines the filtered pieces of a tdo. # Explicit constructions of tdo's in prime characteristic Suppose $S = \mathop{\mathrm{Spec}}k$, the case of prime characteristic. In this section, we to produce an explicit bijection between tdo's on $X/S$ up to isomorphism and $H^2(X,\mathbb Z_p(1))$, independent of the method of Gaitsgory-Rozenblyum discussed in §[2.2](#subsection: stacky approach){reference-type="ref" reference="subsection: stacky approach"}. We also compare twisted Grothendieck differential operators to twisted crystalline differential operators. ## Differential operators on torsors The ring of Grothendieck differential operators $\mathrm{Diff}_X$ in characteristic $p$ is a union of matrix algebras $\mathop{\mathrm{End}}_{\mathcal O_{X^{(r)}}}(\mathcal O_X)$ [@chase74; @berthelot96]. When $\mathrm{Diff}_X$ is replaced by a tdo, we show instead that $\mathrm{Diff}_X$ is a union of Azumaya algebras. **Definition 24**. Given a tdo $D$ on $X$, define $D^r$ to be the centralizer of $\mathcal O_{X^{(r)}}$ in $D$. By Corollary [Corollary 13](#corollary: cofinal-chains){reference-type="ref" reference="corollary: cofinal-chains"}, we have $D = \cup_{r \geq 1} D^r$. **Remark 25**. The subalgebra $\mathrm{Diff}_X^r = \mathop{\mathrm{End}}_{\mathcal O_{X^{(r)}}}(\mathcal O_X)\subseteq \mathrm{Diff}_X$ is the image of Berthelot's differential operators of level $r-1$ [@berthelot96]. **Proposition 26**. *$D^r$ is an Azumaya algebra over $X^{(r)}$ which splits along the relative Frobenius $X \to X^{(r)}$.* *Proof.* The claim is local. If $X$ has étale coordinates $t_1,\ldots, t_n$ dual to derivations $\partial_1,\ldots, \partial_n$, then $\mathop{\mathrm{gr}}(D^r)$ has a basis of those principal symbols $\partial_1^{(i_1)} \cdots \partial_n^{(i_n)}$ such that $i_j < p^r$ for all $j$. Hence $D^r$ is a locally free $\mathcal O_{X^{(r)}}$ module of rank $p^{2r \dim X}$. Now the map $\mathcal O_X \otimes_{\mathcal O_{X^{(r)}}} D^r \to \mathop{\mathrm{End}}_{\mathcal O_X}(D^r)$ by left and right multiplication may be checked to be an isomorphism by passing to associated graded. ◻ Hence $D^r$ defines an Azumaya algebra on $X^{(r)}$ which contains $\mathcal O_X$ as a maximal commutative subalgebra. As $k$ is perfect, we may identify $X^{(r)}$ with $X$ as a scheme; thus $\mathcal O_X \to D^r$ defines a $\mu_{p^r}$-gerbe on $X_{fppf}$. We now explicitly construct this gerbe using differential geometry. **Lemma 27** ([@illusie79],(5.1.3)). *If $X \to \mathop{\mathrm{Spec}}k$ is a scheme over a perfect field $k$ and $\varepsilon: X_{fppf} \to X_{\acute{e}t}$ is the canonical projection, then $$R\varepsilon_\ast \mu_{p^r} \cong \mathcal O_X^\times/(\mathcal O_X^\times)^{p^r}[-1].$$* *Proof.* Consider the short exact sequence $$\begin{tikzcd} 0 & {\mu_{p^r}} & {\mathbb G_m} & {\mathbb G_m} & 0 \arrow[from=1-1, to=1-2] \arrow[from=1-2, to=1-3] \arrow[from=1-3, to=1-4] \arrow[from=1-4, to=1-5] \end{tikzcd}$$ on $X_{fppf}$. By Hilbert's Theorem 90, $R\varepsilon_\ast \mathbb G_m = \mathbb G_m$, so comparing the triangle $R\varepsilon_\ast \mu_{p^r} \to \mathbb G_m \to \mathbb G_m \to^{+1}$ to the short exact sequence $$\begin{tikzcd} 0 & {\mathcal O_X^\times} & {\mathcal O_X^\times} & {\mathcal O_X^\times/(\mathcal O_X^\times)^{p^r}} & 0 \arrow[from=1-1, to=1-2] \arrow["{p^r}", from=1-2, to=1-3] \arrow[from=1-3, to=1-4] \arrow[from=1-4, to=1-5] \end{tikzcd}$$ on $X_{\acute{e}t}$ gives the result. ◻ Thus, a $\mu_{p^r}$-gerbe on $X_{fppf}$ is the same as an $\mathcal O_X^{\times}/(\mathcal O_X^\times)^{p^r}$-torsor on $X_{\acute{e}t}$. If $\mathcal L$ is a torsor over $\mathcal O_X^\times/(\mathcal O_X^\times)^{p^r}$, then taking $\mathrm{Diff}_{\mathcal L}^r$ makes sense since $(\mathcal O_X^\times)^{p^r}$ centralizes such differential operators. **Proposition 28**. *Consider the isomorphism $$H^2(X_{fppf},\mu_{p^r}) \cong H^1(X_{\acute{e}t},\mathcal O_X^\times/(\mathcal O_X^\times)^{p^r})$$ induced by Lemma [Lemma 27](#lemma: fppf-to-étale){reference-type="ref" reference="lemma: fppf-to-étale"} If $[D^r]$ is mapped to the $\mathcal O_X^\times/(\mathcal O_X^\times)^{p^r}$-torsor $\mathcal L$, then $D^r \cong \mathrm{Diff}_{\mathcal L}^r$ as associative algebras over $\mathcal O_X$.* *Proof.* By Proposition [Proposition 26](#prop: centralizer-is-azumaya){reference-type="ref" reference="prop: centralizer-is-azumaya"}, the algebra $D^r$ is Azumaya and thus splits étale locally. Étale covers of $X^{(r)}$ are of the form $\{U_\alpha^{(r)} \to X^{(r)}\}$ where $\{U_\alpha \to X\}$ is an étale cover. If $D^r$ splits on $U_\alpha^{(r)} \to X^{(r)}$, then $$D^r|_{U_\alpha} \cong End_{\mathcal O_{U_\alpha}^{(r)}}(\mathcal E_\alpha)$$ for some sheaf $\mathcal E$ on $X^{(r)}$. Using $\mathcal O_X \to D^r$ gives $\mathcal E_\alpha$ the structure of a $Fr^r_\ast\mathcal O_X$-module where $Fr^r: X \to X^{(r)}$ is the $r$-fold relative Frobenius. We claim that $$\mathcal E_\alpha = Fr^r_\ast \mathcal L_\alpha$$ for some line bundle $\mathcal L_\alpha$. It suffices to check this locally on $X^{(r)}$, so we may base change to the algebraic closure of $k$ and consider a closed point $p$ with local coordinates $t_1,\ldots, t_n$. We now replace $X$ with $(Fr^r)^{-1}(x)$. If $t_1,\ldots, t_n$ are local coordinates at $p$, then the coordinate ring of $(Fr^r)^{-1}(p)$ is $k[t_1,\ldots, t_n]/(t_1^p,\ldots, t_n^p)$. The sheaf $\mathcal E$ is a faithful $D^r$-module and thus a faithful $\mathcal O_X$-module. Hence $(t_1\cdots t_n)^{p-1}\mathcal E \neq 0$. If $e$ is a section of $\mathcal E$ such that $(t_1\cdots t_n)^{p-1}e \neq 0$, then since $(t_1\cdots t_n)^{p-1}$ generates the minimal nonzero ideal of $k[t_1,\ldots,t_n]/(t_1^p,\ldots, t_n^p)$, we obtain $\mathcal E$ is locally rank 1 with generator $e$. We have shown that on a splitting cover $U_\alpha^{(r)} \to X^{(r)}$, $$D^r|_{U_\alpha} \cong \mathop{\mathrm{End}}_{\mathcal O_{X^{(r)}}}(Fr^r_\ast \mathcal L_\alpha) \cong \mathrm{Diff}_{\mathcal L_\alpha}^r.$$ On overlaps $U_{\alpha\beta}$ an isomorphism $\mathrm{Diff}_{\mathcal L_\alpha}^r|_{U_{\alpha\beta}} \cong \mathrm{Diff}_{\mathcal L_\beta}^r|_{U_{\alpha\beta}}$ implies that $\mathcal L_\alpha$ and $\mathcal L_\beta$ differ by a $p^r$th power on $U_{\alpha\beta}$; hence $\mathcal L_\alpha / (\mathcal O_X^\times)^{(p^r)}$ glue to a $\mathcal O_X^\times/(\mathcal O_X^\times)^{p^r}$-torsor on $X$. Conversely, given a torsor over $\mathcal O_X^\times/(\mathcal O_X^\times)^{p^r}$, it may be locally lifted to line bundles $\mathcal L_\alpha$, and the algebras $\mathrm{Diff}_{\mathcal L_\alpha}^r$ descend to an Azumaya $D^r$. ◻ It is known that $H^2(X,\mathbb Z_p(1)) \cong \varprojlim H^2(X_{fppf},\mu_{p^r})$, as the system\ $\{H^\ast(X_{fppf},\mu_{p^r})\}_{r \geq 1}$ satisfies the Mittag-Leffler condition [@illusie79 p. 627]. **Corollary 29**. *The bijections $$\varprojlim\limits_r H^1(X_{\acute{e}t},\mathcal O_X^\times/(\mathcal O_X^\times)^{p^r})) \leftrightarrow H^2(X,\mathbb Z_p(1)) \leftrightarrow \{\text{tdo's on }X/k\}/\cong$$ send a system $(\mathcal L_r)_{r \geq 1}$ of compatible $\mathcal O_X^\times/(\mathcal O_X^\times)^{p^r}$-torsors to $$D = \cup_r \mathrm{Diff}_{\mathcal L_r}^r.$$* ## Compatibility with twisted crystalline differential operators {#subsection: compatibility} While $\mathrm{Diff}_X$ has divided powers of partial derivatives and is thus not finitely generated, there is a different algebra associated to $X$ known as the *crystalline differential operators* $\mathcal D^{crys}_X$. The algebra of crystalline differential operators $\mathcal D^{crys}_X$ is the enveloping algebra of the Lie algebroid $T_{X/S}$ over $\mathcal O_X$. It has a PBW filtration with associated graded $\mathop{\mathrm{gr}}\mathcal D^{crys}_X = \mathop{\mathrm{Sym}}_{\mathcal O_X} T_{X/S}$. Unlike $\mathrm{Diff}_X$, the algebra $\mathcal D^{crys}_X$ has a large center isomorphic to $\mathcal O_{T^*X^{(1)}}$. The center is parameterized by $s: \mathcal O_{T^*X^{(1)}} = \mathop{\mathrm{Sym}}_{\mathcal O_{X^{(1)}}} T_{X^(1)} \to \mathcal D^{crys}_X$ sending a function $f \in \mathcal O_{X^{(1)}}$ to $Fr(f)$ and a vector field $\partial$ to $\partial^p - \partial^{ {[p]} }$, where $\partial^{ {[p]} }$ is the $p$th power of $\partial$ as a vector field [@bmr §1.3]. The inclusion $\mathcal O_X \oplus T_{X/S} \to \mathrm{Diff}_X$ induces a map $\mathcal D^{crys}_X \to \mathrm{Diff}_X$ whose kernel is generated by the ideal of the zero section in $T^*X^{(1)}$ and whose image is $\mathrm{Diff}_X^1$ [@bmr (2.2.5)]. Associated to any tdo $D$ on $X$, we will construct a map analogous to $\mathcal D^{crys}_X \to \mathrm{Diff}_X$. Instead of crystalline differential operators, the domain will be more general filtered quantizations of $T^*X$ with a Frobenius splitting in the sense of [@bk08]. These have an explicit description in terms of restricted Lie algebroids, which we now recall. **Definition 30**. A *restricted Lie algebroid* on $X/k$ is a Lie algebroid $\tau: \mathcal A \to T_{X/S}$ equipped with an operation $-^ {[p]} : \mathcal A \to \mathcal A$ such that (i) $-^ {[p]}$ makes the Lie algebra $\mathcal A$ into a restricted Lie algebra such that the anchor map $\tau$ is a morphism of restricted Lie algebras; (ii) for $f$ a section of $\mathcal O_X$ and $x$ a section of $\mathcal A$, $$(fx)^ {[p]} = f^px^ {[p]} + \tau(fx)^{p-1}(f)x.$$ **Definition 31** ([@mundinger22], Definition 4.2). A *restricted Atiyah algebra* on $X$ is a restricted Lie algebroid of the form $$0 \to \mathcal O_X \to \mathcal A \to T_{X/S} \to 0$$ such that for all sections $f$ of $\mathcal O_X \subset \mathcal A$, we have $f^ {[p]} = f^p$ and $[x,f] = \tau(x)f$ for $x \in \mathcal A$. **Proposition 32** ([@devalapurkar21], Talk 11, Remark 23). *The functor sending a restricted Atiyah algebra $\mathcal A$ to its enveloping algebra $U\mathcal A$ defines an equivalence between restricted Atiyah algebras on $X$ and Frobenius-split filtered quantizations of $T^*X$.* *Proof.* Filtered quantizations of $T^*X$ are of the form $U\mathcal A$ where $\mathcal A$ is an Atiyah algebra. The algebra $\mathcal A$ has a filtered Frobenius splitting in the sense of Bezrukavnikov-Kaledin [@bk08] if and only if there is $-^{ {[p]} }: \mathcal A \to \mathcal A$ such that $s(\tau(x)) = x^p - x^ {[p]}$ defines a Frobenius splitting $s: \mathop{\mathrm{Sym}}_{\mathcal O_X} T_{X/S} \to U\mathcal A$. The axioms of a Frobenius splitting then hold if and only if $-^ {[p]}$ makes $\mathcal A$ into a restricted Atiyah algebra. ◻ Classification of restricted Atiyah algebras is as follows: there is the exact sequence $$\label{eq: milne-sequence} \begin{tikzcd} 0 & {(\mathcal O_X^\times)^p} & {\mathcal O_X^\times} & {\Omega^1_{X,cl}} & {\Omega^1_X} & 0 \arrow[from=1-1, to=1-2] \arrow[from=1-2, to=1-3] \arrow["d\log", from=1-3, to=1-4] \arrow["{1 - C}", from=1-4, to=1-5] \arrow[from=1-5, to=1-6] \end{tikzcd}$$ of sheaves on $X_{\acute{e}t}$, where $C$ is the Cartier operator [@milne-etale Proposition 4.14]. It was proved in [@mundinger22 Theorem 4.5] that restricted Atiyah algebras are classified by $$H^1(X, (\begin{tikzcd} {\Omega^1_{X,cl}} & {\Omega^1_X} \arrow["{id - C}", from=1-1, to=1-2] \end{tikzcd})).$$ In light of [\[eq: milne-sequence\]](#eq: milne-sequence){reference-type="eqref" reference="eq: milne-sequence"} and Lemma [Lemma 27](#lemma: fppf-to-étale){reference-type="ref" reference="lemma: fppf-to-étale"}, we have $$(\begin{tikzcd} {\Omega^1_{X,cl}} & {\Omega^1_X} \arrow["{id - C}", from=1-1, to=1-2] \end{tikzcd}) \simeq \mathcal O_X^\times/(\mathcal O_X^\times)^p \simeq R\varepsilon_\ast \mu_p [1],$$ on $X_{\acute{e}t}$, where $\varepsilon: X_{fppf} \to X_{\acute{e}t}$ is the canonical projection. Hence isomorphism classes of restricted Atiyah algebras on $X \to \mathop{\mathrm{Spec}}k$ are in bijection with $H^2(X_{fppf},\mu_p)$. This was also known by D. Bragg independently at the time of publication of [@mundinger22]. Additionally associated to a restricted Atiyah algebra $\mathcal A$ is its *restricted enveloping algebra* $u\mathcal A = U\mathcal A / (x^p - x^ {[p]} \mid x \in \mathcal A)$. **Lemma 33**. *Suppose that $X \to \mathop{\mathrm{Spec}}k$ and $\mathcal A$ is a restricted Atiyah algebra. Then:* (i) *$u\mathcal A$ is an Azumaya algebra on $X^{(1)}$ which contains $\mathcal O_X$ as a maximal commutative subalgebra;* (ii) *the class of the associated $\mu_p$-gerbe $[u\mathcal A] \in H^2(X_{fppf},\mu_p)$ agrees with the class of $[\mathcal A]$.* *Proof.* Since $U\mathcal A$ is a Frobenius-split quantization of $T^*X$, the algebra $U\mathcal A$ is Azumaya over its center $\mathcal O_{T^*X^{(1)}}$. Reducing modulo the ideal of the zero section of $T^*X^{(1)} \to X^{(1)}$ gives (i). If $\mathcal L$ is a $\mathcal O_X^\times/(\mathcal O_X^\times)^p$-torsor on $X$, then $$d\log[\mathcal L] \in H^1(X, \begin{tikzcd} {\Omega^1_{X,cl}} & {\Omega^1_X} \arrow["{id - C}", from=1-1, to=1-2] \end{tikzcd})$$ corresponds to the restricted Atiyah algebra $\mathrm{Diff}_{\mathcal L,\leq 1}$. The map $$u\mathrm{Diff}_{\mathcal L, \leq 1} \to \mathrm{Diff}_{\mathcal L}^1$$ induced by $\mathrm{Diff}_{\mathcal L,\leq 1}\to \mathrm{Diff}_{\mathcal L}^1$ is an isomorphism, as can be checked locally. Thus (ii) follows. ◻ **Proposition 34**. *Let $D$ be a tdo on $X \to \mathop{\mathrm{Spec}}k$. Then $D_{\leq 1}$ is a restricted Atiyah algebra via the identifications $\mathop{\mathrm{gr}}^0 D \cong \mathcal O_X$ and $\mathop{\mathrm{gr}}^1 D \cong T_{X/S}$. Further, the restricted enveloping algebra $u D_{\leq 1}$ is isomorphic to $D^1$.* *Proof.* Suppose $x$ is a section of $D_{\leq 1}$. Then $ad(x^p) = ad(x)^p$, so for $f_0,f_1$ sections of $\mathcal O_X$, $$[[x^p,f_0],f_1] = [ad(x)^p(f_0),f_1] = 0$$ since $[Q,f_0] \in \mathcal O_X$. It follows that $D_{\leq 1}$ is a restricted Atiyah algebra with $p$-operation $x^ {[p]} = x^p$. The natural map $UD_{\leq 1} \to D$ thus factors through $uD_{\leq 1}$. As $uD_{\leq 1}$ is generated in degree 1, the image commutes with $\mathcal O_X^p$, and passing to associated graded shows the map $uD_{\leq 1} \to D^1$ is an isomorphism. ◻ We have constructed the desired map $\widetilde{\mathcal D}^{crys} = UD_{\leq 1} \to D$ whose kernel is $(x^p - x^ {[p]} \mid x \in D_{\leq 1})$, whose image is $D^1$, and which agrees with $\mathop{\mathrm{gr}}(\mathcal D^{crys}_X \to \mathrm{Diff}_X)$ on associated graded. # Derived completions {#appendix: completion} Fix a stable $\infty$-category $\mathcal C$ with countable limits. We define the derived completion of an object of $\mathcal C$ at an endomorphism and give a universal property for it. Our treatment is a dual version of Arinkin-Gaitsgory's cocompletion in triangulated categories [@ag15 §3.1]. Given $\mathcal C$, the category of endomorphisms in $\mathcal C$ is the $\infty$-category $$\mathcal C^{end} = \mathop{\mathrm{Fun}}(B\mathbb N, \mathcal C),$$ where $\mathbb N$ is the free monoid on one generator $1 \in \mathbb N$. We will denote an object of $\mathcal C^{end}$ by $(A,a)$ where $A \in \mathcal C$ and $a \in \mathop{\mathrm{End}}(A)$ is the image of $1 \in \mathbb N$, although an object of $\mathcal C^{end}$ also involves coherent choices of powers of $a$. We also have the full subcategory $\mathcal C^{aut} \to \mathcal C^{end}$ of those $(A,a)$ where $a$ is an equivalence. **Definition 35**. Let $(A,a) \in \mathcal C^{end}$. Then the *derived colocalization* of $(A,a)$ is the inverse limit $$\label{eq: derived-colocalization} T(A,a) = \varprojlim \left( \begin{tikzcd} \cdots & {(A,a)} & {(A,a)} & {(A,a)} \arrow["a", from=1-1, to=1-2] \arrow["a", from=1-2, to=1-3] \arrow["a", from=1-3, to=1-4] \end{tikzcd} \right)$$ in $\mathcal C^{end}$. The endomorphism of $T(A,a)$ is induced by $a: (A,a) \to (A,a)$. **Proposition 36**. *Derived colocalization is right adjoint to the inclusion $\mathcal C^{aut} \to \mathcal C^{end}$.* *Proof.* Let $p: T(A,a) \to (A,a)$ be the projection onto the first factor in the inverse limit [\[eq: derived-colocalization\]](#eq: derived-colocalization){reference-type="eqref" reference="eq: derived-colocalization"}. It suffices to show that if $(B,b) \in \mathcal C^{aut}$, then $$p_*: \mathop{\mathrm{Hom}}((B,b),T(A,a)) \to \mathop{\mathrm{Hom}}((B,b),(A,a))$$ is an equivalence. By definition of limit, $$\label{eq: universal property of limit} \mathop{\mathrm{Hom}}((B,b),T(A,a)) \simeq \varprojlim_a \mathop{\mathrm{Hom}}((B,b),(A,a)),$$ but the action of $a$ on $\mathop{\mathrm{Hom}}((B,b),(A,a))$ is equivalent to the action of $b$ since morphisms in $\mathcal C^{end}$ are intertwiners. The morphism $b$ is an equivalence, so $a_*: \mathop{\mathrm{Hom}}((B,b),(A,a)) \to \mathop{\mathrm{Hom}}((B,b),(A,a))$ is also an equivalence. Now a filtered limit along equivalences gives an equivalence $$\varprojlim_a \mathop{\mathrm{Hom}}((B,b),(A,a)) \to \mathop{\mathrm{Hom}}((B,b),(A,a)),$$ by e.g. [@mayponto]\[Proposition 2.2.9\]. ◻ Thus the inclusion $\mathcal C^{aut} \to \mathcal C^{end}$ is a colocalization, so that we have a short exact sequence of categories $$\begin{tikzcd} {\mathcal C^{aut}} & {\mathcal C^{end}} & {\mathcal C^{end, c}} \arrow[shift left, from=1-1, to=1-2] \arrow[shift left, from=1-2, to=1-1] \arrow[shift left, from=1-2, to=1-3] \arrow[shift left, from=1-3, to=1-2] \end{tikzcd}$$ where $(\mathcal C^{end})^\perp = \mathcal C^{end,c}$ will be the *complete objects* in $\mathcal C^{end}$: **Definition 37**. Let $(A,a) \in \mathcal C^{end}$. The *completion* of $(A,a)$ is $$A^\wedge_a = \mathop{\mathrm{cone}}(T(A,a) \to (A,a)).$$ $(A,a)$ is *complete* if $T(A,a)$ is contractible, that is, $(A,a) \to A^\wedge_a$ is an equivalence. By definition, the full subcategory $\mathcal C^{end,c}$ on complete objects of $\mathcal C^{end}$ is the right orthogonal of $\mathcal C^{aut}$, which immediately implies: **Corollary 38**. *The inclusion $\mathcal C^{end,c} \to \mathcal C^{end}$ is right adjoint to completion $(A,a) \mapsto A^\wedge_a$.* Consider now the case when $\mathcal C = D(R)$ for a ring $R$. On abelian groups, $R\lim$ over an $\mathbb N$-indexed diagram has cohomological dimension one [@weibel Corollary 3.5.4]. Hence in $D(R)$ the cohomology of $A^\wedge_a$ may be expressed in terms of short exact sequences involving $H^*(A)$ and the action of $H^*(a)$. If $M$ is a classical $\mathbb Z[t]$-module, recall the classical *Tate module* $T_tM = \lim_r B[t^r]$, where the transition maps are given by multiplication by $t$. **Lemma 39**. *Let $R$ be a ring and $(A,a) \in D(R)^{end}$. For all $i \in \mathbb Z$, there are short exact sequences $$\begin{tikzcd} 0 & {H^0(H^i(A)^\wedge_a)} & {H^i(A^\wedge_a)} & {T_aH^{i+1}(A)} & 0 \arrow[from=1-1, to=1-2] \arrow[from=1-2, to=1-3] \arrow[from=1-3, to=1-4] \arrow[from=1-4, to=1-5] \end{tikzcd}$$* *Proof.* The category $D(R)^{end}$ is equivalent to $D(R[a])$. Now [@stacks-project 15.93.5, [Tag 0BKF](https://stacks.math.columbia.edu/tag/0BKF)] gives the above short exact sequence for complexes of $R[a]$-modules completed at $a$. ◻

arxiv_math

--- abstract: | Low-rank matrix approximation (LRMA) has been arisen in many applications, such as dynamic MRI, recommendation system and so on. The alternating direction method of multipliers (ADMM) has been designed for the nuclear norm regularized least squares problem or rank constrained form of LRMA with specialized measurement operators and shows a good performance. However, due to the lack of guarantees for the nonconvex ADMM, there are no ADMM algorithms designed directly for the rank constraint form of the general LRMA. Therefore, in this paper, we propose an ADMM-based algorithm for the reformulated rank constraint matrix approximation problem, which works well for noiseless and noisy case. Based on the Kurdyka-Lojasiewicz (KL) property, we prove that our proposed nonconvex ADMM algorithm globally converges to the optimal solution when the original LRMA has the unique optimal solution. And we discuss a specific case: the matrix completion problem. Numerical experiments show that spcialized for matrix completion, the proposed algorithm performs better when the sampling rate is really low in noisy case, which is the key in matrix completion.\ **keywords** Low-rank matrix approximation, ADMM, rank constraint, matrix completion. author: - | Zekun Liu\ School of Mathematical Sciences, Shanghai Jiao Tong University bibliography: - references.bib title: "**Low-Rank Matrix Approximation via Nonconvex ADMM**" --- # Introduction {#sec:1} Low-rank matrices arise widely in the field of computer science and applied mathematics, such as the Netflix problem [@netflix], quantum state tomography [@quantum], sensor localization [@sensor1; @sensor2] and natural language processing [@nlp1; @nlp2]. In all of these applications, the data size can be extremely large, hence it is expensive and impossible to fully sample the entire data, which leads to the incomplete observations. A natural question is how can we recover the original data from these incomplete observations. In general, such recovery is not guaranteed to be possible. However, with the prior information that the matrix is low-rank [@lrcc], it is possible to recover the original matrix from incomplete observations in an efficient way. One of the most common and widely used model in matrix recovery is the low-rank matrix approximation [@lrma1; @lrma2]. For the general low-rank matrix approximation problem (LRMA), we will only observe $b=\mathcal{A}(X)+e$ where $e$ denotes noise and $\mathcal{A}:\mathbb{R}^{m\times n}\to \mathbb{R}^{d}$ is the linear measurement operator which transforms $X$ to $(\left \langle X,A_1 \right \rangle,\left \langle X,A_2 \right \rangle,\cdots,\left \langle X,A_d \right \rangle)^{\top}$ with $A_1,A_2,\cdots,A_d \in \mathbb{R}^{m\times n}$ given and $\left \langle \cdot,\cdot \right \rangle$ being matrix inner product. In this case, the formulation of the low-rank approximation problem is $$\label{equ:1.1} \begin{aligned} \min_{X\in \mathbb{R}^{m\times n}} &\left \| \mathcal{A}(X)-b \right \|_{2}^{2} \\ s.t. \enspace &rank(X)\le r, \end{aligned}$$ where $r<\min\left \{ m,n \right \}$ denotes the upper tight estimation of the rank of the target matrix. As stated in [@ADMiRA], the formulation ([\[equ:1.1\]](#equ:1.1){reference-type="ref" reference="equ:1.1"}) can handle both noiseless and noisy case in a single form. And it can also work for the case that the original matrix is not exactly low-rank but can be approximated accurately by a low-rank matrix. Besides, comparing to the fixed-rank constraint, low-rank constraint can avoid the uncertainty about the existence of the solutions and the complexity of the algorithm convergence analysis which are both arisen from the closure of the fixed-rank constraint [@fixrank]. Due to the combinatorial characteristic of rank, problem ([\[equ:1.1\]](#equ:1.1){reference-type="ref" reference="equ:1.1"}) belongs to the NP-hard combinatorial optimization problem, which is difficult to analyse in theory and design algorithms [@fixrank]. A widely used convex relaxation to the rank minimization problem is the nulear norm $\left \| \cdot \right \|_\ast$ given by [@nuclear]. Based on this convex relaxation, singular value thresholding (SVT) [@SVT] is proposed to solve the specific low-rank matrix completion problem (LRMC) where the operator $\mathcal{A}$ only takes a subset of the original matrix [@lrmc]. However, the efficiency of SVT is only guaranteed in the noiseless case with linear equality constraints. Inspired by the similarity and connection between rank minimization for matrices and $\ell_0$-norm minimization for vectors [@relation], atomic decomposition for minimum rank approximation (ADMiRA) [@ADMiRA] is proposed, which is a generalization of the compressive sampling matching pursuit (CoSaMP) [@CoSaMP] to matrices. And it has strong theoretical guarantees. But for LRMC, because the linear operator does not satisfies the R-RIP [@lrcc], ADMiRA has no theoretical guarantees. Iterative hard thresholding (IHT) [@IHT1; @IHT2] can also work for the convex nuclear norm minimization. And it converges really fast when the rank of the original matrix is very low. Similarly, the alternating direction method of multipliers (ADMM) [@NADMM; @ADMM1; @ADMM2] is also efficient for the nuclear norm regularized least squares problem or LRMC. And specialized for LRMC, some manifold optimization algorithms such as [@mani1; @mani2] are efficient and theoretically sound. Since the rank constraint of matrices is an extension of $\ell_{0}$-norm constraint of vectors, we desire to generalize our previous research on the multiple measurement vector (MMV) problem with the $\ell_{2,0}$-norm [@MMVADMM] to the LRMA. In this paper, we propose an ADMM [@ADMM] algorithm for the general LRMA, where the measurement operator $\mathcal{A}$ is linear. First, we reformulate the rank constraint problem ([\[equ:1.1\]](#equ:1.1){reference-type="ref" reference="equ:1.1"}) to convert it to a two-block problem with a linear equality constraint. Then we apply ADMM to solve the reformulated problem and call the algorithm LRMA-ADMM. Specialized for LRMC, we also give the specific form of LRMA-ADMM, and call it LRMC-ADMM for the significant matrix completion problem. Theoretical analysis shows that LRMA-ADMM globally converges to the unique optimal solution. For the specialized matrix completion problem, compared with the similar algorithms, experiments show that the recovery accuracy of the proposed algorithm is almost the same with NIHT [@NIHT] in noiseless case, both better than ADMM [@ADMM1]. But for the noisy case, our algorithm shows higher reconstruction accuracy when the sampling rate is really low, which is the key of matrix completion. The rest of this paper is organized as follows. In Section [2](#sec:2){reference-type="ref" reference="sec:2"}, we introduce some notations and preliminaries used for further analysis. Section [3](#sec:3){reference-type="ref" reference="sec:3"} reformulates the original problem. In Section [4](#sec:4){reference-type="ref" reference="sec:4"}, we propose our LRMA-ADMM for solving the low-rank matrix approximation problem, and analyse its time complexity. Section [4](#sec:4){reference-type="ref" reference="sec:4"} discusses the algorithm for a specific case: the low-rank matrix completion. Section [6](#sec:6){reference-type="ref" reference="sec:6"} proves the global convergence of the proposed algorithm. And Section [7](#sec:7){reference-type="ref" reference="sec:7"} gives the numerical experiments. At last, Section [8](#sec:8){reference-type="ref" reference="sec:8"} concludes the paper. # Preliminaries {#sec:2} In this section, we list some notations and definitions which are used for further analysis. Notations: $rank(X)$, $X^{\top}$ and $\left \| X \right \|_F$ represent the rank, the transpose and the Frobenius-norm of the matrix $X$ respectively. $\left \langle \cdot,\cdot \right \rangle$ denotes the inner product of two matrices of equal size. $I_k$ represents the $k\times k$ identity matrix. While $I$ denotes the identity reflection. $\mathbf{1}$ represents the matrix of all ones. $vec(X)$ denotes the vector given by concatenating each column of the matrix $X$ in order. $x_m$ represents the $m$-th element of the vector $x$. And $a_{ij}$ denotes the $(i,j)$-element of the matrix $A$ . $A\odot B$ denotes the element-wise product of two matrices of equal size. While $A\oslash B$ denotes the element-wise division of two matrices of equal size. For two composable operators $\mathcal{A}$ and $\mathcal{B}$, $\mathcal{A}\mathcal{B}$ represents their composition $\mathcal{A}\circ \mathcal{B}$. ***Definition* 1** ([@view1]). *For the linear operator $$\begin{aligned} \mathcal{A}:&\mathbb{R}^{m\times n}\to \mathbb{R}^d \\ &X\mapsto (\left \langle A_1,X \right \rangle,\left \langle A_2,X \right \rangle,\cdots,\left \langle A_d,X \right \rangle)^{\top}, \end{aligned}$$ the adjoint of $\mathcal{A}$ is defined as $\mathcal{A}^{\ast}(w)=\sum_{i=1}^d w_i A_i$, where $w\in \mathbb{R}^d$.* ***Definition* 2**. *For the function $F(i,j)$ where $i=1,2,\cdots,m; j=1,2,\cdots,n$, denote $\sum_{(i,j)\ne (p,q)} F(i,j)\\ =\sum_{i=1}^m \sum_{j=1}^n F(i,j)-F(p,q)$ to simplify the right hand side.* ***Definition* 3** ([@cop]). *For the generalized real function $f:\mathbb{R}^{m\times n}\to \mathbb{R}\cup \left \{ \pm \infty \right \}$.* (i) *Given a nonempty set $\mathcal{X}$, call $f$ proper to $\mathcal{X}$ if $\exists x\in \mathcal{X}$ such that $f(x)<+\infty$, and $\forall x\in \mathcal{X}, f(x)>-\infty$.* (ii) *$f$ is lower semicontinuous if $\forall x\in \mathbb{R}^{m\times n}, \liminf_{y\to x}f(y)\ge f(x)$.* (iii) *$f$ is closed if its epigraph $$epif=\left \{ (x,t)\in \mathbb{R}^{m\times n}\times \mathbb{R}|f(x)\le t\right \}$$ is closed.* (iv) *$f$ is Gradient-$L$-Lipschitz continuous if $\exists L>0$, for $\forall x,y\in \mathbb{R}^{m\times n}$, $\left \| \nabla f(x)-\nabla f(y) \right \| \le L\left \| x-y \right \|$.* ***Proposition* 1** ([@cop]). *For the generalized real function $f$, $f$ is lower semicontinuous iff $f$ is closed.* ***Definition* 4** ([@semi]). *$S$ is a semi-algebraic set in $\mathbb{R}^{m\times n}$ if there exist polynomials $g_{ij},h_{ij}:\mathbb{R}^{m\times n}\to \mathbb{R}$ with $1\le i\le p,1\le j\le q$, such that $$S=\bigcup_{i=1}^{p}\bigcap_{j=1}^{q}\left \{x\in \mathbb{R}^{m\times n}:g_{ij}(x)=0,h_{ij}(x)>0 \right \}.$$ And for the proper function $f$, it is semi-algebraic if its graph $$graphf=\left \{(x,t)\in \mathbb{R}^{m\times n}\times\mathbb{R}:f(x)=t \right \}$$ is a semi-algebraic set in $\mathbb{R}^{m\times n}\times\mathbb{R}$.* ***Proposition* 2** ([@semi]). *Semi-algebra has the following properties.* (i) *Semi-algebra is stable under finite unions.* (ii) *Real polynomials are all semi-algebra.* (iii) *The indicator functions of semi-algebraic sets are also semi-algebra.* (iv) *Suppose $A$ is a semi-algebra in $\mathbb{R}^{m\times n}\times\mathbb{R}$, then its projection $\pi (A)$ where $\pi:\mathbb{R}^{m\times n}\times\mathbb{R}\to \mathbb{R}^{m\times n}$ is also a semi-algebra.* ***Proposition* 3** ([@semiKL]). *If $f$ is proper lower semicontinuous and semi-algebraic, then it is also a KL function.* As for the definition of subdifferential, Karush-Kuhn-Tucker (KKT) conditions and Kurdyka-Lojasiewicz (KL) property, we recommend the interested readers to refer to [@cop; @semi; @semiKL]. # Problem formulation {#sec:3} In order to apply ADMM to problem ([\[equ:1.1\]](#equ:1.1){reference-type="ref" reference="equ:1.1"}), we rewrite the low-rank approximation problem ([\[equ:1.1\]](#equ:1.1){reference-type="ref" reference="equ:1.1"}) to make it a two-block optimization problem with a linear equation constraint. First, we introduce the indicator function $$\mathcal{I}_{\mathcal{M}}(X)=\begin{cases} 0, & \text{ \textit{if} } X\in \mathcal{M} \\ +\infty, & \text{ \textit{if} } X\notin \mathcal{M} \end{cases}$$ of the set $\mathcal{M}=\left \{ X\in \mathbb{R}^{m\times n}: rank(X) \le r \right \}$ to convert problem ([\[equ:1.1\]](#equ:1.1){reference-type="ref" reference="equ:1.1"}) to an unconstrained optimization problem as follows: $$\label{equ:3.1} \begin{aligned} \min_{X\in \mathbb{R}^{m\times n}} \left \| \mathcal{A}(X)-b \right \|_{2}^{2} + \mathcal{I}_{\mathcal{M}}(X). \end{aligned}$$ Then introduce $Y\in \mathbb{R}^{m\times n}$ as an auxiliary matrix of $X$ to convert problem ([\[equ:3.1\]](#equ:3.1){reference-type="ref" reference="equ:3.1"}) as the following linear equation constrained problem: $$\label{equ:3.2} \begin{aligned} \min_{X,Y\in \mathbb{R}^{m\times n}} &\left \| \mathcal{A}(X)-b \right \|_{2}^{2} + \mathcal{I}_{\mathcal{M}}(Y)\\ s.t. \quad & X-Y=0. \end{aligned}$$ ([\[equ:3.2\]](#equ:3.2){reference-type="ref" reference="equ:3.2"}) is the final two-block optimization problem with a linear equation constraint to describe the low-rank matrix approximation problem. # Algorithm {#sec:4} In this section, we apply ADMM to problem ([\[equ:3.2\]](#equ:3.2){reference-type="ref" reference="equ:3.2"}) to develop our algorithm for solving the low-rank approximation problem. The augmented Lagrangian function of problem ([\[equ:3.2\]](#equ:3.2){reference-type="ref" reference="equ:3.2"}) is $$\label{equ:4.1} \mathcal{L}_{\rho}(X,Y,L)= \left \| \mathcal{A}(X)-b \right \|_{2}^{2} + \mathcal{I}_{\mathcal{M}}(Y) + \left \langle L,X-Y \right \rangle +\frac{\rho}{2}\left \| X-Y \right \|_{F}^{2},$$ where $L\in \mathbb{R}^{m\times n}$ is the Lagrangian multiplier of the constraint $X-Y=0$, and $\rho>0$ is the penalty parameter. Applying ADMM to problem ([\[equ:3.2\]](#equ:3.2){reference-type="ref" reference="equ:3.2"}), the target matrix can be approximately obtained by minimizing the variables in the augmented Lagrangian function ([\[equ:4.1\]](#equ:4.1){reference-type="ref" reference="equ:4.1"}) with a Gauss-Seidel format as follows: $$\label{equ:4.2} \begin{aligned} \left\{\begin{matrix} Y^{k+1}=&\mathop{\arg\min}\limits_{Y\in \mathbb{R}^{m\times n}}\mathcal{L}_{\rho}(X^k,Y,L^{k}),\\ X^{k+1}=&\mathop{\arg\min}\limits_{X\in \mathbb{R}^{m\times n}}\mathcal{L}_{\rho}(X,Y^{k+1},L^{k}),\\ L^{k+1}=&L^{k}+\rho (X^{k+1}-Y^{k+1}). \end{matrix}\right. \end{aligned}$$ We will solve the subproblems in Equation ([\[equ:4.2\]](#equ:4.2){reference-type="ref" reference="equ:4.2"}) one by one. ## Update Y {#sub:4.1} Fix $X$ and $L$, $$\label{equ:4.3} \begin{split} Y^{k+1}&=\mathop{\arg\min}\limits_{Y\in \mathbb{R}^{m\times n}}\mathcal{L}_{\rho}(X^k,Y,L^{k})\\ &=\mathop{\arg\min}\limits_{Y\in \mathbb{R}^{m\times n}}\left \{ \mathcal{I}_{\mathcal{M}}(Y)+\left \langle L^{k},X^k-Y \right \rangle +\frac{\rho}{2}\left \| X^k-Y \right \|_{F}^{2} \right \} \\ &=\mathop{\arg\min}\limits_{Y\in \mathcal{M}}\left \| X^k-Y+\frac{L^{k}}{\rho} \right \|_{F}^{2}\\ &=\mathcal{P}_{\mathcal{M} }(X^{k}+\frac{L^{k}}{\rho}), \end{split}$$ where $\mathcal{P}_{\mathcal{M}}(\cdot)$ denotes the projection operator on the set $\mathcal{M}=\left \{ X\in \mathbb{R}^{m\times n}:rank(X)\le r \right \}$. In fact, the projection onto the rank-constrained set $\mathcal{M}$ has the explicit solution, which is actually the truncated SVD [@TSVD; @TSVD2]: First calculate the SVD of $X^{k}+\frac{L^{k}}{\rho}$, we have $X^{k}+\frac{L^{k}}{\rho}=\sum_{i=1}^{\min \left \{ m,n \right \}} \sigma_i u_i v_{i}^{\top}$, where $\sigma_1\ge \sigma_2 \ge \cdots \ge \sigma_{\min \left \{ m,n \right \}} \ge 0$ is the singular value, and $u_i, v_i$ are the corresponding left and right singular vectors. Then the projection $$\label{equ:4.4} Y^{k+1}=\mathcal{P}_{\mathcal{M} }(X^{k}+\frac{L^{k}}{\rho})=\sum_{i=1}^{r} \sigma_i u_i v_{i}^{\top}.$$ ## Update X {#sub:4.2} Fix $Y$ and $L$, $$\label{equ:4.5} \begin{split} X^{k+1}&=\mathop{\arg\min}\limits_{X\in \mathbb{R}^{m\times n}}\mathcal{L}_{\rho}(X,Y^{k+1},L^{k})\\ &=\mathop{\arg\min}\limits_{X\in \mathbb{R}^{m\times n}}\left \{ \left \| \mathcal{A}(X)-b \right \|_{2}^{2} + \left \langle L^{k},X-Y^{k+1}\right \rangle +\frac{\rho}{2}\left \| X-Y^{k+1} \right \|_{F}^{2} \right \} \\ &=\mathop{\arg\min}\limits_{X\in \mathbb{R}^{m\times n}}\left \{ \left \| \mathcal{A}(X)-b \right \|_{2}^{2}+\frac{\rho}{2}\left \| X-Y^{k+1}+\frac{L^{k}}{\rho} \right \|_{F}^{2} \right \}. \end{split}$$ Denote $f(X)=\left \| \mathcal{A}(X)-b \right \|_{2}^{2}+\frac{\rho}{2}\left \| X-Y^{k+1}+\frac{L^{k}}{\rho} \right \|_{F}^{2}$. Obviously, it is a continuously differentiable function, hence $$X^{k+1}=\mathop{\arg\min}\limits_{X\in \mathbb{R}^{m\times n}}f(X) \Longleftrightarrow \nabla f(X^{k+1})=0.$$ It is not hard to calculate that $$\nabla f(X^{k+1})=2\mathcal{A}^{\ast}(\mathcal{A}(X^{k+1})-b)+\rho (X^{k+1}-Y^{k+1}+\frac{L^{k}}{\rho})=(2\mathcal{A}^{\ast}\mathcal{A}+\rho I)(X^{k+1})-2\mathcal{A}^{\ast}(b)-\rho Y^{k+1}+L^{k}.$$ Therefore, to update $X$, we just need to solve the matrix equation $\nabla f(X^{k+1})=0$, which is $$\label{equ:4.6} (2\mathcal{A}^{\ast}\mathcal{A}+\rho I)(X^{k+1})=2\mathcal{A}^{\ast}(b)+\rho Y^{k+1}-L^{k}.$$ Next we will prove the existence and uniqueness of the solution to the matrix equation ([\[equ:4.6\]](#equ:4.6){reference-type="ref" reference="equ:4.6"}) by giving its explicit solution directly. ***Proposition* 4**. *The matrix equation ([\[equ:4.6\]](#equ:4.6){reference-type="ref" reference="equ:4.6"}) has the unique solution which can be given explicitly.* ***Proof* 1**. *Denote $A_i=(a_{st}^i)_{m\times n}, \enspace i=1,2,\cdots,d$, and $X^{k+1}=(x_{st})_{m\times n}$, then $$\label{equ:4.7} \begin{split} (2\mathcal{A}^{\ast}\mathcal{A}+\rho I)(X^{k+1})&=2\sum_{i=1}^d \left \langle A_i,X^{k+1} \right \rangle \cdot A_i +\rho X^{k+1} \\ &= 2\sum_{i=1}^d (\sum_{s=1}^m \sum_{t=1}^n a_{st}^i x_{st})\cdot \begin{pmatrix} a_{11}^i & \cdots & a_{1n}^i\\ \vdots & \ddots & \vdots \\ a_{m1}^i & \cdots & a_{mn}^i \end{pmatrix} + \rho \cdot \begin{pmatrix} x_{11} & \cdots & x_{1n}\\ \vdots & \ddots & \vdots \\ x_{m1} & \cdots & x_{mn} \end{pmatrix} \\ &= \begin{pmatrix} (2\sum_{i=1}^d (a_{11}^i)^2+\rho)x_{11}+2\sum_{(s,t)\ne (1,1)} (\sum_{i=1}^d a_{st}^ia_{11}^i)x_{st} & \cdots & \ast \\ \vdots & \ddots & \vdots \\ (2\sum_{i=1}^d (a_{m1}^i)^2+\rho)x_{m1}+2\sum_{(s,t)\ne (m,1)} (\sum_{i=1}^d a_{st}^ia_{m1}^i)x_{st} & \cdots & \ast \end{pmatrix} \\ & \coloneqq W, \end{split}$$ where $W_{pq}=(2\sum_{i=1}^d (a_{pq}^i)^2+\rho)x_{pq}+2\sum_{(s,t)\ne (p,q)} (\sum_{i=1}^d a_{st}^ia_{pq}^i)x_{st}$ for $1\le p\le m,1\le q\le n$, and $\ast$ represents the omitted elements of the matrix $W$.* *From the expression of $(2\mathcal{A}^{\ast}\mathcal{A}+\rho I)(X^{k+1})$ in ([\[equ:4.7\]](#equ:4.7){reference-type="ref" reference="equ:4.7"}), we can see that the matrix equation ([\[equ:4.6\]](#equ:4.6){reference-type="ref" reference="equ:4.6"}) is actually a system of linear equations with variables the elements of $X^{k+1}$.* *Therefore, denote $\tilde{X}=vec((X^{k+1})^{\top})\in \mathbb{R}^{mn}, B=vec((2\mathcal{A}^{\ast}(b)+\rho Y^{k+1}-L^{k})^{\top})\in \mathbb{R}^{mn}$. The matrix equation ([\[equ:4.6\]](#equ:4.6){reference-type="ref" reference="equ:4.6"}) can be converted to a system of linear equations as follows: $$\label{equ:4.8} \left [ 2\begin{pmatrix} \sum_{i=1}^d (a_{11}^i)^2 & \sum_{i=1}^d a_{11}^ia_{12}^i & \cdots & \sum_{i=1}^d a_{11}^ia_{m,n-1}^i & \sum_{i=1}^d a_{11}^ia_{mn}^i\\ \sum_{i=1}^d a_{12}^ia_{11}^i & \sum_{i=1}^d (a_{12}^i)^2 & \cdots & \sum_{i=1}^d a_{12}^ia_{m,n-1}^i & \sum_{i=1}^d a_{12}^ia_{mn}^i\\ \vdots & \vdots & \ddots & \vdots & \vdots\\ \sum_{i=1}^d a_{m,n-1}^ia_{11}^i & \sum_{i=1}^d a_{m,n-1}^ia_{12}^i & \cdots & \sum_{i=1}^d (a_{m,n-1}^i)^2 & \sum_{i=1}^d a_{m,n-1}^ia_{mn}^i\\ \sum_{i=1}^d a_{mn}^ia_{11}^i & \sum_{i=1}^d a_{mn}^ia_{12}^i & \cdots & \sum_{i=1}^d a_{mn}^ia_{m,n-1}^i & \sum_{i=1}^d (a_{mn}^i)^2 \end{pmatrix}+\rho I_{mn} \right ]\Tilde{X}=B.$$* *Moreover, denote $$D=\begin{pmatrix} \sum_{i=1}^d (a_{11}^i)^2 & \sum_{i=1}^d a_{11}^ia_{12}^i & \cdots & \sum_{i=1}^d a_{11}^ia_{m,n-1}^i & \sum_{i=1}^d a_{11}^ia_{mn}^i\\ \sum_{i=1}^d a_{12}^ia_{11}^i & \sum_{i=1}^d (a_{12}^i)^2 & \cdots & \sum_{i=1}^d a_{12}^ia_{m,n-1}^i & \sum_{i=1}^d a_{12}^ia_{mn}^i\\ \vdots & \vdots & \ddots & \vdots & \vdots\\ \sum_{i=1}^d a_{m,n-1}^ia_{11}^i & \sum_{i=1}^d a_{m,n-1}^ia_{12}^i & \cdots & \sum_{i=1}^d (a_{m,n-1}^i)^2 & \sum_{i=1}^d a_{m,n-1}^ia_{mn}^i\\ \sum_{i=1}^d a_{mn}^ia_{11}^i & \sum_{i=1}^d a_{mn}^ia_{12}^i & \cdots & \sum_{i=1}^d a_{mn}^ia_{m,n-1}^i & \sum_{i=1}^d (a_{mn}^i)^2 \end{pmatrix}$$ in the linear equations ([\[equ:4.8\]](#equ:4.8){reference-type="ref" reference="equ:4.8"}). Note that $D$ can be decomposed as the multiplication of two matrices: $$\label{equ:4.9} D=\begin{pmatrix} a_{11}^1 & a_{11}^2 & \cdots & a_{11}^{d-1} & a_{11}^d\\ a_{12}^1 & a_{12}^2 & \cdots & a_{12}^{d-1} & a_{12}^d\\ \vdots & \vdots & \ddots & \vdots & \vdots \\ a_{m,n-1}^1 & a_{m,n-1}^2 & \cdots & a_{m,n-1}^{d-1} & a_{m,n-1}^d\\ a_{mn}^1 & a_{mn}^2 & \cdots & a_{mn}^{d-1} & a_{mn}^d \end{pmatrix}\cdot \begin{pmatrix} a_{11}^1 & a_{12}^1 & \cdots & a_{m,n-1}^1 & a_{mn}^1\\ a_{11}^2 & a_{12}^2 & \cdots & a_{m,n-1}^2 & a_{mn}^2\\ \vdots & \vdots & \ddots & \vdots & \vdots \\ a_{11}^{d-1} & a_{12}^{d-1} & \cdots & a_{m,n-1}^{d-1} & a_{mn}^{d-1}\\ a_{11}^d & a_{12}^d & \cdots & a_{m,n-1}^d & a_{mn}^d \end{pmatrix}.$$* *Therefore, denote $\tilde{A}=(vec(A_1),vec(A_2),\cdots,vec(A_d))\in \mathbb{R}^{(mn)\times d}$, then $D=\tilde{A}\cdot \tilde{A}^{\top}$, and the system of linear equations ([\[equ:4.8\]](#equ:4.8){reference-type="ref" reference="equ:4.8"}) can be expressed as $$\label{equ:4.10} (2\tilde{A}\cdot \tilde{A}^{\top}+\rho I_{mn})\tilde{X}=B,$$ where $2\tilde{A}\cdot \tilde{A}^{\top}+\rho I_{mn}$ is obviously positive definite, equation ([\[equ:4.10\]](#equ:4.10){reference-type="ref" reference="equ:4.10"}) has the unique solution $\tilde{X}=(2\tilde{A}\cdot \tilde{A}^{\top}+\rho I_{mn})^{-1}B$.* *At last, with the solution of equation ([\[equ:4.10\]](#equ:4.10){reference-type="ref" reference="equ:4.10"}) $\tilde{X}=vec((X^{k+1})^{\top})$, we can reverse the vectorization operator to obtain $$\label{equ:4.11} X^{k+1}=\begin{pmatrix} \tilde{X}_1 & \tilde{X}_2 & \cdots & \tilde{X}_{n-1} & \tilde{X}_n\\ \tilde{X}_{n+1} & \tilde{X}_{n+2} & \cdots & \tilde{X}_{2n-1} & \tilde{X}_{2n}\\ \vdots & \vdots & \ddots & \vdots & \vdots \\ \tilde{X}_{(m-2)n+1} & \tilde{X}_{(m-2)n+2} & \cdots & \tilde{X}_{(m-1)n-1} & \tilde{X}_{(m-1)n}\\ \tilde{X}_{(m-1)n+1} & \tilde{X}_{(m-1)n+2} & \cdots & \tilde{X}_{mn-1} & \tilde{X}_{mn} \end{pmatrix},$$ which is the unique solution of the matrix equation ([\[equ:4.6\]](#equ:4.6){reference-type="ref" reference="equ:4.6"}). $\square$* ## Update L {#sub:4.3} Fix $Y$ and $X$. $L$ can be updated by $$\label{equ:4.12} L^{k+1}=L^{k}+\rho (X^{k+1}-Y^{k+1}).$$ ## Complexity analysis {#sub:4.4} Call our algorithm for solving the low-rank matrix approximation problem ([\[equ:1.1\]](#equ:1.1){reference-type="ref" reference="equ:1.1"}) the LRMA-ADMM, and it is summarized in Algorithm [\[alg:1\]](#alg:1){reference-type="ref" reference="alg:1"}. We will describe the parameter initialization in Section [7.1](#sub:7.1){reference-type="ref" reference="sub:7.1"}. initialization:$X^{0},L^{0},\rho$, and let $k=0$; $\Hat{X}=Y^{k}$. For the general linear operator $\mathcal{A}$, $A_i(1\le i\le d)$ are full-rank matrices without any special structure. Therefore, when updating $X$, we need to solve the system of linear equations ([\[equ:4.10\]](#equ:4.10){reference-type="ref" reference="equ:4.10"}) with the order of coefficient matrix $mn$. Note that $2\tilde{A}\cdot \tilde{A}^{\top}+\rho I_{mn}$ does not change in the iteration; therefore, the inverse $(2\tilde{A}\cdot \tilde{A}^{\top}+\rho I_{mn})^{-1}$ can be calculated only once outside the iteration. Though $(2\tilde{A}\cdot \tilde{A}^{\top}+\rho I_{mn})^{-1}$ just need to be calculated once, it will still cost a lot of time with $mn$ increasing. An obvious way to accelerate the calculation of $(2\tilde{A}\cdot \tilde{A}^{\top}+\rho I_{mn})^{-1}$ is using the SMW-formula [@SMW1; @SMW2], converting the inverse of an $(mn)\times (mn)$ matrix to the inverse of a $d\times d$ matrix: $$\label{equ:4.13} (2\tilde{A}\cdot \tilde{A}^{\top}+\rho I_{mn})^{-1}=\frac{I_{mn}}{\rho}-\frac{2\tilde{A}(I_d+\frac{2\tilde{A}^{\top} \tilde{A}}{\rho})^{-1}\tilde{A}^{\top}}{\rho^{2}}.$$ In each iteration, updating $Y$ by SVD of an $m\times n$ matrix ([\[equ:4.4\]](#equ:4.4){reference-type="ref" reference="equ:4.4"}) costs $\mathcal{O}(n^2\cdot \max(m,n))$ time. Since $(2\tilde{A}\cdot \tilde{A}^{\top}+\rho I_{mn})^{-1}$ is calculated only once outside the iteration, updating $X$ by solving ([\[equ:4.10\]](#equ:4.10){reference-type="ref" reference="equ:4.10"}) only involves the multiplication between an $(mn)\times (mn)$ matrix and an $mn$ vector, which will cost $\mathcal{O}(m^2 n^2)$ time. Therefore, when the linear operator $\mathcal{A}$ has no special structure, the time complexity of Algorithm [\[alg:1\]](#alg:1){reference-type="ref" reference="alg:1"} is $\mathcal{O}(m^2 n^2)$ each iteration. # A specific case: low-rank matrix completion {#sec:5} In this section, we consider a special but significant application of low-rank matrix approximation, which is known as the low-rank matrix completion problem. In this specific case, each matrix $A_k=(a_{ij}^k)_{m\times n}(1\le k\le d)$ in the linear operator $$\begin{aligned} \mathcal{A}:&\mathbb{R}^{m\times n}\to \mathbb{R}^d \\ &X\mapsto (\left \langle A_1,X \right \rangle,\left \langle A_2,X \right \rangle,\cdots,\left \langle A_d,X \right \rangle)^{\top}\end{aligned}$$ has exactly one non-zero entry, and $\mathcal{A}$ actually returns a subset of the target matrix. Denote the location matrix $P=(p_{ij})\in \mathbb{R}^{m\times n}$, where $$p_{ij}=\begin{cases} 1, & \text{ \textit{if} } a_{ij}^k\ne 0 \text{ \textit{for some} } 1\le k\le d \\ 0, & \text{ \textit{otherwise} } \end{cases}$$ Let $M=P\odot X$, it is not hard to see that there is a one-to-one correspondence between the $(i,j)$-elements of $M$ such that $P_{ij}=1$ and $b=\mathcal{A}(X)$, which means $M$ is the measurement matrix in this case. Therefore, $\mathcal{A}^{\ast}(b)=\sum_{i=1}^d b_i A_i=M$, and $2\mathcal{A}^{\ast}\mathcal{A}(X^{k+1})=2\sum_{i=1}^d \left \langle A_i,X^{k+1} \right \rangle A_i=2P\odot X^{k+1}$. The matrix equation ([\[equ:4.6\]](#equ:4.6){reference-type="ref" reference="equ:4.6"}) becomes $$(2P+\rho \mathbf{1})\odot X^{k+1}=2M+\rho Y^{k+1}-L^k,$$ whose solution can be directly given by hadamard division: $$\label{equ:5.1} X^{k+1}=(2M+\rho Y^{k+1}-L^k)\oslash (2P+\rho \mathbf{1}).$$ The algorithm for the specific low-rank matrix completion problem is summarized in the following. initialization:$X^{0},L^{0},\rho$, and let $k=0$; $\Hat{X}=Y^{k}$. Unlike the Algorithm [\[alg:1\]](#alg:1){reference-type="ref" reference="alg:1"}, updating $X$ in Algorithm [\[alg:2\]](#alg:2){reference-type="ref" reference="alg:2"} only involves a hadamard division, which costs $\mathcal{O}(mn)$ time. Therefore, the time complexity of Algorithm [\[alg:2\]](#alg:2){reference-type="ref" reference="alg:2"} is $\mathcal{O}(n^2\cdot \max(m,n))$ per iteration. # Convergence analysis {#sec:6} In this section, we will establish the global convergence of the nonconvex Algorithm [\[alg:1\]](#alg:1){reference-type="ref" reference="alg:1"} based on the KL property of functions in problem ([\[equ:3.2\]](#equ:3.2){reference-type="ref" reference="equ:3.2"}). ***Proposition* 5**. *$f(X)=\left \| \mathcal{A}(X)-b \right \|_2^2$ is a Gradient-$L$-Lipschitz continuous and KL function.* ***Proof* 2**. *Since $f(X)=\left \| \mathcal{A}(X)-b \right \|_2^2=\sum_{i=1}^d (\left \langle A_i,X \right \rangle -b_i)^2$ is a real polynomial, it is of course a KL function.* *$\nabla f(X)=2\mathcal{A}^{\ast}(\mathcal{A}(X)-b)=2\sum_{i=1}^d (\left \langle A_i,X \right \rangle -b_i)\cdot A_i$, therefore we have $$\label{equ:6.1} \begin{split} \left \| \nabla f(X)-\nabla f(Y) \right \|_F &= \left \| 2\sum_{i=1}^d \left \langle A_i,X-Y \right \rangle \cdot A_i \right \|_F \\ &\le 2\sum_{i=1}^d \left \| \left \langle A_i,X-Y \right \rangle \cdot A_i \right \|_F \\ &= 2\sum_{i=1}^d \left | \left \langle A_i,X-Y \right \rangle \right | \cdot \left \| A_i \right \|_F \\ &\le (2\sum_{i=1}^d \left \| A_i \right \|_F^2)\cdot \left \| X-Y \right \|_F, \end{split}$$ where the first inequality is due to the triangular inequality; the second equality is because of the norm being positively homogeneous; and the last inequality uses the Cauchy-Schwarz inequality.* *From the inequality ([\[equ:6.1\]](#equ:6.1){reference-type="ref" reference="equ:6.1"}), we know that $f(X)$ is a Gradient-$L$-Lipschitz continuous function with the Lipschitz constant $L=2\sum_{i=1}^d \left \| A_i \right \|_F^2$. $\square$* ***Proposition* 6** ([@rKL]). *The rank function $f(X)=rank(X)$ is proper and lower semicontinuous. And it is also a semi-algebraic, therefore a KL function.* Due to Proposition [*Proposition* 1](#pro:2.1){reference-type="ref" reference="pro:2.1"}, the next Corollary [*Corollary* 1](#cor:6.1){reference-type="ref" reference="cor:6.1"} holds immediately. ***Corollary* 1**. *$f(X)=rank(X)$ is a closed function.* ***Corollary* 2**. *The indicator function $$\mathcal{I}_{\mathcal{M}}(X)=\begin{cases} 0, & \text{ \textit{if} } X\in \mathcal{M} \\ +\infty, & \text{ \textit{if} } X\notin \mathcal{M} \end{cases}$$ to the set $\mathcal{M}=\left \{ X\in \mathbb{R}^{m\times n}:rank(X)\le r \right \}$ is proper and lower semicontinuous, and it is a KL function.* ***Proof* 3**. *From the definition of the proper function ([*Definition* 3](#def:2.3){reference-type="ref" reference="def:2.3"}), it is easy to see that $\mathcal{I}_{\mathcal{M}}(X)$ is proper.* *Next we prove that $\mathcal{I}_{\mathcal{M}}(X)$ is closed, therefore it is lower semicontinuous.* *According to the definition of the closed function ([*Definition* 3](#def:2.3){reference-type="ref" reference="def:2.3"}), we only need to prove that for any sequence $(X_{k},t_{k})\in epi\mathcal{I}_{\mathcal{M}}$, if $(X_{k},t_{k})\to (X,t)$, then $t\ge \mathcal{I}_{\mathcal{M}}(X)$.* *Since $t_{k}\to t$, there are at most two cases when $k$ is large enough:* *If $rank(X_k)>r$, then $t_{k}\ge \mathcal{I}_{\mathcal{M}}(X_{k})=+\infty$. Therefore, $t=\lim_{k \to +\infty} t_k = +\infty \ge \mathcal{I}_{\mathcal{M}}(A)$.* *If $rank(X_k)\le r$, then $t_{k}\ge \mathcal{I}_{\mathcal{M}}(X_{k})=0$. Hence $t=\lim_{k \to +\infty} t_k \ge 0$. Corollary [*Corollary* 1](#cor:6.1){reference-type="ref" reference="cor:6.1"} tells that the rank function is closed. Therefore,* *$\left. \begin{aligned} &(X_{k},r)\in epirank(\cdot) \\ &(X_{k},r)\to (X,r) \end{aligned} \right\} \Longrightarrow (X,r)\in epirank(\cdot)$,* *which means $r\ge rank(X)$. Hence $t\ge 0= \mathcal{I}_{\mathcal{M}}(X)$.* *Above all, $t\ge \mathcal{I}_{\mathcal{M}}(X)$, which tells that $\mathcal{I}_{\mathcal{M}}(X)$ is closed.* *Then we illustrate that $\mathcal{M}$ is semi-algebraic.* *Since $rank(\cdot)\in \mathbb{N}$, we can express $\mathcal{M}$ as $$\mathcal{M}=\left \{ X\in \mathbb{R}^{m\times n}:rank(X)\le r \right \}=\bigcup_{i=0}^{r}\left \{ X\in \mathbb{R}^{m\times n}:rank(X)=i \right \}.$$* *Proposition [*Proposition* 6](#pro:6.2){reference-type="ref" reference="pro:6.2"} tells that $rank(\cdot)$ is semi-algebraic. Therefore, through the definition of the semi-algebra ([*Definition* 4](#def:2.4){reference-type="ref" reference="def:2.4"}), $graphrank(\cdot)=\left \{ (X,i):rank(X)=i \right \}$ is a semi-algebraic set.* *Denote the projection operator $$\begin{aligned} \pi:&\mathbb{R}^{m\times n}\times \mathbb{R}\to \mathbb{R}^{m\times n} \\ &(X,i)\mapsto X, \end{aligned}$$ Then $\mathcal{M}=\bigcup_{i=0}^{r}\pi(graphrank(\cdot))$ is also semi-algebraic due to its stability under the finite union and projection.* *At last, since $\mathcal{I}_{\mathcal{M}}(X)$ is the indicator function to the semi-algebra $\mathcal{M}$, it is also a semi-algebra. From the Proposition [*Proposition* 3](#pro:2.3){reference-type="ref" reference="pro:2.3"}, $\mathcal{I}_{\mathcal{M}}(X)$ is a KL function. $\square$* In the following part of the section, we refer to [@converge] to establish the global convergence of the Algorithm [\[alg:1\]](#alg:1){reference-type="ref" reference="alg:1"}. Consider the special two-block nonconvex optimization problem with a linear equality constraint $$\label{equ:6.2} \begin{aligned} \min_{x,y} \enspace &f(x)+g(y)\\ s.t.\enspace &Ax+y=b \end{aligned}$$ with the following four assumptions. ***Assumption* 1** ([@converge]). *There are four assumptions for problem ([\[equ:6.2\]](#equ:6.2){reference-type="ref" reference="equ:6.2"}):* (i) *$f$ is Gradient-$L$-Lipschitz continuous and $g$ is proper lower semicontinuous,* (ii) *The subproblems in ([\[equ:4.2\]](#equ:4.2){reference-type="ref" reference="equ:4.2"}) has solutions,* (iii) *The penalty parameter $\rho>2L$,* (iv) *$A^{T}A\succeq \mu I_n$ for some $\mu>0$.* Let $f(X)=\left \| \mathcal{A}(X)-b \right \|_2^2$, $g(Y)=\mathcal{I}_{\mathcal{M}}(Y)$, $A=-I$, $b=0$ in problem ([\[equ:6.2\]](#equ:6.2){reference-type="ref" reference="equ:6.2"}), we obtain the problem ([\[equ:3.2\]](#equ:3.2){reference-type="ref" reference="equ:3.2"}). Based on the Proposition [*Proposition* 5](#pro:6.1){reference-type="ref" reference="pro:6.1"}, Corollary [*Corollary* 2](#cor:6.2){reference-type="ref" reference="cor:6.2"} and the analysis in Section [4](#sec:4){reference-type="ref" reference="sec:4"}, it is apparent that problem ([\[equ:3.2\]](#equ:3.2){reference-type="ref" reference="equ:3.2"}) satisfies the Assumption [*Assumption* 1](#ass:6.1){reference-type="ref" reference="ass:6.1"}. The main convergence result about problem ([\[equ:6.2\]](#equ:6.2){reference-type="ref" reference="equ:6.2"}) in [@converge] is shown below. ***Lemma* 1** ([@converge]). *Applying the classic ADMM to problem ([\[equ:6.2\]](#equ:6.2){reference-type="ref" reference="equ:6.2"}), we obtain a sequence $\left \{w^{k}=(x^k,y^k,\lambda^k)\right \}$. If $$\label{equ:6.3} \bar{f}:=\inf_{x}\left \{ f(x)-\frac{1}{2L}\left \| \nabla f(x) \right \|^{2} \right \}>-\infty,$$ $\liminf_{\left \|x \right \|\to +\infty}f(x)=+\infty$ and $\inf_{y}g(y)>-\infty$, then $\left \{w^{k}\right \}$ is bounded. Besides, if $f$ and $g$ are also KL functions, then $\left \{ w^{k} \right \}$ converges to the KKT point of ([\[equ:6.2\]](#equ:6.2){reference-type="ref" reference="equ:6.2"}).* Based on the above Lemma ([*Lemma* 1](#lem:6.1){reference-type="ref" reference="lem:6.1"}), we can conclude that the Algorithm [\[alg:1\]](#alg:1){reference-type="ref" reference="alg:1"} is globally convergent. ***Theorem* 1**. *Algorithm [\[alg:1\]](#alg:1){reference-type="ref" reference="alg:1"} is globally convergent, and the sequence $\left \{ (Y^k,X^k,L^k) \right \}$ generated by it globally converges to the KKT point of the problem ([\[equ:3.2\]](#equ:3.2){reference-type="ref" reference="equ:3.2"}). Moreover, suppose that the low-rank matrix approximation problem ([\[equ:1.1\]](#equ:1.1){reference-type="ref" reference="equ:1.1"}) has the unique optimal solution, then the KKT point which $\left \{ (Y^k,X^k,L^k) \right \}$ converges to is actually the optimal solution.* ***Proof* 4**. *We start by proving the first half of the theorem.* *For $f(X)=\left \| \mathcal{A}(X)-b \right \|_2^2$, we know $\nabla f(X)=2\mathcal{A}^{\ast}(\mathcal{A}(X)-b)$ and $L=2\sum_{i=1}^d \left \| A_i \right \|_F^2$ from Proposition [*Proposition* 5](#pro:6.1){reference-type="ref" reference="pro:6.1"}. Therefore, for any $X\in \mathbb{R}^{m\times n}$, $$\label{equ:6.4} \begin{split} \left \| \mathcal{A}(X)-b \right \|_2^2-\frac{1}{2L}\left \| 2\mathcal{A}^{\ast}(\mathcal{A}(X)-b) \right \|_F^2 &= \sum_{i=1}^d (\left \langle A_i,X \right \rangle-b_i)^2 - \frac{1}{2L}\left \| 2\sum_{i=1}^d (\left \langle A_i,X \right \rangle-b_i)\cdot A_i \right \|_F^2 \\ &\ge \sum_{i=1}^d (\left \langle A_i,X \right \rangle-b_i)^2-\frac{1}{2L}\cdot 4 \sum_{i=1}^d (\left \langle A_i,X \right \rangle-b_i)^2\cdot \left \| A_i \right \|_F^2\\ &\ge \sum_{i=1}^d (\left \langle A_i,X \right \rangle-b_i)^2-\frac{2}{L}(\sum_{i=1}^d (\left \langle A_i,X \right \rangle-b_i)^2)(\sum_{i=1}^d \left \| A_i \right \|_F^2) \\ &= 0, \end{split}$$ where the first inequality is due to the triangular inequality; and the second inequality is because $\sum_{i=1}^n a_i^2 b_i^2 \le (\sum_{i=1}^n a_i^2) (\sum_{i=1}^n b_i^2)$ for any $a_i,b_i\in \mathbb{R}$.* *Hence ([\[equ:6.3\]](#equ:6.3){reference-type="ref" reference="equ:6.3"}) in Lemma [*Lemma* 1](#lem:6.1){reference-type="ref" reference="lem:6.1"} is satisfied.* *It is easy to see that $\liminf_{\left \|X \right \|\to +\infty}f(X)=\lim_{\left \|X \right \|\to +\infty}\sum_{i=1}^d (\left \langle A_i,X \right \rangle-b_i)^2 =+\infty$ and $\inf_{Y}g(Y)=\inf_{Y}\mathcal{I}_{\mathcal{M}}(Y)=0>-\infty$. Besides, according to Proposition [*Proposition* 5](#pro:6.1){reference-type="ref" reference="pro:6.1"} and Corollary [*Corollary* 2](#cor:6.2){reference-type="ref" reference="cor:6.2"} we know $f$ and $g$ are both KL functions. Therefore, by Lemma [*Lemma* 1](#lem:6.1){reference-type="ref" reference="lem:6.1"}, the generated sequence $\left \{ (Y^k,X^k,L^k) \right \}$ is globally convergent to the KKT point of the problem ([\[equ:3.2\]](#equ:3.2){reference-type="ref" reference="equ:3.2"}).* *Next is the second half of the theorem. Assume the unique optimal solution of problem ([\[equ:1.1\]](#equ:1.1){reference-type="ref" reference="equ:1.1"}) being $X^{\ast}$. Then it must satisfy $\mathcal{A}(X^{\ast})=b$, $rank(X^{\ast})\le r$. From the analysis in Section [3](#sec:3){reference-type="ref" reference="sec:3"}, problem ([\[equ:3.2\]](#equ:3.2){reference-type="ref" reference="equ:3.2"}) is equivalent to problem ([\[equ:1.1\]](#equ:1.1){reference-type="ref" reference="equ:1.1"}). Therefore, $X^{\ast}$ is also one of the optimal variables of problem ([\[equ:3.2\]](#equ:3.2){reference-type="ref" reference="equ:3.2"}), which is also unique according to the following analysis.* *Denote $(Y^k,X^k,L^k)\to (\bar{Y},\bar{X},\bar{L})$, from the above proof we have $(\bar{Y},\bar{X},\bar{L})$ is the KKT point of problem ([\[equ:3.2\]](#equ:3.2){reference-type="ref" reference="equ:3.2"}). The Lagrangian function of problem ([\[equ:3.2\]](#equ:3.2){reference-type="ref" reference="equ:3.2"}) is $$\label{equ:6.5} \mathcal{L}(Y,X,L)=\left \| \mathcal{A}(X)-b \right \|_2^2+\mathcal{I}_{\mathcal{M}}(Y)+\left \langle L,X-Y \right \rangle,$$ and it satisfies the KKT conditions at $(\bar{Y},\bar{X},\bar{L})$, which are* *0 = \_X (\|Y,\|X,\|L)=2\^((\|X)-b)+\|L, [\[equ:6.6a\]]{#equ:6.6a label="equ:6.6a"}\ 0 \_Y(\|Y,\|X,\|L)=\_Y\_(\|Y)-\|L, [\[equ:6.6b\]]{#equ:6.6b label="equ:6.6b"}\ 0 =\|X-\|Y. [\[equ:6.6c\]]{#equ:6.6c label="equ:6.6c"}* *From the updating rule of $Y$ in ([\[equ:4.3\]](#equ:4.3){reference-type="ref" reference="equ:4.3"}), we know that $\mathcal{I}_{\mathcal{M}}(Y^k)=0$ always holds in the iterations. Since $\mathcal{I}_{\mathcal{M}}(Y)$ is proper lower semicontinuous by Proposition [*Proposition* 6](#pro:6.2){reference-type="ref" reference="pro:6.2"}, we obtain $0\le \mathcal{I}_{\mathcal{M}}(\bar{Y})\le \liminf_{k\to +\infty}\mathcal{I}_{\mathcal{M}}(Y^{k})=0$. Hence $\mathcal{I}_{\mathcal{M}}(Y)\equiv 0$ when limiting the space on the generated points and their accumulations by Algorithm [\[alg:1\]](#alg:1){reference-type="ref" reference="alg:1"}. Therefore, $\partial_{Y}\mathcal{I}_{\mathcal{M}}(\bar{Y})=0$. From ([\[equ:6.6b\]](#equ:6.6b){reference-type="ref" reference="equ:6.6b"}) we obtain $\bar{L}=0$.* *We can conclude that $\mathcal{A}^{\ast}(\mathcal{A}(\bar{X})-b)=0$ from ([\[equ:6.6a\]](#equ:6.6a){reference-type="ref" reference="equ:6.6a"}) since $\bar{L}=0$. Because $\mathcal{A}^{\ast}(w)=\sum_{i=1}^d w_i A_i$ for $w\in \mathbb{R}^d$, if $\mathcal{A}^{\ast}(w)=0$, there must be $w=0$. Otherwise, there exists some $A_k$ which can be expressed by other $A_i,i\ne k$. Then the measurement vector $b_k=\left \langle A_k,X \right \rangle$ can also be expressed by other measurements $b_i,i\ne k$, which suggests that this measurement is redundant. Hence from $\mathcal{A}^{\ast}(\mathcal{A}(\bar{X})-b)=0$ we have $\mathcal{A}(\bar{X})=b$. From ([\[equ:6.6c\]](#equ:6.6c){reference-type="ref" reference="equ:6.6c"}) we know $\bar{X}=\bar{Y}$, which means that $rank(\bar{X})=rank(\bar{Y})\le r$ since we have proved that $\mathcal{I}_{\mathcal{M}}(\bar{Y})=0$.* *Therefore, $\bar{X}$ also satisfies $\mathcal{A}(\bar{X})=b$, $rank(\bar{X})\le r$, which means that $\bar{X}$ is actually the unique optimal solution $X^{\ast}$. Besides, due to the uniqueness of $X^{\ast}$, we can conclude that $(\bar{Y},\bar{X},\bar{L})=(X^{\ast},X^{\ast},0)$ is also the unique optimal solution of problem ([\[equ:3.2\]](#equ:3.2){reference-type="ref" reference="equ:3.2"}). $\square$* # Numerical simulations {#sec:7} In this section, we consider the specific low-rank matrix completion problem, and design numerical experiments to verify the proposed specialized algorithm [\[alg:2\]](#alg:2){reference-type="ref" reference="alg:2"} by comparing it with the other algorithms. ## Experiment setup {#sub:7.1} The experiments were performed on a PC with an Intel Core i7-13700H 2.4 GHz CPU and 16 GB RAM. The results in Sections [7.2](#sub:7.2){reference-type="ref" reference="sub:7.2"}-[7.4](#sub:7.4){reference-type="ref" reference="sub:7.4"} had been averaged over 10 trials. The original matrix $X\in \mathbb{R}^{m\times n}$ was generated by $X=BC^\top$ where $B\in \mathbb{R}^{m\times r}$ and $C\in \mathbb{R}^{n\times r}$ both have the $i.i.d.$ Gaussian random entries. And the measurement $b$ is $d$ randomly chosen from $X$, which might be added with an extra Gaussian white noise. The recovered matrix was denoted as $\hat{X}$. And we used $SNR_r=20log_{10}(\frac{\left \| X \right \|_F}{\left \| X-\hat{X} \right \|_F})$ and $SNR_m=20log_{10}(\frac{\left \| b \right \|_2}{\left \| e \right \|_2})$ to measure the reconstruction error and measurement noise level, respectively. The computational cost is measured by the number of iterations. Because the projection in ([\[equ:4.3\]](#equ:4.3){reference-type="ref" reference="equ:4.3"}) is a hard thresholding operator, we check the NIHT [@NIHT] for the LRMC. The ADMM [@ADMM1] is the same ADMM scheme algorithm for the nuclear norm regularized least squares problem. We introduce it in our experiments as the most direct comparison and call it ADMM for distinguishing with LRMC-ADMM. The parameter settings are shown in Table [1](#tab:1){reference-type="ref" reference="tab:1"}. For Algorithm [\[alg:2\]](#alg:2){reference-type="ref" reference="alg:2"}, $X^{0}$ and $L^{0}$ are initialized as an $i.i.d.$ Gaussian random matrix and a null matrix, respectively. The maximum iterations $MaxIter$ are all 500; and the stop criterion is $\frac{\left \| \mathcal{A}(\hat{X})-b \right \|_2}{\left \| b \right \|_2}<Tol$ with $Tol=10^{-4}$. Algorithm Parameters --------------- ------------------------------------------------------------------------- NIHT [@NIHT] $r$, $Tol=10^{-4}$, $MaxIter=500$ ADMM [@ADMM1] step size $\mu=10^{-4}$, ratio $\rho=1.1$, $Tol=10^{-4}$, $MaxIter=500$ LRMC-ADMM $r$, $\rho=1$, $Tol=10^{-4}$, $MaxIter=500$ : Parameter settings ## Performance with different sampling rate {#sub:7.2} In the first experiment, we studied how does the sampling rate $d/mn$ affect the reconstruction in both noiseless and noisy cases. Set $m=n=500$, $r=2$, and let the sampling rate $d/mn$ range from 0.05 to 0.30 with step size 0.05. For the noisy case, set $SNR_m=20$. The results are shown in Table [\[tab:2\]](#tab:2){reference-type="ref" reference="tab:2"} and [\[tab:3\]](#tab:3){reference-type="ref" reference="tab:3"}. c\|ccc\|ccc \*$d/mn$ & &\ & NIHT & ADMM & LRMC-ADMM & NIHT & ADMM & LRMC-ADMM\ & 71 & 33 & 71 & 67 & 138 & 403\ 0.10 & 76 & 77 & 75 & 26 & 124 & 102\ 0.15 & 80 & 79 & 77 & 19 & 100 & 48\ 0.20 & 80 & 80 & 79 & 15 & 84 & 32\ 0.25 & 81 & 81 & 80 & 13 & 74 & 30\ 0.30 & 81 & 82 & 80 & 12 & 67 & 28\ c\|ccc\|ccc \*$d/mn$ & &\ & NIHT & ADMM & LRMC-ADMM & NIHT & ADMM & LRMC-ADMM\ & 11 & 19 & 27 & 500 & 112 & 500\ 0.10 & 30 & 22 & 31 & 500 & 106 & 500\ 0.15 & 32 & 23 & 32 & 500 & 102 & 500\ 0.20 & 33 & 23 & 33 & 500 & 100 & 500\ 0.25 & 34 & 23 & 34 & 500 & 99 & 500\ 0.30 & 35 & 23 & 35 & 500 & 98 & 500\ It is easy to see that both NIHT and LRMC-ADMM perform good for the noiseless case, while ADMM cannot recover the original matrix accurately when the sampling rate is sufficiently low. As for the iteration number, NIHT is the most efficient, LRMC-ADMM is more efficient than ADMM. But when the sampling rate is 0.05, the iteration number of LRMC-ADMM is large, which means that it requires more iterations to obtain the accurate solution. However, the measurement is always noisy in applications. For the noisy case, both NIHT and LRMC-ADMM need to iterate to the $MaxIter$. ADMM stopped early for obtaining a suboptimal solution, which means that the improvement in reconstruction accuracy will be extremely slight even if ADMM iterates to $MaxIter$. The reconstruction accuracy of NIHT and LRMC-ADMM is almost the same when the sampling rate is not so low ($i.e.$, $d/mn \ge 0.10$). But when the sampling rate is really low, LRMC-ADMM demonstrated a clear advantage in recovery accuracy than NIHT, which is the key of matrix completion. ## Performance with different target rank {#sub:7.3} The second experiments studied how the target rank $r$ influenced the recovery. Set $m=n=500$, $d/mn=0.20$ and tested the target rank $r=2,5,10,15,20,30$. c\|ccc\|ccc \*$r$ & &\ & NIHT & ADMM & LRMC-ADMM & NIHT & ADMM & LRMC-ADMM\ & 80 & 80 & 80 & 14 & 84 & 32\ 5 & 79 & 78 & 77 & 20 & 84 & 38\ 10 & 75 & 75 & 74 & 29 & 96 & 61\ 15 & 73 & 72 & 73 & 46 & 109 & 89\ 20 & 70 & 25 & 71 & 78 & 98 & 139\ 30 & 66 & 8 & 66 & 197 & 85 & 348\ c\|ccc\|ccc \*$r$ & &\ & NIHT & ADMM & LRMC-ADMM & NIHT & ADMM & LRMC-ADMM\ & 34 & 23 & 34 & 500 & 100 & 500\ 5 & 30 & 21 & 30 & 500 & 96 & 500\ 10 & 26 & 19 & 26 & 500 & 93 & 500\ 15 & 24 & 16 & 24 & 500 & 91 & 500\ 20 & 22 & 13 & 22 & 500 & 89 & 500\ 30 & 18 & 7 & 18 & 500 & 83 & 500\ As shown in Table [\[tab:4\]](#tab:4){reference-type="ref" reference="tab:4"} and [\[tab:5\]](#tab:5){reference-type="ref" reference="tab:5"}, The recovery accuracy of NIHT and LRMC-ADMM is almost the same for the tested $r$ in both noiseless and noisy cases. In noiseless case, when the target rank is not so large ($i.e.$, $r\le 15$), all the algorithms succeed to recover the original matrix, and NIHT is the most efficient. However, when $r\ge 20$, ADMM fails to recover the matrix while NIHT and LRMC-ADMM still perform good. As for the noisy case, the situation of the iteration number is consistent with the results in Table [\[tab:3\]](#tab:3){reference-type="ref" reference="tab:3"}. ADMM converges to the suboptimal solution and performs worse than NIHT and LRMC-ADMM. The experiment implies that LRMC-ADMM performs good even if the original matrix is not really low-rank. ## Performance with different matrix size {#sub:7.4} In the last experiment, we tested the performance of the algorithms for different matrix size. [@lrcc] proves that a sufficient condition to complete an $n\times n$ rank-$r$ matrix is $d=\mathcal{O} (n^{1.2}rlog_{10}n)$. Therefore, we set $r=2$, and tested $m=n=500,1000,1500,2000,2500,3000$ with $d=10 \left \lceil n^{1.2}rlog_{10}n \right \rceil$. c\|ccc\|ccc \*$n$ & &\ & NIHT & ADMM & LRMC-ADMM & NIHT & ADMM & LRMC-ADMM\ & 80 & 81 & 82 & 10 & 58 & 26\ 1000 & 83 & 81 & 81 & 10 & 71 & 30\ 1500 & 84 & 81 & 80 & 9 & 81 & 32\ 2000 & 83 & 80 & 80 & 9 & 89 & 34\ 2500 & 81 & 80 & 81 & 9 & 95 & 36\ 3000 & 81 & 80 & 81 & 9 & 100 & 37\ c\|ccc\|ccc \*$n$ & &\ & NIHT & ADMM & LRMC-ADMM & NIHT & ADMM & LRMC-ADMM\ & 36 & 22 & 36 & 500 & 97 & 500\ 1000 & 37 & 23 & 37 & 500 & 95 & 500\ 1500 & 38 & 24 & 38 & 500 & 94 & 500\ 2000 & 38 & 24 & 38 & 500 & 94 & 500\ 2500 & 39 & 25 & 39 & 500 & 94 & 500\ 3000 & 39 & 25 & 39 & 500 & 94 & 500\ Table [\[tab:6\]](#tab:6){reference-type="ref" reference="tab:6"} shows that all the algorithms perform good in the noiseless case, and NIHT converges really fast in this low-rank cases. While LRMC-ADMM needs less iterations to converge comparing with the ADMM. Table [\[tab:7\]](#tab:7){reference-type="ref" reference="tab:7"} tells that NIHT and LRMC-ADMM can give solutions with the same accuracy for noisy case. But both of them need to converge to the $MaxIter$, while ADMM converges early to obtain a suboptimal solution. This experiment shows that LRMC-ADMM is stable with the matrix size changing. # Conclusions {#sec:8} In this paper, we propose an ADMM algorithm for solving the general rank constrained LRMA in both noiseless and noisy cases. And we prove the global convergence of the proposed algorithm in theory. Specialized for LRMC, numerical experiments show that Algorithm [\[alg:2\]](#alg:2){reference-type="ref" reference="alg:2"} has almost the same reconstruction ablity comparing with NIHT [@NIHT] in the noiseless case, both of which are better than the ADMM [@ADMM1]. Though NIHT is the most efficient among the tested algorithms, Algorithm [\[alg:2\]](#alg:2){reference-type="ref" reference="alg:2"} requires less iterations to converge comparing with the same ADMM scheme algorithm. And for the noisy case, Algorithm [\[alg:2\]](#alg:2){reference-type="ref" reference="alg:2"} shows a clear advantage in recovering when the sampling rate is really low, which is the key in matrix completion. There are still some work to be studied in the future. First, the measurement operator is assumed to be linear, for the nonlinear operator, how to obtain the ADMM algorithm of the rank constrained LRMA, does it still work. Second, when updating $X$, we require to solve a matrix equation ([\[equ:4.6\]](#equ:4.6){reference-type="ref" reference="equ:4.6"}), how can we use the special structure of the operator $\mathcal{A}$ to avoid solve ([\[equ:4.10\]](#equ:4.10){reference-type="ref" reference="equ:4.10"}) directly. Third, when updating $Y$, it is unavoidable to calculate a SVD, which might be the most complex step for some special operators such as matrix completion. We can use some fast methods [@PROPACK] to compute the SVD based on these special structures. And when the original matrix is large, random algorithms can be applied to compute an approximate SVD [@random]. Last, though the algorithm is not sensitive to the penalty parameter $\rho$, we can still accelerate it by applying the self-adaptive penalty technique [@selfad].

arxiv_math

arxiv_math

--- abstract: | We show that the derived category of perverse Nori motives and mixed Hodge modules are the derived categories of their constructible hearts. This enables us to construct $\infty$-categorical lifts of the six operations and therefore to obtain realisation functors from the category of Voevodsky étale motives to the derived categories of perverse Nori motives and mixed Hodge modules that commute with the operations. We give a proof that the realisation induces an equivalence of categories between Artin motives in the category of étale motives and Artin motives in the derived category of Nori motives. We also prove that if a motivic $t$-structure exists then Voevodsky étale motives and the derived category of perverse Nori motives are equivalent. Finally we give a presentation of the indization of the derived category of perverse Nori motives as a category of modules in Voevodsky étale motives that gives a continuity result for perverse Nori motives. author: - Swann Tubach bibliography: - BibSCNet.bib title: - - On the Nori and Hodge realisations of Voevodsky étale motives --- # Introduction {#introduction .unnumbered} Let $k$ be a field of characteristic zero. Following the vision of Beilinson, Deligne, Grothendieck and others there should exist an abelian category of mixed motives $\mathcal{MM}(k)$, target of the universal cohomology theory $$M^*:\mathrm{Var}_k^\mathrm{op}\to\mathcal{MM}(k)$$ on $k$-varieties in the sense that any other reasonable cohomology theory on $k$-varieties would factor uniquely through $M^*$. This is out of reach of the current technology. However two constructions have almost all the required properties to provide a category of mixed motives. The first one is Voevodsky, Levine, Hanamura and others' triangulated category of geometric motives $\mathrm{DM}_\mathrm{gm}(k,\mathbb{Q})$ ([@MR1764197]), a candidate for the bounded derived category of mixed motives. The second is the abelian category of Nori motives $\mathcal{M}_\mathrm{Nori}(k)$ ([@fakhruddinNotesNoriLectures2000]), a candidate for $\mathcal{MM}(k)$. These categories have realisation functors factoring the known cohomology theories of varieties. Indeed, Huber ([@MR1775312]), Levine ([@MR2181828]) and Nori have constructed a Hodge realisation functor $$\mathrm{DM}_\mathrm{gm}( k,\mathbb{Q})\to \mathrm{D}^b(\mathrm{MHS}^p_\mathbb{Q})$$ to the derived category of polarisable rational mixed Hodge structures that realises the Hodge cohomology of varieties and factors through the derived category of the $\mathbb{Q}$-linear abelian category of motives constructed by Nori thanks to the work of Harrer ([@harrerComparisonCategoriesMotives2016] and Choudhury--Gallauer ([@MR3649230])). There exist relative versions of the categories of Voevodsky and Nori. Indeed the triangulated category of constructible rational étale motives $\mathcal{DM}_c(-)$ constructed by Ayoub ([@MR3205601]) and Cisinski and Déglise ([@MR3477640]) gives a triangulated category of étale motivic sheaves generalising Voevodsky's category, and the abelian category of perverse Nori motives ${\mathcal{M}_{\mathrm{perv}}}(X)$ constructed by Ivorra and S. Morel ([@ivorraFourOperationsPerverse2022]) gives a category of motivic sheaves generalising the abelian category of Nori. Both settings afford a 6 functors formalism, the tensor product on perverse Nori motives being constructed by Terenzi ([@TerenziThesis]). If $X$ is a quasi-projective smooth $k$-variety, a realisation functor $$\mathrm{DM}_c(X)\to \mathrm{D}^b(\mathrm{MHM}^p(X))$$ to Saito's derived category of polarisable mixed Hodge modules ([@MR1047415]) has been constructed by Ivorra ([@MR3518311]). It also factors through the derived category of perverse Nori motives and computes the relative Hodge homology of $X$-schemes. Unfortunately, Ivorra's functor has no obvious compatibility with the 6 operations that exist on $\mathrm{DM}_c(-)$ and $\mathrm{D}^b(\mathrm{MHM}^p(-))$. The reason behind this difficulty is that the functor is built by constructing an explicit complex $K^\bullet\in\mathrm{Ch}^b(\mathrm{MHM}^p(X))$ that computes the relative homology sheaf $f!f^!\mathbb{Q}_X\in\mathrm{D}^b(\mathrm{MHM}^p(X))$ of a smooth affine $X$-scheme $f:Y\to X$, and that we do not know if one can construct the 6 operations explicitly on complexes only in terms of objects living in the abelian category $\mathrm{MHM}^p(X)$. On the other hand, we know since Robalo's thesis [@MR3281141] that the $\infty$-category $\mathcal{DM}_c(X)$ (which is a stable $\infty$-category lifting $\mathrm{DM}_c(X)$) affords an universal property that enables one to construct realisations functors to any stable $\infty$-category $\mathcal{C}$ that have reasonable properties. Indeed, he shows that $\mathcal{DM}(X)=\mathrm{Ind}\mathcal{DM}_c(X)$ is the target of the universal symmetric monoidal functor $\mathrm{Sm}/X\to\mathcal{DM}(X)$ to a stable $\mathbb{Q}$-linear presentably symmetric monoidal $\infty$-category, satisfying $\mathbb{A}^1$-invariance, étale hyperdescent and $\mathbb{P}^1$-stability. The work of Drew and Gallauer ([@MR4560376]) shows that this universal property works very well in families, so that the $\infty$-functor $$\mathcal{DM}:\mathrm{Sch}^\mathrm{op}_k\to \mathrm{CAlg}(\mathrm{St}\cap\mathrm{Pr}^L)$$ is the target of the universal natural transformation $\mathrm{Sm}/(-)\Rightarrow \mathcal{DM}$ of functors $\mathrm{Sch}^\mathrm{op}_X\to\mathrm{CAlg}(\mathrm{\mathrm{Cat}_\infty})$ sending $f:Y\to X$ in $\mathrm{Sch}_k$, smooth, to a right adjoint and such that $\mathcal{DM}$ is $\mathbb{Q}$-linear, satisfy non-effective étale descent and $\mathbb{A}^1$-invariance, together with $\mathbb{P}^1$-stability. Moreover Cisinski and Déglise (see also Ayoub's [@MR2423375 Scholie 1.4.2]) proved in [@MR3971240 Theorem 4.4.25] that the full subcategory of constructible objects $\mathcal{C}_{\mathrm{ct}}$ of any such functor $\mathcal{C}$ affords the six operations, and that any natural transformation $\mathcal{DM}\to\mathcal{C}$ between such functors would induce a natural transformation between the categories of constructible objects that commutes with the six operations. In other words, to construct a family of realisation functors $\mathrm{Hdg}^*:\mathcal{DM}_c\to\mathcal{D}^b(\mathrm{MHM}^p(-))$ compatible with the 6 operations, it suffices to lift the 6 operations on mixed Hodge modules to the $\infty$-categorical setting. This is what we do. Our main result is the following: **Theorem 1** (, , and ). *The 6 operations on $\mathrm{D}^b(\mathrm{MHM}^p(-))$ and $\mathrm{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))$ constructed by Saito, Ivorra, S. Morel and Terenzi admit $\infty$-categorical lifts and are defined over any finite type $k$-scheme. For every finite type $k$-scheme there exist essentially unique $\infty$-functors $$\mathrm{Nor}^*: \mathcal{DM}_c(X)\to\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X)),$$ $$R_H:\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))\to \mathcal{D}^b(\mathrm{MHM}^p(X)),$$ and $$\mathrm{Hdg}^*: \mathcal{DM}_c(X)\to\mathcal{D}^b(\mathrm{MHM}^p(X))$$ that commute with the $6$ operations and such that $R_H\circ\mathrm{Nor}^*\simeq \mathrm{Hdg}^*$. Moreover, $\mathrm{Nor}^*$ is compatible with the Betti and the $\ell$-adic cohomology, $R_H$ is $t$-exact and all functors are weight-exact.* How to give $\infty$-categorical lifts of triangulated functors? In general this is a hard question. One general solution would be to use the formalism developed by Liu and Zheng in [@liuGluingRestrictedNerves2015]. In our particular case we have a simpler method. Indeed, we use that it is very easy to construct $\infty$-categorical lifts of derived functors. Most of the $6$ operations are not $t$-exact (even on one side) for the perverse $t$-structure. However, they are $t$-exact (at least on one side) for the constructible $t$-structure, that is the $t$-structure whose heart behaves like the abelian category of constructible sheaves. In [@MR1940678], Nori proves that the triangulated category of cohomologically constructible sheaves $\mathrm{D}^b_c(X(\mathbb{C}),\mathbb{Q})$ on the $\mathbb{C}$-points of an algebraic variety $X$ is the derived category of its constructible heart. We show that one can adapt his argument to make things work for polarisable mixed Hodge modules and perverse Nori motives, and obtain the following result: **Theorem 2** ( and ). *Let $X$ be a quasi-projective $k$-variety. Denote by $\mathrm{MHM}_c(X)$ (*resp*. by ${\mathcal{M}_{\mathrm{ct}}}(X)$) the heart of the constructible $t$-structure on $\mathrm{D}^b(\mathrm{MHM}^p(X))$ (*resp*. on $\mathrm{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))$). The canonical $\infty$-functors $$\mathrm{real}_{\mathrm{MHM}_c(X)}:\mathcal{D}^b(\mathrm{MHM}_c(X))\to\mathcal{D}^b(\mathrm{MHM}^p(X))$$ and $$\mathrm{real}_{{\mathcal{M}_{\mathrm{ct}}}(X)}:\mathcal{D}^b({\mathcal{M}_{\mathrm{ct}}}(X))\to\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))$$ are equivalences of $\infty$-categories.* This theorem enables us to make the handy change of variables $\mathcal{D}^b({\mathcal{M}_{\mathrm{ct}}}(X))\simeq\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))$ and see, for example, pullback functors $f^*$ as derived functors of their restriction to the constructible heart, so that they are $\infty$-functors for a simple reason! This consideration was the starting idea of this paper. Each of the 6 operations can be obtained as a right derived functor on the constructible or the perverse heart. This is how we obtain the $\infty$-categorical lifts of the operations. Also, using Zariski and $h$-descent, we extend perverse Nori motives together with the 6 operations to all finite type $k$-schemes. In contrary to usual sheaves, categories of motivic sheaves are often continuous, that is if one writes a scheme $X$ as the limit of a projective system of schemes $X_i$ with affine transitions, it is often true that the functor $$\mathrm{colim}_i \mathcal{DM}(X_i)\to\mathcal{DM}(X)$$ is an equivalence (see [@MR3205601 Proposition 3.20] or [@MR3971240 Proposition 4.3.4]). This is very useful to spread out properties known over fields to open subsets of schemes. Ivorra and S. Morel have proven that the category ${\mathcal{M}_{\mathrm{perv}}}(-)$ satisfies some continuity ([@ivorraFourOperationsPerverse2022 Proposition 6.8]), but this only gives results for limits of diagrams of schemes for which the pullbacks by transitions are $t$-exact for the perverse $t$-structure, which restricts applications. Let $\mathcal{DN}(X)=\mathrm{Ind}\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))$ be the indization of the bounded derived $\infty$-category of ${\mathcal{M}_{\mathrm{perv}}}(X)$. This is a compactly generated stable presentably symmetric monoidal $\infty$-category whose compact objects consist of $\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))$. The functor $\mathrm{Nor}^*:\mathcal{DM}_c(X)\to\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))$ extends to a functor $$\mathrm{Nor}^*:\mathcal{DM}(X)\to\mathcal{DN}(X)$$ which preserves colimits, hence has a right adjoint $\mathrm{Nor}_*$. For each finite type $k$-scheme, denote by $\mathscr{N}_X:=\mathrm{Nor}_*\mathrm{Nor}^*\mathbb{Q}_X\in\mathcal{DM}(X)$. As $\mathrm{Nor}_*$ is naturally lax symmetric monoidal, we have that $\mathscr{N}_X\in\mathrm{CAlg}(\mathcal{DM}(X))$ is an $\mathbb{E}_\infty$-algebra of $\mathcal{DM}(X)$. Using the proof of the same result for the Betti realisation in [@ayoubAnabelianPresentationMotivic2022 Theorem 1.93], we prove **Proposition 3** (). *For each $X$ of finite type over some field of characteristic zero $k$, denote by $p_X:X\to\mathrm{Spec}\ \mathbb{Q}$ the unique morphism. Then $p_X^*\mathscr{N}_{\mathrm{Spec}\ \mathbb{Q}}\simeq\mathscr{N}_X$ and the natural functor $$\mathrm{Mod}_{\mathscr{N}_X}(\mathcal{DM}(X))\to\mathcal{DN}(X)$$ is an equivalence of categories that commutes with pullback functors.* Although we do not go in the details, the same arguments would prove that the $\infty$-category $\mathcal{DH}^\mathrm{geo}(X)\subset\mathrm{Ind}\mathcal{D}^b(\mathrm{MHM}^p(X))$ of geometric origin objects in the indization of the derived category of polarisable mixed Hodge modules is also the category of modules over some algebra $\mathscr{H}_X$ in $\mathcal{DM}(X)$. Note that in that case it should be false that $\mathscr{H}_X\simeq(\mathscr{H}_{\mathrm{Spec}\ \mathbb{Q}})_{\mid X}$. As the functor $(\mathcal{A},\mathcal{C})\mapsto\mathrm{Mod}_\mathcal{A}(\mathcal{C})$ that sends a symmetric monoidal $\infty$-category pointed at an algebra object $\mathcal{A}\in\mathrm{CAlg}(\mathcal{C})$ to the $\infty$-category of modules over $\mathcal{A}$ preserves colimits, we obtain the following corollary: **Corollary 4** (). *The $\infty$-functor $X\mapsto\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))$ has an unique extension $\mathcal{DN}_c(X)$ to quasi-compact and quasi-separated schemes of characteristic zero such that for all limit $X=\lim_i X_i$ of such schemes with affines transitions the natural functor $$\mathrm{colim}_i\mathcal{DN}_c(X_i)\to\mathcal{DN}_c(X)$$ is an equivalence of $\infty$-categories.* Now that a well behaved comparison functor $\mathrm{Nor}^*:\mathcal{DM}_c(X)\to\mathcal{DN}_c(X)$ is available, one wants to see to what extend we can compare both categories. We have two results in that direction. The first one is conditional to the existence of a motivic $t$-structure on $\mathrm{DM}_\mathrm{gm}(k)$, an out of reach conjecture that we know imply all standard conjectures in characteristic zero ([@MR2953406]). Moreover, by the work of Bondarko in [@MR3347995 Theorem 3.1.4], if such a $t$-structure exists, then for all finite type $k$-schemes, we have a perverse and a constructible $t$-structure on $\mathcal{DM}_c(X)$. **Theorem 5** (). *Assume that a motivic $t$-structure exists for all fields of characteristic $0$. Let $X$ be a finite type $k$ scheme for such a field $k$. Then the heart of the perverse $t$-structure of $\mathcal{DM}_c(X)$ is canonically equivalent to ${\mathcal{M}_{\mathrm{perv}}}(X)$ the category of perverse Nori motives. Moreover, the functor $\mathrm{Nor}^* : \mathcal{DM}_c(X)\to\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))$ is an equivalence of stable $\infty$-categories. This implies that $\mathcal{DM}_c(X)$ is both the derived category of its perverse and constructible hearts.* In particular over a field we would have $\mathrm{DM}_\mathrm{gm}(k,\mathbb{Q})\simeq\mathrm{D}^b(\mathcal{MM}(k))$, showing that the "even more optimistic" expectation of [@MR2115000 21.1.8] is in fact no more optimistic than expecting the standard conjectures to be proven. The second result we have is unconditional and is a comparison of the categories of Artin motives both in $\mathcal{DM}_c(X)$ and $\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))$: we prove that they are equivalent. Those are the full subcategories $\mathcal{DM}^0_c(X)$ and $\mathcal{DN}^0_c(X)$ generated by the $f_*\mathbb{Q}_Y$ for $f:Y\to X$ a finite morphism. Artin motives in Voevodsky's category $\mathrm{DM}_\mathrm{gm}(k)$ have been studied by Orgogozo ([@MR2102056]) who proved that the triangulated category of Artin motives $\mathcal{DM}^0_c(\mathrm{Spec}\ k)$ is the derived category $\mathcal{D}^b(\mathrm{Rep}^A_\mathbb{Q}(\mathrm{Gal}(\overline{k}/k)))$ of Artin representations of the Galois groups of $k$ (those are the representations that factor through a finite quotient.) Ayoub and Barbieri-Viale proved in [@MR3302623 Theorem 4.3] that over a subfield $k$ of $\mathbb{C}$ the abelian category of Artin motives in ${\mathcal{M}_{\mathrm{perv}}}(\mathrm{Spec}\ k)$ (generated by the cohomologies of étales $k$-schemes) and $(\mathcal{DM}_c^0(\mathrm{Spec}\ k))^\heartsuit\simeq \mathrm{Rep}^A_\mathbb{Q}(\mathrm{Gal}(\overline{k}/k))$ coincide naturally, and this result holds with integral coefficients. Over a general basis $S$, Pepin-Lehalleur constructed in [@MR3920833] and [@MR4033829] a $t$-structure on $\mathcal{DM}^0_c(S)$, compatible with the $\ell$-adic realisation. Our result is the following: **Theorem 6** (). *Let $X$ be a finite type $k$-scheme. The functor $$\mathrm{Nor}^*:\mathcal{DM}^0_c(X)\to\mathcal{DN}^0_c(X)$$ is an equivalence of categories. Moreover, the composition $\mathrm{Nor}^* :\mathcal{DM}_c^0(X)\to\mathcal{D}^b({\mathcal{M}_{\mathrm{ct}}}(X))$ is $t$-exact.* One part of the proof consists in proving that the inclusion of Artin motives in cohomological objects of $\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))$ admits a right adjoint. This is done by using ideas of Nair and Vaish in [@MR3293216] going back to [@MR2350050]: this right adjoint can be seen as a weight truncation functor because in the category of cohomological motives, Artin motives are the only objects such that on a stratification each of their cohomology sheaves are pure of weight $0$. One can also consider smooth Artin motives, which are the stable full subcategories spanned by the $f_*\mathbb{Q}_Y$ for $f:Y\to X$ finite and étale. In [@ruimyabelianCategoriesArtin2023] Ruimy constructs a $t$-structure on the category of smooth Artin motives in $\mathcal{DM}_c(S,\mathbb{Z})$ over a regular basis with integral coefficients. Ideas in the thesis of Johann Haas [@HaasThesis] enable us to prove that, when seen as a full subcategory of $\mathcal{DN}_c(X)$ when $X$ is normal, the $\infty$-category of smooth Artin motives is exactly the $\infty$-category of dualisable Artin motives, and this gives a $t$-structure of heart the $\mathbb{Q}$-linear Artin representations of $\pi^{\acute{e}t}_1(X)$, generalising the result of Ruimy (with rational coefficients) to normal schemes. Finally in we give exact sequences relating the Tannakian motivic group of a normal variety $X$ to the analogue for its base change to the algebraic closure, and to the Galois group of the base field. # Organisation of the paper. {#organisation-of-the-paper. .unnumbered} After some recollections on motivic constructions in , in our we review Nori's proof in [@MR1940678] that the category of cohomologically constructible complexes of sheaves on the $\mathbb{C}$-points of a quasi-projective variety $X$ is equivalent to the derived category of constructible sheaves and show that it gives a proof that the derived categories of the perverse and constructible hearts on the categories of polarisable mixed Hodge modules (or of perverse Nori motives) are equivalent. The only idea here was to replace local systems with dualisable objects and cohomology of the global sections with $\mathrm{Hom}_{\mathrm{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))}(\mathbb{Q}_X,-[q])$. The main point of the proof goes by showing effaceability of cohomology of the constant object on affine $n$-spaces, then deduce it for any affine variety thanks to Noether normalisation. We divided in two parts. In the first one we prove general results about the functoriality of the functor $\mathcal{D}^b(-)$ sending a Noetherian abelian category to its bounded derived $\infty$-category. When the abelian category $\mathcal{A}$ is $\mathbb{Q}$-linear and symmetric monoidal with an exact tensor product, we prove that $\mathcal{D}^b(\mathcal{A})$ is endowed with a symmetric monoidal structure, and that if $\mathcal{A}$ is obtained as the heart of a $t$-structure on a stable $\infty$-category satisfying some hypothesis, the canonical functor $\mathcal{D}^b(\mathcal{A})\to\mathcal{C}$ is also symmetric monoidal. In the second part we prove that the $6$ operations on polarisable mixed Hodge modules and on perverse Nori motives can be lifted to $\infty$-categorical functors of stable $\infty$-categories. In , we first prove that the functor sending a quasi-projective $k$-scheme to the bounded derived $\infty$-category of polarisable mixed Hodge modules or perverse Nori motives is an étale hyper sheaf. We use this property to extend those constructions to all finite type $k$-schemes. Then in a second part we recall the results of Drew and Gallauer in [@MR4560376] and prove that they easily imply a universal property for $\mathcal{DM}$. Finally in the third part we construct the realisation and show that it is weight-exact and commutes with all the operations. In the last , after proving with an argument due to Nori that pushforwards and internal $\mathop{\mathrm{\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}}$ are right derived functors on perverse Nori motives, we give some consequences of the existence of a realisation functor, compatible with the operations, between the motivic categories of Voevodsky and Nori. We begin with the proof that the derived category of perverse Nori motives is continuous. As explained earlier, the proof consists in proving that the $\infty$-category of Nori motives can be seen as the category of modules over a $\mathbb{E}_\infty$-algebra in $\mathcal{DM}(-)$. Next, we prove that if a motivic $t$-structure exists, then the category of Voevodsky étale motives is naturally equivalent to the derived category of perverse Nori motives. Finally in we study Artin motives and prove that the stable $\infty$-categories of Artin motives in $\mathcal{DM}_c(X)$ and in $\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))$ are equivalent. We also identify smooth Artin motives with dualisable Artin motives over a normal basis, which proves that they afford a $t$-structure whose heart is the category of Artin representation of the étale fundamental group. ## Forthcoming and future work {#forthcoming-and-future-work .unnumbered} The next goal for the PhD project would be to prove a $1$-motivic analogue of : Prove that the categories of $1$-motives in Voevodsky's and Nori's categories are equivalent. In another direction, in joint work with Raphaël Ruimy we will construct an integral version of the derived category of perverse Nori motives. This will give an Abelian category of motivic sheaves with integral coefficients over any finite type $k$-scheme with $k$ a field of characteristic zero. ## Notations and conventions {#notations-and-conventions .unnumbered} For the whole paper, $k$ is a field of characteristic $\mathrm{char} k = 0$. When $k$ is a subfield of $\mathbb{C}$, if $X$ is a finite type $k$-scheme, we denote by $\mathrm{D}_c^b(X,\mathbb{Q})$ the category of constructible complexes of sheaves on $X(\mathbb{C})$. By [@MR0923133], it is also the derived category of the abelian category of perverse sheaves $Perv(X,\mathbb{Q})$ defined in [@MR0751966]. For a general field $k$, we denote by $\mathrm{D}^b_c(X,\mathbb{Q}_\ell)$ the category of $\mathbb{Q}_\ell$-adic étale sheaves on $X$. Starting with , we will use the language of $\infty$-categories as developed in [@lurieKerodon] by Lurie, after Joyal's work on quasi-categories. We fix an universe and call small an $\infty$-category whose mapping spaces and core are in this universe. We denote by $\mathrm{Cat}_\infty$ the $\infty$-category of non necessarily small $\infty$-categories. If $\mathcal{C}$ is a symmetric monoidal $\infty$-category, and $A\in\mathcal{C}$ is a commutative algebra, we denote by $\mathrm{Mod}_A(\mathcal{C})$ the category of modules over $A$ in $\mathcal{C}$. For example, the category of $H\mathbb{Q}$-modules in spectra $\mathrm{Mod}_\mathbb{Q}:= \mathrm{Mod}_{H\mathbb{Q}}(\mathrm{Sp})$ is the unbounded derived category of $\mathbb{Q}$-vector spaces. We will also have to consider the $\infty$-category of $\mathrm{Mod}_\mathbb{Q}$-modules in the $\infty$-category of stable $\infty$-categories, that we denote by $\mathrm{Mod}_\mathbb{Q}(\mathrm{St})$ and which is equivalent to the category of stable $\mathbb{Q}$-linear $\infty$-categories. We will also denote by $\mathrm{Gpd}_\infty$ the $\infty$-category of $\infty$-groupoids, which is also the $\infty$-category of Kan complexes. We try to always put the index $\mathcal{C}$ when mentioning $\mathrm{Hom}$, as in $\mathrm{Hom}_\mathcal{C}(A,B)$. In the case where $\mathcal{C}$ is an ordinary category, this is the $\mathrm{Hom}$-set, and in the case $\mathcal{C}$ is an $\infty$-category this is the $\pi_0$ of the mapping space $\mathrm{map}_\mathcal{C}(A,B)$. In stable categories, the mapping spectra will be denoted by $\mathrm{Map}_\mathcal{C}(-,-)$. ## Acknowledgements {#acknowledgements .unnumbered} This work was written during my PhD supervised by Sophie Morel at the UMPA in Lyon. I would like to thank her for the faith and constant attention she has for my work and for the immense amount of time she is spending to accompany me in this thesis. I thank also Frédéric Déglise for sharing some ideas with me and Robin Carlier for taking the time of explaining to me so much facts about $\infty$-categories. I had interesting email exchanges with Joseph Ayoub and Marc Hoyois who answered questions I had. I also had useful conversations with Raphaël Ruimy and Luca Terenzi. # Recollections. {#SectionRecoll} ## Categories of étale motives. Ayoub ([@MR2602027] and [@MR3205601]), Cisinski-Déglise ([@MR3971240] and [@MR3477640]) and Voevodsky ([@MR1883180]) have constructed triangulated categories of motivic sheaves, that are the relative versions of $\mathrm{DM}_{\mathrm{gm}}(k)$. In this paper, we are interested in the étale versions. We will present the stable $\infty$-categorical version as this is the one we will be using. Also, we only consider $\mathbb{Q}$-linear categories. We start with Ayoub's construction ([@MR2602027]). Let $S$ be a scheme. One starts with the category $\mathrm{Sm}_S$ of smooth $S$-schemes, and consider presheaves of spectra on it : $\mathrm{PSh}(\mathrm{Sm}_S,\mathrm{Sp})$. The category of rational effective étale motivic sheaves is the full subcategory of $\mathrm{PSh}(\mathrm{Sm}_S,\mathrm{Sp})$ whose objects are the presheaves $\mathcal{F}$ that are $\mathbb{Q}$-linear (*i.e.* whose image lands in $\mathcal{D}(\mathbb{Q})\simeq\mathrm{Mod}_{H\mathbb{Q}}\subset \mathrm{Sp}$), $\mathbb{A}^1$-invariant (*i.e.* the natural map $\mathcal{F}(X)\to\mathcal{F}(\mathbb{A}^1_X)$ is an equivalence for any smooth $S$-scheme $X$), and satisfy étale hyper-descent (*i.e.* for any étale hyper cover $U_\bullet\to X$ of a smooth $S$-scheme $X$, the natural map $\mathcal{F}(X)\to \lim_{\Delta}\mathcal{F}(U_n)$ is an equivalence.) It is a symmetric monoidal stable $\infty$-category, the tensor product being inherited from the Cartesian structure of $\mathrm{Sm}_S$. Inverting the object $\mathrm{cofib}(S\overset{1}{\to}\mathrm{Gm}_S)$ for the tensor product gives the category that we will denote by $\mathcal{DM}(S)$ and call the category of Voevodsky (étale) motives. Note that we could have considered sheaves of spaces instead of spectra if we had inverted $\mathrm{cofib}(S\overset{\infty}{\to}\mathbb{P}^1_S)\simeq\mathrm{cofib}(S\overset{1}{\to}{\mathbb{G}_m}_S)\otimes S^1$ and take $\mathbb{Q}$-linear coefficient after this, obtaining an isomorphism $\mathcal{DM}(S) = \mathcal{SH}_{\acute{e}t}(S)_\mathbb{Q}$ with the rational part of the étale stable motivic homotopy category (see [@MR4224739]). As a formula, we have $$\mathcal{DM}(S):= (\mathrm{L}_{\mathbb{A}^1}\mathrm{Shv}_{\acute{e}t}^\wedge(\mathrm{Sm}_S,\mathcal{D}(\mathbb{Q})))[({\mathbb{G}_m}_S/1_S)^{-1}]$$ Cisinski and Déglise's construction ([@MR3477640]) follows Voevodsky original construction ([@MR2687724]). They choose the $h$-topology instead of the étale topology, whose covers are given by the proper surjective maps. As $\mathrm{Sm}_S$ is not big enough to consider this topology, they make the same construction as Ayoub, but starting with finite type $S$-schemes, to obtain a category $\underline{\mathcal{DM}}_h(S,\mathbb{Q})$: $$\underline{\mathcal{DM}}_h(S,\mathbb{Q}):= (\mathrm{L}_{\mathbb{A}^1}\mathrm{Shv}_h^\wedge(\mathrm{Sch}_S,\mathcal{D}(\mathbb{Q})))[({\mathbb{G}_m}_S/1_S)^{-1}]$$ The category $\mathcal{DM}_h(S)$ of $h$-motives is the presentable stable full subcategory generated by the objects of the form $X\otimes \mathbb{Q}(1)^k$ for $X\in\mathrm{Sm}_S$ , $\mathbb{Q}(1) = \mathrm{cofib}(S\overset{1}{\to}\mathrm{Gm}_S)[-1]$ and $k\in \mathbb{Z}$. They show in [@MR3477640 Corollary 5.5.7] (one needs [@MR4224739] to prove the result in the best generality) that the change of sites map induces an equivalence of categories $$\mathcal{DM}(S)\to\mathcal{DM}_h(S)$$ when $S$ is a Noetherian scheme of finite dimension. Both categories are stable presentably symmetric monoidal $\infty$-categories (for a detailed $\infty$-categorical construction, see [@preisMotivicNearbyCycles2023] or [@MR4061978]). We will denote by $\mathcal{DM}_c(S)$ the full subcategory of étale motives whose objects are compact objects (*i.e.* $M\in\mathcal{DM}(S)$ such that $\mathrm{Hom}_{\mathcal{DM}}(M,-)$ commutes with small filtered colimits.) As we work rationally, one can show ([@MR3205601 Proposition 8.3]) that the subcategory of compact objects coincides with the idempotent complete stable subcategory generated by motives of the form $\mathrm{M}(X)$ for $X$ a smooth $S$-scheme, where we have denoted by $\mathrm{M}(X)$ the image of $X\in\mathrm{Sm}_S$ by the Yoneda functor. As $\mathcal{DM}(S)$ is compactly generated, we have $\mathrm{Ind}\mathcal{DM}_c(S)=\mathcal{DM}(S)$. Moreover, if $S =\mathrm{Spec}\ k$ is the spectrum of a field, then there is a natural equivalence $\mathrm{ho}(\mathcal{DM}_c(\mathrm{Spec}\ k))\simeq \mathrm{DM}_\mathrm{gm}(k,\mathbb{Q})$ between the homotopy category of étale motives and the triangulated now classical category of Voevodsky motives over a field. The categories $\mathcal{DM}(X)$ can be put together to give a functor $\mathrm{Sch}_S\to\mathrm{CAlg}(\mathrm{Pr}^L)$ from the category of finite type $S$-schemes (with $S$ a fixed scheme) to the $\infty$-category of presentable symmetric monoidal $\infty$-categories. Moreover, this functor $\mathcal{DM}$ is a pullback formalism in the sense of Drew and Gallauer (see below), and the $\infty$-category of constructible objects $\mathcal{DM}_c(-)$ is part of a six functors formalism (see [@MR3971240 Section A.5] for a detailed definition), which include that all pullbacks $f^*$ have right adjoints $f_*$, that there exist exceptional functoriality consisting of a $f_!$ left adjoint to $f^!$, such that $f^!\simeq f^*$ for $f$ étale and $p_*\simeq p_!$ for $p$ proper. Ayoub constructed in [@MR2602027] the Betti realisation of $\mathcal{DM}(S)$ with values in the derived category of sheaves on the analytification $S^{an}$ on $S$ when $S$ is of finite type over a subfield of the complex numbers, which restricts to $\mathcal{DM}_c(S)$ and then lands into $\mathcal{D}_c^b(S(\mathbb{C}),\mathbb{Q})$ the stable category of bounded complexes of sheaves with constructible cohomology. This category is both the derived category of perverse sheaves ([@MR0923133]) and of the abelian category of constructible sheaves ([@MR1940678]). One can find in [@MR3205601] and [@MR3477640] the construction of the $\ell$-adic realisation functors with values in the derived category of $\ell$-adic sheaves. An $\infty$-categorical version of this functor can be found in [@MR4061978 2.1.2]. This is a functor $\mathcal{DM}_c(S)\to\mathcal{D}_c^b(S,\mathbb{Q}_\ell)$, the latter category being enhanced using condensed coefficients in [@MR4609461] where it is denoted by $\mathcal{D}_{\mathrm{cons}}(S,\mathbb{Q}_\ell)$. It also has a version on the big category $\mathcal{DM}(S)$ by taking indization. All the realisation commutes with the $6$ operations. ## Perverse Nori motives. Let $X$ be a quasi-projective $k$-scheme with $k$ a field of characteristic zero. In [@ivorraFourOperationsPerverse2022], Ivorra and S. Morel introduced an abelian category of perverse Nori motives ${\mathcal{M}_{\mathrm{perv}}}(X)$. The construction of the category ${\mathcal{M}_{\mathrm{perv}}}(X)$ goes as follows: Let $X$ be a quasi-projective $k$-scheme. Pick your favourite prime number $\ell$. We can compose the $\ell$-adic realisation $$\rho_\ell:\mathcal{DM}_c(X)\to \mathcal{D}^b(X_{\acute{e}t},\mathbb{Q}_\ell)$$ with the perverse truncation functor $${\ ^{\mathrm{p}}\mathrm{H}}^0:\mathcal{D}^b(\acute{e}t,\mathbb{Q}_\ell)\to\mathrm{Perv}(X,\mathbb{Q}_\ell)$$ to obtain a homological functor ${\ ^{\mathrm{p}}\mathrm{H}}^0_ell$ that will factor through $\mathcal{R}(\mathcal{DM}_c(X))$. This latter category is the abelian category of coherent modules over $\mathcal{DM}_c(X)$: it is the full subcategory of additive presheaves $\mathcal{F}:\mathcal{DM}_c(X)^\mathrm{op}\to\mathrm{Ab}$ that fit in exact sequences $$y_M\to y_N\to\mathcal{F}\to 0.$$ Denote by $Z_\ell$ the kernel of $\widetilde{{\ ^{\mathrm{p}}\mathrm{H}}^0_\ell}:\mathcal{R}(\mathcal{DM}_c(X))\to Perv(X,\mathbb{Q}_\ell)$, that is the objects mapping to zero. The quotient category $\mathcal{R}(\mathcal{DM}_c(X))/Z_\ell$ is the category of perverse Nori motives ${\mathcal{M}_{\mathrm{perv}}}(X)$. By construction, ${\mathcal{M}_{\mathrm{perv}}}(X)$ has the following universal property: Any homological functor $H:\mathcal{DM}_c(X)\to\mathcal{A}$ to an abelian category $\mathcal{A}$ such that ${\ ^{\mathrm{p}}\mathrm{H}}^0_\ell(M)=0\Rightarrow H(M)=0$ for all $M\in\mathcal{DM}_c(X)$ will factor uniquely through the universal functor $$\mathcal{DM}_c(X)\overset{H_\mathrm{univ}}{\to}{\mathcal{M}_{\mathrm{perv}}}(X).$$ As we will see in this construction is the best approximation possible of the heart of the conjectural perverse $t$-structure on $\mathcal{DM}_c(X)$. A priori this category depends on the chosen $\ell$, but one can show, using a continuity argument that this is not the case because for finite type fields (or more generally, for fields embeddable into $\mathbb{C}$), the Betti realisation can also be used to define ${\mathcal{M}_{\mathrm{perv}}}(X)$. In fact, every realisation functor existing on $\mathcal{DM}_c(X)$ gives us a realisation functor on ${\mathcal{M}_{\mathrm{perv}}}(X)$: we have the $\ell$-adic realisation $R_\ell$ and the Betti realisation $R_B$. If $S=\mathrm{Spec}\ k$, then Ivorra and S. Morel have proven that ${\mathcal{M}_{\mathrm{perv}}}(\mathrm{Spec}\ k)$ coincide with the category of Nori constructed by Nori, and thus also have an Hodge realisation. Ivorra and S. Morel show in [@ivorraFourOperationsPerverse2022 Theorem 5.1] that the derived category $\mathrm{D}^b({\mathcal{M}_{\mathrm{perv}}}(-))$ is part of a homotopical $2$-functor in the sense of Ayoub [@MR2423375 Definition 1.4.1]. This means that they constructed four of the six operations, namely for $f$ a morphism of quasi-projective varieties we have $f^*,f_*,f_!,f^!$, together with Verdier duality $\mathbb{D}$ , verifying $f^!\simeq \mathbb{D}\circ f^*\mathbb{D}$ and $f_!\simeq\mathbb{D}\circ f_*\circ\mathbb{D}$. In his PhD thesis Terenzi [@TerenziThesis] constructed the remaining two operations: tensor product and internal $\mathop{\mathrm{\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}}$. This proves that $\mathrm{D}^b({\mathcal{M}_{\mathrm{perv}}}(-))$ is part of a full 6 functor formalism. The $6$ operations commute with the realisation $R_\ell$ and $R_B$ that become morphism of symmetric monoidal stable homotopical $2$-functors. The construction of the operations of perverse Nori motives is very similar to Saito's construction of mixed Hodge modules. ## Mixed Hodge modules Let $X$ be a separated finite type $\mathbb{C}$-scheme. In [@MR1047415], Saito constructed an abelian category $\mathrm{MHM}^p(X)$ of polarisable mixed Hodge modules, together with the six operations on $\mathrm{D}^b(\mathrm{MHM}^p(X))$. It has a faithful exact functor $\mathrm{MHM}^p(X)\to Perv(X^\mathrm{an},\mathbb{Q})$ to the abelian category of perverse sheaves on the analytification of $X$, such that the induced functor on the derived categories commutes with the operations. If $X=\mathrm{Spec}\ \mathbb{C}$, then there is a natural equivalence $\mathrm{MHM}^p(\mathrm{Spec}\ \mathbb{C})\simeq\mathrm{MHS}^p_\mathbb{Q}$ between polarisable mixed Hodge structures and polarisable mixed Hodge modules. One of the most interesting property of mixed Hodge modules is that its *lisse* objects, that is the objects such that the underlying perverse sheaf is a shift of a locally constant sheaf are exactly polarisable variations of mixed Hodge structures. We will not go in details in the construction of Saito's category, and we will mostly need to know that the functor taking a complex of polarisable mixed Hodge modules to its underlying complex of perverse sheaves is conservative and commutes with the operations. # Adapting Nori's argument to perverse Nori motives. {#section1} ## Hypothesis on the coefficient system. {#hypothesis-on-the-coefficient-system. .unnumbered} $k$ is a field of characteristic zero. For this section we will call *variety* an element of a fixed subcategory $\mathrm{Var}_k$ of finite type separated $k$-schemes. For example $\mathrm{Var}_k$ could be the full subcategory of quasi-projective $k$-varieties or of separated reduced $k$-schemes of finite type. Choose $\mathrm{D}:\mathrm{Var}_k\to \mathrm{Triang}$ your favourite $\mathbb{Q}$-linear triangulated category of coefficients (*e.g.* the derived category of perverse Nori motives [@ivorraFourOperationsPerverse2022 Theorem 5.1] and [@TerenziThesis Theorem 0.0.1] or of (polarisable) mixed Hodge modules [@MR1047415 Theorem 0.1]) with a $6$ functors formalism and a conservative Betti realisation $R_B:\mathrm{D}\to\mathrm{D}^b_c$ compatible with the operations. Note that in the mixed Hodge module situation the Betti realisation is nothing more that the functor forgetting the structure of a mixed Hodge module, and is called 'taking the $\mathbb{Q}$-structure'' by Saito. It has a constructible $t$-structure for which $R_B$ is $t$-exact when $\mathrm{D}^b_c(X(\mathbb{C}),\mathbb{Q})$ is endowed with its natural $t$-structure of heart the abelian category of constructible sheaves $Cons(X)$. For $X\in\mathrm{Var}_k$ we will call motives the elements of $\mathrm{D}(X)$. Denote by ${\mathcal{M}_{\mathrm{ct}}}(X)$ the heart of the constructible $t$-structure on $\mathrm{D}(X)$. As $R_B$ is $t$-exact and conservative, all known $t$-exactness results about the $6$ functors are true in $\mathrm{D}$. We will call elements of ${\mathcal{M}_{\mathrm{ct}}}(X)$ *constructible motives* . Denote by $\mathop{\mathrm{\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}}(-,-)$ and $\mathbb{Q}_X$ the internal $\mathrm{Hom}$ and unit object of $\mathrm{D}(X)$. If $k$ is too big to be embedded in $\mathbb{C}$, then one needs an $\ell$-adic realisation instead of a Betti realisation. Except for which is different in both cases, in this section we will say Betti realisation even for big fields $k$. We assume that $\mathrm{D}(X)$ has enough structure so that there exists a natural functor $\mathrm{D}^b({\mathcal{M}_{\mathrm{ct}}}(X))\to\mathrm{D}(X)$ (for example if $\mathrm{D}(X)$ is a derived category as shown by Beilinson, Bernstein, Deligne and Gabber in [@MR0751966 section 3.1] or more generally if $\mathrm{D}(X)$ is the homotopy category of some stable $\infty$-category $\mathcal{D}(X)$ as proven in [@bunkeControlledObjectsLeftexact2019 Remark 7.60]), and we assume that the natural functor $\mathrm{D}^b({\mathcal{M}_{\mathrm{ct}}}(\mathrm{Spec}\ k))\to\mathrm{D}(\mathrm{Spec}\ k)$ is an equivalence over the base field. This is true for perverse Nori motives and mixed Hodge modules. **Definition 7**. Let $X$ be a $k$-variety. A motive $L$ is a *lisse* object if its Betti realisation is a local system, that is a locally constant bounded complex of finite dimensional $\mathbb{Q}$-vector spaces. We denote $\mathrm{D}^{liss}(X)$ the full subcategory of lisse objects in $\mathrm{D}(X)$. *Remark 8*. Note that as $\mathrm{D}(X)$ is closed as a tensor category, given an object $M\in\mathrm{D}(X)$, we have a natural candidate for the strong dual of $M$ : it is $\mathop{\mathrm{\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}}(M,\mathbb{Q}_X)$. Therefore, an object $M$ is dualisable if and only if the map $$\label{testdual}\mathop{\mathrm{\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}}(M,\mathbb{Q}_X)\otimes M \to \mathop{\mathrm{\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}}(M,M)$$ obtained by the $\otimes$-$\mathop{\mathrm{\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}}$ adjunction from the map $\mathrm{ev}_M\otimes \mathrm{Id}_M : \mathop{\mathrm{\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}}(M,\mathbb{Q}_X)\otimes M\otimes M\to M$, is an isomorphism. We will denote by $M^\wedge = \mathop{\mathrm{\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}}(M,\mathbb{Q}_X)$ the dual of $M$ when $M$ is dualisable. **Proposition 9**. *Let $X$ be a $k$-variety. An object $L\in \mathrm{D}(X)$ is lisse if and only if it is dualisable and in that case $L\in {\mathcal{M}_{\mathrm{ct}}}(X)$ if and only if $L^\wedge\in{\mathcal{M}_{\mathrm{ct}}}(X)$. In particular, for any variety $Y$ and for every object $N\in \mathrm{D}(Y)$, there exists a dense open on which $N$ is dualisable.* *Proof.* For the first point, dualisable objects in $\mathrm{D}^b_c(X,\mathbb{Q})$ are exactly local systems by Ayoub's [@ayoubAnabelianPresentationMotivic2022 Lemma 1.24] , and the Betti realisation is conservative, so we can test if the map is an isomorphism after applying Betti realisation. If $L\in{\mathcal{M}_{\mathrm{ct}}}(X)$ is dualisable, the functor $\mathop{\mathrm{\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}}(L,-) = -\otimes L^\wedge$ is $t$-exact because it is left and right adjoint to the exact functor $-\otimes L$. Therefore, $L^\wedge=\mathop{\mathrm{\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}}(L,\mathbb{Q})\in{\mathcal{M}_{\mathrm{ct}}}(X)$ lies in the heart. The second point follows from the fact that for $N\in \mathrm{D}(Y)$, the complex of sheaves $R_B(N)$ is constructible so there exists a dense open $U$ of $Y$ on which $R_B(N)$ is locally constant. ◻ ## Cohomology of motives over an affine variety. **Definition 10**. 1. An additive functor $F:\mathcal{A}\to \mathcal{B}$ between abelian categories is called *effaceable* if for any $M\in \mathcal{A}$ there is an injection $M\to N$ in $\mathcal{A}(X)$ such that the induced map $F(M)\to F(N)$ is the zero map. 2. An object $M\in \mathrm{D}(X)$ is called *admissible* if the functor $\mathrm{Hom}_{\mathrm{D}(X)}(M,-[q]) : {\mathcal{M}_{\mathrm{ct}}}(X)\to \mathrm{Vect}_\mathbb{Q}$ is effaceable for every $q>0$. **Lemma 11**. *Let $X$ be a variety and $M$ be a constructible motive on $\mathbb{A}^1_X$. Assume that there is $U\subset \mathbb{A}^1_X$ a smooth open subset on which $M$ is lisse and such that the restriction of $\pi:\mathbb{A}^1_X\to X$ to $Z = \mathbb{A}^1_X\setminus U$ is finite. Assume that ${M}_{\mid {Z}} = 0$. Then there exists $N\in {\mathcal{M}_{\mathrm{ct}}}(\mathbb{A}^1_X)$ such that $\pi_*N = 0$ and $M$ injects into $N$.* *Proof.* Assume that $k$ is a subfield of $\mathbb{C}$. Then the following holds : Denote by $p_i:\mathbb{A}^2_X\to \mathbb{A}^1_X$ the projections, and by $\Delta:\mathbb{A}^1_X\to \mathbb{A}^2_X$ the diagonal. The map $\alpha : p_1^*M\to \Delta_* M$ in ${\mathcal{M}_{\mathrm{ct}}}(X)$ (both functors are $t$-exact) obtained by adjunction from $\mathrm{id}:\Delta^*p_1^*M\simeq M\to M$ is surjective because it is surjective after realisation. We therefore have an exact sequence $$0\to \ker \alpha\to p_1^*M\to \Delta_* M\to 0.$$ Denote $N:=\mathrm{H}^1(p_2)_*\ker \alpha$. Then the long exact sequence of cohomology obtained after applying $(p_2)_*$ to the above sequence gives a map $M = (p_2)_*\Delta_* M = \mathrm{H}^0((p_2)_*\Delta_* M) \to N$, which is injective by [@MR1940678 Proposition 2.2]. Moreover, Nori proves in *loc cit.* that for all $p\geqslant 0$ we have $\mathrm{H}^p(\pi_*N)=0$, which is equivalent to $\pi_*N=0$. In the case $k$ is too big to be embedded in $\mathbb{C}$, on may use [@MR4325954 Proposition 3.2] and the $\ell$-adic realisation, noting that the construction of $N$ is the same. ◻ **Lemma 12**. *Let $\pi:\mathbb{A}^1_X\to X$ be the projection, with $X$ a variety.* 1. *[\[facts:1\]]{#facts:1 label="facts:1"}Let $f:A\to B$ be a map in $\mathrm{D}(\mathbb{A}^1_X)$ such that $\pi_*f$ is an isomorphism. Then for all $q\in\mathbb{Z}$, the induced map $$\mathrm{Hom}_{\mathrm{D}(\mathbb{A}^1_X)}(\mathbb{Q}_{\mathbb{A}^1_X},A[q])\to \mathrm{Hom}_{\mathrm{D}(\mathbb{A}^1_X)}(\mathbb{Q}_{\mathbb{A}^1_X},B[q])$$ is an isomorphism.* 2. *[\[facts:2\]]{#facts:2 label="facts:2"} Let $f:A\to B$ be a map in $\mathrm{D}(X)$ such that the induced map $$\mathrm{Hom}_{\mathrm{D}(X)}(\mathbb{Q}_{X},A[q])\to \mathrm{Hom}_{\mathrm{D}(X)}(\mathbb{Q}_{X},B[q])$$ is zero for some $q\in\mathbb{Z}$. Then the map $$\mathrm{Hom}_{\mathrm{D}(\mathbb{A}^1_X)}(\mathbb{Q}_{\mathbb{A}^1_X},\pi^*A[q])\to \mathrm{Hom}_{\mathrm{D}(\mathbb{A}^1_X)}(\mathbb{Q}_{\mathbb{A}^1_X},\pi^*B[q])$$ is also the zero map.* *Proof.* For the first fact, we look at the following diagram in which $\eta_{X}$ is the unit of the adjunction between $\pi^*$ and $\pi_*$ : $$\begin{tikzcd} {\mathrm{Hom}_{\mathrm{D}(\mathbb{A}^1_X)}(\mathbb{Q}_{\mathbb{A}^1_X},A[q])} & {\mathrm{Hom}_{\mathrm{D}(\mathbb{A}^1_X)}(\mathbb{Q}_{\mathbb{A}^1_X},B[q])} \\ {\mathrm{Hom}_{\mathrm{D}(\mathbb{A}^1_X)}(\pi_*\mathbb{Q}_{\mathbb{A}^1_X},\pi_*A[q])} & {\mathrm{Hom}_{\mathrm{D}(\mathbb{A}^1_X)}(\pi_*\mathbb{Q}_{\mathbb{A}^1_X},\pi_*B[q])} \\ {\mathrm{Hom}_{\mathrm{D}(X)}(\mathbb{Q}_{X},\pi_*A[q])} & {\mathrm{Hom}_{\mathrm{D}(X)}(\mathbb{Q}_{X},\pi_*B[q])} \arrow["{\pi_*}", from=1-1, to=2-1] \arrow["{\pi_*}", from=1-2, to=2-2] \arrow["{\circ \eta_X}", from=2-1, to=3-1] \arrow["{\circ \eta_X}", from=2-2, to=3-2] \arrow["{f[q]\circ}", from=1-1, to=1-2] \arrow["{(\pi_*f[q])\circ}", from=2-1, to=2-2] \arrow["{(\pi_*f[q])\circ}"', from=3-1, to=3-2] \end{tikzcd}.$$ The long vertical compositions are the adjunction isomorphisms, and the second and third horizontal maps are isomorphisms because $\pi_*f$ is, hence we obtain the claim. For the second fact, one has to look at the following diagram : $$\begin{tikzcd} {\mathrm{Hom}_{\mathrm{D}(X)}(\mathbb{Q}_{X},A[q])} & {\mathrm{Hom}_{\mathrm{D}(X)}(\mathbb{Q}_{X},B[q])} \\ {\mathrm{Hom}_{\mathrm{D}(\mathbb{A}^1_X)}(\mathbb{Q}_{\mathbb{A}^1_X},\pi^*A[q])} & {\mathrm{Hom}_{\mathrm{D}(\mathbb{A}^1_X)}(\mathbb{Q}_{\mathbb{A}^1_X},\pi^*B[q])} \arrow["{\pi^*}", from=1-2, to=2-2] \arrow["{\pi^*}", from=1-1, to=2-1] \arrow["{f[q]\circ}", from=1-1, to=1-2] \arrow["{\pi^*f[q]\circ}"', from=2-1, to=2-2] \end{tikzcd}$$ in which the vertical maps are isomorphisms because $\pi^*$ is fully faithful, giving the claim. ◻ Nori's method gives in our setting a slightly less good result than in the case of constructible sheaves. It will thankfully be sufficient for us. **Theorem 13** (*cf.* [@MR1940678] Theorem 1). *For every $n\in\mathbb{N}$, the motive $\mathbb{Q}_{\mathbb{A}^n_k}\in{\mathcal{M}_{\mathrm{ct}}}(\mathbb{A}^n_k)$ is admissible.* *Proof.* We argue as in Nori's paper.\ We proceed by induction on $n$ to prove the theorem. First the case of a point. If $n=0$, and $M\in {\mathcal{M}_{\mathrm{ct}}}(\mathrm{Spec}\ k)$, as $\mathrm{D}(\mathrm{Spec}\ k)= \mathrm{D}^b({\mathcal{M}_{\mathrm{ct}}}(\mathrm{Spec}\ k))$, the functors $\mathrm{Hom}_{\mathrm{D}(\mathrm{Spec}\ k)}(M,-[q])$ are effaceable by [@MR0751966 Proposition 3.1.16]. Suppose that the result is know for $n-1$. Let $M\in {\mathcal{M}_{\mathrm{ct}}}(\mathbb{A}^n_k)$ and $q>0$. Let $\pi:\mathbb{A}^n_k\to\mathbb{A}^{n-1}_k$ be the projection on the $n-1$ first coordinates. Up to change of coordinates, one can choose a smooth open of the form $D(f)$ in $\mathbb{A}^n_k$ such that $M$ is lisse on that open and the natural projection $\pi_{|V(f)}:V(f)\to \mathbb{A}^{n-1}_k$ is finite. Denote by $j$ and $i$ the inclusions of $D(f)=U$ and $V(f)=Z$ in $\mathbb{A}^n_k$.\ By , $j_!{M}_{\mid {U}}$ is a submotive of a constructible motive $N'$ such that $\pi_*N'=0$. Taking $N$ to be the push-out of the maps $j_!{M}_{\mid {U}}\to M$ and $j_!{M}_{\mid {U}}\to N'$ we get a morphism of exact sequences: with $\gamma$ also injective. Also, as $\pi_*N' = 0$, we see that $\pi_*N\to \pi_*i_*{M}_{\mid {Z}}=({\pi}_{\mid {Z}})_*{M}_{\mid {Z}}$ is an isomorphism because of the distinguished triangle $N'\to N\to i_*M_Z\overset{+1}{\to}$. As ${\pi}_{\mid {Z}}$ is finite, its pushforward is $t$-exact, hence $\pi_*N\in {\mathcal{M}_{\mathrm{ct}}}(\mathbb{A}^{n-1}_k)$. By the induction hypothesis, we can find an injection $g:\pi_*N\to K$ with $K$ a constructible motive such that $\mathrm{Hom}_{\mathrm{D}(\mathbb{A}^n_k)}(\mathbb{Q}_{\mathbb{A}^n_k},g[q]) = 0$. Take $L$ the pushout of the map $\pi^*\pi_*N \to N$ and the injection ($\pi^*$ is $t$-exact) $\pi^*g:\pi^*\pi_*N\to \pi^*K$. We get a morphism of exact sequences : $$\label{suitesexact2}\begin{tikzcd} 0\ar[r] & \pi^*\pi_*N \ar[d]\ar[r,"\pi^*g"] & \pi^*K \ar[d,"\iota"] \ar[r] & C \ar[d,"\mathrm{id}_C"] \ar[r] & 0 \\ 0 \ar[r] & N \ar[r,"h"] & L \ar[r] & C \ar[r] & 0 \end{tikzcd}.$$ The adjunction map $\pi_*N \to \pi_*\pi^*\pi_*N$ is an isomorphism by $\mathbb{A}^1$-invariance. Therefore in the diagram above the left and the right vertical maps become isomorphisms after applying $\pi_*$, thus this is also the case for the middle map $\iota:\pi^*K\to L$. By 1., the map $$\mathrm{Hom}_{\mathrm{D}(\mathbb{A}^n_k)}(\mathbb{Q}_{\mathbb{A}^n_k},\pi^*\pi_*N[q])\to \mathrm{Hom}_{\mathrm{D}(\mathbb{A}^n_k)}(\mathbb{Q}_{\mathbb{A}^n_k},N[q])$$ induced by $\varepsilon : \pi^*\pi_*N\to N$ is an isomorphism. Also by 2. , the map $$\mathrm{Hom}_{\mathrm{D}(\mathbb{A}^n_k)}(\mathbb{Q}_{\mathbb{A}^n_k},\pi^*\pi_*N[q])\to \mathrm{Hom}_{\mathrm{D}(\mathbb{A}^n_k)}(\mathbb{Q}_{\mathbb{A}^n_k},\pi^*K[q])$$ induced by $\pi^*g$ is the zero map. Therefore when applying $\mathrm{Hom}_{\mathrm{D}(\mathbb{A}^n_k)}(\mathbb{Q}_{\mathbb{A}^n_k},-[q])$ to the left square of [\[suitesexact2\]](#suitesexact2){reference-type="ref" reference="suitesexact2"}, we see that the map $$\mathrm{Hom}_{\mathrm{D}(\mathbb{A}^n_k)}(\mathbb{Q}_{\mathbb{A}^n_k},N[q])\to \mathrm{Hom}_{\mathrm{D}(\mathbb{A}^n_k)}(\mathbb{Q}_{\mathbb{A}^n_k},L[q])$$ induced by $h$ factors through the zero map. Hence we see that the injection $M\overset{\gamma}{\to} N\overset{h}{\to} L$ induces the zero map after applying the functor $\mathrm{Hom}_{\mathrm{D}(\mathbb{A}^n_k)}(\mathbb{Q}_{\mathbb{A}^n_k},-[q])$, finishing the induction. ◻ **Corollary 14**. *If $X$ is affine, the object $\mathbb{Q}_X$ is admissible.* *Proof.* Let $q>0$. Take $i:X\to \mathbb{A}^n_k$ a closed immersion and $M\in{\mathcal{M}_{\mathrm{ct}}}(X)$. By , there is an injection $i_*M\to K$ with $K$ a constructible motive such that $\mathrm{Hom}_{\mathrm{D}(\mathbb{A}^{n})}({\mathbb{Q}}_{\mid {\mathbb{A}^{n}_k}},i_*M[q])\to \mathrm{Hom}_{\mathrm{D}(\mathbb{A}^{n})}({\mathbb{Q}}_{\mid {\mathbb{A}^{n}_k}},K[q])$ is the zero map. We have an inclusion $f:M\to N:=i^*K$, and the image of the map in the $\mathrm{Hom}$-sets is $$\mathrm{Hom}_{\mathrm{D}(X)}(\mathbb{Q}_X,i^*i_*M[q])\simeq\mathrm{Hom}_{\mathrm{D}(X)}(\mathbb{Q}_X,M[q])\to \mathrm{Hom}_{\mathrm{D}(X)}(\mathbb{Q}_X,i^*K[q])$$which fits in a commutative diagram : giving the result. ◻ ## Admissibility of constant motives on any variety. **Notation 15**. *If $j:U\hookrightarrow X$ is an open subset of a variety $X$, and $M\in {\mathcal{M}_{\mathrm{ct}}}(X)$ is a constructible motive, we will use the notation $M[U]:=j_!{M}_{\mid {U}}$. Note that by localisation we have $M/M[U]=i_*{M}_{\mid {Z}}$ with $i:Z=X\setminus U\to X$ the closed complement.* **Lemma 16** (Stability of admissibility). *Let $$0\to M'\to M\to M''\to 0$$ be an exact sequence of constructible motives on $X$. [\[stadm1\]]{#stadm1 label="stadm1"}* 1. *Assume that $M''$ is admissible and at least one of $M', M$ is admissible. Then all three are admissible. [\[stadm2a\]]{#stadm2a label="stadm2a"}* 2. *Assume that $M'$ and $M$ are admissible. If the functor $$\mathrm{coker}\ (\mathrm{Hom}_{\mathrm{D}(X)}(M,-)\to\mathrm{Hom}_{\mathrm{D}(X)}(M',-))$$ is effaceable. Then $M''$ is admissible. [\[stadm2b\]]{#stadm2b label="stadm2b"}* *Proof.* These are formal properties of abelian categories, proven in [@MR1940678 Lemma 3.2]. ◻ **Corollary 17** ([@MR1940678] Lemma 3.5). *Let $M\in{\mathcal{M}_{\mathrm{ct}}}(X)$* 1. *If $U\subset X$ is open, and if $M$ and $M[U]$ are admissible, then so is $M/M[U]$.* 2. *If $V,W$ are open of $X$, and if $M[V],M[W],M[V\cap W]$ are admissible, then the same is true for $M[V\cup W]$* *Proof.* We use part 2. of . For 1., it suffices to show that any $P\in {\mathcal{M}_{\mathrm{ct}}}(X)$ is a subsheaf of a $Q\in{\mathcal{M}_{\mathrm{ct}}}(X)$ such that the map $\mathrm{Hom}_{\mathrm{D}(X)}(M,Q)\to\mathrm{Hom}_{\mathrm{D}(X)}(M[U],Q)$ is surjective. Denote by $j:U\to X$ the open immersion and by $i:Z\to X$ its closed complement. Then define $Q = i_*i^*P\oplus j_*j^*P$. Localisation ensures that the map $P\to Q$ is injective, and the induced map between $\mathrm{Hom}$'s is $\mathrm{Hom}_{\mathrm{D}(X)}(M,i_*{P}_{\mid {Z}})\oplus \mathrm{Hom}_{\mathrm{D}(U)}({M}_{\mid {U}},{P}_{\mid {U}})=\mathrm{Hom}_{\mathrm{D}(X)}(M,Q)\to \mathrm{Hom}_{\mathrm{D}(X)}(M[U],Q) = 0\oplus \mathrm{Hom}_{\mathrm{D}(X)}(j_!{M}_{\mid {U}},j_*{P}_{\mid {U}}) = \mathrm{Hom}_{\mathrm{D}(U)}({M}_{\mid {U}},{P}_{\mid {U}})$ is surjective as it is just the second projection.\ For 2. first one has the Mayer-Vietoris exact sequence $$0\to M[V\cap W]\to M[V]\oplus M[W] \to M[V\cup W]\to 0.$$ For a given $P$ we take the same $Q$ as above for $U=V\cap W$ that gives surjectivity of $\mathrm{Hom}_{\mathrm{D}(X)}(M,Q)\to \mathrm{Hom}_{\mathrm{D}(X)}(M[U],Q)$. Now the map $a:M[U]\to M$ factors through $M[V]$ thus the map $$\mathrm{Hom}_{\mathrm{D}(X)}(M[V]\oplus M[W],Q)\overset{a^*\oplus 0}{\to} \mathrm{Hom}_{\mathrm{D}(X)}(M[U],Q)$$ is surjective. ◻ **Proposition 18** (Avoiding the use of injective objects, after Nori [@MR1940678] Remark 3.8). *Let $F:\mathcal{A}\to \mathcal{B}$ be an exact functor between abelian categories, and let $G:\mathcal{B}\to\mathcal{A}$ be a right adjoint to $F$. Let $H:\mathcal{A}\to \mathcal{C}$ be an effaceable functor. Then $HG$ is also effaceable.* *Proof.* Take $B$ an object of $\mathcal{B}$. By assumption on $H$, there is an injection $u:G(B)\to A$ such that the induced map $H(u):HG(B)\to H(A)$ vanishes. By exactness of $F$, the morphism $F(u):FG(B)\to F(A)$ is a monomorphism. We also have the counit of the adjunction $\varepsilon:FG(B)\to B$. Take $B'$ the pushout of these two maps : The map $v:B\to B'$ is a monomorphism and we have a commutative diagram (with $\eta$ the unit of the adjunction) : The composition $G(\varepsilon)\circ\eta$ is the identity, therefore, the map $G(v)$ factors as $w\circ u$ with $w = G(t)\circ \eta$. Now, $HG(v)=H(w) \circ H (u) = 0$ hence $HG$ is effaceable. ◻ **Corollary 19**. *Let $F:\mathrm{D}(X)\to \mathrm{D}(Y)$ be a $t$-exact exact functor which is a left adjoint. If $M\in{\mathcal{M}_{\mathrm{ct}}}(X)$ is admissible, then so is $F(M)\in{\mathcal{M}_{\mathrm{ct}}}(Y)$.* *Proof.* Let $q>0$ and let $G$ be a right adjoint of $F$. The functor $H = \mathrm{Hom}_{\mathrm{D}(X)}(M,-[q])$ is effaceable by definition. By , $H\circ G$ is therefore effaceable. We have a natural isomorphism $H\circ G \simeq \mathrm{Hom}_{\mathrm{D}(Y)}(F(M),-[q])$ which gives the claim. ◻ **Corollary 20**. *Let $M\in{\mathcal{M}_{\mathrm{ct}}}(X)$ be an admissible motive.* 1. *Let $j:X\to Y$ be an étale morphism. Then $j_!M$ is admissible.* 2. *Let $i:X\to Y$ be a finite morphism. Then $i_*M$ is admissible.* 3. *Let $L\in{\mathcal{M}_{\mathrm{ct}}}(X)$ be a lisse motive. Then $M\otimes L$ is admissible.* *Proof.* We use . For the first two points, we have the adjunctions $(j_!,j^*)$ and $(i_*,i^!)$ and $j_!$ and $i_*$ are $t$-exact. Let $L\in{\mathcal{M}_{\mathrm{ct}}}(X)$ be a lisse motive. The functor $-\otimes L$ is left adjoint to $\mathop{\mathrm{\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}}(L,-)$ and right adjoint to $\mathop{\mathrm{\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}}(L^\wedge,-)$ hence is exact. ◻ **Proposition 21** ([@MR1940678] Proposition 3.6). *Let $U\subset X$ an open of a variety $X$. Then the constructible motive $\mathbb{Q}_X[U]$ is admissible.* *Proof.* Let $j:U\to X$ be an open immersion. We prove the theorem by induction on the number $n$ of affines needed to cover $U$. If $U$ is affine, ensures that $\mathbb{Q}_U$ is admissible hence by , $j_!\mathbb{Q}_U=\mathbb{Q}_X[U]$ is admissible. For the induction, one can write $U = V\cup W$ with $V$ affine and $W$ covered by $n-1$ affines, and separateness of $X$ gives that $V\cap W$ is covered by $n-1$ affines. Therefore by induction $\mathbb{Q}_X[V]$, $\mathbb{Q}_X[V\cap W]$ and $\mathbb{Q}_X[W]$ are admissible. Then, ensures that $\mathbb{Q}_X[U]$ is admissible. ◻ **Corollary 22** ([@MR1940678] Theorem 2). *Let $X$ be a variety over $k$. Then the constant motive $\mathbb{Q}_X$ is admissible, that is, for every $q>0$, every constructible motive $M\in {\mathcal{M}_{\mathrm{ct}}}(X)$ can be embedded in a constructible motive $N\in {\mathcal{M}_{\mathrm{ct}}}(X)$ such that the map $$\mathrm{Hom}_{\mathrm{D}(X)}(\mathbb{Q}_X,M[q])\to \mathrm{Hom}_{\mathrm{D}(X)}(\mathbb{Q}_X,N[q])$$ vanishes.* *Proof.* One takes $U=X$ in the previous proposition. ◻ ## Lisse motives enter the party. **Proposition 23**. *Let $X$ be a variety. Any lisse motive in ${\mathcal{M}_{\mathrm{ct}}}(X)$ is admissible.* *Proof.* By the unit object $\mathbb{Q}_X$ on $X$ is admissible. By $L \simeq \mathbb{Q}_X\otimes L$ is admissible. ◻ **Theorem 24**. *Let $X$ be a variety. Then any $M\in {\mathcal{M}_{\mathrm{ct}}}(X)$ is admissible, that is, for every $q>0$, the functor $\mathrm{Hom}_{\mathrm{D}(X)}(M,-[q]):{\mathcal{M}_{\mathrm{ct}}}(X)\to \mathrm{Vect}_\mathbb{Q}$ is effaceable.* *Proof.* We prove the result on Noetherian induction on the support $S$ of $M$ in $X$, that is the complement of the largest open $V$ of $X$ such that ${M}_{\mid {V}} = 0$. If the support is empty, then $M=0$ is admissible. By there exists a dense open $Z$ of $S$ such that ${M}_{\mid {Z}}$ is lisse. We choose an open subset $U$ of $X$ such that $S\cap U = Z$. Denote by $i:Z\to X$ the closed immersion, and by $j:W = U\setminus Z \to U$ the open immersion of the complement in $U$. As $S\cap V = \emptyset$, we have that $j^*{M}_{\mid {U}} = {M}_{\mid {V}} = 0$. Therefore by localisation, ${M}_{\mid {U}} = i_*{M}_{\mid {Z}}$. By and , the latter is admissible. Now denote by $\iota : U\to X$ the open immersion. By $\iota_!{M}_{\mid {U}}$ is admissible, and $\iota^*(M/\iota_!\iota^*M)=0$, hence the support of $N:= M/\iota_!\iota^*M$ is strictly smaller than that of $M$. By the induction hypothesis, $N$ is admissible. By localisation for the immersion $\iota$ and by 1. we see that $M$ is admissible. ◻ **Corollary 25**. *The natural functor $\mathrm{D}^b({\mathcal{M}_{\mathrm{ct}}}(X))\to\mathrm{D}(X)$ is an equivalence.* *Proof.* We apply [@MR0751966 Lemme 3.1.16]. The conditions are verified because of the previous . ◻ # Higher categorical enhancements. {#section2} ## Categorical preliminaries. Recall that an abelian category $\mathcal{A}$ is Noetherian if any increasing sequence of subobjects of any object eventually stabilises ([@MR1246272 Definition 1.2].) **Lemma 26**. *Let $F:\mathcal{A}\to\mathcal{B}$ be a faithful exact functor between Noetherian abelian categories. Then $\mathrm{Ind}F : \mathrm{Ind}\mathcal{A}\to\mathrm{Ind}\mathcal{B}$ is still faithful exact.* *Proof.* The functor $\mathrm{Ind}F$ is always exact by [@MR2182076 Corollary 8.6.8]. Now as it is exact, to check that it is faithful, it suffices to show that if $\mathrm{Ind}F(A) = 0$, then $A=0$. We therefore take $A = "\mathrm{colim}_{i\in I}"A_i\in \mathrm{Ind}\mathcal{A}$ such that $\mathrm{Ind}F(A) := "\mathrm{colim}_i" F(A_i) = 0$. Note that by Huber's [@MR1246272 Lemma 1.3], we can choose a system $(A_i)$ such that every map $A_i\to A_j$ is injective, because $\mathcal{A}$ is Noetherian. Exactitude of $F$ gives that $\mathrm{Ind}F(A)$ also have injective transitions. But then as $A_i\to A$ is injective (its kernel is the object $"\mathrm{colim}_{i\geqslant j}"(\ker A_i\to A_j))$ , $F(A_i)\to \mathrm{Ind}F(A)$ is also injective : $F(A_i) = 0$ for all $i\in I$, and faithfulness of $F$ gives that $A_i =0$ for all $i\in I$ : the object $A=0$ is zero. ◻ **Lemma 27**. *Let $$\begin{tikzcd} \mathcal{A}\ar[r,shift left=.5ex,"F"] & \mathcal{B}\ar[l,shift left=.5ex,"G"]\end{tikzcd}$$ be an adjunction between small abelian categories. Indization induces an adjunction $$\begin{tikzcd} \mathrm{Ind}\mathcal{A}\ar[r,shift left=.5ex,"\mathrm{Ind}F"] & \mathrm{Ind}\mathcal{B}\ar[l,shift left=.5ex,"\mathrm{Ind}G"]\end{tikzcd}$$ between Grothendieck abelian categories.* *Proof.* We can see $\mathrm{Ind}$ as an $\infty$-functor from small $\infty$-categories to not necessarily small $\infty$-categories (up to cardinals technicalities it is the left adjoint to the forgetful functor from the $\infty$-category of $\infty$-categories admitting small filtered colimits and functors preserving them to the $\infty$-category of small $\infty$-categories, see [@MR2522659 Proposition 5.3.5.10]). Therefore, if $F$ is left adjoint to $G$, then the unit and counit transformations induce a unit and a counit between $\mathrm{Ind}F$ and $\mathrm{Ind}G$, that satisfy the triangle identities by functoriality of $\mathrm{Ind}$. ◻ **Lemma 28** ([@MR1246272]). *Let $\mathcal{A}$ be a Noetherian abelian category. The natural functor $\mathrm{D}^b(\mathcal{A})\to \mathrm{D}(\mathrm{Ind}(\mathcal{A}))$ is fully faithful and an object is in the essential image if and only if its bounded and if its cohomology objects are Noetherian objects.* *Therefore, the fact that whether $M\in\mathrm{D}(\mathrm{Ind}(\mathcal{A}))$ lies in $\mathrm{D}^b(\mathcal{A})$ or not can be tested after applying a faithful exact functor $F:\mathcal{A}\to \mathcal{B}$ from $\mathcal{A}$ to an Noetherian abelian category $\mathcal{B}$.* *Proof.* The fact that the functor is fully faithful is [@MR1246272 Proposition 2.2] (note that $\mathrm{D}^+(\mathrm{Ind}\mathcal{A})\to\mathrm{D}(\mathrm{Ind}\mathcal{A})$ is fully faithful by [@lurieHigherAlgebra2022 Remark 1.3.5.10]). In this proposition, Huber shows that the essential image consist of objects that have cohomology in $\mathcal{A}$. The identification of the image follows from the remark between Proposition 1.5 and Proposition 1.6 of [@MR1246272] in which she explains that an object $A\in\mathrm{Ind}\mathcal{A}$ is in $\mathcal{A}$ if and only if it is Noetherian. The last point follows from , so that $\mathrm{Ind}F$ is still faithful exact, and then that $\mathrm{H}^q\mathrm{Ind}F(K) = \mathrm{Ind}F(\mathrm{H}^q(K))$ so that $K$ is bounded if and only if $\mathrm{Ind}F K$ is, and if $A_0\subset A_1\subset\cdots\subset \mathrm{H}^q(K)$ is an increasing family of subobject, then $\mathrm{Ind}(F)(A_0)\subset \mathrm{Ind}(F)(A_1)\subset\cdots \mathrm{H}^q(\mathrm{Ind}F(K))$, which , if stabilises, also stabilises in $\mathrm{Ind}\mathcal{A}$ because of the conservativity of $\mathrm{Ind}F$. ◻ Recall the following fundamental definition of Lurie: **Definition 29**. An $\infty$-category $\mathcal{C}$ is *stable* if it has all finite limits and colimits, if the initial objects coincides with the final object and if any commutative square in $\mathcal{C}$ is cocartesian if and only if it is Cartesian. An exact functor between stable $\infty$-categories is an $\infty$-functor preserving finite limits and colimits. The homotopy category $\mathrm{ho}(\mathcal{C})$ is always a triangulated category and exact functors induce triangulated functors. We will use the following result of Bunke, Cisinski, Kasprowski, and Winges: **Theorem 30**. *Let $\mathcal{C}$ be a stable $\infty$-category and let $\mathcal{A}_1,\dots,\mathcal{A}_n$ be abelian categories. Denote by $\mathrm{Fun}^{\mathrm{nex}}(\prod_i\mathcal{D}^b(\mathcal{A}_i)\to\mathcal{C})$ the category of $n$-multi-exact functors $\prod_i \mathcal{D}^b(\mathcal{A}_i)\to\mathcal{C}$ that are exact in each variables (meaning that if one fixes one variable then we obtain an $(n-1)$-multi-exact functor). Denote also by $\mathrm{Fun}^{\amalg,\mathrm{nex}}(\prod_i\mathcal{A}_i\to\mathcal{C})$ the category of finite coproduct preserving $n$-multi-exact functors, that is functors $\prod_i\mathcal{A}_i\to\mathcal{C}$ that are finite coproduct preserving and that in each variable $i$ send short exact sequences $0\to x\to y\to z \to 0$ to cocartesian squares $$\begin{tikzcd} f(x) \arrow[r] \arrow[d] & f(y) \arrow[d] \\ 0 \arrow[r] & \arrow[ul, phantom, very near start, "{\ulcorner}"] f(z) \end{tikzcd}$$ (so fixing one variable gives a coproduct preserving $(n-1)$-multi-exact functor).* *Then the restriction to the hearts functor $$\mathrm{Fun}^{\mathrm{nex}}(\prod_i\mathcal{D}^b(\mathcal{A}_i)\to\mathcal{C})\to \mathrm{Fun}^{\amalg,\mathrm{nex}}(\prod_i\mathcal{A}_i\to\mathcal{C})$$ is an equivalence of $\infty$-categories.* *Proof.* We give the proof for $n=2$, the general case being strictly the same except for a too large number of indices. Recall that we have canonical equivalence of $\infty$-categories $\mathrm{Fun}(\mathcal{D}^b(\mathcal{A}_1)\times\mathcal{D}^b(\mathcal{A}_2),\mathcal{C})\simeq \mathrm{Fun}(\mathcal{D}^b(\mathcal{A}_1),\mathrm{Fun}(\mathcal{D}^b(\mathcal{A}_2),\mathcal{C}))$. By definition, the $\infty$-category $\mathrm{Fun}^{\mathrm{2ex}}(\mathcal{D}^b(\mathcal{A}_1)\times\mathcal{D}^b(\mathcal{A}_2)\to\mathcal{C})$ consists exactly of functors that are sent in $\mathrm{Fun}^\mathrm{ex}(\mathcal{D}(\mathcal{A}_1),\mathrm{Fun}^\mathrm{ex}(\mathcal{D}^b(\mathcal{A}_2),\mathcal{C}))$ by that equivalence. By [@bunkeControlledObjectsLeftexact2019 Corollary 7.56] the latter category is exactly $\mathrm{Fun}^{\amalg,\mathrm{ex}}(\mathcal{A}_1,\mathrm{Fun}^{\amalg,\mathrm{ex}}(\mathcal{A}_2,\mathcal{C}))$, which again is equivalent to $\mathrm{Fun}^{\amalg,\mathrm{2ex}}(\mathcal{A}_1\times\mathcal{A}_2\to\mathcal{C})$. This finishes the proof. ◻ We will mostly use the particular case of $n=1$ : **Corollary 31** ([@bunkeControlledObjectsLeftexact2019]). *Let $\mathcal{C}$ be a stable $\infty$-category and let $\mathcal{A}$ be an abelian categories. Then restriction to the heart gives an equivalence of $\infty$-categories $\mathrm{Fun}^\mathrm{ex}(\mathcal{D}^b(\mathcal{A}),\mathcal{C})\to\mathrm{Fun}^{\amalg,\mathrm{ex}}(\mathcal{A},\mathcal{C})$).* Recall the following definition of [@lurieSpectralAlgebraicGeometry Appendix C], that we write in cohomological conventions. **Definition 32**. A *prestable* $\infty$-category $\mathcal{C}$ is an $\infty$-category such that 1. $\mathcal{C}$ is pointed and admits finite colimits. 2. The suspension functor $\Sigma : \mathcal{C}\to\mathcal{C}$ is fully faithful. 3. For every morphism $f:Y\to\Sigma Z$ in $\mathcal{C}$ there exist a pullback square $$\begin{tikzcd} X \arrow[r] \arrow[d] & Y \arrow[d,"f"] \\ 0 \arrow[r] & \arrow[ul, phantom, "{\square}"] \Sigma Z \end{tikzcd}$$ in $\mathcal{C}$ and this square is also a pushout square. Using the Spanier-Whitehead construction ([@lurieSpectralAlgebraicGeometry Proposition C.1.2.2]), one can always see a prestable $\infty$-category as a full subcategory closed under colimits and extension of a stable $\infty$-category. A typical example of prestable $\infty$-category $\mathcal{C}$ is the cohomologically negative part $\mathcal{D}^{\leqslant 0}$ of a $t$-structure on a stable $\infty$-category $\mathcal{D}$. The prestable $\infty$-categories arising that way are exactly those admitting finite limits by [@lurieSpectralAlgebraicGeometry Proposition C.1.2.8]. One construction of such embedding is $\mathcal{C}\to \mathrm{Sp}(\mathcal{C})=\mathcal{C}\otimes\mathrm{Sp}$. Another examples of prestable $\infty$-category are abelian categories. These are the discrete prestable $\infty$-categories: for every $X,Y\in\mathcal{C}$, the mapping space $\mathrm{map}_\mathcal{C}(X,Y)$ is a discrete space. Every prestable $\infty$-category $\mathcal{C}$ gives us an abelian category $\mathcal{C}^\heartsuit$ by taking discrete objects. The last example of prestable $\infty$-categories is that of stable $\infty$-categories. Note that for those, the only discrete object is the zero object. **Definition 33**. A prestable $\infty$-category is called *Grothendieck* if it is presentable and if filtered colimits are left exact in $\mathcal{C}$. Examples of Grothendieck prestable $\infty$-categories are Grothendieck abelian categories $\mathcal{A}$ and the left part $\mathcal{D}(\mathcal{A})^{\leqslant 0}$ of the $t$-structure on the derived $\infty$-category of such Grothendieck abelian categories. **Definition 34**. An object $C$ of a prestable $\infty$-category with finite limits $\mathcal{C}$ is $\infty$*-connective* if when we embed $\mathcal{C}$ in $\mathrm{Sp}(\mathcal{C})$, then $\tau^{\geqslant n}C = 0$ for all $n\in \mathbb{Z}$, where we have denoted by $\tau^{\geqslant n}$ the truncation functors for the natural $t$-structure. A Grothendieck prestable $\infty$-category is *separated* if every $\infty$-connective object is zero. A colimit preserving functor $F:\mathcal{C}\to\mathcal{D}$ between Grothendieck prestable $\infty$-categories is left exact if $\mathrm{Sp}(F)$ is left $t$-exact. We denote by $\mathrm{Groth}_\infty^{\mathrm{sep},\mathrm{lex}}$ the $\infty$-category of separated Grothendieck prestable $\infty$-categories and left exact functors. We also denote by $\mathrm{Groth}_\mathrm{ab}^{\mathrm{lex}}$ the $\infty$-category (actually, the $2$-category) of Grothendieck abelian categories and left exact functors. Thanks to [@lurieSpectralAlgebraicGeometry Theorem C.4.2.1 and Theorem C.5.4.16] the Lurie tensor product on $\mathrm{Pr}^L$ restricts to those $\infty$-categories and induces a structure of symmetric monoidal $\infty$-categories. We have the following theorem of Lurie ([@lurieSpectralAlgebraicGeometry Theorem C.5.4.9]) **Theorem 35**. *There is an adjunction $$\begin{tikzcd} \mathrm{Groth}_{\mathrm{ab}}^\mathrm{lex} \arrow[r, bend left = 25, "\mathcal{D}(-)^{\leqslant 0}"{name=D}] \arrow[r, leftarrow, bend right = 25, swap, "(-)^\heartsuit"{name=C}] \arrow[d, from=D, to=C, phantom, "{\bot}"] & \mathrm{Groth}_{\infty}^\mathrm{lex,sep} \end{tikzcd}$$ with the left adjoint fully faithful and the right adjoint symmetric monoidal.* Therefore, $\mathcal{D}(-)^{\leqslant 0}$ is oplax monoidal as a left adjoint to a symmetric monoidal functor. For $\mathbb{Q}$-linear categories we have a better result. As any prestable $\infty$-categories can always be fully faithfully embedded in a stable $\infty$-category, all homotopy groups of mapping spaces of prestable $\infty$-categories are abelian groups. **Definition 36**. A prestable $\infty$-category is called $\mathbb{Q}$*-linear* if all the homotopy groups of all its mapping spaces are $\mathbb{Q}$-vector spaces. We denote by $\mathrm{Groth}_{\infty,\mathbb{Q}}^{\mathrm{sep},\mathrm{lex}}$ the full subcategory of separated Grothendieck $\mathbb{Q}$-linear prestable $\infty$-categories. **Proposition 37**. *The prestable $\infty$-category $\mathcal{D}(\mathbb{Q})^{\leqslant 0}$ is a separated Grothendieck $\mathbb{Q}$-linear prestable $\infty$-category which is an idempotent algebra of $\mathrm{Groth}_{\infty}^\mathrm{lex,sep}$. Therefore, the $\infty$-category of modules $\mathrm{Mod}_{\mathcal{D}(\mathbb{Q})^{\leqslant 0}}(\mathrm{Groth}_{\infty}^\mathrm{lex,sep})$ is a full subcategory of $\mathrm{Groth}_{\infty,\mathbb{Q}}^\mathrm{lex,sep}$. Moreover, the inclusion functor $$\mathrm{Mod}_{\mathcal{D}(\mathbb{Q})^{\leqslant 0}}(\mathrm{Groth}_{\infty}^\mathrm{lex,sep}) \to \mathrm{Groth}_{\infty,\mathbb{Q}}^\mathrm{lex,sep}$$ is an equivalence of $\infty$-categories.* *Proof.* The $\infty$-category $\mathcal{D}(\mathbb{Q})^{\leqslant 0}$ is idempotent because $\mathcal{D}(\mathbb{Q})^{\leqslant 0}\otimes \mathcal{D}(\mathbb{Q})^{\leqslant 0}\simeq \mathcal{D}(\mathbb{Q}\otimes_\mathbb{S}\mathbb{Q})^{\leqslant 0}$ and $\mathbb{Q}\otimes_\mathbb{S}\mathbb{Q}\simeq \mathbb{Q}$ because $\mathbb{Q}$ is a localisation of $\mathbb{S}$ and the tensor product commutes with localisation. All claims in the proposition are then obvious except for the last. Let $\mathcal{C}\in \mathrm{Groth}_{\infty,\mathbb{Q}}^\mathrm{lex,sep}$. As every presentable stable $\infty$-category is tensored under $\mathrm{Sp}$ and that the action of $\mathrm{Sp}$ on $\mathrm{Sp}(\mathcal{C})$ is always $t$-exact, we obtain an action of $\mathrm{Sp}^{\leqslant 0}$ on $\mathcal{C}$. We've just shown that $\mathrm{Groth}_{\infty}^\mathrm{lex,sep} = \mathrm{Mod}_{\mathrm{Sp}^{\leqslant 0}}(\mathrm{Groth}_{\infty}^\mathrm{lex,sep})$. We have a commutative square of presentable $\infty$-categories : $$\begin{tikzcd} \mathcal{C}\arrow[r] \arrow[d,"F"] & \mathrm{RFun}(\mathcal{C}^\mathrm{op},\mathrm{Sp}^{\leqslant 0}) \arrow[d,"G"] \ar[r] & \mathrm{Fun}(\mathcal{C}^\mathrm{op},\mathrm{Sp}^{\leqslant 0})\ar[d] \\ \mathcal{C}\otimes\mathcal{D}(\mathbb{Q})^{\leqslant 0} \arrow[r] & \mathrm{RFun}(\mathcal{C}^\mathrm{op},\mathcal{D}(\mathbb{Q})^{\leqslant 0}) \ar[r] & \mathrm{Fun}(\mathcal{C}^\mathrm{op},\mathcal{D}(\mathbb{Q})^{\leqslant 0}) \end{tikzcd}$$ in which the first horizontal arrows are isomorphisms by [@lurieHigherAlgebra2022 Proposition 4.8.1.17], the compositions of the horizontal arrows are the Yoneda embeddings (note that our categories are separated hence the spectral Yoneda indeed lands in $\mathrm{Sp}^{\leqslant 0})$, and the vertical arrows are induced by the unique map of algebras $\mathrm{Sp}^{\leqslant 0}\to\mathcal{D}(\mathbb{Q})^{\leqslant 0}$. The commutation of the outer square shows that $\mathrm{Map}_{\mathcal{C}\otimes\mathcal{D}(\mathbb{Q})^{\leqslant 0} }(F(X),F(Y))\simeq \mathrm{Map}_{\mathcal{C}}(X,Y)\otimes_{\mathbb{S}}\mathbb{Q}$. As $\mathcal{C}$ is $\mathbb{Q}$-linear, this gives that $F$ is fully faithful. But the middle vertical map is obviously essentially surjective : if $u : \mathcal{C}^\mathrm{op}\to\mathcal{D}(\mathbb{Q})^{\leqslant 0}$ is limit preserving, then $u = u\otimes_\mathbb{S}\mathbb{Q}= G(u)$ is the image of a limit preserving functor by $G$. Therefore, $\mathcal{C}\simeq \mathcal{C}\otimes \mathcal{D}(\mathbb{Q})^{\leqslant 0}$ is in $\mathrm{Mod}_{\mathcal{D}(\mathbb{Q})^{\leqslant 0}}(\mathrm{Groth}_{\infty}^\mathrm{lex,sep})$. ◻ **Proposition 38**. *The functor $\mathcal{D}(-)^{\leqslant 0} : \mathrm{Groth}_{\mathrm{ab},\mathbb{Q}}^\mathrm{lex}\to\mathrm{Mod}_{\mathcal{D}(\mathbb{Q})^{\leqslant 0}}(\mathrm{Groth}_{\infty}^\mathrm{lex,sep})$ is symmetric monoidal.* *Proof.* A proof of this fact can be found in [@ayoubAnabelianPresentationMotivic2022 Step 3 of Corollary 2.13]. As the functor is already oplax symmetric monoidal, it suffices to check that for $\mathcal{A}$ and $\mathcal{B}$ two $\mathbb{Q}$-linear Grothendieck abelian categories, the natural functor $\mathcal{D}(\mathcal{A})^{\leqslant 0}\otimes \mathcal{D}(\mathcal{B})^{\leqslant 0}\to\mathcal{D}(\mathcal{A}\otimes\mathcal{B})^{\leqslant 0}$ is an equivalence. This follows from the Gabriel-Popescu theorem that enables us to see $\mathcal{A}$ and $\mathcal{B}$ as exact localisation's of $\mathrm{Mod}_A^\heartsuit$ and $\mathrm{Mod}_B^\heartsuit$ for $A$ and $B$ discrete $\mathbb{Q}$-algebras. Then, as $A$ and $B$ are flat over $\mathbb{Q}$, we obtain that $A\otimes_\mathbb{S}B$ is also discrete, hence that $\mathcal{D}(A)^{\leqslant 0}\otimes \mathcal{D}(B)^{\leqslant 0}\simeq \mathcal{D}(A\otimes_{\mathbb{S}}B)^{\leqslant 0}$ and $\mathcal{A}\otimes \mathcal{B}$ is an exact localisation of $\mathrm{Mod}_{A\otimes_{\mathbb{S}}B}$. As the tensor product commutes with localisation's, we see that $\mathcal{D}(\mathcal{A})^{\leqslant 0}\otimes \mathcal{D}(\mathcal{B})^{\leqslant 0}\simeq\mathcal{D}(\mathcal{A}\otimes\mathcal{B})^{\leqslant 0}$. ◻ As tensoring with $\mathrm{Sp}$ is symmetric monoidal, we obtain : **Corollary 39**. *The functor $\mathcal{D}(-):\mathrm{Groth}_{\mathrm{ab},\mathbb{Q}}^\mathrm{lex} \to \mathrm{St}^L$ is symmetric monoidal.* By if $\mathcal{A}$ is a small Noetherian abelian category, $\mathcal{D}^b(\mathcal{A})\to\mathcal{D}(\mathrm{Ind}(\mathcal{A}))$ is fully faithful. The following propositions surely do not have the optimal hypothesis. However, they will be sufficient for our use. **Corollary 40**. *Let $\mathcal{A}$ be a small Noetherian $\mathbb{Q}$-linear abelian category endowed with a symmetric tensor product exact in both variables. Then the bounded derived category $\mathcal{D}^b(\mathcal{A})$ is canonically a symmetric monoidal $\infty$-category. Moreover, any diagram of such abelian categories with exact transitions give rise to a natural diagram in $\mathrm{CAlg}(\mathrm{St})$ the $\infty$-category of symmetric monoidal stable $\infty$-categories.* *Proof.* To avoid too many indices, we only deal with the case of the index diagram being $\Delta^1$, that is the case of an exact symmetric monoidal functor $F$ between two $\mathbb{Q}$-linear Noetherian small abelian categories $\mathcal{A}$ and $\mathcal{B}$. It induces a symmetric monoidal functor $\mathrm{Ind}\mathcal{A}\to\mathrm{Ind}\mathcal{B}$. By , there is a symmetric monoidal functor between symmetric monoidal $\infty$-categories $\mathcal{D}(\mathrm{Ind}\mathcal{A})\to\mathcal{D}(\mathrm{Ind}\mathcal{B})$ induced by $F$. To prove the proposition it suffices to show that $\mathcal{D}^b(\mathcal{A})\subset\mathcal{D}(\mathrm{Ind}\mathcal{A})$ is preserved by the tensor product, and that $F$ sends $\mathcal{D}^b(\mathcal{A})$ to $\mathcal{D}^b(\mathcal{B})$. As $-\otimes-$ is exact in both variables, if $K\in\mathcal{D}^b(\mathcal{A})$, $K\otimes - : \mathcal{D}(\mathrm{Ind}\mathcal{A})\to\mathcal{D}(\mathrm{Ind}\mathcal{A})$ is an exact functor, hence is determined by its restriction to $\mathrm{Ind}\mathcal{A}$ by the universal properties of stabilisation and of $\mathcal{D}(-)^{\leqslant 0}$. The tensor product being $t$-exact, we see that $K\otimes A$ is cohomologically bounded for $A\in\mathrm{Ind}\mathcal{A}$, the computation $\mathrm{H}^n(K\otimes A)=\mathrm{H}^n(K)\otimes A$ implies that if $A$ is in $\mathcal{A}$, then $K\otimes A\in\mathcal{D}^b(\mathcal{A})$ by . Now, this proves that $K\otimes -:\mathcal{A}\to \mathcal{D}(\mathrm{Ind}\mathcal{A})$ factors through $\mathcal{D}^b(\mathcal{A})$. By this implies that the unique extension of this functor from $\mathcal{A}$ to $\mathcal{D}(\mathrm{Ind}\mathcal{A})$ lands in $\mathcal{D}^b(\mathcal{A})$, thus $\mathcal{D}^b(\mathcal{A})$ is preserved by the tensor product. The fact that $F$ sends $\mathcal{D}^b(\mathcal{A})$ to $\mathcal{D}^b(\mathcal{B})$ is similar : as $F$ is $t$-exact it sends bounded objects to bounded objects, and then it will send cohomologically Noetherian objects to cohomologically Noetherian objects because $F:\mathrm{Ind}\mathcal{A}\to\mathrm{Ind}\mathcal{B}$ sends $\mathcal{A}$ to $\mathcal{B}$. ◻ **Proposition 41**. *Let $\mathcal{C}$ be a small $\mathbb{Q}$-linear stable $\infty$-category with a bounded $t$-structure whose heart is a Noetherian abelian category. Assume that $\mathcal{C}$ is symmetric monoidal, that the tensor product is $t$-exact in each variable and that the $t$-structure induced on $\mathrm{Ind}\mathcal{C}$ is left separated. Then the canonical functor $\mathcal{D}^b(\mathcal{C}^\heartsuit)\to\mathcal{C}$ is symmetric monoidal.* *Moreover, any diagram $\mathcal{C}:I\to\mathrm{CAlg}(\mathrm{St})$ of such stable $\infty$-categories with $t$-exact symmetric monoidal transitions induces a diagram $\mathcal{D}: I\to\mathrm{CAlg}(\mathrm{St})$ of such stable $\infty$-categories with $t$-exact symmetric monoidal transitions with $\mathcal{D}(i)=\mathcal{D}^b(\mathcal{C}(i)^\heartsuit)$ and such that there is a natural transformation $\mathrm{real}:\mathcal{D}\Rightarrow \mathcal{C}$ such that $\mathrm{real}(i):\mathcal{D}(i)\to \mathcal{C}(i)$ is the symmetric monoidal canonical functor $\mathcal{D}^b(\mathcal{C}(i)^\heartsuit)\to \mathcal{C}(i)$.* *Proof.* Let $\mathcal{C}$ be a small $\mathbb{Q}$-linear stable $\infty$-category with a bounded $t$-structure whose heart is a Noetherian abelian category, satisfying the assumptions of the proposition. Denote by $\mathcal{A}=\mathcal{C}^\heartsuit$ the heart of the $t$-structure on $\mathcal{C}$. By , $\mathcal{D}^b(\mathcal{A})$ is symmetric monoidal. Moreover, the inclusion functor $\mathcal{D}^b(\mathcal{A})\to\mathcal{D}(\mathrm{Ind}\mathcal{A})$ is symmetric monoidal. Thanks to the assumption on the left separateness of the $t$-structure on $\mathrm{Ind}\mathcal{C}$, because $(\mathrm{Ind}\mathcal{C})^\heartsuit = \mathrm{Ind}(\mathcal{C}^\heartsuit)$, the identity functor of $\mathrm{Ind}\mathcal{A}$ induces the counit map of the adjunction $\mathcal{D}(\mathrm{Ind}\mathcal{A})^{\leqslant 0}\to\mathrm{Ind}\mathcal{C}^{\leqslant 0}$, which is a symmetric monoidal functor because both functors of the adjunction are symmetric monoidal (indeed the counit is a map of $\mathbb{E}_\infty$-algebras). As the $t$-structure on $\mathcal{C}$ is right bounded, the induced $t$-structure on $\mathrm{Ind}\mathcal{C}$ is right complete by [@lurieSpectralAlgebraicGeometry Lemma C.2.4.3] thus we can identify $\mathrm{Ind}\mathcal{C}$ with $\mathrm{Sp}(\mathrm{Ind}\mathcal{C}^{\leqslant 0})$ and obtain a symmetric monoidal functor $$\mathcal{D}(\mathrm{Ind}\mathcal{A})\to\mathrm{Ind}\mathcal{C}$$ which induces the identity on $\mathcal{A}$. The restriction to $\mathcal{D}^b(\mathcal{A})$ is thus by [@bunkeControlledObjectsLeftexact2019 Corollary 7.59] the unique functor $\mathcal{D}^b(\mathcal{A})\to\mathrm{Ind}\mathcal{C}$ such that its restriction to $\mathcal{A}$ preserves finite coproducts and sends shorts exact sequences to cofiber sequences. Therefore it has to be $\mathcal{D}^b(\mathcal{A})\to\mathcal{C}\to\mathrm{Ind}\mathcal{C}$, so that our functor $\mathcal{D}(\mathrm{Ind}\mathcal{A})\to\mathrm{Ind}\mathcal{C}$ sends $\mathcal{D}^b(\mathcal{A})$ to $\mathcal{C}$, hence that the realisation functor $\mathcal{D}^b(\mathcal{A})\to\mathcal{C}$ is symmetric monoidal. It remains to check the functoriality claim. For readability we will stick to $I=\Delta^1$. Let $\mathcal{C}\to\mathcal{D}$ be a symmetric monoidal $t$-exact functor between $\infty$-categories satisfying the assumptions of the proposition. The only thing one has to check is that the square $$\begin{tikzcd} \mathcal{D}^b(\mathcal{C}^\heartsuit)\ar[r] \ar[d] & \mathcal{C}\ar[d] \\ \mathcal{D}^b(\mathcal{D}^\heartsuit)\ar[r] & \mathcal{D}\end{tikzcd}$$ is commutative. This is the case by [@bunkeControlledObjectsLeftexact2019 Corollary 7.59] (that we gave in ), because the restriction of the two possible compositions to $\mathcal{C}^\heartsuit$ coincide. ◻ Let $\mathcal{A}$ be a Grothendieck abelian category. By the main theorem of Hovey's paper [@MR1814077], the $1$-category of unbounded chain complexes $\mathrm{Ch}(\mathcal{A})$ affords the injective model structure whose weak equivalences are the quasi-isomorphisms. Moreover by [@lurieHigherAlgebra2022 Proposition 1.3.5.15], there is a canonical equivalence $N(\mathrm{Ch}(\mathcal{A}))[W_{\mathrm{qiso}}^{-1}]\simeq \mathcal{D}(\mathcal{A})$. Recall that in [@MR3931682 Section 7.5] Cisinski defines $\infty$-categories with weak equivalences and fibrations (*resp*. and cofibrations) and explains how to derive functor between those. We will use his version of Quillen's theorem on adjunctions of model categories : **Theorem 42** (Theorem 7.5.30 in [@MR3931682]). *Let $$\begin{tikzcd} \mathcal{C}\ar[r,shift left=.5ex,"F"] & \mathcal{D}\ar[l,shift left=.5ex,"G"]\end{tikzcd}$$ be an adjunction between $\infty$-categories. We suppose that $\mathcal{C}$ is an $\infty$-category with weak equivalences and fibrations, and that $\mathcal{D}$ is an $\infty$-category with weak equivalences and cofibrations. If $F$ sends weak equivalences between cofibrant objects to weak equivalences and $G$ sends weak equivalences between fibrant objects to weak equivalences, then the the adjunction can be promoted to a canonical adjunction $$\begin{tikzcd} \mathcal{C}[W^{-1}] \ar[r,shift left=.5ex,"\mathcal{L}F"] & \mathcal{D}[W^{-1}] \ar[l,shift left=.5ex,"\mathcal{R}G"]\end{tikzcd}$$ where $\mathcal{L}F$ is a left derived functor of $F$, and $\mathcal{R}G$ a right derived functor of $G$.* **Lemma 43**. *Let $$\label{adjun1}\begin{tikzcd} \mathcal{A}\ar[r,shift left=.5ex,"F"] & \mathcal{B}\ar[l,shift left=.5ex,"G"]\end{tikzcd}$$ be an adjunction between Grothendieck abelian categories. Assume that $F$ is exact, so that $G$ is left exact. Then there is an adjunction $$\begin{tikzcd} \mathcal{D}(\mathcal{A}) \ar[r,shift left=.5ex,"F"] & \mathcal{D}(\mathcal{B}) \ar[l,shift left=.5ex,"\mathcal{R}G"]\end{tikzcd}$$ of exact functor of presentable stable $\infty$-categories, that induces on the hearts the adjunction [\[adjun1\]](#adjun1){reference-type="ref" reference="adjun1"}, such that $F$ is both the left derived functor of its restriction to the heart, and the functor obtained by functoriality of $\mathcal{D}(-)$, and that $\mathcal{R}G$ is a right derived functor of $G$.* *Proof.* Endow $N(\mathrm{Ch}(\mathcal{A}))$ with the structure of category with weak equivalences and cofibrations induced by the model category of Hovey and $N(\mathrm{Ch}(\mathcal{B}))$ with the structure of category with weak equivalences and fibrations induced by that same model structure. Then it is classical that under the hypothesis of this lemma, the pair $(F,G)$ of functors induced between $\mathrm{Ch}(-)$'s satisfies the hypothesis of . The fact that the obtained functor $\mathcal{L}F$ is indeed the functor obtained by functoriality of $\mathcal{D}(-)$ follows from the universal property of the latter. ◻ **Lemma 44**. *Let $F:\mathcal{A}\to\mathcal{C}$ be a functor from an abelian category to a stable $\infty$-category $\mathcal{C}$, such that $F$ is exact and sends short exact sequences to cofiber sequences. Then :* 1. *Assume that $\mathcal{A}$ is Grothendieck abelian, and that $\mathcal{C}$ is a presentable stable $\infty$-category equipped with a left separated $t$-structure such that $F$ lands in the heart. Then both the left and right derived functors of $F :\mathrm{Ch}(\mathcal{A})\to\mathcal{C}$ with respect to the model category of Hovey coincide with the functor $\mathcal{D}(\mathcal{A})\to\mathcal{C}$ of the universal property of $\mathcal{D}$.* 2. *Assume that $\mathcal{A}$ is small and Noetherian, and that $\mathcal{C}$ is small, equipped with a bounded $t$-structure that induces a left separated $t$-structure on $\mathrm{Ind}\mathcal{C}$ and such that $F$ lands in the heart. Then the following functors coincide:* 1. *The restriction to $\mathcal{D}^b(\mathcal{A})$ of the functor $\mathcal{D}(\mathrm{Ind}\mathcal{A})\to\mathrm{Ind}\mathcal{C}$ obtained as in 1. with $Ind F:\mathrm{Ind}\mathcal{A}\to\mathrm{Ind}\mathcal{C}$.* 2. *The derived functor of $F:\mathrm{Ch}^b(\mathcal{A})\to\mathcal{C}$ with respect to the trivial structure of a category with weak equivalence and fibrations (or cofibrations).* 3. *The canonical functor $F:\mathcal{D}^b(\mathcal{A})\to\mathcal{C}$ obtained with .* *Proof.* The proof is easy and only uses that all functors have the same value on the heart, using the universal property of $\mathcal{D}(\mathcal{A})$ and of $\mathcal{D}^b(\mathcal{A})$ in . ◻ We will combine the two previous lemmas with the following classical proposition : **Proposition 45** (Proposition 7.5.29 [@MR3931682]). *Let $F:\mathcal{C}_0\to \mathcal{C}_1$ and $G:\mathcal{C}_1\to\mathcal{C}_2$ be two functors of $\infty$-categories with weak equivalences and fibrations which send weak equivalences between fibrant objects to weak equivalences. Then there is a natural equivalence $\mathcal{R}G \circ \mathcal{R}F \simeq \mathcal{R}(G\circ F)$.* Recall that by [@lurieKerodon [Tag 01F1](https://kerodon.net/tag/01F1)], we have the following proposition : **Proposition 46**. *Let $F:\mathcal{C}\to\mathcal{D}$ be a functor of $\infty$-categories. Assume that for a collection $A\subset\mathcal{C}$ of objects of $\mathcal{C}$ is given the data, for each $a\in A$ of an isomorphism $f_a:F(a)\to d_a$ with $d_a\in\mathcal{D}$. Then there exist a functor $G:\mathcal{C}\to\mathcal{D}$ and a natural equivalence $g:F\Rightarrow G$ such that for every $a\in A$, we have $g_a = f_a$ (and this is really an equality).* *Proof.* Indeed, denote by $\iota : A\to\mathcal{C}$ the canonical monomorphism of simplicial sets. Then by [@lurieKerodon [Tag 01F1](https://kerodon.net/tag/01F1)], the restriction $$\iota^* : \mathrm{Fun}(\mathcal{C},\mathcal{D})\to\mathrm{Fun}(A,\mathcal{D})$$ is an isofibration of simplicial sets (*cf.* [@MR3931682 3.3.15].) The data of the $f_a$ defines an isomorphism $\gamma$ in $\mathrm{Fun}(A,\mathcal{D})$ between $F_{\mid A}$ and the map of simplicial sets $A\to\mathcal{D}$ defined by $a\mapsto d_a$. By the lifting property of isofibrations, there is a functor $G:\mathcal{C}\to\mathcal{D}$ and an isomorphism $F\simeq G$, that is a natural equivalence $g:F\Rightarrow G$ such that $\iota^*g = \gamma$, which is exactly the statement of the proposition. ◻ Finally, we will use the following two theorems : **Theorem 47** (Theorem 3.3.1 of [@MR4093970]). *Let $\mathcal{D}$ be an $\infty$-category which admits finite limits and let $\mathcal{C}$ be an $\infty$-category. Let $G:\mathcal{D}\to\mathcal{C}$ be a functor which preserves finite limits. Then $G$ admits a left adjoint if and only if $\mathrm{ho}(G):\mathrm{ho}(\mathcal{D})\to\mathrm{ho}(\mathcal{C})$ does.* **Theorem 48** (Theorem 7.6.10 of [@MR3931682]). *Let $F:\mathcal{D}\to\mathcal{C}$ be a functor between $\infty$-categories having finite limits. Assume that $F$ preserves finite limits. Then $F$ is an equivalence of $\infty$-categories if and only if $\mathrm{ho}(F):\mathrm{ho}(\mathcal{D})\to\mathrm{ho}(\mathcal{C})$ is an equivalence of categories.* ## Enhancement of the $6$ functors formalism. In this subsection, we will work with perverse Nori motives or polarisable mixed Hodge modules identically. Therefore we denote by $\mathrm{Var}_k$ the category of quasi-projective $k$-schemes when dealing with perverse Nori motives or the category of separated and reduced finite type $\mathbb{C}$-schemes when dealing with mixed Hodge structures. We will call elements of $\mathrm{Var}_k$ varieties or $k$-varieties. For $X\in\mathrm{Var}_k$, we will denote by ${\mathcal{M}_{\mathrm{perv}}}(X)$ the category of perverse Nori motives constructed in [@ivorraFourOperationsPerverse2022] or the category of Saito's polarisable mixed Hodge modules constructed in [@MR1047415]. We will have to use realisation functors hence we denote by $\mathrm{rat} : \mathrm{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))\to \mathrm{D}^b_c(X)$ either the Betti realisation of perverse Nori motives when $k\subset\mathbb{C}$, the underlying $\mathbb{Q}$-structure functor of mixed Hodge modules, or the $\ell$-adic realisation of perverse Nori motives. Here $\mathrm{D}^b_c(X)$ is the derived category of bounded cohomologically constructible sheaves on $X^\mathrm{an}$ in the two first cases, and the derived category of constructible $\ell$-adic étale sheaves on $X$. The triangulated category $\mathrm{D}^b_c(X)$ is endowed with a perverse and a constructible $t$-structure for which $\mathrm{rat}$ is $t$-exact and conservative. Moreover, $\mathrm{D}^b_c(X)$ is embedded in a larger category $\mathrm{D}(X)$, which is the full derived category of sheaves of $\mathbb{Q}$-vector spaces on $X^{\mathrm{an}}$ in the first two cases, and the derived category of proétale sheaves of $\mathbb{Q}_\ell$-modules on $X$ in the second case. We will use the following common properties : **Proposition 49**. *The triangulated $\mathrm{D}^b_c(X)$ and $\mathrm{D}(X)$ admit canonical $\infty$-categorical stable enhancements $\mathcal{D}^b_c(X)$ and $\mathcal{D}(X)$. The constructible $t$-structure on $\mathcal{D}^b_c(X)$ is induced by the canonical $t$-structure on $\mathcal{D}(X)$, and the latter is left complete of heart $\mathrm{Shv}(X)$. The constructible heart $Cons(X)$ of $\mathrm{D}^b_c(X)$ is a Noetherian $\mathbb{Q}$-linear abelian category. The canonical functor $\mathcal{D}(\mathrm{Shv}(X))\to\mathcal{D}(X)$ is an equivalence of $\infty$-categories.* *If $f:Y\to X$ is a morphism in $\mathrm{Var}_k$, then the functors $f^*:\mathcal{D}^b_c(X)\to\mathcal{D}^b_c(Y)$ and $f_* :\mathcal{D}^b(Y)\to\mathcal{D}^b_c(X)$ are induced by functors on $\mathcal{D}(-)$, which are derived functors of their restriction to $\mathrm{Shv}$. Moreover, all the $f^*$ are symmetric monoidal and can be put together to give functors $\mathcal{D}^b_c,\mathcal{D}:\mathrm{Var}_k^\mathrm{op}\to\mathrm{CAlg}(\mathrm{St})$ to the $\infty$-category of stably symmetric monoidal $\infty$-categories and symmetric monoidal exact functors.* *Proof.* We deal with $\mathrm{D}^b_c(X^{\mathrm{an}},\mathbb{Q})$ and $\mathrm{D}^b_c(X_{\acute{e}t},\mathbb{Q}_\ell)$ separately. First the analytic side. The category $\mathcal{D}(X)$ is the derived category $\mathcal{D}(\mathrm{Shv}(X))$ of sheaves of $\mathbb{Q}$-vector spaces on $X^{\mathrm{an}}$. Equivalently, this is the $\infty$-categories of $\mathcal{D}(\mathbb{Q})$-valued hyper-sheaves on $X^\mathrm{an}$. The $\infty$-categorical enhancement of $\mathrm{D}^b_c(X)$ in that case consist of the full subcategory of bounded cohomologically constructible sheaves. This is also $\mathcal{D}^b(Perv(X))$ and $\mathcal{D}^b(Cons(X))$ thanks to the theorems of Beilinson ([@MR0923133]) and of Nori (). Note that these equivalences holds for the enhancements thanks to . The $t$-structure on $\mathcal{D}(X)$ is left complete because as the family of $x^*$ for $x\in X$ is a conservative family of exact functors, this can be checked on a point, where $\mathcal{D}(\mathbb{Q})$ has a left separated $t$-structure by [@lurieHigherAlgebra2022 Proposition 7.1.1.13]. We prove that $Cons(X)$ is Noetherian. The proof is the same as for étale sheaves: let $(\mathcal{F}_i)$ is an inductive system of subsheaves of a given constructible sheaf $\mathcal{F}$. We want to prove that the system is essentially constant hence we may as well assume that $\mathrm{colim}_i\mathcal{F}_i = \mathcal{F}$. For $i\in I$, let $S_i$ the set of points of $X$ for which $(\mathcal{F}_i)_x\to \mathcal{F}_x$ is not surjective. This is easily a constructible subset of $X$. Indeed, if one write $X=\coprod_j X_j$ as a finite disjoint union of locally closed subset on which $\mathcal{F}$ and $\mathcal{F}_i$ are constant, then $S_i\cap X_j = X_j$ or $\emptyset$. As the category of finite dimension vector spaces is Noetherian, for each $x\in X$ there is a $i_x$ such that for all $i\geqslant i_0$ we have $x\not\in S_i$. This means that $\cap_i S_i = \emptyset$. As the constructible topology on $X$ is quasi-compact by [@stacks-project [Tag 0901](https://stacks.math.columbia.edu/tag/0901)], there exists $i_0$ such that for all $i\geqslant i_0$, $S_i=\emptyset$. Therefore, for $i\geqslant i_0$, $\mathcal{F}_i\to\mathcal{F}$ is surjective, hence an isomorphism. We now deal with $\ell$-adic sheaves. For this we use the paper [@MR4609461] of Hemo, Richarz and Scholbach. In this paper they define $\mathcal{D}(X):=\mathcal{D}(X,\mathbb{Q}_\ell)$ as the category of proétale hyper-sheaves of $\mathbb{Q}_\ell$-modules on $X$. Then they define $\mathcal{D}^b_c(X):=\mathcal{D}_\mathrm{cons}(X)$ as the full subcategory of objects which are dualisable on a finite stratification. They prove (together with a missing piece in [@mR4630128 Corollary 2.4]) that the homotopy category of $\mathcal{D}_c^b(X)$ is the usual derived category of étale $\mathbb{Q}_\ell$-adic sheaves. Therefore $\mathcal{D}^b_c(X)$ indeed has a perverse and a constructible $t$-structure. Moreover, the $t$-structure on $\mathcal{D}(X)$ restricts to $\mathcal{D}^b_c(X)$ thanks to [@MR4609461 Theorem 6.2 and Corollary 6.11]. The $t$-structure on $\mathcal{D}(X)$ is left complete because the proétale site is locally weakly contractible. The category $Cons(X)$ is Noetherian. Indeed assume that there is an index category $I$ and an inductive system $(\mathcal{F}_i)_{i\in I}$ of subsheaves of a given constructible $\ell$-adic sheaf $\mathcal{F}$. By definition (or by [@mR4630128 Corollary 2.4] depending of the definition one chooses), there is an inductive system $(\mathcal{G}_i)$ of subsheaves of a $\mathbb{Z}_\ell$-sheaf $\mathcal{G}$ such that $\mathcal{G}[1/\ell]=\mathcal{F}$ and $(\mathcal{G}_i[1/\ell])_i=(\mathcal{F}_i)_i$. To prove that $(\mathcal{F}_i)$ is essentially constant it suffices to check that $(\mathcal{G}_i)$ is essentially constant. As $\mathcal{H}\mapsto \mathcal{H}\otimes_{\mathbb{Z}_\ell}\mathbb{Z}/\ell\mathbb{Z}$ is conservative on constructible sheaves, we can assume that each $\mathcal{G}_i$ and $\mathcal{G}$ are of $\ell$-torsion. We can now apply [@stacks-project [Tag 09YV](https://stacks.math.columbia.edu/tag/09YV)]. The statements about pullbacks and pushforwards are classical and it suffices to check them on the homotopy categories. The last claim follows from the functoriality of $U\mapsto \mathrm{PShv}(\mathcal{X}/U,\mathrm{Sp})$ for $\mathcal{X}$ any $\infty$-category, which gives the functoriality of $\mathcal{D}$, hence of $\mathcal{D}^b_c$ which is stable by tensor products and pullbacks. ◻ **Corollary 50**. *Let $X\in\mathrm{Var}_k$. The category ${\mathcal{M}_{\mathrm{ct}}}(X)$ is Noetherian.* *Proof.* This can be checked after applying the functor $\mathrm{rat}$. ◻ Thanks to , gives : **Proposition 51**. *Let $X\in\mathrm{Var}_k$. The natural functor $$\mathrm{real}_{{\mathcal{M}_{\mathrm{ct}}}(X)} : \mathcal{D}^b({\mathcal{M}_{\mathrm{ct}}}(X))\to \mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))$$ is an equivalence of $\infty$-categories.* We will constantly use the previous proposition without necessarily stating it. **Proposition 52**. *Let $X\in\mathrm{Var}_k$. There exists a conservative exact $t$-exact functor $$\mathcal{R}: \mathcal{D}(\mathrm{Ind}{\mathcal{M}_{\mathrm{ct}}}(X))\to\mathcal{D}(X)$$ that induces a functor $$\mathcal{R}:\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))\to\mathcal{D}^b_c(X)$$ whose underlying homotopy functor is $\mathrm{rat}:\mathrm{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))\to\mathrm{D}^b_c(X)$.* *Proof.* The functor $\mathrm{rat}:{\mathcal{M}_{\mathrm{ct}}}(X)\to Cons(X)$ induces an exact functor $$\mathrm{Ind}{\mathcal{M}_{\mathrm{ct}}}(X)\to\mathcal{D}(X).$$ By the universal property of $\mathcal{D}(\mathrm{Ind}{\mathcal{M}_{\mathrm{ct}}}(X))$ (note that the $t$-structure on $\mathcal{D}(X)$ is left complete), we obtain the desired functor $\mathcal{R}:\mathcal{D}(\mathrm{Ind}{\mathcal{M}_{\mathrm{ct}}}(X))\to\mathcal{D}(X)$. By , the restriction of this functor to $\mathcal{D}^b({\mathcal{M}_{\mathrm{ct}}}(X))$ is the functor $$\ ^\mathrm{ct}\mathcal{R}:\mathcal{D}^b({\mathcal{M}_{\mathrm{ct}}}(X))\to\mathrm{D}^b_c(X)$$ obtained by the universal property . Now consider the functor $\ ^p\mathcal{R}:\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))\to\mathcal{D}^b_c(X)$ obtained by deriving the functor $$\mathrm{rat}:{\mathcal{M}_{\mathrm{perv}}}(X)\to Perv(X)\to\mathcal{D}^b_c(X).$$ This functor is by construction an $\infty$-categorical lift of $\mathrm{rat}$. This implies that the restriction to ${\mathcal{M}_{\mathrm{ct}}}(X)$ of $\ ^p\mathcal{R}$ is $\mathrm{rat}:{\mathcal{M}_{\mathrm{ct}}}(X)\to \mathcal{D}^b_c(X)$, hence by and , we have identifications $^p\mathcal{R}= {\ ^\mathrm{ct}}\mathcal{R}=\mathcal{R}_{\mid \mathcal{D}^b({\mathcal{M}_{\mathrm{ct}}}(X))}$ and thus the proof is finished. ◻ **Corollary 53**. *Let $X\in\mathrm{Var}_k$. The constructible $t$-structure on $\mathrm{Ind}\mathcal{D}^b({\mathcal{M}_{\mathrm{ct}}}(X))$ is left and right separated.* *Proof.* The right separateness follows from the $t$-structure being bounded and by [@lurieSpectralAlgebraicGeometry Proposition C.2.4.3]. The conservative $t$-exact functor $\mathcal{D}^b({\mathcal{M}_{\mathrm{ct}}}(X))\to\mathcal{D}^b_c(X)$ induces a conservative $t$-exact functor $\mathrm{Ind}\mathcal{D}^b({\mathcal{M}_{\mathrm{ct}}}(X))\to\mathrm{Ind}\mathcal{D}^b_c(X)\to\mathcal{D}(X)$. Let $M\in\mathrm{Ind}\mathcal{D}^b({\mathcal{M}_{\mathrm{ct}}}(X))$ be such that for all $n\in\mathbb{Z}$, $\tau^{\leqslant n}M=0$. Then for all $n\in\mathbb{Z}$, the object $\tau^{\leqslant n}\mathcal{R}(M) = 0$ of $\mathcal{D}(X)$ is zero, hence by left completeness of $\mathcal{D}(X)$, $\mathcal{R}(M)=0$, thus $M=0$. ◻ We deal now with the most important result of this section : **Theorem 54**. *There exists a functor $\mathcal{D}_\mathcal{M}^* : \mathrm{Var}_k^\mathrm{op}\to \mathrm{CAlg}(\mathrm{St})$ with values in the category of stably symmetric monoidal $\infty$-categories and symmetric monoidal exact functors and a natural transformation $\mathcal{R}:\mathcal{D}^*_\mathcal{M}\Rightarrow\mathcal{D}^b_c$ such that :* 1. *For each $X\in\mathrm{Var}_k$, the $\infty$-category $\mathcal{D}_\mathcal{M}^*(X)$ is $\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))$. Moreover, the natural transformation $\mathcal{R}$ at $X$ is the realisation of .* 2. *For each $f:Y\to X$ in $\mathrm{Var}_k$, the triangulated functor $\mathrm{ho}(\mathcal{D}^*_\mathcal{M}(f))$ is the pullback functor $f^*_{\mathscr{M}}$ constructed in [@ivorraFourOperationsPerverse2022] or [@MR1047415]. Moreover, the $\infty$-functor $\mathcal{D}^*_\mathcal{M}(f)$ has a left adjoint $f_*$ which gives the pushforward functor $f_*^\mathscr{M}$ constructed in [@ivorraFourOperationsPerverse2022] and [@MR1047415] on the homotopy categories.* 3. *For each $X\in\mathrm{Var}_k$, the symmetric monoidal structure on $\mathcal{D}^*_\mathcal{M}(X)$ induces the symmetric monoidal structure constructed by Terenzi in [@TerenziThesis Chapter 4] or by Saito in [@MR1047415]. Moreover, the symmetric monoidal structure is closed.* *Proof.* Thanks to [@TerenziThesis Theorem 4.4.4] and to the main theorem of [@MR1047415], $X\mapsto \mathrm{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))$ is part of a symmetric monoidal stable homotopic $2$-functor, and the realisation is a symmetric monoidal morphism of symmetric monoidal stable homotopic $2$-functors. As pullbacks and tensor products are $t$-exact for the constructible $t$-structure, this gives us two diagrams of $\mathbb{Q}$-linear symmetric monoidal Noetherian abelian with exact symmetric monoidal transitions ${\mathcal{M}_{\mathrm{ct}}},Cons:\mathrm{Var}_k\to\mathrm{CAlg}(\mathrm{Add})$ and a natural transformation $\mathrm{rat}:{\mathcal{M}_{\mathrm{ct}}}\Rightarrow Cons$. Therefore by , we obtain a natural transformation of functors $\mathrm{Var}_k^\mathrm{op}\to\mathrm{CAlg}(\mathrm{St})$ of the form $\mathcal{D}^b(\mathrm{rat}):\mathcal{D}^b({\mathcal{M}_{\mathrm{ct}}})\Rightarrow\mathcal{D}^b(Cons)$. Using we also obtain a natural transformation $\mathrm{real}:\mathcal{D}^b(Cons)\Rightarrow\mathcal{D}^b_c$. Composing the two gives a natural transformation $\mathcal{R}: \mathcal{D}^b({\mathcal{M}_{\mathrm{ct}}})\Rightarrow \mathcal{D}^b_c$. By construction, for each $X\in\mathrm{Var}_k$, the functor $\mathcal{R}_X:\mathcal{D}^b({\mathcal{M}_{\mathrm{ct}}}(X))\to\mathcal{D}^b_c(X)$ is the one constructed in , hence is a $\infty$-categorical lift of the triangulated functor $\mathrm{rat}$. Fix $f:Y\to X$ in $\mathrm{Var}_k$. We want to show claims 2. and 3. We now apply to the pair of adjoint functors $(f^*,{\ ^{\mathrm{ct}}\mathrm{H}}^0f_*)$ between $\mathrm{Ind}{\mathcal{M}_{\mathrm{ct}}}(X)$ and $\mathrm{Ind}{\mathcal{M}_{\mathrm{ct}}}(Y)$. This gives an adjunction ($\ ^\mathrm{ct}f^*,\ ^\mathrm{ct}\mathcal{R}f_*$) between the $\infty$-categories $\mathcal{D}(\mathrm{Ind}({\mathcal{M}_{\mathrm{ct}}}(X)))$ and $\mathcal{D}(\mathrm{Ind}({\mathcal{M}_{\mathrm{ct}}}(Y)))$. Because of and the definition of the realisation on $\mathcal{D}({\mathcal{M}_{\mathrm{ct}}}(X))$ in as a derived functor, we see that our adjunction ($\ ^\mathrm{ct}f^*,\ ^\mathrm{ct}\mathcal{R}f_*$) commutes with the realisation $\mathcal{R}$. Therefore, as $\mathcal{R}$ is $t$-exact and conservative, we see that both functors preserve the full subcategories $\mathcal{D}^b({\mathcal{M}_{\mathrm{ct}}})$ hence induce an adjunction ($\ ^\mathrm{ct}f^*,\ ^\mathrm{ct}\mathcal{R}f_*$) between $\mathcal{D}^b({\mathcal{M}_{\mathrm{ct}}}(X))$ and $\mathcal{D}^b({\mathcal{M}_{\mathrm{ct}}}(Y))$, compatible with the realisation. We denote by $\ ^\mathrm{ct}f^*$ and $\ ^\mathrm{ct}f_*$ the two functors of this adjunction. By , $\ ^\mathrm{ct}f^*=\mathcal{D}^*_\mathcal{M}(f)$, hence the formation of both $\ ^\mathrm{ct}f^*$ and $\ ^\mathrm{ct}f_*$ is compatible with composition of morphisms in $\mathrm{Var}_k$. We can now prove that the two functors constructed in the previous paragraph agree with the triangulated functors constructed by Ivorra and Morel in [@ivorraFourOperationsPerverse2022] or by Saito in [@MR1047415]. Remark the following facts: In both cases (perverse Nori motives and mixed Hodge modules), if $i$ is a closed immersion, then $i^\mathscr{M}_*$ is the derived functor of its perverse ${\ ^{\mathrm{p}}\mathrm{H}}^0$. For Nori motives this is the definition in [@ivorraFourOperationsPerverse2022 Section 2.5], and for mixed Hodge modules this is [@MR1047415 (4.2.4) and (4.2.10)]. Also, if $f$ is a smooth morphism of relative dimension $d$ then $f^*_\mathscr{M}[d]$ is the derived functor of its perverse ${\ ^{\mathrm{p}}\mathrm{H}}^0$. For Nori motives this is the definition of the pullback along a smooth morphism. For mixed Hodge modules, $f^*_\mathscr{H}$ is defined as follows : one writes $f:Y\to X$ as the composition of the closed immersion $i:Y\to Y\times X$ that sends $y$ to $(y,f(y))$ with the second projection $\pi : Y\times X\to X$, and $f^*_\mathscr{H}= p^*_\mathscr{H}\circ i^*_\mathscr{H}$. Then, $p_\mathscr{H}= \mathbb{Q}_Y\boxtimes (-)$ is the external product with the constant sheaf, hence is a $dg$-functor because the external product is the derived functor of the external product on polarisable mixed Hodge modules in [@MR1047415 Theorem 3.28]. The functor $i^*_\mathscr{H}$ is a $dg$-functor because the same argument of [@ivorraFourOperationsPerverse2022 Remark 4.5] applies to this case because the construction of $i^*_\mathscr{H}$ is exactly the same in the beginning of the proof of [@MR1047415 Proposition 2.19]. Hence $f^*_\mathscr{H}$ also has a $dg$-enhancement, and the $dg$-functor $f^*_\mathscr{H}[d]$ is $t$-exact for the perverse $t$-structure, hence by [@MR2729639 Theorem 1], it is the derived functor of its perverse ${\ ^{\mathrm{p}}\mathrm{H}}^0$. This is mind, assume first that $f=i:Z\to X$ is a closed immersion. Then the restriction of $i_*$ to the perverse heart is exact, hence by it induces an exact functor $\ ^pi_*:\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(Z))\to\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))$. By the previous paragraph, the underlying triangulated functor on the homotopy category of this $\infty$-functor is $i_*^\mathscr{M}$. In particular by , $\ ^pi_*$ has a left adjoint $\ ^pi^*$ whose homotopy functor is $i^*_\mathscr{M}$. This implies that the restriction to ${\mathcal{M}_{\mathrm{ct}}}(Z)$ of that adjoint $\ ^pi^*$ is $(i^*_\mathscr{M})_{\mid {\mathcal{M}_{\mathrm{ct}}}(Z)}$, which is also, by definition, the restriction to ${\mathcal{M}_{\mathrm{ct}}}(Z)$ of $\ ^\mathrm{ct}i^*$. By we obtain that $\ ^\mathrm{ct}i^*\simeq \ ^pi^*$, hence also that $\ ^\mathrm{ct}i_*\simeq \ ^pi_*$, so that the adjunction ($\ ^\mathrm{ct}i^*,\ ^\mathrm{ct}i_*$) induces the adjunction $(i^*_\mathscr{M},i_*^\mathscr{M})$ on the homotopy categories. We can also play the same game when $f$ is a smooth morphism of pure relative dimension $d$. Indeed $f^*[d]$ is perverse $t$-exact, hence it induces, together with a shift, an exact functor $\ ^pf^*:\mathcal{D}^b({\mathcal{M}_{\mathrm{ct}}}(X))\to\mathcal{D}^b({\mathcal{M}_{\mathrm{ct}}}(Y))$, whose homotopy functor is $f^*_\mathscr{M}$. Therefore the restriction to ${\mathcal{M}_{\mathrm{ct}}}(X)$ of $\ ^pf^*$ is the same as the restriction of $\ ^\mathrm{ct}f^*$, hence by they are isomorphic, this $\mathcal{D}^*_\mathcal{M}(f)$ induces the right functor on homotopy categories, and so does its adjoint. As every $f:Y\to X$ can be factored as a closed immersion in a smooth morphism of relative dimension $d$ for some $d$, compatibility of the constructions with composition give the result. We finish with the proof of the third point : Note that in both [@TerenziThesis] and [@MR1047415] the tensor product is built in the following way : there is an external tensor product $\boxtimes : {\mathcal{M}_{\mathrm{perv}}}(X)\times{\mathcal{M}_{\mathrm{perv}}}(X)\to{\mathcal{M}_{\mathrm{perv}}}(X\times X)$ that is exact in both variable hence induces by an $\infty$-functor $\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))\times \mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))\to \mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X\times X))$ which we compose with the pullback $\Delta^*$ under the diagonal $\Delta :X\to X\times X$ to obtain the tensor product. The other structure morphism of the monoidal structure are obtained similarly. Denote by $\otimes^T : \mathcal{D}^*_\mathcal{M}(X)\times\mathcal{D}^*_\mathcal{M}(X)\to\mathcal{D}^*_\mathcal{M}(X)$ Saito's or Terenzi's tensor product and denote by $\otimes^\infty : \mathcal{D}^*_\mathcal{M}(X)\times\mathcal{D}^*_\mathcal{M}(X)\to\mathcal{D}^*_\mathcal{M}(X)$ the one we just constructed. We want to check that $\mathrm{ho}(\otimes^\infty)=\mathrm{ho}(\otimes^T)$. For this there are several isomorphisms to check (see [@MR1712872 Section XI.1]) : 1. A functorial isomorphism $\otimes^T\Rightarrow \otimes^\infty$. 2. Modulo point 1. isomorphism, an identification of the unit isomorphism $\rho:M\otimes 1\to M$ and $\lambda:1\otimes M\to M$. 3. Modulo point 1. isomorphism, an identification of the associativity isomorphism $c : M\otimes(N\otimes P)\to (M\otimes N)\otimes P$. 4. Modulo point 1. isomorphism, an identification of the commutativity isomorphism $\gamma : M\otimes N\to N\otimes M$. By definition, $\otimes^\infty$ coincides with $\otimes^T$ when restricted to ${\mathcal{M}_{\mathrm{ct}}}(X)\times{\mathcal{M}_{\mathrm{ct}}}(X)$. By this gives 1. We explain how to obtain 3. and the other checks will be left to the reader as they are very similar. Denote by $t_1^\infty : \mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))\times\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))\times\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))\to\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))$ the functor sending $M,N,P$ to $M\otimes^\infty (N\otimes^\infty P)$ and by $t_2^\infty : \mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))\times\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))\times\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))\to\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))$ the functor sending $M,N,P$ to $(M\otimes^\infty N)\otimes^\infty P$. The associativity isomorphism is a natural equivalence $c^\infty:t_1^\infty\Rightarrow t_2^\infty$. We give the same definition for $t_1^T$ , $t_2^T$ and $c^T$. Again by definition, the image by the restriction functor of the morphisms $c^\infty$ and $c^T$ are the same, hence up to the equivalence of 1., we obtain 3. ◻ **Corollary 55**. *For each $X\in\mathrm{Var}_k$, the tensor structure on $\mathcal{D}^b_\mathcal{M}(X)$ is closed.* *Proof.* By 3., for each $M\in\mathcal{D}^b_\mathcal{M}(X)$, the functor $$\mathrm{ho}(-\otimes M) : \mathrm{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))\to\mathrm{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))$$ is the tensor product constructed by [@TerenziThesis] or [@MR1047415 Theorem 0.1], which admits a right adjoint. By of Nguyen, Raptis and Schrade, the right adjoint of $\mathrm{ho}(-\otimes M)$ is the homotopy functor of a right adjoint of $-\otimes M:\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))\to\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))$. It must be the internal $\mathrm{Hom}$. ◻ **Proposition 56**. *Let $X$ be a variety. Duality $$\mathbb{D}_X : \mathrm{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))\to \mathrm{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))^{op}$$ is the underlying homotopy functor of an exact functor of stable categories $$\mathcal{D}_X : \mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))\to\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))^{op}.$$ Moreover, we have natural isomorphisms $\mathcal{D}_X\circ\mathcal{D}_X \simeq Id$ and $\mathcal{D}_X\circ \mathcal{R}\simeq \mathcal{R}\circ\mathcal{D}_X^c$, where $\mathcal{D}_X^c$ is duality of $\mathcal{D}^b_c$.* *Proof.* We consider the adjunction $(\mathbb{D}_X,\mathbb{D}_X)$ between ${\mathcal{M}_{\mathrm{perv}}}(X)$ and ${\mathcal{M}_{\mathrm{perv}}}(X)^{op}$. It induces an adjunction between $\mathrm{Ch}^b({\mathcal{M}_{\mathrm{perv}}}(X))$ and $\mathrm{Ch}^b({\mathcal{M}_{\mathrm{perv}}}(X))^{op}$. On both side we put the trivial structure of category with weak equivalences and fibrations. We therefore get an adjunction $(\mathcal{D}_X,\mathcal{D}_X)$ between the bounded derived categories, that induces the adjunction $(\mathbb{D}_X,\mathbb{D}_X)$ on the homotopy categories (because that is how $\mathbb{D}_X$ is defined in [@ivorraFourOperationsPerverse2022 Proposition 2.6] and in [@MR1047415 (4.2.3)].) The unit of the adjunction $Id\to \mathcal{D}_X\circ\mathcal{D}_X$ is invertible when passing to the homotopy category, hence is invertible in the $\infty$-categories. As $\mathrm{ho}(\mathcal{D}_X)$ is an equivalence, by , the $\infty$-functor $\mathcal{D}_X$ is also an equivalence. The compatibility with the realisation follows from the fact that both $\mathcal{D}_X\circ \mathcal{R}$ and $\mathcal{R}\circ\mathcal{D}_X^c$ are equivalent when restricted to the perverse heart, hence we use . ◻ **Notation 57**. *In the following, we will still write $\mathbb{D}_X$ instead of $\mathcal{D}_X$.* **Proposition 58**. *There exists a functor $\mathcal{D}_\mathcal{M}^! : \mathrm{Var}_k^\mathrm{op}\to \mathrm{St}$ with values in the category of stably symmetric monoidal $\infty$-categories and symmetric monoidal exact functors and a natural transformation $\mathcal{R}:\mathcal{D}^*_\mathcal{M}\Rightarrow(\mathcal{D}^b_c)^!$ such that :* 1. *For each $X\in\mathrm{Var}_k$, the $\infty$-category $\mathcal{D}_\mathcal{M}^!(X)$ is $\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))^\mathrm{op}$ and $(\mathcal{D}^b_c)^!(X)$ is $\mathcal{D}^b_c(X)^\mathrm{op}$. Moreover, the natural transformation $\mathcal{R}$ at $X$ is the opposite of the realisation of .* 2. *For each $f:Y\to X$ in $\mathrm{Var}_k$, the triangulated functor $\mathrm{ho}(\mathcal{D}^!_\mathcal{M}(f))$ is the exceptional pullback functor $f^!_{\mathscr{M}}$ constructed in [@ivorraFourOperationsPerverse2022] or [@MR1047415] and $(\mathcal{D}^b_c)^!(f)=f^!:=\mathbb{D}^c_Y\circ f^*\circ\mathbb{D}^c_X$. Moreover, the $\infty$-functor $\mathcal{D}^!_\mathcal{M}(f)$ has a right adjoint $f_!$ which gives the exceptional pushforward functor $f_!^\mathscr{M}$ constructed in [@ivorraFourOperationsPerverse2022] and [@MR1047415] on the homotopy categories.* 3. *There is a commutative square of functors $\mathrm{Var}_k^\mathrm{op}\to \mathrm{St}$: $$\begin{tikzcd} \mathcal{D}_\mathcal{M}^* \arrow[r] \arrow[d] & \mathcal{D}_c^b \arrow[d] \\ \mathcal{D}_\mathcal{M}^! \arrow[r] & (\mathcal{D}_c^b)^! \end{tikzcd}$$ with the horizontal map induced by Verdier duality and the vertical maps induced by the realisation.* *Proof.* By point 3. we have no choice for the definition. To make things rigorous, we use that Verdier duality gives an isomorphism $\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(-))\simeq\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(-))^\mathrm{op}$ of functors $(\mathrm{Var}_k^\mathrm{op})^{\mathrm{disc}}\to \mathrm{St}$ restricted to the discrete category of objects of $\mathrm{Var}_k^\mathrm{op}$, and use to extend this isomorphism to $\mathrm{Var}^\mathrm{op}_k$. All the properties follow, using that the formulas stating that duality swaps stars and shrieks is valid on the homotopy categories. ◻ **Proposition 59**. *For a smooth morphism $f:X\to Y$ of relative dimension $d$, we have a natural isomorphism $f^! \simeq f^*(d)[2d]$.* *Proof.* Tate twists being perverse $t$-exact, by and they are the derived functor of their restriction to the heart. Thus for Nori motives tensoring by $\mathbb{Q}(d)$ is the same functor as the derived functor of the Tate twist obtained by the universal property of perverse Nori motives because this holds both in $\mathcal{D}^b_c$ and in $\mathcal{DM}$. By [@ivorraFourOperationsPerverse2022 Proposition 2.6 (2)], there is a natural isomorphism of functors from ${\mathcal{M}_{\mathrm{perv}}}(Y)$ to ${\mathcal{M}_{\mathrm{perv}}}(Y)^\mathrm{op}$ of the form : $$\varepsilon : \mathbb{D}_X\circ f^*(d)[d] \to f^*[d]\circ \mathbb{D}_Y.$$ All our $\infty$-enhancements of the functors involved above are obtained by deriving the functors on ${\mathcal{M}_{\mathrm{perv}}}$, thus this identity also holds in $\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(Y))$. Shifting by $-d$ gives that $\mathbb{D}_X\circ f^*(d)[2d] \simeq f^*\circ \mathbb{D}_Y$ on $\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))$. In view of this definition of $f^!$ in , we get a natural equivalence $f^! \simeq f^*(d)[2d]$. For polarisable mixed Hodge modules, by [@MR1047415 p 257], they are stable by Tate twists which, being a tensor product with the twisted $\mathbb{Q}_X(d)$ are $dg$-functors and are exact hence are the derived functors of their perverse ${\ ^{\mathrm{p}}\mathrm{H}}^0$ by [@MR2729639] which shows that they have $\infty$-lifts. ◻ **Corollary 60**. *For a smooth morphism $f:Y\to X$ in $\mathrm{Var}_k$, the functor $f^*:\mathcal{D}^b_\mathcal{M}(X)\to\mathcal{D}^b_\mathcal{M}(Y)$ admits a left adjoint $f_\sharp$, compatible with the realisation $\mathcal{R}$.* *Proof.* Let $f:X\to Y$ be a smooth morphism of relative codimension $d$. Then $f_\sharp := f_!(-d)[-2d]$ is left adjoint to $f^!(-d)[-2d] = f^*$. ◻ # The realisation functor. {#Sectionrealisation} ## Extending to all finite type $k$-schemes. {#extension} **Proposition 61**. *Let $\chi : (\mathcal{C}^\mathrm{op})^{\lhd}\to \mathrm{Cat}_\infty$ be a functor. For any $f:c\to c'$ in $\mathcal{C}$, denote $f^*=\chi(f)$, and assume that any such $f^*$ has a right adjoint $f_*$. For each $c\in\mathcal{C}$, denote $f_c:c\to v$ the unique map from $c$ to the end point $v$ (we have that $(\mathcal{C}^\mathrm{op})^\lhd \simeq (\mathcal{C}^\rhd)^\mathrm{op}$.) Let $\overline{\pi}:\overline{\mathcal{D}}\to (\mathcal{C}^\mathrm{op})^{\lhd}$ be the cocartesian fibration that classifiess $\chi$, and let $\pi:\mathcal{D}\to \mathcal{C}^\mathrm{op}$ be its pullback by the inclusion $\mathcal{C}^\mathrm{op}\to(\mathcal{C}^\mathrm{op})^{\lhd}$.\ Then $\chi$ is a limit diagram if and only if the following two conditions are verified :* 1. *The family $(f_c^*)_{c\in\mathcal{C}}$ is conservative.* 2. *For any cocartesian section $X:\mathcal{C}^\mathrm{op}\to \mathcal{D}$ of $\pi$, the limit $X(v):=\lim_{c\in\mathcal{C}}(f_c)_*X(c)$ exists in $\chi(v)$ and the map $f_c^*X(v)\to X(c)$ adjoint to the canonical map $$\lim_c (f_c)_*X(c)\to (f_c)_*X(c)$$ is an equivalence.* *Proof.* By [@lurieKerodon [Construction 02T0](https://kerodon.net/tag/02T0)], there is a diffraction functor $$\mathrm{Df}:\overline{\mathcal{D}}_v\to \mathrm{Fun}_{/\mathcal{C}}^\mathrm{Cocart}(\mathcal{C},\mathcal{D})$$ that fits in a commutative triangle : $$\label{diffraction} \begin{tikzcd} {\overline{\mathcal{D}}_v} && {\mathrm{Fun}_{/\mathcal{C}}^\mathrm{Cocart}(\mathcal{C},\mathcal{D})} \\ & {\mathrm{Fun}_{/\mathcal{C}^\lhd}^\mathrm{Cocart}(\mathcal{C}^\lhd,\overline{\mathcal{D}})} \arrow["{\mathrm{Df}}", from=1-1, to=1-3] \arrow["{\mathrm{ev}_v}", from=2-2, to=1-1] \arrow["{\mathrm{res}}"', from=2-2, to=1-3] \end{tikzcd}.$$ By the diffraction criterion [@lurieKerodon [Theorem 02T8](https://kerodon.net/tag/02T8)], $\chi$ is a limit diagram if and only if the restriction morphism in [\[diffraction\]](#diffraction){reference-type="ref" reference="diffraction"} is an equivalence. In that case by [@lurieKerodon [Remark 02TF](https://kerodon.net/tag/02TF)], the functor $\mathrm{Df}$ is an equivalence. Assume that for any cocartesian section $X:\mathcal{C}^\mathrm{op}\to \mathcal{D}$ of $\pi$, the functor $c\mapsto (f_c)_*X(c)$ admits a limit in $\overline{D}_v$. By the preservation of limits of any left adjoint in $(\mathcal{C}^\mathrm{op})^\lhd$ and [@lurieKerodon [Corollary 0311](https://kerodon.net/tag/0311) and [Corollary 02KY](https://kerodon.net/tag/02KY)], this is equivalent to saying that for any cocartesian section $X:\mathcal{C}^\mathrm{op}\to\mathcal{D}$ of $\pi$, the lifting problem $$\begin{tikzcd} \mathcal{C}^\mathrm{op}& {\overline{\mathcal{D}}} \\ {(\mathcal{C}^\mathrm{op})^\lhd} & {(\mathcal{C}^\mathrm{op})^\lhd} \arrow["X", from=1-1, to=1-2] \arrow["{\overline{\pi}}", from=1-2, to=2-2] \arrow["i",hookrightarrow, from=1-1, to=2-1] \arrow[equal, from=2-1, to=2-2] \arrow["{\overline{X}}", dashed, from=2-1, to=1-2] \end{tikzcd}$$ admits a solution $\overline{X}$ which is a $\overline{\pi}$-limit ([@lurieKerodon [Definition 02KG](https://kerodon.net/tag/02KG)].) By [@lurieKerodon [Example 02ZA](https://kerodon.net/tag/02ZA)], in that case $\overline{X}$ is the right Kan extension functor along the inclusion $i:\mathcal{C}^\mathrm{op}\to(\mathcal{C}^\mathrm{op})^\lhd$. This easily implies that the restriction functor $\mathrm{res}:\mathrm{Fun}_{/\mathcal{C}^\lhd}^\mathrm{Cocart}(\mathcal{C}^\lhd,\overline{\mathcal{D}})\to \mathrm{Fun}_{/\mathcal{C}}^\mathrm{Cocart}(\mathcal{C},\mathcal{D})$ is fully faithful as the left adjoint of the right Kan extension along $i : \mathcal{C}^\mathrm{op}\to(\mathcal{C}^\mathrm{op})^\lhd$. Therefore, assuming point 2. of the proposition, $\chi$ is a limit diagram if and only if the restriction functor is conservative by [@lurieKerodon [Corollary 03UZ](https://kerodon.net/tag/03UZ)]. But as the family of evaluations $\mathrm{ev}_c : \mathrm{Fun}(\mathcal{C}^\mathrm{op},\mathcal{D})\to\mathcal{D}$ is conservative, this is equivalent to the family of functors $\mathrm{ev}_c\circ\mathrm{res}=\mathrm{ev}_c\circ\mathrm{Df}\circ\mathrm{ev}_v = f_c^*$ being conservative, which is point 1. of the proposition. By [\[diffraction\]](#diffraction){reference-type="ref" reference="diffraction"}, if $\chi$ is a limit diagram, $\mathrm{Df}$ is an equivalence which implies that 1. holds. To finish the proof of the proposition, we therefore only have to show that if $\chi$ is a limit diagram, then 2. holds. Let $\overline{X}$ be a cocartesian section of $\overline{\pi}$. Assume first that $\overline{X}$ is a $\overline{\pi}$-limit diagram. By [@lurieKerodon [Example 03ZA](https://kerodon.net/tag/03ZA)] this is equivalent to suppose that $\overline{X}$ is the $\overline{\pi}$-right Kan extension of its restriction $X$ to $\mathcal{C}^\mathrm{op}$. Then by [@lurieKerodon [Corollary 0307](https://kerodon.net/tag/0307)], the limit of $c\mapsto (f_c)_*X(c)$ exists and the fact that $\overline{X}$ is cocartesian is equivalent to the fact that for every $c\in\mathcal{C}$, the canonical map $$f_c^*(\lim_{c'} (f_{c'})_*X(c))\to X(c)$$ is an equivalence. Therefore, to show that 2. holds, we only have to show that any cocartesian section $\overline{X}$ of $\overline{\pi}$ is a $\overline{\pi}$-limit. This is exactly what is proved in the second paragraph of the proof of [@lurieDerivedAlgebraicGeometry2011 Theorem 5.17]. ◻ **Corollary 62**. *Let $\chi,\chi':(\mathcal{C}^\mathrm{op})^{\lhd}\to\mathrm{Cat}_\infty$ be two functors as in , and assume one has a natural transformation $B:\chi\to\chi'$ which commutes with the rights adjoints. Suppose also that $B_v:\chi(v)\to\chi'(v)$ preserves the limits of point 2. in and that each $B_c$ for $c\in \mathcal{C}$ is conservative. Then if $\chi'$ is a limit diagram, so is $\chi$.* **Lemma 63**. *Let $f:Y\to X$ be either a separated étale map or a proper map. Then the cohomological amplitude of the functor $f_*$ on $\mathbb{Q}_\ell$-sheaves and thus perverse Nori motives is less than $2\dim Y+\dim X+1$.* *Proof.* By conservativity of the $\ell$-adic realisation, it suffices to deal with $\mathbb{Q}_\ell$-sheaves. If $X$ is a point and for $\ell$-torsion sheaves this is Artin's [@SGA4 Exposé X Corollaire 4.3]. By proper base change, this gives the result for $f_!$ and $\ell$-torsion sheaves. Now, the functor $\mathrm{D}^b(X,\mathbb{Z}_\ell)\to\mathrm{D}^b(X,\mathbb{Z}/\ell\mathbb{Z})$ is conservative and of cohomological amplitude $1$ (the short exact sequence $0\to \mathbb{Z}_\ell\overset{\times \ell}{\to}\mathbb{Z}_\ell\to\mathbb{Z}/\ell\mathbb{Z}\to 0$ gives that $\mathrm{H}^{-1}(\mathrm{H}^0(M)\otimes \mathbb{F}_\ell)=\mathrm{H}^0(M)[\ell]$, $\mathrm{H}^{0}(\mathrm{H}^{0}(M)\otimes\mathbb{F}_\ell)=\mathrm{H}^{0}(M)/\ell$ and $\mathrm{H}^i(\mathrm{H}^0(M)\otimes\mathbb{F}_\ell)=0$ for $i>0$) , and tensoring by $\mathbb{Q}_\ell$ at most makes the amplitude be smaller. Hence this gives the cohomological amplitude for $f_!$ on $\mathbb{Q}_\ell$ sheaves. If $f$ is proper, $f_!=f_*$ and we have the lemma. If $f$ is étale, we can factor $f$ as an open immersion in a finite étale morphism. The finite morphism is dealt with the proper case, and the pushforward under an open immersion is $t$-exact. ◻ **Theorem 64**. *The functor $X\mapsto {\mathcal{D}^b_\mathcal{M}}(X)$ is a sheaf with respect to the étale and $h$ topologies.* *Proof.* We first deal with the case of perverse Nori motives: The functor $X\mapsto \mathcal{D}_c^b(X,\mathbb{Q}_\ell)$ is isomorphic to the functor $\mathcal{D}_{\mathrm{cons}}(-,\mathbb{Q}_\ell)$ of [@MR4609461 Definition 3.3] by [@MR4609461 Theorem 3.45]. The functor $\mathcal{D}_{\mathrm{cons}}(-,\mathbb{Q}_\ell)$ is an étale sheaf by [@MR4609461 Theorem 3.13]. By [@MR3971240 Corollary 3.3.38], it is also a $h$-sheaf because of localisation and proper base change (see also [@MR4061978 Theorem 2.1.15]). Moreover, the $\ell$-adic realisation $\mathcal{D}^b({\mathcal{M}_{\mathrm{ct}}})\to \mathcal{D}_{\mathrm{cons}}(-,\mathbb{Q}_\ell)$ is conservative. By one just have to check that it preserves the limits in the for $\mathcal{C}$ the Cech nerve of a morphism of schemes $f:Y\to X$. Let $F$ be a cocartesian section as in the proposition, that is, for each $n\in\mathbb{N}$, $$M_n:=F(\underset{n+1\text{-times}}{\underbrace{Y\times_X\cdots Y}})\in \mathcal{D}^b({\mathcal{M}_{\mathrm{ct}}}(X))$$ which verify in particular that for each $p_n$ one of the projections $\underset{n+2\text{-times}}{\underbrace{Y\times_X\cdots Y}} \to \underset{n+1\text{-times}}{\underbrace{Y\times_X\cdots Y}}$, we have $p_n^*M_n \simeq M_{n+1}$. Let $a\in\mathbb{N}$ be an integer such that $M_0\in \mathcal{D}^{[-a,a]}({\mathcal{M}_{\mathrm{ct}}}(Y))$. Then as each pullback is $t$-exact, each $M_n$ is also in $\mathcal{D}^{[-a,a]}({\mathcal{M}_{\mathrm{ct}}}(Y^{n+1}))$. By the cohomological dimension of $f_* :\mathcal{D}^b_c(Y,\mathbb{Q}_\ell)\to\mathcal{D}^b_c(X,\mathbb{Q}_\ell)$ is $\dim X + 2\dim Y+1$ for $f:Y\to X$ separated étale morphism of finite type $k$-schemes. Hence if $f_n : \underset{n+1\text{-times}}{\underbrace{Y\times_X\cdots Y}} \to X$ is the obvious map, it is étale by base change and hence has $\ell$-adic cohomological dimension $d=3\dim X+1$. Therefore we have $(f_n)_*M_n\in \mathcal{D}^{[-a-d,a+d]}({\mathcal{M}_{\mathrm{ct}}}(X))$. But $\mathcal{D}^{[-a-d,a+d]}({\mathcal{M}_{\mathrm{ct}}}(X))$ and $\mathcal{D}_{\mathrm{cons}}^{[-a-d,a+d]}(X,\mathbb{Q}_\ell)$ are (2(a+d)+1)-categories, in which totalisations (simplicial limits) are finite limits so the realisation commutes with them. For polarisable mixed Hodge modules, one has to do the same thing but with the forgetful functor to constructible sheaves, which is an étale sheaf (note that in the analytic side, an étale morphism is a local isomorphism.) ◻ **Corollary 65**. *Let $X$ be a $k$-variety and $M\in{\mathcal{D}^b_\mathcal{M}}(X)$. Then the functor that sends a morphism (*resp.* a smooth morphism) of varieties $f:Y\to X$ to the object $f_*f^*M\in{\mathcal{D}^b_\mathcal{M}}(X)$ (*resp.* $f_\sharp f^*M\in{\mathcal{D}^b_\mathcal{M}}(X)$) is an étale sheaf (*resp.* an étale cosheaf) and a $h$-sheaf.* *Proof.* The fact that $f\mapsto f_*f^*M$ is an étale sheaf is just point 1. of . Duality is an anti-auto-equivalence of categories ${\mathcal{D}^b_\mathcal{M}}(X)\simeq {\mathcal{D}^b_\mathcal{M}}(X)^\mathrm{op}$, hence sends limits to colimits and sends the étale sheaf $f\mapsto f_*f^*\mathbb{D}M$ to the étale cosheaf $f\mapsto f_!f^!M = f_\sharp f^*M$. ◻ **Notation 66**. *For here and until the end of , we will only deal with perverse Nori motives. For a quasi projective variety $X$ we will denote by $\mathcal{DN}_c(X)$ the derived category $\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))$. We will denote by $\mathrm{Var}_k$ the category of quasi-projective varieties over $k$ and by $\mathrm{Sch}_k$ the category of finite type $k$-schemes. We will denote by $\mathcal{R}_\ell$ the $\ell$-adic realisation of perverse Nori motives.* Recall the following fact : **Proposition 67**. *Let $X$ be a finite type $k$-scheme. Restriction induces an equivalence of categories $$\mathrm{Shv}_{\acute{e}t}((\mathrm{Sch}_k)/X,\mathrm{Grpd}_\infty) \to \mathrm{Shv}_{\acute{e}t}((\mathrm{Var}_k)/X,\mathrm{Grpd}_\infty)$$ In particular, we have an equivalence of categories $$\mathrm{Shv}_{\acute{e}t}((\mathrm{Sch}_k)/X,\mathrm{Cat}_\infty) \to \mathrm{Shv}_{\acute{e}t}((\mathrm{Var}_k)/X,\mathrm{Cat}_\infty)$$* *Proof.* The first fact is an easy consequence of [@MR3302973 Lemma C.3]. The particular case follows from the fact that for any site $\mathcal{C}$ with topology $\tau$ and any presentable category $\mathcal{D}$ (*e.g.:* $\mathrm{Cat}_\infty$), we have a natural equivalence $\mathrm{Shv}_\tau(\mathcal{C},\mathrm{Gprd}_\infty)\otimes \mathcal{D}\simeq \mathrm{Shv}_\tau(\mathcal{C},\mathcal{D})$ by [@lurieSpectralAlgebraicGeometry Proposition 1.3.1.7 and Remark 1.3.1.6]. ◻ By , the bounded category of perverse Nori motives satisfies étale descent. By we can do the following definition : **Notation 68**. *The stable *category of Nori motives* over a finite type $k$-scheme is the value at $X\in \mathrm{Sch}_k$ of the sheaf $\mathcal{DN}_c$ on $(\mathrm{Sch}_k)_{\acute{\mathrm{e}}\mathrm{t}}$ extending the functor $\mathcal{DN}_c^* :\mathrm{Var}_k^\mathrm{op}\to\mathrm{CAlg}(\mathrm{Cat}_\infty)$ of .* By definition, $\mathcal{DN}_c(X)$ is a symmetric monoidal stable $\infty$-category and if $f:Y\to X$ is a morphism of finite type $k$-schemes we have a symmetric monoidal functor $f^*:\mathcal{DN}_c(X)\to\mathcal{DN}_c(Y)$. Now, denote by $\mathcal{DN}_c^!$ the expressing the functoriality of exceptional pullbacks. Note that we have taken the convention $\mathcal{DN}_c^!(X)=\mathcal{DN}_c(X)^\mathrm{op}$. Duality is a morphism of presheaves $\mathcal{DN}_c^*\to\mathcal{DN}_c^!$, hence $\mathcal{DN}_c^!$ is an étale sheaf, and extends to all finite-type $k$-schemes : **Notation 69**. *Let $f : Y\to X$ be a morphism of finite type $k$-schemes. Denote by $f^!:\mathcal{DN}_c(X)^\mathrm{op}\to\mathcal{DN}_c(Y)^\mathrm{op}$ the symmetric monoidal functor obtained when extending the sheaf $\mathcal{DN}_c^!$ to finite-type $k$-schemes (note that for $f^!$ to be symmetric monoidal one has to consider the tensor structure on $\mathcal{DN}_c(X)$ given by $M\otimes'N := \mathbb{D}_X(\mathbb{D}_X(M)\otimes\mathbb{D}_X(N))$.) Let also $\mathbb{D}_X : \mathcal{DN}_c(X)\to\mathcal{DN}_c(X)^\mathrm{op}$ be the auto-equivalence of categories extending duality on quasi-projective $k$-schemes. We have a natural equivalence $\mathbb{D}_Yf^*\mathbb{D}_X\simeq f^!$.* Let $f:Y\to X$ be a proper surjective morphism of finite type $k$-schemes. We would like to define a pushforward under this map. Denote by $\mathcal{D}_X$ (*resp.* $\mathcal{D}_Y$) the restriction of $\mathcal{DN}_c^*$ to quasi-projective $k$-schemes over $X$ (*resp.* over $Y$.) Denote by $f_*\mathcal{D}_Y$ the étale sheaf on quasi-projective $k$-schemes over $Y$ such that $f_*\mathcal{D}_Y(W) = \mathcal{D}_Y(W\times_X Y)$. If a pushforward of $f_*$ would exist, then by proper base change, this would define a morphism of presheaves $f_*\mathcal{D}_Y \to \mathcal{D}_X$. The construction will have three steps. The first is to define pushforwards of maps with quasi-projective target using Chow's lemma. The second is to verify that these map indeed form a morphism of presheaves (that is, that proper base change extends in our case.) The third and last step will be to extend this morphism of sheaves to all finite-type schemes by étale descent. Let $f:Y\to X$ be a proper surjective morphism with $X$ quasi-projective. By Chow's lemma, there exists a diagram : $$\begin{tikzcd} {Y'} & Y \\ & X \arrow["f", from=1-2, to=2-2] \arrow["g"', from=1-1, to=2-2] \arrow["\pi", from=1-1, to=1-2] \end{tikzcd}$$ with $\pi : Y'\to Y$ proper and surjective (also known as a $h$-covering), and $g : Y'\to X$ projective. Therefore, $Y'$ is also quasi-projective, and the map $Y'\to Y$ has a pushforward. By $h$-descent, we know that if $f_*$ and $\pi_*$ would exist, denoting by $\pi_i:(Y')^{i/Y}\to Y$ the projections of the $(i+1)$th-fold Cartesian product of $Y'$ over $Y$ , then $$\mathrm{Id}_{\mathcal{DN}_c(Y)}\simeq \lim_\Delta (\pi_i)_*\pi_i^*$$ by . Therefore we would have ($f_*$ is supposed to be a right adjoint hence has to commute with limits) $$f_* \simeq \lim_\Delta (f\pi_i)_*\pi_i^*.$$ If you look carefully the above formula, you see that on the right hand side there are only already existing functors ! Indeed, we have constructed $\pi_i^*$, and $f\pi_i = f\pi p_i$ with $p_i : (Y')^{i/Y}\to Y'$ one of the projections so by stability of projectivity under base change, $\pi_i$ is projective, hence $f\pi_i$ is a map of quasi-projective varieties. Let $M\in\mathcal{DN}_c(Y)$. Let $a\in\mathbb{N}$ such that $\pi^*M \in \mathcal{D}^{[-a,a]}({\mathcal{M}_{\mathrm{ct}}}(Y'))$. By exactitude of pullbacks for all $i\in \Delta$ we have $\pi_i^*M\in\mathcal{D}^{[-a,a]}({\mathcal{M}_{\mathrm{ct}}}((Y')^{i/Y}))$. By and the fact that the base change of a proper birational map is stile proper birational, we have that there exists $\delta\in\mathbb{N}$ (in fact $\delta = 3\dim X+1$) such for all $i\in\Delta$, $(f\pi_i)_*\pi^*M\in\mathcal{D}^{[-a-\delta,a+\delta]}({\mathcal{M}_{\mathrm{ct}}}(X))$. Therefore, the system $((f\pi_i)_*\pi_i^*M)_{i\in\Delta}$ has a limit because it is a finite limit in a $2(a+\delta)$-category. **Notation 70**. *The *pushforward* of $f$ is defined as $$f_* \simeq \lim_\Delta (f\pi_i)_*\pi_i^*.$$ This formula defines a right adjoint to $f$ and hence does not depends one the choice of $Y'$.* *Proof.* Let $M\in \mathcal{DN}_c(Y)$ and $N\in\mathcal{DN}_c(X)$. We have : $$\begin{aligned} \mathrm{Hom}_{\mathcal{DN}_c(Y)}(N,f_*M) & \simeq & \mathrm{Hom}_{\mathcal{DN}_c(X)}(N,\lim_i (f\pi_i)_*\pi_i^*M) \\ & \simeq & \lim_i \mathrm{Hom}_{\mathcal{DN}_c(X)}(N,(f\pi_i)_*\pi_i^*M) \\ & \simeq & \lim_i \mathrm{Hom}_{\mathcal{DN}_c(Y)}((f\pi_i)^*N,\pi_i^*M) \\ & \simeq & \lim_i \mathrm{Hom}_{\mathcal{DN}_c((Y')^{i/Y})}(\pi_i^*(f^*N),\pi_i^*M) \\ & \simeq & \mathrm{Hom}_{\mathcal{DN}_c(Y)}(f^*,M) \end{aligned}$$ by continuity of $\mathrm{Hom}$'s, adjunction for the $(f\pi_i)$, composition of pullbacks and fully faithfulness in the $h$-descent for the morphism $\pi : Y'\to Y$. ◻ **Proposition 71**. *Let $f: Y\to X$ be a surjective proper morphism of finite type $k$-schemes with $X$ quasi projective. Let $h:W\to X$ be a morphism with $W$ quasi-projective. Form the following Cartesian square $$\begin{tikzcd} Z \arrow[r,"f_Z"] \arrow[d,"h_Z",swap] \arrow[dr, phantom, very near start, "{ \lrcorner }"] & W \arrow[d,"h"] \\ Y \arrow[r,"f"] & X \end{tikzcd}.$$ Then the exchange morphism $$h^*f_*\to (f_Z)_*h_Z^*$$ is an isomorphism.* *Proof.* As above by Chow's lemma take a surjective projective morphism $\pi:Y'\to Y$ such that $g=f\pi$ is also projective. Form the double Cartesian square $$\begin{tikzcd} Z' \arrow[r,"\pi_{Z}"] \arrow[d,"h_{Z'}",swap] \arrow[dr, phantom, very near start, "{ \lrcorner }"] & Z \arrow[r,"f_Z"] \arrow[d,"h_Z",swap] \arrow[dr, phantom, very near start, "{ \lrcorner }"] & W \arrow[d,"h"] \\ Y' \arrow[r,"\pi"] & Y \arrow[r,"f"] & X \end{tikzcd}$$ in which by stability of projectivity by pullbacks the maps $\pi$,$\pi_Z$, $f\pi$ and $f_Z\pi_Z$ are projective, hence $Y'$ and $Z'$ are quasi-projective. Denote by $(Y')^{i/Y}$ and $(Z')^{i/Z}$ the $i$-th fold fiber product of $Y'$ and $Z'$ over $Y$ and $Z$. Denote by $g_i : (Y')^{i/Y} \to X$ and $(g_Z)_i:(Z')^{i/Z}\to W$ the composition of the proper map and the projection. Then with obvious notations we have $f_*=\lim_\Delta (f\pi_i)_*\pi^*$ and $(f_Z)_*=\lim_\Delta (f_Z(\pi_Z)_i)_*(\pi_Z)_i^*$. By the proper base change of perverse Nori motives for each $i$ we have that the exchange map $$h^*(g_i)_*\to (g_{Zi})_*h_{Z'}^*$$ is an isomorphism. Taking the limit of these maps give the claimed isomorphism. ◻ Let $f$ be a proper surjective morphism of finite type $k$-schemes. Consider the étale sheaves on the category $(\mathrm{Var}_k)/X$ of quasi projective $k$-schemes over $X$ given by $$f_*\mathcal{D}_Y : W \mapsto \mathcal{DN}_c(Y\times_X W)$$ and $$\mathcal{D}_X : W\mapsto \mathcal{DN}_c(W).$$ By and for each $W\in (\mathrm{Var}_k)/X$ the pushforward by the map $Y\times_X W\to W$ has a pushforward that satisfy proper base change hence by the next defines a morphism of étale sheaves $$\label{flstar} f_* : f_*\mathcal{D}_Y\to\mathcal{D}_X.$$ We can therefore extend this morphism to a morphism of étale sheaves on the category $(\mathrm{Sch}_k)/X$ of finite type $k$ schemes over $X$. **Proposition 72**. *Let $\mathcal{S}$ be a $1$-category with finite products, together with a distinguished family of arrows $P\subset \mathcal{S}$ stable by isomorphisms and fiber product (that is for any $p:Y\to X$ in $P$ and any $f:W\to X$ in $\mathcal{S}$, the map $Y\times_X W\to W$ is also in $P$.) Let $\mathcal{C}$ be a functor $\mathcal{C}: \mathcal{S}^\mathrm{op}\to\mathrm{Cat}_\infty$ such that for any arrow $f$ of $P$, $\mathcal{C}(f)=:f^*$ has a right adjoint $f_*$. Assume that for any Cartesian square $$\begin{tikzcd} Z \arrow[r,"f'"] \arrow[d,"g'"] \arrow[dr, phantom, very near start, "{ \lrcorner }"] & W \arrow[d,"g"] \\ Y \arrow[r,"f"] & X \end{tikzcd}$$ with $f$ in $X$, the canonical natural transformation $g^*\circ f_*\to f'_*\circ (g')^*$ is an equivalence.* *Let $f:Y\to X$ be an arrow of $P$. Then there exists a natural transformation $\alpha:f_*\mathcal{C}_Y\Rightarrow \mathcal{C}_X$ of functors $(\mathcal{S}/X)^\mathrm{op}\to \mathrm{Cat}_\infty$ such that $\mathcal{C}_X(W\overset{g}{\to} W') = \mathcal{C}(W')\overset{g^*}{\to}\mathcal{C}(W)$, $f_*\mathcal{C}_Y(W\overset{g}{\to}W')=\mathcal{C}(Y\times_X W')\to\mathcal{C}(Y\times_X W)$ and the natural transformation is $\alpha_W=(f\times_X W)_* :\mathcal{C}(Y\times_X W)\to\mathcal{C}(W)$.* *Proof.* For each pair $(f:Y\to X,g:W\to X)\in P/X \times \mathcal{S}/X$ we can associate the morphism $W\times_X Y\to W$. This gives a functor $(P/X)^\mathrm{op}\times (\mathcal{S}/X)^\mathrm{op}\to \mathrm{Fun}(\Delta^1,\mathcal{S}^\mathrm{op})$. Composing with $\mathcal{C}$ gives a functors $(P/X)^\mathrm{op}\times (\mathcal{S}/X)^\mathrm{op}\to \mathrm{Fun}(\Delta^1,\mathrm{Cat}_\infty)$. Now recall that there is a natural equivalence $\mathrm{Fun}(A\times B,C)\simeq\mathrm{Fun}(A,\mathrm{Fun}(B,C))$ when $A,B,C \in\mathrm{Cat}_\infty$. Applying this multiple times gives a functors $(\mathcal{S}/X)^\mathrm{op}\times \Delta^1 \to\mathrm{Fun}((P/X)^\mathrm{op},\mathrm{Cat}_\infty)$. But the hypothesis on $\mathcal{C}$ exactly means that this latter functor factors through the category $\mathrm{Fun}^{\mathrm{RAdj}}((P/X)^\mathrm{op},\mathrm{Cat}_\infty)$ of right adjointable functors (see [@lurieHigherAlgebra2022 Definition 4.7.4.16]). By Luries's [@lurieHigherAlgebra2022 Corollary 4.7.4.18], there is a canonical equivalence $\mathrm{Fun}^{\mathrm{RAdj}}((P/X)^\mathrm{op},\mathrm{Cat}_\infty)\simeq \mathrm{Fun}^\mathrm{LAdj}(P/X,\mathrm{Cat}_\infty)$. Composing with the inclusion $\mathrm{Fun}^\mathrm{LAdj}(P/X,\mathrm{Cat}_\infty)\subset \mathrm{Fun}(P/X,\mathrm{Cat}_\infty)$ give a functor $(\mathcal{S}/X)^\mathrm{op}\times \Delta^1\to \mathrm{Fun}(P/X,\mathrm{Cat}_\infty)$ which, after using the $(\times,\mathrm{Fun})$ adjunction again, gives a functor $P/X\to\mathrm{Fun}(\Delta^1,\mathrm{Fun}((\mathcal{S}/X)^\mathrm{op},\mathrm{Cat}_\infty))$. The value at $(f:Y\to X) \in P/X$ is the wanted natural transformation of functors. ◻ **Notation 73**. *Let $f$ be a proper surjective morphism of finite type $k$-schemes. Denote by $f_* : \mathcal{DN}_c(Y)\to\mathcal{DN}_c(X)$ the functor induced on global sections of the morphism extending the morphism [\[flstar\]](#flstar){reference-type="ref" reference="flstar"}.* **Proposition 74**. *Let $f$ be a proper surjective morphism of finite type $k$-schemes. Then $f_*$ of is the right adjoint of $f^*$ and satisfies proper base change.* *Proof.* The fact that $f_*$ satisfies proper base change is only a reformulation of the fact that [\[flstar\]](#flstar){reference-type="ref" reference="flstar"} is a morphism of presheaves. Let $\mathcal{U}$ be the set of inclusions of affine open subsets of $X$. It is a covering sieve of $X$. For $U\in\mathcal{U}$, denote by $f_U:Y\times_X U\to U$ the proper morphism obtained by pullback. Then if $M\in\mathcal{DN}_c(Y)$ we have by definition $$f_* = \lim_{U\in \mathcal{U}}(f_U)_*f_U^*.$$ Therefore the same proof as in the proof of the adjoint property of gives that $f_*$ is the right adjoint of $f^*$. ◻ Let $i:Z\to X$ be a closed immersion between finite type $k$-schemes. Let $j:U\to X$ be the open complement. If $W$ is a quasi-projective $k$-scheme over $X$, then $Z_W = Z\times_X W$ and $U_W = U\times_X W$ are also quasi-projective $k$-schemes as they are locally closed in $W$. Therefore, the map $i_W : Z_W\to W$ admits a pushforward. **Notation 75**. *We let $\mathcal{C}_{Z,U}$ be the presheaf (it is a presheaf by composition of pullbacks and by proper base change for closed immersions, this is a construction very similar that of ) defined on quasi-projective $k$-schemes over $X$ by the formula $$\mathcal{C}_{Z,U}(W) = \begin{tikzcd} \mathcal{DN}_c(Z_W) \arrow[r,"(i_W)_*"] \arrow[d] & \mathcal{DN}_c(W) \arrow[d,"(j_W)^*"] \\ * \arrow[r] & \mathcal{DN}_c(U_W) \end{tikzcd}$$ It has values in the full subcategory $\mathrm{Sq}^{\mathrm{cart}}(\mathrm{Cat}_\infty)$ of $\mathrm{Fun}(\Delta^1\times\Delta^1,\mathrm{Cat}_\infty)$ consisting of Cartesian squares. This is a reformulation of the localisation property for perverse Nori motives.* The category $\mathrm{Sq}^{\mathrm{cart}}(\mathrm{Cat}_\infty)$ is stable by limits. implies that $\mathcal{C}_{U,Z}$ is an étale sheaf. Therefore we can extend it to all finite type $k$-schemes over $X$. **Proposition 76**. *The pushforward $i_*$ of $i:Z\to X$ exists, and we have the localisation property: $$\begin{tikzcd} \mathcal{DN}_c(Z) \arrow[r,"i_*"] \arrow[d] & \mathcal{DN}_c(X) \arrow[d,"j^*"] \\ * \arrow[r] & \mathcal{DN}_c(U) \end{tikzcd}$$ is a Cartesian square. Moreover $i_*$ is right adjoint to $i^*$ and we have an isomorphism $i_*\simeq \mathbb{D}_X\circ i_* \circ \mathbb{D}_Z$.* *Proof.* The fact that $i_*$ is an adjoint can be checked as in . For $W$ a quasi-projective $k$-scheme above $X$, define $i_*\mathcal{D}_Z^*(W) = \mathcal{DN}_c(Z\times_X W)$ , $\mathcal{D}_X^*(X)= \mathcal{DN}_c(X)$ , $i_*\mathcal{D}_Z^!(W) = \mathcal{DN}_c(Z\times_X W)^\mathrm{op}$ and $\mathcal{D}_X^!(W) = \mathcal{DN}_c(W)^\mathrm{op}$. This gives étale sheaves over $(\mathrm{Var}_k)/X$, the first two having the $f^*$ as transitions, and the last two $f^!$. Proper base change induces as in the following commutative diagram of étale sheaves : $$\begin{tikzcd} i_*\mathcal{D}_Z^* \arrow[r,"i_*"] \arrow[d,"\mathbb{D}"] & \mathcal{D}_X^* \arrow[d,"\mathbb{D}"] \\ i_*\mathcal{D}_Z^! \arrow[r,"i_*"] & \mathcal{D}_X^! \end{tikzcd}$$ which expresses the fact that for closed immersions of quasi projective schemes, $i_*\simeq \mathbb{D}_X\circ i_* \circ \mathbb{D}_Z$. Therefore the extension by étale descent also have that property. ◻ The dual of the previous proof gives : **Proposition 77**. *The pushforward $j_*$ of $j:U\to X$ exists, and we have the localisation property: $$\begin{tikzcd} \mathcal{DN}_c(U)^\mathrm{op}\arrow[r,"j_*"] \arrow[d] & \mathcal{DN}_c(X)^\mathrm{op}\arrow[d,"i^!"] \\ * \arrow[r] & \mathcal{DN}_c(U)^\mathrm{op} \end{tikzcd}$$ is a Cartesian square. Moreover $j_*$ is right adjoint to $j^!$ and we have an isomorphism $j^!\simeq \mathbb{D}_U\circ j^!\circ \mathbb{D}_X$ and therefore $j^! \simeq j^*$.* *Proof.* This time one uses the functoriality of exceptional pullbacks : we have a sheaf of Cartesian squares $$\mathcal{C}_{Z,U}^!(W) = \begin{tikzcd} \mathcal{DN}_c^!(U_W) \arrow[r,"(j_W)_*"] \arrow[d] & \mathcal{DN}_c^!(W) \arrow[d,"i_W^!"] \\ * \arrow[r] & \mathcal{DN}_c^!(U_W) \end{tikzcd}.$$ The fact that this is a presheaf is composition of upper shrieks and the base change for $(j_*,f^!)$ which is dual to $(j_!,f^*)$. The fact that $j_*$ is right adjoint to $j^!$ is checked in the same way as and the isomorphism $j^*\simeq j^!$ can be proven as the proof $i_!\simeq i_*$ in . ◻ **Corollary 78**. *Let $i:Z\to X$ be a closed immersion of finite-type $k$-schemes with open complement $j:U\to X$. The pushforwards $i_*$ and $j_*$ exists and are right adjoints to $i^*$ and $j^*$. Moreover, denoting by $j_! =\mathbb{D}_X\circ j_* \circ \mathbb{D}_U$ we obtain a left adjoint to $j^*$. We are in the following situation (see [@MR0751966 section 1.4]):* 1. *The three functors $i_*$, $j_*$ and $j_!$ are fully faithful.* 2. *The three composites $j^*i_*$, $i^!j_*$ and $i^*j_*$ vanish.* 3. *The pairs $(i^*,j^*)$ and $(i^!,j^*)$ are conservative.* 4. *We have two fiber sequences : $$\begin{aligned} j_!j^* \to \mathrm{Id} \to i_*i^* \\ i_*i^! \to \mathrm{Id} \to j_*j^* \end{aligned}$$* *Proof.* The first part of 1. and 2. are implied by , and the second part are given by . The third part is obtained by duality. Parts 3. and 4. are just reformulations of the fact that the diagrams are limits diagrams, using . ◻ **Corollary 79**. *Let $f:Y\to X$ be a proper morphism of finite type $k$-schemes. Then the functor $f^*:\mathcal{DN}_c(X)\to\mathcal{DN}_c(Y)$ has a right adjoint $f_*$.* *Proof.* We can factor $f:Y\to X$ as $Y\overset{p}{\to} f(Y)\overset{i}{\to} X$ with $p$ a surjective closed morphism and $i$ a closed immersion. Therefore as $f^* \simeq p^*i^*$ we can set $f_* = i_*p_*$. ◻ **Corollary 80**. *Let $f:Y\to X$ be a separated morphism of finite type $k$-schemes. Then the functor $f^*:\mathcal{DN}_c(X)\to\mathcal{DN}_c(Y)$ has a right adjoint $f_*$.* *Proof.* By Nagata compactification $f = p\circ j$ with $p$ proper and $j$ an open immersion. ◻ **Notation 81**. *Let $f:Y\to X$ be a separated morphism of finite type $k$-schemes. We denote by $f_! : \mathcal{DN}_c(Y)\to \mathcal{DN}_c(X)$ the functor defined by $f_!\simeq \mathbb{D}_X\circ f_* \circ \mathbb{D}_Y$. By definition this is a left adjoint to $f^!$.* **Notation 82**. *For each quasi-projective $k$-scheme $X$, there is a $\ell$-adic realisation $R_\ell : \mathcal{DN}_c(X)\to \mathcal{D}_{\mathrm{cons}}(X,\mathbb{Q}_\ell)$ compatible with the operations. This implies that we have a morphism of étale sheaves $$R_\ell : \mathcal{DN}_c^*\to \mathcal{D}^*_{\mathrm{cons}}(-,\mathbb{Q}_\ell)$$ compatible with duality, hence that we have an $\ell$-adic realisation for any quasi-projective $k$-schemes which commutes with all $f^*$, duality and hence all $f^!$.* **Proposition 83**. *The $\ell$-adic realisation commutes with pushforwards and exceptional pushforwards.* *Proof.* We only do the pushforward case, the exceptional one follows by compatibility with duality. Let $f:Y\to X$ be a separated morphism of finite type $k$-schemes. We write $f = i\circ p\circ j$ with $j$ an open immersion, $i$ a closed immersion and $p$ a surjective proper morphism. For the open and closed immersion, one can see the $\ell$-adic realisation as a morphism of sheaves on $(\mathrm{Var}_k)/X$ with values in Cartesian squares of $\infty$-categories as in and . Therefore it will commute with $i_*$ and $j_*$. For the surjective proper map $f:Y\to Z$, assume first that $Z$ is quasi-projective. Then the limit defining $f_*$ in is finite limit hence for $M\in \mathcal{DN}_c(Y)$, $$R_\ell(f_*M) = \lim_\Delta R_\ell((f\pi_i)_*\pi_i^*M)$$ and $\mathcal{D}_\mathrm{cons}(-,\mathbb{Q}_\ell)$ also satisfies $h$-descent (this is Deligne's cohomological descent [@MR0498552 Partie 5.]) thus we have $$f_*R_\ell(M) = \lim_\Delta (f\pi_i)_*\pi_i^*R_\ell(M).$$ As $R_\ell$ commutes with operations on quasi-projective schemes we are done. In the general case now, one just has to contemplate the following diagram of étale sheaves on quasi-projective schemes : $$\begin{tikzcd} f_*\mathcal{D}_Y \arrow[r,"R_\ell"] \arrow[d,"f_*"] & f_*\mathcal{D}_{\mathrm{cons}Y}(-,\mathbb{Q}_\ell) \arrow[d,"f_*"] \\ \mathcal{D}_X \arrow[r,"R_\ell"] & \mathcal{D}_{\mathrm{cons}Y}(-,\mathbb{Q}_\ell) \end{tikzcd}$$ with the notations of [\[flstar\]](#flstar){reference-type="ref" reference="flstar"}, that proves the compatibility with general surjective proper maps. ◻ **Proposition 84**. *The $\ell$-adic realisation is conservative.* *Proof.* As $\mathcal{DN}_c(X)$ is stable and $R_\ell$ is an exact functor, it suffices to show that if $R_\ell(M) = 0$ for some $M\in\mathcal{DN}_c(X)$, then $M = 0$. Take a finite Zariski covering of $X$ by affine open subschemes. This gives a Zariski covering $f:U\to X$ with $U$ affine. We have $R_\ell(f^*M) \simeq f^*(R_\ell(M)) = 0$ hence $f^*M = 0$ by conservativity of the realisation on quasi-projective $k$-schemes. By Zariski descent, we hence have $M = 0$. ◻ *Remark 85*. Using conservativity and commutation with the operations of the $\ell$-adic realisation, all the classical results on six functors formalism come for free, because the morphisms can be constructed easily using formal properties of the formalism, and they can be checked to be isomorphisms after applying the realisation. **Corollary 86**. *Let $X$ be a finite type $k$-scheme. Then there exist a constructible $t$-structure on $\mathcal{DN}_c(X)$. Denote by ${\mathcal{M}_{\mathrm{ct}}}(X)$ its heart. Then the natural functor $$\label{realXtf} \mathrm{real}_{{\mathcal{M}_{\mathrm{ct}}}(X)}:\mathcal{D}^b({\mathcal{M}_{\mathrm{ct}}}(X))\to\mathcal{DN}_c(X)$$ is an equivalence of $\infty$-categories.* *Proof.* The existence of the $t$-structure is the same as [@MR1047415 Remark 4.6]. Let $f:Y\to X$ be a morphism. Then by the functor ${\mathcal{M}_{\mathrm{ct}}}(X)\overset{f^*}{\to} {\mathcal{M}_{\mathrm{ct}}}(Y)\to \mathcal{D}^b(Y)$ can be uniquely extended to an exact functor $f^*:\mathcal{D}^b({\mathcal{M}_{\mathrm{ct}}}(X))\to \mathcal{D}^b({\mathcal{M}_{\mathrm{ct}}}(Y))$. Moreover, the restriction to ${\mathcal{M}_{\mathrm{ct}}}(X)$ composition with $\mathrm{real}_{{\mathcal{M}_{\mathrm{ct}}}(Y)}:\mathcal{D}^b({\mathcal{M}_{\mathrm{ct}}}(Y))\to\mathcal{DN}_c(Y)$ is the composition $${\mathcal{M}_{\mathrm{ct}}}(X)\to \mathcal{DN}_c(X)\overset{f^*}{\to}\mathcal{DN}_c(Y)$$ hence by they are equal. We have proven that $\mathrm{real}_{{\mathcal{M}_{\mathrm{ct}}}(-)}:\mathcal{D}^b({\mathcal{M}_{\mathrm{ct}}}(-))\to\mathcal{DN}_c(-)$ is compatible with pullbacks. Let $f:Y\to X$ be a separated étale map. Then $f_*$ is $t$-exact hence we can do the same thing as in the previous paragraph to prove that $\mathrm{real}_{{\mathcal{M}_{\mathrm{ct}}}(-)}:\mathcal{D}^b({\mathcal{M}_{\mathrm{ct}}}(-))\to\mathcal{DN}_c(-)$ is compatible with étale pushforwards. The $\ell$-adic realisation is conservative on ${\mathcal{M}_{\mathrm{ct}}}(X)$ hence also on $\mathcal{D}^b({\mathcal{M}_{\mathrm{ct}}}(X))$. The same proof as then applies, showing that $\mathcal{D}^b({\mathcal{M}_{\mathrm{ct}}}(-))$ has descent along surjective separated étale maps. Let $M,N\in\mathcal{D}^b({\mathcal{M}_{\mathrm{ct}}}(X))$. We therefore have an étale sheaf $$\label{HHom}H_{\mathcal{D}^b({\mathcal{M}_{\mathrm{ct}}})}:Y/X\mapsto \mathrm{Hom}_{\mathcal{D}^b({\mathcal{M}_{\mathrm{ct}}}(Y))}({M}_{\mid {Y}},{N}_{\mid {X}})$$ and a morphism of étale sheaves $H_{\mathcal{D}^b({\mathcal{M}_{\mathrm{ct}}})}\to H_{\mathcal{DN}_c}$ with $$H_{\mathcal{DN}_c}:Y/X\mapsto \mathrm{Hom}_{\mathcal{DN}_c(Y)}({ \mathrm{real}_{{\mathcal{M}_{\mathrm{ct}}}(Y)}(M)}_{\mid {Y}},{ \mathrm{real}_{{\mathcal{M}_{\mathrm{ct}}}(Y)}(N)}_{\mid {X}}).$$ This morphism is an isomorphism on quasi-projective schemes by , hence is an isomorphism for any finite type $k$-scheme. By [@MR0751966 Lemme 3.1.16] this implies that $\mathrm{real}_{{\mathcal{M}_{\mathrm{ct}}}(X)}$ is an equivalence of categories. ◻ **Corollary 87**. *Let $X$ be a separated finite type $k$-scheme. Then there exist a perverse $t$-structure on $\mathcal{DN}_c(X)$. Denote by ${\mathcal{M}_{\mathrm{perv}}}(X)$ its heart. Then the natural functor $$\label{realpXtf} \mathrm{real}_{{\mathcal{M}_{\mathrm{perv}}}(X)}:\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))\to\mathcal{DN}_c(X)$$ is an equivalence of $\infty$-categories. Moreover, ${\mathcal{M}_{\mathrm{perv}}}(X)$ has the universal property of [@ivorraFourOperationsPerverse2022].* *Proof.* The definition of ${\mathcal{M}_{\mathrm{perv}}}(X)$ as in [@ivorraFourOperationsPerverse2022 Definition 2.10] works also with the $\ell$-adic realisation by [@ivorraFourOperationsPerverse2022 Proposition 6.11], hence works for any separated finite type $k$-scheme. Also, pullbacks and pushforwards under an affine étale morphisms exist in the same way as [@ivorraFourOperationsPerverse2022 Proposition 2.5], and the proof of [@ivorraFourOperationsPerverse2022 Proposition 2.7] works in that setting. Hence if $S$ is a separated finite type $k$-scheme, we have that $X\mapsto {\mathcal{M}_{\mathrm{perv}}}(X)$ is an étale sheaf on affine étale $S$-scheme of finite type. Let $X$ be a separated finite type $k$-scheme. Then the perverse $t$-structure exists on $\mathcal{DN}_c(X)$ by gluing with . Denote by $\mathcal{C}(X)$ its heart. Fix a separated finite type $k$-scheme $S$. For any affine scheme $X$ étale and of finite type above $S$, we have $\mathcal{C}(X) \simeq {\mathcal{M}_{\mathrm{perv}}}(X)$. As $X\mapsto \mathcal{C}(X)$ is also an étale sheaf on affine étale $S$-schemes because it is a subcategory of $\mathcal{DN}_c(X)$ and being in the heart can be tested affine étale locally, we obtain that $\mathcal{C}(X)={\mathcal{M}_{\mathrm{perv}}}(X)$ for any scheme $X$ affine étale over $S$. In particular, ${\mathcal{M}_{\mathrm{perv}}}(S)=\mathcal{C}(S)$. As in the proof of , we obtain pushforwards and pullbacks under affine étale maps on $\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))$. Conservativity of the $\ell$-adic realisation then implies that $X\mapsto \mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))$ is an étale sheaf on the category of affine étale $S$-schemes. Therefore, the same proof as for the constructible heart using the sheaves of $\mathrm{Hom}$ [\[HHom\]](#HHom){reference-type="ref" reference="HHom"} implies that the functor $\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(S))\to\mathcal{DN}_c(S)$ is an equivalence. ◻ **Proposition 88**. *Let $f:Y\to X$ be a morphism of finite type $k$-schemes. Then the pullback functor $f^*:\mathcal{DN}_c(X)\to \mathcal{DN}_c(Y)$ has a right adjoint $f_*$.* *Proof.* Let $\pi : U\to Y$ be a Zariski covering by an affine scheme. Both $\pi$ and $f\circ \pi$ are separated morphisms. Let $U^{\bullet}$ be the Cech nerve of $\pi$, for which we denote by $p_n:U^n\to U$ the projection from the the $(n+1)$-th fold $U$ over $Y$ to $U$ and $\pi_n=\pi\circ p_n$. Let $M\in\mathcal{DN}_c(Y)$ of cohomological amplitude $[-a,a]$ for the constructible $t$-structure. Each $p_n$ is separated by base change, and $f\pi$ is also separated hence $f\pi_n$ is separated for each $n$. Therefore by Deligne's theorem on cohomological amplitude, there is a $\delta$ such that all $(f\pi_n)_*$ have cohomological dimension $\delta$. The limit $\lim_{n\in \Delta} (f\pi_n)_*\pi_n^*M$ is therefore a finite limit, and we can set $f_*M$ to be that limit. The fact that this definition does not depend on the choices and gives indeed a right adjoint to $f^*$ can be checked exactly as in . ◻ **Corollary 89**. *Let $f:Y\to X$ be a morphism of finite type $k$-schemes. Then the functor $f^!:\mathcal{DN}_c(X)\to\mathcal{DN}_c(Y)$ admits a left adjoint $f_!$, having the formula $f_! = \mathbb{D}_X\circ f_* \circ\mathbb{D}_Y$.* ## The universal property of Voevodsky motives. {#UnivDAsection} Recall the following construction of Drew and Gallauer in [@MR4560376] : Let $\mathcal{S}$ be a small $1$-category with finite products, and $P\subset \mathcal{S}$ a subcategory containing all isomorphisms, and stable by pullbacks along all morphisms of $\mathcal{S}$. **Definition 90** ([@MR4560376] Definition 2.11). A *pullback formalism* is a functor $C:\mathcal{S}^\mathrm{op}\to\mathrm{CAlg}(\mathrm{Cat}_\infty)$ such that : 1. For all $f:Y\to X$ in $P$, the induced map $f^*:=C(f):C(X)\to C(Y)$ admits a left adjoint $p_\sharp$. 2. for each Cartesian square in $\mathcal{S}$, with $p\in P$, the exchange transformation $p'_\sharp(f')^*\Rightarrow f^*p_\sharp$ is invertible. 3. For $p\in P$, the exchange transformation $p_\sharp(p^*(-)\otimes - )\Rightarrow -\otimes p_\sharp(-)$ is invertible. A morphism of pullbacks formalism is a natural transformation such that the exchange transformation with the lower sharp functors is an equivalence. This defines a subcategory $\mathrm{PB}$ of $\mathrm{Fun}(\mathcal{S}^\mathrm{op},\mathrm{CAlg}(\mathrm{Cat}_\infty))$. Let $\mathcal{L}$ be a set of diagrams $I^\rhd\to\mathcal{S}_\amalg$ in the subcategory $\mathcal{S}_\amalg \subset \mathrm{PSh}(\mathcal{S},\mathrm{Grpd}_\infty)$ generated under coproduct by the representable presheaves satisfying some reasonable conditions (see [@MR4560376 Convention 5.21], these condition will be satisfied for us.) We may consider the full subcategory $\mathrm{PB}_\mathcal{L}$ of pullback formalism that satisfy non effective $\mathcal{L}$-descent, that is such that for all $u:I^{\lhd}\to\mathcal{S}_\amalg$, the functor $$\overline{C}(u(v))\to\lim_{i\in I^\mathrm{op}}\overline{C}(u(i))$$ is fully faithful, where $\overline{C} : \mathrm{PSh}(\mathcal{S},\mathrm{Grpd}_\infty)^\mathrm{op}\to \mathrm{Cat}_\infty$ is the extension by colimits of $C$ along the inclusion $\mathcal{S}^\mathrm{op}\to\mathrm{PSh}(\mathcal{S},\mathrm{Grpd}_\infty)^\mathrm{op}$ and $v$ is the cone point of the diagram. We may also ask for the pullback formalism to pointed, that is such that for each $X\in\mathcal{S}$, $C(X)$ has an object $0_X$ which is both initial and final. A morphism a pointed pullback formalisms is a morphism that sends $0$ to $0$. This defines $\infty$-categories $\mathrm{PB}^{\mathrm{pt}}$ and $\mathrm{PB}^{\mathrm{pt}}_\mathcal{L}$. It is also useful to ask for the pointed pullback formalism to take values in the $\infty$-category $\mathrm{CAlg}(\mathrm{Pr}^\mathrm{L})$ of symmetric monoidal presentable $\infty$-categories. This defines an $\infty$-category $\mathrm{PB}^{\mathrm{pr},\mathrm{pt}}_{\mathcal{L}}$. Let $\pi:(\mathbb{T},t)\to 1_S$ be a pointed object of $P$ with $1_S$ the final object of $S$. Define $\mathrm{PB}^{\mathrm{pr},\mathrm{pt}}_\mathbb{T}$ to be the subcategory of $\mathrm{PB}^{\mathrm{pr},\mathrm{pt}}$ of pullback formalisms such that for all $X\in\mathcal{S}$, $(\pi_X)_\sharp(\pi_X)^*1_{C(X)}$ is invertible for the tensor product, where $\pi_X$ is the pullback of $\pi$ to $X$. There is a distinguished object in $\mathrm{PB}^{\mathrm{pr},\mathrm{pt}}_{\mathcal{L},\mathbb{T}}$: **Construction 91** ([@MR4560376] Construction 7.2). Let $X\in \mathcal{S}$. Follow the following recipe : 1. Take $P_X$ to be the category $P/X$ of arrows in $P$ that maps to $X$. Endow it with the symmetric monoidal structure induced by the fiber product over $X$. 2. Take the category of presheaves $\mathrm{PSh}(P_X,\mathrm{Grpd}_\infty)$ on it, and give it the symmetric monoidal structure called the Day convolution (see [@lurieHigherAlgebra2022 Example 2.2.6.17]). 3. Restrict to the full subcategory $\mathrm{PSh}(P_X,\mathrm{Grpd}_\infty)_\mathcal{L}$ of objects that satisfy $\mathcal{L}$-descent. We have a localisation functor $\mathrm{PSh}(P_X,\mathrm{Grpd}_\infty)\to \mathrm{PSh}(P_X,\mathrm{Grpd}_\infty)_\mathcal{L}$ which is symmetric monoidal. 4. Take pointed objects in $\mathrm{PSh}(P_X,\mathrm{Grpd}_\infty)_\mathcal{L}$ and make the functor $$\mathrm{PSh}(P_X,\mathrm{Grpd}_\infty)_\mathcal{L}\to \mathrm{PSh}(P_X,\mathrm{Grpd}_\infty)_\mathcal{L}^*$$ monoidal. 5. $\otimes$-invert $\mathbb{T}_X$ : set $(\mathrm{PSh}(P_X,\mathrm{Grpd}_\infty)_\mathcal{L}^*)[\mathbb{T}_X^{-1}]$ by taking $\mathbb{T}$-spectra. The functor $$\mathrm{PSh}(P_X,\mathrm{Grpd}_\infty)_\mathcal{L}^*\to(\mathrm{PSh}(P_X,\mathrm{Grpd}_\infty)_\mathcal{L}^*)[\mathbb{T}_X^{-1}]$$ is symmetric monoidal. Drew and Gallauer show that this construction can be made functorial in $X$, and that all steps are actually witness of localisation's of the $\infty$-categories of pullback formalisms. We obtain an object $\mathrm{L}_{\mathbb{T}}\mathrm{L}_{\mathrm{pt}}\mathrm{L}_\mathcal{L}\widehat{C_{\mathrm{gm}}}\in\mathrm{PB}^{\mathrm{pr},\mathrm{pt}}_{\mathcal{L},\mathbb{T}}$, with $\widehat{C_{\mathrm{gm}}}$ the functor $$\widehat{C_{\mathrm{gm}}}:\mathcal{S}^\mathrm{op}\to \mathrm{CAlg}(\mathrm{Cat}_\infty^{\mathrm{pr}})$$ defined to be the composite of $\mathcal{C}\mapsto \mathrm{PSh}(\mathcal{C},\mathrm{Grpd}_\infty)$ with the functor $\mathcal{S}^{op}\to \mathrm{CAlg}(\mathrm{Cat}_\infty)$ sending $X$ to $P_X$. We have $(\mathrm{PSh}(P_X,\mathrm{Grpd}_\infty)_\mathcal{L}^*)[\mathbb{T}_X^{-1}]=\mathrm{L}_{\mathbb{T}}\mathrm{L}_{\mathrm{pt}}\mathrm{L}_\mathcal{L}\widehat{C_{\mathrm{gm}}}(X)$. Their main theorem is the following : **Theorem 92** ([@MR4560376] Theorem 7.3). *$\mathrm{L}_{\mathbb{T}}\mathrm{L}_{\mathrm{pt}}\mathrm{L}_\mathcal{L}\widehat{C_{\mathrm{gm}}}$ is the initial object of $\mathrm{PB}^{\mathrm{pr},\mathrm{pt}}_{\mathcal{L},\mathbb{T}}$.* We will consider some further localisation's of $\mathrm{L}_{\mathbb{T}}\mathrm{L}_{\mathrm{pt}}\mathrm{L}_\mathcal{L}\widehat{C_{\mathrm{gm}}}$. **Definition 93**. A pullback formalism $C\in \mathrm{PB}^{\mathrm{pr},\mathrm{pt}}_{\mathcal{L},\mathbb{T}}$ is called *stable* if it takes values in *stable* symmetric monoidal categories and exact functors. We denote the subcategory of such pullback formalisms by $\mathrm{PB}^{\mathrm{pr},\mathrm{st}}_{\mathcal{L},\mathbb{T}}$. **Proposition 94**. *$\mathrm{PB}^{\mathrm{pr},\mathrm{st}}_{\mathcal{L},\mathbb{T}}\subset \mathrm{PB}^{\mathrm{pr},\mathrm{pt}}_{\mathcal{L},\mathbb{T}}$ is a reflexive full subcategory which is presentable. This means that there is a localisation functor $$\mathrm{L}_{\mathrm{st}} : \mathrm{PB}^{\mathrm{pr},\mathrm{pt}}_{\mathcal{L},\mathbb{T}} \to \mathrm{PB}^{\mathrm{pr},\mathrm{st}}_{\mathcal{L},\mathbb{T}}$$ which is left adjoint to the inclusion.* *Proof.* By [@lurieHigherAlgebra2022 Proposition 4.8.2.18], the subcategory $\mathrm{St}\subset \mathrm{Pr}^\mathrm{L}$ of stable presentable $\infty$-categories is a reflexive subcategory. The adjoint is given by tensorisation by $\mathrm{Sp}$ the $\infty$-category of spectra. Therefore the inclusion $\mathrm{Fun}(\mathcal{S}^\mathrm{op},\mathrm{St}) \to \mathrm{Fun}(\mathcal{S}^\mathrm{op},\mathrm{Pr}^\mathrm{L})$ is also a reflexive subcategory, the adjoint begin given by post-composition with the left adjoint of $\mathrm{St}\subset \mathrm{Pr}^\mathrm{L}$. Therefore, we have a pullback diagram in $\mathrm{Pr}^\mathrm{L}$ :$$\begin{tikzcd} \mathrm{PB}^{\mathrm{pr},\mathrm{st}}_{\mathcal{L},\mathbb{T}} \arrow[dr, phantom, very near start, "{\lrcorner}"] \arrow[r] \arrow[d] & \mathrm{PB}^{\mathrm{pr},\mathrm{pt}}_{\mathcal{L},\mathbb{T}} \arrow[d] \\ \mathrm{Fun}(\mathcal{S}^\mathrm{op},\mathrm{St}) \arrow[r] & \mathrm{Fun}(\mathcal{S}^\mathrm{op},\mathrm{Pr}^\mathrm{L}) \end{tikzcd}$$ which implies that that the inclusion $\mathrm{PB}^{\mathrm{pr},\mathrm{st}}_{\mathcal{L},\mathbb{T}}\subset \mathrm{PB}^{\mathrm{pr},\mathrm{pt}}_{\mathcal{L},\mathbb{T}}$ has a left adjoint. ◻ **Corollary 95**. *$\mathrm{L}_{\mathrm{st}}\mathrm{L}_{\mathbb{T}}\mathrm{L}_{\mathrm{pt}}\mathrm{L}_\mathcal{L}\widehat{C_{\mathrm{gm}}}$ is the initial object of $\mathrm{PB}^{\mathrm{pr},\mathrm{st}}_{\mathcal{L},\mathbb{T}}$.* *Proof.* Let $C\in \mathrm{PB}^{\mathrm{pr},\mathrm{st}}_{\mathcal{L},\mathbb{T}}$. Denote by $I=\mathrm{L}_{\mathbb{T}}\mathrm{L}_{\mathrm{pt}}\mathrm{L}_\mathcal{L}\widehat{C_{\mathrm{gm}}}$ the initial object of $\mathrm{PB}^{\mathrm{pr},\mathrm{pt}}_{\mathcal{L},\mathbb{T}}$. The pullback formalism $\mathrm{L}_{\mathrm{st}}(I)=\mathrm{L}_{\mathrm{st}}\mathrm{L}_{\mathbb{T}}\mathrm{L}_{\mathrm{pt}}\mathrm{L}_\mathcal{L}\widehat{C_{\mathrm{gm}}}$ is in the category $\mathrm{PB}^{\mathrm{pr},\mathrm{st}}_{\mathcal{L},\mathbb{T}}$ and we have $$\mathrm{Hom}_{\mathrm{PB}^{\mathrm{pr},\mathrm{st}}_{\mathcal{L},\mathbb{T}}}(\mathrm{L}_{\mathrm{st}}(I),C)\simeq \mathrm{Hom}_{\mathrm{PB}^{\mathrm{pr},\mathrm{pt}}_{\mathcal{L},\mathbb{T}}}(I,C)=*$$ ensuring the claim. ◻ The following lemma is well known. **Lemma 96**. *A stable presentable $\infty$-category $D$ is in the essential image of the fully faithful embedding $\mathrm{Mod}_{\mathrm{Mod}_\mathbb{Q}}(\mathrm{St})\overset{\iota}{\to}\mathrm{St}$ if and only if for all $d\in D$ and all $n\in\mathbb{N}^*$, the multiplication $d\overset{\times n}{\to} d$ is invertible.* *Proof.* The proof is exactly the same as the proof of , except that one has to replace $\mathrm{Sp}^{\leqslant 0}$ by $\mathrm{Sp}$. The map $\iota$ is fully faithful. Indeed, by [@lurieHigherAlgebra2022 Section 4.8.4], ◻ **Definition 97**. Let $C\in \mathrm{PB}^{\mathrm{pr},\mathrm{st}}_{\mathcal{L},\mathbb{T}}$. We say that $C$ is $\mathbb{Q}$*-linear* if it takes values in $\mathrm{Mod}_{\mathrm{Mod}_\mathbb{Q}}(\mathrm{St})$. We denote by $\mathrm{PB}^{\mathrm{pr},\mathrm{st}}_{\mathcal{L},\mathbb{T},\mathbb{Q}}$ the category of such $C$. **Proposition 98**. *The $\mathrm{PB}^{\mathrm{pr},\mathrm{st}}_{\mathcal{L},\mathbb{T},\mathbb{Q}}\subset \mathrm{PB}^{\mathrm{pr},\mathrm{st}}_{\mathcal{L},\mathbb{T}}$ is a reflexive full subcategory which is presentable.* *Proof.* We have a pullback square $$\begin{tikzcd} \mathrm{PB}^{\mathrm{pr},\mathrm{st}}_{\mathcal{L},\mathbb{T},\mathbb{Q}} \arrow[dr, phantom, very near start, "{\lrcorner}"] \arrow[r] \arrow[d] & \mathrm{PB}^{\mathrm{pr},\mathrm{st}}_{\mathcal{L},\mathbb{T}} \arrow[d] \\ \mathrm{Fun}(\mathcal{S}^\mathrm{op},\mathrm{Mod}_{\mathrm{Mod}_\mathbb{Q}}(\mathrm{Pr}^\mathrm{L})) \arrow[r] & \mathrm{Fun}(\mathcal{S}^\mathrm{op},\mathrm{Pr}^\mathrm{L}) \end{tikzcd}$$ which implies the result because the bottom functor admits a left adjoint by and the right functor admits a left adjoint by [@MR4560376 Proposition 4.4 and Theorem 7.13]. ◻ **Corollary 99**. *The object $\mathrm{L}_{\mathbb{Q}}\mathrm{L}_{\mathrm{st}}\mathrm{L}_{\mathbb{T}}\mathrm{L}_{\mathrm{pt}}\mathrm{L}_\mathcal{L}\widehat{C_{\mathrm{gm}}}$ is the initial object of $\mathrm{PB}^{\mathrm{pr},\mathrm{st}}_{\mathcal{L},\mathbb{T},\mathbb{Q}}$.* We apply it with the following data, similar to [@MR4560376 Convention 7.10] : $\mathcal{S}= \mathrm{Sch}_k$ the category of finite type $k$-schemes when dealing with Nori motives, and the category of separated finite type $\mathbb{C}$-schemes when dealing with mixed Hodge modules. $P\subset \mathcal{S}$ is the set of smooth morphisms, $\mathcal{L}=(\acute{\mathrm{e}}\mathrm{t},\mathbb{A}^1_k)$ is union of the set of étale hyper-covers in $\mathrm{Var}_k$ with the set of morphisms of the form $\mathbb{A}^1_X\to X$ for $X\in\mathrm{Var}_k$, and $\mathbb{T} = (\mathbb{P}^1_k,\infty_k)$ is the pointed projective line. **Theorem 100** (Drew-Gallauer). *The initial object of $\mathrm{PB}^{\mathrm{pr},\mathrm{st}}_{(\acute{\mathrm{e}}\mathrm{t},\mathbb{A}^1),(\mathbb{P}^1,\infty),\mathbb{Q}}$ is $\mathcal{DM}(-,\mathbb{Q})$.* *Proof.* By , the initial object is $\mathrm{L}_{\mathbb{Q}}\mathrm{L}_{\mathrm{st}}\mathrm{L}_{(\mathbb{P}^1,\infty)}\mathrm{L}_{\mathrm{pt}}\mathrm{L}_{(\acute{\mathrm{e}}\mathrm{t},\mathbb{A}^1)}\widehat{C_{\mathrm{gm}}}$. Note that $$\mathrm{L}_{(\mathbb{P}^1,\infty)}\mathrm{L}_{\mathrm{pt}}\mathrm{L}_{(\acute{\mathrm{e}}\mathrm{t},\mathbb{A}^1)}\widehat{C_{\mathrm{gm}}} \simeq \mathcal{SH}_{\acute{\mathrm{e}}\mathrm{t}}(-)$$ is already stable (see [@MR4224739 Section 5].) Therefore $\mathrm{L}_{\mathrm{st}}\mathcal{SH}_{\acute{\mathrm{e}}\mathrm{t}} = \mathcal{SH}_{\acute{\mathrm{e}}\mathrm{t}}$. The $\mathbb{Q}$-linear version of $\mathcal{SH}_{\acute{\mathrm{e}}\mathrm{t}}$ is $\mathcal{DM}=\mathcal{DM}^{\acute{\mathrm{e}}\mathrm{t}}(-,\mathbb{Q})$ as it follows by definition (see also [@preisMotivicNearbyCycles2023 Section 1]). ◻ ## Construction of the realisation. {#realisationSection} **Notation 101**. *For $X$ a separated non reduced finite type $k$-scheme we define $\mathcal{D}^b(\mathrm{MHM}^p(X)):=\mathcal{D}^b(\mathrm{MHM}^p(X_{\mathrm{red}}))$. By localisation for mixed Hodge structure this is well defined.* **Theorem 102**. *The functor $\mathrm{Ind}{\mathcal{D}^b_\mathcal{M}}:\mathrm{Sch}_k^{op}\to \mathrm{Cat}_{\infty}$ is in $\mathrm{PB}^{\mathrm{pr},\mathrm{st}}_{(\acute{\mathrm{e}}\mathrm{t},\mathbb{A}^1),(\mathbb{P}^1,\infty),\mathbb{Q}}$.* *Proof.* By the functor ${\mathcal{D}^b_\mathcal{M}}: \mathrm{Sch}_k^{op}\to \mathrm{Cat}_{\infty}$ takes values in the category of symmetric monoidal small $\infty$-categories. Therefore by [@lurieHigherAlgebra2022 4.8.1.14], $\mathrm{Ind}{\mathcal{D}^b_\mathcal{M}}$ is a functor that takes values in symmetric monoidal presentable categories. Moreover, for each $X\in \mathrm{Var}_k$, ${\mathcal{D}^b_\mathcal{M}}(X)$ is stable, hence $\mathrm{Ind}{\mathcal{D}^b_\mathcal{M}}$ is also stable by [@lurieHigherAlgebra2022 Proposition 1.1.3.6][^1]. By , for each smooth morphism $f:Y\to X$ in $\mathrm{Var}_k$, the functor $f^*:{\mathcal{D}^b_\mathcal{M}}(X)\to{\mathcal{D}^b_\mathcal{M}}(Y)$ has a left adjoint $f_\sharp$. Adjunctions remains after indization, hence we have for now $\mathrm{Ind}{\mathcal{D}^b_\mathcal{M}}\in \mathrm{PB}^{\mathrm{pr,st}}$. By , ${\mathcal{D}^b_\mathcal{M}}$ satisfy effective étale descent. We will show that $\mathrm{Ind}{\mathcal{D}^b_\mathcal{M}}$ satisfy non effective étale hyper-descent, that is, for $A,B\in\mathrm{Ind}{\mathcal{D}^b_\mathcal{M}}(X)$, the presheaf of complexes of $\mathbb{Q}$-vector spaces on the small étale site on $X$ defined by $U\mapsto\mathcal{F}(U):=\mathrm{Map}_{\mathrm{Ind}{\mathcal{D}^b_\mathcal{M}}(U)}({A}_{\mid {U}},{B}_{\mid {U}})$ is an étale hyper-sheaf. For this we use [@MR3971240 Theorem 3.3.23] that asserts that such a presheaf of complexes of $\mathbb{Q}$-vector spaces is an étale hyper-sheaf if and only if it is a Nisnevich hyper-sheaf and it satisfies $\mathcal{F}(U)\simeq\mathcal{F}(V)^G$ for any Galois cover $V\to U$ of group $G$ (note that in [@MR3971240 Definition 3.2.5] they write "$\tau$-descent" for $\tau$-hyper-descent.) By Clausen and Matthew's [@MR4296353 Corollary 3.27], the Nisnevich site of a finite dimensional scheme is hyper-complete, hence a Nisnevich sheaf is automatically hyper-complete. We will therefore verify Nisnevich excision [@MR1813224 Section 3,Proposition 1.4] and Galois descent. Let $A$ and $B$ be objects of $\mathrm{Ind}{\mathcal{D}^b_\mathcal{M}}(X)$. Let $$\begin{tikzcd} E \arrow[r] \arrow[d] & V \arrow[d] \\ U \arrow[r] & X \end{tikzcd}$$ be a Nisnevich square. Denote by $s:\square\to \mathrm{Var}_k$ the corresponding diagram. We have to show that the following map $$\label{eqindnori1} \mathrm{Map}_{\mathrm{Ind}{\mathcal{D}^b_\mathcal{M}}(X)}(A,B)\to \lim_{i\in\ulcorner^\mathrm{op}}\mathrm{Map}_{\mathrm{Ind}{\mathcal{D}^b_\mathcal{M}}(s(i))}(\mathrm{Ind}(f_i^*)A,\mathrm{Ind}(f_i^*)B)$$ with $f_i : s(i)\to s(v)=X$. Write $A = \mathrm{colim}_jA_j$ and $B=\mathrm{colim}_\ell B_\ell$ with $A_j,B_\ell\in{\mathcal{D}^b_\mathcal{M}}(X)$. Then [\[eqindnori1\]](#eqindnori1){reference-type="ref" reference="eqindnori1"} becomes $$\lim_j\mathrm{colim}_\ell\mathrm{Map}_{{\mathcal{D}^b_\mathcal{M}}(X)}(A_j,B_\ell)\to\lim_{i\in \ulcorner^\mathrm{op}}\lim_j\mathrm{colim}_\ell\mathrm{Map}_{{\mathcal{D}^b_\mathcal{M}}(s(i))}(f_i^*A_k,f_i^*B_j)$$ and $$\lim_{i\in \ulcorner^\mathrm{op}}\lim_j\mathrm{colim}_\ell\mathrm{Map}_{{\mathcal{D}^b_\mathcal{M}}(s(i))}(f_i^*A_k,f_i^*B_j)=\lim_j\mathrm{colim}_\ell\lim_{i\in \ulcorner^\mathrm{op}}\mathrm{Map}_{{\mathcal{D}^b_\mathcal{M}}(s(i))}(f_i^*A_k,f_i^*B_j) .$$ The equality comes from the fact that finite limits commute with limits and filtered colimits. A limit of a colimit of isomorphisms is an isomorphism, and as all the maps $$\mathrm{Map}_{{\mathcal{D}^b_\mathcal{M}}(X)}(A_j,B_\ell)\to\lim_{i\in \ulcorner^\mathrm{op}}\mathrm{Map}_{{\mathcal{D}^b_\mathcal{M}}(s(i))}(f_i^*A_k,f_i^*B_j)$$ are isomorphisms by , we obtain Nisnevich descent. The case of Galois invariants is treated similarly, remarking that taking invariant under a finite group is a finite limit (namely, the equaliser of all the maps induces by $\mathrm{Id}-g$ for $g\in G$). We have shown that $\mathrm{Ind}{\mathcal{D}^b_\mathcal{M}}$ satisfies non effective étale hyper-descent. As ${\mathcal{D}^b_\mathcal{M}}$ is $\mathbb{A}^1$-invariant and $\mathbb{P}^1$-stable, and the indization of an isomorphism is an isomorphism, $\mathrm{Ind}{\mathcal{D}^b_\mathcal{M}}$ is also $\mathbb{A}^1$-invariant and $\mathbb{P}^1$-stable. We have $\mathrm{Ind}{\mathcal{D}^b_\mathcal{M}}\in \mathrm{PB}^{\mathrm{st},\mathrm{pr}}_{(\acute{\mathrm{e}}\mathrm{t},\mathbb{A}^1),(\mathbb{P}^1,\infty)}$. $\mathrm{Ind}{\mathcal{D}^b_\mathcal{M}}$ is obviously $\mathbb{Q}$-linear, and the proof is finished. ◻ **Theorem 103**. *There exists an (essentially unique) morphism of pullback formalisms $\mathcal{DM}\to \mathrm{Ind}{\mathcal{D}^b_\mathcal{M}}$ on the category $\mathrm{Sch}_k$. It restricts to a functor $$\label{reali}\nu:\mathcal{DM}_c\to {\mathcal{D}^b_\mathcal{M}}.$$ The functor [\[reali\]](#reali){reference-type="ref" reference="reali"} commutes with the $6$ operations, that is all the exchange transformations are isomorphisms.* *Proof.* The morphism of pullback formalisms $\tilde{\nu}:\mathcal{DM}\to\mathrm{Ind}{\mathcal{D}^b_\mathcal{M}}$ exists and is essentially unique thanks to and . Denote by $\mathcal{DM}_c(X)$ (*resp.* by $\mathcal{D}^{\mathrm{gm}}_\mathcal{M}(X)$) the thick full subcategory of $\mathcal{DM}(X)$ (*resp.* of $\mathrm{Ind}{\mathcal{D}^b_\mathcal{M}}(X)$) spanned by the $f_\sharp \mathbb{Q}_Y(i)$ for $f:Y\to X$ smooth and $i\in \mathbb{Z}$. As ${\mathcal{D}^b_\mathcal{M}}(X)\subset \mathrm{Ind}{\mathcal{D}^b_\mathcal{M}}(X)$ is thick and each $f_\sharp \mathbb{Q}_Y(i)$ is in ${\mathcal{D}^b_\mathcal{M}}(X)$, we have $\mathcal{D}^{\mathrm{gm}}_\mathcal{M}(X)\subset {\mathcal{D}^b_\mathcal{M}}(X)$. Moreover, $\tilde{\nu}$ commutes with all the $f^*$, $f_\sharp$ and is monoidal. Note that the Tate twist $M(i)=M\otimes \mathbb{Q}_X(1)$ is tensorisation by $\mathbb{Q}_X(1)\simeq \mathrm{fib}(p_\sharp\mathbb{Q}_{\mathbb{P}^1_X}\to \mathbb{Q}_X)[-2]$ with $p:\mathbb{P}^1_X\to X$ and the map $p_\sharp\mathbb{Q}_{\mathbb{P}^1_X}\to \mathbb{Q}_X$ is induced by the inclusion $\infty_X\to \mathbb{P}^1_X$. Therefore, $\tilde{\nu}$ also commute with Tate twists, and induces a morphism of pullback formalisms $$\nu^{\mathrm{gm}}:\mathcal{DM}_c\to \mathcal{D}^{\mathrm{gm}}_\mathcal{M}$$ In particular, at the level of triangulated categories, we obtain a premotivic morphism that verifies all the hypothesis of [@MR3971240 Theorem 4.4.25] : $\mathrm{D}^{\mathrm{gm}}_\mathcal{M}$ is $\tau$-generated for the Tate twist, $\mathcal{DM}_c$ is $\tau$-dualisable by absolute purity, $\mathrm{D}^{\mathrm{gm}}_\mathcal{M}$ is separated because it is a sub pullback formalism of ${\mathcal{D}^b_\mathcal{M}}$ which satisfy étale descent by (which is equivalent to separateness by [@MR3971240 Corollaire 3.3.34] in the $\mathbb{Q}$-linear case) and all Tate twist are invertible in $\mathcal{DM}_c$ by construction. Therefore, the functor $\nu^{\mathrm{gm}}$ commutes with the 6 operations. One has to be a little careful here: a priori, [@MR3971240 Theorem 4.4.25] implies only that $\nu^{\mathrm{gm}}$ commutes with all the operations when $\mathcal{D}^{\mathrm{gm}}_\mathcal{M}$ is endowed with the operations they construct in their book. We know that for $f:Y\to X$ a morphism, $f^*$ is the same for both constructions, hence by adjunction this is also the case for $f_*$. For $j:U\to X$ an open immersion the $j_\sharp$ are the same, but $j_\sharp = j_!$ hence by Nagata compactification (and the fact that for a proper $p$, $p_*=p_!$) and composition both $f_!$ are isomorphic for any $f:Y\to X$ separated. By adjunction we also have an isomorphism between the $f^!$. This defines a isomorphism of étale sheaves of $\infty$-categories, hence we also have compatibility of operations for non separated morphisms (when they are defined for $\mathcal{DM}$). The tensor product is the same, hence also the internal $\mathop{\mathrm{\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}}$ by adjunction. We have obtained that the functor $$\nu : \mathcal{DM}_c\to {\mathcal{D}^b_\mathcal{M}}$$ commute with the 6 operations at the triangulated categories level. But all exchange morphisms exist at the level of $\infty$-categories, and the functor from an $\infty$-category to its homotopy category is conservative, hence $\nu$ commutes with all the operations. ◻ **Corollary 104**. *There exists a Hodge realisation of $R_H:\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}})\to\mathcal{D}^b(\mathrm{MHM}^p)$ compatible with the six operations.* *Proof.* By , there exist a Hodge realisation $\mathcal{DM}_c(-,\mathbb{Q})$ to the derived category of polarisable mixed Hodge modules. Therefore, taking the underlying $\mathbb{Q}$-structure and then perverse ${\ ^{\mathrm{p}}\mathrm{H}}^0$, we obtain a factorisation of the perverse ${\ ^{\mathrm{p}}\mathrm{H}}^0$ of $\mathcal{DM}_c(X)$ through $\mathrm{MHM}^p(X)$. The universal property of perverse Nori motives then gives a faithful exact functor ${\mathcal{M}_{\mathrm{perv}}}(X)\to \mathrm{MHM}^p(X)$, which induces a functor $\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))\to \mathcal{D}^b(\mathrm{MHM}^p(X))$. Now all the constructions of [@ivorraFourOperationsPerverse2022] are the constructions of the operations for mixed Hodge modules, hence this functor is compatible with the operations. ◻ Recall that by [@ivorraFourOperationsPerverse2022 Corollary 6.18], [@bondarkoWeightsTstructuresGeneral2011 Proposition 2.3.1] and [@MR2834728 Theorem 3.3] the categories $\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))$, $\mathcal{D}^b(\mathrm{MHM}^p(X))$ and $\mathcal{DM}_c(X)$ admits bounded weight structures (for the last one, note that Beilinson motives and étale motives are equivalent by [@MR3971240 Theorem 16.1.2].) **Proposition 105**. *Let $X$ be a finite type $k$-scheme. Then the functors $$R_H:\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))\to\mathcal{D}^b(\mathrm{MHM}^p(X)),$$ $$\nu:\mathcal{DM}_c(X)\to\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))$$ and $$\mathrm{Hdg}^*:\mathcal{DM}_c(X)\to\mathcal{D}^b(\mathrm{MHM}^p(X))$$ are weight-exact.* *Proof.* First note that by definition of the weight structure on $\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))$, the $\ell$-adic realisation $R_\ell :\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))\to\mathcal{D}^b_c(X,\mathbb{Q}_\ell)$ is weight exact and reflects weights. As all weight structures considered are bounded, it suffices to show that our functors map zero weight objects to zero weight objects. Let us first consider functors with source $\mathcal{DM}_c(X)$. For those, is suffices to check that the image of the relative motive of a smooth proper $X$-scheme is pure of weight $0$. For the $\ell$-adic realisation this is [@MR0601520] and [@morelMixedAdicComplexes]. Therefore, the functor $\nu : \mathcal{DM}_c(X)\to\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))$ is also weight exact. The fact that $\mathrm{Hdg}^*$ is exact is the fact that the mixed Hodge modules associated to a smooth proper $X$-scheme is pure of weight $0$, see [@saitoFormalismeMixedSheaves2006 Theorem 6.7 and Proposition 6.6]. Now is only left $R_H : \mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))\to\mathcal{D}^b(\mathrm{MHM}^p(X))$. By [@ivorraFourOperationsPerverse2022 Corollary 6.27], any pure complex of perverse Nori motives is the direct sum of shifts of its ${\ ^{\mathrm{p}}\mathrm{H}}^i$, hence to check that $R_H$ is weight exact it suffices to show that it sends pure objects of ${\mathcal{M}_{\mathrm{perv}}}(X)$ to pure objects of $\mathrm{MHM}^p(X)$. By construction of $R_H$, we have a commutative diagram :$$\begin{tikzcd} {\mathcal{DM}_c(X)} & {{\mathcal{M}_{\mathrm{perv}}}(X)} & {\mathrm{MHM}^p(X)} \arrow["{\mathrm{H}_u}", from=1-1, to=1-2] \arrow["{R_H}", from=1-2, to=1-3] \arrow["{\mathrm{Hdg}^*}"', from=1-1, to=1-3, bend left=30] \end{tikzcd}.$$ With $H_u$ the universal functor defining ${\mathcal{M}_{\mathrm{perv}}}(X)$. But by [@ivorraFourOperationsPerverse2022 Corollary 6.27], $H_u$ sends pure objects to pure objects, and any motive $M\in{\mathcal{M}_{\mathrm{perv}}}(X)$ is a quotient of some $\mathrm{H}_u(N)$ for $N\in \mathcal{DM}_c(X)$. Assume that $M$ is pure of some weight $w$. Then $M$ is a direct factor of a quotient of the weight filtration of $\mathrm{H}^0(N)$. As $\mathrm{Hdg}^*$ is weight exact, it follows that $R_H(M)$ is a direct factor of a weight $w$ mixed Hodge module, hence $R_H$ is weight exact. ◻ *Remark 106*. Our construction of the realisation functor $\mathcal{DM}_c(X)$ is very similar to that of Ivorra in [@MR3518311] (which works for smooth varieties). Indeed, given a smooth variety $X$, in both definitions one first gives the image of the smooth $X$-schemes in $\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))$. To be more precise, Ivorra only gives the image of smooth affine $X$-schemes and then extend to smooth $X$-schemes by colimits but as the functor $(f:Y\to X)\mapsto f_!f^!\mathbb{Q}_X$ is a Nisnevich cosheaf, its value on smooth $X$-schemes is indeed determined by its value on smooth affine $X$-schemes. Once one has the value of the functor on smooth $X$-schemes, the rest of the construction is just applying the universal property of $\mathcal{DM}_c(X)$ : one has to check that the functor on smooth $X$-schemes is an $\mathbb{A}^1$-invariant étale sheaf and $\mathbb{P}^1$-stable. Therefore, it is easy to see that if both definitions on smooth affine schemes (ours with the $6$-operations and Ivorra's using cellular complexes and Beilinson basic lemma) coincide, then our functor $\mathcal{DM}_c(X)\to\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))$ and Ivorra's are equivalent. In particular, this would prove that Ivorra's construction is compatible with the Betti realisation constructed by Ayoub on Voevodsky motives, hence that the category of perverse Nori motives as in [@ivorraFourOperationsPerverse2022] is equivalent as the diagram category of relative pairs studied in [@MR3723805] and [@ivorraFourOperationsPerverse2022 Section 2.10]. This can be reduced to check a simple fact, that the author did not manage to do : in [@MR3518311 Proposition 4.15] Ivorra proves that for a given smooth affine map $f:Y\to X$ with $X$ a smooth variety, there is a specific cellular stratification $Y^\bullet$ of $Y$ such that the cellular complex induced on $Y$ is an $\mathrm{H}^0f_!$-acyclic resolution of $f^!\mathbb{Q}_X$, hence that there is an isomorphism in $\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))$ between Ivorra's functor at $Y$ and the homology of $Y$, that is $\mathrm{ra}^{\mathcal{M}_{\mathrm{perv}}}_X(Y,Y_\bullet)\simeq f_!f^!\mathbb{Q}_X$. This isomorphism induces a canonical and functorial isomorphism on each ${\ ^{\mathrm{p}}\mathrm{H}}^i$. The question is whether one can construct such an isomorphism in a functorial way, as this would give the wanted isomorphism of functors. This does not seem easy, and it is unlikely that one can arrange those specific stratification so that they are compatible with a give morphism of smooth affine $X$-schemes. # Some applications. {#consqSect} ## Pushforwards and internal $\mathrm{Hom}$s are derived functors. {#effaceableSection} We begin with an application of Nori's criterion of effaceability in [@MR1940678 Theorem 3.11]. **Proposition 107**. *Let $f:Y\to X$ a morphism of finite type $k$-schemes and $M\in {\mathcal{M}_{\mathrm{ct}}}(X)$. Then the functors ${\ ^{\mathrm{ct}}\mathrm{H}}^qf_*:{\mathcal{M}_{\mathrm{ct}}}(Y)\to {\mathcal{M}_{\mathrm{ct}}}(X)$ and $\mathrm{H}^q\mathop{\mathrm{\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}}(M,-)$ are effaceable.* *Proof.* We verify that the tuple $$({\mathcal{M}_{\mathrm{ct}}}(X), \mathrm{Ind}{\mathcal{M}_{\mathrm{ct}}}(X),{\mathcal{M}_{\mathrm{ct}}}(Y), \mathrm{Ind}{\mathcal{M}_{\mathrm{ct}}}(Y),\mathrm{Ind}({\ ^{\mathrm{ct}}\mathrm{H}}^0f^*),\mathrm{Ind}({\ ^{\mathrm{ct}}\mathrm{H}}^0f_*))$$ has the four properties of Theorem 3.11 for [@MR1940678]. Note that by the proof of Theorem 5 of [@MR1940678], the tuple $$(Cons(X), Shv(X)=\mathrm{Ind}Cons(X),Cons(Y), Shv(Y),\mathrm{Ind}(\mathrm{H}^0f^*),\mathrm{Ind}(\mathrm{H}^0f_*))$$ has the wanted properties, and the Betti (or $\ell$-adic) realisation is compatible with $f^*$, $f_*$ and is still faithful exact on the hearts (we really use Noetherianity here). This implies, as being lisse can be checked after Betti (or $\ell$-adic) realisation, that for a $M\in\mathrm{Ind}{\mathcal{M}_{\mathrm{ct}}}(X)$, $M$ is in ${\mathcal{M}_{\mathrm{ct}}}(X)$ if and only if $R_B(M)$ is constructible. 1. ${\mathcal{M}_{\mathrm{ct}}}(X)$ is a full subcategory of $\mathrm{Ind}({\mathcal{M}_{\mathrm{ct}}}(X))$ because it is identified with the compact objects. It is stable by isomorphisms. $\mathrm{Ind}({\mathcal{M}_{\mathrm{ct}}}(X))$ has enough injective. We know that the functors $\mathrm{Ext}^q_{{\mathcal{M}_{\mathrm{ct}}}(X)}(M,-)$ are effaceable on ${\mathcal{M}_{\mathrm{ct}}}(X)$ for $q>0$, and [@MR4124836 Theorem 2.2.1] says that $\mathrm{Ext}^q_{{\mathcal{M}_{\mathrm{ct}}}(X)}(M,-)=\mathrm{Ext}^q_{\mathrm{Ind}{\mathcal{M}_{\mathrm{ct}}}(X)}(M,-)$. 2. The same is true over $Y$. Denote by $F = \mathrm{Ind}(\mathrm{H}^0f^*)$ and $G = \mathrm{Ind}(\mathrm{H}^0f_*)$, so that by definition, $F((M_i)_i)=(\mathrm{H}^0f^*M_i)_i$. We then have that $F$ is left adjoint to $G$. 1. $G$ is a right adjoint to $F$, so it is left exact. Also, note that as the Betti realisation is $t$-exact, we have, for $M\in\mathrm{D}^b({\mathcal{M}_{\mathrm{ct}}}(X))$, $R_B(R^q(G(M))) = R^qf_*R_B(M)$. Hence, by Proposition 3.4 of [@MR1940678], if $M\in {\mathcal{M}_{\mathrm{ct}}}(X)$, then $R^qG(M)\in\mathrm{Ind}{\mathcal{M}_{\mathrm{ct}}}(Y)$ is in ${\mathcal{M}_{\mathrm{ct}}}(Y)$, hence we have the wanted property. 2. $F$ is a left adjoint to $G$, it is left exact because $\mathrm{H}^0f^*$ is left exact, and kernels are computed degree-wise. $F$ preserves ${\mathcal{M}_{\mathrm{ct}}}$ by construction. The same verification for $F,G = M\otimes-,\mathop{\mathrm{\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}}(M,-)$.\ Now, the assertion about $f_*$ being derived is [@MR0102537 Proposition 2.2.1]. ◻ **Corollary 108**. *The functors $f_*:\mathcal{D}^b({\mathcal{M}_{\mathrm{ct}}}(X))\to\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))$ and $\mathop{\mathrm{\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}}(M,-)$ are derived functors of their ${\ ^{\mathrm{ct}}\mathrm{H}}^0$.* *Proof.* Indeed, we have constructed in $\infty$-categorical lifts of those functors. Using we see that they are even $\mathrm{Perf}_\mathbb{Q}$-linear exact functors of idempotent complete $\mathrm{Perf}_\mathbb{Q}$-linear small stable $\infty$-categories (recall that the compact objects of a $\mathrm{Mod}_\mathbb{Q}$-linear $\infty$-category form an idempotent complete $\mathrm{Perf}_\mathbb{Q}$-linear small stable $\infty$-category). In [@cohnDifferentialGradedCategories2016 Corollary 5.5] Cohn proves that the $\infty$-category of those is the $\infty$-category of $dg$-categories localised at Morita equivalences. In particular, our functors are $dg$-functors of $dg$-categories. By Vologodsky's [@MR2729639 Theorem 1], as our functor are as well effaceable and left exact, they are derived functors of their ${\ ^{\mathrm{ct}}\mathrm{H}}^0$. This implies that the $\infty$-functors $f_*$ and $\mathop{\mathrm{\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}}(M,-)$ are the derived functors of their ${\ ^{\mathrm{ct}}\mathrm{H}}^0$. ◻ *Remark 109*. The same proof applies for the pushforward $$f_* :\mathcal{D}^b(Cons(X))\to\mathcal{D}^b(Cons(Y)),$$ answering positively to a question of Nori in [@fakhruddinNotesNoriLectures2000 Section 7.4]: the functor $f_*$ is a derived functor. **Corollary 110**. *Assume that $k\subset \mathbb{C}$. There is a natural equivalence of $\infty$-categories $\mathrm{Ind}\mathcal{D}^b({\mathcal{M}_{\mathrm{ct}}}(X))\simeq\mathcal{D}(\mathrm{Ind}{\mathcal{M}_{\mathrm{ct}}}(X))$.* *Proof.* By , the $t$-structure on $\mathrm{Ind}\mathcal{D}^b({\mathcal{M}_{\mathrm{ct}}}(X))$ is right separated of heart $\mathrm{Ind}{\mathcal{M}_{\mathrm{ct}}}(X)$ hence there exists an unique $\infty$-functor $$\mathrm{r}_X:\mathcal{D}(\mathrm{Ind}{\mathcal{M}_{\mathrm{ct}}}(X))\to\mathrm{Ind}\mathcal{D}^b({\mathcal{M}_{\mathrm{ct}}}(X))$$ which is the identity on $\mathrm{Ind}{\mathcal{M}_{\mathrm{ct}}}(X)$. In particular, the composition of $\mathrm{r}_X$ with the fully faithful functor $\mathcal{D}^b({\mathcal{M}_{\mathrm{ct}}}(X))\to \mathcal{D}(\mathrm{Ind}{\mathcal{M}_{\mathrm{ct}}}(X))$ is the natural functor $\mathcal{D}^b({\mathcal{M}_{\mathrm{ct}}}(X))\to\mathrm{Ind}\mathcal{D}^b({\mathcal{M}_{\mathrm{ct}}}(X))$. As $\mathcal{D}(\mathrm{Ind}{\mathcal{M}_{\mathrm{ct}}}(X))$ is generated under colimits by the image of the fully faithful functor $\mathcal{D}^b({\mathcal{M}_{\mathrm{ct}}}(X))\to\mathcal{D}(\mathrm{Ind}{\mathcal{M}_{\mathrm{ct}}}(X))$, it suffices to show that any $M\in\mathcal{D}^b({\mathcal{M}_{\mathrm{ct}}}(X))$ is compact in $\mathcal{D}(\mathrm{Ind}{\mathcal{M}_{\mathrm{ct}}}(X))$. Seen as an object of $\mathrm{Ind}\mathcal{D}^b({\mathcal{M}_{\mathrm{ct}}}(X))$ , any $M\in{\mathcal{M}_{\mathrm{ct}}}(X)$ has cohomological dimension (see [@MR4296353 Definition 2.8]) less than some fixed $d$ that does not depend on $M$, as for $M,N\in{\mathcal{M}_{\mathrm{ct}}}(X)$, the complex $(\pi_X)_*\mathop{\mathrm{\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}}(M,N)$ is concentrated in degrees $[0,d]$ as this is true after applying the Betti realisation (where this follows from Artin vanishing that implies that the cohomological dimension of an affine variety is less that its dimension, and then a space covered by affines has finite cohomological dimension. One can probably take $d=2\dim X+1$). By , $(\pi_X)_*\mathop{\mathrm{\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}}(M,-):\mathcal{D}^b({\mathcal{M}_{\mathrm{ct}}}(X))\to\mathcal{D}^b({\mathcal{M}_{\mathrm{ct}}}(\mathrm{Spec}\ k))$ is a derived functor, hence $\mathrm{r}_X$ induces a natural equivalence $$\mathrm{r}_{\mathrm{Spec}\ k}\circ(\pi_X)_*\mathop{\mathrm{\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}}_{\mathcal{D}(\mathrm{Ind}{\mathcal{M}_{\mathrm{ct}}}(X))}(M,-)\to (\pi_X)_*\mathop{\mathrm{\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}}_{\mathrm{Ind}\mathcal{D}^b({\mathcal{M}_{\mathrm{ct}}}(X))}(M,\mathrm{r}_{X}(-))$$ of functors $\mathcal{D}^b({\mathcal{M}_{\mathrm{ct}}}(X))\to\mathcal{D}^b({\mathcal{M}_{\mathrm{ct}}}(\mathrm{Spec}\ k))$. This implies that for $M,N \in{\mathcal{M}_{\mathrm{ct}}}(X)$, the object $(\pi_X)_*\mathop{\mathrm{\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}}_{\mathcal{D}(\mathrm{Ind}{\mathcal{M}_{\mathrm{ct}}}(X))}(M,N)$ of $\mathcal{D}(\mathrm{Ind}{\mathcal{M}_{\mathrm{ct}}}(\mathrm{Spec}\ k))$ is concentrated in degrees $[0,d]$. In particular, every object of ${\mathcal{M}_{\mathrm{ct}}}(X)$ is of ${\mathcal{M}_{\mathrm{ct}}}(X)$-cohomological dimension $\leqslant d$. As coproducts of objects of ${\mathcal{M}_{\mathrm{ct}}}(X)$-cohomological dimension $\leqslant d$ is still of ${\mathcal{M}_{\mathrm{ct}}}(X)$-cohomological dimension $\leqslant d$, we see that every object of $\mathrm{Ind}{\mathcal{M}_{\mathrm{ct}}}(X)$ is of ${\mathcal{M}_{\mathrm{ct}}}(X)$-cohomological dimension $\leqslant d$. This implies that for $M\in{\mathcal{M}_{\mathrm{ct}}}(X)$ the functor $\mathrm{Hom}_{\mathrm{Ind}{\mathcal{M}_{\mathrm{ct}}}(X)}(M,-)$ is of finite cohomological dimension. Therefore by [@MR3477640 Lemma 1.1.7], the functor $\mathrm{Map}_{\mathcal{D}(\mathrm{Ind}{\mathcal{M}_{\mathrm{ct}}}(X))}(M,-):\mathcal{D}(\mathrm{Ind}{\mathcal{M}_{\mathrm{ct}}}(X))\to\mathcal{D}(\mathbb{Q})$ commutes with colimits, hence that $M$ is a compact object of $\mathcal{D}(\mathrm{Ind}{\mathcal{M}_{\mathrm{ct}}}(X))$. ◻ ## Relation with the $t$-structure conjecture. {#tstructCOnj} Let $H:\mathcal{T}\to \mathcal{A}$ be an homological functor from a triangulated category $\mathcal{T}$ to an abelian category $\mathcal{A}$. Let $H_u:\mathcal{T}\to \mathcal{M}_u$ an homological functor and $F_u : \mathcal{M}_u\to\mathcal{A}$ a faithful exact functor such that $H = F_u\circ H_u$. We consider the following universal property : For all factorisation $$\begin{tikzcd} {\mathcal{T}} & \mathcal{A}\\ {\mathcal{B}} \arrow["H", from=1-1, to=1-2] \arrow["C"', from=1-1, to=2-1] \arrow["G"', from=2-1, to=1-2] \end{tikzcd}$$ with $C$ a homological functor and $G$ a faithful exact functor there exists a unique (up to unique natural isomorphism) faithful exact functor $G_u:\mathcal{M}_u \to \mathcal{B}$ such that $G = F_u\circ G_u$ and $C = G_u \circ H_u$. By [@ivorraFourOperationsPerverse2022 Section 1], such an universal factorisation $H = F_u\circ H_u$ always exists, and perverse Nori motives ${\mathcal{M}_{\mathrm{perv}}}(X)$ are defined as the universal factorisation of the perverse ${\ ^{\mathrm{p}}\mathrm{H}}^0$ of the Betti realisation of Voevodsky motives $\mathcal{DM}_c(X)$. **Lemma 111**. *Let $\mathrm{D}$ be a triangulated category with a $t$-structure, and let $r:\mathcal{D}\to \mathrm{D}^b(\mathcal{A})$ be a conservative triangulated $t$-exact functor. Then the pair $(\mathrm{H}^0:\mathrm{D}\to\mathrm{D}^\heartsuit,r_{\mid \mathrm{D}^\heartsuit}:\mathrm{D}^\heartsuit\to\mathcal{\mathcal{A}})$ is the universal factorisation of $\mathrm{H}^0r$ in the above sense.* *Proof.* Indeed, let $$\begin{tikzcd} {\mathrm{D}} & \mathcal{A}\\ {\mathcal{B}} \arrow["\mathrm{H}^0\circ r", from=1-1, to=1-2] \arrow["C"', from=1-1, to=2-1] \arrow["G"', from=2-1, to=1-2] \end{tikzcd}$$ be a factorisation of $\mathrm{H}^0\circ r$ with $C$ homological and $G$ faithful exact. Then if there is a commutative diagram $$\begin{tikzcd} \mathrm{D}&&& {\mathcal{A}} \\ {\mathrm{D}^\heartsuit} & \mathcal{B} \arrow["{\mathrm{H}^0r}"', from=1-1, to=1-4] \arrow["{\mathrm{H}^0}"', from=1-1, to=2-1] \arrow["{F_u}"', from=2-1, to=2-2] \arrow["F"', from=2-2, to=1-4] \arrow["C", from=1-1, to=2-2] \arrow["{r_{\mid \mathrm{D}^\heartsuit}}"', from=2-1, to=1-4,bend right=50] \end{tikzcd}$$ Then denoting by $i:\mathrm{D}^\heartsuit\to\mathrm{D}$ the inclusion, as $\mathrm{H}^0\circ i = \mathrm{Id}$ we have $F_u = C\circ i$. Now as $F_u\circ F = r_{\mid \mathrm{D}^\heartsuit}$, we have that $F_u$ is faithful exact, hence $\mathrm{D}^\heartsuit$ affords the universal property. ◻ Recall the following conjecture ([@MR2115000 Section 21.1.7] and [@MR2953406]): **Conjecture 112**. *There exits a non degenerate $t$-structure on $\mathcal{DM}_c(k)$ compatible with the tensor product and such that the $\ell$-adic realisation is $t$-exact.* In [@MR2953406] Beilinson proves that if the characteristic of $k$ is zero the conjecture implies that the realisation of $\mathcal{DM}_c(k)$ are conservative. In [@MR3347995 Theorem 3.1.4], Bondarko proves that if such a $t$-structure exists for all fields of characteristic $0$, then there exists a perverse $t$-structure on each $\mathcal{DM}_c(X)$ for all $X$ finite type $k$-schemes. The joint conservativity of the family of $x^*$ for $x\in X$ and of the realisation over fields implies that in that case, the $\ell$-adic realisation is conservative and $t$-exact, when the target is endowed with the perverse $t$-structure. **Theorem 113**. *Assume for all fields of characteristic $0$. Let $X$ be a finite type $k$ scheme for such a field $k$. Then the heart of the perverse $t$-structure of $\mathcal{DM}_c(X)$ is canonically equivalent to ${\mathcal{M}_{\mathrm{perv}}}(X)$ the category of perverse Nori motives. Moreover, the functor $\nu : \mathcal{DM}_c(X)\to\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))$ is an equivalence of stable $\infty$-categories. This implies that $\mathcal{DM}_c(X)$ is both the derived category of its perverse and constructible hearts.* We decompose the proof of the theorem in several little lemmas : **Lemma 114**. *Denote by $\mathcal{M}(X)$ the heart of the perverse $t$-structure on $\mathcal{DM}_c(X)$. There is a canonical equivalence $F_X:{\mathcal{M}_{\mathrm{perv}}}(X)\simeq \mathcal{M}(X)$. Denote by $\gamma_X : \mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))\to\mathcal{DM}_c(X)$ the canonical functor induced by the inclusion of the heart. The composition $\nu_X\circ \gamma_X$ is equivalent to the identity functor $\mathrm{Id}_{\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))}$.* *Proof.* By construction and with , with the conservativity of the $\ell$-adic realisation it is immediate that there exist a commutative diagram $$\begin{tikzcd} {\mathcal{DM}_c(X)} \\ {{\mathcal{M}_{\mathrm{perv}}}(X)} & {\mathcal{M}(X)} \arrow["{H_u}"', from=1-1, to=2-1] \arrow["{\mathrm{H}^0}", from=1-1, to=2-2] \arrow["{F_X}"', from=2-1, to=2-2] \end{tikzcd}$$ with $H_u$ the universal homological functor of the definition of perverse Nori motives and $F_X$ an exact equivalence of categories, compatible with the realisation. We identify via $F_X$ the categories ${\mathcal{M}_{\mathrm{perv}}}(X)$ and $\mathcal{M}(X)$. By the universal property of the bounded derived category we have a commutative diagram $$\begin{tikzcd} {\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))} & {\mathcal{DM}_c(X)} & {\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))} \\ {{\mathcal{M}_{\mathrm{perv}}}(X)} & {{\mathcal{M}_{\mathrm{perv}}}(X)} & {{\mathcal{M}_{\mathrm{perv}}}(X)} \arrow["\gamma_X"', from=1-1, to=1-2] \arrow["\nu_X"', from=1-2, to=1-3] \arrow["{\mathrm{Id}}"', from=2-1, to=2-2] \arrow["{H^0\nu_X}"', from=2-2, to=2-3] \arrow["{\mathrm{H}^0}"', from=1-1, to=2-1] \arrow["{\mathrm{H}^0}"', from=1-2, to=2-2] \arrow["{\mathrm{H}^0}"', from=1-3, to=2-3] \end{tikzcd}$$ compatible with the realisation. Applying to the derived category $\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))$ we obtain that $\mathrm{H}^0\nu = \mathrm{Id}$. The functor $\nu_X\circ\gamma_X$ is determined by its restriction to the heart ${\mathcal{M}_{\mathrm{perv}}}(X)$, hence it is the identity. ◻ We now will check that the functor $\gamma_X\circ\nu_X : \mathcal{DM}_c(X)\to\mathcal{DM}_c(X)$ is also the identity (up to a unique invertible natural transformation.) By it suffices to show that $\mathrm{Ind}\gamma_X$ is part of a morphism of $\mathrm{PB}^{\mathrm{pr},\mathrm{st}}_{(\acute{\mathrm{e}}\mathrm{t},\mathbb{A}^1),(\mathbb{P}^1,\infty),\mathbb{Q}}$. As all the presentable categories we consider are compactly generated and all functors preserve compact objects, have to show that on the small categories, all $\gamma_X$ assemble to a natural transformation $\gamma:\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(-))\to\mathcal{DM}_c(-)$ in $\mathrm{Fun}(\mathrm{Sch}_k,\mathrm{Cat}_\infty)$, that each $\gamma_X$ is symmetric monoidal, and commutes with the left adjoint $f_\sharp$ of $f^*$ when $f$ is smooth. First, remark that the realisation on $\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))$ being obtained as derived functors, they are characterised by their value on ${\mathcal{M}_{\mathrm{perv}}}(X)$. This implies that $\gamma_X : \mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))\to\mathcal{DM}_c(X)$ is compatible with the $\ell$-adic realisation. Note that as for perverse Nori motives, $\mathcal{DM}_c(X)$ admits a constructible $t$-structure for which the realisation are $t$-exact. Denote by $\mathcal{CM}(X)$ the heart of this $t$-structure. The commutativity of $\gamma_X$ with the $\ell$-adic realisation implies that $\gamma_X$ restricts to a functor $\gamma_X^{\mathrm{ct}}:{\mathcal{M}_{\mathrm{ct}}}(X)\to \mathcal{CM}(X)$. By the universal property of the bounded derived category, we have that $\gamma_X : \mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))=\mathcal{D}^b({\mathcal{M}_{\mathrm{ct}}}(X))\to\mathcal{DM}_c(X)$ factors as $$\gamma_X = \rho_X\circ \mathcal{R}\gamma_X^{\mathrm{ct}}$$ with $\mathcal{R}\gamma_X^{\mathrm{ct}} = \mathcal{D}^b(\gamma_X^\mathrm{ct})$ and $\rho_X : \mathcal{D}^b(\mathcal{CM}(X))\to\mathcal{DM}_c(X)$ obtained by universal property. **Lemma 115**. *The $\infty$-category $\mathcal{D}^b(\mathcal{CM}(X))$ is symmetric monoidal, the functor $\rho_X : \mathcal{D}^b(\mathcal{CM}(X))\to\mathcal{DM}_c(X)$ is a symmetric monoidal $\infty$-functor. For each morphism $f:Y\to X$ of finite type $k$-schemes, there exists a symmetric monoidal $\infty$-functor $f^*:\mathcal{D}^b(\mathcal{CM}(X))\to \mathcal{D}^b(\mathcal{CM}(Y))$. Moreover all the $f^*$ can be put together to give a functor $\mathcal{D}^b(\mathcal{CM}(-)):\mathrm{Sch}_k^\mathrm{op}\to\mathrm{CAlg}{\mathrm{Cat}_\infty}$ and all the $\rho_X$ can be put together to give a morphism $\gamma:\mathcal{D}^b(\mathcal{CM})\Rightarrow \mathcal{DM}_c$ of functors $\mathrm{Sch}_k^\mathrm{op}\to\mathrm{CAlg}(\mathrm{Cat}_\infty)$.* *Proof.* This is a direct consequence of the more general . ◻ **Lemma 116**. *With the structure of , the functors $\mathcal{R}\gamma_X^\mathrm{ct}$ can be put together to give a morphism $\mathcal{R}\gamma^\mathrm{ct} : \mathcal{D}^b({\mathcal{M}_{\mathrm{ct}}})\to\mathcal{D}^b(\mathcal{CM})$ of functors $\mathrm{Sch}_k^\mathrm{op}\to\mathrm{CAlg}(\mathrm{Cat}_\infty)$.* *Proof.* Let $i:Z\to X$ be a closed immersion with open complement $j:U\to X$. We have that $\gamma_X(j_!j^*\to \mathrm{Id}) = j_!j^*\gamma_X\to\gamma_X$ on the perverse heart (as by definition $\gamma_X$ is the identity on the heart) hence $\gamma_X$ commutes with the counit. This gives the following morphism of cofiber sequences : $$\begin{tikzcd} {\gamma_Xj_!j^*} & {\gamma_X} & {\gamma_Xi_*i^*} \\ {j_!j^*\gamma_X} & {\gamma_X} & {i_*i^*\gamma_X} \arrow[from=1-1, to=1-2] \arrow[from=1-2, to=1-3] \arrow[from=2-2, to=2-3] \arrow[from=2-1, to=2-2] \arrow[from=1-1, to=2-1] \arrow["{\mathrm{Id}}"', from=1-2, to=2-2] \arrow[from=1-3, to=2-3] \end{tikzcd}$$ in which the first vertical map is an equivalence. As $i_*$ is perverse $t$-exact, there is a natural equivalence $\gamma_X\circ i_*\simeq i_*\circ\gamma_X$. We therefore have an equivalence $i_*\gamma_Z i^* =\gamma_Xi_*i^*\simeq i_*i^*\gamma_X$ and as $i_*$ is fully faithful, there is a natural equivalence $\gamma_Z\circ i^*\simeq i^*\circ \gamma_X$. Note that as $f^*[d]$ is perverse $t$-exact for $f:Y\to X$ smooth of pure relative dimension $d$, we also have an natural equivalence $f^*\circ \gamma_X \simeq \gamma_Y\circ f^*$ that works for any smooth morphism. If $f:Y\to X$ is a quasi-projective morphism, we can write it as $g\circ i$ with $i$ a closed immersion and $g$ a smooth morphism. The commutation of $\gamma_X$ with pullback of smooth morphisms and closed immersion then gives a natural equivalence $f^*\circ \gamma_X \simeq \gamma_Y\circ f^*$. We do not claim that this gives a morphisms of functors $\mathrm{QProj}^\mathrm{op}\to\mathrm{Cat}_\infty$. However, the restriction to constructible hearts do give a morphism of functors $\mathrm{QProj}_k^\mathrm{op}\to\mathrm{Add}$ to the $(2,1)$-category of additive categories. Using that the external tensor product is perverse $t$-exact, and that one can recover the internal tensor product from the external one an pullbacks, we obtain a morphism of symmetric monoidal functors $\mathrm{QProj}_k^\mathrm{op}\to\mathrm{CAlg}(\mathrm{Cat}_1)$. As all pullbacks are constructible $t$-exact, the functor $f^*$ in is actually the derived functor of its restriction to the constructible heart. Hence with the same technique as we obtain that $\mathcal{R}\gamma_X^\mathrm{ct}$ assemble together to form a morphism of symmetric monoidal functors $\mathrm{QProj}_k^\mathrm{op}\to\mathrm{CAlg}(\mathrm{Cat}_\infty)$. Now, the $\ell$-adic realisation being conservative, the same proof as gives that $X\mapsto \mathcal{D}^b(\mathcal{CM}(X))$ is an étale sheaf, so that $\mathcal{R}\gamma^\mathrm{ct}$ is a morphism a étale sheaves that extend to all finite type $k$-schemes. ◻ The two previous and combined give the following proposition : **Proposition 117**. *The functors $\gamma_X :\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))\to\mathcal{DM}_c(X)$ assemble to give a morphism $\gamma$ of functors $\mathrm{Sch}_k^\mathrm{op}\to\mathrm{CAlg}(\mathrm{Cat}_\infty)$.* The following lemma will finish the proof of : **Lemma 118**. *Let $f:Y\to X$ be a smooth morphism of finite type $k$-schemes. Then $\gamma$ commutes with $f_\sharp$.* *Proof.* As $\gamma$ commute with the adjoint $f^*$, there exist exchange transformations $f_\sharp\circ \gamma_Y \to \gamma_X\circ f_\sharp$ (Indeed apply $\gamma_Y$ to the unit $\mathrm{Id}\to f^*f_\sharp$, use commutation and then the adjunction). The compatibility of $\gamma_X$ with the conservative $\ell$-adic realisation then ensures that this natural transformation is invertible. ◻ This finishes the proof of . ## On the Nori realisation of Voevodsky motives and continuity. {#continuitySection} **Lemma 119**. *Let $k=\mathrm{colim}_i k_i$ be a filtered colimit of fields of characteristic zero. Then the natural functors $$\mathrm{colim}_i\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(\mathrm{Spec}\ k_i)) \to \mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(\mathrm{Spec}\ k))$$ and $$\mathrm{colim}_i\mathrm{Ind}\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(\mathrm{Spec}\ k_i)) \to \mathrm{Ind}\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(\mathrm{Spec}\ k))$$ are equivalences.* *Proof.* As the functor $\mathrm{Ind}$, from small indempotent complete stable $\infty$-categories to compactly generated presentable stable categories is an equivalence, the second statement follows from the first. By [@ivorraFourOperationsPerverse2022 Proposition 6.8], the diagram $\mathcal{M}:I^{\rhd}\to \mathrm{AbCAt}$ from $I$ to the $(2,1)$-category of small abelian categories with exact functors sending $i\in I$ to ${\mathcal{M}_{\mathrm{perv}}}(\mathrm{Spec}\ k_i)$ and the cone point $v$ to ${\mathcal{M}_{\mathrm{perv}}}(\mathrm{Spec}\ k)$ is a colimit diagram. Therefore, the diagram $\mathrm{Ind}\circ \mathcal{M}: I^\rhd \to\mathrm{Groth}$ to Grothendieck abelian categories is still a colimit diagram. By , $\mathcal{D}^{\leqslant 0}$ is a left adjoint, hence $\mathcal{D}^{\leqslant 0}\circ\mathrm{Ind}\circ\mathcal{M}$ is a colimit diagram of Grothendieck prestable $\infty$-categories. The endofunctor $\otimes \mathrm{Sp}$ of the $\infty$-category $\mathrm{Pr}^L$ preserves colimits, hence $\mathrm{Sp}\circ \mathcal{D}^{\leqslant 0}\circ\mathrm{Ind}\circ\mathcal{M}$ is a colimit diagram. This means that the functor $$\mathrm{colim}_i\mathcal{D}(\mathrm{Ind}{\mathcal{M}_{\mathrm{perv}}}(\mathrm{Spec}\ k_i))\to\mathcal{D}(\mathrm{Ind}{\mathcal{M}_{\mathrm{perv}}}(\mathrm{Spec}\ k))$$ is an equivalence. As each ${\mathcal{M}_{\mathrm{perv}}}(\mathrm{Spec}\ k_i)$ is Noetherian, $\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(\mathrm{Spec}\ k_i)) \to\mathcal{D}(\mathrm{Ind}{\mathcal{M}_{\mathrm{perv}}}(\mathrm{Spec}\ k_i))$ is fully faithful. Therefore, we know that the functor $$\mathrm{colim}_i\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(\mathrm{Spec}\ k_i))\to\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(\mathrm{Spec}\ k))$$ is fully faithful. To prove that is is essentially surjective by devissage it suffices to prove that any object of ${\mathcal{M}_{\mathrm{perv}}}(\mathrm{Spec}\ k)$ is in the colimit. This true again by [@ivorraFourOperationsPerverse2022 Proposition 6.8]. ◻ By construction, the functor $\mathrm{Nor}^*$ preserves colimits, hence it has a right adjoint $\mathrm{Nor}_*:\mathcal{DN}\to \mathcal{DM}$. Denote by $$\label{defN}\mathscr{N}_k = \mathrm{Nor}_*\mathrm{Nor}^*\mathbb{Q}_{\mathrm{Spec}\ k}\in\mathcal{DM}(\mathrm{Spec}\ k,\mathbb{Q}).$$ As in Ayoub's [@ayoubAnabelianPresentationMotivic2022 Construction 1.91], $\mathrm{Nor}^*$ factors canonically through the functor $$\mathrm{Mod}_\mathscr{N}(\mathcal{DM}) : X\mapsto \mathrm{Mod}_{\mathscr{N}_{\mid X}}(\mathcal{DM}(X,\mathbb{Q}))$$ as $$\label{factoModN} \mathrm{Nor}^* : \mathcal{DM}\overset{\mathscr{N}\otimes -}{\to} \mathrm{Mod}_\mathscr{N}(\mathcal{DM}) \overset{\tilde{\mathrm{Nor}^*}}{\to} \mathcal{DN} .$$ **Lemma 120**. *Let $k$ be a field of characteristic zero and let $\pi_X:X\to\mathrm{Spec}\ k$ be a finite type $k$-scheme. The functor $$\tilde{\mathrm{Nor}^*}:\mathrm{Mod}_{\pi_X^*\mathscr{N}_k}(\mathcal{DM}(X))\to \mathcal{DN}(X)$$ is fully faithful. Equivalently, the natural map $\pi_X^*\mathrm{Nor}_*\mathbb{Q}_{\mathrm{Spec}\ k}\to\mathrm{Nor}_*\mathbb{Q}_X$ is an equivalence.* *Proof.* First, if $\tilde{\mathrm{Nor}}^*$ is an equivalence then the adjunction unit $\mathrm{Id}\to \tilde{\mathrm{Nor}_*}\tilde{\mathrm{Nor}}^*$ is an isomorphism. Plugging $\pi_X^*\mathscr{N}_k$, we obtain $\pi_X^*\mathscr{N}_k\simeq \tilde{\mathrm{Nor}_*}\tilde{\mathrm{Nor}}^*\pi_X^*\mathscr{N}_k \simeq \tilde{\mathrm{Nor}_*}\pi_X^*\tilde{\mathrm{Nor}}^*\mathscr{N}_k$ but by definition, $\tilde{\mathrm{Nor}}^*\mathscr{N}_k =\mathbb{Q}_{\mathrm{Spec}\ k}$ and we obtain $\pi^*\mathscr{N}_k\simeq \tilde{\mathrm{Nor}}_*\mathbb{Q}_X$ in $\mathrm{Mod}_{\pi_X^*\mathscr{N}_k}(\mathrm{DM}(X))$ but as forgetting the module structure is conservative, we get $\pi_X^*\mathscr{N}_k \simeq \mathrm{Nor}_*\mathbb{Q}_X$ as claimed. The proof of fully faithfulness can be found in [@ayoubAnabelianPresentationMotivic2022 Remark 1.97]. As $\tilde{\mathrm{Nor}^*}$ preserves compact objects, its right adjoint $\tilde{\mathrm{Nor_*}}$ commutes with colimits. The functor $\tilde{\mathrm{Nor^*}}$ is fully faithful if and only if the unit $\mathrm{Id}\to \tilde{\mathrm{Nor_*}}\tilde{\mathrm{Nor^*}}$ is an equivalence, and the preservation of colimits of all functors used to write the unit ensures that it suffices that for $M$ a compact object of $\mathrm{Mod}_{\mathscr{N}}(\mathcal{DM}(X))$, the map $M\to \tilde{\mathrm{Nor_*}}\tilde{\mathrm{Nor^*}}M$ is an equivalence. A generating set of compact objects is given by the $M\otimes_\mathbb{Q}\mathscr{N}_{\mid X}$ for $M$ a compact object of $\mathcal{DM}(X)$, for which the unit map is $M\otimes\mathscr{N}_{\mid X} \to \tilde{\mathrm{Nor}_*}\mathrm{Nor}^*M$, and as the map $\mathrm{Mod}_\mathscr{N}(\mathcal{DM}(X))$ is conservative, it suffices to check that $M\otimes_\mathbb{Q}\mathscr{N}_{\mid X}\to\mathrm{Nor}_*\mathrm{Nor}^*M$ is an equivalence. Therefore, it suffices to check that for any compact objects $C$ and $M$ of $\mathcal{DM}(X)$, the map $$\label{mapEq1}\mathrm{Map}_{\mathcal{DM}(X)}(C,M\otimes\mathscr{N}_{\mid X})\to \mathrm{Map}_{\mathcal{DM}(X)}(C,\mathrm{Nor}_*\mathrm{Nor}^*M)\simeq \mathrm{Map}_{\mathcal{DN}(X)}(\mathrm{Nor}^*C,\mathrm{Nor}^*M)$$ is an equivalence. The functor $\mathop{\mathrm{\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}}(C,-)$ commutes with colimits because $C$ is compact. Indeed, this can be tested after composing with $\mathrm{Map}_{\mathcal{DM}(X)}(D,-)$ with $D$ a compact object, and as compact objects are preserves by tensor products, this follows from adjunction. As the compact generators of $\mathcal{DM}(\mathrm{Spec}\ k)$ are dualisable, the object $\pi_X^*\mathscr{N}_k$ is a colimit of dualisable objects $(\omega_i)$. hence as $C$ is compact and tensor products commutes with colimits we have $\mathop{\mathrm{\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}}(C,-\otimes \pi_X^*\mathscr{N}_k)\simeq \mathrm{colim}_i\mathop{\mathrm{\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}}(C,-\otimes \omega_i) \simeq \mathrm{colim}_i\mathop{\mathrm{\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}}(C,-)\otimes \omega_i \simeq \mathop{\mathrm{\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}}(C,-)\otimes\pi_X^*\mathscr{N}_k$. Now, applying the functor $\mathrm{Map}_{\mathcal{DM}(X)}(\mathbb{Q}_X,-)$ to that last equivalence and using the commutation of $\mathrm{Nor}^*$ with internal $\mathop{\mathrm{\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}}$ we obtain that [\[mapEq1\]](#mapEq1){reference-type="ref" reference="mapEq1"} is an equivalence if and only if the map $$\mathrm{Map}_{\mathcal{DM}(X)}(\mathbb{Q}_X,\mathop{\mathrm{\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}}(C,M)\otimes \mathscr{N}_{\mid X})\to \mathrm{Map}_{\mathcal{DN}(X)}(\mathbb{Q}_X,\mathrm{Nor}^*\mathop{\mathrm{\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}}(M,N)).$$ Using the adjunction $(\pi_X^*,(\pi_X)_*)$ we see that it suffices to show that the map $$\mathrm{Map}_{\mathcal{DM}(k)}(\mathbb{Q}_k,(\pi_X)_*(\mathop{\mathrm{\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}}(C,M)\otimes \mathscr{N}_{\mid X}))\to \mathrm{Map}_{\mathcal{DN}(k)}(\mathrm{Nor}^*(\mathbb{Q}_k),\mathrm{Nor}^*(\pi_X)_*\mathop{\mathrm{\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}}(M,N))$$ is an equivalence. Therefore thanks to the $(\mathrm{Nor}^*,\mathrm{Nor}_*)$ adjunction and the projection formula for $\pi_X$ we see that it suffices to show that the map $$((\pi_X)_*\mathop{\mathrm{\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}}(C,M))\otimes \mathscr{N}_k\to\mathrm{Nor}_*\mathrm{Nor}^*(\pi_X)_*\mathop{\mathrm{\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}}(M,N)$$ is an equivalence. But for any object $P\in \mathcal{DM}(k)$, the natural map $P\otimes\mathscr{N}_k\to \mathrm{Nor}_*\mathrm{Nor}^*P$ is an equivalence: $\mathscr{N}_k$ is a colimit of dualisable objects, and for those the map is an equivalence thanks to [@MR3259031 Lemma 2.8]. ◻ **Lemma 121**. *Let $f:\mathrm{Spec}\ k\to\mathrm{Spec}\ \mathbb{Q}$ be a map of fields. Then the natural map $f^*\mathscr{N}_{\mathrm{Spec}\ \mathbb{Q}}\to\mathscr{N}_{\mathrm{Spec}\ k}$ is an equivalence. Consequently, for any scheme $X$ of finite type over $k$ if we denote by $p_X:X\to\mathrm{Spec}\ \mathbb{Q}$ the unique map (which has to factor through $f$), then $\mathscr{N}_X\simeq p_X^*\mathscr{N}_{\mathrm{Spec}\ \mathbb{Q}}$.* *Proof.* The second claims follows from the first and the previous lemma. It suffices to show that for any compact object $M$, the map $$\mathrm{Map}_{\mathcal{DM}(\mathrm{Spec}\ k )}(M,f^*\mathrm{Nor}_{\mathbb{Q},*}\mathbb{Q}_{\mathrm{Spec}\ \mathbb{Q}})\to \mathrm{Map}_{\mathcal{DM}(\mathrm{Spec}\ k )}(M,\mathrm{Nor}_{k,*}\mathbb{Q}_{\mathrm{Spec}\ k})$$ is an equivalence. Write $k=\mathrm{colim}_i k_i$ as a filtered colimit of fields of finite type over $\mathbb{Q}$. Denote by $f_i : \mathrm{Spec}\ k\to\mathrm{Spec}\ k_i$ and by $p_i:\mathrm{Spec}\ k_i\to\mathrm{Spec}\ \mathbb{Q}$. By continuity of $\mathcal{DM}$ and of $\mathcal{DN}$ for colimits of fields (), we have $f^*\mathrm{Nor}_{\mathbb{Q},*}\mathbb{Q}_{\mathrm{Spec}\ \mathbb{Q}} \simeq \mathrm{colim}_i f_i^*\mathrm{Nor}_{k_i,*}\mathbb{Q}_{\mathrm{Spec}\ k_i}$. As $M$ is compact we have $$\mathrm{Map}_{\mathcal{DM}(\mathrm{Spec}\ k )}(M,f^*\mathrm{Nor}_{\mathbb{Q},*}\mathbb{Q}_{\mathrm{Spec}\ \mathbb{Q}})\simeq \mathrm{colim}_i \mathrm{Map}_{\mathcal{DM}(\mathrm{Spec}\ k )}(M,f_i^*\mathrm{Nor}_{k_i,*}\mathbb{Q}_{\mathrm{Spec}\ _i}).$$ By continuity of the compact objects of $\mathcal{DM}$, we know that $M = f_0^*M_0$ for some $M_0\in\mathcal{DM}_c(\mathrm{Spec}\ k_0)$ and $0\in I$. But now denoting by $g_{i,j}:\mathrm{Spec}\ k_i\to\mathrm{Spec}\ k_j$ we have $$\mathrm{Map}_{\mathcal{DM}(\mathrm{Spec}\ k )}(M,f_i^*\mathrm{Nor}_{k_i,*}\mathbb{Q}_{\mathrm{Spec}\ _i})\simeq \mathrm{colim}_{j\geqslant i}\mathrm{Map}_{\mathcal{DM}(\mathrm{Spec}\ k_j )}(g_{j,0}^*M_0,g_{j,i}^*\mathrm{Nor}_{k_i,*}\mathbb{Q}_{\mathrm{Spec}\ k_i}).$$ Thus as the colimits are sifted we have $$\mathrm{Map}_{\mathcal{DM}(\mathrm{Spec}\ k )}(M,f^*\mathrm{Nor}_{\mathbb{Q},*}\mathbb{Q}_{\mathrm{Spec}\ \mathbb{Q}})\simeq \mathrm{colim}_{j\geqslant 0}\mathrm{Map}_{\mathcal{DM}(\mathrm{Spec}\ k_j )}(g_{j,0}^*M_0,g_{j,0}^*\mathrm{Nor}_{k_0,*}\mathbb{Q}_{\mathrm{Spec}\ k_0})$$ from which, using that $g_{j,0}$ is smooth hence $g_{j,0}^*$ has a left adjoint commuting with $\mathrm{Nor}^*$ thus commutes with $\mathrm{Nor}_*$, we deduce $$\mathrm{Map}_{\mathcal{DM}(\mathrm{Spec}\ k )}(M,f^*\mathrm{Nor}_{\mathbb{Q},*}\mathbb{Q}_{\mathrm{Spec}\ \mathbb{Q}})\simeq \mathrm{colim}_{j\geqslant 0}\mathrm{Map}_{\mathcal{DM}(\mathrm{Spec}\ k_j )}(g_{j,0}^*M_0,\mathrm{Nor}_{k_j,*}\mathbb{Q}_{\mathrm{Spec}\ k_j}).$$ Finally, by adjunction the latter can be written $$\mathrm{colim}_{j\geqslant 0}\mathrm{Map}_{\mathcal{DN}(\mathrm{Spec}\ k_j )}(g_{j,0}^*\mathrm{Nor}^*M_0,g_{j,0}^*\mathbb{Q}_{\mathrm{Spec}\ k_0})$$ which by is equivalent to $\mathrm{Map}_{\mathcal{DN}(\mathrm{Spec}\ k)}(f_0^*\mathrm{Nor}^*M_0,\mathbb{Q}_{\mathrm{Spec}\ k})$. To finish, commutation of $f_0^*$ and $\mathrm{Nor}_*$ and passing the the right adjoint $\mathrm{Nor}_*$ gives: $$\mathrm{Map}_{\mathcal{DN}(\mathrm{Spec}\ k)}(f_0^*\mathrm{Nor}^*M_0,\mathbb{Q}_{\mathrm{Spec}\ k}) \simeq \mathrm{Map}_{\mathcal{DM}(\mathrm{Spec}\ k)}(M,\mathrm{Nor}_*\mathbb{Q}_{\mathrm{Spec}\ k})$$ which was what we wanted to prove. ◻ From the previous lemma there is no more ambiguity for $\mathscr{N}_X\in\mathcal{DM}(X)$. **Proposition 122**. *Let $X=\lim_i X_i$ be a limit of $\mathbb{Q}$-schemes with affine transitions. The natural morphism $$\mathrm{colim}_i \mathrm{Mod}_\mathscr{N}(\mathcal{DM}(X_i))\to \mathrm{Mod}_\mathscr{N}(\mathcal{DM}(X))$$ is an equivalence.* *Proof.* By [@MR4319065 Lemma 5.1 (ii)], the functor $$\mathrm{colim}_i \mathcal{DM}(X_i,\mathbb{Q}) \to \mathcal{DM}(X,\mathbb{Q})$$ is an equivalence. Therefore, the assignment $i\mapsto (\mathcal{DM}(X_i,\mathbb{Q}),\mathscr{N}_{\mid X_i})$ is a colimit diagram in the category $\mathrm{Pr}^{\mathrm{Alg}}$ of [@lurieHigherAlgebra2022 Notation 4.8.5.10]. By [@lurieHigherAlgebra2022 Theorem 4.8.5.11], the functor $(\mathcal{C},A)$ that sends a presentable symmetric monoidal category $\mathcal{C}$ together with an algebra object $A\in\mathcal{C}$ to the category $\mathrm{Mod}_A(\mathcal{C})$ has a right adjoint, hence preserves colimits. This is why the assignment $i\mapsto \mathrm{Mod}_{\mathscr{N}_{X_i}}(\mathcal{DM}(X_i,\mathbb{Q}))$ is a colimit diagram. We found a precise statement giving the symmetric monoidal structure in the paper by Ayoub, Gallauer and Vezzani The precise statement is [@MR4466640 Lemma 3.5.6]. ◻ Using Ayoub's arguments for [@ayoubAnabelianPresentationMotivic2022 Theorem 1.93 (i) p 53], we can prove : **Proposition 123**. *The functor $$\tilde{\mathrm{Nor}^*} :\mathrm{Mod}_\mathscr{N}(\mathcal{DM}(X)) \to\mathcal{DN}(X)$$ is an equivalence for every finite type $k$-scheme $X$.* *Proof.* The functor is fully faithful by . The essential surjectivity is proven by Noetherian induction. Assume that $X=\mathrm{Spec}\ k$ is the spectrum of a field. As $\mathcal{DN}(\mathrm{Spec}\ k)$ is compactly generated it suffices to show that $\tilde{\mathrm{Nor}^*}$ reaches all compact objects $M\in\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(k))$. As the functor is already fully faithful, by dévissage it suffices to check that any object of the heart $M\in{\mathcal{M}_{\mathrm{perv}}}(k)$ is in the image of $\tilde{\mathrm{Nor}^*}$. Any such object has a weight filtration whose graded pieces are direct factors of $\mathrm{H}^{n+2j}(\pi_X)_*\mathbb{Q}_X(i)$ with $\pi_X:X\to\mathrm{Spec}\ K$ a smooth projective morphism by [@MR3618276 Theorem 10.2.5]. Therefore by dévissage again it suffices to check that any such $\mathrm{H}^{n+2j}(\pi_X)_*\mathbb{Q}_X(i)$ is in the image. But as $(\pi_X)_*\mathbb{Q}_X(i)$ is pure of weight $-2i$, it is the direct sum of its $\mathrm{H}^n[n]$, thus is suffices to show that each $(\pi_X)_*\mathbb{Q}_X(i)$ is in the image, which is obvious by compatibility with the $6$ operations. Assume now that $X$ is such that the statement is true for any proper closed subset of $X$. We take an object $M\in\mathcal{DN}(X)$. Let $\eta$ be a generic point of $X$. By the case of a field, there exists $N\in\mathrm{Mod}_\mathscr{N}(\mathcal{DM}(\eta))$ such that $\tilde{\mathrm{Nor}^*}(N)=M_\eta$. By there exists an open subset $U$ of $X$ and an object $N'\in\mathrm{Mod}_\mathscr{N}(\mathcal{DM}(U))$ such that $(N')_\eta = N$. Also, the fully faithfulness of $\tilde{\mathrm{Nor}^*}$ and the continuity of $\mathrm{Mod}_\mathscr{N}(\mathcal{DM})$ ensures that $\mathcal{DN}$ satisfies the non effective part of continuity, so that $\mathrm{colim}_{\eta\in U}\mathcal{DN}(U)\to\mathcal{DN}(\eta)$ is fully faithful. This imply that the isomorphism $\tilde{\mathrm{Nor}^*}(N)\to M_\eta$ lifts to a smaller open subset $V$ of $U$ : we have $\tilde{\mathrm{Nor}^*}((N')_{\mid V})\simeq M_{\mid V}$. Denote by $Z$ the reduced closed complement of $V$ in $X$. By induction hypothesis, there exists a $N''\in\mathrm{Mod}_\mathscr{N}(\mathcal{DM}(Z))$ such that $\tilde{\mathrm{Nor}^*}(N'')=M_{\mid Z}$. The cofiber sequence $\tilde{\mathrm{Nor}^*}(j_!(N')_{\mid V})\to M \to \tilde{\mathrm{Nor}^*}(i_*N'')$ with $j:U\to X$ and $i:Z\to X$ the immersion then ensures that $M$ is the image of $\tilde{\mathrm{Nor}^*}$, which finishes the proof. ◻ *Remark 124*. The same arguments would probably prove that the $\infty$-category $$\mathcal{DH}^{\mathrm{geo}}(X)\subset\mathrm{Ind}\mathcal{D}^b(\mathrm{MHM}^p(X))$$ of geometric origin objects in the indization of the derived category of polarisable mixed Hodge modules is also the category of modules over some algebra $\mathscr{H}_X$ in $\mathcal{DM}(X)$. Note that the lack of continuity of Mixed Hodge modules would prevent things like $p_X^*\mathscr{H}_{\mathrm{Spec}\ \mathbb{Q}}\simeq \mathscr{H}_{X}$ of happening so one would have to be careful of which algebra is used. Similar ideas are developped in [@drewMotivicHodgeModules2018]. **Corollary 125**. *The $\infty$-functor $X\mapsto\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))$ has an unique extension $\mathcal{DN}_c(X)$ to quasi-compact and quasi-separated schemes of characteristic zero such that for all limit $X=\lim_i X_i$ of such schemes with affines transitions the natural functor $$\mathrm{colim}_i\mathcal{DN}_c(X_i)\to\mathcal{DN}_c(X)$$ is an equivalence of $\infty$-categories.* *Proof.* Let $\mathcal{C}$ be category of schemes that are of finite type over a field of characteristic zero. Let $\mathrm{Sch}_\mathbb{Q}$ be the category of quasi-compact and quasi-separated (qcqs) $\mathbb{Q}$-schemes, and $\iota : \mathcal{C}\to\mathrm{Sch}_\mathbb{Q}$ the inclusion. Nori motives, together with the pullback functoriality, define a functor $\mathcal{C}^\mathrm{op}\to\mathrm{CAlg}(\mathrm{Cat}_\infty)$ which sends limits of schemes with affine transitions to colimits of $\infty$-categories by togenther with the fact that the colimit is taken in $\mathrm{Pr}^L$ hance restricts to compact obejcts. We can denote by $\mathcal{DN}_c$ its left Kan extension along $\iota$. ◻ **Lemma 126**. *Let $f:Y\to X$ be a morphism of $qcqs$ $\mathbb{Q}$-schemes. There exist a filtering index $I$, diagrams of finite type $\mathbb{Q}$-schemes with affine transitions $(X_i)_{i\in I}$ and $(Y_i)_{i\in I}$ such that $X = \lim_i X_i$ and $Y=\lim_i Y_i$. Moreover, there exists a morphism of diagrams $(f_i):(Y_i)\to (X_i)$ such that $f = \lim_i f_i$. That is, $f$ is a limit of morphisms of finite type $\mathbb{Q}$-schemes.* *Proof.* The proof of [@stacks-project [Tag 07RB](https://stacks.math.columbia.edu/tag/07RN)] works *verbatim* if one replaces $\mathbb{Z}$ by $\mathbb{Q}$. Therefore we have that $X$ and $Y$ are limits of finite-type $\mathbb{Q}$-schemes. Next, one applies the proof of [@stacks-project [Tag 09GS1](https://stacks.math.columbia.edu/tag/0GS1)] to obtain that $f$ is a limit of morphisms of finite type over $\mathbb{Q}$. ◻ **Proposition 127**. *Let $f:Y\to X$ be a morphism of qcqs $\mathbb{Q}$-schemes that are of finite type over some compatible fields. Then $f^*:\mathcal{DN}_c(X)\to\mathcal{DN}_c(Y)$ has a right adjoint $f_*$.* *Proof.* As in , write $f$ as a limit of morphisms $f_i : Y_i\to X_i$ of schemes of finite type over $\mathbb{Q}$. Let $M\in\mathcal{DN}_c(Y)$, that we can write in an unique way $M = \mathrm{colim}_i p_i^*M_i$ with $p_i:X\to X_i$ the projection and $M_i\in\mathcal{DN}_c(X_i)$. We define $f_*M$ by $$f_*M:= \mathrm{colim}_i q_i^*(f_i)_*M_i$$ with $q_i:Y\to Y_i$ the projection. Then for $N\in\mathcal{DN}_c(Y)$ that we write $N=\mathrm{colim}_i q_i^*N_i$ we have $$\begin{aligned} \mathrm{Hom}_{\mathcal{DN}_c(X)}(N,f_*M) & =& \mathrm{colim}_i \mathrm{Hom}_{\mathcal{DN}_c(X_i)}(N_i,(f_i)_*M_i)\\ &= &\mathrm{colim}_i \mathrm{Hom}_{\mathcal{DN}_c(Y_i)}(f_i^*N_i,M_i)\\ & = &\mathrm{Hom}_{\mathcal{DN}_c(Y)}(f^*N,M) \end{aligned}$$ which shows that we indeed obtain the right adjoint of $f^*$. ◻ ## Categories of $0$-motives with rational coefficients. {#ArtinMotivesSection} In this section, all equivalences of categories will be equivalences of $\infty$-categories. But as the categories we will study are (pre)stables $\infty$-categories, and functors are exact, it suffices by [@MR3931682 Theorem 7.6.10] to prove that their homotopy categories are equivalent, whence the lack of care of the higher morphisms. We fix a $k$ field of characteristic zero. Recall the definitions of [@MR3920833]. **Definition 128**. Let $n\in \mathbb{N}$ and let $X$ be a finite-type $k$-scheme. Denote by $\mathcal{DN}_c^n(X)$ (*resp.* $\mathcal{DM}_c^n(X)$) the smallest stable idempotent complete full subcategory of $\mathcal{DN}_c(X)$ (*resp.* of $\mathcal{DM}_c(X)$) that contains all the $f_*\mathbb{Q}_Y$ for $f:Y\to X$ a proper morphism of relative dimension $\leqslant n$. These are the *categories of $n$-motives*. If one only allow the $f$'s to be smooth on top of being proper, we denote by $\mathcal{DN}_c^{\mathrm{sm},n}(X)$ (*resp.* $\mathcal{DM}_c^{\mathrm{sm},n}(X)$) the categories of *smooth* $n$-motives. When taking proper maps $f:Y\to X$ of any relative dimension, we denote by $\mathcal{DN}_c^\mathrm{coh}(X)$ (*resp.* of $\mathcal{DM}^\mathrm{coh}_c(X)$) the categories of *cohomological* motives. There are non compact versions of these categories : **Definition 129**. Let $n\in \mathbb{N}$ and let $X$ be a finite-type $k$-scheme. Denote by $\mathcal{DN}^n(X)$ (*resp.* $\mathcal{DM}^n(X)$) the smallest localising full subcategory of $\mathcal{DN}(X)$ (*resp.* of $\mathcal{DM}(X)$) that contains all the $f_*\mathbb{Q}_Y$ for $f:Y\to X$ a proper morphism of relative dimension $\leqslant n$. Similar definitions can be made for $\mathcal{DN}^{\mathrm{sm},n}(X)$ , $\mathcal{DM}^{\mathrm{sm},n}(X)$ $\mathcal{DN}^\mathrm{coh}(X)$ and $\mathcal{DM}^\mathrm{coh}(X)$. For each $\mathcal{C}_c(X)$ of the , we have $\mathrm{Ind}\mathcal{C}_c(X)\simeq \mathcal{C}(X)$ with $\mathcal{C}(X)$ the corresponding category in . Of course as the realisation functor $\nu : \mathcal{DM}(X)\to\mathcal{DN}(X)$ commutes with the $6$ operations, it preserves the categories defined above. By we have continuity for perverse Nori motives. We then have : **Lemma 130**. *Let $X=\lim_{i\in I}X_i$ be a limit of schemes $X_i$ of finite type over a field $k_i$, with affine transition, and $k = \mathrm{colim}_i k_i$. Then $\mathcal{DN}_c^n(X) = \mathrm{colim}_i \mathcal{DN}_c^n(X_i)$ and $\mathcal{DN}_c^{\mathrm{sm},n}(X) = \mathrm{colim}_i \mathcal{DN}_c^{\mathrm{sm},n}(X_i)$.* *Proof.* We only deal with smooth $n$-motives as the other case is identical. By continuity for the full category of motives, we have that $\mathrm{colim}_i \mathcal{DN}_c^{\mathrm{sm},n}(X_i)$ is a stable full subcategory of $\mathcal{DN}_c(X)$. Its image is contained in the stable full subcategory generated by the $f_i^*(h_i)_*\mathbb{Q}_{Y_i}$ with $f_i : X\to X_i$ the projection and $h_i : Y_i\to X_i$ a smooth proper morphism of relative dimension $\leqslant n$. As being proper, smooth, and of relative dimension $\leqslant n$ is stable by base change, proper base change gives that $f_i^*(h_i)_*\mathbb{Q}_{Y_i}$ is in $\mathcal{DN}_c^{\mathrm{sm},n}(X)$, hence $\mathrm{colim}_i \mathcal{DN}_c^{\mathrm{sm},n}(X)(X_i)$ is a stable full subcategory of $\mathcal{DN}_c^{\mathrm{sm},n}(X)(X)$. By [@MR3920833 Lemma 1.24] the generators of $\mathcal{DN}_c^{\mathrm{sm},n}(X)$ are in the colimit, thus the lemma. ◻ The same proofs as [@MR3920833 Proposition 1.19 and Proposition 1.25] give : **Proposition 131**. *Let $X$ be a finite type $k$-scheme and let $M\in\mathcal{DN}_c(X)$. The following are equivalent :* 1. *$M\in \mathcal{DN}_c^\mathrm{coh}(X)$ (*resp.* in $\mathcal{DN}_c^{n}(X)$)* 2. *There exists a closed subset $i:Z\to X$ with open complement $j:U\to X$ such that $j^*M\in \mathcal{DN}_c^\mathrm{coh}(U)$ (*resp.* in $\mathcal{DN}_c^{n}(U)$) and $i^*M \in \mathcal{DN}_c^\mathrm{coh}(Z)$ (*resp.* in $\mathcal{DN}_c^{n}(Z)$).* 3. *For all $x\in X$, $x^*M \in \mathcal{DN}_c^\mathrm{coh}(\kappa(x))$ (*resp.* in $\mathcal{DN}_c^{n}(\kappa(x))$)* **Theorem 132** (Ayoub -- Barbieri-Viale -- Huber -- Müller-Stachs -- Orgogozo -- Pepin-Lehalleur). *Let $k$ be a field a characteristic zero. The induced functor $$\nu_X^0:\mathcal{D}^b(\mathrm{Rep}^A_\mathbb{Q}(\mathrm{Gal}(\overline{k}/k)))\simeq \mathcal{DM}_c^0(\mathrm{Spec}\ k)\to\mathcal{DN}_c^0(\mathrm{Spec}\ k)$$ is an equivalence of stable $\infty$-categories. Moreover, the composition with the inclusion $$\mathcal{D}^b(\mathrm{Rep}^A_\mathbb{Q}(\mathrm{Gal}(\overline{k}/k)))\to\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(\mathrm{Spec}\ k))$$ is $t$-exact.* *Proof.* By [@MR3920833 Proposition 4.19 and Proposition 1.28] we have the $t$-exact equivalence $$\mathcal{D}^b(\mathrm{Rep}^A_\mathbb{Q}(\mathrm{Gal}(\overline{k}/k)))\simeq \mathcal{DM}_c^0(\mathrm{Spec}\ k).$$ By [@MR3618276 Proposition 9.4.3], the smallest full subcategory of ${\mathcal{M}_{\mathrm{perv}}}(\mathrm{Spec}\ k)$ that contains the $\mathrm{H}^0(f_*\mathbb{Q}_X)$ for $f:X\to \mathrm{Spec}\ k$ finite étale is stable under subobjects coincides with the category $\mathrm{Rep}_Q^A(\mathrm{Gal}(\overline{k}/k))$. Indeed, they prove that the universal diagram category $\mathcal{MM}^0_\mathrm{Nori}(k)$ of the diagram of pairs $(X,Y,0)$ with $\dim X = 0$ is $\mathrm{Rep}_\mathbb{Q}^A(\mathrm{Gal}(\overline{k}/k))$ and that the natural faithful exact functor $\mathcal{MM}^0_\mathrm{Nori}(k)\to{\mathcal{M}_{\mathrm{perv}}}(k)$ is fully faithful of image the objects that are subquotients of motives of the form $\mathrm{H}^0(f_*\mathbb{Q}_X)$ for $f:X\to\mathrm{Spec}\ k$ finite étale. Now, all objects of $\mathcal{MM}^0_\mathrm{Nori}(k)$ are pure of weight $0$, hence this category is semi-simple and stable under extension in ${\mathcal{M}_{\mathrm{perv}}}(\mathrm{Spec}\ k)$. This shows that the natural map $\mathcal{D}^b(\mathcal{MM}^0_\mathrm{Nori}(k))\to\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(\mathrm{Spec}\ k))$ is fully faithful. As all generators of $\mathcal{DN}^0_c(\mathrm{Spec}\ k)$ are in the heart ${\mathcal{M}_{\mathrm{perv}}}(\mathrm{Spec}\ k)$ we see that the essential image is exactly $\mathcal{DN}_c^0(\mathrm{Spec}\ k)$. Of course, the functor $\mathcal{DM}^0_c(\mathrm{Spec}\ k)\to\mathcal{DN}^0_c(\mathrm{Spec}\ k)$ is $t$-exact (this can be seen by looking at the realisations) and induces the promised equivalence $$\mathcal{DM}^0_c(\mathrm{Spec}\ k) \simeq\mathcal{D}^b(\mathrm{Rep}^A_\mathbb{Q}(\mathrm{Ga}(\overline{k}/k)))\simeq\mathcal{D}^b(\mathcal{MM}^0_\mathrm{Nori}(k))\simeq \mathcal{DN}^0_c(\mathrm{Spec}\ k).$$ ◻ We use the ideas of Ruimy [@ruimyArtinPerverseSheaves2023], Nair and Vaish [@MR3293216] and S. Morel [@MR2350050] to have an Artin truncation functor on $\mathcal{DN}^\mathrm{coh}_c(\mathrm{Spec}\ k)$. **Construction 133**. Denote by $$\mathcal{DN}_{(c)}(\mathrm{Spec}\ k)^{ \leqslant \mathrm{Id}} := \{M\in\mathcal{DN}_{(c)}(\mathrm{Spec}\ k)\mid \forall i\in\mathbb{Z}, \mathrm{H}^i(M) \text{ is of weights }\leqslant 0\}$$ and by and by $$\mathcal{DN}_{(c)}(\mathrm{Spec}\ k)^{> \mathrm{Id}} := \{M\in\mathcal{DN}_{(c)}(\mathrm{Spec}\ k)\mid \forall i\in\mathbb{Z}, \mathrm{H}^i(M) \text{ is of weights }> 0\} .$$ By [@MR2350050 Proposition 3.1.1], this form a $t$-structure $t^\mathrm{Id}$ on $\mathcal{DN}_{(c)}(\mathrm{Spec}\ k)$. Denote by $\tau^{\leqslant \mathrm{Id}}$ be the truncation functor right adjoint to the inclusion $\mathcal{DN}_{(c)}(\mathrm{Spec}\ k)^{\omega \leqslant 0}\to(\mathrm{Ind})\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(\mathrm{Spec}\ k))$. As the canonical $t$-structure on $\mathcal{DN}(\mathrm{Spec}\ k)$ restricts to $\mathcal{DN}_c(\mathrm{Spec}\ k)$, we see that the $t$-structure $t^\mathrm{Id}$ of Morel on $\mathcal{DN}(\mathrm{Spec}\ k)$ restricts to $\mathcal{DN}_c(\mathrm{Spec}\ k)$. Consider also the colimit preserving inclusion $\mathcal{DN}^0(\mathrm{Spec}\ k)\to\mathcal{DN}^\mathrm{coh}(\mathrm{Spec}\ k)$. We denote by $\omega^0$ its right adjoint. **Proposition 134**. *The intersection $\mathcal{DN}^\mathrm{coh}(\mathrm{Spec}\ k)\cap \mathcal{DN}(\mathrm{Spec}\ k)^{\leqslant \mathrm{Id}}$ is the $\infty$-category $\mathcal{DN}^0(\mathrm{Spec}\ k)$ of Artin motives in $\mathrm{Ind}\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(\mathrm{Spec}\ k))$. Morever, $\omega^0\simeq (\tau^{\leqslant\mathrm{Id}})_{\mid \mathcal{DN}^\mathrm{coh}(\mathrm{Spec}\ k)}$. In particular $\omega^0$ preserves compact objects.* *Proof.* We have an obvious inclusion $\mathcal{DN}^0(\mathrm{Spec}\ k)\subset \mathcal{DN}^\mathrm{coh}(\mathrm{Spec}\ k)\cap \mathcal{DN}(\mathrm{Spec}\ k)^{ \leqslant \mathrm{Id}}$ as any the generators of Artin motives are in the intersection, which is stable under extensions and shifts. Let $M = f_*\mathbb{Q}_X$ where $f:X\to\mathrm{Spec}\ k$ is a smooth and proper morphism. Denote by $X\overset{\alpha}{\to}\pi_0(X)\overset{\pi}{\to}\mathrm{Spec}\ k$ the Stein factorisation of $f$ (see [@MR0217085 Corollaire 4.3.3]). There is a natural map $\pi_*\mathbb{Q}_{\pi_0(X)}\to f_*\mathbb{Q}_X$, which is an isomorphism on $\mathrm{H}^0$, as it can be seen after realisation ($\mathrm{H}^0(\pi_0(X))\simeq \mathrm{H}^0(X)$ is of rank the number of connected components of $X$.) It is also an isomorphism on $\mathrm{H}^i$ for $i<0$ because pushforward are left exact. As $\mathrm{H}^i(\pi_*\mathbb{Q}_{\pi_0(X)}) = 0$ if $i>0$ because $\pi$ is finite étale hence $\pi_*$ is $t$-exact, we have in fact that the cofiber $C$ of the map $\pi_*\mathbb{Q}_{\pi_0(X)}\to f_*\mathbb{Q}_X$ verifies $\mathrm{H}^i(C)=0$ if $i\leqslant 0$ and $\mathrm{H}^i(C)\simeq\mathrm{H}^i(f_*\mathbb{Q}_X)$ for $i>0$. The cofiber sequence $$\pi_*\mathbb{Q}_{\pi_0(X)}\to f_*\mathbb{Q}_X \to C$$ is therefore equivalent to the truncation cofiber sequence $$\tau^{\leqslant 0}f_*\mathbb{Q}_X\to f_*\mathbb{Q}_X \to \tau^{>0}f_*\mathbb{Q}_X$$ for the canonical $t$-structure. For $i\geqslant 1$, the weight of $\mathrm{H}^i(f_*\mathbb{Q}_X)$ is $i$ and the weight of $\pi_*\mathbb{Q}_{\pi_0(X)}$ is zero, hence this cofiber sequence is also the truncation cofiber sequence for the Morel $t$-structure: the map $\pi_*\mathbb{Q}_{\pi_0(X)}\to f_*\mathbb{Q}_X$ coincides with the map $\tau^{\leqslant \mathrm{Id}}f_*\mathbb{Q}_X\to f_*\mathbb{Q}_X$. In particular, $\tau^{\leqslant \mathrm{Id}}M$ is an Artin motive. As all such $M$ are generators under colimits and shifts of the $\infty$-category $\mathcal{DN}^\mathrm{coh}(\mathrm{Spec}\ k)$, we see that the restriction of $\tau^{\leqslant \mathrm{Id}}$ to that $\infty$-category lands in $\mathcal{DN}^0(\mathrm{Spec}\ k)$. Thus, if $M_0\in\mathcal{DN}^0(\mathrm{Spec}\ k)$ and $N\in\mathcal{DN}^\mathrm{coh}(\mathrm{Spec}\ k)$ we have $$\begin{aligned} \mathrm{Map}_{\mathcal{DN}^\mathrm{coh}}(M_0,N) & = & \mathrm{Map}_{\mathcal{DN}(\mathrm{Spec}\ k)^{\leqslant \mathrm{Id}}}(M_0,\tau^{\leqslant \mathrm{Id}}N) \\ & = & \mathrm{Map}_{\mathcal{DN}^0}(M_0,\tau^{\leqslant \mathrm{Id}}N)\\ & = & \mathrm{Map}_{\mathcal{DN}^0}(M_0,\omega^0N) \end{aligned}$$ which proves that $\omega^0\simeq (\tau^{\leqslant\mathrm{Id}})_{\mid \mathcal{DN}^\mathrm{coh}(\mathrm{Spec}\ k)}$ is the right adjoint to the inclusion of Artin motives in cohomological motives. ◻ **Corollary 135**. *Let $\omega^0:\mathcal{DM}^\mathrm{coh}(\mathrm{Spec}\ k)\to\mathcal{DM}^0(\mathrm{Spec}\ k)$ be the right adjoint to the inclusion constructed by [@MR3920833], [@MR4109490] and [@MR2494373]. Then we have a natural equivalence $\nu\circ\omega^0\simeq\omega^0\circ \nu$.* *Proof.* As $\nu$ sends Artin motives to Artin motives we have a natural transformation $\nu\circ\omega^0\to \omega^0\circ \nu$. To show that this functor is an equivalence it suffices to check that this natural transformation is an equivalence when evaluated at the generators $f_*\mathbb{Q}_X$ of cohomological motives, where $f:X\to \mathrm{Spec}\ k$ is proper and smooth. This is the case by commutation of $\nu$ with the operations and because of the computation $\omega^0f_*\mathbb{Q}_X\simeq \pi_*\mathbb{Q}_{\pi_0(X)}$ with $\pi:\pi_0(X)\to\mathrm{Spec}\ k$ the Stein factorisation of $X$. Indeed for Nori motives this has been proven in the previous proposition, and for Voevodsky motives this is [@MR3920833 Proposition 3.7]. ◻ **Lemma 136**. *Let $S$ be a regular connected finite type $k$-scheme. Let $f:X\to S$ and $g:Y\to S$ be two finite étale morphisms and $m\in\mathbb{Z}$. Then the induced map $$\nu : \mathrm{Hom}_{\mathcal{DM}_c(S)}(f_*\mathbb{Q}_X,g_*\mathbb{Q}_Y[m]) \to \mathrm{Hom}_{\mathcal{DN}_c(S)}(f_*\mathbb{Q}_X,g_*\mathbb{Q}_Y[m])$$ is an isomorphism.* *Proof.* Assume first that $m=0$. Note that by [@MR3920833 Proposition 4.1], the $\ell$-adic realisation on the compact $0$-motives is conservative. As $f_*\mathbb{Q}_X$ and $g_*\mathbb{Q}_Y$ are in the heart of the $t$-structure, it follows that the map $\nu$ of the lemma is injective. Let $d$ be the dimension of $S$. As $X$ and $Y$ are then also of dimension $d$, and $f_*$ and $g_*$ are $t$-exact for the perverse $t$-structure, we have that $f_*\mathbb{Q}_X[-d]$ and $g_*\mathbb{Q}_Y[-d]$ are in the perverse heart ${\mathcal{M}_{\mathrm{perv}}}(S)$. They are pure of weight $w=d$, and have strict support $S$ because for all $s\in S$, $s^*f_*\mathbb{Q}_X = (f_s)_*\mathbb{Q}_{X_s}$ by proper base change, and $(f_s)_*\mathbb{Q}_{X_s}\neq 0$ because $S$ is connected and the $\ell$-adic realisation is conservative (the group $\mathrm{H}^0_{\acute{e}t}(X_s,\mathbb{Q}_{\ell})$ is of constant dimension when $s$ varies.) Therefore, with the notations of the proof of [@ivorraFourOperationsPerverse2022 Theorem 6.24], $f_*\mathbb{Q}_X$ and $g_*\mathbb{Q}_Y$ live in ${\mathcal{M}_{\mathrm{perv}}}(S)_w^0$, which satisfies ${\mathcal{M}_{\mathrm{perv}}}(S)_w^0 = {\mathcal{M}_{\mathrm{perv}}}(\eta)_w$ with $\eta$ the generic point of $S$. We have the following diagram : $$\begin{tikzcd} \mathcal{DM}^{0,\mathrm{sm}}_c(S) \arrow[r] \arrow[d] & \mathcal{DN}^{0,\mathrm{sm}}_c(S) \arrow[d] \\ \mathcal{DM}^{0}_c(\eta) \arrow[r] & \mathcal{DN}^{0}_c(\eta) \end{tikzcd}$$ in which the vertical maps are the pullbacks by $\eta$. By [@MR4033829 Theorem 4.1], the category $\mathcal{DM}^0_c(S)$ is endowed with a $t$-structure such that Artin representations of the étale fundamental group $\pi_1^{\acute{e}t}(S)$ embeds fully faithfully in the heart. In particular, our $f_*\mathbb{Q}_X$ are objects of the heart. This implies that on objects of the heart, because $S$ is normal, the left vertical morphism is full. Now, if $\alpha \in \mathrm{Hom}_{\mathcal{DN}_c(S)}(f_*\mathbb{Q}_X,g_*\mathbb{Q}_Y)$, the commutation of the above diagram gives $\beta\in \mathrm{Hom}_{\mathcal{DM}_c(S)}(f_*\mathbb{Q}_X,g_*\mathbb{Q}_Y)$ such that $\eta^*\beta = \nu^{-1}(\eta^*\alpha)$. Therefore, $\nu(\beta) = \alpha$ by ${\mathcal{M}_{\mathrm{perv}}}(S)_w^0 = {\mathcal{M}_{\mathrm{perv}}}(\eta)_w$, and this finishes the proof. Now, for $m\neq 0$, the left hand side is $\mathrm{Hom}_{\mathcal{DM}_c(S)}(f_*\mathbb{Q}_X,g_*\mathbb{Q}_Y[m]) = \mathrm{Hom}_{\mathcal{DM}_c(S)}(\mathbb{Q}_S,\pi_*\mathbb{Q}_{X\times_S Y}[m])$ because each $f_*\mathbb{Q}_X$ is self dual and the Kunneth formula. Therefore, for $m\neq 0$, it is zero as $\mathrm{Hom}_{\mathcal{DM}_c(S)}(\mathbb{Q}_S,\pi_*\mathbb{Q}_{X\times_S Y}[m]) = \mathrm{H}^m_{\acute{e}t}(X\times_S Y,\mathbb{Q}) = 0$ by [@ruimyabelianCategoriesArtin2023 Proposition 1.7.2 and Corollary 2.1.4]. The right hand side is also zero for $m< 0$ because $f_*\mathbb{Q}_X$ and $g_*\mathbb{Q}_Y$ are in the heart. It is also zero for $m>0$ because the category of pure weight $w$ objects of the heart is semi-simple. ◻ **Proposition 137**. *Let $S$ be a regular finite type $k$-scheme. Then the functor $$\nu_S^{0,\mathrm{sm}} :\mathcal{DM}_c^{0,\mathrm{sm}}(S)\to \mathcal{DN}_c^{0,\mathrm{sm}}(S)$$ is an equivalence of categories.* *Proof.* The functor $\nu : \mathcal{DM}(S)\to\mathcal{DN}(S)$ preserves compact objects, an commutes with $f_*$ for any morphism $f$ of finite type. Therefore, if we denote by $\mathcal{DM}^{0,\mathrm{sm}}(S)$ (*resp.* $\mathcal{DM}^{0,\mathrm{sm}}(S)$) the smallest presentable stable subcategory containing the $f_*\mathbb{Q}_X$ for $X$ finite étale the functor $\nu$ restricts to a functor $\nu : \mathcal{DM}^{0,\mathrm{sm}}(S)\to\mathcal{DN}^{0,\mathrm{sm}}(S)$. We are going to prove that this functor is an equivalence of categories, and then as it preserves compact objects we will obtain the result. $\nu$ admits a right adjoint $F$. Now, thanks to the computation of , the proof is the same as [@ruimyabelianCategoriesArtin2023 Theorem 2.1.1]. For completeness we recall the arguments. As $\nu$ sends generators to generators, it suffices to prove that it is fully faithful. Therefore, it suffices to show that the unit $\eta : \mathrm{Id}\to F\nu$ is an equivalence. As $\mathcal{DM}^{0,\mathrm{sm}}(S)$ is generated by the $f_*\mathbb{Q}_X$ for $f$ finite étale, the family of functors $\mathrm{Map}_{\mathcal{DM}(S)}(f_*\mathbb{Q}_X,-)$ for such $f$'s is conservative. We only have to check that for all $M\in\mathcal{DM}^{0,\mathrm{sm}}(S)$, $n\in\mathbb{Z}$ and $f:X\to S$ finite étale, the map $$\eta_* : \mathrm{Hom}_{\mathcal{DM}(S)}(f_*\mathbb{Q}_X[n],M)\to \mathrm{Hom}_{\mathcal{DM}(S)}(f_*\mathbb{Q}_X[n],F\nu M)$$ is an isomorphism. By adjunction it suffices to show that the maps $$\label{} \nu : \mathrm{Hom}_{\mathcal{DM}(S)}(f_*\mathbb{Q}_X[n],M)\to \mathrm{Hom}_{\mathcal{DN}(S)}(\nu f_*\mathbb{Q}_X[n],\nu M)$$ are isomorphisms. Both $f_*\mathbb{Q}_X[n]$ and $\nu f_*\mathbb{Q}_X[n]$ are compact and $\nu$ commutes with colimits hence we can assume that $M = g_*\mathbb{Q}_Y[m]$ for some $m\in\mathbb{Z}$ and $g:Y\to S$ finite étale. The result then follows from . ◻ **Corollary 138**. *Let $S$ be a regular finite-type $k$-scheme. The constructible $t$-structure on $\mathcal{DN}_c(S)$ restricts to smooth $0$-motives, with heart $\mathrm{Rep}^A_\mathbb{Q}(\pi^1_{\acute{e}t}(S))$. Moreover we have a $t$-equivalence of $\infty$-categories to $\mathcal{D}^b(\mathrm{Rep}^A_\mathbb{Q}(\pi^1_{\acute{e}t}(S)))\simeq \mathcal{DN}_c^{0,\mathrm{sm}}(S)$.* *Proof.* Indeed, if we endow smooth $0$-motives with the $t$-structure of Ruimy transported by $\nu$, then as the $\ell$-adic realisation is $t$-exact and conservative, we get the result. ◻ We now deal with non necessarily smooth $0$-motives. For this, we will need the truncation functor of Vaish (see also [@MR4033829]) over a general basis. **Proposition 139**. *Let $S$ be a finite type $k$-scheme. The inclusion $\mathcal{DN}^0_c(S)\subset \mathcal{DN}^{\mathrm{coh}}_c(S)$ admits a right adjoint $\omega^0$. Moreover, $\nu_S$ commutes with $\omega^0$.* *Proof.* The existence of $\omega^0$ for Nori motives is proven exactly in the same way as for étale motives in [@MR4109490 Theorem 5.2.2]. This uses the existence of a truncation functor over a field, a fact ensured by . By definition, $\omega^0$ is a truncation functor for a $m$-structure ([@MR4109490 section 2.5]), obtained by punctual gluing. As $\nu_K$ is $m$-exact for all fields $K$ by , we have that $\nu_S$ is also $m$-exact, hence commutes with truncations. ◻ **Proposition 140**. *The functor $$\nu^0:\mathcal{DM}^0(S)\to\mathcal{DN}^0(S)$$ is fully faithful.* *Proof.* We prove the result by induction on the dimension of $S$. If $S$ is zero dimensional, as we can supposed $S$ reduced, $S$ is then a coproduct of spectrum of fields, for which we know the result. Assume that the result is known for all finite type $k$-schemes of dimension $\leqslant d-1$ and let $S$ be of dimension $d$. Let $M,N\in\mathcal{DM}_c^0(S)$. By the same proof as [@ruimyabelianCategoriesArtin2023 Proposition 2.4.2], we know that there is a dense and smooth open subset $j:U\to S$ on which $M$ and $N$ are smooth $0$-motives. Let $i:Z\to S$ be the reduced closed complement. Applying $\mathrm{Map}(-,N)$ to the cofiber sequence $j_!M_{\mid U}\to M\to i_* M_{\mid Z}$, we obtain the following commutative diagram with rows cofiber sequences of spectra: $$\begin{tikzcd} \mathrm{Map}_{\mathcal{DM}(S)}(i_*M_{\mid Z},N)\ar[d,"\nu_Z"] \ar[r] & \mathrm{Map}_{\mathcal{DM}(S)}(M,N) \ar[d,"\nu_S"]\ar[r] & \mathrm{Map}_{\mathcal{DM}(S)}(j_!M_{\mid U},N)\ar[d,"\nu_U"] \\ \mathrm{Map}_{\mathcal{DN}(S)}(i_*\nu(M)_{\mid Z},\nu(N)) \ar[r] & \mathrm{Map}_{\mathcal{DN}(S)}(\nu(M),\nu(N)) \ar[r] & \mathrm{Map}_{\mathcal{DN}(S)}(j_!\nu(M)_{\mid U},\nu(N)) \end{tikzcd}.$$ By adjunction and , $\nu_U$ is an isomorphism, hence $\nu_S$ is an isomorphism if and only if $\nu_Z$ is. By [@MR3920833 Proposition 1.12 (iv)], $i^!N$ is a cohomological motive, hence we have $$\begin{aligned} \mathrm{Map}_{\mathcal{DM}_c(S)}(i_*M,N)&\simeq& \mathrm{Map}_{\mathcal{DM}_c(Z)}(M,i^!N)\\ &\simeq& \mathrm{Map}_{\mathcal{DM}_c^0(Z)}(M,\omega^0i^!N) \\ &\simeq & \mathrm{Map}_{\mathcal{DN}_c^0(Z)}(M,\omega^0i^!N) \\ & \simeq & \mathrm{Map}_{\mathcal{DN}_c(S)}(i_*M,N) \end{aligned}$$ By adjunctions and the induction hypothesis. ◻ **Theorem 141**. *Let $S$ be a finite type $k$-scheme. The functor $$\nu^0:\mathcal{DM}^0_c(S)\to\mathcal{DN}^0_c(S)$$ is an equivalence of categories. Moreover, the composition $\nu^0 :\mathcal{DM}_c^0(S)\to\mathcal{D}^b({\mathcal{M}_{\mathrm{ct}}}(S))$ is $t$-exact.* *Proof.* By it suffices to show that it is essentially surjective. We prove it by induction on the dimension of $S$. The case of zero dimensional schemes is already done. Let $M\in \mathcal{DN}_c^0(S)$. Let $j:U\to S$ be the inclusion of a dense smooth open subset on which $M$ is smooth, and let $i:Z\to U$ be the inclusion of the reduced closed complement. By the smooth case and the induction hypothesis, there are $N_U \in\mathcal{DM}^{0,\mathrm{sm}}_c(U)$ and $N_Z\in\mathcal{DM}^0_c(Z)$ such that $\nu_U(N_U) = M_{\mid U}$ and $\nu_Z(N_Z) = M_{\mid Z}$. Moreover, $M$ is the cofiber of the map $\delta:i_*M_{\mid Z}[-1]\to j_!M_{\mid U}$. As $\nu_S$ is fully faithful, there exists a map $\alpha : i_*N_Z[-1]\to j_!N_U$ that maps to $\delta$ (note that as $i$ and $j$ are quasi finite, $i_*=i_!$ and $j_!$ preserve $0$-motives by the same proof as [@MR3920833 Proposition 1.17]). Let $N$ be the cofiber of $\alpha$. We have $\nu_S(N) = M$ because $\nu_S$ is an exact $\infty$-functor. The $t$-exactness comes from the $t$-exactness and conservativity of the $\ell$-adic realisation. ◻ Let $S$ be a scheme of finite dimension that lives over $\mathrm{Spec}\ \mathbb{Q}$. We can make the same definition as in to define Artin motives and smooth Artin motives over $S$ (recall that thanks to we have pullbacks and pushforward in that generality.) The same proof as gives that the assignment $S\mapsto \mathcal{DN}_c^{0,\mathrm{(sm)}}(S)$ is still continuous. As the same is true for Artin motives in Voevodsky category, we obtain: **Corollary 142**. *Let $S$ be a Noetherian finite dimensional scheme of characteristic zero. Then the functor $$\nu^0:\mathcal{DM}^0_c(S)\to\mathcal{DN}^0_c(S)$$ is an equivalence of $\infty$-categories. The same statement is true for smooth Artin motives.* **Definition 143**. Let $X$ be a finite type $k$-scheme. For $\mathcal{C}$ a stable symmetric monoidal full subcategory of $\mathcal{DM}(X)$ (*resp*. of $\mathcal{DN}(X)$), we denote by $\mathcal{C}^{\mathrm{rig}}$ the full subcategory of objects $C\in\mathcal{C}$ that are dualisable as objects of $\mathcal{DM}(X)$ (*resp*. of $\mathcal{DN}(X)$). Note that $\mathcal{C}^{\mathrm{rig}}$ is again stable, and is even closed under sub-quotients by Hovey, Palmieri, and Strickland's [@MR1388895 Theorem A.2.5 (a)]. **Proposition 144**. *The constructible $t$-structure on $\mathcal{DN}_c(X)$ restricts to the full subcategory $\mathcal{DN}_c^\mathrm{rig}(X)$ of dualisable objects.* *Proof.* This can be checked after $\ell$-adic realisation. Then this is [@MR4609461 Theorem 6.2 (2)]. ◻ **Lemma 145**. *Let $i:Z\to S$ be a closed immersion of finite type $k$-schemes with $S$ regular, such that the codimension of $Z$ in $S$ is $\geqslant 1$. Then for any $M\in\mathcal{DN}^{0,\mathrm{rig}}_c(S)$ we have $\omega^1(i^!M)=0$.* *Proof.* We argue as in [@MR4033829 Lemma 2.4] : If $i$ was a regular immersion of codimension $c>0$, then by absolute purity as in [@MR3920833 Proposition 1.7] we would have an isomorphism $i^!M\simeq i^*M(-c)[-2c]$. Now, by , $M = \nu_S(M')$ for some $M'\in \mathcal{DM}^{0}_c(S)$. By [@MR3920833 Corollary 3.9 (iii)], $\omega^0(i^*M'(-2c)[2c])=0$ (note that $i^!M'\in\mathcal{DM}^\mathrm{coh}$ by [@MR3920833 Proposition 1.12 (iv)]) hence $\omega^0(i^!M)=\nu_S(0)=0$. Now if $i$ is not a regular immersion, we can find a dense open subset $u:U\to Z$, such that $U$ is regular with reduced complement $i_1:Z_1\to Z$. As $U$ and $S$ are regular, [@stacks-project [Tag 0E9J](https://stacks.math.columbia.edu/tag/0E9J)] ensures that $i\circ u$ is a regular immersion of codimension $c_1\geqslant 1$. We have a cofiber sequence $$\begin{aligned} (i_1)_*(i\circ i_1)^!M \to M\to u_*(i\circ u)^!M \end{aligned}$$ which gives $\omega^0((i_1)_*(i\circ i_1)^!M)\simeq \omega^0(i^!M)$, because $(i\circ u)^!M\simeq (i\circ u)^*M[-2c](-c)$ thus $\omega^0(u_*(i\circ u)^!M) = 0$ by the same argument as above using [@MR3920833 Corollary 3.9 (iii)]. As $i_1$ is quasi-finite and $(i_1)_*\simeq (i_1)_!$, the functors $(i_1)_*$ and $\omega^0$ commute, giving finally that $(i_1)_*\omega^0((i\circ i_1)^!M)\simeq \omega^0(i^!M)$. A Noetherian induction then implies that $\omega^0(i^!M)=0$. ◻ **Lemma 146**. *Let $S$ be a connected normal finite type $k$-scheme with generic point $\eta$. Then if $M\in \mathrm{Rep}^A_\mathbb{Q}(\pi^1_{\acute{e}t}(\eta))$ is such that there is $N\in \mathrm{Rep}^A_{\mathbb{Q}_\ell}(\pi^1_{\acute{e}t}(S))$ whose restriction to $\eta$ is the $\ell$-adic realisation of $M$, then there is a $M'\in \mathrm{Rep}^A_\mathbb{Q}(\pi^1_{\acute{e}t}(S))$ whose restriction to $\eta$ is $M$.* *Proof.* Indeed it even works with integer coefficients : let $L\in \mathrm{Rep}^A_\mathbb{Z}(\pi^1_{\acute{e}t}(\eta))$ such that its Tate module $R_\ell(L)$ extends to $S$. We have to check that the morphism $\pi_1^{\acute{e}t}(\eta)\to\mathrm{GL}(L_{\overline{\eta}})$ factors through $\pi_1^{\acute{e}t}(S)$. But the composition with $\mathrm{GL}(L_{\overline{\eta}})\to\mathrm{GL}(L_{\overline{\eta}}\otimes_\mathbb{Z}\mathbb{Z}_\ell)$ does factor. Now if $L$ has no torsion and also the extension of its Tate module, its extension also is torsion free. ◻ **Proposition 147** (Haas). *Let $S$ be a regular $k$-scheme of finite type. Let $M\in\mathcal{DM}^0_c(S)^\heartsuit$. The following are equivalent :* 1. *$M\in\mathcal{DM}^{0,\mathrm{sm}}_c(S)^\heartsuit \simeq \mathrm{Rep}^A_\mathbb{Q}(\pi^1_{\acute{e}t}(S))$.* 2. *$M$ is dualisable as an object of $\mathcal{DM}(S)$.* 3. *$M$ is dualisable as an object of $\mathcal{DN}(S)$.* 4. *The $\ell$-adic realisation of $M$ is a lisse sheaf.* *That is, the natural functors $$\mathrm{Rep}^A_\mathbb{Q}(\pi^1_{\acute{e}t}(S))\to\mathcal{DM}^{\mathrm{rig}}_c(S)\cap \mathcal{DM}_c^0(S)^\heartsuit \to\mathcal{DN}_c^{0,\mathrm{rig}}(S)^\heartsuit\to \mathrm{Liss}(S_{\acute{e}t},\mathbb{Q}_\ell)\times_{\mathrm{Shv}_{\acute{e}t}(S,\mathbb{Q}_\ell)} \mathcal{DN}^0_c(S)^\heartsuit$$ are equivalences.* *Proof.* Of course, $1.\Rightarrow 2. \Rightarrow 3. \Rightarrow 4.$ because of Poincaré duality for finite étale morphisms and because a symmetric monoidal functor preserves dualisable objects. We prove that $4.\Rightarrow 1.$ by adapting Johann Haas' [@HaasThesis Lemma 6.12] to the simpler case of $0$-motives. We can suppose $S$ connected with generic point $\eta$. We first claim that the functor $\mathcal{DN}^{0,\mathrm{rig}}_c(S)^\heartsuit \to \mathcal{DN}^{0,\mathrm{rig}}_c(\eta)^\heartsuit$ is fully faithful. Indeed for any dense open subset $j:U\to S$, the combination of and of the localisation sequence gives that for $M\in\mathcal{DM}^{0,\mathrm{rig}}(S)$, $M\to \omega^0(j_*j^*M)$ is an equivalence as $\omega^0(i^!M)= 0$. Therefore the unit of the adjunction $(j^*,\tau^{\leqslant 0}\omega^0j_*)$ on $\mathcal{DN}^0_c(S)^\heartsuit$ is an equivalence on $\mathcal{DN}^{0,\mathrm{rig}}(S)^\heartsuit$, this means that $j^*$ is fully faithful on $\mathrm{DN}^{0,\mathrm{rig}}(S)^\heartsuit$. Passing to the colimit, $\eta^*$ is fully faithful. Therefore there is a commutative diagram $$\begin{tikzcd} \mathrm{Rep}^A_\mathbb{Q}(\pi^1_{\acute{e}t}(S)) \ar[r] \ar[d] & \mathcal{DN}^{0,\mathrm{rig}}_c(S)^\heartsuit \ar[d] \\ \mathrm{Rep}^A_\mathbb{Q}(\pi^1_{\acute{e}t}(\eta)) \ar[r] & \mathcal{DN}^{0}_c(\eta)^\heartsuit \end{tikzcd}$$ [\[goodRed\]]{#goodRed label="goodRed"} in which the lower horizontal functor is an equivalence of categories, the top functor is fully faithful by and the right vertical functor is fully faithful by what we've just seen. The left vertical functor is also fully faithful because $S$ is normal, by [@SGA1 Exposé V. Proposition 8.2]. But then by if $M\in\mathcal{DN}_c^{0,\mathrm{rig}}(S)^\heartsuit$, we can lift to to $S$ its pre-image in $\mathrm{Rep}^A_\mathbb{Q}(\pi_{\acute{e}t}^1(\eta))$ and the top functor is also essentially surjective. Therefore if $M\in\mathcal{DM}_c^0(S)^\heartsuit$ is such that its $\ell$-adic realisation is lisse (*i.e.* dualisable), then its image in the category of perverse Nori motives is in $\mathcal{DN}^{0,\mathrm{rig}}_c(S)^\heartsuit$, which is equivalent to $\mathrm{Rep}^A_\mathbb{Q}(\pi^1_{\acute{e}t}(S))$: our object $M$ is in $\mathcal{DM}^{0,\mathrm{sm}}_c(S)^\heartsuit$, and the proof is finished. ◻ **Corollary 148**. *Let $S$ be a regular $k$-scheme of finite type. Let $M\in\mathcal{DM}^0_c(S)=\mathcal{DN}^0_c(S)$. Then the following are equivalent :* 1. *$M\in\mathcal{DM}^{0,\mathrm{sm}}(S)=\mathcal{D}^b(\mathrm{Rep}^A_\mathbb{Q}(\pi^1_{\acute{e}t}(S)))$.* 2. *$M\in\mathcal{DM}(S)$ is dualisable.* 3. *The $\ell$-adic realisation of $M$ is dualisable.* *Moreover, in that case, the dual of $M$ is also a $0$-motive. That is, the inclusion $\mathcal{DM}_c^{0,\mathrm{sm}}(S)\to\mathcal{DM}_c^{0,\mathrm{rig}}(S)$ is an equivalence of categories, with $\mathcal{DM}_c^{0,\mathrm{rig}}(S)$ the full subcategory of $M\in\mathcal{DM}_c^0(S)$ that are dualisable in $\mathcal{DM}(S)$.* *Proof.* $2.\Rightarrow 3.$ is obvious, and the proof of $1.\Rightarrow 2.$ follows by dévissage, using that if $1.$, each $\mathrm{H}^i(M)$ are dualisable, that $M$ is an iterated extension of its $\mathrm{H}^i(M)$ and that by [@MR1388895 Theorem A.2.5 (a)] the full subcategory of dualisable objects in a symmetric monoidal stable $\infty$-category is thick. The proof of $3.\Rightarrow 1.$ goes as follow : the hypothesis on $M$ implies that $M$, seen as an object of $\mathcal{D}^b({\mathcal{M}_{\mathrm{ct}}}(S))$, is dualisable. Then by each $\mathrm{H}^i(M)$ is dualisable hence by the previous proposition, $\mathrm{H}^i(M)\in\mathcal{DM}^{0,\mathrm{sm}}_c(S)$. As the latter category is thick and $M$ is an iterated extension of its ${\ ^{\mathrm{ct}}\mathrm{H}}^i(M)$, we obtain that $M\in\mathcal{DM}^{0,\mathrm{sm}}_c(S)$. ◻ In the case $S$ is only normal, following Haas we also have a result in that direction, but not as strong as the preceding corollary that would be false by [@ruimyabelianCategoriesArtin2023 Remark 2.1.2]. However this enables us to endow $\mathcal{DM}^{0,\mathrm{sm}}_c(S)$ with a $t$-structure. **Corollary 149**. *Let $S$ be a normal finite type $k$-scheme. The natural functors $\mathcal{DM}^{0,\mathrm{sm}}_c(S)\to\mathcal{DM}^{0,\mathrm{rig}}(S) \to\mathcal{DN}^{0,\mathrm{rig}}_c(S)$ are equivalences of $\infty$-categories. If we transport the natural constructible $t$-structure from $\mathcal{DN}^{0,\mathrm{rig}}_c(S)$ to $\mathcal{DM}^{0,\mathrm{sm}}_c(S)$, the inclusion $\mathcal{DM}^{0,\mathrm{sm}}_c(S)\to \mathcal{DM}^{0}_c(S)$ is $t$-exact, and the heart of $\mathcal{DM}^{0,\mathrm{sm}}_c(S)$ is the category $\mathrm{Rep}^A_\mathbb{Q}(\pi^1_{\acute{e}t}(S))$ which embeds fully faithfully in it.* *Proof.* We can assume $S$ to be connected. By the two functors are fully faithful. Therefore it suffices to show that the composition functor is essentially surjective. By dévissage it suffices to show that any object $M\in\mathcal{DN}^{0,\mathrm{rig}}_c(S)^\heartsuit$ is in the image of $\mathcal{DM}_c^{0,\mathrm{sm}}(S)\to\mathcal{DN}_c^{0,\mathrm{rig}}(S)$. For this, we are going to show that the natural functor $\mathrm{Rep}^A_\mathbb{Q}(\pi^1_{\acute{e}t}(S))\to \mathcal{DN}^{0,\mathrm{rig}}_c(S)^\heartsuit$ is an equivalence of categories. Pick a dense regular open subset $U$ of $S$, with immersion $j:U\to S$. We have the following commutative diagram : $$\begin{tikzcd} \mathrm{Rep}^A_\mathbb{Q}(\pi^1_{\acute{e}t}(S)) \ar[r] \ar[d] & \mathcal{DN}^{0,\mathrm{rig}}_c(S)^\heartsuit \ar[d] \\ \mathrm{Rep}^A_\mathbb{Q}(\pi^1_{\acute{e}t}(U)) \ar[r] & \mathcal{DN}^{0,\mathrm{rig}}_c(U)^\heartsuit \end{tikzcd}.$$ In this diagram, the lower horizontal functor is an equivalence of categories. The left vertical functor is fully faithful because both $U$ and $S$ are normal and $j$ is dominant. Therefore, the right vertical functor being faithful (it can be check on the $\ell$-adic realisation), all functors are fully faithful. Now, the top functor is also essentially surjective thanks to . This shows $\mathrm{Rep}^A_\mathbb{Q}(\pi^1_{\acute{e}t}(S)) \to \mathcal{DM}_c^{0,\mathrm{sm}}(S)$ is fully faithful and that $\mathrm{Rep}^A_\mathbb{Q}(\pi^1_{\acute{e}t}(S)) \to \mathcal{DN}_c^{0,\mathrm{rig}}(S)^\heartsuit$ is an equivalence of categories. Therefore, $\mathcal{DM}_c^{0,\mathrm{sm}}(S)$ has a $t$-structure whose heart is $\mathrm{Rep}^A_\mathbb{Q}(\pi^1_{\acute{e}t}(S))$. Now, if $M\in\mathcal{DM}^{0,\mathrm{rig}}_c(S)$, then $\nu_S(M)$ has all its cohomology sheaves in $\mathrm{Rep}^A_\mathbb{Q}(\pi^1_{\acute{e}t}(S))$, hence $M\in\mathcal{DM}_c^{0,\mathrm{sm}}(S)$. This finishes the proof. ◻ We now apply this computation of $0$-motives to motivic Galois groups. **Notation 150**. *Let $\sigma : \overline{k}\to\mathbb{C}$ be an embedding. Let $X$ be a finite type $k$-scheme. We denote by $\mathcal{M}^{\mathrm{rig}}(X)$ the subcategory of ${\mathcal{M}_{\mathrm{ct}}}(X)$ whose objects are the dualisable objects. This means that for $M\in {\mathcal{M}_{\mathrm{ct}}}(X)$, we have $M\in \mathcal{M}^{\mathrm{rig}}(X)$ if and only if $M$ is dualisable and its dual is in ${\mathcal{M}_{\mathrm{perv}}}(X)$ (that last condition is automatic thanks to ).* **Definition 151**. Let $X$ be a connected and normal finite type $k$-scheme with a point $x:\mathrm{Spec}\ L\to X$, with $L/k$ a finite type extension. We define $p:\widetilde{X}\to X$ to be the limit of all finite étale $f:Y\to X$ with $Y$ connected with a point $y$ above $x$. The limit exists because the transitions morphisms have to be affine. There is a morphism $\pi_{\widetilde{X}} : \widetilde{X}\to\mathrm{Spec}\ \overline{k}$ compatible with $p$, which has a section $\widetilde{x}$. Therefore there is also a map $q : \widetilde{X}\to X_{\overline{k}}$. Base change gives a monoidal functor $p^* : \mathcal{DN}_c(X)\to\mathcal{DN}_c(\widetilde{X})$ which sends $\mathcal{M}^\mathrm{rig}(X)$ to $\mathcal{M}^\mathrm{rig}(\widetilde{X})$. **Lemma 152**. *Let $x:\mathrm{Spec}\ K\to X$ be a point of a connected finite type $k$ scheme, with $K/k$ a finite extension. Let $\omega_x : \mathcal{M}^\mathrm{rig}(X)\to\mathrm{Vect}_\mathbb{Q}$ be the composition of $x^*$ with the Betti realisation. Then $\mathcal{M}^\mathrm{rig}(X)$ is a Tannakian category with fiber functor $\omega_x$.* *Proof.* By , the Betti realisation of $M\in\mathcal{M}^\mathrm{rig}(X)$ is a local system, on which $x^*$ is conservative. Therefore $\omega_x = R_Bx^* = x^* R_B$ is conservative, proving the claim. ◻ **Lemma 153**. *Let $X$ and $\widetilde{X}$ as in . The category $\mathcal{M}^\mathrm{rig}(\widetilde{X})$ together with the fiber functor $\omega_{\widetilde{x}}$ is Tannakian.* *Proof.* The category $\mathcal{M}^\mathrm{rig}(\widetilde{X})$ is the colimit of all $\mathcal{M}^\mathrm{rig}(Y)$ with $Y$ finite étale over $X$. Hence it is Tannakian. ◻ **Notation 154**. *Let $X$ be a connected finite type $k$-scheme. We denote by $\mathcal{G}_\mathrm{mot}(X,x,\sigma)$ the Tannakian group of $\mathcal{M}^\mathrm{rig}(X)$.* Let $s:X_{\overline{k}}\to X$ be the projection. **Proposition 155**. *Let $X$ be a normal connected finite type $k$-scheme. The commutative diagram $$\begin{tikzcd} {\mathcal{DN}^{0,\mathrm{rig}\heartsuit}(k)} & {\mathcal{DN}^{0,\mathrm{rig}\heartsuit}(X)} & {\mathcal{DN}^{0,\mathrm{rig}\heartsuit}(X_{\overline{k}})} \\ {\mathcal{DN}^{0,\mathrm{rig}\heartsuit}(k)} & {\mathcal{M}^\mathrm{rig}(X)} & {\mathcal{M}^\mathrm{rig}(X_{\overline{k}})} \\ {*} & {\mathcal{M}^\mathrm{rig}(\widetilde{X})} & {\mathcal{M}^\mathrm{rig}(\widetilde{X})} \arrow[from=1-2, to=2-2] \arrow["{s^*}", from=1-3, to=2-3] \arrow["{p^*}", from=2-2, to=3-2] \arrow["{q^*}", from=2-3, to=3-3] \arrow["{s^*}", from=2-2, to=2-3] \arrow[from=3-2, to=3-3] \arrow["{\pi_X^*}", from=1-1, to=1-2] \arrow[from=1-2, to=1-3] \arrow[from=1-1, to=2-1] \arrow[from=2-1, to=2-2] \arrow[from=2-1, to=3-1] \arrow[from=3-1, to=3-2] \arrow["{\pi_X^*}", from=2-1, to=2-2] \end{tikzcd}$$ of categories gives by taking Tannakian duals in the following commutative diagram $$\begin{tikzcd} & 0 & 0 & {} \\ 0 & { \mathcal{G}_\mathrm{mot}(\widetilde{X},\widetilde{x},\sigma)} & { \mathcal{G}_\mathrm{mot}(\widetilde{X},\widetilde{x},\sigma)} & 0 \\ 0 & { \mathcal{G}_\mathrm{mot}(X_{\overline{k}},\overline{x},\sigma)} & { \mathcal{G}_\mathrm{mot}(X,x,\sigma)} & {\mathrm{Gal}(\overline{k}/k)} & 0 \\ 0 & {\pi_1^{\acute{e}t}(X_{\overline{k}},\overline{x})} & {\pi_1^{\acute{e}t}(X,x)} & {\mathrm{Gal}(\overline{k}/k)} & 0 \\ & 0 & 0 & 0 \arrow[from=2-2, to=2-3] \arrow[from=2-3, to=2-4] \arrow[from=2-1, to=2-2] \arrow[from=1-2, to=2-2] \arrow[from=1-3, to=2-3] \arrow[from=3-1, to=3-2] \arrow[from=2-2, to=3-2] \arrow[from=2-3, to=3-3] \arrow[from=3-2, to=3-3] \arrow[from=4-2, to=4-3] \arrow[from=4-1, to=4-2] \arrow[from=3-2, to=4-2] \arrow[from=3-3, to=4-3] \arrow[from=4-3, to=4-4] \arrow[from=4-4, to=4-5] \arrow[from=3-3, to=3-4] \arrow[from=3-4, to=3-5] \arrow[from=2-4, to=3-4] \arrow[from=3-4, to=4-4] \arrow[from=4-4, to=5-4] \arrow[from=4-2, to=5-2] \arrow[from=4-3, to=5-3] \end{tikzcd}$$ in which rows and columns are exact.* *Proof.* Taking Tannakian duals indeed give the commutative diagram of pro-algebraic $\mathbb{Q}$-groups by . The facts that the second maps in each column and each row is faithfully flat is equivalent to the fact that the maps between Tannakian categories are fully faithful and the image is closed under sub-quotients. Therefore it suffices to check that $\mathrm{Rep}^A_\mathbb{Q}(\mathrm{Gal}(\overline{k}/k))\to\mathrm{Rep}_\mathbb{Q}^A(\pi_{\acute{e}t}^1(X))$ and $\mathrm{Rep}_\mathbb{Q}^A(\pi_{\acute{e}t}^1(S))\to \mathcal{M}^{\mathrm{rig}}(X)$ are fully faithful with image closed under sub-quotients. For the first map, this is [@SGA1]. The second is fully faithful, we have to check that it is closed under sub-quotients. Working dually it suffices to check that the category $\mathrm{Rep}_\mathbb{Q}^A(\pi_{\acute{e}t}^1(X))$ is stable under subobjects in $\mathcal{M}^{\mathrm{rig}}(X)$. Let $M\to N$ be a monomorphism in $\mathcal{M}^{\mathrm{rig}}(X)$ with $N$ a $0$-motive. By to show that $M$ is a $0$-motive it suffices to check that for all $x\in X$ the motive $M_x$ is a $0$-motive. But by [@MR3618276 Theorem 9.1.16] the category of $0$-motives is closed under sub-quotients in the categories of motives over a field. We prove the middle exactitude and injectivity at the same time. As the proofs are the same, we only deal with the sequence $$\mathrm{DN}^{0,\mathrm{sm}\heartsuit}(X)\to\mathcal{M}^\mathrm{rig}(X)\to \mathcal{M}^\mathrm{rig}(\widetilde{X}) .$$ As in the proof of [@MR3618276 Theorem 9.1.16 and Erratum p. 234], we first prove that any motive $M\in \mathcal{M}^\mathrm{rig}(\widetilde{X})$ is a direct factor of the image of a motive $N \in \mathcal{M}^\mathrm{rig}(X)$. Indeed, if $M\in \mathcal{M}^\mathrm{rig}(\widetilde{X})$, there is a finite étale morphism $f:Y\to X$ (that we can assume to be Galois) such that $M$ is defined over $Y$ that is $M \in \mathcal{M}^\mathrm{rig}(Y)$. But as the composition $M\to f^*f_*M = f^!f_!M\to M$ is the multiplication by $\deg f$, $M$ is a direct factor of $f^*N$ with $N = f_*N \in \mathcal{M}^\mathrm{rig}(X)$. Let $\mathcal{U}$ be the Tannakian full subcategory of $\mathcal{M}^\mathrm{rig}(\widetilde{X})$ generated by the unit object $\mathbb{Q}$. We say that elements of $\mathcal{U}$ are *trivial*. To finish the proof we just have to prove that $M\in\mathcal{M}^{\mathrm{rig}}(X)$ has trivial image in $\mathcal{M}^\mathrm{rig}(\widetilde{X})$ if and only if it is in $\mathcal{DN}^{0,\mathrm{sm}\heartsuit}$. Of course, as any representation of $\pi_1^{\acute{e}t}(X)$ is trivialised after a base change to a finite étale cover of $X$, the image of Artin motives in $\mathcal{M}^\mathrm{rig}(\widetilde{X})$ is trivial. Conversely, let $M\in\mathcal{M}^\mathrm{rig}(X)$ with trivial image in $\mathcal{M}^\mathrm{rig}(\widetilde{X})$. This means that there is a finite étale morphism $f:Y\to X$ such that $f^*M$ is trivial. We can assume that $f$ is Galois. We have that $f_*f^*M$ is in the category generated by $f_*\mathbb{Q}_Y$, which is inside $\mathcal{DN}^{0,\mathrm{sm}\heartsuit}(X)$. As $M$ is a direct factor of $f_*f^*M$, we get that $M$ is an Artin motive. ◻ Swann Tubach [swann.tubach\@ens-lyon.fr]([email protected]) E.N.S Lyon, UMPA, 46 Allée d'Italie, 69364 Lyon Cedex 07, France [^1]: Note that for triangulated categories, it was not known in general that $\mathrm{Ind}\mathrm{D}^b(\mathcal{A})$ is triangulated for $\mathcal{A}$ an abelian category, see section 15.4 of [@MR1074006].

arxiv_math

--- abstract: | Monte Carlo (MC) methods, renowned for their dimension-independent convergence rate of $O(N^{-1/2})$, are pivotal in computational problems. However, this rate might not always meet practical requirements. This study delves into amalgamating quasi-Monte Carlo (QMC) methods and importance sampling (IS) to enhance this rate. QMC methods, deterministic counterparts of MC, utilize low-discrepancy sequences and have found extensive applications in finance and statistics over the past three decades. Under specific conditions, QMC methods achieve an error bound of $O(\frac{{\log N}^d}{N})$ for d-dimensional integrals, surpassing the conventional MC rate. The study further explores randomized QMC (RQMC), which maintains the QMC convergence rate and facilitates computational efficiency analysis. Emphasis is laid on integrating randomly shifted lattice rules, a distinct RQMC quadrature, with IS---a classic variance reduction technique. The study underscores the intricacies of establishing a theoretical convergence rate for IS in QMC compared to MC, given the influence of problem dimensions and smoothness on QMC. The research also touches on the significance of IS density selection and its potential implications. The study culminates in examining the error bound of IS with a randomly shifted lattice rule, drawing inspiration from the reproducing kernel Hilbert space (RKHS). In the realm of finance and statistics, many problems boil down to computing expectations, predominantly integrals concerning a Gaussian measure. This study considers optimal drift importance sampling (ODIS) and Laplace importance sampling (LapIS) as common importance densities. Conclusively, the paper establishes that under certain conditions, the IS-randomly shifted lattice rule can achieve a near $O(N^{-1})$ error bound. author: - "Zhan Zheng[^1]" - "Hejin Wang[^2]" - "Xiaoqun Wang[^3]" bibliography: - references.bib title: "On the convergence conditions of Laplace importance sampling with randomized quasi-Monte Carlo[^4]" --- Importance sampling, Quasi-Monte Carlo, Lattice rules 41A63, 65D30, 97N40 # Introduction Monte Carlo (MC) techniques are prevalent in numerous computational challenges. A salient feature of MC is its classic $O(N^{-1/2})$ convergence rate for square-integrable integrands, given a sample size n. While this rate is dimension-agnostic, it might not always meet practical application demands. To address this, we integrate quasi-Monte Carlo (QMC) approaches with importance sampling (IS) to bolster the convergence speed. QMC strategies, deterministic counterparts of MC, leverage low-discrepancy sequences. Over the past three decades, their application has increased in scientific computation, notably within the finance and statistics domains. Under specific regularity conditions, the QMC quadrature achieves an error bound of $O(N^{-1+\epsilon})$ for any $\epsilon>0$ in the dimensional integral, outperforming the standard MC rate. In practice, randomized QMC (RQMC) is frequently employed. It retains QMC's convergence attributes with the added benefit of facilitating computational efficiency analysis, such as error estimation or confidence interval construction. Various RQMC quadratures exist, including scrambled digital nets and randomly shifted lattice rules. Comprehensive insights into QMC and RQMC can be found in referenced works. In prior research, we melded scrambled digital nets with IS, establishing that an approximate $O(N^{-3/2+\epsilon})$ mean square error is achievable given specific integrand boundary growth conditions and optimal IS density selection. This study pivots on randomly shifted lattice rules, a distinct RQMC variant. We focus on the synergy between the randomly shifted lattice rule and IS. Recognized as a pivotal variance reduction technique in MC literature, IS's application in security pricing was explored in earlier studies. Subsequent research over the years underscored its efficacy in addressing rare events. It became evident that IS's utility extended beyond the conventional variance reduction method. Sampling from a desired distribution often differs from the original problem's focus. The efficacy of Importance Sampling (IS) is intrinsically tied to the choice of importance density, underscoring the significance of our selection. A comprehensive review of IS is available in [@Owen2013]. Incorporating IS within the Quasi-Monte Carlo (QMC) quadrature complicates the derivation of a theoretical convergence rate, especially when juxtaposed with Monte Carlo (MC). While MC remains largely indifferent to problem dimensions and smoothness (barring square-integrability), these factors profoundly impact QMC. Kuo [@Kuo2008a] demonstrated QMC's superiority over MC for log-likelihood integrals, and Dick et al.[@Dick2019] elucidated a weighted discrepancy bound for QMC-based IS, deriving explicit error bounds for adequately regular integrands. Our prior work highlighted potential boundary singularities in the unit hypercube due to IS density, an effect traceable to specific matrix eigenvalues. This insight spurred our exploration of error bounds for IS combined with randomly shifted lattice rules. However, our primary theoretical foundations are rooted in the Reproducing Kernel Hilbert Space (RKHS), a concept not applicable to scrambling. Many computational challenges in finance and statistics revolve around expectation calculations, often manifesting as Gaussian measure integrals. Common scenarios include Brownian motion-driven assets in security pricing or Gaussian priors in Bayesian computations. In determining importance density, two prevalent choices emerge, namely the Optimal Drift Importance Sampling (ODIS) and the Laplace Importance Sampling (LapIS). While ODIS employs a multivariate normal density with the original covariance matrix, LapIS opts for a general multivariate normal density, aligning its mean and covariance matrix with the integrand's mode and curvature. We also consider the multivariate t distribution as an IS proposal. Our findings indicate that the IS-randomly shifted lattice rule approximates an error bound under specific conditions. The paper's structure is as follows: Section 2 offers an overview of MC, QMC, IS, ODIS, LapIS, and RKHS. Section 3 delineates the theoretical error rate for RQMC-based IS estimators and extends the IS density to the t distribution. Section 4 presents examples illustrating the effects of ODIS and LapIS, and Section 5 concludes our study. # Preliminaries {#sec:pre} ## Monte Carlo and quasi-Monte Carlo method Consider an integral on the unit hypercube $$I(g) = \int_{(0,1)^d} g(\bm z) d\bm z.$$ For numerical computation, we take $$\label{eq:plainsim} \hat{I}_N(g) = \frac 1N \sum_{i=1}^N g(\bm z_i)$$ to simulate the integral. In MC setting, the $\bm z_i'$s are randomly sampled points in the unit hypercube. If the variance of the integrand $g(\bm z)$ is finite, i.e. $$\label{eq:MCvariance} \sigma^2 := \int_{(0,1)^d} (g(\bm z) -I(g))^2 d\bm z < \infty,$$ then by central limit theorem, the root mean square error (RMSE) of MC is $$\sqrt{\mathbb{E}[(\hat{I}_N(g)-I(g))^2]}=\frac {\sigma}{\sqrt{N}}.$$ In other words, MC has an RMSE rate $O(N^{-1/2})$ if the integrand is square-integrable. To speed up the convergence rate, we use QMC quadrature rule to replace MC. QMC takes the same form of [\[eq:plainsim\]](#eq:plainsim){reference-type="eqref" reference="eq:plainsim"}, whereas $\{\bm z_1,\dots,\bm z_N\}:=\mathcal{P}$ is a deterministic low-discrepancy point set in $(0,1)^d$ rather than independent and identically distributed (i.i.d.) points for MC. In this paper, we use lattice approach to constructing low-discrepancy point sets. The typical error bound of QMC integration is given by Koksma-Hlawka inequality [@Niederreiter1992] $$\label{eq:hk} |\hat{I}_N(g)-I(g)| \leq D^*(\mathcal{P}) V_\mathrm{HK}(g).$$ where $D^*(\mathcal{P})$ denotes the star discrepancy of the point set $\mathcal{P}$, while $V_\mathrm{HK}(\cdot)$ is the variation (in the sense of Hardy and Krause) of a function defined over the unit hypercube. Low-discrepancy sequences achieve a star discrepancy $O(N^{-1}(\log N)^d)$, therefore, as long as the variation is bounded, QMC integration has a deterministic error bound of $O(N^{-1}(\log N)^d)$, which is asymptotically superior to that of MC for a fixed dimension $d$. ## Importance sampling In this paper, we study integrals with respect to the normal density $$\label{eq:targetori} C = \int_{\mathbb{R}^d} G(\bm z)p(\bm z;\bm \mu_0,\bm \Sigma_0)d\bm z,$$ where $p(\bm z;\bm \mu,\bm \Sigma)$ denotes the probability density function of $d$-dimensional normal distribution with the mean $\bm \mu$ and the covariance matrix $\bm \Sigma$. Performing IS is using a proposal function $q$ (under some regular conditions) to change the integral into $$\begin{aligned} \label{eq:ISgenform} C &=& \int_{\mathbb{R}^d} G(\bm z) \frac{p(\bm z;\bm \mu_0,\bm \Sigma_0)}{q(\bm z)} q(\bm z) d\bm z \nonumber \\ &=& \int_{\mathbb{R}^d} G(\bm z) W(\bm z) q(\bm z) d\bm z \nonumber \\ &=& \int_{\mathbb{R}^d} G_{IS}(\bm z) q(\bm z) d\bm z, \end{aligned}$$ where $$\label{eq:LRf} W(\bm z) = \frac{p(\bm z;\bm \mu_0,\bm \Sigma_0)}{q(\bm z)}$$ is the likelihood ratio (LR) function, $G_{IS}(\bm z)=G(\bm z) W(\bm z)$ is the IS integrand. Firstly, we choose IS density among normal densities, i.e., $q(\bm z)=p(\bm z;\bm \mu,\bm \Sigma)$, yielding to $$\label{eq:LRnornorform} W(\bm z) = \sqrt{\frac{\det (\bm \Sigma)}{\det (\bm \Sigma_0)}}\exp(\frac 12 (\bm z-\bm \mu)^T\Sigma^{-1}(\bm z-\bm \mu)-\frac 12 (\bm z-\bm \mu_0)^T\Sigma_0^{-1}(\bm z-\bm \mu_0))$$ To associate IS with QMC method, we need to perform $$\int_{\mathbb{R}^d} F(\bm z) p(\bm z;\bm 0,\bm I_d) d\bm z = \int_{(0,1)^d} F(\Phi^{-1}(\bm u)) d\bm u,$$ where $\Phi(\cdot)$ denotes the cumulative distribution function (CDF) of the $d$-dimensional standard normal distribution, $\Phi^{-1}(\cdot)$ is its inverse (applied componentwise). By the affine transformation property of normal distribution, we write $$\begin{aligned} \label{eq:ISnorform} C &=& \int_{\mathbb{R}^d} G(\bm \mu+\bm L\bm z) W(\bm z)p(\bm z;\bm 0,\bm I_d)d\bm z\nonumber \\ &=& \int_{\mathbb{R}^d} G_{IS}(\bm z) p(\bm z;\bm 0,\bm I_d) d\bm z \nonumber\\ &=& \int_{(0,1)^d} G_{IS}(\Phi^{-1}(\bm u)) d\bm u, \end{aligned}$$ where $G_{IS}(\bm z)=G(\bm \mu+\bm L \bm z)W(\bm z)$,$\bm L$ is any square root decomposition of covariance matrix $\bm \Sigma$ (satisfying $\bm \Sigma=\bm L \bm L^T$), and the LR function is now $$\label{eq:LRstd} W(\bm z) = \sqrt{\frac{\det (\bm \Sigma)}{\det (\bm \Sigma_0)}}\exp(\frac 12 \bm z^T \bm z-\frac 12 (\bm \mu-\bm \mu_0+\bm L \bm z)^T\Sigma_0^{-1}(\bm \mu-\bm \mu_0+\bm L \bm z)).$$ It should be noted that one suffices to take $\bm \mu=\bm \mu_0$ and $\bm \Sigma=\bm \Sigma_0$ if IS is not applied. It is clear that a good choice of parameters $\bm \mu$ and $\bm \Sigma$ may reduce the calculation error. For MC simulation, that is equivalent to reducing the variance $\sigma^2$ given by [\[eq:MCvariance\]](#eq:MCvariance){reference-type="eqref" reference="eq:MCvariance"}. If we suppose that $G$ is nonnegative for all $\bm z$, apparently the zero-variance IS density is $$p_{opt}(\bm z) := \frac{1}{C}G(\bm z)p(\bm z;\bm \mu_0,\bm \Sigma_0).$$ However, it is not practical since the unknown integral $C$ is involved. Nevertheless, we are inspired to select an IS density which mimics the behavior of $p_{opt}$. Let $H(\bm z) =\log (p_{opt}(\bm z))$. Assume that $H(\bm z)$ is differentiable and unimodal. Let $\bm \mu_{\star}$ be the mode of $H(\bm z)$, i.e., $$\bm \mu_{\star} = \arg \max_{\bm z \in \mathbb{R}^d} H(\bm z)= \arg \max_{\bm z \in \mathbb{R}^d} G(\bm z)p(\bm z;\bm \mu_0,\bm \Sigma_0).$$ Taking a second-order Taylor approximation around the mode $\bm \mu_{\star}$ gives $$H(\bm y) \approx H(\bm \mu_{\star}) - \frac{1}{2}(\bm y - \bm \mu_{\star})^T \bm \Sigma_{\star}^{-1}(\bm y - \bm \mu_{\star}),$$ where $$\label{eq:solveoptvar} \bm \Sigma_{\star} = -\nabla^2 H(\bm \mu_{\star})^{-1} = (\bm\Sigma_0 - \nabla^2\log(G(\bm \mu_{\star})))^{-1}.$$ Therefore, we obtain $$p_{opt}(\bm z) = \exp (H(\bm z)) \approx \exp\left(H(\bm \mu_{\star})-\frac{1}{2}(\bm z - \bm \mu_{\star})^T \bm \Sigma_{\star}^{-1}(\bm z - \bm \mu _{\star})\right)\propto p(\bm z;\bm \mu_{\star},\bm \Sigma_{\star}).$$ In other words, $p(\bm z;\bm \mu_{\star},\bm \Sigma_{\star})$ is a density which is, in a sense, close to the optimal one. Then, LapIS just takes $\bm \mu= \bm \mu_{\star}$ and $\bm \Sigma=\bm \Sigma_{\star}$. Meanwhile, ODIS chooses $\bm \mu= \bm \mu_{\star}$ but keeps $\bm \Sigma=\bm \Sigma_0$ unchanged. We should note that LapIS is not necessarily better than ODIS (see e.g.,[@Zhang2021][@He2023]). Generally speaking, in MC setting LapIS tends to effective if $p_{opt}(\bm z)$ is close to a normal density. For a QMC point of view, it becomes more complicated since the performance is in addition affected by other factors, for example, the boundary growth for scrambled digital nets. We try to analyze this question for randomly shifted lattice rule in the following sections. ## Reproducing kernel Hilbert space RKHS is a suitable function space for error analysis, which can be reviewed from Aronszajn[@Aronszajn1950]. We refer to [@Kuo2006; @Waterhouse2006; @Nichols2014] for recent work. Consider integrals with respect to probability density which is multiplied by 1-dimension marginal density. That is $$\label{eq:martype} C(f) = \int_{\mathbb{R}^d} f(\bm x) \prod_{i=1}^d \phi(x_i)d\bm x.$$ Let $\Phi$ be the cumulative probability function of $\phi$: $$\Phi(x) = \int_{-\infty}^x \phi(y) dy,$$ $\Phi^{-1}$ is its inverse. For example, if $\phi(x)=p(x;0,1)$, $\Phi$ is then the cumulative probability function of standard normal distribution. Assume that $\mathcal{F}$ is an Hilbert space defined on functionals over $\mathbb{R}^d$. Then we have an isometric space $\mathcal{G}$ defined on functionals over $(0,1)^d$ by $$\label{eq:isometry} g=f\circ\Phi^{-1}.$$ Suppose that $K_{\mathcal{F}}$ is a reproducing kernel, which means a function $K_{\mathcal{F}}:\mathbb{R}^d \times \mathbb{R}^d \rightarrow \mathbb{R}$ satisfies the following 3 properties: - property of functional: $K_{\mathcal{F}}(\cdot,\bm x) \in \mathcal{F}$ for all $\bm x \in \mathbb{R}^d$.\ - property of symmetry: $K_{\mathcal{F}}(\bm x,\bm y) = K_{\mathcal{F}}(\bm y,\bm x)$ for all $\bm x,\bm y \in \mathbb{R}^d$.\ - property of reproducing:$<f,K_{\mathcal{F}}(\cdot,\bm x)>=f(\bm x)$ for all $\bm x \in \mathbb{R}^d$ and $f \in \mathcal{F}$. Furthermore, we need to assume that $$\label{eq:conditionofRKHS} K_{\mathcal{F}}(\bm x,\bm x) \prod_{i=1}^d \phi(x_i)d\bm x < \infty.$$ This is due to $$\label{eq:RKHSform} C(f) = \int_{\mathbb{R}^d} <f,K_{\mathcal{F}}(\cdot,\bm x)> \prod_{i=1}^d \phi(x_i)d\bm x = <f,h>,$$ where $$\label{eq:representation} h(\bm x)=\int_{\mathbb{R}^d} K_{\mathcal{F}}(\bm x,\bm y) \prod_{i=1}^d \phi(y_i)d\bm y.$$  [\[eq:conditionofRKHS\]](#eq:conditionofRKHS){reference-type="eqref" reference="eq:conditionofRKHS"} implies that [\[eq:RKHSform\]](#eq:RKHSform){reference-type="eqref" reference="eq:RKHSform"} and  [\[eq:representation\]](#eq:representation){reference-type="eqref" reference="eq:representation"} are well-defined. Then the initial error is given by $$e_0(\mathcal{F}) = \sup_{||f||_{\mathcal{F}}||\leq 1} |C(f)|= \sqrt{<h,h>},$$ and square both sides to obtain $$|e_0(\mathcal{F})|^2 = <h,h> \int_{\mathbb{R}^d}\int_{\mathbb{R}^d} K_{\mathcal{F}}(\bm x,\bm y) \prod_{i=1}^d [\phi(x_i)\phi(y_i)] d\bm x d\bm y.$$ Recall the isometry space $\mathcal{G}$ of $\mathcal{F}$ ([\[eq:isometry\]](#eq:isometry){reference-type="ref" reference="eq:isometry"}), the integration remains the same: $$\label{eq:sameint} C(f) = \int_{(0,1)^d} g(\bm u) d\bm u :=I(g).$$ In addition, $\mathcal{G}$ is also a RKHS equipped with reproducing kernel $$K_{\mathcal{G}}(\bm u,\bm v)=K_{\mathcal{F}}(\Phi^{-1}(\bm u),\Phi^{-1}(\bm v))$$ Thus we have the same initial error $$e_0(g) = \sup_{||g||_{\mathcal{G}}||\leq 1} |I(g))| = \sup_{||f||_{\mathcal{F}}||\leq 1} |C(f)| = e_0(f).$$ Now consider a QMC rule to estimate the integral [\[eq:sameint\]](#eq:sameint){reference-type="eqref" reference="eq:sameint"} $$\hat{I}_N(g)=\frac 1N \sum_{i=1}^N g(\bm u_i),$$ then the worst-case error in the space $\mathcal{G}$ is defined as $$e_w(\hat{I}_N,\mathcal{G})=\sup_{||g||_{\mathcal{G}}\leq 1} |\hat{I}_N(g)-I(g)|.$$ Therefore, we derive the relationship between the worst-case error above and the original integral $$|\hat{I}_N(f\circ\Phi^{-1})-C(f)|=|\hat{I}_N(g)-I(g)|\leq e_w(\hat{I}_N,\mathcal{G})||g||_{\mathcal{G}}=e_w(\hat{I}_N,\mathcal{G})||f||_{\mathcal{F}}.$$ Furthermore, randomly shifted lattice rule ia applied in this paper. Thus the points $\bm u_i,i=1,2,...,N$ of the shifted rank-1 lattice rule are $$u_i=\{\frac{iz}{N}+\zeta\},i=1,2,...,N.$$ where $\{\cdot\}$ takes the fractional part (component-wise to a vector), $\bm \zeta \in (0,1)^d$ is sampled from i.i.d. vectors, $\bm z \in \mathcal{Z}_N^d$ denotes the generating vector, the notation $$\mathcal{Z}_N = \{z\in \mathbb{Z}|1\leq z\leq N-1, (z,N)=1\},$$ i.e. the set of positive integers no more than $N-1$ which are relatively prime to $N$. Finally, we express the shift-averaged worst-case error, which only depends on the generating vector (see [@Sloan2002]) $$\begin{aligned} |e_N^{sa}(\bm z)|^2 &=& \int_{(0,1)^d} |e_w(\hat{I}_N,\mathcal{G})|^2 d\bm\zeta \nonumber \\ &=& -\int_{(0,1)^d}\int_{(0,1)^d} K_{\mathcal{G}}(\bm u,\bm v)d\bm ud\bm v+\frac 1N \sum_{i=1}^N K_{\mathcal{G}}^{sik}(\{\frac{i\bm z}{N}\}), \end{aligned}$$ where the last term $K_{\mathcal{G}}^{sik}$ denotes the shift-invariant kernel $$K_{\mathcal{G}}^{sik}(\bm u,\bm v) = \int_{(0,1)^d} K_{\mathcal{G}}(\{\bm u+\bm \zeta\},\{\bm v+\bm \zeta\})d\bm \zeta.$$ Note that it depends on the difference of $\bm u$ and $\bm v$, for simplicity we write $$\label{eq:sik} K_{\mathcal{G}}^{sik}(\{\bm u-\bm v\}):=K_{\mathcal{G}}^{sik}(\{\bm u-\bm v\},\bm 0)=K_{\mathcal{G}}^{sik}(\bm u,\bm v)$$ Shift-averaged worst-case error is the main target for error analysis of randomly shifted lattice rule. Here we introduce Fourier analysis to illustrate how to apply the theory above. For linear functional space $\mathcal{F}$ over $\mathbb{R}^d$, given weight function $\psi$ and weight coefficients $\gamma_k>0$, $k=1,2,...,d$, we define the inner product $$\label{eq:multiinnerprod} <f,g>_{\mathcal{F}}=\sum_{u \subset 1:d} \Big(\prod_{k \in u} \frac{1}{\gamma_k} \int_{\mathbb{R}^{|u|}} \frac{\partial^{|u|}f}{\partial \bm x_u}(\bm x_u,\bm 0) \frac{\partial^{|u|}g}{\partial \bm x_u}(\bm x_u,\bm 0) \prod_{k \in u} \psi^2(x_k) d\bm x_u \Big),$$ where $(\bm x_u,\bm 0)$ denotes a $d$-dimensional vector whose $k$'th component takes $x_k$ if $k \in u$, otherwise takes $0$. Now the reproducing kernel of each component is given by $$\label{eq:conofRKHS} \int_{R} K_{\mathcal{F},j}(y,y) \phi(y) dy < \infty,$$ where $K_{\mathcal{F},j}$ depends on $\gamma_j$ $$\label{eq:rkj} K_{\mathcal{F},j}(x,y)=1+\gamma_j\eta(x,y),$$ $$\label{eq:etaexpression} \eta(x,y)= \begin{cases} \int_0^{\max(x,y)} \frac{1}{\psi^2(t)}dt, & x,y>0,\\ \int_{\max(x,y)}^0 \frac{1}{\psi^2(t)}dt, & x,y<0,\\ 0, & \text{otherwise}. \end{cases}$$ Then the shift-invariant kernel [\[eq:sik\]](#eq:sik){reference-type="ref" reference="eq:sik"} becomes (see [@Kuo2006]) $$\label{eq:sifourierform} K_{\mathcal{G}}^{sa}(\bm u,\bm v) = \sum_{u \in 1:d} \gamma_u \prod_{i \in u} \theta(\{u_i-v_i\})), \quad \bm u,\bm v \in (0,1)^d,$$ where $$\label{eq:fourierform} \theta(\bm x)= \int_{\Phi^{-1}(x)}^0 \frac{\Phi(t)-x}{\psi^2(t)}dt +\int_{\Omega^{-1}(1-x)}^0 \frac{\Phi(t)-1+x}{\psi^2(t)}dt,\quad x \in (0,1).$$ Expand [\[eq:fourierform\]](#eq:fourierform){reference-type="eqref" reference="eq:fourierform"} by Fourier series, we have $$\label{eq:fourierexpansion} \theta(\bm x) = \sum_{h \in \mathbb{Z}} \hat{\theta}(h)\exp(2\pi ihx),$$ where the Fourier coefficients $$\label{eq:invfourierexpansion} \hat{\theta}(h) =\int_0^1 \theta(x)\exp(-2\pi ihx) dx.$$ Set $$\label{eq:C1expansion} C_1:=\int_{-\infty}^0 \frac{\Phi(t)}{\psi^2(t)}dt + \int_0^{\infty} \frac{1-\Phi(t)}{\psi^2(t)}dt.$$ Clearly $$C_1=\theta(0)=\sum_{h \in \mathbb{Z}} \hat{\theta}(h).$$ To avoid tedious statements, we close this section by an associated theorem. [\[thm:conofRKHSequ\]]{#thm:conofRKHSequ label="thm:conofRKHSequ"} The well-defined condition for RKHS [\[eq:conofRKHS\]](#eq:conofRKHS){reference-type="eqref" reference="eq:conofRKHS"} is equivalent to $$\label{eq:conofRKHSequ} C_1 < \infty$$ The proof is plain by applying Fubini theorem. # Main results {#sec:main} ## Estimators of ODIS and LapIS Again, consider the problem of estimating an integral $$\label{eq:target} C = \int_{\mathbb{R}^d} G(\bm z)p(\bm z;\bm \mu,\bm \Sigma)d\bm z.$$ Firstly we assume that $G(\bm z)>0$ for all $\bm z \in \mathbb{R}^d$. Let $g(\bm z) = \log G(\bm z)$. We rewrite [\[eq:target\]](#eq:target){reference-type="eqref" reference="eq:target"} as $$\label{eq:exptype} C = \int_{\mathbb{R}^d} \frac{\exp (H(\bm z))}{\sqrt{(2\pi)^d\det{(\bm \Sigma)}}}d\bm z,$$ where $H(\bm z)=g(\bm z) - \frac 12 \bm z^T \bm \Sigma^{-1} \bm z$. Assume $H$ is a unimodal function. We take $\bm z_{\star} = \arg \max_{\bm z \in \mathbb{R}^d} H(\bm z)$, which solves $$\label{eq:LapISdrift} \nabla H(\bm z) = \nabla g(\bm z) - \bm \Sigma^{-1} \bm z =0.$$ Here we deduce the estimators of ODIS and LapIS, respectively. ODIS takes importance density $p(\bm z;\bm z_{\star},\bm \Sigma)$. Therefore, the integral [\[eq:target\]](#eq:target){reference-type="eqref" reference="eq:target"} becomes $$\label{eq:ODIStype} \int_{\mathbb{R}^d} \frac{\exp (H(\bm z))}{\sqrt{(2\pi)^d\det{(\bm \Sigma)}}}d\bm z = (2\pi)^{-d/2} \int_{\mathbb{R}^d} \exp (H(\bm L\bm x+ \bm z_{\star})) d\bm x,$$ where $\bm L \bm L^T =\bm \Sigma$. Let $\rho(\cdot)$ denote the PDF of 1-dimensional standard normal distribution. Take $$\begin{aligned} \label{eq:fODISform} f_{O}(\bm x) &= (2\pi)^{-d/2} \exp (H(\bm L \bm x+ \bm z_{\star})) \prod_{j=1}^d \frac{1}{\rho(x_j)} \nonumber \\ &= \exp(g(\bm L \bm x+ \bm z_{\star})-\frac 12 (\bm L \bm x+ \bm z_{\star})^T \bm \Sigma^{-1} (\bm L\bm x+ \bm z_{\star}) +\frac 12 \bm x^T\bm x) \nonumber \\ &= \exp(g(\bm L \bm x+ \bm z_{\star})- \bm z_{\star}^T \bm L^{-T} \bm x - \frac 12 \bm z_{\star}^T \bm \Sigma^{-1} \bm z_{\star}) \nonumber \\ \end{aligned}$$ Thus we rewrite the integral as $$\label{eq:ODISform} C = \int_{\mathbb{R}^d} f_O(\bm x) \prod_{j=1}^d \rho(x_j) d \bm x = \int_{(0,1)^d}f_O(\bm \Phi^{-1}(\bm u)) d\bm u,$$ where $\bm \Phi^{-1}(\cdot)$ is the inverse (applied componentwise) of CDF of d-dimensional standard normal distribution. Let $$\label{eq:estimatorofODIS} \hat{I}_N(G_{ODIS}) = \frac 1N \sum_{i=1}^N f_O(\bm \Phi^{-1}(\bm u_i)),$$ where $\bm u_i^{'}s$ are from the unit hypercube. For MC they are i.i.d., and here for QMC associated with lattice rule, they can be constructed by rank-1 randomly shifted lattice rule (see Algorithm 6 of [@Nichols2014], CBC algorithm). Next consider LapIS. Let $\bm \Sigma_{\star} = (-\nabla^2 H(z_{\star}))^{-1}$, where $\nabla^2 H(\bm z)= \nabla^2 g(\bm z) - \bm \Sigma^{-1}$, or equivalently $$\label{eq:LapIScovariance} \bm \Sigma^{-1} = \bm \Sigma_{\star}^{-1} + \nabla^2 g(\bm z)$$ Suppose $\bm L_{\star}$ is a factorization matrix of $\bm \Sigma_{\star}$ satisfies $\bm L_{\star} \bm L_{\star}^T=\bm \Sigma_{\star}$. Therefore, the integral [\[eq:target\]](#eq:target){reference-type="eqref" reference="eq:target"} becomes $$\label{eq:LapIStype} \int_{\mathbb{R}^d} \frac{\exp (H(\bm z))}{\sqrt{(2\pi)^d\det{(\bm \Sigma)}}}d\bm z = (2\pi)^{-d/2} \frac{\det (\bm L_{\star})}{\sqrt{\det{(\bm \Sigma)}}} \int_{\mathbb{R}^d} \exp (H(\bm L_{\star} \bm x+ \bm z_{\star})) d\bm x.$$ Let $\rho(\cdot)$ still denote the PDF of 1-dimensional standard normal distribution. Take $$\begin{aligned} \label{eq:fLapISform} f_{L}(\bm x) &= (2\pi)^{-d/2} \frac{\det (\bm L_{\star})}{\sqrt{\det{(\bm \Sigma)}}}\exp (H(\bm L_{\star} \bm x+ \bm z_{\star})) \prod_{j=1}^d \frac{1}{\rho(x_j)} \nonumber \\ &= \frac{\det (\bm L_{\star})}{\sqrt{\det{(\bm \Sigma)}}} \exp(g(\bm L_{\star} \bm x+ \bm z_{\star})-\frac 12 (\bm L_{\star} \bm x+ \bm z_{\star})^T \bm \Sigma^{-1} (\bm L_{\star} \bm x+ \bm z_{\star}) +\frac 12 \bm x^T\bm x) \nonumber \\ &= \frac{\det (\bm L_{\star})}{\sqrt{\det{(\bm \Sigma)}}} \exp(g(\bm L_{\star} \bm x+ \bm z_{\star})-\frac 12 (\bm L_{\star} \bm x+ \bm z_{\star})^T (\bm \Sigma_{\star}^{-1}+\nabla^2 g(\bm z_{\star})) (\bm L_{\star} \bm x+ \bm z_{\star}) +\frac 12 \bm x^T\bm x) \nonumber \\ &= \frac{\det (\bm L_{\star})}{\sqrt{\det{(\bm \Sigma)}}} \exp(g(\bm L_{\star} \bm x+ \bm z_{\star})-\frac 12 (\bm L_{\star} \bm x+ \bm z_{\star})^T \nabla^2 g(\bm z_{\star}) (\bm L_{\star} \bm x+ \bm z_{\star}) - \bm z_{\star}^T\bm \Sigma_{\star}^{-1}\bm L_{\star}\bm x -\frac 12 \bm z_{\star}^T\bm \Sigma_{\star}^{-1} \bm z_{\star}). \nonumber \\ \end{aligned}$$ The third equation follows from [\[eq:LapIScovariance\]](#eq:LapIScovariance){reference-type="eqref" reference="eq:LapIScovariance"}. Finally, we rewrite the integral as $$\label{eq:LapISform} C = \int_{\mathbb{R}^d} f_L(\bm x) \prod_{j=1}^d \rho(x_j) d \bm x = \int_{(0,1)^d} f_L(\bm \Phi^{-1}(\bm u)) d\bm u.$$ Let $$\label{eq:estimatorofLapIS} \hat{I}_N(G_{LapIS}) = \frac 1N \sum_{i=1}^N f_L(\bm \Phi^{-1}(\bm u_i)),$$ where $\bm u_i^{'}s$ are the same as the previous ODIS part. ## Error bound for importance sampling Nichols[@Nichols2014] connected the Fourier analysis and shift-averaged worst-case error, claimed that (Kuo,2010)[\[lem:Kuo2010\]]{#lem:Kuo2010 label="lem:Kuo2010"} If there exist constants $C_2>0$ and $r_2>1/2$, such that for any $h \in \mathbb{Z}$ and $h \neq 0$ holds $$\label{eq:Kuoupperboundcon} \hat{\theta}(h) \leq \frac{C_2}{|h|^{2r_2}},$$ then the generating vector $\bm z$ constructed by component-by-component algorithm satisfies $$\label{eq:Kuoupperboundsol} e_{N}^{sa}(\bm z)=O(N^{-r_2+\delta}), \quad \delta>0.$$ We fix CBC algorithm and POD weights (see e.g. [@Kuo2011; @Kuo2012]), and leave the research of weight coefficients for future work. For nonzero $h$, after some algebra (which is deferred to the Appendix) we can obtain $$\label{eq:invfourierafter} \hat{\theta}(h) = \frac{1}{\pi^2h^2} \int_0^1 \frac{\sin^2(\pi ht)}{\psi^2(\Phi^{-1}(t))\phi(\Phi^{-1}(t))} dt.$$ For classic ODIS or LapIS, we take $\phi(x)=p(x;0,1)$ and weight function $\psi(x)=\exp(-\frac{x^2}{2\alpha^2})$. Since both of them are symmetric about $x=0$, we can rewrite [\[eq:invfourierafter\]](#eq:invfourierafter){reference-type="eqref" reference="eq:invfourierafter"} as $$\label{eq:invfouriersym} \hat{\theta}(h) = \frac{2}{\pi^2h^2} \int_0^{1/2} \frac{\sin^2(\pi ht)}{\psi^2(\Phi^{-1}(t))\phi(\Phi^{-1}(t))} dt.$$ Substitute $\phi$ and $\psi$, $$\label{eq:invfouriernornor} \hat{\theta}(h) = \frac{2\sqrt{2\pi}}{\pi^2h^2} \int_0^{1/2} \exp\big((1+\frac{2}{\alpha^2})\frac{(\Phi^{-1}(u))^2}{2}\big) \sin^2(\pi ht) dt.$$ Note the fact that for any $0<u\leq 1/2$, $\exp(\frac{[\Phi^{-1}(u)]^2}{2}) \leq \frac 1u$. Therefore, we scale the right hand side of [\[eq:invfouriernornor\]](#eq:invfouriernornor){reference-type="eqref" reference="eq:invfouriernornor"} $$\label{eq:invfouriernornormid} \hat{\theta}(h) \leq \frac{2\sqrt{2\pi}}{\pi^2h^2} \int_0^{1/2} u^{-1-2/\alpha^2} \sin^2(\pi ht) dt$$ Moreover, for any positive integer $h$ and $0<c<1$, we have $$\label{eq:ubofsin2} \int_0^{1/2} u^{-1-c}\sin^2(\pi ht) dt \leq \frac{(2\pi h)^c}{c(2-c)}.$$ Therefore, for $\alpha^2>2$ (i.e. $0<\frac{2}{\alpha^2}<1$) and any positive integer $h$: $$\hat{\theta}(h) \leq \frac{\sqrt{2}\pi^{\frac{2}{\alpha^2}-\frac{3}{2}}}{(\alpha^2-1)} h^{-2(1-1/\alpha^2)}.$$ Finally, by symmetry of Fourier coefficient, we claim that for nonzero integer $h$, the condition  [\[eq:Kuoupperboundcon\]](#eq:Kuoupperboundcon){reference-type="eqref" reference="eq:Kuoupperboundcon"} of lemma [\[lem:Kuo2010\]](#lem:Kuo2010){reference-type="ref" reference="lem:Kuo2010"} holds, where $$C_2 = \frac{\sqrt{2}\pi^{\frac{2}{\alpha^2}-\frac{3}{2}}}{(\alpha^2-1)}, \quad r_2 = 1-1/\alpha^2, \quad \alpha^2>2.$$ Thus, due to [\[eq:Kuoupperboundsol\]](#eq:Kuoupperboundsol){reference-type="eqref" reference="eq:Kuoupperboundsol"}, the shift-averaged worst-case error associated with normal proposal and weight function is given by $$e_{N}^{sa}(\bm z)=O(N^{-(1-1/\alpha^2)+\delta}).$$ The hidden condition remained is that the IS integrand locates in the RKHS. We list up some conditions for the for log original integrand $g$. [\[assum:originalboundary\]]{#assum:originalboundary label="assum:originalboundary"} $$g(\bm z) = O(||\bm z||^{\beta}), \beta < 2,$$ [\[assum:eigenvalue\]]{#assum:eigenvalue label="assum:eigenvalue"} $$-\min_{i \in 1:d} h_i \max_{j \in 1:d} \lambda_j< \frac{1}{\alpha}.$$ where $h_i$ denote eigenvalues of $\nabla^2 g(\bm z_{\star})$, and $\lambda_k$ denote eigenvalues of $\Sigma_{\star}, k=1,2,...,d$. [\[assum:convex\]]{#assum:convex label="assum:convex"} $\nabla^2 g(\bm z_{\star})$ is semi-definite. Now we have prepared to state main results for ODIS and LapIS. [\[thm:alternative\]]{#thm:alternative label="thm:alternative"} Consider the integral [\[eq:target\]](#eq:target){reference-type="eqref" reference="eq:target"}. Apply RQMC associated with rank-1 randomly shifted lattice rule constructed by CBC algorithm and equipped by weight function $\psi(\bm x) \equiv \exp(-\frac{x^2}{2\alpha^2})$, where $\alpha^2>2$ makes the corresponding reproducing kernel Hilbert space well-defined. We claim $$\label{eq:errorbound} \sqrt{\mathbb{E}|(\hat{I}_N(G_{IS})-C|^2}= O(N^{-1+1/\alpha^2+\delta}), \quad \delta>0.$$ For ODIS, a sufficient condition for [\[eq:errorbound\]](#eq:errorbound){reference-type="eqref" reference="eq:errorbound"} is Assumption [\[assum:originalboundary\]](#assum:originalboundary){reference-type="ref" reference="assum:originalboundary"}. For LapIS, sufficient conditions for  [\[eq:errorbound\]](#eq:errorbound){reference-type="eqref" reference="eq:errorbound"} are Assumptuion [\[assum:originalboundary\]](#assum:originalboundary){reference-type="ref" reference="assum:originalboundary"}+Assumptuion [\[assum:eigenvalue\]](#assum:eigenvalue){reference-type="ref" reference="assum:eigenvalue"}. *Proof.* As long as $\beta < 2$, Assumptuion [\[assum:originalboundary\]](#assum:originalboundary){reference-type="ref" reference="assum:originalboundary"} makes $g(\bm L_{\star} \bm x+ \bm z_{\star})$ dominated by $\frac {(\bm x^T \bm x)}{2\alpha}$, implying $\exp(g(\bm L_{\star} \bm x+ \bm z_{\star})-\frac {(\bm x^T \bm x)}{2\alpha})$ vanishes as $\bm x$ goes to infinity. For ODIS, [\[eq:fODISform\]](#eq:fODISform){reference-type="eqref" reference="eq:fODISform"} ensures that Assumption [\[assum:originalboundary\]](#assum:originalboundary){reference-type="ref" reference="assum:originalboundary"} is sufficient to make $f_O$ locate in RKHS. However, for LapIS, there exist some terms more for [\[eq:fLapISform\]](#eq:fLapISform){reference-type="eqref" reference="eq:fLapISform"}. More specifically, we pick up the dominating term $\exp[D(\bm x)]$, where $$\label{eq:domination} D(\bm x) = g(\bm L_{\star} \bm x+ \bm z_{\star}) - \frac 12 (\bm L_{\star} \bm x)^T \nabla^2 g(\bm z_{\star})(\bm L_{\star} \bm x)$$ Assumptuion [\[assum:originalboundary\]](#assum:originalboundary){reference-type="ref" reference="assum:originalboundary"} controls the first term. If Assumptuion [\[assum:eigenvalue\]](#assum:eigenvalue){reference-type="ref" reference="assum:eigenvalue"} is added, let $\nabla^2 g(\bm z_{\star}) = \bm Q^T \bm T \bm Q$ denote the singular value decomposition. Then $\bm Q$ is an orthogonal matrix and $\bm T$ is a diagonal matrix with eigenvalues being diagonal entries. Furthermore, $$- \frac 12 (\bm L_{\star} \bm x)^T \nabla^2 g(\bm z_{\star})(\bm L_{\star} \bm x) = -\frac 12 (\bm Q \bm L_{\star} \bm x)^T \bm T (\bm Q \bm L_{\star} \bm x),$$ which is dominated by $$-\frac{h}{2} ||\bm Q \bm L_{\star} \bm x||^2 \leq -\frac{h}{2} ||\bm Q||^2_2||\bm L_{\star}||^2_2||x||^2 \leq -\frac{h}{2} \max_{j \in 1:d} \lambda_j ||x||^2.$$ The second inequality holds since the spectral norm of orthogonal matrix is 1 and the spectral norm of $\bm L_{\star}$ is the nonnegative square root of maximal eigenvalue of $\bm L_{\star}^T \bm L_{\star}$, which is the same as $\bm L_{\star} \bm L_{\star}^T = \bm \Sigma_{\star}$. Note that Assumptuion [\[assum:eigenvalue\]](#assum:eigenvalue){reference-type="ref" reference="assum:eigenvalue"} is equivalent to $\frac{1}{2\alpha} > -\frac{h}{2} \max_{j \in 1:d} \lambda_j$, then the integrand still belongs to RKHS. Our claim holds. ◻ If Assumption [\[assum:eigenvalue\]](#assum:eigenvalue){reference-type="ref" reference="assum:eigenvalue"} is reckoned to be sophisticated, one can use Assumption [\[assum:convex\]](#assum:convex){reference-type="ref" reference="assum:convex"} to replace it since the latter is sufficient for the former. We consider $\beta$ in Assumptuion [\[assum:originalboundary\]](#assum:originalboundary){reference-type="ref" reference="assum:originalboundary"}. If $\beta =2$, then $$\begin{aligned} g(\bm L_{\star} \bm x+ \bm z_{\star}) &=& O(||\bm L_{\star} \bm x||^2) \nonumber \\ &\leq& O(||\bm L_{\star}||^2||\bm x||^2) \nonumber \\ &=& O((\max_{j \in 1:d} \lambda_j)||\bm x||^2) \nonumber \\ \end{aligned}$$ Since the RKHS contains $\exp(\gamma ||\bm x||^2)$ when $\gamma < \frac{1}{2\alpha}$, we have $\max_{j \in 1:d} \lambda_j < \frac{1}{2\alpha}$. Therefore, a more accurate statement of Assumptuion [\[assum:originalboundary\]](#assum:originalboundary){reference-type="ref" reference="assum:originalboundary"} is that $$g(\bm z) = O(||\bm z||^{\beta}), \beta < 2$$ or $$\label{assum:originalboundaryplus} g(\bm z) = O(||\bm z||^2), \quad \max_{j \in 1:d} \lambda_j < \frac{1}{2\alpha}$$ Let us consider weight functions. For LapIS, in fact we do not have many choices. For example, we cannot take $\psi(\bm x) \equiv \exp(-\frac{|x|}{\alpha})$ as the weight function. Because LapIS brings $\exp(- \frac 12 (\bm L_{\star} \bm x)^T \nabla^2 g(\bm z_{\star})(\bm L_{\star} \bm x))$, which may not belong to the corresponding RKHS, leading the conclusion failed. However, it is not the case for ODIS since the optimal drift does not produce singularities. Hence, we have more choices of weight function for ODIS than LapIS. It turns out that the conditions of convergence for LapIS are more strict than ODIS. ## Multivariate $t$ distribution as the proposal We take multivariate $t$ distribution as the proposal instead of normal distribution in this section. The multivariate $t$ distribution with center $\bm\mu\in\mathbb{R}^{d\times1}$, scale (positive definite) matrix $\bm\Sigma\in\mathbb{R}^{d\times d}$ and $\nu>0$ degrees of freedom, denoted by $t_\nu(\bm\mu,\bm\Sigma)$ has a representation $$\bm t = \bm \mu + \bm L\bm x,$$ where $\bm L\bm L^T = \bm \Sigma$, and $\bm x$ follows a multivariate t-distribution with independent components. $$q(\bm x;\bm 0,\bm I_d,\nu) = c_{\nu}\prod_{j=1}^d(1+\frac{x_j^2}{\nu})^{-\frac{\nu + 1}{2}}$$ where $c_{\nu}$ denotes for normalization constant. By a change of variables, we have $$\begin{aligned} \label{eq:rstrational} C &=& \int_{\mathbb{R}^d} G(\bm t)p(\bm t;\bm \mu_0,\bm \Sigma_0) d\bm t \nonumber \\ &=& \int_{\mathbb{R}^d} G(\bm t)\frac{p(\bm t;\bm \mu_0,\bm \Sigma_0)}{q(\bm t;\bm \mu,\bm \Sigma, \nu)}q(\bm{\mu+Lx}; \bm \mu,\bm \Sigma,\nu) d\bm t \nonumber \\ &=& \int_{\mathbb{R}^d} G(\bm{\mu+Lx})\frac{p(\bm{\mu+Lx};\bm \mu_0,\bm \Sigma_0)}{q(\bm{\mu+Lx}; \bm \mu,\bm \Sigma,\nu)}q(\bm x;\bm 0,\bm I_d,\nu) d\bm x \nonumber \\ &=& \int_{\mathbb{R}^d} G(\bm{\mu+Lx})W(\bm x)q(\bm x;\bm 0,\bm I_d,\nu) d\bm x, \end{aligned}$$ where $\bm L^T \bm L=\bm \Sigma$, the likelihood ratio function is given by $$\begin{aligned} \label{eq:LRoftrational} W(\bm x) &=& \frac{p(\bm{\mu+Lx};\bm \mu_0,\bm \Sigma_0)}{q(\bm{\mu+Lx}; \bm \mu,\bm \Sigma,\nu)} \nonumber \\ &=& \frac{(\frac{\nu}{2})^{d/2}\Gamma(\frac{\nu}{2})}{\Gamma(\frac{\nu+d}{2})(\det{\Sigma_0})^{1/2}}\prod_{j=1}^d(1+\frac{x_j^2}{\nu})^{-\frac{\nu + 1}{2}}\exp\{-(\bm \mu+\bm L \bm x -\bm \mu_0)^T\bm \Sigma_0^{-1}(\bm \mu+\bm L \bm x -\bm \mu_0)/2\} \nonumber \\ &\propto& \prod_{j=1}^d(1+\frac{x_j^2}{\nu})^{-\frac{\nu + 1}{2}}\exp\{-\frac 12 \bm x^T \bm L^T \bm \Sigma_0^{-1} \bm L \bm x-\bm L^T \bm \Sigma_0^{-1}(\bm \mu -\bm \mu_0) \bm x\}, \nonumber \end{aligned}$$ $\Gamma(\cdot)$ is the well-known Gamma function. To generate $t$-variables, we set $$\label{eq:repofz} \bm z= T^{-1}(u).$$ where $T$ denotes for the CDF of multivariate t-distribution with independent components. One can simulate variables from $(0,1)^{d}$ via  [\[eq:repofz\]](#eq:repofz){reference-type="eqref" reference="eq:repofz"}. Hence the estimator for multivariate $t$ distribution IS is $$\label{eq:rstproposal} \hat{I}_N(G_t)=\frac 1N \sum_{i=1}^N f_t(T^{-1}(u_i)),$$ where $f_t(\bm x)=G(\bm \mu +\bm L \bm x)W(\bm x)$, $\bm u_1,...,\bm u_N$ are random points for MC or randomly shifted lattice rule for RQMC on $(0,1)^{d}$. When we analyze the error bound for randomly shifted lattice rule, the choice of weight function is crucial. Here $t$ distribution is taken as the proposal function, we cannnot still take the normal density as the weight function since the corresponding RKHS does not contain the density of $t$ distribution. An alternative selection is the rational function $$\rho_{\lambda}(x)=\frac{\lambda-1}{2}\frac{1}{(1+|x|)^{\lambda}}, \quad \lambda>1.$$ The tail is heavier such that $t$ distribution with degrees of freedom $\nu>2\lambda+1$ locates in the RKHS. We have the following result. [\[thm:tproposal\]]{#thm:tproposal label="thm:tproposal"} Consider the integral [\[eq:target\]](#eq:target){reference-type="eqref" reference="eq:target"}. Apply RQMC associated with rank-1 randomly shifted lattice rule constructed by CBC algorithm and equipped by weight function $\psi(\bm x)=\prod_{i=1}^{d} \rho_{\lambda}(x_i)$ , where $\nu>2\lambda+1$ makes the corresponding reproducing kernel Hilbert space well-defined. Then a sufficient condition for $$\label{eq:errorboundoft} \sqrt{\mathbb{E}|(\hat{I}_N(G_t)-C|^2}= O(N^{-1+\frac{2\lambda+1}{2\nu}+\delta}), \quad \delta>0.$$ is Assumption [\[assum:originalboundary\]](#assum:originalboundary){reference-type="ref" reference="assum:originalboundary"}. *Proof.* We start from the Fourier coefficient [\[eq:invfouriersym\]](#eq:invfouriersym){reference-type="eqref" reference="eq:invfouriersym"}. The difficulty is that there is no analytic expression of CDF of $t$ distribution, therefore we try to give an upper bound of the integrand. Note that $$\min(1,\nu) \leq \frac{(1+|x|)^2}{1+x^2/\nu} \leq 1+\nu,$$ Denote the normalizing constant $c_{\nu}=\frac{\Gamma(\frac{\nu+1}{2})}{\sqrt{\nu\pi}\Gamma(\frac{\nu}{2})}$, we thus have $$\label{eq:studentsppdf} L_{\nu} \leq \frac{\phi(x)}{\rho_{\nu}(x)} \leq U_{\nu},$$ where the lower constant $$L_{\nu}=\frac{2c_{\nu}}{\nu}\big(\min(1,\nu)\big)^{\frac{\nu+1}{2}},$$ and the upper constant $$U_{\nu}=\frac{2c_{\nu}}{\nu}(1+\nu)^{\frac{\nu+1}{2}}$$ are irrelevant to $x$, and obviously $L_{\nu}<1<U_{\nu}$. Take the upper limit integral with respect to [\[eq:studentsppdf\]](#eq:studentsppdf){reference-type="eqref" reference="eq:studentsppdf"} $$\label{eq:studentspcdf} L_{\nu} \leq \frac{\Phi(x)}{P_{\nu}(x)} \leq U_{\nu},$$ where $$P_{\nu}(x) = \int_{-\infty}^{x} \rho_{\nu}(t) dt = \begin{cases} \frac{1}{2(1-x)^2}, & x<0 \\ 1-\frac{1}{2(1+x)^2}, & x\geq 0\\ \end{cases}$$ Let $x=\Phi^{-1}(u),\quad 0<u<1$, $$\frac{u}{U_{\nu}} \leq P_{\nu}(\Phi^{-1}(u)) \leq \frac{u}{L_{\nu}}$$ Take $P_{\nu}^{-1}$ on both sides and note that the right hand side may be larger than 1, we have $$\label{eq:studentspinverse} P_{\nu}^{-1}(\frac{u}{U_{\nu}}) \leq \Phi^{-1}(u) \leq P_{\nu}^{-1}(\min(\frac{u}{L_{\nu}},1))$$ Hence, for $0<t\leq 1/2$: $$\begin{aligned} \label{eq:hatthetaofstudentrational} \psi^2(\Phi^{-1}(t))\phi(\Phi^{-1}(t)) &\geq& \psi^2(P_{\nu}^{-1}(\frac{t}{U_{\nu}}))L_{\nu}\rho_{\nu}(\Phi^{-1}(t)) \nonumber \\ &\geq& \psi^2(P_{\nu}^{-1}(\frac{t}{U_{\nu}}))L_{\nu}\rho_{\nu}(P_{\nu}^{-1}(\frac{t}{U_{\nu}})) \nonumber \\ &=& \Big(\frac{2t}{U_{\nu}}\Big)^{2\lambda/\nu}L_{\nu}\frac{\nu}{2}\Big(\frac{2t}{U_{\nu}}\Big)^{1+1/\nu} \nonumber \\ &=& \frac{\nu L_{\nu}}{2} \Big(\frac{2t}{U_{\nu}}\Big)^{1+(2\lambda+1)/\nu} \nonumber \\ \end{aligned}$$ where the inequality is due to the monotonicity of $\psi$.Combine  [\[eq:invfouriersym\]](#eq:invfouriersym){reference-type="eqref" reference="eq:invfouriersym"} and  [\[eq:hatthetaofstudentrational\]](#eq:hatthetaofstudentrational){reference-type="eqref" reference="eq:hatthetaofstudentrational"}, we finally derive $$\hat{\theta}(h) \leq \frac{C_t}{h^2} \int_0^{1/2} t^{-1-\frac{2\lambda+1}{\nu}} \sin^2(\pi ht) dt.$$ According to [\[eq:ubofsin2\]](#eq:ubofsin2){reference-type="eqref" reference="eq:ubofsin2"}, since we have demonstrated that for any nonzero integer $h$, $\hat{\theta}(h) \leq C(\lambda,\nu)h^{-2r_{\star}}$, where the constant $C(\lambda,\nu)$ only depends on $\lambda$ and $\nu$, $r_{\star}=1-\frac{2\lambda+1}{2\nu}$. By lemma [\[lem:Kuo2010\]](#lem:Kuo2010){reference-type="ref" reference="lem:Kuo2010"}, the error bound holds. We conclude our proof by claiming that $f_t$ locate in the RKHS. This fact is trivial since the likelihood ratio contains a negative quadratic form in the exponent term, whereas the  [\[assum:originalboundary\]](#assum:originalboundary){reference-type="eqref" reference="assum:originalboundary"} ensures the boundedness of $f_t$. ◻ Since $\nu>2\lambda+1$, the convergence rate is no worse than MC. Furthermore, we find out that as the degree of freedom goes to infinity, the error bound gets close to $O(N^{-1+\delta})$, which corresponds with the normal importance density, i.e. the limit case of $t$ distribution. # Examples {#sec:exam} ## Generalized linear mixed model Consider a class of highly structured models in statistics, which is known as generalized linear mixed model (GLMM, see [@Kuo2008a]). To make it short, we express the model as $$\label{eq:GLMM} L(\beta,\kappa,\sigma) = \int_{\mathbb{R}^d} \prod_{j=1}^d \frac{\exp(y_j(\omega_j+\beta)-\exp(\omega_j+\beta))}{y_j}\frac{\exp(-\frac{1}{2} \bm \omega^T\bm \Sigma^{-1}\bm \omega^T)}{\sqrt{(2\pi)^d\det(\bm \Sigma)}} d\bm \omega,$$ where $\bm y = (y_1,y_2,...,y_d)$ denotes the data of nonnegative integers, $\Sigma$ denotes the covariance matrix, whose entries $\Sigma_{i,j}=\frac{\sigma^2 \kappa^{|i-j|}}{1-\kappa^2}$ for $i,j = 1,2,...,d.$ Our goal is to maximize the log-likelihood $\log L(\beta,\kappa,\sigma)$ given $\bm y$ with respect to $(\beta,\kappa,\sigma)$ under restrictions $\kappa \in (0,1), \sigma>0$. Note the integral  [\[eq:GLMM\]](#eq:GLMM){reference-type="eqref" reference="eq:GLMM"} can be rewritten by $\int_{\mathbb{R}^d} \exp(F(\bm \omega)) d\bm \omega$ regardless of a normalizing constant, where $$\label{eq:unimodalF} F(\bm \omega) = \sum_{j=1}^d (y_j(\omega_j+\beta)-\exp(\omega_j+\beta)) - -\frac{1}{2} \bm \omega^T\bm \Sigma^{-1}\bm \omega^T.$$ Obviously $F$ is a unimodal function, thus we take $$\bm \omega_{\star} = \arg \max_{\bm \omega \in \mathbb{R}^d} F(\bm \omega),$$ which solves $$\label{eq:solofdrift} \nabla F(\bm \omega) = \bm y - e^{\beta}\exp(\bm \omega) - \bm \Sigma^{-1} \bm \omega = \bm 0.$$ The Hessian is $$\label{eq:Hessian} \nabla^2 F(\bm \omega^{\star}) = -e^{\beta}\mathrm{diag}(e^{\omega_{\star 1}},e^{\omega_{\star 2}},...,e^{\omega_{\star d}})-\bm \Sigma^{-1}$$ Recall the denotation in LapIS, let $$\label{eq:solofcovariance} \bm \Sigma_{\star}= (-\nabla^2 F(\bm \omega_{\star}))^{-1},$$ and take the decompostion of $\bm \Sigma_{\star}$, that is $\bm \Sigma_{\star}=\bm L_{\star} \bm L_{\star}^T$, then the integral $\int_{\mathbb{R}^d} \exp F(\bm \omega) d\bm \omega$ becomes $$\begin{aligned} \label{eq:LapISofGLMM} \int_{\mathbb{R}^d} \exp(F(\bm \omega)) d\bm \omega &=& \det(\bm L_{\star}) \int_{\mathbb{R}^d} \exp(F(\bm L_{\star}\bm v)+\bm \omega_{\star}) d\bm v \nonumber \\ &=& \det(\bm L_{\star}) \int_{(0,1)^d} \exp(F(\bm L_{\star}\Phi^{-1}(\bm u)+\bm w_{\star}))\prod_{j=1}^d \frac{1}{\rho(\Phi^{-1}(u_j))} d\bm u \end{aligned}$$ where $\rho$ and $\Phi$ denote the PDF and CDF of standard normal distribution, respectively. According to [@Kuo2010] and previous analysis, we take the weight function $\psi(\bm x)= \exp (-\frac{\bm x^2}{2\alpha^2})$ to establish the RKHS with $\alpha^2 > 2$. There exists a randomly shifted lattice rule which attains $O(N^{-1+1/\alpha^2+\delta})$ error bound for this generalized linear mixed model if the integrand belongs to the RKHS, which is equivalent to $$\label{eq:sufofGLMM} \max_{j \in 1:d} \lambda_j< \frac{\exp(-\beta)}{\alpha^2 \exp(\max_{j \in 1:d} \omega_{\star j} )}$$ *Proof.* Let $$\begin{aligned} g(\bm\omega) &=& \log\left(\prod_{j=1}^d \frac{\exp(y_j(\omega_j+\beta)-\exp(\omega_j+\beta))}{y_j}\right) \nonumber \\ &=& \sum_{j=1}^d \left[y_j(\omega_j+\beta)-\exp(\omega_j+\beta)-\log y_j\right]. \end{aligned}$$ Obviously $g$ satisfies Assumptuion [\[assum:originalboundary\]](#assum:originalboundary){reference-type="ref" reference="assum:originalboundary"} since it is upper bounded. Take the derivatives with respect to the components, we have $$\frac{\partial g(\bm\omega)}{\partial \omega_k}= y_k- \exp(\omega_k+\beta),$$ $$\frac{\partial^2 g(\bm\omega)}{\partial \omega_j\omega_k}=-\exp(\omega_k+\beta)\delta_{jk}.$$ Note that $\nabla^2 g$ is a negative definite diagonal matrix, by Theorem [\[thm:alternative\]](#thm:alternative){reference-type="ref" reference="thm:alternative"}, it is sufficient to verify whether [\[assum:eigenvalue\]](#assum:eigenvalue){reference-type="ref" reference="assum:eigenvalue"} is valid. Clearly it is equivalent to [\[eq:sufofGLMM\]](#eq:sufofGLMM){reference-type="eqref" reference="eq:sufofGLMM"}. ◻ Since $\alpha^2>2$ for RKHS well-defined, we transpose some terms in [\[eq:sufofGLMM\]](#eq:sufofGLMM){reference-type="eqref" reference="eq:sufofGLMM"} and immediately obtain a necessary condition: $$2 < \alpha^2 < \frac{\exp(-\beta)}{\max_{j \in 1:d} \lambda_j \exp(\max_{j \in 1:d} \omega_{\star j})}$$ or equivalently $$\label{eq:necofGLMM} 2e^\beta\max_{j \in 1:d} \lambda_j \exp(\max_{j \in 1:d} \omega_{\star j}) <1.$$ This is not always valid, especially when $d$ is large. The reason is that LapIS may make the boundary unbounded, even diverge faster than $\exp(\bm x^T\bm x/2\alpha^2)$, which is unfavorable in QMC computation. ## Randleman-Bartter model Consider the problem of valuing a zero coupon bond, where the interest rates are assumed to follow the Randleman-Bartter model(see [@Hull2005]). A 3-dimensional Gaussian integral is analyzed by Catflisch[@Caflisch1998]. Here we reformulate the model in general. Our goal is to value a fair price of a $(d+1)-$year zero coupon boud with a face value of \$1, which can be expressed as a integral $$\label{eq:RBmodel} P = \int_{\mathbb{R}^d} \prod_{k=0}^d \frac{p(\bm z;\bm 0,\bm I_d)}{1+r_k}d\bm z.$$ The interest rates can be represented by $$r_k = r_0\exp(-k\sigma^2/2+\sigma B_k),\quad k=1,2,...,d,$$ where the Brownian vector $(B_1,B_2,...,B_d)^T$ is zero-mean and the covariance matrix $\bm C$ with entries $C_{i,j}=\min(i,j)$. $r_0$ is the current annually interest rate, $\sigma$ is the volatility. The first step is to generalize the Brownian motion. This is equivalent to find a matrix $\bm A$ such that $\bm A \bm A^T= \bm C$. Here we take the standard construction to make an analysis. In fact, there exist several constructions which are widely used in practice such as Brownian bridge and principal component analysis (see [@Hull2005]). We leave other generation methods for future work. Standard construction gives $$\label{eq:STD} \bm A := (a_{i,j})_{d\times d} = \begin{pmatrix} 1 & 0 & 0 & \cdots & 0 \\ 1 & 1 & 0 & \cdots & 0 \\ 1 & 1 & 1 & \cdots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 1 & 1 & 1 & \cdots & 1 \end{pmatrix}$$ and we take $$\label{eq:BMgeneration} (B_1,B_2,...,B_d)^T = \bm A (z_1,z_2,...,z_d)^T,\quad (z_1,z_2,...,z_d)^T \sim N(\bm 0,\bm I_d).$$ Next we rewrite [\[eq:RBmodel\]](#eq:RBmodel){reference-type="eqref" reference="eq:RBmodel"} as standard form [\[eq:target\]](#eq:target){reference-type="eqref" reference="eq:target"}, that is $$G(\bm z) = \prod_{k=0}^d (1+r_k)^{-1}, \quad \bm \mu = \bm 0, \quad \bm \Sigma = \bm I_d.$$ Note that $a_{i,j}=1$ if and only if $i \geq j$, otherwise $a_{i,j}=0$. Therefore, $$\frac{\partial r_k}{\partial z_i} = \sigma r_k \bm 1\{k \geq i\}.$$ $$\frac{\partial^2 r_k}{\partial z_iz_j} = \sigma^2 r_k \bm 1\{k \geq \max(i,j)\}.$$ Let $g(\bm z) = \log G(\bm z) = -\sum_{k=0}^d \log(1+r_k)$, $H(\bm z)=g(\bm z)-\frac 12 \bm z^T \bm z$. Then $\nabla^2 H = \nabla^2 g - \bm I_d$. Let $\bm z_{\star}$ solves $\nabla H(\bm z) = \bm 0$, $\bm \Sigma_{\star}=(-\nabla^2 H(\bm z_{\star}))^{-1}$. We have $$\frac{\partial g}{\partial z_i} = -\sum_{k=1}^d \frac{1}{1+r_k}\frac{\partial r_k}{\partial z_i} = -\sum_{k=i}^d \frac{\sigma r_k}{1+r_k}.$$ $$\begin{aligned} \frac{\partial^2 g}{\partial z_iz_j} &=& -\sum_{k=i}^d \frac{\sigma[\frac{\partial r_k}{\partial z_j}(1+r_k)-\frac{\partial r_k}{\partial z_j}r_k]}{(1+r_k)^2} \nonumber \\ &=& -\sum_{k=i}^d \frac{\sigma \frac{\partial r_k}{\partial z_j}}{(1+r_k)^2} \nonumber \\ &=& -\sum_{k=i}^d \frac{\sigma^2 r_k \bm 1\{k \geq j\} }{(1+r_k)^2} \nonumber \\ &=& -\sum_{k=\max(i,j)}^d \frac{\sigma^2 r_k}{(1+r_k)^2}. \end{aligned}$$ Denote $R_k = \frac{\sigma^2 r_k}{(1+r_k)^2}$ for simplicity. We obtain $$\label{eq:nabla2g} \nabla^2 g = \begin{pmatrix} -\sum_{k=1}^d R_k & -\sum_{k=2}^d R_k & -\sum_{k=3}^d R_k & \cdots & -R_d \\ -\sum_{k=2}^d R_k & -\sum_{k=2}^d R_k & -\sum_{k=3}^d R_k & \cdots & -R_d \\ -\sum_{k=3}^d R_k & -\sum_{k=3}^d R_k & -\sum_{k=3}^d R_k & \cdots & -R_d \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ -R_d & -R_d & -R_d & \cdots & -R_d \end{pmatrix}$$ This matrix is negative definite, then $-\nabla^2 g = \bm H \bm H^T$, where one choice of decomposition matrix $\bm H$ is upper triangular $$\bm H = \begin{pmatrix} \sqrt{R_1} & \sqrt{R_2} & \sqrt{R_3} & \cdots & \sqrt{R_d} \\ 0 & \sqrt{R_2} & \sqrt{R_3} & \cdots & \sqrt{R_d} \\ 0 & 0 & \sqrt{R_3} & \cdots & \sqrt{R_d} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \cdots & \sqrt{R_d} \end{pmatrix}$$ Denote $r_{\star}=\min_{k \in 1:d} r_k(\bm z_{\star})$. We still take the weight function $\psi(\bm x)= \exp (-\frac{\bm x^2}{2\alpha^2})$ to establish the RKHS with $\alpha^2 > 2$. [\[thm:RBmodel\]]{#thm:RBmodel label="thm:RBmodel"} The CBC algorithm attains $O(N^{-1+1/\alpha^2+\delta})$ error bound for $\alpha^2 < \frac{1}{\kappa}\sqrt{\frac{6}{d^3+2d^2+2d+1}}$ in the Randleman-Bartter model, where $\kappa = \frac{\sigma^2}{2+r_{\star}+1/r_{\star}}$. *Proof.* Clearly Assumption [\[assum:originalboundary\]](#assum:originalboundary){reference-type="ref" reference="assum:originalboundary"} is satified. Since $\nabla^2 g$ is negative definite, it suffices to check Assumption [\[assum:eigenvalue\]](#assum:eigenvalue){reference-type="ref" reference="assum:eigenvalue"} is valid. According to previous theorem, we need to compute the maximal eigenvalue of $\bm \Sigma_{\star}$, and the minimal eigenvalue of $\nabla^2 g(\bm z_{\star})$ which is unlikely to work out in theoretical analysis. Instead, estimating the eigenvalue by inequalities is performed in the following parts. Note that $-\nabla^2 g$ is a normal matrix. Hence we use a classic conclusion in linear algebra: $$\label{eq:normaleq} \sum_{k=1}^d \nu_k^2 = ||-\nabla^2 g||_F^2,$$ where $\nu_k'$s denote all eigenvalues of $-\nabla^2 g$, and $||\cdot||_F$ denote the Frobenius norm, which is equivalent to the nonnegative square root of square sum of all entries. Since $R_k = \frac{\sigma^2 r_k}{(1+r_k)^2} \geq \kappa$, we estimate the Frobenius norm as follows: $$\begin{aligned} \label{eq:estofFnorm} ||-\nabla^2 g||_F^2 &=& \sum_{j=1}^d (2j-1)(\sum_{k=j}^d R_k)^2 \nonumber \\ &\geq& \sum_{j=1}^d (2j-1)(d-j+1)^2\kappa^2 \nonumber \\ &=& d\frac{(d^3+2d^2+2d+1)\kappa^2}{6}) :=S_d \end{aligned}$$ Thus we have $$\max_{k \in 1:d} \nu_k \geq \sqrt{S_d/d}.$$ Finally $$h = -\max_{k \in 1:d} \nu_k \leq -\sqrt{\frac{(d^3+2d^2+2d+1)}{6}}\kappa.$$ For $\max_{k \in 1:d} \lambda_k$, we just use the trivial upper bound 1. We complete the proof by Theorem [\[thm:alternative\]](#thm:alternative){reference-type="ref" reference="thm:alternative"}. ◻ One may worry about the upper bound of $\alpha$ in Theorem [\[thm:RBmodel\]](#thm:RBmodel){reference-type="ref" reference="thm:RBmodel"}, which decreases as the dimension $d$ grows, eventually leading $\alpha > 2$ intractable. This problem is not crucial. Firstly, the estimation of $\max_{k \in 1:d} \lambda_k$ is naive, thus improvement still remains. Secondly, if we take $\sigma=0.1$ for example, $\kappa \leq \sigma^2/4=0.025$, making $d=61$ still valid for the model, which is enough for most practical cases. We report in Figures [1](#Figure1){reference-type="ref" reference="Figure1"} and [2](#Figure2){reference-type="ref" reference="Figure2"} the numerical results for dimension $d=5$ and $d=16$, respectively. The coefficients used are $r_0=0.1$, $\sigma=0.01$, $\gamma_u = \Big((|u|!)^2\prod_{i \in u} \frac{\tilde{\kappa}}{i^{\eta}}\Big)^{\frac{1}{1+\lambda}}$, where $\tilde{\kappa}=0.1$, $\eta=3.1$, $\lambda=0.51$ (see [@Graham2015]). In the setting of MC, ODIS and LapIS are more effective than not applying IS (i.e., NONE), supporting the benefits of using the two IS methods. Moreover, all of them have RMSEs decaying approximately at the canonical MC rate $O(N^{-1/2})$ as the sample size $N$ increases. The situation remains similar in the setting of RQMC. Both ODIS and LapIS converge faster than MC setting, and a nearly $O(N^{-1})$ error rate can be observed, which comforms to theoretical analysis above. As the dimension increases, although tiny fluctuation occurs, the main results keep the same. In conclusion, performing IS is efficient in this example, especially combining LapIS with randomly shifted lattice rule. {#Figure1 height="160pt" width="240pt"} {#Figure2 height="180pt" width="270pt"} # Conclusion {#sec:con} This research delves into augmenting the convergence rate of the Monte Carlo (MC) method, denoted as by amalgamating quasi-Monte Carlo (QMC) techniques and importance sampling (IS). QMC, a deterministic counterpart of MC, offers an error bound of $O(\frac{(logN)^d}{N})$ for d-dimensional integrals. The adoption of randomized QMC (RQMC) enhances computational efficacy. The study underscores the synergy of IS with randomly shifted lattice rules and the intricacies in deducing a theoretical convergence rate for IS within QMC frameworks. Given the ubiquity of Gaussian measure integrals in finance and statistics, the study evaluates optimal drift importance sampling (ODIS) and Laplace importance sampling (LapIS) as significance densities. Preliminary results indicate that the IS-randomly shifted lattice rule can potentially achieve an $O(N^{-1})$. error bound under certain conditions. # Appendix In this appendix, we deduce the simple form of $\hat{\theta}(h)$. Firstly, $$\begin{aligned} \label{eq:invfourierbefore1} \hat{\theta}(h) &=& \int_0^1 \int_{\Phi^{-1}(x)}^0 \frac{\Phi(t)-x}{\psi^2(t)} \exp(-2\pi ihx)dtdx+ \int_0^1 \int_{\Phi^{-1}(1-x)}^0 \frac{\Phi(t)-1+x}{\psi^2(t)} \exp(-2\pi ihx)dtdx \nonumber \\ &=& \int_0^1 \int_{\Phi^{-1}(x)}^0 \frac{\Phi(t)-x}{\psi^2(t)} \exp(-2\pi ihx)dtdx+ \int_0^1 \int_{\Phi^{-1}(x)}^0 \frac{\Phi(t)-x}{\psi^2(t)} \exp(2\pi ihx)dtdx \nonumber \\ &=& 2\int_0^1 \int_{\Phi^{-1}(x)}^0 \frac{\Phi(t)-x}{\psi^2(t)} \cos(2\pi hx)dtdx. \nonumber \end{aligned}$$ Split the outer integral at $\Phi(0)$, by Fubini theorem $$\begin{aligned} \label{eq:invfourierafter1} \hat{\theta}(h) &=& 2\int_0^{\Phi(0)} \int_{\Phi^{-1}(x)}^0 \frac{\Phi(t)-x}{\psi^2(t)} \cos(2\pi hx)dtdx + 2\int_{\Phi(0)}^1 \int_{\Phi^{-1}(x)}^0 \frac{\Phi(t)-x}{\psi^2(t)} \cos(2\pi hx)dtdx \nonumber \\ &=& 2\int_{-\infty}^0 \frac{1}{\psi^2(t)}\int_0^{\Phi(t)}(\Phi(t)-x)\cos(2\pi hx)dxdt+2\int_0^{\infty} \frac{1}{\psi^2(t)}\int_{\Phi(t)}^1(x-\Phi(t))\cos(2\pi hx)dxdt \nonumber \\ &=& 2\int_{-\infty}^0 \frac{1}{\psi^2(t)} \frac{\sin^2(\pi h\Phi(t))}{2\pi^2h^2}+2\int_0^{\infty} \frac{1}{\psi^2(t)} \frac{\sin^2(\pi h\Phi(t))}{2\pi^2h^2} dt \nonumber \\ &=& \frac{1}{\pi^2h^2} \int_{-\infty}^{\infty} \frac{\sin^2(\pi h\Phi(t))}{\psi^2(t)} dt \nonumber \\ &=& \frac{1}{\pi^2h^2} \int_0^1 \frac{\sin^2(\pi ht)}{\psi^2(\Phi^{-1}(t))\phi(\Phi^{-1}(t))} dt. \nonumber \end{aligned}$$ [^1]: Corresponding author. Department of Mathematical Sciences, Tsinghua University, Beijing 100084, People's Republic of China (). [^2]: Department of Mathematical Sciences, Tsinghua University, Beijing 100084, People's Republic of China (). [^3]: Department of Mathematical Sciences, Tsinghua University, Beijing 100084, People's Republic of China (). [^4]: Submitted to the editors DATE.

arxiv_math

--- author: - "Devika Shylaja, Sarvesh Kumar [^1]" bibliography: - VEMBib.bib title: Morley Type Virtual Element Method for Von Kármán Equations --- This paper analyses the nonconforming Morley type virtual element method to approximate a regular solution to the von Kármán equations  that describes bending of very thin elastic plates. Local existence and uniqueness of a discrete solution to the non-linear problem is discussed. A priori error estimate in the energy norm is established under minimal regularity assumptions on the exact solution. Error estimates in piecewise $H^1$ and $L^2$ norm are also derived. A working procedure to find an approximation for the discrete solution using Newtons method is discussed. Numerical results that justify theoretical estimates are presented. # Introduction {#sec:intro} The von Kármán equations  [@CiarletPlates; @Knightly; @BergerFife] model the bending of very thin elastic plates through a system of fourth-order semi-linear elliptic equations defined by: for a given load $f \in L^2(\Omega)$, seek the vertical displacement $u$ and the Airy stress function $v$ such that [\[vke\]]{#vke label="vke"} $$\begin{aligned} & \Delta^2 u =[u,v]+ f \text{ and }\Delta^2 v =-\frac{1}{2}[u,u] \text{ in } \Omega, \label{vke.domain}\\ & u=\frac{\partial u}{\partial n} = v = \frac{\partial v}{\partial n} = 0 \text{ on } \partial\Omega, \label{vke.bdy}\end{aligned}$$ with the von Kármán bracket $\displaystyle [\eta,\chi]:=\eta_{xx}\chi_{yy}+\eta_{yy}\chi_{xx}-2\eta_{xy}\chi_{xy}$ and $n$ is the unit outward normal to the boundary $\partial \Omega$ of the polygonal domain $\Omega \subseteq \mathbb{R}^2$. The major challenges of the problem in its numerical approximation are the non-linearity and the higher order nature of the equations. The results regarding the existence of solutions, regularity and bifurcation phenomena of the von Kármán equations  in [\[vke\]](#vke){reference-type="eqref" reference="vke"} are presented in [@CiarletPlates; @Knightly; @Fife; @Berger; @BergerFife; @BlumRannacher] and the references therein. It is well-known [@BlumRannacher] that the solutions of the von Kármán equations  belong to $H^2_0(\Omega)\cap H^{2+\alpha}(\Omega)$, where $\alpha\in (\frac{1}{2},1]$, referred to as the index of elliptic regularity, is determined by the interior angles of $\Omega$. Note that when $\Omega$ is convex, $\alpha=1$. The numerical methods to approximate the regular solutions of von Kármán equations  has been studied using conforming finite element methods (FEMs) in [@Brezzi; @ng1], nonconforming Morley FEM in [@ng2; @carstensen2017nonconforming], mixed FEMs in [@Miyoshi; @Chen2020AMF; @Reinhart], discontinuous Galerkin methods, $C^0$ interior penalty methods in [@brennernew; @CCGMNN18], and hybrid FEMs in [@Quarteroni]. More recently, a conforming virtual element method is analysed in [@Carlo_VEMvKE]. The Virtual Element Method (VEM) [@Veiga_basicVEM], which is a generalization of the FEM, has got more and more attention in recent years, because it can deal with the polygonal meshes and avoid an explicit construction of the discrete shape function, [@Veiga_HdivHcurlVEM; @Veiga_hitchhikersVEM; @Brenner_errorVEM]. The polytopal meshes can be very useful for a wide range of reasons, including meshing of the domain (such as cracks) and data features, automatic use of hanging nodes, adaptivity. A conforming VEM for plate bending problems is introduced in [@Brezzi_VEM_platebending]. A $C^1$ virtual element for the Cahn-Hilliard equations and the vibration problem of Kirchhoff plates is developed in [@Antonietti_CahnHilliard], `\cite{}`{=latex}. This has been extended to the von Kármán equations  to approximate the regular solutions in [@Carlo_VEMvKE]. In [@Zhao_ncfem], a $C^0$ noncoforming VEM for plate bending problems is constructed for any order of accuracy. This nonconforming method is modified to fully nonconforming Morley type VEM in [@Zhao_Morley; @Antonietti_ncVEM]. Note that both these papers deal with the same degrees of freedom whereas use different definition on the local virtual space. Recently, the Morley type VEM is analysed for the Navier-Stokes equations in stream function vorticity formulation in [@Adak_MorleyVEMNSE]. The aim of this paper is to extend and analyze the nonconforming Morley type VEM presented in [@Zhao_Morley] to approximate a regular solution to the von Kármán equations. Since the discrete space is not a subspace of $H^2_0(\Omega)$, the convergence analysis offers a lot of challenges and novelty for this semilinear problem with trilinear nonlinearity. The trilinear form in [@Adak_MorleyVEMNSE] for the Navier-Stokes equation in stream-function form vanishes whenever the second and third variables are equal, and satisfies the anti-symmetric property with respect to the second and third variables, and this aids the wellposedness of the discrete formulation and error analysis. However, the trilinear form for von Kármán equations  does not satisfy the properties stated above and hence leads to interesting challenges in the analysis. A discrete version of Sobolev embedding is employed for establishing the well-posedness of the discrete linearized problem or equivalently a discrete inf-sup condition. This discrete inf-sup condition allows the proof of local existence and uniqueness of a discrete solution to the non-linear problem with a Banach fixed point theorem. Optimal order error estimate in $H^2$ and $H^1$ norms are established using minimal regularity assumption of the exact solution. The discrete non-linear problem can be solved using the Newton's method by choosing an appropriate initial guess such that there exists a closed sphere in which the approximate solution is unique and the Newton's iterates converge quadratically to the discrete solution. The remaining parts are organised as follows. Section [2](#sec:vke){reference-type="ref" reference="sec:vke"} discusses the weak formulation of the von Kármán equations  and the linearised problem. Section [3](#sec:MorleyVEM){reference-type="ref" reference="sec:MorleyVEM"} deals with the Morley type VEM for the von Kármán equations. Some auxiliary results required for the convergence analysis and the wellposedness for the discrete linearised problem are established in Section [4](#sec:wellposed){reference-type="ref" reference="sec:wellposed"} and is followed by the existence and local uniqueness of the discrete solution using fixed point of a non-linear operator. A priori error control in $H^2$ and $H^1$ norms, and convergence of the Newtons method are derived in Section [5](#sec:error){reference-type="ref" reference="sec:error"}. Section [6](#sec.numericalresults){reference-type="ref" reference="sec.numericalresults"} provides the results of computational experiments that validate the theoretical estimates. Throughout the paper, standard notations on Lebesgue and Sobolev spaces and their norms are employed. The standard semi-norm and norm on $H^{s}(\Omega)$ (resp. $W^{s,p} (\Omega)$) for $s>0$ and $1 \le p \le \infty$ are denoted by $|\cdot|_{s}$ and $\|\cdot\|_{s}$ (resp. $|\cdot|_{s,p}$ and $\|\cdot\|_{s,p}$ ). The norm in $H^{-s}(\Omega)$ is denoted by $\|\cdot\|_{-s}$. The standard $L^2$ inner product and norm are denoted by $(\cdot, \cdot)$ and $\|\cdot\|.$ The notation ${\boldsymbol{H}}^s(\Omega)$ (resp. ${\boldsymbol{L}}^p(\Omega)$) is used to denote the product space $H^{s}(\Omega) \times H^s(\Omega)$ (resp. $L^p(\Omega) \times L^p(\Omega)$). For all $\Phi = (\varphi_1,\varphi_2) \in {\boldsymbol{H}}^s(\Omega) \; ( \text{ resp. } {\boldsymbol L}^2(\Omega))$, the product space is equipped with the norm $\|{\Phi}\|_{s}:=(\| \varphi_1\|_s^2 +\|\varphi_2\|_s^2)^{1/2} \; (\text{ resp. } \|{\Phi}\|:=(\| \varphi_1\|^2 +\|\varphi_2\|^2)^{1/2}). \;$ The notation $a\lesssim b$ (resp. $a \gtrsim b$) means there exists a generic mesh independent constant $C$ such that $a\leq Cb$ (resp. $a\ge Cb$). # Weak formulation {#sec:vke} This section deals with the continuous weak formulation and its linearisation of the von Kármán equations. For all $\eta,\chi, \varphi \in V:=H^2_0(\Omega)$, the weak formulation associated with [\[vke\]](#vke){reference-type="eqref" reference="vke"} seeks $u,v\in \: V$ such that, for all $(\varphi_{1},\varphi_{2}) \in {\mathbf V}=: V \times V$, $$\begin{aligned} & a(u,\varphi_1)+ b(u,v,\varphi_1) + b(v,u,\varphi_1) = f(\varphi_1) \label{vk_weak1}\\%\;\text{ and }\; & a(v,\varphi_2) -b(u,u,\varphi_2) = 0, \label{vk_weak2} \end{aligned}$$ with, for all $\eta,\chi, \varphi \in V$, $$\begin{aligned} & a(\eta,\chi):=\int_\Omega D^2\eta:D^2\chi {\rm\,dx},\quad f(\varphi):=\int_\Omega f\varphi {\rm\,dx}\mbox{ and } \nonumber \\ & b(\eta,\chi,\varphi):=-\frac{1}{2}\int_\Omega[\eta,\chi]\varphi{\rm\,dx}=\frac{1}{2}\int_\Omega {\rm cof}(D^2\eta) \nabla \chi\cdot \nabla \varphi {\rm\,dx}, \label{defn.b}\end{aligned}$$ where $D^2$ is the Hessian matrix, $:$ denotes the scalar product between the matrices and ${\rm cof}(D^2\eta)$ denotes the co-factor matrix of $D^2\eta$. It is known that [\[vk_weak1\]](#vk_weak1){reference-type="eqref" reference="vk_weak1"}-[\[vk_weak2\]](#vk_weak2){reference-type="eqref" reference="vk_weak2"} possesses at least one solution [@Knightly; @Brezzi; @CiarletPlates]. The combined vector form for [\[vk_weak1\]](#vk_weak1){reference-type="eqref" reference="vk_weak1"}-[\[vk_weak2\]](#vk_weak2){reference-type="eqref" reference="vk_weak2"} seeks $\Psi=(u,v)\in {\mathbf V}$ such that $$\label{VKE_weak} %\cA(\Psi;\Phi):= {A}(\Psi,\Phi)+B(\Psi,\Psi,\Phi)- {F}(\Phi)=0\quad \text{for all}\:\Phi\in {\mathbf V},$$ where, for all $\Xi=(\xi_1,\xi_2),\Theta=(\theta_1,\theta_2)$, and $\Phi=(\varphi_1,\varphi_2)\in {\mathbf V}$, $$\begin{aligned} & {A}(\Theta,\Phi):={} a(\theta_1,\varphi_1) + a(\theta_2,\varphi_2), \; \\ &B(\Xi,\Theta,\Phi):={} b(\xi_1,\theta_2,\varphi_1)+b(\xi_2,\theta_1,\varphi_1)-b(\xi_1,\theta_1,\varphi_2),\,\text{ and }\;\\ & {F}(\Phi) := (f(\varphi_1),0).\end{aligned}$$ The trilinear form $b(\bullet,\bullet,\bullet)$ is symmetric in all the three variables and so is $B(\bullet,\bullet,\bullet)$. The boundedness and ellipticity properties stated below hold [@CiarletPlates; @ng2]: $${A}(\Theta,\Phi)\leq \|\Theta\|_2 \: \|\Phi\|_2,\: {A}(\Theta,\Theta) \geq \|\Theta\|_2^2, \;\text{ and }\; B(\Xi, \Theta, \Phi) \lesssim \|\Xi\|_2 \: \|\Theta\|_2 \: \|\Phi\|_2. \label{eqn:a_vk1}$$ **Lemma 1** (*a priori bounds*). *[@BlumRannacher; @ngr][\[lem:aprioriboundcts\]]{#lem:aprioriboundcts label="lem:aprioriboundcts"} For $f\in H^{-1}(\Omega)$, the solution $\Psi$ of [\[VKE_weak\]](#VKE_weak){reference-type="eqref" reference="VKE_weak"} belongs to ${\mathbf V}\cap {\boldsymbol{H}}^{2 + \alpha} (\Omega)$, $\alpha \in (1/2,1]$, and satisfies the $a~priori$ bounds $\|\Psi\|_2 \lesssim \|f\|_{-1}$ and $\|\Psi\|_{2+\alpha} \lesssim \|f\|^3_{-1}+ \|f\|_{-1}.$* Assume that the solution $\Psi=(u,v)$ is *regular* [@Brezzi; @ng2]. That is, the linearised problem defined by: for given $G=(g_1,g_2) \in \boldsymbol{L}^2(\Omega)$, find $\Theta \in {\mathbf V}$ such that, for all $\Phi \in {\mathbf V}$, $$\label{eqn.ctslinearised} \mathcal{A}(\Theta,\Phi):=A(\Theta,\Phi)+B(\Psi,\Theta,\Phi)+B(\Theta,\Psi,\Phi) =(G,\Phi)$$ is well-posed and satisfies the a priori bounds $$\label{eqn.boundlinearised} \| \Theta \|_2 \lesssim \|G\| \mbox{ and } \| \Theta\|_{2+\alpha}\lesssim \|G\|,$$ where $\alpha$ is the index of elliptic regularity. This is equivalent to an inf-sup condition $$\begin{aligned} \label{inf-sup} 0<\beta:=\inf_{\substack{\Theta\in {\mathbf V}\\ |\Theta|_2=1}}\sup_{\substack{\Phi\in {\mathbf V}\\ |\Phi|_2=1}}\mathcal{A}(\Theta,\Phi).%\big{(}A(\Theta,\Phi)+B(\Psi,\Theta,\Phi)+B(\Theta,\Psi,\Phi)\big{)}.\end{aligned}$$It is well-known [@Knightly] that for sufficiently small $f$, the solution is unique and is a regular solution; but this paper aims at a local approximation of an arbitrary regular solution. Let $\Psi$ be a regular solution to [\[VKE_Morleyweak\]](#VKE_Morleyweak){reference-type="eqref" reference="VKE_Morleyweak"}. Then the dual problem defined by: for $Q \in \boldsymbol{H}^{-1}(\Omega),$ find $\boldsymbol{\xi}\in {\mathbf V}$ such that $$\label{eqn.dualcts} \mathcal{A}_{}(\Phi,\boldsymbol{\xi})=(Q,\Phi)\quad \forall\, \Phi \in {\mathbf V}$$ is well-posed and satisfies the a priori bounds [@ng2]: $$\label{eqn.boundduallinearised} \| \boldsymbol{\xi}\|_2 \lesssim \|Q\|_{-1} \mbox{ and } \| \boldsymbol{\xi}\|_{2+\alpha}\lesssim \|Q\|_{-1}.$$ # Morley type virtual element method {#sec:MorleyVEM} This section deals with the Morley type VEM proposed in [@Zhao_Morley] for [\[VKE_weak\]](#VKE_weak){reference-type="eqref" reference="VKE_weak"}. The Morley type virtual element is a nonconforming virtual element which has fewer degrees of freedom and not even $C^0$ continuous; it is a simplified version of the $C^0$ continuous nonconforming virtual element presented in [@Zhao_ncfem]. Let $\mathcal{T}_h$ be a decomposition of $\Omega$ into non-overlapping simple polygons. Let ${\mathcal{E}}_h$ denotes the set of edges $e$ in $\mathcal{T}_h$ and $h_K$ denotes the diameter of the element $K$. Let $h_{\max}$ be the maximum of the diameters of all the elements of the mesh, i.e., $h_{\max}=\max_{K \in \mathcal{T}_h} h_K$. For any $K \in \mathcal{T}_h$, let $n_K$ denotes its unit outward normal vector along the boundary $\partial K$. The unit normal of an edge $e \in {\mathcal{E}}_h$ is denoted by $n_e$, whose orientation is chosen arbitrarily but fixed for internal edges and coinciding with the outward normal of $\Omega$ for boundary edges. Define the jump $\left[\hskip -3.5pt\left[\varphi\right]\hskip -3.5pt\right]_e:=\varphi|_{K_+}-\varphi|_{K_-}$ and the average $\langle \varphi \rangle _e:=\frac{1}{2}\left(\varphi|_{K_+}+\varphi|_{K_-}\right)$ across the interior edge $e$ of $\varphi\in H^1(\mathcal{T}_h)$ of the adjacent triangles $K_+$ and $K_-$. Extend the definition of the jump and the average to an edge on boundary by $\left[\hskip -3.5pt\left[\varphi\right]\hskip -3.5pt\right]_e:=\varphi|_e$ and $\langle \varphi\rangle_e:=\varphi|_e$ for $e\in {\mathcal{E}}(\partial\Omega)$. For any vector function, the jump and the average are understood component-wise. For a non-negative integer $m$ and $D \subseteq \mathbb{R}^2$, ${\mathcal P}_{m}(D)$ denotes the space of polynomials of degree atmost equal to $m$ on $D$ and $${{\mathcal P}}_m(\mathcal{T}_h):=\{q \in L^2(\Omega):q|_K \in {{\mathcal P}}_m(K) \quad \forall\; K \in \mathcal{T}_h\}.$$ For $\Phi =(\varphi_1,\varphi_2) \in W^{m,p}(\mathcal{T}_h)$, where $W^{m,p}(\mathcal{T}_h)$ denotes the broken Sobolev space with respect to $\mathcal{T}_h$, $\| \Phi \|_{m,p,h}^2:=|\varphi_1|_{m,p,h}^2+|\varphi_2|_{m,p,h}^2$, and $|\varphi_i|_{m,p,h}=(\sum_{K\in \mathcal{T}}|\varphi_i|^{p}_{m,p,K})^{1/p}$, $i=1,2$; with $|\cdot|_{m,p,K}$ denoting the usual semi-norm in $W^{m,p}(K)$. When $p=2$, the corresponding norms are denoted by $\|\bullet\|_{m,h}$ and $|\bullet|_{m,h}$. The notation ${\bf X}$ is used to denote the product space $X\times X$. Assume that there exists a positive real number $C_\mathcal{T}$ such that, for every $K \in \mathcal{T}_h$ [@Veiga_basicVEM]: - $K \in \mathcal{T}_h$ is star-shaped with respect to every point of a ball of radius $C_{\mathcal{T}}h_K$, - the ratio between the shortest edge and the diameter $h_K$ of $K$ is larger than $C_\mathcal{T}$. From [@LCJH_errorVEM], we have that if the mesh $\mathcal{T}_h$ fulfilling the assumptions (**A1**) and (**A2**), then the mesh also satisfies the following property: (**P1**): For each $K \in \mathcal{T}_h$, there exists a virtual triangulation $\mathcal{T}_h^K$ of $K$ such that $\mathcal{T}_h^K$ is uniformly shape regular and quasi-uniform. The corresponding mesh size $h_T$ of $\mathcal{T}_h^K$ is proportional to $h_K$. Every edge of $K$ is a side of a certain triangle in $\mathcal{T}_h^K$. For every $K \in \mathcal{T}_h$, the local shape function space $\widetilde{V}_h(K)$ [@Zhao_ncfem; @Antonietti_ncVEM] is defined by $$\label{def.vhtildek} \widetilde{V}_h(K):=\displaystyle\left\{\varphi \in H^2(K);\Delta^2\varphi =0, \varphi_{|e} \in {{\mathcal P}}_2(e), \Delta \varphi_{|e} \in {{\mathcal P}}_{0}(e), e \subseteq \partial K \right\}.$$ Obviously, ${{\mathcal P}}_2(K)\subseteq \widetilde{V}_h(K)$. The degrees of freedom on $\widetilde{V}_h(K)$ are - The values of $\varphi(a_i)$, $\forall$ vertex $a_i$, - The moments $\displaystyle \frac{1}{h_e}\int_e\varphi{\rm\,ds}$, $\forall$ edge $e$, - The moments $\displaystyle \int_e\frac{\partial \varphi}{\partial n_e}{\rm\,ds}$, $\forall$ edge $e$. For each $K$ and any given $\varphi \in \widetilde{V}_h(K)$, define a projection operator $\Pi^K:\widetilde{V}_h(K) \to {{\mathcal P}}_2(K)\subseteq \widetilde{V}_h(K)$ as the solution to [\[eqn.projectionPiK\]]{#eqn.projectionPiK label="eqn.projectionPiK"} $$\begin{aligned} a^K(\Pi^K \varphi,q)&=a^K(\varphi,q) \quad \forall \; q \in P_2(K)\label{eqn.projectionPiK1} \\ \widehat{\Pi^K(\varphi)}&=\widehat{\varphi}\label{eqn.projectionPiK2}\\ \hspace{-2.2cm}\int_{\partial K}\nabla \Pi^K \varphi {\rm\,ds}&=\int_{\partial K}\nabla \varphi {\rm\,ds}\label{eqn.projectionPiK3}\end{aligned}$$ such that $\Pi^K q=q$ for all $q \in {{\mathcal P}}_2(K)$ and $\Pi^K$ is computable from the above degrees of freedom. Here, $a^K(\bullet,\bullet)$ is the restriction of the continuous bilinear form $a(\bullet,\bullet)$ on the element $K$ and $$\label{def.hat} \widehat \varphi=\frac{1}{n}\sum_{i=1}^{n}\varphi(a_i).$$ This $C^0$-nonconforming virtual element is modified to fully nonconforming virtual element in [@Zhao_Morley] such that the dimension of the shape function space and the degrees of freedom are reduced. Define the local shape function space $V_h(K)$ on a polygon $K$ by $$\label{def.vhk} \displaystyle V_h(K):=\left\{\varphi \in \widetilde{V}_h(K):\int_e \Pi^K\varphi{\rm\,ds}=\int_e \varphi{\rm\,ds}\; \forall e \subseteq \partial K\right\}.$$ Note that $\widetilde{V}_h(K) \subseteq V_h(K)$ and since $\Pi^Kq=q$ for all $q \in {{\mathcal P}}_2(K)$, ${{\mathcal P}}_2(K) \subseteq V_h(K)$. The degrees of freedom on ${V}_h(K)$ are - The values of $\varphi(a_i)$, $\forall$ vertex $a_i$, - The moments $\displaystyle \int_e\frac{\partial \varphi}{\partial n_e}{\rm\,ds}$, $\forall$ edge $e$. Comparing with the degrees of freedom associated with $\widetilde{V}_h(K)$, the zero-order moments of $\varphi$ on edges are removed in the above degrees of freedom. Another fully nonconforming virtual element is presented in [@Antonietti_ncVEM] with the same degrees of freedom as above, but with a different local virtual space. **Remark 2**. *The special case of $V_h(K)$ with $K$ as a triangle together with 6 degrees of freedom leads to $V_h(K)={{\mathcal P}}_2(K)$. This shows that, for the (lowest-order) triangular case, the simplified nonconforming virtual element coincides with the Morley nonconforming finite element [@Ciarlet] with the same degrees of freedom. Hence, the simplified nonconforming virtual element can be viewed as the extension of the Morley element to polygonal meshes.* For every decomposition $\mathcal{T}_h$ of $\Omega$ into simple polygons $K$, define the global space $V_h$ by $$\begin{aligned} V_h:=\Big\{&\varphi_h \in L^2(\Omega);\,\varphi_{h|K} \in V_h(K) \,\forall\; K \in \mathcal{T}_h, \varphi_h \text{ is continuous at the internal vertices}\\ &\text{ and vanishes at the boundary vertices, }\int_e\left[\hskip -3.5pt\left[\frac{\partial \varphi_h}{\partial n_e}\right]\hskip -3.5pt\right]{\rm\,ds}=0 \; \forall \;e \in {\mathcal{E}}_h\Big\}.\label{defn.vh}\end{aligned}$$ The global degrees of freedom on ${V}_h$ are - The values of $\varphi_h(a_i)$, $\forall$ internal vertex $a_i$, - The moments $\displaystyle \int_e\frac{\partial \varphi_h}{\partial n_e}{\rm\,ds}$, $\forall$ internal edge $e$. The space $V_h$ is not a subspace of $H^2_0(\Omega)$ and not even $C^0$ continuous over $\Omega$; hence the simplified virtual element is fully nonconforming. Moreover, $\dim(V_h)=N_V+N_E,$ where $N_V$ is the number of internal vertices of $\mathcal{T}_h$ and $N_E$ is the number of internal edges, see [@Zhao_Morley] for more details. For each element $K \in \mathcal{T}_h$, let $\chi_i$ denote the operator associated with the $i$th degree of freedom, $i=1,\cdots,N^K$. The construction of $V_h$ shows that for every smooth enough function $\varphi$ there exists a unique element, usually known as the interpolant of $\varphi$ restricted to $K$, $\varphi_I^K \in V_h(K)$ such that $$\label{def.phiI} \chi_i(\varphi-\varphi_I^K)=0, \quad i=1,\cdots,N^K.$$ **Lemma 3** (Interpolation error). *[@Brenner; @Zhao_Morley][\[lem.vI\]]{#lem.vI label="lem.vI"} For every $K \in \mathcal{T}_h$ and every $\varphi\in H^s(K)$ with $2 \le s\le 3$, it holds that $$\|\varphi-\varphi_I^K\|_{m,K} \lesssim h_K^{s-m}|\varphi|_{s,K},\quad m=0,1,2.$$* **Lemma 4** (Polynomial error). *[@Brenner; @Ciarlet; @Zhao_Morley][\[lem.vpi\]]{#lem.vpi label="lem.vpi"} For every $K \in \mathcal{T}_h$ and every $\varphi\in H^s(K)$ with $2 \le s\le 3$, there exists a polynomial $\varphi_\pi^K \in {{\mathcal P}}_2(K)$ such that $$\|\varphi-\varphi_\pi^K\|_{m,K} \lesssim h_K^{s-m}|\varphi|_{s,K},\quad m=0,1,2.$$ Also, $\|\varphi-\varphi_\pi^K\|_{1,4,K} \lesssim h_K^{s-3/2}|\varphi|_{s,K}.$* For each polygon $K$, define the discrete local bilinear form on $V_h(K) \times V_h(K)$ by $$\label{defn.ahK} a_h^K(\varphi_h,\psi_h):=a^K(\Pi^K\varphi_h,\Pi^K \psi_h)+S^K(\varphi_h-\Pi^K\varphi_h,\psi_h-\Pi^K\psi_h),\, \forall\,\varphi_h,\psi_h \in V_h(K),$$ where $\Pi^K$ is the projection operator defined in [\[eqn.projectionPiK\]](#eqn.projectionPiK){reference-type="eqref" reference="eqn.projectionPiK"} and $S^K(\bullet,\bullet)$ is a symmetric and positive definite bilinear form satisfying $$\label{defn.SK} c_0 a^K(\varphi_h,\varphi_h)\le S^K(\varphi_h,\varphi_h)\le c_1a^K(\varphi_h,\varphi_h) \quad \forall \;\varphi_h \in \ker(\Pi^K)$$ for some positive constants $c_0$ and $c_1$ independent of $K$ and $h_K$. It is clear from [\[defn.SK\]](#defn.SK){reference-type="eqref" reference="defn.SK"} that $S^K$ must scale like $a^K(\bullet,\bullet)$ on the kernel of $\Pi^K$. As in [@Brezzi_VEM_platebending], set $$S^K(\varphi,\psi)=\sum_{i=1}^{N^K}\chi_i(\varphi)\chi_i(\psi)h_i^{-2},$$ where $h_i$ is the characteristic length attached to each degree of freedom $\chi_i$. The standard arguments [@Veiga_basicVEM] reveals the consistency and stability properties of $a_h^K(\bullet,\bullet)$. That is, $$a_h^K(p,\varphi_h)=a^K(p,\varphi_h) \quad \forall\, p \in {{\mathcal P}}_2(K),\,\forall \,\varphi_h \in V_h(K)\label{eqn.consistency}$$ and there exists two positive constants $\alpha_{\ast}$ and $\alpha^{\ast}$ independent of $h$ and $K$ such that $$\alpha_\ast a^K(\varphi_h,\varphi_h)\le a_h^K(\varphi_h,\varphi_h)\le \alpha^\ast a^K(\varphi_h,\varphi_h)\quad \forall \,\varphi_h \in V_h(K). \label{eqn.stability}$$ The global discrete bilinear form $a_h:V_h\times V_h \to \mathbb{R}$ is then defined by $$\label{defn.ah} a_h(\varphi_h,\psi_h):=\sum_{K \in \mathcal{T}_h}a_h^K(\varphi_h,\psi_h)\quad \forall \; \varphi_h,\psi_h \in V_h.$$ The construction of the discrete trilinear form associated with discrete weak formulation is as follows. Define, for all $\varphi_h,\psi_h,\theta_h \in V_h(K)$, $$\label{defn.bhk} b_{h}^K(\varphi_h,\psi_h,\theta_h)=\int_K {\rm cof}(D^2(\Pi^K\varphi_h))\nabla (\Pi^K \psi_h)\cdot \nabla (\Pi^K\theta_h){\rm\,dx}.$$ Then the global discrete trilinear form $b_h:V_h \times V_h \times V_h \to \mathbb{R}$ is defined by $$\label{defn.bh} b_h(\varphi_h,\psi_h,\theta_h):=\sum_{K \in \mathcal{T}_h}b_h^K(\varphi_h,\psi_h,\theta_h)\quad \forall \; \varphi_h,\psi_h,\theta_h \in V_h.$$ For constructing the linear form on the right-hand side, let $P_0^K(f)$ denote the $L^2$ projection of load $f$ onto ${{\mathcal P}}_0(K)$. Then the right hand side is defined by [@Zhao_Morley] $$\label{defn.fh} \displaystyle \langle f_h,\varphi_h\rangle := \sum_{K \in \mathcal{T}_h}(P_0^K(f),\widehat{\varphi_h})$$ with $\widehat{\varphi_h}$ from [\[def.hat\]](#def.hat){reference-type="eqref" reference="def.hat"}. The Morley type nonconforming virtual element discretisation associated with [\[VKE_weak\]](#VKE_weak){reference-type="eqref" reference="VKE_weak"} seeks $\Psi_h:=(u_h,v_h) \in {\mathbf V}_h:=V_h \times V_h$ such that $$\label{VKE_Morleyweak} A_h(\Psi_h,\Phi_h)+B_h(\Psi_h,\Psi_h,\Phi_h)=F_h(\Phi_h) \quad \forall\,\Phi_h\in {\mathbf V}_h,$$ where for all $\Xi_h=(\xi_{1,h},\xi_{2,h}),\Theta_h=(\theta_{1,h},\theta_{2,h})$, and $\Phi=(\varphi_{1,h},\varphi_{2,h})\in {\mathbf V}_h$, $$\hspace{-6.4cm}{A}_h(\Theta_h,\Phi_h):={} a_h(\theta_{1,h},\varphi_{1,h}) + a_h(\theta_{2,h},\varphi_{2,h}), \;\label{defn.Ah}$$ $$B_h(\Xi_h,\Theta_h,\Phi_h):={} b_h(\xi_{1,h},\theta_{2,h},\varphi_{1,h})+b_h(\xi_{2,h},\theta_{1,h},\varphi_{1,h})-b_h(\xi_{1,h},\theta_{1,h},\varphi_{2,h}),\label{defn.Bh}$$ $$\hspace{-9.8cm}F_h(\Phi_h)=(\langle f_h,\varphi_{1,h}\rangle,0)\label{defn.Fh}$$ with $a_h(\bullet,\bullet)$, $b_h(\bullet,\bullet,\bullet)$ and $\langle f_h,\bullet\rangle$ from [\[defn.ah\]](#defn.ah){reference-type="eqref" reference="defn.ah"}, [\[defn.bh\]](#defn.bh){reference-type="eqref" reference="defn.bh"}, and [\[defn.fh\]](#defn.fh){reference-type="eqref" reference="defn.fh"} respectively. For all $\varphi+\varphi_h,\psi+\psi_h, \theta+\theta_h \in V+V_h$, extend the definition of $b_h^K(\bullet,\bullet,\bullet)$ on $V_h(K)$ to $\widehat{b}_h^K(\bullet,\bullet,\bullet)$ on $V+V_h(K)$ as $$\label{defn.bhkhat} \widehat{b}_h^K(\varphi+\varphi_h, \psi+\psi_h,\theta+\theta_h)=\int_K {\rm cof}(D^2(\varphi+\Pi^K\varphi_h))\nabla (\psi+\Pi^K \psi_h)\cdot \nabla (\theta+\Pi^K\theta_h){\rm\,dx}.$$ Then the global discrete trilinear form $\widehat{b}_h:(V+V_h) \times (V+V_h)\times (V+V_h)\to \mathbb{R}$ is defined by $$\label{defn.bhhat} \widehat{b}_h(\varphi+\varphi_h, \psi+\psi_h,\theta+\theta_h):=\sum_{K \in \mathcal{T}_h}\widehat{b}_h^K(\varphi+\varphi_h, \psi+\psi_h,\theta+\theta_h).$$ For all $\Xi_h=(\xi_{1,h},\xi_{2,h}),\Theta_h=(\theta_{1,h},\theta_{2,h})$, and $\Phi=(\varphi_{1,h},\varphi_{2,h})\in {\mathbf V}+{\mathbf V}_h$, $$\label{defn.Bhhat} \widehat{B}_h(\Xi_h,\Theta_h,\Phi_h):={} \widehat{b}_h(\xi_{1,h},\theta_{2,h},\varphi_{1,h})+\widehat{b}_h(\xi_{2,h},\theta_{1,h},\varphi_{1,h})-\widehat{b}_h(\xi_{1,h},\theta_{1,h},\varphi_{2,h}).$$ The Morley type nonconforming virtual element formulation corresponding to the continuous linearised problem [\[eqn.ctslinearised\]](#eqn.ctslinearised){reference-type="eqref" reference="eqn.ctslinearised"} seeks $\Theta_h \in \text{\bf V}_h$ such that $$\label{eqn.discretelinearised} \mathcal{A}_{h}(\Theta_h,\Phi_h):=A_h(\Theta_h,\Phi_h)+\widehat{B}_h(\Psi,\Theta_h,\Phi_h)+\widehat{B}_h(\Theta_h,\Psi,\Phi_h) =G_h(\Phi_h)\quad \forall \, \Phi_h \in \text{\bf V}_h,$$ where $G_h(\Phi_h)=(\langle g_{1,h},\varphi_{1,h}\rangle,\langle g_{2,h},\varphi_{2,h} \rangle)$ with $\langle \bullet,\bullet\rangle$ as in [\[defn.fh\]](#defn.fh){reference-type="eqref" reference="defn.fh"}. Define the broken semi-norm on $V_h$ by $$\label{eqn.norm} |\varphi_h|_{m,h}:=\left(\sum_{K \in \mathcal{T}_h}|\varphi_h|_{m,K}^2\right)^{1/2}, \quad m=1,2.$$ Then, the piecewise version of the energy norm in $H^2(\mathcal{T}_h)\equiv \prod\limits_{K \in \mathcal{T}} H^2(K)$, $|\bullet|_{2,h}$, is a norm on $V_h$ [@Zhao_Morley Lemma 5.1]. This, in particular, implies $$\label{eqn.seminormbound} \|\varphi_h\|^2+|\varphi_h|_{1,h}^2 \lesssim |\varphi_h|_{2,h}^2.$$ # Well-posedness, existence and uniqueness {#sec:wellposed} This section presents some auxiliary results that are useful to establish the convergence analysis and is followed by the well-posedness of the discrete linearised problem in Section [4.1](#sec:wellposedness_subsec){reference-type="ref" reference="sec:wellposedness_subsec"}. Section [4.2](#sec:existence){reference-type="ref" reference="sec:existence"} discusses the existence and local uniqueness of the discrete solution. **Lemma 5** (Boundedness and coercivity). *Any $\Phi_h,\Theta_h \in {\mathbf V}_h$ satisfy* - *$A_h(\Phi_h,\Theta_h)\lesssim |\Phi_h|_{2,h}|\Theta_h|_{2,h}$.* - *$A_h(\Phi_h,\Phi_h)\gtrsim |\Phi_h|_{2,h}^2.$* - *$F_h(\Phi_h) \lesssim \|f\||\Phi_h|_{2,h}.$* The symmetry of $a_h(\bullet,\bullet)$, [\[eqn.stability\]](#eqn.stability){reference-type="eqref" reference="eqn.stability"}, and the definition of $a^K(\bullet,\bullet)$ imply, for all $\varphi_h,\theta_h \in V_h$, $$\begin{aligned} a_h^K(\varphi_h,\theta_h)\le (a_h^K(\varphi_h,\varphi_h))^{1/2} (a_h^K(\theta_h,\theta_h))^{1/2}&\le \alpha^\ast(a^K(\varphi_h,\varphi_h))^{1/2} (a^K(\theta_h,\theta_h))^{1/2}\\ &=\alpha^\ast|\varphi_h|_{2,K}|\theta_h|_{2,K}. \end{aligned}$$ The sum over all $K \in \mathcal{T}_h$, Hölder inequality, the definition of $|\bullet|_{2,h}$ in [\[eqn.norm\]](#eqn.norm){reference-type="eqref" reference="eqn.norm"}, and [\[defn.Ah\]](#defn.Ah){reference-type="eqref" reference="defn.Ah"} conclude the proof of $(a)$. 0◻ The estimate follows from the definition of $A_h(\bullet,\bullet)$ in [\[defn.Ah\]](#defn.Ah){reference-type="eqref" reference="defn.Ah"}, the stability property [\[eqn.stability\]](#eqn.stability){reference-type="eqref" reference="eqn.stability"}, the definition of $a^K(\bullet,\bullet)$, and [\[eqn.norm\]](#eqn.norm){reference-type="eqref" reference="eqn.norm"}.0◻ The definition of $F_h(\bullet)$ in [\[defn.Fh\]](#defn.Fh){reference-type="eqref" reference="defn.Fh"}, [\[defn.fh\]](#defn.fh){reference-type="eqref" reference="defn.fh"}, Hölder inequality, $\|P_0^K f\|_{0,K} \lesssim \|f\|_{0,K}$, $\|\widehat{\varphi_h}-\varphi_h\|_{0,K} \lesssim h_K \|\varphi_h\|_{1,K}$ for $\varphi_h \in V_h$, and [\[eqn.seminormbound\]](#eqn.seminormbound){reference-type="eqref" reference="eqn.seminormbound"} concludes the proof of $(c)$. 0◻ Since the discrete space $V_h$ is not a subspace of $V$, an *enrichment operator ${E}_h$* which maps the nonconforming discrete space to the conforming space plays an important role in establishing the boundedness properties of the discrete trilinear form and hence to derive the local existence and uniqueness of the discrete solution, and a priori error estimates for the solution of von Kármán equations. For that, consider the local finite dimensional space [@Antonietti_CahnHilliard Section 2.2]: $$\begin{aligned} %\label{defn.vhck} \widehat{V}_h^c(K):=\Big\{\varphi_h \in H^2(K);\,&\Delta^2\varphi_h \in {{\mathcal P}}_2(K),\,\varphi_{h}|_{\partial K}\in C^0(\partial K),\, \varphi_{h}|_e \in {{\mathcal P}}_3(e)\;\forall e \subseteq \partial K,\\ &\nabla \varphi_{h|\partial K}\in C^0(\partial K)^2,\, \frac{\partial \varphi_h}{\partial n_e}\big|_e \in {{\mathcal P}}_1(e)\;\forall e \subseteq \partial K\Big\}.\end{aligned}$$ For $K \in \mathcal{T}_h$, the $H^2$ conforming local virtual finite element space $V_h^c(K)$ is then defined by $$\begin{aligned} \label{defn.vhck} V_h^c(K):=\Big\{\varphi_h \in \widehat{V}_h^c(K);\,&(\varphi_h-\Pi^{K,c}\varphi_h,q)_{0,K}=0\,\forall\, q \in {{\mathcal P}}_2(K)\Big\},\end{aligned}$$ where $\Pi^{K,c}:\widehat{V}_h^c(K) \to {{\mathcal P}}_2(K)\subseteq \widehat{V}_h^c(K)$ is the projection operator associated with the conforming virtual element, see [@Antonietti_CahnHilliard Section 2.2] for more details. The global $C^1$ virtual element space is $$\begin{aligned} \label{defn.vhc} V_h^c:=\Big\{\varphi_h \in V; \,\varphi_h|_K \in {V}_h^c(K)\; \forall\; K \in \mathcal{T}_h\Big\}.\end{aligned}$$ Let ${E}_h:V_h \to V_h^c$ be the enrichment operator. The properties of ${E}_h$ [@Adak_MorleyVEMNSE Propostion 4.1], [@JHYY_mediusVEM Lemma 4.2] that are useful in the analysis are stated in the next lemma. **Lemma 6** (Enrichment operator). *For all ${\varphi}_h\in V_h$, ${E}_h {\varphi}_h \in V_h^c$ satisfies $$\sum_{m=0}^{2}h_K^{m-2}|{\varphi}_h-{E}_h{\varphi}_h|_{m,K} \lesssim |{\varphi}_h|_{2,h}.$$* **Lemma 7** (Discrete Sobolev embeddings). *[@Adak_MorleyVEMNSE Theorem 4.1][\[lem:discreteSobolev\]]{#lem:discreteSobolev label="lem:discreteSobolev"} For $2\le q <\infty$, any ${\varphi}_h \in V_h$ satisfies $|{\varphi}_h|_{1,q,h}\lesssim |{\varphi}_h|_{2,h}.$* Recall the definition of $\Pi^K$ from [\[eqn.projectionPiK\]](#eqn.projectionPiK){reference-type="eqref" reference="eqn.projectionPiK"}. Define $\Pi^h$ in $L^2(\Omega)$ as, for all $v \in H^2(\mathcal{T}_h)$, $$\label{eqn.Pih} (\Pi^h v)|_K:=\Pi^K v,\quad \forall\;K \in \mathcal{T}_h.$$ For $\varphi \in V_h$, a choice of $q=\Pi^K\varphi \in {{\mathcal P}}_2(K)$ in [\[eqn.projectionPiK1\]](#eqn.projectionPiK1){reference-type="eqref" reference="eqn.projectionPiK1"} leads to $a^K(\Pi^K \varphi,\Pi^K \varphi)=a^K(\varphi,\Pi^K \varphi)$. The definition of $a^K(\bullet,\bullet)$ and Cauchy inequality imply $|\Pi^K \varphi|_{2,K} \le |\varphi|_{2,K}.$ Consequently, $$\label{eqn.Pihbound} |\Pi^h \varphi|_{2,h} \le |\varphi|_{2,h}.$$ **Lemma 8** (Poincaré-Freidrich inequality and inverse estimates). *[@Brenner_PFH2],[@JHYY_mediusVEM Lemma 3.2],[@Brenner_errorVEM (2.8)] [\[lem:PF\]]{#lem:PF label="lem:PF"} Any $\varphi\in H^2(K)$ satisfies* - *$\displaystyle h_K^{-2}\|\varphi\|_{0,K}\lesssim |\varphi|_{2,K}+h_K^{-2}\left|\int_{\partial K}\varphi{\rm\,ds}\right|+h_K^{-1}\left|\int_{\partial K}\nabla \varphi {\rm\,ds}\right|$,* - *$|\varphi|_{1,K} \lesssim h_K|\varphi|_{2,K}+h_K^{-1}\|\varphi\|_{0,K},$* - *$\|\varphi\|_{0,\infty,K} \lesssim h_K^{-1}\|\varphi\|_{0,K}+|\varphi|_{1,K}+h_K|\varphi|_{2,K}.$* **Lemma 9** (Projection error). *Any $K \in \mathcal{T}_h$ and $\varphi_h \in V_h$ satisfy $$\|\varphi_h-\Pi^K \varphi_h\|_{0,K}+h_K|\varphi_h-\Pi^K \varphi_h|_{1,K} \lesssim h_K^{2}|\varphi_h-\Pi^K \varphi_h|_{2,K}.$$* *Proof.* Since $\varphi_h \in V_h$, the choice $\varphi=\varphi_h-\Pi^K \varphi_h$ in Lemma [\[lem:PF\]](#lem:PF){reference-type="ref" reference="lem:PF"}.a, [\[def.vhk\]](#def.vhk){reference-type="eqref" reference="def.vhk"}, and [\[eqn.projectionPiK3\]](#eqn.projectionPiK3){reference-type="eqref" reference="eqn.projectionPiK3"} lead to $h_K^{-2}\|\varphi_h-\Pi^K \varphi_h\|_{0,K} \lesssim | \varphi_h-\Pi^K \varphi_h|_{2,K}$. A combination of this and Lemma [\[lem:PF\]](#lem:PF){reference-type="ref" reference="lem:PF"}.b shows $|\varphi_h-\Pi^K \varphi_h|_{1,K} \lesssim h_K|\varphi_h-\Pi^K \varphi_h|_{2,K}$. ◻ Recall the definition of $B_h(\bullet,\bullet,\bullet)$ and $\widehat{B}_h(\bullet,\bullet,\bullet)$ from [\[defn.Bh\]](#defn.Bh){reference-type="eqref" reference="defn.Bh"} and [\[defn.Bhhat\]](#defn.Bhhat){reference-type="eqref" reference="defn.Bhhat"}, and the index of elliptic regularity $\alpha \in (\frac{1}{2},1]$. **Lemma 10** (Bounds for $\widehat{B}_h(\bullet,\bullet,\bullet)$). *Any $\boldsymbol{\xi}\in {\mathbf V}\cap \boldsymbol{H}^{2+\alpha}(\Omega)$, $\Phi,\Theta \in {\mathbf V}$, and $\Phi_h,\Theta_h, \boldsymbol{\xi}_h \in {\mathbf V}_h$ satisfy* - *$B_h(\Phi_h,\Theta_h,\boldsymbol{\xi}_h)\lesssim |\Phi_h|_{2,h}|\Theta_h|_{2,h}|\boldsymbol{\xi}_h|_{2,h},$* - *$\widehat{B}_h(\Theta+\Theta_h,\Phi+\Phi_h,\boldsymbol{\xi}+\boldsymbol{\xi}_h) \lesssim |\Theta+\Pi^h\Theta_h|_{2,h}|\Phi+\Pi^h\Phi_h|_{1,4,h}|\boldsymbol{\xi}+\Pi^h\boldsymbol{\xi}_h|_{1,4,h},$* - *$\widehat{B}_h(\boldsymbol{\xi},\Theta,\Phi+\Phi_h) \lesssim \|\boldsymbol{\xi}\|_{2+\alpha}|\Theta|_{2}|\Phi+\Pi^h\Phi|_{1,h},$* - *$\widehat{B}_h(\boldsymbol{\xi},\Theta_h,\Phi+\Phi_h) \lesssim \|\boldsymbol{\xi}\|_{2+\alpha}|\Theta_h|_{2,h}|\Phi+\Pi^h\Phi|_{1,h},$* - *$\widehat{B}_h(\boldsymbol{\xi},\Phi+\Phi_h,\Theta+\Theta_h) \lesssim \|\boldsymbol{\xi}\|_{2+\alpha}|\Phi+\Pi^h\Phi_h|_{1,h}(|\Theta|_{2}+|\Theta_h|_{2,h}),$* - *$\widehat{B}_h(\Theta+\Theta_h,\boldsymbol{\xi},\Phi+\Phi_h) \lesssim |\Theta+\Pi^h\Theta_h|_{2,h}\|\boldsymbol{\xi}\|_{2+\alpha}|\Phi+\Pi^h\Phi_h|_{1,h}.$* For $\varphi_h,\theta_h,\xi_h \in V_h$, the definition of $b_h(\bullet,\bullet,\bullet)$ in [\[defn.bh\]](#defn.bh){reference-type="eqref" reference="defn.bh"} and a Hölder inequality show $$b_h(\varphi_h,\theta_h,\xi_h)\lesssim |\Pi^h \varphi_h|_{2,h}|\Pi^h \theta_h|_{1,4,h}|\Pi^h \xi_h|_{1,4,h}.$$ Lemma [\[lem:discreteSobolev\]](#lem:discreteSobolev){reference-type="ref" reference="lem:discreteSobolev"} for $q=4$ and [\[eqn.Pihbound\]](#eqn.Pihbound){reference-type="eqref" reference="eqn.Pihbound"} read $b_h(\varphi_h,\theta_h,\xi_h)\lesssim | \varphi_h|_{2,h}| \theta_h|_{2,h}|\xi_h|_{2,h}$. This and the definition of $B_h(\bullet,\bullet,\bullet)$ in [\[defn.Bh\]](#defn.Bh){reference-type="eqref" reference="defn.Bh"} concludes the proof of $(a)$. 0◻ The result follows from the definition of $\widehat{b}_h(\bullet,\bullet,\bullet)$ in [\[defn.bhhat\]](#defn.bhhat){reference-type="eqref" reference="defn.bhhat"}, a Hölder inequality (same as in $(a)$), and [\[defn.Bhhat\]](#defn.Bhhat){reference-type="eqref" reference="defn.Bhhat"}. 0◻ The definition of $\widehat{b}_h(\bullet,\bullet,\bullet)$ in [\[defn.bhhat\]](#defn.bhhat){reference-type="eqref" reference="defn.bhhat"} and a Hölder inequality lead to, for all $\xi \in V\cap H^{2+\alpha}(\Omega)$, $\varphi,\theta \in V$ and $\varphi_h \in V_h$, $$\widehat{b}_h(\xi,\theta,\varphi+\varphi_h)\lesssim |\xi|_{2,4}|\theta|_{1,4}|\varphi+\Pi^h\varphi_h|_{1,h}.$$ The Sobolev embeddings $H^{2+\alpha}(\Omega) \hookrightarrow W^{2,4}(\Omega)$ for $\alpha>1/2$ and $H^2(\Omega) \hookrightarrow W^{1,4}(\Omega)$ [@brezis], and [\[defn.Bhhat\]](#defn.Bhhat){reference-type="eqref" reference="defn.Bhhat"} concludes the proof of $(c)$. 0◻ A Hölder inequality reveals, for all $\xi \in V\cap H^{2+\alpha}(\Omega)$, $\varphi \in V$ and $\theta_h,\varphi_h \in V_h$, $$\widehat{b}_h(\xi,\theta_h,\varphi+\varphi_h)\lesssim |\xi|_{2,4}|\Pi^h\theta_h|_{1,4,h}|\varphi+\Pi^h\varphi_h|_{1,h}.$$ The Sobolev embedding $H^{2+\alpha}(\Omega) \hookrightarrow W^{2,4}(\Omega)$, Lemma [\[lem:discreteSobolev\]](#lem:discreteSobolev){reference-type="ref" reference="lem:discreteSobolev"}, [\[eqn.Pihbound\]](#eqn.Pihbound){reference-type="eqref" reference="eqn.Pihbound"}, and [\[defn.Bhhat\]](#defn.Bhhat){reference-type="eqref" reference="defn.Bhhat"} proves the assertion of $(d)$. 0◻ The Hölder inequality, the Sobolev embeddings $H^{2+\alpha}(\Omega) \hookrightarrow W^{2,4}(\Omega)$, $H^2(\Omega) \hookrightarrow W^{1,4}(\Omega)$, and Lemma [\[lem:discreteSobolev\]](#lem:discreteSobolev){reference-type="ref" reference="lem:discreteSobolev"} provide $$\begin{aligned} \widehat{b}_h(\xi,\varphi+\varphi_h,\theta+\theta_h)&\lesssim |\xi|_{2,4}|\varphi+\Pi^h\varphi_h|_{1,h}(|\theta|_{1,4}+|\Pi^h\theta_h|_{1,4,h})\\ &\lesssim \|\xi\|_{2+\alpha}|\varphi+\Pi^h\varphi_h|_{1,h}(|\theta|_{2}+|\Pi^h\theta_h|_{2,h}).\end{aligned}$$ This, [\[eqn.Pihbound\]](#eqn.Pihbound){reference-type="eqref" reference="eqn.Pihbound"}, and [\[defn.Bhhat\]](#defn.Bhhat){reference-type="eqref" reference="defn.Bhhat"} concludes the proof of $(e)$.0◻ The Hölder inequality and the Sobolev embedding $H^{2+\alpha}(\Omega) \hookrightarrow W^{1,\infty}(\Omega)$ [@brezis] show $$\begin{aligned} \label{eqn.bhhatd} \widehat{b}_h(\theta+\theta_h,\xi,\varphi+\varphi_h)&\lesssim |\theta+\Pi^h\theta_h|_{2,h}|\xi|_{1,\infty}|\varphi+\Pi^h\varphi_h|_{1,h}\nonumber\\ &\lesssim |\theta+\Pi^h\theta_h|_{2,h}\|\xi\|_{2+\alpha}|\varphi+\Pi^h\varphi_h|_{1,h}.\end{aligned}$$ A combination of [\[eqn.bhhatd\]](#eqn.bhhatd){reference-type="eqref" reference="eqn.bhhatd"} and [\[defn.Bhhat\]](#defn.Bhhat){reference-type="eqref" reference="defn.Bhhat"} implies the assertion. 0◻ Define, for all $\Theta_h=(\theta_{1,h},\theta_{2,h})$, and $\Phi=(\varphi_{1,h},\varphi_{2,h})\in {\mathbf V}+{\mathbf V}_h$, $$\label{defn.Apw} A_{\rm {pw}}(\Theta_h,\Phi_h):=\sum_{K \in \mathcal{T}_h} (a^K(\theta_{1,h},\varphi_{1,h}) + a^K(\theta_{2,h},\varphi_{2,h})).$$ Define $\varphi_\pi^h \in {{\mathcal P}}_2(\mathcal{T}_h)$ and $\varphi_I^h \in V_h$ by $$\varphi_\pi^h|_K :=\varphi_\pi^K \quad \mbox{ and }\quad \varphi_I^h|_K :=\varphi_I^K$$ for all $K \in \mathcal{T}_h$, where $\varphi_\pi^K$ and $\varphi_I^K$ are as in Lemma [\[lem.vpi\]](#lem.vpi){reference-type="ref" reference="lem.vpi"} and [\[lem.vI\]](#lem.vI){reference-type="ref" reference="lem.vI"}. **Lemma 11** (Bounds for $A_{\rm {pw}}(\bullet,\bullet)$). *[@Adak_Morleywinddriven],[@ScbSungZhang Lemmas 4.2, 4.3] [\[lem:Apw\]]{#lem:Apw label="lem:Apw"} Any $\boldsymbol{\xi}\in \boldsymbol{H}^{2+\alpha}(\Omega)$ for $\alpha \in (\frac{1}{2},1]$, $\Phi \in {\mathbf V}\cap \boldsymbol{H}^{2+\alpha}(\Omega)$, and $\Phi_h \in {\mathbf V}_h$ satisfy* - *$A_{\rm {pw}}(\boldsymbol{\xi},E_h\Phi_h-\Phi_h)\lesssim h_{\max}^{\alpha}\|\boldsymbol{\xi}\|_{2+\alpha}|\Phi_h|_{2,h},$* - *$A_{\rm {pw}}(\boldsymbol{\xi},\Phi-\Phi_I^h)\lesssim h_{\max}^{2\alpha}\|\boldsymbol{\xi}\|_{2+\alpha}\|\Phi\|_{2+\alpha}$, where $\Phi_I^h \in \text{\bf V}_h$ is the interpolant of $\Phi$.* **Remark 12** (consequences of Lemma [\[lem.vI\]](#lem.vI){reference-type="ref" reference="lem.vI"}, [\[lem.vpi\]](#lem.vpi){reference-type="ref" reference="lem.vpi"}, and [Lemma 6](#lem:Eh){reference-type="ref" reference="lem:Eh"}). *The estimates in Lemma [\[lem.vI\]](#lem.vI){reference-type="ref" reference="lem.vI"}, [\[lem.vpi\]](#lem.vpi){reference-type="ref" reference="lem.vpi"}, and [Lemma 6](#lem:Eh){reference-type="ref" reference="lem:Eh"} give rise some typical estimates utilised throughout the analysis in this paper. For $\varphi \in V \cap H^{2+\alpha}(\Omega)$, a triangle inequality with $\varphi$, Lemma [\[lem.vI\]](#lem.vI){reference-type="ref" reference="lem.vI"}, and Lemma [\[lem.vpi\]](#lem.vpi){reference-type="ref" reference="lem.vpi"} show $$\label{eqn.vIvpi} \|\varphi_I^h-\varphi_\pi^h\|_{2,h} \le \|\varphi-\varphi_I^h\|_{2,h}+ \|\varphi-\varphi_\pi^h\|_{2,h} \lesssim h_{\max}^{\alpha}\|\varphi\|_{2+\alpha}.$$ For $\varphi_h \in V_h$, triangle inequality with $\varphi_h$, Lemma [Lemma 6](#lem:Eh){reference-type="ref" reference="lem:Eh"}, [Lemma 9](#lem:vh-PiKvh){reference-type="ref" reference="lem:vh-PiKvh"}, and [\[eqn.Pihbound\]](#eqn.Pihbound){reference-type="eqref" reference="eqn.Pihbound"} provide $$\label{eqn.Eh} |E_h\varphi_h-\Pi^h\varphi_h|_{1,h} \le |E_h\varphi_h-\varphi_h|_{1,h}+ |\varphi_h-\Pi^h\varphi_h|_{1,h}\lesssim h_{\max}|\varphi_h|_{2,h}.$$ Analog arguments lead to $|E_h\varphi_h-\Pi^h\varphi_h|_{2,h}\lesssim |\varphi_h|_{2,h}$. Lemma [Lemma 9](#lem:vh-PiKvh){reference-type="ref" reference="lem:vh-PiKvh"}, $\Pi^h\varphi_\pi^h=\varphi_\pi^h$, and [\[eqn.Pihbound\]](#eqn.Pihbound){reference-type="eqref" reference="eqn.Pihbound"} read $$\begin{aligned} |\varphi_I^h-\Pi^h\varphi_I^h |_{1,h} &\lesssim h_{\max}|\varphi_I^h-\Pi^h\varphi_I^h |_{2,h} \le h_{\max}(|\varphi_I^h-\varphi_\pi^h|_{2,h}+|\Pi^h(\varphi_\pi^h-\varphi_I^h) |_{2,h})\\ &\lesssim h_{\max}|\varphi_I^h-\varphi_\pi^h |_{2,h}\lesssim h_{\max}^{1+\alpha}\|\varphi\|_{2+\alpha}\label{t3}\end{aligned}$$ with [\[eqn.vIvpi\]](#eqn.vIvpi){reference-type="eqref" reference="eqn.vIvpi"} in the last step. This and Lemma [\[lem.vI\]](#lem.vI){reference-type="ref" reference="lem.vI"} lead to $$\label{eqn.xiPixiIh1h} |\varphi-\Pi^h\varphi_I^h |_{1,h}\lesssim h_{\max}^{1+\alpha}\|\varphi\|_{2+\alpha}\; \mbox{ and }\; |\varphi-\Pi^h\varphi_I^h |_{2,h}\lesssim h_{\max}^{\alpha}\|\varphi\|_{2+\alpha}.$$* ## Well-posedness of the discrete problem {#sec:wellposedness_subsec} This section establishes the well-posedness of the discrete linearised problem [\[eqn.discretelinearised\]](#eqn.discretelinearised){reference-type="eqref" reference="eqn.discretelinearised"}. **Theorem 13** (Well-posedness of discrete linearised problem). *Let $\Psi \in {\mathbf V}\cap \boldsymbol{H}^{2+\alpha}(\Omega)$ be a regular solution to [\[VKE_Morleyweak\]](#VKE_Morleyweak){reference-type="eqref" reference="VKE_Morleyweak"}. Then for sufficiently small $h_{\max}$, the discrete linearised problem [\[eqn.discretelinearised\]](#eqn.discretelinearised){reference-type="eqref" reference="eqn.discretelinearised"} is well-posed.* *Proof.* Since ${\mathbf V}_h$ is finite dimensional and [\[eqn.discretelinearised\]](#eqn.discretelinearised){reference-type="eqref" reference="eqn.discretelinearised"} is linear, establishing an a priori bound is sufficient to prove that [\[eqn.discretelinearised\]](#eqn.discretelinearised){reference-type="eqref" reference="eqn.discretelinearised"} has a unique solution. The choice $\Phi_h =\Theta_h$ in [\[eqn.discretelinearised\]](#eqn.discretelinearised){reference-type="eqref" reference="eqn.discretelinearised"}, Lemma [Lemma 5](#lem:Ahproperties){reference-type="ref" reference="lem:Ahproperties"}.b, [Lemma 10](#lem:boundednessBh){reference-type="ref" reference="lem:boundednessBh"}.d and .f, and [\[eqn.Pihbound\]](#eqn.Pihbound){reference-type="eqref" reference="eqn.Pihbound"} lead to $$\begin{aligned} G_h(\Theta_h)&=\mathcal{A}_{h}(\Theta_h,\Theta_h)=A_h(\Theta_h,\Theta_h)+\widehat{B}_h(\Psi,\Theta_h,\Theta_h)+\widehat{B}_h(\Theta_h,\Psi,\Theta_h)\nonumber\\ %&\gtrsim |\Theta_h|_{2,h}^2-|\Psi|_{2,4}|\Pi^h\Theta_h|_{1,4,h}|\Pi^h\Theta_h|_{1,2,h}-|\Pi^h\Theta_h|_{2,h}|\Psi|_{1,\infty}|\Pi^h\Theta_h|_{1,2,h}\\ &\gtrsim |\Theta_h|_{2,h}^2-\|\Psi\|_{2+\alpha}|\Theta_h|_{2,h}|\Pi^h\Theta_h|_{1,h}.\label{eqn.gh}\end{aligned}$$ The arguments in the proof of Lemma [Lemma 5](#lem:Ahproperties){reference-type="ref" reference="lem:Ahproperties"}.c show $G_h(\Theta_h) \lesssim \|G\||\Theta_h|_{2,h}$. This with the above displayed inequality results in $$\label{eqn.thetah} |\Theta_h|_{2,h}\lesssim \|\Psi\|_{2+\alpha}|\Pi^h\Theta_h|_{1,h}+\|G\|.$$ The triangle inequality and [\[eqn.Eh\]](#eqn.Eh){reference-type="eqref" reference="eqn.Eh"} provide $$\label{eqn.tri.thetah} |\Pi^h\Theta_h|_{1,h}\le |\Pi^h\Theta_h-E_h\Theta_h|_{1,h}+|E_h\Theta_h|_{1}\lesssim h_{\max}|\Theta_h|_{2,h}+|E_h\Theta_h|_{1}.%\lesssim h_{\max}|\Theta_h|_{2,h}+|E_h\Pi^h\Theta_h|_{1,2,h}.$$ To estimate $|E_h\Theta_h|_{1},$ choose $Q=-\Delta E_h\Theta_h$ and $\Phi=E_h\Theta_h$ in the dual problem [\[eqn.dualcts\]](#eqn.dualcts){reference-type="eqref" reference="eqn.dualcts"}. The definition of $\mathcal{A}(\bullet,\bullet)$ in [\[eqn.ctslinearised\]](#eqn.ctslinearised){reference-type="eqref" reference="eqn.ctslinearised"} and [\[eqn.discretelinearised\]](#eqn.discretelinearised){reference-type="eqref" reference="eqn.discretelinearised"} for $\Phi_h=\boldsymbol{\xi}_I^h$ read $$\begin{aligned} |E_h\Theta_h|_{1}^2%&=\cA_{}(E_h\Theta_h,\bxi)\\ &=A(E_h\Theta_h,\boldsymbol{\xi})+B(\Psi,E_h\Theta_h,\boldsymbol{\xi})+B(E_h\Theta_h,\Psi,\boldsymbol{\xi})\\ %&=A_{\pw}(E_h\Pi^h\Theta_h,\bxi-\bxi_I^h)+A_{\pw}(E_h\Pi^h\Theta_h,\bxi_I^h)+\hB_h(\Psi,E_h\Pi^h\Theta_h,\bxi-\bxi_I^h)\nonumber\\ %&\quad+\hB_h(E_h\Pi^h\Theta_h,\Psi,\bxi-\bxi_I^h)+\hB_h(\Psi,E_h\Pi^h\Theta_h,\bxi_I^h)+\hB_h(E_h\Pi^h\Theta_h,\Psi,\bxi_I^h)\nonumber\\ %&\quad -A_h(\Theta_h,\bxi_I^h)-\hB_h(\Psi,\Theta_h,\bxi_h)-\hB_h(\Theta_h,\Psi,\bxi_h)+(G,\bxi_h)\\ &=A_{\rm {pw}}(E_h\Theta_h,\boldsymbol{\xi})+\widehat{B}_h(\Psi,E_h\Theta_h,\boldsymbol{\xi}-\boldsymbol{\xi}_I^h)+\widehat{B}_h(E_h\Theta_h,\Psi,\boldsymbol{\xi}-\boldsymbol{\xi}_I^h)\nonumber\\ &\quad+\widehat{B}_h(\Psi,E_h\Theta_h,\boldsymbol{\xi}_I^h)+\widehat{B}_h(E_h\Theta_h,\Psi,\boldsymbol{\xi}_I^h)-A_h(\Theta_h,\boldsymbol{\xi}_I^h)\nonumber\\ &\quad -\widehat{B}_h(\Psi,\Theta_h,\boldsymbol{\xi}_I^h)-\widehat{B}_h(\Theta_h,\Psi,\boldsymbol{\xi}_I^h)+G_h(\boldsymbol{\xi}_I^h).\end{aligned}$$ Since $\boldsymbol{\xi}_\pi^h \in {{\mathcal P}}_2(\mathcal{T}_h)$, the consistency property in [\[eqn.consistency\]](#eqn.consistency){reference-type="eqref" reference="eqn.consistency"} shows $A_h(\Theta_h,\boldsymbol{\xi}_\pi^h)=A_{\rm {pw}}(\Theta_h,\boldsymbol{\xi}_\pi^h).$ This and elementary algebra reveal $$\begin{aligned} |E_h\Theta_h|_{1,2,h}^2 %&=A_{\pw}(E_h\Theta_h-\Theta_h,\bxi)+A_{\pw}(\Theta_h,\bxi-\bxi_\pi^h)-A_h(\Theta_h,\bxi_I^h-\bxi_\pi^h)\nonumber\\ %&\quad+\hB_h(\Psi,E_h\Theta_h,\bxi-\bxi_I^h)+\hB_h(E_h\Theta_h,\Psi,\bxi-\bxi_I^h)+\hB_h(\Psi,E_h\Theta_h,\bxi_I^h)\nonumber\\ %&\quad+\hB_h(E_h\Theta_h,\Psi,\bxi_I^h) -\hB_h(\Psi,\Theta_h,\bxi_I^h)-\hB_h(\Theta_h,\Psi,\bxi_I^h)+G_h(\bxi_I^h)\nonumber\\ %%&=A_{\pw}(E_h\Theta_h-\Theta_h,\bxi)+A_{\pw}(\Theta_h,\bxi-\bxi_\pi^h)+A_h(\Theta_h,\bxi_\pi^h-\bxi_I^h)\nonumber\\ %%&\quad+\hB_h(\Psi,E_h\Theta_h,\bxi-\bxi_I^h)+\hB_h(E_h\Theta_h,\Psi,\bxi-\bxi_I^h)\nonumber\\ %%&\quad+\hB_h(\Psi,E_h\Theta_h-\Theta_h,\bxi_I^h)+\hB_h(E_h\Theta_h-\Theta_h,\Psi,\bxi_I^h) +G_h(\bxi_I^h)\\ &=A_{\rm {pw}}(E_h\Theta_h-\Theta_h,\boldsymbol{\xi})+A_{\rm {pw}}(\Theta_h,\boldsymbol{\xi}-\boldsymbol{\xi}_\pi^h)+A_h(\Theta_h,\boldsymbol{\xi}_\pi^h-\boldsymbol{\xi}_I^h)\nonumber\\ &\quad+\widehat{B}_h(\Psi,E_h\Theta_h,\boldsymbol{\xi}-\boldsymbol{\xi}_I^h)+\widehat{B}_h(E_h\Theta_h,\Psi,\boldsymbol{\xi}-\boldsymbol{\xi}_I^h)\nonumber\\ &\quad+\widehat{B}_h(\Psi,E_h\Theta_h-\Theta_h,\boldsymbol{\xi}_I^h)+\widehat{B}_h(E_h\Theta_h-\Theta_h,\Psi,\boldsymbol{\xi}_I^h-\boldsymbol{\xi})\nonumber\\ &\quad+\widehat{B}_h(E_h\Theta_h-\Theta_h,\Psi,\boldsymbol{\xi}) +G_h(\boldsymbol{\xi}_I^h):=T_1+\cdots+T_9.\label{eqn.t1tot9}\end{aligned}$$ Lemma [\[lem:Apw\]](#lem:Apw){reference-type="ref" reference="lem:Apw"} leads to $T_1 \lesssim h_{\max}^\alpha \|\boldsymbol{\xi}\|_{2+\alpha}|\Theta_h|_{2,h}$. The Cauchy inequality and Lemma [\[lem.vpi\]](#lem.vpi){reference-type="ref" reference="lem.vpi"} show $T_2\lesssim h_{\max}^\alpha \|\boldsymbol{\xi}\|_{2+\alpha}|\Theta_h|_{2,h}$. Lemma [Lemma 5](#lem:Ahproperties){reference-type="ref" reference="lem:Ahproperties"}.a, and [\[eqn.vIvpi\]](#eqn.vIvpi){reference-type="eqref" reference="eqn.vIvpi"} provide $T_3\lesssim h_{\max}^\alpha \|\boldsymbol{\xi}\|_{2+\alpha}|\Theta_h|_{2,h}$. Lemma [Lemma 10](#lem:boundednessBh){reference-type="ref" reference="lem:boundednessBh"}.c, triangle inequality with $\Theta_h$, Lemma [Lemma 6](#lem:Eh){reference-type="ref" reference="lem:Eh"} and [\[eqn.xiPixiIh1h\]](#eqn.xiPixiIh1h){reference-type="eqref" reference="eqn.xiPixiIh1h"} read $T_4 \lesssim h_{\max}^{1+\alpha}\|\boldsymbol{\xi}\|_{2+\alpha}|\Theta_h|_{2,h}$. Lemma [Lemma 10](#lem:boundednessBh){reference-type="ref" reference="lem:boundednessBh"}.f shows $T_5 \lesssim |E_h \Theta_h|_{2,h}\|\Psi\|_{2+\alpha}|\boldsymbol{\xi}-\Pi^h\boldsymbol{\xi}_I^h|_{1,h}$. Lemma [Lemma 6](#lem:Eh){reference-type="ref" reference="lem:Eh"} and [\[eqn.xiPixiIh1h\]](#eqn.xiPixiIh1h){reference-type="eqref" reference="eqn.xiPixiIh1h"} provide $T_5 \lesssim h_{\max}^{1+\alpha}\|\boldsymbol{\xi}\|_{2+\alpha}|\Theta_h|_{2,h}$. Lemma [Lemma 10](#lem:boundednessBh){reference-type="ref" reference="lem:boundednessBh"}.e, [\[eqn.Eh\]](#eqn.Eh){reference-type="eqref" reference="eqn.Eh"}, and Lemma [\[lem.vI\]](#lem.vI){reference-type="ref" reference="lem.vI"} reveal $T_6 \lesssim h_{\max}\|\boldsymbol{\xi}\|_{2+\alpha}|\Theta_h|_{2,h}$. Lemma [Lemma 10](#lem:boundednessBh){reference-type="ref" reference="lem:boundednessBh"}.f, Remark [Remark 12](#remark.consequence){reference-type="ref" reference="remark.consequence"}, and [\[eqn.xiPixiIh1h\]](#eqn.xiPixiIh1h){reference-type="eqref" reference="eqn.xiPixiIh1h"} lead to $T_7 \lesssim h_{\max}^{1+\alpha}\|\boldsymbol{\xi}\|_{2+\alpha}|\Theta_h|_{2,h}$. Since $\Psi,\boldsymbol{\xi}\in \text{\bf V}\cap \boldsymbol{H}^{2+\alpha}(\Omega)$, the Sobolev embedding $H^{2+\alpha}(\Omega) \hookrightarrow W^{2,4}(\Omega)$ shows $$|\nabla \Psi \cdot \nabla \boldsymbol{\xi}|_{1} \le |\nabla \Psi|_{1,4}|\nabla \boldsymbol{\xi}|_{1,4}\lesssim \|\Psi\|_{2+\alpha}\|\boldsymbol{\xi}\|_{2+\alpha}.$$ Hence, $\nabla \Psi \cdot \nabla \boldsymbol{\xi}\in H^1(\Omega)$ and $T_8 \lesssim \sum_{K \in \mathcal{T}_h}|D^2(E_h \Theta_h- \Pi^h\Theta_h)|_{-1,K}|\nabla \Psi \cdot \nabla \boldsymbol{\xi}|_{1,K}$. The definition of the dual norm and integration by parts lead to $|D^2(E_h \Theta_h- \Pi^h\Theta_h)|_{-1,K}\lesssim |E_h \Theta_h-\Pi^h\Theta_h|_{1,K}$. The combination of the above estimates and Hölder inequality provides $T_8 \lesssim |E_h \Theta_h- \Pi^h\Theta_h|_{1,h}\|\Psi\|_{2+\alpha}\|\boldsymbol{\xi}\|_{2+\alpha}$. This and [\[eqn.Eh\]](#eqn.Eh){reference-type="eqref" reference="eqn.Eh"} imply $T_8 \lesssim h_{\max}\|\boldsymbol{\xi}\|_{2+\alpha}|\Theta_h|_{2,h}$. The same arguments in the proof of Lemma [Lemma 5](#lem:Ahproperties){reference-type="ref" reference="lem:Ahproperties"}.c provides $T_9 \lesssim \|G\||\boldsymbol{\xi}_I^h|_{2,h}\lesssim \|G\|\|\boldsymbol{\xi}\|_{2+\alpha}$ with Lemma [\[lem.vI\]](#lem.vI){reference-type="ref" reference="lem.vI"} in the last step. A substitution of the estimates $T_1,\cdots,T_9$ in [\[eqn.t1tot9\]](#eqn.t1tot9){reference-type="eqref" reference="eqn.t1tot9"} reads $$|E_h\Theta_h|_{1}^2\lesssim (h_{\max}^\alpha|\Theta_h|_{2,h}+\|G\|)\|\boldsymbol{\xi}\|_{2+\alpha}.$$ Since $\|Q\|_{-1}=|E_h\Theta_h|_{1,2,h}$, the above displayed inequality and [\[eqn.boundduallinearised\]](#eqn.boundduallinearised){reference-type="eqref" reference="eqn.boundduallinearised"} imply $$|E_h\Theta_h|_{1}\lesssim h_{\max}^\alpha|\Theta_h|_{2,h}+\|G\|.$$ The combination of the above displayed inequality, [\[eqn.tri.thetah\]](#eqn.tri.thetah){reference-type="eqref" reference="eqn.tri.thetah"}, and [\[eqn.thetah\]](#eqn.thetah){reference-type="eqref" reference="eqn.thetah"} results in $$|\Theta_h|_{2,h}\le C_1 h_{\max}^\alpha |\Theta_h|_{2,h}+C\|G\|.$$ For a choice of $h_{\max}\le h_1=\left(\frac{1}{2C_1}\right)^{\frac{1}{\alpha}}$, $|\Theta_h|_{2,h}\lesssim \|G\|$ and this implies the assertion. ◻ **Remark 14**. *Let $\Psi$ be a regular solution to [\[eqn.discretelinearised\]](#eqn.discretelinearised){reference-type="eqref" reference="eqn.discretelinearised"}. Then for a sufficiently small $h_{\max}$, the discrete linearised dual problem: given $Q \in \boldsymbol{L}^2(\Omega)$, find $\boldsymbol{\xi}_h \in {\mathbf V}_h$ such that $$\mathcal{A}_{h}(\Phi_h,\boldsymbol{\xi}_h)=Q_h(\Phi_h), \forall\, \Phi_h \in \text{\bf V}_h$$ is well-posed, where $\mathcal{A}_{h}(\bullet,\bullet)$ is defined as in [\[eqn.discretelinearised\]](#eqn.discretelinearised){reference-type="eqref" reference="eqn.discretelinearised"}. The proof is similar to Theorem [Theorem 13](#thm.discretelinearised){reference-type="ref" reference="thm.discretelinearised"} and hence is skipped.* Since the discrete linearised problem and the dual problem are well-posed, $\mathcal{A}_{h}:{\mathbf V}_h\times {\mathbf V}_h \to \mathbb{R}$ defined by $$\label{defn.Apsih} \mathcal{A}_{h}(\Theta_h,\Phi_h):=A_h(\Theta_h,\Phi_h)+\widehat{B}_h(\Psi,\Theta_h,\Phi_h)+\widehat{B}_h(\Theta_h,\Psi,\Phi_h)$$ satisifies *discrete inf-sup condition* on ${\mathbf V}_h \times {\mathbf V}_h$ [@Brezzi; @ng2], that is, there exists a constant $\widehat{\beta}>0$ such that $$\label{defn.beta} \sup_{|\Theta_h|_{2,h}=1}\mathcal{A}_{h}(\Theta_h,\Phi_h)\ge \widehat{\beta} |\Phi_h|_{2,h}\mbox{ and } \sup_{|\Phi_h|_{2,h}=1}\mathcal{A}_{h}(\Theta_h,\Phi_h)\ge \widehat{\beta} |\Theta_h|_{2,h}.$$ Define the perturbed bilinear form by, for all $\Theta_h,\Phi_h \in {\mathbf V}_h$, $$\label{defn.perturbed} \widehat{\mathcal A}_{h}(\Theta_h,\Phi_h):=A_h(\Theta_h,\Phi_h)+B_h(\Psi_I^h,\Theta_h,\Phi_h)+B_h(\Theta_h,\Psi_I^h,\Phi_h),$$ where $\Psi_I^h$ is the interpolant of $\Psi$. **Lemma 15**. *Let $\Psi$ be a regular solution to [\[VKE_weak\]](#VKE_weak){reference-type="eqref" reference="VKE_weak"}. Then for a sufficiently small $h_{\max}$, the perturbed bilinear form $\widehat{\mathcal A}_h(\bullet,\bullet)$ in [\[defn.perturbed\]](#defn.perturbed){reference-type="eqref" reference="defn.perturbed"} satisfies discrete inf-sup condition on ${\mathbf V}_h \times {\mathbf V}_h$.* *Proof.* Elementary algebra and [\[defn.beta\]](#defn.beta){reference-type="eqref" reference="defn.beta"} show $$\begin{aligned} \sup_{|\Theta_h|_{2,h}=1}\widehat{\mathcal A}_{h}&(\Theta_h,\Phi_h)=\sup_{|\Theta_h|_{2,h}=1}(A_h(\Theta_h,\Phi_h)+B_h(\Psi_I^h,\Theta_h,\Phi_h)+B_h(\Theta_h,\Psi_I^h,\Phi_h))\\ %\sup_{|\Theta_h|_{2,h}=1}&(A_h(\Theta_h,\Phi_h)+\hB_h(\Psi-\widetilde{\Psi},\Theta_h,\Phi_h)+\hB_h(\Theta_h,\Psi-\widetilde{\Psi},\Phi_h))\\ &\ge \sup_{|\Theta_h|_{2,h}=1}\mathcal{A}_h(\Theta_h,\Phi_h)-\sup_{|\Theta_h|_{2,h}=1}(\widehat{B}_h({\Psi}-\Psi_I^h,\Theta_h,\Phi_h)+\widehat{B}_h(\Theta_h,{\Psi}-\Psi_I^h,\Phi_h))\\ &\ge \widehat{\beta} |\Phi_h|_{2,h}-\sup_{|\Theta_h|_{2,h}=1}(\widehat{B}_h({\Psi}-\Psi_I^h,\Theta_h,\Phi_h)+\widehat{B}_h(\Theta_h,{\Psi}-\Psi_I^h,\Phi_h)).\label{eqn.sup}%\\ %&\ge \widehat{\beta} |\Phi_h|_{2,h}-2C_b|{\Psi}-\Pi^h\Psi_I^h|_{2,h}|\Phi_h|_{2,h}\label{eqn.sup}\end{aligned}$$ Lemma [Lemma 10](#lem:boundednessBh){reference-type="ref" reference="lem:boundednessBh"}.b, [\[lem:discreteSobolev\]](#lem:discreteSobolev){reference-type="ref" reference="lem:discreteSobolev"}, and [\[eqn.Pihbound\]](#eqn.Pihbound){reference-type="eqref" reference="eqn.Pihbound"} imply $\widehat{B}_h({\Psi}-\Psi_I^h,\Theta_h,\Phi_h) \lesssim |{\Psi}-\Pi^h\Psi_I^h|_{2,h}|\Phi_h|_{2,h}$. This and [\[eqn.xiPixiIh1h\]](#eqn.xiPixiIh1h){reference-type="eqref" reference="eqn.xiPixiIh1h"} provide $\widehat{B}_h({\Psi}-\Psi_I^h,\Theta_h,\Phi_h) \le C_b^1h_{\max}^\alpha\|\Psi\|_{2+\alpha}|\Phi_h|_{2,h}$. Lemma [Lemma 10](#lem:boundednessBh){reference-type="ref" reference="lem:boundednessBh"}.b, [\[lem:discreteSobolev\]](#lem:discreteSobolev){reference-type="ref" reference="lem:discreteSobolev"}, and [\[eqn.Pihbound\]](#eqn.Pihbound){reference-type="eqref" reference="eqn.Pihbound"} imply $\widehat{B}_h(\Theta_h,{\Psi}-\Psi_I^h,\Phi_h) \lesssim |{\Psi}-\Pi^h\Psi_I^h|_{1,4,h}|\Phi_h|_{2,h}$. The relation $\Pi^h\Psi_\pi^h=\Psi_\pi^h$, inverse estimate for $\Pi^h({\Psi_\pi^h}-\Psi_I^h )\in {{\mathcal P}}_2(\mathcal{T}_h)$[@Brenner], and Lemma [\[lem.vpi\]](#lem.vpi){reference-type="ref" reference="lem.vpi"} show $$\begin{aligned} \label{eqn.PsiPiPsiIh14h} |{\Psi}-\Pi^h\Psi_I^h|_{1,4,h}&\le |{\Psi}-\Psi_\pi^h|_{1,4,h}+|\Pi^h{(\Psi_\pi^h}-\Psi_I^h)|_{1,4,h} \nonumber \\&\lesssim h_{\max}^{\alpha+1/2}\|\Psi\|_{2+\alpha}+h_{\max}^{-1/2}|\Pi^h({\Psi_\pi^h}-\Psi_I^h)|_{1,h}\nonumber\\ &\lesssim h_{\max}^{\alpha+1/2}\|\Psi\|_{2+\alpha}+h_{\max}^{-1/2}(|{\Psi_\pi^h}-\Psi_I^h|_{1,h}+|\Psi_I^h-\Pi^h\Psi_I^h|_{1,h})\nonumber\\ &\lesssim h_{\max}^{\alpha+1/2}\|\Psi\|_{2+\alpha}+h_{\max}^{-1/2}|\Psi_I^h-\Pi^h\Psi_I^h|_{1,h}\end{aligned}$$ with Lemma [\[lem.vpi\]](#lem.vpi){reference-type="ref" reference="lem.vpi"} and [\[lem.vI\]](#lem.vI){reference-type="ref" reference="lem.vI"} in the last step. This and [\[t3\]](#t3){reference-type="eqref" reference="t3"} reveal $$\label{eqn.psipihppsiIh14h} |{\Psi}-\Pi^h\Psi_I^h|_{1,4,h} \le h_{\max}^{\alpha+1/2}\|\Psi\|_{2+\alpha}.$$ Consequently, $\widehat{B}_h(\Theta_h,{\Psi}-\Psi_I^h,\Phi_h) \le C_b^2h_{\max}^\alpha\|\Psi\|_{2+\alpha}|\Phi_h|_{2,h}$. Let $C_b:=\max\{C_b^1,C_b^2\}$. Then, [\[eqn.sup\]](#eqn.sup){reference-type="eqref" reference="eqn.sup"} shows $$\begin{aligned} \sup_{|\Theta_h|_{2,h}=1}\widehat{\mathcal A}_{h}&(\Theta_h,\Phi_h)\ge \widehat{\beta} |\Phi_h|_{2,h}-2C_bh_{\max}^\alpha\|\Psi\|_{2+\alpha}|\Phi_h|_{2,h}.\end{aligned}$$ A choice of $\displaystyle h_{\max}\le h_2=\left(\frac{\widehat{\beta}}{4C_b\|\Psi\|_{2+\alpha}}\right)^{1/\alpha}$ provides $$\sup_{|\Theta_h|_{2,h}=1}\widehat{\mathcal A}_{h}(\Theta_h,\Phi_h)\ge ({\widehat{\beta}}/{2}) |\Phi_h|_{2,h}.$$ The analog arguments leads to $\displaystyle \sup_{|\Phi_h|_{2,h}=1}\widehat{\mathcal A}_{h}(\Phi_h,\Theta_h)\ge ({\widehat{\beta}}/{2}) |\Theta_h|_{2,h}.$ ◻ ## Existence and uniqueness {#sec:existence} This section establishes the existence and uniqueness of the discrete solution and is an application of Brouwer's fixed point theorem. **Theorem 16** (Existence and uniqueness of a discrete solution). *Let $\Psi$ be a regular solution to [\[VKE_weak\]](#VKE_weak){reference-type="eqref" reference="VKE_weak"}. For a sufficiently small choice of $h_{\max}$, there exists a unique solution $\Psi_h$ to the discrete [\[VKE_Morleyweak\]](#VKE_Morleyweak){reference-type="eqref" reference="VKE_Morleyweak"} that satisfies $|\Psi_h-\Psi_I^h|_{2,h}\le R(h_{\max})$ for some positive constant $R(h_{\max})$ depending on $h_{\max}$.* *Proof.* Consider the non-linear mapping $T_h:\text{\bf V}_h \to\text{\bf V}_h$ defined by, for all $\Phi_h \in \text{\bf V}_h$, $$\begin{aligned} \label{eqn.Th} &\widehat{\mathcal A}_h(T_h(\Theta_h),\Phi_h)=F_h(\Phi_h)+B_h(\Psi_I^h,\Theta_h,\Phi_h)+B_h(\Theta_h,\Psi_I^h,\Phi_h)-B_h(\Theta_h,\Theta_h,\Phi_h).\end{aligned}$$ Lemma [Lemma 15](#lem:perturbed){reference-type="ref" reference="lem:perturbed"} shows that the operator $T_h$ is well defined and continuous. It is easy to check that the any discrete solution $\Psi_h$ to [\[VKE_Morleyweak\]](#VKE_Morleyweak){reference-type="eqref" reference="VKE_Morleyweak"} is a fixed point of $T_h$ and vice versa. Hence, in order to show the existence of a solution to [\[VKE_Morleyweak\]](#VKE_Morleyweak){reference-type="eqref" reference="VKE_Morleyweak"}, it is enough to prove that $T_h$ has a fixed point. For that, define $$B_R(\Psi_I^h):=\{\Phi_h \in \text{\bf V}_h:|\Phi_h-\Psi_I^h|_{2,h}\le R\}.$$ *Step 1 establishes mapping of ball to ball.* Lemma [Lemma 15](#lem:perturbed){reference-type="ref" reference="lem:perturbed"} provides, for any $\Phi_h \in {\mathbf V}_h$ with $|\Phi_h|_{2,h}=1,$ $$\label{eqn.beta4} {\widehat{\beta}}|T_h(\Theta_h)-\Psi_I^h|_{2,h} \le \widehat{\mathcal A}_h(T_h(\Theta_h)-\Psi_I^h,\Phi_h).$$ The definition of $\widehat{\mathcal A}_h(\bullet,\bullet)$ (resp. $\widehat{\mathcal A}_h(T_h(\bullet),\bullet)$) in [\[defn.perturbed\]](#defn.perturbed){reference-type="eqref" reference="defn.perturbed"} (resp. [\[eqn.Th\]](#eqn.Th){reference-type="eqref" reference="eqn.Th"}) and [\[VKE_weak\]](#VKE_weak){reference-type="eqref" reference="VKE_weak"} with $\Phi=E_h\Phi_h$ show $$\begin{aligned} \widehat{\mathcal A}_h(T_h(\Theta_h)-\Psi_I^h,\Phi_h)&=\widehat{\mathcal A}_h(T_h(\Theta_h),\Phi_h)-\widehat{\mathcal A}_h(\Psi_I^h,\Phi_h)\nonumber\\ &=F_h(\Phi_h)+B_h(\Psi_I^h,\Theta_h,\Phi_h)+B_h(\Theta_h,\Psi_I^h,\Phi_h)-B_h(\Theta_h,\Theta_h,\Phi_h)\nonumber\\ &\quad -A_h(\Psi_I^h,\Phi_h)-2B_h(\Psi_I^h,\Psi_I^h,\Phi_h)\nonumber\\ &=(F_h(\Phi_h)-F(E_h\Phi_h))+(A(\Psi,E_h\Phi_h)-A_h(\Psi_I^h,\Phi_h))\nonumber\\ &\quad +(B(\Psi,\Psi,E_h\Phi_h)-B_h(\Psi_I^h,\Psi_I^h,\Phi_h))+B_h(\Psi_I^h-\Theta_h,\Theta_h-\Psi_I^h,\Phi_h)\nonumber\\ &=:T_1+T_2+T_3+T_4.\label{t1tot4}\end{aligned}$$ An introduction of $(f,\varphi_{1,h})$ for $\Phi_h=(\varphi_{1,h},\varphi_{2,h})$, $|\langle f_h,\varphi_{1,h}\rangle_K -(f,\varphi_{1,h})_K| \lesssim h_{K}\|f\|_{0,K}|\varphi_{1,h}|_{1,K}$ [@Zhao_Morley Lemma 5.2] for $K \in \mathcal{T}_h$, Cauchy inequality, and Lemma [Lemma 6](#lem:Eh){reference-type="ref" reference="lem:Eh"} lead to $$T_1 = \langle f_h,\varphi_{1,h}\rangle -(f,\varphi_{1,h})+(f,\varphi_{1,h}-E_h\varphi_{1,h})\lesssim h_{\rm max}\|f\|.$$ The consistency property $A_h(\Psi_\pi^h,\Phi_h)=A_{\rm {pw}}(\Psi_\pi^h,\Phi_h)$ from [\[eqn.consistency\]](#eqn.consistency){reference-type="eqref" reference="eqn.consistency"} reveals $T_2 = A_{\rm {pw}}(\Psi,E_h\Phi_h-\Phi_h)-A_h(\Psi_I^h-\Psi_\pi^h,\Phi_h)+A_{\rm {pw}}(\Psi-\Psi_\pi^h,\Phi_h)$. Lemma [\[lem:Apw\]](#lem:Apw){reference-type="ref" reference="lem:Apw"}.a and [Lemma 5](#lem:Ahproperties){reference-type="ref" reference="lem:Ahproperties"}.a, Cauchy inequality, [\[eqn.vIvpi\]](#eqn.vIvpi){reference-type="eqref" reference="eqn.vIvpi"}, and Lemma [\[lem.vpi\]](#lem.vpi){reference-type="ref" reference="lem.vpi"} read $T_2\lesssim h_{\rm max}^\alpha \|\Psi\|_{2+\alpha}.$ Lemma [Lemma 10](#lem:boundednessBh){reference-type="ref" reference="lem:boundednessBh"}.c, b, e, [\[lem:discreteSobolev\]](#lem:discreteSobolev){reference-type="ref" reference="lem:discreteSobolev"}, and [\[eqn.Pihbound\]](#eqn.Pihbound){reference-type="eqref" reference="eqn.Pihbound"} show $$\begin{aligned} T_3&=\widehat{B}_h(\Psi,\Psi,E_h\Phi_h-\Phi_h)+\widehat{B}_h(\Psi-\Psi_I^h,\Psi_I^h,\Phi_h)+\widehat{B}_h(\Psi,\Psi-\Psi_I^h,\Phi_h)\nonumber\\ &\lesssim \|\Psi\|_{2+\alpha}\|\Psi\|_2|E_h\Phi_h-\Pi^h\Phi_h|_{1,h}+|\Psi-\Pi^h\Psi_I^h|_{2,h}|\Psi_I^h|_{2,h}+\|\Psi\|_{2+\alpha}|\Psi-\Pi^h\Psi_I^h|_{1,h}\nonumber .%\\ %&\lesssim h_{\max}\|\Psi\|_{2+\alpha}\|\Psi\|_2\|E_h\Phi_h-\Pi^h\Phi_h\|_{1,h}+h_{\max}^\alpha \|\Psi\|_{2+\alpha}\|\Psi\|_2\lesssim h_{\max}^\alpha\|\Psi\|_{2+\alpha}\|\Psi\|_2\end{aligned}$$ The estimates in [\[eqn.xiPixiIh1h\]](#eqn.xiPixiIh1h){reference-type="eqref" reference="eqn.xiPixiIh1h"}, [\[eqn.Eh\]](#eqn.Eh){reference-type="eqref" reference="eqn.Eh"}, and Lemma [\[lem.vI\]](#lem.vI){reference-type="ref" reference="lem.vI"} in the above displayed inequality shows $T_3 \lesssim h_{\max}^\alpha\|\Psi\|_{2+\alpha}^2$. Lemma [Lemma 10](#lem:boundednessBh){reference-type="ref" reference="lem:boundednessBh"}.a (with hidden constant $C_b$) implies $T_4 \le C_b |\Psi_I^h-\Theta_h|_{2,h}^2.$ A combination of $T_1,\cdots,T_4$ in [\[t1tot4\]](#t1tot4){reference-type="eqref" reference="t1tot4"} and then in [\[eqn.beta4\]](#eqn.beta4){reference-type="eqref" reference="eqn.beta4"} leads to $$|T_h(\Theta_h)-\Psi_I^h|_{2,h} \le Ch_{\rm max}^\alpha+C_b|\Psi_I^h-\Theta_h|_{2,h}^2$$ for some positive constant $C$ independent of $h$ but dependent on $\|\Psi\|_{2+\alpha}$ and $\|f\|$. A choice of $h_{\max}\le h_3=\left(\frac{1}{4C C_b}\right)^{\frac{1}{\alpha}}$ yields $4CC_bh_{\max}^\alpha \le 1$. Since $|\Theta_h-\Psi_I^h|_{2,h}\le R(h_{\max})$, a choice of $R(h_{\max})=2Ch_{\max}^\alpha$ leads to $$|\Theta_h-\Psi_I^h|_{2,h}\le Ch_{\rm max}^\alpha(1+4CC_bh_{\rm max}^\alpha)\le R(h_{\max}).$$ Hence, $T_h$ maps the ball $B_R(\Psi_I^h)$ into itself. *Step 2 establishes the existence of a discrete solution.* Since $T_h$ maps $B_R(\Psi_I^h)$ to itself from Step 1, the Brouwer fixed point theorem yields that the mapping $T_h$ has a fixed point, say $\Psi_h$. Hence, [\[eqn.Th\]](#eqn.Th){reference-type="eqref" reference="eqn.Th"} and [\[defn.perturbed\]](#defn.perturbed){reference-type="eqref" reference="defn.perturbed"} reveal that $\Psi_h$ is a solution to [\[VKE_Morleyweak\]](#VKE_Morleyweak){reference-type="eqref" reference="VKE_Morleyweak"} with $|\Psi_h-\Psi_I^h|_{2,h}\le R(h_{\max})$. *Step 3 establishes that $T_h$ is a contraction.* For $\Theta_1,\Theta_2 \in B_{R(h_{\max})}(\Psi_I^h)$ and for all $\Phi_h \in \text{\bf V}_h$, let $T_h(\Theta_i)$, $i=1,2$ be the solutions of $$\begin{aligned} \widehat{\mathcal A}_h(T_h(\Theta_i),\Phi_h)&=F_h(\Phi_h)+B_h(\Psi_I^h,\Theta_i,\Phi_h)+B_h(\Theta_i,\Psi_I^h,\Phi_h)-B_h(\Theta_i,\Theta_i,\Phi_h). \end{aligned}$$ Lemma [Lemma 15](#lem:perturbed){reference-type="ref" reference="lem:perturbed"} shows, for any $\Phi_h \in {\mathbf V}_h$ with $|\Phi_h|_{2,h}=1,$ $$\begin{aligned} {\widehat{\beta}}&|T_h(\Theta_1)-T_h(\Theta_2)|_{2,h} \le \widehat{\mathcal A}_h(T_h(\Theta_1)-T_h(\Theta_2),\Phi_h)\\ &=B_h(\Psi_I^h,\Theta_1-\Theta_2,\Phi_h)+B_h(\Theta_1-\Theta_2,\Psi_I^h,\Phi_h)+B_h(\Theta_2,\Theta_2,\Phi_h)-B_h(\Theta_1,\Theta_1,\Phi_h)\\ &=B_h(\Theta_2-\Theta_1,\Theta_1-\Psi_I^h,\Phi_h)+B_h(\Theta_2-\Psi_I^h,\Theta_2-\Theta_1,\Phi_h)\\ &\lesssim |\Theta_2-\Theta_1|_{2,h}(|\Theta_1-\Psi_I^h|_{2,h}+|\Theta_2-\Psi_I^h|_{2,h})\end{aligned}$$ with Lemma [Lemma 10](#lem:boundednessBh){reference-type="ref" reference="lem:boundednessBh"}.a in the last step. Since $\Theta_1,\Theta_2 \in B_{R(h_{\max})}(\Psi_I^h)$, the above displayed estimate provides $$|T_h(\Theta_1)-T_h(\Theta_2)|_{2,h} \le Ch_{\max}^\alpha|\Theta_1-\Theta_2|_{2,h}$$ for some positive constant $C$ independent of $h_{\max}$. *Step 4 establishes local uniqueness of a discrete solution.* Let $\Psi$ be a regular solution to [\[VKE_weak\]](#VKE_weak){reference-type="eqref" reference="VKE_weak"}. For a sufficiently small choice of $h_{\max}$, the contraction mapping theorem establishes the local uniquenes of the discrete solution $\Psi_h$ to [\[VKE_Morleyweak\]](#VKE_Morleyweak){reference-type="eqref" reference="VKE_Morleyweak"}. ◻ **Lemma 17** (Bound for $\Psi_h$). *Let $\Psi_h$ be a discrete solution to [\[VKE_Morleyweak\]](#VKE_Morleyweak){reference-type="eqref" reference="VKE_Morleyweak"} that satisfies $|\Psi_h-\Psi_I^h|_{2,h}\le R(h_{\max})$. Then $|\Psi_h|_{2,h} \lesssim \|f\|$.* *Proof.* Triangle inequalities, $|\Psi_h-\Psi_I^h|_{2,h}\le R(h_{\max})$, and Lemma [\[lem.vI\]](#lem.vI){reference-type="ref" reference="lem.vI"} provide $$\begin{aligned} |\Psi_h|_{2,h}&\le |\Psi_h-\Psi_I^h|_{2,h}+|\Psi-\Psi_I^h|_{2,h}+|\Psi|_2\lesssim R(h_{\max})+ h_{\max}^\alpha\|\Psi\|_{2+\alpha}+|\Psi|_2. \end{aligned}$$ Since $R(h)=2Ch_{\max}^\alpha$ with $C$ depends on $\|\Psi\|_{2+\alpha}$ and $\|f\|$ from Step1 of Theorem [Theorem 16](#thm:existence){reference-type="ref" reference="thm:existence"}, a combination of Lemma [\[lem:aprioriboundcts\]](#lem:aprioriboundcts){reference-type="ref" reference="lem:aprioriboundcts"} and the above displayed estimate concludes the proof. ◻ # A priori error control {#sec:error} This section deals with the a priori error control under minimal regularity assumption on the exact solution and is followed by the convergence of the Newtons method. ## Error estimates This section proves an a priori error estimate in $H^2$ and $H^1$ norms. **Theorem 18** (Energy norm estimate). *Let $\Psi$ be a regular solution to [\[VKE_weak\]](#VKE_weak){reference-type="eqref" reference="VKE_weak"}. For a sufficiently small $h_{\max}$, the discrete solution $\Psi_h$ to [\[VKE_Morleyweak\]](#VKE_Morleyweak){reference-type="eqref" reference="VKE_Morleyweak"} satisfies $$\|\Psi-\Psi_h\|_{2,h} \lesssim h_{\max}^\alpha,$$ where $\alpha \in (\frac{1}{2},1]$ is the index of elliptic regularity.* *Proof.* The triangle inequality, Lemma [\[lem.vI\]](#lem.vI){reference-type="ref" reference="lem.vI"}, [\[eqn.seminormbound\]](#eqn.seminormbound){reference-type="eqref" reference="eqn.seminormbound"}, and Theorem [Theorem 16](#thm:existence){reference-type="ref" reference="thm:existence"} for sufficiently small $h_{\max}$ with $R(h_{\max}) \lesssim h_{\max}^\alpha$ show $$\begin{aligned} \|\Psi-\Psi_h\|_{2,h}&\le \|\Psi-\Psi_I^h\|_{2,h}+\|\Psi_I^h-\Psi_h\|_{2,h} \lesssim h_{\max}^\alpha. \end{aligned}$$ ◻ To prove the lower order error estimates, define the augmented local space [@Zhao_Morley] $$\begin{aligned} \label{def.whk} W_h(K):=\displaystyle\bigg\{&\varphi \in H^2(K);\Delta^2\varphi \in {{\mathcal P}}_0(K), \varphi_{|e} \in {{\mathcal P}}_2(e), \Delta \varphi_{|e} \in {{\mathcal P}}_{0}(e), \int_e \Pi^K\varphi{\rm\,ds}=\int_e \varphi{\rm\,ds}\\ & \; \forall e \subseteq \partial K,\int_K \Pi^K\varphi{\rm\,dx}=\int_K \varphi{\rm\,dx}\bigg\}.\end{aligned}$$ The global space $W_h$ can be then assembled in the same way as $V_h$ is defined. As in [@Zhao_Morley Section 6], reformulate the nonconforming VEM [\[VKE_Morleyweak\]](#VKE_Morleyweak){reference-type="eqref" reference="VKE_Morleyweak"} as $$\label{VKE_Morleyweak_modified} A_h(\Psi_h,\Phi_h)+B_h(\Psi_h,\Psi_h,\Phi_h)=(f_h,\varphi_{1,h})=: L_h(\Phi_h) \quad \forall\,\Phi_h\in \boldsymbol{W}_h$$ with the virtual element space $W_h$ instead of $V_h$ and $f_h|_K:=P_0^K f$. The same arguments in Theorem [Theorem 18](#thm:energyerror){reference-type="ref" reference="thm:energyerror"} leads to $$\label{eqn.energyerror.modified} \|\Psi-\Psi_h\|_{2,h} \lesssim h_{\max}^\alpha.$$ Also, for $f \in H^1(\Omega)$ and $\varphi_h \in W_h$, the following approximation property hold [@Zhao_Morley (6.2)]: $$\label{eqn.ffh} (f-f_h,\varphi_{I}^{h})\lesssim h_{\max}^2\|f\|_1\|\varphi_I^h\|_{1,h}.$$ **Theorem 19** ($H^1$ estimate). *Let $\Psi$ be a regular solution to [\[VKE_weak\]](#VKE_weak){reference-type="eqref" reference="VKE_weak"} and $f \in H^1(\Omega)$. For a sufficiently small $h_{\max}$, the discrete solution $\Psi_h$ to [\[VKE_Morleyweak_modified\]](#VKE_Morleyweak_modified){reference-type="eqref" reference="VKE_Morleyweak_modified"} satisfies $$\|\Psi-\Psi_h\|_{1,h} \lesssim h_{\max}^{2\alpha},$$ where $\alpha \in (\frac{1}{2},1]$ is the index of elliptic regularity.* *Proof.* The triangle inequality and Lemma [\[lem.vI\]](#lem.vI){reference-type="ref" reference="lem.vI"} lead to $$\begin{aligned} \label{eqn.H1tri} \|\Psi-\Psi_h\|_{1,h}&\le \|\Psi-\Psi_I^h\|_{1,h}+\|\Psi_I^h-\Psi_h\|_{1,h}\nonumber\\ &\lesssim h_{\max}^{1+\alpha}+\|\rho_h-E_h\rho_h\|_{1,h}+\|E_h\rho_h\|_{1}\end{aligned}$$ with $\rho_h:=\Psi_I^h-\Psi_h \in {\mathbf V}_h$ in the last step. The triangle inequality, Lemma [\[lem.vI\]](#lem.vI){reference-type="ref" reference="lem.vI"}, and [\[eqn.energyerror.modified\]](#eqn.energyerror.modified){reference-type="eqref" reference="eqn.energyerror.modified"} show $$\label{eqn.rhoh} \|\rho_h\|_{2,h}\le \|\Psi_I^h-\Psi\|_{2,h}+\|\Psi-\Psi_h\|_{2,h}\lesssim h_{\max}^\alpha.$$ A combination of [\[eqn.H1tri\]](#eqn.H1tri){reference-type="eqref" reference="eqn.H1tri"}, Lemma [Lemma 6](#lem:Eh){reference-type="ref" reference="lem:Eh"}, and [\[eqn.rhoh\]](#eqn.rhoh){reference-type="eqref" reference="eqn.rhoh"} reads $$\begin{aligned} \label{eqn.H1tri1} \|\Psi-\Psi_h\|_{1,h}&\lesssim h_{\max}^{1+\alpha}+\|E_h\rho_h\|_{1}.\end{aligned}$$ The choice $Q=-\Delta E_h\rho_h$ and $\Phi=E_h\rho_h$ in [\[eqn.dualcts\]](#eqn.dualcts){reference-type="eqref" reference="eqn.dualcts"} and elementary algebra reveal $$\begin{aligned} \label{eqn.H1nabla} \|\nabla E_h\rho_h\|_{}^2 &=\mathcal{A}(E_h\rho_h,\boldsymbol{\xi})=A(E_h\rho_h,\boldsymbol{\xi})+B(\Psi,E_h\rho_h,\boldsymbol{\xi})+B(E_h\rho_h,\Psi,\boldsymbol{\xi})\nonumber\\ &=A_{\rm {pw}}(E_h\rho_h-\rho_h,\boldsymbol{\xi})+A_{\rm {pw}}(\rho_h,\boldsymbol{\xi})+\widehat{B}_h(\Psi,E_h\rho_h-\rho_h,\boldsymbol{\xi})\nonumber\\ &\quad +\widehat{B}_h(E_h\rho_h-\rho_h,\Psi,\boldsymbol{\xi})+\widehat{B}_h(\Psi,\rho_h,\boldsymbol{\xi})+\widehat{B}_h(\rho_h,\Psi,\boldsymbol{\xi})\nonumber\\ &=A_{\rm {pw}}(E_h\rho_h-\rho_h,\boldsymbol{\xi})+\widehat{B}_h(\Psi,E_h\rho_h-\rho_h,\boldsymbol{\xi}) +\widehat{B}_h(E_h\rho_h-\rho_h,\Psi,\boldsymbol{\xi})\nonumber\\ &\quad + A_{\rm {pw}}(\Psi_I^h-\Psi,\boldsymbol{\xi})+A_{\rm {pw}}(\Psi-\Psi_h,\boldsymbol{\xi}-\boldsymbol{\xi}_I^h)+A_{\rm {pw}}(\Psi,\boldsymbol{\xi}_I^h-\boldsymbol{\xi})\nonumber\\ &\quad +A(\Psi,\boldsymbol{\xi})-A_{\rm {pw}}(\Psi_h,\boldsymbol{\xi}_I^h)+\widehat{B}_h(\Psi,\rho_h,\boldsymbol{\xi}) +\widehat{B}_h(\rho_h,\Psi,\boldsymbol{\xi}).\end{aligned}$$ Consider the third last term on the right-hand side of [\[eqn.H1nabla\]](#eqn.H1nabla){reference-type="eqref" reference="eqn.H1nabla"}. Since $\Psi_\pi^h, \boldsymbol{\xi}_\pi^h \in {{\mathcal P}}_2(\mathcal{T}_h)$, the consistency property [\[eqn.consistency\]](#eqn.consistency){reference-type="eqref" reference="eqn.consistency"} reads $A_{\rm {pw}}(\Psi_\pi^h,\boldsymbol{\xi}_I^h-\boldsymbol{\xi}_\pi^h)=A_{h}(\Psi_\pi^h,\boldsymbol{\xi}_I^h-\boldsymbol{\xi}_\pi^h)$ and $A_{\rm {pw}}(\Psi_h,\boldsymbol{\xi}_\pi^h)=A_{h}(\Psi_h,\boldsymbol{\xi}_\pi^h)$. Hence, $$\begin{aligned} A_{\rm {pw}}(\Psi_h,\boldsymbol{\xi}_I^h)&=A_{\rm {pw}}(\Psi_h-\Psi_\pi^h,\boldsymbol{\xi}_I^h-\boldsymbol{\xi}_\pi^h)+A_{\rm {pw}}(\Psi_\pi^h,\boldsymbol{\xi}_I^h-\boldsymbol{\xi}_\pi^h)+A_{\rm {pw}}(\Psi_h,\boldsymbol{\xi}_\pi^h)\\ %&=A_{\pw}(\Psi_h-\Psi_\pi^h,\bxi_I^h-\bxi_\pi^h)+A_{h}(\Psi_\pi^h,\bxi_I^h-\bxi_\pi^h)+A_{h}(\Psi_h,\bxi_\pi^h)\\ &=A_{\rm {pw}}(\Psi_h-\Psi_\pi^h,\boldsymbol{\xi}_I^h-\boldsymbol{\xi}_\pi^h)+A_{h}(\Psi_\pi^h-\Psi_h,\boldsymbol{\xi}_I^h-\boldsymbol{\xi}_\pi^h)+A_{h}(\Psi_h,\boldsymbol{\xi}_I^h)\\ &=A_{\rm {pw}}(\Psi_h-\Psi_\pi^h,\boldsymbol{\xi}_I^h-\boldsymbol{\xi}_\pi^h)+A_{h}(\Psi_\pi^h-\Psi_h,\boldsymbol{\xi}_I^h-\boldsymbol{\xi}_\pi^h)+L_{h}(\boldsymbol{\xi}_I^h)-B_h(\Psi_h,\Psi_h,\boldsymbol{\xi}_I^h)\end{aligned}$$ with [\[VKE_Morleyweak_modified\]](#VKE_Morleyweak_modified){reference-type="eqref" reference="VKE_Morleyweak_modified"} for $\Phi_h=\boldsymbol{\xi}_I^h$ in the last step. This, [\[eqn.H1nabla\]](#eqn.H1nabla){reference-type="eqref" reference="eqn.H1nabla"}, and [\[VKE_weak\]](#VKE_weak){reference-type="eqref" reference="VKE_weak"} for $\Phi=\boldsymbol{\xi}$ result in $$\begin{aligned} \label{eqn.H1nabla.1} \|\nabla E_h\rho_h\|_{}^2 &=A_{\rm {pw}}(E_h\rho_h-\rho_h,\boldsymbol{\xi})+\widehat{B}_h(\Psi,E_h\rho_h-\rho_h,\boldsymbol{\xi}) +\widehat{B}_h(E_h\rho_h-\rho_h,\Psi,\boldsymbol{\xi})\nonumber\\ &\quad + A_{\rm {pw}}(\Psi_I^h-\Psi,\boldsymbol{\xi})+A_{\rm {pw}}(\Psi-\Psi_h,\boldsymbol{\xi}-\boldsymbol{\xi}_I^h)+A_{\rm {pw}}(\Psi,\boldsymbol{\xi}_I^h-\boldsymbol{\xi}) +(F(\boldsymbol{\xi})-L_{h}(\boldsymbol{\xi}_I^h))\nonumber\\ &\quad-A_{\rm {pw}}(\Psi_h-\Psi_\pi^h,\boldsymbol{\xi}_I^h-\boldsymbol{\xi}_\pi^h)-A_{h}(\Psi_\pi^h-\Psi_h,\boldsymbol{\xi}_I^h-\boldsymbol{\xi}_\pi^h) +(B_h(\Psi_h,\Psi_h,\boldsymbol{\xi}_I^h)\nonumber\\ &\quad-B(\Psi,\Psi,\boldsymbol{\xi})+\widehat{B}_h(\Psi,\Psi_I^h-\Psi_h,\boldsymbol{\xi}) +\widehat{B}_h(\Psi_I^h-\Psi_h,\Psi,\boldsymbol{\xi})):=\sum_{i=1}^{10}T_i.\end{aligned}$$ Lemma [\[lem:Apw\]](#lem:Apw){reference-type="ref" reference="lem:Apw"}.a and [\[eqn.rhoh\]](#eqn.rhoh){reference-type="eqref" reference="eqn.rhoh"} provide $T_1 \lesssim h_{\max}^\alpha |\rho_h|_{2,h}\|\boldsymbol{\xi}\|_{2+\alpha}\lesssim h_{\max}^{2\alpha}\|\boldsymbol{\xi}\|_{2+\alpha}$. Lemma [Lemma 10](#lem:boundednessBh){reference-type="ref" reference="lem:boundednessBh"}.e and [\[eqn.Eh\]](#eqn.Eh){reference-type="eqref" reference="eqn.Eh"} imply $$\begin{aligned} T_2 &\lesssim \|\Psi\|_{2+\alpha}|E_h\rho_h-\Pi^h\rho_h|_{1,h}|\boldsymbol{\xi}|_2 %\lesssim (|E_h \rho_h-\rho_h|_{1,h}+|\rho_h-\Pi^h\rho_h|_{1,h})|\bxi|_2\\ %& \lesssim h_{\max}|\rho_h|_{2,h}|\boldsymbol{\xi}|_2 \lesssim h_{\max}^{1+\alpha}|\boldsymbol{\xi}|_2\end{aligned}$$ with [\[eqn.rhoh\]](#eqn.rhoh){reference-type="eqref" reference="eqn.rhoh"} in the last step. Since $\Psi,\boldsymbol{\xi}\in \text{\bf V}\cap \boldsymbol{H}^{2+\alpha}(\Omega)$, the Sobolev embedding $H^{2+\alpha}(\Omega) \hookrightarrow W^{2,4}(\Omega)$ shows $$|\nabla \Psi \cdot \nabla \boldsymbol{\xi}|_{1} \le |\nabla \Psi|_{1,4}|\nabla \boldsymbol{\xi}|_{1,4}\lesssim \|\Psi\|_{2+\alpha}\|\boldsymbol{\xi}\|_{2+\alpha}.$$ Hence, $\nabla \Psi \cdot \nabla \boldsymbol{\xi}\in H^1(\Omega)$ and $T_3 \lesssim |D^2(E_h\rho_h-\Pi^h\rho_h)|_{-1,h}|\nabla \Psi \cdot \nabla \boldsymbol{\xi}|_{1,h}$. The definition of the dual norm and integration by parts lead to $|D^2(E_h\rho_h-\Pi^h\rho_h)|_{-1,h}\lesssim |E_h\rho_h-\Pi^h\rho_h|_{1,h}$. The combination of the above estimates, [\[eqn.Eh\]](#eqn.Eh){reference-type="eqref" reference="eqn.Eh"}, and [\[eqn.rhoh\]](#eqn.rhoh){reference-type="eqref" reference="eqn.rhoh"} implies $T_3 \lesssim h_{\max}^{1+\alpha}\|\boldsymbol{\xi}\|_{2+\alpha}$. Lemma [\[lem:Apw\]](#lem:Apw){reference-type="ref" reference="lem:Apw"}.b reads $T_4+T_6 \lesssim h_{\rm max}^{2\alpha}\|\boldsymbol{\xi}\|_{2+\alpha}$. The boundedness of $A_{\rm {pw}}(\bullet,\bullet)$, [\[eqn.energyerror.modified\]](#eqn.energyerror.modified){reference-type="eqref" reference="eqn.energyerror.modified"}, and Lemma [\[lem.vI\]](#lem.vI){reference-type="ref" reference="lem.vI"} provide $T_5 \lesssim h_{\rm max}^{2\alpha}\|\boldsymbol{\xi}\|_{2+\alpha}$. The Cauchy inequality, Lemma [\[lem.vI\]](#lem.vI){reference-type="ref" reference="lem.vI"}, and [\[eqn.ffh\]](#eqn.ffh){reference-type="eqref" reference="eqn.ffh"} show $$\begin{aligned} T_7&=(f,\boldsymbol{\xi}_1-\boldsymbol{\xi}_{1,I}^{h})+(f-f_h,\boldsymbol{\xi}_{1,I}^{h})\lesssim h_{\max}^{2+\alpha}\|\boldsymbol{\xi}\|_{2+\alpha}+h_{\max}^2\|\boldsymbol{\xi}_I^h\|_{1,h} \lesssim h_{\max}^{2}\|\boldsymbol{\xi}\|_{2+\alpha}.\end{aligned}$$ Triangle inequality with $\Psi$, [\[eqn.energyerror.modified\]](#eqn.energyerror.modified){reference-type="eqref" reference="eqn.energyerror.modified"}, and Lemma [\[lem.vpi\]](#lem.vpi){reference-type="ref" reference="lem.vpi"} provide $|\Psi_h-\Psi_\pi^h|_{2,h} \lesssim h_{\max}^{\alpha}$. This, the boundedness of $A_{\rm {pw}}(\bullet,\bullet)$, Lemma [Lemma 5](#lem:Ahproperties){reference-type="ref" reference="lem:Ahproperties"}.a, and [\[eqn.vIvpi\]](#eqn.vIvpi){reference-type="eqref" reference="eqn.vIvpi"} imply $T_8+T_9 \lesssim h_{\max}^{2\alpha}\|\boldsymbol{\xi}\|_{2+\alpha}$. Elementary algebra results in $$\begin{aligned} T_{10}%&=B_h(\Psi_h,\Psi_h,\bxi_I^h)-B(\Psi,\Psi,\bxi)+\hB_h(\Psi,\Psi_I^h-\Psi_h,\bxi) +\hB_h(\Psi_I^h-\Psi_h,\Psi,\bxi)\nonumber\\ &=B_h(\Psi_h,\Psi_h,\boldsymbol{\xi}_I^h)-B(\Psi,\Psi,\boldsymbol{\xi})+\widehat{B}_h(\Psi,\Psi_I^h-\Psi,\boldsymbol{\xi}) +\widehat{B}_h(\Psi_I^h-\Psi,\Psi,\boldsymbol{\xi})\nonumber\\ &\qquad +\widehat{B}_h(\Psi,\Psi-\Psi_h,\boldsymbol{\xi}) +\widehat{B}_h(\Psi-\Psi_h,\Psi,\boldsymbol{\xi})\\ &=\widehat{B}_h(\Psi,\Psi_I^h-\Psi,\boldsymbol{\xi}) +\widehat{B}_h(\Psi_I^h-\Psi,\Psi,\boldsymbol{\xi}) +\widehat{B}_h(\Psi-\Psi_h,\Psi-\Psi_h,\boldsymbol{\xi})\nonumber\\ &\quad +\widehat{B}_h(\Psi_h-\Psi,\Psi_h,\boldsymbol{\xi}_I^h-\boldsymbol{\xi})+\widehat{B}_h(\Psi,\Psi_h,\boldsymbol{\xi}_I^h-\boldsymbol{\xi}).\label{t10}\end{aligned}$$ Lemma [Lemma 10](#lem:boundednessBh){reference-type="ref" reference="lem:boundednessBh"}.e and [\[eqn.xiPixiIh1h\]](#eqn.xiPixiIh1h){reference-type="eqref" reference="eqn.xiPixiIh1h"} reveal $\widehat{B}_h(\Psi,\Psi_I^h-\Psi,\boldsymbol{\xi}) \lesssim h_{\max}^{1+\alpha}\|\boldsymbol{\xi}\|_{2}$. Arguments analogous to $T_3$ together with [\[eqn.xiPixiIh1h\]](#eqn.xiPixiIh1h){reference-type="eqref" reference="eqn.xiPixiIh1h"} lead to $\widehat{B}_h(\Psi_I^h-\Psi,\Psi,\boldsymbol{\xi})\lesssim h_{\max}^{1+\alpha}\|\boldsymbol{\xi}\|_{2+\alpha}$. The relation $\Psi_\pi^h=\Pi^h\Psi_\pi^h$, Lemma [\[lem.vpi\]](#lem.vpi){reference-type="ref" reference="lem.vpi"}, and Lemma [\[lem:discreteSobolev\]](#lem:discreteSobolev){reference-type="ref" reference="lem:discreteSobolev"} imply $$\label{eqn.psipihpsih2h} |\Psi-\Pi^h\Psi_h |_{1,h} \le |\Psi-\Psi_\pi^h |_{1,h}+|\Pi^h(\Psi_\pi^h-\Psi_h) |_{1,h}\lesssim h_{\max}^{1+\alpha}+|\Pi^h(\Psi_\pi^h-\Psi_h) |_{2,h}\lesssim h_{\max}^\alpha$$ with [\[eqn.Pihbound\]](#eqn.Pihbound){reference-type="eqref" reference="eqn.Pihbound"}, Lemma [\[lem.vpi\]](#lem.vpi){reference-type="ref" reference="lem.vpi"}, and [\[eqn.energyerror.modified\]](#eqn.energyerror.modified){reference-type="eqref" reference="eqn.energyerror.modified"} in the last step. The same arguments provides $|\Psi-\Pi^h\Psi_h |_{2,h} \lesssim h_{\max}^\alpha$. This, [\[eqn.psipihpsih2h\]](#eqn.psipihpsih2h){reference-type="eqref" reference="eqn.psipihpsih2h"}, the symmetry of the $\widehat{B}_h(\bullet,\bullet,\bullet)$ with respect to the second and third arguments, and Lemma [Lemma 10](#lem:boundednessBh){reference-type="ref" reference="lem:boundednessBh"}.f show $\widehat{B}_h(\Psi-\Psi_h,\Psi-\Psi_h,\boldsymbol{\xi}) \lesssim h_{\max}^{2\alpha}\|\boldsymbol{\xi}\|_{2+\alpha}$. Lemma [Lemma 10](#lem:boundednessBh){reference-type="ref" reference="lem:boundednessBh"}.b, the estimate $|\Psi-\Pi^h\Psi_h |_{2,h} \lesssim h_{\max}^\alpha$ from the previous term, Lemma [\[lem:discreteSobolev\]](#lem:discreteSobolev){reference-type="ref" reference="lem:discreteSobolev"},[\[eqn.Pihbound\]](#eqn.Pihbound){reference-type="eqref" reference="eqn.Pihbound"}, Lemma [Lemma 17](#lem:Psih){reference-type="ref" reference="lem:Psih"}, and the same arguments in [\[eqn.psipihppsiIh14h\]](#eqn.psipihppsiIh14h){reference-type="eqref" reference="eqn.psipihppsiIh14h"} for $\boldsymbol{\xi}$ lead to $\widehat{B}_h(\Psi_h-\Psi,\Psi_h,\boldsymbol{\xi}_I^h-\boldsymbol{\xi})\lesssim h_{\max}^{2\alpha+1/2}\|\boldsymbol{\xi}\|_{2+\alpha}.$ Lemma [Lemma 10](#lem:boundednessBh){reference-type="ref" reference="lem:boundednessBh"}.d, [Lemma 17](#lem:Psih){reference-type="ref" reference="lem:Psih"}, and [\[eqn.xiPixiIh1h\]](#eqn.xiPixiIh1h){reference-type="eqref" reference="eqn.xiPixiIh1h"} read $\widehat{B}_h(\Psi,\Psi_h,\boldsymbol{\xi}_I^h-\boldsymbol{\xi}) \lesssim h_{\max}^{1+\alpha}\|\boldsymbol{\xi}\|_{2+\alpha}$. A combination of these estimates in [\[t10\]](#t10){reference-type="eqref" reference="t10"} leads to $T_{10} \lesssim h_{\max}^{2\alpha}\|\boldsymbol{\xi}\|_{2+\alpha}.$ The substitution of $T_1,\cdots,T_{10}$ in [\[eqn.H1nabla.1\]](#eqn.H1nabla.1){reference-type="eqref" reference="eqn.H1nabla.1"} provides $$\|\nabla E_h\rho_h\|_{}^2\lesssim h_{\max}^{2\alpha}\|\boldsymbol{\xi}\|_{2+\alpha}.$$ Since $\|\boldsymbol{\xi}\|_{2+\alpha} \lesssim \|\Delta E_h\rho_h\|_{-1} \lesssim \|\nabla E_h\rho_h\|$ from [\[eqn.boundduallinearised\]](#eqn.boundduallinearised){reference-type="eqref" reference="eqn.boundduallinearised"} and integration by parts, $\|\nabla E_h\rho_h\|_{}\lesssim h_{\max}^{2\alpha}$. This with [\[eqn.H1tri1\]](#eqn.H1tri1){reference-type="eqref" reference="eqn.H1tri1"} concludes the proof. ◻ **Theorem 20** ($L^2$ estimate). *Let $\Psi$ be a regular solution to [\[VKE_weak\]](#VKE_weak){reference-type="eqref" reference="VKE_weak"} and $f \in H^1(\Omega)$. For sufficiently small $h_{\max}$, the discrete solution $\Psi_h$ to [\[VKE_Morleyweak_modified\]](#VKE_Morleyweak_modified){reference-type="eqref" reference="VKE_Morleyweak_modified"} satisfies $$\|\Psi-\Psi_h\| \lesssim h_{\max}^{2\alpha},$$ where $\alpha \in (\frac{1}{2},1]$ is the index of elliptic regularity.* *Proof.* The triangle inequalities lead to $$\begin{aligned} \|\Psi-\Psi_h\|&\le \|\Psi-\Psi_I^h\|+\|\Psi_I^h-\Psi_h\|\le \|\Psi-\Psi_I^h\|+\|\rho_h-E_h\rho_h\|+\|E_h\rho_h\|\end{aligned}$$ with $\rho_h= \Psi_I^h-\Psi_h$ in the last step. Lemma [\[lem.vI\]](#lem.vI){reference-type="ref" reference="lem.vI"} shows $\|\Psi-\Psi_I^h\|\lesssim h_{\max}^{2+\alpha}$. Lemma [Lemma 6](#lem:Eh){reference-type="ref" reference="lem:Eh"} and [\[eqn.rhoh\]](#eqn.rhoh){reference-type="eqref" reference="eqn.rhoh"} provide $\|\rho_h-E_h\rho_h\| \lesssim h_{\max}^2|\rho_h|_{2,h} \lesssim h_{\max}^{2+\alpha}$. Since $E_h\rho_h \in H^1_0(\Omega)$, $\|E_h\rho_h\| \lesssim \|\nabla E_h\rho_h\|$. This and Theorem [Theorem 19](#thm:lowererror){reference-type="ref" reference="thm:lowererror"} reveal $\|E_h\rho_h\|\lesssim h_{\max}^{2\alpha}$. A combination of these estimates concludes the proof. ◻ **Remark 21** ($L^2$ error estimate). *It is well-known [@HuShi; @ng2] that for the Morley nonconforming finite element method for the biharmonic problem and von Kármán equations, the $L^2$ error estimate cannot be improved compared to that of $H^1$ error estimate. Since Morley finite element method is a special case of simplified fully nonconforming VEM (see Remark [Remark 2](#rem.morley){reference-type="ref" reference="rem.morley"}), it is expected that using a lower order VEM, the order of convergence in $L^2$ norm cannot be improved than that of the $H^1$ norm. Also, the same result for this VEM for the biharmonic problem is presented in [@Zhao_Morley Theorem 6.2].* ## Convergence of the Newtons method {#sec:Newton} The discrete solution $\Psi_{h}$ of [\[VKE_Morleyweak\]](#VKE_Morleyweak){reference-type="eqref" reference="VKE_Morleyweak"} is characterized as the fixed point of [\[eqn.Th\]](#eqn.Th){reference-type="eqref" reference="eqn.Th"} and so depends on the unknown $\Psi_I^h$. The approximate solution to [\[VKE_Morleyweak\]](#VKE_Morleyweak){reference-type="eqref" reference="VKE_Morleyweak"} is computed with the Newton method, where the iterates $\Psi_{h}^{j}$ solve $$\label{NewtonIterate} A_{h}(\Psi_{h}^{j},\Phi_{h})+B_{h}(\Psi_{h}^{j-1},\Psi_{h}^{j},\Phi_{h})+B_{h}(\Psi_{h}^{j},\Psi_{h}^{j-1},\Phi_{h})=B_{h}(\Psi_{h}^{j-1},\Psi_{h}^{j-1},\Phi_{h})+F_{h}(\Phi_{h})$$ for all $\Phi_{h}\in {\mathbf V}_h$. The Newton method has locally quadratic convergence. **Theorem 22** (Convergence of Newton method). *Let $\Psi$ be a regular solution to [\[VKE_weak\]](#VKE_weak){reference-type="eqref" reference="VKE_weak"} and let $\Psi_{h}$ solve [\[VKE_Morleyweak\]](#VKE_Morleyweak){reference-type="eqref" reference="VKE_Morleyweak"}. There exists a positive constant $R$ independent of $h$, such that for any initial guess $\Psi_{h}^0$ with $\displaystyle | \Psi_{h}- \Psi_{h}^0|_{2,h}\leq R$, it follows $|\Psi_{h}- \Psi_{h}^{j}|_{2,h} \leq R\quad \text{for all}\:j=0,1,2,\ldots$ and the iterates of the Newton method in [\[NewtonIterate\]](#NewtonIterate){reference-type="eqref" reference="NewtonIterate"} are well defined and converges quadratically to $\Psi_{h}$.* *Proof.* The proof follows the lines of [@ng2 Theorem 6.2]. However, for the sake of completeness, we provide a detailed proof. Lemma [Lemma 15](#lem:perturbed){reference-type="ref" reference="lem:perturbed"} shows that there exists a positive constant $\epsilon$ (sufficiently small) independent of $h$ such that for each $\boldsymbol{z}_h \in {\mathbf V}_h$ with $| \boldsymbol{z}_h-\Psi_I^h|_{2,h}\leq \epsilon$, the bilinear form $$\label{NewtonNonsingular} A_{h}(\bullet,\bullet)+B_{h}(\boldsymbol{z}_h,\bullet,\bullet)+B_{h}(\bullet,\boldsymbol{z}_h,\bullet)$$ satisfies discrete inf-sup condition on ${\mathbf V}_h \times {\mathbf V}_h$. For sufficiently small $h_{\max}$, Theorem [Theorem 16](#thm:existence){reference-type="ref" reference="thm:existence"} with $R(h_{\max})\lesssim h_{\max}^\alpha$ implies $|\Psi_I^h-\Psi_h|_{2,h} \lesssim h_{\max}^\alpha$. Thus $h_{\max}$ can be chosen sufficiently small so that $|\Psi_I^h-\Psi_{h}|_{2,h}\leq \epsilon/2$. Recall ${\widehat{\beta}}$ from [\[defn.beta\]](#defn.beta){reference-type="eqref" reference="defn.beta"}. Let ${C_{b}}$ be the hidden positive constant in Lemma [Lemma 10](#lem:boundednessBh){reference-type="ref" reference="lem:boundednessBh"}.a. Set $$R:=\min\left\{\epsilon/2,{\widehat{\beta}}/8{C_{b}}\right\}.$$ Assume that the initial guess $\Psi_{h}^0$ satisfies $|\Psi_{h}-\Psi_{h}^0|_{2,h}\leq R$. Then, $$|\Psi_I^h-\Psi_{h}^0|_{2,h}\leq |\Psi_I^h-\Psi_{h}|_{2,h}+|\Psi_{h}-\Psi_{h}^0|_{2,h}\leq \epsilon.$$ This implies $|\Psi_{h}-\Psi_{h}^{j-1}|_{2,h}\leq R$ and $|\Psi_I^h-\Psi_{h}^{j-1}|_{2,h} \leq \epsilon$ for $j=1$ and suppose for mathematical induction that this holds for some $j\in\mathbb N$. Then $\boldsymbol{z}_{h}:=\Psi_{h}^{j-1}$ in [\[NewtonNonsingular\]](#NewtonNonsingular){reference-type="eqref" reference="NewtonNonsingular"} leads to an discrete inf-sup condition of $A_{h}(\bullet,\bullet)+B_{h}(\Psi^{j-1}_{h},\bullet,\bullet)+B_{h}(\bullet,\Psi^{j-1}_{h},\bullet)$ and so to an unique solution $\Psi_{h}^j$ to [\[NewtonIterate\]](#NewtonIterate){reference-type="eqref" reference="NewtonIterate"} in step $j$ of the Newton scheme. The discrete inf-sup condition [\[NewtonNonsingular\]](#NewtonNonsingular){reference-type="eqref" reference="NewtonNonsingular"} implies the existence of $\Phi_{h}\in{\mathbf V}_h$ with $| \Phi_{h}|_{2,h}=1$ and $$\frac{{\widehat{\beta}}}{4}|\Psi_h-\Psi_h^{j}|_{2,h}\leq A_{h}(\Psi_h-\Psi_h^{j}, \Phi_{h})+B_{h}(\Psi_h^{j-1},\Psi_h-\Psi_{h}^{j}, \Phi_{h})+B_{h}(\Psi_h-\Psi_{h}^{j},\Psi_h^{j-1}, \Phi_{h}).$$ The application of [\[NewtonIterate\]](#NewtonIterate){reference-type="eqref" reference="NewtonIterate"}, [\[VKE_Morleyweak\]](#VKE_Morleyweak){reference-type="eqref" reference="VKE_Morleyweak"}, and Lemma [Lemma 10](#lem:boundednessBh){reference-type="ref" reference="lem:boundednessBh"}.a result in $$\begin{aligned} &A_{h}(\Psi_h-\Psi_h^{j}, \Phi_{h})+B_{h}(\Psi_h^{j-1},\Psi_h-\Psi_{h}^{j}, \Phi_{h})+B_{h}(\Psi_h-\Psi_{h}^{j},\Psi_h^{j-1}, \Phi_{h})\notag\\ &=A_{h}(\Psi_h, \Phi_{h})+B_{h}(\Psi_h^{j-1},\Psi_h, \Phi_{h})+B_{h}(\Psi_h, \Psi_h^{j-1}, \Phi_{h})-B_{h}(\Psi_h^{j-1},\Psi_h^{j-1}, \Phi_{h})-F_{h}( \Phi_{h})\notag\\ &=-B_{h}(\Psi_h,\Psi_h,\Phi_{{\rm dG}})+B_{h}(\Psi_h^{j-1},\Psi_h, \Phi_{h})+B_{h}(\Psi_h, \Psi_h^{j-1}, \Phi_{h})-B_{h}(\Psi_h^{j-1},\Psi_h^{j-1}, \Phi_{h})\notag\\ &=B_{h}(\Psi_h-\Psi_h^{j-1},\Psi_h^{j-1}-\Psi_h, \Phi_{h})\leq {C_{b}}| \Psi_h-\Psi_h^{j-1}|_{2,h}^2.\notag \end{aligned}$$ This implies $$\label{induction0} |\Psi_h-\Psi_h^{j}|_{2,h}\leq \left(4{C_{b}}/{\widehat{\beta}}\right)|\Psi_h-\Psi_h^{j-1}|_{2,h}^2$$ and establishes the quadratic convergence of the Newton method to $\Psi_h$. The definition of $R$, $|\Psi_{h}-\Psi_{h}^{j-1}|_{2,h}\leq R$, and [\[induction0\]](#induction0){reference-type="eqref" reference="induction0"} guarantee $|\Psi_h-\Psi_h^{j}|_{2,h}\leq \frac{1}{2}|\Psi_h-\Psi_h^{j-1}|_{2,h}<R$ to allow an induction step $j\to j+1$ to conclude the proof. ◻ Hence, the discrete non-linear problem can be solved using the Newton's method by choosing an appropriate initial guess such that there exists a closed sphere in which the approximate solution is unique and the Newton's iterates converge quadratically to the discrete solution. # Numerical Results {#sec.numericalresults} This section presents a few examples on general polygonal meshes to illustrate the theoretical estimates in the previous section. The solution $\Psi_h=(u_h,v_h)$ to [\[VKE_Morleyweak\]](#VKE_Morleyweak){reference-type="eqref" reference="VKE_Morleyweak"} is computed using Newtons method where the initial value for $\Psi_h$ in the iterative scheme is the discrete solution to the corresponding biharmonic problem without the trilinear term. The convergence of Newtons' method for the VEM scheme is proved in Theorem [Theorem 22](#NewtonThm){reference-type="ref" reference="NewtonThm"}. The implementation associated with the trilinear term was done following the ideas in [@ng1 Section 5.1] taking into account of the VEM approximation [@Veiga_hitchhikersVEM]. The numerical results are presented for square domain and L-shaped domain in Subsections [6.1](#sec.square){reference-type="ref" reference="sec.square"} and [6.2](#sec.Lshaped){reference-type="ref" reference="sec.Lshaped"}. Let the errors in $L^2(\Omega)$, $H^1(\Omega)$, and $H^2(\Omega)$ norms be denoted by $$\begin{aligned} {\rm err}(u):=\|u-\Pi^hu_h\|_{0,h},\, {\rm err}(\nabla u):=|u-\Pi^hu_h|_{1,h},\, \mbox{ and } {\rm err}(Hu)=|u-\Pi^hu_h|_{2,h},\end{aligned}$$ where $\Pi^h$ is the elliptic projection operator in [\[eqn.Pih\]](#eqn.Pih){reference-type="eqref" reference="eqn.Pih"}. The model problem is constructed in such a way that the exact solution is known. {#fig.Lshaped width="8cm"} {#fig.Lshaped width="8cm"} {#fig.Lshaped width="8cm"} {#fig.Lshaped width="8cm"} {#fig.Lshaped width="8cm"} {#fig.Lshaped width="8cm"} ## Example on the square domain {#sec.square} Let the computational domain be $\Omega=(0,1)^2$ and the exact solution be given by $u=x^2y^2(1-x)^2(1-y)^2$ and $v=\sin^2(\pi x)\sin^2(\pi y)$. Then the right hand side load functions are computed as $f=\Delta^2u-[u,v]$ and $g=\Delta^2v+\frac{1}{2}[u,u]$. A series of triangular, square, concave, structured Voronoi, and random Voronoi meshes (see Figure [\[fig.Triangle\]](#fig.Triangle){reference-type="ref" reference="fig.Triangle"}-[\[fig.RV\]](#fig.RV){reference-type="ref" reference="fig.RV"}) are employed to test the convergence results for the VEM. We observe in this example that the Newtons' method converges in three iterations with a tolerance level of $10^{-8}$. Table [2](#table.eg2.Triangle){reference-type="ref" reference="table.eg2.Triangle"}-[10](#table.eg2.RV){reference-type="ref" reference="table.eg2.RV"} show errors and orders of convergence for the displacement $u$ and the Airy-stress function $v$ for the aforementioned five types of meshes. Observe that linear order of convergences are obtained for $u$ and $v$ in the energy norm, and quadratic order of convergence in $L^2$ and $H^1$ norms, see Tables [2](#table.eg2.Triangle){reference-type="ref" reference="table.eg2.Triangle"}-[8](#table.eg2.SV){reference-type="ref" reference="table.eg2.SV"}. These numerical order of convergence clearly matches the expected order of convergence given in [\[eqn.energyerror.modified\]](#eqn.energyerror.modified){reference-type="eqref" reference="eqn.energyerror.modified"}, Theorems [Theorem 19](#thm:lowererror){reference-type="ref" reference="thm:lowererror"}, and [Theorem 20](#thm:lowererrorl2){reference-type="ref" reference="thm:lowererrorl2"}. $h$ ${\rm err}(u)$ Order ${\rm err}(\nabla u)$ Order ${\rm err}(Hu)$ Order ---------- ---------------- -------- ----------------------- -------- ----------------- -------- 0.500000 0.003848 \- 0.010163 \- 0.087728 \- 0.250000 0.000919 2.0659 0.002565 1.9864 0.040578 1.1123 0.125000 0.000248 1.8892 0.000730 1.8124 0.020991 0.9509 0.062500 0.000064 1.9633 0.000191 1.9382 0.010621 0.9829 0.031250 0.000016 1.9900 0.000048 1.9831 0.005328 0.9954 : Convergence results, Square domain, Triangular mesh $h$ ${\rm err}(v)$ Order ${\rm err}(\nabla v)$ Order ${\rm err}(Hv)$ Order ---------- ---------------- -------- ----------------------- -------- ----------------- -------- 0.500000 0.767727 \- 2.122015 \- 19.371564 \- 0.250000 0.177680 2.1113 0.567581 1.9025 9.503684 1.0274 0.125000 0.048263 1.8803 0.161082 1.8170 5.054876 0.9108 0.062500 0.012392 1.9615 0.041987 1.9398 2.575889 0.9726 0.031250 0.003121 1.9895 0.010620 1.9832 1.294492 0.9927 : Convergence results, Square domain, Triangular mesh [\[table.eg2.Triangle\]]{#table.eg2.Triangle label="table.eg2.Triangle"} $h$ ${\rm err}(u)$ Order ${\rm err}(\nabla u)$ Order ${\rm err}(Hu)$ Order ---------- ---------------- -------- ----------------------- -------- ----------------- -------- 0.176777 0.002259 \- 0.004788 \- 0.037658 \- 0.088388 0.000728 1.6342 0.001567 1.6117 0.014181 1.4089 0.044194 0.000198 1.8802 0.000439 1.8353 0.005473 1.3736 0.022097 0.000051 1.9632 0.000115 1.9388 0.002382 1.1200 0.011049 0.000013 1.9899 0.000029 1.9818 0.001134 1.0706 : Convergence results, Square domain, Square mesh $h$ ${\rm err}(v)$ Order ${\rm err}(\nabla v)$ Order ${\rm err}(Hv)$ Order ---------- ---------------- -------- ----------------------- -------- ----------------- -------- 0.176777 0.340448 \- 0.647508 \- 7.335924 \- 0.088388 0.109373 1.6382 0.194961 1.7317 2.871195 1.3533 0.044194 0.029214 1.9045 0.050934 1.9365 1.130682 1.3445 0.022097 0.007421 1.9769 0.012843 1.9876 0.507089 1.1569 0.011049 0.001862 1.9944 0.003216 1.9975 0.245155 1.0485 : Convergence results, Square domain, Square mesh [\[table.eg2.Square\]]{#table.eg2.Square label="table.eg2.Square"} $h$ ${\rm err}(u)$ Order ${\rm err}(\nabla u)$ Order ${\rm err}(Hu)$ Order ---------- ---------------- -------- ----------------------- -------- ----------------- -------- 0.242956 0.006516 \- 0.011816 \- 0.043874 \- 0.121478 0.001881 1.7929 0.003695 1.6772 0.020415 1.1037 0.060739 0.000520 1.8534 0.001120 1.7215 0.008999 1.1818 0.030370 0.000138 1.9118 0.000319 1.8141 0.003882 1.2130 0.015185 0.000036 1.9570 0.000085 1.9089 0.001727 1.1686 : Convergence results, Square domain, Concave mesh $h$ ${\rm err}(v)$ Order ${\rm err}(\nabla v)$ Order ${\rm err}(Hv)$ Order ---------- ---------------- -------- ----------------------- -------- ----------------- -------- 0.242956 0.985669 \- 2.021104 \- 7.247490 \- 0.121478 0.313766 1.6514 0.644574 1.6487 3.459449 1.0669 0.060739 0.086675 1.8560 0.179839 1.8416 1.548383 1.1598 0.030370 0.022635 1.9370 0.047895 1.9088 0.721287 1.1021 0.015185 0.005770 1.9720 0.012390 1.9507 0.346804 1.0565 : Convergence results, Square domain, Concave mesh [\[table.eg2.concave\]]{#table.eg2.concave label="table.eg2.concave"} $h$ ${\rm err}(u)$ Order ${\rm err}(\nabla u)$ Order ${\rm err}(Hu)$ Order ---------- ---------------- -------- ----------------------- -------- ----------------- -------- 0.340697 0.005826 \- 0.011623 \- 0.071067 \- 0.171923 0.002225 1.4075 0.004481 1.3935 0.030928 1.2164 0.083555 0.000650 1.7061 0.001476 1.5390 0.013413 1.1578 0.047445 0.000213 1.9673 0.000537 1.7855 0.006463 1.2900 0.027786 0.000074 1.9693 0.000195 1.8969 0.003328 1.2404 : Convergence results, Square domain, Structured Voronoi Mesh $h$ ${\rm err}(v)$ Order ${\rm err}(\nabla v)$ Order ${\rm err}(Hv)$ Order ---------- ---------------- -------- ----------------------- -------- ----------------- -------- 0.340697 0.906489 \- 2.598308 \- 10.88671 \- 0.171923 0.370324 1.3089 0.935058 1.4943 4.857832 1.1798 0.083555 0.110610 1.6747 0.267744 1.7332 2.300755 1.0358 0.047445 0.035905 1.9881 0.091163 1.9037 1.187114 1.1692 0.027786 0.012727 1.9384 0.033555 1.8680 0.654228 1.1136 : Convergence results, Square domain, Structured Voronoi Mesh [\[table.eg2.SV\]]{#table.eg2.SV label="table.eg2.SV"} $h$ ${\rm err}(u)$ Order ${\rm err}(\nabla u)$ Order ${\rm err}(Hu)$ Order ---------- ---------------- -------- ----------------------- -------- ----------------- -------- 0.373676 0.006044 \- 0.012864 \- 0.072629 \- 0.174941 0.002051 1.4240 0.004305 1.4422 0.028624 1.2269 0.089478 0.000502 2.1000 0.001204 1.9008 0.011996 1.2972 0.041643 0.000109 1.9973 0.000285 1.8855 0.004432 1.3017 0.020068 0.000035 1.5357 0.000095 1.4965 0.002194 0.9631 : Convergence results, Square domain, Random Voronoi Mesh $h$ ${\rm err}(v)$ Order ${\rm err}(\nabla v)$ Order ${\rm err}(Hv)$ Order ---------- ---------------- -------- ----------------------- -------- ----------------- -------- 0.373676 0.871569 \- 2.513214 -- 10.518042 \- 0.174941 0.350708 1.1995 0.912492 1.3349 4.654239 1.0743 0.089478 0.086986 2.0795 0.223585 2.0977 2.118548 1.1739 0.041643 0.018494 2.0243 0.049397 1.9741 0.856600 1.1839 0.020068 0.006145 1.5093 0.016285 1.5201 0.447808 0.8885 : Convergence results, Square domain, Random Voronoi Mesh [\[table.eg2.RV\]]{#table.eg2.RV label="table.eg2.RV"} ## Example on the L-shaped domain {#sec.Lshaped} Consider the L-shaped domain $\Omega=(-1,1)^2 \setminus\big{(}[0,1)\times(-1,0]\big{)}$. Choose the right hand functions such that the exact singular solution [@Grisvard] in polar coordinates is given by $$\begin{aligned} u=v=(r^2 \cos^2\theta-1)^2 (r^2 \sin^2\theta-1)^2 r^{1+ \alpha}g_{\alpha,\omega}(\theta),\end{aligned}$$ where $\alpha\approx 0.5444837367$ is a non-characteristic root of $\sin^2( \alpha\omega) = \alpha^2\sin^2(\omega)$, $\omega=\frac{3\pi}{2}$, and $g_{\alpha,\omega}(\theta)=(\frac{1}{\alpha-1}\sin ((\alpha-1)\omega)-\frac{1}{ \alpha+1}\sin(( \alpha+1)\omega))(\cos(( \alpha-1)\theta)-\cos(( \alpha+1)\theta))$ $-(\frac{1}{\alpha-1}\sin(( \alpha-1)\theta)-\frac{1}{ \alpha+1}\sin(( \alpha+1)\theta)) (\cos(( \alpha-1)\omega)-\cos(( \alpha+1)\omega)).$ The computation of the discrete solution is executed utilizing triangular meshes, as depicted in Figure [6](#fig.Lshaped){reference-type="ref" reference="fig.Lshaped"}. In this example, the Newtons' method exhibits convergence within four iterations, while maintaining a tolerance threshold of $10^{-8}$. This example is particularly interesting since the solution is less regular due to the corner singularity. Since $\Omega$ is non-convex, we expect only sub-optimal order of convergences in the energy, $H^1$ and $L^2$ norms. Table [12](#table.Lshaped){reference-type="ref" reference="table.Lshaped"} confirms these estimates numerically. A similar observation for Morley nonconforming FEM is present in [@ng2 Section 5] and [@JDNNDS_ACOM Section 6.2.2] for a different weak formulation. $h$ ${\rm err}(u)$ Order ${\rm err}(\nabla u)$ Order ${\rm err}(Hu)$ Order ---------- ---------------- -------- ----------------------- -------- ----------------- -------- 0.707107 1.515408 \- 3.748752 \- 20.463977 \- 0.353553 0.474713 1.6746 1.189979 1.6555 11.578325 0.8217 0.176777 0.135893 1.8046 0.347020 1.7778 6.166005 0.9090 0.088388 0.039031 1.7998 0.101701 1.7707 3.209604 0.9419 0.044194 0.011978 1.7042 0.033175 1.6162 1.682817 0.9315 0.022097 0.004070 1.5574 0.012555 1.4018 0.905976 0.8933 : Convergence results, L-shaped domain, Triangular mesh $h$ ${\rm err}(v)$ Order ${\rm err}(\nabla v)$ Order ${\rm err}(Hv)$ Order ---------- ---------------- -------- ----------------------- -------- ----------------- -------- 0.707107 1.035284 \- 2.458626 \- 15.791769 \- 0.353553 0.431224 1.2635 1.084192 1.1812 11.359452 0.4753 0.176777 0.124537 1.7919 0.319836 1.7612 6.205400 0.8723 0.088388 0.035117 1.8263 0.091430 1.8066 3.236073 0.9393 0.044194 0.010510 1.7405 0.029067 1.6533 1.696020 0.9321 0.022097 0.003484 1.5929 0.010896 1.4156 0.912122 0.8949 : Convergence results, L-shaped domain, Triangular mesh [\[table.Lshaped\]]{#table.Lshaped label="table.Lshaped"} The first author thanks Indian Institute of Space Science and Technology (IIST) for the financial support towards the research work. The second author thanks the Department of Science and Technology (DST-SERB), India, for supporting this work through the core research grant CRG/2021/002410. [^1]: Department of Mathematics, Indian Institute of Space Science and Technology, Thiruvanathapuram 695547, India. devikas.pdf\@iist.ac.in, sarvesh\@iist.ac.in

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--- abstract: | In this paper, we discuss a general framework for multicontinuum homogenization. Multicontinuum models are widely used in many applications and some derivations for these models are established. In these models, several macroscopic variables at each macroscale point are defined and the resulting multicontinuum equations are formulated. In this paper, we propose a general formulation and associated ingredients that allow performing multicontinuum homogenization. Our derivation consists of several main parts. In the first part, we propose a general expansion, where the solution is expressed via the product of multiple macro variables and associated cell problems. The second part consists of formulating the cell problems. The cell problems are formulated as saddle point problems with constraints for each continua. Defining the continua via test functions, we set the constraints as an integral representation. Finally, substituting the expansion to the original system, we obtain multicontinuum systems. We present an application to the mixed formulation of elliptic equations. This is a challenging system as the system does not have symmetry. We discuss the local problems and various macroscale representations for the solution and its gradient. Using various order approximations, one can obtain different systems of equations. We discuss the applicability of multicontinuum homogenization and relate this to high contrast in the cell problem. Numerical results are presented. author: - "E. Chung[^1], Y. Efendiev[^2], J. Galvis[^3], W.T. Leung[^4]" bibliography: - references.bib - references4.bib - references1.bib - references2.bib - references3.bib - decSol.bib title: Multicontinuum homogenization. General theory and applications. --- # Introduction Many problems have multiscale nature. For example, the flow in porous media occurs in multiscale media with heterogeneities at multiple scales and high contrast. The simulations of these problems are often performed on a coarse computational grid, where the grid size is much larger compared to the scales of heterogeneities. In these simulations, we distinguish two cases in the paper. The first is the case with no-scale separation and the second is the case with scale separation. In the first case, approaches use the information within the entire computational grid or beyond to derive macroscopic equations. We will not discuss this case in the paper. In the second case, the representative volume-based information (which is much smaller compared to the target coarse block) is used in deriving macroscopic equations. For the case of no-scale separation, many approaches are developed to account for subgrid effects. These approaches, e.g., [@chung2016adaptiveJCP; @GMsFEM13; @eh09; @hw97; @jennylt03; @chung2018constraint; @oz06_1; @altmann2021numerical; @fish2013practical; @new2023; @contreras2023exponential; @abreu2020conservation; @abreu2019convergence], include the construction of multiscale basis functions that are supported in domain larger than the target coarse block. Among these approaches, the CEM-GMsFEM [@chung2018constraint] is related to the approaches presented in this paper. In these approaches, the multiscale basis functions are computed in oversampled regions. There are several basis functions in each target coarse block representing different continua effects. These concepts will further be used in multicontinua homogenization. In the case of scale separation, one uses information in representative volume (which is much smaller compared to the coarse block) to derive effective properties. The well-known approach includes the homogenization technique [@bensoussan2011asymptotic; @blanc2023homogenization; @bakhvalov2012homogenisation; @allaire1992homogenization], which is widely used in many applications. The main idea of this approach is to assume that the solution in each macroscopic point, can be represented by its average. The homogenization method provides a systematic expansion, which allows for deriving the equations. In this derivation, the small-scale $\epsilon$ is the RVE size. In the derivation, all terms depending on different powers of $\epsilon$ are separated. The latter is one of the limitations in extending these methods to problems where the media properties can depend on $\epsilon$ (high-contrast case). In this paper, we introduce a general homogenization method, where we assume that the media properties can have high contrast. In our expansion, we consider that each macroscopic point has several macroscopic variables associated with it. The macroscopic variables are defined via auxiliary functions and assumed to be smooth functions. The expansion of the solution via macroscopic variables uses the solution of local microscopic problems posed in RVE, called solutions of cell problems. These local problems account for the micro-scale behavior of the solution given certain constraints. These constraints are related to the definition of macroscopic variables. In particular, our first cell problem imposes constraints to represent the constants in the average behavior of each continua. The consequent cell problems impose constraints to represent the high-order polynomials in the average behavior of each continua. The multi-continuum homogenization expansion is substituted into the fine-scale equations. Our next assumptions include the fact that the integrals in the macroscopic variational formulation can be written in terms of the integrals over RVE and macroscopic variables are smooth. Using these assumptions, we derive a system of equations on a coarse grid. The resulting system of equation include additional terms and can involve higher-order derivatives. These equations share similarities to other models derived earlier and some terms can be negligible due to high contrast in the media properties. Our approaches share some common ingredients with mixture theories [@rajagopal1995mechanics; @truesdell1984thermodynamics; @malek20]. In mixture theories, the conservation of mass and momentum are written for each component. This model can be used in deriving a general set of macroscopic equations. However, these models do not make any specific assumptions on exchange terms. Our models generalize some earlier derived model equations related to works [@efendiev2023multicontinuum], dual-permeability models [@rubin48; @barenblatt1960basic; @showalter1991micro; @aifantis1979continuum; @iecsan1997theory; @bunoiu2019upscaling; @arbogast1990derivation; @bedford1972multi; @chai2018efficient; @alotaibi2022generalized] and we establish a general tool for deriving multicontinuum homogenization models. One of the challenging aspects of multicontinuum homogenization is in formulating cell problems correctly. We consider large oversampled regions, where we can impose higher-order polynomial constraints. By imposing averages in each RVE within the oversampled region, our main test is to guarantee that the solution of the cell problem converges to zero. To achieve this, one needs a careful formulation of cell problems. For example, in a carefully studied example of mixed Darcy equations, we show how one can achieve this. We obtain a generalized Darcy approximation on the coarse grid. We present numerical examples. In this numerical example, we consider a mixed formulation between velocity and pressure in Darcy's equation. Because pressure and velocity are treated separately, their relation at the microscale will not necessarily preserve at the macroscale as in the standard homogenization. We note that there is a linear relation between the velocity and the gradient of the pressure via the multiscale permeability field. Because the mixed formulation is not symmetric, this causes further challenges that are addressed in numerical examples when imposing local constraint problems. Our numerical results show a good convergence as we decrease the mesh size. The paper is organized as follows. In the next section, we present preliminaries and a simple derivation of multicontinuum homogenization for zero-order equations. In Section 3, we present a general theory for multicontinuum homogenization and also discuss the relation to mixture theory. Section 4 is devoted to mixed-order systems. In Section 5, we present numerical experiments. # Preliminaries and zero order equation To present preliminaries, we consider zero-order equations (following [@blanc2023homogenization]). We consider the following zero-order equation $$\label{eq:zero} A(x) u(x) = f(x),$$ where $A(x)$ is a scalar function with multiple scales and high contrast. For example, we assume $A$ is a periodic function where the period consists of two distinct regions with highly varying coefficients. We denote by $\psi_i$ the characteristic function for the region $i$, called the $i$th continua. {#fig:ill} It is assumed that the problem is solved on a computational grid consisting of grid blocks, denoted $\omega$, that are much larger than heterogeneities. We assume some type of periodicity within each computational block represented by Representative Volume Element $R_\omega$ that corresponds to a computational element $\omega$ (see Figure [1](#fig:ill){reference-type="ref" reference="fig:ill"}) (more precise meaning will be defined later). We assume that within each $R_\omega$, there are several distinct average states (known as multicontinua). We denote the characteristic function for the continuum $i$ within $R_\omega$ by $\psi_i^\omega$ ($\omega$ will be omitted since local computations are restricted to a coarse block), i.e., $\psi_i=1$ within continuum $i$ (can be irregularly shaped regions consisting of several parts, in general) and $0$ otherwise. We introduce oversampled $R_\omega^+$ that contains several $R_\omega^p$'s, where $p$ denotes different $R_\omega$'s. We denote the central (target) RVE by, simply, $R_\omega$. We denote $\psi^p_i$, the characteristic function for $R_\omega^p$ and will omit the index $p$ for simplicity if it is clear which region we are referring to. We consider the expansion of $u$ in each RVE as (for simplicity, we use equal sign instead of approximation) $$\label{eq:u11} u=\phi_i U_i,$$ where $\phi_i$ is a microscopic function for each $i$ and $U_i(x)$ is a smooth function for each $i$. The summation over repeated indices is taken. To obtain the microscopic function $\phi_i$, we formulate the following cell problems in each RVE within $\omega$ and use $y$ dependence to denote microscopic nature: $$\begin{split} A(y) \phi_i(y) = D_{ij}\psi_j\ \ \text{in} \ R_\omega\\ \int_{R_\omega} \phi_i \psi_j = \delta_{ij} \int_{R_\omega} \psi_j, \end{split}$$ where $D_{ij}$ are constants and can be shown that $D_{ij}\psi_j=C_i \psi_i$. Moreover, it can be easily computed that $$C_j ={\int_{R_\omega} \psi_j \over \int_{R_\omega} \psi^2_j A^{-1} }.$$ Next, we derive macroscopic equations. For this, we first write an integral form (for any test function $v$) $$\begin{split} \int_D f v &= \int_D A(x) u(x) v(x)= \sum_\omega \int_\omega A(x) u(x) v(x) \\ &\approx \sum_\omega {|\omega|\over |R_\omega|} \int_{R_\omega} A(y) u(y) v(y). \end{split}$$ Substituting $u$ from ([\[eq:u11\]](#eq:u11){reference-type="ref" reference="eq:u11"}) into the equation and writing $v=\phi_i V_i$, we get $$\begin{split} \int_{R_\omega} A(y) u(y) v(x)dy \approx U_i(x_\omega) V_j(x_\omega) \int_{R_\omega} A(y)\phi_i(y) \phi_j(y) \; dy, \end{split}$$ where $x_\omega$ is a mid-point of $R_\omega$. We will omit the microscopic dependence of macroscale variables (e.g., $U_i$) and simply use $U_i$ notation $$\begin{split} \int_{R_\omega} A(y) u(y) v(x)dy =U_i V_j\int_{R_\omega} A(y)\phi_i(y) \phi_j(y) \; dy. \end{split}$$ We denote $$\begin{split} \alpha_{ij}= \int_{R_\omega} A(y)\phi_i(y) \phi_j(y) \; dy. \end{split}$$ It can be shown that $$\begin{split} \alpha_{ij} =\delta_{ij} C_i \int_{R_\omega}\psi_j. \end{split}$$ From the above, we see that the macroscopic equation has the form $$\alpha_{ij} U_i = b_j,$$ where $$b_j=\int_{R_\omega} f \phi_j.$$ Taking into account that $\alpha$ is a diagonal matrix, we have $$U_i = {b_i\over C_i \int_{R_\omega}\psi_i}.$$ We note that, in single continua homogenization, we obtain $$U_1 = b_1{1\over |R_\omega|} \int_{R_\omega} A^{-1}.$$ # General case. A formal derivation. In this section, we present a formal derivation of generalized multicontinuum homogenization. The derivation makes several assumptions, which may or may not hold depending on particular problems. We will make these assumptions as we go along. We consider a general linear system given by $$\label{eq:main0} \begin{split} {\partial u \over \partial t} + Au = f, \ \text{in}\ D, \end{split}$$ where $A$ is a differential operator, $u$ is a vector valued solution and $D$ is the domain. The problem ([\[eq:main0\]](#eq:main0){reference-type="ref" reference="eq:main0"}) is supplemented with some appropriate initial and boundary conditions. We next present several examples. **Example 1.** In the scalar case, $Au = -div(\kappa(x)\nabla u)$, where $\kappa(x)$ is a multiscale and high-contrast coefficient. **Example 2.** In a vector case, one can consider the elasticity problem with $Au=-\nabla_i C_{ijkl}(x) e_{kl}(u)$, where $C_{ijkl}(x)$'s represent heterogeneous and high-contrast media properties, $e_{kl}(u)=(\nabla_k u_l + \nabla_l u_k)/2$, and $u$ is the displacement vector. **Example 3.** We can consider Example 1 in a mixed formulation as a first-order system. In this case, $u=(p,v)$, where $p$ and $v$ solve $\kappa^{-1} v + \nabla p=0$, $div(v)=f$. **Example 4.** One can consider the first order systems, $A u = v(x) \cdot\nabla u + a(x) u$, where $v(x)$ and $a(x)$ are highly heterogeneous fields. We write ([\[eq:main0\]](#eq:main0){reference-type="ref" reference="eq:main0"}) as a variational problem $$\begin{split} ({\partial u \over \partial t},v) + a_D(u,v) = (f,v), \end{split}$$ where $a_D(u,v)=\int_D (A u) v$, e.g., in Example 1, $a_D(u,v)=\int_D \kappa \nabla u\cdot \nabla v$ (assuming zero Dirichlet boundary conditions). In the multicontinuum homogenization, we assume that in each RVE, $R_\omega$, there exist functions $\psi_i$ ($i$ refers to continua, $\psi_i$ can be a characteristic function of subregion), such that $$U_i(x_\omega^*)={\int_{R_\omega} u \psi_i \over \int_{R_\omega}\psi_i }$$ are macroscopic variables, where $x_\omega^*$ is a point in $R_\omega$. **One main assumption is that $U_i$'s are smooth functions if we consider them over all RVEs.** Next, we present the steps in deriving macroscopic equations. **Step 1. Expansion.** The first step consists of expanding the solution $u$ in terms of macroscopic variables. The coefficients in front of them, denoted by $\phi_i$'s, represent the local microscopic solution in RVE. We consider the expansion of the solution $u$ as $$\label{eq:expansion1} \begin{split} u = \phi_m U_m +\phi_m^j \nabla_j U_m + \phi_m^{ij} \nabla_{ij}^2 U_m + ..., \end{split}$$ where $\nabla_j$ refers to ${\partial \over \partial x_j}$. In this expansion, we will discuss the functions $\phi$, which are defined as the solutions of local problems in RVE, $R_\omega$. **Step 2. Cell problems.** Next, we introduce equations for $\phi_i$'s. These equations are written in each RVE subject to some constraints. These constraints are related to definitions of macroscopic variables. We use Taylor's expansion concepts in defining the local functions such that they solve local problems with constraints that their averages with respect to $\psi_i$ behave as constants, linear functions, and quadratic functions. Our first cell problem imposes constraints to represent the constants in the average behavior of each continua (continua $m$ in ([\[eq:cell1\]](#eq:cell1){reference-type="ref" reference="eq:cell1"})) We consider the cell problem in oversampled regions $R_\omega^+$ that contain several $R_\omega$, denoted by $R_\omega^p$. $$\label{eq:cell1} \begin{split} A\phi_m^i &= \Gamma_{mn}^{ijp}\psi_n^pe^j \; \text{in} \; R_\omega^+, \\ \int_{R_\omega^p} \phi_m^i \psi_n^p &= \delta_{mn} e^i \int_{R_\omega^p} \psi_n^p, \ \forall p, \end{split}$$ where $e^i$ is the unit vector (solution $u$ is vector valued) and $\psi_n^p$ is the characteristic function in $R_\omega^p$. This cell problem corresponds to appropriate energy minimizing solution subject to the constraints. Here and later, by $\Gamma$, we denote the Lagrange multipliers due to constraints. We denote $\phi_m$ the matrix spanned by $\phi_m^i$ (as columns). Our second cell problem imposes constraints to represent the linear functions in the average behavior of each continua. $$\label{eq:cell2} \begin{split} A\phi_m^{il} &= \Gamma_{mn}^{ijpl}\psi_n^pe^j \; \text{in} \; R_\omega^+, \\ \int_{R_\omega^p} \phi_m^{il} \psi_n^p &= \delta_{mn} e^i\int_{R_\omega^p} (x_l -c_l) \psi_n^p,\ \forall p, \end{split}$$ where $c_l$ (later on also) is chosen such that $\int_{R_\omega} (x_l -c_l)=0$, where $R_\omega$ is the RVE defined in the middle of $R_\omega^+$. Similarly, we denote $\phi_m^l$ the matrix spanned by $\phi_m^{il}$ (as columns). We can also define higher-order cell problems. The next cell problem imposes constraints to represent the quadratics in the average behavior of each continua. $$\begin{split} A\phi_m^{ijl} &= \Gamma_{mn}^{ijlps}\psi_n^pe^s \; \text{in} \; R_\omega^+, \\ \int_{R_\omega^p} \phi_m^{ijl} \psi_n^p &= \delta_{mn} e^i\int_{R_\omega^p} (x_jx_l -c_{jl}) \psi_n^p,\ \forall p. \end{split}$$ Similarly, we denote $\phi_m^{lj}$ the matrix spanned by $\phi_m^{ijl}$ (as columns). We note that in our cell problems, we solve for each component of the vector solutions. In some applications, one can lump some components if some relations between components of the vector are known apriori. **The decay of cell solutions.** Existence and uniqueness can be shown in most cases for positive symmetric operators with appropriate norms. In general, we need inf-sup condition for well-posedness of cell problems [@brezzi1974existence]. We note that the decay of local solution away from $R_\omega$ (middle RVE) is important. The latter indicates a correct computation of the local problems. In some cases, one can use appropriate local boundary conditions when the information is available about the global solution. **Step 3. Substitution in the variational formulation.** In this step, we use $u$ and $v$ expansion in the fine-grid formulation of the problem. In particular, we have $$\begin{split} u = \phi_m^i U_m^i + \phi_m^{il} \nabla_l U_m^i + \phi_m^{ilp} \nabla^2_{lp} U_m^i\\ v = \phi_m^i V_m^i + \phi_m^{il} \nabla_l V_m^i+ \phi_m^{ilp} \nabla^2_{lp} V_m^i. \end{split}$$ We substitute and get the following equation (we use matrix notations for $\phi$'s) $$\label{eq:expan1} \begin{split} & \int_D {\partial \over \partial t}(\phi_m U_m + \phi_m^{l} \nabla_l U_m + \phi_m^{lp} \nabla^2_{lp} U_m) (\phi_n V_n + \phi_n^k \nabla_k V_n+\phi_n^{ks} \nabla_{ks}^2 V_n)\\ & a_D(\phi_m U_m, \phi_n V_n) + a_D (\phi_m U_m, \phi_n^k \nabla_k V_n) + a_D (\phi_m U_m, \phi_n^{ks} \nabla_{ks}^2 V_n) + \\ & a_D(\phi_m^l \nabla_l U_m, \phi_n V_n) + a_D (\phi_m^l \nabla_l U_m, \phi_n^k \nabla_k V_n) + a_D (\phi_m^l\nabla_l U_m, \phi_n^{ks} \nabla_{ks}^2 V_n) + \\ & a_D(\phi_m^{lp} \nabla_{lp}^2 U_m, \phi_n V_n) + a_D (\phi_m^{lp} \nabla_{lp}^2 U_m, \phi_n^k \nabla_k V_n) + a_D(\phi_m^{lp}\nabla_{lp}^2 U_m,\phi_n^{ks} \nabla_{ks}^2 V_n) \\ = &\int_D f(\phi_n V_n + \phi_n^k \nabla_k V_n+\phi_n^{ks} \nabla_{ks}^2 V_n). \end{split}$$ Our next two steps include using RVE concepts and taking into account that $U_i$ and $V_i$ are smooth functions. **Step 4. Integral localization.** Our next step includes dividing the integral over the coarse partition $\omega$ and then using the RVE concept. More precisely, for each integral and a smooth function $F$, we have $$\begin{split} \int_D F =\sum_\omega F \approx \sum_\omega {|\omega|\over |R_\omega|} \int_{R_\omega}F. \end{split}$$ **Step 5. Piecesmooth approximation of macroscopic terms.** In this step, the macroscopic terms, $U_i$ and $V_i$ assumed to be smooth functions and the operator $A$ acts only on cell problem solutions. As before (in zero-order equation case), we take the macroscopic variables out of the integrals over $R_\omega$. To demonstrate this step, we consider only two term expansion in ([\[eq:expansion1\]](#eq:expansion1){reference-type="ref" reference="eq:expansion1"}) writing the integrals over RVE. More precisely, the terms in the equation ([\[eq:expan1\]](#eq:expan1){reference-type="ref" reference="eq:expan1"}) have the following forms in $R_\omega$. $$\label{eq:expan2} \begin{split} &\sum_\omega{|\omega|\over |R_\omega|} (\int_{R_\omega} \phi_m\phi_n){\partial \over \partial t} U_m V_n + \sum_\omega{|\omega|\over |R_\omega|} (\int_{R_\omega} \phi_m\phi_n^k){\partial \over \partial t} U_m \nabla_k V_n + \\ &\sum_\omega{|\omega|\over |R_\omega|} (\int_{R_\omega} \phi_m^l\phi_n){\partial \over \partial t} \nabla_l U_m V_n + \sum_\omega{|\omega|\over |R_\omega|} (\int_{R_\omega} \phi_m^{l}\phi_n^k){\partial \over \partial t} \nabla_l U_m \nabla_k V_n + \\ &+ \sum_\omega{|\omega|\over |R_\omega|} a_{R_\omega}(\phi_m,\phi_n) U_mV_n + \sum_\omega{|\omega|\over |R_\omega|} a_{R_\omega} (\phi_m, \phi_n^k ) U_m\nabla_k V_n+ \\ &\sum_\omega{|\omega|\over |R_\omega|} a_{R_\omega}(\phi_m^l , \phi_n ) \nabla_l U_m V_n+ \sum_\omega{|\omega|\over |R_\omega|} a_{R_\omega} (\phi_m^l, \phi_n^k ) \nabla_l U_m\nabla_k V_n + \\ =&\sum_\omega{|\omega|\over |R_\omega|} (\int_{R_\omega} f\phi_n) V_n + \sum_\omega{|\omega|\over |R_\omega|} (\int_{R_\omega}f\phi_n^k)\nabla_k V_n. %\int_D f(\phi_n V_n + \phi_n^k \nabla_k V_n+\phi_n^{ks} \nabla_{ks}^2 V_n) \end{split}$$ In Equation ([\[eq:expan2\]](#eq:expan2){reference-type="ref" reference="eq:expan2"}), we further take into account that $U_i$ and $V_i$ are smooth functions defined in $D$ and get the following macroscopic equation for $U_i$ (in strong form) $$\begin{split} A_{nm}{\partial \over \partial t} U_m+ B_{nm}U_m + B_{nm}^i\nabla_iU_m - \nabla_k\overline{B}^k_{nm}U_m - \nabla_k(B_{nm}^{ik}\nabla_iU_m)=b_n. \end{split}$$ Here, we neglect the second, third, and fourth terms in ([\[eq:expan2\]](#eq:expan2){reference-type="ref" reference="eq:expan2"}). The latter is because $\phi_n^k$ is of order RVE size, while $\phi_n$ is of order $O(1)$, in general. Because the coefficients in the operator $A$ have high-contrast properties, we can not neglect these terms. We will remark on this later. The coefficients $A$'s and $B$'s are defined from ([\[eq:expan2\]](#eq:expan2){reference-type="ref" reference="eq:expan2"}). More precisely, $$\begin{split} &A_{nm}={1 \over |R_\omega|}\int_{R_\omega} \phi_m\phi_n, \ \ b_n ={1 \over |R_\omega|}\int_{R_\omega}f\phi_n^k,\\ &B_{nm}= {1 \over |R_\omega|}a_{R_\omega}(\phi_m,\phi_n),\ \ B_{nm}^i= {1 \over |R_\omega|}a_{R_\omega}(\phi^i_m,\phi_n),\\ &\overline{B}^k_{nm}={1 \over |R_\omega|}a_{R_\omega}(\phi_m,\phi_n^k), \ \ B_{nm}^{ik}= {1 \over |R_\omega|}a_{R_\omega}(\phi^i_m,\phi^k_n).\\ \end{split}$$ If we use the second-order expansion, the macroscopic equation will have the following form $$\begin{split} &A_{nm}{\partial \over \partial t} U_m+\\ &B_{nm}U_m + B_{nm}^i\nabla_iU_m + B_{nm}^{ij}\nabla^2_{ij}U_m-\\ &\nabla_k(B^k_{nm}U_m) - \nabla_k(B_{nm}^{ik}\nabla_iU_m) - \nabla_k (B_{nm}^{ijk}\nabla^2_{ij}U_m)+\\ &\nabla^2_{kp}(B^{kp}_{nm}U_m) + \nabla^2_{kp}(B_{nm}^{ikp}\nabla_iU_m) + \nabla^2_{kp}(B_{nm}^{ijkp}\nabla^2_{ij}U_m)=b_n. \end{split}$$ Next, we make several remarks. First, different terms in the macroscopic equation can have negligible weights. In general, $\phi_k$'s (the cell solutions accounting for the averages) are of order $O(1)$, while $\phi_k^n$'s (the cell solutions accounting for the gradients) are of order $O(\epsilon)$, where $\epsilon$ is the RVE size (see [@efendiev2023multicontinuum]). For this reason, we have neglected some terms in the time derivative terms and source terms. However, because of high-contrast coefficients, one can not neglect different terms that stem from $\phi_k$'s or from $\phi_k^n$'s. In [@efendiev2023multicontinuum], we show that the zero-order terms are important when there is high contrast. More precisely, the reaction terms scale as the inverse of the RVE size. If the effective diffusivity is high, then the reaction and diffusion terms balance each other. Otherwise, one can show that there is no multicontinuum and our macroscopic equations result to single continuum homogenization. Our second remark is regarding the definition of the continua. Throughout the paper, we assume that $\psi_i$'s are associated with subregions defined apriori. In general, one can use spatial functions for $\psi_i$, for example, defined via local spectral problems as it is done in nonlocal multicontinua approach or GMsFEM [@GMsFEM13; @vasilyeva2019nonlocal]. ## Example. A scalar elliptic equation This example is discussed in [@efendiev2023multicontinuum]. We briefly mention it here. We will focus on multicontinuum expansion, macroscopic equations, and constraints, for simplicity, and do not write down the cell problem equations (cf. [\[eq:cell1\]](#eq:cell1){reference-type="ref" reference="eq:cell1"}). The multicontinuum expansion is $u=\phi_i U_i+\phi^m_{i}\nabla_m U_i$, where cell solutions have constraints for $\phi_m$ $$\label{eq:cell_ex1} \begin{split} \int_{R_\omega^p} \phi_m \psi_n^p = \delta_{mn} \int_{R_\omega^p} \psi_n^p \end{split}$$ and for $\phi_m^l$ $$\label{eq:cell_ex12} \begin{split} \int_{R_\omega^p} \phi_m^{l} \psi_n^p = \delta_{mn}\int_{R_\omega^p} (x_l -c_l) \psi_n^p. \end{split}$$ Note that the equations for $\phi_m$ and for $\phi_m^{l}$ are solved separately. The macroscopic equations have the following form $$\label{eq:macro11} \begin{split} B_{nm}U_m + B_{nm}^i\nabla_iU_m - \nabla_k(\overline{B}^k_{nm}U_m) - \nabla_k(B_{nm}^{ik}\nabla_iU_m) =b_n. \end{split}$$ ## Example. A system of elliptic equations We consider $$-{\partial \over \partial x_k} (A^{kl}_{ji} {\partial \over \partial x_l} u_i) = f_j.$$ The multicontinuum expansion has the following form $$u_i = \phi_{mij}U_{mj} + \phi_{mij}^k\nabla_k U_{mj},$$ where the cell problems have the constraints for $\phi_{imn}$ $$\label{eq:cell_ex21} \begin{split} \int_{R_\omega^p} \phi_{mij} \psi_n^p = \delta_{mn}\delta_{ij} \int_{R_\omega^p} \psi_n^p \end{split}$$ and for $\phi_{mij}^{l}$ $$\label{eq:cell_ex22} \begin{split} \int_{R_\omega^p} \phi_{mij}^{l} \psi_n^p = \delta_{mn}\delta_{ij}\int_{R_\omega^p} (x_l -c_l) \psi_n^p . \end{split}$$ For example, for two equations, we have $$\begin{bmatrix} u_1 \\ u_2\\ \end{bmatrix} = \begin{bmatrix} \phi_{j11} & \phi_{j12} \\ \phi_{j21} & \phi_{j22} \\ \end{bmatrix} \begin{bmatrix} U_{j1} \\ U_{j2}\\ \end{bmatrix} + \begin{bmatrix} \phi^m_{j11} &\phi^m_{j12} \\ \phi^m_{j21} & \phi^m_{j22} \\ \end{bmatrix} \begin{bmatrix} \nabla_m U_{j1}\\ \nabla_m U_{j2} \end{bmatrix}.$$ The constraints are the following $$\label{eq:cell_ex23} \begin{split} \int_{R_\omega^p} \begin{bmatrix} \phi_{m11} \\ \phi_{m21} \\ \end{bmatrix} \psi_n^p = \delta_{mn} \begin{bmatrix} 1 \\ 0\\ \end{bmatrix} \int_{R_\omega^p} \psi_n^p, \ \ \ \int_{R_\omega^p} \begin{bmatrix} \phi_{m12} \\ \phi_{m22} \\ \end{bmatrix} \psi_n^p = \delta_{mn} \begin{bmatrix} 0 \\ 1\\ \end{bmatrix} \int_{R_\omega^p} \psi_n^p, \end{split}$$ $$\label{eq:cell_ex231} \begin{split} \int_{R_\omega^p} \begin{bmatrix} \phi^k_{m11} \\ \phi^k_{m21} \\ \end{bmatrix} \psi_n^p = \delta_{mn} \begin{bmatrix} 1 \\ 0\\ \end{bmatrix} \int_{R_\omega^p} (x_k-c_k)\psi_n^p, \\\ \int_{R_\omega^p} \begin{bmatrix} \phi^k_{m12} \\ \phi^k_{m22} \\ \end{bmatrix} \psi_n^p = \delta_{mn} \begin{bmatrix} 0 \\ 1\\ \end{bmatrix} \int_{R_\omega^p} (x_k-c_k)\psi_n^p. \end{split}$$ The macroscopic equations have the following form $$\begin{split} B_{ijnm}U_{jm} + B_{ijnm}^l\nabla_lU_{jm} - \nabla_k(B^k_{ijnm}U_{jm}) - \nabla_k(B_{ijnm}^{lk}\nabla_lU_{jm}) =b_{in}. \end{split}$$ It can be shown that the second and third terms cancel each other and the scaling of $B_{nm}$ is of order $1/\epsilon^2$, where $\epsilon$ is RVE size. Because of high contrast, this term can balance with the diffusion term. ## Mixture theory and its relation Here, we briefly note that one can also derive general multicontinuum equations using mixture theory [@rajagopal1995mechanics; @truesdell1984thermodynamics; @malek20]; however, precise micro and macro relations can not be derived from this theory. Mixture theory specifies several model classes [@malek20]. One that is suitable for our models is Class II, where $N$ balances of mass for N components of the mixture and also $N$ balances of linear momentum for N components of mixture are formulated. In this case, the equations have the following form $$\begin{split} {\partial \rho_i\over \partial t} + div (\rho_i v_i) = m_i,\ \ \sum_i m_i =0,\\ {\partial \rho_i v_i \over \partial t} + div (\rho_i v_i \otimes v_i)=div(\mathcal{T}_i) + \mathcal{Q}_i+ m_i v_i,\ \ \sum_i (\mathcal{Q}_i + m_i v_i) =0. \end{split}$$ Here, we use a simplified formulation from [@malek20], and use the notations from [@malek20], where $\rho_i$ is the density of $i$th component, $v_i$ is the velocity, $m_i$ is the exchange terms for mass conservation, $\mathcal{T}_i$ is the stress tensor, and $\mathcal{Q}_i$ is the exchange terms for momentum. To derive a multicontinuum equations, we consider solid and two fluid continua mixture. For momentum equations, we have (ignoring gravity) $$\begin{split} {\partial \rho_1^f v_1^f \over \partial t} + div (\rho_1^f v_1^f \otimes v_1^f)=div(\mathcal{T}^f_1) + \mathcal{Q}^f_1,\\ {\partial \rho_2^f v_2^f \over \partial t} + div (\rho_2^f v_2^f \otimes v_2^f)=div(\mathcal{T}^f_2) + \mathcal{Q}^f_2,\\\ {\partial \rho^s v^s \over \partial t} + div (\rho^s v^s \otimes v^s)=div(\mathcal{T}_s) +\mathcal{Q}^s, \end{split}$$ where $\mathcal{Q}^s=-\mathcal{Q}^f_1-\mathcal{Q}^f_2$, $s$ denotes the solid and $f$ denotes the fluid. It is assumed that $v^s \approx 0$, $\mathcal{T}^f_i=p_i I$, $i=1,2$, $\mathcal{Q}^f_i=\kappa_i^{-1} v_i$, and the flow is steady-state and slow. In the mass conservation equations, $$\begin{split} {\partial \rho_1^f\over \partial t} + div (\rho_1^f v_1^f )= m^f_1,\\ {\partial \rho_2^f \over \partial t} + div (\rho_2^f v_2^f)=m^f_2,\\\ {\partial \rho^s \over \partial t} + div (\rho^s v^s)=m^s. \end{split}$$ We have $v^s\approx 0$, $m^s \approx 0$, and take $$\begin{split} m^f_1 = \alpha \rho_1^f (p_2-p_1)\\ m^f_2 = \alpha \rho_2^f (p_1-p_2).\\ \end{split}$$ The resulting equations have the form of multicontinuum equations ([\[eq:macro11\]](#eq:macro11){reference-type="ref" reference="eq:macro11"}). # First-order mixed system We consider a first-order mixed system as an example of a system, where the variables are coupled. $$\label{eq:mixed_main} \begin{split} \kappa^{-1} v + \nabla u = 0\\ div( v) = f. \end{split}$$ This equation is a non-symmetric system with the solution vector $(v,u)$ and the operator $$A= \begin{bmatrix} \kappa^{-1} & \nabla \\ div & 0 \\ \end{bmatrix}.$$ The local cell problems and constraints require special attention to achieve a decay property. We omit this part to numerical results. We consider the derivation of macroscopic equations. In general, as before, one can use various constraints and derive various macroscopic equations. We consider piecewise constant velocity and piecewise linear type pressure approximations at the RVE level. We use different notations because differing notations for variables. In this case, we have the following expansion $$\label{eq:expan11} \begin{split} &v_s=\phi_{is}^{vu} U_i + \phi_{ism}^{vu} \nabla_m U_i + \phi_{isk}^{vv} V_{ik}\\ &u = \phi^{uu}_i U_i + \phi^{uu}_{im} \nabla_m U_i + \phi_{ik}^{uv} V_{ik}.\\ \end{split}$$ Here, $i$ refers to the continua, $(\phi^{uv},\phi^{uu})$ represents the cell solutions with zero constraints on $v$ and $(\phi^{vu},\phi^{vv})$ represents cell solutions with zero constraints on $u$ (see Section 5, ([\[eq:mixed_c1\]](#eq:mixed_c1){reference-type="ref" reference="eq:mixed_c1"})-([\[eq:mixed_c3\]](#eq:mixed_c3){reference-type="ref" reference="eq:mixed_c3"})). We multiple the mixed system ([\[eq:mixed_main\]](#eq:mixed_main){reference-type="ref" reference="eq:mixed_main"}) by $$\begin{bmatrix} \phi^{vu}_j Q_j + \phi_{jl}^{vu}\nabla_l Q_j\\ \phi^{uu}_j Q_j + \phi_{jl}^{uu}\nabla_l Q_j \end{bmatrix}$$ and sum up the equations (use vector notations for simplicity) $$\begin{split} &\int_{R_\omega} (\phi^{vu}_j Q_j + \phi_{jl}^{vu}\nabla_l Q_j)\kappa^{-1}(\phi^{vu}_i U_i + \phi_{im}^{vu}\nabla_m U_i+\phi_{i}^{vv} V_{i}) \\ &+\int_{R_\omega} (\phi^{vu}_j Q_j + \phi_{jl}^{vu}\nabla_l Q_j) \nabla(\phi^{uu}_i U_i + \phi_{im}^{uu}\nabla_m U_i+\phi_{i}^{uv} V_{i})\\ &+ \int_{R_\omega} div(\phi^{vu}_i U_i + \phi_{im}^{vu}\nabla_m U_i+\phi_{i}^{vv} V_{i})(\phi^{uu}_j Q_j + \phi_{jl}^{uu}\nabla_l Q_j) \\ =&\int_{R_\omega} f (\phi^{uu}_j Q_j + \phi_{jl}^{uu}\nabla_l Q_j). \end{split}$$ In the global form, the equation has the form $$\label{eq:mix21} \begin{split} \alpha^{u}_{ij}U_i + \alpha^{u}_{ijm}\nabla_m U_i - \nabla_m(\overline{\alpha}^{u}_{ijm} U_i)-\nabla_n(\alpha^{u}_{ijnm}\nabla_m U_i)+\beta^{u}_{ji} V_{i} + \beta^{u}_{jim} \nabla_m V_{i} = f^u_j. \end{split}$$ In our numerical simulations, we observe that the sum of two convection terms (the second and third terms in the equation) is small and can be neglected. Next, we multiply the system ([\[eq:mixed_main\]](#eq:mixed_main){reference-type="ref" reference="eq:mixed_main"}) by $$\begin{bmatrix} \phi_{j}^{vv} Q_{j}\\ \phi_{j}^{uv} Q_{j} \end{bmatrix}$$ and sum up (use vector notations for simplicity) $$\begin{split} &\int_{R_\omega} \phi_{j}^{vv} Q_{j}\kappa^{-1}(\phi^{vu}_i U_i + \phi_{im}^{vu}\nabla_m U_i+\phi_{i}^{vv} V_{i}) \\ &+\int_{R_\omega} \phi_{j}^{vv} Q_{j}\nabla(\phi^{uu}_i U_i + \phi_{im}^{uu}\nabla_m U_i+\phi_{i}^{uv} V_{i})\\ &+ \int_{R_\omega} div(\phi^{vu}_i U_i + \phi_{im}^{vu}\nabla_m U_i+\phi_{i}^{vv} V_{i})\phi_{j}^{uv} Q_{j} \\ =&\int_{R_\omega} f \phi_{j}^{uv} Q_{j}. \end{split}$$ In the global form, the equation has the form $$\label{eq:mix22} \begin{split} \alpha^{v}_{ij}U_i + \alpha^{v}_{ijm}\nabla_m U_i + \beta^{v}_{ji} V_{i} = f^v_j. \end{split}$$ In our numerical simulations, we observe that $\alpha^{v}_{ij}$ is small and can be neglected. In general, one can choose a more general representation of the velocity via piecewise linear functions and obtain general models with higher order derivatives. Note that the polynomial constraints in the approximation of velocity and pressures in ([\[eq:expan11\]](#eq:expan11){reference-type="ref" reference="eq:expan11"}) is for homogenization and is not related to stable polynomial approximation in finite element methods. # Numerical example In this section, we will present some numerical examples to demonstrate the performance of the method in a mixed formulation. As we mentioned earlier that for nonsymmetric problems, it is challenging to guarantee the decay of local solutions. Here, we propose the following local problems for velocity $v$ and pressure $u$ in the equation $$\begin{split} \kappa^{-1} v + \nabla u &=0\\ div(v)&=q. \end{split}$$ Next, we describe the local solutions for velocity and pressure. We will only write down the constraints, the formulation of the local problem follows from equations ([\[eq:cell1\]](#eq:cell1){reference-type="ref" reference="eq:cell1"}) and ([\[eq:cell2\]](#eq:cell2){reference-type="ref" reference="eq:cell2"}). For the velocity constraints, we impose an intermediate domain $R_\omega^V$, where $R_\omega^V$ is a subset of $R_\omega^+$ and contains $R_\omega$. Moreover, we assume that $R_\omega^V$ consists of $R_\omega^p$, where $p$ is a numeration of local domains, one of them being $R_\omega$. We remind that the local solution has the following matrix form. $$\begin{bmatrix} u \\ v_s\\ \end{bmatrix} = \begin{bmatrix} \phi^{uu}_i & \phi^{uu}_{im} & \phi^{uv}_{ik} \\ \phi^{vu}_{is} & \phi^{vu}_{ism} & \phi^{vv}_{isk} \\ \end{bmatrix} \begin{bmatrix} U_i \\ \nabla_m U_i\\ V_{ik} \end{bmatrix}$$ The local constraints for $\phi$'s are imposed column by column. The constraints are the following $$\label{eq:mixed_c1} \begin{split} \int_{R_\omega^p}\phi^{uu}_i \psi_j &= \delta_{ij} \int_{R_\omega^p}\psi_j, \ \forall R_\omega^p\subset R_\omega^+,\\ \int_{R_\omega^p}\phi^{vu}_{is} \psi_j &= 0, \ \forall R_\omega^p\subset R_\omega^V,\\ \end{split}$$ and $$\label{eq:mixed_c2} \begin{split} \int_{R_\omega^p}\phi^{uu}_{im} \psi_j &= \delta_{ij} \int_{R_\omega^p}(x_m -c_m)\psi_j, \ \forall R_\omega^p\subset R_\omega^+,\\ \int_{R_\omega^p}\phi^{vu}_{ism} \psi_j &= 0, \ \forall R_\omega^p\subset R_\omega^V,\\ \end{split}$$ and $$\label{eq:mixed_c3} \begin{split} \int_{R_\omega^p}\phi^{uv}_{ik} \psi_j &= 0, \ \forall R_\omega^p\subset R_\omega^+,\\ \int_{R_\omega^p} \phi^{vv}_{isk} \psi_j &= \delta_{ij} \delta_{sk} \int_{R_\omega^p} \psi_j, \ \forall R_\omega^p\subset R_\omega^V. \end{split}$$ In the calculations of macroscopic domains, we use another intermediate domain $R_\omega^I$, which is a subset of $R_\omega^+$ and contains $R_\omega^V$. The local expansion is given by ([\[eq:expan11\]](#eq:expan11){reference-type="ref" reference="eq:expan11"}). In the first example, we consider the layered medium depicted on Figure [3](#fig:perm1){reference-type="ref" reference="fig:perm1"}. The permeability field $\kappa$ has a period denoted by $\epsilon$. We denote the low conductivity region and the high conductivity region of $\kappa$ by $\Omega_{1}$ and $\Omega_{2}$, respectively. The source term $f$ and conductivity $\kappa$ are as follows $$f(x)=\begin{cases} 1000\min\{\kappa\}e^{-40|(x-0.5)^{2}+(y-0.5)^{2}|} & x\in\Omega_{1}\\ e^{-40|(x-0.5)^{2}+(y-0.5)^{2}|} & x\in\Omega_{2} \end{cases}$$ and $$\kappa(x)=\begin{cases} \cfrac{\epsilon}{10000} & x\in\Omega_{1}\\ \cfrac{1}{100\epsilon} & x\in\Omega_{2} \end{cases}$$ We divide the computational domain $\Omega$ into $M\times M$ coarse grid. The coarse mesh size $H$ is defined as $H=1/M$. We consider the whole coarse grid element as an RVE for the corresponding coarse element. The oversampling RVE $R_\omega^{+}$ (or $\omega^+$) for each coarse RVE $\omega$ is defined as an extension of $K$ (target coarse block) by $l$ layers of coarse grid element, where $l$ will be changed in simulations. We define the relative $L^{2}$- error in $\Omega_{1}$ and the relative $L^{2}$- error in $\Omega_{2}$ by $$e_{2}^{(i)}=\cfrac{\sum_{K}|\cfrac{1}{|K|}\int_{K}U_{i}-\cfrac{1}{|K\cap\Omega_{i}|}\int_{K\cap\Omega_{i}}u|^{2}}{\sum_{K}|\cfrac{1}{K\cap\Omega_{i}}\int_{K\cap\Omega_{i}}u|^{2}}.$$ $K$ denotes the RVE, which is taken to be $\omega$. For the first case, we take the fine-mesh size to be $H\epsilon$. We present $e_2^{(i)}$ in Table [2](#tab:case1){reference-type="ref" reference="tab:case1"}. First, we observe that the proposed approach provides an accurate approximation of the averaged solution as we decrease the mesh size. In Figure [7](#fig:compare_case1){reference-type="ref" reference="fig:compare_case1"}, we depict upscaled solutions and corresponding averaged fine-scale solutions. We observe that these solutions are very close. In the first table, we decrease the coarse-mesh size and the period size. In standard numerical homogenization methods, this gives a resonance error (stagnating errors). Here, by choosing an appropriate number of layers, we observe that the error remains small. In the second table, we observe convergence as we decrease the mesh size and fix $\epsilon$. In general, we expect a certain threshold error due to fine-scale discretization, which is used to compute the solution. {#fig:perm1} {#fig:perm1} {#fig:compare_case1} {#fig:compare_case1} {#fig:compare_case1} {#fig:compare_case1} $H$ $\epsilon$ $e_{2}^{(1)}$ $e_{2}^{(2)}$ -------- ------------ --------------- --------------- $1/10$ $1/10$ $27.19\%$ $6.21\%$ $1/20$ $1/20$ $11.63\%$ $1.19\%$ $1/40$ $1/40$ $3.25\%$ $0.88\%$ : Error comparison for Case 1. $H$ $\epsilon$ $e_{2}^{(1)}$ $e_{2}^{(2)}$ -------- ------------ --------------- --------------- $1/10$ $1/10$ $27.19\%$ $6.21\%$ $1/10$ $1/20$ $12.79\%$ $2.43\%$ $1/10$ $1/40$ $6.32\%$ $1.70\%$ : Error comparison for Case 1. For the second case, we change the permeability field to the one shown in Figure [9](#fig:case2_perm){reference-type="ref" reference="fig:case2_perm"}. We present $e_2^{(i)}$ in Table [4](#tab:case2){reference-type="ref" reference="tab:case2"}. Again, we observe that the proposed approach provides an accurate approximation of the averaged solution as we decrease the mesh size. In Figure [13](#fig:case2_compare){reference-type="ref" reference="fig:case2_compare"}, we depict upscaled solutions and corresponding averaged fine-scale solutions. We observe a good agreement between coarse- and fine-grid solutions. In the first table, we decrease the coarse-mesh size and the period size at the same time. We observe that the error decreases as the mesh size decreases. Here, by choosing an appropriate number of layers, we observe that the error remains small. In the second table, we observe the convergence as we decrease the mesh size and fix $\epsilon$. Again, the error decreases as we decrease the mesh size. {#fig:case2_perm} {#fig:case2_perm} {#fig:case2_compare} {#fig:case2_compare} {#fig:case2_compare} {#fig:case2_compare} $H$ $\epsilon$ $e_{2}^{(1)}$ $e_{2}^{(2)}$ -------- ------------ --------------- --------------- $1/10$ $1/10$ $11.74\%$ $1.61\%$ $1/20$ $1/20$ $4.18\%$ $0.97\%$ $1/40$ $1/40$ $1.86\%$ $1.08\%$ : Error comparison for Case 2. $H$ $\epsilon$ $e_{2}^{(1)}$ $e_{2}^{(2)}$ -------- ------------ --------------- --------------- $1/10$ $1/10$ $11.74\%$ $1.61\%$ $1/10$ $1/20$ $9.12\%$ $6.13\%$ $1/10$ $1/40$ $8.05\%$ $7.27\%$ : Error comparison for Case 2. # Conclusions In this paper, we propose a general framework for multicontinuum homogenization. The method introduces several macroscopic variables at each macroscale point using characteristic functions associated with subdomains. The homogenization expansion is written using macroscale variables and associated local cell problems. The local cell problems are formulated as constraint problems in oversampled regions. A decay of local cell solutions is needed for accurate approximations. This is not an easy task, in general, since the constraints are formulated in a spatially localized fashion. We present an example of a mixed formulation of the elliptic equation, where we use some special formulations for cell problems. The proposed general framework shows that one can obtain various macroscale equations. We briefly discuss the relation to mixture theories. [^1]: Department of Mathematics, The Chinese University of Hong Kong, Sha Tin, Hong Kong [^2]: Department of Mathematics, Texas A&M University, College Station, TX 77843, USA [^3]: Departamento de Matemáticas, Universidad Nacional de Colombia, Carrera 45 No. 26-85, Edificio Uriel Gutiérrez, Bogotá D.C., Colombia [^4]: Department of Mathematics, City University of Hong Kong, Hong Kong

arxiv_math

--- abstract: | We show the convolution equivalence property of univariate tempered stable distributions in the sense of [@ros:07]. This makes rigorous various classic heuristic arguments on the asymptotic similarity between the probability and Lévy densities of such distributions. Some specific examples from the literature are discussed. author: - "Lorenzo Torricelli[^1]" bibliography: - Bibliography.bib title: "On the convolution equivalence of tempered stable distributions on the real line[^2]" --- : Tempered stable distributions, convolution equivalence, subexponentiality, long tails, heavy tails. : 60E07 # Tempered stable distributions and tail properties We discuss here some tail properties of a certain infinitely divisible (i.d.) distribution class on the real line called tempered stable distributions. For a distribution $\mu$ on $\mathbb R$ denote by $F$ (or $F_\mu$ if necessary) its cumulative distribution function (c.d.f.) $F(x)=\mu((-\infty,x])$ and by $\overline F(x)=1-F(x)=\mu([x, \infty))$ (resp. $\overline {F_\mu}$) its survival function (s.f.). Let further $\hat \mu(z)=\int_{(-\infty,\infty)} e^{zx}\mu(dx)$ for all $z \in \mathbb R$ such that the integral converges, be the moment generating function (m.g.f.) of $\mu$, i.e. $\hat \mu(z) = \mathscr{L}(-z;\mu)$, where $\mathscr{L}(\cdot;\mu)$ indicates the Laplace transform of the measure $\mu$. Saying that $\mu$ is *infinitely divisible* (i.d.) amounts to the property that its characteristic function (c.f.) has the so-called Lévy-Khintchine representation $$\label{eq:LK} \int_{-\infty}^\infty e^{izx }\mu(dx)= \exp\left(i z b - z^2 \sigma /2+\int_{\mathbb R} \left( e^{i z x} -1 - i z x\mathds{1}_{\{| x |<1\}}\right)\nu( d x)\right), \qquad z \in \mathbb R$$ with $b \in \mathbb R$, $\sigma>0$ and $\nu$ is a measure on $\mathbb R$ such that $\nu(\{0 \})=0$, $\int_{\mathbb R} (| x |^2 \wedge 1) dx <\infty$, called the Lévy measure. With abuse of notation, if $\nu$ is absolutely continuous we indicate a Lévy density using the same expression as the corresponding measure, and always understand $\nu(\{0 \})=0$. Classically, "tempering" means tilting the c.d.f. or the probability density function (p.d.f.) of a distribution by multiplication with an exponential of negative argument. In their pioneering works, @man+sta:95 and @kop:95 applied tempering to Lévy measures as well. In particular, the approach followed by the latter of applying an exponential smoothing factor to the stable Lévy density proved to be particularly consequential, since it solves the the infinite variance issue of stable random walk models in physics, while retaining analytical expressions for the c.f.s. It was later established in @ros:07 that such approach could be extended from exponential to general completely monotone tempering functions, and that multivariate versions of the procedure can be envisaged as well. We recall that a *completely monotone* (c.m.) function $f: [0,\infty) \rightarrow \mathbb R$ is an infinitely-differentiable function whose derivatives satisfy $$(-1)^{n}f^{(n)}(x) \geq 0, \qquad x \geq 0, n \in \mathbb N.$$ An important theorem by S.N. Bernstein states that $f$ is c.m. if and only if $$\label{eq:bernst} f(x)=\mathscr{L}(x;\eta)$$ for some positively-supported positive Borel measure $\eta$. Let $q_+,q_-:[0,\infty) \rightarrow \mathbb R$ be c.m. functions with corresponding Bernstein representing measures $Q_\pm$ in [\[eq:bernst\]](#eq:bernst){reference-type="eqref" reference="eq:bernst"} such that $\lim_{x \rightarrow \infty}q_\pm(x)=0$ and let $$\label{eq:rosrep} \nu(x)=\delta_+\frac{q_+(x)}{x^{1+\alpha}}\mathds{1}_{\left\{x>0\right\}}+\delta_-\frac{q_-(|x|)}{x^{1+\alpha}}\mathds{1}_{\left\{x<0\right\}}, \qquad \delta_+, \delta_- \geq 0, \, \alpha \in (0,2).$$ A *tempered $\alpha$-stable* (TS$_\alpha$) law $\mu$ on $\mathbb R$ in the sense of @ros:07 is an i.d. distribution with Lévy triplet $(b, 0, \nu)$ with $\nu$ given by [\[eq:rosrep\]](#eq:rosrep){reference-type="eqref" reference="eq:rosrep"}. If additionally $q_\pm(0)=1$ then we say that $\mu$ is *proper*. In other words a proper TS$_\alpha$ law is one for which $q_\pm$ are probability s.f.s of some probability distribution on $\mathbb R_+$, and the corresponding Bernstein representing measures $Q_\pm$ in [\[eq:bernst\]](#eq:bernst){reference-type="eqref" reference="eq:bernst"} are positive p.d.f.s.. In the applied literature it is often stated or implied (e.g. @kop:95) that asymptotically the tails of i.d. p.d.f.s "behave as" those of their Lévy densities. A first indication of some kind of similarity between $\overline \mu(x)$ and those of $\overline \nu(x)$ for large $x$ is provided by the fact that absolute/exponential moments of $\mu$ are finite if and only if those of $\nu\mathds{1}_{\{x>1\}}/\bar \nu(1)$ are (e.g. @sat:99, Corollary 25.8). For instance if $q_\pm(x)=e^{-\lambda x}$ , it follows from [\[eq:rosrep\]](#eq:rosrep){reference-type="eqref" reference="eq:rosrep"} that exponential moments of order $\theta< \lambda$ exist, and thus all moments also do. In applications, this is precisely the idea motivating Lévy tempering as a solution for resolving issues related to non-existence of moments. It is also known that (e.g. @wat+yam:10) that the tails of probability densities of an i.d. correspond to those of the Lévy densities in a weak sense i.e. there exist $c,C>0$ such that for all $\delta \in (0,1)$ there exists $x_\delta>0$ such that $$c(1-\delta) \overline \nu(x) < \overline F(x) < C(1+\delta) \overline \nu(x)$$ for all $x>x_\delta$ provided that the ratio $\overline F(x)/\overline \nu(x)$ stays bounded. However in principle $c \neq C$, and the stronger statement $\overline F(x) \sim D \, \overline \nu(x)$, $D=c=C$ does not need to hold. A known sufficient condition for this asymptotic equivalence to hold is the so called convolution equivalence property of the given probability distribution. For $\gamma \geq 0$, a distribution $\mu$ on $\mathbb R$ is said to belong to the class $\mathcal L(\gamma)$ if for all for all $x,y \in \mathbb R$ we have $\overline F(x)>0$ and $$\frac{\overline {F}(x+y)}{ \overline F(x)}=e^{- y \gamma}, \qquad x \rightarrow + \infty .$$ It is instead said to be *convolution equivalent*, or of class $\mathcal S(\gamma)$, if for some $\gamma$ we have $\mu \in \mathcal L(\gamma)$ and $$\label{eq:yammain} \lim_{x\rightarrow \infty}\frac{\overline{F*F}(x)}{\overline{F}(x)}= 2 \hat \mu ( \gamma)<\infty$$ (i.e. @wat:08, Equations (1.1) and (1.2), @wat+yam:10, Definition 1.1). In general it holds $\mathcal S(\gamma) \subsetneq \mathcal L(\gamma)$ for all $\gamma\geq 0$. For example an Exp$(\lambda)$ distribution belongs to $\mathcal L(\lambda)$ but not to $\mathcal S(\lambda)$ since $\hat \mu(\lambda)=\infty$ and the limit [\[eq:yammain\]](#eq:yammain){reference-type="eqref" reference="eq:yammain"} is infinite. For $\gamma=0$, examples in $\mathcal L(0) \setminus \mathcal S(0)$ are given in @pitman1980subexponential and @embrechts1980closure. The class $\mathcal L(0)$ is called that of the *long-tailed* distributions and $\mathcal S(0)$ that of the *subexponential* distributions. It is easy to show that a long-tailed distribution is a particular instance of a heavy-tailed distribution -- the latter meaning that no exponential moment exists -- but these two classes do not coincide. Qualitatively, subexponentiality amounts to the property that the sum of two independent random variables distributes asymptotically as their maximum. For full details and properties the classes $\mathcal L(\gamma)$ and $\mathcal S(\gamma)$ subexponentiality, long-tailedness, related concepts and applications we refer the reader to @embrechts1980closure, @embrechts1982convolution, @embrechts2013modelling, @cline1986convolution, @pakes2004convolution, @wat:08, @wat+yam:10 and references therein. We show in this note that TS$_\alpha$ laws are convolution equivalent. The consequence for applications is that their probability and Lévy densities are asymptotically equivalent, and the proportionality constant can be explicitly determined, so long as the characteristic function of the model is known. # Convolution equivalence of tempered stable distributions {#sec:dens} The following Lemma looks natural, but has seemingly not appeared before. **Lemma 1**. *Let $\mu$ be a probability distribution on $\mathbb R_+$ such that $\overline {F_\mu}$ is completely monotone. Then $\mu \in \mathcal L(\gamma)$ with $\gamma = \inf_{[0,\infty)} \mbox{{\upshape supp}}(\eta)$ where $\eta$ is the Bernstein representing probability measure of $\overline{F_\mu}$. In particular $\mu \in \mathcal L(0)$ if and only if $0 \in \mbox{{\upshape supp}}(\eta)$* *Proof.* We drop the subscript from the s.f.. Assume $0 \notin \mbox{supp}(\eta)$. Then by definition supp$(\eta) \cap [0,\gamma) =\emptyset$ but supp$(\eta) \cap [0,\gamma+\epsilon) \neq \emptyset$ for all $\epsilon>0$. The shifted measure measure $\eta^\gamma$ given by $\eta^\gamma([0,x))=\eta([\gamma, x+\gamma))$, $x \geq 0$ is then again a Borel measure on $\mathbb R_+$ with $0 \in$ supp$(\eta^\gamma)$, and we have by the usual properties of the Laplace transform that ${F^\gamma}(x)=e^{\gamma x}\overline F(x)$ and ${F^\gamma}(x): =\mathscr{L}(x; \eta^\gamma )$. Observe that $F^\gamma$ does not need to be a s.f. (unless $\gamma=0$); in any case, to prove the claim it is enough to show that $$\alpha(x;y):= \frac{ F^\gamma(x)}{F^\gamma(x+y)} \rightarrow 1, \qquad x \rightarrow \infty.$$ Let $h>0$ be fixed and set $I_0(x)=\int_{[0,h)}e^{-s x} \eta^\gamma(ds)$, $I_\infty(x)=\int_{[h,\infty)}e^{-s x} \eta^\gamma(ds)$. We have $$\label{eq:alphaineq} \alpha(x;y) <\frac{I_0(x)}{I_{0}(x+y)}\left(1 +\frac{I_\infty(x)}{I_{0}(x)}\right).$$ Now notice that $$I_0(x) \geq \int_{ [0,h/2)} e^{-s x} \eta^\gamma(ds) > e^{-x h/2}\eta^\gamma( [0, h/2)) := c e^{-x h/2}$$ and $c>0$ because $0\in$ supp$(\eta^\gamma)$. Furthermore $$I_\infty(x) < e^{-x h}\eta^\gamma( [h,\infty)) <e^{-x h}$$ since $Q(\mathbb R_+)=q(0)=1$, so that $$\label{eq:eps} \frac{I_\infty(x)}{I_{0}(x)} < \frac{e^{-xh/2}}{c}:=\epsilon(x).$$ Moreover $$\label{eq:Iest} I_0(x+y) > e^{-y h} I_0(x)$$ and using [\[eq:eps\]](#eq:eps){reference-type="eqref" reference="eq:eps"} and [\[eq:Iest\]](#eq:Iest){reference-type="eqref" reference="eq:Iest"} in [\[eq:alphaineq\]](#eq:alphaineq){reference-type="eqref" reference="eq:alphaineq"} we obtain $$\alpha(x;y) < e^{y h} (1 + \epsilon(x))$$ for all $h > 0$. Hence for all $x, y \geq 0$ $$\alpha(x;y) \leq 1 + \epsilon(x) \rightarrow 1, \qquad x \rightarrow \infty.$$ and therefore $\limsup_{x\rightarrow \infty} \alpha(x;y) \leq 1$. But on the other hand $\alpha(x ; y) \geq 1$ for all $y \geq 0$ since $F^\gamma$ is nonincreasing, so we conclude $\alpha(x;y) \rightarrow 1$ as $x\rightarrow \infty$ for all $y \geq 0$ as claimed. ◻ We can now prove convolution equivalence of TS$_\alpha$ distributions, and explicitly identify the tail parameter. The following Theorem generalizes @kuc+tap:08, Lemmas 7.3 to 7.5, to arbitrary stability parameters and tempering functions. **Theorem 1**. *Let $\mu$ be a proper TS$_\alpha$ distribution. We have $\mu \in \mathcal S(\gamma)$ where $\gamma = \inf_{[0,\infty)} \mbox{{\upshape supp}}(Q_+)$.* *Proof.* We drop the $+$ subscripts. If $\nu$ is the Lévy measure of $\mu$, let $\overline \nu(x)=\nu([x, \infty))$, $x \in \mathbb R$. For $x \in \mathbb R$ define the probability density $$\label{eq:normlaw} \nu_1(x):=\frac{ \nu(x) }{\overline \nu(1)}\mathds{1}_{\{x>1\}}=\frac{\delta}{\overline \nu(1)} \frac{q(x)}{ x^{1+\alpha}} \mathds{1}_{\{x>1\}}.$$ According to @wat:08, Theorem B (whose original statement is due to @pakes2004convolution), we know $\mu \in \mathcal S(\gamma)$ if and only if $\nu_1 \in \mathcal S(\gamma)$, and we will prove the latter. First we have to show $\nu_1 \in \mathcal L(\gamma)$. Reasoning on the densities, for all $y >0$ it holds $$\begin{aligned} \label{eq:longtailedTGS} & \lim_{x \rightarrow \infty} \frac{\overline{ \nu_1}(x+y)}{\overline{ \nu_1}(x)}= \lim_{x\rightarrow \infty}\left( \frac{x}{x+y} \right)^{1+\alpha} \frac{q(x+y)} {q(x)}=e^{-\gamma y}\end{aligned}$$ having used Lemma [Lemma 1](#lemma){reference-type="ref" reference="lemma"} on the probability distribution $\epsilon$ induced by the c.m. function $q$, which is a positive s.f. being $\mu$ proper. Next, interpreting the convolution in [\[eq:yammain\]](#eq:yammain){reference-type="eqref" reference="eq:yammain"} on the Lévy densities we obtain $$\begin{aligned} \label{tailslim} \lim_{x\rightarrow \infty}\frac{\overline{\nu_1* \nu_1}(x)}{\overline{\nu_1}(x)}&= \lim_{x\rightarrow \infty} \frac{1}{\nu_1(x)} \displaystyle{ \int_0^x \nu_1(x-z)\nu_1(z) dz} \nonumber \\ &=\frac{1}{\overline \nu(1)}\lim_{x\rightarrow \infty} \frac{1}{\nu_1(x)} \displaystyle{ \int_0^{x}\mathds{1}_{\{z\geq 1\}}\mathds{1}_{\{ x-z \geq 1 \}} \nu(x-z)\nu(z) dz}\nonumber \\ &= \frac{ \delta}{\overline \nu(1)} \lim_{x\rightarrow \infty}\int_{1}^{x-1} \left(\frac{x}{(x-z)z}\right)^{1+\alpha} \frac{q(x-z)q(z)}{q(x)} dz .\end{aligned}$$ We can show that for any given $0 \leq z<x$ the ratio $$r(x;z)=\frac{q (x-z)}{q(x)}$$ is decreasing in $x$. To establish this, observe that from the quotient rule it follows $$\frac{d}{dx} \frac{ q (x-z)}{q(x)} =\frac{q(x)q'(x-z)-q( x)q'(x-z)}{q( x)^2}.$$ Such an expression is negative if and only if $$\frac{q' (x-z)}{q (x-z)} \leq \frac{q' (x)}{q(x)}, \qquad z \geq 0, \, x > z$$ which is implied by $$\label{eq:logconv} \frac{d}{dx}\log q(x) \geq 0, \qquad x>0.$$ But $q$ is c.m., and thus also log-convex (@ste+vh:03, A.3, Proposition 3.8) and hence [\[eq:logconv\]](#eq:logconv){reference-type="eqref" reference="eq:logconv"} holds true. Therefore, for all $z \geq 0$ and $x > z$, there is a maximum for $r(x;z)$ in $x=z$ and we further have $r(x;z) \leq 1$, for all $z$. Combining this bound with the fact that the integrand in the last line of [\[tailslim\]](#tailslim){reference-type="eqref" reference="tailslim"} is symmetric in $z$ and $x-z$ about $x/2$ leads to $$\begin{aligned} \label{sim}\int_{1}^{x-1} \left(\frac{x}{(x-z)z}\right)^{1+\alpha} \frac{q(x-z)q(z)}{q(x)} dz = &2 \int_{1}^{x/2} \left(\frac{x}{(x-z)z}\right)^{1+\alpha} \frac{q(x-z)q(z)}{q(x)} dz \nonumber \\ & < 2 \int_1^{\infty}\left( \frac{x} {(x-z)z}\right)^{1+\alpha} q(z)\mathds{1}_{\{1 \leq z \leq x/2\}} dz. \end{aligned}$$ Now for $z \in (1,x/2)$ we have $x/(x-z) \leq 2$, so that $$\left( \frac{x} {(x-z)z}\right)^{1+\alpha}\mathds{1}_{\{1 \leq z \leq x/2 \} }q(z) < \left(\frac{2} {z}\right)^{1+\alpha}q(z), \qquad z \geq 0, x >0$$ and the last expression is integrable for all $\alpha \in (0,2)$. We can therefore apply the dominated convergence theorem and continue [\[tailslim\]](#tailslim){reference-type="eqref" reference="tailslim"} by taking the limit inside the integral. Furthermore $q(x-z) \sim e^{\gamma z}q(x)$ from Lemma [Lemma 1](#lemma){reference-type="ref" reference="lemma"}, so that $$\begin{aligned} \label{eq:final} \lim_{x \rightarrow \infty}&\frac{\overline{\nu_1*\nu_1}(x)}{\overline{\nu_1}(x)}=\frac{2 \delta}{\overline \nu(1)} \int_1^{\infty}\lim_{x\rightarrow \infty} \left( \frac{x}{(x-z)z}\right)^{1+\alpha} \frac{q(x-z) q( z) }{q( x)}\mathds{1}_{\{1 \leq z \leq x/ 2 \} } dz \nonumber \\&= \frac{2 \delta}{\overline \nu(1)} \int_1^\infty \frac{e^{\gamma z} q(z) }{z^{1+\alpha}} dz= 2 \hat {{\nu_1}}( \gamma). \end{aligned}$$ It remains to be shown that this last expression is finite whatever the value of $\alpha$, which happens if and only if $q^\gamma(z):=e^{\gamma z}q(z)=O(1)$. But this follows since $q^\gamma(z)=\mathscr {L}(z, Q^\gamma)$, where $Q^\gamma$ is the full-support shift of $Q$, i.e. $Q^{\gamma}([0,x))=Q([\gamma, x+\gamma))$, $x \geq 0$. Therefore $q^\gamma$ is completely monotone and hence nonincreasing. This completes the proof that $\nu_1 \in \mathcal S(\gamma)$, and hence that $\mu \in \mathcal S(\gamma)$. ◻ Convolution equivalence is a sufficient condition to establish that the Lévy tail function and the s.f. are of the same order at infinity, from which we can deduce the asymptotics of the tail of a TS$_\alpha$ p.d.f.. Furthermore, the explicit proportionality constant is known. **Corollary 1**. *Let $\mu$ be a proper TS$_\alpha$ distribution and let $\gamma_\pm=\inf_{[0,\infty)} \mbox{\upshape supp} (Q_\pm)$ where $Q_\pm$ are the Bernstein representing measures of $q_\pm$. Then the p.d.f. $p(x)$ of $\mu$ satisfies $$\label{eq:pasympt} p(x) \sim \delta_\pm \hat \mu^\pm_q( \gamma_\pm) \frac{q_\pm(|x|)}{|x|^{1+\alpha}}, \qquad x \rightarrow \pm \infty.$$ where $\mu^\pm_q$ are the probability laws with s.f. $q_\pm$.* *Proof.* When $x \rightarrow \infty$ the claim follows by Theorem [Theorem 1](#prop:tailsinfty){reference-type="ref" reference="prop:tailsinfty"}, together with Theorem B in @wat:08, and differentiation of the s.f.s. The case $x \rightarrow -\infty$ is obtained considering the TS$_\alpha$ distribution with s.f. $F_\mu(-x)$, whose Lévy measure is obtained by interchanging $q_+$ with $q_-$, and $\delta_+$ with $\delta_-$ in [\[eq:rosrep\]](#eq:rosrep){reference-type="eqref" reference="eq:rosrep"}, and applying again Theorem [Theorem 1](#prop:tailsinfty){reference-type="ref" reference="prop:tailsinfty"}. ◻ The key counterexample is a $\Gamma(a,b)$ distribution with shape $a$ and rate $b$. Then it is known that its Lévy measure is of the form [\[eq:rosrep\]](#eq:rosrep){reference-type="eqref" reference="eq:rosrep"} with $q_+(x)=e^{-b x}$, $\delta_+=a$, $\delta_-=0$, but with $\alpha=0$. Hence a Gamma law is not a TS$_\alpha$ law. In fact, Gamma laws are not convolution equivalent: accordingly $b^{-a} x^{a-1}e^{-bx} \Gamma(a)^{-1} \not \sim a C x^{-1}e^{-bx}$, $x, C,a, b>0$. We discuss some explicit application of Corollary [Corollary 1](#cor:tailGTGS){reference-type="ref" reference="cor:tailGTGS"} in the last Section. # Some examples For many TS$_\alpha$ laws in the literature $\hat \mu_q^\pm$ can be determined analytically, which in light of Corollary [Corollary 1](#cor:tailGTGS){reference-type="ref" reference="cor:tailGTGS"} makes it possible to determine explicit leading orders of the probability density tails. The most popular TS$_\alpha$ model is the one with exponential tempering, i.e. where $q_\pm(x)=e^{-\theta_\pm x}\mathds{1}_{\{x>0 \}}$, $\theta_\pm >0$. This idea was originally suggested by @kop:95 and found widespread application in finance, e.g. @boy+lev:00 and @car+al:02. A thorough analytical investigation -- limited to the case $\alpha \in (0,1)$ -- is found in @kuc+tap:13, which we refer to for more details. In this case $\mu^\pm_q$ are exponential distributions of parameter $\theta_\pm$ and $\mu^\pm_q \in \mathcal L(\theta_\pm).$ The values of the m.g.f.s $\hat {\mu}^\pm_q$ are well-known in the literature and take different forms in the case $\alpha=1$, and $\alpha\neq1$ (@con+tan:03). From [\[eq:pasympt\]](#eq:pasympt){reference-type="eqref" reference="eq:pasympt"} we see that $p(x) \sim C_\pm \delta_\pm e^{-\theta_\pm x}/x^{1+\alpha}$, $x \rightarrow \pm \infty$ where $C_\pm=\hat \mu^\pm_q(\theta_\pm)$. This analysis recovers @kuc+tap:13, Theorem 7.10. In order to show such result the authors did prove Theorem [Theorem 1](#prop:tailsinfty){reference-type="ref" reference="prop:tailsinfty"} in the simplest case exponential functions for $q_\pm$. Another example is the KR TS$_\alpha$ law based on upper gamma-incomplete tempering introduced in @kim+al:08. The authors set (in an equivalent formulation) $$\label{eq:KRtemp} q_\pm(x)= (\alpha+p_\pm) \Gamma(-\alpha-p_\pm) \Gamma^*\left( -\alpha - p_\pm, \frac{x}{r_\pm} \right)\mathds{1}_{\{x>0 \}}, \quad p_\pm >-\alpha, \, r_\pm >0$$ where $$\label{eq:gammastar}\Gamma^*(s,x)=\frac{x^{-s}}{\Gamma(s)} \int_x^\infty e^{- t} t^{s-1}dt=\frac{1}{\Gamma(s)}\int_{1}^\infty e^{- x t} t^{s-1}dt$$ is the modified upper incomplete Gamma function. Of course $\Gamma(s,x) \rightarrow 0$, for all $s$ and $x \rightarrow \infty$. Moreover, from the series representation of the lower incomplete gamma function $\gamma(s,x)$ (@abr+ste:64, 6.5.29) and $\gamma(s,x)+\Gamma(s,x)=\Gamma(s)$ it follows $\Gamma(s,x)x^{-s} \rightarrow -1/s$, $s <0$, $x \rightarrow 0$, so that the KR law is proper. Furthermore on the right hand side of [\[eq:gammastar\]](#eq:gammastar){reference-type="eqref" reference="eq:gammastar"} we can apply Theorem 4 of @mil+sam:01, to establish complete monotonicity. Then, from $\Gamma(s,x)\sim e^{-x}x^{s-1}$, $x \rightarrow \infty$ (@abr+ste:64, 6.5.32) applied to [\[eq:KRtemp\]](#eq:KRtemp){reference-type="eqref" reference="eq:KRtemp"} we obtain $$\frac{ q_\pm(x+y)}{q_\pm(x)} \sim \left(\frac{x+y}{x}\right)^{\gamma_\pm} e^{-y/r_\pm} \rightarrow e^{-y/r_\pm}$$ as $x \rightarrow \infty$ and thus $\mu^\pm_q \in \mathcal L(1/r_\pm).$ Applying Corollary [Corollary 1](#cor:tailGTGS){reference-type="ref" reference="cor:tailGTGS"} then shows that the KR TS$_\alpha$ p.d.f. tails differ from those of exponential tempering only by a constant, and more precisely we have $p(x) \sim \delta_\pm r_\pm^\alpha C_\pm e^{-x/r_\pm}/x^{1+\alpha}$. where $C_\pm=\hat \mu^\pm_q(1/r_\pm)$ can be explicitly calculated from from @kim+al:08, Theorem 3.4. Finally a recent TS$_\alpha$ suggestion in @GTGS takes into account the following tempering functions $$q_\pm(x)=e^{-\theta_\pm x}E_{\gamma_\pm}(-\lambda_\pm x^{\gamma_\pm})\mathds{1}_{\{x>0 \}}, \qquad \lambda_\pm >0, \theta_\pm \geq 0, \gamma_\pm \in (0,1).$$ Here $$E_\gamma(z)=\sum_{k=0}^\infty\frac{z^{k}}{\Gamma(k+\gamma)}, \qquad z \in \mathbb C, \quad \gamma \geq 0$$ is the so-called Mittag-Leffler function, a type of heavy-tailed exponential which is ubiquitous in fractional calculus. It is known that $E_\gamma(-x)$ for $\gamma \in [0,1]$ is c.m. (@pol:48). The functions $q_\pm$ are then c.m. being product of c.m. functions, one of which attained as composition of a c.m. function with a third one having c.m. derivative. That $q(0)=1, q(\infty)=0$ is clear. The c.f. of such laws is given in @GTGS, Section 3, Theorems 3 and 4. Furthermore, it is well-known (e.g. @hau+al:11) that $$\begin{aligned} \label{eq:MLasymptInf} E_\gamma(- x^\gamma) \sim &\frac{x^{-\gamma}}{\Gamma(1-\gamma) }, \quad x \rightarrow \infty, \qquad \gamma \in (0,1). \end{aligned}$$ From the above it follows that for all $y>0$ $$\frac{q_\pm(x+y)}{q_\pm(x)}\sim e^{-\theta_\pm y} \left(\frac{x}{x+y}\right)^\gamma \rightarrow e^{-\theta_\pm y}, \qquad x \rightarrow \infty,$$ so that $\mu^\pm_q \in \mathcal L(\theta_\pm)$. The tilted Mittag-Leffler laws $\mu^\pm_q$ are discussed in @tor+al:21. Combining [\[eq:MLasymptInf\]](#eq:MLasymptInf){reference-type="eqref" reference="eq:MLasymptInf"} with [\[eq:pasympt\]](#eq:pasympt){reference-type="eqref" reference="eq:pasympt"} we are led to the asymptotics $$\label{eq:infttail} p(x) \sim C_\pm \frac{\delta_\pm}{\lambda_\pm \Gamma(1-\gamma_\pm)} \frac{e^{-\theta_\pm x}}{x^{1+\gamma_\pm+\alpha}}, \qquad x \rightarrow \pm \infty.$$ If $\theta_\pm>0$, then $C_\pm=\hat \mu^\pm_q( \theta_\pm)$ is given by @GTGS, Theorem 3, whereas if $\theta_\pm=0$ we have $C_\pm=1$ . The conclusion is that mixed exponential/Mittag-Leffler tempered distributions, termed *generalized tempered geometric* stable (GTGS) in @GTGS -- because the radial part of the Lévy measure extends that of a geometric tempered (or "tempered Linnik") distribution @bar+al:16a -- retain exponential tails if and only if $\theta_\pm>0$. Instead when $\theta_\pm=0$ we see that $\mu$ is heavy (and long)-tailed, which is unlike the previous two examples. In particular the expectation and variance are finite if and only if respectively $\alpha+\gamma_\pm >1$ and $\alpha+\gamma_\pm >2$. In conclusion GTGS laws capture power law tails while retaining analytical expressions, which could be useful for e.g. analyzing financial data. See the discussion in @GTGS for further details. [^1]: University of Bologna, Department of Statistical Sciences "P. Fortunati". Email: lorenzo.torricelli2\@unibo.it [^2]: The author gratefully acknowledges the comments of Claudia Klüppelberg and Iosif Pinelis, and the contribution to Lemma [\[lemma\]](#lemma){reference-type="ref" reference="lemma"} of Iosif Pinelis. They are not responsible of any errors.

arxiv_math

--- abstract: | We study the natural representation of the topological full group of an ample Hausdorff groupoid into the groupoid's complex Steinberg algebra. We characterise precisely when this representation is injective and show that it is rarely surjective. We then restrict our attention to discrete groupoids, which provide unexpected insight into the behaviour of the extension of the representation of the topological full group into the full and reduced groupoid C\*-algebras. We show that the extension into the full groupoid C\*-algebra is not surjective unless the groupoid is a group, and we provide an example showing that the extension may still surject onto the reduced groupoid C\*-algebra even when the groupoid is not a group. address: - School of Mathematics and Statistics, Victoria University of Wellington, PO Box 600, Wellington 6140, NEW ZEALAND - Department of Mathematical Sciences, University of Delaware, Newark, DE, 19716, USA - Research Institute of Mathematics, Seoul National University, Seoul 08826, KOREA - Department of Mathematics and Statistics, University of Windsor, Windsor, ON N9B 3P4, CANADA author: - Becky Armstrong - Lisa Orloff Clark - Mahya Ghandehari - Eun Ji Kang - Dilian Yang bibliography: - references.bib title: Representing topological full groups in Steinberg algebras and C\*-algebras --- Topological full groups of ample Hausdorff groupoids were introduced by Matui [@Matui2012] as a generalisation of the topological full groups studied by Giordano, Putnam, and Skau in the context of Cantor minimal systems [@GPS1999]. Matui showed in [@Matui2015 Theorem 3.10] that for any two minimal effective Hausdorff étale groupoids whose unit spaces are Cantor sets, the groupoids are isomorphic if and only if their topological full groups are isomorphic. This is equivalent to there being a diagonal-preserving isomorphism of the Steinberg algebras of the groupoids; see [@ABHS2017 Theorem 3.1]. It is therefore clear that there are strong connections between the topological full groups and Steinberg algebras of ample Hausdorff groupoids. In addition to being a groupoid invariant, topological full groups have enticing connections to some infamous open questions. For example, they give presentations of Thompson's groups [@LV2020; @MM2017; @Matui2015; @Yang2022], and have already been used to solve several important problems in group theory; see [@BHM2022; @JM2013; @JNdlS2016; @Nek2018; @SWZ2019]. Recent results also reveal interesting connections between topological full groups and the elusive simplicity problem for group C\*-algebras; see [@BS2019; @LBMB2018; @Scarparo2023]. It is this latter problem that motivates our study. For every ample Hausdorff groupoid $\mathcal{G}$ with compact unit space, there are natural representations of the topological full group of $\mathcal{G}$ in the complex Steinberg algebra of $\mathcal{G}$ and in the full and reduced C\*-algebras of $\mathcal{G}$. It is known that these representations often fail to be injective. We make this statement precise by showing that the representation of the topological full group taking values in the Steinberg algebra of the groupoid is almost never injective. In particular, we show in [Theorem 4](#thm: main){reference-type="ref" reference="thm: main"} that injectivity fails when 1. the groupoid is all isotropy and has at least $2$ nontrivial isotropy groups; or 2. the groupoid is not all isotropy and has at least $3$ non-unit elements. We then show that this representation is almost never surjective as a map into the complex Steinberg algebra. In fact, we show in [Corollary 7](#cor: surj iff group){reference-type="ref" reference="cor: surj iff group"} that the representation is surjective onto the Steinberg algebra if and only if $\mathcal{G}$ is a group. However, strangely, the extensions of this representation of the topological full group into the full or reduced groupoid C\*-algebras may still be surjective. For example, surjectivity holds for the representation of the topological full group associated to the Cuntz groupoid (that is, the boundary-path groupoid of the directed graph with a single vertex and two edges) into the Cuntz algebra $\mathcal{O}_2$; see [@BS2019 Remark 4.7] and [@HO2017 Proposition 5.3]. provides another such example. Our proof techniques for the results in [\[sec: injectivity,sec: surjectivity\]](#sec: injectivity,sec: surjectivity){reference-type="ref" reference="sec: injectivity,sec: surjectivity"} were developed by first considering these questions for discrete groupoids. The arguments in our proof of [Theorem 4](#thm: main){reference-type="ref" reference="thm: main"} in particular are quite combinatorial in nature. In [\[sec: discrete groupoids\]](#sec: discrete groupoids){reference-type="ref" reference="sec: discrete groupoids"} we demonstrate that surprising things can happen in the setting of discrete groupoids. In [Theorem 11](#thm: discrete full iff group){reference-type="ref" reference="thm: discrete full iff group"}, we show that the image of the representation of the topological full group of a discrete groupoid with finite unit space is dense in the full groupoid C\*-algebra if and only if the groupoid is a group. (Note that the Cuntz groupoid mentioned above is not discrete, and thus this result does not hold for ample Hausdorff groupoids in general; see [Remark 12](#rem: Cuntz groupoid){reference-type="ref" reference="rem: Cuntz groupoid"}.) In [Example 13](#eg: F_2 sqcup F_2){reference-type="ref" reference="eg: F_2 sqcup F_2"} we demonstrate that it is possible for the image of the representation of the topological full group of a discrete groupoid to be dense in the reduced groupoid C\*-algebra even when the groupoid is not a group. Finally, in [Corollary 14](#cor: isomorphism){reference-type="ref" reference="cor: isomorphism"} we combine our results from [\[sec: injectivity,sec: surjectivity,sec: discrete groupoids\]](#sec: injectivity,sec: surjectivity,sec: discrete groupoids){reference-type="ref" reference="sec: injectivity,sec: surjectivity,sec: discrete groupoids"} to show that the representation of the topological full group of an ample Hausdorff groupoid $\mathcal{G}$ with compact unit space is an isomorphism into the Steinberg algebra of $\mathcal{G}$ if and only if $\mathcal{G}$ is a group, and that when $\mathcal{G}$ is discrete, the same result holds for the extension of this representation to the full C\*-algebra. ## Groupoids A [*groupoid*]{style="color: magenta"} $\mathcal{G}$ is a small category in which every morphism $\gamma \in \mathcal{G}$ has a unique inverse $\gamma^{-1} \in \mathcal{G}$. Throughout, we assume that all groupoids are nonempty. We define the [*range*]{style="color: magenta"} and [*source*]{style="color: magenta"} of each $\gamma \in \mathcal{G}$ by $r(\gamma) \coloneqq \gamma \gamma^{-1}$ and $s(\gamma) \coloneqq \gamma^{-1} \gamma$, respectively, where composition is read from right to left. We write $$\mathcal{G}^{(2)}= \{ (\alpha,\beta) \in \mathcal{G}\times \mathcal{G}\mid s(\alpha) = r(\beta) \}$$ for the set of [*composable pairs*]{style="color: magenta"} in $\mathcal{G}$, and we write $\mathcal{G}^{(0)}= r(\mathcal{G}) = s(\mathcal{G})$ for the [*unit space*]{style="color: magenta"} of $\mathcal{G}$. Note that a groupoid $\mathcal{G}$ is a group if and only if $\mathcal{G}^{(0)}$ is a singleton. A [*topological groupoid*]{style="color: magenta"} is a groupoid endowed with a topology under which composition and inversion are continuous. A [*Hausdorff groupoid*]{style="color: magenta"} is a topological groupoid with a locally compact Hausdorff topology. If $\mathcal{G}$ is a Hausdorff groupoid, then $\mathcal{G}^{(0)}$ is closed in $\mathcal{G}$. A topological groupoid $\mathcal{G}$ is [*étale*]{style="color: magenta"} if the range and source maps $r,s\colon \mathcal{G}\to \mathcal{G}^{(0)}$ are local homeomorphisms. A subset $B \subseteq \mathcal{G}$ is called a [*bisection*]{style="color: magenta"} of $\mathcal{G}$ if $r\ensuremath{\vert_{B}}$ and $s\ensuremath{\vert_{B}}$ are injective. If $B$ is an open bisection of an étale groupoid $\mathcal{G}$, then $r\ensuremath{\vert_{B}}$ and $s\ensuremath{\vert_{B}}$ are homeomorphisms onto open subsets of $\mathcal{G}^{(0)}$. Every étale groupoid has a basis consisting of open bisections; see [@Exel2008 Proposition 3.5]. We say that an étale groupoid is [*ample*]{style="color: magenta"} if it has a basis of *compact* open bisections. By [@Exel2010 Proposition 4.1], a Hausdorff étale groupoid is ample if and only if its unit space is totally disconnected. If $\mathcal{G}$ is an étale groupoid, then the unit space $\mathcal{G}^{(0)}$ is open in $\mathcal{G}$, and for all $u, v \in \mathcal{G}^{(0)}$, each of the sets $$\mathcal{G}^u \coloneqq r^{-1}(u), \ \mathcal{G}_v \coloneqq s^{-1}(v), \text{ and } \ \mathcal{G}_v^u \coloneqq \mathcal{G}^u \cap \mathcal{G}_v$$ is discrete with respect to the relative topology induced by $\mathcal{G}$. The [*isotropy group*]{style="color: magenta"} of a unit $u \in \mathcal{G}^{(0)}$ is the group $$\mathcal{G}^u_u = \{\gamma \in \mathcal{G}\mid r(\gamma) = s(\gamma) = u\},$$ and the [*isotropy subgroupoid*]{style="color: magenta"} of $\mathcal{G}$ is the collection $$\operatorname{Iso}(\mathcal{G}) \coloneqq \bigcup_{u \in \mathcal{G}^{(0)}} \, \mathcal{G}^u_u = \{ \gamma \in \mathcal{G}\mid r(\gamma) = s(\gamma) \}.$$ Let $\mathcal{G}$ be a Hausdorff étale groupoid. For each continuous function $f\colon \mathcal{G}\to \mathbb{C}$, we define $\operatorname{supp}(f) \coloneqq \overline{\{ \gamma \in \mathcal{G}: f(\gamma) \ne 0 \}}$. We write $C_c(\mathcal{G})$ for the collection of continuous compactly supported complex-valued functions on $\mathcal{G}$. This is a $*$-algebra with respect to the convolution product $$(f * g)(\gamma) = \sum_{\alpha\beta = \gamma} f(\alpha) g(\beta)$$ and $*$-involution $f^*(\gamma) = \overline{f(\gamma^{-1})}$ for $f,g \in C_c(\mathcal{G})$ and $\gamma \in \mathcal{G}$. Given a Hilbert space $\mathcal{H}$, we write $B(\mathcal{H})$ for the C\*-algebra of bounded linear operators on $\mathcal{H}$. The [*full groupoid C\*-algebra*]{style="color: magenta"} $C^*(\mathcal{G})$ is the completion of $C_c(\mathcal{G})$ with respect to the [*full C\*-norm*]{style="color: magenta"} $$\ensuremath{\norm{f}_{\mathrm{max}}} \coloneqq \sup\{ \norm{\pi(f)} \mid \pi\colon C_c(\mathcal{G}) \to B(\mathcal{H}) \text{ is a $*$-representation for some } \mathcal{H}\}.$$ For each $u \in \mathcal{G}^{(0)}$, there is a $*$-representation $\pi_u\colon C_c(\mathcal{G}) \to B(\ell^2(\mathcal{G}_u))$, called the [*regular representation*]{style="color: magenta"} of $C_c(\mathcal{G})$ associated to $u$, such that $$\pi_u(f) \delta_\gamma = \sum_{\alpha \in \mathcal{G}_{r(\gamma)}} f(\alpha) \delta_{\alpha\gamma} \quad \text{for } f \in C_c(\mathcal{G}) \text{ and } \gamma \in \mathcal{G}_u.$$ The [*reduced groupoid C\*-algebra*]{style="color: magenta"} $C_r^*(\mathcal{G})$ is the completion of $C_c(\mathcal{G})$ with respect to the [*reduced C\*-norm*]{style="color: magenta"} $$\ensuremath{\norm{f}_r} \coloneqq \sup\{ \norm{\pi_u(f)} \mid u \in \mathcal{G}^{(0)}\}.$$ See [@Renault1980 Chapter II] or [@Sims2020 Chapter 9] for details. The [*(complex) Steinberg algebra*]{style="color: magenta"} of an ample Hausdorff groupoid $\mathcal{G}$ is the collection $$\begin{aligned} A(\mathcal{G}) \coloneqq&~\operatorname{span}\{ 1_U\colon \mathcal{G}\to \mathbb{C}\mid U \text{ is a compact open bisection of } \mathcal{G}\} \\ =&~\{ f \in C_c(\mathcal{G}) \mid f \text{ is locally constant} \}\end{aligned}$$ equipped with the convolution product and $*$-involution defined above. If $\mathcal{G}$ is discrete, then $A(\mathcal{G}) = C_c(\mathcal{G})$. In general, $A(\mathcal{G})$ is dense in $C_c(\mathcal{G})$ with respect to both the full and reduced C\*-norms (see [@CFST2014 Proposition 4.2]), and for all $f \in A(\mathcal{G})$, we have $$\label{eqn: Steinberg *-reps are bounded} \ensuremath{\norm{f}_{\mathrm{max}}} \coloneqq \sup\{ \norm{\pi(f)} \mid \pi\colon A(\mathcal{G}) \to B(\mathcal{H}) \text{ is a $*$-representation for some } \mathcal{H}\}$$ (see [@CZ2022 Theorem 7.1]). Note that a discrete group $G$ may be viewed as an ample Hausdorff groupoid, and in this case the singletons in $G$ are all compact open bisections, and so the Steinberg algebra $A(G)$ is just the complex group ring $\mathbb{C}G$, which is generated by the point-mass functions $\delta_g \coloneqq 1_{\{g\}}$ for $g \in G$. See [@CFST2014; @Steinberg2010] for further details on Steinberg algebras. ## Topological full groups Let $\mathcal{G}$ be an ample groupoid with compact unit space $\mathcal{G}^{(0)}$. We write $B^\mathrm{co}(\mathcal{G})$ for the group of compact open bisections of $\mathcal{G}$, and we say that a bisection $B$ of $\mathcal{G}$ is [*full*]{style="color: magenta"} if $r(B) = s(B) = \mathcal{G}^{(0)}$. We define the [*topological full group*]{style="color: magenta"} of $\mathcal{G}$ to be the (discrete) group $$F(\mathcal{G}) \coloneqq \{ B \in B^\mathrm{co}(\mathcal{G})\mid B \text{ is full} \}$$ equipped with the operations $$AB \coloneqq \{ \alpha\beta \mid (\alpha, \beta) \in (A \times B) \cap \mathcal{G}^{(2)}\} \ \text{ and } \ B^{-1} \coloneqq \{ \gamma^{-1} \mid \gamma \in B \}$$ for all $A, B\in F(\mathcal{G})$. Note that if $\mathcal{G}$ is a discrete group, then $$F(\mathcal{G}) = B^\mathrm{co}(\mathcal{G})= \{ \{g\} \mid g \in \mathcal{G}\} \cong \mathcal{G}.$$ See [@Matui2017; @Nek2019] for further details on topological full groups. Let $\mathcal{G}$ be an ample Hausdorff groupoid with compact unit space. Given compact open bisections $U$ and $V$ of $\mathcal{G}$, we have $$1_U * 1_V = 1_{UV} \quad \text{ and } \quad (1_U)^* = 1_{U^{-1}}.$$ It follows that there is a $*$-homomorphism $\pi\colon \mathbb{C}F(\mathcal{G}) \to A(\mathcal{G})$ satisfying $\pi(\delta_U) = 1_U$, which we call the [*representation*]{style="color: magenta"} of $F(\mathcal{G})$ in $A(\mathcal{G})$. This representation is studied extensively in [@BS2019], as are the C\*-completions $\overline{\pi(\mathbb{C}F(\mathcal{G}))}^{\ensuremath{\norm{\cdot}_{\mathrm{max}}}}$ and $\overline{\pi(\mathbb{C}F(\mathcal{G}))}^{\ensuremath{\norm{\cdot}_r}}$. In this paper we investigate the necessary and sufficient conditions under which $\pi$ is injective and surjective. **Remark 1**. In [@NO2019 Definition 3.2] Nyland and Ortega define the topological full group of an (effective) ample Hausdorff groupoid with a unit space that is not necessarily compact. Since the Steinberg algebra of an ample Hausdorff groupoid $\mathcal{G}$ is unital (with unit $1_{\mathcal{G}^{(0)}}$) if and only if the unit space $\mathcal{G}^{(0)}$ is compact, it is impossible to represent the topological full group of $\mathcal{G}$ in $A(\mathcal{G})$ (or in $C^*(\mathcal{G})$ or $C_r^*(\mathcal{G})$) unless $\mathcal{G}^{(0)}$ is compact. It is for this reason that we restrict our attention in this paper to ample Hausdorff groupoids with compact unit space. [\[sec: injectivity\]]{#sec: injectivity label="sec: injectivity"} In this section we characterise precisely when the representation $\pi\colon \delta_U \mapsto 1_U$ of $\mathbb{C}F(\mathcal{G})$ in $A(\mathcal{G})$ is injective. In particular, we show in [Theorem 4](#thm: main){reference-type="ref" reference="thm: main"} that $\pi$ is injective if and only if $\mathcal{G}$ either is isomorphic to a group or contains exactly $2$ non-unit elements outside its isotropy. **Proposition 2**. *Let $\mathcal{G}$ be an ample Hausdorff groupoid with compact unit space $\mathcal{G}^{(0)}$. Suppose that either* *Then the representation $\pi\colon \mathbb{C}F(\mathcal{G}) \to A(\mathcal{G})$ is not injective, and so $\mathbb{C}F(\mathcal{G})$ is not simple.* *Proof.* We first assume that [\[cond: iso rep not inj\]](#cond: iso rep not inj){reference-type="ref" reference="cond: iso rep not inj"} holds. Fix $\gamma_1, \gamma_2 \in \mathcal{G}\setminus \mathcal{G}^{(0)}$ such that $r(\gamma_1) \ne r(\gamma_2)$. Since $\mathcal{G}^{(0)}$ is Hausdorff and $\mathcal{G}$ is all isotropy, we can find disjoint compact open bisections $B_1$ and $B_2$ containing $\gamma_1$ and $\gamma_2$, respectively, such that $$r(B_1) = s(B_1), \quad r(B_2) = s(B_2), \quad \text{and } \quad r(B_1) \cap r(B_2) = \varnothing.$$ Set $R \coloneqq \mathcal{G}^{(0)}\setminus \big(r(B_1) \cup r(B_2)\big)$, and note that $R$ is a compact open bisection of $\mathcal{G}$. Consider the following disjoint unions: $$\begin{aligned} U_1 &\coloneqq B_1 \cup r(B_2) \cup R, \\ U_2 &\coloneqq B_2 \cup r(B_1) \cup R, \text{ and} \\ U_3 &\coloneqq B_1 \cup B_2 \cup R.\end{aligned}$$ It is straightforward to verify that $U_1$, $U_2$, and $U_3$ are distinct elements of $F(\mathcal{G})$. Define $a \coloneqq \delta_{U_1} + \delta_{U_2} - \delta_{U_3} - \delta_{\mathcal{G}^{(0)}}$. Then $$\pi(a) = 1_{U_1} + 1_{U_2} - 1_{U_3} - 1_{\mathcal{G}^{(0)}} = 1_{r(B_2)} + 1_{r(B_1)} + 1_R - 1_{\mathcal{G}^{(0)}} = 0,$$ and so $0 \ne a \in \ker{\pi}$. Hence $\pi$ is not injective, and $\mathbb{C}F(\mathcal{G})$ is not simple because it contains the nontrivial proper ideal $\ker{\pi}$. We now assume that [\[cond: non-iso rep not inj\]](#cond: non-iso rep not inj){reference-type="ref" reference="cond: non-iso rep not inj"} holds instead. Then there exist $$\label[condition]{cond: not inj} \text{$\gamma_1 \in \mathcal{G}\setminus \mathcal{G}^{(0)}$ and $\gamma_2 \in \mathcal{G}\setminus \operatorname{Iso}(\mathcal{G})$ such that $\gamma_1 \ne \gamma_2$ and $\gamma_1 \ne \gamma_2^{-1}$.}$$ For any $\gamma_1, \gamma_2$ satisfying [\[cond: not inj\]](#cond: not inj){reference-type="ref" reference="cond: not inj"}, we have $r(\gamma_2) \ne s(\gamma_2)$, and either $\gamma_1 \notin \operatorname{Iso}(\mathcal{G})$ or $\gamma_1 \in \operatorname{Iso}(\mathcal{G})$. By replacing $\gamma_1$ with $\gamma_1^{-1}$ or $\gamma_2$ with $\gamma_2^{-1}$ if necessary, we can summarise all possible cases as follows: Moreover, we can reduce [\[case: parallel arrows\]](#case: parallel arrows){reference-type="ref" reference="case: parallel arrows"} to [\[case: loop with tail\]](#case: loop with tail){reference-type="ref" reference="case: loop with tail"} by replacing $\gamma_1$ with $\gamma_2 \gamma_1$. Therefore, it suffices to show that $\ker{\pi}$ is nontrivial in each of the . **:** Suppose that the hypotheses of [\[case: joined arrows\]](#case: joined arrows){reference-type="ref" reference="case: joined arrows"} hold, and let $B_1$ and $B_2$ be compact open bisections containing $\gamma_1$ and $\gamma_2$, respectively. Since $\mathcal{G}^{(0)}$ is Hausdorff and $r(\gamma_1)$, $s(\gamma_1)$, and $s(\gamma_2)$ are all distinct, we may assume that $r(B_1)$, $s(B_1)$, and $s(B_2)$ are mutually disjoint by shrinking $B_1$ and $B_2$ if necessary. Moreover, since $s(\gamma_1) = r(\gamma_2)$, we can replace $B_1$ with $B_1 \big(s(B_1) \cap r(B_2)\big)$ and $B_2$ with $\big(s(B_1) \cap r(B_2)\big) B_2$, and thus without loss of generality we may assume that $s(B_1) = r(B_2)$. Set $R \coloneqq \mathcal{G}^{(0)}\setminus \big(r(B_1) \cup s(B_1) \cup s(B_2)\big)$, and note that $R$ is a compact open bisection of $\mathcal{G}$. Consider the following disjoint unions that are distinct elements of $F(\mathcal{G})$: $$\begin{aligned} U \coloneqq&~B_1 \cup B_2 \cup (B_1 B_2)^{-1} \cup R, \\ U^{-1} =&~B_1^{-1} \cup B_2^{-1} \cup (B_1 B_2) \cup R, \\ U_1 \coloneqq&~B_1 \cup B_1^{-1} \cup s(B_2) \cup R, \\ U_2 \coloneqq&~B_2 \cup B_2^{-1} \cup r(B_1) \cup R, \text{ and} \\ U_3 \coloneqq&~B_1 B_2 \cup (B_1 B_2)^{-1} \cup s(B_1) \cup R.\end{aligned}$$ Define $a \coloneqq \delta_U + \delta_{U^{-1}} - \delta_{U_1} - \delta_{U_2} - \delta_{U_3} + \delta_{\mathcal{G}^{(0)}}$. Then $$\begin{aligned} \pi(a) = 1_U + 1_{U^{-1}} - 1_{U_1} - 1_{U_2} - 1_{U_3} + 1_{\mathcal{G}^{(0)}} = -1_{s(B_2)} - 1_{r(B_1)} - 1_{s(B_1)} - 1_R + 1_{\mathcal{G}^{(0)}} = 0,\end{aligned}$$ and so $a \in \ker{\pi} {\setminus} \{0\}$. Hence $\pi$ is not injective, and $\mathbb{C}F(\mathcal{G})$ is not simple because it contains the nontrivial proper ideal $\ker{\pi}$. **:** Now suppose that the hypotheses of [\[case: separate arrows\]](#case: separate arrows){reference-type="ref" reference="case: separate arrows"} hold. Choose compact open bisections $B_1$ and $B_2$ containing $\gamma_1$ and $\gamma_2$, respectively, such that $r(B_1)$, $s(B_1)$, $r(B_2)$, and $s(B_2)$ are mutually disjoint. Set $R \coloneqq \mathcal{G}^{(0)}\setminus \big(r(B_1) \cup s(B_1) \cup r(B_2) \cup s(B_2)\big)$, and note that $R$ is a compact open bisection of $\mathcal{G}$. Consider the following disjoint unions that are distinct elements of $F(\mathcal{G})$: $$\begin{aligned} U_1 &\coloneqq B_1 \cup B_1^{-1} \cup r(B_2) \cup s(B_2) \cup R, \\ U_2 &\coloneqq B_2 \cup B_2^{-1} \cup r(B_1) \cup s(B_1) \cup R, \text{ and} \\ U_3 &\coloneqq B_1 \cup B_1^{-1} \cup B_2 \cup B_2^{-1} \cup R.\end{aligned}$$ It is straightforward to verify that $a \coloneqq \delta_{U_1} + \delta_{U_2} - \delta_{U_3} - \delta_{\mathcal{G}^{(0)}} \in \ker{\pi} {\setminus} \{0\}$, and hence $\pi$ is not injective. **:** Next, suppose that the hypotheses of [\[case: separate loop and arrow\]](#case: separate loop and arrow){reference-type="ref" reference="case: separate loop and arrow"} hold. Choose compact open bisections $B_1$ and $B_2$ containing $\gamma_1$ and $\gamma_2$, respectively, such that $r(B_1) = s(B_1)$, and such that $r(B_1)$, $r(B_2)$, and $s(B_2)$ are mutually disjoint. Set $R \coloneqq \mathcal{G}^{(0)}\setminus \big(r(B_1) \cup r(B_2) \cup s(B_2)\big)$, and note that $R$ is a compact open bisection of $\mathcal{G}$. Consider the following disjoint unions that are distinct elements of $F(\mathcal{G})$: $$\begin{aligned} U_1 &\coloneqq r(B_1) \cup B_2 \cup B_2^{-1} \cup R, \\ U_2 &\coloneqq B_1 \cup r(B_2) \cup s(B_2) \cup R, \text{ and} \\ U_3 &\coloneqq B_1 \cup B_2 \cup B_2^{-1} \cup R.\end{aligned}$$ It is straightforward to verify that $a \coloneqq \delta_{U_1} + \delta_{U_2} - \delta_{U_3} - \delta_{\mathcal{G}^{(0)}} \in \ker{\pi} {\setminus} \{0\}$, and hence $\pi$ is not injective in this case either. **:** Finally, suppose that the hypotheses of [\[case: loop with tail\]](#case: loop with tail){reference-type="ref" reference="case: loop with tail"} hold, and let $B_1'$ and $B_2'$ be compact open bisections containing $\gamma_1$ and $\gamma_2$, respectively. Since $\mathcal{G}^{(0)}$ is Hausdorff and $r(\gamma_2) \ne s(\gamma_2)$, we may assume that $r(B_2') \cap s(B_2') = \varnothing$. Let $V \coloneqq r(B_1') \cap s(B_1') \cap r(B_2')$. Then $B_1 \coloneqq V B_1' V$ and $B_2 \coloneqq V B_2'$ are compact open bisections containing $\gamma_1$ and $\gamma_2$, respectively. We have $r(B_1) = s(B_1) = r(B_2) = V$ and $r(B_2) \cap s(B_2) = \varnothing$. Set $R \coloneqq \mathcal{G}^{(0)}\setminus \big(r(B_2) \cup s(B_2)\big)$, and note that $R$ is a compact open bisection of $\mathcal{G}$. Consider the following disjoint unions that are elements of $F(\mathcal{G})$: $$\begin{aligned} U_1 &\coloneqq B_2 \cup (B_1 B_2)^{-1} \cup R, \\ U_2 &\coloneqq B_1 B_2 \cup B_2^{-1} \cup R, \\ U_3 &\coloneqq B_2 \cup B_2^{-1} \cup R, \text{ and} \\ U_4 &\coloneqq B_1 B_2 \cup (B_1 B_2)^{-1} \cup R.\end{aligned}$$ To see that $U_1$, $U_2$, $U_3$, and $U_4$ are distinct elements of $F(\mathcal{G})$, note that since $\gamma_1 \notin \mathcal{G}^{(0)}$, we have $\gamma_2 \ne \gamma_1 \gamma_2$, and hence $\gamma_2 \notin B_1 B_2$, because $\gamma_1 \gamma_2$ is the unique element of the bisection $B_1 B_2$ with source $s(\gamma_2)$. It is straightforward to verify that $a \coloneqq \delta_{U_1} + \delta_{U_2} - \delta_{U_3} - \delta_{U_4} \in \ker{\pi} {\setminus} \{0\}$. Thus $\pi$ is not injective, and $\mathbb{C}F(\mathcal{G})$ is not simple, and this completes the proof. ◻ **Remark 3**. It follows immediately from the above proof that under the hypotheses of [Proposition 2](#prop: rep not inj){reference-type="ref" reference="prop: rep not inj"}, the extensions $$\overline{\pi}_{\max}\colon C^*(F(\mathcal{G})) \to C^*(\mathcal{G}) \quad \text{and} \quad \overline{\pi}_r\colon C_r^*(F(\mathcal{G})) \to C_r^*(\mathcal{G})$$ of $\pi\colon \mathbb{C}F(\mathcal{G}) \to A(\mathcal{G})$ are also not injective, and so neither $C^*(F(\mathcal{G}))$ nor $C_r^*(F(\mathcal{G}))$ is simple. We conclude this section by proving that the converse of [Proposition 2](#prop: rep not inj){reference-type="ref" reference="prop: rep not inj"} also holds. **Theorem 4**. *Let $\mathcal{G}$ be an ample Hausdorff groupoid with compact unit space. The representation $\pi\colon \mathbb{C}F(\mathcal{G}) \to A(\mathcal{G})$ is injective if and only if* *Proof.* If $\pi$ is injective, then the result follows by the contrapositive of [Proposition 2](#prop: rep not inj){reference-type="ref" reference="prop: rep not inj"}. For the converse, first suppose that [\[cond: all isotropy\]](#cond: all isotropy){reference-type="ref" reference="cond: all isotropy"} holds. If $\mathcal{G}= \mathcal{G}^{(0)}$, then $F(\mathcal{G}) = \{\mathcal{G}^{(0)}\}$, and so $\mathbb{C}F(\mathcal{G}) \cong \mathbb{C}$, and hence $\pi$ is injective. Suppose that $\mathcal{G}\ne \mathcal{G}^{(0)}$. Then there exists a nontrivial discrete group $\Gamma$ with identity $e_\Gamma$ such that $\mathcal{G}= \Gamma \sqcup X$, where $X = \mathcal{G}^{(0)}{\setminus} \{e_\Gamma\}$. Since $\mathcal{G}$ is Hausdorff and étale, $X = (\mathcal{G}{\setminus} \{e_\Gamma\}) \cap \mathcal{G}^{(0)}$ is open in $\mathcal{G}$. We claim that $X$ is compact. To see this, first observe that since $\mathcal{G}$ is Hausdorff, $\mathcal{G}^{(0)}$ is closed, and so $\Gamma {\setminus} \{e_\Gamma\} = \mathcal{G}{\setminus} \mathcal{G}^{(0)}$ is open in $\mathcal{G}$. Thus, since $\mathcal{G}$ is étale, $\{e_\Gamma\} = r(\Gamma {\setminus} \{e_\Gamma\})$ is open in $\mathcal{G}$, and so $X = (\mathcal{G}{\setminus} \{e_\Gamma\}) \cap \mathcal{G}^{(0)}$ is closed. Now, since $X \subseteq \mathcal{G}^{(0)}$ and $\mathcal{G}^{(0)}$ is compact by hypothesis, $X$ must also be compact, as claimed. For each $\gamma \in \Gamma$, choose a compact open bisection $U_\gamma$ of $\mathcal{G}$ containing $\gamma$. Then $U_\gamma \cap \Gamma = \{\gamma\}$. Since $X = \mathcal{G}^{(0)}{\setminus} \{e_\Gamma\}$ is compact and open in $\mathcal{G}$, we have $V_\gamma \coloneqq U_\gamma \cup X = \{\gamma\} \sqcup X \in F(\mathcal{G})$, and it follows that $F(\mathcal{G}) = \big\{ \{\gamma\} \sqcup X \mid \gamma \in \Gamma \big\}$. Now, let $f \in \ker{\pi} \subseteq \mathbb{C}F(\mathcal{G})$. Then for some $m \in \mathbb{N}$, there exist $c_1, \dotsc, c_m \in \mathbb{C}$ and $\gamma_1, \dotsc, \gamma_m \in \Gamma$ such that $\gamma_i \ne \gamma_j$ whenever $i \ne j$, and $f = \sum_{i=1}^m c_i \, \delta_{\{\gamma_i\} \sqcup X}$. Since $\pi(f) = 0$, we have $$c_k = \Big( \sum_{i=1}^m c_i 1_{\{\gamma_i\}} + \big( \sum_{i=1}^m c_i \big) 1_X \Big)(\gamma_k) = \pi(f)(\gamma_k) = 0$$ for each $k \in \{1, \dotsc, m\}$, and so $f = 0$. Thus $\pi$ is injective. Now suppose that [\[cond: not all isotropy\]](#cond: not all isotropy){reference-type="ref" reference="cond: not all isotropy"} holds. Since $\mathcal{G}\ne \operatorname{Iso}(\mathcal{G})$, there exists $\gamma \in \mathcal{G}\setminus \operatorname{Iso}(\mathcal{G})$, and it follows that $\gamma$ and $\gamma^{-1}$ are distinct elements of $\mathcal{G}\setminus \mathcal{G}^{(0)}$. Thus $\abs{\mathcal{G}\setminus \mathcal{G}^{(0)}} = 2$, and so $\mathcal{G}= \mathcal{G}^{(0)}\sqcup \{\gamma, \gamma^{-1}\}$. In particular, $\mathcal{G}$ is compact. Since $\mathcal{G}$ is Hausdorff, $\mathcal{G}^{(0)}$ is closed, and so $\{\gamma, \gamma^{-1}\} = \mathcal{G}\setminus \mathcal{G}^{(0)}$ is open. Thus $\{r(\gamma), s(\gamma)\} = r(\{\gamma, \gamma^{-1}\})$ is open since $\mathcal{G}$ is étale. Therefore, $U \coloneqq \mathcal{G}\setminus \{r(\gamma), s(\gamma)\}$ is a closed subset of $\mathcal{G}$, and by the compactness of $\mathcal{G}$ it follows that $U$ is a full compact open bisection containing $\gamma$ and $\gamma^{-1}$. In fact, given $V \in F(\mathcal{G})$ with $\gamma \in V$, we must have $r(\gamma), s(\gamma) \notin V$, and so $V = U$. It follows that $F(\mathcal{G}) = \{U, \mathcal{G}^{(0)}\}$. Suppose that $f = a \delta_U + b \delta_{\mathcal{G}^{(0)}} \in \ker(\pi)$ for some $a, b \in \mathbb{C}$. Then $a = \pi(f)(\gamma) = 0$ and $b = \pi(f)(r(\gamma)) = 0$, and so $f = 0$. Thus $\pi$ is injective. ◻ [\[sec: surjectivity\]]{#sec: surjectivity label="sec: surjectivity"} In this section we study the image of the representation $\pi\colon \delta_U \mapsto 1_U$ of $\mathbb{C}F(\mathcal{G})$ in $A(\mathcal{G})$. In particular, we show in [Corollary 7](#cor: surj iff group){reference-type="ref" reference="cor: surj iff group"} that $\pi$ is surjective if and only if $\mathcal{G}$ is a group. In order to prove this, we first prove the following result. **Proposition 5**. *Let $\mathcal{G}$ be an ample Hausdorff groupoid with compact unit space $\mathcal{G}^{(0)}$. If $B$ is a nonempty compact open bisection of $\mathcal{G}$ such that $1_B \in \pi(\mathbb{C}F(\mathcal{G}))$, then $B \in F(\mathcal{G})$.* *Proof.* Suppose for contradiction that there exists a nonempty compact open bisection $B$ of $\mathcal{G}$ such that $B \notin F(\mathcal{G})$ and $1_B \in \pi(\mathbb{C}F(\mathcal{G}))$. We may assume that $s(B) \ne \mathcal{G}^{(0)}$, because if not, then $s(B^{-1}) = r(B) \ne \mathcal{G}^{(0)}$, so $B^{-1} \notin F(\mathcal{G})$, and we can replace $B$ by $B^{-1}$ in the following argument since $1_{B^{-1}} = (1_B)^* \in \pi(\mathbb{C}F(\mathcal{G}))$. Since $1_B \in \pi(\mathbb{C}F(\mathcal{G}))$, there exist $c_1, \dotsc, c_m \in \mathbb{C}$ and $U_1, \dotsc, U_m$ in $F(\mathcal{G})$ such that $$\label{eqn: char fn in image} 1_B = \pi\Big( \sum_{i=1}^m c_i \, \delta_{U_i} \Big) = \sum_{i=1}^m c_i 1_{U_i}.$$ Fix $x \in \mathcal{G}^{(0)}{\setminus} s(B)$. Then for each $i \in \{1, \dotsc, m\}$, there exists $\gamma_i \in U_i {\setminus} B$ with $s(\gamma_i) = x$. Let $S \coloneqq \{\gamma_1, \dotsc, \gamma_m\}$. Note that it is possible that $\gamma_i = \gamma_j$ for some $i \ne j$, and so $\abs{S}$ may be less than $m$. Using that $S \subseteq \mathcal{G}{\setminus} B$ for the first equality, [\[eqn: char fn in image\]](#eqn: char fn in image){reference-type="ref" reference="eqn: char fn in image"} for the second equality, and that $U_i \cap S = \{\gamma_i\}$ for each $i \in \{1, \dotsc, m\}$ for the last equality, we compute $$\label{eqn: c_i's sum to 0} 0 = \sum_{\gamma \in S} 1_B(\gamma) = \sum_{\gamma \in S} \sum_{i=1}^m c_i 1_{U_i}(\gamma) = \sum_{\gamma \in S} \sum_{i : \gamma \in U_i} c_i = \sum_{i=1}^m c_i.$$ Now fix $\beta \in B$. Then for each $i \in \{1, \dotsc, m\}$, there exists $\alpha_i \in U_i$ such that $s(\alpha_i) = s(\beta)$. Let $T \coloneqq \{\alpha_1, \dotsc, \alpha_m\}$. It follows from [\[eqn: char fn in image\]](#eqn: char fn in image){reference-type="ref" reference="eqn: char fn in image"} that $\beta \in U_k$ for some $k \in \{1, \dotsc, m\}$, and so since each $U_i$ is a bisection, we must have $\beta = \alpha_k \in T$. Moreover, since $U_1, \dotsc, U_m$ and $B$ are bisections, we have that for each $i \in \{1, \dotsc, m\}$, $U_i \cap T = \{\alpha_i\}$ and either $\alpha_i = \beta$ or $\alpha_i \notin B$. Using these facts and [\[eqn: char fn in image\]](#eqn: char fn in image){reference-type="ref" reference="eqn: char fn in image"}, we compute $$0 < \abs{\{ i \in \{1, \dotsc, m\} \mid \alpha_i = \beta \}} = \sum_{\alpha \in T}1_B(\alpha) = \sum_{\alpha \in T}\sum_{i=1}^m c_i 1_{U_i}(\alpha) = \sum_{\alpha \in T} \sum_{i : \alpha \in U_i} c_i = \sum_{i=1}^m c_i.$$ This contradicts [\[eqn: c_i\'s sum to 0\]](#eqn: c_i's sum to 0){reference-type="ref" reference="eqn: c_i's sum to 0"}, and thus completes the proof. ◻ The following result is an immediate corollary of [Proposition 5](#prop: TFG and im(pi)){reference-type="ref" reference="prop: TFG and im(pi)"}, because $A(\mathcal{G})$ is the span of characteristic functions on compact open bisections of $\mathcal{G}$. **Corollary 6**. *Let $\mathcal{G}$ be an ample Hausdorff groupoid with compact unit space $\mathcal{G}^{(0)}$. If there exists a nonempty compact open bisection $B$ of $\mathcal{G}$ such that $B \notin F(\mathcal{G})$, then the representation $\pi\colon \mathbb{C}F(\mathcal{G}) \to A(\mathcal{G})$ is not surjective.* As the following corollary shows, it turns out that the hypothesis of [Corollary 6](#cor: ample rep not surj){reference-type="ref" reference="cor: ample rep not surj"} is very easily satisfied, as it holds whenever $\mathcal{G}$ is not a group. **Corollary 7**. *Let $\mathcal{G}$ be an ample Hausdorff groupoid with compact unit space $\mathcal{G}^{(0)}$. The representation $\pi\colon \mathbb{C}F(\mathcal{G}) \to A(\mathcal{G})$ is surjective if and only if $\mathcal{G}$ is a group.* *Proof.* If $\mathcal{G}$ is a group, then $F(\mathcal{G}) \cong \mathcal{G}$, so $\mathbb{C}F(\mathcal{G}) = A(\mathcal{G})$, and $\pi\colon \mathbb{C}F(\mathcal{G}) \to A(\mathcal{G})$ is the identity map and hence is surjective. For the converse, suppose that $\mathcal{G}$ is not a group, and fix distinct units $u, v \in \mathcal{G}^{(0)}$. Since $\mathcal{G}$ is an ample Hausdorff groupoid, there exist disjoint compact open sets $U, V \subseteq \mathcal{G}^{(0)}$ containing $u$ and $v$, respectively. But then $v \notin U$, so $U \in B^\mathrm{co}(\mathcal{G}){\setminus} F(\mathcal{G})$, and hence [Corollary 6](#cor: ample rep not surj){reference-type="ref" reference="cor: ample rep not surj"} implies that $\pi\colon \mathbb{C}F(\mathcal{G}) \to A(\mathcal{G})$ is not surjective. ◻ For the next result, recall (for instance, from [@Matui2016 Section 2.2]) that there are linear maps $r_*, s_*\colon A(\mathcal{G}) \to A(\mathcal{G}^{(0)})$ given by $$r_*f(u) \coloneqq \sum_{\gamma \in \mathcal{G}^u} f(\gamma) \quad \text{and} \quad s_*f(u) \coloneqq \sum_{\gamma \in \mathcal{G}_u} f(\gamma), \ \text{ for all } f \in A(\mathcal{G}) \text{ and } u \in \mathcal{G}^{(0)};$$ and there is a linear map $\delta_1\colon A(\mathcal{G}) \to A(\mathcal{G}^{(0)})$ given by $\delta_1 \coloneqq s_* - r_*$. **Proposition 8**. *Let $\mathcal{G}$ be an ample Hausdorff groupoid with compact unit space $\mathcal{G}^{(0)}$. Then $$\pi(\mathbb{C}F(\mathcal{G})) \subseteq \{ f \in A(\mathcal{G}) \mid r_*f(u) = s_*f(v) \text{ for all } u, v \in \mathcal{G}^{(0)}\} \subseteq \ker{\delta_1}.$$* *Proof.* Fix $f \in \pi(\mathbb{C}F(\mathcal{G}))$. Then there exist $U_1, \dotsc, U_m \in F(\mathcal{G})$ and $c_1, \dotsc, c_m \in \mathbb{C}$ such that $$f = \pi\Big( \sum_{i=1}^m c_i \, \delta_{U_i} \Big) = \sum_{i=1}^m c_i 1_{U_i}.$$ Fix $u, v \in \mathcal{G}^{(0)}$. For each $i \in \{1, \dotsc, m\}$, the sets $U_i \cap \mathcal{G}^u$ and $U_i \cap \mathcal{G}_v$ are singletons because $U_i$ is a full bisection of $\mathcal{G}$. Thus $$r_*f(u) \,=\, \sum_{\gamma \in \mathcal{G}^u} f(\gamma) \,=\, \sum_{\gamma \in \mathcal{G}^u} \ \sum_{i : \gamma \in U_i} c_i \,=\, \sum_{i=1}^m c_i \,=\, \sum_{\gamma \in \mathcal{G}_v} \ \sum_{i : \gamma \in U_i} c_i \,=\, \sum_{\gamma \in \mathcal{G}_v } f(\gamma) \,=\, s_*f(v).$$ It follows that $r_*f(x) = s_*f(x)$ for all $x \in \mathcal{G}^{(0)}$, and so $\delta_1(f) = 0$. ◻ [\[sec: discrete groupoids\]]{#sec: discrete groupoids label="sec: discrete groupoids"} In this section we restrict our attention to discrete groupoids, and to the extension of the representations of their topological full groups with respect to the full and reduced C\*-norms. In particular, we prove an analogue of [Corollary 7](#cor: surj iff group){reference-type="ref" reference="cor: surj iff group"} for the extension of the representation $\pi$ with respect to the full C\*-norm (see [Theorem 11](#thm: discrete full iff group){reference-type="ref" reference="thm: discrete full iff group"}), and we show in [Example 13](#eg: F_2 sqcup F_2){reference-type="ref" reference="eg: F_2 sqcup F_2"} that this result does not hold in the reduced setting. We conclude the section by connecting our results from [\[sec: injectivity,sec: surjectivity,sec: discrete groupoids\]](#sec: injectivity,sec: surjectivity,sec: discrete groupoids){reference-type="ref" reference="sec: injectivity,sec: surjectivity,sec: discrete groupoids"} in [Corollary 14](#cor: isomorphism){reference-type="ref" reference="cor: isomorphism"}. Let $\mathcal{G}$ be a discrete groupoid with finite unit space $\mathcal{G}^{(0)}= \{a_1, \dotsc, a_n\}$. Recall that, for a groupoid $\mathcal{G}$ and $a, b \in \mathcal{G}^{(0)}$, we define $\mathcal{G}_b^a \coloneqq \{ \gamma \in \mathcal{G}\mid r(\gamma) = a \text{ and } s(\gamma) = b \}$. Thus $\mathcal{P}_\mathcal{G}\coloneqq \big\{ \mathcal{G}_{a_j}^{a_i} : i, j \in \{1, \dotsc, n\} \big\}$ is a partition of $\mathcal{G}$ into disjoint sets. For $\gamma \in \mathcal{G}$, write $1_\gamma \coloneqq 1_{\{\gamma\}} \in A(\mathcal{G})$. Given $f \in A(\mathcal{G})$ and $i, j \in \{1, \dotsc, n\}$, we define a map $f_{i,j}\colon \mathcal{G}\to \mathbb{C}$ by $$f_{i,j}(\gamma) \coloneqq \begin{cases} f(\gamma) & \text{if } \gamma \in \mathcal{G}_{a_j}^{a_i} \\ 0 & \text{otherwise}. \end{cases}$$ Then each $f_{i,j} \in A(\mathcal{G})$, and since $\mathcal{P}_\mathcal{G}$ is a partition of $\mathcal{G}$, it follows that $f = \displaystyle\sum_{i,j=1}^n f_{i,j}$. Define $T\colon A(\mathcal{G}) \to M_n(\mathbb{C})$ by $$T(f)_{ij} \coloneqq \sum_{\gamma \in \mathcal{G}_{a_j}^{a_i}} f(\gamma), \ \text{ for each } i, j \in \{1, \dotsc, n\}.$$ We will use this map $T$ to study the image of the representation $\pi\colon \mathbb{C}F(\mathcal{G}) \to A(\mathcal{G})$. We first show that $T$ is a $*$-representation of $A(\mathcal{G})$. **Lemma 9**. *Let $\mathcal{G}$ be a discrete groupoid with finite unit space $\mathcal{G}^{(0)}\coloneqq \{a_1, \dotsc, a_n\}$. The map $T\colon A(\mathcal{G}) \to M_n(\mathbb{C})$ defined above is a $*$-representation of $A(\mathcal{G})$.* *Proof.* It is straightforward to verify that $T$ is linear. Fix $f, g \in A(\mathcal{G})$. For all $i, j \in \{1, \dotsc, n\}$, we have $$\begin{aligned} T(f * g)_{ij} = \sum_{\gamma \in \mathcal{G}_{a_j}^{a_i}} (f * g)(\gamma) = \sum_{\gamma \in \mathcal{G}_{a_j}^{a_i}} \sum_{\alpha\beta = \gamma} f(\alpha) g(\beta) &= \sum_{k=1}^n \sum_{\alpha \in \mathcal{G}_{a_k}^{a_i}} f(\alpha) \sum_{\beta \in \mathcal{G}_{a_j}^{a_k}} g(\beta) \\ &= \sum_{k=1}^n T(f)_{ik} \, T(g)_{kj} = \big(T(f) T(g)\big)_{ij},\end{aligned}$$ and $$T(f^*)_{ij} = \sum_{\gamma \in \mathcal{G}_{a_j}^{a_i}} f^*(\gamma) = \sum_{\gamma \in \mathcal{G}_{a_j}^{a_i}} \overline{f(\gamma^{-1})} = \overline{\sum_{\eta \in \mathcal{G}_{a_i}^{a_j}} f(\eta)} = \overline{T(f)_{ji}} = \big(T(f)^*\big)_{ij}.$$ Thus $T(f * g) = T(f) T(g)$ and $T(f^*) = T(f)^*$, and so $T$ is a $*$-homomorphism. ◻ The following result is a corollary of [Proposition 8](#prop: ample rep image){reference-type="ref" reference="prop: ample rep image"}. **Corollary 10**. *Let $\mathcal{G}$ be a discrete groupoid with finite unit space $\mathcal{G}^{(0)}\coloneqq \{a_1, \dotsc, a_n\}$. Then $$\pi(\mathbb{C}F(\mathcal{G})) \subseteq \big\{ f \in A(\mathcal{G}) \mid \exists \, c_f \in \mathbb{C}\text{ such that all row and column sums of } T(f) \text{ are } c_f \big\}.$$* *Proof.* Fix $f = \displaystyle\sum_{i,j=1}^n f_{i,j} \in \pi(\mathbb{C}F(\mathcal{G}))$. Then, for each $i, j \in \{1, \dotsc, n\}$, we have $$\text{$i$\textsuperscript{th} row sum of } T(f) = \sum_{k=1}^n T(f)_{ik} = \sum_{k=1}^n \sum_{\gamma \in \mathcal{G}_{a_k}^{a_i}} f(\gamma) = \sum_{\gamma \in \mathcal{G}^{a_i}} f(\gamma) = r_*f(a_i),$$ and $$\text{$j$\textsuperscript{th} column sum of } T(f) = \sum_{k=1}^n T(f)_{kj} = \sum_{k=1}^n \sum_{\gamma \in \mathcal{G}_{a_j}^{a_k}} f(\gamma) = \sum_{\gamma \in \mathcal{G}_{a_j}} f(\gamma) = s_*f(a_j).$$ By [Proposition 8](#prop: ample rep image){reference-type="ref" reference="prop: ample rep image"}, it follows that for all $i, j \in \{1, \dotsc, n\}$, we have $$\text{$i$\textsuperscript{th} row sum of } T(f) \,=\, \text{$j$\textsuperscript{th} column sum of } T(f). \qedhere$$ ◻ We now use [Corollary 10](#cor: discrete image rep){reference-type="ref" reference="cor: discrete image rep"} to study the completions of $\pi(\mathbb{C}F(\mathcal{G}))$ in the full and reduced groupoid C\*-algebras. In [Theorem 11](#thm: discrete full iff group){reference-type="ref" reference="thm: discrete full iff group"} we prove that for a discrete groupoid $\mathcal{G}$, an analogue of [Corollary 7](#cor: surj iff group){reference-type="ref" reference="cor: surj iff group"} holds for the full groupoid C\*-algebra $C^*(\mathcal{G})$. **Theorem 11**. *Let $\mathcal{G}$ be a discrete groupoid with finite unit space $\mathcal{G}^{(0)}$. Then $$\overline{\pi(\mathbb{C}F(\mathcal{G}))}^{\ensuremath{\norm{\cdot}_{\mathrm{max}}}} = C^*(\mathcal{G})$$ if and only if $\mathcal{G}$ is a group.* *Proof.* If $\mathcal{G}$ is a group, then $F(\mathcal{G}) \cong \mathcal{G}$, so $\mathbb{C}F(\mathcal{G}) = A(\mathcal{G})$, and hence $$\overline{\pi(\mathbb{C}F(\mathcal{G}))}^{\ensuremath{\norm{\cdot}_{\mathrm{max}}}} = \overline{A(\mathcal{G})}^{\ensuremath{\norm{\cdot}_{\mathrm{max}}}} = C^*(\mathcal{G}).$$ Suppose that $\mathcal{G}$ is not a group. We show that $\overline{\pi(\mathbb{C}F(\mathcal{G}))}^{\ensuremath{\norm{\cdot}_{\mathrm{max}}}} \ne C^*(\mathcal{G})$ by proving an even stronger result: that $1_\gamma \notin \overline{\pi(\mathbb{C}F(\mathcal{G}))}^{\ensuremath{\norm{\cdot}_{\mathrm{max}}}}$ for each $\gamma \in \mathcal{G}$. Write $\mathcal{G}^{(0)}= \{a_1, \dotsc, a_n\}$, and note that $n \ge 2$ since $\mathcal{G}$ is not a group. Fix $\gamma \in \mathcal{G}$, and suppose for contradiction that $1_\gamma \in \overline{\pi(\mathbb{C}F(\mathcal{G}))}^{\ensuremath{\norm{\cdot}_{\mathrm{max}}}}$. Then there exists a sequence $(\varphi_m)_{m=0}^\infty$ of functions in $\pi(\mathbb{C}F(\mathcal{G}))$ such that $\ensuremath{\norm{\varphi_m - 1_\gamma}_{\mathrm{max}}} \to 0$ as $m \to \infty$. By [Lemma 9](#lem: T *-rep){reference-type="ref" reference="lem: T *-rep"}, $T\colon A(\mathcal{G}) \to M_n(\mathbb{C})$ is a $*$-representation of $A(\mathcal{G})$, and hence [\[eqn: Steinberg \*-reps are bounded\]](#eqn: Steinberg *-reps are bounded){reference-type="ref" reference="eqn: Steinberg *-reps are bounded"} on implies that $T$ is bounded. Thus $$\label{eqn: convergence of T(phi_m)} \norm{T(\varphi_m) - T(1_\gamma)}_{M_n(\mathbb{C})} \,=\, \norm{T(\varphi_m - 1_\gamma)}_{M_n(\mathbb{C})} \,\to\, 0 \quad \text{as } m \to \infty.$$ Let $\ell$ and $k$ be the unique elements of $\{1, \dotsc, n\}$ such that $\gamma \in \mathcal{G}_{a_k}^{a_\ell}$. Note that each $T(\varphi_m)$ has $n \ge 2$ rows, and it follows from [\[eqn: convergence of T(phi_m)\]](#eqn: convergence of T(phi_m)){reference-type="ref" reference="eqn: convergence of T(phi_m)"} that for each $i \in \{1, \dotsc, n\}$, we have $$i^\textsuperscript{th} \text{ row sum of } T(\varphi_m) \,\to\, i^\textsuperscript{th} \text{ row sum of } T(1_\gamma) \,=\, \sum_{j=1}^n T(1_\gamma)_{ij} \,=\, T(1_\gamma)_{ik} \,=\, \begin{cases} 1 & \text{if } i = \ell \\ 0 & \text{otherwise} \end{cases}$$ as $m \to \infty$. But this contradicts [Corollary 10](#cor: discrete image rep){reference-type="ref" reference="cor: discrete image rep"}, which says that for each $m \in \mathbb{N}$, all of the row (and column) sums of $T(\varphi_m)$ are equal. So we must have $1_\gamma \notin \overline{\pi(\mathbb{C}F(\mathcal{G}))}^{\ensuremath{\norm{\cdot}_{\mathrm{max}}}}$. ◻ **Remark 12**. It is known that [Theorem 11](#thm: discrete full iff group){reference-type="ref" reference="thm: discrete full iff group"} does not hold for ample Hausdorff groupoids in general. For example, if $\mathcal{G}$ is the Cuntz groupoid (that is, the boundary-path groupoid of the directed graph with a single vertex and two edges), then $F(\mathcal{G})$ is Thompson's group $V_2$, and the representation $\pi\colon \mathbb{C}(F(\mathcal{G})) \to A(\mathcal{G})$ extends to a surjective representation of $F(\mathcal{G})$ in the Cuntz algebra $\mathcal{O}_2$; see [@BS2019 Remark 4.7] and [@HO2017 Proposition 5.3]. It turns out that [Theorem 11](#thm: discrete full iff group){reference-type="ref" reference="thm: discrete full iff group"} does not hold in the reduced setting. We provide an example demonstrating this fact below. **Example 13**. Let $\mathcal{G}= \mathbb{F}_2 \sqcup \mathbb{F}_2$. Then each element of $\mathcal{G}$ is of the form $(g,k)$, where $g \in \mathbb{F}_2$, and $k \in \{1, 2\}$ identifies whether $g$ belongs to the first or the second copy of $\mathbb{F}_2$. Since $\mathcal{G}$ is not a group, we know by [Theorem 11](#thm: discrete full iff group){reference-type="ref" reference="thm: discrete full iff group"} that $\overline{\pi(\mathbb{C}F(\mathcal{G}))}^{\ensuremath{\norm{\cdot}_{\mathrm{max}}}} \ne C^*(\mathcal{G})$. We show that despite this, we still have $\overline{\pi(\mathbb{C}F(\mathcal{G}))}^{\ensuremath{\norm{\cdot}_r}} = C_r^*(\mathcal{G})$. To do so, it suffices to show that for each $g \in \mathbb{F}_2$, we have $1_{(g,1)} \in \overline{\pi(\mathbb{C}F(\mathcal{G}))}^{\ensuremath{\norm{\cdot}_r}}$, because a symmetric argument then shows that $1_{(g,2)} \in \overline{\pi(\mathbb{C}F(\mathcal{G}))}^{\ensuremath{\norm{\cdot}_r}}$. Fix $t \in \mathbb{F}_2$, and for each $m \in \mathbb{N}$, let $E_m$ denote the set of (reduced) elements of $\mathbb{F}_2$ with length $m$. List elements of $\mathbb{F}_2$ in increasing order of their lengths. That is, write $\mathbb{F}_2 = \{g_1, g_2, g_3, \dotsc \}$, with $\abs{g_i} \le \abs{g_{i+1}}$ for all $i \ge 1$. Now define a sequence of functions $(\phi_n)_{n=1}^\infty \subseteq \pi(\mathbb{C}F(\mathcal{G})) \subseteq A(\mathcal{G})$ by $$\phi_n \coloneqq \pi\Big( \delta_{(t,1)} + \sum_{i=1}^n \tfrac{1}{n} \, \delta_{(g_i,2)} \Big) = 1_{(t,1)} + \frac{1}{n} \Big( \sum_{i=1}^n 1_{(g_i, 2)} \Big).$$ We claim that $\phi_n \to 1_{(t,1)}$ in $C^*_r(\mathcal{G})$. Since the map $1_{g_i} \mapsto 1_{(g_i,2)}$ extends to an embedding of $C_r^*(\mathbb{F}_2)$ in $C_r^*(\mathcal{G})$, it suffices to show that $\psi_n \coloneqq \dfrac{1}{n} \Big( \displaystyle\sum_{i=1}^n 1_{g_i} \Big) \to 0$ in $C_r^*(\mathbb{F}_2)$. By [@Haagerup1978 Lemma 1.5], we know that for all $f \in C_c(\mathbb{F}_2)$, $$\label[inequality]{ineq: Haagerup's inequality} \ensuremath{\norm{f}_r} \,\le\, 2 \, \Big( \sum_{s \in \mathbb{F}_2} \abs{f(s)}^2 \, \big( 1 + \abs{s}^4 \big) \Big)^{\tfrac{1}{2}}.$$ For each $m \ge 1$, we have $\abs{E_m} = 4 \times 3^{m-1}$. Thus, for each $n \ge 1$, we have $$\sum_{m=0}^{\ceil{\log_3{n}}} \abs{E_m} \,=\, \abs{E_0} + 4 \sum_{m=1}^{\ceil{\log_3{n}}} 3^{m-1} \,=\, 1 + \frac{4\big(3^{\ceil{\log_3{n}}} - 1\big)}{3 - 1} \,\ge\, 1 + \frac{4(n - 1)}{2} \,\ge\, n,$$ and it follows that $\operatorname{supp}(\psi_n) = \{g_1, \dotsc, g_n\} \subseteq \bigcup_{m=0}^{\ceil{\log_3{n}}} E_m$. Now, for each $n \ge 1$, [\[ineq: Haagerup\'s inequality\]](#ineq: Haagerup's inequality){reference-type="ref" reference="ineq: Haagerup's inequality"} implies that $$\begin{aligned} \ensuremath{\norm{\psi_n}_r} \,&\le\, 2 \, \Big( \sum_{s \in \mathbb{F}_2} \abs{\psi_n(s)}^2 \, \big( 1 + \abs{s}^4 \big) \Big)^{\tfrac{1}{2}} \,=\, 2 \, \Big( \sum_{m=0}^{\ceil{\log_3{n}}} \sum_{s \in E_m} \abs{\psi_n(s)}^2 \, \big( 1 + \abs{s}^4 \big) \Big)^{\tfrac{1}{2}} \\ &\le\, 2 \, \Big( \sum_{m=0}^{\ceil{\log_3{n}}} \frac{\abs{E_m}}{n^2} \, (1 + m^4) \Big)^{\tfrac{1}{2}} \,\le\, 2 \, \Big( \sum_{m=0}^{\ceil{\log_3{n}}} \frac{4 \times 3^{m-1} \times 2m^4}{n^2} \Big)^{\tfrac{1}{2}} \\ &\le\, 2 \, \Big( \frac{8 \ceil{\log_3{n}}^4}{n^2} \sum_{m=0}^{\ceil{\log_3{n}}} 3^{m-1} \Big)^{\tfrac{1}{2}} \,=\, 2 \, \Big( \frac{8 \ceil{\log_3{n}}^4 \, \big( 3^{1 + \ceil{\log_3{n}}} - \tfrac{1}{3} \big)}{n^2 \, (3 - 1)} \Big)^{\tfrac{1}{2}} \\ &\le\, \frac{4 \ceil{\log_3{n}}^2}{n} \big( 3^{2 + \log_3{n}}\big)^{\tfrac{1}{2}} \,=\, \frac{12 \ceil{\log_3{n}}^2}{\sqrt{n}}.\end{aligned}$$ Since $\dfrac{12 \ceil{\log_3{n}}^2}{\sqrt{n}} \,\to\, 0$ as $n \to \infty$, we deduce that $\psi_n \to 0$ in $C_r^*(\mathbb{F}_2)$, as required. We conclude the paper with a corollary of [\[thm: main,cor: surj iff group,thm: discrete full iff group\]](#thm: main,cor: surj iff group,thm: discrete full iff group){reference-type="ref" reference="thm: main,cor: surj iff group,thm: discrete full iff group"}. **Corollary 14**. *Let $\mathcal{G}$ be an ample Hausdorff groupoid with compact unit space $\mathcal{G}^{(0)}$. The representation $\pi\colon \mathbb{C}F(\mathcal{G}) \to A(\mathcal{G})$ is an isomorphism if and only if $\mathcal{G}$ is a group. Similarly, if $\mathcal{G}$ is discrete, then the representation $\overline{\pi}_{\max}\colon C^*(F(\mathcal{G})) \to C^*(\mathcal{G})$ is an isomorphism if and only if $\mathcal{G}$ is a group.* *Proof.* If $\mathcal{G}$ is a group, then $\mathcal{G}$ satisfies [\[cond: all isotropy\]](#cond: all isotropy){reference-type="ref" reference="cond: all isotropy"} of [Theorem 4](#thm: main){reference-type="ref" reference="thm: main"}, so [\[thm: main,cor: surj iff group\]](#thm: main,cor: surj iff group){reference-type="ref" reference="thm: main,cor: surj iff group"} together imply that $\pi\colon \mathbb{C}F(\mathcal{G}) \to A(\mathcal{G})$ is an isomorphism. If $\mathcal{G}$ is not a group, then [Corollary 7](#cor: surj iff group){reference-type="ref" reference="cor: surj iff group"} implies that $\pi$ is not an isomorphism. Now suppose that $\mathcal{G}$ is discrete. If $\mathcal{G}$ is a group, then $F(\mathcal{G}) \cong \mathcal{G}$, and so the representation $\pi\colon \mathbb{C}F(\mathcal{G}) \to A(\mathcal{G})$ is the identity map, and hence it extends to an isomorphism $\overline{\pi}_{\max}\colon C^*(F(\mathcal{G})) \to C^*(\mathcal{G})$. If $\mathcal{G}$ is not a group, then [Theorem 11](#thm: discrete full iff group){reference-type="ref" reference="thm: discrete full iff group"} implies that $\overline{\pi}_{\max}$ is not an isomorphism. ◻

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